LMFDB - Embedded newform 7623.2.a.db.1.16

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			1.16
Root			$2.68084$ 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)

$f(q)$	$=$	$q+2.68084 q^{2} +5.18691 q^{4} +1.17039 q^{5} -1.00000 q^{7} +8.54361 q^{8} +O(q^{10})$	$q+2.68084 q^{2} +5.18691 q^{4} +1.17039 q^{5} -1.00000 q^{7} +8.54361 q^{8} +3.13763 q^{10} +1.68268 q^{13} -2.68084 q^{14} +12.5302 q^{16} +5.59675 q^{17} -4.35416 q^{19} +6.07072 q^{20} -0.119612 q^{23} -3.63018 q^{25} +4.51101 q^{26} -5.18691 q^{28} +4.39599 q^{29} -6.31294 q^{31} +16.5044 q^{32} +15.0040 q^{34} -1.17039 q^{35} +5.85418 q^{37} -11.6728 q^{38} +9.99937 q^{40} +11.0165 q^{41} -5.97227 q^{43} -0.320661 q^{46} +12.5812 q^{47} +1.00000 q^{49} -9.73195 q^{50} +8.72794 q^{52} +0.417004 q^{53} -8.54361 q^{56} +11.7849 q^{58} -0.723544 q^{59} +8.85799 q^{61} -16.9240 q^{62} +19.1851 q^{64} +1.96940 q^{65} +8.78741 q^{67} +29.0299 q^{68} -3.13763 q^{70} +3.21286 q^{71} +14.5891 q^{73} +15.6941 q^{74} -22.5846 q^{76} -14.4761 q^{79} +14.6653 q^{80} +29.5335 q^{82} +3.45931 q^{83} +6.55039 q^{85} -16.0107 q^{86} -12.0621 q^{89} -1.68268 q^{91} -0.620417 q^{92} +33.7283 q^{94} -5.09607 q^{95} -5.99575 q^{97} +2.68084 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.68084	1.89564	0.947821	−	0.318804i	$-0.103281\pi$
$2$	2.68084	1.89564	0.947821	+	0.318804i	$0.103281\pi$
$3$	0	0
$3$	0	0
$4$	5.18691	2.59346
$5$	1.17039	0.523415	0.261708	−	0.965147i	$-0.415714\pi$
$5$	1.17039	0.523415	0.261708	+	0.965147i	$0.415714\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	8.54361	3.02062
$9$	0	0
$10$	3.13763	0.992207
$11$	0	0
$11$	0	0
$12$	0	0
$13$	1.68268	0.466693	0.233346	−	0.972394i	$-0.425032\pi$
$13$	1.68268	0.466693	0.233346	+	0.972394i	$0.425032\pi$
$14$	−2.68084	−0.716485
$15$	0	0
$16$	12.5302	3.13256
$17$	5.59675	1.35741	0.678706	−	0.734410i	$-0.262541\pi$
$17$	5.59675	1.35741	0.678706	+	0.734410i	$0.262541\pi$
$18$	0	0
$19$	−4.35416	−0.998913	−0.499456	−	0.866339i	$-0.666467\pi$
$19$	−4.35416	−0.998913	−0.499456	+	0.866339i	$0.666467\pi$
$20$	6.07072	1.35745
$21$	0	0
$22$	0	0
$23$	−0.119612	−0.0249408	−0.0124704	−	0.999922i	$-0.503970\pi$
$23$	−0.119612	−0.0249408	−0.0124704	+	0.999922i	$0.503970\pi$
$24$	0	0
$25$	−3.63018	−0.726037
$26$	4.51101	0.884682
$27$	0	0
$28$	−5.18691	−0.980234
$29$	4.39599	0.816315	0.408157	−	0.912912i	$-0.366171\pi$
$29$	4.39599	0.816315	0.408157	+	0.912912i	$0.366171\pi$
$30$	0	0
$31$	−6.31294	−1.13384	−0.566919	−	0.823774i	$-0.691865\pi$
$31$	−6.31294	−1.13384	−0.566919	+	0.823774i	$0.691865\pi$
$32$	16.5044	2.91759
$33$	0	0
$34$	15.0040	2.57317
$35$	−1.17039	−0.197832
$36$	0	0
$37$	5.85418	0.962421	0.481210	−	0.876605i	$-0.340197\pi$
$37$	5.85418	0.962421	0.481210	+	0.876605i	$0.340197\pi$
$38$	−11.6728	−1.89358
$39$	0	0
$40$	9.99937	1.58104
$41$	11.0165	1.72049	0.860244	−	0.509883i	$-0.170311\pi$
$41$	11.0165	1.72049	0.860244	+	0.509883i	$0.170311\pi$
$42$	0	0
$43$	−5.97227	−0.910762	−0.455381	−	0.890297i	$-0.650497\pi$
$43$	−5.97227	−0.910762	−0.455381	+	0.890297i	$0.650497\pi$
$44$	0	0
$45$	0	0
$46$	−0.320661	−0.0472789
$47$	12.5812	1.83516	0.917580	−	0.397552i	$-0.130140\pi$
$47$	12.5812	1.83516	0.917580	+	0.397552i	$0.130140\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−9.73195	−1.37631
$51$	0	0
$52$	8.72794	1.21035
$53$	0.417004	0.0572800	0.0286400	−	0.999590i	$-0.490882\pi$
$53$	0.417004	0.0572800	0.0286400	+	0.999590i	$0.490882\pi$
$54$	0	0
$55$	0	0
$56$	−8.54361	−1.14169
$57$	0	0
$58$	11.7849	1.54744
$59$	−0.723544	−0.0941974	−0.0470987	−	0.998890i	$-0.514998\pi$
$59$	−0.723544	−0.0941974	−0.0470987	+	0.998890i	$0.514998\pi$
$60$	0	0
$61$	8.85799	1.13415	0.567075	−	0.823666i	$-0.308075\pi$
$61$	8.85799	1.13415	0.567075	+	0.823666i	$0.308075\pi$
$62$	−16.9240	−2.14935
$63$	0	0
$64$	19.1851	2.39814
$65$	1.96940	0.244274
$66$	0	0
$67$	8.78741	1.07355	0.536777	−	0.843724i	$-0.319642\pi$
$67$	8.78741	1.07355	0.536777	+	0.843724i	$0.319642\pi$
$68$	29.0299	3.52039
$69$	0	0
$70$	−3.13763	−0.375019
$71$	3.21286	0.381297	0.190648	−	0.981658i	$-0.438941\pi$
$71$	3.21286	0.381297	0.190648	+	0.981658i	$0.438941\pi$
$72$	0	0
$73$	14.5891	1.70752	0.853762	−	0.520664i	$-0.174315\pi$
$73$	14.5891	1.70752	0.853762	+	0.520664i	$0.174315\pi$
$74$	15.6941	1.82440
$75$	0	0
$76$	−22.5846	−2.59064
$77$	0	0
$78$	0	0
$79$	−14.4761	−1.62869	−0.814347	−	0.580378i	$-0.802905\pi$
$79$	−14.4761	−1.62869	−0.814347	+	0.580378i	$0.802905\pi$
$80$	14.6653	1.63963
$81$	0	0
$82$	29.5335	3.26143
$83$	3.45931	0.379709	0.189854	−	0.981812i	$-0.439198\pi$
$83$	3.45931	0.379709	0.189854	+	0.981812i	$0.439198\pi$
$84$	0	0
$85$	6.55039	0.710490
$86$	−16.0107	−1.72648
$87$	0	0
$88$	0	0
