LMFDB - Embedded newform 8016.2.a.be.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8016,2,Mod(1,8016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8016, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8016 = 2^{4} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8016.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:	$64.0080822603$
Analytic rank:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - x^{10} - 33 x^{9} + 22 x^{8} + 417 x^{7} - 151 x^{6} - 2470 x^{5} + 272 x^{4} + 6584 x^{3} + \cdots + 242 $ x^11 - x^10 - 33x^9 + 22x^8 + 417x^7 - 151x^6 - 2470x^5 + 272x^4 + 6584x^3 + 292x^2 - 5687*x + 242
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 4008)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.80306$ of defining polynomial
Character	$\chi$	$=$	8016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.80306 q^{5} +4.10861 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.80306 q^{5} +4.10861 q^{7} +1.00000 q^{9} +2.59144 q^{11} -3.46748 q^{13} -1.80306 q^{15} +1.68470 q^{17} +2.49774 q^{19} +4.10861 q^{21} +8.91932 q^{23} -1.74898 q^{25} +1.00000 q^{27} -4.42158 q^{29} +2.95084 q^{31} +2.59144 q^{33} -7.40807 q^{35} +0.918154 q^{37} -3.46748 q^{39} -1.83849 q^{41} -0.850997 q^{43} -1.80306 q^{45} +6.74458 q^{47} +9.88072 q^{49} +1.68470 q^{51} +4.65455 q^{53} -4.67252 q^{55} +2.49774 q^{57} -1.23850 q^{59} -7.97208 q^{61} +4.10861 q^{63} +6.25208 q^{65} +1.65708 q^{67} +8.91932 q^{69} +6.24057 q^{71} -4.76409 q^{73} -1.74898 q^{75} +10.6472 q^{77} +10.7193 q^{79} +1.00000 q^{81} +3.69008 q^{83} -3.03761 q^{85} -4.42158 q^{87} -12.0739 q^{89} -14.2466 q^{91} +2.95084 q^{93} -4.50357 q^{95} +14.8662 q^{97} +2.59144 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 11 q^{3} + 10 q^{5} + q^{7} + 11 q^{9}+O(q^{10}) $ 11 * q + 11 * q^3 + 10 * q^5 + q^7 + 11 * q^9	$ 11 q + 11 q^{3} + 10 q^{5} + q^{7} + 11 q^{9} + q^{11} + 10 q^{13} + 10 q^{15} + 17 q^{17} - 2 q^{19} + q^{21} + 3 q^{23} + 21 q^{25} + 11 q^{27} + 17 q^{29} + 15 q^{31} + q^{33} - 11 q^{35} + 4 q^{37} + 10 q^{39} + 16 q^{41} - 10 q^{43} + 10 q^{45} + 16 q^{47} + 22 q^{49} + 17 q^{51} + 42 q^{53} + 5 q^{55} - 2 q^{57} + 2 q^{59} + 12 q^{61} + q^{63} + 10 q^{65} + q^{67} + 3 q^{69} + 9 q^{71} + 24 q^{73} + 21 q^{75} + 22 q^{77} + 30 q^{79} + 11 q^{81} - 16 q^{83} + 25 q^{85} + 17 q^{87} + 37 q^{89} - q^{91} + 15 q^{93} - 5 q^{95} + 4 q^{97} + q^{99}+O(q^{100}) $ 11 * q + 11 * q^3 + 10 * q^5 + q^7 + 11 * q^9 + q^11 + 10 * q^13 + 10 * q^15 + 17 * q^17 - 2 * q^19 + q^21 + 3 * q^23 + 21 * q^25 + 11 * q^27 + 17 * q^29 + 15 * q^31 + q^33 - 11 * q^35 + 4 * q^37 + 10 * q^39 + 16 * q^41 - 10 * q^43 + 10 * q^45 + 16 * q^47 + 22 * q^49 + 17 * q^51 + 42 * q^53 + 5 * q^55 - 2 * q^57 + 2 * q^59 + 12 * q^61 + q^63 + 10 * q^65 + q^67 + 3 * q^69 + 9 * q^71 + 24 * q^73 + 21 * q^75 + 22 * q^77 + 30 * q^79 + 11 * q^81 - 16 * q^83 + 25 * q^85 + 17 * q^87 + 37 * q^89 - q^91 + 15 * q^93 - 5 * q^95 + 4 * q^97 + 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−1.80306	−0.806352	−0.403176	−	0.915122i	$-0.632094\pi$
$5$	−1.80306	−0.806352	−0.403176	+	0.915122i	$0.632094\pi$
$6$	0	0
$7$	4.10861	1.55291	0.776455	−	0.630172i	$-0.217016\pi$
$7$	4.10861	1.55291	0.776455	+	0.630172i	$0.217016\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.59144	0.781349	0.390675	−	0.920529i	$-0.372242\pi$
$11$	2.59144	0.781349	0.390675	+	0.920529i	$0.372242\pi$
$12$	0	0
$13$	−3.46748	−0.961707	−0.480853	−	0.876801i	$-0.659673\pi$
$13$	−3.46748	−0.961707	−0.480853	+	0.876801i	$0.659673\pi$
$14$	0	0
$15$	−1.80306	−0.465548
$16$	0	0
$17$	1.68470	0.408600	0.204300	−	0.978908i	$-0.434508\pi$
$17$	1.68470	0.408600	0.204300	+	0.978908i	$0.434508\pi$
$18$	0	0
$19$	2.49774	0.573021	0.286510	−	0.958077i	$-0.407505\pi$
$19$	2.49774	0.573021	0.286510	+	0.958077i	$0.407505\pi$
$20$	0	0
$21$	4.10861	0.896573
$22$	0	0
$23$	8.91932	1.85981	0.929904	−	0.367803i	$-0.119890\pi$
$23$	8.91932	1.85981	0.929904	+	0.367803i	$0.119890\pi$
$24$	0	0
$25$	−1.74898	−0.349796
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−4.42158	−0.821067	−0.410534	−	0.911845i	$-0.634658\pi$
$29$	−4.42158	−0.821067	−0.410534	+	0.911845i	$0.634658\pi$
$30$	0	0
$31$	2.95084	0.529987	0.264993	−	0.964250i	$-0.414630\pi$
$31$	2.95084	0.529987	0.264993	+	0.964250i	$0.414630\pi$
$32$	0	0
$33$	2.59144	0.451112
$34$	0	0
$35$	−7.40807	−1.25219
$36$	0	0
$37$	0.918154	0.150944	0.0754718	−	0.997148i	$-0.475954\pi$
$37$	0.918154	0.150944	0.0754718	+	0.997148i	$0.475954\pi$
$38$	0	0
$39$	−3.46748	−0.555242
$40$	0	0
$41$	−1.83849	−0.287125	−0.143562	−	0.989641i	$-0.545856\pi$
$41$	−1.83849	−0.287125	−0.143562	+	0.989641i	$0.545856\pi$
$42$	0	0
$43$	−0.850997	−0.129776	−0.0648879	−	0.997893i	$-0.520669\pi$
$43$	−0.850997	−0.129776	−0.0648879	+	0.997893i	$0.520669\pi$
