LMFDB - Embedded newform 7623.2.a.cd.1.1

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:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.254102$ 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)

$f(q)$	$=$	$q-1.93543 q^{2} +1.74590 q^{4} -4.18953 q^{5} +1.00000 q^{7} +0.491797 q^{8} +O(q^{10})$	$q-1.93543 q^{2} +1.74590 q^{4} -4.18953 q^{5} +1.00000 q^{7} +0.491797 q^{8} +8.10856 q^{10} +3.17313 q^{13} -1.93543 q^{14} -4.44364 q^{16} +6.85446 q^{17} +0.318669 q^{19} -7.31450 q^{20} +1.87086 q^{23} +12.5522 q^{25} -6.14137 q^{26} +1.74590 q^{28} -3.17313 q^{29} +9.23353 q^{31} +7.61676 q^{32} -13.2663 q^{34} -4.18953 q^{35} -7.55220 q^{37} -0.616763 q^{38} -2.06040 q^{40} +9.36266 q^{41} +10.8873 q^{43} -3.62093 q^{46} +8.06040 q^{47} +1.00000 q^{49} -24.2939 q^{50} +5.53996 q^{52} -0.508203 q^{53} +0.491797 q^{56} +6.14137 q^{58} +7.04399 q^{59} +2.00000 q^{61} -17.8709 q^{62} -5.85446 q^{64} -13.2939 q^{65} -2.66492 q^{67} +11.9672 q^{68} +8.10856 q^{70} +5.01641 q^{71} +4.82687 q^{73} +14.6168 q^{74} +0.556364 q^{76} -5.01641 q^{79} +18.6168 q^{80} -18.1208 q^{82} +3.52461 q^{83} -28.7170 q^{85} -21.0716 q^{86} +1.74173 q^{89} +3.17313 q^{91} +3.26634 q^{92} -15.6004 q^{94} -1.33508 q^{95} -12.2499 q^{97} -1.93543 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{2} + 6 q^{4} - 4 q^{5} + 3 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q + 2 * q^2 + 6 * q^4 - 4 * q^5 + 3 * q^7 + 3 * q^8	$ 3 q + 2 q^{2} + 6 q^{4} - 4 q^{5} + 3 q^{7} + 3 q^{8} + 11 q^{10} + 4 q^{13} + 2 q^{14} - 4 q^{16} + 8 q^{17} + 8 q^{19} + 3 q^{20} - 10 q^{23} + 15 q^{25} + q^{26} + 6 q^{28} - 4 q^{29} - 2 q^{31} + 8 q^{32} - 4 q^{34} - 4 q^{35} + 13 q^{38} + 18 q^{40} + 14 q^{41} + 14 q^{43} - 28 q^{46} + 3 q^{49} - 19 q^{50} + 29 q^{52} + 3 q^{56} - q^{58} + 6 q^{61} - 38 q^{62} - 5 q^{64} + 14 q^{65} - 4 q^{67} + 42 q^{68} + 11 q^{70} + 12 q^{71} + 20 q^{73} + 29 q^{74} + 11 q^{76} - 12 q^{79} + 41 q^{80} - 6 q^{82} + 6 q^{83} + 6 q^{85} - 24 q^{86} - 26 q^{89} + 4 q^{91} - 26 q^{92} - 35 q^{94} - 8 q^{95} - 4 q^{97} + 2 q^{98}+O(q^{100}) $ 3 * q + 2 * q^2 + 6 * q^4 - 4 * q^5 + 3 * q^7 + 3 * q^8 + 11 * q^10 + 4 * q^13 + 2 * q^14 - 4 * q^16 + 8 * q^17 + 8 * q^19 + 3 * q^20 - 10 * q^23 + 15 * q^25 + q^26 + 6 * q^28 - 4 * q^29 - 2 * q^31 + 8 * q^32 - 4 * q^34 - 4 * q^35 + 13 * q^38 + 18 * q^40 + 14 * q^41 + 14 * q^43 - 28 * q^46 + 3 * q^49 - 19 * q^50 + 29 * q^52 + 3 * q^56 - q^58 + 6 * q^61 - 38 * q^62 - 5 * q^64 + 14 * q^65 - 4 * q^67 + 42 * q^68 + 11 * q^70 + 12 * q^71 + 20 * q^73 + 29 * q^74 + 11 * q^76 - 12 * q^79 + 41 * q^80 - 6 * q^82 + 6 * q^83 + 6 * q^85 - 24 * q^86 - 26 * q^89 + 4 * q^91 - 26 * q^92 - 35 * q^94 - 8 * q^95 - 4 * q^97 + 2 * 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 $

$2$	−1.93543	−1.36856	−0.684279	−	0.729221i	$-0.739883\pi$
$2$	−1.93543	−1.36856	−0.684279	+	0.729221i	$0.739883\pi$
$3$	0	0
$3$	0	0
$4$	1.74590	0.872949
$5$	−4.18953	−1.87362	−0.936808	−	0.349843i	$-0.886235\pi$
$5$	−4.18953	−1.87362	−0.936808	+	0.349843i	$0.886235\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0.491797	0.173876
$9$	0	0
$10$	8.10856	2.56415
$11$	0	0
$11$	0	0
$12$	0	0
$13$	3.17313	0.880067	0.440034	−	0.897981i	$-0.354967\pi$
$13$	3.17313	0.880067	0.440034	+	0.897981i	$0.354967\pi$
$14$	−1.93543	−0.517266
$15$	0	0
$16$	−4.44364	−1.11091
$17$	6.85446	1.66245	0.831225	−	0.555936i	$-0.187640\pi$
$17$	6.85446	1.66245	0.831225	+	0.555936i	$0.187640\pi$
$18$	0	0
$19$	0.318669	0.0731078	0.0365539	−	0.999332i	$-0.488362\pi$
$19$	0.318669	0.0731078	0.0365539	+	0.999332i	$0.488362\pi$
$20$	−7.31450	−1.63557
$21$	0	0
$22$	0	0
$23$	1.87086	0.390102	0.195051	−	0.980793i	$-0.437513\pi$
$23$	1.87086	0.390102	0.195051	+	0.980793i	$0.437513\pi$
$24$	0	0
$25$	12.5522	2.51044
$26$	−6.14137	−1.20442
$27$	0	0
$28$	1.74590	0.329944
$29$	−3.17313	−0.589235	−0.294617	−	0.955615i	$-0.595192\pi$
$29$	−3.17313	−0.589235	−0.294617	+	0.955615i	$0.595192\pi$
$30$	0	0
$31$	9.23353	1.65839	0.829195	−	0.558959i	$-0.188799\pi$
$31$	9.23353	1.65839	0.829195	+	0.558959i	$0.188799\pi$
$32$	7.61676	1.34647
$33$	0	0
$34$	−13.2663	−2.27516
$35$	−4.18953	−0.708161
$36$	0	0
$37$	−7.55220	−1.24157	−0.620787	−	0.783980i	$-0.713187\pi$
$37$	−7.55220	−1.24157	−0.620787	+	0.783980i	$0.713187\pi$
$38$	−0.616763	−0.100052
$39$	0	0
$40$	−2.06040	−0.325778
$41$	9.36266	1.46220	0.731101	−	0.682269i	$-0.239007\pi$
$41$	9.36266	1.46220	0.731101	+	0.682269i	$0.239007\pi$
$42$	0	0
$43$	10.8873	1.66029	0.830147	−	0.557545i	$-0.188257\pi$
$43$	10.8873	1.66029	0.830147	+	0.557545i	$0.188257\pi$
$44$	0	0
$45$	0	0
$46$	−3.62093	−0.533877
$47$	8.06040	1.17573	0.587865	−	0.808959i	$-0.299969\pi$
$47$	8.06040	1.17573	0.587865	+	0.808959i	$0.299969\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−24.2939	−3.43568
$51$	0	0
$52$	5.53996	0.768254
$53$	−0.508203	−0.0698071	−0.0349036	−	0.999391i	$-0.511112\pi$
$53$	−0.508203	−0.0698071	−0.0349036	+	0.999391i	$0.511112\pi$
$54$	0	0
$55$	0	0
