LMFDB - Embedded newform 4550.2.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4550,2,Mod(1,4550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4550, 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("4550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4550 = 2 \cdot 5^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4550.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:	$36.3319329197$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 182)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4550.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} -3.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +6.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} -3.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +6.00000 q^{9} -5.00000 q^{11} -3.00000 q^{12} +1.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} +6.00000 q^{18} +2.00000 q^{19} +3.00000 q^{21} -5.00000 q^{22} -5.00000 q^{23} -3.00000 q^{24} +1.00000 q^{26} -9.00000 q^{27} -1.00000 q^{28} +4.00000 q^{29} +1.00000 q^{31} +1.00000 q^{32} +15.0000 q^{33} +4.00000 q^{34} +6.00000 q^{36} -7.00000 q^{37} +2.00000 q^{38} -3.00000 q^{39} -9.00000 q^{41} +3.00000 q^{42} +12.0000 q^{43} -5.00000 q^{44} -5.00000 q^{46} +7.00000 q^{47} -3.00000 q^{48} +1.00000 q^{49} -12.0000 q^{51} +1.00000 q^{52} +4.00000 q^{53} -9.00000 q^{54} -1.00000 q^{56} -6.00000 q^{57} +4.00000 q^{58} -6.00000 q^{59} +13.0000 q^{61} +1.00000 q^{62} -6.00000 q^{63} +1.00000 q^{64} +15.0000 q^{66} -11.0000 q^{67} +4.00000 q^{68} +15.0000 q^{69} +6.00000 q^{72} -7.00000 q^{73} -7.00000 q^{74} +2.00000 q^{76} +5.00000 q^{77} -3.00000 q^{78} -17.0000 q^{79} +9.00000 q^{81} -9.00000 q^{82} -4.00000 q^{83} +3.00000 q^{84} +12.0000 q^{86} -12.0000 q^{87} -5.00000 q^{88} +14.0000 q^{89} -1.00000 q^{91} -5.00000 q^{92} -3.00000 q^{93} +7.00000 q^{94} -3.00000 q^{96} -5.00000 q^{97} +1.00000 q^{98} -30.0000 q^{99} +O(q^{100})\)

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−3.00000	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$3$	−3.00000	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−3.00000	−1.22474
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	6.00000	2.00000
$10$	0	0
$11$	−5.00000	−1.50756	−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000	−1.50756	−0.753778	+	0.657129i	$0.771771\pi$
$12$	−3.00000	−0.866025
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	6.00000	1.41421
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	3.00000	0.654654
$22$	−5.00000	−1.06600
$23$	−5.00000	−1.04257	−0.521286	−	0.853382i	$-0.674548\pi$
$23$	−5.00000	−1.04257	−0.521286	+	0.853382i	$0.674548\pi$
$24$	−3.00000	−0.612372
$25$	0	0
$26$	1.00000	0.196116
$27$	−9.00000	−1.73205
$28$	−1.00000	−0.188982
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	0	0
$31$	1.00000	0.179605	0.0898027	−	0.995960i	$-0.471376\pi$
$31$	1.00000	0.179605	0.0898027	+	0.995960i	$0.471376\pi$
$32$	1.00000	0.176777
$33$	15.0000	2.61116
$34$	4.00000	0.685994
$35$	0	0
$36$	6.00000	1.00000
$37$	−7.00000	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$38$	2.00000	0.324443
$39$	−3.00000	−0.480384
$40$	0	0
$41$	−9.00000	−1.40556	−0.702782	−	0.711405i	$-0.748059\pi$
$41$	−9.00000	−1.40556	−0.702782	+	0.711405i	$0.748059\pi$
$42$	3.00000	0.462910
$43$	12.0000	1.82998	0.914991	−	0.403473i	$-0.132197\pi$
$43$	12.0000	1.82998	0.914991	+	0.403473i	$0.132197\pi$
$44$	−5.00000	−0.753778
$45$	0	0
$46$	−5.00000	−0.737210
$47$	7.00000	1.02105	0.510527	−	0.859861i	$-0.329450\pi$
$47$	7.00000	1.02105	0.510527	+	0.859861i	$0.329450\pi$
$48$	−3.00000	−0.433013
$49$	1.00000	0.142857
$50$	0	0
$51$	−12.0000	−1.68034
$52$	1.00000	0.138675
$53$	4.00000	0.549442	0.274721	−	0.961524i	$-0.411414\pi$
$53$	4.00000	0.549442	0.274721	+	0.961524i	$0.411414\pi$
$54$	−9.00000	−1.22474
$55$	0	0
$56$	−1.00000	−0.133631
$57$	−6.00000	−0.794719
$58$	4.00000	0.525226
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	13.0000	1.66448	0.832240	−	0.554416i	$-0.187058\pi$
$61$	13.0000	1.66448	0.832240	+	0.554416i	$0.187058\pi$
$62$	1.00000	0.127000
$63$	−6.00000	−0.755929
$64$	1.00000	0.125000
$65$	0	0
$66$	15.0000	1.84637
$67$	−11.0000	−1.34386	−0.671932	−	0.740613i	$-0.734535\pi$
$67$	−11.0000	−1.34386	−0.671932	+	0.740613i	$0.734535\pi$
$68$	4.00000	0.485071
$69$	15.0000	1.80579
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	6.00000	0.707107
$73$	−7.00000	−0.819288	−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−7.00000	−0.819288	−0.409644	+	0.912245i	$0.634347\pi$
$74$	−7.00000	−0.813733
$75$	0	0
$76$	2.00000	0.229416
$77$	5.00000	0.569803
$78$	−3.00000	−0.339683
