LMFDB - Embedded newform 9702.2.a.cx.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9702,2,Mod(1,9702)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9702, 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("9702.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9702.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:	$77.4708600410$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 154)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	9702.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +3.41421 q^{5} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} +3.41421 q^{5} -1.00000 q^{8} -3.41421 q^{10} -1.00000 q^{11} +1.82843 q^{13} +1.00000 q^{16} +7.65685 q^{17} -3.41421 q^{19} +3.41421 q^{20} +1.00000 q^{22} -2.24264 q^{23} +6.65685 q^{25} -1.82843 q^{26} +8.65685 q^{29} -4.00000 q^{31} -1.00000 q^{32} -7.65685 q^{34} -6.58579 q^{37} +3.41421 q^{38} -3.41421 q^{40} +2.58579 q^{41} +5.65685 q^{43} -1.00000 q^{44} +2.24264 q^{46} -6.48528 q^{47} -6.65685 q^{50} +1.82843 q^{52} +11.8995 q^{53} -3.41421 q^{55} -8.65685 q^{58} +8.41421 q^{59} +6.17157 q^{61} +4.00000 q^{62} +1.00000 q^{64} +6.24264 q^{65} +11.2426 q^{67} +7.65685 q^{68} -3.07107 q^{71} -6.58579 q^{73} +6.58579 q^{74} -3.41421 q^{76} -4.75736 q^{79} +3.41421 q^{80} -2.58579 q^{82} -16.1421 q^{83} +26.1421 q^{85} -5.65685 q^{86} +1.00000 q^{88} +4.48528 q^{89} -2.24264 q^{92} +6.48528 q^{94} -11.6569 q^{95} +1.82843 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + 4 q^{5} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 + 4 * q^5 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} + 4 q^{5} - 2 q^{8} - 4 q^{10} - 2 q^{11} - 2 q^{13} + 2 q^{16} + 4 q^{17} - 4 q^{19} + 4 q^{20} + 2 q^{22} + 4 q^{23} + 2 q^{25} + 2 q^{26} + 6 q^{29} - 8 q^{31} - 2 q^{32} - 4 q^{34} - 16 q^{37} + 4 q^{38} - 4 q^{40} + 8 q^{41} - 2 q^{44} - 4 q^{46} + 4 q^{47} - 2 q^{50} - 2 q^{52} + 4 q^{53} - 4 q^{55} - 6 q^{58} + 14 q^{59} + 18 q^{61} + 8 q^{62} + 2 q^{64} + 4 q^{65} + 14 q^{67} + 4 q^{68} + 8 q^{71} - 16 q^{73} + 16 q^{74} - 4 q^{76} - 18 q^{79} + 4 q^{80} - 8 q^{82} - 4 q^{83} + 24 q^{85} + 2 q^{88} - 8 q^{89} + 4 q^{92} - 4 q^{94} - 12 q^{95} - 2 q^{97}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + 4 * q^5 - 2 * q^8 - 4 * q^10 - 2 * q^11 - 2 * q^13 + 2 * q^16 + 4 * q^17 - 4 * q^19 + 4 * q^20 + 2 * q^22 + 4 * q^23 + 2 * q^25 + 2 * q^26 + 6 * q^29 - 8 * q^31 - 2 * q^32 - 4 * q^34 - 16 * q^37 + 4 * q^38 - 4 * q^40 + 8 * q^41 - 2 * q^44 - 4 * q^46 + 4 * q^47 - 2 * q^50 - 2 * q^52 + 4 * q^53 - 4 * q^55 - 6 * q^58 + 14 * q^59 + 18 * q^61 + 8 * q^62 + 2 * q^64 + 4 * q^65 + 14 * q^67 + 4 * q^68 + 8 * q^71 - 16 * q^73 + 16 * q^74 - 4 * q^76 - 18 * q^79 + 4 * q^80 - 8 * q^82 - 4 * q^83 + 24 * q^85 + 2 * q^88 - 8 * q^89 + 4 * q^92 - 4 * q^94 - 12 * q^95 - 2 * q^97

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
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	3.41421	1.52688	0.763441	−	0.645877i	$-0.223508\pi$
$5$	3.41421	1.52688	0.763441	+	0.645877i	$0.223508\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−3.41421	−1.07967
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	1.82843	0.507114	0.253557	−	0.967320i	$-0.418399\pi$
$13$	1.82843	0.507114	0.253557	+	0.967320i	$0.418399\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	7.65685	1.85706	0.928530	−	0.371257i	$-0.121073\pi$
$17$	7.65685	1.85706	0.928530	+	0.371257i	$0.121073\pi$
$18$	0	0
$19$	−3.41421	−0.783274	−0.391637	−	0.920120i	$-0.628091\pi$
$19$	−3.41421	−0.783274	−0.391637	+	0.920120i	$0.628091\pi$
$20$	3.41421	0.763441
$21$	0	0
$22$	1.00000	0.213201
$23$	−2.24264	−0.467623	−0.233811	−	0.972282i	$-0.575120\pi$
$23$	−2.24264	−0.467623	−0.233811	+	0.972282i	$0.575120\pi$
$24$	0	0
$25$	6.65685	1.33137
$26$	−1.82843	−0.358584
$27$	0	0
$28$	0	0
$29$	8.65685	1.60754	0.803769	−	0.594942i	$-0.202825\pi$
$29$	8.65685	1.60754	0.803769	+	0.594942i	$0.202825\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−7.65685	−1.31314
$35$	0	0
$36$	0	0
$37$	−6.58579	−1.08270	−0.541348	−	0.840798i	$-0.682086\pi$
$37$	−6.58579	−1.08270	−0.541348	+	0.840798i	$0.682086\pi$
$38$	3.41421	0.553859
$39$	0	0
$40$	−3.41421	−0.539835
$41$	2.58579	0.403832	0.201916	−	0.979403i	$-0.435283\pi$
$41$	2.58579	0.403832	0.201916	+	0.979403i	$0.435283\pi$
$42$	0	0
$43$	5.65685	0.862662	0.431331	−	0.902194i	$-0.358044\pi$
$43$	5.65685	0.862662	0.431331	+	0.902194i	$0.358044\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	2.24264	0.330659
$47$	−6.48528	−0.945976	−0.472988	−	0.881069i	$-0.656825\pi$
$47$	−6.48528	−0.945976	−0.472988	+	0.881069i	$0.656825\pi$
$48$	0	0
$49$	0	0
$50$	−6.65685	−0.941421
$51$	0	0
$52$	1.82843	0.253557
$53$	11.8995	1.63452	0.817261	−	0.576268i	$-0.195492\pi$
$53$	11.8995	1.63452	0.817261	+	0.576268i	$0.195492\pi$
$54$	0	0
$55$	−3.41421	−0.460372
$56$	0	0
$57$	0	0
$58$	−8.65685	−1.13670
$59$	8.41421	1.09544	0.547719	−	0.836663i	$-0.315496\pi$
