LMFDB - Embedded newform 2541.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2541,2,Mod(1,2541)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2541, 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("2541.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2541 = 3 \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2541.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:	$20.2899871536$
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	$\chi$	$=$	2541.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +1.00000 q^{12} -5.00000 q^{13} -1.00000 q^{14} -1.00000 q^{15} -1.00000 q^{16} +7.00000 q^{17} +1.00000 q^{18} +6.00000 q^{19} -1.00000 q^{20} +1.00000 q^{21} -4.00000 q^{23} +3.00000 q^{24} -4.00000 q^{25} -5.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} -9.00000 q^{29} -1.00000 q^{30} -2.00000 q^{31} +5.00000 q^{32} +7.00000 q^{34} -1.00000 q^{35} -1.00000 q^{36} +9.00000 q^{37} +6.00000 q^{38} +5.00000 q^{39} -3.00000 q^{40} +7.00000 q^{41} +1.00000 q^{42} +6.00000 q^{43} +1.00000 q^{45} -4.00000 q^{46} +2.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} -4.00000 q^{50} -7.00000 q^{51} +5.00000 q^{52} +3.00000 q^{53} -1.00000 q^{54} +3.00000 q^{56} -6.00000 q^{57} -9.00000 q^{58} +2.00000 q^{59} +1.00000 q^{60} -6.00000 q^{61} -2.00000 q^{62} -1.00000 q^{63} +7.00000 q^{64} -5.00000 q^{65} +8.00000 q^{67} -7.00000 q^{68} +4.00000 q^{69} -1.00000 q^{70} +6.00000 q^{71} -3.00000 q^{72} +10.0000 q^{73} +9.00000 q^{74} +4.00000 q^{75} -6.00000 q^{76} +5.00000 q^{78} +14.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +7.00000 q^{82} -1.00000 q^{84} +7.00000 q^{85} +6.00000 q^{86} +9.00000 q^{87} -9.00000 q^{89} +1.00000 q^{90} +5.00000 q^{91} +4.00000 q^{92} +2.00000 q^{93} +2.00000 q^{94} +6.00000 q^{95} -5.00000 q^{96} +17.0000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.00000	−0.500000
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	−1.00000	−0.408248
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−3.00000	−1.06066
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	0	0
$11$	0	0
$12$	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	$0.743877\pi$
$14$	−1.00000	−0.267261
$15$	−1.00000	−0.258199
$16$	−1.00000	−0.250000
$17$	7.00000	1.69775	0.848875	−	0.528594i	$-0.177281\pi$
$17$	7.00000	1.69775	0.848875	+	0.528594i	$0.177281\pi$
$18$	1.00000	0.235702
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	−1.00000	−0.223607
$21$	1.00000	0.218218
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	3.00000	0.612372
$25$	−4.00000	−0.800000
$26$	−5.00000	−0.980581
$27$	−1.00000	−0.192450
$28$	1.00000	0.188982
$29$	−9.00000	−1.67126	−0.835629	−	0.549294i	$-0.814897\pi$
$29$	−9.00000	−1.67126	−0.835629	+	0.549294i	$0.814897\pi$
$30$	−1.00000	−0.182574
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	5.00000	0.883883
$33$	0	0
$34$	7.00000	1.20049
$35$	−1.00000	−0.169031
$36$	−1.00000	−0.166667
$37$	9.00000	1.47959	0.739795	−	0.672832i	$-0.234922\pi$
$37$	9.00000	1.47959	0.739795	+	0.672832i	$0.234922\pi$
$38$	6.00000	0.973329
$39$	5.00000	0.800641
$40$	−3.00000	−0.474342
$41$	7.00000	1.09322	0.546608	−	0.837389i	$-0.315919\pi$
$41$	7.00000	1.09322	0.546608	+	0.837389i	$0.315919\pi$
$42$	1.00000	0.154303
$43$	6.00000	0.914991	0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000	0.914991	0.457496	+	0.889212i	$0.348747\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	−4.00000	−0.589768
$47$	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
$49$	1.00000	0.142857
$50$	−4.00000	−0.565685
$51$	−7.00000	−0.980196
$52$	5.00000	0.693375
$53$	3.00000	0.412082	0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000	0.412082	0.206041	+	0.978543i	$0.433942\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	3.00000	0.400892
$57$	−6.00000	−0.794719
$58$	−9.00000	−1.18176
$59$	2.00000	0.260378	0.130189	−	0.991489i	$-0.458442\pi$
$59$	2.00000	0.260378	0.130189	+	0.991489i	$0.458442\pi$
$60$	1.00000	0.129099
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	−2.00000	−0.254000
$63$	−1.00000	−0.125988
$64$	7.00000	0.875000
$65$	−5.00000	−0.620174
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	−7.00000	−0.848875
$69$	4.00000	0.481543
$70$	−1.00000	−0.119523
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	−3.00000	−0.353553
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	9.00000	1.04623
$75$	4.00000	0.461880
$76$	−6.00000	−0.688247
$77$	0	0
$78$	5.00000	0.566139
$79$	14.0000	1.57512	0.787562	−	0.616236i	$-0.211343\pi$
$79$	14.0000	1.57512	0.787562	+	0.616236i	$0.211343\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	7.00000	0.773021
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	−1.00000	−0.109109
$85$	7.00000	0.759257
$86$	6.00000	0.646997
$87$	9.00000	0.964901
$88$	0	0