$79$	−17.0000	−1.91265	−0.956325	−	0.292306i	$-0.905577\pi$
$79$	−17.0000	−1.91265	−0.956325	+	0.292306i	$0.905577\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	−9.00000	−0.993884
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	3.00000	0.327327
$85$	0	0
$86$	12.0000	1.29399
$87$	−12.0000	−1.28654
$88$	−5.00000	−0.533002
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	−5.00000	−0.521286
$93$	−3.00000	−0.311086
$94$	7.00000	0.721995
$95$	0	0
$96$	−3.00000	−0.306186
$97$	−5.00000	−0.507673	−0.253837	−	0.967247i	$-0.581693\pi$
$97$	−5.00000	−0.507673	−0.253837	+	0.967247i	$0.581693\pi$
$98$	1.00000	0.101015
$99$	−30.0000	−3.01511
$100$	0	0
$101$	15.0000	1.49256	0.746278	−	0.665635i	$-0.231839\pi$
$101$	15.0000	1.49256	0.746278	+	0.665635i	$0.231839\pi$
$102$	−12.0000	−1.18818
$103$	−6.00000	−0.591198	−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000	−0.591198	−0.295599	+	0.955312i	$0.595519\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	4.00000	0.388514
$107$	8.00000	0.773389	0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000	0.773389	0.386695	+	0.922208i	$0.373617\pi$
$108$	−9.00000	−0.866025
$109$	−18.0000	−1.72409	−0.862044	−	0.506834i	$-0.830816\pi$
$109$	−18.0000	−1.72409	−0.862044	+	0.506834i	$0.830816\pi$
$110$	0	0
$111$	21.0000	1.99323
$112$	−1.00000	−0.0944911
$113$	−1.00000	−0.0940721	−0.0470360	−	0.998893i	$-0.514978\pi$
$113$	−1.00000	−0.0940721	−0.0470360	+	0.998893i	$0.514978\pi$
$114$	−6.00000	−0.561951
$115$	0	0
$116$	4.00000	0.371391
$117$	6.00000	0.554700
$118$	−6.00000	−0.552345
$119$	−4.00000	−0.366679
$120$	0	0
$121$	14.0000	1.27273
$122$	13.0000	1.17696
$123$	27.0000	2.43451
$124$	1.00000	0.0898027
$125$	0	0
$126$	−6.00000	−0.534522
$127$	−9.00000	−0.798621	−0.399310	−	0.916816i	$-0.630750\pi$
$127$	−9.00000	−0.798621	−0.399310	+	0.916816i	$0.630750\pi$
$128$	1.00000	0.0883883
$129$	−36.0000	−3.16962
$130$	0	0
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	15.0000	1.30558
$133$	−2.00000	−0.173422
$134$	−11.0000	−0.950255
$135$	0	0
$136$	4.00000	0.342997
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	15.0000	1.27688
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	−21.0000	−1.76852
$142$	0	0
$143$	−5.00000	−0.418121
$144$	6.00000	0.500000
$145$	0	0
$146$	−7.00000	−0.579324
$147$	−3.00000	−0.247436
$148$	−7.00000	−0.575396
$149$	7.00000	0.573462	0.286731	−	0.958011i	$-0.407431\pi$
$149$	7.00000	0.573462	0.286731	+	0.958011i	$0.407431\pi$
$150$	0	0
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	2.00000	0.162221
$153$	24.0000	1.94029
$154$	5.00000	0.402911
$155$	0	0
$156$	−3.00000	−0.240192
$157$	−1.00000	−0.0798087	−0.0399043	−	0.999204i	$-0.512705\pi$
$157$	−1.00000	−0.0798087	−0.0399043	+	0.999204i	$0.512705\pi$
$158$	−17.0000	−1.35245
$159$	−12.0000	−0.951662
$160$	0	0
$161$	5.00000	0.394055
$162$	9.00000	0.707107
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	−9.00000	−0.702782
$165$	0	0
$166$	−4.00000	−0.310460
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	3.00000	0.231455
$169$	1.00000	0.0769231
$170$	0	0
$171$	12.0000	0.917663
$172$	12.0000	0.914991
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	−12.0000	−0.909718
$175$	0	0
$176$	−5.00000	−0.376889
$177$	18.0000	1.35296
$178$	14.0000	1.04934
$179$	−2.00000	−0.149487	−0.0747435	−	0.997203i	$-0.523814\pi$
$179$	−2.00000	−0.149487	−0.0747435	+	0.997203i	$0.523814\pi$
$180$	0	0
$181$	−5.00000	−0.371647	−0.185824	−	0.982583i	$-0.559495\pi$
$181$	−5.00000	−0.371647	−0.185824	+	0.982583i	$0.559495\pi$
$182$	−1.00000	−0.0741249
$183$	−39.0000	−2.88296
$184$	−5.00000	−0.368605
$185$	0	0
$186$	−3.00000	−0.219971
$187$	−20.0000	−1.46254
$188$	7.00000	0.510527
$189$	9.00000	0.654654
$190$	0	0
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	−3.00000	−0.216506
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	−5.00000	−0.358979
$195$	0	0
$196$	1.00000	0.0714286
$197$	−27.0000	−1.92367	−0.961835	−	0.273629i	$-0.911776\pi$
$197$	−27.0000	−1.92367	−0.961835	+	0.273629i	$0.911776\pi$
$198$	−30.0000	−2.13201
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	0	0
$201$	33.0000	2.32764
$202$	15.0000	1.05540
$203$	−4.00000	−0.280745
$204$	−12.0000	−0.840168
$205$	0	0
$206$	−6.00000	−0.418040
$207$	−30.0000	−2.08514
$208$	1.00000	0.0693375
$209$	−10.0000	−0.691714
$210$	0	0
$211$	−14.0000	−0.963800	−0.481900	−	0.876226i	$-0.660053\pi$
$211$	−14.0000	−0.963800	−0.481900	+	0.876226i	$0.660053\pi$
$212$	4.00000	0.274721
$213$	0	0
$214$	8.00000	0.546869
$215$	0	0
$216$	−9.00000	−0.612372
$217$	−1.00000	−0.0678844
$218$	−18.0000	−1.21911
$219$	21.0000	1.41905
