LMFDB - Embedded newform 8624.2.a.ca.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8624,2,Mod(1,8624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8624, 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("8624.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8624 = 2^{4} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8624.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:	$68.8629867032$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 154)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.64575$ of defining polynomial
Character	$\chi$	$=$	8624.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.64575 q^{3} +3.64575 q^{5} +4.00000 q^{9} +O(q^{10})$	$q+2.64575 q^{3} +3.64575 q^{5} +4.00000 q^{9} -1.00000 q^{11} +5.00000 q^{13} +9.64575 q^{15} +6.00000 q^{17} +0.354249 q^{19} +3.64575 q^{23} +8.29150 q^{25} +2.64575 q^{27} -4.29150 q^{29} +4.00000 q^{31} -2.64575 q^{33} -1.64575 q^{37} +13.2288 q^{39} -4.93725 q^{41} +4.00000 q^{43} +14.5830 q^{45} -13.2915 q^{47} +15.8745 q^{51} -3.64575 q^{53} -3.64575 q^{55} +0.937254 q^{57} +0.645751 q^{59} +3.70850 q^{61} +18.2288 q^{65} -3.93725 q^{67} +9.64575 q^{69} -9.64575 q^{71} +5.64575 q^{73} +21.9373 q^{75} -2.64575 q^{79} -5.00000 q^{81} -13.2915 q^{83} +21.8745 q^{85} -11.3542 q^{87} -14.5830 q^{89} +10.5830 q^{93} +1.29150 q^{95} -5.70850 q^{97} -4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} + 8 q^{9}+O(q^{10}) $ 2 * q + 2 * q^5 + 8 * q^9	$ 2 q + 2 q^{5} + 8 q^{9} - 2 q^{11} + 10 q^{13} + 14 q^{15} + 12 q^{17} + 6 q^{19} + 2 q^{23} + 6 q^{25} + 2 q^{29} + 8 q^{31} + 2 q^{37} + 6 q^{41} + 8 q^{43} + 8 q^{45} - 16 q^{47} - 2 q^{53} - 2 q^{55} - 14 q^{57} - 4 q^{59} + 18 q^{61} + 10 q^{65} + 8 q^{67} + 14 q^{69} - 14 q^{71} + 6 q^{73} + 28 q^{75} - 10 q^{81} - 16 q^{83} + 12 q^{85} - 28 q^{87} - 8 q^{89} - 8 q^{95} - 22 q^{97} - 8 q^{99}+O(q^{100}) $ 2 * q + 2 * q^5 + 8 * q^9 - 2 * q^11 + 10 * q^13 + 14 * q^15 + 12 * q^17 + 6 * q^19 + 2 * q^23 + 6 * q^25 + 2 * q^29 + 8 * q^31 + 2 * q^37 + 6 * q^41 + 8 * q^43 + 8 * q^45 - 16 * q^47 - 2 * q^53 - 2 * q^55 - 14 * q^57 - 4 * q^59 + 18 * q^61 + 10 * q^65 + 8 * q^67 + 14 * q^69 - 14 * q^71 + 6 * q^73 + 28 * q^75 - 10 * q^81 - 16 * q^83 + 12 * q^85 - 28 * q^87 - 8 * q^89 - 8 * q^95 - 22 * q^97 - 8 * 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
$3$	2.64575	1.52753	0.763763	−	0.645497i	$-0.223350\pi$
$3$	2.64575	1.52753	0.763763	+	0.645497i	$0.223350\pi$
$4$	0	0
$5$	3.64575	1.63043	0.815215	−	0.579159i	$-0.196619\pi$
$5$	3.64575	1.63043	0.815215	+	0.579159i	$0.196619\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	4.00000	1.33333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	5.00000	1.38675	0.693375	−	0.720577i	$-0.256123\pi$
$13$	5.00000	1.38675	0.693375	+	0.720577i	$0.256123\pi$
$14$	0	0
$15$	9.64575	2.49052
$16$	0	0
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	0	0
$19$	0.354249	0.0812702	0.0406351	−	0.999174i	$-0.487062\pi$
$19$	0.354249	0.0812702	0.0406351	+	0.999174i	$0.487062\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.64575	0.760192	0.380096	−	0.924947i	$-0.375891\pi$
$23$	3.64575	0.760192	0.380096	+	0.924947i	$0.375891\pi$
$24$	0	0
$25$	8.29150	1.65830
$26$	0	0
$27$	2.64575	0.509175
$28$	0	0
$29$	−4.29150	−0.796912	−0.398456	−	0.917187i	$-0.630454\pi$
$29$	−4.29150	−0.796912	−0.398456	+	0.917187i	$0.630454\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	−2.64575	−0.460566
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−1.64575	−0.270560	−0.135280	−	0.990807i	$-0.543193\pi$
$37$	−1.64575	−0.270560	−0.135280	+	0.990807i	$0.543193\pi$
$38$	0	0
$39$	13.2288	2.11830
$40$	0	0
$41$	−4.93725	−0.771070	−0.385535	−	0.922693i	$-0.625983\pi$
$41$	−4.93725	−0.771070	−0.385535	+	0.922693i	$0.625983\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	14.5830	2.17391
$46$	0	0
$47$	−13.2915	−1.93876	−0.969382	−	0.245556i	$-0.921030\pi$
$47$	−13.2915	−1.93876	−0.969382	+	0.245556i	$0.921030\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	15.8745	2.22288
$52$	0	0
$53$	−3.64575	−0.500782	−0.250391	−	0.968145i	$-0.580559\pi$
$53$	−3.64575	−0.500782	−0.250391	+	0.968145i	$0.580559\pi$
$54$	0	0
$55$	−3.64575	−0.491593
$56$	0	0
$57$	0.937254	0.124142
$58$	0	0
$59$	0.645751	0.0840697	0.0420348	−	0.999116i	$-0.486616\pi$
$59$	0.645751	0.0840697	0.0420348	+	0.999116i	$0.486616\pi$
$60$	0	0
$61$	3.70850	0.474824	0.237412	−	0.971409i	$-0.423701\pi$
$61$	3.70850	0.474824	0.237412	+	0.971409i	$0.423701\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	18.2288	2.26100
$66$	0	0
$67$	−3.93725	−0.481012	−0.240506	−	0.970648i	$-0.577313\pi$
