LMFDB - Embedded newform 7623.2.a.dc.1.10

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			1.10
Root			$0.759363$ 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)

$f(q)$	$=$	$q+0.759363 q^{2} -1.42337 q^{4} -2.86525 q^{5} +1.00000 q^{7} -2.59958 q^{8} +O(q^{10})$	$q+0.759363 q^{2} -1.42337 q^{4} -2.86525 q^{5} +1.00000 q^{7} -2.59958 q^{8} -2.17577 q^{10} -6.96619 q^{13} +0.759363 q^{14} +0.872709 q^{16} -6.65872 q^{17} -5.75106 q^{19} +4.07830 q^{20} +0.724581 q^{23} +3.20966 q^{25} -5.28987 q^{26} -1.42337 q^{28} -7.56740 q^{29} +2.81968 q^{31} +5.86186 q^{32} -5.05639 q^{34} -2.86525 q^{35} -1.72211 q^{37} -4.36715 q^{38} +7.44844 q^{40} -1.12086 q^{41} -11.7092 q^{43} +0.550220 q^{46} -5.18956 q^{47} +1.00000 q^{49} +2.43729 q^{50} +9.91544 q^{52} +6.25771 q^{53} -2.59958 q^{56} -5.74641 q^{58} -3.98228 q^{59} -5.97074 q^{61} +2.14116 q^{62} +2.70587 q^{64} +19.9599 q^{65} +10.3611 q^{67} +9.47781 q^{68} -2.17577 q^{70} +0.995183 q^{71} +3.53684 q^{73} -1.30771 q^{74} +8.18588 q^{76} -0.669794 q^{79} -2.50053 q^{80} -0.851138 q^{82} -13.4240 q^{83} +19.0789 q^{85} -8.89157 q^{86} -16.6006 q^{89} -6.96619 q^{91} -1.03135 q^{92} -3.94076 q^{94} +16.4782 q^{95} +10.1121 q^{97} +0.759363 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.759363	0.536951	0.268475	−	0.963287i	$-0.413480\pi$
$2$	0.759363	0.536951	0.268475	+	0.963287i	$0.413480\pi$
$3$	0	0
$3$	0	0
$4$	−1.42337	−0.711684
$5$	−2.86525	−1.28138	−0.640689	−	0.767800i	$-0.721351\pi$
$5$	−2.86525	−1.28138	−0.640689	+	0.767800i	$0.721351\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.59958	−0.919090
$9$	0	0
$10$	−2.17577	−0.688038
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−6.96619	−1.93207	−0.966036	−	0.258407i	$-0.916803\pi$
$13$	−6.96619	−1.93207	−0.966036	+	0.258407i	$0.916803\pi$
$14$	0.759363	0.202948
$15$	0	0
$16$	0.872709	0.218177
$17$	−6.65872	−1.61498	−0.807489	−	0.589883i	$-0.799174\pi$
$17$	−6.65872	−1.61498	−0.807489	+	0.589883i	$0.799174\pi$
$18$	0	0
$19$	−5.75106	−1.31938	−0.659692	−	0.751536i	$-0.729313\pi$
$19$	−5.75106	−1.31938	−0.659692	+	0.751536i	$0.729313\pi$
$20$	4.07830	0.911936
$21$	0	0
$22$	0	0
$23$	0.724581	0.151086	0.0755428	−	0.997143i	$-0.475931\pi$
$23$	0.724581	0.151086	0.0755428	+	0.997143i	$0.475931\pi$
$24$	0	0
$25$	3.20966	0.641931
$26$	−5.28987	−1.03743
$27$	0	0
$28$	−1.42337	−0.268991
$29$	−7.56740	−1.40523	−0.702616	−	0.711570i	$-0.747985\pi$
$29$	−7.56740	−1.40523	−0.702616	+	0.711570i	$0.747985\pi$
$30$	0	0
$31$	2.81968	0.506430	0.253215	−	0.967410i	$-0.418512\pi$
$31$	2.81968	0.506430	0.253215	+	0.967410i	$0.418512\pi$
$32$	5.86186	1.03624
$33$	0	0
$34$	−5.05639	−0.867164
$35$	−2.86525	−0.484316
$36$	0	0
$37$	−1.72211	−0.283113	−0.141557	−	0.989930i	$-0.545211\pi$
$37$	−1.72211	−0.283113	−0.141557	+	0.989930i	$0.545211\pi$
$38$	−4.36715	−0.708445
$39$	0	0
$40$	7.44844	1.17770
$41$	−1.12086	−0.175048	−0.0875242	−	0.996162i	$-0.527896\pi$
$41$	−1.12086	−0.175048	−0.0875242	+	0.996162i	$0.527896\pi$
$42$	0	0
$43$	−11.7092	−1.78564	−0.892821	−	0.450411i	$-0.851277\pi$
$43$	−11.7092	−1.78564	−0.892821	+	0.450411i	$0.851277\pi$
$44$	0	0
$45$	0	0
$46$	0.550220	0.0811256
$47$	−5.18956	−0.756975	−0.378488	−	0.925606i	$-0.623556\pi$
$47$	−5.18956	−0.756975	−0.378488	+	0.925606i	$0.623556\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	2.43729	0.344686
$51$	0	0
$52$	9.91544	1.37502
$53$	6.25771	0.859562	0.429781	−	0.902933i	$-0.358591\pi$
$53$	6.25771	0.859562	0.429781	+	0.902933i	$0.358591\pi$
$54$	0	0
$55$	0	0
$56$	−2.59958	−0.347383
$57$	0	0
$58$	−5.74641	−0.754540
$59$	−3.98228	−0.518448	−0.259224	−	0.965817i	$-0.583467\pi$
$59$	−3.98228	−0.518448	−0.259224	+	0.965817i	$0.583467\pi$
$60$	0	0
$61$	−5.97074	−0.764475	−0.382237	−	0.924064i	$-0.624846\pi$
$61$	−5.97074	−0.764475	−0.382237	+	0.924064i	$0.624846\pi$
$62$	2.14116	0.271928
$63$	0	0
$64$	2.70587	0.338233
$65$	19.9599	2.47572
$66$	0	0
$67$	10.3611	1.26581	0.632906	−	0.774229i	$-0.281862\pi$
$67$	10.3611	1.26581	0.632906	+	0.774229i	$0.281862\pi$
$68$	9.47781	1.14935
$69$	0	0
$70$	−2.17577	−0.260054
$71$	0.995183	0.118107	0.0590533	−	0.998255i	$-0.481192\pi$
$71$	0.995183	0.118107	0.0590533	+	0.998255i	$0.481192\pi$
$72$	0	0
$73$	3.53684	0.413956	0.206978	−	0.978346i	$-0.433637\pi$
$73$	3.53684	0.413956	0.206978	+	0.978346i	$0.433637\pi$
$74$	−1.30771	−0.152018
$75$	0	0
$76$	8.18588	0.938984
$77$	0	0
$78$	0	0
$79$	−0.669794	−0.0753578	−0.0376789	−	0.999290i	$-0.511996\pi$
$79$	−0.669794	−0.0753578	−0.0376789	+	0.999290i	$0.511996\pi$
$80$	−2.50053	−0.279568
$81$	0	0
$82$	−0.851138	−0.0939924
$83$	−13.4240	−1.47348	−0.736739	−	0.676177i	$-0.763635\pi$
$83$	−13.4240	−1.47348	−0.736739	+	0.676177i	$0.763635\pi$
