LMFDB - Embedded newform 309.5.d.a.205.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [309,5,Mod(205,309)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(309, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 5, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("309.205");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 309 = 3 \cdot 103 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	309.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$31.9413185929$
Analytic rank:	$0$
Dimension:	$70$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			205.11
Character	$\chi$	$=$	309.205
Dual form			309.5.d.a.205.12

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.84937 q^{2} -5.19615i q^{3} +18.2151 q^{4} +37.7904i q^{5} +30.3942i q^{6} +2.46368 q^{7} -12.9570 q^{8} -27.0000 q^{9} +O(q^{10})$	\(q-5.84937 q^{2} -5.19615i q^{3} +18.2151 q^{4} +37.7904i q^{5} +30.3942i q^{6} +2.46368 q^{7} -12.9570 q^{8} -27.0000 q^{9} -221.050i q^{10} +109.589i q^{11} -94.6485i q^{12} -276.332 q^{13} -14.4109 q^{14} +196.365 q^{15} -215.651 q^{16} -508.064 q^{17} +157.933 q^{18} +172.650 q^{19} +688.357i q^{20} -12.8016i q^{21} -641.029i q^{22} +36.7949 q^{23} +67.3268i q^{24} -803.114 q^{25} +1616.37 q^{26} +140.296i q^{27} +44.8761 q^{28} +1167.58 q^{29} -1148.61 q^{30} +1006.60i q^{31} +1468.74 q^{32} +569.443 q^{33} +2971.85 q^{34} +93.1033i q^{35} -491.808 q^{36} -1046.30i q^{37} -1009.89 q^{38} +1435.86i q^{39} -489.652i q^{40} -514.193 q^{41} +74.8815i q^{42} +425.698i q^{43} +1996.18i q^{44} -1020.34i q^{45} -215.227 q^{46} -1531.62i q^{47} +1120.56i q^{48} -2394.93 q^{49} +4697.71 q^{50} +2639.98i q^{51} -5033.42 q^{52} +1757.90i q^{53} -820.644i q^{54} -4141.43 q^{55} -31.9219 q^{56} -897.114i q^{57} -6829.59 q^{58} +5502.21 q^{59} +3576.81 q^{60} -511.770 q^{61} -5887.98i q^{62} -66.5192 q^{63} -5140.76 q^{64} -10442.7i q^{65} -3330.88 q^{66} +3160.96i q^{67} -9254.44 q^{68} -191.192i q^{69} -544.595i q^{70} -4567.41i q^{71} +349.840 q^{72} -7638.39i q^{73} +6120.17i q^{74} +4173.10i q^{75} +3144.83 q^{76} +269.993i q^{77} -8398.89i q^{78} +3753.49 q^{79} -8149.55i q^{80} +729.000 q^{81} +3007.71 q^{82} +1970.28 q^{83} -233.183i q^{84} -19199.9i q^{85} -2490.06i q^{86} -6066.91i q^{87} -1419.95i q^{88} -13999.4i q^{89} +5968.35i q^{90} -680.792 q^{91} +670.223 q^{92} +5230.45 q^{93} +8959.02i q^{94} +6524.50i q^{95} -7631.78i q^{96} -1150.62 q^{97} +14008.8 q^{98} -2958.91i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 70 q + 576 q^{4} - 56 q^{7} - 180 q^{8} - 1890 q^{9}+O(q^{10}) $ 70 * q + 576 * q^4 - 56 * q^7 - 180 * q^8 - 1890 * q^9	$ 70 q + 576 q^{4} - 56 q^{7} - 180 q^{8} - 1890 q^{9} + 708 q^{13} - 204 q^{14} + 108 q^{15} + 4584 q^{16} - 312 q^{17} + 492 q^{19} + 1008 q^{23} - 9262 q^{25} + 1410 q^{26} - 4046 q^{28} + 204 q^{29} + 2664 q^{30} + 330 q^{32} + 3384 q^{33} + 5136 q^{34} - 15552 q^{36} + 5130 q^{38} - 5724 q^{41} - 9006 q^{46} + 28518 q^{49} - 4332 q^{50} + 14116 q^{52} + 5672 q^{55} - 22986 q^{56} - 16772 q^{58} + 10008 q^{59} + 4608 q^{60} - 7572 q^{61} + 1512 q^{63} + 27464 q^{64} + 13896 q^{66} - 7014 q^{68} + 4860 q^{72} + 9204 q^{76} - 2920 q^{79} + 51030 q^{81} + 19642 q^{82} + 11244 q^{83} + 40192 q^{91} + 57660 q^{92} + 8424 q^{93} + 40776 q^{97} + 55530 q^{98}+O(q^{100}) $ 70 * q + 576 * q^4 - 56 * q^7 - 180 * q^8 - 1890 * q^9 + 708 * q^13 - 204 * q^14 + 108 * q^15 + 4584 * q^16 - 312 * q^17 + 492 * q^19 + 1008 * q^23 - 9262 * q^25 + 1410 * q^26 - 4046 * q^28 + 204 * q^29 + 2664 * q^30 + 330 * q^32 + 3384 * q^33 + 5136 * q^34 - 15552 * q^36 + 5130 * q^38 - 5724 * q^41 - 9006 * q^46 + 28518 * q^49 - 4332 * q^50 + 14116 * q^52 + 5672 * q^55 - 22986 * q^56 - 16772 * q^58 + 10008 * q^59 + 4608 * q^60 - 7572 * q^61 + 1512 * q^63 + 27464 * q^64 + 13896 * q^66 - 7014 * q^68 + 4860 * q^72 + 9204 * q^76 - 2920 * q^79 + 51030 * q^81 + 19642 * q^82 + 11244 * q^83 + 40192 * q^91 + 57660 * q^92 + 8424 * q^93 + 40776 * q^97 + 55530 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/309\mathbb{Z}\right)^\times$.

$n$	$104$	$211$
$\chi(n)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−5.84937			−1.46234			−0.731171	−	0.682194i	$-0.761026\pi$
$2$	−5.84937			−1.46234			−0.731171	+	0.682194i	$0.761026\pi$
$3$		−	5.19615i		−	0.577350i
$3$		−	5.19615i		−	0.577350i
$4$	18.2151			1.13844
$5$			37.7904i			1.51162i	0.654793	+	0.755808i	$0.272756\pi$
$5$			37.7904i			1.51162i	−0.654793	+	0.755808i	$0.727244\pi$
$6$			30.3942i			0.844284i
$7$	2.46368			0.0502791			0.0251395	−	0.999684i	$-0.491997\pi$
$7$	2.46368			0.0502791			0.0251395	+	0.999684i	$0.491997\pi$
$8$	−12.9570			−0.202454
$9$	−27.0000			−0.333333
$10$		−	221.050i		−	2.21050i
$11$			109.589i			0.905698i	0.891587	+	0.452849i	$0.149592\pi$
$11$			109.589i			0.905698i	−0.891587	+	0.452849i	$0.850408\pi$
$12$		−	94.6485i		−	0.657281i
$13$	−276.332			−1.63510			−0.817550	−	0.575858i	$-0.804668\pi$
$13$	−276.332			−1.63510			−0.817550	+	0.575858i	$0.804668\pi$
$14$	−14.4109			−0.0735252
$15$	196.365			0.872732
$16$	−215.651			−0.842388
$17$	−508.064			−1.75801			−0.879003	−	0.476817i	$-0.841791\pi$
$17$	−508.064			−1.75801			−0.879003	+	0.476817i	$0.841791\pi$
$18$	157.933			0.487447
$19$	172.650			0.478254			0.239127	−	0.970988i	$-0.423139\pi$
$19$	172.650			0.478254			0.239127	+	0.970988i	$0.423139\pi$
$20$			688.357i			1.72089i
$21$		−	12.8016i		−	0.0290286i
$22$		−	641.029i		−	1.32444i
$23$	36.7949			0.0695555			0.0347778	−	0.999395i	$-0.488928\pi$
$23$	36.7949			0.0695555			0.0347778	+	0.999395i	$0.488928\pi$
$24$			67.3268i			0.116887i
$25$	−803.114			−1.28498
$26$	1616.37			2.39108
$27$			140.296i			0.192450i
$28$	44.8761			0.0572400
$29$	1167.58			1.38832			0.694160	−	0.719821i	$-0.255776\pi$
$29$	1167.58			1.38832			0.694160	+	0.719821i	$0.255776\pi$
$30$	−1148.61			−1.27623
$31$			1006.60i			1.04745i	0.851887	+	0.523726i	$0.175458\pi$
$31$			1006.60i			1.04745i	−0.851887	+	0.523726i	$0.824542\pi$
$32$	1468.74			1.43431
$33$	569.443			0.522905
$34$	2971.85			2.57081
$35$			93.1033i			0.0760027i
$36$	−491.808			−0.379482
$37$		−	1046.30i		−	0.764277i	−0.924105	−	0.382139i	$-0.875188\pi$
$37$		−	1046.30i		−	0.764277i	0.924105	−	0.382139i	$-0.124812\pi$
$38$	−1009.89			−0.699371
$39$			1435.86i			0.944025i
$40$		−	489.652i		−	0.306032i
$41$	−514.193			−0.305885			−0.152943	−	0.988235i	$-0.548875\pi$
$41$	−514.193			−0.305885			−0.152943	+	0.988235i	$0.548875\pi$
$42$			74.8815i			0.0424498i
$43$			425.698i			0.230231i	0.993352	+	0.115116i	$0.0367239\pi$
$43$			425.698i			0.230231i	−0.993352	+	0.115116i	$0.963276\pi$
$44$			1996.18i			1.03109i
$45$		−	1020.34i		−	0.503872i
$46$	−215.227			−0.101714