$89$	−9.00000	−0.953998	−0.476999	−	0.878904i	$-0.658275\pi$
$89$	−9.00000	−0.953998	−0.476999	+	0.878904i	$0.658275\pi$
$90$	1.00000	0.105409
$91$	5.00000	0.524142
$92$	4.00000	0.417029
$93$	2.00000	0.207390
$94$	2.00000	0.206284
$95$	6.00000	0.615587
$96$	−5.00000	−0.510310
$97$	17.0000	1.72609	0.863044	−	0.505128i	$-0.168555\pi$
$97$	17.0000	1.72609	0.863044	+	0.505128i	$0.168555\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	4.00000	0.400000
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	−7.00000	−0.693103
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	15.0000	1.47087
$105$	1.00000	0.0975900
$106$	3.00000	0.291386
$107$	18.0000	1.74013	0.870063	−	0.492941i	$-0.164078\pi$
$107$	18.0000	1.74013	0.870063	+	0.492941i	$0.164078\pi$
$108$	1.00000	0.0962250
$109$	1.00000	0.0957826	0.0478913	−	0.998853i	$-0.484750\pi$
$109$	1.00000	0.0957826	0.0478913	+	0.998853i	$0.484750\pi$
$110$	0	0
$111$	−9.00000	−0.854242
$112$	1.00000	0.0944911
$113$	−15.0000	−1.41108	−0.705541	−	0.708669i	$-0.749296\pi$
$113$	−15.0000	−1.41108	−0.705541	+	0.708669i	$0.749296\pi$
$114$	−6.00000	−0.561951
$115$	−4.00000	−0.373002
$116$	9.00000	0.835629
$117$	−5.00000	−0.462250
$118$	2.00000	0.184115
$119$	−7.00000	−0.641689
$120$	3.00000	0.273861
$121$	0	0
$122$	−6.00000	−0.543214
$123$	−7.00000	−0.631169
$124$	2.00000	0.179605
$125$	−9.00000	−0.804984
$126$	−1.00000	−0.0890871
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	−3.00000	−0.265165
$129$	−6.00000	−0.528271
$130$	−5.00000	−0.438529
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	−6.00000	−0.520266
$134$	8.00000	0.691095
$135$	−1.00000	−0.0860663
$136$	−21.0000	−1.80074
$137$	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	$0.822307\pi$
$140$	1.00000	0.0845154
$141$	−2.00000	−0.168430
$142$	6.00000	0.503509
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	−9.00000	−0.747409
$146$	10.0000	0.827606
$147$	−1.00000	−0.0824786
$148$	−9.00000	−0.739795
$149$	−13.0000	−1.06500	−0.532501	−	0.846430i	$-0.678748\pi$
$149$	−13.0000	−1.06500	−0.532501	+	0.846430i	$0.678748\pi$
$150$	4.00000	0.326599
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	−18.0000	−1.45999
$153$	7.00000	0.565916
$154$	0	0
$155$	−2.00000	−0.160644
$156$	−5.00000	−0.400320
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	14.0000	1.11378
$159$	−3.00000	−0.237915
$160$	5.00000	0.395285
$161$	4.00000	0.315244
$162$	1.00000	0.0785674
$163$	10.0000	0.783260	0.391630	−	0.920123i	$-0.371911\pi$
$163$	10.0000	0.783260	0.391630	+	0.920123i	$0.371911\pi$
$164$	−7.00000	−0.546608
$165$	0	0
$166$	0	0
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	−3.00000	−0.231455
$169$	12.0000	0.923077
$170$	7.00000	0.536875
$171$	6.00000	0.458831
$172$	−6.00000	−0.457496
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	9.00000	0.682288
$175$	4.00000	0.302372
$176$	0	0
$177$	−2.00000	−0.150329
$178$	−9.00000	−0.674579
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	−1.00000	−0.0745356
$181$	−5.00000	−0.371647	−0.185824	−	0.982583i	$-0.559495\pi$
$181$	−5.00000	−0.371647	−0.185824	+	0.982583i	$0.559495\pi$
$182$	5.00000	0.370625
$183$	6.00000	0.443533
$184$	12.0000	0.884652
$185$	9.00000	0.661693
$186$	2.00000	0.146647
$187$	0	0
$188$	−2.00000	−0.145865
$189$	1.00000	0.0727393
$190$	6.00000	0.435286
$191$	−10.0000	−0.723575	−0.361787	−	0.932261i	$-0.617833\pi$
$191$	−10.0000	−0.723575	−0.361787	+	0.932261i	$0.617833\pi$
$192$	−7.00000	−0.505181
$193$	−5.00000	−0.359908	−0.179954	−	0.983675i	$-0.557595\pi$
$193$	−5.00000	−0.359908	−0.179954	+	0.983675i	$0.557595\pi$
$194$	17.0000	1.22053
$195$	5.00000	0.358057
$196$	−1.00000	−0.0714286
$197$	7.00000	0.498729	0.249365	−	0.968410i	$-0.419778\pi$
$197$	7.00000	0.498729	0.249365	+	0.968410i	$0.419778\pi$
$198$	0	0
$199$	6.00000	0.425329	0.212664	−	0.977125i	$-0.431786\pi$
$199$	6.00000	0.425329	0.212664	+	0.977125i	$0.431786\pi$
$200$	12.0000	0.848528
$201$	−8.00000	−0.564276
$202$	14.0000	0.985037
$203$	9.00000	0.631676
$204$	7.00000	0.490098
$205$	7.00000	0.488901
$206$	−16.0000	−1.11477
$207$	−4.00000	−0.278019
$208$	5.00000	0.346688
$209$	0	0
$210$	1.00000	0.0690066
$211$	24.0000	1.65223	0.826114	−	0.563503i	$-0.190547\pi$
$211$	24.0000	1.65223	0.826114	+	0.563503i	$0.190547\pi$
$212$	−3.00000	−0.206041
$213$	−6.00000	−0.411113
$214$	18.0000	1.23045
$215$	6.00000	0.409197
$216$	3.00000	0.204124
$217$	2.00000	0.135769
$218$	1.00000	0.0677285
$219$	−10.0000	−0.675737
$220$	0	0
$221$	−35.0000	−2.35435
$222$	−9.00000	−0.604040
$223$	−8.00000	−0.535720	−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000	−0.535720	−0.267860	+	0.963458i	$0.586316\pi$
$224$	−5.00000	−0.334077
$225$	−4.00000	−0.266667
$226$	−15.0000	−0.997785
$227$	−6.00000	−0.398234	−0.199117	−	0.979976i	$-0.563807\pi$
$227$	−6.00000	−0.398234	−0.199117	+	0.979976i	$0.563807\pi$