$220$	0	0
$221$	4.00000	0.269069
$222$	21.0000	1.40943
$223$	−3.00000	−0.200895	−0.100447	−	0.994942i	$-0.532027\pi$
$223$	−3.00000	−0.200895	−0.100447	+	0.994942i	$0.532027\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−1.00000	−0.0665190
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	−6.00000	−0.397360
$229$	16.0000	1.05731	0.528655	−	0.848837i	$-0.322697\pi$
$229$	16.0000	1.05731	0.528655	+	0.848837i	$0.322697\pi$
$230$	0	0
$231$	−15.0000	−0.986928
$232$	4.00000	0.262613
$233$	21.0000	1.37576	0.687878	−	0.725826i	$-0.258542\pi$
$233$	21.0000	1.37576	0.687878	+	0.725826i	$0.258542\pi$
$234$	6.00000	0.392232
$235$	0	0
$236$	−6.00000	−0.390567
$237$	51.0000	3.31281
$238$	−4.00000	−0.259281
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	−30.0000	−1.93247	−0.966235	−	0.257663i	$-0.917048\pi$
$241$	−30.0000	−1.93247	−0.966235	+	0.257663i	$0.917048\pi$
$242$	14.0000	0.899954
$243$	0	0
$244$	13.0000	0.832240
$245$	0	0
$246$	27.0000	1.72146
$247$	2.00000	0.127257
$248$	1.00000	0.0635001
$249$	12.0000	0.760469
$250$	0	0
$251$	−13.0000	−0.820553	−0.410276	−	0.911961i	$-0.634568\pi$
$251$	−13.0000	−0.820553	−0.410276	+	0.911961i	$0.634568\pi$
$252$	−6.00000	−0.377964
$253$	25.0000	1.57174
$254$	−9.00000	−0.564710
$255$	0	0
$256$	1.00000	0.0625000
$257$	−28.0000	−1.74659	−0.873296	−	0.487190i	$-0.838022\pi$
$257$	−28.0000	−1.74659	−0.873296	+	0.487190i	$0.838022\pi$
$258$	−36.0000	−2.24126
$259$	7.00000	0.434959
$260$	0	0
$261$	24.0000	1.48556
$262$	8.00000	0.494242
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	15.0000	0.923186
$265$	0	0
$266$	−2.00000	−0.122628
$267$	−42.0000	−2.57036
$268$	−11.0000	−0.671932
$269$	13.0000	0.792624	0.396312	−	0.918116i	$-0.370290\pi$
$269$	13.0000	0.792624	0.396312	+	0.918116i	$0.370290\pi$
$270$	0	0
$271$	−1.00000	−0.0607457	−0.0303728	−	0.999539i	$-0.509669\pi$
$271$	−1.00000	−0.0607457	−0.0303728	+	0.999539i	$0.509669\pi$
$272$	4.00000	0.242536
$273$	3.00000	0.181568
$274$	−18.0000	−1.08742
$275$	0	0
$276$	15.0000	0.902894
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	4.00000	0.239904
$279$	6.00000	0.359211
$280$	0	0
$281$	−16.0000	−0.954480	−0.477240	−	0.878773i	$-0.658363\pi$
$281$	−16.0000	−0.954480	−0.477240	+	0.878773i	$0.658363\pi$
$282$	−21.0000	−1.25053
$283$	−3.00000	−0.178331	−0.0891657	−	0.996017i	$-0.528420\pi$
$283$	−3.00000	−0.178331	−0.0891657	+	0.996017i	$0.528420\pi$
$284$	0	0
$285$	0	0
$286$	−5.00000	−0.295656
$287$	9.00000	0.531253
$288$	6.00000	0.353553
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	15.0000	0.879316
$292$	−7.00000	−0.409644
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	−3.00000	−0.174964
$295$	0	0
$296$	−7.00000	−0.406867
$297$	45.0000	2.61116
$298$	7.00000	0.405499
$299$	−5.00000	−0.289157
$300$	0	0
$301$	−12.0000	−0.691669
$302$	−12.0000	−0.690522
$303$	−45.0000	−2.58518
$304$	2.00000	0.114708
$305$	0	0
$306$	24.0000	1.37199
$307$	−32.0000	−1.82634	−0.913168	−	0.407583i	$-0.866372\pi$
$307$	−32.0000	−1.82634	−0.913168	+	0.407583i	$0.866372\pi$
$308$	5.00000	0.284901
$309$	18.0000	1.02398
$310$	0	0
$311$	26.0000	1.47432	0.737162	−	0.675716i	$-0.236165\pi$
$311$	26.0000	1.47432	0.737162	+	0.675716i	$0.236165\pi$
$312$	−3.00000	−0.169842
$313$	30.0000	1.69570	0.847850	−	0.530236i	$-0.177897\pi$
$313$	30.0000	1.69570	0.847850	+	0.530236i	$0.177897\pi$
$314$	−1.00000	−0.0564333
$315$	0	0
$316$	−17.0000	−0.956325
$317$	−11.0000	−0.617822	−0.308911	−	0.951091i	$-0.599964\pi$
$317$	−11.0000	−0.617822	−0.308911	+	0.951091i	$0.599964\pi$
$318$	−12.0000	−0.672927
$319$	−20.0000	−1.11979
$320$	0	0
$321$	−24.0000	−1.33955
$322$	5.00000	0.278639
$323$	8.00000	0.445132
$324$	9.00000	0.500000
$325$	0	0
$326$	−4.00000	−0.221540
$327$	54.0000	2.98621
$328$	−9.00000	−0.496942
$329$	−7.00000	−0.385922
$330$	0	0
$331$	3.00000	0.164895	0.0824475	−	0.996595i	$-0.473726\pi$
$331$	3.00000	0.164895	0.0824475	+	0.996595i	$0.473726\pi$
$332$	−4.00000	−0.219529
$333$	−42.0000	−2.30159
$334$	−8.00000	−0.437741
$335$	0	0
$336$	3.00000	0.163663
$337$	−9.00000	−0.490261	−0.245131	−	0.969490i	$-0.578831\pi$
$337$	−9.00000	−0.490261	−0.245131	+	0.969490i	$0.578831\pi$
$338$	1.00000	0.0543928
$339$	3.00000	0.162938
$340$	0	0
$341$	−5.00000	−0.270765
$342$	12.0000	0.648886
$343$	−1.00000	−0.0539949
$344$	12.0000	0.646997
$345$	0	0
$346$	−18.0000	−0.967686
$347$	−16.0000	−0.858925	−0.429463	−	0.903085i	$-0.641297\pi$
$347$	−16.0000	−0.858925	−0.429463	+	0.903085i	$0.641297\pi$
$348$	−12.0000	−0.643268
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	0	0
$351$	−9.00000	−0.480384
$352$	−5.00000	−0.266501