$56$	0.491797	0.0657191
$57$	0	0
$58$	6.14137	0.806402
$59$	7.04399	0.917050	0.458525	−	0.888682i	$-0.348378\pi$
$59$	7.04399	0.917050	0.458525	+	0.888682i	$0.348378\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	−17.8709	−2.26960
$63$	0	0
$64$	−5.85446	−0.731807
$65$	−13.2939	−1.64891
$66$	0	0
$67$	−2.66492	−0.325572	−0.162786	−	0.986661i	$-0.552048\pi$
$67$	−2.66492	−0.325572	−0.162786	+	0.986661i	$0.552048\pi$
$68$	11.9672	1.45123
$69$	0	0
$70$	8.10856	0.969158
$71$	5.01641	0.595338	0.297669	−	0.954669i	$-0.403791\pi$
$71$	5.01641	0.595338	0.297669	+	0.954669i	$0.403791\pi$
$72$	0	0
$73$	4.82687	0.564943	0.282471	−	0.959276i	$-0.408846\pi$
$73$	4.82687	0.564943	0.282471	+	0.959276i	$0.408846\pi$
$74$	14.6168	1.69916
$75$	0	0
$76$	0.556364	0.0638194
$77$	0	0
$78$	0	0
$79$	−5.01641	−0.564390	−0.282195	−	0.959357i	$-0.591062\pi$
$79$	−5.01641	−0.564390	−0.282195	+	0.959357i	$0.591062\pi$
$80$	18.6168	2.08142
$81$	0	0
$82$	−18.1208	−2.00111
$83$	3.52461	0.386876	0.193438	−	0.981112i	$-0.438036\pi$
$83$	3.52461	0.386876	0.193438	+	0.981112i	$0.438036\pi$
$84$	0	0
$85$	−28.7170	−3.11479
$86$	−21.0716	−2.27221
$87$	0	0
$88$	0	0
$89$	1.74173	0.184623	0.0923115	−	0.995730i	$-0.470574\pi$
$89$	1.74173	0.184623	0.0923115	+	0.995730i	$0.470574\pi$
$90$	0	0
$91$	3.17313	0.332634
$92$	3.26634	0.340539
$93$	0	0
$94$	−15.6004	−1.60905
$95$	−1.33508	−0.136976
$96$	0	0
$97$	−12.2499	−1.24379	−0.621896	−	0.783100i	$-0.713637\pi$
$97$	−12.2499	−1.24379	−0.621896	+	0.783100i	$0.713637\pi$
$98$	−1.93543	−0.195508
$99$	0	0
$100$	21.9149	2.19149
$101$	4.88727	0.486302	0.243151	−	0.969988i	$-0.421819\pi$
$101$	4.88727	0.486302	0.243151	+	0.969988i	$0.421819\pi$
$102$	0	0
$103$	−0.637339	−0.0627988	−0.0313994	−	0.999507i	$-0.509996\pi$
$103$	−0.637339	−0.0627988	−0.0313994	+	0.999507i	$0.509996\pi$
$104$	1.56053	0.153023
$105$	0	0
$106$	0.983593	0.0955350
$107$	0.956008	0.0924208	0.0462104	−	0.998932i	$-0.485286\pi$
$107$	0.956008	0.0924208	0.0462104	+	0.998932i	$0.485286\pi$
$108$	0	0
$109$	7.61259	0.729154	0.364577	−	0.931173i	$-0.381214\pi$
$109$	7.61259	0.729154	0.364577	+	0.931173i	$0.381214\pi$
$110$	0	0
$111$	0	0
$112$	−4.44364	−0.419884
$113$	7.70892	0.725194	0.362597	−	0.931946i	$-0.381890\pi$
$113$	7.70892	0.725194	0.362597	+	0.931946i	$0.381890\pi$
$114$	0	0
$115$	−7.83805	−0.730902
$116$	−5.53996	−0.514372
$117$	0	0
$118$	−13.6332	−1.25504
$119$	6.85446	0.628347
$120$	0	0
$121$	0	0
$122$	−3.87086	−0.350452
$123$	0	0
$124$	16.1208	1.44769
$125$	−31.6402	−2.82998
$126$	0	0
$127$	5.49180	0.487318	0.243659	−	0.969861i	$-0.421652\pi$
$127$	5.49180	0.487318	0.243659	+	0.969861i	$0.421652\pi$
$128$	−3.90262	−0.344946
$129$	0	0
$130$	25.7295	2.25663
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	0	0
$133$	0.318669	0.0276321
$134$	5.15778	0.445564
$135$	0	0
$136$	3.37100	0.289061
$137$	−15.6126	−1.33387	−0.666937	−	0.745114i	$-0.732395\pi$
$137$	−15.6126	−1.33387	−0.666937	+	0.745114i	$0.732395\pi$
$138$	0	0
$139$	9.01641	0.764762	0.382381	−	0.924005i	$-0.375104\pi$
$139$	9.01641	0.764762	0.382381	+	0.924005i	$0.375104\pi$
$140$	−7.31450	−0.618188
$141$	0	0
$142$	−9.70892	−0.814754
$143$	0	0
$144$	0	0
$145$	13.2939	1.10400
$146$	−9.34209	−0.773157
$147$	0	0
$148$	−13.1854	−1.08383
$149$	−5.20594	−0.426487	−0.213244	−	0.976999i	$-0.568403\pi$
$149$	−5.20594	−0.426487	−0.213244	+	0.976999i	$0.568403\pi$
$150$	0	0
$151$	−6.24993	−0.508612	−0.254306	−	0.967124i	$-0.581847\pi$
$151$	−6.24993	−0.508612	−0.254306	+	0.967124i	$0.581847\pi$
$152$	0.156721	0.0127117
$153$	0	0
$154$	0	0
$155$	−38.6842	−3.10719
$156$	0	0
$157$	−18.1208	−1.44620	−0.723099	−	0.690745i	$-0.757283\pi$
$157$	−18.1208	−1.44620	−0.723099	+	0.690745i	$0.757283\pi$
$158$	9.70892	0.772400
$159$	0	0
$160$	−31.9107	−2.52276
$161$	1.87086	0.147445
$162$	0	0
$163$	−2.66492	−0.208733	−0.104366	−	0.994539i	$-0.533281\pi$
$163$	−2.66492	−0.208733	−0.104366	+	0.994539i	$0.533281\pi$
$164$	16.3463	1.27643
$165$	0	0
$166$	−6.82164	−0.529462
$167$	−11.1455	−0.862468	−0.431234	−	0.902240i	$-0.641922\pi$
$167$	−11.1455	−0.862468	−0.431234	+	0.902240i	$0.641922\pi$
$168$	0	0
$169$	−2.93126	−0.225482
$170$	55.5798	4.26277
$171$	0	0
$172$	19.0081	1.44935
$173$	−24.8461	−1.88902	−0.944508	−	0.328489i	$-0.893461\pi$
$173$	−24.8461	−1.88902	−0.944508	+	0.328489i	$0.893461\pi$
$174$	0	0
$175$	12.5522	0.948857
$176$	0	0
$177$	0	0
$178$	−3.37100	−0.252667
$179$	12.7581	0.953588	0.476794	−	0.879015i	$-0.341799\pi$
$179$	12.7581	0.953588	0.476794	+	0.879015i	$0.341799\pi$
$180$	0	0
$181$	−3.23353	−0.240346	−0.120173	−	0.992753i	$-0.538345\pi$
$181$	−3.23353	−0.240346	−0.120173	+	0.992753i	$0.538345\pi$
$182$	−6.14137	−0.455229
$183$	0	0
$184$	0.920085	0.0678296
$185$	31.6402	2.32623
$186$	0	0
$187$	0	0
$188$	14.0726	1.02635
$189$	0	0
$190$	2.58395	0.187459
$191$	−20.9753	−1.51772	−0.758858	−	0.651256i	$-0.774242\pi$
$191$	−20.9753	−1.51772	−0.758858	+	0.651256i	$0.774242\pi$
$192$	0	0
$193$	−0.249933	−0.0179905	−0.00899527	−	0.999960i	$-0.502863\pi$
