LMFDB - Embedded newform 7623.2.a.cx.1.4

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:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 19x^{8} - x^{7} + 124x^{6} + 6x^{5} - 316x^{4} + 17x^{3} + 253x^{2} - 70x - 11 $ x^10 - 19x^8 - x^7 + 124x^6 + 6x^5 - 316x^4 + 17x^3 + 253x^2 - 70*x - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.871604$ 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)

$f(q)$	$=$	$q-0.871604 q^{2} -1.24031 q^{4} +4.06436 q^{5} -1.00000 q^{7} +2.82426 q^{8} +O(q^{10})$	$q-0.871604 q^{2} -1.24031 q^{4} +4.06436 q^{5} -1.00000 q^{7} +2.82426 q^{8} -3.54252 q^{10} -6.09006 q^{13} +0.871604 q^{14} +0.0189705 q^{16} -1.70162 q^{17} +5.45712 q^{19} -5.04106 q^{20} +6.39153 q^{23} +11.5191 q^{25} +5.30813 q^{26} +1.24031 q^{28} -4.74935 q^{29} -4.36309 q^{31} -5.66506 q^{32} +1.48314 q^{34} -4.06436 q^{35} +2.43952 q^{37} -4.75645 q^{38} +11.4788 q^{40} +3.21037 q^{41} +0.127191 q^{43} -5.57089 q^{46} -8.46784 q^{47} +1.00000 q^{49} -10.0401 q^{50} +7.55354 q^{52} +4.71404 q^{53} -2.82426 q^{56} +4.13955 q^{58} -4.63368 q^{59} +5.31081 q^{61} +3.80289 q^{62} +4.89975 q^{64} -24.7522 q^{65} +14.0686 q^{67} +2.11054 q^{68} +3.54252 q^{70} -2.21883 q^{71} -5.57393 q^{73} -2.12630 q^{74} -6.76849 q^{76} +6.27134 q^{79} +0.0771030 q^{80} -2.79817 q^{82} +0.127722 q^{83} -6.91602 q^{85} -0.110860 q^{86} +8.12736 q^{89} +6.09006 q^{91} -7.92745 q^{92} +7.38061 q^{94} +22.1797 q^{95} -2.59721 q^{97} -0.871604 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 18 q^{4} - 5 q^{5} - 10 q^{7} - 3 q^{8}+O(q^{10}) $ 10 * q + 18 * q^4 - 5 * q^5 - 10 * q^7 - 3 * q^8	$ 10 q + 18 q^{4} - 5 q^{5} - 10 q^{7} - 3 q^{8} + 6 q^{10} - 6 q^{13} + 38 q^{16} + 8 q^{17} - 7 q^{20} + 31 q^{25} - q^{26} - 18 q^{28} - 14 q^{29} + 26 q^{31} - 41 q^{32} + 21 q^{34} + 5 q^{35} + 24 q^{37} - 8 q^{38} + 5 q^{40} + 19 q^{41} + 6 q^{43} + q^{46} - 15 q^{47} + 10 q^{49} - q^{50} + 25 q^{52} + q^{53} + 3 q^{56} + 11 q^{58} - 23 q^{59} + 11 q^{62} + 53 q^{64} - 29 q^{65} + 38 q^{67} + 87 q^{68} - 6 q^{70} - 26 q^{71} + q^{73} - 39 q^{74} + 2 q^{76} - 5 q^{79} - 6 q^{80} + 5 q^{82} + 6 q^{83} + q^{85} + 41 q^{86} + 9 q^{89} + 6 q^{91} + 48 q^{92} - 42 q^{95} + 24 q^{97}+O(q^{100}) $ 10 * q + 18 * q^4 - 5 * q^5 - 10 * q^7 - 3 * q^8 + 6 * q^10 - 6 * q^13 + 38 * q^16 + 8 * q^17 - 7 * q^20 + 31 * q^25 - q^26 - 18 * q^28 - 14 * q^29 + 26 * q^31 - 41 * q^32 + 21 * q^34 + 5 * q^35 + 24 * q^37 - 8 * q^38 + 5 * q^40 + 19 * q^41 + 6 * q^43 + q^46 - 15 * q^47 + 10 * q^49 - q^50 + 25 * q^52 + q^53 + 3 * q^56 + 11 * q^58 - 23 * q^59 + 11 * q^62 + 53 * q^64 - 29 * q^65 + 38 * q^67 + 87 * q^68 - 6 * q^70 - 26 * q^71 + q^73 - 39 * q^74 + 2 * q^76 - 5 * q^79 - 6 * q^80 + 5 * q^82 + 6 * q^83 + q^85 + 41 * q^86 + 9 * q^89 + 6 * q^91 + 48 * q^92 - 42 * q^95 + 24 * 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 $

$2$	−0.871604	−0.616317	−0.308159	−	0.951335i	$-0.599713\pi$
$2$	−0.871604	−0.616317	−0.308159	+	0.951335i	$0.599713\pi$
$3$	0	0
$3$	0	0
$4$	−1.24031	−0.620153
$5$	4.06436	1.81764	0.908820	−	0.417189i	$-0.136985\pi$
$5$	4.06436	1.81764	0.908820	+	0.417189i	$0.136985\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	2.82426	0.998528
$9$	0	0
$10$	−3.54252	−1.12024
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−6.09006	−1.68908	−0.844540	−	0.535492i	$-0.820126\pi$
$13$	−6.09006	−1.68908	−0.844540	+	0.535492i	$0.820126\pi$
$14$	0.871604	0.232946
$15$	0	0
$16$	0.0189705	0.00474262
$17$	−1.70162	−0.412705	−0.206352	−	0.978478i	$-0.566159\pi$
$17$	−1.70162	−0.412705	−0.206352	+	0.978478i	$0.566159\pi$
$18$	0	0
$19$	5.45712	1.25195	0.625974	−	0.779844i	$-0.284702\pi$
$19$	5.45712	1.25195	0.625974	+	0.779844i	$0.284702\pi$
$20$	−5.04106	−1.12721
$21$	0	0
$22$	0	0
$23$	6.39153	1.33273	0.666363	−	0.745627i	$-0.267850\pi$
$23$	6.39153	1.33273	0.666363	+	0.745627i	$0.267850\pi$
$24$	0	0
$25$	11.5191	2.30381
$26$	5.30813	1.04101
$27$	0	0
$28$	1.24031	0.234396
$29$	−4.74935	−0.881931	−0.440966	−	0.897524i	$-0.645364\pi$
$29$	−4.74935	−0.881931	−0.440966	+	0.897524i	$0.645364\pi$
$30$	0	0
$31$	−4.36309	−0.783634	−0.391817	−	0.920043i	$-0.628153\pi$
$31$	−4.36309	−0.783634	−0.391817	+	0.920043i	$0.628153\pi$
$32$	−5.66506	−1.00145
$33$	0	0
$34$	1.48314	0.254357
$35$	−4.06436	−0.687003
$36$	0	0
$37$	2.43952	0.401055	0.200528	−	0.979688i	$-0.435734\pi$
$37$	2.43952	0.401055	0.200528	+	0.979688i	$0.435734\pi$
$38$	−4.75645	−0.771598
$39$	0	0
$40$	11.4788	1.81496
$41$	3.21037	0.501375	0.250688	−	0.968068i	$-0.419343\pi$
$41$	3.21037	0.501375	0.250688	+	0.968068i	$0.419343\pi$
$42$	0	0
$43$	0.127191	0.0193964	0.00969821	−	0.999953i	$-0.496913\pi$
$43$	0.127191	0.0193964	0.00969821	+	0.999953i	$0.496913\pi$
$44$	0	0
$45$	0	0
$46$	−5.57089	−0.821382
$47$	−8.46784	−1.23516	−0.617581	−	0.786507i	$-0.711887\pi$
$47$	−8.46784	−1.23516	−0.617581	+	0.786507i	$0.711887\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−10.0401	−1.41988
$51$	0	0
$52$	7.55354	1.04749
$53$	4.71404	0.647523	0.323761	−	0.946139i	$-0.395052\pi$
