LMFDB - Embedded newform 350.4.c.e.99.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [350,4,Mod(99,350)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("350.99"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(350, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 350 = 2 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	350.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-8,0,-12,0,0,36,0,74] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$20.6506685020$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			99.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	350.99
Dual form			350.4.c.e.99.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.00000i q^{2} +3.00000i q^{3} -4.00000 q^{4} -6.00000 q^{6} +7.00000i q^{7} -8.00000i q^{8} +18.0000 q^{9} +37.0000 q^{11} -12.0000i q^{12} +18.0000i q^{13} -14.0000 q^{14} +16.0000 q^{16} +121.000i q^{17} +36.0000i q^{18} +45.0000 q^{19} -21.0000 q^{21} +74.0000i q^{22} -72.0000i q^{23} +24.0000 q^{24} -36.0000 q^{26} +135.000i q^{27} -28.0000i q^{28} -210.000 q^{29} -148.000 q^{31} +32.0000i q^{32} +111.000i q^{33} -242.000 q^{34} -72.0000 q^{36} +136.000i q^{37} +90.0000i q^{38} -54.0000 q^{39} +227.000 q^{41} -42.0000i q^{42} -32.0000i q^{43} -148.000 q^{44} +144.000 q^{46} +346.000i q^{47} +48.0000i q^{48} -49.0000 q^{49} -363.000 q^{51} -72.0000i q^{52} -452.000i q^{53} -270.000 q^{54} +56.0000 q^{56} +135.000i q^{57} -420.000i q^{58} +140.000 q^{59} -578.000 q^{61} -296.000i q^{62} +126.000i q^{63} -64.0000 q^{64} -222.000 q^{66} +801.000i q^{67} -484.000i q^{68} +216.000 q^{69} -478.000 q^{71} -144.000i q^{72} -247.000i q^{73} -272.000 q^{74} -180.000 q^{76} +259.000i q^{77} -108.000i q^{78} -610.000 q^{79} +81.0000 q^{81} +454.000i q^{82} +653.000i q^{83} +84.0000 q^{84} +64.0000 q^{86} -630.000i q^{87} -296.000i q^{88} +1115.00 q^{89} -126.000 q^{91} +288.000i q^{92} -444.000i q^{93} -692.000 q^{94} -96.0000 q^{96} -614.000i q^{97} -98.0000i q^{98} +666.000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{4} - 12 q^{6} + 36 q^{9} + 74 q^{11} - 28 q^{14} + 32 q^{16} + 90 q^{19} - 42 q^{21} + 48 q^{24} - 72 q^{26} - 420 q^{29} - 296 q^{31} - 484 q^{34} - 144 q^{36} - 108 q^{39} + 454 q^{41} - 296 q^{44}+ \cdots + 1332 q^{99}+O(q^{100}) $ 2 * q - 8 * q^4 - 12 * q^6 + 36 * q^9 + 74 * q^11 - 28 * q^14 + 32 * q^16 + 90 * q^19 - 42 * q^21 + 48 * q^24 - 72 * q^26 - 420 * q^29 - 296 * q^31 - 484 * q^34 - 144 * q^36 - 108 * q^39 + 454 * q^41 - 296 * q^44 + 288 * q^46 - 98 * q^49 - 726 * q^51 - 540 * q^54 + 112 * q^56 + 280 * q^59 - 1156 * q^61 - 128 * q^64 - 444 * q^66 + 432 * q^69 - 956 * q^71 - 544 * q^74 - 360 * q^76 - 1220 * q^79 + 162 * q^81 + 168 * q^84 + 128 * q^86 + 2230 * q^89 - 252 * q^91 - 1384 * q^94 - 192 * q^96 + 1332 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/350\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$-1$

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 $

$2$	2.00000i	0.707107i
$2$	2.00000i	0.707107i
$3$	3.00000i	0.577350i	0.957427	+	0.288675i	$0.0932147\pi$
$3$	3.00000i	0.577350i	−0.957427	+	0.288675i	$0.906785\pi$
$4$	−4.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−6.00000	−0.408248
$7$	7.00000i	0.377964i
$7$	7.00000i	0.377964i
$8$	− 8.00000i	− 0.353553i
$9$	18.0000	0.666667
$10$	0	0
$11$	37.0000	1.01417	0.507087	−	0.861895i	$-0.330722\pi$
$11$	37.0000	1.01417	0.507087	+	0.861895i	$0.330722\pi$
$12$	− 12.0000i	− 0.288675i
$13$	18.0000i	0.384023i	0.981393	+	0.192012i	$0.0615011\pi$
$13$	18.0000i	0.384023i	−0.981393	+	0.192012i	$0.938499\pi$
$14$	−14.0000	−0.267261
$15$	0	0
$16$	16.0000	0.250000
$17$	121.000i	1.72628i	0.504962	+	0.863141i	$0.331506\pi$
$17$	121.000i	1.72628i	−0.504962	+	0.863141i	$0.668494\pi$
$18$	36.0000i	0.471405i
$19$	45.0000	0.543353	0.271677	−	0.962389i	$-0.412422\pi$
$19$	45.0000	0.543353	0.271677	+	0.962389i	$0.412422\pi$
$20$	0	0
$21$	−21.0000	−0.218218
$22$	74.0000i	0.717130i
$23$	− 72.0000i	− 0.652741i	−0.945242	−	0.326370i	$-0.894174\pi$
$23$	− 72.0000i	− 0.652741i	0.945242	−	0.326370i	$-0.105826\pi$
$24$	24.0000	0.204124
$25$	0	0
$26$	−36.0000	−0.271545
$27$	135.000i	0.962250i
$28$	− 28.0000i	− 0.188982i
$29$	−210.000	−1.34469	−0.672345	−	0.740238i	$-0.734713\pi$
$29$	−210.000	−1.34469	−0.672345	+	0.740238i	$0.734713\pi$
$30$	0	0
$31$	−148.000	−0.857470	−0.428735	−	0.903430i	$-0.641041\pi$
$31$	−148.000	−0.857470	−0.428735	+	0.903430i	$0.641041\pi$
$32$	32.0000i	0.176777i
$33$	111.000i	0.585534i
$34$	−242.000	−1.22067
$35$	0	0
$36$	−72.0000	−0.333333
$37$	136.000i	0.604277i	0.953264	+	0.302139i	$0.0977006\pi$
$37$	136.000i	0.604277i	−0.953264	+	0.302139i	$0.902299\pi$
$38$	90.0000i	0.384209i
$39$	−54.0000	−0.221716
$40$	0	0
$41$	227.000	0.864669	0.432335	−	0.901713i	$-0.357690\pi$
$41$	227.000	0.864669	0.432335	+	0.901713i	$0.357690\pi$
$42$	− 42.0000i	− 0.154303i
$43$	− 32.0000i	− 0.113487i	−0.998389	−	0.0567437i	$-0.981928\pi$
$43$	− 32.0000i	− 0.113487i	0.998389	−	0.0567437i	$-0.0180718\pi$
$44$	−148.000	−0.507087
$45$	0	0
$46$	144.000	0.461557
$47$	346.000i	1.07381i	0.843641	+	0.536907i	$0.180408\pi$
$47$	346.000i	1.07381i	−0.843641	+	0.536907i	$0.819592\pi$
$48$	48.0000i	0.144338i
$49$	−49.0000	−0.142857
$50$	0	0
$51$	−363.000	−0.996670
$52$	− 72.0000i	− 0.192012i
$53$	− 452.000i	− 1.17145i	−0.810509	−	0.585726i	$-0.800809\pi$
$53$	− 452.000i	− 1.17145i	0.810509	−	0.585726i	$-0.199191\pi$
$54$	−270.000	−0.680414
$55$	0	0
$56$	56.0000	0.133631
$57$	135.000i	0.313705i
$58$	− 420.000i	− 0.950840i
$59$	140.000	0.308923	0.154461	−	0.987999i	$-0.450636\pi$
