LMFDB - Embedded newform 7623.2.a.ci.1.3

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			1.3
Root			$-1.35567$ 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)

$f(q)$	$=$	$q-0.162147 q^{2} -1.97371 q^{4} +4.38705 q^{5} -1.00000 q^{7} +0.644326 q^{8} +O(q^{10})$	$q-0.162147 q^{2} -1.97371 q^{4} +4.38705 q^{5} -1.00000 q^{7} +0.644326 q^{8} -0.711349 q^{10} +1.67571 q^{13} +0.162147 q^{14} +3.84294 q^{16} -6.18663 q^{17} -3.47528 q^{19} -8.65877 q^{20} -0.568595 q^{23} +14.2462 q^{25} -0.271711 q^{26} +1.97371 q^{28} +8.86742 q^{29} +4.33447 q^{31} -1.91177 q^{32} +1.00314 q^{34} -4.38705 q^{35} +0.969445 q^{37} +0.563507 q^{38} +2.82669 q^{40} -5.77725 q^{41} -5.04388 q^{43} +0.0921961 q^{46} -4.67256 q^{47} +1.00000 q^{49} -2.30999 q^{50} -3.30735 q^{52} +2.37882 q^{53} -0.644326 q^{56} -1.43783 q^{58} +2.84901 q^{59} +9.29883 q^{61} -0.702822 q^{62} -7.37589 q^{64} +7.35141 q^{65} -7.14275 q^{67} +12.2106 q^{68} +0.711349 q^{70} +0.794487 q^{71} +2.38705 q^{73} -0.157193 q^{74} +6.85919 q^{76} -0.670617 q^{79} +16.8592 q^{80} +0.936765 q^{82} +9.28503 q^{83} -27.1411 q^{85} +0.817850 q^{86} +12.2106 q^{89} -1.67571 q^{91} +1.12224 q^{92} +0.757643 q^{94} -15.2462 q^{95} +15.0596 q^{97} -0.162147 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.162147	−0.114655	−0.0573277	−	0.998355i	$-0.518258\pi$
$2$	−0.162147	−0.114655	−0.0573277	+	0.998355i	$0.518258\pi$
$3$	0	0
$3$	0	0
$4$	−1.97371	−0.986854
$5$	4.38705	1.96195	0.980975	−	0.194133i	$-0.0621895\pi$
$5$	4.38705	1.96195	0.980975	+	0.194133i	$0.0621895\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0.644326	0.227804
$9$	0	0
$10$	−0.711349	−0.224948
$11$	0	0
$11$	0	0
$12$	0	0
$13$	1.67571	0.464757	0.232379	−	0.972625i	$-0.425349\pi$
$13$	1.67571	0.464757	0.232379	+	0.972625i	$0.425349\pi$
$14$	0.162147	0.0433357
$15$	0	0
$16$	3.84294	0.960735
$17$	−6.18663	−1.50048	−0.750239	−	0.661167i	$-0.770062\pi$
$17$	−6.18663	−1.50048	−0.750239	+	0.661167i	$0.770062\pi$
$18$	0	0
$19$	−3.47528	−0.797284	−0.398642	−	0.917107i	$-0.630518\pi$
$19$	−3.47528	−0.797284	−0.398642	+	0.917107i	$0.630518\pi$
$20$	−8.65877	−1.93616
$21$	0	0
$22$	0	0
$23$	−0.568595	−0.118560	−0.0592801	−	0.998241i	$-0.518881\pi$
$23$	−0.568595	−0.118560	−0.0592801	+	0.998241i	$0.518881\pi$
$24$	0	0
$25$	14.2462	2.84925
$26$	−0.271711	−0.0532869
$27$	0	0
$28$	1.97371	0.372996
$29$	8.86742	1.64664	0.823320	−	0.567578i	$-0.192120\pi$
$29$	8.86742	1.64664	0.823320	+	0.567578i	$0.192120\pi$
$30$	0	0
$31$	4.33447	0.778494	0.389247	−	0.921133i	$-0.372735\pi$
$31$	4.33447	0.778494	0.389247	+	0.921133i	$0.372735\pi$
$32$	−1.91177	−0.337957
$33$	0	0
$34$	1.00314	0.172038
$35$	−4.38705	−0.741547
$36$	0	0
$37$	0.969445	0.159376	0.0796879	−	0.996820i	$-0.474608\pi$
$37$	0.969445	0.159376	0.0796879	+	0.996820i	$0.474608\pi$
$38$	0.563507	0.0914129
$39$	0	0
$40$	2.82669	0.446939
$41$	−5.77725	−0.902255	−0.451128	−	0.892459i	$-0.648978\pi$
$41$	−5.77725	−0.902255	−0.451128	+	0.892459i	$0.648978\pi$
$42$	0	0
$43$	−5.04388	−0.769184	−0.384592	−	0.923087i	$-0.625658\pi$
$43$	−5.04388	−0.769184	−0.384592	+	0.923087i	$0.625658\pi$
$44$	0	0
$45$	0	0
$46$	0.0921961	0.0135936
$47$	−4.67256	−0.681563	−0.340782	−	0.940143i	$-0.610692\pi$
$47$	−4.67256	−0.681563	−0.340782	+	0.940143i	$0.610692\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−2.30999	−0.326682
$51$	0	0
$52$	−3.30735	−0.458647
$53$	2.37882	0.326756	0.163378	−	0.986564i	$-0.447761\pi$
$53$	2.37882	0.326756	0.163378	+	0.986564i	$0.447761\pi$
$54$	0	0
$55$	0	0
$56$	−0.644326	−0.0861016
$57$	0	0
$58$	−1.43783	−0.188796
$59$	2.84901	0.370910	0.185455	−	0.982653i	$-0.440624\pi$
$59$	2.84901	0.370910	0.185455	+	0.982653i	$0.440624\pi$
$60$	0	0
$61$	9.29883	1.19059	0.595296	−	0.803506i	$-0.297035\pi$
$61$	9.29883	1.19059	0.595296	+	0.803506i	$0.297035\pi$
$62$	−0.702822	−0.0892585
$63$	0	0
$64$	−7.37589	−0.921987
$65$	7.35141	0.911830
$66$	0	0
$67$	−7.14275	−0.872626	−0.436313	−	0.899795i	$-0.643716\pi$
$67$	−7.14275	−0.872626	−0.436313	+	0.899795i	$0.643716\pi$
$68$	12.2106	1.48075
$69$	0	0
$70$	0.711349	0.0850224
$71$	0.794487	0.0942882	0.0471441	−	0.998888i	$-0.484988\pi$
$71$	0.794487	0.0942882	0.0471441	+	0.998888i	$0.484988\pi$
$72$	0	0
$73$	2.38705	0.279384	0.139692	−	0.990195i	$-0.455389\pi$
$73$	2.38705	0.279384	0.139692	+	0.990195i	$0.455389\pi$
$74$	−0.157193	−0.0182733
$75$	0	0
$76$	6.85919	0.786803
$77$	0	0
$78$	0	0
$79$	−0.670617	−0.0754504	−0.0377252	−	0.999288i	$-0.512011\pi$
$79$	−0.670617	−0.0754504	−0.0377252	+	0.999288i	$0.512011\pi$
$80$	16.8592	1.88491
$81$	0	0
$82$	0.936765	0.103448
$83$	9.28503	1.01916	0.509582	−	0.860422i	$-0.329800\pi$
$83$	9.28503	1.01916	0.509582	+	0.860422i	$0.329800\pi$
