LMFDB - Embedded newform 7623.2.a.dc.1.11

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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 26x^{14} + 268x^{12} - 1395x^{10} + 3876x^{8} - 5635x^{6} + 4042x^{4} - 1272x^{2} + 121 $ x^16 - 26x^14 + 268x^12 - 1395x^10 + 3876x^8 - 5635x^6 + 4042x^4 - 1272*x^2 + 121
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 693)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$0.899827$ 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)

$f(q)$	$=$	$q+0.899827 q^{2} -1.19031 q^{4} -0.803961 q^{5} +1.00000 q^{7} -2.87073 q^{8} +O(q^{10})$	$q+0.899827 q^{2} -1.19031 q^{4} -0.803961 q^{5} +1.00000 q^{7} -2.87073 q^{8} -0.723426 q^{10} +2.94705 q^{13} +0.899827 q^{14} -0.202537 q^{16} +3.30399 q^{17} +7.93896 q^{19} +0.956964 q^{20} -7.53481 q^{23} -4.35365 q^{25} +2.65184 q^{26} -1.19031 q^{28} -0.234450 q^{29} +3.98395 q^{31} +5.55921 q^{32} +2.97302 q^{34} -0.803961 q^{35} -0.536441 q^{37} +7.14369 q^{38} +2.30795 q^{40} -4.78449 q^{41} +3.16645 q^{43} -6.78003 q^{46} -11.0114 q^{47} +1.00000 q^{49} -3.91753 q^{50} -3.50791 q^{52} +11.7317 q^{53} -2.87073 q^{56} -0.210965 q^{58} -5.25351 q^{59} +3.65442 q^{61} +3.58486 q^{62} +5.40740 q^{64} -2.36931 q^{65} -13.6736 q^{67} -3.93278 q^{68} -0.723426 q^{70} -10.3371 q^{71} +8.90937 q^{73} -0.482705 q^{74} -9.44983 q^{76} -8.37651 q^{79} +0.162832 q^{80} -4.30522 q^{82} -7.26234 q^{83} -2.65628 q^{85} +2.84925 q^{86} +8.03460 q^{89} +2.94705 q^{91} +8.96877 q^{92} -9.90831 q^{94} -6.38261 q^{95} +19.2711 q^{97} +0.899827 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 20 q^{4} + 16 q^{7}+O(q^{10}) $ 16 * q + 20 * q^4 + 16 * q^7	$ 16 q + 20 q^{4} + 16 q^{7} - 6 q^{10} + 32 q^{16} + 10 q^{19} + 44 q^{25} + 20 q^{28} + 30 q^{31} + 12 q^{34} + 38 q^{37} - 68 q^{40} + 16 q^{43} + 80 q^{46} + 16 q^{49} + 2 q^{52} + 18 q^{58} - 28 q^{61} + 34 q^{64} + 52 q^{67} - 6 q^{70} - 14 q^{73} - 14 q^{76} + 54 q^{79} + 64 q^{82} + 30 q^{85} - 60 q^{94} + 108 q^{97}+O(q^{100}) $ 16 * q + 20 * q^4 + 16 * q^7 - 6 * q^10 + 32 * q^16 + 10 * q^19 + 44 * q^25 + 20 * q^28 + 30 * q^31 + 12 * q^34 + 38 * q^37 - 68 * q^40 + 16 * q^43 + 80 * q^46 + 16 * q^49 + 2 * q^52 + 18 * q^58 - 28 * q^61 + 34 * q^64 + 52 * q^67 - 6 * q^70 - 14 * q^73 - 14 * q^76 + 54 * q^79 + 64 * q^82 + 30 * q^85 - 60 * q^94 + 108 * 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 $

$2$	0.899827	0.636274	0.318137	−	0.948045i	$-0.396943\pi$
$2$	0.899827	0.636274	0.318137	+	0.948045i	$0.396943\pi$
$3$	0	0
$3$	0	0
$4$	−1.19031	−0.595156
$5$	−0.803961	−0.359542	−0.179771	−	0.983708i	$-0.557536\pi$
$5$	−0.803961	−0.359542	−0.179771	+	0.983708i	$0.557536\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.87073	−1.01496
$9$	0	0
$10$	−0.723426	−0.228767
$11$	0	0
$11$	0	0
$12$	0	0
$13$	2.94705	0.817365	0.408682	−	0.912677i	$-0.365988\pi$
$13$	2.94705	0.817365	0.408682	+	0.912677i	$0.365988\pi$
$14$	0.899827	0.240489
$15$	0	0
$16$	−0.202537	−0.0506344
$17$	3.30399	0.801336	0.400668	−	0.916223i	$-0.368778\pi$
$17$	3.30399	0.801336	0.400668	+	0.916223i	$0.368778\pi$
$18$	0	0
$19$	7.93896	1.82132	0.910661	−	0.413155i	$-0.135573\pi$
$19$	7.93896	1.82132	0.910661	+	0.413155i	$0.135573\pi$
$20$	0.956964	0.213984
$21$	0	0
$22$	0	0
$23$	−7.53481	−1.57112	−0.785558	−	0.618788i	$-0.787624\pi$
$23$	−7.53481	−1.57112	−0.785558	+	0.618788i	$0.787624\pi$
$24$	0	0
$25$	−4.35365	−0.870729
$26$	2.65184	0.520068
$27$	0	0
$28$	−1.19031	−0.224948
$29$	−0.234450	−0.0435364	−0.0217682	−	0.999763i	$-0.506930\pi$
$29$	−0.234450	−0.0435364	−0.0217682	+	0.999763i	$0.506930\pi$
$30$	0	0
$31$	3.98395	0.715538	0.357769	−	0.933810i	$-0.383538\pi$
$31$	3.98395	0.715538	0.357769	+	0.933810i	$0.383538\pi$
$32$	5.55921	0.982738
$33$	0	0
$34$	2.97302	0.509869
$35$	−0.803961	−0.135894
$36$	0	0
$37$	−0.536441	−0.0881904	−0.0440952	−	0.999027i	$-0.514040\pi$
$37$	−0.536441	−0.0881904	−0.0440952	+	0.999027i	$0.514040\pi$
$38$	7.14369	1.15886
$39$	0	0
$40$	2.30795	0.364920
$41$	−4.78449	−0.747212	−0.373606	−	0.927587i	$-0.621879\pi$
$41$	−4.78449	−0.747212	−0.373606	+	0.927587i	$0.621879\pi$
$42$	0	0
$43$	3.16645	0.482878	0.241439	−	0.970416i	$-0.422381\pi$
$43$	3.16645	0.482878	0.241439	+	0.970416i	$0.422381\pi$
$44$	0	0
$45$	0	0
$46$	−6.78003	−0.999660
$47$	−11.0114	−1.60617	−0.803085	−	0.595864i	$-0.796810\pi$
$47$	−11.0114	−1.60617	−0.803085	+	0.595864i	$0.796810\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−3.91753	−0.554022
$51$	0	0
$52$	−3.50791	−0.486459
$53$	11.7317	1.61148	0.805738	−	0.592273i	$-0.201769\pi$
$53$	11.7317	1.61148	0.805738	+	0.592273i	$0.201769\pi$
$54$	0	0
$55$	0	0
$56$	−2.87073	−0.383617
$57$	0	0
$58$	−0.210965	−0.0277011
$59$	−5.25351	−0.683949	−0.341974	−	0.939709i	$-0.611096\pi$
$59$	−5.25351	−0.683949	−0.341974	+	0.939709i	$0.611096\pi$
$60$	0	0
$61$	3.65442	0.467901	0.233950	−	0.972249i	$-0.424835\pi$
$61$	3.65442	0.467901	0.233950	+	0.972249i	$0.424835\pi$