$353$	−3.00000	−0.159674	−0.0798369	−	0.996808i	$-0.525440\pi$
$353$	−3.00000	−0.159674	−0.0798369	+	0.996808i	$0.525440\pi$
$354$	18.0000	0.956689
$355$	0	0
$356$	14.0000	0.741999
$357$	12.0000	0.635107
$358$	−2.00000	−0.105703
$359$	−4.00000	−0.211112	−0.105556	−	0.994413i	$-0.533662\pi$
$359$	−4.00000	−0.211112	−0.105556	+	0.994413i	$0.533662\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	−5.00000	−0.262794
$363$	−42.0000	−2.20443
$364$	−1.00000	−0.0524142
$365$	0	0
$366$	−39.0000	−2.03856
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	−5.00000	−0.260643
$369$	−54.0000	−2.81113
$370$	0	0
$371$	−4.00000	−0.207670
$372$	−3.00000	−0.155543
$373$	−4.00000	−0.207112	−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000	−0.207112	−0.103556	+	0.994624i	$0.533022\pi$
$374$	−20.0000	−1.03418
$375$	0	0
$376$	7.00000	0.360997
$377$	4.00000	0.206010
$378$	9.00000	0.462910
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	0	0
$381$	27.0000	1.38325
$382$	−16.0000	−0.818631
$383$	9.00000	0.459879	0.229939	−	0.973205i	$-0.426147\pi$
$383$	9.00000	0.459879	0.229939	+	0.973205i	$0.426147\pi$
$384$	−3.00000	−0.153093
$385$	0	0
$386$	−4.00000	−0.203595
$387$	72.0000	3.65997
$388$	−5.00000	−0.253837
$389$	8.00000	0.405616	0.202808	−	0.979219i	$-0.434993\pi$
$389$	8.00000	0.405616	0.202808	+	0.979219i	$0.434993\pi$
$390$	0	0
$391$	−20.0000	−1.01144
$392$	1.00000	0.0505076
$393$	−24.0000	−1.21064
$394$	−27.0000	−1.36024
$395$	0	0
$396$	−30.0000	−1.50756
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	−20.0000	−1.00251
$399$	6.00000	0.300376
$400$	0	0
$401$	−28.0000	−1.39825	−0.699127	−	0.714998i	$-0.746428\pi$
$401$	−28.0000	−1.39825	−0.699127	+	0.714998i	$0.746428\pi$
$402$	33.0000	1.64589
$403$	1.00000	0.0498135
$404$	15.0000	0.746278
$405$	0	0
$406$	−4.00000	−0.198517
$407$	35.0000	1.73489
$408$	−12.0000	−0.594089
$409$	−6.00000	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$410$	0	0
$411$	54.0000	2.66362
$412$	−6.00000	−0.295599
$413$	6.00000	0.295241
$414$	−30.0000	−1.47442
$415$	0	0
$416$	1.00000	0.0490290
$417$	−12.0000	−0.587643
$418$	−10.0000	−0.489116
$419$	−9.00000	−0.439679	−0.219839	−	0.975536i	$-0.570553\pi$
$419$	−9.00000	−0.439679	−0.219839	+	0.975536i	$0.570553\pi$
$420$	0	0
$421$	−35.0000	−1.70580	−0.852898	−	0.522078i	$-0.825157\pi$
$421$	−35.0000	−1.70580	−0.852898	+	0.522078i	$0.825157\pi$
$422$	−14.0000	−0.681509
$423$	42.0000	2.04211
$424$	4.00000	0.194257
$425$	0	0
$426$	0	0
$427$	−13.0000	−0.629114
$428$	8.00000	0.386695
$429$	15.0000	0.724207
$430$	0	0
$431$	−6.00000	−0.289010	−0.144505	−	0.989504i	$-0.546159\pi$
$431$	−6.00000	−0.289010	−0.144505	+	0.989504i	$0.546159\pi$
$432$	−9.00000	−0.433013
$433$	−26.0000	−1.24948	−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000	−1.24948	−0.624740	+	0.780833i	$0.714795\pi$
$434$	−1.00000	−0.0480015
$435$	0	0
$436$	−18.0000	−0.862044
$437$	−10.0000	−0.478365
$438$	21.0000	1.00342
$439$	−2.00000	−0.0954548	−0.0477274	−	0.998860i	$-0.515198\pi$
$439$	−2.00000	−0.0954548	−0.0477274	+	0.998860i	$0.515198\pi$
$440$	0	0
$441$	6.00000	0.285714
$442$	4.00000	0.190261
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	21.0000	0.996616
$445$	0	0
$446$	−3.00000	−0.142054
$447$	−21.0000	−0.993266
$448$	−1.00000	−0.0472456
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	0	0
$451$	45.0000	2.11897
$452$	−1.00000	−0.0470360
$453$	36.0000	1.69143
$454$	12.0000	0.563188
$455$	0	0
$456$	−6.00000	−0.280976
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	16.0000	0.747631
$459$	−36.0000	−1.68034
$460$	0	0
$461$	12.0000	0.558896	0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000	0.558896	0.279448	+	0.960161i	$0.409849\pi$
$462$	−15.0000	−0.697863
$463$	−12.0000	−0.557687	−0.278844	−	0.960337i	$-0.589951\pi$
$463$	−12.0000	−0.557687	−0.278844	+	0.960337i	$0.589951\pi$
$464$	4.00000	0.185695
$465$	0	0
$466$	21.0000	0.972806
$467$	36.0000	1.66588	0.832941	−	0.553362i	$-0.186655\pi$
$467$	36.0000	1.66588	0.832941	+	0.553362i	$0.186655\pi$
$468$	6.00000	0.277350
$469$	11.0000	0.507933
$470$	0	0
$471$	3.00000	0.138233
$472$	−6.00000	−0.276172
$473$	−60.0000	−2.75880
$474$	51.0000	2.34251
$475$	0	0
$476$	−4.00000	−0.183340
$477$	24.0000	1.09888
$478$	−6.00000	−0.274434
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	0	0
$481$	−7.00000	−0.319173
$482$	−30.0000	−1.36646
$483$	−15.0000	−0.682524
$484$	14.0000	0.636364
$485$	0	0
$486$	0	0
$487$	26.0000	1.17817	0.589086	−	0.808070i	$-0.299488\pi$
$487$	26.0000	1.17817	0.589086	+	0.808070i	$0.299488\pi$
$488$	13.0000	0.588482
$489$	12.0000	0.542659
$490$	0	0