$84$	0	0
$85$	−27.1411	−2.94386
$86$	0.817850	0.0881911
$87$	0	0
$88$	0	0
$89$	12.2106	1.29432	0.647161	−	0.762354i	$-0.275956\pi$
$89$	12.2106	1.29432	0.647161	+	0.762354i	$0.275956\pi$
$90$	0	0
$91$	−1.67571	−0.175662
$92$	1.12224	0.117002
$93$	0	0
$94$	0.757643	0.0781449
$95$	−15.2462	−1.56423
$96$	0	0
$97$	15.0596	1.52907	0.764536	−	0.644581i	$-0.222968\pi$
$97$	15.0596	1.52907	0.764536	+	0.644581i	$0.222968\pi$
$98$	−0.162147	−0.0163793
$99$	0	0
$100$	−28.1179	−2.81179
$101$	−5.76079	−0.573220	−0.286610	−	0.958047i	$-0.592528\pi$
$101$	−5.76079	−0.573220	−0.286610	+	0.958047i	$0.592528\pi$
$102$	0	0
$103$	7.71449	0.760132	0.380066	−	0.924959i	$-0.375901\pi$
$103$	7.71449	0.760132	0.380066	+	0.924959i	$0.375901\pi$
$104$	1.07970	0.105873
$105$	0	0
$106$	−0.385719	−0.0374644
$107$	16.4634	1.59158	0.795790	−	0.605573i	$-0.207056\pi$
$107$	16.4634	1.59158	0.795790	+	0.605573i	$0.207056\pi$
$108$	0	0
$109$	9.74355	0.933263	0.466632	−	0.884452i	$-0.345467\pi$
$109$	9.74355	0.933263	0.466632	+	0.884452i	$0.345467\pi$
$110$	0	0
$111$	0	0
$112$	−3.84294	−0.363124
$113$	2.85725	0.268787	0.134394	−	0.990928i	$-0.457091\pi$
$113$	2.85725	0.268787	0.134394	+	0.990928i	$0.457091\pi$
$114$	0	0
$115$	−2.49446	−0.232609
$116$	−17.5017	−1.62499
$117$	0	0
$118$	−0.461960	−0.0425268
$119$	6.18663	0.567127
$120$	0	0
$121$	0	0
$122$	−1.50778	−0.136508
$123$	0	0
$124$	−8.55498	−0.768260
$125$	40.5638	3.62813
$126$	0	0
$127$	−16.0514	−1.42433	−0.712165	−	0.702012i	$-0.752285\pi$
$127$	−16.0514	−1.42433	−0.712165	+	0.702012i	$0.752285\pi$
$128$	5.01953	0.443668
$129$	0	0
$130$	−1.19201	−0.104546
$131$	−0.750136	−0.0655397	−0.0327699	−	0.999463i	$-0.510433\pi$
$131$	−0.750136	−0.0655397	−0.0327699	+	0.999463i	$0.510433\pi$
$132$	0	0
$133$	3.47528	0.301345
$134$	1.15818	0.100051
$135$	0	0
$136$	−3.98620	−0.341814
$137$	13.0082	1.11137	0.555684	−	0.831393i	$-0.312456\pi$
$137$	13.0082	1.11137	0.555684	+	0.831393i	$0.312456\pi$
$138$	0	0
$139$	3.90112	0.330889	0.165444	−	0.986219i	$-0.447094\pi$
$139$	3.90112	0.330889	0.165444	+	0.986219i	$0.447094\pi$
$140$	8.65877	0.731799
$141$	0	0
$142$	−0.128824	−0.0108107
$143$	0	0
$144$	0	0
$145$	38.9019	3.23062
$146$	−0.387054	−0.0320328
$147$	0	0
$148$	−1.91340	−0.157281
$149$	−18.5434	−1.51914	−0.759568	−	0.650428i	$-0.774590\pi$
$149$	−18.5434	−1.51914	−0.759568	+	0.650428i	$0.774590\pi$
$150$	0	0
$151$	20.7979	1.69251	0.846255	−	0.532779i	$-0.178852\pi$
$151$	20.7979	1.69251	0.846255	+	0.532779i	$0.178852\pi$
$152$	−2.23921	−0.181624
$153$	0	0
$154$	0	0
$155$	19.0156	1.52737
$156$	0	0
$157$	5.80975	0.463669	0.231834	−	0.972755i	$-0.425527\pi$
$157$	5.80975	0.463669	0.231834	+	0.972755i	$0.425527\pi$
$158$	0.108739	0.00865079
$159$	0	0
$160$	−8.38705	−0.663055
$161$	0.568595	0.0448116
$162$	0	0
$163$	−8.90307	−0.697342	−0.348671	−	0.937245i	$-0.613367\pi$
$163$	−8.90307	−0.697342	−0.348671	+	0.937245i	$0.613367\pi$
$164$	11.4026	0.890394
$165$	0	0
$166$	−1.50554	−0.116853
$167$	3.82355	0.295875	0.147937	−	0.988997i	$-0.452737\pi$
$167$	3.82355	0.295875	0.147937	+	0.988997i	$0.452737\pi$
$168$	0	0
$169$	−10.1920	−0.784001
$170$	4.40085	0.337530
$171$	0	0
$172$	9.95514	0.759072
$173$	−1.75061	−0.133096	−0.0665482	−	0.997783i	$-0.521199\pi$
$173$	−1.75061	−0.133096	−0.0665482	+	0.997783i	$0.521199\pi$
$174$	0	0
$175$	−14.2462	−1.07691
$176$	0	0
$177$	0	0
$178$	−1.97991	−0.148401
$179$	12.8541	0.960761	0.480380	−	0.877060i	$-0.340499\pi$
$179$	12.8541	0.960761	0.480380	+	0.877060i	$0.340499\pi$
$180$	0	0
$181$	−22.0831	−1.64142	−0.820712	−	0.571342i	$-0.806423\pi$
$181$	−22.0831	−1.64142	−0.820712	+	0.571342i	$0.806423\pi$
$182$	0.271711	0.0201406
$183$	0	0
$184$	−0.366361	−0.0270085
$185$	4.25301	0.312687
$186$	0	0
$187$	0	0
$188$	9.22227	0.672603
$189$	0	0
$190$	2.47214	0.179348
$191$	−14.7160	−1.06481	−0.532405	−	0.846490i	$-0.678712\pi$
$191$	−14.7160	−1.06481	−0.532405	+	0.846490i	$0.678712\pi$
$192$	0	0
$193$	8.51287	0.612770	0.306385	−	0.951908i	$-0.400881\pi$
$193$	8.51287	0.612770	0.306385	+	0.951908i	$0.400881\pi$
$194$	−2.44187	−0.175316
$195$	0	0
$196$	−1.97371	−0.140979
$197$	4.72273	0.336480	0.168240	−	0.985746i	$-0.446192\pi$
$197$	4.72273	0.336480	0.168240	+	0.985746i	$0.446192\pi$
$198$	0	0
$199$	16.7620	1.18822	0.594112	−	0.804382i	$-0.297504\pi$
$199$	16.7620	1.18822	0.594112	+	0.804382i	$0.297504\pi$
$200$	9.17922	0.649069
$201$	0	0
$202$	0.934096	0.0657227
$203$	−8.86742	−0.622371
$204$	0	0
$205$	−25.3451	−1.77018
$206$	−1.25088	−0.0871532
$207$	0	0
$208$	6.43964	0.446509
$209$	0	0
$210$	0	0
$211$	−10.1454	−0.698440	−0.349220	−	0.937041i	$-0.613553\pi$
$211$	−10.1454	−0.698440	−0.349220	+	0.937041i	$0.613553\pi$
$212$	−4.69510	−0.322461
$213$	0	0
$214$	−2.66950	−0.182483
