LMFDB - Embedded newform 7623.2.a.bz.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7623,2,Mod(1,7623)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [3,-2,0,8,-1,0,3,-6,0,-4,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$60.8699614608$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.568.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x - 2 $ x^3 - x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 847)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.12489$ 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)

$f(q)$	$=$	$q+2.12489 q^{2} +2.51514 q^{4} -0.484862 q^{5} +1.00000 q^{7} +1.09461 q^{8} -1.03028 q^{10} +5.60975 q^{13} +2.12489 q^{14} -2.70436 q^{16} -5.60975 q^{17} -5.28005 q^{19} -1.21949 q^{20} -2.48486 q^{23} -4.76491 q^{25} +11.9201 q^{26} +2.51514 q^{28} -5.28005 q^{29} -7.12489 q^{31} -7.93567 q^{32} -11.9201 q^{34} -0.484862 q^{35} -0.235091 q^{37} -11.2195 q^{38} -0.530734 q^{40} -2.39025 q^{41} +1.03028 q^{43} -5.28005 q^{46} +1.60975 q^{47} +1.00000 q^{49} -10.1249 q^{50} +14.1093 q^{52} +3.03028 q^{53} +1.09461 q^{56} -11.2195 q^{58} -3.12489 q^{59} -2.39025 q^{61} -15.1396 q^{62} -11.4537 q^{64} -2.71995 q^{65} +10.0147 q^{67} -14.1093 q^{68} -1.03028 q^{70} -12.0752 q^{71} +2.39025 q^{73} -0.499542 q^{74} -13.2800 q^{76} +9.03028 q^{79} +1.31124 q^{80} -5.07901 q^{82} +3.21949 q^{83} +2.71995 q^{85} +2.18922 q^{86} +1.26537 q^{89} +5.60975 q^{91} -6.24977 q^{92} +3.42053 q^{94} +2.56009 q^{95} -8.79518 q^{97} +2.12489 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{2} + 8 q^{4} - q^{5} + 3 q^{7} - 6 q^{8} - 4 q^{10} + 8 q^{13} - 2 q^{14} + 10 q^{16} - 8 q^{17} + 14 q^{20} - 7 q^{23} + 2 q^{25} + 12 q^{26} + 8 q^{28} - 13 q^{31} - 34 q^{32} - 12 q^{34}+ \cdots - 2 q^{98}+O(q^{100}) $ 3 * q - 2 * q^2 + 8 * q^4 - q^5 + 3 * q^7 - 6 * q^8 - 4 * q^10 + 8 * q^13 - 2 * q^14 + 10 * q^16 - 8 * q^17 + 14 * q^20 - 7 * q^23 + 2 * q^25 + 12 * q^26 + 8 * q^28 - 13 * q^31 - 34 * q^32 - 12 * q^34 - q^35 - 17 * q^37 - 16 * q^38 - 36 * q^40 - 16 * q^41 + 4 * q^43 - 4 * q^47 + 3 * q^49 - 22 * q^50 + 10 * q^53 - 6 * q^56 - 16 * q^58 - q^59 - 16 * q^61 - 4 * q^62 + 34 * q^64 - 24 * q^65 - 3 * q^67 - 4 * q^70 - 5 * q^71 + 16 * q^73 + 32 * q^74 - 24 * q^76 + 28 * q^79 + 56 * q^80 + 28 * q^82 - 8 * q^83 + 24 * q^85 - 12 * q^86 + 21 * q^89 + 8 * q^91 - 2 * q^92 + 20 * q^94 - 24 * q^95 - 11 * q^97 - 2 * q^98

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 $

$2$	2.12489	1.50252	0.751260	−	0.660006i	$-0.229446\pi$
$2$	2.12489	1.50252	0.751260	+	0.660006i	$0.229446\pi$
$3$	0	0
$3$	0	0
$4$	2.51514	1.25757
$5$	−0.484862	−0.216837	−0.108418	−	0.994105i	$-0.534579\pi$
$5$	−0.484862	−0.216837	−0.108418	+	0.994105i	$0.534579\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.09461	0.387003
$9$	0	0
$10$	−1.03028	−0.325802
$11$	0	0
$11$	0	0
$12$	0	0
$13$	5.60975	1.55586	0.777932	−	0.628348i	$-0.216269\pi$
$13$	5.60975	1.55586	0.777932	+	0.628348i	$0.216269\pi$
$14$	2.12489	0.567900
$15$	0	0
$16$	−2.70436	−0.676089
$17$	−5.60975	−1.36056	−0.680282	−	0.732951i	$-0.738143\pi$
$17$	−5.60975	−1.36056	−0.680282	+	0.732951i	$0.738143\pi$
$18$	0	0
$19$	−5.28005	−1.21133	−0.605663	−	0.795721i	$-0.707092\pi$
$19$	−5.28005	−1.21133	−0.605663	+	0.795721i	$0.707092\pi$
$20$	−1.21949	−0.272687
$21$	0	0
$22$	0	0
$23$	−2.48486	−0.518130	−0.259065	−	0.965860i	$-0.583414\pi$
$23$	−2.48486	−0.518130	−0.259065	+	0.965860i	$0.583414\pi$
$24$	0	0
$25$	−4.76491	−0.952982
$26$	11.9201	2.33772
$27$	0	0
$28$	2.51514	0.475316
$29$	−5.28005	−0.980480	−0.490240	−	0.871587i	$-0.663091\pi$
$29$	−5.28005	−0.980480	−0.490240	+	0.871587i	$0.663091\pi$
$30$	0	0
$31$	−7.12489	−1.27967	−0.639834	−	0.768513i	$-0.720997\pi$
$31$	−7.12489	−1.27967	−0.639834	+	0.768513i	$0.720997\pi$
$32$	−7.93567	−1.40284
$33$	0	0
$34$	−11.9201	−2.04428
$35$	−0.484862	−0.0819566
$36$	0	0
$37$	−0.235091	−0.0386487	−0.0193244	−	0.999813i	$-0.506152\pi$
$37$	−0.235091	−0.0386487	−0.0193244	+	0.999813i	$0.506152\pi$
$38$	−11.2195	−1.82004
$39$	0	0
$40$	−0.530734	−0.0839165
$41$	−2.39025	−0.373295	−0.186647	−	0.982427i	$-0.559762\pi$
$41$	−2.39025	−0.373295	−0.186647	+	0.982427i	$0.559762\pi$
$42$	0	0
$43$	1.03028	0.157116	0.0785578	−	0.996910i	$-0.474968\pi$
$43$	1.03028	0.157116	0.0785578	+	0.996910i	$0.474968\pi$
$44$	0	0
$45$	0	0
$46$	−5.28005	−0.778500
$47$	1.60975	0.234806	0.117403	−	0.993084i	$-0.462543\pi$
$47$	1.60975	0.234806	0.117403	+	0.993084i	$0.462543\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−10.1249	−1.43188
$51$	0	0
$52$	14.1093	1.95661
$53$	3.03028	0.416240	0.208120	−	0.978103i	$-0.433265\pi$
$53$	3.03028	0.416240	0.208120	+	0.978103i	$0.433265\pi$
$54$	0	0
$55$	0	0
$56$	1.09461	0.146273
$57$	0	0
$58$	−11.2195	−1.47319
$59$	−3.12489	−0.406825	−0.203413	−	0.979093i	$-0.565203\pi$
$59$	−3.12489	−0.406825	−0.203413	+	0.979093i	$0.565203\pi$
$60$	0	0
$61$	−2.39025	−0.306040	−0.153020	−	0.988223i	$-0.548900\pi$