$67$	−3.93725	−0.481012	−0.240506	+	0.970648i	$0.577313\pi$
$68$	0	0
$69$	9.64575	1.16121
$70$	0	0
$71$	−9.64575	−1.14474	−0.572370	−	0.819995i	$-0.693976\pi$
$71$	−9.64575	−1.14474	−0.572370	+	0.819995i	$0.693976\pi$
$72$	0	0
$73$	5.64575	0.660785	0.330393	−	0.943844i	$-0.392819\pi$
$73$	5.64575	0.660785	0.330393	+	0.943844i	$0.392819\pi$
$74$	0	0
$75$	21.9373	2.53310
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−2.64575	−0.297670	−0.148835	−	0.988862i	$-0.547552\pi$
$79$	−2.64575	−0.297670	−0.148835	+	0.988862i	$0.547552\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	−13.2915	−1.45893	−0.729466	−	0.684017i	$-0.760231\pi$
$83$	−13.2915	−1.45893	−0.729466	+	0.684017i	$0.760231\pi$
$84$	0	0
$85$	21.8745	2.37262
$86$	0	0
$87$	−11.3542	−1.21730
$88$	0	0
$89$	−14.5830	−1.54580	−0.772898	−	0.634531i	$-0.781193\pi$
$89$	−14.5830	−1.54580	−0.772898	+	0.634531i	$0.781193\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	10.5830	1.09741
$94$	0	0
$95$	1.29150	0.132505
$96$	0	0
$97$	−5.70850	−0.579610	−0.289805	−	0.957086i	$-0.593590\pi$
$97$	−5.70850	−0.579610	−0.289805	+	0.957086i	$0.593590\pi$
$98$	0	0
$99$	−4.00000	−0.402015
$100$	0	0
$101$	3.00000	0.298511	0.149256	−	0.988799i	$-0.452312\pi$
$101$	3.00000	0.298511	0.149256	+	0.988799i	$0.452312\pi$
$102$	0	0
$103$	−12.9373	−1.27475	−0.637373	−	0.770556i	$-0.719979\pi$
$103$	−12.9373	−1.27475	−0.637373	+	0.770556i	$0.719979\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−4.93725	−0.477302	−0.238651	−	0.971105i	$-0.576705\pi$
$107$	−4.93725	−0.477302	−0.238651	+	0.971105i	$0.576705\pi$
$108$	0	0
$109$	10.5830	1.01367	0.506834	−	0.862044i	$-0.330816\pi$
$109$	10.5830	1.01367	0.506834	+	0.862044i	$0.330816\pi$
$110$	0	0
$111$	−4.35425	−0.413287
$112$	0	0
$113$	−7.70850	−0.725154	−0.362577	−	0.931954i	$-0.618103\pi$
$113$	−7.70850	−0.725154	−0.362577	+	0.931954i	$0.618103\pi$
$114$	0	0
$115$	13.2915	1.23944
$116$	0	0
$117$	20.0000	1.84900
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−13.0627	−1.17783
$124$	0	0
$125$	12.0000	1.07331
$126$	0	0
$127$	−0.0627461	−0.00556781	−0.00278391	−	0.999996i	$-0.500886\pi$
$127$	−0.0627461	−0.00556781	−0.00278391	+	0.999996i	$0.500886\pi$
$128$	0	0
$129$	10.5830	0.931782
$130$	0	0
$131$	−15.6458	−1.36698	−0.683488	−	0.729962i	$-0.739538\pi$
$131$	−15.6458	−1.36698	−0.683488	+	0.729962i	$0.739538\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	9.64575	0.830174
$136$	0	0
$137$	18.8745	1.61256	0.806279	−	0.591535i	$-0.201478\pi$
$137$	18.8745	1.61256	0.806279	+	0.591535i	$0.201478\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	−35.1660	−2.96151
$142$	0	0
$143$	−5.00000	−0.418121
$144$	0	0
$145$	−15.6458	−1.29931
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−4.70850	−0.385735	−0.192868	−	0.981225i	$-0.561779\pi$
$149$	−4.70850	−0.385735	−0.192868	+	0.981225i	$0.561779\pi$
$150$	0	0
$151$	3.35425	0.272965	0.136482	−	0.990642i	$-0.456420\pi$
$151$	3.35425	0.272965	0.136482	+	0.990642i	$0.456420\pi$
$152$	0	0
$153$	24.0000	1.94029
$154$	0	0
$155$	14.5830	1.17134
$156$	0	0
$157$	−21.1660	−1.68923	−0.844616	−	0.535373i	$-0.820171\pi$
$157$	−21.1660	−1.68923	−0.844616	+	0.535373i	$0.820171\pi$
$158$	0	0
$159$	−9.64575	−0.764958
$160$	0	0
$161$	0	0
$162$	0	0
$163$	4.64575	0.363883	0.181942	−	0.983309i	$-0.441762\pi$
$163$	4.64575	0.363883	0.181942	+	0.983309i	$0.441762\pi$
$164$	0	0
$165$	−9.64575	−0.750921
$166$	0	0
$167$	−15.2288	−1.17844	−0.589218	−	0.807974i	$-0.700564\pi$
$167$	−15.2288	−1.17844	−0.589218	+	0.807974i	$0.700564\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	1.41699	0.108360
$172$	0	0
$173$	10.2915	0.782448	0.391224	−	0.920295i	$-0.372052\pi$
$173$	10.2915	0.782448	0.391224	+	0.920295i	$0.372052\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1.70850	0.128419
$178$	0	0
$179$	4.06275	0.303664	0.151832	−	0.988406i	$-0.451483\pi$
$179$	4.06275	0.303664	0.151832	+	0.988406i	$0.451483\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	9.81176	0.725306
$184$	0	0
$185$	−6.00000	−0.441129
$186$	0	0
$187$	−6.00000	−0.438763
$188$	0	0
$189$	0	0
$190$	0	0
$191$	13.2915	0.961739	0.480870	−	0.876792i	$-0.340321\pi$
$191$	13.2915	0.961739	0.480870	+	0.876792i	$0.340321\pi$
$192$	0	0
$193$	−11.5203	−0.829246	−0.414623	−	0.909993i	$-0.636087\pi$
$193$	−11.5203	−0.829246	−0.414623	+	0.909993i	$0.636087\pi$
$194$	0	0
$195$	48.2288	3.45373
$196$	0	0
$197$	18.8745	1.34475	0.672377	−	0.740209i	$-0.265274\pi$
$197$	18.8745	1.34475	0.672377	+	0.740209i	$0.265274\pi$
$198$	0	0
$199$	22.2288	1.57575	0.787877	−	0.615832i	$-0.211180\pi$
$199$	22.2288	1.57575	0.787877	+	0.615832i	$0.211180\pi$