$44$	0	0
$45$	−1.80306	−0.268784
$46$	0	0
$47$	6.74458	0.983798	0.491899	−	0.870652i	$-0.336303\pi$
$47$	6.74458	0.983798	0.491899	+	0.870652i	$0.336303\pi$
$48$	0	0
$49$	9.88072	1.41153
$50$	0	0
$51$	1.68470	0.235905
$52$	0	0
$53$	4.65455	0.639351	0.319676	−	0.947527i	$-0.396426\pi$
$53$	4.65455	0.639351	0.319676	+	0.947527i	$0.396426\pi$
$54$	0	0
$55$	−4.67252	−0.630043
$56$	0	0
$57$	2.49774	0.330834
$58$	0	0
$59$	−1.23850	−0.161239	−0.0806194	−	0.996745i	$-0.525690\pi$
$59$	−1.23850	−0.161239	−0.0806194	+	0.996745i	$0.525690\pi$
$60$	0	0
$61$	−7.97208	−1.02072	−0.510360	−	0.859961i	$-0.670488\pi$
$61$	−7.97208	−1.02072	−0.510360	+	0.859961i	$0.670488\pi$
$62$	0	0
$63$	4.10861	0.517637
$64$	0	0
$65$	6.25208	0.775475
$66$	0	0
$67$	1.65708	0.202445	0.101223	−	0.994864i	$-0.467725\pi$
$67$	1.65708	0.202445	0.101223	+	0.994864i	$0.467725\pi$
$68$	0	0
$69$	8.91932	1.07376
$70$	0	0
$71$	6.24057	0.740620	0.370310	−	0.928908i	$-0.379251\pi$
$71$	6.24057	0.740620	0.370310	+	0.928908i	$0.379251\pi$
$72$	0	0
$73$	−4.76409	−0.557594	−0.278797	−	0.960350i	$-0.589936\pi$
$73$	−4.76409	−0.557594	−0.278797	+	0.960350i	$0.589936\pi$
$74$	0	0
$75$	−1.74898	−0.201955
$76$	0	0
$77$	10.6472	1.21337
$78$	0	0
$79$	10.7193	1.20601	0.603007	−	0.797736i	$-0.293969\pi$
$79$	10.7193	1.20601	0.603007	+	0.797736i	$0.293969\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	3.69008	0.405039	0.202519	−	0.979278i	$-0.435087\pi$
$83$	3.69008	0.405039	0.202519	+	0.979278i	$0.435087\pi$
$84$	0	0
$85$	−3.03761	−0.329475
$86$	0	0
$87$	−4.42158	−0.474044
$88$	0	0
$89$	−12.0739	−1.27983	−0.639916	−	0.768445i	$-0.721031\pi$
$89$	−12.0739	−1.27983	−0.639916	+	0.768445i	$0.721031\pi$
$90$	0	0
$91$	−14.2466	−1.49344
$92$	0	0
$93$	2.95084	0.305988
$94$	0	0
$95$	−4.50357	−0.462057
$96$	0	0
$97$	14.8662	1.50944	0.754719	−	0.656048i	$-0.227773\pi$
$97$	14.8662	1.50944	0.754719	+	0.656048i	$0.227773\pi$
$98$	0	0
$99$	2.59144	0.260450
$100$	0	0
$101$	1.27675	0.127041	0.0635206	−	0.997981i	$-0.479767\pi$
$101$	1.27675	0.127041	0.0635206	+	0.997981i	$0.479767\pi$
$102$	0	0
$103$	−5.26805	−0.519077	−0.259538	−	0.965733i	$-0.583570\pi$
$103$	−5.26805	−0.519077	−0.259538	+	0.965733i	$0.583570\pi$
$104$	0	0
$105$	−7.40807	−0.722954
$106$	0	0
$107$	−7.81076	−0.755094	−0.377547	−	0.925990i	$-0.623232\pi$
$107$	−7.81076	−0.755094	−0.377547	+	0.925990i	$0.623232\pi$
$108$	0	0
$109$	−2.01235	−0.192748	−0.0963739	−	0.995345i	$-0.530724\pi$
$109$	−2.01235	−0.192748	−0.0963739	+	0.995345i	$0.530724\pi$
$110$	0	0
$111$	0.918154	0.0871473
$112$	0	0
$113$	8.20964	0.772298	0.386149	−	0.922436i	$-0.373805\pi$
$113$	8.20964	0.772298	0.386149	+	0.922436i	$0.373805\pi$
$114$	0	0
$115$	−16.0821	−1.49966
$116$	0	0
$117$	−3.46748	−0.320569
$118$	0	0
$119$	6.92178	0.634519
$120$	0	0
$121$	−4.28443	−0.389494
$122$	0	0
$123$	−1.83849	−0.165771
$124$	0	0
$125$	12.1688	1.08841
$126$	0	0
$127$	−9.38074	−0.832406	−0.416203	−	0.909272i	$-0.636639\pi$
$127$	−9.38074	−0.832406	−0.416203	+	0.909272i	$0.636639\pi$
$128$	0	0
$129$	−0.850997	−0.0749261
$130$	0	0
$131$	−19.8572	−1.73493	−0.867465	−	0.497498i	$-0.834252\pi$
$131$	−19.8572	−1.73493	−0.867465	+	0.497498i	$0.834252\pi$
$132$	0	0
$133$	10.2623	0.889850
$134$	0	0
$135$	−1.80306	−0.155183
$136$	0	0
$137$	5.44295	0.465023	0.232511	−	0.972594i	$-0.425306\pi$
$137$	5.44295	0.465023	0.232511	+	0.972594i	$0.425306\pi$
$138$	0	0
$139$	13.0858	1.10992	0.554960	−	0.831877i	$-0.312734\pi$
$139$	13.0858	1.10992	0.554960	+	0.831877i	$0.312734\pi$
$140$	0	0
$141$	6.74458	0.567996
$142$	0	0
$143$	−8.98578	−0.751429
$144$	0	0
$145$	7.97238	0.662070
$146$	0	0
$147$	9.88072	0.814948
$148$	0	0
$149$	−0.961230	−0.0787470	−0.0393735	−	0.999225i	$-0.512536\pi$
$149$	−0.961230	−0.0787470	−0.0393735	+	0.999225i	$0.512536\pi$
$150$	0	0
$151$	8.62272	0.701707	0.350853	−	0.936430i	$-0.385892\pi$
$151$	8.62272	0.701707	0.350853	+	0.936430i	$0.385892\pi$
$152$	0	0
$153$	1.68470	0.136200
$154$	0	0
$155$	−5.32054	−0.427356
$156$	0	0
$157$	3.40121	0.271446	0.135723	−	0.990747i	$-0.456664\pi$
$157$	3.40121	0.271446	0.135723	+	0.990747i	$0.456664\pi$
$158$	0	0
$159$	4.65455	0.369130
$160$	0	0
$161$	36.6461	2.88811
$162$	0	0
$163$	2.70291	0.211708	0.105854	−	0.994382i	$-0.466242\pi$
$163$	2.70291	0.211708	0.105854	+	0.994382i	$0.466242\pi$
$164$	0	0
$165$	−4.67252	−0.363755
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−0.976559	−0.0751200
$170$	0	0
$171$	2.49774	0.191007
$172$	0	0
$173$	13.7489	1.04531	0.522654	−	0.852545i	$-0.324942\pi$
$173$	13.7489	1.04531	0.522654	+	0.852545i	$0.324942\pi$
$174$	0	0
$175$	−7.18588	−0.543201
$176$	0	0
$177$	−1.23850	−0.0930912
$178$	0	0
$179$	−0.148844	−0.0111251	−0.00556257	−	0.999985i	$-0.501771\pi$
$179$	−0.148844	−0.0111251	−0.00556257	+	0.999985i	$0.501771\pi$
$180$	0	0