$228$	6.00000	0.397360
$229$	15.0000	0.991228	0.495614	−	0.868543i	$-0.334943\pi$
$229$	15.0000	0.991228	0.495614	+	0.868543i	$0.334943\pi$
$230$	−4.00000	−0.263752
$231$	0	0
$232$	27.0000	1.77264
$233$	9.00000	0.589610	0.294805	−	0.955557i	$-0.404745\pi$
$233$	9.00000	0.589610	0.294805	+	0.955557i	$0.404745\pi$
$234$	−5.00000	−0.326860
$235$	2.00000	0.130466
$236$	−2.00000	−0.130189
$237$	−14.0000	−0.909398
$238$	−7.00000	−0.453743
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	1.00000	0.0645497
$241$	−2.00000	−0.128831	−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000	−0.128831	−0.0644157	+	0.997923i	$0.520518\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	6.00000	0.384111
$245$	1.00000	0.0638877
$246$	−7.00000	−0.446304
$247$	−30.0000	−1.90885
$248$	6.00000	0.381000
$249$	0	0
$250$	−9.00000	−0.569210
$251$	4.00000	0.252478	0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000	0.252478	0.126239	+	0.992000i	$0.459709\pi$
$252$	1.00000	0.0629941
$253$	0	0
$254$	8.00000	0.501965
$255$	−7.00000	−0.438357
$256$	−17.0000	−1.06250
$257$	−17.0000	−1.06043	−0.530215	−	0.847863i	$-0.677889\pi$
$257$	−17.0000	−1.06043	−0.530215	+	0.847863i	$0.677889\pi$
$258$	−6.00000	−0.373544
$259$	−9.00000	−0.559233
$260$	5.00000	0.310087
$261$	−9.00000	−0.557086
$262$	12.0000	0.741362
$263$	8.00000	0.493301	0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000	0.493301	0.246651	+	0.969104i	$0.420670\pi$
$264$	0	0
$265$	3.00000	0.184289
$266$	−6.00000	−0.367884
$267$	9.00000	0.550791
$268$	−8.00000	−0.488678
$269$	−11.0000	−0.670682	−0.335341	−	0.942097i	$-0.608852\pi$
$269$	−11.0000	−0.670682	−0.335341	+	0.942097i	$0.608852\pi$
$270$	−1.00000	−0.0608581
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	−7.00000	−0.424437
$273$	−5.00000	−0.302614
$274$	2.00000	0.120824
$275$	0	0
$276$	−4.00000	−0.240772
$277$	−11.0000	−0.660926	−0.330463	−	0.943819i	$-0.607205\pi$
$277$	−11.0000	−0.660926	−0.330463	+	0.943819i	$0.607205\pi$
$278$	−20.0000	−1.19952
$279$	−2.00000	−0.119737
$280$	3.00000	0.179284
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	−2.00000	−0.119098
$283$	−14.0000	−0.832214	−0.416107	−	0.909316i	$-0.636606\pi$
$283$	−14.0000	−0.832214	−0.416107	+	0.909316i	$0.636606\pi$
$284$	−6.00000	−0.356034
$285$	−6.00000	−0.355409
$286$	0	0
$287$	−7.00000	−0.413197
$288$	5.00000	0.294628
$289$	32.0000	1.88235
$290$	−9.00000	−0.528498
$291$	−17.0000	−0.996558
$292$	−10.0000	−0.585206
$293$	9.00000	0.525786	0.262893	−	0.964825i	$-0.415323\pi$
$293$	9.00000	0.525786	0.262893	+	0.964825i	$0.415323\pi$
$294$	−1.00000	−0.0583212
$295$	2.00000	0.116445
$296$	−27.0000	−1.56934
$297$	0	0
$298$	−13.0000	−0.753070
$299$	20.0000	1.15663
$300$	−4.00000	−0.230940
$301$	−6.00000	−0.345834
$302$	2.00000	0.115087
$303$	−14.0000	−0.804279
$304$	−6.00000	−0.344124
$305$	−6.00000	−0.343559
$306$	7.00000	0.400163
$307$	−28.0000	−1.59804	−0.799022	−	0.601302i	$-0.794649\pi$
$307$	−28.0000	−1.59804	−0.799022	+	0.601302i	$0.794649\pi$
$308$	0	0
$309$	16.0000	0.910208
$310$	−2.00000	−0.113592
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	−15.0000	−0.849208
$313$	−19.0000	−1.07394	−0.536972	−	0.843600i	$-0.680432\pi$
$313$	−19.0000	−1.07394	−0.536972	+	0.843600i	$0.680432\pi$
$314$	2.00000	0.112867
$315$	−1.00000	−0.0563436
$316$	−14.0000	−0.787562
$317$	−14.0000	−0.786318	−0.393159	−	0.919470i	$-0.628618\pi$
$317$	−14.0000	−0.786318	−0.393159	+	0.919470i	$0.628618\pi$
$318$	−3.00000	−0.168232
$319$	0	0
$320$	7.00000	0.391312
$321$	−18.0000	−1.00466
$322$	4.00000	0.222911
$323$	42.0000	2.33694
$324$	−1.00000	−0.0555556
$325$	20.0000	1.10940
$326$	10.0000	0.553849
$327$	−1.00000	−0.0553001
$328$	−21.0000	−1.15953
$329$	−2.00000	−0.110264
$330$	0	0
$331$	34.0000	1.86881	0.934405	−	0.356214i	$-0.115932\pi$
$331$	34.0000	1.86881	0.934405	+	0.356214i	$0.115932\pi$
$332$	0	0
$333$	9.00000	0.493197
$334$	12.0000	0.656611
$335$	8.00000	0.437087
$336$	−1.00000	−0.0545545
$337$	−13.0000	−0.708155	−0.354078	−	0.935216i	$-0.615205\pi$
$337$	−13.0000	−0.708155	−0.354078	+	0.935216i	$0.615205\pi$
$338$	12.0000	0.652714
$339$	15.0000	0.814688
$340$	−7.00000	−0.379628
$341$	0	0
$342$	6.00000	0.324443
$343$	−1.00000	−0.0539949
$344$	−18.0000	−0.970495
$345$	4.00000	0.215353
$346$	6.00000	0.322562
$347$	10.0000	0.536828	0.268414	−	0.963304i	$-0.413500\pi$
$347$	10.0000	0.536828	0.268414	+	0.963304i	$0.413500\pi$
$348$	−9.00000	−0.482451
$349$	15.0000	0.802932	0.401466	−	0.915874i	$-0.368501\pi$
$349$	15.0000	0.802932	0.401466	+	0.915874i	$0.368501\pi$
$350$	4.00000	0.213809
$351$	5.00000	0.266880
$352$	0	0
$353$	−33.0000	−1.75641	−0.878206	−	0.478282i	$-0.841260\pi$
$353$	−33.0000	−1.75641	−0.878206	+	0.478282i	$0.841260\pi$
$354$	−2.00000	−0.106299
$355$	6.00000	0.318447