$89$	−12.0621	−1.27858	−0.639292	−	0.768964i	$-0.720772\pi$
$89$	−12.0621	−1.27858	−0.639292	+	0.768964i	$0.720772\pi$
$90$	0	0
$91$	−1.68268	−0.176393
$92$	−0.620417	−0.0646830
$93$	0	0
$94$	33.7283	3.47880
$95$	−5.09607	−0.522846
$96$	0	0
$97$	−5.99575	−0.608776	−0.304388	−	0.952548i	$-0.598452\pi$
$97$	−5.99575	−0.608776	−0.304388	+	0.952548i	$0.598452\pi$
$98$	2.68084	0.270806
$99$	0	0
$100$	−18.8294	−1.88294
$101$	−0.101612	−0.0101108	−0.00505540	−	0.999987i	$-0.501609\pi$
$101$	−0.101612	−0.0101108	−0.00505540	+	0.999987i	$0.501609\pi$
$102$	0	0
$103$	−1.73990	−0.171438	−0.0857188	−	0.996319i	$-0.527319\pi$
$103$	−1.73990	−0.171438	−0.0857188	+	0.996319i	$0.527319\pi$
$104$	14.3762	1.40970
$105$	0	0
$106$	1.11792	0.108582
$107$	−11.0032	−1.06372	−0.531861	−	0.846831i	$-0.678507\pi$
$107$	−11.0032	−1.06372	−0.531861	+	0.846831i	$0.678507\pi$
$108$	0	0
$109$	−16.5040	−1.58080	−0.790400	−	0.612592i	$-0.790127\pi$
$109$	−16.5040	−1.58080	−0.790400	+	0.612592i	$0.790127\pi$
$110$	0	0
$111$	0	0
$112$	−12.5302	−1.18400
$113$	−6.28108	−0.590875	−0.295437	−	0.955362i	$-0.595465\pi$
$113$	−6.28108	−0.590875	−0.295437	+	0.955362i	$0.595465\pi$
$114$	0	0
$115$	−0.139993	−0.0130544
$116$	22.8016	2.11708
$117$	0	0
$118$	−1.93971	−0.178564
$119$	−5.59675	−0.513053
$120$	0	0
$121$	0	0
$122$	23.7469	2.14994
$123$	0	0
$124$	−32.7447	−2.94056
$125$	−10.1007	−0.903434
$126$	0	0
$127$	−17.2379	−1.52962	−0.764809	−	0.644257i	$-0.777167\pi$
$127$	−17.2379	−1.52962	−0.764809	+	0.644257i	$0.777167\pi$
$128$	18.4235	1.62843
$129$	0	0
$130$	5.27965	0.463056
$131$	0.324063	0.0283135	0.0141567	−	0.999900i	$-0.495494\pi$
$131$	0.324063	0.0283135	0.0141567	+	0.999900i	$0.495494\pi$
$132$	0	0
$133$	4.35416	0.377553
$134$	23.5577	2.03507
$135$	0	0
$136$	47.8165	4.10023
$137$	10.3726	0.886188	0.443094	−	0.896475i	$-0.353881\pi$
$137$	10.3726	0.886188	0.443094	+	0.896475i	$0.353881\pi$
$138$	0	0
$139$	−13.3776	−1.13467	−0.567336	−	0.823486i	$-0.692026\pi$
$139$	−13.3776	−1.13467	−0.567336	+	0.823486i	$0.692026\pi$
$140$	−6.07072	−0.513069
$141$	0	0
$142$	8.61317	0.722801
$143$	0	0
$144$	0	0
$145$	5.14503	0.427271
$146$	39.1110	3.23685
$147$	0	0
$148$	30.3651	2.49600
$149$	−9.66988	−0.792188	−0.396094	−	0.918210i	$-0.629635\pi$
$149$	−9.66988	−0.792188	−0.396094	+	0.918210i	$0.629635\pi$
$150$	0	0
$151$	−2.56082	−0.208396	−0.104198	−	0.994557i	$-0.533228\pi$
$151$	−2.56082	−0.208396	−0.104198	+	0.994557i	$0.533228\pi$
$152$	−37.2002	−3.01734
$153$	0	0
$154$	0	0
$155$	−7.38862	−0.593468
$156$	0	0
$157$	18.5092	1.47719	0.738597	−	0.674148i	$-0.235489\pi$
$157$	18.5092	1.47719	0.738597	+	0.674148i	$0.235489\pi$
$158$	−38.8083	−3.08742
$159$	0	0
$160$	19.3166	1.52711
$161$	0.119612	0.00942675
$162$	0	0
$163$	6.47823	0.507414	0.253707	−	0.967281i	$-0.418350\pi$
$163$	6.47823	0.507414	0.253707	+	0.967281i	$0.418350\pi$
$164$	57.1416	4.46201
$165$	0	0
$166$	9.27386	0.719791
$167$	−25.2567	−1.95442	−0.977212	−	0.212266i	$-0.931916\pi$
$167$	−25.2567	−1.95442	−0.977212	+	0.212266i	$0.931916\pi$
$168$	0	0
$169$	−10.1686	−0.782198
$170$	17.5606	1.34683
$171$	0	0
$172$	−30.9776	−2.36202
$173$	−9.26804	−0.704636	−0.352318	−	0.935880i	$-0.614606\pi$
$173$	−9.26804	−0.704636	−0.352318	+	0.935880i	$0.614606\pi$
$174$	0	0
$175$	3.63018	0.274416
$176$	0	0
$177$	0	0
$178$	−32.3367	−2.42373
$179$	15.4438	1.15432	0.577161	−	0.816630i	$-0.304160\pi$
$179$	15.4438	1.15432	0.577161	+	0.816630i	$0.304160\pi$
$180$	0	0
$181$	10.5318	0.782824	0.391412	−	0.920215i	$-0.371987\pi$
$181$	10.5318	0.782824	0.391412	+	0.920215i	$0.371987\pi$
$182$	−4.51101	−0.334378
$183$	0	0
$184$	−1.02192	−0.0753369
$185$	6.85168	0.503746
$186$	0	0
$187$	0	0
$188$	65.2577	4.75940
$189$	0	0
$190$	−13.6618	−0.991128
$191$	6.86336	0.496615	0.248308	−	0.968681i	$-0.420126\pi$
$191$	6.86336	0.496615	0.248308	+	0.968681i	$0.420126\pi$
$192$	0	0
$193$	−22.3297	−1.60733	−0.803663	−	0.595085i	$-0.797119\pi$
$193$	−22.3297	−1.60733	−0.803663	+	0.595085i	$0.797119\pi$
$194$	−16.0737	−1.15402
$195$	0	0
$196$	5.18691	0.370494
$197$	4.45203	0.317194	0.158597	−	0.987343i	$-0.449303\pi$
$197$	4.45203	0.317194	0.158597	+	0.987343i	$0.449303\pi$
$198$	0	0
$199$	−15.0993	−1.07036	−0.535180	−	0.844738i	$-0.679756\pi$
$199$	−15.0993	−1.07036	−0.535180	+	0.844738i	$0.679756\pi$
$200$	−31.0149	−2.19308
$201$	0	0
$202$	−0.272406	−0.0191665
$203$	−4.39599	−0.308538
$204$	0	0
$205$	12.8936	0.900529
$206$	−4.66440	−0.324984
$207$	0	0
$208$	21.0844	1.46194
$209$	0	0
$210$	0	0
$211$	22.8486	1.57296	0.786480	−	0.617616i	$-0.211901\pi$
$211$	22.8486	1.57296	0.786480	+	0.617616i	$0.211901\pi$
$212$	2.16297	0.148553
$213$	0	0
$214$	−29.4979	−2.01644
$215$	−6.98989	−0.476707
