LMFDB - Embedded newform 574.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [574,2,Mod(1,574)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(574, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("574.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 574 = 2 \cdot 7 \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	574.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:	$4.58341307602$
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	$\chi$	$=$	574.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +3.00000 q^{10} +1.00000 q^{12} +2.00000 q^{13} -1.00000 q^{14} -3.00000 q^{15} +1.00000 q^{16} -3.00000 q^{17} +2.00000 q^{18} -4.00000 q^{19} -3.00000 q^{20} +1.00000 q^{21} -6.00000 q^{23} -1.00000 q^{24} +4.00000 q^{25} -2.00000 q^{26} -5.00000 q^{27} +1.00000 q^{28} -3.00000 q^{29} +3.00000 q^{30} -1.00000 q^{31} -1.00000 q^{32} +3.00000 q^{34} -3.00000 q^{35} -2.00000 q^{36} -4.00000 q^{37} +4.00000 q^{38} +2.00000 q^{39} +3.00000 q^{40} +1.00000 q^{41} -1.00000 q^{42} -1.00000 q^{43} +6.00000 q^{45} +6.00000 q^{46} -6.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} -4.00000 q^{50} -3.00000 q^{51} +2.00000 q^{52} +9.00000 q^{53} +5.00000 q^{54} -1.00000 q^{56} -4.00000 q^{57} +3.00000 q^{58} -6.00000 q^{59} -3.00000 q^{60} -1.00000 q^{61} +1.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} -6.00000 q^{65} +8.00000 q^{67} -3.00000 q^{68} -6.00000 q^{69} +3.00000 q^{70} +3.00000 q^{71} +2.00000 q^{72} -10.0000 q^{73} +4.00000 q^{74} +4.00000 q^{75} -4.00000 q^{76} -2.00000 q^{78} -1.00000 q^{79} -3.00000 q^{80} +1.00000 q^{81} -1.00000 q^{82} -12.0000 q^{83} +1.00000 q^{84} +9.00000 q^{85} +1.00000 q^{86} -3.00000 q^{87} +3.00000 q^{89} -6.00000 q^{90} +2.00000 q^{91} -6.00000 q^{92} -1.00000 q^{93} +6.00000 q^{94} +12.0000 q^{95} -1.00000 q^{96} -1.00000 q^{97} -1.00000 q^{98} +O(q^{100})\)

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
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	1.00000	0.500000
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	−1.00000	−0.408248
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	−2.00000	−0.666667
$10$	3.00000	0.948683
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	1.00000	0.288675
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	−1.00000	−0.267261
$15$	−3.00000	−0.774597
$16$	1.00000	0.250000
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	2.00000	0.471405
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	−3.00000	−0.670820
$21$	1.00000	0.218218
$22$	0	0
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	−1.00000	−0.204124
$25$	4.00000	0.800000
$26$	−2.00000	−0.392232
$27$	−5.00000	−0.962250
$28$	1.00000	0.188982
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	3.00000	0.547723
$31$	−1.00000	−0.179605	−0.0898027	−	0.995960i	$-0.528624\pi$
$31$	−1.00000	−0.179605	−0.0898027	+	0.995960i	$0.528624\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	3.00000	0.514496
$35$	−3.00000	−0.507093
$36$	−2.00000	−0.333333
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	4.00000	0.648886
$39$	2.00000	0.320256
$40$	3.00000	0.474342
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	−1.00000	−0.154303
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	0	0
$45$	6.00000	0.894427
$46$	6.00000	0.884652
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	1.00000	0.144338
$49$	1.00000	0.142857
$50$	−4.00000	−0.565685
$51$	−3.00000	−0.420084
$52$	2.00000	0.277350
$53$	9.00000	1.23625	0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000	1.23625	0.618123	+	0.786082i	$0.287894\pi$
$54$	5.00000	0.680414
$55$	0	0
$56$	−1.00000	−0.133631
$57$	−4.00000	−0.529813
$58$	3.00000	0.393919
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	−3.00000	−0.387298
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	1.00000	0.127000
$63$	−2.00000	−0.251976
$64$	1.00000	0.125000
$65$	−6.00000	−0.744208
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	−3.00000	−0.363803
$69$	−6.00000	−0.722315
$70$	3.00000	0.358569
$71$	3.00000	0.356034	0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000	0.356034	0.178017	+	0.984027i	$0.443032\pi$
$72$	2.00000	0.235702
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	4.00000	0.464991
$75$	4.00000	0.461880
$76$	−4.00000	−0.458831
$77$	0	0
$78$	−2.00000	−0.226455
$79$	−1.00000	−0.112509	−0.0562544	−	0.998416i	$-0.517916\pi$
$79$	−1.00000	−0.112509	−0.0562544	+	0.998416i	$0.517916\pi$
$80$	−3.00000	−0.335410
$81$	1.00000	0.111111
$82$	−1.00000	−0.110432
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	1.00000	0.109109
$85$	9.00000	0.976187
