LMFDB - Embedded newform 8036.2.a.q.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8036,2,Mod(1,8036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8036, 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("8036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8036 = 2^{2} \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8036.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:	$64.1677830643$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - x^{14} - 37 x^{13} + 39 x^{12} + 537 x^{11} - 616 x^{10} - 3853 x^{9} + 4929 x^{8} + \cdots + 3381 $ x^15 - x^14 - 37x^13 + 39x^12 + 537x^11 - 616x^10 - 3853x^9 + 4929x^8 + 13971x^7 - 20311x^6 - 22309x^5 + 38415x^4 + 8429x^3 - 22584x^2 - 399*x + 3381
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1148)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.10047$ of defining polynomial
Character	$\chi$	$=$	8036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.10047 q^{3} -3.31369 q^{5} +6.61290 q^{9} +O(q^{10})$	$q-3.10047 q^{3} -3.31369 q^{5} +6.61290 q^{9} -0.792186 q^{11} +6.10265 q^{13} +10.2740 q^{15} +0.423014 q^{17} +0.177816 q^{19} +5.55791 q^{23} +5.98052 q^{25} -11.2017 q^{27} +6.87240 q^{29} +1.03742 q^{31} +2.45615 q^{33} +10.9371 q^{37} -18.9211 q^{39} -1.00000 q^{41} +7.60020 q^{43} -21.9131 q^{45} -0.719310 q^{47} -1.31154 q^{51} +9.70480 q^{53} +2.62506 q^{55} -0.551312 q^{57} +8.31526 q^{59} +9.24684 q^{61} -20.2223 q^{65} -14.6528 q^{67} -17.2321 q^{69} +6.50677 q^{71} +15.7518 q^{73} -18.5424 q^{75} +0.0105912 q^{79} +14.8917 q^{81} +7.04482 q^{83} -1.40174 q^{85} -21.3076 q^{87} -16.3892 q^{89} -3.21648 q^{93} -0.589225 q^{95} -12.9888 q^{97} -5.23864 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - q^{3} + 3 q^{5} + 30 q^{9}+O(q^{10}) $ 15 * q - q^3 + 3 * q^5 + 30 * q^9	$ 15 q - q^{3} + 3 q^{5} + 30 q^{9} + 9 q^{11} + 7 q^{13} + 2 q^{15} + 3 q^{17} + 7 q^{19} - q^{23} + 32 q^{25} + 11 q^{27} + 18 q^{29} + 30 q^{31} - 16 q^{33} + 23 q^{37} + 5 q^{39} - 15 q^{41} + 12 q^{43} - 13 q^{45} - 16 q^{47} + 29 q^{51} + 33 q^{53} + 37 q^{55} + 16 q^{57} - 10 q^{59} + q^{61} + 16 q^{65} + 20 q^{67} + 21 q^{69} + 5 q^{71} - 3 q^{73} - 51 q^{75} + 25 q^{79} + 43 q^{81} + 18 q^{83} + 36 q^{85} - 53 q^{87} - 11 q^{89} + 65 q^{93} - 30 q^{95} + 16 q^{97} - 18 q^{99}+O(q^{100}) $ 15 * q - q^3 + 3 * q^5 + 30 * q^9 + 9 * q^11 + 7 * q^13 + 2 * q^15 + 3 * q^17 + 7 * q^19 - q^23 + 32 * q^25 + 11 * q^27 + 18 * q^29 + 30 * q^31 - 16 * q^33 + 23 * q^37 + 5 * q^39 - 15 * q^41 + 12 * q^43 - 13 * q^45 - 16 * q^47 + 29 * q^51 + 33 * q^53 + 37 * q^55 + 16 * q^57 - 10 * q^59 + q^61 + 16 * q^65 + 20 * q^67 + 21 * q^69 + 5 * q^71 - 3 * q^73 - 51 * q^75 + 25 * q^79 + 43 * q^81 + 18 * q^83 + 36 * q^85 - 53 * q^87 - 11 * q^89 + 65 * q^93 - 30 * q^95 + 16 * q^97 - 18 * 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 $

$2$	0	0
$2$	0	0
$3$	−3.10047	−1.79006	−0.895028	−	0.446010i	$-0.852844\pi$
$3$	−3.10047	−1.79006	−0.895028	+	0.446010i	$0.852844\pi$
$4$	0	0
$5$	−3.31369	−1.48193	−0.740963	−	0.671546i	$-0.765631\pi$
$5$	−3.31369	−1.48193	−0.740963	+	0.671546i	$0.765631\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	6.61290	2.20430
$10$	0	0
$11$	−0.792186	−0.238853	−0.119426	−	0.992843i	$-0.538106\pi$
$11$	−0.792186	−0.238853	−0.119426	+	0.992843i	$0.538106\pi$
$12$	0	0
$13$	6.10265	1.69257	0.846286	−	0.532729i	$-0.178834\pi$
$13$	6.10265	1.69257	0.846286	+	0.532729i	$0.178834\pi$
$14$	0	0
$15$	10.2740	2.65273
$16$	0	0
$17$	0.423014	0.102596	0.0512980	−	0.998683i	$-0.483664\pi$
$17$	0.423014	0.102596	0.0512980	+	0.998683i	$0.483664\pi$
$18$	0	0
$19$	0.177816	0.0407937	0.0203969	−	0.999792i	$-0.493507\pi$
$19$	0.177816	0.0407937	0.0203969	+	0.999792i	$0.493507\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.55791	1.15890	0.579452	−	0.815007i	$-0.303267\pi$
$23$	5.55791	1.15890	0.579452	+	0.815007i	$0.303267\pi$
$24$	0	0
$25$	5.98052	1.19610
$26$	0	0
$27$	−11.2017	−2.15576
$28$	0	0
$29$	6.87240	1.27617	0.638086	−	0.769965i	$-0.279726\pi$
$29$	6.87240	1.27617	0.638086	+	0.769965i	$0.279726\pi$
$30$	0	0
$31$	1.03742	0.186326	0.0931630	−	0.995651i	$-0.470302\pi$
$31$	1.03742	0.186326	0.0931630	+	0.995651i	$0.470302\pi$
$32$	0	0
$33$	2.45615	0.427560
$34$	0	0
$35$	0	0
$36$	0	0
$37$	10.9371	1.79805	0.899026	−	0.437896i	$-0.144276\pi$
$37$	10.9371	1.79805	0.899026	+	0.437896i	$0.144276\pi$
$38$	0	0
$39$	−18.9211	−3.02980
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	7.60020	1.15902	0.579509	−	0.814965i	$-0.303244\pi$
$43$	7.60020	1.15902	0.579509	+	0.814965i	$0.303244\pi$
$44$	0	0
$45$	−21.9131	−3.26661
$46$	0	0
$47$	−0.719310	−0.104922	−0.0524611	−	0.998623i	$-0.516707\pi$