$216$	0	0
$217$	6.31294	0.428551
$218$	−44.2447	−2.99663
$219$	0	0
$220$	0	0
$221$	9.41757	0.633494
$222$	0	0
$223$	9.71068	0.650275	0.325137	−	0.945667i	$-0.394589\pi$
$223$	9.71068	0.650275	0.325137	+	0.945667i	$0.394589\pi$
$224$	−16.5044	−1.10274
$225$	0	0
$226$	−16.8386	−1.12009
$227$	−18.6174	−1.23568	−0.617838	−	0.786305i	$-0.711991\pi$
$227$	−18.6174	−1.23568	−0.617838	+	0.786305i	$0.711991\pi$
$228$	0	0
$229$	9.04273	0.597561	0.298780	−	0.954322i	$-0.403420\pi$
$229$	9.04273	0.597561	0.298780	+	0.954322i	$0.403420\pi$
$230$	−0.375299	−0.0247465
$231$	0	0
$232$	37.5576	2.46578
$233$	8.62319	0.564924	0.282462	−	0.959278i	$-0.408849\pi$
$233$	8.62319	0.564924	0.282462	+	0.959278i	$0.408849\pi$
$234$	0	0
$235$	14.7250	0.960550
$236$	−3.75296	−0.244297
$237$	0	0
$238$	−15.0040	−0.972565
$239$	−20.6870	−1.33813	−0.669067	−	0.743202i	$-0.733306\pi$
$239$	−20.6870	−1.33813	−0.669067	+	0.743202i	$0.733306\pi$
$240$	0	0
$241$	24.8415	1.60018	0.800090	−	0.599880i	$-0.204785\pi$
$241$	24.8415	1.60018	0.800090	+	0.599880i	$0.204785\pi$
$242$	0	0
$243$	0	0
$244$	45.9456	2.94137
$245$	1.17039	0.0747736
$246$	0	0
$247$	−7.32668	−0.466185
$248$	−53.9353	−3.42490
$249$	0	0
$250$	−27.0784	−1.71259
$251$	−10.5076	−0.663236	−0.331618	−	0.943414i	$-0.607595\pi$
$251$	−10.5076	−0.663236	−0.331618	+	0.943414i	$0.607595\pi$
$252$	0	0
$253$	0	0
$254$	−46.2121	−2.89961
$255$	0	0
$256$	11.0203	0.688771
$257$	3.68093	0.229610	0.114805	−	0.993388i	$-0.463376\pi$
$257$	3.68093	0.229610	0.114805	+	0.993388i	$0.463376\pi$
$258$	0	0
$259$	−5.85418	−0.363761
$260$	10.2151	0.633514
$261$	0	0
$262$	0.868760	0.0536722
$263$	−10.1816	−0.627824	−0.313912	−	0.949452i	$-0.601640\pi$
$263$	−10.1816	−0.627824	−0.313912	+	0.949452i	$0.601640\pi$
$264$	0	0
$265$	0.488059	0.0299812
$266$	11.6728	0.715706
$267$	0	0
$268$	45.5795	2.78421
$269$	32.0196	1.95227	0.976135	−	0.217163i	$-0.0696802\pi$
$269$	32.0196	1.95227	0.976135	+	0.217163i	$0.0696802\pi$
$270$	0	0
$271$	20.7478	1.26034	0.630169	−	0.776458i	$-0.282986\pi$
$271$	20.7478	1.26034	0.630169	+	0.776458i	$0.282986\pi$
$272$	70.1286	4.25217
$273$	0	0
$274$	27.8072	1.67990
$275$	0	0
$276$	0	0
$277$	−15.1464	−0.910057	−0.455028	−	0.890477i	$-0.650371\pi$
$277$	−15.1464	−0.910057	−0.455028	+	0.890477i	$0.650371\pi$
$278$	−35.8632	−2.15093
$279$	0	0
$280$	−9.99937	−0.597577
$281$	25.7257	1.53467	0.767335	−	0.641247i	$-0.221583\pi$
$281$	25.7257	1.53467	0.767335	+	0.641247i	$0.221583\pi$
$282$	0	0
$283$	−17.9944	−1.06966	−0.534828	−	0.844961i	$-0.679624\pi$
$283$	−17.9944	−1.06966	−0.534828	+	0.844961i	$0.679624\pi$
$284$	16.6648	0.988876
$285$	0	0
$286$	0	0
$287$	−11.0165	−0.650283
$288$	0	0
$289$	14.3236	0.842567
$290$	13.7930	0.809953
$291$	0	0
$292$	75.6723	4.42839
$293$	−30.9599	−1.80870	−0.904349	−	0.426794i	$-0.859643\pi$
$293$	−30.9599	−1.80870	−0.904349	+	0.426794i	$0.859643\pi$
$294$	0	0
$295$	−0.846830	−0.0493043
$296$	50.0158	2.90711
$297$	0	0
$298$	−25.9234	−1.50170
$299$	−0.201269	−0.0116397
$300$	0	0
$301$	5.97227	0.344236
$302$	−6.86514	−0.395045
$303$	0	0
$304$	−54.5586	−3.12915
$305$	10.3673	0.593631
$306$	0	0
$307$	10.0193	0.571830	0.285915	−	0.958255i	$-0.407702\pi$
$307$	10.0193	0.571830	0.285915	+	0.958255i	$0.407702\pi$
$308$	0	0
$309$	0	0
$310$	−19.8077	−1.12500
$311$	16.3344	0.926240	0.463120	−	0.886296i	$-0.346730\pi$
$311$	16.3344	0.926240	0.463120	+	0.886296i	$0.346730\pi$
$312$	0	0
$313$	4.47058	0.252692	0.126346	−	0.991986i	$-0.459675\pi$
$313$	4.47058	0.252692	0.126346	+	0.991986i	$0.459675\pi$
$314$	49.6202	2.80023
$315$	0	0
$316$	−75.0865	−4.22395
$317$	−11.9782	−0.672765	−0.336382	−	0.941725i	$-0.609203\pi$
$317$	−11.9782	−0.672765	−0.336382	+	0.941725i	$0.609203\pi$
$318$	0	0
$319$	0	0
$320$	22.4541	1.25522
$321$	0	0
$322$	0.320661	0.0178697
$323$	−24.3692	−1.35594
$324$	0	0
$325$	−6.10845	−0.338836
$326$	17.3671	0.961876
$327$	0	0
$328$	94.1206	5.19694
$329$	−12.5812	−0.693625
$330$	0	0
$331$	−18.0873	−0.994168	−0.497084	−	0.867702i	$-0.665596\pi$
$331$	−18.0873	−0.994168	−0.497084	+	0.867702i	$0.665596\pi$
$332$	17.9431	0.984758
$333$	0	0
$334$	−67.7093	−3.70489
$335$	10.2847	0.561914
$336$	0	0
$337$	−19.0858	−1.03967	−0.519836	−	0.854266i	$-0.674007\pi$
$337$	−19.0858	−1.03967	−0.519836	+	0.854266i	$0.674007\pi$
$338$	−27.2603	−1.48277
$339$	0	0
$340$	33.9763	1.84262
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−51.0247	−2.75107
$345$	0	0
$346$	−24.8462	−1.33574
$347$	−5.31766	−0.285467	−0.142733	−	0.989761i	$-0.545589\pi$
$347$	−5.31766	−0.285467	−0.142733	+	0.989761i	$0.545589\pi$
$348$	0	0
$349$	20.6262	1.10410	0.552048	−	0.833813i	$-0.313847\pi$
$349$	20.6262	1.10410	0.552048	+	0.833813i	$0.313847\pi$
$350$	9.73195	0.520194