$193$	−0.249933	−0.0179905	−0.00899527	+	0.999960i	$0.502863\pi$
$194$	23.7089	1.70220
$195$	0	0
$196$	1.74590	0.124707
$197$	18.4999	1.31806	0.659030	−	0.752116i	$-0.270967\pi$
$197$	18.4999	1.31806	0.659030	+	0.752116i	$0.270967\pi$
$198$	0	0
$199$	9.87086	0.699727	0.349864	−	0.936801i	$-0.386228\pi$
$199$	9.87086	0.699727	0.349864	+	0.936801i	$0.386228\pi$
$200$	6.17313	0.436506
$201$	0	0
$202$	−9.45898	−0.665532
$203$	−3.17313	−0.222710
$204$	0	0
$205$	−39.2252	−2.73961
$206$	1.23353	0.0859438
$207$	0	0
$208$	−14.1002	−0.977674
$209$	0	0
$210$	0	0
$211$	4.63734	0.319248	0.159624	−	0.987178i	$-0.448972\pi$
$211$	4.63734	0.319248	0.159624	+	0.987178i	$0.448972\pi$
$212$	−0.887271	−0.0609381
$213$	0	0
$214$	−1.85029	−0.126483
$215$	−45.6126	−3.11075
$216$	0	0
$217$	9.23353	0.626813
$218$	−14.7337	−0.997889
$219$	0	0
$220$	0	0
$221$	21.7501	1.46307
$222$	0	0
$223$	−18.3463	−1.22856	−0.614278	−	0.789090i	$-0.710553\pi$
$223$	−18.3463	−1.22856	−0.614278	+	0.789090i	$0.710553\pi$
$224$	7.61676	0.508916
$225$	0	0
$226$	−14.9201	−0.992469
$227$	0.379068	0.0251596	0.0125798	−	0.999921i	$-0.495996\pi$
$227$	0.379068	0.0251596	0.0125798	+	0.999921i	$0.495996\pi$
$228$	0	0
$229$	24.9424	1.64824	0.824121	−	0.566413i	$-0.191669\pi$
$229$	24.9424	1.64824	0.824121	+	0.566413i	$0.191669\pi$
$230$	15.1700	1.00028
$231$	0	0
$232$	−1.56053	−0.102454
$233$	23.4506	1.53630	0.768151	−	0.640268i	$-0.221177\pi$
$233$	23.4506	1.53630	0.768151	+	0.640268i	$0.221177\pi$
$234$	0	0
$235$	−33.7693	−2.20287
$236$	12.2981	0.800538
$237$	0	0
$238$	−13.2663	−0.859929
$239$	5.07681	0.328391	0.164196	−	0.986428i	$-0.447497\pi$
$239$	5.07681	0.328391	0.164196	+	0.986428i	$0.447497\pi$
$240$	0	0
$241$	−19.2939	−1.24283	−0.621415	−	0.783481i	$-0.713442\pi$
$241$	−19.2939	−1.24283	−0.621415	+	0.783481i	$0.713442\pi$
$242$	0	0
$243$	0	0
$244$	3.49180	0.223539
$245$	−4.18953	−0.267660
$246$	0	0
$247$	1.01118	0.0643397
$248$	4.54102	0.288355
$249$	0	0
$250$	61.2374	3.87299
$251$	−23.8021	−1.50238	−0.751188	−	0.660088i	$-0.770519\pi$
$251$	−23.8021	−1.50238	−0.751188	+	0.660088i	$0.770519\pi$
$252$	0	0
$253$	0	0
$254$	−10.6290	−0.666923
$255$	0	0
$256$	19.2622	1.20389
$257$	−14.9149	−0.930363	−0.465182	−	0.885215i	$-0.654011\pi$
$257$	−14.9149	−0.930363	−0.465182	+	0.885215i	$0.654011\pi$
$258$	0	0
$259$	−7.55220	−0.469271
$260$	−23.2098	−1.43941
$261$	0	0
$262$	−7.74173	−0.478286
$263$	2.92319	0.180252	0.0901259	−	0.995930i	$-0.471273\pi$
$263$	2.92319	0.180252	0.0901259	+	0.995930i	$0.471273\pi$
$264$	0	0
$265$	2.12914	0.130792
$266$	−0.616763	−0.0378162
$267$	0	0
$268$	−4.65269	−0.284208
$269$	−11.9672	−0.729652	−0.364826	−	0.931076i	$-0.618872\pi$
$269$	−11.9672	−0.729652	−0.364826	+	0.931076i	$0.618872\pi$
$270$	0	0
$271$	20.3187	1.23427	0.617136	−	0.786857i	$-0.288293\pi$
$271$	20.3187	1.23427	0.617136	+	0.786857i	$0.288293\pi$
$272$	−30.4587	−1.84683
$273$	0	0
$274$	30.2171	1.82548
$275$	0	0
$276$	0	0
$277$	−18.0552	−1.08483	−0.542415	−	0.840111i	$-0.682490\pi$
$277$	−18.0552	−1.08483	−0.542415	+	0.840111i	$0.682490\pi$
$278$	−17.4506	−1.04662
$279$	0	0
$280$	−2.06040	−0.123132
$281$	27.2939	1.62822	0.814110	−	0.580711i	$-0.197226\pi$
$281$	27.2939	1.62822	0.814110	+	0.580711i	$0.197226\pi$
$282$	0	0
$283$	−29.8901	−1.77678	−0.888391	−	0.459087i	$-0.848177\pi$
$283$	−29.8901	−1.77678	−0.888391	+	0.459087i	$0.848177\pi$
$284$	8.75814	0.519700
$285$	0	0
$286$	0	0
$287$	9.36266	0.552660
$288$	0	0
$289$	29.9836	1.76374
$290$	−25.7295	−1.51089
$291$	0	0
$292$	8.42723	0.493166
$293$	−4.34625	−0.253911	−0.126955	−	0.991908i	$-0.540521\pi$
$293$	−4.34625	−0.253911	−0.126955	+	0.991908i	$0.540521\pi$
$294$	0	0
$295$	−29.5110	−1.71820
$296$	−3.71414	−0.215880
$297$	0	0
$298$	10.0757	0.583672
$299$	5.93649	0.343316
$300$	0	0
$301$	10.8873	0.627532
$302$	12.0963	0.696065
$303$	0	0
$304$	−1.41605	−0.0812161
$305$	−8.37907	−0.479784
$306$	0	0
$307$	20.7581	1.18473	0.592365	−	0.805670i	$-0.298194\pi$
$307$	20.7581	1.18473	0.592365	+	0.805670i	$0.298194\pi$
$308$	0	0
$309$	0	0
$310$	74.8706	4.25236
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	8.12914	0.459486	0.229743	−	0.973251i	$-0.426211\pi$
$313$	8.12914	0.459486	0.229743	+	0.973251i	$0.426211\pi$
$314$	35.0716	1.97920
$315$	0	0
$316$	−8.75814	−0.492684
$317$	19.9917	1.12284	0.561422	−	0.827530i	$-0.310255\pi$
$317$	19.9917	1.12284	0.561422	+	0.827530i	$0.310255\pi$
$318$	0	0
$319$	0	0
$320$	24.5275	1.37113
$321$	0	0
$322$	−3.62093	−0.201787
$323$	2.18431	0.121538
$324$	0	0
$325$	39.8297	2.20935
$326$	5.15778	0.285663
$327$	0	0
$328$	4.60453	0.254242
$329$	8.06040	0.444384
$330$	0	0
$331$	17.0164	0.935306	0.467653	−	0.883912i	$-0.345100\pi$
$331$	17.0164	0.935306	0.467653	+	0.883912i	$0.345100\pi$
$332$	6.15361	0.337723
$333$	0	0
$334$	21.5714	1.18034
$335$	11.1648	0.609998
$336$	0	0
$337$	−1.52461	−0.0830508	−0.0415254	−	0.999137i	$-0.513222\pi$
$337$	−1.52461	−0.0830508	−0.0415254	+	0.999137i	$0.513222\pi$
$338$	5.67326	0.308585