$59$	8.41421	1.09544	0.547719	+	0.836663i	$0.315496\pi$
$60$	0	0
$61$	6.17157	0.790189	0.395094	−	0.918640i	$-0.370712\pi$
$61$	6.17157	0.790189	0.395094	+	0.918640i	$0.370712\pi$
$62$	4.00000	0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	6.24264	0.774304
$66$	0	0
$67$	11.2426	1.37351	0.686754	−	0.726890i	$-0.259035\pi$
$67$	11.2426	1.37351	0.686754	+	0.726890i	$0.259035\pi$
$68$	7.65685	0.928530
$69$	0	0
$70$	0	0
$71$	−3.07107	−0.364469	−0.182234	−	0.983255i	$-0.558333\pi$
$71$	−3.07107	−0.364469	−0.182234	+	0.983255i	$0.558333\pi$
$72$	0	0
$73$	−6.58579	−0.770808	−0.385404	−	0.922748i	$-0.625938\pi$
$73$	−6.58579	−0.770808	−0.385404	+	0.922748i	$0.625938\pi$
$74$	6.58579	0.765582
$75$	0	0
$76$	−3.41421	−0.391637
$77$	0	0
$78$	0	0
$79$	−4.75736	−0.535245	−0.267622	−	0.963524i	$-0.586238\pi$
$79$	−4.75736	−0.535245	−0.267622	+	0.963524i	$0.586238\pi$
$80$	3.41421	0.381721
$81$	0	0
$82$	−2.58579	−0.285552
$83$	−16.1421	−1.77183	−0.885915	−	0.463848i	$-0.846468\pi$
$83$	−16.1421	−1.77183	−0.885915	+	0.463848i	$0.846468\pi$
$84$	0	0
$85$	26.1421	2.83551
$86$	−5.65685	−0.609994
$87$	0	0
$88$	1.00000	0.106600
$89$	4.48528	0.475439	0.237719	−	0.971334i	$-0.423600\pi$
$89$	4.48528	0.475439	0.237719	+	0.971334i	$0.423600\pi$
$90$	0	0
$91$	0	0
$92$	−2.24264	−0.233811
$93$	0	0
$94$	6.48528	0.668906
$95$	−11.6569	−1.19597
$96$	0	0
$97$	1.82843	0.185649	0.0928243	−	0.995683i	$-0.470411\pi$
$97$	1.82843	0.185649	0.0928243	+	0.995683i	$0.470411\pi$
$98$	0	0
$99$	0	0
$100$	6.65685	0.665685
$101$	−11.8284	−1.17697	−0.588486	−	0.808507i	$-0.700276\pi$
$101$	−11.8284	−1.17697	−0.588486	+	0.808507i	$0.700276\pi$
$102$	0	0
$103$	10.5858	1.04305	0.521524	−	0.853236i	$-0.325364\pi$
$103$	10.5858	1.04305	0.521524	+	0.853236i	$0.325364\pi$
$104$	−1.82843	−0.179292
$105$	0	0
$106$	−11.8995	−1.15578
$107$	−11.0711	−1.07028	−0.535140	−	0.844763i	$-0.679741\pi$
$107$	−11.0711	−1.07028	−0.535140	+	0.844763i	$0.679741\pi$
$108$	0	0
$109$	−0.485281	−0.0464815	−0.0232408	−	0.999730i	$-0.507398\pi$
$109$	−0.485281	−0.0464815	−0.0232408	+	0.999730i	$0.507398\pi$
$110$	3.41421	0.325532
$111$	0	0
$112$	0	0
$113$	13.8284	1.30087	0.650434	−	0.759562i	$-0.274587\pi$
$113$	13.8284	1.30087	0.650434	+	0.759562i	$0.274587\pi$
$114$	0	0
$115$	−7.65685	−0.714005
$116$	8.65685	0.803769
$117$	0	0
$118$	−8.41421	−0.774591
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	−6.17157	−0.558748
$123$	0	0
$124$	−4.00000	−0.359211
$125$	5.65685	0.505964
$126$	0	0
$127$	−9.72792	−0.863213	−0.431607	−	0.902062i	$-0.642053\pi$
$127$	−9.72792	−0.863213	−0.431607	+	0.902062i	$0.642053\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−6.24264	−0.547516
$131$	3.41421	0.298301	0.149151	−	0.988814i	$-0.452346\pi$
$131$	3.41421	0.298301	0.149151	+	0.988814i	$0.452346\pi$
$132$	0	0
$133$	0	0
$134$	−11.2426	−0.971216
$135$	0	0
$136$	−7.65685	−0.656570
$137$	5.34315	0.456496	0.228248	−	0.973603i	$-0.426700\pi$
$137$	5.34315	0.456496	0.228248	+	0.973603i	$0.426700\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	0	0
$142$	3.07107	0.257718
$143$	−1.82843	−0.152901
$144$	0	0
$145$	29.5563	2.45452
$146$	6.58579	0.545044
$147$	0	0
$148$	−6.58579	−0.541348
$149$	−6.34315	−0.519651	−0.259825	−	0.965656i	$-0.583665\pi$
$149$	−6.34315	−0.519651	−0.259825	+	0.965656i	$0.583665\pi$
$150$	0	0
$151$	9.72792	0.791647	0.395824	−	0.918327i	$-0.370459\pi$
$151$	9.72792	0.791647	0.395824	+	0.918327i	$0.370459\pi$
$152$	3.41421	0.276929
$153$	0	0
$154$	0	0
$155$	−13.6569	−1.09694
$156$	0	0
$157$	6.34315	0.506238	0.253119	−	0.967435i	$-0.418544\pi$
$157$	6.34315	0.506238	0.253119	+	0.967435i	$0.418544\pi$
$158$	4.75736	0.378475
$159$	0	0
$160$	−3.41421	−0.269917
$161$	0	0
$162$	0	0
$163$	−15.7279	−1.23191	−0.615953	−	0.787783i	$-0.711229\pi$
$163$	−15.7279	−1.23191	−0.615953	+	0.787783i	$0.711229\pi$
$164$	2.58579	0.201916
$165$	0	0
$166$	16.1421	1.25287
$167$	11.7279	0.907534	0.453767	−	0.891120i	$-0.350080\pi$
$167$	11.7279	0.907534	0.453767	+	0.891120i	$0.350080\pi$
$168$	0	0
$169$	−9.65685	−0.742835
$170$	−26.1421	−2.00501
$171$	0	0
$172$	5.65685	0.431331
$173$	4.17157	0.317159	0.158579	−	0.987346i	$-0.449309\pi$
$173$	4.17157	0.317159	0.158579	+	0.987346i	$0.449309\pi$
$174$	0	0
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	−4.48528	−0.336186
$179$	18.8995	1.41261	0.706307	−	0.707905i	$-0.250360\pi$
$179$	18.8995	1.41261	0.706307	+	0.707905i	$0.250360\pi$
$180$	0	0
$181$	3.65685	0.271812	0.135906	−	0.990722i	$-0.456606\pi$
$181$	3.65685	0.271812	0.135906	+	0.990722i	$0.456606\pi$
$182$	0	0
$183$	0	0
$184$	2.24264	0.165330
$185$	−22.4853	−1.65315
$186$	0	0
$187$	−7.65685	−0.559925
$188$	−6.48528	−0.472988
$189$	0	0
$190$	11.6569	0.845677
$191$	12.8284	0.928232	0.464116	−	0.885774i	$-0.346372\pi$
$191$	12.8284	0.928232	0.464116	+	0.885774i	$0.346372\pi$
$192$	0	0
$193$	2.10051	0.151198	0.0755988	−	0.997138i	$-0.475913\pi$
$193$	2.10051	0.151198	0.0755988	+	0.997138i	$0.475913\pi$
$194$	−1.82843	−0.131273
