LMFDB - Embedded newform 3450.2.a.s.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:	$1$
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)

$f(q)$

$=$

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

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.00000 q^{11} -1.00000 q^{12} -3.00000 q^{13} +3.00000 q^{14} +1.00000 q^{16} -4.00000 q^{17} +1.00000 q^{18} -3.00000 q^{19} -3.00000 q^{21} -3.00000 q^{22} -1.00000 q^{23} -1.00000 q^{24} -3.00000 q^{26} -1.00000 q^{27} +3.00000 q^{28} +3.00000 q^{29} -10.0000 q^{31} +1.00000 q^{32} +3.00000 q^{33} -4.00000 q^{34} +1.00000 q^{36} -8.00000 q^{37} -3.00000 q^{38} +3.00000 q^{39} -9.00000 q^{41} -3.00000 q^{42} +1.00000 q^{43} -3.00000 q^{44} -1.00000 q^{46} +2.00000 q^{47} -1.00000 q^{48} +2.00000 q^{49} +4.00000 q^{51} -3.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} +3.00000 q^{56} +3.00000 q^{57} +3.00000 q^{58} +8.00000 q^{59} +12.0000 q^{61} -10.0000 q^{62} +3.00000 q^{63} +1.00000 q^{64} +3.00000 q^{66} +8.00000 q^{67} -4.00000 q^{68} +1.00000 q^{69} -14.0000 q^{71} +1.00000 q^{72} +13.0000 q^{73} -8.00000 q^{74} -3.00000 q^{76} -9.00000 q^{77} +3.00000 q^{78} -17.0000 q^{79} +1.00000 q^{81} -9.00000 q^{82} +1.00000 q^{83} -3.00000 q^{84} +1.00000 q^{86} -3.00000 q^{87} -3.00000 q^{88} -6.00000 q^{89} -9.00000 q^{91} -1.00000 q^{92} +10.0000 q^{93} +2.00000 q^{94} -1.00000 q^{96} +6.00000 q^{97} +2.00000 q^{98} -3.00000 q^{99} +O(q^{100})\)

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	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	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
$13$	−3.00000	−0.832050	−0.416025	−	0.909353i	$-0.636577\pi$
$13$	−3.00000	−0.832050	−0.416025	+	0.909353i	$0.636577\pi$
$14$	3.00000	0.801784
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	1.00000	0.235702
$19$	−3.00000	−0.688247	−0.344124	−	0.938924i	$-0.611824\pi$
$19$	−3.00000	−0.688247	−0.344124	+	0.938924i	$0.611824\pi$
$20$	0	0
$21$	−3.00000	−0.654654
$22$	−3.00000	−0.639602
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−3.00000	−0.588348
$27$	−1.00000	−0.192450
$28$	3.00000	0.566947
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	1.00000	0.176777
$33$	3.00000	0.522233
$34$	−4.00000	−0.685994
$35$	0	0
$36$	1.00000	0.166667
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	−3.00000	−0.486664
$39$	3.00000	0.480384
$40$	0	0
$41$	−9.00000	−1.40556	−0.702782	−	0.711405i	$-0.748059\pi$
$41$	−9.00000	−1.40556	−0.702782	+	0.711405i	$0.748059\pi$
$42$	−3.00000	−0.462910
$43$	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
$46$	−1.00000	−0.147442
$47$	2.00000	0.291730	0.145865	−	0.989305i	$-0.453403\pi$
$47$	2.00000	0.291730	0.145865	+	0.989305i	$0.453403\pi$
$48$	−1.00000	−0.144338
$49$	2.00000	0.285714
$50$	0	0
$51$	4.00000	0.560112
$52$	−3.00000	−0.416025
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	3.00000	0.400892
$57$	3.00000	0.397360
$58$	3.00000	0.393919
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	12.0000	1.53644	0.768221	−	0.640184i	$-0.221142\pi$
$61$	12.0000	1.53644	0.768221	+	0.640184i	$0.221142\pi$
$62$	−10.0000	−1.27000
$63$	3.00000	0.377964
$64$	1.00000	0.125000
$65$	0	0
$66$	3.00000	0.369274
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	−4.00000	−0.485071
$69$	1.00000	0.120386
$70$	0	0
$71$	−14.0000	−1.66149	−0.830747	−	0.556650i	$-0.812086\pi$
$71$	−14.0000	−1.66149	−0.830747	+	0.556650i	$0.812086\pi$
$72$	1.00000	0.117851
$73$	13.0000	1.52153	0.760767	−	0.649025i	$-0.224823\pi$
$73$	13.0000	1.52153	0.760767	+	0.649025i	$0.224823\pi$
$74$	−8.00000	−0.929981
$75$	0	0
$76$	−3.00000	−0.344124
$77$	−9.00000	−1.02565
$78$	3.00000	0.339683
$79$	−17.0000	−1.91265	−0.956325	−	0.292306i	$-0.905577\pi$
$79$	−17.0000	−1.91265	−0.956325	+	0.292306i	$0.905577\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−9.00000	−0.993884
$83$	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
$86$	1.00000	0.107833
$87$	−3.00000	−0.321634
$88$	−3.00000	−0.319801
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	−9.00000	−0.943456
$92$	−1.00000	−0.104257
$93$	10.0000	1.03695
$94$	2.00000	0.206284
$95$	0	0
$96$	−1.00000	−0.102062
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	2.00000	0.202031
$99$	−3.00000	−0.301511
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	4.00000	0.396059
