LMFDB - Embedded newform 6012.2.a.g.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6012,2,Mod(1,6012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6012, 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("6012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6012 = 2^{2} \cdot 3^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6012.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:	$48.0060616952$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 11x^{5} - 7x^{4} + 21x^{3} + 17x^{2} - 4x - 4 $ x^7 - 11x^5 - 7x^4 + 21x^3 + 17x^2 - 4*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 668)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.27771$ of defining polynomial
Character	$\chi$	$=$	6012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.610181 q^{5} -0.184306 q^{7} +O(q^{10})$	$q-0.610181 q^{5} -0.184306 q^{7} +5.07449 q^{11} +2.65594 q^{13} +0.128237 q^{17} -7.41417 q^{19} -3.99471 q^{23} -4.62768 q^{25} +1.33375 q^{29} -8.19083 q^{31} +0.112460 q^{35} +3.63984 q^{37} +2.15173 q^{41} -2.76015 q^{43} -4.94015 q^{47} -6.96603 q^{49} +0.756443 q^{53} -3.09636 q^{55} +6.98791 q^{59} -0.248451 q^{61} -1.62060 q^{65} -15.5507 q^{67} -4.03018 q^{71} -1.57460 q^{73} -0.935260 q^{77} +8.12950 q^{79} +15.0509 q^{83} -0.0782477 q^{85} -3.67917 q^{89} -0.489505 q^{91} +4.52399 q^{95} +10.2997 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 2 q^{5} - 12 q^{7}+O(q^{10}) $ 7 * q + 2 * q^5 - 12 * q^7	$ 7 q + 2 q^{5} - 12 q^{7} + 7 q^{11} - 9 q^{13} + q^{17} - 11 q^{19} + 19 q^{23} + 3 q^{25} + 5 q^{29} - 13 q^{31} + 7 q^{35} - 26 q^{37} + 2 q^{41} - 24 q^{43} + 11 q^{47} + 19 q^{49} - 4 q^{53} - 4 q^{55} + 4 q^{59} - 5 q^{61} - 13 q^{65} - 42 q^{67} - 9 q^{71} + 27 q^{73} - 12 q^{77} - 8 q^{79} - 16 q^{83} - 27 q^{85} - 9 q^{89} - 2 q^{91} - 10 q^{95} - 8 q^{97}+O(q^{100}) $ 7 * q + 2 * q^5 - 12 * q^7 + 7 * q^11 - 9 * q^13 + q^17 - 11 * q^19 + 19 * q^23 + 3 * q^25 + 5 * q^29 - 13 * q^31 + 7 * q^35 - 26 * q^37 + 2 * q^41 - 24 * q^43 + 11 * q^47 + 19 * q^49 - 4 * q^53 - 4 * q^55 + 4 * q^59 - 5 * q^61 - 13 * q^65 - 42 * q^67 - 9 * q^71 + 27 * q^73 - 12 * q^77 - 8 * q^79 - 16 * q^83 - 27 * q^85 - 9 * q^89 - 2 * q^91 - 10 * q^95 - 8 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−0.610181	−0.272881	−0.136441	−	0.990648i	$-0.543566\pi$
$5$	−0.610181	−0.272881	−0.136441	+	0.990648i	$0.543566\pi$
$6$	0	0
$7$	−0.184306	−0.0696611	−0.0348306	−	0.999393i	$-0.511089\pi$
$7$	−0.184306	−0.0696611	−0.0348306	+	0.999393i	$0.511089\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.07449	1.53002	0.765009	−	0.644020i	$-0.222735\pi$
$11$	5.07449	1.53002	0.765009	+	0.644020i	$0.222735\pi$
$12$	0	0
$13$	2.65594	0.736624	0.368312	−	0.929702i	$-0.379936\pi$
$13$	2.65594	0.736624	0.368312	+	0.929702i	$0.379936\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.128237	0.0311020	0.0155510	−	0.999879i	$-0.495050\pi$
$17$	0.128237	0.0311020	0.0155510	+	0.999879i	$0.495050\pi$
$18$	0	0
$19$	−7.41417	−1.70093	−0.850464	−	0.526033i	$-0.823679\pi$
$19$	−7.41417	−1.70093	−0.850464	+	0.526033i	$0.823679\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.99471	−0.832955	−0.416477	−	0.909146i	$-0.636736\pi$
$23$	−3.99471	−0.832955	−0.416477	+	0.909146i	$0.636736\pi$
$24$	0	0
$25$	−4.62768	−0.925536
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.33375	0.247671	0.123836	−	0.992303i	$-0.460480\pi$
$29$	1.33375	0.247671	0.123836	+	0.992303i	$0.460480\pi$
$30$	0	0
$31$	−8.19083	−1.47112	−0.735559	−	0.677461i	$-0.763080\pi$
$31$	−8.19083	−1.47112	−0.735559	+	0.677461i	$0.763080\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.112460	0.0190092
$36$	0	0
$37$	3.63984	0.598385	0.299193	−	0.954193i	$-0.403283\pi$
$37$	3.63984	0.598385	0.299193	+	0.954193i	$0.403283\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.15173	0.336044	0.168022	−	0.985783i	$-0.446262\pi$
$41$	2.15173	0.336044	0.168022	+	0.985783i	$0.446262\pi$
$42$	0	0
$43$	−2.76015	−0.420919	−0.210460	−	0.977603i	$-0.567496\pi$
$43$	−2.76015	−0.420919	−0.210460	+	0.977603i	$0.567496\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.94015	−0.720595	−0.360298	−	0.932837i	$-0.617325\pi$
$47$	−4.94015	−0.720595	−0.360298	+	0.932837i	$0.617325\pi$
$48$	0	0
$49$	−6.96603	−0.995147
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0.756443	0.103905	0.0519527	−	0.998650i	$-0.483455\pi$
$53$	0.756443	0.103905	0.0519527	+	0.998650i	$0.483455\pi$
$54$	0	0
$55$	−3.09636	−0.417513
$56$	0	0
$57$	0	0
$58$	0	0
$59$	6.98791	0.909748	0.454874	−	0.890556i	$-0.349684\pi$
$59$	6.98791	0.909748	0.454874	+	0.890556i	$0.349684\pi$
