LMFDB - Embedded newform 7623.2.a.da.1.2

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 22x^{10} + 181x^{8} - 692x^{6} + 1240x^{4} - 936x^{2} + 244 $ x^12 - 22x^10 + 181x^8 - 692x^6 + 1240x^4 - 936*x^2 + 244
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.41714$ of defining polynomial
Character	$\chi$	$=$	7623.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.41714 q^{2} +3.84255 q^{4} -4.01054 q^{5} +1.00000 q^{7} -4.45369 q^{8} +O(q^{10})$	$q-2.41714 q^{2} +3.84255 q^{4} -4.01054 q^{5} +1.00000 q^{7} -4.45369 q^{8} +9.69401 q^{10} -0.690498 q^{13} -2.41714 q^{14} +3.08008 q^{16} -5.78000 q^{17} -1.80100 q^{19} -15.4107 q^{20} -2.03655 q^{23} +11.0844 q^{25} +1.66903 q^{26} +3.84255 q^{28} -7.98110 q^{29} -10.4676 q^{31} +1.46241 q^{32} +13.9711 q^{34} -4.01054 q^{35} -7.44454 q^{37} +4.35325 q^{38} +17.8617 q^{40} +5.66440 q^{41} +3.44102 q^{43} +4.92263 q^{46} -7.32704 q^{47} +1.00000 q^{49} -26.7925 q^{50} -2.65327 q^{52} +1.79792 q^{53} -4.45369 q^{56} +19.2914 q^{58} -10.9949 q^{59} -6.95019 q^{61} +25.3017 q^{62} -9.69500 q^{64} +2.76927 q^{65} -11.8438 q^{67} -22.2099 q^{68} +9.69401 q^{70} +13.5523 q^{71} +7.20252 q^{73} +17.9945 q^{74} -6.92041 q^{76} -5.33847 q^{79} -12.3528 q^{80} -13.6916 q^{82} +8.59090 q^{83} +23.1809 q^{85} -8.31742 q^{86} -13.0692 q^{89} -0.690498 q^{91} -7.82555 q^{92} +17.7104 q^{94} +7.22296 q^{95} -9.50344 q^{97} -2.41714 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 20 q^{4} + 12 q^{7}+O(q^{10}) $ 12 * q + 20 * q^4 + 12 * q^7	$ 12 q + 20 q^{4} + 12 q^{7} + 20 q^{10} + 20 q^{13} + 28 q^{16} + 12 q^{19} + 32 q^{25} + 20 q^{28} - 16 q^{31} - 24 q^{34} + 4 q^{37} + 48 q^{40} + 16 q^{43} + 24 q^{46} + 12 q^{49} + 96 q^{52} + 20 q^{58} + 44 q^{61} + 76 q^{64} + 20 q^{70} + 52 q^{73} - 8 q^{79} - 68 q^{82} + 72 q^{85} + 20 q^{91} + 20 q^{94} - 32 q^{97}+O(q^{100}) $ 12 * q + 20 * q^4 + 12 * q^7 + 20 * q^10 + 20 * q^13 + 28 * q^16 + 12 * q^19 + 32 * q^25 + 20 * q^28 - 16 * q^31 - 24 * q^34 + 4 * q^37 + 48 * q^40 + 16 * q^43 + 24 * q^46 + 12 * q^49 + 96 * q^52 + 20 * q^58 + 44 * q^61 + 76 * q^64 + 20 * q^70 + 52 * q^73 - 8 * q^79 - 68 * q^82 + 72 * q^85 + 20 * q^91 + 20 * q^94 - 32 * 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 $

$2$	−2.41714	−1.70917	−0.854587	−	0.519309i	$-0.826189\pi$
$2$	−2.41714	−1.70917	−0.854587	+	0.519309i	$0.826189\pi$
$3$	0	0
$3$	0	0
$4$	3.84255	1.92127
$5$	−4.01054	−1.79357	−0.896783	−	0.442470i	$-0.854102\pi$
$5$	−4.01054	−1.79357	−0.896783	+	0.442470i	$0.854102\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−4.45369	−1.57462
$9$	0	0
$10$	9.69401	3.06552
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−0.690498	−0.191510	−0.0957549	−	0.995405i	$-0.530526\pi$
$13$	−0.690498	−0.191510	−0.0957549	+	0.995405i	$0.530526\pi$
$14$	−2.41714	−0.646007
$15$	0	0
$16$	3.08008	0.770020
$17$	−5.78000	−1.40186	−0.700928	−	0.713232i	$-0.747231\pi$
$17$	−5.78000	−1.40186	−0.700928	+	0.713232i	$0.747231\pi$
$18$	0	0
$19$	−1.80100	−0.413177	−0.206588	−	0.978428i	$-0.566236\pi$
$19$	−1.80100	−0.413177	−0.206588	+	0.978428i	$0.566236\pi$
$20$	−15.4107	−3.44593
$21$	0	0
$22$	0	0
$23$	−2.03655	−0.424651	−0.212325	−	0.977199i	$-0.568104\pi$
$23$	−2.03655	−0.424651	−0.212325	+	0.977199i	$0.568104\pi$
$24$	0	0
$25$	11.0844	2.21688
$26$	1.66903	0.327323
$27$	0	0
$28$	3.84255	0.726173
$29$	−7.98110	−1.48205	−0.741027	−	0.671475i	$-0.765661\pi$
$29$	−7.98110	−1.48205	−0.741027	+	0.671475i	$0.765661\pi$
$30$	0	0
$31$	−10.4676	−1.88004	−0.940020	−	0.341120i	$-0.889194\pi$
$31$	−10.4676	−1.88004	−0.940020	+	0.341120i	$0.889194\pi$
$32$	1.46241	0.258520
$33$	0	0
$34$	13.9711	2.39602
$35$	−4.01054	−0.677904
$36$	0	0
$37$	−7.44454	−1.22387	−0.611937	−	0.790906i	$-0.709610\pi$
$37$	−7.44454	−1.22387	−0.611937	+	0.790906i	$0.709610\pi$
$38$	4.35325	0.706191
$39$	0	0
$40$	17.8617	2.82418
$41$	5.66440	0.884631	0.442316	−	0.896859i	$-0.354157\pi$
$41$	5.66440	0.884631	0.442316	+	0.896859i	$0.354157\pi$
$42$	0	0
$43$	3.44102	0.524751	0.262376	−	0.964966i	$-0.415494\pi$
$43$	3.44102	0.524751	0.262376	+	0.964966i	$0.415494\pi$
$44$	0	0
$45$	0	0
$46$	4.92263	0.725802
$47$	−7.32704	−1.06876	−0.534379	−	0.845245i	$-0.679455\pi$
$47$	−7.32704	−1.06876	−0.534379	+	0.845245i	$0.679455\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−26.7925	−3.78903
$51$	0	0
$52$	−2.65327	−0.367943
$53$	1.79792	0.246963	0.123481	−	0.992347i	$-0.460594\pi$
$53$	1.79792	0.246963	0.123481	+	0.992347i	$0.460594\pi$
$54$	0	0
$55$	0	0
$56$	−4.45369	−0.595149
$57$	0	0
$58$	19.2914	2.53309
$59$	−10.9949	−1.43141	−0.715704	−	0.698403i	$-0.753894\pi$
$59$	−10.9949	−1.43141	−0.715704	+	0.698403i	$0.753894\pi$
$60$	0	0
$61$	−6.95019	−0.889881	−0.444940	−	0.895560i	$-0.646775\pi$
$61$	−6.95019	−0.889881	−0.444940	+	0.895560i	$0.646775\pi$
$62$	25.3017	3.21331
$63$	0	0
$64$	−9.69500	−1.21187
