LMFDB - Embedded newform 2352.3.f.k.97.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2352,3,Mod(97,2352)] 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("2352.97"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2352, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2352.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,-24,0,28,0,0,0,12,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)

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

Self dual:	no
Analytic conductor:	$64.0873581775$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.126303473664.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 9x^{6} + 2x^{5} + 92x^{4} + 14x^{3} - 441x^{2} - 686x + 2401 $ x^8 - 2x^7 - 9x^6 + 2x^5 + 92x^4 + 14x^3 - 441x^2 - 686*x + 2401
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			97.1
Root			$-2.38781 - 1.13946i$ of defining polynomial
Character	$\chi$	$=$	2352.97
Dual form			2352.3.f.k.97.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} -2.67601i q^{5} -3.00000 q^{9} +11.7009 q^{11} -20.4983i q^{13} -4.63498 q^{15} +16.4314i q^{17} -31.1143i q^{19} +5.65685 q^{23} +17.8390 q^{25} +5.19615i q^{27} +21.6493 q^{29} -5.31354i q^{31} -20.2666i q^{33} +19.2846 q^{37} -35.5040 q^{39} +28.0340i q^{41} -73.0145 q^{43} +8.02802i q^{45} +32.3284i q^{47} +28.4599 q^{51} +94.7811 q^{53} -31.3118i q^{55} -53.8916 q^{57} -112.950i q^{59} -34.6410i q^{61} -54.8535 q^{65} +7.68731 q^{67} -9.79796i q^{69} -22.9843 q^{71} +6.74200i q^{73} -30.8980i q^{75} +47.1902 q^{79} +9.00000 q^{81} -39.2763i q^{83} +43.9704 q^{85} -37.4977i q^{87} +101.074i q^{89} -9.20332 q^{93} -83.2622 q^{95} -30.0955i q^{97} -35.1028 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 24 q^{9} + 28 q^{11} + 12 q^{15} + 12 q^{25} - 4 q^{29} - 100 q^{37} - 12 q^{39} + 20 q^{43} - 24 q^{51} + 100 q^{53} - 156 q^{57} + 296 q^{65} + 68 q^{67} - 424 q^{71} - 80 q^{79} + 72 q^{81} - 232 q^{85}+ \cdots - 84 q^{99}+O(q^{100}) $ 8 * q - 24 * q^9 + 28 * q^11 + 12 * q^15 + 12 * q^25 - 4 * q^29 - 100 * q^37 - 12 * q^39 + 20 * q^43 - 24 * q^51 + 100 * q^53 - 156 * q^57 + 296 * q^65 + 68 * q^67 - 424 * q^71 - 80 * q^79 + 72 * q^81 - 232 * q^85 - 48 * q^93 + 288 * q^95 - 84 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times$.

$n$	$785$	$1471$	$1765$	$2257$
$\chi(n)$	$1$	$1$	$1$	$-1$

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 $

$2$	0	0
$2$	0	0
$3$	− 1.73205i	− 0.577350i
$3$	− 1.73205i	− 0.577350i
$4$	0	0
$5$	− 2.67601i	− 0.535202i	−0.963530	−	0.267601i	$-0.913769\pi$
$5$	− 2.67601i	− 0.535202i	0.963530	−	0.267601i	$-0.0862308\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	11.7009	1.06372	0.531861	−	0.846832i	$-0.321493\pi$
$11$	11.7009	1.06372	0.531861	+	0.846832i	$0.321493\pi$
$12$	0	0
$13$	− 20.4983i	− 1.57679i	−0.615170	−	0.788395i	$-0.710913\pi$
$13$	− 20.4983i	− 1.57679i	0.615170	−	0.788395i	$-0.289087\pi$
$14$	0	0
$15$	−4.63498	−0.308999
$16$	0	0
$17$	16.4314i	0.966550i	0.875469	+	0.483275i	$0.160553\pi$
$17$	16.4314i	0.966550i	−0.875469	+	0.483275i	$0.839447\pi$
$18$	0	0
$19$	− 31.1143i	− 1.63760i	−0.574082	−	0.818798i	$-0.694641\pi$
$19$	− 31.1143i	− 1.63760i	0.574082	−	0.818798i	$-0.305359\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.65685	0.245950	0.122975	−	0.992410i	$-0.460756\pi$
$23$	5.65685	0.245950	0.122975	+	0.992410i	$0.460756\pi$
$24$	0	0
$25$	17.8390	0.713559
$26$	0	0
$27$	5.19615i	0.192450i
$28$	0	0
$29$	21.6493	0.746527	0.373264	−	0.927725i	$-0.378239\pi$
$29$	21.6493	0.746527	0.373264	+	0.927725i	$0.378239\pi$
$30$	0	0
$31$	− 5.31354i	− 0.171404i	−0.996321	−	0.0857022i	$-0.972687\pi$
$31$	− 5.31354i	− 0.171404i	0.996321	−	0.0857022i	$-0.0273134\pi$
$32$	0	0
$33$	− 20.2666i	− 0.614140i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	19.2846	0.521206	0.260603	−	0.965446i	$-0.416079\pi$
$37$	19.2846	0.521206	0.260603	+	0.965446i	$0.416079\pi$
$38$	0	0
$39$	−35.5040	−0.910360
$40$	0	0
$41$	28.0340i	0.683756i	0.939744	+	0.341878i	$0.111063\pi$
$41$	28.0340i	0.683756i	−0.939744	+	0.341878i	$0.888937\pi$
$42$	0	0
$43$	−73.0145	−1.69801	−0.849006	−	0.528383i	$-0.822798\pi$
$43$	−73.0145	−1.69801	−0.849006	+	0.528383i	$0.822798\pi$
$44$	0	0
$45$	8.02802i	0.178401i
$46$	0	0
$47$	32.3284i	0.687838i	0.938999	+	0.343919i	$0.111755\pi$
$47$	32.3284i	0.687838i	−0.938999	+	0.343919i	$0.888245\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	28.4599	0.558038
$52$	0	0
$53$	94.7811	1.78832	0.894161	−	0.447745i	$-0.147773\pi$
$53$	94.7811	1.78832	0.894161	+	0.447745i	$0.147773\pi$
$54$	0	0
$55$	− 31.3118i	− 0.569306i
$56$	0	0
$57$	−53.8916	−0.945467
$58$	0	0
$59$	− 112.950i	− 1.91440i	−0.289430	−	0.957199i	$-0.593466\pi$
$59$	− 112.950i	− 1.91440i	0.289430	−	0.957199i	$-0.406534\pi$
$60$	0	0
$61$	− 34.6410i	− 0.567886i	−0.958841	−	0.283943i	$-0.908357\pi$
