LMFDB - Embedded newform 2550.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2550,2,Mod(1,2550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2550, 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("2550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2550.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:	$20.3618525154$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2550.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -3.00000 q^{11} -1.00000 q^{12} -4.00000 q^{13} +3.00000 q^{14} +1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{18} -5.00000 q^{19} +3.00000 q^{21} +3.00000 q^{22} -4.00000 q^{23} +1.00000 q^{24} +4.00000 q^{26} -1.00000 q^{27} -3.00000 q^{28} +7.00000 q^{31} -1.00000 q^{32} +3.00000 q^{33} +1.00000 q^{34} +1.00000 q^{36} -3.00000 q^{37} +5.00000 q^{38} +4.00000 q^{39} +2.00000 q^{41} -3.00000 q^{42} +1.00000 q^{43} -3.00000 q^{44} +4.00000 q^{46} -3.00000 q^{47} -1.00000 q^{48} +2.00000 q^{49} +1.00000 q^{51} -4.00000 q^{52} +11.0000 q^{53} +1.00000 q^{54} +3.00000 q^{56} +5.00000 q^{57} +2.00000 q^{61} -7.00000 q^{62} -3.00000 q^{63} +1.00000 q^{64} -3.00000 q^{66} -13.0000 q^{67} -1.00000 q^{68} +4.00000 q^{69} +2.00000 q^{71} -1.00000 q^{72} +6.00000 q^{73} +3.00000 q^{74} -5.00000 q^{76} +9.00000 q^{77} -4.00000 q^{78} -5.00000 q^{79} +1.00000 q^{81} -2.00000 q^{82} +16.0000 q^{83} +3.00000 q^{84} -1.00000 q^{86} +3.00000 q^{88} -10.0000 q^{89} +12.0000 q^{91} -4.00000 q^{92} -7.00000 q^{93} +3.00000 q^{94} +1.00000 q^{96} +2.00000 q^{97} -2.00000 q^{98} -3.00000 q^{99} +O(q^{100})\)

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
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	−1.00000	−0.288675
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	3.00000	0.801784
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−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
$22$	3.00000	0.639602
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	4.00000	0.784465
$27$	−1.00000	−0.192450
$28$	−3.00000	−0.566947
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	7.00000	1.25724	0.628619	−	0.777714i	$-0.283621\pi$
$31$	7.00000	1.25724	0.628619	+	0.777714i	$0.283621\pi$
$32$	−1.00000	−0.176777
$33$	3.00000	0.522233
$34$	1.00000	0.171499
$35$	0	0
$36$	1.00000	0.166667
$37$	−3.00000	−0.493197	−0.246598	−	0.969118i	$-0.579313\pi$
$37$	−3.00000	−0.493197	−0.246598	+	0.969118i	$0.579313\pi$
$38$	5.00000	0.811107
$39$	4.00000	0.640513
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	−3.00000	−0.462910
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	−3.00000	−0.452267
$45$	0	0
$46$	4.00000	0.589768
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	−1.00000	−0.144338
$49$	2.00000	0.285714
$50$	0	0
$51$	1.00000	0.140028
$52$	−4.00000	−0.554700
$53$	11.0000	1.51097	0.755483	−	0.655168i	$-0.227402\pi$
$53$	11.0000	1.51097	0.755483	+	0.655168i	$0.227402\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	3.00000	0.400892
$57$	5.00000	0.662266
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	−7.00000	−0.889001
$63$	−3.00000	−0.377964
$64$	1.00000	0.125000
$65$	0	0
$66$	−3.00000	−0.369274
$67$	−13.0000	−1.58820	−0.794101	−	0.607785i	$-0.792058\pi$
$67$	−13.0000	−1.58820	−0.794101	+	0.607785i	$0.792058\pi$
$68$	−1.00000	−0.121268
$69$	4.00000	0.481543
$70$	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
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	3.00000	0.348743
$75$	0	0
$76$	−5.00000	−0.573539
$77$	9.00000	1.02565
$78$	−4.00000	−0.452911
$79$	−5.00000	−0.562544	−0.281272	−	0.959628i	$-0.590756\pi$
$79$	−5.00000	−0.562544	−0.281272	+	0.959628i	$0.590756\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−2.00000	−0.220863
$83$	16.0000	1.75623	0.878114	−	0.478451i	$-0.158802\pi$
$83$	16.0000	1.75623	0.878114	+	0.478451i	$0.158802\pi$
$84$	3.00000	0.327327
$85$	0	0
$86$	−1.00000	−0.107833
$87$	0	0
$88$	3.00000	0.319801
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	12.0000	1.25794
$92$	−4.00000	−0.417029
$93$	−7.00000	−0.725866
$94$	3.00000	0.309426
$95$	0	0
$96$	1.00000	0.102062
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	−2.00000	−0.202031
$99$	−3.00000	−0.301511
$100$	0	0
$101$	7.00000	0.696526	0.348263	−	0.937397i	$-0.386772\pi$
$101$	7.00000	0.696526	0.348263	+	0.937397i	$0.386772\pi$
$102$	−1.00000	−0.0990148
$103$	−14.0000	−1.37946	−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000	−1.37946	−0.689730	+	0.724066i	$0.742271\pi$