$351$	0	0
$352$	0	0
$353$	−7.31745	−0.389468	−0.194734	−	0.980856i	$-0.562384\pi$
$353$	−7.31745	−0.389468	−0.194734	+	0.980856i	$0.562384\pi$
$354$	0	0
$355$	3.76031	0.199576
$356$	−62.5652	−3.31595
$357$	0	0
$358$	41.4023	2.18818
$359$	24.2290	1.27876	0.639378	−	0.768892i	$-0.279192\pi$
$359$	24.2290	1.27876	0.639378	+	0.768892i	$0.279192\pi$
$360$	0	0
$361$	−0.0412986	−0.00217361
$362$	28.2342	1.48395
$363$	0	0
$364$	−8.72794	−0.457468
$365$	17.0749	0.893744
$366$	0	0
$367$	−22.5059	−1.17480	−0.587399	−	0.809298i	$-0.699848\pi$
$367$	−22.5059	−1.17480	−0.587399	+	0.809298i	$0.699848\pi$
$368$	−1.49877	−0.0781287
$369$	0	0
$370$	18.3683	0.954921
$371$	−0.417004	−0.0216498
$372$	0	0
$373$	−6.25290	−0.323763	−0.161881	−	0.986810i	$-0.551756\pi$
$373$	−6.25290	−0.323763	−0.161881	+	0.986810i	$0.551756\pi$
$374$	0	0
$375$	0	0
$376$	107.489	5.54332
$377$	7.39706	0.380968
$378$	0	0
$379$	12.8192	0.658480	0.329240	−	0.944246i	$-0.393207\pi$
$379$	12.8192	0.658480	0.329240	+	0.944246i	$0.393207\pi$
$380$	−26.4329	−1.35598
$381$	0	0
$382$	18.3996	0.941405
$383$	−6.46736	−0.330467	−0.165233	−	0.986255i	$-0.552838\pi$
$383$	−6.46736	−0.330467	−0.165233	+	0.986255i	$0.552838\pi$
$384$	0	0
$385$	0	0
$386$	−59.8624	−3.04691
$387$	0	0
$388$	−31.0994	−1.57883
$389$	14.8967	0.755295	0.377647	−	0.925949i	$-0.376733\pi$
$389$	14.8967	0.755295	0.377647	+	0.925949i	$0.376733\pi$
$390$	0	0
$391$	−0.669439	−0.0338550
$392$	8.54361	0.431517
$393$	0	0
$394$	11.9352	0.601286
$395$	−16.9428	−0.852483
$396$	0	0
$397$	1.85862	0.0932813	0.0466407	−	0.998912i	$-0.485148\pi$
$397$	1.85862	0.0932813	0.0466407	+	0.998912i	$0.485148\pi$
$398$	−40.4788	−2.02902
$399$	0	0
$400$	−45.4871	−2.27435
$401$	8.74249	0.436579	0.218290	−	0.975884i	$-0.429952\pi$
$401$	8.74249	0.436579	0.218290	+	0.975884i	$0.429952\pi$
$402$	0	0
$403$	−10.6227	−0.529154
$404$	−0.527054	−0.0262219
$405$	0	0
$406$	−11.7849	−0.584877
$407$	0	0
$408$	0	0
$409$	−12.7198	−0.628952	−0.314476	−	0.949265i	$-0.601829\pi$
$409$	−12.7198	−0.628952	−0.314476	+	0.949265i	$0.601829\pi$
$410$	34.5657	1.70708
$411$	0	0
$412$	−9.02472	−0.444616
$413$	0.723544	0.0356033
$414$	0	0
$415$	4.04875	0.198745
$416$	27.7716	1.36162
$417$	0	0
$418$	0	0
$419$	8.09885	0.395655	0.197827	−	0.980237i	$-0.436611\pi$
$419$	8.09885	0.395655	0.197827	+	0.980237i	$0.436611\pi$
$420$	0	0
$421$	−26.4786	−1.29049	−0.645243	−	0.763977i	$-0.723244\pi$
$421$	−26.4786	−1.29049	−0.645243	+	0.763977i	$0.723244\pi$
$422$	61.2534	2.98177
$423$	0	0
$424$	3.56272	0.173021
$425$	−20.3172	−0.985531
$426$	0	0
$427$	−8.85799	−0.428668
$428$	−57.0728	−2.75872
$429$	0	0
$430$	−18.7388	−0.903665
$431$	−0.356446	−0.0171694	−0.00858471	−	0.999963i	$-0.502733\pi$
$431$	−0.356446	−0.0171694	−0.00858471	+	0.999963i	$0.502733\pi$
$432$	0	0
$433$	−18.2884	−0.878886	−0.439443	−	0.898270i	$-0.644824\pi$
$433$	−18.2884	−0.878886	−0.439443	+	0.898270i	$0.644824\pi$
$434$	16.9240	0.812378
$435$	0	0
$436$	−85.6049	−4.09973
$437$	0.520810	0.0249137
$438$	0	0
$439$	−6.67633	−0.318644	−0.159322	−	0.987227i	$-0.550931\pi$
$439$	−6.67633	−0.318644	−0.159322	+	0.987227i	$0.550931\pi$
$440$	0	0
$441$	0	0
$442$	25.2470	1.20088
$443$	20.4403	0.971146	0.485573	−	0.874196i	$-0.338611\pi$
$443$	20.4403	0.971146	0.485573	+	0.874196i	$0.338611\pi$
$444$	0	0
$445$	−14.1174	−0.669230
$446$	26.0328	1.23269
$447$	0	0
$448$	−19.1851	−0.906411
$449$	−5.51429	−0.260236	−0.130118	−	0.991499i	$-0.541536\pi$
$449$	−5.51429	−0.260236	−0.130118	+	0.991499i	$0.541536\pi$
$450$	0	0
$451$	0	0
$452$	−32.5794	−1.53241
$453$	0	0
$454$	−49.9102	−2.34240
$455$	−1.96940	−0.0923269
$456$	0	0
$457$	−2.00275	−0.0936849	−0.0468424	−	0.998902i	$-0.514916\pi$
$457$	−2.00275	−0.0936849	−0.0468424	+	0.998902i	$0.514916\pi$
$458$	24.2421	1.13276
$459$	0	0
$460$	−0.726131	−0.0338561
$461$	−8.17436	−0.380718	−0.190359	−	0.981715i	$-0.560965\pi$
$461$	−8.17436	−0.380718	−0.190359	+	0.981715i	$0.560965\pi$
$462$	0	0
$463$	−4.26686	−0.198298	−0.0991488	−	0.995073i	$-0.531612\pi$
$463$	−4.26686	−0.198298	−0.0991488	+	0.995073i	$0.531612\pi$
$464$	55.0828	2.55715
$465$	0	0
$466$	23.1174	1.07089
$467$	−40.5138	−1.87476	−0.937378	−	0.348315i	$-0.886754\pi$
$467$	−40.5138	−1.87476	−0.937378	+	0.348315i	$0.886754\pi$
$468$	0	0
$469$	−8.78741	−0.405765
$470$	39.4753	1.82086
$471$	0	0
$472$	−6.18168	−0.284535
$473$	0	0
$474$	0	0
$475$	15.8064	0.725247
$476$	−29.0299	−1.33058
$477$	0	0
$478$	−55.4587	−2.53662
$479$	8.24083	0.376533	0.188267	−	0.982118i	$-0.439713\pi$
$479$	8.24083	0.376533	0.188267	+	0.982118i	$0.439713\pi$
$480$	0	0
$481$	9.85073	0.449155
$482$	66.5960	3.03337
$483$	0	0
$484$	0	0
$485$	−7.01738	−0.318643
$486$	0	0