$200$	0	0
$201$	−10.4170	−0.734758
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−18.0000	−1.25717
$206$	0	0
$207$	14.5830	1.01359
$208$	0	0
$209$	−0.354249	−0.0245039
$210$	0	0
$211$	14.9373	1.02832	0.514161	−	0.857693i	$-0.328103\pi$
$211$	14.9373	1.02832	0.514161	+	0.857693i	$0.328103\pi$
$212$	0	0
$213$	−25.5203	−1.74862
$214$	0	0
$215$	14.5830	0.994553
$216$	0	0
$217$	0	0
$218$	0	0
$219$	14.9373	1.00937
$220$	0	0
$221$	30.0000	2.01802
$222$	0	0
$223$	12.3542	0.827302	0.413651	−	0.910436i	$-0.364253\pi$
$223$	12.3542	0.827302	0.413651	+	0.910436i	$0.364253\pi$
$224$	0	0
$225$	33.1660	2.21107
$226$	0	0
$227$	13.2915	0.882188	0.441094	−	0.897461i	$-0.354591\pi$
$227$	13.2915	0.882188	0.441094	+	0.897461i	$0.354591\pi$
$228$	0	0
$229$	−16.0000	−1.05731	−0.528655	−	0.848837i	$-0.677303\pi$
$229$	−16.0000	−1.05731	−0.528655	+	0.848837i	$0.677303\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	16.9373	1.10960	0.554798	−	0.831985i	$-0.312795\pi$
$233$	16.9373	1.10960	0.554798	+	0.831985i	$0.312795\pi$
$234$	0	0
$235$	−48.4575	−3.16102
$236$	0	0
$237$	−7.00000	−0.454699
$238$	0	0
$239$	−9.22876	−0.596959	−0.298479	−	0.954416i	$-0.596479\pi$
$239$	−9.22876	−0.596959	−0.298479	+	0.954416i	$0.596479\pi$
$240$	0	0
$241$	22.8118	1.46943	0.734717	−	0.678373i	$-0.237315\pi$
$241$	22.8118	1.46943	0.734717	+	0.678373i	$0.237315\pi$
$242$	0	0
$243$	−21.1660	−1.35780
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.77124	0.112702
$248$	0	0
$249$	−35.1660	−2.22856
$250$	0	0
$251$	−7.29150	−0.460236	−0.230118	−	0.973163i	$-0.573911\pi$
$251$	−7.29150	−0.460236	−0.230118	+	0.973163i	$0.573911\pi$
$252$	0	0
$253$	−3.64575	−0.229206
$254$	0	0
$255$	57.8745	3.62424
$256$	0	0
$257$	−0.416995	−0.0260114	−0.0130057	−	0.999915i	$-0.504140\pi$
$257$	−0.416995	−0.0260114	−0.0130057	+	0.999915i	$0.504140\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−17.1660	−1.06255
$262$	0	0
$263$	−4.06275	−0.250520	−0.125260	−	0.992124i	$-0.539976\pi$
$263$	−4.06275	−0.250520	−0.125260	+	0.992124i	$0.539976\pi$
$264$	0	0
$265$	−13.2915	−0.816491
$266$	0	0
$267$	−38.5830	−2.36124
$268$	0	0
$269$	26.5830	1.62079	0.810397	−	0.585881i	$-0.199251\pi$
$269$	26.5830	1.62079	0.810397	+	0.585881i	$0.199251\pi$
$270$	0	0
$271$	17.9373	1.08961	0.544805	−	0.838563i	$-0.316604\pi$
$271$	17.9373	1.08961	0.544805	+	0.838563i	$0.316604\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−8.29150	−0.499996
$276$	0	0
$277$	−11.7085	−0.703495	−0.351748	−	0.936095i	$-0.614413\pi$
$277$	−11.7085	−0.703495	−0.351748	+	0.936095i	$0.614413\pi$
$278$	0	0
$279$	16.0000	0.957895
$280$	0	0
$281$	−7.06275	−0.421328	−0.210664	−	0.977559i	$-0.567563\pi$
$281$	−7.06275	−0.421328	−0.210664	+	0.977559i	$0.567563\pi$
$282$	0	0
$283$	7.64575	0.454493	0.227246	−	0.973837i	$-0.427028\pi$
$283$	7.64575	0.454493	0.227246	+	0.973837i	$0.427028\pi$
$284$	0	0
$285$	3.41699	0.202405
$286$	0	0
$287$	0	0
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	−15.1033	−0.885369
$292$	0	0
$293$	12.0000	0.701047	0.350524	−	0.936554i	$-0.386004\pi$
$293$	12.0000	0.701047	0.350524	+	0.936554i	$0.386004\pi$
$294$	0	0
$295$	2.35425	0.137070
$296$	0	0
$297$	−2.64575	−0.153522
$298$	0	0
$299$	18.2288	1.05420
$300$	0	0
$301$	0	0
$302$	0	0
$303$	7.93725	0.455983
$304$	0	0
$305$	13.5203	0.774168
$306$	0	0
$307$	4.22876	0.241348	0.120674	−	0.992692i	$-0.461494\pi$
$307$	4.22876	0.241348	0.120674	+	0.992692i	$0.461494\pi$
$308$	0	0
$309$	−34.2288	−1.94721
$310$	0	0
$311$	−16.9373	−0.960424	−0.480212	−	0.877153i	$-0.659440\pi$
$311$	−16.9373	−0.960424	−0.480212	+	0.877153i	$0.659440\pi$
$312$	0	0
$313$	2.41699	0.136617	0.0683083	−	0.997664i	$-0.478240\pi$
$313$	2.41699	0.136617	0.0683083	+	0.997664i	$0.478240\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	12.0000	0.673987	0.336994	−	0.941507i	$-0.390590\pi$
$317$	12.0000	0.673987	0.336994	+	0.941507i	$0.390590\pi$
$318$	0	0
$319$	4.29150	0.240278
$320$	0	0
$321$	−13.0627	−0.729091
$322$	0	0
$323$	2.12549	0.118266
$324$	0	0
$325$	41.4575	2.29965
$326$	0	0
$327$	28.0000	1.54840
$328$	0	0
$329$	0	0
$330$	0	0
$331$	15.3542	0.843946	0.421973	−	0.906608i	$-0.361338\pi$
$331$	15.3542	0.843946	0.421973	+	0.906608i	$0.361338\pi$
$332$	0	0
$333$	−6.58301	−0.360746
$334$	0	0
$335$	−14.3542	−0.784256
$336$	0	0
$337$	24.9373	1.35842	0.679209	−	0.733945i	$-0.262323\pi$
$337$	24.9373	1.35842	0.679209	+	0.733945i	$0.262323\pi$
$338$	0	0
$339$	−20.3948	−1.10769
$340$	0	0
$341$	−4.00000	−0.216612
$342$	0	0
$343$	0	0
$344$	0	0
$345$	35.1660	1.89327
$346$	0	0
$347$	−26.8118	−1.43933	−0.719665	−	0.694321i	$-0.755705\pi$
