LMFDB - Embedded newform 6292.2.a.u.1.10

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:	$1$
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			1.10
Root			$-3.21342$ 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)

$f(q)$	$=$	$q+3.21342 q^{3} -0.170915 q^{5} -3.45180 q^{7} +7.32609 q^{9} +O(q^{10})$	$q+3.21342 q^{3} -0.170915 q^{5} -3.45180 q^{7} +7.32609 q^{9} +1.00000 q^{13} -0.549222 q^{15} -1.86720 q^{17} -4.48340 q^{19} -11.0921 q^{21} -7.66994 q^{23} -4.97079 q^{25} +13.9016 q^{27} -8.18392 q^{29} -6.08945 q^{31} +0.589964 q^{35} +7.47260 q^{37} +3.21342 q^{39} -4.52128 q^{41} +6.84596 q^{43} -1.25214 q^{45} -4.59225 q^{47} +4.91495 q^{49} -6.00009 q^{51} +11.2126 q^{53} -14.4071 q^{57} +5.42948 q^{59} -10.9496 q^{61} -25.2882 q^{63} -0.170915 q^{65} -10.7591 q^{67} -24.6468 q^{69} +0.589030 q^{71} +0.565405 q^{73} -15.9732 q^{75} -0.460282 q^{79} +22.6933 q^{81} -3.51756 q^{83} +0.319132 q^{85} -26.2984 q^{87} -12.8589 q^{89} -3.45180 q^{91} -19.5680 q^{93} +0.766280 q^{95} -2.97415 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	3.21342	1.85527	0.927635	−	0.373487i	$-0.121838\pi$
$3$	3.21342	1.85527	0.927635	+	0.373487i	$0.121838\pi$
$4$	0	0
$5$	−0.170915	−0.0764354	−0.0382177	−	0.999269i	$-0.512168\pi$
$5$	−0.170915	−0.0764354	−0.0382177	+	0.999269i	$0.512168\pi$
$6$	0	0
$7$	−3.45180	−1.30466	−0.652330	−	0.757935i	$-0.726208\pi$
$7$	−3.45180	−1.30466	−0.652330	+	0.757935i	$0.726208\pi$
$8$	0	0
$9$	7.32609	2.44203
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−0.549222	−0.141808
$16$	0	0
$17$	−1.86720	−0.452862	−0.226431	−	0.974027i	$-0.572706\pi$
$17$	−1.86720	−0.452862	−0.226431	+	0.974027i	$0.572706\pi$
$18$	0	0
$19$	−4.48340	−1.02856	−0.514281	−	0.857621i	$-0.671941\pi$
$19$	−4.48340	−1.02856	−0.514281	+	0.857621i	$0.671941\pi$
$20$	0	0
$21$	−11.0921	−2.42050
$22$	0	0
$23$	−7.66994	−1.59929	−0.799646	−	0.600472i	$-0.794980\pi$
$23$	−7.66994	−1.59929	−0.799646	+	0.600472i	$0.794980\pi$
$24$	0	0
$25$	−4.97079	−0.994158
$26$	0	0
$27$	13.9016	2.67536
$28$	0	0
$29$	−8.18392	−1.51972	−0.759858	−	0.650089i	$-0.774732\pi$
$29$	−8.18392	−1.51972	−0.759858	+	0.650089i	$0.774732\pi$
$30$	0	0
$31$	−6.08945	−1.09370	−0.546849	−	0.837231i	$-0.684173\pi$
$31$	−6.08945	−1.09370	−0.546849	+	0.837231i	$0.684173\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.589964	0.0997222
$36$	0	0
$37$	7.47260	1.22849	0.614244	−	0.789116i	$-0.289461\pi$
$37$	7.47260	1.22849	0.614244	+	0.789116i	$0.289461\pi$
$38$	0	0
$39$	3.21342	0.514560
$40$	0	0
$41$	−4.52128	−0.706105	−0.353053	−	0.935603i	$-0.614856\pi$
$41$	−4.52128	−0.706105	−0.353053	+	0.935603i	$0.614856\pi$
$42$	0	0
$43$	6.84596	1.04400	0.521999	−	0.852946i	$-0.325186\pi$
$43$	6.84596	1.04400	0.521999	+	0.852946i	$0.325186\pi$
$44$	0	0
$45$	−1.25214	−0.186658
$46$	0	0
$47$	−4.59225	−0.669849	−0.334924	−	0.942245i	$-0.608711\pi$
$47$	−4.59225	−0.669849	−0.334924	+	0.942245i	$0.608711\pi$
$48$	0	0
$49$	4.91495	0.702135
$50$	0	0
$51$	−6.00009	−0.840181
$52$	0	0
$53$	11.2126	1.54017	0.770085	−	0.637941i	$-0.220214\pi$
$53$	11.2126	1.54017	0.770085	+	0.637941i	$0.220214\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−14.4071	−1.90826
$58$	0	0
$59$	5.42948	0.706858	0.353429	−	0.935461i	$-0.385016\pi$
$59$	5.42948	0.706858	0.353429	+	0.935461i	$0.385016\pi$
$60$	0	0
$61$	−10.9496	−1.40195	−0.700975	−	0.713186i	$-0.747252\pi$
$61$	−10.9496	−1.40195	−0.700975	+	0.713186i	$0.747252\pi$
$62$	0	0
$63$	−25.2882	−3.18602
$64$	0	0
$65$	−0.170915	−0.0211994
$66$	0	0
$67$	−10.7591	−1.31443	−0.657215	−	0.753703i	$-0.728266\pi$
$67$	−10.7591	−1.31443	−0.657215	+	0.753703i	$0.728266\pi$
$68$	0	0
$69$	−24.6468	−2.96712
$70$	0	0
$71$	0.589030	0.0699050	0.0349525	−	0.999389i	$-0.488872\pi$
$71$	0.589030	0.0699050	0.0349525	+	0.999389i	$0.488872\pi$
$72$	0	0
$73$	0.565405	0.0661757	0.0330878	−	0.999452i	$-0.489466\pi$
$73$	0.565405	0.0661757	0.0330878	+	0.999452i	$0.489466\pi$
$74$	0	0
$75$	−15.9732	−1.84443
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−0.460282	−0.0517858	−0.0258929	−	0.999665i	$-0.508243\pi$
$79$	−0.460282	−0.0517858	−0.0258929	+	0.999665i	$0.508243\pi$
$80$	0	0
$81$	22.6933	2.52148
$82$	0	0
$83$	−3.51756	−0.386103	−0.193051	−	0.981189i	$-0.561838\pi$
$83$	−3.51756	−0.386103	−0.193051	+	0.981189i	$0.561838\pi$
$84$	0	0
$85$	0.319132	0.0346147
$86$	0	0
$87$	−26.2984	−2.81949
$88$	0	0