$339$	0	0
$340$	−50.1369	−2.71906
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	5.35432	0.288686
$345$	0	0
$346$	48.0880	2.58523
$347$	17.4506	0.936800	0.468400	−	0.883517i	$-0.344831\pi$
$347$	17.4506	0.936800	0.468400	+	0.883517i	$0.344831\pi$
$348$	0	0
$349$	−6.85969	−0.367191	−0.183595	−	0.983002i	$-0.558774\pi$
$349$	−6.85969	−0.367191	−0.183595	+	0.983002i	$0.558774\pi$
$350$	−24.2939	−1.29856
$351$	0	0
$352$	0	0
$353$	31.9313	1.69953	0.849765	−	0.527162i	$-0.176744\pi$
$353$	31.9313	1.69953	0.849765	+	0.527162i	$0.176744\pi$
$354$	0	0
$355$	−21.0164	−1.11544
$356$	3.04088	0.161166
$357$	0	0
$358$	−24.6925	−1.30504
$359$	−24.4342	−1.28959	−0.644795	−	0.764356i	$-0.723057\pi$
$359$	−24.4342	−1.28959	−0.644795	+	0.764356i	$0.723057\pi$
$360$	0	0
$361$	−18.8984	−0.994655
$362$	6.25827	0.328927
$363$	0	0
$364$	5.53996	0.290373
$365$	−20.2223	−1.05849
$366$	0	0
$367$	17.0716	0.891129	0.445565	−	0.895250i	$-0.353003\pi$
$367$	17.0716	0.891129	0.445565	+	0.895250i	$0.353003\pi$
$368$	−8.31344	−0.433368
$369$	0	0
$370$	−61.2374	−3.18358
$371$	−0.508203	−0.0263846
$372$	0	0
$373$	−3.17836	−0.164569	−0.0822845	−	0.996609i	$-0.526222\pi$
$373$	−3.17836	−0.164569	−0.0822845	+	0.996609i	$0.526222\pi$
$374$	0	0
$375$	0	0
$376$	3.96408	0.204432
$377$	−10.0687	−0.518566
$378$	0	0
$379$	3.93960	0.202364	0.101182	−	0.994868i	$-0.467738\pi$
$379$	3.93960	0.202364	0.101182	+	0.994868i	$0.467738\pi$
$380$	−2.33091	−0.119573
$381$	0	0
$382$	40.5962	2.07708
$383$	24.7581	1.26508	0.632541	−	0.774527i	$-0.282012\pi$
$383$	24.7581	1.26508	0.632541	+	0.774527i	$0.282012\pi$
$384$	0	0
$385$	0	0
$386$	0.483728	0.0246211
$387$	0	0
$388$	−21.3871	−1.08577
$389$	−3.65375	−0.185252	−0.0926261	−	0.995701i	$-0.529526\pi$
$389$	−3.65375	−0.185252	−0.0926261	+	0.995701i	$0.529526\pi$
$390$	0	0
$391$	12.8238	0.648526
$392$	0.491797	0.0248395
$393$	0	0
$394$	−35.8052	−1.80384
$395$	21.0164	1.05745
$396$	0	0
$397$	4.56337	0.229029	0.114515	−	0.993422i	$-0.463469\pi$
$397$	4.56337	0.229029	0.114515	+	0.993422i	$0.463469\pi$
$398$	−19.1044	−0.957617
$399$	0	0
$400$	−55.7774	−2.78887
$401$	7.23353	0.361225	0.180613	−	0.983554i	$-0.442192\pi$
$401$	7.23353	0.361225	0.180613	+	0.983554i	$0.442192\pi$
$402$	0	0
$403$	29.2992	1.45949
$404$	8.53268	0.424517
$405$	0	0
$406$	6.14137	0.304791
$407$	0	0
$408$	0	0
$409$	−5.30749	−0.262439	−0.131219	−	0.991353i	$-0.541889\pi$
$409$	−5.30749	−0.262439	−0.131219	+	0.991353i	$0.541889\pi$
$410$	75.9177	3.74931
$411$	0	0
$412$	−1.11273	−0.0548202
$413$	7.04399	0.346612
$414$	0	0
$415$	−14.7665	−0.724858
$416$	24.1690	1.18498
$417$	0	0
$418$	0	0
$419$	11.4231	0.558053	0.279026	−	0.960283i	$-0.409988\pi$
$419$	11.4231	0.558053	0.279026	+	0.960283i	$0.409988\pi$
$420$	0	0
$421$	−27.4147	−1.33611	−0.668056	−	0.744111i	$-0.732873\pi$
$421$	−27.4147	−1.33611	−0.668056	+	0.744111i	$0.732873\pi$
$422$	−8.97526	−0.436909
$423$	0	0
$424$	−0.249933	−0.0121378
$425$	86.0385	4.17348
$426$	0	0
$427$	2.00000	0.0967868
$428$	1.66909	0.0806786
$429$	0	0
$430$	88.2801	4.25724
$431$	−2.28586	−0.110106	−0.0550529	−	0.998483i	$-0.517533\pi$
$431$	−2.28586	−0.110106	−0.0550529	+	0.998483i	$0.517533\pi$
$432$	0	0
$433$	31.5714	1.51723	0.758613	−	0.651541i	$-0.225877\pi$
$433$	31.5714	1.51723	0.758613	+	0.651541i	$0.225877\pi$
$434$	−17.8709	−0.857829
$435$	0	0
$436$	13.2908	0.636515
$437$	0.596187	0.0285195
$438$	0	0
$439$	18.5275	0.884267	0.442133	−	0.896949i	$-0.354222\pi$
$439$	18.5275	0.884267	0.442133	+	0.896949i	$0.354222\pi$
$440$	0	0
$441$	0	0
$442$	−42.0958	−2.00229
$443$	−28.4342	−1.35095	−0.675476	−	0.737382i	$-0.736062\pi$
$443$	−28.4342	−1.35095	−0.675476	+	0.737382i	$0.736062\pi$
$444$	0	0
$445$	−7.29703	−0.345913
$446$	35.5079	1.68135
$447$	0	0
$448$	−5.85446	−0.276597
$449$	5.68656	0.268365	0.134183	−	0.990957i	$-0.457159\pi$
$449$	5.68656	0.268365	0.134183	+	0.990957i	$0.457159\pi$
$450$	0	0
$451$	0	0
$452$	13.4590	0.633057
$453$	0	0
$454$	−0.733661	−0.0344324
$455$	−13.2939	−0.623229
$456$	0	0
$457$	13.0081	0.608492	0.304246	−	0.952594i	$-0.401596\pi$
$457$	13.0081	0.608492	0.304246	+	0.952594i	$0.401596\pi$
$458$	−48.2744	−2.25571
$459$	0	0
$460$	−13.6844	−0.638040
$461$	−6.37907	−0.297103	−0.148551	−	0.988905i	$-0.547461\pi$
$461$	−6.37907	−0.297103	−0.148551	+	0.988905i	$0.547461\pi$
$462$	0	0
$463$	34.0932	1.58445	0.792223	−	0.610232i	$-0.208924\pi$
$463$	34.0932	1.58445	0.792223	+	0.610232i	$0.208924\pi$
$464$	14.1002	0.654586
$465$	0	0
$466$	−45.3871	−2.10252
$467$	−18.1484	−0.839807	−0.419903	−	0.907569i	$-0.637936\pi$
$467$	−18.1484	−0.839807	−0.419903	+	0.907569i	$0.637936\pi$
$468$	0	0
$469$	−2.66492	−0.123055
$470$	65.3582	3.01475
$471$	0	0
$472$	3.46421	0.159453
$473$	0	0
$474$	0	0
$475$	4.00000	0.183533
$476$	11.9672	0.548515
$477$	0	0
$478$	−9.82581	−0.449422
$479$	−42.7498	−1.95329	−0.976644	−	0.214864i	$-0.931069\pi$
$479$	−42.7498	−1.95329	−0.976644	+	0.214864i	$0.931069\pi$
$480$	0	0
$481$	−23.9641	−1.09267
$482$	37.3421	1.70089
$483$	0	0