$356$	9.00000	0.476999
$357$	7.00000	0.370479
$358$	6.00000	0.317110
$359$	28.0000	1.47778	0.738892	−	0.673824i	$-0.235349\pi$
$359$	28.0000	1.47778	0.738892	+	0.673824i	$0.235349\pi$
$360$	−3.00000	−0.158114
$361$	17.0000	0.894737
$362$	−5.00000	−0.262794
$363$	0	0
$364$	−5.00000	−0.262071
$365$	10.0000	0.523424
$366$	6.00000	0.313625
$367$	−14.0000	−0.730794	−0.365397	−	0.930852i	$-0.619067\pi$
$367$	−14.0000	−0.730794	−0.365397	+	0.930852i	$0.619067\pi$
$368$	4.00000	0.208514
$369$	7.00000	0.364405
$370$	9.00000	0.467888
$371$	−3.00000	−0.155752
$372$	−2.00000	−0.103695
$373$	−26.0000	−1.34623	−0.673114	−	0.739538i	$-0.735044\pi$
$373$	−26.0000	−1.34623	−0.673114	+	0.739538i	$0.735044\pi$
$374$	0	0
$375$	9.00000	0.464758
$376$	−6.00000	−0.309426
$377$	45.0000	2.31762
$378$	1.00000	0.0514344
$379$	−32.0000	−1.64373	−0.821865	−	0.569683i	$-0.807066\pi$
$379$	−32.0000	−1.64373	−0.821865	+	0.569683i	$0.807066\pi$
$380$	−6.00000	−0.307794
$381$	−8.00000	−0.409852
$382$	−10.0000	−0.511645
$383$	−16.0000	−0.817562	−0.408781	−	0.912633i	$-0.634046\pi$
$383$	−16.0000	−0.817562	−0.408781	+	0.912633i	$0.634046\pi$
$384$	3.00000	0.153093
$385$	0	0
$386$	−5.00000	−0.254493
$387$	6.00000	0.304997
$388$	−17.0000	−0.863044
$389$	−9.00000	−0.456318	−0.228159	−	0.973624i	$-0.573271\pi$
$389$	−9.00000	−0.456318	−0.228159	+	0.973624i	$0.573271\pi$
$390$	5.00000	0.253185
$391$	−28.0000	−1.41602
$392$	−3.00000	−0.151523
$393$	−12.0000	−0.605320
$394$	7.00000	0.352655
$395$	14.0000	0.704416
$396$	0	0
$397$	−29.0000	−1.45547	−0.727734	−	0.685859i	$-0.759427\pi$
$397$	−29.0000	−1.45547	−0.727734	+	0.685859i	$0.759427\pi$
$398$	6.00000	0.300753
$399$	6.00000	0.300376
$400$	4.00000	0.200000
$401$	17.0000	0.848939	0.424470	−	0.905442i	$-0.360461\pi$
$401$	17.0000	0.848939	0.424470	+	0.905442i	$0.360461\pi$
$402$	−8.00000	−0.399004
$403$	10.0000	0.498135
$404$	−14.0000	−0.696526
$405$	1.00000	0.0496904
$406$	9.00000	0.446663
$407$	0	0
$408$	21.0000	1.03965
$409$	9.00000	0.445021	0.222511	−	0.974930i	$-0.428575\pi$
$409$	9.00000	0.445021	0.222511	+	0.974930i	$0.428575\pi$
$410$	7.00000	0.345705
$411$	−2.00000	−0.0986527
$412$	16.0000	0.788263
$413$	−2.00000	−0.0984136
$414$	−4.00000	−0.196589
$415$	0	0
$416$	−25.0000	−1.22573
$417$	20.0000	0.979404
$418$	0	0
$419$	32.0000	1.56330	0.781651	−	0.623716i	$-0.214378\pi$
$419$	32.0000	1.56330	0.781651	+	0.623716i	$0.214378\pi$
$420$	−1.00000	−0.0487950
$421$	1.00000	0.0487370	0.0243685	−	0.999703i	$-0.492242\pi$
$421$	1.00000	0.0487370	0.0243685	+	0.999703i	$0.492242\pi$
$422$	24.0000	1.16830
$423$	2.00000	0.0972433
$424$	−9.00000	−0.437079
$425$	−28.0000	−1.35820
$426$	−6.00000	−0.290701
$427$	6.00000	0.290360
$428$	−18.0000	−0.870063
$429$	0	0
$430$	6.00000	0.289346
$431$	−30.0000	−1.44505	−0.722525	−	0.691345i	$-0.757018\pi$
$431$	−30.0000	−1.44505	−0.722525	+	0.691345i	$0.757018\pi$
$432$	1.00000	0.0481125
$433$	−11.0000	−0.528626	−0.264313	−	0.964437i	$-0.585145\pi$
$433$	−11.0000	−0.528626	−0.264313	+	0.964437i	$0.585145\pi$
$434$	2.00000	0.0960031
$435$	9.00000	0.431517
$436$	−1.00000	−0.0478913
$437$	−24.0000	−1.14808
$438$	−10.0000	−0.477818
$439$	4.00000	0.190910	0.0954548	−	0.995434i	$-0.469569\pi$
$439$	4.00000	0.190910	0.0954548	+	0.995434i	$0.469569\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	−35.0000	−1.66478
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	9.00000	0.427121
$445$	−9.00000	−0.426641
$446$	−8.00000	−0.378811
$447$	13.0000	0.614879
$448$	−7.00000	−0.330719
$449$	−19.0000	−0.896665	−0.448333	−	0.893867i	$-0.647982\pi$
$449$	−19.0000	−0.896665	−0.448333	+	0.893867i	$0.647982\pi$
$450$	−4.00000	−0.188562
$451$	0	0
$452$	15.0000	0.705541
$453$	−2.00000	−0.0939682
$454$	−6.00000	−0.281594
$455$	5.00000	0.234404
$456$	18.0000	0.842927
$457$	−33.0000	−1.54367	−0.771837	−	0.635820i	$-0.780662\pi$
$457$	−33.0000	−1.54367	−0.771837	+	0.635820i	$0.780662\pi$
$458$	15.0000	0.700904
$459$	−7.00000	−0.326732
$460$	4.00000	0.186501
$461$	21.0000	0.978068	0.489034	−	0.872265i	$-0.337349\pi$
$461$	21.0000	0.978068	0.489034	+	0.872265i	$0.337349\pi$
$462$	0	0
$463$	−26.0000	−1.20832	−0.604161	−	0.796862i	$-0.706492\pi$
$463$	−26.0000	−1.20832	−0.604161	+	0.796862i	$0.706492\pi$
$464$	9.00000	0.417815
$465$	2.00000	0.0927478
$466$	9.00000	0.416917
$467$	24.0000	1.11059	0.555294	−	0.831654i	$-0.312606\pi$
$467$	24.0000	1.11059	0.555294	+	0.831654i	$0.312606\pi$
$468$	5.00000	0.231125
$469$	−8.00000	−0.369406
$470$	2.00000	0.0922531
$471$	−2.00000	−0.0921551
$472$	−6.00000	−0.276172
$473$	0	0
$474$	−14.0000	−0.643041
$475$	−24.0000	−1.10120
$476$	7.00000	0.320844
$477$	3.00000	0.137361
$478$	12.0000	0.548867
$479$	14.0000	0.639676	0.319838	−	0.947472i	$-0.396371\pi$
$479$	14.0000	0.639676	0.319838	+	0.947472i	$0.396371\pi$
$480$	−5.00000	−0.228218