$89$	−12.8589	−1.36304	−0.681519	−	0.731800i	$-0.738680\pi$
$89$	−12.8589	−1.36304	−0.681519	+	0.731800i	$0.738680\pi$
$90$	0	0
$91$	−3.45180	−0.361847
$92$	0	0
$93$	−19.5680	−2.02911
$94$	0	0
$95$	0.766280	0.0786186
$96$	0	0
$97$	−2.97415	−0.301980	−0.150990	−	0.988535i	$-0.548246\pi$
$97$	−2.97415	−0.301980	−0.150990	+	0.988535i	$0.548246\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1.06282	0.105755	0.0528774	−	0.998601i	$-0.483161\pi$
$101$	1.06282	0.105755	0.0528774	+	0.998601i	$0.483161\pi$
$102$	0	0
$103$	7.58573	0.747444	0.373722	−	0.927541i	$-0.378081\pi$
$103$	7.58573	0.747444	0.373722	+	0.927541i	$0.378081\pi$
$104$	0	0
$105$	1.89581	0.185012
$106$	0	0
$107$	16.7903	1.62318	0.811589	−	0.584229i	$-0.198603\pi$
$107$	16.7903	1.62318	0.811589	+	0.584229i	$0.198603\pi$
$108$	0	0
$109$	−15.6454	−1.49856	−0.749278	−	0.662256i	$-0.769599\pi$
$109$	−15.6454	−1.49856	−0.749278	+	0.662256i	$0.769599\pi$
$110$	0	0
$111$	24.0126	2.27918
$112$	0	0
$113$	7.92403	0.745430	0.372715	−	0.927946i	$-0.378427\pi$
$113$	7.92403	0.745430	0.372715	+	0.927946i	$0.378427\pi$
$114$	0	0
$115$	1.31091	0.122243
$116$	0	0
$117$	7.32609	0.677297
$118$	0	0
$119$	6.44520	0.590830
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−14.5288	−1.31002
$124$	0	0
$125$	1.70416	0.152424
$126$	0	0
$127$	8.57875	0.761241	0.380620	−	0.924731i	$-0.375710\pi$
$127$	8.57875	0.761241	0.380620	+	0.924731i	$0.375710\pi$
$128$	0	0
$129$	21.9990	1.93690
$130$	0	0
$131$	−1.11416	−0.0973443	−0.0486722	−	0.998815i	$-0.515499\pi$
$131$	−1.11416	−0.0973443	−0.0486722	+	0.998815i	$0.515499\pi$
$132$	0	0
$133$	15.4758	1.34192
$134$	0	0
$135$	−2.37598	−0.204492
$136$	0	0
$137$	−20.4432	−1.74658	−0.873290	−	0.487201i	$-0.838018\pi$
$137$	−20.4432	−1.74658	−0.873290	+	0.487201i	$0.838018\pi$
$138$	0	0
$139$	2.96363	0.251371	0.125686	−	0.992070i	$-0.459887\pi$
$139$	2.96363	0.251371	0.125686	+	0.992070i	$0.459887\pi$
$140$	0	0
$141$	−14.7568	−1.24275
$142$	0	0
$143$	0	0
$144$	0	0
$145$	1.39875	0.116160
$146$	0	0
$147$	15.7938	1.30265
$148$	0	0
$149$	−18.7585	−1.53676	−0.768379	−	0.639995i	$-0.778937\pi$
$149$	−18.7585	−1.53676	−0.768379	+	0.639995i	$0.778937\pi$
$150$	0	0
$151$	18.1299	1.47539	0.737696	−	0.675132i	$-0.235914\pi$
$151$	18.1299	1.47539	0.737696	+	0.675132i	$0.235914\pi$
$152$	0	0
$153$	−13.6793	−1.10590
$154$	0	0
$155$	1.04078	0.0835973
$156$	0	0
$157$	−6.14413	−0.490355	−0.245177	−	0.969478i	$-0.578846\pi$
$157$	−6.14413	−0.490355	−0.245177	+	0.969478i	$0.578846\pi$
$158$	0	0
$159$	36.0309	2.85743
$160$	0	0
$161$	26.4751	2.08653
$162$	0	0
$163$	18.0960	1.41739	0.708693	−	0.705517i	$-0.249285\pi$
$163$	18.0960	1.41739	0.708693	+	0.705517i	$0.249285\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	14.7069	1.13805	0.569025	−	0.822320i	$-0.307321\pi$
$167$	14.7069	1.13805	0.569025	+	0.822320i	$0.307321\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−32.8458	−2.51178
$172$	0	0
$173$	−12.4812	−0.948925	−0.474463	−	0.880276i	$-0.657358\pi$
$173$	−12.4812	−0.948925	−0.474463	+	0.880276i	$0.657358\pi$
$174$	0	0
$175$	17.1582	1.29704
$176$	0	0
$177$	17.4472	1.31141
$178$	0	0
$179$	−8.02669	−0.599943	−0.299972	−	0.953948i	$-0.596977\pi$
$179$	−8.02669	−0.599943	−0.299972	+	0.953948i	$0.596977\pi$
$180$	0	0
$181$	5.12070	0.380619	0.190310	−	0.981724i	$-0.439051\pi$
$181$	5.12070	0.380619	0.190310	+	0.981724i	$0.439051\pi$
$182$	0	0
$183$	−35.1856	−2.60100
$184$	0	0
$185$	−1.27718	−0.0939000
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−47.9854	−3.49043
$190$	0	0
$191$	−6.84743	−0.495462	−0.247731	−	0.968829i	$-0.579685\pi$
$191$	−6.84743	−0.495462	−0.247731	+	0.968829i	$0.579685\pi$
$192$	0	0
$193$	21.4825	1.54634	0.773172	−	0.634196i	$-0.218669\pi$
$193$	21.4825	1.54634	0.773172	+	0.634196i	$0.218669\pi$
$194$	0	0
$195$	−0.549222	−0.0393306
$196$	0	0
$197$	−22.3745	−1.59411	−0.797057	−	0.603904i	$-0.793611\pi$
$197$	−22.3745	−1.59411	−0.797057	+	0.603904i	$0.793611\pi$
$198$	0	0
$199$	−10.4206	−0.738698	−0.369349	−	0.929291i	$-0.620419\pi$
$199$	−10.4206	−0.738698	−0.369349	+	0.929291i	$0.620419\pi$
$200$	0	0
$201$	−34.5734	−2.43862
$202$	0	0
$203$	28.2493	1.98271
$204$	0	0
$205$	0.772754	0.0539714
$206$	0	0
$207$	−56.1906	−3.90552
$208$	0	0
$209$	0	0
$210$	0	0
$211$	27.8012	1.91392	0.956958	−	0.290227i	$-0.0937309\pi$
$211$	27.8012	1.91392	0.956958	+	0.290227i	$0.0937309\pi$
$212$	0	0
$213$	1.89280	0.129693
$214$	0	0
$215$	−1.17008	−0.0797985