$47$		−	1531.62i		−	0.693355i	−0.937984	−	0.346678i	$-0.887310\pi$
$47$		−	1531.62i		−	0.693355i	0.937984	−	0.346678i	$-0.112690\pi$
$48$			1120.56i			0.486353i
$49$	−2394.93			−0.997472
$50$	4697.71			1.87908
$51$			2639.98i			1.01499i
$52$	−5033.42			−1.86147
$53$			1757.90i			0.625809i	0.949785	+	0.312904i	$0.101302\pi$
$53$			1757.90i			0.625809i	−0.949785	+	0.312904i	$0.898698\pi$
$54$		−	820.644i		−	0.281428i
$55$	−4141.43			−1.36907
$56$	−31.9219			−0.0101792
$57$		−	897.114i		−	0.276120i
$58$	−6829.59			−2.03020
$59$	5502.21			1.58064			0.790320	−	0.612694i	$-0.209914\pi$
$59$	5502.21			1.58064			0.790320	+	0.612694i	$0.209914\pi$
$60$	3576.81			0.993557
$61$	−511.770			−0.137536			−0.0687679	−	0.997633i	$-0.521907\pi$
$61$	−511.770			−0.137536			−0.0687679	+	0.997633i	$0.521907\pi$
$62$		−	5887.98i		−	1.53173i
$63$	−66.5192			−0.0167597
$64$	−5140.76			−1.25507
$65$		−	10442.7i		−	2.47164i
$66$	−3330.88			−0.764666
$67$			3160.96i			0.704158i	0.935970	+	0.352079i	$0.114525\pi$
$67$			3160.96i			0.704158i	−0.935970	+	0.352079i	$0.885475\pi$
$68$	−9254.44			−2.00139
$69$		−	191.192i		−	0.0401579i
$70$		−	544.595i		−	0.111142i
$71$		−	4567.41i		−	0.906052i	−0.891497	−	0.453026i	$-0.850344\pi$
$71$		−	4567.41i		−	0.906052i	0.891497	−	0.453026i	$-0.149656\pi$
$72$	349.840			0.0674846
$73$		−	7638.39i		−	1.43336i	−0.697401	−	0.716681i	$-0.745660\pi$
$73$		−	7638.39i		−	1.43336i	0.697401	−	0.716681i	$-0.254340\pi$
$74$			6120.17i			1.11763i
$75$			4173.10i			0.741885i
$76$	3144.83			0.544466
$77$			269.993i			0.0455377i
$78$		−	8398.89i		−	1.38049i
$79$	3753.49			0.601424			0.300712	−	0.953715i	$-0.402776\pi$
$79$	3753.49			0.601424			0.300712	+	0.953715i	$0.402776\pi$
$80$		−	8149.55i		−	1.27337i
$81$	729.000			0.111111
$82$	3007.71			0.447309
$83$	1970.28			0.286004			0.143002	−	0.989722i	$-0.454325\pi$
$83$	1970.28			0.286004			0.143002	+	0.989722i	$0.454325\pi$
$84$		−	233.183i		−	0.0330475i
$85$		−	19199.9i		−	2.65743i
$86$		−	2490.06i		−	0.336677i
$87$		−	6066.91i		−	0.801547i
$88$		−	1419.95i		−	0.183362i
$89$		−	13999.4i		−	1.76738i	−0.468074	−	0.883689i	$-0.655052\pi$
$89$		−	13999.4i		−	1.76738i	0.468074	−	0.883689i	$-0.344948\pi$
$90$			5968.35i			0.736833i
$91$	−680.792			−0.0822113
$92$	670.223			0.0791851
$93$	5230.45			0.604747
$94$			8959.02i			1.01392i
$95$			6524.50i			0.722936i
$96$		−	7631.78i		−	0.828101i
$97$	−1150.62			−0.122290			−0.0611448	−	0.998129i	$-0.519475\pi$
$97$	−1150.62			−0.122290			−0.0611448	+	0.998129i	$0.519475\pi$
$98$	14008.8			1.45865
$99$		−	2958.91i		−	0.301899i
$100$	−14628.8			−1.46288
$101$			7298.17i			0.715436i	0.933830	+	0.357718i	$0.116445\pi$
$101$			7298.17i			0.715436i	−0.933830	+	0.357718i	$0.883555\pi$
$102$		−	15442.2i		−	1.48426i
$103$	−10439.4	−	1889.29i	−0.984015	−	0.178084i
$103$	−10439.4	−	1889.29i	−0.984015	−	0.178084i
$104$	3580.44			0.331032
$105$	483.779			0.0438802
$106$		−	10282.6i		−	0.915147i
$107$	−6690.43			−0.584368			−0.292184	−	0.956362i	$-0.594382\pi$
$107$	−6690.43			−0.584368			−0.292184	+	0.956362i	$0.594382\pi$
$108$			2555.51i			0.219094i
$109$		−	9376.13i		−	0.789170i	−0.918859	−	0.394585i	$-0.870888\pi$
$109$		−	9376.13i		−	0.789170i	0.918859	−	0.394585i	$-0.129112\pi$
$110$	24224.7			2.00204
$111$	−5436.71			−0.441256
$112$	−531.295			−0.0423545
$113$			714.851i			0.0559833i	0.999608	+	0.0279917i	$0.00891119\pi$
$113$			714.851i			0.0559833i	−0.999608	+	0.0279917i	$0.991089\pi$
$114$			5247.55i			0.403782i
$115$			1390.49i			0.105141i
$116$	21267.5			1.58053
$117$	7460.96			0.545033
$118$	−32184.5			−2.31144
$119$	−1251.70			−0.0883909
$120$	−2544.31			−0.176688
$121$	2631.16			0.179712
$122$	2993.53			0.201124
$123$			2671.83i			0.176603i
$124$			18335.4i			1.19247i
$125$		−	6731.01i		−	0.430785i
$126$	389.096			0.0245084
$127$		−	11308.9i		−	0.701151i	−0.936534	−	0.350576i	$-0.885986\pi$
$127$		−	11308.9i		−	0.701151i	0.936534	−	0.350576i	$-0.114014\pi$
$128$	6570.43			0.401027
$129$	2211.99			0.132924
$130$			61083.2i			3.61439i
$131$	19100.8			1.11303			0.556517	−	0.830836i	$-0.312137\pi$
$131$	19100.8			1.11303			0.556517	+	0.830836i	$0.312137\pi$
$132$	10372.5			0.595298
$133$	425.353			0.0240462
$134$		−	18489.6i		−	1.02972i
$135$	−5301.85			−0.290911
$136$	6583.00			0.355915
$137$	−14363.5			−0.765279			−0.382640	−	0.923898i	$-0.624985\pi$
$137$	−14363.5			−0.765279			−0.382640	+	0.923898i	$0.624985\pi$
$138$			1118.35i			0.0587246i
$139$	−7209.37			−0.373137			−0.186568	−	0.982442i	$-0.559737\pi$
$139$	−7209.37			−0.373137			−0.186568	+	0.982442i	$0.559737\pi$
$140$			1695.89i			0.0865249i
$141$	−7958.54			−0.400309
$142$			26716.5i			1.32496i
$143$		−	30283.1i		−	1.48091i
$144$	5822.59			0.280796
$145$			44123.2i			2.09861i
$146$			44679.8i			2.09607i
$147$			12444.4i			0.575891i
$148$		−	19058.4i		−	0.870087i
$149$	−18216.2			−0.820513			−0.410256	−	0.911970i	$-0.634561\pi$
$149$	−18216.2			−0.820513			−0.410256	+	0.911970i	$0.634561\pi$
$150$		−	24410.0i		−	1.08489i
$151$		−	8386.29i		−	0.367804i	−0.982945	−	0.183902i	$-0.941127\pi$
$151$		−	8386.29i		−	0.367804i	0.982945	−	0.183902i	$-0.0588729\pi$
$152$	−2237.03			−0.0968243
$153$	13717.7			0.586002
$154$		−	1579.29i		−	0.0665916i
$155$	−38039.9			−1.58335
$156$			26154.4i			1.07472i
$157$		−	21627.6i		−	0.877422i	−0.898628	−	0.438711i	$-0.855435\pi$
$157$		−	21627.6i		−	0.877422i	0.898628	−	0.438711i	$-0.144565\pi$
$158$	−21955.5			−0.879488
$159$	9134.30			0.361311
$160$			55504.2i			2.16813i
$161$	90.6506			0.00349719
$162$	−4264.19			−0.162482
$163$	−33476.1			−1.25997			−0.629985	−	0.776607i	$-0.716939\pi$
$163$	−33476.1			−1.25997			−0.629985	+	0.776607i	$0.716939\pi$
$164$	−9366.09			−0.348234
$165$			21519.5i			0.790431i
$166$	−11524.9			−0.418235
$167$	46193.4			1.65633			0.828166	−	0.560483i	$-0.189385\pi$
$167$	46193.4			1.65633			0.828166	+	0.560483i	$0.189385\pi$
$168$			165.871i			0.00587696i
$169$	47798.3			1.67355
$170$			112307.i			3.88607i
$171$	−4661.54			−0.159418
$172$			7754.14i			0.262106i
$173$			23480.0i			0.784523i	0.919854	+	0.392262i	$0.128307\pi$
$173$			23480.0i			0.784523i	−0.919854	+	0.392262i	$0.871693\pi$
$174$			35487.6i			1.17214i
$175$	−1978.61			−0.0646078
$176$		−	23633.1i		−	0.762949i
$177$		−	28590.3i		−	0.912583i
$178$			81887.7i			2.58451i
$179$	32050.2			1.00029			0.500143	−	0.865943i	$-0.333281\pi$
$179$	32050.2			1.00029			0.500143	+	0.865943i	$0.333281\pi$
$180$		−	18585.6i		−	0.573630i
$181$		−	9908.10i		−	0.302436i	−0.988500	−	0.151218i	$-0.951680\pi$
$181$		−	9908.10i		−	0.302436i	0.988500	−	0.151218i	$-0.0483195\pi$
$182$	3982.20			0.120221
$183$			2659.24i			0.0794063i