$59$	140.000	0.308923	0.154461	+	0.987999i	$0.450636\pi$
$60$	0	0
$61$	−578.000	−1.21320	−0.606601	−	0.795006i	$-0.707467\pi$
$61$	−578.000	−1.21320	−0.606601	+	0.795006i	$0.707467\pi$
$62$	− 296.000i	− 0.606323i
$63$	126.000i	0.251976i
$64$	−64.0000	−0.125000
$65$	0	0
$66$	−222.000	−0.414035
$67$	801.000i	1.46056i	0.683146	+	0.730282i	$0.260611\pi$
$67$	801.000i	1.46056i	−0.683146	+	0.730282i	$0.739389\pi$
$68$	− 484.000i	− 0.863141i
$69$	216.000	0.376860
$70$	0	0
$71$	−478.000	−0.798988	−0.399494	−	0.916736i	$-0.630814\pi$
$71$	−478.000	−0.798988	−0.399494	+	0.916736i	$0.630814\pi$
$72$	− 144.000i	− 0.235702i
$73$	− 247.000i	− 0.396016i	−0.980200	−	0.198008i	$-0.936553\pi$
$73$	− 247.000i	− 0.396016i	0.980200	−	0.198008i	$-0.0634472\pi$
$74$	−272.000	−0.427289
$75$	0	0
$76$	−180.000	−0.271677
$77$	259.000i	0.383322i
$78$	− 108.000i	− 0.156777i
$79$	−610.000	−0.868739	−0.434369	−	0.900735i	$-0.643029\pi$
$79$	−610.000	−0.868739	−0.434369	+	0.900735i	$0.643029\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	454.000i	0.611414i
$83$	653.000i	0.863567i	0.901977	+	0.431784i	$0.142116\pi$
$83$	653.000i	0.863567i	−0.901977	+	0.431784i	$0.857884\pi$
$84$	84.0000	0.109109
$85$	0	0
$86$	64.0000	0.0802476
$87$	− 630.000i	− 0.776357i
$88$	− 296.000i	− 0.358565i
$89$	1115.00	1.32797	0.663987	−	0.747744i	$-0.268863\pi$
$89$	1115.00	1.32797	0.663987	+	0.747744i	$0.268863\pi$
$90$	0	0
$91$	−126.000	−0.145147
$92$	288.000i	0.326370i
$93$	− 444.000i	− 0.495061i
$94$	−692.000	−0.759302
$95$	0	0
$96$	−96.0000	−0.102062
$97$	− 614.000i	− 0.642704i	−0.946960	−	0.321352i	$-0.895863\pi$
$97$	− 614.000i	− 0.642704i	0.946960	−	0.321352i	$-0.104137\pi$
$98$	− 98.0000i	− 0.101015i
$99$	666.000	0.676116
$100$	0	0
$101$	−768.000	−0.756622	−0.378311	−	0.925678i	$-0.623495\pi$
$101$	−768.000	−0.756622	−0.378311	+	0.925678i	$0.623495\pi$
$102$	− 726.000i	− 0.704752i
$103$	148.000i	0.141581i	0.997491	+	0.0707906i	$0.0225522\pi$
$103$	148.000i	0.141581i	−0.997491	+	0.0707906i	$0.977448\pi$
$104$	144.000	0.135773
$105$	0	0
$106$	904.000	0.828342
$107$	531.000i	0.479754i	0.970803	+	0.239877i	$0.0771072\pi$
$107$	531.000i	0.479754i	−0.970803	+	0.239877i	$0.922893\pi$
$108$	− 540.000i	− 0.481125i
$109$	1650.00	1.44992	0.724960	−	0.688791i	$-0.241858\pi$
$109$	1650.00	1.44992	0.724960	+	0.688791i	$0.241858\pi$
$110$	0	0
$111$	−408.000	−0.348880
$112$	112.000i	0.0944911i
$113$	2063.00i	1.71744i	0.512445	+	0.858720i	$0.328740\pi$
$113$	2063.00i	1.71744i	−0.512445	+	0.858720i	$0.671260\pi$
$114$	−270.000	−0.221823
$115$	0	0
$116$	840.000	0.672345
$117$	324.000i	0.256015i
$118$	280.000i	0.218441i
$119$	−847.000	−0.652474
$120$	0	0
$121$	38.0000	0.0285500
$122$	− 1156.00i	− 0.857863i
$123$	681.000i	0.499217i
$124$	592.000	0.428735
$125$	0	0
$126$	−252.000	−0.178174
$127$	1706.00i	1.19199i	0.802987	+	0.595996i	$0.203243\pi$
$127$	1706.00i	1.19199i	−0.802987	+	0.595996i	$0.796757\pi$
$128$	− 128.000i	− 0.0883883i
$129$	96.0000	0.0655219
$130$	0	0
$131$	1712.00	1.14182	0.570909	−	0.821013i	$-0.306591\pi$
$131$	1712.00	1.14182	0.570909	+	0.821013i	$0.306591\pi$
$132$	− 444.000i	− 0.292767i
$133$	315.000i	0.205368i
$134$	−1602.00	−1.03277
$135$	0	0
$136$	968.000	0.610333
$137$	− 2949.00i	− 1.83905i	−0.393030	−	0.919526i	$-0.628573\pi$
$137$	− 2949.00i	− 1.83905i	0.393030	−	0.919526i	$-0.371427\pi$
$138$	432.000i	0.266480i
$139$	1685.00	1.02820	0.514100	−	0.857730i	$-0.328126\pi$
$139$	1685.00	1.02820	0.514100	+	0.857730i	$0.328126\pi$
$140$	0	0
$141$	−1038.00	−0.619967
$142$	− 956.000i	− 0.564970i
$143$	666.000i	0.389467i
$144$	288.000	0.166667
$145$	0	0
$146$	494.000	0.280026
$147$	− 147.000i	− 0.0824786i
$148$	− 544.000i	− 0.302139i
$149$	−420.000	−0.230924	−0.115462	−	0.993312i	$-0.536835\pi$
$149$	−420.000	−0.230924	−0.115462	+	0.993312i	$0.536835\pi$
$150$	0	0
$151$	3032.00	1.63404	0.817022	−	0.576606i	$-0.195623\pi$
$151$	3032.00	1.63404	0.817022	+	0.576606i	$0.195623\pi$
$152$	− 360.000i	− 0.192104i
$153$	2178.00i	1.15086i
$154$	−518.000	−0.271050
$155$	0	0
$156$	216.000	0.110858
$157$	− 1084.00i	− 0.551036i	−0.961296	−	0.275518i	$-0.911151\pi$
$157$	− 1084.00i	− 0.551036i	0.961296	−	0.275518i	$-0.0888493\pi$
$158$	− 1220.00i	− 0.614291i
$159$	1356.00	0.676338
$160$	0	0
$161$	504.000	0.246713
$162$	162.000i	0.0785674i
$163$	− 517.000i	− 0.248433i	−0.992255	−	0.124216i	$-0.960358\pi$
$163$	− 517.000i	− 0.248433i	0.992255	−	0.124216i	$-0.0396417\pi$
$164$	−908.000	−0.432335
$165$	0	0
$166$	−1306.00	−0.610634
$167$	− 2844.00i	− 1.31782i	−0.752223	−	0.658908i	$-0.771019\pi$
$167$	− 2844.00i	− 1.31782i	0.752223	−	0.658908i	$-0.228981\pi$
$168$	168.000i	0.0771517i
$169$	1873.00	0.852526
$170$	0	0
$171$	810.000	0.362235
$172$	128.000i	0.0567437i
$173$	818.000i	0.359488i	0.983713	+	0.179744i	$0.0575269\pi$
$173$	818.000i	0.359488i	−0.983713	+	0.179744i	$0.942473\pi$
$174$	1260.00	0.548968
$175$	0	0
$176$	592.000	0.253544
$177$	420.000i	0.178357i
$178$	2230.00i	0.939020i
$179$	−3405.00	−1.42180	−0.710898	−	0.703295i	$-0.751711\pi$
$179$	−3405.00	−1.42180	−0.710898	+	0.703295i	$0.751711\pi$
$180$	0	0
$181$	3442.00	1.41349	0.706745	−	0.707468i	$-0.250163\pi$
$181$	3442.00	1.41349	0.706745	+	0.707468i	$0.250163\pi$
$182$	− 252.000i	− 0.102635i
$183$	− 1734.00i	− 0.700442i
$184$	−576.000	−0.230779
$185$	0	0
$186$	888.000	0.350061
$187$	4477.00i	1.75075i
$188$	− 1384.00i	− 0.536907i
$189$	−945.000	−0.363696
$190$	0	0
$191$	2862.00	1.08423	0.542113	−	0.840306i	$-0.317625\pi$
$191$	2862.00	1.08423	0.542113	+	0.840306i	$0.317625\pi$
$192$	− 192.000i	− 0.0721688i
