LMFDB - Embedded newform 7623.2.a.ci.1.4

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:	$4$
Coefficient field:	4.4.725.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 3x^{2} + x + 1 $ x^4 - x^3 - 3*x^2 + x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.09529$ 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)

$f(q)$	$=$	$q+2.39026 q^{2} +3.71333 q^{4} +2.58993 q^{5} -1.00000 q^{7} +4.09529 q^{8} +O(q^{10})$	$q+2.39026 q^{2} +3.71333 q^{4} +2.58993 q^{5} -1.00000 q^{7} +4.09529 q^{8} +6.19059 q^{10} +6.78051 q^{13} -2.39026 q^{14} +2.36215 q^{16} +3.14511 q^{17} -1.04548 q^{19} +9.61724 q^{20} +6.52707 q^{23} +1.70772 q^{25} +16.2072 q^{26} -3.71333 q^{28} +0.607298 q^{29} -8.83673 q^{31} -2.54445 q^{32} +7.51762 q^{34} -2.58993 q^{35} +8.94299 q^{37} -2.49897 q^{38} +10.6065 q^{40} -8.69747 q^{41} +4.48159 q^{43} +15.6014 q^{46} -3.26290 q^{47} +1.00000 q^{49} +4.08188 q^{50} +25.1783 q^{52} -1.89958 q^{53} -4.09529 q^{56} +1.45160 q^{58} +0.174006 q^{59} +8.13437 q^{61} -21.1221 q^{62} -10.8062 q^{64} +17.5610 q^{65} -7.33649 q^{67} +11.6788 q^{68} -6.19059 q^{70} -2.70693 q^{71} +0.589926 q^{73} +21.3761 q^{74} -3.88221 q^{76} -9.80862 q^{79} +6.11779 q^{80} -20.7892 q^{82} -8.74577 q^{83} +8.14560 q^{85} +10.7122 q^{86} +11.6788 q^{89} -6.78051 q^{91} +24.2372 q^{92} -7.79916 q^{94} -2.70772 q^{95} +11.8528 q^{97} +2.39026 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} + 3 q^{4} + 4 q^{5} - 4 q^{7} + 9 q^{8}+O(q^{10}) $ 4 * q - q^2 + 3 * q^4 + 4 * q^5 - 4 * q^7 + 9 * q^8	$ 4 q - q^{2} + 3 q^{4} + 4 q^{5} - 4 q^{7} + 9 q^{8} + 10 q^{10} + 6 q^{13} + q^{14} - 3 q^{16} - 8 q^{17} - 10 q^{19} + 10 q^{23} + 12 q^{25} + 20 q^{26} - 3 q^{28} - 18 q^{31} + 2 q^{32} + 18 q^{34} - 4 q^{35} - 2 q^{37} + 8 q^{38} + 6 q^{40} - 10 q^{41} - 4 q^{43} + 11 q^{46} - 4 q^{47} + 4 q^{49} + 9 q^{50} + 20 q^{52} - 9 q^{56} + 14 q^{58} + 16 q^{59} + 14 q^{61} - 11 q^{64} + 28 q^{65} - 28 q^{67} + 16 q^{68} - 10 q^{70} + 18 q^{71} - 4 q^{73} + 41 q^{74} - 4 q^{76} - 20 q^{79} + 36 q^{80} - 24 q^{82} - 6 q^{83} - 20 q^{85} + 20 q^{86} + 16 q^{89} - 6 q^{91} + 22 q^{92} - 16 q^{94} - 16 q^{95} + 32 q^{97} - q^{98}+O(q^{100}) $ 4 * q - q^2 + 3 * q^4 + 4 * q^5 - 4 * q^7 + 9 * q^8 + 10 * q^10 + 6 * q^13 + q^14 - 3 * q^16 - 8 * q^17 - 10 * q^19 + 10 * q^23 + 12 * q^25 + 20 * q^26 - 3 * q^28 - 18 * q^31 + 2 * q^32 + 18 * q^34 - 4 * q^35 - 2 * q^37 + 8 * q^38 + 6 * q^40 - 10 * q^41 - 4 * q^43 + 11 * q^46 - 4 * q^47 + 4 * q^49 + 9 * q^50 + 20 * q^52 - 9 * q^56 + 14 * q^58 + 16 * q^59 + 14 * q^61 - 11 * q^64 + 28 * q^65 - 28 * q^67 + 16 * q^68 - 10 * q^70 + 18 * q^71 - 4 * q^73 + 41 * q^74 - 4 * q^76 - 20 * q^79 + 36 * q^80 - 24 * q^82 - 6 * q^83 - 20 * q^85 + 20 * q^86 + 16 * q^89 - 6 * q^91 + 22 * q^92 - 16 * q^94 - 16 * q^95 + 32 * 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 $

$2$	2.39026	1.69017	0.845083	−	0.534634i	$-0.179551\pi$
$2$	2.39026	1.69017	0.845083	+	0.534634i	$0.179551\pi$
$3$	0	0
$3$	0	0
$4$	3.71333	1.85666
$5$	2.58993	1.15825	0.579125	−	0.815239i	$-0.303394\pi$
$5$	2.58993	1.15825	0.579125	+	0.815239i	$0.303394\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	4.09529	1.44791
$9$	0	0
$10$	6.19059	1.95764
$11$	0	0
$11$	0	0
$12$	0	0
$13$	6.78051	1.88058	0.940288	−	0.340380i	$-0.110556\pi$
$13$	6.78051	1.88058	0.940288	+	0.340380i	$0.110556\pi$
$14$	−2.39026	−0.638823
$15$	0	0
$16$	2.36215	0.590537
$17$	3.14511	0.762801	0.381400	−	0.924410i	$-0.375442\pi$
$17$	3.14511	0.762801	0.381400	+	0.924410i	$0.375442\pi$
$18$	0	0
$19$	−1.04548	−0.239850	−0.119925	−	0.992783i	$-0.538265\pi$
$19$	−1.04548	−0.239850	−0.119925	+	0.992783i	$0.538265\pi$
$20$	9.61724	2.15048
$21$	0	0
$22$	0	0
$23$	6.52707	1.36099	0.680495	−	0.732753i	$-0.261765\pi$
$23$	6.52707	1.36099	0.680495	+	0.732753i	$0.261765\pi$
$24$	0	0
$25$	1.70772	0.341543
$26$	16.2072	3.17849
$27$	0	0
$28$	−3.71333	−0.701753
$29$	0.607298	0.112772	0.0563862	−	0.998409i	$-0.482042\pi$
$29$	0.607298	0.112772	0.0563862	+	0.998409i	$0.482042\pi$
$30$	0	0
$31$	−8.83673	−1.58712	−0.793562	−	0.608490i	$-0.791776\pi$
$31$	−8.83673	−1.58712	−0.793562	+	0.608490i	$0.791776\pi$
$32$	−2.54445	−0.449799
$33$	0	0
$34$	7.51762	1.28926
$35$	−2.58993	−0.437777
$36$	0	0
$37$	8.94299	1.47022	0.735110	−	0.677948i	$-0.237131\pi$
$37$	8.94299	1.47022	0.735110	+	0.677948i	$0.237131\pi$
$38$	−2.49897	−0.405386
$39$	0	0
$40$	10.6065	1.67704
$41$	−8.69747	−1.35832	−0.679158	−	0.733992i	$-0.737655\pi$
$41$	−8.69747	−1.35832	−0.679158	+	0.733992i	$0.737655\pi$
$42$	0	0
$43$	4.48159	0.683437	0.341718	−	0.939802i	$-0.388991\pi$
$43$	4.48159	0.683437	0.341718	+	0.939802i	$0.388991\pi$
$44$	0	0
$45$	0	0
$46$	15.6014	2.30030
$47$	−3.26290	−0.475943	−0.237971	−	0.971272i	$-0.576482\pi$
$47$	−3.26290	−0.475943	−0.237971	+	0.971272i	$0.576482\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	4.08188	0.577265
$51$	0	0
$52$	25.1783	3.49160
$53$	−1.89958	−0.260928	−0.130464	−	0.991453i	$-0.541647\pi$
$53$	−1.89958	−0.260928	−0.130464	+	0.991453i	$0.541647\pi$
$54$	0	0