$61$	− 34.6410i	− 0.567886i	0.958841	−	0.283943i	$-0.0916426\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−54.8535	−0.843900
$66$	0	0
$67$	7.68731	0.114736	0.0573680	−	0.998353i	$-0.481729\pi$
$67$	7.68731	0.114736	0.0573680	+	0.998353i	$0.481729\pi$
$68$	0	0
$69$	− 9.79796i	− 0.141999i
$70$	0	0
$71$	−22.9843	−0.323723	−0.161861	−	0.986813i	$-0.551750\pi$
$71$	−22.9843	−0.323723	−0.161861	+	0.986813i	$0.551750\pi$
$72$	0	0
$73$	6.74200i	0.0923562i	0.998933	+	0.0461781i	$0.0147042\pi$
$73$	6.74200i	0.0923562i	−0.998933	+	0.0461781i	$0.985296\pi$
$74$	0	0
$75$	− 30.8980i	− 0.411974i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	47.1902	0.597345	0.298672	−	0.954356i	$-0.403456\pi$
$79$	47.1902	0.597345	0.298672	+	0.954356i	$0.403456\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	− 39.2763i	− 0.473208i	−0.971606	−	0.236604i	$-0.923966\pi$
$83$	− 39.2763i	− 0.473208i	0.971606	−	0.236604i	$-0.0760345\pi$
$84$	0	0
$85$	43.9704	0.517299
$86$	0	0
$87$	− 37.4977i	− 0.431008i
$88$	0	0
$89$	101.074i	1.13567i	0.823144	+	0.567833i	$0.192218\pi$
$89$	101.074i	1.13567i	−0.823144	+	0.567833i	$0.807782\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−9.20332	−0.0989604
$94$	0	0
$95$	−83.2622	−0.876444
$96$	0	0
$97$	− 30.0955i	− 0.310263i	−0.987894	−	0.155132i	$-0.950420\pi$
$97$	− 30.0955i	− 0.310263i	0.987894	−	0.155132i	$-0.0495801\pi$
$98$	0	0
$99$	−35.1028	−0.354574
$100$	0	0
$101$	− 75.6987i	− 0.749493i	−0.927127	−	0.374746i	$-0.877730\pi$
$101$	− 75.6987i	− 0.749493i	0.927127	−	0.374746i	$-0.122270\pi$
$102$	0	0
$103$	65.5187i	0.636104i	0.948073	+	0.318052i	$0.103029\pi$
$103$	65.5187i	0.636104i	−0.948073	+	0.318052i	$0.896971\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−127.945	−1.19575	−0.597873	−	0.801591i	$-0.703987\pi$
$107$	−127.945	−1.19575	−0.597873	+	0.801591i	$0.703987\pi$
$108$	0	0
$109$	79.0931	0.725625	0.362813	−	0.931862i	$-0.381817\pi$
$109$	79.0931	0.725625	0.362813	+	0.931862i	$0.381817\pi$
$110$	0	0
$111$	− 33.4019i	− 0.300918i
$112$	0	0
$113$	117.995	1.04420	0.522101	−	0.852884i	$-0.325148\pi$
$113$	117.995	1.04420	0.522101	+	0.852884i	$0.325148\pi$
$114$	0	0
$115$	− 15.1378i	− 0.131633i
$116$	0	0
$117$	61.4948i	0.525596i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	15.9120	0.131504
$122$	0	0
$123$	48.5563	0.394767
$124$	0	0
$125$	− 114.637i	− 0.917100i
$126$	0	0
$127$	−37.9759	−0.299022	−0.149511	−	0.988760i	$-0.547770\pi$
$127$	−37.9759	−0.299022	−0.149511	+	0.988760i	$0.547770\pi$
$128$	0	0
$129$	126.465i	0.980348i
$130$	0	0
$131$	− 257.874i	− 1.96850i	−0.176772	−	0.984252i	$-0.556566\pi$
$131$	− 257.874i	− 1.96850i	0.176772	−	0.984252i	$-0.443434\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	13.9049	0.103000
$136$	0	0
$137$	−104.908	−0.765753	−0.382876	−	0.923800i	$-0.625067\pi$
$137$	−104.908	−0.765753	−0.382876	+	0.923800i	$0.625067\pi$
$138$	0	0
$139$	− 143.051i	− 1.02914i	−0.857447	−	0.514572i	$-0.827951\pi$
$139$	− 143.051i	− 1.02914i	0.857447	−	0.514572i	$-0.172049\pi$
$140$	0	0
$141$	55.9944	0.397124
$142$	0	0
$143$	− 239.849i	− 1.67727i
$144$	0	0
$145$	− 57.9337i	− 0.399543i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−284.205	−1.90742	−0.953709	−	0.300730i	$-0.902770\pi$
$149$	−284.205	−1.90742	−0.953709	+	0.300730i	$0.902770\pi$
$150$	0	0
$151$	−189.213	−1.25306	−0.626532	−	0.779395i	$-0.715526\pi$
$151$	−189.213	−1.25306	−0.626532	+	0.779395i	$0.715526\pi$
$152$	0	0
$153$	− 49.2941i	− 0.322183i
$154$	0	0
$155$	−14.2191	−0.0917359
$156$	0	0
$157$	151.395i	0.964300i	0.876089	+	0.482150i	$0.160144\pi$
$157$	151.395i	0.964300i	−0.876089	+	0.482150i	$0.839856\pi$
$158$	0	0
$159$	− 164.166i	− 1.03249i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−168.597	−1.03434	−0.517170	−	0.855883i	$-0.673014\pi$
$163$	−168.597	−1.03434	−0.517170	+	0.855883i	$0.673014\pi$
$164$	0	0
$165$	−54.2336	−0.328689
$166$	0	0
$167$	− 141.148i	− 0.845196i	−0.906317	−	0.422598i	$-0.861118\pi$
$167$	− 141.148i	− 0.845196i	0.906317	−	0.422598i	$-0.138882\pi$
$168$	0	0
$169$	−251.179	−1.48626
$170$	0	0
$171$	93.3430i	0.545865i
$172$	0	0
$173$	− 62.7309i	− 0.362606i	−0.983427	−	0.181303i	$-0.941969\pi$
$173$	− 62.7309i	− 0.362606i	0.983427	−	0.181303i	$-0.0580315\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−195.634	−1.10528
$178$	0	0
$179$	84.0076	0.469316	0.234658	−	0.972078i	$-0.424603\pi$
$179$	84.0076	0.469316	0.234658	+	0.972078i	$0.424603\pi$
$180$	0	0
$181$	289.875i	1.60152i	0.598987	+	0.800759i	$0.295570\pi$
$181$	289.875i	1.60152i	−0.598987	+	0.800759i	$0.704430\pi$
$182$	0	0
$183$	−60.0000	−0.327869
$184$	0	0
$185$	− 51.6058i	− 0.278950i
$186$	0	0
$187$	192.262i	1.02814i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	217.558	1.13905	0.569523	−	0.821975i	$-0.307128\pi$
$191$	217.558	1.13905	0.569523	+	0.821975i	$0.307128\pi$
$192$	0	0
$193$	320.668	1.66149	0.830747	−	0.556650i	$-0.187913\pi$