$61$	−2.39025	−0.306040	−0.153020	+	0.988223i	$0.548900\pi$
$62$	−15.1396	−1.92273
$63$	0	0
$64$	−11.4537	−1.43171
$65$	−2.71995	−0.337369
$66$	0	0
$67$	10.0147	1.22349	0.611744	−	0.791056i	$-0.290468\pi$
$67$	10.0147	1.22349	0.611744	+	0.791056i	$0.290468\pi$
$68$	−14.1093	−1.71100
$69$	0	0
$70$	−1.03028	−0.123142
$71$	−12.0752	−1.43307	−0.716533	−	0.697553i	$-0.754272\pi$
$71$	−12.0752	−1.43307	−0.716533	+	0.697553i	$0.754272\pi$
$72$	0	0
$73$	2.39025	0.279758	0.139879	−	0.990169i	$-0.455329\pi$
$73$	2.39025	0.279758	0.139879	+	0.990169i	$0.455329\pi$
$74$	−0.499542	−0.0580705
$75$	0	0
$76$	−13.2800	−1.52333
$77$	0	0
$78$	0	0
$79$	9.03028	1.01599	0.507993	−	0.861361i	$-0.330388\pi$
$79$	9.03028	1.01599	0.507993	+	0.861361i	$0.330388\pi$
$80$	1.31124	0.146601
$81$	0	0
$82$	−5.07901	−0.560883
$83$	3.21949	0.353385	0.176693	−	0.984266i	$-0.443460\pi$
$83$	3.21949	0.353385	0.176693	+	0.984266i	$0.443460\pi$
$84$	0	0
$85$	2.71995	0.295020
$86$	2.18922	0.236070
$87$	0	0
$88$	0	0
$89$	1.26537	0.134129	0.0670643	−	0.997749i	$-0.478637\pi$
$89$	1.26537	0.134129	0.0670643	+	0.997749i	$0.478637\pi$
$90$	0	0
$91$	5.60975	0.588061
$92$	−6.24977	−0.651584
$93$	0	0
$94$	3.42053	0.352801
$95$	2.56009	0.262660
$96$	0	0
$97$	−8.79518	−0.893016	−0.446508	−	0.894780i	$-0.647333\pi$
$97$	−8.79518	−0.893016	−0.446508	+	0.894780i	$0.647333\pi$
$98$	2.12489	0.214646
$99$	0	0
$100$	−11.9844	−1.19844
$101$	12.9503	1.28861	0.644304	−	0.764770i	$-0.277147\pi$
$101$	12.9503	1.28861	0.644304	+	0.764770i	$0.277147\pi$
$102$	0	0
$103$	12.1698	1.19913	0.599565	−	0.800326i	$-0.295340\pi$
$103$	12.1698	1.19913	0.599565	+	0.800326i	$0.295340\pi$
$104$	6.14048	0.602124
$105$	0	0
$106$	6.43899	0.625410
$107$	−10.5601	−1.02088	−0.510441	−	0.859913i	$-0.670518\pi$
$107$	−10.5601	−1.02088	−0.510441	+	0.859913i	$0.670518\pi$
$108$	0	0
$109$	−7.34060	−0.703102	−0.351551	−	0.936169i	$-0.614346\pi$
$109$	−7.34060	−0.703102	−0.351551	+	0.936169i	$0.614346\pi$
$110$	0	0
$111$	0	0
$112$	−2.70436	−0.255538
$113$	−13.7649	−1.29489	−0.647447	−	0.762111i	$-0.724163\pi$
$113$	−13.7649	−1.29489	−0.647447	+	0.762111i	$0.724163\pi$
$114$	0	0
$115$	1.20482	0.112350
$116$	−13.2800	−1.23302
$117$	0	0
$118$	−6.64002	−0.611264
$119$	−5.60975	−0.514245
$120$	0	0
$121$	0	0
$122$	−5.07901	−0.459832
$123$	0	0
$124$	−17.9201	−1.60927
$125$	4.73463	0.423478
$126$	0	0
$127$	2.56009	0.227172	0.113586	−	0.993528i	$-0.463766\pi$
$127$	2.56009	0.227172	0.113586	+	0.993528i	$0.463766\pi$
$128$	−8.46640	−0.748331
$129$	0	0
$130$	−5.77959	−0.506903
$131$	2.71995	0.237643	0.118822	−	0.992916i	$-0.462088\pi$
$131$	2.71995	0.237643	0.118822	+	0.992916i	$0.462088\pi$
$132$	0	0
$133$	−5.28005	−0.457838
$134$	21.2800	1.83832
$135$	0	0
$136$	−6.14048	−0.526542
$137$	9.70436	0.829099	0.414550	−	0.910027i	$-0.363939\pi$
$137$	9.70436	0.829099	0.414550	+	0.910027i	$0.363939\pi$
$138$	0	0
$139$	−19.7190	−1.67255	−0.836273	−	0.548313i	$-0.815270\pi$
$139$	−19.7190	−1.67255	−0.836273	+	0.548313i	$0.815270\pi$
$140$	−1.21949	−0.103066
$141$	0	0
$142$	−25.6585	−2.15321
$143$	0	0
$144$	0	0
$145$	2.56009	0.212604
$146$	5.07901	0.420342
$147$	0	0
$148$	−0.591287	−0.0486035
$149$	−10.5601	−0.865117	−0.432558	−	0.901606i	$-0.642389\pi$
$149$	−10.5601	−0.865117	−0.432558	+	0.901606i	$0.642389\pi$
$150$	0	0
$151$	15.4693	1.25887	0.629435	−	0.777053i	$-0.283286\pi$
$151$	15.4693	1.25887	0.629435	+	0.777053i	$0.283286\pi$
$152$	−5.77959	−0.468787
$153$	0	0
$154$	0	0
$155$	3.45459	0.277479
$156$	0	0
$157$	9.10551	0.726699	0.363349	−	0.931653i	$-0.381633\pi$
$157$	9.10551	0.726699	0.363349	+	0.931653i	$0.381633\pi$
$158$	19.1883	1.52654
$159$	0	0
$160$	3.84770	0.304188
$161$	−2.48486	−0.195835
$162$	0	0
$163$	−13.2800	−1.04017	−0.520087	−	0.854113i	$-0.674100\pi$
$163$	−13.2800	−1.04017	−0.520087	+	0.854113i	$0.674100\pi$
$164$	−6.01182	−0.469444
$165$	0	0
$166$	6.84106	0.530969
$167$	−10.0606	−0.778509	−0.389254	−	0.921130i	$-0.627267\pi$
$167$	−10.0606	−0.778509	−0.389254	+	0.921130i	$0.627267\pi$
$168$	0	0
$169$	18.4693	1.42071
$170$	5.77959	0.443274
$171$	0	0
$172$	2.59129	0.197584
$173$	8.16984	0.621142	0.310571	−	0.950550i	$-0.399480\pi$
$173$	8.16984	0.621142	0.310571	+	0.950550i	$0.399480\pi$
$174$	0	0
$175$	−4.76491	−0.360193
$176$	0	0
$177$	0	0
$178$	2.68876	0.201531
$179$	12.2645	0.916688	0.458344	−	0.888775i	$-0.348443\pi$
$179$	12.2645	0.916688	0.458344	+	0.888775i	$0.348443\pi$
$180$	0	0
$181$	−7.04496	−0.523647	−0.261824	−	0.965116i	$-0.584324\pi$
$181$	−7.04496	−0.523647	−0.261824	+	0.965116i	$0.584324\pi$
$182$	11.9201	0.883574
$183$	0	0
$184$	−2.71995	−0.200518
$185$	0.113987	0.00838047
$186$	0	0
$187$	0	0
$188$	4.04874	0.295284
$189$	0	0
$190$	5.43991	0.394652
$191$	−18.7952	−1.35997	−0.679986	−	0.733225i	$-0.738014\pi$
$191$	−18.7952	−1.35997	−0.679986	+	0.733225i	$0.738014\pi$
$192$	0	0
$193$	16.4995	1.18766	0.593831	−	0.804589i	$-0.297615\pi$
$193$	16.4995	1.18766	0.593831	+	0.804589i	$0.297615\pi$
$194$	−18.6888	−1.34177
$195$	0	0
