LMFDB - Embedded newform 7623.2.a.bn.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7623,2,Mod(1,7623)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7623, 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("7623.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7623 = 3^{2} \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7623.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:	$60.8699614608$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2541)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	7623.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.61803 q^{2} +0.618034 q^{4} +2.23607 q^{5} -1.00000 q^{7} -2.23607 q^{8} +O(q^{10})$	$q+1.61803 q^{2} +0.618034 q^{4} +2.23607 q^{5} -1.00000 q^{7} -2.23607 q^{8} +3.61803 q^{10} -0.236068 q^{13} -1.61803 q^{14} -4.85410 q^{16} +2.00000 q^{17} +1.47214 q^{19} +1.38197 q^{20} +2.00000 q^{23} -0.381966 q^{26} -0.618034 q^{28} +5.00000 q^{29} +10.4721 q^{31} -3.38197 q^{32} +3.23607 q^{34} -2.23607 q^{35} -1.47214 q^{37} +2.38197 q^{38} -5.00000 q^{40} +2.47214 q^{41} +8.47214 q^{43} +3.23607 q^{46} -9.47214 q^{47} +1.00000 q^{49} -0.145898 q^{52} -6.47214 q^{53} +2.23607 q^{56} +8.09017 q^{58} -7.94427 q^{59} -12.4721 q^{61} +16.9443 q^{62} +4.23607 q^{64} -0.527864 q^{65} +9.18034 q^{67} +1.23607 q^{68} -3.61803 q^{70} +2.47214 q^{71} +10.7082 q^{73} -2.38197 q^{74} +0.909830 q^{76} +0.472136 q^{79} -10.8541 q^{80} +4.00000 q^{82} +7.52786 q^{83} +4.47214 q^{85} +13.7082 q^{86} +0.472136 q^{89} +0.236068 q^{91} +1.23607 q^{92} -15.3262 q^{94} +3.29180 q^{95} +9.41641 q^{97} +1.61803 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - q^{4} - 2 q^{7}+O(q^{10}) $ 2 * q + q^2 - q^4 - 2 * q^7	$ 2 q + q^{2} - q^{4} - 2 q^{7} + 5 q^{10} + 4 q^{13} - q^{14} - 3 q^{16} + 4 q^{17} - 6 q^{19} + 5 q^{20} + 4 q^{23} - 3 q^{26} + q^{28} + 10 q^{29} + 12 q^{31} - 9 q^{32} + 2 q^{34} + 6 q^{37} + 7 q^{38} - 10 q^{40} - 4 q^{41} + 8 q^{43} + 2 q^{46} - 10 q^{47} + 2 q^{49} - 7 q^{52} - 4 q^{53} + 5 q^{58} + 2 q^{59} - 16 q^{61} + 16 q^{62} + 4 q^{64} - 10 q^{65} - 4 q^{67} - 2 q^{68} - 5 q^{70} - 4 q^{71} + 8 q^{73} - 7 q^{74} + 13 q^{76} - 8 q^{79} - 15 q^{80} + 8 q^{82} + 24 q^{83} + 14 q^{86} - 8 q^{89} - 4 q^{91} - 2 q^{92} - 15 q^{94} + 20 q^{95} - 8 q^{97} + q^{98}+O(q^{100}) $ 2 * q + q^2 - q^4 - 2 * q^7 + 5 * q^10 + 4 * q^13 - q^14 - 3 * q^16 + 4 * q^17 - 6 * q^19 + 5 * q^20 + 4 * q^23 - 3 * q^26 + q^28 + 10 * q^29 + 12 * q^31 - 9 * q^32 + 2 * q^34 + 6 * q^37 + 7 * q^38 - 10 * q^40 - 4 * q^41 + 8 * q^43 + 2 * q^46 - 10 * q^47 + 2 * q^49 - 7 * q^52 - 4 * q^53 + 5 * q^58 + 2 * q^59 - 16 * q^61 + 16 * q^62 + 4 * q^64 - 10 * q^65 - 4 * q^67 - 2 * q^68 - 5 * q^70 - 4 * q^71 + 8 * q^73 - 7 * q^74 + 13 * q^76 - 8 * q^79 - 15 * q^80 + 8 * q^82 + 24 * q^83 + 14 * q^86 - 8 * q^89 - 4 * q^91 - 2 * q^92 - 15 * q^94 + 20 * q^95 - 8 * q^97 + q^98

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 $

$2$	1.61803	1.14412	0.572061	−	0.820211i	$-0.306144\pi$
$2$	1.61803	1.14412	0.572061	+	0.820211i	$0.306144\pi$
$3$	0	0
$3$	0	0
$4$	0.618034	0.309017
$5$	2.23607	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$5$	2.23607	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−2.23607	−0.790569
$9$	0	0
$10$	3.61803	1.14412
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−0.236068	−0.0654735	−0.0327367	−	0.999464i	$-0.510422\pi$
$13$	−0.236068	−0.0654735	−0.0327367	+	0.999464i	$0.510422\pi$
$14$	−1.61803	−0.432438
$15$	0	0
$16$	−4.85410	−1.21353
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	1.47214	0.337731	0.168866	−	0.985639i	$-0.445990\pi$
$19$	1.47214	0.337731	0.168866	+	0.985639i	$0.445990\pi$
$20$	1.38197	0.309017
$21$	0	0
$22$	0	0
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	0	0
$25$	0	0
$26$	−0.381966	−0.0749097
$27$	0	0
$28$	−0.618034	−0.116797
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	0	0
$31$	10.4721	1.88085	0.940426	−	0.340000i	$-0.110427\pi$
$31$	10.4721	1.88085	0.940426	+	0.340000i	$0.110427\pi$
$32$	−3.38197	−0.597853
$33$	0	0
$34$	3.23607	0.554981
$35$	−2.23607	−0.377964
$36$	0	0
$37$	−1.47214	−0.242018	−0.121009	−	0.992651i	$-0.538613\pi$
$37$	−1.47214	−0.242018	−0.121009	+	0.992651i	$0.538613\pi$
$38$	2.38197	0.386406
$39$	0	0
$40$	−5.00000	−0.790569
$41$	2.47214	0.386083	0.193041	−	0.981191i	$-0.438165\pi$
$41$	2.47214	0.386083	0.193041	+	0.981191i	$0.438165\pi$
$42$	0	0
$43$	8.47214	1.29199	0.645994	−	0.763342i	$-0.276443\pi$
$43$	8.47214	1.29199	0.645994	+	0.763342i	$0.276443\pi$
$44$	0	0
$45$	0	0
$46$	3.23607	0.477132
$47$	−9.47214	−1.38165	−0.690827	−	0.723021i	$-0.742753\pi$
$47$	−9.47214	−1.38165	−0.690827	+	0.723021i	$0.742753\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	−0.145898	−0.0202324
$53$	−6.47214	−0.889016	−0.444508	−	0.895775i	$-0.646622\pi$
$53$	−6.47214	−0.889016	−0.444508	+	0.895775i	$0.646622\pi$
$54$	0	0
$55$	0	0
$56$	2.23607	0.298807
$57$	0	0