$181$	7.10022	0.527755	0.263877	−	0.964556i	$-0.414999\pi$
$181$	7.10022	0.527755	0.263877	+	0.964556i	$0.414999\pi$
$182$	0	0
$183$	−7.97208	−0.589313
$184$	0	0
$185$	−1.65549	−0.121714
$186$	0	0
$187$	4.36580	0.319259
$188$	0	0
$189$	4.10861	0.298858
$190$	0	0
$191$	5.56545	0.402702	0.201351	−	0.979519i	$-0.435467\pi$
$191$	5.56545	0.402702	0.201351	+	0.979519i	$0.435467\pi$
$192$	0	0
$193$	−16.6812	−1.20074	−0.600371	−	0.799721i	$-0.704981\pi$
$193$	−16.6812	−1.20074	−0.600371	+	0.799721i	$0.704981\pi$
$194$	0	0
$195$	6.25208	0.447721
$196$	0	0
$197$	10.0656	0.717145	0.358573	−	0.933502i	$-0.383264\pi$
$197$	10.0656	0.717145	0.358573	+	0.933502i	$0.383264\pi$
$198$	0	0
$199$	14.2270	1.00853	0.504263	−	0.863550i	$-0.331764\pi$
$199$	14.2270	1.00853	0.504263	+	0.863550i	$0.331764\pi$
$200$	0	0
$201$	1.65708	0.116882
$202$	0	0
$203$	−18.1666	−1.27504
$204$	0	0
$205$	3.31491	0.231524
$206$	0	0
$207$	8.91932	0.619936
$208$	0	0
$209$	6.47275	0.447729
$210$	0	0
$211$	−19.4006	−1.33560	−0.667798	−	0.744343i	$-0.732763\pi$
$211$	−19.4006	−1.33560	−0.667798	+	0.744343i	$0.732763\pi$
$212$	0	0
$213$	6.24057	0.427597
$214$	0	0
$215$	1.53440	0.104645
$216$	0	0
$217$	12.1239	0.823022
$218$	0	0
$219$	−4.76409	−0.321927
$220$	0	0
$221$	−5.84167	−0.392953
$222$	0	0
$223$	5.50496	0.368640	0.184320	−	0.982866i	$-0.440992\pi$
$223$	5.50496	0.368640	0.184320	+	0.982866i	$0.440992\pi$
$224$	0	0
$225$	−1.74898	−0.116599
$226$	0	0
$227$	24.7959	1.64576	0.822880	−	0.568215i	$-0.192366\pi$
$227$	24.7959	1.64576	0.822880	+	0.568215i	$0.192366\pi$
$228$	0	0
$229$	−22.8293	−1.50860	−0.754302	−	0.656528i	$-0.772024\pi$
$229$	−22.8293	−1.50860	−0.754302	+	0.656528i	$0.772024\pi$
$230$	0	0
$231$	10.6472	0.700537
$232$	0	0
$233$	12.2217	0.800672	0.400336	−	0.916368i	$-0.368893\pi$
$233$	12.2217	0.800672	0.400336	+	0.916368i	$0.368893\pi$
$234$	0	0
$235$	−12.1609	−0.793288
$236$	0	0
$237$	10.7193	0.696293
$238$	0	0
$239$	−7.79185	−0.504013	−0.252007	−	0.967726i	$-0.581090\pi$
$239$	−7.79185	−0.504013	−0.252007	+	0.967726i	$0.581090\pi$
$240$	0	0
$241$	−22.1396	−1.42613	−0.713067	−	0.701096i	$-0.752695\pi$
$241$	−22.1396	−1.42613	−0.713067	+	0.701096i	$0.752695\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−17.8155	−1.13819
$246$	0	0
$247$	−8.66087	−0.551078
$248$	0	0
$249$	3.69008	0.233849
$250$	0	0
$251$	16.1687	1.02056	0.510279	−	0.860009i	$-0.329542\pi$
$251$	16.1687	1.02056	0.510279	+	0.860009i	$0.329542\pi$
$252$	0	0
$253$	23.1139	1.45316
$254$	0	0
$255$	−3.03761	−0.190223
$256$	0	0
$257$	22.3740	1.39565	0.697825	−	0.716268i	$-0.254151\pi$
$257$	22.3740	1.39565	0.697825	+	0.716268i	$0.254151\pi$
$258$	0	0
$259$	3.77234	0.234402
$260$	0	0
$261$	−4.42158	−0.273689
$262$	0	0
$263$	20.4376	1.26023	0.630117	−	0.776500i	$-0.283007\pi$
$263$	20.4376	1.26023	0.630117	+	0.776500i	$0.283007\pi$
$264$	0	0
$265$	−8.39242	−0.515542
$266$	0	0
$267$	−12.0739	−0.738911
$268$	0	0
$269$	16.8017	1.02442	0.512209	−	0.858861i	$-0.328827\pi$
$269$	16.8017	1.02442	0.512209	+	0.858861i	$0.328827\pi$
$270$	0	0
$271$	8.49263	0.515890	0.257945	−	0.966160i	$-0.416955\pi$
$271$	8.49263	0.515890	0.257945	+	0.966160i	$0.416955\pi$
$272$	0	0
$273$	−14.2466	−0.862241
$274$	0	0
$275$	−4.53238	−0.273313
$276$	0	0
$277$	−12.1608	−0.730670	−0.365335	−	0.930876i	$-0.619046\pi$
$277$	−12.1608	−0.730670	−0.365335	+	0.930876i	$0.619046\pi$
$278$	0	0
$279$	2.95084	0.176662
$280$	0	0
$281$	−14.5701	−0.869178	−0.434589	−	0.900629i	$-0.643106\pi$
$281$	−14.5701	−0.869178	−0.434589	+	0.900629i	$0.643106\pi$
$282$	0	0
$283$	−8.16542	−0.485384	−0.242692	−	0.970103i	$-0.578030\pi$
$283$	−8.16542	−0.485384	−0.242692	+	0.970103i	$0.578030\pi$
$284$	0	0
$285$	−4.50357	−0.266769
$286$	0	0
$287$	−7.55366	−0.445879
$288$	0	0
$289$	−14.1618	−0.833046
$290$	0	0
$291$	14.8662	0.871474
$292$	0	0
$293$	29.2346	1.70791	0.853953	−	0.520350i	$-0.174199\pi$
$293$	29.2346	1.70791	0.853953	+	0.520350i	$0.174199\pi$
$294$	0	0
$295$	2.23309	0.130015
$296$	0	0
$297$	2.59144	0.150371
$298$	0	0
$299$	−30.9276	−1.78859
$300$	0	0
$301$	−3.49642	−0.201530
$302$	0	0
$303$	1.27675	0.0733473
$304$	0	0
$305$	14.3741	0.823061
$306$	0	0
$307$	−23.8192	−1.35943	−0.679716	−	0.733475i	$-0.737897\pi$
$307$	−23.8192	−1.35943	−0.679716	+	0.733475i	$0.737897\pi$
$308$	0	0
$309$	−5.26805	−0.299689
$310$	0	0
$311$	16.4270	0.931492	0.465746	−	0.884918i	$-0.345786\pi$
$311$	16.4270	0.931492	0.465746	+	0.884918i	$0.345786\pi$
$312$	0	0
$313$	12.4875	0.705832	0.352916	−	0.935655i	$-0.385190\pi$
$313$	12.4875	0.705832	0.352916	+	0.935655i	$0.385190\pi$
$314$	0	0
$315$	−7.40807	−0.417398
$316$	0	0
$317$	−3.38861	−0.190323	−0.0951617	−	0.995462i	$-0.530337\pi$
$317$	−3.38861	−0.190323	−0.0951617	+	0.995462i	$0.530337\pi$
$318$	0	0
$319$	−11.4583	−0.641540
$320$	0	0
$321$	−7.81076	−0.435954
$322$	0	0
$323$	4.20794	0.234136
