LMFDB - Embedded newform 5550.2.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5550,2,Mod(1,5550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5550, 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("5550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5550 = 2 \cdot 3 \cdot 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5550.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:	$44.3169731218$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1110)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	5550.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{12} -1.00000 q^{13} +3.00000 q^{14} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} -5.00000 q^{19} -3.00000 q^{21} -1.00000 q^{22} -7.00000 q^{23} -1.00000 q^{24} +1.00000 q^{26} +1.00000 q^{27} -3.00000 q^{28} -2.00000 q^{29} +2.00000 q^{31} -1.00000 q^{32} +1.00000 q^{33} -1.00000 q^{34} +1.00000 q^{36} -1.00000 q^{37} +5.00000 q^{38} -1.00000 q^{39} +4.00000 q^{41} +3.00000 q^{42} +12.0000 q^{43} +1.00000 q^{44} +7.00000 q^{46} -12.0000 q^{47} +1.00000 q^{48} +2.00000 q^{49} +1.00000 q^{51} -1.00000 q^{52} +9.00000 q^{53} -1.00000 q^{54} +3.00000 q^{56} -5.00000 q^{57} +2.00000 q^{58} +10.0000 q^{59} +14.0000 q^{61} -2.00000 q^{62} -3.00000 q^{63} +1.00000 q^{64} -1.00000 q^{66} -2.00000 q^{67} +1.00000 q^{68} -7.00000 q^{69} +2.00000 q^{71} -1.00000 q^{72} +9.00000 q^{73} +1.00000 q^{74} -5.00000 q^{76} -3.00000 q^{77} +1.00000 q^{78} +4.00000 q^{79} +1.00000 q^{81} -4.00000 q^{82} +1.00000 q^{83} -3.00000 q^{84} -12.0000 q^{86} -2.00000 q^{87} -1.00000 q^{88} +1.00000 q^{89} +3.00000 q^{91} -7.00000 q^{92} +2.00000 q^{93} +12.0000 q^{94} -1.00000 q^{96} +8.00000 q^{97} -2.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	1.00000	0.288675
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	3.00000	0.801784
$15$	0	0
$16$	1.00000	0.250000
$17$	1.00000	0.242536	0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000	0.242536	0.121268	+	0.992620i	$0.461304\pi$
$18$	−1.00000	−0.235702
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	0	0
$21$	−3.00000	−0.654654
$22$	−1.00000	−0.213201
$23$	−7.00000	−1.45960	−0.729800	−	0.683660i	$-0.760387\pi$
$23$	−7.00000	−1.45960	−0.729800	+	0.683660i	$0.760387\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	1.00000	0.196116
$27$	1.00000	0.192450
$28$	−3.00000	−0.566947
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	−1.00000	−0.176777
$33$	1.00000	0.174078
$34$	−1.00000	−0.171499
$35$	0	0
$36$	1.00000	0.166667
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	5.00000	0.811107
$39$	−1.00000	−0.160128
$40$	0	0
$41$	4.00000	0.624695	0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000	0.624695	0.312348	+	0.949968i	$0.398885\pi$
$42$	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	$0.132197\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	7.00000	1.03209
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	1.00000	0.144338
$49$	2.00000	0.285714
$50$	0	0
$51$	1.00000	0.140028
$52$	−1.00000	−0.138675
$53$	9.00000	1.23625	0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000	1.23625	0.618123	+	0.786082i	$0.287894\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	3.00000	0.400892
$57$	−5.00000	−0.662266
$58$	2.00000	0.262613
$59$	10.0000	1.30189	0.650945	−	0.759125i	$-0.274373\pi$
$59$	10.0000	1.30189	0.650945	+	0.759125i	$0.274373\pi$
$60$	0	0
$61$	14.0000	1.79252	0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000	1.79252	0.896258	+	0.443533i	$0.146275\pi$
$62$	−2.00000	−0.254000
$63$	−3.00000	−0.377964
$64$	1.00000	0.125000
$65$	0	0
$66$	−1.00000	−0.123091
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	1.00000	0.121268
$69$	−7.00000	−0.842701
$70$	0	0
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	−1.00000	−0.117851
$73$	9.00000	1.05337	0.526685	−	0.850060i	$-0.323435\pi$
$73$	9.00000	1.05337	0.526685	+	0.850060i	$0.323435\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	−5.00000	−0.573539
$77$	−3.00000	−0.341882
$78$	1.00000	0.113228
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−4.00000	−0.441726
$83$	1.00000	0.109764	0.0548821	−	0.998493i	$-0.482522\pi$
$83$	1.00000	0.109764	0.0548821	+	0.998493i	$0.482522\pi$
$84$	−3.00000	−0.327327
$85$	0	0
$86$	−12.0000	−1.29399
$87$	−2.00000	−0.214423
$88$	−1.00000	−0.106600
$89$	1.00000	0.106000	0.0529999	−	0.998595i	$-0.483122\pi$
$89$	1.00000	0.106000	0.0529999	+	0.998595i	$0.483122\pi$
$90$	0	0
$91$	3.00000	0.314485
$92$	−7.00000	−0.729800
$93$	2.00000	0.207390
$94$	12.0000	1.23771
