LMFDB - Embedded newform 7098.2.a.x.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7098,2,Mod(1,7098)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7098, 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("7098.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7098.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:	$56.6778153547$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 546)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	7098.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.00000 q^{10} +5.00000 q^{11} +1.00000 q^{12} -1.00000 q^{14} -3.00000 q^{15} +1.00000 q^{16} -3.00000 q^{17} +1.00000 q^{18} -1.00000 q^{19} -3.00000 q^{20} -1.00000 q^{21} +5.00000 q^{22} +1.00000 q^{23} +1.00000 q^{24} +4.00000 q^{25} +1.00000 q^{27} -1.00000 q^{28} +5.00000 q^{29} -3.00000 q^{30} +1.00000 q^{32} +5.00000 q^{33} -3.00000 q^{34} +3.00000 q^{35} +1.00000 q^{36} +7.00000 q^{37} -1.00000 q^{38} -3.00000 q^{40} -1.00000 q^{42} +1.00000 q^{43} +5.00000 q^{44} -3.00000 q^{45} +1.00000 q^{46} -8.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} +4.00000 q^{50} -3.00000 q^{51} +14.0000 q^{53} +1.00000 q^{54} -15.0000 q^{55} -1.00000 q^{56} -1.00000 q^{57} +5.00000 q^{58} -14.0000 q^{59} -3.00000 q^{60} -3.00000 q^{61} -1.00000 q^{63} +1.00000 q^{64} +5.00000 q^{66} +8.00000 q^{67} -3.00000 q^{68} +1.00000 q^{69} +3.00000 q^{70} -10.0000 q^{71} +1.00000 q^{72} -11.0000 q^{73} +7.00000 q^{74} +4.00000 q^{75} -1.00000 q^{76} -5.00000 q^{77} -3.00000 q^{80} +1.00000 q^{81} +6.00000 q^{83} -1.00000 q^{84} +9.00000 q^{85} +1.00000 q^{86} +5.00000 q^{87} +5.00000 q^{88} +16.0000 q^{89} -3.00000 q^{90} +1.00000 q^{92} -8.00000 q^{94} +3.00000 q^{95} +1.00000 q^{96} -2.00000 q^{97} +1.00000 q^{98} +5.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	−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
$9$	1.00000	0.333333
$10$	−3.00000	−0.948683
$11$	5.00000	1.50756	0.753778	−	0.657129i	$-0.228229\pi$
$11$	5.00000	1.50756	0.753778	+	0.657129i	$0.228229\pi$
$12$	1.00000	0.288675
$13$	0	0
$13$	0	0
$14$	−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	$0.618522\pi$
$18$	1.00000	0.235702
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	−3.00000	−0.670820
$21$	−1.00000	−0.218218
$22$	5.00000	1.06600
$23$	1.00000	0.208514	0.104257	−	0.994550i	$-0.466753\pi$
$23$	1.00000	0.208514	0.104257	+	0.994550i	$0.466753\pi$
$24$	1.00000	0.204124
$25$	4.00000	0.800000
$26$	0	0
$27$	1.00000	0.192450
$28$	−1.00000	−0.188982
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	−3.00000	−0.547723
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	1.00000	0.176777
$33$	5.00000	0.870388
$34$	−3.00000	−0.514496
$35$	3.00000	0.507093
$36$	1.00000	0.166667
$37$	7.00000	1.15079	0.575396	−	0.817875i	$-0.304848\pi$
$37$	7.00000	1.15079	0.575396	+	0.817875i	$0.304848\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	−3.00000	−0.474342
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	−1.00000	−0.154303
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	5.00000	0.753778
$45$	−3.00000	−0.447214
$46$	1.00000	0.147442
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	1.00000	0.144338
$49$	1.00000	0.142857
$50$	4.00000	0.565685
$51$	−3.00000	−0.420084
$52$	0	0
$53$	14.0000	1.92305	0.961524	−	0.274721i	$-0.0885855\pi$
$53$	14.0000	1.92305	0.961524	+	0.274721i	$0.0885855\pi$
$54$	1.00000	0.136083
$55$	−15.0000	−2.02260
$56$	−1.00000	−0.133631
$57$	−1.00000	−0.132453
$58$	5.00000	0.656532
$59$	−14.0000	−1.82264	−0.911322	−	0.411693i	$-0.864937\pi$
$59$	−14.0000	−1.82264	−0.911322	+	0.411693i	$0.864937\pi$
$60$	−3.00000	−0.387298
$61$	−3.00000	−0.384111	−0.192055	−	0.981384i	$-0.561515\pi$
$61$	−3.00000	−0.384111	−0.192055	+	0.981384i	$0.561515\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	5.00000	0.615457
$67$	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
$69$	1.00000	0.120386
$70$	3.00000	0.358569
$71$	−10.0000	−1.18678	−0.593391	−	0.804914i	$-0.702211\pi$
$71$	−10.0000	−1.18678	−0.593391	+	0.804914i	$0.702211\pi$
$72$	1.00000	0.117851
$73$	−11.0000	−1.28745	−0.643726	−	0.765256i	$-0.722612\pi$
$73$	−11.0000	−1.28745	−0.643726	+	0.765256i	$0.722612\pi$
$74$	7.00000	0.813733
$75$	4.00000	0.461880
$76$	−1.00000	−0.114708
$77$	−5.00000	−0.569803
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	−3.00000	−0.335410
$81$	1.00000	0.111111
$82$	0	0
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	−1.00000	−0.109109
$85$	9.00000	0.976187
$86$	1.00000	0.107833
$87$	5.00000	0.536056
$88$	5.00000	0.533002
$89$	16.0000	1.69600	0.847998	−	0.529999i	$-0.177808\pi$