$84$	0	0
$85$	19.0789	2.06940
$86$	−8.89157	−0.958803
$87$	0	0
$88$	0	0
$89$	−16.6006	−1.75966	−0.879828	−	0.475292i	$-0.842343\pi$
$89$	−16.6006	−1.75966	−0.879828	+	0.475292i	$0.842343\pi$
$90$	0	0
$91$	−6.96619	−0.730255
$92$	−1.03135	−0.107525
$93$	0	0
$94$	−3.94076	−0.406459
$95$	16.4782	1.69063
$96$	0	0
$97$	10.1121	1.02672	0.513362	−	0.858172i	$-0.328400\pi$
$97$	10.1121	1.02672	0.513362	+	0.858172i	$0.328400\pi$
$98$	0.759363	0.0767073
$99$	0	0
$100$	−4.56852	−0.456852
$101$	−6.42353	−0.639165	−0.319582	−	0.947559i	$-0.603543\pi$
$101$	−6.42353	−0.639165	−0.319582	+	0.947559i	$0.603543\pi$
$102$	0	0
$103$	−8.95787	−0.882645	−0.441323	−	0.897348i	$-0.645491\pi$
$103$	−8.95787	−0.882645	−0.441323	+	0.897348i	$0.645491\pi$
$104$	18.1092	1.77575
$105$	0	0
$106$	4.75188	0.461543
$107$	−4.52891	−0.437826	−0.218913	−	0.975744i	$-0.570251\pi$
$107$	−4.52891	−0.437826	−0.218913	+	0.975744i	$0.570251\pi$
$108$	0	0
$109$	−1.02744	−0.0984106	−0.0492053	−	0.998789i	$-0.515669\pi$
$109$	−1.02744	−0.0984106	−0.0492053	+	0.998789i	$0.515669\pi$
$110$	0	0
$111$	0	0
$112$	0.872709	0.0824632
$113$	−13.2380	−1.24533	−0.622664	−	0.782490i	$-0.713950\pi$
$113$	−13.2380	−1.24533	−0.622664	+	0.782490i	$0.713950\pi$
$114$	0	0
$115$	−2.07611	−0.193598
$116$	10.7712	1.00008
$117$	0	0
$118$	−3.02399	−0.278381
$119$	−6.65872	−0.610404
$120$	0	0
$121$	0	0
$122$	−4.53396	−0.410486
$123$	0	0
$124$	−4.01344	−0.360418
$125$	5.12978	0.458822
$126$	0	0
$127$	16.2934	1.44581	0.722903	−	0.690949i	$-0.242807\pi$
$127$	16.2934	1.44581	0.722903	+	0.690949i	$0.242807\pi$
$128$	−9.66899	−0.854626
$129$	0	0
$130$	15.1568	1.32934
$131$	−6.19831	−0.541549	−0.270774	−	0.962643i	$-0.587280\pi$
$131$	−6.19831	−0.541549	−0.270774	+	0.962643i	$0.587280\pi$
$132$	0	0
$133$	−5.75106	−0.498680
$134$	7.86785	0.679679
$135$	0	0
$136$	17.3099	1.48431
$137$	−16.0095	−1.36779	−0.683893	−	0.729582i	$-0.739714\pi$
$137$	−16.0095	−1.36779	−0.683893	+	0.729582i	$0.739714\pi$
$138$	0	0
$139$	4.61401	0.391355	0.195678	−	0.980668i	$-0.437309\pi$
$139$	4.61401	0.391355	0.195678	+	0.980668i	$0.437309\pi$
$140$	4.07830	0.344679
$141$	0	0
$142$	0.755706	0.0634174
$143$	0	0
$144$	0	0
$145$	21.6825	1.80063
$146$	2.68575	0.222274
$147$	0	0
$148$	2.45119	0.201487
$149$	7.21841	0.591355	0.295678	−	0.955288i	$-0.404455\pi$
$149$	7.21841	0.591355	0.295678	+	0.955288i	$0.404455\pi$
$150$	0	0
$151$	0.626349	0.0509716	0.0254858	−	0.999675i	$-0.491887\pi$
$151$	0.626349	0.0509716	0.0254858	+	0.999675i	$0.491887\pi$
$152$	14.9503	1.21263
$153$	0	0
$154$	0	0
$155$	−8.07909	−0.648928
$156$	0	0
$157$	−6.14752	−0.490626	−0.245313	−	0.969444i	$-0.578891\pi$
$157$	−6.14752	−0.490626	−0.245313	+	0.969444i	$0.578891\pi$
$158$	−0.508617	−0.0404634
$159$	0	0
$160$	−16.7957	−1.32782
$161$	0.724581	0.0571050
$162$	0	0
$163$	21.6415	1.69510	0.847548	−	0.530719i	$-0.178078\pi$
$163$	21.6415	1.69510	0.847548	+	0.530719i	$0.178078\pi$
$164$	1.59539	0.124579
$165$	0	0
$166$	−10.1937	−0.791186
$167$	23.9947	1.85677	0.928384	−	0.371622i	$-0.121198\pi$
$167$	23.9947	1.85677	0.928384	+	0.371622i	$0.121198\pi$
$168$	0	0
$169$	35.5278	2.73290
$170$	14.4878	1.11117
$171$	0	0
$172$	16.6666	1.27081
$173$	10.8109	0.821935	0.410968	−	0.911650i	$-0.365191\pi$
$173$	10.8109	0.821935	0.410968	+	0.911650i	$0.365191\pi$
$174$	0	0
$175$	3.20966	0.242627
$176$	0	0
$177$	0	0
$178$	−12.6059	−0.944849
$179$	−5.92839	−0.443109	−0.221554	−	0.975148i	$-0.571113\pi$
$179$	−5.92839	−0.443109	−0.221554	+	0.975148i	$0.571113\pi$
$180$	0	0
$181$	−4.84617	−0.360213	−0.180106	−	0.983647i	$-0.557644\pi$
$181$	−4.84617	−0.360213	−0.180106	+	0.983647i	$0.557644\pi$
$182$	−5.28987	−0.392111
$183$	0	0
$184$	−1.88361	−0.138861
$185$	4.93427	0.362775
$186$	0	0
$187$	0	0
$188$	7.38665	0.538727
$189$	0	0
$190$	12.5130	0.907786
$191$	15.6923	1.13546	0.567729	−	0.823216i	$-0.307822\pi$
$191$	15.6923	1.13546	0.567729	+	0.823216i	$0.307822\pi$
$192$	0	0
$193$	−22.7548	−1.63793	−0.818963	−	0.573846i	$-0.805451\pi$
$193$	−22.7548	−1.63793	−0.818963	+	0.573846i	$0.805451\pi$
$194$	7.67873	0.551301
$195$	0	0
$196$	−1.42337	−0.101669
$197$	−15.3025	−1.09026	−0.545129	−	0.838352i	$-0.683519\pi$
$197$	−15.3025	−1.09026	−0.545129	+	0.838352i	$0.683519\pi$
$198$	0	0
$199$	−0.547814	−0.0388335	−0.0194168	−	0.999811i	$-0.506181\pi$
$199$	−0.547814	−0.0388335	−0.0194168	+	0.999811i	$0.506181\pi$
$200$	−8.34376	−0.589993
$201$	0	0
$202$	−4.87779	−0.343200
$203$	−7.56740	−0.531128
$204$	0	0
$205$	3.21153	0.224303
$206$	−6.80228	−0.473937
$207$	0	0
$208$	−6.07945	−0.421534
$209$	0	0
$210$	0	0
$211$	−13.8726	−0.955030	−0.477515	−	0.878624i	$-0.658462\pi$
$211$	−13.8726	−0.955030	−0.477515	+	0.878624i	$0.658462\pi$
$212$	−8.90702	−0.611737
