LMFDB - Embedded newform 7942.2.a.y.1.2

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			1.2
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)

$f(q)$	$=$	$q+1.00000 q^{2} +2.44949 q^{3} +1.00000 q^{4} +2.44949 q^{6} -2.44949 q^{7} +1.00000 q^{8} +3.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +2.44949 q^{3} +1.00000 q^{4} +2.44949 q^{6} -2.44949 q^{7} +1.00000 q^{8} +3.00000 q^{9} -1.00000 q^{11} +2.44949 q^{12} -3.89898 q^{13} -2.44949 q^{14} +1.00000 q^{16} -4.44949 q^{17} +3.00000 q^{18} -6.00000 q^{21} -1.00000 q^{22} +0.898979 q^{23} +2.44949 q^{24} -5.00000 q^{25} -3.89898 q^{26} -2.44949 q^{28} +1.89898 q^{29} -2.00000 q^{31} +1.00000 q^{32} -2.44949 q^{33} -4.44949 q^{34} +3.00000 q^{36} -8.44949 q^{37} -9.55051 q^{39} +7.55051 q^{41} -6.00000 q^{42} +9.44949 q^{43} -1.00000 q^{44} +0.898979 q^{46} -6.55051 q^{47} +2.44949 q^{48} -1.00000 q^{49} -5.00000 q^{50} -10.8990 q^{51} -3.89898 q^{52} +7.79796 q^{53} -2.44949 q^{56} +1.89898 q^{58} -2.89898 q^{59} +7.89898 q^{61} -2.00000 q^{62} -7.34847 q^{63} +1.00000 q^{64} -2.44949 q^{66} -9.55051 q^{67} -4.44949 q^{68} +2.20204 q^{69} -7.44949 q^{71} +3.00000 q^{72} -14.4495 q^{73} -8.44949 q^{74} -12.2474 q^{75} +2.44949 q^{77} -9.55051 q^{78} -4.44949 q^{79} -9.00000 q^{81} +7.55051 q^{82} -5.44949 q^{83} -6.00000 q^{84} +9.44949 q^{86} +4.65153 q^{87} -1.00000 q^{88} +9.00000 q^{89} +9.55051 q^{91} +0.898979 q^{92} -4.89898 q^{93} -6.55051 q^{94} +2.44949 q^{96} -3.89898 q^{97} -1.00000 q^{98} -3.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.250000\pi$
$3$	2.44949	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$4$	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	$-0.653195\pi$
$7$	−2.44949	−0.925820	−0.462910	+	0.886405i	$0.653195\pi$
$8$	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
$13$	−3.89898	−1.08138	−0.540691	−	0.841221i	$-0.681837\pi$
$13$	−3.89898	−1.08138	−0.540691	+	0.841221i	$0.681837\pi$
$14$	−2.44949	−0.654654
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.44949	−1.07916	−0.539580	−	0.841934i	$-0.681417\pi$
$17$	−4.44949	−1.07916	−0.539580	+	0.841934i	$0.681417\pi$
$18$	3.00000	0.707107
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−6.00000	−1.30931
$22$	−1.00000	−0.213201
$23$	0.898979	0.187450	0.0937251	−	0.995598i	$-0.470123\pi$
$23$	0.898979	0.187450	0.0937251	+	0.995598i	$0.470123\pi$
$24$	2.44949	0.500000
$25$	−5.00000	−1.00000
$26$	−3.89898	−0.764653
$27$	0	0
$28$	−2.44949	−0.462910
$29$	1.89898	0.352632	0.176316	−	0.984334i	$-0.443582\pi$
$29$	1.89898	0.352632	0.176316	+	0.984334i	$0.443582\pi$
$30$	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
$34$	−4.44949	−0.763081
$35$	0	0
$36$	3.00000	0.500000
$37$	−8.44949	−1.38909	−0.694544	−	0.719450i	$-0.744394\pi$
$37$	−8.44949	−1.38909	−0.694544	+	0.719450i	$0.744394\pi$
$38$	0	0
$39$	−9.55051	−1.52931
$40$	0	0
$41$	7.55051	1.17919	0.589596	−	0.807698i	$-0.299287\pi$
$41$	7.55051	1.17919	0.589596	+	0.807698i	$0.299287\pi$
$42$	−6.00000	−0.925820
$43$	9.44949	1.44103	0.720517	−	0.693437i	$-0.243905\pi$
$43$	9.44949	1.44103	0.720517	+	0.693437i	$0.243905\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	0.898979	0.132547
$47$	−6.55051	−0.955490	−0.477745	−	0.878499i	$-0.658546\pi$
$47$	−6.55051	−0.955490	−0.477745	+	0.878499i	$0.658546\pi$
$48$	2.44949	0.353553
$49$	−1.00000	−0.142857
$50$	−5.00000	−0.707107
$51$	−10.8990	−1.52616
$52$	−3.89898	−0.540691
$53$	7.79796	1.07113	0.535566	−	0.844493i	$-0.320098\pi$
$53$	7.79796	1.07113	0.535566	+	0.844493i	$0.320098\pi$
$54$	0	0
$55$	0	0
$56$	−2.44949	−0.327327
$57$	0	0
$58$	1.89898	0.249348
$59$	−2.89898	−0.377415	−0.188707	−	0.982033i	$-0.560430\pi$
$59$	−2.89898	−0.377415	−0.188707	+	0.982033i	$0.560430\pi$
$60$	0	0
$61$	7.89898	1.01136	0.505680	−	0.862721i	$-0.331242\pi$
$61$	7.89898	1.01136	0.505680	+	0.862721i	$0.331242\pi$
$62$	−2.00000	−0.254000
$63$	−7.34847	−0.925820
$64$	1.00000	0.125000
$65$	0	0
$66$	−2.44949	−0.301511
$67$	−9.55051	−1.16678	−0.583390	−	0.812192i	$-0.698274\pi$
$67$	−9.55051	−1.16678	−0.583390	+	0.812192i	$0.698274\pi$
$68$	−4.44949	−0.539580
$69$	2.20204	0.265095
$70$	0	0
$71$	−7.44949	−0.884092	−0.442046	−	0.896992i	$-0.645747\pi$
$71$	−7.44949	−0.884092	−0.442046	+	0.896992i	$0.645747\pi$
$72$	3.00000	0.353553
$73$	−14.4495	−1.69118	−0.845592	−	0.533829i	$-0.820753\pi$
$73$	−14.4495	−1.69118	−0.845592	+	0.533829i	$0.820753\pi$
$74$	−8.44949	−0.982233
$75$	−12.2474	−1.41421
$76$	0	0
$77$	2.44949	0.279145
$78$	−9.55051	−1.08138
$79$	−4.44949	−0.500607	−0.250303	−	0.968167i	$-0.580530\pi$
$79$	−4.44949	−0.500607	−0.250303	+	0.968167i	$0.580530\pi$
$80$	0	0
$81$	−9.00000	−1.00000
$82$	7.55051	0.833814
$83$	−5.44949	−0.598159	−0.299080	−	0.954228i	$-0.596680\pi$
$83$	−5.44949	−0.598159	−0.299080	+	0.954228i	$0.596680\pi$
$84$	−6.00000	−0.654654
$85$	0	0
$86$	9.44949	1.01896
$87$	4.65153	0.498696