$89$	16.0000	1.69600	0.847998	+	0.529999i	$0.177808\pi$
$90$	−3.00000	−0.316228
$91$	0	0
$92$	1.00000	0.104257
$93$	0	0
$94$	−8.00000	−0.825137
$95$	3.00000	0.307794
$96$	1.00000	0.102062
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	1.00000	0.101015
$99$	5.00000	0.502519
$100$	4.00000	0.400000
$101$	18.0000	1.79107	0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000	1.79107	0.895533	+	0.444994i	$0.146794\pi$
$102$	−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	$0.317693\pi$
$104$	0	0
$105$	3.00000	0.292770
$106$	14.0000	1.35980
$107$	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
$109$	9.00000	0.862044	0.431022	−	0.902342i	$-0.358153\pi$
$109$	9.00000	0.862044	0.431022	+	0.902342i	$0.358153\pi$
$110$	−15.0000	−1.43019
$111$	7.00000	0.664411
$112$	−1.00000	−0.0944911
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	−1.00000	−0.0936586
$115$	−3.00000	−0.279751
$116$	5.00000	0.464238
$117$	0	0
$118$	−14.0000	−1.28880
$119$	3.00000	0.275010
$120$	−3.00000	−0.273861
$121$	14.0000	1.27273
$122$	−3.00000	−0.271607
$123$	0	0
$124$	0	0
$125$	3.00000	0.268328
$126$	−1.00000	−0.0890871
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	1.00000	0.0883883
$129$	1.00000	0.0880451
$130$	0	0
$131$	7.00000	0.611593	0.305796	−	0.952097i	$-0.401077\pi$
$131$	7.00000	0.611593	0.305796	+	0.952097i	$0.401077\pi$
$132$	5.00000	0.435194
$133$	1.00000	0.0867110
$134$	8.00000	0.691095
$135$	−3.00000	−0.258199
$136$	−3.00000	−0.257248
$137$	−3.00000	−0.256307	−0.128154	−	0.991754i	$-0.540905\pi$
$137$	−3.00000	−0.256307	−0.128154	+	0.991754i	$0.540905\pi$
$138$	1.00000	0.0851257
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	3.00000	0.253546
$141$	−8.00000	−0.673722
$142$	−10.0000	−0.839181
$143$	0	0
$144$	1.00000	0.0833333
$145$	−15.0000	−1.24568
$146$	−11.0000	−0.910366
$147$	1.00000	0.0824786
$148$	7.00000	0.575396
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	4.00000	0.326599
$151$	−15.0000	−1.22068	−0.610341	−	0.792139i	$-0.708968\pi$
$151$	−15.0000	−1.22068	−0.610341	+	0.792139i	$0.708968\pi$
$152$	−1.00000	−0.0811107
$153$	−3.00000	−0.242536
$154$	−5.00000	−0.402911
$155$	0	0
$156$	0	0
$157$	13.0000	1.03751	0.518756	−	0.854922i	$-0.326395\pi$
$157$	13.0000	1.03751	0.518756	+	0.854922i	$0.326395\pi$
$158$	0	0
$159$	14.0000	1.11027
$160$	−3.00000	−0.237171
$161$	−1.00000	−0.0788110
$162$	1.00000	0.0785674
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	0	0
$165$	−15.0000	−1.16775
$166$	6.00000	0.465690
$167$	7.00000	0.541676	0.270838	−	0.962625i	$-0.412699\pi$
$167$	7.00000	0.541676	0.270838	+	0.962625i	$0.412699\pi$
$168$	−1.00000	−0.0771517
$169$	0	0
$170$	9.00000	0.690268
$171$	−1.00000	−0.0764719
$172$	1.00000	0.0762493
$173$	26.0000	1.97674	0.988372	−	0.152057i	$-0.0485898\pi$
$173$	26.0000	1.97674	0.988372	+	0.152057i	$0.0485898\pi$
$174$	5.00000	0.379049
$175$	−4.00000	−0.302372
$176$	5.00000	0.376889
$177$	−14.0000	−1.05230
$178$	16.0000	1.19925
$179$	−10.0000	−0.747435	−0.373718	−	0.927543i	$-0.621917\pi$
$179$	−10.0000	−0.747435	−0.373718	+	0.927543i	$0.621917\pi$
$180$	−3.00000	−0.223607
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	−3.00000	−0.221766
$184$	1.00000	0.0737210
$185$	−21.0000	−1.54395
$186$	0	0
$187$	−15.0000	−1.09691
$188$	−8.00000	−0.583460
$189$	−1.00000	−0.0727393
$190$	3.00000	0.217643
$191$	17.0000	1.23008	0.615038	−	0.788497i	$-0.289140\pi$
$191$	17.0000	1.23008	0.615038	+	0.788497i	$0.289140\pi$
$192$	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	$0.695330\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	1.00000	0.0714286
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	5.00000	0.355335
$199$	5.00000	0.354441	0.177220	−	0.984171i	$-0.443289\pi$
$199$	5.00000	0.354441	0.177220	+	0.984171i	$0.443289\pi$
$200$	4.00000	0.282843
$201$	8.00000	0.564276
$202$	18.0000	1.26648
$203$	−5.00000	−0.350931
$204$	−3.00000	−0.210042
$205$	0	0
$206$	11.0000	0.766406
$207$	1.00000	0.0695048
$208$	0	0
$209$	−5.00000	−0.345857
$210$	3.00000	0.207020
$211$	−13.0000	−0.894957	−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000	−0.894957	−0.447478	+	0.894295i	$0.647678\pi$
$212$	14.0000	0.961524
$213$	−10.0000	−0.685189
$214$	18.0000	1.23045
$215$	−3.00000	−0.204598
$216$	1.00000	0.0680414
$217$	0	0
$218$	9.00000	0.609557
$219$	−11.0000	−0.743311
$220$	−15.0000	−1.01130
$221$	0	0
$222$	7.00000	0.469809
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	−1.00000	−0.0668153
$225$	4.00000	0.266667
$226$	−6.00000	−0.399114
$227$	8.00000	0.530979	0.265489	−	0.964114i	$-0.414466\pi$
$227$	8.00000	0.530979	0.265489	+	0.964114i	$0.414466\pi$