$213$	0	0
$214$	−3.43908	−0.235091
$215$	33.5499	2.28808
$216$	0	0
$217$	2.81968	0.191412
$218$	−0.780198	−0.0528416
$219$	0	0
$220$	0	0
$221$	46.3859	3.12025
$222$	0	0
$223$	−7.60179	−0.509053	−0.254527	−	0.967066i	$-0.581920\pi$
$223$	−7.60179	−0.509053	−0.254527	+	0.967066i	$0.581920\pi$
$224$	5.86186	0.391662
$225$	0	0
$226$	−10.0525	−0.668680
$227$	0.295156	0.0195902	0.00979511	−	0.999952i	$-0.496882\pi$
$227$	0.295156	0.0195902	0.00979511	+	0.999952i	$0.496882\pi$
$228$	0	0
$229$	7.11072	0.469890	0.234945	−	0.972009i	$-0.424509\pi$
$229$	7.11072	0.469890	0.234945	+	0.972009i	$0.424509\pi$
$230$	−1.57652	−0.103953
$231$	0	0
$232$	19.6721	1.29153
$233$	−10.2537	−0.671743	−0.335872	−	0.941908i	$-0.609031\pi$
$233$	−10.2537	−0.671743	−0.335872	+	0.941908i	$0.609031\pi$
$234$	0	0
$235$	14.8694	0.969972
$236$	5.66824	0.368971
$237$	0	0
$238$	−5.05639	−0.327757
$239$	19.8042	1.28103	0.640513	−	0.767947i	$-0.278722\pi$
$239$	19.8042	1.28103	0.640513	+	0.767947i	$0.278722\pi$
$240$	0	0
$241$	−20.0305	−1.29028	−0.645138	−	0.764066i	$-0.723200\pi$
$241$	−20.0305	−1.29028	−0.645138	+	0.764066i	$0.723200\pi$
$242$	0	0
$243$	0	0
$244$	8.49856	0.544064
$245$	−2.86525	−0.183054
$246$	0	0
$247$	40.0630	2.54915
$248$	−7.32999	−0.465455
$249$	0	0
$250$	3.89537	0.246365
$251$	2.69309	0.169986	0.0849932	−	0.996382i	$-0.472913\pi$
$251$	2.69309	0.169986	0.0849932	+	0.996382i	$0.472913\pi$
$252$	0	0
$253$	0	0
$254$	12.3726	0.776327
$255$	0	0
$256$	−12.7540	−0.797126
$257$	−25.3591	−1.58186	−0.790929	−	0.611908i	$-0.790402\pi$
$257$	−25.3591	−1.58186	−0.790929	+	0.611908i	$0.790402\pi$
$258$	0	0
$259$	−1.72211	−0.107007
$260$	−28.4102	−1.76193
$261$	0	0
$262$	−4.70677	−0.290785
$263$	9.04980	0.558034	0.279017	−	0.960286i	$-0.409991\pi$
$263$	9.04980	0.558034	0.279017	+	0.960286i	$0.409991\pi$
$264$	0	0
$265$	−17.9299	−1.10142
$266$	−4.36715	−0.267767
$267$	0	0
$268$	−14.7477	−0.900857
$269$	−2.73289	−0.166627	−0.0833136	−	0.996523i	$-0.526550\pi$
$269$	−2.73289	−0.166627	−0.0833136	+	0.996523i	$0.526550\pi$
$270$	0	0
$271$	17.9520	1.09051	0.545254	−	0.838271i	$-0.316433\pi$
$271$	17.9520	1.09051	0.545254	+	0.838271i	$0.316433\pi$
$272$	−5.81113	−0.352351
$273$	0	0
$274$	−12.1570	−0.734434
$275$	0	0
$276$	0	0
$277$	−11.1503	−0.669958	−0.334979	−	0.942226i	$-0.608729\pi$
$277$	−11.1503	−0.669958	−0.334979	+	0.942226i	$0.608729\pi$
$278$	3.50371	0.210139
$279$	0	0
$280$	7.44844	0.445130
$281$	−3.96239	−0.236377	−0.118188	−	0.992991i	$-0.537709\pi$
$281$	−3.96239	−0.236377	−0.118188	+	0.992991i	$0.537709\pi$
$282$	0	0
$283$	5.74911	0.341749	0.170875	−	0.985293i	$-0.445341\pi$
$283$	5.74911	0.341749	0.170875	+	0.985293i	$0.445341\pi$
$284$	−1.41651	−0.0840545
$285$	0	0
$286$	0	0
$287$	−1.12086	−0.0661621
$288$	0	0
$289$	27.3386	1.60815
$290$	16.4649	0.966852
$291$	0	0
$292$	−5.03422	−0.294606
$293$	−23.2438	−1.35792	−0.678958	−	0.734177i	$-0.737568\pi$
$293$	−23.2438	−1.35792	−0.678958	+	0.734177i	$0.737568\pi$
$294$	0	0
$295$	11.4102	0.664328
$296$	4.47676	0.260206
$297$	0	0
$298$	5.48140	0.317529
$299$	−5.04757	−0.291908
$300$	0	0
$301$	−11.7092	−0.674909
$302$	0.475627	0.0273692
$303$	0	0
$304$	−5.01900	−0.287860
$305$	17.1077	0.979582
$306$	0	0
$307$	3.87940	0.221409	0.110705	−	0.993853i	$-0.464689\pi$
$307$	3.87940	0.221409	0.110705	+	0.993853i	$0.464689\pi$
$308$	0	0
$309$	0	0
$310$	−6.13497	−0.348443
$311$	−3.74658	−0.212449	−0.106225	−	0.994342i	$-0.533876\pi$
$311$	−3.74658	−0.212449	−0.106225	+	0.994342i	$0.533876\pi$
$312$	0	0
$313$	18.0642	1.02105	0.510523	−	0.859864i	$-0.329452\pi$
$313$	18.0642	1.02105	0.510523	+	0.859864i	$0.329452\pi$
$314$	−4.66820	−0.263442
$315$	0	0
$316$	0.953364	0.0536309
$317$	−7.41444	−0.416436	−0.208218	−	0.978082i	$-0.566766\pi$
$317$	−7.41444	−0.416436	−0.208218	+	0.978082i	$0.566766\pi$
$318$	0	0
$319$	0	0
$320$	−7.75298	−0.433405
$321$	0	0
$322$	0.550220	0.0306626
$323$	38.2947	2.13078
$324$	0	0
$325$	−22.3591	−1.24026
$326$	16.4338	0.910183
$327$	0	0
$328$	2.91376	0.160885
$329$	−5.18956	−0.286110
$330$	0	0
$331$	14.5172	0.797937	0.398968	−	0.916965i	$-0.369368\pi$
$331$	14.5172	0.797937	0.398968	+	0.916965i	$0.369368\pi$
$332$	19.1073	1.04865
$333$	0	0
$334$	18.2207	0.996993
$335$	−29.6872	−1.62198
$336$	0	0
$337$	10.5543	0.574930	0.287465	−	0.957791i	$-0.407187\pi$
$337$	10.5543	0.574930	0.287465	+	0.957791i	$0.407187\pi$
$338$	26.9785	1.46744
$339$	0	0
$340$	−27.1563	−1.47276
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	30.4391	1.64117
$345$	0	0
$346$	8.20938	0.441339
$347$	22.5100	1.20840	0.604199	−	0.796833i	$-0.293493\pi$
$347$	22.5100	1.20840	0.604199	+	0.796833i	$0.293493\pi$
$348$	0	0
$349$	−20.7206	−1.10915	−0.554574	−	0.832135i	$-0.687119\pi$
