LMFDB - Embedded newform 3450.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3450,2,Mod(1,3450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3450, 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("3450.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3450.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:	$27.5483886973$
Analytic rank:	$2$
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	$\chi$	$=$	3450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -5.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -5.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -3.00000 q^{11} -1.00000 q^{12} -5.00000 q^{13} +5.00000 q^{14} +1.00000 q^{16} -6.00000 q^{17} -1.00000 q^{18} -1.00000 q^{19} +5.00000 q^{21} +3.00000 q^{22} -1.00000 q^{23} +1.00000 q^{24} +5.00000 q^{26} -1.00000 q^{27} -5.00000 q^{28} -5.00000 q^{29} -8.00000 q^{31} -1.00000 q^{32} +3.00000 q^{33} +6.00000 q^{34} +1.00000 q^{36} -4.00000 q^{37} +1.00000 q^{38} +5.00000 q^{39} -7.00000 q^{41} -5.00000 q^{42} +7.00000 q^{43} -3.00000 q^{44} +1.00000 q^{46} +6.00000 q^{47} -1.00000 q^{48} +18.0000 q^{49} +6.00000 q^{51} -5.00000 q^{52} -8.00000 q^{53} +1.00000 q^{54} +5.00000 q^{56} +1.00000 q^{57} +5.00000 q^{58} -10.0000 q^{59} -12.0000 q^{61} +8.00000 q^{62} -5.00000 q^{63} +1.00000 q^{64} -3.00000 q^{66} +12.0000 q^{67} -6.00000 q^{68} +1.00000 q^{69} +10.0000 q^{71} -1.00000 q^{72} -15.0000 q^{73} +4.00000 q^{74} -1.00000 q^{76} +15.0000 q^{77} -5.00000 q^{78} -5.00000 q^{79} +1.00000 q^{81} +7.00000 q^{82} +9.00000 q^{83} +5.00000 q^{84} -7.00000 q^{86} +5.00000 q^{87} +3.00000 q^{88} +14.0000 q^{89} +25.0000 q^{91} -1.00000 q^{92} +8.00000 q^{93} -6.00000 q^{94} +1.00000 q^{96} -16.0000 q^{97} -18.0000 q^{98} -3.00000 q^{99} +O(q^{100})\)

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
$7$	−5.00000	−1.88982	−0.944911	−	0.327327i	$-0.893852\pi$
$7$	−5.00000	−1.88982	−0.944911	+	0.327327i	$0.893852\pi$
$8$	−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
$13$	−5.00000	−1.38675	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−5.00000	−1.38675	−0.693375	+	0.720577i	$0.743877\pi$
$14$	5.00000	1.33631
$15$	0	0
$16$	1.00000	0.250000
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	−1.00000	−0.235702
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	5.00000	1.09109
$22$	3.00000	0.639602
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	1.00000	0.204124
$25$	0	0
$26$	5.00000	0.980581
$27$	−1.00000	−0.192450
$28$	−5.00000	−0.944911
$29$	−5.00000	−0.928477	−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000	−0.928477	−0.464238	+	0.885710i	$0.653672\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	−1.00000	−0.176777
$33$	3.00000	0.522233
$34$	6.00000	1.02899
$35$	0	0
$36$	1.00000	0.166667
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	1.00000	0.162221
$39$	5.00000	0.800641
$40$	0	0
$41$	−7.00000	−1.09322	−0.546608	−	0.837389i	$-0.684081\pi$
$41$	−7.00000	−1.09322	−0.546608	+	0.837389i	$0.684081\pi$
$42$	−5.00000	−0.771517
$43$	7.00000	1.06749	0.533745	−	0.845645i	$-0.320784\pi$
$43$	7.00000	1.06749	0.533745	+	0.845645i	$0.320784\pi$
$44$	−3.00000	−0.452267
$45$	0	0
$46$	1.00000	0.147442
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	−1.00000	−0.144338
$49$	18.0000	2.57143
$50$	0	0
$51$	6.00000	0.840168
$52$	−5.00000	−0.693375
$53$	−8.00000	−1.09888	−0.549442	−	0.835532i	$-0.685160\pi$
$53$	−8.00000	−1.09888	−0.549442	+	0.835532i	$0.685160\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	5.00000	0.668153
$57$	1.00000	0.132453
$58$	5.00000	0.656532
$59$	−10.0000	−1.30189	−0.650945	−	0.759125i	$-0.725627\pi$
$59$	−10.0000	−1.30189	−0.650945	+	0.759125i	$0.725627\pi$
$60$	0	0
$61$	−12.0000	−1.53644	−0.768221	−	0.640184i	$-0.778858\pi$
$61$	−12.0000	−1.53644	−0.768221	+	0.640184i	$0.778858\pi$
$62$	8.00000	1.01600
$63$	−5.00000	−0.629941
$64$	1.00000	0.125000
$65$	0	0
$66$	−3.00000	−0.369274
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	−6.00000	−0.727607
$69$	1.00000	0.120386
$70$	0	0
$71$	10.0000	1.18678	0.593391	−	0.804914i	$-0.297789\pi$
$71$	10.0000	1.18678	0.593391	+	0.804914i	$0.297789\pi$
$72$	−1.00000	−0.117851
$73$	−15.0000	−1.75562	−0.877809	−	0.479012i	$-0.840995\pi$
$73$	−15.0000	−1.75562	−0.877809	+	0.479012i	$0.840995\pi$
$74$	4.00000	0.464991
$75$	0	0
$76$	−1.00000	−0.114708
$77$	15.0000	1.70941
$78$	−5.00000	−0.566139
$79$	−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
$82$	7.00000	0.773021
$83$	9.00000	0.987878	0.493939	−	0.869496i	$-0.335557\pi$
$83$	9.00000	0.987878	0.493939	+	0.869496i	$0.335557\pi$