$195$	0	0
$196$	0	0
$197$	17.4853	1.24577	0.622887	−	0.782312i	$-0.285959\pi$
$197$	17.4853	1.24577	0.622887	+	0.782312i	$0.285959\pi$
$198$	0	0
$199$	19.8995	1.41064	0.705319	−	0.708890i	$-0.250804\pi$
$199$	19.8995	1.41064	0.705319	+	0.708890i	$0.250804\pi$
$200$	−6.65685	−0.470711
$201$	0	0
$202$	11.8284	0.832245
$203$	0	0
$204$	0	0
$205$	8.82843	0.616604
$206$	−10.5858	−0.737547
$207$	0	0
$208$	1.82843	0.126779
$209$	3.41421	0.236166
$210$	0	0
$211$	4.58579	0.315699	0.157849	−	0.987463i	$-0.449544\pi$
$211$	4.58579	0.315699	0.157849	+	0.987463i	$0.449544\pi$
$212$	11.8995	0.817261
$213$	0	0
$214$	11.0711	0.756803
$215$	19.3137	1.31718
$216$	0	0
$217$	0	0
$218$	0.485281	0.0328674
$219$	0	0
$220$	−3.41421	−0.230186
$221$	14.0000	0.941742
$222$	0	0
$223$	11.4142	0.764352	0.382176	−	0.924089i	$-0.375175\pi$
$223$	11.4142	0.764352	0.382176	+	0.924089i	$0.375175\pi$
$224$	0	0
$225$	0	0
$226$	−13.8284	−0.919853
$227$	23.1716	1.53795	0.768976	−	0.639278i	$-0.220767\pi$
$227$	23.1716	1.53795	0.768976	+	0.639278i	$0.220767\pi$
$228$	0	0
$229$	0.686292	0.0453514	0.0226757	−	0.999743i	$-0.492781\pi$
$229$	0.686292	0.0453514	0.0226757	+	0.999743i	$0.492781\pi$
$230$	7.65685	0.504878
$231$	0	0
$232$	−8.65685	−0.568350
$233$	1.41421	0.0926482	0.0463241	−	0.998926i	$-0.485249\pi$
$233$	1.41421	0.0926482	0.0463241	+	0.998926i	$0.485249\pi$
$234$	0	0
$235$	−22.1421	−1.44439
$236$	8.41421	0.547719
$237$	0	0
$238$	0	0
$239$	22.2132	1.43685	0.718426	−	0.695603i	$-0.244863\pi$
$239$	22.2132	1.43685	0.718426	+	0.695603i	$0.244863\pi$
$240$	0	0
$241$	3.75736	0.242033	0.121016	−	0.992651i	$-0.461385\pi$
$241$	3.75736	0.242033	0.121016	+	0.992651i	$0.461385\pi$
$242$	−1.00000	−0.0642824
$243$	0	0
$244$	6.17157	0.395094
$245$	0	0
$246$	0	0
$247$	−6.24264	−0.397210
$248$	4.00000	0.254000
$249$	0	0
$250$	−5.65685	−0.357771
$251$	−2.14214	−0.135210	−0.0676052	−	0.997712i	$-0.521536\pi$
$251$	−2.14214	−0.135210	−0.0676052	+	0.997712i	$0.521536\pi$
$252$	0	0
$253$	2.24264	0.140994
$254$	9.72792	0.610384
$255$	0	0
$256$	1.00000	0.0625000
$257$	−3.14214	−0.196001	−0.0980005	−	0.995186i	$-0.531245\pi$
$257$	−3.14214	−0.196001	−0.0980005	+	0.995186i	$0.531245\pi$
$258$	0	0
$259$	0	0
$260$	6.24264	0.387152
$261$	0	0
$262$	−3.41421	−0.210931
$263$	−31.0416	−1.91411	−0.957054	−	0.289908i	$-0.906375\pi$
$263$	−31.0416	−1.91411	−0.957054	+	0.289908i	$0.906375\pi$
$264$	0	0
$265$	40.6274	2.49572
$266$	0	0
$267$	0	0
$268$	11.2426	0.686754
$269$	−13.6569	−0.832673	−0.416337	−	0.909211i	$-0.636686\pi$
$269$	−13.6569	−0.832673	−0.416337	+	0.909211i	$0.636686\pi$
$270$	0	0
$271$	−26.5563	−1.61318	−0.806592	−	0.591109i	$-0.798690\pi$
$271$	−26.5563	−1.61318	−0.806592	+	0.591109i	$0.798690\pi$
$272$	7.65685	0.464265
$273$	0	0
$274$	−5.34315	−0.322791
$275$	−6.65685	−0.401423
$276$	0	0
$277$	−3.82843	−0.230028	−0.115014	−	0.993364i	$-0.536691\pi$
$277$	−3.82843	−0.230028	−0.115014	+	0.993364i	$0.536691\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	16.7279	0.997904	0.498952	−	0.866630i	$-0.333718\pi$
$281$	16.7279	0.997904	0.498952	+	0.866630i	$0.333718\pi$
$282$	0	0
$283$	−20.5858	−1.22370	−0.611849	−	0.790975i	$-0.709574\pi$
$283$	−20.5858	−1.22370	−0.611849	+	0.790975i	$0.709574\pi$
$284$	−3.07107	−0.182234
$285$	0	0
$286$	1.82843	0.108117
$287$	0	0
$288$	0	0
$289$	41.6274	2.44867
$290$	−29.5563	−1.73561
$291$	0	0
$292$	−6.58579	−0.385404
$293$	5.17157	0.302127	0.151063	−	0.988524i	$-0.451730\pi$
$293$	5.17157	0.302127	0.151063	+	0.988524i	$0.451730\pi$
$294$	0	0
$295$	28.7279	1.67260
$296$	6.58579	0.382791
$297$	0	0
$298$	6.34315	0.367449
$299$	−4.10051	−0.237138
$300$	0	0
$301$	0	0
$302$	−9.72792	−0.559779
$303$	0	0
$304$	−3.41421	−0.195819
$305$	21.0711	1.20653
$306$	0	0
$307$	−9.89949	−0.564994	−0.282497	−	0.959268i	$-0.591163\pi$
$307$	−9.89949	−0.564994	−0.282497	+	0.959268i	$0.591163\pi$
$308$	0	0
$309$	0	0
$310$	13.6569	0.775657
$311$	8.72792	0.494915	0.247458	−	0.968899i	$-0.420405\pi$
$311$	8.72792	0.494915	0.247458	+	0.968899i	$0.420405\pi$
$312$	0	0
$313$	−9.34315	−0.528106	−0.264053	−	0.964508i	$-0.585059\pi$
$313$	−9.34315	−0.528106	−0.264053	+	0.964508i	$0.585059\pi$
$314$	−6.34315	−0.357964
$315$	0	0
$316$	−4.75736	−0.267622
$317$	−31.3137	−1.75875	−0.879377	−	0.476127i	$-0.842040\pi$
$317$	−31.3137	−1.75875	−0.879377	+	0.476127i	$0.842040\pi$
$318$	0	0
$319$	−8.65685	−0.484691
$320$	3.41421	0.190860
$321$	0	0
$322$	0	0
$323$	−26.1421	−1.45459
$324$	0	0
$325$	12.1716	0.675157
$326$	15.7279	0.871089
$327$	0	0
$328$	−2.58579	−0.142776
$329$	0	0
$330$	0	0
$331$	−9.92893	−0.545743	−0.272872	−	0.962050i	$-0.587973\pi$
$331$	−9.92893	−0.545743	−0.272872	+	0.962050i	$0.587973\pi$
$332$	−16.1421	−0.885915
$333$	0	0
$334$	−11.7279	−0.641723
$335$	38.3848	2.09718
$336$	0	0
$337$	−19.7574	−1.07625	−0.538126	−	0.842864i	$-0.680868\pi$
$337$	−19.7574	−1.07625	−0.538126	+	0.842864i	$0.680868\pi$
$338$	9.65685	0.525264
$339$	0	0
$340$	26.1421	1.41776