$349$	−20.7206	−1.10915	−0.554574	+	0.832135i	$0.687119\pi$
$350$	2.43729	0.130279
$351$	0	0
$352$	0	0
$353$	−6.01460	−0.320125	−0.160062	−	0.987107i	$-0.551170\pi$
$353$	−6.01460	−0.320125	−0.160062	+	0.987107i	$0.551170\pi$
$354$	0	0
$355$	−2.85145	−0.151339
$356$	23.6287	1.25232
$357$	0	0
$358$	−4.50180	−0.237928
$359$	18.4700	0.974810	0.487405	−	0.873176i	$-0.337944\pi$
$359$	18.4700	0.974810	0.487405	+	0.873176i	$0.337944\pi$
$360$	0	0
$361$	14.0747	0.740775
$362$	−3.68000	−0.193417
$363$	0	0
$364$	9.91544	0.519710
$365$	−10.1339	−0.530434
$366$	0	0
$367$	−4.02990	−0.210359	−0.105180	−	0.994453i	$-0.533542\pi$
$367$	−4.02990	−0.210359	−0.105180	+	0.994453i	$0.533542\pi$
$368$	0.632349	0.0329634
$369$	0	0
$370$	3.74691	0.194792
$371$	6.25771	0.324884
$372$	0	0
$373$	5.06561	0.262287	0.131144	−	0.991363i	$-0.458135\pi$
$373$	5.06561	0.262287	0.131144	+	0.991363i	$0.458135\pi$
$374$	0	0
$375$	0	0
$376$	13.4907	0.695728
$377$	52.7159	2.71501
$378$	0	0
$379$	2.19657	0.112830	0.0564152	−	0.998407i	$-0.482033\pi$
$379$	2.19657	0.112830	0.0564152	+	0.998407i	$0.482033\pi$
$380$	−23.4546	−1.20319
$381$	0	0
$382$	11.9162	0.609685
$383$	24.4857	1.25116	0.625579	−	0.780161i	$-0.284863\pi$
$383$	24.4857	1.25116	0.625579	+	0.780161i	$0.284863\pi$
$384$	0	0
$385$	0	0
$386$	−17.2792	−0.879487
$387$	0	0
$388$	−14.3932	−0.730703
$389$	−34.3208	−1.74013	−0.870066	−	0.492936i	$-0.835924\pi$
$389$	−34.3208	−1.74013	−0.870066	+	0.492936i	$0.835924\pi$
$390$	0	0
$391$	−4.82479	−0.244000
$392$	−2.59958	−0.131299
$393$	0	0
$394$	−11.6202	−0.585415
$395$	1.91913	0.0965618
$396$	0	0
$397$	8.46742	0.424968	0.212484	−	0.977165i	$-0.431845\pi$
$397$	8.46742	0.424968	0.212484	+	0.977165i	$0.431845\pi$
$398$	−0.415990	−0.0208517
$399$	0	0
$400$	2.80110	0.140055
$401$	−28.9079	−1.44359	−0.721797	−	0.692105i	$-0.756684\pi$
$401$	−28.9079	−1.44359	−0.721797	+	0.692105i	$0.756684\pi$
$402$	0	0
$403$	−19.6424	−0.978459
$404$	9.14304	0.454883
$405$	0	0
$406$	−5.74641	−0.285189
$407$	0	0
$408$	0	0
$409$	−1.59556	−0.0788955	−0.0394478	−	0.999222i	$-0.512560\pi$
$409$	−1.59556	−0.0788955	−0.0394478	+	0.999222i	$0.512560\pi$
$410$	2.43872	0.120440
$411$	0	0
$412$	12.7503	0.628164
$413$	−3.98228	−0.195955
$414$	0	0
$415$	38.4632	1.88808
$416$	−40.8348	−2.00209
$417$	0	0
$418$	0	0
$419$	0.224640	0.0109744	0.00548719	−	0.999985i	$-0.498253\pi$
$419$	0.224640	0.0109744	0.00548719	+	0.999985i	$0.498253\pi$
$420$	0	0
$421$	11.5297	0.561924	0.280962	−	0.959719i	$-0.409346\pi$
$421$	11.5297	0.561924	0.280962	+	0.959719i	$0.409346\pi$
$422$	−10.5344	−0.512804
$423$	0	0
$424$	−16.2674	−0.790015
$425$	−21.3722	−1.03670
$426$	0	0
$427$	−5.97074	−0.288944
$428$	6.44630	0.311593
$429$	0	0
$430$	25.4766	1.22859
$431$	−27.8911	−1.34347	−0.671733	−	0.740794i	$-0.734450\pi$
$431$	−27.8911	−1.34347	−0.671733	+	0.740794i	$0.734450\pi$
$432$	0	0
$433$	25.9669	1.24789	0.623944	−	0.781469i	$-0.285529\pi$
$433$	25.9669	1.24789	0.623944	+	0.781469i	$0.285529\pi$
$434$	2.14116	0.102779
$435$	0	0
$436$	1.46242	0.0700372
$437$	−4.16711	−0.199340
$438$	0	0
$439$	23.5551	1.12422	0.562112	−	0.827061i	$-0.309989\pi$
$439$	23.5551	1.12422	0.562112	+	0.827061i	$0.309989\pi$
$440$	0	0
$441$	0	0
$442$	35.2238	1.67542
$443$	−24.8412	−1.18024	−0.590120	−	0.807315i	$-0.700920\pi$
$443$	−24.8412	−1.18024	−0.590120	+	0.807315i	$0.700920\pi$
$444$	0	0
$445$	47.5648	2.25479
$446$	−5.77252	−0.273337
$447$	0	0
$448$	2.70587	0.127840
$449$	18.4127	0.868947	0.434473	−	0.900685i	$-0.356935\pi$
$449$	18.4127	0.868947	0.434473	+	0.900685i	$0.356935\pi$
$450$	0	0
$451$	0	0
$452$	18.8426	0.886279
$453$	0	0
$454$	0.224131	0.0105190
$455$	19.9599	0.935733
$456$	0	0
$457$	4.61369	0.215820	0.107910	−	0.994161i	$-0.465584\pi$
$457$	4.61369	0.215820	0.107910	+	0.994161i	$0.465584\pi$
$458$	5.39962	0.252308
$459$	0	0
$460$	2.95506	0.137780
$461$	−33.5148	−1.56094	−0.780470	−	0.625193i	$-0.785020\pi$
$461$	−33.5148	−1.56094	−0.780470	+	0.625193i	$0.785020\pi$
$462$	0	0
$463$	−16.5338	−0.768392	−0.384196	−	0.923252i	$-0.625521\pi$
$463$	−16.5338	−0.768392	−0.384196	+	0.923252i	$0.625521\pi$
$464$	−6.60414	−0.306590
$465$	0	0
$466$	−7.78630	−0.360693
$467$	19.4446	0.899788	0.449894	−	0.893082i	$-0.351462\pi$
$467$	19.4446	0.899788	0.449894	+	0.893082i	$0.351462\pi$
$468$	0	0
$469$	10.3611	0.478432
$470$	11.2913	0.520827
$471$	0	0
$472$	10.3522	0.476501
$473$	0	0
$474$	0	0
$475$	−18.4589	−0.846954
$476$	9.47781	0.434415
$477$	0	0
$478$	15.0386	0.687848
$479$	−39.2757	−1.79455	−0.897276	−	0.441469i	$-0.854457\pi$
$479$	−39.2757	−1.79455	−0.897276	+	0.441469i	$0.854457\pi$
$480$	0	0
$481$	11.9965	0.546995
$482$	−15.2104	−0.692815
$483$	0	0
$484$	0	0
$485$	−28.9736	−1.31562
$486$	0	0