$196$	2.51514	0.179653
$197$	−24.4995	−1.74552	−0.872760	−	0.488149i	$-0.837672\pi$
$197$	−24.4995	−1.74552	−0.872760	+	0.488149i	$0.837672\pi$
$198$	0	0
$199$	−15.3893	−1.09092	−0.545461	−	0.838136i	$-0.683645\pi$
$199$	−15.3893	−1.09092	−0.545461	+	0.838136i	$0.683645\pi$
$200$	−5.21571	−0.368807
$201$	0	0
$202$	27.5180	1.93616
$203$	−5.28005	−0.370587
$204$	0	0
$205$	1.15894	0.0809441
$206$	25.8595	1.80172
$207$	0	0
$208$	−15.1708	−1.05190
$209$	0	0
$210$	0	0
$211$	−14.4390	−0.994021	−0.497011	−	0.867745i	$-0.665569\pi$
$211$	−14.4390	−0.994021	−0.497011	+	0.867745i	$0.665569\pi$
$212$	7.62156	0.523451
$213$	0	0
$214$	−22.4390	−1.53390
$215$	−0.499542	−0.0340685
$216$	0	0
$217$	−7.12489	−0.483669
$218$	−15.5979	−1.05643
$219$	0	0
$220$	0	0
$221$	−31.4693	−2.11685
$222$	0	0
$223$	17.2536	1.15538	0.577692	−	0.816255i	$-0.303954\pi$
$223$	17.2536	1.15538	0.577692	+	0.816255i	$0.303954\pi$
$224$	−7.93567	−0.530224
$225$	0	0
$226$	−29.2489	−1.94560
$227$	−10.7200	−0.711508	−0.355754	−	0.934580i	$-0.615776\pi$
$227$	−10.7200	−0.711508	−0.355754	+	0.934580i	$0.615776\pi$
$228$	0	0
$229$	5.45459	0.360449	0.180225	−	0.983625i	$-0.442318\pi$
$229$	5.45459	0.360449	0.180225	+	0.983625i	$0.442318\pi$
$230$	2.56009	0.168808
$231$	0	0
$232$	−5.77959	−0.379449
$233$	−29.7796	−1.95093	−0.975463	−	0.220164i	$-0.929341\pi$
$233$	−29.7796	−1.95093	−0.975463	+	0.220164i	$0.929341\pi$
$234$	0	0
$235$	−0.780505	−0.0509145
$236$	−7.85952	−0.511611
$237$	0	0
$238$	−11.9201	−0.772663
$239$	−2.56009	−0.165599	−0.0827994	−	0.996566i	$-0.526386\pi$
$239$	−2.56009	−0.165599	−0.0827994	+	0.996566i	$0.526386\pi$
$240$	0	0
$241$	27.3893	1.76430	0.882151	−	0.470966i	$-0.156095\pi$
$241$	27.3893	1.76430	0.882151	+	0.470966i	$0.156095\pi$
$242$	0	0
$243$	0	0
$244$	−6.01182	−0.384867
$245$	−0.484862	−0.0309767
$246$	0	0
$247$	−29.6197	−1.88466
$248$	−7.79897	−0.495235
$249$	0	0
$250$	10.0606	0.636285
$251$	−9.84484	−0.621401	−0.310700	−	0.950508i	$-0.600564\pi$
$251$	−9.84484	−0.621401	−0.310700	+	0.950508i	$0.600564\pi$
$252$	0	0
$253$	0	0
$254$	5.43991	0.341330
$255$	0	0
$256$	4.91721	0.307325
$257$	22.4995	1.40348	0.701741	−	0.712432i	$-0.252406\pi$
$257$	22.4995	1.40348	0.701741	+	0.712432i	$0.252406\pi$
$258$	0	0
$259$	−0.235091	−0.0146079
$260$	−6.84106	−0.424264
$261$	0	0
$262$	5.77959	0.357064
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	0	0
$265$	−1.46927	−0.0902563
$266$	−11.2195	−0.687911
$267$	0	0
$268$	25.1883	1.53862
$269$	−9.21949	−0.562123	−0.281061	−	0.959690i	$-0.590686\pi$
$269$	−9.21949	−0.562123	−0.281061	+	0.959690i	$0.590686\pi$
$270$	0	0
$271$	10.5601	0.641480	0.320740	−	0.947167i	$-0.396068\pi$
$271$	10.5601	0.641480	0.320740	+	0.947167i	$0.396068\pi$
$272$	15.1708	0.919862
$273$	0	0
$274$	20.6206	1.24574
$275$	0	0
$276$	0	0
$277$	−18.5601	−1.11517	−0.557584	−	0.830121i	$-0.688272\pi$
$277$	−18.5601	−1.11517	−0.557584	+	0.830121i	$0.688272\pi$
$278$	−41.9007	−2.51304
$279$	0	0
$280$	−0.530734	−0.0317174
$281$	−25.6585	−1.53066	−0.765328	−	0.643640i	$-0.777423\pi$
$281$	−25.6585	−1.53066	−0.765328	+	0.643640i	$0.777423\pi$
$282$	0	0
$283$	30.2791	1.79991	0.899954	−	0.435985i	$-0.143600\pi$
$283$	30.2791	1.79991	0.899954	+	0.435985i	$0.143600\pi$
$284$	−30.3709	−1.80218
$285$	0	0
$286$	0	0
$287$	−2.39025	−0.141092
$288$	0	0
$289$	14.4693	0.851133
$290$	5.43991	0.319442
$291$	0	0
$292$	6.01182	0.351815
$293$	−3.04965	−0.178163	−0.0890813	−	0.996024i	$-0.528393\pi$
$293$	−3.04965	−0.178163	−0.0890813	+	0.996024i	$0.528393\pi$
$294$	0	0
$295$	1.51514	0.0882147
$296$	−0.257333	−0.0149572
$297$	0	0
$298$	−22.4390	−1.29986
$299$	−13.9394	−0.806139
$300$	0	0
$301$	1.03028	0.0593841
$302$	32.8704	1.89148
$303$	0	0
$304$	14.2791	0.818964
$305$	1.15894	0.0663609
$306$	0	0
$307$	3.71904	0.212257	0.106128	−	0.994352i	$-0.466155\pi$
$307$	3.71904	0.212257	0.106128	+	0.994352i	$0.466155\pi$
$308$	0	0
$309$	0	0
$310$	7.34060	0.416918
$311$	−4.16984	−0.236450	−0.118225	−	0.992987i	$-0.537720\pi$
$311$	−4.16984	−0.236450	−0.118225	+	0.992987i	$0.537720\pi$
$312$	0	0
$313$	−11.7044	−0.661569	−0.330785	−	0.943706i	$-0.607313\pi$
$313$	−11.7044	−0.661569	−0.330785	+	0.943706i	$0.607313\pi$
$314$	19.3482	1.09188
$315$	0	0
$316$	22.7124	1.27767
$317$	2.23509	0.125535	0.0627676	−	0.998028i	$-0.480007\pi$
$317$	2.23509	0.125535	0.0627676	+	0.998028i	$0.480007\pi$
$318$	0	0
$319$	0	0
$320$	5.55345	0.310447
$321$	0	0
$322$	−5.28005	−0.294246
$323$	29.6197	1.64809
$324$	0	0
$325$	−26.7299	−1.48271
$326$	−28.2186	−1.56288
$327$	0	0
$328$	−2.61639	−0.144466
$329$	1.60975	0.0887482
$330$	0	0
$331$	22.3250	1.22709	0.613547	−	0.789659i	$-0.289742\pi$
$331$	22.3250	1.22709	0.613547	+	0.789659i	$0.289742\pi$
$332$	8.09747	0.444407
$333$	0	0
$334$	−21.3775	−1.16973
$335$	−4.85574	−0.265297
$336$	0	0
$337$	−13.2800	−0.723410	−0.361705	−	0.932293i	$-0.617805\pi$
$337$	−13.2800	−0.723410	−0.361705	+	0.932293i	$0.617805\pi$
$338$	39.2451	2.13465
$339$	0	0
$340$	6.84106	0.371008