$58$	8.09017	1.06229
$59$	−7.94427	−1.03426	−0.517128	−	0.855908i	$-0.672999\pi$
$59$	−7.94427	−1.03426	−0.517128	+	0.855908i	$0.672999\pi$
$60$	0	0
$61$	−12.4721	−1.59689	−0.798447	−	0.602066i	$-0.794345\pi$
$61$	−12.4721	−1.59689	−0.798447	+	0.602066i	$0.794345\pi$
$62$	16.9443	2.15192
$63$	0	0
$64$	4.23607	0.529508
$65$	−0.527864	−0.0654735
$66$	0	0
$67$	9.18034	1.12156	0.560779	−	0.827966i	$-0.310502\pi$
$67$	9.18034	1.12156	0.560779	+	0.827966i	$0.310502\pi$
$68$	1.23607	0.149895
$69$	0	0
$70$	−3.61803	−0.432438
$71$	2.47214	0.293389	0.146694	−	0.989182i	$-0.453137\pi$
$71$	2.47214	0.293389	0.146694	+	0.989182i	$0.453137\pi$
$72$	0	0
$73$	10.7082	1.25330	0.626650	−	0.779301i	$-0.284425\pi$
$73$	10.7082	1.25330	0.626650	+	0.779301i	$0.284425\pi$
$74$	−2.38197	−0.276898
$75$	0	0
$76$	0.909830	0.104365
$77$	0	0
$78$	0	0
$79$	0.472136	0.0531194	0.0265597	−	0.999647i	$-0.491545\pi$
$79$	0.472136	0.0531194	0.0265597	+	0.999647i	$0.491545\pi$
$80$	−10.8541	−1.21353
$81$	0	0
$82$	4.00000	0.441726
$83$	7.52786	0.826290	0.413145	−	0.910665i	$-0.364430\pi$
$83$	7.52786	0.826290	0.413145	+	0.910665i	$0.364430\pi$
$84$	0	0
$85$	4.47214	0.485071
$86$	13.7082	1.47819
$87$	0	0
$88$	0	0
$89$	0.472136	0.0500463	0.0250232	−	0.999687i	$-0.492034\pi$
$89$	0.472136	0.0500463	0.0250232	+	0.999687i	$0.492034\pi$
$90$	0	0
$91$	0.236068	0.0247466
$92$	1.23607	0.128869
$93$	0	0
$94$	−15.3262	−1.58078
$95$	3.29180	0.337731
$96$	0	0
$97$	9.41641	0.956091	0.478046	−	0.878335i	$-0.341345\pi$
$97$	9.41641	0.956091	0.478046	+	0.878335i	$0.341345\pi$
$98$	1.61803	0.163446
$99$	0	0
$100$	0	0
$101$	9.52786	0.948058	0.474029	−	0.880509i	$-0.342799\pi$
$101$	9.52786	0.948058	0.474029	+	0.880509i	$0.342799\pi$
$102$	0	0
$103$	−10.4721	−1.03185	−0.515925	−	0.856634i	$-0.672552\pi$
$103$	−10.4721	−1.03185	−0.515925	+	0.856634i	$0.672552\pi$
$104$	0.527864	0.0517613
$105$	0	0
$106$	−10.4721	−1.01714
$107$	19.6525	1.89988	0.949938	−	0.312438i	$-0.101146\pi$
$107$	19.6525	1.89988	0.949938	+	0.312438i	$0.101146\pi$
$108$	0	0
$109$	10.4721	1.00305	0.501524	−	0.865144i	$-0.332773\pi$
$109$	10.4721	1.00305	0.501524	+	0.865144i	$0.332773\pi$
$110$	0	0
$111$	0	0
$112$	4.85410	0.458670
$113$	5.52786	0.520018	0.260009	−	0.965606i	$-0.416275\pi$
$113$	5.52786	0.520018	0.260009	+	0.965606i	$0.416275\pi$
$114$	0	0
$115$	4.47214	0.417029
$116$	3.09017	0.286915
$117$	0	0
$118$	−12.8541	−1.18332
$119$	−2.00000	−0.183340
$120$	0	0
$121$	0	0
$122$	−20.1803	−1.82704
$123$	0	0
$124$	6.47214	0.581215
$125$	−11.1803	−1.00000
$126$	0	0
$127$	9.41641	0.835571	0.417786	−	0.908546i	$-0.362806\pi$
$127$	9.41641	0.835571	0.417786	+	0.908546i	$0.362806\pi$
$128$	13.6180	1.20368
$129$	0	0
$130$	−0.854102	−0.0749097
$131$	19.4164	1.69642	0.848210	−	0.529661i	$-0.177681\pi$
$131$	19.4164	1.69642	0.848210	+	0.529661i	$0.177681\pi$
$132$	0	0
$133$	−1.47214	−0.127650
$134$	14.8541	1.28320
$135$	0	0
$136$	−4.47214	−0.383482
$137$	9.41641	0.804498	0.402249	−	0.915530i	$-0.368229\pi$
$137$	9.41641	0.804498	0.402249	+	0.915530i	$0.368229\pi$
$138$	0	0
$139$	8.94427	0.758643	0.379322	−	0.925265i	$-0.376157\pi$
$139$	8.94427	0.758643	0.379322	+	0.925265i	$0.376157\pi$
$140$	−1.38197	−0.116797
$141$	0	0
$142$	4.00000	0.335673
$143$	0	0
$144$	0	0
$145$	11.1803	0.928477
$146$	17.3262	1.43393
$147$	0	0
$148$	−0.909830	−0.0747876
$149$	21.0000	1.72039	0.860194	−	0.509968i	$-0.170343\pi$
$149$	21.0000	1.72039	0.860194	+	0.509968i	$0.170343\pi$
$150$	0	0
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	−3.29180	−0.267000
$153$	0	0
$154$	0	0
$155$	23.4164	1.88085
$156$	0	0
$157$	3.52786	0.281554	0.140777	−	0.990041i	$-0.455040\pi$
$157$	3.52786	0.281554	0.140777	+	0.990041i	$0.455040\pi$
$158$	0.763932	0.0607752
$159$	0	0
$160$	−7.56231	−0.597853
$161$	−2.00000	−0.157622
$162$	0	0
$163$	−3.76393	−0.294814	−0.147407	−	0.989076i	$-0.547093\pi$
$163$	−3.76393	−0.294814	−0.147407	+	0.989076i	$0.547093\pi$
$164$	1.52786	0.119306
$165$	0	0
$166$	12.1803	0.945378
$167$	−5.41641	−0.419134	−0.209567	−	0.977794i	$-0.567205\pi$
$167$	−5.41641	−0.419134	−0.209567	+	0.977794i	$0.567205\pi$
$168$	0	0
$169$	−12.9443	−0.995713
$170$	7.23607	0.554981
$171$	0	0
$172$	5.23607	0.399246
$173$	−7.52786	−0.572333	−0.286166	−	0.958180i	$-0.592381\pi$
$173$	−7.52786	−0.572333	−0.286166	+	0.958180i	$0.592381\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0.763932	0.0572591
$179$	26.3607	1.97029	0.985145	−	0.171725i	$-0.0549342\pi$
$179$	26.3607	1.97029	0.985145	+	0.171725i	$0.0549342\pi$
$180$	0	0
$181$	12.9443	0.962140	0.481070	−	0.876682i	$-0.340248\pi$
$181$	12.9443	0.962140	0.481070	+	0.876682i	$0.340248\pi$
$182$	0.381966	0.0283132
$183$	0	0
$184$	−4.47214	−0.329690
$185$	−3.29180	−0.242018
$186$	0	0
$187$	0	0
$188$	−5.85410	−0.426954
$189$	0	0
$190$	5.32624	0.386406
$191$	−15.5279	−1.12356	−0.561778	−	0.827288i	$-0.689883\pi$
$191$	−15.5279	−1.12356	−0.561778	+	0.827288i	$0.689883\pi$
$192$	0	0