$60$	0	0
$61$	−0.248451	−0.0318109	−0.0159055	−	0.999874i	$-0.505063\pi$
$61$	−0.248451	−0.0318109	−0.0159055	+	0.999874i	$0.505063\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.62060	−0.201011
$66$	0	0
$67$	−15.5507	−1.89982	−0.949908	−	0.312529i	$-0.898824\pi$
$67$	−15.5507	−1.89982	−0.949908	+	0.312529i	$0.898824\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−4.03018	−0.478295	−0.239147	−	0.970983i	$-0.576868\pi$
$71$	−4.03018	−0.478295	−0.239147	+	0.970983i	$0.576868\pi$
$72$	0	0
$73$	−1.57460	−0.184292	−0.0921462	−	0.995745i	$-0.529373\pi$
$73$	−1.57460	−0.184292	−0.0921462	+	0.995745i	$0.529373\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.935260	−0.106583
$78$	0	0
$79$	8.12950	0.914640	0.457320	−	0.889302i	$-0.348809\pi$
$79$	8.12950	0.914640	0.457320	+	0.889302i	$0.348809\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	15.0509	1.65205	0.826027	−	0.563631i	$-0.190596\pi$
$83$	15.0509	1.65205	0.826027	+	0.563631i	$0.190596\pi$
$84$	0	0
$85$	−0.0782477	−0.00848715
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−3.67917	−0.389991	−0.194996	−	0.980804i	$-0.562469\pi$
$89$	−3.67917	−0.389991	−0.194996	+	0.980804i	$0.562469\pi$
$90$	0	0
$91$	−0.489505	−0.0513141
$92$	0	0
$93$	0	0
$94$	0	0
$95$	4.52399	0.464152
$96$	0	0
$97$	10.2997	1.04577	0.522886	−	0.852402i	$-0.324855\pi$
$97$	10.2997	1.04577	0.522886	+	0.852402i	$0.324855\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−16.1953	−1.61149	−0.805745	−	0.592263i	$-0.798235\pi$
$101$	−16.1953	−1.61149	−0.805745	+	0.592263i	$0.798235\pi$
$102$	0	0
$103$	2.33475	0.230049	0.115025	−	0.993363i	$-0.463305\pi$
$103$	2.33475	0.230049	0.115025	+	0.993363i	$0.463305\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−5.75331	−0.556194	−0.278097	−	0.960553i	$-0.589704\pi$
$107$	−5.75331	−0.556194	−0.278097	+	0.960553i	$0.589704\pi$
$108$	0	0
$109$	12.2756	1.17578	0.587892	−	0.808939i	$-0.299958\pi$
$109$	12.2756	1.17578	0.587892	+	0.808939i	$0.299958\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−14.0646	−1.32308	−0.661542	−	0.749908i	$-0.730098\pi$
$113$	−14.0646	−1.32308	−0.661542	+	0.749908i	$0.730098\pi$
$114$	0	0
$115$	2.43750	0.227298
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.0236348	−0.00216660
$120$	0	0
$121$	14.7505	1.34095
$122$	0	0
$123$	0	0
$124$	0	0
$125$	5.87463	0.525443
$126$	0	0
$127$	−22.3104	−1.97973	−0.989866	−	0.142004i	$-0.954646\pi$
$127$	−22.3104	−1.97973	−0.989866	+	0.142004i	$0.954646\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−3.05168	−0.266627	−0.133313	−	0.991074i	$-0.542562\pi$
$131$	−3.05168	−0.266627	−0.133313	+	0.991074i	$0.542562\pi$
$132$	0	0
$133$	1.36648	0.118489
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1.65628	−0.141505	−0.0707526	−	0.997494i	$-0.522540\pi$
$137$	−1.65628	−0.141505	−0.0707526	+	0.997494i	$0.522540\pi$
$138$	0	0
$139$	9.85669	0.836034	0.418017	−	0.908439i	$-0.362725\pi$
$139$	9.85669	0.836034	0.418017	+	0.908439i	$0.362725\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	13.4775	1.12705
$144$	0	0
$145$	−0.813829	−0.0675848
$146$	0	0
$147$	0	0
$148$	0	0
$149$	15.9076	1.30320	0.651599	−	0.758564i	$-0.274099\pi$
$149$	15.9076	1.30320	0.651599	+	0.758564i	$0.274099\pi$
$150$	0	0
$151$	14.6612	1.19311	0.596557	−	0.802571i	$-0.296535\pi$
$151$	14.6612	1.19311	0.596557	+	0.802571i	$0.296535\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.99789	0.401440
$156$	0	0
$157$	−4.30181	−0.343322	−0.171661	−	0.985156i	$-0.554913\pi$
$157$	−4.30181	−0.343322	−0.171661	+	0.985156i	$0.554913\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0.736250	0.0580246
$162$	0	0
$163$	18.6533	1.46104	0.730520	−	0.682891i	$-0.239278\pi$
$163$	18.6533	1.46104	0.730520	+	0.682891i	$0.239278\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−5.94600	−0.457385
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−22.4896	−1.70985	−0.854927	−	0.518748i	$-0.826398\pi$
$173$	−22.4896	−1.70985	−0.854927	+	0.518748i	$0.826398\pi$
$174$	0	0
$175$	0.852909	0.0644739
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−6.85112	−0.512077	−0.256038	−	0.966667i	$-0.582417\pi$
$179$	−6.85112	−0.512077	−0.256038	+	0.966667i	$0.582417\pi$
$180$	0	0
$181$	−15.3603	−1.14172	−0.570862	−	0.821046i	$-0.693391\pi$
$181$	−15.3603	−1.14172	−0.570862	+	0.821046i	$0.693391\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−2.22096	−0.163288
$186$	0	0
$187$	0.650737	0.0475866
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−14.9438	−1.08130	−0.540648	−	0.841249i	$-0.681821\pi$
$191$	−14.9438	−1.08130	−0.540648	+	0.841249i	$0.681821\pi$
