LMFDB - Embedded newform 7942.2.a.y.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7942,2,Mod(1,7942)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7942, 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("7942.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7942 = 2 \cdot 11 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7942.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:	$63.4171892853$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 418)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.44949$ of defining polynomial
Character	$\chi$	$=$	7942.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.44949 q^{3} +1.00000 q^{4} -2.44949 q^{6} +2.44949 q^{7} +1.00000 q^{8} +3.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.44949 q^{3} +1.00000 q^{4} -2.44949 q^{6} +2.44949 q^{7} +1.00000 q^{8} +3.00000 q^{9} -1.00000 q^{11} -2.44949 q^{12} +5.89898 q^{13} +2.44949 q^{14} +1.00000 q^{16} +0.449490 q^{17} +3.00000 q^{18} -6.00000 q^{21} -1.00000 q^{22} -8.89898 q^{23} -2.44949 q^{24} -5.00000 q^{25} +5.89898 q^{26} +2.44949 q^{28} -7.89898 q^{29} -2.00000 q^{31} +1.00000 q^{32} +2.44949 q^{33} +0.449490 q^{34} +3.00000 q^{36} -3.55051 q^{37} -14.4495 q^{39} +12.4495 q^{41} -6.00000 q^{42} +4.55051 q^{43} -1.00000 q^{44} -8.89898 q^{46} -11.4495 q^{47} -2.44949 q^{48} -1.00000 q^{49} -5.00000 q^{50} -1.10102 q^{51} +5.89898 q^{52} -11.7980 q^{53} +2.44949 q^{56} -7.89898 q^{58} +6.89898 q^{59} -1.89898 q^{61} -2.00000 q^{62} +7.34847 q^{63} +1.00000 q^{64} +2.44949 q^{66} -14.4495 q^{67} +0.449490 q^{68} +21.7980 q^{69} -2.55051 q^{71} +3.00000 q^{72} -9.55051 q^{73} -3.55051 q^{74} +12.2474 q^{75} -2.44949 q^{77} -14.4495 q^{78} +0.449490 q^{79} -9.00000 q^{81} +12.4495 q^{82} -0.550510 q^{83} -6.00000 q^{84} +4.55051 q^{86} +19.3485 q^{87} -1.00000 q^{88} +9.00000 q^{89} +14.4495 q^{91} -8.89898 q^{92} +4.89898 q^{93} -11.4495 q^{94} -2.44949 q^{96} +5.89898 q^{97} -1.00000 q^{98} -3.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 6 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 + 6 * q^9	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 6 q^{9} - 2 q^{11} + 2 q^{13} + 2 q^{16} - 4 q^{17} + 6 q^{18} - 12 q^{21} - 2 q^{22} - 8 q^{23} - 10 q^{25} + 2 q^{26} - 6 q^{29} - 4 q^{31} + 2 q^{32} - 4 q^{34} + 6 q^{36} - 12 q^{37} - 24 q^{39} + 20 q^{41} - 12 q^{42} + 14 q^{43} - 2 q^{44} - 8 q^{46} - 18 q^{47} - 2 q^{49} - 10 q^{50} - 12 q^{51} + 2 q^{52} - 4 q^{53} - 6 q^{58} + 4 q^{59} + 6 q^{61} - 4 q^{62} + 2 q^{64} - 24 q^{67} - 4 q^{68} + 24 q^{69} - 10 q^{71} + 6 q^{72} - 24 q^{73} - 12 q^{74} - 24 q^{78} - 4 q^{79} - 18 q^{81} + 20 q^{82} - 6 q^{83} - 12 q^{84} + 14 q^{86} + 24 q^{87} - 2 q^{88} + 18 q^{89} + 24 q^{91} - 8 q^{92} - 18 q^{94} + 2 q^{97} - 2 q^{98} - 6 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 + 6 * q^9 - 2 * q^11 + 2 * q^13 + 2 * q^16 - 4 * q^17 + 6 * q^18 - 12 * q^21 - 2 * q^22 - 8 * q^23 - 10 * q^25 + 2 * q^26 - 6 * q^29 - 4 * q^31 + 2 * q^32 - 4 * q^34 + 6 * q^36 - 12 * q^37 - 24 * q^39 + 20 * q^41 - 12 * q^42 + 14 * q^43 - 2 * q^44 - 8 * q^46 - 18 * q^47 - 2 * q^49 - 10 * q^50 - 12 * q^51 + 2 * q^52 - 4 * q^53 - 6 * q^58 + 4 * q^59 + 6 * q^61 - 4 * q^62 + 2 * q^64 - 24 * q^67 - 4 * q^68 + 24 * q^69 - 10 * q^71 + 6 * q^72 - 24 * q^73 - 12 * q^74 - 24 * q^78 - 4 * q^79 - 18 * q^81 + 20 * q^82 - 6 * q^83 - 12 * q^84 + 14 * q^86 + 24 * q^87 - 2 * q^88 + 18 * q^89 + 24 * q^91 - 8 * q^92 - 18 * q^94 + 2 * q^97 - 2 * q^98 - 6 * q^99

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
$3$	−2.44949	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$3$	−2.44949	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$4$	1.00000	0.500000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	−2.44949	−1.00000
$7$	2.44949	0.925820	0.462910	−	0.886405i	$-0.346805\pi$
$7$	2.44949	0.925820	0.462910	+	0.886405i	$0.346805\pi$
$8$	1.00000	0.353553
$9$	3.00000	1.00000
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−2.44949	−0.707107
$13$	5.89898	1.63608	0.818041	−	0.575160i	$-0.195060\pi$
$13$	5.89898	1.63608	0.818041	+	0.575160i	$0.195060\pi$
$14$	2.44949	0.654654
$15$	0	0
$16$	1.00000	0.250000
$17$	0.449490	0.109017	0.0545086	−	0.998513i	$-0.482641\pi$
$17$	0.449490	0.109017	0.0545086	+	0.998513i	$0.482641\pi$
$18$	3.00000	0.707107
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−6.00000	−1.30931
$22$	−1.00000	−0.213201
$23$	−8.89898	−1.85557	−0.927783	−	0.373121i	$-0.878288\pi$
$23$	−8.89898	−1.85557	−0.927783	+	0.373121i	$0.878288\pi$
$24$	−2.44949	−0.500000
$25$	−5.00000	−1.00000
$26$	5.89898	1.15689
$27$	0	0
$28$	2.44949	0.462910
$29$	−7.89898	−1.46680	−0.733402	−	0.679795i	$-0.762069\pi$
$29$	−7.89898	−1.46680	−0.733402	+	0.679795i	$0.762069\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	1.00000	0.176777
$33$	2.44949	0.426401
$34$	0.449490	0.0770869
$35$	0	0
$36$	3.00000	0.500000
$37$	−3.55051	−0.583700	−0.291850	−	0.956464i	$-0.594271\pi$
$37$	−3.55051	−0.583700	−0.291850	+	0.956464i	$0.594271\pi$
$38$	0	0
$39$	−14.4495	−2.31377
$40$	0	0
$41$	12.4495	1.94428	0.972142	−	0.234393i	$-0.0753104\pi$
$41$	12.4495	1.94428	0.972142	+	0.234393i	$0.0753104\pi$
$42$	−6.00000	−0.925820
$43$	4.55051	0.693946	0.346973	−	0.937875i	$-0.387209\pi$
$43$	4.55051	0.693946	0.346973	+	0.937875i	$0.387209\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−8.89898	−1.31208
$47$	−11.4495	−1.67008	−0.835040	−	0.550189i	$-0.814555\pi$
