LMFDB - Embedded newform 7623.2.a.cq.1.5

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:	$1$
Dimension:	$6$
Coefficient field:	6.6.3829849.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 6x^{4} + 15x^{3} - 5x^{2} - 3x + 1 $ x^6 - 2x^5 - 6x^4 + 15x^3 - 5x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.300853$ 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)

$f(q)$	$=$	$q+1.36768 q^{2} -0.129461 q^{4} -0.636509 q^{5} -1.00000 q^{7} -2.91241 q^{8} +O(q^{10})$	$q+1.36768 q^{2} -0.129461 q^{4} -0.636509 q^{5} -1.00000 q^{7} -2.91241 q^{8} -0.870539 q^{10} +3.59486 q^{13} -1.36768 q^{14} -3.72432 q^{16} +1.46233 q^{17} +5.85378 q^{19} +0.0824033 q^{20} -8.56018 q^{23} -4.59486 q^{25} +4.91660 q^{26} +0.129461 q^{28} +7.73436 q^{29} -9.44863 q^{31} +0.731167 q^{32} +2.00000 q^{34} +0.636509 q^{35} +5.59486 q^{37} +8.00607 q^{38} +1.85378 q^{40} +8.37086 q^{41} -1.74108 q^{43} -11.7076 q^{46} +4.99900 q^{47} +1.00000 q^{49} -6.28428 q^{50} -0.465395 q^{52} -11.1062 q^{53} +2.91241 q^{56} +10.5781 q^{58} +13.3944 q^{59} -11.1897 q^{61} -12.9227 q^{62} +8.44863 q^{64} -2.28816 q^{65} -1.85378 q^{67} -0.189316 q^{68} +0.870539 q^{70} -9.66839 q^{71} -11.5949 q^{73} +7.65195 q^{74} -0.757838 q^{76} -4.51785 q^{79} +2.37056 q^{80} +11.4486 q^{82} +3.08948 q^{83} -0.930789 q^{85} -2.38123 q^{86} -4.19769 q^{89} -3.59486 q^{91} +1.10821 q^{92} +6.83702 q^{94} -3.72598 q^{95} +7.44863 q^{97} +1.36768 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{4} - 6 q^{7}+O(q^{10}) $ 6 * q + 4 * q^4 - 6 * q^7	$ 6 q + 4 q^{4} - 6 q^{7} - 10 q^{10} - 4 q^{13} + 8 q^{16} - 2 q^{25} - 4 q^{28} + 4 q^{31} + 12 q^{34} + 8 q^{37} - 24 q^{40} - 20 q^{43} + 6 q^{49} + 18 q^{52} - 2 q^{58} - 16 q^{61} - 10 q^{64} + 24 q^{67} + 10 q^{70} - 44 q^{73} - 54 q^{76} - 8 q^{79} + 8 q^{82} + 36 q^{85} + 4 q^{91} - 34 q^{94} - 16 q^{97}+O(q^{100}) $ 6 * q + 4 * q^4 - 6 * q^7 - 10 * q^10 - 4 * q^13 + 8 * q^16 - 2 * q^25 - 4 * q^28 + 4 * q^31 + 12 * q^34 + 8 * q^37 - 24 * q^40 - 20 * q^43 + 6 * q^49 + 18 * q^52 - 2 * q^58 - 16 * q^61 - 10 * q^64 + 24 * q^67 + 10 * q^70 - 44 * q^73 - 54 * q^76 - 8 * q^79 + 8 * q^82 + 36 * q^85 + 4 * q^91 - 34 * q^94 - 16 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.36768	0.967093	0.483547	−	0.875319i	$-0.339348\pi$
$2$	1.36768	0.967093	0.483547	+	0.875319i	$0.339348\pi$
$3$	0	0
$3$	0	0
$4$	−0.129461	−0.0647306
$5$	−0.636509	−0.284656	−0.142328	−	0.989820i	$-0.545459\pi$
$5$	−0.636509	−0.284656	−0.142328	+	0.989820i	$0.545459\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−2.91241	−1.02969
$9$	0	0
$10$	−0.870539	−0.275289
$11$	0	0
$11$	0	0
$12$	0	0
$13$	3.59486	0.997034	0.498517	−	0.866880i	$-0.333878\pi$
$13$	3.59486	0.997034	0.498517	+	0.866880i	$0.333878\pi$
$14$	−1.36768	−0.365527
$15$	0	0
$16$	−3.72432	−0.931079
$17$	1.46233	0.354668	0.177334	−	0.984151i	$-0.443253\pi$
$17$	1.46233	0.354668	0.177334	+	0.984151i	$0.443253\pi$
$18$	0	0
$19$	5.85378	1.34295	0.671474	−	0.741028i	$-0.265661\pi$
$19$	5.85378	1.34295	0.671474	+	0.741028i	$0.265661\pi$
$20$	0.0824033	0.0184259
$21$	0	0
$22$	0	0
$23$	−8.56018	−1.78492	−0.892461	−	0.451125i	$-0.851023\pi$
$23$	−8.56018	−1.78492	−0.892461	+	0.451125i	$0.851023\pi$
$24$	0	0
$25$	−4.59486	−0.918971
$26$	4.91660	0.964225
$27$	0	0
$28$	0.129461	0.0244659
$29$	7.73436	1.43623	0.718117	−	0.695922i	$-0.245004\pi$
$29$	7.73436	1.43623	0.718117	+	0.695922i	$0.245004\pi$
$30$	0	0
$31$	−9.44863	−1.69702	−0.848512	−	0.529175i	$-0.822501\pi$
$31$	−9.44863	−1.69702	−0.848512	+	0.529175i	$0.822501\pi$
$32$	0.731167	0.129253
$33$	0	0
$34$	2.00000	0.342997
$35$	0.636509	0.107590
$36$	0	0
$37$	5.59486	0.919789	0.459894	−	0.887974i	$-0.347887\pi$
$37$	5.59486	0.919789	0.459894	+	0.887974i	$0.347887\pi$
$38$	8.00607	1.29876
$39$	0	0
$40$	1.85378	0.293108
$41$	8.37086	1.30731	0.653655	−	0.756793i	$-0.273235\pi$
$41$	8.37086	1.30731	0.653655	+	0.756793i	$0.273235\pi$
$42$	0	0
$43$	−1.74108	−0.265512	−0.132756	−	0.991149i	$-0.542383\pi$
$43$	−1.74108	−0.265512	−0.132756	+	0.991149i	$0.542383\pi$
$44$	0	0
$45$	0	0
$46$	−11.7076	−1.72619
$47$	4.99900	0.729180	0.364590	−	0.931168i	$-0.381209\pi$
$47$	4.99900	0.729180	0.364590	+	0.931168i	$0.381209\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−6.28428	−0.888731
$51$	0	0
$52$	−0.465395	−0.0645386
$53$	−11.1062	−1.52556	−0.762778	−	0.646660i	$-0.776165\pi$
$53$	−11.1062	−1.52556	−0.762778	+	0.646660i	$0.776165\pi$
$54$	0	0
$55$	0	0
$56$	2.91241	0.389188
$57$	0	0
$58$	10.5781	1.38897
$59$	13.3944	1.74380	0.871900	−	0.489685i	$-0.162888\pi$
$59$	13.3944	1.74380	0.871900	+	0.489685i	$0.162888\pi$
$60$	0	0
$61$	−11.1897	−1.43270	−0.716348	−	0.697743i	$-0.754188\pi$
$61$	−11.1897	−1.43270	−0.716348	+	0.697743i	$0.754188\pi$
$62$	−12.9227	−1.64118
$63$	0	0
$64$	8.44863	1.05608
$65$	−2.28816	−0.283811
$66$	0	0
$67$	−1.85378	−0.226475	−0.113238	−	0.993568i	$-0.536122\pi$