$491$	28.0000	1.26362	0.631811	−	0.775122i	$-0.282312\pi$
$491$	28.0000	1.26362	0.631811	+	0.775122i	$0.282312\pi$
$492$	27.0000	1.21725
$493$	16.0000	0.720604
$494$	2.00000	0.0899843
$495$	0	0
$496$	1.00000	0.0449013
$497$	0	0
$498$	12.0000	0.537733
$499$	1.00000	0.0447661	0.0223831	−	0.999749i	$-0.492875\pi$
$499$	1.00000	0.0447661	0.0223831	+	0.999749i	$0.492875\pi$
$500$	0	0
$501$	24.0000	1.07224
$502$	−13.0000	−0.580218
$503$	40.0000	1.78351	0.891756	−	0.452517i	$-0.149474\pi$
$503$	40.0000	1.78351	0.891756	+	0.452517i	$0.149474\pi$
$504$	−6.00000	−0.267261
$505$	0	0
$506$	25.0000	1.11139
$507$	−3.00000	−0.133235
$508$	−9.00000	−0.399310
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	0	0
$511$	7.00000	0.309662
$512$	1.00000	0.0441942
$513$	−18.0000	−0.794719
$514$	−28.0000	−1.23503
$515$	0	0
$516$	−36.0000	−1.58481
$517$	−35.0000	−1.53930
$518$	7.00000	0.307562
$519$	54.0000	2.37034
$520$	0	0
$521$	−14.0000	−0.613351	−0.306676	−	0.951814i	$-0.599217\pi$
$521$	−14.0000	−0.613351	−0.306676	+	0.951814i	$0.599217\pi$
$522$	24.0000	1.05045
$523$	13.0000	0.568450	0.284225	−	0.958758i	$-0.408264\pi$
$523$	13.0000	0.568450	0.284225	+	0.958758i	$0.408264\pi$
$524$	8.00000	0.349482
$525$	0	0
$526$	0	0
$527$	4.00000	0.174243
$528$	15.0000	0.652791
$529$	2.00000	0.0869565
$530$	0	0
$531$	−36.0000	−1.56227
$532$	−2.00000	−0.0867110
$533$	−9.00000	−0.389833
$534$	−42.0000	−1.81752
$535$	0	0
$536$	−11.0000	−0.475128
$537$	6.00000	0.258919
$538$	13.0000	0.560470
$539$	−5.00000	−0.215365
$540$	0	0
$541$	18.0000	0.773880	0.386940	−	0.922105i	$-0.373532\pi$
$541$	18.0000	0.773880	0.386940	+	0.922105i	$0.373532\pi$
$542$	−1.00000	−0.0429537
$543$	15.0000	0.643712
$544$	4.00000	0.171499
$545$	0	0
$546$	3.00000	0.128388
$547$	32.0000	1.36822	0.684111	−	0.729378i	$-0.260191\pi$
$547$	32.0000	1.36822	0.684111	+	0.729378i	$0.260191\pi$
$548$	−18.0000	−0.768922
$549$	78.0000	3.32896
$550$	0	0
$551$	8.00000	0.340811
$552$	15.0000	0.638442
$553$	17.0000	0.722914
$554$	10.0000	0.424859
$555$	0	0
$556$	4.00000	0.169638
$557$	9.00000	0.381342	0.190671	−	0.981654i	$-0.438934\pi$
$557$	9.00000	0.381342	0.190671	+	0.981654i	$0.438934\pi$
$558$	6.00000	0.254000
$559$	12.0000	0.507546
$560$	0	0
$561$	60.0000	2.53320
$562$	−16.0000	−0.674919
$563$	−35.0000	−1.47507	−0.737537	−	0.675307i	$-0.764011\pi$
$563$	−35.0000	−1.47507	−0.737537	+	0.675307i	$0.764011\pi$
$564$	−21.0000	−0.884260
$565$	0	0
$566$	−3.00000	−0.126099
$567$	−9.00000	−0.377964
$568$	0	0
$569$	21.0000	0.880366	0.440183	−	0.897908i	$-0.354914\pi$
$569$	21.0000	0.880366	0.440183	+	0.897908i	$0.354914\pi$
$570$	0	0
$571$	22.0000	0.920671	0.460336	−	0.887745i	$-0.347729\pi$
$571$	22.0000	0.920671	0.460336	+	0.887745i	$0.347729\pi$
$572$	−5.00000	−0.209061
$573$	48.0000	2.00523
$574$	9.00000	0.375653
$575$	0	0
$576$	6.00000	0.250000
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	−1.00000	−0.0415945
$579$	12.0000	0.498703
$580$	0	0
$581$	4.00000	0.165948
$582$	15.0000	0.621770
$583$	−20.0000	−0.828315
$584$	−7.00000	−0.289662
$585$	0	0
$586$	6.00000	0.247858
$587$	−18.0000	−0.742940	−0.371470	−	0.928445i	$-0.621146\pi$
$587$	−18.0000	−0.742940	−0.371470	+	0.928445i	$0.621146\pi$
$588$	−3.00000	−0.123718
$589$	2.00000	0.0824086
$590$	0	0
$591$	81.0000	3.33189
$592$	−7.00000	−0.287698
$593$	46.0000	1.88899	0.944497	−	0.328521i	$-0.106550\pi$
$593$	46.0000	1.88899	0.944497	+	0.328521i	$0.106550\pi$
$594$	45.0000	1.84637
$595$	0	0
$596$	7.00000	0.286731
$597$	60.0000	2.45564
$598$	−5.00000	−0.204465
$599$	−9.00000	−0.367730	−0.183865	−	0.982952i	$-0.558861\pi$
$599$	−9.00000	−0.367730	−0.183865	+	0.982952i	$0.558861\pi$
$600$	0	0
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	−12.0000	−0.489083
$603$	−66.0000	−2.68773
$604$	−12.0000	−0.488273
$605$	0	0
$606$	−45.0000	−1.82800
$607$	−22.0000	−0.892952	−0.446476	−	0.894795i	$-0.647321\pi$
$607$	−22.0000	−0.892952	−0.446476	+	0.894795i	$0.647321\pi$
$608$	2.00000	0.0811107
$609$	12.0000	0.486265
$610$	0	0
$611$	7.00000	0.283190
$612$	24.0000	0.970143
$613$	23.0000	0.928961	0.464481	−	0.885583i	$-0.346241\pi$
$613$	23.0000	0.928961	0.464481	+	0.885583i	$0.346241\pi$
$614$	−32.0000	−1.29141
$615$	0	0
$616$	5.00000	0.201456
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	18.0000	0.724066
$619$	6.00000	0.241160	0.120580	−	0.992704i	$-0.461525\pi$
$619$	6.00000	0.241160	0.120580	+	0.992704i	$0.461525\pi$
$620$	0	0
$621$	45.0000	1.80579
$622$	26.0000	1.04251
$623$	−14.0000	−0.560898
$624$	−3.00000	−0.120096
$625$	0	0
$626$	30.0000	1.19904
$627$	30.0000	1.19808
$628$	−1.00000	−0.0399043
$629$	−28.0000	−1.11643
$630$	0	0