$47$	−11.4495	−1.67008	−0.835040	+	0.550189i	$0.814555\pi$
$48$	−2.44949	−0.353553
$49$	−1.00000	−0.142857
$50$	−5.00000	−0.707107
$51$	−1.10102	−0.154174
$52$	5.89898	0.818041
$53$	−11.7980	−1.62057	−0.810287	−	0.586033i	$-0.800689\pi$
$53$	−11.7980	−1.62057	−0.810287	+	0.586033i	$0.800689\pi$
$54$	0	0
$55$	0	0
$56$	2.44949	0.327327
$57$	0	0
$58$	−7.89898	−1.03719
$59$	6.89898	0.898171	0.449085	−	0.893489i	$-0.351750\pi$
$59$	6.89898	0.898171	0.449085	+	0.893489i	$0.351750\pi$
$60$	0	0
$61$	−1.89898	−0.243139	−0.121570	−	0.992583i	$-0.538793\pi$
$61$	−1.89898	−0.243139	−0.121570	+	0.992583i	$0.538793\pi$
$62$	−2.00000	−0.254000
$63$	7.34847	0.925820
$64$	1.00000	0.125000
$65$	0	0
$66$	2.44949	0.301511
$67$	−14.4495	−1.76529	−0.882643	−	0.470044i	$-0.844238\pi$
$67$	−14.4495	−1.76529	−0.882643	+	0.470044i	$0.844238\pi$
$68$	0.449490	0.0545086
$69$	21.7980	2.62417
$70$	0	0
$71$	−2.55051	−0.302690	−0.151345	−	0.988481i	$-0.548360\pi$
$71$	−2.55051	−0.302690	−0.151345	+	0.988481i	$0.548360\pi$
$72$	3.00000	0.353553
$73$	−9.55051	−1.11780	−0.558901	−	0.829234i	$-0.688777\pi$
$73$	−9.55051	−1.11780	−0.558901	+	0.829234i	$0.688777\pi$
$74$	−3.55051	−0.412738
$75$	12.2474	1.41421
$76$	0	0
$77$	−2.44949	−0.279145
$78$	−14.4495	−1.63608
$79$	0.449490	0.0505715	0.0252858	−	0.999680i	$-0.491950\pi$
$79$	0.449490	0.0505715	0.0252858	+	0.999680i	$0.491950\pi$
$80$	0	0
$81$	−9.00000	−1.00000
$82$	12.4495	1.37482
$83$	−0.550510	−0.0604264	−0.0302132	−	0.999543i	$-0.509619\pi$
$83$	−0.550510	−0.0604264	−0.0302132	+	0.999543i	$0.509619\pi$
$84$	−6.00000	−0.654654
$85$	0	0
$86$	4.55051	0.490694
$87$	19.3485	2.07437
$88$	−1.00000	−0.106600
$89$	9.00000	0.953998	0.476999	−	0.878904i	$-0.341725\pi$
$89$	9.00000	0.953998	0.476999	+	0.878904i	$0.341725\pi$
$90$	0	0
$91$	14.4495	1.51472
$92$	−8.89898	−0.927783
$93$	4.89898	0.508001
$94$	−11.4495	−1.18092
$95$	0	0
$96$	−2.44949	−0.250000
$97$	5.89898	0.598951	0.299475	−	0.954104i	$-0.403188\pi$
$97$	5.89898	0.598951	0.299475	+	0.954104i	$0.403188\pi$
$98$	−1.00000	−0.101015
$99$	−3.00000	−0.301511
$100$	−5.00000	−0.500000
$101$	5.00000	0.497519	0.248759	−	0.968565i	$-0.419977\pi$
$101$	5.00000	0.497519	0.248759	+	0.968565i	$0.419977\pi$
$102$	−1.10102	−0.109017
$103$	−6.34847	−0.625533	−0.312767	−	0.949830i	$-0.601256\pi$
$103$	−6.34847	−0.625533	−0.312767	+	0.949830i	$0.601256\pi$
$104$	5.89898	0.578443
$105$	0	0
$106$	−11.7980	−1.14592
$107$	4.34847	0.420382	0.210191	−	0.977660i	$-0.432591\pi$
$107$	4.34847	0.420382	0.210191	+	0.977660i	$0.432591\pi$
$108$	0	0
$109$	−6.10102	−0.584372	−0.292186	−	0.956362i	$-0.594383\pi$
$109$	−6.10102	−0.584372	−0.292186	+	0.956362i	$0.594383\pi$
$110$	0	0
$111$	8.69694	0.825477
$112$	2.44949	0.231455
$113$	13.6969	1.28850	0.644250	−	0.764815i	$-0.277170\pi$
$113$	13.6969	1.28850	0.644250	+	0.764815i	$0.277170\pi$
$114$	0	0
$115$	0	0
$116$	−7.89898	−0.733402
$117$	17.6969	1.63608
$118$	6.89898	0.635103
$119$	1.10102	0.100930
$120$	0	0
$121$	1.00000	0.0909091
$122$	−1.89898	−0.171926
$123$	−30.4949	−2.74963
$124$	−2.00000	−0.179605
$125$	0	0
$126$	7.34847	0.654654
$127$	16.6969	1.48161	0.740807	−	0.671718i	$-0.234443\pi$
$127$	16.6969	1.48161	0.740807	+	0.671718i	$0.234443\pi$
$128$	1.00000	0.0883883
$129$	−11.1464	−0.981388
$130$	0	0
$131$	−10.3485	−0.904150	−0.452075	−	0.891980i	$-0.649316\pi$
$131$	−10.3485	−0.904150	−0.452075	+	0.891980i	$0.649316\pi$
$132$	2.44949	0.213201
$133$	0	0
$134$	−14.4495	−1.24825
$135$	0	0
$136$	0.449490	0.0385434
$137$	1.89898	0.162241	0.0811204	−	0.996704i	$-0.474150\pi$
$137$	1.89898	0.162241	0.0811204	+	0.996704i	$0.474150\pi$
$138$	21.7980	1.85557
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	28.0454	2.36185
$142$	−2.55051	−0.214034
$143$	−5.89898	−0.493297
$144$	3.00000	0.250000
$145$	0	0
$146$	−9.55051	−0.790406
$147$	2.44949	0.202031
$148$	−3.55051	−0.291850
$149$	−10.6969	−0.876327	−0.438164	−	0.898895i	$-0.644371\pi$
$149$	−10.6969	−0.876327	−0.438164	+	0.898895i	$0.644371\pi$
$150$	12.2474	1.00000
$151$	3.55051	0.288936	0.144468	−	0.989509i	$-0.453853\pi$
$151$	3.55051	0.288936	0.144468	+	0.989509i	$0.453853\pi$
$152$	0	0
$153$	1.34847	0.109017
$154$	−2.44949	−0.197386
$155$	0	0
$156$	−14.4495	−1.15689
$157$	−6.00000	−0.478852	−0.239426	−	0.970915i	$-0.576959\pi$
$157$	−6.00000	−0.478852	−0.239426	+	0.970915i	$0.576959\pi$
$158$	0.449490	0.0357595
$159$	28.8990	2.29184
$160$	0	0
$161$	−21.7980	−1.71792
$162$	−9.00000	−0.707107
$163$	−6.89898	−0.540370	−0.270185	−	0.962808i	$-0.587085\pi$
$163$	−6.89898	−0.540370	−0.270185	+	0.962808i	$0.587085\pi$
$164$	12.4495	0.972142
$165$	0	0
$166$	−0.550510	−0.0427279
$167$	−21.1464	−1.63636	−0.818180	−	0.574962i	$-0.805017\pi$
$167$	−21.1464	−1.63636	−0.818180	+	0.574962i	$0.805017\pi$
$168$	−6.00000	−0.462910
$169$	21.7980	1.67677
$170$	0	0
$171$	0	0
$172$	4.55051	0.346973
$173$	9.79796	0.744925	0.372463	−	0.928047i	$-0.378514\pi$
$173$	9.79796	0.744925	0.372463	+	0.928047i	$0.378514\pi$
$174$	19.3485	1.46680
$175$	−12.2474	−0.925820
$176$	−1.00000	−0.0753778
$177$	−16.8990	−1.27021
$178$	9.00000	0.674579
$179$	−9.79796	−0.732334	−0.366167	−	0.930549i	$-0.619330\pi$
$179$	−9.79796	−0.732334	−0.366167	+	0.930549i	$0.619330\pi$
$180$	0	0
$181$	20.2474	1.50498	0.752491	−	0.658603i	$-0.228852\pi$
$181$	20.2474	1.50498	0.752491	+	0.658603i	$0.228852\pi$
$182$	14.4495	1.07107
$183$	4.65153	0.343851
$184$	−8.89898	−0.656041