$86$	1.00000	0.107833
$87$	−3.00000	−0.321634
$88$	0	0
$89$	3.00000	0.317999	0.159000	−	0.987279i	$-0.449173\pi$
$89$	3.00000	0.317999	0.159000	+	0.987279i	$0.449173\pi$
$90$	−6.00000	−0.632456
$91$	2.00000	0.209657
$92$	−6.00000	−0.625543
$93$	−1.00000	−0.103695
$94$	6.00000	0.618853
$95$	12.0000	1.23117
$96$	−1.00000	−0.102062
$97$	−1.00000	−0.101535	−0.0507673	−	0.998711i	$-0.516167\pi$
$97$	−1.00000	−0.101535	−0.0507673	+	0.998711i	$0.516167\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	4.00000	0.400000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	3.00000	0.297044
$103$	11.0000	1.08386	0.541931	−	0.840423i	$-0.317693\pi$
$103$	11.0000	1.08386	0.541931	+	0.840423i	$0.317693\pi$
$104$	−2.00000	−0.196116
$105$	−3.00000	−0.292770
$106$	−9.00000	−0.874157
$107$	3.00000	0.290021	0.145010	−	0.989430i	$-0.453678\pi$
$107$	3.00000	0.290021	0.145010	+	0.989430i	$0.453678\pi$
$108$	−5.00000	−0.481125
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	1.00000	0.0944911
$113$	9.00000	0.846649	0.423324	−	0.905978i	$-0.360863\pi$
$113$	9.00000	0.846649	0.423324	+	0.905978i	$0.360863\pi$
$114$	4.00000	0.374634
$115$	18.0000	1.67851
$116$	−3.00000	−0.278543
$117$	−4.00000	−0.369800
$118$	6.00000	0.552345
$119$	−3.00000	−0.275010
$120$	3.00000	0.273861
$121$	−11.0000	−1.00000
$122$	1.00000	0.0905357
$123$	1.00000	0.0901670
$124$	−1.00000	−0.0898027
$125$	3.00000	0.268328
$126$	2.00000	0.178174
$127$	14.0000	1.24230	0.621150	−	0.783692i	$-0.286666\pi$
$127$	14.0000	1.24230	0.621150	+	0.783692i	$0.286666\pi$
$128$	−1.00000	−0.0883883
$129$	−1.00000	−0.0880451
$130$	6.00000	0.526235
$131$	18.0000	1.57267	0.786334	−	0.617802i	$-0.211977\pi$
$131$	18.0000	1.57267	0.786334	+	0.617802i	$0.211977\pi$
$132$	0	0
$133$	−4.00000	−0.346844
$134$	−8.00000	−0.691095
$135$	15.0000	1.29099
$136$	3.00000	0.257248
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	6.00000	0.510754
$139$	−10.0000	−0.848189	−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000	−0.848189	−0.424094	+	0.905618i	$0.639408\pi$
$140$	−3.00000	−0.253546
$141$	−6.00000	−0.505291
$142$	−3.00000	−0.251754
$143$	0	0
$144$	−2.00000	−0.166667
$145$	9.00000	0.747409
$146$	10.0000	0.827606
$147$	1.00000	0.0824786
$148$	−4.00000	−0.328798
$149$	−3.00000	−0.245770	−0.122885	−	0.992421i	$-0.539215\pi$
$149$	−3.00000	−0.245770	−0.122885	+	0.992421i	$0.539215\pi$
$150$	−4.00000	−0.326599
$151$	17.0000	1.38344	0.691720	−	0.722166i	$-0.256853\pi$
$151$	17.0000	1.38344	0.691720	+	0.722166i	$0.256853\pi$
$152$	4.00000	0.324443
$153$	6.00000	0.485071
$154$	0	0
$155$	3.00000	0.240966
$156$	2.00000	0.160128
$157$	−4.00000	−0.319235	−0.159617	−	0.987179i	$-0.551026\pi$
$157$	−4.00000	−0.319235	−0.159617	+	0.987179i	$0.551026\pi$
$158$	1.00000	0.0795557
$159$	9.00000	0.713746
$160$	3.00000	0.237171
$161$	−6.00000	−0.472866
$162$	−1.00000	−0.0785674
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	1.00000	0.0780869
$165$	0	0
$166$	12.0000	0.931381
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	−1.00000	−0.0771517
$169$	−9.00000	−0.692308
$170$	−9.00000	−0.690268
$171$	8.00000	0.611775
$172$	−1.00000	−0.0762493
$173$	21.0000	1.59660	0.798300	−	0.602260i	$-0.205733\pi$
$173$	21.0000	1.59660	0.798300	+	0.602260i	$0.205733\pi$
$174$	3.00000	0.227429
$175$	4.00000	0.302372
$176$	0	0
$177$	−6.00000	−0.450988
$178$	−3.00000	−0.224860
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	6.00000	0.447214
$181$	8.00000	0.594635	0.297318	−	0.954779i	$-0.403908\pi$
$181$	8.00000	0.594635	0.297318	+	0.954779i	$0.403908\pi$
$182$	−2.00000	−0.148250
$183$	−1.00000	−0.0739221
$184$	6.00000	0.442326
$185$	12.0000	0.882258
$186$	1.00000	0.0733236
$187$	0	0
$188$	−6.00000	−0.437595
$189$	−5.00000	−0.363696
$190$	−12.0000	−0.870572
$191$	3.00000	0.217072	0.108536	−	0.994092i	$-0.465384\pi$
$191$	3.00000	0.217072	0.108536	+	0.994092i	$0.465384\pi$
$192$	1.00000	0.0721688
$193$	−16.0000	−1.15171	−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000	−1.15171	−0.575853	+	0.817554i	$0.695330\pi$
$194$	1.00000	0.0717958
$195$	−6.00000	−0.429669
$196$	1.00000	0.0714286
$197$	24.0000	1.70993	0.854965	−	0.518686i	$-0.173579\pi$
$197$	24.0000	1.70993	0.854965	+	0.518686i	$0.173579\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	−4.00000	−0.282843
$201$	8.00000	0.564276
$202$	6.00000	0.422159
$203$	−3.00000	−0.210559
$204$	−3.00000	−0.210042
$205$	−3.00000	−0.209529
$206$	−11.0000	−0.766406
$207$	12.0000	0.834058
$208$	2.00000	0.138675
$209$	0	0
$210$	3.00000	0.207020
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	9.00000	0.618123
$213$	3.00000	0.205557
$214$	−3.00000	−0.205076