$228$	−1.00000	−0.0662266
$229$	26.0000	1.71813	0.859064	−	0.511868i	$-0.171046\pi$
$229$	26.0000	1.71813	0.859064	+	0.511868i	$0.171046\pi$
$230$	−3.00000	−0.197814
$231$	−5.00000	−0.328976
$232$	5.00000	0.328266
$233$	−14.0000	−0.917170	−0.458585	−	0.888650i	$-0.651644\pi$
$233$	−14.0000	−0.917170	−0.458585	+	0.888650i	$0.651644\pi$
$234$	0	0
$235$	24.0000	1.56559
$236$	−14.0000	−0.911322
$237$	0	0
$238$	3.00000	0.194461
$239$	4.00000	0.258738	0.129369	−	0.991596i	$-0.458705\pi$
$239$	4.00000	0.258738	0.129369	+	0.991596i	$0.458705\pi$
$240$	−3.00000	−0.193649
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	14.0000	0.899954
$243$	1.00000	0.0641500
$244$	−3.00000	−0.192055
$245$	−3.00000	−0.191663
$246$	0	0
$247$	0	0
$248$	0	0
$249$	6.00000	0.380235
$250$	3.00000	0.189737
$251$	−17.0000	−1.07303	−0.536515	−	0.843891i	$-0.680260\pi$
$251$	−17.0000	−1.07303	−0.536515	+	0.843891i	$0.680260\pi$
$252$	−1.00000	−0.0629941
$253$	5.00000	0.314347
$254$	12.0000	0.752947
$255$	9.00000	0.563602
$256$	1.00000	0.0625000
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	1.00000	0.0622573
$259$	−7.00000	−0.434959
$260$	0	0
$261$	5.00000	0.309492
$262$	7.00000	0.432461
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	5.00000	0.307729
$265$	−42.0000	−2.58004
$266$	1.00000	0.0613139
$267$	16.0000	0.979184
$268$	8.00000	0.488678
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	−3.00000	−0.182574
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	−3.00000	−0.181902
$273$	0	0
$274$	−3.00000	−0.181237
$275$	20.0000	1.20605
$276$	1.00000	0.0601929
$277$	−28.0000	−1.68236	−0.841178	−	0.540758i	$-0.818138\pi$
$277$	−28.0000	−1.68236	−0.841178	+	0.540758i	$0.818138\pi$
$278$	20.0000	1.19952
$279$	0	0
$280$	3.00000	0.179284
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	−8.00000	−0.476393
$283$	−14.0000	−0.832214	−0.416107	−	0.909316i	$-0.636606\pi$
$283$	−14.0000	−0.832214	−0.416107	+	0.909316i	$0.636606\pi$
$284$	−10.0000	−0.593391
$285$	3.00000	0.177705
$286$	0	0
$287$	0	0
$288$	1.00000	0.0589256
$289$	−8.00000	−0.470588
$290$	−15.0000	−0.880830
$291$	−2.00000	−0.117242
$292$	−11.0000	−0.643726
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	1.00000	0.0583212
$295$	42.0000	2.44533
$296$	7.00000	0.406867
$297$	5.00000	0.290129
$298$	−6.00000	−0.347571
$299$	0	0
$300$	4.00000	0.230940
$301$	−1.00000	−0.0576390
$302$	−15.0000	−0.863153
$303$	18.0000	1.03407
$304$	−1.00000	−0.0573539
$305$	9.00000	0.515339
$306$	−3.00000	−0.171499
$307$	−8.00000	−0.456584	−0.228292	−	0.973593i	$-0.573314\pi$
$307$	−8.00000	−0.456584	−0.228292	+	0.973593i	$0.573314\pi$
$308$	−5.00000	−0.284901
$309$	11.0000	0.625768
$310$	0	0
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	13.0000	0.733632
$315$	3.00000	0.169031
$316$	0	0
$317$	8.00000	0.449325	0.224662	−	0.974437i	$-0.427872\pi$
$317$	8.00000	0.449325	0.224662	+	0.974437i	$0.427872\pi$
$318$	14.0000	0.785081
$319$	25.0000	1.39973
$320$	−3.00000	−0.167705
$321$	18.0000	1.00466
$322$	−1.00000	−0.0557278
$323$	3.00000	0.166924
$324$	1.00000	0.0555556
$325$	0	0
$326$	4.00000	0.221540
$327$	9.00000	0.497701
$328$	0	0
$329$	8.00000	0.441054
$330$	−15.0000	−0.825723
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	6.00000	0.329293
$333$	7.00000	0.383598
$334$	7.00000	0.383023
$335$	−24.0000	−1.31126
$336$	−1.00000	−0.0545545
$337$	−33.0000	−1.79762	−0.898812	−	0.438334i	$-0.855569\pi$
$337$	−33.0000	−1.79762	−0.898812	+	0.438334i	$0.855569\pi$
$338$	0	0
$339$	−6.00000	−0.325875
$340$	9.00000	0.488094
$341$	0	0
$342$	−1.00000	−0.0540738
$343$	−1.00000	−0.0539949
$344$	1.00000	0.0539164
$345$	−3.00000	−0.161515
$346$	26.0000	1.39777
$347$	28.0000	1.50312	0.751559	−	0.659665i	$-0.229302\pi$
$347$	28.0000	1.50312	0.751559	+	0.659665i	$0.229302\pi$
$348$	5.00000	0.268028
$349$	16.0000	0.856460	0.428230	−	0.903670i	$-0.359137\pi$
$349$	16.0000	0.856460	0.428230	+	0.903670i	$0.359137\pi$
$350$	−4.00000	−0.213809
$351$	0	0
$352$	5.00000	0.266501
$353$	−24.0000	−1.27739	−0.638696	−	0.769460i	$-0.720526\pi$
$353$	−24.0000	−1.27739	−0.638696	+	0.769460i	$0.720526\pi$
$354$	−14.0000	−0.744092
$355$	30.0000	1.59223
$356$	16.0000	0.847998
$357$	3.00000	0.158777
$358$	−10.0000	−0.528516
$359$	−34.0000	−1.79445	−0.897226	−	0.441572i	$-0.854421\pi$
$359$	−34.0000	−1.79445	−0.897226	+	0.441572i	$0.854421\pi$
$360$	−3.00000	−0.158114
$361$	−18.0000	−0.947368
$362$	−2.00000	−0.105118
$363$	14.0000	0.734809
$364$	0	0
$365$	33.0000	1.72730
$366$	−3.00000	−0.156813
$367$	−32.0000	−1.67039	−0.835193	−	0.549957i	$-0.814644\pi$