$185$	0	0
$186$	4.89898	0.359211
$187$	−0.449490	−0.0328699
$188$	−11.4495	−0.835040
$189$	0	0
$190$	0	0
$191$	−22.3485	−1.61708	−0.808539	−	0.588442i	$-0.799741\pi$
$191$	−22.3485	−1.61708	−0.808539	+	0.588442i	$0.799741\pi$
$192$	−2.44949	−0.176777
$193$	8.65153	0.622751	0.311375	−	0.950287i	$-0.399210\pi$
$193$	8.65153	0.622751	0.311375	+	0.950287i	$0.399210\pi$
$194$	5.89898	0.423522
$195$	0	0
$196$	−1.00000	−0.0714286
$197$	15.8990	1.13276	0.566378	−	0.824146i	$-0.308344\pi$
$197$	15.8990	1.13276	0.566378	+	0.824146i	$0.308344\pi$
$198$	−3.00000	−0.213201
$199$	5.65153	0.400626	0.200313	−	0.979732i	$-0.435804\pi$
$199$	5.65153	0.400626	0.200313	+	0.979732i	$0.435804\pi$
$200$	−5.00000	−0.353553
$201$	35.3939	2.49649
$202$	5.00000	0.351799
$203$	−19.3485	−1.35800
$204$	−1.10102	−0.0770869
$205$	0	0
$206$	−6.34847	−0.442319
$207$	−26.6969	−1.85557
$208$	5.89898	0.409021
$209$	0	0
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	−11.7980	−0.810287
$213$	6.24745	0.428068
$214$	4.34847	0.297255
$215$	0	0
$216$	0	0
$217$	−4.89898	−0.332564
$218$	−6.10102	−0.413213
$219$	23.3939	1.58081
$220$	0	0
$221$	2.65153	0.178361
$222$	8.69694	0.583700
$223$	23.2474	1.55676	0.778382	−	0.627791i	$-0.216041\pi$
$223$	23.2474	1.55676	0.778382	+	0.627791i	$0.216041\pi$
$224$	2.44949	0.163663
$225$	−15.0000	−1.00000
$226$	13.6969	0.911107
$227$	−23.7980	−1.57953	−0.789763	−	0.613412i	$-0.789796\pi$
$227$	−23.7980	−1.57953	−0.789763	+	0.613412i	$0.789796\pi$
$228$	0	0
$229$	−15.3485	−1.01426	−0.507128	−	0.861871i	$-0.669293\pi$
$229$	−15.3485	−1.01426	−0.507128	+	0.861871i	$0.669293\pi$
$230$	0	0
$231$	6.00000	0.394771
$232$	−7.89898	−0.518593
$233$	−5.55051	−0.363626	−0.181813	−	0.983333i	$-0.558197\pi$
$233$	−5.55051	−0.363626	−0.181813	+	0.983333i	$0.558197\pi$
$234$	17.6969	1.15689
$235$	0	0
$236$	6.89898	0.449085
$237$	−1.10102	−0.0715190
$238$	1.10102	0.0713686
$239$	2.20204	0.142438	0.0712191	−	0.997461i	$-0.477311\pi$
$239$	2.20204	0.142438	0.0712191	+	0.997461i	$0.477311\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	1.00000	0.0642824
$243$	22.0454	1.41421
$244$	−1.89898	−0.121570
$245$	0	0
$246$	−30.4949	−1.94428
$247$	0	0
$248$	−2.00000	−0.127000
$249$	1.34847	0.0854558
$250$	0	0
$251$	−0.247449	−0.0156188	−0.00780941	−	0.999970i	$-0.502486\pi$
$251$	−0.247449	−0.0156188	−0.00780941	+	0.999970i	$0.502486\pi$
$252$	7.34847	0.462910
$253$	8.89898	0.559474
$254$	16.6969	1.04766
$255$	0	0
$256$	1.00000	0.0625000
$257$	22.0000	1.37232	0.686161	−	0.727450i	$-0.259294\pi$
$257$	22.0000	1.37232	0.686161	+	0.727450i	$0.259294\pi$
$258$	−11.1464	−0.693946
$259$	−8.69694	−0.540401
$260$	0	0
$261$	−23.6969	−1.46680
$262$	−10.3485	−0.639331
$263$	−17.7980	−1.09747	−0.548735	−	0.835997i	$-0.684890\pi$
$263$	−17.7980	−1.09747	−0.548735	+	0.835997i	$0.684890\pi$
$264$	2.44949	0.150756
$265$	0	0
$266$	0	0
$267$	−22.0454	−1.34916
$268$	−14.4495	−0.882643
$269$	−15.5505	−0.948131	−0.474066	−	0.880489i	$-0.657214\pi$
$269$	−15.5505	−0.948131	−0.474066	+	0.880489i	$0.657214\pi$
$270$	0	0
$271$	−4.69694	−0.285319	−0.142659	−	0.989772i	$-0.545565\pi$
$271$	−4.69694	−0.285319	−0.142659	+	0.989772i	$0.545565\pi$
$272$	0.449490	0.0272543
$273$	−35.3939	−2.14213
$274$	1.89898	0.114722
$275$	5.00000	0.301511
$276$	21.7980	1.31208
$277$	17.4949	1.05117	0.525583	−	0.850742i	$-0.323847\pi$
$277$	17.4949	1.05117	0.525583	+	0.850742i	$0.323847\pi$
$278$	−4.00000	−0.239904
$279$	−6.00000	−0.359211
$280$	0	0
$281$	−17.1464	−1.02287	−0.511435	−	0.859322i	$-0.670886\pi$
$281$	−17.1464	−1.02287	−0.511435	+	0.859322i	$0.670886\pi$
$282$	28.0454	1.67008
$283$	12.8990	0.766765	0.383382	−	0.923590i	$-0.374759\pi$
$283$	12.8990	0.766765	0.383382	+	0.923590i	$0.374759\pi$
$284$	−2.55051	−0.151345
$285$	0	0
$286$	−5.89898	−0.348814
$287$	30.4949	1.80006
$288$	3.00000	0.176777
$289$	−16.7980	−0.988115
$290$	0	0
$291$	−14.4495	−0.847044
$292$	−9.55051	−0.558901
$293$	−1.89898	−0.110940	−0.0554698	−	0.998460i	$-0.517666\pi$
$293$	−1.89898	−0.110940	−0.0554698	+	0.998460i	$0.517666\pi$
$294$	2.44949	0.142857
$295$	0	0
$296$	−3.55051	−0.206369
$297$	0	0
$298$	−10.6969	−0.619657
$299$	−52.4949	−3.03586
$300$	12.2474	0.707107
$301$	11.1464	0.642469
$302$	3.55051	0.204309
$303$	−12.2474	−0.703598
$304$	0	0
$305$	0	0
$306$	1.34847	0.0770869
$307$	−5.24745	−0.299488	−0.149744	−	0.988725i	$-0.547845\pi$
$307$	−5.24745	−0.299488	−0.149744	+	0.988725i	$0.547845\pi$
$308$	−2.44949	−0.139573
$309$	15.5505	0.884638
$310$	0	0
$311$	−17.2474	−0.978013	−0.489007	−	0.872280i	$-0.662641\pi$
$311$	−17.2474	−0.978013	−0.489007	+	0.872280i	$0.662641\pi$
$312$	−14.4495	−0.818041
$313$	9.69694	0.548103	0.274052	−	0.961715i	$-0.411636\pi$
$313$	9.69694	0.548103	0.274052	+	0.961715i	$0.411636\pi$
$314$	−6.00000	−0.338600
$315$	0	0
$316$	0.449490	0.0252858
$317$	−20.0454	−1.12586	−0.562931	−	0.826504i	$-0.690326\pi$
$317$	−20.0454	−1.12586	−0.562931	+	0.826504i	$0.690326\pi$
$318$	28.8990	1.62057
$319$	7.89898	0.442258
$320$	0	0
$321$	−10.6515	−0.594510
$322$	−21.7980	−1.21475
$323$	0	0
$324$	−9.00000	−0.500000
$325$	−29.4949	−1.63608
$326$	−6.89898	−0.382099
$327$	14.9444	0.826426
$328$	12.4495	0.687408
$329$	−28.0454	−1.54619
$330$	0	0
$331$	9.34847	0.513838	0.256919	−	0.966433i	$-0.417293\pi$
$331$	9.34847	0.513838	0.256919	+	0.966433i	$0.417293\pi$
$332$	−0.550510	−0.0302132
$333$	−10.6515	−0.583700
$334$	−21.1464	−1.15708
$335$	0	0