$193$	−1.52786	−0.109978	−0.0549890	−	0.998487i	$-0.517512\pi$
$193$	−1.52786	−0.109978	−0.0549890	+	0.998487i	$0.517512\pi$
$194$	15.2361	1.09389
$195$	0	0
$196$	0.618034	0.0441453
$197$	−23.8885	−1.70199	−0.850994	−	0.525175i	$-0.824000\pi$
$197$	−23.8885	−1.70199	−0.850994	+	0.525175i	$0.824000\pi$
$198$	0	0
$199$	−11.8885	−0.842757	−0.421378	−	0.906885i	$-0.638454\pi$
$199$	−11.8885	−0.842757	−0.421378	+	0.906885i	$0.638454\pi$
$200$	0	0
$201$	0	0
$202$	15.4164	1.08469
$203$	−5.00000	−0.350931
$204$	0	0
$205$	5.52786	0.386083
$206$	−16.9443	−1.18056
$207$	0	0
$208$	1.14590	0.0794537
$209$	0	0
$210$	0	0
$211$	−26.3607	−1.81474	−0.907372	−	0.420328i	$-0.861915\pi$
$211$	−26.3607	−1.81474	−0.907372	+	0.420328i	$0.861915\pi$
$212$	−4.00000	−0.274721
$213$	0	0
$214$	31.7984	2.17369
$215$	18.9443	1.29199
$216$	0	0
$217$	−10.4721	−0.710895
$218$	16.9443	1.14761
$219$	0	0
$220$	0	0
$221$	−0.472136	−0.0317593
$222$	0	0
$223$	−23.8885	−1.59970	−0.799848	−	0.600203i	$-0.795086\pi$
$223$	−23.8885	−1.59970	−0.799848	+	0.600203i	$0.795086\pi$
$224$	3.38197	0.225967
$225$	0	0
$226$	8.94427	0.594964
$227$	3.52786	0.234153	0.117076	−	0.993123i	$-0.462648\pi$
$227$	3.52786	0.234153	0.117076	+	0.993123i	$0.462648\pi$
$228$	0	0
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	7.23607	0.477132
$231$	0	0
$232$	−11.1803	−0.734025
$233$	14.9443	0.979032	0.489516	−	0.871994i	$-0.337174\pi$
$233$	14.9443	0.979032	0.489516	+	0.871994i	$0.337174\pi$
$234$	0	0
$235$	−21.1803	−1.38165
$236$	−4.90983	−0.319603
$237$	0	0
$238$	−3.23607	−0.209763
$239$	−26.1246	−1.68986	−0.844930	−	0.534876i	$-0.820358\pi$
$239$	−26.1246	−1.68986	−0.844930	+	0.534876i	$0.820358\pi$
$240$	0	0
$241$	19.1803	1.23551	0.617757	−	0.786369i	$-0.288041\pi$
$241$	19.1803	1.23551	0.617757	+	0.786369i	$0.288041\pi$
$242$	0	0
$243$	0	0
$244$	−7.70820	−0.493467
$245$	2.23607	0.142857
$246$	0	0
$247$	−0.347524	−0.0221124
$248$	−23.4164	−1.48694
$249$	0	0
$250$	−18.0902	−1.14412
$251$	11.0000	0.694314	0.347157	−	0.937807i	$-0.387147\pi$
$251$	11.0000	0.694314	0.347157	+	0.937807i	$0.387147\pi$
$252$	0	0
$253$	0	0
$254$	15.2361	0.955996
$255$	0	0
$256$	13.5623	0.847644
$257$	13.6525	0.851618	0.425809	−	0.904813i	$-0.359990\pi$
$257$	13.6525	0.851618	0.425809	+	0.904813i	$0.359990\pi$
$258$	0	0
$259$	1.47214	0.0914741
$260$	−0.326238	−0.0202324
$261$	0	0
$262$	31.4164	1.94091
$263$	11.2918	0.696282	0.348141	−	0.937442i	$-0.386813\pi$
$263$	11.2918	0.696282	0.348141	+	0.937442i	$0.386813\pi$
$264$	0	0
$265$	−14.4721	−0.889016
$266$	−2.38197	−0.146048
$267$	0	0
$268$	5.67376	0.346580
$269$	−13.4164	−0.818013	−0.409006	−	0.912532i	$-0.634125\pi$
$269$	−13.4164	−0.818013	−0.409006	+	0.912532i	$0.634125\pi$
$270$	0	0
$271$	−6.88854	−0.418449	−0.209225	−	0.977868i	$-0.567094\pi$
$271$	−6.88854	−0.418449	−0.209225	+	0.977868i	$0.567094\pi$
$272$	−9.70820	−0.588646
$273$	0	0
$274$	15.2361	0.920445
$275$	0	0
$276$	0	0
$277$	−30.3607	−1.82420	−0.912098	−	0.409972i	$-0.865539\pi$
$277$	−30.3607	−1.82420	−0.912098	+	0.409972i	$0.865539\pi$
$278$	14.4721	0.867981
$279$	0	0
$280$	5.00000	0.298807
$281$	11.4721	0.684370	0.342185	−	0.939633i	$-0.388833\pi$
$281$	11.4721	0.684370	0.342185	+	0.939633i	$0.388833\pi$
$282$	0	0
$283$	18.4164	1.09474	0.547371	−	0.836890i	$-0.315629\pi$
$283$	18.4164	1.09474	0.547371	+	0.836890i	$0.315629\pi$
$284$	1.52786	0.0906621
$285$	0	0
$286$	0	0
$287$	−2.47214	−0.145926
$288$	0	0
$289$	−13.0000	−0.764706
$290$	18.0902	1.06229
$291$	0	0
$292$	6.61803	0.387291
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	−17.7639	−1.03426
$296$	3.29180	0.191332
$297$	0	0
$298$	33.9787	1.96833
$299$	−0.472136	−0.0273043
$300$	0	0
$301$	−8.47214	−0.488326
$302$	−16.1803	−0.931074
$303$	0	0
$304$	−7.14590	−0.409845
$305$	−27.8885	−1.59689
$306$	0	0
$307$	−29.8885	−1.70583	−0.852915	−	0.522050i	$-0.825167\pi$
$307$	−29.8885	−1.70583	−0.852915	+	0.522050i	$0.825167\pi$
$308$	0	0
$309$	0	0
$310$	37.8885	2.15192
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	14.9443	0.844700	0.422350	−	0.906433i	$-0.361205\pi$
$313$	14.9443	0.844700	0.422350	+	0.906433i	$0.361205\pi$
$314$	5.70820	0.322133
$315$	0	0
$316$	0.291796	0.0164148
$317$	3.05573	0.171627	0.0858134	−	0.996311i	$-0.472651\pi$
$317$	3.05573	0.171627	0.0858134	+	0.996311i	$0.472651\pi$
$318$	0	0
$319$	0	0
$320$	9.47214	0.529508
$321$	0	0
$322$	−3.23607	−0.180339
$323$	2.94427	0.163824
$324$	0	0
$325$	0	0
$326$	−6.09017	−0.337303
$327$	0	0
$328$	−5.52786	−0.305225
$329$	9.47214	0.522216
$330$	0	0
$331$	−16.0000	−0.879440	−0.439720	−	0.898135i	$-0.644922\pi$
$331$	−16.0000	−0.879440	−0.439720	+	0.898135i	$0.644922\pi$
$332$	4.65248	0.255338
$333$	0	0
$334$	−8.76393	−0.479541
$335$	20.5279	1.12156
$336$	0	0
$337$	27.5279	1.49954	0.749769	−	0.661699i	$-0.230165\pi$
$337$	27.5279	1.49954	0.749769	+	0.661699i	$0.230165\pi$
$338$	−20.9443	−1.13922
$339$	0	0
$340$	2.76393	0.149895