$88$	−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
$91$	9.55051	1.00117
$92$	0.898979	0.0937251
$93$	−4.89898	−0.508001
$94$	−6.55051	−0.675634
$95$	0	0
$96$	2.44949	0.250000
$97$	−3.89898	−0.395881	−0.197941	−	0.980214i	$-0.563425\pi$
$97$	−3.89898	−0.395881	−0.197941	+	0.980214i	$0.563425\pi$
$98$	−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	$0.419977\pi$
$102$	−10.8990	−1.07916
$103$	8.34847	0.822599	0.411300	−	0.911500i	$-0.365075\pi$
$103$	8.34847	0.822599	0.411300	+	0.911500i	$0.365075\pi$
$104$	−3.89898	−0.382326
$105$	0	0
$106$	7.79796	0.757405
$107$	−10.3485	−1.00042	−0.500212	−	0.865903i	$-0.666745\pi$
$107$	−10.3485	−1.00042	−0.500212	+	0.865903i	$0.666745\pi$
$108$	0	0
$109$	−15.8990	−1.52285	−0.761423	−	0.648255i	$-0.775499\pi$
$109$	−15.8990	−1.52285	−0.761423	+	0.648255i	$0.775499\pi$
$110$	0	0
$111$	−20.6969	−1.96447
$112$	−2.44949	−0.231455
$113$	−15.6969	−1.47664	−0.738322	−	0.674449i	$-0.764381\pi$
$113$	−15.6969	−1.47664	−0.738322	+	0.674449i	$0.764381\pi$
$114$	0	0
$115$	0	0
$116$	1.89898	0.176316
$117$	−11.6969	−1.08138
$118$	−2.89898	−0.266873
$119$	10.8990	0.999108
$120$	0	0
$121$	1.00000	0.0909091
$122$	7.89898	0.715140
$123$	18.4949	1.66763
$124$	−2.00000	−0.179605
$125$	0	0
$126$	−7.34847	−0.654654
$127$	−12.6969	−1.12667	−0.563336	−	0.826228i	$-0.690482\pi$
$127$	−12.6969	−1.12667	−0.563336	+	0.826228i	$0.690482\pi$
$128$	1.00000	0.0883883
$129$	23.1464	2.03793
$130$	0	0
$131$	4.34847	0.379928	0.189964	−	0.981791i	$-0.439163\pi$
$131$	4.34847	0.379928	0.189964	+	0.981791i	$0.439163\pi$
$132$	−2.44949	−0.213201
$133$	0	0
$134$	−9.55051	−0.825038
$135$	0	0
$136$	−4.44949	−0.381541
$137$	−7.89898	−0.674855	−0.337428	−	0.941351i	$-0.609557\pi$
$137$	−7.89898	−0.674855	−0.337428	+	0.941351i	$0.609557\pi$
$138$	2.20204	0.187450
$139$	−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
$141$	−16.0454	−1.35127
$142$	−7.44949	−0.625147
$143$	3.89898	0.326049
$144$	3.00000	0.250000
$145$	0	0
$146$	−14.4495	−1.19585
$147$	−2.44949	−0.202031
$148$	−8.44949	−0.694544
$149$	18.6969	1.53171	0.765856	−	0.643012i	$-0.222315\pi$
$149$	18.6969	1.53171	0.765856	+	0.643012i	$0.222315\pi$
$150$	−12.2474	−1.00000
$151$	8.44949	0.687610	0.343805	−	0.939041i	$-0.388284\pi$
$151$	8.44949	0.687610	0.343805	+	0.939041i	$0.388284\pi$
$152$	0	0
$153$	−13.3485	−1.07916
$154$	2.44949	0.197386
$155$	0	0
$156$	−9.55051	−0.764653
$157$	−6.00000	−0.478852	−0.239426	−	0.970915i	$-0.576959\pi$
$157$	−6.00000	−0.478852	−0.239426	+	0.970915i	$0.576959\pi$
$158$	−4.44949	−0.353982
$159$	19.1010	1.51481
$160$	0	0
$161$	−2.20204	−0.173545
$162$	−9.00000	−0.707107
$163$	2.89898	0.227066	0.113533	−	0.993534i	$-0.463783\pi$
$163$	2.89898	0.227066	0.113533	+	0.993534i	$0.463783\pi$
$164$	7.55051	0.589596
$165$	0	0
$166$	−5.44949	−0.422962
$167$	13.1464	1.01730	0.508651	−	0.860973i	$-0.330145\pi$
$167$	13.1464	1.01730	0.508651	+	0.860973i	$0.330145\pi$
$168$	−6.00000	−0.462910
$169$	2.20204	0.169388
$170$	0	0
$171$	0	0
$172$	9.44949	0.720517
$173$	−9.79796	−0.744925	−0.372463	−	0.928047i	$-0.621486\pi$
$173$	−9.79796	−0.744925	−0.372463	+	0.928047i	$0.621486\pi$
$174$	4.65153	0.352632
$175$	12.2474	0.925820
$176$	−1.00000	−0.0753778
$177$	−7.10102	−0.533745
$178$	9.00000	0.674579
$179$	9.79796	0.732334	0.366167	−	0.930549i	$-0.380670\pi$
$179$	9.79796	0.732334	0.366167	+	0.930549i	$0.380670\pi$
$180$	0	0
$181$	−4.24745	−0.315710	−0.157855	−	0.987462i	$-0.550458\pi$
$181$	−4.24745	−0.315710	−0.157855	+	0.987462i	$0.550458\pi$
$182$	9.55051	0.707931
$183$	19.3485	1.43028
$184$	0.898979	0.0662736
$185$	0	0
$186$	−4.89898	−0.359211
$187$	4.44949	0.325379
$188$	−6.55051	−0.477745
$189$	0	0
$190$	0	0
$191$	−7.65153	−0.553645	−0.276823	−	0.960921i	$-0.589281\pi$
$191$	−7.65153	−0.553645	−0.276823	+	0.960921i	$0.589281\pi$
$192$	2.44949	0.176777
$193$	23.3485	1.68066	0.840330	−	0.542075i	$-0.182361\pi$
$193$	23.3485	1.68066	0.840330	+	0.542075i	$0.182361\pi$
$194$	−3.89898	−0.279930
$195$	0	0
$196$	−1.00000	−0.0714286
$197$	6.10102	0.434680	0.217340	−	0.976096i	$-0.430262\pi$
$197$	6.10102	0.434680	0.217340	+	0.976096i	$0.430262\pi$
$198$	−3.00000	−0.213201
$199$	20.3485	1.44246	0.721232	−	0.692693i	$-0.243576\pi$
$199$	20.3485	1.44246	0.721232	+	0.692693i	$0.243576\pi$
$200$	−5.00000	−0.353553
$201$	−23.3939	−1.65008
$202$	5.00000	0.351799
$203$	−4.65153	−0.326473
$204$	−10.8990	−0.763081
$205$	0	0
$206$	8.34847	0.581665
$207$	2.69694	0.187450
$208$	−3.89898	−0.270346
$209$	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	$0.741699\pi$
$212$	7.79796	0.535566
$213$	−18.2474	−1.25029
$214$	−10.3485	−0.707407
$215$	0	0
$216$	0	0
$217$	4.89898	0.332564
$218$	−15.8990	−1.07681
$219$	−35.3939	−2.39170
$220$	0	0
$221$	17.3485	1.16698
$222$	−20.6969	−1.38909
$223$	−1.24745	−0.0835353	−0.0417677	−	0.999127i	$-0.513299\pi$
$223$	−1.24745	−0.0835353	−0.0417677	+	0.999127i	$0.513299\pi$