$336$	−6.00000	−0.327327
$337$	−31.7980	−1.73215	−0.866073	−	0.499918i	$-0.833363\pi$
$337$	−31.7980	−1.73215	−0.866073	+	0.499918i	$0.833363\pi$
$338$	21.7980	1.18565
$339$	−33.5505	−1.82221
$340$	0	0
$341$	2.00000	0.108306
$342$	0	0
$343$	−19.5959	−1.05808
$344$	4.55051	0.245347
$345$	0	0
$346$	9.79796	0.526742
$347$	−2.55051	−0.136919	−0.0684593	−	0.997654i	$-0.521808\pi$
$347$	−2.55051	−0.136919	−0.0684593	+	0.997654i	$0.521808\pi$
$348$	19.3485	1.03719
$349$	31.0000	1.65939	0.829696	−	0.558216i	$-0.188514\pi$
$349$	31.0000	1.65939	0.829696	+	0.558216i	$0.188514\pi$
$350$	−12.2474	−0.654654
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	−21.4949	−1.14406	−0.572029	−	0.820233i	$-0.693844\pi$
$353$	−21.4949	−1.14406	−0.572029	+	0.820233i	$0.693844\pi$
$354$	−16.8990	−0.898171
$355$	0	0
$356$	9.00000	0.476999
$357$	−2.69694	−0.142737
$358$	−9.79796	−0.517838
$359$	−4.69694	−0.247895	−0.123947	−	0.992289i	$-0.539555\pi$
$359$	−4.69694	−0.247895	−0.123947	+	0.992289i	$0.539555\pi$
$360$	0	0
$361$	0	0
$362$	20.2474	1.06418
$363$	−2.44949	−0.128565
$364$	14.4495	0.757359
$365$	0	0
$366$	4.65153	0.243139
$367$	−20.5505	−1.07273	−0.536364	−	0.843987i	$-0.680203\pi$
$367$	−20.5505	−1.07273	−0.536364	+	0.843987i	$0.680203\pi$
$368$	−8.89898	−0.463891
$369$	37.3485	1.94428
$370$	0	0
$371$	−28.8990	−1.50036
$372$	4.89898	0.254000
$373$	−1.79796	−0.0930948	−0.0465474	−	0.998916i	$-0.514822\pi$
$373$	−1.79796	−0.0930948	−0.0465474	+	0.998916i	$0.514822\pi$
$374$	−0.449490	−0.0232426
$375$	0	0
$376$	−11.4495	−0.590462
$377$	−46.5959	−2.39981
$378$	0	0
$379$	−11.3485	−0.582932	−0.291466	−	0.956581i	$-0.594143\pi$
$379$	−11.3485	−0.582932	−0.291466	+	0.956581i	$0.594143\pi$
$380$	0	0
$381$	−40.8990	−2.09532
$382$	−22.3485	−1.14345
$383$	−2.69694	−0.137807	−0.0689036	−	0.997623i	$-0.521950\pi$
$383$	−2.69694	−0.137807	−0.0689036	+	0.997623i	$0.521950\pi$
$384$	−2.44949	−0.125000
$385$	0	0
$386$	8.65153	0.440351
$387$	13.6515	0.693946
$388$	5.89898	0.299475
$389$	16.2474	0.823778	0.411889	−	0.911234i	$-0.364869\pi$
$389$	16.2474	0.823778	0.411889	+	0.911234i	$0.364869\pi$
$390$	0	0
$391$	−4.00000	−0.202289
$392$	−1.00000	−0.0505076
$393$	25.3485	1.27866
$394$	15.8990	0.800979
$395$	0	0
$396$	−3.00000	−0.150756
$397$	−10.6515	−0.534585	−0.267293	−	0.963615i	$-0.586129\pi$
$397$	−10.6515	−0.534585	−0.267293	+	0.963615i	$0.586129\pi$
$398$	5.65153	0.283286
$399$	0	0
$400$	−5.00000	−0.250000
$401$	−9.30306	−0.464573	−0.232286	−	0.972647i	$-0.574621\pi$
$401$	−9.30306	−0.464573	−0.232286	+	0.972647i	$0.574621\pi$
$402$	35.3939	1.76529
$403$	−11.7980	−0.587698
$404$	5.00000	0.248759
$405$	0	0
$406$	−19.3485	−0.960248
$407$	3.55051	0.175992
$408$	−1.10102	−0.0545086
$409$	−30.9444	−1.53010	−0.765051	−	0.643970i	$-0.777286\pi$
$409$	−30.9444	−1.53010	−0.765051	+	0.643970i	$0.777286\pi$
$410$	0	0
$411$	−4.65153	−0.229443
$412$	−6.34847	−0.312767
$413$	16.8990	0.831544
$414$	−26.6969	−1.31208
$415$	0	0
$416$	5.89898	0.289221
$417$	9.79796	0.479808
$418$	0	0
$419$	−2.65153	−0.129536	−0.0647679	−	0.997900i	$-0.520631\pi$
$419$	−2.65153	−0.129536	−0.0647679	+	0.997900i	$0.520631\pi$
$420$	0	0
$421$	20.6515	1.00649	0.503247	−	0.864143i	$-0.332139\pi$
$421$	20.6515	1.00649	0.503247	+	0.864143i	$0.332139\pi$
$422$	−20.0000	−0.973585
$423$	−34.3485	−1.67008
$424$	−11.7980	−0.572960
$425$	−2.24745	−0.109017
$426$	6.24745	0.302690
$427$	−4.65153	−0.225103
$428$	4.34847	0.210191
$429$	14.4495	0.697628
$430$	0	0
$431$	28.0454	1.35090	0.675450	−	0.737406i	$-0.263949\pi$
$431$	28.0454	1.35090	0.675450	+	0.737406i	$0.263949\pi$
$432$	0	0
$433$	−2.30306	−0.110678	−0.0553390	−	0.998468i	$-0.517624\pi$
$433$	−2.30306	−0.110678	−0.0553390	+	0.998468i	$0.517624\pi$
$434$	−4.89898	−0.235159
$435$	0	0
$436$	−6.10102	−0.292186
$437$	0	0
$438$	23.3939	1.11780
$439$	27.5959	1.31708	0.658541	−	0.752545i	$-0.271174\pi$
$439$	27.5959	1.31708	0.658541	+	0.752545i	$0.271174\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	2.65153	0.126120
$443$	−6.65153	−0.316024	−0.158012	−	0.987437i	$-0.550508\pi$
$443$	−6.65153	−0.316024	−0.158012	+	0.987437i	$0.550508\pi$
$444$	8.69694	0.412738
$445$	0	0
$446$	23.2474	1.10080
$447$	26.2020	1.23931
$448$	2.44949	0.115728
$449$	−29.7980	−1.40625	−0.703126	−	0.711065i	$-0.748213\pi$
$449$	−29.7980	−1.40625	−0.703126	+	0.711065i	$0.748213\pi$
$450$	−15.0000	−0.707107
$451$	−12.4495	−0.586224
$452$	13.6969	0.644250
$453$	−8.69694	−0.408618
$454$	−23.7980	−1.11689
$455$	0	0
$456$	0	0
$457$	30.9444	1.44752	0.723759	−	0.690053i	$-0.242413\pi$
$457$	30.9444	1.44752	0.723759	+	0.690053i	$0.242413\pi$
$458$	−15.3485	−0.717187
$459$	0	0
$460$	0	0
$461$	6.79796	0.316613	0.158306	−	0.987390i	$-0.449397\pi$
$461$	6.79796	0.316613	0.158306	+	0.987390i	$0.449397\pi$
$462$	6.00000	0.279145
$463$	5.85357	0.272039	0.136019	−	0.990706i	$-0.456569\pi$
$463$	5.85357	0.272039	0.136019	+	0.990706i	$0.456569\pi$
$464$	−7.89898	−0.366701
$465$	0	0
$466$	−5.55051	−0.257122
$467$	−21.3485	−0.987889	−0.493945	−	0.869493i	$-0.664445\pi$
$467$	−21.3485	−0.987889	−0.493945	+	0.869493i	$0.664445\pi$
$468$	17.6969	0.818041
$469$	−35.3939	−1.63434
$470$	0	0
$471$	14.6969	0.677199
$472$	6.89898	0.317551
$473$	−4.55051	−0.209233
$474$	−1.10102	−0.0505715
$475$	0	0
$476$	1.10102	0.0504652
$477$	−35.3939	−1.62057
$478$	2.20204	0.100719
$479$	−8.85357	−0.404530	−0.202265	−	0.979331i	$-0.564830\pi$
$479$	−8.85357	−0.404530	−0.202265	+	0.979331i	$0.564830\pi$
