LMFDB - Embedded newform 6240.2.a.bx.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6240,2,Mod(1,6240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6240, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6240.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6240.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:	$49.8266508613$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.11491$ of defining polynomial
Character	$\chi$	$=$	6240.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} -4.22982 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.00000 q^{5} -4.22982 q^{7} +1.00000 q^{9} -1.28415 q^{11} -1.00000 q^{13} +1.00000 q^{15} -0.945668 q^{17} -4.22982 q^{19} +4.22982 q^{21} +6.94567 q^{23} +1.00000 q^{25} -1.00000 q^{27} -4.94567 q^{29} -7.17548 q^{31} +1.28415 q^{33} +4.22982 q^{35} +7.89134 q^{37} +1.00000 q^{39} -10.4596 q^{41} -8.00000 q^{43} -1.00000 q^{45} -9.74378 q^{47} +10.8913 q^{49} +0.945668 q^{51} -6.45963 q^{53} +1.28415 q^{55} +4.22982 q^{57} -9.28415 q^{59} +6.00000 q^{61} -4.22982 q^{63} +1.00000 q^{65} -11.1755 q^{67} -6.94567 q^{69} -8.60719 q^{71} +0.945668 q^{73} -1.00000 q^{75} +5.43171 q^{77} +14.3510 q^{79} +1.00000 q^{81} -4.60719 q^{83} +0.945668 q^{85} +4.94567 q^{87} +2.00000 q^{89} +4.22982 q^{91} +7.17548 q^{93} +4.22982 q^{95} -10.8370 q^{97} -1.28415 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 3 q^{5} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 - 3 * q^5 + 3 * q^9	$ 3 q - 3 q^{3} - 3 q^{5} + 3 q^{9} - 2 q^{11} - 3 q^{13} + 3 q^{15} + 8 q^{17} + 10 q^{23} + 3 q^{25} - 3 q^{27} - 4 q^{29} + 2 q^{31} + 2 q^{33} + 2 q^{37} + 3 q^{39} - 6 q^{41} - 24 q^{43} - 3 q^{45} - 2 q^{47} + 11 q^{49} - 8 q^{51} + 6 q^{53} + 2 q^{55} - 26 q^{59} + 18 q^{61} + 3 q^{65} - 10 q^{67} - 10 q^{69} - 6 q^{71} - 8 q^{73} - 3 q^{75} + 20 q^{77} - 4 q^{79} + 3 q^{81} + 6 q^{83} - 8 q^{85} + 4 q^{87} + 6 q^{89} - 2 q^{93} - 2 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - 3 * q^5 + 3 * q^9 - 2 * q^11 - 3 * q^13 + 3 * q^15 + 8 * q^17 + 10 * q^23 + 3 * q^25 - 3 * q^27 - 4 * q^29 + 2 * q^31 + 2 * q^33 + 2 * q^37 + 3 * q^39 - 6 * q^41 - 24 * q^43 - 3 * q^45 - 2 * q^47 + 11 * q^49 - 8 * q^51 + 6 * q^53 + 2 * q^55 - 26 * q^59 + 18 * q^61 + 3 * q^65 - 10 * q^67 - 10 * q^69 - 6 * q^71 - 8 * q^73 - 3 * q^75 + 20 * q^77 - 4 * q^79 + 3 * q^81 + 6 * q^83 - 8 * q^85 + 4 * q^87 + 6 * q^89 - 2 * q^93 - 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−4.22982	−1.59872	−0.799360	−	0.600853i	$-0.794828\pi$
$7$	−4.22982	−1.59872	−0.799360	+	0.600853i	$0.794828\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.28415	−0.387185	−0.193592	−	0.981082i	$-0.562014\pi$