$367$	−32.0000	−1.67039	−0.835193	+	0.549957i	$0.814644\pi$
$368$	1.00000	0.0521286
$369$	0	0
$370$	−21.0000	−1.09174
$371$	−14.0000	−0.726844
$372$	0	0
$373$	−26.0000	−1.34623	−0.673114	−	0.739538i	$-0.735044\pi$
$373$	−26.0000	−1.34623	−0.673114	+	0.739538i	$0.735044\pi$
$374$	−15.0000	−0.775632
$375$	3.00000	0.154919
$376$	−8.00000	−0.412568
$377$	0	0
$378$	−1.00000	−0.0514344
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	3.00000	0.153897
$381$	12.0000	0.614779
$382$	17.0000	0.869796
$383$	31.0000	1.58403	0.792013	−	0.610504i	$-0.209033\pi$
$383$	31.0000	1.58403	0.792013	+	0.610504i	$0.209033\pi$
$384$	1.00000	0.0510310
$385$	15.0000	0.764471
$386$	−16.0000	−0.814379
$387$	1.00000	0.0508329
$388$	−2.00000	−0.101535
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	0	0
$391$	−3.00000	−0.151717
$392$	1.00000	0.0505076
$393$	7.00000	0.353103
$394$	−2.00000	−0.100759
$395$	0	0
$396$	5.00000	0.251259
$397$	12.0000	0.602263	0.301131	−	0.953583i	$-0.402636\pi$
$397$	12.0000	0.602263	0.301131	+	0.953583i	$0.402636\pi$
$398$	5.00000	0.250627
$399$	1.00000	0.0500626
$400$	4.00000	0.200000
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	8.00000	0.399004
$403$	0	0
$404$	18.0000	0.895533
$405$	−3.00000	−0.149071
$406$	−5.00000	−0.248146
$407$	35.0000	1.73489
$408$	−3.00000	−0.148522
$409$	19.0000	0.939490	0.469745	−	0.882802i	$-0.344346\pi$
$409$	19.0000	0.939490	0.469745	+	0.882802i	$0.344346\pi$
$410$	0	0
$411$	−3.00000	−0.147979
$412$	11.0000	0.541931
$413$	14.0000	0.688895
$414$	1.00000	0.0491473
$415$	−18.0000	−0.883585
$416$	0	0
$417$	20.0000	0.979404
$418$	−5.00000	−0.244558
$419$	−35.0000	−1.70986	−0.854931	−	0.518742i	$-0.826401\pi$
$419$	−35.0000	−1.70986	−0.854931	+	0.518742i	$0.826401\pi$
$420$	3.00000	0.146385
$421$	−30.0000	−1.46211	−0.731055	−	0.682318i	$-0.760972\pi$
$421$	−30.0000	−1.46211	−0.731055	+	0.682318i	$0.760972\pi$
$422$	−13.0000	−0.632830
$423$	−8.00000	−0.388973
$424$	14.0000	0.679900
$425$	−12.0000	−0.582086
$426$	−10.0000	−0.484502
$427$	3.00000	0.145180
$428$	18.0000	0.870063
$429$	0	0
$430$	−3.00000	−0.144673
$431$	−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
$433$	−4.00000	−0.192228	−0.0961139	−	0.995370i	$-0.530641\pi$
$433$	−4.00000	−0.192228	−0.0961139	+	0.995370i	$0.530641\pi$
$434$	0	0
$435$	−15.0000	−0.719195
$436$	9.00000	0.431022
$437$	−1.00000	−0.0478365
$438$	−11.0000	−0.525600
$439$	15.0000	0.715911	0.357955	−	0.933739i	$-0.383474\pi$
$439$	15.0000	0.715911	0.357955	+	0.933739i	$0.383474\pi$
$440$	−15.0000	−0.715097
$441$	1.00000	0.0476190
$442$	0	0
$443$	−6.00000	−0.285069	−0.142534	−	0.989790i	$-0.545525\pi$
$443$	−6.00000	−0.285069	−0.142534	+	0.989790i	$0.545525\pi$
$444$	7.00000	0.332205
$445$	−48.0000	−2.27542
$446$	16.0000	0.757622
$447$	−6.00000	−0.283790
$448$	−1.00000	−0.0472456
$449$	21.0000	0.991051	0.495526	−	0.868593i	$-0.334975\pi$
$449$	21.0000	0.991051	0.495526	+	0.868593i	$0.334975\pi$
$450$	4.00000	0.188562
$451$	0	0
$452$	−6.00000	−0.282216
$453$	−15.0000	−0.704761
$454$	8.00000	0.375459
$455$	0	0
$456$	−1.00000	−0.0468293
$457$	8.00000	0.374224	0.187112	−	0.982339i	$-0.440087\pi$
$457$	8.00000	0.374224	0.187112	+	0.982339i	$0.440087\pi$
$458$	26.0000	1.21490
$459$	−3.00000	−0.140028
$460$	−3.00000	−0.139876
$461$	15.0000	0.698620	0.349310	−	0.937007i	$-0.386416\pi$
$461$	15.0000	0.698620	0.349310	+	0.937007i	$0.386416\pi$
$462$	−5.00000	−0.232621
$463$	−11.0000	−0.511213	−0.255607	−	0.966781i	$-0.582275\pi$
$463$	−11.0000	−0.511213	−0.255607	+	0.966781i	$0.582275\pi$
$464$	5.00000	0.232119
$465$	0	0
$466$	−14.0000	−0.648537
$467$	7.00000	0.323921	0.161961	−	0.986797i	$-0.448218\pi$
$467$	7.00000	0.323921	0.161961	+	0.986797i	$0.448218\pi$
$468$	0	0
$469$	−8.00000	−0.369406
$470$	24.0000	1.10704
$471$	13.0000	0.599008
$472$	−14.0000	−0.644402
$473$	5.00000	0.229900
$474$	0	0
$475$	−4.00000	−0.183533
$476$	3.00000	0.137505
$477$	14.0000	0.641016
$478$	4.00000	0.182956
$479$	21.0000	0.959514	0.479757	−	0.877401i	$-0.340725\pi$
$479$	21.0000	0.959514	0.479757	+	0.877401i	$0.340725\pi$
$480$	−3.00000	−0.136931
$481$	0	0
$482$	−10.0000	−0.455488
$483$	−1.00000	−0.0455016
$484$	14.0000	0.636364
$485$	6.00000	0.272446
$486$	1.00000	0.0453609
$487$	−32.0000	−1.45006	−0.725029	−	0.688718i	$-0.758174\pi$
$487$	−32.0000	−1.45006	−0.725029	+	0.688718i	$0.758174\pi$
$488$	−3.00000	−0.135804
$489$	4.00000	0.180886
$490$	−3.00000	−0.135526
$491$	28.0000	1.26362	0.631811	−	0.775122i	$-0.282312\pi$
$491$	28.0000	1.26362	0.631811	+	0.775122i	$0.282312\pi$
$492$	0	0
$493$	−15.0000	−0.675566
$494$	0	0
$495$	−15.0000	−0.674200
$496$	0	0
$497$	10.0000	0.448561