$487$	−0.964882	−0.0437230	−0.0218615	−	0.999761i	$-0.506959\pi$
$487$	−0.964882	−0.0437230	−0.0218615	+	0.999761i	$0.506959\pi$
$488$	75.6792	3.42584
$489$	0	0
$490$	3.13763	0.141744
$491$	−30.4284	−1.37321	−0.686607	−	0.727029i	$-0.740901\pi$
$491$	−30.4284	−1.37321	−0.686607	+	0.727029i	$0.740901\pi$
$492$	0	0
$493$	24.6033	1.10808
$494$	−19.6417	−0.883720
$495$	0	0
$496$	−79.1027	−3.55181
$497$	−3.21286	−0.144117
$498$	0	0
$499$	−37.5590	−1.68137	−0.840687	−	0.541522i	$-0.817848\pi$
$499$	−37.5590	−1.68137	−0.840687	+	0.541522i	$0.817848\pi$
$500$	−52.3914	−2.34302
$501$	0	0
$502$	−28.1693	−1.25726
$503$	22.9897	1.02506	0.512530	−	0.858669i	$-0.328708\pi$
$503$	22.9897	1.02506	0.512530	+	0.858669i	$0.328708\pi$
$504$	0	0
$505$	−0.118926	−0.00529214
$506$	0	0
$507$	0	0
$508$	−89.4116	−3.96700
$509$	−4.19559	−0.185966	−0.0929831	−	0.995668i	$-0.529640\pi$
$509$	−4.19559	−0.185966	−0.0929831	+	0.995668i	$0.529640\pi$
$510$	0	0
$511$	−14.5891	−0.645383
$512$	−7.30328	−0.322763
$513$	0	0
$514$	9.86798	0.435258
$515$	−2.03637	−0.0897330
$516$	0	0
$517$	0	0
$518$	−15.6941	−0.689560
$519$	0	0
$520$	16.8258	0.737859
$521$	36.9418	1.61845	0.809225	−	0.587499i	$-0.199887\pi$
$521$	36.9418	1.61845	0.809225	+	0.587499i	$0.199887\pi$
$522$	0	0
$523$	1.84645	0.0807395	0.0403697	−	0.999185i	$-0.487146\pi$
$523$	1.84645	0.0807395	0.0403697	+	0.999185i	$0.487146\pi$
$524$	1.68088	0.0734298
$525$	0	0
$526$	−27.2953	−1.19013
$527$	−35.3320	−1.53909
$528$	0	0
$529$	−22.9857	−0.999378
$530$	1.30841	0.0568336
$531$	0	0
$532$	22.5846	0.979168
$533$	18.5373	0.802939
$534$	0	0
$535$	−12.8781	−0.556769
$536$	75.0762	3.24280
$537$	0	0
$538$	85.8395	3.70081
$539$	0	0
$540$	0	0
$541$	9.73546	0.418560	0.209280	−	0.977856i	$-0.432888\pi$
$541$	9.73546	0.418560	0.209280	+	0.977856i	$0.432888\pi$
$542$	55.6215	2.38915
$543$	0	0
$544$	92.3708	3.96037
$545$	−19.3162	−0.827414
$546$	0	0
$547$	1.27745	0.0546199	0.0273100	−	0.999627i	$-0.491306\pi$
$547$	1.27745	0.0546199	0.0273100	+	0.999627i	$0.491306\pi$
$548$	53.8016	2.29829
$549$	0	0
$550$	0	0
$551$	−19.1408	−0.815427
$552$	0	0
$553$	14.4761	0.615588
$554$	−40.6050	−1.72514
$555$	0	0
$556$	−69.3884	−2.94272
$557$	−36.7874	−1.55873	−0.779366	−	0.626569i	$-0.784459\pi$
$557$	−36.7874	−1.55873	−0.779366	+	0.626569i	$0.784459\pi$
$558$	0	0
$559$	−10.0494	−0.425046
$560$	−14.6653	−0.619721
$561$	0	0
$562$	68.9667	2.90918
$563$	14.6601	0.617849	0.308924	−	0.951087i	$-0.400031\pi$
$563$	14.6601	0.617849	0.308924	+	0.951087i	$0.400031\pi$
$564$	0	0
$565$	−7.35133	−0.309273
$566$	−48.2402	−2.02769
$567$	0	0
$568$	27.4494	1.15175
$569$	−27.8442	−1.16729	−0.583645	−	0.812009i	$-0.698374\pi$
$569$	−27.8442	−1.16729	−0.583645	+	0.812009i	$0.698374\pi$
$570$	0	0
$571$	−14.5879	−0.610485	−0.305243	−	0.952275i	$-0.598738\pi$
$571$	−14.5879	−0.610485	−0.305243	+	0.952275i	$0.598738\pi$
$572$	0	0
$573$	0	0
$574$	−29.5335	−1.23270
$575$	0.434214	0.0181080
$576$	0	0
$577$	−26.1147	−1.08717	−0.543584	−	0.839355i	$-0.682933\pi$
$577$	−26.1147	−1.08717	−0.543584	+	0.839355i	$0.682933\pi$
$578$	38.3994	1.59721
$579$	0	0
$580$	26.6868	1.10811
$581$	−3.45931	−0.143516
$582$	0	0
$583$	0	0
$584$	124.643	5.15778
$585$	0	0
$586$	−82.9987	−3.42864
$587$	14.6634	0.605222	0.302611	−	0.953114i	$-0.402142\pi$
$587$	14.6634	0.605222	0.302611	+	0.953114i	$0.402142\pi$
$588$	0	0
$589$	27.4876	1.13261
$590$	−2.27022	−0.0934633
$591$	0	0
$592$	73.3542	3.01484
$593$	−17.7060	−0.727100	−0.363550	−	0.931575i	$-0.618435\pi$
$593$	−17.7060	−0.727100	−0.363550	+	0.931575i	$0.618435\pi$
$594$	0	0
$595$	−6.55039	−0.268540
$596$	−50.1568	−2.05450
$597$	0	0
$598$	−0.539572	−0.0220647
$599$	20.2930	0.829148	0.414574	−	0.910016i	$-0.363931\pi$
$599$	20.2930	0.829148	0.414574	+	0.910016i	$0.363931\pi$
$600$	0	0
$601$	10.9477	0.446566	0.223283	−	0.974754i	$-0.428323\pi$
$601$	10.9477	0.446566	0.223283	+	0.974754i	$0.428323\pi$
$602$	16.0107	0.652548
$603$	0	0
$604$	−13.2827	−0.540467
$605$	0	0
$606$	0	0
$607$	33.9487	1.37793	0.688967	−	0.724793i	$-0.258064\pi$
$607$	33.9487	1.37793	0.688967	+	0.724793i	$0.258064\pi$
$608$	−71.8626	−2.91441
$609$	0	0
$610$	27.7931	1.12531
$611$	21.1702	0.856455
$612$	0	0
$613$	−7.46975	−0.301700	−0.150850	−	0.988557i	$-0.548201\pi$
$613$	−7.46975	−0.301700	−0.150850	+	0.988557i	$0.548201\pi$
$614$	26.8601	1.08399
$615$	0	0
$616$	0	0
$617$	−29.3727	−1.18250	−0.591250	−	0.806489i	$-0.701365\pi$
$617$	−29.3727	−1.18250	−0.591250	+	0.806489i	$0.701365\pi$
$618$	0	0
$619$	36.1344	1.45236	0.726181	−	0.687503i	$-0.241293\pi$
$619$	36.1344	1.45236	0.726181	+	0.687503i	$0.241293\pi$
$620$	−38.3241	−1.53913
$621$	0	0
$622$	43.7900	1.75582
$623$	12.0621	0.483259
$624$	0	0
$625$	6.32915	0.253166