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−18.9443	−1.02141
$345$	0	0
$346$	−12.1803	−0.654819
$347$	−4.94427	−0.265422	−0.132711	−	0.991155i	$-0.542368\pi$
$347$	−4.94427	−0.265422	−0.132711	+	0.991155i	$0.542368\pi$
$348$	0	0
$349$	−18.1246	−0.970188	−0.485094	−	0.874462i	$-0.661215\pi$
$349$	−18.1246	−0.970188	−0.485094	+	0.874462i	$0.661215\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	5.18034	0.275722	0.137861	−	0.990452i	$-0.455977\pi$
$353$	5.18034	0.275722	0.137861	+	0.990452i	$0.455977\pi$
$354$	0	0
$355$	5.52786	0.293389
$356$	0.291796	0.0154652
$357$	0	0
$358$	42.6525	2.25425
$359$	28.9443	1.52762	0.763810	−	0.645441i	$-0.223326\pi$
$359$	28.9443	1.52762	0.763810	+	0.645441i	$0.223326\pi$
$360$	0	0
$361$	−16.8328	−0.885938
$362$	20.9443	1.10081
$363$	0	0
$364$	0.145898	0.00764713
$365$	23.9443	1.25330
$366$	0	0
$367$	−16.0000	−0.835193	−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000	−0.835193	−0.417597	+	0.908633i	$0.637127\pi$
$368$	−9.70820	−0.506075
$369$	0	0
$370$	−5.32624	−0.276898
$371$	6.47214	0.336017
$372$	0	0
$373$	−6.00000	−0.310668	−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000	−0.310668	−0.155334	+	0.987862i	$0.549645\pi$
$374$	0	0
$375$	0	0
$376$	21.1803	1.09229
$377$	−1.18034	−0.0607906
$378$	0	0
$379$	11.7639	0.604273	0.302136	−	0.953265i	$-0.402300\pi$
$379$	11.7639	0.604273	0.302136	+	0.953265i	$0.402300\pi$
$380$	2.03444	0.104365
$381$	0	0
$382$	−25.1246	−1.28549
$383$	0.944272	0.0482500	0.0241250	−	0.999709i	$-0.492320\pi$
$383$	0.944272	0.0482500	0.0241250	+	0.999709i	$0.492320\pi$
$384$	0	0
$385$	0	0
$386$	−2.47214	−0.125828
$387$	0	0
$388$	5.81966	0.295448
$389$	25.5279	1.29431	0.647157	−	0.762357i	$-0.275958\pi$
$389$	25.5279	1.29431	0.647157	+	0.762357i	$0.275958\pi$
$390$	0	0
$391$	4.00000	0.202289
$392$	−2.23607	−0.112938
$393$	0	0
$394$	−38.6525	−1.94728
$395$	1.05573	0.0531194
$396$	0	0
$397$	26.9443	1.35229	0.676147	−	0.736767i	$-0.263648\pi$
$397$	26.9443	1.35229	0.676147	+	0.736767i	$0.263648\pi$
$398$	−19.2361	−0.964217
$399$	0	0
$400$	0	0
$401$	26.0000	1.29838	0.649189	−	0.760627i	$-0.275108\pi$
$401$	26.0000	1.29838	0.649189	+	0.760627i	$0.275108\pi$
$402$	0	0
$403$	−2.47214	−0.123146
$404$	5.88854	0.292966
$405$	0	0
$406$	−8.09017	−0.401508
$407$	0	0
$408$	0	0
$409$	7.52786	0.372229	0.186114	−	0.982528i	$-0.440410\pi$
$409$	7.52786	0.372229	0.186114	+	0.982528i	$0.440410\pi$
$410$	8.94427	0.441726
$411$	0	0
$412$	−6.47214	−0.318859
$413$	7.94427	0.390912
$414$	0	0
$415$	16.8328	0.826290
$416$	0.798374	0.0391435
$417$	0	0
$418$	0	0
$419$	−17.0000	−0.830504	−0.415252	−	0.909706i	$-0.636307\pi$
$419$	−17.0000	−0.830504	−0.415252	+	0.909706i	$0.636307\pi$
$420$	0	0
$421$	−37.3607	−1.82085	−0.910424	−	0.413676i	$-0.864245\pi$
$421$	−37.3607	−1.82085	−0.910424	+	0.413676i	$0.864245\pi$
$422$	−42.6525	−2.07629
$423$	0	0
$424$	14.4721	0.702829
$425$	0	0
$426$	0	0
$427$	12.4721	0.603569
$428$	12.1459	0.587094
$429$	0	0
$430$	30.6525	1.47819
$431$	20.2361	0.974737	0.487369	−	0.873196i	$-0.337957\pi$
$431$	20.2361	0.974737	0.487369	+	0.873196i	$0.337957\pi$
$432$	0	0
$433$	6.58359	0.316387	0.158194	−	0.987408i	$-0.449433\pi$
$433$	6.58359	0.316387	0.158194	+	0.987408i	$0.449433\pi$
$434$	−16.9443	−0.813351
$435$	0	0
$436$	6.47214	0.309959
$437$	2.94427	0.140844
$438$	0	0
$439$	−26.8885	−1.28332	−0.641660	−	0.766989i	$-0.721754\pi$
$439$	−26.8885	−1.28332	−0.641660	+	0.766989i	$0.721754\pi$
$440$	0	0
$441$	0	0
$442$	−0.763932	−0.0363365
$443$	−14.0000	−0.665160	−0.332580	−	0.943075i	$-0.607919\pi$
$443$	−14.0000	−0.665160	−0.332580	+	0.943075i	$0.607919\pi$
$444$	0	0
$445$	1.05573	0.0500463
$446$	−38.6525	−1.83025
$447$	0	0
$448$	−4.23607	−0.200135
$449$	−0.472136	−0.0222815	−0.0111407	−	0.999938i	$-0.503546\pi$
$449$	−0.472136	−0.0222815	−0.0111407	+	0.999938i	$0.503546\pi$
$450$	0	0
$451$	0	0
$452$	3.41641	0.160694
$453$	0	0
$454$	5.70820	0.267899
$455$	0.527864	0.0247466
$456$	0	0
$457$	−6.36068	−0.297540	−0.148770	−	0.988872i	$-0.547531\pi$
$457$	−6.36068	−0.297540	−0.148770	+	0.988872i	$0.547531\pi$
$458$	22.6525	1.05848
$459$	0	0
$460$	2.76393	0.128869
$461$	18.9443	0.882323	0.441161	−	0.897428i	$-0.354567\pi$
$461$	18.9443	0.882323	0.441161	+	0.897428i	$0.354567\pi$
$462$	0	0
$463$	6.70820	0.311757	0.155878	−	0.987776i	$-0.450179\pi$
$463$	6.70820	0.311757	0.155878	+	0.987776i	$0.450179\pi$
$464$	−24.2705	−1.12673
$465$	0	0
$466$	24.1803	1.12013
$467$	−10.0557	−0.465324	−0.232662	−	0.972558i	$-0.574744\pi$
$467$	−10.0557	−0.465324	−0.232662	+	0.972558i	$0.574744\pi$
$468$	0	0
$469$	−9.18034	−0.423909
$470$	−34.2705	−1.58078
$471$	0	0
$472$	17.7639	0.817651
$473$	0	0
$474$	0	0
$475$	0	0
$476$	−1.23607	−0.0566551
$477$	0	0
$478$	−42.2705	−1.93341
$479$	−35.8885	−1.63979	−0.819895	−	0.572514i	$-0.805968\pi$
$479$	−35.8885	−1.63979	−0.819895	+	0.572514i	$0.805968\pi$
$480$	0	0
$481$	0.347524	0.0158457
$482$	31.0344	1.41358
$483$	0	0