$487$	−17.1008	−0.774911	−0.387455	−	0.921889i	$-0.626646\pi$
$487$	−17.1008	−0.774911	−0.387455	+	0.921889i	$0.626646\pi$
$488$	15.5214	0.702621
$489$	0	0
$490$	−2.17577	−0.0982911
$491$	34.7859	1.56986	0.784932	−	0.619582i	$-0.212698\pi$
$491$	34.7859	1.56986	0.784932	+	0.619582i	$0.212698\pi$
$492$	0	0
$493$	50.3892	2.26942
$494$	30.4224	1.36877
$495$	0	0
$496$	2.46076	0.110491
$497$	0.995183	0.0446401
$498$	0	0
$499$	15.6577	0.700935	0.350467	−	0.936575i	$-0.386023\pi$
$499$	15.6577	0.700935	0.350467	+	0.936575i	$0.386023\pi$
$500$	−7.30157	−0.326536
$501$	0	0
$502$	2.04504	0.0912744
$503$	−21.8981	−0.976386	−0.488193	−	0.872736i	$-0.662344\pi$
$503$	−21.8981	−0.976386	−0.488193	+	0.872736i	$0.662344\pi$
$504$	0	0
$505$	18.4050	0.819012
$506$	0	0
$507$	0	0
$508$	−23.1915	−1.02896
$509$	−26.8438	−1.18983	−0.594915	−	0.803789i	$-0.702814\pi$
$509$	−26.8438	−1.18983	−0.594915	+	0.803789i	$0.702814\pi$
$510$	0	0
$511$	3.53684	0.156461
$512$	9.65305	0.426609
$513$	0	0
$514$	−19.2568	−0.849380
$515$	25.6665	1.13100
$516$	0	0
$517$	0	0
$518$	−1.30771	−0.0574574
$519$	0	0
$520$	−51.8873	−2.27541
$521$	−15.9670	−0.699528	−0.349764	−	0.936838i	$-0.613738\pi$
$521$	−15.9670	−0.699528	−0.349764	+	0.936838i	$0.613738\pi$
$522$	0	0
$523$	13.2422	0.579040	0.289520	−	0.957172i	$-0.406504\pi$
$523$	13.2422	0.579040	0.289520	+	0.957172i	$0.406504\pi$
$524$	8.82247	0.385411
$525$	0	0
$526$	6.87208	0.299637
$527$	−18.7755	−0.817873
$528$	0	0
$529$	−22.4750	−0.977173
$530$	−13.6153	−0.591411
$531$	0	0
$532$	8.18588	0.354903
$533$	7.80810	0.338206
$534$	0	0
$535$	12.9764	0.561021
$536$	−26.9345	−1.16339
$537$	0	0
$538$	−2.07526	−0.0894706
$539$	0	0
$540$	0	0
$541$	−25.5950	−1.10042	−0.550209	−	0.835027i	$-0.685452\pi$
$541$	−25.5950	−1.10042	−0.550209	+	0.835027i	$0.685452\pi$
$542$	13.6321	0.585550
$543$	0	0
$544$	−39.0325	−1.67351
$545$	2.94386	0.126101
$546$	0	0
$547$	30.1551	1.28934	0.644670	−	0.764461i	$-0.276995\pi$
$547$	30.1551	1.28934	0.644670	+	0.764461i	$0.276995\pi$
$548$	22.7874	0.973431
$549$	0	0
$550$	0	0
$551$	43.5206	1.85404
$552$	0	0
$553$	−0.669794	−0.0284826
$554$	−8.46715	−0.359735
$555$	0	0
$556$	−6.56743	−0.278521
$557$	0.733579	0.0310827	0.0155414	−	0.999879i	$-0.495053\pi$
$557$	0.733579	0.0310827	0.0155414	+	0.999879i	$0.495053\pi$
$558$	0	0
$559$	81.5688	3.44999
$560$	−2.50053	−0.105667
$561$	0	0
$562$	−3.00890	−0.126923
$563$	20.9036	0.880982	0.440491	−	0.897757i	$-0.354804\pi$
$563$	20.9036	0.880982	0.440491	+	0.897757i	$0.354804\pi$
$564$	0	0
$565$	37.9302	1.59574
$566$	4.36566	0.183502
$567$	0	0
$568$	−2.58706	−0.108551
$569$	−28.2526	−1.18441	−0.592205	−	0.805787i	$-0.701742\pi$
$569$	−28.2526	−1.18441	−0.592205	+	0.805787i	$0.701742\pi$
$570$	0	0
$571$	38.7512	1.62169	0.810843	−	0.585264i	$-0.199009\pi$
$571$	38.7512	1.62169	0.810843	+	0.585264i	$0.199009\pi$
$572$	0	0
$573$	0	0
$574$	−0.851138	−0.0355258
$575$	2.32566	0.0969866
$576$	0	0
$577$	12.7178	0.529450	0.264725	−	0.964324i	$-0.414719\pi$
$577$	12.7178	0.529450	0.264725	+	0.964324i	$0.414719\pi$
$578$	20.7599	0.863499
$579$	0	0
$580$	−30.8622	−1.28148
$581$	−13.4240	−0.556922
$582$	0	0
$583$	0	0
$584$	−9.19430	−0.380463
$585$	0	0
$586$	−17.6505	−0.729134
$587$	−7.77003	−0.320704	−0.160352	−	0.987060i	$-0.551263\pi$
$587$	−7.77003	−0.320704	−0.160352	+	0.987060i	$0.551263\pi$
$588$	0	0
$589$	−16.2162	−0.668175
$590$	8.66450	0.356712
$591$	0	0
$592$	−1.50290	−0.0617688
$593$	25.2949	1.03874	0.519368	−	0.854551i	$-0.326168\pi$
$593$	25.2949	1.03874	0.519368	+	0.854551i	$0.326168\pi$
$594$	0	0
$595$	19.0789	0.782159
$596$	−10.2744	−0.420858
$597$	0	0
$598$	−3.83294	−0.156740
$599$	11.9194	0.487014	0.243507	−	0.969899i	$-0.421702\pi$
$599$	11.9194	0.487014	0.243507	+	0.969899i	$0.421702\pi$
$600$	0	0
$601$	21.0106	0.857041	0.428521	−	0.903532i	$-0.359035\pi$
$601$	21.0106	0.857041	0.428521	+	0.903532i	$0.359035\pi$
$602$	−8.89157	−0.362393
$603$	0	0
$604$	−0.891525	−0.0362756
$605$	0	0
$606$	0	0
$607$	25.7817	1.04645	0.523223	−	0.852196i	$-0.324729\pi$
$607$	25.7817	1.04645	0.523223	+	0.852196i	$0.324729\pi$
$608$	−33.7119	−1.36720
$609$	0	0
$610$	12.9909	0.525987
$611$	36.1514	1.46253
$612$	0	0
$613$	−31.9173	−1.28913	−0.644563	−	0.764551i	$-0.722961\pi$
$613$	−31.9173	−1.28913	−0.644563	+	0.764551i	$0.722961\pi$
$614$	2.94588	0.118886
$615$	0	0
$616$	0	0
$617$	−13.9772	−0.562701	−0.281350	−	0.959605i	$-0.590782\pi$
$617$	−13.9772	−0.562701	−0.281350	+	0.959605i	$0.590782\pi$
$618$	0	0
$619$	22.4745	0.903326	0.451663	−	0.892189i	$-0.350831\pi$
$619$	22.4745	0.903326	0.451663	+	0.892189i	$0.350831\pi$
$620$	11.4995	0.461832
$621$	0	0
$622$	−2.84502	−0.114075
$623$	−16.6006	−0.665088
$624$	0	0
$625$	−30.7464	−1.22986
$626$	13.7173	0.548252
