LMFDB - Embedded newform 2850.2.a.bg.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2850,2,Mod(1,2850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2850, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2850.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2850.a (trivial)

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:	yes
Analytic conductor:	$22.7573645761$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{10}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 10 $ x^2 - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.16228$ of defining polynomial
Character	$\chi$	$=$	2850.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -3.16228 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -3.16228 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.16228 q^{11} -1.00000 q^{12} -1.16228 q^{13} -3.16228 q^{14} +1.00000 q^{16} +7.16228 q^{17} +1.00000 q^{18} -1.00000 q^{19} +3.16228 q^{21} -2.16228 q^{22} +7.32456 q^{23} -1.00000 q^{24} -1.16228 q^{26} -1.00000 q^{27} -3.16228 q^{28} -10.1623 q^{29} -7.32456 q^{31} +1.00000 q^{32} +2.16228 q^{33} +7.16228 q^{34} +1.00000 q^{36} +10.0000 q^{37} -1.00000 q^{38} +1.16228 q^{39} +6.32456 q^{41} +3.16228 q^{42} -2.83772 q^{43} -2.16228 q^{44} +7.32456 q^{46} +6.00000 q^{47} -1.00000 q^{48} +3.00000 q^{49} -7.16228 q^{51} -1.16228 q^{52} +8.16228 q^{53} -1.00000 q^{54} -3.16228 q^{56} +1.00000 q^{57} -10.1623 q^{58} +11.4868 q^{59} +2.16228 q^{61} -7.32456 q^{62} -3.16228 q^{63} +1.00000 q^{64} +2.16228 q^{66} +12.4868 q^{67} +7.16228 q^{68} -7.32456 q^{69} -5.16228 q^{71} +1.00000 q^{72} -1.00000 q^{73} +10.0000 q^{74} -1.00000 q^{76} +6.83772 q^{77} +1.16228 q^{78} +7.32456 q^{79} +1.00000 q^{81} +6.32456 q^{82} -12.4868 q^{83} +3.16228 q^{84} -2.83772 q^{86} +10.1623 q^{87} -2.16228 q^{88} +5.32456 q^{89} +3.67544 q^{91} +7.32456 q^{92} +7.32456 q^{93} +6.00000 q^{94} -1.00000 q^{96} +19.4868 q^{97} +3.00000 q^{98} -2.16228 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^6 + 2 * q^8 + 2 * q^9	$ 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} + 2 q^{8} + 2 q^{9} + 2 q^{11} - 2 q^{12} + 4 q^{13} + 2 q^{16} + 8 q^{17} + 2 q^{18} - 2 q^{19} + 2 q^{22} + 2 q^{23} - 2 q^{24} + 4 q^{26} - 2 q^{27} - 14 q^{29} - 2 q^{31} + 2 q^{32} - 2 q^{33} + 8 q^{34} + 2 q^{36} + 20 q^{37} - 2 q^{38} - 4 q^{39} - 12 q^{43} + 2 q^{44} + 2 q^{46} + 12 q^{47} - 2 q^{48} + 6 q^{49} - 8 q^{51} + 4 q^{52} + 10 q^{53} - 2 q^{54} + 2 q^{57} - 14 q^{58} + 4 q^{59} - 2 q^{61} - 2 q^{62} + 2 q^{64} - 2 q^{66} + 6 q^{67} + 8 q^{68} - 2 q^{69} - 4 q^{71} + 2 q^{72} - 2 q^{73} + 20 q^{74} - 2 q^{76} + 20 q^{77} - 4 q^{78} + 2 q^{79} + 2 q^{81} - 6 q^{83} - 12 q^{86} + 14 q^{87} + 2 q^{88} - 2 q^{89} + 20 q^{91} + 2 q^{92} + 2 q^{93} + 12 q^{94} - 2 q^{96} + 20 q^{97} + 6 q^{98} + 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^6 + 2 * q^8 + 2 * q^9 + 2 * q^11 - 2 * q^12 + 4 * q^13 + 2 * q^16 + 8 * q^17 + 2 * q^18 - 2 * q^19 + 2 * q^22 + 2 * q^23 - 2 * q^24 + 4 * q^26 - 2 * q^27 - 14 * q^29 - 2 * q^31 + 2 * q^32 - 2 * q^33 + 8 * q^34 + 2 * q^36 + 20 * q^37 - 2 * q^38 - 4 * q^39 - 12 * q^43 + 2 * q^44 + 2 * q^46 + 12 * q^47 - 2 * q^48 + 6 * q^49 - 8 * q^51 + 4 * q^52 + 10 * q^53 - 2 * q^54 + 2 * q^57 - 14 * q^58 + 4 * q^59 - 2 * q^61 - 2 * q^62 + 2 * q^64 - 2 * q^66 + 6 * q^67 + 8 * q^68 - 2 * q^69 - 4 * q^71 + 2 * q^72 - 2 * q^73 + 20 * q^74 - 2 * q^76 + 20 * q^77 - 4 * q^78 + 2 * q^79 + 2 * q^81 - 6 * q^83 - 12 * q^86 + 14 * q^87 + 2 * q^88 - 2 * q^89 + 20 * q^91 + 2 * q^92 + 2 * q^93 + 12 * q^94 - 2 * q^96 + 20 * q^97 + 6 * q^98 + 2 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	−3.16228	−1.19523	−0.597614	−	0.801784i	$-0.703885\pi$
$7$	−3.16228	−1.19523	−0.597614	+	0.801784i	$0.703885\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.16228	−0.651951	−0.325976	−	0.945378i	$-0.605693\pi$
$11$	−2.16228	−0.651951	−0.325976	+	0.945378i	$0.605693\pi$
$12$	−1.00000	−0.288675
$13$	−1.16228	−0.322358	−0.161179	−	0.986925i	$-0.551530\pi$
$13$	−1.16228	−0.322358	−0.161179	+	0.986925i	$0.551530\pi$
$14$	−3.16228	−0.845154
$15$	0	0
$16$	1.00000	0.250000
$17$	7.16228	1.73711	0.868554	−	0.495595i	$-0.165050\pi$
$17$	7.16228	1.73711	0.868554	+	0.495595i	$0.165050\pi$
$18$	1.00000	0.235702
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	3.16228	0.690066
$22$	−2.16228	−0.460999
$23$	7.32456	1.52728	0.763638	−	0.645645i	$-0.223411\pi$
$23$	7.32456	1.52728	0.763638	+	0.645645i	$0.223411\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−1.16228	−0.227941
$27$	−1.00000	−0.192450
$28$	−3.16228	−0.597614
$29$	−10.1623	−1.88709	−0.943544	−	0.331248i	$-0.892530\pi$
$29$	−10.1623	−1.88709	−0.943544	+	0.331248i	$0.892530\pi$
$30$	0	0
$31$	−7.32456	−1.31553	−0.657764	−	0.753224i	$-0.728498\pi$
$31$	−7.32456	−1.31553	−0.657764	+	0.753224i	$0.728498\pi$
$32$	1.00000	0.176777
$33$	2.16228	0.376404
$34$	7.16228	1.22832
$35$	0	0
$36$	1.00000	0.166667
$37$	10.0000	1.64399	0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000	1.64399	0.821995	+	0.569495i	$0.192861\pi$
$38$	−1.00000	−0.162221
$39$	1.16228	0.186113
$40$	0	0
$41$	6.32456	0.987730	0.493865	−	0.869539i	$-0.335584\pi$
$41$	6.32456	0.987730	0.493865	+	0.869539i	$0.335584\pi$
$42$	3.16228	0.487950
$43$	−2.83772	−0.432749	−0.216374	−	0.976310i	$-0.569423\pi$
$43$	−2.83772	−0.432749	−0.216374	+	0.976310i	$0.569423\pi$
$44$	−2.16228	−0.325976
$45$	0	0
$46$	7.32456	1.07995
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	−1.00000	−0.144338
$49$	3.00000	0.428571