$480$	0	0
$481$	−20.9444	−0.954982
$482$	−10.0000	−0.455488
$483$	53.3939	2.42951
$484$	1.00000	0.0454545
$485$	0	0
$486$	22.0454	1.00000
$487$	22.3485	1.01271	0.506353	−	0.862326i	$-0.330993\pi$
$487$	22.3485	1.01271	0.506353	+	0.862326i	$0.330993\pi$
$488$	−1.89898	−0.0859628
$489$	16.8990	0.764198
$490$	0	0
$491$	5.79796	0.261658	0.130829	−	0.991405i	$-0.458236\pi$
$491$	5.79796	0.261658	0.130829	+	0.991405i	$0.458236\pi$
$492$	−30.4949	−1.37482
$493$	−3.55051	−0.159907
$494$	0	0
$495$	0	0
$496$	−2.00000	−0.0898027
$497$	−6.24745	−0.280236
$498$	1.34847	0.0604264
$499$	28.6969	1.28465	0.642326	−	0.766432i	$-0.277970\pi$
$499$	28.6969	1.28465	0.642326	+	0.766432i	$0.277970\pi$
$500$	0	0
$501$	51.7980	2.31416
$502$	−0.247449	−0.0110442
$503$	−1.10102	−0.0490921	−0.0245460	−	0.999699i	$-0.507814\pi$
$503$	−1.10102	−0.0490921	−0.0245460	+	0.999699i	$0.507814\pi$
$504$	7.34847	0.327327
$505$	0	0
$506$	8.89898	0.395608
$507$	−53.3939	−2.37131
$508$	16.6969	0.740807
$509$	15.5959	0.691277	0.345638	−	0.938368i	$-0.387662\pi$
$509$	15.5959	0.691277	0.345638	+	0.938368i	$0.387662\pi$
$510$	0	0
$511$	−23.3939	−1.03488
$512$	1.00000	0.0441942
$513$	0	0
$514$	22.0000	0.970378
$515$	0	0
$516$	−11.1464	−0.490694
$517$	11.4495	0.503548
$518$	−8.69694	−0.382122
$519$	−24.0000	−1.05348
$520$	0	0
$521$	12.3031	0.539007	0.269503	−	0.962999i	$-0.413140\pi$
$521$	12.3031	0.539007	0.269503	+	0.962999i	$0.413140\pi$
$522$	−23.6969	−1.03719
$523$	13.6515	0.596940	0.298470	−	0.954419i	$-0.403524\pi$
$523$	13.6515	0.596940	0.298470	+	0.954419i	$0.403524\pi$
$524$	−10.3485	−0.452075
$525$	30.0000	1.30931
$526$	−17.7980	−0.776028
$527$	−0.898979	−0.0391602
$528$	2.44949	0.106600
$529$	56.1918	2.44312
$530$	0	0
$531$	20.6969	0.898171
$532$	0	0
$533$	73.4393	3.18101
$534$	−22.0454	−0.953998
$535$	0	0
$536$	−14.4495	−0.624123
$537$	24.0000	1.03568
$538$	−15.5505	−0.670430
$539$	1.00000	0.0430730
$540$	0	0
$541$	18.2020	0.782567	0.391283	−	0.920270i	$-0.372031\pi$
$541$	18.2020	0.782567	0.391283	+	0.920270i	$0.372031\pi$
$542$	−4.69694	−0.201751
$543$	−49.5959	−2.12836
$544$	0.449490	0.0192717
$545$	0	0
$546$	−35.3939	−1.51472
$547$	−30.3485	−1.29761	−0.648803	−	0.760956i	$-0.724730\pi$
$547$	−30.3485	−1.29761	−0.648803	+	0.760956i	$0.724730\pi$
$548$	1.89898	0.0811204
$549$	−5.69694	−0.243139
$550$	5.00000	0.213201
$551$	0	0
$552$	21.7980	0.927783
$553$	1.10102	0.0468202
$554$	17.4949	0.743287
$555$	0	0
$556$	−4.00000	−0.169638
$557$	−22.5959	−0.957420	−0.478710	−	0.877973i	$-0.658895\pi$
$557$	−22.5959	−0.957420	−0.478710	+	0.877973i	$0.658895\pi$
$558$	−6.00000	−0.254000
$559$	26.8434	1.13535
$560$	0	0
$561$	1.10102	0.0464851
$562$	−17.1464	−0.723278
$563$	8.89898	0.375047	0.187524	−	0.982260i	$-0.439954\pi$
$563$	8.89898	0.375047	0.187524	+	0.982260i	$0.439954\pi$
$564$	28.0454	1.18092
$565$	0	0
$566$	12.8990	0.542185
$567$	−22.0454	−0.925820
$568$	−2.55051	−0.107017
$569$	−5.59592	−0.234593	−0.117297	−	0.993097i	$-0.537423\pi$
$569$	−5.59592	−0.234593	−0.117297	+	0.993097i	$0.537423\pi$
$570$	0	0
$571$	33.9444	1.42053	0.710264	−	0.703935i	$-0.248575\pi$
$571$	33.9444	1.42053	0.710264	+	0.703935i	$0.248575\pi$
$572$	−5.89898	−0.246649
$573$	54.7423	2.28689
$574$	30.4949	1.27283
$575$	44.4949	1.85557
$576$	3.00000	0.125000
$577$	−18.2020	−0.757761	−0.378880	−	0.925446i	$-0.623691\pi$
$577$	−18.2020	−0.757761	−0.378880	+	0.925446i	$0.623691\pi$
$578$	−16.7980	−0.698703
$579$	−21.1918	−0.880703
$580$	0	0
$581$	−1.34847	−0.0559439
$582$	−14.4495	−0.598951
$583$	11.7980	0.488622
$584$	−9.55051	−0.395203
$585$	0	0
$586$	−1.89898	−0.0784461
$587$	−12.6515	−0.522185	−0.261092	−	0.965314i	$-0.584083\pi$
$587$	−12.6515	−0.522185	−0.261092	+	0.965314i	$0.584083\pi$
$588$	2.44949	0.101015
$589$	0	0
$590$	0	0
$591$	−38.9444	−1.60196
$592$	−3.55051	−0.145925
$593$	−33.3939	−1.37132	−0.685661	−	0.727921i	$-0.740487\pi$
$593$	−33.3939	−1.37132	−0.685661	+	0.727921i	$0.740487\pi$
$594$	0	0
$595$	0	0
$596$	−10.6969	−0.438164
$597$	−13.8434	−0.566571
$598$	−52.4949	−2.14668
$599$	−19.6515	−0.802940	−0.401470	−	0.915872i	$-0.631501\pi$
$599$	−19.6515	−0.802940	−0.401470	+	0.915872i	$0.631501\pi$
$600$	12.2474	0.500000
$601$	−43.5959	−1.77831	−0.889157	−	0.457602i	$-0.848709\pi$
$601$	−43.5959	−1.77831	−0.889157	+	0.457602i	$0.848709\pi$
$602$	11.1464	0.454294
$603$	−43.3485	−1.76529
$604$	3.55051	0.144468
$605$	0	0
$606$	−12.2474	−0.497519
$607$	−15.1464	−0.614775	−0.307387	−	0.951585i	$-0.599455\pi$
$607$	−15.1464	−0.614775	−0.307387	+	0.951585i	$0.599455\pi$
$608$	0	0
$609$	47.3939	1.92050
$610$	0	0
$611$	−67.5403	−2.73239
$612$	1.34847	0.0545086
$613$	−7.20204	−0.290888	−0.145444	−	0.989367i	$-0.546461\pi$
$613$	−7.20204	−0.290888	−0.145444	+	0.989367i	$0.546461\pi$
$614$	−5.24745	−0.211770
$615$	0	0
$616$	−2.44949	−0.0986928
$617$	20.3939	0.821027	0.410513	−	0.911855i	$-0.365350\pi$
$617$	20.3939	0.821027	0.410513	+	0.911855i	$0.365350\pi$
$618$	15.5505	0.625533
$619$	4.69694	0.188786	0.0943929	−	0.995535i	$-0.469909\pi$
$619$	4.69694	0.188786	0.0943929	+	0.995535i	$0.469909\pi$
$620$	0	0
$621$	0	0
$622$	−17.2474	−0.691560
$623$	22.0454	0.883231
$624$	−14.4495	−0.578443
$625$	25.0000	1.00000
$626$	9.69694	0.387568
$627$	0	0
$628$	−6.00000	−0.239426
$629$	−1.59592	−0.0636334
$630$	0	0
$631$	45.0454	1.79323	0.896615	−	0.442812i	$-0.146019\pi$
$631$	45.0454	1.79323	0.896615	+	0.442812i	$0.146019\pi$