$84$	5.00000	0.545545
$85$	0	0
$86$	−7.00000	−0.754829
$87$	5.00000	0.536056
$88$	3.00000	0.319801
$89$	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
$91$	25.0000	2.62071
$92$	−1.00000	−0.104257
$93$	8.00000	0.829561
$94$	−6.00000	−0.618853
$95$	0	0
$96$	1.00000	0.102062
$97$	−16.0000	−1.62455	−0.812277	−	0.583272i	$-0.801772\pi$
$97$	−16.0000	−1.62455	−0.812277	+	0.583272i	$0.801772\pi$
$98$	−18.0000	−1.81827
$99$	−3.00000	−0.301511
$100$	0	0
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	−6.00000	−0.594089
$103$	−19.0000	−1.87213	−0.936063	−	0.351833i	$-0.885559\pi$
$103$	−19.0000	−1.87213	−0.936063	+	0.351833i	$0.885559\pi$
$104$	5.00000	0.490290
$105$	0	0
$106$	8.00000	0.777029
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	−1.00000	−0.0962250
$109$	12.0000	1.14939	0.574696	−	0.818367i	$-0.305120\pi$
$109$	12.0000	1.14939	0.574696	+	0.818367i	$0.305120\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	−5.00000	−0.472456
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	−1.00000	−0.0936586
$115$	0	0
$116$	−5.00000	−0.464238
$117$	−5.00000	−0.462250
$118$	10.0000	0.920575
$119$	30.0000	2.75010
$120$	0	0
$121$	−2.00000	−0.181818
$122$	12.0000	1.08643
$123$	7.00000	0.631169
$124$	−8.00000	−0.718421
$125$	0	0
$126$	5.00000	0.445435
$127$	−6.00000	−0.532414	−0.266207	−	0.963916i	$-0.585770\pi$
$127$	−6.00000	−0.532414	−0.266207	+	0.963916i	$0.585770\pi$
$128$	−1.00000	−0.0883883
$129$	−7.00000	−0.616316
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	3.00000	0.261116
$133$	5.00000	0.433555
$134$	−12.0000	−1.03664
$135$	0	0
$136$	6.00000	0.514496
$137$	8.00000	0.683486	0.341743	−	0.939793i	$-0.388983\pi$
$137$	8.00000	0.683486	0.341743	+	0.939793i	$0.388983\pi$
$138$	−1.00000	−0.0851257
$139$	2.00000	0.169638	0.0848189	−	0.996396i	$-0.472969\pi$
$139$	2.00000	0.169638	0.0848189	+	0.996396i	$0.472969\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	−10.0000	−0.839181
$143$	15.0000	1.25436
$144$	1.00000	0.0833333
$145$	0	0
$146$	15.0000	1.24141
$147$	−18.0000	−1.48461
$148$	−4.00000	−0.328798
$149$	−20.0000	−1.63846	−0.819232	−	0.573462i	$-0.805600\pi$
$149$	−20.0000	−1.63846	−0.819232	+	0.573462i	$0.805600\pi$
$150$	0	0
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	1.00000	0.0811107
$153$	−6.00000	−0.485071
$154$	−15.0000	−1.20873
$155$	0	0
$156$	5.00000	0.400320
$157$	−4.00000	−0.319235	−0.159617	−	0.987179i	$-0.551026\pi$
$157$	−4.00000	−0.319235	−0.159617	+	0.987179i	$0.551026\pi$
$158$	5.00000	0.397779
$159$	8.00000	0.634441
$160$	0	0
$161$	5.00000	0.394055
$162$	−1.00000	−0.0785674
$163$	−12.0000	−0.939913	−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000	−0.939913	−0.469956	+	0.882690i	$0.655730\pi$
$164$	−7.00000	−0.546608
$165$	0	0
$166$	−9.00000	−0.698535
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	−5.00000	−0.385758
$169$	12.0000	0.923077
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	7.00000	0.533745
$173$	−21.0000	−1.59660	−0.798300	−	0.602260i	$-0.794267\pi$
$173$	−21.0000	−1.59660	−0.798300	+	0.602260i	$0.794267\pi$
$174$	−5.00000	−0.379049
$175$	0	0
$176$	−3.00000	−0.226134
$177$	10.0000	0.751646
$178$	−14.0000	−1.04934
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	0	0
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	−25.0000	−1.85312
$183$	12.0000	0.887066
$184$	1.00000	0.0737210
$185$	0	0
$186$	−8.00000	−0.586588
$187$	18.0000	1.31629
$188$	6.00000	0.437595
$189$	5.00000	0.363696
$190$	0	0
$191$	−13.0000	−0.940647	−0.470323	−	0.882494i	$-0.655863\pi$
$191$	−13.0000	−0.940647	−0.470323	+	0.882494i	$0.655863\pi$
$192$	−1.00000	−0.0721688
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	16.0000	1.14873
$195$	0	0
$196$	18.0000	1.28571
$197$	11.0000	0.783718	0.391859	−	0.920025i	$-0.371832\pi$
$197$	11.0000	0.783718	0.391859	+	0.920025i	$0.371832\pi$
$198$	3.00000	0.213201
$199$	−1.00000	−0.0708881	−0.0354441	−	0.999372i	$-0.511285\pi$
$199$	−1.00000	−0.0708881	−0.0354441	+	0.999372i	$0.511285\pi$
$200$	0	0
$201$	−12.0000	−0.846415
$202$	10.0000	0.703598
$203$	25.0000	1.75466
$204$	6.00000	0.420084
$205$	0	0
$206$	19.0000	1.32379
$207$	−1.00000	−0.0695048
$208$	−5.00000	−0.346688
$209$	3.00000	0.207514
$210$	0	0
$211$	10.0000	0.688428	0.344214	−	0.938891i	$-0.388145\pi$
$211$	10.0000	0.688428	0.344214	+	0.938891i	$0.388145\pi$
$212$	−8.00000	−0.549442
$213$	−10.0000	−0.685189
$214$	−12.0000	−0.820303
$215$	0	0
$216$	1.00000	0.0680414
$217$	40.0000	2.71538
$218$	−12.0000	−0.812743
$219$	15.0000	1.01361
$220$	0	0
$221$	30.0000	2.01802
$222$	−4.00000	−0.268462