$50$	0	0
$51$	−7.16228	−1.00292
$52$	−1.16228	−0.161179
$53$	8.16228	1.12118	0.560588	−	0.828095i	$-0.310575\pi$
$53$	8.16228	1.12118	0.560588	+	0.828095i	$0.310575\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	−3.16228	−0.422577
$57$	1.00000	0.132453
$58$	−10.1623	−1.33437
$59$	11.4868	1.49546	0.747729	−	0.664004i	$-0.231144\pi$
$59$	11.4868	1.49546	0.747729	+	0.664004i	$0.231144\pi$
$60$	0	0
$61$	2.16228	0.276851	0.138426	−	0.990373i	$-0.455796\pi$
$61$	2.16228	0.276851	0.138426	+	0.990373i	$0.455796\pi$
$62$	−7.32456	−0.930219
$63$	−3.16228	−0.398410
$64$	1.00000	0.125000
$65$	0	0
$66$	2.16228	0.266158
$67$	12.4868	1.52551	0.762755	−	0.646688i	$-0.223846\pi$
$67$	12.4868	1.52551	0.762755	+	0.646688i	$0.223846\pi$
$68$	7.16228	0.868554
$69$	−7.32456	−0.881773
$70$	0	0
$71$	−5.16228	−0.612650	−0.306325	−	0.951927i	$-0.599099\pi$
$71$	−5.16228	−0.612650	−0.306325	+	0.951927i	$0.599099\pi$
$72$	1.00000	0.117851
$73$	−1.00000	−0.117041	−0.0585206	−	0.998286i	$-0.518638\pi$
$73$	−1.00000	−0.117041	−0.0585206	+	0.998286i	$0.518638\pi$
$74$	10.0000	1.16248
$75$	0	0
$76$	−1.00000	−0.114708
$77$	6.83772	0.779231
$78$	1.16228	0.131602
$79$	7.32456	0.824077	0.412038	−	0.911166i	$-0.364817\pi$
$79$	7.32456	0.824077	0.412038	+	0.911166i	$0.364817\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	6.32456	0.698430
$83$	−12.4868	−1.37061	−0.685304	−	0.728257i	$-0.740331\pi$
$83$	−12.4868	−1.37061	−0.685304	+	0.728257i	$0.740331\pi$
$84$	3.16228	0.345033
$85$	0	0
$86$	−2.83772	−0.305999
$87$	10.1623	1.08951
$88$	−2.16228	−0.230500
$89$	5.32456	0.564402	0.282201	−	0.959355i	$-0.408936\pi$
$89$	5.32456	0.564402	0.282201	+	0.959355i	$0.408936\pi$
$90$	0	0
$91$	3.67544	0.385291
$92$	7.32456	0.763638
$93$	7.32456	0.759521
$94$	6.00000	0.618853
$95$	0	0
$96$	−1.00000	−0.102062
$97$	19.4868	1.97859	0.989294	−	0.145936i	$-0.0466193\pi$
$97$	19.4868	1.97859	0.989294	+	0.145936i	$0.0466193\pi$
$98$	3.00000	0.303046
$99$	−2.16228	−0.217317
$100$	0	0
$101$	17.8114	1.77230	0.886150	−	0.463399i	$-0.153370\pi$
$101$	17.8114	1.77230	0.886150	+	0.463399i	$0.153370\pi$
$102$	−7.16228	−0.709171
$103$	−9.64911	−0.950755	−0.475378	−	0.879782i	$-0.657689\pi$
$103$	−9.64911	−0.950755	−0.475378	+	0.879782i	$0.657689\pi$
$104$	−1.16228	−0.113971
$105$	0	0
$106$	8.16228	0.792790
$107$	19.1623	1.85249	0.926244	−	0.376925i	$-0.123019\pi$
$107$	19.1623	1.85249	0.926244	+	0.376925i	$0.123019\pi$
$108$	−1.00000	−0.0962250
$109$	20.3246	1.94674	0.973370	−	0.229241i	$-0.0736244\pi$
$109$	20.3246	1.94674	0.973370	+	0.229241i	$0.0736244\pi$
$110$	0	0
$111$	−10.0000	−0.949158
$112$	−3.16228	−0.298807
$113$	−15.6491	−1.47214	−0.736072	−	0.676903i	$-0.763322\pi$
$113$	−15.6491	−1.47214	−0.736072	+	0.676903i	$0.763322\pi$
$114$	1.00000	0.0936586
$115$	0	0
$116$	−10.1623	−0.943544
$117$	−1.16228	−0.107453
$118$	11.4868	1.05745
$119$	−22.6491	−2.07624
$120$	0	0
$121$	−6.32456	−0.574960
$122$	2.16228	0.195763
$123$	−6.32456	−0.570266
$124$	−7.32456	−0.657764
$125$	0	0
$126$	−3.16228	−0.281718
$127$	−9.64911	−0.856220	−0.428110	−	0.903727i	$-0.640820\pi$
$127$	−9.64911	−0.856220	−0.428110	+	0.903727i	$0.640820\pi$
$128$	1.00000	0.0883883
$129$	2.83772	0.249848
$130$	0	0
$131$	6.48683	0.566757	0.283379	−	0.959008i	$-0.408545\pi$
$131$	6.48683	0.566757	0.283379	+	0.959008i	$0.408545\pi$
$132$	2.16228	0.188202
$133$	3.16228	0.274204
$134$	12.4868	1.07870
$135$	0	0
$136$	7.16228	0.614160
$137$	−10.3246	−0.882086	−0.441043	−	0.897486i	$-0.645391\pi$
$137$	−10.3246	−0.882086	−0.441043	+	0.897486i	$0.645391\pi$
$138$	−7.32456	−0.623508
$139$	2.51317	0.213164	0.106582	−	0.994304i	$-0.466009\pi$
$139$	2.51317	0.213164	0.106582	+	0.994304i	$0.466009\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	−5.16228	−0.433209
$143$	2.51317	0.210162
$144$	1.00000	0.0833333
$145$	0	0
$146$	−1.00000	−0.0827606
$147$	−3.00000	−0.247436
$148$	10.0000	0.821995
$149$	−8.64911	−0.708563	−0.354281	−	0.935139i	$-0.615274\pi$
$149$	−8.64911	−0.708563	−0.354281	+	0.935139i	$0.615274\pi$
$150$	0	0
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	−1.00000	−0.0811107
$153$	7.16228	0.579036
$154$	6.83772	0.550999
$155$	0	0
$156$	1.16228	0.0930567
$157$	5.67544	0.452950	0.226475	−	0.974017i	$-0.427280\pi$
$157$	5.67544	0.452950	0.226475	+	0.974017i	$0.427280\pi$
$158$	7.32456	0.582710
$159$	−8.16228	−0.647311
$160$	0	0
$161$	−23.1623	−1.82544
$162$	1.00000	0.0785674
$163$	2.64911	0.207494	0.103747	−	0.994604i	$-0.466917\pi$
$163$	2.64911	0.207494	0.103747	+	0.994604i	$0.466917\pi$
$164$	6.32456	0.493865
$165$	0	0
$166$	−12.4868	−0.969166
$167$	−22.6491	−1.75264	−0.876320	−	0.481729i	$-0.840009\pi$
$167$	−22.6491	−1.75264	−0.876320	+	0.481729i	$0.840009\pi$
$168$	3.16228	0.243975
$169$	−11.6491	−0.896085
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	−2.83772	−0.216374
$173$	4.16228	0.316452	0.158226	−	0.987403i	$-0.449423\pi$
$173$	4.16228	0.316452	0.158226	+	0.987403i	$0.449423\pi$
$174$	10.1623	0.770400
$175$	0	0
$176$	−2.16228	−0.162988
$177$	−11.4868	−0.863403
$178$	5.32456	0.399092
$179$	4.64911	0.347491	0.173745	−	0.984791i	$-0.444413\pi$
$179$	4.64911	0.347491	0.173745	+	0.984791i	$0.444413\pi$
$180$	0	0
$181$	2.83772	0.210926	0.105463	−	0.994423i	$-0.466368\pi$
$181$	2.83772	0.210926	0.105463	+	0.994423i	$0.466368\pi$
$182$	3.67544	0.272442
$183$	−2.16228	−0.159840
$184$	7.32456	0.539973
$185$	0	0
$186$	7.32456	0.537062
$187$	−15.4868	−1.13251
$188$	6.00000	0.437595
$189$	3.16228	0.230022
$190$	0	0