$67$	−1.85378	−0.226475	−0.113238	+	0.993568i	$0.536122\pi$
$68$	−0.189316	−0.0229579
$69$	0	0
$70$	0.870539	0.104049
$71$	−9.66839	−1.14743	−0.573714	−	0.819056i	$-0.694498\pi$
$71$	−9.66839	−1.14743	−0.573714	+	0.819056i	$0.694498\pi$
$72$	0	0
$73$	−11.5949	−1.35708	−0.678538	−	0.734566i	$-0.737386\pi$
$73$	−11.5949	−1.35708	−0.678538	+	0.734566i	$0.737386\pi$
$74$	7.65195	0.889521
$75$	0	0
$76$	−0.757838	−0.0869299
$77$	0	0
$78$	0	0
$79$	−4.51785	−0.508297	−0.254149	−	0.967165i	$-0.581795\pi$
$79$	−4.51785	−0.508297	−0.254149	+	0.967165i	$0.581795\pi$
$80$	2.37056	0.265037
$81$	0	0
$82$	11.4486	1.26429
$83$	3.08948	0.339114	0.169557	−	0.985520i	$-0.445766\pi$
$83$	3.08948	0.339114	0.169557	+	0.985520i	$0.445766\pi$
$84$	0	0
$85$	−0.930789	−0.100958
$86$	−2.38123	−0.256775
$87$	0	0
$88$	0	0
$89$	−4.19769	−0.444954	−0.222477	−	0.974938i	$-0.571414\pi$
$89$	−4.19769	−0.444954	−0.222477	+	0.974938i	$0.571414\pi$
$90$	0	0
$91$	−3.59486	−0.376843
$92$	1.10821	0.115539
$93$	0	0
$94$	6.83702	0.705185
$95$	−3.72598	−0.382278
$96$	0	0
$97$	7.44863	0.756294	0.378147	−	0.925746i	$-0.376561\pi$
$97$	7.44863	0.756294	0.378147	+	0.925746i	$0.376561\pi$
$98$	1.36768	0.138156
$99$	0	0
$100$	0.594856	0.0594856
$101$	−6.93304	−0.689863	−0.344932	−	0.938628i	$-0.612098\pi$
$101$	−6.93304	−0.689863	−0.344932	+	0.938628i	$0.612098\pi$
$102$	0	0
$103$	−6.67187	−0.657399	−0.328699	−	0.944435i	$-0.606610\pi$
$103$	−6.67187	−0.657399	−0.328699	+	0.944435i	$0.606610\pi$
$104$	−10.4697	−1.02664
$105$	0	0
$106$	−15.1897	−1.47536
$107$	−12.1214	−1.17182	−0.585908	−	0.810378i	$-0.699262\pi$
$107$	−12.1214	−1.17182	−0.585908	+	0.810378i	$0.699262\pi$
$108$	0	0
$109$	−13.4486	−1.28815	−0.644073	−	0.764964i	$-0.722757\pi$
$109$	−13.4486	−1.28815	−0.644073	+	0.764964i	$0.722757\pi$
$110$	0	0
$111$	0	0
$112$	3.72432	0.351915
$113$	−9.99801	−0.940533	−0.470267	−	0.882524i	$-0.655842\pi$
$113$	−9.99801	−0.940533	−0.470267	+	0.882524i	$0.655842\pi$
$114$	0	0
$115$	5.44863	0.508088
$116$	−1.00130	−0.0929683
$117$	0	0
$118$	18.3192	1.68642
$119$	−1.46233	−0.134052
$120$	0	0
$121$	0	0
$122$	−15.3039	−1.38555
$123$	0	0
$124$	1.22323	0.109850
$125$	6.10721	0.546246
$126$	0	0
$127$	12.6383	1.12147	0.560736	−	0.827995i	$-0.310518\pi$
$127$	12.6383	1.12147	0.560736	+	0.827995i	$0.310518\pi$
$128$	10.0927	0.892074
$129$	0	0
$130$	−3.12946	−0.274472
$131$	4.19769	0.366754	0.183377	−	0.983043i	$-0.441297\pi$
$131$	4.19769	0.366754	0.183377	+	0.983043i	$0.441297\pi$
$132$	0	0
$133$	−5.85378	−0.507587
$134$	−2.53537	−0.219023
$135$	0	0
$136$	−4.25892	−0.365200
$137$	−16.9556	−1.44861	−0.724305	−	0.689479i	$-0.757839\pi$
$137$	−16.9556	−1.44861	−0.724305	+	0.689479i	$0.757839\pi$
$138$	0	0
$139$	−12.5178	−1.06175	−0.530875	−	0.847450i	$-0.678137\pi$
$139$	−12.5178	−1.06175	−0.530875	+	0.847450i	$0.678137\pi$
$140$	−0.0824033	−0.00696435
$141$	0	0
$142$	−13.2232	−1.10967
$143$	0	0
$144$	0	0
$145$	−4.92299	−0.408832
$146$	−15.8580	−1.31242
$147$	0	0
$148$	−0.724317	−0.0595385
$149$	−3.91530	−0.320754	−0.160377	−	0.987056i	$-0.551271\pi$
$149$	−3.91530	−0.320754	−0.160377	+	0.987056i	$0.551271\pi$
$150$	0	0
$151$	−0.930789	−0.0757466	−0.0378733	−	0.999283i	$-0.512058\pi$
$151$	−0.930789	−0.0757466	−0.0378733	+	0.999283i	$0.512058\pi$
$152$	−17.0486	−1.38283
$153$	0	0
$154$	0	0
$155$	6.01414	0.483068
$156$	0	0
$157$	5.70756	0.455513	0.227756	−	0.973718i	$-0.426861\pi$
$157$	5.70756	0.455513	0.227756	+	0.973718i	$0.426861\pi$
$158$	−6.17895	−0.491571
$159$	0	0
$160$	−0.465395	−0.0367927
$161$	8.56018	0.674637
$162$	0	0
$163$	−1.85378	−0.145199	−0.0725996	−	0.997361i	$-0.523130\pi$
$163$	−1.85378	−0.145199	−0.0725996	+	0.997361i	$0.523130\pi$
$164$	−1.08370	−0.0846230
$165$	0	0
$166$	4.22540	0.327955
$167$	23.1345	1.79020	0.895101	−	0.445864i	$-0.147103\pi$
$167$	23.1345	1.79020	0.895101	+	0.445864i	$0.147103\pi$
$168$	0	0
$169$	−0.0770108	−0.00592391
$170$	−1.27302	−0.0976361
$171$	0	0
$172$	0.225402	0.0171868
$173$	−7.09785	−0.539639	−0.269820	−	0.962911i	$-0.586964\pi$
$173$	−7.09785	−0.539639	−0.269820	+	0.962911i	$0.586964\pi$
$174$	0	0
$175$	4.59486	0.347338
$176$	0	0
$177$	0	0
$178$	−5.74108	−0.430312
$179$	5.09207	0.380599	0.190300	−	0.981726i	$-0.439054\pi$
$179$	5.09207	0.380599	0.190300	+	0.981726i	$0.439054\pi$
$180$	0	0
$181$	−19.4486	−1.44561	−0.722803	−	0.691054i	$-0.757147\pi$
$181$	−19.4486	−1.44561	−0.722803	+	0.691054i	$0.757147\pi$
$182$	−4.91660	−0.364443
$183$	0	0
$184$	24.9308	1.83792
$185$	−3.56118	−0.261823
$186$	0	0
$187$	0	0
$188$	−0.647177	−0.0472003
$189$	0	0
$190$	−5.09594	−0.369698
$191$	20.7746	1.50320	0.751599	−	0.659620i	$-0.229283\pi$
$191$	20.7746	1.50320	0.751599	+	0.659620i	$0.229283\pi$
$192$	0	0
$193$	−21.4486	−1.54391	−0.771953	−	0.635679i	$-0.780720\pi$
$193$	−21.4486	−1.54391	−0.771953	+	0.635679i	$0.780720\pi$
$194$	10.1873	0.731407
$195$	0	0