$65$	2.76927	0.343485
$66$	0	0
$67$	−11.8438	−1.44696	−0.723478	−	0.690347i	$-0.757458\pi$
$67$	−11.8438	−1.44696	−0.723478	+	0.690347i	$0.757458\pi$
$68$	−22.2099	−2.69335
$69$	0	0
$70$	9.69401	1.15866
$71$	13.5523	1.60836	0.804181	−	0.594385i	$-0.202604\pi$
$71$	13.5523	1.60836	0.804181	+	0.594385i	$0.202604\pi$
$72$	0	0
$73$	7.20252	0.842991	0.421496	−	0.906830i	$-0.361505\pi$
$73$	7.20252	0.842991	0.421496	+	0.906830i	$0.361505\pi$
$74$	17.9945	2.09181
$75$	0	0
$76$	−6.92041	−0.793826
$77$	0	0
$78$	0	0
$79$	−5.33847	−0.600625	−0.300313	−	0.953841i	$-0.597091\pi$
$79$	−5.33847	−0.600625	−0.300313	+	0.953841i	$0.597091\pi$
$80$	−12.3528	−1.38108
$81$	0	0
$82$	−13.6916	−1.51199
$83$	8.59090	0.942974	0.471487	−	0.881873i	$-0.343717\pi$
$83$	8.59090	0.942974	0.471487	+	0.881873i	$0.343717\pi$
$84$	0	0
$85$	23.1809	2.51432
$86$	−8.31742	−0.896891
$87$	0	0
$88$	0	0
$89$	−13.0692	−1.38533	−0.692665	−	0.721260i	$-0.743563\pi$
$89$	−13.0692	−1.38533	−0.692665	+	0.721260i	$0.743563\pi$
$90$	0	0
$91$	−0.690498	−0.0723839
$92$	−7.82555	−0.815870
$93$	0	0
$94$	17.7104	1.82669
$95$	7.22296	0.741060
$96$	0	0
$97$	−9.50344	−0.964928	−0.482464	−	0.875916i	$-0.660258\pi$
$97$	−9.50344	−0.964928	−0.482464	+	0.875916i	$0.660258\pi$
$98$	−2.41714	−0.244168
$99$	0	0
$100$	42.5924	4.25924
$101$	9.70420	0.965604	0.482802	−	0.875729i	$-0.339619\pi$
$101$	9.70420	0.965604	0.482802	+	0.875729i	$0.339619\pi$
$102$	0	0
$103$	7.32767	0.722017	0.361009	−	0.932563i	$-0.382433\pi$
$103$	7.32767	0.722017	0.361009	+	0.932563i	$0.382433\pi$
$104$	3.07526	0.301554
$105$	0	0
$106$	−4.34581	−0.422103
$107$	−12.4348	−1.20212	−0.601058	−	0.799205i	$-0.705254\pi$
$107$	−12.4348	−1.20212	−0.601058	+	0.799205i	$0.705254\pi$
$108$	0	0
$109$	12.5955	1.20643	0.603213	−	0.797580i	$-0.293887\pi$
$109$	12.5955	1.20643	0.603213	+	0.797580i	$0.293887\pi$
$110$	0	0
$111$	0	0
$112$	3.08008	0.291040
$113$	−11.8186	−1.11180	−0.555900	−	0.831249i	$-0.687626\pi$
$113$	−11.8186	−1.11180	−0.555900	+	0.831249i	$0.687626\pi$
$114$	0	0
$115$	8.16767	0.761639
$116$	−30.6678	−2.84743
$117$	0	0
$118$	26.5761	2.44653
$119$	−5.78000	−0.529852
$120$	0	0
$121$	0	0
$122$	16.7996	1.52096
$123$	0	0
$124$	−40.2223	−3.61207
$125$	−24.4017	−2.18256
$126$	0	0
$127$	−9.59200	−0.851152	−0.425576	−	0.904923i	$-0.639928\pi$
$127$	−9.59200	−0.851152	−0.425576	+	0.904923i	$0.639928\pi$
$128$	20.5093	1.81278
$129$	0	0
$130$	−6.69370	−0.587076
$131$	1.48293	0.129564	0.0647819	−	0.997899i	$-0.479365\pi$
$131$	1.48293	0.129564	0.0647819	+	0.997899i	$0.479365\pi$
$132$	0	0
$133$	−1.80100	−0.156166
$134$	28.6282	2.47310
$135$	0	0
$136$	25.7423	2.20739
$137$	17.6212	1.50548	0.752740	−	0.658318i	$-0.228732\pi$
$137$	17.6212	1.50548	0.752740	+	0.658318i	$0.228732\pi$
$138$	0	0
$139$	−2.36252	−0.200387	−0.100193	−	0.994968i	$-0.531946\pi$
$139$	−2.36252	−0.200387	−0.100193	+	0.994968i	$0.531946\pi$
$140$	−15.4107	−1.30244
$141$	0	0
$142$	−32.7577	−2.74897
$143$	0	0
$144$	0	0
$145$	32.0085	2.65816
$146$	−17.4095	−1.44082
$147$	0	0
$148$	−28.6060	−2.35140
$149$	−18.4024	−1.50758	−0.753791	−	0.657114i	$-0.771777\pi$
$149$	−18.4024	−1.50758	−0.753791	+	0.657114i	$0.771777\pi$
$150$	0	0
$151$	5.35792	0.436021	0.218011	−	0.975946i	$-0.430043\pi$
$151$	5.35792	0.436021	0.218011	+	0.975946i	$0.430043\pi$
$152$	8.02107	0.650595
$153$	0	0
$154$	0	0
$155$	41.9808	3.37198
$156$	0	0
$157$	−5.46584	−0.436222	−0.218111	−	0.975924i	$-0.569989\pi$
$157$	−5.46584	−0.436222	−0.218111	+	0.975924i	$0.569989\pi$
$158$	12.9038	1.02657
$159$	0	0
$160$	−5.86505	−0.463673
$161$	−2.03655	−0.160503
$162$	0	0
$163$	−17.4528	−1.36701	−0.683505	−	0.729946i	$-0.739545\pi$
$163$	−17.4528	−1.36701	−0.683505	+	0.729946i	$0.739545\pi$
$164$	21.7657	1.69962
$165$	0	0
$166$	−20.7654	−1.61171
$167$	−11.8534	−0.917247	−0.458623	−	0.888631i	$-0.651657\pi$
$167$	−11.8534	−0.917247	−0.458623	+	0.888631i	$0.651657\pi$
$168$	0	0
$169$	−12.5232	−0.963324
$170$	−56.0314	−4.29741
$171$	0	0
$172$	13.2223	1.00819
$173$	−15.1122	−1.14896	−0.574481	−	0.818518i	$-0.694796\pi$
$173$	−15.1122	−1.14896	−0.574481	+	0.818518i	$0.694796\pi$
$174$	0	0
$175$	11.0844	0.837902
$176$	0	0
$177$	0	0
$178$	31.5900	2.36777
$179$	−20.3747	−1.52287	−0.761436	−	0.648240i	$-0.775506\pi$
$179$	−20.3747	−1.52287	−0.761436	+	0.648240i	$0.775506\pi$
$180$	0	0
$181$	−25.0574	−1.86250	−0.931251	−	0.364379i	$-0.881281\pi$
$181$	−25.0574	−1.86250	−0.931251	+	0.364379i	$0.881281\pi$
$182$	1.66903	0.123717
$183$	0	0
$184$	9.07017	0.668662
$185$	29.8566	2.19510
$186$	0	0
$187$	0	0
$188$	−28.1545	−2.05338
$189$	0	0
$190$	−17.4589	−1.26660
$191$	7.44391	0.538622	0.269311	−	0.963053i	$-0.413204\pi$
$191$	7.44391	0.538622	0.269311	+	0.963053i	$0.413204\pi$
$192$	0	0
$193$	23.0191	1.65695	0.828477	−	0.560023i	$-0.189208\pi$
$193$	23.0191	1.65695	0.828477	+	0.560023i	$0.189208\pi$
$194$	22.9711	1.64923
$195$	0	0