$223$	14.0000	0.937509	0.468755	−	0.883328i	$-0.344703\pi$
$223$	14.0000	0.937509	0.468755	+	0.883328i	$0.344703\pi$
$224$	5.00000	0.334077
$225$	0	0
$226$	−10.0000	−0.665190
$227$	−8.00000	−0.530979	−0.265489	−	0.964114i	$-0.585534\pi$
$227$	−8.00000	−0.530979	−0.265489	+	0.964114i	$0.585534\pi$
$228$	1.00000	0.0662266
$229$	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
$232$	5.00000	0.328266
$233$	−7.00000	−0.458585	−0.229293	−	0.973358i	$-0.573641\pi$
$233$	−7.00000	−0.458585	−0.229293	+	0.973358i	$0.573641\pi$
$234$	5.00000	0.326860
$235$	0	0
$236$	−10.0000	−0.650945
$237$	5.00000	0.324785
$238$	−30.0000	−1.94461
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	−18.0000	−1.15948	−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000	−1.15948	−0.579741	+	0.814801i	$0.696846\pi$
$242$	2.00000	0.128565
$243$	−1.00000	−0.0641500
$244$	−12.0000	−0.768221
$245$	0	0
$246$	−7.00000	−0.446304
$247$	5.00000	0.318142
$248$	8.00000	0.508001
$249$	−9.00000	−0.570352
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	−5.00000	−0.314970
$253$	3.00000	0.188608
$254$	6.00000	0.376473
$255$	0	0
$256$	1.00000	0.0625000
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	7.00000	0.435801
$259$	20.0000	1.24274
$260$	0	0
$261$	−5.00000	−0.309492
$262$	4.00000	0.247121
$263$	4.00000	0.246651	0.123325	−	0.992366i	$-0.460644\pi$
$263$	4.00000	0.246651	0.123325	+	0.992366i	$0.460644\pi$
$264$	−3.00000	−0.184637
$265$	0	0
$266$	−5.00000	−0.306570
$267$	−14.0000	−0.856786
$268$	12.0000	0.733017
$269$	3.00000	0.182913	0.0914566	−	0.995809i	$-0.470848\pi$
$269$	3.00000	0.182913	0.0914566	+	0.995809i	$0.470848\pi$
$270$	0	0
$271$	−28.0000	−1.70088	−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000	−1.70088	−0.850439	+	0.526073i	$0.823664\pi$
$272$	−6.00000	−0.363803
$273$	−25.0000	−1.51307
$274$	−8.00000	−0.483298
$275$	0	0
$276$	1.00000	0.0601929
$277$	1.00000	0.0600842	0.0300421	−	0.999549i	$-0.490436\pi$
$277$	1.00000	0.0600842	0.0300421	+	0.999549i	$0.490436\pi$
$278$	−2.00000	−0.119952
$279$	−8.00000	−0.478947
$280$	0	0
$281$	2.00000	0.119310	0.0596550	−	0.998219i	$-0.481000\pi$
$281$	2.00000	0.119310	0.0596550	+	0.998219i	$0.481000\pi$
$282$	6.00000	0.357295
$283$	−20.0000	−1.18888	−0.594438	−	0.804141i	$-0.702626\pi$
$283$	−20.0000	−1.18888	−0.594438	+	0.804141i	$0.702626\pi$
$284$	10.0000	0.593391
$285$	0	0
$286$	−15.0000	−0.886969
$287$	35.0000	2.06598
$288$	−1.00000	−0.0589256
$289$	19.0000	1.11765
$290$	0	0
$291$	16.0000	0.937937
$292$	−15.0000	−0.877809
$293$	28.0000	1.63578	0.817889	−	0.575376i	$-0.195144\pi$
$293$	28.0000	1.63578	0.817889	+	0.575376i	$0.195144\pi$
$294$	18.0000	1.04978
$295$	0	0
$296$	4.00000	0.232495
$297$	3.00000	0.174078
$298$	20.0000	1.15857
$299$	5.00000	0.289157
$300$	0	0
$301$	−35.0000	−2.01737
$302$	10.0000	0.575435
$303$	10.0000	0.574485
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	6.00000	0.342997
$307$	−12.0000	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	15.0000	0.854704
$309$	19.0000	1.08087
$310$	0	0
$311$	28.0000	1.58773	0.793867	−	0.608091i	$-0.208065\pi$
$311$	28.0000	1.58773	0.793867	+	0.608091i	$0.208065\pi$
$312$	−5.00000	−0.283069
$313$	0	0		−	1.00000i	$-0.5\pi$
$313$	0	0			1.00000i	$0.5\pi$
$314$	4.00000	0.225733
$315$	0	0
$316$	−5.00000	−0.281272
$317$	−9.00000	−0.505490	−0.252745	−	0.967533i	$-0.581333\pi$
$317$	−9.00000	−0.505490	−0.252745	+	0.967533i	$0.581333\pi$
$318$	−8.00000	−0.448618
$319$	15.0000	0.839839
$320$	0	0
$321$	−12.0000	−0.669775
$322$	−5.00000	−0.278639
$323$	6.00000	0.333849
$324$	1.00000	0.0555556
$325$	0	0
$326$	12.0000	0.664619
$327$	−12.0000	−0.663602
$328$	7.00000	0.386510
$329$	−30.0000	−1.65395
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	9.00000	0.493939
$333$	−4.00000	−0.219199
$334$	−12.0000	−0.656611
$335$	0	0
$336$	5.00000	0.272772
$337$	−8.00000	−0.435788	−0.217894	−	0.975972i	$-0.569919\pi$
$337$	−8.00000	−0.435788	−0.217894	+	0.975972i	$0.569919\pi$
$338$	−12.0000	−0.652714
$339$	−10.0000	−0.543125
$340$	0	0
$341$	24.0000	1.29967
$342$	1.00000	0.0540738
$343$	−55.0000	−2.96972
$344$	−7.00000	−0.377415
$345$	0	0
$346$	21.0000	1.12897
$347$	4.00000	0.214731	0.107366	−	0.994220i	$-0.465758\pi$
$347$	4.00000	0.214731	0.107366	+	0.994220i	$0.465758\pi$
$348$	5.00000	0.268028
$349$	1.00000	0.0535288	0.0267644	−	0.999642i	$-0.491480\pi$
$349$	1.00000	0.0535288	0.0267644	+	0.999642i	$0.491480\pi$
$350$	0	0
$351$	5.00000	0.266880
$352$	3.00000	0.159901
$353$	−9.00000	−0.479022	−0.239511	−	0.970894i	$-0.576987\pi$
$353$	−9.00000	−0.479022	−0.239511	+	0.970894i	$0.576987\pi$