$53$	4.71404	0.647523	0.323761	+	0.946139i	$0.395052\pi$
$54$	0	0
$55$	0	0
$56$	−2.82426	−0.377408
$57$	0	0
$58$	4.13955	0.543550
$59$	−4.63368	−0.603253	−0.301627	−	0.953426i	$-0.597530\pi$
$59$	−4.63368	−0.603253	−0.301627	+	0.953426i	$0.597530\pi$
$60$	0	0
$61$	5.31081	0.679980	0.339990	−	0.940429i	$-0.389576\pi$
$61$	5.31081	0.679980	0.339990	+	0.940429i	$0.389576\pi$
$62$	3.80289	0.482967
$63$	0	0
$64$	4.89975	0.612469
$65$	−24.7522	−3.07014
$66$	0	0
$67$	14.0686	1.71876	0.859379	−	0.511340i	$-0.170851\pi$
$67$	14.0686	1.71876	0.859379	+	0.511340i	$0.170851\pi$
$68$	2.11054	0.255940
$69$	0	0
$70$	3.54252	0.423412
$71$	−2.21883	−0.263327	−0.131664	−	0.991294i	$-0.542032\pi$
$71$	−2.21883	−0.263327	−0.131664	+	0.991294i	$0.542032\pi$
$72$	0	0
$73$	−5.57393	−0.652379	−0.326189	−	0.945304i	$-0.605765\pi$
$73$	−5.57393	−0.652379	−0.326189	+	0.945304i	$0.605765\pi$
$74$	−2.12630	−0.247177
$75$	0	0
$76$	−6.76849	−0.776400
$77$	0	0
$78$	0	0
$79$	6.27134	0.705581	0.352790	−	0.935702i	$-0.385233\pi$
$79$	6.27134	0.705581	0.352790	+	0.935702i	$0.385233\pi$
$80$	0.0771030	0.00862037
$81$	0	0
$82$	−2.79817	−0.309006
$83$	0.127722	0.0140193	0.00700965	−	0.999975i	$-0.497769\pi$
$83$	0.127722	0.0140193	0.00700965	+	0.999975i	$0.497769\pi$
$84$	0	0
$85$	−6.91602	−0.750148
$86$	−0.110860	−0.0119543
$87$	0	0
$88$	0	0
$89$	8.12736	0.861498	0.430749	−	0.902472i	$-0.358249\pi$
$89$	8.12736	0.861498	0.430749	+	0.902472i	$0.358249\pi$
$90$	0	0
$91$	6.09006	0.638412
$92$	−7.92745	−0.826494
$93$	0	0
$94$	7.38061	0.761252
$95$	22.1797	2.27559
$96$	0	0
$97$	−2.59721	−0.263706	−0.131853	−	0.991269i	$-0.542093\pi$
$97$	−2.59721	−0.263706	−0.131853	+	0.991269i	$0.542093\pi$
$98$	−0.871604	−0.0880453
$99$	0	0
$100$	−14.2872	−1.42872
$101$	6.02573	0.599583	0.299791	−	0.954005i	$-0.403083\pi$
$101$	6.02573	0.599583	0.299791	+	0.954005i	$0.403083\pi$
$102$	0	0
$103$	12.8938	1.27046	0.635232	−	0.772322i	$-0.280905\pi$
$103$	12.8938	1.27046	0.635232	+	0.772322i	$0.280905\pi$
$104$	−17.2000	−1.68659
$105$	0	0
$106$	−4.10878	−0.399080
$107$	5.34760	0.516972	0.258486	−	0.966015i	$-0.416776\pi$
$107$	5.34760	0.516972	0.258486	+	0.966015i	$0.416776\pi$
$108$	0	0
$109$	−1.44173	−0.138093	−0.0690465	−	0.997613i	$-0.521996\pi$
$109$	−1.44173	−0.138093	−0.0690465	+	0.997613i	$0.521996\pi$
$110$	0	0
$111$	0	0
$112$	−0.0189705	−0.00179254
$113$	−19.5174	−1.83604	−0.918021	−	0.396532i	$-0.870214\pi$
$113$	−19.5174	−1.83604	−0.918021	+	0.396532i	$0.870214\pi$
$114$	0	0
$115$	25.9775	2.42242
$116$	5.89064	0.546932
$117$	0	0
$118$	4.03873	0.371795
$119$	1.70162	0.155988
$120$	0	0
$121$	0	0
$122$	−4.62893	−0.419083
$123$	0	0
$124$	5.41156	0.485973
$125$	26.4958	2.36986
$126$	0	0
$127$	−0.641713	−0.0569428	−0.0284714	−	0.999595i	$-0.509064\pi$
$127$	−0.641713	−0.0569428	−0.0284714	+	0.999595i	$0.509064\pi$
$128$	7.05948	0.623976
$129$	0	0
$130$	21.5742	1.89218
$131$	13.9987	1.22307	0.611537	−	0.791216i	$-0.290552\pi$
$131$	13.9987	1.22307	0.611537	+	0.791216i	$0.290552\pi$
$132$	0	0
$133$	−5.45712	−0.473192
$134$	−12.2623	−1.05930
$135$	0	0
$136$	−4.80584	−0.412097
$137$	2.70017	0.230691	0.115346	−	0.993325i	$-0.463202\pi$
$137$	2.70017	0.230691	0.115346	+	0.993325i	$0.463202\pi$
$138$	0	0
$139$	−2.06192	−0.174890	−0.0874448	−	0.996169i	$-0.527870\pi$
$139$	−2.06192	−0.174890	−0.0874448	+	0.996169i	$0.527870\pi$
$140$	5.04106	0.426047
$141$	0	0
$142$	1.93395	0.162293
$143$	0	0
$144$	0	0
$145$	−19.3031	−1.60303
$146$	4.85826	0.402072
$147$	0	0
$148$	−3.02576	−0.248716
$149$	−10.0829	−0.826027	−0.413013	−	0.910725i	$-0.635524\pi$
$149$	−10.0829	−0.826027	−0.413013	+	0.910725i	$0.635524\pi$
$150$	0	0
$151$	−14.9639	−1.21775	−0.608873	−	0.793268i	$-0.708378\pi$
$151$	−14.9639	−1.21775	−0.608873	+	0.793268i	$0.708378\pi$
$152$	15.4123	1.25011
$153$	0	0
$154$	0	0
$155$	−17.7332	−1.42436
$156$	0	0
$157$	19.4540	1.55260	0.776300	−	0.630364i	$-0.217094\pi$
$157$	19.4540	1.55260	0.776300	+	0.630364i	$0.217094\pi$
$158$	−5.46613	−0.434862
$159$	0	0
$160$	−23.0249	−1.82028
$161$	−6.39153	−0.503723
$162$	0	0
$163$	12.1104	0.948562	0.474281	−	0.880374i	$-0.342708\pi$
$163$	12.1104	0.948562	0.474281	+	0.880374i	$0.342708\pi$
$164$	−3.98184	−0.310929
$165$	0	0
$166$	−0.111323	−0.00864034
$167$	13.1680	1.01897	0.509484	−	0.860480i	$-0.329836\pi$
$167$	13.1680	1.01897	0.509484	+	0.860480i	$0.329836\pi$
$168$	0	0
$169$	24.0889	1.85299
$170$	6.02804	0.462329
$171$	0	0
$172$	−0.157755	−0.0120287
$173$	14.7179	1.11898	0.559492	−	0.828836i	$-0.310996\pi$
$173$	14.7179	1.11898	0.559492	+	0.828836i	$0.310996\pi$
$174$	0	0
$175$	−11.5191	−0.870759
$176$	0	0
$177$	0	0
$178$	−7.08384	−0.530956
$179$	11.6904	0.873780	0.436890	−	0.899515i	$-0.356080\pi$
$179$	11.6904	0.873780	0.436890	+	0.899515i	$0.356080\pi$
$180$	0	0
$181$	19.7002	1.46431	0.732154	−	0.681139i	$-0.238515\pi$
$181$	19.7002	1.46431	0.732154	+	0.681139i	$0.238515\pi$
$182$	−5.30813	−0.393465
$183$	0	0
$184$	18.0514	1.33077
$185$	9.91511	0.728974
$186$	0	0
$187$	0	0
$188$	10.5027	0.765989
$189$	0	0
$190$	−19.3319	−1.40249
$191$	−6.96194	−0.503748	−0.251874	−	0.967760i	$-0.581047\pi$