$95$	0	0
$96$	−1.00000	−0.102062
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	−2.00000	−0.202031
$99$	1.00000	0.100504
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	−1.00000	−0.0990148
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	−9.00000	−0.874157
$107$	19.0000	1.83680	0.918400	−	0.395654i	$-0.129482\pi$
$107$	19.0000	1.83680	0.918400	+	0.395654i	$0.129482\pi$
$108$	1.00000	0.0962250
$109$	−19.0000	−1.81987	−0.909935	−	0.414751i	$-0.863869\pi$
$109$	−19.0000	−1.81987	−0.909935	+	0.414751i	$0.863869\pi$
$110$	0	0
$111$	−1.00000	−0.0949158
$112$	−3.00000	−0.283473
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	5.00000	0.468293
$115$	0	0
$116$	−2.00000	−0.185695
$117$	−1.00000	−0.0924500
$118$	−10.0000	−0.920575
$119$	−3.00000	−0.275010
$120$	0	0
$121$	−10.0000	−0.909091
$122$	−14.0000	−1.26750
$123$	4.00000	0.360668
$124$	2.00000	0.179605
$125$	0	0
$126$	3.00000	0.267261
$127$	−13.0000	−1.15356	−0.576782	−	0.816898i	$-0.695692\pi$
$127$	−13.0000	−1.15356	−0.576782	+	0.816898i	$0.695692\pi$
$128$	−1.00000	−0.0883883
$129$	12.0000	1.05654
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	1.00000	0.0870388
$133$	15.0000	1.30066
$134$	2.00000	0.172774
$135$	0	0
$136$	−1.00000	−0.0857493
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	7.00000	0.595880
$139$	−6.00000	−0.508913	−0.254457	−	0.967084i	$-0.581897\pi$
$139$	−6.00000	−0.508913	−0.254457	+	0.967084i	$0.581897\pi$
$140$	0	0
$141$	−12.0000	−1.01058
$142$	−2.00000	−0.167836
$143$	−1.00000	−0.0836242
$144$	1.00000	0.0833333
$145$	0	0
$146$	−9.00000	−0.744845
$147$	2.00000	0.164957
$148$	−1.00000	−0.0821995
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	−17.0000	−1.38344	−0.691720	−	0.722166i	$-0.743147\pi$
$151$	−17.0000	−1.38344	−0.691720	+	0.722166i	$0.743147\pi$
$152$	5.00000	0.405554
$153$	1.00000	0.0808452
$154$	3.00000	0.241747
$155$	0	0
$156$	−1.00000	−0.0800641
$157$	8.00000	0.638470	0.319235	−	0.947676i	$-0.396574\pi$
$157$	8.00000	0.638470	0.319235	+	0.947676i	$0.396574\pi$
$158$	−4.00000	−0.318223
$159$	9.00000	0.713746
$160$	0	0
$161$	21.0000	1.65503
$162$	−1.00000	−0.0785674
$163$	9.00000	0.704934	0.352467	−	0.935824i	$-0.385343\pi$
$163$	9.00000	0.704934	0.352467	+	0.935824i	$0.385343\pi$
$164$	4.00000	0.312348
$165$	0	0
$166$	−1.00000	−0.0776151
$167$	7.00000	0.541676	0.270838	−	0.962625i	$-0.412699\pi$
$167$	7.00000	0.541676	0.270838	+	0.962625i	$0.412699\pi$
$168$	3.00000	0.231455
$169$	−12.0000	−0.923077
$170$	0	0
$171$	−5.00000	−0.382360
$172$	12.0000	0.914991
$173$	23.0000	1.74866	0.874329	−	0.485334i	$-0.161302\pi$
$173$	23.0000	1.74866	0.874329	+	0.485334i	$0.161302\pi$
$174$	2.00000	0.151620
$175$	0	0
$176$	1.00000	0.0753778
$177$	10.0000	0.751646
$178$	−1.00000	−0.0749532
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	−3.00000	−0.222375
$183$	14.0000	1.03491
$184$	7.00000	0.516047
$185$	0	0
$186$	−2.00000	−0.146647
$187$	1.00000	0.0731272
$188$	−12.0000	−0.875190
$189$	−3.00000	−0.218218
$190$	0	0
$191$	5.00000	0.361787	0.180894	−	0.983503i	$-0.442101\pi$
$191$	5.00000	0.361787	0.180894	+	0.983503i	$0.442101\pi$
$192$	1.00000	0.0721688
$193$	22.0000	1.58359	0.791797	−	0.610784i	$-0.209146\pi$
$193$	22.0000	1.58359	0.791797	+	0.610784i	$0.209146\pi$
$194$	−8.00000	−0.574367
$195$	0	0
$196$	2.00000	0.142857
$197$	−13.0000	−0.926212	−0.463106	−	0.886303i	$-0.653265\pi$
$197$	−13.0000	−0.926212	−0.463106	+	0.886303i	$0.653265\pi$
$198$	−1.00000	−0.0710669
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	0	0
$201$	−2.00000	−0.141069
$202$	−6.00000	−0.422159
$203$	6.00000	0.421117
$204$	1.00000	0.0700140
$205$	0	0
$206$	8.00000	0.557386
$207$	−7.00000	−0.486534
$208$	−1.00000	−0.0693375
$209$	−5.00000	−0.345857
$210$	0	0
$211$	−10.0000	−0.688428	−0.344214	−	0.938891i	$-0.611855\pi$
$211$	−10.0000	−0.688428	−0.344214	+	0.938891i	$0.611855\pi$
$212$	9.00000	0.618123
$213$	2.00000	0.137038
$214$	−19.0000	−1.29881
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	−6.00000	−0.407307
$218$	19.0000	1.28684
$219$	9.00000	0.608164
$220$	0	0
$221$	−1.00000	−0.0672673
$222$	1.00000	0.0671156
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	3.00000	0.200446
$225$	0	0
$226$	−2.00000	−0.133038
$227$	4.00000	0.265489	0.132745	−	0.991150i	$-0.457621\pi$
$227$	4.00000	0.265489	0.132745	+	0.991150i	$0.457621\pi$
$228$	−5.00000	−0.331133
$229$	26.0000	1.71813	0.859064	−	0.511868i	$-0.171046\pi$
$229$	26.0000	1.71813	0.859064	+	0.511868i	$0.171046\pi$
$230$	0	0
$231$	−3.00000	−0.197386