$104$	4.00000	0.392232
$105$	0	0
$106$	−11.0000	−1.06841
$107$	17.0000	1.64345	0.821726	−	0.569883i	$-0.193011\pi$
$107$	17.0000	1.64345	0.821726	+	0.569883i	$0.193011\pi$
$108$	−1.00000	−0.0962250
$109$	5.00000	0.478913	0.239457	−	0.970907i	$-0.423031\pi$
$109$	5.00000	0.478913	0.239457	+	0.970907i	$0.423031\pi$
$110$	0	0
$111$	3.00000	0.284747
$112$	−3.00000	−0.283473
$113$	1.00000	0.0940721	0.0470360	−	0.998893i	$-0.485022\pi$
$113$	1.00000	0.0940721	0.0470360	+	0.998893i	$0.485022\pi$
$114$	−5.00000	−0.468293
$115$	0	0
$116$	0	0
$117$	−4.00000	−0.369800
$118$	0	0
$119$	3.00000	0.275010
$120$	0	0
$121$	−2.00000	−0.181818
$122$	−2.00000	−0.181071
$123$	−2.00000	−0.180334
$124$	7.00000	0.628619
$125$	0	0
$126$	3.00000	0.267261
$127$	−18.0000	−1.59724	−0.798621	−	0.601834i	$-0.794437\pi$
$127$	−18.0000	−1.59724	−0.798621	+	0.601834i	$0.794437\pi$
$128$	−1.00000	−0.0883883
$129$	−1.00000	−0.0880451
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	3.00000	0.261116
$133$	15.0000	1.30066
$134$	13.0000	1.12303
$135$	0	0
$136$	1.00000	0.0857493
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	−4.00000	−0.340503
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	0	0
$141$	3.00000	0.252646
$142$	−2.00000	−0.167836
$143$	12.0000	1.00349
$144$	1.00000	0.0833333
$145$	0	0
$146$	−6.00000	−0.496564
$147$	−2.00000	−0.164957
$148$	−3.00000	−0.246598
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	5.00000	0.405554
$153$	−1.00000	−0.0808452
$154$	−9.00000	−0.725241
$155$	0	0
$156$	4.00000	0.320256
$157$	12.0000	0.957704	0.478852	−	0.877896i	$-0.341053\pi$
$157$	12.0000	0.957704	0.478852	+	0.877896i	$0.341053\pi$
$158$	5.00000	0.397779
$159$	−11.0000	−0.872357
$160$	0	0
$161$	12.0000	0.945732
$162$	−1.00000	−0.0785674
$163$	6.00000	0.469956	0.234978	−	0.972001i	$-0.424498\pi$
$163$	6.00000	0.469956	0.234978	+	0.972001i	$0.424498\pi$
$164$	2.00000	0.156174
$165$	0	0
$166$	−16.0000	−1.24184
$167$	−18.0000	−1.39288	−0.696441	−	0.717614i	$-0.745234\pi$
$167$	−18.0000	−1.39288	−0.696441	+	0.717614i	$0.745234\pi$
$168$	−3.00000	−0.231455
$169$	3.00000	0.230769
$170$	0	0
$171$	−5.00000	−0.382360
$172$	1.00000	0.0762493
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	0	0
$176$	−3.00000	−0.226134
$177$	0	0
$178$	10.0000	0.749532
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	7.00000	0.520306	0.260153	−	0.965567i	$-0.416227\pi$
$181$	7.00000	0.520306	0.260153	+	0.965567i	$0.416227\pi$
$182$	−12.0000	−0.889499
$183$	−2.00000	−0.147844
$184$	4.00000	0.294884
$185$	0	0
$186$	7.00000	0.513265
$187$	3.00000	0.219382
$188$	−3.00000	−0.218797
$189$	3.00000	0.218218
$190$	0	0
$191$	−3.00000	−0.217072	−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000	−0.217072	−0.108536	+	0.994092i	$0.534616\pi$
$192$	−1.00000	−0.0721688
$193$	6.00000	0.431889	0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000	0.431889	0.215945	+	0.976406i	$0.430717\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	2.00000	0.142857
$197$	12.0000	0.854965	0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000	0.854965	0.427482	+	0.904024i	$0.359401\pi$
$198$	3.00000	0.213201
$199$	−25.0000	−1.77220	−0.886102	−	0.463491i	$-0.846597\pi$
$199$	−25.0000	−1.77220	−0.886102	+	0.463491i	$0.846597\pi$
$200$	0	0
$201$	13.0000	0.916949
$202$	−7.00000	−0.492518
$203$	0	0
$204$	1.00000	0.0700140
$205$	0	0
$206$	14.0000	0.975426
$207$	−4.00000	−0.278019
$208$	−4.00000	−0.277350
$209$	15.0000	1.03757
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	11.0000	0.755483
$213$	−2.00000	−0.137038
$214$	−17.0000	−1.16210
$215$	0	0
$216$	1.00000	0.0680414
$217$	−21.0000	−1.42557
$218$	−5.00000	−0.338643
$219$	−6.00000	−0.405442
$220$	0	0
$221$	4.00000	0.269069
$222$	−3.00000	−0.201347
$223$	26.0000	1.74109	0.870544	−	0.492090i	$-0.163767\pi$
$223$	26.0000	1.74109	0.870544	+	0.492090i	$0.163767\pi$
$224$	3.00000	0.200446
$225$	0	0
$226$	−1.00000	−0.0665190
$227$	27.0000	1.79205	0.896026	−	0.444001i	$-0.146441\pi$
$227$	27.0000	1.79205	0.896026	+	0.444001i	$0.146441\pi$
$228$	5.00000	0.331133
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	0	0
$231$	−9.00000	−0.592157
$232$	0	0
$233$	26.0000	1.70332	0.851658	−	0.524097i	$-0.175597\pi$
$233$	26.0000	1.70332	0.851658	+	0.524097i	$0.175597\pi$
$234$	4.00000	0.261488
$235$	0	0
$236$	0	0
$237$	5.00000	0.324785
$238$	−3.00000	−0.194461