$627$	0	0
$628$	8.75018	0.349170
$629$	11.4671	0.457221
$630$	0	0
$631$	−41.0383	−1.63371	−0.816854	−	0.576845i	$-0.804284\pi$
$631$	−41.0383	−1.63371	−0.816854	+	0.576845i	$0.804284\pi$
$632$	1.74118	0.0692606
$633$	0	0
$634$	−5.63025	−0.223606
$635$	−46.6847	−1.85262
$636$	0	0
$637$	−6.96619	−0.276010
$638$	0	0
$639$	0	0
$640$	27.7041	1.09510
$641$	−13.8318	−0.546323	−0.273162	−	0.961968i	$-0.588069\pi$
$641$	−13.8318	−0.546323	−0.273162	+	0.961968i	$0.588069\pi$
$642$	0	0
$643$	−33.5125	−1.32161	−0.660803	−	0.750560i	$-0.729784\pi$
$643$	−33.5125	−1.32161	−0.660803	+	0.750560i	$0.729784\pi$
$644$	−1.03135	−0.0406407
$645$	0	0
$646$	29.0796	1.14412
$647$	8.58616	0.337557	0.168778	−	0.985654i	$-0.446018\pi$
$647$	8.58616	0.337557	0.168778	+	0.985654i	$0.446018\pi$
$648$	0	0
$649$	0	0
$650$	−16.9787	−0.665957
$651$	0	0
$652$	−30.8039	−1.20637
$653$	−47.4478	−1.85678	−0.928388	−	0.371613i	$-0.878805\pi$
$653$	−47.4478	−1.85678	−0.928388	+	0.371613i	$0.878805\pi$
$654$	0	0
$655$	17.7597	0.693929
$656$	−0.978182	−0.0381916
$657$	0	0
$658$	−3.94076	−0.153627
$659$	−47.4501	−1.84839	−0.924197	−	0.381916i	$-0.875264\pi$
$659$	−47.4501	−1.84839	−0.924197	+	0.381916i	$0.875264\pi$
$660$	0	0
$661$	−24.3627	−0.947599	−0.473800	−	0.880633i	$-0.657118\pi$
$661$	−24.3627	−0.947599	−0.473800	+	0.880633i	$0.657118\pi$
$662$	11.0238	0.428453
$663$	0	0
$664$	34.8968	1.35426
$665$	16.4782	0.638998
$666$	0	0
$667$	−5.48320	−0.212310
$668$	−34.1533	−1.32143
$669$	0	0
$670$	−22.5433	−0.870926
$671$	0	0
$672$	0	0
$673$	31.9467	1.23146	0.615728	−	0.787959i	$-0.288862\pi$
$673$	31.9467	1.23146	0.615728	+	0.787959i	$0.288862\pi$
$674$	8.01456	0.308709
$675$	0	0
$676$	−50.5690	−1.94496
$677$	−1.01464	−0.0389957	−0.0194979	−	0.999810i	$-0.506207\pi$
$677$	−1.01464	−0.0389957	−0.0194979	+	0.999810i	$0.506207\pi$
$678$	0	0
$679$	10.1121	0.388065
$680$	−49.5971	−1.90196
$681$	0	0
$682$	0	0
$683$	0.904878	0.0346242	0.0173121	−	0.999850i	$-0.494489\pi$
$683$	0.904878	0.0346242	0.0173121	+	0.999850i	$0.494489\pi$
$684$	0	0
$685$	45.8713	1.75265
$686$	0.759363	0.0289926
$687$	0	0
$688$	−10.2188	−0.389587
$689$	−43.5924	−1.66074
$690$	0	0
$691$	−21.4535	−0.816130	−0.408065	−	0.912953i	$-0.633796\pi$
$691$	−21.4535	−0.816130	−0.408065	+	0.912953i	$0.633796\pi$
$692$	−15.3878	−0.584958
$693$	0	0
$694$	17.0932	0.648851
$695$	−13.2203	−0.501474
$696$	0	0
$697$	7.46347	0.282699
$698$	−15.7345	−0.595558
$699$	0	0
$700$	−4.56852	−0.172674
$701$	−19.0750	−0.720453	−0.360227	−	0.932865i	$-0.617301\pi$
$701$	−19.0750	−0.720453	−0.360227	+	0.932865i	$0.617301\pi$
$702$	0	0
$703$	9.90396	0.373535
$704$	0	0
$705$	0	0
$706$	−4.56727	−0.171891
$707$	−6.42353	−0.241582
$708$	0	0
$709$	−9.71578	−0.364884	−0.182442	−	0.983217i	$-0.558400\pi$
$709$	−9.71578	−0.364884	−0.182442	+	0.983217i	$0.558400\pi$
$710$	−2.16529	−0.0812617
$711$	0	0
$712$	43.1545	1.61728
$713$	2.04309	0.0765142
$714$	0	0
$715$	0	0
$716$	8.43828	0.315353
$717$	0	0
$718$	14.0255	0.523425
$719$	−12.2204	−0.455743	−0.227871	−	0.973691i	$-0.573177\pi$
$719$	−12.2204	−0.455743	−0.227871	+	0.973691i	$0.573177\pi$
$720$	0	0
$721$	−8.95787	−0.333609
$722$	10.6878	0.397760
$723$	0	0
$724$	6.89788	0.256358
$725$	−24.2888	−0.902062
$726$	0	0
$727$	0.0225849	0.000837628 0	0.000418814	−	1.00000i	$-0.499867\pi$
$727$	0.0225849	0.000837628 0	0.000418814		1.00000i	$0.499867\pi$
$728$	18.1092	0.671170
$729$	0	0
$730$	−7.69534	−0.284817
$731$	77.9686	2.88377
$732$	0	0
$733$	5.28222	0.195103	0.0975516	−	0.995230i	$-0.468899\pi$
$733$	5.28222	0.195103	0.0975516	+	0.995230i	$0.468899\pi$
$734$	−3.06016	−0.112953
$735$	0	0
$736$	4.24740	0.156561
$737$	0	0
$738$	0	0
$739$	−36.0831	−1.32734	−0.663670	−	0.748025i	$-0.731002\pi$
$739$	−36.0831	−1.32734	−0.663670	+	0.748025i	$0.731002\pi$
$740$	−7.02329	−0.258181
$741$	0	0
$742$	4.75188	0.174447
$743$	−34.2049	−1.25486	−0.627429	−	0.778674i	$-0.715893\pi$
$743$	−34.2049	−1.25486	−0.627429	+	0.778674i	$0.715893\pi$
$744$	0	0
$745$	−20.6825	−0.757750
$746$	3.84664	0.140835
$747$	0	0
$748$	0	0
$749$	−4.52891	−0.165483
$750$	0	0
$751$	7.33500	0.267658	0.133829	−	0.991004i	$-0.457273\pi$
$751$	7.33500	0.267658	0.133829	+	0.991004i	$0.457273\pi$
$752$	−4.52898	−0.165155
$753$	0	0
$754$	40.0306	1.45783
$755$	−1.79465	−0.0653139
$756$	0	0
$757$	−34.8941	−1.26825	−0.634124	−	0.773231i	$-0.718639\pi$
$757$	−34.8941	−1.26825	−0.634124	+	0.773231i	$0.718639\pi$
$758$	1.66800	0.0605844
$759$	0	0
$760$	−42.8365	−1.55384
$761$	2.28870	0.0829655	0.0414827	−	0.999139i	$-0.486792\pi$
$761$	2.28870	0.0829655	0.0414827	+	0.999139i	$0.486792\pi$
$762$	0	0
$763$	−1.02744	−0.0371957
$764$	−22.3360	−0.808087
$765$	0	0
$766$	18.5935	0.671811
$767$	27.7413	1.00168