$324$	0	0
$325$	6.06455	0.336401
$326$	0	0
$327$	−2.01235	−0.111283
$328$	0	0
$329$	27.7109	1.52775
$330$	0	0
$331$	−7.83764	−0.430795	−0.215398	−	0.976526i	$-0.569105\pi$
$331$	−7.83764	−0.430795	−0.215398	+	0.976526i	$0.569105\pi$
$332$	0	0
$333$	0.918154	0.0503145
$334$	0	0
$335$	−2.98782	−0.163242
$336$	0	0
$337$	−19.0047	−1.03525	−0.517627	−	0.855606i	$-0.673185\pi$
$337$	−19.0047	−1.03525	−0.517627	+	0.855606i	$0.673185\pi$
$338$	0	0
$339$	8.20964	0.445887
$340$	0	0
$341$	7.64694	0.414105
$342$	0	0
$343$	11.8358	0.639071
$344$	0	0
$345$	−16.0821	−0.865829
$346$	0	0
$347$	−25.2208	−1.35392	−0.676961	−	0.736019i	$-0.736703\pi$
$347$	−25.2208	−1.35392	−0.676961	+	0.736019i	$0.736703\pi$
$348$	0	0
$349$	1.06370	0.0569387	0.0284693	−	0.999595i	$-0.490937\pi$
$349$	1.06370	0.0569387	0.0284693	+	0.999595i	$0.490937\pi$
$350$	0	0
$351$	−3.46748	−0.185081
$352$	0	0
$353$	19.0049	1.01153	0.505764	−	0.862672i	$-0.331211\pi$
$353$	19.0049	1.01153	0.505764	+	0.862672i	$0.331211\pi$
$354$	0	0
$355$	−11.2521	−0.597201
$356$	0	0
$357$	6.92178	0.366340
$358$	0	0
$359$	−22.4615	−1.18547	−0.592736	−	0.805397i	$-0.701952\pi$
$359$	−22.4615	−1.18547	−0.592736	+	0.805397i	$0.701952\pi$
$360$	0	0
$361$	−12.7613	−0.671647
$362$	0	0
$363$	−4.28443	−0.224874
$364$	0	0
$365$	8.58993	0.449618
$366$	0	0
$367$	−6.29449	−0.328569	−0.164285	−	0.986413i	$-0.552532\pi$
$367$	−6.29449	−0.328569	−0.164285	+	0.986413i	$0.552532\pi$
$368$	0	0
$369$	−1.83849	−0.0957082
$370$	0	0
$371$	19.1237	0.992855
$372$	0	0
$373$	−11.9702	−0.619794	−0.309897	−	0.950770i	$-0.600295\pi$
$373$	−11.9702	−0.619794	−0.309897	+	0.950770i	$0.600295\pi$
$374$	0	0
$375$	12.1688	0.628394
$376$	0	0
$377$	15.3318	0.789626
$378$	0	0
$379$	18.5730	0.954032	0.477016	−	0.878895i	$-0.341718\pi$
$379$	18.5730	0.954032	0.477016	+	0.878895i	$0.341718\pi$
$380$	0	0
$381$	−9.38074	−0.480590
$382$	0	0
$383$	29.3120	1.49777	0.748886	−	0.662698i	$-0.230589\pi$
$383$	29.3120	1.49777	0.748886	+	0.662698i	$0.230589\pi$
$384$	0	0
$385$	−19.1976	−0.978400
$386$	0	0
$387$	−0.850997	−0.0432586
$388$	0	0
$389$	15.9476	0.808573	0.404287	−	0.914632i	$-0.367520\pi$
$389$	15.9476	0.808573	0.404287	+	0.914632i	$0.367520\pi$
$390$	0	0
$391$	15.0264	0.759917
$392$	0	0
$393$	−19.8572	−1.00166
$394$	0	0
$395$	−19.3275	−0.972473
$396$	0	0
$397$	−22.9256	−1.15060	−0.575302	−	0.817941i	$-0.695116\pi$
$397$	−22.9256	−1.15060	−0.575302	+	0.817941i	$0.695116\pi$
$398$	0	0
$399$	10.2623	0.513755
$400$	0	0
$401$	21.0031	1.04884	0.524421	−	0.851459i	$-0.324282\pi$
$401$	21.0031	1.04884	0.524421	+	0.851459i	$0.324282\pi$
$402$	0	0
$403$	−10.2320	−0.509692
$404$	0	0
$405$	−1.80306	−0.0895947
$406$	0	0
$407$	2.37934	0.117940
$408$	0	0
$409$	34.4707	1.70447	0.852234	−	0.523160i	$-0.175247\pi$
$409$	34.4707	1.70447	0.852234	+	0.523160i	$0.175247\pi$
$410$	0	0
$411$	5.44295	0.268481
$412$	0	0
$413$	−5.08851	−0.250389
$414$	0	0
$415$	−6.65343	−0.326604
$416$	0	0
$417$	13.0858	0.640812
$418$	0	0
$419$	7.80419	0.381260	0.190630	−	0.981662i	$-0.438947\pi$
$419$	7.80419	0.381260	0.190630	+	0.981662i	$0.438947\pi$
$420$	0	0
$421$	−28.5378	−1.39085	−0.695424	−	0.718599i	$-0.744783\pi$
$421$	−28.5378	−1.39085	−0.695424	+	0.718599i	$0.744783\pi$
$422$	0	0
$423$	6.74458	0.327933
$424$	0	0
$425$	−2.94650	−0.142926
$426$	0	0
$427$	−32.7542	−1.58509
$428$	0	0
$429$	−8.98578	−0.433838
$430$	0	0
$431$	−10.9732	−0.528559	−0.264279	−	0.964446i	$-0.585134\pi$
$431$	−10.9732	−0.528559	−0.264279	+	0.964446i	$0.585134\pi$
$432$	0	0
$433$	18.8851	0.907558	0.453779	−	0.891114i	$-0.350076\pi$
$433$	18.8851	0.907558	0.453779	+	0.891114i	$0.350076\pi$
$434$	0	0
$435$	7.97238	0.382246
$436$	0	0
$437$	22.2781	1.06571
$438$	0	0
$439$	25.2125	1.20333	0.601665	−	0.798749i	$-0.294504\pi$
$439$	25.2125	1.20333	0.601665	+	0.798749i	$0.294504\pi$
$440$	0	0
$441$	9.88072	0.470510
$442$	0	0
$443$	20.2876	0.963893	0.481947	−	0.876201i	$-0.339930\pi$
$443$	20.2876	0.963893	0.481947	+	0.876201i	$0.339930\pi$
$444$	0	0
$445$	21.7700	1.03200
$446$	0	0
$447$	−0.961230	−0.0454646
$448$	0	0
$449$	−26.3700	−1.24448	−0.622240	−	0.782827i	$-0.713777\pi$
$449$	−26.3700	−1.24448	−0.622240	+	0.782827i	$0.713777\pi$
$450$	0	0
$451$	−4.76435	−0.224345
$452$	0	0
$453$	8.62272	0.405131
$454$	0	0
$455$	25.6874	1.20424
$456$	0	0
$457$	−20.0483	−0.937818	−0.468909	−	0.883247i	$-0.655353\pi$
$457$	−20.0483	−0.937818	−0.468909	+	0.883247i	$0.655353\pi$
$458$	0	0
$459$	1.68470	0.0786351
$460$	0	0
$461$	−28.0956	−1.30854	−0.654272	−	0.756260i	$-0.727025\pi$
$461$	−28.0956	−1.30854	−0.654272	+	0.756260i	$0.727025\pi$
$462$	0	0
$463$	23.8772	1.10967	0.554834	−	0.831961i	$-0.312782\pi$
$463$	23.8772	1.10967	0.554834	+	0.831961i	$0.312782\pi$
$464$	0	0
$465$	−5.32054	−0.246734
$466$	0	0
$467$	33.4385	1.54735	0.773674	−	0.633584i	$-0.218417\pi$
