LMFDB - Embedded newform 7623.2.a.ca.1.3

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:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2541)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.81361$ 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)

$f(q)$	$=$	$q+1.81361 q^{2} +1.28917 q^{4} +2.10278 q^{5} +1.00000 q^{7} -1.28917 q^{8} +O(q^{10})$	$q+1.81361 q^{2} +1.28917 q^{4} +2.10278 q^{5} +1.00000 q^{7} -1.28917 q^{8} +3.81361 q^{10} +0.186393 q^{13} +1.81361 q^{14} -4.91638 q^{16} +2.10278 q^{17} +4.52444 q^{19} +2.71083 q^{20} +1.94610 q^{23} -0.578337 q^{25} +0.338044 q^{26} +1.28917 q^{28} -0.186393 q^{29} +4.20555 q^{31} -6.33804 q^{32} +3.81361 q^{34} +2.10278 q^{35} -1.28917 q^{37} +8.20555 q^{38} -2.71083 q^{40} -8.91638 q^{41} +10.9653 q^{43} +3.52946 q^{46} +9.01916 q^{47} +1.00000 q^{49} -1.04888 q^{50} +0.240293 q^{52} +6.71083 q^{53} -1.28917 q^{56} -0.338044 q^{58} +4.44082 q^{59} +11.3083 q^{61} +7.62721 q^{62} -1.66196 q^{64} +0.391944 q^{65} -5.07306 q^{67} +2.71083 q^{68} +3.81361 q^{70} -6.72999 q^{71} -11.2005 q^{73} -2.33804 q^{74} +5.83276 q^{76} -7.45998 q^{79} -10.3380 q^{80} -16.1708 q^{82} +12.1758 q^{83} +4.42166 q^{85} +19.8867 q^{86} -0.946101 q^{89} +0.186393 q^{91} +2.50885 q^{92} +16.3572 q^{94} +9.51388 q^{95} +7.27358 q^{97} +1.81361 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 3 q^{4} - q^{5} + 3 q^{7} - 3 q^{8}+O(q^{10}) $ 3 * q - q^2 + 3 * q^4 - q^5 + 3 * q^7 - 3 * q^8	$ 3 q - q^{2} + 3 q^{4} - q^{5} + 3 q^{7} - 3 q^{8} + 5 q^{10} + 7 q^{13} - q^{14} - q^{16} - q^{17} + 8 q^{19} + 9 q^{20} + 2 q^{23} - 11 q^{26} + 3 q^{28} - 7 q^{29} - 2 q^{31} - 7 q^{32} + 5 q^{34} - q^{35} - 3 q^{37} + 10 q^{38} - 9 q^{40} - 13 q^{41} + 8 q^{43} + 20 q^{46} + 6 q^{47} + 3 q^{49} + 8 q^{50} + 11 q^{52} + 21 q^{53} - 3 q^{56} + 11 q^{58} - 6 q^{59} + 12 q^{61} + 10 q^{62} - 17 q^{64} - 7 q^{65} + 2 q^{67} + 9 q^{68} + 5 q^{70} - 4 q^{73} + 5 q^{74} - 10 q^{76} + 18 q^{79} - 19 q^{80} - 9 q^{82} + 12 q^{83} + 15 q^{85} + 36 q^{86} + q^{89} + 7 q^{91} - 44 q^{92} + 16 q^{94} - 8 q^{95} - 25 q^{97} - q^{98}+O(q^{100}) $ 3 * q - q^2 + 3 * q^4 - q^5 + 3 * q^7 - 3 * q^8 + 5 * q^10 + 7 * q^13 - q^14 - q^16 - q^17 + 8 * q^19 + 9 * q^20 + 2 * q^23 - 11 * q^26 + 3 * q^28 - 7 * q^29 - 2 * q^31 - 7 * q^32 + 5 * q^34 - q^35 - 3 * q^37 + 10 * q^38 - 9 * q^40 - 13 * q^41 + 8 * q^43 + 20 * q^46 + 6 * q^47 + 3 * q^49 + 8 * q^50 + 11 * q^52 + 21 * q^53 - 3 * q^56 + 11 * q^58 - 6 * q^59 + 12 * q^61 + 10 * q^62 - 17 * q^64 - 7 * q^65 + 2 * q^67 + 9 * q^68 + 5 * q^70 - 4 * q^73 + 5 * q^74 - 10 * q^76 + 18 * q^79 - 19 * q^80 - 9 * q^82 + 12 * q^83 + 15 * q^85 + 36 * q^86 + q^89 + 7 * q^91 - 44 * q^92 + 16 * q^94 - 8 * q^95 - 25 * q^97 - q^98

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 $

$2$	1.81361	1.28241	0.641207	−	0.767368i	$-0.278434\pi$
$2$	1.81361	1.28241	0.641207	+	0.767368i	$0.278434\pi$
$3$	0	0
$3$	0	0
$4$	1.28917	0.644584
$5$	2.10278	0.940390	0.470195	−	0.882563i	$-0.344184\pi$
$5$	2.10278	0.940390	0.470195	+	0.882563i	$0.344184\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.28917	−0.455790
$9$	0	0
$10$	3.81361	1.20597
$11$	0	0
$11$	0	0
$12$	0	0
$13$	0.186393	0.0516963	0.0258481	−	0.999666i	$-0.491771\pi$
$13$	0.186393	0.0516963	0.0258481	+	0.999666i	$0.491771\pi$
$14$	1.81361	0.484707
$15$	0	0
$16$	−4.91638	−1.22910
$17$	2.10278	0.509998	0.254999	−	0.966941i	$-0.417925\pi$
$17$	2.10278	0.509998	0.254999	+	0.966941i	$0.417925\pi$
$18$	0	0
$19$	4.52444	1.03798	0.518989	−	0.854781i	$-0.326309\pi$
$19$	4.52444	1.03798	0.518989	+	0.854781i	$0.326309\pi$
$20$	2.71083	0.606160
$21$	0	0
$22$	0	0
$23$	1.94610	0.405790	0.202895	−	0.979200i	$-0.434965\pi$
$23$	1.94610	0.405790	0.202895	+	0.979200i	$0.434965\pi$
$24$	0	0
$25$	−0.578337	−0.115667
$26$	0.338044	0.0662960
$27$	0	0
$28$	1.28917	0.243630
$29$	−0.186393	−0.0346124	−0.0173062	−	0.999850i	$-0.505509\pi$
$29$	−0.186393	−0.0346124	−0.0173062	+	0.999850i	$0.505509\pi$
$30$	0	0
$31$	4.20555	0.755339	0.377670	−	0.925940i	$-0.376726\pi$
$31$	4.20555	0.755339	0.377670	+	0.925940i	$0.376726\pi$
$32$	−6.33804	−1.12042
$33$	0	0
$34$	3.81361	0.654028
$35$	2.10278	0.355434
$36$	0	0
$37$	−1.28917	−0.211938	−0.105969	−	0.994369i	$-0.533794\pi$
$37$	−1.28917	−0.211938	−0.105969	+	0.994369i	$0.533794\pi$
$38$	8.20555	1.33112
$39$	0	0
$40$	−2.71083	−0.428620
$41$	−8.91638	−1.39250	−0.696252	−	0.717797i	$-0.745151\pi$
$41$	−8.91638	−1.39250	−0.696252	+	0.717797i	$0.745151\pi$
$42$	0	0
$43$	10.9653	1.67219	0.836093	−	0.548588i	$-0.184834\pi$
$43$	10.9653	1.67219	0.836093	+	0.548588i	$0.184834\pi$
$44$	0	0
$45$	0	0
$46$	3.52946	0.520391
$47$	9.01916	1.31558	0.657790	−	0.753202i	$-0.271492\pi$
$47$	9.01916	1.31558	0.657790	+	0.753202i	$0.271492\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−1.04888	−0.148333
$51$	0	0
$52$	0.240293	0.0333226
$53$	6.71083	0.921804	0.460902	−	0.887451i	$-0.347526\pi$
$53$	6.71083	0.921804	0.460902	+	0.887451i	$0.347526\pi$
$54$	0	0
$55$	0	0
$56$	−1.28917	−0.172272
$57$	0	0
