LMFDB - Embedded newform 6292.2.a.v.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6292,2,Mod(1,6292)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6292, 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("6292.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6292 = 2^{2} \cdot 11^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6292.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:	$50.2418729518$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 19x^{8} + 40x^{7} + 106x^{6} - 244x^{5} - 154x^{4} + 488x^{3} - 107x^{2} - 138x + 33 $ x^10 - 2x^9 - 19x^8 + 40x^7 + 106x^6 - 244x^5 - 154x^4 + 488x^3 - 107x^2 - 138*x + 33
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.52003$ of defining polynomial
Character	$\chi$	$=$	6292.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.52003 q^{3} +2.97755 q^{5} -1.33685 q^{7} +3.35054 q^{9} +O(q^{10})$	$q-2.52003 q^{3} +2.97755 q^{5} -1.33685 q^{7} +3.35054 q^{9} -1.00000 q^{13} -7.50351 q^{15} +5.23353 q^{17} +0.681336 q^{19} +3.36891 q^{21} +1.66546 q^{23} +3.86581 q^{25} -0.883378 q^{27} +0.221782 q^{29} +3.34668 q^{31} -3.98054 q^{35} -6.62484 q^{37} +2.52003 q^{39} +8.87303 q^{41} -5.13759 q^{43} +9.97641 q^{45} +4.12859 q^{47} -5.21283 q^{49} -13.1887 q^{51} -4.66558 q^{53} -1.71699 q^{57} -9.81901 q^{59} +12.1158 q^{61} -4.47918 q^{63} -2.97755 q^{65} +3.01937 q^{67} -4.19700 q^{69} -14.2973 q^{71} +13.9670 q^{73} -9.74194 q^{75} +2.29035 q^{79} -7.82549 q^{81} +8.54868 q^{83} +15.5831 q^{85} -0.558898 q^{87} +12.7557 q^{89} +1.33685 q^{91} -8.43373 q^{93} +2.02871 q^{95} +3.62624 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 2 q^{3} - 6 q^{5} + 4 q^{7} + 12 q^{9}+O(q^{10}) $ 10 * q - 2 * q^3 - 6 * q^5 + 4 * q^7 + 12 * q^9	$ 10 q - 2 q^{3} - 6 q^{5} + 4 q^{7} + 12 q^{9} - 10 q^{13} - 4 q^{15} + 20 q^{17} + 14 q^{19} + 2 q^{21} - 2 q^{23} + 2 q^{25} + 10 q^{27} + 16 q^{29} - 2 q^{31} - 20 q^{35} + 2 q^{39} + 22 q^{41} - 12 q^{43} - 8 q^{45} + 6 q^{47} - 2 q^{49} - 8 q^{51} + 6 q^{53} + 12 q^{57} - 4 q^{59} + 24 q^{61} + 68 q^{63} + 6 q^{65} - 6 q^{67} - 2 q^{69} - 8 q^{71} - 6 q^{73} - 18 q^{75} - 24 q^{79} + 10 q^{81} + 22 q^{83} + 34 q^{85} - 4 q^{87} - 42 q^{89} - 4 q^{91} - 38 q^{93} + 14 q^{97}+O(q^{100}) $ 10 * q - 2 * q^3 - 6 * q^5 + 4 * q^7 + 12 * q^9 - 10 * q^13 - 4 * q^15 + 20 * q^17 + 14 * q^19 + 2 * q^21 - 2 * q^23 + 2 * q^25 + 10 * q^27 + 16 * q^29 - 2 * q^31 - 20 * q^35 + 2 * q^39 + 22 * q^41 - 12 * q^43 - 8 * q^45 + 6 * q^47 - 2 * q^49 - 8 * q^51 + 6 * q^53 + 12 * q^57 - 4 * q^59 + 24 * q^61 + 68 * q^63 + 6 * q^65 - 6 * q^67 - 2 * q^69 - 8 * q^71 - 6 * q^73 - 18 * q^75 - 24 * q^79 + 10 * q^81 + 22 * q^83 + 34 * q^85 - 4 * q^87 - 42 * q^89 - 4 * q^91 - 38 * q^93 + 14 * 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
$3$	−2.52003	−1.45494	−0.727470	−	0.686140i	$-0.759304\pi$
$3$	−2.52003	−1.45494	−0.727470	+	0.686140i	$0.759304\pi$
$4$	0	0
$5$	2.97755	1.33160	0.665801	−	0.746130i	$-0.268090\pi$
$5$	2.97755	1.33160	0.665801	+	0.746130i	$0.268090\pi$
$6$	0	0
$7$	−1.33685	−0.505283	−0.252641	−	0.967560i	$-0.581299\pi$
$7$	−1.33685	−0.505283	−0.252641	+	0.967560i	$0.581299\pi$
$8$	0	0
$9$	3.35054	1.11685
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−7.50351	−1.93740
$16$	0	0
$17$	5.23353	1.26932	0.634659	−	0.772792i	$-0.281140\pi$
$17$	5.23353	1.26932	0.634659	+	0.772792i	$0.281140\pi$
$18$	0	0
$19$	0.681336	0.156309	0.0781546	−	0.996941i	$-0.475097\pi$
$19$	0.681336	0.156309	0.0781546	+	0.996941i	$0.475097\pi$
$20$	0	0
$21$	3.36891	0.735155
$22$	0	0
$23$	1.66546	0.347272	0.173636	−	0.984810i	$-0.444448\pi$
$23$	1.66546	0.347272	0.173636	+	0.984810i	$0.444448\pi$
$24$	0	0
$25$	3.86581	0.773161
$26$	0	0
$27$	−0.883378	−0.170006
$28$	0	0
$29$	0.221782	0.0411839	0.0205920	−	0.999788i	$-0.493445\pi$
$29$	0.221782	0.0411839	0.0205920	+	0.999788i	$0.493445\pi$
$30$	0	0
$31$	3.34668	0.601082	0.300541	−	0.953769i	$-0.402833\pi$
$31$	3.34668	0.601082	0.300541	+	0.953769i	$0.402833\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.98054	−0.672835
$36$	0	0
$37$	−6.62484	−1.08912	−0.544559	−	0.838723i	$-0.683303\pi$
$37$	−6.62484	−1.08912	−0.544559	+	0.838723i	$0.683303\pi$
$38$	0	0
$39$	2.52003	0.403527
$40$	0	0
$41$	8.87303	1.38573	0.692867	−	0.721065i	$-0.256347\pi$
$41$	8.87303	1.38573	0.692867	+	0.721065i	$0.256347\pi$
$42$	0	0
$43$	−5.13759	−0.783476	−0.391738	−	0.920077i	$-0.628126\pi$
$43$	−5.13759	−0.783476	−0.391738	+	0.920077i	$0.628126\pi$
$44$	0	0
$45$	9.97641	1.48720
$46$	0	0
$47$	4.12859	0.602216	0.301108	−	0.953590i	$-0.402644\pi$
$47$	4.12859	0.602216	0.301108	+	0.953590i	$0.402644\pi$
$48$	0	0
$49$	−5.21283	−0.744690
$50$	0	0
$51$	−13.1887	−1.84678
$52$	0	0