$354$	−10.0000	−0.531494
$355$	0	0
$356$	14.0000	0.741999
$357$	−30.0000	−1.58777
$358$	−6.00000	−0.317110
$359$	−7.00000	−0.369446	−0.184723	−	0.982791i	$-0.559139\pi$
$359$	−7.00000	−0.369446	−0.184723	+	0.982791i	$0.559139\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	14.0000	0.735824
$363$	2.00000	0.104973
$364$	25.0000	1.31036
$365$	0	0
$366$	−12.0000	−0.627250
$367$	−1.00000	−0.0521996	−0.0260998	−	0.999659i	$-0.508309\pi$
$367$	−1.00000	−0.0521996	−0.0260998	+	0.999659i	$0.508309\pi$
$368$	−1.00000	−0.0521286
$369$	−7.00000	−0.364405
$370$	0	0
$371$	40.0000	2.07670
$372$	8.00000	0.414781
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	−18.0000	−0.930758
$375$	0	0
$376$	−6.00000	−0.309426
$377$	25.0000	1.28757
$378$	−5.00000	−0.257172
$379$	12.0000	0.616399	0.308199	−	0.951322i	$-0.400274\pi$
$379$	12.0000	0.616399	0.308199	+	0.951322i	$0.400274\pi$
$380$	0	0
$381$	6.00000	0.307389
$382$	13.0000	0.665138
$383$	9.00000	0.459879	0.229939	−	0.973205i	$-0.426147\pi$
$383$	9.00000	0.459879	0.229939	+	0.973205i	$0.426147\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	6.00000	0.305392
$387$	7.00000	0.355830
$388$	−16.0000	−0.812277
$389$	−38.0000	−1.92668	−0.963338	−	0.268290i	$-0.913542\pi$
$389$	−38.0000	−1.92668	−0.963338	+	0.268290i	$0.913542\pi$
$390$	0	0
$391$	6.00000	0.303433
$392$	−18.0000	−0.909137
$393$	4.00000	0.201773
$394$	−11.0000	−0.554172
$395$	0	0
$396$	−3.00000	−0.150756
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	1.00000	0.0501255
$399$	−5.00000	−0.250313
$400$	0	0
$401$	20.0000	0.998752	0.499376	−	0.866385i	$-0.333563\pi$
$401$	20.0000	0.998752	0.499376	+	0.866385i	$0.333563\pi$
$402$	12.0000	0.598506
$403$	40.0000	1.99254
$404$	−10.0000	−0.497519
$405$	0	0
$406$	−25.0000	−1.24073
$407$	12.0000	0.594818
$408$	−6.00000	−0.297044
$409$	17.0000	0.840596	0.420298	−	0.907386i	$-0.361926\pi$
$409$	17.0000	0.840596	0.420298	+	0.907386i	$0.361926\pi$
$410$	0	0
$411$	−8.00000	−0.394611
$412$	−19.0000	−0.936063
$413$	50.0000	2.46034
$414$	1.00000	0.0491473
$415$	0	0
$416$	5.00000	0.245145
$417$	−2.00000	−0.0979404
$418$	−3.00000	−0.146735
$419$	−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
$421$	−18.0000	−0.877266	−0.438633	−	0.898666i	$-0.644537\pi$
$421$	−18.0000	−0.877266	−0.438633	+	0.898666i	$0.644537\pi$
$422$	−10.0000	−0.486792
$423$	6.00000	0.291730
$424$	8.00000	0.388514
$425$	0	0
$426$	10.0000	0.484502
$427$	60.0000	2.90360
$428$	12.0000	0.580042
$429$	−15.0000	−0.724207
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	−1.00000	−0.0481125
$433$	28.0000	1.34559	0.672797	−	0.739827i	$-0.265093\pi$
$433$	28.0000	1.34559	0.672797	+	0.739827i	$0.265093\pi$
$434$	−40.0000	−1.92006
$435$	0	0
$436$	12.0000	0.574696
$437$	1.00000	0.0478365
$438$	−15.0000	−0.716728
$439$	−12.0000	−0.572729	−0.286364	−	0.958121i	$-0.592447\pi$
$439$	−12.0000	−0.572729	−0.286364	+	0.958121i	$0.592447\pi$
$440$	0	0
$441$	18.0000	0.857143
$442$	−30.0000	−1.42695
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	4.00000	0.189832
$445$	0	0
$446$	−14.0000	−0.662919
$447$	20.0000	0.945968
$448$	−5.00000	−0.236228
$449$	−2.00000	−0.0943858	−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000	−0.0943858	−0.0471929	+	0.998886i	$0.515028\pi$
$450$	0	0
$451$	21.0000	0.988851
$452$	10.0000	0.470360
$453$	10.0000	0.469841
$454$	8.00000	0.375459
$455$	0	0
$456$	−1.00000	−0.0468293
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	−16.0000	−0.747631
$459$	6.00000	0.280056
$460$	0	0
$461$	7.00000	0.326023	0.163011	−	0.986624i	$-0.447879\pi$
$461$	7.00000	0.326023	0.163011	+	0.986624i	$0.447879\pi$
$462$	15.0000	0.697863
$463$	−4.00000	−0.185896	−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000	−0.185896	−0.0929479	+	0.995671i	$0.529629\pi$
$464$	−5.00000	−0.232119
$465$	0	0
$466$	7.00000	0.324269
$467$	−21.0000	−0.971764	−0.485882	−	0.874024i	$-0.661502\pi$
$467$	−21.0000	−0.971764	−0.485882	+	0.874024i	$0.661502\pi$
$468$	−5.00000	−0.231125
$469$	−60.0000	−2.77054
$470$	0	0
$471$	4.00000	0.184310
$472$	10.0000	0.460287
$473$	−21.0000	−0.965581
$474$	−5.00000	−0.229658
$475$	0	0
$476$	30.0000	1.37505
$477$	−8.00000	−0.366295
$478$	−24.0000	−1.09773
$479$	−25.0000	−1.14228	−0.571140	−	0.820853i	$-0.693499\pi$
$479$	−25.0000	−1.14228	−0.571140	+	0.820853i	$0.693499\pi$
$480$	0	0
$481$	20.0000	0.911922
$482$	18.0000	0.819878
$483$	−5.00000	−0.227508
$484$	−2.00000	−0.0909091
$485$	0	0
$486$	1.00000	0.0453609
$487$	−2.00000	−0.0906287	−0.0453143	−	0.998973i	$-0.514429\pi$
$487$	−2.00000	−0.0906287	−0.0453143	+	0.998973i	$0.514429\pi$