$58$	−0.338044	−0.0443874
$59$	4.44082	0.578145	0.289073	−	0.957307i	$-0.406653\pi$
$59$	4.44082	0.578145	0.289073	+	0.957307i	$0.406653\pi$
$60$	0	0
$61$	11.3083	1.44788	0.723941	−	0.689862i	$-0.242329\pi$
$61$	11.3083	1.44788	0.723941	+	0.689862i	$0.242329\pi$
$62$	7.62721	0.968657
$63$	0	0
$64$	−1.66196	−0.207744
$65$	0.391944	0.0486146
$66$	0	0
$67$	−5.07306	−0.619772	−0.309886	−	0.950774i	$-0.600291\pi$
$67$	−5.07306	−0.619772	−0.309886	+	0.950774i	$0.600291\pi$
$68$	2.71083	0.328737
$69$	0	0
$70$	3.81361	0.455813
$71$	−6.72999	−0.798703	−0.399351	−	0.916798i	$-0.630765\pi$
$71$	−6.72999	−0.798703	−0.399351	+	0.916798i	$0.630765\pi$
$72$	0	0
$73$	−11.2005	−1.31092	−0.655461	−	0.755229i	$-0.727526\pi$
$73$	−11.2005	−1.31092	−0.655461	+	0.755229i	$0.727526\pi$
$74$	−2.33804	−0.271792
$75$	0	0
$76$	5.83276	0.669064
$77$	0	0
$78$	0	0
$79$	−7.45998	−0.839313	−0.419656	−	0.907683i	$-0.637849\pi$
$79$	−7.45998	−0.839313	−0.419656	+	0.907683i	$0.637849\pi$
$80$	−10.3380	−1.15583
$81$	0	0
$82$	−16.1708	−1.78577
$83$	12.1758	1.33647	0.668236	−	0.743950i	$-0.267050\pi$
$83$	12.1758	1.33647	0.668236	+	0.743950i	$0.267050\pi$
$84$	0	0
$85$	4.42166	0.479597
$86$	19.8867	2.14443
$87$	0	0
$88$	0	0
$89$	−0.946101	−0.100286	−0.0501432	−	0.998742i	$-0.515968\pi$
$89$	−0.946101	−0.100286	−0.0501432	+	0.998742i	$0.515968\pi$
$90$	0	0
$91$	0.186393	0.0195393
$92$	2.50885	0.261566
$93$	0	0
$94$	16.3572	1.68712
$95$	9.51388	0.976103
$96$	0	0
$97$	7.27358	0.738520	0.369260	−	0.929326i	$-0.379611\pi$
$97$	7.27358	0.738520	0.369260	+	0.929326i	$0.379611\pi$
$98$	1.81361	0.183202
$99$	0	0
$100$	−0.745574	−0.0745574
$101$	15.3919	1.53156	0.765778	−	0.643105i	$-0.222354\pi$
$101$	15.3919	1.53156	0.765778	+	0.643105i	$0.222354\pi$
$102$	0	0
$103$	10.5244	1.03700	0.518502	−	0.855077i	$-0.326490\pi$
$103$	10.5244	1.03700	0.518502	+	0.855077i	$0.326490\pi$
$104$	−0.240293	−0.0235626
$105$	0	0
$106$	12.1708	1.18213
$107$	−13.5733	−1.31218	−0.656091	−	0.754682i	$-0.727791\pi$
$107$	−13.5733	−1.31218	−0.656091	+	0.754682i	$0.727791\pi$
$108$	0	0
$109$	17.6167	1.68737	0.843685	−	0.536839i	$-0.180382\pi$
$109$	17.6167	1.68737	0.843685	+	0.536839i	$0.180382\pi$
$110$	0	0
$111$	0	0
$112$	−4.91638	−0.464554
$113$	−10.5436	−0.991858	−0.495929	−	0.868363i	$-0.665172\pi$
$113$	−10.5436	−0.991858	−0.495929	+	0.868363i	$0.665172\pi$
$114$	0	0
$115$	4.09221	0.381601
$116$	−0.240293	−0.0223106
$117$	0	0
$118$	8.05390	0.741422
$119$	2.10278	0.192761
$120$	0	0
$121$	0	0
$122$	20.5089	1.85678
$123$	0	0
$124$	5.42166	0.486880
$125$	−11.7300	−1.04916
$126$	0	0
$127$	21.2091	1.88201	0.941003	−	0.338399i	$-0.109885\pi$
$127$	21.2091	1.88201	0.941003	+	0.338399i	$0.109885\pi$
$128$	9.66196	0.854004
$129$	0	0
$130$	0.710831	0.0623440
$131$	−5.55918	−0.485708	−0.242854	−	0.970063i	$-0.578084\pi$
$131$	−5.55918	−0.485708	−0.242854	+	0.970063i	$0.578084\pi$
$132$	0	0
$133$	4.52444	0.392319
$134$	−9.20053	−0.794804
$135$	0	0
$136$	−2.71083	−0.232452
$137$	−22.4550	−1.91846	−0.959228	−	0.282633i	$-0.908792\pi$
$137$	−22.4550	−1.91846	−0.959228	+	0.282633i	$0.908792\pi$
$138$	0	0
$139$	3.27001	0.277359	0.138679	−	0.990337i	$-0.455714\pi$
$139$	3.27001	0.277359	0.138679	+	0.990337i	$0.455714\pi$
$140$	2.71083	0.229107
$141$	0	0
$142$	−12.2056	−1.02427
$143$	0	0
$144$	0	0
$145$	−0.391944	−0.0325491
$146$	−20.3133	−1.68114
$147$	0	0
$148$	−1.66196	−0.136612
$149$	2.65693	0.217664	0.108832	−	0.994060i	$-0.465289\pi$
$149$	2.65693	0.217664	0.108832	+	0.994060i	$0.465289\pi$
$150$	0	0
$151$	−16.5330	−1.34544	−0.672720	−	0.739898i	$-0.734874\pi$
$151$	−16.5330	−1.34544	−0.672720	+	0.739898i	$0.734874\pi$
$152$	−5.83276	−0.473100
$153$	0	0
$154$	0	0
$155$	8.84333	0.710313
$156$	0	0
$157$	17.1950	1.37231	0.686155	−	0.727456i	$-0.259297\pi$
$157$	17.1950	1.37231	0.686155	+	0.727456i	$0.259297\pi$
$158$	−13.5295	−1.07635
$159$	0	0
$160$	−13.3275	−1.05363
$161$	1.94610	0.153374
$162$	0	0
$163$	−4.20555	−0.329404	−0.164702	−	0.986343i	$-0.552666\pi$
$163$	−4.20555	−0.329404	−0.164702	+	0.986343i	$0.552666\pi$
$164$	−11.4947	−0.897587
$165$	0	0
$166$	22.0822	1.71391
$167$	−0.646370	−0.0500176	−0.0250088	−	0.999687i	$-0.507961\pi$
$167$	−0.646370	−0.0500176	−0.0250088	+	0.999687i	$0.507961\pi$
$168$	0	0
$169$	−12.9653	−0.997327
$170$	8.01916	0.615041
$171$	0	0
$172$	14.1361	1.07786
$173$	1.35363	0.102915	0.0514573	−	0.998675i	$-0.483613\pi$
$173$	1.35363	0.102915	0.0514573	+	0.998675i	$0.483613\pi$
$174$	0	0
$175$	−0.578337	−0.0437182
$176$	0	0
$177$	0	0
$178$	−1.71585	−0.128609
$179$	13.9406	1.04197	0.520983	−	0.853567i	$-0.325565\pi$
$179$	13.9406	1.04197	0.520983	+	0.853567i	$0.325565\pi$
$180$	0	0
$181$	1.39697	0.103836	0.0519179	−	0.998651i	$-0.483467\pi$
$181$	1.39697	0.103836	0.0519179	+	0.998651i	$0.483467\pi$
$182$	0.338044	0.0250575
$183$	0	0
$184$	−2.50885	−0.184955
$185$	−2.71083	−0.199304
$186$	0	0
$187$	0	0
$188$	11.6272	0.848002
$189$	0	0
$190$	17.2544	1.25177
$191$	−13.4061	−0.970030	−0.485015	−	0.874506i	$-0.661186\pi$
$191$	−13.4061	−0.970030	−0.485015	+	0.874506i	$0.661186\pi$