$196$	3.84255	0.274468
$197$	6.34515	0.452073	0.226037	−	0.974119i	$-0.427423\pi$
$197$	6.34515	0.452073	0.226037	+	0.974119i	$0.427423\pi$
$198$	0	0
$199$	−21.3877	−1.51614	−0.758069	−	0.652175i	$-0.773857\pi$
$199$	−21.3877	−1.51614	−0.758069	+	0.652175i	$0.773857\pi$
$200$	−49.3665	−3.49074
$201$	0	0
$202$	−23.4564	−1.65039
$203$	−7.98110	−0.560164
$204$	0	0
$205$	−22.7173	−1.58665
$206$	−17.7120	−1.23405
$207$	0	0
$208$	−2.12679	−0.147466
$209$	0	0
$210$	0	0
$211$	−3.37623	−0.232429	−0.116215	−	0.993224i	$-0.537076\pi$
$211$	−3.37623	−0.232429	−0.116215	+	0.993224i	$0.537076\pi$
$212$	6.90858	0.474484
$213$	0	0
$214$	30.0566	2.05463
$215$	−13.8003	−0.941176
$216$	0	0
$217$	−10.4676	−0.710588
$218$	−30.4449	−2.06199
$219$	0	0
$220$	0	0
$221$	3.99108	0.268469
$222$	0	0
$223$	−1.94696	−0.130378	−0.0651889	−	0.997873i	$-0.520765\pi$
$223$	−1.94696	−0.130378	−0.0651889	+	0.997873i	$0.520765\pi$
$224$	1.46241	0.0977113
$225$	0	0
$226$	28.5672	1.90026
$227$	−4.59774	−0.305163	−0.152581	−	0.988291i	$-0.548759\pi$
$227$	−4.59774	−0.305163	−0.152581	+	0.988291i	$0.548759\pi$
$228$	0	0
$229$	1.42275	0.0940178	0.0470089	−	0.998894i	$-0.485031\pi$
$229$	1.42275	0.0940178	0.0470089	+	0.998894i	$0.485031\pi$
$230$	−19.7424	−1.30177
$231$	0	0
$232$	35.5454	2.33367
$233$	9.75278	0.638926	0.319463	−	0.947599i	$-0.396497\pi$
$233$	9.75278	0.638926	0.319463	+	0.947599i	$0.396497\pi$
$234$	0	0
$235$	29.3854	1.91689
$236$	−42.2483	−2.75013
$237$	0	0
$238$	13.9711	0.905609
$239$	−0.842040	−0.0544670	−0.0272335	−	0.999629i	$-0.508670\pi$
$239$	−0.842040	−0.0544670	−0.0272335	+	0.999629i	$0.508670\pi$
$240$	0	0
$241$	26.8301	1.72828	0.864141	−	0.503250i	$-0.167863\pi$
$241$	26.8301	1.72828	0.864141	+	0.503250i	$0.167863\pi$
$242$	0	0
$243$	0	0
$244$	−26.7064	−1.70971
$245$	−4.01054	−0.256224
$246$	0	0
$247$	1.24358	0.0791274
$248$	46.6195	2.96034
$249$	0	0
$250$	58.9823	3.73037
$251$	13.8130	0.871867	0.435933	−	0.899979i	$-0.356418\pi$
$251$	13.8130	0.871867	0.435933	+	0.899979i	$0.356418\pi$
$252$	0	0
$253$	0	0
$254$	23.1852	1.45477
$255$	0	0
$256$	−30.1838	−1.88649
$257$	4.15222	0.259008	0.129504	−	0.991579i	$-0.458661\pi$
$257$	4.15222	0.259008	0.129504	+	0.991579i	$0.458661\pi$
$258$	0	0
$259$	−7.44454	−0.462581
$260$	10.6410	0.659930
$261$	0	0
$262$	−3.58443	−0.221447
$263$	6.34086	0.390994	0.195497	−	0.980704i	$-0.437368\pi$
$263$	6.34086	0.390994	0.195497	+	0.980704i	$0.437368\pi$
$264$	0	0
$265$	−7.21061	−0.442945
$266$	4.35325	0.266915
$267$	0	0
$268$	−45.5106	−2.78000
$269$	−7.00736	−0.427246	−0.213623	−	0.976916i	$-0.568526\pi$
$269$	−7.00736	−0.427246	−0.213623	+	0.976916i	$0.568526\pi$
$270$	0	0
$271$	−28.1196	−1.70815	−0.854073	−	0.520153i	$-0.825875\pi$
$271$	−28.1196	−1.70815	−0.854073	+	0.520153i	$0.825875\pi$
$272$	−17.8029	−1.07946
$273$	0	0
$274$	−42.5928	−2.57313
$275$	0	0
$276$	0	0
$277$	−16.1096	−0.967932	−0.483966	−	0.875087i	$-0.660804\pi$
$277$	−16.1096	−0.967932	−0.483966	+	0.875087i	$0.660804\pi$
$278$	5.71054	0.342495
$279$	0	0
$280$	17.8617	1.06744
$281$	−9.29098	−0.554254	−0.277127	−	0.960833i	$-0.589382\pi$
$281$	−9.29098	−0.554254	−0.277127	+	0.960833i	$0.589382\pi$
$282$	0	0
$283$	−15.9612	−0.948796	−0.474398	−	0.880310i	$-0.657334\pi$
$283$	−15.9612	−0.948796	−0.474398	+	0.880310i	$0.657334\pi$
$284$	52.0753	3.09010
$285$	0	0
$286$	0	0
$287$	5.66440	0.334359
$288$	0	0
$289$	16.4084	0.965202
$290$	−77.3689	−4.54326
$291$	0	0
$292$	27.6760	1.61962
$293$	7.55261	0.441228	0.220614	−	0.975361i	$-0.429194\pi$
$293$	7.55261	0.441228	0.220614	+	0.975361i	$0.429194\pi$
$294$	0	0
$295$	44.0953	2.56733
$296$	33.1557	1.92713
$297$	0	0
$298$	44.4811	2.57672
$299$	1.40624	0.0813247
$300$	0	0
$301$	3.44102	0.198337
$302$	−12.9508	−0.745236
$303$	0	0
$304$	−5.54721	−0.318154
$305$	27.8740	1.59606
$306$	0	0
$307$	−0.903714	−0.0515777	−0.0257888	−	0.999667i	$-0.508210\pi$
$307$	−0.903714	−0.0515777	−0.0257888	+	0.999667i	$0.508210\pi$
$308$	0	0
$309$	0	0
$310$	−101.473	−5.76329
$311$	−6.62277	−0.375543	−0.187771	−	0.982213i	$-0.560126\pi$
$311$	−6.62277	−0.375543	−0.187771	+	0.982213i	$0.560126\pi$
$312$	0	0
$313$	−9.09175	−0.513896	−0.256948	−	0.966425i	$-0.582717\pi$
$313$	−9.09175	−0.513896	−0.256948	+	0.966425i	$0.582717\pi$
$314$	13.2117	0.745578
$315$	0	0
$316$	−20.5133	−1.15397
$317$	20.9949	1.17919	0.589596	−	0.807698i	$-0.299287\pi$
$317$	20.9949	1.17919	0.589596	+	0.807698i	$0.299287\pi$
$318$	0	0
$319$	0	0
$320$	38.8822	2.17358
$321$	0	0
$322$	4.92263	0.274327
$323$	10.4098	0.579214
$324$	0	0
$325$	−7.65376	−0.424554
$326$	42.1858	2.33646
$327$	0	0
$328$	−25.2275	−1.39296
$329$	−7.32704	−0.403953
$330$	0	0
$331$	34.8916	1.91782	0.958909	−	0.283715i	$-0.0915670\pi$
$331$	34.8916	1.91782	0.958909	+	0.283715i	$0.0915670\pi$
$332$	33.0110	1.81171
$333$	0	0
$334$	28.6514	1.56773
$335$	47.5002	2.59521
$336$	0	0