$488$	12.0000	0.543214
$489$	12.0000	0.542659
$490$	0	0
$491$	−32.0000	−1.44414	−0.722070	−	0.691820i	$-0.756809\pi$
$491$	−32.0000	−1.44414	−0.722070	+	0.691820i	$0.756809\pi$
$492$	7.00000	0.315584
$493$	30.0000	1.35113
$494$	−5.00000	−0.224961
$495$	0	0
$496$	−8.00000	−0.359211
$497$	−50.0000	−2.24281
$498$	9.00000	0.403300
$499$	−6.00000	−0.268597	−0.134298	−	0.990941i	$-0.542878\pi$
$499$	−6.00000	−0.268597	−0.134298	+	0.990941i	$0.542878\pi$
$500$	0	0
$501$	−12.0000	−0.536120
$502$	−12.0000	−0.535586
$503$	19.0000	0.847168	0.423584	−	0.905857i	$-0.360772\pi$
$503$	19.0000	0.847168	0.423584	+	0.905857i	$0.360772\pi$
$504$	5.00000	0.222718
$505$	0	0
$506$	−3.00000	−0.133366
$507$	−12.0000	−0.532939
$508$	−6.00000	−0.266207
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	0	0
$511$	75.0000	3.31780
$512$	−1.00000	−0.0441942
$513$	1.00000	0.0441511
$514$	6.00000	0.264649
$515$	0	0
$516$	−7.00000	−0.308158
$517$	−18.0000	−0.791639
$518$	−20.0000	−0.878750
$519$	21.0000	0.921798
$520$	0	0
$521$	−4.00000	−0.175243	−0.0876216	−	0.996154i	$-0.527927\pi$
$521$	−4.00000	−0.175243	−0.0876216	+	0.996154i	$0.527927\pi$
$522$	5.00000	0.218844
$523$	1.00000	0.0437269	0.0218635	−	0.999761i	$-0.493040\pi$
$523$	1.00000	0.0437269	0.0218635	+	0.999761i	$0.493040\pi$
$524$	−4.00000	−0.174741
$525$	0	0
$526$	−4.00000	−0.174408
$527$	48.0000	2.09091
$528$	3.00000	0.130558
$529$	1.00000	0.0434783
$530$	0	0
$531$	−10.0000	−0.433963
$532$	5.00000	0.216777
$533$	35.0000	1.51602
$534$	14.0000	0.605839
$535$	0	0
$536$	−12.0000	−0.518321
$537$	−6.00000	−0.258919
$538$	−3.00000	−0.129339
$539$	−54.0000	−2.32594
$540$	0	0
$541$	−39.0000	−1.67674	−0.838370	−	0.545101i	$-0.816491\pi$
$541$	−39.0000	−1.67674	−0.838370	+	0.545101i	$0.816491\pi$
$542$	28.0000	1.20270
$543$	14.0000	0.600798
$544$	6.00000	0.257248
$545$	0	0
$546$	25.0000	1.06990
$547$	−30.0000	−1.28271	−0.641354	−	0.767245i	$-0.721627\pi$
$547$	−30.0000	−1.28271	−0.641354	+	0.767245i	$0.721627\pi$
$548$	8.00000	0.341743
$549$	−12.0000	−0.512148
$550$	0	0
$551$	5.00000	0.213007
$552$	−1.00000	−0.0425628
$553$	25.0000	1.06311
$554$	−1.00000	−0.0424859
$555$	0	0
$556$	2.00000	0.0848189
$557$	10.0000	0.423714	0.211857	−	0.977301i	$-0.432049\pi$
$557$	10.0000	0.423714	0.211857	+	0.977301i	$0.432049\pi$
$558$	8.00000	0.338667
$559$	−35.0000	−1.48034
$560$	0	0
$561$	−18.0000	−0.759961
$562$	−2.00000	−0.0843649
$563$	21.0000	0.885044	0.442522	−	0.896758i	$-0.354084\pi$
$563$	21.0000	0.885044	0.442522	+	0.896758i	$0.354084\pi$
$564$	−6.00000	−0.252646
$565$	0	0
$566$	20.0000	0.840663
$567$	−5.00000	−0.209980
$568$	−10.0000	−0.419591
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	15.0000	0.627182
$573$	13.0000	0.543083
$574$	−35.0000	−1.46087
$575$	0	0
$576$	1.00000	0.0416667
$577$	−23.0000	−0.957503	−0.478751	−	0.877951i	$-0.658910\pi$
$577$	−23.0000	−0.957503	−0.478751	+	0.877951i	$0.658910\pi$
$578$	−19.0000	−0.790296
$579$	6.00000	0.249351
$580$	0	0
$581$	−45.0000	−1.86691
$582$	−16.0000	−0.663221
$583$	24.0000	0.993978
$584$	15.0000	0.620704
$585$	0	0
$586$	−28.0000	−1.15667
$587$	−42.0000	−1.73353	−0.866763	−	0.498721i	$-0.833803\pi$
$587$	−42.0000	−1.73353	−0.866763	+	0.498721i	$0.833803\pi$
$588$	−18.0000	−0.742307
$589$	8.00000	0.329634
$590$	0	0
$591$	−11.0000	−0.452480
$592$	−4.00000	−0.164399
$593$	−3.00000	−0.123195	−0.0615976	−	0.998101i	$-0.519620\pi$
$593$	−3.00000	−0.123195	−0.0615976	+	0.998101i	$0.519620\pi$
$594$	−3.00000	−0.123091
$595$	0	0
$596$	−20.0000	−0.819232
$597$	1.00000	0.0409273
$598$	−5.00000	−0.204465
$599$	−44.0000	−1.79779	−0.898896	−	0.438163i	$-0.855629\pi$
$599$	−44.0000	−1.79779	−0.898896	+	0.438163i	$0.855629\pi$
$600$	0	0
$601$	−38.0000	−1.55005	−0.775026	−	0.631929i	$-0.782263\pi$
$601$	−38.0000	−1.55005	−0.775026	+	0.631929i	$0.782263\pi$
$602$	35.0000	1.42649
$603$	12.0000	0.488678
$604$	−10.0000	−0.406894
$605$	0	0
$606$	−10.0000	−0.406222
$607$	22.0000	0.892952	0.446476	−	0.894795i	$-0.352679\pi$
$607$	22.0000	0.892952	0.446476	+	0.894795i	$0.352679\pi$
$608$	1.00000	0.0405554
$609$	−25.0000	−1.01305
$610$	0	0
$611$	−30.0000	−1.21367
$612$	−6.00000	−0.242536
$613$	−10.0000	−0.403896	−0.201948	−	0.979396i	$-0.564727\pi$
$613$	−10.0000	−0.403896	−0.201948	+	0.979396i	$0.564727\pi$
$614$	12.0000	0.484281
$615$	0	0
$616$	−15.0000	−0.604367
$617$	2.00000	0.0805170	0.0402585	−	0.999189i	$-0.487182\pi$
$617$	2.00000	0.0805170	0.0402585	+	0.999189i	$0.487182\pi$
$618$	−19.0000	−0.764292
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	−28.0000	−1.12270
$623$	−70.0000	−2.80449