$239$	15.0000	0.970269	0.485135	−	0.874439i	$-0.338771\pi$
$239$	15.0000	0.970269	0.485135	+	0.874439i	$0.338771\pi$
$240$	0	0
$241$	12.0000	0.772988	0.386494	−	0.922292i	$-0.373686\pi$
$241$	12.0000	0.772988	0.386494	+	0.922292i	$0.373686\pi$
$242$	2.00000	0.128565
$243$	−1.00000	−0.0641500
$244$	2.00000	0.128037
$245$	0	0
$246$	2.00000	0.127515
$247$	20.0000	1.27257
$248$	−7.00000	−0.444500
$249$	−16.0000	−1.01396
$250$	0	0
$251$	−18.0000	−1.13615	−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000	−1.13615	−0.568075	+	0.822977i	$0.692312\pi$
$252$	−3.00000	−0.188982
$253$	12.0000	0.754434
$254$	18.0000	1.12942
$255$	0	0
$256$	1.00000	0.0625000
$257$	−8.00000	−0.499026	−0.249513	−	0.968371i	$-0.580271\pi$
$257$	−8.00000	−0.499026	−0.249513	+	0.968371i	$0.580271\pi$
$258$	1.00000	0.0622573
$259$	9.00000	0.559233
$260$	0	0
$261$	0	0
$262$	−12.0000	−0.741362
$263$	−19.0000	−1.17159	−0.585795	−	0.810459i	$-0.699218\pi$
$263$	−19.0000	−1.17159	−0.585795	+	0.810459i	$0.699218\pi$
$264$	−3.00000	−0.184637
$265$	0	0
$266$	−15.0000	−0.919709
$267$	10.0000	0.611990
$268$	−13.0000	−0.794101
$269$	20.0000	1.21942	0.609711	−	0.792624i	$-0.291286\pi$
$269$	20.0000	1.21942	0.609711	+	0.792624i	$0.291286\pi$
$270$	0	0
$271$	32.0000	1.94386	0.971931	−	0.235267i	$-0.0755965\pi$
$271$	32.0000	1.94386	0.971931	+	0.235267i	$0.0755965\pi$
$272$	−1.00000	−0.0606339
$273$	−12.0000	−0.726273
$274$	−2.00000	−0.120824
$275$	0	0
$276$	4.00000	0.240772
$277$	−23.0000	−1.38194	−0.690968	−	0.722885i	$-0.742815\pi$
$277$	−23.0000	−1.38194	−0.690968	+	0.722885i	$0.742815\pi$
$278$	20.0000	1.19952
$279$	7.00000	0.419079
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	−3.00000	−0.178647
$283$	−14.0000	−0.832214	−0.416107	−	0.909316i	$-0.636606\pi$
$283$	−14.0000	−0.832214	−0.416107	+	0.909316i	$0.636606\pi$
$284$	2.00000	0.118678
$285$	0	0
$286$	−12.0000	−0.709575
$287$	−6.00000	−0.354169
$288$	−1.00000	−0.0589256
$289$	1.00000	0.0588235
$290$	0	0
$291$	−2.00000	−0.117242
$292$	6.00000	0.351123
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	2.00000	0.116642
$295$	0	0
$296$	3.00000	0.174371
$297$	3.00000	0.174078
$298$	−10.0000	−0.579284
$299$	16.0000	0.925304
$300$	0	0
$301$	−3.00000	−0.172917
$302$	−12.0000	−0.690522
$303$	−7.00000	−0.402139
$304$	−5.00000	−0.286770
$305$	0	0
$306$	1.00000	0.0571662
$307$	32.0000	1.82634	0.913168	−	0.407583i	$-0.133628\pi$
$307$	32.0000	1.82634	0.913168	+	0.407583i	$0.133628\pi$
$308$	9.00000	0.512823
$309$	14.0000	0.796432
$310$	0	0
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	−4.00000	−0.226455
$313$	−4.00000	−0.226093	−0.113047	−	0.993590i	$-0.536061\pi$
$313$	−4.00000	−0.226093	−0.113047	+	0.993590i	$0.536061\pi$
$314$	−12.0000	−0.677199
$315$	0	0
$316$	−5.00000	−0.281272
$317$	32.0000	1.79730	0.898650	−	0.438667i	$-0.144549\pi$
$317$	32.0000	1.79730	0.898650	+	0.438667i	$0.144549\pi$
$318$	11.0000	0.616849
$319$	0	0
$320$	0	0
$321$	−17.0000	−0.948847
$322$	−12.0000	−0.668734
$323$	5.00000	0.278207
$324$	1.00000	0.0555556
$325$	0	0
$326$	−6.00000	−0.332309
$327$	−5.00000	−0.276501
$328$	−2.00000	−0.110432
$329$	9.00000	0.496186
$330$	0	0
$331$	−23.0000	−1.26419	−0.632097	−	0.774889i	$-0.717806\pi$
$331$	−23.0000	−1.26419	−0.632097	+	0.774889i	$0.717806\pi$
$332$	16.0000	0.878114
$333$	−3.00000	−0.164399
$334$	18.0000	0.984916
$335$	0	0
$336$	3.00000	0.163663
$337$	12.0000	0.653682	0.326841	−	0.945079i	$-0.394016\pi$
$337$	12.0000	0.653682	0.326841	+	0.945079i	$0.394016\pi$
$338$	−3.00000	−0.163178
$339$	−1.00000	−0.0543125
$340$	0	0
$341$	−21.0000	−1.13721
$342$	5.00000	0.270369
$343$	15.0000	0.809924
$344$	−1.00000	−0.0539164
$345$	0	0
$346$	−6.00000	−0.322562
$347$	27.0000	1.44944	0.724718	−	0.689046i	$-0.241970\pi$
$347$	27.0000	1.44944	0.724718	+	0.689046i	$0.241970\pi$
$348$	0	0
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	0	0
$351$	4.00000	0.213504
$352$	3.00000	0.159901
$353$	16.0000	0.851594	0.425797	−	0.904819i	$-0.359994\pi$
$353$	16.0000	0.851594	0.425797	+	0.904819i	$0.359994\pi$
$354$	0	0
$355$	0	0
$356$	−10.0000	−0.529999
$357$	−3.00000	−0.158777
$358$	0	0
$359$	5.00000	0.263890	0.131945	−	0.991257i	$-0.457878\pi$
$359$	5.00000	0.263890	0.131945	+	0.991257i	$0.457878\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	−7.00000	−0.367912
$363$	2.00000	0.104973
$364$	12.0000	0.628971
$365$	0	0
$366$	2.00000	0.104542
$367$	7.00000	0.365397	0.182699	−	0.983169i	$-0.441517\pi$