$498$	6.00000	0.268866
$499$	14.0000	0.626726	0.313363	−	0.949633i	$-0.398544\pi$
$499$	14.0000	0.626726	0.313363	+	0.949633i	$0.398544\pi$
$500$	3.00000	0.134164
$501$	7.00000	0.312737
$502$	−17.0000	−0.758747
$503$	−6.00000	−0.267527	−0.133763	−	0.991013i	$-0.542706\pi$
$503$	−6.00000	−0.267527	−0.133763	+	0.991013i	$0.542706\pi$
$504$	−1.00000	−0.0445435
$505$	−54.0000	−2.40297
$506$	5.00000	0.222277
$507$	0	0
$508$	12.0000	0.532414
$509$	−1.00000	−0.0443242	−0.0221621	−	0.999754i	$-0.507055\pi$
$509$	−1.00000	−0.0443242	−0.0221621	+	0.999754i	$0.507055\pi$
$510$	9.00000	0.398527
$511$	11.0000	0.486611
$512$	1.00000	0.0441942
$513$	−1.00000	−0.0441511
$514$	−18.0000	−0.793946
$515$	−33.0000	−1.45415
$516$	1.00000	0.0440225
$517$	−40.0000	−1.75920
$518$	−7.00000	−0.307562
$519$	26.0000	1.14127
$520$	0	0
$521$	27.0000	1.18289	0.591446	−	0.806345i	$-0.298557\pi$
$521$	27.0000	1.18289	0.591446	+	0.806345i	$0.298557\pi$
$522$	5.00000	0.218844
$523$	−16.0000	−0.699631	−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000	−0.699631	−0.349816	+	0.936819i	$0.613756\pi$
$524$	7.00000	0.305796
$525$	−4.00000	−0.174574
$526$	24.0000	1.04645
$527$	0	0
$528$	5.00000	0.217597
$529$	−22.0000	−0.956522
$530$	−42.0000	−1.82436
$531$	−14.0000	−0.607548
$532$	1.00000	0.0433555
$533$	0	0
$534$	16.0000	0.692388
$535$	−54.0000	−2.33462
$536$	8.00000	0.345547
$537$	−10.0000	−0.431532
$538$	0	0
$539$	5.00000	0.215365
$540$	−3.00000	−0.129099
$541$	−15.0000	−0.644900	−0.322450	−	0.946586i	$-0.604506\pi$
$541$	−15.0000	−0.644900	−0.322450	+	0.946586i	$0.604506\pi$
$542$	0	0
$543$	−2.00000	−0.0858282
$544$	−3.00000	−0.128624
$545$	−27.0000	−1.15655
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	−3.00000	−0.128154
$549$	−3.00000	−0.128037
$550$	20.0000	0.852803
$551$	−5.00000	−0.213007
$552$	1.00000	0.0425628
$553$	0	0
$554$	−28.0000	−1.18961
$555$	−21.0000	−0.891400
$556$	20.0000	0.848189
$557$	22.0000	0.932170	0.466085	−	0.884740i	$-0.345664\pi$
$557$	22.0000	0.932170	0.466085	+	0.884740i	$0.345664\pi$
$558$	0	0
$559$	0	0
$560$	3.00000	0.126773
$561$	−15.0000	−0.633300
$562$	10.0000	0.421825
$563$	−9.00000	−0.379305	−0.189652	−	0.981851i	$-0.560736\pi$
$563$	−9.00000	−0.379305	−0.189652	+	0.981851i	$0.560736\pi$
$564$	−8.00000	−0.336861
$565$	18.0000	0.757266
$566$	−14.0000	−0.588464
$567$	−1.00000	−0.0419961
$568$	−10.0000	−0.419591
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	3.00000	0.125656
$571$	−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
$573$	17.0000	0.710185
$574$	0	0
$575$	4.00000	0.166812
$576$	1.00000	0.0416667
$577$	38.0000	1.58196	0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000	1.58196	0.790980	+	0.611842i	$0.209571\pi$
$578$	−8.00000	−0.332756
$579$	−16.0000	−0.664937
$580$	−15.0000	−0.622841
$581$	−6.00000	−0.248922
$582$	−2.00000	−0.0829027
$583$	70.0000	2.89910
$584$	−11.0000	−0.455183
$585$	0	0
$586$	−6.00000	−0.247858
$587$	8.00000	0.330195	0.165098	−	0.986277i	$-0.447206\pi$
$587$	8.00000	0.330195	0.165098	+	0.986277i	$0.447206\pi$
$588$	1.00000	0.0412393
$589$	0	0
$590$	42.0000	1.72911
$591$	−2.00000	−0.0822690
$592$	7.00000	0.287698
$593$	24.0000	0.985562	0.492781	−	0.870153i	$-0.335980\pi$
$593$	24.0000	0.985562	0.492781	+	0.870153i	$0.335980\pi$
$594$	5.00000	0.205152
$595$	−9.00000	−0.368964
$596$	−6.00000	−0.245770
$597$	5.00000	0.204636
$598$	0	0
$599$	−45.0000	−1.83865	−0.919325	−	0.393499i	$-0.871265\pi$
$599$	−45.0000	−1.83865	−0.919325	+	0.393499i	$0.871265\pi$
$600$	4.00000	0.163299
$601$	−48.0000	−1.95796	−0.978980	−	0.203954i	$-0.934621\pi$
$601$	−48.0000	−1.95796	−0.978980	+	0.203954i	$0.934621\pi$
$602$	−1.00000	−0.0407570
$603$	8.00000	0.325785
$604$	−15.0000	−0.610341
$605$	−42.0000	−1.70754
$606$	18.0000	0.731200
$607$	23.0000	0.933541	0.466771	−	0.884378i	$-0.345417\pi$
$607$	23.0000	0.933541	0.466771	+	0.884378i	$0.345417\pi$
$608$	−1.00000	−0.0405554
$609$	−5.00000	−0.202610
$610$	9.00000	0.364399
$611$	0	0
$612$	−3.00000	−0.121268
$613$	−39.0000	−1.57520	−0.787598	−	0.616190i	$-0.788675\pi$
$613$	−39.0000	−1.57520	−0.787598	+	0.616190i	$0.788675\pi$
$614$	−8.00000	−0.322854
$615$	0	0
$616$	−5.00000	−0.201456
$617$	−17.0000	−0.684394	−0.342197	−	0.939628i	$-0.611171\pi$
$617$	−17.0000	−0.684394	−0.342197	+	0.939628i	$0.611171\pi$
$618$	11.0000	0.442485
$619$	11.0000	0.442127	0.221064	−	0.975259i	$-0.429047\pi$
$619$	11.0000	0.442127	0.221064	+	0.975259i	$0.429047\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	8.00000	0.320771
$623$	−16.0000	−0.641026
$624$	0	0
$625$	−29.0000	−1.16000
$626$	−6.00000	−0.239808
$627$	−5.00000	−0.199681