$215$	−22.1278	−1.50910
$216$	0	0
$217$	−4.33447	−0.294243
$218$	−1.57989	−0.107004
$219$	0	0
$220$	0	0
$221$	−10.3670	−0.697358
$222$	0	0
$223$	−6.11172	−0.409271	−0.204636	−	0.978838i	$-0.565601\pi$
$223$	−6.11172	−0.409271	−0.204636	+	0.978838i	$0.565601\pi$
$224$	1.91177	0.127736
$225$	0	0
$226$	−0.463295	−0.0308179
$227$	−14.2090	−0.943081	−0.471541	−	0.881844i	$-0.656302\pi$
$227$	−14.2090	−0.943081	−0.471541	+	0.881844i	$0.656302\pi$
$228$	0	0
$229$	−2.72781	−0.180259	−0.0901295	−	0.995930i	$-0.528728\pi$
$229$	−2.72781	−0.180259	−0.0901295	+	0.995930i	$0.528728\pi$
$230$	0.404469	0.0266699
$231$	0	0
$232$	5.71351	0.375110
$233$	17.7897	1.16544	0.582720	−	0.812673i	$-0.301988\pi$
$233$	17.7897	1.16544	0.582720	+	0.812673i	$0.301988\pi$
$234$	0	0
$235$	−20.4988	−1.33719
$236$	−5.62312	−0.366034
$237$	0	0
$238$	−1.00314	−0.0650242
$239$	−12.7058	−0.821869	−0.410934	−	0.911665i	$-0.634798\pi$
$239$	−12.7058	−0.821869	−0.410934	+	0.911665i	$0.634798\pi$
$240$	0	0
$241$	−18.6604	−1.20202	−0.601012	−	0.799240i	$-0.705235\pi$
$241$	−18.6604	−1.20202	−0.601012	+	0.799240i	$0.705235\pi$
$242$	0	0
$243$	0	0
$244$	−18.3532	−1.17494
$245$	4.38705	0.280279
$246$	0	0
$247$	−5.82355	−0.370543
$248$	2.79281	0.177344
$249$	0	0
$250$	−6.57730	−0.415985
$251$	−18.8493	−1.18976	−0.594878	−	0.803816i	$-0.702800\pi$
$251$	−18.8493	−1.18976	−0.594878	+	0.803816i	$0.702800\pi$
$252$	0	0
$253$	0	0
$254$	2.60269	0.163307
$255$	0	0
$256$	13.9379	0.871118
$257$	10.3345	0.644647	0.322323	−	0.946630i	$-0.395536\pi$
$257$	10.3345	0.644647	0.322323	+	0.946630i	$0.395536\pi$
$258$	0	0
$259$	−0.969445	−0.0602384
$260$	−14.5095	−0.899844
$261$	0	0
$262$	0.121632	0.00751448
$263$	4.14979	0.255887	0.127943	−	0.991781i	$-0.459162\pi$
$263$	4.14979	0.255887	0.127943	+	0.991781i	$0.459162\pi$
$264$	0	0
$265$	10.4360	0.641079
$266$	−0.563507	−0.0345508
$267$	0	0
$268$	14.0977	0.861155
$269$	5.06228	0.308653	0.154326	−	0.988020i	$-0.450679\pi$
$269$	5.06228	0.308653	0.154326	+	0.988020i	$0.450679\pi$
$270$	0	0
$271$	−26.3160	−1.59859	−0.799293	−	0.600942i	$-0.794792\pi$
$271$	−26.3160	−1.59859	−0.799293	+	0.600942i	$0.794792\pi$
$272$	−23.7749	−1.44156
$273$	0	0
$274$	−2.10925	−0.127424
$275$	0	0
$276$	0	0
$277$	25.8187	1.55130	0.775648	−	0.631165i	$-0.217423\pi$
$277$	25.8187	1.55130	0.775648	+	0.631165i	$0.217423\pi$
$278$	−0.632556	−0.0379382
$279$	0	0
$280$	−2.82669	−0.168927
$281$	5.43844	0.324430	0.162215	−	0.986755i	$-0.448136\pi$
$281$	5.43844	0.324430	0.162215	+	0.986755i	$0.448136\pi$
$282$	0	0
$283$	25.3839	1.50892	0.754458	−	0.656348i	$-0.227900\pi$
$283$	25.3839	1.50892	0.754458	+	0.656348i	$0.227900\pi$
$284$	−1.56809	−0.0930487
$285$	0	0
$286$	0	0
$287$	5.77725	0.341020
$288$	0	0
$289$	21.2744	1.25143
$290$	−6.30783	−0.370408
$291$	0	0
$292$	−4.71135	−0.275711
$293$	−6.75402	−0.394574	−0.197287	−	0.980346i	$-0.563213\pi$
$293$	−6.75402	−0.394574	−0.197287	+	0.980346i	$0.563213\pi$
$294$	0	0
$295$	12.4988	0.727707
$296$	0.624638	0.0363064
$297$	0	0
$298$	3.00676	0.174177
$299$	−0.952798	−0.0551017
$300$	0	0
$301$	5.04388	0.290724
$302$	−3.37232	−0.194055
$303$	0	0
$304$	−13.3553	−0.765979
$305$	40.7945	2.33588
$306$	0	0
$307$	−17.6396	−1.00674	−0.503372	−	0.864070i	$-0.667908\pi$
$307$	−17.6396	−1.00674	−0.503372	+	0.864070i	$0.667908\pi$
$308$	0	0
$309$	0	0
$310$	−3.08332	−0.175121
$311$	12.2414	0.694149	0.347074	−	0.937838i	$-0.387175\pi$
$311$	12.2414	0.694149	0.347074	+	0.937838i	$0.387175\pi$
$312$	0	0
$313$	3.15951	0.178586	0.0892931	−	0.996005i	$-0.471539\pi$
$313$	3.15951	0.178586	0.0892931	+	0.996005i	$0.471539\pi$
$314$	−0.942035	−0.0531621
$315$	0	0
$316$	1.32360	0.0744585
$317$	−8.43364	−0.473681	−0.236840	−	0.971549i	$-0.576112\pi$
$317$	−8.43364	−0.473681	−0.236840	+	0.971549i	$0.576112\pi$
$318$	0	0
$319$	0	0
$320$	−32.3584	−1.80889
$321$	0	0
$322$	−0.0921961	−0.00513789
$323$	21.5003	1.19631
$324$	0	0
$325$	23.8725	1.32421
$326$	1.44361	0.0799540
$327$	0	0
$328$	−3.72243	−0.205537
$329$	4.67256	0.257607
$330$	0	0
$331$	6.13284	0.337092	0.168546	−	0.985694i	$-0.446093\pi$
$331$	6.13284	0.337092	0.168546	+	0.985694i	$0.446093\pi$
$332$	−18.3259	−1.00577
$333$	0	0
$334$	−0.619977	−0.0339237
$335$	−31.3356	−1.71205
$336$	0	0
$337$	−29.8792	−1.62763	−0.813813	−	0.581127i	$-0.802612\pi$
$337$	−29.8792	−1.62763	−0.813813	+	0.581127i	$0.802612\pi$
$338$	1.65261	0.0898899
$339$	0	0
$340$	53.5686	2.90516
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−3.24990	−0.175223
$345$	0	0
$346$	0.283857	0.0152602
$347$	29.1716	1.56602	0.783008	−	0.622012i	$-0.213685\pi$
$347$	29.1716	1.56602	0.783008	+	0.622012i	$0.213685\pi$
$348$	0	0
$349$	13.3924	0.716881	0.358440	−	0.933553i	$-0.383309\pi$