$193$	320.668	1.66149	0.830747	+	0.556650i	$0.187913\pi$
$194$	0	0
$195$	95.0090i	0.487226i
$196$	0	0
$197$	−344.420	−1.74833	−0.874164	−	0.485632i	$-0.838590\pi$
$197$	−344.420	−1.74833	−0.874164	+	0.485632i	$0.838590\pi$
$198$	0	0
$199$	148.743i	0.747453i	0.927539	+	0.373726i	$0.121920\pi$
$199$	148.743i	0.747453i	−0.927539	+	0.373726i	$0.878080\pi$
$200$	0	0
$201$	− 13.3148i	− 0.0662429i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	75.0192	0.365947
$206$	0	0
$207$	−16.9706	−0.0819834
$208$	0	0
$209$	− 364.067i	− 1.74195i
$210$	0	0
$211$	−299.342	−1.41868	−0.709342	−	0.704864i	$-0.751008\pi$
$211$	−299.342	−1.41868	−0.709342	+	0.704864i	$0.751008\pi$
$212$	0	0
$213$	39.8100i	0.186902i
$214$	0	0
$215$	195.387i	0.908779i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	11.6775	0.0533219
$220$	0	0
$221$	336.814	1.52405
$222$	0	0
$223$	− 55.5613i	− 0.249154i	−0.992210	−	0.124577i	$-0.960243\pi$
$223$	− 55.5613i	− 0.249154i	0.992210	−	0.124577i	$-0.0397574\pi$
$224$	0	0
$225$	−53.5169	−0.237853
$226$	0	0
$227$	11.8899i	0.0523786i	0.999657	+	0.0261893i	$0.00833726\pi$
$227$	11.8899i	0.0523786i	−0.999657	+	0.0261893i	$0.991663\pi$
$228$	0	0
$229$	− 147.629i	− 0.644669i	−0.946626	−	0.322335i	$-0.895532\pi$
$229$	− 147.629i	− 0.644669i	0.946626	−	0.322335i	$-0.104468\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	45.8809	0.196914	0.0984569	−	0.995141i	$-0.468609\pi$
$233$	45.8809	0.196914	0.0984569	+	0.995141i	$0.468609\pi$
$234$	0	0
$235$	86.5110	0.368132
$236$	0	0
$237$	− 81.7359i	− 0.344877i
$238$	0	0
$239$	−423.448	−1.77175	−0.885875	−	0.463923i	$-0.846441\pi$
$239$	−423.448	−1.77175	−0.885875	+	0.463923i	$0.846441\pi$
$240$	0	0
$241$	− 347.561i	− 1.44216i	−0.692852	−	0.721080i	$-0.743646\pi$
$241$	− 347.561i	− 1.44216i	0.692852	−	0.721080i	$-0.256354\pi$
$242$	0	0
$243$	− 15.5885i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−637.790	−2.58214
$248$	0	0
$249$	−68.0286	−0.273207
$250$	0	0
$251$	− 166.117i	− 0.661822i	−0.943662	−	0.330911i	$-0.892644\pi$
$251$	− 166.117i	− 0.661822i	0.943662	−	0.330911i	$-0.107356\pi$
$252$	0	0
$253$	66.1905	0.261623
$254$	0	0
$255$	− 76.1590i	− 0.298663i
$256$	0	0
$257$	209.647i	0.815746i	0.913039	+	0.407873i	$0.133729\pi$
$257$	209.647i	0.815746i	−0.913039	+	0.407873i	$0.866271\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−64.9479	−0.248842
$262$	0	0
$263$	465.242	1.76898	0.884490	−	0.466559i	$-0.154506\pi$
$263$	465.242	1.76898	0.884490	+	0.466559i	$0.154506\pi$
$264$	0	0
$265$	− 253.635i	− 0.957113i
$266$	0	0
$267$	175.066	0.655677
$268$	0	0
$269$	373.877i	1.38988i	0.719069	+	0.694939i	$0.244568\pi$
$269$	373.877i	1.38988i	−0.719069	+	0.694939i	$0.755432\pi$
$270$	0	0
$271$	176.123i	0.649901i	0.945731	+	0.324950i	$0.105348\pi$
$271$	176.123i	0.649901i	−0.945731	+	0.324950i	$0.894652\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	208.733	0.759029
$276$	0	0
$277$	−436.839	−1.57704	−0.788518	−	0.615012i	$-0.789151\pi$
$277$	−436.839	−1.57704	−0.788518	+	0.615012i	$0.789151\pi$
$278$	0	0
$279$	15.9406i	0.0571348i
$280$	0	0
$281$	−275.670	−0.981034	−0.490517	−	0.871432i	$-0.663192\pi$
$281$	−275.670	−0.981034	−0.490517	+	0.871432i	$0.663192\pi$
$282$	0	0
$283$	297.552i	1.05142i	0.850663	+	0.525711i	$0.176201\pi$
$283$	297.552i	1.05142i	−0.850663	+	0.525711i	$0.823799\pi$
$284$	0	0
$285$	144.214i	0.506015i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	19.0106	0.0657808
$290$	0	0
$291$	−52.1270	−0.179130
$292$	0	0
$293$	− 196.735i	− 0.671449i	−0.941960	−	0.335724i	$-0.891019\pi$
$293$	− 196.735i	− 0.671449i	0.941960	−	0.335724i	$-0.108981\pi$
$294$	0	0
$295$	−302.254	−1.02459
$296$	0	0
$297$	60.7999i	0.204713i
$298$	0	0
$299$	− 115.956i	− 0.387812i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−131.114	−0.432720
$304$	0	0
$305$	−92.6996	−0.303933
$306$	0	0
$307$	− 84.9923i	− 0.276848i	−0.990373	−	0.138424i	$-0.955796\pi$
$307$	− 84.9923i	− 0.276848i	0.990373	−	0.138424i	$-0.0442036\pi$
$308$	0	0
$309$	113.482	0.367255
$310$	0	0
$311$	146.079i	0.469708i	0.972031	+	0.234854i	$0.0754613\pi$
$311$	146.079i	0.469708i	−0.972031	+	0.234854i	$0.924539\pi$
$312$	0	0
$313$	− 423.926i	− 1.35440i	−0.735801	−	0.677198i	$-0.763194\pi$
$313$	− 423.926i	− 1.35440i	0.735801	−	0.677198i	$-0.236806\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	292.164	0.921652	0.460826	−	0.887490i	$-0.347553\pi$
$317$	292.164	0.921652	0.460826	+	0.887490i	$0.347553\pi$
$318$	0	0
$319$	253.317	0.794097
$320$	0	0
$321$	221.607i	0.690364i
$322$	0	0
$323$	511.251	1.58282
$324$	0	0
$325$	− 365.668i	− 1.12513i
$326$	0	0
$327$	− 136.993i	− 0.418940i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−178.495	−0.539260	−0.269630	−	0.962964i	$-0.586901\pi$
$331$	−178.495	−0.539260	−0.269630	+	0.962964i	$0.586901\pi$
$332$	0	0
$333$	−57.8538	−0.173735
$334$	0	0
$335$	− 20.5713i	− 0.0614069i