$62$	3.58486	0.455278
$63$	0	0
$64$	5.40740	0.675925
$65$	−2.36931	−0.293877
$66$	0	0
$67$	−13.6736	−1.67050	−0.835250	−	0.549871i	$-0.814677\pi$
$67$	−13.6736	−1.67050	−0.835250	+	0.549871i	$0.814677\pi$
$68$	−3.93278	−0.476920
$69$	0	0
$70$	−0.723426	−0.0864660
$71$	−10.3371	−1.22678	−0.613392	−	0.789779i	$-0.710195\pi$
$71$	−10.3371	−1.22678	−0.613392	+	0.789779i	$0.710195\pi$
$72$	0	0
$73$	8.90937	1.04276	0.521382	−	0.853324i	$-0.325417\pi$
$73$	8.90937	1.04276	0.521382	+	0.853324i	$0.325417\pi$
$74$	−0.482705	−0.0561133
$75$	0	0
$76$	−9.44983	−1.08397
$77$	0	0
$78$	0	0
$79$	−8.37651	−0.942431	−0.471215	−	0.882018i	$-0.656185\pi$
$79$	−8.37651	−0.942431	−0.471215	+	0.882018i	$0.656185\pi$
$80$	0.162832	0.0182052
$81$	0	0
$82$	−4.30522	−0.475432
$83$	−7.26234	−0.797145	−0.398573	−	0.917137i	$-0.630494\pi$
$83$	−7.26234	−0.797145	−0.398573	+	0.917137i	$0.630494\pi$
$84$	0	0
$85$	−2.65628	−0.288114
$86$	2.84925	0.307243
$87$	0	0
$88$	0	0
$89$	8.03460	0.851666	0.425833	−	0.904802i	$-0.359981\pi$
$89$	8.03460	0.851666	0.425833	+	0.904802i	$0.359981\pi$
$90$	0	0
$91$	2.94705	0.308935
$92$	8.96877	0.935058
$93$	0	0
$94$	−9.90831	−1.02196
$95$	−6.38261	−0.654842
$96$	0	0
$97$	19.2711	1.95669	0.978343	−	0.206992i	$-0.0663675\pi$
$97$	19.2711	1.95669	0.978343	+	0.206992i	$0.0663675\pi$
$98$	0.899827	0.0908963
$99$	0	0
$100$	5.18219	0.518219
$101$	17.2794	1.71936	0.859682	−	0.510830i	$-0.170662\pi$
$101$	17.2794	1.71936	0.859682	+	0.510830i	$0.170662\pi$
$102$	0	0
$103$	14.3573	1.41467	0.707334	−	0.706880i	$-0.249898\pi$
$103$	14.3573	1.41467	0.707334	+	0.706880i	$0.249898\pi$
$104$	−8.46018	−0.829589
$105$	0	0
$106$	10.5565	1.02534
$107$	15.3202	1.48106	0.740529	−	0.672024i	$-0.234575\pi$
$107$	15.3202	1.48106	0.740529	+	0.672024i	$0.234575\pi$
$108$	0	0
$109$	−5.50237	−0.527032	−0.263516	−	0.964655i	$-0.584882\pi$
$109$	−5.50237	−0.527032	−0.263516	+	0.964655i	$0.584882\pi$
$110$	0	0
$111$	0	0
$112$	−0.202537	−0.0191380
$113$	−6.69477	−0.629791	−0.314895	−	0.949126i	$-0.601969\pi$
$113$	−6.69477	−0.629791	−0.314895	+	0.949126i	$0.601969\pi$
$114$	0	0
$115$	6.05769	0.564883
$116$	0.279069	0.0259109
$117$	0	0
$118$	−4.72725	−0.435179
$119$	3.30399	0.302877
$120$	0	0
$121$	0	0
$122$	3.28835	0.297713
$123$	0	0
$124$	−4.74214	−0.425856
$125$	7.51997	0.672607
$126$	0	0
$127$	−11.5436	−1.02433	−0.512165	−	0.858887i	$-0.671156\pi$
$127$	−11.5436	−1.02433	−0.512165	+	0.858887i	$0.671156\pi$
$128$	−6.25269	−0.552665
$129$	0	0
$130$	−2.13197	−0.186986
$131$	21.2008	1.85232	0.926161	−	0.377128i	$-0.123088\pi$
$131$	21.2008	1.85232	0.926161	+	0.377128i	$0.123088\pi$
$132$	0	0
$133$	7.93896	0.688395
$134$	−12.3039	−1.06290
$135$	0	0
$136$	−9.48487	−0.813321
$137$	0.815166	0.0696443	0.0348222	−	0.999394i	$-0.488914\pi$
$137$	0.815166	0.0696443	0.0348222	+	0.999394i	$0.488914\pi$
$138$	0	0
$139$	9.27263	0.786495	0.393247	−	0.919433i	$-0.371352\pi$
$139$	9.27263	0.786495	0.393247	+	0.919433i	$0.371352\pi$
$140$	0.956964	0.0808782
$141$	0	0
$142$	−9.30157	−0.780570
$143$	0	0
$144$	0	0
$145$	0.188489	0.0156532
$146$	8.01690	0.663483
$147$	0	0
$148$	0.638532	0.0524870
$149$	6.94847	0.569241	0.284621	−	0.958640i	$-0.408132\pi$
$149$	6.94847	0.569241	0.284621	+	0.958640i	$0.408132\pi$
$150$	0	0
$151$	14.0283	1.14161	0.570803	−	0.821087i	$-0.306632\pi$
$151$	14.0283	1.14161	0.570803	+	0.821087i	$0.306632\pi$
$152$	−22.7906	−1.84856
$153$	0	0
$154$	0	0
$155$	−3.20294	−0.257266
$156$	0	0
$157$	0.733382	0.0585302	0.0292651	−	0.999572i	$-0.490683\pi$
$157$	0.733382	0.0585302	0.0292651	+	0.999572i	$0.490683\pi$
$158$	−7.53741	−0.599644
$159$	0	0
$160$	−4.46939	−0.353336
$161$	−7.53481	−0.593826
$162$	0	0
$163$	2.96358	0.232125	0.116063	−	0.993242i	$-0.462973\pi$
$163$	2.96358	0.232125	0.116063	+	0.993242i	$0.462973\pi$
$164$	5.69504	0.444708
$165$	0	0
$166$	−6.53485	−0.507203
$167$	8.04008	0.622160	0.311080	−	0.950384i	$-0.399309\pi$
$167$	8.04008	0.622160	0.311080	+	0.950384i	$0.399309\pi$
$168$	0	0
$169$	−4.31490	−0.331915
$170$	−2.39020	−0.183320
$171$	0	0
$172$	−3.76906	−0.287388
$173$	5.15560	0.391973	0.195986	−	0.980607i	$-0.437209\pi$
$173$	5.15560	0.391973	0.195986	+	0.980607i	$0.437209\pi$
$174$	0	0
$175$	−4.35365	−0.329105
$176$	0	0
$177$	0	0
$178$	7.22975	0.541893
$179$	6.01273	0.449412	0.224706	−	0.974427i	$-0.427858\pi$
$179$	6.01273	0.449412	0.224706	+	0.974427i	$0.427858\pi$
$180$	0	0
$181$	2.45046	0.182141	0.0910707	−	0.995844i	$-0.470971\pi$
$181$	2.45046	0.182141	0.0910707	+	0.995844i	$0.470971\pi$
$182$	2.65184	0.196567
$183$	0	0
$184$	21.6304	1.59461
$185$	0.431278	0.0317082
$186$	0	0
$187$	0	0
$188$	13.1069	0.955922
$189$	0	0
$190$	−5.74325	−0.416659
$191$	−21.2746	−1.53938	−0.769689	−	0.638419i	$-0.779589\pi$
$191$	−21.2746	−1.53938	−0.769689	+	0.638419i	$0.779589\pi$
$192$	0	0
$193$	−8.39924	−0.604590	−0.302295	−	0.953214i	$-0.597753\pi$
$193$	−8.39924	−0.604590	−0.302295	+	0.953214i	$0.597753\pi$
$194$	17.3407	1.24499