$191$	−3.64911	−0.264040	−0.132020	−	0.991247i	$-0.542146\pi$
$191$	−3.64911	−0.264040	−0.132020	+	0.991247i	$0.542146\pi$
$192$	−1.00000	−0.0721688
$193$	−2.83772	−0.204264	−0.102132	−	0.994771i	$-0.532566\pi$
$193$	−2.83772	−0.204264	−0.102132	+	0.994771i	$0.532566\pi$
$194$	19.4868	1.39907
$195$	0	0
$196$	3.00000	0.214286
$197$	20.1359	1.43463	0.717313	−	0.696751i	$-0.245372\pi$
$197$	20.1359	1.43463	0.717313	+	0.696751i	$0.245372\pi$
$198$	−2.16228	−0.153666
$199$	6.64911	0.471343	0.235671	−	0.971833i	$-0.424271\pi$
$199$	6.64911	0.471343	0.235671	+	0.971833i	$0.424271\pi$
$200$	0	0
$201$	−12.4868	−0.880753
$202$	17.8114	1.25320
$203$	32.1359	2.25550
$204$	−7.16228	−0.501460
$205$	0	0
$206$	−9.64911	−0.672285
$207$	7.32456	0.509092
$208$	−1.16228	−0.0805895
$209$	2.16228	0.149568
$210$	0	0
$211$	12.8114	0.881972	0.440986	−	0.897514i	$-0.354629\pi$
$211$	12.8114	0.881972	0.440986	+	0.897514i	$0.354629\pi$
$212$	8.16228	0.560588
$213$	5.16228	0.353713
$214$	19.1623	1.30991
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	23.1623	1.57236
$218$	20.3246	1.37655
$219$	1.00000	0.0675737
$220$	0	0
$221$	−8.32456	−0.559970
$222$	−10.0000	−0.671156
$223$	−0.350889	−0.0234973	−0.0117486	−	0.999931i	$-0.503740\pi$
$223$	−0.350889	−0.0234973	−0.0117486	+	0.999931i	$0.503740\pi$
$224$	−3.16228	−0.211289
$225$	0	0
$226$	−15.6491	−1.04096
$227$	−2.00000	−0.132745	−0.0663723	−	0.997795i	$-0.521143\pi$
$227$	−2.00000	−0.132745	−0.0663723	+	0.997795i	$0.521143\pi$
$228$	1.00000	0.0662266
$229$	1.83772	0.121440	0.0607201	−	0.998155i	$-0.480660\pi$
$229$	1.83772	0.121440	0.0607201	+	0.998155i	$0.480660\pi$
$230$	0	0
$231$	−6.83772	−0.449889
$232$	−10.1623	−0.667186
$233$	19.8114	1.29789	0.648944	−	0.760837i	$-0.275211\pi$
$233$	19.8114	1.29789	0.648944	+	0.760837i	$0.275211\pi$
$234$	−1.16228	−0.0759805
$235$	0	0
$236$	11.4868	0.747729
$237$	−7.32456	−0.475781
$238$	−22.6491	−1.46812
$239$	−11.6754	−0.755222	−0.377611	−	0.925964i	$-0.623254\pi$
$239$	−11.6754	−0.755222	−0.377611	+	0.925964i	$0.623254\pi$
$240$	0	0
$241$	29.4868	1.89941	0.949707	−	0.313140i	$-0.101381\pi$
$241$	29.4868	1.89941	0.949707	+	0.313140i	$0.101381\pi$
$242$	−6.32456	−0.406558
$243$	−1.00000	−0.0641500
$244$	2.16228	0.138426
$245$	0	0
$246$	−6.32456	−0.403239
$247$	1.16228	0.0739540
$248$	−7.32456	−0.465110
$249$	12.4868	0.791321
$250$	0	0
$251$	14.6491	0.924644	0.462322	−	0.886712i	$-0.347016\pi$
$251$	14.6491	0.924644	0.462322	+	0.886712i	$0.347016\pi$
$252$	−3.16228	−0.199205
$253$	−15.8377	−0.995709
$254$	−9.64911	−0.605439
$255$	0	0
$256$	1.00000	0.0625000
$257$	−6.67544	−0.416403	−0.208201	−	0.978086i	$-0.566761\pi$
$257$	−6.67544	−0.416403	−0.208201	+	0.978086i	$0.566761\pi$
$258$	2.83772	0.176669
$259$	−31.6228	−1.96494
$260$	0	0
$261$	−10.1623	−0.629029
$262$	6.48683	0.400758
$263$	−5.97367	−0.368352	−0.184176	−	0.982893i	$-0.558962\pi$
$263$	−5.97367	−0.368352	−0.184176	+	0.982893i	$0.558962\pi$
$264$	2.16228	0.133079
$265$	0	0
$266$	3.16228	0.193892
$267$	−5.32456	−0.325857
$268$	12.4868	0.762755
$269$	−0.324555	−0.0197885	−0.00989424	−	0.999951i	$-0.503149\pi$
$269$	−0.324555	−0.0197885	−0.00989424	+	0.999951i	$0.503149\pi$
$270$	0	0
$271$	5.16228	0.313586	0.156793	−	0.987631i	$-0.449884\pi$
$271$	5.16228	0.313586	0.156793	+	0.987631i	$0.449884\pi$
$272$	7.16228	0.434277
$273$	−3.67544	−0.222448
$274$	−10.3246	−0.623729
$275$	0	0
$276$	−7.32456	−0.440886
$277$	−6.81139	−0.409257	−0.204628	−	0.978840i	$-0.565599\pi$
$277$	−6.81139	−0.409257	−0.204628	+	0.978840i	$0.565599\pi$
$278$	2.51317	0.150730
$279$	−7.32456	−0.438510
$280$	0	0
$281$	−17.6491	−1.05286	−0.526429	−	0.850219i	$-0.676469\pi$
$281$	−17.6491	−1.05286	−0.526429	+	0.850219i	$0.676469\pi$
$282$	−6.00000	−0.357295
$283$	−0.649111	−0.0385856	−0.0192928	−	0.999814i	$-0.506141\pi$
$283$	−0.649111	−0.0385856	−0.0192928	+	0.999814i	$0.506141\pi$
$284$	−5.16228	−0.306325
$285$	0	0
$286$	2.51317	0.148607
$287$	−20.0000	−1.18056
$288$	1.00000	0.0589256
$289$	34.2982	2.01754
$290$	0	0
$291$	−19.4868	−1.14234
$292$	−1.00000	−0.0585206
$293$	−2.48683	−0.145282	−0.0726412	−	0.997358i	$-0.523143\pi$
$293$	−2.48683	−0.145282	−0.0726412	+	0.997358i	$0.523143\pi$
$294$	−3.00000	−0.174964
$295$	0	0
$296$	10.0000	0.581238
$297$	2.16228	0.125468
$298$	−8.64911	−0.501030
$299$	−8.51317	−0.492329
$300$	0	0
$301$	8.97367	0.517234
$302$	−10.0000	−0.575435
$303$	−17.8114	−1.02324
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	7.16228	0.409440
$307$	32.4868	1.85412	0.927061	−	0.374911i	$-0.122327\pi$
$307$	32.4868	1.85412	0.927061	+	0.374911i	$0.122327\pi$
$308$	6.83772	0.389615
$309$	9.64911	0.548919
$310$	0	0
$311$	−26.0000	−1.47432	−0.737162	−	0.675716i	$-0.763835\pi$
$311$	−26.0000	−1.47432	−0.737162	+	0.675716i	$0.763835\pi$
$312$	1.16228	0.0658010
$313$	−23.3246	−1.31838	−0.659191	−	0.751976i	$-0.729101\pi$
$313$	−23.3246	−1.31838	−0.659191	+	0.751976i	$0.729101\pi$
$314$	5.67544	0.320284
$315$	0	0
$316$	7.32456	0.412038
$317$	−2.16228	−0.121446	−0.0607228	−	0.998155i	$-0.519341\pi$
$317$	−2.16228	−0.121446	−0.0607228	+	0.998155i	$0.519341\pi$
$318$	−8.16228	−0.457718
$319$	21.9737	1.23029
$320$	0	0
$321$	−19.1623	−1.06953
$322$	−23.1623	−1.29078
$323$	−7.16228	−0.398520
$324$	1.00000	0.0555556
$325$	0	0
$326$	2.64911	0.146721
$327$	−20.3246	−1.12395
$328$	6.32456	0.349215
$329$	−18.9737	−1.04605
$330$	0	0
$331$	−34.8114	−1.91341	−0.956703	−	0.291064i	$-0.905991\pi$
$331$	−34.8114	−1.91341	−0.956703	+	0.291064i	$0.905991\pi$
$332$	−12.4868	−0.685304