$53$	−4.66558	−0.640867	−0.320434	−	0.947271i	$-0.603829\pi$
$53$	−4.66558	−0.640867	−0.320434	+	0.947271i	$0.603829\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.71699	−0.227420
$58$	0	0
$59$	−9.81901	−1.27833	−0.639163	−	0.769071i	$-0.720719\pi$
$59$	−9.81901	−1.27833	−0.639163	+	0.769071i	$0.720719\pi$
$60$	0	0
$61$	12.1158	1.55127	0.775635	−	0.631181i	$-0.217430\pi$
$61$	12.1158	1.55127	0.775635	+	0.631181i	$0.217430\pi$
$62$	0	0
$63$	−4.47918	−0.564324
$64$	0	0
$65$	−2.97755	−0.369320
$66$	0	0
$67$	3.01937	0.368875	0.184438	−	0.982844i	$-0.440954\pi$
$67$	3.01937	0.368875	0.184438	+	0.982844i	$0.440954\pi$
$68$	0	0
$69$	−4.19700	−0.505259
$70$	0	0
$71$	−14.2973	−1.69677	−0.848387	−	0.529376i	$-0.822426\pi$
$71$	−14.2973	−1.69677	−0.848387	+	0.529376i	$0.822426\pi$
$72$	0	0
$73$	13.9670	1.63471	0.817354	−	0.576135i	$-0.195440\pi$
$73$	13.9670	1.63471	0.817354	+	0.576135i	$0.195440\pi$
$74$	0	0
$75$	−9.74194	−1.12490
$76$	0	0
$77$	0	0
$78$	0	0
$79$	2.29035	0.257684	0.128842	−	0.991665i	$-0.458874\pi$
$79$	2.29035	0.257684	0.128842	+	0.991665i	$0.458874\pi$
$80$	0	0
$81$	−7.82549	−0.869499
$82$	0	0
$83$	8.54868	0.938340	0.469170	−	0.883108i	$-0.344553\pi$
$83$	8.54868	0.938340	0.469170	+	0.883108i	$0.344553\pi$
$84$	0	0
$85$	15.5831	1.69023
$86$	0	0
$87$	−0.558898	−0.0599201
$88$	0	0
$89$	12.7557	1.35211	0.676053	−	0.736853i	$-0.263689\pi$
$89$	12.7557	1.35211	0.676053	+	0.736853i	$0.263689\pi$
$90$	0	0
$91$	1.33685	0.140140
$92$	0	0
$93$	−8.43373	−0.874537
$94$	0	0
$95$	2.02871	0.208141
$96$	0	0
$97$	3.62624	0.368189	0.184094	−	0.982909i	$-0.441065\pi$
$97$	3.62624	0.368189	0.184094	+	0.982909i	$0.441065\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−3.07880	−0.306352	−0.153176	−	0.988199i	$-0.548950\pi$
$101$	−3.07880	−0.306352	−0.153176	+	0.988199i	$0.548950\pi$
$102$	0	0
$103$	12.6887	1.25025	0.625127	−	0.780523i	$-0.285047\pi$
$103$	12.6887	1.25025	0.625127	+	0.780523i	$0.285047\pi$
$104$	0	0
$105$	10.0311	0.978934
$106$	0	0
$107$	0.523618	0.0506201	0.0253101	−	0.999680i	$-0.491943\pi$
$107$	0.523618	0.0506201	0.0253101	+	0.999680i	$0.491943\pi$
$108$	0	0
$109$	2.82754	0.270830	0.135415	−	0.990789i	$-0.456763\pi$
$109$	2.82754	0.270830	0.135415	+	0.990789i	$0.456763\pi$
$110$	0	0
$111$	16.6948	1.58460
$112$	0	0
$113$	15.0073	1.41177	0.705884	−	0.708327i	$-0.250550\pi$
$113$	15.0073	1.41177	0.705884	+	0.708327i	$0.250550\pi$
$114$	0	0
$115$	4.95898	0.462427
$116$	0	0
$117$	−3.35054	−0.309758
$118$	0	0
$119$	−6.99646	−0.641365
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−22.3603	−2.01616
$124$	0	0
$125$	−3.37712	−0.302058
$126$	0	0
$127$	−13.1576	−1.16755	−0.583776	−	0.811915i	$-0.698425\pi$
$127$	−13.1576	−1.16755	−0.583776	+	0.811915i	$0.698425\pi$
$128$	0	0
$129$	12.9469	1.13991
$130$	0	0
$131$	−5.67452	−0.495785	−0.247893	−	0.968788i	$-0.579738\pi$
$131$	−5.67452	−0.495785	−0.247893	+	0.968788i	$0.579738\pi$
$132$	0	0
$133$	−0.910845	−0.0789803
$134$	0	0
$135$	−2.63030	−0.226380
$136$	0	0
$137$	−11.6122	−0.992096	−0.496048	−	0.868295i	$-0.665216\pi$
$137$	−11.6122	−0.992096	−0.496048	+	0.868295i	$0.665216\pi$
$138$	0	0
$139$	−12.9088	−1.09491	−0.547456	−	0.836834i	$-0.684404\pi$
$139$	−12.9088	−1.09491	−0.547456	+	0.836834i	$0.684404\pi$
$140$	0	0
$141$	−10.4042	−0.876188
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0.660368	0.0548406
$146$	0	0
$147$	13.1365	1.08348
$148$	0	0
$149$	−3.70802	−0.303773	−0.151886	−	0.988398i	$-0.548535\pi$
$149$	−3.70802	−0.303773	−0.151886	+	0.988398i	$0.548535\pi$
$150$	0	0
$151$	−9.96353	−0.810820	−0.405410	−	0.914135i	$-0.632871\pi$
$151$	−9.96353	−0.810820	−0.405410	+	0.914135i	$0.632871\pi$
$152$	0	0
$153$	17.5352	1.41764
$154$	0	0
$155$	9.96491	0.800401
$156$	0	0
$157$	−4.64667	−0.370845	−0.185422	−	0.982659i	$-0.559365\pi$
$157$	−4.64667	−0.370845	−0.185422	+	0.982659i	$0.559365\pi$
$158$	0	0
$159$	11.7574	0.932423
$160$	0	0
$161$	−2.22647	−0.175470
$162$	0	0
$163$	−0.747119	−0.0585189	−0.0292594	−	0.999572i	$-0.509315\pi$
$163$	−0.747119	−0.0585189	−0.0292594	+	0.999572i	$0.509315\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0.167021	0.0129244	0.00646222	−	0.999979i	$-0.497943\pi$
$167$	0.167021	0.0129244	0.00646222	+	0.999979i	$0.497943\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	2.28285	0.174574
$172$	0	0
$173$	5.83405	0.443555	0.221777	−	0.975097i	$-0.428814\pi$
$173$	5.83405	0.443555	0.221777	+	0.975097i	$0.428814\pi$
$174$	0	0
$175$	−5.16801	−0.390665
$176$	0	0
$177$	24.7442	1.85989
$178$	0	0
$179$	−22.6896	−1.69590	−0.847951	−	0.530075i	$-0.822164\pi$
$179$	−22.6896	−1.69590	−0.847951	+	0.530075i	$0.822164\pi$
$180$	0	0
$181$	17.5253	1.30265	0.651324	−	0.758800i	$-0.274214\pi$