$481$	−45.0000	−2.05182
$482$	−2.00000	−0.0910975
$483$	−4.00000	−0.182006
$484$	0	0
$485$	17.0000	0.771930
$486$	−1.00000	−0.0453609
$487$	−22.0000	−0.996915	−0.498458	−	0.866914i	$-0.666100\pi$
$487$	−22.0000	−0.996915	−0.498458	+	0.866914i	$0.666100\pi$
$488$	18.0000	0.814822
$489$	−10.0000	−0.452216
$490$	1.00000	0.0451754
$491$	28.0000	1.26362	0.631811	−	0.775122i	$-0.282312\pi$
$491$	28.0000	1.26362	0.631811	+	0.775122i	$0.282312\pi$
$492$	7.00000	0.315584
$493$	−63.0000	−2.83738
$494$	−30.0000	−1.34976
$495$	0	0
$496$	2.00000	0.0898027
$497$	−6.00000	−0.269137
$498$	0	0
$499$	−16.0000	−0.716258	−0.358129	−	0.933672i	$-0.616585\pi$
$499$	−16.0000	−0.716258	−0.358129	+	0.933672i	$0.616585\pi$
$500$	9.00000	0.402492
$501$	−12.0000	−0.536120
$502$	4.00000	0.178529
$503$	16.0000	0.713405	0.356702	−	0.934218i	$-0.383901\pi$
$503$	16.0000	0.713405	0.356702	+	0.934218i	$0.383901\pi$
$504$	3.00000	0.133631
$505$	14.0000	0.622992
$506$	0	0
$507$	−12.0000	−0.532939
$508$	−8.00000	−0.354943
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	−7.00000	−0.309965
$511$	−10.0000	−0.442374
$512$	−11.0000	−0.486136
$513$	−6.00000	−0.264906
$514$	−17.0000	−0.749838
$515$	−16.0000	−0.705044
$516$	6.00000	0.264135
$517$	0	0
$518$	−9.00000	−0.395437
$519$	−6.00000	−0.263371
$520$	15.0000	0.657794
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	−9.00000	−0.393919
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	−12.0000	−0.524222
$525$	−4.00000	−0.174574
$526$	8.00000	0.348817
$527$	−14.0000	−0.609850
$528$	0	0
$529$	−7.00000	−0.304348
$530$	3.00000	0.130312
$531$	2.00000	0.0867926
$532$	6.00000	0.260133
$533$	−35.0000	−1.51602
$534$	9.00000	0.389468
$535$	18.0000	0.778208
$536$	−24.0000	−1.03664
$537$	−6.00000	−0.258919
$538$	−11.0000	−0.474244
$539$	0	0
$540$	1.00000	0.0430331
$541$	−2.00000	−0.0859867	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000	−0.0859867	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	8.00000	0.343629
$543$	5.00000	0.214571
$544$	35.0000	1.50061
$545$	1.00000	0.0428353
$546$	−5.00000	−0.213980
$547$	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
$549$	−6.00000	−0.256074
$550$	0	0
$551$	−54.0000	−2.30048
$552$	−12.0000	−0.510754
$553$	−14.0000	−0.595341
$554$	−11.0000	−0.467345
$555$	−9.00000	−0.382029
$556$	20.0000	0.848189
$557$	−38.0000	−1.61011	−0.805056	−	0.593199i	$-0.797865\pi$
$557$	−38.0000	−1.61011	−0.805056	+	0.593199i	$0.797865\pi$
$558$	−2.00000	−0.0846668
$559$	−30.0000	−1.26886
$560$	1.00000	0.0422577
$561$	0	0
$562$	18.0000	0.759284
$563$	22.0000	0.927189	0.463595	−	0.886047i	$-0.346559\pi$
$563$	22.0000	0.927189	0.463595	+	0.886047i	$0.346559\pi$
$564$	2.00000	0.0842152
$565$	−15.0000	−0.631055
$566$	−14.0000	−0.588464
$567$	−1.00000	−0.0419961
$568$	−18.0000	−0.755263
$569$	42.0000	1.76073	0.880366	−	0.474295i	$-0.157297\pi$
$569$	42.0000	1.76073	0.880366	+	0.474295i	$0.157297\pi$
$570$	−6.00000	−0.251312
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	0	0
$573$	10.0000	0.417756
$574$	−7.00000	−0.292174
$575$	16.0000	0.667246
$576$	7.00000	0.291667
$577$	−15.0000	−0.624458	−0.312229	−	0.950007i	$-0.601076\pi$
$577$	−15.0000	−0.624458	−0.312229	+	0.950007i	$0.601076\pi$
$578$	32.0000	1.33102
$579$	5.00000	0.207793
$580$	9.00000	0.373705
$581$	0	0
$582$	−17.0000	−0.704673
$583$	0	0
$584$	−30.0000	−1.24141
$585$	−5.00000	−0.206725
$586$	9.00000	0.371787
$587$	−2.00000	−0.0825488	−0.0412744	−	0.999148i	$-0.513142\pi$
$587$	−2.00000	−0.0825488	−0.0412744	+	0.999148i	$0.513142\pi$
$588$	1.00000	0.0412393
$589$	−12.0000	−0.494451
$590$	2.00000	0.0823387
$591$	−7.00000	−0.287942
$592$	−9.00000	−0.369898
$593$	−37.0000	−1.51941	−0.759704	−	0.650269i	$-0.774656\pi$
$593$	−37.0000	−1.51941	−0.759704	+	0.650269i	$0.774656\pi$
$594$	0	0
$595$	−7.00000	−0.286972
$596$	13.0000	0.532501
$597$	−6.00000	−0.245564
$598$	20.0000	0.817861
$599$	10.0000	0.408589	0.204294	−	0.978909i	$-0.434510\pi$
$599$	10.0000	0.408589	0.204294	+	0.978909i	$0.434510\pi$
$600$	−12.0000	−0.489898
$601$	5.00000	0.203954	0.101977	−	0.994787i	$-0.467483\pi$
$601$	5.00000	0.203954	0.101977	+	0.994787i	$0.467483\pi$
$602$	−6.00000	−0.244542
$603$	8.00000	0.325785
$604$	−2.00000	−0.0813788
$605$	0	0
$606$	−14.0000	−0.568711
$607$	−28.0000	−1.13648	−0.568242	−	0.822861i	$-0.692376\pi$
$607$	−28.0000	−1.13648	−0.568242	+	0.822861i	$0.692376\pi$
$608$	30.0000	1.21666
$609$	−9.00000	−0.364698
$610$	−6.00000	−0.242933
$611$	−10.0000	−0.404557
$612$	−7.00000	−0.282958
$613$	−31.0000	−1.25208	−0.626039	−	0.779792i	$-0.715325\pi$
$613$	−31.0000	−1.25208	−0.626039	+	0.779792i	$0.715325\pi$
$614$	−28.0000	−1.12999
$615$	−7.00000	−0.282267
$616$	0	0
$617$	45.0000	1.81163	0.905816	−	0.423672i	$-0.139259\pi$
$617$	45.0000	1.81163	0.905816	+	0.423672i	$0.139259\pi$
$618$	16.0000	0.643614