$224$	−2.44949	−0.163663
$225$	−15.0000	−1.00000
$226$	−15.6969	−1.04414
$227$	−4.20204	−0.278899	−0.139450	−	0.990229i	$-0.544533\pi$
$227$	−4.20204	−0.278899	−0.139450	+	0.990229i	$0.544533\pi$
$228$	0	0
$229$	−0.651531	−0.0430544	−0.0215272	−	0.999768i	$-0.506853\pi$
$229$	−0.651531	−0.0430544	−0.0215272	+	0.999768i	$0.506853\pi$
$230$	0	0
$231$	6.00000	0.394771
$232$	1.89898	0.124674
$233$	−10.4495	−0.684569	−0.342284	−	0.939596i	$-0.611201\pi$
$233$	−10.4495	−0.684569	−0.342284	+	0.939596i	$0.611201\pi$
$234$	−11.6969	−0.764653
$235$	0	0
$236$	−2.89898	−0.188707
$237$	−10.8990	−0.707965
$238$	10.8990	0.706476
$239$	21.7980	1.40999	0.704996	−	0.709211i	$-0.250949\pi$
$239$	21.7980	1.40999	0.704996	+	0.709211i	$0.250949\pi$
$240$	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
$244$	7.89898	0.505680
$245$	0	0
$246$	18.4949	1.17919
$247$	0	0
$248$	−2.00000	−0.127000
$249$	−13.3485	−0.845925
$250$	0	0
$251$	24.2474	1.53049	0.765243	−	0.643742i	$-0.222619\pi$
$251$	24.2474	1.53049	0.765243	+	0.643742i	$0.222619\pi$
$252$	−7.34847	−0.462910
$253$	−0.898979	−0.0565184
$254$	−12.6969	−0.796677
$255$	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	$0.259294\pi$
$258$	23.1464	1.44103
$259$	20.6969	1.28605
$260$	0	0
$261$	5.69694	0.352632
$262$	4.34847	0.268649
$263$	1.79796	0.110867	0.0554334	−	0.998462i	$-0.482346\pi$
$263$	1.79796	0.110867	0.0554334	+	0.998462i	$0.482346\pi$
$264$	−2.44949	−0.150756
$265$	0	0
$266$	0	0
$267$	22.0454	1.34916
$268$	−9.55051	−0.583390
$269$	−20.4495	−1.24683	−0.623414	−	0.781892i	$-0.714255\pi$
$269$	−20.4495	−1.24683	−0.623414	+	0.781892i	$0.714255\pi$
$270$	0	0
$271$	24.6969	1.50023	0.750116	−	0.661306i	$-0.229998\pi$
$271$	24.6969	1.50023	0.750116	+	0.661306i	$0.229998\pi$
$272$	−4.44949	−0.269790
$273$	23.3939	1.41586
$274$	−7.89898	−0.477195
$275$	5.00000	0.301511
$276$	2.20204	0.132547
$277$	−31.4949	−1.89234	−0.946172	−	0.323663i	$-0.895086\pi$
$277$	−31.4949	−1.89234	−0.946172	+	0.323663i	$0.895086\pi$
$278$	−4.00000	−0.239904
$279$	−6.00000	−0.359211
$280$	0	0
$281$	17.1464	1.02287	0.511435	−	0.859322i	$-0.329114\pi$
$281$	17.1464	1.02287	0.511435	+	0.859322i	$0.329114\pi$
$282$	−16.0454	−0.955490
$283$	3.10102	0.184337	0.0921683	−	0.995743i	$-0.470620\pi$
$283$	3.10102	0.184337	0.0921683	+	0.995743i	$0.470620\pi$
$284$	−7.44949	−0.442046
$285$	0	0
$286$	3.89898	0.230551
$287$	−18.4949	−1.09172
$288$	3.00000	0.176777
$289$	2.79796	0.164586
$290$	0	0
$291$	−9.55051	−0.559861
$292$	−14.4495	−0.845592
$293$	7.89898	0.461463	0.230732	−	0.973017i	$-0.425888\pi$
$293$	7.89898	0.461463	0.230732	+	0.973017i	$0.425888\pi$
$294$	−2.44949	−0.142857
$295$	0	0
$296$	−8.44949	−0.491117
$297$	0	0
$298$	18.6969	1.08308
$299$	−3.50510	−0.202705
$300$	−12.2474	−0.707107
$301$	−23.1464	−1.33414
$302$	8.44949	0.486213
$303$	12.2474	0.703598
$304$	0	0
$305$	0	0
$306$	−13.3485	−0.763081
$307$	19.2474	1.09851	0.549255	−	0.835655i	$-0.314912\pi$
$307$	19.2474	1.09851	0.549255	+	0.835655i	$0.314912\pi$
$308$	2.44949	0.139573
$309$	20.4495	1.16333
$310$	0	0
$311$	7.24745	0.410965	0.205483	−	0.978661i	$-0.434124\pi$
$311$	7.24745	0.410965	0.205483	+	0.978661i	$0.434124\pi$
$312$	−9.55051	−0.540691
$313$	−19.6969	−1.11334	−0.556668	−	0.830735i	$-0.687921\pi$
$313$	−19.6969	−1.11334	−0.556668	+	0.830735i	$0.687921\pi$
$314$	−6.00000	−0.338600
$315$	0	0
$316$	−4.44949	−0.250303
$317$	24.0454	1.35052	0.675262	−	0.737578i	$-0.264030\pi$
$317$	24.0454	1.35052	0.675262	+	0.737578i	$0.264030\pi$
$318$	19.1010	1.07113
$319$	−1.89898	−0.106322
$320$	0	0
$321$	−25.3485	−1.41481
$322$	−2.20204	−0.122715
$323$	0	0
$324$	−9.00000	−0.500000
$325$	19.4949	1.08138
$326$	2.89898	0.160560
$327$	−38.9444	−2.15363
$328$	7.55051	0.416907
$329$	16.0454	0.884612
$330$	0	0
$331$	−5.34847	−0.293978	−0.146989	−	0.989138i	$-0.546958\pi$
$331$	−5.34847	−0.293978	−0.146989	+	0.989138i	$0.546958\pi$
$332$	−5.44949	−0.299080
$333$	−25.3485	−1.38909
$334$	13.1464	0.719341
$335$	0	0
$336$	−6.00000	−0.327327
$337$	−12.2020	−0.664688	−0.332344	−	0.943158i	$-0.607839\pi$
$337$	−12.2020	−0.664688	−0.332344	+	0.943158i	$0.607839\pi$
$338$	2.20204	0.119775
$339$	−38.4495	−2.08829
$340$	0	0
$341$	2.00000	0.108306
$342$	0	0
$343$	19.5959	1.05808
$344$	9.44949	0.509482
$345$	0	0
$346$	−9.79796	−0.526742
$347$	−7.44949	−0.399910	−0.199955	−	0.979805i	$-0.564080\pi$
$347$	−7.44949	−0.399910	−0.199955	+	0.979805i	$0.564080\pi$
$348$	4.65153	0.249348
$349$	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
$353$	27.4949	1.46341	0.731703	−	0.681624i	$-0.238726\pi$
$353$	27.4949	1.46341	0.731703	+	0.681624i	$0.238726\pi$
$354$	−7.10102	−0.377415
$355$	0	0
$356$	9.00000	0.476999
$357$	26.6969	1.41295
$358$	9.79796	0.517838
$359$	24.6969	1.30345	0.651727	−	0.758453i	$-0.274045\pi$
$359$	24.6969	1.30345	0.651727	+	0.758453i	$0.274045\pi$
$360$	0	0
$361$	0	0
$362$	−4.24745	−0.223241
$363$	2.44949	0.128565