$367$	7.00000	0.365397	0.182699	+	0.983169i	$0.441517\pi$
$368$	−4.00000	−0.208514
$369$	2.00000	0.104116
$370$	0	0
$371$	−33.0000	−1.71327
$372$	−7.00000	−0.362933
$373$	−14.0000	−0.724893	−0.362446	−	0.932005i	$-0.618058\pi$
$373$	−14.0000	−0.724893	−0.362446	+	0.932005i	$0.618058\pi$
$374$	−3.00000	−0.155126
$375$	0	0
$376$	3.00000	0.154713
$377$	0	0
$378$	−3.00000	−0.154303
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	18.0000	0.922168
$382$	3.00000	0.153493
$383$	−4.00000	−0.204390	−0.102195	−	0.994764i	$-0.532587\pi$
$383$	−4.00000	−0.204390	−0.102195	+	0.994764i	$0.532587\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−6.00000	−0.305392
$387$	1.00000	0.0508329
$388$	2.00000	0.101535
$389$	−5.00000	−0.253510	−0.126755	−	0.991934i	$-0.540456\pi$
$389$	−5.00000	−0.253510	−0.126755	+	0.991934i	$0.540456\pi$
$390$	0	0
$391$	4.00000	0.202289
$392$	−2.00000	−0.101015
$393$	−12.0000	−0.605320
$394$	−12.0000	−0.604551
$395$	0	0
$396$	−3.00000	−0.150756
$397$	−3.00000	−0.150566	−0.0752828	−	0.997162i	$-0.523986\pi$
$397$	−3.00000	−0.150566	−0.0752828	+	0.997162i	$0.523986\pi$
$398$	25.0000	1.25314
$399$	−15.0000	−0.750939
$400$	0	0
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	−13.0000	−0.648381
$403$	−28.0000	−1.39478
$404$	7.00000	0.348263
$405$	0	0
$406$	0	0
$407$	9.00000	0.446113
$408$	−1.00000	−0.0495074
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	0	0
$411$	−2.00000	−0.0986527
$412$	−14.0000	−0.689730
$413$	0	0
$414$	4.00000	0.196589
$415$	0	0
$416$	4.00000	0.196116
$417$	20.0000	0.979404
$418$	−15.0000	−0.733674
$419$	−20.0000	−0.977064	−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000	−0.977064	−0.488532	+	0.872546i	$0.662467\pi$
$420$	0	0
$421$	−28.0000	−1.36464	−0.682318	−	0.731055i	$-0.739028\pi$
$421$	−28.0000	−1.36464	−0.682318	+	0.731055i	$0.739028\pi$
$422$	−12.0000	−0.584151
$423$	−3.00000	−0.145865
$424$	−11.0000	−0.534207
$425$	0	0
$426$	2.00000	0.0969003
$427$	−6.00000	−0.290360
$428$	17.0000	0.821726
$429$	−12.0000	−0.579365
$430$	0	0
$431$	2.00000	0.0963366	0.0481683	−	0.998839i	$-0.484662\pi$
$431$	2.00000	0.0963366	0.0481683	+	0.998839i	$0.484662\pi$
$432$	−1.00000	−0.0481125
$433$	21.0000	1.00920	0.504598	−	0.863355i	$-0.331641\pi$
$433$	21.0000	1.00920	0.504598	+	0.863355i	$0.331641\pi$
$434$	21.0000	1.00803
$435$	0	0
$436$	5.00000	0.239457
$437$	20.0000	0.956730
$438$	6.00000	0.286691
$439$	20.0000	0.954548	0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000	0.954548	0.477274	+	0.878755i	$0.341625\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	−4.00000	−0.190261
$443$	−14.0000	−0.665160	−0.332580	−	0.943075i	$-0.607919\pi$
$443$	−14.0000	−0.665160	−0.332580	+	0.943075i	$0.607919\pi$
$444$	3.00000	0.142374
$445$	0	0
$446$	−26.0000	−1.23114
$447$	−10.0000	−0.472984
$448$	−3.00000	−0.141737
$449$	5.00000	0.235965	0.117982	−	0.993016i	$-0.462357\pi$
$449$	5.00000	0.235965	0.117982	+	0.993016i	$0.462357\pi$
$450$	0	0
$451$	−6.00000	−0.282529
$452$	1.00000	0.0470360
$453$	−12.0000	−0.563809
$454$	−27.0000	−1.26717
$455$	0	0
$456$	−5.00000	−0.234146
$457$	−13.0000	−0.608114	−0.304057	−	0.952654i	$-0.598341\pi$
$457$	−13.0000	−0.608114	−0.304057	+	0.952654i	$0.598341\pi$
$458$	0	0
$459$	1.00000	0.0466760
$460$	0	0
$461$	37.0000	1.72326	0.861631	−	0.507535i	$-0.169443\pi$
$461$	37.0000	1.72326	0.861631	+	0.507535i	$0.169443\pi$
$462$	9.00000	0.418718
$463$	−24.0000	−1.11537	−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000	−1.11537	−0.557687	+	0.830051i	$0.688311\pi$
$464$	0	0
$465$	0	0
$466$	−26.0000	−1.20443
$467$	−18.0000	−0.832941	−0.416470	−	0.909149i	$-0.636733\pi$
$467$	−18.0000	−0.832941	−0.416470	+	0.909149i	$0.636733\pi$
$468$	−4.00000	−0.184900
$469$	39.0000	1.80085
$470$	0	0
$471$	−12.0000	−0.552931
$472$	0	0
$473$	−3.00000	−0.137940
$474$	−5.00000	−0.229658
$475$	0	0
$476$	3.00000	0.137505
$477$	11.0000	0.503655
$478$	−15.0000	−0.686084
$479$	−10.0000	−0.456912	−0.228456	−	0.973554i	$-0.573368\pi$
$479$	−10.0000	−0.456912	−0.228456	+	0.973554i	$0.573368\pi$
$480$	0	0
$481$	12.0000	0.547153
$482$	−12.0000	−0.546585
$483$	−12.0000	−0.546019
$484$	−2.00000	−0.0909091
$485$	0	0
$486$	1.00000	0.0453609
$487$	32.0000	1.45006	0.725029	−	0.688718i	$-0.241826\pi$
$487$	32.0000	1.45006	0.725029	+	0.688718i	$0.241826\pi$
$488$	−2.00000	−0.0905357
$489$	−6.00000	−0.271329
$490$	0	0
$491$	2.00000	0.0902587	0.0451294	−	0.998981i	$-0.485630\pi$
