LMFDB - Embedded newform 7623.2.a.ca.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7623,2,Mod(1,7623)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7623, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("7623.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$60.8699614608$
Analytic rank:	$0$
Dimension:	$3$
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			1.2
Root			$0.470683$ of defining polynomial
Character	$\chi$	$=$	7623.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.470683 q^{2} -1.77846 q^{4} -3.24914 q^{5} +1.00000 q^{7} +1.77846 q^{8} +O(q^{10})$	$q-0.470683 q^{2} -1.77846 q^{4} -3.24914 q^{5} +1.00000 q^{7} +1.77846 q^{8} +1.52932 q^{10} +2.47068 q^{13} -0.470683 q^{14} +2.71982 q^{16} -3.24914 q^{17} +5.30777 q^{19} +5.77846 q^{20} +8.86469 q^{23} +5.55691 q^{25} -1.16291 q^{26} -1.77846 q^{28} -2.47068 q^{29} -6.49828 q^{31} -4.83709 q^{32} +1.52932 q^{34} -3.24914 q^{35} +1.77846 q^{37} -2.49828 q^{38} -5.77846 q^{40} -1.28018 q^{41} +4.89572 q^{43} -4.17246 q^{46} -3.96896 q^{47} +1.00000 q^{49} -2.61555 q^{50} -4.39400 q^{52} +9.77846 q^{53} +1.77846 q^{56} +1.16291 q^{58} -2.41205 q^{59} -4.74742 q^{61} +3.05863 q^{62} -3.16291 q^{64} -8.02760 q^{65} +14.8337 q^{67} +5.77846 q^{68} +1.52932 q^{70} +3.19051 q^{71} -8.98195 q^{73} -0.837090 q^{74} -9.43965 q^{76} +12.3810 q^{79} -8.83709 q^{80} +0.602558 q^{82} -13.0828 q^{83} +10.5569 q^{85} -2.30434 q^{86} -7.86469 q^{89} +2.47068 q^{91} -15.7655 q^{92} +1.86813 q^{94} -17.2457 q^{95} -14.8517 q^{97} -0.470683 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.470683	−0.332823	−0.166412	−	0.986056i	$-0.553218\pi$
$2$	−0.470683	−0.332823	−0.166412	+	0.986056i	$0.553218\pi$
$3$	0	0
$3$	0	0
$4$	−1.77846	−0.889229
$5$	−3.24914	−1.45306	−0.726530	−	0.687135i	$-0.758868\pi$
$5$	−3.24914	−1.45306	−0.726530	+	0.687135i	$0.758868\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.77846	0.628780
$9$	0	0
$10$	1.52932	0.483612
$11$	0	0
$11$	0	0
$12$	0	0
$13$	2.47068	0.685244	0.342622	−	0.939473i	$-0.388685\pi$
$13$	2.47068	0.685244	0.342622	+	0.939473i	$0.388685\pi$
$14$	−0.470683	−0.125795
$15$	0	0
$16$	2.71982	0.679956
$17$	−3.24914	−0.788032	−0.394016	−	0.919104i	$-0.628915\pi$
$17$	−3.24914	−0.788032	−0.394016	+	0.919104i	$0.628915\pi$
$18$	0	0
$19$	5.30777	1.21769	0.608843	−	0.793290i	$-0.291634\pi$
$19$	5.30777	1.21769	0.608843	+	0.793290i	$0.291634\pi$
$20$	5.77846	1.29210
$21$	0	0
$22$	0	0
$23$	8.86469	1.84842	0.924208	−	0.381890i	$-0.124727\pi$
$23$	8.86469	1.84842	0.924208	+	0.381890i	$0.124727\pi$
$24$	0	0
$25$	5.55691	1.11138
$26$	−1.16291	−0.228065
$27$	0	0
$28$	−1.77846	−0.336097
$29$	−2.47068	−0.458794	−0.229397	−	0.973333i	$-0.573675\pi$
$29$	−2.47068	−0.458794	−0.229397	+	0.973333i	$0.573675\pi$
$30$	0	0
$31$	−6.49828	−1.16713	−0.583563	−	0.812068i	$-0.698342\pi$
$31$	−6.49828	−1.16713	−0.583563	+	0.812068i	$0.698342\pi$
$32$	−4.83709	−0.855085
$33$	0	0
$34$	1.52932	0.262276
$35$	−3.24914	−0.549205
$36$	0	0
$37$	1.77846	0.292377	0.146188	−	0.989257i	$-0.453299\pi$
$37$	1.77846	0.292377	0.146188	+	0.989257i	$0.453299\pi$
$38$	−2.49828	−0.405275
$39$	0	0
$40$	−5.77846	−0.913654
$41$	−1.28018	−0.199930	−0.0999650	−	0.994991i	$-0.531873\pi$
$41$	−1.28018	−0.199930	−0.0999650	+	0.994991i	$0.531873\pi$
$42$	0	0
$43$	4.89572	0.746591	0.373295	−	0.927713i	$-0.378228\pi$
$43$	4.89572	0.746591	0.373295	+	0.927713i	$0.378228\pi$
$44$	0	0
$45$	0	0
$46$	−4.17246	−0.615196
$47$	−3.96896	−0.578933	−0.289466	−	0.957188i	$-0.593478\pi$
$47$	−3.96896	−0.578933	−0.289466	+	0.957188i	$0.593478\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−2.61555	−0.369894
$51$	0	0
$52$	−4.39400	−0.609339
$53$	9.77846	1.34317	0.671587	−	0.740926i	$-0.265613\pi$
$53$	9.77846	1.34317	0.671587	+	0.740926i	$0.265613\pi$
$54$	0	0
$55$	0	0
$56$	1.77846	0.237656
$57$	0	0
$58$	1.16291	0.152698
$59$	−2.41205	−0.314022	−0.157011	−	0.987597i	$-0.550186\pi$
$59$	−2.41205	−0.314022	−0.157011	+	0.987597i	$0.550186\pi$
$60$	0	0
$61$	−4.74742	−0.607845	−0.303923	−	0.952697i	$-0.598296\pi$
$61$	−4.74742	−0.607845	−0.303923	+	0.952697i	$0.598296\pi$
$62$	3.05863	0.388447
$63$	0	0
$64$	−3.16291	−0.395364
$65$	−8.02760	−0.995701
$66$	0	0
$67$	14.8337	1.81222	0.906110	−	0.423043i	$-0.139038\pi$
$67$	14.8337	1.81222	0.906110	+	0.423043i	$0.139038\pi$
$68$	5.77846	0.700741
$69$	0	0
$70$	1.52932	0.182788
$71$	3.19051	0.378644	0.189322	−	0.981915i	$-0.439371\pi$
$71$	3.19051	0.378644	0.189322	+	0.981915i	$0.439371\pi$
$72$	0	0
$73$	−8.98195	−1.05126	−0.525629	−	0.850714i	$-0.676170\pi$
$73$	−8.98195	−1.05126	−0.525629	+	0.850714i	$0.676170\pi$
$74$	−0.837090	−0.0973098
$75$	0	0
$76$	−9.43965	−1.08280
$77$	0	0
$78$	0	0