$215$	3.00000	0.204598
$216$	5.00000	0.340207
$217$	−1.00000	−0.0678844
$218$	−2.00000	−0.135457
$219$	−10.0000	−0.675737
$220$	0	0
$221$	−6.00000	−0.403604
$222$	4.00000	0.268462
$223$	−1.00000	−0.0669650	−0.0334825	−	0.999439i	$-0.510660\pi$
$223$	−1.00000	−0.0669650	−0.0334825	+	0.999439i	$0.510660\pi$
$224$	−1.00000	−0.0668153
$225$	−8.00000	−0.533333
$226$	−9.00000	−0.598671
$227$	−15.0000	−0.995585	−0.497792	−	0.867296i	$-0.665856\pi$
$227$	−15.0000	−0.995585	−0.497792	+	0.867296i	$0.665856\pi$
$228$	−4.00000	−0.264906
$229$	−16.0000	−1.05731	−0.528655	−	0.848837i	$-0.677303\pi$
$229$	−16.0000	−1.05731	−0.528655	+	0.848837i	$0.677303\pi$
$230$	−18.0000	−1.18688
$231$	0	0
$232$	3.00000	0.196960
$233$	−24.0000	−1.57229	−0.786146	−	0.618041i	$-0.787927\pi$
$233$	−24.0000	−1.57229	−0.786146	+	0.618041i	$0.787927\pi$
$234$	4.00000	0.261488
$235$	18.0000	1.17419
$236$	−6.00000	−0.390567
$237$	−1.00000	−0.0649570
$238$	3.00000	0.194461
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	−3.00000	−0.193649
$241$	−28.0000	−1.80364	−0.901819	−	0.432113i	$-0.857768\pi$
$241$	−28.0000	−1.80364	−0.901819	+	0.432113i	$0.857768\pi$
$242$	11.0000	0.707107
$243$	16.0000	1.02640
$244$	−1.00000	−0.0640184
$245$	−3.00000	−0.191663
$246$	−1.00000	−0.0637577
$247$	−8.00000	−0.509028
$248$	1.00000	0.0635001
$249$	−12.0000	−0.760469
$250$	−3.00000	−0.189737
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	−2.00000	−0.125988
$253$	0	0
$254$	−14.0000	−0.878438
$255$	9.00000	0.563602
$256$	1.00000	0.0625000
$257$	−27.0000	−1.68421	−0.842107	−	0.539311i	$-0.818685\pi$
$257$	−27.0000	−1.68421	−0.842107	+	0.539311i	$0.818685\pi$
$258$	1.00000	0.0622573
$259$	−4.00000	−0.248548
$260$	−6.00000	−0.372104
$261$	6.00000	0.371391
$262$	−18.0000	−1.11204
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	0	0
$265$	−27.0000	−1.65860
$266$	4.00000	0.245256
$267$	3.00000	0.183597
$268$	8.00000	0.488678
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	−15.0000	−0.912871
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	−3.00000	−0.181902
$273$	2.00000	0.121046
$274$	6.00000	0.362473
$275$	0	0
$276$	−6.00000	−0.361158
$277$	8.00000	0.480673	0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000	0.480673	0.240337	+	0.970690i	$0.422742\pi$
$278$	10.0000	0.599760
$279$	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	$0.680403\pi$
$282$	6.00000	0.357295
$283$	8.00000	0.475551	0.237775	−	0.971320i	$-0.423582\pi$
$283$	8.00000	0.475551	0.237775	+	0.971320i	$0.423582\pi$
$284$	3.00000	0.178017
$285$	12.0000	0.710819
$286$	0	0
$287$	1.00000	0.0590281
$288$	2.00000	0.117851
$289$	−8.00000	−0.470588
$290$	−9.00000	−0.528498
$291$	−1.00000	−0.0586210
$292$	−10.0000	−0.585206
$293$	−24.0000	−1.40209	−0.701047	−	0.713115i	$-0.747284\pi$
$293$	−24.0000	−1.40209	−0.701047	+	0.713115i	$0.747284\pi$
$294$	−1.00000	−0.0583212
$295$	18.0000	1.04800
$296$	4.00000	0.232495
$297$	0	0
$298$	3.00000	0.173785
$299$	−12.0000	−0.693978
$300$	4.00000	0.230940
$301$	−1.00000	−0.0576390
$302$	−17.0000	−0.978240
$303$	−6.00000	−0.344691
$304$	−4.00000	−0.229416
$305$	3.00000	0.171780
$306$	−6.00000	−0.342997
$307$	8.00000	0.456584	0.228292	−	0.973593i	$-0.426686\pi$
$307$	8.00000	0.456584	0.228292	+	0.973593i	$0.426686\pi$
$308$	0	0
$309$	11.0000	0.625768
$310$	−3.00000	−0.170389
$311$	6.00000	0.340229	0.170114	−	0.985424i	$-0.445586\pi$
$311$	6.00000	0.340229	0.170114	+	0.985424i	$0.445586\pi$
$312$	−2.00000	−0.113228
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	4.00000	0.225733
$315$	6.00000	0.338062
$316$	−1.00000	−0.0562544
$317$	30.0000	1.68497	0.842484	−	0.538721i	$-0.181092\pi$
$317$	30.0000	1.68497	0.842484	+	0.538721i	$0.181092\pi$
$318$	−9.00000	−0.504695
$319$	0	0
$320$	−3.00000	−0.167705
$321$	3.00000	0.167444
$322$	6.00000	0.334367
$323$	12.0000	0.667698
$324$	1.00000	0.0555556
$325$	8.00000	0.443760
$326$	−20.0000	−1.10770
$327$	2.00000	0.110600
$328$	−1.00000	−0.0552158
$329$	−6.00000	−0.330791
$330$	0	0
$331$	2.00000	0.109930	0.0549650	−	0.998488i	$-0.482495\pi$
$331$	2.00000	0.109930	0.0549650	+	0.998488i	$0.482495\pi$
$332$	−12.0000	−0.658586
$333$	8.00000	0.438397
$334$	0	0
$335$	−24.0000	−1.31126
$336$	1.00000	0.0545545
$337$	−19.0000	−1.03500	−0.517498	−	0.855684i	$-0.673136\pi$
$337$	−19.0000	−1.03500	−0.517498	+	0.855684i	$0.673136\pi$
$338$	9.00000	0.489535
$339$	9.00000	0.488813
$340$	9.00000	0.488094
$341$	0	0
$342$	−8.00000	−0.432590
$343$	1.00000	0.0539949
$344$	1.00000	0.0539164
$345$	18.0000	0.969087
$346$	−21.0000	−1.12897
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	−3.00000	−0.160817