$631$	−26.0000	−1.03504	−0.517522	−	0.855670i	$-0.673145\pi$
$631$	−26.0000	−1.03504	−0.517522	+	0.855670i	$0.673145\pi$
$632$	−17.0000	−0.676224
$633$	42.0000	1.66935
$634$	−11.0000	−0.436866
$635$	0	0
$636$	−12.0000	−0.475831
$637$	1.00000	0.0396214
$638$	−20.0000	−0.791808
$639$	0	0
$640$	0	0
$641$	−33.0000	−1.30342	−0.651711	−	0.758468i	$-0.725948\pi$
$641$	−33.0000	−1.30342	−0.651711	+	0.758468i	$0.725948\pi$
$642$	−24.0000	−0.947204
$643$	36.0000	1.41970	0.709851	−	0.704352i	$-0.248762\pi$
$643$	36.0000	1.41970	0.709851	+	0.704352i	$0.248762\pi$
$644$	5.00000	0.197028
$645$	0	0
$646$	8.00000	0.314756
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	9.00000	0.353553
$649$	30.0000	1.17760
$650$	0	0
$651$	3.00000	0.117579
$652$	−4.00000	−0.156652
$653$	12.0000	0.469596	0.234798	−	0.972044i	$-0.424557\pi$
$653$	12.0000	0.469596	0.234798	+	0.972044i	$0.424557\pi$
$654$	54.0000	2.11157
$655$	0	0
$656$	−9.00000	−0.351391
$657$	−42.0000	−1.63858
$658$	−7.00000	−0.272888
$659$	26.0000	1.01282	0.506408	−	0.862294i	$-0.330973\pi$
$659$	26.0000	1.01282	0.506408	+	0.862294i	$0.330973\pi$
$660$	0	0
$661$	−20.0000	−0.777910	−0.388955	−	0.921257i	$-0.627164\pi$
$661$	−20.0000	−0.777910	−0.388955	+	0.921257i	$0.627164\pi$
$662$	3.00000	0.116598
$663$	−12.0000	−0.466041
$664$	−4.00000	−0.155230
$665$	0	0
$666$	−42.0000	−1.62747
$667$	−20.0000	−0.774403
$668$	−8.00000	−0.309529
$669$	9.00000	0.347960
$670$	0	0
$671$	−65.0000	−2.50930
$672$	3.00000	0.115728
$673$	9.00000	0.346925	0.173462	−	0.984841i	$-0.444505\pi$
$673$	9.00000	0.346925	0.173462	+	0.984841i	$0.444505\pi$
$674$	−9.00000	−0.346667
$675$	0	0
$676$	1.00000	0.0384615
$677$	−3.00000	−0.115299	−0.0576497	−	0.998337i	$-0.518361\pi$
$677$	−3.00000	−0.115299	−0.0576497	+	0.998337i	$0.518361\pi$
$678$	3.00000	0.115214
$679$	5.00000	0.191882
$680$	0	0
$681$	−36.0000	−1.37952
$682$	−5.00000	−0.191460
$683$	−3.00000	−0.114792	−0.0573959	−	0.998351i	$-0.518280\pi$
$683$	−3.00000	−0.114792	−0.0573959	+	0.998351i	$0.518280\pi$
$684$	12.0000	0.458831
$685$	0	0
$686$	−1.00000	−0.0381802
$687$	−48.0000	−1.83131
$688$	12.0000	0.457496
$689$	4.00000	0.152388
$690$	0	0
$691$	−36.0000	−1.36950	−0.684752	−	0.728776i	$-0.740090\pi$
$691$	−36.0000	−1.36950	−0.684752	+	0.728776i	$0.740090\pi$
$692$	−18.0000	−0.684257
$693$	30.0000	1.13961
$694$	−16.0000	−0.607352
$695$	0	0
$696$	−12.0000	−0.454859
$697$	−36.0000	−1.36360
$698$	2.00000	0.0757011
$699$	−63.0000	−2.38288
$700$	0	0
$701$	−12.0000	−0.453234	−0.226617	−	0.973984i	$-0.572767\pi$
$701$	−12.0000	−0.453234	−0.226617	+	0.973984i	$0.572767\pi$
$702$	−9.00000	−0.339683
$703$	−14.0000	−0.528020
$704$	−5.00000	−0.188445
$705$	0	0
$706$	−3.00000	−0.112906
$707$	−15.0000	−0.564133
$708$	18.0000	0.676481
$709$	29.0000	1.08912	0.544559	−	0.838723i	$-0.316697\pi$
$709$	29.0000	1.08912	0.544559	+	0.838723i	$0.316697\pi$
$710$	0	0
$711$	−102.000	−3.82530
$712$	14.0000	0.524672
$713$	−5.00000	−0.187251
$714$	12.0000	0.449089
$715$	0	0
$716$	−2.00000	−0.0747435
$717$	18.0000	0.672222
$718$	−4.00000	−0.149279
$719$	−18.0000	−0.671287	−0.335643	−	0.941989i	$-0.608954\pi$
$719$	−18.0000	−0.671287	−0.335643	+	0.941989i	$0.608954\pi$
$720$	0	0
$721$	6.00000	0.223452
$722$	−15.0000	−0.558242
$723$	90.0000	3.34714
$724$	−5.00000	−0.185824
$725$	0	0
$726$	−42.0000	−1.55877
$727$	−14.0000	−0.519231	−0.259616	−	0.965712i	$-0.583596\pi$
$727$	−14.0000	−0.519231	−0.259616	+	0.965712i	$0.583596\pi$
$728$	−1.00000	−0.0370625
$729$	−27.0000	−1.00000
$730$	0	0
$731$	48.0000	1.77534
$732$	−39.0000	−1.44148
$733$	−20.0000	−0.738717	−0.369358	−	0.929287i	$-0.620423\pi$
$733$	−20.0000	−0.738717	−0.369358	+	0.929287i	$0.620423\pi$
$734$	−8.00000	−0.295285
$735$	0	0
$736$	−5.00000	−0.184302
$737$	55.0000	2.02595
$738$	−54.0000	−1.98777
$739$	−16.0000	−0.588570	−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000	−0.588570	−0.294285	+	0.955718i	$0.595081\pi$
$740$	0	0
$741$	−6.00000	−0.220416
$742$	−4.00000	−0.146845
$743$	36.0000	1.32071	0.660356	−	0.750953i	$-0.270405\pi$
$743$	36.0000	1.32071	0.660356	+	0.750953i	$0.270405\pi$
$744$	−3.00000	−0.109985
$745$	0	0
$746$	−4.00000	−0.146450
$747$	−24.0000	−0.878114
$748$	−20.0000	−0.731272
$749$	−8.00000	−0.292314
$750$	0	0
$751$	25.0000	0.912263	0.456131	−	0.889912i	$-0.349235\pi$
$751$	25.0000	0.912263	0.456131	+	0.889912i	$0.349235\pi$
$752$	7.00000	0.255264
$753$	39.0000	1.42124
$754$	4.00000	0.145671
$755$	0	0
$756$	9.00000	0.327327
$757$	18.0000	0.654221	0.327111	−	0.944986i	$-0.393925\pi$
$757$	18.0000	0.654221	0.327111	+	0.944986i	$0.393925\pi$
$758$	−16.0000	−0.581146
$759$	−75.0000	−2.72233
$760$	0	0