$333$	10.0000	0.547997
$334$	−22.6491	−1.23930
$335$	0	0
$336$	3.16228	0.172516
$337$	34.9737	1.90514	0.952568	−	0.304324i	$-0.0984306\pi$
$337$	34.9737	1.90514	0.952568	+	0.304324i	$0.0984306\pi$
$338$	−11.6491	−0.633628
$339$	15.6491	0.849943
$340$	0	0
$341$	15.8377	0.857661
$342$	−1.00000	−0.0540738
$343$	12.6491	0.682988
$344$	−2.83772	−0.153000
$345$	0	0
$346$	4.16228	0.223765
$347$	−20.0000	−1.07366	−0.536828	−	0.843692i	$-0.680378\pi$
$347$	−20.0000	−1.07366	−0.536828	+	0.843692i	$0.680378\pi$
$348$	10.1623	0.544755
$349$	−24.1623	−1.29338	−0.646689	−	0.762754i	$-0.723847\pi$
$349$	−24.1623	−1.29338	−0.646689	+	0.762754i	$0.723847\pi$
$350$	0	0
$351$	1.16228	0.0620378
$352$	−2.16228	−0.115250
$353$	−7.81139	−0.415758	−0.207879	−	0.978155i	$-0.566656\pi$
$353$	−7.81139	−0.415758	−0.207879	+	0.978155i	$0.566656\pi$
$354$	−11.4868	−0.610518
$355$	0	0
$356$	5.32456	0.282201
$357$	22.6491	1.19872
$358$	4.64911	0.245713
$359$	−24.3246	−1.28380	−0.641900	−	0.766788i	$-0.721854\pi$
$359$	−24.3246	−1.28380	−0.641900	+	0.766788i	$0.721854\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	2.83772	0.149147
$363$	6.32456	0.331953
$364$	3.67544	0.192646
$365$	0	0
$366$	−2.16228	−0.113024
$367$	11.4868	0.599608	0.299804	−	0.954001i	$-0.403079\pi$
$367$	11.4868	0.599608	0.299804	+	0.954001i	$0.403079\pi$
$368$	7.32456	0.381819
$369$	6.32456	0.329243
$370$	0	0
$371$	−25.8114	−1.34006
$372$	7.32456	0.379761
$373$	28.1359	1.45682	0.728412	−	0.685139i	$-0.240259\pi$
$373$	28.1359	1.45682	0.728412	+	0.685139i	$0.240259\pi$
$374$	−15.4868	−0.800805
$375$	0	0
$376$	6.00000	0.309426
$377$	11.8114	0.608317
$378$	3.16228	0.162650
$379$	−14.9737	−0.769146	−0.384573	−	0.923095i	$-0.625651\pi$
$379$	−14.9737	−0.769146	−0.384573	+	0.923095i	$0.625651\pi$
$380$	0	0
$381$	9.64911	0.494339
$382$	−3.64911	−0.186705
$383$	15.4868	0.791340	0.395670	−	0.918393i	$-0.370512\pi$
$383$	15.4868	0.791340	0.395670	+	0.918393i	$0.370512\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−2.83772	−0.144436
$387$	−2.83772	−0.144250
$388$	19.4868	0.989294
$389$	−9.35089	−0.474109	−0.237054	−	0.971496i	$-0.576182\pi$
$389$	−9.35089	−0.474109	−0.237054	+	0.971496i	$0.576182\pi$
$390$	0	0
$391$	52.4605	2.65304
$392$	3.00000	0.151523
$393$	−6.48683	−0.327217
$394$	20.1359	1.01443
$395$	0	0
$396$	−2.16228	−0.108659
$397$	−27.1359	−1.36191	−0.680957	−	0.732323i	$-0.738436\pi$
$397$	−27.1359	−1.36191	−0.680957	+	0.732323i	$0.738436\pi$
$398$	6.64911	0.333290
$399$	−3.16228	−0.158312
$400$	0	0
$401$	−11.6491	−0.581729	−0.290864	−	0.956764i	$-0.593943\pi$
$401$	−11.6491	−0.581729	−0.290864	+	0.956764i	$0.593943\pi$
$402$	−12.4868	−0.622787
$403$	8.51317	0.424071
$404$	17.8114	0.886150
$405$	0	0
$406$	32.1359	1.59488
$407$	−21.6228	−1.07180
$408$	−7.16228	−0.354586
$409$	−20.3246	−1.00498	−0.502492	−	0.864582i	$-0.667583\pi$
$409$	−20.3246	−1.00498	−0.502492	+	0.864582i	$0.667583\pi$
$410$	0	0
$411$	10.3246	0.509273
$412$	−9.64911	−0.475378
$413$	−36.3246	−1.78741
$414$	7.32456	0.359982
$415$	0	0
$416$	−1.16228	−0.0569854
$417$	−2.51317	−0.123070
$418$	2.16228	0.105760
$419$	8.00000	0.390826	0.195413	−	0.980721i	$-0.437395\pi$
$419$	8.00000	0.390826	0.195413	+	0.980721i	$0.437395\pi$
$420$	0	0
$421$	−6.64911	−0.324058	−0.162029	−	0.986786i	$-0.551804\pi$
$421$	−6.64911	−0.324058	−0.162029	+	0.986786i	$0.551804\pi$
$422$	12.8114	0.623649
$423$	6.00000	0.291730
$424$	8.16228	0.396395
$425$	0	0
$426$	5.16228	0.250113
$427$	−6.83772	−0.330901
$428$	19.1623	0.926244
$429$	−2.51317	−0.121337
$430$	0	0
$431$	12.3246	0.593653	0.296826	−	0.954931i	$-0.404072\pi$
$431$	12.3246	0.593653	0.296826	+	0.954931i	$0.404072\pi$
$432$	−1.00000	−0.0481125
$433$	−16.1359	−0.775444	−0.387722	−	0.921776i	$-0.626738\pi$
$433$	−16.1359	−0.775444	−0.387722	+	0.921776i	$0.626738\pi$
$434$	23.1623	1.11182
$435$	0	0
$436$	20.3246	0.973370
$437$	−7.32456	−0.350381
$438$	1.00000	0.0477818
$439$	−11.6491	−0.555982	−0.277991	−	0.960584i	$-0.589668\pi$
$439$	−11.6491	−0.555982	−0.277991	+	0.960584i	$0.589668\pi$
$440$	0	0
$441$	3.00000	0.142857
$442$	−8.32456	−0.395959
$443$	−18.1623	−0.862916	−0.431458	−	0.902133i	$-0.642001\pi$
$443$	−18.1623	−0.862916	−0.431458	+	0.902133i	$0.642001\pi$
$444$	−10.0000	−0.474579
$445$	0	0
$446$	−0.350889	−0.0166151
$447$	8.64911	0.409089
$448$	−3.16228	−0.149404
$449$	−11.0000	−0.519122	−0.259561	−	0.965727i	$-0.583578\pi$
$449$	−11.0000	−0.519122	−0.259561	+	0.965727i	$0.583578\pi$
$450$	0	0
$451$	−13.6754	−0.643952
$452$	−15.6491	−0.736072
$453$	10.0000	0.469841
$454$	−2.00000	−0.0938647
$455$	0	0
$456$	1.00000	0.0468293
$457$	22.3246	1.04430	0.522149	−	0.852854i	$-0.325130\pi$
$457$	22.3246	1.04430	0.522149	+	0.852854i	$0.325130\pi$
$458$	1.83772	0.0858711
$459$	−7.16228	−0.334306
$460$	0	0
$461$	12.0000	0.558896	0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000	0.558896	0.279448	+	0.960161i	$0.409849\pi$
$462$	−6.83772	−0.318120
$463$	−34.3246	−1.59520	−0.797599	−	0.603188i	$-0.793897\pi$
$463$	−34.3246	−1.59520	−0.797599	+	0.603188i	$0.793897\pi$
$464$	−10.1623	−0.471772
$465$	0	0
$466$	19.8114	0.917745
$467$	7.13594	0.330212	0.165106	−	0.986276i	$-0.447203\pi$
$467$	7.13594	0.330212	0.165106	+	0.986276i	$0.447203\pi$
$468$	−1.16228	−0.0537263
$469$	−39.4868	−1.82333
$470$	0	0
$471$	−5.67544	−0.261511
$472$	11.4868	0.528724
$473$	6.13594	0.282131
$474$	−7.32456	−0.336428
$475$	0	0
$476$	−22.6491	−1.03812
$477$	8.16228	0.373725
$478$	−11.6754	−0.534022
$479$	−2.02633	−0.0925856	−0.0462928	−	0.998928i	$-0.514741\pi$