$491$	2.00000	0.0902587	0.0451294	+	0.998981i	$0.485630\pi$
$492$	−2.00000	−0.0901670
$493$	0	0
$494$	−20.0000	−0.899843
$495$	0	0
$496$	7.00000	0.314309
$497$	−6.00000	−0.269137
$498$	16.0000	0.716977
$499$	10.0000	0.447661	0.223831	−	0.974628i	$-0.428144\pi$
$499$	10.0000	0.447661	0.223831	+	0.974628i	$0.428144\pi$
$500$	0	0
$501$	18.0000	0.804181
$502$	18.0000	0.803379
$503$	−14.0000	−0.624229	−0.312115	−	0.950044i	$-0.601037\pi$
$503$	−14.0000	−0.624229	−0.312115	+	0.950044i	$0.601037\pi$
$504$	3.00000	0.133631
$505$	0	0
$506$	−12.0000	−0.533465
$507$	−3.00000	−0.133235
$508$	−18.0000	−0.798621
$509$	−15.0000	−0.664863	−0.332432	−	0.943127i	$-0.607869\pi$
$509$	−15.0000	−0.664863	−0.332432	+	0.943127i	$0.607869\pi$
$510$	0	0
$511$	−18.0000	−0.796273
$512$	−1.00000	−0.0441942
$513$	5.00000	0.220755
$514$	8.00000	0.352865
$515$	0	0
$516$	−1.00000	−0.0440225
$517$	9.00000	0.395820
$518$	−9.00000	−0.395437
$519$	−6.00000	−0.263371
$520$	0	0
$521$	−3.00000	−0.131432	−0.0657162	−	0.997838i	$-0.520933\pi$
$521$	−3.00000	−0.131432	−0.0657162	+	0.997838i	$0.520933\pi$
$522$	0	0
$523$	−44.0000	−1.92399	−0.961993	−	0.273075i	$-0.911959\pi$
$523$	−44.0000	−1.92399	−0.961993	+	0.273075i	$0.911959\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	19.0000	0.828439
$527$	−7.00000	−0.304925
$528$	3.00000	0.130558
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	15.0000	0.650332
$533$	−8.00000	−0.346518
$534$	−10.0000	−0.432742
$535$	0	0
$536$	13.0000	0.561514
$537$	0	0
$538$	−20.0000	−0.862261
$539$	−6.00000	−0.258438
$540$	0	0
$541$	−13.0000	−0.558914	−0.279457	−	0.960158i	$-0.590154\pi$
$541$	−13.0000	−0.558914	−0.279457	+	0.960158i	$0.590154\pi$
$542$	−32.0000	−1.37452
$543$	−7.00000	−0.300399
$544$	1.00000	0.0428746
$545$	0	0
$546$	12.0000	0.513553
$547$	2.00000	0.0855138	0.0427569	−	0.999086i	$-0.486386\pi$
$547$	2.00000	0.0855138	0.0427569	+	0.999086i	$0.486386\pi$
$548$	2.00000	0.0854358
$549$	2.00000	0.0853579
$550$	0	0
$551$	0	0
$552$	−4.00000	−0.170251
$553$	15.0000	0.637865
$554$	23.0000	0.977176
$555$	0	0
$556$	−20.0000	−0.848189
$557$	−23.0000	−0.974541	−0.487271	−	0.873251i	$-0.662007\pi$
$557$	−23.0000	−0.974541	−0.487271	+	0.873251i	$0.662007\pi$
$558$	−7.00000	−0.296334
$559$	−4.00000	−0.169182
$560$	0	0
$561$	−3.00000	−0.126660
$562$	18.0000	0.759284
$563$	−24.0000	−1.01148	−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$564$	3.00000	0.126323
$565$	0	0
$566$	14.0000	0.588464
$567$	−3.00000	−0.125988
$568$	−2.00000	−0.0839181
$569$	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
$571$	2.00000	0.0836974	0.0418487	−	0.999124i	$-0.486675\pi$
$571$	2.00000	0.0836974	0.0418487	+	0.999124i	$0.486675\pi$
$572$	12.0000	0.501745
$573$	3.00000	0.125327
$574$	6.00000	0.250435
$575$	0	0
$576$	1.00000	0.0416667
$577$	−43.0000	−1.79011	−0.895057	−	0.445952i	$-0.852865\pi$
$577$	−43.0000	−1.79011	−0.895057	+	0.445952i	$0.852865\pi$
$578$	−1.00000	−0.0415945
$579$	−6.00000	−0.249351
$580$	0	0
$581$	−48.0000	−1.99138
$582$	2.00000	0.0829027
$583$	−33.0000	−1.36672
$584$	−6.00000	−0.248282
$585$	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	$0.621146\pi$
$588$	−2.00000	−0.0824786
$589$	−35.0000	−1.44215
$590$	0	0
$591$	−12.0000	−0.493614
$592$	−3.00000	−0.123299
$593$	36.0000	1.47834	0.739171	−	0.673517i	$-0.235217\pi$
$593$	36.0000	1.47834	0.739171	+	0.673517i	$0.235217\pi$
$594$	−3.00000	−0.123091
$595$	0	0
$596$	10.0000	0.409616
$597$	25.0000	1.02318
$598$	−16.0000	−0.654289
$599$	−35.0000	−1.43006	−0.715031	−	0.699093i	$-0.753587\pi$
$599$	−35.0000	−1.43006	−0.715031	+	0.699093i	$0.753587\pi$
$600$	0	0
$601$	−28.0000	−1.14214	−0.571072	−	0.820900i	$-0.693472\pi$
$601$	−28.0000	−1.14214	−0.571072	+	0.820900i	$0.693472\pi$
$602$	3.00000	0.122271
$603$	−13.0000	−0.529401
$604$	12.0000	0.488273
$605$	0	0
$606$	7.00000	0.284356
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	5.00000	0.202777
$609$	0	0
$610$	0	0
$611$	12.0000	0.485468
$612$	−1.00000	−0.0404226
$613$	46.0000	1.85792	0.928961	−	0.370177i	$-0.120703\pi$
$613$	46.0000	1.85792	0.928961	+	0.370177i	$0.120703\pi$
$614$	−32.0000	−1.29141
$615$	0	0
$616$	−9.00000	−0.362620
$617$	−33.0000	−1.32853	−0.664265	−	0.747497i	$-0.731255\pi$
$617$	−33.0000	−1.32853	−0.664265	+	0.747497i	$0.731255\pi$
$618$	−14.0000	−0.563163
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	18.0000	0.721734
$623$	30.0000	1.20192
$624$	4.00000	0.160128
$625$	0	0