$624$	5.00000	0.200160
$625$	0	0
$626$	0	0
$627$	−3.00000	−0.119808
$628$	−4.00000	−0.159617
$629$	24.0000	0.956943
$630$	0	0
$631$	29.0000	1.15447	0.577236	−	0.816577i	$-0.304131\pi$
$631$	29.0000	1.15447	0.577236	+	0.816577i	$0.304131\pi$
$632$	5.00000	0.198889
$633$	−10.0000	−0.397464
$634$	9.00000	0.357436
$635$	0	0
$636$	8.00000	0.317221
$637$	−90.0000	−3.56593
$638$	−15.0000	−0.593856
$639$	10.0000	0.395594
$640$	0	0
$641$	−36.0000	−1.42191	−0.710957	−	0.703235i	$-0.751738\pi$
$641$	−36.0000	−1.42191	−0.710957	+	0.703235i	$0.751738\pi$
$642$	12.0000	0.473602
$643$	−1.00000	−0.0394362	−0.0197181	−	0.999806i	$-0.506277\pi$
$643$	−1.00000	−0.0394362	−0.0197181	+	0.999806i	$0.506277\pi$
$644$	5.00000	0.197028
$645$	0	0
$646$	−6.00000	−0.236067
$647$	−24.0000	−0.943537	−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000	−0.943537	−0.471769	+	0.881722i	$0.656384\pi$
$648$	−1.00000	−0.0392837
$649$	30.0000	1.17760
$650$	0	0
$651$	−40.0000	−1.56772
$652$	−12.0000	−0.469956
$653$	−11.0000	−0.430463	−0.215232	−	0.976563i	$-0.569051\pi$
$653$	−11.0000	−0.430463	−0.215232	+	0.976563i	$0.569051\pi$
$654$	12.0000	0.469237
$655$	0	0
$656$	−7.00000	−0.273304
$657$	−15.0000	−0.585206
$658$	30.0000	1.16952
$659$	3.00000	0.116863	0.0584317	−	0.998291i	$-0.481390\pi$
$659$	3.00000	0.116863	0.0584317	+	0.998291i	$0.481390\pi$
$660$	0	0
$661$	40.0000	1.55582	0.777910	−	0.628376i	$-0.216280\pi$
$661$	40.0000	1.55582	0.777910	+	0.628376i	$0.216280\pi$
$662$	−4.00000	−0.155464
$663$	−30.0000	−1.16510
$664$	−9.00000	−0.349268
$665$	0	0
$666$	4.00000	0.154997
$667$	5.00000	0.193601
$668$	12.0000	0.464294
$669$	−14.0000	−0.541271
$670$	0	0
$671$	36.0000	1.38976
$672$	−5.00000	−0.192879
$673$	−45.0000	−1.73462	−0.867311	−	0.497766i	$-0.834154\pi$
$673$	−45.0000	−1.73462	−0.867311	+	0.497766i	$0.834154\pi$
$674$	8.00000	0.308148
$675$	0	0
$676$	12.0000	0.461538
$677$	−24.0000	−0.922395	−0.461197	−	0.887298i	$-0.652580\pi$
$677$	−24.0000	−0.922395	−0.461197	+	0.887298i	$0.652580\pi$
$678$	10.0000	0.384048
$679$	80.0000	3.07012
$680$	0	0
$681$	8.00000	0.306561
$682$	−24.0000	−0.919007
$683$	10.0000	0.382639	0.191320	−	0.981528i	$-0.438723\pi$
$683$	10.0000	0.382639	0.191320	+	0.981528i	$0.438723\pi$
$684$	−1.00000	−0.0382360
$685$	0	0
$686$	55.0000	2.09991
$687$	−16.0000	−0.610438
$688$	7.00000	0.266872
$689$	40.0000	1.52388
$690$	0	0
$691$	−10.0000	−0.380418	−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000	−0.380418	−0.190209	+	0.981744i	$0.560917\pi$
$692$	−21.0000	−0.798300
$693$	15.0000	0.569803
$694$	−4.00000	−0.151838
$695$	0	0
$696$	−5.00000	−0.189525
$697$	42.0000	1.59086
$698$	−1.00000	−0.0378506
$699$	7.00000	0.264764
$700$	0	0
$701$	12.0000	0.453234	0.226617	−	0.973984i	$-0.427233\pi$
$701$	12.0000	0.453234	0.226617	+	0.973984i	$0.427233\pi$
$702$	−5.00000	−0.188713
$703$	4.00000	0.150863
$704$	−3.00000	−0.113067
$705$	0	0
$706$	9.00000	0.338719
$707$	50.0000	1.88044
$708$	10.0000	0.375823
$709$	12.0000	0.450669	0.225335	−	0.974281i	$-0.427652\pi$
$709$	12.0000	0.450669	0.225335	+	0.974281i	$0.427652\pi$
$710$	0	0
$711$	−5.00000	−0.187515
$712$	−14.0000	−0.524672
$713$	8.00000	0.299602
$714$	30.0000	1.12272
$715$	0	0
$716$	6.00000	0.224231
$717$	−24.0000	−0.896296
$718$	7.00000	0.261238
$719$	−22.0000	−0.820462	−0.410231	−	0.911982i	$-0.634552\pi$
$719$	−22.0000	−0.820462	−0.410231	+	0.911982i	$0.634552\pi$
$720$	0	0
$721$	95.0000	3.53798
$722$	18.0000	0.669891
$723$	18.0000	0.669427
$724$	−14.0000	−0.520306
$725$	0	0
$726$	−2.00000	−0.0742270
$727$	−32.0000	−1.18681	−0.593407	−	0.804902i	$-0.702218\pi$
$727$	−32.0000	−1.18681	−0.593407	+	0.804902i	$0.702218\pi$
$728$	−25.0000	−0.926562
$729$	1.00000	0.0370370
$730$	0	0
$731$	−42.0000	−1.55343
$732$	12.0000	0.443533
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	1.00000	0.0369107
$735$	0	0
$736$	1.00000	0.0368605
$737$	−36.0000	−1.32608
$738$	7.00000	0.257674
$739$	−24.0000	−0.882854	−0.441427	−	0.897297i	$-0.645528\pi$
$739$	−24.0000	−0.882854	−0.441427	+	0.897297i	$0.645528\pi$
$740$	0	0
$741$	−5.00000	−0.183680
$742$	−40.0000	−1.46845
$743$	−11.0000	−0.403551	−0.201775	−	0.979432i	$-0.564671\pi$
$743$	−11.0000	−0.403551	−0.201775	+	0.979432i	$0.564671\pi$
$744$	−8.00000	−0.293294
$745$	0	0
$746$	−26.0000	−0.951928
$747$	9.00000	0.329293
$748$	18.0000	0.658145
$749$	−60.0000	−2.19235
$750$	0	0
$751$	−43.0000	−1.56909	−0.784546	−	0.620070i	$-0.787104\pi$
$751$	−43.0000	−1.56909	−0.784546	+	0.620070i	$0.787104\pi$
$752$	6.00000	0.218797
$753$	−12.0000	−0.437304
$754$	−25.0000	−0.910446
$755$	0	0
$756$	5.00000	0.181848