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	1.12775	0.0608042
$345$	0	0
$346$	17.3600	0.933278
$347$	18.0294	0.967867	0.483933	−	0.875105i	$-0.339208\pi$
$347$	18.0294	0.967867	0.483933	+	0.875105i	$0.339208\pi$
$348$	0	0
$349$	−21.9494	−1.17493	−0.587463	−	0.809251i	$-0.699873\pi$
$349$	−21.9494	−1.17493	−0.587463	+	0.809251i	$0.699873\pi$
$350$	−10.1249	−0.541198
$351$	0	0
$352$	0	0
$353$	−18.8557	−1.00359	−0.501795	−	0.864987i	$-0.667327\pi$
$353$	−18.8557	−1.00359	−0.501795	+	0.864987i	$0.667327\pi$
$354$	0	0
$355$	5.85482	0.310742
$356$	3.18257	0.168676
$357$	0	0
$358$	26.0606	1.37734
$359$	−1.52982	−0.0807407	−0.0403703	−	0.999185i	$-0.512854\pi$
$359$	−1.52982	−0.0807407	−0.0403703	+	0.999185i	$0.512854\pi$
$360$	0	0
$361$	8.87890	0.467310
$362$	−14.9697	−0.786791
$363$	0	0
$364$	14.1093	0.739528
$365$	−1.15894	−0.0606618
$366$	0	0
$367$	14.6547	0.764969	0.382485	−	0.923962i	$-0.375068\pi$
$367$	14.6547	0.764969	0.382485	+	0.923962i	$0.375068\pi$
$368$	6.71995	0.350302
$369$	0	0
$370$	0.242209	0.0125918
$371$	3.03028	0.157324
$372$	0	0
$373$	10.0606	0.520916	0.260458	−	0.965485i	$-0.416127\pi$
$373$	10.0606	0.520916	0.260458	+	0.965485i	$0.416127\pi$
$374$	0	0
$375$	0	0
$376$	1.76204	0.0908705
$377$	−29.6197	−1.52549
$378$	0	0
$379$	−17.8936	−0.919131	−0.459566	−	0.888144i	$-0.651995\pi$
$379$	−17.8936	−0.919131	−0.459566	+	0.888144i	$0.651995\pi$
$380$	6.43899	0.330313
$381$	0	0
$382$	−39.9376	−2.04339
$383$	18.9650	0.969068	0.484534	−	0.874773i	$-0.338989\pi$
$383$	18.9650	0.969068	0.484534	+	0.874773i	$0.338989\pi$
$384$	0	0
$385$	0	0
$386$	35.0596	1.78449
$387$	0	0
$388$	−22.1211	−1.12303
$389$	−34.2645	−1.73728	−0.868638	−	0.495447i	$-0.835004\pi$
$389$	−34.2645	−1.73728	−0.868638	+	0.495447i	$0.835004\pi$
$390$	0	0
$391$	13.9394	0.704948
$392$	1.09461	0.0552861
$393$	0	0
$394$	−52.0587	−2.62268
$395$	−4.37844	−0.220303
$396$	0	0
$397$	16.2791	0.817026	0.408513	−	0.912752i	$-0.366047\pi$
$397$	16.2791	0.817026	0.408513	+	0.912752i	$0.366047\pi$
$398$	−32.7006	−1.63913
$399$	0	0
$400$	12.8860	0.644301
$401$	−28.9385	−1.44512	−0.722561	−	0.691308i	$-0.757035\pi$
$401$	−28.9385	−1.44512	−0.722561	+	0.691308i	$0.757035\pi$
$402$	0	0
$403$	−39.9688	−1.99099
$404$	32.5719	1.62051
$405$	0	0
$406$	−11.2195	−0.556814
$407$	0	0
$408$	0	0
$409$	15.5104	0.766942	0.383471	−	0.923553i	$-0.374729\pi$
$409$	15.5104	0.766942	0.383471	+	0.923553i	$0.374729\pi$
$410$	2.46262	0.121620
$411$	0	0
$412$	30.6088	1.50799
$413$	−3.12489	−0.153766
$414$	0	0
$415$	−1.56101	−0.0766270
$416$	−44.5171	−2.18263
$417$	0	0
$418$	0	0
$419$	12.9503	0.632666	0.316333	−	0.948648i	$-0.397548\pi$
$419$	12.9503	0.632666	0.316333	+	0.948648i	$0.397548\pi$
$420$	0	0
$421$	−17.0908	−0.832956	−0.416478	−	0.909146i	$-0.636736\pi$
$421$	−17.0908	−0.832956	−0.416478	+	0.909146i	$0.636736\pi$
$422$	−30.6812	−1.49354
$423$	0	0
$424$	3.31697	0.161086
$425$	26.7299	1.29659
$426$	0	0
$427$	−2.39025	−0.115672
$428$	−26.5601	−1.28383
$429$	0	0
$430$	−1.06147	−0.0511886
$431$	−19.3482	−0.931968	−0.465984	−	0.884793i	$-0.654300\pi$
$431$	−19.3482	−0.931968	−0.465984	+	0.884793i	$0.654300\pi$
$432$	0	0
$433$	15.8255	0.760523	0.380262	−	0.924879i	$-0.375834\pi$
$433$	15.8255	0.760523	0.380262	+	0.924879i	$0.375834\pi$
$434$	−15.1396	−0.726722
$435$	0	0
$436$	−18.4626	−0.884199
$437$	13.1202	0.627624
$438$	0	0
$439$	−11.0596	−0.527848	−0.263924	−	0.964544i	$-0.585017\pi$
$439$	−11.0596	−0.527848	−0.263924	+	0.964544i	$0.585017\pi$
$440$	0	0
$441$	0	0
$442$	−66.8686	−3.18061
$443$	34.7034	1.64881	0.824405	−	0.566000i	$-0.191510\pi$
$443$	34.7034	1.64881	0.824405	+	0.566000i	$0.191510\pi$
$444$	0	0
$445$	−0.613528	−0.0290840
$446$	36.6618	1.73599
$447$	0	0
$448$	−11.4537	−0.541135
$449$	36.2938	1.71281	0.856405	−	0.516304i	$-0.172693\pi$
$449$	36.2938	1.71281	0.856405	+	0.516304i	$0.172693\pi$
$450$	0	0
$451$	0	0
$452$	−34.6206	−1.62842
$453$	0	0
$454$	−22.7787	−1.06906
$455$	−2.71995	−0.127513
$456$	0	0
$457$	−2.06055	−0.0963886	−0.0481943	−	0.998838i	$-0.515347\pi$
$457$	−2.06055	−0.0963886	−0.0481943	+	0.998838i	$0.515347\pi$
$458$	11.5904	0.541582
$459$	0	0
$460$	3.03028	0.141287
$461$	7.17076	0.333975	0.166988	−	0.985959i	$-0.446596\pi$
$461$	7.17076	0.333975	0.166988	+	0.985959i	$0.446596\pi$
$462$	0	0
$463$	3.45459	0.160548	0.0802741	−	0.996773i	$-0.474420\pi$
$463$	3.45459	0.160548	0.0802741	+	0.996773i	$0.474420\pi$
$464$	14.2791	0.662892
$465$	0	0
$466$	−63.2782	−2.93131
$467$	13.4958	0.624509	0.312255	−	0.949998i	$-0.398916\pi$
$467$	13.4958	0.624509	0.312255	+	0.949998i	$0.398916\pi$
$468$	0	0
$469$	10.0147	0.462435
$470$	−1.65848	−0.0765002
$471$	0	0
$472$	−3.42053	−0.157443
$473$	0	0
$474$	0	0
$475$	25.1589	1.15437
$476$	−14.1093	−0.646698
$477$	0	0
$478$	−5.43991	−0.248816
$479$	−35.0596	−1.60192	−0.800958	−	0.598721i	$-0.795676\pi$
$479$	−35.0596	−1.60192	−0.800958	+	0.598721i	$0.795676\pi$
$480$	0	0
$481$	−1.31880	−0.0601322
$482$	58.1992	2.65090