$11$	−1.28415	−0.387185	−0.193592	+	0.981082i	$0.562014\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−0.945668	−0.229358	−0.114679	−	0.993403i	$-0.536584\pi$
$17$	−0.945668	−0.229358	−0.114679	+	0.993403i	$0.536584\pi$
$18$	0	0
$19$	−4.22982	−0.970386	−0.485193	−	0.874407i	$-0.661251\pi$
$19$	−4.22982	−0.970386	−0.485193	+	0.874407i	$0.661251\pi$
$20$	0	0
$21$	4.22982	0.923021
$22$	0	0
$23$	6.94567	1.44827	0.724136	−	0.689657i	$-0.242239\pi$
$23$	6.94567	1.44827	0.724136	+	0.689657i	$0.242239\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−4.94567	−0.918387	−0.459194	−	0.888336i	$-0.651862\pi$
$29$	−4.94567	−0.918387	−0.459194	+	0.888336i	$0.651862\pi$
$30$	0	0
$31$	−7.17548	−1.28875	−0.644377	−	0.764708i	$-0.722883\pi$
$31$	−7.17548	−1.28875	−0.644377	+	0.764708i	$0.722883\pi$
$32$	0	0
$33$	1.28415	0.223541
$34$	0	0
$35$	4.22982	0.714969
$36$	0	0
$37$	7.89134	1.29733	0.648664	−	0.761075i	$-0.275328\pi$
$37$	7.89134	1.29733	0.648664	+	0.761075i	$0.275328\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	−10.4596	−1.63352	−0.816760	−	0.576978i	$-0.804232\pi$
$41$	−10.4596	−1.63352	−0.816760	+	0.576978i	$0.804232\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−9.74378	−1.42128	−0.710638	−	0.703558i	$-0.751594\pi$
$47$	−9.74378	−1.42128	−0.710638	+	0.703558i	$0.751594\pi$
$48$	0	0
$49$	10.8913	1.55591
$50$	0	0
$51$	0.945668	0.132420
$52$	0	0
$53$	−6.45963	−0.887298	−0.443649	−	0.896201i	$-0.646316\pi$
$53$	−6.45963	−0.887298	−0.443649	+	0.896201i	$0.646316\pi$
$54$	0	0
$55$	1.28415	0.173154
$56$	0	0
$57$	4.22982	0.560253
$58$	0	0
$59$	−9.28415	−1.20869	−0.604347	−	0.796722i	$-0.706566\pi$
$59$	−9.28415	−1.20869	−0.604347	+	0.796722i	$0.706566\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	−4.22982	−0.532907
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	−11.1755	−1.36530	−0.682651	−	0.730744i	$-0.739173\pi$
$67$	−11.1755	−1.36530	−0.682651	+	0.730744i	$0.739173\pi$
$68$	0	0
$69$	−6.94567	−0.836160
$70$	0	0
$71$	−8.60719	−1.02149	−0.510743	−	0.859734i	$-0.670630\pi$
$71$	−8.60719	−1.02149	−0.510743	+	0.859734i	$0.670630\pi$
$72$	0	0
$73$	0.945668	0.110682	0.0553410	−	0.998468i	$-0.482375\pi$
$73$	0.945668	0.110682	0.0553410	+	0.998468i	$0.482375\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	5.43171	0.619000
$78$	0	0
$79$	14.3510	1.61461	0.807305	−	0.590135i	$-0.200925\pi$
$79$	14.3510	1.61461	0.807305	+	0.590135i	$0.200925\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−4.60719	−0.505705	−0.252852	−	0.967505i	$-0.581369\pi$
$83$	−4.60719	−0.505705	−0.252852	+	0.967505i	$0.581369\pi$
$84$	0	0
$85$	0.945668	0.102572
$86$	0	0
$87$	4.94567	0.530231
$88$	0	0
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	4.22982	0.443405
$92$	0	0
$93$	7.17548	0.744063
$94$	0	0
$95$	4.22982	0.433970
$96$	0	0
$97$	−10.8370	−1.10033	−0.550165	−	0.835056i	$-0.685435\pi$
