LMFDB - Embedded newform 950.3.d.b.949.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [950,3,Mod(949,950)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(950, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("950.949");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 950 = 2 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	950.d (of order $2$, degree $1$, not 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:	$25.8856251142$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			949.20
Character	$\chi$	$=$	950.949
Dual form			950.3.d.b.949.17

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} -3.53663 q^{3} +2.00000 q^{4} -5.00155 q^{6} -0.146462i q^{7} +2.82843 q^{8} +3.50774 q^{9} +O(q^{10})$	\(q+1.41421 q^{2} -3.53663 q^{3} +2.00000 q^{4} -5.00155 q^{6} -0.146462i q^{7} +2.82843 q^{8} +3.50774 q^{9} +15.8960 q^{11} -7.07326 q^{12} -16.9258 q^{13} -0.207129i q^{14} +4.00000 q^{16} -27.0986i q^{17} +4.96069 q^{18} +(-11.5499 + 15.0864i) q^{19} +0.517983i q^{21} +22.4804 q^{22} +20.1654i q^{23} -10.0031 q^{24} -23.9366 q^{26} +19.4241 q^{27} -0.292925i q^{28} -5.45043i q^{29} -9.15415i q^{31} +5.65685 q^{32} -56.2183 q^{33} -38.3232i q^{34} +7.01547 q^{36} -7.37605 q^{37} +(-16.3340 + 21.3354i) q^{38} +59.8601 q^{39} -24.0093i q^{41} +0.732538i q^{42} -40.3144i q^{43} +31.7920 q^{44} +28.5181i q^{46} -85.7192i q^{47} -14.1465 q^{48} +48.9785 q^{49} +95.8376i q^{51} -33.8515 q^{52} -59.0413 q^{53} +27.4698 q^{54} -0.414258i q^{56} +(40.8477 - 53.3550i) q^{57} -7.70807i q^{58} -86.4540i q^{59} -15.3251 q^{61} -12.9459i q^{62} -0.513751i q^{63} +8.00000 q^{64} -79.5047 q^{66} -100.455 q^{67} -54.1971i q^{68} -71.3174i q^{69} -93.5509i q^{71} +9.92138 q^{72} -4.45910i q^{73} -10.4313 q^{74} +(-23.0998 + 30.1728i) q^{76} -2.32817i q^{77} +84.6550 q^{78} -15.2517i q^{79} -100.265 q^{81} -33.9543i q^{82} -145.028i q^{83} +1.03597i q^{84} -57.0132i q^{86} +19.2761i q^{87} +44.9607 q^{88} -24.1281i q^{89} +2.47899i q^{91} +40.3308i q^{92} +32.3748i q^{93} -121.225i q^{94} -20.0062 q^{96} -3.08788 q^{97} +69.2661 q^{98} +55.7591 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 48 q^{4} - 16 q^{6} + 56 q^{9}+O(q^{10}) $ 24 * q + 48 * q^4 - 16 * q^6 + 56 * q^9	$ 24 q + 48 q^{4} - 16 q^{6} + 56 q^{9} + 8 q^{11} + 96 q^{16} + 24 q^{19} - 32 q^{24} - 32 q^{26} + 112 q^{36} - 200 q^{39} + 16 q^{44} + 216 q^{49} - 224 q^{54} - 152 q^{61} + 192 q^{64} + 16 q^{66} - 144 q^{74} + 48 q^{76} - 264 q^{81} - 64 q^{96} - 256 q^{99}+O(q^{100}) $ 24 * q + 48 * q^4 - 16 * q^6 + 56 * q^9 + 8 * q^11 + 96 * q^16 + 24 * q^19 - 32 * q^24 - 32 * q^26 + 112 * q^36 - 200 * q^39 + 16 * q^44 + 216 * q^49 - 224 * q^54 - 152 * q^61 + 192 * q^64 + 16 * q^66 - 144 * q^74 + 48 * q^76 - 264 * q^81 - 64 * q^96 - 256 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/950\mathbb{Z}\right)^\times$.

$n$	$77$	$401$
$\chi(n)$	$-1$	$-1$

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 $

$2$	1.41421			0.707107
$2$	1.41421			0.707107
$3$	−3.53663			−1.17888			−0.589438	−	0.807814i	$-0.700651\pi$
$3$	−3.53663			−1.17888			−0.589438	+	0.807814i	$0.700651\pi$
$4$	2.00000			0.500000
$5$		0			0
$5$		0			0
$6$	−5.00155			−0.833591
$7$		−	0.146462i		−	0.0209232i	−0.999945	−	0.0104616i	$-0.996670\pi$
$7$		−	0.146462i		−	0.0209232i	0.999945	−	0.0104616i	$-0.00333009\pi$
$8$	2.82843			0.353553
$9$	3.50774			0.389749
$10$		0			0
$11$	15.8960			1.44509			0.722546	−	0.691322i	$-0.242972\pi$
$11$	15.8960			1.44509			0.722546	+	0.691322i	$0.242972\pi$
$12$	−7.07326			−0.589438
$13$	−16.9258			−1.30198			−0.650991	−	0.759086i	$-0.725646\pi$
$13$	−16.9258			−1.30198			−0.650991	+	0.759086i	$0.725646\pi$
$14$		−	0.207129i		−	0.0147949i
$15$		0			0
$16$	4.00000			0.250000
$17$		−	27.0986i		−	1.59403i	−0.603957	−	0.797017i	$-0.706410\pi$
$17$		−	27.0986i		−	1.59403i	0.603957	−	0.797017i	$-0.293590\pi$
$18$	4.96069			0.275594
$19$	−11.5499	+	15.0864i	−0.607889	+	0.794022i
$19$	−11.5499	+	15.0864i	−0.607889	+	0.794022i
$20$		0			0
$21$			0.517983i			0.0246658i
$22$	22.4804			1.02183
$23$			20.1654i			0.876755i	0.898791	+	0.438378i	$0.144447\pi$
$23$			20.1654i			0.876755i	−0.898791	+	0.438378i	$0.855553\pi$
$24$	−10.0031			−0.416796
$25$		0			0
$26$	−23.9366			−0.920640
$27$	19.4241			0.719411
$28$		−	0.292925i		−	0.0104616i
$29$		−	5.45043i		−	0.187946i	−0.995575	−	0.0939730i	$-0.970043\pi$
$29$		−	5.45043i		−	0.187946i	0.995575	−	0.0939730i	$-0.0299567\pi$
$30$		0			0
$31$		−	9.15415i		−	0.295295i	−0.989040	−	0.147648i	$-0.952830\pi$
$31$		−	9.15415i		−	0.295295i	0.989040	−	0.147648i	$-0.0471701\pi$
$32$	5.65685			0.176777
$33$	−56.2183			−1.70359
$34$		−	38.3232i		−	1.12715i
$35$		0			0
$36$	7.01547			0.194874
$37$	−7.37605			−0.199353			−0.0996764	−	0.995020i	$-0.531781\pi$
$37$	−7.37605			−0.199353			−0.0996764	+	0.995020i	$0.531781\pi$
$38$	−16.3340	+	21.3354i	−0.429843	+	0.561458i
$39$	59.8601			1.53487
$40$		0			0
$41$		−	24.0093i		−	0.585593i	−0.956175	−	0.292797i	$-0.905414\pi$
$41$		−	24.0093i		−	0.585593i	0.956175	−	0.292797i	$-0.0945859\pi$
$42$			0.732538i			0.0174414i
$43$		−	40.3144i		−	0.937544i	−0.883319	−	0.468772i	$-0.844697\pi$
$43$		−	40.3144i		−	0.937544i	0.883319	−	0.468772i	$-0.155303\pi$
$44$	31.7920			0.722546
$45$		0			0
$46$			28.5181i			0.619960i
$47$		−	85.7192i		−	1.82381i	−0.410398	−	0.911906i	$-0.634610\pi$
$47$		−	85.7192i		−	1.82381i	0.410398	−	0.911906i	$-0.365390\pi$
$48$	−14.1465			−0.294719
$49$	48.9785			0.999562
$50$		0			0
$51$			95.8376i			1.87917i
$52$	−33.8515			−0.650991
$53$	−59.0413			−1.11399			−0.556994	−	0.830517i	$-0.688045\pi$
$53$	−59.0413			−1.11399			−0.556994	+	0.830517i	$0.688045\pi$
$54$	27.4698			0.508700
$55$		0			0
$56$		−	0.414258i		−	0.00739747i
$57$	40.8477	−	53.3550i	0.716626	−	0.936053i
$58$		−	7.70807i		−	0.132898i
$59$		−	86.4540i		−	1.46532i	−0.680593	−	0.732661i	$-0.738278\pi$
$59$		−	86.4540i		−	1.46532i	0.680593	−	0.732661i	$-0.261722\pi$
$60$		0			0
$61$	−15.3251			−0.251231			−0.125615	−	0.992079i	$-0.540091\pi$
$61$	−15.3251			−0.251231			−0.125615	+	0.992079i	$0.540091\pi$
$62$		−	12.9459i		−	0.208805i