$192$	0	0
$193$	15.0978	1.08676	0.543380	−	0.839487i	$-0.317144\pi$
$193$	15.0978	1.08676	0.543380	+	0.839487i	$0.317144\pi$
$194$	13.1914	0.947089
$195$	0	0
$196$	1.28917	0.0920835
$197$	3.96526	0.282513	0.141256	−	0.989973i	$-0.454886\pi$
$197$	3.96526	0.282513	0.141256	+	0.989973i	$0.454886\pi$
$198$	0	0
$199$	22.5628	1.59943	0.799716	−	0.600379i	$-0.204984\pi$
$199$	22.5628	1.59943	0.799716	+	0.600379i	$0.204984\pi$
$200$	0.745574	0.0527200
$201$	0	0
$202$	27.9149	1.96409
$203$	−0.186393	−0.0130823
$204$	0	0
$205$	−18.7491	−1.30950
$206$	19.0872	1.32987
$207$	0	0
$208$	−0.916382	−0.0635396
$209$	0	0
$210$	0	0
$211$	6.18137	0.425543	0.212772	−	0.977102i	$-0.431751\pi$
$211$	6.18137	0.425543	0.212772	+	0.977102i	$0.431751\pi$
$212$	8.65139	0.594180
$213$	0	0
$214$	−24.6167	−1.68276
$215$	23.0575	1.57251
$216$	0	0
$217$	4.20555	0.285491
$218$	31.9497	2.16390
$219$	0	0
$220$	0	0
$221$	0.391944	0.0263650
$222$	0	0
$223$	−11.2544	−0.753652	−0.376826	−	0.926284i	$-0.622985\pi$
$223$	−11.2544	−0.753652	−0.376826	+	0.926284i	$0.622985\pi$
$224$	−6.33804	−0.423478
$225$	0	0
$226$	−19.1219	−1.27197
$227$	−21.2630	−1.41128	−0.705638	−	0.708572i	$-0.749340\pi$
$227$	−21.2630	−1.41128	−0.705638	+	0.708572i	$0.749340\pi$
$228$	0	0
$229$	−6.07306	−0.401319	−0.200659	−	0.979661i	$-0.564308\pi$
$229$	−6.07306	−0.401319	−0.200659	+	0.979661i	$0.564308\pi$
$230$	7.42166	0.489370
$231$	0	0
$232$	0.240293	0.0157760
$233$	−1.60806	−0.105347	−0.0526736	−	0.998612i	$-0.516774\pi$
$233$	−1.60806	−0.105347	−0.0526736	+	0.998612i	$0.516774\pi$
$234$	0	0
$235$	18.9653	1.23716
$236$	5.72496	0.372663
$237$	0	0
$238$	3.81361	0.247199
$239$	−0.578337	−0.0374095	−0.0187048	−	0.999825i	$-0.505954\pi$
$239$	−0.578337	−0.0374095	−0.0187048	+	0.999825i	$0.505954\pi$
$240$	0	0
$241$	20.5244	1.32210	0.661048	−	0.750344i	$-0.270112\pi$
$241$	20.5244	1.32210	0.661048	+	0.750344i	$0.270112\pi$
$242$	0	0
$243$	0	0
$244$	14.5783	0.933282
$245$	2.10278	0.134341
$246$	0	0
$247$	0.843326	0.0536595
$248$	−5.42166	−0.344276
$249$	0	0
$250$	−21.2736	−1.34546
$251$	−12.8222	−0.809330	−0.404665	−	0.914465i	$-0.632612\pi$
$251$	−12.8222	−0.809330	−0.404665	+	0.914465i	$0.632612\pi$
$252$	0	0
$253$	0	0
$254$	38.4650	2.41351
$255$	0	0
$256$	20.8469	1.30293
$257$	−24.4444	−1.52480	−0.762400	−	0.647107i	$-0.775979\pi$
$257$	−24.4444	−1.52480	−0.762400	+	0.647107i	$0.775979\pi$
$258$	0	0
$259$	−1.28917	−0.0801050
$260$	0.505281	0.0313362
$261$	0	0
$262$	−10.0822	−0.622878
$263$	22.7683	1.40395	0.701977	−	0.712200i	$-0.252301\pi$
$263$	22.7683	1.40395	0.701977	+	0.712200i	$0.252301\pi$
$264$	0	0
$265$	14.1114	0.866855
$266$	8.20555	0.503115
$267$	0	0
$268$	−6.54002	−0.399496
$269$	−19.5925	−1.19457	−0.597287	−	0.802028i	$-0.703755\pi$
$269$	−19.5925	−1.19457	−0.597287	+	0.802028i	$0.703755\pi$
$270$	0	0
$271$	10.9511	0.665233	0.332617	−	0.943062i	$-0.392068\pi$
$271$	10.9511	0.665233	0.332617	+	0.943062i	$0.392068\pi$
$272$	−10.3380	−0.626836
$273$	0	0
$274$	−40.7244	−2.46025
$275$	0	0
$276$	0	0
$277$	9.89169	0.594334	0.297167	−	0.954826i	$-0.403958\pi$
$277$	9.89169	0.594334	0.297167	+	0.954826i	$0.403958\pi$
$278$	5.93051	0.355689
$279$	0	0
$280$	−2.71083	−0.162003
$281$	−31.5960	−1.88486	−0.942431	−	0.334401i	$-0.891466\pi$
$281$	−31.5960	−1.88486	−0.942431	+	0.334401i	$0.891466\pi$
$282$	0	0
$283$	5.52946	0.328692	0.164346	−	0.986403i	$-0.447449\pi$
$283$	5.52946	0.328692	0.164346	+	0.986403i	$0.447449\pi$
$284$	−8.67609	−0.514831
$285$	0	0
$286$	0	0
$287$	−8.91638	−0.526317
$288$	0	0
$289$	−12.5783	−0.739902
$290$	−0.710831	−0.0417415
$291$	0	0
$292$	−14.4394	−0.845000
$293$	−8.94610	−0.522637	−0.261318	−	0.965253i	$-0.584157\pi$
$293$	−8.94610	−0.522637	−0.261318	+	0.965253i	$0.584157\pi$
$294$	0	0
$295$	9.33804	0.543682
$296$	1.66196	0.0965992
$297$	0	0
$298$	4.81863	0.279136
$299$	0.362741	0.0209778
$300$	0	0
$301$	10.9653	0.632027
$302$	−29.9844	−1.72541
$303$	0	0
$304$	−22.2439	−1.27577
$305$	23.7789	1.36157
$306$	0	0
$307$	−10.2494	−0.584964	−0.292482	−	0.956271i	$-0.594481\pi$
$307$	−10.2494	−0.584964	−0.292482	+	0.956271i	$0.594481\pi$
$308$	0	0
$309$	0	0
$310$	16.0383	0.910915
$311$	5.82417	0.330258	0.165129	−	0.986272i	$-0.447196\pi$
$311$	5.82417	0.330258	0.165129	+	0.986272i	$0.447196\pi$
$312$	0	0
$313$	−18.8569	−1.06586	−0.532929	−	0.846160i	$-0.678909\pi$
$313$	−18.8569	−1.06586	−0.532929	+	0.846160i	$0.678909\pi$
$314$	31.1849	1.75987
$315$	0	0
$316$	−9.61717	−0.541008
$317$	33.9844	1.90875	0.954377	−	0.298603i	$-0.0965206\pi$
$317$	33.9844	1.90875	0.954377	+	0.298603i	$0.0965206\pi$
$318$	0	0
$319$	0	0
$320$	−3.49472	−0.195361
$321$	0	0
$322$	3.52946	0.196689
$323$	9.51388	0.529366
$324$	0	0
$325$	−0.107798	−0.00597957
$326$	−7.62721	−0.422432
$327$	0	0
$328$	11.4947	0.634690
$329$	9.01916	0.497242
$330$	0	0
$331$	17.4741	0.960464	0.480232	−	0.877142i	$-0.340552\pi$
$331$	17.4741	0.960464	0.480232	+	0.877142i	$0.340552\pi$
$332$	15.6967	0.861468
$333$	0	0
$334$	−1.17226	−0.0641432
$335$	−10.6675	−0.582828
$336$	0	0