$337$	22.6494	1.23379	0.616897	−	0.787044i	$-0.288390\pi$
$337$	22.6494	1.23379	0.616897	+	0.787044i	$0.288390\pi$
$338$	30.2703	1.64649
$339$	0	0
$340$	89.0738	4.83070
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	−15.3252	−0.826282
$345$	0	0
$346$	36.5283	1.96377
$347$	−15.5034	−0.832268	−0.416134	−	0.909303i	$-0.636615\pi$
$347$	−15.5034	−0.832268	−0.416134	+	0.909303i	$0.636615\pi$
$348$	0	0
$349$	−16.3823	−0.876927	−0.438463	−	0.898749i	$-0.644477\pi$
$349$	−16.3823	−0.876927	−0.438463	+	0.898749i	$0.644477\pi$
$350$	−26.7925	−1.43212
$351$	0	0
$352$	0	0
$353$	−19.9280	−1.06066	−0.530330	−	0.847792i	$-0.677932\pi$
$353$	−19.9280	−1.06066	−0.530330	+	0.847792i	$0.677932\pi$
$354$	0	0
$355$	−54.3520	−2.88470
$356$	−50.2189	−2.66160
$357$	0	0
$358$	49.2483	2.60285
$359$	−21.2044	−1.11913	−0.559563	−	0.828788i	$-0.689031\pi$
$359$	−21.2044	−1.11913	−0.559563	+	0.828788i	$0.689031\pi$
$360$	0	0
$361$	−15.7564	−0.829285
$362$	60.5671	3.18334
$363$	0	0
$364$	−2.65327	−0.139069
$365$	−28.8860	−1.51196
$366$	0	0
$367$	−3.65357	−0.190715	−0.0953575	−	0.995443i	$-0.530399\pi$
$367$	−3.65357	−0.190715	−0.0953575	+	0.995443i	$0.530399\pi$
$368$	−6.27274	−0.326989
$369$	0	0
$370$	−72.1675	−3.75181
$371$	1.79792	0.0933432
$372$	0	0
$373$	19.1169	0.989837	0.494919	−	0.868939i	$-0.335198\pi$
$373$	19.1169	0.989837	0.494919	+	0.868939i	$0.335198\pi$
$374$	0	0
$375$	0	0
$376$	32.6323	1.68288
$377$	5.51094	0.283828
$378$	0	0
$379$	33.3753	1.71438	0.857188	−	0.515003i	$-0.172209\pi$
$379$	33.3753	1.71438	0.857188	+	0.515003i	$0.172209\pi$
$380$	27.7546	1.42378
$381$	0	0
$382$	−17.9929	−0.920599
$383$	34.3561	1.75552	0.877758	−	0.479105i	$-0.159039\pi$
$383$	34.3561	1.75552	0.877758	+	0.479105i	$0.159039\pi$
$384$	0	0
$385$	0	0
$386$	−55.6404	−2.83202
$387$	0	0
$388$	−36.5174	−1.85389
$389$	−9.01029	−0.456840	−0.228420	−	0.973563i	$-0.573356\pi$
$389$	−9.01029	−0.456840	−0.228420	+	0.973563i	$0.573356\pi$
$390$	0	0
$391$	11.7713	0.595299
$392$	−4.45369	−0.224945
$393$	0	0
$394$	−15.3371	−0.772672
$395$	21.4101	1.07726
$396$	0	0
$397$	−4.43406	−0.222539	−0.111269	−	0.993790i	$-0.535492\pi$
$397$	−4.43406	−0.222539	−0.111269	+	0.993790i	$0.535492\pi$
$398$	51.6971	2.59134
$399$	0	0
$400$	34.1408	1.70704
$401$	6.23579	0.311401	0.155700	−	0.987804i	$-0.450237\pi$
$401$	6.23579	0.311401	0.155700	+	0.987804i	$0.450237\pi$
$402$	0	0
$403$	7.22787	0.360046
$404$	37.2889	1.85519
$405$	0	0
$406$	19.2914	0.957417
$407$	0	0
$408$	0	0
$409$	−13.1954	−0.652471	−0.326236	−	0.945289i	$-0.605780\pi$
$409$	−13.1954	−0.652471	−0.326236	+	0.945289i	$0.605780\pi$
$410$	54.9108	2.71185
$411$	0	0
$412$	28.1569	1.38719
$413$	−10.9949	−0.541022
$414$	0	0
$415$	−34.4541	−1.69129
$416$	−1.00979	−0.0495091
$417$	0	0
$418$	0	0
$419$	0.552355	0.0269843	0.0134922	−	0.999909i	$-0.495705\pi$
$419$	0.552355	0.0269843	0.0134922	+	0.999909i	$0.495705\pi$
$420$	0	0
$421$	−6.61758	−0.322521	−0.161260	−	0.986912i	$-0.551556\pi$
$421$	−6.61758	−0.322521	−0.161260	+	0.986912i	$0.551556\pi$
$422$	8.16081	0.397262
$423$	0	0
$424$	−8.00737	−0.388872
$425$	−64.0679	−3.10775
$426$	0	0
$427$	−6.95019	−0.336343
$428$	−47.7813	−2.30960
$429$	0	0
$430$	33.3573	1.60863
$431$	0.520604	0.0250766	0.0125383	−	0.999921i	$-0.496009\pi$
$431$	0.520604	0.0250766	0.0125383	+	0.999921i	$0.496009\pi$
$432$	0	0
$433$	0.949704	0.0456398	0.0228199	−	0.999740i	$-0.492736\pi$
$433$	0.949704	0.0456398	0.0228199	+	0.999740i	$0.492736\pi$
$434$	25.3017	1.21452
$435$	0	0
$436$	48.3987	2.31788
$437$	3.66782	0.175456
$438$	0	0
$439$	−36.6885	−1.75105	−0.875524	−	0.483174i	$-0.839484\pi$
$439$	−36.6885	−1.75105	−0.875524	+	0.483174i	$0.839484\pi$
$440$	0	0
$441$	0	0
$442$	−9.64699	−0.458860
$443$	10.2133	0.485249	0.242624	−	0.970120i	$-0.421992\pi$
$443$	10.2133	0.485249	0.242624	+	0.970120i	$0.421992\pi$
$444$	0	0
$445$	52.4144	2.48468
$446$	4.70606	0.222838
$447$	0	0
$448$	−9.69500	−0.458046
$449$	−20.0890	−0.948059	−0.474030	−	0.880509i	$-0.657201\pi$
$449$	−20.0890	−0.948059	−0.474030	+	0.880509i	$0.657201\pi$
$450$	0	0
$451$	0	0
$452$	−45.4135	−2.13607
$453$	0	0
$454$	11.1134	0.521576
$455$	2.76927	0.129825
$456$	0	0
$457$	−0.566344	−0.0264925	−0.0132462	−	0.999912i	$-0.504217\pi$
$457$	−0.566344	−0.0264925	−0.0132462	+	0.999912i	$0.504217\pi$
$458$	−3.43897	−0.160693
$459$	0	0
$460$	31.3847	1.46332
$461$	−12.2429	−0.570209	−0.285105	−	0.958496i	$-0.592028\pi$
$461$	−12.2429	−0.570209	−0.285105	+	0.958496i	$0.592028\pi$
$462$	0	0
$463$	−23.5815	−1.09593	−0.547963	−	0.836502i	$-0.684597\pi$
$463$	−23.5815	−1.09593	−0.547963	+	0.836502i	$0.684597\pi$
$464$	−24.5824	−1.14121
$465$	0	0
$466$	−23.5738	−1.09204
$467$	14.4991	0.670937	0.335468	−	0.942051i	$-0.391105\pi$
$467$	14.4991	0.670937	0.335468	+	0.942051i	$0.391105\pi$
$468$	0	0
$469$	−11.8438	−0.546898
$470$	−71.0284	−3.27630
$471$	0	0
$472$	48.9677	2.25392
$473$	0	0
$474$	0	0