$347$	−26.8118	−1.43933	−0.719665	+	0.694321i	$0.755705\pi$
$348$	0	0
$349$	29.8745	1.59915	0.799573	−	0.600569i	$-0.205059\pi$
$349$	29.8745	1.59915	0.799573	+	0.600569i	$0.205059\pi$
$350$	0	0
$351$	13.2288	0.706099
$352$	0	0
$353$	−35.1660	−1.87170	−0.935849	−	0.352401i	$-0.885365\pi$
$353$	−35.1660	−1.87170	−0.935849	+	0.352401i	$0.885365\pi$
$354$	0	0
$355$	−35.1660	−1.86642
$356$	0	0
$357$	0	0
$358$	0	0
$359$	10.0627	0.531091	0.265546	−	0.964098i	$-0.414448\pi$
$359$	10.0627	0.531091	0.265546	+	0.964098i	$0.414448\pi$
$360$	0	0
$361$	−18.8745	−0.993395
$362$	0	0
$363$	2.64575	0.138866
$364$	0	0
$365$	20.5830	1.07736
$366$	0	0
$367$	9.77124	0.510055	0.255027	−	0.966934i	$-0.417916\pi$
$367$	9.77124	0.510055	0.255027	+	0.966934i	$0.417916\pi$
$368$	0	0
$369$	−19.7490	−1.02809
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−10.8745	−0.563061	−0.281530	−	0.959552i	$-0.590842\pi$
$373$	−10.8745	−0.563061	−0.281530	+	0.959552i	$0.590842\pi$
$374$	0	0
$375$	31.7490	1.63951
$376$	0	0
$377$	−21.4575	−1.10512
$378$	0	0
$379$	−21.9373	−1.12684	−0.563421	−	0.826170i	$-0.690515\pi$
$379$	−21.9373	−1.12684	−0.563421	+	0.826170i	$0.690515\pi$
$380$	0	0
$381$	−0.166010	−0.00850497
$382$	0	0
$383$	34.1033	1.74260	0.871298	−	0.490755i	$-0.163279\pi$
$383$	34.1033	1.74260	0.871298	+	0.490755i	$0.163279\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	16.0000	0.813326
$388$	0	0
$389$	−20.8118	−1.05520	−0.527599	−	0.849493i	$-0.676908\pi$
$389$	−20.8118	−1.05520	−0.527599	+	0.849493i	$0.676908\pi$
$390$	0	0
$391$	21.8745	1.10624
$392$	0	0
$393$	−41.3948	−2.08809
$394$	0	0
$395$	−9.64575	−0.485330
$396$	0	0
$397$	31.1660	1.56418	0.782089	−	0.623167i	$-0.214154\pi$
$397$	31.1660	1.56418	0.782089	+	0.623167i	$0.214154\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0.416995	0.0208237	0.0104119	−	0.999946i	$-0.496686\pi$
$401$	0.416995	0.0208237	0.0104119	+	0.999946i	$0.496686\pi$
$402$	0	0
$403$	20.0000	0.996271
$404$	0	0
$405$	−18.2288	−0.905794
$406$	0	0
$407$	1.64575	0.0815769
$408$	0	0
$409$	18.9373	0.936387	0.468193	−	0.883626i	$-0.344905\pi$
$409$	18.9373	0.936387	0.468193	+	0.883626i	$0.344905\pi$
$410$	0	0
$411$	49.9373	2.46322
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−48.4575	−2.37869
$416$	0	0
$417$	10.5830	0.518252
$418$	0	0
$419$	−21.8745	−1.06864	−0.534320	−	0.845282i	$-0.679432\pi$
$419$	−21.8745	−1.06864	−0.534320	+	0.845282i	$0.679432\pi$
$420$	0	0
$421$	−33.1660	−1.61641	−0.808206	−	0.588900i	$-0.799561\pi$
$421$	−33.1660	−1.61641	−0.808206	+	0.588900i	$0.799561\pi$
$422$	0	0
$423$	−53.1660	−2.58502
$424$	0	0
$425$	49.7490	2.41318
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−13.2288	−0.638690
$430$	0	0
$431$	−2.77124	−0.133486	−0.0667430	−	0.997770i	$-0.521261\pi$
$431$	−2.77124	−0.133486	−0.0667430	+	0.997770i	$0.521261\pi$
$432$	0	0
$433$	−16.0000	−0.768911	−0.384455	−	0.923144i	$-0.625611\pi$
$433$	−16.0000	−0.768911	−0.384455	+	0.923144i	$0.625611\pi$
$434$	0	0
$435$	−41.3948	−1.98473
$436$	0	0
$437$	1.29150	0.0617809
$438$	0	0
$439$	11.9373	0.569734	0.284867	−	0.958567i	$-0.408051\pi$
$439$	11.9373	0.569734	0.284867	+	0.958567i	$0.408051\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	18.4575	0.876943	0.438471	−	0.898745i	$-0.355520\pi$
$443$	18.4575	0.876943	0.438471	+	0.898745i	$0.355520\pi$
$444$	0	0
$445$	−53.1660	−2.52031
$446$	0	0
$447$	−12.4575	−0.589220
$448$	0	0
$449$	9.87451	0.466007	0.233003	−	0.972476i	$-0.425145\pi$
$449$	9.87451	0.466007	0.233003	+	0.972476i	$0.425145\pi$
$450$	0	0
$451$	4.93725	0.232486
$452$	0	0
$453$	8.87451	0.416961
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−39.1660	−1.83211	−0.916054	−	0.401054i	$-0.868644\pi$
$457$	−39.1660	−1.83211	−0.916054	+	0.401054i	$0.868644\pi$
$458$	0	0
$459$	15.8745	0.740959
$460$	0	0
$461$	−32.1660	−1.49812	−0.749060	−	0.662502i	$-0.769495\pi$
$461$	−32.1660	−1.49812	−0.749060	+	0.662502i	$0.769495\pi$
$462$	0	0
$463$	22.4575	1.04369	0.521845	−	0.853041i	$-0.325244\pi$
$463$	22.4575	1.04369	0.521845	+	0.853041i	$0.325244\pi$
$464$	0	0
$465$	38.5830	1.78924
$466$	0	0
$467$	−10.7085	−0.495530	−0.247765	−	0.968820i	$-0.579696\pi$
$467$	−10.7085	−0.495530	−0.247765	+	0.968820i	$0.579696\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−56.0000	−2.58034
$472$	0	0
$473$	−4.00000	−0.183920
$474$	0	0
$475$	2.93725	0.134770
$476$	0	0
$477$	−14.5830	−0.667710
$478$	0	0
$479$	25.9373	1.18510	0.592552	−	0.805532i	$-0.298121\pi$
$479$	25.9373	1.18510	0.592552	+	0.805532i	$0.298121\pi$
$480$	0	0
$481$	−8.22876	−0.375199
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−20.8118	−0.945013
$486$	0	0