$191$	−6.96194	−0.503748	−0.251874	+	0.967760i	$0.581047\pi$
$192$	0	0
$193$	16.8633	1.21385	0.606923	−	0.794761i	$-0.292404\pi$
$193$	16.8633	1.21385	0.606923	+	0.794761i	$0.292404\pi$
$194$	2.26374	0.162527
$195$	0	0
$196$	−1.24031	−0.0885933
$197$	−6.42145	−0.457510	−0.228755	−	0.973484i	$-0.573465\pi$
$197$	−6.42145	−0.457510	−0.228755	+	0.973484i	$0.573465\pi$
$198$	0	0
$199$	−13.2645	−0.940297	−0.470149	−	0.882587i	$-0.655800\pi$
$199$	−13.2645	−0.940297	−0.470149	+	0.882587i	$0.655800\pi$
$200$	32.5329	2.30042
$201$	0	0
$202$	−5.25205	−0.369533
$203$	4.74935	0.333339
$204$	0	0
$205$	13.0481	0.911319
$206$	−11.2383	−0.783008
$207$	0	0
$208$	−0.115531	−0.00801067
$209$	0	0
$210$	0	0
$211$	2.91158	0.200442	0.100221	−	0.994965i	$-0.468045\pi$
$211$	2.91158	0.200442	0.100221	+	0.994965i	$0.468045\pi$
$212$	−5.84685	−0.401563
$213$	0	0
$214$	−4.66099	−0.318619
$215$	0.516950	0.0352557
$216$	0	0
$217$	4.36309	0.296186
$218$	1.25662	0.0851091
$219$	0	0
$220$	0	0
$221$	10.3630	0.697091
$222$	0	0
$223$	−3.35570	−0.224714	−0.112357	−	0.993668i	$-0.535840\pi$
$223$	−3.35570	−0.224714	−0.112357	+	0.993668i	$0.535840\pi$
$224$	5.66506	0.378513
$225$	0	0
$226$	17.0114	1.13158
$227$	−3.33791	−0.221545	−0.110772	−	0.993846i	$-0.535332\pi$
$227$	−3.33791	−0.221545	−0.110772	+	0.993846i	$0.535332\pi$
$228$	0	0
$229$	6.80143	0.449451	0.224726	−	0.974422i	$-0.427851\pi$
$229$	6.80143	0.449451	0.224726	+	0.974422i	$0.427851\pi$
$230$	−22.6421	−1.49298
$231$	0	0
$232$	−13.4134	−0.880634
$233$	−6.18553	−0.405228	−0.202614	−	0.979259i	$-0.564944\pi$
$233$	−6.18553	−0.405228	−0.202614	+	0.979259i	$0.564944\pi$
$234$	0	0
$235$	−34.4164	−2.24508
$236$	5.74718	0.374109
$237$	0	0
$238$	−1.48314	−0.0961379
$239$	−20.4784	−1.32464	−0.662319	−	0.749222i	$-0.730427\pi$
$239$	−20.4784	−1.32464	−0.662319	+	0.749222i	$0.730427\pi$
$240$	0	0
$241$	1.04113	0.0670653	0.0335326	−	0.999438i	$-0.489324\pi$
$241$	1.04113	0.0670653	0.0335326	+	0.999438i	$0.489324\pi$
$242$	0	0
$243$	0	0
$244$	−6.58703	−0.421691
$245$	4.06436	0.259663
$246$	0	0
$247$	−33.2342	−2.11464
$248$	−12.3225	−0.782481
$249$	0	0
$250$	−23.0939	−1.46059
$251$	−6.63300	−0.418672	−0.209336	−	0.977844i	$-0.567130\pi$
$251$	−6.63300	−0.418672	−0.209336	+	0.977844i	$0.567130\pi$
$252$	0	0
$253$	0	0
$254$	0.559320	0.0350948
$255$	0	0
$256$	−15.9526	−0.997036
$257$	−2.88246	−0.179803	−0.0899015	−	0.995951i	$-0.528655\pi$
$257$	−2.88246	−0.179803	−0.0899015	+	0.995951i	$0.528655\pi$
$258$	0	0
$259$	−2.43952	−0.151585
$260$	30.7004	1.90396
$261$	0	0
$262$	−12.2013	−0.753801
$263$	21.1018	1.30119	0.650595	−	0.759425i	$-0.274520\pi$
$263$	21.1018	1.30119	0.650595	+	0.759425i	$0.274520\pi$
$264$	0	0
$265$	19.1596	1.17696
$266$	4.75645	0.291636
$267$	0	0
$268$	−17.4494	−1.06589
$269$	17.1315	1.04453	0.522263	−	0.852784i	$-0.325088\pi$
$269$	17.1315	1.04453	0.522263	+	0.852784i	$0.325088\pi$
$270$	0	0
$271$	16.3599	0.993790	0.496895	−	0.867811i	$-0.334473\pi$
$271$	16.3599	0.993790	0.496895	+	0.867811i	$0.334473\pi$
$272$	−0.0322806	−0.00195730
$273$	0	0
$274$	−2.35348	−0.142179
$275$	0	0
$276$	0	0
$277$	−2.47852	−0.148920	−0.0744599	−	0.997224i	$-0.523723\pi$
$277$	−2.47852	−0.148920	−0.0744599	+	0.997224i	$0.523723\pi$
$278$	1.79718	0.107787
$279$	0	0
$280$	−11.4788	−0.685992
$281$	−24.1948	−1.44334	−0.721671	−	0.692236i	$-0.756626\pi$
$281$	−24.1948	−1.44334	−0.721671	+	0.692236i	$0.756626\pi$
$282$	0	0
$283$	−29.1859	−1.73492	−0.867460	−	0.497508i	$-0.834249\pi$
$283$	−29.1859	−1.73492	−0.867460	+	0.497508i	$0.834249\pi$
$284$	2.75203	0.163303
$285$	0	0
$286$	0	0
$287$	−3.21037	−0.189502
$288$	0	0
$289$	−14.1045	−0.829675
$290$	16.8246	0.987977
$291$	0	0
$292$	6.91338	0.404575
$293$	27.1918	1.58856	0.794280	−	0.607551i	$-0.207848\pi$
$293$	27.1918	1.58856	0.794280	+	0.607551i	$0.207848\pi$
$294$	0	0
$295$	−18.8330	−1.09650
$296$	6.88986	0.400465
$297$	0	0
$298$	8.78834	0.509095
$299$	−38.9248	−2.25108
$300$	0	0
$301$	−0.127191	−0.00733115
$302$	13.0426	0.750518
$303$	0	0
$304$	0.103524	0.00593752
$305$	21.5851	1.23596
$306$	0	0
$307$	−9.95364	−0.568084	−0.284042	−	0.958812i	$-0.591676\pi$
$307$	−9.95364	−0.568084	−0.284042	+	0.958812i	$0.591676\pi$
$308$	0	0
$309$	0	0
$310$	15.4563	0.877860
$311$	−16.2407	−0.920923	−0.460462	−	0.887680i	$-0.652316\pi$
$311$	−16.2407	−0.920923	−0.460462	+	0.887680i	$0.652316\pi$
$312$	0	0
$313$	16.6148	0.939125	0.469562	−	0.882899i	$-0.344412\pi$
$313$	16.6148	0.939125	0.469562	+	0.882899i	$0.344412\pi$
$314$	−16.9562	−0.956894
$315$	0	0
$316$	−7.77838	−0.437568
$317$	−8.10002	−0.454942	−0.227471	−	0.973785i	$-0.573046\pi$
$317$	−8.10002	−0.454942	−0.227471	+	0.973785i	$0.573046\pi$
$318$	0	0
$319$	0	0
$320$	19.9144	1.11325
$321$	0	0
$322$	5.57089	0.310453
$323$	−9.28597	−0.516685
$324$	0	0
$325$	−70.1518	−3.89132
$326$	−10.5555	−0.584615
$327$	0	0
$328$	9.06693	0.500637
$329$	8.46784	0.466847
$330$	0	0
$331$	−6.64605	−0.365300	−0.182650	−	0.983178i	$-0.558468\pi$
$331$	−6.64605	−0.365300	−0.182650	+	0.983178i	$0.558468\pi$
$332$	−0.158414	−0.00869411