$619$	14.0000	0.562708	0.281354	−	0.959604i	$-0.409217\pi$
$619$	14.0000	0.562708	0.281354	+	0.959604i	$0.409217\pi$
$620$	2.00000	0.0803219
$621$	4.00000	0.160514
$622$	−18.0000	−0.721734
$623$	9.00000	0.360577
$624$	−5.00000	−0.200160
$625$	11.0000	0.440000
$626$	−19.0000	−0.759393
$627$	0	0
$628$	−2.00000	−0.0798087
$629$	63.0000	2.51197
$630$	−1.00000	−0.0398410
$631$	46.0000	1.83123	0.915616	−	0.402055i	$-0.131704\pi$
$631$	46.0000	1.83123	0.915616	+	0.402055i	$0.131704\pi$
$632$	−42.0000	−1.67067
$633$	−24.0000	−0.953914
$634$	−14.0000	−0.556011
$635$	8.00000	0.317470
$636$	3.00000	0.118958
$637$	−5.00000	−0.198107
$638$	0	0
$639$	6.00000	0.237356
$640$	−3.00000	−0.118585
$641$	9.00000	0.355479	0.177739	−	0.984078i	$-0.443122\pi$
$641$	9.00000	0.355479	0.177739	+	0.984078i	$0.443122\pi$
$642$	−18.0000	−0.710403
$643$	−4.00000	−0.157745	−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000	−0.157745	−0.0788723	+	0.996885i	$0.525132\pi$
$644$	−4.00000	−0.157622
$645$	−6.00000	−0.236250
$646$	42.0000	1.65247
$647$	32.0000	1.25805	0.629025	−	0.777385i	$-0.283454\pi$
$647$	32.0000	1.25805	0.629025	+	0.777385i	$0.283454\pi$
$648$	−3.00000	−0.117851
$649$	0	0
$650$	20.0000	0.784465
$651$	−2.00000	−0.0783862
$652$	−10.0000	−0.391630
$653$	46.0000	1.80012	0.900060	−	0.435767i	$-0.143523\pi$
$653$	46.0000	1.80012	0.900060	+	0.435767i	$0.143523\pi$
$654$	−1.00000	−0.0391031
$655$	12.0000	0.468879
$656$	−7.00000	−0.273304
$657$	10.0000	0.390137
$658$	−2.00000	−0.0779681
$659$	22.0000	0.856998	0.428499	−	0.903542i	$-0.359042\pi$
$659$	22.0000	0.856998	0.428499	+	0.903542i	$0.359042\pi$
$660$	0	0
$661$	−17.0000	−0.661223	−0.330612	−	0.943767i	$-0.607255\pi$
$661$	−17.0000	−0.661223	−0.330612	+	0.943767i	$0.607255\pi$
$662$	34.0000	1.32145
$663$	35.0000	1.35929
$664$	0	0
$665$	−6.00000	−0.232670
$666$	9.00000	0.348743
$667$	36.0000	1.39393
$668$	−12.0000	−0.464294
$669$	8.00000	0.309298
$670$	8.00000	0.309067
$671$	0	0
$672$	5.00000	0.192879
$673$	14.0000	0.539660	0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000	0.539660	0.269830	+	0.962908i	$0.413032\pi$
$674$	−13.0000	−0.500741
$675$	4.00000	0.153960
$676$	−12.0000	−0.461538
$677$	−51.0000	−1.96009	−0.980045	−	0.198778i	$-0.936303\pi$
$677$	−51.0000	−1.96009	−0.980045	+	0.198778i	$0.936303\pi$
$678$	15.0000	0.576072
$679$	−17.0000	−0.652400
$680$	−21.0000	−0.805313
$681$	6.00000	0.229920
$682$	0	0
$683$	44.0000	1.68361	0.841807	−	0.539779i	$-0.181492\pi$
$683$	44.0000	1.68361	0.841807	+	0.539779i	$0.181492\pi$
$684$	−6.00000	−0.229416
$685$	2.00000	0.0764161
$686$	−1.00000	−0.0381802
$687$	−15.0000	−0.572286
$688$	−6.00000	−0.228748
$689$	−15.0000	−0.571454
$690$	4.00000	0.152277
$691$	−34.0000	−1.29342	−0.646710	−	0.762736i	$-0.723856\pi$
$691$	−34.0000	−1.29342	−0.646710	+	0.762736i	$0.723856\pi$
$692$	−6.00000	−0.228086
$693$	0	0
$694$	10.0000	0.379595
$695$	−20.0000	−0.758643
$696$	−27.0000	−1.02343
$697$	49.0000	1.85601
$698$	15.0000	0.567758
$699$	−9.00000	−0.340411
$700$	−4.00000	−0.151186
$701$	35.0000	1.32193	0.660966	−	0.750416i	$-0.270147\pi$
$701$	35.0000	1.32193	0.660966	+	0.750416i	$0.270147\pi$
$702$	5.00000	0.188713
$703$	54.0000	2.03665
$704$	0	0
$705$	−2.00000	−0.0753244
$706$	−33.0000	−1.24197
$707$	−14.0000	−0.526524
$708$	2.00000	0.0751646
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	6.00000	0.225176
$711$	14.0000	0.525041
$712$	27.0000	1.01187
$713$	8.00000	0.299602
$714$	7.00000	0.261968
$715$	0	0
$716$	−6.00000	−0.224231
$717$	−12.0000	−0.448148
$718$	28.0000	1.04495
$719$	6.00000	0.223762	0.111881	−	0.993722i	$-0.464312\pi$
$719$	6.00000	0.223762	0.111881	+	0.993722i	$0.464312\pi$
$720$	−1.00000	−0.0372678
$721$	16.0000	0.595871
$722$	17.0000	0.632674
$723$	2.00000	0.0743808
$724$	5.00000	0.185824
$725$	36.0000	1.33701
$726$	0	0
$727$	−6.00000	−0.222528	−0.111264	−	0.993791i	$-0.535490\pi$
$727$	−6.00000	−0.222528	−0.111264	+	0.993791i	$0.535490\pi$
$728$	−15.0000	−0.555937
$729$	1.00000	0.0370370
$730$	10.0000	0.370117
$731$	42.0000	1.55343
$732$	−6.00000	−0.221766
$733$	3.00000	0.110808	0.0554038	−	0.998464i	$-0.482355\pi$
$733$	3.00000	0.110808	0.0554038	+	0.998464i	$0.482355\pi$
$734$	−14.0000	−0.516749
$735$	−1.00000	−0.0368856
$736$	−20.0000	−0.737210
$737$	0	0
$738$	7.00000	0.257674
$739$	2.00000	0.0735712	0.0367856	−	0.999323i	$-0.488288\pi$
$739$	2.00000	0.0735712	0.0367856	+	0.999323i	$0.488288\pi$
$740$	−9.00000	−0.330847
$741$	30.0000	1.10208
$742$	−3.00000	−0.110133
$743$	10.0000	0.366864	0.183432	−	0.983032i	$-0.441279\pi$
$743$	10.0000	0.366864	0.183432	+	0.983032i	$0.441279\pi$
$744$	−6.00000	−0.219971
$745$	−13.0000	−0.476283
$746$	−26.0000	−0.951928
$747$	0	0
$748$	0	0
$749$	−18.0000	−0.657706
$750$	9.00000	0.328634
$751$	46.0000	1.67856	0.839282	−	0.543696i	$-0.182976\pi$