$79$	12.3810	1.39297	0.696486	−	0.717570i	$-0.254746\pi$
$79$	12.3810	1.39297	0.696486	+	0.717570i	$0.254746\pi$
$80$	−8.83709	−0.988017
$81$	0	0
$82$	0.602558	0.0665414
$83$	−13.0828	−1.43602	−0.718012	−	0.696031i	$-0.754948\pi$
$83$	−13.0828	−1.43602	−0.718012	+	0.696031i	$0.754948\pi$
$84$	0	0
$85$	10.5569	1.14506
$86$	−2.30434	−0.248483
$87$	0	0
$88$	0	0
$89$	−7.86469	−0.833655	−0.416828	−	0.908986i	$-0.636858\pi$
$89$	−7.86469	−0.833655	−0.416828	+	0.908986i	$0.636858\pi$
$90$	0	0
$91$	2.47068	0.258998
$92$	−15.7655	−1.64366
$93$	0	0
$94$	1.86813	0.192682
$95$	−17.2457	−1.76937
$96$	0	0
$97$	−14.8517	−1.50796	−0.753981	−	0.656897i	$-0.771869\pi$
$97$	−14.8517	−1.50796	−0.753981	+	0.656897i	$0.771869\pi$
$98$	−0.470683	−0.0475462
$99$	0	0
$100$	−9.88273	−0.988273
$101$	6.97240	0.693780	0.346890	−	0.937906i	$-0.387238\pi$
$101$	6.97240	0.693780	0.346890	+	0.937906i	$0.387238\pi$
$102$	0	0
$103$	11.3078	1.11419	0.557094	−	0.830449i	$-0.311916\pi$
$103$	11.3078	1.11419	0.557094	+	0.830449i	$0.311916\pi$
$104$	4.39400	0.430868
$105$	0	0
$106$	−4.60256	−0.447040
$107$	−15.9233	−1.53937	−0.769683	−	0.638427i	$-0.779586\pi$
$107$	−15.9233	−1.53937	−0.769683	+	0.638427i	$0.779586\pi$
$108$	0	0
$109$	−14.4948	−1.38835	−0.694177	−	0.719804i	$-0.744232\pi$
$109$	−14.4948	−1.38835	−0.694177	+	0.719804i	$0.744232\pi$
$110$	0	0
$111$	0	0
$112$	2.71982	0.256999
$113$	1.66119	0.156272	0.0781358	−	0.996943i	$-0.475103\pi$
$113$	1.66119	0.156272	0.0781358	+	0.996943i	$0.475103\pi$
$114$	0	0
$115$	−28.8026	−2.68586
$116$	4.39400	0.407973
$117$	0	0
$118$	1.13531	0.104514
$119$	−3.24914	−0.297848
$120$	0	0
$121$	0	0
$122$	2.23453	0.202305
$123$	0	0
$124$	11.5569	1.03784
$125$	−1.80949	−0.161846
$126$	0	0
$127$	−21.5405	−1.91141	−0.955705	−	0.294328i	$-0.904904\pi$
$127$	−21.5405	−1.91141	−0.955705	+	0.294328i	$0.904904\pi$
$128$	11.1629	0.986671
$129$	0	0
$130$	3.77846	0.331393
$131$	−12.4121	−1.08445	−0.542223	−	0.840235i	$-0.682417\pi$
$131$	−12.4121	−1.08445	−0.542223	+	0.840235i	$0.682417\pi$
$132$	0	0
$133$	5.30777	0.460242
$134$	−6.98195	−0.603149
$135$	0	0
$136$	−5.77846	−0.495499
$137$	−11.0992	−0.948270	−0.474135	−	0.880452i	$-0.657239\pi$
$137$	−11.0992	−0.948270	−0.474135	+	0.880452i	$0.657239\pi$
$138$	0	0
$139$	13.1905	1.11880	0.559402	−	0.828896i	$-0.311031\pi$
$139$	13.1905	1.11880	0.559402	+	0.828896i	$0.311031\pi$
$140$	5.77846	0.488369
$141$	0	0
$142$	−1.50172	−0.126021
$143$	0	0
$144$	0	0
$145$	8.02760	0.666656
$146$	4.22766	0.349883
$147$	0	0
$148$	−3.16291	−0.259990
$149$	12.6431	1.03577	0.517883	−	0.855451i	$-0.326720\pi$
$149$	12.6431	1.03577	0.517883	+	0.855451i	$0.326720\pi$
$150$	0	0
$151$	23.2147	1.88918	0.944591	−	0.328248i	$-0.106458\pi$
$151$	23.2147	1.88918	0.944591	+	0.328248i	$0.106458\pi$
$152$	9.43965	0.765657
$153$	0	0
$154$	0	0
$155$	21.1138	1.69590
$156$	0	0
$157$	−21.0518	−1.68011	−0.840057	−	0.542499i	$-0.817478\pi$
$157$	−21.0518	−1.68011	−0.840057	+	0.542499i	$0.817478\pi$
$158$	−5.82754	−0.463614
$159$	0	0
$160$	15.7164	1.24249
$161$	8.86469	0.698635
$162$	0	0
$163$	6.49828	0.508985	0.254492	−	0.967075i	$-0.418092\pi$
$163$	6.49828	0.508985	0.254492	+	0.967075i	$0.418092\pi$
$164$	2.27674	0.177783
$165$	0	0
$166$	6.15785	0.477942
$167$	16.9103	1.30856	0.654280	−	0.756252i	$-0.272972\pi$
$167$	16.9103	1.30856	0.654280	+	0.756252i	$0.272972\pi$
$168$	0	0
$169$	−6.89572	−0.530440
$170$	−4.96896	−0.381102
$171$	0	0
$172$	−8.70683	−0.663890
$173$	18.9103	1.43773	0.718863	−	0.695152i	$-0.244663\pi$
$173$	18.9103	1.43773	0.718863	+	0.695152i	$0.244663\pi$
$174$	0	0
$175$	5.55691	0.420063
$176$	0	0
$177$	0	0
$178$	3.70178	0.277460
$179$	−15.1690	−1.13379	−0.566893	−	0.823791i	$-0.691855\pi$
$179$	−15.1690	−1.13379	−0.566893	+	0.823791i	$0.691855\pi$
$180$	0	0
$181$	−15.5078	−1.15269	−0.576344	−	0.817207i	$-0.695521\pi$
$181$	−15.5078	−1.15269	−0.576344	+	0.817207i	$0.695521\pi$
$182$	−1.16291	−0.0862006
$183$	0	0
$184$	15.7655	1.16225
$185$	−5.77846	−0.424841
$186$	0	0
$187$	0	0
$188$	7.05863	0.514804
$189$	0	0
$190$	8.11727	0.588888
$191$	−0.483673	−0.0349974	−0.0174987	−	0.999847i	$-0.505570\pi$
$191$	−0.483673	−0.0349974	−0.0174987	+	0.999847i	$0.505570\pi$
$192$	0	0
$193$	18.2311	1.31230	0.656151	−	0.754629i	$-0.272183\pi$
$193$	18.2311	1.31230	0.656151	+	0.754629i	$0.272183\pi$
$194$	6.99045	0.501885
$195$	0	0
$196$	−1.77846	−0.127033
$197$	−2.10428	−0.149923	−0.0749617	−	0.997186i	$-0.523883\pi$
$197$	−2.10428	−0.149923	−0.0749617	+	0.997186i	$0.523883\pi$
$198$	0	0
$199$	−2.63016	−0.186447	−0.0932234	−	0.995645i	$-0.529717\pi$
$199$	−2.63016	−0.186447	−0.0932234	+	0.995645i	$0.529717\pi$
$200$	9.88273	0.698815
$201$	0	0
$202$	−3.28179	−0.230906
$203$	−2.47068	−0.173408
$204$	0	0
$205$	4.15947	0.290510