$193$	− 2827.00i	− 1.05436i	−0.849753	−	0.527181i	$-0.823249\pi$
$193$	− 2827.00i	− 1.05436i	0.849753	−	0.527181i	$-0.176751\pi$
$194$	1228.00	0.454460
$195$	0	0
$196$	196.000	0.0714286
$197$	− 1634.00i	− 0.590953i	−0.955350	−	0.295476i	$-0.904522\pi$
$197$	− 1634.00i	− 0.590953i	0.955350	−	0.295476i	$-0.0954784\pi$
$198$	1332.00i	0.478086i
$199$	1270.00	0.452402	0.226201	−	0.974081i	$-0.427369\pi$
$199$	1270.00	0.452402	0.226201	+	0.974081i	$0.427369\pi$
$200$	0	0
$201$	−2403.00	−0.843256
$202$	− 1536.00i	− 0.535013i
$203$	− 1470.00i	− 0.508245i
$204$	1452.00	0.498335
$205$	0	0
$206$	−296.000	−0.100113
$207$	− 1296.00i	− 0.435161i
$208$	288.000i	0.0960058i
$209$	1665.00	0.551055
$210$	0	0
$211$	−2393.00	−0.780763	−0.390381	−	0.920653i	$-0.627657\pi$
$211$	−2393.00	−0.780763	−0.390381	+	0.920653i	$0.627657\pi$
$212$	1808.00i	0.585726i
$213$	− 1434.00i	− 0.461296i
$214$	−1062.00	−0.339238
$215$	0	0
$216$	1080.00	0.340207
$217$	− 1036.00i	− 0.324093i
$218$	3300.00i	1.02525i
$219$	741.000	0.228640
$220$	0	0
$221$	−2178.00	−0.662933
$222$	− 816.000i	− 0.246695i
$223$	5338.00i	1.60295i	0.598025	+	0.801477i	$0.295952\pi$
$223$	5338.00i	1.60295i	−0.598025	+	0.801477i	$0.704048\pi$
$224$	−224.000	−0.0668153
$225$	0	0
$226$	−4126.00	−1.21441
$227$	− 3284.00i	− 0.960206i	−0.877212	−	0.480103i	$-0.840599\pi$
$227$	− 3284.00i	− 0.960206i	0.877212	−	0.480103i	$-0.159401\pi$
$228$	− 540.000i	− 0.156853i
$229$	5150.00	1.48612	0.743060	−	0.669225i	$-0.233374\pi$
$229$	5150.00	1.48612	0.743060	+	0.669225i	$0.233374\pi$
$230$	0	0
$231$	−777.000	−0.221311
$232$	1680.00i	0.475420i
$233$	− 3642.00i	− 1.02401i	−0.858981	−	0.512007i	$-0.828902\pi$
$233$	− 3642.00i	− 1.02401i	0.858981	−	0.512007i	$-0.171098\pi$
$234$	−648.000	−0.181030
$235$	0	0
$236$	−560.000	−0.154461
$237$	− 1830.00i	− 0.501567i
$238$	− 1694.00i	− 0.461369i
$239$	−1540.00	−0.416796	−0.208398	−	0.978044i	$-0.566825\pi$
$239$	−1540.00	−0.416796	−0.208398	+	0.978044i	$0.566825\pi$
$240$	0	0
$241$	3187.00	0.851837	0.425918	−	0.904762i	$-0.359951\pi$
$241$	3187.00	0.851837	0.425918	+	0.904762i	$0.359951\pi$
$242$	76.0000i	0.0201879i
$243$	3888.00i	1.02640i
$244$	2312.00	0.606601
$245$	0	0
$246$	−1362.00	−0.353000
$247$	810.000i	0.208660i
$248$	1184.00i	0.303162i
$249$	−1959.00	−0.498581
$250$	0	0
$251$	−2883.00	−0.724993	−0.362497	−	0.931985i	$-0.618076\pi$
$251$	−2883.00	−0.724993	−0.362497	+	0.931985i	$0.618076\pi$
$252$	− 504.000i	− 0.125988i
$253$	− 2664.00i	− 0.661993i
$254$	−3412.00	−0.842866
$255$	0	0
$256$	256.000	0.0625000
$257$	5326.00i	1.29271i	0.763036	+	0.646356i	$0.223708\pi$
$257$	5326.00i	1.29271i	−0.763036	+	0.646356i	$0.776292\pi$
$258$	192.000i	0.0463310i
$259$	−952.000	−0.228395
$260$	0	0
$261$	−3780.00	−0.896460
$262$	3424.00i	0.807387i
$263$	− 382.000i	− 0.0895632i	−0.998997	−	0.0447816i	$-0.985741\pi$
$263$	− 382.000i	− 0.0895632i	0.998997	−	0.0447816i	$-0.0142592\pi$
$264$	888.000	0.207018
$265$	0	0
$266$	−630.000	−0.145217
$267$	3345.00i	0.766707i
$268$	− 3204.00i	− 0.730282i
$269$	−5250.00	−1.18996	−0.594978	−	0.803742i	$-0.702839\pi$
$269$	−5250.00	−1.18996	−0.594978	+	0.803742i	$0.702839\pi$
$270$	0	0
$271$	−3378.00	−0.757191	−0.378596	−	0.925562i	$-0.623593\pi$
$271$	−3378.00	−0.757191	−0.378596	+	0.925562i	$0.623593\pi$
$272$	1936.00i	0.431571i
$273$	− 378.000i	− 0.0838007i
$274$	5898.00	1.30041
$275$	0	0
$276$	−864.000	−0.188430
$277$	− 5994.00i	− 1.30016i	−0.759865	−	0.650080i	$-0.774735\pi$
$277$	− 5994.00i	− 1.30016i	0.759865	−	0.650080i	$-0.225265\pi$
$278$	3370.00i	0.727047i
$279$	−2664.00	−0.571647
$280$	0	0
$281$	−7858.00	−1.66822	−0.834109	−	0.551600i	$-0.814017\pi$
$281$	−7858.00	−1.66822	−0.834109	+	0.551600i	$0.814017\pi$
$282$	− 2076.00i	− 0.438383i
$283$	− 7387.00i	− 1.55163i	−0.630960	−	0.775815i	$-0.717339\pi$
$283$	− 7387.00i	− 1.55163i	0.630960	−	0.775815i	$-0.282661\pi$
$284$	1912.00	0.399494
$285$	0	0
$286$	−1332.00	−0.275394
$287$	1589.00i	0.326814i
$288$	576.000i	0.117851i
$289$	−9728.00	−1.98005
$290$	0	0
$291$	1842.00	0.371065
$292$	988.000i	0.198008i
$293$	− 5102.00i	− 1.01728i	−0.860980	−	0.508638i	$-0.830149\pi$
$293$	− 5102.00i	− 1.01728i	0.860980	−	0.508638i	$-0.169851\pi$
$294$	294.000	0.0583212
$295$	0	0
$296$	1088.00	0.213644
$297$	4995.00i	0.975890i
$298$	− 840.000i	− 0.163288i
$299$	1296.00	0.250668
$300$	0	0
$301$	224.000	0.0428942
$302$	6064.00i	1.15544i
$303$	− 2304.00i	− 0.436836i
$304$	720.000	0.135838
$305$	0	0
$306$	−4356.00	−0.813778
$307$	9341.00i	1.73654i	0.496088	+	0.868272i	$0.334769\pi$
$307$	9341.00i	1.73654i	−0.496088	+	0.868272i	$0.665231\pi$
$308$	− 1036.00i	− 0.191661i
$309$	−444.000	−0.0817420
$310$	0	0
$311$	8682.00	1.58299	0.791497	−	0.611173i	$-0.209302\pi$
$311$	8682.00	1.58299	0.791497	+	0.611173i	$0.209302\pi$
$312$	432.000i	0.0783884i
$313$	998.000i	0.180225i	0.995932	+	0.0901123i	$0.0287226\pi$
$313$	998.000i	0.180225i	−0.995932	+	0.0901123i	$0.971277\pi$
$314$	2168.00	0.389641
$315$	0	0
$316$	2440.00	0.434369
$317$	− 4084.00i	− 0.723597i	−0.932256	−	0.361799i	$-0.882163\pi$
$317$	− 4084.00i	− 0.723597i	0.932256	−	0.361799i	$-0.117837\pi$
$318$	2712.00i	0.478243i
$319$	−7770.00	−1.36375
$320$	0	0
$321$	−1593.00	−0.276986
$322$	1008.00i	0.174452i
$323$	5445.00i	0.937981i
$324$	−324.000	−0.0555556
$325$	0	0
$326$	1034.00	0.175669
$327$	4950.00i	0.837112i
$328$	− 1816.00i	− 0.305707i
$329$	−2422.00	−0.405864
$330$	0	0
$331$	11857.0	1.96894	0.984471	−	0.175548i	$-0.0561699\pi$
$331$	11857.0	1.96894	0.984471	+	0.175548i	$0.0561699\pi$
$332$	− 2612.00i	− 0.431784i