$55$	0	0
$56$	−4.09529	−0.547257
$57$	0	0
$58$	1.45160	0.190604
$59$	0.174006	0.0226537	0.0113268	−	0.999936i	$-0.496394\pi$
$59$	0.174006	0.0226537	0.0113268	+	0.999936i	$0.496394\pi$
$60$	0	0
$61$	8.13437	1.04150	0.520750	−	0.853709i	$-0.325652\pi$
$61$	8.13437	1.04150	0.520750	+	0.853709i	$0.325652\pi$
$62$	−21.1221	−2.68250
$63$	0	0
$64$	−10.8062	−1.35077
$65$	17.5610	2.17818
$66$	0	0
$67$	−7.33649	−0.896294	−0.448147	−	0.893960i	$-0.647916\pi$
$67$	−7.33649	−0.896294	−0.448147	+	0.893960i	$0.647916\pi$
$68$	11.6788	1.41626
$69$	0	0
$70$	−6.19059	−0.739917
$71$	−2.70693	−0.321253	−0.160626	−	0.987015i	$-0.551351\pi$
$71$	−2.70693	−0.321253	−0.160626	+	0.987015i	$0.551351\pi$
$72$	0	0
$73$	0.589926	0.0690456	0.0345228	−	0.999404i	$-0.489009\pi$
$73$	0.589926	0.0690456	0.0345228	+	0.999404i	$0.489009\pi$
$74$	21.3761	2.48492
$75$	0	0
$76$	−3.88221	−0.445320
$77$	0	0
$78$	0	0
$79$	−9.80862	−1.10356	−0.551778	−	0.833991i	$-0.686050\pi$
$79$	−9.80862	−1.10356	−0.551778	+	0.833991i	$0.686050\pi$
$80$	6.11779	0.683990
$81$	0	0
$82$	−20.7892	−2.29578
$83$	−8.74577	−0.959973	−0.479986	−	0.877276i	$-0.659358\pi$
$83$	−8.74577	−0.959973	−0.479986	+	0.877276i	$0.659358\pi$
$84$	0	0
$85$	8.14560	0.883514
$86$	10.7122	1.15512
$87$	0	0
$88$	0	0
$89$	11.6788	1.23795	0.618976	−	0.785410i	$-0.287548\pi$
$89$	11.6788	1.23795	0.618976	+	0.785410i	$0.287548\pi$
$90$	0	0
$91$	−6.78051	−0.710791
$92$	24.2372	2.52690
$93$	0	0
$94$	−7.79916	−0.804422
$95$	−2.70772	−0.277806
$96$	0	0
$97$	11.8528	1.20347	0.601736	−	0.798695i	$-0.294476\pi$
$97$	11.8528	1.20347	0.601736	+	0.798695i	$0.294476\pi$
$98$	2.39026	0.241452
$99$	0	0
$100$	6.34131	0.634131
$101$	−3.71845	−0.370000	−0.185000	−	0.982739i	$-0.559228\pi$
$101$	−3.71845	−0.370000	−0.185000	+	0.982739i	$0.559228\pi$
$102$	0	0
$103$	7.32703	0.721954	0.360977	−	0.932575i	$-0.382443\pi$
$103$	7.32703	0.721954	0.360977	+	0.932575i	$0.382443\pi$
$104$	27.7682	2.72290
$105$	0	0
$106$	−4.54049	−0.441011
$107$	−13.3804	−1.29353	−0.646765	−	0.762689i	$-0.723879\pi$
$107$	−13.3804	−1.29353	−0.646765	+	0.762689i	$0.723879\pi$
$108$	0	0
$109$	14.1228	1.35272	0.676362	−	0.736570i	$-0.263556\pi$
$109$	14.1228	1.35272	0.676362	+	0.736570i	$0.263556\pi$
$110$	0	0
$111$	0	0
$112$	−2.36215	−0.223202
$113$	2.66351	0.250562	0.125281	−	0.992121i	$-0.460017\pi$
$113$	2.66351	0.250562	0.125281	+	0.992121i	$0.460017\pi$
$114$	0	0
$115$	16.9046	1.57637
$116$	2.25510	0.209380
$117$	0	0
$118$	0.415920	0.0382885
$119$	−3.14511	−0.288312
$120$	0	0
$121$	0	0
$122$	19.4432	1.76031
$123$	0	0
$124$	−32.8137	−2.94676
$125$	−8.52677	−0.762658
$126$	0	0
$127$	−10.3633	−0.919596	−0.459798	−	0.888024i	$-0.652078\pi$
$127$	−10.3633	−0.919596	−0.459798	+	0.888024i	$0.652078\pi$
$128$	−20.7406	−1.83323
$129$	0	0
$130$	41.9754	3.68148
$131$	11.6441	1.01735	0.508674	−	0.860959i	$-0.330136\pi$
$131$	11.6441	1.01735	0.508674	+	0.860959i	$0.330136\pi$
$132$	0	0
$133$	1.04548	0.0906546
$134$	−17.5361	−1.51489
$135$	0	0
$136$	12.8801	1.10446
$137$	15.4895	1.32336	0.661679	−	0.749787i	$-0.269844\pi$
$137$	15.4895	1.32336	0.661679	+	0.749787i	$0.269844\pi$
$138$	0	0
$139$	−5.81808	−0.493483	−0.246742	−	0.969081i	$-0.579360\pi$
$139$	−5.81808	−0.493483	−0.246742	+	0.969081i	$0.579360\pi$
$140$	−9.61724	−0.812805
$141$	0	0
$142$	−6.47025	−0.542971
$143$	0	0
$144$	0	0
$145$	1.57286	0.130619
$146$	1.41007	0.116699
$147$	0	0
$148$	33.2083	2.72970
$149$	14.4143	1.18087	0.590434	−	0.807086i	$-0.298956\pi$
$149$	14.4143	1.18087	0.590434	+	0.807086i	$0.298956\pi$
$150$	0	0
$151$	−22.2171	−1.80800	−0.904002	−	0.427529i	$-0.859384\pi$
$151$	−22.2171	−1.80800	−0.904002	+	0.427529i	$0.859384\pi$
$152$	−4.28155	−0.347279
$153$	0	0
$154$	0	0
$155$	−22.8865	−1.83829
$156$	0	0
$157$	−9.79125	−0.781427	−0.390713	−	0.920512i	$-0.627772\pi$
$157$	−9.79125	−0.781427	−0.390713	+	0.920512i	$0.627772\pi$
$158$	−23.4451	−1.86519
$159$	0	0
$160$	−6.58993	−0.520979
$161$	−6.52707	−0.514405
$162$	0	0
$163$	11.3638	0.890082	0.445041	−	0.895510i	$-0.353189\pi$
$163$	11.3638	0.890082	0.445041	+	0.895510i	$0.353189\pi$
$164$	−32.2965	−2.52194
$165$	0	0
$166$	−20.9046	−1.62251
$167$	5.08889	0.393790	0.196895	−	0.980425i	$-0.436914\pi$
$167$	5.08889	0.393790	0.196895	+	0.980425i	$0.436914\pi$
$168$	0	0
$169$	32.9754	2.53657
$170$	19.4701	1.49329
$171$	0	0
$172$	16.6416	1.26891
$173$	−7.77467	−0.591097	−0.295549	−	0.955328i	$-0.595502\pi$
$173$	−7.77467	−0.591097	−0.295549	+	0.955328i	$0.595502\pi$
$174$	0	0
$175$	−1.70772	−0.129091
$176$	0	0
$177$	0	0
$178$	27.9154	2.09235
$179$	6.14590	0.459366	0.229683	−	0.973265i	$-0.426231\pi$
$179$	6.14590	0.459366	0.229683	+	0.973265i	$0.426231\pi$
$180$	0	0
$181$	−9.25801	−0.688142	−0.344071	−	0.938944i	$-0.611806\pi$
$181$	−9.25801	−0.688142	−0.344071	+	0.938944i	$0.611806\pi$
$182$	−16.2072	−1.20136
$183$	0	0
$184$	26.7303	1.97058
$185$	23.1617	1.70288
$186$	0	0
$187$	0	0
$188$	−12.1162	−0.883665
$189$	0	0
$190$	−6.47214	−0.469538
$191$	14.6374	1.05913	0.529564	−	0.848270i	$-0.322356\pi$
$191$	14.6374	1.05913	0.529564	+	0.848270i	$0.322356\pi$