$216$	0	0
$217$	21.0196	1.42690
$218$	0	0
$219$	1.81689	0.122774
$220$	0	0
$221$	−1.86720	−0.125601
$222$	0	0
$223$	−4.85526	−0.325133	−0.162566	−	0.986698i	$-0.551977\pi$
$223$	−4.85526	−0.325133	−0.162566	+	0.986698i	$0.551977\pi$
$224$	0	0
$225$	−36.4164	−2.42776
$226$	0	0
$227$	21.8200	1.44824	0.724121	−	0.689673i	$-0.242246\pi$
$227$	21.8200	1.44824	0.724121	+	0.689673i	$0.242246\pi$
$228$	0	0
$229$	−28.5815	−1.88872	−0.944359	−	0.328916i	$-0.893317\pi$
$229$	−28.5815	−1.88872	−0.944359	+	0.328916i	$0.893317\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0.309638	0.0202851	0.0101425	−	0.999949i	$-0.496771\pi$
$233$	0.309638	0.0202851	0.0101425	+	0.999949i	$0.496771\pi$
$234$	0	0
$235$	0.784884	0.0512002
$236$	0	0
$237$	−1.47908	−0.0960766
$238$	0	0
$239$	−10.7857	−0.697670	−0.348835	−	0.937184i	$-0.613423\pi$
$239$	−10.7857	−0.697670	−0.348835	+	0.937184i	$0.613423\pi$
$240$	0	0
$241$	26.5412	1.70967	0.854835	−	0.518899i	$-0.173658\pi$
$241$	26.5412	1.70967	0.854835	+	0.518899i	$0.173658\pi$
$242$	0	0
$243$	31.2186	2.00267
$244$	0	0
$245$	−0.840037	−0.0536680
$246$	0	0
$247$	−4.48340	−0.285272
$248$	0	0
$249$	−11.3034	−0.716325
$250$	0	0
$251$	17.1840	1.08464	0.542321	−	0.840171i	$-0.317546\pi$
$251$	17.1840	1.08464	0.542321	+	0.840171i	$0.317546\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	1.02550	0.0642196
$256$	0	0
$257$	24.5037	1.52850	0.764249	−	0.644921i	$-0.223110\pi$
$257$	24.5037	1.52850	0.764249	+	0.644921i	$0.223110\pi$
$258$	0	0
$259$	−25.7939	−1.60276
$260$	0	0
$261$	−59.9561	−3.71119
$262$	0	0
$263$	5.21836	0.321778	0.160889	−	0.986973i	$-0.448564\pi$
$263$	5.21836	0.321778	0.160889	+	0.986973i	$0.448564\pi$
$264$	0	0
$265$	−1.91640	−0.117724
$266$	0	0
$267$	−41.3210	−2.52881
$268$	0	0
$269$	−1.52588	−0.0930346	−0.0465173	−	0.998917i	$-0.514812\pi$
$269$	−1.52588	−0.0930346	−0.0465173	+	0.998917i	$0.514812\pi$
$270$	0	0
$271$	−15.3497	−0.932431	−0.466215	−	0.884671i	$-0.654383\pi$
$271$	−15.3497	−0.932431	−0.466215	+	0.884671i	$0.654383\pi$
$272$	0	0
$273$	−11.0921	−0.671325
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−10.6433	−0.639496	−0.319748	−	0.947503i	$-0.603598\pi$
$277$	−10.6433	−0.639496	−0.319748	+	0.947503i	$0.603598\pi$
$278$	0	0
$279$	−44.6119	−2.67084
$280$	0	0
$281$	−14.9134	−0.889657	−0.444828	−	0.895616i	$-0.646735\pi$
$281$	−14.9134	−0.889657	−0.444828	+	0.895616i	$0.646735\pi$
$282$	0	0
$283$	−3.03706	−0.180535	−0.0902673	−	0.995918i	$-0.528772\pi$
$283$	−3.03706	−0.180535	−0.0902673	+	0.995918i	$0.528772\pi$
$284$	0	0
$285$	2.46238	0.145859
$286$	0	0
$287$	15.6066	0.921227
$288$	0	0
$289$	−13.5136	−0.794916
$290$	0	0
$291$	−9.55721	−0.560254
$292$	0	0
$293$	25.3503	1.48098	0.740489	−	0.672068i	$-0.234594\pi$
$293$	25.3503	1.48098	0.740489	+	0.672068i	$0.234594\pi$
$294$	0	0
$295$	−0.927978	−0.0540290
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−7.66994	−0.443564
$300$	0	0
$301$	−23.6309	−1.36206
$302$	0	0
$303$	3.41530	0.196204
$304$	0	0
$305$	1.87145	0.107159
$306$	0	0
$307$	−17.2415	−0.984024	−0.492012	−	0.870588i	$-0.663738\pi$
$307$	−17.2415	−0.984024	−0.492012	+	0.870588i	$0.663738\pi$
$308$	0	0
$309$	24.3762	1.38671
$310$	0	0
$311$	20.0938	1.13941	0.569707	−	0.821848i	$-0.307057\pi$
$311$	20.0938	1.13941	0.569707	+	0.821848i	$0.307057\pi$
$312$	0	0
$313$	11.7181	0.662344	0.331172	−	0.943570i	$-0.392556\pi$
$313$	11.7181	0.662344	0.331172	+	0.943570i	$0.392556\pi$
$314$	0	0
$315$	4.32213	0.243525
$316$	0	0
$317$	17.4284	0.978878	0.489439	−	0.872037i	$-0.337202\pi$
$317$	17.4284	0.978878	0.489439	+	0.872037i	$0.337202\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	53.9543	3.01144
$322$	0	0
$323$	8.37139	0.465797
$324$	0	0
$325$	−4.97079	−0.275730
$326$	0	0
$327$	−50.2752	−2.78023
$328$	0	0
$329$	15.8515	0.873924
$330$	0	0
$331$	−19.0628	−1.04778	−0.523892	−	0.851785i	$-0.675520\pi$
$331$	−19.0628	−1.04778	−0.523892	+	0.851785i	$0.675520\pi$
$332$	0	0
$333$	54.7449	3.00000
$334$	0	0
$335$	1.83888	0.100469
$336$	0	0
$337$	−11.1802	−0.609025	−0.304512	−	0.952508i	$-0.598493\pi$
$337$	−11.1802	−0.609025	−0.304512	+	0.952508i	$0.598493\pi$
$338$	0	0
$339$	25.4633	1.38297
$340$	0	0
$341$	0	0
$342$	0	0
$343$	7.19719	0.388612
$344$	0	0
$345$	4.21249	0.226793
$346$	0	0
$347$	−11.7760	−0.632169	−0.316084	−	0.948731i	$-0.602368\pi$
$347$	−11.7760	−0.632169	−0.316084	+	0.948731i	$0.602368\pi$
$348$	0	0