$341$	4.00000	0.216612
$342$	0	0
$343$	0	0
$344$	−5.65685	−0.304997
$345$	0	0
$346$	−4.17157	−0.224265
$347$	−14.5858	−0.783006	−0.391503	−	0.920177i	$-0.628045\pi$
$347$	−14.5858	−0.783006	−0.391503	+	0.920177i	$0.628045\pi$
$348$	0	0
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	0	0
$352$	1.00000	0.0533002
$353$	25.3137	1.34731	0.673656	−	0.739045i	$-0.264723\pi$
$353$	25.3137	1.34731	0.673656	+	0.739045i	$0.264723\pi$
$354$	0	0
$355$	−10.4853	−0.556501
$356$	4.48528	0.237719
$357$	0	0
$358$	−18.8995	−0.998869
$359$	−10.7574	−0.567752	−0.283876	−	0.958861i	$-0.591620\pi$
$359$	−10.7574	−0.567752	−0.283876	+	0.958861i	$0.591620\pi$
$360$	0	0
$361$	−7.34315	−0.386481
$362$	−3.65685	−0.192200
$363$	0	0
$364$	0	0
$365$	−22.4853	−1.17693
$366$	0	0
$367$	14.7279	0.768791	0.384396	−	0.923168i	$-0.374410\pi$
$367$	14.7279	0.768791	0.384396	+	0.923168i	$0.374410\pi$
$368$	−2.24264	−0.116906
$369$	0	0
$370$	22.4853	1.16895
$371$	0	0
$372$	0	0
$373$	13.9706	0.723368	0.361684	−	0.932301i	$-0.382202\pi$
$373$	13.9706	0.723368	0.361684	+	0.932301i	$0.382202\pi$
$374$	7.65685	0.395927
$375$	0	0
$376$	6.48528	0.334453
$377$	15.8284	0.815205
$378$	0	0
$379$	−25.8701	−1.32886	−0.664428	−	0.747352i	$-0.731325\pi$
$379$	−25.8701	−1.32886	−0.664428	+	0.747352i	$0.731325\pi$
$380$	−11.6569	−0.597984
$381$	0	0
$382$	−12.8284	−0.656359
$383$	−30.3848	−1.55259	−0.776295	−	0.630370i	$-0.782903\pi$
$383$	−30.3848	−1.55259	−0.776295	+	0.630370i	$0.782903\pi$
$384$	0	0
$385$	0	0
$386$	−2.10051	−0.106913
$387$	0	0
$388$	1.82843	0.0928243
$389$	−2.72792	−0.138311	−0.0691556	−	0.997606i	$-0.522030\pi$
$389$	−2.72792	−0.138311	−0.0691556	+	0.997606i	$0.522030\pi$
$390$	0	0
$391$	−17.1716	−0.868404
$392$	0	0
$393$	0	0
$394$	−17.4853	−0.880896
$395$	−16.2426	−0.817256
$396$	0	0
$397$	22.0000	1.10415	0.552074	−	0.833795i	$-0.313837\pi$
$397$	22.0000	1.10415	0.552074	+	0.833795i	$0.313837\pi$
$398$	−19.8995	−0.997472
$399$	0	0
$400$	6.65685	0.332843
$401$	4.31371	0.215416	0.107708	−	0.994183i	$-0.465649\pi$
$401$	4.31371	0.215416	0.107708	+	0.994183i	$0.465649\pi$
$402$	0	0
$403$	−7.31371	−0.364322
$404$	−11.8284	−0.588486
$405$	0	0
$406$	0	0
$407$	6.58579	0.326445
$408$	0	0
$409$	−22.7279	−1.12382	−0.561912	−	0.827197i	$-0.689934\pi$
$409$	−22.7279	−1.12382	−0.561912	+	0.827197i	$0.689934\pi$
$410$	−8.82843	−0.436005
$411$	0	0
$412$	10.5858	0.521524
$413$	0	0
$414$	0	0
$415$	−55.1127	−2.70538
$416$	−1.82843	−0.0896460
$417$	0	0
$418$	−3.41421	−0.166995
$419$	−2.14214	−0.104650	−0.0523251	−	0.998630i	$-0.516663\pi$
$419$	−2.14214	−0.104650	−0.0523251	+	0.998630i	$0.516663\pi$
$420$	0	0
$421$	23.3137	1.13624	0.568120	−	0.822946i	$-0.307671\pi$
$421$	23.3137	1.13624	0.568120	+	0.822946i	$0.307671\pi$
$422$	−4.58579	−0.223233
$423$	0	0
$424$	−11.8995	−0.577891
$425$	50.9706	2.47244
$426$	0	0
$427$	0	0
$428$	−11.0711	−0.535140
$429$	0	0
$430$	−19.3137	−0.931390
$431$	20.4142	0.983318	0.491659	−	0.870788i	$-0.336391\pi$
$431$	20.4142	0.983318	0.491659	+	0.870788i	$0.336391\pi$
$432$	0	0
$433$	−2.14214	−0.102944	−0.0514722	−	0.998674i	$-0.516391\pi$
$433$	−2.14214	−0.102944	−0.0514722	+	0.998674i	$0.516391\pi$
$434$	0	0
$435$	0	0
$436$	−0.485281	−0.0232408
$437$	7.65685	0.366277
$438$	0	0
$439$	9.38478	0.447911	0.223955	−	0.974599i	$-0.428103\pi$
$439$	9.38478	0.447911	0.223955	+	0.974599i	$0.428103\pi$
$440$	3.41421	0.162766
$441$	0	0
$442$	−14.0000	−0.665912
$443$	32.6274	1.55018	0.775088	−	0.631854i	$-0.217706\pi$
$443$	32.6274	1.55018	0.775088	+	0.631854i	$0.217706\pi$
$444$	0	0
$445$	15.3137	0.725939
$446$	−11.4142	−0.540479
$447$	0	0
$448$	0	0
$449$	−33.6569	−1.58837	−0.794183	−	0.607679i	$-0.792101\pi$
$449$	−33.6569	−1.58837	−0.794183	+	0.607679i	$0.792101\pi$
$450$	0	0
$451$	−2.58579	−0.121760
$452$	13.8284	0.650434
$453$	0	0
$454$	−23.1716	−1.08750
$455$	0	0
$456$	0	0
$457$	−0.343146	−0.0160517	−0.00802584	−	0.999968i	$-0.502555\pi$
$457$	−0.343146	−0.0160517	−0.00802584	+	0.999968i	$0.502555\pi$
$458$	−0.686292	−0.0320683
$459$	0	0
$460$	−7.65685	−0.357003
$461$	−14.3137	−0.666656	−0.333328	−	0.942811i	$-0.608172\pi$
$461$	−14.3137	−0.666656	−0.333328	+	0.942811i	$0.608172\pi$
$462$	0	0
$463$	7.17157	0.333291	0.166646	−	0.986017i	$-0.446706\pi$
$463$	7.17157	0.333291	0.166646	+	0.986017i	$0.446706\pi$
$464$	8.65685	0.401884
$465$	0	0
$466$	−1.41421	−0.0655122
$467$	34.0000	1.57333	0.786666	−	0.617379i	$-0.211805\pi$
$467$	34.0000	1.57333	0.786666	+	0.617379i	$0.211805\pi$
$468$	0	0
$469$	0	0
$470$	22.1421	1.02134
$471$	0	0
$472$	−8.41421	−0.387296
$473$	−5.65685	−0.260102
$474$	0	0
$475$	−22.7279	−1.04283
$476$	0	0
$477$	0	0
$478$	−22.2132	−1.01601
$479$	26.0711	1.19122	0.595609	−	0.803275i	$-0.296911\pi$
$479$	26.0711	1.19122	0.595609	+	0.803275i	$0.296911\pi$
$480$	0	0
$481$	−12.0416	−0.549051
$482$	−3.75736	−0.171143
$483$	0	0
$484$	1.00000	0.0454545
$485$	6.24264	0.283464
$486$	0	0
$487$	1.65685	0.0750792	0.0375396	−	0.999295i	$-0.488048\pi$