$206$	−5.32238	−0.370828
$207$	0	0
$208$	6.71982	0.465936
$209$	0	0
$210$	0	0
$211$	16.9509	1.16695	0.583475	−	0.812131i	$-0.301693\pi$
$211$	16.9509	1.16695	0.583475	+	0.812131i	$0.301693\pi$
$212$	−17.3906	−1.19439
$213$	0	0
$214$	7.49484	0.512337
$215$	−15.9069	−1.08484
$216$	0	0
$217$	−6.49828	−0.441132
$218$	6.82248	0.462077
$219$	0	0
$220$	0	0
$221$	−8.02760	−0.539995
$222$	0	0
$223$	−2.11727	−0.141783	−0.0708913	−	0.997484i	$-0.522584\pi$
$223$	−2.11727	−0.141783	−0.0708913	+	0.997484i	$0.522584\pi$
$224$	−4.83709	−0.323192
$225$	0	0
$226$	−0.781895	−0.0520109
$227$	28.4052	1.88532	0.942659	−	0.333758i	$-0.108317\pi$
$227$	28.4052	1.88532	0.942659	+	0.333758i	$0.108317\pi$
$228$	0	0
$229$	13.8337	0.914153	0.457077	−	0.889427i	$-0.348896\pi$
$229$	13.8337	0.914153	0.457077	+	0.889427i	$0.348896\pi$
$230$	13.5569	0.893916
$231$	0	0
$232$	−4.39400	−0.288481
$233$	−10.0276	−0.656930	−0.328465	−	0.944516i	$-0.606531\pi$
$233$	−10.0276	−0.656930	−0.328465	+	0.944516i	$0.606531\pi$
$234$	0	0
$235$	12.8957	0.841224
$236$	4.28973	0.279238
$237$	0	0
$238$	1.52932	0.0991309
$239$	5.55691	0.359447	0.179723	−	0.983717i	$-0.442480\pi$
$239$	5.55691	0.359447	0.179723	+	0.983717i	$0.442480\pi$
$240$	0	0
$241$	21.3078	1.37255	0.686277	−	0.727340i	$-0.259244\pi$
$241$	21.3078	1.37255	0.686277	+	0.727340i	$0.259244\pi$
$242$	0	0
$243$	0	0
$244$	8.44309	0.540513
$245$	−3.24914	−0.207580
$246$	0	0
$247$	13.1138	0.834413
$248$	−11.5569	−0.733865
$249$	0	0
$250$	0.851698	0.0538661
$251$	29.9931	1.89315	0.946575	−	0.322485i	$-0.104518\pi$
$251$	29.9931	1.89315	0.946575	+	0.322485i	$0.104518\pi$
$252$	0	0
$253$	0	0
$254$	10.1387	0.636162
$255$	0	0
$256$	1.07162	0.0669764
$257$	14.4543	0.901632	0.450816	−	0.892617i	$-0.351133\pi$
$257$	14.4543	0.901632	0.450816	+	0.892617i	$0.351133\pi$
$258$	0	0
$259$	1.77846	0.110508
$260$	14.2767	0.885406
$261$	0	0
$262$	5.84215	0.360929
$263$	−13.1284	−0.809534	−0.404767	−	0.914420i	$-0.632647\pi$
$263$	−13.1284	−0.809534	−0.404767	+	0.914420i	$0.632647\pi$
$264$	0	0
$265$	−31.7716	−1.95171
$266$	−2.49828	−0.153179
$267$	0	0
$268$	−26.3810	−1.61148
$269$	−8.95436	−0.545957	−0.272978	−	0.962020i	$-0.588009\pi$
$269$	−8.95436	−0.545957	−0.272978	+	0.962020i	$0.588009\pi$
$270$	0	0
$271$	9.38445	0.570065	0.285032	−	0.958518i	$-0.407996\pi$
$271$	9.38445	0.570065	0.285032	+	0.958518i	$0.407996\pi$
$272$	−8.83709	−0.535827
$273$	0	0
$274$	5.22422	0.315607
$275$	0	0
$276$	0	0
$277$	−20.7846	−1.24882	−0.624412	−	0.781095i	$-0.714661\pi$
$277$	−20.7846	−1.24882	−0.624412	+	0.781095i	$0.714661\pi$
$278$	−6.20855	−0.372364
$279$	0	0
$280$	−5.77846	−0.345329
$281$	11.0878	0.661446	0.330723	−	0.943728i	$-0.392707\pi$
$281$	11.0878	0.661446	0.330723	+	0.943728i	$0.392707\pi$
$282$	0	0
$283$	−2.17246	−0.129139	−0.0645697	−	0.997913i	$-0.520567\pi$
$283$	−2.17246	−0.129139	−0.0645697	+	0.997913i	$0.520567\pi$
$284$	−5.67418	−0.336701
$285$	0	0
$286$	0	0
$287$	−1.28018	−0.0755664
$288$	0	0
$289$	−6.44309	−0.379005
$290$	−3.77846	−0.221879
$291$	0	0
$292$	15.9740	0.934809
$293$	−15.8647	−0.926825	−0.463412	−	0.886143i	$-0.653375\pi$
$293$	−15.8647	−0.926825	−0.463412	+	0.886143i	$0.653375\pi$
$294$	0	0
$295$	7.83709	0.456293
$296$	3.16291	0.183840
$297$	0	0
$298$	−5.95092	−0.344727
$299$	21.9018	1.26662
$300$	0	0
$301$	4.89572	0.282185
$302$	−10.9268	−0.628764
$303$	0	0
$304$	14.4362	0.827973
$305$	15.4250	0.883235
$306$	0	0
$307$	−9.59750	−0.547758	−0.273879	−	0.961764i	$-0.588307\pi$
$307$	−9.59750	−0.547758	−0.273879	+	0.961764i	$0.588307\pi$
$308$	0	0
$309$	0	0
$310$	−9.93793	−0.564436
$311$	31.0828	1.76254	0.881272	−	0.472610i	$-0.156688\pi$
$311$	31.0828	1.76254	0.881272	+	0.472610i	$0.156688\pi$
$312$	0	0
$313$	17.8888	1.01114	0.505569	−	0.862786i	$-0.331283\pi$
$313$	17.8888	1.01114	0.505569	+	0.862786i	$0.331283\pi$
$314$	9.90871	0.559181
$315$	0	0
$316$	−22.0191	−1.23867
$317$	14.9268	0.838370	0.419185	−	0.907901i	$-0.362316\pi$
$317$	14.9268	0.838370	0.419185	+	0.907901i	$0.362316\pi$
$318$	0	0
$319$	0	0
$320$	10.2767	0.574487
$321$	0	0
$322$	−4.17246	−0.232522
$323$	−17.2457	−0.959577
$324$	0	0
$325$	13.7294	0.761569
$326$	−3.05863	−0.169402
$327$	0	0
$328$	−2.27674	−0.125712
$329$	−3.96896	−0.218816
$330$	0	0
$331$	−6.86974	−0.377595	−0.188798	−	0.982016i	$-0.560459\pi$
$331$	−6.86974	−0.377595	−0.188798	+	0.982016i	$0.560459\pi$
$332$	23.2672	1.27695
$333$	0	0
$334$	−7.95941	−0.435520
$335$	−48.1966	−2.63326
$336$	0	0
$337$	14.5113	0.790479	0.395240	−	0.918578i	$-0.370662\pi$
$337$	14.5113	0.790479	0.395240	+	0.918578i	$0.370662\pi$
$338$	3.24570	0.176543
$339$	0	0
$340$	−18.7750	−1.01822
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	8.70683	0.469441
$345$	0	0