$337$	19.0141	1.03577	0.517883	−	0.855452i	$-0.326720\pi$
$337$	19.0141	1.03577	0.517883	+	0.855452i	$0.326720\pi$
$338$	−23.5139	−1.27899
$339$	0	0
$340$	5.70027	0.309140
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	−14.1361	−0.762166
$345$	0	0
$346$	2.45495	0.131979
$347$	−19.8272	−1.06438	−0.532191	−	0.846625i	$-0.678631\pi$
$347$	−19.8272	−1.06438	−0.532191	+	0.846625i	$0.678631\pi$
$348$	0	0
$349$	10.9547	0.586391	0.293196	−	0.956052i	$-0.405281\pi$
$349$	10.9547	0.586391	0.293196	+	0.956052i	$0.405281\pi$
$350$	−1.04888	−0.0560648
$351$	0	0
$352$	0	0
$353$	−22.9164	−1.21972	−0.609858	−	0.792511i	$-0.708774\pi$
$353$	−22.9164	−1.21972	−0.609858	+	0.792511i	$0.708774\pi$
$354$	0	0
$355$	−14.1517	−0.751092
$356$	−1.21968	−0.0646431
$357$	0	0
$358$	25.2827	1.33623
$359$	−13.4161	−0.708076	−0.354038	−	0.935231i	$-0.615192\pi$
$359$	−13.4161	−0.708076	−0.354038	+	0.935231i	$0.615192\pi$
$360$	0	0
$361$	1.47054	0.0773968
$362$	2.53355	0.133160
$363$	0	0
$364$	0.240293	0.0125948
$365$	−23.5522	−1.23278
$366$	0	0
$367$	−33.6499	−1.75651	−0.878256	−	0.478190i	$-0.841293\pi$
$367$	−33.6499	−1.75651	−0.878256	+	0.478190i	$0.841293\pi$
$368$	−9.56777	−0.498755
$369$	0	0
$370$	−4.91638	−0.255591
$371$	6.71083	0.348409
$372$	0	0
$373$	−14.4947	−0.750508	−0.375254	−	0.926922i	$-0.622445\pi$
$373$	−14.4947	−0.750508	−0.375254	+	0.926922i	$0.622445\pi$
$374$	0	0
$375$	0	0
$376$	−11.6272	−0.599628
$377$	−0.0347425	−0.00178933
$378$	0	0
$379$	28.0524	1.44096	0.720479	−	0.693477i	$-0.243922\pi$
$379$	28.0524	1.44096	0.720479	+	0.693477i	$0.243922\pi$
$380$	12.2650	0.629181
$381$	0	0
$382$	−24.3133	−1.24398
$383$	−11.7053	−0.598112	−0.299056	−	0.954235i	$-0.596672\pi$
$383$	−11.7053	−0.598112	−0.299056	+	0.954235i	$0.596672\pi$
$384$	0	0
$385$	0	0
$386$	27.3814	1.39368
$387$	0	0
$388$	9.37687	0.476039
$389$	23.1219	1.17233	0.586164	−	0.810192i	$-0.300637\pi$
$389$	23.1219	1.17233	0.586164	+	0.810192i	$0.300637\pi$
$390$	0	0
$391$	4.09221	0.206952
$392$	−1.28917	−0.0651128
$393$	0	0
$394$	7.19142	0.362298
$395$	−15.6867	−0.789281
$396$	0	0
$397$	17.8675	0.896744	0.448372	−	0.893847i	$-0.352004\pi$
$397$	17.8675	0.896744	0.448372	+	0.893847i	$0.352004\pi$
$398$	40.9200	2.05113
$399$	0	0
$400$	2.84333	0.142166
$401$	−13.6620	−0.682246	−0.341123	−	0.940019i	$-0.610807\pi$
$401$	−13.6620	−0.682246	−0.341123	+	0.940019i	$0.610807\pi$
$402$	0	0
$403$	0.783887	0.0390482
$404$	19.8428	0.987217
$405$	0	0
$406$	−0.338044	−0.0167769
$407$	0	0
$408$	0	0
$409$	12.8030	0.633070	0.316535	−	0.948581i	$-0.397481\pi$
$409$	12.8030	0.633070	0.316535	+	0.948581i	$0.397481\pi$
$410$	−34.0036	−1.67932
$411$	0	0
$412$	13.5678	0.668436
$413$	4.44082	0.218518
$414$	0	0
$415$	25.6030	1.25680
$416$	−1.18137	−0.0579214
$417$	0	0
$418$	0	0
$419$	−6.91136	−0.337642	−0.168821	−	0.985647i	$-0.553996\pi$
$419$	−6.91136	−0.337642	−0.168821	+	0.985647i	$0.553996\pi$
$420$	0	0
$421$	5.41110	0.263721	0.131860	−	0.991268i	$-0.457905\pi$
$421$	5.41110	0.263721	0.131860	+	0.991268i	$0.457905\pi$
$422$	11.2106	0.545722
$423$	0	0
$424$	−8.65139	−0.420149
$425$	−1.21611	−0.0589901
$426$	0	0
$427$	11.3083	0.547248
$428$	−17.4983	−0.845812
$429$	0	0
$430$	41.8172	2.01660
$431$	18.4705	0.889695	0.444847	−	0.895606i	$-0.353258\pi$
$431$	18.4705	0.889695	0.444847	+	0.895606i	$0.353258\pi$
$432$	0	0
$433$	−13.3330	−0.640744	−0.320372	−	0.947292i	$-0.603808\pi$
$433$	−13.3330	−0.640744	−0.320372	+	0.947292i	$0.603808\pi$
$434$	7.62721	0.366118
$435$	0	0
$436$	22.7108	1.08765
$437$	8.80501	0.421201
$438$	0	0
$439$	9.51941	0.454337	0.227168	−	0.973855i	$-0.427053\pi$
$439$	9.51941	0.454337	0.227168	+	0.973855i	$0.427053\pi$
$440$	0	0
$441$	0	0
$442$	0.710831	0.0338108
$443$	37.2005	1.76745	0.883725	−	0.468006i	$-0.155028\pi$
$443$	37.2005	1.76745	0.883725	+	0.468006i	$0.155028\pi$
$444$	0	0
$445$	−1.98944	−0.0943084
$446$	−20.4111	−0.966494
$447$	0	0
$448$	−1.66196	−0.0785200
$449$	−15.7003	−0.740941	−0.370471	−	0.928844i	$-0.620804\pi$
$449$	−15.7003	−0.740941	−0.370471	+	0.928844i	$0.620804\pi$
$450$	0	0
$451$	0	0
$452$	−13.5925	−0.639336
$453$	0	0
$454$	−38.5628	−1.80984
$455$	0.391944	0.0183746
$456$	0	0
$457$	−31.7738	−1.48632	−0.743159	−	0.669115i	$-0.766673\pi$
$457$	−31.7738	−1.48632	−0.743159	+	0.669115i	$0.766673\pi$
$458$	−11.0141	−0.514657
$459$	0	0
$460$	5.27555	0.245974
$461$	9.04385	0.421214	0.210607	−	0.977571i	$-0.432456\pi$
$461$	9.04385	0.421214	0.210607	+	0.977571i	$0.432456\pi$
$462$	0	0
$463$	−38.3799	−1.78367	−0.891833	−	0.452364i	$-0.850581\pi$
$463$	−38.3799	−1.78367	−0.891833	+	0.452364i	$0.850581\pi$
$464$	0.916382	0.0425419
$465$	0	0
$466$	−2.91638	−0.135099
$467$	−31.1950	−1.44353	−0.721766	−	0.692137i	$-0.756669\pi$
$467$	−31.1950	−1.44353	−0.721766	+	0.692137i	$0.756669\pi$
$468$	0	0
$469$	−5.07306	−0.234252
$470$	34.3955	1.58655
$471$	0	0
$472$	−5.72496	−0.263513
$473$	0	0
$474$	0	0
$475$	−2.61665	−0.120060
$476$	2.71083	0.124251
$477$	0	0
$478$	−1.04888	−0.0479745
$479$	−31.9391	−1.45934	−0.729668	−	0.683802i	$-0.760325\pi$