$192$	0	0
$193$	−21.3141	−1.53422	−0.767111	−	0.641514i	$-0.778307\pi$
$193$	−21.3141	−1.53422	−0.767111	+	0.641514i	$0.778307\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.98752	0.497840	0.248920	−	0.968524i	$-0.419924\pi$
$197$	6.98752	0.497840	0.248920	+	0.968524i	$0.419924\pi$
$198$	0	0
$199$	−8.93797	−0.633596	−0.316798	−	0.948493i	$-0.602608\pi$
$199$	−8.93797	−0.633596	−0.316798	+	0.948493i	$0.602608\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−0.245818	−0.0172531
$204$	0	0
$205$	−1.31295	−0.0917002
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−37.6232	−2.60245
$210$	0	0
$211$	−26.6715	−1.83614	−0.918070	−	0.396419i	$-0.870253\pi$
$211$	−26.6715	−1.83614	−0.918070	+	0.396419i	$0.870253\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1.68419	0.114861
$216$	0	0
$217$	1.50962	0.102480
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0.340589	0.0229105
$222$	0	0
$223$	−3.46480	−0.232020	−0.116010	−	0.993248i	$-0.537011\pi$
$223$	−3.46480	−0.232020	−0.116010	+	0.993248i	$0.537011\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	20.8856	1.38622	0.693112	−	0.720830i	$-0.256239\pi$
$227$	20.8856	1.38622	0.693112	+	0.720830i	$0.256239\pi$
$228$	0	0
$229$	−10.4052	−0.687593	−0.343796	−	0.939044i	$-0.611713\pi$
$229$	−10.4052	−0.687593	−0.343796	+	0.939044i	$0.611713\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−5.91067	−0.387221	−0.193611	−	0.981078i	$-0.562020\pi$
$233$	−5.91067	−0.387221	−0.193611	+	0.981078i	$0.562020\pi$
$234$	0	0
$235$	3.01439	0.196637
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−10.2644	−0.663946	−0.331973	−	0.943289i	$-0.607714\pi$
$239$	−10.2644	−0.663946	−0.331973	+	0.943289i	$0.607714\pi$
$240$	0	0
$241$	−0.544851	−0.0350970	−0.0175485	−	0.999846i	$-0.505586\pi$
$241$	−0.544851	−0.0350970	−0.0175485	+	0.999846i	$0.505586\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	4.25054	0.271557
$246$	0	0
$247$	−19.6916	−1.25294
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−1.46231	−0.0923003	−0.0461501	−	0.998935i	$-0.514695\pi$
$251$	−1.46231	−0.0923003	−0.0461501	+	0.998935i	$0.514695\pi$
$252$	0	0
$253$	−20.2711	−1.27444
$254$	0	0
$255$	0	0
$256$	0	0
$257$	5.88199	0.366909	0.183454	−	0.983028i	$-0.441272\pi$
$257$	5.88199	0.366909	0.183454	+	0.983028i	$0.441272\pi$
$258$	0	0
$259$	−0.670844	−0.0416842
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−25.6134	−1.57939	−0.789694	−	0.613501i	$-0.789760\pi$
$263$	−25.6134	−1.57939	−0.789694	+	0.613501i	$0.789760\pi$
$264$	0	0
$265$	−0.461567	−0.0283538
$266$	0	0
$267$	0	0
$268$	0	0
$269$	32.1214	1.95847	0.979237	−	0.202717i	$-0.0649772\pi$
$269$	32.1214	1.95847	0.979237	+	0.202717i	$0.0649772\pi$
$270$	0	0
$271$	−19.4060	−1.17883	−0.589415	−	0.807831i	$-0.700642\pi$
$271$	−19.4060	−1.17883	−0.589415	+	0.807831i	$0.700642\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−23.4831	−1.41609
$276$	0	0
$277$	14.5086	0.871739	0.435869	−	0.900010i	$-0.356441\pi$
$277$	14.5086	0.871739	0.435869	+	0.900010i	$0.356441\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−14.7215	−0.878212	−0.439106	−	0.898435i	$-0.644705\pi$
$281$	−14.7215	−0.878212	−0.439106	+	0.898435i	$0.644705\pi$
$282$	0	0
$283$	−20.5585	−1.22207	−0.611037	−	0.791602i	$-0.709247\pi$
$283$	−20.5585	−1.22207	−0.611037	+	0.791602i	$0.709247\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−0.396577	−0.0234092
$288$	0	0
$289$	−16.9836	−0.999033
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−14.3988	−0.841188	−0.420594	−	0.907249i	$-0.638178\pi$
$293$	−14.3988	−0.841188	−0.420594	+	0.907249i	$0.638178\pi$
$294$	0	0
$295$	−4.26389	−0.248253
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−10.6097	−0.613575
$300$	0	0
$301$	0.508713	0.0293217
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0.151600	0.00868060
$306$	0	0
$307$	1.89578	0.108198	0.0540991	−	0.998536i	$-0.482771\pi$
$307$	1.89578	0.108198	0.0540991	+	0.998536i	$0.482771\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−15.4944	−0.878609	−0.439305	−	0.898338i	$-0.644775\pi$
$311$	−15.4944	−0.878609	−0.439305	+	0.898338i	$0.644775\pi$
$312$	0	0
$313$	20.2090	1.14228	0.571139	−	0.820854i	$-0.306502\pi$
$313$	20.2090	1.14228	0.571139	+	0.820854i	$0.306502\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	23.5767	1.32420	0.662099	−	0.749417i	$-0.269666\pi$
$317$	23.5767	1.32420	0.662099	+	0.749417i	$0.269666\pi$
$318$	0	0
$319$	6.76810	0.378941
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−0.950770	−0.0529023
$324$	0	0
$325$	−12.2908	−0.681772