$632$	0.449490	0.0178797
$633$	48.9898	1.94717
$634$	−20.0454	−0.796105
$635$	0	0
$636$	28.8990	1.14592
$637$	−5.89898	−0.233726
$638$	7.89898	0.312724
$639$	−7.65153	−0.302690
$640$	0	0
$641$	−5.49490	−0.217035	−0.108518	−	0.994095i	$-0.534610\pi$
$641$	−5.49490	−0.217035	−0.108518	+	0.994095i	$0.534610\pi$
$642$	−10.6515	−0.420382
$643$	11.3485	0.447540	0.223770	−	0.974642i	$-0.428164\pi$
$643$	11.3485	0.447540	0.223770	+	0.974642i	$0.428164\pi$
$644$	−21.7980	−0.858960
$645$	0	0
$646$	0	0
$647$	−14.7526	−0.579983	−0.289991	−	0.957029i	$-0.593652\pi$
$647$	−14.7526	−0.579983	−0.289991	+	0.957029i	$0.593652\pi$
$648$	−9.00000	−0.353553
$649$	−6.89898	−0.270809
$650$	−29.4949	−1.15689
$651$	12.0000	0.470317
$652$	−6.89898	−0.270185
$653$	−11.3485	−0.444100	−0.222050	−	0.975035i	$-0.571275\pi$
$653$	−11.3485	−0.444100	−0.222050	+	0.975035i	$0.571275\pi$
$654$	14.9444	0.584372
$655$	0	0
$656$	12.4495	0.486071
$657$	−28.6515	−1.11780
$658$	−28.0454	−1.09332
$659$	−31.9444	−1.24438	−0.622188	−	0.782868i	$-0.713756\pi$
$659$	−31.9444	−1.24438	−0.622188	+	0.782868i	$0.713756\pi$
$660$	0	0
$661$	25.1464	0.978083	0.489041	−	0.872261i	$-0.337347\pi$
$661$	25.1464	0.978083	0.489041	+	0.872261i	$0.337347\pi$
$662$	9.34847	0.363339
$663$	−6.49490	−0.252241
$664$	−0.550510	−0.0213639
$665$	0	0
$666$	−10.6515	−0.412738
$667$	70.2929	2.72175
$668$	−21.1464	−0.818180
$669$	−56.9444	−2.20160
$670$	0	0
$671$	1.89898	0.0733093
$672$	−6.00000	−0.231455
$673$	−30.7423	−1.18503	−0.592515	−	0.805559i	$-0.701865\pi$
$673$	−30.7423	−1.18503	−0.592515	+	0.805559i	$0.701865\pi$
$674$	−31.7980	−1.22481
$675$	0	0
$676$	21.7980	0.838383
$677$	15.6969	0.603282	0.301641	−	0.953422i	$-0.402466\pi$
$677$	15.6969	0.603282	0.301641	+	0.953422i	$0.402466\pi$
$678$	−33.5505	−1.28850
$679$	14.4495	0.554521
$680$	0	0
$681$	58.2929	2.23379
$682$	2.00000	0.0765840
$683$	0.853572	0.0326610	0.0163305	−	0.999867i	$-0.494802\pi$
$683$	0.853572	0.0326610	0.0163305	+	0.999867i	$0.494802\pi$
$684$	0	0
$685$	0	0
$686$	−19.5959	−0.748176
$687$	37.5959	1.43437
$688$	4.55051	0.173487
$689$	−69.5959	−2.65139
$690$	0	0
$691$	43.6413	1.66019	0.830097	−	0.557619i	$-0.188285\pi$
$691$	43.6413	1.66019	0.830097	+	0.557619i	$0.188285\pi$
$692$	9.79796	0.372463
$693$	−7.34847	−0.279145
$694$	−2.55051	−0.0968160
$695$	0	0
$696$	19.3485	0.733402
$697$	5.59592	0.211961
$698$	31.0000	1.17337
$699$	13.5959	0.514245
$700$	−12.2474	−0.462910
$701$	−7.40408	−0.279648	−0.139824	−	0.990176i	$-0.544654\pi$
$701$	−7.40408	−0.279648	−0.139824	+	0.990176i	$0.544654\pi$
$702$	0	0
$703$	0	0
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−21.4949	−0.808971
$707$	12.2474	0.460613
$708$	−16.8990	−0.635103
$709$	−43.3939	−1.62969	−0.814846	−	0.579678i	$-0.803178\pi$
$709$	−43.3939	−1.62969	−0.814846	+	0.579678i	$0.803178\pi$
$710$	0	0
$711$	1.34847	0.0505715
$712$	9.00000	0.337289
$713$	17.7980	0.666539
$714$	−2.69694	−0.100930
$715$	0	0
$716$	−9.79796	−0.366167
$717$	−5.39388	−0.201438
$718$	−4.69694	−0.175288
$719$	−40.1464	−1.49721	−0.748605	−	0.663017i	$-0.769276\pi$
$719$	−40.1464	−1.49721	−0.748605	+	0.663017i	$0.769276\pi$
$720$	0	0
$721$	−15.5505	−0.579131
$722$	0	0
$723$	24.4949	0.910975
$724$	20.2474	0.752491
$725$	39.4949	1.46680
$726$	−2.44949	−0.0909091
$727$	−43.9444	−1.62981	−0.814904	−	0.579597i	$-0.803210\pi$
$727$	−43.9444	−1.62981	−0.814904	+	0.579597i	$0.803210\pi$
$728$	14.4495	0.535534
$729$	−27.0000	−1.00000
$730$	0	0
$731$	2.04541	0.0756521
$732$	4.65153	0.171926
$733$	40.8990	1.51064	0.755319	−	0.655357i	$-0.227482\pi$
$733$	40.8990	1.51064	0.755319	+	0.655357i	$0.227482\pi$
$734$	−20.5505	−0.758533
$735$	0	0
$736$	−8.89898	−0.328021
$737$	14.4495	0.532254
$738$	37.3485	1.37482
$739$	−41.7423	−1.53552	−0.767759	−	0.640739i	$-0.778628\pi$
$739$	−41.7423	−1.53552	−0.767759	+	0.640739i	$0.778628\pi$
$740$	0	0
$741$	0	0
$742$	−28.8990	−1.06091
$743$	2.85357	0.104687	0.0523437	−	0.998629i	$-0.483331\pi$
$743$	2.85357	0.104687	0.0523437	+	0.998629i	$0.483331\pi$
$744$	4.89898	0.179605
$745$	0	0
$746$	−1.79796	−0.0658280
$747$	−1.65153	−0.0604264
$748$	−0.449490	−0.0164350
$749$	10.6515	0.389198
$750$	0	0
$751$	−45.5403	−1.66179	−0.830895	−	0.556430i	$-0.812171\pi$
$751$	−45.5403	−1.66179	−0.830895	+	0.556430i	$0.812171\pi$
$752$	−11.4495	−0.417520
$753$	0.606123	0.0220884
$754$	−46.5959	−1.69692
$755$	0	0
$756$	0	0
$757$	35.7980	1.30110	0.650549	−	0.759464i	$-0.274539\pi$
$757$	35.7980	1.30110	0.650549	+	0.759464i	$0.274539\pi$
$758$	−11.3485	−0.412195
$759$	−21.7980	−0.791216
$760$	0	0
$761$	−47.1918	−1.71070	−0.855351	−	0.518048i	$-0.826659\pi$
$761$	−47.1918	−1.71070	−0.855351	+	0.518048i	$0.826659\pi$
$762$	−40.8990	−1.48161
$763$	−14.9444	−0.541023
$764$	−22.3485	−0.808539
$765$	0	0
$766$	−2.69694	−0.0974443
$767$	40.6969	1.46948
$768$	−2.44949	−0.0883883
$769$	19.3939	0.699361	0.349681	−	0.936869i	$-0.386290\pi$
$769$	19.3939	0.699361	0.349681	+	0.936869i	$0.386290\pi$
$770$	0	0
$771$	−53.8888	−1.94076
$772$	8.65153	0.311375
$773$	23.1464	0.832519	0.416260	−	0.909246i	$-0.363341\pi$
$773$	23.1464	0.832519	0.416260	+	0.909246i	$0.363341\pi$
$774$	13.6515	0.490694
$775$	10.0000	0.359211
$776$	5.89898	0.211761
$777$	21.3031	0.764243
$778$	16.2474	0.582499
$779$	0	0
$780$	0	0
$781$	2.55051	0.0912644
$782$	−4.00000	−0.143040
$783$	0	0
$784$	−1.00000	−0.0357143
$785$	0	0
$786$	25.3485	0.904150
$787$	−0.752551	−0.0268256	−0.0134128	−	0.999910i	$-0.504270\pi$