$333$	0	0
$334$	−11.4773	−0.628008
$335$	57.1801	3.12408
$336$	0	0
$337$	−20.9521	−1.14133	−0.570665	−	0.821183i	$-0.693315\pi$
$337$	−20.9521	−1.14133	−0.570665	+	0.821183i	$0.693315\pi$
$338$	−20.9960	−1.14203
$339$	0	0
$340$	8.57798	0.465207
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0.359220	0.0193679
$345$	0	0
$346$	−12.8282	−0.689649
$347$	17.7871	0.954862	0.477431	−	0.878669i	$-0.341568\pi$
$347$	17.7871	0.954862	0.477431	+	0.878669i	$0.341568\pi$
$348$	0	0
$349$	11.7341	0.628110	0.314055	−	0.949405i	$-0.398312\pi$
$349$	11.7341	0.628110	0.314055	+	0.949405i	$0.398312\pi$
$350$	10.0401	0.536664
$351$	0	0
$352$	0	0
$353$	−15.2682	−0.812642	−0.406321	−	0.913730i	$-0.633189\pi$
$353$	−15.2682	−0.812642	−0.406321	+	0.913730i	$0.633189\pi$
$354$	0	0
$355$	−9.01815	−0.478634
$356$	−10.0804	−0.534261
$357$	0	0
$358$	−10.1894	−0.538526
$359$	−6.74198	−0.355828	−0.177914	−	0.984046i	$-0.556935\pi$
$359$	−6.74198	−0.355828	−0.177914	+	0.984046i	$0.556935\pi$
$360$	0	0
$361$	10.7801	0.567375
$362$	−17.1708	−0.902478
$363$	0	0
$364$	−7.55354	−0.395913
$365$	−22.6545	−1.18579
$366$	0	0
$367$	−20.9743	−1.09485	−0.547424	−	0.836855i	$-0.684392\pi$
$367$	−20.9743	−1.09485	−0.547424	+	0.836855i	$0.684392\pi$
$368$	0.121250	0.00632062
$369$	0	0
$370$	−8.64205	−0.449279
$371$	−4.71404	−0.244741
$372$	0	0
$373$	29.1253	1.50805	0.754027	−	0.656844i	$-0.228109\pi$
$373$	29.1253	1.50805	0.754027	+	0.656844i	$0.228109\pi$
$374$	0	0
$375$	0	0
$376$	−23.9154	−1.23334
$377$	28.9238	1.48965
$378$	0	0
$379$	31.9353	1.64041	0.820203	−	0.572072i	$-0.193860\pi$
$379$	31.9353	1.64041	0.820203	+	0.572072i	$0.193860\pi$
$380$	−27.5096	−1.41121
$381$	0	0
$382$	6.06806	0.310469
$383$	−3.58968	−0.183424	−0.0917121	−	0.995786i	$-0.529234\pi$
$383$	−3.58968	−0.183424	−0.0917121	+	0.995786i	$0.529234\pi$
$384$	0	0
$385$	0	0
$386$	−14.6981	−0.748115
$387$	0	0
$388$	3.22133	0.163538
$389$	−10.4710	−0.530901	−0.265451	−	0.964124i	$-0.585521\pi$
$389$	−10.4710	−0.530901	−0.265451	+	0.964124i	$0.585521\pi$
$390$	0	0
$391$	−10.8760	−0.550022
$392$	2.82426	0.142647
$393$	0	0
$394$	5.59697	0.281971
$395$	25.4890	1.28249
$396$	0	0
$397$	17.7269	0.889686	0.444843	−	0.895609i	$-0.353259\pi$
$397$	17.7269	0.889686	0.444843	+	0.895609i	$0.353259\pi$
$398$	11.5614	0.579522
$399$	0	0
$400$	0.218522	0.0109261
$401$	−8.08035	−0.403513	−0.201757	−	0.979436i	$-0.564665\pi$
$401$	−8.08035	−0.403513	−0.201757	+	0.979436i	$0.564665\pi$
$402$	0	0
$403$	26.5715	1.32362
$404$	−7.47375	−0.371833
$405$	0	0
$406$	−4.13955	−0.205442
$407$	0	0
$408$	0	0
$409$	39.6889	1.96249	0.981245	−	0.192765i	$-0.0617456\pi$
$409$	39.6889	1.96249	0.981245	+	0.192765i	$0.0617456\pi$
$410$	−11.3728	−0.561662
$411$	0	0
$412$	−15.9922	−0.787881
$413$	4.63368	0.228008
$414$	0	0
$415$	0.519109	0.0254820
$416$	34.5006	1.69153
$417$	0	0
$418$	0	0
$419$	−16.1715	−0.790028	−0.395014	−	0.918675i	$-0.629260\pi$
$419$	−16.1715	−0.790028	−0.395014	+	0.918675i	$0.629260\pi$
$420$	0	0
$421$	24.8084	1.20909	0.604544	−	0.796572i	$-0.293355\pi$
$421$	24.8084	1.20909	0.604544	+	0.796572i	$0.293355\pi$
$422$	−2.53775	−0.123536
$423$	0	0
$424$	13.3137	0.646570
$425$	−19.6011	−0.950794
$426$	0	0
$427$	−5.31081	−0.257008
$428$	−6.63265	−0.320601
$429$	0	0
$430$	−0.450576	−0.0217287
$431$	−21.8821	−1.05402	−0.527012	−	0.849858i	$-0.676688\pi$
$431$	−21.8821	−1.05402	−0.527012	+	0.849858i	$0.676688\pi$
$432$	0	0
$433$	39.3439	1.89075	0.945374	−	0.325989i	$-0.105697\pi$
$433$	39.3439	1.89075	0.945374	+	0.325989i	$0.105697\pi$
$434$	−3.80289	−0.182544
$435$	0	0
$436$	1.78819	0.0856388
$437$	34.8793	1.66850
$438$	0	0
$439$	9.59194	0.457799	0.228899	−	0.973450i	$-0.426487\pi$
$439$	9.59194	0.457799	0.228899	+	0.973450i	$0.426487\pi$
$440$	0	0
$441$	0	0
$442$	−9.03244	−0.429629
$443$	0.629136	0.0298911	0.0149456	−	0.999888i	$-0.495242\pi$
$443$	0.629136	0.0298911	0.0149456	+	0.999888i	$0.495242\pi$
$444$	0	0
$445$	33.0325	1.56589
$446$	2.92484	0.138495
$447$	0	0
$448$	−4.89975	−0.231492
$449$	17.3364	0.818154	0.409077	−	0.912500i	$-0.365851\pi$
$449$	17.3364	0.818154	0.409077	+	0.912500i	$0.365851\pi$
$450$	0	0
$451$	0	0
$452$	24.2075	1.13863
$453$	0	0
$454$	2.90934	0.136542
$455$	24.7522	1.16040
$456$	0	0
$457$	−29.1154	−1.36196	−0.680980	−	0.732302i	$-0.738446\pi$
$457$	−29.1154	−1.36196	−0.680980	+	0.732302i	$0.738446\pi$
$458$	−5.92816	−0.277005
$459$	0	0
$460$	−32.2201	−1.50227
$461$	−15.1850	−0.707236	−0.353618	−	0.935390i	$-0.615049\pi$
$461$	−15.1850	−0.707236	−0.353618	+	0.935390i	$0.615049\pi$
$462$	0	0
$463$	17.8886	0.831355	0.415677	−	0.909512i	$-0.363545\pi$
$463$	17.8886	0.831355	0.415677	+	0.909512i	$0.363545\pi$
$464$	−0.0900974	−0.00418267
$465$	0	0
$466$	5.39134	0.249749
$467$	30.3101	1.40259	0.701293	−	0.712874i	$-0.252607\pi$
$467$	30.3101	1.40259	0.701293	+	0.712874i	$0.252607\pi$
$468$	0	0
$469$	−14.0686	−0.649629
$470$	29.9975	1.38368
$471$	0	0
$472$	−13.0867	−0.602366
$473$	0	0
$474$	0	0
$475$	62.8609	2.88425
$476$	−2.11054	−0.0967362
$477$	0	0
$478$	17.8491	0.816397