$232$	2.00000	0.131306
$233$	14.0000	0.917170	0.458585	−	0.888650i	$-0.348356\pi$
$233$	14.0000	0.917170	0.458585	+	0.888650i	$0.348356\pi$
$234$	1.00000	0.0653720
$235$	0	0
$236$	10.0000	0.650945
$237$	4.00000	0.259828
$238$	3.00000	0.194461
$239$	20.0000	1.29369	0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000	1.29369	0.646846	+	0.762620i	$0.276088\pi$
$240$	0	0
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	10.0000	0.642824
$243$	1.00000	0.0641500
$244$	14.0000	0.896258
$245$	0	0
$246$	−4.00000	−0.255031
$247$	5.00000	0.318142
$248$	−2.00000	−0.127000
$249$	1.00000	0.0633724
$250$	0	0
$251$	18.0000	1.13615	0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000	1.13615	0.568075	+	0.822977i	$0.307688\pi$
$252$	−3.00000	−0.188982
$253$	−7.00000	−0.440086
$254$	13.0000	0.815693
$255$	0	0
$256$	1.00000	0.0625000
$257$	13.0000	0.810918	0.405459	−	0.914113i	$-0.367112\pi$
$257$	13.0000	0.810918	0.405459	+	0.914113i	$0.367112\pi$
$258$	−12.0000	−0.747087
$259$	3.00000	0.186411
$260$	0	0
$261$	−2.00000	−0.123797
$262$	0	0
$263$	−8.00000	−0.493301	−0.246651	−	0.969104i	$-0.579330\pi$
$263$	−8.00000	−0.493301	−0.246651	+	0.969104i	$0.579330\pi$
$264$	−1.00000	−0.0615457
$265$	0	0
$266$	−15.0000	−0.919709
$267$	1.00000	0.0611990
$268$	−2.00000	−0.122169
$269$	−5.00000	−0.304855	−0.152428	−	0.988315i	$-0.548709\pi$
$269$	−5.00000	−0.304855	−0.152428	+	0.988315i	$0.548709\pi$
$270$	0	0
$271$	−28.0000	−1.70088	−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000	−1.70088	−0.850439	+	0.526073i	$0.823664\pi$
$272$	1.00000	0.0606339
$273$	3.00000	0.181568
$274$	−10.0000	−0.604122
$275$	0	0
$276$	−7.00000	−0.421350
$277$	17.0000	1.02143	0.510716	−	0.859750i	$-0.329381\pi$
$277$	17.0000	1.02143	0.510716	+	0.859750i	$0.329381\pi$
$278$	6.00000	0.359856
$279$	2.00000	0.119737
$280$	0	0
$281$	−11.0000	−0.656205	−0.328102	−	0.944642i	$-0.606409\pi$
$281$	−11.0000	−0.656205	−0.328102	+	0.944642i	$0.606409\pi$
$282$	12.0000	0.714590
$283$	21.0000	1.24832	0.624160	−	0.781296i	$-0.285441\pi$
$283$	21.0000	1.24832	0.624160	+	0.781296i	$0.285441\pi$
$284$	2.00000	0.118678
$285$	0	0
$286$	1.00000	0.0591312
$287$	−12.0000	−0.708338
$288$	−1.00000	−0.0589256
$289$	−16.0000	−0.941176
$290$	0	0
$291$	8.00000	0.468968
$292$	9.00000	0.526685
$293$	9.00000	0.525786	0.262893	−	0.964825i	$-0.415323\pi$
$293$	9.00000	0.525786	0.262893	+	0.964825i	$0.415323\pi$
$294$	−2.00000	−0.116642
$295$	0	0
$296$	1.00000	0.0581238
$297$	1.00000	0.0580259
$298$	−6.00000	−0.347571
$299$	7.00000	0.404820
$300$	0	0
$301$	−36.0000	−2.07501
$302$	17.0000	0.978240
$303$	6.00000	0.344691
$304$	−5.00000	−0.286770
$305$	0	0
$306$	−1.00000	−0.0571662
$307$	−18.0000	−1.02731	−0.513657	−	0.857996i	$-0.671710\pi$
$307$	−18.0000	−1.02731	−0.513657	+	0.857996i	$0.671710\pi$
$308$	−3.00000	−0.170941
$309$	−8.00000	−0.455104
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	1.00000	0.0566139
$313$	−8.00000	−0.452187	−0.226093	−	0.974106i	$-0.572595\pi$
$313$	−8.00000	−0.452187	−0.226093	+	0.974106i	$0.572595\pi$
$314$	−8.00000	−0.451466
$315$	0	0
$316$	4.00000	0.225018
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	−9.00000	−0.504695
$319$	−2.00000	−0.111979
$320$	0	0
$321$	19.0000	1.06048
$322$	−21.0000	−1.17028
$323$	−5.00000	−0.278207
$324$	1.00000	0.0555556
$325$	0	0
$326$	−9.00000	−0.498464
$327$	−19.0000	−1.05070
$328$	−4.00000	−0.220863
$329$	36.0000	1.98474
$330$	0	0
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	1.00000	0.0548821
$333$	−1.00000	−0.0547997
$334$	−7.00000	−0.383023
$335$	0	0
$336$	−3.00000	−0.163663
$337$	1.00000	0.0544735	0.0272367	−	0.999629i	$-0.491329\pi$
$337$	1.00000	0.0544735	0.0272367	+	0.999629i	$0.491329\pi$
$338$	12.0000	0.652714
$339$	2.00000	0.108625
$340$	0	0
$341$	2.00000	0.108306
$342$	5.00000	0.270369
$343$	15.0000	0.809924
$344$	−12.0000	−0.646997
$345$	0	0
$346$	−23.0000	−1.23649
$347$	−24.0000	−1.28839	−0.644194	−	0.764862i	$-0.722807\pi$
$347$	−24.0000	−1.28839	−0.644194	+	0.764862i	$0.722807\pi$
$348$	−2.00000	−0.107211
$349$	−4.00000	−0.214115	−0.107058	−	0.994253i	$-0.534143\pi$
$349$	−4.00000	−0.214115	−0.107058	+	0.994253i	$0.534143\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	−1.00000	−0.0533002
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	−10.0000	−0.531494
$355$	0	0
$356$	1.00000	0.0529999
$357$	−3.00000	−0.158777
$358$	−12.0000	−0.634220
$359$	4.00000	0.211112	0.105556	−	0.994413i	$-0.466338\pi$
$359$	4.00000	0.211112	0.105556	+	0.994413i	$0.466338\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	−2.00000	−0.105118
