LMFDB - Embedded newform 2541.2.a.d.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:	$1$
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	$\chi$	$=$	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	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	−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	$-0.256123\pi$
$13$	5.00000	1.38675	0.693375	+	0.720577i	$0.256123\pi$
$14$	−1.00000	−0.267261
$15$	−1.00000	−0.258199
$16$	−1.00000	−0.250000
$17$	−7.00000	−1.69775	−0.848875	−	0.528594i	$-0.822719\pi$
$17$	−7.00000	−1.69775	−0.848875	+	0.528594i	$0.822719\pi$
$18$	−1.00000	−0.235702
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	−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	$-0.185103\pi$
$29$	9.00000	1.67126	0.835629	+	0.549294i	$0.185103\pi$
$30$	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	$-0.684081\pi$
$41$	−7.00000	−1.09322	−0.546608	+	0.837389i	$0.684081\pi$
$42$	1.00000	0.154303
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	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	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	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	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	−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	$-0.788657\pi$
$79$	−14.0000	−1.57512	−0.787562	+	0.616236i	$0.788657\pi$
$80$	−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	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	−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	$-0.835922\pi$
$107$	−18.0000	−1.74013	−0.870063	+	0.492941i	$0.835922\pi$
$108$	1.00000	0.0962250
$109$	−1.00000	−0.0957826	−0.0478913	−	0.998853i	$-0.515250\pi$
$109$	−1.00000	−0.0957826	−0.0478913	+	0.998853i	$0.515250\pi$
$110$	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	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	3.00000	0.265165
$129$	6.00000	0.528271
$130$	−5.00000	−0.438529
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	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	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	−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	$-0.321252\pi$
$149$	13.0000	1.06500	0.532501	+	0.846430i	$0.321252\pi$
$150$	−4.00000	−0.326599
$151$	−2.00000	−0.162758	−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000	−0.162758	−0.0813788	+	0.996683i	$0.525932\pi$
$152$	−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	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	−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	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	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	$-0.442405\pi$
$193$	5.00000	0.359908	0.179954	+	0.983675i	$0.442405\pi$
$194$	−17.0000	−1.22053
$195$	−5.00000	−0.358057
$196$	−1.00000	−0.0714286
$197$	−7.00000	−0.498729	−0.249365	−	0.968410i	$-0.580222\pi$
$197$	−7.00000	−0.498729	−0.249365	+	0.968410i	$0.580222\pi$
$198$	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	$-0.809453\pi$
$211$	−24.0000	−1.65223	−0.826114	+	0.563503i	$0.809453\pi$
$212$	−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	$-0.436193\pi$
$227$	6.00000	0.398234	0.199117	+	0.979976i	$0.436193\pi$
$228$	−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	$-0.595255\pi$
$233$	−9.00000	−0.589610	−0.294805	+	0.955557i	$0.595255\pi$
$234$	−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	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	1.00000	0.0645497
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	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	$-0.579330\pi$
$263$	−8.00000	−0.493301	−0.246651	+	0.969104i	$0.579330\pi$
$264$	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	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	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	$-0.392795\pi$
$277$	11.0000	0.660926	0.330463	+	0.943819i	$0.392795\pi$
$278$	−20.0000	−1.19952
$279$	−2.00000	−0.119737
$280$	3.00000	0.179284
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	2.00000	0.119098
$283$	14.0000	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$284$	−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	$-0.584677\pi$
$293$	−9.00000	−0.525786	−0.262893	+	0.964825i	$0.584677\pi$
$294$	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	$-0.205351\pi$
$307$	28.0000	1.59804	0.799022	+	0.601302i	$0.205351\pi$
$308$	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	$-0.384795\pi$
$337$	13.0000	0.708155	0.354078	+	0.935216i	$0.384795\pi$
$338$	−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	$-0.586500\pi$
$347$	−10.0000	−0.536828	−0.268414	+	0.963304i	$0.586500\pi$
$348$	9.00000	0.482451
$349$	−15.0000	−0.802932	−0.401466	−	0.915874i	$-0.631499\pi$
$349$	−15.0000	−0.802932	−0.401466	+	0.915874i	$0.631499\pi$
$350$	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	$-0.764651\pi$
$359$	−28.0000	−1.47778	−0.738892	+	0.673824i	$0.764651\pi$
$360$	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	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	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	$-0.571425\pi$
$409$	−9.00000	−0.445021	−0.222511	+	0.974930i	$0.571425\pi$
$410$	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	$-0.242982\pi$
$431$	30.0000	1.44505	0.722525	+	0.691345i	$0.242982\pi$
$432$	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	$-0.530431\pi$
$439$	−4.00000	−0.190910	−0.0954548	+	0.995434i	$0.530431\pi$
$440$	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	$-0.219338\pi$
$457$	33.0000	1.54367	0.771837	+	0.635820i	$0.219338\pi$
$458$	−15.0000	−0.700904
$459$	7.00000	0.326732
$460$	4.00000	0.186501
$461$	−21.0000	−0.978068	−0.489034	−	0.872265i	$-0.662651\pi$