$484$	0	0
$485$	51.3215	2.33039
$486$	0	0
$487$	26.7909	1.21401	0.607007	−	0.794697i	$-0.292370\pi$
$487$	26.7909	1.21401	0.607007	+	0.794697i	$0.292370\pi$
$488$	0.983593	0.0445252
$489$	0	0
$490$	8.10856	0.366307
$491$	31.6813	1.42976	0.714879	−	0.699248i	$-0.246482\pi$
$491$	31.6813	1.42976	0.714879	+	0.699248i	$0.246482\pi$
$492$	0	0
$493$	−21.7501	−0.979574
$494$	−1.95707	−0.0880526
$495$	0	0
$496$	−41.0304	−1.84232
$497$	5.01641	0.225017
$498$	0	0
$499$	−24.1260	−1.08003	−0.540015	−	0.841656i	$-0.681581\pi$
$499$	−24.1260	−1.08003	−0.540015	+	0.841656i	$0.681581\pi$
$500$	−55.2405	−2.47043
$501$	0	0
$502$	46.0674	2.05609
$503$	30.8873	1.37720	0.688598	−	0.725144i	$-0.258227\pi$
$503$	30.8873	1.37720	0.688598	+	0.725144i	$0.258227\pi$
$504$	0	0
$505$	−20.4754	−0.911143
$506$	0	0
$507$	0	0
$508$	9.58812	0.425404
$509$	−28.2088	−1.25033	−0.625166	−	0.780492i	$-0.714969\pi$
$509$	−28.2088	−1.25033	−0.625166	+	0.780492i	$0.714969\pi$
$510$	0	0
$511$	4.82687	0.213528
$512$	−29.4754	−1.30264
$513$	0	0
$514$	28.8667	1.27326
$515$	2.67015	0.117661
$516$	0	0
$517$	0	0
$518$	14.6168	0.642224
$519$	0	0
$520$	−6.53791	−0.286706
$521$	33.7610	1.47910	0.739548	−	0.673104i	$-0.235039\pi$
$521$	33.7610	1.47910	0.739548	+	0.673104i	$0.235039\pi$
$522$	0	0
$523$	−12.8185	−0.560515	−0.280258	−	0.959925i	$-0.590420\pi$
$523$	−12.8185	−0.560515	−0.280258	+	0.959925i	$0.590420\pi$
$524$	6.98359	0.305080
$525$	0	0
$526$	−5.65765	−0.246685
$527$	63.2908	2.75699
$528$	0	0
$529$	−19.4999	−0.847820
$530$	−4.12080	−0.178996
$531$	0	0
$532$	0.556364	0.0241215
$533$	29.7089	1.28684
$534$	0	0
$535$	−4.00523	−0.173161
$536$	−1.31060	−0.0566093
$537$	0	0
$538$	23.1617	0.998571
$539$	0	0
$540$	0	0
$541$	21.8625	0.939943	0.469972	−	0.882681i	$-0.344264\pi$
$541$	21.8625	0.939943	0.469972	+	0.882681i	$0.344264\pi$
$542$	−39.3254	−1.68917
$543$	0	0
$544$	52.2088	2.23843
$545$	−31.8932	−1.36616
$546$	0	0
$547$	−23.4178	−1.00127	−0.500637	−	0.865657i	$-0.666901\pi$
$547$	−23.4178	−1.00127	−0.500637	+	0.865657i	$0.666901\pi$
$548$	−27.2580	−1.16440
$549$	0	0
$550$	0	0
$551$	−1.01118	−0.0430776
$552$	0	0
$553$	−5.01641	−0.213319
$554$	34.9446	1.48465
$555$	0	0
$556$	15.7417	0.667598
$557$	−6.91486	−0.292992	−0.146496	−	0.989211i	$-0.546800\pi$
$557$	−6.91486	−0.292992	−0.146496	+	0.989211i	$0.546800\pi$
$558$	0	0
$559$	34.5467	1.46117
$560$	18.6168	0.786702
$561$	0	0
$562$	−52.8255	−2.22831
$563$	11.8381	0.498914	0.249457	−	0.968386i	$-0.419748\pi$
$563$	11.8381	0.498914	0.249457	+	0.968386i	$0.419748\pi$
$564$	0	0
$565$	−32.2968	−1.35874
$566$	57.8503	2.43163
$567$	0	0
$568$	2.46705	0.103515
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	0	0
$571$	15.2252	0.637154	0.318577	−	0.947897i	$-0.396795\pi$
$571$	15.2252	0.637154	0.318577	+	0.947897i	$0.396795\pi$
$572$	0	0
$573$	0	0
$574$	−18.1208	−0.756347
$575$	23.4835	0.979328
$576$	0	0
$577$	17.3215	0.721104	0.360552	−	0.932739i	$-0.382588\pi$
$577$	17.3215	0.721104	0.360552	+	0.932739i	$0.382588\pi$
$578$	−58.0312	−2.41378
$579$	0	0
$580$	23.2098	0.963736
$581$	3.52461	0.146225
$582$	0	0
$583$	0	0
$584$	2.37384	0.0982302
$585$	0	0
$586$	8.41188	0.347492
$587$	−27.8678	−1.15023	−0.575113	−	0.818074i	$-0.695042\pi$
$587$	−27.8678	−1.15023	−0.575113	+	0.818074i	$0.695042\pi$
$588$	0	0
$589$	2.94244	0.121241
$590$	57.1166	2.35145
$591$	0	0
$592$	33.5592	1.37927
$593$	−24.3463	−0.999781	−0.499890	−	0.866089i	$-0.666626\pi$
$593$	−24.3463	−0.999781	−0.499890	+	0.866089i	$0.666626\pi$
$594$	0	0
$595$	−28.7170	−1.17728
$596$	−9.08904	−0.372302
$597$	0	0
$598$	−11.4897	−0.469848
$599$	−21.0164	−0.858707	−0.429354	−	0.903136i	$-0.641259\pi$
$599$	−21.0164	−0.858707	−0.429354	+	0.903136i	$0.641259\pi$
$600$	0	0
$601$	−40.9477	−1.67029	−0.835145	−	0.550030i	$-0.814616\pi$
$601$	−40.9477	−1.67029	−0.835145	+	0.550030i	$0.814616\pi$
$602$	−21.0716	−0.858813
$603$	0	0
$604$	−10.9117	−0.443993
$605$	0	0
$606$	0	0
$607$	14.0276	0.569362	0.284681	−	0.958622i	$-0.408112\pi$
$607$	14.0276	0.569362	0.284681	+	0.958622i	$0.408112\pi$
$608$	2.42723	0.0984371
$609$	0	0
$610$	16.2171	0.656612
$611$	25.5767	1.03472
$612$	0	0
$613$	−18.5306	−0.748442	−0.374221	−	0.927339i	$-0.622090\pi$
$613$	−18.5306	−0.748442	−0.374221	+	0.927339i	$0.622090\pi$
$614$	−40.1760	−1.62137
$615$	0	0
$616$	0	0
$617$	−2.43424	−0.0979987	−0.0489994	−	0.998799i	$-0.515603\pi$
$617$	−2.43424	−0.0979987	−0.0489994	+	0.998799i	$0.515603\pi$
$618$	0	0
$619$	−30.4259	−1.22292	−0.611460	−	0.791275i	$-0.709418\pi$
$619$	−30.4259	−1.22292	−0.611460	+	0.791275i	$0.709418\pi$
$620$	−67.5386	−2.71242
$621$	0	0
$622$	15.4835	0.620830
$623$	1.74173	0.0697809
$624$	0	0
$625$	69.7966	2.79187
$626$	−15.7334	−0.628833
$627$	0	0
$628$	−31.6371	−1.26246
$629$	−51.7662	−2.06405
$630$	0	0
$631$	34.9836	1.39267	0.696337	−	0.717715i	$-0.254812\pi$
$631$	34.9836	1.39267	0.696337	+	0.717715i	$0.254812\pi$
$632$	−2.46705	−0.0981341
$633$	0	0
$634$	−38.6925	−1.53668
$635$	−23.0081	−0.913047
$636$	0	0
$637$	3.17313	0.125724