$467$	33.4385	1.54735	0.773674	+	0.633584i	$0.218417\pi$
$468$	0	0
$469$	6.80832	0.314379
$470$	0	0
$471$	3.40121	0.156720
$472$	0	0
$473$	−2.20531	−0.101400
$474$	0	0
$475$	−4.36849	−0.200440
$476$	0	0
$477$	4.65455	0.213117
$478$	0	0
$479$	3.13119	0.143068	0.0715340	−	0.997438i	$-0.477211\pi$
$479$	3.13119	0.143068	0.0715340	+	0.997438i	$0.477211\pi$
$480$	0	0
$481$	−3.18368	−0.145163
$482$	0	0
$483$	36.6461	1.66745
$484$	0	0
$485$	−26.8047	−1.21714
$486$	0	0
$487$	26.2378	1.18895	0.594475	−	0.804114i	$-0.297360\pi$
$487$	26.2378	1.18895	0.594475	+	0.804114i	$0.297360\pi$
$488$	0	0
$489$	2.70291	0.122230
$490$	0	0
$491$	5.20065	0.234702	0.117351	−	0.993090i	$-0.462560\pi$
$491$	5.20065	0.234702	0.117351	+	0.993090i	$0.462560\pi$
$492$	0	0
$493$	−7.44904	−0.335488
$494$	0	0
$495$	−4.67252	−0.210014
$496$	0	0
$497$	25.6401	1.15012
$498$	0	0
$499$	33.0631	1.48011	0.740053	−	0.672548i	$-0.234800\pi$
$499$	33.0631	1.48011	0.740053	+	0.672548i	$0.234800\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	−14.7807	−0.659040	−0.329520	−	0.944149i	$-0.606887\pi$
$503$	−14.7807	−0.659040	−0.329520	+	0.944149i	$0.606887\pi$
$504$	0	0
$505$	−2.30205	−0.102440
$506$	0	0
$507$	−0.976559	−0.0433705
$508$	0	0
$509$	31.0505	1.37629	0.688145	−	0.725573i	$-0.258425\pi$
$509$	31.0505	1.37629	0.688145	+	0.725573i	$0.258425\pi$
$510$	0	0
$511$	−19.5738	−0.865894
$512$	0	0
$513$	2.49774	0.110278
$514$	0	0
$515$	9.49861	0.418559
$516$	0	0
$517$	17.4782	0.768689
$518$	0	0
$519$	13.7489	0.603509
$520$	0	0
$521$	3.63406	0.159211	0.0796056	−	0.996826i	$-0.474634\pi$
$521$	3.63406	0.159211	0.0796056	+	0.996826i	$0.474634\pi$
$522$	0	0
$523$	−26.1659	−1.14416	−0.572078	−	0.820199i	$-0.693863\pi$
$523$	−26.1659	−1.14416	−0.572078	+	0.820199i	$0.693863\pi$
$524$	0	0
$525$	−7.18588	−0.313617
$526$	0	0
$527$	4.97128	0.216553
$528$	0	0
$529$	56.5543	2.45888
$530$	0	0
$531$	−1.23850	−0.0537462
$532$	0	0
$533$	6.37495	0.276130
$534$	0	0
$535$	14.0833	0.608872
$536$	0	0
$537$	−0.148844	−0.00642310
$538$	0	0
$539$	25.6053	1.10290
$540$	0	0
$541$	0.260024	0.0111793	0.00558966	−	0.999984i	$-0.498221\pi$
$541$	0.260024	0.0111793	0.00558966	+	0.999984i	$0.498221\pi$
$542$	0	0
$543$	7.10022	0.304699
$544$	0	0
$545$	3.62838	0.155423
$546$	0	0
$547$	27.8527	1.19090	0.595448	−	0.803394i	$-0.296975\pi$
$547$	27.8527	1.19090	0.595448	+	0.803394i	$0.296975\pi$
$548$	0	0
$549$	−7.97208	−0.340240
$550$	0	0
$551$	−11.0440	−0.470489
$552$	0	0
$553$	44.0414	1.87283
$554$	0	0
$555$	−1.65549	−0.0702714
$556$	0	0
$557$	8.13555	0.344714	0.172357	−	0.985035i	$-0.444862\pi$
$557$	8.13555	0.344714	0.172357	+	0.985035i	$0.444862\pi$
$558$	0	0
$559$	2.95082	0.124806
$560$	0	0
$561$	4.36580	0.184324
$562$	0	0
$563$	44.9622	1.89493	0.947466	−	0.319856i	$-0.103634\pi$
$563$	44.9622	1.89493	0.947466	+	0.319856i	$0.103634\pi$
$564$	0	0
$565$	−14.8025	−0.622745
$566$	0	0
$567$	4.10861	0.172546
$568$	0	0
$569$	8.14797	0.341581	0.170790	−	0.985307i	$-0.445368\pi$
$569$	8.14797	0.341581	0.170790	+	0.985307i	$0.445368\pi$
$570$	0	0
$571$	−5.71162	−0.239024	−0.119512	−	0.992833i	$-0.538133\pi$
$571$	−5.71162	−0.239024	−0.119512	+	0.992833i	$0.538133\pi$
$572$	0	0
$573$	5.56545	0.232500
$574$	0	0
$575$	−15.5997	−0.650553
$576$	0	0
$577$	8.28848	0.345054	0.172527	−	0.985005i	$-0.444807\pi$
$577$	8.28848	0.345054	0.172527	+	0.985005i	$0.444807\pi$
$578$	0	0
$579$	−16.6812	−0.693249
$580$	0	0
$581$	15.1611	0.628989
$582$	0	0
$583$	12.0620	0.499556
$584$	0	0
$585$	6.25208	0.258492
$586$	0	0
$587$	−32.2682	−1.33185	−0.665925	−	0.746019i	$-0.731963\pi$
$587$	−32.2682	−1.33185	−0.665925	+	0.746019i	$0.731963\pi$
$588$	0	0
$589$	7.37043	0.303693
$590$	0	0
$591$	10.0656	0.414044
$592$	0	0
$593$	−26.2913	−1.07965	−0.539826	−	0.841776i	$-0.681510\pi$
$593$	−26.2913	−1.07965	−0.539826	+	0.841776i	$0.681510\pi$
$594$	0	0
$595$	−12.4804	−0.511646
$596$	0	0
$597$	14.2270	0.582273
$598$	0	0
$599$	−1.95363	−0.0798233	−0.0399117	−	0.999203i	$-0.512708\pi$
$599$	−1.95363	−0.0798233	−0.0399117	+	0.999203i	$0.512708\pi$
$600$	0	0
$601$	−0.855179	−0.0348835	−0.0174417	−	0.999848i	$-0.505552\pi$
$601$	−0.855179	−0.0348835	−0.0174417	+	0.999848i	$0.505552\pi$
$602$	0	0
$603$	1.65708	0.0674817
$604$	0	0
$605$	7.72508	0.314069
$606$	0	0
$607$	17.8511	0.724553	0.362276	−	0.932071i	$-0.382000\pi$
$607$	17.8511	0.724553	0.362276	+	0.932071i	$0.382000\pi$
$608$	0	0
$609$	−18.1666	−0.736147
$610$	0	0
$611$	−23.3867	−0.946125
$612$	0	0
$613$	21.0354	0.849611	0.424806	−	0.905285i	$-0.360342\pi$
$613$	21.0354	0.849611	0.424806	+	0.905285i	$0.360342\pi$
$614$	0	0
$615$	3.31491	0.133670
$616$	0	0
$617$	−29.7077	−1.19599	−0.597993	−	0.801501i	$-0.704035\pi$
$617$	−29.7077	−1.19599	−0.597993	+	0.801501i	$0.704035\pi$
$618$	0	0
$619$	7.12602	0.286419	0.143209	−	0.989692i	$-0.454258\pi$