$626$	11.9849	0.479014
$627$	0	0
$628$	96.0055	3.83104
$629$	32.7644	1.30640
$630$	0	0
$631$	39.1360	1.55798	0.778990	−	0.627036i	$-0.215732\pi$
$631$	39.1360	1.55798	0.778990	+	0.627036i	$0.215732\pi$
$632$	−123.679	−4.91967
$633$	0	0
$634$	−32.1118	−1.27532
$635$	−20.1751	−0.800625
$636$	0	0
$637$	1.68268	0.0666704
$638$	0	0
$639$	0	0
$640$	21.5627	0.852342
$641$	−27.8586	−1.10035	−0.550174	−	0.835050i	$-0.685439\pi$
$641$	−27.8586	−1.10035	−0.550174	+	0.835050i	$0.685439\pi$
$642$	0	0
$643$	13.1417	0.518258	0.259129	−	0.965843i	$-0.416565\pi$
$643$	13.1417	0.518258	0.259129	+	0.965843i	$0.416565\pi$
$644$	0.620417	0.0244479
$645$	0	0
$646$	−65.3298	−2.57037
$647$	−3.36604	−0.132333	−0.0661663	−	0.997809i	$-0.521077\pi$
$647$	−3.36604	−0.132333	−0.0661663	+	0.997809i	$0.521077\pi$
$648$	0	0
$649$	0	0
$650$	−16.3758	−0.642312
$651$	0	0
$652$	33.6020	1.31596
$653$	−24.8798	−0.973622	−0.486811	−	0.873507i	$-0.661840\pi$
$653$	−24.8798	−0.973622	−0.486811	+	0.873507i	$0.661840\pi$
$654$	0	0
$655$	0.379280	0.0148197
$656$	138.039	5.38953
$657$	0	0
$658$	−33.7283	−1.31486
$659$	−30.5058	−1.18834	−0.594169	−	0.804340i	$-0.702519\pi$
$659$	−30.5058	−1.18834	−0.594169	+	0.804340i	$0.702519\pi$
$660$	0	0
$661$	10.2042	0.396899	0.198449	−	0.980111i	$-0.436409\pi$
$661$	10.2042	0.396899	0.198449	+	0.980111i	$0.436409\pi$
$662$	−48.4892	−1.88459
$663$	0	0
$664$	29.5550	1.14696
$665$	5.09607	0.197617
$666$	0	0
$667$	−0.525813	−0.0203596
$668$	−131.004	−5.06871
$669$	0	0
$670$	27.5717	1.06519
$671$	0	0
$672$	0	0
$673$	48.7196	1.87800	0.939001	−	0.343914i	$-0.111753\pi$
$673$	48.7196	1.87800	0.939001	+	0.343914i	$0.111753\pi$
$674$	−51.1661	−1.97085
$675$	0	0
$676$	−52.7435	−2.02860
$677$	−16.0041	−0.615087	−0.307543	−	0.951534i	$-0.599507\pi$
$677$	−16.0041	−0.615087	−0.307543	+	0.951534i	$0.599507\pi$
$678$	0	0
$679$	5.99575	0.230096
$680$	55.9640	2.14612
$681$	0	0
$682$	0	0
$683$	−26.6991	−1.02161	−0.510805	−	0.859696i	$-0.670653\pi$
$683$	−26.6991	−1.02161	−0.510805	+	0.859696i	$0.670653\pi$
$684$	0	0
$685$	12.1400	0.463844
$686$	−2.68084	−0.102355
$687$	0	0
$688$	−74.8339	−2.85302
$689$	0.701687	0.0267321
$690$	0	0
$691$	15.7496	0.599143	0.299571	−	0.954074i	$-0.403156\pi$
$691$	15.7496	0.599143	0.299571	+	0.954074i	$0.403156\pi$
$692$	−48.0725	−1.82744
$693$	0	0
$694$	−14.2558	−0.541143
$695$	−15.6570	−0.593905
$696$	0	0
$697$	61.6566	2.33541
$698$	55.2956	2.09297
$699$	0	0
$700$	18.8294	0.711686
$701$	−32.1718	−1.21511	−0.607557	−	0.794276i	$-0.707850\pi$
$701$	−32.1718	−1.21511	−0.607557	+	0.794276i	$0.707850\pi$
$702$	0	0
$703$	−25.4900	−0.961374
$704$	0	0
$705$	0	0
$706$	−19.6169	−0.738293
$707$	0.101612	0.00382152
$708$	0	0
$709$	−3.52842	−0.132513	−0.0662564	−	0.997803i	$-0.521106\pi$
$709$	−3.52842	−0.132513	−0.0662564	+	0.997803i	$0.521106\pi$
$710$	10.0808	0.378325
$711$	0	0
$712$	−103.054	−3.86212
$713$	0.755104	0.0282789
$714$	0	0
$715$	0	0
$716$	80.1055	2.99368
$717$	0	0
$718$	64.9540	2.42406
$719$	−37.1145	−1.38414	−0.692069	−	0.721831i	$-0.743301\pi$
$719$	−37.1145	−1.38414	−0.692069	+	0.721831i	$0.743301\pi$
$720$	0	0
$721$	1.73990	0.0647973
$722$	−0.110715	−0.00412039
$723$	0	0
$724$	54.6277	2.03022
$725$	−15.9582	−0.592674
$726$	0	0
$727$	−7.85215	−0.291220	−0.145610	−	0.989342i	$-0.546514\pi$
$727$	−7.85215	−0.291220	−0.145610	+	0.989342i	$0.546514\pi$
$728$	−14.3762	−0.532817
$729$	0	0
$730$	45.7752	1.69422
$731$	−33.4253	−1.23628
$732$	0	0
$733$	29.0960	1.07468	0.537342	−	0.843365i	$-0.319429\pi$
$733$	29.0960	1.07468	0.537342	+	0.843365i	$0.319429\pi$
$734$	−60.3347	−2.22699
$735$	0	0
$736$	−1.97412	−0.0727671
$737$	0	0
$738$	0	0
$739$	7.18580	0.264334	0.132167	−	0.991227i	$-0.457806\pi$
$739$	7.18580	0.264334	0.132167	+	0.991227i	$0.457806\pi$
$740$	35.5391	1.30644
$741$	0	0
$742$	−1.11792	−0.0410402
$743$	10.6220	0.389684	0.194842	−	0.980835i	$-0.437581\pi$
$743$	10.6220	0.389684	0.194842	+	0.980835i	$0.437581\pi$
$744$	0	0
$745$	−11.3175	−0.414643
$746$	−16.7630	−0.613739
$747$	0	0
$748$	0	0
$749$	11.0032	0.402049
$750$	0	0
$751$	16.0045	0.584014	0.292007	−	0.956416i	$-0.405677\pi$
$751$	16.0045	0.584014	0.292007	+	0.956416i	$0.405677\pi$
$752$	157.646	5.74874
$753$	0	0
$754$	19.8304	0.722179
$755$	−2.99716	−0.109078
$756$	0	0
$757$	−8.61194	−0.313006	−0.156503	−	0.987677i	$-0.550022\pi$
$757$	−8.61194	−0.313006	−0.156503	+	0.987677i	$0.550022\pi$
$758$	34.3664	1.24824
$759$	0	0
$760$	−43.5388	−1.57932
$761$	15.5310	0.563000	0.281500	−	0.959561i	$-0.409168\pi$
$761$	15.5310	0.563000	0.281500	+	0.959561i	$0.409168\pi$
$762$	0	0
$763$	16.5040	0.597486
$764$	35.5997	1.28795
$765$	0	0
$766$	−17.3380	−0.626446
$767$	−1.21750	−0.0439612
$768$	0	0