$103$	13.0000	1.28093	0.640464	−	0.767988i	$-0.278742\pi$
$103$	13.0000	1.28093	0.640464	+	0.767988i	$0.278742\pi$
$104$	−3.00000	−0.294174
$105$	0	0
$106$	−6.00000	−0.582772
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	−1.00000	−0.0962250
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	0	0
$111$	8.00000	0.759326
$112$	3.00000	0.283473
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	3.00000	0.280976
$115$	0	0
$116$	3.00000	0.278543
$117$	−3.00000	−0.277350
$118$	8.00000	0.736460
$119$	−12.0000	−1.10004
$120$	0	0
$121$	−2.00000	−0.181818
$122$	12.0000	1.08643
$123$	9.00000	0.811503
$124$	−10.0000	−0.898027
$125$	0	0
$126$	3.00000	0.267261
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	1.00000	0.0883883
$129$	−1.00000	−0.0880451
$130$	0	0
$131$	10.0000	0.873704	0.436852	−	0.899533i	$-0.356093\pi$
$131$	10.0000	0.873704	0.436852	+	0.899533i	$0.356093\pi$
$132$	3.00000	0.261116
$133$	−9.00000	−0.780399
$134$	8.00000	0.691095
$135$	0	0
$136$	−4.00000	−0.342997
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	1.00000	0.0851257
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	0	0
$141$	−2.00000	−0.168430
$142$	−14.0000	−1.17485
$143$	9.00000	0.752618
$144$	1.00000	0.0833333
$145$	0	0
$146$	13.0000	1.07589
$147$	−2.00000	−0.164957
$148$	−8.00000	−0.657596
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	−22.0000	−1.79033	−0.895167	−	0.445730i	$-0.852944\pi$
$151$	−22.0000	−1.79033	−0.895167	+	0.445730i	$0.852944\pi$
$152$	−3.00000	−0.243332
$153$	−4.00000	−0.323381
$154$	−9.00000	−0.725241
$155$	0	0
$156$	3.00000	0.240192
$157$	6.00000	0.478852	0.239426	−	0.970915i	$-0.423041\pi$
$157$	6.00000	0.478852	0.239426	+	0.970915i	$0.423041\pi$
$158$	−17.0000	−1.35245
$159$	6.00000	0.475831
$160$	0	0
$161$	−3.00000	−0.236433
$162$	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	$0.424498\pi$
$164$	−9.00000	−0.702782
$165$	0	0
$166$	1.00000	0.0776151
$167$	2.00000	0.154765	0.0773823	−	0.997001i	$-0.475344\pi$
$167$	2.00000	0.154765	0.0773823	+	0.997001i	$0.475344\pi$
$168$	−3.00000	−0.231455
$169$	−4.00000	−0.307692
$170$	0	0
$171$	−3.00000	−0.229416
$172$	1.00000	0.0762493
$173$	15.0000	1.14043	0.570214	−	0.821496i	$-0.306860\pi$
$173$	15.0000	1.14043	0.570214	+	0.821496i	$0.306860\pi$
$174$	−3.00000	−0.227429
$175$	0	0
$176$	−3.00000	−0.226134
$177$	−8.00000	−0.601317
$178$	−6.00000	−0.449719
$179$	−10.0000	−0.747435	−0.373718	−	0.927543i	$-0.621917\pi$
$179$	−10.0000	−0.747435	−0.373718	+	0.927543i	$0.621917\pi$
$180$	0	0
$181$	−16.0000	−1.18927	−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000	−1.18927	−0.594635	+	0.803996i	$0.702704\pi$
$182$	−9.00000	−0.667124
$183$	−12.0000	−0.887066
$184$	−1.00000	−0.0737210
$185$	0	0
$186$	10.0000	0.733236
$187$	12.0000	0.877527
$188$	2.00000	0.145865
$189$	−3.00000	−0.218218
$190$	0	0
$191$	−15.0000	−1.08536	−0.542681	−	0.839939i	$-0.682591\pi$
$191$	−15.0000	−1.08536	−0.542681	+	0.839939i	$0.682591\pi$
$192$	−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	$0.569283\pi$
$194$	6.00000	0.430775
$195$	0	0
$196$	2.00000	0.142857
$197$	−17.0000	−1.21120	−0.605600	−	0.795769i	$-0.707067\pi$
$197$	−17.0000	−1.21120	−0.605600	+	0.795769i	$0.707067\pi$
$198$	−3.00000	−0.213201
$199$	11.0000	0.779769	0.389885	−	0.920864i	$-0.372515\pi$
$199$	11.0000	0.779769	0.389885	+	0.920864i	$0.372515\pi$
$200$	0	0
$201$	−8.00000	−0.564276
$202$	−6.00000	−0.422159
$203$	9.00000	0.631676
$204$	4.00000	0.280056
$205$	0	0
$206$	13.0000	0.905753
$207$	−1.00000	−0.0695048
$208$	−3.00000	−0.208013
$209$	9.00000	0.622543
$210$	0	0
$211$	22.0000	1.51454	0.757271	−	0.653101i	$-0.226532\pi$
$211$	22.0000	1.51454	0.757271	+	0.653101i	$0.226532\pi$
$212$	−6.00000	−0.412082
$213$	14.0000	0.959264
$214$	−12.0000	−0.820303
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	−30.0000	−2.03653
$218$	0	0
$219$	−13.0000	−0.878459
$220$	0	0
$221$	12.0000	0.807207
$222$	8.00000	0.536925
$223$	−6.00000	−0.401790	−0.200895	−	0.979613i	$-0.564385\pi$
$223$	−6.00000	−0.401790	−0.200895	+	0.979613i	$0.564385\pi$
$224$	3.00000	0.200446
$225$	0	0
$226$	6.00000	0.399114
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	3.00000	0.198680
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	0	0
$231$	9.00000	0.592157