$626$	4.00000	0.159872
$627$	−15.0000	−0.599042
$628$	12.0000	0.478852
$629$	3.00000	0.119618
$630$	0	0
$631$	32.0000	1.27390	0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000	1.27390	0.636950	+	0.770905i	$0.280196\pi$
$632$	5.00000	0.198889
$633$	−12.0000	−0.476957
$634$	−32.0000	−1.27088
$635$	0	0
$636$	−11.0000	−0.436178
$637$	−8.00000	−0.316972
$638$	0	0
$639$	2.00000	0.0791188
$640$	0	0
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	17.0000	0.670936
$643$	−4.00000	−0.157745	−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000	−0.157745	−0.0788723	+	0.996885i	$0.525132\pi$
$644$	12.0000	0.472866
$645$	0	0
$646$	−5.00000	−0.196722
$647$	−8.00000	−0.314512	−0.157256	−	0.987558i	$-0.550265\pi$
$647$	−8.00000	−0.314512	−0.157256	+	0.987558i	$0.550265\pi$
$648$	−1.00000	−0.0392837
$649$	0	0
$650$	0	0
$651$	21.0000	0.823055
$652$	6.00000	0.234978
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	5.00000	0.195515
$655$	0	0
$656$	2.00000	0.0780869
$657$	6.00000	0.234082
$658$	−9.00000	−0.350857
$659$	−20.0000	−0.779089	−0.389545	−	0.921008i	$-0.627368\pi$
$659$	−20.0000	−0.779089	−0.389545	+	0.921008i	$0.627368\pi$
$660$	0	0
$661$	2.00000	0.0777910	0.0388955	−	0.999243i	$-0.487616\pi$
$661$	2.00000	0.0777910	0.0388955	+	0.999243i	$0.487616\pi$
$662$	23.0000	0.893920
$663$	−4.00000	−0.155347
$664$	−16.0000	−0.620920
$665$	0	0
$666$	3.00000	0.116248
$667$	0	0
$668$	−18.0000	−0.696441
$669$	−26.0000	−1.00522
$670$	0	0
$671$	−6.00000	−0.231627
$672$	−3.00000	−0.115728
$673$	−44.0000	−1.69608	−0.848038	−	0.529936i	$-0.822216\pi$
$673$	−44.0000	−1.69608	−0.848038	+	0.529936i	$0.822216\pi$
$674$	−12.0000	−0.462223
$675$	0	0
$676$	3.00000	0.115385
$677$	42.0000	1.61419	0.807096	−	0.590421i	$-0.201038\pi$
$677$	42.0000	1.61419	0.807096	+	0.590421i	$0.201038\pi$
$678$	1.00000	0.0384048
$679$	−6.00000	−0.230259
$680$	0	0
$681$	−27.0000	−1.03464
$682$	21.0000	0.804132
$683$	−24.0000	−0.918334	−0.459167	−	0.888350i	$-0.651852\pi$
$683$	−24.0000	−0.918334	−0.459167	+	0.888350i	$0.651852\pi$
$684$	−5.00000	−0.191180
$685$	0	0
$686$	−15.0000	−0.572703
$687$	0	0
$688$	1.00000	0.0381246
$689$	−44.0000	−1.67627
$690$	0	0
$691$	42.0000	1.59776	0.798878	−	0.601494i	$-0.205427\pi$
$691$	42.0000	1.59776	0.798878	+	0.601494i	$0.205427\pi$
$692$	6.00000	0.228086
$693$	9.00000	0.341882
$694$	−27.0000	−1.02491
$695$	0	0
$696$	0	0
$697$	−2.00000	−0.0757554
$698$	10.0000	0.378506
$699$	−26.0000	−0.983410
$700$	0	0
$701$	−38.0000	−1.43524	−0.717620	−	0.696435i	$-0.754769\pi$
$701$	−38.0000	−1.43524	−0.717620	+	0.696435i	$0.754769\pi$
$702$	−4.00000	−0.150970
$703$	15.0000	0.565736
$704$	−3.00000	−0.113067
$705$	0	0
$706$	−16.0000	−0.602168
$707$	−21.0000	−0.789786
$708$	0	0
$709$	5.00000	0.187779	0.0938895	−	0.995583i	$-0.470070\pi$
$709$	5.00000	0.187779	0.0938895	+	0.995583i	$0.470070\pi$
$710$	0	0
$711$	−5.00000	−0.187515
$712$	10.0000	0.374766
$713$	−28.0000	−1.04861
$714$	3.00000	0.112272
$715$	0	0
$716$	0	0
$717$	−15.0000	−0.560185
$718$	−5.00000	−0.186598
$719$	30.0000	1.11881	0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000	1.11881	0.559406	+	0.828894i	$0.311029\pi$
$720$	0	0
$721$	42.0000	1.56416
$722$	−6.00000	−0.223297
$723$	−12.0000	−0.446285
$724$	7.00000	0.260153
$725$	0	0
$726$	−2.00000	−0.0742270
$727$	−28.0000	−1.03846	−0.519231	−	0.854634i	$-0.673782\pi$
$727$	−28.0000	−1.03846	−0.519231	+	0.854634i	$0.673782\pi$
$728$	−12.0000	−0.444750
$729$	1.00000	0.0370370
$730$	0	0
$731$	−1.00000	−0.0369863
$732$	−2.00000	−0.0739221
$733$	26.0000	0.960332	0.480166	−	0.877178i	$-0.340576\pi$
$733$	26.0000	0.960332	0.480166	+	0.877178i	$0.340576\pi$
$734$	−7.00000	−0.258375
$735$	0	0
$736$	4.00000	0.147442
$737$	39.0000	1.43658
$738$	−2.00000	−0.0736210
$739$	35.0000	1.28750	0.643748	−	0.765238i	$-0.277379\pi$
$739$	35.0000	1.28750	0.643748	+	0.765238i	$0.277379\pi$
$740$	0	0
$741$	−20.0000	−0.734718
$742$	33.0000	1.21147
$743$	26.0000	0.953847	0.476924	−	0.878945i	$-0.341752\pi$
$743$	26.0000	0.953847	0.476924	+	0.878945i	$0.341752\pi$
$744$	7.00000	0.256632
$745$	0	0
$746$	14.0000	0.512576
$747$	16.0000	0.585409
$748$	3.00000	0.109691
$749$	−51.0000	−1.86350
$750$	0	0
$751$	12.0000	0.437886	0.218943	−	0.975738i	$-0.429739\pi$
$751$	12.0000	0.437886	0.218943	+	0.975738i	$0.429739\pi$
$752$	−3.00000	−0.109399
$753$	18.0000	0.655956
$754$	0	0
$755$	0	0
$756$	3.00000	0.109109