$769$	−51.4797	−1.85640	−0.928202	−	0.372076i	$-0.878646\pi$
$769$	−51.4797	−1.85640	−0.928202	+	0.372076i	$0.878646\pi$
$770$	0	0
$771$	0	0
$772$	−115.822	−4.16853
$773$	20.2078	0.726823	0.363411	−	0.931629i	$-0.381612\pi$
$773$	20.2078	0.726823	0.363411	+	0.931629i	$0.381612\pi$
$774$	0	0
$775$	22.9171	0.823208
$776$	−51.2253	−1.83888
$777$	0	0
$778$	39.9358	1.43177
$779$	−47.9676	−1.71862
$780$	0	0
$781$	0	0
$782$	−1.79466	−0.0641769
$783$	0	0
$784$	12.5302	0.447508
$785$	21.6630	0.773185
$786$	0	0
$787$	−49.5330	−1.76566	−0.882831	−	0.469692i	$-0.844365\pi$
$787$	−49.5330	−1.76566	−0.882831	+	0.469692i	$0.844365\pi$
$788$	23.0923	0.822629
$789$	0	0
$790$	−45.4209	−1.61600
$791$	6.28108	0.223330
$792$	0	0
$793$	14.9052	0.529299
$794$	4.98266	0.176828
$795$	0	0
$796$	−78.3187	−2.77593
$797$	26.6619	0.944414	0.472207	−	0.881488i	$-0.343457\pi$
$797$	26.6619	0.944414	0.472207	+	0.881488i	$0.343457\pi$
$798$	0	0
$799$	70.4140	2.49107
$800$	−59.9139	−2.11827
$801$	0	0
$802$	23.4372	0.827597
$803$	0	0
$804$	0	0
$805$	0.139993	0.00493411
$806$	−28.4778	−1.00309
$807$	0	0
$808$	−0.868135	−0.0305409
$809$	51.2793	1.80288	0.901442	−	0.432900i	$-0.142510\pi$
$809$	51.2793	1.80288	0.901442	+	0.432900i	$0.142510\pi$
$810$	0	0
$811$	−26.9340	−0.945782	−0.472891	−	0.881121i	$-0.656790\pi$
$811$	−26.9340	−0.945782	−0.472891	+	0.881121i	$0.656790\pi$
$812$	−22.8016	−0.800180
$813$	0	0
$814$	0	0
$815$	7.58207	0.265588
$816$	0	0
$817$	26.0042	0.909772
$818$	−34.0997	−1.19227
$819$	0	0
$820$	66.8781	2.33548
$821$	−36.3229	−1.26768	−0.633839	−	0.773465i	$-0.718522\pi$
$821$	−36.3229	−1.26768	−0.633839	+	0.773465i	$0.718522\pi$
$822$	0	0
$823$	−11.4540	−0.399262	−0.199631	−	0.979871i	$-0.563974\pi$
$823$	−11.4540	−0.399262	−0.199631	+	0.979871i	$0.563974\pi$
$824$	−14.8650	−0.517848
$825$	0	0
$826$	1.93971	0.0674910
$827$	−18.8743	−0.656324	−0.328162	−	0.944621i	$-0.606429\pi$
$827$	−18.8743	−0.656324	−0.328162	+	0.944621i	$0.606429\pi$
$828$	0	0
$829$	−12.0160	−0.417331	−0.208666	−	0.977987i	$-0.566912\pi$
$829$	−12.0160	−0.417331	−0.208666	+	0.977987i	$0.566912\pi$
$830$	10.8541	0.376750
$831$	0	0
$832$	32.2825	1.11919
$833$	5.59675	0.193916
$834$	0	0
$835$	−29.5603	−1.02297
$836$	0	0
$837$	0	0
$838$	21.7117	0.750019
$839$	−5.34302	−0.184461	−0.0922307	−	0.995738i	$-0.529400\pi$
$839$	−5.34302	−0.184461	−0.0922307	+	0.995738i	$0.529400\pi$
$840$	0	0
$841$	−9.67528	−0.333630
$842$	−70.9849	−2.44630
$843$	0	0
$844$	118.513	4.07940
$845$	−11.9012	−0.409414
$846$	0	0
$847$	0	0
$848$	5.22516	0.179433
$849$	0	0
$850$	−54.4673	−1.86821
$851$	−0.700230	−0.0240036
$852$	0	0
$853$	−1.05041	−0.0359655	−0.0179827	−	0.999838i	$-0.505724\pi$
$853$	−1.05041	−0.0359655	−0.0179827	+	0.999838i	$0.505724\pi$
$854$	−23.7469	−0.812601
$855$	0	0
$856$	−94.0073	−3.21310
$857$	13.9506	0.476542	0.238271	−	0.971199i	$-0.423419\pi$
$857$	13.9506	0.476542	0.238271	+	0.971199i	$0.423419\pi$
$858$	0	0
$859$	51.5493	1.75884	0.879419	−	0.476048i	$-0.157931\pi$
$859$	51.5493	1.75884	0.879419	+	0.476048i	$0.157931\pi$
$860$	−36.2560	−1.23632
$861$	0	0
$862$	−0.955576	−0.0325470
$863$	33.1894	1.12978	0.564890	−	0.825166i	$-0.308918\pi$
$863$	33.1894	1.12978	0.564890	+	0.825166i	$0.308918\pi$
$864$	0	0
$865$	−10.8472	−0.368817
$866$	−49.0284	−1.66605
$867$	0	0
$868$	32.7447	1.11143
$869$	0	0
$870$	0	0
$871$	14.7864	0.501020
$872$	−141.004	−4.77500
$873$	0	0
$874$	1.39621	0.0472275
$875$	10.1007	0.341466
$876$	0	0
$877$	50.2380	1.69642	0.848208	−	0.529663i	$-0.177682\pi$
$877$	50.2380	1.69642	0.848208	+	0.529663i	$0.177682\pi$
$878$	−17.8982	−0.604034
$879$	0	0
$880$	0	0
$881$	−38.9209	−1.31128	−0.655639	−	0.755074i	$-0.727601\pi$
$881$	−38.9209	−1.31128	−0.655639	+	0.755074i	$0.727601\pi$
$882$	0	0
$883$	−10.3534	−0.348420	−0.174210	−	0.984709i	$-0.555737\pi$
$883$	−10.3534	−0.348420	−0.174210	+	0.984709i	$0.555737\pi$
$884$	48.8481	1.64294
$885$	0	0
$886$	54.7971	1.84094
$887$	−24.6430	−0.827432	−0.413716	−	0.910406i	$-0.635769\pi$
$887$	−24.6430	−0.827432	−0.413716	+	0.910406i	$0.635769\pi$
$888$	0	0
$889$	17.2379	0.578141
$890$	−37.8466	−1.26862
$891$	0	0
$892$	50.3684	1.68646
$893$	−54.7806	−1.83316
$894$	0	0
$895$	18.0753	0.604190
$896$	−18.4235	−0.615487
$897$	0	0
$898$	−14.7829	−0.493313
$899$	−27.7516	−0.925569
$900$	0	0
$901$	2.33387	0.0777525
$902$	0	0
$903$	0	0
$904$	−53.6631	−1.78481
$905$	12.3264	0.409742
$906$	0	0
$907$	43.0708	1.43014	0.715071	−	0.699052i	$-0.246394\pi$
$907$	43.0708	1.43014	0.715071	+	0.699052i	$0.246394\pi$
$908$	−96.5666	−3.20467
$909$	0	0
$910$	−5.27965	−0.175019
$911$	45.4296	1.50515	0.752574	−	0.658508i	$-0.228812\pi$
$911$	45.4296	1.50515	0.752574	+	0.658508i	$0.228812\pi$
$912$	0	0
$913$	0	0