$196$	−0.129461	−0.00924723
$197$	−3.27879	−0.233604	−0.116802	−	0.993155i	$-0.537264\pi$
$197$	−3.27879	−0.233604	−0.116802	+	0.993155i	$0.537264\pi$
$198$	0	0
$199$	16.6383	1.17946	0.589731	−	0.807600i	$-0.299234\pi$
$199$	16.6383	1.17946	0.589731	+	0.807600i	$0.299234\pi$
$200$	13.3821	0.946259
$201$	0	0
$202$	−9.48215	−0.667162
$203$	−7.73436	−0.542845
$204$	0	0
$205$	−5.32813	−0.372133
$206$	−9.12495	−0.635766
$207$	0	0
$208$	−13.3884	−0.928317
$209$	0	0
$210$	0	0
$211$	−22.3794	−1.54066	−0.770332	−	0.637644i	$-0.779909\pi$
$211$	−22.3794	−1.54066	−0.770332	+	0.637644i	$0.779909\pi$
$212$	1.43783	0.0987502
$213$	0	0
$214$	−16.5781	−1.13326
$215$	1.10821	0.0755794
$216$	0	0
$217$	9.44863	0.641415
$218$	−18.3934	−1.24576
$219$	0	0
$220$	0	0
$221$	5.25688	0.353616
$222$	0	0
$223$	−8.51785	−0.570397	−0.285199	−	0.958468i	$-0.592060\pi$
$223$	−8.51785	−0.570397	−0.285199	+	0.958468i	$0.592060\pi$
$224$	−0.731167	−0.0488532
$225$	0	0
$226$	−13.6740	−0.909583
$227$	−23.8641	−1.58391	−0.791957	−	0.610576i	$-0.790938\pi$
$227$	−23.8641	−1.58391	−0.791957	+	0.610576i	$0.790938\pi$
$228$	0	0
$229$	4.77677	0.315658	0.157829	−	0.987466i	$-0.449551\pi$
$229$	4.77677	0.315658	0.157829	+	0.987466i	$0.449551\pi$
$230$	7.45197	0.491368
$231$	0	0
$232$	−22.5256	−1.47888
$233$	−22.5666	−1.47838	−0.739192	−	0.673495i	$-0.764792\pi$
$233$	−22.5666	−1.47838	−0.739192	+	0.673495i	$0.764792\pi$
$234$	0	0
$235$	−3.18191	−0.207565
$236$	−1.73405	−0.112877
$237$	0	0
$238$	−2.00000	−0.129641
$239$	20.1381	1.30263	0.651313	−	0.758809i	$-0.274219\pi$
$239$	20.1381	1.30263	0.651313	+	0.758809i	$0.274219\pi$
$240$	0	0
$241$	−0.922989	−0.0594550	−0.0297275	−	0.999558i	$-0.509464\pi$
$241$	−0.922989	−0.0594550	−0.0297275	+	0.999558i	$0.509464\pi$
$242$	0	0
$243$	0	0
$244$	1.44863	0.0927393
$245$	−0.636509	−0.0406651
$246$	0	0
$247$	21.0435	1.33897
$248$	27.5183	1.74742
$249$	0	0
$250$	8.35269	0.528271
$251$	−10.8483	−0.684741	−0.342371	−	0.939565i	$-0.611230\pi$
$251$	−10.8483	−0.684741	−0.342371	+	0.939565i	$0.611230\pi$
$252$	0	0
$253$	0	0
$254$	17.2852	1.08457
$255$	0	0
$256$	−3.09377	−0.193361
$257$	−6.81546	−0.425137	−0.212568	−	0.977146i	$-0.568183\pi$
$257$	−6.81546	−0.425137	−0.212568	+	0.977146i	$0.568183\pi$
$258$	0	0
$259$	−5.59486	−0.347647
$260$	0.296228	0.0183713
$261$	0	0
$262$	5.74108	0.354685
$263$	7.92367	0.488595	0.244297	−	0.969700i	$-0.421443\pi$
$263$	7.92367	0.488595	0.244297	+	0.969700i	$0.421443\pi$
$264$	0	0
$265$	7.06921	0.434258
$266$	−8.00607	−0.490884
$267$	0	0
$268$	0.239993	0.0146599
$269$	−26.4101	−1.61025	−0.805127	−	0.593103i	$-0.797903\pi$
$269$	−26.4101	−1.61025	−0.805127	+	0.593103i	$0.797903\pi$
$270$	0	0
$271$	−18.1462	−1.10230	−0.551152	−	0.834405i	$-0.685812\pi$
$271$	−18.1462	−1.10230	−0.551152	+	0.834405i	$0.685812\pi$
$272$	−5.44620	−0.330224
$273$	0	0
$274$	−23.1897	−1.40094
$275$	0	0
$276$	0	0
$277$	−4.29244	−0.257908	−0.128954	−	0.991651i	$-0.541162\pi$
$277$	−4.29244	−0.257908	−0.128954	+	0.991651i	$0.541162\pi$
$278$	−17.1204	−1.02681
$279$	0	0
$280$	−1.85378	−0.110784
$281$	−12.3107	−0.734393	−0.367197	−	0.930143i	$-0.619682\pi$
$281$	−12.3107	−0.734393	−0.367197	+	0.930143i	$0.619682\pi$
$282$	0	0
$283$	18.3716	1.09208	0.546040	−	0.837759i	$-0.316135\pi$
$283$	18.3716	1.09208	0.546040	+	0.837759i	$0.316135\pi$
$284$	1.25168	0.0742737
$285$	0	0
$286$	0	0
$287$	−8.37086	−0.494117
$288$	0	0
$289$	−14.8616	−0.874211
$290$	−6.73306	−0.395379
$291$	0	0
$292$	1.50108	0.0878443
$293$	3.46493	0.202424	0.101212	−	0.994865i	$-0.467728\pi$
$293$	3.46493	0.202424	0.101212	+	0.994865i	$0.467728\pi$
$294$	0	0
$295$	−8.52565	−0.496382
$296$	−16.2945	−0.947101
$297$	0	0
$298$	−5.35486	−0.310199
$299$	−30.7726	−1.77963
$300$	0	0
$301$	1.74108	0.100354
$302$	−1.27302	−0.0732540
$303$	0	0
$304$	−21.8013	−1.25039
$305$	7.12236	0.407825
$306$	0	0
$307$	−9.03569	−0.515694	−0.257847	−	0.966186i	$-0.583013\pi$
$307$	−9.03569	−0.515694	−0.257847	+	0.966186i	$0.583013\pi$
$308$	0	0
$309$	0	0
$310$	8.22540	0.467171
$311$	−8.72499	−0.494749	−0.247374	−	0.968920i	$-0.579568\pi$
$311$	−8.72499	−0.494749	−0.247374	+	0.968920i	$0.579568\pi$
$312$	0	0
$313$	−14.1205	−0.798138	−0.399069	−	0.916921i	$-0.630667\pi$
$313$	−14.1205	−0.798138	−0.399069	+	0.916921i	$0.630667\pi$
$314$	7.80609	0.440523
$315$	0	0
$316$	0.584886	0.0329024
$317$	−16.0121	−0.899332	−0.449666	−	0.893197i	$-0.648457\pi$
$317$	−16.0121	−0.899332	−0.449666	+	0.893197i	$0.648457\pi$
$318$	0	0
$319$	0	0
$320$	−5.37763	−0.300619
$321$	0	0
$322$	11.7076	0.652437
$323$	8.56018	0.476301
$324$	0	0
$325$	−16.5178	−0.916245
$326$	−2.53537	−0.140421
$327$	0	0
$328$	−24.3794	−1.34613
$329$	−4.99900	−0.275604
$330$	0	0
$331$	−13.0357	−0.716506	−0.358253	−	0.933624i	$-0.616628\pi$
$331$	−13.0357	−0.716506	−0.358253	+	0.933624i	$0.616628\pi$
$332$	−0.399967	−0.0219511
$333$	0	0
$334$	31.6405	1.73129