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0.910500	0.0501975
$330$	0	0
$331$	−18.2487	−1.00304	−0.501521	−	0.865146i	$-0.667226\pi$
$331$	−18.2487	−1.00304	−0.501521	+	0.865146i	$0.667226\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	9.48873	0.518424
$336$	0	0
$337$	−13.1625	−0.717009	−0.358504	−	0.933528i	$-0.616713\pi$
$337$	−13.1625	−0.717009	−0.358504	+	0.933528i	$0.616713\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−41.5643	−2.25083
$342$	0	0
$343$	2.57402	0.138984
$344$	0	0
$345$	0	0
$346$	0	0
$347$	20.2699	1.08814	0.544072	−	0.839038i	$-0.316882\pi$
$347$	20.2699	1.08814	0.544072	+	0.839038i	$0.316882\pi$
$348$	0	0
$349$	−15.5167	−0.830591	−0.415296	−	0.909686i	$-0.636322\pi$
$349$	−15.5167	−0.830591	−0.415296	+	0.909686i	$0.636322\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	10.3568	0.551237	0.275619	−	0.961267i	$-0.411117\pi$
$353$	10.3568	0.551237	0.275619	+	0.961267i	$0.411117\pi$
$354$	0	0
$355$	2.45914	0.130518
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1.12973	−0.0596249	−0.0298125	−	0.999556i	$-0.509491\pi$
$359$	−1.12973	−0.0596249	−0.0298125	+	0.999556i	$0.509491\pi$
$360$	0	0
$361$	35.9700	1.89316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0.960789	0.0502900
$366$	0	0
$367$	8.33352	0.435006	0.217503	−	0.976060i	$-0.430209\pi$
$367$	8.33352	0.435006	0.217503	+	0.976060i	$0.430209\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−0.139417	−0.00723817
$372$	0	0
$373$	−33.1243	−1.71511	−0.857556	−	0.514391i	$-0.828018\pi$
$373$	−33.1243	−1.71511	−0.857556	+	0.514391i	$0.828018\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.54235	0.182440
$378$	0	0
$379$	1.19041	0.0611472	0.0305736	−	0.999533i	$-0.490267\pi$
$379$	1.19041	0.0611472	0.0305736	+	0.999533i	$0.490267\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	5.60495	0.286399	0.143200	−	0.989694i	$-0.454261\pi$
$383$	5.60495	0.286399	0.143200	+	0.989694i	$0.454261\pi$
$384$	0	0
$385$	0.570678	0.0290844
$386$	0	0
$387$	0	0
$388$	0	0
$389$	29.7226	1.50700	0.753498	−	0.657450i	$-0.228365\pi$
$389$	29.7226	1.50700	0.753498	+	0.657450i	$0.228365\pi$
$390$	0	0
$391$	−0.512269	−0.0259066
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−4.96047	−0.249588
$396$	0	0
$397$	1.62739	0.0816765	0.0408382	−	0.999166i	$-0.486997\pi$
$397$	1.62739	0.0816765	0.0408382	+	0.999166i	$0.486997\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	15.9783	0.797918	0.398959	−	0.916969i	$-0.369372\pi$
$401$	15.9783	0.797918	0.398959	+	0.916969i	$0.369372\pi$
$402$	0	0
$403$	−21.7543	−1.08366
$404$	0	0
$405$	0	0
$406$	0	0
$407$	18.4703	0.915540
$408$	0	0
$409$	6.78228	0.335362	0.167681	−	0.985841i	$-0.446372\pi$
$409$	6.78228	0.335362	0.167681	+	0.985841i	$0.446372\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1.28791	−0.0633741
$414$	0	0
$415$	−9.18379	−0.450815
$416$	0	0
$417$	0	0
$418$	0	0
$419$	23.3434	1.14040	0.570200	−	0.821506i	$-0.306866\pi$
$419$	23.3434	1.14040	0.570200	+	0.821506i	$0.306866\pi$
$420$	0	0
$421$	−14.2208	−0.693078	−0.346539	−	0.938036i	$-0.612643\pi$
$421$	−14.2208	−0.693078	−0.346539	+	0.938036i	$0.612643\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−0.593439	−0.0287860
$426$	0	0
$427$	0.0457911	0.00221598
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−25.9317	−1.24909	−0.624543	−	0.780990i	$-0.714715\pi$
$431$	−25.9317	−1.24909	−0.624543	+	0.780990i	$0.714715\pi$
$432$	0	0
$433$	28.9884	1.39309	0.696546	−	0.717512i	$-0.254719\pi$
$433$	28.9884	1.39309	0.696546	+	0.717512i	$0.254719\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	29.6175	1.41680
$438$	0	0
$439$	−10.8487	−0.517778	−0.258889	−	0.965907i	$-0.583356\pi$
$439$	−10.8487	−0.517778	−0.258889	+	0.965907i	$0.583356\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−8.31119	−0.394876	−0.197438	−	0.980315i	$-0.563262\pi$
$443$	−8.31119	−0.394876	−0.197438	+	0.980315i	$0.563262\pi$
$444$	0	0
$445$	2.24496	0.106421
$446$	0	0
$447$	0	0
$448$	0	0
$449$	25.5362	1.20513	0.602563	−	0.798071i	$-0.294146\pi$
$449$	25.5362	1.20513	0.602563	+	0.798071i	$0.294146\pi$
$450$	0	0
$451$	10.9189	0.514153
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0.298687	0.0140027
$456$	0	0
$457$	−15.4963	−0.724888	−0.362444	−	0.932006i	$-0.618058\pi$
$457$	−15.4963	−0.724888	−0.362444	+	0.932006i	$0.618058\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	2.39851	0.111710	0.0558549	−	0.998439i	$-0.482212\pi$
$461$	2.39851	0.111710	0.0558549	+	0.998439i	$0.482212\pi$
$462$	0	0
$463$	26.6280	1.23751	0.618753	−	0.785585i	$-0.287638\pi$