$47$	−0.719310	−0.104922	−0.0524611	+	0.998623i	$0.516707\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−1.31154	−0.183653
$52$	0	0
$53$	9.70480	1.33306	0.666529	−	0.745479i	$-0.267779\pi$
$53$	9.70480	1.33306	0.666529	+	0.745479i	$0.267779\pi$
$54$	0	0
$55$	2.62506	0.353962
$56$	0	0
$57$	−0.551312	−0.0730230
$58$	0	0
$59$	8.31526	1.08255	0.541277	−	0.840844i	$-0.317941\pi$
$59$	8.31526	1.08255	0.541277	+	0.840844i	$0.317941\pi$
$60$	0	0
$61$	9.24684	1.18394	0.591968	−	0.805961i	$-0.298351\pi$
$61$	9.24684	1.18394	0.591968	+	0.805961i	$0.298351\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−20.2223	−2.50827
$66$	0	0
$67$	−14.6528	−1.79013	−0.895064	−	0.445937i	$-0.852870\pi$
$67$	−14.6528	−1.79013	−0.895064	+	0.445937i	$0.852870\pi$
$68$	0	0
$69$	−17.2321	−2.07450
$70$	0	0
$71$	6.50677	0.772211	0.386106	−	0.922455i	$-0.373820\pi$
$71$	6.50677	0.772211	0.386106	+	0.922455i	$0.373820\pi$
$72$	0	0
$73$	15.7518	1.84360	0.921802	−	0.387661i	$-0.126717\pi$
$73$	15.7518	1.84360	0.921802	+	0.387661i	$0.126717\pi$
$74$	0	0
$75$	−18.5424	−2.14109
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0.0105912	0.00119160	0.000595801	−	1.00000i	$-0.499810\pi$
$79$	0.0105912	0.00119160	0.000595801		1.00000i	$0.499810\pi$
$80$	0	0
$81$	14.8917	1.65464
$82$	0	0
$83$	7.04482	0.773269	0.386635	−	0.922233i	$-0.373637\pi$
$83$	7.04482	0.773269	0.386635	+	0.922233i	$0.373637\pi$
$84$	0	0
$85$	−1.40174	−0.152040
$86$	0	0
$87$	−21.3076	−2.28442
$88$	0	0
$89$	−16.3892	−1.73725	−0.868625	−	0.495470i	$-0.834996\pi$
$89$	−16.3892	−1.73725	−0.868625	+	0.495470i	$0.834996\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−3.21648	−0.333534
$94$	0	0
$95$	−0.589225	−0.0604533
$96$	0	0
$97$	−12.9888	−1.31881	−0.659407	−	0.751786i	$-0.729192\pi$
$97$	−12.9888	−1.31881	−0.659407	+	0.751786i	$0.729192\pi$
$98$	0	0
$99$	−5.23864	−0.526503
$100$	0	0
$101$	−3.90959	−0.389019	−0.194510	−	0.980901i	$-0.562312\pi$
$101$	−3.90959	−0.389019	−0.194510	+	0.980901i	$0.562312\pi$
$102$	0	0
$103$	−4.71318	−0.464403	−0.232202	−	0.972668i	$-0.574593\pi$
$103$	−4.71318	−0.464403	−0.232202	+	0.972668i	$0.574593\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−0.411875	−0.0398174	−0.0199087	−	0.999802i	$-0.506338\pi$
$107$	−0.411875	−0.0398174	−0.0199087	+	0.999802i	$0.506338\pi$
$108$	0	0
$109$	1.17123	0.112184	0.0560918	−	0.998426i	$-0.482136\pi$
$109$	1.17123	0.112184	0.0560918	+	0.998426i	$0.482136\pi$
$110$	0	0
$111$	−33.9102	−3.21861
$112$	0	0
$113$	−8.91233	−0.838401	−0.419201	−	0.907894i	$-0.637690\pi$
$113$	−8.91233	−0.838401	−0.419201	+	0.907894i	$0.637690\pi$
$114$	0	0
$115$	−18.4172	−1.71741
$116$	0	0
$117$	40.3562	3.73093
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.3724	−0.942949
$122$	0	0
$123$	3.10047	0.279560
$124$	0	0
$125$	−3.24914	−0.290612
$126$	0	0
$127$	−11.1494	−0.989351	−0.494675	−	0.869078i	$-0.664713\pi$
$127$	−11.1494	−0.989351	−0.494675	+	0.869078i	$0.664713\pi$
$128$	0	0
$129$	−23.5642	−2.07471
$130$	0	0
$131$	−14.6291	−1.27815	−0.639074	−	0.769146i	$-0.720682\pi$
$131$	−14.6291	−1.27815	−0.639074	+	0.769146i	$0.720682\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	37.1188	3.19468
$136$	0	0
$137$	−0.269536	−0.0230280	−0.0115140	−	0.999934i	$-0.503665\pi$
$137$	−0.269536	−0.0230280	−0.0115140	+	0.999934i	$0.503665\pi$
$138$	0	0
$139$	3.94454	0.334572	0.167286	−	0.985908i	$-0.446500\pi$
$139$	3.94454	0.334572	0.167286	+	0.985908i	$0.446500\pi$
$140$	0	0
$141$	2.23020	0.187817
$142$	0	0
$143$	−4.83443	−0.404276
$144$	0	0
$145$	−22.7730	−1.89119
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−5.84100	−0.478513	−0.239256	−	0.970956i	$-0.576904\pi$
$149$	−5.84100	−0.478513	−0.239256	+	0.970956i	$0.576904\pi$
$150$	0	0
$151$	3.95499	0.321853	0.160926	−	0.986966i	$-0.448552\pi$
$151$	3.95499	0.321853	0.160926	+	0.986966i	$0.448552\pi$
$152$	0	0
$153$	2.79735	0.226152
$154$	0	0
$155$	−3.43768	−0.276121
$156$	0	0
$157$	12.1983	0.973527	0.486764	−	0.873534i	$-0.338177\pi$
$157$	12.1983	0.973527	0.486764	+	0.873534i	$0.338177\pi$
$158$	0	0
$159$	−30.0894	−2.38625
$160$	0	0
$161$	0	0
$162$	0	0
$163$	12.2328	0.958151	0.479075	−	0.877774i	$-0.340972\pi$
$163$	12.2328	0.958151	0.479075	+	0.877774i	$0.340972\pi$
$164$	0	0
$165$	−8.13890	−0.633612
$166$	0	0
$167$	−9.89569	−0.765751	−0.382876	−	0.923800i	$-0.625066\pi$
$167$	−9.89569	−0.765751	−0.382876	+	0.923800i	$0.625066\pi$
$168$	0	0
$169$	24.2424	1.86480
$170$	0	0
$171$	1.17588	0.0899216
$172$	0	0
$173$	3.45189	0.262443	0.131221	−	0.991353i	$-0.458110\pi$
$173$	3.45189	0.262443	0.131221	+	0.991353i	$0.458110\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−25.7812	−1.93783
$178$	0	0
$179$	23.3124	1.74245	0.871226	−	0.490881i	$-0.163325\pi$
$179$	23.3124	1.74245	0.871226	+	0.490881i	$0.163325\pi$