$479$	−31.9391	−1.45934	−0.729668	+	0.683802i	$0.760325\pi$
$480$	0	0
$481$	−0.240293	−0.0109564
$482$	37.2233	1.69547
$483$	0	0
$484$	0	0
$485$	15.2947	0.694497
$486$	0	0
$487$	−26.3869	−1.19571	−0.597853	−	0.801606i	$-0.703979\pi$
$487$	−26.3869	−1.19571	−0.597853	+	0.801606i	$0.703979\pi$
$488$	−14.5783	−0.659930
$489$	0	0
$490$	3.81361	0.172281
$491$	14.3572	0.647931	0.323966	−	0.946069i	$-0.394984\pi$
$491$	14.3572	0.647931	0.323966	+	0.946069i	$0.394984\pi$
$492$	0	0
$493$	−0.391944	−0.0176523
$494$	1.52946	0.0688137
$495$	0	0
$496$	−20.6761	−0.928384
$497$	−6.72999	−0.301881
$498$	0	0
$499$	−30.3174	−1.35719	−0.678597	−	0.734510i	$-0.737412\pi$
$499$	−30.3174	−1.35719	−0.678597	+	0.734510i	$0.737412\pi$
$500$	−15.1219	−0.676273
$501$	0	0
$502$	−23.2544	−1.03790
$503$	12.0086	0.535437	0.267718	−	0.963497i	$-0.413730\pi$
$503$	12.0086	0.535437	0.267718	+	0.963497i	$0.413730\pi$
$504$	0	0
$505$	32.3658	1.44026
$506$	0	0
$507$	0	0
$508$	27.3421	1.21311
$509$	9.54913	0.423258	0.211629	−	0.977350i	$-0.432123\pi$
$509$	9.54913	0.423258	0.211629	+	0.977350i	$0.432123\pi$
$510$	0	0
$511$	−11.2005	−0.495482
$512$	18.4842	0.816892
$513$	0	0
$514$	−44.3325	−1.95542
$515$	22.1305	0.975187
$516$	0	0
$517$	0	0
$518$	−2.33804	−0.102728
$519$	0	0
$520$	−0.505281	−0.0221581
$521$	26.7542	1.17212	0.586061	−	0.810267i	$-0.300678\pi$
$521$	26.7542	1.17212	0.586061	+	0.810267i	$0.300678\pi$
$522$	0	0
$523$	−24.9739	−1.09203	−0.546015	−	0.837775i	$-0.683856\pi$
$523$	−24.9739	−1.09203	−0.546015	+	0.837775i	$0.683856\pi$
$524$	−7.16672	−0.313080
$525$	0	0
$526$	41.2927	1.80045
$527$	8.84333	0.385221
$528$	0	0
$529$	−19.2127	−0.835334
$530$	25.5925	1.11167
$531$	0	0
$532$	5.83276	0.252882
$533$	−1.66196	−0.0719873
$534$	0	0
$535$	−28.5416	−1.23396
$536$	6.54002	0.282486
$537$	0	0
$538$	−35.5330	−1.53194
$539$	0	0
$540$	0	0
$541$	9.03474	0.388434	0.194217	−	0.980959i	$-0.437783\pi$
$541$	9.03474	0.388434	0.194217	+	0.980959i	$0.437783\pi$
$542$	19.8610	0.853104
$543$	0	0
$544$	−13.3275	−0.571411
$545$	37.0439	1.58678
$546$	0	0
$547$	11.3522	0.485384	0.242692	−	0.970103i	$-0.421970\pi$
$547$	11.3522	0.485384	0.242692	+	0.970103i	$0.421970\pi$
$548$	−28.9482	−1.23661
$549$	0	0
$550$	0	0
$551$	−0.843326	−0.0359269
$552$	0	0
$553$	−7.45998	−0.317230
$554$	17.9396	0.762182
$555$	0	0
$556$	4.21560	0.178781
$557$	16.5089	0.699503	0.349751	−	0.936843i	$-0.386266\pi$
$557$	16.5089	0.699503	0.349751	+	0.936843i	$0.386266\pi$
$558$	0	0
$559$	2.04385	0.0864458
$560$	−10.3380	−0.436862
$561$	0	0
$562$	−57.3028	−2.41717
$563$	2.51030	0.105797	0.0528984	−	0.998600i	$-0.483154\pi$
$563$	2.51030	0.105797	0.0528984	+	0.998600i	$0.483154\pi$
$564$	0	0
$565$	−22.1708	−0.932733
$566$	10.0283	0.421519
$567$	0	0
$568$	8.67609	0.364041
$569$	−5.92498	−0.248388	−0.124194	−	0.992258i	$-0.539634\pi$
$569$	−5.92498	−0.248388	−0.124194	+	0.992258i	$0.539634\pi$
$570$	0	0
$571$	25.7633	1.07816	0.539080	−	0.842255i	$-0.318772\pi$
$571$	25.7633	1.07816	0.539080	+	0.842255i	$0.318772\pi$
$572$	0	0
$573$	0	0
$574$	−16.1708	−0.674956
$575$	−1.12550	−0.0469367
$576$	0	0
$577$	22.4997	0.936677	0.468338	−	0.883549i	$-0.344853\pi$
$577$	22.4997	0.936677	0.468338	+	0.883549i	$0.344853\pi$
$578$	−22.8122	−0.948861
$579$	0	0
$580$	−0.505281	−0.0209807
$581$	12.1758	0.505139
$582$	0	0
$583$	0	0
$584$	14.4394	0.597505
$585$	0	0
$586$	−16.2247	−0.670236
$587$	16.2736	0.671683	0.335841	−	0.941919i	$-0.390979\pi$
$587$	16.2736	0.671683	0.335841	+	0.941919i	$0.390979\pi$
$588$	0	0
$589$	19.0278	0.784025
$590$	16.9355	0.697225
$591$	0	0
$592$	6.33804	0.260492
$593$	−36.1794	−1.48571	−0.742855	−	0.669452i	$-0.766529\pi$
$593$	−36.1794	−1.48571	−0.742855	+	0.669452i	$0.766529\pi$
$594$	0	0
$595$	4.42166	0.181271
$596$	3.42523	0.140303
$597$	0	0
$598$	0.657869	0.0269022
$599$	−1.69670	−0.0693252	−0.0346626	−	0.999399i	$-0.511036\pi$
$599$	−1.69670	−0.0693252	−0.0346626	+	0.999399i	$0.511036\pi$
$600$	0	0
$601$	−2.61862	−0.106816	−0.0534079	−	0.998573i	$-0.517008\pi$
$601$	−2.61862	−0.106816	−0.0534079	+	0.998573i	$0.517008\pi$
$602$	19.8867	0.810520
$603$	0	0
$604$	−21.3139	−0.867249
$605$	0	0
$606$	0	0
$607$	−37.0333	−1.50313	−0.751567	−	0.659656i	$-0.770702\pi$
$607$	−37.0333	−1.50313	−0.751567	+	0.659656i	$0.770702\pi$
$608$	−28.6761	−1.16297
$609$	0	0
$610$	43.1255	1.74610
$611$	1.68111	0.0680105
$612$	0	0
$613$	18.7733	0.758247	0.379124	−	0.925346i	$-0.376225\pi$
$613$	18.7733	0.758247	0.379124	+	0.925346i	$0.376225\pi$
$614$	−18.5884	−0.750166
$615$	0	0
$616$	0	0
$617$	28.3764	1.14239	0.571195	−	0.820815i	$-0.306480\pi$
$617$	28.3764	1.14239	0.571195	+	0.820815i	$0.306480\pi$
$618$	0	0
$619$	8.09775	0.325476	0.162738	−	0.986669i	$-0.447967\pi$
$619$	8.09775	0.325476	0.162738	+	0.986669i	$0.447967\pi$
$620$	11.4005	0.457857
$621$	0	0
$622$	10.5628	0.423528
$623$	−0.946101	−0.0379047
$624$	0	0
$625$	−21.7738	−0.870954
$626$	−34.1991	−1.36687
$627$	0	0
$628$	22.1672	0.884569
$629$	−2.71083	−0.108088
$630$	0	0
$631$	−20.5925	−0.819773	−0.409887	−	0.912136i	$-0.634432\pi$