$914$	−5.36907	−0.177593
$915$	0	0
$916$	46.9039	1.54975
$917$	−0.324063	−0.0107015
$918$	0	0
$919$	−16.7647	−0.553015	−0.276508	−	0.961012i	$-0.589177\pi$
$919$	−16.7647	−0.553015	−0.276508	+	0.961012i	$0.589177\pi$
$920$	−1.19605	−0.0394324
$921$	0	0
$922$	−21.9142	−0.721705
$923$	5.40623	0.177948
$924$	0	0
$925$	−21.2517	−0.698753
$926$	−11.4388	−0.375901
$927$	0	0
$928$	72.5530	2.38167
$929$	−22.5588	−0.740131	−0.370065	−	0.929006i	$-0.620665\pi$
$929$	−22.5588	−0.740131	−0.370065	+	0.929006i	$0.620665\pi$
$930$	0	0
$931$	−4.35416	−0.142702
$932$	44.7278	1.46511
$933$	0	0
$934$	−108.611	−3.55386
$935$	0	0
$936$	0	0
$937$	−7.56458	−0.247124	−0.123562	−	0.992337i	$-0.539432\pi$
$937$	−7.56458	−0.247124	−0.123562	+	0.992337i	$0.539432\pi$
$938$	−23.5577	−0.769185
$939$	0	0
$940$	76.3770	2.49114
$941$	−48.7593	−1.58951	−0.794754	−	0.606932i	$-0.792400\pi$
$941$	−48.7593	−1.58951	−0.794754	+	0.606932i	$0.792400\pi$
$942$	0	0
$943$	−1.31771	−0.0429104
$944$	−9.06618	−0.295079
$945$	0	0
$946$	0	0
$947$	14.4056	0.468119	0.234060	−	0.972222i	$-0.424799\pi$
$947$	14.4056	0.468119	0.234060	+	0.972222i	$0.424799\pi$
$948$	0	0
$949$	24.5488	0.796889
$950$	42.3744	1.37481
$951$	0	0
$952$	−47.8165	−1.54974
$953$	−31.4716	−1.01947	−0.509733	−	0.860333i	$-0.670256\pi$
$953$	−31.4716	−1.01947	−0.509733	+	0.860333i	$0.670256\pi$
$954$	0	0
$955$	8.03282	0.259936
$956$	−107.302	−3.47039
$957$	0	0
$958$	22.0924	0.713772
$959$	−10.3726	−0.334948
$960$	0	0
$961$	8.85325	0.285589
$962$	26.4083	0.851436
$963$	0	0
$964$	128.851	4.15000
$965$	−26.1345	−0.841299
$966$	0	0
$967$	−19.1109	−0.614566	−0.307283	−	0.951618i	$-0.599420\pi$
$967$	−19.1109	−0.614566	−0.307283	+	0.951618i	$0.599420\pi$
$968$	0	0
$969$	0	0
$970$	−18.8125	−0.604032
$971$	42.7809	1.37291	0.686453	−	0.727174i	$-0.259167\pi$
$971$	42.7809	1.37291	0.686453	+	0.727174i	$0.259167\pi$
$972$	0	0
$973$	13.3776	0.428866
$974$	−2.58670	−0.0828831
$975$	0	0
$976$	110.993	3.55279
$977$	−40.2273	−1.28699	−0.643493	−	0.765452i	$-0.722515\pi$
$977$	−40.2273	−1.28699	−0.643493	+	0.765452i	$0.722515\pi$
$978$	0	0
$979$	0	0
$980$	6.07072	0.193922
$981$	0	0
$982$	−81.5737	−2.60312
$983$	23.9559	0.764074	0.382037	−	0.924147i	$-0.375223\pi$
$983$	23.9559	0.764074	0.382037	+	0.924147i	$0.375223\pi$
$984$	0	0
$985$	5.21062	0.166024
$986$	65.9574	2.10051
$987$	0	0
$988$	−38.0028	−1.20903
$989$	0.714356	0.0227152
$990$	0	0
$991$	−19.7275	−0.626665	−0.313332	−	0.949644i	$-0.601445\pi$
$991$	−19.7275	−0.626665	−0.313332	+	0.949644i	$0.601445\pi$
$992$	−104.191	−3.30807
$993$	0	0
$994$	−8.61317	−0.273193
$995$	−17.6721	−0.560243
$996$	0	0
$997$	−1.90849	−0.0604426	−0.0302213	−	0.999543i	$-0.509621\pi$
$997$	−1.90849	−0.0604426	−0.0302213	+	0.999543i	$0.509621\pi$
$998$	−100.690	−3.18728
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.db.1.16		16
3.2	odd	2	inner	7623.2.a.db.1.1		16
11.2	odd	10		693.2.m.k.631.8	yes	32
11.6	odd	10		693.2.m.k.190.8	yes	32
11.10	odd	2		7623.2.a.dc.1.1		16
33.2	even	10		693.2.m.k.631.1	yes	32
33.17	even	10		693.2.m.k.190.1	✓	32
33.32	even	2		7623.2.a.dc.1.16		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
693.2.m.k.190.1	✓	32	33.17	even	10
693.2.m.k.190.8	yes	32	11.6	odd	10
693.2.m.k.631.1	yes	32	33.2	even	10
693.2.m.k.631.8	yes	32	11.2	odd	10
7623.2.a.db.1.1		16	3.2	odd	2	inner
7623.2.a.db.1.16		16	1.1	even	1	trivial
7623.2.a.dc.1.1		16	11.10	odd	2
7623.2.a.dc.1.16		16	33.32	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+2.68084 q^{2} +5.18691 q^{4} +1.17039 q^{5} -1.00000 q^{7} +8.54361 q^{8} +O(q^{10})\)	\(q+2.68084 q^{2} +5.18691 q^{4} +1.17039 q^{5} -1.00000 q^{7} +8.54361 q^{8} +3.13763 q^{10} +1.68268 q^{13} -2.68084 q^{14} +12.5302 q^{16} +5.59675 q^{17} -4.35416 q^{19} +6.07072 q^{20} -0.119612 q^{23} -3.63018 q^{25} +4.51101 q^{26} -5.18691 q^{28} +4.39599 q^{29} -6.31294 q^{31} +16.5044 q^{32} +15.0040 q^{34} -1.17039 q^{35} +5.85418 q^{37} -11.6728 q^{38} +9.99937 q^{40} +11.0165 q^{41} -5.97227 q^{43} -0.320661 q^{46} +12.5812 q^{47} +1.00000 q^{49} -9.73195 q^{50} +8.72794 q^{52} +0.417004 q^{53} -8.54361 q^{56} +11.7849 q^{58} -0.723544 q^{59} +8.85799 q^{61} -16.9240 q^{62} +19.1851 q^{64} +1.96940 q^{65} +8.78741 q^{67} +29.0299 q^{68} -3.13763 q^{70} +3.21286 q^{71} +14.5891 q^{73} +15.6941 q^{74} -22.5846 q^{76} -14.4761 q^{79} +14.6653 q^{80} +29.5335 q^{82} +3.45931 q^{83} +6.55039 q^{85} -16.0107 q^{86} -12.0621 q^{89} -1.68268 q^{91} -0.620417 q^{92} +33.7283 q^{94} -5.09607 q^{95} -5.99575 q^{97} +2.68084 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 20 q^{4} - 16 q^{7}+O(q^{10}) \) 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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