$638$	0	0
$639$	0	0
$640$	16.3502	0.646297
$641$	−8.56337	−0.338233	−0.169116	−	0.985596i	$-0.554091\pi$
$641$	−8.56337	−0.338233	−0.169116	+	0.985596i	$0.554091\pi$
$642$	0	0
$643$	−5.11273	−0.201626	−0.100813	−	0.994905i	$-0.532144\pi$
$643$	−5.11273	−0.201626	−0.100813	+	0.994905i	$0.532144\pi$
$644$	3.26634	0.128712
$645$	0	0
$646$	−4.22758	−0.166332
$647$	−25.7693	−1.01310	−0.506548	−	0.862212i	$-0.669079\pi$
$647$	−25.7693	−1.01310	−0.506548	+	0.862212i	$0.669079\pi$
$648$	0	0
$649$	0	0
$650$	−77.0877	−3.02363
$651$	0	0
$652$	−4.65269	−0.182213
$653$	47.8953	1.87429	0.937145	−	0.348941i	$-0.113459\pi$
$653$	47.8953	1.87429	0.937145	+	0.348941i	$0.113459\pi$
$654$	0	0
$655$	−16.7581	−0.654795
$656$	−41.6043	−1.62437
$657$	0	0
$658$	−15.6004	−0.608165
$659$	8.95601	0.348877	0.174438	−	0.984668i	$-0.444189\pi$
$659$	8.95601	0.348877	0.174438	+	0.984668i	$0.444189\pi$
$660$	0	0
$661$	−10.7993	−0.420044	−0.210022	−	0.977697i	$-0.567353\pi$
$661$	−10.7993	−0.420044	−0.210022	+	0.977697i	$0.567353\pi$
$662$	−32.9341	−1.28002
$663$	0	0
$664$	1.73339	0.0672686
$665$	−1.33508	−0.0517720
$666$	0	0
$667$	−5.93649	−0.229862
$668$	−19.4590	−0.752891
$669$	0	0
$670$	−21.6087	−0.834817
$671$	0	0
$672$	0	0
$673$	−32.2499	−1.24314	−0.621572	−	0.783357i	$-0.713506\pi$
$673$	−32.2499	−1.24314	−0.621572	+	0.783357i	$0.713506\pi$
$674$	2.95078	0.113660
$675$	0	0
$676$	−5.11769	−0.196834
$677$	−27.7089	−1.06494	−0.532470	−	0.846449i	$-0.678736\pi$
$677$	−27.7089	−1.06494	−0.532470	+	0.846449i	$0.678736\pi$
$678$	0	0
$679$	−12.2499	−0.470109
$680$	−14.1229	−0.541589
$681$	0	0
$682$	0	0
$683$	−22.3156	−0.853881	−0.426941	−	0.904280i	$-0.640409\pi$
$683$	−22.3156	−0.853881	−0.426941	+	0.904280i	$0.640409\pi$
$684$	0	0
$685$	65.4095	2.49917
$686$	−1.93543	−0.0738951
$687$	0	0
$688$	−48.3791	−1.84443
$689$	−1.61259	−0.0614349
$690$	0	0
$691$	25.1372	0.956264	0.478132	−	0.878288i	$-0.341314\pi$
$691$	25.1372	0.956264	0.478132	+	0.878288i	$0.341314\pi$
$692$	−43.3788	−1.64901
$693$	0	0
$694$	−33.7745	−1.28206
$695$	−37.7745	−1.43287
$696$	0	0
$697$	64.1760	2.43084
$698$	13.2765	0.502521
$699$	0	0
$700$	21.9149	0.828304
$701$	−19.2747	−0.727995	−0.363997	−	0.931400i	$-0.618588\pi$
$701$	−19.2747	−0.727995	−0.363997	+	0.931400i	$0.618588\pi$
$702$	0	0
$703$	−2.40665	−0.0907686
$704$	0	0
$705$	0	0
$706$	−61.8008	−2.32590
$707$	4.88727	0.183805
$708$	0	0
$709$	9.76098	0.366581	0.183291	−	0.983059i	$-0.441325\pi$
$709$	9.76098	0.366581	0.183291	+	0.983059i	$0.441325\pi$
$710$	40.6758	1.52654
$711$	0	0
$712$	0.856577	0.0321016
$713$	17.2747	0.646942
$714$	0	0
$715$	0	0
$716$	22.2744	0.832434
$717$	0	0
$718$	47.2908	1.76488
$719$	−14.0276	−0.523141	−0.261570	−	0.965184i	$-0.584240\pi$
$719$	−14.0276	−0.523141	−0.261570	+	0.965184i	$0.584240\pi$
$720$	0	0
$721$	−0.637339	−0.0237357
$722$	36.5767	1.36124
$723$	0	0
$724$	−5.64541	−0.209810
$725$	−39.8297	−1.47924
$726$	0	0
$727$	−34.3051	−1.27231	−0.636153	−	0.771563i	$-0.719475\pi$
$727$	−34.3051	−1.27231	−0.636153	+	0.771563i	$0.719475\pi$
$728$	1.56053	0.0578372
$729$	0	0
$730$	39.1390	1.44860
$731$	74.6263	2.76016
$732$	0	0
$733$	10.7581	0.397361	0.198680	−	0.980064i	$-0.436334\pi$
$733$	10.7581	0.397361	0.198680	+	0.980064i	$0.436334\pi$
$734$	−33.0409	−1.21956
$735$	0	0
$736$	14.2499	0.525259
$737$	0	0
$738$	0	0
$739$	6.38741	0.234965	0.117482	−	0.993075i	$-0.462518\pi$
$739$	6.38741	0.234965	0.117482	+	0.993075i	$0.462518\pi$
$740$	55.2405	2.03068
$741$	0	0
$742$	0.983593	0.0361088
$743$	−23.1096	−0.847810	−0.423905	−	0.905707i	$-0.639341\pi$
$743$	−23.1096	−0.847810	−0.423905	+	0.905707i	$0.639341\pi$
$744$	0	0
$745$	21.8105	0.799074
$746$	6.15149	0.225222
$747$	0	0
$748$	0	0
$749$	0.956008	0.0349318
$750$	0	0
$751$	31.8678	1.16287	0.581435	−	0.813593i	$-0.302491\pi$
$751$	31.8678	1.16287	0.581435	+	0.813593i	$0.302491\pi$
$752$	−35.8175	−1.30613
$753$	0	0
$754$	19.4874	0.709688
$755$	26.1843	0.952944
$756$	0	0
$757$	−48.3103	−1.75587	−0.877934	−	0.478781i	$-0.841079\pi$
$757$	−48.3103	−1.75587	−0.877934	+	0.478781i	$0.841079\pi$
$758$	−7.62483	−0.276946
$759$	0	0
$760$	−0.656586	−0.0238169
$761$	−17.8625	−0.647516	−0.323758	−	0.946140i	$-0.604946\pi$
$761$	−17.8625	−0.647516	−0.323758	+	0.946140i	$0.604946\pi$
$762$	0	0
$763$	7.61259	0.275594
$764$	−36.6207	−1.32489
$765$	0	0
$766$	−47.9177	−1.73134
$767$	22.3515	0.807065
$768$	0	0
$769$	24.8820	0.897269	0.448635	−	0.893715i	$-0.351910\pi$
$769$	24.8820	0.897269	0.448635	+	0.893715i	$0.351910\pi$
$770$	0	0
$771$	0	0
$772$	−0.436357	−0.0157048
$773$	34.9700	1.25778	0.628892	−	0.777492i	$-0.283509\pi$
$773$	34.9700	1.25778	0.628892	+	0.777492i	$0.283509\pi$
$774$	0	0
$775$	115.901	4.16329
$776$	−6.02448	−0.216266
$777$	0	0
$778$	7.07158	0.253528
$779$	2.98359	0.106898
$780$	0	0
$781$	0	0
$782$	−24.8195	−0.887544
$783$	0	0
$784$	−4.44364	−0.158701
$785$	75.9177	2.70962
$786$	0	0
$787$	15.1648	0.540566	0.270283	−	0.962781i	$-0.412883\pi$
$787$	15.1648	0.540566	0.270283	+	0.962781i	$0.412883\pi$