$483$	0	0
$484$	0	0
$485$	4.26445	0.193639
$486$	0	0
$487$	29.6656	1.34428	0.672138	−	0.740426i	$-0.265376\pi$
$487$	29.6656	1.34428	0.672138	+	0.740426i	$0.265376\pi$
$488$	−2.61639	−0.118439
$489$	0	0
$490$	−1.03028	−0.0465431
$491$	−25.5298	−1.15214	−0.576072	−	0.817399i	$-0.695415\pi$
$491$	−25.5298	−1.15214	−0.576072	+	0.817399i	$0.695415\pi$
$492$	0	0
$493$	29.6197	1.33401
$494$	−62.9385	−2.83174
$495$	0	0
$496$	19.2682	0.865169
$497$	−12.0752	−0.541648
$498$	0	0
$499$	−25.6585	−1.14863	−0.574316	−	0.818634i	$-0.694732\pi$
$499$	−25.6585	−1.14863	−0.574316	+	0.818634i	$0.694732\pi$
$500$	11.9083	0.532553
$501$	0	0
$502$	−20.9192	−0.933668
$503$	−2.56009	−0.114149	−0.0570745	−	0.998370i	$-0.518177\pi$
$503$	−2.56009	−0.114149	−0.0570745	+	0.998370i	$0.518177\pi$
$504$	0	0
$505$	−6.27913	−0.279418
$506$	0	0
$507$	0	0
$508$	6.43899	0.285684
$509$	−13.4546	−0.596364	−0.298182	−	0.954509i	$-0.596380\pi$
$509$	−13.4546	−0.596364	−0.298182	+	0.954509i	$0.596380\pi$
$510$	0	0
$511$	2.39025	0.105739
$512$	27.3813	1.21009
$513$	0	0
$514$	47.8089	2.10876
$515$	−5.90069	−0.260016
$516$	0	0
$517$	0	0
$518$	−0.499542	−0.0219486
$519$	0	0
$520$	−2.97729	−0.130563
$521$	32.2333	1.41216	0.706082	−	0.708130i	$-0.250461\pi$
$521$	32.2333	1.41216	0.706082	+	0.708130i	$0.250461\pi$
$522$	0	0
$523$	−32.3397	−1.41412	−0.707058	−	0.707156i	$-0.749978\pi$
$523$	−32.3397	−1.41412	−0.707058	+	0.707156i	$0.749978\pi$
$524$	6.84106	0.298853
$525$	0	0
$526$	33.9982	1.48239
$527$	39.9688	1.74107
$528$	0	0
$529$	−16.8255	−0.731542
$530$	−3.12202	−0.135612
$531$	0	0
$532$	−13.2800	−0.575763
$533$	−13.4087	−0.580796
$534$	0	0
$535$	5.12019	0.221365
$536$	10.9622	0.473493
$537$	0	0
$538$	−19.5904	−0.844601
$539$	0	0
$540$	0	0
$541$	−23.5005	−1.01036	−0.505182	−	0.863013i	$-0.668575\pi$
$541$	−23.5005	−1.01036	−0.505182	+	0.863013i	$0.668575\pi$
$542$	22.4390	0.963837
$543$	0	0
$544$	44.5171	1.90865
$545$	3.55918	0.152458
$546$	0	0
$547$	−4.12110	−0.176206	−0.0881028	−	0.996111i	$-0.528080\pi$
$547$	−4.12110	−0.176206	−0.0881028	+	0.996111i	$0.528080\pi$
$548$	24.4078	1.04265
$549$	0	0
$550$	0	0
$551$	27.8789	1.18768
$552$	0	0
$553$	9.03028	0.384006
$554$	−39.4381	−1.67556
$555$	0	0
$556$	−49.5961	−2.10334
$557$	13.9394	0.590633	0.295317	−	0.955399i	$-0.404575\pi$
$557$	13.9394	0.590633	0.295317	+	0.955399i	$0.404575\pi$
$558$	0	0
$559$	5.77959	0.244451
$560$	1.31124	0.0554100
$561$	0	0
$562$	−54.5213	−2.29984
$563$	−2.71995	−0.114632	−0.0573162	−	0.998356i	$-0.518254\pi$
$563$	−2.71995	−0.114632	−0.0573162	+	0.998356i	$0.518254\pi$
$564$	0	0
$565$	6.67408	0.280781
$566$	64.3397	2.70440
$567$	0	0
$568$	−13.2177	−0.554601
$569$	27.7190	1.16204	0.581021	−	0.813888i	$-0.302653\pi$
$569$	27.7190	1.16204	0.581021	+	0.813888i	$0.302653\pi$
$570$	0	0
$571$	38.1193	1.59524	0.797621	−	0.603159i	$-0.206092\pi$
$571$	38.1193	1.59524	0.797621	+	0.603159i	$0.206092\pi$
$572$	0	0
$573$	0	0
$574$	−5.07901	−0.211994
$575$	11.8401	0.493768
$576$	0	0
$577$	23.2048	0.966029	0.483015	−	0.875612i	$-0.339542\pi$
$577$	23.2048	0.966029	0.483015	+	0.875612i	$0.339542\pi$
$578$	30.7455	1.27885
$579$	0	0
$580$	6.43899	0.267364
$581$	3.21949	0.133567
$582$	0	0
$583$	0	0
$584$	2.61639	0.108267
$585$	0	0
$586$	−6.48016	−0.267693
$587$	11.0497	0.456068	0.228034	−	0.973653i	$-0.426770\pi$
$587$	11.0497	0.456068	0.228034	+	0.973653i	$0.426770\pi$
$588$	0	0
$589$	37.6197	1.55009
$590$	3.21949	0.132545
$591$	0	0
$592$	0.635770	0.0261300
$593$	36.3884	1.49429	0.747147	−	0.664659i	$-0.231423\pi$
$593$	36.3884	1.49429	0.747147	+	0.664659i	$0.231423\pi$
$594$	0	0
$595$	2.71995	0.111507
$596$	−26.5601	−1.08794
$597$	0	0
$598$	−29.6197	−1.21124
$599$	−29.6585	−1.21181	−0.605906	−	0.795536i	$-0.707189\pi$
$599$	−29.6585	−1.21181	−0.605906	+	0.795536i	$0.707189\pi$
$600$	0	0
$601$	1.39117	0.0567470	0.0283735	−	0.999597i	$-0.490967\pi$
$601$	1.39117	0.0567470	0.0283735	+	0.999597i	$0.490967\pi$
$602$	2.18922	0.0892259
$603$	0	0
$604$	38.9073	1.58312
$605$	0	0
$606$	0	0
$607$	30.9385	1.25576	0.627878	−	0.778312i	$-0.283924\pi$
$607$	30.9385	1.25576	0.627878	+	0.778312i	$0.283924\pi$
$608$	41.9007	1.69930
$609$	0	0
$610$	2.46262	0.0997086
$611$	9.03028	0.365326
$612$	0	0
$613$	8.15986	0.329574	0.164787	−	0.986329i	$-0.447306\pi$
$613$	8.15986	0.329574	0.164787	+	0.986329i	$0.447306\pi$
$614$	7.90253	0.318920
$615$	0	0
$616$	0	0
$617$	27.5298	1.10831	0.554154	−	0.832414i	$-0.313042\pi$
$617$	27.5298	1.10831	0.554154	+	0.832414i	$0.313042\pi$
$618$	0	0
$619$	19.9735	0.802803	0.401401	−	0.915902i	$-0.368523\pi$
$619$	19.9735	0.802803	0.401401	+	0.915902i	$0.368523\pi$
$620$	8.68876	0.348949
$621$	0	0
$622$	−8.86043	−0.355271
$623$	1.26537	0.0506959
$624$	0	0
$625$	21.5289	0.861156
$626$	−24.8704	−0.994022
$627$	0	0
$628$	22.9016	0.913874
$629$	1.31880	0.0525841
$630$	0	0
$631$	−17.2342	−0.686082	−0.343041	−	0.939320i	$-0.611457\pi$
$631$	−17.2342	−0.686082	−0.343041	+	0.939320i	$0.611457\pi$
$632$	9.88462	0.393189