$184$	−476.753			−0.0140818
$185$	39539.9			1.15529
$186$	−30594.9			−0.884347
$187$		−	55678.4i		−	1.59222i
$188$		−	27898.7i		−	0.789347i
$189$			345.644i			0.00967622i
$190$		−	38164.2i		−	1.05718i
$191$		−	63660.3i		−	1.74503i	−0.488591	−	0.872513i	$-0.662489\pi$
$191$		−	63660.3i		−	1.74503i	0.488591	−	0.872513i	$-0.337511\pi$
$192$			26712.2i			0.724615i
$193$		−	5411.54i		−	0.145280i	−0.997358	−	0.0726401i	$-0.976858\pi$
$193$		−	5411.54i		−	0.145280i	0.997358	−	0.0726401i	$-0.0231424\pi$
$194$	6730.41			0.178829
$195$	−54261.8			−1.42700
$196$	−43623.9			−1.13557
$197$			43643.4i			1.12457i	0.826944	+	0.562285i	$0.190077\pi$
$197$			43643.4i			1.12457i	−0.826944	+	0.562285i	$0.809923\pi$
$198$			17307.8i			0.441480i
$199$		−	78982.6i		−	1.99446i	−0.0743784	−	0.997230i	$-0.523697\pi$
$199$		−	78982.6i		−	1.99446i	0.0743784	−	0.997230i	$-0.476303\pi$
$200$	10406.0			0.260150
$201$	16424.8			0.406546
$202$		−	42689.7i		−	1.04621i
$203$	2876.53			0.0698034
$204$			48087.5i			1.15550i
$205$		−	19431.6i		−	0.462381i
$206$	61064.0	+	11051.1i	1.43897	+	0.260419i
$207$	−993.461			−0.0231852
$208$	59591.4			1.37739
$209$			18920.6i			0.433154i
$210$	−2829.80			−0.0641678
$211$			36457.1i			0.818873i	0.912339	+	0.409437i	$0.134275\pi$
$211$			36457.1i			0.818873i	−0.912339	+	0.409437i	$0.865725\pi$
$212$			32020.3i			0.712449i
$213$	−23733.0			−0.523110
$214$	39134.8			0.854547
$215$	−16087.3			−0.348022
$216$		−	1817.82i		−	0.0389622i
$217$			2479.94i			0.0526649i
$218$			54844.5i			1.15404i
$219$	−39690.2			−0.827552
$220$	−75436.6			−1.55861
$221$	140394.			2.87452
$222$	31801.3			0.645267
$223$	−40474.1			−0.813893			−0.406947	−	0.913452i	$-0.633407\pi$
$223$	−40474.1			−0.813893			−0.406947	+	0.913452i	$0.633407\pi$
$224$	3618.49			0.0721160
$225$	21684.1			0.428328
$226$		−	4181.43i		−	0.0818668i
$227$			43230.5i			0.838956i	0.907766	+	0.419478i	$0.137787\pi$
$227$			43230.5i			0.838956i	−0.907766	+	0.419478i	$0.862213\pi$
$228$		−	16341.0i		−	0.314348i
$229$	47437.8			0.904593			0.452296	−	0.891868i	$-0.350605\pi$
$229$	47437.8			0.904593			0.452296	+	0.891868i	$0.350605\pi$
$230$		−	8133.50i		−	0.153752i
$231$	1402.92			0.0262912
$232$	−15128.3			−0.281071
$233$			25456.3i			0.468904i	0.972128	+	0.234452i	$0.0753295\pi$
$233$			25456.3i			0.468904i	−0.972128	+	0.234452i	$0.924670\pi$
$234$	−43641.9			−0.797025
$235$	57880.6			1.04809
$236$	100223.			1.79947
$237$		−	19503.7i		−	0.347232i
$238$	7321.68			0.129258
$239$	−102594.			−1.79609			−0.898043	−	0.439908i	$-0.855011\pi$
$239$	−102594.			−1.79609			−0.898043	+	0.439908i	$0.855011\pi$
$240$	−42346.3			−0.735179
$241$			32530.6i			0.560089i	0.959987	+	0.280045i	$0.0903493\pi$
$241$			32530.6i			0.560089i	−0.959987	+	0.280045i	$0.909651\pi$
$242$	−15390.6			−0.262800
$243$		−	3788.00i		−	0.0641500i
$244$	−9321.96			−0.156577
$245$		−	90505.4i		−	1.50779i
$246$		−	15628.5i		−	0.258254i
$247$	−47708.6			−0.781993
$248$		−	13042.6i		−	0.212061i
$249$		−	10237.9i		−	0.165124i
$250$			39372.2i			0.629955i
$251$			47303.1i			0.750830i	0.926857	+	0.375415i	$0.122500\pi$
$251$			47303.1i			0.750830i	−0.926857	+	0.375415i	$0.877500\pi$
$252$	−1211.66			−0.0190800
$253$			4032.33i			0.0629963i
$254$			66149.7i			1.02532i
$255$	−99765.8			−1.53427
$256$	43819.4			0.668630
$257$		−	88775.0i		−	1.34408i	−0.740516	−	0.672039i	$-0.765419\pi$
$257$		−	88775.0i		−	1.34408i	0.740516	−	0.672039i	$-0.234581\pi$
$258$	−12938.8			−0.194381
$259$		−	2577.73i		−	0.0384272i
$260$		−	190215.i		−	2.81383i
$261$	−31524.6			−0.462773
$262$	−111728.			−1.62764
$263$			104454.i			1.51013i	0.655649	+	0.755065i	$0.272395\pi$
$263$			104454.i			1.51013i	−0.655649	+	0.755065i	$0.727605\pi$
$264$	−7378.30			−0.105864
$265$	−66431.6			−0.945983
$266$	−2488.05			−0.0351637
$267$	−72743.1			−1.02040
$268$			57577.3i			0.801645i
$269$	−113622.			−1.57020			−0.785102	−	0.619366i	$-0.787390\pi$
$269$	−113622.			−1.57020			−0.785102	+	0.619366i	$0.787390\pi$
$270$	31012.5			0.425411
$271$		−	114161.i		−	1.55446i	−0.629220	−	0.777228i	$-0.716625\pi$
$271$		−	114161.i		−	1.55446i	0.629220	−	0.777228i	$-0.283375\pi$
$272$	109565.			1.48092
$273$			3537.50i			0.0474647i
$274$	84017.6			1.11910
$275$		−	88012.8i		−	1.16381i
$276$		−	3482.58i		−	0.0457175i
$277$		−	118456.i		−	1.54382i	−0.635730	−	0.771912i	$-0.719301\pi$
$277$		−	118456.i		−	1.54382i	0.635730	−	0.771912i	$-0.280699\pi$
$278$	42170.3			0.545654
$279$		−	27178.2i		−	0.349151i
$280$		−	1206.34i		−	0.0153870i
$281$			102401.i			1.29685i	0.761277	+	0.648426i	$0.224573\pi$
$281$			102401.i			1.29685i	−0.761277	+	0.648426i	$0.775427\pi$
$282$	46552.4			0.585389
$283$		−	135378.i		−	1.69035i	−0.534490	−	0.845175i	$-0.679496\pi$
$283$		−	135378.i		−	1.69035i	0.534490	−	0.845175i	$-0.320504\pi$
$284$		−	83195.9i		−	1.03149i
$285$	33902.3			0.417388
$286$			177137.i			2.16559i
$287$	−1266.81			−0.0153796
$288$	−39655.9			−0.478105
$289$	174608.			2.09058
$290$		−	258093.i		−	3.06888i
$291$			5978.81i			0.0706039i
$292$		−	139134.i		−	1.63180i
$293$		−	110744.i		−	1.28998i	−0.764190	−	0.644991i	$-0.776861\pi$
$293$		−	110744.i		−	1.28998i	0.764190	−	0.644991i	$-0.223139\pi$
$294$		−	72792.0i		−	0.842149i
$295$			207931.i			2.38932i
$296$			13556.9i			0.154731i
$297$	−15375.0			−0.174302
$298$	106553.			1.19987
$299$	−10167.6			−0.113730
$300$			76013.6i			0.844595i
$301$			1048.78i			0.0115758i
$302$			49054.5i			0.537855i
$303$	37922.4			0.413057
$304$	−37232.1			−0.402876
$305$		−	19340.0i		−	0.207901i
$306$	−80240.0			−0.856935
$307$		−	22638.3i		−	0.240197i	−0.992762	−	0.120098i	$-0.961679\pi$
$307$		−	22638.3i		−	0.240197i	0.992762	−	0.120098i	$-0.0383210\pi$
$308$			4917.95i			0.0518421i
$309$	−9817.03	+	54244.8i	−0.102817	+	0.568122i
$310$	222509.			2.31539
$311$	−106298.			−1.09902			−0.549509	−	0.835488i	$-0.685185\pi$
$311$	−106298.			−1.09902			−0.549509	+	0.835488i	$0.685185\pi$
$312$		−	18604.5i		−	0.191122i
$313$	−159999.			−1.63315			−0.816577	−	0.577236i	$-0.804131\pi$
$313$	−159999.			−1.63315			−0.816577	+	0.577236i	$0.804131\pi$
$314$			126508.i			1.28309i
$315$		−	2513.79i		−	0.0253342i
$316$	68370.2			0.684688
$317$	31772.9			0.316183			0.158092	−	0.987424i	$-0.449466\pi$
$317$	31772.9			0.316183			0.158092	+	0.987424i	$0.449466\pi$
$318$	−53429.9			−0.528360
$319$			127954.i			1.25740i
$320$		−	194272.i		−	1.89718i
$321$			34764.5i			0.337385i
$322$	−530.249			−0.00511408
$323$	−87717.0			−0.840773
$324$	13278.8			0.126494
$325$	221926.			2.10108
$326$	195814.			1.84251
$327$	−48719.8			−0.455628
$328$	6662.42			0.0619277
$329$		−	3773.42i		−	0.0348613i
$330$		−	125875.i		−	1.15588i
$331$			20238.8i			0.184727i	0.995725	+	0.0923634i	$0.0294421\pi$