$619$	7.12602	0.286419	0.143209	+	0.989692i	$0.454258\pi$
$620$	0	0
$621$	8.91932	0.357920
$622$	0	0
$623$	−49.6070	−1.98746
$624$	0	0
$625$	−13.1962	−0.527847
$626$	0	0
$627$	6.47275	0.258497
$628$	0	0
$629$	1.54681	0.0616755
$630$	0	0
$631$	10.8881	0.433447	0.216724	−	0.976233i	$-0.430463\pi$
$631$	10.8881	0.433447	0.216724	+	0.976233i	$0.430463\pi$
$632$	0	0
$633$	−19.4006	−0.771106
$634$	0	0
$635$	16.9140	0.671213
$636$	0	0
$637$	−34.2612	−1.35748
$638$	0	0
$639$	6.24057	0.246873
$640$	0	0
$641$	6.00715	0.237268	0.118634	−	0.992938i	$-0.462148\pi$
$641$	6.00715	0.237268	0.118634	+	0.992938i	$0.462148\pi$
$642$	0	0
$643$	−22.8736	−0.902045	−0.451023	−	0.892513i	$-0.648941\pi$
$643$	−22.8736	−0.902045	−0.451023	+	0.892513i	$0.648941\pi$
$644$	0	0
$645$	1.53440	0.0604169
$646$	0	0
$647$	−19.1587	−0.753205	−0.376602	−	0.926375i	$-0.622908\pi$
$647$	−19.1587	−0.753205	−0.376602	+	0.926375i	$0.622908\pi$
$648$	0	0
$649$	−3.20950	−0.125984
$650$	0	0
$651$	12.1239	0.475172
$652$	0	0
$653$	−29.0547	−1.13700	−0.568499	−	0.822684i	$-0.692475\pi$
$653$	−29.0547	−1.13700	−0.568499	+	0.822684i	$0.692475\pi$
$654$	0	0
$655$	35.8037	1.39897
$656$	0	0
$657$	−4.76409	−0.185865
$658$	0	0
$659$	−18.7083	−0.728771	−0.364386	−	0.931248i	$-0.618721\pi$
$659$	−18.7083	−0.728771	−0.364386	+	0.931248i	$0.618721\pi$
$660$	0	0
$661$	−6.65833	−0.258979	−0.129490	−	0.991581i	$-0.541334\pi$
$661$	−6.65833	−0.258979	−0.129490	+	0.991581i	$0.541334\pi$
$662$	0	0
$663$	−5.84167	−0.226872
$664$	0	0
$665$	−18.5034	−0.717533
$666$	0	0
$667$	−39.4375	−1.52703
$668$	0	0
$669$	5.50496	0.212834
$670$	0	0
$671$	−20.6592	−0.797539
$672$	0	0
$673$	−25.7547	−0.992772	−0.496386	−	0.868102i	$-0.665340\pi$
$673$	−25.7547	−0.992772	−0.496386	+	0.868102i	$0.665340\pi$
$674$	0	0
$675$	−1.74898	−0.0673182
$676$	0	0
$677$	11.7352	0.451019	0.225510	−	0.974241i	$-0.427595\pi$
$677$	11.7352	0.451019	0.225510	+	0.974241i	$0.427595\pi$
$678$	0	0
$679$	61.0797	2.34402
$680$	0	0
$681$	24.7959	0.950180
$682$	0	0
$683$	30.4318	1.16444	0.582220	−	0.813031i	$-0.302184\pi$
$683$	30.4318	1.16444	0.582220	+	0.813031i	$0.302184\pi$
$684$	0	0
$685$	−9.81397	−0.374972
$686$	0	0
$687$	−22.8293	−0.870993
$688$	0	0
$689$	−16.1396	−0.614868
$690$	0	0
$691$	35.5839	1.35368	0.676838	−	0.736132i	$-0.263350\pi$
$691$	35.5839	1.35368	0.676838	+	0.736132i	$0.263350\pi$
$692$	0	0
$693$	10.6472	0.404455
$694$	0	0
$695$	−23.5944	−0.894986
$696$	0	0
$697$	−3.09731	−0.117319
$698$	0	0
$699$	12.2217	0.462268
$700$	0	0
$701$	−11.4767	−0.433468	−0.216734	−	0.976231i	$-0.569540\pi$
$701$	−11.4767	−0.433468	−0.216734	+	0.976231i	$0.569540\pi$
$702$	0	0
$703$	2.29331	0.0864938
$704$	0	0
$705$	−12.1609	−0.458005
$706$	0	0
$707$	5.24567	0.197284
$708$	0	0
$709$	2.41968	0.0908729	0.0454365	−	0.998967i	$-0.485532\pi$
$709$	2.41968	0.0908729	0.0454365	+	0.998967i	$0.485532\pi$
$710$	0	0
$711$	10.7193	0.402005
$712$	0	0
$713$	26.3195	0.985673
$714$	0	0
$715$	16.2019	0.605916
$716$	0	0
$717$	−7.79185	−0.290992
$718$	0	0
$719$	−26.9803	−1.00620	−0.503098	−	0.864229i	$-0.667806\pi$
$719$	−26.9803	−1.00620	−0.503098	+	0.864229i	$0.667806\pi$
$720$	0	0
$721$	−21.6444	−0.806080
$722$	0	0
$723$	−22.1396	−0.823379
$724$	0	0
$725$	7.73325	0.287206
$726$	0	0
$727$	−8.04133	−0.298237	−0.149118	−	0.988819i	$-0.547644\pi$
$727$	−8.04133	−0.298237	−0.149118	+	0.988819i	$0.547644\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−1.43368	−0.0530264
$732$	0	0
$733$	−48.1812	−1.77961	−0.889807	−	0.456336i	$-0.849161\pi$
$733$	−48.1812	−1.77961	−0.889807	+	0.456336i	$0.849161\pi$
$734$	0	0
$735$	−17.8155	−0.657135
$736$	0	0
$737$	4.29424	0.158180
$738$	0	0
$739$	−21.9605	−0.807829	−0.403915	−	0.914797i	$-0.632351\pi$
$739$	−21.9605	−0.807829	−0.403915	+	0.914797i	$0.632351\pi$
$740$	0	0
$741$	−8.66087	−0.318165
$742$	0	0
$743$	−4.55455	−0.167090	−0.0835451	−	0.996504i	$-0.526624\pi$
$743$	−4.55455	−0.167090	−0.0835451	+	0.996504i	$0.526624\pi$
$744$	0	0
$745$	1.73315	0.0634979
$746$	0	0
$747$	3.69008	0.135013
$748$	0	0
$749$	−32.0914	−1.17259
$750$	0	0
$751$	−24.3222	−0.887530	−0.443765	−	0.896143i	$-0.646357\pi$
$751$	−24.3222	−0.887530	−0.443765	+	0.896143i	$0.646357\pi$
$752$	0	0
$753$	16.1687	0.589219
$754$	0	0
$755$	−15.5473	−0.565823
$756$	0	0
$757$	9.86234	0.358453	0.179226	−	0.983808i	$-0.442641\pi$
$757$	9.86234	0.358453	0.179226	+	0.983808i	$0.442641\pi$
$758$	0	0
$759$	23.1139	0.838982
$760$	0	0
$761$	−1.04767	−0.0379781	−0.0189891	−	0.999820i	$-0.506045\pi$
$761$	−1.04767	−0.0379781	−0.0189891	+	0.999820i	$0.506045\pi$
$762$	0	0
$763$	−8.26796	−0.299320
$764$	0	0
$765$	−3.03761	−0.109825
$766$	0	0
$767$	4.29447	0.155064
$768$	0	0
$769$	−11.2840	−0.406912	−0.203456	−	0.979084i	$-0.565217\pi$
$769$	−11.2840	−0.406912	−0.203456	+	0.979084i	$0.565217\pi$
$770$	0	0
$771$	22.3740	0.805779