$180$	0	0
$181$	21.9722	1.63318	0.816591	−	0.577216i	$-0.195861\pi$
$181$	21.9722	1.63318	0.816591	+	0.577216i	$0.195861\pi$
$182$	0	0
$183$	−28.6695	−2.11931
$184$	0	0
$185$	−36.2422	−2.66458
$186$	0	0
$187$	−0.335106	−0.0245054
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−13.4033	−0.969829	−0.484914	−	0.874562i	$-0.661149\pi$
$191$	−13.4033	−0.969829	−0.484914	+	0.874562i	$0.661149\pi$
$192$	0	0
$193$	13.4914	0.971130	0.485565	−	0.874200i	$-0.338614\pi$
$193$	13.4914	0.971130	0.485565	+	0.874200i	$0.338614\pi$
$194$	0	0
$195$	62.6985	4.48994
$196$	0	0
$197$	1.13551	0.0809021	0.0404510	−	0.999182i	$-0.487121\pi$
$197$	1.13551	0.0809021	0.0404510	+	0.999182i	$0.487121\pi$
$198$	0	0
$199$	6.42701	0.455599	0.227799	−	0.973708i	$-0.426847\pi$
$199$	6.42701	0.455599	0.227799	+	0.973708i	$0.426847\pi$
$200$	0	0
$201$	45.4306	3.20443
$202$	0	0
$203$	0	0
$204$	0	0
$205$	3.31369	0.231438
$206$	0	0
$207$	36.7539	2.55457
$208$	0	0
$209$	−0.140863	−0.00974370
$210$	0	0
$211$	−6.69852	−0.461145	−0.230573	−	0.973055i	$-0.574060\pi$
$211$	−6.69852	−0.461145	−0.230573	+	0.973055i	$0.574060\pi$
$212$	0	0
$213$	−20.1740	−1.38230
$214$	0	0
$215$	−25.1847	−1.71758
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−48.8378	−3.30015
$220$	0	0
$221$	2.58151	0.173651
$222$	0	0
$223$	15.8949	1.06440	0.532201	−	0.846618i	$-0.321365\pi$
$223$	15.8949	1.06440	0.532201	+	0.846618i	$0.321365\pi$
$224$	0	0
$225$	39.5486	2.63657
$226$	0	0
$227$	−26.1890	−1.73822	−0.869112	−	0.494614i	$-0.835309\pi$
$227$	−26.1890	−1.73822	−0.869112	+	0.494614i	$0.835309\pi$
$228$	0	0
$229$	9.66874	0.638929	0.319464	−	0.947598i	$-0.396497\pi$
$229$	9.66874	0.638929	0.319464	+	0.947598i	$0.396497\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	9.61564	0.629942	0.314971	−	0.949101i	$-0.398005\pi$
$233$	9.61564	0.629942	0.314971	+	0.949101i	$0.398005\pi$
$234$	0	0
$235$	2.38357	0.155487
$236$	0	0
$237$	−0.0328376	−0.00213303
$238$	0	0
$239$	−11.3337	−0.733119	−0.366559	−	0.930395i	$-0.619464\pi$
$239$	−11.3337	−0.733119	−0.366559	+	0.930395i	$0.619464\pi$
$240$	0	0
$241$	13.4507	0.866435	0.433218	−	0.901289i	$-0.357378\pi$
$241$	13.4507	0.866435	0.433218	+	0.901289i	$0.357378\pi$
$242$	0	0
$243$	−12.5663	−0.806129
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.08515	0.0690463
$248$	0	0
$249$	−21.8422	−1.38420
$250$	0	0
$251$	23.7920	1.50174	0.750870	−	0.660450i	$-0.229635\pi$
$251$	23.7920	1.50174	0.750870	+	0.660450i	$0.229635\pi$
$252$	0	0
$253$	−4.40289	−0.276808
$254$	0	0
$255$	4.34604	0.272159
$256$	0	0
$257$	−10.8482	−0.676690	−0.338345	−	0.941022i	$-0.609867\pi$
$257$	−10.8482	−0.676690	−0.338345	+	0.941022i	$0.609867\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	45.4465	2.81307
$262$	0	0
$263$	0.397372	0.0245030	0.0122515	−	0.999925i	$-0.496100\pi$
$263$	0.397372	0.0245030	0.0122515	+	0.999925i	$0.496100\pi$
$264$	0	0
$265$	−32.1587	−1.97549
$266$	0	0
$267$	50.8141	3.10977
$268$	0	0
$269$	−20.4031	−1.24400	−0.622000	−	0.783017i	$-0.713680\pi$
$269$	−20.4031	−1.24400	−0.622000	+	0.783017i	$0.713680\pi$
$270$	0	0
$271$	8.26540	0.502087	0.251044	−	0.967976i	$-0.419226\pi$
$271$	8.26540	0.502087	0.251044	+	0.967976i	$0.419226\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−4.73768	−0.285693
$276$	0	0
$277$	12.6537	0.760287	0.380143	−	0.924928i	$-0.375875\pi$
$277$	12.6537	0.760287	0.380143	+	0.924928i	$0.375875\pi$
$278$	0	0
$279$	6.86034	0.410718
$280$	0	0
$281$	4.90459	0.292583	0.146292	−	0.989242i	$-0.453266\pi$
$281$	4.90459	0.292583	0.146292	+	0.989242i	$0.453266\pi$
$282$	0	0
$283$	19.4702	1.15738	0.578692	−	0.815546i	$-0.303563\pi$
$283$	19.4702	1.15738	0.578692	+	0.815546i	$0.303563\pi$
$284$	0	0
$285$	1.82687	0.108215
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−16.8211	−0.989474
$290$	0	0
$291$	40.2714	2.36075
$292$	0	0
$293$	25.0432	1.46304	0.731521	−	0.681819i	$-0.238811\pi$
$293$	25.0432	1.46304	0.731521	+	0.681819i	$0.238811\pi$
$294$	0	0
$295$	−27.5542	−1.60427
$296$	0	0
$297$	8.87380	0.514910
$298$	0	0
$299$	33.9180	1.96153
$300$	0	0
$301$	0	0
$302$	0	0
$303$	12.1216	0.696366
$304$	0	0
$305$	−30.6411	−1.75451
$306$	0	0
$307$	−18.1513	−1.03595	−0.517973	−	0.855397i	$-0.673313\pi$
$307$	−18.1513	−1.03595	−0.517973	+	0.855397i	$0.673313\pi$
$308$	0	0
$309$	14.6131	0.831308
$310$	0	0
$311$	33.4499	1.89677	0.948386	−	0.317119i	$-0.102715\pi$
$311$	33.4499	1.89677	0.948386	+	0.317119i	$0.102715\pi$
$312$	0	0
$313$	−21.5440	−1.21774	−0.608869	−	0.793271i	$-0.708376\pi$
$313$	−21.5440	−1.21774	−0.608869	+	0.793271i	$0.708376\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−7.64818	−0.429565	−0.214782	−	0.976662i	$-0.568904\pi$
$317$	−7.64818	−0.429565	−0.214782	+	0.976662i	$0.568904\pi$
$318$	0	0