$195$	0	0
$196$	−1.19031	−0.0850222
$197$	10.9646	0.781194	0.390597	−	0.920562i	$-0.372269\pi$
$197$	10.9646	0.781194	0.390597	+	0.920562i	$0.372269\pi$
$198$	0	0
$199$	6.26130	0.443852	0.221926	−	0.975064i	$-0.428766\pi$
$199$	6.26130	0.443852	0.221926	+	0.975064i	$0.428766\pi$
$200$	12.4981	0.883752
$201$	0	0
$202$	15.5485	1.09399
$203$	−0.234450	−0.0164552
$204$	0	0
$205$	3.84655	0.268655
$206$	12.9191	0.900116
$207$	0	0
$208$	−0.596888	−0.0413867
$209$	0	0
$210$	0	0
$211$	16.7203	1.15107	0.575537	−	0.817776i	$-0.304793\pi$
$211$	16.7203	1.15107	0.575537	+	0.817776i	$0.304793\pi$
$212$	−13.9644	−0.959078
$213$	0	0
$214$	13.7855	0.942359
$215$	−2.54570	−0.173615
$216$	0	0
$217$	3.98395	0.270448
$218$	−4.95118	−0.335337
$219$	0	0
$220$	0	0
$221$	9.73704	0.654984
$222$	0	0
$223$	13.0968	0.877026	0.438513	−	0.898725i	$-0.355505\pi$
$223$	13.0968	0.877026	0.438513	+	0.898725i	$0.355505\pi$
$224$	5.55921	0.371440
$225$	0	0
$226$	−6.02413	−0.400719
$227$	−15.9025	−1.05548	−0.527742	−	0.849405i	$-0.676961\pi$
$227$	−15.9025	−1.05548	−0.527742	+	0.849405i	$0.676961\pi$
$228$	0	0
$229$	9.01594	0.595790	0.297895	−	0.954599i	$-0.403715\pi$
$229$	9.01594	0.595790	0.297895	+	0.954599i	$0.403715\pi$
$230$	5.45088	0.359420
$231$	0	0
$232$	0.673044	0.0441875
$233$	2.54200	0.166532	0.0832659	−	0.996527i	$-0.473465\pi$
$233$	2.54200	0.166532	0.0832659	+	0.996527i	$0.473465\pi$
$234$	0	0
$235$	8.85270	0.577487
$236$	6.25331	0.407056
$237$	0	0
$238$	2.97302	0.192713
$239$	19.1749	1.24032	0.620161	−	0.784475i	$-0.287067\pi$
$239$	19.1749	1.24032	0.620161	+	0.784475i	$0.287067\pi$
$240$	0	0
$241$	−7.50063	−0.483158	−0.241579	−	0.970381i	$-0.577665\pi$
$241$	−7.50063	−0.483158	−0.241579	+	0.970381i	$0.577665\pi$
$242$	0	0
$243$	0	0
$244$	−4.34990	−0.278474
$245$	−0.803961	−0.0513632
$246$	0	0
$247$	23.3965	1.48868
$248$	−11.4368	−0.726239
$249$	0	0
$250$	6.76667	0.427962
$251$	14.6903	0.927245	0.463622	−	0.886033i	$-0.346549\pi$
$251$	14.6903	0.927245	0.463622	+	0.886033i	$0.346549\pi$
$252$	0	0
$253$	0	0
$254$	−10.3872	−0.651754
$255$	0	0
$256$	−16.4411	−1.02757
$257$	20.7501	1.29436	0.647178	−	0.762339i	$-0.275949\pi$
$257$	20.7501	1.29436	0.647178	+	0.762339i	$0.275949\pi$
$258$	0	0
$259$	−0.536441	−0.0333328
$260$	2.82022	0.174903
$261$	0	0
$262$	19.0771	1.17858
$263$	−27.3462	−1.68624	−0.843118	−	0.537729i	$-0.819283\pi$
$263$	−27.3462	−1.68624	−0.843118	+	0.537729i	$0.819283\pi$
$264$	0	0
$265$	−9.43185	−0.579394
$266$	7.14369	0.438008
$267$	0	0
$268$	16.2759	0.994207
$269$	−1.57905	−0.0962763	−0.0481381	−	0.998841i	$-0.515329\pi$
$269$	−1.57905	−0.0962763	−0.0481381	+	0.998841i	$0.515329\pi$
$270$	0	0
$271$	15.1014	0.917342	0.458671	−	0.888606i	$-0.348326\pi$
$271$	15.1014	0.917342	0.458671	+	0.888606i	$0.348326\pi$
$272$	−0.669183	−0.0405752
$273$	0	0
$274$	0.733509	0.0443129
$275$	0	0
$276$	0	0
$277$	−16.2348	−0.975453	−0.487727	−	0.872996i	$-0.662174\pi$
$277$	−16.2348	−0.975453	−0.487727	+	0.872996i	$0.662174\pi$
$278$	8.34377	0.500426
$279$	0	0
$280$	2.30795	0.137927
$281$	17.0866	1.01930	0.509651	−	0.860381i	$-0.329774\pi$
$281$	17.0866	1.01930	0.509651	+	0.860381i	$0.329774\pi$
$282$	0	0
$283$	−0.115027	−0.00683762	−0.00341881	−	0.999994i	$-0.501088\pi$
$283$	−0.115027	−0.00683762	−0.00341881	+	0.999994i	$0.501088\pi$
$284$	12.3043	0.730127
$285$	0	0
$286$	0	0
$287$	−4.78449	−0.282420
$288$	0	0
$289$	−6.08362	−0.357860
$290$	0.169608	0.00995970
$291$	0	0
$292$	−10.6049	−0.620606
$293$	18.6601	1.09013	0.545067	−	0.838392i	$-0.316504\pi$
$293$	18.6601	1.09013	0.545067	+	0.838392i	$0.316504\pi$
$294$	0	0
$295$	4.22362	0.245909
$296$	1.53998	0.0895094
$297$	0	0
$298$	6.25243	0.362193
$299$	−22.2055	−1.28417
$300$	0	0
$301$	3.16645	0.182511
$302$	12.6230	0.726374
$303$	0	0
$304$	−1.60794	−0.0922215
$305$	−2.93802	−0.168230
$306$	0	0
$307$	13.9263	0.794817	0.397408	−	0.917642i	$-0.369910\pi$
$307$	13.9263	0.794817	0.397408	+	0.917642i	$0.369910\pi$
$308$	0	0
$309$	0	0
$310$	−2.88209	−0.163692
$311$	−10.1568	−0.575939	−0.287970	−	0.957640i	$-0.592980\pi$
$311$	−10.1568	−0.575939	−0.287970	+	0.957640i	$0.592980\pi$
$312$	0	0
$313$	4.81885	0.272377	0.136189	−	0.990683i	$-0.456515\pi$
$313$	4.81885	0.272377	0.136189	+	0.990683i	$0.456515\pi$
$314$	0.659917	0.0372413
$315$	0	0
$316$	9.97065	0.560893
$317$	8.07318	0.453435	0.226718	−	0.973961i	$-0.427201\pi$
$317$	8.07318	0.453435	0.226718	+	0.973961i	$0.427201\pi$
$318$	0	0
$319$	0	0
$320$	−4.34734	−0.243024
$321$	0	0
$322$	−6.78003	−0.377836
$323$	26.2303	1.45949
$324$	0	0
$325$	−12.8304	−0.711703
$326$	2.66671	0.147695
$327$	0	0
$328$	13.7350	0.758387
$329$	−11.0114	−0.607076
$330$	0	0
$331$	0.776780	0.0426957	0.0213479	−	0.999772i	$-0.493204\pi$
$331$	0.776780	0.0426957	0.0213479	+	0.999772i	$0.493204\pi$
$332$	8.64444	0.474425
$333$	0	0
$334$	7.23468	0.395864
$335$	10.9931	0.600615
$336$	0	0
$337$	0.209600	0.0114176	0.00570882	−	0.999984i	$-0.498183\pi$