$232$	3.00000	0.196960
$233$	−1.00000	−0.0655122	−0.0327561	−	0.999463i	$-0.510428\pi$
$233$	−1.00000	−0.0655122	−0.0327561	+	0.999463i	$0.510428\pi$
$234$	−3.00000	−0.196116
$235$	0	0
$236$	8.00000	0.520756
$237$	17.0000	1.10427
$238$	−12.0000	−0.777844
$239$	−10.0000	−0.646846	−0.323423	−	0.946254i	$-0.604834\pi$
$239$	−10.0000	−0.646846	−0.323423	+	0.946254i	$0.604834\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	−2.00000	−0.128565
$243$	−1.00000	−0.0641500
$244$	12.0000	0.768221
$245$	0	0
$246$	9.00000	0.573819
$247$	9.00000	0.572656
$248$	−10.0000	−0.635001
$249$	−1.00000	−0.0633724
$250$	0	0
$251$	−16.0000	−1.00991	−0.504956	−	0.863145i	$-0.668491\pi$
$251$	−16.0000	−1.00991	−0.504956	+	0.863145i	$0.668491\pi$
$252$	3.00000	0.188982
$253$	3.00000	0.188608
$254$	16.0000	1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	−1.00000	−0.0622573
$259$	−24.0000	−1.49129
$260$	0	0
$261$	3.00000	0.185695
$262$	10.0000	0.617802
$263$	−8.00000	−0.493301	−0.246651	−	0.969104i	$-0.579330\pi$
$263$	−8.00000	−0.493301	−0.246651	+	0.969104i	$0.579330\pi$
$264$	3.00000	0.184637
$265$	0	0
$266$	−9.00000	−0.551825
$267$	6.00000	0.367194
$268$	8.00000	0.488678
$269$	−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
$271$	22.0000	1.33640	0.668202	−	0.743980i	$-0.267064\pi$
$271$	22.0000	1.33640	0.668202	+	0.743980i	$0.267064\pi$
$272$	−4.00000	−0.242536
$273$	9.00000	0.544705
$274$	−18.0000	−1.08742
$275$	0	0
$276$	1.00000	0.0601929
$277$	−13.0000	−0.781094	−0.390547	−	0.920583i	$-0.627714\pi$
$277$	−13.0000	−0.781094	−0.390547	+	0.920583i	$0.627714\pi$
$278$	8.00000	0.479808
$279$	−10.0000	−0.598684
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	−2.00000	−0.119098
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	−14.0000	−0.830747
$285$	0	0
$286$	9.00000	0.532181
$287$	−27.0000	−1.59376
$288$	1.00000	0.0589256
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−6.00000	−0.351726
$292$	13.0000	0.760767
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	−2.00000	−0.116642
$295$	0	0
$296$	−8.00000	−0.464991
$297$	3.00000	0.174078
$298$	−6.00000	−0.347571
$299$	3.00000	0.173494
$300$	0	0
$301$	3.00000	0.172917
$302$	−22.0000	−1.26596
$303$	6.00000	0.344691
$304$	−3.00000	−0.172062
$305$	0	0
$306$	−4.00000	−0.228665
$307$	8.00000	0.456584	0.228292	−	0.973593i	$-0.426686\pi$
$307$	8.00000	0.456584	0.228292	+	0.973593i	$0.426686\pi$
$308$	−9.00000	−0.512823
$309$	−13.0000	−0.739544
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	3.00000	0.169842
$313$	24.0000	1.35656	0.678280	−	0.734803i	$-0.262726\pi$
$313$	24.0000	1.35656	0.678280	+	0.734803i	$0.262726\pi$
$314$	6.00000	0.338600
$315$	0	0
$316$	−17.0000	−0.956325
$317$	31.0000	1.74113	0.870567	−	0.492050i	$-0.163752\pi$
$317$	31.0000	1.74113	0.870567	+	0.492050i	$0.163752\pi$
$318$	6.00000	0.336463
$319$	−9.00000	−0.503903
$320$	0	0
$321$	12.0000	0.669775
$322$	−3.00000	−0.167183
$323$	12.0000	0.667698
$324$	1.00000	0.0555556
$325$	0	0
$326$	6.00000	0.332309
$327$	0	0
$328$	−9.00000	−0.496942
$329$	6.00000	0.330791
$330$	0	0
$331$	−26.0000	−1.42909	−0.714545	−	0.699590i	$-0.753366\pi$
$331$	−26.0000	−1.42909	−0.714545	+	0.699590i	$0.753366\pi$
$332$	1.00000	0.0548821
$333$	−8.00000	−0.438397
$334$	2.00000	0.109435
$335$	0	0
$336$	−3.00000	−0.163663
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	−4.00000	−0.217571
$339$	−6.00000	−0.325875
$340$	0	0
$341$	30.0000	1.62459
$342$	−3.00000	−0.162221
$343$	−15.0000	−0.809924
$344$	1.00000	0.0539164
$345$	0	0
$346$	15.0000	0.806405
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	−3.00000	−0.160817
$349$	−5.00000	−0.267644	−0.133822	−	0.991005i	$-0.542725\pi$
$349$	−5.00000	−0.267644	−0.133822	+	0.991005i	$0.542725\pi$
$350$	0	0
$351$	3.00000	0.160128
$352$	−3.00000	−0.159901
$353$	9.00000	0.479022	0.239511	−	0.970894i	$-0.423013\pi$
$353$	9.00000	0.479022	0.239511	+	0.970894i	$0.423013\pi$
$354$	−8.00000	−0.425195
$355$	0	0
$356$	−6.00000	−0.317999
$357$	12.0000	0.635107
$358$	−10.0000	−0.528516
$359$	31.0000	1.63612	0.818059	−	0.575135i	$-0.195050\pi$
$359$	31.0000	1.63612	0.818059	+	0.575135i	$0.195050\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	−16.0000	−0.840941
$363$	2.00000	0.104973
$364$	−9.00000	−0.471728
$365$	0	0
$366$	−12.0000	−0.627250