$479$	−2.02633	−0.0925856	−0.0462928	+	0.998928i	$0.514741\pi$
$480$	0	0
$481$	−11.6228	−0.529953
$482$	29.4868	1.34309
$483$	23.1623	1.05392
$484$	−6.32456	−0.287480
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	−3.35089	−0.151843	−0.0759216	−	0.997114i	$-0.524190\pi$
$487$	−3.35089	−0.151843	−0.0759216	+	0.997114i	$0.524190\pi$
$488$	2.16228	0.0978817
$489$	−2.64911	−0.119797
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	−6.32456	−0.285133
$493$	−72.7851	−3.27807
$494$	1.16228	0.0522933
$495$	0	0
$496$	−7.32456	−0.328882
$497$	16.3246	0.732256
$498$	12.4868	0.559548
$499$	25.8114	1.15548	0.577738	−	0.816222i	$-0.303935\pi$
$499$	25.8114	1.15548	0.577738	+	0.816222i	$0.303935\pi$
$500$	0	0
$501$	22.6491	1.01189
$502$	14.6491	0.653822
$503$	21.2982	0.949641	0.474820	−	0.880083i	$-0.342513\pi$
$503$	21.2982	0.949641	0.474820	+	0.880083i	$0.342513\pi$
$504$	−3.16228	−0.140859
$505$	0	0
$506$	−15.8377	−0.704073
$507$	11.6491	0.517355
$508$	−9.64911	−0.428110
$509$	−14.8114	−0.656503	−0.328252	−	0.944590i	$-0.606459\pi$
$509$	−14.8114	−0.656503	−0.328252	+	0.944590i	$0.606459\pi$
$510$	0	0
$511$	3.16228	0.139891
$512$	1.00000	0.0441942
$513$	1.00000	0.0441511
$514$	−6.67544	−0.294441
$515$	0	0
$516$	2.83772	0.124924
$517$	−12.9737	−0.570581
$518$	−31.6228	−1.38943
$519$	−4.16228	−0.182704
$520$	0	0
$521$	−17.9737	−0.787441	−0.393720	−	0.919230i	$-0.628812\pi$
$521$	−17.9737	−0.787441	−0.393720	+	0.919230i	$0.628812\pi$
$522$	−10.1623	−0.444791
$523$	8.64911	0.378199	0.189100	−	0.981958i	$-0.439443\pi$
$523$	8.64911	0.378199	0.189100	+	0.981958i	$0.439443\pi$
$524$	6.48683	0.283379
$525$	0	0
$526$	−5.97367	−0.260464
$527$	−52.4605	−2.28522
$528$	2.16228	0.0941011
$529$	30.6491	1.33257
$530$	0	0
$531$	11.4868	0.498486
$532$	3.16228	0.137102
$533$	−7.35089	−0.318402
$534$	−5.32456	−0.230416
$535$	0	0
$536$	12.4868	0.539349
$537$	−4.64911	−0.200624
$538$	−0.324555	−0.0139926
$539$	−6.48683	−0.279408
$540$	0	0
$541$	6.16228	0.264937	0.132469	−	0.991187i	$-0.457710\pi$
$541$	6.16228	0.264937	0.132469	+	0.991187i	$0.457710\pi$
$542$	5.16228	0.221739
$543$	−2.83772	−0.121778
$544$	7.16228	0.307080
$545$	0	0
$546$	−3.67544	−0.157295
$547$	14.8114	0.633289	0.316645	−	0.948544i	$-0.397444\pi$
$547$	14.8114	0.633289	0.316645	+	0.948544i	$0.397444\pi$
$548$	−10.3246	−0.441043
$549$	2.16228	0.0922838
$550$	0	0
$551$	10.1623	0.432928
$552$	−7.32456	−0.311754
$553$	−23.1623	−0.984960
$554$	−6.81139	−0.289388
$555$	0	0
$556$	2.51317	0.106582
$557$	−29.1623	−1.23565	−0.617823	−	0.786317i	$-0.711985\pi$
$557$	−29.1623	−1.23565	−0.617823	+	0.786317i	$0.711985\pi$
$558$	−7.32456	−0.310073
$559$	3.29822	0.139500
$560$	0	0
$561$	15.4868	0.653855
$562$	−17.6491	−0.744483
$563$	24.8377	1.04679	0.523393	−	0.852092i	$-0.324666\pi$
$563$	24.8377	1.04679	0.523393	+	0.852092i	$0.324666\pi$
$564$	−6.00000	−0.252646
$565$	0	0
$566$	−0.649111	−0.0272842
$567$	−3.16228	−0.132803
$568$	−5.16228	−0.216604
$569$	22.0000	0.922288	0.461144	−	0.887325i	$-0.347439\pi$
$569$	22.0000	0.922288	0.461144	+	0.887325i	$0.347439\pi$
$570$	0	0
$571$	15.8114	0.661686	0.330843	−	0.943686i	$-0.392667\pi$
$571$	15.8114	0.661686	0.330843	+	0.943686i	$0.392667\pi$
$572$	2.51317	0.105081
$573$	3.64911	0.152444
$574$	−20.0000	−0.834784
$575$	0	0
$576$	1.00000	0.0416667
$577$	−29.3246	−1.22080	−0.610399	−	0.792094i	$-0.708991\pi$
$577$	−29.3246	−1.22080	−0.610399	+	0.792094i	$0.708991\pi$
$578$	34.2982	1.42662
$579$	2.83772	0.117932
$580$	0	0
$581$	39.4868	1.63819
$582$	−19.4868	−0.807755
$583$	−17.6491	−0.730951
$584$	−1.00000	−0.0413803
$585$	0	0
$586$	−2.48683	−0.102730
$587$	12.8114	0.528783	0.264391	−	0.964415i	$-0.414829\pi$
$587$	12.8114	0.528783	0.264391	+	0.964415i	$0.414829\pi$
$588$	−3.00000	−0.123718
$589$	7.32456	0.301803
$590$	0	0
$591$	−20.1359	−0.828282
$592$	10.0000	0.410997
$593$	10.3246	0.423979	0.211989	−	0.977272i	$-0.432006\pi$
$593$	10.3246	0.423979	0.211989	+	0.977272i	$0.432006\pi$
$594$	2.16228	0.0887193
$595$	0	0
$596$	−8.64911	−0.354281
$597$	−6.64911	−0.272130
$598$	−8.51317	−0.348129
$599$	−35.8114	−1.46321	−0.731607	−	0.681727i	$-0.761229\pi$
$599$	−35.8114	−1.46321	−0.731607	+	0.681727i	$0.761229\pi$
$600$	0	0
$601$	14.1886	0.578766	0.289383	−	0.957213i	$-0.406550\pi$
$601$	14.1886	0.578766	0.289383	+	0.957213i	$0.406550\pi$
$602$	8.97367	0.365739
$603$	12.4868	0.508503
$604$	−10.0000	−0.406894
$605$	0	0
$606$	−17.8114	−0.723538
$607$	39.9737	1.62248	0.811241	−	0.584713i	$-0.198793\pi$
$607$	39.9737	1.62248	0.811241	+	0.584713i	$0.198793\pi$
$608$	−1.00000	−0.0405554
$609$	−32.1359	−1.30221
$610$	0	0
$611$	−6.97367	−0.282124
$612$	7.16228	0.289518
$613$	23.2982	0.941006	0.470503	−	0.882398i	$-0.344072\pi$
$613$	23.2982	0.941006	0.470503	+	0.882398i	$0.344072\pi$
$614$	32.4868	1.31106
$615$	0	0
$616$	6.83772	0.275500
$617$	−24.3246	−0.979270	−0.489635	−	0.871928i	$-0.662870\pi$
$617$	−24.3246	−0.979270	−0.489635	+	0.871928i	$0.662870\pi$
$618$	9.64911	0.388144
$619$	17.1623	0.689810	0.344905	−	0.938638i	$-0.387911\pi$
$619$	17.1623	0.689810	0.344905	+	0.938638i	$0.387911\pi$
$620$	0	0
$621$	−7.32456	−0.293924
$622$	−26.0000	−1.04251
$623$	−16.8377	−0.674589
$624$	1.16228	0.0465283
$625$	0	0
$626$	−23.3246	−0.932237
$627$	−2.16228	−0.0863531
$628$	5.67544	0.226475
$629$	71.6228	2.85579
$630$	0	0
$631$	−5.29822	−0.210919	−0.105459	−	0.994424i	$-0.533631\pi$
$631$	−5.29822	−0.210919	−0.105459	+	0.994424i	$0.533631\pi$
$632$	7.32456	0.291355
$633$	−12.8114	−0.509207