$363$	−10.0000	−0.524864
$364$	3.00000	0.157243
$365$	0	0
$366$	−14.0000	−0.731792
$367$	−35.0000	−1.82699	−0.913493	−	0.406855i	$-0.866625\pi$
$367$	−35.0000	−1.82699	−0.913493	+	0.406855i	$0.866625\pi$
$368$	−7.00000	−0.364900
$369$	4.00000	0.208232
$370$	0	0
$371$	−27.0000	−1.40177
$372$	2.00000	0.103695
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	−1.00000	−0.0517088
$375$	0	0
$376$	12.0000	0.618853
$377$	2.00000	0.103005
$378$	3.00000	0.154303
$379$	−2.00000	−0.102733	−0.0513665	−	0.998680i	$-0.516358\pi$
$379$	−2.00000	−0.102733	−0.0513665	+	0.998680i	$0.516358\pi$
$380$	0	0
$381$	−13.0000	−0.666010
$382$	−5.00000	−0.255822
$383$	−13.0000	−0.664269	−0.332134	−	0.943232i	$-0.607769\pi$
$383$	−13.0000	−0.664269	−0.332134	+	0.943232i	$0.607769\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−22.0000	−1.11977
$387$	12.0000	0.609994
$388$	8.00000	0.406138
$389$	14.0000	0.709828	0.354914	−	0.934899i	$-0.384510\pi$
$389$	14.0000	0.709828	0.354914	+	0.934899i	$0.384510\pi$
$390$	0	0
$391$	−7.00000	−0.354005
$392$	−2.00000	−0.101015
$393$	0	0
$394$	13.0000	0.654931
$395$	0	0
$396$	1.00000	0.0502519
$397$	16.0000	0.803017	0.401508	−	0.915855i	$-0.368486\pi$
$397$	16.0000	0.803017	0.401508	+	0.915855i	$0.368486\pi$
$398$	−4.00000	−0.200502
$399$	15.0000	0.750939
$400$	0	0
$401$	25.0000	1.24844	0.624220	−	0.781248i	$-0.285417\pi$
$401$	25.0000	1.24844	0.624220	+	0.781248i	$0.285417\pi$
$402$	2.00000	0.0997509
$403$	−2.00000	−0.0996271
$404$	6.00000	0.298511
$405$	0	0
$406$	−6.00000	−0.297775
$407$	−1.00000	−0.0495682
$408$	−1.00000	−0.0495074
$409$	−8.00000	−0.395575	−0.197787	−	0.980245i	$-0.563376\pi$
$409$	−8.00000	−0.395575	−0.197787	+	0.980245i	$0.563376\pi$
$410$	0	0
$411$	10.0000	0.493264
$412$	−8.00000	−0.394132
$413$	−30.0000	−1.47620
$414$	7.00000	0.344031
$415$	0	0
$416$	1.00000	0.0490290
$417$	−6.00000	−0.293821
$418$	5.00000	0.244558
$419$	−1.00000	−0.0488532	−0.0244266	−	0.999702i	$-0.507776\pi$
$419$	−1.00000	−0.0488532	−0.0244266	+	0.999702i	$0.507776\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	10.0000	0.486792
$423$	−12.0000	−0.583460
$424$	−9.00000	−0.437079
$425$	0	0
$426$	−2.00000	−0.0969003
$427$	−42.0000	−2.03252
$428$	19.0000	0.918400
$429$	−1.00000	−0.0482805
$430$	0	0
$431$	3.00000	0.144505	0.0722525	−	0.997386i	$-0.476981\pi$
$431$	3.00000	0.144505	0.0722525	+	0.997386i	$0.476981\pi$
$432$	1.00000	0.0481125
$433$	19.0000	0.913082	0.456541	−	0.889702i	$-0.349088\pi$
$433$	19.0000	0.913082	0.456541	+	0.889702i	$0.349088\pi$
$434$	6.00000	0.288009
$435$	0	0
$436$	−19.0000	−0.909935
$437$	35.0000	1.67428
$438$	−9.00000	−0.430037
$439$	24.0000	1.14546	0.572729	−	0.819745i	$-0.305885\pi$
$439$	24.0000	1.14546	0.572729	+	0.819745i	$0.305885\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	1.00000	0.0475651
$443$	36.0000	1.71041	0.855206	−	0.518289i	$-0.173431\pi$
$443$	36.0000	1.71041	0.855206	+	0.518289i	$0.173431\pi$
$444$	−1.00000	−0.0474579
$445$	0	0
$446$	0	0
$447$	6.00000	0.283790
$448$	−3.00000	−0.141737
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	0	0
$451$	4.00000	0.188353
$452$	2.00000	0.0940721
$453$	−17.0000	−0.798730
$454$	−4.00000	−0.187729
$455$	0	0
$456$	5.00000	0.234146
$457$	−24.0000	−1.12267	−0.561336	−	0.827588i	$-0.689713\pi$
$457$	−24.0000	−1.12267	−0.561336	+	0.827588i	$0.689713\pi$
$458$	−26.0000	−1.21490
$459$	1.00000	0.0466760
$460$	0	0
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	3.00000	0.139573
$463$	36.0000	1.67306	0.836531	−	0.547920i	$-0.184580\pi$
$463$	36.0000	1.67306	0.836531	+	0.547920i	$0.184580\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	−14.0000	−0.648537
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	−1.00000	−0.0462250
$469$	6.00000	0.277054
$470$	0	0
$471$	8.00000	0.368621
$472$	−10.0000	−0.460287
$473$	12.0000	0.551761
$474$	−4.00000	−0.183726
$475$	0	0
$476$	−3.00000	−0.137505
$477$	9.00000	0.412082
$478$	−20.0000	−0.914779
$479$	−9.00000	−0.411220	−0.205610	−	0.978634i	$-0.565918\pi$
$479$	−9.00000	−0.411220	−0.205610	+	0.978634i	$0.565918\pi$
$480$	0	0
$481$	1.00000	0.0455961
$482$	14.0000	0.637683
$483$	21.0000	0.955533
$484$	−10.0000	−0.454545
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	−6.00000	−0.271886	−0.135943	−	0.990717i	$-0.543406\pi$
$487$	−6.00000	−0.271886	−0.135943	+	0.990717i	$0.543406\pi$
$488$	−14.0000	−0.633750
$489$	9.00000	0.406994
$490$	0	0
$491$	−35.0000	−1.57953	−0.789764	−	0.613411i	$-0.789797\pi$
$491$	−35.0000	−1.57953	−0.789764	+	0.613411i	$0.789797\pi$