$63$		−	0.513751i		−	0.00815478i
$64$	8.00000			0.125000
$65$		0			0
$66$	−79.5047			−1.20462
$67$	−100.455			−1.49933			−0.749665	−	0.661818i	$-0.769785\pi$
$67$	−100.455			−1.49933			−0.749665	+	0.661818i	$0.769785\pi$
$68$		−	54.1971i		−	0.797017i
$69$		−	71.3174i		−	1.03359i
$70$		0			0
$71$		−	93.5509i		−	1.31762i	−0.752310	−	0.658809i	$-0.771060\pi$
$71$		−	93.5509i		−	1.31762i	0.752310	−	0.658809i	$-0.228940\pi$
$72$	9.92138			0.137797
$73$		−	4.45910i		−	0.0610836i	−0.999533	−	0.0305418i	$-0.990277\pi$
$73$		−	4.45910i		−	0.0610836i	0.999533	−	0.0305418i	$-0.00972327\pi$
$74$	−10.4313			−0.140964
$75$		0			0
$76$	−23.0998	+	30.1728i	−0.303945	+	0.397011i
$77$		−	2.32817i		−	0.0302360i
$78$	84.6550			1.08532
$79$		−	15.2517i		−	0.193059i	−0.995330	−	0.0965295i	$-0.969226\pi$
$79$		−	15.2517i		−	0.193059i	0.995330	−	0.0965295i	$-0.0307742\pi$
$80$		0			0
$81$	−100.265			−1.23784
$82$		−	33.9543i		−	0.414077i
$83$		−	145.028i		−	1.74732i	−0.486535	−	0.873661i	$-0.661739\pi$
$83$		−	145.028i		−	1.74732i	0.486535	−	0.873661i	$-0.338261\pi$
$84$			1.03597i			0.0123329i
$85$		0			0
$86$		−	57.0132i		−	0.662944i
$87$			19.2761i			0.221565i
$88$	44.9607			0.510917
$89$		−	24.1281i		−	0.271102i	−0.990770	−	0.135551i	$-0.956720\pi$
$89$		−	24.1281i		−	0.271102i	0.990770	−	0.135551i	$-0.0432804\pi$
$90$		0			0
$91$			2.47899i			0.0272416i
$92$			40.3308i			0.438378i
$93$			32.3748i			0.348116i
$94$		−	121.225i		−	1.28963i
$95$		0			0
$96$	−20.0062			−0.208398
$97$	−3.08788			−0.0318338			−0.0159169	−	0.999873i	$-0.505067\pi$
$97$	−3.08788			−0.0318338			−0.0159169	+	0.999873i	$0.505067\pi$
$98$	69.2661			0.706797
$99$	55.7591			0.563223
$100$		0			0
$101$	103.059			1.02039			0.510193	−	0.860060i	$-0.329574\pi$
$101$	103.059			1.02039			0.510193	+	0.860060i	$0.329574\pi$
$102$			135.535i			1.32877i
$103$	−126.615			−1.22927			−0.614637	−	0.788810i	$-0.710698\pi$
$103$	−126.615			−1.22927			−0.614637	+	0.788810i	$0.710698\pi$
$104$	−47.8733			−0.460320
$105$		0			0
$106$	−83.4970			−0.787708
$107$	−22.0909			−0.206457			−0.103228	−	0.994658i	$-0.532917\pi$
$107$	−22.0909			−0.206457			−0.103228	+	0.994658i	$0.532917\pi$
$108$	38.8482			0.359705
$109$			91.1256i			0.836015i	0.908444	+	0.418007i	$0.137271\pi$
$109$			91.1256i			0.836015i	−0.908444	+	0.418007i	$0.862729\pi$
$110$		0			0
$111$	26.0864			0.235012
$112$		−	0.585849i		−	0.00523080i
$113$	170.895			1.51234			0.756172	−	0.654373i	$-0.227067\pi$
$113$	170.895			1.51234			0.756172	+	0.654373i	$0.227067\pi$
$114$	57.7674	−	75.4554i	0.506731	−	0.661889i
$115$		0			0
$116$		−	10.9009i		−	0.0939730i
$117$	−59.3711			−0.507445
$118$		−	122.264i		−	1.03614i
$119$	−3.96892			−0.0333523
$120$		0			0
$121$	131.683			1.08829
$122$	−21.6729			−0.177647
$123$			84.9120i			0.690342i
$124$		−	18.3083i		−	0.147648i
$125$		0			0
$126$		−	0.726554i		−	0.00576630i
$127$	64.0896			0.504643			0.252321	−	0.967644i	$-0.418806\pi$
$127$	64.0896			0.504643			0.252321	+	0.967644i	$0.418806\pi$
$128$	11.3137			0.0883883
$129$			142.577i			1.10525i
$130$		0			0
$131$	−141.977			−1.08380			−0.541898	−	0.840444i	$-0.682294\pi$
$131$	−141.977			−1.08380			−0.541898	+	0.840444i	$0.682294\pi$
$132$	−112.437			−0.851793
$133$	2.20959	+	1.69163i	0.0166135	+	0.0127190i
$134$	−142.065			−1.06019
$135$		0			0
$136$		−	76.6463i		−	0.563576i
$137$		−	96.3583i		−	0.703345i	−0.936123	−	0.351673i	$-0.885613\pi$
$137$		−	96.3583i		−	0.703345i	0.936123	−	0.351673i	$-0.114387\pi$
$138$		−	100.858i		−	0.730856i
$139$	182.574			1.31348			0.656739	−	0.754118i	$-0.271935\pi$
$139$	182.574			1.31348			0.656739	+	0.754118i	$0.271935\pi$
$140$		0			0
$141$			303.157i			2.15005i
$142$		−	132.301i		−	0.931697i
$143$	−269.052			−1.88148
$144$	14.0309			0.0974371
$145$		0			0
$146$		−	6.30612i		−	0.0431926i
$147$	−173.219			−1.17836
$148$	−14.7521			−0.0996764
$149$	15.4418			0.103636			0.0518182	−	0.998657i	$-0.483498\pi$
$149$	15.4418			0.103636			0.0518182	+	0.998657i	$0.483498\pi$
$150$		0			0
$151$			185.156i			1.22620i	0.790006	+	0.613099i	$0.210077\pi$
$151$			185.156i			1.22620i	−0.790006	+	0.613099i	$0.789923\pi$
$152$	−32.6680	+	42.6708i	−0.214921	+	0.280729i
$153$		−	95.0547i		−	0.621272i
$154$		−	3.29253i		−	0.0213800i
$155$		0			0
$156$	119.720			0.767437
$157$			29.2193i			0.186110i	0.995661	+	0.0930552i	$0.0296633\pi$
$157$			29.2193i			0.186110i	−0.995661	+	0.0930552i	$0.970337\pi$
$158$		−	21.5691i		−	0.136513i
$159$	208.807			1.31325
$160$		0			0
$161$	2.95347			0.0183445
$162$	−141.797			−0.875288
$163$			94.1505i			0.577611i	0.957388	+	0.288805i	$0.0932580\pi$
$163$			94.1505i			0.577611i	−0.957388	+	0.288805i	$0.906742\pi$
$164$		−	48.0186i		−	0.292797i
$165$		0			0
$166$		−	205.100i		−	1.23554i
$167$	56.3258			0.337280			0.168640	−	0.985678i	$-0.446062\pi$
$167$	56.3258			0.337280			0.168640	+	0.985678i	$0.446062\pi$
$168$			1.46508i			0.00872069i
$169$	117.481			0.695156
$170$		0			0
$171$	−40.5140	+	52.9192i	−0.236924	+	0.309469i
$172$		−	80.6288i		−	0.468772i
$173$	45.4392			0.262654			0.131327	−	0.991339i	$-0.458076\pi$
$173$	45.4392			0.262654			0.131327	+	0.991339i	$0.458076\pi$
$174$			27.2606i			0.156670i
$175$		0			0
$176$	63.5841			0.361273
$177$			305.756i			1.72743i
$178$		−	34.1222i		−	0.191698i
$179$		−	147.183i		−	0.822252i	−0.911579	−	0.411126i	$-0.865136\pi$
$179$		−	147.183i		−	0.822252i	0.911579	−	0.411126i	$-0.134864\pi$
$180$		0			0
$181$		−	243.383i		−	1.34466i	−0.740254	−	0.672328i	$-0.765295\pi$
$181$		−	243.383i		−	1.34466i	0.740254	−	0.672328i	$-0.234705\pi$
$182$			3.50582i			0.0192627i
$183$	54.1991			0.296170
$184$			57.0363i			0.309980i
$185$		0			0
$186$			45.7849i			0.246155i
$187$		−	430.759i		−	2.30353i
$188$		−	171.438i		−	0.911906i
$189$		−	2.84490i		−	0.0150524i
$190$		0			0
$191$	−73.8128			−0.386454			−0.193227	−	0.981154i	$-0.561895\pi$
$191$	−73.8128			−0.386454			−0.193227	+	0.981154i	$0.561895\pi$
$192$	−28.2930			−0.147359
$193$	−99.7983			−0.517089			−0.258545	−	0.965999i	$-0.583243\pi$
$193$	−99.7983			−0.517089			−0.258545	+	0.965999i	$0.583243\pi$
$194$	−4.36693			−0.0225099
$195$		0			0