$757$	22.0000	0.799604	0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000	0.799604	0.399802	+	0.916602i	$0.369079\pi$
$758$	20.0000	0.726433
$759$	−12.0000	−0.435572
$760$	0	0
$761$	−48.0000	−1.74000	−0.869999	−	0.493053i	$-0.835881\pi$
$761$	−48.0000	−1.74000	−0.869999	+	0.493053i	$0.835881\pi$
$762$	−18.0000	−0.652071
$763$	−15.0000	−0.543036
$764$	−3.00000	−0.108536
$765$	0	0
$766$	4.00000	0.144526
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	35.0000	1.26213	0.631066	−	0.775729i	$-0.282618\pi$
$769$	35.0000	1.26213	0.631066	+	0.775729i	$0.282618\pi$
$770$	0	0
$771$	8.00000	0.288113
$772$	6.00000	0.215945
$773$	6.00000	0.215805	0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000	0.215805	0.107903	+	0.994161i	$0.465587\pi$
$774$	−1.00000	−0.0359443
$775$	0	0
$776$	−2.00000	−0.0717958
$777$	−9.00000	−0.322873
$778$	5.00000	0.179259
$779$	−10.0000	−0.358287
$780$	0	0
$781$	−6.00000	−0.214697
$782$	−4.00000	−0.143040
$783$	0	0
$784$	2.00000	0.0714286
$785$	0	0
$786$	12.0000	0.428026
$787$	42.0000	1.49714	0.748569	−	0.663057i	$-0.230741\pi$
$787$	42.0000	1.49714	0.748569	+	0.663057i	$0.230741\pi$
$788$	12.0000	0.427482
$789$	19.0000	0.676418
$790$	0	0
$791$	−3.00000	−0.106668
$792$	3.00000	0.106600
$793$	−8.00000	−0.284088
$794$	3.00000	0.106466
$795$	0	0
$796$	−25.0000	−0.886102
$797$	17.0000	0.602171	0.301085	−	0.953597i	$-0.402651\pi$
$797$	17.0000	0.602171	0.301085	+	0.953597i	$0.402651\pi$
$798$	15.0000	0.530994
$799$	3.00000	0.106132
$800$	0	0
$801$	−10.0000	−0.353333
$802$	18.0000	0.635602
$803$	−18.0000	−0.635206
$804$	13.0000	0.458475
$805$	0	0
$806$	28.0000	0.986258
$807$	−20.0000	−0.704033
$808$	−7.00000	−0.246259
$809$	45.0000	1.58212	0.791058	−	0.611741i	$-0.209531\pi$
$809$	45.0000	1.58212	0.791058	+	0.611741i	$0.209531\pi$
$810$	0	0
$811$	2.00000	0.0702295	0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000	0.0702295	0.0351147	+	0.999383i	$0.488820\pi$
$812$	0	0
$813$	−32.0000	−1.12229
$814$	−9.00000	−0.315450
$815$	0	0
$816$	1.00000	0.0350070
$817$	−5.00000	−0.174928
$818$	10.0000	0.349642
$819$	12.0000	0.419314
$820$	0	0
$821$	42.0000	1.46581	0.732905	−	0.680331i	$-0.238164\pi$
$821$	42.0000	1.46581	0.732905	+	0.680331i	$0.238164\pi$
$822$	2.00000	0.0697580
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	14.0000	0.487713
$825$	0	0
$826$	0	0
$827$	17.0000	0.591148	0.295574	−	0.955320i	$-0.404489\pi$
$827$	17.0000	0.591148	0.295574	+	0.955320i	$0.404489\pi$
$828$	−4.00000	−0.139010
$829$	−20.0000	−0.694629	−0.347314	−	0.937749i	$-0.612906\pi$
$829$	−20.0000	−0.694629	−0.347314	+	0.937749i	$0.612906\pi$
$830$	0	0
$831$	23.0000	0.797861
$832$	−4.00000	−0.138675
$833$	−2.00000	−0.0692959
$834$	−20.0000	−0.692543
$835$	0	0
$836$	15.0000	0.518786
$837$	−7.00000	−0.241955
$838$	20.0000	0.690889
$839$	−30.0000	−1.03572	−0.517858	−	0.855467i	$-0.673270\pi$
$839$	−30.0000	−1.03572	−0.517858	+	0.855467i	$0.673270\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	28.0000	0.964944
$843$	18.0000	0.619953
$844$	12.0000	0.413057
$845$	0	0
$846$	3.00000	0.103142
$847$	6.00000	0.206162
$848$	11.0000	0.377742
$849$	14.0000	0.480479
$850$	0	0
$851$	12.0000	0.411355
$852$	−2.00000	−0.0685189
$853$	51.0000	1.74621	0.873103	−	0.487535i	$-0.162104\pi$
$853$	51.0000	1.74621	0.873103	+	0.487535i	$0.162104\pi$
$854$	6.00000	0.205316
$855$	0	0
$856$	−17.0000	−0.581048
$857$	47.0000	1.60549	0.802745	−	0.596323i	$-0.203372\pi$
$857$	47.0000	1.60549	0.802745	+	0.596323i	$0.203372\pi$
$858$	12.0000	0.409673
$859$	35.0000	1.19418	0.597092	−	0.802173i	$-0.296323\pi$
$859$	35.0000	1.19418	0.597092	+	0.802173i	$0.296323\pi$
$860$	0	0
$861$	6.00000	0.204479
$862$	−2.00000	−0.0681203
$863$	−19.0000	−0.646768	−0.323384	−	0.946268i	$-0.604820\pi$
$863$	−19.0000	−0.646768	−0.323384	+	0.946268i	$0.604820\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−21.0000	−0.713609
$867$	−1.00000	−0.0339618
$868$	−21.0000	−0.712786
$869$	15.0000	0.508840
$870$	0	0
$871$	52.0000	1.76195
$872$	−5.00000	−0.169321
$873$	2.00000	0.0676897
$874$	−20.0000	−0.676510
$875$	0	0
$876$	−6.00000	−0.202721
$877$	−38.0000	−1.28317	−0.641584	−	0.767052i	$-0.721723\pi$
$877$	−38.0000	−1.28317	−0.641584	+	0.767052i	$0.721723\pi$
$878$	−20.0000	−0.674967
$879$	−6.00000	−0.202375
$880$	0	0
$881$	7.00000	0.235836	0.117918	−	0.993023i	$-0.462378\pi$
$881$	7.00000	0.235836	0.117918	+	0.993023i	$0.462378\pi$
$882$	−2.00000	−0.0673435
$883$	−24.0000	−0.807664	−0.403832	−	0.914833i	$-0.632322\pi$