$487$	1.65685	0.0750792	0.0375396	+	0.999295i	$0.488048\pi$
$488$	−6.17157	−0.279374
$489$	0	0
$490$	0	0
$491$	−24.8284	−1.12049	−0.560246	−	0.828327i	$-0.689293\pi$
$491$	−24.8284	−1.12049	−0.560246	+	0.828327i	$0.689293\pi$
$492$	0	0
$493$	66.2843	2.98529
$494$	6.24264	0.280870
$495$	0	0
$496$	−4.00000	−0.179605
$497$	0	0
$498$	0	0
$499$	−6.14214	−0.274960	−0.137480	−	0.990505i	$-0.543900\pi$
$499$	−6.14214	−0.274960	−0.137480	+	0.990505i	$0.543900\pi$
$500$	5.65685	0.252982
$501$	0	0
$502$	2.14214	0.0956082
$503$	38.2132	1.70384	0.851921	−	0.523670i	$-0.175437\pi$
$503$	38.2132	1.70384	0.851921	+	0.523670i	$0.175437\pi$
$504$	0	0
$505$	−40.3848	−1.79710
$506$	−2.24264	−0.0996975
$507$	0	0
$508$	−9.72792	−0.431607
$509$	13.3137	0.590120	0.295060	−	0.955479i	$-0.404660\pi$
$509$	13.3137	0.590120	0.295060	+	0.955479i	$0.404660\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	3.14214	0.138594
$515$	36.1421	1.59261
$516$	0	0
$517$	6.48528	0.285222
$518$	0	0
$519$	0	0
$520$	−6.24264	−0.273758
$521$	−28.2843	−1.23916	−0.619578	−	0.784935i	$-0.712696\pi$
$521$	−28.2843	−1.23916	−0.619578	+	0.784935i	$0.712696\pi$
$522$	0	0
$523$	12.7279	0.556553	0.278277	−	0.960501i	$-0.410237\pi$
$523$	12.7279	0.556553	0.278277	+	0.960501i	$0.410237\pi$
$524$	3.41421	0.149151
$525$	0	0
$526$	31.0416	1.35348
$527$	−30.6274	−1.33415
$528$	0	0
$529$	−17.9706	−0.781329
$530$	−40.6274	−1.76474
$531$	0	0
$532$	0	0
$533$	4.72792	0.204789
$534$	0	0
$535$	−37.7990	−1.63419
$536$	−11.2426	−0.485608
$537$	0	0
$538$	13.6569	0.588789
$539$	0	0
$540$	0	0
$541$	−41.1421	−1.76884	−0.884419	−	0.466693i	$-0.845445\pi$
$541$	−41.1421	−1.76884	−0.884419	+	0.466693i	$0.845445\pi$
$542$	26.5563	1.14069
$543$	0	0
$544$	−7.65685	−0.328285
$545$	−1.65685	−0.0709718
$546$	0	0
$547$	−18.8701	−0.806825	−0.403413	−	0.915018i	$-0.632176\pi$
$547$	−18.8701	−0.806825	−0.403413	+	0.915018i	$0.632176\pi$
$548$	5.34315	0.228248
$549$	0	0
$550$	6.65685	0.283849
$551$	−29.5563	−1.25914
$552$	0	0
$553$	0	0
$554$	3.82843	0.162654
$555$	0	0
$556$	0	0
$557$	−24.4853	−1.03747	−0.518737	−	0.854934i	$-0.673598\pi$
$557$	−24.4853	−1.03747	−0.518737	+	0.854934i	$0.673598\pi$
$558$	0	0
$559$	10.3431	0.437468
$560$	0	0
$561$	0	0
$562$	−16.7279	−0.705625
$563$	5.07107	0.213720	0.106860	−	0.994274i	$-0.465920\pi$
$563$	5.07107	0.213720	0.106860	+	0.994274i	$0.465920\pi$
$564$	0	0
$565$	47.2132	1.98627
$566$	20.5858	0.865285
$567$	0	0
$568$	3.07107	0.128859
$569$	4.00000	0.167689	0.0838444	−	0.996479i	$-0.473280\pi$
$569$	4.00000	0.167689	0.0838444	+	0.996479i	$0.473280\pi$
$570$	0	0
$571$	−10.3848	−0.434589	−0.217295	−	0.976106i	$-0.569723\pi$
$571$	−10.3848	−0.434589	−0.217295	+	0.976106i	$0.569723\pi$
$572$	−1.82843	−0.0764504
$573$	0	0
$574$	0	0
$575$	−14.9289	−0.622580
$576$	0	0
$577$	−32.3137	−1.34524	−0.672619	−	0.739989i	$-0.734831\pi$
$577$	−32.3137	−1.34524	−0.672619	+	0.739989i	$0.734831\pi$
$578$	−41.6274	−1.73147
$579$	0	0
$580$	29.5563	1.22726
$581$	0	0
$582$	0	0
$583$	−11.8995	−0.492827
$584$	6.58579	0.272522
$585$	0	0
$586$	−5.17157	−0.213636
$587$	44.8995	1.85320	0.926600	−	0.376048i	$-0.122717\pi$
$587$	44.8995	1.85320	0.926600	+	0.376048i	$0.122717\pi$
$588$	0	0
$589$	13.6569	0.562721
$590$	−28.7279	−1.18271
$591$	0	0
$592$	−6.58579	−0.270674
$593$	35.6985	1.46596	0.732981	−	0.680250i	$-0.238129\pi$
$593$	35.6985	1.46596	0.732981	+	0.680250i	$0.238129\pi$
$594$	0	0
$595$	0	0
$596$	−6.34315	−0.259825
$597$	0	0
$598$	4.10051	0.167682
$599$	2.62742	0.107353	0.0536767	−	0.998558i	$-0.482906\pi$
$599$	2.62742	0.107353	0.0536767	+	0.998558i	$0.482906\pi$
$600$	0	0
$601$	−35.9411	−1.46607	−0.733035	−	0.680191i	$-0.761897\pi$
$601$	−35.9411	−1.46607	−0.733035	+	0.680191i	$0.761897\pi$
$602$	0	0
$603$	0	0
$604$	9.72792	0.395824
$605$	3.41421	0.138808
$606$	0	0
$607$	−9.02944	−0.366494	−0.183247	−	0.983067i	$-0.558661\pi$
$607$	−9.02944	−0.366494	−0.183247	+	0.983067i	$0.558661\pi$
$608$	3.41421	0.138465
$609$	0	0
$610$	−21.0711	−0.853143
$611$	−11.8579	−0.479718
$612$	0	0
$613$	−16.6274	−0.671575	−0.335788	−	0.941938i	$-0.609002\pi$
$613$	−16.6274	−0.671575	−0.335788	+	0.941938i	$0.609002\pi$
$614$	9.89949	0.399511
$615$	0	0
$616$	0	0
$617$	8.02944	0.323253	0.161626	−	0.986852i	$-0.448326\pi$
$617$	8.02944	0.323253	0.161626	+	0.986852i	$0.448326\pi$
$618$	0	0
$619$	45.9411	1.84653	0.923265	−	0.384164i	$-0.125510\pi$
$619$	45.9411	1.84653	0.923265	+	0.384164i	$0.125510\pi$
$620$	−13.6569	−0.548472
$621$	0	0
$622$	−8.72792	−0.349958
$623$	0	0
$624$	0	0
$625$	−13.9706	−0.558823
$626$	9.34315	0.373427
$627$	0	0
$628$	6.34315	0.253119
$629$	−50.4264	−2.01063
$630$	0	0
$631$	48.7279	1.93983	0.969914	−	0.243448i	$-0.0782785\pi$
$631$	48.7279	1.93983	0.969914	+	0.243448i	$0.0782785\pi$
$632$	4.75736	0.189238
$633$	0	0
$634$	31.3137	1.24363
$635$	−33.2132	−1.31803
$636$	0	0
$637$	0	0
$638$	8.65685	0.342728
$639$	0	0
$640$	−3.41421	−0.134959
$641$	41.2843	1.63063	0.815315	−	0.579017i	$-0.196564\pi$