$337$	0.209600	0.0114176	0.00570882	+	0.999984i	$0.498183\pi$
$338$	−3.88266	−0.211189
$339$	0	0
$340$	3.16180	0.171473
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	−9.09001	−0.490100
$345$	0	0
$346$	4.63915	0.249402
$347$	22.7970	1.22380	0.611902	−	0.790933i	$-0.290405\pi$
$347$	22.7970	1.22380	0.611902	+	0.790933i	$0.290405\pi$
$348$	0	0
$349$	−14.0294	−0.750976	−0.375488	−	0.926827i	$-0.622525\pi$
$349$	−14.0294	−0.750976	−0.375488	+	0.926827i	$0.622525\pi$
$350$	−3.91753	−0.209401
$351$	0	0
$352$	0	0
$353$	−22.3703	−1.19065	−0.595327	−	0.803484i	$-0.702977\pi$
$353$	−22.3703	−1.19065	−0.595327	+	0.803484i	$0.702977\pi$
$354$	0	0
$355$	8.31060	0.441081
$356$	−9.56367	−0.506874
$357$	0	0
$358$	5.41042	0.285949
$359$	−2.70211	−0.142612	−0.0713060	−	0.997454i	$-0.522717\pi$
$359$	−2.70211	−0.142612	−0.0713060	+	0.997454i	$0.522717\pi$
$360$	0	0
$361$	44.0270	2.31721
$362$	2.20499	0.115892
$363$	0	0
$364$	−3.50791	−0.183864
$365$	−7.16279	−0.374918
$366$	0	0
$367$	28.2820	1.47631	0.738155	−	0.674631i	$-0.235697\pi$
$367$	28.2820	1.47631	0.738155	+	0.674631i	$0.235697\pi$
$368$	1.52608	0.0795525
$369$	0	0
$370$	0.388076	0.0201751
$371$	11.7317	0.609080
$372$	0	0
$373$	8.53857	0.442110	0.221055	−	0.975261i	$-0.429050\pi$
$373$	8.53857	0.442110	0.221055	+	0.975261i	$0.429050\pi$
$374$	0	0
$375$	0	0
$376$	31.6106	1.63019
$377$	−0.690937	−0.0355851
$378$	0	0
$379$	34.6825	1.78152	0.890759	−	0.454475i	$-0.150173\pi$
$379$	34.6825	1.78152	0.890759	+	0.454475i	$0.150173\pi$
$380$	7.59729	0.389733
$381$	0	0
$382$	−19.1435	−0.979466
$383$	12.7173	0.649826	0.324913	−	0.945744i	$-0.394665\pi$
$383$	12.7173	0.649826	0.324913	+	0.945744i	$0.394665\pi$
$384$	0	0
$385$	0	0
$386$	−7.55786	−0.384685
$387$	0	0
$388$	−22.9386	−1.16453
$389$	2.48749	0.126121	0.0630605	−	0.998010i	$-0.479914\pi$
$389$	2.48749	0.126121	0.0630605	+	0.998010i	$0.479914\pi$
$390$	0	0
$391$	−24.8950	−1.25899
$392$	−2.87073	−0.144994
$393$	0	0
$394$	9.86622	0.497053
$395$	6.73439	0.338844
$396$	0	0
$397$	19.1634	0.961781	0.480891	−	0.876781i	$-0.340313\pi$
$397$	19.1634	0.961781	0.480891	+	0.876781i	$0.340313\pi$
$398$	5.63409	0.282411
$399$	0	0
$400$	0.881777	0.0440888
$401$	−9.17005	−0.457930	−0.228965	−	0.973435i	$-0.573534\pi$
$401$	−9.17005	−0.457930	−0.228965	+	0.973435i	$0.573534\pi$
$402$	0	0
$403$	11.7409	0.584855
$404$	−20.5679	−1.02329
$405$	0	0
$406$	−0.210965	−0.0104700
$407$	0	0
$408$	0	0
$409$	−29.6366	−1.46544	−0.732718	−	0.680533i	$-0.761748\pi$
$409$	−29.6366	−1.46544	−0.732718	+	0.680533i	$0.761748\pi$
$410$	3.46123	0.170938
$411$	0	0
$412$	−17.0897	−0.841947
$413$	−5.25351	−0.258508
$414$	0	0
$415$	5.83864	0.286607
$416$	16.3833	0.803256
$417$	0	0
$418$	0	0
$419$	31.7616	1.55166	0.775828	−	0.630944i	$-0.217332\pi$
$419$	31.7616	1.55166	0.775828	+	0.630944i	$0.217332\pi$
$420$	0	0
$421$	5.94862	0.289918	0.144959	−	0.989438i	$-0.453695\pi$
$421$	5.94862	0.289918	0.144959	+	0.989438i	$0.453695\pi$
$422$	15.0454	0.732399
$423$	0	0
$424$	−33.6786	−1.63558
$425$	−14.3844	−0.697747
$426$	0	0
$427$	3.65442	0.176850
$428$	−18.2358	−0.881460
$429$	0	0
$430$	−2.29069	−0.110467
$431$	11.2906	0.543847	0.271923	−	0.962319i	$-0.412340\pi$
$431$	11.2906	0.543847	0.271923	+	0.962319i	$0.412340\pi$
$432$	0	0
$433$	17.1930	0.826243	0.413122	−	0.910676i	$-0.364438\pi$
$433$	17.1930	0.826243	0.413122	+	0.910676i	$0.364438\pi$
$434$	3.58486	0.172079
$435$	0	0
$436$	6.54953	0.313666
$437$	−59.8185	−2.86151
$438$	0	0
$439$	−33.6983	−1.60833	−0.804167	−	0.594404i	$-0.797388\pi$
$439$	−33.6983	−1.60833	−0.804167	+	0.594404i	$0.797388\pi$
$440$	0	0
$441$	0	0
$442$	8.76165	0.416749
$443$	−1.38268	−0.0656930	−0.0328465	−	0.999460i	$-0.510457\pi$
$443$	−1.38268	−0.0656930	−0.0328465	+	0.999460i	$0.510457\pi$
$444$	0	0
$445$	−6.45951	−0.306210
$446$	11.7848	0.558029
$447$	0	0
$448$	5.40740	0.255476
$449$	−21.4531	−1.01243	−0.506217	−	0.862406i	$-0.668956\pi$
$449$	−21.4531	−1.01243	−0.506217	+	0.862406i	$0.668956\pi$
$450$	0	0
$451$	0	0
$452$	7.96886	0.374823
$453$	0	0
$454$	−14.3095	−0.671577
$455$	−2.36931	−0.111075
$456$	0	0
$457$	−3.54337	−0.165752	−0.0828760	−	0.996560i	$-0.526411\pi$
$457$	−3.54337	−0.165752	−0.0828760	+	0.996560i	$0.526411\pi$
$458$	8.11279	0.379086
$459$	0	0
$460$	−7.21054	−0.336193
$461$	−20.0540	−0.934006	−0.467003	−	0.884256i	$-0.654666\pi$
$461$	−20.0540	−0.934006	−0.467003	+	0.884256i	$0.654666\pi$
$462$	0	0
$463$	−9.80547	−0.455699	−0.227849	−	0.973696i	$-0.573169\pi$
$463$	−9.80547	−0.455699	−0.227849	+	0.973696i	$0.573169\pi$
$464$	0.0474850	0.00220444
$465$	0	0
$466$	2.28736	0.105960
$467$	−18.1244	−0.838697	−0.419349	−	0.907825i	$-0.637742\pi$
$467$	−18.1244	−0.838697	−0.419349	+	0.907825i	$0.637742\pi$
$468$	0	0
$469$	−13.6736	−0.631389
$470$	7.96590	0.367440
$471$	0	0
$472$	15.0814	0.694178
$473$	0	0
$474$	0	0
$475$	−34.5634	−1.58588
$476$	−3.93278	−0.180259
$477$	0	0
$478$	17.2541	0.789185
$479$	−38.6169	−1.76445	−0.882227	−	0.470825i	$-0.843956\pi$