$319$	−5.44421	−0.304817
$320$	0	0
$321$	1.27700	0.0712754
$322$	0	0
$323$	0.0752185	0.00418527
$324$	0	0
$325$	36.4971	2.02449
$326$	0	0
$327$	−3.63136	−0.200815
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−7.12802	−0.391791	−0.195896	−	0.980625i	$-0.562761\pi$
$331$	−7.12802	−0.391791	−0.195896	+	0.980625i	$0.562761\pi$
$332$	0	0
$333$	72.3261	3.96344
$334$	0	0
$335$	48.5549	2.65284
$336$	0	0
$337$	−18.0930	−0.985589	−0.492795	−	0.870146i	$-0.664025\pi$
$337$	−18.0930	−0.985589	−0.492795	+	0.870146i	$0.664025\pi$
$338$	0	0
$339$	27.6324	1.50079
$340$	0	0
$341$	−0.821828	−0.0445045
$342$	0	0
$343$	0	0
$344$	0	0
$345$	57.1018	3.07426
$346$	0	0
$347$	−28.4867	−1.52925	−0.764624	−	0.644476i	$-0.777075\pi$
$347$	−28.4867	−1.52925	−0.764624	+	0.644476i	$0.777075\pi$
$348$	0	0
$349$	17.3830	0.930488	0.465244	−	0.885182i	$-0.345967\pi$
$349$	17.3830	0.930488	0.465244	+	0.885182i	$0.345967\pi$
$350$	0	0
$351$	−68.3599	−3.64878
$352$	0	0
$353$	−26.5923	−1.41537	−0.707683	−	0.706530i	$-0.750260\pi$
$353$	−26.5923	−1.41537	−0.707683	+	0.706530i	$0.750260\pi$
$354$	0	0
$355$	−21.5614	−1.14436
$356$	0	0
$357$	0	0
$358$	0	0
$359$	21.5703	1.13843	0.569217	−	0.822187i	$-0.307246\pi$
$359$	21.5703	1.13843	0.569217	+	0.822187i	$0.307246\pi$
$360$	0	0
$361$	−18.9684	−0.998336
$362$	0	0
$363$	32.1594	1.68793
$364$	0	0
$365$	−52.1964	−2.73208
$366$	0	0
$367$	−2.36326	−0.123361	−0.0616807	−	0.998096i	$-0.519646\pi$
$367$	−2.36326	−0.123361	−0.0616807	+	0.998096i	$0.519646\pi$
$368$	0	0
$369$	−6.61290	−0.344254
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−17.0641	−0.883546	−0.441773	−	0.897127i	$-0.645650\pi$
$373$	−17.0641	−0.883546	−0.441773	+	0.897127i	$0.645650\pi$
$374$	0	0
$375$	10.0739	0.520212
$376$	0	0
$377$	41.9399	2.16001
$378$	0	0
$379$	−12.0302	−0.617947	−0.308974	−	0.951071i	$-0.599985\pi$
$379$	−12.0302	−0.617947	−0.308974	+	0.951071i	$0.599985\pi$
$380$	0	0
$381$	34.5684	1.77099
$382$	0	0
$383$	8.73714	0.446447	0.223223	−	0.974767i	$-0.428342\pi$
$383$	8.73714	0.446447	0.223223	+	0.974767i	$0.428342\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	50.2593	2.55482
$388$	0	0
$389$	23.2242	1.17752	0.588758	−	0.808310i	$-0.299617\pi$
$389$	23.2242	1.17752	0.588758	+	0.808310i	$0.299617\pi$
$390$	0	0
$391$	2.35107	0.118899
$392$	0	0
$393$	45.3569	2.28795
$394$	0	0
$395$	−0.0350959	−0.00176587
$396$	0	0
$397$	34.9679	1.75499	0.877495	−	0.479585i	$-0.159213\pi$
$397$	34.9679	1.75499	0.877495	+	0.479585i	$0.159213\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	28.0528	1.40089	0.700444	−	0.713707i	$-0.252985\pi$
$401$	28.0528	1.40089	0.700444	+	0.713707i	$0.252985\pi$
$402$	0	0
$403$	6.33101	0.315370
$404$	0	0
$405$	−49.3465	−2.45205
$406$	0	0
$407$	−8.66423	−0.429470
$408$	0	0
$409$	39.0591	1.93135	0.965675	−	0.259754i	$-0.0836414\pi$
$409$	39.0591	1.93135	0.965675	+	0.259754i	$0.0836414\pi$
$410$	0	0
$411$	0.835688	0.0412215
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−23.3443	−1.14593
$416$	0	0
$417$	−12.2299	−0.598902
$418$	0	0
$419$	−9.90234	−0.483761	−0.241880	−	0.970306i	$-0.577764\pi$
$419$	−9.90234	−0.483761	−0.241880	+	0.970306i	$0.577764\pi$
$420$	0	0
$421$	13.5351	0.659660	0.329830	−	0.944040i	$-0.393008\pi$
$421$	13.5351	0.659660	0.329830	+	0.944040i	$0.393008\pi$
$422$	0	0
$423$	−4.75673	−0.231280
$424$	0	0
$425$	2.52984	0.122715
$426$	0	0
$427$	0	0
$428$	0	0
$429$	14.9890	0.723676
$430$	0	0
$431$	4.60848	0.221983	0.110991	−	0.993821i	$-0.464597\pi$
$431$	4.60848	0.221983	0.110991	+	0.993821i	$0.464597\pi$
$432$	0	0
$433$	12.0521	0.579188	0.289594	−	0.957150i	$-0.406480\pi$
$433$	12.0521	0.579188	0.289594	+	0.957150i	$0.406480\pi$
$434$	0	0
$435$	70.6069	3.38534
$436$	0	0
$437$	0.988283	0.0472760
$438$	0	0
$439$	30.9186	1.47566	0.737832	−	0.674984i	$-0.235850\pi$
$439$	30.9186	1.47566	0.737832	+	0.674984i	$0.235850\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−16.6901	−0.792971	−0.396486	−	0.918041i	$-0.629770\pi$
$443$	−16.6901	−0.792971	−0.396486	+	0.918041i	$0.629770\pi$
$444$	0	0
$445$	54.3086	2.57448
$446$	0	0
$447$	18.1098	0.856565
$448$	0	0
$449$	20.5726	0.970883	0.485441	−	0.874269i	$-0.338659\pi$
$449$	20.5726	0.970883	0.485441	+	0.874269i	$0.338659\pi$
$450$	0	0
$451$	0.792186	0.0373026
$452$	0	0
$453$	−12.2623	−0.576134
$454$	0	0
$455$	0	0
$456$	0	0
$457$	28.1632	1.31742	0.658709	−	0.752397i	$-0.271103\pi$
$457$	28.1632	1.31742	0.658709	+	0.752397i	$0.271103\pi$
$458$	0	0
$459$	−4.73847	−0.221173
$460$	0	0
$461$	−24.0754	−1.12130	−0.560651	−	0.828052i	$-0.689449\pi$
$461$	−24.0754	−1.12130	−0.560651	+	0.828052i	$0.689449\pi$
$462$	0	0
$463$	−27.4976	−1.27792	−0.638960	−	0.769240i	$-0.720635\pi$
$463$	−27.4976	−1.27792	−0.638960	+	0.769240i	$0.720635\pi$