$181$	17.5253	1.30265	0.651324	+	0.758800i	$0.274214\pi$
$182$	0	0
$183$	−30.5322	−2.25700
$184$	0	0
$185$	−19.7258	−1.45027
$186$	0	0
$187$	0	0
$188$	0	0
$189$	1.18095	0.0859011
$190$	0	0
$191$	−13.3822	−0.968300	−0.484150	−	0.874985i	$-0.660871\pi$
$191$	−13.3822	−0.968300	−0.484150	+	0.874985i	$0.660871\pi$
$192$	0	0
$193$	23.0222	1.65717	0.828586	−	0.559862i	$-0.189146\pi$
$193$	23.0222	1.65717	0.828586	+	0.559862i	$0.189146\pi$
$194$	0	0
$195$	7.50351	0.537338
$196$	0	0
$197$	11.8910	0.847202	0.423601	−	0.905849i	$-0.360766\pi$
$197$	11.8910	0.847202	0.423601	+	0.905849i	$0.360766\pi$
$198$	0	0
$199$	9.53933	0.676225	0.338113	−	0.941106i	$-0.390212\pi$
$199$	9.53933	0.676225	0.338113	+	0.941106i	$0.390212\pi$
$200$	0	0
$201$	−7.60890	−0.536691
$202$	0	0
$203$	−0.296490	−0.0208095
$204$	0	0
$205$	26.4199	1.84525
$206$	0	0
$207$	5.58018	0.387850
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−7.70836	−0.530665	−0.265333	−	0.964157i	$-0.585482\pi$
$211$	−7.70836	−0.530665	−0.265333	+	0.964157i	$0.585482\pi$
$212$	0	0
$213$	36.0295	2.46870
$214$	0	0
$215$	−15.2974	−1.04328
$216$	0	0
$217$	−4.47402	−0.303716
$218$	0	0
$219$	−35.1971	−2.37840
$220$	0	0
$221$	−5.23353	−0.352046
$222$	0	0
$223$	−20.0277	−1.34115	−0.670576	−	0.741841i	$-0.733953\pi$
$223$	−20.0277	−1.34115	−0.670576	+	0.741841i	$0.733953\pi$
$224$	0	0
$225$	12.9526	0.863504
$226$	0	0
$227$	−1.59850	−0.106096	−0.0530481	−	0.998592i	$-0.516894\pi$
$227$	−1.59850	−0.106096	−0.0530481	+	0.998592i	$0.516894\pi$
$228$	0	0
$229$	5.91144	0.390639	0.195319	−	0.980740i	$-0.437426\pi$
$229$	5.91144	0.390639	0.195319	+	0.980740i	$0.437426\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−14.6504	−0.959782	−0.479891	−	0.877328i	$-0.659324\pi$
$233$	−14.6504	−0.959782	−0.479891	+	0.877328i	$0.659324\pi$
$234$	0	0
$235$	12.2931	0.801912
$236$	0	0
$237$	−5.77174	−0.374915
$238$	0	0
$239$	12.8248	0.829566	0.414783	−	0.909920i	$-0.363858\pi$
$239$	12.8248	0.829566	0.414783	+	0.909920i	$0.363858\pi$
$240$	0	0
$241$	−3.02466	−0.194836	−0.0974178	−	0.995244i	$-0.531058\pi$
$241$	−3.02466	−0.194836	−0.0974178	+	0.995244i	$0.531058\pi$
$242$	0	0
$243$	22.3706	1.43507
$244$	0	0
$245$	−15.5215	−0.991629
$246$	0	0
$247$	−0.681336	−0.0433524
$248$	0	0
$249$	−21.5429	−1.36523
$250$	0	0
$251$	13.9357	0.879615	0.439808	−	0.898092i	$-0.355047\pi$
$251$	13.9357	0.879615	0.439808	+	0.898092i	$0.355047\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−39.2699	−2.45918
$256$	0	0
$257$	−18.7052	−1.16680	−0.583399	−	0.812186i	$-0.698278\pi$
$257$	−18.7052	−1.16680	−0.583399	+	0.812186i	$0.698278\pi$
$258$	0	0
$259$	8.85644	0.550312
$260$	0	0
$261$	0.743091	0.0459962
$262$	0	0
$263$	18.7121	1.15384	0.576920	−	0.816801i	$-0.304254\pi$
$263$	18.7121	1.15384	0.576920	+	0.816801i	$0.304254\pi$
$264$	0	0
$265$	−13.8920	−0.853379
$266$	0	0
$267$	−32.1448	−1.96723
$268$	0	0
$269$	18.0774	1.10220	0.551098	−	0.834440i	$-0.314209\pi$
$269$	18.0774	1.10220	0.551098	+	0.834440i	$0.314209\pi$
$270$	0	0
$271$	18.9799	1.15295	0.576474	−	0.817115i	$-0.304428\pi$
$271$	18.9799	1.15295	0.576474	+	0.817115i	$0.304428\pi$
$272$	0	0
$273$	−3.36891	−0.203895
$274$	0	0
$275$	0	0
$276$	0	0
$277$	27.8211	1.67161	0.835803	−	0.549029i	$-0.185003\pi$
$277$	27.8211	1.67161	0.835803	+	0.549029i	$0.185003\pi$
$278$	0	0
$279$	11.2132	0.671317
$280$	0	0
$281$	−16.3559	−0.975709	−0.487855	−	0.872925i	$-0.662220\pi$
$281$	−16.3559	−0.975709	−0.487855	+	0.872925i	$0.662220\pi$
$282$	0	0
$283$	20.3548	1.20997	0.604983	−	0.796238i	$-0.293180\pi$
$283$	20.3548	1.20997	0.604983	+	0.796238i	$0.293180\pi$
$284$	0	0
$285$	−5.11241	−0.302833
$286$	0	0
$287$	−11.8619	−0.700188
$288$	0	0
$289$	10.3899	0.611169
$290$	0	0
$291$	−9.13823	−0.535692
$292$	0	0
$293$	23.0828	1.34851	0.674257	−	0.738497i	$-0.264464\pi$
$293$	23.0828	1.34851	0.674257	+	0.738497i	$0.264464\pi$
$294$	0	0
$295$	−29.2366	−1.70222
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1.66546	−0.0963158
$300$	0	0
$301$	6.86820	0.395877
$302$	0	0
$303$	7.75867	0.445724
$304$	0	0
$305$	36.0754	2.06567
$306$	0	0
$307$	22.2577	1.27031	0.635156	−	0.772384i	$-0.280936\pi$
$307$	22.2577	1.27031	0.635156	+	0.772384i	$0.280936\pi$
$308$	0	0
$309$	−31.9759	−1.81904
$310$	0	0
$311$	32.3337	1.83347	0.916737	−	0.399492i	$-0.130814\pi$
$311$	32.3337	1.83347	0.916737	+	0.399492i	$0.130814\pi$
$312$	0	0
$313$	24.7782	1.40055	0.700275	−	0.713874i	$-0.253061\pi$
$313$	24.7782	1.40055	0.700275	+	0.713874i	$0.253061\pi$
$314$	0	0
$315$	−13.3370	−0.751454
$316$	0	0
$317$	1.48680	0.0835072	0.0417536	−	0.999128i	$-0.486706\pi$
$317$	1.48680	0.0835072	0.0417536	+	0.999128i	$0.486706\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−1.31953	−0.0736492
$322$	0	0