$631$	−20.5925	−0.819773	−0.409887	+	0.912136i	$0.634432\pi$
$632$	9.61717	0.382550
$633$	0	0
$634$	61.6344	2.44781
$635$	44.5980	1.76982
$636$	0	0
$637$	0.186393	0.00738518
$638$	0	0
$639$	0	0
$640$	20.3169	0.803097
$641$	−16.8242	−0.664515	−0.332257	−	0.943189i	$-0.607810\pi$
$641$	−16.8242	−0.664515	−0.332257	+	0.943189i	$0.607810\pi$
$642$	0	0
$643$	−27.6655	−1.09102	−0.545511	−	0.838104i	$-0.683664\pi$
$643$	−27.6655	−1.09102	−0.545511	+	0.838104i	$0.683664\pi$
$644$	2.50885	0.0988626
$645$	0	0
$646$	17.2544	0.678866
$647$	−37.2530	−1.46457	−0.732283	−	0.681001i	$-0.761545\pi$
$647$	−37.2530	−1.46457	−0.732283	+	0.681001i	$0.761545\pi$
$648$	0	0
$649$	0	0
$650$	−0.195504	−0.00766828
$651$	0	0
$652$	−5.42166	−0.212329
$653$	0.280575	0.0109797	0.00548987	−	0.999985i	$-0.498253\pi$
$653$	0.280575	0.0109797	0.00548987	+	0.999985i	$0.498253\pi$
$654$	0	0
$655$	−11.6897	−0.456755
$656$	43.8363	1.71152
$657$	0	0
$658$	16.3572	0.637670
$659$	−25.9250	−1.00989	−0.504947	−	0.863150i	$-0.668488\pi$
$659$	−25.9250	−1.00989	−0.504947	+	0.863150i	$0.668488\pi$
$660$	0	0
$661$	30.9446	1.20361	0.601804	−	0.798644i	$-0.294449\pi$
$661$	30.9446	1.20361	0.601804	+	0.798644i	$0.294449\pi$
$662$	31.6912	1.23171
$663$	0	0
$664$	−15.6967	−0.609150
$665$	9.51388	0.368932
$666$	0	0
$667$	−0.362741	−0.0140454
$668$	−0.833279	−0.0322405
$669$	0	0
$670$	−19.3466	−0.747426
$671$	0	0
$672$	0	0
$673$	10.9653	0.422680	0.211340	−	0.977413i	$-0.432217\pi$
$673$	10.9653	0.422680	0.211340	+	0.977413i	$0.432217\pi$
$674$	34.4842	1.32828
$675$	0	0
$676$	−16.7144	−0.642862
$677$	32.4544	1.24733	0.623663	−	0.781694i	$-0.285644\pi$
$677$	32.4544	1.24733	0.623663	+	0.781694i	$0.285644\pi$
$678$	0	0
$679$	7.27358	0.279134
$680$	−5.70027	−0.218595
$681$	0	0
$682$	0	0
$683$	−17.0972	−0.654208	−0.327104	−	0.944988i	$-0.606073\pi$
$683$	−17.0972	−0.654208	−0.327104	+	0.944988i	$0.606073\pi$
$684$	0	0
$685$	−47.2177	−1.80410
$686$	1.81361	0.0692438
$687$	0	0
$688$	−53.9094	−2.05528
$689$	1.25086	0.0476538
$690$	0	0
$691$	−22.8816	−0.870459	−0.435229	−	0.900320i	$-0.643333\pi$
$691$	−22.8816	−0.870459	−0.435229	+	0.900320i	$0.643333\pi$
$692$	1.74506	0.0663371
$693$	0	0
$694$	−35.9588	−1.36498
$695$	6.87610	0.260825
$696$	0	0
$697$	−18.7491	−0.710174
$698$	19.8675	0.751996
$699$	0	0
$700$	−0.745574	−0.0281800
$701$	34.8414	1.31594	0.657970	−	0.753044i	$-0.271415\pi$
$701$	34.8414	1.31594	0.657970	+	0.753044i	$0.271415\pi$
$702$	0	0
$703$	−5.83276	−0.219987
$704$	0	0
$705$	0	0
$706$	−41.5613	−1.56418
$707$	15.3919	0.578874
$708$	0	0
$709$	−7.78746	−0.292464	−0.146232	−	0.989250i	$-0.546715\pi$
$709$	−7.78746	−0.292464	−0.146232	+	0.989250i	$0.546715\pi$
$710$	−25.6655	−0.963210
$711$	0	0
$712$	1.21968	0.0457096
$713$	8.18442	0.306509
$714$	0	0
$715$	0	0
$716$	17.9717	0.671635
$717$	0	0
$718$	−24.3316	−0.908046
$719$	−36.0383	−1.34400	−0.672001	−	0.740550i	$-0.734565\pi$
$719$	−36.0383	−1.34400	−0.672001	+	0.740550i	$0.734565\pi$
$720$	0	0
$721$	10.5244	0.391951
$722$	2.66698	0.0992547
$723$	0	0
$724$	1.80093	0.0669309
$725$	0.107798	0.00400353
$726$	0	0
$727$	−46.6988	−1.73196	−0.865982	−	0.500076i	$-0.833305\pi$
$727$	−46.6988	−1.73196	−0.865982	+	0.500076i	$0.833305\pi$
$728$	−0.240293	−0.00890584
$729$	0	0
$730$	−42.7144	−1.58093
$731$	23.0575	0.852811
$732$	0	0
$733$	−0.116908	−0.00431811	−0.00215906	−	0.999998i	$-0.500687\pi$
$733$	−0.116908	−0.00431811	−0.00215906	+	0.999998i	$0.500687\pi$
$734$	−61.0278	−2.25258
$735$	0	0
$736$	−12.3345	−0.454655
$737$	0	0
$738$	0	0
$739$	−46.3799	−1.70611	−0.853057	−	0.521818i	$-0.825254\pi$
$739$	−46.3799	−1.70611	−0.853057	+	0.521818i	$0.825254\pi$
$740$	−3.49472	−0.128468
$741$	0	0
$742$	12.1708	0.446804
$743$	45.3466	1.66361	0.831803	−	0.555070i	$-0.187309\pi$
$743$	45.3466	1.66361	0.831803	+	0.555070i	$0.187309\pi$
$744$	0	0
$745$	5.58693	0.204689
$746$	−26.2877	−0.962462
$747$	0	0
$748$	0	0
$749$	−13.5733	−0.495958
$750$	0	0
$751$	34.2680	1.25046	0.625229	−	0.780441i	$-0.285005\pi$
$751$	34.2680	1.25046	0.625229	+	0.780441i	$0.285005\pi$
$752$	−44.3416	−1.61697
$753$	0	0
$754$	−0.0630093	−0.00229466
$755$	−34.7652	−1.26524
$756$	0	0
$757$	−22.0106	−0.799988	−0.399994	−	0.916518i	$-0.630988\pi$
$757$	−22.0106	−0.799988	−0.399994	+	0.916518i	$0.630988\pi$
$758$	50.8761	1.84790
$759$	0	0
$760$	−12.2650	−0.444898
$761$	11.2877	0.409179	0.204590	−	0.978848i	$-0.434414\pi$
$761$	11.2877	0.409179	0.204590	+	0.978848i	$0.434414\pi$
$762$	0	0
$763$	17.6167	0.637766
$764$	−17.2827	−0.625266
$765$	0	0
$766$	−21.2288	−0.767027
$767$	0.827740	0.0298880
$768$	0	0
$769$	−12.3864	−0.446665	−0.223333	−	0.974742i	$-0.571694\pi$
$769$	−12.3864	−0.446665	−0.223333	+	0.974742i	$0.571694\pi$
$770$	0	0
$771$	0	0
$772$	19.4635	0.700508
$773$	−26.9013	−0.967573	−0.483786	−	0.875186i	$-0.660739\pi$
$773$	−26.9013	−0.967573	−0.483786	+	0.875186i	$0.660739\pi$
$774$	0	0
$775$	−2.43223	−0.0873681
$776$	−9.37687	−0.336610
$777$	0	0
$778$	41.9341	1.50341
$779$	−40.3416	−1.44539
$780$	0	0
$781$	0	0
$782$	7.42166	0.265398
$783$	0	0