$475$	−19.9630	−0.915964
$476$	−22.2099	−1.01799
$477$	0	0
$478$	2.03533	0.0930936
$479$	29.8824	1.36536	0.682681	−	0.730717i	$-0.260814\pi$
$479$	29.8824	1.36536	0.682681	+	0.730717i	$0.260814\pi$
$480$	0	0
$481$	5.14044	0.234384
$482$	−64.8521	−2.95393
$483$	0	0
$484$	0	0
$485$	38.1139	1.73066
$486$	0	0
$487$	23.4828	1.06411	0.532053	−	0.846711i	$-0.321421\pi$
$487$	23.4828	1.06411	0.532053	+	0.846711i	$0.321421\pi$
$488$	30.9540	1.40122
$489$	0	0
$490$	9.69401	0.437931
$491$	−14.1496	−0.638561	−0.319281	−	0.947660i	$-0.603441\pi$
$491$	−14.1496	−0.638561	−0.319281	+	0.947660i	$0.603441\pi$
$492$	0	0
$493$	46.1308	2.07763
$494$	−3.00591	−0.135242
$495$	0	0
$496$	−32.2411	−1.44767
$497$	13.5523	0.607903
$498$	0	0
$499$	4.41009	0.197423	0.0987114	−	0.995116i	$-0.468528\pi$
$499$	4.41009	0.197423	0.0987114	+	0.995116i	$0.468528\pi$
$500$	−93.7648	−4.19329
$501$	0	0
$502$	−33.3878	−1.49017
$503$	13.9458	0.621812	0.310906	−	0.950441i	$-0.399368\pi$
$503$	13.9458	0.621812	0.310906	+	0.950441i	$0.399368\pi$
$504$	0	0
$505$	−38.9191	−1.73188
$506$	0	0
$507$	0	0
$508$	−36.8577	−1.63530
$509$	−10.1933	−0.451812	−0.225906	−	0.974149i	$-0.572534\pi$
$509$	−10.1933	−0.451812	−0.225906	+	0.974149i	$0.572534\pi$
$510$	0	0
$511$	7.20252	0.318621
$512$	31.9398	1.41155
$513$	0	0
$514$	−10.0365	−0.442690
$515$	−29.3879	−1.29499
$516$	0	0
$517$	0	0
$518$	17.9945	0.790631
$519$	0	0
$520$	−12.3335	−0.540858
$521$	−0.311793	−0.0136599	−0.00682994	−	0.999977i	$-0.502174\pi$
$521$	−0.311793	−0.0136599	−0.00682994	+	0.999977i	$0.502174\pi$
$522$	0	0
$523$	−1.83936	−0.0804296	−0.0402148	−	0.999191i	$-0.512804\pi$
$523$	−1.83936	−0.0804296	−0.0402148	+	0.999191i	$0.512804\pi$
$524$	5.69821	0.248928
$525$	0	0
$526$	−15.3267	−0.668277
$527$	60.5029	2.63555
$528$	0	0
$529$	−18.8525	−0.819672
$530$	17.4290	0.757069
$531$	0	0
$532$	−6.92041	−0.300038
$533$	−3.91126	−0.169416
$534$	0	0
$535$	49.8702	2.15608
$536$	52.7488	2.27840
$537$	0	0
$538$	16.9377	0.730238
$539$	0	0
$540$	0	0
$541$	1.05220	0.0452374	0.0226187	−	0.999744i	$-0.492800\pi$
$541$	1.05220	0.0452374	0.0226187	+	0.999744i	$0.492800\pi$
$542$	67.9690	2.91952
$543$	0	0
$544$	−8.45273	−0.362408
$545$	−50.5146	−2.16381
$546$	0	0
$547$	46.4795	1.98732	0.993659	−	0.112434i	$-0.0358647\pi$
$547$	46.4795	1.98732	0.993659	+	0.112434i	$0.0358647\pi$
$548$	67.7103	2.89244
$549$	0	0
$550$	0	0
$551$	14.3739	0.612350
$552$	0	0
$553$	−5.33847	−0.227015
$554$	38.9391	1.65436
$555$	0	0
$556$	−9.07811	−0.384997
$557$	−11.5830	−0.490789	−0.245395	−	0.969423i	$-0.578918\pi$
$557$	−11.5830	−0.490789	−0.245395	+	0.969423i	$0.578918\pi$
$558$	0	0
$559$	−2.37602	−0.100495
$560$	−12.3528	−0.522000
$561$	0	0
$562$	22.4576	0.947316
$563$	35.4172	1.49266	0.746328	−	0.665578i	$-0.231815\pi$
$563$	35.4172	1.49266	0.746328	+	0.665578i	$0.231815\pi$
$564$	0	0
$565$	47.3989	1.99409
$566$	38.5805	1.62166
$567$	0	0
$568$	−60.3577	−2.53255
$569$	−36.2759	−1.52077	−0.760383	−	0.649475i	$-0.774989\pi$
$569$	−36.2759	−1.52077	−0.760383	+	0.649475i	$0.774989\pi$
$570$	0	0
$571$	−7.31960	−0.306316	−0.153158	−	0.988202i	$-0.548944\pi$
$571$	−7.31960	−0.306316	−0.153158	+	0.988202i	$0.548944\pi$
$572$	0	0
$573$	0	0
$574$	−13.6916	−0.571478
$575$	−22.5740	−0.941400
$576$	0	0
$577$	14.7700	0.614883	0.307441	−	0.951567i	$-0.400527\pi$
$577$	14.7700	0.614883	0.307441	+	0.951567i	$0.400527\pi$
$578$	−39.6614	−1.64970
$579$	0	0
$580$	122.994	5.10706
$581$	8.59090	0.356411
$582$	0	0
$583$	0	0
$584$	−32.0778	−1.32739
$585$	0	0
$586$	−18.2557	−0.754136
$587$	9.27769	0.382931	0.191466	−	0.981499i	$-0.438676\pi$
$587$	9.27769	0.382931	0.191466	+	0.981499i	$0.438676\pi$
$588$	0	0
$589$	18.8521	0.776788
$590$	−106.584	−4.38801
$591$	0	0
$592$	−22.9298	−0.942407
$593$	−22.1174	−0.908252	−0.454126	−	0.890937i	$-0.650048\pi$
$593$	−22.1174	−0.908252	−0.454126	+	0.890937i	$0.650048\pi$
$594$	0	0
$595$	23.1809	0.950325
$596$	−70.7120	−2.89648
$597$	0	0
$598$	−3.39906	−0.138998
$599$	5.17361	0.211388	0.105694	−	0.994399i	$-0.466294\pi$
$599$	5.17361	0.211388	0.105694	+	0.994399i	$0.466294\pi$
$600$	0	0
$601$	9.56657	0.390228	0.195114	−	0.980781i	$-0.437492\pi$
$601$	9.56657	0.390228	0.195114	+	0.980781i	$0.437492\pi$
$602$	−8.31742	−0.338993
$603$	0	0
$604$	20.5881	0.837716
$605$	0	0
$606$	0	0
$607$	24.6208	0.999328	0.499664	−	0.866219i	$-0.333457\pi$
$607$	24.6208	0.999328	0.499664	+	0.866219i	$0.333457\pi$
$608$	−2.63379	−0.106814
$609$	0	0
$610$	−67.3753	−2.72794
$611$	5.05931	0.204678
$612$	0	0
$613$	−39.7281	−1.60460	−0.802301	−	0.596920i	$-0.796391\pi$
$613$	−39.7281	−1.60460	−0.802301	+	0.596920i	$0.796391\pi$
$614$	2.18440	0.0881552
$615$	0	0
$616$	0	0
$617$	−35.9804	−1.44852	−0.724258	−	0.689529i	$-0.757817\pi$
$617$	−35.9804	−1.44852	−0.724258	+	0.689529i	$0.757817\pi$
$618$	0	0
$619$	−0.447180	−0.0179737	−0.00898683	−	0.999960i	$-0.502861\pi$