$335$	1.17995	0.0644674
$336$	0	0
$337$	−22.4843	−1.22480	−0.612400	−	0.790548i	$-0.709796\pi$
$337$	−22.4843	−1.22480	−0.612400	+	0.790548i	$0.709796\pi$
$338$	−0.105326	−0.00572897
$339$	0	0
$340$	0.120501	0.00653509
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	5.07074	0.273396
$345$	0	0
$346$	−9.70756	−0.521881
$347$	2.87565	0.154373	0.0771865	−	0.997017i	$-0.475406\pi$
$347$	2.87565	0.154373	0.0771865	+	0.997017i	$0.475406\pi$
$348$	0	0
$349$	7.07701	0.378824	0.189412	−	0.981898i	$-0.439342\pi$
$349$	7.07701	0.378824	0.189412	+	0.981898i	$0.439342\pi$
$350$	6.28428	0.335909
$351$	0	0
$352$	0	0
$353$	3.93981	0.209695	0.104847	−	0.994488i	$-0.466565\pi$
$353$	3.93981	0.209695	0.104847	+	0.994488i	$0.466565\pi$
$354$	0	0
$355$	6.15402	0.326622
$356$	0.543438	0.0288022
$357$	0	0
$358$	6.96431	0.368075
$359$	−10.5628	−0.557482	−0.278741	−	0.960366i	$-0.589917\pi$
$359$	−10.5628	−0.557482	−0.278741	+	0.960366i	$0.589917\pi$
$360$	0	0
$361$	15.2667	0.803512
$362$	−26.5994	−1.39804
$363$	0	0
$364$	0.465395	0.0243933
$365$	7.38023	0.386299
$366$	0	0
$367$	21.8616	1.14117	0.570583	−	0.821240i	$-0.306717\pi$
$367$	21.8616	1.14117	0.570583	+	0.821240i	$0.306717\pi$
$368$	31.8808	1.66190
$369$	0	0
$370$	−4.87054	−0.253207
$371$	11.1062	0.576606
$372$	0	0
$373$	−6.25892	−0.324075	−0.162037	−	0.986785i	$-0.551807\pi$
$373$	−6.25892	−0.324075	−0.162037	+	0.986785i	$0.551807\pi$
$374$	0	0
$375$	0	0
$376$	−14.5592	−0.750832
$377$	27.8039	1.43197
$378$	0	0
$379$	22.3716	1.14915	0.574577	−	0.818451i	$-0.305167\pi$
$379$	22.3716	1.14915	0.574577	+	0.818451i	$0.305167\pi$
$380$	0.482371	0.0247451
$381$	0	0
$382$	28.4129	1.45373
$383$	22.9697	1.17370	0.586848	−	0.809697i	$-0.300368\pi$
$383$	22.9697	1.17370	0.586848	+	0.809697i	$0.300368\pi$
$384$	0	0
$385$	0	0
$386$	−29.3348	−1.49310
$387$	0	0
$388$	−0.964310	−0.0489554
$389$	38.0598	1.92971	0.964854	−	0.262788i	$-0.0846419\pi$
$389$	38.0598	1.92971	0.964854	+	0.262788i	$0.0846419\pi$
$390$	0	0
$391$	−12.5178	−0.633055
$392$	−2.91241	−0.147099
$393$	0	0
$394$	−4.48432	−0.225917
$395$	2.87565	0.144690
$396$	0	0
$397$	−17.6027	−0.883452	−0.441726	−	0.897150i	$-0.645634\pi$
$397$	−17.6027	−0.883452	−0.441726	+	0.897150i	$0.645634\pi$
$398$	22.7559	1.14065
$399$	0	0
$400$	17.1127	0.855635
$401$	−6.95755	−0.347443	−0.173722	−	0.984795i	$-0.555579\pi$
$401$	−6.95755	−0.347443	−0.173722	+	0.984795i	$0.555579\pi$
$402$	0	0
$403$	−33.9665	−1.69199
$404$	0.897560	0.0446553
$405$	0	0
$406$	−10.5781	−0.524982
$407$	0	0
$408$	0	0
$409$	−14.6719	−0.725477	−0.362739	−	0.931891i	$-0.618158\pi$
$409$	−14.6719	−0.725477	−0.362739	+	0.931891i	$0.618158\pi$
$410$	−7.28716	−0.359887
$411$	0	0
$412$	0.863748	0.0425538
$413$	−13.3944	−0.659094
$414$	0	0
$415$	−1.96648	−0.0965307
$416$	2.62844	0.128870
$417$	0	0
$418$	0	0
$419$	−16.2700	−0.794843	−0.397421	−	0.917636i	$-0.630095\pi$
$419$	−16.2700	−0.794843	−0.397421	+	0.917636i	$0.630095\pi$
$420$	0	0
$421$	20.7846	1.01298	0.506489	−	0.862246i	$-0.330943\pi$
$421$	20.7846	1.01298	0.506489	+	0.862246i	$0.330943\pi$
$422$	−30.6078	−1.48996
$423$	0	0
$424$	32.3459	1.57086
$425$	−6.71922	−0.325930
$426$	0	0
$427$	11.1897	0.541508
$428$	1.56925	0.0758524
$429$	0	0
$430$	1.51568	0.0730924
$431$	33.4394	1.61072	0.805360	−	0.592786i	$-0.201972\pi$
$431$	33.4394	1.61072	0.805360	+	0.592786i	$0.201972\pi$
$432$	0	0
$433$	−29.7076	−1.42765	−0.713827	−	0.700322i	$-0.753040\pi$
$433$	−29.7076	−1.42765	−0.713827	+	0.700322i	$0.753040\pi$
$434$	12.9227	0.620308
$435$	0	0
$436$	1.74108	0.0833825
$437$	−50.1094	−2.39706
$438$	0	0
$439$	4.81809	0.229955	0.114977	−	0.993368i	$-0.463320\pi$
$439$	4.81809	0.229955	0.114977	+	0.993368i	$0.463320\pi$
$440$	0	0
$441$	0	0
$442$	7.18971	0.341980
$443$	7.12236	0.338393	0.169197	−	0.985582i	$-0.445883\pi$
$443$	7.12236	0.338393	0.169197	+	0.985582i	$0.445883\pi$
$444$	0	0
$445$	2.67187	0.126659
$446$	−11.6497	−0.551627
$447$	0	0
$448$	−8.44863	−0.399160
$449$	32.2104	1.52010	0.760052	−	0.649862i	$-0.225173\pi$
$449$	32.2104	1.52010	0.760052	+	0.649862i	$0.225173\pi$
$450$	0	0
$451$	0	0
$452$	1.29435	0.0608813
$453$	0	0
$454$	−32.6383	−1.53179
$455$	2.28816	0.107271
$456$	0	0
$457$	19.8281	0.927517	0.463759	−	0.885962i	$-0.346500\pi$
$457$	19.8281	0.927517	0.463759	+	0.885962i	$0.346500\pi$
$458$	6.53307	0.305270
$459$	0	0
$460$	−0.705387	−0.0328888
$461$	33.8866	1.57826	0.789128	−	0.614229i	$-0.210533\pi$
$461$	33.8866	1.57826	0.789128	+	0.614229i	$0.210533\pi$
$462$	0	0
$463$	1.56134	0.0725614	0.0362807	−	0.999342i	$-0.488449\pi$
$463$	1.56134	0.0725614	0.0362807	+	0.999342i	$0.488449\pi$
$464$	−28.8052	−1.33725
$465$	0	0
$466$	−30.8637	−1.42974
$467$	−5.94241	−0.274982	−0.137491	−	0.990503i	$-0.543904\pi$
$467$	−5.94241	−0.274982	−0.137491	+	0.990503i	$0.543904\pi$
$468$	0	0
$469$	1.85378	0.0855995
$470$	−4.35183	−0.200735
$471$	0	0
$472$	−39.0100	−1.79558
$473$	0	0