$349$	26.0000	1.39175	0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000	1.39175	0.695874	+	0.718164i	$0.255017\pi$
$350$	−4.00000	−0.213809
$351$	−10.0000	−0.533761
$352$	0	0
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	6.00000	0.318896
$355$	−9.00000	−0.477670
$356$	3.00000	0.159000
$357$	−3.00000	−0.158777
$358$	0	0
$359$	36.0000	1.90001	0.950004	−	0.312239i	$-0.101079\pi$
$359$	36.0000	1.90001	0.950004	+	0.312239i	$0.101079\pi$
$360$	−6.00000	−0.316228
$361$	−3.00000	−0.157895
$362$	−8.00000	−0.420471
$363$	−11.0000	−0.577350
$364$	2.00000	0.104828
$365$	30.0000	1.57027
$366$	1.00000	0.0522708
$367$	−7.00000	−0.365397	−0.182699	−	0.983169i	$-0.558483\pi$
$367$	−7.00000	−0.365397	−0.182699	+	0.983169i	$0.558483\pi$
$368$	−6.00000	−0.312772
$369$	−2.00000	−0.104116
$370$	−12.0000	−0.623850
$371$	9.00000	0.467257
$372$	−1.00000	−0.0518476
$373$	32.0000	1.65690	0.828449	−	0.560065i	$-0.189224\pi$
$373$	32.0000	1.65690	0.828449	+	0.560065i	$0.189224\pi$
$374$	0	0
$375$	3.00000	0.154919
$376$	6.00000	0.309426
$377$	−6.00000	−0.309016
$378$	5.00000	0.257172
$379$	−7.00000	−0.359566	−0.179783	−	0.983706i	$-0.557540\pi$
$379$	−7.00000	−0.359566	−0.179783	+	0.983706i	$0.557540\pi$
$380$	12.0000	0.615587
$381$	14.0000	0.717242
$382$	−3.00000	−0.153493
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	16.0000	0.814379
$387$	2.00000	0.101666
$388$	−1.00000	−0.0507673
$389$	−36.0000	−1.82527	−0.912636	−	0.408773i	$-0.865957\pi$
$389$	−36.0000	−1.82527	−0.912636	+	0.408773i	$0.865957\pi$
$390$	6.00000	0.303822
$391$	18.0000	0.910299
$392$	−1.00000	−0.0505076
$393$	18.0000	0.907980
$394$	−24.0000	−1.20910
$395$	3.00000	0.150946
$396$	0	0
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	16.0000	0.802008
$399$	−4.00000	−0.200250
$400$	4.00000	0.200000
$401$	3.00000	0.149813	0.0749064	−	0.997191i	$-0.476134\pi$
$401$	3.00000	0.149813	0.0749064	+	0.997191i	$0.476134\pi$
$402$	−8.00000	−0.399004
$403$	−2.00000	−0.0996271
$404$	−6.00000	−0.298511
$405$	−3.00000	−0.149071
$406$	3.00000	0.148888
$407$	0	0
$408$	3.00000	0.148522
$409$	−28.0000	−1.38451	−0.692255	−	0.721653i	$-0.743383\pi$
$409$	−28.0000	−1.38451	−0.692255	+	0.721653i	$0.743383\pi$
$410$	3.00000	0.148159
$411$	−6.00000	−0.295958
$412$	11.0000	0.541931
$413$	−6.00000	−0.295241
$414$	−12.0000	−0.589768
$415$	36.0000	1.76717
$416$	−2.00000	−0.0980581
$417$	−10.0000	−0.489702
$418$	0	0
$419$	−30.0000	−1.46560	−0.732798	−	0.680446i	$-0.761786\pi$
$419$	−30.0000	−1.46560	−0.732798	+	0.680446i	$0.761786\pi$
$420$	−3.00000	−0.146385
$421$	11.0000	0.536107	0.268054	−	0.963404i	$-0.413620\pi$
$421$	11.0000	0.536107	0.268054	+	0.963404i	$0.413620\pi$
$422$	4.00000	0.194717
$423$	12.0000	0.583460
$424$	−9.00000	−0.437079
$425$	−12.0000	−0.582086
$426$	−3.00000	−0.145350
$427$	−1.00000	−0.0483934
$428$	3.00000	0.145010
$429$	0	0
$430$	−3.00000	−0.144673
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	−5.00000	−0.240563
$433$	14.0000	0.672797	0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000	0.672797	0.336399	+	0.941720i	$0.390791\pi$
$434$	1.00000	0.0480015
$435$	9.00000	0.431517
$436$	2.00000	0.0957826
$437$	24.0000	1.14808
$438$	10.0000	0.477818
$439$	−28.0000	−1.33637	−0.668184	−	0.743996i	$-0.732928\pi$
$439$	−28.0000	−1.33637	−0.668184	+	0.743996i	$0.732928\pi$
$440$	0	0
$441$	−2.00000	−0.0952381
$442$	6.00000	0.285391
$443$	15.0000	0.712672	0.356336	−	0.934358i	$-0.384026\pi$
$443$	15.0000	0.712672	0.356336	+	0.934358i	$0.384026\pi$
$444$	−4.00000	−0.189832
$445$	−9.00000	−0.426641
$446$	1.00000	0.0473514
$447$	−3.00000	−0.141895
$448$	1.00000	0.0472456
$449$	15.0000	0.707894	0.353947	−	0.935266i	$-0.384839\pi$
$449$	15.0000	0.707894	0.353947	+	0.935266i	$0.384839\pi$
$450$	8.00000	0.377124
$451$	0	0
$452$	9.00000	0.423324
$453$	17.0000	0.798730
$454$	15.0000	0.703985
$455$	−6.00000	−0.281284
$456$	4.00000	0.187317
$457$	−34.0000	−1.59045	−0.795226	−	0.606313i	$-0.792648\pi$
$457$	−34.0000	−1.59045	−0.795226	+	0.606313i	$0.792648\pi$
$458$	16.0000	0.747631
$459$	15.0000	0.700140
$460$	18.0000	0.839254
$461$	−15.0000	−0.698620	−0.349310	−	0.937007i	$-0.613584\pi$
$461$	−15.0000	−0.698620	−0.349310	+	0.937007i	$0.613584\pi$
$462$	0	0
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	−3.00000	−0.139272
$465$	3.00000	0.139122
$466$	24.0000	1.11178
$467$	−24.0000	−1.11059	−0.555294	−	0.831654i	$-0.687394\pi$
$467$	−24.0000	−1.11059	−0.555294	+	0.831654i	$0.687394\pi$
$468$	−4.00000	−0.184900
$469$	8.00000	0.369406
$470$	−18.0000	−0.830278
$471$	−4.00000	−0.184310
$472$	6.00000	0.276172
$473$	0	0
$474$	1.00000	0.0459315
$475$	−16.0000	−0.734130
$476$	−3.00000	−0.137505