$364$	9.55051	0.500583
$365$	0	0
$366$	19.3485	1.01136
$367$	−25.4495	−1.32845	−0.664226	−	0.747532i	$-0.731239\pi$
$367$	−25.4495	−1.32845	−0.664226	+	0.747532i	$0.731239\pi$
$368$	0.898979	0.0468625
$369$	22.6515	1.17919
$370$	0	0
$371$	−19.1010	−0.991676
$372$	−4.89898	−0.254000
$373$	17.7980	0.921543	0.460772	−	0.887519i	$-0.347573\pi$
$373$	17.7980	0.921543	0.460772	+	0.887519i	$0.347573\pi$
$374$	4.44949	0.230078
$375$	0	0
$376$	−6.55051	−0.337817
$377$	−7.40408	−0.381330
$378$	0	0
$379$	3.34847	0.171999	0.0859997	−	0.996295i	$-0.472592\pi$
$379$	3.34847	0.171999	0.0859997	+	0.996295i	$0.472592\pi$
$380$	0	0
$381$	−31.1010	−1.59335
$382$	−7.65153	−0.391486
$383$	26.6969	1.36415	0.682075	−	0.731282i	$-0.261078\pi$
$383$	26.6969	1.36415	0.682075	+	0.731282i	$0.261078\pi$
$384$	2.44949	0.125000
$385$	0	0
$386$	23.3485	1.18841
$387$	28.3485	1.44103
$388$	−3.89898	−0.197941
$389$	−8.24745	−0.418162	−0.209081	−	0.977898i	$-0.567047\pi$
$389$	−8.24745	−0.418162	−0.209081	+	0.977898i	$0.567047\pi$
$390$	0	0
$391$	−4.00000	−0.202289
$392$	−1.00000	−0.0505076
$393$	10.6515	0.537299
$394$	6.10102	0.307365
$395$	0	0
$396$	−3.00000	−0.150756
$397$	−25.3485	−1.27220	−0.636102	−	0.771605i	$-0.719454\pi$
$397$	−25.3485	−1.27220	−0.636102	+	0.771605i	$0.719454\pi$
$398$	20.3485	1.01998
$399$	0	0
$400$	−5.00000	−0.250000
$401$	−38.6969	−1.93243	−0.966216	−	0.257732i	$-0.917025\pi$
$401$	−38.6969	−1.93243	−0.966216	+	0.257732i	$0.917025\pi$
$402$	−23.3939	−1.16678
$403$	7.79796	0.388444
$404$	5.00000	0.248759
$405$	0	0
$406$	−4.65153	−0.230852
$407$	8.44949	0.418826
$408$	−10.8990	−0.539580
$409$	22.9444	1.13453	0.567263	−	0.823536i	$-0.308002\pi$
$409$	22.9444	1.13453	0.567263	+	0.823536i	$0.308002\pi$
$410$	0	0
$411$	−19.3485	−0.954390
$412$	8.34847	0.411300
$413$	7.10102	0.349418
$414$	2.69694	0.132547
$415$	0	0
$416$	−3.89898	−0.191163
$417$	−9.79796	−0.479808
$418$	0	0
$419$	−17.3485	−0.847528	−0.423764	−	0.905773i	$-0.639291\pi$
$419$	−17.3485	−0.847528	−0.423764	+	0.905773i	$0.639291\pi$
$420$	0	0
$421$	35.3485	1.72278	0.861389	−	0.507945i	$-0.169595\pi$
$421$	35.3485	1.72278	0.861389	+	0.507945i	$0.169595\pi$
$422$	−20.0000	−0.973585
$423$	−19.6515	−0.955490
$424$	7.79796	0.378702
$425$	22.2474	1.07916
$426$	−18.2474	−0.884092
$427$	−19.3485	−0.936338
$428$	−10.3485	−0.500212
$429$	9.55051	0.461103
$430$	0	0
$431$	−16.0454	−0.772880	−0.386440	−	0.922315i	$-0.626295\pi$
$431$	−16.0454	−0.772880	−0.386440	+	0.922315i	$0.626295\pi$
$432$	0	0
$433$	−31.6969	−1.52326	−0.761629	−	0.648014i	$-0.775600\pi$
$433$	−31.6969	−1.52326	−0.761629	+	0.648014i	$0.775600\pi$
$434$	4.89898	0.235159
$435$	0	0
$436$	−15.8990	−0.761423
$437$	0	0
$438$	−35.3939	−1.69118
$439$	−11.5959	−0.553443	−0.276721	−	0.960950i	$-0.589248\pi$
$439$	−11.5959	−0.553443	−0.276721	+	0.960950i	$0.589248\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	17.3485	0.825183
$443$	−21.3485	−1.01430	−0.507148	−	0.861859i	$-0.669300\pi$
$443$	−21.3485	−1.01430	−0.507148	+	0.861859i	$0.669300\pi$
$444$	−20.6969	−0.982233
$445$	0	0
$446$	−1.24745	−0.0590684
$447$	45.7980	2.16617
$448$	−2.44949	−0.115728
$449$	−10.2020	−0.481464	−0.240732	−	0.970592i	$-0.577388\pi$
$449$	−10.2020	−0.481464	−0.240732	+	0.970592i	$0.577388\pi$
$450$	−15.0000	−0.707107
$451$	−7.55051	−0.355540
$452$	−15.6969	−0.738322
$453$	20.6969	0.972427
$454$	−4.20204	−0.197212
$455$	0	0
$456$	0	0
$457$	−22.9444	−1.07329	−0.536647	−	0.843807i	$-0.680309\pi$
$457$	−22.9444	−1.07329	−0.536647	+	0.843807i	$0.680309\pi$
$458$	−0.651531	−0.0304440
$459$	0	0
$460$	0	0
$461$	−12.7980	−0.596060	−0.298030	−	0.954556i	$-0.596330\pi$
$461$	−12.7980	−0.596060	−0.298030	+	0.954556i	$0.596330\pi$
$462$	6.00000	0.279145
$463$	40.1464	1.86576	0.932881	−	0.360184i	$-0.117286\pi$
$463$	40.1464	1.86576	0.932881	+	0.360184i	$0.117286\pi$
$464$	1.89898	0.0881579
$465$	0	0
$466$	−10.4495	−0.484063
$467$	−6.65153	−0.307796	−0.153898	−	0.988087i	$-0.549183\pi$
$467$	−6.65153	−0.307796	−0.153898	+	0.988087i	$0.549183\pi$
$468$	−11.6969	−0.540691
$469$	23.3939	1.08023
$470$	0	0
$471$	−14.6969	−0.677199
$472$	−2.89898	−0.133436
$473$	−9.44949	−0.434488
$474$	−10.8990	−0.500607
$475$	0	0
$476$	10.8990	0.499554
$477$	23.3939	1.07113
$478$	21.7980	0.997015
$479$	−43.1464	−1.97141	−0.985705	−	0.168479i	$-0.946115\pi$
$479$	−43.1464	−1.97141	−0.985705	+	0.168479i	$0.946115\pi$
$480$	0	0
$481$	32.9444	1.50213
$482$	−10.0000	−0.455488
$483$	−5.39388	−0.245430
$484$	1.00000	0.0454545
$485$	0	0
$486$	−22.0454	−1.00000
$487$	7.65153	0.346724	0.173362	−	0.984858i	$-0.444537\pi$
$487$	7.65153	0.346724	0.173362	+	0.984858i	$0.444537\pi$
$488$	7.89898	0.357570
$489$	7.10102	0.321119
$490$	0	0
$491$	−13.7980	−0.622693	−0.311347	−	0.950296i	$-0.600780\pi$
$491$	−13.7980	−0.622693	−0.311347	+	0.950296i	$0.600780\pi$
$492$	18.4949	0.833814
$493$	−8.44949	−0.380546
$494$	0	0
$495$	0	0