$474$	0	0
$475$	−26.8973	−1.23413
$476$	0.189316	0.00867727
$477$	0	0
$478$	27.5424	1.25976
$479$	29.1700	1.33281	0.666405	−	0.745590i	$-0.267832\pi$
$479$	29.1700	1.33281	0.666405	+	0.745590i	$0.267832\pi$
$480$	0	0
$481$	20.1127	0.917060
$482$	−1.26235	−0.0574985
$483$	0	0
$484$	0	0
$485$	−4.74112	−0.215283
$486$	0	0
$487$	15.4151	0.698525	0.349263	−	0.937025i	$-0.386432\pi$
$487$	15.4151	0.698525	0.349263	+	0.937025i	$0.386432\pi$
$488$	32.5891	1.47524
$489$	0	0
$490$	−0.870539	−0.0393269
$491$	24.6654	1.11313	0.556567	−	0.830803i	$-0.312118\pi$
$491$	24.6654	1.11313	0.556567	+	0.830803i	$0.312118\pi$
$492$	0	0
$493$	11.3102	0.509386
$494$	28.7807	1.29490
$495$	0	0
$496$	35.1897	1.58006
$497$	9.66839	0.433687
$498$	0	0
$499$	2.43866	0.109170	0.0545848	−	0.998509i	$-0.482616\pi$
$499$	2.43866	0.109170	0.0545848	+	0.998509i	$0.482616\pi$
$500$	−0.790648	−0.0353588
$501$	0	0
$502$	−14.8370	−0.662209
$503$	18.2286	0.812772	0.406386	−	0.913702i	$-0.366789\pi$
$503$	18.2286	0.812772	0.406386	+	0.913702i	$0.366789\pi$
$504$	0	0
$505$	4.41294	0.196373
$506$	0	0
$507$	0	0
$508$	−1.63618	−0.0725936
$509$	20.2312	0.896731	0.448365	−	0.893850i	$-0.352006\pi$
$509$	20.2312	0.896731	0.448365	+	0.893850i	$0.352006\pi$
$510$	0	0
$511$	11.5949	0.512926
$512$	−24.4166	−1.07907
$513$	0	0
$514$	−9.32134	−0.411147
$515$	4.24671	0.187132
$516$	0	0
$517$	0	0
$518$	−7.65195	−0.336207
$519$	0	0
$520$	6.66407	0.292239
$521$	−33.7904	−1.48038	−0.740191	−	0.672396i	$-0.765265\pi$
$521$	−33.7904	−1.48038	−0.740191	+	0.672396i	$0.765265\pi$
$522$	0	0
$523$	−14.8895	−0.651071	−0.325536	−	0.945530i	$-0.605545\pi$
$523$	−14.8895	−0.651071	−0.325536	+	0.945530i	$0.605545\pi$
$524$	−0.543438	−0.0237402
$525$	0	0
$526$	10.8370	0.472516
$527$	−13.8171	−0.601881
$528$	0	0
$529$	50.2767	2.18594
$530$	9.66839	0.419968
$531$	0	0
$532$	0.757838	0.0328564
$533$	30.0921	1.30343
$534$	0	0
$535$	7.71536	0.333564
$536$	5.39897	0.233200
$537$	0	0
$538$	−36.1205	−1.55727
$539$	0	0
$540$	0	0
$541$	−24.2254	−1.04153	−0.520766	−	0.853700i	$-0.674353\pi$
$541$	−24.2254	−1.04153	−0.520766	+	0.853700i	$0.674353\pi$
$542$	−24.8182	−1.06603
$543$	0	0
$544$	1.06921	0.0458420
$545$	8.56018	0.366678
$546$	0	0
$547$	18.0870	0.773343	0.386672	−	0.922217i	$-0.373625\pi$
$547$	18.0870	0.773343	0.386672	+	0.922217i	$0.373625\pi$
$548$	2.19509	0.0937695
$549$	0	0
$550$	0	0
$551$	45.2752	1.92879
$552$	0	0
$553$	4.51785	0.192118
$554$	−5.87067	−0.249421
$555$	0	0
$556$	1.62058	0.0687277
$557$	24.4761	1.03709	0.518543	−	0.855052i	$-0.326475\pi$
$557$	24.4761	1.03709	0.518543	+	0.855052i	$0.326475\pi$
$558$	0	0
$559$	−6.25892	−0.264724
$560$	−2.37056	−0.100175
$561$	0	0
$562$	−16.8370	−0.710227
$563$	−37.5163	−1.58113	−0.790563	−	0.612381i	$-0.790212\pi$
$563$	−37.5163	−1.58113	−0.790563	+	0.612381i	$0.790212\pi$
$564$	0	0
$565$	6.36382	0.267728
$566$	25.1264	1.05614
$567$	0	0
$568$	28.1584	1.18150
$569$	−19.3123	−0.809613	−0.404806	−	0.914402i	$-0.632661\pi$
$569$	−19.3123	−0.809613	−0.404806	+	0.914402i	$0.632661\pi$
$570$	0	0
$571$	8.81029	0.368699	0.184350	−	0.982861i	$-0.440982\pi$
$571$	8.81029	0.368699	0.184350	+	0.982861i	$0.440982\pi$
$572$	0	0
$573$	0	0
$574$	−11.4486	−0.477857
$575$	39.3328	1.64029
$576$	0	0
$577$	−31.6740	−1.31861	−0.659304	−	0.751877i	$-0.729149\pi$
$577$	−31.6740	−1.31861	−0.659304	+	0.751877i	$0.729149\pi$
$578$	−20.3258	−0.845443
$579$	0	0
$580$	0.637336	0.0264640
$581$	−3.08948	−0.128173
$582$	0	0
$583$	0	0
$584$	33.7690	1.39737
$585$	0	0
$586$	4.73891	0.195762
$587$	−13.7240	−0.566450	−0.283225	−	0.959054i	$-0.591404\pi$
$587$	−13.7240	−0.566450	−0.283225	+	0.959054i	$0.591404\pi$
$588$	0	0
$589$	−55.3102	−2.27902
$590$	−11.6603	−0.480048
$591$	0	0
$592$	−20.8370	−0.856396
$593$	4.73795	0.194564	0.0972822	−	0.995257i	$-0.468985\pi$
$593$	4.73795	0.194564	0.0972822	+	0.995257i	$0.468985\pi$
$594$	0	0
$595$	0.930789	0.0381586
$596$	0.506880	0.0207626
$597$	0	0
$598$	−42.0870	−1.72106
$599$	19.6664	0.803547	0.401774	−	0.915739i	$-0.368394\pi$
$599$	19.6664	0.803547	0.401774	+	0.915739i	$0.368394\pi$
$600$	0	0
$601$	−0.697587	−0.0284552	−0.0142276	−	0.999899i	$-0.504529\pi$
$601$	−0.697587	−0.0284552	−0.0142276	+	0.999899i	$0.504529\pi$
$602$	2.38123	0.0970517
$603$	0	0
$604$	0.120501	0.00490312
$605$	0	0
$606$	0	0
$607$	37.2689	1.51270	0.756349	−	0.654169i	$-0.226981\pi$
$607$	37.2689	1.51270	0.756349	+	0.654169i	$0.226981\pi$
$608$	4.28009	0.173581
$609$	0	0
$610$	9.74108	0.394405
$611$	17.9707	0.727017
$612$	0	0
$613$	−5.22323	−0.210964	−0.105482	−	0.994421i	$-0.533639\pi$
$613$	−5.22323	−0.210964	−0.105482	+	0.994421i	$0.533639\pi$
$614$	−12.3579	−0.498724
$615$	0	0
$616$	0	0
$617$	−6.17895	−0.248755	−0.124378	−	0.992235i	$-0.539693\pi$
$617$	−6.17895	−0.248755	−0.124378	+	0.992235i	$0.539693\pi$
$618$	0	0
$619$	−17.1562	−0.689566	−0.344783	−	0.938683i	$-0.612047\pi$