$477$	−18.0000	−0.824163
$478$	24.0000	1.09773
$479$	6.00000	0.274147	0.137073	−	0.990561i	$-0.456230\pi$
$479$	6.00000	0.274147	0.137073	+	0.990561i	$0.456230\pi$
$480$	3.00000	0.136931
$481$	−8.00000	−0.364769
$482$	28.0000	1.27537
$483$	−6.00000	−0.273009
$484$	−11.0000	−0.500000
$485$	3.00000	0.136223
$486$	−16.0000	−0.725775
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	1.00000	0.0452679
$489$	20.0000	0.904431
$490$	3.00000	0.135526
$491$	−3.00000	−0.135388	−0.0676941	−	0.997706i	$-0.521564\pi$
$491$	−3.00000	−0.135388	−0.0676941	+	0.997706i	$0.521564\pi$
$492$	1.00000	0.0450835
$493$	9.00000	0.405340
$494$	8.00000	0.359937
$495$	0	0
$496$	−1.00000	−0.0449013
$497$	3.00000	0.134568
$498$	12.0000	0.537733
$499$	−34.0000	−1.52205	−0.761025	−	0.648723i	$-0.775303\pi$
$499$	−34.0000	−1.52205	−0.761025	+	0.648723i	$0.775303\pi$
$500$	3.00000	0.134164
$501$	0	0
$502$	12.0000	0.535586
$503$	30.0000	1.33763	0.668817	−	0.743427i	$-0.266801\pi$
$503$	30.0000	1.33763	0.668817	+	0.743427i	$0.266801\pi$
$504$	2.00000	0.0890871
$505$	18.0000	0.800989
$506$	0	0
$507$	−9.00000	−0.399704
$508$	14.0000	0.621150
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	−9.00000	−0.398527
$511$	−10.0000	−0.442374
$512$	−1.00000	−0.0441942
$513$	20.0000	0.883022
$514$	27.0000	1.19092
$515$	−33.0000	−1.45415
$516$	−1.00000	−0.0440225
$517$	0	0
$518$	4.00000	0.175750
$519$	21.0000	0.921798
$520$	6.00000	0.263117
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	−6.00000	−0.262613
$523$	38.0000	1.66162	0.830812	−	0.556553i	$-0.187876\pi$
$523$	38.0000	1.66162	0.830812	+	0.556553i	$0.187876\pi$
$524$	18.0000	0.786334
$525$	4.00000	0.174574
$526$	−24.0000	−1.04645
$527$	3.00000	0.130682
$528$	0	0
$529$	13.0000	0.565217
$530$	27.0000	1.17281
$531$	12.0000	0.520756
$532$	−4.00000	−0.173422
$533$	2.00000	0.0866296
$534$	−3.00000	−0.129823
$535$	−9.00000	−0.389104
$536$	−8.00000	−0.345547
$537$	0	0
$538$	6.00000	0.258678
$539$	0	0
$540$	15.0000	0.645497
$541$	−28.0000	−1.20381	−0.601907	−	0.798566i	$-0.705592\pi$
$541$	−28.0000	−1.20381	−0.601907	+	0.798566i	$0.705592\pi$
$542$	16.0000	0.687259
$543$	8.00000	0.343313
$544$	3.00000	0.128624
$545$	−6.00000	−0.257012
$546$	−2.00000	−0.0855921
$547$	44.0000	1.88130	0.940652	−	0.339372i	$-0.110215\pi$
$547$	44.0000	1.88130	0.940652	+	0.339372i	$0.110215\pi$
$548$	−6.00000	−0.256307
$549$	2.00000	0.0853579
$550$	0	0
$551$	12.0000	0.511217
$552$	6.00000	0.255377
$553$	−1.00000	−0.0425243
$554$	−8.00000	−0.339887
$555$	12.0000	0.509372
$556$	−10.0000	−0.424094
$557$	−9.00000	−0.381342	−0.190671	−	0.981654i	$-0.561066\pi$
$557$	−9.00000	−0.381342	−0.190671	+	0.981654i	$0.561066\pi$
$558$	−2.00000	−0.0846668
$559$	−2.00000	−0.0845910
$560$	−3.00000	−0.126773
$561$	0	0
$562$	18.0000	0.759284
$563$	12.0000	0.505740	0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000	0.505740	0.252870	+	0.967500i	$0.418626\pi$
$564$	−6.00000	−0.252646
$565$	−27.0000	−1.13590
$566$	−8.00000	−0.336265
$567$	1.00000	0.0419961
$568$	−3.00000	−0.125877
$569$	−21.0000	−0.880366	−0.440183	−	0.897908i	$-0.645086\pi$
$569$	−21.0000	−0.880366	−0.440183	+	0.897908i	$0.645086\pi$
$570$	−12.0000	−0.502625
$571$	−10.0000	−0.418487	−0.209243	−	0.977864i	$-0.567100\pi$
$571$	−10.0000	−0.418487	−0.209243	+	0.977864i	$0.567100\pi$
$572$	0	0
$573$	3.00000	0.125327
$574$	−1.00000	−0.0417392
$575$	−24.0000	−1.00087
$576$	−2.00000	−0.0833333
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	8.00000	0.332756
$579$	−16.0000	−0.664937
$580$	9.00000	0.373705
$581$	−12.0000	−0.497844
$582$	1.00000	0.0414513
$583$	0	0
$584$	10.0000	0.413803
$585$	12.0000	0.496139
$586$	24.0000	0.991431
$587$	−33.0000	−1.36206	−0.681028	−	0.732257i	$-0.738467\pi$
$587$	−33.0000	−1.36206	−0.681028	+	0.732257i	$0.738467\pi$
$588$	1.00000	0.0412393
$589$	4.00000	0.164817
$590$	−18.0000	−0.741048
$591$	24.0000	0.987228
$592$	−4.00000	−0.164399
$593$	−39.0000	−1.60154	−0.800769	−	0.598973i	$-0.795576\pi$
$593$	−39.0000	−1.60154	−0.800769	+	0.598973i	$0.795576\pi$
$594$	0	0
$595$	9.00000	0.368964
$596$	−3.00000	−0.122885
$597$	−16.0000	−0.654836
$598$	12.0000	0.490716
$599$	30.0000	1.22577	0.612883	−	0.790173i	$-0.290010\pi$
$599$	30.0000	1.22577	0.612883	+	0.790173i	$0.290010\pi$
$600$	−4.00000	−0.163299
$601$	29.0000	1.18293	0.591467	−	0.806329i	$-0.298549\pi$
$601$	29.0000	1.18293	0.591467	+	0.806329i	$0.298549\pi$
$602$	1.00000	0.0407570
$603$	−16.0000	−0.651570
$604$	17.0000	0.691720
$605$	33.0000	1.34164
$606$	6.00000	0.243733
$607$	−25.0000	−1.01472	−0.507359	−	0.861735i	$-0.669378\pi$
$607$	−25.0000	−1.01472	−0.507359	+	0.861735i	$0.669378\pi$