$492$	4.00000	0.180334
$493$	−2.00000	−0.0900755
$494$	−5.00000	−0.224961
$495$	0	0
$496$	2.00000	0.0898027
$497$	−6.00000	−0.269137
$498$	−1.00000	−0.0448111
$499$	−11.0000	−0.492428	−0.246214	−	0.969216i	$-0.579187\pi$
$499$	−11.0000	−0.492428	−0.246214	+	0.969216i	$0.579187\pi$
$500$	0	0
$501$	7.00000	0.312737
$502$	−18.0000	−0.803379
$503$	−36.0000	−1.60516	−0.802580	−	0.596544i	$-0.796540\pi$
$503$	−36.0000	−1.60516	−0.802580	+	0.596544i	$0.796540\pi$
$504$	3.00000	0.133631
$505$	0	0
$506$	7.00000	0.311188
$507$	−12.0000	−0.532939
$508$	−13.0000	−0.576782
$509$	−11.0000	−0.487566	−0.243783	−	0.969830i	$-0.578389\pi$
$509$	−11.0000	−0.487566	−0.243783	+	0.969830i	$0.578389\pi$
$510$	0	0
$511$	−27.0000	−1.19441
$512$	−1.00000	−0.0441942
$513$	−5.00000	−0.220755
$514$	−13.0000	−0.573405
$515$	0	0
$516$	12.0000	0.528271
$517$	−12.0000	−0.527759
$518$	−3.00000	−0.131812
$519$	23.0000	1.00959
$520$	0	0
$521$	36.0000	1.57719	0.788594	−	0.614914i	$-0.210809\pi$
$521$	36.0000	1.57719	0.788594	+	0.614914i	$0.210809\pi$
$522$	2.00000	0.0875376
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	0	0
$525$	0	0
$526$	8.00000	0.348817
$527$	2.00000	0.0871214
$528$	1.00000	0.0435194
$529$	26.0000	1.13043
$530$	0	0
$531$	10.0000	0.433963
$532$	15.0000	0.650332
$533$	−4.00000	−0.173259
$534$	−1.00000	−0.0432742
$535$	0	0
$536$	2.00000	0.0863868
$537$	12.0000	0.517838
$538$	5.00000	0.215565
$539$	2.00000	0.0861461
$540$	0	0
$541$	3.00000	0.128980	0.0644900	−	0.997918i	$-0.479458\pi$
$541$	3.00000	0.128980	0.0644900	+	0.997918i	$0.479458\pi$
$542$	28.0000	1.20270
$543$	2.00000	0.0858282
$544$	−1.00000	−0.0428746
$545$	0	0
$546$	−3.00000	−0.128388
$547$	13.0000	0.555840	0.277920	−	0.960604i	$-0.410355\pi$
$547$	13.0000	0.555840	0.277920	+	0.960604i	$0.410355\pi$
$548$	10.0000	0.427179
$549$	14.0000	0.597505
$550$	0	0
$551$	10.0000	0.426014
$552$	7.00000	0.297940
$553$	−12.0000	−0.510292
$554$	−17.0000	−0.722261
$555$	0	0
$556$	−6.00000	−0.254457
$557$	12.0000	0.508456	0.254228	−	0.967144i	$-0.418179\pi$
$557$	12.0000	0.508456	0.254228	+	0.967144i	$0.418179\pi$
$558$	−2.00000	−0.0846668
$559$	−12.0000	−0.507546
$560$	0	0
$561$	1.00000	0.0422200
$562$	11.0000	0.464007
$563$	6.00000	0.252870	0.126435	−	0.991975i	$-0.459647\pi$
$563$	6.00000	0.252870	0.126435	+	0.991975i	$0.459647\pi$
$564$	−12.0000	−0.505291
$565$	0	0
$566$	−21.0000	−0.882696
$567$	−3.00000	−0.125988
$568$	−2.00000	−0.0839181
$569$	−1.00000	−0.0419222	−0.0209611	−	0.999780i	$-0.506673\pi$
$569$	−1.00000	−0.0419222	−0.0209611	+	0.999780i	$0.506673\pi$
$570$	0	0
$571$	18.0000	0.753277	0.376638	−	0.926360i	$-0.377080\pi$
$571$	18.0000	0.753277	0.376638	+	0.926360i	$0.377080\pi$
$572$	−1.00000	−0.0418121
$573$	5.00000	0.208878
$574$	12.0000	0.500870
$575$	0	0
$576$	1.00000	0.0416667
$577$	−22.0000	−0.915872	−0.457936	−	0.888985i	$-0.651411\pi$
$577$	−22.0000	−0.915872	−0.457936	+	0.888985i	$0.651411\pi$
$578$	16.0000	0.665512
$579$	22.0000	0.914289
$580$	0	0
$581$	−3.00000	−0.124461
$582$	−8.00000	−0.331611
$583$	9.00000	0.372742
$584$	−9.00000	−0.372423
$585$	0	0
$586$	−9.00000	−0.371787
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	2.00000	0.0824786
$589$	−10.0000	−0.412043
$590$	0	0
$591$	−13.0000	−0.534749
$592$	−1.00000	−0.0410997
$593$	8.00000	0.328521	0.164260	−	0.986417i	$-0.447476\pi$
$593$	8.00000	0.328521	0.164260	+	0.986417i	$0.447476\pi$
$594$	−1.00000	−0.0410305
$595$	0	0
$596$	6.00000	0.245770
$597$	4.00000	0.163709
$598$	−7.00000	−0.286251
$599$	36.0000	1.47092	0.735460	−	0.677568i	$-0.236966\pi$
$599$	36.0000	1.47092	0.735460	+	0.677568i	$0.236966\pi$
$600$	0	0
$601$	−13.0000	−0.530281	−0.265141	−	0.964210i	$-0.585418\pi$
$601$	−13.0000	−0.530281	−0.265141	+	0.964210i	$0.585418\pi$
$602$	36.0000	1.46725
$603$	−2.00000	−0.0814463
$604$	−17.0000	−0.691720
$605$	0	0
$606$	−6.00000	−0.243733
$607$	14.0000	0.568242	0.284121	−	0.958788i	$-0.408298\pi$
$607$	14.0000	0.568242	0.284121	+	0.958788i	$0.408298\pi$
$608$	5.00000	0.202777
$609$	6.00000	0.243132
$610$	0	0
$611$	12.0000	0.485468
$612$	1.00000	0.0404226
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	18.0000	0.726421
$615$	0	0
$616$	3.00000	0.120873
$617$	42.0000	1.69086	0.845428	−	0.534089i	$-0.179345\pi$
$617$	42.0000	1.69086	0.845428	+	0.534089i	$0.179345\pi$
$618$	8.00000	0.321807
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	−7.00000	−0.280900
$622$	−24.0000	−0.962312
$623$	−3.00000	−0.120192
$624$	−1.00000	−0.0400320
$625$	0	0
$626$	8.00000	0.319744