$751$	46.0000	1.67856	0.839282	+	0.543696i	$0.182976\pi$
$752$	−2.00000	−0.0729325
$753$	−4.00000	−0.145768
$754$	45.0000	1.63880
$755$	2.00000	0.0727875
$756$	−1.00000	−0.0363696
$757$	17.0000	0.617876	0.308938	−	0.951082i	$-0.400027\pi$
$757$	17.0000	0.617876	0.308938	+	0.951082i	$0.400027\pi$
$758$	−32.0000	−1.16229
$759$	0	0
$760$	−18.0000	−0.652929
$761$	−45.0000	−1.63125	−0.815624	−	0.578582i	$-0.803606\pi$
$761$	−45.0000	−1.63125	−0.815624	+	0.578582i	$0.803606\pi$
$762$	−8.00000	−0.289809
$763$	−1.00000	−0.0362024
$764$	10.0000	0.361787
$765$	7.00000	0.253086
$766$	−16.0000	−0.578103
$767$	−10.0000	−0.361079
$768$	17.0000	0.613435
$769$	−19.0000	−0.685158	−0.342579	−	0.939489i	$-0.611300\pi$
$769$	−19.0000	−0.685158	−0.342579	+	0.939489i	$0.611300\pi$
$770$	0	0
$771$	17.0000	0.612240
$772$	5.00000	0.179954
$773$	30.0000	1.07903	0.539513	−	0.841978i	$-0.318609\pi$
$773$	30.0000	1.07903	0.539513	+	0.841978i	$0.318609\pi$
$774$	6.00000	0.215666
$775$	8.00000	0.287368
$776$	−51.0000	−1.83079
$777$	9.00000	0.322873
$778$	−9.00000	−0.322666
$779$	42.0000	1.50481
$780$	−5.00000	−0.179029
$781$	0	0
$782$	−28.0000	−1.00128
$783$	9.00000	0.321634
$784$	−1.00000	−0.0357143
$785$	2.00000	0.0713831
$786$	−12.0000	−0.428026
$787$	40.0000	1.42585	0.712923	−	0.701242i	$-0.247371\pi$
$787$	40.0000	1.42585	0.712923	+	0.701242i	$0.247371\pi$
$788$	−7.00000	−0.249365
$789$	−8.00000	−0.284808
$790$	14.0000	0.498098
$791$	15.0000	0.533339
$792$	0	0
$793$	30.0000	1.06533
$794$	−29.0000	−1.02917
$795$	−3.00000	−0.106399
$796$	−6.00000	−0.212664
$797$	2.00000	0.0708436	0.0354218	−	0.999372i	$-0.488723\pi$
$797$	2.00000	0.0708436	0.0354218	+	0.999372i	$0.488723\pi$
$798$	6.00000	0.212398
$799$	14.0000	0.495284
$800$	−20.0000	−0.707107
$801$	−9.00000	−0.317999
$802$	17.0000	0.600291
$803$	0	0
$804$	8.00000	0.282138
$805$	4.00000	0.140981
$806$	10.0000	0.352235
$807$	11.0000	0.387218
$808$	−42.0000	−1.47755
$809$	18.0000	0.632846	0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000	0.632846	0.316423	+	0.948618i	$0.397518\pi$
$810$	1.00000	0.0351364
$811$	−2.00000	−0.0702295	−0.0351147	−	0.999383i	$-0.511180\pi$
$811$	−2.00000	−0.0702295	−0.0351147	+	0.999383i	$0.511180\pi$
$812$	−9.00000	−0.315838
$813$	−8.00000	−0.280572
$814$	0	0
$815$	10.0000	0.350285
$816$	7.00000	0.245049
$817$	36.0000	1.25948
$818$	9.00000	0.314678
$819$	5.00000	0.174714
$820$	−7.00000	−0.244451
$821$	10.0000	0.349002	0.174501	−	0.984657i	$-0.444169\pi$
$821$	10.0000	0.349002	0.174501	+	0.984657i	$0.444169\pi$
$822$	−2.00000	−0.0697580
$823$	12.0000	0.418294	0.209147	−	0.977884i	$-0.432931\pi$
$823$	12.0000	0.418294	0.209147	+	0.977884i	$0.432931\pi$
$824$	48.0000	1.67216
$825$	0	0
$826$	−2.00000	−0.0695889
$827$	−34.0000	−1.18230	−0.591148	−	0.806563i	$-0.701325\pi$
$827$	−34.0000	−1.18230	−0.591148	+	0.806563i	$0.701325\pi$
$828$	4.00000	0.139010
$829$	−29.0000	−1.00721	−0.503606	−	0.863934i	$-0.667994\pi$
$829$	−29.0000	−1.00721	−0.503606	+	0.863934i	$0.667994\pi$
$830$	0	0
$831$	11.0000	0.381586
$832$	−35.0000	−1.21341
$833$	7.00000	0.242536
$834$	20.0000	0.692543
$835$	12.0000	0.415277
$836$	0	0
$837$	2.00000	0.0691301
$838$	32.0000	1.10542
$839$	−26.0000	−0.897620	−0.448810	−	0.893627i	$-0.648152\pi$
$839$	−26.0000	−0.897620	−0.448810	+	0.893627i	$0.648152\pi$
$840$	−3.00000	−0.103510
$841$	52.0000	1.79310
$842$	1.00000	0.0344623
$843$	−18.0000	−0.619953
$844$	−24.0000	−0.826114
$845$	12.0000	0.412813
$846$	2.00000	0.0687614
$847$	0	0
$848$	−3.00000	−0.103020
$849$	14.0000	0.480479
$850$	−28.0000	−0.960392
$851$	−36.0000	−1.23406
$852$	6.00000	0.205557
$853$	−25.0000	−0.855984	−0.427992	−	0.903783i	$-0.640779\pi$
$853$	−25.0000	−0.855984	−0.427992	+	0.903783i	$0.640779\pi$
$854$	6.00000	0.205316
$855$	6.00000	0.205196
$856$	−54.0000	−1.84568
$857$	38.0000	1.29806	0.649028	−	0.760765i	$-0.275176\pi$
$857$	38.0000	1.29806	0.649028	+	0.760765i	$0.275176\pi$
$858$	0	0
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	−6.00000	−0.204598
$861$	7.00000	0.238559
$862$	−30.0000	−1.02180
$863$	−30.0000	−1.02121	−0.510606	−	0.859815i	$-0.670579\pi$
$863$	−30.0000	−1.02121	−0.510606	+	0.859815i	$0.670579\pi$
$864$	−5.00000	−0.170103
$865$	6.00000	0.204006
$866$	−11.0000	−0.373795
$867$	−32.0000	−1.08678
$868$	−2.00000	−0.0678844
$869$	0	0
$870$	9.00000	0.305129
$871$	−40.0000	−1.35535
$872$	−3.00000	−0.101593
$873$	17.0000	0.575363
$874$	−24.0000	−0.811812
$875$	9.00000	0.304256
$876$	10.0000	0.337869
$877$	−27.0000	−0.911725	−0.455863	−	0.890050i	$-0.650669\pi$
$877$	−27.0000	−0.911725	−0.455863	+	0.890050i	$0.650669\pi$
$878$	4.00000	0.134993
$879$	−9.00000	−0.303562
$880$	0	0
$881$	−49.0000	−1.65085	−0.825426	−	0.564510i	$-0.809065\pi$
$881$	−49.0000	−1.65085	−0.825426	+	0.564510i	$0.809065\pi$
$882$	1.00000	0.0336718