$464$	0	0
$465$	10.6584	0.494272
$466$	0	0
$467$	−40.7584	−1.88608	−0.943038	−	0.332685i	$-0.892045\pi$
$467$	−40.7584	−1.88608	−0.943038	+	0.332685i	$0.892045\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−37.8203	−1.74267
$472$	0	0
$473$	−6.02077	−0.276835
$474$	0	0
$475$	1.06343	0.0487935
$476$	0	0
$477$	64.1769	2.93846
$478$	0	0
$479$	5.97986	0.273227	0.136613	−	0.990624i	$-0.456378\pi$
$479$	5.97986	0.273227	0.136613	+	0.990624i	$0.456378\pi$
$480$	0	0
$481$	66.7455	3.04333
$482$	0	0
$483$	0	0
$484$	0	0
$485$	43.0408	1.95438
$486$	0	0
$487$	−17.2220	−0.780405	−0.390202	−	0.920729i	$-0.627595\pi$
$487$	−17.2220	−0.780405	−0.390202	+	0.920729i	$0.627595\pi$
$488$	0	0
$489$	−37.9276	−1.71514
$490$	0	0
$491$	−30.4029	−1.37207	−0.686033	−	0.727571i	$-0.740649\pi$
$491$	−30.4029	−1.37207	−0.686033	+	0.727571i	$0.740649\pi$
$492$	0	0
$493$	2.90712	0.130930
$494$	0	0
$495$	17.3592	0.780239
$496$	0	0
$497$	0	0
$498$	0	0
$499$	3.05907	0.136943	0.0684713	−	0.997653i	$-0.478188\pi$
$499$	3.05907	0.136943	0.0684713	+	0.997653i	$0.478188\pi$
$500$	0	0
$501$	30.6813	1.37074
$502$	0	0
$503$	−13.3163	−0.593745	−0.296872	−	0.954917i	$-0.595944\pi$
$503$	−13.3163	−0.593745	−0.296872	+	0.954917i	$0.595944\pi$
$504$	0	0
$505$	12.9552	0.576498
$506$	0	0
$507$	−75.1627	−3.33809
$508$	0	0
$509$	−19.2352	−0.852585	−0.426293	−	0.904585i	$-0.640181\pi$
$509$	−19.2352	−0.852585	−0.426293	+	0.904585i	$0.640181\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−1.99183	−0.0879416
$514$	0	0
$515$	15.6180	0.688211
$516$	0	0
$517$	0.569827	0.0250610
$518$	0	0
$519$	−10.7025	−0.469787
$520$	0	0
$521$	−23.8725	−1.04587	−0.522936	−	0.852372i	$-0.675163\pi$
$521$	−23.8725	−1.04587	−0.522936	+	0.852372i	$0.675163\pi$
$522$	0	0
$523$	−32.9618	−1.44132	−0.720660	−	0.693288i	$-0.756161\pi$
$523$	−32.9618	−1.44132	−0.720660	+	0.693288i	$0.756161\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.438843	0.0191163
$528$	0	0
$529$	7.89032	0.343057
$530$	0	0
$531$	54.9880	2.38628
$532$	0	0
$533$	−6.10265	−0.264335
$534$	0	0
$535$	1.36482	0.0590065
$536$	0	0
$537$	−72.2794	−3.11909
$538$	0	0
$539$	0	0
$540$	0	0
$541$	6.56283	0.282158	0.141079	−	0.989998i	$-0.454943\pi$
$541$	6.56283	0.282158	0.141079	+	0.989998i	$0.454943\pi$
$542$	0	0
$543$	−68.1242	−2.92349
$544$	0	0
$545$	−3.88109	−0.166248
$546$	0	0
$547$	9.98754	0.427036	0.213518	−	0.976939i	$-0.431508\pi$
$547$	9.98754	0.427036	0.213518	+	0.976939i	$0.431508\pi$
$548$	0	0
$549$	61.1484	2.60975
$550$	0	0
$551$	1.22202	0.0520598
$552$	0	0
$553$	0	0
$554$	0	0
$555$	112.368	4.76974
$556$	0	0
$557$	−22.6717	−0.960632	−0.480316	−	0.877096i	$-0.659478\pi$
$557$	−22.6717	−0.960632	−0.480316	+	0.877096i	$0.659478\pi$
$558$	0	0
$559$	46.3814	1.96172
$560$	0	0
$561$	1.03898	0.0438659
$562$	0	0
$563$	−6.55561	−0.276286	−0.138143	−	0.990412i	$-0.544113\pi$
$563$	−6.55561	−0.276286	−0.138143	+	0.990412i	$0.544113\pi$
$564$	0	0
$565$	29.5327	1.24245
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−10.7593	−0.451054	−0.225527	−	0.974237i	$-0.572410\pi$
$569$	−10.7593	−0.451054	−0.225527	+	0.974237i	$0.572410\pi$
$570$	0	0
$571$	25.3975	1.06285	0.531427	−	0.847104i	$-0.321656\pi$
$571$	25.3975	1.06285	0.531427	+	0.847104i	$0.321656\pi$
$572$	0	0
$573$	41.5565	1.73605
$574$	0	0
$575$	33.2392	1.38617
$576$	0	0
$577$	−1.65595	−0.0689381	−0.0344690	−	0.999406i	$-0.510974\pi$
$577$	−1.65595	−0.0689381	−0.0344690	+	0.999406i	$0.510974\pi$
$578$	0	0
$579$	−41.8296	−1.73838
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−7.68801	−0.318405
$584$	0	0
$585$	−133.728	−5.52897
$586$	0	0
$587$	−39.9868	−1.65043	−0.825215	−	0.564818i	$-0.808946\pi$
$587$	−39.9868	−1.65043	−0.825215	+	0.564818i	$0.808946\pi$
$588$	0	0
$589$	0.184469	0.00760093
$590$	0	0
$591$	−3.52063	−0.144819
$592$	0	0
$593$	1.35910	0.0558115	0.0279057	−	0.999611i	$-0.491116\pi$
$593$	1.35910	0.0558115	0.0279057	+	0.999611i	$0.491116\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−19.9267	−0.815547
$598$	0	0
$599$	−28.6150	−1.16918	−0.584588	−	0.811330i	$-0.698744\pi$
$599$	−28.6150	−1.16918	−0.584588	+	0.811330i	$0.698744\pi$
$600$	0	0
$601$	−20.9899	−0.856194	−0.428097	−	0.903733i	$-0.640816\pi$
$601$	−20.9899	−0.856194	−0.428097	+	0.903733i	$0.640816\pi$
$602$	0	0
$603$	−96.8977	−3.94598
$604$	0	0
$605$	34.3710	1.39738
$606$	0	0
$607$	41.4598	1.68280	0.841401	−	0.540411i	$-0.181731\pi$
$607$	41.4598	1.68280	0.841401	+	0.540411i	$0.181731\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−4.38970	−0.177588
$612$	0	0
$613$	−3.80139	−0.153537	−0.0767683	−	0.997049i	$-0.524460\pi$
$613$	−3.80139	−0.153537	−0.0767683	+	0.997049i	$0.524460\pi$
$614$	0	0
$615$	−10.2740	−0.414287
$616$	0	0
$617$	0.818128	0.0329366	0.0164683	−	0.999864i	$-0.494758\pi$