$336$	0	0
$337$	−27.2722	−0.0809263	−0.0404631	−	0.999181i	$-0.512883\pi$
$337$	−27.2722	−0.0809263	−0.0404631	+	0.999181i	$0.512883\pi$
$338$	0	0
$339$	− 204.373i	− 0.602870i
$340$	0	0
$341$	− 62.1734i	− 0.182327i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−26.2194	−0.0759983
$346$	0	0
$347$	125.475	0.361600	0.180800	−	0.983520i	$-0.442131\pi$
$347$	125.475	0.361600	0.180800	+	0.983520i	$0.442131\pi$
$348$	0	0
$349$	− 413.962i	− 1.18614i	−0.805152	−	0.593068i	$-0.797916\pi$
$349$	− 413.962i	− 1.18614i	0.805152	−	0.593068i	$-0.202084\pi$
$350$	0	0
$351$	106.512	0.303453
$352$	0	0
$353$	− 691.677i	− 1.95942i	−0.200412	−	0.979712i	$-0.564228\pi$
$353$	− 691.677i	− 1.95942i	0.200412	−	0.979712i	$-0.435772\pi$
$354$	0	0
$355$	61.5062i	0.173257i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−378.387	−1.05400	−0.527002	−	0.849864i	$-0.676684\pi$
$359$	−378.387	−1.05400	−0.527002	+	0.849864i	$0.676684\pi$
$360$	0	0
$361$	−607.101	−1.68172
$362$	0	0
$363$	− 27.5604i	− 0.0759241i
$364$	0	0
$365$	18.0416	0.0494292
$366$	0	0
$367$	174.324i	0.474998i	0.971388	+	0.237499i	$0.0763276\pi$
$367$	174.324i	0.474998i	−0.971388	+	0.237499i	$0.923672\pi$
$368$	0	0
$369$	− 84.1020i	− 0.227919i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−166.429	−0.446191	−0.223096	−	0.974797i	$-0.571616\pi$
$373$	−166.429	−0.446191	−0.223096	+	0.974797i	$0.571616\pi$
$374$	0	0
$375$	−198.558	−0.529488
$376$	0	0
$377$	− 443.773i	− 1.17712i
$378$	0	0
$379$	484.521	1.27842	0.639210	−	0.769033i	$-0.279262\pi$
$379$	484.521	1.27842	0.639210	+	0.769033i	$0.279262\pi$
$380$	0	0
$381$	65.7761i	0.172641i
$382$	0	0
$383$	70.7512i	0.184729i	0.995725	+	0.0923645i	$0.0294425\pi$
$383$	70.7512i	0.184729i	−0.995725	+	0.0923645i	$0.970558\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	219.044	0.566004
$388$	0	0
$389$	716.247	1.84125	0.920626	−	0.390445i	$-0.127679\pi$
$389$	716.247	1.84125	0.920626	+	0.390445i	$0.127679\pi$
$390$	0	0
$391$	92.9498i	0.237723i
$392$	0	0
$393$	−446.651	−1.13652
$394$	0	0
$395$	− 126.281i	− 0.319700i
$396$	0	0
$397$	124.225i	0.312909i	0.987685	+	0.156454i	$0.0500064\pi$
$397$	124.225i	0.312909i	−0.987685	+	0.156454i	$0.949994\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−326.557	−0.814358	−0.407179	−	0.913348i	$-0.633487\pi$
$401$	−326.557	−0.814358	−0.407179	+	0.913348i	$0.633487\pi$
$402$	0	0
$403$	−108.918	−0.270269
$404$	0	0
$405$	− 24.0841i	− 0.0594668i
$406$	0	0
$407$	225.648	0.554418
$408$	0	0
$409$	− 507.240i	− 1.24019i	−0.784525	−	0.620097i	$-0.787093\pi$
$409$	− 507.240i	− 1.24019i	0.784525	−	0.620097i	$-0.212907\pi$
$410$	0	0
$411$	181.706i	0.442108i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−105.104	−0.253262
$416$	0	0
$417$	−247.772	−0.594177
$418$	0	0
$419$	− 367.851i	− 0.877926i	−0.898505	−	0.438963i	$-0.855346\pi$
$419$	− 367.851i	− 0.877926i	0.898505	−	0.438963i	$-0.144654\pi$
$420$	0	0
$421$	311.187	0.739162	0.369581	−	0.929199i	$-0.379501\pi$
$421$	311.187	0.739162	0.369581	+	0.929199i	$0.379501\pi$
$422$	0	0
$423$	− 96.9852i	− 0.229279i
$424$	0	0
$425$	293.119i	0.689691i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−415.431	−0.968370
$430$	0	0
$431$	581.546	1.34930	0.674648	−	0.738140i	$-0.264296\pi$
$431$	581.546	1.34930	0.674648	+	0.738140i	$0.264296\pi$
$432$	0	0
$433$	786.154i	1.81560i	0.419406	+	0.907799i	$0.362238\pi$
$433$	786.154i	1.81560i	−0.419406	+	0.907799i	$0.637762\pi$
$434$	0	0
$435$	−100.344	−0.230676
$436$	0	0
$437$	− 176.009i	− 0.402767i
$438$	0	0
$439$	4.25672i	0.00969640i	0.999988	+	0.00484820i	$0.00154324\pi$
$439$	4.25672i	0.00969640i	−0.999988	+	0.00484820i	$0.998457\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	480.379	1.08438	0.542188	−	0.840257i	$-0.317596\pi$
$443$	480.379	1.08438	0.542188	+	0.840257i	$0.317596\pi$
$444$	0	0
$445$	270.476	0.607810
$446$	0	0
$447$	492.258i	1.10125i
$448$	0	0
$449$	−343.206	−0.764379	−0.382189	−	0.924084i	$-0.624830\pi$
$449$	−343.206	−0.764379	−0.382189	+	0.924084i	$0.624830\pi$
$450$	0	0
$451$	328.024i	0.727326i
$452$	0	0
$453$	327.726i	0.723457i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	758.607	1.65997	0.829986	−	0.557784i	$-0.188348\pi$
$457$	758.607	1.65997	0.829986	+	0.557784i	$0.188348\pi$
$458$	0	0
$459$	−85.3798	−0.186013
$460$	0	0
$461$	395.110i	0.857073i	0.903525	+	0.428536i	$0.140971\pi$
$461$	395.110i	0.857073i	−0.903525	+	0.428536i	$0.859029\pi$
$462$	0	0
$463$	466.432	1.00741	0.503706	−	0.863875i	$-0.331969\pi$
$463$	466.432	1.00741	0.503706	+	0.863875i	$0.331969\pi$
$464$	0	0
$465$	24.6282i	0.0529638i
$466$	0	0
$467$	357.640i	0.765825i	0.923785	+	0.382912i	$0.125079\pi$
$467$	357.640i	0.765825i	−0.923785	+	0.382912i	$0.874921\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	262.224	0.556739
$472$	0	0
$473$	−854.339	−1.80621
$474$	0	0
$475$	− 555.048i	− 1.16852i
$476$	0	0