$608$	4.00000	0.162221
$609$	−3.00000	−0.121566
$610$	−3.00000	−0.121466
$611$	−12.0000	−0.485468
$612$	6.00000	0.242536
$613$	−16.0000	−0.646234	−0.323117	−	0.946359i	$-0.604731\pi$
$613$	−16.0000	−0.646234	−0.323117	+	0.946359i	$0.604731\pi$
$614$	−8.00000	−0.322854
$615$	−3.00000	−0.120972
$616$	0	0
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	−11.0000	−0.442485
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	3.00000	0.120483
$621$	30.0000	1.20386
$622$	−6.00000	−0.240578
$623$	3.00000	0.120192
$624$	2.00000	0.0800641
$625$	−29.0000	−1.16000
$626$	−14.0000	−0.559553
$627$	0	0
$628$	−4.00000	−0.159617
$629$	12.0000	0.478471
$630$	−6.00000	−0.239046
$631$	−34.0000	−1.35352	−0.676759	−	0.736204i	$-0.736616\pi$
$631$	−34.0000	−1.35352	−0.676759	+	0.736204i	$0.736616\pi$
$632$	1.00000	0.0397779
$633$	−4.00000	−0.158986
$634$	−30.0000	−1.19145
$635$	−42.0000	−1.66672
$636$	9.00000	0.356873
$637$	2.00000	0.0792429
$638$	0	0
$639$	−6.00000	−0.237356
$640$	3.00000	0.118585
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	−3.00000	−0.118401
$643$	29.0000	1.14365	0.571824	−	0.820376i	$-0.306236\pi$
$643$	29.0000	1.14365	0.571824	+	0.820376i	$0.306236\pi$
$644$	−6.00000	−0.236433
$645$	3.00000	0.118125
$646$	−12.0000	−0.472134
$647$	−45.0000	−1.76913	−0.884566	−	0.466415i	$-0.845546\pi$
$647$	−45.0000	−1.76913	−0.884566	+	0.466415i	$0.845546\pi$
$648$	−1.00000	−0.0392837
$649$	0	0
$650$	−8.00000	−0.313786
$651$	−1.00000	−0.0391931
$652$	20.0000	0.783260
$653$	21.0000	0.821794	0.410897	−	0.911682i	$-0.365216\pi$
$653$	21.0000	0.821794	0.410897	+	0.911682i	$0.365216\pi$
$654$	−2.00000	−0.0782062
$655$	−54.0000	−2.10995
$656$	1.00000	0.0390434
$657$	20.0000	0.780274
$658$	6.00000	0.233904
$659$	42.0000	1.63609	0.818044	−	0.575156i	$-0.195059\pi$
$659$	42.0000	1.63609	0.818044	+	0.575156i	$0.195059\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	−2.00000	−0.0777322
$663$	−6.00000	−0.233021
$664$	12.0000	0.465690
$665$	12.0000	0.465340
$666$	−8.00000	−0.309994
$667$	18.0000	0.696963
$668$	0	0
$669$	−1.00000	−0.0386622
$670$	24.0000	0.927201
$671$	0	0
$672$	−1.00000	−0.0385758
$673$	8.00000	0.308377	0.154189	−	0.988041i	$-0.450724\pi$
$673$	8.00000	0.308377	0.154189	+	0.988041i	$0.450724\pi$
$674$	19.0000	0.731853
$675$	−20.0000	−0.769800
$676$	−9.00000	−0.346154
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	−9.00000	−0.345643
$679$	−1.00000	−0.0383765
$680$	−9.00000	−0.345134
$681$	−15.0000	−0.574801
$682$	0	0
$683$	24.0000	0.918334	0.459167	−	0.888350i	$-0.348148\pi$
$683$	24.0000	0.918334	0.459167	+	0.888350i	$0.348148\pi$
$684$	8.00000	0.305888
$685$	18.0000	0.687745
$686$	−1.00000	−0.0381802
$687$	−16.0000	−0.610438
$688$	−1.00000	−0.0381246
$689$	18.0000	0.685745
$690$	−18.0000	−0.685248
$691$	5.00000	0.190209	0.0951045	−	0.995467i	$-0.469681\pi$
$691$	5.00000	0.190209	0.0951045	+	0.995467i	$0.469681\pi$
$692$	21.0000	0.798300
$693$	0	0
$694$	0	0
$695$	30.0000	1.13796
$696$	3.00000	0.113715
$697$	−3.00000	−0.113633
$698$	−26.0000	−0.984115
$699$	−24.0000	−0.907763
$700$	4.00000	0.151186
$701$	42.0000	1.58632	0.793159	−	0.609015i	$-0.208435\pi$
$701$	42.0000	1.58632	0.793159	+	0.609015i	$0.208435\pi$
$702$	10.0000	0.377426
$703$	16.0000	0.603451
$704$	0	0
$705$	18.0000	0.677919
$706$	−6.00000	−0.225813
$707$	−6.00000	−0.225653
$708$	−6.00000	−0.225494
$709$	−19.0000	−0.713560	−0.356780	−	0.934188i	$-0.616125\pi$
$709$	−19.0000	−0.713560	−0.356780	+	0.934188i	$0.616125\pi$
$710$	9.00000	0.337764
$711$	2.00000	0.0750059
$712$	−3.00000	−0.112430
$713$	6.00000	0.224702
$714$	3.00000	0.112272
$715$	0	0
$716$	0	0
$717$	−24.0000	−0.896296
$718$	−36.0000	−1.34351
$719$	−6.00000	−0.223762	−0.111881	−	0.993722i	$-0.535688\pi$
$719$	−6.00000	−0.223762	−0.111881	+	0.993722i	$0.535688\pi$
$720$	6.00000	0.223607
$721$	11.0000	0.409661
$722$	3.00000	0.111648
$723$	−28.0000	−1.04133
$724$	8.00000	0.297318
$725$	−12.0000	−0.445669
$726$	11.0000	0.408248
$727$	44.0000	1.63187	0.815935	−	0.578144i	$-0.196223\pi$
$727$	44.0000	1.63187	0.815935	+	0.578144i	$0.196223\pi$
$728$	−2.00000	−0.0741249
$729$	13.0000	0.481481
$730$	−30.0000	−1.11035
$731$	3.00000	0.110959
$732$	−1.00000	−0.0369611
$733$	−25.0000	−0.923396	−0.461698	−	0.887037i	$-0.652760\pi$
$733$	−25.0000	−0.923396	−0.461698	+	0.887037i	$0.652760\pi$
$734$	7.00000	0.258375
$735$	−3.00000	−0.110657
$736$	6.00000	0.221163
$737$	0	0
$738$	2.00000	0.0736210
$739$	−37.0000	−1.36107	−0.680534	−	0.732717i	$-0.738252\pi$
$739$	−37.0000	−1.36107	−0.680534	+	0.732717i	$0.738252\pi$
$740$	12.0000	0.441129
$741$	−8.00000	−0.293887
$742$	−9.00000	−0.330400