$487$	30.5830	1.38585	0.692924	−	0.721011i	$-0.256322\pi$
$487$	30.5830	1.38585	0.692924	+	0.721011i	$0.256322\pi$
$488$	0	0
$489$	12.2915	0.555841
$490$	0	0
$491$	−10.7085	−0.483268	−0.241634	−	0.970367i	$-0.577683\pi$
$491$	−10.7085	−0.483268	−0.241634	+	0.970367i	$0.577683\pi$
$492$	0	0
$493$	−25.7490	−1.15968
$494$	0	0
$495$	−14.5830	−0.655457
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−17.8745	−0.800173	−0.400086	−	0.916477i	$-0.631020\pi$
$499$	−17.8745	−0.800173	−0.400086	+	0.916477i	$0.631020\pi$
$500$	0	0
$501$	−40.2915	−1.80009
$502$	0	0
$503$	19.9373	0.888958	0.444479	−	0.895789i	$-0.353389\pi$
$503$	19.9373	0.888958	0.444479	+	0.895789i	$0.353389\pi$
$504$	0	0
$505$	10.9373	0.486701
$506$	0	0
$507$	31.7490	1.41002
$508$	0	0
$509$	20.5830	0.912326	0.456163	−	0.889896i	$-0.349223\pi$
$509$	20.5830	0.912326	0.456163	+	0.889896i	$0.349223\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0.937254	0.0413808
$514$	0	0
$515$	−47.1660	−2.07838
$516$	0	0
$517$	13.2915	0.584560
$518$	0	0
$519$	27.2288	1.19521
$520$	0	0
$521$	−2.12549	−0.0931195	−0.0465598	−	0.998916i	$-0.514826\pi$
$521$	−2.12549	−0.0931195	−0.0465598	+	0.998916i	$0.514826\pi$
$522$	0	0
$523$	−15.5203	−0.678654	−0.339327	−	0.940669i	$-0.610199\pi$
$523$	−15.5203	−0.678654	−0.339327	+	0.940669i	$0.610199\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	24.0000	1.04546
$528$	0	0
$529$	−9.70850	−0.422109
$530$	0	0
$531$	2.58301	0.112093
$532$	0	0
$533$	−24.6863	−1.06928
$534$	0	0
$535$	−18.0000	−0.778208
$536$	0	0
$537$	10.7490	0.463854
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−8.29150	−0.356480	−0.178240	−	0.983987i	$-0.557040\pi$
$541$	−8.29150	−0.356480	−0.178240	+	0.983987i	$0.557040\pi$
$542$	0	0
$543$	−26.4575	−1.13540
$544$	0	0
$545$	38.5830	1.65271
$546$	0	0
$547$	−27.5203	−1.17668	−0.588341	−	0.808613i	$-0.700219\pi$
$547$	−27.5203	−1.17668	−0.588341	+	0.808613i	$0.700219\pi$
$548$	0	0
$549$	14.8340	0.633099
$550$	0	0
$551$	−1.52026	−0.0647652
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−15.8745	−0.673835
$556$	0	0
$557$	31.7490	1.34525	0.672624	−	0.739984i	$-0.265167\pi$
$557$	31.7490	1.34525	0.672624	+	0.739984i	$0.265167\pi$
$558$	0	0
$559$	20.0000	0.845910
$560$	0	0
$561$	−15.8745	−0.670222
$562$	0	0
$563$	−13.0627	−0.550529	−0.275265	−	0.961369i	$-0.588765\pi$
$563$	−13.0627	−0.550529	−0.275265	+	0.961369i	$0.588765\pi$
$564$	0	0
$565$	−28.1033	−1.18231
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−41.1660	−1.72577	−0.862884	−	0.505401i	$-0.831344\pi$
$569$	−41.1660	−1.72577	−0.862884	+	0.505401i	$0.831344\pi$
$570$	0	0
$571$	44.9373	1.88057	0.940283	−	0.340394i	$-0.110561\pi$
$571$	44.9373	1.88057	0.940283	+	0.340394i	$0.110561\pi$
$572$	0	0
$573$	35.1660	1.46908
$574$	0	0
$575$	30.2288	1.26063
$576$	0	0
$577$	−25.4575	−1.05981	−0.529905	−	0.848057i	$-0.677772\pi$
$577$	−25.4575	−1.05981	−0.529905	+	0.848057i	$0.677772\pi$
$578$	0	0
$579$	−30.4797	−1.26669
$580$	0	0
$581$	0	0
$582$	0	0
$583$	3.64575	0.150992
$584$	0	0
$585$	72.9150	3.01467
$586$	0	0
$587$	7.93725	0.327606	0.163803	−	0.986493i	$-0.447624\pi$
$587$	7.93725	0.327606	0.163803	+	0.986493i	$0.447624\pi$
$588$	0	0
$589$	1.41699	0.0583863
$590$	0	0
$591$	49.9373	2.05414
$592$	0	0
$593$	22.9373	0.941920	0.470960	−	0.882155i	$-0.343908\pi$
$593$	22.9373	0.941920	0.470960	+	0.882155i	$0.343908\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	58.8118	2.40701
$598$	0	0
$599$	19.7490	0.806923	0.403461	−	0.914997i	$-0.367807\pi$
$599$	19.7490	0.806923	0.403461	+	0.914997i	$0.367807\pi$
$600$	0	0
$601$	−24.5830	−1.00276	−0.501381	−	0.865227i	$-0.667174\pi$
$601$	−24.5830	−1.00276	−0.501381	+	0.865227i	$0.667174\pi$
$602$	0	0
$603$	−15.7490	−0.641350
$604$	0	0
$605$	3.64575	0.148221
$606$	0	0
$607$	−21.2915	−0.864195	−0.432098	−	0.901827i	$-0.642226\pi$
$607$	−21.2915	−0.864195	−0.432098	+	0.901827i	$0.642226\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−66.4575	−2.68858
$612$	0	0
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	0	0
$615$	−47.6235	−1.92037
$616$	0	0
$617$	−16.2915	−0.655871	−0.327936	−	0.944700i	$-0.606353\pi$
$617$	−16.2915	−0.655871	−0.327936	+	0.944700i	$0.606353\pi$
$618$	0	0
$619$	−34.5830	−1.39001	−0.695004	−	0.719006i	$-0.744597\pi$
$619$	−34.5830	−1.39001	−0.695004	+	0.719006i	$0.744597\pi$
$620$	0	0
$621$	9.64575	0.387071
$622$	0	0
$623$	0	0
$624$	0	0
$625$	2.29150	0.0916601
$626$	0	0
$627$	−0.937254	−0.0374303
$628$	0	0
$629$	−9.87451	−0.393722
$630$	0	0
$631$	−34.8118	−1.38583	−0.692917	−	0.721017i	$-0.743675\pi$
$631$	−34.8118	−1.38583	−0.692917	+	0.721017i	$0.743675\pi$
$632$	0	0
$633$	39.5203	1.57079
$634$	0	0