$496$	−2.00000	−0.0898027
$497$	18.2474	0.818510
$498$	−13.3485	−0.598159
$499$	−0.696938	−0.0311993	−0.0155996	−	0.999878i	$-0.504966\pi$
$499$	−0.696938	−0.0311993	−0.0155996	+	0.999878i	$0.504966\pi$
$500$	0	0
$501$	32.2020	1.43868
$502$	24.2474	1.08222
$503$	−10.8990	−0.485961	−0.242981	−	0.970031i	$-0.578125\pi$
$503$	−10.8990	−0.485961	−0.242981	+	0.970031i	$0.578125\pi$
$504$	−7.34847	−0.327327
$505$	0	0
$506$	−0.898979	−0.0399645
$507$	5.39388	0.239550
$508$	−12.6969	−0.563336
$509$	−23.5959	−1.04587	−0.522935	−	0.852372i	$-0.675163\pi$
$509$	−23.5959	−1.04587	−0.522935	+	0.852372i	$0.675163\pi$
$510$	0	0
$511$	35.3939	1.56573
$512$	1.00000	0.0441942
$513$	0	0
$514$	22.0000	0.970378
$515$	0	0
$516$	23.1464	1.01896
$517$	6.55051	0.288091
$518$	20.6969	0.909371
$519$	−24.0000	−1.05348
$520$	0	0
$521$	41.6969	1.82678	0.913388	−	0.407090i	$-0.133456\pi$
$521$	41.6969	1.82678	0.913388	+	0.407090i	$0.133456\pi$
$522$	5.69694	0.249348
$523$	28.3485	1.23959	0.619796	−	0.784763i	$-0.287215\pi$
$523$	28.3485	1.23959	0.619796	+	0.784763i	$0.287215\pi$
$524$	4.34847	0.189964
$525$	30.0000	1.30931
$526$	1.79796	0.0783947
$527$	8.89898	0.387646
$528$	−2.44949	−0.106600
$529$	−22.1918	−0.964862
$530$	0	0
$531$	−8.69694	−0.377415
$532$	0	0
$533$	−29.4393	−1.27516
$534$	22.0454	0.953998
$535$	0	0
$536$	−9.55051	−0.412519
$537$	24.0000	1.03568
$538$	−20.4495	−0.881640
$539$	1.00000	0.0430730
$540$	0	0
$541$	37.7980	1.62506	0.812531	−	0.582919i	$-0.198089\pi$
$541$	37.7980	1.62506	0.812531	+	0.582919i	$0.198089\pi$
$542$	24.6969	1.06082
$543$	−10.4041	−0.446482
$544$	−4.44949	−0.190770
$545$	0	0
$546$	23.3939	1.00117
$547$	−15.6515	−0.669211	−0.334606	−	0.942358i	$-0.608603\pi$
$547$	−15.6515	−0.669211	−0.334606	+	0.942358i	$0.608603\pi$
$548$	−7.89898	−0.337428
$549$	23.6969	1.01136
$550$	5.00000	0.213201
$551$	0	0
$552$	2.20204	0.0937251
$553$	10.8990	0.463472
$554$	−31.4949	−1.33809
$555$	0	0
$556$	−4.00000	−0.169638
$557$	16.5959	0.703192	0.351596	−	0.936152i	$-0.385639\pi$
$557$	16.5959	0.703192	0.351596	+	0.936152i	$0.385639\pi$
$558$	−6.00000	−0.254000
$559$	−36.8434	−1.55831
$560$	0	0
$561$	10.8990	0.460155
$562$	17.1464	0.723278
$563$	−0.898979	−0.0378875	−0.0189437	−	0.999821i	$-0.506030\pi$
$563$	−0.898979	−0.0378875	−0.0189437	+	0.999821i	$0.506030\pi$
$564$	−16.0454	−0.675634
$565$	0	0
$566$	3.10102	0.130346
$567$	22.0454	0.925820
$568$	−7.44949	−0.312574
$569$	33.5959	1.40841	0.704207	−	0.709995i	$-0.251303\pi$
$569$	33.5959	1.40841	0.704207	+	0.709995i	$0.251303\pi$
$570$	0	0
$571$	−19.9444	−0.834647	−0.417323	−	0.908758i	$-0.637032\pi$
$571$	−19.9444	−0.834647	−0.417323	+	0.908758i	$0.637032\pi$
$572$	3.89898	0.163025
$573$	−18.7423	−0.782973
$574$	−18.4949	−0.771962
$575$	−4.49490	−0.187450
$576$	3.00000	0.125000
$577$	−37.7980	−1.57355	−0.786775	−	0.617240i	$-0.788251\pi$
$577$	−37.7980	−1.57355	−0.786775	+	0.617240i	$0.788251\pi$
$578$	2.79796	0.116380
$579$	57.1918	2.37681
$580$	0	0
$581$	13.3485	0.553788
$582$	−9.55051	−0.395881
$583$	−7.79796	−0.322958
$584$	−14.4495	−0.597924
$585$	0	0
$586$	7.89898	0.326304
$587$	−27.3485	−1.12879	−0.564396	−	0.825504i	$-0.690891\pi$
$587$	−27.3485	−1.12879	−0.564396	+	0.825504i	$0.690891\pi$
$588$	−2.44949	−0.101015
$589$	0	0
$590$	0	0
$591$	14.9444	0.614730
$592$	−8.44949	−0.347272
$593$	25.3939	1.04280	0.521401	−	0.853312i	$-0.325410\pi$
$593$	25.3939	1.04280	0.521401	+	0.853312i	$0.325410\pi$
$594$	0	0
$595$	0	0
$596$	18.6969	0.765856
$597$	49.8434	2.03995
$598$	−3.50510	−0.143334
$599$	−34.3485	−1.40344	−0.701720	−	0.712453i	$-0.747584\pi$
$599$	−34.3485	−1.40344	−0.701720	+	0.712453i	$0.747584\pi$
$600$	−12.2474	−0.500000
$601$	−4.40408	−0.179646	−0.0898231	−	0.995958i	$-0.528630\pi$
$601$	−4.40408	−0.179646	−0.0898231	+	0.995958i	$0.528630\pi$
$602$	−23.1464	−0.943378
$603$	−28.6515	−1.16678
$604$	8.44949	0.343805
$605$	0	0
$606$	12.2474	0.497519
$607$	19.1464	0.777130	0.388565	−	0.921421i	$-0.372971\pi$
$607$	19.1464	0.777130	0.388565	+	0.921421i	$0.372971\pi$
$608$	0	0
$609$	−11.3939	−0.461703
$610$	0	0
$611$	25.5403	1.03325
$612$	−13.3485	−0.539580
$613$	−26.7980	−1.08236	−0.541180	−	0.840907i	$-0.682022\pi$
$613$	−26.7980	−1.08236	−0.541180	+	0.840907i	$0.682022\pi$
$614$	19.2474	0.776764
$615$	0	0
$616$	2.44949	0.0986928
$617$	−38.3939	−1.54568	−0.772840	−	0.634601i	$-0.781164\pi$
$617$	−38.3939	−1.54568	−0.772840	+	0.634601i	$0.781164\pi$
$618$	20.4495	0.822599
$619$	−24.6969	−0.992654	−0.496327	−	0.868136i	$-0.665318\pi$
$619$	−24.6969	−0.992654	−0.496327	+	0.868136i	$0.665318\pi$
$620$	0	0
$621$	0	0
$622$	7.24745	0.290596
$623$	−22.0454	−0.883231
$624$	−9.55051	−0.382326
$625$	25.0000	1.00000
$626$	−19.6969	−0.787248
$627$	0	0
$628$	−6.00000	−0.239426
$629$	37.5959	1.49905
$630$	0	0
$631$	0.954592	0.0380017	0.0190009	−	0.999819i	$-0.493951\pi$
$631$	0.954592	0.0380017	0.0190009	+	0.999819i	$0.493951\pi$