$757$	−28.0000	−1.01768	−0.508839	−	0.860862i	$-0.669925\pi$
$757$	−28.0000	−1.01768	−0.508839	+	0.860862i	$0.669925\pi$
$758$	−12.0000	−0.435860
$759$	−3.00000	−0.108893
$760$	0	0
$761$	51.0000	1.84875	0.924374	−	0.381487i	$-0.124588\pi$
$761$	51.0000	1.84875	0.924374	+	0.381487i	$0.124588\pi$
$762$	−6.00000	−0.217357
$763$	−60.0000	−2.17215
$764$	−13.0000	−0.470323
$765$	0	0
$766$	−9.00000	−0.325183
$767$	50.0000	1.80540
$768$	−1.00000	−0.0360844
$769$	−2.00000	−0.0721218	−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000	−0.0721218	−0.0360609	+	0.999350i	$0.511481\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	−6.00000	−0.215945
$773$	12.0000	0.431610	0.215805	−	0.976436i	$-0.430762\pi$
$773$	12.0000	0.431610	0.215805	+	0.976436i	$0.430762\pi$
$774$	−7.00000	−0.251610
$775$	0	0
$776$	16.0000	0.574367
$777$	−20.0000	−0.717496
$778$	38.0000	1.36237
$779$	7.00000	0.250801
$780$	0	0
$781$	−30.0000	−1.07348
$782$	−6.00000	−0.214560
$783$	5.00000	0.178685
$784$	18.0000	0.642857
$785$	0	0
$786$	−4.00000	−0.142675
$787$	17.0000	0.605985	0.302992	−	0.952993i	$-0.402014\pi$
$787$	17.0000	0.605985	0.302992	+	0.952993i	$0.402014\pi$
$788$	11.0000	0.391859
$789$	−4.00000	−0.142404
$790$	0	0
$791$	−50.0000	−1.77780
$792$	3.00000	0.106600
$793$	60.0000	2.13066
$794$	−2.00000	−0.0709773
$795$	0	0
$796$	−1.00000	−0.0354441
$797$	−42.0000	−1.48772	−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000	−1.48772	−0.743858	+	0.668338i	$0.767006\pi$
$798$	5.00000	0.176998
$799$	−36.0000	−1.27359
$800$	0	0
$801$	14.0000	0.494666
$802$	−20.0000	−0.706225
$803$	45.0000	1.58802
$804$	−12.0000	−0.423207
$805$	0	0
$806$	−40.0000	−1.40894
$807$	−3.00000	−0.105605
$808$	10.0000	0.351799
$809$	19.0000	0.668004	0.334002	−	0.942572i	$-0.391601\pi$
$809$	19.0000	0.668004	0.334002	+	0.942572i	$0.391601\pi$
$810$	0	0
$811$	−30.0000	−1.05344	−0.526721	−	0.850038i	$-0.676579\pi$
$811$	−30.0000	−1.05344	−0.526721	+	0.850038i	$0.676579\pi$
$812$	25.0000	0.877328
$813$	28.0000	0.982003
$814$	−12.0000	−0.420600
$815$	0	0
$816$	6.00000	0.210042
$817$	−7.00000	−0.244899
$818$	−17.0000	−0.594391
$819$	25.0000	0.873571
$820$	0	0
$821$	9.00000	0.314102	0.157051	−	0.987590i	$-0.449801\pi$
$821$	9.00000	0.314102	0.157051	+	0.987590i	$0.449801\pi$
$822$	8.00000	0.279032
$823$	−50.0000	−1.74289	−0.871445	−	0.490493i	$-0.836817\pi$
$823$	−50.0000	−1.74289	−0.871445	+	0.490493i	$0.836817\pi$
$824$	19.0000	0.661896
$825$	0	0
$826$	−50.0000	−1.73972
$827$	27.0000	0.938882	0.469441	−	0.882964i	$-0.344455\pi$
$827$	27.0000	0.938882	0.469441	+	0.882964i	$0.344455\pi$
$828$	−1.00000	−0.0347524
$829$	−31.0000	−1.07667	−0.538337	−	0.842729i	$-0.680947\pi$
$829$	−31.0000	−1.07667	−0.538337	+	0.842729i	$0.680947\pi$
$830$	0	0
$831$	−1.00000	−0.0346896
$832$	−5.00000	−0.173344
$833$	−108.000	−3.74198
$834$	2.00000	0.0692543
$835$	0	0
$836$	3.00000	0.103757
$837$	8.00000	0.276520
$838$	1.00000	0.0345444
$839$	7.00000	0.241667	0.120833	−	0.992673i	$-0.461443\pi$
$839$	7.00000	0.241667	0.120833	+	0.992673i	$0.461443\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	18.0000	0.620321
$843$	−2.00000	−0.0688837
$844$	10.0000	0.344214
$845$	0	0
$846$	−6.00000	−0.206284
$847$	10.0000	0.343604
$848$	−8.00000	−0.274721
$849$	20.0000	0.686398
$850$	0	0
$851$	4.00000	0.137118
$852$	−10.0000	−0.342594
$853$	−9.00000	−0.308154	−0.154077	−	0.988059i	$-0.549240\pi$
$853$	−9.00000	−0.308154	−0.154077	+	0.988059i	$0.549240\pi$
$854$	−60.0000	−2.05316
$855$	0	0
$856$	−12.0000	−0.410152
$857$	22.0000	0.751506	0.375753	−	0.926720i	$-0.377384\pi$
$857$	22.0000	0.751506	0.375753	+	0.926720i	$0.377384\pi$
$858$	15.0000	0.512092
$859$	−30.0000	−1.02359	−0.511793	−	0.859109i	$-0.671019\pi$
$859$	−30.0000	−1.02359	−0.511793	+	0.859109i	$0.671019\pi$
$860$	0	0
$861$	−35.0000	−1.19280
$862$	0	0
$863$	−18.0000	−0.612727	−0.306364	−	0.951915i	$-0.599112\pi$
$863$	−18.0000	−0.612727	−0.306364	+	0.951915i	$0.599112\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−28.0000	−0.951479
$867$	−19.0000	−0.645274
$868$	40.0000	1.35769
$869$	15.0000	0.508840
$870$	0	0
$871$	−60.0000	−2.03302
$872$	−12.0000	−0.406371
$873$	−16.0000	−0.541518
$874$	−1.00000	−0.0338255
$875$	0	0
$876$	15.0000	0.506803
$877$	−42.0000	−1.41824	−0.709120	−	0.705088i	$-0.750907\pi$
$877$	−42.0000	−1.41824	−0.709120	+	0.705088i	$0.750907\pi$
$878$	12.0000	0.404980
$879$	−28.0000	−0.944417
$880$	0	0
$881$	−24.0000	−0.808581	−0.404290	−	0.914631i	$-0.632481\pi$
$881$	−24.0000	−0.808581	−0.404290	+	0.914631i	$0.632481\pi$
$882$	−18.0000	−0.606092
$883$	−8.00000	−0.269221	−0.134611	−	0.990899i	$-0.542978\pi$