$641$	41.2843	1.63063	0.815315	+	0.579017i	$0.196564\pi$
$642$	0	0
$643$	4.41421	0.174080	0.0870398	−	0.996205i	$-0.472259\pi$
$643$	4.41421	0.174080	0.0870398	+	0.996205i	$0.472259\pi$
$644$	0	0
$645$	0	0
$646$	26.1421	1.02855
$647$	46.1838	1.81567	0.907836	−	0.419326i	$-0.137734\pi$
$647$	46.1838	1.81567	0.907836	+	0.419326i	$0.137734\pi$
$648$	0	0
$649$	−8.41421	−0.330287
$650$	−12.1716	−0.477408
$651$	0	0
$652$	−15.7279	−0.615953
$653$	−18.3848	−0.719452	−0.359726	−	0.933058i	$-0.617130\pi$
$653$	−18.3848	−0.719452	−0.359726	+	0.933058i	$0.617130\pi$
$654$	0	0
$655$	11.6569	0.455471
$656$	2.58579	0.100958
$657$	0	0
$658$	0	0
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	0	0
$661$	−14.9706	−0.582287	−0.291144	−	0.956679i	$-0.594036\pi$
$661$	−14.9706	−0.582287	−0.291144	+	0.956679i	$0.594036\pi$
$662$	9.92893	0.385899
$663$	0	0
$664$	16.1421	0.626436
$665$	0	0
$666$	0	0
$667$	−19.4142	−0.751721
$668$	11.7279	0.453767
$669$	0	0
$670$	−38.3848	−1.48293
$671$	−6.17157	−0.238251
$672$	0	0
$673$	−25.5563	−0.985125	−0.492562	−	0.870277i	$-0.663940\pi$
$673$	−25.5563	−0.985125	−0.492562	+	0.870277i	$0.663940\pi$
$674$	19.7574	0.761025
$675$	0	0
$676$	−9.65685	−0.371417
$677$	12.6863	0.487574	0.243787	−	0.969829i	$-0.421610\pi$
$677$	12.6863	0.487574	0.243787	+	0.969829i	$0.421610\pi$
$678$	0	0
$679$	0	0
$680$	−26.1421	−1.00251
$681$	0	0
$682$	−4.00000	−0.153168
$683$	−16.4142	−0.628072	−0.314036	−	0.949411i	$-0.601681\pi$
$683$	−16.4142	−0.628072	−0.314036	+	0.949411i	$0.601681\pi$
$684$	0	0
$685$	18.2426	0.697015
$686$	0	0
$687$	0	0
$688$	5.65685	0.215666
$689$	21.7574	0.828889
$690$	0	0
$691$	13.9289	0.529882	0.264941	−	0.964265i	$-0.414648\pi$
$691$	13.9289	0.529882	0.264941	+	0.964265i	$0.414648\pi$
$692$	4.17157	0.158579
$693$	0	0
$694$	14.5858	0.553669
$695$	0	0
$696$	0	0
$697$	19.7990	0.749940
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−36.1127	−1.36396	−0.681979	−	0.731372i	$-0.738880\pi$
$701$	−36.1127	−1.36396	−0.681979	+	0.731372i	$0.738880\pi$
$702$	0	0
$703$	22.4853	0.848048
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−25.3137	−0.952694
$707$	0	0
$708$	0	0
$709$	36.0416	1.35357	0.676786	−	0.736180i	$-0.263372\pi$
$709$	36.0416	1.35357	0.676786	+	0.736180i	$0.263372\pi$
$710$	10.4853	0.393506
$711$	0	0
$712$	−4.48528	−0.168093
$713$	8.97056	0.335950
$714$	0	0
$715$	−6.24264	−0.233462
$716$	18.8995	0.706307
$717$	0	0
$718$	10.7574	0.401461
$719$	18.4853	0.689385	0.344692	−	0.938716i	$-0.387983\pi$
$719$	18.4853	0.689385	0.344692	+	0.938716i	$0.387983\pi$
$720$	0	0
$721$	0	0
$722$	7.34315	0.273284
$723$	0	0
$724$	3.65685	0.135906
$725$	57.6274	2.14023
$726$	0	0
$727$	−48.4264	−1.79604	−0.898018	−	0.439959i	$-0.854993\pi$
$727$	−48.4264	−1.79604	−0.898018	+	0.439959i	$0.854993\pi$
$728$	0	0
$729$	0	0
$730$	22.4853	0.832218
$731$	43.3137	1.60202
$732$	0	0
$733$	−13.0000	−0.480166	−0.240083	−	0.970752i	$-0.577175\pi$
$733$	−13.0000	−0.480166	−0.240083	+	0.970752i	$0.577175\pi$
$734$	−14.7279	−0.543618
$735$	0	0
$736$	2.24264	0.0826648
$737$	−11.2426	−0.414128
$738$	0	0
$739$	32.4264	1.19282	0.596412	−	0.802678i	$-0.296592\pi$
$739$	32.4264	1.19282	0.596412	+	0.802678i	$0.296592\pi$
$740$	−22.4853	−0.826575
$741$	0	0
$742$	0	0
$743$	−9.31371	−0.341687	−0.170843	−	0.985298i	$-0.554649\pi$
$743$	−9.31371	−0.341687	−0.170843	+	0.985298i	$0.554649\pi$
$744$	0	0
$745$	−21.6569	−0.793446
$746$	−13.9706	−0.511499
$747$	0	0
$748$	−7.65685	−0.279962
$749$	0	0
$750$	0	0
$751$	34.3431	1.25320	0.626600	−	0.779341i	$-0.284446\pi$
$751$	34.3431	1.25320	0.626600	+	0.779341i	$0.284446\pi$
$752$	−6.48528	−0.236494
$753$	0	0
$754$	−15.8284	−0.576437
$755$	33.2132	1.20875
$756$	0	0
$757$	19.6569	0.714441	0.357220	−	0.934020i	$-0.383725\pi$
$757$	19.6569	0.714441	0.357220	+	0.934020i	$0.383725\pi$
$758$	25.8701	0.939643
$759$	0	0
$760$	11.6569	0.422839
$761$	2.97056	0.107683	0.0538414	−	0.998549i	$-0.482853\pi$
$761$	2.97056	0.107683	0.0538414	+	0.998549i	$0.482853\pi$
$762$	0	0
$763$	0	0
$764$	12.8284	0.464116
$765$	0	0
$766$	30.3848	1.09785
$767$	15.3848	0.555512
$768$	0	0
$769$	−10.9706	−0.395609	−0.197804	−	0.980242i	$-0.563381\pi$
$769$	−10.9706	−0.395609	−0.197804	+	0.980242i	$0.563381\pi$
$770$	0	0
$771$	0	0
$772$	2.10051	0.0755988
$773$	−45.9411	−1.65239	−0.826194	−	0.563386i	$-0.809498\pi$
$773$	−45.9411	−1.65239	−0.826194	+	0.563386i	$0.809498\pi$
$774$	0	0
$775$	−26.6274	−0.956485
$776$	−1.82843	−0.0656367
$777$	0	0
$778$	2.72792	0.0978007
$779$	−8.82843	−0.316311
$780$	0	0
$781$	3.07107	0.109891
$782$	17.1716	0.614054
$783$	0	0
$784$	0	0
$785$	21.6569	0.772966
$786$	0	0
$787$	−17.5563	−0.625816	−0.312908	−	0.949783i	$-0.601303\pi$
$787$	−17.5563	−0.625816	−0.312908	+	0.949783i	$0.601303\pi$
$788$	17.4853	0.622887
$789$	0	0
$790$	16.2426	0.577887
$791$	0	0
$792$	0	0
$793$	11.2843	0.400716
$794$	−22.0000	−0.780751
$795$	0	0
$796$	19.8995	0.705319
$797$	−3.65685	−0.129532	−0.0647662	−	0.997900i	$-0.520630\pi$