$367$	27.0000	1.40939	0.704694	−	0.709511i	$-0.251084\pi$
$367$	27.0000	1.40939	0.704694	+	0.709511i	$0.251084\pi$
$368$	−1.00000	−0.0521286
$369$	−9.00000	−0.468521
$370$	0	0
$371$	−18.0000	−0.934513
$372$	10.0000	0.518476
$373$	−16.0000	−0.828449	−0.414224	−	0.910175i	$-0.635947\pi$
$373$	−16.0000	−0.828449	−0.414224	+	0.910175i	$0.635947\pi$
$374$	12.0000	0.620505
$375$	0	0
$376$	2.00000	0.103142
$377$	−9.00000	−0.463524
$378$	−3.00000	−0.154303
$379$	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
$381$	−16.0000	−0.819705
$382$	−15.0000	−0.767467
$383$	−29.0000	−1.48183	−0.740915	−	0.671598i	$-0.765608\pi$
$383$	−29.0000	−1.48183	−0.740915	+	0.671598i	$0.765608\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−6.00000	−0.305392
$387$	1.00000	0.0508329
$388$	6.00000	0.304604
$389$	38.0000	1.92668	0.963338	−	0.268290i	$-0.0864585\pi$
$389$	38.0000	1.92668	0.963338	+	0.268290i	$0.0864585\pi$
$390$	0	0
$391$	4.00000	0.202289
$392$	2.00000	0.101015
$393$	−10.0000	−0.504433
$394$	−17.0000	−0.856448
$395$	0	0
$396$	−3.00000	−0.150756
$397$	30.0000	1.50566	0.752828	−	0.658217i	$-0.228689\pi$
$397$	30.0000	1.50566	0.752828	+	0.658217i	$0.228689\pi$
$398$	11.0000	0.551380
$399$	9.00000	0.450564
$400$	0	0
$401$	−28.0000	−1.39825	−0.699127	−	0.714998i	$-0.746428\pi$
$401$	−28.0000	−1.39825	−0.699127	+	0.714998i	$0.746428\pi$
$402$	−8.00000	−0.399004
$403$	30.0000	1.49441
$404$	−6.00000	−0.298511
$405$	0	0
$406$	9.00000	0.446663
$407$	24.0000	1.18964
$408$	4.00000	0.198030
$409$	−23.0000	−1.13728	−0.568638	−	0.822588i	$-0.692530\pi$
$409$	−23.0000	−1.13728	−0.568638	+	0.822588i	$0.692530\pi$
$410$	0	0
$411$	18.0000	0.887875
$412$	13.0000	0.640464
$413$	24.0000	1.18096
$414$	−1.00000	−0.0491473
$415$	0	0
$416$	−3.00000	−0.147087
$417$	−8.00000	−0.391762
$418$	9.00000	0.440204
$419$	19.0000	0.928211	0.464105	−	0.885780i	$-0.346376\pi$
$419$	19.0000	0.928211	0.464105	+	0.885780i	$0.346376\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	22.0000	1.07094
$423$	2.00000	0.0972433
$424$	−6.00000	−0.291386
$425$	0	0
$426$	14.0000	0.678302
$427$	36.0000	1.74216
$428$	−12.0000	−0.580042
$429$	−9.00000	−0.434524
$430$	0	0
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	−1.00000	−0.0481125
$433$	4.00000	0.192228	0.0961139	−	0.995370i	$-0.469359\pi$
$433$	4.00000	0.192228	0.0961139	+	0.995370i	$0.469359\pi$
$434$	−30.0000	−1.44005
$435$	0	0
$436$	0	0
$437$	3.00000	0.143509
$438$	−13.0000	−0.621164
$439$	−4.00000	−0.190910	−0.0954548	−	0.995434i	$-0.530431\pi$
$439$	−4.00000	−0.190910	−0.0954548	+	0.995434i	$0.530431\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	12.0000	0.570782
$443$	26.0000	1.23530	0.617649	−	0.786454i	$-0.288085\pi$
$443$	26.0000	1.23530	0.617649	+	0.786454i	$0.288085\pi$
$444$	8.00000	0.379663
$445$	0	0
$446$	−6.00000	−0.284108
$447$	6.00000	0.283790
$448$	3.00000	0.141737
$449$	−26.0000	−1.22702	−0.613508	−	0.789689i	$-0.710242\pi$
$449$	−26.0000	−1.22702	−0.613508	+	0.789689i	$0.710242\pi$
$450$	0	0
$451$	27.0000	1.27138
$452$	6.00000	0.282216
$453$	22.0000	1.03365
$454$	−12.0000	−0.563188
$455$	0	0
$456$	3.00000	0.140488
$457$	22.0000	1.02912	0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000	1.02912	0.514558	+	0.857455i	$0.327956\pi$
$458$	6.00000	0.280362
$459$	4.00000	0.186704
$460$	0	0
$461$	−9.00000	−0.419172	−0.209586	−	0.977790i	$-0.567212\pi$
$461$	−9.00000	−0.419172	−0.209586	+	0.977790i	$0.567212\pi$
$462$	9.00000	0.418718
$463$	−4.00000	−0.185896	−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000	−0.185896	−0.0929479	+	0.995671i	$0.529629\pi$
$464$	3.00000	0.139272
$465$	0	0
$466$	−1.00000	−0.0463241
$467$	11.0000	0.509019	0.254510	−	0.967070i	$-0.418086\pi$
$467$	11.0000	0.509019	0.254510	+	0.967070i	$0.418086\pi$
$468$	−3.00000	−0.138675
$469$	24.0000	1.10822
$470$	0	0
$471$	−6.00000	−0.276465
$472$	8.00000	0.368230
$473$	−3.00000	−0.137940
$474$	17.0000	0.780836
$475$	0	0
$476$	−12.0000	−0.550019
$477$	−6.00000	−0.274721
$478$	−10.0000	−0.457389
$479$	33.0000	1.50781	0.753904	−	0.656984i	$-0.228168\pi$
$479$	33.0000	1.50781	0.753904	+	0.656984i	$0.228168\pi$
$480$	0	0
$481$	24.0000	1.09431
$482$	−10.0000	−0.455488
$483$	3.00000	0.136505
$484$	−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
$489$	−6.00000	−0.271329