$192$	0	0
$193$	−16.4713	−1.18563	−0.592817	−	0.805337i	$-0.701984\pi$
$193$	−16.4713	−1.18563	−0.592817	+	0.805337i	$0.701984\pi$
$194$	28.3313	2.03407
$195$	0	0
$196$	3.71333	0.265238
$197$	6.81654	0.485658	0.242829	−	0.970069i	$-0.421925\pi$
$197$	6.81654	0.485658	0.242829	+	0.970069i	$0.421925\pi$
$198$	0	0
$199$	15.5561	1.10275	0.551373	−	0.834259i	$-0.314104\pi$
$199$	15.5561	1.10275	0.551373	+	0.834259i	$0.314104\pi$
$200$	6.99360	0.494522
$201$	0	0
$202$	−8.88806	−0.625361
$203$	−0.607298	−0.0426239
$204$	0	0
$205$	−22.5258	−1.57327
$206$	17.5135	1.22022
$207$	0	0
$208$	16.0166	1.11055
$209$	0	0
$210$	0	0
$211$	1.56464	0.107714	0.0538571	−	0.998549i	$-0.482848\pi$
$211$	1.56464	0.107714	0.0538571	+	0.998549i	$0.482848\pi$
$212$	−7.05377	−0.484455
$213$	0	0
$214$	−31.9826	−2.18628
$215$	11.6070	0.791591
$216$	0	0
$217$	8.83673	0.599876
$218$	33.7572	2.28633
$219$	0	0
$220$	0	0
$221$	21.3254	1.43450
$222$	0	0
$223$	4.13926	0.277186	0.138593	−	0.990349i	$-0.455742\pi$
$223$	4.13926	0.277186	0.138593	+	0.990349i	$0.455742\pi$
$224$	2.54445	0.170008
$225$	0	0
$226$	6.36648	0.423492
$227$	5.57761	0.370199	0.185099	−	0.982720i	$-0.440739\pi$
$227$	5.57761	0.370199	0.185099	+	0.982720i	$0.440739\pi$
$228$	0	0
$229$	−0.788428	−0.0521008	−0.0260504	−	0.999661i	$-0.508293\pi$
$229$	−0.788428	−0.0521008	−0.0260504	+	0.999661i	$0.508293\pi$
$230$	40.4064	2.66432
$231$	0	0
$232$	2.48706	0.163284
$233$	−27.7066	−1.81512	−0.907561	−	0.419921i	$-0.862058\pi$
$233$	−27.7066	−1.81512	−0.907561	+	0.419921i	$0.862058\pi$
$234$	0	0
$235$	−8.45066	−0.551260
$236$	0.646142	0.0420603
$237$	0	0
$238$	−7.51762	−0.487295
$239$	8.58122	0.555073	0.277537	−	0.960715i	$-0.410482\pi$
$239$	8.58122	0.555073	0.277537	+	0.960715i	$0.410482\pi$
$240$	0	0
$241$	−19.6392	−1.26507	−0.632535	−	0.774531i	$-0.717986\pi$
$241$	−19.6392	−1.26507	−0.632535	+	0.774531i	$0.717986\pi$
$242$	0	0
$243$	0	0
$244$	30.2056	1.93371
$245$	2.58993	0.165464
$246$	0	0
$247$	−7.08889	−0.451055
$248$	−36.1890	−2.29800
$249$	0	0
$250$	−20.3812	−1.28902
$251$	29.8538	1.88436	0.942178	−	0.335114i	$-0.108775\pi$
$251$	29.8538	1.88436	0.942178	+	0.335114i	$0.108775\pi$
$252$	0	0
$253$	0	0
$254$	−24.7710	−1.55427
$255$	0	0
$256$	−27.9631	−1.74769
$257$	−2.83673	−0.176950	−0.0884752	−	0.996078i	$-0.528199\pi$
$257$	−2.83673	−0.176950	−0.0884752	+	0.996078i	$0.528199\pi$
$258$	0	0
$259$	−8.94299	−0.555691
$260$	65.2099	4.04414
$261$	0	0
$262$	27.8323	1.71949
$263$	−10.2373	−0.631262	−0.315631	−	0.948882i	$-0.602216\pi$
$263$	−10.2373	−0.631262	−0.315631	+	0.948882i	$0.602216\pi$
$264$	0	0
$265$	−4.91978	−0.302219
$266$	2.49897	0.153221
$267$	0	0
$268$	−27.2428	−1.66412
$269$	−10.0483	−0.612656	−0.306328	−	0.951926i	$-0.599100\pi$
$269$	−10.0483	−0.612656	−0.306328	+	0.951926i	$0.599100\pi$
$270$	0	0
$271$	−2.50432	−0.152127	−0.0760634	−	0.997103i	$-0.524235\pi$
$271$	−2.50432	−0.152127	−0.0760634	+	0.997103i	$0.524235\pi$
$272$	7.42921	0.450462
$273$	0	0
$274$	37.0239	2.23670
$275$	0	0
$276$	0	0
$277$	−14.9108	−0.895904	−0.447952	−	0.894058i	$-0.647846\pi$
$277$	−14.9108	−0.895904	−0.447952	+	0.894058i	$0.647846\pi$
$278$	−13.9067	−0.834069
$279$	0	0
$280$	−10.6065	−0.633860
$281$	−2.04676	−0.122099	−0.0610497	−	0.998135i	$-0.519445\pi$
$281$	−2.04676	−0.122099	−0.0610497	+	0.998135i	$0.519445\pi$
$282$	0	0
$283$	17.0723	1.01484	0.507422	−	0.861698i	$-0.330599\pi$
$283$	17.0723	1.01484	0.507422	+	0.861698i	$0.330599\pi$
$284$	−10.0517	−0.596459
$285$	0	0
$286$	0	0
$287$	8.69747	0.513395
$288$	0	0
$289$	−7.10830	−0.418135
$290$	3.75953	0.220767
$291$	0	0
$292$	2.19059	0.128194
$293$	26.7355	1.56191	0.780953	−	0.624590i	$-0.214734\pi$
$293$	26.7355	1.56191	0.780953	+	0.624590i	$0.214734\pi$
$294$	0	0
$295$	0.450663	0.0262386
$296$	36.6242	2.12874
$297$	0	0
$298$	34.4540	1.99587
$299$	44.2569	2.55944
$300$	0	0
$301$	−4.48159	−0.258315
$302$	−53.1046	−3.05583
$303$	0	0
$304$	−2.46958	−0.141640
$305$	21.0674	1.20632
$306$	0	0
$307$	−16.3329	−0.932166	−0.466083	−	0.884741i	$-0.654335\pi$
$307$	−16.3329	−0.932166	−0.466083	+	0.884741i	$0.654335\pi$
$308$	0	0
$309$	0	0
$310$	−54.7046	−3.10701
$311$	−26.0663	−1.47809	−0.739043	−	0.673658i	$-0.764722\pi$
$311$	−26.0663	−1.47809	−0.739043	+	0.673658i	$0.764722\pi$
$312$	0	0
$313$	−21.4866	−1.21450	−0.607249	−	0.794512i	$-0.707727\pi$
$313$	−21.4866	−1.21450	−0.607249	+	0.794512i	$0.707727\pi$
$314$	−23.4036	−1.32074
$315$	0	0
$316$	−36.4226	−2.04893
$317$	24.8208	1.39408	0.697038	−	0.717035i	$-0.254501\pi$
$317$	24.8208	1.39408	0.697038	+	0.717035i	$0.254501\pi$
$318$	0	0
$319$	0	0
$320$	−27.9872	−1.56453
$321$	0	0
$322$	−15.6014	−0.869431
$323$	−3.28815	−0.182957
$324$	0	0
$325$	11.5792	0.642298
$326$	27.1624	1.50439
$327$	0	0
$328$	−35.6187	−1.96671
$329$	3.26290	0.179889
$330$	0	0
$331$	−31.6351	−1.73882	−0.869411	−	0.494089i	$-0.835502\pi$
$331$	−31.6351	−1.73882	−0.869411	+	0.494089i	$0.835502\pi$
$332$	−32.4759	−1.78235
$333$	0	0
$334$	12.1638	0.665571
$335$	−19.0010	−1.03813
$336$	0	0