$349$	13.3924	0.716881	0.358440	+	0.933553i	$0.383309\pi$
$350$	2.30999	0.123474
$351$	0	0
$352$	0	0
$353$	21.3175	1.13462	0.567309	−	0.823505i	$-0.307985\pi$
$353$	21.3175	1.13462	0.567309	+	0.823505i	$0.307985\pi$
$354$	0	0
$355$	3.48546	0.184989
$356$	−24.1002	−1.27731
$357$	0	0
$358$	−2.08426	−0.110156
$359$	20.6234	1.08846	0.544231	−	0.838935i	$-0.316821\pi$
$359$	20.6234	1.08846	0.544231	+	0.838935i	$0.316821\pi$
$360$	0	0
$361$	−6.92242	−0.364338
$362$	3.58072	0.188198
$363$	0	0
$364$	3.30735	0.173352
$365$	10.4721	0.548137
$366$	0	0
$367$	32.8546	1.71499	0.857497	−	0.514489i	$-0.172018\pi$
$367$	32.8546	1.71499	0.857497	+	0.514489i	$0.172018\pi$
$368$	−2.18508	−0.113905
$369$	0	0
$370$	−0.689613	−0.0358513
$371$	−2.37882	−0.123502
$372$	0	0
$373$	26.0979	1.35130	0.675650	−	0.737223i	$-0.263863\pi$
$373$	26.0979	1.35130	0.675650	+	0.737223i	$0.263863\pi$
$374$	0	0
$375$	0	0
$376$	−3.01065	−0.155262
$377$	14.8592	0.765287
$378$	0	0
$379$	−11.6733	−0.599616	−0.299808	−	0.953999i	$-0.596923\pi$
$379$	−11.6733	−0.599616	−0.299808	+	0.953999i	$0.596923\pi$
$380$	30.0916	1.54367
$381$	0	0
$382$	2.38615	0.122086
$383$	15.3701	0.785376	0.392688	−	0.919672i	$-0.371545\pi$
$383$	15.3701	0.785376	0.392688	+	0.919672i	$0.371545\pi$
$384$	0	0
$385$	0	0
$386$	−1.38034	−0.0702573
$387$	0	0
$388$	−29.7233	−1.50897
$389$	9.64424	0.488983	0.244491	−	0.969651i	$-0.421379\pi$
$389$	9.64424	0.488983	0.244491	+	0.969651i	$0.421379\pi$
$390$	0	0
$391$	3.51769	0.177897
$392$	0.644326	0.0325434
$393$	0	0
$394$	−0.765777	−0.0385793
$395$	−2.94203	−0.148030
$396$	0	0
$397$	20.2855	1.01810	0.509050	−	0.860737i	$-0.329997\pi$
$397$	20.2855	1.01810	0.509050	+	0.860737i	$0.329997\pi$
$398$	−2.71791	−0.136236
$399$	0	0
$400$	54.7475	2.73737
$401$	8.15655	0.407319	0.203659	−	0.979042i	$-0.434717\pi$
$401$	8.15655	0.407319	0.203659	+	0.979042i	$0.434717\pi$
$402$	0	0
$403$	7.26330	0.361811
$404$	11.3701	0.565684
$405$	0	0
$406$	1.43783	0.0713582
$407$	0	0
$408$	0	0
$409$	−9.67209	−0.478254	−0.239127	−	0.970988i	$-0.576861\pi$
$409$	−9.67209	−0.478254	−0.239127	+	0.970988i	$0.576861\pi$
$410$	4.10964	0.202961
$411$	0	0
$412$	−15.2262	−0.750139
$413$	−2.84901	−0.140191
$414$	0	0
$415$	40.7339	1.99955
$416$	−3.20357	−0.157068
$417$	0	0
$418$	0	0
$419$	18.9357	0.925072	0.462536	−	0.886601i	$-0.346940\pi$
$419$	18.9357	0.925072	0.462536	+	0.886601i	$0.346940\pi$
$420$	0	0
$421$	−13.0950	−0.638210	−0.319105	−	0.947719i	$-0.603382\pi$
$421$	−13.0950	−0.638210	−0.319105	+	0.947719i	$0.603382\pi$
$422$	1.64505	0.0800799
$423$	0	0
$424$	1.53274	0.0744362
$425$	−88.1362	−4.27524
$426$	0	0
$427$	−9.29883	−0.450002
$428$	−32.4940	−1.57066
$429$	0	0
$430$	3.58795	0.173027
$431$	16.3034	0.785309	0.392655	−	0.919686i	$-0.371557\pi$
$431$	16.3034	0.785309	0.392655	+	0.919686i	$0.371557\pi$
$432$	0	0
$433$	11.6842	0.561508	0.280754	−	0.959780i	$-0.409415\pi$
$433$	11.6842	0.561508	0.280754	+	0.959780i	$0.409415\pi$
$434$	0.702822	0.0337366
$435$	0	0
$436$	−19.2309	−0.920995
$437$	1.97603	0.0945262
$438$	0	0
$439$	11.7172	0.559230	0.279615	−	0.960112i	$-0.409793\pi$
$439$	11.7172	0.559230	0.279615	+	0.960112i	$0.409793\pi$
$440$	0	0
$441$	0	0
$442$	1.68098	0.0799558
$443$	12.5671	0.597081	0.298540	−	0.954397i	$-0.403500\pi$
$443$	12.5671	0.597081	0.298540	+	0.954397i	$0.403500\pi$
$444$	0	0
$445$	53.5686	2.53939
$446$	0.990999	0.0469252
$447$	0	0
$448$	7.37589	0.348478
$449$	15.9360	0.752068	0.376034	−	0.926606i	$-0.377288\pi$
$449$	15.9360	0.752068	0.376034	+	0.926606i	$0.377288\pi$
$450$	0	0
$451$	0	0
$452$	−5.63937	−0.265254
$453$	0	0
$454$	2.30394	0.108129
$455$	−7.35141	−0.344639
$456$	0	0
$457$	−23.7945	−1.11306	−0.556530	−	0.830828i	$-0.687867\pi$
$457$	−23.7945	−1.11306	−0.556530	+	0.830828i	$0.687867\pi$
$458$	0.442307	0.0206677
$459$	0	0
$460$	4.92333	0.229552
$461$	22.6035	1.05275	0.526375	−	0.850252i	$-0.323551\pi$
$461$	22.6035	1.05275	0.526375	+	0.850252i	$0.323551\pi$
$462$	0	0
$463$	3.03759	0.141169	0.0705843	−	0.997506i	$-0.477514\pi$
$463$	3.03759	0.141169	0.0705843	+	0.997506i	$0.477514\pi$
$464$	34.0770	1.58198
$465$	0	0
$466$	−2.88454	−0.133624
$467$	23.0051	1.06455	0.532274	−	0.846572i	$-0.321338\pi$
$467$	23.0051	1.06455	0.532274	+	0.846572i	$0.321338\pi$
$468$	0	0
$469$	7.14275	0.329822
$470$	3.32382	0.153316
$471$	0	0
$472$	1.83569	0.0844946
$473$	0	0
$474$	0	0
$475$	−49.5097	−2.27166
$476$	−12.2106	−0.559672
$477$	0	0
$478$	2.06021	0.0942317
$479$	−8.17283	−0.373426	−0.186713	−	0.982414i	$-0.559783\pi$
$479$	−8.17283	−0.373426	−0.186713	+	0.982414i	$0.559783\pi$
$480$	0	0
$481$	1.62450	0.0740710
$482$	3.02573	0.137818
$483$	0	0
$484$	0	0
$485$	66.0673	2.99996
$486$	0	0
$487$	−0.999232	−0.0452795	−0.0226398	−	0.999744i	$-0.507207\pi$