$788$	32.2989	1.15060
$789$	0	0
$790$	−40.6758	−1.44718
$791$	7.70892	0.274097
$792$	0	0
$793$	6.34625	0.225362
$794$	−8.83210	−0.313440
$795$	0	0
$796$	17.2335	0.610826
$797$	−8.24470	−0.292042	−0.146021	−	0.989281i	$-0.546647\pi$
$797$	−8.24470	−0.292042	−0.146021	+	0.989281i	$0.546647\pi$
$798$	0	0
$799$	55.2497	1.95459
$800$	95.6071	3.38022
$801$	0	0
$802$	−14.0000	−0.494357
$803$	0	0
$804$	0	0
$805$	−7.83805	−0.276255
$806$	−56.7065	−1.99740
$807$	0	0
$808$	2.40354	0.0845564
$809$	29.8433	1.04923	0.524617	−	0.851338i	$-0.324209\pi$
$809$	29.8433	1.04923	0.524617	+	0.851338i	$0.324209\pi$
$810$	0	0
$811$	16.6321	0.584032	0.292016	−	0.956413i	$-0.405674\pi$
$811$	16.6321	0.584032	0.292016	+	0.956413i	$0.405674\pi$
$812$	−5.53996	−0.194414
$813$	0	0
$814$	0	0
$815$	11.1648	0.391086
$816$	0	0
$817$	3.46944	0.121380
$818$	10.2723	0.359162
$819$	0	0
$820$	−68.4832	−2.39154
$821$	30.3327	1.05862	0.529309	−	0.848429i	$-0.322451\pi$
$821$	30.3327	1.05862	0.529309	+	0.848429i	$0.322451\pi$
$822$	0	0
$823$	−18.4067	−0.641616	−0.320808	−	0.947144i	$-0.603954\pi$
$823$	−18.4067	−0.641616	−0.320808	+	0.947144i	$0.603954\pi$
$824$	−0.313441	−0.0109192
$825$	0	0
$826$	−13.6332	−0.474359
$827$	45.4559	1.58066	0.790328	−	0.612684i	$-0.209910\pi$
$827$	45.4559	1.58066	0.790328	+	0.612684i	$0.209910\pi$
$828$	0	0
$829$	−20.3463	−0.706655	−0.353327	−	0.935500i	$-0.614950\pi$
$829$	−20.3463	−0.706655	−0.353327	+	0.935500i	$0.614950\pi$
$830$	28.5795	0.992009
$831$	0	0
$832$	−18.5769	−0.644040
$833$	6.85446	0.237493
$834$	0	0
$835$	46.6946	1.61593
$836$	0	0
$837$	0	0
$838$	−22.1086	−0.763728
$839$	−16.1812	−0.558637	−0.279318	−	0.960199i	$-0.590109\pi$
$839$	−16.1812	−0.558637	−0.279318	+	0.960199i	$0.590109\pi$
$840$	0	0
$841$	−18.9313	−0.652802
$842$	53.0593	1.82855
$843$	0	0
$844$	8.09632	0.278687
$845$	12.2806	0.422466
$846$	0	0
$847$	0	0
$848$	2.25827	0.0775493
$849$	0	0
$850$	−166.522	−5.71165
$851$	−14.1291	−0.484341
$852$	0	0
$853$	20.9836	0.718465	0.359232	−	0.933248i	$-0.383039\pi$
$853$	20.9836	0.718465	0.359232	+	0.933248i	$0.383039\pi$
$854$	−3.87086	−0.132458
$855$	0	0
$856$	0.470162	0.0160698
$857$	4.79095	0.163656	0.0818279	−	0.996646i	$-0.473924\pi$
$857$	4.79095	0.163656	0.0818279	+	0.996646i	$0.473924\pi$
$858$	0	0
$859$	−9.96719	−0.340076	−0.170038	−	0.985438i	$-0.554389\pi$
$859$	−9.96719	−0.340076	−0.170038	+	0.985438i	$0.554389\pi$
$860$	−79.6350	−2.71553
$861$	0	0
$862$	4.42412	0.150686
$863$	−25.5470	−0.869629	−0.434814	−	0.900520i	$-0.643186\pi$
$863$	−25.5470	−0.869629	−0.434814	+	0.900520i	$0.643186\pi$
$864$	0	0
$865$	104.094	3.53929
$866$	−61.1044	−2.07641
$867$	0	0
$868$	16.1208	0.547176
$869$	0	0
$870$	0	0
$871$	−8.45614	−0.286525
$872$	3.74385	0.126783
$873$	0	0
$874$	−1.15388	−0.0390306
$875$	−31.6402	−1.06963
$876$	0	0
$877$	50.9893	1.72179	0.860893	−	0.508787i	$-0.169906\pi$
$877$	50.9893	1.72179	0.860893	+	0.508787i	$0.169906\pi$
$878$	−35.8586	−1.21017
$879$	0	0
$880$	0	0
$881$	−32.1895	−1.08449	−0.542246	−	0.840219i	$-0.682426\pi$
$881$	−32.1895	−1.08449	−0.542246	+	0.840219i	$0.682426\pi$
$882$	0	0
$883$	36.6154	1.23221	0.616104	−	0.787665i	$-0.288710\pi$
$883$	36.6154	1.23221	0.616104	+	0.787665i	$0.288710\pi$
$884$	37.9734	1.27718
$885$	0	0
$886$	55.0325	1.84885
$887$	−48.2004	−1.61841	−0.809206	−	0.587525i	$-0.800103\pi$
$887$	−48.2004	−1.61841	−0.809206	+	0.587525i	$0.800103\pi$
$888$	0	0
$889$	5.49180	0.184189
$890$	14.1229	0.473401
$891$	0	0
$892$	−32.0307	−1.07247
$893$	2.56860	0.0859550
$894$	0	0
$895$	−53.4506	−1.78666
$896$	−3.90262	−0.130377
$897$	0	0
$898$	−11.0060	−0.367273
$899$	−29.2992	−0.977181
$900$	0	0
$901$	−3.48346	−0.116051
$902$	0	0
$903$	0	0
$904$	3.79122	0.126094
$905$	13.5470	0.450316
$906$	0	0
$907$	22.9013	0.760425	0.380212	−	0.924899i	$-0.375851\pi$
$907$	22.9013	0.760425	0.380212	+	0.924899i	$0.375851\pi$
$908$	0.661814	0.0219631
$909$	0	0
$910$	25.7295	0.852924
$911$	36.0552	1.19456	0.597281	−	0.802032i	$-0.296248\pi$
$911$	36.0552	1.19456	0.597281	+	0.802032i	$0.296248\pi$
$912$	0	0
$913$	0	0
$914$	−25.1762	−0.832756
$915$	0	0
$916$	43.5470	1.43883
$917$	4.00000	0.132092
$918$	0	0
$919$	28.3379	0.934782	0.467391	−	0.884051i	$-0.345194\pi$
$919$	28.3379	0.934782	0.467391	+	0.884051i	$0.345194\pi$
$920$	−3.85473	−0.127087
$921$	0	0
$922$	12.3463	0.406602
$923$	15.9177	0.523937
$924$	0	0
$925$	−94.7966	−3.11689
$926$	−65.9851	−2.16841
$927$	0	0
$928$	−24.1690	−0.793385
$929$	−5.08514	−0.166838	−0.0834191	−	0.996515i	$-0.526584\pi$
$929$	−5.08514	−0.166838	−0.0834191	+	0.996515i	$0.526584\pi$
$930$	0	0
$931$	0.318669	0.0104440
$932$	40.9424	1.34111
$933$	0	0
$934$	35.1250	1.14932
$935$	0	0
$936$	0	0
$937$	−32.2088	−1.05222	−0.526108	−	0.850418i	$-0.676349\pi$
$937$	−32.2088	−1.05222	−0.526108	+	0.850418i	$0.676349\pi$
$938$	5.15778	0.168407
$939$	0	0
$940$	−58.9578	−1.92299
$941$	32.7805	1.06861	0.534307	−	0.845291i	$-0.320573\pi$
$941$	32.7805	1.06861	0.534307	+	0.845291i	$0.320573\pi$
$942$	0	0