$323$	3.56579	0.198406
$324$	0	0
$325$	−3.86581	−0.214436
$326$	0	0
$327$	−7.12549	−0.394041
$328$	0	0
$329$	−5.51931	−0.304289
$330$	0	0
$331$	7.80162	0.428816	0.214408	−	0.976744i	$-0.431218\pi$
$331$	7.80162	0.428816	0.214408	+	0.976744i	$0.431218\pi$
$332$	0	0
$333$	−22.1968	−1.21638
$334$	0	0
$335$	8.99033	0.491194
$336$	0	0
$337$	24.3920	1.32872	0.664359	−	0.747413i	$-0.268704\pi$
$337$	24.3920	1.32872	0.664359	+	0.747413i	$0.268704\pi$
$338$	0	0
$339$	−37.8188	−2.05404
$340$	0	0
$341$	0	0
$342$	0	0
$343$	16.3267	0.881561
$344$	0	0
$345$	−12.4968	−0.672804
$346$	0	0
$347$	26.5740	1.42657	0.713283	−	0.700877i	$-0.247208\pi$
$347$	26.5740	1.42657	0.713283	+	0.700877i	$0.247208\pi$
$348$	0	0
$349$	−31.4362	−1.68274	−0.841371	−	0.540459i	$-0.818251\pi$
$349$	−31.4362	−1.68274	−0.841371	+	0.540459i	$0.818251\pi$
$350$	0	0
$351$	0.883378	0.0471512
$352$	0	0
$353$	−10.5109	−0.559437	−0.279718	−	0.960082i	$-0.590241\pi$
$353$	−10.5109	−0.559437	−0.279718	+	0.960082i	$0.590241\pi$
$354$	0	0
$355$	−42.5709	−2.25943
$356$	0	0
$357$	17.6313	0.933146
$358$	0	0
$359$	21.3556	1.12710	0.563552	−	0.826080i	$-0.309434\pi$
$359$	21.3556	1.12710	0.563552	+	0.826080i	$0.309434\pi$
$360$	0	0
$361$	−18.5358	−0.975567
$362$	0	0
$363$	0	0
$364$	0	0
$365$	41.5873	2.17678
$366$	0	0
$367$	18.8100	0.981872	0.490936	−	0.871196i	$-0.336655\pi$
$367$	18.8100	0.981872	0.490936	+	0.871196i	$0.336655\pi$
$368$	0	0
$369$	29.7295	1.54765
$370$	0	0
$371$	6.23719	0.323819
$372$	0	0
$373$	−0.517259	−0.0267826	−0.0133913	−	0.999910i	$-0.504263\pi$
$373$	−0.517259	−0.0267826	−0.0133913	+	0.999910i	$0.504263\pi$
$374$	0	0
$375$	8.51043	0.439477
$376$	0	0
$377$	−0.221782	−0.0114224
$378$	0	0
$379$	−24.4314	−1.25496	−0.627479	−	0.778633i	$-0.715913\pi$
$379$	−24.4314	−1.25496	−0.627479	+	0.778633i	$0.715913\pi$
$380$	0	0
$381$	33.1576	1.69872
$382$	0	0
$383$	8.20768	0.419393	0.209696	−	0.977767i	$-0.432752\pi$
$383$	8.20768	0.419393	0.209696	+	0.977767i	$0.432752\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−17.2137	−0.875023
$388$	0	0
$389$	36.2073	1.83578	0.917890	−	0.396834i	$-0.129891\pi$
$389$	36.2073	1.83578	0.917890	+	0.396834i	$0.129891\pi$
$390$	0	0
$391$	8.71622	0.440798
$392$	0	0
$393$	14.2999	0.721337
$394$	0	0
$395$	6.81962	0.343132
$396$	0	0
$397$	−29.8708	−1.49917	−0.749586	−	0.661907i	$-0.769747\pi$
$397$	−29.8708	−1.49917	−0.749586	+	0.661907i	$0.769747\pi$
$398$	0	0
$399$	2.29536	0.114912
$400$	0	0
$401$	−15.3179	−0.764940	−0.382470	−	0.923968i	$-0.624927\pi$
$401$	−15.3179	−0.764940	−0.382470	+	0.923968i	$0.624927\pi$
$402$	0	0
$403$	−3.34668	−0.166710
$404$	0	0
$405$	−23.3008	−1.15783
$406$	0	0
$407$	0	0
$408$	0	0
$409$	31.5141	1.55827	0.779137	−	0.626854i	$-0.215658\pi$
$409$	31.5141	1.55827	0.779137	+	0.626854i	$0.215658\pi$
$410$	0	0
$411$	29.2630	1.44344
$412$	0	0
$413$	13.1266	0.645916
$414$	0	0
$415$	25.4541	1.24949
$416$	0	0
$417$	32.5306	1.59303
$418$	0	0
$419$	32.6998	1.59749	0.798745	−	0.601670i	$-0.205498\pi$
$419$	32.6998	1.59749	0.798745	+	0.601670i	$0.205498\pi$
$420$	0	0
$421$	20.2825	0.988511	0.494255	−	0.869317i	$-0.335441\pi$
$421$	20.2825	0.988511	0.494255	+	0.869317i	$0.335441\pi$
$422$	0	0
$423$	13.8330	0.672584
$424$	0	0
$425$	20.2318	0.981388
$426$	0	0
$427$	−16.1970	−0.783830
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−2.56165	−0.123390	−0.0616952	−	0.998095i	$-0.519651\pi$
$431$	−2.56165	−0.123390	−0.0616952	+	0.998095i	$0.519651\pi$
$432$	0	0
$433$	10.8684	0.522302	0.261151	−	0.965298i	$-0.415898\pi$
$433$	10.8684	0.522302	0.261151	+	0.965298i	$0.415898\pi$
$434$	0	0
$435$	−1.66415	−0.0797897
$436$	0	0
$437$	1.13474	0.0542818
$438$	0	0
$439$	33.5495	1.60123	0.800616	−	0.599178i	$-0.204506\pi$
$439$	33.5495	1.60123	0.800616	+	0.599178i	$0.204506\pi$
$440$	0	0
$441$	−17.4658	−0.831705
$442$	0	0
$443$	−1.79396	−0.0852338	−0.0426169	−	0.999091i	$-0.513569\pi$
$443$	−1.79396	−0.0852338	−0.0426169	+	0.999091i	$0.513569\pi$
$444$	0	0
$445$	37.9808	1.80046
$446$	0	0
$447$	9.34431	0.441971
$448$	0	0
$449$	7.85776	0.370831	0.185415	−	0.982660i	$-0.440637\pi$
$449$	7.85776	0.370831	0.185415	+	0.982660i	$0.440637\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	25.1084	1.17969
$454$	0	0
$455$	3.98054	0.186611
$456$	0	0
$457$	−19.1139	−0.894109	−0.447055	−	0.894507i	$-0.647527\pi$
$457$	−19.1139	−0.894109	−0.447055	+	0.894507i	$0.647527\pi$
$458$	0	0
$459$	−4.62319	−0.215792
$460$	0	0
$461$	−22.8205	−1.06286	−0.531428	−	0.847104i	$-0.678344\pi$
$461$	−22.8205	−1.06286	−0.531428	+	0.847104i	$0.678344\pi$
$462$	0	0
$463$	22.3939	1.04073	0.520367	−	0.853942i	$-0.325795\pi$
$463$	22.3939	1.04073	0.520367	+	0.853942i	$0.325795\pi$
$464$	0	0
$465$	−25.1119	−1.16453
$466$	0	0