$477$	−284.343	−0.596107
$478$	0	0
$479$	− 119.585i	− 0.249656i	−0.992178	−	0.124828i	$-0.960162\pi$
$479$	− 119.585i	− 0.249656i	0.992178	−	0.124828i	$-0.0398380\pi$
$480$	0	0
$481$	− 395.301i	− 0.821832i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−80.5358	−0.166053
$486$	0	0
$487$	−293.027	−0.601698	−0.300849	−	0.953672i	$-0.597270\pi$
$487$	−293.027	−0.601698	−0.300849	+	0.953672i	$0.597270\pi$
$488$	0	0
$489$	292.019i	0.597176i
$490$	0	0
$491$	−95.4062	−0.194310	−0.0971549	−	0.995269i	$-0.530974\pi$
$491$	−95.4062	−0.194310	−0.0971549	+	0.995269i	$0.530974\pi$
$492$	0	0
$493$	355.727i	0.721556i
$494$	0	0
$495$	93.9354i	0.189769i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−692.931	−1.38864	−0.694319	−	0.719667i	$-0.744294\pi$
$499$	−692.931	−1.38864	−0.694319	+	0.719667i	$0.744294\pi$
$500$	0	0
$501$	−244.475	−0.487974
$502$	0	0
$503$	− 11.1075i	− 0.0220824i	−0.999939	−	0.0110412i	$-0.996485\pi$
$503$	− 11.1075i	− 0.0220824i	0.999939	−	0.0110412i	$-0.00351460\pi$
$504$	0	0
$505$	−202.570	−0.401130
$506$	0	0
$507$	435.054i	0.858095i
$508$	0	0
$509$	165.938i	0.326007i	0.986625	+	0.163004i	$0.0521183\pi$
$509$	165.938i	0.326007i	−0.986625	+	0.163004i	$0.947882\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	161.675	0.315156
$514$	0	0
$515$	175.329	0.340444
$516$	0	0
$517$	378.273i	0.731669i
$518$	0	0
$519$	−108.653	−0.209351
$520$	0	0
$521$	69.6797i	0.133742i	0.997762	+	0.0668712i	$0.0213016\pi$
$521$	69.6797i	0.133742i	−0.997762	+	0.0668712i	$0.978698\pi$
$522$	0	0
$523$	− 74.3157i	− 0.142095i	−0.997473	−	0.0710475i	$-0.977366\pi$
$523$	− 74.3157i	− 0.142095i	0.997473	−	0.0710475i	$-0.0226342\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	87.3086	0.165671
$528$	0	0
$529$	−497.000	−0.939509
$530$	0	0
$531$	338.849i	0.638133i
$532$	0	0
$533$	574.648	1.07814
$534$	0	0
$535$	342.381i	0.639965i
$536$	0	0
$537$	− 145.505i	− 0.270960i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	192.543	0.355903	0.177951	−	0.984039i	$-0.443053\pi$
$541$	192.543	0.355903	0.177951	+	0.984039i	$0.443053\pi$
$542$	0	0
$543$	502.078	0.924637
$544$	0	0
$545$	− 211.654i	− 0.388356i
$546$	0	0
$547$	43.1121	0.0788156	0.0394078	−	0.999223i	$-0.487453\pi$
$547$	43.1121	0.0788156	0.0394078	+	0.999223i	$0.487453\pi$
$548$	0	0
$549$	103.923i	0.189295i
$550$	0	0
$551$	− 673.603i	− 1.22251i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−89.3838	−0.161052
$556$	0	0
$557$	214.985	0.385969	0.192985	−	0.981202i	$-0.438183\pi$
$557$	214.985	0.385969	0.192985	+	0.981202i	$0.438183\pi$
$558$	0	0
$559$	1496.67i	2.67741i
$560$	0	0
$561$	333.008	0.593597
$562$	0	0
$563$	− 499.699i	− 0.887565i	−0.896134	−	0.443783i	$-0.853636\pi$
$563$	− 499.699i	− 0.887565i	0.896134	−	0.443783i	$-0.146364\pi$
$564$	0	0
$565$	− 315.755i	− 0.558859i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1037.20	1.82285	0.911426	−	0.411464i	$-0.134983\pi$
$569$	1037.20	1.82285	0.911426	+	0.411464i	$0.134983\pi$
$570$	0	0
$571$	−425.614	−0.745384	−0.372692	−	0.927955i	$-0.621565\pi$
$571$	−425.614	−0.745384	−0.372692	+	0.927955i	$0.621565\pi$
$572$	0	0
$573$	− 376.821i	− 0.657629i
$574$	0	0
$575$	100.913	0.175500
$576$	0	0
$577$	210.399i	0.364643i	0.983239	+	0.182321i	$0.0583611\pi$
$577$	210.399i	0.364643i	−0.983239	+	0.182321i	$0.941639\pi$
$578$	0	0
$579$	− 555.414i	− 0.959264i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	1109.03	1.90228
$584$	0	0
$585$	164.560	0.281300
$586$	0	0
$587$	440.365i	0.750196i	0.926985	+	0.375098i	$0.122391\pi$
$587$	440.365i	0.750196i	−0.926985	+	0.375098i	$0.877609\pi$
$588$	0	0
$589$	−165.327	−0.280691
$590$	0	0
$591$	596.554i	1.00940i
$592$	0	0
$593$	− 45.9442i	− 0.0774775i	−0.999249	−	0.0387388i	$-0.987666\pi$
$593$	− 45.9442i	− 0.0774775i	0.999249	−	0.0387388i	$-0.0123340\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	257.631	0.431542
$598$	0	0
$599$	300.755	0.502095	0.251047	−	0.967975i	$-0.419225\pi$
$599$	300.755	0.502095	0.251047	+	0.967975i	$0.419225\pi$
$600$	0	0
$601$	55.0039i	0.0915206i	0.998952	+	0.0457603i	$0.0145710\pi$
$601$	55.0039i	0.0915206i	−0.998952	+	0.0457603i	$0.985429\pi$
$602$	0	0
$603$	−23.0619	−0.0382453
$604$	0	0
$605$	− 42.5807i	− 0.0703813i
$606$	0	0
$607$	− 454.240i	− 0.748336i	−0.927361	−	0.374168i	$-0.877928\pi$
$607$	− 454.240i	− 0.748336i	0.927361	−	0.374168i	$-0.122072\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	662.676	1.08458
$612$	0	0
$613$	809.383	1.32036	0.660182	−	0.751106i	$-0.270479\pi$
$613$	809.383	1.32036	0.660182	+	0.751106i	$0.270479\pi$
$614$	0	0
$615$	− 129.937i	− 0.211280i
$616$	0	0
$617$	398.646	0.646104	0.323052	−	0.946381i	$-0.395291\pi$
$617$	398.646	0.646104	0.323052	+	0.946381i	$0.395291\pi$
$618$	0	0
$619$	725.195i	1.17156i	0.810471	+	0.585779i	$0.199212\pi$
$619$	725.195i	1.17156i	−0.810471	+	0.585779i	$0.800788\pi$
$620$	0	0