$632$	−4.44949	−0.176991
$633$	−48.9898	−1.94717
$634$	24.0454	0.954965
$635$	0	0
$636$	19.1010	0.757405
$637$	3.89898	0.154483
$638$	−1.89898	−0.0751813
$639$	−22.3485	−0.884092
$640$	0	0
$641$	43.4949	1.71795	0.858973	−	0.512022i	$-0.171103\pi$
$641$	43.4949	1.71795	0.858973	+	0.512022i	$0.171103\pi$
$642$	−25.3485	−1.00042
$643$	−3.34847	−0.132051	−0.0660254	−	0.997818i	$-0.521032\pi$
$643$	−3.34847	−0.132051	−0.0660254	+	0.997818i	$0.521032\pi$
$644$	−2.20204	−0.0867726
$645$	0	0
$646$	0	0
$647$	−39.2474	−1.54298	−0.771488	−	0.636244i	$-0.780487\pi$
$647$	−39.2474	−1.54298	−0.771488	+	0.636244i	$0.780487\pi$
$648$	−9.00000	−0.353553
$649$	2.89898	0.113795
$650$	19.4949	0.764653
$651$	12.0000	0.470317
$652$	2.89898	0.113533
$653$	3.34847	0.131036	0.0655179	−	0.997851i	$-0.479130\pi$
$653$	3.34847	0.131036	0.0655179	+	0.997851i	$0.479130\pi$
$654$	−38.9444	−1.52285
$655$	0	0
$656$	7.55051	0.294798
$657$	−43.3485	−1.69118
$658$	16.0454	0.625515
$659$	21.9444	0.854832	0.427416	−	0.904055i	$-0.359424\pi$
$659$	21.9444	0.854832	0.427416	+	0.904055i	$0.359424\pi$
$660$	0	0
$661$	−9.14643	−0.355755	−0.177877	−	0.984053i	$-0.556923\pi$
$661$	−9.14643	−0.355755	−0.177877	+	0.984053i	$0.556923\pi$
$662$	−5.34847	−0.207874
$663$	42.4949	1.65037
$664$	−5.44949	−0.211481
$665$	0	0
$666$	−25.3485	−0.982233
$667$	1.70714	0.0661009
$668$	13.1464	0.508651
$669$	−3.05561	−0.118137
$670$	0	0
$671$	−7.89898	−0.304937
$672$	−6.00000	−0.231455
$673$	42.7423	1.64760	0.823798	−	0.566883i	$-0.191851\pi$
$673$	42.7423	1.64760	0.823798	+	0.566883i	$0.191851\pi$
$674$	−12.2020	−0.470005
$675$	0	0
$676$	2.20204	0.0846939
$677$	−13.6969	−0.526416	−0.263208	−	0.964739i	$-0.584781\pi$
$677$	−13.6969	−0.526416	−0.263208	+	0.964739i	$0.584781\pi$
$678$	−38.4495	−1.47664
$679$	9.55051	0.366515
$680$	0	0
$681$	−10.2929	−0.394423
$682$	2.00000	0.0765840
$683$	35.1464	1.34484	0.672420	−	0.740169i	$-0.265255\pi$
$683$	35.1464	1.34484	0.672420	+	0.740169i	$0.265255\pi$
$684$	0	0
$685$	0	0
$686$	19.5959	0.748176
$687$	−1.59592	−0.0608881
$688$	9.44949	0.360258
$689$	−30.4041	−1.15830
$690$	0	0
$691$	−39.6413	−1.50803	−0.754014	−	0.656859i	$-0.771885\pi$
$691$	−39.6413	−1.50803	−0.754014	+	0.656859i	$0.771885\pi$
$692$	−9.79796	−0.372463
$693$	7.34847	0.279145
$694$	−7.44949	−0.282779
$695$	0	0
$696$	4.65153	0.176316
$697$	−33.5959	−1.27254
$698$	31.0000	1.17337
$699$	−25.5959	−0.968127
$700$	12.2474	0.462910
$701$	−46.5959	−1.75990	−0.879952	−	0.475063i	$-0.842425\pi$
$701$	−46.5959	−1.75990	−0.879952	+	0.475063i	$0.842425\pi$
$702$	0	0
$703$	0	0
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	27.4949	1.03478
$707$	−12.2474	−0.460613
$708$	−7.10102	−0.266873
$709$	15.3939	0.578129	0.289065	−	0.957310i	$-0.406656\pi$
$709$	15.3939	0.578129	0.289065	+	0.957310i	$0.406656\pi$
$710$	0	0
$711$	−13.3485	−0.500607
$712$	9.00000	0.337289
$713$	−1.79796	−0.0673341
$714$	26.6969	0.999108
$715$	0	0
$716$	9.79796	0.366167
$717$	53.3939	1.99403
$718$	24.6969	0.921682
$719$	−5.85357	−0.218301	−0.109151	−	0.994025i	$-0.534813\pi$
$719$	−5.85357	−0.218301	−0.109151	+	0.994025i	$0.534813\pi$
$720$	0	0
$721$	−20.4495	−0.761579
$722$	0	0
$723$	−24.4949	−0.910975
$724$	−4.24745	−0.157855
$725$	−9.49490	−0.352632
$726$	2.44949	0.0909091
$727$	9.94439	0.368817	0.184408	−	0.982850i	$-0.440963\pi$
$727$	9.94439	0.368817	0.184408	+	0.982850i	$0.440963\pi$
$728$	9.55051	0.353965
$729$	−27.0000	−1.00000
$730$	0	0
$731$	−42.0454	−1.55511
$732$	19.3485	0.715140
$733$	31.1010	1.14874	0.574371	−	0.818595i	$-0.305247\pi$
$733$	31.1010	1.14874	0.574371	+	0.818595i	$0.305247\pi$
$734$	−25.4495	−0.939358
$735$	0	0
$736$	0.898979	0.0331368
$737$	9.55051	0.351798
$738$	22.6515	0.833814
$739$	31.7423	1.16766	0.583831	−	0.811876i	$-0.301553\pi$
$739$	31.7423	1.16766	0.583831	+	0.811876i	$0.301553\pi$
$740$	0	0
$741$	0	0
$742$	−19.1010	−0.701221
$743$	37.1464	1.36277	0.681385	−	0.731925i	$-0.261378\pi$
$743$	37.1464	1.36277	0.681385	+	0.731925i	$0.261378\pi$
$744$	−4.89898	−0.179605
$745$	0	0
$746$	17.7980	0.651630
$747$	−16.3485	−0.598159
$748$	4.44949	0.162689
$749$	25.3485	0.926213
$750$	0	0
$751$	47.5403	1.73477	0.867385	−	0.497637i	$-0.165799\pi$
$751$	47.5403	1.73477	0.867385	+	0.497637i	$0.165799\pi$
$752$	−6.55051	−0.238873
$753$	59.3939	2.16443
$754$	−7.40408	−0.269641
$755$	0	0
$756$	0	0
$757$	16.2020	0.588873	0.294437	−	0.955671i	$-0.404868\pi$
$757$	16.2020	0.588873	0.294437	+	0.955671i	$0.404868\pi$
$758$	3.34847	0.121622
$759$	−2.20204	−0.0799290
$760$	0	0
$761$	31.1918	1.13070	0.565352	−	0.824850i	$-0.308741\pi$
$761$	31.1918	1.13070	0.565352	+	0.824850i	$0.308741\pi$
$762$	−31.1010	−1.12667
$763$	38.9444	1.40988
$764$	−7.65153	−0.276823
$765$	0	0
$766$	26.6969	0.964600
$767$	11.3031	0.408130
$768$	2.44949	0.0883883
$769$	−39.3939	−1.42058	−0.710290	−	0.703909i	$-0.751436\pi$
$769$	−39.3939	−1.42058	−0.710290	+	0.703909i	$0.751436\pi$