$349$	−20.0136	−1.07130	−0.535651	−	0.844440i	$-0.679934\pi$
$349$	−20.0136	−1.07130	−0.535651	+	0.844440i	$0.679934\pi$
$350$	0	0
$351$	13.9016	0.742010
$352$	0	0
$353$	−9.42588	−0.501689	−0.250844	−	0.968027i	$-0.580708\pi$
$353$	−9.42588	−0.501689	−0.250844	+	0.968027i	$0.580708\pi$
$354$	0	0
$355$	−0.100674	−0.00534322
$356$	0	0
$357$	20.7111	1.09615
$358$	0	0
$359$	−22.1309	−1.16803	−0.584013	−	0.811744i	$-0.698518\pi$
$359$	−22.1309	−1.16803	−0.584013	+	0.811744i	$0.698518\pi$
$360$	0	0
$361$	1.10089	0.0579415
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−0.0966361	−0.00505816
$366$	0	0
$367$	−25.9853	−1.35642	−0.678211	−	0.734867i	$-0.737245\pi$
$367$	−25.9853	−1.35642	−0.678211	+	0.734867i	$0.737245\pi$
$368$	0	0
$369$	−33.1233	−1.72433
$370$	0	0
$371$	−38.7037	−2.00940
$372$	0	0
$373$	−31.3513	−1.62331	−0.811655	−	0.584137i	$-0.801433\pi$
$373$	−31.3513	−1.62331	−0.811655	+	0.584137i	$0.801433\pi$
$374$	0	0
$375$	5.47617	0.282788
$376$	0	0
$377$	−8.18392	−0.421493
$378$	0	0
$379$	3.20474	0.164617	0.0823083	−	0.996607i	$-0.473771\pi$
$379$	3.20474	0.164617	0.0823083	+	0.996607i	$0.473771\pi$
$380$	0	0
$381$	27.5671	1.41231
$382$	0	0
$383$	−6.24242	−0.318973	−0.159486	−	0.987200i	$-0.550984\pi$
$383$	−6.24242	−0.318973	−0.159486	+	0.987200i	$0.550984\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	50.1541	2.54948
$388$	0	0
$389$	−24.1234	−1.22310	−0.611551	−	0.791205i	$-0.709454\pi$
$389$	−24.1234	−1.22310	−0.611551	+	0.791205i	$0.709454\pi$
$390$	0	0
$391$	14.3213	0.724258
$392$	0	0
$393$	−3.58026	−0.180600
$394$	0	0
$395$	0.0786690	0.00395827
$396$	0	0
$397$	2.94088	0.147598	0.0737992	−	0.997273i	$-0.476488\pi$
$397$	2.94088	0.147598	0.0737992	+	0.997273i	$0.476488\pi$
$398$	0	0
$399$	49.7304	2.48963
$400$	0	0
$401$	−2.15938	−0.107834	−0.0539171	−	0.998545i	$-0.517171\pi$
$401$	−2.15938	−0.107834	−0.0539171	+	0.998545i	$0.517171\pi$
$402$	0	0
$403$	−6.08945	−0.303337
$404$	0	0
$405$	−3.87862	−0.192730
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−1.10337	−0.0545580	−0.0272790	−	0.999628i	$-0.508684\pi$
$409$	−1.10337	−0.0545580	−0.0272790	+	0.999628i	$0.508684\pi$
$410$	0	0
$411$	−65.6926	−3.24038
$412$	0	0
$413$	−18.7415	−0.922208
$414$	0	0
$415$	0.601203	0.0295119
$416$	0	0
$417$	9.52339	0.466362
$418$	0	0
$419$	23.8509	1.16519	0.582597	−	0.812761i	$-0.302036\pi$
$419$	23.8509	1.16519	0.582597	+	0.812761i	$0.302036\pi$
$420$	0	0
$421$	−14.1016	−0.687270	−0.343635	−	0.939103i	$-0.611658\pi$
$421$	−14.1016	−0.687270	−0.343635	+	0.939103i	$0.611658\pi$
$422$	0	0
$423$	−33.6432	−1.63579
$424$	0	0
$425$	9.28144	0.450216
$426$	0	0
$427$	37.7958	1.82907
$428$	0	0
$429$	0	0
$430$	0	0
$431$	1.07983	0.0520135	0.0260067	−	0.999662i	$-0.491721\pi$
$431$	1.07983	0.0520135	0.0260067	+	0.999662i	$0.491721\pi$
$432$	0	0
$433$	−0.343331	−0.0164994	−0.00824972	−	0.999966i	$-0.502626\pi$
$433$	−0.343331	−0.0164994	−0.00824972	+	0.999966i	$0.502626\pi$
$434$	0	0
$435$	4.49479	0.215509
$436$	0	0
$437$	34.3874	1.64497
$438$	0	0
$439$	−2.10855	−0.100635	−0.0503177	−	0.998733i	$-0.516023\pi$
$439$	−2.10855	−0.100635	−0.0503177	+	0.998733i	$0.516023\pi$
$440$	0	0
$441$	36.0073	1.71464
$442$	0	0
$443$	26.7949	1.27306	0.636532	−	0.771250i	$-0.280368\pi$
$443$	26.7949	1.27306	0.636532	+	0.771250i	$0.280368\pi$
$444$	0	0
$445$	2.19777	0.104184
$446$	0	0
$447$	−60.2791	−2.85110
$448$	0	0
$449$	−39.5264	−1.86537	−0.932683	−	0.360697i	$-0.882539\pi$
$449$	−39.5264	−1.86537	−0.932683	+	0.360697i	$0.882539\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	58.2591	2.73725
$454$	0	0
$455$	0.589964	0.0276580
$456$	0	0
$457$	17.3715	0.812604	0.406302	−	0.913739i	$-0.366818\pi$
$457$	17.3715	0.812604	0.406302	+	0.913739i	$0.366818\pi$
$458$	0	0
$459$	−25.9569	−1.21157
$460$	0	0
$461$	37.1824	1.73176	0.865878	−	0.500256i	$-0.166761\pi$
$461$	37.1824	1.73176	0.865878	+	0.500256i	$0.166761\pi$
$462$	0	0
$463$	21.0138	0.976593	0.488297	−	0.872678i	$-0.337618\pi$
$463$	21.0138	0.976593	0.488297	+	0.872678i	$0.337618\pi$
$464$	0	0
$465$	3.34446	0.155096
$466$	0	0
$467$	14.6826	0.679429	0.339714	−	0.940529i	$-0.389670\pi$
$467$	14.6826	0.679429	0.339714	+	0.940529i	$0.389670\pi$
$468$	0	0
$469$	37.1382	1.71488
$470$	0	0
$471$	−19.7437	−0.909741
$472$	0	0
$473$	0	0
$474$	0	0
$475$	22.2860	1.02255
$476$	0	0
$477$	82.1446	3.76114
$478$	0	0
$479$	−11.2122	−0.512299	−0.256150	−	0.966637i	$-0.582454\pi$