$479$	−38.6169	−1.76445	−0.882227	+	0.470825i	$0.843956\pi$
$480$	0	0
$481$	−1.58092	−0.0720837
$482$	−6.74927	−0.307421
$483$	0	0
$484$	0	0
$485$	−15.4932	−0.703511
$486$	0	0
$487$	16.4315	0.744581	0.372290	−	0.928116i	$-0.378573\pi$
$487$	16.4315	0.744581	0.372290	+	0.928116i	$0.378573\pi$
$488$	−10.4909	−0.474899
$489$	0	0
$490$	−0.723426	−0.0326811
$491$	−26.0846	−1.17718	−0.588590	−	0.808431i	$-0.700317\pi$
$491$	−26.0846	−1.17718	−0.588590	+	0.808431i	$0.700317\pi$
$492$	0	0
$493$	−0.774623	−0.0348873
$494$	21.0528	0.947210
$495$	0	0
$496$	−0.806899	−0.0362308
$497$	−10.3371	−0.463680
$498$	0	0
$499$	−32.7320	−1.46528	−0.732642	−	0.680615i	$-0.761713\pi$
$499$	−32.7320	−1.46528	−0.732642	+	0.680615i	$0.761713\pi$
$500$	−8.95110	−0.400306
$501$	0	0
$502$	13.2187	0.589982
$503$	−18.6428	−0.831241	−0.415621	−	0.909538i	$-0.636436\pi$
$503$	−18.6428	−0.831241	−0.415621	+	0.909538i	$0.636436\pi$
$504$	0	0
$505$	−13.8920	−0.618184
$506$	0	0
$507$	0	0
$508$	13.7405	0.609635
$509$	−23.8913	−1.05896	−0.529482	−	0.848321i	$-0.677614\pi$
$509$	−23.8913	−1.05896	−0.529482	+	0.848321i	$0.677614\pi$
$510$	0	0
$511$	8.90937	0.394127
$512$	−2.28881	−0.101152
$513$	0	0
$514$	18.6715	0.823565
$515$	−11.5427	−0.508633
$516$	0	0
$517$	0	0
$518$	−0.482705	−0.0212088
$519$	0	0
$520$	6.80166	0.298272
$521$	−36.0222	−1.57816	−0.789080	−	0.614290i	$-0.789443\pi$
$521$	−36.0222	−1.57816	−0.789080	+	0.614290i	$0.789443\pi$
$522$	0	0
$523$	28.3402	1.23923	0.619615	−	0.784906i	$-0.287289\pi$
$523$	28.3402	1.23923	0.619615	+	0.784906i	$0.287289\pi$
$524$	−25.2355	−1.10242
$525$	0	0
$526$	−24.6068	−1.07291
$527$	13.1629	0.573387
$528$	0	0
$529$	33.7733	1.46841
$530$	−8.48703	−0.368653
$531$	0	0
$532$	−9.44983	−0.409702
$533$	−14.1001	−0.610745
$534$	0	0
$535$	−12.3168	−0.532503
$536$	39.2533	1.69548
$537$	0	0
$538$	−1.42087	−0.0612581
$539$	0	0
$540$	0	0
$541$	25.7194	1.10576	0.552882	−	0.833259i	$-0.313528\pi$
$541$	25.7194	1.10576	0.552882	+	0.833259i	$0.313528\pi$
$542$	13.5886	0.583681
$543$	0	0
$544$	18.3676	0.787504
$545$	4.42369	0.189490
$546$	0	0
$547$	33.1803	1.41869	0.709343	−	0.704864i	$-0.248992\pi$
$547$	33.1803	1.41869	0.709343	+	0.704864i	$0.248992\pi$
$548$	−0.970301	−0.0414492
$549$	0	0
$550$	0	0
$551$	−1.86129	−0.0792937
$552$	0	0
$553$	−8.37651	−0.356205
$554$	−14.6085	−0.620656
$555$	0	0
$556$	−11.0373	−0.468087
$557$	32.3919	1.37249	0.686245	−	0.727370i	$-0.259258\pi$
$557$	32.3919	1.37249	0.686245	+	0.727370i	$0.259258\pi$
$558$	0	0
$559$	9.33167	0.394688
$560$	0.162832	0.00688092
$561$	0	0
$562$	15.3750	0.648555
$563$	−34.6650	−1.46096	−0.730478	−	0.682937i	$-0.760702\pi$
$563$	−34.6650	−1.46096	−0.730478	+	0.682937i	$0.760702\pi$
$564$	0	0
$565$	5.38233	0.226437
$566$	−0.103504	−0.00435060
$567$	0	0
$568$	29.6749	1.24513
$569$	−18.4658	−0.774127	−0.387064	−	0.922053i	$-0.626511\pi$
$569$	−18.4658	−0.774127	−0.387064	+	0.922053i	$0.626511\pi$
$570$	0	0
$571$	14.2034	0.594394	0.297197	−	0.954816i	$-0.403948\pi$
$571$	14.2034	0.594394	0.297197	+	0.954816i	$0.403948\pi$
$572$	0	0
$573$	0	0
$574$	−4.30522	−0.179696
$575$	32.8039	1.36802
$576$	0	0
$577$	16.9012	0.703606	0.351803	−	0.936074i	$-0.385569\pi$
$577$	16.9012	0.703606	0.351803	+	0.936074i	$0.385569\pi$
$578$	−5.47421	−0.227697
$579$	0	0
$580$	−0.224361	−0.00931607
$581$	−7.26234	−0.301292
$582$	0	0
$583$	0	0
$584$	−25.5764	−1.05836
$585$	0	0
$586$	16.7909	0.693624
$587$	34.7412	1.43392	0.716962	−	0.697113i	$-0.245532\pi$
$587$	34.7412	1.43392	0.716962	+	0.697113i	$0.245532\pi$
$588$	0	0
$589$	31.6284	1.30322
$590$	3.80053	0.156465
$591$	0	0
$592$	0.108649	0.00446547
$593$	2.43045	0.0998068	0.0499034	−	0.998754i	$-0.484109\pi$
$593$	2.43045	0.0998068	0.0499034	+	0.998754i	$0.484109\pi$
$594$	0	0
$595$	−2.65628	−0.108897
$596$	−8.27085	−0.338787
$597$	0	0
$598$	−19.9811	−0.817087
$599$	−16.7195	−0.683142	−0.341571	−	0.939856i	$-0.610959\pi$
$599$	−16.7195	−0.683142	−0.341571	+	0.939856i	$0.610959\pi$
$600$	0	0
$601$	0.429964	0.0175386	0.00876929	−	0.999962i	$-0.497209\pi$
$601$	0.429964	0.0175386	0.00876929	+	0.999962i	$0.497209\pi$
$602$	2.84925	0.116127
$603$	0	0
$604$	−16.6980	−0.679433
$605$	0	0
$606$	0	0
$607$	11.5440	0.468557	0.234279	−	0.972169i	$-0.424727\pi$
$607$	11.5440	0.468557	0.234279	+	0.972169i	$0.424727\pi$
$608$	44.1343	1.78988
$609$	0	0
$610$	−2.64371	−0.107041
$611$	−32.4510	−1.31283
$612$	0	0
$613$	−18.0048	−0.727209	−0.363604	−	0.931554i	$-0.618454\pi$
$613$	−18.0048	−0.727209	−0.363604	+	0.931554i	$0.618454\pi$
$614$	12.5313	0.505721
$615$	0	0
$616$	0	0
$617$	32.1061	1.29254	0.646272	−	0.763107i	$-0.276327\pi$
$617$	32.1061	1.29254	0.646272	+	0.763107i	$0.276327\pi$
$618$	0	0
$619$	−0.280294	−0.0112660	−0.00563298	−	0.999984i	$-0.501793\pi$
$619$	−0.280294	−0.0112660	−0.00563298	+	0.999984i	$0.501793\pi$
$620$	3.81249	0.153113
$621$	0	0
$622$	−9.13936	−0.366455
$623$	8.03460	0.321899
$624$	0	0
$625$	15.7225	0.628899