$331$			20238.8i			0.184727i	−0.995725	+	0.0923634i	$0.970558\pi$
$332$	35888.9			0.325599
$333$			28250.0i			0.254759i
$334$	−270202.			−2.42212
$335$	−119454.			−1.06442
$336$			2760.69i			0.0244534i
$337$	−107287.			−0.944686			−0.472343	−	0.881415i	$-0.656592\pi$
$337$	−107287.			−0.944686			−0.472343	+	0.881415i	$0.656592\pi$
$338$	−279590.			−2.44731
$339$	3714.47			0.0323220
$340$		−	349729.i		−	3.02534i
$341$	−110313.			−0.948675
$342$	27267.1			0.233124
$343$	−11815.6			−0.100431
$344$		−	5515.79i		−	0.0466112i
$345$	7225.21			0.0607033
$346$		−	137343.i		−	1.14724i
$347$	110904.			0.921065			0.460532	−	0.887643i	$-0.347659\pi$
$347$	110904.			0.921065			0.460532	+	0.887643i	$0.347659\pi$
$348$		−	110509.i		−	0.912517i
$349$			4221.65i			0.0346602i	0.999850	+	0.0173301i	$0.00551662\pi$
$349$			4221.65i			0.0346602i	−0.999850	+	0.0173301i	$0.994483\pi$
$350$	11573.6			0.0944787
$351$		−	38768.3i		−	0.314675i
$352$			160958.i			1.29905i
$353$		−	131272.i		−	1.05347i	−0.850030	−	0.526734i	$-0.823416\pi$
$353$		−	131272.i		−	1.05347i	0.850030	−	0.526734i	$-0.176584\pi$
$354$			167235.i			1.33451i
$355$	172604.			1.36960
$356$		−	255001.i		−	2.01206i
$357$			6504.04i			0.0510325i
$358$	−187473.			−1.46276
$359$	−63970.1			−0.496350			−0.248175	−	0.968715i	$-0.579831\pi$
$359$	−63970.1			−0.496350			−0.248175	+	0.968715i	$0.579831\pi$
$360$			13220.6i			0.102011i
$361$	−100513.			−0.771273
$362$			57956.1i			0.442265i
$363$		−	13671.9i		−	0.103757i
$364$	−12400.7			−0.0935931
$365$	288658.			2.16669
$366$		−	15554.9i		−	0.116119i
$367$	153292.			1.13812			0.569060	−	0.822296i	$-0.307307\pi$
$367$	153292.			1.13812			0.569060	+	0.822296i	$0.307307\pi$
$368$	−7934.86			−0.0585927
$369$	13883.2			0.101962
$370$	−231284.			−1.68943
$371$			4330.89i			0.0314651i
$372$	95273.3			0.688471
$373$	245924.			1.76760			0.883800	−	0.467865i	$-0.154977\pi$
$373$	245924.			1.76760			0.883800	+	0.467865i	$0.154977\pi$
$374$			325684.i			2.32837i
$375$	−34975.4			−0.248714
$376$			19845.3i			0.140372i
$377$	−322639.			−2.27004
$378$		−	2021.80i		−	0.0141499i
$379$			182232.i			1.26867i	0.773060	+	0.634333i	$0.218725\pi$
$379$			182232.i			1.26867i	−0.773060	+	0.634333i	$0.781275\pi$
$380$			118845.i			0.823023i
$381$	−58762.6			−0.404810
$382$			372372.i			2.55182i
$383$		−	168248.i		−	1.14697i	−0.819215	−	0.573486i	$-0.805591\pi$
$383$		−	168248.i		−	1.14697i	0.819215	−	0.573486i	$-0.194409\pi$
$384$		−	34140.9i		−	0.231533i
$385$	−10203.1			−0.0688355
$386$			31654.1i			0.212449i
$387$		−	11493.8i		−	0.0767438i
$388$	−20958.7			−0.139220
$389$			252806.i			1.67066i	0.549748	+	0.835330i	$0.314724\pi$
$389$			252806.i			1.67066i	−0.549748	+	0.835330i	$0.685276\pi$
$390$	317397.			2.08677
$391$	−18694.1			−0.122279
$392$	31031.2			0.201942
$393$		−	99250.6i		−	0.642611i
$394$		−	255286.i		−	1.64451i
$395$			141846.i			0.909122i
$396$		−	53897.0i		−	0.343696i
$397$			113845.i			0.722324i	0.932503	+	0.361162i	$0.117620\pi$
$397$			113845.i			0.722324i	−0.932503	+	0.361162i	$0.882380\pi$
$398$			461998.i			2.91658i
$399$		−	2210.20i		−	0.0138831i
$400$	173193.			1.08245
$401$	−9830.49			−0.0611345			−0.0305672	−	0.999533i	$-0.509731\pi$
$401$	−9830.49			−0.0611345			−0.0305672	+	0.999533i	$0.509731\pi$
$402$	−96075.0			−0.594509
$403$		−	278156.i		−	1.71269i
$404$			132937.i			0.814485i
$405$			27549.2i			0.167957i
$406$	−16825.9			−0.102077
$407$	114663.			0.692204
$408$		−	34206.3i		−	0.205488i
$409$	228979.			1.36883			0.684414	−	0.729093i	$-0.260058\pi$
$409$	228979.			1.36883			0.684414	+	0.729093i	$0.260058\pi$
$410$			113662.i			0.676160i
$411$			74635.1i			0.441834i
$412$	−190155.	−	34413.6i	−1.12025	−	0.202738i
$413$	13555.7			0.0794732
$414$	5811.12			0.0339047
$415$			74457.6i			0.432328i
$416$	−405859.			−2.34525
$417$			37461.0i			0.215431i
$418$		−	110673.i		−	0.633419i
$419$	93692.1			0.533673			0.266836	−	0.963742i	$-0.414022\pi$
$419$	93692.1			0.533673			0.266836	+	0.963742i	$0.414022\pi$
$420$	8812.09			0.0499551
$421$	−31827.7			−0.179573			−0.0897866	−	0.995961i	$-0.528619\pi$
$421$	−31827.7			−0.179573			−0.0897866	+	0.995961i	$0.528619\pi$
$422$		−	213251.i		−	1.19747i
$423$			41353.8i			0.231118i
$424$		−	22777.1i		−	0.126697i
$425$	408033.			2.25901
$426$	138823.			0.764965
$427$	−1260.84			−0.00691517
$428$	−121867.			−0.665271
$429$	−157355.			−0.855002
$430$	94100.5			0.508927
$431$	−1180.81			−0.00635663			−0.00317832	−	0.999995i	$-0.501012\pi$
$431$	−1180.81			−0.00635663			−0.00317832	+	0.999995i	$0.501012\pi$
$432$		−	30255.0i		−	0.162118i
$433$		−	60787.9i		−	0.324221i	−0.986773	−	0.162111i	$-0.948170\pi$
$433$		−	60787.9i		−	0.324221i	0.986773	−	0.162111i	$-0.0518301\pi$
$434$		−	14506.1i		−	0.0770142i
$435$	229271.			1.21163
$436$		−	170787.i		−	0.898427i
$437$	6352.62			0.0332652
$438$	232163.			1.21016
$439$			174126.i			0.903512i	0.892141	+	0.451756i	$0.149202\pi$
$439$			174126.i			0.903512i	−0.892141	+	0.451756i	$0.850798\pi$
$440$	53660.7			0.277173
$441$	64663.1			0.332491
$442$	−821217.			−4.20352
$443$		−	324337.i		−	1.65268i	−0.563170	−	0.826341i	$-0.690418\pi$
$443$		−	324337.i		−	1.65268i	0.563170	−	0.826341i	$-0.309582\pi$
$444$	−99030.3			−0.502345
$445$	529043.			2.67160
$446$	236748.			1.19019
$447$			94654.1i			0.473723i
$448$	−12665.2			−0.0631037
$449$			359695.i			1.78419i	0.451845	+	0.892096i	$0.350766\pi$
$449$			359695.i			1.78419i	−0.451845	+	0.892096i	$0.649234\pi$
$450$	−126838.			−0.626362
$451$		−	56350.2i		−	0.277040i
$452$			13021.1i			0.0637339i
$453$	−43576.5			−0.212352
$454$		−	252871.i		−	1.22684i
$455$		−	25727.4i		−	0.124272i
$456$			11623.9i			0.0559016i
$457$		−	242372.i		−	1.16051i	−0.814433	−	0.580257i	$-0.802952\pi$
$457$		−	242372.i		−	1.16051i	0.814433	−	0.580257i	$-0.197048\pi$
$458$	−277481.			−1.32282
$459$		−	71279.4i		−	0.338328i
$460$			25328.0i			0.119697i
$461$	−193110.			−0.908662			−0.454331	−	0.890833i	$-0.650122\pi$
$461$	−193110.			−0.908662			−0.454331	+	0.890833i	$0.650122\pi$
$462$	−8206.22			−0.0384467
$463$			374279.i			1.74596i	0.487760	+	0.872978i	$0.337814\pi$
$463$			374279.i			1.74596i	−0.487760	+	0.872978i	$0.662186\pi$
$464$	−251789.			−1.16950
$465$			197661.i			0.914145i
$466$		−	148903.i		−	0.685698i
$467$	−85584.5			−0.392429			−0.196215	−	0.980561i	$-0.562865\pi$
$467$	−85584.5			−0.392429			−0.196215	+	0.980561i	$0.562865\pi$
$468$	135902.			0.620490
$469$			7787.59i			0.0354044i
$470$	−338565.			−1.53266
$471$	−112380.			−0.506580
$472$	−71292.4			−0.320007
$473$	−46652.0			−0.208520
$474$			114084.i			0.507773i
$475$	−138657.			−0.614548
$476$	−22799.9			−0.100628