$635$	−0.228757	−0.00907793
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−38.5830	−1.52632
$640$	0	0
$641$	−18.8745	−0.745498	−0.372749	−	0.927932i	$-0.621585\pi$
$641$	−18.8745	−0.745498	−0.372749	+	0.927932i	$0.621585\pi$
$642$	0	0
$643$	−6.52026	−0.257134	−0.128567	−	0.991701i	$-0.541038\pi$
$643$	−6.52026	−0.257134	−0.128567	+	0.991701i	$0.541038\pi$
$644$	0	0
$645$	38.5830	1.51920
$646$	0	0
$647$	−38.8118	−1.52585	−0.762924	−	0.646488i	$-0.776237\pi$
$647$	−38.8118	−1.52585	−0.762924	+	0.646488i	$0.776237\pi$
$648$	0	0
$649$	−0.645751	−0.0253480
$650$	0	0
$651$	0	0
$652$	0	0
$653$	2.35425	0.0921289	0.0460644	−	0.998938i	$-0.485332\pi$
$653$	2.35425	0.0921289	0.0460644	+	0.998938i	$0.485332\pi$
$654$	0	0
$655$	−57.0405	−2.22876
$656$	0	0
$657$	22.5830	0.881047
$658$	0	0
$659$	−14.5830	−0.568073	−0.284037	−	0.958813i	$-0.591674\pi$
$659$	−14.5830	−0.568073	−0.284037	+	0.958813i	$0.591674\pi$
$660$	0	0
$661$	28.5830	1.11175	0.555875	−	0.831266i	$-0.312383\pi$
$661$	28.5830	1.11175	0.555875	+	0.831266i	$0.312383\pi$
$662$	0	0
$663$	79.3725	3.08257
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−15.6458	−0.605806
$668$	0	0
$669$	32.6863	1.26372
$670$	0	0
$671$	−3.70850	−0.143165
$672$	0	0
$673$	−14.9373	−0.575789	−0.287894	−	0.957662i	$-0.592955\pi$
$673$	−14.9373	−0.575789	−0.287894	+	0.957662i	$0.592955\pi$
$674$	0	0
$675$	21.9373	0.844365
$676$	0	0
$677$	−2.12549	−0.0816893	−0.0408446	−	0.999166i	$-0.513005\pi$
$677$	−2.12549	−0.0816893	−0.0408446	+	0.999166i	$0.513005\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	35.1660	1.34756
$682$	0	0
$683$	13.9373	0.533294	0.266647	−	0.963794i	$-0.414084\pi$
$683$	13.9373	0.533294	0.266647	+	0.963794i	$0.414084\pi$
$684$	0	0
$685$	68.8118	2.62916
$686$	0	0
$687$	−42.3320	−1.61507
$688$	0	0
$689$	−18.2288	−0.694460
$690$	0	0
$691$	18.7712	0.714092	0.357046	−	0.934087i	$-0.383784\pi$
$691$	18.7712	0.714092	0.357046	+	0.934087i	$0.383784\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	14.5830	0.553165
$696$	0	0
$697$	−29.6235	−1.12207
$698$	0	0
$699$	44.8118	1.69494
$700$	0	0
$701$	−6.87451	−0.259647	−0.129823	−	0.991537i	$-0.541441\pi$
$701$	−6.87451	−0.259647	−0.129823	+	0.991537i	$0.541441\pi$
$702$	0	0
$703$	−0.583005	−0.0219885
$704$	0	0
$705$	−128.207	−4.82854
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−6.81176	−0.255821	−0.127911	−	0.991786i	$-0.540827\pi$
$709$	−6.81176	−0.255821	−0.127911	+	0.991786i	$0.540827\pi$
$710$	0	0
$711$	−10.5830	−0.396894
$712$	0	0
$713$	14.5830	0.546138
$714$	0	0
$715$	−18.2288	−0.681717
$716$	0	0
$717$	−24.4170	−0.911869
$718$	0	0
$719$	3.87451	0.144495	0.0722474	−	0.997387i	$-0.476983\pi$
$719$	3.87451	0.144495	0.0722474	+	0.997387i	$0.476983\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	60.3542	2.24460
$724$	0	0
$725$	−35.5830	−1.32152
$726$	0	0
$727$	17.2915	0.641306	0.320653	−	0.947197i	$-0.396098\pi$
$727$	17.2915	0.641306	0.320653	+	0.947197i	$0.396098\pi$
$728$	0	0
$729$	−41.0000	−1.51852
$730$	0	0
$731$	24.0000	0.887672
$732$	0	0
$733$	41.4575	1.53127	0.765634	−	0.643276i	$-0.222425\pi$
$733$	41.4575	1.53127	0.765634	+	0.643276i	$0.222425\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.93725	0.145031
$738$	0	0
$739$	7.87451	0.289668	0.144834	−	0.989456i	$-0.453735\pi$
$739$	7.87451	0.289668	0.144834	+	0.989456i	$0.453735\pi$
$740$	0	0
$741$	4.68627	0.172154
$742$	0	0
$743$	34.7085	1.27333	0.636666	−	0.771140i	$-0.280313\pi$
$743$	34.7085	1.27333	0.636666	+	0.771140i	$0.280313\pi$
$744$	0	0
$745$	−17.1660	−0.628914
$746$	0	0
$747$	−53.1660	−1.94524
$748$	0	0
$749$	0	0
$750$	0	0
$751$	16.0000	0.583848	0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000	0.583848	0.291924	+	0.956441i	$0.405705\pi$
$752$	0	0
$753$	−19.2915	−0.703021
$754$	0	0
$755$	12.2288	0.445050
$756$	0	0
$757$	19.1660	0.696600	0.348300	−	0.937383i	$-0.386759\pi$
$757$	19.1660	0.696600	0.348300	+	0.937383i	$0.386759\pi$
$758$	0	0
$759$	−9.64575	−0.350119
$760$	0	0
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	87.4980	3.16350
$766$	0	0
$767$	3.22876	0.116584
$768$	0	0
$769$	−15.1660	−0.546900	−0.273450	−	0.961886i	$-0.588165\pi$
$769$	−15.1660	−0.546900	−0.273450	+	0.961886i	$0.588165\pi$
$770$	0	0
$771$	−1.10326	−0.0397331
$772$	0	0
$773$	2.58301	0.0929042	0.0464521	−	0.998921i	$-0.485209\pi$
$773$	2.58301	0.0929042	0.0464521	+	0.998921i	$0.485209\pi$
$774$	0	0
$775$	33.1660	1.19136
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1.74902	−0.0626650
$780$	0	0
$781$	9.64575	0.345152
$782$	0	0
$783$	−11.3542	−0.405768
$784$	0	0
$785$	−77.1660	−2.75417
$786$	0	0
$787$	0.811762	0.0289362	0.0144681	−	0.999895i	$-0.495395\pi$