$772$	0	0
$773$	2.32588	0.0836561	0.0418280	−	0.999125i	$-0.486682\pi$
$773$	2.32588	0.0836561	0.0418280	+	0.999125i	$0.486682\pi$
$774$	0	0
$775$	−5.16096	−0.185387
$776$	0	0
$777$	3.77234	0.135332
$778$	0	0
$779$	−4.59208	−0.164528
$780$	0	0
$781$	16.1721	0.578683
$782$	0	0
$783$	−4.42158	−0.158015
$784$	0	0
$785$	−6.13259	−0.218882
$786$	0	0
$787$	9.04798	0.322526	0.161263	−	0.986912i	$-0.448443\pi$
$787$	9.04798	0.322526	0.161263	+	0.986912i	$0.448443\pi$
$788$	0	0
$789$	20.4376	0.727596
$790$	0	0
$791$	33.7303	1.19931
$792$	0	0
$793$	27.6431	0.981634
$794$	0	0
$795$	−8.39242	−0.297649
$796$	0	0
$797$	5.67319	0.200955	0.100477	−	0.994939i	$-0.467963\pi$
$797$	5.67319	0.200955	0.100477	+	0.994939i	$0.467963\pi$
$798$	0	0
$799$	11.3626	0.401979
$800$	0	0
$801$	−12.0739	−0.426611
$802$	0	0
$803$	−12.3459	−0.435676
$804$	0	0
$805$	−66.0750	−2.32884
$806$	0	0
$807$	16.8017	0.591448
$808$	0	0
$809$	−37.7932	−1.32874	−0.664369	−	0.747405i	$-0.731300\pi$
$809$	−37.7932	−1.32874	−0.664369	+	0.747405i	$0.731300\pi$
$810$	0	0
$811$	−17.2974	−0.607394	−0.303697	−	0.952769i	$-0.598221\pi$
$811$	−17.2974	−0.607394	−0.303697	+	0.952769i	$0.598221\pi$
$812$	0	0
$813$	8.49263	0.297849
$814$	0	0
$815$	−4.87350	−0.170711
$816$	0	0
$817$	−2.12557	−0.0743643
$818$	0	0
$819$	−14.2466	−0.497815
$820$	0	0
$821$	13.1038	0.457325	0.228662	−	0.973506i	$-0.426565\pi$
$821$	13.1038	0.457325	0.228662	+	0.973506i	$0.426565\pi$
$822$	0	0
$823$	5.47594	0.190879	0.0954396	−	0.995435i	$-0.469574\pi$
$823$	5.47594	0.190879	0.0954396	+	0.995435i	$0.469574\pi$
$824$	0	0
$825$	−4.53238	−0.157797
$826$	0	0
$827$	2.81367	0.0978411	0.0489205	−	0.998803i	$-0.484422\pi$
$827$	2.81367	0.0978411	0.0489205	+	0.998803i	$0.484422\pi$
$828$	0	0
$829$	11.7545	0.408250	0.204125	−	0.978945i	$-0.434565\pi$
$829$	11.7545	0.408250	0.204125	+	0.978945i	$0.434565\pi$
$830$	0	0
$831$	−12.1608	−0.421852
$832$	0	0
$833$	16.6460	0.576751
$834$	0	0
$835$	1.80306	0.0623974
$836$	0	0
$837$	2.95084	0.101996
$838$	0	0
$839$	35.1585	1.21381	0.606903	−	0.794776i	$-0.292412\pi$
$839$	35.1585	1.21381	0.606903	+	0.794776i	$0.292412\pi$
$840$	0	0
$841$	−9.44960	−0.325848
$842$	0	0
$843$	−14.5701	−0.501820
$844$	0	0
$845$	1.76079	0.0605732
$846$	0	0
$847$	−17.6031	−0.604849
$848$	0	0
$849$	−8.16542	−0.280236
$850$	0	0
$851$	8.18931	0.280726
$852$	0	0
$853$	−5.19930	−0.178021	−0.0890104	−	0.996031i	$-0.528370\pi$
$853$	−5.19930	−0.178021	−0.0890104	+	0.996031i	$0.528370\pi$
$854$	0	0
$855$	−4.50357	−0.154019
$856$	0	0
$857$	1.20592	0.0411933	0.0205966	−	0.999788i	$-0.493443\pi$
$857$	1.20592	0.0411933	0.0205966	+	0.999788i	$0.493443\pi$
$858$	0	0
$859$	39.2254	1.33835	0.669176	−	0.743104i	$-0.266647\pi$
$859$	39.2254	1.33835	0.669176	+	0.743104i	$0.266647\pi$
$860$	0	0
$861$	−7.55366	−0.257428
$862$	0	0
$863$	−17.4008	−0.592329	−0.296164	−	0.955137i	$-0.595708\pi$
$863$	−17.4008	−0.592329	−0.296164	+	0.955137i	$0.595708\pi$
$864$	0	0
$865$	−24.7900	−0.842886
$866$	0	0
$867$	−14.1618	−0.480959
$868$	0	0
$869$	27.7784	0.942319
$870$	0	0
$871$	−5.74591	−0.194693
$872$	0	0
$873$	14.8662	0.503146
$874$	0	0
$875$	49.9969	1.69020
$876$	0	0
$877$	−51.8185	−1.74979	−0.874893	−	0.484316i	$-0.839068\pi$
$877$	−51.8185	−1.74979	−0.874893	+	0.484316i	$0.839068\pi$
$878$	0	0
$879$	29.2346	0.986060
$880$	0	0
$881$	−55.7905	−1.87963	−0.939814	−	0.341686i	$-0.889002\pi$
$881$	−55.7905	−1.87963	−0.939814	+	0.341686i	$0.889002\pi$
$882$	0	0
$883$	−34.6270	−1.16529	−0.582645	−	0.812727i	$-0.697982\pi$
$883$	−34.6270	−1.16529	−0.582645	+	0.812727i	$0.697982\pi$
$884$	0	0
$885$	2.23309	0.0750643
$886$	0	0
$887$	3.25650	0.109343	0.0546713	−	0.998504i	$-0.482589\pi$
$887$	3.25650	0.109343	0.0546713	+	0.998504i	$0.482589\pi$
$888$	0	0
$889$	−38.5418	−1.29265
$890$	0	0
$891$	2.59144	0.0868166
$892$	0	0
$893$	16.8462	0.563736
$894$	0	0
$895$	0.268375	0.00897079
$896$	0	0
$897$	−30.9276	−1.03264
$898$	0	0
$899$	−13.0474	−0.435155
$900$	0	0
$901$	7.84151	0.261239
$902$	0	0
$903$	−3.49642	−0.116354
$904$	0	0
$905$	−12.8021	−0.425556
$906$	0	0
$907$	−1.93813	−0.0643547	−0.0321773	−	0.999482i	$-0.510244\pi$
$907$	−1.93813	−0.0643547	−0.0321773	+	0.999482i	$0.510244\pi$
$908$	0	0
$909$	1.27675	0.0423471
$910$	0	0
$911$	22.6161	0.749304	0.374652	−	0.927165i	$-0.377762\pi$
$911$	22.6161	0.749304	0.374652	+	0.927165i	$0.377762\pi$
$912$	0	0
$913$	9.56262	0.316477
$914$	0	0
$915$	14.3741	0.475194
$916$	0	0
$917$	−81.5855	−2.69419
$918$	0	0
$919$	0.152045	0.00501549	0.00250774	−	0.999997i	$-0.499202\pi$
$919$	0.152045	0.00501549	0.00250774	+	0.999997i	$0.499202\pi$
$920$	0	0
$921$	−23.8192	−0.784869
$922$	0	0
$923$	−21.6391	−0.712259
$924$	0	0
$925$	−1.60583	−0.0527994
$926$	0	0
$927$	−5.26805	−0.173026
$928$	0	0
$929$	−40.4885	−1.32838	−0.664192	−	0.747562i	$-0.731224\pi$