$627$	−5.00000	−0.199681
$628$	8.00000	0.319235
$629$	−1.00000	−0.0398726
$630$	0	0
$631$	32.0000	1.27390	0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000	1.27390	0.636950	+	0.770905i	$0.280196\pi$
$632$	−4.00000	−0.159111
$633$	−10.0000	−0.397464
$634$	−6.00000	−0.238290
$635$	0	0
$636$	9.00000	0.356873
$637$	−2.00000	−0.0792429
$638$	2.00000	0.0791808
$639$	2.00000	0.0791188
$640$	0	0
$641$	44.0000	1.73790	0.868948	−	0.494904i	$-0.164797\pi$
$641$	44.0000	1.73790	0.868948	+	0.494904i	$0.164797\pi$
$642$	−19.0000	−0.749870
$643$	7.00000	0.276053	0.138027	−	0.990429i	$-0.455924\pi$
$643$	7.00000	0.276053	0.138027	+	0.990429i	$0.455924\pi$
$644$	21.0000	0.827516
$645$	0	0
$646$	5.00000	0.196722
$647$	17.0000	0.668339	0.334169	−	0.942513i	$-0.391544\pi$
$647$	17.0000	0.668339	0.334169	+	0.942513i	$0.391544\pi$
$648$	−1.00000	−0.0392837
$649$	10.0000	0.392534
$650$	0	0
$651$	−6.00000	−0.235159
$652$	9.00000	0.352467
$653$	−24.0000	−0.939193	−0.469596	−	0.882881i	$-0.655601\pi$
$653$	−24.0000	−0.939193	−0.469596	+	0.882881i	$0.655601\pi$
$654$	19.0000	0.742959
$655$	0	0
$656$	4.00000	0.156174
$657$	9.00000	0.351123
$658$	−36.0000	−1.40343
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	7.00000	0.272268	0.136134	−	0.990690i	$-0.456532\pi$
$661$	7.00000	0.272268	0.136134	+	0.990690i	$0.456532\pi$
$662$	−12.0000	−0.466393
$663$	−1.00000	−0.0388368
$664$	−1.00000	−0.0388075
$665$	0	0
$666$	1.00000	0.0387492
$667$	14.0000	0.542082
$668$	7.00000	0.270838
$669$	0	0
$670$	0	0
$671$	14.0000	0.540464
$672$	3.00000	0.115728
$673$	−19.0000	−0.732396	−0.366198	−	0.930537i	$-0.619341\pi$
$673$	−19.0000	−0.732396	−0.366198	+	0.930537i	$0.619341\pi$
$674$	−1.00000	−0.0385186
$675$	0	0
$676$	−12.0000	−0.461538
$677$	3.00000	0.115299	0.0576497	−	0.998337i	$-0.481639\pi$
$677$	3.00000	0.115299	0.0576497	+	0.998337i	$0.481639\pi$
$678$	−2.00000	−0.0768095
$679$	−24.0000	−0.921035
$680$	0	0
$681$	4.00000	0.153280
$682$	−2.00000	−0.0765840
$683$	−40.0000	−1.53056	−0.765279	−	0.643699i	$-0.777399\pi$
$683$	−40.0000	−1.53056	−0.765279	+	0.643699i	$0.777399\pi$
$684$	−5.00000	−0.191180
$685$	0	0
$686$	−15.0000	−0.572703
$687$	26.0000	0.991962
$688$	12.0000	0.457496
$689$	−9.00000	−0.342873
$690$	0	0
$691$	−2.00000	−0.0760836	−0.0380418	−	0.999276i	$-0.512112\pi$
$691$	−2.00000	−0.0760836	−0.0380418	+	0.999276i	$0.512112\pi$
$692$	23.0000	0.874329
$693$	−3.00000	−0.113961
$694$	24.0000	0.911028
$695$	0	0
$696$	2.00000	0.0758098
$697$	4.00000	0.151511
$698$	4.00000	0.151402
$699$	14.0000	0.529529
$700$	0	0
$701$	−40.0000	−1.51078	−0.755390	−	0.655276i	$-0.772552\pi$
$701$	−40.0000	−1.51078	−0.755390	+	0.655276i	$0.772552\pi$
$702$	1.00000	0.0377426
$703$	5.00000	0.188579
$704$	1.00000	0.0376889
$705$	0	0
$706$	14.0000	0.526897
$707$	−18.0000	−0.676960
$708$	10.0000	0.375823
$709$	27.0000	1.01401	0.507003	−	0.861944i	$-0.330753\pi$
$709$	27.0000	1.01401	0.507003	+	0.861944i	$0.330753\pi$
$710$	0	0
$711$	4.00000	0.150012
$712$	−1.00000	−0.0374766
$713$	−14.0000	−0.524304
$714$	3.00000	0.112272
$715$	0	0
$716$	12.0000	0.448461
$717$	20.0000	0.746914
$718$	−4.00000	−0.149279
$719$	10.0000	0.372937	0.186469	−	0.982461i	$-0.440296\pi$
$719$	10.0000	0.372937	0.186469	+	0.982461i	$0.440296\pi$
$720$	0	0
$721$	24.0000	0.893807
$722$	−6.00000	−0.223297
$723$	−14.0000	−0.520666
$724$	2.00000	0.0743294
$725$	0	0
$726$	10.0000	0.371135
$727$	−30.0000	−1.11264	−0.556319	−	0.830969i	$-0.687787\pi$
$727$	−30.0000	−1.11264	−0.556319	+	0.830969i	$0.687787\pi$
$728$	−3.00000	−0.111187
$729$	1.00000	0.0370370
$730$	0	0
$731$	12.0000	0.443836
$732$	14.0000	0.517455
$733$	−26.0000	−0.960332	−0.480166	−	0.877178i	$-0.659424\pi$
$733$	−26.0000	−0.960332	−0.480166	+	0.877178i	$0.659424\pi$
$734$	35.0000	1.29187
$735$	0	0
$736$	7.00000	0.258023
$737$	−2.00000	−0.0736709
$738$	−4.00000	−0.147242
$739$	54.0000	1.98642	0.993211	−	0.116326i	$-0.0371118\pi$
$739$	54.0000	1.98642	0.993211	+	0.116326i	$0.0371118\pi$
$740$	0	0
$741$	5.00000	0.183680
$742$	27.0000	0.991201
$743$	−14.0000	−0.513610	−0.256805	−	0.966463i	$-0.582670\pi$
$743$	−14.0000	−0.513610	−0.256805	+	0.966463i	$0.582670\pi$
$744$	−2.00000	−0.0733236
$745$	0	0
$746$	10.0000	0.366126
$747$	1.00000	0.0365881
$748$	1.00000	0.0365636
$749$	−57.0000	−2.08273
$750$	0	0
$751$	44.0000	1.60558	0.802791	−	0.596260i	$-0.203347\pi$
$751$	44.0000	1.60558	0.802791	+	0.596260i	$0.203347\pi$
$752$	−12.0000	−0.437595
$753$	18.0000	0.655956
$754$	−2.00000	−0.0728357
$755$	0	0
$756$	−3.00000	−0.109109