$784$	−4.91638	−0.175585
$785$	36.1572	1.29051
$786$	0	0
$787$	−16.9583	−0.604497	−0.302248	−	0.953229i	$-0.597737\pi$
$787$	−16.9583	−0.604497	−0.302248	+	0.953229i	$0.597737\pi$
$788$	5.11189	0.182103
$789$	0	0
$790$	−28.4494	−1.01218
$791$	−10.5436	−0.374887
$792$	0	0
$793$	2.10780	0.0748501
$794$	32.4046	1.15000
$795$	0	0
$796$	29.0872	1.03097
$797$	−5.99141	−0.212226	−0.106113	−	0.994354i	$-0.533841\pi$
$797$	−5.99141	−0.212226	−0.106113	+	0.994354i	$0.533841\pi$
$798$	0	0
$799$	18.9653	0.670943
$800$	3.66553	0.129596
$801$	0	0
$802$	−24.7774	−0.874921
$803$	0	0
$804$	0	0
$805$	4.09221	0.144232
$806$	1.42166	0.0500759
$807$	0	0
$808$	−19.8428	−0.698068
$809$	30.2877	1.06486	0.532430	−	0.846474i	$-0.321279\pi$
$809$	30.2877	1.06486	0.532430	+	0.846474i	$0.321279\pi$
$810$	0	0
$811$	25.2388	0.886256	0.443128	−	0.896458i	$-0.353869\pi$
$811$	25.2388	0.886256	0.443128	+	0.896458i	$0.353869\pi$
$812$	−0.240293	−0.00843262
$813$	0	0
$814$	0	0
$815$	−8.84333	−0.309768
$816$	0	0
$817$	49.6116	1.73569
$818$	23.2197	0.811857
$819$	0	0
$820$	−24.1708	−0.844081
$821$	−0.0977518	−0.00341156	−0.00170578	−	0.999999i	$-0.500543\pi$
$821$	−0.0977518	−0.00341156	−0.00170578	+	0.999999i	$0.500543\pi$
$822$	0	0
$823$	38.7839	1.35192	0.675961	−	0.736938i	$-0.263729\pi$
$823$	38.7839	1.35192	0.675961	+	0.736938i	$0.263729\pi$
$824$	−13.5678	−0.472656
$825$	0	0
$826$	8.05390	0.280231
$827$	−27.8555	−0.968630	−0.484315	−	0.874894i	$-0.660931\pi$
$827$	−27.8555	−0.968630	−0.484315	+	0.874894i	$0.660931\pi$
$828$	0	0
$829$	−40.4585	−1.40518	−0.702591	−	0.711594i	$-0.747974\pi$
$829$	−40.4585	−1.40518	−0.702591	+	0.711594i	$0.747974\pi$
$830$	46.4338	1.61174
$831$	0	0
$832$	−0.309778	−0.0107396
$833$	2.10278	0.0728568
$834$	0	0
$835$	−1.35917	−0.0470360
$836$	0	0
$837$	0	0
$838$	−12.5345	−0.432997
$839$	53.6741	1.85304	0.926518	−	0.376250i	$-0.122787\pi$
$839$	53.6741	1.85304	0.926518	+	0.376250i	$0.122787\pi$
$840$	0	0
$841$	−28.9653	−0.998802
$842$	9.81361	0.338199
$843$	0	0
$844$	7.96883	0.274298
$845$	−27.2630	−0.937876
$846$	0	0
$847$	0	0
$848$	−32.9930	−1.13298
$849$	0	0
$850$	−2.20555	−0.0756497
$851$	−2.50885	−0.0860023
$852$	0	0
$853$	11.6408	0.398574	0.199287	−	0.979941i	$-0.436137\pi$
$853$	11.6408	0.398574	0.199287	+	0.979941i	$0.436137\pi$
$854$	20.5089	0.701798
$855$	0	0
$856$	17.4983	0.598079
$857$	34.9497	1.19386	0.596929	−	0.802294i	$-0.296387\pi$
$857$	34.9497	1.19386	0.596929	+	0.802294i	$0.296387\pi$
$858$	0	0
$859$	−48.1049	−1.64132	−0.820659	−	0.571418i	$-0.806393\pi$
$859$	−48.1049	−1.64132	−0.820659	+	0.571418i	$0.806393\pi$
$860$	29.7250	1.01361
$861$	0	0
$862$	33.4983	1.14096
$863$	−33.4499	−1.13865	−0.569324	−	0.822113i	$-0.692795\pi$
$863$	−33.4499	−1.13865	−0.569324	+	0.822113i	$0.692795\pi$
$864$	0	0
$865$	2.84638	0.0967798
$866$	−24.1809	−0.821699
$867$	0	0
$868$	5.42166	0.184023
$869$	0	0
$870$	0	0
$871$	−0.945585	−0.0320399
$872$	−22.7108	−0.769086
$873$	0	0
$874$	15.9688	0.540154
$875$	−11.7300	−0.396546
$876$	0	0
$877$	27.7980	0.938672	0.469336	−	0.883020i	$-0.344493\pi$
$877$	27.7980	0.938672	0.469336	+	0.883020i	$0.344493\pi$
$878$	17.2645	0.582648
$879$	0	0
$880$	0	0
$881$	−33.8363	−1.13998	−0.569988	−	0.821653i	$-0.693052\pi$
$881$	−33.8363	−1.13998	−0.569988	+	0.821653i	$0.693052\pi$
$882$	0	0
$883$	−37.4630	−1.26073	−0.630366	−	0.776298i	$-0.717095\pi$
$883$	−37.4630	−1.26073	−0.630366	+	0.776298i	$0.717095\pi$
$884$	0.505281	0.0169945
$885$	0	0
$886$	67.4671	2.26660
$887$	3.48970	0.117173	0.0585863	−	0.998282i	$-0.481341\pi$
$887$	3.48970	0.117173	0.0585863	+	0.998282i	$0.481341\pi$
$888$	0	0
$889$	21.2091	0.711331
$890$	−3.60806	−0.120942
$891$	0	0
$892$	−14.5089	−0.485792
$893$	40.8066	1.36554
$894$	0	0
$895$	29.3139	0.979854
$896$	9.66196	0.322783
$897$	0	0
$898$	−28.4741	−0.950193
$899$	−0.783887	−0.0261441
$900$	0	0
$901$	14.1114	0.470118
$902$	0	0
$903$	0	0
$904$	13.5925	0.452079
$905$	2.93751	0.0976460
$906$	0	0
$907$	25.6655	0.852210	0.426105	−	0.904674i	$-0.359885\pi$
$907$	25.6655	0.852210	0.426105	+	0.904674i	$0.359885\pi$
$908$	−27.4116	−0.909686
$909$	0	0
$910$	0.710831	0.0235638
$911$	14.8716	0.492718	0.246359	−	0.969179i	$-0.420766\pi$
$911$	14.8716	0.492718	0.246359	+	0.969179i	$0.420766\pi$
$912$	0	0
$913$	0	0
$914$	−57.6252	−1.90607
$915$	0	0
$916$	−7.82919	−0.258684
$917$	−5.55918	−0.183580
$918$	0	0
$919$	−30.7144	−1.01317	−0.506587	−	0.862189i	$-0.669093\pi$
$919$	−30.7144	−1.01317	−0.506587	+	0.862189i	$0.669093\pi$
$920$	−5.27555	−0.173930
$921$	0	0
$922$	16.4020	0.540171
$923$	−1.25443	−0.0412899
$924$	0	0
$925$	0.745574	0.0245143
$926$	−69.6061	−2.28740
$927$	0	0
$928$	1.18137	0.0387804
$929$	21.2111	0.695913	0.347957	−	0.937511i	$-0.386876\pi$
$929$	21.2111	0.695913	0.347957	+	0.937511i	$0.386876\pi$
$930$	0	0
$931$	4.52444	0.148282
$932$	−2.07306	−0.0679052
$933$	0	0
$934$	−56.5754	−1.85120
$935$	0	0
$936$	0	0
$937$	58.9008	1.92421	0.962103	−	0.272688i	$-0.0879126\pi$
$937$	58.9008	1.92421	0.962103	+	0.272688i	$0.0879126\pi$
$938$	−9.20053	−0.300408
$939$	0	0