$196$	97.9571			0.499781
$197$			333.125i			1.69099i	0.533985	+	0.845494i	$0.320694\pi$
$197$			333.125i			1.69099i	−0.533985	+	0.845494i	$0.679306\pi$
$198$	78.8552			0.398259
$199$	−80.1431			−0.402729			−0.201365	−	0.979516i	$-0.564538\pi$
$199$	−80.1431			−0.402729			−0.201365	+	0.979516i	$0.564538\pi$
$200$		0			0
$201$	355.272			1.76752
$202$	145.747			0.721522
$203$	−0.798283			−0.00393243
$204$			191.675i			0.939584i
$205$		0			0
$206$	−179.061			−0.869229
$207$			70.7348i			0.341714i
$208$	−67.7030			−0.325495
$209$	−183.597	+	239.814i	−0.878457	+	1.14744i
$210$		0			0
$211$		−	72.6207i		−	0.344174i	−0.985082	−	0.172087i	$-0.944949\pi$
$211$		−	72.6207i		−	0.344174i	0.985082	−	0.172087i	$-0.0550510\pi$
$212$	−118.083			−0.556994
$213$			330.855i			1.55331i
$214$	−31.2412			−0.145987
$215$		0			0
$216$	54.9396			0.254350
$217$	−1.34074			−0.00617852
$218$			128.871i			0.591152i
$219$			15.7702i			0.0720100i
$220$		0			0
$221$			458.664i			2.07540i
$222$	36.8917			0.166179
$223$	−323.793			−1.45199			−0.725993	−	0.687702i	$-0.758620\pi$
$223$	−323.793			−1.45199			−0.725993	+	0.687702i	$0.758620\pi$
$224$		−	0.828516i		−	0.00369873i
$225$		0			0
$226$	241.682			1.06939
$227$	433.568			1.90999			0.954996	−	0.296618i	$-0.0958589\pi$
$227$	433.568			1.90999			0.954996	+	0.296618i	$0.0958589\pi$
$228$	81.6954	−	106.710i	0.358313	−	0.468027i
$229$	−206.994			−0.903904			−0.451952	−	0.892042i	$-0.649272\pi$
$229$	−206.994			−0.903904			−0.451952	+	0.892042i	$0.649272\pi$
$230$		0			0
$231$			8.23387i			0.0356444i
$232$		−	15.4161i		−	0.0664489i
$233$			290.836i			1.24822i	0.781335	+	0.624112i	$0.214539\pi$
$233$			290.836i			1.24822i	−0.781335	+	0.624112i	$0.785461\pi$
$234$	−83.9634			−0.358818
$235$		0			0
$236$		−	172.908i		−	0.732661i
$237$			53.9395i			0.227593i
$238$	−5.61290			−0.0235836
$239$	90.0708			0.376865			0.188433	−	0.982086i	$-0.439659\pi$
$239$	90.0708			0.376865			0.188433	+	0.982086i	$0.439659\pi$
$240$		0			0
$241$			85.6209i			0.355273i	0.984096	+	0.177637i	$0.0568452\pi$
$241$			85.6209i			0.355273i	−0.984096	+	0.177637i	$0.943155\pi$
$242$	186.229			0.769540
$243$	179.785			0.739854
$244$	−30.6501			−0.125615
$245$		0			0
$246$			120.084i			0.488145i
$247$	195.491	−	255.349i	0.791461	−	1.03380i
$248$		−	25.8918i		−	0.104403i
$249$			512.909i			2.05988i
$250$		0			0
$251$	−418.090			−1.66570			−0.832849	−	0.553500i	$-0.813292\pi$
$251$	−418.090			−1.66570			−0.832849	+	0.553500i	$0.813292\pi$
$252$		−	1.02750i		−	0.00407739i
$253$			320.549i			1.26699i
$254$	90.6364			0.356836
$255$		0			0
$256$	16.0000			0.0625000
$257$	−81.0271			−0.315281			−0.157640	−	0.987497i	$-0.550389\pi$
$257$	−81.0271			−0.315281			−0.157640	+	0.987497i	$0.550389\pi$
$258$			201.634i			0.781528i
$259$			1.08031i			0.00417110i
$260$		0			0
$261$		−	19.1187i		−	0.0732516i
$262$	−200.786			−0.766360
$263$			281.473i			1.07024i	0.844776	+	0.535120i	$0.179733\pi$
$263$			281.473i			1.07024i	−0.844776	+	0.535120i	$0.820267\pi$
$264$	−159.009			−0.602308
$265$		0			0
$266$	3.12483	+	2.39232i	0.0117475	+	0.00899368i
$267$			85.3320i			0.319596i
$268$	−200.910			−0.749665
$269$			392.714i			1.45990i	0.683499	+	0.729951i	$0.260457\pi$
$269$			392.714i			1.45990i	−0.683499	+	0.729951i	$0.739543\pi$
$270$		0			0
$271$	435.861			1.60834			0.804172	−	0.594397i	$-0.202609\pi$
$271$	435.861			1.60834			0.804172	+	0.594397i	$0.202609\pi$
$272$		−	108.394i		−	0.398508i
$273$		−	8.76725i		−	0.0321145i
$274$		−	136.271i		−	0.497340i
$275$		0			0
$276$		−	142.635i		−	0.516793i
$277$			306.929i			1.10805i	0.832501	+	0.554024i	$0.186908\pi$
$277$			306.929i			1.10805i	−0.832501	+	0.554024i	$0.813092\pi$
$278$	258.198			0.928770
$279$		−	32.1103i		−	0.115091i
$280$		0			0
$281$		−	158.309i		−	0.563376i	−0.959506	−	0.281688i	$-0.909106\pi$
$281$		−	158.309i		−	0.563376i	0.959506	−	0.281688i	$-0.0908944\pi$
$282$			428.729i			1.52031i
$283$		−	109.390i		−	0.386539i	−0.981146	−	0.193269i	$-0.938091\pi$
$283$		−	109.390i		−	0.386539i	0.981146	−	0.193269i	$-0.0619091\pi$
$284$		−	187.102i		−	0.658809i
$285$		0			0
$286$	−380.497			−1.33041
$287$	−3.51646			−0.0122525
$288$	19.8428			0.0688985
$289$	−445.333			−1.54094
$290$		0			0
$291$	10.9207			0.0375282
$292$		−	8.91820i		−	0.0305418i
$293$	332.451			1.13464			0.567322	−	0.823496i	$-0.307980\pi$
$293$	332.451			1.13464			0.567322	+	0.823496i	$0.307980\pi$
$294$	−244.969			−0.833226
$295$		0			0
$296$	−20.8626			−0.0704818
$297$	308.766			1.03962
$298$	21.8380			0.0732819
$299$		−	341.314i		−	1.14152i
$300$		0			0
$301$	−5.90454			−0.0196164
$302$			261.850i			0.867053i
$303$	−364.481			−1.20291
$304$	−46.1996	+	60.3456i	−0.151972	+	0.198505i
$305$		0			0
$306$		−	134.428i		−	0.439306i
$307$	469.296			1.52865			0.764326	−	0.644830i	$-0.223072\pi$
$307$	469.296			1.52865			0.764326	+	0.644830i	$0.223072\pi$
$308$		−	4.65634i		−	0.0151180i
$309$	447.791			1.44916
$310$		0			0
$311$	50.0531			0.160942			0.0804712	−	0.996757i	$-0.474357\pi$
$311$	50.0531			0.160942			0.0804712	+	0.996757i	$0.474357\pi$
$312$	169.310			0.542660
$313$		−	214.496i		−	0.685290i	−0.939465	−	0.342645i	$-0.888677\pi$
$313$		−	214.496i		−	0.685290i	0.939465	−	0.342645i	$-0.111323\pi$
$314$			41.3224i			0.131600i
$315$		0			0
$316$		−	30.5033i		−	0.0965295i
$317$	119.302			0.376348			0.188174	−	0.982136i	$-0.439743\pi$
$317$	119.302			0.376348			0.188174	+	0.982136i	$0.439743\pi$
$318$	295.298			0.928610
$319$		−	86.6402i		−	0.271599i
$320$		0			0
$321$	78.1272			0.243387
$322$	4.17683			0.0129715
$323$	408.820	+	312.986i	1.26570	+	0.968996i
$324$	−200.531			−0.618922
$325$		0			0
$326$			133.149i			0.408432i
$327$		−	322.277i		−	0.985557i
$328$		−	67.9086i		−	0.207038i
$329$	−12.5546			−0.0381600
$330$		0			0
$331$		−	474.154i		−	1.43249i	−0.697849	−	0.716245i	$-0.745859\pi$
$331$		−	474.154i		−	1.43249i	0.697849	−	0.716245i	$-0.254141\pi$
$332$		−	290.056i		−	0.873661i
$333$	−25.8733			−0.0776975
$334$	79.6567			0.238493
$335$		0			0
$336$			2.07193i			0.00616646i
$337$	−237.394			−0.704433			−0.352216	−	0.935919i	$-0.614572\pi$