$634$	−2.16228	−0.0858750
$635$	0	0
$636$	−8.16228	−0.323655
$637$	−3.48683	−0.138153
$638$	21.9737	0.869946
$639$	−5.16228	−0.204217
$640$	0	0
$641$	13.2982	0.525248	0.262624	−	0.964898i	$-0.415412\pi$
$641$	13.2982	0.525248	0.262624	+	0.964898i	$0.415412\pi$
$642$	−19.1623	−0.756275
$643$	−18.6491	−0.735449	−0.367725	−	0.929935i	$-0.619863\pi$
$643$	−18.6491	−0.735449	−0.367725	+	0.929935i	$0.619863\pi$
$644$	−23.1623	−0.912722
$645$	0	0
$646$	−7.16228	−0.281796
$647$	−9.00000	−0.353827	−0.176913	−	0.984226i	$-0.556611\pi$
$647$	−9.00000	−0.353827	−0.176913	+	0.984226i	$0.556611\pi$
$648$	1.00000	0.0392837
$649$	−24.8377	−0.974966
$650$	0	0
$651$	−23.1623	−0.907801
$652$	2.64911	0.103747
$653$	−30.9737	−1.21209	−0.606047	−	0.795429i	$-0.707246\pi$
$653$	−30.9737	−1.21209	−0.606047	+	0.795429i	$0.707246\pi$
$654$	−20.3246	−0.794753
$655$	0	0
$656$	6.32456	0.246932
$657$	−1.00000	−0.0390137
$658$	−18.9737	−0.739671
$659$	−18.1886	−0.708528	−0.354264	−	0.935146i	$-0.615269\pi$
$659$	−18.1886	−0.708528	−0.354264	+	0.935146i	$0.615269\pi$
$660$	0	0
$661$	−13.2982	−0.517241	−0.258620	−	0.965979i	$-0.583268\pi$
$661$	−13.2982	−0.517241	−0.258620	+	0.965979i	$0.583268\pi$
$662$	−34.8114	−1.35298
$663$	8.32456	0.323299
$664$	−12.4868	−0.484583
$665$	0	0
$666$	10.0000	0.387492
$667$	−74.4342	−2.88210
$668$	−22.6491	−0.876320
$669$	0.350889	0.0135662
$670$	0	0
$671$	−4.67544	−0.180494
$672$	3.16228	0.121988
$673$	24.4605	0.942883	0.471441	−	0.881897i	$-0.343734\pi$
$673$	24.4605	0.942883	0.471441	+	0.881897i	$0.343734\pi$
$674$	34.9737	1.34714
$675$	0	0
$676$	−11.6491	−0.448043
$677$	−0.486833	−0.0187105	−0.00935526	−	0.999956i	$-0.502978\pi$
$677$	−0.486833	−0.0187105	−0.00935526	+	0.999956i	$0.502978\pi$
$678$	15.6491	0.601000
$679$	−61.6228	−2.36487
$680$	0	0
$681$	2.00000	0.0766402
$682$	15.8377	0.606458
$683$	45.1096	1.72607	0.863036	−	0.505143i	$-0.168560\pi$
$683$	45.1096	1.72607	0.863036	+	0.505143i	$0.168560\pi$
$684$	−1.00000	−0.0382360
$685$	0	0
$686$	12.6491	0.482945
$687$	−1.83772	−0.0701135
$688$	−2.83772	−0.108187
$689$	−9.48683	−0.361420
$690$	0	0
$691$	−5.48683	−0.208729	−0.104364	−	0.994539i	$-0.533281\pi$
$691$	−5.48683	−0.208729	−0.104364	+	0.994539i	$0.533281\pi$
$692$	4.16228	0.158226
$693$	6.83772	0.259744
$694$	−20.0000	−0.759190
$695$	0	0
$696$	10.1623	0.385200
$697$	45.2982	1.71579
$698$	−24.1623	−0.914556
$699$	−19.8114	−0.749336
$700$	0	0
$701$	6.97367	0.263392	0.131696	−	0.991290i	$-0.457958\pi$
$701$	6.97367	0.263392	0.131696	+	0.991290i	$0.457958\pi$
$702$	1.16228	0.0438673
$703$	−10.0000	−0.377157
$704$	−2.16228	−0.0814939
$705$	0	0
$706$	−7.81139	−0.293985
$707$	−56.3246	−2.11830
$708$	−11.4868	−0.431702
$709$	42.4868	1.59563	0.797813	−	0.602905i	$-0.205990\pi$
$709$	42.4868	1.59563	0.797813	+	0.602905i	$0.205990\pi$
$710$	0	0
$711$	7.32456	0.274692
$712$	5.32456	0.199546
$713$	−53.6491	−2.00917
$714$	22.6491	0.847622
$715$	0	0
$716$	4.64911	0.173745
$717$	11.6754	0.436027
$718$	−24.3246	−0.907784
$719$	6.67544	0.248952	0.124476	−	0.992223i	$-0.460275\pi$
$719$	6.67544	0.248952	0.124476	+	0.992223i	$0.460275\pi$
$720$	0	0
$721$	30.5132	1.13637
$722$	1.00000	0.0372161
$723$	−29.4868	−1.09663
$724$	2.83772	0.105463
$725$	0	0
$726$	6.32456	0.234726
$727$	13.6228	0.505241	0.252620	−	0.967565i	$-0.418708\pi$
$727$	13.6228	0.505241	0.252620	+	0.967565i	$0.418708\pi$
$728$	3.67544	0.136221
$729$	1.00000	0.0370370
$730$	0	0
$731$	−20.3246	−0.751731
$732$	−2.16228	−0.0799201
$733$	31.5132	1.16397	0.581983	−	0.813201i	$-0.302277\pi$
$733$	31.5132	1.16397	0.581983	+	0.813201i	$0.302277\pi$
$734$	11.4868	0.423987
$735$	0	0
$736$	7.32456	0.269987
$737$	−27.0000	−0.994558
$738$	6.32456	0.232810
$739$	−6.32456	−0.232653	−0.116326	−	0.993211i	$-0.537112\pi$
$739$	−6.32456	−0.232653	−0.116326	+	0.993211i	$0.537112\pi$
$740$	0	0
$741$	−1.16228	−0.0426973
$742$	−25.8114	−0.947566
$743$	−20.6491	−0.757542	−0.378771	−	0.925490i	$-0.623653\pi$
$743$	−20.6491	−0.757542	−0.378771	+	0.925490i	$0.623653\pi$
$744$	7.32456	0.268531
$745$	0	0
$746$	28.1359	1.03013
$747$	−12.4868	−0.456869
$748$	−15.4868	−0.566255
$749$	−60.5964	−2.21415
$750$	0	0
$751$	−26.0000	−0.948753	−0.474377	−	0.880322i	$-0.657327\pi$
$751$	−26.0000	−0.948753	−0.474377	+	0.880322i	$0.657327\pi$
$752$	6.00000	0.218797
$753$	−14.6491	−0.533843
$754$	11.8114	0.430145
$755$	0	0
$756$	3.16228	0.115011
$757$	−49.1359	−1.78588	−0.892938	−	0.450179i	$-0.851360\pi$
$757$	−49.1359	−1.78588	−0.892938	+	0.450179i	$0.851360\pi$
$758$	−14.9737	−0.543868
$759$	15.8377	0.574873
$760$	0	0
$761$	−20.5132	−0.743602	−0.371801	−	0.928313i	$-0.621260\pi$
$761$	−20.5132	−0.743602	−0.371801	+	0.928313i	$0.621260\pi$
$762$	9.64911	0.349550
$763$	−64.2719	−2.32680
$764$	−3.64911	−0.132020
$765$	0	0
$766$	15.4868	0.559562
$767$	−13.3509	−0.482073
$768$	−1.00000	−0.0360844
$769$	47.3246	1.70657	0.853284	−	0.521447i	$-0.174608\pi$
$769$	47.3246	1.70657	0.853284	+	0.521447i	$0.174608\pi$
$770$	0	0
$771$	6.67544	0.240410
$772$	−2.83772	−0.102132
$773$	−28.9737	−1.04211	−0.521055	−	0.853523i	$-0.674461\pi$
$773$	−28.9737	−1.04211	−0.521055	+	0.853523i	$0.674461\pi$
$774$	−2.83772	−0.102000
$775$	0	0
$776$	19.4868	0.699537
$777$	31.6228	1.13446
$778$	−9.35089	−0.335246
$779$	−6.32456	−0.226601
$780$	0	0
$781$	11.1623	0.399418
$782$	52.4605	1.87598
$783$	10.1623	0.363170
$784$	3.00000	0.107143
$785$	0	0
$786$	−6.48683	−0.231378
$787$	15.5132	0.552985	0.276492	−	0.961016i	$-0.410828\pi$
$787$	15.5132	0.552985	0.276492	+	0.961016i	$0.410828\pi$