$333$	2448.00i	0.402852i
$334$	5688.00	0.931837
$335$	0	0
$336$	−336.000	−0.0545545
$337$	− 1219.00i	− 0.197042i	−0.995135	−	0.0985210i	$-0.968589\pi$
$337$	− 1219.00i	− 0.197042i	0.995135	−	0.0985210i	$-0.0314112\pi$
$338$	3746.00i	0.602827i
$339$	−6189.00	−0.991564
$340$	0	0
$341$	−5476.00	−0.869625
$342$	1620.00i	0.256139i
$343$	− 343.000i	− 0.0539949i
$344$	−256.000	−0.0401238
$345$	0	0
$346$	−1636.00	−0.254196
$347$	− 6589.00i	− 1.01935i	−0.860366	−	0.509677i	$-0.829765\pi$
$347$	− 6589.00i	− 1.01935i	0.860366	−	0.509677i	$-0.170235\pi$
$348$	2520.00i	0.388179i
$349$	5240.00	0.803698	0.401849	−	0.915706i	$-0.368368\pi$
$349$	5240.00	0.803698	0.401849	+	0.915706i	$0.368368\pi$
$350$	0	0
$351$	−2430.00	−0.369527
$352$	1184.00i	0.179282i
$353$	− 10482.0i	− 1.58045i	−0.612814	−	0.790227i	$-0.709962\pi$
$353$	− 10482.0i	− 1.58045i	0.612814	−	0.790227i	$-0.290038\pi$
$354$	−840.000	−0.126117
$355$	0	0
$356$	−4460.00	−0.663987
$357$	− 2541.00i	− 0.376706i
$358$	− 6810.00i	− 1.00536i
$359$	−11750.0	−1.72741	−0.863707	−	0.503995i	$-0.831863\pi$
$359$	−11750.0	−1.72741	−0.863707	+	0.503995i	$0.831863\pi$
$360$	0	0
$361$	−4834.00	−0.704767
$362$	6884.00i	0.999489i
$363$	114.000i	0.0164833i
$364$	504.000	0.0725736
$365$	0	0
$366$	3468.00	0.495288
$367$	56.0000i	0.00796506i	0.999992	+	0.00398253i	$0.00126768\pi$
$367$	56.0000i	0.00796506i	−0.999992	+	0.00398253i	$0.998732\pi$
$368$	− 1152.00i	− 0.163185i
$369$	4086.00	0.576446
$370$	0	0
$371$	3164.00	0.442767
$372$	1776.00i	0.247530i
$373$	− 6892.00i	− 0.956714i	−0.878166	−	0.478357i	$-0.841233\pi$
$373$	− 6892.00i	− 0.956714i	0.878166	−	0.478357i	$-0.158767\pi$
$374$	−8954.00	−1.23797
$375$	0	0
$376$	2768.00	0.379651
$377$	− 3780.00i	− 0.516392i
$378$	− 1890.00i	− 0.257172i
$379$	6465.00	0.876213	0.438107	−	0.898923i	$-0.355649\pi$
$379$	6465.00	0.876213	0.438107	+	0.898923i	$0.355649\pi$
$380$	0	0
$381$	−5118.00	−0.688197
$382$	5724.00i	0.766663i
$383$	11958.0i	1.59537i	0.603077	+	0.797683i	$0.293941\pi$
$383$	11958.0i	1.59537i	−0.603077	+	0.797683i	$0.706059\pi$
$384$	384.000	0.0510310
$385$	0	0
$386$	5654.00	0.745547
$387$	− 576.000i	− 0.0756582i
$388$	2456.00i	0.321352i
$389$	−9920.00	−1.29297	−0.646483	−	0.762928i	$-0.723761\pi$
$389$	−9920.00	−1.29297	−0.646483	+	0.762928i	$0.723761\pi$
$390$	0	0
$391$	8712.00	1.12682
$392$	392.000i	0.0505076i
$393$	5136.00i	0.659229i
$394$	3268.00	0.417867
$395$	0	0
$396$	−2664.00	−0.338058
$397$	4886.00i	0.617686i	0.951113	+	0.308843i	$0.0999418\pi$
$397$	4886.00i	0.617686i	−0.951113	+	0.308843i	$0.900058\pi$
$398$	2540.00i	0.319896i
$399$	−945.000	−0.118569
$400$	0	0
$401$	−1763.00	−0.219551	−0.109776	−	0.993956i	$-0.535013\pi$
$401$	−1763.00	−0.219551	−0.109776	+	0.993956i	$0.535013\pi$
$402$	− 4806.00i	− 0.596272i
$403$	− 2664.00i	− 0.329289i
$404$	3072.00	0.378311
$405$	0	0
$406$	2940.00	0.359384
$407$	5032.00i	0.612843i
$408$	2904.00i	0.352376i
$409$	9645.00	1.16605	0.583025	−	0.812454i	$-0.301869\pi$
$409$	9645.00	1.16605	0.583025	+	0.812454i	$0.301869\pi$
$410$	0	0
$411$	8847.00	1.06178
$412$	− 592.000i	− 0.0707906i
$413$	980.000i	0.116762i
$414$	2592.00	0.307705
$415$	0	0
$416$	−576.000	−0.0678864
$417$	5055.00i	0.593632i
$418$	3330.00i	0.389655i
$419$	−4215.00	−0.491447	−0.245723	−	0.969340i	$-0.579025\pi$
$419$	−4215.00	−0.491447	−0.245723	+	0.969340i	$0.579025\pi$
$420$	0	0
$421$	5012.00	0.580214	0.290107	−	0.956994i	$-0.406309\pi$
$421$	5012.00	0.580214	0.290107	+	0.956994i	$0.406309\pi$
$422$	− 4786.00i	− 0.552083i
$423$	6228.00i	0.715876i
$424$	−3616.00	−0.414171
$425$	0	0
$426$	2868.00	0.326186
$427$	− 4046.00i	− 0.458547i
$428$	− 2124.00i	− 0.239877i
$429$	−1998.00	−0.224859
$430$	0	0
$431$	−7548.00	−0.843560	−0.421780	−	0.906698i	$-0.638595\pi$
$431$	−7548.00	−0.843560	−0.421780	+	0.906698i	$0.638595\pi$
$432$	2160.00i	0.240563i
$433$	− 7177.00i	− 0.796546i	−0.917267	−	0.398273i	$-0.869610\pi$
$433$	− 7177.00i	− 0.796546i	0.917267	−	0.398273i	$-0.130390\pi$
$434$	2072.00	0.229169
$435$	0	0
$436$	−6600.00	−0.724960
$437$	− 3240.00i	− 0.354669i
$438$	1482.00i	0.161673i
$439$	16580.0	1.80255	0.901276	−	0.433246i	$-0.142632\pi$
$439$	16580.0	1.80255	0.901276	+	0.433246i	$0.142632\pi$
$440$	0	0
$441$	−882.000	−0.0952381
$442$	− 4356.00i	− 0.468764i
$443$	− 557.000i	− 0.0597379i	−0.999554	−	0.0298689i	$-0.990491\pi$
$443$	− 557.000i	− 0.0597379i	0.999554	−	0.0298689i	$-0.00950899\pi$
$444$	1632.00	0.174440
$445$	0	0
$446$	−10676.0	−1.13346
$447$	− 1260.00i	− 0.133324i
$448$	− 448.000i	− 0.0472456i
$449$	10935.0	1.14934	0.574671	−	0.818385i	$-0.305130\pi$
$449$	10935.0	1.14934	0.574671	+	0.818385i	$0.305130\pi$
$450$	0	0
$451$	8399.00	0.876926
$452$	− 8252.00i	− 0.858720i
$453$	9096.00i	0.943416i
$454$	6568.00	0.678968
$455$	0	0
$456$	1080.00	0.110911
$457$	− 7489.00i	− 0.766566i	−0.923631	−	0.383283i	$-0.874793\pi$
$457$	− 7489.00i	− 0.766566i	0.923631	−	0.383283i	$-0.125207\pi$
$458$	10300.0i	1.05085i
$459$	−16335.0	−1.66112
$460$	0	0
$461$	−13718.0	−1.38592	−0.692962	−	0.720974i	$-0.743695\pi$
$461$	−13718.0	−1.38592	−0.692962	+	0.720974i	$0.743695\pi$
$462$	− 1554.00i	− 0.156491i
$463$	12008.0i	1.20531i	0.798001	+	0.602656i	$0.205891\pi$
$463$	12008.0i	1.20531i	−0.798001	+	0.602656i	$0.794109\pi$
$464$	−3360.00	−0.336173
$465$	0	0
$466$	7284.00	0.724088
$467$	− 5324.00i	− 0.527549i	−0.964584	−	0.263774i	$-0.915033\pi$
$467$	− 5324.00i	− 0.527549i	0.964584	−	0.263774i	$-0.0849675\pi$
$468$	− 1296.00i	− 0.128008i
$469$	−5607.00	−0.552041
$470$	0	0
$471$	3252.00	0.318141
$472$	− 1120.00i	− 0.109221i
$473$	− 1184.00i	− 0.115096i