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	9.00000	0.405751
$493$	−12.0000	−0.540453
$494$	9.00000	0.404929
$495$	0	0
$496$	−10.0000	−0.449013
$497$	−42.0000	−1.88396
$498$	−1.00000	−0.0448111
$499$	34.0000	1.52205	0.761025	−	0.648723i	$-0.224697\pi$
$499$	34.0000	1.52205	0.761025	+	0.648723i	$0.224697\pi$
$500$	0	0
$501$	−2.00000	−0.0893534
$502$	−16.0000	−0.714115
$503$	−35.0000	−1.56057	−0.780286	−	0.625422i	$-0.784927\pi$
$503$	−35.0000	−1.56057	−0.780286	+	0.625422i	$0.784927\pi$
$504$	3.00000	0.133631
$505$	0	0
$506$	3.00000	0.133366
$507$	4.00000	0.177646
$508$	16.0000	0.709885
$509$	34.0000	1.50702	0.753512	−	0.657434i	$-0.228358\pi$
$509$	34.0000	1.50702	0.753512	+	0.657434i	$0.228358\pi$
$510$	0	0
$511$	39.0000	1.72526
$512$	1.00000	0.0441942
$513$	3.00000	0.132453
$514$	−18.0000	−0.793946
$515$	0	0
$516$	−1.00000	−0.0440225
$517$	−6.00000	−0.263880
$518$	−24.0000	−1.05450
$519$	−15.0000	−0.658427
$520$	0	0
$521$	4.00000	0.175243	0.0876216	−	0.996154i	$-0.472073\pi$
$521$	4.00000	0.175243	0.0876216	+	0.996154i	$0.472073\pi$
$522$	3.00000	0.131306
$523$	11.0000	0.480996	0.240498	−	0.970650i	$-0.422689\pi$
$523$	11.0000	0.480996	0.240498	+	0.970650i	$0.422689\pi$
$524$	10.0000	0.436852
$525$	0	0
$526$	−8.00000	−0.348817
$527$	40.0000	1.74243
$528$	3.00000	0.130558
$529$	1.00000	0.0434783
$530$	0	0
$531$	8.00000	0.347170
$532$	−9.00000	−0.390199
$533$	27.0000	1.16950
$534$	6.00000	0.259645
$535$	0	0
$536$	8.00000	0.345547
$537$	10.0000	0.431532
$538$	−5.00000	−0.215565
$539$	−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	$0.590154\pi$
$542$	22.0000	0.944981
$543$	16.0000	0.686626
$544$	−4.00000	−0.171499
$545$	0	0
$546$	9.00000	0.385164
$547$	22.0000	0.940652	0.470326	−	0.882493i	$-0.344136\pi$
$547$	22.0000	0.940652	0.470326	+	0.882493i	$0.344136\pi$
$548$	−18.0000	−0.768922
$549$	12.0000	0.512148
$550$	0	0
$551$	−9.00000	−0.383413
$552$	1.00000	0.0425628
$553$	−51.0000	−2.16874
$554$	−13.0000	−0.552317
$555$	0	0
$556$	8.00000	0.339276
$557$	−32.0000	−1.35588	−0.677942	−	0.735116i	$-0.737128\pi$
$557$	−32.0000	−1.35588	−0.677942	+	0.735116i	$0.737128\pi$
$558$	−10.0000	−0.423334
$559$	−3.00000	−0.126886
$560$	0	0
$561$	−12.0000	−0.506640
$562$	−10.0000	−0.421825
$563$	5.00000	0.210725	0.105362	−	0.994434i	$-0.466400\pi$
$563$	5.00000	0.210725	0.105362	+	0.994434i	$0.466400\pi$
$564$	−2.00000	−0.0842152
$565$	0	0
$566$	0	0
$567$	3.00000	0.125988
$568$	−14.0000	−0.587427
$569$	−20.0000	−0.838444	−0.419222	−	0.907884i	$-0.637697\pi$
$569$	−20.0000	−0.838444	−0.419222	+	0.907884i	$0.637697\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	9.00000	0.376309
$573$	15.0000	0.626634
$574$	−27.0000	−1.12696
$575$	0	0
$576$	1.00000	0.0416667
$577$	−7.00000	−0.291414	−0.145707	−	0.989328i	$-0.546546\pi$
$577$	−7.00000	−0.291414	−0.145707	+	0.989328i	$0.546546\pi$
$578$	−1.00000	−0.0415945
$579$	6.00000	0.249351
$580$	0	0
$581$	3.00000	0.124461
$582$	−6.00000	−0.248708
$583$	18.0000	0.745484
$584$	13.0000	0.537944
$585$	0	0
$586$	0	0
$587$	24.0000	0.990586	0.495293	−	0.868726i	$-0.335061\pi$
$587$	24.0000	0.990586	0.495293	+	0.868726i	$0.335061\pi$
$588$	−2.00000	−0.0824786
$589$	30.0000	1.23613
$590$	0	0
$591$	17.0000	0.699287
$592$	−8.00000	−0.328798
$593$	−21.0000	−0.862367	−0.431183	−	0.902264i	$-0.641904\pi$
$593$	−21.0000	−0.862367	−0.431183	+	0.902264i	$0.641904\pi$
$594$	3.00000	0.123091
$595$	0	0
$596$	−6.00000	−0.245770
$597$	−11.0000	−0.450200
$598$	3.00000	0.122679
$599$	6.00000	0.245153	0.122577	−	0.992459i	$-0.460884\pi$
$599$	6.00000	0.245153	0.122577	+	0.992459i	$0.460884\pi$
$600$	0	0
$601$	−14.0000	−0.571072	−0.285536	−	0.958368i	$-0.592172\pi$
$601$	−14.0000	−0.571072	−0.285536	+	0.958368i	$0.592172\pi$
$602$	3.00000	0.122271
$603$	8.00000	0.325785
$604$	−22.0000	−0.895167
$605$	0	0
$606$	6.00000	0.243733
$607$	2.00000	0.0811775	0.0405887	−	0.999176i	$-0.487077\pi$
$607$	2.00000	0.0811775	0.0405887	+	0.999176i	$0.487077\pi$
$608$	−3.00000	−0.121666
$609$	−9.00000	−0.364698
$610$	0	0
$611$	−6.00000	−0.242734
$612$	−4.00000	−0.161690
$613$	20.0000	0.807792	0.403896	−	0.914805i	$-0.367656\pi$
$613$	20.0000	0.807792	0.403896	+	0.914805i	$0.367656\pi$
$614$	8.00000	0.322854
$615$	0	0
$616$	−9.00000	−0.362620
$617$	−16.0000	−0.644136	−0.322068	−	0.946717i	$-0.604378\pi$