$487$	−0.999232	−0.0452795	−0.0226398	+	0.999744i	$0.507207\pi$
$488$	5.99147	0.271221
$489$	0	0
$490$	−0.711349	−0.0321355
$491$	−24.2957	−1.09645	−0.548224	−	0.836331i	$-0.684696\pi$
$491$	−24.2957	−1.09645	−0.548224	+	0.836331i	$0.684696\pi$
$492$	0	0
$493$	−54.8595	−2.47075
$494$	0.944272	0.0424848
$495$	0	0
$496$	16.6571	0.747927
$497$	−0.794487	−0.0356376
$498$	0	0
$499$	−41.0807	−1.83902	−0.919512	−	0.393061i	$-0.871416\pi$
$499$	−41.0807	−1.83902	−0.919512	+	0.393061i	$0.871416\pi$
$500$	−80.0611	−3.58044
$501$	0	0
$502$	3.05636	0.136412
$503$	10.3270	0.460457	0.230228	−	0.973137i	$-0.426053\pi$
$503$	10.3270	0.460457	0.230228	+	0.973137i	$0.426053\pi$
$504$	0	0
$505$	−25.2729	−1.12463
$506$	0	0
$507$	0	0
$508$	31.6807	1.40561
$509$	40.6105	1.80003	0.900014	−	0.435860i	$-0.143556\pi$
$509$	40.6105	1.80003	0.900014	+	0.435860i	$0.143556\pi$
$510$	0	0
$511$	−2.38705	−0.105597
$512$	−12.2990	−0.543546
$513$	0	0
$514$	−1.67571	−0.0739122
$515$	33.8439	1.49134
$516$	0	0
$517$	0	0
$518$	0.157193	0.00690666
$519$	0	0
$520$	4.73670	0.207718
$521$	22.2145	0.973234	0.486617	−	0.873615i	$-0.338231\pi$
$521$	22.2145	0.973234	0.486617	+	0.873615i	$0.338231\pi$
$522$	0	0
$523$	−22.6653	−0.991085	−0.495543	−	0.868584i	$-0.665031\pi$
$523$	−22.6653	−0.991085	−0.495543	+	0.868584i	$0.665031\pi$
$524$	1.48055	0.0646781
$525$	0	0
$526$	−0.672876	−0.0293388
$527$	−26.8158	−1.16811
$528$	0	0
$529$	−22.6767	−0.985943
$530$	−1.69217	−0.0735032
$531$	0	0
$532$	−6.85919	−0.297384
$533$	−9.68098	−0.419330
$534$	0	0
$535$	72.2260	3.12260
$536$	−4.60226	−0.198787
$537$	0	0
$538$	−0.820835	−0.0353887
$539$	0	0
$540$	0	0
$541$	−6.26662	−0.269423	−0.134712	−	0.990885i	$-0.543011\pi$
$541$	−6.26662	−0.269423	−0.134712	+	0.990885i	$0.543011\pi$
$542$	4.26707	0.183286
$543$	0	0
$544$	11.8274	0.507097
$545$	42.7455	1.83102
$546$	0	0
$547$	34.0916	1.45765	0.728827	−	0.684698i	$-0.240066\pi$
$547$	34.0916	1.45765	0.728827	+	0.684698i	$0.240066\pi$
$548$	−25.6745	−1.09676
$549$	0	0
$550$	0	0
$551$	−30.8168	−1.31284
$552$	0	0
$553$	0.670617	0.0285176
$554$	−4.18643	−0.177865
$555$	0	0
$556$	−7.69968	−0.326539
$557$	26.3992	1.11857	0.559284	−	0.828976i	$-0.311076\pi$
$557$	26.3992	1.11857	0.559284	+	0.828976i	$0.311076\pi$
$558$	0	0
$559$	−8.45205	−0.357484
$560$	−16.8592	−0.712431
$561$	0	0
$562$	−0.881827	−0.0371976
$563$	6.85125	0.288746	0.144373	−	0.989523i	$-0.453884\pi$
$563$	6.85125	0.288746	0.144373	+	0.989523i	$0.453884\pi$
$564$	0	0
$565$	12.5349	0.527347
$566$	−4.11593	−0.173005
$567$	0	0
$568$	0.511908	0.0214792
$569$	−18.4008	−0.771404	−0.385702	−	0.922623i	$-0.626041\pi$
$569$	−18.4008	−0.771404	−0.385702	+	0.922623i	$0.626041\pi$
$570$	0	0
$571$	0.336889	0.0140984	0.00704918	−	0.999975i	$-0.497756\pi$
$571$	0.336889	0.0140984	0.00704918	+	0.999975i	$0.497756\pi$
$572$	0	0
$573$	0	0
$574$	−0.936765	−0.0390998
$575$	−8.10035	−0.337808
$576$	0	0
$577$	−25.8347	−1.07551	−0.537757	−	0.843100i	$-0.680728\pi$
$577$	−25.8347	−1.07551	−0.537757	+	0.843100i	$0.680728\pi$
$578$	−3.44958	−0.143484
$579$	0	0
$580$	−76.7809	−3.18815
$581$	−9.28503	−0.385208
$582$	0	0
$583$	0	0
$584$	1.53804	0.0636446
$585$	0	0
$586$	1.09515	0.0452401
$587$	3.73415	0.154125	0.0770623	−	0.997026i	$-0.475446\pi$
$587$	3.73415	0.154125	0.0770623	+	0.997026i	$0.475446\pi$
$588$	0	0
$589$	−15.0635	−0.620681
$590$	−2.02664	−0.0834355
$591$	0	0
$592$	3.72552	0.153118
$593$	−17.5647	−0.721295	−0.360648	−	0.932702i	$-0.617444\pi$
$593$	−17.5647	−0.721295	−0.360648	+	0.932702i	$0.617444\pi$
$594$	0	0
$595$	27.1411	1.11268
$596$	36.5993	1.49917
$597$	0	0
$598$	0.154494	0.00631771
$599$	−11.3088	−0.462066	−0.231033	−	0.972946i	$-0.574211\pi$
$599$	−11.3088	−0.462066	−0.231033	+	0.972946i	$0.574211\pi$
$600$	0	0
$601$	−28.9821	−1.18220	−0.591102	−	0.806596i	$-0.701307\pi$
$601$	−28.9821	−1.18220	−0.591102	+	0.806596i	$0.701307\pi$
$602$	−0.817850	−0.0333331
$603$	0	0
$604$	−41.0490	−1.67026
$605$	0	0
$606$	0	0
$607$	7.34347	0.298062	0.149031	−	0.988832i	$-0.452385\pi$
$607$	7.34347	0.298062	0.149031	+	0.988832i	$0.452385\pi$
$608$	6.64395	0.269448
$609$	0	0
$610$	−6.61471	−0.267822
$611$	−7.82984	−0.316761
$612$	0	0
$613$	18.9535	0.765526	0.382763	−	0.923847i	$-0.374973\pi$
$613$	18.9535	0.765526	0.382763	+	0.923847i	$0.374973\pi$
$614$	2.86021	0.115429
$615$	0	0
$616$	0	0
$617$	−25.2571	−1.01681	−0.508407	−	0.861117i	$-0.669766\pi$
$617$	−25.2571	−1.01681	−0.508407	+	0.861117i	$0.669766\pi$
$618$	0	0
$619$	−15.5725	−0.625912	−0.312956	−	0.949768i	$-0.601319\pi$
$619$	−15.5725	−0.625912	−0.312956	+	0.949768i	$0.601319\pi$
$620$	−37.5312	−1.50729
$621$	0	0
$622$	−1.98492	−0.0795879
$623$	−12.2106	−0.489207
$624$	0	0
$625$	106.724	4.26897