$628$	13.0000	0.518756
$629$	−21.0000	−0.837325
$630$	3.00000	0.119523
$631$	−35.0000	−1.39333	−0.696664	−	0.717398i	$-0.745333\pi$
$631$	−35.0000	−1.39333	−0.696664	+	0.717398i	$0.745333\pi$
$632$	0	0
$633$	−13.0000	−0.516704
$634$	8.00000	0.317721
$635$	−36.0000	−1.42862
$636$	14.0000	0.555136
$637$	0	0
$638$	25.0000	0.989759
$639$	−10.0000	−0.395594
$640$	−3.00000	−0.118585
$641$	8.00000	0.315981	0.157991	−	0.987441i	$-0.449498\pi$
$641$	8.00000	0.315981	0.157991	+	0.987441i	$0.449498\pi$
$642$	18.0000	0.710403
$643$	−19.0000	−0.749287	−0.374643	−	0.927169i	$-0.622235\pi$
$643$	−19.0000	−0.749287	−0.374643	+	0.927169i	$0.622235\pi$
$644$	−1.00000	−0.0394055
$645$	−3.00000	−0.118125
$646$	3.00000	0.118033
$647$	2.00000	0.0786281	0.0393141	−	0.999227i	$-0.487483\pi$
$647$	2.00000	0.0786281	0.0393141	+	0.999227i	$0.487483\pi$
$648$	1.00000	0.0392837
$649$	−70.0000	−2.74774
$650$	0	0
$651$	0	0
$652$	4.00000	0.156652
$653$	−1.00000	−0.0391330	−0.0195665	−	0.999809i	$-0.506229\pi$
$653$	−1.00000	−0.0391330	−0.0195665	+	0.999809i	$0.506229\pi$
$654$	9.00000	0.351928
$655$	−21.0000	−0.820538
$656$	0	0
$657$	−11.0000	−0.429151
$658$	8.00000	0.311872
$659$	−30.0000	−1.16863	−0.584317	−	0.811525i	$-0.698638\pi$
$659$	−30.0000	−1.16863	−0.584317	+	0.811525i	$0.698638\pi$
$660$	−15.0000	−0.583874
$661$	−50.0000	−1.94477	−0.972387	−	0.233373i	$-0.925024\pi$
$661$	−50.0000	−1.94477	−0.972387	+	0.233373i	$0.925024\pi$
$662$	20.0000	0.777322
$663$	0	0
$664$	6.00000	0.232845
$665$	−3.00000	−0.116335
$666$	7.00000	0.271244
$667$	5.00000	0.193601
$668$	7.00000	0.270838
$669$	16.0000	0.618596
$670$	−24.0000	−0.927201
$671$	−15.0000	−0.579069
$672$	−1.00000	−0.0385758
$673$	11.0000	0.424019	0.212009	−	0.977268i	$-0.431999\pi$
$673$	11.0000	0.424019	0.212009	+	0.977268i	$0.431999\pi$
$674$	−33.0000	−1.27111
$675$	4.00000	0.153960
$676$	0	0
$677$	8.00000	0.307465	0.153732	−	0.988113i	$-0.450871\pi$
$677$	8.00000	0.307465	0.153732	+	0.988113i	$0.450871\pi$
$678$	−6.00000	−0.230429
$679$	2.00000	0.0767530
$680$	9.00000	0.345134
$681$	8.00000	0.306561
$682$	0	0
$683$	−1.00000	−0.0382639	−0.0191320	−	0.999817i	$-0.506090\pi$
$683$	−1.00000	−0.0382639	−0.0191320	+	0.999817i	$0.506090\pi$
$684$	−1.00000	−0.0382360
$685$	9.00000	0.343872
$686$	−1.00000	−0.0381802
$687$	26.0000	0.991962
$688$	1.00000	0.0381246
$689$	0	0
$690$	−3.00000	−0.114208
$691$	−40.0000	−1.52167	−0.760836	−	0.648944i	$-0.775211\pi$
$691$	−40.0000	−1.52167	−0.760836	+	0.648944i	$0.775211\pi$
$692$	26.0000	0.988372
$693$	−5.00000	−0.189934
$694$	28.0000	1.06287
$695$	−60.0000	−2.27593
$696$	5.00000	0.189525
$697$	0	0
$698$	16.0000	0.605609
$699$	−14.0000	−0.529529
$700$	−4.00000	−0.151186
$701$	−2.00000	−0.0755390	−0.0377695	−	0.999286i	$-0.512025\pi$
$701$	−2.00000	−0.0755390	−0.0377695	+	0.999286i	$0.512025\pi$
$702$	0	0
$703$	−7.00000	−0.264010
$704$	5.00000	0.188445
$705$	24.0000	0.903892
$706$	−24.0000	−0.903252
$707$	−18.0000	−0.676960
$708$	−14.0000	−0.526152
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	30.0000	1.12588
$711$	0	0
$712$	16.0000	0.599625
$713$	0	0
$714$	3.00000	0.112272
$715$	0	0
$716$	−10.0000	−0.373718
$717$	4.00000	0.149383
$718$	−34.0000	−1.26887
$719$	20.0000	0.745874	0.372937	−	0.927857i	$-0.378351\pi$
$719$	20.0000	0.745874	0.372937	+	0.927857i	$0.378351\pi$
$720$	−3.00000	−0.111803
$721$	−11.0000	−0.409661
$722$	−18.0000	−0.669891
$723$	−10.0000	−0.371904
$724$	−2.00000	−0.0743294
$725$	20.0000	0.742781
$726$	14.0000	0.519589
$727$	7.00000	0.259616	0.129808	−	0.991539i	$-0.458564\pi$
$727$	7.00000	0.259616	0.129808	+	0.991539i	$0.458564\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	33.0000	1.22138
$731$	−3.00000	−0.110959
$732$	−3.00000	−0.110883
$733$	36.0000	1.32969	0.664845	−	0.746981i	$-0.268498\pi$
$733$	36.0000	1.32969	0.664845	+	0.746981i	$0.268498\pi$
$734$	−32.0000	−1.18114
$735$	−3.00000	−0.110657
$736$	1.00000	0.0368605
$737$	40.0000	1.47342
$738$	0	0
$739$	−24.0000	−0.882854	−0.441427	−	0.897297i	$-0.645528\pi$
$739$	−24.0000	−0.882854	−0.441427	+	0.897297i	$0.645528\pi$
$740$	−21.0000	−0.771975
$741$	0	0
$742$	−14.0000	−0.513956
$743$	−4.00000	−0.146746	−0.0733729	−	0.997305i	$-0.523376\pi$
$743$	−4.00000	−0.146746	−0.0733729	+	0.997305i	$0.523376\pi$
$744$	0	0
$745$	18.0000	0.659469
$746$	−26.0000	−0.951928
$747$	6.00000	0.219529
$748$	−15.0000	−0.548454
$749$	−18.0000	−0.657706
$750$	3.00000	0.109545
$751$	48.0000	1.75154	0.875772	−	0.482724i	$-0.160353\pi$
$751$	48.0000	1.75154	0.875772	+	0.482724i	$0.160353\pi$
$752$	−8.00000	−0.291730
$753$	−17.0000	−0.619514
$754$	0	0
$755$	45.0000	1.63772