$787$	−0.752551	−0.0268256	−0.0134128	+	0.999910i	$0.504270\pi$
$788$	15.8990	0.566378
$789$	43.5959	1.55206
$790$	0	0
$791$	33.5505	1.19292
$792$	−3.00000	−0.106600
$793$	−11.2020	−0.397796
$794$	−10.6515	−0.378009
$795$	0	0
$796$	5.65153	0.200313
$797$	42.0454	1.48932	0.744662	−	0.667441i	$-0.232611\pi$
$797$	42.0454	1.48932	0.744662	+	0.667441i	$0.232611\pi$
$798$	0	0
$799$	−5.14643	−0.182068
$800$	−5.00000	−0.176777
$801$	27.0000	0.953998
$802$	−9.30306	−0.328503
$803$	9.55051	0.337030
$804$	35.3939	1.24825
$805$	0	0
$806$	−11.7980	−0.415565
$807$	38.0908	1.34086
$808$	5.00000	0.175899
$809$	35.1464	1.23568	0.617841	−	0.786303i	$-0.288007\pi$
$809$	35.1464	1.23568	0.617841	+	0.786303i	$0.288007\pi$
$810$	0	0
$811$	7.44949	0.261587	0.130793	−	0.991410i	$-0.458248\pi$
$811$	7.44949	0.261587	0.130793	+	0.991410i	$0.458248\pi$
$812$	−19.3485	−0.678998
$813$	11.5051	0.403502
$814$	3.55051	0.124445
$815$	0	0
$816$	−1.10102	−0.0385434
$817$	0	0
$818$	−30.9444	−1.08195
$819$	43.3485	1.51472
$820$	0	0
$821$	−22.2929	−0.778026	−0.389013	−	0.921232i	$-0.627184\pi$
$821$	−22.2929	−0.778026	−0.389013	+	0.921232i	$0.627184\pi$
$822$	−4.65153	−0.162241
$823$	−4.55051	−0.158621	−0.0793104	−	0.996850i	$-0.525272\pi$
$823$	−4.55051	−0.158621	−0.0793104	+	0.996850i	$0.525272\pi$
$824$	−6.34847	−0.221159
$825$	−12.2474	−0.426401
$826$	16.8990	0.587991
$827$	−34.8434	−1.21162	−0.605811	−	0.795608i	$-0.707151\pi$
$827$	−34.8434	−1.21162	−0.605811	+	0.795608i	$0.707151\pi$
$828$	−26.6969	−0.927783
$829$	31.1464	1.08176	0.540880	−	0.841100i	$-0.318091\pi$
$829$	31.1464	1.08176	0.540880	+	0.841100i	$0.318091\pi$
$830$	0	0
$831$	−42.8536	−1.48657
$832$	5.89898	0.204510
$833$	−0.449490	−0.0155739
$834$	9.79796	0.339276
$835$	0	0
$836$	0	0
$837$	0	0
$838$	−2.65153	−0.0915956
$839$	−45.2474	−1.56212	−0.781058	−	0.624459i	$-0.785320\pi$
$839$	−45.2474	−1.56212	−0.781058	+	0.624459i	$0.785320\pi$
$840$	0	0
$841$	33.3939	1.15151
$842$	20.6515	0.711699
$843$	42.0000	1.44656
$844$	−20.0000	−0.688428
$845$	0	0
$846$	−34.3485	−1.18092
$847$	2.44949	0.0841655
$848$	−11.7980	−0.405144
$849$	−31.5959	−1.08437
$850$	−2.24745	−0.0770869
$851$	31.5959	1.08309
$852$	6.24745	0.214034
$853$	26.6969	0.914086	0.457043	−	0.889445i	$-0.348909\pi$
$853$	26.6969	0.914086	0.457043	+	0.889445i	$0.348909\pi$
$854$	−4.65153	−0.159172
$855$	0	0
$856$	4.34847	0.148628
$857$	23.1464	0.790667	0.395333	−	0.918538i	$-0.370629\pi$
$857$	23.1464	0.790667	0.395333	+	0.918538i	$0.370629\pi$
$858$	14.4495	0.493297
$859$	−10.8990	−0.371868	−0.185934	−	0.982562i	$-0.559531\pi$
$859$	−10.8990	−0.371868	−0.185934	+	0.982562i	$0.559531\pi$
$860$	0	0
$861$	−74.6969	−2.54566
$862$	28.0454	0.955230
$863$	26.3485	0.896912	0.448456	−	0.893805i	$-0.351974\pi$
$863$	26.3485	0.896912	0.448456	+	0.893805i	$0.351974\pi$
$864$	0	0
$865$	0	0
$866$	−2.30306	−0.0782612
$867$	41.1464	1.39741
$868$	−4.89898	−0.166282
$869$	−0.449490	−0.0152479
$870$	0	0
$871$	−85.2372	−2.88815
$872$	−6.10102	−0.206607
$873$	17.6969	0.598951
$874$	0	0
$875$	0	0
$876$	23.3939	0.790406
$877$	1.59592	0.0538903	0.0269452	−	0.999637i	$-0.491422\pi$
$877$	1.59592	0.0538903	0.0269452	+	0.999637i	$0.491422\pi$
$878$	27.5959	0.931317
$879$	4.65153	0.156892
$880$	0	0
$881$	3.10102	0.104476	0.0522380	−	0.998635i	$-0.483365\pi$
$881$	3.10102	0.104476	0.0522380	+	0.998635i	$0.483365\pi$
$882$	−3.00000	−0.101015
$883$	−7.55051	−0.254095	−0.127047	−	0.991897i	$-0.540550\pi$
$883$	−7.55051	−0.254095	−0.127047	+	0.991897i	$0.540550\pi$
$884$	2.65153	0.0891806
$885$	0	0
$886$	−6.65153	−0.223463
$887$	1.34847	0.0452772	0.0226386	−	0.999744i	$-0.492793\pi$
$887$	1.34847	0.0452772	0.0226386	+	0.999744i	$0.492793\pi$
$888$	8.69694	0.291850
$889$	40.8990	1.37171
$890$	0	0
$891$	9.00000	0.301511
$892$	23.2474	0.778382
$893$	0	0
$894$	26.2020	0.876327
$895$	0	0
$896$	2.44949	0.0818317
$897$	128.586	4.29335
$898$	−29.7980	−0.994371
$899$	15.7980	0.526891
$900$	−15.0000	−0.500000
$901$	−5.30306	−0.176671
$902$	−12.4495	−0.414523
$903$	−27.3031	−0.908589
$904$	13.6969	0.455553
$905$	0	0
$906$	−8.69694	−0.288936
$907$	−29.1010	−0.966284	−0.483142	−	0.875542i	$-0.660504\pi$
$907$	−29.1010	−0.966284	−0.483142	+	0.875542i	$0.660504\pi$
$908$	−23.7980	−0.789763
$909$	15.0000	0.497519
$910$	0	0
$911$	−24.4949	−0.811552	−0.405776	−	0.913973i	$-0.632999\pi$
$911$	−24.4949	−0.811552	−0.405776	+	0.913973i	$0.632999\pi$
$912$	0	0
$913$	0.550510	0.0182192
$914$	30.9444	1.02355
$915$	0	0
$916$	−15.3485	−0.507128
$917$	−25.3485	−0.837080
$918$	0	0
$919$	−10.6515	−0.351362	−0.175681	−	0.984447i	$-0.556213\pi$
$919$	−10.6515	−0.351362	−0.175681	+	0.984447i	$0.556213\pi$
$920$	0	0
$921$	12.8536	0.423540
$922$	6.79796	0.223879
$923$	−15.0454	−0.495226
$924$	6.00000	0.197386
$925$	17.7526	0.583700
$926$	5.85357	0.192360
$927$	−19.0454	−0.625533
$928$	−7.89898	−0.259297
$929$	11.4949	0.377135	0.188568	−	0.982060i	$-0.439616\pi$
$929$	11.4949	0.377135	0.188568	+	0.982060i	$0.439616\pi$
$930$	0	0
$931$	0	0
$932$	−5.55051	−0.181813
$933$	42.2474	1.38312
$934$	−21.3485	−0.698543
$935$	0	0
$936$	17.6969	0.578443
$937$	5.79796	0.189411	0.0947055	−	0.995505i	$-0.469809\pi$
$937$	5.79796	0.189411	0.0947055	+	0.995505i	$0.469809\pi$
$938$	−35.3939	−1.15565
$939$	−23.7526	−0.775135
$940$	0	0
$941$	26.2020	0.854162	0.427081	−	0.904213i	$-0.359542\pi$
$941$	26.2020	0.854162	0.427081	+	0.904213i	$0.359542\pi$
$942$	14.6969	0.478852