$479$	7.94750	0.363130	0.181565	−	0.983379i	$-0.441884\pi$
$479$	7.94750	0.363130	0.181565	+	0.983379i	$0.441884\pi$
$480$	0	0
$481$	−14.8569	−0.677414
$482$	−0.907456	−0.0413335
$483$	0	0
$484$	0	0
$485$	−10.5560	−0.479323
$486$	0	0
$487$	−10.1711	−0.460896	−0.230448	−	0.973085i	$-0.574019\pi$
$487$	−10.1711	−0.460896	−0.230448	+	0.973085i	$0.574019\pi$
$488$	14.9991	0.678979
$489$	0	0
$490$	−3.54252	−0.160035
$491$	−1.07736	−0.0486206	−0.0243103	−	0.999704i	$-0.507739\pi$
$491$	−1.07736	−0.0486206	−0.0243103	+	0.999704i	$0.507739\pi$
$492$	0	0
$493$	8.08161	0.363977
$494$	28.9671	1.30329
$495$	0	0
$496$	−0.0827699	−0.00371648
$497$	2.21883	0.0995283
$498$	0	0
$499$	34.6159	1.54962	0.774810	−	0.632194i	$-0.217845\pi$
$499$	34.6159	1.54962	0.774810	+	0.632194i	$0.217845\pi$
$500$	−32.8629	−1.46968
$501$	0	0
$502$	5.78136	0.258035
$503$	−2.75919	−0.123026	−0.0615130	−	0.998106i	$-0.519593\pi$
$503$	−2.75919	−0.123026	−0.0615130	+	0.998106i	$0.519593\pi$
$504$	0	0
$505$	24.4908	1.08982
$506$	0	0
$507$	0	0
$508$	0.795920	0.0353132
$509$	−14.5914	−0.646753	−0.323377	−	0.946270i	$-0.604818\pi$
$509$	−14.5914	−0.646753	−0.323377	+	0.946270i	$0.604818\pi$
$510$	0	0
$511$	5.57393	0.246576
$512$	−0.214624	−0.00948514
$513$	0	0
$514$	2.51237	0.110816
$515$	52.4051	2.30924
$516$	0	0
$517$	0	0
$518$	2.12630	0.0934242
$519$	0	0
$520$	−69.9069	−3.06562
$521$	13.7317	0.601599	0.300799	−	0.953687i	$-0.402747\pi$
$521$	13.7317	0.601599	0.300799	+	0.953687i	$0.402747\pi$
$522$	0	0
$523$	7.54609	0.329967	0.164984	−	0.986296i	$-0.447243\pi$
$523$	7.54609	0.329967	0.164984	+	0.986296i	$0.447243\pi$
$524$	−17.3627	−0.758493
$525$	0	0
$526$	−18.3924	−0.801946
$527$	7.42434	0.323409
$528$	0	0
$529$	17.8517	0.776160
$530$	−16.6996	−0.725383
$531$	0	0
$532$	6.76849	0.293451
$533$	−19.5513	−0.846863
$534$	0	0
$535$	21.7346	0.939668
$536$	39.7335	1.71623
$537$	0	0
$538$	−14.9319	−0.643760
$539$	0	0
$540$	0	0
$541$	−2.01679	−0.0867084	−0.0433542	−	0.999060i	$-0.513804\pi$
$541$	−2.01679	−0.0867084	−0.0433542	+	0.999060i	$0.513804\pi$
$542$	−14.2593	−0.612490
$543$	0	0
$544$	9.63981	0.413304
$545$	−5.85973	−0.251003
$546$	0	0
$547$	41.1514	1.75951	0.879754	−	0.475429i	$-0.157707\pi$
$547$	41.1514	1.75951	0.879754	+	0.475429i	$0.157707\pi$
$548$	−3.34904	−0.143064
$549$	0	0
$550$	0	0
$551$	−25.9177	−1.10413
$552$	0	0
$553$	−6.27134	−0.266684
$554$	2.16029	0.0917818
$555$	0	0
$556$	2.55741	0.108458
$557$	18.4609	0.782214	0.391107	−	0.920345i	$-0.372092\pi$
$557$	18.4609	0.782214	0.391107	+	0.920345i	$0.372092\pi$
$558$	0	0
$559$	−0.774600	−0.0327621
$560$	−0.0771030	−0.00325819
$561$	0	0
$562$	21.0883	0.889557
$563$	28.5154	1.20178	0.600890	−	0.799332i	$-0.294813\pi$
$563$	28.5154	1.20178	0.600890	+	0.799332i	$0.294813\pi$
$564$	0	0
$565$	−79.3258	−3.33726
$566$	25.4385	1.06926
$567$	0	0
$568$	−6.26658	−0.262940
$569$	−17.7384	−0.743631	−0.371816	−	0.928307i	$-0.621265\pi$
$569$	−17.7384	−0.743631	−0.371816	+	0.928307i	$0.621265\pi$
$570$	0	0
$571$	−20.8488	−0.872496	−0.436248	−	0.899827i	$-0.643693\pi$
$571$	−20.8488	−0.872496	−0.436248	+	0.899827i	$0.643693\pi$
$572$	0	0
$573$	0	0
$574$	2.79817	0.116793
$575$	73.6244	3.07035
$576$	0	0
$577$	21.2988	0.886680	0.443340	−	0.896353i	$-0.353793\pi$
$577$	21.2988	0.886680	0.443340	+	0.896353i	$0.353793\pi$
$578$	12.2935	0.511343
$579$	0	0
$580$	23.9417	0.994126
$581$	−0.127722	−0.00529880
$582$	0	0
$583$	0	0
$584$	−15.7422	−0.651419
$585$	0	0
$586$	−23.7005	−0.979058
$587$	−29.1874	−1.20469	−0.602346	−	0.798235i	$-0.705767\pi$
$587$	−29.1874	−1.20469	−0.602346	+	0.798235i	$0.705767\pi$
$588$	0	0
$589$	−23.8099	−0.981069
$590$	16.4149	0.675790
$591$	0	0
$592$	0.0462789	0.00190205
$593$	37.3094	1.53212	0.766058	−	0.642772i	$-0.222216\pi$
$593$	37.3094	1.53212	0.766058	+	0.642772i	$0.222216\pi$
$594$	0	0
$595$	6.91602	0.283529
$596$	12.5059	0.512263
$597$	0	0
$598$	33.9271	1.38738
$599$	4.95368	0.202402	0.101201	−	0.994866i	$-0.467731\pi$
$599$	4.95368	0.202402	0.101201	+	0.994866i	$0.467731\pi$
$600$	0	0
$601$	29.4572	1.20158	0.600792	−	0.799405i	$-0.294852\pi$
$601$	29.4572	1.20158	0.600792	+	0.799405i	$0.294852\pi$
$602$	0.110860	0.00451832
$603$	0	0
$604$	18.5598	0.755189
$605$	0	0
$606$	0	0
$607$	−26.3426	−1.06921	−0.534606	−	0.845101i	$-0.679540\pi$
$607$	−26.3426	−1.06921	−0.534606	+	0.845101i	$0.679540\pi$
$608$	−30.9149	−1.25377
$609$	0	0
$610$	−18.8136	−0.761742
$611$	51.5697	2.08629
$612$	0	0
$613$	48.2089	1.94714	0.973569	−	0.228391i	$-0.0733465\pi$
$613$	48.2089	1.94714	0.973569	+	0.228391i	$0.0733465\pi$
$614$	8.67563	0.350120
$615$	0	0
$616$	0	0
$617$	29.2376	1.17706	0.588532	−	0.808474i	$-0.299706\pi$
$617$	29.2376	1.17706	0.588532	+	0.808474i	$0.299706\pi$
$618$	0	0
$619$	6.14388	0.246943	0.123472	−	0.992348i	$-0.460597\pi$
$619$	6.14388	0.246943	0.123472	+	0.992348i	$0.460597\pi$
$620$	21.9946	0.883323
$621$	0	0
$622$	14.1554	0.567581
$623$	−8.12736	−0.325616
$624$	0	0
$625$	50.0934	2.00374
$626$	−14.4815	−0.578799
$627$	0	0
$628$	−24.1289	−0.962849
$629$	−4.15115	−0.165517
$630$	0	0
$631$	40.5646	1.61485	0.807425	−	0.589971i	$-0.200861\pi$