$479$	−11.2122	−0.512299	−0.256150	+	0.966637i	$0.582454\pi$
$480$	0	0
$481$	7.47260	0.340721
$482$	0	0
$483$	85.0757	3.87108
$484$	0	0
$485$	0.508327	0.0230819
$486$	0	0
$487$	−6.95648	−0.315228	−0.157614	−	0.987501i	$-0.550380\pi$
$487$	−6.95648	−0.315228	−0.157614	+	0.987501i	$0.550380\pi$
$488$	0	0
$489$	58.1500	2.62964
$490$	0	0
$491$	22.6966	1.02428	0.512142	−	0.858901i	$-0.328852\pi$
$491$	22.6966	1.02428	0.512142	+	0.858901i	$0.328852\pi$
$492$	0	0
$493$	15.2810	0.688221
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2.03322	−0.0912022
$498$	0	0
$499$	−41.3092	−1.84925	−0.924627	−	0.380874i	$-0.875623\pi$
$499$	−41.3092	−1.84925	−0.924627	+	0.380874i	$0.875623\pi$
$500$	0	0
$501$	47.2593	2.11139
$502$	0	0
$503$	−6.91185	−0.308184	−0.154092	−	0.988057i	$-0.549245\pi$
$503$	−6.91185	−0.308184	−0.154092	+	0.988057i	$0.549245\pi$
$504$	0	0
$505$	−0.181652	−0.00808341
$506$	0	0
$507$	3.21342	0.142713
$508$	0	0
$509$	−40.0058	−1.77323	−0.886613	−	0.462511i	$-0.846949\pi$
$509$	−40.0058	−1.77323	−0.886613	+	0.462511i	$0.846949\pi$
$510$	0	0
$511$	−1.95167	−0.0863367
$512$	0	0
$513$	−62.3263	−2.75177
$514$	0	0
$515$	−1.29651	−0.0571312
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−40.1073	−1.76051
$520$	0	0
$521$	23.9281	1.04831	0.524155	−	0.851623i	$-0.324381\pi$
$521$	23.9281	1.04831	0.524155	+	0.851623i	$0.324381\pi$
$522$	0	0
$523$	5.04650	0.220668	0.110334	−	0.993895i	$-0.464808\pi$
$523$	5.04650	0.220668	0.110334	+	0.993895i	$0.464808\pi$
$524$	0	0
$525$	55.1365	2.40635
$526$	0	0
$527$	11.3702	0.495294
$528$	0	0
$529$	35.8279	1.55774
$530$	0	0
$531$	39.7768	1.72617
$532$	0	0
$533$	−4.52128	−0.195838
$534$	0	0
$535$	−2.86971	−0.124068
$536$	0	0
$537$	−25.7932	−1.11306
$538$	0	0
$539$	0	0
$540$	0	0
$541$	22.6464	0.973643	0.486821	−	0.873501i	$-0.338156\pi$
$541$	22.6464	0.973643	0.486821	+	0.873501i	$0.338156\pi$
$542$	0	0
$543$	16.4550	0.706151
$544$	0	0
$545$	2.67403	0.114543
$546$	0	0
$547$	−43.5664	−1.86276	−0.931382	−	0.364044i	$-0.881396\pi$
$547$	−43.5664	−1.86276	−0.931382	+	0.364044i	$0.881396\pi$
$548$	0	0
$549$	−80.2176	−3.42360
$550$	0	0
$551$	36.6918	1.56312
$552$	0	0
$553$	1.58880	0.0675628
$554$	0	0
$555$	−4.10411	−0.174210
$556$	0	0
$557$	−1.30479	−0.0552856	−0.0276428	−	0.999618i	$-0.508800\pi$
$557$	−1.30479	−0.0552856	−0.0276428	+	0.999618i	$0.508800\pi$
$558$	0	0
$559$	6.84596	0.289553
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−17.8286	−0.751387	−0.375693	−	0.926744i	$-0.622595\pi$
$563$	−17.8286	−0.751387	−0.375693	+	0.926744i	$0.622595\pi$
$564$	0	0
$565$	−1.35433	−0.0569772
$566$	0	0
$567$	−78.3329	−3.28967
$568$	0	0
$569$	19.3343	0.810536	0.405268	−	0.914198i	$-0.367178\pi$
$569$	19.3343	0.810536	0.405268	+	0.914198i	$0.367178\pi$
$570$	0	0
$571$	5.84549	0.244626	0.122313	−	0.992492i	$-0.460969\pi$
$571$	5.84549	0.244626	0.122313	+	0.992492i	$0.460969\pi$
$572$	0	0
$573$	−22.0037	−0.919217
$574$	0	0
$575$	38.1256	1.58995
$576$	0	0
$577$	−8.14028	−0.338884	−0.169442	−	0.985540i	$-0.554197\pi$
$577$	−8.14028	−0.338884	−0.169442	+	0.985540i	$0.554197\pi$
$578$	0	0
$579$	69.0324	2.86889
$580$	0	0
$581$	12.1419	0.503732
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−1.25214	−0.0517695
$586$	0	0
$587$	11.3391	0.468013	0.234006	−	0.972235i	$-0.424816\pi$
$587$	11.3391	0.468013	0.234006	+	0.972235i	$0.424816\pi$
$588$	0	0
$589$	27.3015	1.12494
$590$	0	0
$591$	−71.8986	−2.95751
$592$	0	0
$593$	−34.0428	−1.39797	−0.698984	−	0.715137i	$-0.746364\pi$
$593$	−34.0428	−1.39797	−0.698984	+	0.715137i	$0.746364\pi$
$594$	0	0
$595$	−1.10158	−0.0451604
$596$	0	0
$597$	−33.4859	−1.37049
$598$	0	0
$599$	−0.781579	−0.0319344	−0.0159672	−	0.999873i	$-0.505083\pi$
$599$	−0.781579	−0.0319344	−0.0159672	+	0.999873i	$0.505083\pi$
$600$	0	0
$601$	22.5836	0.921203	0.460601	−	0.887607i	$-0.347634\pi$
$601$	22.5836	0.921203	0.460601	+	0.887607i	$0.347634\pi$
$602$	0	0
$603$	−78.8219	−3.20988
$604$	0	0
$605$	0	0
$606$	0	0
$607$	32.4576	1.31741	0.658706	−	0.752400i	$-0.271104\pi$
$607$	32.4576	1.31741	0.658706	+	0.752400i	$0.271104\pi$
$608$	0	0
$609$	90.7769	3.67847
$610$	0	0
$611$	−4.59225	−0.185783
$612$	0	0
$613$	−1.64261	−0.0663445	−0.0331723	−	0.999450i	$-0.510561\pi$
$613$	−1.64261	−0.0663445	−0.0331723	+	0.999450i	$0.510561\pi$
$614$	0	0
$615$	2.48318	0.100132
$616$	0	0
$617$	16.8508	0.678387	0.339193	−	0.940717i	$-0.389846\pi$
$617$	16.8508	0.678387	0.339193	+	0.940717i	$0.389846\pi$
$618$	0	0