$617$	−16.0000	−0.644136	−0.322068	+	0.946717i	$0.604378\pi$
$618$	−13.0000	−0.522937
$619$	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
$621$	1.00000	0.0401286
$622$	24.0000	0.962312
$623$	−18.0000	−0.721155
$624$	3.00000	0.120096
$625$	0	0
$626$	24.0000	0.959233
$627$	−9.00000	−0.359425
$628$	6.00000	0.239426
$629$	32.0000	1.27592
$630$	0	0
$631$	5.00000	0.199047	0.0995234	−	0.995035i	$-0.468268\pi$
$631$	5.00000	0.199047	0.0995234	+	0.995035i	$0.468268\pi$
$632$	−17.0000	−0.676224
$633$	−22.0000	−0.874421
$634$	31.0000	1.23117
$635$	0	0
$636$	6.00000	0.237915
$637$	−6.00000	−0.237729
$638$	−9.00000	−0.356313
$639$	−14.0000	−0.553831
$640$	0	0
$641$	−16.0000	−0.631962	−0.315981	−	0.948766i	$-0.602334\pi$
$641$	−16.0000	−0.631962	−0.315981	+	0.948766i	$0.602334\pi$
$642$	12.0000	0.473602
$643$	−31.0000	−1.22252	−0.611260	−	0.791430i	$-0.709337\pi$
$643$	−31.0000	−1.22252	−0.611260	+	0.791430i	$0.709337\pi$
$644$	−3.00000	−0.118217
$645$	0	0
$646$	12.0000	0.472134
$647$	−18.0000	−0.707653	−0.353827	−	0.935311i	$-0.615120\pi$
$647$	−18.0000	−0.707653	−0.353827	+	0.935311i	$0.615120\pi$
$648$	1.00000	0.0392837
$649$	−24.0000	−0.942082
$650$	0	0
$651$	30.0000	1.17579
$652$	6.00000	0.234978
$653$	−3.00000	−0.117399	−0.0586995	−	0.998276i	$-0.518695\pi$
$653$	−3.00000	−0.117399	−0.0586995	+	0.998276i	$0.518695\pi$
$654$	0	0
$655$	0	0
$656$	−9.00000	−0.351391
$657$	13.0000	0.507178
$658$	6.00000	0.233904
$659$	−33.0000	−1.28550	−0.642749	−	0.766077i	$-0.722206\pi$
$659$	−33.0000	−1.28550	−0.642749	+	0.766077i	$0.722206\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	−26.0000	−1.01052
$663$	−12.0000	−0.466041
$664$	1.00000	0.0388075
$665$	0	0
$666$	−8.00000	−0.309994
$667$	−3.00000	−0.116160
$668$	2.00000	0.0773823
$669$	6.00000	0.231973
$670$	0	0
$671$	−36.0000	−1.38976
$672$	−3.00000	−0.115728
$673$	3.00000	0.115642	0.0578208	−	0.998327i	$-0.481585\pi$
$673$	3.00000	0.115642	0.0578208	+	0.998327i	$0.481585\pi$
$674$	14.0000	0.539260
$675$	0	0
$676$	−4.00000	−0.153846
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	−6.00000	−0.230429
$679$	18.0000	0.690777
$680$	0	0
$681$	12.0000	0.459841
$682$	30.0000	1.14876
$683$	38.0000	1.45403	0.727015	−	0.686622i	$-0.240907\pi$
$683$	38.0000	1.45403	0.727015	+	0.686622i	$0.240907\pi$
$684$	−3.00000	−0.114708
$685$	0	0
$686$	−15.0000	−0.572703
$687$	−6.00000	−0.228914
$688$	1.00000	0.0381246
$689$	18.0000	0.685745
$690$	0	0
$691$	−48.0000	−1.82601	−0.913003	−	0.407953i	$-0.866243\pi$
$691$	−48.0000	−1.82601	−0.913003	+	0.407953i	$0.866243\pi$
$692$	15.0000	0.570214
$693$	−9.00000	−0.341882
$694$	0	0
$695$	0	0
$696$	−3.00000	−0.113715
$697$	36.0000	1.36360
$698$	−5.00000	−0.189253
$699$	1.00000	0.0378235
$700$	0	0
$701$	−36.0000	−1.35970	−0.679851	−	0.733351i	$-0.737955\pi$
$701$	−36.0000	−1.35970	−0.679851	+	0.733351i	$0.737955\pi$
$702$	3.00000	0.113228
$703$	24.0000	0.905177
$704$	−3.00000	−0.113067
$705$	0	0
$706$	9.00000	0.338719
$707$	−18.0000	−0.676960
$708$	−8.00000	−0.300658
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	0	0
$711$	−17.0000	−0.637550
$712$	−6.00000	−0.224860
$713$	10.0000	0.374503
$714$	12.0000	0.449089
$715$	0	0
$716$	−10.0000	−0.373718
$717$	10.0000	0.373457
$718$	31.0000	1.15691
$719$	16.0000	0.596699	0.298350	−	0.954457i	$-0.403564\pi$
$719$	16.0000	0.596699	0.298350	+	0.954457i	$0.403564\pi$
$720$	0	0
$721$	39.0000	1.45244
$722$	−10.0000	−0.372161
$723$	10.0000	0.371904
$724$	−16.0000	−0.594635
$725$	0	0
$726$	2.00000	0.0742270
$727$	−16.0000	−0.593407	−0.296704	−	0.954970i	$-0.595887\pi$
$727$	−16.0000	−0.593407	−0.296704	+	0.954970i	$0.595887\pi$
$728$	−9.00000	−0.333562
$729$	1.00000	0.0370370
$730$	0	0
$731$	−4.00000	−0.147945
$732$	−12.0000	−0.443533
$733$	14.0000	0.517102	0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000	0.517102	0.258551	+	0.965998i	$0.416755\pi$
$734$	27.0000	0.996588
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	−24.0000	−0.884051
$738$	−9.00000	−0.331295
$739$	50.0000	1.83928	0.919640	−	0.392763i	$-0.128481\pi$
$739$	50.0000	1.83928	0.919640	+	0.392763i	$0.128481\pi$
$740$	0	0
$741$	−9.00000	−0.330623
$742$	−18.0000	−0.660801
$743$	−29.0000	−1.06391	−0.531953	−	0.846774i	$-0.678542\pi$
$743$	−29.0000	−1.06391	−0.531953	+	0.846774i	$0.678542\pi$
$744$	10.0000	0.366618