$477$		−	47463.2i		−	0.208603i
$478$	600111.			2.62649
$479$			27574.8i			0.120183i	0.998193	+	0.0600914i	$0.0191392\pi$
$479$			27574.8i			0.120183i	−0.998193	+	0.0600914i	$0.980861\pi$
$480$	288408.			1.25177
$481$			289125.i			1.24967i
$482$		−	190283.i		−	0.819043i
$483$		−	471.034i		−	0.00201910i
$484$	47926.8			0.204592
$485$		−	43482.5i		−	0.184855i
$486$			22157.4i			0.0938093i
$487$		−	2052.87i		−	0.00865574i	−0.999991	−	0.00432787i	$-0.998622\pi$
$487$		−	2052.87i		−	0.00865574i	0.999991	−	0.00432787i	$-0.00137761\pi$
$488$	6631.03			0.0278446
$489$			173947.i			0.727444i
$490$			529399.i			2.20491i
$491$	319868.			1.32681			0.663403	−	0.748262i	$-0.269112\pi$
$491$	319868.			1.32681			0.663403	+	0.748262i	$0.269112\pi$
$492$			48667.6i			0.201053i
$493$	−593203.			−2.44067
$494$	279065.			1.14354
$495$	111819.			0.456356
$496$		−	217075.i		−	0.882361i
$497$		−	11252.6i		−	0.0455555i
$498$			59885.1i			0.241468i
$499$			161998.i			0.650590i	0.945613	+	0.325295i	$0.105464\pi$
$499$			161998.i			0.650590i	−0.945613	+	0.325295i	$0.894536\pi$
$500$		−	122606.i		−	0.490425i
$501$		−	240028.i		−	0.956284i
$502$		−	276693.i		−	1.09797i
$503$	123138.			0.486694			0.243347	−	0.969939i	$-0.421755\pi$
$503$	123138.			0.486694			0.243347	+	0.969939i	$0.421755\pi$
$504$	861.893			0.00339306
$505$	−275801.			−1.08147
$506$		−	23586.6i		−	0.0921221i
$507$		−	248367.i		−	0.966226i
$508$		−	205992.i		−	0.798222i
$509$	−257206.			−0.992763			−0.496381	−	0.868105i	$-0.665338\pi$
$509$	−257206.			−0.992763			−0.496381	+	0.868105i	$0.665338\pi$
$510$	583567.			2.24362
$511$		−	18818.5i		−	0.0720682i
$512$	−361442.			−1.37879
$513$			24222.1i			0.0920400i
$514$			519278.i			1.96550i
$515$	71397.0	−	394510.i	0.269194	−	1.48745i
$516$	40291.7			0.151327
$517$	167850.			0.627970
$518$			15078.1i			0.0561936i
$519$	122006.			0.452945
$520$			135306.i			0.500394i
$521$		−	464400.i		−	1.71087i	−0.517911	−	0.855435i	$-0.673290\pi$
$521$		−	464400.i		−	1.71087i	0.517911	−	0.855435i	$-0.326710\pi$
$522$	184399.			0.676733
$523$	118508.			0.433256			0.216628	−	0.976254i	$-0.430494\pi$
$523$	118508.			0.433256			0.216628	+	0.976254i	$0.430494\pi$
$524$	347923.			1.26713
$525$			10281.2i			0.0373013i
$526$		−	610991.i		−	2.20833i
$527$		−	511418.i		−	1.84143i
$528$	−122801.			−0.440489
$529$	−278487.			−0.995162
$530$	388583.			1.38335
$531$	−148560.			−0.526880
$532$	7747.85			0.0273752
$533$	142088.			0.500153
$534$	425501.			1.49217
$535$		−	252834.i		−	0.883341i
$536$		−	40956.7i		−	0.142559i
$537$		−	166538.i		−	0.577516i
$538$	664615.			2.29618
$539$		−	262459.i		−	0.903408i
$540$	−96573.8			−0.331186
$541$	−171892.			−0.587302			−0.293651	−	0.955913i	$-0.594870\pi$
$541$	−171892.			−0.587302			−0.293651	+	0.955913i	$0.594870\pi$
$542$			667768.i			2.27315i
$543$	−51484.0			−0.174611
$544$	−746212.			−2.52153
$545$	354328.			1.19292
$546$		−	20692.1i		−	0.0694097i
$547$	−306920.			−1.02577			−0.512886	−	0.858456i	$-0.671424\pi$
$547$	−306920.			−1.02577			−0.512886	+	0.858456i	$0.671424\pi$
$548$	−261633.			−0.871228
$549$	13817.8			0.0458452
$550$			514820.i			1.70188i
$551$	201582.			0.663969
$552$			2477.28i			0.00813012i
$553$	9247.38			0.0302391
$554$			692893.i			2.25760i
$555$		−	205455.i		−	0.667009i
$556$	−131320.			−0.424796
$557$		−	125671.i		−	0.405064i	−0.979276	−	0.202532i	$-0.935083\pi$
$557$		−	125671.i		−	0.405064i	0.979276	−	0.202532i	$-0.0649171\pi$
$558$			158976.i			0.510578i
$559$		−	117634.i		−	0.376451i
$560$		−	20077.8i		−	0.0640238i
$561$	−289313.			−0.919270
$562$		−	598980.i		−	1.89644i
$563$			143691.i			0.453327i	0.973973	+	0.226663i	$0.0727817\pi$
$563$			143691.i			0.453327i	−0.973973	+	0.226663i	$0.927218\pi$
$564$	−144966.			−0.455730
$565$	−27014.5			−0.0846253
$566$			791878.i			2.47187i
$567$	1796.02			0.00558657
$568$			59180.1i			0.183434i
$569$			578029.i			1.78536i	0.450694	+	0.892678i	$0.351177\pi$
$569$			578029.i			1.78536i	−0.450694	+	0.892678i	$0.648823\pi$
$570$	−198307.			−0.610363
$571$	−344768.			−1.05744			−0.528718	−	0.848797i	$-0.677327\pi$
$571$	−344768.			−1.05744			−0.528718	+	0.848797i	$0.677327\pi$
$572$		−	551609.i		−	1.68593i
$573$	−330789.			−1.00749
$574$	7410.01			0.0224903
$575$	−29550.5			−0.0893776
$576$	138801.			0.418356
$577$		−	400794.i		−	1.20384i	−0.798555	−	0.601922i	$-0.794402\pi$
$577$		−	400794.i		−	1.20384i	0.798555	−	0.601922i	$-0.205598\pi$
$578$	−1.02134e6			−3.05715
$579$	−28119.2			−0.0838775
$580$			803709.i			2.38915i
$581$	4854.13			0.0143800
$582$		−	34972.3i		−	0.103247i
$583$	−192647.			−0.566794
$584$			98970.9i			0.290190i
$585$			281953.i			0.823881i
$586$			647781.i			1.88640i
$587$	−19140.4			−0.0555489			−0.0277745	−	0.999614i	$-0.508842\pi$
$587$	−19140.4			−0.0555489			−0.0277745	+	0.999614i	$0.508842\pi$
$588$			226677.i			0.655620i
$589$			173789.i			0.500948i
$590$		−	1.21626e6i		−	3.49401i
$591$	226778.			0.649270
$592$			225635.i			0.643818i
$593$			295168.i			0.839382i	0.907667	+	0.419691i	$0.137862\pi$
$593$			295168.i			0.839382i	−0.907667	+	0.419691i	$0.862138\pi$
$594$	89933.9			0.254889
$595$		−	47302.4i		−	0.133613i
$596$	−331810.			−0.934108
$597$	−410406.			−1.15150
$598$	59474.0			0.166312
$599$			584288.i			1.62844i	0.580553	+	0.814222i	$0.302836\pi$
$599$			584288.i			1.62844i	−0.580553	+	0.814222i	$0.697164\pi$
$600$		−	54071.1i		−	0.150197i
$601$			401263.i			1.11091i	0.831546	+	0.555456i	$0.187456\pi$
$601$			401263.i			1.11091i	−0.831546	+	0.555456i	$0.812544\pi$
$602$		−	6134.71i		−	0.0169278i
$603$		−	85346.0i		−	0.234719i
$604$		−	152757.i		−	0.418724i
$605$			99432.5i			0.271655i
$606$	−221822.			−0.604031
$607$	325232.			0.882705			0.441353	−	0.897334i	$-0.354499\pi$
$607$	325232.			0.882705			0.441353	+	0.897334i	$0.354499\pi$
$608$	253577.			0.685966
$609$		−	14946.9i		−	0.0403010i
$610$			113127.i			0.304023i
$611$			423236.i			1.13371i
$612$	249870.			0.667131
$613$	282740.			0.752432			0.376216	−	0.926532i	$-0.377225\pi$
$613$	282740.			0.752432			0.376216	+	0.926532i	$0.377225\pi$
$614$			132420.i			0.351250i
$615$	−100969.			−0.266956
$616$		−	3498.31i		−	0.00921927i
$617$			465221.i			1.22205i	0.791611	+	0.611025i	$0.209243\pi$
$617$			465221.i			1.22205i	−0.791611	+	0.611025i	$0.790757\pi$
$618$	57423.4	−	317298.i	0.150353	−	0.830788i
$619$	298621.			0.779361			0.389680	−	0.920950i	$-0.372585\pi$
$619$	298621.			0.779361			0.389680	+	0.920950i	$0.372585\pi$
$620$	−692901.			−1.80255
$621$			5162.18i			0.0133860i
$622$	621777.			1.60714
$623$		−	34490.0i		−	0.0888622i
$624$		−	309646.i		−	0.795236i
$625$	−247579.			−0.633802
$626$	935890.			2.38823
$627$	98314.2			0.250081