$619$	−45.2114	−1.81720	−0.908600	−	0.417667i	$-0.862848\pi$
$619$	−45.2114	−1.81720	−0.908600	+	0.417667i	$0.862848\pi$
$620$	0	0
$621$	−106.624	−4.27868
$622$	0	0
$623$	44.3863	1.77830
$624$	0	0
$625$	24.5627	0.982507
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−13.9528	−0.556335
$630$	0	0
$631$	−16.0068	−0.637219	−0.318610	−	0.947886i	$-0.603216\pi$
$631$	−16.0068	−0.637219	−0.318610	+	0.947886i	$0.603216\pi$
$632$	0	0
$633$	89.3371	3.55083
$634$	0	0
$635$	−1.46623	−0.0581857
$636$	0	0
$637$	4.91495	0.194737
$638$	0	0
$639$	4.31529	0.170710
$640$	0	0
$641$	−37.0693	−1.46415	−0.732074	−	0.681225i	$-0.761448\pi$
$641$	−37.0693	−1.46415	−0.732074	+	0.681225i	$0.761448\pi$
$642$	0	0
$643$	−27.8613	−1.09874	−0.549372	−	0.835578i	$-0.685133\pi$
$643$	−27.8613	−1.09874	−0.549372	+	0.835578i	$0.685133\pi$
$644$	0	0
$645$	−3.75995	−0.148048
$646$	0	0
$647$	42.3817	1.66620	0.833099	−	0.553125i	$-0.186565\pi$
$647$	42.3817	1.66620	0.833099	+	0.553125i	$0.186565\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	67.5449	2.64729
$652$	0	0
$653$	−46.8326	−1.83270	−0.916351	−	0.400376i	$-0.868880\pi$
$653$	−46.8326	−1.83270	−0.916351	+	0.400376i	$0.868880\pi$
$654$	0	0
$655$	0.190426	0.00744055
$656$	0	0
$657$	4.14221	0.161603
$658$	0	0
$659$	−13.0254	−0.507396	−0.253698	−	0.967284i	$-0.581647\pi$
$659$	−13.0254	−0.507396	−0.253698	+	0.967284i	$0.581647\pi$
$660$	0	0
$661$	40.9311	1.59204	0.796018	−	0.605273i	$-0.206936\pi$
$661$	40.9311	1.59204	0.796018	+	0.605273i	$0.206936\pi$
$662$	0	0
$663$	−6.00009	−0.233024
$664$	0	0
$665$	−2.64505	−0.102571
$666$	0	0
$667$	62.7702	2.43047
$668$	0	0
$669$	−15.6020	−0.603209
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−28.1473	−1.08500	−0.542500	−	0.840056i	$-0.682522\pi$
$673$	−28.1473	−1.08500	−0.542500	+	0.840056i	$0.682522\pi$
$674$	0	0
$675$	−69.1017	−2.65973
$676$	0	0
$677$	−19.6784	−0.756301	−0.378151	−	0.925744i	$-0.623440\pi$
$677$	−19.6784	−0.756301	−0.378151	+	0.925744i	$0.623440\pi$
$678$	0	0
$679$	10.2662	0.393980
$680$	0	0
$681$	70.1168	2.68688
$682$	0	0
$683$	−29.3823	−1.12428	−0.562140	−	0.827042i	$-0.690022\pi$
$683$	−29.3823	−1.12428	−0.562140	+	0.827042i	$0.690022\pi$
$684$	0	0
$685$	3.49404	0.133501
$686$	0	0
$687$	−91.8444	−3.50408
$688$	0	0
$689$	11.2126	0.427166
$690$	0	0
$691$	23.1507	0.880694	0.440347	−	0.897828i	$-0.354856\pi$
$691$	23.1507	0.880694	0.440347	+	0.897828i	$0.354856\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−0.506528	−0.0192137
$696$	0	0
$697$	8.44212	0.319768
$698$	0	0
$699$	0.994998	0.0376343
$700$	0	0
$701$	29.1491	1.10095	0.550473	−	0.834853i	$-0.314447\pi$
$701$	29.1491	1.10095	0.550473	+	0.834853i	$0.314447\pi$
$702$	0	0
$703$	−33.5027	−1.26358
$704$	0	0
$705$	2.52216	0.0949902
$706$	0	0
$707$	−3.66865	−0.137974
$708$	0	0
$709$	24.4793	0.919340	0.459670	−	0.888090i	$-0.347968\pi$
$709$	24.4793	0.919340	0.459670	+	0.888090i	$0.347968\pi$
$710$	0	0
$711$	−3.37207	−0.126462
$712$	0	0
$713$	46.7057	1.74914
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−34.6591	−1.29437
$718$	0	0
$719$	−11.1803	−0.416956	−0.208478	−	0.978027i	$-0.566851\pi$
$719$	−11.1803	−0.416956	−0.208478	+	0.978027i	$0.566851\pi$
$720$	0	0
$721$	−26.1845	−0.975160
$722$	0	0
$723$	85.2882	3.17190
$724$	0	0
$725$	40.6805	1.51084
$726$	0	0
$727$	−29.4341	−1.09165	−0.545826	−	0.837899i	$-0.683784\pi$
$727$	−29.4341	−1.09165	−0.545826	+	0.837899i	$0.683784\pi$
$728$	0	0
$729$	32.2385	1.19402
$730$	0	0
$731$	−12.7827	−0.472787
$732$	0	0
$733$	9.70564	0.358486	0.179243	−	0.983805i	$-0.442635\pi$
$733$	9.70564	0.358486	0.179243	+	0.983805i	$0.442635\pi$
$734$	0	0
$735$	−2.69940	−0.0995687
$736$	0	0
$737$	0	0
$738$	0	0
$739$	22.8236	0.839581	0.419791	−	0.907621i	$-0.362104\pi$
$739$	22.8236	0.839581	0.419791	+	0.907621i	$0.362104\pi$
$740$	0	0
$741$	−14.4071	−0.529257
$742$	0	0
$743$	−1.38016	−0.0506333	−0.0253166	−	0.999679i	$-0.508059\pi$
$743$	−1.38016	−0.0506333	−0.0253166	+	0.999679i	$0.508059\pi$
$744$	0	0
$745$	3.20611	0.117463
$746$	0	0
$747$	−25.7700	−0.942874
$748$	0	0
$749$	−57.9568	−2.11769
$750$	0	0
$751$	6.21365	0.226739	0.113370	−	0.993553i	$-0.463836\pi$
$751$	6.21365	0.226739	0.113370	+	0.993553i	$0.463836\pi$
$752$	0	0
$753$	55.2193	2.01230
$754$	0	0
$755$	−3.09867	−0.112772
$756$	0	0
$757$	−21.7796	−0.791594	−0.395797	−	0.918338i	$-0.629532\pi$