$633$	0	0
$634$	4.74931	0.188619
$635$	−1.24129	−0.0492592
$636$	0	0
$637$	5.60975	0.222266
$638$	0	0
$639$	0	0
$640$	4.10504	0.162266
$641$	44.4149	1.75428	0.877142	−	0.480231i	$-0.159447\pi$
$641$	44.4149	1.75428	0.877142	+	0.480231i	$0.159447\pi$
$642$	0	0
$643$	−22.5336	−0.888638	−0.444319	−	0.895869i	$-0.646554\pi$
$643$	−22.5336	−0.888638	−0.444319	+	0.895869i	$0.646554\pi$
$644$	−6.24977	−0.246275
$645$	0	0
$646$	62.9385	2.47628
$647$	18.7758	0.738153	0.369077	−	0.929399i	$-0.379674\pi$
$647$	18.7758	0.738153	0.369077	+	0.929399i	$0.379674\pi$
$648$	0	0
$649$	0	0
$650$	−56.7980	−2.22780
$651$	0	0
$652$	−33.4012	−1.30809
$653$	−31.6126	−1.23710	−0.618549	−	0.785747i	$-0.712279\pi$
$653$	−31.6126	−1.23710	−0.618549	+	0.785747i	$0.712279\pi$
$654$	0	0
$655$	−1.31880	−0.0515298
$656$	6.46410	0.252381
$657$	0	0
$658$	3.42053	0.133346
$659$	−19.8477	−0.773157	−0.386578	−	0.922257i	$-0.626343\pi$
$659$	−19.8477	−0.773157	−0.386578	+	0.922257i	$0.626343\pi$
$660$	0	0
$661$	31.4234	1.22223	0.611114	−	0.791542i	$-0.290722\pi$
$661$	31.4234	1.22223	0.611114	+	0.791542i	$0.290722\pi$
$662$	47.4381	1.84373
$663$	0	0
$664$	3.52409	0.136761
$665$	2.56009	0.0992762
$666$	0	0
$667$	13.1202	0.508016
$668$	−25.3037	−0.979029
$669$	0	0
$670$	−10.3179	−0.398615
$671$	0	0
$672$	0	0
$673$	−19.2195	−0.740857	−0.370429	−	0.928861i	$-0.620789\pi$
$673$	−19.2195	−0.740857	−0.370429	+	0.928861i	$0.620789\pi$
$674$	−28.2186	−1.08694
$675$	0	0
$676$	46.4528	1.78664
$677$	−25.7309	−0.988917	−0.494458	−	0.869201i	$-0.664634\pi$
$677$	−25.7309	−0.988917	−0.494458	+	0.869201i	$0.664634\pi$
$678$	0	0
$679$	−8.79518	−0.337528
$680$	2.97729	0.114174
$681$	0	0
$682$	0	0
$683$	11.0596	0.423185	0.211593	−	0.977358i	$-0.432135\pi$
$683$	11.0596	0.423185	0.211593	+	0.977358i	$0.432135\pi$
$684$	0	0
$685$	−4.70527	−0.179779
$686$	2.12489	0.0811285
$687$	0	0
$688$	−2.78623	−0.106224
$689$	16.9991	0.647613
$690$	0	0
$691$	45.0256	1.71285	0.856427	−	0.516268i	$-0.172679\pi$
$691$	45.0256	1.71285	0.856427	+	0.516268i	$0.172679\pi$
$692$	20.5483	0.781128
$693$	0	0
$694$	38.3103	1.45424
$695$	9.56101	0.362670
$696$	0	0
$697$	13.4087	0.507891
$698$	−46.6400	−1.76535
$699$	0	0
$700$	−11.9844	−0.452968
$701$	−46.7787	−1.76681	−0.883403	−	0.468614i	$-0.844754\pi$
$701$	−46.7787	−1.76681	−0.883403	+	0.468614i	$0.844754\pi$
$702$	0	0
$703$	1.24129	0.0468162
$704$	0	0
$705$	0	0
$706$	−40.0663	−1.50791
$707$	12.9503	0.487048
$708$	0	0
$709$	−9.14426	−0.343420	−0.171710	−	0.985148i	$-0.554929\pi$
$709$	−9.14426	−0.343420	−0.171710	+	0.985148i	$0.554929\pi$
$710$	12.4408	0.466896
$711$	0	0
$712$	1.38508	0.0519082
$713$	17.7044	0.663033
$714$	0	0
$715$	0	0
$716$	30.8468	1.15280
$717$	0	0
$718$	−3.25069	−0.121315
$719$	30.4655	1.13617	0.568085	−	0.822970i	$-0.307684\pi$
$719$	30.4655	1.13617	0.568085	+	0.822970i	$0.307684\pi$
$720$	0	0
$721$	12.1698	0.453229
$722$	18.8666	0.702143
$723$	0	0
$724$	−17.7190	−0.658523
$725$	25.1589	0.934380
$726$	0	0
$727$	−6.12580	−0.227193	−0.113597	−	0.993527i	$-0.536237\pi$
$727$	−6.12580	−0.227193	−0.113597	+	0.993527i	$0.536237\pi$
$728$	6.14048	0.227581
$729$	0	0
$730$	−2.46262	−0.0911457
$731$	−5.77959	−0.213766
$732$	0	0
$733$	−42.0705	−1.55391	−0.776955	−	0.629556i	$-0.783237\pi$
$733$	−42.0705	−1.55391	−0.776955	+	0.629556i	$0.783237\pi$
$734$	31.1396	1.14938
$735$	0	0
$736$	19.7190	0.726853
$737$	0	0
$738$	0	0
$739$	−28.8780	−1.06229	−0.531147	−	0.847280i	$-0.678239\pi$
$739$	−28.8780	−1.06229	−0.531147	+	0.847280i	$0.678239\pi$
$740$	0.286692	0.0105390
$741$	0	0
$742$	6.43899	0.236383
$743$	14.6812	0.538601	0.269300	−	0.963056i	$-0.413208\pi$
$743$	14.6812	0.538601	0.269300	+	0.963056i	$0.413208\pi$
$744$	0	0
$745$	5.12019	0.187589
$746$	21.3775	0.782687
$747$	0	0
$748$	0	0
$749$	−10.5601	−0.385857
$750$	0	0
$751$	−6.60597	−0.241055	−0.120528	−	0.992710i	$-0.538459\pi$
$751$	−6.60597	−0.241055	−0.120528	+	0.992710i	$0.538459\pi$
$752$	−4.35333	−0.158750
$753$	0	0
$754$	−62.9385	−2.29209
$755$	−7.50046	−0.272970
$756$	0	0
$757$	2.49954	0.0908474	0.0454237	−	0.998968i	$-0.485536\pi$
$757$	2.49954	0.0908474	0.0454237	+	0.998968i	$0.485536\pi$
$758$	−38.0218	−1.38101
$759$	0	0
$760$	2.80230	0.101650
$761$	12.0487	0.436766	0.218383	−	0.975863i	$-0.429922\pi$
$761$	12.0487	0.436766	0.218383	+	0.975863i	$0.429922\pi$
$762$	0	0
$763$	−7.34060	−0.265748
$764$	−47.2725	−1.71026
$765$	0	0
$766$	40.2985	1.45604
$767$	−17.5298	−0.632965
$768$	0	0
$769$	0.489560	0.0176540	0.00882699	−	0.999961i	$-0.497190\pi$
$769$	0.489560	0.0176540	0.00882699	+	0.999961i	$0.497190\pi$
$770$	0	0
$771$	0	0
$772$	41.4986	1.49357
$773$	46.7181	1.68033	0.840167	−	0.542328i	$-0.182457\pi$
$773$	46.7181	1.68033	0.840167	+	0.542328i	$0.182457\pi$
$774$	0	0
$775$	33.9494	1.21950
$776$	−9.62729	−0.345600
$777$	0	0
$778$	−72.8080	−2.61029
$779$	12.6206	0.452182
$780$	0	0
$781$	0	0
$782$	29.6197	1.05920
$783$	0	0
$784$	−2.70436	−0.0965842
$785$	−4.41491	−0.157575
$786$	0	0