$619$	−17.1562	−0.689566	−0.344783	+	0.938683i	$0.612047\pi$
$620$	−0.778599	−0.0312693
$621$	0	0
$622$	−11.9330	−0.478468
$623$	4.19769	0.168177
$624$	0	0
$625$	19.0870	0.763479
$626$	−19.3123	−0.771874
$627$	0	0
$628$	−0.738908	−0.0294856
$629$	8.18155	0.326220
$630$	0	0
$631$	9.03569	0.359705	0.179853	−	0.983694i	$-0.442438\pi$
$631$	9.03569	0.359705	0.179853	+	0.983694i	$0.442438\pi$
$632$	13.1578	0.523391
$633$	0	0
$634$	−21.8994	−0.869738
$635$	−8.04442	−0.319233
$636$	0	0
$637$	3.59486	0.142433
$638$	0	0
$639$	0	0
$640$	−6.42407	−0.253934
$641$	5.25688	0.207634	0.103817	−	0.994596i	$-0.466894\pi$
$641$	5.25688	0.207634	0.103817	+	0.994596i	$0.466894\pi$
$642$	0	0
$643$	−45.6740	−1.80121	−0.900604	−	0.434640i	$-0.856875\pi$
$643$	−45.6740	−1.80121	−0.900604	+	0.434640i	$0.856875\pi$
$644$	−1.10821	−0.0436697
$645$	0	0
$646$	11.7076	0.460628
$647$	2.63911	0.103754	0.0518770	−	0.998653i	$-0.483480\pi$
$647$	2.63911	0.103754	0.0518770	+	0.998653i	$0.483480\pi$
$648$	0	0
$649$	0	0
$650$	−22.5911	−0.886095
$651$	0	0
$652$	0.239993	0.00939883
$653$	−3.30330	−0.129268	−0.0646341	−	0.997909i	$-0.520588\pi$
$653$	−3.30330	−0.129268	−0.0646341	+	0.997909i	$0.520588\pi$
$654$	0	0
$655$	−2.67187	−0.104398
$656$	−31.1758	−1.21721
$657$	0	0
$658$	−6.83702	−0.266535
$659$	−3.72598	−0.145144	−0.0725719	−	0.997363i	$-0.523121\pi$
$659$	−3.72598	−0.145144	−0.0725719	+	0.997363i	$0.523121\pi$
$660$	0	0
$661$	−29.5356	−1.14880	−0.574401	−	0.818574i	$-0.694765\pi$
$661$	−29.5356	−1.14880	−0.574401	+	0.818574i	$0.694765\pi$
$662$	−17.8286	−0.692928
$663$	0	0
$664$	−8.99783	−0.349184
$665$	3.72598	0.144487
$666$	0	0
$667$	−66.2075	−2.56356
$668$	−2.99502	−0.115881
$669$	0	0
$670$	1.61379	0.0623460
$671$	0	0
$672$	0	0
$673$	−6.84381	−0.263809	−0.131905	−	0.991262i	$-0.542109\pi$
$673$	−6.84381	−0.263809	−0.131905	+	0.991262i	$0.542109\pi$
$674$	−30.7513	−1.18450
$675$	0	0
$676$	0.00996992	0.000383458 0
$677$	33.5080	1.28782	0.643908	−	0.765103i	$-0.277312\pi$
$677$	33.5080	1.28782	0.643908	+	0.765103i	$0.277312\pi$
$678$	0	0
$679$	−7.44863	−0.285852
$680$	2.71084	0.103956
$681$	0	0
$682$	0	0
$683$	3.98386	0.152438	0.0762191	−	0.997091i	$-0.475715\pi$
$683$	3.98386	0.152438	0.0762191	+	0.997091i	$0.475715\pi$
$684$	0	0
$685$	10.7924	0.412355
$686$	−1.36768	−0.0522181
$687$	0	0
$688$	6.48432	0.247213
$689$	−39.9253	−1.52103
$690$	0	0
$691$	−17.8616	−0.679486	−0.339743	−	0.940518i	$-0.610340\pi$
$691$	−17.8616	−0.679486	−0.339743	+	0.940518i	$0.610340\pi$
$692$	0.918896	0.0349312
$693$	0	0
$694$	3.93296	0.149293
$695$	7.96772	0.302233
$696$	0	0
$697$	12.2410	0.463661
$698$	9.67906	0.366358
$699$	0	0
$700$	−0.594856	−0.0224834
$701$	50.6773	1.91406	0.957029	−	0.289994i	$-0.0936533\pi$
$701$	50.6773	1.91406	0.957029	+	0.289994i	$0.0936533\pi$
$702$	0	0
$703$	32.7510	1.23523
$704$	0	0
$705$	0	0
$706$	5.38838	0.202795
$707$	6.93304	0.260744
$708$	0	0
$709$	−3.97428	−0.149257	−0.0746286	−	0.997211i	$-0.523777\pi$
$709$	−3.97428	−0.149257	−0.0746286	+	0.997211i	$0.523777\pi$
$710$	8.41671	0.315874
$711$	0	0
$712$	12.2254	0.458166
$713$	80.8820	3.02906
$714$	0	0
$715$	0	0
$716$	−0.659226	−0.0246364
$717$	0	0
$718$	−14.4465	−0.539137
$719$	−44.0022	−1.64100	−0.820502	−	0.571643i	$-0.806306\pi$
$719$	−44.0022	−1.64100	−0.820502	+	0.571643i	$0.806306\pi$
$720$	0	0
$721$	6.67187	0.248473
$722$	20.8799	0.777071
$723$	0	0
$724$	2.51785	0.0935750
$725$	−35.5383	−1.31986
$726$	0	0
$727$	39.0022	1.44651	0.723255	−	0.690581i	$-0.242645\pi$
$727$	39.0022	1.44651	0.723255	+	0.690581i	$0.242645\pi$
$728$	10.4697	0.388033
$729$	0	0
$730$	10.0938	0.373587
$731$	−2.54604	−0.0941686
$732$	0	0
$733$	−3.18971	−0.117815	−0.0589073	−	0.998263i	$-0.518762\pi$
$733$	−3.18971	−0.117815	−0.0589073	+	0.998263i	$0.518762\pi$
$734$	29.8996	1.10361
$735$	0	0
$736$	−6.25892	−0.230707
$737$	0	0
$738$	0	0
$739$	49.1718	1.80881	0.904407	−	0.426671i	$-0.140314\pi$
$739$	49.1718	1.80881	0.904407	+	0.426671i	$0.140314\pi$
$740$	0.461035	0.0169480
$741$	0	0
$742$	15.1897	0.557632
$743$	−42.3505	−1.55369	−0.776845	−	0.629692i	$-0.783181\pi$
$743$	−42.3505	−1.55369	−0.776845	+	0.629692i	$0.783181\pi$
$744$	0	0
$745$	2.49212	0.0913044
$746$	−8.56018	−0.313410
$747$	0	0
$748$	0	0
$749$	12.1214	0.442905
$750$	0	0
$751$	38.0792	1.38953	0.694765	−	0.719237i	$-0.255509\pi$
$751$	38.0792	1.38953	0.694765	+	0.719237i	$0.255509\pi$
$752$	−18.6179	−0.678924
$753$	0	0
$754$	38.0267	1.38485
$755$	0.592456	0.0215617
$756$	0	0
$757$	−19.6818	−0.715349	−0.357674	−	0.933846i	$-0.616430\pi$
$757$	−19.6818	−0.715349	−0.357674	+	0.933846i	$0.616430\pi$
$758$	30.5971	1.11134
$759$	0	0
$760$	10.8516	0.393629
$761$	21.6232	0.783839	0.391919	−	0.920000i	$-0.371811\pi$
$761$	21.6232	0.783839	0.391919	+	0.920000i	$0.371811\pi$
$762$	0	0
$763$	13.4486	0.486873
$764$	−2.68951	−0.0973030
$765$	0	0
$766$	31.4151	1.13507
$767$	48.1509	1.73863
$768$	0	0