$337$	−237.394			−0.704433			−0.352216	+	0.935919i	$0.614572\pi$
$338$	166.144			0.491549
$339$	−604.392			−1.78287
$340$		0			0
$341$		−	145.515i		−	0.426729i
$342$	−57.2955	+	74.8390i	−0.167531	+	0.218827i
$343$		−	14.3502i		−	0.0418372i
$344$		−	114.026i		−	0.331472i
$345$		0			0
$346$	64.2607			0.185724
$347$			612.413i			1.76488i	0.470426	+	0.882440i	$0.344100\pi$
$347$			612.413i			1.76488i	−0.470426	+	0.882440i	$0.655900\pi$
$348$			38.5523i			0.110782i
$349$	−625.895			−1.79340			−0.896698	−	0.442644i	$-0.854041\pi$
$349$	−625.895			−1.79340			−0.896698	+	0.442644i	$0.854041\pi$
$350$		0			0
$351$	−328.767			−0.936659
$352$	89.9215			0.255459
$353$		−	218.281i		−	0.618359i	−0.951004	−	0.309179i	$-0.899946\pi$
$353$		−	218.281i		−	0.618359i	0.951004	−	0.309179i	$-0.100054\pi$
$354$			432.404i			1.22148i
$355$		0			0
$356$		−	48.2561i		−	0.135551i
$357$	14.0366			0.0393182
$358$		−	208.148i		−	0.581420i
$359$	−316.232			−0.880870			−0.440435	−	0.897784i	$-0.645176\pi$
$359$	−316.232			−0.880870			−0.440435	+	0.897784i	$0.645176\pi$
$360$		0			0
$361$	−94.1997	−	348.493i	−0.260941	−	0.965355i
$362$		−	344.195i		−	0.950815i
$363$	−465.716			−1.28296
$364$			4.95797i			0.0136208i
$365$		0			0
$366$	76.6491			0.209424
$367$		−	600.077i		−	1.63509i	−0.575867	−	0.817543i	$-0.695335\pi$
$367$		−	600.077i		−	1.63509i	0.575867	−	0.817543i	$-0.304665\pi$
$368$			80.6615i			0.219189i
$369$		−	84.2184i		−	0.228234i
$370$		0			0
$371$			8.64733i			0.0233082i
$372$			64.7496i			0.174058i
$373$	60.9072			0.163290			0.0816451	−	0.996661i	$-0.473983\pi$
$373$	60.9072			0.163290			0.0816451	+	0.996661i	$0.473983\pi$
$374$		−	609.186i		−	1.62884i
$375$		0			0
$376$		−	242.450i		−	0.644815i
$377$			92.2527i			0.244702i
$378$		−	4.02329i		−	0.0106436i
$379$			162.907i			0.429835i	0.976632	+	0.214917i	$0.0689482\pi$
$379$			162.907i			0.429835i	−0.976632	+	0.214917i	$0.931052\pi$
$380$		0			0
$381$	−226.661			−0.594911
$382$	−104.387			−0.273264
$383$	638.343			1.66669			0.833346	−	0.552752i	$-0.186422\pi$
$383$	638.343			1.66669			0.833346	+	0.552752i	$0.186422\pi$
$384$	−40.0124			−0.104199
$385$		0			0
$386$	−141.136			−0.365637
$387$		−	141.412i		−	0.365406i
$388$	−6.17577			−0.0159169
$389$	−175.089			−0.450100			−0.225050	−	0.974347i	$-0.572254\pi$
$389$	−175.089			−0.450100			−0.225050	+	0.974347i	$0.572254\pi$
$390$		0			0
$391$	546.453			1.39758
$392$	138.532			0.353399
$393$	502.121			1.27766
$394$			471.109i			1.19571i
$395$		0			0
$396$	111.518			0.281611
$397$			725.042i			1.82630i	0.407622	+	0.913151i	$0.366358\pi$
$397$			725.042i			1.82630i	−0.407622	+	0.913151i	$0.633642\pi$
$398$	−113.339			−0.284773
$399$	−7.81450	−	5.98265i	−0.0195852	−	0.0149941i
$400$		0			0
$401$		−	566.987i		−	1.41393i	−0.707248	−	0.706966i	$-0.750063\pi$
$401$		−	566.987i		−	1.41393i	0.707248	−	0.706966i	$-0.249937\pi$
$402$	502.431			1.24983
$403$			154.941i			0.384469i
$404$	206.118			0.510193
$405$		0			0
$406$	−1.12894			−0.00278065
$407$	−117.250			−0.288083
$408$			271.070i			0.664386i
$409$		−	216.398i		−	0.529091i	−0.964373	−	0.264545i	$-0.914778\pi$
$409$		−	216.398i		−	0.529091i	0.964373	−	0.264545i	$-0.0852219\pi$
$410$		0			0
$411$			340.783i			0.829157i
$412$	−253.231			−0.614637
$413$	−12.6623			−0.0306592
$414$			100.034i			0.241628i
$415$		0			0
$416$	−95.7465			−0.230160
$417$	−645.695			−1.54843
$418$	−259.646	+	339.148i	−0.621163	+	0.811359i
$419$	−484.914			−1.15731			−0.578656	−	0.815572i	$-0.696423\pi$
$419$	−484.914			−1.15731			−0.578656	+	0.815572i	$0.696423\pi$
$420$		0			0
$421$			413.265i			0.981626i	0.871265	+	0.490813i	$0.163300\pi$
$421$			413.265i			0.981626i	−0.871265	+	0.490813i	$0.836700\pi$
$422$		−	102.701i		−	0.243368i
$423$		−	300.680i		−	0.710828i
$424$	−166.994			−0.393854
$425$		0			0
$426$			467.899i			1.09836i
$427$			2.24455i			0.00525655i
$428$	−44.1818			−0.103228
$429$	951.538			2.21804
$430$		0			0
$431$		−	450.677i		−	1.04565i	−0.852439	−	0.522827i	$-0.824877\pi$
$431$		−	450.677i		−	1.04565i	0.852439	−	0.522827i	$-0.175123\pi$
$432$	77.6964			0.179853
$433$	2.50133			0.00577673			0.00288837	−	0.999996i	$-0.499081\pi$
$433$	2.50133			0.00577673			0.00288837	+	0.999996i	$0.499081\pi$
$434$	−1.89609			−0.00436887
$435$		0			0
$436$			182.251i			0.418007i
$437$	−304.223	−	232.908i	−0.696163	−	0.532970i
$438$			22.3024i			0.0509187i
$439$			491.152i			1.11880i	0.828898	+	0.559399i	$0.188968\pi$
$439$			491.152i			1.11880i	−0.828898	+	0.559399i	$0.811032\pi$
$440$		0			0
$441$	171.804			0.389578
$442$			648.649i			1.46753i
$443$		−	238.895i		−	0.539267i	−0.962963	−	0.269634i	$-0.913097\pi$
$443$		−	238.895i		−	0.539267i	0.962963	−	0.269634i	$-0.0869026\pi$
$444$	52.1727			0.117506
$445$		0			0
$446$	−457.912			−1.02671
$447$	−54.6119			−0.122174
$448$		−	1.17170i		−	0.00261540i
$449$		−	452.733i		−	1.00832i	−0.863612	−	0.504158i	$-0.831803\pi$
$449$		−	452.733i		−	1.00832i	0.863612	−	0.504158i	$-0.168197\pi$
$450$		0			0
$451$		−	381.653i		−	0.846236i
$452$	341.790			0.756172
$453$		−	654.828i		−	1.44554i
$454$	613.158			1.35057
$455$		0			0
$456$	115.535	−	150.911i	0.253366	−	0.330945i
$457$		−	697.754i		−	1.52681i	−0.645918	−	0.763407i	$-0.723525\pi$
$457$		−	697.754i		−	1.52681i	0.645918	−	0.763407i	$-0.276475\pi$
$458$	−292.734			−0.639156
$459$		−	526.365i		−	1.14677i
$460$		0			0
$461$	420.342			0.911804			0.455902	−	0.890030i	$-0.349317\pi$
$461$	420.342			0.911804			0.455902	+	0.890030i	$0.349317\pi$
$462$			11.6444i			0.0252044i
$463$		−	480.580i		−	1.03797i	−0.854783	−	0.518985i	$-0.826310\pi$
$463$		−	480.580i		−	1.03797i	0.854783	−	0.518985i	$-0.173690\pi$
$464$		−	21.8017i		−	0.0469865i
$465$		0			0
$466$			411.304i			0.882627i
$467$			559.285i			1.19761i	0.800894	+	0.598806i	$0.204358\pi$
$467$			559.285i			1.19761i	−0.800894	+	0.598806i	$0.795642\pi$
$468$	−118.742			−0.253723
$469$			14.7129i			0.0313708i
$470$		0			0
$471$		−	103.338i		−	0.219401i
$472$		−	244.529i		−	0.518070i
$473$		−	640.838i		−	1.35484i
$474$			76.2819i			0.160932i
$475$		0			0
$476$	−7.93784			−0.0166761
$477$	−207.101			−0.434175
$478$	127.379			0.266484