$626$	4.33613	0.173307
$627$	0	0
$628$	−0.872952	−0.0348346
$629$	−1.77240	−0.0706702
$630$	0	0
$631$	30.3354	1.20763	0.603816	−	0.797124i	$-0.293646\pi$
$631$	30.3354	1.20763	0.603816	+	0.797124i	$0.293646\pi$
$632$	24.0467	0.956525
$633$	0	0
$634$	7.26447	0.288509
$635$	9.28061	0.368290
$636$	0	0
$637$	2.94705	0.116766
$638$	0	0
$639$	0	0
$640$	5.02692	0.198706
$641$	3.93047	0.155244	0.0776222	−	0.996983i	$-0.475267\pi$
$641$	3.93047	0.155244	0.0776222	+	0.996983i	$0.475267\pi$
$642$	0	0
$643$	−13.7243	−0.541234	−0.270617	−	0.962687i	$-0.587228\pi$
$643$	−13.7243	−0.541234	−0.270617	+	0.962687i	$0.587228\pi$
$644$	8.96877	0.353419
$645$	0	0
$646$	23.6027	0.928636
$647$	−2.01034	−0.0790345	−0.0395173	−	0.999219i	$-0.512582\pi$
$647$	−2.01034	−0.0790345	−0.0395173	+	0.999219i	$0.512582\pi$
$648$	0	0
$649$	0	0
$650$	−11.5452	−0.452838
$651$	0	0
$652$	−3.52758	−0.138151
$653$	−0.795349	−0.0311244	−0.0155622	−	0.999879i	$-0.504954\pi$
$653$	−0.795349	−0.0311244	−0.0155622	+	0.999879i	$0.504954\pi$
$654$	0	0
$655$	−17.0446	−0.665988
$656$	0.969039	0.0378346
$657$	0	0
$658$	−9.90831	−0.386266
$659$	−21.9135	−0.853628	−0.426814	−	0.904340i	$-0.640364\pi$
$659$	−21.9135	−0.853628	−0.426814	+	0.904340i	$0.640364\pi$
$660$	0	0
$661$	−19.4952	−0.758276	−0.379138	−	0.925340i	$-0.623779\pi$
$661$	−19.4952	−0.758276	−0.379138	+	0.925340i	$0.623779\pi$
$662$	0.698968	0.0271662
$663$	0	0
$664$	20.8482	0.809067
$665$	−6.38261	−0.247507
$666$	0	0
$667$	1.76654	0.0684007
$668$	−9.57020	−0.370282
$669$	0	0
$670$	9.89186	0.382156
$671$	0	0
$672$	0	0
$673$	−7.00748	−0.270118	−0.135059	−	0.990838i	$-0.543122\pi$
$673$	−7.00748	−0.270118	−0.135059	+	0.990838i	$0.543122\pi$
$674$	0.188604	0.00726474
$675$	0	0
$676$	5.13607	0.197541
$677$	−5.85627	−0.225075	−0.112537	−	0.993647i	$-0.535898\pi$
$677$	−5.85627	−0.225075	−0.112537	+	0.993647i	$0.535898\pi$
$678$	0	0
$679$	19.2711	0.739557
$680$	7.62547	0.292423
$681$	0	0
$682$	0	0
$683$	34.2255	1.30960	0.654802	−	0.755801i	$-0.272752\pi$
$683$	34.2255	1.30960	0.654802	+	0.755801i	$0.272752\pi$
$684$	0	0
$685$	−0.655362	−0.0250401
$686$	0.899827	0.0343556
$687$	0	0
$688$	−0.641324	−0.0244502
$689$	34.5740	1.31716
$690$	0	0
$691$	15.0086	0.570954	0.285477	−	0.958386i	$-0.407848\pi$
$691$	15.0086	0.570954	0.285477	+	0.958386i	$0.407848\pi$
$692$	−6.13676	−0.233285
$693$	0	0
$694$	20.5133	0.778675
$695$	−7.45484	−0.282778
$696$	0	0
$697$	−15.8079	−0.598768
$698$	−12.6240	−0.477826
$699$	0	0
$700$	5.18219	0.195868
$701$	11.9907	0.452884	0.226442	−	0.974025i	$-0.427291\pi$
$701$	11.9907	0.452884	0.226442	+	0.974025i	$0.427291\pi$
$702$	0	0
$703$	−4.25878	−0.160623
$704$	0	0
$705$	0	0
$706$	−20.1294	−0.757582
$707$	17.2794	0.649858
$708$	0	0
$709$	39.5551	1.48552	0.742762	−	0.669556i	$-0.233515\pi$
$709$	39.5551	1.48552	0.742762	+	0.669556i	$0.233515\pi$
$710$	7.47810	0.280648
$711$	0	0
$712$	−23.0651	−0.864403
$713$	−30.0183	−1.12419
$714$	0	0
$715$	0	0
$716$	−7.15702	−0.267470
$717$	0	0
$718$	−2.43143	−0.0907403
$719$	−44.3811	−1.65514	−0.827568	−	0.561366i	$-0.810276\pi$
$719$	−44.3811	−1.65514	−0.827568	+	0.561366i	$0.810276\pi$
$720$	0	0
$721$	14.3573	0.534694
$722$	39.6167	1.47438
$723$	0	0
$724$	−2.91681	−0.108402
$725$	1.02071	0.0379084
$726$	0	0
$727$	−45.1325	−1.67387	−0.836935	−	0.547302i	$-0.815655\pi$
$727$	−45.1325	−1.67387	−0.836935	+	0.547302i	$0.815655\pi$
$728$	−8.46018	−0.313555
$729$	0	0
$730$	−6.44527	−0.238550
$731$	10.4619	0.386948
$732$	0	0
$733$	13.7971	0.509608	0.254804	−	0.966993i	$-0.417989\pi$
$733$	13.7971	0.509608	0.254804	+	0.966993i	$0.417989\pi$
$734$	25.4490	0.939338
$735$	0	0
$736$	−41.8876	−1.54400
$737$	0	0
$738$	0	0
$739$	−16.7453	−0.615985	−0.307993	−	0.951389i	$-0.599657\pi$
$739$	−16.7453	−0.615985	−0.307993	+	0.951389i	$0.599657\pi$
$740$	−0.513355	−0.0188713
$741$	0	0
$742$	10.5565	0.387542
$743$	−10.8932	−0.399631	−0.199815	−	0.979834i	$-0.564034\pi$
$743$	−10.8932	−0.399631	−0.199815	+	0.979834i	$0.564034\pi$
$744$	0	0
$745$	−5.58631	−0.204666
$746$	7.68323	0.281303
$747$	0	0
$748$	0	0
$749$	15.3202	0.559788
$750$	0	0
$751$	−27.0918	−0.988596	−0.494298	−	0.869293i	$-0.664575\pi$
$751$	−27.0918	−0.988596	−0.494298	+	0.869293i	$0.664575\pi$
$752$	2.23021	0.0813275
$753$	0	0
$754$	−0.621724	−0.0226419
$755$	−11.2782	−0.410456
$756$	0	0
$757$	30.2302	1.09873	0.549367	−	0.835581i	$-0.314869\pi$
$757$	30.2302	1.09873	0.549367	+	0.835581i	$0.314869\pi$
$758$	31.2082	1.13353
$759$	0	0
$760$	18.3227	0.664636
$761$	38.5957	1.39909	0.699547	−	0.714587i	$-0.253385\pi$
$761$	38.5957	1.39909	0.699547	+	0.714587i	$0.253385\pi$
$762$	0	0
$763$	−5.50237	−0.199199
$764$	25.3234	0.916170
$765$	0	0
$766$	11.4434	0.413467
$767$	−15.4824	−0.559035
$768$	0	0
$769$	40.5122	1.46091	0.730454	−	0.682962i	$-0.239309\pi$
$769$	40.5122	1.46091	0.730454	+	0.682962i	$0.239309\pi$
$770$	0	0
$771$	0	0
$772$	9.99770	0.359825
$773$	−22.1942	−0.798270	−0.399135	−	0.916892i	$-0.630690\pi$