$346$	−8.90078	−0.478509
$347$	31.4734	1.68958	0.844789	−	0.535099i	$-0.179726\pi$
$347$	31.4734	1.68958	0.844789	+	0.535099i	$0.179726\pi$
$348$	0	0
$349$	−22.6578	−1.21284	−0.606421	−	0.795144i	$-0.707395\pi$
$349$	−22.6578	−1.21284	−0.606421	+	0.795144i	$0.707395\pi$
$350$	−2.61555	−0.139807
$351$	0	0
$352$	0	0
$353$	−15.2802	−0.813282	−0.406641	−	0.913588i	$-0.633300\pi$
$353$	−15.2802	−0.813282	−0.406641	+	0.913588i	$0.633300\pi$
$354$	0	0
$355$	−10.3664	−0.550192
$356$	13.9870	0.741310
$357$	0	0
$358$	7.13981	0.377351
$359$	16.4768	0.869612	0.434806	−	0.900524i	$-0.356817\pi$
$359$	16.4768	0.869612	0.434806	+	0.900524i	$0.356817\pi$
$360$	0	0
$361$	9.17246	0.482761
$362$	7.29928	0.383642
$363$	0	0
$364$	−4.39400	−0.230308
$365$	29.1836	1.52754
$366$	0	0
$367$	15.9525	0.832716	0.416358	−	0.909201i	$-0.363306\pi$
$367$	15.9525	0.832716	0.416358	+	0.909201i	$0.363306\pi$
$368$	24.1104	1.25684
$369$	0	0
$370$	2.71982	0.141397
$371$	9.77846	0.507672
$372$	0	0
$373$	−0.723262	−0.0374491	−0.0187245	−	0.999825i	$-0.505961\pi$
$373$	−0.723262	−0.0374491	−0.0187245	+	0.999825i	$0.505961\pi$
$374$	0	0
$375$	0	0
$376$	−7.05863	−0.364021
$377$	−6.10428	−0.314386
$378$	0	0
$379$	−2.42666	−0.124649	−0.0623245	−	0.998056i	$-0.519851\pi$
$379$	−2.42666	−0.124649	−0.0623245	+	0.998056i	$0.519851\pi$
$380$	30.6707	1.57338
$381$	0	0
$382$	0.227657	0.0116479
$383$	21.2553	1.08609	0.543046	−	0.839703i	$-0.317271\pi$
$383$	21.2553	1.08609	0.543046	+	0.839703i	$0.317271\pi$
$384$	0	0
$385$	0	0
$386$	−8.58107	−0.436765
$387$	0	0
$388$	26.4131	1.34092
$389$	4.78189	0.242452	0.121226	−	0.992625i	$-0.461317\pi$
$389$	4.78189	0.242452	0.121226	+	0.992625i	$0.461317\pi$
$390$	0	0
$391$	−28.8026	−1.45661
$392$	1.77846	0.0898256
$393$	0	0
$394$	0.990448	0.0498981
$395$	−40.2277	−2.02407
$396$	0	0
$397$	8.66463	0.434865	0.217433	−	0.976075i	$-0.430232\pi$
$397$	8.66463	0.434865	0.217433	+	0.976075i	$0.430232\pi$
$398$	1.23797	0.0620539
$399$	0	0
$400$	15.1138	0.755691
$401$	−15.1629	−0.757200	−0.378600	−	0.925560i	$-0.623594\pi$
$401$	−15.1629	−0.757200	−0.378600	+	0.925560i	$0.623594\pi$
$402$	0	0
$403$	−16.0552	−0.799766
$404$	−12.4001	−0.616929
$405$	0	0
$406$	1.16291	0.0577142
$407$	0	0
$408$	0	0
$409$	−17.0242	−0.841791	−0.420895	−	0.907109i	$-0.638284\pi$
$409$	−17.0242	−0.841791	−0.420895	+	0.907109i	$0.638284\pi$
$410$	−1.95779	−0.0966886
$411$	0	0
$412$	−20.1104	−0.990768
$413$	−2.41205	−0.118689
$414$	0	0
$415$	42.5078	2.08663
$416$	−11.9509	−0.585942
$417$	0	0
$418$	0	0
$419$	−7.76041	−0.379121	−0.189560	−	0.981869i	$-0.560706\pi$
$419$	−7.76041	−0.379121	−0.189560	+	0.981869i	$0.560706\pi$
$420$	0	0
$421$	−15.9966	−0.779625	−0.389812	−	0.920894i	$-0.627460\pi$
$421$	−15.9966	−0.779625	−0.389812	+	0.920894i	$0.627460\pi$
$422$	−7.97852	−0.388388
$423$	0	0
$424$	17.3906	0.844561
$425$	−18.0552	−0.875806
$426$	0	0
$427$	−4.74742	−0.229744
$428$	28.3189	1.36885
$429$	0	0
$430$	7.48711	0.361061
$431$	26.1725	1.26068	0.630342	−	0.776318i	$-0.282915\pi$
$431$	26.1725	1.26068	0.630342	+	0.776318i	$0.282915\pi$
$432$	0	0
$433$	−20.3173	−0.976388	−0.488194	−	0.872735i	$-0.662344\pi$
$433$	−20.3173	−0.976388	−0.488194	+	0.872735i	$0.662344\pi$
$434$	3.05863	0.146819
$435$	0	0
$436$	25.7785	1.23456
$437$	47.0518	2.25079
$438$	0	0
$439$	18.7880	0.896703	0.448351	−	0.893857i	$-0.352011\pi$
$439$	18.7880	0.896703	0.448351	+	0.893857i	$0.352011\pi$
$440$	0	0
$441$	0	0
$442$	3.77846	0.179723
$443$	34.9820	1.66204	0.831021	−	0.556240i	$-0.187757\pi$
$443$	34.9820	1.66204	0.831021	+	0.556240i	$0.187757\pi$
$444$	0	0
$445$	25.5535	1.21135
$446$	0.996562	0.0471886
$447$	0	0
$448$	−3.16291	−0.149433
$449$	8.77502	0.414119	0.207059	−	0.978328i	$-0.433611\pi$
$449$	8.77502	0.414119	0.207059	+	0.978328i	$0.433611\pi$
$450$	0	0
$451$	0	0
$452$	−2.95436	−0.138961
$453$	0	0
$454$	−13.3698	−0.627478
$455$	−8.02760	−0.376340
$456$	0	0
$457$	−31.9053	−1.49247	−0.746233	−	0.665685i	$-0.768139\pi$
$457$	−31.9053	−1.49247	−0.746233	+	0.665685i	$0.768139\pi$
$458$	−6.51127	−0.304252
$459$	0	0
$460$	51.2242	2.38834
$461$	19.0958	0.889379	0.444690	−	0.895685i	$-0.353314\pi$
$461$	19.0958	0.889379	0.444690	+	0.895685i	$0.353314\pi$
$462$	0	0
$463$	21.1430	0.982601	0.491300	−	0.870990i	$-0.336522\pi$
$463$	21.1430	0.982601	0.491300	+	0.870990i	$0.336522\pi$
$464$	−6.71982	−0.311960
$465$	0	0
$466$	4.71982	0.218642
$467$	7.05176	0.326316	0.163158	−	0.986600i	$-0.447832\pi$
$467$	7.05176	0.326316	0.163158	+	0.986600i	$0.447832\pi$
$468$	0	0
$469$	14.8337	0.684954
$470$	−6.06980	−0.279979
$471$	0	0
$472$	−4.28973	−0.197451
$473$	0	0
$474$	0	0
$475$	29.4948	1.35332
$476$	5.77846	0.264855
$477$	0	0
$478$	−2.61555	−0.119632
$479$	20.7310	0.947223	0.473612	−	0.880734i	$-0.342950\pi$