$788$	20.1359	0.717313
$789$	5.97367	0.212668
$790$	0	0
$791$	49.4868	1.75955
$792$	−2.16228	−0.0768332
$793$	−2.51317	−0.0892452
$794$	−27.1359	−0.963019
$795$	0	0
$796$	6.64911	0.235671
$797$	−15.6228	−0.553387	−0.276694	−	0.960958i	$-0.589239\pi$
$797$	−15.6228	−0.553387	−0.276694	+	0.960958i	$0.589239\pi$
$798$	−3.16228	−0.111943
$799$	42.9737	1.52030
$800$	0	0
$801$	5.32456	0.188134
$802$	−11.6491	−0.411344
$803$	2.16228	0.0763051
$804$	−12.4868	−0.440377
$805$	0	0
$806$	8.51317	0.299864
$807$	0.324555	0.0114249
$808$	17.8114	0.626602
$809$	−16.8377	−0.591983	−0.295991	−	0.955191i	$-0.595650\pi$
$809$	−16.8377	−0.591983	−0.295991	+	0.955191i	$0.595650\pi$
$810$	0	0
$811$	42.8114	1.50331	0.751656	−	0.659556i	$-0.229256\pi$
$811$	42.8114	1.50331	0.751656	+	0.659556i	$0.229256\pi$
$812$	32.1359	1.12775
$813$	−5.16228	−0.181049
$814$	−21.6228	−0.757878
$815$	0	0
$816$	−7.16228	−0.250730
$817$	2.83772	0.0992793
$818$	−20.3246	−0.710631
$819$	3.67544	0.128430
$820$	0	0
$821$	43.1623	1.50637	0.753187	−	0.657807i	$-0.228516\pi$
$821$	43.1623	1.50637	0.753187	+	0.657807i	$0.228516\pi$
$822$	10.3246	0.360110
$823$	23.2982	0.812125	0.406062	−	0.913845i	$-0.366902\pi$
$823$	23.2982	0.812125	0.406062	+	0.913845i	$0.366902\pi$
$824$	−9.64911	−0.336143
$825$	0	0
$826$	−36.3246	−1.26389
$827$	−1.16228	−0.0404164	−0.0202082	−	0.999796i	$-0.506433\pi$
$827$	−1.16228	−0.0404164	−0.0202082	+	0.999796i	$0.506433\pi$
$828$	7.32456	0.254546
$829$	46.7851	1.62491	0.812456	−	0.583022i	$-0.198130\pi$
$829$	46.7851	1.62491	0.812456	+	0.583022i	$0.198130\pi$
$830$	0	0
$831$	6.81139	0.236284
$832$	−1.16228	−0.0402947
$833$	21.4868	0.744475
$834$	−2.51317	−0.0870239
$835$	0	0
$836$	2.16228	0.0747839
$837$	7.32456	0.253174
$838$	8.00000	0.276355
$839$	−19.1623	−0.661555	−0.330778	−	0.943709i	$-0.607311\pi$
$839$	−19.1623	−0.661555	−0.330778	+	0.943709i	$0.607311\pi$
$840$	0	0
$841$	74.2719	2.56110
$842$	−6.64911	−0.229143
$843$	17.6491	0.607868
$844$	12.8114	0.440986
$845$	0	0
$846$	6.00000	0.206284
$847$	20.0000	0.687208
$848$	8.16228	0.280294
$849$	0.649111	0.0222774
$850$	0	0
$851$	73.2456	2.51083
$852$	5.16228	0.176857
$853$	19.6754	0.673674	0.336837	−	0.941563i	$-0.390643\pi$
$853$	19.6754	0.673674	0.336837	+	0.941563i	$0.390643\pi$
$854$	−6.83772	−0.233982
$855$	0	0
$856$	19.1623	0.654953
$857$	38.3246	1.30914	0.654571	−	0.756001i	$-0.272849\pi$
$857$	38.3246	1.30914	0.654571	+	0.756001i	$0.272849\pi$
$858$	−2.51317	−0.0857981
$859$	−30.6491	−1.04573	−0.522867	−	0.852414i	$-0.675138\pi$
$859$	−30.6491	−1.04573	−0.522867	+	0.852414i	$0.675138\pi$
$860$	0	0
$861$	20.0000	0.681598
$862$	12.3246	0.419776
$863$	17.0263	0.579583	0.289792	−	0.957090i	$-0.406414\pi$
$863$	17.0263	0.579583	0.289792	+	0.957090i	$0.406414\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−16.1359	−0.548322
$867$	−34.2982	−1.16483
$868$	23.1623	0.786179
$869$	−15.8377	−0.537258
$870$	0	0
$871$	−14.5132	−0.491760
$872$	20.3246	0.688276
$873$	19.4868	0.659529
$874$	−7.32456	−0.247757
$875$	0	0
$876$	1.00000	0.0337869
$877$	−17.6228	−0.595079	−0.297539	−	0.954710i	$-0.596166\pi$
$877$	−17.6228	−0.595079	−0.297539	+	0.954710i	$0.596166\pi$
$878$	−11.6491	−0.393138
$879$	2.48683	0.0838788
$880$	0	0
$881$	1.67544	0.0564472	0.0282236	−	0.999602i	$-0.491015\pi$
$881$	1.67544	0.0564472	0.0282236	+	0.999602i	$0.491015\pi$
$882$	3.00000	0.101015
$883$	−10.5132	−0.353796	−0.176898	−	0.984229i	$-0.556606\pi$
$883$	−10.5132	−0.353796	−0.176898	+	0.984229i	$0.556606\pi$
$884$	−8.32456	−0.279985
$885$	0	0
$886$	−18.1623	−0.610174
$887$	34.3246	1.15251	0.576253	−	0.817271i	$-0.304514\pi$
$887$	34.3246	1.15251	0.576253	+	0.817271i	$0.304514\pi$
$888$	−10.0000	−0.335578
$889$	30.5132	1.02338
$890$	0	0
$891$	−2.16228	−0.0724390
$892$	−0.350889	−0.0117486
$893$	−6.00000	−0.200782
$894$	8.64911	0.289270
$895$	0	0
$896$	−3.16228	−0.105644
$897$	8.51317	0.284246
$898$	−11.0000	−0.367075
$899$	74.4342	2.48252
$900$	0	0
$901$	58.4605	1.94760
$902$	−13.6754	−0.455343
$903$	−8.97367	−0.298625
$904$	−15.6491	−0.520482
$905$	0	0
$906$	10.0000	0.332228
$907$	−13.3509	−0.443309	−0.221655	−	0.975125i	$-0.571146\pi$
$907$	−13.3509	−0.443309	−0.221655	+	0.975125i	$0.571146\pi$
$908$	−2.00000	−0.0663723
$909$	17.8114	0.590766
$910$	0	0
$911$	−44.9737	−1.49004	−0.745022	−	0.667040i	$-0.767561\pi$
$911$	−44.9737	−1.49004	−0.745022	+	0.667040i	$0.767561\pi$
$912$	1.00000	0.0331133
$913$	27.0000	0.893570
$914$	22.3246	0.738431
$915$	0	0
$916$	1.83772	0.0607201
$917$	−20.5132	−0.677404
$918$	−7.16228	−0.236390
$919$	−25.1623	−0.830027	−0.415013	−	0.909815i	$-0.636223\pi$
$919$	−25.1623	−0.830027	−0.415013	+	0.909815i	$0.636223\pi$
$920$	0	0
$921$	−32.4868	−1.07048
$922$	12.0000	0.395199
$923$	6.00000	0.197492
$924$	−6.83772	−0.224945
$925$	0	0
$926$	−34.3246	−1.12797
$927$	−9.64911	−0.316918
$928$	−10.1623	−0.333593
$929$	13.2982	0.436300	0.218150	−	0.975915i	$-0.429998\pi$
$929$	13.2982	0.436300	0.218150	+	0.975915i	$0.429998\pi$
$930$	0	0
$931$	−3.00000	−0.0983210
$932$	19.8114	0.648944
$933$	26.0000	0.851202
$934$	7.13594	0.233495
$935$	0	0
$936$	−1.16228	−0.0379902
$937$	−1.35089	−0.0441316	−0.0220658	−	0.999757i	$-0.507024\pi$
$937$	−1.35089	−0.0441316	−0.0220658	+	0.999757i	$0.507024\pi$
$938$	−39.4868	−1.28929
$939$	23.3246	0.761168
$940$	0	0
$941$	1.18861	0.0387476	0.0193738	−	0.999812i	$-0.493833\pi$
$941$	1.18861	0.0387476	0.0193738	+	0.999812i	$0.493833\pi$
$942$	−5.67544	−0.184916
$943$	46.3246	1.50854
$944$	11.4868	0.373865
$945$	0	0