$770$	0	0
$771$	53.8888	1.94076
$772$	23.3485	0.840330
$773$	−11.1464	−0.400909	−0.200455	−	0.979703i	$-0.564242\pi$
$773$	−11.1464	−0.400909	−0.200455	+	0.979703i	$0.564242\pi$
$774$	28.3485	1.01896
$775$	10.0000	0.359211
$776$	−3.89898	−0.139965
$777$	50.6969	1.81874
$778$	−8.24745	−0.295685
$779$	0	0
$780$	0	0
$781$	7.44949	0.266564
$782$	−4.00000	−0.143040
$783$	0	0
$784$	−1.00000	−0.0357143
$785$	0	0
$786$	10.6515	0.379928
$787$	−25.2474	−0.899974	−0.449987	−	0.893035i	$-0.648571\pi$
$787$	−25.2474	−0.899974	−0.449987	+	0.893035i	$0.648571\pi$
$788$	6.10102	0.217340
$789$	4.40408	0.156789
$790$	0	0
$791$	38.4495	1.36711
$792$	−3.00000	−0.106600
$793$	−30.7980	−1.09367
$794$	−25.3485	−0.899584
$795$	0	0
$796$	20.3485	0.721232
$797$	−2.04541	−0.0724521	−0.0362260	−	0.999344i	$-0.511534\pi$
$797$	−2.04541	−0.0724521	−0.0362260	+	0.999344i	$0.511534\pi$
$798$	0	0
$799$	29.1464	1.03113
$800$	−5.00000	−0.176777
$801$	27.0000	0.953998
$802$	−38.6969	−1.36644
$803$	14.4495	0.509911
$804$	−23.3939	−0.825038
$805$	0	0
$806$	7.79796	0.274671
$807$	−50.0908	−1.76328
$808$	5.00000	0.175899
$809$	0.853572	0.0300100	0.0150050	−	0.999887i	$-0.495224\pi$
$809$	0.853572	0.0300100	0.0150050	+	0.999887i	$0.495224\pi$
$810$	0	0
$811$	2.55051	0.0895605	0.0447803	−	0.998997i	$-0.485741\pi$
$811$	2.55051	0.0895605	0.0447803	+	0.998997i	$0.485741\pi$
$812$	−4.65153	−0.163237
$813$	60.4949	2.12165
$814$	8.44949	0.296154
$815$	0	0
$816$	−10.8990	−0.381541
$817$	0	0
$818$	22.9444	0.802232
$819$	28.6515	1.00117
$820$	0	0
$821$	46.2929	1.61563	0.807816	−	0.589435i	$-0.200650\pi$
$821$	46.2929	1.61563	0.807816	+	0.589435i	$0.200650\pi$
$822$	−19.3485	−0.674855
$823$	−9.44949	−0.329389	−0.164694	−	0.986345i	$-0.552664\pi$
$823$	−9.44949	−0.329389	−0.164694	+	0.986345i	$0.552664\pi$
$824$	8.34847	0.290833
$825$	12.2474	0.426401
$826$	7.10102	0.247076
$827$	28.8434	1.00298	0.501491	−	0.865163i	$-0.332785\pi$
$827$	28.8434	1.00298	0.501491	+	0.865163i	$0.332785\pi$
$828$	2.69694	0.0937251
$829$	−3.14643	−0.109280	−0.0546400	−	0.998506i	$-0.517401\pi$
$829$	−3.14643	−0.109280	−0.0546400	+	0.998506i	$0.517401\pi$
$830$	0	0
$831$	−77.1464	−2.67618
$832$	−3.89898	−0.135173
$833$	4.44949	0.154166
$834$	−9.79796	−0.339276
$835$	0	0
$836$	0	0
$837$	0	0
$838$	−17.3485	−0.599293
$839$	−20.7526	−0.716458	−0.358229	−	0.933634i	$-0.616619\pi$
$839$	−20.7526	−0.716458	−0.358229	+	0.933634i	$0.616619\pi$
$840$	0	0
$841$	−25.3939	−0.875651
$842$	35.3485	1.21819
$843$	42.0000	1.44656
$844$	−20.0000	−0.688428
$845$	0	0
$846$	−19.6515	−0.675634
$847$	−2.44949	−0.0841655
$848$	7.79796	0.267783
$849$	7.59592	0.260691
$850$	22.2474	0.763081
$851$	−7.59592	−0.260385
$852$	−18.2474	−0.625147
$853$	−2.69694	−0.0923414	−0.0461707	−	0.998934i	$-0.514702\pi$
$853$	−2.69694	−0.0923414	−0.0461707	+	0.998934i	$0.514702\pi$
$854$	−19.3485	−0.662091
$855$	0	0
$856$	−10.3485	−0.353703
$857$	−11.1464	−0.380755	−0.190377	−	0.981711i	$-0.560971\pi$
$857$	−11.1464	−0.380755	−0.190377	+	0.981711i	$0.560971\pi$
$858$	9.55051	0.326049
$859$	−1.10102	−0.0375663	−0.0187832	−	0.999824i	$-0.505979\pi$
$859$	−1.10102	−0.0375663	−0.0187832	+	0.999824i	$0.505979\pi$
$860$	0	0
$861$	−45.3031	−1.54392
$862$	−16.0454	−0.546509
$863$	11.6515	0.396623	0.198311	−	0.980139i	$-0.436454\pi$
$863$	11.6515	0.396623	0.198311	+	0.980139i	$0.436454\pi$
$864$	0	0
$865$	0	0
$866$	−31.6969	−1.07711
$867$	6.85357	0.232760
$868$	4.89898	0.166282
$869$	4.44949	0.150939
$870$	0	0
$871$	37.2372	1.26174
$872$	−15.8990	−0.538407
$873$	−11.6969	−0.395881
$874$	0	0
$875$	0	0
$876$	−35.3939	−1.19585
$877$	−37.5959	−1.26952	−0.634762	−	0.772708i	$-0.718902\pi$
$877$	−37.5959	−1.26952	−0.634762	+	0.772708i	$0.718902\pi$
$878$	−11.5959	−0.391343
$879$	19.3485	0.652608
$880$	0	0
$881$	12.8990	0.434578	0.217289	−	0.976107i	$-0.430279\pi$
$881$	12.8990	0.434578	0.217289	+	0.976107i	$0.430279\pi$
$882$	−3.00000	−0.101015
$883$	−12.4495	−0.418959	−0.209479	−	0.977813i	$-0.567177\pi$
$883$	−12.4495	−0.418959	−0.209479	+	0.977813i	$0.567177\pi$
$884$	17.3485	0.583492
$885$	0	0
$886$	−21.3485	−0.717216
$887$	−13.3485	−0.448198	−0.224099	−	0.974566i	$-0.571944\pi$
$887$	−13.3485	−0.448198	−0.224099	+	0.974566i	$0.571944\pi$
$888$	−20.6969	−0.694544
$889$	31.1010	1.04309
$890$	0	0
$891$	9.00000	0.301511
$892$	−1.24745	−0.0417677
$893$	0	0
$894$	45.7980	1.53171
$895$	0	0
$896$	−2.44949	−0.0818317
$897$	−8.58571	−0.286669
$898$	−10.2020	−0.340447
$899$	−3.79796	−0.126669
$900$	−15.0000	−0.500000
$901$	−34.6969	−1.15592
$902$	−7.55051	−0.251404
$903$	−56.6969	−1.88676
$904$	−15.6969	−0.522072
$905$	0	0
$906$	20.6969	0.687610
$907$	−38.8990	−1.29162	−0.645810	−	0.763498i	$-0.723480\pi$
$907$	−38.8990	−1.29162	−0.645810	+	0.763498i	$0.723480\pi$
$908$	−4.20204	−0.139450
$909$	15.0000	0.497519
$910$	0	0
$911$	24.4949	0.811552	0.405776	−	0.913973i	$-0.367001\pi$
$911$	24.4949	0.811552	0.405776	+	0.913973i	$0.367001\pi$