$461$	−21.0000	−0.978068	−0.489034	+	0.872265i	$0.662651\pi$
$462$	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	$-0.603629\pi$
$479$	−14.0000	−0.639676	−0.319838	+	0.947472i	$0.603629\pi$
$480$	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	$-0.717688\pi$
$491$	−28.0000	−1.26362	−0.631811	+	0.775122i	$0.717688\pi$
$492$	−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	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	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	$-0.644054\pi$
$523$	−20.0000	−0.874539	−0.437269	+	0.899331i	$0.644054\pi$
$524$	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	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	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	$-0.513614\pi$
$547$	−2.00000	−0.0855138	−0.0427569	+	0.999086i	$0.513614\pi$
$548$	−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	$-0.202135\pi$
$557$	38.0000	1.61011	0.805056	+	0.593199i	$0.202135\pi$
$558$	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	$-0.653441\pi$
$563$	−22.0000	−0.927189	−0.463595	+	0.886047i	$0.653441\pi$
$564$	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	$-0.842703\pi$
$569$	−42.0000	−1.76073	−0.880366	+	0.474295i	$0.842703\pi$
$570$	−6.00000	−0.251312
$571$	−32.0000	−1.33916	−0.669579	−	0.742741i	$-0.733526\pi$
$571$	−32.0000	−1.33916	−0.669579	+	0.742741i	$0.733526\pi$
$572$	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	$-0.225344\pi$
$593$	37.0000	1.51941	0.759704	+	0.650269i	$0.225344\pi$
$594$	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	$-0.532517\pi$
$601$	−5.00000	−0.203954	−0.101977	+	0.994787i	$0.532517\pi$
$602$	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	$-0.307624\pi$
$607$	28.0000	1.13648	0.568242	+	0.822861i	$0.307624\pi$
$608$	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	$-0.284675\pi$
$613$	31.0000	1.25208	0.626039	+	0.779792i	$0.284675\pi$
$614$	−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	$-0.640958\pi$
$659$	−22.0000	−0.856998	−0.428499	+	0.903542i	$0.640958\pi$
$660$	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	$-0.586968\pi$
$673$	−14.0000	−0.539660	−0.269830	+	0.962908i	$0.586968\pi$
$674$	−13.0000	−0.500741
$675$	4.00000	0.153960
$676$	−12.0000	−0.461538
$677$	51.0000	1.96009	0.980045	−	0.198778i	$-0.0636972\pi$
$677$	51.0000	1.96009	0.980045	+	0.198778i	$0.0636972\pi$
$678$	−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	$-0.729853\pi$
$701$	−35.0000	−1.32193	−0.660966	+	0.750416i	$0.729853\pi$
$702$	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	$-0.517645\pi$
$733$	−3.00000	−0.110808	−0.0554038	+	0.998464i	$0.517645\pi$
$734$	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	$-0.511712\pi$
$739$	−2.00000	−0.0735712	−0.0367856	+	0.999323i	$0.511712\pi$
$740$	−9.00000	−0.330847
$741$	30.0000	1.10208
$742$	−3.00000	−0.110133
$743$	−10.0000	−0.366864	−0.183432	−	0.983032i	$-0.558721\pi$
$743$	−10.0000	−0.366864	−0.183432	+	0.983032i	$0.558721\pi$
$744$	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	$-0.196394\pi$
$761$	45.0000	1.63125	0.815624	+	0.578582i	$0.196394\pi$
$762$	−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	$-0.388700\pi$
$769$	19.0000	0.685158	0.342579	+	0.939489i	$0.388700\pi$
$770$	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	$-0.752629\pi$
$787$	−40.0000	−1.42585	−0.712923	+	0.701242i	$0.752629\pi$
$788$	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	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	−1.00000	−0.0351364
$811$	2.00000	0.0702295	0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000	0.0702295	0.0351147	+	0.999383i	$0.488820\pi$
$812$	−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	$-0.555831\pi$
$821$	−10.0000	−0.349002	−0.174501	+	0.984657i	$0.555831\pi$
$822$	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	$-0.298675\pi$
$827$	34.0000	1.18230	0.591148	+	0.806563i	$0.298675\pi$
$828$	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	$-0.359221\pi$
$853$	25.0000	0.855984	0.427992	+	0.903783i	$0.359221\pi$
$854$	−6.00000	−0.205316
$855$	−6.00000	−0.205196
$856$	−54.0000	−1.84568
$857$	−38.0000	−1.29806	−0.649028	−	0.760765i	$-0.724824\pi$
$857$	−38.0000	−1.29806	−0.649028	+	0.760765i	$0.724824\pi$
$858$	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	$-0.349331\pi$
$877$	27.0000	0.911725	0.455863	+	0.890050i	$0.349331\pi$
$878$	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	$-0.542881\pi$
$887$	−8.00000	−0.268614	−0.134307	+	0.990940i	$0.542881\pi$
$888$	−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	$-0.425830\pi$
$919$	14.0000	0.461817	0.230909	+	0.972975i	$0.425830\pi$
$920$	−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	$-0.657080\pi$
$937$	−29.0000	−0.947389	−0.473694	+	0.880689i	$0.657080\pi$
$938$	−8.00000	−0.261209
$939$	19.0000	0.620042
$940$	−2.00000	−0.0652328
$941$	3.00000	0.0977972	0.0488986	−	0.998804i	$-0.484429\pi$
$941$	3.00000	0.0977972	0.0488986	+	0.998804i	$0.484429\pi$
$942$	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	$-0.704544\pi$
$953$	−37.0000	−1.19855	−0.599274	+	0.800544i	$0.704544\pi$
$954$	−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	$-0.202730\pi$
$967$	50.0000	1.60789	0.803946	+	0.594703i	$0.202730\pi$
$968$	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	$-0.767194\pi$
$997$	−47.0000	−1.48850	−0.744252	+	0.667898i	$0.767194\pi$
$998$	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.d.1.1	✓	1
3.2	odd	2		7623.2.a.m.1.1		1
11.10	odd	2		2541.2.a.i.1.1	yes	1
33.32	even	2		7623.2.a.e.1.1		1

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2541 = 3 \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2541.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more