$757$	−21.7796	−0.791594	−0.395797	+	0.918338i	$0.629532\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−11.1685	−0.404856	−0.202428	−	0.979297i	$-0.564883\pi$
$761$	−11.1685	−0.404856	−0.202428	+	0.979297i	$0.564883\pi$
$762$	0	0
$763$	54.0048	1.95510
$764$	0	0
$765$	2.33799	0.0845301
$766$	0	0
$767$	5.42948	0.196047
$768$	0	0
$769$	38.9852	1.40584	0.702921	−	0.711268i	$-0.251879\pi$
$769$	38.9852	1.40584	0.702921	+	0.711268i	$0.251879\pi$
$770$	0	0
$771$	78.7407	2.83578
$772$	0	0
$773$	36.4918	1.31252	0.656259	−	0.754535i	$-0.272138\pi$
$773$	36.4918	1.31252	0.656259	+	0.754535i	$0.272138\pi$
$774$	0	0
$775$	30.2694	1.08731
$776$	0	0
$777$	−82.8869	−2.97355
$778$	0	0
$779$	20.2707	0.726274
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−113.769	−4.06578
$784$	0	0
$785$	1.05012	0.0374805
$786$	0	0
$787$	25.6729	0.915139	0.457569	−	0.889174i	$-0.348720\pi$
$787$	25.6729	0.915139	0.457569	+	0.889174i	$0.348720\pi$
$788$	0	0
$789$	16.7688	0.596985
$790$	0	0
$791$	−27.3522	−0.972532
$792$	0	0
$793$	−10.9496	−0.388831
$794$	0	0
$795$	−6.15821	−0.218409
$796$	0	0
$797$	3.15078	0.111606	0.0558032	−	0.998442i	$-0.482228\pi$
$797$	3.15078	0.111606	0.0558032	+	0.998442i	$0.482228\pi$
$798$	0	0
$799$	8.57464	0.303349
$800$	0	0
$801$	−94.2053	−3.32858
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−4.52499	−0.159485
$806$	0	0
$807$	−4.90330	−0.172604
$808$	0	0
$809$	29.3417	1.03160	0.515800	−	0.856709i	$-0.327495\pi$
$809$	29.3417	1.03160	0.515800	+	0.856709i	$0.327495\pi$
$810$	0	0
$811$	20.1217	0.706569	0.353285	−	0.935516i	$-0.385065\pi$
$811$	20.1217	0.706569	0.353285	+	0.935516i	$0.385065\pi$
$812$	0	0
$813$	−49.3252	−1.72991
$814$	0	0
$815$	−3.09287	−0.108338
$816$	0	0
$817$	−30.6932	−1.07382
$818$	0	0
$819$	−25.2882	−0.883642
$820$	0	0
$821$	−4.68172	−0.163393	−0.0816966	−	0.996657i	$-0.526034\pi$
$821$	−4.68172	−0.163393	−0.0816966	+	0.996657i	$0.526034\pi$
$822$	0	0
$823$	−5.59989	−0.195200	−0.0975999	−	0.995226i	$-0.531117\pi$
$823$	−5.59989	−0.195200	−0.0975999	+	0.995226i	$0.531117\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−17.9201	−0.623144	−0.311572	−	0.950223i	$-0.600856\pi$
$827$	−17.9201	−0.623144	−0.311572	+	0.950223i	$0.600856\pi$
$828$	0	0
$829$	42.5515	1.47787	0.738937	−	0.673774i	$-0.235328\pi$
$829$	42.5515	1.47787	0.738937	+	0.673774i	$0.235328\pi$
$830$	0	0
$831$	−34.2015	−1.18644
$832$	0	0
$833$	−9.17718	−0.317970
$834$	0	0
$835$	−2.51362	−0.0869874
$836$	0	0
$837$	−84.6529	−2.92603
$838$	0	0
$839$	−23.3031	−0.804514	−0.402257	−	0.915527i	$-0.631774\pi$
$839$	−23.3031	−0.804514	−0.402257	+	0.915527i	$0.631774\pi$
$840$	0	0
$841$	37.9766	1.30954
$842$	0	0
$843$	−47.9230	−1.65055
$844$	0	0
$845$	−0.170915	−0.00587965
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−9.75937	−0.334941
$850$	0	0
$851$	−57.3144	−1.96471
$852$	0	0
$853$	−17.3929	−0.595521	−0.297760	−	0.954641i	$-0.596240\pi$
$853$	−17.3929	−0.595521	−0.297760	+	0.954641i	$0.596240\pi$
$854$	0	0
$855$	5.61383	0.191989
$856$	0	0
$857$	−53.3154	−1.82122	−0.910610	−	0.413268i	$-0.864387\pi$
$857$	−53.3154	−1.82122	−0.910610	+	0.413268i	$0.864387\pi$
$858$	0	0
$859$	−42.2093	−1.44016	−0.720081	−	0.693890i	$-0.755895\pi$
$859$	−42.2093	−1.44016	−0.720081	+	0.693890i	$0.755895\pi$
$860$	0	0
$861$	50.1505	1.70912
$862$	0	0
$863$	29.8663	1.01666	0.508331	−	0.861162i	$-0.330263\pi$
$863$	29.8663	1.01666	0.508331	+	0.861162i	$0.330263\pi$
$864$	0	0
$865$	2.13322	0.0725315
$866$	0	0
$867$	−43.4248	−1.47478
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−10.7591	−0.364557
$872$	0	0
$873$	−21.7889	−0.737443
$874$	0	0
$875$	−5.88241	−0.198862
$876$	0	0
$877$	56.9240	1.92219	0.961093	−	0.276225i	$-0.0890834\pi$
$877$	56.9240	1.92219	0.961093	+	0.276225i	$0.0890834\pi$
$878$	0	0
$879$	81.4611	2.74762
$880$	0	0
$881$	5.35988	0.180579	0.0902895	−	0.995916i	$-0.471221\pi$
$881$	5.35988	0.180579	0.0902895	+	0.995916i	$0.471221\pi$
$882$	0	0
$883$	6.70223	0.225548	0.112774	−	0.993621i	$-0.464026\pi$
$883$	6.70223	0.225548	0.112774	+	0.993621i	$0.464026\pi$
$884$	0	0
$885$	−2.98199	−0.100238
$886$	0	0
$887$	15.7327	0.528251	0.264126	−	0.964488i	$-0.414917\pi$
$887$	15.7327	0.528251	0.264126	+	0.964488i	$0.414917\pi$
$888$	0	0
$889$	−29.6121	−0.993160
$890$	0	0
$891$	0	0
$892$	0	0
$893$	20.5889	0.688981
$894$	0	0
$895$	1.37188	0.0458569
$896$	0	0
$897$	−24.6468	−0.822931
$898$	0	0
$899$	49.8356	1.66211
$900$	0	0
$901$	−20.9362	−0.697484
$902$	0	0
$903$	−75.9361	−2.52699