$773$	−22.1942	−0.798270	−0.399135	+	0.916892i	$0.630690\pi$
$774$	0	0
$775$	−17.3447	−0.623040
$776$	−55.3221	−1.98595
$777$	0	0
$778$	2.23831	0.0802474
$779$	−37.9839	−1.36091
$780$	0	0
$781$	0	0
$782$	−22.4012	−0.801064
$783$	0	0
$784$	−0.202537	−0.00723348
$785$	−0.589610	−0.0210441
$786$	0	0
$787$	15.0781	0.537478	0.268739	−	0.963213i	$-0.413393\pi$
$787$	15.0781	0.537478	0.268739	+	0.963213i	$0.413393\pi$
$788$	−13.0513	−0.464932
$789$	0	0
$790$	6.05978	0.215597
$791$	−6.69477	−0.238039
$792$	0	0
$793$	10.7698	0.382446
$794$	17.2437	0.611956
$795$	0	0
$796$	−7.45290	−0.264161
$797$	22.9577	0.813202	0.406601	−	0.913606i	$-0.366714\pi$
$797$	22.9577	0.813202	0.406601	+	0.913606i	$0.366714\pi$
$798$	0	0
$799$	−36.3815	−1.28708
$800$	−24.2028	−0.855699
$801$	0	0
$802$	−8.25146	−0.291369
$803$	0	0
$804$	0	0
$805$	6.05769	0.213506
$806$	10.5648	0.372128
$807$	0	0
$808$	−49.6044	−1.74508
$809$	27.6048	0.970531	0.485266	−	0.874367i	$-0.338723\pi$
$809$	27.6048	0.970531	0.485266	+	0.874367i	$0.338723\pi$
$810$	0	0
$811$	−50.0296	−1.75678	−0.878389	−	0.477947i	$-0.841381\pi$
$811$	−50.0296	−1.75678	−0.878389	+	0.477947i	$0.841381\pi$
$812$	0.279069	0.00979340
$813$	0	0
$814$	0	0
$815$	−2.38260	−0.0834589
$816$	0	0
$817$	25.1383	0.879477
$818$	−26.6678	−0.932418
$819$	0	0
$820$	−4.57859	−0.159891
$821$	−27.1692	−0.948213	−0.474107	−	0.880467i	$-0.657229\pi$
$821$	−27.1692	−0.948213	−0.474107	+	0.880467i	$0.657229\pi$
$822$	0	0
$823$	17.4262	0.607439	0.303720	−	0.952761i	$-0.401771\pi$
$823$	17.4262	0.607439	0.303720	+	0.952761i	$0.401771\pi$
$824$	−41.2159	−1.43583
$825$	0	0
$826$	−4.72725	−0.164482
$827$	30.4545	1.05901	0.529503	−	0.848308i	$-0.322378\pi$
$827$	30.4545	1.05901	0.529503	+	0.848308i	$0.322378\pi$
$828$	0	0
$829$	27.4137	0.952118	0.476059	−	0.879413i	$-0.342065\pi$
$829$	27.4137	0.952118	0.476059	+	0.879413i	$0.342065\pi$
$830$	5.25377	0.182361
$831$	0	0
$832$	15.9359	0.552477
$833$	3.30399	0.114477
$834$	0	0
$835$	−6.46391	−0.223693
$836$	0	0
$837$	0	0
$838$	28.5800	0.987278
$839$	−29.3429	−1.01303	−0.506514	−	0.862231i	$-0.669066\pi$
$839$	−29.3429	−1.01303	−0.506514	+	0.862231i	$0.669066\pi$
$840$	0	0
$841$	−28.9450	−0.998105
$842$	5.35273	0.184467
$843$	0	0
$844$	−19.9024	−0.685068
$845$	3.46901	0.119338
$846$	0	0
$847$	0	0
$848$	−2.37611	−0.0815960
$849$	0	0
$850$	−12.9435	−0.443958
$851$	4.04198	0.138557
$852$	0	0
$853$	−19.5073	−0.667916	−0.333958	−	0.942588i	$-0.608384\pi$
$853$	−19.5073	−0.667916	−0.333958	+	0.942588i	$0.608384\pi$
$854$	3.28835	0.112525
$855$	0	0
$856$	−43.9801	−1.50321
$857$	10.9432	0.373811	0.186906	−	0.982378i	$-0.440154\pi$
$857$	10.9432	0.373811	0.186906	+	0.982378i	$0.440154\pi$
$858$	0	0
$859$	−15.3191	−0.522680	−0.261340	−	0.965247i	$-0.584164\pi$
$859$	−15.3191	−0.522680	−0.261340	+	0.965247i	$0.584164\pi$
$860$	3.03017	0.103328
$861$	0	0
$862$	10.1595	0.346036
$863$	13.4613	0.458227	0.229114	−	0.973400i	$-0.426417\pi$
$863$	13.4613	0.458227	0.229114	+	0.973400i	$0.426417\pi$
$864$	0	0
$865$	−4.14490	−0.140931
$866$	15.4707	0.525717
$867$	0	0
$868$	−4.74214	−0.160959
$869$	0	0
$870$	0	0
$871$	−40.2969	−1.36541
$872$	15.7958	0.534914
$873$	0	0
$874$	−53.8263	−1.82070
$875$	7.51997	0.254221
$876$	0	0
$877$	3.22081	0.108759	0.0543796	−	0.998520i	$-0.482682\pi$
$877$	3.22081	0.108759	0.0543796	+	0.998520i	$0.482682\pi$
$878$	−30.3227	−1.02334
$879$	0	0
$880$	0	0
$881$	−19.3721	−0.652661	−0.326331	−	0.945256i	$-0.605812\pi$
$881$	−19.3721	−0.652661	−0.326331	+	0.945256i	$0.605812\pi$
$882$	0	0
$883$	−2.93248	−0.0986858	−0.0493429	−	0.998782i	$-0.515713\pi$
$883$	−2.93248	−0.0986858	−0.0493429	+	0.998782i	$0.515713\pi$
$884$	−11.5901	−0.389817
$885$	0	0
$886$	−1.24417	−0.0417987
$887$	7.68647	0.258087	0.129043	−	0.991639i	$-0.458809\pi$
$887$	7.68647	0.258087	0.129043	+	0.991639i	$0.458809\pi$
$888$	0	0
$889$	−11.5436	−0.387160
$890$	−5.81244	−0.194833
$891$	0	0
$892$	−15.5893	−0.521967
$893$	−87.4186	−2.92535
$894$	0	0
$895$	−4.83400	−0.161583
$896$	−6.25269	−0.208888
$897$	0	0
$898$	−19.3041	−0.644185
$899$	−0.934038	−0.0311519
$900$	0	0
$901$	38.7615	1.29133
$902$	0	0
$903$	0	0
$904$	19.2189	0.639210
$905$	−1.97008	−0.0654875
$906$	0	0
$907$	−12.7746	−0.424172	−0.212086	−	0.977251i	$-0.568026\pi$
$907$	−12.7746	−0.424172	−0.212086	+	0.977251i	$0.568026\pi$
$908$	18.9289	0.628177
$909$	0	0
$910$	−2.13197	−0.0706742
$911$	11.0831	0.367200	0.183600	−	0.983001i	$-0.441225\pi$
$911$	11.0831	0.367200	0.183600	+	0.983001i	$0.441225\pi$
$912$	0	0
$913$	0	0
$914$	−3.18843	−0.105464
$915$	0	0
$916$	−10.7318	−0.354588
$917$	21.2008	0.700112
$918$	0	0
$919$	13.5522	0.447046	0.223523	−	0.974699i	$-0.428244\pi$
$919$	13.5522	0.447046	0.223523	+	0.974699i	$0.428244\pi$
$920$	−17.3900	−0.573331
$921$	0	0
$922$	−18.0451	−0.594284
$923$	−30.4638	−1.00273
$924$	0	0
$925$	2.33548	0.0767900
$926$	−8.82323	−0.289949
$927$	0	0
$928$	−1.30336	−0.0427849