$337$	−32.8075	−1.78714	−0.893569	−	0.448925i	$-0.851807\pi$
$337$	−32.8075	−1.78714	−0.893569	+	0.448925i	$0.851807\pi$
$338$	78.8196	4.28722
$339$	0	0
$340$	30.2473	1.64039
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	18.3534	0.989551
$345$	0	0
$346$	−18.5835	−0.999053
$347$	−14.0886	−0.756315	−0.378158	−	0.925741i	$-0.623442\pi$
$347$	−14.0886	−0.756315	−0.378158	+	0.925741i	$0.623442\pi$
$348$	0	0
$349$	−22.2403	−1.19050	−0.595249	−	0.803541i	$-0.702947\pi$
$349$	−22.2403	−1.19050	−0.595249	+	0.803541i	$0.702947\pi$
$350$	−4.08188	−0.218186
$351$	0	0
$352$	0	0
$353$	−15.2345	−0.810850	−0.405425	−	0.914128i	$-0.632876\pi$
$353$	−15.2345	−0.810850	−0.405425	+	0.914128i	$0.632876\pi$
$354$	0	0
$355$	−7.01074	−0.372091
$356$	43.3673	2.29846
$357$	0	0
$358$	14.6903	0.776405
$359$	−15.4483	−0.815330	−0.407665	−	0.913132i	$-0.633657\pi$
$359$	−15.4483	−0.815330	−0.407665	+	0.913132i	$0.633657\pi$
$360$	0	0
$361$	−17.9070	−0.942472
$362$	−22.1290	−1.16308
$363$	0	0
$364$	−25.1783	−1.31970
$365$	1.52786	0.0799721
$366$	0	0
$367$	28.3390	1.47928	0.739641	−	0.673001i	$-0.234995\pi$
$367$	28.3390	1.47928	0.739641	+	0.673001i	$0.234995\pi$
$368$	15.4179	0.803715
$369$	0	0
$370$	55.3624	2.87815
$371$	1.89958	0.0986214
$372$	0	0
$373$	−1.01940	−0.0527828	−0.0263914	−	0.999652i	$-0.508402\pi$
$373$	−1.01940	−0.0527828	−0.0263914	+	0.999652i	$0.508402\pi$
$374$	0	0
$375$	0	0
$376$	−13.3625	−0.689120
$377$	4.11779	0.212077
$378$	0	0
$379$	−8.90750	−0.457547	−0.228774	−	0.973480i	$-0.573472\pi$
$379$	−8.90750	−0.457547	−0.228774	+	0.973480i	$0.573472\pi$
$380$	−10.0546	−0.515792
$381$	0	0
$382$	34.9872	1.79010
$383$	−9.80783	−0.501157	−0.250578	−	0.968096i	$-0.580621\pi$
$383$	−9.80783	−0.501157	−0.250578	+	0.968096i	$0.580621\pi$
$384$	0	0
$385$	0	0
$386$	−39.3707	−2.00392
$387$	0	0
$388$	44.0134	2.23444
$389$	−24.1420	−1.22405	−0.612024	−	0.790840i	$-0.709644\pi$
$389$	−24.1420	−1.22405	−0.612024	+	0.790840i	$0.709644\pi$
$390$	0	0
$391$	20.5284	1.03816
$392$	4.09529	0.206844
$393$	0	0
$394$	16.2933	0.820843
$395$	−25.4036	−1.27819
$396$	0	0
$397$	20.6730	1.03755	0.518773	−	0.854912i	$-0.326389\pi$
$397$	20.6730	1.03755	0.518773	+	0.854912i	$0.326389\pi$
$398$	37.1832	1.86382
$399$	0	0
$400$	4.03388	0.201694
$401$	25.2166	1.25926	0.629629	−	0.776896i	$-0.283207\pi$
$401$	25.2166	1.25926	0.629629	+	0.776896i	$0.283207\pi$
$402$	0	0
$403$	−59.9176	−2.98471
$404$	−13.8078	−0.686965
$405$	0	0
$406$	−1.45160	−0.0720416
$407$	0	0
$408$	0	0
$409$	10.1558	0.502174	0.251087	−	0.967964i	$-0.419212\pi$
$409$	10.1558	0.502174	0.251087	+	0.967964i	$0.419212\pi$
$410$	−53.8424	−2.65909
$411$	0	0
$412$	27.2077	1.34043
$413$	−0.174006	−0.00856229
$414$	0	0
$415$	−22.6509	−1.11189
$416$	−17.2526	−0.845881
$417$	0	0
$418$	0	0
$419$	28.3684	1.38589	0.692943	−	0.720993i	$-0.256314\pi$
$419$	28.3684	1.38589	0.692943	+	0.720993i	$0.256314\pi$
$420$	0	0
$421$	−27.6839	−1.34923	−0.674615	−	0.738170i	$-0.735690\pi$
$421$	−27.6839	−1.34923	−0.674615	+	0.738170i	$0.735690\pi$
$422$	3.73989	0.182055
$423$	0	0
$424$	−7.77935	−0.377798
$425$	5.37095	0.260529
$426$	0	0
$427$	−8.13437	−0.393650
$428$	−49.6858	−2.40165
$429$	0	0
$430$	27.7437	1.33792
$431$	−7.31248	−0.352230	−0.176115	−	0.984370i	$-0.556353\pi$
$431$	−7.31248	−0.352230	−0.176115	+	0.984370i	$0.556353\pi$
$432$	0	0
$433$	−10.5321	−0.506142	−0.253071	−	0.967448i	$-0.581441\pi$
$433$	−10.5321	−0.506142	−0.253071	+	0.967448i	$0.581441\pi$
$434$	21.1221	1.01389
$435$	0	0
$436$	52.4428	2.51155
$437$	−6.82393	−0.326433
$438$	0	0
$439$	−0.574098	−0.0274002	−0.0137001	−	0.999906i	$-0.504361\pi$
$439$	−0.574098	−0.0274002	−0.0137001	+	0.999906i	$0.504361\pi$
$440$	0	0
$441$	0	0
$442$	50.9733	2.42455
$443$	18.2117	0.865266	0.432633	−	0.901570i	$-0.357585\pi$
$443$	18.2117	0.865266	0.432633	+	0.901570i	$0.357585\pi$
$444$	0	0
$445$	30.2473	1.43386
$446$	9.89390	0.468490
$447$	0	0
$448$	10.8062	0.510544
$449$	−4.43378	−0.209243	−0.104622	−	0.994512i	$-0.533363\pi$
$449$	−4.43378	−0.209243	−0.104622	+	0.994512i	$0.533363\pi$
$450$	0	0
$451$	0	0
$452$	9.89050	0.465210
$453$	0	0
$454$	13.3319	0.625698
$455$	−17.5610	−0.823274
$456$	0	0
$457$	−20.2931	−0.949270	−0.474635	−	0.880183i	$-0.657420\pi$
$457$	−20.2931	−0.949270	−0.474635	+	0.880183i	$0.657420\pi$
$458$	−1.88454	−0.0880590
$459$	0	0
$460$	62.7725	2.92678
$461$	4.85723	0.226224	0.113112	−	0.993582i	$-0.463918\pi$
$461$	4.85723	0.226224	0.113112	+	0.993582i	$0.463918\pi$
$462$	0	0
$463$	−19.5168	−0.907024	−0.453512	−	0.891250i	$-0.649829\pi$
$463$	−19.5168	−0.907024	−0.453512	+	0.891250i	$0.649829\pi$
$464$	1.43453	0.0665963
$465$	0	0
$466$	−66.2259	−3.06786
$467$	2.74624	0.127081	0.0635403	−	0.997979i	$-0.479761\pi$
$467$	2.74624	0.127081	0.0635403	+	0.997979i	$0.479761\pi$
$468$	0	0
$469$	7.33649	0.338767
$470$	−20.1993	−0.931722
$471$	0	0
$472$	0.712607	0.0328004
$473$	0	0
$474$	0	0
$475$	−1.78538	−0.0819190
$476$	−11.6788	−0.535298
$477$	0	0
$478$	20.5113	0.938166
$479$	18.0252	0.823595	0.411797	−	0.911275i	$-0.364901\pi$
$479$	18.0252	0.823595	0.411797	+	0.911275i	$0.364901\pi$