$883$	−8.00000	−0.269221	−0.134611	+	0.990899i	$0.542978\pi$
$884$	30.0000	1.00901
$885$	0	0
$886$	−4.00000	−0.134383
$887$	−22.0000	−0.738688	−0.369344	−	0.929293i	$-0.620418\pi$
$887$	−22.0000	−0.738688	−0.369344	+	0.929293i	$0.620418\pi$
$888$	−4.00000	−0.134231
$889$	30.0000	1.00617
$890$	0	0
$891$	−3.00000	−0.100504
$892$	14.0000	0.468755
$893$	−6.00000	−0.200782
$894$	−20.0000	−0.668900
$895$	0	0
$896$	5.00000	0.167038
$897$	−5.00000	−0.166945
$898$	2.00000	0.0667409
$899$	40.0000	1.33407
$900$	0	0
$901$	48.0000	1.59911
$902$	−21.0000	−0.699224
$903$	35.0000	1.16473
$904$	−10.0000	−0.332595
$905$	0	0
$906$	−10.0000	−0.332228
$907$	37.0000	1.22856	0.614282	−	0.789086i	$-0.289446\pi$
$907$	37.0000	1.22856	0.614282	+	0.789086i	$0.289446\pi$
$908$	−8.00000	−0.265489
$909$	−10.0000	−0.331679
$910$	0	0
$911$	−15.0000	−0.496972	−0.248486	−	0.968635i	$-0.579933\pi$
$911$	−15.0000	−0.496972	−0.248486	+	0.968635i	$0.579933\pi$
$912$	1.00000	0.0331133
$913$	−27.0000	−0.893570
$914$	−18.0000	−0.595387
$915$	0	0
$916$	16.0000	0.528655
$917$	20.0000	0.660458
$918$	−6.00000	−0.198030
$919$	−12.0000	−0.395843	−0.197922	−	0.980218i	$-0.563419\pi$
$919$	−12.0000	−0.395843	−0.197922	+	0.980218i	$0.563419\pi$
$920$	0	0
$921$	12.0000	0.395413
$922$	−7.00000	−0.230533
$923$	−50.0000	−1.64577
$924$	−15.0000	−0.493464
$925$	0	0
$926$	4.00000	0.131448
$927$	−19.0000	−0.624042
$928$	5.00000	0.164133
$929$	−43.0000	−1.41078	−0.705392	−	0.708817i	$-0.749229\pi$
$929$	−43.0000	−1.41078	−0.705392	+	0.708817i	$0.749229\pi$
$930$	0	0
$931$	−18.0000	−0.589926
$932$	−7.00000	−0.229293
$933$	−28.0000	−0.916679
$934$	21.0000	0.687141
$935$	0	0
$936$	5.00000	0.163430
$937$	20.0000	0.653372	0.326686	−	0.945133i	$-0.394068\pi$
$937$	20.0000	0.653372	0.326686	+	0.945133i	$0.394068\pi$
$938$	60.0000	1.95907
$939$	0	0
$940$	0	0
$941$	−48.0000	−1.56476	−0.782378	−	0.622804i	$-0.785993\pi$
$941$	−48.0000	−1.56476	−0.782378	+	0.622804i	$0.785993\pi$
$942$	−4.00000	−0.130327
$943$	7.00000	0.227951
$944$	−10.0000	−0.325472
$945$	0	0
$946$	21.0000	0.682769
$947$	−32.0000	−1.03986	−0.519930	−	0.854209i	$-0.674042\pi$
$947$	−32.0000	−1.03986	−0.519930	+	0.854209i	$0.674042\pi$
$948$	5.00000	0.162392
$949$	75.0000	2.43460
$950$	0	0
$951$	9.00000	0.291845
$952$	−30.0000	−0.972306
$953$	12.0000	0.388718	0.194359	−	0.980930i	$-0.437737\pi$
$953$	12.0000	0.388718	0.194359	+	0.980930i	$0.437737\pi$
$954$	8.00000	0.259010
$955$	0	0
$956$	24.0000	0.776215
$957$	−15.0000	−0.484881
$958$	25.0000	0.807713
$959$	−40.0000	−1.29167
$960$	0	0
$961$	33.0000	1.06452
$962$	−20.0000	−0.644826
$963$	12.0000	0.386695
$964$	−18.0000	−0.579741
$965$	0	0
$966$	5.00000	0.160872
$967$	8.00000	0.257263	0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000	0.257263	0.128631	+	0.991692i	$0.458942\pi$
$968$	2.00000	0.0642824
$969$	−6.00000	−0.192748
$970$	0	0
$971$	−49.0000	−1.57248	−0.786242	−	0.617918i	$-0.787976\pi$
$971$	−49.0000	−1.57248	−0.786242	+	0.617918i	$0.787976\pi$
$972$	−1.00000	−0.0320750
$973$	−10.0000	−0.320585
$974$	2.00000	0.0640841
$975$	0	0
$976$	−12.0000	−0.384111
$977$	−8.00000	−0.255943	−0.127971	−	0.991778i	$-0.540847\pi$
$977$	−8.00000	−0.255943	−0.127971	+	0.991778i	$0.540847\pi$
$978$	−12.0000	−0.383718
$979$	−42.0000	−1.34233
$980$	0	0
$981$	12.0000	0.383131
$982$	32.0000	1.02116
$983$	31.0000	0.988746	0.494373	−	0.869250i	$-0.335398\pi$
$983$	31.0000	0.988746	0.494373	+	0.869250i	$0.335398\pi$
$984$	−7.00000	−0.223152
$985$	0	0
$986$	−30.0000	−0.955395
$987$	30.0000	0.954911
$988$	5.00000	0.159071
$989$	−7.00000	−0.222587
$990$	0	0
$991$	4.00000	0.127064	0.0635321	−	0.997980i	$-0.479763\pi$
$991$	4.00000	0.127064	0.0635321	+	0.997980i	$0.479763\pi$
$992$	8.00000	0.254000
$993$	−4.00000	−0.126936
$994$	50.0000	1.58590
$995$	0	0
$996$	−9.00000	−0.285176
$997$	1.00000	0.0316703	0.0158352	−	0.999875i	$-0.494959\pi$
$997$	1.00000	0.0316703	0.0158352	+	0.999875i	$0.494959\pi$
$998$	6.00000	0.189927
$999$	4.00000	0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3450.2.a.a.1.1	✓	1
5.2	odd	4		3450.2.d.l.2899.1		2
5.3	odd	4		3450.2.d.l.2899.2		2
5.4	even	2		3450.2.a.bb.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3450.2.a.a.1.1	✓	1	1.1	even	1	trivial
3450.2.a.bb.1.1	yes	1	5.4	even	2
3450.2.d.l.2899.1		2	5.2	odd	4
3450.2.d.l.2899.2		2	5.3	odd	4

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:	\( N \)	\(=\)	\( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3450.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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