$756$	−1.00000	−0.0363696
$757$	18.0000	0.654221	0.327111	−	0.944986i	$-0.393925\pi$
$757$	18.0000	0.654221	0.327111	+	0.944986i	$0.393925\pi$
$758$	4.00000	0.145287
$759$	5.00000	0.181489
$760$	3.00000	0.108821
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	12.0000	0.434714
$763$	−9.00000	−0.325822
$764$	17.0000	0.615038
$765$	9.00000	0.325396
$766$	31.0000	1.12008
$767$	0	0
$768$	1.00000	0.0360844
$769$	−1.00000	−0.0360609	−0.0180305	−	0.999837i	$-0.505740\pi$
$769$	−1.00000	−0.0360609	−0.0180305	+	0.999837i	$0.505740\pi$
$770$	15.0000	0.540562
$771$	−18.0000	−0.648254
$772$	−16.0000	−0.575853
$773$	−9.00000	−0.323708	−0.161854	−	0.986815i	$-0.551747\pi$
$773$	−9.00000	−0.323708	−0.161854	+	0.986815i	$0.551747\pi$
$774$	1.00000	0.0359443
$775$	0	0
$776$	−2.00000	−0.0717958
$777$	−7.00000	−0.251124
$778$	30.0000	1.07555
$779$	0	0
$780$	0	0
$781$	−50.0000	−1.78914
$782$	−3.00000	−0.107280
$783$	5.00000	0.178685
$784$	1.00000	0.0357143
$785$	−39.0000	−1.39197
$786$	7.00000	0.249682
$787$	−13.0000	−0.463400	−0.231700	−	0.972787i	$-0.574429\pi$
$787$	−13.0000	−0.463400	−0.231700	+	0.972787i	$0.574429\pi$
$788$	−2.00000	−0.0712470
$789$	24.0000	0.854423
$790$	0	0
$791$	6.00000	0.213335
$792$	5.00000	0.177667
$793$	0	0
$794$	12.0000	0.425864
$795$	−42.0000	−1.48959
$796$	5.00000	0.177220
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	1.00000	0.0353996
$799$	24.0000	0.849059
$800$	4.00000	0.141421
$801$	16.0000	0.565332
$802$	30.0000	1.05934
$803$	−55.0000	−1.94091
$804$	8.00000	0.282138
$805$	3.00000	0.105736
$806$	0	0
$807$	0	0
$808$	18.0000	0.633238
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	−3.00000	−0.105409
$811$	35.0000	1.22902	0.614508	−	0.788911i	$-0.289355\pi$
$811$	35.0000	1.22902	0.614508	+	0.788911i	$0.289355\pi$
$812$	−5.00000	−0.175466
$813$	0	0
$814$	35.0000	1.22675
$815$	−12.0000	−0.420342
$816$	−3.00000	−0.105021
$817$	−1.00000	−0.0349856
$818$	19.0000	0.664319
$819$	0	0
$820$	0	0
$821$	−20.0000	−0.698005	−0.349002	−	0.937122i	$-0.613479\pi$
$821$	−20.0000	−0.698005	−0.349002	+	0.937122i	$0.613479\pi$
$822$	−3.00000	−0.104637
$823$	−44.0000	−1.53374	−0.766872	−	0.641800i	$-0.778188\pi$
$823$	−44.0000	−1.53374	−0.766872	+	0.641800i	$0.778188\pi$
$824$	11.0000	0.383203
$825$	20.0000	0.696311
$826$	14.0000	0.487122
$827$	−23.0000	−0.799788	−0.399894	−	0.916561i	$-0.630953\pi$
$827$	−23.0000	−0.799788	−0.399894	+	0.916561i	$0.630953\pi$
$828$	1.00000	0.0347524
$829$	25.0000	0.868286	0.434143	−	0.900844i	$-0.357051\pi$
$829$	25.0000	0.868286	0.434143	+	0.900844i	$0.357051\pi$
$830$	−18.0000	−0.624789
$831$	−28.0000	−0.971309
$832$	0	0
$833$	−3.00000	−0.103944
$834$	20.0000	0.692543
$835$	−21.0000	−0.726735
$836$	−5.00000	−0.172929
$837$	0	0
$838$	−35.0000	−1.20905
$839$	36.0000	1.24286	0.621429	−	0.783470i	$-0.286552\pi$
$839$	36.0000	1.24286	0.621429	+	0.783470i	$0.286552\pi$
$840$	3.00000	0.103510
$841$	−4.00000	−0.137931
$842$	−30.0000	−1.03387
$843$	10.0000	0.344418
$844$	−13.0000	−0.447478
$845$	0	0
$846$	−8.00000	−0.275046
$847$	−14.0000	−0.481046
$848$	14.0000	0.480762
$849$	−14.0000	−0.480479
$850$	−12.0000	−0.411597
$851$	7.00000	0.239957
$852$	−10.0000	−0.342594
$853$	−16.0000	−0.547830	−0.273915	−	0.961754i	$-0.588319\pi$
$853$	−16.0000	−0.547830	−0.273915	+	0.961754i	$0.588319\pi$
$854$	3.00000	0.102658
$855$	3.00000	0.102598
$856$	18.0000	0.615227
$857$	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
$859$	−50.0000	−1.70598	−0.852989	−	0.521929i	$-0.825213\pi$
$859$	−50.0000	−1.70598	−0.852989	+	0.521929i	$0.825213\pi$
$860$	−3.00000	−0.102299
$861$	0	0
$862$	−30.0000	−1.02180
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	1.00000	0.0340207
$865$	−78.0000	−2.65208
$866$	−4.00000	−0.135926
$867$	−8.00000	−0.271694
$868$	0	0
$869$	0	0
$870$	−15.0000	−0.508548
$871$	0	0
$872$	9.00000	0.304778
$873$	−2.00000	−0.0676897
$874$	−1.00000	−0.0338255
$875$	−3.00000	−0.101419
$876$	−11.0000	−0.371656
$877$	38.0000	1.28317	0.641584	−	0.767052i	$-0.278277\pi$
$877$	38.0000	1.28317	0.641584	+	0.767052i	$0.278277\pi$
$878$	15.0000	0.506225
$879$	−6.00000	−0.202375
$880$	−15.0000	−0.505650
$881$	−7.00000	−0.235836	−0.117918	−	0.993023i	$-0.537622\pi$
$881$	−7.00000	−0.235836	−0.117918	+	0.993023i	$0.537622\pi$
$882$	1.00000	0.0336718
$883$	41.0000	1.37976	0.689880	−	0.723924i	$-0.257663\pi$
$883$	41.0000	1.37976	0.689880	+	0.723924i	$0.257663\pi$
$884$	0	0
$885$	42.0000	1.41181
$886$	−6.00000	−0.201574
$887$	−2.00000	−0.0671534	−0.0335767	−	0.999436i	$-0.510690\pi$
$887$	−2.00000	−0.0671534	−0.0335767	+	0.999436i	$0.510690\pi$
$888$	7.00000	0.234905