$797$	−3.65685	−0.129532	−0.0647662	+	0.997900i	$0.520630\pi$
$798$	0	0
$799$	−49.6569	−1.75673
$800$	−6.65685	−0.235355
$801$	0	0
$802$	−4.31371	−0.152322
$803$	6.58579	0.232407
$804$	0	0
$805$	0	0
$806$	7.31371	0.257614
$807$	0	0
$808$	11.8284	0.416123
$809$	1.37258	0.0482574	0.0241287	−	0.999709i	$-0.492319\pi$
$809$	1.37258	0.0482574	0.0241287	+	0.999709i	$0.492319\pi$
$810$	0	0
$811$	32.2843	1.13365	0.566827	−	0.823837i	$-0.308171\pi$
$811$	32.2843	1.13365	0.566827	+	0.823837i	$0.308171\pi$
$812$	0	0
$813$	0	0
$814$	−6.58579	−0.230832
$815$	−53.6985	−1.88098
$816$	0	0
$817$	−19.3137	−0.675701
$818$	22.7279	0.794663
$819$	0	0
$820$	8.82843	0.308302
$821$	12.5147	0.436767	0.218383	−	0.975863i	$-0.429922\pi$
$821$	12.5147	0.436767	0.218383	+	0.975863i	$0.429922\pi$
$822$	0	0
$823$	7.45584	0.259894	0.129947	−	0.991521i	$-0.458519\pi$
$823$	7.45584	0.259894	0.129947	+	0.991521i	$0.458519\pi$
$824$	−10.5858	−0.368773
$825$	0	0
$826$	0	0
$827$	−49.3553	−1.71625	−0.858127	−	0.513438i	$-0.828372\pi$
$827$	−49.3553	−1.71625	−0.858127	+	0.513438i	$0.828372\pi$
$828$	0	0
$829$	30.2426	1.05037	0.525185	−	0.850988i	$-0.323996\pi$
$829$	30.2426	1.05037	0.525185	+	0.850988i	$0.323996\pi$
$830$	55.1127	1.91299
$831$	0	0
$832$	1.82843	0.0633893
$833$	0	0
$834$	0	0
$835$	40.0416	1.38570
$836$	3.41421	0.118083
$837$	0	0
$838$	2.14214	0.0739988
$839$	1.51472	0.0522939	0.0261469	−	0.999658i	$-0.491676\pi$
$839$	1.51472	0.0522939	0.0261469	+	0.999658i	$0.491676\pi$
$840$	0	0
$841$	45.9411	1.58418
$842$	−23.3137	−0.803443
$843$	0	0
$844$	4.58579	0.157849
$845$	−32.9706	−1.13422
$846$	0	0
$847$	0	0
$848$	11.8995	0.408630
$849$	0	0
$850$	−50.9706	−1.74828
$851$	14.7696	0.506294
$852$	0	0
$853$	14.0000	0.479351	0.239675	−	0.970853i	$-0.422959\pi$
$853$	14.0000	0.479351	0.239675	+	0.970853i	$0.422959\pi$
$854$	0	0
$855$	0	0
$856$	11.0711	0.378401
$857$	−10.6274	−0.363026	−0.181513	−	0.983389i	$-0.558099\pi$
$857$	−10.6274	−0.363026	−0.181513	+	0.983389i	$0.558099\pi$
$858$	0	0
$859$	19.7279	0.673108	0.336554	−	0.941664i	$-0.390739\pi$
$859$	19.7279	0.673108	0.336554	+	0.941664i	$0.390739\pi$
$860$	19.3137	0.658592
$861$	0	0
$862$	−20.4142	−0.695311
$863$	7.21320	0.245540	0.122770	−	0.992435i	$-0.460822\pi$
$863$	7.21320	0.245540	0.122770	+	0.992435i	$0.460822\pi$
$864$	0	0
$865$	14.2426	0.484264
$866$	2.14214	0.0727927
$867$	0	0
$868$	0	0
$869$	4.75736	0.161382
$870$	0	0
$871$	20.5563	0.696525
$872$	0.485281	0.0164337
$873$	0	0
$874$	−7.65685	−0.258997
$875$	0	0
$876$	0	0
$877$	−25.6274	−0.865376	−0.432688	−	0.901544i	$-0.642435\pi$
$877$	−25.6274	−0.865376	−0.432688	+	0.901544i	$0.642435\pi$
$878$	−9.38478	−0.316721
$879$	0	0
$880$	−3.41421	−0.115093
$881$	44.4558	1.49776	0.748878	−	0.662708i	$-0.230593\pi$
$881$	44.4558	1.49776	0.748878	+	0.662708i	$0.230593\pi$
$882$	0	0
$883$	−9.72792	−0.327371	−0.163685	−	0.986513i	$-0.552338\pi$
$883$	−9.72792	−0.327371	−0.163685	+	0.986513i	$0.552338\pi$
$884$	14.0000	0.470871
$885$	0	0
$886$	−32.6274	−1.09614
$887$	22.8995	0.768890	0.384445	−	0.923148i	$-0.374393\pi$
$887$	22.8995	0.768890	0.384445	+	0.923148i	$0.374393\pi$
$888$	0	0
$889$	0	0
$890$	−15.3137	−0.513317
$891$	0	0
$892$	11.4142	0.382176
$893$	22.1421	0.740958
$894$	0	0
$895$	64.5269	2.15690
$896$	0	0
$897$	0	0
$898$	33.6569	1.12314
$899$	−34.6274	−1.15489
$900$	0	0
$901$	91.1127	3.03540
$902$	2.58579	0.0860973
$903$	0	0
$904$	−13.8284	−0.459927
$905$	12.4853	0.415025
$906$	0	0
$907$	33.3137	1.10616	0.553082	−	0.833127i	$-0.313452\pi$
$907$	33.3137	1.10616	0.553082	+	0.833127i	$0.313452\pi$
$908$	23.1716	0.768976
$909$	0	0
$910$	0	0
$911$	13.5147	0.447763	0.223881	−	0.974616i	$-0.428127\pi$
$911$	13.5147	0.447763	0.223881	+	0.974616i	$0.428127\pi$
$912$	0	0
$913$	16.1421	0.534227
$914$	0.343146	0.0113503
$915$	0	0
$916$	0.686292	0.0226757
$917$	0	0
$918$	0	0
$919$	−6.14214	−0.202610	−0.101305	−	0.994855i	$-0.532302\pi$
$919$	−6.14214	−0.202610	−0.101305	+	0.994855i	$0.532302\pi$
$920$	7.65685	0.252439
$921$	0	0
$922$	14.3137	0.471397
$923$	−5.61522	−0.184827
$924$	0	0
$925$	−43.8406	−1.44147
$926$	−7.17157	−0.235673
$927$	0	0
$928$	−8.65685	−0.284175
$929$	−10.4558	−0.343045	−0.171523	−	0.985180i	$-0.554869\pi$
$929$	−10.4558	−0.343045	−0.171523	+	0.985180i	$0.554869\pi$
$930$	0	0
$931$	0	0
$932$	1.41421	0.0463241
$933$	0	0
$934$	−34.0000	−1.11251
$935$	−26.1421	−0.854939
$936$	0	0
$937$	−16.5858	−0.541834	−0.270917	−	0.962603i	$-0.587327\pi$
$937$	−16.5858	−0.541834	−0.270917	+	0.962603i	$0.587327\pi$
$938$	0	0
$939$	0	0
$940$	−22.1421	−0.722197
$941$	−13.3431	−0.434974	−0.217487	−	0.976063i	$-0.569786\pi$
$941$	−13.3431	−0.434974	−0.217487	+	0.976063i	$0.569786\pi$
$942$	0	0
$943$	−5.79899	−0.188841
$944$	8.41421	0.273859
$945$	0	0
$946$	5.65685	0.183920
$947$	−29.1716	−0.947949	−0.473974	−	0.880539i	$-0.657181\pi$
$947$	−29.1716	−0.947949	−0.473974	+	0.880539i	$0.657181\pi$
$948$	0	0
$949$	−12.0416	−0.390888
$950$	22.7279	0.737391