$479$	−246.678			−0.514986			−0.257493	−	0.966280i	$-0.582896\pi$
$479$	−246.678			−0.514986			−0.257493	+	0.966280i	$0.582896\pi$
$480$		0			0
$481$	124.845			0.259554
$482$			121.086i			0.251216i
$483$	−10.4453			−0.0216259
$484$	263.367			0.544147
$485$		0			0
$486$	254.254			0.523156
$487$	418.804			0.859968			0.429984	−	0.902837i	$-0.358519\pi$
$487$	418.804			0.859968			0.429984	+	0.902837i	$0.358519\pi$
$488$	−43.3459			−0.0888235
$489$		−	332.975i		−	0.680931i
$490$		0			0
$491$	−514.657			−1.04818			−0.524091	−	0.851662i	$-0.675595\pi$
$491$	−514.657			−1.04818			−0.524091	+	0.851662i	$0.675595\pi$
$492$			169.824i			0.345171i
$493$	−147.699			−0.299592
$494$	276.466	−	361.118i	0.559647	−	0.731008i
$495$		0			0
$496$		−	36.6166i		−	0.0738238i
$497$	−13.7017			−0.0275688
$498$			725.363i			1.45655i
$499$	152.709			0.306031			0.153015	−	0.988224i	$-0.451102\pi$
$499$	152.709			0.306031			0.153015	+	0.988224i	$0.451102\pi$
$500$		0			0
$501$	−199.203			−0.397612
$502$	−591.269			−1.17783
$503$		−	173.503i		−	0.344937i	−0.985015	−	0.172469i	$-0.944826\pi$
$503$		−	173.503i		−	0.344937i	0.985015	−	0.172469i	$-0.0551743\pi$
$504$		−	1.45311i		−	0.00288315i
$505$		0			0
$506$			453.325i			0.895899i
$507$	−415.488			−0.819502
$508$	128.179			0.252321
$509$			856.653i			1.68301i	0.540247	+	0.841506i	$0.318331\pi$
$509$			856.653i			1.68301i	−0.540247	+	0.841506i	$0.681669\pi$
$510$		0			0
$511$	−0.653090			−0.00127806
$512$	22.6274			0.0441942
$513$	−224.346	+	293.040i	−0.437322	+	0.571228i
$514$	−114.590			−0.222937
$515$		0			0
$516$			285.154i			0.552624i
$517$		−	1362.59i		−	2.63558i
$518$			1.52779i			0.00294941i
$519$	−160.701			−0.309637
$520$		0			0
$521$			99.1018i			0.190215i	0.995467	+	0.0951073i	$0.0303194\pi$
$521$			99.1018i			0.190215i	−0.995467	+	0.0951073i	$0.969681\pi$
$522$		−	27.0379i		−	0.0517967i
$523$	56.3564			0.107756			0.0538780	−	0.998548i	$-0.482842\pi$
$523$	56.3564			0.107756			0.0538780	+	0.998548i	$0.482842\pi$
$524$	−283.955			−0.541898
$525$		0			0
$526$			398.063i			0.756773i
$527$	−248.064			−0.470710
$528$	−224.873			−0.425896
$529$	122.358			0.231300
$530$		0			0
$531$		−	303.258i		−	0.571107i
$532$	4.41918	+	3.38325i	0.00830673	+	0.00635949i
$533$			406.376i			0.762431i
$534$			120.678i			0.225988i
$535$		0			0
$536$	−284.130			−0.530093
$537$			520.532i			0.969333i
$538$			555.381i			1.03231i
$539$	778.564			1.44446
$540$		0			0
$541$	909.463			1.68108			0.840538	−	0.541752i	$-0.182239\pi$
$541$	909.463			1.68108			0.840538	+	0.541752i	$0.182239\pi$
$542$	616.401			1.13727
$543$			860.754i			1.58518i
$544$		−	153.293i		−	0.281788i
$545$		0			0
$546$		−	12.3988i		−	0.0227084i
$547$	174.393			0.318817			0.159408	−	0.987213i	$-0.449041\pi$
$547$	174.393			0.318817			0.159408	+	0.987213i	$0.449041\pi$
$548$		−	192.717i		−	0.351673i
$549$	−53.7563			−0.0979168
$550$		0			0
$551$	82.2275	+	62.9519i	0.149233	+	0.114250i
$552$		−	201.716i		−	0.365428i
$553$	−2.23379			−0.00403941
$554$			434.064i			0.783508i
$555$		0			0
$556$	365.147			0.656739
$557$			134.156i			0.240854i	0.992722	+	0.120427i	$0.0384264\pi$
$557$			134.156i			0.240854i	−0.992722	+	0.120427i	$0.961574\pi$
$558$		−	45.4109i		−	0.0813815i
$559$			682.352i			1.22066i
$560$		0			0
$561$			1523.44i			2.71557i
$562$		−	223.882i		−	0.398367i
$563$	178.310			0.316715			0.158357	−	0.987382i	$-0.449380\pi$
$563$	178.310			0.316715			0.158357	+	0.987382i	$0.449380\pi$
$564$			606.314i			1.07502i
$565$		0			0
$566$		−	154.702i		−	0.273324i
$567$			14.6851i			0.0258997i
$568$		−	264.602i		−	0.465849i
$569$			189.835i			0.333629i	0.985988	+	0.166815i	$0.0533482\pi$
$569$			189.835i			0.333629i	−0.985988	+	0.166815i	$0.946652\pi$
$570$		0			0
$571$	−604.730			−1.05907			−0.529536	−	0.848287i	$-0.677634\pi$
$571$	−604.730			−1.05907			−0.529536	+	0.848287i	$0.677634\pi$
$572$	−538.104			−0.940742
$573$	261.048			0.455582
$574$	−4.97303			−0.00866381
$575$		0			0
$576$	28.0619			0.0487186
$577$			1138.28i			1.97276i	0.164488	+	0.986379i	$0.447403\pi$
$577$			1138.28i			1.97276i	−0.164488	+	0.986379i	$0.552597\pi$
$578$	−629.795			−1.08961
$579$	352.949			0.609584
$580$		0			0
$581$	−21.2411			−0.0365596
$582$	15.4442			0.0265364
$583$	−938.522			−1.60982
$584$		−	12.6122i		−	0.0215963i
$585$		0			0
$586$	470.156			0.802314
$587$			103.008i			0.175482i	0.996143	+	0.0877410i	$0.0279648\pi$
$587$			103.008i			0.175482i	−0.996143	+	0.0877410i	$0.972035\pi$
$588$	−346.438			−0.589180
$589$	138.103	+	105.730i	0.234471	+	0.179507i
$590$		0			0
$591$		−	1178.14i		−	1.99346i
$592$	−29.5042			−0.0498382
$593$			354.332i			0.597525i	0.954327	+	0.298763i	$0.0965739\pi$
$593$			354.332i			0.597525i	−0.954327	+	0.298763i	$0.903426\pi$
$594$	436.661			0.735119
$595$		0			0
$596$	30.8836			0.0518182
$597$	283.436			0.474768
$598$		−	482.691i		−	0.807176i
$599$		−	1112.64i		−	1.85750i	−0.370704	−	0.928751i	$-0.620884\pi$
$599$		−	1112.64i		−	1.85750i	0.370704	−	0.928751i	$-0.379116\pi$
$600$		0			0
$601$		−	44.8624i		−	0.0746463i	−0.999303	−	0.0373231i	$-0.988117\pi$
$601$		−	44.8624i		−	0.0746463i	0.999303	−	0.0373231i	$-0.0118831\pi$
$602$	−8.35028			−0.0138709
$603$	−352.370			−0.584361
$604$			370.312i			0.613099i
$605$		0			0
$606$	−515.455			−0.850585
$607$	−740.250			−1.21952			−0.609762	−	0.792585i	$-0.708735\pi$
$607$	−740.250			−1.21952			−0.609762	+	0.792585i	$0.708735\pi$
$608$	−65.3361	+	85.3416i	−0.107461	+	0.140365i
$609$	2.82323			0.00463585
$610$		0			0
$611$			1450.86i			2.37457i
$612$		−	190.109i		−	0.310636i
$613$		−	758.605i		−	1.23753i	−0.785577	−	0.618764i	$-0.787634\pi$
$613$		−	758.605i		−	1.23753i	0.785577	−	0.618764i	$-0.212366\pi$
$614$	663.685			1.08092
$615$		0			0
$616$		−	6.58505i		−	0.0106900i
$617$			617.701i			1.00114i	0.865697	+	0.500568i	$0.166875\pi$
$617$			617.701i			1.00114i	−0.865697	+	0.500568i	$0.833125\pi$
$618$	633.272			1.02471
$619$	434.677			0.702224			0.351112	−	0.936333i	$-0.385804\pi$
$619$	434.677			0.702224			0.351112	+	0.936333i	$0.385804\pi$
$620$		0			0
$621$			391.694i			0.630747i
$622$	70.7858			0.113803
$623$	−3.53385			−0.00567232