$97$	−10.8370	−1.10033	−0.550165	+	0.835056i	$0.685435\pi$
$98$	0	0
$99$	−1.28415	−0.129062
$100$	0	0
$101$	3.51396	0.349652	0.174826	−	0.984599i	$-0.444064\pi$
$101$	3.51396	0.349652	0.174826	+	0.984599i	$0.444064\pi$
$102$	0	0
$103$	18.3510	1.80817	0.904087	−	0.427348i	$-0.140552\pi$
$103$	18.3510	1.80817	0.904087	+	0.427348i	$0.140552\pi$
$104$	0	0
$105$	−4.22982	−0.412788
$106$	0	0
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	0	0
$109$	−0.486038	−0.0465540	−0.0232770	−	0.999729i	$-0.507410\pi$
$109$	−0.486038	−0.0465540	−0.0232770	+	0.999729i	$0.507410\pi$
$110$	0	0
$111$	−7.89134	−0.749012
$112$	0	0
$113$	4.94567	0.465249	0.232625	−	0.972567i	$-0.425269\pi$
$113$	4.94567	0.465249	0.232625	+	0.972567i	$0.425269\pi$
$114$	0	0
$115$	−6.94567	−0.647687
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	−9.35097	−0.850088
$122$	0	0
$123$	10.4596	0.943113
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−1.43171	−0.127043	−0.0635217	−	0.997980i	$-0.520233\pi$
$127$	−1.43171	−0.127043	−0.0635217	+	0.997980i	$0.520233\pi$
$128$	0	0
$129$	8.00000	0.704361
$130$	0	0
$131$	−4.37737	−0.382453	−0.191226	−	0.981546i	$-0.561246\pi$
$131$	−4.37737	−0.382453	−0.191226	+	0.981546i	$0.561246\pi$
$132$	0	0
$133$	17.8913	1.55138
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	22.4596	1.91886	0.959428	−	0.281954i	$-0.0909826\pi$
$137$	22.4596	1.91886	0.959428	+	0.281954i	$0.0909826\pi$
$138$	0	0
$139$	−9.89134	−0.838972	−0.419486	−	0.907762i	$-0.637790\pi$
$139$	−9.89134	−0.838972	−0.419486	+	0.907762i	$0.637790\pi$
$140$	0	0
$141$	9.74378	0.820574
$142$	0	0
$143$	1.28415	0.107386
$144$	0	0
$145$	4.94567	0.410715
$146$	0	0
$147$	−10.8913	−0.898302
$148$	0	0
$149$	15.8913	1.30187	0.650934	−	0.759134i	$-0.274377\pi$
$149$	15.8913	1.30187	0.650934	+	0.759134i	$0.274377\pi$
$150$	0	0
$151$	15.1755	1.23496	0.617482	−	0.786585i	$-0.288153\pi$
$151$	15.1755	1.23496	0.617482	+	0.786585i	$0.288153\pi$
$152$	0	0
$153$	−0.945668	−0.0764527
$154$	0	0
$155$	7.17548	0.576349
$156$	0	0
$157$	−5.32304	−0.424825	−0.212412	−	0.977180i	$-0.568132\pi$
$157$	−5.32304	−0.424825	−0.212412	+	0.977180i	$0.568132\pi$
$158$	0	0
$159$	6.45963	0.512282
$160$	0	0
$161$	−29.3789	−2.31538
$162$	0	0
$163$	−7.85244	−0.615051	−0.307525	−	0.951540i	$-0.599501\pi$
$163$	−7.85244	−0.615051	−0.307525	+	0.951540i	$0.599501\pi$
$164$	0	0
$165$	−1.28415	−0.0999707
$166$	0	0
$167$	4.14756	0.320948	0.160474	−	0.987040i	$-0.448698\pi$
$167$	4.14756	0.320948	0.160474	+	0.987040i	$0.448698\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−4.22982	−0.323462
$172$	0	0
$173$	15.5962	1.18576	0.592879	−	0.805291i	$-0.297991\pi$
$173$	15.5962	1.18576	0.592879	+	0.805291i	$0.297991\pi$
$174$	0	0
$175$	−4.22982	−0.319744