$619$	−0.447180	−0.0179737	−0.00898683	+	0.999960i	$0.502861\pi$
$620$	161.313	6.47849
$621$	0	0
$622$	16.0081	0.641868
$623$	−13.0692	−0.523605
$624$	0	0
$625$	42.4420	1.69768
$626$	21.9760	0.878337
$627$	0	0
$628$	−21.0028	−0.838101
$629$	43.0295	1.71570
$630$	0	0
$631$	−6.09051	−0.242459	−0.121230	−	0.992624i	$-0.538684\pi$
$631$	−6.09051	−0.242459	−0.121230	+	0.992624i	$0.538684\pi$
$632$	23.7759	0.945754
$633$	0	0
$634$	−50.7476	−2.01544
$635$	38.4691	1.52660
$636$	0	0
$637$	−0.690498	−0.0273585
$638$	0	0
$639$	0	0
$640$	−82.2534	−3.25135
$641$	35.5955	1.40594	0.702969	−	0.711220i	$-0.251857\pi$
$641$	35.5955	1.40594	0.702969	+	0.711220i	$0.251857\pi$
$642$	0	0
$643$	−29.2126	−1.15203	−0.576017	−	0.817438i	$-0.695394\pi$
$643$	−29.2126	−1.15203	−0.576017	+	0.817438i	$0.695394\pi$
$644$	−7.82555	−0.308370
$645$	0	0
$646$	−25.1618	−0.989978
$647$	−9.44881	−0.371471	−0.185736	−	0.982600i	$-0.559467\pi$
$647$	−9.44881	−0.371471	−0.185736	+	0.982600i	$0.559467\pi$
$648$	0	0
$649$	0	0
$650$	18.5002	0.725637
$651$	0	0
$652$	−67.0633	−2.62640
$653$	−24.2269	−0.948073	−0.474036	−	0.880505i	$-0.657204\pi$
$653$	−24.2269	−0.948073	−0.474036	+	0.880505i	$0.657204\pi$
$654$	0	0
$655$	−5.94733	−0.232381
$656$	17.4468	0.681184
$657$	0	0
$658$	17.7104	0.690425
$659$	−19.2004	−0.747943	−0.373971	−	0.927440i	$-0.622004\pi$
$659$	−19.2004	−0.747943	−0.373971	+	0.927440i	$0.622004\pi$
$660$	0	0
$661$	6.95611	0.270561	0.135281	−	0.990807i	$-0.456806\pi$
$661$	6.95611	0.270561	0.135281	+	0.990807i	$0.456806\pi$
$662$	−84.3378	−3.27788
$663$	0	0
$664$	−38.2612	−1.48482
$665$	7.22296	0.280094
$666$	0	0
$667$	16.2539	0.629355
$668$	−45.5474	−1.76228
$669$	0	0
$670$	−114.814	−4.43567
$671$	0	0
$672$	0	0
$673$	49.2066	1.89677	0.948387	−	0.317114i	$-0.102714\pi$
$673$	49.2066	1.89677	0.948387	+	0.317114i	$0.102714\pi$
$674$	−54.7468	−2.10877
$675$	0	0
$676$	−48.1210	−1.85081
$677$	−14.8594	−0.571093	−0.285546	−	0.958365i	$-0.592175\pi$
$677$	−14.8594	−0.571093	−0.285546	+	0.958365i	$0.592175\pi$
$678$	0	0
$679$	−9.50344	−0.364708
$680$	−103.241	−3.95910
$681$	0	0
$682$	0	0
$683$	22.6395	0.866275	0.433138	−	0.901328i	$-0.357406\pi$
$683$	22.6395	0.866275	0.433138	+	0.901328i	$0.357406\pi$
$684$	0	0
$685$	−70.6705	−2.70018
$686$	−2.41714	−0.0922867
$687$	0	0
$688$	10.5986	0.404069
$689$	−1.24146	−0.0472958
$690$	0	0
$691$	13.0444	0.496232	0.248116	−	0.968730i	$-0.420189\pi$
$691$	13.0444	0.496232	0.248116	+	0.968730i	$0.420189\pi$
$692$	−58.0695	−2.20747
$693$	0	0
$694$	37.4739	1.42249
$695$	9.47498	0.359407
$696$	0	0
$697$	−32.7403	−1.24013
$698$	39.5984	1.49882
$699$	0	0
$700$	42.5924	1.60984
$701$	−0.471163	−0.0177956	−0.00889778	−	0.999960i	$-0.502832\pi$
$701$	−0.471163	−0.0177956	−0.00889778	+	0.999960i	$0.502832\pi$
$702$	0	0
$703$	13.4076	0.505676
$704$	0	0
$705$	0	0
$706$	48.1686	1.81285
$707$	9.70420	0.364964
$708$	0	0
$709$	32.8816	1.23490	0.617448	−	0.786612i	$-0.288167\pi$
$709$	32.8816	1.23490	0.617448	+	0.786612i	$0.288167\pi$
$710$	131.376	4.93046
$711$	0	0
$712$	58.2060	2.18136
$713$	21.3179	0.798360
$714$	0	0
$715$	0	0
$716$	−78.2906	−2.92586
$717$	0	0
$718$	51.2539	1.91278
$719$	−16.6079	−0.619372	−0.309686	−	0.950839i	$-0.600224\pi$
$719$	−16.6079	−0.619372	−0.309686	+	0.950839i	$0.600224\pi$
$720$	0	0
$721$	7.32767	0.272897
$722$	38.0854	1.41739
$723$	0	0
$724$	−96.2843	−3.57838
$725$	−88.4658	−3.28554
$726$	0	0
$727$	8.21481	0.304670	0.152335	−	0.988329i	$-0.451321\pi$
$727$	8.21481	0.304670	0.152335	+	0.988329i	$0.451321\pi$
$728$	3.07526	0.113977
$729$	0	0
$730$	69.8213	2.58420
$731$	−19.8891	−0.735626
$732$	0	0
$733$	−9.37274	−0.346190	−0.173095	−	0.984905i	$-0.555377\pi$
$733$	−9.37274	−0.346190	−0.173095	+	0.984905i	$0.555377\pi$
$734$	8.83119	0.325965
$735$	0	0
$736$	−2.97827	−0.109781
$737$	0	0
$738$	0	0
$739$	39.0743	1.43737	0.718685	−	0.695336i	$-0.244744\pi$
$739$	39.0743	1.43737	0.718685	+	0.695336i	$0.244744\pi$
$740$	114.725	4.21739
$741$	0	0
$742$	−4.34581	−0.159540
$743$	35.1057	1.28790	0.643952	−	0.765066i	$-0.277294\pi$
$743$	35.1057	1.28790	0.643952	+	0.765066i	$0.277294\pi$
$744$	0	0
$745$	73.8034	2.70395
$746$	−46.2082	−1.69180
$747$	0	0
$748$	0	0
$749$	−12.4348	−0.454357
$750$	0	0
$751$	−0.0765251	−0.00279244	−0.00139622	−	0.999999i	$-0.500444\pi$
$751$	−0.0765251	−0.00279244	−0.00139622	+	0.999999i	$0.500444\pi$
$752$	−22.5678	−0.822965
$753$	0	0
$754$	−13.3207	−0.485111
$755$	−21.4881	−0.782033
$756$	0	0
$757$	−9.60692	−0.349170	−0.174585	−	0.984642i	$-0.555858\pi$
$757$	−9.60692	−0.349170	−0.174585	+	0.984642i	$0.555858\pi$
$758$	−80.6728	−2.93017
$759$	0	0
$760$	−32.1688	−1.16689
$761$	31.3129	1.13509	0.567545	−	0.823342i	$-0.307893\pi$
$761$	31.3129	1.13509	0.567545	+	0.823342i	$0.307893\pi$
$762$	0	0
$763$	12.5955	0.455986
$764$	28.6036	1.03484
$765$	0	0
$766$	−83.0434	−3.00048
$767$	7.59193	0.274129
$768$	0	0