$883$	−24.0000	−0.807664	−0.403832	+	0.914833i	$0.632322\pi$
$884$	4.00000	0.134535
$885$	0	0
$886$	14.0000	0.470339
$887$	12.0000	0.402921	0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000	0.402921	0.201460	+	0.979497i	$0.435431\pi$
$888$	−3.00000	−0.100673
$889$	54.0000	1.81110
$890$	0	0
$891$	−3.00000	−0.100504
$892$	26.0000	0.870544
$893$	15.0000	0.501956
$894$	10.0000	0.334450
$895$	0	0
$896$	3.00000	0.100223
$897$	−16.0000	−0.534224
$898$	−5.00000	−0.166852
$899$	0	0
$900$	0	0
$901$	−11.0000	−0.366463
$902$	6.00000	0.199778
$903$	3.00000	0.0998337
$904$	−1.00000	−0.0332595
$905$	0	0
$906$	12.0000	0.398673
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	27.0000	0.896026
$909$	7.00000	0.232175
$910$	0	0
$911$	−8.00000	−0.265052	−0.132526	−	0.991180i	$-0.542309\pi$
$911$	−8.00000	−0.265052	−0.132526	+	0.991180i	$0.542309\pi$
$912$	5.00000	0.165567
$913$	−48.0000	−1.58857
$914$	13.0000	0.430002
$915$	0	0
$916$	0	0
$917$	−36.0000	−1.18882
$918$	−1.00000	−0.0330049
$919$	10.0000	0.329870	0.164935	−	0.986304i	$-0.447259\pi$
$919$	10.0000	0.329870	0.164935	+	0.986304i	$0.447259\pi$
$920$	0	0
$921$	−32.0000	−1.05444
$922$	−37.0000	−1.21853
$923$	−8.00000	−0.263323
$924$	−9.00000	−0.296078
$925$	0	0
$926$	24.0000	0.788689
$927$	−14.0000	−0.459820
$928$	0	0
$929$	35.0000	1.14831	0.574156	−	0.818746i	$-0.305330\pi$
$929$	35.0000	1.14831	0.574156	+	0.818746i	$0.305330\pi$
$930$	0	0
$931$	−10.0000	−0.327737
$932$	26.0000	0.851658
$933$	18.0000	0.589294
$934$	18.0000	0.588978
$935$	0	0
$936$	4.00000	0.130744
$937$	22.0000	0.718709	0.359354	−	0.933201i	$-0.382997\pi$
$937$	22.0000	0.718709	0.359354	+	0.933201i	$0.382997\pi$
$938$	−39.0000	−1.27340
$939$	4.00000	0.130535
$940$	0	0
$941$	−38.0000	−1.23876	−0.619382	−	0.785090i	$-0.712617\pi$
$941$	−38.0000	−1.23876	−0.619382	+	0.785090i	$0.712617\pi$
$942$	12.0000	0.390981
$943$	−8.00000	−0.260516
$944$	0	0
$945$	0	0
$946$	3.00000	0.0975384
$947$	27.0000	0.877382	0.438691	−	0.898638i	$-0.355442\pi$
$947$	27.0000	0.877382	0.438691	+	0.898638i	$0.355442\pi$
$948$	5.00000	0.162392
$949$	−24.0000	−0.779073
$950$	0	0
$951$	−32.0000	−1.03767
$952$	−3.00000	−0.0972306
$953$	16.0000	0.518291	0.259145	−	0.965838i	$-0.416559\pi$
$953$	16.0000	0.518291	0.259145	+	0.965838i	$0.416559\pi$
$954$	−11.0000	−0.356138
$955$	0	0
$956$	15.0000	0.485135
$957$	0	0
$958$	10.0000	0.323085
$959$	−6.00000	−0.193750
$960$	0	0
$961$	18.0000	0.580645
$962$	−12.0000	−0.386896
$963$	17.0000	0.547817
$964$	12.0000	0.386494
$965$	0	0
$966$	12.0000	0.386094
$967$	−48.0000	−1.54358	−0.771788	−	0.635880i	$-0.780637\pi$
$967$	−48.0000	−1.54358	−0.771788	+	0.635880i	$0.780637\pi$
$968$	2.00000	0.0642824
$969$	−5.00000	−0.160623
$970$	0	0
$971$	42.0000	1.34784	0.673922	−	0.738802i	$-0.264608\pi$
$971$	42.0000	1.34784	0.673922	+	0.738802i	$0.264608\pi$
$972$	−1.00000	−0.0320750
$973$	60.0000	1.92351
$974$	−32.0000	−1.02535
$975$	0	0
$976$	2.00000	0.0640184
$977$	−28.0000	−0.895799	−0.447900	−	0.894084i	$-0.647828\pi$
$977$	−28.0000	−0.895799	−0.447900	+	0.894084i	$0.647828\pi$
$978$	6.00000	0.191859
$979$	30.0000	0.958804
$980$	0	0
$981$	5.00000	0.159638
$982$	−2.00000	−0.0638226
$983$	−44.0000	−1.40338	−0.701691	−	0.712481i	$-0.747571\pi$
$983$	−44.0000	−1.40338	−0.701691	+	0.712481i	$0.747571\pi$
$984$	2.00000	0.0637577
$985$	0	0
$986$	0	0
$987$	−9.00000	−0.286473
$988$	20.0000	0.636285
$989$	−4.00000	−0.127193
$990$	0	0
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	−7.00000	−0.222250
$993$	23.0000	0.729883
$994$	6.00000	0.190308
$995$	0	0
$996$	−16.0000	−0.506979
$997$	37.0000	1.17180	0.585901	−	0.810383i	$-0.300741\pi$
$997$	37.0000	1.17180	0.585901	+	0.810383i	$0.300741\pi$
$998$	−10.0000	−0.316544
$999$	3.00000	0.0949158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2550.2.a.a.1.1	✓	1
3.2	odd	2		7650.2.a.bk.1.1		1
5.2	odd	4		2550.2.d.o.2449.1		2
5.3	odd	4		2550.2.d.o.2449.2		2
5.4	even	2		2550.2.a.bf.1.1	yes	1
15.14	odd	2		7650.2.a.bd.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2550.2.a.a.1.1	✓	1	1.1	even	1	trivial
2550.2.a.bf.1.1	yes	1	5.4	even	2
2550.2.d.o.2449.1		2	5.2	odd	4
2550.2.d.o.2449.2		2	5.3	odd	4
7650.2.a.bd.1.1		1	15.14	odd	2
7650.2.a.bk.1.1		1	3.2	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:	\( N \)	\(=\)	\( 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2550.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more