$628$		−	393949.i		−	0.998897i
$629$			531585.i			1.34360i
$630$			14704.1i			0.0370473i
$631$	−81643.7			−0.205052			−0.102526	−	0.994730i	$-0.532692\pi$
$631$	−81643.7			−0.205052			−0.102526	+	0.994730i	$0.532692\pi$
$632$	−48634.1			−0.121761
$633$	189436.			0.472777
$634$	−185852.			−0.462368
$635$	427367.			1.05987
$636$	166382.			0.411333
$637$	661796.			1.63097
$638$		−	748450.i		−	1.83875i
$639$			123320.i			0.302017i
$640$			248299.i			0.606199i
$641$	533125.			1.29752			0.648758	−	0.760995i	$-0.275289\pi$
$641$	533125.			1.29752			0.648758	+	0.760995i	$0.275289\pi$
$642$		−	203350.i		−	0.493373i
$643$	−29056.4			−0.0702780			−0.0351390	−	0.999382i	$-0.511187\pi$
$643$	−29056.4			−0.0702780			−0.0351390	+	0.999382i	$0.511187\pi$
$644$	1651.21			0.00398136
$645$			83592.0i			0.200930i
$646$	513089.			1.22950
$647$	−705752.			−1.68594			−0.842972	−	0.537957i	$-0.819196\pi$
$647$	−705752.			−1.68594			−0.842972	+	0.537957i	$0.819196\pi$
$648$	−9445.68			−0.0224949
$649$			602984.i			1.43158i
$650$	−1.29813e6			−3.07249
$651$	12886.1			0.0304061
$652$	−609772.			−1.43441
$653$			454088.i			1.06491i	0.846458	+	0.532456i	$0.178731\pi$
$653$			454088.i			1.06491i	−0.846458	+	0.532456i	$0.821269\pi$
$654$	284980.			0.666284
$655$			721826.i			1.68248i
$656$	110886.			0.257674
$657$			206237.i			0.477788i
$658$			22072.1i			0.0509791i
$659$	−239663.			−0.551861			−0.275931	−	0.961178i	$-0.588986\pi$
$659$	−239663.			−0.551861			−0.275931	+	0.961178i	$0.588986\pi$
$660$			391980.i			0.899862i
$661$		−	127238.i		−	0.291215i	−0.989342	−	0.145608i	$-0.953486\pi$
$661$		−	127238.i		−	0.291215i	0.989342	−	0.145608i	$-0.0465137\pi$
$662$		−	118384.i		−	0.270134i
$663$		−	729510.i		−	1.65960i
$664$	−25529.0			−0.0579025
$665$			16074.3i			0.0363486i
$666$		−	165245.i		−	0.372545i
$667$	42960.8			0.0965653
$668$	841419.			1.88564
$669$			210310.i			0.469902i
$670$	698731.			1.55654
$671$		−	56084.6i		−	0.124566i
$672$		−	18802.2i		−	0.0416362i
$673$	−769932.			−1.69990			−0.849948	−	0.526866i	$-0.823367\pi$
$673$	−769932.			−1.69990			−0.849948	+	0.526866i	$0.823367\pi$
$674$	627562.			1.38145
$675$		−	112674.i		−	0.247295i
$676$	870652.			1.90525
$677$	−295000.			−0.643642			−0.321821	−	0.946800i	$-0.604295\pi$
$677$	−295000.			−0.643642			−0.321821	+	0.946800i	$0.604295\pi$
$678$	−21727.3			−0.0472658
$679$	−2834.76			−0.00614861
$680$			248774.i			0.538007i
$681$	224633.			0.484371
$682$	645261.			1.38729
$683$		−	764789.i		−	1.63946i	−0.572751	−	0.819729i	$-0.694124\pi$
$683$		−	764789.i		−	1.63946i	0.572751	−	0.819729i	$-0.305876\pi$
$684$	−84910.5			−0.181489
$685$		−	542804.i		−	1.15681i
$686$	69113.9			0.146865
$687$		−	246494.i		−	0.522267i
$688$		−	91802.3i		−	0.193944i
$689$		−	485763.i		−	1.02326i
$690$	−42262.9			−0.0887690
$691$		−	412793.i		−	0.864523i	−0.901748	−	0.432262i	$-0.857716\pi$
$691$		−	412793.i		−	0.864523i	0.901748	−	0.432262i	$-0.142284\pi$
$692$			427691.i			0.893137i
$693$		−	7289.80i		−	0.0151792i
$694$	−648721.			−1.34691
$695$		−	272445.i		−	0.564039i
$696$			78609.2i			0.162276i
$697$	261243.			0.537748
$698$		−	24694.0i		−	0.0506851i
$699$	132275.			0.270722
$700$	−36040.7			−0.0735524
$701$	−165871.			−0.337547			−0.168774	−	0.985655i	$-0.553981\pi$
$701$	−165871.			−0.337547			−0.168774	+	0.985655i	$0.553981\pi$
$702$			226770.i			0.460163i
$703$		−	180643.i		−	0.365519i
$704$		−	563373.i		−	1.13671i
$705$		−	300756.i		−	0.605113i
$706$			767856.i			1.54053i
$707$			17980.3i			0.0359715i
$708$		−	520776.i		−	1.03893i
$709$	−5067.44			−0.0100808			−0.00504041	−	0.999987i	$-0.501604\pi$
$709$	−5067.44			−0.0100808			−0.00504041	+	0.999987i	$0.501604\pi$
$710$	−1.00963e6			−2.00283
$711$	−101344.			−0.200475
$712$			181391.i			0.357813i
$713$			37037.8i			0.0728561i
$714$		−	38044.6i		−	0.0746270i
$715$	1.14441e6			2.23856
$716$	583798.			1.13877
$717$			533095.i			1.03697i
$718$	374185.			0.725834
$719$			746876.i			1.44474i	0.691505	+	0.722371i	$0.256948\pi$
$719$			746876.i			1.44474i	−0.691505	+	0.722371i	$0.743052\pi$
$720$			220038.i			0.424456i
$721$	−25719.3	−	4654.59i	−0.0494754	−	0.00895388i
$722$	587938.			1.12787
$723$	169034.			0.323368
$724$		−	180477.i		−	0.344306i
$725$	−937698.			−1.78397
$726$			79972.0i			0.151728i
$727$		−	91807.3i		−	0.173703i	−0.996221	−	0.0868517i	$-0.972319\pi$
$727$		−	91807.3i		−	0.173703i	0.996221	−	0.0868517i	$-0.0276806\pi$
$728$	8821.05			0.0166440
$729$	−19683.0			−0.0370370
$730$	−1.68847e6			−3.16845
$731$		−	216282.i		−	0.404748i
$732$			48438.3i			0.0903997i
$733$		−	292160.i		−	0.543766i	−0.962330	−	0.271883i	$-0.912354\pi$
$733$		−	292160.i		−	0.543766i	0.962330	−	0.271883i	$-0.0876464\pi$
$734$	−896663.			−1.66432
$735$	−470280.			−0.870526
$736$	54042.0			0.0997644
$737$	−346408.			−0.637754
$738$	−81208.1			−0.149103
$739$	505583.			0.925770			0.462885	−	0.886418i	$-0.346814\pi$
$739$	505583.			0.925770			0.462885	+	0.886418i	$0.346814\pi$
$740$	720224.			1.31524
$741$			247901.i			0.451484i
$742$		−	25333.0i		−	0.0460127i
$743$		−	312404.i		−	0.565899i	−0.959135	−	0.282949i	$-0.908687\pi$
$743$		−	312404.i		−	0.565899i	0.959135	−	0.282949i	$-0.0913129\pi$
$744$	−67771.2			−0.122433
$745$		−	688397.i		−	1.24030i
$746$	−1.43850e6			−2.58484
$747$	−53197.5			−0.0953345
$748$		−	1.01419e6i		−	1.81266i
$749$	−16483.1			−0.0293815
$750$	204584.			0.363705
$751$	−572878.			−1.01574			−0.507869	−	0.861434i	$-0.669567\pi$
$751$	−572878.			−1.01574			−0.507869	+	0.861434i	$0.669567\pi$
$752$			330296.i			0.584074i
$753$	245794.			0.433492
$754$	1.88723e6			3.31958
$755$	316921.			0.555978
$756$			6295.95i			0.0110158i
$757$	−639913.			−1.11668			−0.558341	−	0.829612i	$-0.688562\pi$
$757$	−639913.			−1.11668			−0.558341	+	0.829612i	$0.688562\pi$
$758$		−	1.06594e6i		−	1.85522i
$759$	20952.6			0.0363709
$760$		−	84538.2i		−	0.146361i
$761$		−	520139.i		−	0.898152i	−0.893494	−	0.449076i	$-0.851753\pi$
$761$		−	520139.i		−	0.898152i	0.893494	−	0.449076i	$-0.148247\pi$
$762$	343724.			0.591971
$763$		−	23099.8i		−	0.0396788i
$764$		−	1.15958e6i		−	1.98662i
$765$			518398.i			0.885810i
$766$			984146.i			1.67727i
$767$	−1.52044e6			−2.58450
$768$		−	227692.i		−	0.386034i
$769$		−	583882.i		−	0.987353i	−0.869646	−	0.493677i	$-0.835653\pi$
$769$		−	583882.i		−	0.987353i	0.869646	−	0.493677i	$-0.164347\pi$
$770$	59681.9			0.100661
$771$	−461289.			−0.776004
$772$		−	98571.8i		−	0.165393i
$773$	−373608.			−0.625256			−0.312628	−	0.949876i	$-0.601209\pi$
$773$	−373608.			−0.625256			−0.312628	+	0.949876i	$0.601209\pi$
$774$			67231.7i			0.112226i
$775$		−	808416.i		−	1.34596i
$776$	14908.7			0.0247580
$777$	−13394.3			−0.0221859