$929$	43.0169	1.41134	0.705670	−	0.708541i	$-0.250646\pi$
$929$	43.0169	1.41134	0.705670	+	0.708541i	$0.250646\pi$
$930$	0	0
$931$	7.93896	0.260189
$932$	−3.02577	−0.0991124
$933$	0	0
$934$	−16.3088	−0.533641
$935$	0	0
$936$	0	0
$937$	−2.56484	−0.0837896	−0.0418948	−	0.999122i	$-0.513339\pi$
$937$	−2.56484	−0.0837896	−0.0418948	+	0.999122i	$0.513339\pi$
$938$	−12.3039	−0.401737
$939$	0	0
$940$	−10.5375	−0.343694
$941$	16.5640	0.539972	0.269986	−	0.962864i	$-0.412981\pi$
$941$	16.5640	0.539972	0.269986	+	0.962864i	$0.412981\pi$
$942$	0	0
$943$	36.0502	1.17396
$944$	1.06403	0.0346313
$945$	0	0
$946$	0	0
$947$	46.5682	1.51326	0.756632	−	0.653841i	$-0.226843\pi$
$947$	46.5682	1.51326	0.756632	+	0.653841i	$0.226843\pi$
$948$	0	0
$949$	26.2564	0.852318
$950$	−31.1011	−1.00905
$951$	0	0
$952$	−9.48487	−0.307406
$953$	−39.9493	−1.29409	−0.647043	−	0.762454i	$-0.723994\pi$
$953$	−39.9493	−1.29409	−0.647043	+	0.762454i	$0.723994\pi$
$954$	0	0
$955$	17.1040	0.553472
$956$	−22.8241	−0.738185
$957$	0	0
$958$	−34.7486	−1.12268
$959$	0.815166	0.0263231
$960$	0	0
$961$	−15.1282	−0.488005
$962$	−1.42255	−0.0458650
$963$	0	0
$964$	8.92809	0.287554
$965$	6.75266	0.217376
$966$	0	0
$967$	35.2572	1.13380	0.566898	−	0.823788i	$-0.308143\pi$
$967$	35.2572	1.13380	0.566898	+	0.823788i	$0.308143\pi$
$968$	0	0
$969$	0	0
$970$	−13.9412	−0.447626
$971$	34.8319	1.11781	0.558905	−	0.829232i	$-0.311222\pi$
$971$	34.8319	1.11781	0.558905	+	0.829232i	$0.311222\pi$
$972$	0	0
$973$	9.27263	0.297267
$974$	14.7855	0.473757
$975$	0	0
$976$	−0.740158	−0.0236919
$977$	−23.2137	−0.742672	−0.371336	−	0.928499i	$-0.621100\pi$
$977$	−23.2137	−0.742672	−0.371336	+	0.928499i	$0.621100\pi$
$978$	0	0
$979$	0	0
$980$	0.956964	0.0305691
$981$	0	0
$982$	−23.4716	−0.749010
$983$	−50.8764	−1.62271	−0.811353	−	0.584557i	$-0.801268\pi$
$983$	−50.8764	−1.62271	−0.811353	+	0.584557i	$0.801268\pi$
$984$	0	0
$985$	−8.81509	−0.280872
$986$	−0.697027	−0.0221979
$987$	0	0
$988$	−27.8491	−0.885998
$989$	−23.8586	−0.758658
$990$	0	0
$991$	−49.0338	−1.55761	−0.778805	−	0.627266i	$-0.784174\pi$
$991$	−49.0338	−1.55761	−0.778805	+	0.627266i	$0.784174\pi$
$992$	22.1476	0.703187
$993$	0	0
$994$	−9.30157	−0.295028
$995$	−5.03384	−0.159584
$996$	0	0
$997$	−4.64075	−0.146974	−0.0734870	−	0.997296i	$-0.523413\pi$
$997$	−4.64075	−0.146974	−0.0734870	+	0.997296i	$0.523413\pi$
$998$	−29.4531	−0.932322
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.dc.1.11		16
3.2	odd	2	inner	7623.2.a.dc.1.6		16
11.5	even	5		693.2.m.k.190.3	✓	32
11.9	even	5		693.2.m.k.631.3	yes	32
11.10	odd	2		7623.2.a.db.1.6		16
33.5	odd	10		693.2.m.k.190.6	yes	32
33.20	odd	10		693.2.m.k.631.6	yes	32
33.32	even	2		7623.2.a.db.1.11		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
693.2.m.k.190.3	✓	32	11.5	even	5
693.2.m.k.190.6	yes	32	33.5	odd	10
693.2.m.k.631.3	yes	32	11.9	even	5
693.2.m.k.631.6	yes	32	33.20	odd	10
7623.2.a.db.1.6		16	11.10	odd	2
7623.2.a.db.1.11		16	33.32	even	2
7623.2.a.dc.1.6		16	3.2	odd	2	inner
7623.2.a.dc.1.11		16	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+0.899827 q^{2} -1.19031 q^{4} -0.803961 q^{5} +1.00000 q^{7} -2.87073 q^{8} +O(q^{10})\)	\(q+0.899827 q^{2} -1.19031 q^{4} -0.803961 q^{5} +1.00000 q^{7} -2.87073 q^{8} -0.723426 q^{10} +2.94705 q^{13} +0.899827 q^{14} -0.202537 q^{16} +3.30399 q^{17} +7.93896 q^{19} +0.956964 q^{20} -7.53481 q^{23} -4.35365 q^{25} +2.65184 q^{26} -1.19031 q^{28} -0.234450 q^{29} +3.98395 q^{31} +5.55921 q^{32} +2.97302 q^{34} -0.803961 q^{35} -0.536441 q^{37} +7.14369 q^{38} +2.30795 q^{40} -4.78449 q^{41} +3.16645 q^{43} -6.78003 q^{46} -11.0114 q^{47} +1.00000 q^{49} -3.91753 q^{50} -3.50791 q^{52} +11.7317 q^{53} -2.87073 q^{56} -0.210965 q^{58} -5.25351 q^{59} +3.65442 q^{61} +3.58486 q^{62} +5.40740 q^{64} -2.36931 q^{65} -13.6736 q^{67} -3.93278 q^{68} -0.723426 q^{70} -10.3371 q^{71} +8.90937 q^{73} -0.482705 q^{74} -9.44983 q^{76} -8.37651 q^{79} +0.162832 q^{80} -4.30522 q^{82} -7.26234 q^{83} -2.65628 q^{85} +2.84925 q^{86} +8.03460 q^{89} +2.94705 q^{91} +8.96877 q^{92} -9.90831 q^{94} -6.38261 q^{95} +19.2711 q^{97} +0.899827 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 20 q^{4} + 16 q^{7}+O(q^{10}) \) 16 * q + 20 * q^4 + 16 * q^7	\( 16 q + 20 q^{4} + 16 q^{7} - 6 q^{10} + 32 q^{16} + 10 q^{19} + 44 q^{25} + 20 q^{28} + 30 q^{31} + 12 q^{34} + 38 q^{37} - 68 q^{40} + 16 q^{43} + 80 q^{46} + 16 q^{49} + 2 q^{52} + 18 q^{58} - 28 q^{61} + 34 q^{64} + 52 q^{67} - 6 q^{70} - 14 q^{73} - 14 q^{76} + 54 q^{79} + 64 q^{82} + 30 q^{85} - 60 q^{94} + 108 q^{97}+O(q^{100}) \) 16 * q + 20 * q^4 + 16 * q^7 - 6 * q^10 + 32 * q^16 + 10 * q^19 + 44 * q^25 + 20 * q^28 + 30 * q^31 + 12 * q^34 + 38 * q^37 - 68 * q^40 + 16 * q^43 + 80 * q^46 + 16 * q^49 + 2 * q^52 + 18 * q^58 - 28 * q^61 + 34 * q^64 + 52 * q^67 - 6 * q^70 - 14 * q^73 - 14 * q^76 + 54 * q^79 + 64 * q^82 + 30 * q^85 - 60 * q^94 + 108 * q^97

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