$768$	0	0
$769$	−9.59977	−0.346176	−0.173088	−	0.984906i	$-0.555375\pi$
$769$	−9.59977	−0.346176	−0.173088	+	0.984906i	$0.555375\pi$
$770$	0	0
$771$	0	0
$772$	32.3885	1.16569
$773$	13.7761	0.495492	0.247746	−	0.968825i	$-0.420310\pi$
$773$	13.7761	0.495492	0.247746	+	0.968825i	$0.420310\pi$
$774$	0	0
$775$	9.05021	0.325093
$776$	−26.2871	−0.943652
$777$	0	0
$778$	−26.0619	−0.934365
$779$	6.44612	0.230956
$780$	0	0
$781$	0	0
$782$	−3.66377	−0.131016
$783$	0	0
$784$	0.872709	0.0311682
$785$	17.6142	0.628677
$786$	0	0
$787$	−2.24272	−0.0799443	−0.0399721	−	0.999201i	$-0.512727\pi$
$787$	−2.24272	−0.0799443	−0.0399721	+	0.999201i	$0.512727\pi$
$788$	21.7811	0.775919
$789$	0	0
$790$	1.45732	0.0518490
$791$	−13.2380	−0.470689
$792$	0	0
$793$	41.5933	1.47702
$794$	6.42985	0.228187
$795$	0	0
$796$	0.779741	0.0276372
$797$	−18.1909	−0.644353	−0.322177	−	0.946680i	$-0.604414\pi$
$797$	−18.1909	−0.644353	−0.322177	+	0.946680i	$0.604414\pi$
$798$	0	0
$799$	34.5558	1.22250
$800$	18.8146	0.665195
$801$	0	0
$802$	−21.9516	−0.775139
$803$	0	0
$804$	0	0
$805$	−2.07611	−0.0731731
$806$	−14.9157	−0.525384
$807$	0	0
$808$	16.6985	0.587450
$809$	31.2009	1.09696	0.548482	−	0.836162i	$-0.315206\pi$
$809$	31.2009	1.09696	0.548482	+	0.836162i	$0.315206\pi$
$810$	0	0
$811$	−25.2002	−0.884899	−0.442449	−	0.896793i	$-0.645890\pi$
$811$	−25.2002	−0.884899	−0.442449	+	0.896793i	$0.645890\pi$
$812$	10.7712	0.377995
$813$	0	0
$814$	0	0
$815$	−62.0084	−2.17206
$816$	0	0
$817$	67.3406	2.35595
$818$	−1.21161	−0.0423630
$819$	0	0
$820$	−4.57119	−0.159633
$821$	11.8863	0.414836	0.207418	−	0.978252i	$-0.433494\pi$
$821$	11.8863	0.414836	0.207418	+	0.978252i	$0.433494\pi$
$822$	0	0
$823$	7.62865	0.265918	0.132959	−	0.991122i	$-0.457552\pi$
$823$	7.62865	0.265918	0.132959	+	0.991122i	$0.457552\pi$
$824$	23.2867	0.811231
$825$	0	0
$826$	−3.02399	−0.105218
$827$	11.4070	0.396661	0.198330	−	0.980135i	$-0.436448\pi$
$827$	11.4070	0.396661	0.198330	+	0.980135i	$0.436448\pi$
$828$	0	0
$829$	−44.6474	−1.55067	−0.775334	−	0.631552i	$-0.782418\pi$
$829$	−44.6474	−1.55067	−0.775334	+	0.631552i	$0.782418\pi$
$830$	29.2075	1.01381
$831$	0	0
$832$	−18.8496	−0.653491
$833$	−6.65872	−0.230711
$834$	0	0
$835$	−68.7509	−2.37922
$836$	0	0
$837$	0	0
$838$	0.170583	0.00589270
$839$	−10.0848	−0.348166	−0.174083	−	0.984731i	$-0.555696\pi$
$839$	−10.0848	−0.348166	−0.174083	+	0.984731i	$0.555696\pi$
$840$	0	0
$841$	28.2656	0.974675
$842$	8.75525	0.301726
$843$	0	0
$844$	19.7458	0.679679
$845$	−101.796	−3.50188
$846$	0	0
$847$	0	0
$848$	5.46116	0.187537
$849$	0	0
$850$	−16.2293	−0.556659
$851$	−1.24781	−0.0427743
$852$	0	0
$853$	−19.0126	−0.650981	−0.325490	−	0.945545i	$-0.605529\pi$
$853$	−19.0126	−0.650981	−0.325490	+	0.945545i	$0.605529\pi$
$854$	−4.53396	−0.155149
$855$	0	0
$856$	11.7733	0.402401
$857$	29.8781	1.02062	0.510308	−	0.859992i	$-0.329531\pi$
$857$	29.8781	1.02062	0.510308	+	0.859992i	$0.329531\pi$
$858$	0	0
$859$	−16.6976	−0.569714	−0.284857	−	0.958570i	$-0.591946\pi$
$859$	−16.6976	−0.569714	−0.284857	+	0.958570i	$0.591946\pi$
$860$	−47.7538	−1.62839
$861$	0	0
$862$	−21.1795	−0.721375
$863$	−16.7819	−0.571261	−0.285631	−	0.958340i	$-0.592203\pi$
$863$	−16.7819	−0.571261	−0.285631	+	0.958340i	$0.592203\pi$
$864$	0	0
$865$	−30.9758	−1.05321
$866$	19.7183	0.670055
$867$	0	0
$868$	−4.01344	−0.136225
$869$	0	0
$870$	0	0
$871$	−72.1774	−2.44564
$872$	2.67090	0.0904482
$873$	0	0
$874$	−3.16435	−0.107036
$875$	5.12978	0.173418
$876$	0	0
$877$	5.53815	0.187010	0.0935050	−	0.995619i	$-0.470193\pi$
$877$	5.53815	0.187010	0.0935050	+	0.995619i	$0.470193\pi$
$878$	17.8869	0.603653
$879$	0	0
$880$	0	0
$881$	6.77615	0.228294	0.114147	−	0.993464i	$-0.463586\pi$
$881$	6.77615	0.228294	0.114147	+	0.993464i	$0.463586\pi$
$882$	0	0
$883$	−23.9604	−0.806331	−0.403165	−	0.915127i	$-0.632090\pi$
$883$	−23.9604	−0.806331	−0.403165	+	0.915127i	$0.632090\pi$
$884$	−66.0242	−2.22063
$885$	0	0
$886$	−18.8635	−0.633731
$887$	27.6881	0.929676	0.464838	−	0.885396i	$-0.346113\pi$
$887$	27.6881	0.929676	0.464838	+	0.885396i	$0.346113\pi$
$888$	0	0
$889$	16.2934	0.546463
$890$	36.1189	1.21071
$891$	0	0
$892$	10.8201	0.362285
$893$	29.8455	0.998741
$894$	0	0
$895$	16.9863	0.567790
$896$	−9.66899	−0.323018
$897$	0	0
$898$	13.9819	0.466582
$899$	−21.3377	−0.711651
$900$	0	0
$901$	−41.6684	−1.38817
$902$	0	0
$903$	0	0
$904$	34.4133	1.14457
$905$	13.8855	0.461569
$906$	0	0
$907$	14.9565	0.496623	0.248311	−	0.968680i	$-0.420124\pi$
$907$	14.9565	0.496623	0.248311	+	0.968680i	$0.420124\pi$
$908$	−0.420116	−0.0139420
$909$	0	0
$910$	15.1568	0.502443
$911$	−25.1916	−0.834635	−0.417318	−	0.908761i	$-0.637030\pi$
$911$	−25.1916	−0.834635	−0.417318	+	0.908761i	$0.637030\pi$
$912$	0	0