$769$	35.0100	1.26249	0.631246	−	0.775583i	$-0.282544\pi$
$769$	35.0100	1.26249	0.631246	+	0.775583i	$0.282544\pi$
$770$	0	0
$771$	0	0
$772$	2.77677	0.0999381
$773$	40.9127	1.47153	0.735764	−	0.677238i	$-0.236823\pi$
$773$	40.9127	1.47153	0.735764	+	0.677238i	$0.236823\pi$
$774$	0	0
$775$	43.4151	1.55952
$776$	−21.6935	−0.778752
$777$	0	0
$778$	52.0535	1.86621
$779$	49.0012	1.75565
$780$	0	0
$781$	0	0
$782$	−17.1204	−0.612223
$783$	0	0
$784$	−3.72432	−0.133011
$785$	−3.63291	−0.129664
$786$	0	0
$787$	−25.3359	−0.903128	−0.451564	−	0.892239i	$-0.649134\pi$
$787$	−25.3359	−0.903128	−0.451564	+	0.892239i	$0.649134\pi$
$788$	0.424476	0.0151213
$789$	0	0
$790$	3.93296	0.139928
$791$	9.99801	0.355488
$792$	0	0
$793$	−40.2254	−1.42845
$794$	−24.0747	−0.854380
$795$	0	0
$796$	−2.15402	−0.0763473
$797$	−16.4839	−0.583888	−0.291944	−	0.956435i	$-0.594302\pi$
$797$	−16.4839	−0.583888	−0.291944	+	0.956435i	$0.594302\pi$
$798$	0	0
$799$	7.31021	0.258617
$800$	−3.35961	−0.118780
$801$	0	0
$802$	−9.51568	−0.336010
$803$	0	0
$804$	0	0
$805$	−5.44863	−0.192039
$806$	−46.4552	−1.63631
$807$	0	0
$808$	20.1919	0.710348
$809$	−41.7826	−1.46900	−0.734499	−	0.678610i	$-0.762583\pi$
$809$	−41.7826	−1.46900	−0.734499	+	0.678610i	$0.762583\pi$
$810$	0	0
$811$	−9.33593	−0.327829	−0.163914	−	0.986475i	$-0.552412\pi$
$811$	−9.33593	−0.327829	−0.163914	+	0.986475i	$0.552412\pi$
$812$	1.00130	0.0351387
$813$	0	0
$814$	0	0
$815$	1.17995	0.0413317
$816$	0	0
$817$	−10.1919	−0.356569
$818$	−20.0664	−0.701604
$819$	0	0
$820$	0.689787	0.0240884
$821$	11.5534	0.403217	0.201608	−	0.979466i	$-0.435383\pi$
$821$	11.5534	0.403217	0.201608	+	0.979466i	$0.435383\pi$
$822$	0	0
$823$	−21.2689	−0.741387	−0.370693	−	0.928755i	$-0.620880\pi$
$823$	−21.2689	−0.741387	−0.370693	+	0.928755i	$0.620880\pi$
$824$	19.4312	0.676919
$825$	0	0
$826$	−18.3192	−0.637406
$827$	7.54504	0.262367	0.131183	−	0.991358i	$-0.458122\pi$
$827$	7.54504	0.262367	0.131183	+	0.991358i	$0.458122\pi$
$828$	0	0
$829$	−32.6048	−1.13241	−0.566206	−	0.824264i	$-0.691589\pi$
$829$	−32.6048	−1.13241	−0.566206	+	0.824264i	$0.691589\pi$
$830$	−2.68951	−0.0933542
$831$	0	0
$832$	30.3716	1.05295
$833$	1.46233	0.0506669
$834$	0	0
$835$	−14.7253	−0.509591
$836$	0	0
$837$	0	0
$838$	−22.2521	−0.768687
$839$	−51.7838	−1.78777	−0.893887	−	0.448292i	$-0.852032\pi$
$839$	−51.7838	−1.78777	−0.893887	+	0.448292i	$0.852032\pi$
$840$	0	0
$841$	30.8203	1.06277
$842$	28.4266	0.979644
$843$	0	0
$844$	2.89727	0.0997281
$845$	0.0490181	0.00168627
$846$	0	0
$847$	0	0
$848$	41.3631	1.42041
$849$	0	0
$850$	−9.18971	−0.315205
$851$	−47.8930	−1.64175
$852$	0	0
$853$	32.9842	1.12936	0.564680	−	0.825310i	$-0.309000\pi$
$853$	32.9842	1.12936	0.564680	+	0.825310i	$0.309000\pi$
$854$	15.3039	0.523689
$855$	0	0
$856$	35.3024	1.20661
$857$	−35.1596	−1.20103	−0.600515	−	0.799614i	$-0.705038\pi$
$857$	−35.1596	−1.20103	−0.600515	+	0.799614i	$0.705038\pi$
$858$	0	0
$859$	−46.3794	−1.58245	−0.791223	−	0.611528i	$-0.790555\pi$
$859$	−46.3794	−1.58245	−0.791223	+	0.611528i	$0.790555\pi$
$860$	−0.143471	−0.00489230
$861$	0	0
$862$	45.7343	1.55772
$863$	−16.0121	−0.545060	−0.272530	−	0.962147i	$-0.587860\pi$
$863$	−16.0121	−0.545060	−0.272530	+	0.962147i	$0.587860\pi$
$864$	0	0
$865$	4.51785	0.153611
$866$	−40.6303	−1.38067
$867$	0	0
$868$	−1.22323	−0.0415192
$869$	0	0
$870$	0	0
$871$	−6.66407	−0.225803
$872$	39.1680	1.32640
$873$	0	0
$874$	−68.5334	−2.31818
$875$	−6.10721	−0.206462
$876$	0	0
$877$	−40.2410	−1.35884	−0.679421	−	0.733749i	$-0.737769\pi$
$877$	−40.2410	−1.35884	−0.679421	+	0.733749i	$0.737769\pi$
$878$	6.58959	0.222388
$879$	0	0
$880$	0	0
$881$	41.9996	1.41500	0.707501	−	0.706712i	$-0.249822\pi$
$881$	41.9996	1.41500	0.707501	+	0.706712i	$0.249822\pi$
$882$	0	0
$883$	13.3359	0.448790	0.224395	−	0.974498i	$-0.427959\pi$
$883$	13.3359	0.448790	0.224395	+	0.974498i	$0.427959\pi$
$884$	−0.680563	−0.0228898
$885$	0	0
$886$	9.74108	0.327258
$887$	38.2246	1.28346	0.641728	−	0.766932i	$-0.278218\pi$
$887$	38.2246	1.28346	0.641728	+	0.766932i	$0.278218\pi$
$888$	0	0
$889$	−12.6383	−0.423877
$890$	3.65425	0.122491
$891$	0	0
$892$	1.10273	0.0369222
$893$	29.2631	0.979251
$894$	0	0
$895$	−3.24115	−0.108340
$896$	−10.0927	−0.337172
$897$	0	0
$898$	44.0535	1.47008
$899$	−73.0791	−2.43732
$900$	0	0
$901$	−16.2410	−0.541066
$902$	0	0
$903$	0	0
$904$	29.1183	0.968461
$905$	12.3792	0.411500
$906$	0	0
$907$	−39.9330	−1.32595	−0.662976	−	0.748641i	$-0.730707\pi$
$907$	−39.9330	−1.32595	−0.662976	+	0.748641i	$0.730707\pi$
$908$	3.08948	0.102528
$909$	0	0
$910$	3.12946	0.103741
$911$	9.33878	0.309408	0.154704	−	0.987961i	$-0.450558\pi$
$911$	9.33878	0.309408	0.154704	+	0.987961i	$0.450558\pi$
$912$	0	0
$913$	0	0
$914$	27.1184	0.896996
$915$	0	0
$916$	−0.618406	−0.0204327
$917$	−4.19769	−0.138620
$918$	0	0
$919$	−34.7924	−1.14769	−0.573847	−	0.818962i	$-0.694550\pi$