$626$	−0.512306	−0.0204759
$627$	0	0
$628$	−11.4668	−0.457573
$629$	−5.99760	−0.239140
$630$	0	0
$631$	−36.0169	−1.43381	−0.716905	−	0.697170i	$-0.754442\pi$
$631$	−36.0169	−1.43381	−0.716905	+	0.697170i	$0.754442\pi$
$632$	−0.432096	−0.0171879
$633$	0	0
$634$	1.36749	0.0543100
$635$	−70.4183	−2.79446
$636$	0	0
$637$	1.67571	0.0663939
$638$	0	0
$639$	0	0
$640$	22.0209	0.870454
$641$	26.7410	1.05620	0.528102	−	0.849181i	$-0.322904\pi$
$641$	26.7410	1.05620	0.528102	+	0.849181i	$0.322904\pi$
$642$	0	0
$643$	7.31364	0.288422	0.144211	−	0.989547i	$-0.453936\pi$
$643$	7.31364	0.288422	0.144211	+	0.989547i	$0.453936\pi$
$644$	−1.12224	−0.0442225
$645$	0	0
$646$	−3.48621	−0.137163
$647$	35.0221	1.37686	0.688431	−	0.725302i	$-0.258300\pi$
$647$	35.0221	1.37686	0.688431	+	0.725302i	$0.258300\pi$
$648$	0	0
$649$	0	0
$650$	−3.87086	−0.151828
$651$	0	0
$652$	17.5721	0.688175
$653$	4.05138	0.158543	0.0792714	−	0.996853i	$-0.474741\pi$
$653$	4.05138	0.158543	0.0792714	+	0.996853i	$0.474741\pi$
$654$	0	0
$655$	−3.29089	−0.128586
$656$	−22.2016	−0.866828
$657$	0	0
$658$	−0.757643	−0.0295360
$659$	−28.6360	−1.11550	−0.557750	−	0.830009i	$-0.688335\pi$
$659$	−28.6360	−1.11550	−0.557750	+	0.830009i	$0.688335\pi$
$660$	0	0
$661$	45.2741	1.76096	0.880479	−	0.474085i	$-0.157221\pi$
$661$	45.2741	1.76096	0.880479	+	0.474085i	$0.157221\pi$
$662$	−0.994424	−0.0386494
$663$	0	0
$664$	5.98258	0.232169
$665$	15.2462	0.591224
$666$	0	0
$667$	−5.04197	−0.195226
$668$	−7.54657	−0.291985
$669$	0	0
$670$	5.08099	0.196296
$671$	0	0
$672$	0	0
$673$	0.0509075	0.00196234	0.000981170	−	1.00000i	$-0.499688\pi$
$673$	0.0509075	0.00196234	0.000981170		1.00000i	$0.499688\pi$
$674$	4.84484	0.186616
$675$	0	0
$676$	20.1161	0.773694
$677$	−39.1437	−1.50442	−0.752208	−	0.658925i	$-0.771011\pi$
$677$	−39.1437	−1.50442	−0.752208	+	0.658925i	$0.771011\pi$
$678$	0	0
$679$	−15.0596	−0.577935
$680$	−17.4877	−0.670622
$681$	0	0
$682$	0	0
$683$	−15.4140	−0.589800	−0.294900	−	0.955528i	$-0.595286\pi$
$683$	−15.4140	−0.589800	−0.294900	+	0.955528i	$0.595286\pi$
$684$	0	0
$685$	57.0678	2.18045
$686$	0.162147	0.00619081
$687$	0	0
$688$	−19.3833	−0.738982
$689$	3.98620	0.151862
$690$	0	0
$691$	−38.2746	−1.45603	−0.728017	−	0.685559i	$-0.759558\pi$
$691$	−38.2746	−1.45603	−0.728017	+	0.685559i	$0.759558\pi$
$692$	3.45520	0.131347
$693$	0	0
$694$	−4.73010	−0.179552
$695$	17.1144	0.649188
$696$	0	0
$697$	35.7417	1.35381
$698$	−2.17155	−0.0821942
$699$	0	0
$700$	28.1179	1.06276
$701$	1.38873	0.0524516	0.0262258	−	0.999656i	$-0.491651\pi$
$701$	1.38873	0.0524516	0.0262258	+	0.999656i	$0.491651\pi$
$702$	0	0
$703$	−3.36909	−0.127068
$704$	0	0
$705$	0	0
$706$	−3.45658	−0.130090
$707$	5.76079	0.216657
$708$	0	0
$709$	−12.8361	−0.482071	−0.241036	−	0.970516i	$-0.577487\pi$
$709$	−12.8361	−0.482071	−0.241036	+	0.970516i	$0.577487\pi$
$710$	−0.565157	−0.0212100
$711$	0	0
$712$	7.86760	0.294851
$713$	−2.46456	−0.0922985
$714$	0	0
$715$	0	0
$716$	−25.3702	−0.948131
$717$	0	0
$718$	−3.34403	−0.124798
$719$	21.5055	0.802021	0.401011	−	0.916073i	$-0.368659\pi$
$719$	21.5055	0.802021	0.401011	+	0.916073i	$0.368659\pi$
$720$	0	0
$721$	−7.71449	−0.287303
$722$	1.12245	0.0417733
$723$	0	0
$724$	43.5856	1.61985
$725$	126.327	4.69168
$726$	0	0
$727$	−5.35770	−0.198706	−0.0993531	−	0.995052i	$-0.531677\pi$
$727$	−5.35770	−0.198706	−0.0993531	+	0.995052i	$0.531677\pi$
$728$	−1.07970	−0.0400164
$729$	0	0
$730$	−1.69803	−0.0628468
$731$	31.2046	1.15414
$732$	0	0
$733$	41.4506	1.53101	0.765506	−	0.643429i	$-0.222489\pi$
$733$	41.4506	1.53101	0.765506	+	0.643429i	$0.222489\pi$
$734$	−5.32727	−0.196633
$735$	0	0
$736$	1.08703	0.0400683
$737$	0	0
$738$	0	0
$739$	42.6086	1.56738	0.783691	−	0.621151i	$-0.213335\pi$
$739$	42.6086	1.56738	0.783691	+	0.621151i	$0.213335\pi$
$740$	−8.39420	−0.308577
$741$	0	0
$742$	0.385719	0.0141602
$743$	30.4479	1.11702	0.558512	−	0.829496i	$-0.311372\pi$
$743$	30.4479	1.11702	0.558512	+	0.829496i	$0.311372\pi$
$744$	0	0
$745$	−81.3510	−2.98047
$746$	−4.23171	−0.154934
$747$	0	0
$748$	0	0
$749$	−16.4634	−0.601561
$750$	0	0
$751$	9.06635	0.330836	0.165418	−	0.986224i	$-0.447103\pi$
$751$	9.06635	0.330836	0.165418	+	0.986224i	$0.447103\pi$
$752$	−17.9564	−0.654802
$753$	0	0
$754$	−2.40938	−0.0877443
$755$	91.2415	3.32062
$756$	0	0
$757$	7.83076	0.284614	0.142307	−	0.989823i	$-0.454548\pi$
$757$	7.83076	0.284614	0.142307	+	0.989823i	$0.454548\pi$
$758$	1.89279	0.0687493
$759$	0	0
$760$	−9.82355	−0.356338
$761$	−26.4983	−0.960564	−0.480282	−	0.877114i	$-0.659466\pi$
$761$	−26.4983	−0.960564	−0.480282	+	0.877114i	$0.659466\pi$
$762$	0	0
$763$	−9.74355	−0.352740
$764$	29.0450	1.05081
$765$	0	0
$766$	−2.49222	−0.0900476
$767$	4.77411	0.172383
$768$	0	0