$929$	−40.4885	−1.32838	−0.664192	+	0.747562i	$0.731224\pi$
$930$	0	0
$931$	24.6795	0.808836
$932$	0	0
$933$	16.4270	0.537797
$934$	0	0
$935$	−7.87180	−0.257435
$936$	0	0
$937$	−49.3072	−1.61079	−0.805397	−	0.592735i	$-0.798048\pi$
$937$	−49.3072	−1.61079	−0.805397	+	0.592735i	$0.798048\pi$
$938$	0	0
$939$	12.4875	0.407513
$940$	0	0
$941$	−22.8417	−0.744618	−0.372309	−	0.928109i	$-0.621434\pi$
$941$	−22.8417	−0.744618	−0.372309	+	0.928109i	$0.621434\pi$
$942$	0	0
$943$	−16.3981	−0.533996
$944$	0	0
$945$	−7.40807	−0.240985
$946$	0	0
$947$	−20.8527	−0.677622	−0.338811	−	0.940854i	$-0.610025\pi$
$947$	−20.8527	−0.677622	−0.338811	+	0.940854i	$0.610025\pi$
$948$	0	0
$949$	16.5194	0.536242
$950$	0	0
$951$	−3.38861	−0.109883
$952$	0	0
$953$	−20.6396	−0.668581	−0.334291	−	0.942470i	$-0.608497\pi$
$953$	−20.6396	−0.668581	−0.334291	+	0.942470i	$0.608497\pi$
$954$	0	0
$955$	−10.0348	−0.324719
$956$	0	0
$957$	−11.4583	−0.370393
$958$	0	0
$959$	22.3630	0.722139
$960$	0	0
$961$	−22.2925	−0.719114
$962$	0	0
$963$	−7.81076	−0.251698
$964$	0	0
$965$	30.0773	0.968222
$966$	0	0
$967$	−43.8774	−1.41100	−0.705501	−	0.708709i	$-0.749278\pi$
$967$	−43.8774	−1.41100	−0.705501	+	0.708709i	$0.749278\pi$
$968$	0	0
$969$	4.20794	0.135179
$970$	0	0
$971$	−9.54975	−0.306466	−0.153233	−	0.988190i	$-0.548969\pi$
$971$	−9.54975	−0.306466	−0.153233	+	0.988190i	$0.548969\pi$
$972$	0	0
$973$	53.7643	1.72361
$974$	0	0
$975$	6.06455	0.194221
$976$	0	0
$977$	11.9627	0.382721	0.191360	−	0.981520i	$-0.438710\pi$
$977$	11.9627	0.382721	0.191360	+	0.981520i	$0.438710\pi$
$978$	0	0
$979$	−31.2888	−0.999995
$980$	0	0
$981$	−2.01235	−0.0642493
$982$	0	0
$983$	37.2845	1.18919	0.594595	−	0.804025i	$-0.297312\pi$
$983$	37.2845	1.18919	0.594595	+	0.804025i	$0.297312\pi$
$984$	0	0
$985$	−18.1489	−0.578272
$986$	0	0
$987$	27.7109	0.882047
$988$	0	0
$989$	−7.59032	−0.241358
$990$	0	0
$991$	46.0389	1.46247	0.731237	−	0.682123i	$-0.238943\pi$
$991$	46.0389	1.46247	0.731237	+	0.682123i	$0.238943\pi$
$992$	0	0
$993$	−7.83764	−0.248720
$994$	0	0
$995$	−25.6521	−0.813228
$996$	0	0
$997$	−6.30573	−0.199705	−0.0998523	−	0.995002i	$-0.531837\pi$
$997$	−6.30573	−0.199705	−0.0998523	+	0.995002i	$0.531837\pi$
$998$	0	0
$999$	0.918154	0.0290491

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.be.1.3		11
4.3	odd	2		4008.2.a.k.1.3	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.k.1.3	✓	11	4.3	odd	2
8016.2.a.be.1.3		11	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:	\( N \)	\(=\)	\( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8016.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.80306 q^{5} +4.10861 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.80306 q^{5} +4.10861 q^{7} +1.00000 q^{9} +2.59144 q^{11} -3.46748 q^{13} -1.80306 q^{15} +1.68470 q^{17} +2.49774 q^{19} +4.10861 q^{21} +8.91932 q^{23} -1.74898 q^{25} +1.00000 q^{27} -4.42158 q^{29} +2.95084 q^{31} +2.59144 q^{33} -7.40807 q^{35} +0.918154 q^{37} -3.46748 q^{39} -1.83849 q^{41} -0.850997 q^{43} -1.80306 q^{45} +6.74458 q^{47} +9.88072 q^{49} +1.68470 q^{51} +4.65455 q^{53} -4.67252 q^{55} +2.49774 q^{57} -1.23850 q^{59} -7.97208 q^{61} +4.10861 q^{63} +6.25208 q^{65} +1.65708 q^{67} +8.91932 q^{69} +6.24057 q^{71} -4.76409 q^{73} -1.74898 q^{75} +10.6472 q^{77} +10.7193 q^{79} +1.00000 q^{81} +3.69008 q^{83} -3.03761 q^{85} -4.42158 q^{87} -12.0739 q^{89} -14.2466 q^{91} +2.95084 q^{93} -4.50357 q^{95} +14.8662 q^{97} +2.59144 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 11 q^{3} + 10 q^{5} + q^{7} + 11 q^{9}+O(q^{10}) \) 11 * q + 11 * q^3 + 10 * q^5 + q^7 + 11 * q^9	\( 11 q + 11 q^{3} + 10 q^{5} + q^{7} + 11 q^{9} + q^{11} + 10 q^{13} + 10 q^{15} + 17 q^{17} - 2 q^{19} + q^{21} + 3 q^{23} + 21 q^{25} + 11 q^{27} + 17 q^{29} + 15 q^{31} + q^{33} - 11 q^{35} + 4 q^{37} + 10 q^{39} + 16 q^{41} - 10 q^{43} + 10 q^{45} + 16 q^{47} + 22 q^{49} + 17 q^{51} + 42 q^{53} + 5 q^{55} - 2 q^{57} + 2 q^{59} + 12 q^{61} + q^{63} + 10 q^{65} + q^{67} + 3 q^{69} + 9 q^{71} + 24 q^{73} + 21 q^{75} + 22 q^{77} + 30 q^{79} + 11 q^{81} - 16 q^{83} + 25 q^{85} + 17 q^{87} + 37 q^{89} - q^{91} + 15 q^{93} - 5 q^{95} + 4 q^{97} + q^{99}+O(q^{100}) \) 11 * q + 11 * q^3 + 10 * q^5 + q^7 + 11 * q^9 + q^11 + 10 * q^13 + 10 * q^15 + 17 * q^17 - 2 * q^19 + q^21 + 3 * q^23 + 21 * q^25 + 11 * q^27 + 17 * q^29 + 15 * q^31 + q^33 - 11 * q^35 + 4 * q^37 + 10 * q^39 + 16 * q^41 - 10 * q^43 + 10 * q^45 + 16 * q^47 + 22 * q^49 + 17 * q^51 + 42 * q^53 + 5 * q^55 - 2 * q^57 + 2 * q^59 + 12 * q^61 + q^63 + 10 * q^65 + q^67 + 3 * q^69 + 9 * q^71 + 24 * q^73 + 21 * q^75 + 22 * q^77 + 30 * q^79 + 11 * q^81 - 16 * q^83 + 25 * q^85 + 17 * q^87 + 37 * q^89 - q^91 + 15 * q^93 - 5 * q^95 + 4 * q^97 + q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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