$943$	17.5163	0.570408
$944$	−31.3009	−1.01876
$945$	0	0
$946$	0	0
$947$	27.3627	0.889167	0.444584	−	0.895737i	$-0.353352\pi$
$947$	27.3627	0.889167	0.444584	+	0.895737i	$0.353352\pi$
$948$	0	0
$949$	15.3163	0.497188
$950$	−7.74173	−0.251175
$951$	0	0
$952$	3.37100	0.109255
$953$	−24.6894	−0.799768	−0.399884	−	0.916566i	$-0.630950\pi$
$953$	−24.6894	−0.799768	−0.399884	+	0.916566i	$0.630950\pi$
$954$	0	0
$955$	87.8765	2.84362
$956$	8.86359	0.286669
$957$	0	0
$958$	82.7393	2.67319
$959$	−15.6126	−0.504157
$960$	0	0
$961$	54.2580	1.75026
$962$	46.3808	1.49538
$963$	0	0
$964$	−33.6852	−1.08493
$965$	1.04710	0.0337074
$966$	0	0
$967$	−21.3298	−0.685922	−0.342961	−	0.939350i	$-0.611430\pi$
$967$	−21.3298	−0.685922	−0.342961	+	0.939350i	$0.611430\pi$
$968$	0	0
$969$	0	0
$970$	−99.3293	−3.18927
$971$	−28.4946	−0.914436	−0.457218	−	0.889355i	$-0.651154\pi$
$971$	−28.4946	−0.914436	−0.457218	+	0.889355i	$0.651154\pi$
$972$	0	0
$973$	9.01641	0.289053
$974$	−51.8521	−1.66145
$975$	0	0
$976$	−8.88727	−0.284475
$977$	19.8157	0.633960	0.316980	−	0.948432i	$-0.397331\pi$
$977$	19.8157	0.633960	0.316980	+	0.948432i	$0.397331\pi$
$978$	0	0
$979$	0	0
$980$	−7.31450	−0.233653
$981$	0	0
$982$	−61.3171	−1.95671
$983$	53.7745	1.71514	0.857571	−	0.514366i	$-0.171973\pi$
$983$	53.7745	1.71514	0.857571	+	0.514366i	$0.171973\pi$
$984$	0	0
$985$	−77.5058	−2.46954
$986$	42.0958	1.34060
$987$	0	0
$988$	1.76541	0.0561653
$989$	20.3686	0.647684
$990$	0	0
$991$	49.7693	1.58097	0.790487	−	0.612479i	$-0.209827\pi$
$991$	49.7693	1.58097	0.790487	+	0.612479i	$0.209827\pi$
$992$	70.3296	2.23297
$993$	0	0
$994$	−9.70892	−0.307948
$995$	−41.3543	−1.31102
$996$	0	0
$997$	−0.659696	−0.0208928	−0.0104464	−	0.999945i	$-0.503325\pi$
$997$	−0.659696	−0.0208928	−0.0104464	+	0.999945i	$0.503325\pi$
$998$	46.6943	1.47808
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.cd.1.1		3
3.2	odd	2		2541.2.a.bg.1.3		3
11.10	odd	2		693.2.a.l.1.3		3
33.32	even	2		231.2.a.e.1.1	✓	3
77.76	even	2		4851.2.a.bi.1.3		3
132.131	odd	2		3696.2.a.bo.1.3		3
165.164	even	2		5775.2.a.bp.1.3		3
231.230	odd	2		1617.2.a.t.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.a.e.1.1	✓	3	33.32	even	2
693.2.a.l.1.3		3	11.10	odd	2
1617.2.a.t.1.1		3	231.230	odd	2
2541.2.a.bg.1.3		3	3.2	odd	2
3696.2.a.bo.1.3		3	132.131	odd	2
4851.2.a.bi.1.3		3	77.76	even	2
5775.2.a.bp.1.3		3	165.164	even	2
7623.2.a.cd.1.1		3	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q-1.93543 q^{2} +1.74590 q^{4} -4.18953 q^{5} +1.00000 q^{7} +0.491797 q^{8} +O(q^{10})\)	\(q-1.93543 q^{2} +1.74590 q^{4} -4.18953 q^{5} +1.00000 q^{7} +0.491797 q^{8} +8.10856 q^{10} +3.17313 q^{13} -1.93543 q^{14} -4.44364 q^{16} +6.85446 q^{17} +0.318669 q^{19} -7.31450 q^{20} +1.87086 q^{23} +12.5522 q^{25} -6.14137 q^{26} +1.74590 q^{28} -3.17313 q^{29} +9.23353 q^{31} +7.61676 q^{32} -13.2663 q^{34} -4.18953 q^{35} -7.55220 q^{37} -0.616763 q^{38} -2.06040 q^{40} +9.36266 q^{41} +10.8873 q^{43} -3.62093 q^{46} +8.06040 q^{47} +1.00000 q^{49} -24.2939 q^{50} +5.53996 q^{52} -0.508203 q^{53} +0.491797 q^{56} +6.14137 q^{58} +7.04399 q^{59} +2.00000 q^{61} -17.8709 q^{62} -5.85446 q^{64} -13.2939 q^{65} -2.66492 q^{67} +11.9672 q^{68} +8.10856 q^{70} +5.01641 q^{71} +4.82687 q^{73} +14.6168 q^{74} +0.556364 q^{76} -5.01641 q^{79} +18.6168 q^{80} -18.1208 q^{82} +3.52461 q^{83} -28.7170 q^{85} -21.0716 q^{86} +1.74173 q^{89} +3.17313 q^{91} +3.26634 q^{92} -15.6004 q^{94} -1.33508 q^{95} -12.2499 q^{97} -1.93543 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{2} + 6 q^{4} - 4 q^{5} + 3 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q + 2 * q^2 + 6 * q^4 - 4 * q^5 + 3 * q^7 + 3 * q^8	\( 3 q + 2 q^{2} + 6 q^{4} - 4 q^{5} + 3 q^{7} + 3 q^{8} + 11 q^{10} + 4 q^{13} + 2 q^{14} - 4 q^{16} + 8 q^{17} + 8 q^{19} + 3 q^{20} - 10 q^{23} + 15 q^{25} + q^{26} + 6 q^{28} - 4 q^{29} - 2 q^{31} + 8 q^{32} - 4 q^{34} - 4 q^{35} + 13 q^{38} + 18 q^{40} + 14 q^{41} + 14 q^{43} - 28 q^{46} + 3 q^{49} - 19 q^{50} + 29 q^{52} + 3 q^{56} - q^{58} + 6 q^{61} - 38 q^{62} - 5 q^{64} + 14 q^{65} - 4 q^{67} + 42 q^{68} + 11 q^{70} + 12 q^{71} + 20 q^{73} + 29 q^{74} + 11 q^{76} - 12 q^{79} + 41 q^{80} - 6 q^{82} + 6 q^{83} + 6 q^{85} - 24 q^{86} - 26 q^{89} + 4 q^{91} - 26 q^{92} - 35 q^{94} - 8 q^{95} - 4 q^{97} + 2 q^{98}+O(q^{100}) \) 3 * q + 2 * q^2 + 6 * q^4 - 4 * q^5 + 3 * q^7 + 3 * q^8 + 11 * q^10 + 4 * q^13 + 2 * q^14 - 4 * q^16 + 8 * q^17 + 8 * q^19 + 3 * q^20 - 10 * q^23 + 15 * q^25 + q^26 + 6 * q^28 - 4 * q^29 - 2 * q^31 + 8 * q^32 - 4 * q^34 - 4 * q^35 + 13 * q^38 + 18 * q^40 + 14 * q^41 + 14 * q^43 - 28 * q^46 + 3 * q^49 - 19 * q^50 + 29 * q^52 + 3 * q^56 - q^58 + 6 * q^61 - 38 * q^62 - 5 * q^64 + 14 * q^65 - 4 * q^67 + 42 * q^68 + 11 * q^70 + 12 * q^71 + 20 * q^73 + 29 * q^74 + 11 * q^76 - 12 * q^79 + 41 * q^80 - 6 * q^82 + 6 * q^83 + 6 * q^85 - 24 * q^86 - 26 * q^89 + 4 * q^91 - 26 * q^92 - 35 * q^94 - 8 * q^95 - 4 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more