$463$	26.6280	1.23751	0.618753	+	0.785585i	$0.287638\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−22.6995	−1.05041	−0.525204	−	0.850976i	$-0.676011\pi$
$467$	−22.6995	−1.05041	−0.525204	+	0.850976i	$0.676011\pi$
$468$	0	0
$469$	2.86608	0.132343
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−14.0064	−0.644014
$474$	0	0
$475$	34.3104	1.57427
$476$	0	0
$477$	0	0
$478$	0	0
$479$	25.2174	1.15221	0.576107	−	0.817374i	$-0.304571\pi$
$479$	25.2174	1.15221	0.576107	+	0.817374i	$0.304571\pi$
$480$	0	0
$481$	9.66717	0.440785
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6.28466	−0.285372
$486$	0	0
$487$	−10.2524	−0.464580	−0.232290	−	0.972647i	$-0.574622\pi$
$487$	−10.2524	−0.464580	−0.232290	+	0.972647i	$0.574622\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−19.6373	−0.886221	−0.443110	−	0.896467i	$-0.646125\pi$
$491$	−19.6373	−0.886221	−0.443110	+	0.896467i	$0.646125\pi$
$492$	0	0
$493$	0.171036	0.00770306
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0.742787	0.0333185
$498$	0	0
$499$	14.3837	0.643902	0.321951	−	0.946756i	$-0.395661\pi$
$499$	14.3837	0.643902	0.321951	+	0.946756i	$0.395661\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−37.4799	−1.67115	−0.835573	−	0.549380i	$-0.814864\pi$
$503$	−37.4799	−1.67115	−0.835573	+	0.549380i	$0.814864\pi$
$504$	0	0
$505$	9.88205	0.439745
$506$	0	0
$507$	0	0
$508$	0	0
$509$	21.0508	0.933060	0.466530	−	0.884505i	$-0.345504\pi$
$509$	21.0508	0.933060	0.466530	+	0.884505i	$0.345504\pi$
$510$	0	0
$511$	0.290208	0.0128380
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−1.42462	−0.0627762
$516$	0	0
$517$	−25.0688	−1.10252
$518$	0	0
$519$	0	0
$520$	0	0
$521$	13.2544	0.580685	0.290342	−	0.956923i	$-0.406231\pi$
$521$	13.2544	0.580685	0.290342	+	0.956923i	$0.406231\pi$
$522$	0	0
$523$	−4.16816	−0.182261	−0.0911304	−	0.995839i	$-0.529048\pi$
$523$	−4.16816	−0.182261	−0.0911304	+	0.995839i	$0.529048\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.05037	−0.0457547
$528$	0	0
$529$	−7.04228	−0.306186
$530$	0	0
$531$	0	0
$532$	0	0
$533$	5.71486	0.247538
$534$	0	0
$535$	3.51056	0.151775
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−35.3491	−1.52259
$540$	0	0
$541$	−20.0822	−0.863402	−0.431701	−	0.902017i	$-0.642086\pi$
$541$	−20.0822	−0.863402	−0.431701	+	0.902017i	$0.642086\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−7.49031	−0.320850
$546$	0	0
$547$	−20.6504	−0.882949	−0.441474	−	0.897274i	$-0.645544\pi$
$547$	−20.6504	−0.882949	−0.441474	+	0.897274i	$0.645544\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−9.88865	−0.421271
$552$	0	0
$553$	−1.49832	−0.0637149
$554$	0	0
$555$	0	0
$556$	0	0
$557$	31.4316	1.33180	0.665900	−	0.746041i	$-0.268048\pi$
$557$	31.4316	1.33180	0.665900	+	0.746041i	$0.268048\pi$
$558$	0	0
$559$	−7.33079	−0.310059
$560$	0	0
$561$	0	0
$562$	0	0
$563$	5.67831	0.239312	0.119656	−	0.992815i	$-0.461821\pi$
$563$	5.67831	0.239312	0.119656	+	0.992815i	$0.461821\pi$
$564$	0	0
$565$	8.58194	0.361045
$566$	0	0
$567$	0	0
$568$	0	0
$569$	5.58704	0.234221	0.117110	−	0.993119i	$-0.462637\pi$
$569$	5.58704	0.234221	0.117110	+	0.993119i	$0.462637\pi$
$570$	0	0
$571$	−5.21501	−0.218241	−0.109121	−	0.994029i	$-0.534803\pi$
$571$	−5.21501	−0.218241	−0.109121	+	0.994029i	$0.534803\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	18.4862	0.770930
$576$	0	0
$577$	20.4310	0.850554	0.425277	−	0.905063i	$-0.360177\pi$
$577$	20.4310	0.850554	0.425277	+	0.905063i	$0.360177\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−2.77398	−0.115084
$582$	0	0
$583$	3.83856	0.158977
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−41.9436	−1.73120	−0.865599	−	0.500739i	$-0.833062\pi$
$587$	−41.9436	−1.73120	−0.865599	+	0.500739i	$0.833062\pi$
$588$	0	0
$589$	60.7283	2.50226
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−23.5669	−0.967775	−0.483887	−	0.875130i	$-0.660776\pi$
$593$	−23.5669	−0.967775	−0.483887	+	0.875130i	$0.660776\pi$
$594$	0	0
$595$	0.0144215	0.000591225 0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−24.8325	−1.01463	−0.507314	−	0.861761i	$-0.669362\pi$
$599$	−24.8325	−1.01463	−0.507314	+	0.861761i	$0.669362\pi$
$600$	0	0
$601$	33.1836	1.35359	0.676795	−	0.736172i	$-0.263369\pi$
$601$	33.1836	1.35359	0.676795	+	0.736172i	$0.263369\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−9.00046	−0.365921
$606$	0	0
$607$	14.5976	0.592500	0.296250	−	0.955110i	$-0.404264\pi$
$607$	14.5976	0.592500	0.296250	+	0.955110i	$0.404264\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−13.1207	−0.530808
$612$	0	0
$613$	6.58591	0.266002	0.133001	−	0.991116i	$-0.457539\pi$