$479$	20.7310	0.947223	0.473612	+	0.880734i	$0.342950\pi$
$480$	0	0
$481$	4.39400	0.200349
$482$	−10.0292	−0.456818
$483$	0	0
$484$	0	0
$485$	48.2553	2.19116
$486$	0	0
$487$	−26.4526	−1.19868	−0.599342	−	0.800493i	$-0.704571\pi$
$487$	−26.4526	−1.19868	−0.599342	+	0.800493i	$0.704571\pi$
$488$	−8.44309	−0.382201
$489$	0	0
$490$	1.52932	0.0690875
$491$	−0.131874	−0.00595140	−0.00297570	−	0.999996i	$-0.500947\pi$
$491$	−0.131874	−0.00595140	−0.00297570	+	0.999996i	$0.500947\pi$
$492$	0	0
$493$	8.02760	0.361545
$494$	−6.17246	−0.277712
$495$	0	0
$496$	−17.6742	−0.793594
$497$	3.19051	0.143114
$498$	0	0
$499$	−18.2441	−0.816717	−0.408359	−	0.912822i	$-0.633899\pi$
$499$	−18.2441	−0.816717	−0.408359	+	0.912822i	$0.633899\pi$
$500$	3.21811	0.143918
$501$	0	0
$502$	−14.1173	−0.630084
$503$	−28.5224	−1.27175	−0.635876	−	0.771791i	$-0.719361\pi$
$503$	−28.5224	−1.27175	−0.635876	+	0.771791i	$0.719361\pi$
$504$	0	0
$505$	−22.6543	−1.00810
$506$	0	0
$507$	0	0
$508$	38.3088	1.69968
$509$	33.3725	1.47921	0.739605	−	0.673041i	$-0.235012\pi$
$509$	33.3725	1.47921	0.739605	+	0.673041i	$0.235012\pi$
$510$	0	0
$511$	−8.98195	−0.397338
$512$	−22.8302	−1.00896
$513$	0	0
$514$	−6.80338	−0.300084
$515$	−36.7405	−1.61898
$516$	0	0
$517$	0	0
$518$	−0.837090	−0.0367796
$519$	0	0
$520$	−14.2767	−0.626076
$521$	−4.63971	−0.203269	−0.101635	−	0.994822i	$-0.532407\pi$
$521$	−4.63971	−0.203269	−0.101635	+	0.994822i	$0.532407\pi$
$522$	0	0
$523$	21.6267	0.945670	0.472835	−	0.881151i	$-0.343231\pi$
$523$	21.6267	0.945670	0.472835	+	0.881151i	$0.343231\pi$
$524$	22.0743	0.964320
$525$	0	0
$526$	6.17934	0.269432
$527$	21.1138	0.919733
$528$	0	0
$529$	55.5827	2.41664
$530$	14.9544	0.649576
$531$	0	0
$532$	−9.43965	−0.409261
$533$	−3.16291	−0.137001
$534$	0	0
$535$	51.7371	2.23679
$536$	26.3810	1.13949
$537$	0	0
$538$	4.21467	0.181707
$539$	0	0
$540$	0	0
$541$	15.1043	0.649384	0.324692	−	0.945820i	$-0.394739\pi$
$541$	15.1043	0.649384	0.324692	+	0.945820i	$0.394739\pi$
$542$	−4.41711	−0.189731
$543$	0	0
$544$	15.7164	0.673834
$545$	47.0958	2.01736
$546$	0	0
$547$	5.34836	0.228679	0.114340	−	0.993442i	$-0.463525\pi$
$547$	5.34836	0.228679	0.114340	+	0.993442i	$0.463525\pi$
$548$	19.7395	0.843229
$549$	0	0
$550$	0	0
$551$	−13.1138	−0.558668
$552$	0	0
$553$	12.3810	0.526494
$554$	9.78295	0.415638
$555$	0	0
$556$	−23.4588	−0.994873
$557$	−1.76547	−0.0748053	−0.0374026	−	0.999300i	$-0.511908\pi$
$557$	−1.76547	−0.0748053	−0.0374026	+	0.999300i	$0.511908\pi$
$558$	0	0
$559$	12.0958	0.511597
$560$	−8.83709	−0.373435
$561$	0	0
$562$	−5.21887	−0.220145
$563$	7.79650	0.328583	0.164292	−	0.986412i	$-0.447466\pi$
$563$	7.79650	0.328583	0.164292	+	0.986412i	$0.447466\pi$
$564$	0	0
$565$	−5.39744	−0.227072
$566$	1.02254	0.0429806
$567$	0	0
$568$	5.67418	0.238083
$569$	42.2423	1.77089	0.885444	−	0.464746i	$-0.153854\pi$
$569$	42.2423	1.77089	0.885444	+	0.464746i	$0.153854\pi$
$570$	0	0
$571$	−1.64820	−0.0689751	−0.0344875	−	0.999405i	$-0.510980\pi$
$571$	−1.64820	−0.0689751	−0.0344875	+	0.999405i	$0.510980\pi$
$572$	0	0
$573$	0	0
$574$	0.602558	0.0251503
$575$	49.2603	2.05430
$576$	0	0
$577$	0.243026	0.0101173	0.00505866	−	0.999987i	$-0.498390\pi$
$577$	0.243026	0.0101173	0.00505866	+	0.999987i	$0.498390\pi$
$578$	3.03265	0.126142
$579$	0	0
$580$	−14.2767	−0.592809
$581$	−13.0828	−0.542766
$582$	0	0
$583$	0	0
$584$	−15.9740	−0.661010
$585$	0	0
$586$	7.46725	0.308469
$587$	−5.85170	−0.241525	−0.120763	−	0.992681i	$-0.538534\pi$
$587$	−5.85170	−0.241525	−0.120763	+	0.992681i	$0.538534\pi$
$588$	0	0
$589$	−34.4914	−1.42119
$590$	−3.68879	−0.151865
$591$	0	0
$592$	4.83709	0.198803
$593$	21.1250	0.867500	0.433750	−	0.901033i	$-0.357190\pi$
$593$	21.1250	0.867500	0.433750	+	0.901033i	$0.357190\pi$
$594$	0	0
$595$	10.5569	0.432791
$596$	−22.4853	−0.921033
$597$	0	0
$598$	−10.3088	−0.421559
$599$	−9.26719	−0.378647	−0.189323	−	0.981915i	$-0.560629\pi$
$599$	−9.26719	−0.378647	−0.189323	+	0.981915i	$0.560629\pi$
$600$	0	0
$601$	−38.5811	−1.57375	−0.786877	−	0.617109i	$-0.788304\pi$
$601$	−38.5811	−1.57375	−0.786877	+	0.617109i	$0.788304\pi$
$602$	−2.30434	−0.0939177
$603$	0	0
$604$	−41.2863	−1.67992
$605$	0	0
$606$	0	0
$607$	−19.5423	−0.793198	−0.396599	−	0.917992i	$-0.629810\pi$
$607$	−19.5423	−0.793198	−0.396599	+	0.917992i	$0.629810\pi$
$608$	−25.6742	−1.04123
$609$	0	0
$610$	−7.26031	−0.293961
$611$	−9.80605	−0.396711
$612$	0	0
$613$	−25.6087	−1.03432	−0.517162	−	0.855887i	$-0.673012\pi$
$613$	−25.6087	−1.03432	−0.517162	+	0.855887i	$0.673012\pi$
$614$	4.51738	0.182307
$615$	0	0
$616$	0	0
$617$	0.899161	0.0361989	0.0180994	−	0.999836i	$-0.494238\pi$
$617$	0.899161	0.0361989	0.0180994	+	0.999836i	$0.494238\pi$
$618$	0	0