$617$	0.818128	0.0329366	0.0164683	+	0.999864i	$0.494758\pi$
$618$	0	0
$619$	−8.72606	−0.350730	−0.175365	−	0.984503i	$-0.556111\pi$
$619$	−8.72606	−0.350730	−0.175365	+	0.984503i	$0.556111\pi$
$620$	0	0
$621$	−62.2578	−2.49832
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−19.1360	−0.765439
$626$	0	0
$627$	0.436741	0.0174418
$628$	0	0
$629$	4.62656	0.184473
$630$	0	0
$631$	−32.1325	−1.27917	−0.639587	−	0.768719i	$-0.720895\pi$
$631$	−32.1325	−1.27917	−0.639587	+	0.768719i	$0.720895\pi$
$632$	0	0
$633$	20.7686	0.825476
$634$	0	0
$635$	36.9457	1.46614
$636$	0	0
$637$	0	0
$638$	0	0
$639$	43.0286	1.70218
$640$	0	0
$641$	−12.9983	−0.513402	−0.256701	−	0.966491i	$-0.582636\pi$
$641$	−12.9983	−0.513402	−0.256701	+	0.966491i	$0.582636\pi$
$642$	0	0
$643$	−3.06891	−0.121026	−0.0605131	−	0.998167i	$-0.519274\pi$
$643$	−3.06891	−0.121026	−0.0605131	+	0.998167i	$0.519274\pi$
$644$	0	0
$645$	78.0842	3.07456
$646$	0	0
$647$	−13.2665	−0.521561	−0.260780	−	0.965398i	$-0.583980\pi$
$647$	−13.2665	−0.521561	−0.260780	+	0.965398i	$0.583980\pi$
$648$	0	0
$649$	−6.58723	−0.258571
$650$	0	0
$651$	0	0
$652$	0	0
$653$	5.01002	0.196057	0.0980287	−	0.995184i	$-0.468746\pi$
$653$	5.01002	0.196057	0.0980287	+	0.995184i	$0.468746\pi$
$654$	0	0
$655$	48.4761	1.89412
$656$	0	0
$657$	104.165	4.06386
$658$	0	0
$659$	−19.0067	−0.740397	−0.370198	−	0.928953i	$-0.620710\pi$
$659$	−19.0067	−0.740397	−0.370198	+	0.928953i	$0.620710\pi$
$660$	0	0
$661$	−9.64577	−0.375177	−0.187588	−	0.982248i	$-0.560067\pi$
$661$	−9.64577	−0.375177	−0.187588	+	0.982248i	$0.560067\pi$
$662$	0	0
$663$	−8.00388	−0.310845
$664$	0	0
$665$	0	0
$666$	0	0
$667$	38.1961	1.47896
$668$	0	0
$669$	−49.2816	−1.90534
$670$	0	0
$671$	−7.32521	−0.282787
$672$	0	0
$673$	−18.4596	−0.711564	−0.355782	−	0.934569i	$-0.615785\pi$
$673$	−18.4596	−0.711564	−0.355782	+	0.934569i	$0.615785\pi$
$674$	0	0
$675$	−66.9919	−2.57852
$676$	0	0
$677$	37.0226	1.42289	0.711447	−	0.702740i	$-0.248040\pi$
$677$	37.0226	1.42289	0.711447	+	0.702740i	$0.248040\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	81.1981	3.11152
$682$	0	0
$683$	−30.1788	−1.15476	−0.577380	−	0.816475i	$-0.695925\pi$
$683$	−30.1788	−1.15476	−0.577380	+	0.816475i	$0.695925\pi$
$684$	0	0
$685$	0.893159	0.0341258
$686$	0	0
$687$	−29.9776	−1.14372
$688$	0	0
$689$	59.2251	2.25630
$690$	0	0
$691$	41.8766	1.59306	0.796530	−	0.604599i	$-0.206667\pi$
$691$	41.8766	1.59306	0.796530	+	0.604599i	$0.206667\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−13.0710	−0.495810
$696$	0	0
$697$	−0.423014	−0.0160228
$698$	0	0
$699$	−29.8130	−1.12763
$700$	0	0
$701$	4.68425	0.176922	0.0884609	−	0.996080i	$-0.471805\pi$
$701$	4.68425	0.176922	0.0884609	+	0.996080i	$0.471805\pi$
$702$	0	0
$703$	1.94479	0.0733492
$704$	0	0
$705$	−7.39018	−0.278330
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−13.6768	−0.513642	−0.256821	−	0.966459i	$-0.582675\pi$
$709$	−13.6768	−0.513642	−0.256821	+	0.966459i	$0.582675\pi$
$710$	0	0
$711$	0.0700385	0.00262665
$712$	0	0
$713$	5.76588	0.215934
$714$	0	0
$715$	16.0198	0.599107
$716$	0	0
$717$	35.1399	1.31232
$718$	0	0
$719$	31.4037	1.17116	0.585580	−	0.810614i	$-0.300867\pi$
$719$	31.4037	1.17116	0.585580	+	0.810614i	$0.300867\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−41.7034	−1.55097
$724$	0	0
$725$	41.1005	1.52643
$726$	0	0
$727$	28.8878	1.07139	0.535694	−	0.844412i	$-0.320050\pi$
$727$	28.8878	1.07139	0.535694	+	0.844412i	$0.320050\pi$
$728$	0	0
$729$	−5.71378	−0.211622
$730$	0	0
$731$	3.21499	0.118911
$732$	0	0
$733$	26.0897	0.963645	0.481822	−	0.876269i	$-0.339975\pi$
$733$	26.0897	0.963645	0.481822	+	0.876269i	$0.339975\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	11.6078	0.427577
$738$	0	0
$739$	50.5825	1.86071	0.930354	−	0.366662i	$-0.119499\pi$
$739$	50.5825	1.86071	0.930354	+	0.366662i	$0.119499\pi$
$740$	0	0
$741$	−3.36446	−0.123597
$742$	0	0
$743$	−32.2795	−1.18422	−0.592110	−	0.805857i	$-0.701705\pi$
$743$	−32.2795	−1.18422	−0.592110	+	0.805857i	$0.701705\pi$
$744$	0	0
$745$	19.3552	0.709121
$746$	0	0
$747$	46.5867	1.70452
$748$	0	0
$749$	0	0
$750$	0	0
$751$	27.2583	0.994669	0.497335	−	0.867559i	$-0.334312\pi$
$751$	27.2583	0.994669	0.497335	+	0.867559i	$0.334312\pi$
$752$	0	0
$753$	−73.7664	−2.68820
$754$	0	0
$755$	−13.1056	−0.476962
$756$	0	0
$757$	−8.70767	−0.316486	−0.158243	−	0.987400i	$-0.550583\pi$
$757$	−8.70767	−0.316486	−0.158243	+	0.987400i	$0.550583\pi$
$758$	0	0
$759$	13.6510	0.495501
$760$	0	0
$761$	−13.9333	−0.505081	−0.252540	−	0.967586i	$-0.581266\pi$
$761$	−13.9333	−0.505081	−0.252540	+	0.967586i	$0.581266\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−9.26954	−0.335141
$766$	0	0
$767$	50.7452	1.83230
$768$	0	0