$480$	0	0
$481$	60.6381	2.76486
$482$	−46.9427	−2.13818
$483$	0	0
$484$	0	0
$485$	30.6979	1.39392
$486$	0	0
$487$	−12.3834	−0.561146	−0.280573	−	0.959833i	$-0.590524\pi$
$487$	−12.3834	−0.561146	−0.280573	+	0.959833i	$0.590524\pi$
$488$	33.3126	1.50799
$489$	0	0
$490$	6.19059	0.279662
$491$	−16.6168	−0.749904	−0.374952	−	0.927044i	$-0.622341\pi$
$491$	−16.6168	−0.749904	−0.374952	+	0.927044i	$0.622341\pi$
$492$	0	0
$493$	1.91002	0.0860228
$494$	−16.9443	−0.762359
$495$	0	0
$496$	−20.8737	−0.937255
$497$	2.70693	0.121422
$498$	0	0
$499$	−20.3850	−0.912557	−0.456279	−	0.889837i	$-0.650818\pi$
$499$	−20.3850	−0.912557	−0.456279	+	0.889837i	$0.650818\pi$
$500$	−31.6627	−1.41600
$501$	0	0
$502$	71.3583	3.18487
$503$	−6.68164	−0.297920	−0.148960	−	0.988843i	$-0.547593\pi$
$503$	−6.68164	−0.297920	−0.148960	+	0.988843i	$0.547593\pi$
$504$	0	0
$505$	−9.63051	−0.428552
$506$	0	0
$507$	0	0
$508$	−38.4824	−1.70738
$509$	18.3114	0.811639	0.405819	−	0.913953i	$-0.366986\pi$
$509$	18.3114	0.811639	0.405819	+	0.913953i	$0.366986\pi$
$510$	0	0
$511$	−0.589926	−0.0260968
$512$	−25.3577	−1.12066
$513$	0	0
$514$	−6.78051	−0.299076
$515$	18.9765	0.836203
$516$	0	0
$517$	0	0
$518$	−21.3761	−0.939210
$519$	0	0
$520$	71.9176	3.15379
$521$	0.587369	0.0257331	0.0128666	−	0.999917i	$-0.495904\pi$
$521$	0.587369	0.0257331	0.0128666	+	0.999917i	$0.495904\pi$
$522$	0	0
$523$	28.6098	1.25102	0.625510	−	0.780216i	$-0.284891\pi$
$523$	28.6098	1.25102	0.625510	+	0.780216i	$0.284891\pi$
$524$	43.2383	1.88887
$525$	0	0
$526$	−24.4699	−1.06694
$527$	−27.7925	−1.21066
$528$	0	0
$529$	19.6027	0.852291
$530$	−11.7595	−0.510801
$531$	0	0
$532$	3.88221	0.168315
$533$	−58.9733	−2.55442
$534$	0	0
$535$	−34.6542	−1.49823
$536$	−30.0451	−1.29775
$537$	0	0
$538$	−24.0180	−1.03549
$539$	0	0
$540$	0	0
$541$	6.17906	0.265659	0.132829	−	0.991139i	$-0.457594\pi$
$541$	6.17906	0.265659	0.132829	+	0.991139i	$0.457594\pi$
$542$	−5.98597	−0.257120
$543$	0	0
$544$	−8.00256	−0.343107
$545$	36.5771	1.56679
$546$	0	0
$547$	−6.05464	−0.258878	−0.129439	−	0.991587i	$-0.541318\pi$
$547$	−6.05464	−0.258878	−0.129439	+	0.991587i	$0.541318\pi$
$548$	57.5176	2.45703
$549$	0	0
$550$	0	0
$551$	−0.634918	−0.0270484
$552$	0	0
$553$	9.80862	0.417105
$554$	−35.6407	−1.51423
$555$	0	0
$556$	−21.6044	−0.916232
$557$	5.98799	0.253719	0.126860	−	0.991921i	$-0.459510\pi$
$557$	5.98799	0.253719	0.126860	+	0.991921i	$0.459510\pi$
$558$	0	0
$559$	30.3875	1.28525
$560$	−6.11779	−0.258524
$561$	0	0
$562$	−4.89228	−0.206368
$563$	−36.1739	−1.52455	−0.762273	−	0.647255i	$-0.775917\pi$
$563$	−36.1739	−1.52455	−0.762273	+	0.647255i	$0.775917\pi$
$564$	0	0
$565$	6.89830	0.290214
$566$	40.8072	1.71525
$567$	0	0
$568$	−11.0857	−0.465144
$569$	−33.4701	−1.40314	−0.701569	−	0.712601i	$-0.747517\pi$
$569$	−33.4701	−1.40314	−0.701569	+	0.712601i	$0.747517\pi$
$570$	0	0
$571$	−4.96371	−0.207725	−0.103862	−	0.994592i	$-0.533120\pi$
$571$	−4.96371	−0.207725	−0.103862	+	0.994592i	$0.533120\pi$
$572$	0	0
$573$	0	0
$574$	20.7892	0.867724
$575$	11.1464	0.464836
$576$	0	0
$577$	12.1249	0.504765	0.252383	−	0.967628i	$-0.418786\pi$
$577$	12.1249	0.504765	0.252383	+	0.967628i	$0.418786\pi$
$578$	−16.9907	−0.706718
$579$	0	0
$580$	5.84053	0.242515
$581$	8.74577	0.362836
$582$	0	0
$583$	0	0
$584$	2.41592	0.0999715
$585$	0	0
$586$	63.9048	2.63988
$587$	4.79565	0.197938	0.0989689	−	0.995091i	$-0.468446\pi$
$587$	4.79565	0.197938	0.0989689	+	0.995091i	$0.468446\pi$
$588$	0	0
$589$	9.23862	0.380671
$590$	1.07720	0.0443477
$591$	0	0
$592$	21.1247	0.868219
$593$	−15.3387	−0.629886	−0.314943	−	0.949111i	$-0.601985\pi$
$593$	−15.3387	−0.629886	−0.314943	+	0.949111i	$0.601985\pi$
$594$	0	0
$595$	−8.14560	−0.333937
$596$	53.5252	2.19248
$597$	0	0
$598$	105.785	4.32589
$599$	46.1427	1.88534	0.942671	−	0.333725i	$-0.108306\pi$
$599$	46.1427	1.88534	0.942671	+	0.333725i	$0.108306\pi$
$600$	0	0
$601$	31.2352	1.27411	0.637056	−	0.770817i	$-0.280152\pi$
$601$	31.2352	1.27411	0.637056	+	0.770817i	$0.280152\pi$
$602$	−10.7122	−0.436595
$603$	0	0
$604$	−82.4994	−3.35685
$605$	0	0
$606$	0	0
$607$	−14.7306	−0.597898	−0.298949	−	0.954269i	$-0.596636\pi$
$607$	−14.7306	−0.597898	−0.298949	+	0.954269i	$0.596636\pi$
$608$	2.66017	0.107884
$609$	0	0
$610$	50.3565	2.03888
$611$	−22.1241	−0.895046
$612$	0	0
$613$	−27.6123	−1.11525	−0.557625	−	0.830093i	$-0.688287\pi$
$613$	−27.6123	−1.11525	−0.557625	+	0.830093i	$0.688287\pi$
$614$	−39.0398	−1.57552
$615$	0	0
$616$	0	0
$617$	−3.29610	−0.132696	−0.0663479	−	0.997797i	$-0.521135\pi$
$617$	−3.29610	−0.132696	−0.0663479	+	0.997797i	$0.521135\pi$
$618$	0	0
$619$	−3.60713	−0.144983	−0.0724915	−	0.997369i	$-0.523095\pi$
$619$	−3.60713	−0.144983	−0.0724915	+	0.997369i	$0.523095\pi$
$620$	−84.9850	−3.41308
$621$	0	0
$622$	−62.3052	−2.49821
$623$	−11.6788	−0.467902
$624$	0	0
$625$	−30.6223	−1.22489
$626$	−51.3586	−2.05270
$627$	0	0
$628$	−36.3581	−1.45085
$629$	28.1267	1.12148
$630$	0	0
$631$	−43.1721	−1.71865	−0.859327	−	0.511426i	$-0.829117\pi$
$631$	−43.1721	−1.71865	−0.859327	+	0.511426i	$0.829117\pi$