$913$	0	0
$914$	3.50347	0.115884
$915$	0	0
$916$	−10.1212	−0.334413
$917$	−6.19831	−0.204686
$918$	0	0
$919$	26.7796	0.883378	0.441689	−	0.897168i	$-0.354379\pi$
$919$	26.7796	0.883378	0.441689	+	0.897168i	$0.354379\pi$
$920$	5.39700	0.177934
$921$	0	0
$922$	−25.4499	−0.838148
$923$	−6.93263	−0.228190
$924$	0	0
$925$	−5.52738	−0.181739
$926$	−12.5552	−0.412589
$927$	0	0
$928$	−44.3591	−1.45616
$929$	5.65027	0.185379	0.0926896	−	0.995695i	$-0.470454\pi$
$929$	5.65027	0.185379	0.0926896	+	0.995695i	$0.470454\pi$
$930$	0	0
$931$	−5.75106	−0.188483
$932$	14.5948	0.478069
$933$	0	0
$934$	14.7655	0.483142
$935$	0	0
$936$	0	0
$937$	−45.2807	−1.47925	−0.739627	−	0.673017i	$-0.764998\pi$
$937$	−45.2807	−1.47925	−0.739627	+	0.673017i	$0.764998\pi$
$938$	7.86785	0.256894
$939$	0	0
$940$	−21.1646	−0.690313
$941$	−32.6876	−1.06558	−0.532792	−	0.846246i	$-0.678857\pi$
$941$	−32.6876	−1.06558	−0.532792	+	0.846246i	$0.678857\pi$
$942$	0	0
$943$	−0.812152	−0.0264473
$944$	−3.47537	−0.113114
$945$	0	0
$946$	0	0
$947$	−39.8004	−1.29334	−0.646669	−	0.762771i	$-0.723838\pi$
$947$	−39.8004	−1.29334	−0.646669	+	0.762771i	$0.723838\pi$
$948$	0	0
$949$	−24.6383	−0.799793
$950$	−14.0170	−0.454773
$951$	0	0
$952$	17.3099	0.561016
$953$	−12.5174	−0.405479	−0.202739	−	0.979233i	$-0.564984\pi$
$953$	−12.5174	−0.405479	−0.202739	+	0.979233i	$0.564984\pi$
$954$	0	0
$955$	−44.9625	−1.45495
$956$	−28.1886	−0.911686
$957$	0	0
$958$	−29.8245	−0.963587
$959$	−16.0095	−0.516974
$960$	0	0
$961$	−23.0494	−0.743529
$962$	9.10973	0.293710
$963$	0	0
$964$	28.5107	0.918269
$965$	65.1982	2.09880
$966$	0	0
$967$	12.7139	0.408850	0.204425	−	0.978882i	$-0.434468\pi$
$967$	12.7139	0.408850	0.204425	+	0.978882i	$0.434468\pi$
$968$	0	0
$969$	0	0
$970$	−22.0015	−0.706425
$971$	22.6126	0.725673	0.362837	−	0.931853i	$-0.381808\pi$
$971$	22.6126	0.725673	0.362837	+	0.931853i	$0.381808\pi$
$972$	0	0
$973$	4.61401	0.147918
$974$	−12.9857	−0.416089
$975$	0	0
$976$	−5.21072	−0.166791
$977$	17.9694	0.574893	0.287446	−	0.957797i	$-0.407194\pi$
$977$	17.9694	0.574893	0.287446	+	0.957797i	$0.407194\pi$
$978$	0	0
$979$	0	0
$980$	4.07830	0.130277
$981$	0	0
$982$	26.4151	0.842940
$983$	−48.2065	−1.53755	−0.768775	−	0.639520i	$-0.779133\pi$
$983$	−48.2065	−1.53755	−0.768775	+	0.639520i	$0.779133\pi$
$984$	0	0
$985$	43.8455	1.39703
$986$	38.2637	1.21857
$987$	0	0
$988$	−57.0243	−1.81419
$989$	−8.48430	−0.269785
$990$	0	0
$991$	−6.62493	−0.210448	−0.105224	−	0.994449i	$-0.533556\pi$
$991$	−6.62493	−0.210448	−0.105224	+	0.994449i	$0.533556\pi$
$992$	16.5286	0.524783
$993$	0	0
$994$	0.755706	0.0239695
$995$	1.56962	0.0497605
$996$	0	0
$997$	−28.1261	−0.890762	−0.445381	−	0.895341i	$-0.646932\pi$
$997$	−28.1261	−0.890762	−0.445381	+	0.895341i	$0.646932\pi$
$998$	11.8899	0.376368
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.dc.1.10		16
3.2	odd	2	inner	7623.2.a.dc.1.7		16
11.5	even	5		693.2.m.k.190.4	✓	32
11.9	even	5		693.2.m.k.631.4	yes	32
11.10	odd	2		7623.2.a.db.1.7		16
33.5	odd	10		693.2.m.k.190.5	yes	32
33.20	odd	10		693.2.m.k.631.5	yes	32
33.32	even	2		7623.2.a.db.1.10		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
693.2.m.k.190.4	✓	32	11.5	even	5
693.2.m.k.190.5	yes	32	33.5	odd	10
693.2.m.k.631.4	yes	32	11.9	even	5
693.2.m.k.631.5	yes	32	33.20	odd	10
7623.2.a.db.1.7		16	11.10	odd	2
7623.2.a.db.1.10		16	33.32	even	2
7623.2.a.dc.1.7		16	3.2	odd	2	inner
7623.2.a.dc.1.10		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)

\(f(q)\)	\(=\)	\(q+0.759363 q^{2} -1.42337 q^{4} -2.86525 q^{5} +1.00000 q^{7} -2.59958 q^{8} +O(q^{10})\)	\(q+0.759363 q^{2} -1.42337 q^{4} -2.86525 q^{5} +1.00000 q^{7} -2.59958 q^{8} -2.17577 q^{10} -6.96619 q^{13} +0.759363 q^{14} +0.872709 q^{16} -6.65872 q^{17} -5.75106 q^{19} +4.07830 q^{20} +0.724581 q^{23} +3.20966 q^{25} -5.28987 q^{26} -1.42337 q^{28} -7.56740 q^{29} +2.81968 q^{31} +5.86186 q^{32} -5.05639 q^{34} -2.86525 q^{35} -1.72211 q^{37} -4.36715 q^{38} +7.44844 q^{40} -1.12086 q^{41} -11.7092 q^{43} +0.550220 q^{46} -5.18956 q^{47} +1.00000 q^{49} +2.43729 q^{50} +9.91544 q^{52} +6.25771 q^{53} -2.59958 q^{56} -5.74641 q^{58} -3.98228 q^{59} -5.97074 q^{61} +2.14116 q^{62} +2.70587 q^{64} +19.9599 q^{65} +10.3611 q^{67} +9.47781 q^{68} -2.17577 q^{70} +0.995183 q^{71} +3.53684 q^{73} -1.30771 q^{74} +8.18588 q^{76} -0.669794 q^{79} -2.50053 q^{80} -0.851138 q^{82} -13.4240 q^{83} +19.0789 q^{85} -8.89157 q^{86} -16.6006 q^{89} -6.96619 q^{91} -1.03135 q^{92} -3.94076 q^{94} +16.4782 q^{95} +10.1121 q^{97} +0.759363 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 20 q^{4} + 16 q^{7}+O(q^{10}) \) 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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