$778$		−	1.47876e6i		−	2.44308i
$779$	−88775.3			−0.146291
$780$	−988386.			−1.62457
$781$	500540.			0.820610
$782$	109349.			0.178814
$783$			163806.i			0.267182i
$784$	516470.			0.840259
$785$	817315.			1.32633
$786$			580553.i			0.939717i
$787$	479065.			0.773473			0.386737	−	0.922190i	$-0.373602\pi$
$787$	479065.			0.773473			0.386737	+	0.922190i	$0.373602\pi$
$788$			794970.i			1.28026i
$789$	542760.			0.871874
$790$		−	829708.i		−	1.32945i
$791$			1761.16i			0.00281479i
$792$			38338.8i			0.0611206i
$793$	141418.			0.224885
$794$		−	665920.i		−	1.05628i
$795$			345189.i			0.546163i
$796$		−	1.43868e6i		−	2.27058i
$797$	−143251.			−0.225518			−0.112759	−	0.993622i	$-0.535969\pi$
$797$	−143251.			−0.225518			−0.112759	+	0.993622i	$0.535969\pi$
$798$			12928.3i			0.0203018i
$799$			778161.i			1.21892i
$800$	−1.17956e6			−1.84307
$801$			377984.i			0.589126i
$802$	57502.1			0.0893995
$803$	837087.			1.29819
$804$	299181.			0.462830
$805$			3425.72i			0.00528640i
$806$			1.62704e6i			2.50454i
$807$			590395.i			0.906558i
$808$		−	94562.7i		−	0.144843i
$809$		−	371378.i		−	0.567439i	−0.958907	−	0.283719i	$-0.908432\pi$
$809$		−	371378.i		−	0.567439i	0.958907	−	0.283719i	$-0.0915684\pi$
$810$		−	161145.i		−	0.245611i
$811$		−	1.00561e6i		−	1.52893i	−0.644665	−	0.764465i	$-0.723003\pi$
$811$		−	1.00561e6i		−	1.52893i	0.644665	−	0.764465i	$-0.276997\pi$
$812$	52396.3			0.0794674
$813$	−593197.			−0.897465
$814$	−670706.			−1.01224
$815$		−	1.26508e6i		−	1.90459i
$816$		−	569314.i		−	0.855011i
$817$			73496.6i			0.110109i
$818$	−1.33938e6			−2.00170
$819$	18381.4			0.0274038
$820$		−	353948.i		−	0.526396i
$821$	1.24017e6			1.83990			0.919948	−	0.392039i	$-0.128230\pi$
$821$	1.24017e6			1.83990			0.919948	+	0.392039i	$0.128230\pi$
$822$		−	436568.i		−	0.646113i
$823$			234344.i			0.345982i	0.984923	+	0.172991i	$0.0553431\pi$
$823$			234344.i			0.345982i	−0.984923	+	0.172991i	$0.944657\pi$
$824$	135264.	+	24479.6i	0.199218	+	0.0360537i
$825$	−457328.			−0.671924
$826$	−79292.0			−0.116217
$827$			295102.i			0.431481i	0.976451	+	0.215740i	$0.0692165\pi$
$827$			295102.i			0.431481i	−0.976451	+	0.215740i	$0.930784\pi$
$828$	−18096.0			−0.0263950
$829$		−	1.08651e6i		−	1.58098i	−0.612478	−	0.790488i	$-0.709827\pi$
$829$		−	1.08651e6i		−	1.58098i	0.612478	−	0.790488i	$-0.290173\pi$
$830$		−	435530.i		−	0.632211i
$831$	−615515.			−0.891327
$832$	1.42056e6			2.05216
$833$	1.21678e6			1.75356
$834$		−	219123.i		−	0.315033i
$835$			1.74567e6i			2.50374i
$836$			344641.i			0.493121i
$837$	−141222.			−0.201582
$838$	−548040.			−0.780412
$839$	159200.			0.226161			0.113081	−	0.993586i	$-0.463928\pi$
$839$	159200.			0.226161			0.113081	+	0.993586i	$0.463928\pi$
$840$	−6268.34			−0.00888371
$841$	655954.			0.927431
$842$	186172.			0.262598
$843$	532090.			0.748738
$844$			664070.i			0.932242i
$845$			1.80632e6i			2.52977i
$846$		−	241894.i		−	0.337974i
$847$	6482.32			0.00903574
$848$		−	379093.i		−	0.527174i
$849$	−703447.			−0.975924
$850$	−2.38674e6			−3.30344
$851$		−	38498.3i		−	0.0531597i
$852$	−432299.			−0.595532
$853$	−1.25883e6			−1.73010			−0.865048	−	0.501689i	$-0.832712\pi$
$853$	−1.25883e6			−1.73010			−0.865048	+	0.501689i	$0.832712\pi$
$854$	7375.10			0.0101123
$855$		−	176162.i		−	0.240979i
$856$	86688.2			0.118308
$857$	317337.			0.432076			0.216038	−	0.976385i	$-0.430687\pi$
$857$	317337.			0.432076			0.216038	+	0.976385i	$0.430687\pi$
$858$	920430.			1.25031
$859$			31163.9i			0.0422343i	0.999777	+	0.0211172i	$0.00672230\pi$
$859$			31163.9i			0.0422343i	−0.999777	+	0.0211172i	$0.993278\pi$
$860$	−293032.			−0.396203
$861$			6582.51i			0.00887944i
$862$	6907.02			0.00929557
$863$		−	1.21307e6i		−	1.62879i	−0.580311	−	0.814395i	$-0.697069\pi$
$863$		−	1.21307e6i		−	1.62879i	0.580311	−	0.814395i	$-0.302931\pi$
$864$			206058.i			0.276034i
$865$	−887318.			−1.18590
$866$			355571.i			0.474122i
$867$		−	907288.i		−	1.20700i
$868$			45172.4i			0.0599561i
$869$			411343.i			0.544708i
$870$	−1.34109e6			−1.77182
$871$		−	873475.i		−	1.15137i
$872$			121487.i			0.159771i
$873$	31066.8			0.0407632
$874$	−37158.8			−0.0486451
$875$		−	16583.0i		−	0.0216595i
$876$	−722962.			−0.942123
$877$		−	221859.i		−	0.288455i	−0.989545	−	0.144228i	$-0.953930\pi$
$877$		−	221859.i		−	0.288455i	0.989545	−	0.144228i	$-0.0460698\pi$
$878$		−	1.01853e6i		−	1.32124i
$879$	−575441.			−0.744772
$880$	893105.			1.15329
$881$			603333.i			0.777330i	0.921379	+	0.388665i	$0.127064\pi$
$881$			603333.i			0.777330i	−0.921379	+	0.388665i	$0.872936\pi$
$882$	−378238.			−0.486215
$883$	−807896.			−1.03618			−0.518088	−	0.855327i	$-0.673356\pi$
$883$	−807896.			−1.03618			−0.518088	+	0.855327i	$0.673356\pi$
$884$	2.55730e6			3.27248
$885$	1.08044e6			1.37948
$886$			1.89717e6i			2.41679i
$887$	−28404.7			−0.0361030			−0.0180515	−	0.999837i	$-0.505746\pi$
$887$	−28404.7			−0.0361030			−0.0180515	+	0.999837i	$0.505746\pi$
$888$	70443.7			0.0893339
$889$		−	27861.4i		−	0.0352532i
$890$	−3.09457e6			−3.90679
$891$			79890.7i			0.100633i
$892$	−737241.			−0.926573
$893$		−	264434.i		−	0.331600i
$894$		−	553667.i		−	0.692745i
$895$			1.21119e6i			1.51205i
$896$	16187.4			0.0201633
$897$			52832.4i			0.0656622i
$898$		−	2.10399e6i		−	2.60910i
$899$			1.17528e6i			1.45420i
$900$	394978.			0.487627
$901$		−	893124.i		−	1.10018i
$902$			329613.i			0.405127i
$903$	5449.63			0.00668331
$904$		−	9262.36i		−	0.0113340i
$905$	374431.			0.457167
$906$	254895.			0.310531
$907$	−106139.			−0.129021			−0.0645106	−	0.997917i	$-0.520549\pi$
$907$	−106139.			−0.129021			−0.0645106	+	0.997917i	$0.520549\pi$
$908$			787450.i			0.955105i
$909$		−	197051.i		−	0.238479i
$910$			150489.i			0.181728i
$911$			64014.3i			0.0771330i	0.999256	+	0.0385665i	$0.0122792\pi$
$911$			64014.3i			0.0771330i	−0.999256	+	0.0385665i	$0.987721\pi$
$912$			193464.i			0.232600i
$913$			215922.i			0.259033i
$914$			1.41773e6i			1.69707i
$915$	−100494.			−0.120032
$916$	864084.			1.02983
$917$	47058.1			0.0559624
$918$			416939.i			0.494752i
$919$		−	367848.i		−	0.435549i	−0.975999	−	0.217774i	$-0.930120\pi$
$919$		−	367848.i		−	0.435549i	0.975999	−	0.217774i	$-0.0698797\pi$
$920$		−	18016.7i		−	0.0212862i
$921$	−117632.			−0.138678
$922$	1.12957e6			1.32878
$923$			1.26212e6i			1.48149i
$924$	25554.4			0.0299311
$925$			840295.i			0.982083i
$926$		−	2.18929e6i		−	2.55318i
$927$	281864.	+	51010.8i	0.328005	+	0.0593612i
$928$	1.71486e6			1.99129
$929$	907959.			1.05205			0.526023	−	0.850470i	$-0.323683\pi$
$929$	907959.			1.05205			0.526023	+	0.850470i	$0.323683\pi$
$930$		−	1.15619e6i		−	1.33679i
$931$	−413484.			−0.477045
$932$			463690.i			0.533821i
$933$			552341.i			0.634518i