$919$	−34.7924	−1.14769	−0.573847	+	0.818962i	$0.694550\pi$
$920$	−15.8687	−0.523175
$921$	0	0
$922$	46.3459	1.52632
$923$	−34.7565	−1.14402
$924$	0	0
$925$	−25.7076	−0.845259
$926$	2.13540	0.0701737
$927$	0	0
$928$	5.65511	0.185638
$929$	31.5739	1.03591	0.517954	−	0.855409i	$-0.326694\pi$
$929$	31.5739	1.03591	0.517954	+	0.855409i	$0.326694\pi$
$930$	0	0
$931$	5.85378	0.191850
$932$	2.92150	0.0956968
$933$	0	0
$934$	−8.12729	−0.265933
$935$	0	0
$936$	0	0
$937$	−50.0199	−1.63408	−0.817040	−	0.576581i	$-0.804387\pi$
$937$	−50.0199	−1.63408	−0.817040	+	0.576581i	$0.804387\pi$
$938$	2.53537	0.0827827
$939$	0	0
$940$	0.411934	0.0134358
$941$	51.5717	1.68119	0.840595	−	0.541664i	$-0.182205\pi$
$941$	51.5717	1.68119	0.840595	+	0.541664i	$0.182205\pi$
$942$	0	0
$943$	−71.6561	−2.33344
$944$	−49.8849	−1.62362
$945$	0	0
$946$	0	0
$947$	46.9709	1.52635	0.763175	−	0.646192i	$-0.223640\pi$
$947$	46.9709	1.52635	0.763175	+	0.646192i	$0.223640\pi$
$948$	0	0
$949$	−41.6818	−1.35305
$950$	−36.7868	−1.19352
$951$	0	0
$952$	4.25892	0.138032
$953$	−4.62354	−0.149771	−0.0748856	−	0.997192i	$-0.523859\pi$
$953$	−4.62354	−0.149771	−0.0748856	+	0.997192i	$0.523859\pi$
$954$	0	0
$955$	−13.2232	−0.427894
$956$	−2.60710	−0.0843198
$957$	0	0
$958$	39.8951	1.28895
$959$	16.9556	0.547523
$960$	0	0
$961$	58.2767	1.87989
$962$	27.5077	0.886883
$963$	0	0
$964$	0.119491	0.00384856
$965$	13.6523	0.439482
$966$	0	0
$967$	28.7432	0.924321	0.462160	−	0.886796i	$-0.347074\pi$
$967$	28.7432	0.924321	0.462160	+	0.886796i	$0.347074\pi$
$968$	0	0
$969$	0	0
$970$	−6.48432	−0.208199
$971$	8.48845	0.272407	0.136204	−	0.990681i	$-0.456510\pi$
$971$	8.48845	0.272407	0.136204	+	0.990681i	$0.456510\pi$
$972$	0	0
$973$	12.5178	0.401304
$974$	21.0829	0.675539
$975$	0	0
$976$	41.6740	1.33395
$977$	19.5016	0.623911	0.311956	−	0.950097i	$-0.399016\pi$
$977$	19.5016	0.623911	0.311956	+	0.950097i	$0.399016\pi$
$978$	0	0
$979$	0	0
$980$	0.0824033	0.00263228
$981$	0	0
$982$	33.7343	1.07650
$983$	30.6078	0.976238	0.488119	−	0.872777i	$-0.337683\pi$
$983$	30.6078	0.976238	0.488119	+	0.872777i	$0.337683\pi$
$984$	0	0
$985$	2.08698	0.0664967
$986$	15.4687	0.492624
$987$	0	0
$988$	−2.72432	−0.0866721
$989$	14.9039	0.473918
$990$	0	0
$991$	−12.2332	−0.388600	−0.194300	−	0.980942i	$-0.562244\pi$
$991$	−12.2332	−0.388600	−0.194300	+	0.980942i	$0.562244\pi$
$992$	−6.90853	−0.219346
$993$	0	0
$994$	13.2232	0.419415
$995$	−10.5905	−0.335740
$996$	0	0
$997$	0.292443	0.00926176	0.00463088	−	0.999989i	$-0.498526\pi$
$997$	0.292443	0.00926176	0.00463088	+	0.999989i	$0.498526\pi$
$998$	3.33530	0.105577
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.cq.1.5	yes	6
3.2	odd	2	inner	7623.2.a.cq.1.2	✓	6
11.10	odd	2		7623.2.a.cr.1.2	yes	6
33.32	even	2		7623.2.a.cr.1.5	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7623.2.a.cq.1.2	✓	6	3.2	odd	2	inner
7623.2.a.cq.1.5	yes	6	1.1	even	1	trivial
7623.2.a.cr.1.2	yes	6	11.10	odd	2
7623.2.a.cr.1.5	yes	6	33.32	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.36768 q^{2} -0.129461 q^{4} -0.636509 q^{5} -1.00000 q^{7} -2.91241 q^{8} +O(q^{10})\)	\(q+1.36768 q^{2} -0.129461 q^{4} -0.636509 q^{5} -1.00000 q^{7} -2.91241 q^{8} -0.870539 q^{10} +3.59486 q^{13} -1.36768 q^{14} -3.72432 q^{16} +1.46233 q^{17} +5.85378 q^{19} +0.0824033 q^{20} -8.56018 q^{23} -4.59486 q^{25} +4.91660 q^{26} +0.129461 q^{28} +7.73436 q^{29} -9.44863 q^{31} +0.731167 q^{32} +2.00000 q^{34} +0.636509 q^{35} +5.59486 q^{37} +8.00607 q^{38} +1.85378 q^{40} +8.37086 q^{41} -1.74108 q^{43} -11.7076 q^{46} +4.99900 q^{47} +1.00000 q^{49} -6.28428 q^{50} -0.465395 q^{52} -11.1062 q^{53} +2.91241 q^{56} +10.5781 q^{58} +13.3944 q^{59} -11.1897 q^{61} -12.9227 q^{62} +8.44863 q^{64} -2.28816 q^{65} -1.85378 q^{67} -0.189316 q^{68} +0.870539 q^{70} -9.66839 q^{71} -11.5949 q^{73} +7.65195 q^{74} -0.757838 q^{76} -4.51785 q^{79} +2.37056 q^{80} +11.4486 q^{82} +3.08948 q^{83} -0.930789 q^{85} -2.38123 q^{86} -4.19769 q^{89} -3.59486 q^{91} +1.10821 q^{92} +6.83702 q^{94} -3.72598 q^{95} +7.44863 q^{97} +1.36768 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{4} - 6 q^{7}+O(q^{10}) \) 6 * q + 4 * q^4 - 6 * q^7	\( 6 q + 4 q^{4} - 6 q^{7} - 10 q^{10} - 4 q^{13} + 8 q^{16} - 2 q^{25} - 4 q^{28} + 4 q^{31} + 12 q^{34} + 8 q^{37} - 24 q^{40} - 20 q^{43} + 6 q^{49} + 18 q^{52} - 2 q^{58} - 16 q^{61} - 10 q^{64} + 24 q^{67} + 10 q^{70} - 44 q^{73} - 54 q^{76} - 8 q^{79} + 8 q^{82} + 36 q^{85} + 4 q^{91} - 34 q^{94} - 16 q^{97}+O(q^{100}) \) 6 * q + 4 * q^4 - 6 * q^7 - 10 * q^10 - 4 * q^13 + 8 * q^16 - 2 * q^25 - 4 * q^28 + 4 * q^31 + 12 * q^34 + 8 * q^37 - 24 * q^40 - 20 * q^43 + 6 * q^49 + 18 * q^52 - 2 * q^58 - 16 * q^61 - 10 * q^64 + 24 * q^67 + 10 * q^70 - 44 * q^73 - 54 * q^76 - 8 * q^79 + 8 * q^82 + 36 * q^85 + 4 * q^91 - 34 * q^94 - 16 * q^97

Introduction

L-functions

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