$624$	239.440			0.383719
$625$		0			0
$626$		−	303.343i		−	0.484573i
$627$	649.316	−	848.133i	1.03559	−	1.35268i
$628$			58.4387i			0.0930552i
$629$			199.881i			0.317775i
$630$		0			0
$631$	−406.725			−0.644571			−0.322286	−	0.946642i	$-0.604451\pi$
$631$	−406.725			−0.644571			−0.322286	+	0.946642i	$0.604451\pi$
$632$		−	43.1382i		−	0.0682567i
$633$			256.832i			0.405738i
$634$	168.719			0.266118
$635$		0			0
$636$	417.614			0.656626
$637$	−828.999			−1.30141
$638$		−	122.528i		−	0.192050i
$639$		−	328.152i		−	0.513540i
$640$		0			0
$641$			994.339i			1.55123i	0.631206	+	0.775616i	$0.282560\pi$
$641$			994.339i			1.55123i	−0.631206	+	0.775616i	$0.717440\pi$
$642$	110.489			0.172101
$643$			847.680i			1.31832i	0.752003	+	0.659160i	$0.229088\pi$
$643$			847.680i			1.31832i	−0.752003	+	0.659160i	$0.770912\pi$
$644$	5.90694			0.00917226
$645$		0			0
$646$	578.159	+	442.629i	0.894983	+	0.685184i
$647$		−	731.308i		−	1.13031i	−0.824986	−	0.565153i	$-0.808817\pi$
$647$		−	731.308i		−	1.13031i	0.824986	−	0.565153i	$-0.191183\pi$
$648$	−283.593			−0.437644
$649$		−	1374.28i		−	2.11753i
$650$		0			0
$651$	4.74169			0.00728370
$652$			188.301i			0.288805i
$653$		−	1111.60i		−	1.70230i	−0.524922	−	0.851150i	$-0.675906\pi$
$653$		−	1111.60i		−	1.70230i	0.524922	−	0.851150i	$-0.324094\pi$
$654$		−	455.769i		−	0.696894i
$655$		0			0
$656$		−	96.0373i		−	0.146398i
$657$		−	15.6414i		−	0.0238072i
$658$	−17.7549			−0.0269832
$659$		−	363.690i		−	0.551881i	−0.961175	−	0.275941i	$-0.911011\pi$
$659$		−	363.690i		−	0.551881i	0.961175	−	0.275941i	$-0.0889893\pi$
$660$		0			0
$661$		−	1069.74i		−	1.61836i	−0.587560	−	0.809180i	$-0.699911\pi$
$661$		−	1069.74i		−	1.61836i	0.587560	−	0.809180i	$-0.300089\pi$
$662$		−	670.555i		−	1.01292i
$663$		−	1622.12i		−	2.44664i
$664$		−	410.200i		−	0.617772i
$665$		0			0
$666$	−36.5903			−0.0549404
$667$	109.910			0.164783
$668$	112.652			0.168640
$669$	1145.14			1.71171
$670$		0			0
$671$	−243.608			−0.363052
$672$			2.93015i			0.00436035i
$673$	270.027			0.401228			0.200614	−	0.979670i	$-0.435706\pi$
$673$	270.027			0.401228			0.200614	+	0.979670i	$0.435706\pi$
$674$	−335.726			−0.498109
$675$		0			0
$676$	234.963			0.347578
$677$	−826.165			−1.22033			−0.610166	−	0.792274i	$-0.708897\pi$
$677$	−826.165			−1.22033			−0.610166	+	0.792274i	$0.708897\pi$
$678$	−854.739			−1.26068
$679$			0.452259i			0.000666066i
$680$		0			0
$681$	−1533.37			−2.25164
$682$		−	205.789i		−	0.301743i
$683$	865.671			1.26745			0.633727	−	0.773557i	$-0.281524\pi$
$683$	865.671			1.26745			0.633727	+	0.773557i	$0.281524\pi$
$684$	−81.0280	+	105.838i	−0.118462	+	0.154734i
$685$		0			0
$686$		−	20.2942i		−	0.0295834i
$687$	732.060			1.06559
$688$		−	161.258i		−	0.234386i
$689$	999.319			1.45039
$690$		0			0
$691$	28.2977			0.0409518			0.0204759	−	0.999790i	$-0.493482\pi$
$691$	28.2977			0.0409518			0.0204759	+	0.999790i	$0.493482\pi$
$692$	90.8783			0.131327
$693$		−	8.16660i		−	0.0117844i
$694$			866.083i			1.24796i
$695$		0			0
$696$			54.5212i			0.0783350i
$697$	−650.618			−0.933455
$698$	−885.149			−1.26812
$699$		−	1028.58i		−	1.47150i
$700$		0			0
$701$	134.912			0.192456			0.0962282	−	0.995359i	$-0.469322\pi$
$701$	134.912			0.192456			0.0962282	+	0.995359i	$0.469322\pi$
$702$	−464.947			−0.662318
$703$	85.1927	−	111.278i	0.121184	−	0.158290i
$704$	127.168			0.180637
$705$		0			0
$706$		−	308.695i		−	0.437246i
$707$		−	15.0943i		−	0.0213497i
$708$			611.512i			0.863717i
$709$	328.768			0.463707			0.231853	−	0.972751i	$-0.425521\pi$
$709$	328.768			0.463707			0.231853	+	0.972751i	$0.425521\pi$
$710$		0			0
$711$		−	53.4988i		−	0.0752445i
$712$		−	68.2445i		−	0.0958490i
$713$	184.597			0.258902
$714$	19.8507			0.0278022
$715$		0			0
$716$		−	294.366i		−	0.411126i
$717$	−318.547			−0.444277
$718$	−447.220			−0.622869
$719$	−132.152			−0.183799			−0.0918997	−	0.995768i	$-0.529294\pi$
$719$	−132.152			−0.183799			−0.0918997	+	0.995768i	$0.529294\pi$
$720$		0			0
$721$			18.5444i			0.0257204i
$722$	−133.218	−	492.844i	−0.184513	−	0.682609i
$723$		−	302.809i		−	0.418823i
$724$		−	486.765i		−	0.672328i
$725$		0			0
$726$	−658.621			−0.907192
$727$		−	370.952i		−	0.510250i	−0.966908	−	0.255125i	$-0.917883\pi$
$727$		−	370.952i		−	0.510250i	0.966908	−	0.255125i	$-0.0821166\pi$
$728$			7.01163i			0.00963136i
$729$	266.557			0.365648
$730$		0			0
$731$	−1092.46			−1.49448
$732$	108.398			0.148085
$733$		−	778.543i		−	1.06213i	−0.847331	−	0.531066i	$-0.821792\pi$
$733$		−	778.543i		−	1.06213i	0.847331	−	0.531066i	$-0.178208\pi$
$734$		−	848.637i		−	1.15618i
$735$		0			0
$736$			114.073i			0.154990i
$737$	−1596.84			−2.16667
$738$		−	119.103i		−	0.161386i
$739$	−686.378			−0.928793			−0.464396	−	0.885627i	$-0.653729\pi$
$739$	−686.378			−0.928793			−0.464396	+	0.885627i	$0.653729\pi$
$740$		0			0
$741$	−691.378	+	903.074i	−0.933034	+	1.21872i
$742$			12.2292i			0.0164814i
$743$	944.989			1.27186			0.635928	−	0.771748i	$-0.280617\pi$
$743$	944.989			1.27186			0.635928	+	0.771748i	$0.280617\pi$
$744$			91.5698i			0.123078i
$745$		0			0
$746$	86.1358			0.115464
$747$		−	508.719i		−	0.681016i
$748$		−	861.519i		−	1.15176i
$749$			3.23548i			0.00431974i
$750$		0			0
$751$		−	372.272i		−	0.495701i	−0.968798	−	0.247851i	$-0.920276\pi$
$751$		−	372.272i		−	0.495701i	0.968798	−	0.247851i	$-0.0797243\pi$
$752$		−	342.877i		−	0.455953i
$753$	1478.63			1.96365
$754$			130.465i			0.173031i
$755$		0			0
$756$		−	5.68980i		−	0.00752618i
$757$		−	65.8572i		−	0.0869977i	−0.999053	−	0.0434988i	$-0.986150\pi$
$757$		−	65.8572i		−	0.0869977i	0.999053	−	0.0434988i	$-0.0138505\pi$
$758$			230.386i			0.303939i
$759$		−	1133.66i		−	1.49363i
$760$		0			0
$761$	−1305.80			−1.71590			−0.857951	−	0.513732i	$-0.828263\pi$
$761$	−1305.80			−1.71590			−0.857951	+	0.513732i	$0.828263\pi$
$762$	−320.547			−0.420666
$763$	13.3465			0.0174921
$764$	−147.626			−0.193227
$765$		0			0
$766$	902.753			1.17853
$767$			1463.30i			1.90782i
$768$	−56.5860			−0.0736797
$769$	−320.091			−0.416244			−0.208122	−	0.978103i	$-0.566735\pi$
$769$	−320.091			−0.416244			−0.208122	+	0.978103i	$0.566735\pi$
$770$		0			0