$632$	−40.1692	−1.59784
$633$	0	0
$634$	59.3281	2.35622
$635$	−26.8402	−1.06512
$636$	0	0
$637$	6.78051	0.268654
$638$	0	0
$639$	0	0
$640$	−53.7167	−2.12334
$641$	−24.9711	−0.986298	−0.493149	−	0.869945i	$-0.664154\pi$
$641$	−24.9711	−0.986298	−0.493149	+	0.869945i	$0.664154\pi$
$642$	0	0
$643$	−8.14304	−0.321130	−0.160565	−	0.987025i	$-0.551332\pi$
$643$	−8.14304	−0.321130	−0.160565	+	0.987025i	$0.551332\pi$
$644$	−24.2372	−0.955078
$645$	0	0
$646$	−7.85952	−0.309229
$647$	−39.8790	−1.56781	−0.783904	−	0.620883i	$-0.786774\pi$
$647$	−39.8790	−1.56781	−0.783904	+	0.620883i	$0.786774\pi$
$648$	0	0
$649$	0	0
$650$	27.6772	1.08559
$651$	0	0
$652$	42.1975	1.65258
$653$	−1.63668	−0.0640484	−0.0320242	−	0.999487i	$-0.510195\pi$
$653$	−1.63668	−0.0640484	−0.0320242	+	0.999487i	$0.510195\pi$
$654$	0	0
$655$	30.1573	1.17834
$656$	−20.5447	−0.802136
$657$	0	0
$658$	7.79916	0.304043
$659$	7.63149	0.297281	0.148640	−	0.988891i	$-0.452510\pi$
$659$	7.63149	0.297281	0.148640	+	0.988891i	$0.452510\pi$
$660$	0	0
$661$	20.4402	0.795031	0.397516	−	0.917595i	$-0.369872\pi$
$661$	20.4402	0.795031	0.397516	+	0.917595i	$0.369872\pi$
$662$	−75.6160	−2.93890
$663$	0	0
$664$	−35.8165	−1.38995
$665$	2.70772	0.105001
$666$	0	0
$667$	3.96388	0.153482
$668$	18.8967	0.731136
$669$	0	0
$670$	−45.4172	−1.75462
$671$	0	0
$672$	0	0
$673$	−24.0554	−0.927269	−0.463634	−	0.886027i	$-0.653455\pi$
$673$	−24.0554	−0.927269	−0.463634	+	0.886027i	$0.653455\pi$
$674$	−78.4184	−3.02056
$675$	0	0
$676$	122.448	4.70955
$677$	8.04672	0.309261	0.154630	−	0.987972i	$-0.450581\pi$
$677$	8.04672	0.309261	0.154630	+	0.987972i	$0.450581\pi$
$678$	0	0
$679$	−11.8528	−0.454870
$680$	33.3586	1.27924
$681$	0	0
$682$	0	0
$683$	19.2894	0.738089	0.369045	−	0.929412i	$-0.379685\pi$
$683$	19.2894	0.738089	0.369045	+	0.929412i	$0.379685\pi$
$684$	0	0
$685$	40.1167	1.53278
$686$	−2.39026	−0.0912604
$687$	0	0
$688$	10.5862	0.403595
$689$	−12.8801	−0.490694
$690$	0	0
$691$	−31.8589	−1.21197	−0.605985	−	0.795476i	$-0.707221\pi$
$691$	−31.8589	−1.21197	−0.605985	+	0.795476i	$0.707221\pi$
$692$	−28.8699	−1.09747
$693$	0	0
$694$	−33.6753	−1.27830
$695$	−15.0684	−0.571577
$696$	0	0
$697$	−27.3545	−1.03612
$698$	−53.1601	−2.01214
$699$	0	0
$700$	−6.34131	−0.239679
$701$	35.0720	1.32465	0.662326	−	0.749216i	$-0.269570\pi$
$701$	35.0720	1.32465	0.662326	+	0.749216i	$0.269570\pi$
$702$	0	0
$703$	−9.34972	−0.352631
$704$	0	0
$705$	0	0
$706$	−36.4143	−1.37047
$707$	3.71845	0.139847
$708$	0	0
$709$	−13.9057	−0.522239	−0.261120	−	0.965306i	$-0.584092\pi$
$709$	−13.9057	−0.522239	−0.261120	+	0.965306i	$0.584092\pi$
$710$	−16.7575	−0.628896
$711$	0	0
$712$	47.8282	1.79244
$713$	−57.6780	−2.16006
$714$	0	0
$715$	0	0
$716$	22.8217	0.852888
$717$	0	0
$718$	−36.9254	−1.37804
$719$	40.9046	1.52549	0.762743	−	0.646702i	$-0.223852\pi$
$719$	40.9046	1.52549	0.762743	+	0.646702i	$0.223852\pi$
$720$	0	0
$721$	−7.32703	−0.272873
$722$	−42.8023	−1.59294
$723$	0	0
$724$	−34.3780	−1.27765
$725$	1.03709	0.0385166
$726$	0	0
$727$	−28.5963	−1.06058	−0.530288	−	0.847817i	$-0.677916\pi$
$727$	−28.5963	−1.06058	−0.530288	+	0.847817i	$0.677916\pi$
$728$	−27.7682	−1.02916
$729$	0	0
$730$	3.65199	0.135166
$731$	14.0951	0.521326
$732$	0	0
$733$	15.3513	0.567013	0.283507	−	0.958970i	$-0.408502\pi$
$733$	15.3513	0.567013	0.283507	+	0.958970i	$0.408502\pi$
$734$	67.7375	2.50024
$735$	0	0
$736$	−16.6078	−0.612171
$737$	0	0
$738$	0	0
$739$	30.8571	1.13510	0.567549	−	0.823340i	$-0.307892\pi$
$739$	30.8571	1.13510	0.567549	+	0.823340i	$0.307892\pi$
$740$	86.0070	3.16168
$741$	0	0
$742$	4.54049	0.166687
$743$	42.5061	1.55940	0.779699	−	0.626155i	$-0.215372\pi$
$743$	42.5061	1.55940	0.779699	+	0.626155i	$0.215372\pi$
$744$	0	0
$745$	37.3321	1.36774
$746$	−2.43664	−0.0892117
$747$	0	0
$748$	0	0
$749$	13.3804	0.488909
$750$	0	0
$751$	21.0811	0.769262	0.384631	−	0.923070i	$-0.374329\pi$
$751$	21.0811	0.769262	0.384631	+	0.923070i	$0.374329\pi$
$752$	−7.70745	−0.281062
$753$	0	0
$754$	9.84258	0.358445
$755$	−57.5407	−2.09412
$756$	0	0
$757$	42.7360	1.55326	0.776632	−	0.629954i	$-0.216926\pi$
$757$	42.7360	1.55326	0.776632	+	0.629954i	$0.216926\pi$
$758$	−21.2912	−0.773331
$759$	0	0
$760$	−11.0889	−0.402236
$761$	13.9961	0.507358	0.253679	−	0.967288i	$-0.418359\pi$
$761$	13.9961	0.507358	0.253679	+	0.967288i	$0.418359\pi$
$762$	0	0
$763$	−14.1228	−0.511281
$764$	54.3536	1.96644
$765$	0	0
$766$	−23.4432	−0.847039
$767$	1.17985	0.0426020
$768$	0	0
$769$	23.8142	0.858761	0.429380	−	0.903124i	$-0.358732\pi$
$769$	23.8142	0.858761	0.429380	+	0.903124i	$0.358732\pi$
$770$	0	0
$771$	0	0
$772$	−61.1635	−2.20132
$773$	43.0321	1.54776	0.773878	−	0.633335i	$-0.218314\pi$
$773$	43.0321	1.54776	0.773878	+	0.633335i	$0.218314\pi$
$774$	0	0
$775$	−15.0906	−0.542071
$776$	48.5408	1.74251
$777$	0	0
$778$	−57.7055	−2.06884
$779$	9.09303	0.325792
$780$	0	0
$781$	0	0
$782$	49.0680	1.75467
$783$	0	0
$784$	2.36215	0.0843625
$785$	−25.3586	−0.905088
$786$	0	0
$787$	31.3512	1.11755	0.558774	−	0.829320i	$-0.311272\pi$