$619$	11.2311	0.451416	0.225708	−	0.974195i	$-0.427531\pi$
$619$	11.2311	0.451416	0.225708	+	0.974195i	$0.427531\pi$
$620$	−37.5500	−1.50805
$621$	0	0
$622$	−14.6302	−0.586616
$623$	−7.86469	−0.315092
$624$	0	0
$625$	−21.9053	−0.876211
$626$	−8.41998	−0.336530
$627$	0	0
$628$	37.4396	1.49400
$629$	−5.77846	−0.230402
$630$	0	0
$631$	−9.95436	−0.396277	−0.198138	−	0.980174i	$-0.563490\pi$
$631$	−9.95436	−0.396277	−0.198138	+	0.980174i	$0.563490\pi$
$632$	22.0191	0.875873
$633$	0	0
$634$	−7.02578	−0.279029
$635$	69.9881	2.77739
$636$	0	0
$637$	2.47068	0.0978920
$638$	0	0
$639$	0	0
$640$	−36.2699	−1.43369
$641$	−42.0828	−1.66217	−0.831085	−	0.556145i	$-0.812280\pi$
$641$	−42.0828	−1.66217	−0.831085	+	0.556145i	$0.812280\pi$
$642$	0	0
$643$	2.87930	0.113548	0.0567742	−	0.998387i	$-0.481918\pi$
$643$	2.87930	0.113548	0.0567742	+	0.998387i	$0.481918\pi$
$644$	−15.7655	−0.621246
$645$	0	0
$646$	8.11727	0.319370
$647$	−4.55530	−0.179087	−0.0895436	−	0.995983i	$-0.528541\pi$
$647$	−4.55530	−0.179087	−0.0895436	+	0.995983i	$0.528541\pi$
$648$	0	0
$649$	0	0
$650$	−6.46219	−0.253468
$651$	0	0
$652$	−11.5569	−0.452604
$653$	37.7440	1.47704	0.738518	−	0.674234i	$-0.235526\pi$
$653$	37.7440	1.47704	0.738518	+	0.674234i	$0.235526\pi$
$654$	0	0
$655$	40.3285	1.57576
$656$	−3.48185	−0.135944
$657$	0	0
$658$	1.86813	0.0728271
$659$	22.2423	0.866436	0.433218	−	0.901289i	$-0.357378\pi$
$659$	22.2423	0.866436	0.433218	+	0.901289i	$0.357378\pi$
$660$	0	0
$661$	14.3027	0.556311	0.278156	−	0.960536i	$-0.410277\pi$
$661$	14.3027	0.556311	0.278156	+	0.960536i	$0.410277\pi$
$662$	3.23347	0.125673
$663$	0	0
$664$	−23.2672	−0.902942
$665$	−17.2457	−0.668760
$666$	0	0
$667$	−21.9018	−0.848043
$668$	−30.0743	−1.16361
$669$	0	0
$670$	22.6854	0.876412
$671$	0	0
$672$	0	0
$673$	4.89572	0.188716	0.0943581	−	0.995538i	$-0.469920\pi$
$673$	4.89572	0.188716	0.0943581	+	0.995538i	$0.469920\pi$
$674$	−6.83021	−0.263090
$675$	0	0
$676$	12.2637	0.471683
$677$	−23.4147	−0.899901	−0.449951	−	0.893053i	$-0.648558\pi$
$677$	−23.4147	−0.899901	−0.449951	+	0.893053i	$0.648558\pi$
$678$	0	0
$679$	−14.8517	−0.569956
$680$	18.7750	0.719989
$681$	0	0
$682$	0	0
$683$	24.2829	0.929158	0.464579	−	0.885532i	$-0.346206\pi$
$683$	24.2829	0.929158	0.464579	+	0.885532i	$0.346206\pi$
$684$	0	0
$685$	36.0629	1.37789
$686$	−0.470683	−0.0179708
$687$	0	0
$688$	13.3155	0.507649
$689$	24.1595	0.920403
$690$	0	0
$691$	−9.17590	−0.349068	−0.174534	−	0.984651i	$-0.555842\pi$
$691$	−9.17590	−0.349068	−0.174534	+	0.984651i	$0.555842\pi$
$692$	−33.6312	−1.27847
$693$	0	0
$694$	−14.8140	−0.562331
$695$	−42.8578	−1.62569
$696$	0	0
$697$	4.15947	0.157551
$698$	10.6646	0.403662
$699$	0	0
$700$	−9.88273	−0.373532
$701$	−20.9621	−0.791727	−0.395864	−	0.918309i	$-0.629555\pi$
$701$	−20.9621	−0.791727	−0.395864	+	0.918309i	$0.629555\pi$
$702$	0	0
$703$	9.43965	0.356023
$704$	0	0
$705$	0	0
$706$	7.19213	0.270679
$707$	6.97240	0.262224
$708$	0	0
$709$	41.0974	1.54345	0.771723	−	0.635959i	$-0.219395\pi$
$709$	41.0974	1.54345	0.771723	+	0.635959i	$0.219395\pi$
$710$	4.87930	0.183117
$711$	0	0
$712$	−13.9870	−0.524185
$713$	−57.6052	−2.15733
$714$	0	0
$715$	0	0
$716$	26.9775	1.00819
$717$	0	0
$718$	−7.75536	−0.289427
$719$	−10.0621	−0.375252	−0.187626	−	0.982241i	$-0.560079\pi$
$719$	−10.0621	−0.375252	−0.187626	+	0.982241i	$0.560079\pi$
$720$	0	0
$721$	11.3078	0.421123
$722$	−4.31733	−0.160674
$723$	0	0
$724$	27.5800	1.02500
$725$	−13.7294	−0.509896
$726$	0	0
$727$	1.33699	0.0495862	0.0247931	−	0.999693i	$-0.492107\pi$
$727$	1.33699	0.0495862	0.0247931	+	0.999693i	$0.492107\pi$
$728$	4.39400	0.162853
$729$	0	0
$730$	−13.7363	−0.508401
$731$	−15.9069	−0.588338
$732$	0	0
$733$	9.73787	0.359676	0.179838	−	0.983696i	$-0.442443\pi$
$733$	9.73787	0.359676	0.179838	+	0.983696i	$0.442443\pi$
$734$	−7.50859	−0.277147
$735$	0	0
$736$	−42.8793	−1.58055
$737$	0	0
$738$	0	0
$739$	13.1430	0.483475	0.241737	−	0.970342i	$-0.422283\pi$
$739$	13.1430	0.483475	0.241737	+	0.970342i	$0.422283\pi$
$740$	10.2767	0.377780
$741$	0	0
$742$	−4.60256	−0.168965
$743$	3.31465	0.121603	0.0608013	−	0.998150i	$-0.480634\pi$
$743$	3.31465	0.121603	0.0608013	+	0.998150i	$0.480634\pi$
$744$	0	0
$745$	−41.0794	−1.50503
$746$	0.340427	0.0124639
$747$	0	0
$748$	0	0
$749$	−15.9233	−0.581825
$750$	0	0
$751$	−23.8854	−0.871591	−0.435795	−	0.900046i	$-0.643533\pi$
$751$	−23.8854	−0.871591	−0.435795	+	0.900046i	$0.643533\pi$
$752$	−10.7949	−0.393649
$753$	0	0
$754$	2.87318	0.104635
$755$	−75.4277	−2.74510
$756$	0	0
$757$	−49.5535	−1.80105	−0.900526	−	0.434802i	$-0.856818\pi$
$757$	−49.5535	−1.80105	−0.900526	+	0.434802i	$0.856818\pi$
$758$	1.14219	0.0414861
$759$	0	0
$760$	−30.6707	−1.11254
$761$	−15.3404	−0.556090	−0.278045	−	0.960568i	$-0.589686\pi$