$474$	3660.00	0.354661
$475$	0	0
$476$	3388.00	0.326237
$477$	− 8136.00i	− 0.780968i
$478$	− 3080.00i	− 0.294719i
$479$	16290.0	1.55388	0.776941	−	0.629574i	$-0.216771\pi$
$479$	16290.0	1.55388	0.776941	+	0.629574i	$0.216771\pi$
$480$	0	0
$481$	−2448.00	−0.232057
$482$	6374.00i	0.602340i
$483$	1512.00i	0.142440i
$484$	−152.000	−0.0142750
$485$	0	0
$486$	−7776.00	−0.725775
$487$	− 8954.00i	− 0.833151i	−0.909101	−	0.416575i	$-0.863230\pi$
$487$	− 8954.00i	− 0.833151i	0.909101	−	0.416575i	$-0.136770\pi$
$488$	4624.00i	0.428932i
$489$	1551.00	0.143433
$490$	0	0
$491$	5892.00	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	5892.00	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	− 2724.00i	− 0.249609i
$493$	− 25410.0i	− 2.32132i
$494$	−1620.00	−0.147545
$495$	0	0
$496$	−2368.00	−0.214368
$497$	− 3346.00i	− 0.301989i
$498$	− 3918.00i	− 0.352550i
$499$	16620.0	1.49101	0.745504	−	0.666501i	$-0.232209\pi$
$499$	16620.0	1.49101	0.745504	+	0.666501i	$0.232209\pi$
$500$	0	0
$501$	8532.00	0.760842
$502$	− 5766.00i	− 0.512648i
$503$	8238.00i	0.730247i	0.930959	+	0.365124i	$0.118973\pi$
$503$	8238.00i	0.730247i	−0.930959	+	0.365124i	$0.881027\pi$
$504$	1008.00	0.0890871
$505$	0	0
$506$	5328.00	0.468100
$507$	5619.00i	0.492206i
$508$	− 6824.00i	− 0.595996i
$509$	20870.0	1.81738	0.908690	−	0.417471i	$-0.137084\pi$
$509$	20870.0	1.81738	0.908690	+	0.417471i	$0.137084\pi$
$510$	0	0
$511$	1729.00	0.149680
$512$	512.000i	0.0441942i
$513$	6075.00i	0.522842i
$514$	−10652.0	−0.914085
$515$	0	0
$516$	−384.000	−0.0327610
$517$	12802.0i	1.08904i
$518$	− 1904.00i	− 0.161500i
$519$	−2454.00	−0.207550
$520$	0	0
$521$	−1213.00	−0.102001	−0.0510005	−	0.998699i	$-0.516241\pi$
$521$	−1213.00	−0.102001	−0.0510005	+	0.998699i	$0.516241\pi$
$522$	− 7560.00i	− 0.633893i
$523$	− 16637.0i	− 1.39099i	−0.718533	−	0.695493i	$-0.755186\pi$
$523$	− 16637.0i	− 1.39099i	0.718533	−	0.695493i	$-0.244814\pi$
$524$	−6848.00	−0.570909
$525$	0	0
$526$	764.000	0.0633308
$527$	− 17908.0i	− 1.48024i
$528$	1776.00i	0.146383i
$529$	6983.00	0.573929
$530$	0	0
$531$	2520.00	0.205949
$532$	− 1260.00i	− 0.102684i
$533$	4086.00i	0.332053i
$534$	−6690.00	−0.542143
$535$	0	0
$536$	6408.00	0.516387
$537$	− 10215.0i	− 0.820875i
$538$	− 10500.0i	− 0.841426i
$539$	−1813.00	−0.144882
$540$	0	0
$541$	7302.00	0.580291	0.290146	−	0.956983i	$-0.406296\pi$
$541$	7302.00	0.580291	0.290146	+	0.956983i	$0.406296\pi$
$542$	− 6756.00i	− 0.535415i
$543$	10326.0i	0.816079i
$544$	−3872.00	−0.305167
$545$	0	0
$546$	756.000	0.0592561
$547$	− 5589.00i	− 0.436871i	−0.975851	−	0.218435i	$-0.929905\pi$
$547$	− 5589.00i	− 0.436871i	0.975851	−	0.218435i	$-0.0700953\pi$
$548$	11796.0i	0.919526i
$549$	−10404.0	−0.808801
$550$	0	0
$551$	−9450.00	−0.730642
$552$	− 1728.00i	− 0.133240i
$553$	− 4270.00i	− 0.328352i
$554$	11988.0	0.919353
$555$	0	0
$556$	−6740.00	−0.514100
$557$	2276.00i	0.173137i	0.996246	+	0.0865684i	$0.0275901\pi$
$557$	2276.00i	0.173137i	−0.996246	+	0.0865684i	$0.972410\pi$
$558$	− 5328.00i	− 0.404215i
$559$	576.000	0.0435818
$560$	0	0
$561$	−13431.0	−1.01080
$562$	− 15716.0i	− 1.17961i
$563$	− 8332.00i	− 0.623716i	−0.950129	−	0.311858i	$-0.899049\pi$
$563$	− 8332.00i	− 0.623716i	0.950129	−	0.311858i	$-0.100951\pi$
$564$	4152.00	0.309984
$565$	0	0
$566$	14774.0	1.09717
$567$	567.000i	0.0419961i
$568$	3824.00i	0.282485i
$569$	21555.0	1.58811	0.794053	−	0.607848i	$-0.207967\pi$
$569$	21555.0	1.58811	0.794053	+	0.607848i	$0.207967\pi$
$570$	0	0
$571$	10332.0	0.757234	0.378617	−	0.925553i	$-0.376400\pi$
$571$	10332.0	0.757234	0.378617	+	0.925553i	$0.376400\pi$
$572$	− 2664.00i	− 0.194733i
$573$	8586.00i	0.625978i
$574$	−3178.00	−0.231093
$575$	0	0
$576$	−1152.00	−0.0833333
$577$	− 7549.00i	− 0.544660i	−0.962204	−	0.272330i	$-0.912206\pi$
$577$	− 7549.00i	− 0.544660i	0.962204	−	0.272330i	$-0.0877943\pi$
$578$	− 19456.0i	− 1.40011i
$579$	8481.00	0.608736
$580$	0	0
$581$	−4571.00	−0.326398
$582$	3684.00i	0.262383i
$583$	− 16724.0i	− 1.18806i
$584$	−1976.00	−0.140013
$585$	0	0
$586$	10204.0	0.719323
$587$	15451.0i	1.08642i	0.839595	+	0.543212i	$0.182792\pi$
$587$	15451.0i	1.08642i	−0.839595	+	0.543212i	$0.817208\pi$
$588$	588.000i	0.0412393i
$589$	−6660.00	−0.465909
$590$	0	0
$591$	4902.00	0.341187
$592$	2176.00i	0.151069i
$593$	20073.0i	1.39005i	0.718986	+	0.695025i	$0.244607\pi$
$593$	20073.0i	1.39005i	−0.718986	+	0.695025i	$0.755393\pi$
$594$	−9990.00	−0.690058
$595$	0	0
$596$	1680.00	0.115462
$597$	3810.00i	0.261194i
$598$	2592.00i	0.177249i
$599$	−1800.00	−0.122781	−0.0613907	−	0.998114i	$-0.519554\pi$
$599$	−1800.00	−0.122781	−0.0613907	+	0.998114i	$0.519554\pi$
$600$	0	0
$601$	−17403.0	−1.18117	−0.590585	−	0.806975i	$-0.701103\pi$
$601$	−17403.0	−1.18117	−0.590585	+	0.806975i	$0.701103\pi$
$602$	448.000i	0.0303308i
$603$	14418.0i	0.973709i
$604$	−12128.0	−0.817022
$605$	0	0
$606$	4608.00	0.308890
$607$	21596.0i	1.44408i	0.691853	+	0.722038i	$0.256795\pi$
$607$	21596.0i	1.44408i	−0.691853	+	0.722038i	$0.743205\pi$
$608$	1440.00i	0.0960522i
$609$	4410.00	0.293435
$610$	0	0
$611$	−6228.00	−0.412370
$612$	− 8712.00i	− 0.575428i
$613$	10938.0i	0.720688i	0.932820	+	0.360344i	$0.117341\pi$
$613$	10938.0i	0.720688i	−0.932820	+	0.360344i	$0.882659\pi$
$614$	−18682.0	−1.22792
$615$	0	0
$616$	2072.00	0.135525
$617$	− 11294.0i	− 0.736919i	−0.929644	−	0.368460i	$-0.879885\pi$
$617$	− 11294.0i	− 0.736919i	0.929644	−	0.368460i	$-0.120115\pi$
$618$	− 888.000i	− 0.0578003i
$619$	−10700.0	−0.694781	−0.347390	−	0.937721i	$-0.612932\pi$
$619$	−10700.0	−0.694781	−0.347390	+	0.937721i	$0.612932\pi$