$621$	29.3939i	0.0473331i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	139.204	0.222726
$626$	0	0
$627$	−630.582	−1.00571
$628$	0	0
$629$	316.872i	0.503771i
$630$	0	0
$631$	124.396	0.197140	0.0985702	−	0.995130i	$-0.468573\pi$
$631$	124.396	0.197140	0.0985702	+	0.995130i	$0.468573\pi$
$632$	0	0
$633$	518.476i	0.819078i
$634$	0	0
$635$	101.624i	0.160037i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	68.9530	0.107908
$640$	0	0
$641$	632.855	0.987293	0.493647	−	0.869663i	$-0.335664\pi$
$641$	632.855	0.987293	0.493647	+	0.869663i	$0.335664\pi$
$642$	0	0
$643$	− 172.306i	− 0.267973i	−0.990983	−	0.133986i	$-0.957222\pi$
$643$	− 172.306i	− 0.267973i	0.990983	−	0.133986i	$-0.0427778\pi$
$644$	0	0
$645$	338.421	0.524684
$646$	0	0
$647$	− 831.259i	− 1.28479i	−0.766374	−	0.642395i	$-0.777941\pi$
$647$	− 831.259i	− 1.28479i	0.766374	−	0.642395i	$-0.222059\pi$
$648$	0	0
$649$	− 1321.62i	− 2.03639i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	177.508	0.271835	0.135917	−	0.990720i	$-0.456602\pi$
$653$	177.508	0.271835	0.135917	+	0.990720i	$0.456602\pi$
$654$	0	0
$655$	−690.073	−1.05355
$656$	0	0
$657$	− 20.2260i	− 0.0307854i
$658$	0	0
$659$	−363.665	−0.551843	−0.275922	−	0.961180i	$-0.588983\pi$
$659$	−363.665	−0.551843	−0.275922	+	0.961180i	$0.588983\pi$
$660$	0	0
$661$	1220.43i	1.84633i	0.384399	+	0.923167i	$0.374409\pi$
$661$	1220.43i	1.84633i	−0.384399	+	0.923167i	$0.625591\pi$
$662$	0	0
$663$	− 583.379i	− 0.879908i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	122.467	0.183609
$668$	0	0
$669$	−96.2350	−0.143849
$670$	0	0
$671$	− 405.332i	− 0.604072i
$672$	0	0
$673$	−56.4549	−0.0838855	−0.0419427	−	0.999120i	$-0.513355\pi$
$673$	−56.4549	−0.0838855	−0.0419427	+	0.999120i	$0.513355\pi$
$674$	0	0
$675$	92.6941i	0.137325i
$676$	0	0
$677$	587.164i	0.867302i	0.901081	+	0.433651i	$0.142775\pi$
$677$	587.164i	0.867302i	−0.901081	+	0.433651i	$0.857225\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	20.5940	0.0302408
$682$	0	0
$683$	597.109	0.874245	0.437122	−	0.899402i	$-0.355998\pi$
$683$	597.109	0.874245	0.437122	+	0.899402i	$0.355998\pi$
$684$	0	0
$685$	280.735i	0.409832i
$686$	0	0
$687$	−255.701	−0.372200
$688$	0	0
$689$	− 1942.85i	− 2.81981i
$690$	0	0
$691$	688.035i	0.995708i	0.867261	+	0.497854i	$0.165879\pi$
$691$	688.035i	0.995708i	−0.867261	+	0.497854i	$0.834121\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−382.806	−0.550800
$696$	0	0
$697$	−460.636	−0.660884
$698$	0	0
$699$	− 79.4681i	− 0.113688i
$700$	0	0
$701$	868.040	1.23829	0.619144	−	0.785278i	$-0.287480\pi$
$701$	868.040	1.23829	0.619144	+	0.785278i	$0.287480\pi$
$702$	0	0
$703$	− 600.028i	− 0.853525i
$704$	0	0
$705$	− 149.842i	− 0.212541i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−785.908	−1.10847	−0.554237	−	0.832359i	$-0.686990\pi$
$709$	−785.908	−1.10847	−0.554237	+	0.832359i	$0.686990\pi$
$710$	0	0
$711$	−141.571	−0.199115
$712$	0	0
$713$	− 30.0579i	− 0.0421570i
$714$	0	0
$715$	−641.838	−0.897675
$716$	0	0
$717$	733.434i	1.02292i
$718$	0	0
$719$	1306.42i	1.81699i	0.417894	+	0.908496i	$0.362768\pi$
$719$	1306.42i	1.81699i	−0.417894	+	0.908496i	$0.637232\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−601.993	−0.832632
$724$	0	0
$725$	386.201	0.532691
$726$	0	0
$727$	− 577.725i	− 0.794670i	−0.917674	−	0.397335i	$-0.869935\pi$
$727$	− 577.725i	− 0.794670i	0.917674	−	0.397335i	$-0.130065\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	− 1199.73i	− 1.64121i
$732$	0	0
$733$	571.147i	0.779190i	0.920986	+	0.389595i	$0.127385\pi$
$733$	571.147i	0.779190i	−0.920986	+	0.389595i	$0.872615\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	89.9488	0.122047
$738$	0	0
$739$	−202.526	−0.274055	−0.137027	−	0.990567i	$-0.543755\pi$
$739$	−202.526	−0.274055	−0.137027	+	0.990567i	$0.543755\pi$
$740$	0	0
$741$	1104.68i	1.49080i
$742$	0	0
$743$	968.312	1.30325	0.651623	−	0.758543i	$-0.274089\pi$
$743$	968.312	1.30325	0.651623	+	0.758543i	$0.274089\pi$
$744$	0	0
$745$	760.536i	1.02085i
$746$	0	0
$747$	117.829i	0.157736i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	727.089	0.968162	0.484081	−	0.875023i	$-0.339154\pi$
$751$	727.089	0.968162	0.484081	+	0.875023i	$0.339154\pi$
$752$	0	0
$753$	−287.724	−0.382103
$754$	0	0
$755$	506.335i	0.670642i
$756$	0	0
$757$	1414.08	1.86800	0.934001	−	0.357269i	$-0.116292\pi$
$757$	1414.08	1.86800	0.934001	+	0.357269i	$0.116292\pi$
$758$	0	0
$759$	− 114.645i	− 0.151048i
$760$	0	0
$761$	166.125i	0.218299i	0.994025	+	0.109149i	$0.0348127\pi$
$761$	166.125i	0.218299i	−0.994025	+	0.109149i	$0.965187\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−131.911	−0.172433
$766$	0	0
$767$	−2315.27	−3.01860
$768$	0	0
$769$	361.961i	0.470690i	0.971912	+	0.235345i	$0.0756221\pi$
$769$	361.961i	0.470690i	−0.971912	+	0.235345i	$0.924378\pi$
$770$	0	0
$771$	363.119	0.470971
$772$	0	0
$773$	1300.28i	1.68212i	0.540943	+	0.841059i	$0.318067\pi$