$943$	−110.788	−3.60775
$944$	6.89898	0.224543
$945$	0	0
$946$	−4.55051	−0.147950
$947$	20.2929	0.659429	0.329715	−	0.944081i	$-0.393047\pi$
$947$	20.2929	0.659429	0.329715	+	0.944081i	$0.393047\pi$
$948$	−1.10102	−0.0357595
$949$	−56.3383	−1.82882
$950$	0	0
$951$	49.1010	1.59221
$952$	1.10102	0.0356843
$953$	−40.0000	−1.29573	−0.647864	−	0.761756i	$-0.724337\pi$
$953$	−40.0000	−1.29573	−0.647864	+	0.761756i	$0.724337\pi$
$954$	−35.3939	−1.14592
$955$	0	0
$956$	2.20204	0.0712191
$957$	−19.3485	−0.625447
$958$	−8.85357	−0.286046
$959$	4.65153	0.150206
$960$	0	0
$961$	−27.0000	−0.870968
$962$	−20.9444	−0.675274
$963$	13.0454	0.420382
$964$	−10.0000	−0.322078
$965$	0	0
$966$	53.3939	1.71792
$967$	−18.2020	−0.585338	−0.292669	−	0.956214i	$-0.594543\pi$
$967$	−18.2020	−0.585338	−0.292669	+	0.956214i	$0.594543\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	0	0
$971$	−19.8434	−0.636804	−0.318402	−	0.947956i	$-0.603146\pi$
$971$	−19.8434	−0.636804	−0.318402	+	0.947956i	$0.603146\pi$
$972$	22.0454	0.707107
$973$	−9.79796	−0.314108
$974$	22.3485	0.716091
$975$	72.2474	2.31377
$976$	−1.89898	−0.0607849
$977$	57.0000	1.82359	0.911796	−	0.410644i	$-0.134696\pi$
$977$	57.0000	1.82359	0.911796	+	0.410644i	$0.134696\pi$
$978$	16.8990	0.540370
$979$	−9.00000	−0.287641
$980$	0	0
$981$	−18.3031	−0.584372
$982$	5.79796	0.185020
$983$	27.7423	0.884843	0.442422	−	0.896807i	$-0.354119\pi$
$983$	27.7423	0.884843	0.442422	+	0.896807i	$0.354119\pi$
$984$	−30.4949	−0.972142
$985$	0	0
$986$	−3.55051	−0.113071
$987$	68.6969	2.18665
$988$	0	0
$989$	−40.4949	−1.28766
$990$	0	0
$991$	3.44949	0.109577	0.0547883	−	0.998498i	$-0.482552\pi$
$991$	3.44949	0.109577	0.0547883	+	0.998498i	$0.482552\pi$
$992$	−2.00000	−0.0635001
$993$	−22.8990	−0.726677
$994$	−6.24745	−0.198157
$995$	0	0
$996$	1.34847	0.0427279
$997$	49.1918	1.55792	0.778961	−	0.627073i	$-0.215747\pi$
$997$	49.1918	1.55792	0.778961	+	0.627073i	$0.215747\pi$
$998$	28.6969	0.908386
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7942.2.a.y.1.1		2
19.8	odd	6		418.2.e.g.45.1	✓	4
19.12	odd	6		418.2.e.g.353.1	yes	4
19.18	odd	2		7942.2.a.v.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
418.2.e.g.45.1	✓	4	19.8	odd	6
418.2.e.g.353.1	yes	4	19.12	odd	6
7942.2.a.v.1.2		2	19.18	odd	2
7942.2.a.y.1.1		2	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:	\( N \)	\(=\)	\( 7942 = 2 \cdot 11 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7942.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.44949 q^{3} +1.00000 q^{4} -2.44949 q^{6} +2.44949 q^{7} +1.00000 q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.44949 q^{3} +1.00000 q^{4} -2.44949 q^{6} +2.44949 q^{7} +1.00000 q^{8} +3.00000 q^{9} -1.00000 q^{11} -2.44949 q^{12} +5.89898 q^{13} +2.44949 q^{14} +1.00000 q^{16} +0.449490 q^{17} +3.00000 q^{18} -6.00000 q^{21} -1.00000 q^{22} -8.89898 q^{23} -2.44949 q^{24} -5.00000 q^{25} +5.89898 q^{26} +2.44949 q^{28} -7.89898 q^{29} -2.00000 q^{31} +1.00000 q^{32} +2.44949 q^{33} +0.449490 q^{34} +3.00000 q^{36} -3.55051 q^{37} -14.4495 q^{39} +12.4495 q^{41} -6.00000 q^{42} +4.55051 q^{43} -1.00000 q^{44} -8.89898 q^{46} -11.4495 q^{47} -2.44949 q^{48} -1.00000 q^{49} -5.00000 q^{50} -1.10102 q^{51} +5.89898 q^{52} -11.7980 q^{53} +2.44949 q^{56} -7.89898 q^{58} +6.89898 q^{59} -1.89898 q^{61} -2.00000 q^{62} +7.34847 q^{63} +1.00000 q^{64} +2.44949 q^{66} -14.4495 q^{67} +0.449490 q^{68} +21.7980 q^{69} -2.55051 q^{71} +3.00000 q^{72} -9.55051 q^{73} -3.55051 q^{74} +12.2474 q^{75} -2.44949 q^{77} -14.4495 q^{78} +0.449490 q^{79} -9.00000 q^{81} +12.4495 q^{82} -0.550510 q^{83} -6.00000 q^{84} +4.55051 q^{86} +19.3485 q^{87} -1.00000 q^{88} +9.00000 q^{89} +14.4495 q^{91} -8.89898 q^{92} +4.89898 q^{93} -11.4495 q^{94} -2.44949 q^{96} +5.89898 q^{97} -1.00000 q^{98} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 6 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 + 6 * q^9	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 6 q^{9} - 2 q^{11} + 2 q^{13} + 2 q^{16} - 4 q^{17} + 6 q^{18} - 12 q^{21} - 2 q^{22} - 8 q^{23} - 10 q^{25} + 2 q^{26} - 6 q^{29} - 4 q^{31} + 2 q^{32} - 4 q^{34} + 6 q^{36} - 12 q^{37} - 24 q^{39} + 20 q^{41} - 12 q^{42} + 14 q^{43} - 2 q^{44} - 8 q^{46} - 18 q^{47} - 2 q^{49} - 10 q^{50} - 12 q^{51} + 2 q^{52} - 4 q^{53} - 6 q^{58} + 4 q^{59} + 6 q^{61} - 4 q^{62} + 2 q^{64} - 24 q^{67} - 4 q^{68} + 24 q^{69} - 10 q^{71} + 6 q^{72} - 24 q^{73} - 12 q^{74} - 24 q^{78} - 4 q^{79} - 18 q^{81} + 20 q^{82} - 6 q^{83} - 12 q^{84} + 14 q^{86} + 24 q^{87} - 2 q^{88} + 18 q^{89} + 24 q^{91} - 8 q^{92} - 18 q^{94} + 2 q^{97} - 2 q^{98} - 6 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 + 6 * q^9 - 2 * q^11 + 2 * q^13 + 2 * q^16 - 4 * q^17 + 6 * q^18 - 12 * q^21 - 2 * q^22 - 8 * q^23 - 10 * q^25 + 2 * q^26 - 6 * q^29 - 4 * q^31 + 2 * q^32 - 4 * q^34 + 6 * q^36 - 12 * q^37 - 24 * q^39 + 20 * q^41 - 12 * q^42 + 14 * q^43 - 2 * q^44 - 8 * q^46 - 18 * q^47 - 2 * q^49 - 10 * q^50 - 12 * q^51 + 2 * q^52 - 4 * q^53 - 6 * q^58 + 4 * q^59 + 6 * q^61 - 4 * q^62 + 2 * q^64 - 24 * q^67 - 4 * q^68 + 24 * q^69 - 10 * q^71 + 6 * q^72 - 24 * q^73 - 12 * q^74 - 24 * q^78 - 4 * q^79 - 18 * q^81 + 20 * q^82 - 6 * q^83 - 12 * q^84 + 14 * q^86 + 24 * q^87 - 2 * q^88 + 18 * q^89 + 24 * q^91 - 8 * q^92 - 18 * q^94 + 2 * q^97 - 2 * q^98 - 6 * q^99

Introduction

L-functions

Modular forms

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