$620$	0	0
$621$	9720.00	0.628100
$622$	17364.0i	1.11935i
$623$	7805.00i	0.501927i
$624$	−864.000	−0.0554290
$625$	0	0
$626$	−1996.00	−0.127438
$627$	4995.00i	0.318152i
$628$	4336.00i	0.275518i
$629$	−16456.0	−1.04315
$630$	0	0
$631$	17432.0	1.09977	0.549887	−	0.835239i	$-0.314671\pi$
$631$	17432.0	1.09977	0.549887	+	0.835239i	$0.314671\pi$
$632$	4880.00i	0.307146i
$633$	− 7179.00i	− 0.450774i
$634$	8168.00	0.511660
$635$	0	0
$636$	−5424.00	−0.338169
$637$	− 882.000i	− 0.0548605i
$638$	− 15540.0i	− 0.964317i
$639$	−8604.00	−0.532659
$640$	0	0
$641$	−8638.00	−0.532263	−0.266131	−	0.963937i	$-0.585746\pi$
$641$	−8638.00	−0.532263	−0.266131	+	0.963937i	$0.585746\pi$
$642$	− 3186.00i	− 0.195859i
$643$	25288.0i	1.55095i	0.631378	+	0.775475i	$0.282490\pi$
$643$	25288.0i	1.55095i	−0.631378	+	0.775475i	$0.717510\pi$
$644$	−2016.00	−0.123356
$645$	0	0
$646$	−10890.0	−0.663253
$647$	17576.0i	1.06798i	0.845490	+	0.533991i	$0.179308\pi$
$647$	17576.0i	1.06798i	−0.845490	+	0.533991i	$0.820692\pi$
$648$	− 648.000i	− 0.0392837i
$649$	5180.00	0.313302
$650$	0	0
$651$	3108.00	0.187115
$652$	2068.00i	0.124216i
$653$	− 11692.0i	− 0.700679i	−0.936623	−	0.350339i	$-0.886066\pi$
$653$	− 11692.0i	− 0.700679i	0.936623	−	0.350339i	$-0.113934\pi$
$654$	−9900.00	−0.591928
$655$	0	0
$656$	3632.00	0.216167
$657$	− 4446.00i	− 0.264011i
$658$	− 4844.00i	− 0.286989i
$659$	11465.0	0.677713	0.338857	−	0.940838i	$-0.389960\pi$
$659$	11465.0	0.677713	0.338857	+	0.940838i	$0.389960\pi$
$660$	0	0
$661$	2182.00	0.128396	0.0641982	−	0.997937i	$-0.479551\pi$
$661$	2182.00	0.128396	0.0641982	+	0.997937i	$0.479551\pi$
$662$	23714.0i	1.39225i
$663$	− 6534.00i	− 0.382744i
$664$	5224.00	0.305317
$665$	0	0
$666$	−4896.00	−0.284859
$667$	15120.0i	0.877734i
$668$	11376.0i	0.658908i
$669$	−16014.0	−0.925466
$670$	0	0
$671$	−21386.0	−1.23040
$672$	− 672.000i	− 0.0385758i
$673$	13858.0i	0.793739i	0.917875	+	0.396870i	$0.129904\pi$
$673$	13858.0i	0.793739i	−0.917875	+	0.396870i	$0.870096\pi$
$674$	2438.00	0.139330
$675$	0	0
$676$	−7492.00	−0.426263
$677$	− 7824.00i	− 0.444167i	−0.975028	−	0.222083i	$-0.928714\pi$
$677$	− 7824.00i	− 0.444167i	0.975028	−	0.222083i	$-0.0712857\pi$
$678$	− 12378.0i	− 0.701142i
$679$	4298.00	0.242919
$680$	0	0
$681$	9852.00	0.554375
$682$	− 10952.0i	− 0.614918i
$683$	− 13267.0i	− 0.743262i	−0.928381	−	0.371631i	$-0.878799\pi$
$683$	− 13267.0i	− 0.743262i	0.928381	−	0.371631i	$-0.121201\pi$
$684$	−3240.00	−0.181118
$685$	0	0
$686$	686.000	0.0381802
$687$	15450.0i	0.858012i
$688$	− 512.000i	− 0.0283718i
$689$	8136.00	0.449865
$690$	0	0
$691$	−1553.00	−0.0854977	−0.0427488	−	0.999086i	$-0.513612\pi$
$691$	−1553.00	−0.0854977	−0.0427488	+	0.999086i	$0.513612\pi$
$692$	− 3272.00i	− 0.179744i
$693$	4662.00i	0.255548i
$694$	13178.0	0.720793
$695$	0	0
$696$	−5040.00	−0.274484
$697$	27467.0i	1.49266i
$698$	10480.0i	0.568301i
$699$	10926.0	0.591215
$700$	0	0
$701$	−9368.00	−0.504742	−0.252371	−	0.967630i	$-0.581210\pi$
$701$	−9368.00	−0.504742	−0.252371	+	0.967630i	$0.581210\pi$
$702$	− 4860.00i	− 0.261295i
$703$	6120.00i	0.328336i
$704$	−2368.00	−0.126772
$705$	0	0
$706$	20964.0	1.11755
$707$	− 5376.00i	− 0.285976i
$708$	− 1680.00i	− 0.0891783i
$709$	3740.00	0.198108	0.0990541	−	0.995082i	$-0.468418\pi$
$709$	3740.00	0.198108	0.0990541	+	0.995082i	$0.468418\pi$
$710$	0	0
$711$	−10980.0	−0.579159
$712$	− 8920.00i	− 0.469510i
$713$	10656.0i	0.559706i
$714$	5082.00	0.266371
$715$	0	0
$716$	13620.0	0.710898
$717$	− 4620.00i	− 0.240637i
$718$	− 23500.0i	− 1.22147i
$719$	−20640.0	−1.07057	−0.535287	−	0.844671i	$-0.679796\pi$
$719$	−20640.0	−1.07057	−0.535287	+	0.844671i	$0.679796\pi$
$720$	0	0
$721$	−1036.00	−0.0535127
$722$	− 9668.00i	− 0.498346i
$723$	9561.00i	0.491808i
$724$	−13768.0	−0.706745
$725$	0	0
$726$	−228.000	−0.0116555
$727$	10686.0i	0.545147i	0.962135	+	0.272573i	$0.0878748\pi$
$727$	10686.0i	0.545147i	−0.962135	+	0.272573i	$0.912125\pi$
$728$	1008.00i	0.0513173i
$729$	−9477.00	−0.481481
$730$	0	0
$731$	3872.00	0.195911
$732$	6936.00i	0.350221i
$733$	− 35552.0i	− 1.79146i	−0.444594	−	0.895732i	$-0.646652\pi$
$733$	− 35552.0i	− 1.79146i	0.444594	−	0.895732i	$-0.353348\pi$
$734$	−112.000	−0.00563215
$735$	0	0
$736$	2304.00	0.115389
$737$	29637.0i	1.48127i
$738$	8172.00i	0.407609i
$739$	5660.00	0.281741	0.140870	−	0.990028i	$-0.455010\pi$
$739$	5660.00	0.281741	0.140870	+	0.990028i	$0.455010\pi$
$740$	0	0
$741$	−2430.00	−0.120470
$742$	6328.00i	0.313084i
$743$	18528.0i	0.914840i	0.889251	+	0.457420i	$0.151226\pi$
$743$	18528.0i	0.914840i	−0.889251	+	0.457420i	$0.848774\pi$
$744$	−3552.00	−0.175030
$745$	0	0
$746$	13784.0	0.676499
$747$	11754.0i	0.575711i
$748$	− 17908.0i	− 0.875376i
$749$	−3717.00	−0.181330
$750$	0	0
$751$	11222.0	0.545268	0.272634	−	0.962118i	$-0.412105\pi$
$751$	11222.0	0.545268	0.272634	+	0.962118i	$0.412105\pi$
$752$	5536.00i	0.268454i
$753$	− 8649.00i	− 0.418575i
$754$	7560.00	0.365145
$755$	0	0
$756$	3780.00	0.181848
$757$	22466.0i	1.07865i	0.842097	+	0.539327i	$0.181321\pi$
$757$	22466.0i	1.07865i	−0.842097	+	0.539327i	$0.818679\pi$
$758$	12930.0i	0.619576i
$759$	7992.00	0.382202
$760$	0	0
$761$	−20973.0	−0.999042	−0.499521	−	0.866302i	$-0.666491\pi$
$761$	−20973.0	−0.999042	−0.499521	+	0.866302i	$0.666491\pi$
$762$	− 10236.0i	− 0.486629i
$763$	11550.0i	0.548018i
$764$	−11448.0	−0.542113
$765$	0	0
$766$	−23916.0	−1.12809
$767$	2520.00i	0.118634i
$768$	768.000i	0.0360844i
$769$	−20155.0	−0.945134	−0.472567	−	0.881295i	$-0.656672\pi$
$769$	−20155.0	−0.945134	−0.472567	+	0.881295i	$0.656672\pi$
$770$	0	0