$773$	1300.28i	1.68212i	−0.540943	+	0.841059i	$0.681933\pi$
$774$	0	0
$775$	− 94.7881i	− 0.122307i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	872.259	1.11972
$780$	0	0
$781$	−268.938	−0.344351
$782$	0	0
$783$	112.493i	0.143669i
$784$	0	0
$785$	405.135	0.516095
$786$	0	0
$787$	− 876.584i	− 1.11383i	−0.830569	−	0.556915i	$-0.811985\pi$
$787$	− 876.584i	− 1.11383i	0.830569	−	0.556915i	$-0.188015\pi$
$788$	0	0
$789$	− 805.823i	− 1.02132i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−710.081	−0.895436
$794$	0	0
$795$	−439.308	−0.552589
$796$	0	0
$797$	− 101.354i	− 0.127170i	−0.997976	−	0.0635850i	$-0.979747\pi$
$797$	− 101.354i	− 0.127170i	0.997976	−	0.0635850i	$-0.0202534\pi$
$798$	0	0
$799$	−531.199	−0.664830
$800$	0	0
$801$	− 303.223i	− 0.378555i
$802$	0	0
$803$	78.8878i	0.0982413i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	647.574	0.802446
$808$	0	0
$809$	514.729	0.636253	0.318127	−	0.948048i	$-0.396946\pi$
$809$	514.729	0.636253	0.318127	+	0.948048i	$0.396946\pi$
$810$	0	0
$811$	1599.34i	1.97206i	0.166576	+	0.986029i	$0.446729\pi$
$811$	1599.34i	1.97206i	−0.166576	+	0.986029i	$0.553271\pi$
$812$	0	0
$813$	305.054	0.375220
$814$	0	0
$815$	451.168i	0.553580i
$816$	0	0
$817$	2271.80i	2.78066i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1203.45	−1.46584	−0.732920	−	0.680315i	$-0.761843\pi$
$821$	−1203.45	−1.46584	−0.732920	+	0.680315i	$0.761843\pi$
$822$	0	0
$823$	−196.504	−0.238765	−0.119383	−	0.992848i	$-0.538092\pi$
$823$	−196.504	−0.238765	−0.119383	+	0.992848i	$0.538092\pi$
$824$	0	0
$825$	− 361.536i	− 0.438225i
$826$	0	0
$827$	921.653	1.11445	0.557227	−	0.830360i	$-0.311865\pi$
$827$	921.653	1.11445	0.557227	+	0.830360i	$0.311865\pi$
$828$	0	0
$829$	246.958i	0.297899i	0.988845	+	0.148950i	$0.0475892\pi$
$829$	246.958i	0.297899i	−0.988845	+	0.148950i	$0.952411\pi$
$830$	0	0
$831$	756.627i	0.910502i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−377.712	−0.452350
$836$	0	0
$837$	27.6100	0.0329868
$838$	0	0
$839$	132.800i	0.158284i	0.996863	+	0.0791418i	$0.0252180\pi$
$839$	132.800i	0.158284i	−0.996863	+	0.0791418i	$0.974782\pi$
$840$	0	0
$841$	−372.308	−0.442697
$842$	0	0
$843$	477.475i	0.566400i
$844$	0	0
$845$	672.156i	0.795451i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	515.376	0.607039
$850$	0	0
$851$	109.090	0.128191
$852$	0	0
$853$	387.617i	0.454416i	0.973846	+	0.227208i	$0.0729598\pi$
$853$	387.617i	0.454416i	−0.973846	+	0.227208i	$0.927040\pi$
$854$	0	0
$855$	249.787	0.292148
$856$	0	0
$857$	− 541.516i	− 0.631874i	−0.948780	−	0.315937i	$-0.897681\pi$
$857$	− 541.516i	− 0.631874i	0.948780	−	0.315937i	$-0.102319\pi$
$858$	0	0
$859$	− 1192.39i	− 1.38812i	−0.719919	−	0.694059i	$-0.755821\pi$
$859$	− 1192.39i	− 1.38812i	0.719919	−	0.694059i	$-0.244179\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−499.496	−0.578790	−0.289395	−	0.957210i	$-0.593454\pi$
$863$	−499.496	−0.578790	−0.289395	+	0.957210i	$0.593454\pi$
$864$	0	0
$865$	−167.868	−0.194067
$866$	0	0
$867$	− 32.9274i	− 0.0379786i
$868$	0	0
$869$	552.170	0.635409
$870$	0	0
$871$	− 157.577i	− 0.180915i
$872$	0	0
$873$	90.2865i	0.103421i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	675.621	0.770377	0.385189	−	0.922838i	$-0.374136\pi$
$877$	675.621	0.770377	0.385189	+	0.922838i	$0.374136\pi$
$878$	0	0
$879$	−340.754	−0.387661
$880$	0	0
$881$	98.9362i	0.112300i	0.998422	+	0.0561500i	$0.0178825\pi$
$881$	98.9362i	0.112300i	−0.998422	+	0.0561500i	$0.982118\pi$
$882$	0	0
$883$	−550.179	−0.623079	−0.311540	−	0.950233i	$-0.600845\pi$
$883$	−550.179	−0.623079	−0.311540	+	0.950233i	$0.600845\pi$
$884$	0	0
$885$	523.519i	0.591547i
$886$	0	0
$887$	287.638i	0.324282i	0.986768	+	0.162141i	$0.0518399\pi$
$887$	287.638i	0.324282i	−0.986768	+	0.162141i	$0.948160\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	105.308	0.118191
$892$	0	0
$893$	1005.88	1.12640
$894$	0	0
$895$	− 224.805i	− 0.251179i
$896$	0	0
$897$	−200.841	−0.223903
$898$	0	0
$899$	− 115.034i	− 0.127958i
$900$	0	0
$901$	1557.38i	1.72850i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	775.707	0.857135
$906$	0	0
$907$	218.301	0.240684	0.120342	−	0.992732i	$-0.461601\pi$
$907$	218.301	0.240684	0.120342	+	0.992732i	$0.461601\pi$
$908$	0	0
$909$	227.096i	0.249831i
$910$	0	0
$911$	8.50926	0.00934057	0.00467029	−	0.999989i	$-0.498513\pi$
$911$	8.50926	0.00934057	0.00467029	+	0.999989i	$0.498513\pi$
$912$	0	0
$913$	− 459.570i	− 0.503362i
$914$	0	0
$915$	160.560i	0.175476i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	430.164	0.468078	0.234039	−	0.972227i	$-0.424806\pi$
$919$	430.164	0.468078	0.234039	+	0.972227i	$0.424806\pi$
$920$	0	0
$921$	−147.211	−0.159838
$922$	0	0
$923$	471.139i	0.510443i
$924$	0	0
$925$	344.018	0.371911
$926$	0	0
$927$	− 196.556i	− 0.212035i
$928$	0	0
$929$	− 544.559i	− 0.586178i	−0.956085	−	0.293089i	$-0.905317\pi$