$484$	0	0
$485$	21.0557	0.956091
$486$	0	0
$487$	5.88854	0.266835	0.133418	−	0.991060i	$-0.457405\pi$
$487$	5.88854	0.266835	0.133418	+	0.991060i	$0.457405\pi$
$488$	27.8885	1.26246
$489$	0	0
$490$	3.61803	0.163446
$491$	−12.1246	−0.547176	−0.273588	−	0.961847i	$-0.588210\pi$
$491$	−12.1246	−0.547176	−0.273588	+	0.961847i	$0.588210\pi$
$492$	0	0
$493$	10.0000	0.450377
$494$	−0.562306	−0.0252993
$495$	0	0
$496$	−50.8328	−2.28246
$497$	−2.47214	−0.110890
$498$	0	0
$499$	19.7639	0.884755	0.442378	−	0.896829i	$-0.354135\pi$
$499$	19.7639	0.884755	0.442378	+	0.896829i	$0.354135\pi$
$500$	−6.90983	−0.309017
$501$	0	0
$502$	17.7984	0.794380
$503$	9.88854	0.440908	0.220454	−	0.975397i	$-0.429246\pi$
$503$	9.88854	0.440908	0.220454	+	0.975397i	$0.429246\pi$
$504$	0	0
$505$	21.3050	0.948058
$506$	0	0
$507$	0	0
$508$	5.81966	0.258206
$509$	−33.4164	−1.48116	−0.740578	−	0.671970i	$-0.765448\pi$
$509$	−33.4164	−1.48116	−0.740578	+	0.671970i	$0.765448\pi$
$510$	0	0
$511$	−10.7082	−0.473703
$512$	−5.29180	−0.233867
$513$	0	0
$514$	22.0902	0.974356
$515$	−23.4164	−1.03185
$516$	0	0
$517$	0	0
$518$	2.38197	0.104658
$519$	0	0
$520$	1.18034	0.0517613
$521$	−10.1246	−0.443567	−0.221784	−	0.975096i	$-0.571188\pi$
$521$	−10.1246	−0.443567	−0.221784	+	0.975096i	$0.571188\pi$
$522$	0	0
$523$	1.58359	0.0692456	0.0346228	−	0.999400i	$-0.488977\pi$
$523$	1.58359	0.0692456	0.0346228	+	0.999400i	$0.488977\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	18.2705	0.796632
$527$	20.9443	0.912347
$528$	0	0
$529$	−19.0000	−0.826087
$530$	−23.4164	−1.01714
$531$	0	0
$532$	−0.909830	−0.0394461
$533$	−0.583592	−0.0252782
$534$	0	0
$535$	43.9443	1.89988
$536$	−20.5279	−0.886669
$537$	0	0
$538$	−21.7082	−0.935907
$539$	0	0
$540$	0	0
$541$	24.3607	1.04735	0.523674	−	0.851919i	$-0.324561\pi$
$541$	24.3607	1.04735	0.523674	+	0.851919i	$0.324561\pi$
$542$	−11.1459	−0.478757
$543$	0	0
$544$	−6.76393	−0.290001
$545$	23.4164	1.00305
$546$	0	0
$547$	−26.0000	−1.11168	−0.555840	−	0.831289i	$-0.687603\pi$
$547$	−26.0000	−1.11168	−0.555840	+	0.831289i	$0.687603\pi$
$548$	5.81966	0.248604
$549$	0	0
$550$	0	0
$551$	7.36068	0.313576
$552$	0	0
$553$	−0.472136	−0.0200773
$554$	−49.1246	−2.08710
$555$	0	0
$556$	5.52786	0.234434
$557$	15.8328	0.670858	0.335429	−	0.942066i	$-0.391119\pi$
$557$	15.8328	0.670858	0.335429	+	0.942066i	$0.391119\pi$
$558$	0	0
$559$	−2.00000	−0.0845910
$560$	10.8541	0.458670
$561$	0	0
$562$	18.5623	0.783004
$563$	21.8885	0.922492	0.461246	−	0.887272i	$-0.347403\pi$
$563$	21.8885	0.922492	0.461246	+	0.887272i	$0.347403\pi$
$564$	0	0
$565$	12.3607	0.520018
$566$	29.7984	1.25252
$567$	0	0
$568$	−5.52786	−0.231944
$569$	−2.00000	−0.0838444	−0.0419222	−	0.999121i	$-0.513348\pi$
$569$	−2.00000	−0.0838444	−0.0419222	+	0.999121i	$0.513348\pi$
$570$	0	0
$571$	−44.8328	−1.87619	−0.938097	−	0.346371i	$-0.887414\pi$
$571$	−44.8328	−1.87619	−0.938097	+	0.346371i	$0.887414\pi$
$572$	0	0
$573$	0	0
$574$	−4.00000	−0.166957
$575$	0	0
$576$	0	0
$577$	26.4721	1.10205	0.551025	−	0.834489i	$-0.314237\pi$
$577$	26.4721	1.10205	0.551025	+	0.834489i	$0.314237\pi$
$578$	−21.0344	−0.874917
$579$	0	0
$580$	6.90983	0.286915
$581$	−7.52786	−0.312308
$582$	0	0
$583$	0	0
$584$	−23.9443	−0.990821
$585$	0	0
$586$	0	0
$587$	34.7771	1.43540	0.717702	−	0.696350i	$-0.245194\pi$
$587$	34.7771	1.43540	0.717702	+	0.696350i	$0.245194\pi$
$588$	0	0
$589$	15.4164	0.635222
$590$	−28.7426	−1.18332
$591$	0	0
$592$	7.14590	0.293695
$593$	4.11146	0.168837	0.0844186	−	0.996430i	$-0.473097\pi$
$593$	4.11146	0.168837	0.0844186	+	0.996430i	$0.473097\pi$
$594$	0	0
$595$	−4.47214	−0.183340
$596$	12.9787	0.531629
$597$	0	0
$598$	−0.763932	−0.0312395
$599$	−4.47214	−0.182727	−0.0913633	−	0.995818i	$-0.529122\pi$
$599$	−4.47214	−0.182727	−0.0913633	+	0.995818i	$0.529122\pi$
$600$	0	0
$601$	18.7082	0.763124	0.381562	−	0.924343i	$-0.375386\pi$
$601$	18.7082	0.763124	0.381562	+	0.924343i	$0.375386\pi$
$602$	−13.7082	−0.558705
$603$	0	0
$604$	−6.18034	−0.251474
$605$	0	0
$606$	0	0
$607$	27.9443	1.13422	0.567112	−	0.823641i	$-0.308061\pi$
$607$	27.9443	1.13422	0.567112	+	0.823641i	$0.308061\pi$
$608$	−4.97871	−0.201914
$609$	0	0
$610$	−45.1246	−1.82704
$611$	2.23607	0.0904616
$612$	0	0
$613$	24.0000	0.969351	0.484675	−	0.874694i	$-0.338938\pi$
$613$	24.0000	0.969351	0.484675	+	0.874694i	$0.338938\pi$
$614$	−48.3607	−1.95168
$615$	0	0
$616$	0	0
$617$	−26.8328	−1.08025	−0.540124	−	0.841585i	$-0.681623\pi$
$617$	−26.8328	−1.08025	−0.540124	+	0.841585i	$0.681623\pi$
$618$	0	0
$619$	5.88854	0.236681	0.118340	−	0.992973i	$-0.462243\pi$
$619$	5.88854	0.236681	0.118340	+	0.992973i	$0.462243\pi$
$620$	14.4721	0.581215
$621$	0	0
$622$	0	0
$623$	−0.472136	−0.0189157
$624$	0	0
$625$	−25.0000	−1.00000
$626$	24.1803	0.966441
$627$	0	0
$628$	2.18034	0.0870050
$629$	−2.94427	−0.117396
$630$	0	0
$631$	26.8328	1.06820	0.534099	−	0.845422i	$-0.320651\pi$
$631$	26.8328	1.06820	0.534099	+	0.845422i	$0.320651\pi$