$771$	−15978.0	−0.746347
$772$	11308.0i	0.527181i
$773$	5388.00i	0.250702i	0.992112	+	0.125351i	$0.0400058\pi$
$773$	5388.00i	0.250702i	−0.992112	+	0.125351i	$0.959994\pi$
$774$	1152.00	0.0534984
$775$	0	0
$776$	−4912.00	−0.227230
$777$	− 2856.00i	− 0.131864i
$778$	− 19840.0i	− 0.914265i
$779$	10215.0	0.469821
$780$	0	0
$781$	−17686.0	−0.810313
$782$	17424.0i	0.796779i
$783$	− 28350.0i	− 1.29393i
$784$	−784.000	−0.0357143
$785$	0	0
$786$	−10272.0	−0.466145
$787$	32396.0i	1.46734i	0.679509	+	0.733668i	$0.262193\pi$
$787$	32396.0i	1.46734i	−0.679509	+	0.733668i	$0.737807\pi$
$788$	6536.00i	0.295476i
$789$	1146.00	0.0517094
$790$	0	0
$791$	−14441.0	−0.649131
$792$	− 5328.00i	− 0.239043i
$793$	− 10404.0i	− 0.465898i
$794$	−9772.00	−0.436770
$795$	0	0
$796$	−5080.00	−0.226201
$797$	− 23394.0i	− 1.03972i	−0.854251	−	0.519861i	$-0.825984\pi$
$797$	− 23394.0i	− 1.03972i	0.854251	−	0.519861i	$-0.174016\pi$
$798$	− 1890.00i	− 0.0838412i
$799$	−41866.0	−1.85371
$800$	0	0
$801$	20070.0	0.885317
$802$	− 3526.00i	− 0.155246i
$803$	− 9139.00i	− 0.401629i
$804$	9612.00	0.421628
$805$	0	0
$806$	5328.00	0.232842
$807$	− 15750.0i	− 0.687021i
$808$	6144.00i	0.267506i
$809$	30570.0	1.32853	0.664267	−	0.747495i	$-0.268744\pi$
$809$	30570.0	1.32853	0.664267	+	0.747495i	$0.268744\pi$
$810$	0	0
$811$	−21868.0	−0.946843	−0.473421	−	0.880836i	$-0.656981\pi$
$811$	−21868.0	−0.946843	−0.473421	+	0.880836i	$0.656981\pi$
$812$	5880.00i	0.254123i
$813$	− 10134.0i	− 0.437165i
$814$	−10064.0	−0.433345
$815$	0	0
$816$	−5808.00	−0.249167
$817$	− 1440.00i	− 0.0616637i
$818$	19290.0i	0.824522i
$819$	−2268.00	−0.0967648
$820$	0	0
$821$	−14448.0	−0.614176	−0.307088	−	0.951681i	$-0.599355\pi$
$821$	−14448.0	−0.614176	−0.307088	+	0.951681i	$0.599355\pi$
$822$	17694.0i	0.750790i
$823$	15958.0i	0.675894i	0.941165	+	0.337947i	$0.109732\pi$
$823$	15958.0i	0.675894i	−0.941165	+	0.337947i	$0.890268\pi$
$824$	1184.00	0.0500565
$825$	0	0
$826$	−1960.00	−0.0825631
$827$	− 17999.0i	− 0.756816i	−0.925639	−	0.378408i	$-0.876472\pi$
$827$	− 17999.0i	− 0.756816i	0.925639	−	0.378408i	$-0.123528\pi$
$828$	5184.00i	0.217580i
$829$	32140.0	1.34652	0.673262	−	0.739404i	$-0.264893\pi$
$829$	32140.0	1.34652	0.673262	+	0.739404i	$0.264893\pi$
$830$	0	0
$831$	17982.0	0.750648
$832$	− 1152.00i	− 0.0480029i
$833$	− 5929.00i	− 0.246612i
$834$	−10110.0	−0.419761
$835$	0	0
$836$	−6660.00	−0.275527
$837$	− 19980.0i	− 0.825101i
$838$	− 8430.00i	− 0.347505i
$839$	−23870.0	−0.982222	−0.491111	−	0.871097i	$-0.663409\pi$
$839$	−23870.0	−0.982222	−0.491111	+	0.871097i	$0.663409\pi$
$840$	0	0
$841$	19711.0	0.808192
$842$	10024.0i	0.410273i
$843$	− 23574.0i	− 0.963146i
$844$	9572.00	0.390381
$845$	0	0
$846$	−12456.0	−0.506201
$847$	266.000i	0.0107909i
$848$	− 7232.00i	− 0.292863i
$849$	22161.0	0.895835
$850$	0	0
$851$	9792.00	0.394436
$852$	5736.00i	0.230648i
$853$	− 24442.0i	− 0.981100i	−0.871413	−	0.490550i	$-0.836796\pi$
$853$	− 24442.0i	− 0.981100i	0.871413	−	0.490550i	$-0.163204\pi$
$854$	8092.00	0.324242
$855$	0	0
$856$	4248.00	0.169619
$857$	15381.0i	0.613075i	0.951859	+	0.306537i	$0.0991704\pi$
$857$	15381.0i	0.613075i	−0.951859	+	0.306537i	$0.900830\pi$
$858$	− 3996.00i	− 0.158999i
$859$	−46995.0	−1.86665	−0.933323	−	0.359038i	$-0.883105\pi$
$859$	−46995.0	−1.86665	−0.933323	+	0.359038i	$0.883105\pi$
$860$	0	0
$861$	−4767.00	−0.188686
$862$	− 15096.0i	− 0.596487i
$863$	− 16652.0i	− 0.656826i	−0.944534	−	0.328413i	$-0.893486\pi$
$863$	− 16652.0i	− 0.656826i	0.944534	−	0.328413i	$-0.106514\pi$
$864$	−4320.00	−0.170103
$865$	0	0
$866$	14354.0	0.563243
$867$	− 29184.0i	− 1.14318i
$868$	4144.00i	0.162047i
$869$	−22570.0	−0.881053
$870$	0	0
$871$	−14418.0	−0.560890
$872$	− 13200.0i	− 0.512624i
$873$	− 11052.0i	− 0.428469i
$874$	6480.00	0.250789
$875$	0	0
$876$	−2964.00	−0.114320
$877$	− 25424.0i	− 0.978914i	−0.872027	−	0.489457i	$-0.837195\pi$
$877$	− 25424.0i	− 0.978914i	0.872027	−	0.489457i	$-0.162805\pi$
$878$	33160.0i	1.27460i
$879$	15306.0	0.587325
$880$	0	0
$881$	21242.0	0.812328	0.406164	−	0.913800i	$-0.366866\pi$
$881$	21242.0	0.812328	0.406164	+	0.913800i	$0.366866\pi$
$882$	− 1764.00i	− 0.0673435i
$883$	− 25887.0i	− 0.986599i	−0.869860	−	0.493299i	$-0.835791\pi$
$883$	− 25887.0i	− 0.986599i	0.869860	−	0.493299i	$-0.164209\pi$
$884$	8712.00	0.331466
$885$	0	0
$886$	1114.00	0.0422410
$887$	− 24434.0i	− 0.924931i	−0.886637	−	0.462465i	$-0.846965\pi$
$887$	− 24434.0i	− 0.924931i	0.886637	−	0.462465i	$-0.153035\pi$
$888$	3264.00i	0.123348i
$889$	−11942.0	−0.450531
$890$	0	0
$891$	2997.00	0.112686
$892$	− 21352.0i	− 0.801477i
$893$	15570.0i	0.583460i
$894$	2520.00	0.0942745
$895$	0	0
$896$	896.000	0.0334077
$897$	3888.00i	0.144723i
$898$	21870.0i	0.812708i
$899$	31080.0	1.15303
$900$	0	0
$901$	54692.0	2.02226
$902$	16798.0i	0.620080i
$903$	672.000i	0.0247650i
$904$	16504.0	0.607207
$905$	0	0
$906$	−18192.0	−0.667096
$907$	− 45164.0i	− 1.65341i	−0.562633	−	0.826707i	$-0.690212\pi$
$907$	− 45164.0i	− 1.65341i	0.562633	−	0.826707i	$-0.309788\pi$
$908$	13136.0i	0.480103i
$909$	−13824.0	−0.504415
$910$	0	0
$911$	−43668.0	−1.58813	−0.794064	−	0.607834i	$-0.792039\pi$
$911$	−43668.0	−1.58813	−0.794064	+	0.607834i	$0.792039\pi$
$912$	2160.00i	0.0784263i
$913$	24161.0i	0.875808i
$914$	14978.0	0.542044
$915$	0	0
$916$	−20600.0	−0.743060
$917$	11984.0i	0.431567i
$918$	− 32670.0i	− 1.17459i
$919$	−27770.0	−0.996788	−0.498394	−	0.866951i	$-0.666077\pi$
$919$	−27770.0	−0.996788	−0.498394	+	0.866951i	$0.666077\pi$
$920$	0	0
$921$	−28023.0	−1.00259
$922$	− 27436.0i	− 0.979996i
$923$	− 8604.00i	− 0.306830i