$940$	24.4494	0.797452
$941$	−50.8852	−1.65881	−0.829405	−	0.558647i	$-0.811320\pi$
$941$	−50.8852	−1.65881	−0.829405	+	0.558647i	$0.811320\pi$
$942$	0	0
$943$	−17.3522	−0.565065
$944$	−21.8328	−0.710596
$945$	0	0
$946$	0	0
$947$	12.6277	0.410346	0.205173	−	0.978726i	$-0.434224\pi$
$947$	12.6277	0.410346	0.205173	+	0.978726i	$0.434224\pi$
$948$	0	0
$949$	−2.08771	−0.0677698
$950$	−4.74557	−0.153967
$951$	0	0
$952$	−2.71083	−0.0878586
$953$	44.4458	1.43974	0.719871	−	0.694108i	$-0.244201\pi$
$953$	44.4458	1.43974	0.719871	+	0.694108i	$0.244201\pi$
$954$	0	0
$955$	−28.1900	−0.912206
$956$	−0.745574	−0.0241136
$957$	0	0
$958$	−57.9250	−1.87147
$959$	−22.4550	−0.725108
$960$	0	0
$961$	−13.3133	−0.429463
$962$	−0.435796	−0.0140506
$963$	0	0
$964$	26.4595	0.852202
$965$	31.7472	1.02198
$966$	0	0
$967$	22.7214	0.730671	0.365335	−	0.930876i	$-0.380954\pi$
$967$	22.7214	0.730671	0.365335	+	0.930876i	$0.380954\pi$
$968$	0	0
$969$	0	0
$970$	27.7386	0.890632
$971$	−18.2736	−0.586427	−0.293214	−	0.956047i	$-0.594725\pi$
$971$	−18.2736	−0.586427	−0.293214	+	0.956047i	$0.594725\pi$
$972$	0	0
$973$	3.27001	0.104832
$974$	−47.8555	−1.53339
$975$	0	0
$976$	−55.5960	−1.77959
$977$	31.6152	1.01146	0.505730	−	0.862692i	$-0.331223\pi$
$977$	31.6152	1.01146	0.505730	+	0.862692i	$0.331223\pi$
$978$	0	0
$979$	0	0
$980$	2.71083	0.0865943
$981$	0	0
$982$	26.0383	0.830916
$983$	−33.8328	−1.07910	−0.539549	−	0.841954i	$-0.681405\pi$
$983$	−33.8328	−1.07910	−0.539549	+	0.841954i	$0.681405\pi$
$984$	0	0
$985$	8.33804	0.265672
$986$	−0.710831	−0.0226375
$987$	0	0
$988$	1.08719	0.0345881
$989$	21.3395	0.678557
$990$	0	0
$991$	−2.71440	−0.0862258	−0.0431129	−	0.999070i	$-0.513728\pi$
$991$	−2.71440	−0.0862258	−0.0431129	+	0.999070i	$0.513728\pi$
$992$	−26.6550	−0.846296
$993$	0	0
$994$	−12.2056	−0.387137
$995$	47.4444	1.50409
$996$	0	0
$997$	−43.2197	−1.36878	−0.684391	−	0.729116i	$-0.739932\pi$
$997$	−43.2197	−1.36878	−0.684391	+	0.729116i	$0.739932\pi$
$998$	−54.9839	−1.74048
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.ca.1.3		3
3.2	odd	2		2541.2.a.bj.1.1	yes	3
11.10	odd	2		7623.2.a.cc.1.1		3
33.32	even	2		2541.2.a.bh.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2541.2.a.bh.1.3	✓	3	33.32	even	2
2541.2.a.bj.1.1	yes	3	3.2	odd	2
7623.2.a.ca.1.3		3	1.1	even	1	trivial
7623.2.a.cc.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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.81361 q^{2} +1.28917 q^{4} +2.10278 q^{5} +1.00000 q^{7} -1.28917 q^{8} +O(q^{10})\)	\(q+1.81361 q^{2} +1.28917 q^{4} +2.10278 q^{5} +1.00000 q^{7} -1.28917 q^{8} +3.81361 q^{10} +0.186393 q^{13} +1.81361 q^{14} -4.91638 q^{16} +2.10278 q^{17} +4.52444 q^{19} +2.71083 q^{20} +1.94610 q^{23} -0.578337 q^{25} +0.338044 q^{26} +1.28917 q^{28} -0.186393 q^{29} +4.20555 q^{31} -6.33804 q^{32} +3.81361 q^{34} +2.10278 q^{35} -1.28917 q^{37} +8.20555 q^{38} -2.71083 q^{40} -8.91638 q^{41} +10.9653 q^{43} +3.52946 q^{46} +9.01916 q^{47} +1.00000 q^{49} -1.04888 q^{50} +0.240293 q^{52} +6.71083 q^{53} -1.28917 q^{56} -0.338044 q^{58} +4.44082 q^{59} +11.3083 q^{61} +7.62721 q^{62} -1.66196 q^{64} +0.391944 q^{65} -5.07306 q^{67} +2.71083 q^{68} +3.81361 q^{70} -6.72999 q^{71} -11.2005 q^{73} -2.33804 q^{74} +5.83276 q^{76} -7.45998 q^{79} -10.3380 q^{80} -16.1708 q^{82} +12.1758 q^{83} +4.42166 q^{85} +19.8867 q^{86} -0.946101 q^{89} +0.186393 q^{91} +2.50885 q^{92} +16.3572 q^{94} +9.51388 q^{95} +7.27358 q^{97} +1.81361 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 3 q^{4} - q^{5} + 3 q^{7} - 3 q^{8}+O(q^{10}) \) 3 * q - q^2 + 3 * q^4 - q^5 + 3 * q^7 - 3 * q^8	\( 3 q - q^{2} + 3 q^{4} - q^{5} + 3 q^{7} - 3 q^{8} + 5 q^{10} + 7 q^{13} - q^{14} - q^{16} - q^{17} + 8 q^{19} + 9 q^{20} + 2 q^{23} - 11 q^{26} + 3 q^{28} - 7 q^{29} - 2 q^{31} - 7 q^{32} + 5 q^{34} - q^{35} - 3 q^{37} + 10 q^{38} - 9 q^{40} - 13 q^{41} + 8 q^{43} + 20 q^{46} + 6 q^{47} + 3 q^{49} + 8 q^{50} + 11 q^{52} + 21 q^{53} - 3 q^{56} + 11 q^{58} - 6 q^{59} + 12 q^{61} + 10 q^{62} - 17 q^{64} - 7 q^{65} + 2 q^{67} + 9 q^{68} + 5 q^{70} - 4 q^{73} + 5 q^{74} - 10 q^{76} + 18 q^{79} - 19 q^{80} - 9 q^{82} + 12 q^{83} + 15 q^{85} + 36 q^{86} + q^{89} + 7 q^{91} - 44 q^{92} + 16 q^{94} - 8 q^{95} - 25 q^{97} - q^{98}+O(q^{100}) \) 3 * q - q^2 + 3 * q^4 - q^5 + 3 * q^7 - 3 * q^8 + 5 * q^10 + 7 * q^13 - q^14 - q^16 - q^17 + 8 * q^19 + 9 * q^20 + 2 * q^23 - 11 * q^26 + 3 * q^28 - 7 * q^29 - 2 * q^31 - 7 * q^32 + 5 * q^34 - q^35 - 3 * q^37 + 10 * q^38 - 9 * q^40 - 13 * q^41 + 8 * q^43 + 20 * q^46 + 6 * q^47 + 3 * q^49 + 8 * q^50 + 11 * q^52 + 21 * q^53 - 3 * q^56 + 11 * q^58 - 6 * q^59 + 12 * q^61 + 10 * q^62 - 17 * q^64 - 7 * q^65 + 2 * q^67 + 9 * q^68 + 5 * q^70 - 4 * q^73 + 5 * q^74 - 10 * q^76 + 18 * q^79 - 19 * q^80 - 9 * q^82 + 12 * q^83 + 15 * q^85 + 36 * q^86 + q^89 + 7 * q^91 - 44 * q^92 + 16 * q^94 - 8 * q^95 - 25 * q^97 - q^98

Introduction

L-functions

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Database

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