$934$	500615.			0.573866
$935$	2.10411e6			2.40683
$936$	−96672.0			−0.110344
$937$			765350.i			0.871728i	0.900012	+	0.435864i	$0.143557\pi$
$937$			765350.i			0.871728i	−0.900012	+	0.435864i	$0.856443\pi$
$938$		−	45552.5i		−	0.0517734i
$939$			831377.i			0.942902i
$940$	1.05430e6			1.19319
$941$	605326.			0.683613			0.341806	−	0.939770i	$-0.388961\pi$
$941$	605326.			0.683613			0.341806	+	0.939770i	$0.388961\pi$
$942$	657353.			0.740793
$943$	−18919.7			−0.0212760
$944$	−1.18656e6			−1.33151
$945$	−13062.0			−0.0146267
$946$	272885.			0.304928
$947$		−	620073.i		−	0.691421i	−0.938341	−	0.345711i	$-0.887638\pi$
$947$		−	620073.i		−	0.691421i	0.938341	−	0.345711i	$-0.112362\pi$
$948$		−	355262.i		−	0.395305i
$949$			2.11073e6i			2.34369i
$950$	811059.			0.898680
$951$		−	165097.i		−	0.182548i
$952$	16218.4			0.0178951
$953$	742637.			0.817694			0.408847	−	0.912603i	$-0.365931\pi$
$953$	742637.			0.817694			0.408847	+	0.912603i	$0.365931\pi$
$954$			277630.i			0.305049i
$955$	2.40575e6			2.63781
$956$	−1.86877e6			−2.04474
$957$	664869.			0.725959
$958$		−	161295.i		−	0.175748i
$959$	−35387.1			−0.0384776
$960$	−1.00946e6			−1.09534
$961$	−89725.4			−0.0971558
$962$		−	1.69120e6i		−	1.82744i
$963$	180642.			0.194789
$964$			592548.i			0.637631i
$965$	204504.			0.219608
$966$			2755.25i			0.00295262i
$967$			54764.8i			0.0585664i	0.999571	+	0.0292832i	$0.00932247\pi$
$967$			54764.8i			0.0585664i	−0.999571	+	0.0292832i	$0.990678\pi$
$968$	−34092.0			−0.0363833
$969$			455791.i			0.485421i
$970$			254345.i			0.270321i
$971$		−	1.41752e6i		−	1.50345i	−0.659475	−	0.751727i	$-0.729221\pi$
$971$		−	1.41752e6i		−	1.50345i	0.659475	−	0.751727i	$-0.270779\pi$
$972$		−	68998.8i		−	0.0730313i
$973$	−17761.6			−0.0187610
$974$			12008.0i			0.0126577i
$975$		−	1.15316e6i		−	1.21306i
$976$	110364.			0.115858
$977$	875882.			0.917606			0.458803	−	0.888538i	$-0.348278\pi$
$977$	875882.			0.917606			0.458803	+	0.888538i	$0.348278\pi$
$978$		−	1.01748e6i		−	1.06377i
$979$	1.53419e6			1.60071
$980$		−	1.64857e6i		−	1.71654i
$981$			253156.i			0.263057i
$982$	−1.87102e6			−1.94024
$983$	−763022.			−0.789642			−0.394821	−	0.918758i	$-0.629193\pi$
$983$	−763022.			−0.789642			−0.394821	+	0.918758i	$0.629193\pi$
$984$		−	34619.0i		−	0.0357539i
$985$	−1.64930e6			−1.69992
$986$	3.46986e6			3.56910
$987$	−19607.3			−0.0201272
$988$	−869018.			−0.890256
$989$			15663.5i			0.0160139i
$990$	−654068.			−0.667348
$991$	−1.05943e6			−1.07876			−0.539378	−	0.842064i	$-0.681341\pi$
$991$	−1.05943e6			−1.07876			−0.539378	+	0.842064i	$0.681341\pi$
$992$			1.47843e6i			1.50237i
$993$	105164.			0.106652
$994$			65820.7i			0.0666177i
$995$	2.98478e6			3.01486
$996$		−	186484.i		−	0.187985i
$997$			1.16815e6i			1.17519i	0.809154	+	0.587597i	$0.199926\pi$
$997$			1.16815e6i			1.17519i	−0.809154	+	0.587597i	$0.800074\pi$
$998$		−	947584.i		−	0.951386i
$999$	146791.			0.147085

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	309.5.d.a.205.11	✓	70
103.102	odd	2	inner	309.5.d.a.205.12	yes	70

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
309.5.d.a.205.11	✓	70	1.1	even	1	trivial
309.5.d.a.205.12	yes	70	103.102	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 309 = 3 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	309.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.84937 q^{2} -5.19615i q^{3} +18.2151 q^{4} +37.7904i q^{5} +30.3942i q^{6} +2.46368 q^{7} -12.9570 q^{8} -27.0000 q^{9} +O(q^{10})\)	\(q-5.84937 q^{2} -5.19615i q^{3} +18.2151 q^{4} +37.7904i q^{5} +30.3942i q^{6} +2.46368 q^{7} -12.9570 q^{8} -27.0000 q^{9} -221.050i q^{10} +109.589i q^{11} -94.6485i q^{12} -276.332 q^{13} -14.4109 q^{14} +196.365 q^{15} -215.651 q^{16} -508.064 q^{17} +157.933 q^{18} +172.650 q^{19} +688.357i q^{20} -12.8016i q^{21} -641.029i q^{22} +36.7949 q^{23} +67.3268i q^{24} -803.114 q^{25} +1616.37 q^{26} +140.296i q^{27} +44.8761 q^{28} +1167.58 q^{29} -1148.61 q^{30} +1006.60i q^{31} +1468.74 q^{32} +569.443 q^{33} +2971.85 q^{34} +93.1033i q^{35} -491.808 q^{36} -1046.30i q^{37} -1009.89 q^{38} +1435.86i q^{39} -489.652i q^{40} -514.193 q^{41} +74.8815i q^{42} +425.698i q^{43} +1996.18i q^{44} -1020.34i q^{45} -215.227 q^{46} -1531.62i q^{47} +1120.56i q^{48} -2394.93 q^{49} +4697.71 q^{50} +2639.98i q^{51} -5033.42 q^{52} +1757.90i q^{53} -820.644i q^{54} -4141.43 q^{55} -31.9219 q^{56} -897.114i q^{57} -6829.59 q^{58} +5502.21 q^{59} +3576.81 q^{60} -511.770 q^{61} -5887.98i q^{62} -66.5192 q^{63} -5140.76 q^{64} -10442.7i q^{65} -3330.88 q^{66} +3160.96i q^{67} -9254.44 q^{68} -191.192i q^{69} -544.595i q^{70} -4567.41i q^{71} +349.840 q^{72} -7638.39i q^{73} +6120.17i q^{74} +4173.10i q^{75} +3144.83 q^{76} +269.993i q^{77} -8398.89i q^{78} +3753.49 q^{79} -8149.55i q^{80} +729.000 q^{81} +3007.71 q^{82} +1970.28 q^{83} -233.183i q^{84} -19199.9i q^{85} -2490.06i q^{86} -6066.91i q^{87} -1419.95i q^{88} -13999.4i q^{89} +5968.35i q^{90} -680.792 q^{91} +670.223 q^{92} +5230.45 q^{93} +8959.02i q^{94} +6524.50i q^{95} -7631.78i q^{96} -1150.62 q^{97} +14008.8 q^{98} -2958.91i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 70 q + 576 q^{4} - 56 q^{7} - 180 q^{8} - 1890 q^{9}+O(q^{10}) \) 70 * q + 576 * q^4 - 56 * q^7 - 180 * q^8 - 1890 * q^9	\( 70 q + 576 q^{4} - 56 q^{7} - 180 q^{8} - 1890 q^{9} + 708 q^{13} - 204 q^{14} + 108 q^{15} + 4584 q^{16} - 312 q^{17} + 492 q^{19} + 1008 q^{23} - 9262 q^{25} + 1410 q^{26} - 4046 q^{28} + 204 q^{29} + 2664 q^{30} + 330 q^{32} + 3384 q^{33} + 5136 q^{34} - 15552 q^{36} + 5130 q^{38} - 5724 q^{41} - 9006 q^{46} + 28518 q^{49} - 4332 q^{50} + 14116 q^{52} + 5672 q^{55} - 22986 q^{56} - 16772 q^{58} + 10008 q^{59} + 4608 q^{60} - 7572 q^{61} + 1512 q^{63} + 27464 q^{64} + 13896 q^{66} - 7014 q^{68} + 4860 q^{72} + 9204 q^{76} - 2920 q^{79} + 51030 q^{81} + 19642 q^{82} + 11244 q^{83} + 40192 q^{91} + 57660 q^{92} + 8424 q^{93} + 40776 q^{97} + 55530 q^{98}+O(q^{100}) \) 70 * q + 576 * q^4 - 56 * q^7 - 180 * q^8 - 1890 * q^9 + 708 * q^13 - 204 * q^14 + 108 * q^15 + 4584 * q^16 - 312 * q^17 + 492 * q^19 + 1008 * q^23 - 9262 * q^25 + 1410 * q^26 - 4046 * q^28 + 204 * q^29 + 2664 * q^30 + 330 * q^32 + 3384 * q^33 + 5136 * q^34 - 15552 * q^36 + 5130 * q^38 - 5724 * q^41 - 9006 * q^46 + 28518 * q^49 - 4332 * q^50 + 14116 * q^52 + 5672 * q^55 - 22986 * q^56 - 16772 * q^58 + 10008 * q^59 + 4608 * q^60 - 7572 * q^61 + 1512 * q^63 + 27464 * q^64 + 13896 * q^66 - 7014 * q^68 + 4860 * q^72 + 9204 * q^76 - 2920 * q^79 + 51030 * q^81 + 19642 * q^82 + 11244 * q^83 + 40192 * q^91 + 57660 * q^92 + 8424 * q^93 + 40776 * q^97 + 55530 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more