$771$	286.563			0.371677
$772$	−199.597			−0.258545
$773$	−892.160			−1.15415			−0.577076	−	0.816690i	$-0.695806\pi$
$773$	−892.160			−1.15415			−0.577076	+	0.816690i	$0.695806\pi$
$774$		−	199.987i		−	0.258381i
$775$		0			0
$776$	−8.73385			−0.0112550
$777$		−	3.82067i		−	0.00491721i
$778$	−247.613			−0.318269
$779$	362.214	+	277.305i	0.464974	+	0.355976i
$780$		0			0
$781$		−	1487.09i		−	1.90408i
$782$	772.801			0.988237
$783$		−	105.870i		−	0.135210i
$784$	195.914			0.249891
$785$		0			0
$786$	710.107			0.903444
$787$	1088.23			1.38276			0.691378	−	0.722493i	$-0.257004\pi$
$787$	1088.23			1.38276			0.691378	+	0.722493i	$0.257004\pi$
$788$			666.249i			0.845494i
$789$		−	995.465i		−	1.26168i
$790$		0			0
$791$		−	25.0297i		−	0.0316431i
$792$	157.710			0.199129
$793$	259.388			0.327098
$794$			1025.36i			1.29139i
$795$		0			0
$796$	−160.286			−0.201365
$797$	1371.00			1.72020			0.860102	−	0.510123i	$-0.170400\pi$
$797$	1371.00			1.72020			0.860102	+	0.510123i	$0.170400\pi$
$798$	−11.0514	−	8.46074i	−0.0138488	−	0.0106024i
$799$	−2322.87			−2.90722
$800$		0			0
$801$		−	84.6349i		−	0.105662i
$802$		−	801.840i		−	0.999801i
$803$		−	70.8820i		−	0.0882714i
$804$	710.544			0.883762
$805$		0			0
$806$			219.120i			0.271860i
$807$		−	1388.88i		−	1.72104i
$808$	291.495			0.360761
$809$	580.084			0.717038			0.358519	−	0.933522i	$-0.383282\pi$
$809$	580.084			0.717038			0.358519	+	0.933522i	$0.383282\pi$
$810$		0			0
$811$		−	1339.70i		−	1.65191i	−0.563738	−	0.825954i	$-0.690637\pi$
$811$		−	1339.70i		−	1.65191i	0.563738	−	0.825954i	$-0.309363\pi$
$812$	−1.59657			−0.00196621
$813$	−1541.48			−1.89604
$814$	−165.816			−0.203706
$815$		0			0
$816$			383.350i			0.469792i
$817$	608.199	+	465.627i	0.744430	+	0.569923i
$818$		−	306.033i		−	0.374124i
$819$			8.69563i			0.0106174i
$820$		0			0
$821$	1526.64			1.85949			0.929747	−	0.368199i	$-0.120025\pi$
$821$	1526.64			1.85949			0.929747	+	0.368199i	$0.120025\pi$
$822$			481.941i			0.586302i
$823$			238.601i			0.289916i	0.989438	+	0.144958i	$0.0463048\pi$
$823$			238.601i			0.289916i	−0.989438	+	0.144958i	$0.953695\pi$
$824$	−358.122			−0.434614
$825$		0			0
$826$	−17.9071			−0.0216793
$827$	1190.01			1.43895			0.719473	−	0.694521i	$-0.244383\pi$
$827$	1190.01			1.43895			0.719473	+	0.694521i	$0.244383\pi$
$828$			141.470i			0.170857i
$829$			1208.18i			1.45739i	0.684838	+	0.728696i	$0.259873\pi$
$829$			1208.18i			1.45739i	−0.684838	+	0.728696i	$0.740127\pi$
$830$		0			0
$831$		−	1085.49i		−	1.30625i
$832$	−135.406			−0.162748
$833$		−	1327.25i		−	1.59334i
$834$	−913.150			−1.09490
$835$		0			0
$836$	−367.195	+	479.628i	−0.439228	+	0.573718i
$837$		−	177.811i		−	0.212439i
$838$	−685.771			−0.818343
$839$		−	360.467i		−	0.429639i	−0.976654	−	0.214819i	$-0.931084\pi$
$839$		−	360.467i		−	0.429639i	0.976654	−	0.214819i	$-0.0689163\pi$
$840$		0			0
$841$	811.293			0.964676
$842$			584.444i			0.694114i
$843$			559.879i			0.664151i
$844$		−	145.241i		−	0.172087i
$845$		0			0
$846$		−	425.226i		−	0.502632i
$847$		−	19.2867i		−	0.0227706i
$848$	−236.165			−0.278497
$849$			386.873i			0.455681i
$850$		0			0
$851$		−	148.741i		−	0.174784i
$852$			661.710i			0.776655i
$853$			625.741i			0.733577i	0.930304	+	0.366788i	$0.119543\pi$
$853$			625.741i			0.733577i	−0.930304	+	0.366788i	$0.880457\pi$
$854$			3.17427i			0.00371694i
$855$		0			0
$856$	−62.4825			−0.0729935
$857$	−1576.70			−1.83979			−0.919894	−	0.392167i	$-0.871726\pi$
$857$	−1576.70			−1.83979			−0.919894	+	0.392167i	$0.871726\pi$
$858$	1345.68			1.56839
$859$	−954.288			−1.11093			−0.555464	−	0.831540i	$-0.687460\pi$
$859$	−954.288			−1.11093			−0.555464	+	0.831540i	$0.687460\pi$
$860$		0			0
$861$	12.4364			0.0144441
$862$		−	637.353i		−	0.739389i
$863$	363.759			0.421505			0.210752	−	0.977539i	$-0.432409\pi$
$863$	363.759			0.421505			0.210752	+	0.977539i	$0.432409\pi$
$864$	109.879			0.127175
$865$		0			0
$866$	3.53741			0.00408477
$867$	1574.98			1.81658
$868$	−2.68148			−0.00308926
$869$		−	242.441i		−	0.278988i
$870$		0			0
$871$	1700.28			1.95210
$872$			257.742i			0.295576i
$873$	−10.8315			−0.0124072
$874$	−430.237	−	329.382i	−0.492261	−	0.376867i
$875$		0			0
$876$			31.5404i			0.0360050i
$877$	−1186.98			−1.35346			−0.676730	−	0.736231i	$-0.736604\pi$
$877$	−1186.98			−1.35346			−0.676730	+	0.736231i	$0.736604\pi$
$878$			694.594i			0.791110i
$879$	−1175.75			−1.33760
$880$		0			0
$881$	316.952			0.359764			0.179882	−	0.983688i	$-0.442428\pi$
$881$	316.952			0.359764			0.179882	+	0.983688i	$0.442428\pi$
$882$	242.967			0.275473
$883$			846.927i			0.959148i	0.877502	+	0.479574i	$0.159209\pi$
$883$			846.927i			0.959148i	−0.877502	+	0.479574i	$0.840791\pi$
$884$			917.328i			1.03770i
$885$		0			0
$886$		−	337.849i		−	0.381319i
$887$	−443.640			−0.500157			−0.250079	−	0.968226i	$-0.580456\pi$
$887$	−443.640			−0.500157			−0.250079	+	0.968226i	$0.580456\pi$
$888$	73.7834			0.0830894
$889$		−	9.38671i		−	0.0105587i
$890$		0			0
$891$	−1593.82			−1.78880
$892$	−647.586			−0.725993
$893$	1293.20	+	990.048i	1.44815	+	1.10868i
$894$	−77.2329			−0.0863903
$895$		0			0
$896$		−	1.65703i		−	0.00184937i
$897$			1207.10i			1.34571i
$898$		−	640.262i		−	0.712986i
$899$	−49.8941			−0.0554995
$900$		0			0
$901$			1599.94i			1.77573i
$902$		−	539.738i		−	0.598379i
$903$	20.8822			0.0231253
$904$	483.364			0.534694
$905$		0			0
$906$		−	926.066i		−	1.02215i
$907$	−5.46337			−0.00602356			−0.00301178	−	0.999995i	$-0.500959\pi$
$907$	−5.46337			−0.00602356			−0.00301178	+	0.999995i	$0.500959\pi$
$908$	867.137			0.954996
$909$	361.504			0.397694
$910$		0			0
$911$		−	1403.59i		−	1.54071i	−0.637614	−	0.770356i	$-0.720078\pi$
$911$		−	1403.59i		−	1.54071i	0.637614	−	0.770356i	$-0.279922\pi$
$912$	163.391	−	213.420i	0.179157	−	0.234013i
$913$		−	2305.36i		−	2.52504i
$914$		−	986.773i		−	1.07962i
$915$		0			0
$916$	−413.988			−0.451952
$917$			20.7943i			0.0226765i
$918$		−	744.393i		−	0.810885i
$919$	357.449			0.388954			0.194477	−	0.980907i	$-0.437699\pi$
$919$	357.449			0.388954			0.194477	+	0.980907i	$0.437699\pi$
$920$		0			0
$921$	−1659.73			−1.80209
$922$	594.453			0.644743