$787$	−8.33968	−0.297278	−0.148639	−	0.988892i	$-0.547489\pi$
$787$	−8.33968	−0.297278	−0.148639	+	0.988892i	$0.547489\pi$
$788$	−61.6197	−2.19511
$789$	0	0
$790$	−9.30368	−0.331010
$791$	−13.7649	−0.489424
$792$	0	0
$793$	−13.4087	−0.476157
$794$	34.5913	1.22760
$795$	0	0
$796$	−38.7063	−1.37191
$797$	31.5151	1.11632	0.558162	−	0.829732i	$-0.311507\pi$
$797$	31.5151	1.11632	0.558162	+	0.829732i	$0.311507\pi$
$798$	0	0
$799$	−9.03028	−0.319468
$800$	37.8127	1.33688
$801$	0	0
$802$	−61.4911	−2.17132
$803$	0	0
$804$	0	0
$805$	1.20482	0.0424641
$806$	−84.9291	−2.99150
$807$	0	0
$808$	14.1756	0.498695
$809$	−15.5005	−0.544967	−0.272484	−	0.962160i	$-0.587845\pi$
$809$	−15.5005	−0.544967	−0.272484	+	0.962160i	$0.587845\pi$
$810$	0	0
$811$	−48.3397	−1.69744	−0.848718	−	0.528846i	$-0.822625\pi$
$811$	−48.3397	−1.69744	−0.848718	+	0.528846i	$0.822625\pi$
$812$	−13.2800	−0.466038
$813$	0	0
$814$	0	0
$815$	6.43899	0.225548
$816$	0	0
$817$	−5.43991	−0.190318
$818$	32.9579	1.15235
$819$	0	0
$820$	2.91490	0.101793
$821$	10.5601	0.368550	0.184275	−	0.982875i	$-0.441006\pi$
$821$	10.5601	0.368550	0.184275	+	0.982875i	$0.441006\pi$
$822$	0	0
$823$	19.3241	0.673595	0.336798	−	0.941577i	$-0.390656\pi$
$823$	19.3241	0.673595	0.336798	+	0.941577i	$0.390656\pi$
$824$	13.3212	0.464067
$825$	0	0
$826$	−6.64002	−0.231036
$827$	0.530734	0.0184554	0.00922772	−	0.999957i	$-0.497063\pi$
$827$	0.530734	0.0184554	0.00922772	+	0.999957i	$0.497063\pi$
$828$	0	0
$829$	−47.1954	−1.63916	−0.819582	−	0.572961i	$-0.805794\pi$
$829$	−47.1954	−1.63916	−0.819582	+	0.572961i	$0.805794\pi$
$830$	−3.31697	−0.115134
$831$	0	0
$832$	−64.2522	−2.22754
$833$	−5.60975	−0.194366
$834$	0	0
$835$	4.87798	0.168809
$836$	0	0
$837$	0	0
$838$	27.5180	0.950594
$839$	37.9036	1.30858	0.654288	−	0.756245i	$-0.272968\pi$
$839$	37.9036	1.30858	0.654288	+	0.756245i	$0.272968\pi$
$840$	0	0
$841$	−1.12110	−0.0386588
$842$	−36.3161	−1.25153
$843$	0	0
$844$	−36.3161	−1.25005
$845$	−8.95504	−0.308063
$846$	0	0
$847$	0	0
$848$	−8.19495	−0.281416
$849$	0	0
$850$	56.7980	1.94816
$851$	0.584169	0.0200251
$852$	0	0
$853$	−6.26915	−0.214652	−0.107326	−	0.994224i	$-0.534229\pi$
$853$	−6.26915	−0.214652	−0.107326	+	0.994224i	$0.534229\pi$
$854$	−5.07901	−0.173800
$855$	0	0
$856$	−11.5592	−0.395085
$857$	−36.9503	−1.26220	−0.631100	−	0.775702i	$-0.717396\pi$
$857$	−36.9503	−1.26220	−0.631100	+	0.775702i	$0.717396\pi$
$858$	0	0
$859$	8.90447	0.303817	0.151908	−	0.988395i	$-0.451458\pi$
$859$	8.90447	0.303817	0.151908	+	0.988395i	$0.451458\pi$
$860$	−1.25642	−0.0428434
$861$	0	0
$862$	−41.1126	−1.40030
$863$	42.1193	1.43376	0.716878	−	0.697198i	$-0.245570\pi$
$863$	42.1193	1.43376	0.716878	+	0.697198i	$0.245570\pi$
$864$	0	0
$865$	−3.96125	−0.134686
$866$	33.6273	1.14270
$867$	0	0
$868$	−17.9201	−0.608247
$869$	0	0
$870$	0	0
$871$	56.1798	1.90358
$872$	−8.03509	−0.272102
$873$	0	0
$874$	27.8789	0.943018
$875$	4.73463	0.160060
$876$	0	0
$877$	24.3397	0.821893	0.410946	−	0.911660i	$-0.365198\pi$
$877$	24.3397	0.821893	0.410946	+	0.911660i	$0.365198\pi$
$878$	−23.5005	−0.793102
$879$	0	0
$880$	0	0
$881$	−5.64380	−0.190145	−0.0950723	−	0.995470i	$-0.530308\pi$
$881$	−5.64380	−0.190145	−0.0950723	+	0.995470i	$0.530308\pi$
$882$	0	0
$883$	15.1807	0.510873	0.255436	−	0.966826i	$-0.417781\pi$
$883$	15.1807	0.510873	0.255436	+	0.966826i	$0.417781\pi$
$884$	−79.1495	−2.66209
$885$	0	0
$886$	73.7408	2.47737
$887$	−26.8174	−0.900441	−0.450221	−	0.892917i	$-0.648655\pi$
$887$	−26.8174	−0.900441	−0.450221	+	0.892917i	$0.648655\pi$
$888$	0	0
$889$	2.56009	0.0858628
$890$	−1.30368	−0.0436994
$891$	0	0
$892$	43.3951	1.45297
$893$	−8.49954	−0.284426
$894$	0	0
$895$	−5.94657	−0.198772
$896$	−8.46640	−0.282843
$897$	0	0
$898$	77.1202	2.57353
$899$	37.6197	1.25469
$900$	0	0
$901$	−16.9991	−0.566322
$902$	0	0
$903$	0	0
$904$	−15.0672	−0.501128
$905$	3.41583	0.113546
$906$	0	0
$907$	32.0975	1.06578	0.532890	−	0.846185i	$-0.321106\pi$
$907$	32.0975	1.06578	0.532890	+	0.846185i	$0.321106\pi$
$908$	−26.9622	−0.894771
$909$	0	0
$910$	−5.77959	−0.191591
$911$	−12.1599	−0.402874	−0.201437	−	0.979501i	$-0.564561\pi$
$911$	−12.1599	−0.402874	−0.201437	+	0.979501i	$0.564561\pi$
$912$	0	0
$913$	0	0
$914$	−4.37844	−0.144826
$915$	0	0
$916$	13.7190	0.453290
$917$	2.71995	0.0898208
$918$	0	0
$919$	−40.0000	−1.31948	−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000	−1.31948	−0.659739	+	0.751495i	$0.729333\pi$
$920$	1.31880	0.0434796
$921$	0	0
$922$	15.2370	0.501805
$923$	−67.7390	−2.22966
$924$	0	0
$925$	1.12019	0.0368315
$926$	7.34060	0.241227
$927$	0	0
$928$	41.9007	1.37546
$929$	21.3794	0.701434	0.350717	−	0.936482i	$-0.385938\pi$
$929$	21.3794	0.701434	0.350717	+	0.936482i	$0.385938\pi$
$930$	0	0
$931$	−5.28005	−0.173047
$932$	−74.8998	−2.45342
$933$	0	0
$934$	28.6769	0.938338
$935$	0	0
$936$	0	0
$937$	−31.5104	−1.02940	−0.514701	−	0.857370i	$-0.672097\pi$
$937$	−31.5104	−1.02940	−0.514701	+	0.857370i	$0.672097\pi$
$938$	21.2800	0.694818
$939$	0	0