$613$	6.58591	0.266002	0.133001	+	0.991116i	$0.457539\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	35.0859	1.41250	0.706252	−	0.707960i	$-0.250384\pi$
$617$	35.0859	1.41250	0.706252	+	0.707960i	$0.250384\pi$
$618$	0	0
$619$	18.3113	0.735994	0.367997	−	0.929827i	$-0.380044\pi$
$619$	18.3113	0.735994	0.367997	+	0.929827i	$0.380044\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0.678093	0.0271672
$624$	0	0
$625$	19.5538	0.782152
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0.466761	0.0186110
$630$	0	0
$631$	−24.0740	−0.958373	−0.479186	−	0.877713i	$-0.659068\pi$
$631$	−24.0740	−0.958373	−0.479186	+	0.877713i	$0.659068\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	13.6134	0.540232
$636$	0	0
$637$	−18.5013	−0.733049
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−4.77393	−0.188559	−0.0942795	−	0.995546i	$-0.530055\pi$
$641$	−4.77393	−0.188559	−0.0942795	+	0.995546i	$0.530055\pi$
$642$	0	0
$643$	0.669908	0.0264186	0.0132093	−	0.999913i	$-0.495795\pi$
$643$	0.669908	0.0264186	0.0132093	+	0.999913i	$0.495795\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−24.8779	−0.978049	−0.489025	−	0.872270i	$-0.662647\pi$
$647$	−24.8779	−0.978049	−0.489025	+	0.872270i	$0.662647\pi$
$648$	0	0
$649$	35.4601	1.39193
$650$	0	0
$651$	0	0
$652$	0	0
$653$	30.7916	1.20497	0.602484	−	0.798131i	$-0.294178\pi$
$653$	30.7916	1.20497	0.602484	+	0.798131i	$0.294178\pi$
$654$	0	0
$655$	1.86208	0.0727575
$656$	0	0
$657$	0	0
$658$	0	0
$659$	33.8998	1.32055	0.660274	−	0.751025i	$-0.270440\pi$
$659$	33.8998	1.32055	0.660274	+	0.751025i	$0.270440\pi$
$660$	0	0
$661$	33.4469	1.30094	0.650468	−	0.759534i	$-0.274573\pi$
$661$	33.4469	1.30094	0.650468	+	0.759534i	$0.274573\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−0.833799	−0.0323333
$666$	0	0
$667$	−5.32794	−0.206299
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−1.26076	−0.0486712
$672$	0	0
$673$	−20.2228	−0.779532	−0.389766	−	0.920914i	$-0.627444\pi$
$673$	−20.2228	−0.779532	−0.389766	+	0.920914i	$0.627444\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−31.5078	−1.21094	−0.605471	−	0.795868i	$-0.707015\pi$
$677$	−31.5078	−1.21094	−0.605471	+	0.795868i	$0.707015\pi$
$678$	0	0
$679$	−1.89829	−0.0728497
$680$	0	0
$681$	0	0
$682$	0	0
$683$	14.9258	0.571119	0.285560	−	0.958361i	$-0.407821\pi$
$683$	14.9258	0.571119	0.285560	+	0.958361i	$0.407821\pi$
$684$	0	0
$685$	1.01063	0.0386141
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.00906	0.0765392
$690$	0	0
$691$	−7.34883	−0.279563	−0.139781	−	0.990182i	$-0.544640\pi$
$691$	−7.34883	−0.279563	−0.139781	+	0.990182i	$0.544640\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−6.01437	−0.228138
$696$	0	0
$697$	0.275931	0.0104516
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−37.5538	−1.41839	−0.709193	−	0.705014i	$-0.750941\pi$
$701$	−37.5538	−1.41839	−0.709193	+	0.705014i	$0.750941\pi$
$702$	0	0
$703$	−26.9864	−1.01781
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.98489	0.112258
$708$	0	0
$709$	−43.3729	−1.62890	−0.814451	−	0.580232i	$-0.802962\pi$
$709$	−43.3729	−1.62890	−0.814451	+	0.580232i	$0.802962\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	32.7200	1.22537
$714$	0	0
$715$	−8.22373	−0.307550
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−36.6696	−1.36755	−0.683774	−	0.729694i	$-0.739662\pi$
$719$	−36.6696	−1.36755	−0.683774	+	0.729694i	$0.739662\pi$
$720$	0	0
$721$	−0.430308	−0.0160255
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−6.17216	−0.229228
$726$	0	0
$727$	40.9797	1.51985	0.759926	−	0.650009i	$-0.225235\pi$
$727$	40.9797	1.51985	0.759926	+	0.650009i	$0.225235\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−0.353953	−0.0130914
$732$	0	0
$733$	−21.8632	−0.807535	−0.403768	−	0.914862i	$-0.632300\pi$
$733$	−21.8632	−0.807535	−0.403768	+	0.914862i	$0.632300\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−78.9117	−2.90675
$738$	0	0
$739$	−23.2901	−0.856741	−0.428370	−	0.903603i	$-0.640912\pi$
$739$	−23.2901	−0.856741	−0.428370	+	0.903603i	$0.640912\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−9.64619	−0.353884	−0.176942	−	0.984221i	$-0.556621\pi$
$743$	−9.64619	−0.353884	−0.176942	+	0.984221i	$0.556621\pi$
$744$	0	0
$745$	−9.70649	−0.355618
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1.06037	0.0387451
$750$	0	0
$751$	−12.2736	−0.447870	−0.223935	−	0.974604i	$-0.571890\pi$
$751$	−12.2736	−0.447870	−0.223935	+	0.974604i	$0.571890\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−8.94600	−0.325578
$756$	0	0
$757$	−25.4007	−0.923206	−0.461603	−	0.887087i	$-0.652725\pi$