$769$	−19.9808	−0.720528	−0.360264	−	0.932850i	$-0.617313\pi$
$769$	−19.9808	−0.720528	−0.360264	+	0.932850i	$0.617313\pi$
$770$	0	0
$771$	33.6344	1.21131
$772$	0	0
$773$	19.4581	0.699858	0.349929	−	0.936776i	$-0.386206\pi$
$773$	19.4581	0.699858	0.349929	+	0.936776i	$0.386206\pi$
$774$	0	0
$775$	6.20431	0.222865
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−0.177816	−0.00637091
$780$	0	0
$781$	−5.15457	−0.184445
$782$	0	0
$783$	−76.9823	−2.75112
$784$	0	0
$785$	−40.4212	−1.44270
$786$	0	0
$787$	−2.22135	−0.0791825	−0.0395913	−	0.999216i	$-0.512606\pi$
$787$	−2.22135	−0.0791825	−0.0395913	+	0.999216i	$0.512606\pi$
$788$	0	0
$789$	−1.23204	−0.0438617
$790$	0	0
$791$	0	0
$792$	0	0
$793$	56.4303	2.00390
$794$	0	0
$795$	99.7070	3.53624
$796$	0	0
$797$	44.0244	1.55942	0.779712	−	0.626139i	$-0.215366\pi$
$797$	44.0244	1.55942	0.779712	+	0.626139i	$0.215366\pi$
$798$	0	0
$799$	−0.304278	−0.0107646
$800$	0	0
$801$	−108.380	−3.82942
$802$	0	0
$803$	−12.4783	−0.440350
$804$	0	0
$805$	0	0
$806$	0	0
$807$	63.2592	2.22683
$808$	0	0
$809$	−12.5660	−0.441798	−0.220899	−	0.975297i	$-0.570899\pi$
$809$	−12.5660	−0.441798	−0.220899	+	0.975297i	$0.570899\pi$
$810$	0	0
$811$	−36.4864	−1.28121	−0.640606	−	0.767870i	$-0.721317\pi$
$811$	−36.4864	−1.28121	−0.640606	+	0.767870i	$0.721317\pi$
$812$	0	0
$813$	−25.6266	−0.898764
$814$	0	0
$815$	−40.5358	−1.41991
$816$	0	0
$817$	1.35143	0.0472807
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.85719	0.0648164	0.0324082	−	0.999475i	$-0.489682\pi$
$821$	1.85719	0.0648164	0.0324082	+	0.999475i	$0.489682\pi$
$822$	0	0
$823$	−37.2988	−1.30016	−0.650078	−	0.759868i	$-0.725264\pi$
$823$	−37.2988	−1.30016	−0.650078	+	0.759868i	$0.725264\pi$
$824$	0	0
$825$	14.6890	0.511406
$826$	0	0
$827$	−3.33099	−0.115830	−0.0579150	−	0.998322i	$-0.518445\pi$
$827$	−3.33099	−0.115830	−0.0579150	+	0.998322i	$0.518445\pi$
$828$	0	0
$829$	−42.5501	−1.47783	−0.738913	−	0.673801i	$-0.764661\pi$
$829$	−42.5501	−1.47783	−0.738913	+	0.673801i	$0.764661\pi$
$830$	0	0
$831$	−39.2324	−1.36096
$832$	0	0
$833$	0	0
$834$	0	0
$835$	32.7912	1.13479
$836$	0	0
$837$	−11.6208	−0.401675
$838$	0	0
$839$	−17.2324	−0.594929	−0.297465	−	0.954733i	$-0.596141\pi$
$839$	−17.2324	−0.594929	−0.297465	+	0.954733i	$0.596141\pi$
$840$	0	0
$841$	18.2298	0.628615
$842$	0	0
$843$	−15.2065	−0.523740
$844$	0	0
$845$	−80.3317	−2.76349
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−60.3668	−2.07178
$850$	0	0
$851$	60.7875	2.08377
$852$	0	0
$853$	14.8844	0.509633	0.254817	−	0.966989i	$-0.417985\pi$
$853$	14.8844	0.509633	0.254817	+	0.966989i	$0.417985\pi$
$854$	0	0
$855$	−3.89649	−0.133257
$856$	0	0
$857$	42.7726	1.46108	0.730542	−	0.682868i	$-0.239268\pi$
$857$	42.7726	1.46108	0.730542	+	0.682868i	$0.239268\pi$
$858$	0	0
$859$	22.8541	0.779773	0.389886	−	0.920863i	$-0.372514\pi$
$859$	22.8541	0.779773	0.389886	+	0.920863i	$0.372514\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	9.93663	0.338247	0.169123	−	0.985595i	$-0.445906\pi$
$863$	9.93663	0.338247	0.169123	+	0.985595i	$0.445906\pi$
$864$	0	0
$865$	−11.4385	−0.388921
$866$	0	0
$867$	52.1531	1.77121
$868$	0	0
$869$	−0.00839019	−0.000284618 0
$870$	0	0
$871$	−89.4212	−3.02992
$872$	0	0
$873$	−85.8936	−2.90706
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−4.22010	−0.142503	−0.0712514	−	0.997458i	$-0.522699\pi$
$877$	−4.22010	−0.142503	−0.0712514	+	0.997458i	$0.522699\pi$
$878$	0	0
$879$	−77.6457	−2.61893
$880$	0	0
$881$	37.1986	1.25325	0.626626	−	0.779320i	$-0.284435\pi$
$881$	37.1986	1.25325	0.626626	+	0.779320i	$0.284435\pi$
$882$	0	0
$883$	−5.39445	−0.181538	−0.0907689	−	0.995872i	$-0.528932\pi$
$883$	−5.39445	−0.181538	−0.0907689	+	0.995872i	$0.528932\pi$
$884$	0	0
$885$	85.4308	2.87173
$886$	0	0
$887$	−22.7727	−0.764633	−0.382316	−	0.924031i	$-0.624874\pi$
$887$	−22.7727	−0.764633	−0.382316	+	0.924031i	$0.624874\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−11.7970	−0.395215
$892$	0	0
$893$	−0.127905	−0.00428017
$894$	0	0
$895$	−77.2501	−2.58219
$896$	0	0
$897$	−105.162	−3.51124
$898$	0	0
$899$	7.12955	0.237784
$900$	0	0
$901$	4.10527	0.136766
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−72.8091	−2.42026
$906$	0	0
$907$	−37.2836	−1.23798	−0.618991	−	0.785398i	$-0.712458\pi$
$907$	−37.2836	−1.23798	−0.618991	+	0.785398i	$0.712458\pi$
$908$	0	0
$909$	−25.8537	−0.857515
$910$	0	0
$911$	−5.99739	−0.198703	−0.0993513	−	0.995052i	$-0.531677\pi$
$911$	−5.99739	−0.198703	−0.0993513	+	0.995052i	$0.531677\pi$
$912$	0	0
$913$	−5.58080	−0.184698
$914$	0	0
$915$	95.0018	3.14066
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−32.4009	−1.06881	−0.534403	−	0.845230i	$-0.679464\pi$
$919$	−32.4009	−1.06881	−0.534403	+	0.845230i	$0.679464\pi$
$920$	0	0