$632$	−1.05573	−0.0419946
$633$	0	0
$634$	4.94427	0.196362
$635$	21.0557	0.835571
$636$	0	0
$637$	−0.236068	−0.00935335
$638$	0	0
$639$	0	0
$640$	30.4508	1.20368
$641$	−15.4164	−0.608912	−0.304456	−	0.952526i	$-0.598475\pi$
$641$	−15.4164	−0.608912	−0.304456	+	0.952526i	$0.598475\pi$
$642$	0	0
$643$	−37.7771	−1.48978	−0.744891	−	0.667186i	$-0.767499\pi$
$643$	−37.7771	−1.48978	−0.744891	+	0.667186i	$0.767499\pi$
$644$	−1.23607	−0.0487079
$645$	0	0
$646$	4.76393	0.187434
$647$	−50.3050	−1.97769	−0.988846	−	0.148942i	$-0.952413\pi$
$647$	−50.3050	−1.97769	−0.988846	+	0.148942i	$0.952413\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	−2.32624	−0.0911025
$653$	−36.9443	−1.44574	−0.722871	−	0.690983i	$-0.757178\pi$
$653$	−36.9443	−1.44574	−0.722871	+	0.690983i	$0.757178\pi$
$654$	0	0
$655$	43.4164	1.69642
$656$	−12.0000	−0.468521
$657$	0	0
$658$	15.3262	0.597479
$659$	−24.1246	−0.939761	−0.469881	−	0.882730i	$-0.655703\pi$
$659$	−24.1246	−0.939761	−0.469881	+	0.882730i	$0.655703\pi$
$660$	0	0
$661$	−25.3050	−0.984249	−0.492124	−	0.870525i	$-0.663779\pi$
$661$	−25.3050	−0.984249	−0.492124	+	0.870525i	$0.663779\pi$
$662$	−25.8885	−1.00619
$663$	0	0
$664$	−16.8328	−0.653240
$665$	−3.29180	−0.127650
$666$	0	0
$667$	10.0000	0.387202
$668$	−3.34752	−0.129520
$669$	0	0
$670$	33.2148	1.28320
$671$	0	0
$672$	0	0
$673$	−24.9443	−0.961531	−0.480766	−	0.876849i	$-0.659641\pi$
$673$	−24.9443	−0.961531	−0.480766	+	0.876849i	$0.659641\pi$
$674$	44.5410	1.71566
$675$	0	0
$676$	−8.00000	−0.307692
$677$	17.5279	0.673651	0.336825	−	0.941567i	$-0.390647\pi$
$677$	17.5279	0.673651	0.336825	+	0.941567i	$0.390647\pi$
$678$	0	0
$679$	−9.41641	−0.361369
$680$	−10.0000	−0.383482
$681$	0	0
$682$	0	0
$683$	−44.9443	−1.71974	−0.859872	−	0.510509i	$-0.829457\pi$
$683$	−44.9443	−1.71974	−0.859872	+	0.510509i	$0.829457\pi$
$684$	0	0
$685$	21.0557	0.804498
$686$	−1.61803	−0.0617768
$687$	0	0
$688$	−41.1246	−1.56786
$689$	1.52786	0.0582070
$690$	0	0
$691$	−14.8328	−0.564267	−0.282133	−	0.959375i	$-0.591042\pi$
$691$	−14.8328	−0.564267	−0.282133	+	0.959375i	$0.591042\pi$
$692$	−4.65248	−0.176861
$693$	0	0
$694$	−8.00000	−0.303676
$695$	20.0000	0.758643
$696$	0	0
$697$	4.94427	0.187278
$698$	−29.3262	−1.11001
$699$	0	0
$700$	0	0
$701$	−12.1115	−0.457443	−0.228722	−	0.973492i	$-0.573455\pi$
$701$	−12.1115	−0.457443	−0.228722	+	0.973492i	$0.573455\pi$
$702$	0	0
$703$	−2.16718	−0.0817369
$704$	0	0
$705$	0	0
$706$	8.38197	0.315459
$707$	−9.52786	−0.358332
$708$	0	0
$709$	−34.5279	−1.29672	−0.648361	−	0.761333i	$-0.724545\pi$
$709$	−34.5279	−1.29672	−0.648361	+	0.761333i	$0.724545\pi$
$710$	8.94427	0.335673
$711$	0	0
$712$	−1.05573	−0.0395651
$713$	20.9443	0.784369
$714$	0	0
$715$	0	0
$716$	16.2918	0.608853
$717$	0	0
$718$	46.8328	1.74779
$719$	−21.3607	−0.796619	−0.398309	−	0.917251i	$-0.630403\pi$
$719$	−21.3607	−0.796619	−0.398309	+	0.917251i	$0.630403\pi$
$720$	0	0
$721$	10.4721	0.390003
$722$	−27.2361	−1.01362
$723$	0	0
$724$	8.00000	0.297318
$725$	0	0
$726$	0	0
$727$	−3.52786	−0.130841	−0.0654206	−	0.997858i	$-0.520839\pi$
$727$	−3.52786	−0.130841	−0.0654206	+	0.997858i	$0.520839\pi$
$728$	−0.527864	−0.0195639
$729$	0	0
$730$	38.7426	1.43393
$731$	16.9443	0.626707
$732$	0	0
$733$	26.3607	0.973654	0.486827	−	0.873498i	$-0.338154\pi$
$733$	26.3607	0.973654	0.486827	+	0.873498i	$0.338154\pi$
$734$	−25.8885	−0.955564
$735$	0	0
$736$	−6.76393	−0.249322
$737$	0	0
$738$	0	0
$739$	−51.8885	−1.90875	−0.954375	−	0.298609i	$-0.903477\pi$
$739$	−51.8885	−1.90875	−0.954375	+	0.298609i	$0.903477\pi$
$740$	−2.03444	−0.0747876
$741$	0	0
$742$	10.4721	0.384444
$743$	−30.5967	−1.12249	−0.561243	−	0.827651i	$-0.689677\pi$
$743$	−30.5967	−1.12249	−0.561243	+	0.827651i	$0.689677\pi$
$744$	0	0
$745$	46.9574	1.72039
$746$	−9.70820	−0.355443
$747$	0	0
$748$	0	0
$749$	−19.6525	−0.718086
$750$	0	0
$751$	−7.18034	−0.262014	−0.131007	−	0.991381i	$-0.541821\pi$
$751$	−7.18034	−0.262014	−0.131007	+	0.991381i	$0.541821\pi$
$752$	45.9787	1.67667
$753$	0	0
$754$	−1.90983	−0.0695519
$755$	−22.3607	−0.813788
$756$	0	0
$757$	−15.5836	−0.566395	−0.283198	−	0.959062i	$-0.591395\pi$
$757$	−15.5836	−0.566395	−0.283198	+	0.959062i	$0.591395\pi$
$758$	19.0344	0.691362
$759$	0	0
$760$	−7.36068	−0.267000
$761$	24.4721	0.887114	0.443557	−	0.896246i	$-0.353716\pi$
$761$	24.4721	0.887114	0.443557	+	0.896246i	$0.353716\pi$
$762$	0	0
$763$	−10.4721	−0.379117
$764$	−9.59675	−0.347198
$765$	0	0
$766$	1.52786	0.0552040
$767$	1.87539	0.0677163
$768$	0	0
$769$	15.1803	0.547417	0.273709	−	0.961813i	$-0.411750\pi$
$769$	15.1803	0.547417	0.273709	+	0.961813i	$0.411750\pi$
$770$	0	0
$771$	0	0
$772$	−0.944272	−0.0339851
$773$	19.1803	0.689869	0.344934	−	0.938627i	$-0.387901\pi$
$773$	19.1803	0.689869	0.344934	+	0.938627i	$0.387901\pi$
$774$	0	0
$775$	0	0
$776$	−21.0557	−0.755857
$777$	0	0
$778$	41.3050	1.48085
$779$	3.63932	0.130392
$780$	0	0
$781$	0	0
$782$	6.47214	0.231443