$631$	40.5646	1.61485	0.807425	+	0.589971i	$0.200861\pi$
$632$	17.7119	0.704542
$633$	0	0
$634$	7.06001	0.280389
$635$	−2.60815	−0.103501
$636$	0	0
$637$	−6.09006	−0.241297
$638$	0	0
$639$	0	0
$640$	28.6923	1.13416
$641$	−23.5266	−0.929244	−0.464622	−	0.885509i	$-0.653810\pi$
$641$	−23.5266	−0.929244	−0.464622	+	0.885509i	$0.653810\pi$
$642$	0	0
$643$	−21.0143	−0.828724	−0.414362	−	0.910112i	$-0.635995\pi$
$643$	−21.0143	−0.828724	−0.414362	+	0.910112i	$0.635995\pi$
$644$	7.92745	0.312385
$645$	0	0
$646$	8.09369	0.318442
$647$	−0.452722	−0.0177983	−0.00889917	−	0.999960i	$-0.502833\pi$
$647$	−0.452722	−0.0177983	−0.00889917	+	0.999960i	$0.502833\pi$
$648$	0	0
$649$	0	0
$650$	61.1446	2.39829
$651$	0	0
$652$	−15.0206	−0.588253
$653$	22.0408	0.862522	0.431261	−	0.902227i	$-0.358069\pi$
$653$	22.0408	0.862522	0.431261	+	0.902227i	$0.358069\pi$
$654$	0	0
$655$	56.8959	2.22311
$656$	0.0609022	0.00237783
$657$	0	0
$658$	−7.38061	−0.287726
$659$	−18.3859	−0.716215	−0.358107	−	0.933680i	$-0.616578\pi$
$659$	−18.3859	−0.716215	−0.358107	+	0.933680i	$0.616578\pi$
$660$	0	0
$661$	−45.9605	−1.78766	−0.893829	−	0.448408i	$-0.851991\pi$
$661$	−45.9605	−1.78766	−0.893829	+	0.448408i	$0.851991\pi$
$662$	5.79273	0.225141
$663$	0	0
$664$	0.360721	0.0139987
$665$	−22.1797	−0.860092
$666$	0	0
$667$	−30.3556	−1.17537
$668$	−16.3323	−0.631917
$669$	0	0
$670$	−49.8384	−1.92542
$671$	0	0
$672$	0	0
$673$	29.7402	1.14640	0.573200	−	0.819415i	$-0.305702\pi$
$673$	29.7402	1.14640	0.573200	+	0.819415i	$0.305702\pi$
$674$	18.2619	0.703422
$675$	0	0
$676$	−29.8776	−1.14914
$677$	−27.7828	−1.06778	−0.533889	−	0.845554i	$-0.679270\pi$
$677$	−27.7828	−1.06778	−0.533889	+	0.845554i	$0.679270\pi$
$678$	0	0
$679$	2.59721	0.0996716
$680$	−19.5327	−0.749044
$681$	0	0
$682$	0	0
$683$	−40.8297	−1.56231	−0.781153	−	0.624339i	$-0.785368\pi$
$683$	−40.8297	−1.56231	−0.781153	+	0.624339i	$0.785368\pi$
$684$	0	0
$685$	10.9745	0.419313
$686$	0.871604	0.0332780
$687$	0	0
$688$	0.00241287	9.19898e−5 0
$689$	−28.7088	−1.09372
$690$	0	0
$691$	37.8156	1.43857	0.719286	−	0.694714i	$-0.244469\pi$
$691$	37.8156	1.43857	0.719286	+	0.694714i	$0.244469\pi$
$692$	−18.2547	−0.693941
$693$	0	0
$694$	−15.5033	−0.588498
$695$	−8.38039	−0.317886
$696$	0	0
$697$	−5.46284	−0.206920
$698$	−10.2275	−0.387115
$699$	0	0
$700$	14.2872	0.540004
$701$	−27.4256	−1.03585	−0.517926	−	0.855426i	$-0.673296\pi$
$701$	−27.4256	−1.03585	−0.517926	+	0.855426i	$0.673296\pi$
$702$	0	0
$703$	13.3128	0.502100
$704$	0	0
$705$	0	0
$706$	13.3078	0.500845
$707$	−6.02573	−0.226621
$708$	0	0
$709$	−31.8439	−1.19592	−0.597962	−	0.801525i	$-0.704022\pi$
$709$	−31.8439	−1.19592	−0.597962	+	0.801525i	$0.704022\pi$
$710$	7.86026	0.294990
$711$	0	0
$712$	22.9538	0.860230
$713$	−27.8868	−1.04437
$714$	0	0
$715$	0	0
$716$	−14.4996	−0.541877
$717$	0	0
$718$	5.87633	0.219303
$719$	15.2468	0.568608	0.284304	−	0.958734i	$-0.408237\pi$
$719$	15.2468	0.568608	0.284304	+	0.958734i	$0.408237\pi$
$720$	0	0
$721$	−12.8938	−0.480190
$722$	−9.39600	−0.349683
$723$	0	0
$724$	−24.4343	−0.908095
$725$	−54.7080	−2.03180
$726$	0	0
$727$	−33.7158	−1.25045	−0.625225	−	0.780445i	$-0.714993\pi$
$727$	−33.7158	−1.25045	−0.625225	+	0.780445i	$0.714993\pi$
$728$	17.2000	0.637473
$729$	0	0
$730$	19.7457	0.730823
$731$	−0.216431	−0.00800499
$732$	0	0
$733$	−4.51648	−0.166820	−0.0834101	−	0.996515i	$-0.526581\pi$
$733$	−4.51648	−0.166820	−0.0834101	+	0.996515i	$0.526581\pi$
$734$	18.2813	0.674774
$735$	0	0
$736$	−36.2084	−1.33466
$737$	0	0
$738$	0	0
$739$	10.0400	0.369327	0.184664	−	0.982802i	$-0.440880\pi$
$739$	10.0400	0.369327	0.184664	+	0.982802i	$0.440880\pi$
$740$	−12.2978	−0.452075
$741$	0	0
$742$	4.10878	0.150838
$743$	15.7027	0.576076	0.288038	−	0.957619i	$-0.406997\pi$
$743$	15.7027	0.576076	0.288038	+	0.957619i	$0.406997\pi$
$744$	0	0
$745$	−40.9808	−1.50142
$746$	−25.3858	−0.929439
$747$	0	0
$748$	0	0
$749$	−5.34760	−0.195397
$750$	0	0
$751$	19.2780	0.703466	0.351733	−	0.936100i	$-0.385593\pi$
$751$	19.2780	0.703466	0.351733	+	0.936100i	$0.385593\pi$
$752$	−0.160639	−0.00585790
$753$	0	0
$754$	−25.2101	−0.918099
$755$	−60.8188	−2.21342
$756$	0	0
$757$	37.5431	1.36453	0.682263	−	0.731106i	$-0.260996\pi$
$757$	37.5431	1.36453	0.682263	+	0.731106i	$0.260996\pi$
$758$	−27.8350	−1.01101
$759$	0	0
$760$	62.6414	2.27224
$761$	38.7469	1.40458	0.702288	−	0.711893i	$-0.252162\pi$
$761$	38.7469	1.40458	0.702288	+	0.711893i	$0.252162\pi$
$762$	0	0
$763$	1.44173	0.0521942
$764$	8.63494	0.312401
$765$	0	0
$766$	3.12878	0.113048
$767$	28.2194	1.01894
$768$	0	0
$769$	−24.0014	−0.865513	−0.432756	−	0.901511i	$-0.642459\pi$
$769$	−24.0014	−0.865513	−0.432756	+	0.901511i	$0.642459\pi$
$770$	0	0
$771$	0	0
$772$	−20.9156	−0.752770
$773$	39.2714	1.41250	0.706248	−	0.707965i	$-0.250387\pi$
$773$	39.2714	1.41250	0.706248	+	0.707965i	$0.250387\pi$
$774$	0	0
$775$	−50.2587	−1.80534
$776$	−7.33520	−0.263318
$777$	0	0
$778$	9.12658	0.327204
$779$	17.5194	0.627696
$780$	0	0
$781$	0	0
$782$	9.47956	0.338988
$783$	0	0
$784$	0.0189705	0.000677517 0