$761$	−15.3404	−0.556090	−0.278045	+	0.960568i	$0.589686\pi$
$762$	0	0
$763$	−14.4948	−0.524749
$764$	0.860192	0.0311207
$765$	0	0
$766$	−10.0045	−0.361477
$767$	−5.95941	−0.215182
$768$	0	0
$769$	32.0613	1.15616	0.578080	−	0.815980i	$-0.303802\pi$
$769$	32.0613	1.15616	0.578080	+	0.815980i	$0.303802\pi$
$770$	0	0
$771$	0	0
$772$	−32.4232	−1.16694
$773$	−44.7209	−1.60850	−0.804249	−	0.594292i	$-0.797432\pi$
$773$	−44.7209	−1.60850	−0.804249	+	0.594292i	$0.797432\pi$
$774$	0	0
$775$	−36.1104	−1.29712
$776$	−26.4131	−0.948175
$777$	0	0
$778$	−2.25076	−0.0806936
$779$	−6.79488	−0.243452
$780$	0	0
$781$	0	0
$782$	13.5569	0.484794
$783$	0	0
$784$	2.71982	0.0971366
$785$	68.4001	2.44130
$786$	0	0
$787$	48.7000	1.73597	0.867983	−	0.496594i	$-0.165416\pi$
$787$	48.7000	1.73597	0.867983	+	0.496594i	$0.165416\pi$
$788$	3.74237	0.133316
$789$	0	0
$790$	18.9345	0.673659
$791$	1.66119	0.0590651
$792$	0	0
$793$	−11.7294	−0.416522
$794$	−4.07830	−0.144733
$795$	0	0
$796$	4.67762	0.165794
$797$	−46.5224	−1.64791	−0.823955	−	0.566656i	$-0.808237\pi$
$797$	−46.5224	−1.64791	−0.823955	+	0.566656i	$0.808237\pi$
$798$	0	0
$799$	12.8957	0.456218
$800$	−26.8793	−0.950327
$801$	0	0
$802$	7.13693	0.252014
$803$	0	0
$804$	0	0
$805$	−28.8026	−1.01516
$806$	7.55691	0.266181
$807$	0	0
$808$	12.4001	0.436235
$809$	3.65957	0.128664	0.0643319	−	0.997929i	$-0.479508\pi$
$809$	3.65957	0.128664	0.0643319	+	0.997929i	$0.479508\pi$
$810$	0	0
$811$	−2.95597	−0.103798	−0.0518992	−	0.998652i	$-0.516527\pi$
$811$	−2.95597	−0.103798	−0.0518992	+	0.998652i	$0.516527\pi$
$812$	4.39400	0.154199
$813$	0	0
$814$	0	0
$815$	−21.1138	−0.739585
$816$	0	0
$817$	25.9854	0.909114
$818$	8.01299	0.280168
$819$	0	0
$820$	−7.39744	−0.258330
$821$	−3.23109	−0.112766	−0.0563830	−	0.998409i	$-0.517957\pi$
$821$	−3.23109	−0.112766	−0.0563830	+	0.998409i	$0.517957\pi$
$822$	0	0
$823$	21.9448	0.764948	0.382474	−	0.923966i	$-0.375072\pi$
$823$	21.9448	0.764948	0.382474	+	0.923966i	$0.375072\pi$
$824$	20.1104	0.700579
$825$	0	0
$826$	1.13531	0.0395026
$827$	32.4508	1.12843	0.564213	−	0.825629i	$-0.309180\pi$
$827$	32.4508	1.12843	0.564213	+	0.825629i	$0.309180\pi$
$828$	0	0
$829$	2.94298	0.102214	0.0511070	−	0.998693i	$-0.483725\pi$
$829$	2.94298	0.102214	0.0511070	+	0.998693i	$0.483725\pi$
$830$	−20.0077	−0.694479
$831$	0	0
$832$	−7.81455	−0.270921
$833$	−3.24914	−0.112576
$834$	0	0
$835$	−54.9440	−1.90142
$836$	0	0
$837$	0	0
$838$	3.65270	0.126180
$839$	−17.4017	−0.600775	−0.300387	−	0.953817i	$-0.597116\pi$
$839$	−17.4017	−0.600775	−0.300387	+	0.953817i	$0.597116\pi$
$840$	0	0
$841$	−22.8957	−0.789508
$842$	7.52932	0.259477
$843$	0	0
$844$	−30.1465	−1.03768
$845$	22.4052	0.770761
$846$	0	0
$847$	0	0
$848$	26.5957	0.913299
$849$	0	0
$850$	8.49828	0.291489
$851$	15.7655	0.540433
$852$	0	0
$853$	−41.9440	−1.43614	−0.718068	−	0.695973i	$-0.754973\pi$
$853$	−41.9440	−1.43614	−0.718068	+	0.695973i	$0.754973\pi$
$854$	2.23453	0.0764641
$855$	0	0
$856$	−28.3189	−0.967922
$857$	9.82248	0.335530	0.167765	−	0.985827i	$-0.446345\pi$
$857$	9.82248	0.335530	0.167765	+	0.985827i	$0.446345\pi$
$858$	0	0
$859$	12.8533	0.438549	0.219275	−	0.975663i	$-0.429631\pi$
$859$	12.8533	0.438549	0.219275	+	0.975663i	$0.429631\pi$
$860$	28.2897	0.964672
$861$	0	0
$862$	−12.3189	−0.419585
$863$	−30.5795	−1.04094	−0.520468	−	0.853881i	$-0.674243\pi$
$863$	−30.5795	−1.04094	−0.520468	+	0.853881i	$0.674243\pi$
$864$	0	0
$865$	−61.4423	−2.08910
$866$	9.56303	0.324965
$867$	0	0
$868$	11.5569	0.392267
$869$	0	0
$870$	0	0
$871$	36.6493	1.24181
$872$	−25.7785	−0.872969
$873$	0	0
$874$	−22.1465	−0.749116
$875$	−1.80949	−0.0611720
$876$	0	0
$877$	6.45608	0.218006	0.109003	−	0.994041i	$-0.465234\pi$
$877$	6.45608	0.218006	0.109003	+	0.994041i	$0.465234\pi$
$878$	−8.84320	−0.298444
$879$	0	0
$880$	0	0
$881$	13.4819	0.454215	0.227108	−	0.973870i	$-0.427073\pi$
$881$	13.4819	0.454215	0.227108	+	0.973870i	$0.427073\pi$
$882$	0	0
$883$	58.9372	1.98339	0.991697	−	0.128598i	$-0.0410477\pi$
$883$	58.9372	1.98339	0.991697	+	0.128598i	$0.0410477\pi$
$884$	14.2767	0.480179
$885$	0	0
$886$	−16.4654	−0.553167
$887$	−1.79650	−0.0603207	−0.0301603	−	0.999545i	$-0.509602\pi$
$887$	−1.79650	−0.0603207	−0.0301603	+	0.999545i	$0.509602\pi$
$888$	0	0
$889$	−21.5405	−0.722445
$890$	−12.0276	−0.403166
$891$	0	0
$892$	3.76547	0.126077
$893$	−21.0664	−0.704959
$894$	0	0
$895$	49.2863	1.64746
$896$	11.1629	0.372927
$897$	0	0
$898$	−4.13026	−0.137828
$899$	16.0552	0.535471
$900$	0	0
$901$	−31.7716	−1.05846
$902$	0	0
$903$	0	0
$904$	2.95436	0.0982604
$905$	50.3871	1.67492
$906$	0	0
$907$	−4.87930	−0.162014	−0.0810072	−	0.996714i	$-0.525814\pi$
$907$	−4.87930	−0.162014	−0.0810072	+	0.996714i	$0.525814\pi$