$467$	−10.3247	−0.477771	−0.238886	−	0.971048i	$-0.576782\pi$
$467$	−10.3247	−0.477771	−0.238886	+	0.971048i	$0.576782\pi$
$468$	0	0
$469$	−4.03645	−0.186386
$470$	0	0
$471$	11.7097	0.539556
$472$	0	0
$473$	0	0
$474$	0	0
$475$	2.63391	0.120852
$476$	0	0
$477$	−15.6322	−0.715751
$478$	0	0
$479$	28.6376	1.30848	0.654242	−	0.756285i	$-0.272988\pi$
$479$	28.6376	1.30848	0.654242	+	0.756285i	$0.272988\pi$
$480$	0	0
$481$	6.62484	0.302067
$482$	0	0
$483$	5.61077	0.255299
$484$	0	0
$485$	10.7973	0.490281
$486$	0	0
$487$	5.54414	0.251229	0.125614	−	0.992079i	$-0.459910\pi$
$487$	5.54414	0.251229	0.125614	+	0.992079i	$0.459910\pi$
$488$	0	0
$489$	1.88276	0.0851414
$490$	0	0
$491$	−33.4478	−1.50948	−0.754740	−	0.656024i	$-0.772237\pi$
$491$	−33.4478	−1.50948	−0.754740	+	0.656024i	$0.772237\pi$
$492$	0	0
$493$	1.16071	0.0522755
$494$	0	0
$495$	0	0
$496$	0	0
$497$	19.1133	0.857351
$498$	0	0
$499$	17.3755	0.777835	0.388917	−	0.921273i	$-0.372849\pi$
$499$	17.3755	0.777835	0.388917	+	0.921273i	$0.372849\pi$
$500$	0	0
$501$	−0.420896	−0.0188043
$502$	0	0
$503$	−12.2046	−0.544178	−0.272089	−	0.962272i	$-0.587714\pi$
$503$	−12.2046	−0.544178	−0.272089	+	0.962272i	$0.587714\pi$
$504$	0	0
$505$	−9.16729	−0.407939
$506$	0	0
$507$	−2.52003	−0.111918
$508$	0	0
$509$	−5.71828	−0.253458	−0.126729	−	0.991937i	$-0.540448\pi$
$509$	−5.71828	−0.253458	−0.126729	+	0.991937i	$0.540448\pi$
$510$	0	0
$511$	−18.6718	−0.825990
$512$	0	0
$513$	−0.601877	−0.0265735
$514$	0	0
$515$	37.7812	1.66484
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−14.7020	−0.645345
$520$	0	0
$521$	−40.6700	−1.78178	−0.890892	−	0.454215i	$-0.849920\pi$
$521$	−40.6700	−1.78178	−0.890892	+	0.454215i	$0.849920\pi$
$522$	0	0
$523$	−10.2401	−0.447769	−0.223884	−	0.974616i	$-0.571874\pi$
$523$	−10.2401	−0.447769	−0.223884	+	0.974616i	$0.571874\pi$
$524$	0	0
$525$	13.0235	0.568394
$526$	0	0
$527$	17.5150	0.762964
$528$	0	0
$529$	−20.2263	−0.879402
$530$	0	0
$531$	−32.8990	−1.42770
$532$	0	0
$533$	−8.87303	−0.384334
$534$	0	0
$535$	1.55910	0.0674058
$536$	0	0
$537$	57.1785	2.46743
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−12.6116	−0.542213	−0.271107	−	0.962549i	$-0.587390\pi$
$541$	−12.6116	−0.542213	−0.271107	+	0.962549i	$0.587390\pi$
$542$	0	0
$543$	−44.1643	−1.89527
$544$	0	0
$545$	8.41916	0.360637
$546$	0	0
$547$	−30.4362	−1.30136	−0.650680	−	0.759352i	$-0.725516\pi$
$547$	−30.4362	−1.30136	−0.650680	+	0.759352i	$0.725516\pi$
$548$	0	0
$549$	40.5945	1.73253
$550$	0	0
$551$	0.151108	0.00643743
$552$	0	0
$553$	−3.06185	−0.130203
$554$	0	0
$555$	49.7096	2.11005
$556$	0	0
$557$	3.47081	0.147063	0.0735315	−	0.997293i	$-0.476573\pi$
$557$	3.47081	0.147063	0.0735315	+	0.997293i	$0.476573\pi$
$558$	0	0
$559$	5.13759	0.217297
$560$	0	0
$561$	0	0
$562$	0	0
$563$	39.4034	1.66066	0.830328	−	0.557275i	$-0.188153\pi$
$563$	39.4034	1.66066	0.830328	+	0.557275i	$0.188153\pi$
$564$	0	0
$565$	44.6850	1.87991
$566$	0	0
$567$	10.4615	0.439343
$568$	0	0
$569$	−37.5899	−1.57585	−0.787926	−	0.615770i	$-0.788845\pi$
$569$	−37.5899	−1.57585	−0.787926	+	0.615770i	$0.788845\pi$
$570$	0	0
$571$	42.0519	1.75982	0.879909	−	0.475141i	$-0.157603\pi$
$571$	42.0519	1.75982	0.879909	+	0.475141i	$0.157603\pi$
$572$	0	0
$573$	33.7235	1.40882
$574$	0	0
$575$	6.43833	0.268497
$576$	0	0
$577$	9.86270	0.410590	0.205295	−	0.978700i	$-0.434185\pi$
$577$	9.86270	0.410590	0.205295	+	0.978700i	$0.434185\pi$
$578$	0	0
$579$	−58.0165	−2.41108
$580$	0	0
$581$	−11.4283	−0.474127
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−9.97641	−0.412474
$586$	0	0
$587$	−6.41916	−0.264947	−0.132474	−	0.991187i	$-0.542292\pi$
$587$	−6.41916	−0.264947	−0.132474	+	0.991187i	$0.542292\pi$
$588$	0	0
$589$	2.28021	0.0939546
$590$	0	0
$591$	−29.9658	−1.23263
$592$	0	0
$593$	29.7287	1.22081	0.610405	−	0.792089i	$-0.291007\pi$
$593$	29.7287	1.22081	0.610405	+	0.792089i	$0.291007\pi$
$594$	0	0
$595$	−20.8323	−0.854042
$596$	0	0
$597$	−24.0394	−0.983866
$598$	0	0
$599$	−44.5224	−1.81914	−0.909568	−	0.415555i	$-0.863588\pi$
$599$	−44.5224	−1.81914	−0.909568	+	0.415555i	$0.863588\pi$
$600$	0	0
$601$	−30.5987	−1.24815	−0.624073	−	0.781366i	$-0.714523\pi$
$601$	−30.5987	−1.24815	−0.624073	+	0.781366i	$0.714523\pi$
$602$	0	0
$603$	10.1165	0.411977
$604$	0	0
$605$	0	0
$606$	0	0
$607$	40.2714	1.63456	0.817282	−	0.576238i	$-0.195480\pi$
$607$	40.2714	1.63456	0.817282	+	0.576238i	$0.195480\pi$
$608$	0	0
$609$	0.747164	0.0302766
$610$	0	0
$611$	−4.12859	−0.167025
$612$	0	0
$613$	41.6337	1.68157	0.840785	−	0.541370i	$-0.182094\pi$
$613$	41.6337	1.68157	0.840785	+	0.541370i	$0.182094\pi$
$614$	0	0
$615$	−66.5789	−2.68472
$616$	0	0
$617$	13.9076	0.559899	0.279950	−	0.960015i	$-0.409682\pi$
$617$	13.9076	0.559899	0.279950	+	0.960015i	$0.409682\pi$