$787$	0.811762	0.0289362	0.0144681	+	0.999895i	$0.495395\pi$
$788$	0	0
$789$	−10.7490	−0.382675
$790$	0	0
$791$	0	0
$792$	0	0
$793$	18.5425	0.658463
$794$	0	0
$795$	−35.1660	−1.24721
$796$	0	0
$797$	−35.1660	−1.24564	−0.622822	−	0.782364i	$-0.714014\pi$
$797$	−35.1660	−1.24564	−0.622822	+	0.782364i	$0.714014\pi$
$798$	0	0
$799$	−79.7490	−2.82132
$800$	0	0
$801$	−58.3320	−2.06106
$802$	0	0
$803$	−5.64575	−0.199234
$804$	0	0
$805$	0	0
$806$	0	0
$807$	70.3320	2.47580
$808$	0	0
$809$	50.5830	1.77840	0.889202	−	0.457515i	$-0.151260\pi$
$809$	50.5830	1.77840	0.889202	+	0.457515i	$0.151260\pi$
$810$	0	0
$811$	−27.7490	−0.974400	−0.487200	−	0.873290i	$-0.661982\pi$
$811$	−27.7490	−0.974400	−0.487200	+	0.873290i	$0.661982\pi$
$812$	0	0
$813$	47.4575	1.66441
$814$	0	0
$815$	16.9373	0.593286
$816$	0	0
$817$	1.41699	0.0495744
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−7.70850	−0.269028	−0.134514	−	0.990912i	$-0.542947\pi$
$821$	−7.70850	−0.269028	−0.134514	+	0.990912i	$0.542947\pi$
$822$	0	0
$823$	43.8745	1.52937	0.764685	−	0.644405i	$-0.222895\pi$
$823$	43.8745	1.52937	0.764685	+	0.644405i	$0.222895\pi$
$824$	0	0
$825$	−21.9373	−0.763757
$826$	0	0
$827$	35.3948	1.23080	0.615398	−	0.788216i	$-0.288995\pi$
$827$	35.3948	1.23080	0.615398	+	0.788216i	$0.288995\pi$
$828$	0	0
$829$	43.3948	1.50716	0.753581	−	0.657355i	$-0.228325\pi$
$829$	43.3948	1.50716	0.753581	+	0.657355i	$0.228325\pi$
$830$	0	0
$831$	−30.9778	−1.07461
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−55.5203	−1.92136
$836$	0	0
$837$	10.5830	0.365802
$838$	0	0
$839$	−27.8745	−0.962335	−0.481167	−	0.876629i	$-0.659787\pi$
$839$	−27.8745	−0.962335	−0.481167	+	0.876629i	$0.659787\pi$
$840$	0	0
$841$	−10.5830	−0.364931
$842$	0	0
$843$	−18.6863	−0.643589
$844$	0	0
$845$	43.7490	1.50501
$846$	0	0
$847$	0	0
$848$	0	0
$849$	20.2288	0.694249
$850$	0	0
$851$	−6.00000	−0.205677
$852$	0	0
$853$	−3.16601	−0.108402	−0.0542011	−	0.998530i	$-0.517261\pi$
$853$	−3.16601	−0.108402	−0.0542011	+	0.998530i	$0.517261\pi$
$854$	0	0
$855$	5.16601	0.176674
$856$	0	0
$857$	36.0000	1.22974	0.614868	−	0.788630i	$-0.289209\pi$
$857$	36.0000	1.22974	0.614868	+	0.788630i	$0.289209\pi$
$858$	0	0
$859$	−37.8118	−1.29012	−0.645060	−	0.764132i	$-0.723168\pi$
$859$	−37.8118	−1.29012	−0.645060	+	0.764132i	$0.723168\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	49.5203	1.68569	0.842845	−	0.538157i	$-0.180879\pi$
$863$	49.5203	1.68569	0.842845	+	0.538157i	$0.180879\pi$
$864$	0	0
$865$	37.5203	1.27573
$866$	0	0
$867$	50.2693	1.70723
$868$	0	0
$869$	2.64575	0.0897510
$870$	0	0
$871$	−19.6863	−0.667044
$872$	0	0
$873$	−22.8340	−0.772813
$874$	0	0
$875$	0	0
$876$	0	0
$877$	8.87451	0.299671	0.149835	−	0.988711i	$-0.452126\pi$
$877$	8.87451	0.299671	0.149835	+	0.988711i	$0.452126\pi$
$878$	0	0
$879$	31.7490	1.07087
$880$	0	0
$881$	6.87451	0.231608	0.115804	−	0.993272i	$-0.463056\pi$
$881$	6.87451	0.231608	0.115804	+	0.993272i	$0.463056\pi$
$882$	0	0
$883$	−6.06275	−0.204028	−0.102014	−	0.994783i	$-0.532529\pi$
$883$	−6.06275	−0.204028	−0.102014	+	0.994783i	$0.532529\pi$
$884$	0	0
$885$	6.22876	0.209377
$886$	0	0
$887$	13.1033	0.439965	0.219982	−	0.975504i	$-0.429400\pi$
$887$	13.1033	0.439965	0.219982	+	0.975504i	$0.429400\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	5.00000	0.167506
$892$	0	0
$893$	−4.70850	−0.157564
$894$	0	0
$895$	14.8118	0.495103
$896$	0	0
$897$	48.2288	1.61031
$898$	0	0
$899$	−17.1660	−0.572519
$900$	0	0
$901$	−21.8745	−0.728746
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−36.4575	−1.21189
$906$	0	0
$907$	10.4575	0.347236	0.173618	−	0.984813i	$-0.444454\pi$
$907$	10.4575	0.347236	0.173618	+	0.984813i	$0.444454\pi$
$908$	0	0
$909$	12.0000	0.398015
$910$	0	0
$911$	1.29150	0.0427894	0.0213947	−	0.999771i	$-0.493189\pi$
$911$	1.29150	0.0427894	0.0213947	+	0.999771i	$0.493189\pi$
$912$	0	0
$913$	13.2915	0.439885
$914$	0	0
$915$	35.7712	1.18256
$916$	0	0
$917$	0	0
$918$	0	0
$919$	35.2915	1.16416	0.582080	−	0.813132i	$-0.302239\pi$
$919$	35.2915	1.16416	0.582080	+	0.813132i	$0.302239\pi$
$920$	0	0
$921$	11.1882	0.368665
$922$	0	0
$923$	−48.2288	−1.58747
$924$	0	0
$925$	−13.6458	−0.448670
$926$	0	0
$927$	−51.7490	−1.69966
$928$	0	0
$929$	35.5830	1.16744	0.583720	−	0.811955i	$-0.301596\pi$
$929$	35.5830	1.16744	0.583720	+	0.811955i	$0.301596\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−44.8118	−1.46707
$934$	0	0
$935$	−21.8745	−0.715373
$936$	0	0
$937$	−46.6863	−1.52517	−0.762587	−	0.646886i	$-0.776071\pi$
$937$	−46.6863	−1.52517	−0.762587	+	0.646886i	$0.776071\pi$
$938$	0	0
$939$	6.39477	0.208685
$940$	0	0