$783$	0	0
$784$	−4.85410	−0.173361
$785$	7.88854	0.281554
$786$	0	0
$787$	−45.2492	−1.61296	−0.806480	−	0.591261i	$-0.798630\pi$
$787$	−45.2492	−1.61296	−0.806480	+	0.591261i	$0.798630\pi$
$788$	−14.7639	−0.525943
$789$	0	0
$790$	1.70820	0.0607752
$791$	−5.52786	−0.196548
$792$	0	0
$793$	2.94427	0.104554
$794$	43.5967	1.54719
$795$	0	0
$796$	−7.34752	−0.260426
$797$	37.7639	1.33767	0.668834	−	0.743412i	$-0.266794\pi$
$797$	37.7639	1.33767	0.668834	+	0.743412i	$0.266794\pi$
$798$	0	0
$799$	−18.9443	−0.670200
$800$	0	0
$801$	0	0
$802$	42.0689	1.48550
$803$	0	0
$804$	0	0
$805$	−4.47214	−0.157622
$806$	−4.00000	−0.140894
$807$	0	0
$808$	−21.3050	−0.749506
$809$	49.3607	1.73543	0.867715	−	0.497063i	$-0.165588\pi$
$809$	49.3607	1.73543	0.867715	+	0.497063i	$0.165588\pi$
$810$	0	0
$811$	−40.5279	−1.42313	−0.711563	−	0.702622i	$-0.752012\pi$
$811$	−40.5279	−1.42313	−0.711563	+	0.702622i	$0.752012\pi$
$812$	−3.09017	−0.108444
$813$	0	0
$814$	0	0
$815$	−8.41641	−0.294814
$816$	0	0
$817$	12.4721	0.436345
$818$	12.1803	0.425876
$819$	0	0
$820$	3.41641	0.119306
$821$	1.83282	0.0639657	0.0319829	−	0.999488i	$-0.489818\pi$
$821$	1.83282	0.0639657	0.0319829	+	0.999488i	$0.489818\pi$
$822$	0	0
$823$	−33.0689	−1.15271	−0.576354	−	0.817200i	$-0.695525\pi$
$823$	−33.0689	−1.15271	−0.576354	+	0.817200i	$0.695525\pi$
$824$	23.4164	0.815749
$825$	0	0
$826$	12.8541	0.447251
$827$	−26.2361	−0.912317	−0.456159	−	0.889898i	$-0.650775\pi$
$827$	−26.2361	−0.912317	−0.456159	+	0.889898i	$0.650775\pi$
$828$	0	0
$829$	−20.9443	−0.727425	−0.363712	−	0.931511i	$-0.618491\pi$
$829$	−20.9443	−0.727425	−0.363712	+	0.931511i	$0.618491\pi$
$830$	27.2361	0.945378
$831$	0	0
$832$	−1.00000	−0.0346688
$833$	2.00000	0.0692959
$834$	0	0
$835$	−12.1115	−0.419134
$836$	0	0
$837$	0	0
$838$	−27.5066	−0.950199
$839$	−29.2492	−1.00980	−0.504898	−	0.863179i	$-0.668470\pi$
$839$	−29.2492	−1.00980	−0.504898	+	0.863179i	$0.668470\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	−60.4508	−2.08327
$843$	0	0
$844$	−16.2918	−0.560787
$845$	−28.9443	−0.995713
$846$	0	0
$847$	0	0
$848$	31.4164	1.07884
$849$	0	0
$850$	0	0
$851$	−2.94427	−0.100928
$852$	0	0
$853$	29.4164	1.00720	0.503599	−	0.863937i	$-0.332009\pi$
$853$	29.4164	1.00720	0.503599	+	0.863937i	$0.332009\pi$
$854$	20.1803	0.690557
$855$	0	0
$856$	−43.9443	−1.50198
$857$	−14.5836	−0.498166	−0.249083	−	0.968482i	$-0.580129\pi$
$857$	−14.5836	−0.498166	−0.249083	+	0.968482i	$0.580129\pi$
$858$	0	0
$859$	9.05573	0.308977	0.154489	−	0.987995i	$-0.450627\pi$
$859$	9.05573	0.308977	0.154489	+	0.987995i	$0.450627\pi$
$860$	11.7082	0.399246
$861$	0	0
$862$	32.7426	1.11522
$863$	−25.8885	−0.881256	−0.440628	−	0.897690i	$-0.645244\pi$
$863$	−25.8885	−0.881256	−0.440628	+	0.897690i	$0.645244\pi$
$864$	0	0
$865$	−16.8328	−0.572333
$866$	10.6525	0.361986
$867$	0	0
$868$	−6.47214	−0.219679
$869$	0	0
$870$	0	0
$871$	−2.16718	−0.0734322
$872$	−23.4164	−0.792980
$873$	0	0
$874$	4.76393	0.161142
$875$	11.1803	0.377964
$876$	0	0
$877$	28.0000	0.945493	0.472746	−	0.881199i	$-0.343263\pi$
$877$	28.0000	0.945493	0.472746	+	0.881199i	$0.343263\pi$
$878$	−43.5066	−1.46828
$879$	0	0
$880$	0	0
$881$	−40.2361	−1.35559	−0.677794	−	0.735252i	$-0.737064\pi$
$881$	−40.2361	−1.35559	−0.677794	+	0.735252i	$0.737064\pi$
$882$	0	0
$883$	−25.1803	−0.847386	−0.423693	−	0.905806i	$-0.639266\pi$
$883$	−25.1803	−0.847386	−0.423693	+	0.905806i	$0.639266\pi$
$884$	−0.291796	−0.00981416
$885$	0	0
$886$	−22.6525	−0.761025
$887$	16.9443	0.568933	0.284466	−	0.958686i	$-0.408184\pi$
$887$	16.9443	0.568933	0.284466	+	0.958686i	$0.408184\pi$
$888$	0	0
$889$	−9.41641	−0.315816
$890$	1.70820	0.0572591
$891$	0	0
$892$	−14.7639	−0.494333
$893$	−13.9443	−0.466627
$894$	0	0
$895$	58.9443	1.97029
$896$	−13.6180	−0.454947
$897$	0	0
$898$	−0.763932	−0.0254927
$899$	52.3607	1.74633
$900$	0	0
$901$	−12.9443	−0.431236
$902$	0	0
$903$	0	0
$904$	−12.3607	−0.411110
$905$	28.9443	0.962140
$906$	0	0
$907$	55.7771	1.85205	0.926024	−	0.377465i	$-0.123204\pi$
$907$	55.7771	1.85205	0.926024	+	0.377465i	$0.123204\pi$
$908$	2.18034	0.0723571
$909$	0	0
$910$	0.854102	0.0283132
$911$	54.2492	1.79736	0.898678	−	0.438608i	$-0.144528\pi$
$911$	54.2492	1.79736	0.898678	+	0.438608i	$0.144528\pi$
$912$	0	0
$913$	0	0
$914$	−10.2918	−0.340422
$915$	0	0
$916$	8.65248	0.285886
$917$	−19.4164	−0.641186
$918$	0	0
$919$	−22.4721	−0.741287	−0.370644	−	0.928775i	$-0.620863\pi$
$919$	−22.4721	−0.741287	−0.370644	+	0.928775i	$0.620863\pi$
$920$	−10.0000	−0.329690
$921$	0	0
$922$	30.6525	1.00949
$923$	−0.583592	−0.0192092
$924$	0	0
$925$	0	0
$926$	10.8541	0.356688
$927$	0	0
$928$	−16.9098	−0.555092
$929$	−12.7082	−0.416943	−0.208471	−	0.978028i	$-0.566849\pi$
$929$	−12.7082	−0.416943	−0.208471	+	0.978028i	$0.566849\pi$
$930$	0	0
$931$	1.47214	0.0482473
$932$	9.23607	0.302537
$933$	0	0
$934$	−16.2705	−0.532387
$935$	0	0
$936$	0	0
$937$	38.3607	1.25319	0.626594	−	0.779346i	$-0.284448\pi$