$618$	0	0
$619$	19.4204	0.780573	0.390287	−	0.920693i	$-0.372376\pi$
$619$	19.4204	0.780573	0.390287	+	0.920693i	$0.372376\pi$
$620$	0	0
$621$	−1.47123	−0.0590383
$622$	0	0
$623$	−17.0525	−0.683195
$624$	0	0
$625$	−29.3846	−1.17538
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−34.6713	−1.38244
$630$	0	0
$631$	5.79062	0.230521	0.115261	−	0.993335i	$-0.463230\pi$
$631$	5.79062	0.230521	0.115261	+	0.993335i	$0.463230\pi$
$632$	0	0
$633$	19.4253	0.772085
$634$	0	0
$635$	−39.1776	−1.55471
$636$	0	0
$637$	5.21283	0.206540
$638$	0	0
$639$	−47.9036	−1.89504
$640$	0	0
$641$	2.65307	0.104790	0.0523951	−	0.998626i	$-0.483314\pi$
$641$	2.65307	0.104790	0.0523951	+	0.998626i	$0.483314\pi$
$642$	0	0
$643$	−45.2351	−1.78390	−0.891948	−	0.452138i	$-0.850662\pi$
$643$	−45.2351	−1.78390	−0.891948	+	0.452138i	$0.850662\pi$
$644$	0	0
$645$	38.5500	1.51790
$646$	0	0
$647$	−26.2824	−1.03327	−0.516635	−	0.856206i	$-0.672815\pi$
$647$	−26.2824	−1.03327	−0.516635	+	0.856206i	$0.672815\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	11.2747	0.441888
$652$	0	0
$653$	2.01689	0.0789269	0.0394635	−	0.999221i	$-0.487435\pi$
$653$	2.01689	0.0789269	0.0394635	+	0.999221i	$0.487435\pi$
$654$	0	0
$655$	−16.8962	−0.660188
$656$	0	0
$657$	46.7969	1.82572
$658$	0	0
$659$	11.9621	0.465977	0.232988	−	0.972480i	$-0.425150\pi$
$659$	11.9621	0.465977	0.232988	+	0.972480i	$0.425150\pi$
$660$	0	0
$661$	−19.7699	−0.768961	−0.384481	−	0.923133i	$-0.625619\pi$
$661$	−19.7699	−0.768961	−0.384481	+	0.923133i	$0.625619\pi$
$662$	0	0
$663$	13.1887	0.512205
$664$	0	0
$665$	−2.71209	−0.105170
$666$	0	0
$667$	0.369369	0.0143020
$668$	0	0
$669$	50.4703	1.95130
$670$	0	0
$671$	0	0
$672$	0	0
$673$	9.87915	0.380813	0.190407	−	0.981705i	$-0.439019\pi$
$673$	9.87915	0.380813	0.190407	+	0.981705i	$0.439019\pi$
$674$	0	0
$675$	−3.41497	−0.131442
$676$	0	0
$677$	−26.5004	−1.01849	−0.509246	−	0.860621i	$-0.670076\pi$
$677$	−26.5004	−1.01849	−0.509246	+	0.860621i	$0.670076\pi$
$678$	0	0
$679$	−4.84775	−0.186039
$680$	0	0
$681$	4.02826	0.154363
$682$	0	0
$683$	46.7575	1.78913	0.894563	−	0.446941i	$-0.147487\pi$
$683$	46.7575	1.78913	0.894563	+	0.446941i	$0.147487\pi$
$684$	0	0
$685$	−34.5759	−1.32108
$686$	0	0
$687$	−14.8970	−0.568356
$688$	0	0
$689$	4.66558	0.177745
$690$	0	0
$691$	−9.59930	−0.365174	−0.182587	−	0.983190i	$-0.558447\pi$
$691$	−9.59930	−0.365174	−0.182587	+	0.983190i	$0.558447\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−38.4367	−1.45799
$696$	0	0
$697$	46.4373	1.75894
$698$	0	0
$699$	36.9195	1.39642
$700$	0	0
$701$	5.38755	0.203485	0.101743	−	0.994811i	$-0.467558\pi$
$701$	5.38755	0.203485	0.101743	+	0.994811i	$0.467558\pi$
$702$	0	0
$703$	−4.51374	−0.170239
$704$	0	0
$705$	−30.9789	−1.16673
$706$	0	0
$707$	4.11590	0.154794
$708$	0	0
$709$	24.8991	0.935105	0.467552	−	0.883965i	$-0.345136\pi$
$709$	24.8991	0.935105	0.467552	+	0.883965i	$0.345136\pi$
$710$	0	0
$711$	7.67390	0.287794
$712$	0	0
$713$	5.57375	0.208739
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−32.3188	−1.20697
$718$	0	0
$719$	−18.1950	−0.678558	−0.339279	−	0.940686i	$-0.610183\pi$
$719$	−18.1950	−0.678558	−0.339279	+	0.940686i	$0.610183\pi$
$720$	0	0
$721$	−16.9629	−0.631732
$722$	0	0
$723$	7.62223	0.283474
$724$	0	0
$725$	0.857368	0.0318418
$726$	0	0
$727$	11.5896	0.429836	0.214918	−	0.976632i	$-0.431052\pi$
$727$	11.5896	0.429836	0.214918	+	0.976632i	$0.431052\pi$
$728$	0	0
$729$	−32.8981	−1.21845
$730$	0	0
$731$	−26.8878	−0.994480
$732$	0	0
$733$	−0.586534	−0.0216641	−0.0108321	−	0.999941i	$-0.503448\pi$
$733$	−0.586534	−0.0216641	−0.0108321	+	0.999941i	$0.503448\pi$
$734$	0	0
$735$	39.1145	1.44276
$736$	0	0
$737$	0	0
$738$	0	0
$739$	22.1370	0.814321	0.407160	−	0.913357i	$-0.366519\pi$
$739$	22.1370	0.814321	0.407160	+	0.913357i	$0.366519\pi$
$740$	0	0
$741$	1.71699	0.0630750
$742$	0	0
$743$	22.9703	0.842699	0.421350	−	0.906898i	$-0.361557\pi$
$743$	22.9703	0.842699	0.421350	+	0.906898i	$0.361557\pi$
$744$	0	0
$745$	−11.0408	−0.404504
$746$	0	0
$747$	28.6427	1.04798
$748$	0	0
$749$	−0.700000	−0.0255775
$750$	0	0
$751$	51.6781	1.88576	0.942881	−	0.333130i	$-0.108105\pi$
$751$	51.6781	1.88576	0.942881	+	0.333130i	$0.108105\pi$
$752$	0	0
$753$	−35.1184	−1.27979
$754$	0	0
$755$	−29.6669	−1.07969
$756$	0	0
$757$	7.88254	0.286496	0.143248	−	0.989687i	$-0.454245\pi$
$757$	7.88254	0.286496	0.143248	+	0.989687i	$0.454245\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	37.9611	1.37609	0.688044	−	0.725669i	$-0.258469\pi$
$761$	37.9611	1.37609	0.688044	+	0.725669i	$0.258469\pi$
$762$	0	0
$763$	−3.78001	−0.136845
$764$	0	0
$765$	52.2119	1.88772
$766$	0	0
$767$	9.81901	0.354544
$768$	0	0
$769$	28.8464	1.04023	0.520115	−	0.854097i	$-0.325889\pi$