$923$			1583.42i			1.71552i
$924$			16.4677i			0.0178222i
$925$		0			0
$926$		−	679.643i		−	0.733956i
$927$	−444.133			−0.479108
$928$		−	30.8323i		−	0.0332245i
$929$	261.924			0.281942			0.140971	−	0.990014i	$-0.454978\pi$
$929$	261.924			0.281942			0.140971	+	0.990014i	$0.454978\pi$
$930$		0			0
$931$	−565.697	+	738.911i	−0.607623	+	0.793674i
$932$			581.672i			0.624112i
$933$	−177.019			−0.189731
$934$			790.948i			0.846839i
$935$		0			0
$936$	−167.927			−0.179409
$937$		−	1286.88i		−	1.37340i	−0.726941	−	0.686700i	$-0.759059\pi$
$937$		−	1286.88i		−	1.37340i	0.726941	−	0.686700i	$-0.240941\pi$
$938$			20.8072i			0.0221825i
$939$			758.592i			0.807872i
$940$		0			0
$941$			1118.46i			1.18858i	0.804249	+	0.594292i	$0.202568\pi$
$941$			1118.46i			1.18858i	−0.804249	+	0.594292i	$0.797432\pi$
$942$		−	146.142i		−	0.155140i
$943$	484.157			0.513422
$944$		−	345.816i		−	0.366331i
$945$		0			0
$946$		−	906.282i		−	0.958015i
$947$			1040.70i			1.09895i	0.835512	+	0.549473i	$0.185171\pi$
$947$			1040.70i			1.09895i	−0.835512	+	0.549473i	$0.814829\pi$
$948$			107.879i			0.113796i
$949$			75.4737i			0.0795297i
$950$		0			0
$951$	−421.928			−0.443667
$952$	−11.2258			−0.0117918
$953$	−158.536			−0.166354			−0.0831771	−	0.996535i	$-0.526507\pi$
$953$	−158.536			−0.166354			−0.0831771	+	0.996535i	$0.526507\pi$
$954$	−292.886			−0.307008
$955$		0			0
$956$	180.142			0.188433
$957$			306.414i			0.320182i
$958$	−348.856			−0.364150
$959$	−14.1129			−0.0147162
$960$		0			0
$961$	877.202			0.912801
$962$	176.558			0.183532
$963$	−77.4890			−0.0804663
$964$			171.242i			0.177637i
$965$		0			0
$966$	−14.7719			−0.0152918
$967$		−	462.170i		−	0.477942i	−0.971027	−	0.238971i	$-0.923190\pi$
$967$		−	462.170i		−	0.477942i	0.971027	−	0.238971i	$-0.0768101\pi$
$968$	372.457			0.384770
$969$	−1445.85	−	1106.91i	−1.49210	−	1.14233i
$970$		0			0
$971$			1501.52i			1.54636i	0.634184	+	0.773182i	$0.281336\pi$
$971$			1501.52i			1.54636i	−0.634184	+	0.773182i	$0.718664\pi$
$972$	359.569			0.369927
$973$		−	26.7401i		−	0.0274822i
$974$	592.279			0.608089
$975$		0			0
$976$	−61.3003			−0.0628077
$977$	1097.99			1.12384			0.561921	−	0.827191i	$-0.310063\pi$
$977$	1097.99			1.12384			0.561921	+	0.827191i	$0.310063\pi$
$978$		−	470.898i		−	0.481491i
$979$		−	383.540i		−	0.391767i
$980$		0			0
$981$			319.645i			0.325835i
$982$	−727.835			−0.741176
$983$	−938.538			−0.954769			−0.477385	−	0.878694i	$-0.658415\pi$
$983$	−938.538			−0.954769			−0.477385	+	0.878694i	$0.658415\pi$
$984$			240.167i			0.244073i
$985$		0			0
$986$	−208.878			−0.211844
$987$	44.4011			0.0449859
$988$	390.982	−	510.698i	0.395730	−	0.516901i
$989$	812.955			0.821997
$990$		0			0
$991$			540.016i			0.544920i	0.962167	+	0.272460i	$0.0878372\pi$
$991$			540.016i			0.544920i	−0.962167	+	0.272460i	$0.912163\pi$
$992$		−	51.7837i		−	0.0522013i
$993$			1676.91i			1.68873i
$994$	−19.3771			−0.0194941
$995$		0			0
$996$			1025.82i			1.02994i
$997$		−	406.644i		−	0.407868i	−0.978985	−	0.203934i	$-0.934627\pi$
$997$		−	406.644i		−	0.407868i	0.978985	−	0.203934i	$-0.0653728\pi$
$998$	215.964			0.216396
$999$	−143.273			−0.143417

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.3.d.b.949.20		24
5.2	odd	4		950.3.c.b.151.11	yes	12
5.3	odd	4		950.3.c.c.151.2	yes	12
5.4	even	2	inner	950.3.d.b.949.5		24
19.18	odd	2	inner	950.3.d.b.949.8		24
95.18	even	4		950.3.c.c.151.11	yes	12
95.37	even	4		950.3.c.b.151.2	✓	12
95.94	odd	2	inner	950.3.d.b.949.17		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.3.c.b.151.2	✓	12	95.37	even	4
950.3.c.b.151.11	yes	12	5.2	odd	4
950.3.c.c.151.2	yes	12	5.3	odd	4
950.3.c.c.151.11	yes	12	95.18	even	4
950.3.d.b.949.5		24	5.4	even	2	inner
950.3.d.b.949.8		24	19.18	odd	2	inner
950.3.d.b.949.17		24	95.94	odd	2	inner
950.3.d.b.949.20		24	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	950.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} -3.53663 q^{3} +2.00000 q^{4} -5.00155 q^{6} -0.146462i q^{7} +2.82843 q^{8} +3.50774 q^{9} +O(q^{10})\)	\(q+1.41421 q^{2} -3.53663 q^{3} +2.00000 q^{4} -5.00155 q^{6} -0.146462i q^{7} +2.82843 q^{8} +3.50774 q^{9} +15.8960 q^{11} -7.07326 q^{12} -16.9258 q^{13} -0.207129i q^{14} +4.00000 q^{16} -27.0986i q^{17} +4.96069 q^{18} +(-11.5499 + 15.0864i) q^{19} +0.517983i q^{21} +22.4804 q^{22} +20.1654i q^{23} -10.0031 q^{24} -23.9366 q^{26} +19.4241 q^{27} -0.292925i q^{28} -5.45043i q^{29} -9.15415i q^{31} +5.65685 q^{32} -56.2183 q^{33} -38.3232i q^{34} +7.01547 q^{36} -7.37605 q^{37} +(-16.3340 + 21.3354i) q^{38} +59.8601 q^{39} -24.0093i q^{41} +0.732538i q^{42} -40.3144i q^{43} +31.7920 q^{44} +28.5181i q^{46} -85.7192i q^{47} -14.1465 q^{48} +48.9785 q^{49} +95.8376i q^{51} -33.8515 q^{52} -59.0413 q^{53} +27.4698 q^{54} -0.414258i q^{56} +(40.8477 - 53.3550i) q^{57} -7.70807i q^{58} -86.4540i q^{59} -15.3251 q^{61} -12.9459i q^{62} -0.513751i q^{63} +8.00000 q^{64} -79.5047 q^{66} -100.455 q^{67} -54.1971i q^{68} -71.3174i q^{69} -93.5509i q^{71} +9.92138 q^{72} -4.45910i q^{73} -10.4313 q^{74} +(-23.0998 + 30.1728i) q^{76} -2.32817i q^{77} +84.6550 q^{78} -15.2517i q^{79} -100.265 q^{81} -33.9543i q^{82} -145.028i q^{83} +1.03597i q^{84} -57.0132i q^{86} +19.2761i q^{87} +44.9607 q^{88} -24.1281i q^{89} +2.47899i q^{91} +40.3308i q^{92} +32.3748i q^{93} -121.225i q^{94} -20.0062 q^{96} -3.08788 q^{97} +69.2661 q^{98} +55.7591 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 48 q^{4} - 16 q^{6} + 56 q^{9}+O(q^{10}) \) 24 * q + 48 * q^4 - 16 * q^6 + 56 * q^9	\( 24 q + 48 q^{4} - 16 q^{6} + 56 q^{9} + 8 q^{11} + 96 q^{16} + 24 q^{19} - 32 q^{24} - 32 q^{26} + 112 q^{36} - 200 q^{39} + 16 q^{44} + 216 q^{49} - 224 q^{54} - 152 q^{61} + 192 q^{64} + 16 q^{66} - 144 q^{74} + 48 q^{76} - 264 q^{81} - 64 q^{96} - 256 q^{99}+O(q^{100}) \) 24 * q + 48 * q^4 - 16 * q^6 + 56 * q^9 + 8 * q^11 + 96 * q^16 + 24 * q^19 - 32 * q^24 - 32 * q^26 + 112 * q^36 - 200 * q^39 + 16 * q^44 + 216 * q^49 - 224 * q^54 - 152 * q^61 + 192 * q^64 + 16 * q^66 - 144 * q^74 + 48 * q^76 - 264 * q^81 - 64 * q^96 - 256 * q^99

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