$946$	6.13594	0.199497
$947$	−48.0000	−1.55979	−0.779895	−	0.625910i	$-0.784728\pi$
$947$	−48.0000	−1.55979	−0.779895	+	0.625910i	$0.784728\pi$
$948$	−7.32456	−0.237890
$949$	1.16228	0.0377291
$950$	0	0
$951$	2.16228	0.0701167
$952$	−22.6491	−0.734062
$953$	−53.3246	−1.72735	−0.863676	−	0.504048i	$-0.831844\pi$
$953$	−53.3246	−1.72735	−0.863676	+	0.504048i	$0.831844\pi$
$954$	8.16228	0.264263
$955$	0	0
$956$	−11.6754	−0.377611
$957$	−21.9737	−0.710308
$958$	−2.02633	−0.0654679
$959$	32.6491	1.05429
$960$	0	0
$961$	22.6491	0.730616
$962$	−11.6228	−0.374733
$963$	19.1623	0.617496
$964$	29.4868	0.949707
$965$	0	0
$966$	23.1623	0.745234
$967$	11.6754	0.375457	0.187728	−	0.982221i	$-0.439887\pi$
$967$	11.6754	0.375457	0.187728	+	0.982221i	$0.439887\pi$
$968$	−6.32456	−0.203279
$969$	7.16228	0.230086
$970$	0	0
$971$	31.4868	1.01046	0.505230	−	0.862985i	$-0.331408\pi$
$971$	31.4868	1.01046	0.505230	+	0.862985i	$0.331408\pi$
$972$	−1.00000	−0.0320750
$973$	−7.94733	−0.254780
$974$	−3.35089	−0.107369
$975$	0	0
$976$	2.16228	0.0692128
$977$	21.9473	0.702157	0.351079	−	0.936346i	$-0.385815\pi$
$977$	21.9473	0.702157	0.351079	+	0.936346i	$0.385815\pi$
$978$	−2.64911	−0.0847092
$979$	−11.5132	−0.367962
$980$	0	0
$981$	20.3246	0.648913
$982$	0	0
$983$	8.13594	0.259496	0.129748	−	0.991547i	$-0.458583\pi$
$983$	8.13594	0.259496	0.129748	+	0.991547i	$0.458583\pi$
$984$	−6.32456	−0.201619
$985$	0	0
$986$	−72.7851	−2.31795
$987$	18.9737	0.603938
$988$	1.16228	0.0369770
$989$	−20.7851	−0.660926
$990$	0	0
$991$	−26.2982	−0.835391	−0.417695	−	0.908587i	$-0.637162\pi$
$991$	−26.2982	−0.835391	−0.417695	+	0.908587i	$0.637162\pi$
$992$	−7.32456	−0.232555
$993$	34.8114	1.10471
$994$	16.3246	0.517783
$995$	0	0
$996$	12.4868	0.395660
$997$	32.4868	1.02887	0.514434	−	0.857530i	$-0.328002\pi$
$997$	32.4868	1.02887	0.514434	+	0.857530i	$0.328002\pi$
$998$	25.8114	0.817045
$999$	−10.0000	−0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2850.2.a.bg.1.1	yes	2
3.2	odd	2		8550.2.a.bq.1.1		2
5.2	odd	4		2850.2.d.v.799.3		4
5.3	odd	4		2850.2.d.v.799.2		4
5.4	even	2		2850.2.a.bf.1.2	✓	2
15.14	odd	2		8550.2.a.ca.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2850.2.a.bf.1.2	✓	2	5.4	even	2
2850.2.a.bg.1.1	yes	2	1.1	even	1	trivial
2850.2.d.v.799.2		4	5.3	odd	4
2850.2.d.v.799.3		4	5.2	odd	4
8550.2.a.bq.1.1		2	3.2	odd	2
8550.2.a.ca.1.2		2	15.14	odd	2

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:	\( N \)	\(=\)	\( 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2850.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -3.16228 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -3.16228 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.16228 q^{11} -1.00000 q^{12} -1.16228 q^{13} -3.16228 q^{14} +1.00000 q^{16} +7.16228 q^{17} +1.00000 q^{18} -1.00000 q^{19} +3.16228 q^{21} -2.16228 q^{22} +7.32456 q^{23} -1.00000 q^{24} -1.16228 q^{26} -1.00000 q^{27} -3.16228 q^{28} -10.1623 q^{29} -7.32456 q^{31} +1.00000 q^{32} +2.16228 q^{33} +7.16228 q^{34} +1.00000 q^{36} +10.0000 q^{37} -1.00000 q^{38} +1.16228 q^{39} +6.32456 q^{41} +3.16228 q^{42} -2.83772 q^{43} -2.16228 q^{44} +7.32456 q^{46} +6.00000 q^{47} -1.00000 q^{48} +3.00000 q^{49} -7.16228 q^{51} -1.16228 q^{52} +8.16228 q^{53} -1.00000 q^{54} -3.16228 q^{56} +1.00000 q^{57} -10.1623 q^{58} +11.4868 q^{59} +2.16228 q^{61} -7.32456 q^{62} -3.16228 q^{63} +1.00000 q^{64} +2.16228 q^{66} +12.4868 q^{67} +7.16228 q^{68} -7.32456 q^{69} -5.16228 q^{71} +1.00000 q^{72} -1.00000 q^{73} +10.0000 q^{74} -1.00000 q^{76} +6.83772 q^{77} +1.16228 q^{78} +7.32456 q^{79} +1.00000 q^{81} +6.32456 q^{82} -12.4868 q^{83} +3.16228 q^{84} -2.83772 q^{86} +10.1623 q^{87} -2.16228 q^{88} +5.32456 q^{89} +3.67544 q^{91} +7.32456 q^{92} +7.32456 q^{93} +6.00000 q^{94} -1.00000 q^{96} +19.4868 q^{97} +3.00000 q^{98} -2.16228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^6 + 2 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} + 2 q^{8} + 2 q^{9} + 2 q^{11} - 2 q^{12} + 4 q^{13} + 2 q^{16} + 8 q^{17} + 2 q^{18} - 2 q^{19} + 2 q^{22} + 2 q^{23} - 2 q^{24} + 4 q^{26} - 2 q^{27} - 14 q^{29} - 2 q^{31} + 2 q^{32} - 2 q^{33} + 8 q^{34} + 2 q^{36} + 20 q^{37} - 2 q^{38} - 4 q^{39} - 12 q^{43} + 2 q^{44} + 2 q^{46} + 12 q^{47} - 2 q^{48} + 6 q^{49} - 8 q^{51} + 4 q^{52} + 10 q^{53} - 2 q^{54} + 2 q^{57} - 14 q^{58} + 4 q^{59} - 2 q^{61} - 2 q^{62} + 2 q^{64} - 2 q^{66} + 6 q^{67} + 8 q^{68} - 2 q^{69} - 4 q^{71} + 2 q^{72} - 2 q^{73} + 20 q^{74} - 2 q^{76} + 20 q^{77} - 4 q^{78} + 2 q^{79} + 2 q^{81} - 6 q^{83} - 12 q^{86} + 14 q^{87} + 2 q^{88} - 2 q^{89} + 20 q^{91} + 2 q^{92} + 2 q^{93} + 12 q^{94} - 2 q^{96} + 20 q^{97} + 6 q^{98} + 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^6 + 2 * q^8 + 2 * q^9 + 2 * q^11 - 2 * q^12 + 4 * q^13 + 2 * q^16 + 8 * q^17 + 2 * q^18 - 2 * q^19 + 2 * q^22 + 2 * q^23 - 2 * q^24 + 4 * q^26 - 2 * q^27 - 14 * q^29 - 2 * q^31 + 2 * q^32 - 2 * q^33 + 8 * q^34 + 2 * q^36 + 20 * q^37 - 2 * q^38 - 4 * q^39 - 12 * q^43 + 2 * q^44 + 2 * q^46 + 12 * q^47 - 2 * q^48 + 6 * q^49 - 8 * q^51 + 4 * q^52 + 10 * q^53 - 2 * q^54 + 2 * q^57 - 14 * q^58 + 4 * q^59 - 2 * q^61 - 2 * q^62 + 2 * q^64 - 2 * q^66 + 6 * q^67 + 8 * q^68 - 2 * q^69 - 4 * q^71 + 2 * q^72 - 2 * q^73 + 20 * q^74 - 2 * q^76 + 20 * q^77 - 4 * q^78 + 2 * q^79 + 2 * q^81 - 6 * q^83 - 12 * q^86 + 14 * q^87 + 2 * q^88 - 2 * q^89 + 20 * q^91 + 2 * q^92 + 2 * q^93 + 12 * q^94 - 2 * q^96 + 20 * q^97 + 6 * q^98 + 2 * q^99

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