$743$	42.0000	1.54083	0.770415	−	0.637542i	$-0.220049\pi$
$743$	42.0000	1.54083	0.770415	+	0.637542i	$0.220049\pi$
$744$	1.00000	0.0366618
$745$	9.00000	0.329734
$746$	−32.0000	−1.17160
$747$	24.0000	0.878114
$748$	0	0
$749$	3.00000	0.109618
$750$	−3.00000	−0.109545
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	−6.00000	−0.218797
$753$	−12.0000	−0.437304
$754$	6.00000	0.218507
$755$	−51.0000	−1.85608
$756$	−5.00000	−0.181848
$757$	−43.0000	−1.56286	−0.781431	−	0.623992i	$-0.785510\pi$
$757$	−43.0000	−1.56286	−0.781431	+	0.623992i	$0.785510\pi$
$758$	7.00000	0.254251
$759$	0	0
$760$	−12.0000	−0.435286
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	−14.0000	−0.507166
$763$	2.00000	0.0724049
$764$	3.00000	0.108536
$765$	−18.0000	−0.650791
$766$	24.0000	0.867155
$767$	−12.0000	−0.433295
$768$	1.00000	0.0360844
$769$	2.00000	0.0721218	0.0360609	−	0.999350i	$-0.488519\pi$
$769$	2.00000	0.0721218	0.0360609	+	0.999350i	$0.488519\pi$
$770$	0	0
$771$	−27.0000	−0.972381
$772$	−16.0000	−0.575853
$773$	18.0000	0.647415	0.323708	−	0.946157i	$-0.395071\pi$
$773$	18.0000	0.647415	0.323708	+	0.946157i	$0.395071\pi$
$774$	−2.00000	−0.0718885
$775$	−4.00000	−0.143684
$776$	1.00000	0.0358979
$777$	−4.00000	−0.143499
$778$	36.0000	1.29066
$779$	−4.00000	−0.143315
$780$	−6.00000	−0.214834
$781$	0	0
$782$	−18.0000	−0.643679
$783$	15.0000	0.536056
$784$	1.00000	0.0357143
$785$	12.0000	0.428298
$786$	−18.0000	−0.642039
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	24.0000	0.854965
$789$	24.0000	0.854423
$790$	−3.00000	−0.106735
$791$	9.00000	0.320003
$792$	0	0
$793$	−2.00000	−0.0710221
$794$	22.0000	0.780751
$795$	−27.0000	−0.957591
$796$	−16.0000	−0.567105
$797$	51.0000	1.80651	0.903256	−	0.429101i	$-0.141170\pi$
$797$	51.0000	1.80651	0.903256	+	0.429101i	$0.141170\pi$
$798$	4.00000	0.141598
$799$	18.0000	0.636794
$800$	−4.00000	−0.141421
$801$	−6.00000	−0.212000
$802$	−3.00000	−0.105934
$803$	0	0
$804$	8.00000	0.282138
$805$	18.0000	0.634417
$806$	2.00000	0.0704470
$807$	−6.00000	−0.211210
$808$	6.00000	0.211079
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	3.00000	0.105409
$811$	2.00000	0.0702295	0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000	0.0702295	0.0351147	+	0.999383i	$0.488820\pi$
$812$	−3.00000	−0.105279
$813$	−16.0000	−0.561144
$814$	0	0
$815$	−60.0000	−2.10171
$816$	−3.00000	−0.105021
$817$	4.00000	0.139942
$818$	28.0000	0.978997
$819$	−4.00000	−0.139771
$820$	−3.00000	−0.104765
$821$	−24.0000	−0.837606	−0.418803	−	0.908077i	$-0.637550\pi$
$821$	−24.0000	−0.837606	−0.418803	+	0.908077i	$0.637550\pi$
$822$	6.00000	0.209274
$823$	23.0000	0.801730	0.400865	−	0.916137i	$-0.368710\pi$
$823$	23.0000	0.801730	0.400865	+	0.916137i	$0.368710\pi$
$824$	−11.0000	−0.383203
$825$	0	0
$826$	6.00000	0.208767
$827$	30.0000	1.04320	0.521601	−	0.853189i	$-0.325335\pi$
$827$	30.0000	1.04320	0.521601	+	0.853189i	$0.325335\pi$
$828$	12.0000	0.417029
$829$	41.0000	1.42399	0.711994	−	0.702185i	$-0.247792\pi$
$829$	41.0000	1.42399	0.711994	+	0.702185i	$0.247792\pi$
$830$	−36.0000	−1.24958
$831$	8.00000	0.277517
$832$	2.00000	0.0693375
$833$	−3.00000	−0.103944
$834$	10.0000	0.346272
$835$	0	0
$836$	0	0
$837$	5.00000	0.172825
$838$	30.0000	1.03633
$839$	−12.0000	−0.414286	−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000	−0.414286	−0.207143	+	0.978311i	$0.566417\pi$
$840$	3.00000	0.103510
$841$	−20.0000	−0.689655
$842$	−11.0000	−0.379085
$843$	−18.0000	−0.619953
$844$	−4.00000	−0.137686
$845$	27.0000	0.928828
$846$	−12.0000	−0.412568
$847$	−11.0000	−0.377964
$848$	9.00000	0.309061
$849$	8.00000	0.274559
$850$	12.0000	0.411597
$851$	24.0000	0.822709
$852$	3.00000	0.102778
$853$	−19.0000	−0.650548	−0.325274	−	0.945620i	$-0.605456\pi$
$853$	−19.0000	−0.650548	−0.325274	+	0.945620i	$0.605456\pi$
$854$	1.00000	0.0342193
$855$	−24.0000	−0.820783
$856$	−3.00000	−0.102538
$857$	42.0000	1.43469	0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000	1.43469	0.717346	+	0.696717i	$0.245357\pi$
$858$	0	0
$859$	14.0000	0.477674	0.238837	−	0.971060i	$-0.423234\pi$
$859$	14.0000	0.477674	0.238837	+	0.971060i	$0.423234\pi$
$860$	3.00000	0.102299
$861$	1.00000	0.0340799
$862$	−12.0000	−0.408722
$863$	−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
$865$	−63.0000	−2.14206
$866$	−14.0000	−0.475739
$867$	−8.00000	−0.271694
$868$	−1.00000	−0.0339422
$869$	0	0
$870$	−9.00000	−0.305129
$871$	16.0000	0.542139
$872$	−2.00000	−0.0677285
$873$	2.00000	0.0676897
$874$	−24.0000	−0.811812
$875$	3.00000	0.101419
$876$	−10.0000	−0.337869
$877$	32.0000	1.08056	0.540282	−	0.841484i	$-0.318318\pi$