$785$	79.0682	2.82207
$786$	0	0
$787$	25.4235	0.906251	0.453125	−	0.891447i	$-0.350309\pi$
$787$	25.4235	0.906251	0.453125	+	0.891447i	$0.350309\pi$
$788$	7.96457	0.283726
$789$	0	0
$790$	−22.2163	−0.790422
$791$	19.5174	0.693959
$792$	0	0
$793$	−32.3432	−1.14854
$794$	−15.4508	−0.548329
$795$	0	0
$796$	16.4521	0.583128
$797$	36.7626	1.30220	0.651099	−	0.758993i	$-0.274308\pi$
$797$	36.7626	1.30220	0.651099	+	0.758993i	$0.274308\pi$
$798$	0	0
$799$	14.4091	0.509757
$800$	−65.2562	−2.30716
$801$	0	0
$802$	7.04287	0.248692
$803$	0	0
$804$	0	0
$805$	−25.9775	−0.915587
$806$	−23.1598	−0.815770
$807$	0	0
$808$	17.0183	0.598700
$809$	40.4785	1.42315	0.711574	−	0.702612i	$-0.247983\pi$
$809$	40.4785	1.42315	0.711574	+	0.702612i	$0.247983\pi$
$810$	0	0
$811$	−9.55021	−0.335353	−0.167677	−	0.985842i	$-0.553626\pi$
$811$	−9.55021	−0.335353	−0.167677	+	0.985842i	$0.553626\pi$
$812$	−5.89064	−0.206721
$813$	0	0
$814$	0	0
$815$	49.2212	1.72414
$816$	0	0
$817$	0.694095	0.0242833
$818$	−34.5930	−1.20952
$819$	0	0
$820$	−16.1836	−0.565157
$821$	36.2589	1.26544	0.632722	−	0.774379i	$-0.281938\pi$
$821$	36.2589	1.26544	0.632722	+	0.774379i	$0.281938\pi$
$822$	0	0
$823$	−41.7383	−1.45491	−0.727454	−	0.686157i	$-0.759296\pi$
$823$	−41.7383	−1.45491	−0.727454	+	0.686157i	$0.759296\pi$
$824$	36.4155	1.26859
$825$	0	0
$826$	−4.03873	−0.140525
$827$	−28.2659	−0.982901	−0.491450	−	0.870906i	$-0.663533\pi$
$827$	−28.2659	−0.982901	−0.491450	+	0.870906i	$0.663533\pi$
$828$	0	0
$829$	−2.01036	−0.0698226	−0.0349113	−	0.999390i	$-0.511115\pi$
$829$	−2.01036	−0.0698226	−0.0349113	+	0.999390i	$0.511115\pi$
$830$	−0.452457	−0.0157050
$831$	0	0
$832$	−29.8398	−1.03451
$833$	−1.70162	−0.0589578
$834$	0	0
$835$	53.5195	1.85212
$836$	0	0
$837$	0	0
$838$	14.0951	0.486908
$839$	24.2099	0.835818	0.417909	−	0.908489i	$-0.362763\pi$
$839$	24.2099	0.835818	0.417909	+	0.908489i	$0.362763\pi$
$840$	0	0
$841$	−6.44371	−0.222197
$842$	−21.6231	−0.745182
$843$	0	0
$844$	−3.61125	−0.124304
$845$	97.9060	3.36807
$846$	0	0
$847$	0	0
$848$	0.0894276	0.00307096
$849$	0	0
$850$	17.0844	0.585991
$851$	15.5923	0.534497
$852$	0	0
$853$	20.1429	0.689680	0.344840	−	0.938662i	$-0.387933\pi$
$853$	20.1429	0.689680	0.344840	+	0.938662i	$0.387933\pi$
$854$	4.62893	0.158399
$855$	0	0
$856$	15.1030	0.516211
$857$	26.6092	0.908953	0.454477	−	0.890759i	$-0.349826\pi$
$857$	26.6092	0.908953	0.454477	+	0.890759i	$0.349826\pi$
$858$	0	0
$859$	14.4645	0.493522	0.246761	−	0.969076i	$-0.420634\pi$
$859$	14.4645	0.493522	0.246761	+	0.969076i	$0.420634\pi$
$860$	−0.641176	−0.0218639
$861$	0	0
$862$	19.0725	0.649614
$863$	−28.3624	−0.965466	−0.482733	−	0.875768i	$-0.660356\pi$
$863$	−28.3624	−0.965466	−0.482733	+	0.875768i	$0.660356\pi$
$864$	0	0
$865$	59.8190	2.03391
$866$	−34.2923	−1.16530
$867$	0	0
$868$	−5.41156	−0.183680
$869$	0	0
$870$	0	0
$871$	−85.6789	−2.90312
$872$	−4.07184	−0.137890
$873$	0	0
$874$	−30.4010	−1.02833
$875$	−26.4958	−0.895723
$876$	0	0
$877$	10.1417	0.342461	0.171231	−	0.985231i	$-0.445226\pi$
$877$	10.1417	0.342461	0.171231	+	0.985231i	$0.445226\pi$
$878$	−8.36038	−0.282149
$879$	0	0
$880$	0	0
$881$	3.71883	0.125291	0.0626453	−	0.998036i	$-0.480046\pi$
$881$	3.71883	0.125291	0.0626453	+	0.998036i	$0.480046\pi$
$882$	0	0
$883$	17.3305	0.583218	0.291609	−	0.956538i	$-0.405809\pi$
$883$	17.3305	0.583218	0.291609	+	0.956538i	$0.405809\pi$
$884$	−12.8533	−0.432303
$885$	0	0
$886$	−0.548358	−0.0184224
$887$	5.40696	0.181548	0.0907740	−	0.995872i	$-0.471066\pi$
$887$	5.40696	0.181548	0.0907740	+	0.995872i	$0.471066\pi$
$888$	0	0
$889$	0.641713	0.0215224
$890$	−28.7913	−0.965087
$891$	0	0
$892$	4.16210	0.139357
$893$	−46.2100	−1.54636
$894$	0	0
$895$	47.5139	1.58822
$896$	−7.05948	−0.235841
$897$	0	0
$898$	−15.1105	−0.504242
$899$	20.7218	0.691111
$900$	0	0
$901$	−8.02152	−0.267236
$902$	0	0
$903$	0	0
$904$	−55.1223	−1.83334
$905$	80.0690	2.66158
$906$	0	0
$907$	−10.7020	−0.355353	−0.177677	−	0.984089i	$-0.556858\pi$
$907$	−10.7020	−0.355353	−0.177677	+	0.984089i	$0.556858\pi$
$908$	4.14003	0.137392
$909$	0	0
$910$	−21.5742	−0.715177
$911$	27.5708	0.913463	0.456732	−	0.889605i	$-0.349020\pi$
$911$	27.5708	0.913463	0.456732	+	0.889605i	$0.349020\pi$
$912$	0	0
$913$	0	0
$914$	25.3771	0.839400
$915$	0	0
$916$	−8.43586	−0.278729
$917$	−13.9987	−0.462278
$918$	0	0
$919$	36.4021	1.20080	0.600398	−	0.799702i	$-0.295009\pi$
$919$	36.4021	1.20080	0.600398	+	0.799702i	$0.295009\pi$
$920$	73.3674	2.41885
$921$	0	0
$922$	13.2353	0.435882
$923$	13.5128	0.444781
$924$	0	0
$925$	28.1010	0.923956
$926$	−15.5918	−0.512378
$927$	0	0
$928$	26.9054	0.883211
$929$	−26.2401	−0.860910	−0.430455	−	0.902612i	$-0.641647\pi$
$929$	−26.2401	−0.860910	−0.430455	+	0.902612i	$0.641647\pi$
$930$	0	0
$931$	5.45712	0.178850
$932$	7.67195	0.251303
$933$	0	0
$934$	−26.4184	−0.864437
$935$	0	0
$936$	0	0
$937$	23.8799	0.780121	0.390061	−	0.920789i	$-0.372454\pi$
$937$	23.8799	0.780121	0.390061	+	0.920789i	$0.372454\pi$
$938$	12.2623	0.400378
$939$	0	0
$940$	42.6869	1.39229
$941$	−17.7793	−0.579588	−0.289794	−	0.957089i	$-0.593587\pi$