$769$	28.8464	1.04023	0.520115	+	0.854097i	$0.325889\pi$
$770$	0	0
$771$	47.1376	1.69762
$772$	0	0
$773$	−28.9455	−1.04110	−0.520549	−	0.853832i	$-0.674273\pi$
$773$	−28.9455	−1.04110	−0.520549	+	0.853832i	$0.674273\pi$
$774$	0	0
$775$	12.9376	0.464733
$776$	0	0
$777$	−22.3185	−0.800671
$778$	0	0
$779$	6.04552	0.216603
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−0.195918	−0.00700152
$784$	0	0
$785$	−13.8357	−0.493817
$786$	0	0
$787$	−41.0071	−1.46175	−0.730873	−	0.682513i	$-0.760887\pi$
$787$	−41.0071	−1.46175	−0.730873	+	0.682513i	$0.760887\pi$
$788$	0	0
$789$	−47.1551	−1.67877
$790$	0	0
$791$	−20.0625	−0.713342
$792$	0	0
$793$	−12.1158	−0.430245
$794$	0	0
$795$	35.0083	1.24161
$796$	0	0
$797$	−16.2290	−0.574862	−0.287431	−	0.957801i	$-0.592801\pi$
$797$	−16.2290	−0.574862	−0.287431	+	0.957801i	$0.592801\pi$
$798$	0	0
$799$	21.6071	0.764404
$800$	0	0
$801$	42.7386	1.51010
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−6.62942	−0.233657
$806$	0	0
$807$	−45.5555	−1.60363
$808$	0	0
$809$	25.1988	0.885944	0.442972	−	0.896536i	$-0.353924\pi$
$809$	25.1988	0.885944	0.442972	+	0.896536i	$0.353924\pi$
$810$	0	0
$811$	20.5293	0.720880	0.360440	−	0.932782i	$-0.382627\pi$
$811$	20.5293	0.720880	0.360440	+	0.932782i	$0.382627\pi$
$812$	0	0
$813$	−47.8300	−1.67747
$814$	0	0
$815$	−2.22458	−0.0779238
$816$	0	0
$817$	−3.50043	−0.122464
$818$	0	0
$819$	4.47918	0.156515
$820$	0	0
$821$	−23.2060	−0.809894	−0.404947	−	0.914340i	$-0.632710\pi$
$821$	−23.2060	−0.809894	−0.404947	+	0.914340i	$0.632710\pi$
$822$	0	0
$823$	−15.5056	−0.540490	−0.270245	−	0.962792i	$-0.587105\pi$
$823$	−15.5056	−0.540490	−0.270245	+	0.962792i	$0.587105\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−8.50834	−0.295864	−0.147932	−	0.988998i	$-0.547262\pi$
$827$	−8.50834	−0.295864	−0.147932	+	0.988998i	$0.547262\pi$
$828$	0	0
$829$	42.8443	1.48805	0.744023	−	0.668155i	$-0.232916\pi$
$829$	42.8443	1.48805	0.744023	+	0.668155i	$0.232916\pi$
$830$	0	0
$831$	−70.1099	−2.43208
$832$	0	0
$833$	−27.2815	−0.945248
$834$	0	0
$835$	0.497312	0.0172102
$836$	0	0
$837$	−2.95638	−0.102188
$838$	0	0
$839$	50.5829	1.74632	0.873158	−	0.487438i	$-0.162068\pi$
$839$	50.5829	1.74632	0.873158	+	0.487438i	$0.162068\pi$
$840$	0	0
$841$	−28.9508	−0.998304
$842$	0	0
$843$	41.2172	1.41960
$844$	0	0
$845$	2.97755	0.102431
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−51.2946	−1.76043
$850$	0	0
$851$	−11.0334	−0.378220
$852$	0	0
$853$	−18.0788	−0.619007	−0.309503	−	0.950898i	$-0.600163\pi$
$853$	−18.0788	−0.619007	−0.309503	+	0.950898i	$0.600163\pi$
$854$	0	0
$855$	6.79729	0.232462
$856$	0	0
$857$	40.2967	1.37651	0.688254	−	0.725469i	$-0.258377\pi$
$857$	40.2967	1.37651	0.688254	+	0.725469i	$0.258377\pi$
$858$	0	0
$859$	−12.0227	−0.410209	−0.205105	−	0.978740i	$-0.565753\pi$
$859$	−12.0227	−0.410209	−0.205105	+	0.978740i	$0.565753\pi$
$860$	0	0
$861$	29.8924	1.01873
$862$	0	0
$863$	−38.1801	−1.29967	−0.649833	−	0.760077i	$-0.725161\pi$
$863$	−38.1801	−1.29967	−0.649833	+	0.760077i	$0.725161\pi$
$864$	0	0
$865$	17.3712	0.590638
$866$	0	0
$867$	−26.1828	−0.889214
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−3.01937	−0.102308
$872$	0	0
$873$	12.1499	0.411211
$874$	0	0
$875$	4.51470	0.152625
$876$	0	0
$877$	−6.88033	−0.232332	−0.116166	−	0.993230i	$-0.537060\pi$
$877$	−6.88033	−0.232332	−0.116166	+	0.993230i	$0.537060\pi$
$878$	0	0
$879$	−58.1694	−1.96200
$880$	0	0
$881$	3.70403	0.124792	0.0623960	−	0.998051i	$-0.480126\pi$
$881$	3.70403	0.124792	0.0623960	+	0.998051i	$0.480126\pi$
$882$	0	0
$883$	23.8831	0.803729	0.401865	−	0.915699i	$-0.368362\pi$
$883$	23.8831	0.803729	0.401865	+	0.915699i	$0.368362\pi$
$884$	0	0
$885$	73.6770	2.47663
$886$	0	0
$887$	16.8560	0.565969	0.282984	−	0.959125i	$-0.408676\pi$
$887$	16.8560	0.565969	0.282984	+	0.959125i	$0.408676\pi$
$888$	0	0
$889$	17.5898	0.589944
$890$	0	0
$891$	0	0
$892$	0	0
$893$	2.81295	0.0941319
$894$	0	0
$895$	−67.5595	−2.25826
$896$	0	0
$897$	4.19700	0.140134
$898$	0	0
$899$	0.742235	0.0247549
$900$	0	0
$901$	−24.4175	−0.813464
$902$	0	0
$903$	−17.3081	−0.575976
$904$	0	0
$905$	52.1826	1.73461
$906$	0	0
$907$	−35.6320	−1.18314	−0.591570	−	0.806254i	$-0.701492\pi$
$907$	−35.6320	−1.18314	−0.591570	+	0.806254i	$0.701492\pi$
$908$	0	0
$909$	−10.3157	−0.342149
$910$	0	0
$911$	−12.3749	−0.409997	−0.204999	−	0.978762i	$-0.565719\pi$
$911$	−12.3749	−0.409997	−0.204999	+	0.978762i	$0.565719\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−90.9111	−3.00543
$916$	0	0
$917$	7.58599	0.250512
$918$	0	0
$919$	−20.4257	−0.673780	−0.336890	−	0.941544i	$-0.609375\pi$
$919$	−20.4257	−0.673780	−0.336890	+	0.941544i	$0.609375\pi$
$920$	0	0
$921$	−56.0900	−1.84823
$922$	0	0