$745$	0	0
$746$	−16.0000	−0.585802
$747$	1.00000	0.0365881
$748$	12.0000	0.438763
$749$	−36.0000	−1.31541
$750$	0	0
$751$	45.0000	1.64207	0.821037	−	0.570875i	$-0.193396\pi$
$751$	45.0000	1.64207	0.821037	+	0.570875i	$0.193396\pi$
$752$	2.00000	0.0729325
$753$	16.0000	0.583072
$754$	−9.00000	−0.327761
$755$	0	0
$756$	−3.00000	−0.109109
$757$	−20.0000	−0.726912	−0.363456	−	0.931611i	$-0.618403\pi$
$757$	−20.0000	−0.726912	−0.363456	+	0.931611i	$0.618403\pi$
$758$	12.0000	0.435860
$759$	−3.00000	−0.108893
$760$	0	0
$761$	−27.0000	−0.978749	−0.489375	−	0.872074i	$-0.662775\pi$
$761$	−27.0000	−0.978749	−0.489375	+	0.872074i	$0.662775\pi$
$762$	−16.0000	−0.579619
$763$	0	0
$764$	−15.0000	−0.542681
$765$	0	0
$766$	−29.0000	−1.04781
$767$	−24.0000	−0.866590
$768$	−1.00000	−0.0360844
$769$	36.0000	1.29819	0.649097	−	0.760706i	$-0.275147\pi$
$769$	36.0000	1.29819	0.649097	+	0.760706i	$0.275147\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	−6.00000	−0.215945
$773$	−40.0000	−1.43870	−0.719350	−	0.694648i	$-0.755560\pi$
$773$	−40.0000	−1.43870	−0.719350	+	0.694648i	$0.755560\pi$
$774$	1.00000	0.0359443
$775$	0	0
$776$	6.00000	0.215387
$777$	24.0000	0.860995
$778$	38.0000	1.36237
$779$	27.0000	0.967375
$780$	0	0
$781$	42.0000	1.50288
$782$	4.00000	0.143040
$783$	−3.00000	−0.107211
$784$	2.00000	0.0714286
$785$	0	0
$786$	−10.0000	−0.356688
$787$	−5.00000	−0.178231	−0.0891154	−	0.996021i	$-0.528404\pi$
$787$	−5.00000	−0.178231	−0.0891154	+	0.996021i	$0.528404\pi$
$788$	−17.0000	−0.605600
$789$	8.00000	0.284808
$790$	0	0
$791$	18.0000	0.640006
$792$	−3.00000	−0.106600
$793$	−36.0000	−1.27840
$794$	30.0000	1.06466
$795$	0	0
$796$	11.0000	0.389885
$797$	24.0000	0.850124	0.425062	−	0.905164i	$-0.360252\pi$
$797$	24.0000	0.850124	0.425062	+	0.905164i	$0.360252\pi$
$798$	9.00000	0.318597
$799$	−8.00000	−0.283020
$800$	0	0
$801$	−6.00000	−0.212000
$802$	−28.0000	−0.988714
$803$	−39.0000	−1.37628
$804$	−8.00000	−0.282138
$805$	0	0
$806$	30.0000	1.05670
$807$	5.00000	0.176008
$808$	−6.00000	−0.211079
$809$	21.0000	0.738321	0.369160	−	0.929366i	$-0.379645\pi$
$809$	21.0000	0.738321	0.369160	+	0.929366i	$0.379645\pi$
$810$	0	0
$811$	16.0000	0.561836	0.280918	−	0.959732i	$-0.409361\pi$
$811$	16.0000	0.561836	0.280918	+	0.959732i	$0.409361\pi$
$812$	9.00000	0.315838
$813$	−22.0000	−0.771574
$814$	24.0000	0.841200
$815$	0	0
$816$	4.00000	0.140028
$817$	−3.00000	−0.104957
$818$	−23.0000	−0.804176
$819$	−9.00000	−0.314485
$820$	0	0
$821$	−55.0000	−1.91951	−0.959757	−	0.280833i	$-0.909389\pi$
$821$	−55.0000	−1.91951	−0.959757	+	0.280833i	$0.909389\pi$
$822$	18.0000	0.627822
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	13.0000	0.452876
$825$	0	0
$826$	24.0000	0.835067
$827$	−9.00000	−0.312961	−0.156480	−	0.987681i	$-0.550015\pi$
$827$	−9.00000	−0.312961	−0.156480	+	0.987681i	$0.550015\pi$
$828$	−1.00000	−0.0347524
$829$	11.0000	0.382046	0.191023	−	0.981586i	$-0.438820\pi$
$829$	11.0000	0.382046	0.191023	+	0.981586i	$0.438820\pi$
$830$	0	0
$831$	13.0000	0.450965
$832$	−3.00000	−0.104006
$833$	−8.00000	−0.277184
$834$	−8.00000	−0.277017
$835$	0	0
$836$	9.00000	0.311272
$837$	10.0000	0.345651
$838$	19.0000	0.656344
$839$	33.0000	1.13929	0.569643	−	0.821892i	$-0.307081\pi$
$839$	33.0000	1.13929	0.569643	+	0.821892i	$0.307081\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	−10.0000	−0.344623
$843$	10.0000	0.344418
$844$	22.0000	0.757271
$845$	0	0
$846$	2.00000	0.0687614
$847$	−6.00000	−0.206162
$848$	−6.00000	−0.206041
$849$	0	0
$850$	0	0
$851$	8.00000	0.274236
$852$	14.0000	0.479632
$853$	−23.0000	−0.787505	−0.393753	−	0.919216i	$-0.628823\pi$
$853$	−23.0000	−0.787505	−0.393753	+	0.919216i	$0.628823\pi$
$854$	36.0000	1.23189
$855$	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	$0.377384\pi$
$858$	−9.00000	−0.307255
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	0	0
$861$	27.0000	0.920158
$862$	16.0000	0.544962
$863$	54.0000	1.83818	0.919091	−	0.394046i	$-0.128925\pi$
$863$	54.0000	1.83818	0.919091	+	0.394046i	$0.128925\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	4.00000	0.135926
$867$	1.00000	0.0339618
$868$	−30.0000	−1.01827
$869$	51.0000	1.73006
$870$	0	0
$871$	−24.0000	−0.813209
$872$	0	0
$873$	6.00000	0.203069
$874$	3.00000	0.101477
$875$	0	0
$876$	−13.0000	−0.439229