$941$	−17.7793	−0.579588	−0.289794	+	0.957089i	$0.593587\pi$
$942$	0	0
$943$	20.5192	0.668196
$944$	−0.0879031	−0.00286100
$945$	0	0
$946$	0	0
$947$	−31.7071	−1.03034	−0.515171	−	0.857088i	$-0.672271\pi$
$947$	−31.7071	−1.03034	−0.515171	+	0.857088i	$0.672271\pi$
$948$	0	0
$949$	33.9456	1.10192
$950$	−54.7898	−1.77762
$951$	0	0
$952$	4.80584	0.155758
$953$	3.28213	0.106319	0.0531593	−	0.998586i	$-0.483071\pi$
$953$	3.28213	0.106319	0.0531593	+	0.998586i	$0.483071\pi$
$954$	0	0
$955$	−28.2959	−0.915633
$956$	25.3995	0.821478
$957$	0	0
$958$	−6.92707	−0.223804
$959$	−2.70017	−0.0871931
$960$	0	0
$961$	−11.9635	−0.385918
$962$	12.9493	0.417502
$963$	0	0
$964$	−1.29132	−0.0415907
$965$	68.5386	2.20633
$966$	0	0
$967$	−3.83683	−0.123384	−0.0616921	−	0.998095i	$-0.519650\pi$
$967$	−3.83683	−0.123384	−0.0616921	+	0.998095i	$0.519650\pi$
$968$	0	0
$969$	0	0
$970$	9.20065	0.295415
$971$	−44.3268	−1.42251	−0.711257	−	0.702932i	$-0.751874\pi$
$971$	−44.3268	−1.42251	−0.711257	+	0.702932i	$0.751874\pi$
$972$	0	0
$973$	2.06192	0.0661021
$974$	8.86517	0.284058
$975$	0	0
$976$	0.100749	0.00322489
$977$	−9.69774	−0.310258	−0.155129	−	0.987894i	$-0.549579\pi$
$977$	−9.69774	−0.310258	−0.155129	+	0.987894i	$0.549579\pi$
$978$	0	0
$979$	0	0
$980$	−5.04106	−0.161031
$981$	0	0
$982$	0.939032	0.0299657
$983$	−28.6752	−0.914597	−0.457298	−	0.889313i	$-0.651183\pi$
$983$	−28.6752	−0.914597	−0.457298	+	0.889313i	$0.651183\pi$
$984$	0	0
$985$	−26.0991	−0.831587
$986$	−7.04396	−0.224325
$987$	0	0
$988$	41.2206	1.31140
$989$	0.812944	0.0258501
$990$	0	0
$991$	−14.0455	−0.446170	−0.223085	−	0.974799i	$-0.571613\pi$
$991$	−14.0455	−0.446170	−0.223085	+	0.974799i	$0.571613\pi$
$992$	24.7172	0.784771
$993$	0	0
$994$	−1.93395	−0.0613410
$995$	−53.9119	−1.70912
$996$	0	0
$997$	−36.2102	−1.14679	−0.573394	−	0.819280i	$-0.694374\pi$
$997$	−36.2102	−1.14679	−0.573394	+	0.819280i	$0.694374\pi$
$998$	−30.1714	−0.955058
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.cx.1.4		10
3.2	odd	2		2541.2.a.bq.1.7		10
11.5	even	5		693.2.m.j.190.4		20
11.9	even	5		693.2.m.j.631.4		20
11.10	odd	2		7623.2.a.cy.1.7		10
33.5	odd	10		231.2.j.g.190.2	yes	20
33.20	odd	10		231.2.j.g.169.2	✓	20
33.32	even	2		2541.2.a.br.1.4		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.j.g.169.2	✓	20	33.20	odd	10
231.2.j.g.190.2	yes	20	33.5	odd	10
693.2.m.j.190.4		20	11.5	even	5
693.2.m.j.631.4		20	11.9	even	5
2541.2.a.bq.1.7		10	3.2	odd	2
2541.2.a.br.1.4		10	33.32	even	2
7623.2.a.cx.1.4		10	1.1	even	1	trivial
7623.2.a.cy.1.7		10	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-0.871604 q^{2} -1.24031 q^{4} +4.06436 q^{5} -1.00000 q^{7} +2.82426 q^{8} +O(q^{10})\)	\(q-0.871604 q^{2} -1.24031 q^{4} +4.06436 q^{5} -1.00000 q^{7} +2.82426 q^{8} -3.54252 q^{10} -6.09006 q^{13} +0.871604 q^{14} +0.0189705 q^{16} -1.70162 q^{17} +5.45712 q^{19} -5.04106 q^{20} +6.39153 q^{23} +11.5191 q^{25} +5.30813 q^{26} +1.24031 q^{28} -4.74935 q^{29} -4.36309 q^{31} -5.66506 q^{32} +1.48314 q^{34} -4.06436 q^{35} +2.43952 q^{37} -4.75645 q^{38} +11.4788 q^{40} +3.21037 q^{41} +0.127191 q^{43} -5.57089 q^{46} -8.46784 q^{47} +1.00000 q^{49} -10.0401 q^{50} +7.55354 q^{52} +4.71404 q^{53} -2.82426 q^{56} +4.13955 q^{58} -4.63368 q^{59} +5.31081 q^{61} +3.80289 q^{62} +4.89975 q^{64} -24.7522 q^{65} +14.0686 q^{67} +2.11054 q^{68} +3.54252 q^{70} -2.21883 q^{71} -5.57393 q^{73} -2.12630 q^{74} -6.76849 q^{76} +6.27134 q^{79} +0.0771030 q^{80} -2.79817 q^{82} +0.127722 q^{83} -6.91602 q^{85} -0.110860 q^{86} +8.12736 q^{89} +6.09006 q^{91} -7.92745 q^{92} +7.38061 q^{94} +22.1797 q^{95} -2.59721 q^{97} -0.871604 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 18 q^{4} - 5 q^{5} - 10 q^{7} - 3 q^{8}+O(q^{10}) \) 10 * q + 18 * q^4 - 5 * q^5 - 10 * q^7 - 3 * q^8	\( 10 q + 18 q^{4} - 5 q^{5} - 10 q^{7} - 3 q^{8} + 6 q^{10} - 6 q^{13} + 38 q^{16} + 8 q^{17} - 7 q^{20} + 31 q^{25} - q^{26} - 18 q^{28} - 14 q^{29} + 26 q^{31} - 41 q^{32} + 21 q^{34} + 5 q^{35} + 24 q^{37} - 8 q^{38} + 5 q^{40} + 19 q^{41} + 6 q^{43} + q^{46} - 15 q^{47} + 10 q^{49} - q^{50} + 25 q^{52} + q^{53} + 3 q^{56} + 11 q^{58} - 23 q^{59} + 11 q^{62} + 53 q^{64} - 29 q^{65} + 38 q^{67} + 87 q^{68} - 6 q^{70} - 26 q^{71} + q^{73} - 39 q^{74} + 2 q^{76} - 5 q^{79} - 6 q^{80} + 5 q^{82} + 6 q^{83} + q^{85} + 41 q^{86} + 9 q^{89} + 6 q^{91} + 48 q^{92} - 42 q^{95} + 24 q^{97}+O(q^{100}) \) 10 * q + 18 * q^4 - 5 * q^5 - 10 * q^7 - 3 * q^8 + 6 * q^10 - 6 * q^13 + 38 * q^16 + 8 * q^17 - 7 * q^20 + 31 * q^25 - q^26 - 18 * q^28 - 14 * q^29 + 26 * q^31 - 41 * q^32 + 21 * q^34 + 5 * q^35 + 24 * q^37 - 8 * q^38 + 5 * q^40 + 19 * q^41 + 6 * q^43 + q^46 - 15 * q^47 + 10 * q^49 - q^50 + 25 * q^52 + q^53 + 3 * q^56 + 11 * q^58 - 23 * q^59 + 11 * q^62 + 53 * q^64 - 29 * q^65 + 38 * q^67 + 87 * q^68 - 6 * q^70 - 26 * q^71 + q^73 - 39 * q^74 + 2 * q^76 - 5 * q^79 - 6 * q^80 + 5 * q^82 + 6 * q^83 + q^85 + 41 * q^86 + 9 * q^89 + 6 * q^91 + 48 * q^92 - 42 * q^95 + 24 * q^97

Introduction

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