$940$	−1.96308	−0.0640286
$941$	−42.7299	−1.39296	−0.696478	−	0.717578i	$-0.745251\pi$
$941$	−42.7299	−1.39296	−0.696478	+	0.717578i	$0.745251\pi$
$942$	0	0
$943$	5.93945	0.193415
$944$	8.45080	0.275050
$945$	0	0
$946$	0	0
$947$	−3.29473	−0.107064	−0.0535321	−	0.998566i	$-0.517048\pi$
$947$	−3.29473	−0.107064	−0.0535321	+	0.998566i	$0.517048\pi$
$948$	0	0
$949$	13.4087	0.435265
$950$	53.4599	1.73447
$951$	0	0
$952$	−6.14048	−0.199014
$953$	−38.6812	−1.25301	−0.626503	−	0.779419i	$-0.715515\pi$
$953$	−38.6812	−1.25301	−0.626503	+	0.779419i	$0.715515\pi$
$954$	0	0
$955$	9.11307	0.294892
$956$	−6.43899	−0.208252
$957$	0	0
$958$	−74.4977	−2.40691
$959$	9.70436	0.313370
$960$	0	0
$961$	19.7640	0.637548
$962$	−2.80230	−0.0903499
$963$	0	0
$964$	68.8880	2.21873
$965$	−8.00000	−0.257529
$966$	0	0
$967$	−3.34816	−0.107670	−0.0538348	−	0.998550i	$-0.517144\pi$
$967$	−3.34816	−0.107670	−0.0538348	+	0.998550i	$0.517144\pi$
$968$	0	0
$969$	0	0
$970$	9.06147	0.290946
$971$	−58.7134	−1.88420	−0.942102	−	0.335327i	$-0.891153\pi$
$971$	−58.7134	−1.88420	−0.942102	+	0.335327i	$0.891153\pi$
$972$	0	0
$973$	−19.7190	−0.632163
$974$	63.0360	2.01980
$975$	0	0
$976$	6.46410	0.206911
$977$	−13.7649	−0.440378	−0.220189	−	0.975457i	$-0.570667\pi$
$977$	−13.7649	−0.440378	−0.220189	+	0.975457i	$0.570667\pi$
$978$	0	0
$979$	0	0
$980$	−1.21949	−0.0389553
$981$	0	0
$982$	−54.2479	−1.73112
$983$	−50.2139	−1.60157	−0.800787	−	0.598949i	$-0.795585\pi$
$983$	−50.2139	−1.60157	−0.800787	+	0.598949i	$0.795585\pi$
$984$	0	0
$985$	11.8789	0.378493
$986$	62.9385	2.00437
$987$	0	0
$988$	−74.4977	−2.37009
$989$	−2.56009	−0.0814062
$990$	0	0
$991$	−4.65940	−0.148011	−0.0740054	−	0.997258i	$-0.523578\pi$
$991$	−4.65940	−0.148011	−0.0740054	+	0.997258i	$0.523578\pi$
$992$	56.5407	1.79517
$993$	0	0
$994$	−25.6585	−0.813838
$995$	7.46170	0.236552
$996$	0	0
$997$	3.38934	0.107341	0.0536707	−	0.998559i	$-0.482908\pi$
$997$	3.38934	0.107341	0.0536707	+	0.998559i	$0.482908\pi$
$998$	−54.5213	−1.72584
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.bz.1.3		3
3.2	odd	2		847.2.a.j.1.1	yes	3
11.10	odd	2		7623.2.a.ce.1.1		3
21.20	even	2		5929.2.a.y.1.1		3
33.2	even	10		847.2.f.u.323.1		12
33.5	odd	10		847.2.f.t.729.3		12
33.8	even	10		847.2.f.u.372.3		12
33.14	odd	10		847.2.f.t.372.1		12
33.17	even	10		847.2.f.u.729.1		12
33.20	odd	10		847.2.f.t.323.3		12
33.26	odd	10		847.2.f.t.148.1		12
33.29	even	10		847.2.f.u.148.3		12
33.32	even	2		847.2.a.i.1.3	✓	3
231.230	odd	2		5929.2.a.t.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
847.2.a.i.1.3	✓	3	33.32	even	2
847.2.a.j.1.1	yes	3	3.2	odd	2
847.2.f.t.148.1		12	33.26	odd	10
847.2.f.t.323.3		12	33.20	odd	10
847.2.f.t.372.1		12	33.14	odd	10
847.2.f.t.729.3		12	33.5	odd	10
847.2.f.u.148.3		12	33.29	even	10
847.2.f.u.323.1		12	33.2	even	10
847.2.f.u.372.3		12	33.8	even	10
847.2.f.u.729.1		12	33.17	even	10
5929.2.a.t.1.3		3	231.230	odd	2
5929.2.a.y.1.1		3	21.20	even	2
7623.2.a.bz.1.3		3	1.1	even	1	trivial
7623.2.a.ce.1.1		3	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+2.12489 q^{2} +2.51514 q^{4} -0.484862 q^{5} +1.00000 q^{7} +1.09461 q^{8} -1.03028 q^{10} +5.60975 q^{13} +2.12489 q^{14} -2.70436 q^{16} -5.60975 q^{17} -5.28005 q^{19} -1.21949 q^{20} -2.48486 q^{23} -4.76491 q^{25} +11.9201 q^{26} +2.51514 q^{28} -5.28005 q^{29} -7.12489 q^{31} -7.93567 q^{32} -11.9201 q^{34} -0.484862 q^{35} -0.235091 q^{37} -11.2195 q^{38} -0.530734 q^{40} -2.39025 q^{41} +1.03028 q^{43} -5.28005 q^{46} +1.60975 q^{47} +1.00000 q^{49} -10.1249 q^{50} +14.1093 q^{52} +3.03028 q^{53} +1.09461 q^{56} -11.2195 q^{58} -3.12489 q^{59} -2.39025 q^{61} -15.1396 q^{62} -11.4537 q^{64} -2.71995 q^{65} +10.0147 q^{67} -14.1093 q^{68} -1.03028 q^{70} -12.0752 q^{71} +2.39025 q^{73} -0.499542 q^{74} -13.2800 q^{76} +9.03028 q^{79} +1.31124 q^{80} -5.07901 q^{82} +3.21949 q^{83} +2.71995 q^{85} +2.18922 q^{86} +1.26537 q^{89} +5.60975 q^{91} -6.24977 q^{92} +3.42053 q^{94} +2.56009 q^{95} -8.79518 q^{97} +2.12489 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{2} + 8 q^{4} - q^{5} + 3 q^{7} - 6 q^{8} - 4 q^{10} + 8 q^{13} - 2 q^{14} + 10 q^{16} - 8 q^{17} + 14 q^{20} - 7 q^{23} + 2 q^{25} + 12 q^{26} + 8 q^{28} - 13 q^{31} - 34 q^{32} - 12 q^{34}+ \cdots - 2 q^{98}+O(q^{100}) \) 3 * q - 2 * q^2 + 8 * q^4 - q^5 + 3 * q^7 - 6 * q^8 - 4 * q^10 + 8 * q^13 - 2 * q^14 + 10 * q^16 - 8 * q^17 + 14 * q^20 - 7 * q^23 + 2 * q^25 + 12 * q^26 + 8 * q^28 - 13 * q^31 - 34 * q^32 - 12 * q^34 - q^35 - 17 * q^37 - 16 * q^38 - 36 * q^40 - 16 * q^41 + 4 * q^43 - 4 * q^47 + 3 * q^49 - 22 * q^50 + 10 * q^53 - 6 * q^56 - 16 * q^58 - q^59 - 16 * q^61 - 4 * q^62 + 34 * q^64 - 24 * q^65 - 3 * q^67 - 4 * q^70 - 5 * q^71 + 16 * q^73 + 32 * q^74 - 24 * q^76 + 28 * q^79 + 56 * q^80 + 28 * q^82 - 8 * q^83 + 24 * q^85 - 12 * q^86 + 21 * q^89 + 8 * q^91 - 2 * q^92 + 20 * q^94 - 24 * q^95 - 11 * q^97 - 2 * q^98

Introduction

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