$757$	−43.0000	−1.56286	−0.781431	−	0.623992i	$-0.785510\pi$
$757$	−43.0000	−1.56286	−0.781431	+	0.623992i	$0.785510\pi$
$758$	2.00000	0.0726433
$759$	−7.00000	−0.254084
$760$	0	0
$761$	−14.0000	−0.507500	−0.253750	−	0.967270i	$-0.581664\pi$
$761$	−14.0000	−0.507500	−0.253750	+	0.967270i	$0.581664\pi$
$762$	13.0000	0.470940
$763$	57.0000	2.06354
$764$	5.00000	0.180894
$765$	0	0
$766$	13.0000	0.469709
$767$	−10.0000	−0.361079
$768$	1.00000	0.0360844
$769$	38.0000	1.37032	0.685158	−	0.728395i	$-0.259733\pi$
$769$	38.0000	1.37032	0.685158	+	0.728395i	$0.259733\pi$
$770$	0	0
$771$	13.0000	0.468184
$772$	22.0000	0.791797
$773$	−9.00000	−0.323708	−0.161854	−	0.986815i	$-0.551747\pi$
$773$	−9.00000	−0.323708	−0.161854	+	0.986815i	$0.551747\pi$
$774$	−12.0000	−0.431331
$775$	0	0
$776$	−8.00000	−0.287183
$777$	3.00000	0.107624
$778$	−14.0000	−0.501924
$779$	−20.0000	−0.716574
$780$	0	0
$781$	2.00000	0.0715656
$782$	7.00000	0.250319
$783$	−2.00000	−0.0714742
$784$	2.00000	0.0714286
$785$	0	0
$786$	0	0
$787$	−14.0000	−0.499046	−0.249523	−	0.968369i	$-0.580274\pi$
$787$	−14.0000	−0.499046	−0.249523	+	0.968369i	$0.580274\pi$
$788$	−13.0000	−0.463106
$789$	−8.00000	−0.284808
$790$	0	0
$791$	−6.00000	−0.213335
$792$	−1.00000	−0.0355335
$793$	−14.0000	−0.497155
$794$	−16.0000	−0.567819
$795$	0	0
$796$	4.00000	0.141776
$797$	46.0000	1.62940	0.814702	−	0.579880i	$-0.196901\pi$
$797$	46.0000	1.62940	0.814702	+	0.579880i	$0.196901\pi$
$798$	−15.0000	−0.530994
$799$	−12.0000	−0.424529
$800$	0	0
$801$	1.00000	0.0353333
$802$	−25.0000	−0.882781
$803$	9.00000	0.317603
$804$	−2.00000	−0.0705346
$805$	0	0
$806$	2.00000	0.0704470
$807$	−5.00000	−0.176008
$808$	−6.00000	−0.211079
$809$	5.00000	0.175791	0.0878953	−	0.996130i	$-0.471986\pi$
$809$	5.00000	0.175791	0.0878953	+	0.996130i	$0.471986\pi$
$810$	0	0
$811$	−16.0000	−0.561836	−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000	−0.561836	−0.280918	+	0.959732i	$0.590639\pi$
$812$	6.00000	0.210559
$813$	−28.0000	−0.982003
$814$	1.00000	0.0350500
$815$	0	0
$816$	1.00000	0.0350070
$817$	−60.0000	−2.09913
$818$	8.00000	0.279713
$819$	3.00000	0.104828
$820$	0	0
$821$	−3.00000	−0.104701	−0.0523504	−	0.998629i	$-0.516671\pi$
$821$	−3.00000	−0.104701	−0.0523504	+	0.998629i	$0.516671\pi$
$822$	−10.0000	−0.348790
$823$	−13.0000	−0.453152	−0.226576	−	0.973994i	$-0.572753\pi$
$823$	−13.0000	−0.453152	−0.226576	+	0.973994i	$0.572753\pi$
$824$	8.00000	0.278693
$825$	0	0
$826$	30.0000	1.04383
$827$	−36.0000	−1.25184	−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000	−1.25184	−0.625921	+	0.779886i	$0.715277\pi$
$828$	−7.00000	−0.243267
$829$	−7.00000	−0.243120	−0.121560	−	0.992584i	$-0.538790\pi$
$829$	−7.00000	−0.243120	−0.121560	+	0.992584i	$0.538790\pi$
$830$	0	0
$831$	17.0000	0.589723
$832$	−1.00000	−0.0346688
$833$	2.00000	0.0692959
$834$	6.00000	0.207763
$835$	0	0
$836$	−5.00000	−0.172929
$837$	2.00000	0.0691301
$838$	1.00000	0.0345444
$839$	−54.0000	−1.86429	−0.932144	−	0.362089i	$-0.882064\pi$
$839$	−54.0000	−1.86429	−0.932144	+	0.362089i	$0.882064\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−22.0000	−0.758170
$843$	−11.0000	−0.378860
$844$	−10.0000	−0.344214
$845$	0	0
$846$	12.0000	0.412568
$847$	30.0000	1.03081
$848$	9.00000	0.309061
$849$	21.0000	0.720718
$850$	0	0
$851$	7.00000	0.239957
$852$	2.00000	0.0685189
$853$	−21.0000	−0.719026	−0.359513	−	0.933140i	$-0.617057\pi$
$853$	−21.0000	−0.719026	−0.359513	+	0.933140i	$0.617057\pi$
$854$	42.0000	1.43721
$855$	0	0
$856$	−19.0000	−0.649407
$857$	27.0000	0.922302	0.461151	−	0.887322i	$-0.347437\pi$
$857$	27.0000	0.922302	0.461151	+	0.887322i	$0.347437\pi$
$858$	1.00000	0.0341394
$859$	−39.0000	−1.33066	−0.665331	−	0.746548i	$-0.731710\pi$
$859$	−39.0000	−1.33066	−0.665331	+	0.746548i	$0.731710\pi$
$860$	0	0
$861$	−12.0000	−0.408959
$862$	−3.00000	−0.102180
$863$	−12.0000	−0.408485	−0.204242	−	0.978920i	$-0.565473\pi$
$863$	−12.0000	−0.408485	−0.204242	+	0.978920i	$0.565473\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−19.0000	−0.645646
$867$	−16.0000	−0.543388
$868$	−6.00000	−0.203653
$869$	4.00000	0.135691
$870$	0	0
$871$	2.00000	0.0677674
$872$	19.0000	0.643421
$873$	8.00000	0.270759
$874$	−35.0000	−1.18389
$875$	0	0
$876$	9.00000	0.304082
$877$	−12.0000	−0.405211	−0.202606	−	0.979260i	$-0.564941\pi$
$877$	−12.0000	−0.405211	−0.202606	+	0.979260i	$0.564941\pi$
$878$	−24.0000	−0.809961
$879$	9.00000	0.303562
$880$	0	0
$881$	36.0000	1.21287	0.606435	−	0.795133i	$-0.292599\pi$
$881$	36.0000	1.21287	0.606435	+	0.795133i	$0.292599\pi$
$882$	−2.00000	−0.0673435