$889$	−12.0000	−0.402467
$890$	−48.0000	−1.60896
$891$	5.00000	0.167506
$892$	16.0000	0.535720
$893$	8.00000	0.267710
$894$	−6.00000	−0.200670
$895$	30.0000	1.00279
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	21.0000	0.700779
$899$	0	0
$900$	4.00000	0.133333
$901$	−42.0000	−1.39922
$902$	0	0
$903$	−1.00000	−0.0332779
$904$	−6.00000	−0.199557
$905$	6.00000	0.199447
$906$	−15.0000	−0.498342
$907$	12.0000	0.398453	0.199227	−	0.979953i	$-0.436157\pi$
$907$	12.0000	0.398453	0.199227	+	0.979953i	$0.436157\pi$
$908$	8.00000	0.265489
$909$	18.0000	0.597022
$910$	0	0
$911$	47.0000	1.55718	0.778590	−	0.627533i	$-0.215935\pi$
$911$	47.0000	1.55718	0.778590	+	0.627533i	$0.215935\pi$
$912$	−1.00000	−0.0331133
$913$	30.0000	0.992855
$914$	8.00000	0.264616
$915$	9.00000	0.297531
$916$	26.0000	0.859064
$917$	−7.00000	−0.231160
$918$	−3.00000	−0.0990148
$919$	10.0000	0.329870	0.164935	−	0.986304i	$-0.447259\pi$
$919$	10.0000	0.329870	0.164935	+	0.986304i	$0.447259\pi$
$920$	−3.00000	−0.0989071
$921$	−8.00000	−0.263609
$922$	15.0000	0.493999
$923$	0	0
$924$	−5.00000	−0.164488
$925$	28.0000	0.920634
$926$	−11.0000	−0.361482
$927$	11.0000	0.361287
$928$	5.00000	0.164133
$929$	54.0000	1.77168	0.885841	−	0.463988i	$-0.153582\pi$
$929$	54.0000	1.77168	0.885841	+	0.463988i	$0.153582\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	−14.0000	−0.458585
$933$	8.00000	0.261908
$934$	7.00000	0.229047
$935$	45.0000	1.47166
$936$	0	0
$937$	−32.0000	−1.04539	−0.522697	−	0.852518i	$-0.675074\pi$
$937$	−32.0000	−1.04539	−0.522697	+	0.852518i	$0.675074\pi$
$938$	−8.00000	−0.261209
$939$	−6.00000	−0.195803
$940$	24.0000	0.782794
$941$	−30.0000	−0.977972	−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000	−0.977972	−0.488986	+	0.872292i	$0.662633\pi$
$942$	13.0000	0.423563
$943$	0	0
$944$	−14.0000	−0.455661
$945$	3.00000	0.0975900
$946$	5.00000	0.162564
$947$	57.0000	1.85225	0.926126	−	0.377215i	$-0.123118\pi$
$947$	57.0000	1.85225	0.926126	+	0.377215i	$0.123118\pi$
$948$	0	0
$949$	0	0
$950$	−4.00000	−0.129777
$951$	8.00000	0.259418
$952$	3.00000	0.0972306
$953$	−54.0000	−1.74923	−0.874616	−	0.484817i	$-0.838886\pi$
$953$	−54.0000	−1.74923	−0.874616	+	0.484817i	$0.838886\pi$
$954$	14.0000	0.453267
$955$	−51.0000	−1.65032
$956$	4.00000	0.129369
$957$	25.0000	0.808135
$958$	21.0000	0.678479
$959$	3.00000	0.0968751
$960$	−3.00000	−0.0968246
$961$	−31.0000	−1.00000
$962$	0	0
$963$	18.0000	0.580042
$964$	−10.0000	−0.322078
$965$	48.0000	1.54517
$966$	−1.00000	−0.0321745
$967$	−17.0000	−0.546683	−0.273342	−	0.961917i	$-0.588129\pi$
$967$	−17.0000	−0.546683	−0.273342	+	0.961917i	$0.588129\pi$
$968$	14.0000	0.449977
$969$	3.00000	0.0963739
$970$	6.00000	0.192648
$971$	−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
$974$	−32.0000	−1.02535
$975$	0	0
$976$	−3.00000	−0.0960277
$977$	3.00000	0.0959785	0.0479893	−	0.998848i	$-0.484719\pi$
$977$	3.00000	0.0959785	0.0479893	+	0.998848i	$0.484719\pi$
$978$	4.00000	0.127906
$979$	80.0000	2.55681
$980$	−3.00000	−0.0958315
$981$	9.00000	0.287348
$982$	28.0000	0.893516
$983$	−11.0000	−0.350846	−0.175423	−	0.984493i	$-0.556129\pi$
$983$	−11.0000	−0.350846	−0.175423	+	0.984493i	$0.556129\pi$
$984$	0	0
$985$	6.00000	0.191176
$986$	−15.0000	−0.477697
$987$	8.00000	0.254643
$988$	0	0
$989$	1.00000	0.0317982
$990$	−15.0000	−0.476731
$991$	−28.0000	−0.889449	−0.444725	−	0.895667i	$-0.646698\pi$
$991$	−28.0000	−0.889449	−0.444725	+	0.895667i	$0.646698\pi$
$992$	0	0
$993$	20.0000	0.634681
$994$	10.0000	0.317181
$995$	−15.0000	−0.475532
$996$	6.00000	0.190117
$997$	38.0000	1.20347	0.601736	−	0.798695i	$-0.294476\pi$
$997$	38.0000	1.20347	0.601736	+	0.798695i	$0.294476\pi$
$998$	14.0000	0.443162
$999$	7.00000	0.221470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7098.2.a.x.1.1		1
13.5	odd	4		546.2.c.d.337.1	✓	2
13.8	odd	4		546.2.c.d.337.2	yes	2
13.12	even	2		7098.2.a.p.1.1		1
39.5	even	4		1638.2.c.g.883.2		2
39.8	even	4		1638.2.c.g.883.1		2
52.31	even	4		4368.2.h.b.337.2		2
52.47	even	4		4368.2.h.b.337.1		2
91.34	even	4		3822.2.c.a.883.2		2
91.83	even	4		3822.2.c.a.883.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.c.d.337.1	✓	2	13.5	odd	4
546.2.c.d.337.2	yes	2	13.8	odd	4
1638.2.c.g.883.1		2	39.8	even	4
1638.2.c.g.883.2		2	39.5	even	4
3822.2.c.a.883.1		2	91.83	even	4
3822.2.c.a.883.2		2	91.34	even	4
4368.2.h.b.337.1		2	52.47	even	4
4368.2.h.b.337.2		2	52.31	even	4
7098.2.a.p.1.1		1	13.12	even	2
7098.2.a.x.1.1		1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7098.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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