$877$	−22.0000	−0.742887	−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000	−0.742887	−0.371444	+	0.928456i	$0.621137\pi$
$878$	−4.00000	−0.134993
$879$	0	0
$880$	0	0
$881$	−40.0000	−1.34763	−0.673817	−	0.738898i	$-0.735346\pi$
$881$	−40.0000	−1.34763	−0.673817	+	0.738898i	$0.735346\pi$
$882$	2.00000	0.0673435
$883$	−48.0000	−1.61533	−0.807664	−	0.589643i	$-0.799269\pi$
$883$	−48.0000	−1.61533	−0.807664	+	0.589643i	$0.799269\pi$
$884$	12.0000	0.403604
$885$	0	0
$886$	26.0000	0.873487
$887$	−8.00000	−0.268614	−0.134307	−	0.990940i	$-0.542881\pi$
$887$	−8.00000	−0.268614	−0.134307	+	0.990940i	$0.542881\pi$
$888$	8.00000	0.268462
$889$	48.0000	1.60987
$890$	0	0
$891$	−3.00000	−0.100504
$892$	−6.00000	−0.200895
$893$	−6.00000	−0.200782
$894$	6.00000	0.200670
$895$	0	0
$896$	3.00000	0.100223
$897$	−3.00000	−0.100167
$898$	−26.0000	−0.867631
$899$	−30.0000	−1.00056
$900$	0	0
$901$	24.0000	0.799556
$902$	27.0000	0.899002
$903$	−3.00000	−0.0998337
$904$	6.00000	0.199557
$905$	0	0
$906$	22.0000	0.730901
$907$	−1.00000	−0.0332045	−0.0166022	−	0.999862i	$-0.505285\pi$
$907$	−1.00000	−0.0332045	−0.0166022	+	0.999862i	$0.505285\pi$
$908$	−12.0000	−0.398234
$909$	−6.00000	−0.199007
$910$	0	0
$911$	55.0000	1.82223	0.911116	−	0.412151i	$-0.135222\pi$
$911$	55.0000	1.82223	0.911116	+	0.412151i	$0.135222\pi$
$912$	3.00000	0.0993399
$913$	−3.00000	−0.0992855
$914$	22.0000	0.727695
$915$	0	0
$916$	6.00000	0.198246
$917$	30.0000	0.990687
$918$	4.00000	0.132020
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	0	0
$921$	−8.00000	−0.263609
$922$	−9.00000	−0.296399
$923$	42.0000	1.38245
$924$	9.00000	0.296078
$925$	0	0
$926$	−4.00000	−0.131448
$927$	13.0000	0.426976
$928$	3.00000	0.0984798
$929$	−41.0000	−1.34517	−0.672583	−	0.740022i	$-0.734815\pi$
$929$	−41.0000	−1.34517	−0.672583	+	0.740022i	$0.734815\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	−1.00000	−0.0327561
$933$	−24.0000	−0.785725
$934$	11.0000	0.359931
$935$	0	0
$936$	−3.00000	−0.0980581
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	24.0000	0.783628
$939$	−24.0000	−0.783210
$940$	0	0
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	−6.00000	−0.195491
$943$	9.00000	0.293080
$944$	8.00000	0.260378
$945$	0	0
$946$	−3.00000	−0.0975384
$947$	2.00000	0.0649913	0.0324956	−	0.999472i	$-0.489654\pi$
$947$	2.00000	0.0649913	0.0324956	+	0.999472i	$0.489654\pi$
$948$	17.0000	0.552134
$949$	−39.0000	−1.26599
$950$	0	0
$951$	−31.0000	−1.00524
$952$	−12.0000	−0.388922
$953$	−24.0000	−0.777436	−0.388718	−	0.921357i	$-0.627082\pi$
$953$	−24.0000	−0.777436	−0.388718	+	0.921357i	$0.627082\pi$
$954$	−6.00000	−0.194257
$955$	0	0
$956$	−10.0000	−0.323423
$957$	9.00000	0.290929
$958$	33.0000	1.06618
$959$	−54.0000	−1.74375
$960$	0	0
$961$	69.0000	2.22581
$962$	24.0000	0.773791
$963$	−12.0000	−0.386695
$964$	−10.0000	−0.322078
$965$	0	0
$966$	3.00000	0.0965234
$967$	−44.0000	−1.41494	−0.707472	−	0.706741i	$-0.750165\pi$
$967$	−44.0000	−1.41494	−0.707472	+	0.706741i	$0.750165\pi$
$968$	−2.00000	−0.0642824
$969$	−12.0000	−0.385496
$970$	0	0
$971$	43.0000	1.37994	0.689968	−	0.723840i	$-0.257625\pi$
$971$	43.0000	1.37994	0.689968	+	0.723840i	$0.257625\pi$
$972$	−1.00000	−0.0320750
$973$	24.0000	0.769405
$974$	−2.00000	−0.0640841
$975$	0	0
$976$	12.0000	0.384111
$977$	38.0000	1.21573	0.607864	−	0.794041i	$-0.292027\pi$
$977$	38.0000	1.21573	0.607864	+	0.794041i	$0.292027\pi$
$978$	−6.00000	−0.191859
$979$	18.0000	0.575282
$980$	0	0
$981$	0	0
$982$	12.0000	0.382935
$983$	−39.0000	−1.24391	−0.621953	−	0.783054i	$-0.713661\pi$
$983$	−39.0000	−1.24391	−0.621953	+	0.783054i	$0.713661\pi$
$984$	9.00000	0.286910
$985$	0	0
$986$	−12.0000	−0.382158
$987$	−6.00000	−0.190982
$988$	9.00000	0.286328
$989$	−1.00000	−0.0317982
$990$	0	0
$991$	−26.0000	−0.825917	−0.412959	−	0.910750i	$-0.635505\pi$
$991$	−26.0000	−0.825917	−0.412959	+	0.910750i	$0.635505\pi$
$992$	−10.0000	−0.317500
$993$	26.0000	0.825085
$994$	−42.0000	−1.33216
$995$	0	0
$996$	−1.00000	−0.0316862
$997$	−49.0000	−1.55185	−0.775923	−	0.630828i	$-0.782715\pi$
$997$	−49.0000	−1.55185	−0.775923	+	0.630828i	$0.782715\pi$
$998$	34.0000	1.07625
$999$	8.00000	0.253109

Twists

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

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

Downloads

Learn more