$937$	38.3607	1.25319	0.626594	+	0.779346i	$0.284448\pi$
$938$	−14.8541	−0.485004
$939$	0	0
$940$	−13.0902	−0.426954
$941$	26.5836	0.866600	0.433300	−	0.901250i	$-0.357349\pi$
$941$	26.5836	0.866600	0.433300	+	0.901250i	$0.357349\pi$
$942$	0	0
$943$	4.94427	0.161008
$944$	38.5623	1.25510
$945$	0	0
$946$	0	0
$947$	44.0000	1.42981	0.714904	−	0.699223i	$-0.246470\pi$
$947$	44.0000	1.42981	0.714904	+	0.699223i	$0.246470\pi$
$948$	0	0
$949$	−2.52786	−0.0820579
$950$	0	0
$951$	0	0
$952$	4.47214	0.144943
$953$	8.41641	0.272634	0.136317	−	0.990665i	$-0.456473\pi$
$953$	8.41641	0.272634	0.136317	+	0.990665i	$0.456473\pi$
$954$	0	0
$955$	−34.7214	−1.12356
$956$	−16.1459	−0.522196
$957$	0	0
$958$	−58.0689	−1.87612
$959$	−9.41641	−0.304072
$960$	0	0
$961$	78.6656	2.53760
$962$	0.562306	0.0181295
$963$	0	0
$964$	11.8541	0.381795
$965$	−3.41641	−0.109978
$966$	0	0
$967$	−13.3050	−0.427858	−0.213929	−	0.976849i	$-0.568626\pi$
$967$	−13.3050	−0.427858	−0.213929	+	0.976849i	$0.568626\pi$
$968$	0	0
$969$	0	0
$970$	34.0689	1.09389
$971$	−12.0557	−0.386887	−0.193443	−	0.981111i	$-0.561966\pi$
$971$	−12.0557	−0.386887	−0.193443	+	0.981111i	$0.561966\pi$
$972$	0	0
$973$	−8.94427	−0.286740
$974$	9.52786	0.305292
$975$	0	0
$976$	60.5410	1.93787
$977$	−16.4721	−0.526990	−0.263495	−	0.964661i	$-0.584875\pi$
$977$	−16.4721	−0.526990	−0.263495	+	0.964661i	$0.584875\pi$
$978$	0	0
$979$	0	0
$980$	1.38197	0.0441453
$981$	0	0
$982$	−19.6180	−0.626037
$983$	40.0000	1.27580	0.637901	−	0.770118i	$-0.279803\pi$
$983$	40.0000	1.27580	0.637901	+	0.770118i	$0.279803\pi$
$984$	0	0
$985$	−53.4164	−1.70199
$986$	16.1803	0.515287
$987$	0	0
$988$	−0.214782	−0.00683312
$989$	16.9443	0.538797
$990$	0	0
$991$	−11.1803	−0.355155	−0.177578	−	0.984107i	$-0.556826\pi$
$991$	−11.1803	−0.355155	−0.177578	+	0.984107i	$0.556826\pi$
$992$	−35.4164	−1.12447
$993$	0	0
$994$	−4.00000	−0.126872
$995$	−26.5836	−0.842757
$996$	0	0
$997$	57.1935	1.81134	0.905668	−	0.423987i	$-0.139370\pi$
$997$	57.1935	1.81134	0.905668	+	0.423987i	$0.139370\pi$
$998$	31.9787	1.01227
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.bn.1.2		2
3.2	odd	2		2541.2.a.q.1.1	✓	2
11.10	odd	2		7623.2.a.y.1.1		2
33.32	even	2		2541.2.a.y.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2541.2.a.q.1.1	✓	2	3.2	odd	2
2541.2.a.y.1.2	yes	2	33.32	even	2
7623.2.a.y.1.1		2	11.10	odd	2
7623.2.a.bn.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:	\( N \)	\(=\)	\( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7623.a (trivial)

\(f(q)\)	\(=\)	\(q+1.61803 q^{2} +0.618034 q^{4} +2.23607 q^{5} -1.00000 q^{7} -2.23607 q^{8} +O(q^{10})\)	\(q+1.61803 q^{2} +0.618034 q^{4} +2.23607 q^{5} -1.00000 q^{7} -2.23607 q^{8} +3.61803 q^{10} -0.236068 q^{13} -1.61803 q^{14} -4.85410 q^{16} +2.00000 q^{17} +1.47214 q^{19} +1.38197 q^{20} +2.00000 q^{23} -0.381966 q^{26} -0.618034 q^{28} +5.00000 q^{29} +10.4721 q^{31} -3.38197 q^{32} +3.23607 q^{34} -2.23607 q^{35} -1.47214 q^{37} +2.38197 q^{38} -5.00000 q^{40} +2.47214 q^{41} +8.47214 q^{43} +3.23607 q^{46} -9.47214 q^{47} +1.00000 q^{49} -0.145898 q^{52} -6.47214 q^{53} +2.23607 q^{56} +8.09017 q^{58} -7.94427 q^{59} -12.4721 q^{61} +16.9443 q^{62} +4.23607 q^{64} -0.527864 q^{65} +9.18034 q^{67} +1.23607 q^{68} -3.61803 q^{70} +2.47214 q^{71} +10.7082 q^{73} -2.38197 q^{74} +0.909830 q^{76} +0.472136 q^{79} -10.8541 q^{80} +4.00000 q^{82} +7.52786 q^{83} +4.47214 q^{85} +13.7082 q^{86} +0.472136 q^{89} +0.236068 q^{91} +1.23607 q^{92} -15.3262 q^{94} +3.29180 q^{95} +9.41641 q^{97} +1.61803 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - q^{4} - 2 q^{7}+O(q^{10}) \) 2 * q + q^2 - q^4 - 2 * q^7	\( 2 q + q^{2} - q^{4} - 2 q^{7} + 5 q^{10} + 4 q^{13} - q^{14} - 3 q^{16} + 4 q^{17} - 6 q^{19} + 5 q^{20} + 4 q^{23} - 3 q^{26} + q^{28} + 10 q^{29} + 12 q^{31} - 9 q^{32} + 2 q^{34} + 6 q^{37} + 7 q^{38} - 10 q^{40} - 4 q^{41} + 8 q^{43} + 2 q^{46} - 10 q^{47} + 2 q^{49} - 7 q^{52} - 4 q^{53} + 5 q^{58} + 2 q^{59} - 16 q^{61} + 16 q^{62} + 4 q^{64} - 10 q^{65} - 4 q^{67} - 2 q^{68} - 5 q^{70} - 4 q^{71} + 8 q^{73} - 7 q^{74} + 13 q^{76} - 8 q^{79} - 15 q^{80} + 8 q^{82} + 24 q^{83} + 14 q^{86} - 8 q^{89} - 4 q^{91} - 2 q^{92} - 15 q^{94} + 20 q^{95} - 8 q^{97} + q^{98}+O(q^{100}) \) 2 * q + q^2 - q^4 - 2 * q^7 + 5 * q^10 + 4 * q^13 - q^14 - 3 * q^16 + 4 * q^17 - 6 * q^19 + 5 * q^20 + 4 * q^23 - 3 * q^26 + q^28 + 10 * q^29 + 12 * q^31 - 9 * q^32 + 2 * q^34 + 6 * q^37 + 7 * q^38 - 10 * q^40 - 4 * q^41 + 8 * q^43 + 2 * q^46 - 10 * q^47 + 2 * q^49 - 7 * q^52 - 4 * q^53 + 5 * q^58 + 2 * q^59 - 16 * q^61 + 16 * q^62 + 4 * q^64 - 10 * q^65 - 4 * q^67 - 2 * q^68 - 5 * q^70 - 4 * q^71 + 8 * q^73 - 7 * q^74 + 13 * q^76 - 8 * q^79 - 15 * q^80 + 8 * q^82 + 24 * q^83 + 14 * q^86 - 8 * q^89 - 4 * q^91 - 2 * q^92 - 15 * q^94 + 20 * q^95 - 8 * q^97 + q^98

Introduction

L-functions

Modular forms

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