$877$	32.0000	1.08056	0.540282	+	0.841484i	$0.318318\pi$
$878$	28.0000	0.944954
$879$	−24.0000	−0.809500
$880$	0	0
$881$	−12.0000	−0.404290	−0.202145	−	0.979356i	$-0.564791\pi$
$881$	−12.0000	−0.404290	−0.202145	+	0.979356i	$0.564791\pi$
$882$	2.00000	0.0673435
$883$	8.00000	0.269221	0.134611	−	0.990899i	$-0.457022\pi$
$883$	8.00000	0.269221	0.134611	+	0.990899i	$0.457022\pi$
$884$	−6.00000	−0.201802
$885$	18.0000	0.605063
$886$	−15.0000	−0.503935
$887$	30.0000	1.00730	0.503651	−	0.863907i	$-0.331990\pi$
$887$	30.0000	1.00730	0.503651	+	0.863907i	$0.331990\pi$
$888$	4.00000	0.134231
$889$	14.0000	0.469545
$890$	9.00000	0.301681
$891$	0	0
$892$	−1.00000	−0.0334825
$893$	24.0000	0.803129
$894$	3.00000	0.100335
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	−12.0000	−0.400668
$898$	−15.0000	−0.500556
$899$	3.00000	0.100056
$900$	−8.00000	−0.266667
$901$	−27.0000	−0.899500
$902$	0	0
$903$	−1.00000	−0.0332779
$904$	−9.00000	−0.299336
$905$	−24.0000	−0.797787
$906$	−17.0000	−0.564787
$907$	−13.0000	−0.431658	−0.215829	−	0.976431i	$-0.569245\pi$
$907$	−13.0000	−0.431658	−0.215829	+	0.976431i	$0.569245\pi$
$908$	−15.0000	−0.497792
$909$	12.0000	0.398015
$910$	6.00000	0.198898
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	−4.00000	−0.132453
$913$	0	0
$914$	34.0000	1.12462
$915$	3.00000	0.0991769
$916$	−16.0000	−0.528655
$917$	18.0000	0.594412
$918$	−15.0000	−0.495074
$919$	5.00000	0.164935	0.0824674	−	0.996594i	$-0.473720\pi$
$919$	5.00000	0.164935	0.0824674	+	0.996594i	$0.473720\pi$
$920$	−18.0000	−0.593442
$921$	8.00000	0.263609
$922$	15.0000	0.493999
$923$	6.00000	0.197492
$924$	0	0
$925$	−16.0000	−0.526077
$926$	16.0000	0.525793
$927$	−22.0000	−0.722575
$928$	3.00000	0.0984798
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	−3.00000	−0.0983739
$931$	−4.00000	−0.131095
$932$	−24.0000	−0.786146
$933$	6.00000	0.196431
$934$	24.0000	0.785304
$935$	0	0
$936$	4.00000	0.130744
$937$	47.0000	1.53542	0.767712	−	0.640796i	$-0.221395\pi$
$937$	47.0000	1.53542	0.767712	+	0.640796i	$0.221395\pi$
$938$	−8.00000	−0.261209
$939$	14.0000	0.456873
$940$	18.0000	0.587095
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	4.00000	0.130327
$943$	−6.00000	−0.195387
$944$	−6.00000	−0.195283
$945$	15.0000	0.487950
$946$	0	0
$947$	36.0000	1.16984	0.584921	−	0.811090i	$-0.301125\pi$
$947$	36.0000	1.16984	0.584921	+	0.811090i	$0.301125\pi$
$948$	−1.00000	−0.0324785
$949$	−20.0000	−0.649227
$950$	16.0000	0.519109
$951$	30.0000	0.972817
$952$	3.00000	0.0972306
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	18.0000	0.582772
$955$	−9.00000	−0.291233
$956$	−24.0000	−0.776215
$957$	0	0
$958$	−6.00000	−0.193851
$959$	−6.00000	−0.193750
$960$	−3.00000	−0.0968246
$961$	−30.0000	−0.967742
$962$	8.00000	0.257930
$963$	−6.00000	−0.193347
$964$	−28.0000	−0.901819
$965$	48.0000	1.54517
$966$	6.00000	0.193047
$967$	−49.0000	−1.57573	−0.787867	−	0.615846i	$-0.788815\pi$
$967$	−49.0000	−1.57573	−0.787867	+	0.615846i	$0.788815\pi$
$968$	11.0000	0.353553
$969$	12.0000	0.385496
$970$	−3.00000	−0.0963242
$971$	45.0000	1.44412	0.722059	−	0.691831i	$-0.243196\pi$
$971$	45.0000	1.44412	0.722059	+	0.691831i	$0.243196\pi$
$972$	16.0000	0.513200
$973$	−10.0000	−0.320585
$974$	−2.00000	−0.0640841
$975$	8.00000	0.256205
$976$	−1.00000	−0.0320092
$977$	−42.0000	−1.34370	−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000	−1.34370	−0.671850	+	0.740688i	$0.734500\pi$
$978$	−20.0000	−0.639529
$979$	0	0
$980$	−3.00000	−0.0958315
$981$	−4.00000	−0.127710
$982$	3.00000	0.0957338
$983$	−33.0000	−1.05254	−0.526268	−	0.850319i	$-0.676409\pi$
$983$	−33.0000	−1.05254	−0.526268	+	0.850319i	$0.676409\pi$
$984$	−1.00000	−0.0318788
$985$	−72.0000	−2.29411
$986$	−9.00000	−0.286618
$987$	−6.00000	−0.190982
$988$	−8.00000	−0.254514
$989$	6.00000	0.190789
$990$	0	0
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	1.00000	0.0317500
$993$	2.00000	0.0634681
$994$	−3.00000	−0.0951542
$995$	48.0000	1.52170
$996$	−12.0000	−0.380235
$997$	−58.0000	−1.83688	−0.918439	−	0.395562i	$-0.870550\pi$
$997$	−58.0000	−1.83688	−0.918439	+	0.395562i	$0.870550\pi$
$998$	34.0000	1.07625
$999$	20.0000	0.632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	574.2.a.c.1.1	✓	1
3.2	odd	2		5166.2.a.bm.1.1		1
4.3	odd	2		4592.2.a.d.1.1		1
7.6	odd	2		4018.2.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
574.2.a.c.1.1	✓	1	1.1	even	1	trivial
4018.2.a.f.1.1		1	7.6	odd	2
4592.2.a.d.1.1		1	4.3	odd	2
5166.2.a.bm.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:	\( N \)	\(=\)	\( 574 = 2 \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	574.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more