$769$	−9.43619	−0.340278	−0.170139	−	0.985420i	$-0.554422\pi$
$769$	−9.43619	−0.340278	−0.170139	+	0.985420i	$0.554422\pi$
$770$	0	0
$771$	0	0
$772$	88.4521	3.18346
$773$	−11.2386	−0.404226	−0.202113	−	0.979362i	$-0.564781\pi$
$773$	−11.2386	−0.404226	−0.202113	+	0.979362i	$0.564781\pi$
$774$	0	0
$775$	−116.027	−4.16782
$776$	42.3254	1.51939
$777$	0	0
$778$	21.7791	0.780818
$779$	−10.2016	−0.365509
$780$	0	0
$781$	0	0
$782$	−28.4528	−1.01747
$783$	0	0
$784$	3.08008	0.110003
$785$	21.9210	0.782392
$786$	0	0
$787$	26.8005	0.955333	0.477667	−	0.878541i	$-0.341483\pi$
$787$	26.8005	0.955333	0.477667	+	0.878541i	$0.341483\pi$
$788$	24.3815	0.868557
$789$	0	0
$790$	−51.7512	−1.84123
$791$	−11.8186	−0.420221
$792$	0	0
$793$	4.79910	0.170421
$794$	10.7177	0.380358
$795$	0	0
$796$	−82.1835	−2.91292
$797$	47.4878	1.68211	0.841053	−	0.540953i	$-0.181936\pi$
$797$	47.4878	1.68211	0.841053	+	0.540953i	$0.181936\pi$
$798$	0	0
$799$	42.3503	1.49825
$800$	16.2099	0.573108
$801$	0	0
$802$	−15.0728	−0.532238
$803$	0	0
$804$	0	0
$805$	8.16767	0.287873
$806$	−17.4707	−0.615381
$807$	0	0
$808$	−43.2195	−1.52046
$809$	−23.5047	−0.826382	−0.413191	−	0.910644i	$-0.635586\pi$
$809$	−23.5047	−0.826382	−0.413191	+	0.910644i	$0.635586\pi$
$810$	0	0
$811$	−49.9468	−1.75387	−0.876934	−	0.480610i	$-0.840415\pi$
$811$	−49.9468	−1.75387	−0.876934	+	0.480610i	$0.840415\pi$
$812$	−30.6678	−1.07623
$813$	0	0
$814$	0	0
$815$	69.9952	2.45182
$816$	0	0
$817$	−6.19727	−0.216815
$818$	31.8951	1.11519
$819$	0	0
$820$	−87.2923	−3.04838
$821$	44.3034	1.54620	0.773100	−	0.634284i	$-0.218705\pi$
$821$	44.3034	1.54620	0.773100	+	0.634284i	$0.218705\pi$
$822$	0	0
$823$	25.4653	0.887666	0.443833	−	0.896109i	$-0.353618\pi$
$823$	25.4653	0.887666	0.443833	+	0.896109i	$0.353618\pi$
$824$	−32.6352	−1.13690
$825$	0	0
$826$	26.5761	0.924700
$827$	11.4958	0.399748	0.199874	−	0.979822i	$-0.435947\pi$
$827$	11.4958	0.399748	0.199874	+	0.979822i	$0.435947\pi$
$828$	0	0
$829$	−1.58988	−0.0552187	−0.0276094	−	0.999619i	$-0.508789\pi$
$829$	−1.58988	−0.0552187	−0.0276094	+	0.999619i	$0.508789\pi$
$830$	83.2803	2.89070
$831$	0	0
$832$	6.69438	0.232086
$833$	−5.78000	−0.200265
$834$	0	0
$835$	47.5387	1.64514
$836$	0	0
$837$	0	0
$838$	−1.33512	−0.0461209
$839$	5.98145	0.206503	0.103251	−	0.994655i	$-0.467075\pi$
$839$	5.98145	0.206503	0.103251	+	0.994655i	$0.467075\pi$
$840$	0	0
$841$	34.6980	1.19648
$842$	15.9956	0.551244
$843$	0	0
$844$	−12.9733	−0.446560
$845$	50.2248	1.72779
$846$	0	0
$847$	0	0
$848$	5.53773	0.190166
$849$	0	0
$850$	154.861	5.31168
$851$	15.1612	0.519719
$852$	0	0
$853$	−8.89990	−0.304727	−0.152363	−	0.988325i	$-0.548688\pi$
$853$	−8.89990	−0.304727	−0.152363	+	0.988325i	$0.548688\pi$
$854$	16.7996	0.574869
$855$	0	0
$856$	55.3807	1.89287
$857$	52.2243	1.78395	0.891973	−	0.452088i	$-0.149321\pi$
$857$	52.2243	1.78395	0.891973	+	0.452088i	$0.149321\pi$
$858$	0	0
$859$	5.76678	0.196760	0.0983800	−	0.995149i	$-0.468634\pi$
$859$	5.76678	0.196760	0.0983800	+	0.995149i	$0.468634\pi$
$860$	−53.0285	−1.80826
$861$	0	0
$862$	−1.25837	−0.0428603
$863$	26.9515	0.917440	0.458720	−	0.888581i	$-0.348308\pi$
$863$	26.9515	0.917440	0.458720	+	0.888581i	$0.348308\pi$
$864$	0	0
$865$	60.6082	2.06074
$866$	−2.29556	−0.0780064
$867$	0	0
$868$	−40.2223	−1.36523
$869$	0	0
$870$	0	0
$871$	8.17816	0.277106
$872$	−56.0963	−1.89966
$873$	0	0
$874$	−8.86563	−0.299884
$875$	−24.4017	−0.824929
$876$	0	0
$877$	−23.0226	−0.777419	−0.388709	−	0.921360i	$-0.627079\pi$
$877$	−23.0226	−0.777419	−0.388709	+	0.921360i	$0.627079\pi$
$878$	88.6812	2.99285
$879$	0	0
$880$	0	0
$881$	−9.80782	−0.330434	−0.165217	−	0.986257i	$-0.552832\pi$
$881$	−9.80782	−0.330434	−0.165217	+	0.986257i	$0.552832\pi$
$882$	0	0
$883$	−15.0392	−0.506109	−0.253054	−	0.967452i	$-0.581435\pi$
$883$	−15.0392	−0.506109	−0.253054	+	0.967452i	$0.581435\pi$
$884$	15.3359	0.515803
$885$	0	0
$886$	−24.6870	−0.829374
$887$	11.0331	0.370455	0.185227	−	0.982696i	$-0.440698\pi$
$887$	11.0331	0.370455	0.185227	+	0.982696i	$0.440698\pi$
$888$	0	0
$889$	−9.59200	−0.321705
$890$	−126.693	−4.24675
$891$	0	0
$892$	−7.48127	−0.250491
$893$	13.1960	0.441586
$894$	0	0
$895$	81.7133	2.73137
$896$	20.5093	0.685168
$897$	0	0
$898$	48.5579	1.62040
$899$	83.5431	2.78632
$900$	0	0
$901$	−10.3920	−0.346207
$902$	0	0
$903$	0	0
$904$	52.6364	1.75066
$905$	100.494	3.34052
$906$	0	0
$907$	−14.5803	−0.484131	−0.242066	−	0.970260i	$-0.577825\pi$
$907$	−14.5803	−0.484131	−0.242066	+	0.970260i	$0.577825\pi$
$908$	−17.6670	−0.586301
$909$	0	0
$910$	−6.69370	−0.221894
$911$	−3.29983	−0.109328	−0.0546642	−	0.998505i	$-0.517409\pi$
$911$	−3.29983	−0.109328	−0.0546642	+	0.998505i	$0.517409\pi$
$912$	0	0
$913$	0	0
$914$	1.36893	0.0452802
$915$	0	0
$916$	5.46697	0.180634
$917$	1.48293	0.0489705
$918$	0	0
$919$	−26.6221	−0.878181	−0.439091	−	0.898443i	$-0.644699\pi$