$883$	37.0000	1.24515	0.622575	−	0.782560i	$-0.286087\pi$
$883$	37.0000	1.24515	0.622575	+	0.782560i	$0.286087\pi$
$884$	−1.00000	−0.0336336
$885$	0	0
$886$	−36.0000	−1.20944
$887$	−18.0000	−0.604381	−0.302190	−	0.953248i	$-0.597718\pi$
$887$	−18.0000	−0.604381	−0.302190	+	0.953248i	$0.597718\pi$
$888$	1.00000	0.0335578
$889$	39.0000	1.30802
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	60.0000	2.00782
$894$	−6.00000	−0.200670
$895$	0	0
$896$	3.00000	0.100223
$897$	7.00000	0.233723
$898$	18.0000	0.600668
$899$	−4.00000	−0.133407
$900$	0	0
$901$	9.00000	0.299833
$902$	−4.00000	−0.133185
$903$	−36.0000	−1.19800
$904$	−2.00000	−0.0665190
$905$	0	0
$906$	17.0000	0.564787
$907$	−27.0000	−0.896520	−0.448260	−	0.893903i	$-0.647956\pi$
$907$	−27.0000	−0.896520	−0.448260	+	0.893903i	$0.647956\pi$
$908$	4.00000	0.132745
$909$	6.00000	0.199007
$910$	0	0
$911$	−40.0000	−1.32526	−0.662630	−	0.748947i	$-0.730560\pi$
$911$	−40.0000	−1.32526	−0.662630	+	0.748947i	$0.730560\pi$
$912$	−5.00000	−0.165567
$913$	1.00000	0.0330952
$914$	24.0000	0.793849
$915$	0	0
$916$	26.0000	0.859064
$917$	0	0
$918$	−1.00000	−0.0330049
$919$	−24.0000	−0.791687	−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000	−0.791687	−0.395843	+	0.918318i	$0.629548\pi$
$920$	0	0
$921$	−18.0000	−0.593120
$922$	−6.00000	−0.197599
$923$	−2.00000	−0.0658308
$924$	−3.00000	−0.0986928
$925$	0	0
$926$	−36.0000	−1.18303
$927$	−8.00000	−0.262754
$928$	2.00000	0.0656532
$929$	46.0000	1.50921	0.754606	−	0.656179i	$-0.227828\pi$
$929$	46.0000	1.50921	0.754606	+	0.656179i	$0.227828\pi$
$930$	0	0
$931$	−10.0000	−0.327737
$932$	14.0000	0.458585
$933$	24.0000	0.785725
$934$	−12.0000	−0.392652
$935$	0	0
$936$	1.00000	0.0326860
$937$	−14.0000	−0.457360	−0.228680	−	0.973502i	$-0.573441\pi$
$937$	−14.0000	−0.457360	−0.228680	+	0.973502i	$0.573441\pi$
$938$	−6.00000	−0.195907
$939$	−8.00000	−0.261070
$940$	0	0
$941$	30.0000	0.977972	0.488986	−	0.872292i	$-0.337367\pi$
$941$	30.0000	0.977972	0.488986	+	0.872292i	$0.337367\pi$
$942$	−8.00000	−0.260654
$943$	−28.0000	−0.911805
$944$	10.0000	0.325472
$945$	0	0
$946$	−12.0000	−0.390154
$947$	44.0000	1.42981	0.714904	−	0.699223i	$-0.246470\pi$
$947$	44.0000	1.42981	0.714904	+	0.699223i	$0.246470\pi$
$948$	4.00000	0.129914
$949$	−9.00000	−0.292152
$950$	0	0
$951$	6.00000	0.194563
$952$	3.00000	0.0972306
$953$	22.0000	0.712650	0.356325	−	0.934362i	$-0.384030\pi$
$953$	22.0000	0.712650	0.356325	+	0.934362i	$0.384030\pi$
$954$	−9.00000	−0.291386
$955$	0	0
$956$	20.0000	0.646846
$957$	−2.00000	−0.0646508
$958$	9.00000	0.290777
$959$	−30.0000	−0.968751
$960$	0	0
$961$	−27.0000	−0.870968
$962$	−1.00000	−0.0322413
$963$	19.0000	0.612266
$964$	−14.0000	−0.450910
$965$	0	0
$966$	−21.0000	−0.675664
$967$	58.0000	1.86515	0.932577	−	0.360971i	$-0.117555\pi$
$967$	58.0000	1.86515	0.932577	+	0.360971i	$0.117555\pi$
$968$	10.0000	0.321412
$969$	−5.00000	−0.160623
$970$	0	0
$971$	−4.00000	−0.128366	−0.0641831	−	0.997938i	$-0.520444\pi$
$971$	−4.00000	−0.128366	−0.0641831	+	0.997938i	$0.520444\pi$
$972$	1.00000	0.0320750
$973$	18.0000	0.577054
$974$	6.00000	0.192252
$975$	0	0
$976$	14.0000	0.448129
$977$	61.0000	1.95156	0.975781	−	0.218748i	$-0.0701972\pi$
$977$	61.0000	1.95156	0.975781	+	0.218748i	$0.0701972\pi$
$978$	−9.00000	−0.287788
$979$	1.00000	0.0319601
$980$	0	0
$981$	−19.0000	−0.606623
$982$	35.0000	1.11689
$983$	42.0000	1.33959	0.669796	−	0.742545i	$-0.266382\pi$
$983$	42.0000	1.33959	0.669796	+	0.742545i	$0.266382\pi$
$984$	−4.00000	−0.127515
$985$	0	0
$986$	2.00000	0.0636930
$987$	36.0000	1.14589
$988$	5.00000	0.159071
$989$	−84.0000	−2.67104
$990$	0	0
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	−2.00000	−0.0635001
$993$	12.0000	0.380808
$994$	6.00000	0.190308
$995$	0	0
$996$	1.00000	0.0316862
$997$	−39.0000	−1.23514	−0.617571	−	0.786515i	$-0.711883\pi$
$997$	−39.0000	−1.23514	−0.617571	+	0.786515i	$0.711883\pi$
$998$	11.0000	0.348199
$999$	−1.00000	−0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5550.2.a.m.1.1		1
5.4	even	2		1110.2.a.l.1.1	✓	1
15.14	odd	2		3330.2.a.e.1.1		1
20.19	odd	2		8880.2.a.y.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1110.2.a.l.1.1	✓	1	5.4	even	2
3330.2.a.e.1.1		1	15.14	odd	2
5550.2.a.m.1.1		1	1.1	even	1	trivial
8880.2.a.y.1.1		1	20.19	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:	\( N \)	\(=\)	\( 5550 = 2 \cdot 3 \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5550.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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