$787$	31.3512	1.11755	0.558774	+	0.829320i	$0.311272\pi$
$788$	25.3120	0.901704
$789$	0	0
$790$	−60.7211	−2.16036
$791$	−2.66351	−0.0947037
$792$	0	0
$793$	55.1552	1.95862
$794$	49.4137	1.75363
$795$	0	0
$796$	57.7650	2.04743
$797$	6.95708	0.246432	0.123216	−	0.992380i	$-0.460679\pi$
$797$	6.95708	0.246432	0.123216	+	0.992380i	$0.460679\pi$
$798$	0	0
$799$	−10.2622	−0.363049
$800$	−4.34519	−0.153626
$801$	0	0
$802$	60.2742	2.12836
$803$	0	0
$804$	0	0
$805$	−16.9046	−0.595810
$806$	−143.218	−5.04465
$807$	0	0
$808$	−15.2282	−0.535725
$809$	−49.7873	−1.75043	−0.875214	−	0.483736i	$-0.839279\pi$
$809$	−49.7873	−1.75043	−0.875214	+	0.483736i	$0.839279\pi$
$810$	0	0
$811$	−4.75368	−0.166924	−0.0834622	−	0.996511i	$-0.526598\pi$
$811$	−4.75368	−0.166924	−0.0834622	+	0.996511i	$0.526598\pi$
$812$	−2.25510	−0.0791383
$813$	0	0
$814$	0	0
$815$	29.4314	1.03094
$816$	0	0
$817$	−4.68542	−0.163922
$818$	24.2751	0.848758
$819$	0	0
$820$	−83.6457	−2.92103
$821$	−17.6949	−0.617557	−0.308779	−	0.951134i	$-0.599920\pi$
$821$	−17.6949	−0.617557	−0.308779	+	0.951134i	$0.599920\pi$
$822$	0	0
$823$	−25.6614	−0.894502	−0.447251	−	0.894409i	$-0.647597\pi$
$823$	−25.6614	−0.894502	−0.447251	+	0.894409i	$0.647597\pi$
$824$	30.0063	1.04532
$825$	0	0
$826$	−0.415920	−0.0144717
$827$	−41.2901	−1.43580	−0.717898	−	0.696148i	$-0.754896\pi$
$827$	−41.2901	−1.43580	−0.717898	+	0.696148i	$0.754896\pi$
$828$	0	0
$829$	22.3238	0.775339	0.387670	−	0.921798i	$-0.373280\pi$
$829$	22.3238	0.775339	0.387670	+	0.921798i	$0.373280\pi$
$830$	−54.1415	−1.87928
$831$	0	0
$832$	−73.2714	−2.54023
$833$	3.14511	0.108972
$834$	0	0
$835$	13.1799	0.456108
$836$	0	0
$837$	0	0
$838$	67.8077	2.34238
$839$	3.18377	0.109916	0.0549579	−	0.998489i	$-0.482498\pi$
$839$	3.18377	0.109916	0.0549579	+	0.998489i	$0.482498\pi$
$840$	0	0
$841$	−28.6312	−0.987282
$842$	−66.1716	−2.28042
$843$	0	0
$844$	5.81002	0.199989
$845$	85.4038	2.93798
$846$	0	0
$847$	0	0
$848$	−4.48710	−0.154087
$849$	0	0
$850$	12.8380	0.440338
$851$	58.3716	2.00095
$852$	0	0
$853$	−7.23751	−0.247808	−0.123904	−	0.992294i	$-0.539541\pi$
$853$	−7.23751	−0.247808	−0.123904	+	0.992294i	$0.539541\pi$
$854$	−19.4432	−0.665334
$855$	0	0
$856$	−54.7966	−1.87291
$857$	−9.16216	−0.312973	−0.156487	−	0.987680i	$-0.550017\pi$
$857$	−9.16216	−0.312973	−0.156487	+	0.987680i	$0.550017\pi$
$858$	0	0
$859$	−18.2015	−0.621028	−0.310514	−	0.950569i	$-0.600501\pi$
$859$	−18.2015	−0.621028	−0.310514	+	0.950569i	$0.600501\pi$
$860$	43.1006	1.46972
$861$	0	0
$862$	−17.4787	−0.595327
$863$	−19.2933	−0.656750	−0.328375	−	0.944547i	$-0.606501\pi$
$863$	−19.2933	−0.656750	−0.328375	+	0.944547i	$0.606501\pi$
$864$	0	0
$865$	−20.1358	−0.684638
$866$	−25.1745	−0.855464
$867$	0	0
$868$	32.8137	1.11377
$869$	0	0
$870$	0	0
$871$	−49.7451	−1.68555
$872$	57.8372	1.95861
$873$	0	0
$874$	−16.3109	−0.551726
$875$	8.52677	0.288258
$876$	0	0
$877$	4.04722	0.136665	0.0683325	−	0.997663i	$-0.478232\pi$
$877$	4.04722	0.136665	0.0683325	+	0.997663i	$0.478232\pi$
$878$	−1.37224	−0.0463109
$879$	0	0
$880$	0	0
$881$	−6.68797	−0.225324	−0.112662	−	0.993633i	$-0.535938\pi$
$881$	−6.68797	−0.225324	−0.112662	+	0.993633i	$0.535938\pi$
$882$	0	0
$883$	−1.85027	−0.0622664	−0.0311332	−	0.999515i	$-0.509912\pi$
$883$	−1.85027	−0.0622664	−0.0311332	+	0.999515i	$0.509912\pi$
$884$	79.1884	2.66339
$885$	0	0
$886$	43.5307	1.46244
$887$	−14.1006	−0.473452	−0.236726	−	0.971576i	$-0.576074\pi$
$887$	−14.1006	−0.473452	−0.236726	+	0.971576i	$0.576074\pi$
$888$	0	0
$889$	10.3633	0.347574
$890$	72.2987	2.42346
$891$	0	0
$892$	15.3704	0.514640
$893$	3.41129	0.114155
$894$	0	0
$895$	15.9174	0.532061
$896$	20.7406	0.692896
$897$	0	0
$898$	−10.5979	−0.353656
$899$	−5.36653	−0.178984
$900$	0	0
$901$	−5.97439	−0.199036
$902$	0	0
$903$	0	0
$904$	10.9079	0.362790
$905$	−23.9776	−0.797041
$906$	0	0
$907$	−2.00946	−0.0667230	−0.0333615	−	0.999443i	$-0.510621\pi$
$907$	−2.00946	−0.0667230	−0.0333615	+	0.999443i	$0.510621\pi$
$908$	20.7115	0.687335
$909$	0	0
$910$	−41.9754	−1.39147
$911$	−41.5512	−1.37665	−0.688327	−	0.725400i	$-0.741655\pi$
$911$	−41.5512	−1.37665	−0.688327	+	0.725400i	$0.741655\pi$
$912$	0	0
$913$	0	0
$914$	−48.5057	−1.60442
$915$	0	0
$916$	−2.92769	−0.0967336
$917$	−11.6441	−0.384521
$918$	0	0
$919$	35.0390	1.15583	0.577915	−	0.816097i	$-0.303867\pi$
$919$	35.0390	1.15583	0.577915	+	0.816097i	$0.303867\pi$
$920$	69.2295	2.28243
$921$	0	0
$922$	11.6100	0.382356
$923$	−18.3543	−0.604141
$924$	0	0
$925$	15.2721	0.502143
$926$	−46.6502	−1.53302
$927$	0	0
$928$	−1.54524	−0.0507249
$929$	1.54300	0.0506243	0.0253121	−	0.999680i	$-0.491942\pi$
$929$	1.54300	0.0506243	0.0253121	+	0.999680i	$0.491942\pi$
$930$	0	0
$931$	−1.04548	−0.0342642
$932$	−102.884	−3.37007
$933$	0	0
$934$	6.56421	0.214788
$935$	0	0
$936$	0	0
$937$	−42.4690	−1.38740	−0.693700	−	0.720264i	$-0.744021\pi$
$937$	−42.4690	−1.38740	−0.693700	+	0.720264i	$0.744021\pi$
$938$	17.5361	0.572574
$939$	0	0
$940$	−31.3801	−1.02351
$941$	−4.81759	−0.157049	−0.0785245	−	0.996912i	$-0.525021\pi$
$941$	−4.81759	−0.157049	−0.0785245	+	0.996912i	$0.525021\pi$