$757$	−25.4007	−0.923206	−0.461603	+	0.887087i	$0.652725\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	46.1364	1.67244	0.836221	−	0.548392i	$-0.184760\pi$
$761$	46.1364	1.67244	0.836221	+	0.548392i	$0.184760\pi$
$762$	0	0
$763$	−2.26246	−0.0819065
$764$	0	0
$765$	0	0
$766$	0	0
$767$	18.5594	0.670143
$768$	0	0
$769$	2.04597	0.0737797	0.0368899	−	0.999319i	$-0.488255\pi$
$769$	2.04597	0.0737797	0.0368899	+	0.999319i	$0.488255\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−33.1862	−1.19362	−0.596812	−	0.802381i	$-0.703566\pi$
$773$	−33.1862	−1.19362	−0.596812	+	0.802381i	$0.703566\pi$
$774$	0	0
$775$	37.9045	1.36157
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−15.9533	−0.571587
$780$	0	0
$781$	−20.4511	−0.731799
$782$	0	0
$783$	0	0
$784$	0	0
$785$	2.62488	0.0936861
$786$	0	0
$787$	−12.0389	−0.429140	−0.214570	−	0.976709i	$-0.568835\pi$
$787$	−12.0389	−0.429140	−0.214570	+	0.976709i	$0.568835\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	2.59219	0.0921676
$792$	0	0
$793$	−0.659870	−0.0234327
$794$	0	0
$795$	0	0
$796$	0	0
$797$	26.5238	0.939520	0.469760	−	0.882794i	$-0.344341\pi$
$797$	26.5238	0.939520	0.469760	+	0.882794i	$0.344341\pi$
$798$	0	0
$799$	−0.633509	−0.0224119
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−7.99027	−0.281971
$804$	0	0
$805$	−0.449246	−0.0158338
$806$	0	0
$807$	0	0
$808$	0	0
$809$	38.4463	1.35170	0.675850	−	0.737039i	$-0.263777\pi$
$809$	38.4463	1.35170	0.675850	+	0.737039i	$0.263777\pi$
$810$	0	0
$811$	21.5375	0.756283	0.378142	−	0.925748i	$-0.376563\pi$
$811$	21.5375	0.756283	0.378142	+	0.925748i	$0.376563\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−11.3819	−0.398691
$816$	0	0
$817$	20.4642	0.715953
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−17.2545	−0.602185	−0.301093	−	0.953595i	$-0.597351\pi$
$821$	−17.2545	−0.602185	−0.301093	+	0.953595i	$0.597351\pi$
$822$	0	0
$823$	5.77925	0.201452	0.100726	−	0.994914i	$-0.467883\pi$
$823$	5.77925	0.201452	0.100726	+	0.994914i	$0.467883\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−6.80055	−0.236478	−0.118239	−	0.992985i	$-0.537725\pi$
$827$	−6.80055	−0.236478	−0.118239	+	0.992985i	$0.537725\pi$
$828$	0	0
$829$	17.1422	0.595374	0.297687	−	0.954664i	$-0.403785\pi$
$829$	17.1422	0.595374	0.297687	+	0.954664i	$0.403785\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−0.893302	−0.0309511
$834$	0	0
$835$	0.610181	0.0211162
$836$	0	0
$837$	0	0
$838$	0	0
$839$	16.8702	0.582425	0.291212	−	0.956658i	$-0.405941\pi$
$839$	16.8702	0.582425	0.291212	+	0.956658i	$0.405941\pi$
$840$	0	0
$841$	−27.2211	−0.938659
$842$	0	0
$843$	0	0
$844$	0	0
$845$	3.62814	0.124812
$846$	0	0
$847$	−2.71860	−0.0934123
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−14.5401	−0.498428
$852$	0	0
$853$	26.8145	0.918112	0.459056	−	0.888407i	$-0.348188\pi$
$853$	26.8145	0.918112	0.459056	+	0.888407i	$0.348188\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−17.3952	−0.594209	−0.297105	−	0.954845i	$-0.596021\pi$
$857$	−17.3952	−0.594209	−0.297105	+	0.954845i	$0.596021\pi$
$858$	0	0
$859$	38.7412	1.32183	0.660916	−	0.750460i	$-0.270168\pi$
$859$	38.7412	1.32183	0.660916	+	0.750460i	$0.270168\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−32.1233	−1.09349	−0.546745	−	0.837299i	$-0.684133\pi$
$863$	−32.1233	−1.09349	−0.546745	+	0.837299i	$0.684133\pi$
$864$	0	0
$865$	13.7227	0.466587
$866$	0	0
$867$	0	0
$868$	0	0
$869$	41.2531	1.39941
$870$	0	0
$871$	−41.3016	−1.39945
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1.08273	−0.0366029
$876$	0	0
$877$	−27.3296	−0.922854	−0.461427	−	0.887178i	$-0.652662\pi$
$877$	−27.3296	−0.922854	−0.461427	+	0.887178i	$0.652662\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−4.90030	−0.165095	−0.0825476	−	0.996587i	$-0.526306\pi$
$881$	−4.90030	−0.165095	−0.0825476	+	0.996587i	$0.526306\pi$
$882$	0	0
$883$	−53.6493	−1.80544	−0.902722	−	0.430225i	$-0.858434\pi$
$883$	−53.6493	−1.80544	−0.902722	+	0.430225i	$0.858434\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	52.1938	1.75250	0.876248	−	0.481861i	$-0.160039\pi$
$887$	52.1938	1.75250	0.876248	+	0.481861i	$0.160039\pi$
$888$	0	0
$889$	4.11195	0.137910
$890$	0	0
$891$	0	0
$892$	0	0
$893$	36.6271	1.22568
$894$	0	0
$895$	4.18043	0.139736
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−10.9245	−0.364353
$900$	0	0
$901$	0.0970038	0.00323167
$902$	0	0
$903$	0	0
$904$	0	0
$905$	9.37259	0.311555
$906$	0	0
$907$	−8.77307	−0.291305	−0.145653	−	0.989336i	$-0.546528\pi$
$907$	−8.77307	−0.291305	−0.145653	+	0.989336i	$0.546528\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	54.0083	1.78937	0.894687	−	0.446693i	$-0.147398\pi$