$921$	56.2774	1.85440
$922$	0	0
$923$	39.7086	1.30702
$924$	0	0
$925$	65.4097	2.15066
$926$	0	0
$927$	−31.1678	−1.02368
$928$	0	0
$929$	−34.0590	−1.11744	−0.558721	−	0.829356i	$-0.688708\pi$
$929$	−34.0590	−1.11744	−0.558721	+	0.829356i	$0.688708\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−103.710	−3.39533
$934$	0	0
$935$	1.11044	0.0363151
$936$	0	0
$937$	27.7733	0.907315	0.453658	−	0.891176i	$-0.350119\pi$
$937$	27.7733	0.907315	0.453658	+	0.891176i	$0.350119\pi$
$938$	0	0
$939$	66.7964	2.17982
$940$	0	0
$941$	−54.4091	−1.77369	−0.886843	−	0.462072i	$-0.847106\pi$
$941$	−54.4091	−1.77369	−0.886843	+	0.462072i	$0.847106\pi$
$942$	0	0
$943$	−5.55791	−0.180990
$944$	0	0
$945$	0	0
$946$	0	0
$947$	28.5414	0.927469	0.463735	−	0.885974i	$-0.346509\pi$
$947$	28.5414	0.927469	0.463735	+	0.885974i	$0.346509\pi$
$948$	0	0
$949$	96.1276	3.12043
$950$	0	0
$951$	23.7129	0.768945
$952$	0	0
$953$	−16.2810	−0.527395	−0.263697	−	0.964605i	$-0.584942\pi$
$953$	−16.2810	−0.527395	−0.263697	+	0.964605i	$0.584942\pi$
$954$	0	0
$955$	44.4143	1.43721
$956$	0	0
$957$	16.8796	0.545640
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−29.9238	−0.965283
$962$	0	0
$963$	−2.72369	−0.0877695
$964$	0	0
$965$	−44.7062	−1.43914
$966$	0	0
$967$	−29.7976	−0.958225	−0.479112	−	0.877754i	$-0.659041\pi$
$967$	−29.7976	−0.958225	−0.479112	+	0.877754i	$0.659041\pi$
$968$	0	0
$969$	−0.233213	−0.00749187
$970$	0	0
$971$	15.4946	0.497245	0.248622	−	0.968601i	$-0.420022\pi$
$971$	15.4946	0.497245	0.248622	+	0.968601i	$0.420022\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−113.158	−3.62395
$976$	0	0
$977$	34.8448	1.11478	0.557392	−	0.830249i	$-0.311802\pi$
$977$	34.8448	1.11478	0.557392	+	0.830249i	$0.311802\pi$
$978$	0	0
$979$	12.9833	0.414947
$980$	0	0
$981$	7.74523	0.247286
$982$	0	0
$983$	0.361484	0.0115296	0.00576478	−	0.999983i	$-0.498165\pi$
$983$	0.361484	0.0115296	0.00576478	+	0.999983i	$0.498165\pi$
$984$	0	0
$985$	−3.76274	−0.119891
$986$	0	0
$987$	0	0
$988$	0	0
$989$	42.2412	1.34319
$990$	0	0
$991$	28.3356	0.900110	0.450055	−	0.893001i	$-0.351404\pi$
$991$	28.3356	0.900110	0.450055	+	0.893001i	$0.351404\pi$
$992$	0	0
$993$	22.1002	0.701329
$994$	0	0
$995$	−21.2971	−0.675163
$996$	0	0
$997$	−31.3530	−0.992958	−0.496479	−	0.868049i	$-0.665374\pi$
$997$	−31.3530	−0.992958	−0.496479	+	0.868049i	$0.665374\pi$
$998$	0	0
$999$	−122.514	−3.87617

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.q.1.2		15
7.2	even	3		1148.2.i.e.165.14	✓	30
7.4	even	3		1148.2.i.e.821.14	yes	30
7.6	odd	2		8036.2.a.r.1.14		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.2.i.e.165.14	✓	30	7.2	even	3
1148.2.i.e.821.14	yes	30	7.4	even	3
8036.2.a.q.1.2		15	1.1	even	1	trivial
8036.2.a.r.1.14		15	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8036.a (trivial)

\(f(q)\)	\(=\)	\(q-3.10047 q^{3} -3.31369 q^{5} +6.61290 q^{9} +O(q^{10})\)	\(q-3.10047 q^{3} -3.31369 q^{5} +6.61290 q^{9} -0.792186 q^{11} +6.10265 q^{13} +10.2740 q^{15} +0.423014 q^{17} +0.177816 q^{19} +5.55791 q^{23} +5.98052 q^{25} -11.2017 q^{27} +6.87240 q^{29} +1.03742 q^{31} +2.45615 q^{33} +10.9371 q^{37} -18.9211 q^{39} -1.00000 q^{41} +7.60020 q^{43} -21.9131 q^{45} -0.719310 q^{47} -1.31154 q^{51} +9.70480 q^{53} +2.62506 q^{55} -0.551312 q^{57} +8.31526 q^{59} +9.24684 q^{61} -20.2223 q^{65} -14.6528 q^{67} -17.2321 q^{69} +6.50677 q^{71} +15.7518 q^{73} -18.5424 q^{75} +0.0105912 q^{79} +14.8917 q^{81} +7.04482 q^{83} -1.40174 q^{85} -21.3076 q^{87} -16.3892 q^{89} -3.21648 q^{93} -0.589225 q^{95} -12.9888 q^{97} -5.23864 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - q^{3} + 3 q^{5} + 30 q^{9}+O(q^{10}) \) 15 * q - q^3 + 3 * q^5 + 30 * q^9	\( 15 q - q^{3} + 3 q^{5} + 30 q^{9} + 9 q^{11} + 7 q^{13} + 2 q^{15} + 3 q^{17} + 7 q^{19} - q^{23} + 32 q^{25} + 11 q^{27} + 18 q^{29} + 30 q^{31} - 16 q^{33} + 23 q^{37} + 5 q^{39} - 15 q^{41} + 12 q^{43} - 13 q^{45} - 16 q^{47} + 29 q^{51} + 33 q^{53} + 37 q^{55} + 16 q^{57} - 10 q^{59} + q^{61} + 16 q^{65} + 20 q^{67} + 21 q^{69} + 5 q^{71} - 3 q^{73} - 51 q^{75} + 25 q^{79} + 43 q^{81} + 18 q^{83} + 36 q^{85} - 53 q^{87} - 11 q^{89} + 65 q^{93} - 30 q^{95} + 16 q^{97} - 18 q^{99}+O(q^{100}) \) 15 * q - q^3 + 3 * q^5 + 30 * q^9 + 9 * q^11 + 7 * q^13 + 2 * q^15 + 3 * q^17 + 7 * q^19 - q^23 + 32 * q^25 + 11 * q^27 + 18 * q^29 + 30 * q^31 - 16 * q^33 + 23 * q^37 + 5 * q^39 - 15 * q^41 + 12 * q^43 - 13 * q^45 - 16 * q^47 + 29 * q^51 + 33 * q^53 + 37 * q^55 + 16 * q^57 - 10 * q^59 + q^61 + 16 * q^65 + 20 * q^67 + 21 * q^69 + 5 * q^71 - 3 * q^73 - 51 * q^75 + 25 * q^79 + 43 * q^81 + 18 * q^83 + 36 * q^85 - 53 * q^87 - 11 * q^89 + 65 * q^93 - 30 * q^95 + 16 * q^97 - 18 * q^99

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