$919$	−26.6221	−0.878181	−0.439091	+	0.898443i	$0.644699\pi$
$920$	−36.3763	−1.19929
$921$	0	0
$922$	29.5928	0.974587
$923$	−9.35784	−0.308017
$924$	0	0
$925$	−82.5183	−2.71318
$926$	56.9998	1.87313
$927$	0	0
$928$	−11.6716	−0.383140
$929$	4.34942	0.142700	0.0713500	−	0.997451i	$-0.477269\pi$
$929$	4.34942	0.142700	0.0713500	+	0.997451i	$0.477269\pi$
$930$	0	0
$931$	−1.80100	−0.0590252
$932$	37.4755	1.22755
$933$	0	0
$934$	−35.0462	−1.14675
$935$	0	0
$936$	0	0
$937$	48.1141	1.57182	0.785909	−	0.618342i	$-0.212195\pi$
$937$	48.1141	1.57182	0.785909	+	0.618342i	$0.212195\pi$
$938$	28.6282	0.934744
$939$	0	0
$940$	112.915	3.68287
$941$	19.5441	0.637118	0.318559	−	0.947903i	$-0.396801\pi$
$941$	19.5441	0.637118	0.318559	+	0.947903i	$0.396801\pi$
$942$	0	0
$943$	−11.5359	−0.375659
$944$	−33.8650	−1.10221
$945$	0	0
$946$	0	0
$947$	54.2546	1.76304	0.881519	−	0.472149i	$-0.156522\pi$
$947$	54.2546	1.76304	0.881519	+	0.472149i	$0.156522\pi$
$948$	0	0
$949$	−4.97333	−0.161441
$950$	48.2532	1.56554
$951$	0	0
$952$	25.7423	0.834314
$953$	−26.0971	−0.845368	−0.422684	−	0.906277i	$-0.638912\pi$
$953$	−26.0971	−0.845368	−0.422684	+	0.906277i	$0.638912\pi$
$954$	0	0
$955$	−29.8541	−0.966055
$956$	−3.23558	−0.104646
$957$	0	0
$958$	−72.2298	−2.33364
$959$	17.6212	0.569018
$960$	0	0
$961$	78.5710	2.53455
$962$	−12.4251	−0.400603
$963$	0	0
$964$	103.096	3.32050
$965$	−92.3191	−2.97186
$966$	0	0
$967$	13.7935	0.443567	0.221784	−	0.975096i	$-0.428812\pi$
$967$	13.7935	0.443567	0.221784	+	0.975096i	$0.428812\pi$
$968$	0	0
$969$	0	0
$970$	−92.1265	−2.95800
$971$	−15.2502	−0.489403	−0.244701	−	0.969599i	$-0.578690\pi$
$971$	−15.2502	−0.489403	−0.244701	+	0.969599i	$0.578690\pi$
$972$	0	0
$973$	−2.36252	−0.0757390
$974$	−56.7611	−1.81874
$975$	0	0
$976$	−21.4071	−0.685226
$977$	12.2867	0.393088	0.196544	−	0.980495i	$-0.437028\pi$
$977$	12.2867	0.393088	0.196544	+	0.980495i	$0.437028\pi$
$978$	0	0
$979$	0	0
$980$	−15.4107	−0.492276
$981$	0	0
$982$	34.2014	1.09141
$983$	12.7524	0.406737	0.203369	−	0.979102i	$-0.434811\pi$
$983$	12.7524	0.406737	0.203369	+	0.979102i	$0.434811\pi$
$984$	0	0
$985$	−25.4475	−0.810824
$986$	−111.504	−3.55102
$987$	0	0
$988$	4.77853	0.152025
$989$	−7.00783	−0.222836
$990$	0	0
$991$	49.0744	1.55890	0.779449	−	0.626465i	$-0.215499\pi$
$991$	49.0744	1.55890	0.779449	+	0.626465i	$0.215499\pi$
$992$	−15.3079	−0.486028
$993$	0	0
$994$	−32.7577	−1.03901
$995$	85.7764	2.71929
$996$	0	0
$997$	26.9368	0.853098	0.426549	−	0.904464i	$-0.359729\pi$
$997$	26.9368	0.853098	0.426549	+	0.904464i	$0.359729\pi$
$998$	−10.6598	−0.337430
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.da.1.2	yes	12
3.2	odd	2	inner	7623.2.a.da.1.11	yes	12
11.10	odd	2		7623.2.a.cz.1.11	yes	12
33.32	even	2		7623.2.a.cz.1.2	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7623.2.a.cz.1.2	✓	12	33.32	even	2
7623.2.a.cz.1.11	yes	12	11.10	odd	2
7623.2.a.da.1.2	yes	12	1.1	even	1	trivial
7623.2.a.da.1.11	yes	12	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.41714 q^{2} +3.84255 q^{4} -4.01054 q^{5} +1.00000 q^{7} -4.45369 q^{8} +O(q^{10})\)	\(q-2.41714 q^{2} +3.84255 q^{4} -4.01054 q^{5} +1.00000 q^{7} -4.45369 q^{8} +9.69401 q^{10} -0.690498 q^{13} -2.41714 q^{14} +3.08008 q^{16} -5.78000 q^{17} -1.80100 q^{19} -15.4107 q^{20} -2.03655 q^{23} +11.0844 q^{25} +1.66903 q^{26} +3.84255 q^{28} -7.98110 q^{29} -10.4676 q^{31} +1.46241 q^{32} +13.9711 q^{34} -4.01054 q^{35} -7.44454 q^{37} +4.35325 q^{38} +17.8617 q^{40} +5.66440 q^{41} +3.44102 q^{43} +4.92263 q^{46} -7.32704 q^{47} +1.00000 q^{49} -26.7925 q^{50} -2.65327 q^{52} +1.79792 q^{53} -4.45369 q^{56} +19.2914 q^{58} -10.9949 q^{59} -6.95019 q^{61} +25.3017 q^{62} -9.69500 q^{64} +2.76927 q^{65} -11.8438 q^{67} -22.2099 q^{68} +9.69401 q^{70} +13.5523 q^{71} +7.20252 q^{73} +17.9945 q^{74} -6.92041 q^{76} -5.33847 q^{79} -12.3528 q^{80} -13.6916 q^{82} +8.59090 q^{83} +23.1809 q^{85} -8.31742 q^{86} -13.0692 q^{89} -0.690498 q^{91} -7.82555 q^{92} +17.7104 q^{94} +7.22296 q^{95} -9.50344 q^{97} -2.41714 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 20 q^{4} + 12 q^{7}+O(q^{10}) \) 12 * q + 20 * q^4 + 12 * q^7	\( 12 q + 20 q^{4} + 12 q^{7} + 20 q^{10} + 20 q^{13} + 28 q^{16} + 12 q^{19} + 32 q^{25} + 20 q^{28} - 16 q^{31} - 24 q^{34} + 4 q^{37} + 48 q^{40} + 16 q^{43} + 24 q^{46} + 12 q^{49} + 96 q^{52} + 20 q^{58} + 44 q^{61} + 76 q^{64} + 20 q^{70} + 52 q^{73} - 8 q^{79} - 68 q^{82} + 72 q^{85} + 20 q^{91} + 20 q^{94} - 32 q^{97}+O(q^{100}) \) 12 * q + 20 * q^4 + 12 * q^7 + 20 * q^10 + 20 * q^13 + 28 * q^16 + 12 * q^19 + 32 * q^25 + 20 * q^28 - 16 * q^31 - 24 * q^34 + 4 * q^37 + 48 * q^40 + 16 * q^43 + 24 * q^46 + 12 * q^49 + 96 * q^52 + 20 * q^58 + 44 * q^61 + 76 * q^64 + 20 * q^70 + 52 * q^73 - 8 * q^79 - 68 * q^82 + 72 * q^85 + 20 * q^91 + 20 * q^94 - 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more