$951$	0	0
$952$	0	0
$953$	25.5563	0.827851	0.413926	−	0.910311i	$-0.364157\pi$
$953$	25.5563	0.827851	0.413926	+	0.910311i	$0.364157\pi$
$954$	0	0
$955$	43.7990	1.41730
$956$	22.2132	0.718426
$957$	0	0
$958$	−26.0711	−0.842318
$959$	0	0
$960$	0	0
$961$	−15.0000	−0.483871
$962$	12.0416	0.388238
$963$	0	0
$964$	3.75736	0.121016
$965$	7.17157	0.230861
$966$	0	0
$967$	20.2843	0.652298	0.326149	−	0.945318i	$-0.394249\pi$
$967$	20.2843	0.652298	0.326149	+	0.945318i	$0.394249\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	−6.24264	−0.200439
$971$	−47.7279	−1.53166	−0.765831	−	0.643042i	$-0.777672\pi$
$971$	−47.7279	−1.53166	−0.765831	+	0.643042i	$0.777672\pi$
$972$	0	0
$973$	0	0
$974$	−1.65685	−0.0530890
$975$	0	0
$976$	6.17157	0.197547
$977$	−40.0000	−1.27971	−0.639857	−	0.768494i	$-0.721006\pi$
$977$	−40.0000	−1.27971	−0.639857	+	0.768494i	$0.721006\pi$
$978$	0	0
$979$	−4.48528	−0.143350
$980$	0	0
$981$	0	0
$982$	24.8284	0.792307
$983$	52.4264	1.67214	0.836071	−	0.548621i	$-0.184847\pi$
$983$	52.4264	1.67214	0.836071	+	0.548621i	$0.184847\pi$
$984$	0	0
$985$	59.6985	1.90215
$986$	−66.2843	−2.11092
$987$	0	0
$988$	−6.24264	−0.198605
$989$	−12.6863	−0.403401
$990$	0	0
$991$	−38.3848	−1.21933	−0.609666	−	0.792658i	$-0.708697\pi$
$991$	−38.3848	−1.21933	−0.609666	+	0.792658i	$0.708697\pi$
$992$	4.00000	0.127000
$993$	0	0
$994$	0	0
$995$	67.9411	2.15388
$996$	0	0
$997$	18.8284	0.596302	0.298151	−	0.954519i	$-0.403630\pi$
$997$	18.8284	0.596302	0.298151	+	0.954519i	$0.403630\pi$
$998$	6.14214	0.194426
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9702.2.a.cx.1.2		2
3.2	odd	2		1078.2.a.t.1.2		2
7.2	even	3		1386.2.k.t.991.1		4
7.4	even	3		1386.2.k.t.793.1		4
7.6	odd	2		9702.2.a.ch.1.1		2
12.11	even	2		8624.2.a.cc.1.1		2
21.2	odd	6		154.2.e.e.67.1	yes	4
21.5	even	6		1078.2.e.m.67.2		4
21.11	odd	6		154.2.e.e.23.1	✓	4
21.17	even	6		1078.2.e.m.177.2		4
21.20	even	2		1078.2.a.x.1.1		2
84.11	even	6		1232.2.q.f.177.2		4
84.23	even	6		1232.2.q.f.529.2		4
84.83	odd	2		8624.2.a.bh.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
154.2.e.e.23.1	✓	4	21.11	odd	6
154.2.e.e.67.1	yes	4	21.2	odd	6
1078.2.a.t.1.2		2	3.2	odd	2
1078.2.a.x.1.1		2	21.20	even	2
1078.2.e.m.67.2		4	21.5	even	6
1078.2.e.m.177.2		4	21.17	even	6
1232.2.q.f.177.2		4	84.11	even	6
1232.2.q.f.529.2		4	84.23	even	6
1386.2.k.t.793.1		4	7.4	even	3
1386.2.k.t.991.1		4	7.2	even	3
8624.2.a.bh.1.2		2	84.83	odd	2
8624.2.a.cc.1.1		2	12.11	even	2
9702.2.a.ch.1.1		2	7.6	odd	2
9702.2.a.cx.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9702.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +3.41421 q^{5} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +3.41421 q^{5} -1.00000 q^{8} -3.41421 q^{10} -1.00000 q^{11} +1.82843 q^{13} +1.00000 q^{16} +7.65685 q^{17} -3.41421 q^{19} +3.41421 q^{20} +1.00000 q^{22} -2.24264 q^{23} +6.65685 q^{25} -1.82843 q^{26} +8.65685 q^{29} -4.00000 q^{31} -1.00000 q^{32} -7.65685 q^{34} -6.58579 q^{37} +3.41421 q^{38} -3.41421 q^{40} +2.58579 q^{41} +5.65685 q^{43} -1.00000 q^{44} +2.24264 q^{46} -6.48528 q^{47} -6.65685 q^{50} +1.82843 q^{52} +11.8995 q^{53} -3.41421 q^{55} -8.65685 q^{58} +8.41421 q^{59} +6.17157 q^{61} +4.00000 q^{62} +1.00000 q^{64} +6.24264 q^{65} +11.2426 q^{67} +7.65685 q^{68} -3.07107 q^{71} -6.58579 q^{73} +6.58579 q^{74} -3.41421 q^{76} -4.75736 q^{79} +3.41421 q^{80} -2.58579 q^{82} -16.1421 q^{83} +26.1421 q^{85} -5.65685 q^{86} +1.00000 q^{88} +4.48528 q^{89} -2.24264 q^{92} +6.48528 q^{94} -11.6569 q^{95} +1.82843 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 4 q^{5} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 + 4 * q^5 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} + 4 q^{5} - 2 q^{8} - 4 q^{10} - 2 q^{11} - 2 q^{13} + 2 q^{16} + 4 q^{17} - 4 q^{19} + 4 q^{20} + 2 q^{22} + 4 q^{23} + 2 q^{25} + 2 q^{26} + 6 q^{29} - 8 q^{31} - 2 q^{32} - 4 q^{34} - 16 q^{37} + 4 q^{38} - 4 q^{40} + 8 q^{41} - 2 q^{44} - 4 q^{46} + 4 q^{47} - 2 q^{50} - 2 q^{52} + 4 q^{53} - 4 q^{55} - 6 q^{58} + 14 q^{59} + 18 q^{61} + 8 q^{62} + 2 q^{64} + 4 q^{65} + 14 q^{67} + 4 q^{68} + 8 q^{71} - 16 q^{73} + 16 q^{74} - 4 q^{76} - 18 q^{79} + 4 q^{80} - 8 q^{82} - 4 q^{83} + 24 q^{85} + 2 q^{88} - 8 q^{89} + 4 q^{92} - 4 q^{94} - 12 q^{95} - 2 q^{97}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + 4 * q^5 - 2 * q^8 - 4 * q^10 - 2 * q^11 - 2 * q^13 + 2 * q^16 + 4 * q^17 - 4 * q^19 + 4 * q^20 + 2 * q^22 + 4 * q^23 + 2 * q^25 + 2 * q^26 + 6 * q^29 - 8 * q^31 - 2 * q^32 - 4 * q^34 - 16 * q^37 + 4 * q^38 - 4 * q^40 + 8 * q^41 - 2 * q^44 - 4 * q^46 + 4 * q^47 - 2 * q^50 - 2 * q^52 + 4 * q^53 - 4 * q^55 - 6 * q^58 + 14 * q^59 + 18 * q^61 + 8 * q^62 + 2 * q^64 + 4 * q^65 + 14 * q^67 + 4 * q^68 + 8 * q^71 - 16 * q^73 + 16 * q^74 - 4 * q^76 - 18 * q^79 + 4 * q^80 - 8 * q^82 - 4 * q^83 + 24 * q^85 + 2 * q^88 - 8 * q^89 + 4 * q^92 - 4 * q^94 - 12 * q^95 - 2 * q^97

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