$929$	− 544.559i	− 0.586178i	0.956085	−	0.293089i	$-0.0946832\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	253.017	0.271186
$934$	0	0
$935$	514.495	0.550262
$936$	0	0
$937$	152.767i	0.163038i	0.996672	+	0.0815190i	$0.0259771\pi$
$937$	152.767i	0.163038i	−0.996672	+	0.0815190i	$0.974023\pi$
$938$	0	0
$939$	−734.261	−0.781961
$940$	0	0
$941$	561.934i	0.597167i	0.954384	+	0.298583i	$0.0965141\pi$
$941$	561.934i	0.597167i	−0.954384	+	0.298583i	$0.903486\pi$
$942$	0	0
$943$	158.584i	0.168170i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	82.5966	0.0872192	0.0436096	−	0.999049i	$-0.486114\pi$
$947$	82.5966	0.0872192	0.0436096	+	0.999049i	$0.486114\pi$
$948$	0	0
$949$	138.199	0.145626
$950$	0	0
$951$	− 506.042i	− 0.532116i
$952$	0	0
$953$	1171.84	1.22963	0.614817	−	0.788670i	$-0.289230\pi$
$953$	1171.84	1.22963	0.614817	+	0.788670i	$0.289230\pi$
$954$	0	0
$955$	− 582.186i	− 0.609619i
$956$	0	0
$957$	− 438.758i	− 0.458472i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	932.766	0.970620
$962$	0	0
$963$	383.834	0.398582
$964$	0	0
$965$	− 858.111i	− 0.889234i
$966$	0	0
$967$	1587.63	1.64181	0.820906	−	0.571063i	$-0.193469\pi$
$967$	1587.63	1.64181	0.820906	+	0.571063i	$0.193469\pi$
$968$	0	0
$969$	− 885.512i	− 0.913841i
$970$	0	0
$971$	− 234.133i	− 0.241126i	−0.992706	−	0.120563i	$-0.961530\pi$
$971$	− 234.133i	− 0.241126i	0.992706	−	0.120563i	$-0.0384699\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−633.356	−0.649596
$976$	0	0
$977$	73.4446	0.0751736	0.0375868	−	0.999293i	$-0.488033\pi$
$977$	73.4446	0.0751736	0.0375868	+	0.999293i	$0.488033\pi$
$978$	0	0
$979$	1182.66i	1.20803i
$980$	0	0
$981$	−237.279	−0.241875
$982$	0	0
$983$	690.972i	0.702922i	0.936203	+	0.351461i	$0.114315\pi$
$983$	690.972i	0.702922i	−0.936203	+	0.351461i	$0.885685\pi$
$984$	0	0
$985$	921.672i	0.935707i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−413.032	−0.417626
$990$	0	0
$991$	972.428	0.981260	0.490630	−	0.871368i	$-0.336767\pi$
$991$	972.428	0.981260	0.490630	+	0.871368i	$0.336767\pi$
$992$	0	0
$993$	309.163i	0.311342i
$994$	0	0
$995$	398.038	0.400038
$996$	0	0
$997$	1561.15i	1.56584i	0.622120	+	0.782922i	$0.286272\pi$
$997$	1561.15i	1.56584i	−0.622120	+	0.782922i	$0.713728\pi$
$998$	0	0
$999$	100.206i	0.100306i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.3.f.k.97.1		8
4.3	odd	2		1176.3.f.a.97.5		8
7.2	even	3		336.3.bh.h.241.1		8
7.3	odd	6		336.3.bh.h.145.1		8
7.6	odd	2	inner	2352.3.f.k.97.8		8
12.11	even	2		3528.3.f.f.2449.7		8
21.2	odd	6		1008.3.cg.n.577.4		8
21.17	even	6		1008.3.cg.n.145.4		8
28.3	even	6		168.3.z.a.145.1	yes	8
28.11	odd	6		1176.3.z.d.313.4		8
28.19	even	6		1176.3.z.d.913.4		8
28.23	odd	6		168.3.z.a.73.1	✓	8
28.27	even	2		1176.3.f.a.97.4		8
84.23	even	6		504.3.by.a.73.4		8
84.59	odd	6		504.3.by.a.145.4		8
84.83	odd	2		3528.3.f.f.2449.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.3.z.a.73.1	✓	8	28.23	odd	6
168.3.z.a.145.1	yes	8	28.3	even	6
336.3.bh.h.145.1		8	7.3	odd	6
336.3.bh.h.241.1		8	7.2	even	3
504.3.by.a.73.4		8	84.23	even	6
504.3.by.a.145.4		8	84.59	odd	6
1008.3.cg.n.145.4		8	21.17	even	6
1008.3.cg.n.577.4		8	21.2	odd	6
1176.3.f.a.97.4		8	28.27	even	2
1176.3.f.a.97.5		8	4.3	odd	2
1176.3.z.d.313.4		8	28.11	odd	6
1176.3.z.d.913.4		8	28.19	even	6
2352.3.f.k.97.1		8	1.1	even	1	trivial
2352.3.f.k.97.8		8	7.6	odd	2	inner
3528.3.f.f.2449.2		8	84.83	odd	2
3528.3.f.f.2449.7		8	12.11	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2352.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} -2.67601i q^{5} -3.00000 q^{9} +11.7009 q^{11} -20.4983i q^{13} -4.63498 q^{15} +16.4314i q^{17} -31.1143i q^{19} +5.65685 q^{23} +17.8390 q^{25} +5.19615i q^{27} +21.6493 q^{29} -5.31354i q^{31} -20.2666i q^{33} +19.2846 q^{37} -35.5040 q^{39} +28.0340i q^{41} -73.0145 q^{43} +8.02802i q^{45} +32.3284i q^{47} +28.4599 q^{51} +94.7811 q^{53} -31.3118i q^{55} -53.8916 q^{57} -112.950i q^{59} -34.6410i q^{61} -54.8535 q^{65} +7.68731 q^{67} -9.79796i q^{69} -22.9843 q^{71} +6.74200i q^{73} -30.8980i q^{75} +47.1902 q^{79} +9.00000 q^{81} -39.2763i q^{83} +43.9704 q^{85} -37.4977i q^{87} +101.074i q^{89} -9.20332 q^{93} -83.2622 q^{95} -30.0955i q^{97} -35.1028 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 24 q^{9} + 28 q^{11} + 12 q^{15} + 12 q^{25} - 4 q^{29} - 100 q^{37} - 12 q^{39} + 20 q^{43} - 24 q^{51} + 100 q^{53} - 156 q^{57} + 296 q^{65} + 68 q^{67} - 424 q^{71} - 80 q^{79} + 72 q^{81} - 232 q^{85}+ \cdots - 84 q^{99}+O(q^{100}) \) 8 * q - 24 * q^9 + 28 * q^11 + 12 * q^15 + 12 * q^25 - 4 * q^29 - 100 * q^37 - 12 * q^39 + 20 * q^43 - 24 * q^51 + 100 * q^53 - 156 * q^57 + 296 * q^65 + 68 * q^67 - 424 * q^71 - 80 * q^79 + 72 * q^81 - 232 * q^85 - 48 * q^93 + 288 * q^95 - 84 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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