$941$	−44.6235	−1.45469	−0.727343	−	0.686274i	$-0.759245\pi$
$941$	−44.6235	−1.45469	−0.727343	+	0.686274i	$0.759245\pi$
$942$	0	0
$943$	−18.0000	−0.586161
$944$	0	0
$945$	0	0
$946$	0	0
$947$	33.8745	1.10077	0.550387	−	0.834910i	$-0.314480\pi$
$947$	33.8745	1.10077	0.550387	+	0.834910i	$0.314480\pi$
$948$	0	0
$949$	28.2288	0.916344
$950$	0	0
$951$	31.7490	1.02953
$952$	0	0
$953$	−25.5203	−0.826682	−0.413341	−	0.910576i	$-0.635638\pi$
$953$	−25.5203	−0.826682	−0.413341	+	0.910576i	$0.635638\pi$
$954$	0	0
$955$	48.4575	1.56805
$956$	0	0
$957$	11.3542	0.367031
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	−19.7490	−0.636403
$964$	0	0
$965$	−42.0000	−1.35203
$966$	0	0
$967$	26.3320	0.846781	0.423390	−	0.905947i	$-0.360840\pi$
$967$	26.3320	0.846781	0.423390	+	0.905947i	$0.360840\pi$
$968$	0	0
$969$	5.62352	0.180654
$970$	0	0
$971$	25.9373	0.832366	0.416183	−	0.909281i	$-0.363368\pi$
$971$	25.9373	0.832366	0.416183	+	0.909281i	$0.363368\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	109.686	3.51277
$976$	0	0
$977$	36.4575	1.16638	0.583190	−	0.812336i	$-0.301804\pi$
$977$	36.4575	1.16638	0.583190	+	0.812336i	$0.301804\pi$
$978$	0	0
$979$	14.5830	0.466075
$980$	0	0
$981$	42.3320	1.35156
$982$	0	0
$983$	37.2915	1.18941	0.594707	−	0.803942i	$-0.297268\pi$
$983$	37.2915	1.18941	0.594707	+	0.803942i	$0.297268\pi$
$984$	0	0
$985$	68.8118	2.19253
$986$	0	0
$987$	0	0
$988$	0	0
$989$	14.5830	0.463713
$990$	0	0
$991$	−56.6863	−1.80070	−0.900349	−	0.435168i	$-0.856689\pi$
$991$	−56.6863	−1.80070	−0.900349	+	0.435168i	$0.856689\pi$
$992$	0	0
$993$	40.6235	1.28915
$994$	0	0
$995$	81.0405	2.56916
$996$	0	0
$997$	−42.5830	−1.34862	−0.674309	−	0.738450i	$-0.735558\pi$
$997$	−42.5830	−1.34862	−0.674309	+	0.738450i	$0.735558\pi$
$998$	0	0
$999$	−4.35425	−0.137762

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8624.2.a.ca.1.2		2
4.3	odd	2		1078.2.a.s.1.1		2
7.2	even	3		1232.2.q.g.529.1		4
7.4	even	3		1232.2.q.g.177.1		4
7.6	odd	2		8624.2.a.bk.1.1		2
12.11	even	2		9702.2.a.cz.1.1		2
28.3	even	6		1078.2.e.v.177.1		4
28.11	odd	6		154.2.e.f.23.2	✓	4
28.19	even	6		1078.2.e.v.67.1		4
28.23	odd	6		154.2.e.f.67.2	yes	4
28.27	even	2		1078.2.a.n.1.2		2
84.11	even	6		1386.2.k.s.793.2		4
84.23	even	6		1386.2.k.s.991.2		4
84.83	odd	2		9702.2.a.dr.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
154.2.e.f.23.2	✓	4	28.11	odd	6
154.2.e.f.67.2	yes	4	28.23	odd	6
1078.2.a.n.1.2		2	28.27	even	2
1078.2.a.s.1.1		2	4.3	odd	2
1078.2.e.v.67.1		4	28.19	even	6
1078.2.e.v.177.1		4	28.3	even	6
1232.2.q.g.177.1		4	7.4	even	3
1232.2.q.g.529.1		4	7.2	even	3
1386.2.k.s.793.2		4	84.11	even	6
1386.2.k.s.991.2		4	84.23	even	6
8624.2.a.bk.1.1		2	7.6	odd	2
8624.2.a.ca.1.2		2	1.1	even	1	trivial
9702.2.a.cz.1.1		2	12.11	even	2
9702.2.a.dr.1.2		2	84.83	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:	\( N \)	\(=\)	\( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8624.a (trivial)

\(f(q)\)	\(=\)	\(q+2.64575 q^{3} +3.64575 q^{5} +4.00000 q^{9} +O(q^{10})\)	\(q+2.64575 q^{3} +3.64575 q^{5} +4.00000 q^{9} -1.00000 q^{11} +5.00000 q^{13} +9.64575 q^{15} +6.00000 q^{17} +0.354249 q^{19} +3.64575 q^{23} +8.29150 q^{25} +2.64575 q^{27} -4.29150 q^{29} +4.00000 q^{31} -2.64575 q^{33} -1.64575 q^{37} +13.2288 q^{39} -4.93725 q^{41} +4.00000 q^{43} +14.5830 q^{45} -13.2915 q^{47} +15.8745 q^{51} -3.64575 q^{53} -3.64575 q^{55} +0.937254 q^{57} +0.645751 q^{59} +3.70850 q^{61} +18.2288 q^{65} -3.93725 q^{67} +9.64575 q^{69} -9.64575 q^{71} +5.64575 q^{73} +21.9373 q^{75} -2.64575 q^{79} -5.00000 q^{81} -13.2915 q^{83} +21.8745 q^{85} -11.3542 q^{87} -14.5830 q^{89} +10.5830 q^{93} +1.29150 q^{95} -5.70850 q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} + 8 q^{9}+O(q^{10}) \) 2 * q + 2 * q^5 + 8 * q^9	\( 2 q + 2 q^{5} + 8 q^{9} - 2 q^{11} + 10 q^{13} + 14 q^{15} + 12 q^{17} + 6 q^{19} + 2 q^{23} + 6 q^{25} + 2 q^{29} + 8 q^{31} + 2 q^{37} + 6 q^{41} + 8 q^{43} + 8 q^{45} - 16 q^{47} - 2 q^{53} - 2 q^{55} - 14 q^{57} - 4 q^{59} + 18 q^{61} + 10 q^{65} + 8 q^{67} + 14 q^{69} - 14 q^{71} + 6 q^{73} + 28 q^{75} - 10 q^{81} - 16 q^{83} + 12 q^{85} - 28 q^{87} - 8 q^{89} - 8 q^{95} - 22 q^{97} - 8 q^{99}+O(q^{100}) \) 2 * q + 2 * q^5 + 8 * q^9 - 2 * q^11 + 10 * q^13 + 14 * q^15 + 12 * q^17 + 6 * q^19 + 2 * q^23 + 6 * q^25 + 2 * q^29 + 8 * q^31 + 2 * q^37 + 6 * q^41 + 8 * q^43 + 8 * q^45 - 16 * q^47 - 2 * q^53 - 2 * q^55 - 14 * q^57 - 4 * q^59 + 18 * q^61 + 10 * q^65 + 8 * q^67 + 14 * q^69 - 14 * q^71 + 6 * q^73 + 28 * q^75 - 10 * q^81 - 16 * q^83 + 12 * q^85 - 28 * q^87 - 8 * q^89 - 8 * q^95 - 22 * q^97 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more