$904$	0	0
$905$	−0.875204	−0.0290928
$906$	0	0
$907$	39.2605	1.30362	0.651812	−	0.758381i	$-0.274009\pi$
$907$	39.2605	1.30362	0.651812	+	0.758381i	$0.274009\pi$
$908$	0	0
$909$	7.78633	0.258256
$910$	0	0
$911$	21.2081	0.702655	0.351327	−	0.936253i	$-0.385730\pi$
$911$	21.2081	0.702655	0.351327	+	0.936253i	$0.385730\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	6.01375	0.198808
$916$	0	0
$917$	3.84585	0.127001
$918$	0	0
$919$	37.3202	1.23108	0.615540	−	0.788106i	$-0.288938\pi$
$919$	37.3202	1.23108	0.615540	+	0.788106i	$0.288938\pi$
$920$	0	0
$921$	−55.4042	−1.82563
$922$	0	0
$923$	0.589030	0.0193882
$924$	0	0
$925$	−37.1447	−1.22131
$926$	0	0
$927$	55.5738	1.82528
$928$	0	0
$929$	50.3085	1.65057	0.825285	−	0.564717i	$-0.191015\pi$
$929$	50.3085	1.65057	0.825285	+	0.564717i	$0.191015\pi$
$930$	0	0
$931$	−22.0357	−0.722190
$932$	0	0
$933$	64.5699	2.11392
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0.496275	0.0162126	0.00810630	−	0.999967i	$-0.497420\pi$
$937$	0.496275	0.0162126	0.00810630	+	0.999967i	$0.497420\pi$
$938$	0	0
$939$	37.6551	1.22883
$940$	0	0
$941$	−36.3922	−1.18635	−0.593176	−	0.805073i	$-0.702126\pi$
$941$	−36.3922	−1.18635	−0.593176	+	0.805073i	$0.702126\pi$
$942$	0	0
$943$	34.6779	1.12927
$944$	0	0
$945$	8.20142	0.266792
$946$	0	0
$947$	−32.7123	−1.06301	−0.531503	−	0.847056i	$-0.678373\pi$
$947$	−32.7123	−1.06301	−0.531503	+	0.847056i	$0.678373\pi$
$948$	0	0
$949$	0.565405	0.0183538
$950$	0	0
$951$	56.0049	1.81608
$952$	0	0
$953$	−47.9486	−1.55321	−0.776603	−	0.629990i	$-0.783059\pi$
$953$	−47.9486	−1.55321	−0.776603	+	0.629990i	$0.783059\pi$
$954$	0	0
$955$	1.17033	0.0378709
$956$	0	0
$957$	0	0
$958$	0	0
$959$	70.5659	2.27869
$960$	0	0
$961$	6.08145	0.196176
$962$	0	0
$963$	123.007	3.96385
$964$	0	0
$965$	−3.67168	−0.118196
$966$	0	0
$967$	26.0356	0.837250	0.418625	−	0.908159i	$-0.362512\pi$
$967$	26.0356	0.837250	0.418625	+	0.908159i	$0.362512\pi$
$968$	0	0
$969$	26.9008	0.864179
$970$	0	0
$971$	−51.0214	−1.63736	−0.818678	−	0.574253i	$-0.805293\pi$
$971$	−51.0214	−1.63736	−0.818678	+	0.574253i	$0.805293\pi$
$972$	0	0
$973$	−10.2299	−0.327954
$974$	0	0
$975$	−15.9732	−0.511553
$976$	0	0
$977$	−35.5900	−1.13863	−0.569313	−	0.822121i	$-0.692791\pi$
$977$	−35.5900	−1.13863	−0.569313	+	0.822121i	$0.692791\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−114.619	−3.65952
$982$	0	0
$983$	−34.2987	−1.09396	−0.546979	−	0.837146i	$-0.684222\pi$
$983$	−34.2987	−1.09396	−0.546979	+	0.837146i	$0.684222\pi$
$984$	0	0
$985$	3.82413	0.121847
$986$	0	0
$987$	50.9377	1.62137
$988$	0	0
$989$	−52.5080	−1.66966
$990$	0	0
$991$	38.9423	1.23704	0.618521	−	0.785768i	$-0.287732\pi$
$991$	38.9423	1.23704	0.618521	+	0.785768i	$0.287732\pi$
$992$	0	0
$993$	−61.2567	−1.94392
$994$	0	0
$995$	1.78104	0.0564627
$996$	0	0
$997$	−54.9901	−1.74155	−0.870777	−	0.491678i	$-0.836384\pi$
$997$	−54.9901	−1.74155	−0.870777	+	0.491678i	$0.836384\pi$
$998$	0	0
$999$	103.881	3.28664

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6292.2.a.u.1.10	✓	10
11.10	odd	2		6292.2.a.v.1.10	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6292.2.a.u.1.10	✓	10	1.1	even	1	trivial
6292.2.a.v.1.10	yes	10	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6292 = 2^{2} \cdot 11^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6292.a (trivial)

\(f(q)\)	\(=\)	\(q+3.21342 q^{3} -0.170915 q^{5} -3.45180 q^{7} +7.32609 q^{9} +O(q^{10})\)	\(q+3.21342 q^{3} -0.170915 q^{5} -3.45180 q^{7} +7.32609 q^{9} +1.00000 q^{13} -0.549222 q^{15} -1.86720 q^{17} -4.48340 q^{19} -11.0921 q^{21} -7.66994 q^{23} -4.97079 q^{25} +13.9016 q^{27} -8.18392 q^{29} -6.08945 q^{31} +0.589964 q^{35} +7.47260 q^{37} +3.21342 q^{39} -4.52128 q^{41} +6.84596 q^{43} -1.25214 q^{45} -4.59225 q^{47} +4.91495 q^{49} -6.00009 q^{51} +11.2126 q^{53} -14.4071 q^{57} +5.42948 q^{59} -10.9496 q^{61} -25.2882 q^{63} -0.170915 q^{65} -10.7591 q^{67} -24.6468 q^{69} +0.589030 q^{71} +0.565405 q^{73} -15.9732 q^{75} -0.460282 q^{79} +22.6933 q^{81} -3.51756 q^{83} +0.319132 q^{85} -26.2984 q^{87} -12.8589 q^{89} -3.45180 q^{91} -19.5680 q^{93} +0.766280 q^{95} -2.97415 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 2 q^{3} - 6 q^{5} - 4 q^{7} + 12 q^{9}+O(q^{10}) \) 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

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Database

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