$912$	0	0
$913$	5.44949	0.180352
$914$	−22.9444	−0.758933
$915$	0	0
$916$	−0.651531	−0.0215272
$917$	−10.6515	−0.351745
$918$	0	0
$919$	−25.3485	−0.836169	−0.418084	−	0.908408i	$-0.637298\pi$
$919$	−25.3485	−0.836169	−0.418084	+	0.908408i	$0.637298\pi$
$920$	0	0
$921$	47.1464	1.55353
$922$	−12.7980	−0.421478
$923$	29.0454	0.956041
$924$	6.00000	0.197386
$925$	42.2474	1.38909
$926$	40.1464	1.31929
$927$	25.0454	0.822599
$928$	1.89898	0.0623371
$929$	−37.4949	−1.23017	−0.615084	−	0.788462i	$-0.710878\pi$
$929$	−37.4949	−1.23017	−0.615084	+	0.788462i	$0.710878\pi$
$930$	0	0
$931$	0	0
$932$	−10.4495	−0.342284
$933$	17.7526	0.581192
$934$	−6.65153	−0.217645
$935$	0	0
$936$	−11.6969	−0.382326
$937$	−13.7980	−0.450760	−0.225380	−	0.974271i	$-0.572362\pi$
$937$	−13.7980	−0.450760	−0.225380	+	0.974271i	$0.572362\pi$
$938$	23.3939	0.763837
$939$	−48.2474	−1.57450
$940$	0	0
$941$	45.7980	1.49297	0.746485	−	0.665402i	$-0.231740\pi$
$941$	45.7980	1.49297	0.746485	+	0.665402i	$0.231740\pi$
$942$	−14.6969	−0.478852
$943$	6.78775	0.221040
$944$	−2.89898	−0.0943537
$945$	0	0
$946$	−9.44949	−0.307229
$947$	−48.2929	−1.56931	−0.784653	−	0.619935i	$-0.787159\pi$
$947$	−48.2929	−1.56931	−0.784653	+	0.619935i	$0.787159\pi$
$948$	−10.8990	−0.353982
$949$	56.3383	1.82882
$950$	0	0
$951$	58.8990	1.90993
$952$	10.8990	0.353238
$953$	−40.0000	−1.29573	−0.647864	−	0.761756i	$-0.724337\pi$
$953$	−40.0000	−1.29573	−0.647864	+	0.761756i	$0.724337\pi$
$954$	23.3939	0.757405
$955$	0	0
$956$	21.7980	0.704996
$957$	−4.65153	−0.150363
$958$	−43.1464	−1.39400
$959$	19.3485	0.624795
$960$	0	0
$961$	−27.0000	−0.870968
$962$	32.9444	1.06217
$963$	−31.0454	−1.00042
$964$	−10.0000	−0.322078
$965$	0	0
$966$	−5.39388	−0.173545
$967$	−37.7980	−1.21550	−0.607750	−	0.794128i	$-0.707928\pi$
$967$	−37.7980	−1.21550	−0.607750	+	0.794128i	$0.707928\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	0	0
$971$	43.8434	1.40700	0.703500	−	0.710695i	$-0.251619\pi$
$971$	43.8434	1.40700	0.703500	+	0.710695i	$0.251619\pi$
$972$	−22.0454	−0.707107
$973$	9.79796	0.314108
$974$	7.65153	0.245171
$975$	47.7526	1.52931
$976$	7.89898	0.252840
$977$	57.0000	1.82359	0.911796	−	0.410644i	$-0.134696\pi$
$977$	57.0000	1.82359	0.911796	+	0.410644i	$0.134696\pi$
$978$	7.10102	0.227066
$979$	−9.00000	−0.287641
$980$	0	0
$981$	−47.6969	−1.52285
$982$	−13.7980	−0.440311
$983$	−45.7423	−1.45895	−0.729477	−	0.684005i	$-0.760237\pi$
$983$	−45.7423	−1.45895	−0.729477	+	0.684005i	$0.760237\pi$
$984$	18.4949	0.589596
$985$	0	0
$986$	−8.44949	−0.269087
$987$	39.3031	1.25103
$988$	0	0
$989$	8.49490	0.270122
$990$	0	0
$991$	−1.44949	−0.0460446	−0.0230223	−	0.999735i	$-0.507329\pi$
$991$	−1.44949	−0.0460446	−0.0230223	+	0.999735i	$0.507329\pi$
$992$	−2.00000	−0.0635001
$993$	−13.1010	−0.415748
$994$	18.2474	0.578774
$995$	0	0
$996$	−13.3485	−0.422962
$997$	−29.1918	−0.924515	−0.462257	−	0.886746i	$-0.652960\pi$
$997$	−29.1918	−0.924515	−0.462257	+	0.886746i	$0.652960\pi$
$998$	−0.696938	−0.0220612
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7942.2.a.y.1.2		2
19.8	odd	6		418.2.e.g.45.2	✓	4
19.12	odd	6		418.2.e.g.353.2	yes	4
19.18	odd	2		7942.2.a.v.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
418.2.e.g.45.2	✓	4	19.8	odd	6
418.2.e.g.353.2	yes	4	19.12	odd	6
7942.2.a.v.1.1		2	19.18	odd	2
7942.2.a.y.1.2		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +2.44949 q^{3} +1.00000 q^{4} +2.44949 q^{6} -2.44949 q^{7} +1.00000 q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +2.44949 q^{3} +1.00000 q^{4} +2.44949 q^{6} -2.44949 q^{7} +1.00000 q^{8} +3.00000 q^{9} -1.00000 q^{11} +2.44949 q^{12} -3.89898 q^{13} -2.44949 q^{14} +1.00000 q^{16} -4.44949 q^{17} +3.00000 q^{18} -6.00000 q^{21} -1.00000 q^{22} +0.898979 q^{23} +2.44949 q^{24} -5.00000 q^{25} -3.89898 q^{26} -2.44949 q^{28} +1.89898 q^{29} -2.00000 q^{31} +1.00000 q^{32} -2.44949 q^{33} -4.44949 q^{34} +3.00000 q^{36} -8.44949 q^{37} -9.55051 q^{39} +7.55051 q^{41} -6.00000 q^{42} +9.44949 q^{43} -1.00000 q^{44} +0.898979 q^{46} -6.55051 q^{47} +2.44949 q^{48} -1.00000 q^{49} -5.00000 q^{50} -10.8990 q^{51} -3.89898 q^{52} +7.79796 q^{53} -2.44949 q^{56} +1.89898 q^{58} -2.89898 q^{59} +7.89898 q^{61} -2.00000 q^{62} -7.34847 q^{63} +1.00000 q^{64} -2.44949 q^{66} -9.55051 q^{67} -4.44949 q^{68} +2.20204 q^{69} -7.44949 q^{71} +3.00000 q^{72} -14.4495 q^{73} -8.44949 q^{74} -12.2474 q^{75} +2.44949 q^{77} -9.55051 q^{78} -4.44949 q^{79} -9.00000 q^{81} +7.55051 q^{82} -5.44949 q^{83} -6.00000 q^{84} +9.44949 q^{86} +4.65153 q^{87} -1.00000 q^{88} +9.00000 q^{89} +9.55051 q^{91} +0.898979 q^{92} -4.89898 q^{93} -6.55051 q^{94} +2.44949 q^{96} -3.89898 q^{97} -1.00000 q^{98} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 6 q^{9}+O(q^{10}) \) 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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