$908$	−50.5174	−1.67648
$909$	0	0
$910$	3.77846	0.125255
$911$	18.1364	0.600885	0.300442	−	0.953800i	$-0.402866\pi$
$911$	18.1364	0.600885	0.300442	+	0.953800i	$0.402866\pi$
$912$	0	0
$913$	0	0
$914$	15.0173	0.496728
$915$	0	0
$916$	−24.6026	−0.812891
$917$	−12.4121	−0.409882
$918$	0	0
$919$	−1.73625	−0.0572737	−0.0286368	−	0.999590i	$-0.509117\pi$
$919$	−1.73625	−0.0572737	−0.0286368	+	0.999590i	$0.509117\pi$
$920$	−51.2242	−1.68881
$921$	0	0
$922$	−8.98807	−0.296006
$923$	7.88273	0.259463
$924$	0	0
$925$	9.88273	0.324942
$926$	−9.95168	−0.327033
$927$	0	0
$928$	11.9509	0.392308
$929$	46.5354	1.52678	0.763389	−	0.645939i	$-0.223534\pi$
$929$	46.5354	1.52678	0.763389	+	0.645939i	$0.223534\pi$
$930$	0	0
$931$	5.30777	0.173955
$932$	17.8337	0.584161
$933$	0	0
$934$	−3.31915	−0.108606
$935$	0	0
$936$	0	0
$937$	32.2069	1.05215	0.526077	−	0.850437i	$-0.323662\pi$
$937$	32.2069	1.05215	0.526077	+	0.850437i	$0.323662\pi$
$938$	−6.98195	−0.227969
$939$	0	0
$940$	−22.9345	−0.748041
$941$	−5.13369	−0.167354	−0.0836768	−	0.996493i	$-0.526666\pi$
$941$	−5.13369	−0.167354	−0.0836768	+	0.996493i	$0.526666\pi$
$942$	0	0
$943$	−11.3484	−0.369553
$944$	−6.56035	−0.213521
$945$	0	0
$946$	0	0
$947$	52.5726	1.70838	0.854190	−	0.519962i	$-0.174054\pi$
$947$	52.5726	1.70838	0.854190	+	0.519962i	$0.174054\pi$
$948$	0	0
$949$	−22.1916	−0.720369
$950$	−13.8827	−0.450415
$951$	0	0
$952$	−5.77846	−0.187281
$953$	29.1077	0.942891	0.471446	−	0.881895i	$-0.343732\pi$
$953$	29.1077	0.942891	0.471446	+	0.881895i	$0.343732\pi$
$954$	0	0
$955$	1.57152	0.0508533
$956$	−9.88273	−0.319630
$957$	0	0
$958$	−9.75774	−0.315258
$959$	−11.0992	−0.358413
$960$	0	0
$961$	11.2277	0.362182
$962$	−2.06819	−0.0666810
$963$	0	0
$964$	−37.8950	−1.22051
$965$	−59.2354	−1.90685
$966$	0	0
$967$	53.3319	1.71504	0.857520	−	0.514451i	$-0.172004\pi$
$967$	53.3319	1.71504	0.857520	+	0.514451i	$0.172004\pi$
$968$	0	0
$969$	0	0
$970$	−22.7129	−0.729269
$971$	3.85170	0.123607	0.0618034	−	0.998088i	$-0.480315\pi$
$971$	3.85170	0.123607	0.0618034	+	0.998088i	$0.480315\pi$
$972$	0	0
$973$	13.1905	0.422868
$974$	12.4508	0.398950
$975$	0	0
$976$	−12.9122	−0.413308
$977$	−24.0568	−0.769646	−0.384823	−	0.922990i	$-0.625737\pi$
$977$	−24.0568	−0.769646	−0.384823	+	0.922990i	$0.625737\pi$
$978$	0	0
$979$	0	0
$980$	5.77846	0.184586
$981$	0	0
$982$	0.0620710	0.00198077
$983$	−18.5604	−0.591983	−0.295992	−	0.955191i	$-0.595650\pi$
$983$	−18.5604	−0.591983	−0.295992	+	0.955191i	$0.595650\pi$
$984$	0	0
$985$	6.83709	0.217848
$986$	−3.77846	−0.120331
$987$	0	0
$988$	−23.3224	−0.741984
$989$	43.3991	1.38001
$990$	0	0
$991$	26.2637	0.834295	0.417148	−	0.908839i	$-0.363030\pi$
$991$	26.2637	0.834295	0.417148	+	0.908839i	$0.363030\pi$
$992$	31.4328	0.997992
$993$	0	0
$994$	−1.50172	−0.0476316
$995$	8.54574	0.270918
$996$	0	0
$997$	−28.0130	−0.887180	−0.443590	−	0.896230i	$-0.646295\pi$
$997$	−28.0130	−0.887180	−0.443590	+	0.896230i	$0.646295\pi$
$998$	8.58719	0.271823
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.ca.1.2		3
3.2	odd	2		2541.2.a.bj.1.2	yes	3
11.10	odd	2		7623.2.a.cc.1.2		3
33.32	even	2		2541.2.a.bh.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2541.2.a.bh.1.2	✓	3	33.32	even	2
2541.2.a.bj.1.2	yes	3	3.2	odd	2
7623.2.a.ca.1.2		3	1.1	even	1	trivial
7623.2.a.cc.1.2		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)

\(f(q)\)	\(=\)	\(q-0.470683 q^{2} -1.77846 q^{4} -3.24914 q^{5} +1.00000 q^{7} +1.77846 q^{8} +O(q^{10})\)	\(q-0.470683 q^{2} -1.77846 q^{4} -3.24914 q^{5} +1.00000 q^{7} +1.77846 q^{8} +1.52932 q^{10} +2.47068 q^{13} -0.470683 q^{14} +2.71982 q^{16} -3.24914 q^{17} +5.30777 q^{19} +5.77846 q^{20} +8.86469 q^{23} +5.55691 q^{25} -1.16291 q^{26} -1.77846 q^{28} -2.47068 q^{29} -6.49828 q^{31} -4.83709 q^{32} +1.52932 q^{34} -3.24914 q^{35} +1.77846 q^{37} -2.49828 q^{38} -5.77846 q^{40} -1.28018 q^{41} +4.89572 q^{43} -4.17246 q^{46} -3.96896 q^{47} +1.00000 q^{49} -2.61555 q^{50} -4.39400 q^{52} +9.77846 q^{53} +1.77846 q^{56} +1.16291 q^{58} -2.41205 q^{59} -4.74742 q^{61} +3.05863 q^{62} -3.16291 q^{64} -8.02760 q^{65} +14.8337 q^{67} +5.77846 q^{68} +1.52932 q^{70} +3.19051 q^{71} -8.98195 q^{73} -0.837090 q^{74} -9.43965 q^{76} +12.3810 q^{79} -8.83709 q^{80} +0.602558 q^{82} -13.0828 q^{83} +10.5569 q^{85} -2.30434 q^{86} -7.86469 q^{89} +2.47068 q^{91} -15.7655 q^{92} +1.86813 q^{94} -17.2457 q^{95} -14.8517 q^{97} -0.470683 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 3 q^{4} - q^{5} + 3 q^{7} - 3 q^{8}+O(q^{10}) \) 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

Modular forms

Varieties

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Groups

Database

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