$769$	−26.3995	−0.951989	−0.475995	−	0.879448i	$-0.657912\pi$
$769$	−26.3995	−0.951989	−0.475995	+	0.879448i	$0.657912\pi$
$770$	0	0
$771$	0	0
$772$	−16.8019	−0.604714
$773$	−34.1566	−1.22853	−0.614264	−	0.789100i	$-0.710547\pi$
$773$	−34.1566	−1.22853	−0.614264	+	0.789100i	$0.710547\pi$
$774$	0	0
$775$	61.7499	2.21812
$776$	9.70330	0.348328
$777$	0	0
$778$	−1.56379	−0.0560645
$779$	20.0776	0.719354
$780$	0	0
$781$	0	0
$782$	−0.570383	−0.0203969
$783$	0	0
$784$	3.84294	0.137248
$785$	25.4877	0.909695
$786$	0	0
$787$	11.7503	0.418853	0.209426	−	0.977824i	$-0.432840\pi$
$787$	11.7503	0.418853	0.209426	+	0.977824i	$0.432840\pi$
$788$	−9.32128	−0.332057
$789$	0	0
$790$	0.477043	0.0169724
$791$	−2.85725	−0.101592
$792$	0	0
$793$	15.5821	0.553337
$794$	−3.28924	−0.116731
$795$	0	0
$796$	−33.0832	−1.17260
$797$	−15.3027	−0.542050	−0.271025	−	0.962572i	$-0.587363\pi$
$797$	−15.3027	−0.542050	−0.271025	+	0.962572i	$0.587363\pi$
$798$	0	0
$799$	28.9074	1.02267
$800$	−27.2356	−0.962924
$801$	0	0
$802$	−1.32256	−0.0467013
$803$	0	0
$804$	0	0
$805$	2.49446	0.0879181
$806$	−1.17772	−0.0414835
$807$	0	0
$808$	−3.71182	−0.130581
$809$	−18.2311	−0.640973	−0.320486	−	0.947253i	$-0.603846\pi$
$809$	−18.2311	−0.640973	−0.320486	+	0.947253i	$0.603846\pi$
$810$	0	0
$811$	6.23292	0.218868	0.109434	−	0.993994i	$-0.465096\pi$
$811$	6.23292	0.218868	0.109434	+	0.993994i	$0.465096\pi$
$812$	17.5017	0.614189
$813$	0	0
$814$	0	0
$815$	−39.0582	−1.36815
$816$	0	0
$817$	17.5289	0.613258
$818$	1.56830	0.0548344
$819$	0	0
$820$	50.0239	1.74691
$821$	29.6399	1.03444	0.517219	−	0.855853i	$-0.326967\pi$
$821$	29.6399	1.03444	0.517219	+	0.855853i	$0.326967\pi$
$822$	0	0
$823$	−19.7302	−0.687753	−0.343876	−	0.939015i	$-0.611740\pi$
$823$	−19.7302	−0.687753	−0.343876	+	0.939015i	$0.611740\pi$
$824$	4.97065	0.173161
$825$	0	0
$826$	0.461960	0.0160736
$827$	−41.1710	−1.43166	−0.715828	−	0.698276i	$-0.753951\pi$
$827$	−41.1710	−1.43166	−0.715828	+	0.698276i	$0.753951\pi$
$828$	0	0
$829$	22.7961	0.791742	0.395871	−	0.918306i	$-0.370443\pi$
$829$	22.7961	0.791742	0.395871	+	0.918306i	$0.370443\pi$
$830$	−6.60489	−0.229259
$831$	0	0
$832$	−12.3598	−0.428500
$833$	−6.18663	−0.214354
$834$	0	0
$835$	16.7741	0.580492
$836$	0	0
$837$	0	0
$838$	−3.07038	−0.106064
$839$	47.2725	1.63203	0.816013	−	0.578033i	$-0.196180\pi$
$839$	47.2725	1.63203	0.816013	+	0.578033i	$0.196180\pi$
$840$	0	0
$841$	49.6312	1.71142
$842$	2.12331	0.0731742
$843$	0	0
$844$	20.0241	0.689258
$845$	−44.7129	−1.53817
$846$	0	0
$847$	0	0
$848$	9.14167	0.313926
$849$	0	0
$850$	14.2910	0.490179
$851$	−0.551222	−0.0188956
$852$	0	0
$853$	10.8549	0.371664	0.185832	−	0.982582i	$-0.440502\pi$
$853$	10.8549	0.371664	0.185832	+	0.982582i	$0.440502\pi$
$854$	1.50778	0.0515951
$855$	0	0
$856$	10.6078	0.362567
$857$	−28.5431	−0.975014	−0.487507	−	0.873119i	$-0.662094\pi$
$857$	−28.5431	−0.975014	−0.487507	+	0.873119i	$0.662094\pi$
$858$	0	0
$859$	−36.8034	−1.25572	−0.627858	−	0.778328i	$-0.716068\pi$
$859$	−36.8034	−1.25572	−0.627858	+	0.778328i	$0.716068\pi$
$860$	43.6737	1.48926
$861$	0	0
$862$	−2.64356	−0.0900399
$863$	45.1551	1.53710	0.768549	−	0.639790i	$-0.220979\pi$
$863$	45.1551	1.53710	0.768549	+	0.639790i	$0.220979\pi$
$864$	0	0
$865$	−7.68003	−0.261129
$866$	−1.89457	−0.0643800
$867$	0	0
$868$	8.55498	0.290375
$869$	0	0
$870$	0	0
$871$	−11.9692	−0.405559
$872$	6.27802	0.212601
$873$	0	0
$874$	−0.320407	−0.0108379
$875$	−40.5638	−1.37131
$876$	0	0
$877$	34.8517	1.17686	0.588428	−	0.808549i	$-0.299747\pi$
$877$	34.8517	1.17686	0.588428	+	0.808549i	$0.299747\pi$
$878$	−1.89991	−0.0641187
$879$	0	0
$880$	0	0
$881$	35.3563	1.19118	0.595592	−	0.803287i	$-0.296917\pi$
$881$	35.3563	1.19118	0.595592	+	0.803287i	$0.296917\pi$
$882$	0	0
$883$	−24.8870	−0.837516	−0.418758	−	0.908098i	$-0.637534\pi$
$883$	−24.8870	−0.837516	−0.418758	+	0.908098i	$0.637534\pi$
$884$	20.4614	0.688190
$885$	0	0
$886$	−2.03772	−0.0684586
$887$	49.5708	1.66442	0.832212	−	0.554457i	$-0.187074\pi$
$887$	49.5708	1.66442	0.832212	+	0.554457i	$0.187074\pi$
$888$	0	0
$889$	16.0514	0.538346
$890$	−8.68599	−0.291155
$891$	0	0
$892$	12.0628	0.403891
$893$	16.2385	0.543399
$894$	0	0
$895$	56.3916	1.88496
$896$	−5.01953	−0.167691
$897$	0	0
$898$	−2.58398	−0.0862287
$899$	38.4356	1.28190
$900$	0	0
$901$	−14.7169	−0.490291
$902$	0	0
$903$	0	0
$904$	1.84100	0.0612307
$905$	−96.8798	−3.22039
$906$	0	0
$907$	−1.42826	−0.0474246	−0.0237123	−	0.999719i	$-0.507549\pi$
$907$	−1.42826	−0.0474246	−0.0237123	+	0.999719i	$0.507549\pi$
$908$	28.0443	0.930684
$909$	0	0
$910$	1.19201	0.0395148
$911$	−54.1725	−1.79482	−0.897408	−	0.441202i	$-0.854552\pi$
$911$	−54.1725	−1.79482	−0.897408	+	0.441202i	$0.854552\pi$
$912$	0	0
$913$	0	0