$176$	0	0
$177$	9.28415	0.697839
$178$	0	0
$179$	23.8649	1.78375	0.891874	−	0.452283i	$-0.149390\pi$
$179$	23.8649	1.78375	0.891874	+	0.452283i	$0.149390\pi$
$180$	0	0
$181$	−10.4596	−0.777458	−0.388729	−	0.921352i	$-0.627086\pi$
$181$	−10.4596	−0.777458	−0.388729	+	0.921352i	$0.627086\pi$
$182$	0	0
$183$	−6.00000	−0.443533
$184$	0	0
$185$	−7.89134	−0.580183
$186$	0	0
$187$	1.21438	0.0888040
$188$	0	0
$189$	4.22982	0.307674
$190$	0	0
$191$	13.4317	0.971884	0.485942	−	0.873991i	$-0.338477\pi$
$191$	13.4317	0.971884	0.485942	+	0.873991i	$0.338477\pi$
$192$	0	0
$193$	−10.0823	−0.725737	−0.362868	−	0.931840i	$-0.618203\pi$
$193$	−10.0823	−0.725737	−0.362868	+	0.931840i	$0.618203\pi$
$194$	0	0
$195$	−1.00000	−0.0716115
$196$	0	0
$197$	5.54037	0.394735	0.197368	−	0.980330i	$-0.436761\pi$
$197$	5.54037	0.394735	0.197368	+	0.980330i	$0.436761\pi$
$198$	0	0
$199$	−19.7827	−1.40236	−0.701178	−	0.712986i	$-0.747342\pi$
$199$	−19.7827	−1.40236	−0.701178	+	0.712986i	$0.747342\pi$
$200$	0	0
$201$	11.1755	0.788258
$202$	0	0
$203$	20.9193	1.46824
$204$	0	0
$205$	10.4596	0.730532
$206$	0	0
$207$	6.94567	0.482757
$208$	0	0
$209$	5.43171	0.375719
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	0	0
$213$	8.60719	0.589755
$214$	0	0
$215$	8.00000	0.545595
$216$	0	0
$217$	30.3510	2.06036
$218$	0	0
$219$	−0.945668	−0.0639023
$220$	0	0
$221$	0.945668	0.0636125
$222$	0	0
$223$	13.1491	0.880527	0.440264	−	0.897869i	$-0.354885\pi$
$223$	13.1491	0.880527	0.440264	+	0.897869i	$0.354885\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	21.7438	1.44319	0.721593	−	0.692318i	$-0.243410\pi$
$227$	21.7438	1.44319	0.721593	+	0.692318i	$0.243410\pi$
$228$	0	0
$229$	−1.40530	−0.0928647	−0.0464324	−	0.998921i	$-0.514785\pi$
$229$	−1.40530	−0.0928647	−0.0464324	+	0.998921i	$0.514785\pi$
$230$	0	0
$231$	−5.43171	−0.357380
$232$	0	0
$233$	21.4053	1.40231	0.701154	−	0.713010i	$-0.252669\pi$
$233$	21.4053	1.40231	0.701154	+	0.713010i	$0.252669\pi$
$234$	0	0
$235$	9.74378	0.635614
$236$	0	0
$237$	−14.3510	−0.932195
$238$	0	0
$239$	−12.3121	−0.796402	−0.398201	−	0.917298i	$-0.630365\pi$
$239$	−12.3121	−0.796402	−0.398201	+	0.917298i	$0.630365\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−10.8913	−0.695822
$246$	0	0
$247$	4.22982	0.269137
$248$	0	0
$249$	4.60719	0.291969
$250$	0	0
$251$	−29.7563	−1.87820	−0.939099	−	0.343646i	$-0.888338\pi$
$251$	−29.7563	−1.87820	−0.939099	+	0.343646i	$0.888338\pi$
$252$	0	0
$253$	−8.91926	−0.560749
$254$	0	0
$255$	−0.945668	−0.0592200
$256$	0	0
$257$	2.37737	0.148296	0.0741482	−	0.997247i	$-0.476376\pi$
$257$	2.37737	0.148296	0.0741482	+	0.997247i	$0.476376\pi$
$258$	0	0
$259$	−33.3789	−2.07406
$260$	0	0
$261$	−4.94567	−0.306129
$262$	0	0
$263$	20.5419	1.26667	0.633334	−	0.773879i	$-0.281686\pi$