$883$	22.0000	0.740359	0.370179	−	0.928960i	$-0.379296\pi$
$883$	22.0000	0.740359	0.370179	+	0.928960i	$0.379296\pi$
$884$	35.0000	1.17718
$885$	−2.00000	−0.0672293
$886$	4.00000	0.134383
$887$	8.00000	0.268614	0.134307	−	0.990940i	$-0.457119\pi$
$887$	8.00000	0.268614	0.134307	+	0.990940i	$0.457119\pi$
$888$	27.0000	0.906061
$889$	−8.00000	−0.268311
$890$	−9.00000	−0.301681
$891$	0	0
$892$	8.00000	0.267860
$893$	12.0000	0.401565
$894$	13.0000	0.434785
$895$	6.00000	0.200558
$896$	3.00000	0.100223
$897$	−20.0000	−0.667781
$898$	−19.0000	−0.634038
$899$	18.0000	0.600334
$900$	4.00000	0.133333
$901$	21.0000	0.699611
$902$	0	0
$903$	6.00000	0.199667
$904$	45.0000	1.49668
$905$	−5.00000	−0.166206
$906$	−2.00000	−0.0664455
$907$	−30.0000	−0.996134	−0.498067	−	0.867139i	$-0.665957\pi$
$907$	−30.0000	−0.996134	−0.498067	+	0.867139i	$0.665957\pi$
$908$	6.00000	0.199117
$909$	14.0000	0.464351
$910$	5.00000	0.165748
$911$	−36.0000	−1.19273	−0.596367	−	0.802712i	$-0.703390\pi$
$911$	−36.0000	−1.19273	−0.596367	+	0.802712i	$0.703390\pi$
$912$	6.00000	0.198680
$913$	0	0
$914$	−33.0000	−1.09154
$915$	6.00000	0.198354
$916$	−15.0000	−0.495614
$917$	−12.0000	−0.396275
$918$	−7.00000	−0.231034
$919$	−14.0000	−0.461817	−0.230909	−	0.972975i	$-0.574170\pi$
$919$	−14.0000	−0.461817	−0.230909	+	0.972975i	$0.574170\pi$
$920$	12.0000	0.395628
$921$	28.0000	0.922631
$922$	21.0000	0.691598
$923$	−30.0000	−0.987462
$924$	0	0
$925$	−36.0000	−1.18367
$926$	−26.0000	−0.854413
$927$	−16.0000	−0.525509
$928$	−45.0000	−1.47720
$929$	−33.0000	−1.08269	−0.541347	−	0.840799i	$-0.682086\pi$
$929$	−33.0000	−1.08269	−0.541347	+	0.840799i	$0.682086\pi$
$930$	2.00000	0.0655826
$931$	6.00000	0.196642
$932$	−9.00000	−0.294805
$933$	18.0000	0.589294
$934$	24.0000	0.785304
$935$	0	0
$936$	15.0000	0.490290
$937$	29.0000	0.947389	0.473694	−	0.880689i	$-0.342920\pi$
$937$	29.0000	0.947389	0.473694	+	0.880689i	$0.342920\pi$
$938$	−8.00000	−0.261209
$939$	19.0000	0.620042
$940$	−2.00000	−0.0652328
$941$	−3.00000	−0.0977972	−0.0488986	−	0.998804i	$-0.515571\pi$
$941$	−3.00000	−0.0977972	−0.0488986	+	0.998804i	$0.515571\pi$
$942$	−2.00000	−0.0651635
$943$	−28.0000	−0.911805
$944$	−2.00000	−0.0650945
$945$	1.00000	0.0325300
$946$	0	0
$947$	42.0000	1.36482	0.682408	−	0.730971i	$-0.260933\pi$
$947$	42.0000	1.36482	0.682408	+	0.730971i	$0.260933\pi$
$948$	14.0000	0.454699
$949$	−50.0000	−1.62307
$950$	−24.0000	−0.778663
$951$	14.0000	0.453981
$952$	21.0000	0.680614
$953$	37.0000	1.19855	0.599274	−	0.800544i	$-0.295456\pi$
$953$	37.0000	1.19855	0.599274	+	0.800544i	$0.295456\pi$
$954$	3.00000	0.0971286
$955$	−10.0000	−0.323592
$956$	−12.0000	−0.388108
$957$	0	0
$958$	14.0000	0.452319
$959$	−2.00000	−0.0645834
$960$	−7.00000	−0.225924
$961$	−27.0000	−0.870968
$962$	−45.0000	−1.45086
$963$	18.0000	0.580042
$964$	2.00000	0.0644157
$965$	−5.00000	−0.160956
$966$	−4.00000	−0.128698
$967$	−50.0000	−1.60789	−0.803946	−	0.594703i	$-0.797270\pi$
$967$	−50.0000	−1.60789	−0.803946	+	0.594703i	$0.797270\pi$
$968$	0	0
$969$	−42.0000	−1.34923
$970$	17.0000	0.545837
$971$	−28.0000	−0.898563	−0.449281	−	0.893390i	$-0.648320\pi$
$971$	−28.0000	−0.898563	−0.449281	+	0.893390i	$0.648320\pi$
$972$	1.00000	0.0320750
$973$	20.0000	0.641171
$974$	−22.0000	−0.704925
$975$	−20.0000	−0.640513
$976$	6.00000	0.192055
$977$	57.0000	1.82359	0.911796	−	0.410644i	$-0.134696\pi$
$977$	57.0000	1.82359	0.911796	+	0.410644i	$0.134696\pi$
$978$	−10.0000	−0.319765
$979$	0	0
$980$	−1.00000	−0.0319438
$981$	1.00000	0.0319275
$982$	28.0000	0.893516
$983$	−30.0000	−0.956851	−0.478426	−	0.878128i	$-0.658792\pi$
$983$	−30.0000	−0.956851	−0.478426	+	0.878128i	$0.658792\pi$
$984$	21.0000	0.669456
$985$	7.00000	0.223039
$986$	−63.0000	−2.00633
$987$	2.00000	0.0636607
$988$	30.0000	0.954427
$989$	−24.0000	−0.763156
$990$	0	0
$991$	−14.0000	−0.444725	−0.222362	−	0.974964i	$-0.571377\pi$
$991$	−14.0000	−0.444725	−0.222362	+	0.974964i	$0.571377\pi$
$992$	−10.0000	−0.317500
$993$	−34.0000	−1.07896
$994$	−6.00000	−0.190308
$995$	6.00000	0.190213
$996$	0	0
$997$	47.0000	1.48850	0.744252	−	0.667898i	$-0.232806\pi$
$997$	47.0000	1.48850	0.744252	+	0.667898i	$0.232806\pi$
$998$	−16.0000	−0.506471
$999$	−9.00000	−0.284747

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2541.2.a.i.1.1	yes	1
3.2	odd	2		7623.2.a.e.1.1		1
11.10	odd	2		2541.2.a.d.1.1	✓	1
33.32	even	2		7623.2.a.m.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2541.2.a.d.1.1	✓	1	11.10	odd	2
2541.2.a.i.1.1	yes	1	1.1	even	1	trivial
7623.2.a.e.1.1		1	3.2	odd	2
7623.2.a.m.1.1		1	33.32	even	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:	\( N \)	\(=\)	\( 2541 = 3 \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2541.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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