$923$	14.2973	0.470601
$924$	0	0
$925$	−25.6104	−0.842064
$926$	0	0
$927$	42.5140	1.39634
$928$	0	0
$929$	−39.8950	−1.30891	−0.654456	−	0.756100i	$-0.727102\pi$
$929$	−39.8950	−1.30891	−0.654456	+	0.756100i	$0.727102\pi$
$930$	0	0
$931$	−3.55169	−0.116402
$932$	0	0
$933$	−81.4817	−2.66759
$934$	0	0
$935$	0	0
$936$	0	0
$937$	42.7361	1.39613	0.698064	−	0.716036i	$-0.254045\pi$
$937$	42.7361	1.39613	0.698064	+	0.716036i	$0.254045\pi$
$938$	0	0
$939$	−62.4419	−2.03771
$940$	0	0
$941$	−51.1071	−1.66604	−0.833022	−	0.553241i	$-0.813391\pi$
$941$	−51.1071	−1.66604	−0.833022	+	0.553241i	$0.813391\pi$
$942$	0	0
$943$	14.7777	0.481227
$944$	0	0
$945$	3.51632	0.114386
$946$	0	0
$947$	23.1946	0.753723	0.376862	−	0.926270i	$-0.377003\pi$
$947$	23.1946	0.753723	0.376862	+	0.926270i	$0.377003\pi$
$948$	0	0
$949$	−13.9670	−0.453387
$950$	0	0
$951$	−3.74679	−0.121498
$952$	0	0
$953$	32.6350	1.05715	0.528575	−	0.848887i	$-0.322727\pi$
$953$	32.6350	1.05715	0.528575	+	0.848887i	$0.322727\pi$
$954$	0	0
$955$	−39.8461	−1.28939
$956$	0	0
$957$	0	0
$958$	0	0
$959$	15.5238	0.501289
$960$	0	0
$961$	−19.7997	−0.638701
$962$	0	0
$963$	1.75441	0.0565349
$964$	0	0
$965$	68.5496	2.20669
$966$	0	0
$967$	−54.4605	−1.75133	−0.875665	−	0.482919i	$-0.839577\pi$
$967$	−54.4605	−1.75133	−0.875665	+	0.482919i	$0.839577\pi$
$968$	0	0
$969$	−8.98590	−0.288669
$970$	0	0
$971$	23.0058	0.738290	0.369145	−	0.929372i	$-0.379651\pi$
$971$	23.0058	0.738290	0.369145	+	0.929372i	$0.379651\pi$
$972$	0	0
$973$	17.2572	0.553240
$974$	0	0
$975$	9.74194	0.311992
$976$	0	0
$977$	−19.9406	−0.637955	−0.318978	−	0.947762i	$-0.603339\pi$
$977$	−19.9406	−0.637955	−0.318978	+	0.947762i	$0.603339\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	9.47381	0.302475
$982$	0	0
$983$	57.1310	1.82220	0.911098	−	0.412189i	$-0.135236\pi$
$983$	57.1310	1.82220	0.911098	+	0.412189i	$0.135236\pi$
$984$	0	0
$985$	35.4062	1.12813
$986$	0	0
$987$	13.9088	0.442722
$988$	0	0
$989$	−8.55644	−0.272079
$990$	0	0
$991$	11.1313	0.353598	0.176799	−	0.984247i	$-0.443426\pi$
$991$	11.1313	0.353598	0.176799	+	0.984247i	$0.443426\pi$
$992$	0	0
$993$	−19.6603	−0.623901
$994$	0	0
$995$	28.4038	0.900462
$996$	0	0
$997$	5.64042	0.178634	0.0893170	−	0.996003i	$-0.471532\pi$
$997$	5.64042	0.178634	0.0893170	+	0.996003i	$0.471532\pi$
$998$	0	0
$999$	5.85224	0.185157

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6292.2.a.v.1.2	yes	10
11.10	odd	2		6292.2.a.u.1.2	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6292.2.a.u.1.2	✓	10	11.10	odd	2
6292.2.a.v.1.2	yes	10	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:	\( N \)	\(=\)	\( 6292 = 2^{2} \cdot 11^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6292.a (trivial)

\(f(q)\)	\(=\)	\(q-2.52003 q^{3} +2.97755 q^{5} -1.33685 q^{7} +3.35054 q^{9} +O(q^{10})\)	\(q-2.52003 q^{3} +2.97755 q^{5} -1.33685 q^{7} +3.35054 q^{9} -1.00000 q^{13} -7.50351 q^{15} +5.23353 q^{17} +0.681336 q^{19} +3.36891 q^{21} +1.66546 q^{23} +3.86581 q^{25} -0.883378 q^{27} +0.221782 q^{29} +3.34668 q^{31} -3.98054 q^{35} -6.62484 q^{37} +2.52003 q^{39} +8.87303 q^{41} -5.13759 q^{43} +9.97641 q^{45} +4.12859 q^{47} -5.21283 q^{49} -13.1887 q^{51} -4.66558 q^{53} -1.71699 q^{57} -9.81901 q^{59} +12.1158 q^{61} -4.47918 q^{63} -2.97755 q^{65} +3.01937 q^{67} -4.19700 q^{69} -14.2973 q^{71} +13.9670 q^{73} -9.74194 q^{75} +2.29035 q^{79} -7.82549 q^{81} +8.54868 q^{83} +15.5831 q^{85} -0.558898 q^{87} +12.7557 q^{89} +1.33685 q^{91} -8.43373 q^{93} +2.02871 q^{95} +3.62624 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 2 q^{3} - 6 q^{5} + 4 q^{7} + 12 q^{9}+O(q^{10}) \) 10 * q - 2 * q^3 - 6 * q^5 + 4 * q^7 + 12 * q^9	\( 10 q - 2 q^{3} - 6 q^{5} + 4 q^{7} + 12 q^{9} - 10 q^{13} - 4 q^{15} + 20 q^{17} + 14 q^{19} + 2 q^{21} - 2 q^{23} + 2 q^{25} + 10 q^{27} + 16 q^{29} - 2 q^{31} - 20 q^{35} + 2 q^{39} + 22 q^{41} - 12 q^{43} - 8 q^{45} + 6 q^{47} - 2 q^{49} - 8 q^{51} + 6 q^{53} + 12 q^{57} - 4 q^{59} + 24 q^{61} + 68 q^{63} + 6 q^{65} - 6 q^{67} - 2 q^{69} - 8 q^{71} - 6 q^{73} - 18 q^{75} - 24 q^{79} + 10 q^{81} + 22 q^{83} + 34 q^{85} - 4 q^{87} - 42 q^{89} - 4 q^{91} - 38 q^{93} + 14 q^{97}+O(q^{100}) \) 10 * q - 2 * q^3 - 6 * q^5 + 4 * q^7 + 12 * q^9 - 10 * q^13 - 4 * q^15 + 20 * q^17 + 14 * q^19 + 2 * q^21 - 2 * q^23 + 2 * q^25 + 10 * q^27 + 16 * q^29 - 2 * q^31 - 20 * q^35 + 2 * q^39 + 22 * q^41 - 12 * q^43 - 8 * q^45 + 6 * q^47 - 2 * q^49 - 8 * q^51 + 6 * q^53 + 12 * q^57 - 4 * q^59 + 24 * q^61 + 68 * q^63 + 6 * q^65 - 6 * q^67 - 2 * q^69 - 8 * q^71 - 6 * q^73 - 18 * q^75 - 24 * q^79 + 10 * q^81 + 22 * q^83 + 34 * q^85 - 4 * q^87 - 42 * q^89 - 4 * q^91 - 38 * q^93 + 14 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more