$914$	3.85821	0.127618
$915$	0	0
$916$	5.38391	0.177889
$917$	0.750136	0.0247717
$918$	0	0
$919$	−22.9009	−0.755432	−0.377716	−	0.925921i	$-0.623290\pi$
$919$	−22.9009	−0.755432	−0.377716	+	0.925921i	$0.623290\pi$
$920$	−1.60724	−0.0529892
$921$	0	0
$922$	−3.66510	−0.120704
$923$	1.33133	0.0438211
$924$	0	0
$925$	13.8110	0.454101
$926$	−0.492536	−0.0161857
$927$	0	0
$928$	−16.9525	−0.556493
$929$	14.5306	0.476733	0.238366	−	0.971175i	$-0.423388\pi$
$929$	14.5306	0.476733	0.238366	+	0.971175i	$0.423388\pi$
$930$	0	0
$931$	−3.47528	−0.113898
$932$	−35.1116	−1.15012
$933$	0	0
$934$	−3.73021	−0.122056
$935$	0	0
$936$	0	0
$937$	30.6351	1.00080	0.500402	−	0.865793i	$-0.333186\pi$
$937$	30.6351	1.00080	0.500402	+	0.865793i	$0.333186\pi$
$938$	−1.15818	−0.0378158
$939$	0	0
$940$	40.4586	1.31961
$941$	−21.0533	−0.686319	−0.343159	−	0.939277i	$-0.611497\pi$
$941$	−21.0533	−0.686319	−0.343159	+	0.939277i	$0.611497\pi$
$942$	0	0
$943$	3.28492	0.106972
$944$	10.9486	0.356346
$945$	0	0
$946$	0	0
$947$	−0.729464	−0.0237044	−0.0118522	−	0.999930i	$-0.503773\pi$
$947$	−0.729464	−0.0237044	−0.0118522	+	0.999930i	$0.503773\pi$
$948$	0	0
$949$	4.00000	0.129845
$950$	8.02786	0.260458
$951$	0	0
$952$	3.98620	0.129194
$953$	−39.0151	−1.26382	−0.631911	−	0.775041i	$-0.717729\pi$
$953$	−39.0151	−1.26382	−0.631911	+	0.775041i	$0.717729\pi$
$954$	0	0
$955$	−64.5597	−2.08910
$956$	25.0775	0.811065
$957$	0	0
$958$	1.32520	0.0428153
$959$	−13.0082	−0.420058
$960$	0	0
$961$	−12.2124	−0.393947
$962$	−0.263409	−0.00849264
$963$	0	0
$964$	36.8302	1.18622
$965$	37.3464	1.20222
$966$	0	0
$967$	2.46386	0.0792324	0.0396162	−	0.999215i	$-0.487386\pi$
$967$	2.46386	0.0792324	0.0396162	+	0.999215i	$0.487386\pi$
$968$	0	0
$969$	0	0
$970$	−10.7126	−0.343962
$971$	−34.6794	−1.11291	−0.556457	−	0.830876i	$-0.687840\pi$
$971$	−34.6794	−1.11291	−0.556457	+	0.830876i	$0.687840\pi$
$972$	0	0
$973$	−3.90112	−0.125064
$974$	0.162023	0.00519154
$975$	0	0
$976$	35.7348	1.14384
$977$	25.1338	0.804100	0.402050	−	0.915618i	$-0.368298\pi$
$977$	25.1338	0.804100	0.402050	+	0.915618i	$0.368298\pi$
$978$	0	0
$979$	0	0
$980$	−8.65877	−0.276594
$981$	0	0
$982$	3.93948	0.125714
$983$	−30.3900	−0.969290	−0.484645	−	0.874711i	$-0.661051\pi$
$983$	−30.3900	−0.969290	−0.484645	+	0.874711i	$0.661051\pi$
$984$	0	0
$985$	20.7189	0.660158
$986$	8.89531	0.283284
$987$	0	0
$988$	11.4940	0.365672
$989$	2.86792	0.0911947
$990$	0	0
$991$	−8.09793	−0.257239	−0.128620	−	0.991694i	$-0.541055\pi$
$991$	−8.09793	−0.257239	−0.128620	+	0.991694i	$0.541055\pi$
$992$	−8.28653	−0.263097
$993$	0	0
$994$	0.128824	0.00408604
$995$	73.5356	2.33124
$996$	0	0
$997$	−45.8866	−1.45324	−0.726621	−	0.687038i	$-0.758910\pi$
$997$	−45.8866	−1.45324	−0.726621	+	0.687038i	$0.758910\pi$
$998$	6.66112	0.210854
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.ci.1.3		4
3.2	odd	2		2541.2.a.bn.1.2		4
11.3	even	5		693.2.m.f.64.2		8
11.4	even	5		693.2.m.f.379.2		8
11.10	odd	2		7623.2.a.cl.1.2		4
33.14	odd	10		231.2.j.f.64.1	✓	8
33.26	odd	10		231.2.j.f.148.1	yes	8
33.32	even	2		2541.2.a.bm.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.j.f.64.1	✓	8	33.14	odd	10
231.2.j.f.148.1	yes	8	33.26	odd	10
693.2.m.f.64.2		8	11.3	even	5
693.2.m.f.379.2		8	11.4	even	5
2541.2.a.bm.1.3		4	33.32	even	2
2541.2.a.bn.1.2		4	3.2	odd	2
7623.2.a.ci.1.3		4	1.1	even	1	trivial
7623.2.a.cl.1.2		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)

\(f(q)\)	\(=\)	\(q-0.162147 q^{2} -1.97371 q^{4} +4.38705 q^{5} -1.00000 q^{7} +0.644326 q^{8} +O(q^{10})\)	\(q-0.162147 q^{2} -1.97371 q^{4} +4.38705 q^{5} -1.00000 q^{7} +0.644326 q^{8} -0.711349 q^{10} +1.67571 q^{13} +0.162147 q^{14} +3.84294 q^{16} -6.18663 q^{17} -3.47528 q^{19} -8.65877 q^{20} -0.568595 q^{23} +14.2462 q^{25} -0.271711 q^{26} +1.97371 q^{28} +8.86742 q^{29} +4.33447 q^{31} -1.91177 q^{32} +1.00314 q^{34} -4.38705 q^{35} +0.969445 q^{37} +0.563507 q^{38} +2.82669 q^{40} -5.77725 q^{41} -5.04388 q^{43} +0.0921961 q^{46} -4.67256 q^{47} +1.00000 q^{49} -2.30999 q^{50} -3.30735 q^{52} +2.37882 q^{53} -0.644326 q^{56} -1.43783 q^{58} +2.84901 q^{59} +9.29883 q^{61} -0.702822 q^{62} -7.37589 q^{64} +7.35141 q^{65} -7.14275 q^{67} +12.2106 q^{68} +0.711349 q^{70} +0.794487 q^{71} +2.38705 q^{73} -0.157193 q^{74} +6.85919 q^{76} -0.670617 q^{79} +16.8592 q^{80} +0.936765 q^{82} +9.28503 q^{83} -27.1411 q^{85} +0.817850 q^{86} +12.2106 q^{89} -1.67571 q^{91} +1.12224 q^{92} +0.757643 q^{94} -15.2462 q^{95} +15.0596 q^{97} -0.162147 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} + 3 q^{4} + 4 q^{5} - 4 q^{7} + 9 q^{8}+O(q^{10}) \) 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

L-functions

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