$911$	54.0083	1.78937	0.894687	+	0.446693i	$0.147398\pi$
$912$	0	0
$913$	76.3758	2.52767
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0.562444	0.0185735
$918$	0	0
$919$	4.59108	0.151446	0.0757228	−	0.997129i	$-0.475874\pi$
$919$	4.59108	0.151446	0.0757228	+	0.997129i	$0.475874\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−10.7039	−0.352323
$924$	0	0
$925$	−16.8440	−0.553827
$926$	0	0
$927$	0	0
$928$	0	0
$929$	20.6593	0.677808	0.338904	−	0.940821i	$-0.389944\pi$
$929$	20.6593	0.677808	0.338904	+	0.940821i	$0.389944\pi$
$930$	0	0
$931$	51.6474	1.69267
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−0.397067	−0.0129855
$936$	0	0
$937$	−6.25885	−0.204468	−0.102234	−	0.994760i	$-0.532599\pi$
$937$	−6.25885	−0.204468	−0.102234	+	0.994760i	$0.532599\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	56.1332	1.82989	0.914945	−	0.403579i	$-0.132234\pi$
$941$	56.1332	1.82989	0.914945	+	0.403579i	$0.132234\pi$
$942$	0	0
$943$	−8.59555	−0.279910
$944$	0	0
$945$	0	0
$946$	0	0
$947$	40.8312	1.32684	0.663418	−	0.748249i	$-0.269105\pi$
$947$	40.8312	1.32684	0.663418	+	0.748249i	$0.269105\pi$
$948$	0	0
$949$	−4.18203	−0.135754
$950$	0	0
$951$	0	0
$952$	0	0
$953$	48.9462	1.58552	0.792762	−	0.609531i	$-0.208642\pi$
$953$	48.9462	1.58552	0.792762	+	0.609531i	$0.208642\pi$
$954$	0	0
$955$	9.11843	0.295066
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0.305262	0.00985741
$960$	0	0
$961$	36.0898	1.16419
$962$	0	0
$963$	0	0
$964$	0	0
$965$	13.0055	0.418661
$966$	0	0
$967$	−21.9422	−0.705614	−0.352807	−	0.935696i	$-0.614773\pi$
$967$	−21.9422	−0.705614	−0.352807	+	0.935696i	$0.614773\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−14.8431	−0.476338	−0.238169	−	0.971224i	$-0.576547\pi$
$971$	−14.8431	−0.476338	−0.238169	+	0.971224i	$0.576547\pi$
$972$	0	0
$973$	−1.81665	−0.0582391
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−10.7071	−0.342551	−0.171275	−	0.985223i	$-0.554789\pi$
$977$	−10.7071	−0.342551	−0.171275	+	0.985223i	$0.554789\pi$
$978$	0	0
$979$	−18.6699	−0.596693
$980$	0	0
$981$	0	0
$982$	0	0
$983$	4.05032	0.129185	0.0645925	−	0.997912i	$-0.479425\pi$
$983$	4.05032	0.129185	0.0645925	+	0.997912i	$0.479425\pi$
$984$	0	0
$985$	−4.26365	−0.135851
$986$	0	0
$987$	0	0
$988$	0	0
$989$	11.0260	0.350607
$990$	0	0
$991$	−23.0339	−0.731696	−0.365848	−	0.930675i	$-0.619221\pi$
$991$	−23.0339	−0.731696	−0.365848	+	0.930675i	$0.619221\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	5.45378	0.172896
$996$	0	0
$997$	33.8866	1.07320	0.536600	−	0.843837i	$-0.319709\pi$
$997$	33.8866	1.07320	0.536600	+	0.843837i	$0.319709\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6012.2.a.g.1.3		7
3.2	odd	2		668.2.a.c.1.4	✓	7
12.11	even	2		2672.2.a.k.1.4		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
668.2.a.c.1.4	✓	7	3.2	odd	2
2672.2.a.k.1.4		7	12.11	even	2
6012.2.a.g.1.3		7	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:	\( N \)	\(=\)	\( 6012 = 2^{2} \cdot 3^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6012.a (trivial)

\(f(q)\)	\(=\)	\(q-0.610181 q^{5} -0.184306 q^{7} +O(q^{10})\)	\(q-0.610181 q^{5} -0.184306 q^{7} +5.07449 q^{11} +2.65594 q^{13} +0.128237 q^{17} -7.41417 q^{19} -3.99471 q^{23} -4.62768 q^{25} +1.33375 q^{29} -8.19083 q^{31} +0.112460 q^{35} +3.63984 q^{37} +2.15173 q^{41} -2.76015 q^{43} -4.94015 q^{47} -6.96603 q^{49} +0.756443 q^{53} -3.09636 q^{55} +6.98791 q^{59} -0.248451 q^{61} -1.62060 q^{65} -15.5507 q^{67} -4.03018 q^{71} -1.57460 q^{73} -0.935260 q^{77} +8.12950 q^{79} +15.0509 q^{83} -0.0782477 q^{85} -3.67917 q^{89} -0.489505 q^{91} +4.52399 q^{95} +10.2997 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 2 q^{5} - 12 q^{7}+O(q^{10}) \) 7 * q + 2 * q^5 - 12 * q^7	\( 7 q + 2 q^{5} - 12 q^{7} + 7 q^{11} - 9 q^{13} + q^{17} - 11 q^{19} + 19 q^{23} + 3 q^{25} + 5 q^{29} - 13 q^{31} + 7 q^{35} - 26 q^{37} + 2 q^{41} - 24 q^{43} + 11 q^{47} + 19 q^{49} - 4 q^{53} - 4 q^{55} + 4 q^{59} - 5 q^{61} - 13 q^{65} - 42 q^{67} - 9 q^{71} + 27 q^{73} - 12 q^{77} - 8 q^{79} - 16 q^{83} - 27 q^{85} - 9 q^{89} - 2 q^{91} - 10 q^{95} - 8 q^{97}+O(q^{100}) \) 7 * q + 2 * q^5 - 12 * q^7 + 7 * q^11 - 9 * q^13 + q^17 - 11 * q^19 + 19 * q^23 + 3 * q^25 + 5 * q^29 - 13 * q^31 + 7 * q^35 - 26 * q^37 + 2 * q^41 - 24 * q^43 + 11 * q^47 + 19 * q^49 - 4 * q^53 - 4 * q^55 + 4 * q^59 - 5 * q^61 - 13 * q^65 - 42 * q^67 - 9 * q^71 + 27 * q^73 - 12 * q^77 - 8 * q^79 - 16 * q^83 - 27 * q^85 - 9 * q^89 - 2 * q^91 - 10 * q^95 - 8 * q^97

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