$761$	−33.0000	−1.19625	−0.598125	−	0.801403i	$-0.704087\pi$
$761$	−33.0000	−1.19625	−0.598125	+	0.801403i	$0.704087\pi$
$762$	27.0000	0.978107
$763$	18.0000	0.651644
$764$	−16.0000	−0.578860
$765$	0	0
$766$	9.00000	0.325183
$767$	−6.00000	−0.216647
$768$	−3.00000	−0.108253
$769$	−1.00000	−0.0360609	−0.0180305	−	0.999837i	$-0.505740\pi$
$769$	−1.00000	−0.0360609	−0.0180305	+	0.999837i	$0.505740\pi$
$770$	0	0
$771$	84.0000	3.02519
$772$	−4.00000	−0.143963
$773$	−34.0000	−1.22290	−0.611448	−	0.791285i	$-0.709412\pi$
$773$	−34.0000	−1.22290	−0.611448	+	0.791285i	$0.709412\pi$
$774$	72.0000	2.58799
$775$	0	0
$776$	−5.00000	−0.179490
$777$	−21.0000	−0.753371
$778$	8.00000	0.286814
$779$	−18.0000	−0.644917
$780$	0	0
$781$	0	0
$782$	−20.0000	−0.715199
$783$	−36.0000	−1.28654
$784$	1.00000	0.0357143
$785$	0	0
$786$	−24.0000	−0.856052
$787$	−8.00000	−0.285169	−0.142585	−	0.989783i	$-0.545541\pi$
$787$	−8.00000	−0.285169	−0.142585	+	0.989783i	$0.545541\pi$
$788$	−27.0000	−0.961835
$789$	0	0
$790$	0	0
$791$	1.00000	0.0355559
$792$	−30.0000	−1.06600
$793$	13.0000	0.461644
$794$	0	0
$795$	0	0
$796$	−20.0000	−0.708881
$797$	−3.00000	−0.106265	−0.0531327	−	0.998587i	$-0.516921\pi$
$797$	−3.00000	−0.106265	−0.0531327	+	0.998587i	$0.516921\pi$
$798$	6.00000	0.212398
$799$	28.0000	0.990569
$800$	0	0
$801$	84.0000	2.96799
$802$	−28.0000	−0.988714
$803$	35.0000	1.23512
$804$	33.0000	1.16382
$805$	0	0
$806$	1.00000	0.0352235
$807$	−39.0000	−1.37287
$808$	15.0000	0.527698
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	52.0000	1.82597	0.912983	−	0.407997i	$-0.133772\pi$
$811$	52.0000	1.82597	0.912983	+	0.407997i	$0.133772\pi$
$812$	−4.00000	−0.140372
$813$	3.00000	0.105215
$814$	35.0000	1.22675
$815$	0	0
$816$	−12.0000	−0.420084
$817$	24.0000	0.839654
$818$	−6.00000	−0.209785
$819$	−6.00000	−0.209657
$820$	0	0
$821$	6.00000	0.209401	0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	54.0000	1.88347
$823$	15.0000	0.522867	0.261434	−	0.965221i	$-0.415805\pi$
$823$	15.0000	0.522867	0.261434	+	0.965221i	$0.415805\pi$
$824$	−6.00000	−0.209020
$825$	0	0
$826$	6.00000	0.208767
$827$	36.0000	1.25184	0.625921	−	0.779886i	$-0.284723\pi$
$827$	36.0000	1.25184	0.625921	+	0.779886i	$0.284723\pi$
$828$	−30.0000	−1.04257
$829$	−54.0000	−1.87550	−0.937749	−	0.347314i	$-0.887094\pi$
$829$	−54.0000	−1.87550	−0.937749	+	0.347314i	$0.887094\pi$
$830$	0	0
$831$	−30.0000	−1.04069
$832$	1.00000	0.0346688
$833$	4.00000	0.138592
$834$	−12.0000	−0.415526
$835$	0	0
$836$	−10.0000	−0.345857
$837$	−9.00000	−0.311086
$838$	−9.00000	−0.310900
$839$	47.0000	1.62262	0.811310	−	0.584616i	$-0.198755\pi$
$839$	47.0000	1.62262	0.811310	+	0.584616i	$0.198755\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	−35.0000	−1.20618
$843$	48.0000	1.65321
$844$	−14.0000	−0.481900
$845$	0	0
$846$	42.0000	1.44399
$847$	−14.0000	−0.481046
$848$	4.00000	0.137361
$849$	9.00000	0.308879
$850$	0	0
$851$	35.0000	1.19978
$852$	0	0
$853$	16.0000	0.547830	0.273915	−	0.961754i	$-0.411681\pi$
$853$	16.0000	0.547830	0.273915	+	0.961754i	$0.411681\pi$
$854$	−13.0000	−0.444851
$855$	0	0
$856$	8.00000	0.273434
$857$	−2.00000	−0.0683187	−0.0341593	−	0.999416i	$-0.510875\pi$
$857$	−2.00000	−0.0683187	−0.0341593	+	0.999416i	$0.510875\pi$
$858$	15.0000	0.512092
$859$	29.0000	0.989467	0.494734	−	0.869045i	$-0.335266\pi$
$859$	29.0000	0.989467	0.494734	+	0.869045i	$0.335266\pi$
$860$	0	0
$861$	−27.0000	−0.920158
$862$	−6.00000	−0.204361
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	−9.00000	−0.306186
$865$	0	0
$866$	−26.0000	−0.883516
$867$	3.00000	0.101885
$868$	−1.00000	−0.0339422
$869$	85.0000	2.88343
$870$	0	0
$871$	−11.0000	−0.372721
$872$	−18.0000	−0.609557
$873$	−30.0000	−1.01535
$874$	−10.0000	−0.338255
$875$	0	0
$876$	21.0000	0.709524
$877$	−13.0000	−0.438979	−0.219489	−	0.975615i	$-0.570439\pi$
$877$	−13.0000	−0.438979	−0.219489	+	0.975615i	$0.570439\pi$
$878$	−2.00000	−0.0674967
$879$	−18.0000	−0.607125
$880$	0	0
$881$	6.00000	0.202145	0.101073	−	0.994879i	$-0.467773\pi$
$881$	6.00000	0.202145	0.101073	+	0.994879i	$0.467773\pi$
$882$	6.00000	0.202031
$883$	−30.0000	−1.00958	−0.504790	−	0.863242i	$-0.668430\pi$
$883$	−30.0000	−1.00958	−0.504790	+	0.863242i	$0.668430\pi$
$884$	4.00000	0.134535
$885$	0	0
$886$	12.0000	0.403148
$887$	−54.0000	−1.81314	−0.906571	−	0.422053i	$-0.861310\pi$
$887$	−54.0000	−1.81314	−0.906571	+	0.422053i	$0.861310\pi$
$888$	21.0000	0.704714
$889$	9.00000	0.301850
$890$	0	0
$891$	−45.0000	−1.50756
$892$	−3.00000	−0.100447
$893$	14.0000	0.468492
$894$	−21.0000	−0.702345
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	15.0000	0.500835