$924$	3108.00	0.110656
$925$	0	0
$926$	−24016.0	−0.852284
$927$	2664.00i	0.0943875i
$928$	− 6720.00i	− 0.237710i
$929$	2930.00	0.103477	0.0517385	−	0.998661i	$-0.483524\pi$
$929$	2930.00	0.103477	0.0517385	+	0.998661i	$0.483524\pi$
$930$	0	0
$931$	−2205.00	−0.0776219
$932$	14568.0i	0.512007i
$933$	26046.0i	0.913942i
$934$	10648.0	0.373033
$935$	0	0
$936$	2592.00	0.0905151
$937$	− 17849.0i	− 0.622307i	−0.950360	−	0.311153i	$-0.899285\pi$
$937$	− 17849.0i	− 0.622307i	0.950360	−	0.311153i	$-0.100715\pi$
$938$	− 11214.0i	− 0.390352i
$939$	−2994.00	−0.104053
$940$	0	0
$941$	47552.0	1.64734	0.823672	−	0.567066i	$-0.191922\pi$
$941$	47552.0	1.64734	0.823672	+	0.567066i	$0.191922\pi$
$942$	6504.00i	0.224959i
$943$	− 16344.0i	− 0.564405i
$944$	2240.00	0.0772307
$945$	0	0
$946$	2368.00	0.0813851
$947$	31876.0i	1.09380i	0.837197	+	0.546901i	$0.184193\pi$
$947$	31876.0i	1.09380i	−0.837197	+	0.546901i	$0.815807\pi$
$948$	7320.00i	0.250783i
$949$	4446.00	0.152079
$950$	0	0
$951$	12252.0	0.417769
$952$	6776.00i	0.230684i
$953$	26673.0i	0.906635i	0.891349	+	0.453318i	$0.149760\pi$
$953$	26673.0i	0.906635i	−0.891349	+	0.453318i	$0.850240\pi$
$954$	16272.0	0.552228
$955$	0	0
$956$	6160.00	0.208398
$957$	− 23310.0i	− 0.787362i
$958$	32580.0i	1.09876i
$959$	20643.0	0.695096
$960$	0	0
$961$	−7887.00	−0.264744
$962$	− 4896.00i	− 0.164089i
$963$	9558.00i	0.319836i
$964$	−12748.0	−0.425918
$965$	0	0
$966$	−3024.00	−0.100720
$967$	− 31714.0i	− 1.05466i	−0.849661	−	0.527329i	$-0.823194\pi$
$967$	− 31714.0i	− 1.05466i	0.849661	−	0.527329i	$-0.176806\pi$
$968$	− 304.000i	− 0.0100939i
$969$	−16335.0	−0.541544
$970$	0	0
$971$	−26013.0	−0.859729	−0.429865	−	0.902893i	$-0.641439\pi$
$971$	−26013.0	−0.859729	−0.429865	+	0.902893i	$0.641439\pi$
$972$	− 15552.0i	− 0.513200i
$973$	11795.0i	0.388623i
$974$	17908.0	0.589127
$975$	0	0
$976$	−9248.00	−0.303300
$977$	29571.0i	0.968332i	0.874976	+	0.484166i	$0.160877\pi$
$977$	29571.0i	0.968332i	−0.874976	+	0.484166i	$0.839123\pi$
$978$	3102.00i	0.101422i
$979$	41255.0	1.34680
$980$	0	0
$981$	29700.0	0.966614
$982$	11784.0i	0.382935i
$983$	− 21052.0i	− 0.683067i	−0.939870	−	0.341533i	$-0.889054\pi$
$983$	− 21052.0i	− 0.683067i	0.939870	−	0.341533i	$-0.110946\pi$
$984$	5448.00	0.176500
$985$	0	0
$986$	50820.0	1.64142
$987$	− 7266.00i	− 0.234326i
$988$	− 3240.00i	− 0.104330i
$989$	−2304.00	−0.0740778
$990$	0	0
$991$	45772.0	1.46720	0.733600	−	0.679581i	$-0.237839\pi$
$991$	45772.0	1.46720	0.733600	+	0.679581i	$0.237839\pi$
$992$	− 4736.00i	− 0.151581i
$993$	35571.0i	1.13677i
$994$	6692.00	0.213539
$995$	0	0
$996$	7836.00	0.249290
$997$	− 5894.00i	− 0.187227i	−0.995609	−	0.0936133i	$-0.970158\pi$
$997$	− 5894.00i	− 0.187227i	0.995609	−	0.0936133i	$-0.0298417\pi$
$998$	33240.0i	1.05430i
$999$	−18360.0	−0.581466

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.4.c.e.99.2		2
5.2	odd	4		350.4.a.g.1.1	✓	1
5.3	odd	4		350.4.a.p.1.1	yes	1
5.4	even	2	inner	350.4.c.e.99.1		2
35.13	even	4		2450.4.a.bj.1.1		1
35.27	even	4		2450.4.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.4.a.g.1.1	✓	1	5.2	odd	4
350.4.a.p.1.1	yes	1	5.3	odd	4
350.4.c.e.99.1		2	5.4	even	2	inner
350.4.c.e.99.2		2	1.1	even	1	trivial
2450.4.a.g.1.1		1	35.27	even	4
2450.4.a.bj.1.1		1	35.13	even	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	350.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} +3.00000i q^{3} -4.00000 q^{4} -6.00000 q^{6} +7.00000i q^{7} -8.00000i q^{8} +18.0000 q^{9} +37.0000 q^{11} -12.0000i q^{12} +18.0000i q^{13} -14.0000 q^{14} +16.0000 q^{16} +121.000i q^{17} +36.0000i q^{18} +45.0000 q^{19} -21.0000 q^{21} +74.0000i q^{22} -72.0000i q^{23} +24.0000 q^{24} -36.0000 q^{26} +135.000i q^{27} -28.0000i q^{28} -210.000 q^{29} -148.000 q^{31} +32.0000i q^{32} +111.000i q^{33} -242.000 q^{34} -72.0000 q^{36} +136.000i q^{37} +90.0000i q^{38} -54.0000 q^{39} +227.000 q^{41} -42.0000i q^{42} -32.0000i q^{43} -148.000 q^{44} +144.000 q^{46} +346.000i q^{47} +48.0000i q^{48} -49.0000 q^{49} -363.000 q^{51} -72.0000i q^{52} -452.000i q^{53} -270.000 q^{54} +56.0000 q^{56} +135.000i q^{57} -420.000i q^{58} +140.000 q^{59} -578.000 q^{61} -296.000i q^{62} +126.000i q^{63} -64.0000 q^{64} -222.000 q^{66} +801.000i q^{67} -484.000i q^{68} +216.000 q^{69} -478.000 q^{71} -144.000i q^{72} -247.000i q^{73} -272.000 q^{74} -180.000 q^{76} +259.000i q^{77} -108.000i q^{78} -610.000 q^{79} +81.0000 q^{81} +454.000i q^{82} +653.000i q^{83} +84.0000 q^{84} +64.0000 q^{86} -630.000i q^{87} -296.000i q^{88} +1115.00 q^{89} -126.000 q^{91} +288.000i q^{92} -444.000i q^{93} -692.000 q^{94} -96.0000 q^{96} -614.000i q^{97} -98.0000i q^{98} +666.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{4} - 12 q^{6} + 36 q^{9} + 74 q^{11} - 28 q^{14} + 32 q^{16} + 90 q^{19} - 42 q^{21} + 48 q^{24} - 72 q^{26} - 420 q^{29} - 296 q^{31} - 484 q^{34} - 144 q^{36} - 108 q^{39} + 454 q^{41} - 296 q^{44}+ \cdots + 1332 q^{99}+O(q^{100}) \) 2 * q - 8 * q^4 - 12 * q^6 + 36 * q^9 + 74 * q^11 - 28 * q^14 + 32 * q^16 + 90 * q^19 - 42 * q^21 + 48 * q^24 - 72 * q^26 - 420 * q^29 - 296 * q^31 - 484 * q^34 - 144 * q^36 - 108 * q^39 + 454 * q^41 - 296 * q^44 + 288 * q^46 - 98 * q^49 - 726 * q^51 - 540 * q^54 + 112 * q^56 + 280 * q^59 - 1156 * q^61 - 128 * q^64 - 444 * q^66 + 432 * q^69 - 956 * q^71 - 544 * q^74 - 360 * q^76 - 1220 * q^79 + 162 * q^81 + 168 * q^84 + 128 * q^86 + 2230 * q^89 - 252 * q^91 - 1384 * q^94 - 192 * q^96 + 1332 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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