$942$	0	0
$943$	−56.7690	−1.84865
$944$	0.411029	0.0133778
$945$	0	0
$946$	0	0
$947$	−18.0449	−0.586379	−0.293189	−	0.956054i	$-0.594717\pi$
$947$	−18.0449	−0.586379	−0.293189	+	0.956054i	$0.594717\pi$
$948$	0	0
$949$	4.00000	0.129845
$950$	−4.26752	−0.138457
$951$	0	0
$952$	−12.8801	−0.417448
$953$	37.5309	1.21574	0.607872	−	0.794035i	$-0.292023\pi$
$953$	37.5309	1.21574	0.607872	+	0.794035i	$0.292023\pi$
$954$	0	0
$955$	37.9099	1.22673
$956$	31.8649	1.03058
$957$	0	0
$958$	43.0850	1.39201
$959$	−15.4895	−0.500182
$960$	0	0
$961$	47.0878	1.51896
$962$	144.941	4.67307
$963$	0	0
$964$	−72.9267	−2.34881
$965$	−42.6596	−1.37326
$966$	0	0
$967$	1.06636	0.0342919	0.0171460	−	0.999853i	$-0.494542\pi$
$967$	1.06636	0.0342919	0.0171460	+	0.999853i	$0.494542\pi$
$968$	0	0
$969$	0	0
$970$	73.3759	2.35596
$971$	45.7575	1.46843	0.734214	−	0.678918i	$-0.237551\pi$
$971$	45.7575	1.46843	0.734214	+	0.678918i	$0.237551\pi$
$972$	0	0
$973$	5.81808	0.186519
$974$	−29.5995	−0.948430
$975$	0	0
$976$	19.2146	0.615044
$977$	34.2304	1.09513	0.547564	−	0.836764i	$-0.315555\pi$
$977$	34.2304	1.09513	0.547564	+	0.836764i	$0.315555\pi$
$978$	0	0
$979$	0	0
$980$	9.61724	0.307212
$981$	0	0
$982$	−39.7183	−1.26746
$983$	29.3390	0.935769	0.467884	−	0.883790i	$-0.345016\pi$
$983$	29.3390	0.935769	0.467884	+	0.883790i	$0.345016\pi$
$984$	0	0
$985$	17.6543	0.562513
$986$	4.56543	0.145393
$987$	0	0
$988$	−26.3234	−0.837458
$989$	29.2517	0.930150
$990$	0	0
$991$	19.0194	0.604171	0.302086	−	0.953281i	$-0.402317\pi$
$991$	19.0194	0.604171	0.302086	+	0.953281i	$0.402317\pi$
$992$	22.4846	0.713886
$993$	0	0
$994$	6.47025	0.205224
$995$	40.2892	1.27725
$996$	0	0
$997$	−4.43152	−0.140348	−0.0701739	−	0.997535i	$-0.522355\pi$
$997$	−4.43152	−0.140348	−0.0701739	+	0.997535i	$0.522355\pi$
$998$	−48.7254	−1.54237
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.ci.1.4		4
3.2	odd	2		2541.2.a.bn.1.1		4
11.5	even	5		693.2.m.f.190.1		8
11.9	even	5		693.2.m.f.631.1		8
11.10	odd	2		7623.2.a.cl.1.1		4
33.5	odd	10		231.2.j.f.190.2	yes	8
33.20	odd	10		231.2.j.f.169.2	✓	8
33.32	even	2		2541.2.a.bm.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.j.f.169.2	✓	8	33.20	odd	10
231.2.j.f.190.2	yes	8	33.5	odd	10
693.2.m.f.190.1		8	11.5	even	5
693.2.m.f.631.1		8	11.9	even	5
2541.2.a.bm.1.4		4	33.32	even	2
2541.2.a.bn.1.1		4	3.2	odd	2
7623.2.a.ci.1.4		4	1.1	even	1	trivial
7623.2.a.cl.1.1		4	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7623.a (trivial)

\(f(q)\)	\(=\)	\(q+2.39026 q^{2} +3.71333 q^{4} +2.58993 q^{5} -1.00000 q^{7} +4.09529 q^{8} +O(q^{10})\)	\(q+2.39026 q^{2} +3.71333 q^{4} +2.58993 q^{5} -1.00000 q^{7} +4.09529 q^{8} +6.19059 q^{10} +6.78051 q^{13} -2.39026 q^{14} +2.36215 q^{16} +3.14511 q^{17} -1.04548 q^{19} +9.61724 q^{20} +6.52707 q^{23} +1.70772 q^{25} +16.2072 q^{26} -3.71333 q^{28} +0.607298 q^{29} -8.83673 q^{31} -2.54445 q^{32} +7.51762 q^{34} -2.58993 q^{35} +8.94299 q^{37} -2.49897 q^{38} +10.6065 q^{40} -8.69747 q^{41} +4.48159 q^{43} +15.6014 q^{46} -3.26290 q^{47} +1.00000 q^{49} +4.08188 q^{50} +25.1783 q^{52} -1.89958 q^{53} -4.09529 q^{56} +1.45160 q^{58} +0.174006 q^{59} +8.13437 q^{61} -21.1221 q^{62} -10.8062 q^{64} +17.5610 q^{65} -7.33649 q^{67} +11.6788 q^{68} -6.19059 q^{70} -2.70693 q^{71} +0.589926 q^{73} +21.3761 q^{74} -3.88221 q^{76} -9.80862 q^{79} +6.11779 q^{80} -20.7892 q^{82} -8.74577 q^{83} +8.14560 q^{85} +10.7122 q^{86} +11.6788 q^{89} -6.78051 q^{91} +24.2372 q^{92} -7.79916 q^{94} -2.70772 q^{95} +11.8528 q^{97} +2.39026 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} + 3 q^{4} + 4 q^{5} - 4 q^{7} + 9 q^{8}+O(q^{10}) \) 4 * q - q^2 + 3 * q^4 + 4 * q^5 - 4 * q^7 + 9 * q^8	\( 4 q - q^{2} + 3 q^{4} + 4 q^{5} - 4 q^{7} + 9 q^{8} + 10 q^{10} + 6 q^{13} + q^{14} - 3 q^{16} - 8 q^{17} - 10 q^{19} + 10 q^{23} + 12 q^{25} + 20 q^{26} - 3 q^{28} - 18 q^{31} + 2 q^{32} + 18 q^{34} - 4 q^{35} - 2 q^{37} + 8 q^{38} + 6 q^{40} - 10 q^{41} - 4 q^{43} + 11 q^{46} - 4 q^{47} + 4 q^{49} + 9 q^{50} + 20 q^{52} - 9 q^{56} + 14 q^{58} + 16 q^{59} + 14 q^{61} - 11 q^{64} + 28 q^{65} - 28 q^{67} + 16 q^{68} - 10 q^{70} + 18 q^{71} - 4 q^{73} + 41 q^{74} - 4 q^{76} - 20 q^{79} + 36 q^{80} - 24 q^{82} - 6 q^{83} - 20 q^{85} + 20 q^{86} + 16 q^{89} - 6 q^{91} + 22 q^{92} - 16 q^{94} - 16 q^{95} + 32 q^{97} - q^{98}+O(q^{100}) \) 4 * q - q^2 + 3 * q^4 + 4 * q^5 - 4 * q^7 + 9 * q^8 + 10 * q^10 + 6 * q^13 + q^14 - 3 * q^16 - 8 * q^17 - 10 * q^19 + 10 * q^23 + 12 * q^25 + 20 * q^26 - 3 * q^28 - 18 * q^31 + 2 * q^32 + 18 * q^34 - 4 * q^35 - 2 * q^37 + 8 * q^38 + 6 * q^40 - 10 * q^41 - 4 * q^43 + 11 * q^46 - 4 * q^47 + 4 * q^49 + 9 * q^50 + 20 * q^52 - 9 * q^56 + 14 * q^58 + 16 * q^59 + 14 * q^61 - 11 * q^64 + 28 * q^65 - 28 * q^67 + 16 * q^68 - 10 * q^70 + 18 * q^71 - 4 * q^73 + 41 * q^74 - 4 * q^76 - 20 * q^79 + 36 * q^80 - 24 * q^82 - 6 * q^83 - 20 * q^85 + 20 * q^86 + 16 * q^89 - 6 * q^91 + 22 * q^92 - 16 * q^94 - 16 * q^95 + 32 * q^97 - q^98

Introduction

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