$898$	18.0000	0.600668
$899$	4.00000	0.133407
$900$	0	0
$901$	16.0000	0.533037
$902$	45.0000	1.49834
$903$	36.0000	1.19800
$904$	−1.00000	−0.0332595
$905$	0	0
$906$	36.0000	1.19602
$907$	−14.0000	−0.464862	−0.232431	−	0.972613i	$-0.574668\pi$
$907$	−14.0000	−0.464862	−0.232431	+	0.972613i	$0.574668\pi$
$908$	12.0000	0.398234
$909$	90.0000	2.98511
$910$	0	0
$911$	28.0000	0.927681	0.463841	−	0.885919i	$-0.346471\pi$
$911$	28.0000	0.927681	0.463841	+	0.885919i	$0.346471\pi$
$912$	−6.00000	−0.198680
$913$	20.0000	0.661903
$914$	10.0000	0.330771
$915$	0	0
$916$	16.0000	0.528655
$917$	−8.00000	−0.264183
$918$	−36.0000	−1.18818
$919$	31.0000	1.02260	0.511298	−	0.859404i	$-0.329165\pi$
$919$	31.0000	1.02260	0.511298	+	0.859404i	$0.329165\pi$
$920$	0	0
$921$	96.0000	3.16331
$922$	12.0000	0.395199
$923$	0	0
$924$	−15.0000	−0.493464
$925$	0	0
$926$	−12.0000	−0.394344
$927$	−36.0000	−1.18240
$928$	4.00000	0.131306
$929$	−29.0000	−0.951459	−0.475730	−	0.879592i	$-0.657816\pi$
$929$	−29.0000	−0.951459	−0.475730	+	0.879592i	$0.657816\pi$
$930$	0	0
$931$	2.00000	0.0655474
$932$	21.0000	0.687878
$933$	−78.0000	−2.55361
$934$	36.0000	1.17796
$935$	0	0
$936$	6.00000	0.196116
$937$	8.00000	0.261349	0.130674	−	0.991425i	$-0.458286\pi$
$937$	8.00000	0.261349	0.130674	+	0.991425i	$0.458286\pi$
$938$	11.0000	0.359163
$939$	−90.0000	−2.93704
$940$	0	0
$941$	6.00000	0.195594	0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000	0.195594	0.0977972	+	0.995206i	$0.468820\pi$
$942$	3.00000	0.0977453
$943$	45.0000	1.46540
$944$	−6.00000	−0.195283
$945$	0	0
$946$	−60.0000	−1.95077
$947$	16.0000	0.519930	0.259965	−	0.965618i	$-0.416289\pi$
$947$	16.0000	0.519930	0.259965	+	0.965618i	$0.416289\pi$
$948$	51.0000	1.65640
$949$	−7.00000	−0.227230
$950$	0	0
$951$	33.0000	1.07010
$952$	−4.00000	−0.129641
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	24.0000	0.777029
$955$	0	0
$956$	−6.00000	−0.194054
$957$	60.0000	1.93952
$958$	−16.0000	−0.516937
$959$	18.0000	0.581250
$960$	0	0
$961$	−30.0000	−0.967742
$962$	−7.00000	−0.225689
$963$	48.0000	1.54678
$964$	−30.0000	−0.966235
$965$	0	0
$966$	−15.0000	−0.482617
$967$	−2.00000	−0.0643157	−0.0321578	−	0.999483i	$-0.510238\pi$
$967$	−2.00000	−0.0643157	−0.0321578	+	0.999483i	$0.510238\pi$
$968$	14.0000	0.449977
$969$	−24.0000	−0.770991
$970$	0	0
$971$	9.00000	0.288824	0.144412	−	0.989518i	$-0.453871\pi$
$971$	9.00000	0.288824	0.144412	+	0.989518i	$0.453871\pi$
$972$	0	0
$973$	−4.00000	−0.128234
$974$	26.0000	0.833094
$975$	0	0
$976$	13.0000	0.416120
$977$	26.0000	0.831814	0.415907	−	0.909407i	$-0.363464\pi$
$977$	26.0000	0.831814	0.415907	+	0.909407i	$0.363464\pi$
$978$	12.0000	0.383718
$979$	−70.0000	−2.23721
$980$	0	0
$981$	−108.000	−3.44817
$982$	28.0000	0.893516
$983$	56.0000	1.78612	0.893061	−	0.449935i	$-0.148553\pi$
$983$	56.0000	1.78612	0.893061	+	0.449935i	$0.148553\pi$
$984$	27.0000	0.860729
$985$	0	0
$986$	16.0000	0.509544
$987$	21.0000	0.668437
$988$	2.00000	0.0636285
$989$	−60.0000	−1.90789
$990$	0	0
$991$	59.0000	1.87420	0.937098	−	0.349065i	$-0.113501\pi$
$991$	59.0000	1.87420	0.937098	+	0.349065i	$0.113501\pi$
$992$	1.00000	0.0317500
$993$	−9.00000	−0.285606
$994$	0	0
$995$	0	0
$996$	12.0000	0.380235
$997$	−33.0000	−1.04512	−0.522560	−	0.852602i	$-0.675023\pi$
$997$	−33.0000	−1.04512	−0.522560	+	0.852602i	$0.675023\pi$
$998$	1.00000	0.0316544
$999$	63.0000	1.99323

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4550.2.a.o.1.1		1
5.4	even	2		182.2.a.b.1.1	✓	1
15.14	odd	2		1638.2.a.q.1.1		1
20.19	odd	2		1456.2.a.b.1.1		1
35.4	even	6		1274.2.f.m.79.1		2
35.9	even	6		1274.2.f.m.1145.1		2
35.19	odd	6		1274.2.f.u.1145.1		2
35.24	odd	6		1274.2.f.u.79.1		2
35.34	odd	2		1274.2.a.a.1.1		1
40.19	odd	2		5824.2.a.be.1.1		1
40.29	even	2		5824.2.a.a.1.1		1
65.34	odd	4		2366.2.d.i.337.1		2
65.44	odd	4		2366.2.d.i.337.2		2
65.64	even	2		2366.2.a.o.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
182.2.a.b.1.1	✓	1	5.4	even	2
1274.2.a.a.1.1		1	35.34	odd	2
1274.2.f.m.79.1		2	35.4	even	6
1274.2.f.m.1145.1		2	35.9	even	6
1274.2.f.u.79.1		2	35.24	odd	6
1274.2.f.u.1145.1		2	35.19	odd	6
1456.2.a.b.1.1		1	20.19	odd	2
1638.2.a.q.1.1		1	15.14	odd	2
2366.2.a.o.1.1		1	65.64	even	2
2366.2.d.i.337.1		2	65.34	odd	4
2366.2.d.i.337.2		2	65.44	odd	4
4550.2.a.o.1.1		1	1.1	even	1	trivial
5824.2.a.a.1.1		1	40.29	even	2
5824.2.a.be.1.1		1	40.19	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:	\( N \)	\(=\)	\( 4550 = 2 \cdot 5^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4550.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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