$263$	20.5419	1.26667	0.633334	+	0.773879i	$0.281686\pi$
$264$	0	0
$265$	6.45963	0.396812
$266$	0	0
$267$	−2.00000	−0.122398
$268$	0	0
$269$	−18.8370	−1.14851	−0.574256	−	0.818676i	$-0.694709\pi$
$269$	−18.8370	−1.14851	−0.574256	+	0.818676i	$0.694709\pi$
$270$	0	0
$271$	−9.74378	−0.591892	−0.295946	−	0.955205i	$-0.595635\pi$
$271$	−9.74378	−0.591892	−0.295946	+	0.955205i	$0.595635\pi$
$272$	0	0
$273$	−4.22982	−0.256000
$274$	0	0
$275$	−1.28415	−0.0774370
$276$	0	0
$277$	19.8913	1.19515	0.597577	−	0.801811i	$-0.296130\pi$
$277$	19.8913	1.19515	0.597577	+	0.801811i	$0.296130\pi$
$278$	0	0
$279$	−7.17548	−0.429585
$280$	0	0
$281$	18.9193	1.12863	0.564314	−	0.825560i	$-0.309141\pi$
$281$	18.9193	1.12863	0.564314	+	0.825560i	$0.309141\pi$
$282$	0	0
$283$	0.919260	0.0546444	0.0273222	−	0.999627i	$-0.491302\pi$
$283$	0.919260	0.0546444	0.0273222	+	0.999627i	$0.491302\pi$
$284$	0	0
$285$	−4.22982	−0.250553
$286$	0	0
$287$	44.2423	2.61154
$288$	0	0
$289$	−16.1057	−0.947395
$290$	0	0
$291$	10.8370	0.635276
$292$	0	0
$293$	−14.2423	−0.832044	−0.416022	−	0.909355i	$-0.636576\pi$
$293$	−14.2423	−0.832044	−0.416022	+	0.909355i	$0.636576\pi$
$294$	0	0
$295$	9.28415	0.540544
$296$	0	0
$297$	1.28415	0.0745138
$298$	0	0
$299$	−6.94567	−0.401678
$300$	0	0
$301$	33.8385	1.95042
$302$	0	0
$303$	−3.51396	−0.201872
$304$	0	0
$305$	−6.00000	−0.343559
$306$	0	0
$307$	2.25622	0.128769	0.0643847	−	0.997925i	$-0.479492\pi$
$307$	2.25622	0.128769	0.0643847	+	0.997925i	$0.479492\pi$
$308$	0	0
$309$	−18.3510	−1.04395
$310$	0	0
$311$	−16.9193	−0.959403	−0.479702	−	0.877432i	$-0.659255\pi$
$311$	−16.9193	−0.959403	−0.479702	+	0.877432i	$0.659255\pi$
$312$	0	0
$313$	26.4596	1.49559	0.747793	−	0.663932i	$-0.231113\pi$
$313$	26.4596	1.49559	0.747793	+	0.663932i	$0.231113\pi$
$314$	0	0
$315$	4.22982	0.238323
$316$	0	0
$317$	−3.13659	−0.176168	−0.0880842	−	0.996113i	$-0.528074\pi$
$317$	−3.13659	−0.176168	−0.0880842	+	0.996113i	$0.528074\pi$
$318$	0	0
$319$	6.35097	0.355586
$320$	0	0
$321$	−4.00000	−0.223258
$322$	0	0
$323$	4.00000	0.222566
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	0.486038	0.0268779
$328$	0	0
$329$	41.2144	2.27222
$330$	0	0
$331$	−24.0125	−1.31985	−0.659923	−	0.751333i	$-0.729411\pi$
$331$	−24.0125	−1.31985	−0.659923	+	0.751333i	$0.729411\pi$
$332$	0	0
$333$	7.89134	0.432443
$334$	0	0
$335$	11.1755	0.610582
$336$	0	0
$337$	23.5962	1.28537	0.642684	−	0.766131i	$-0.277821\pi$
$337$	23.5962	1.28537	0.642684	+	0.766131i	$0.277821\pi$
$338$	0	0
$339$	−4.94567	−0.268612
$340$	0	0
$341$	9.21438	0.498986
$342$	0	0
$343$	−16.4596	−0.888736
$344$	0	0
$345$	6.94567	0.373942
$346$	0	0
$347$	−17.8913	−0.960457	−0.480229	−	0.877143i	$-0.659446\pi$
$347$	−17.8913	−0.960457	−0.480229	+	0.877143i	$0.659446\pi$
$348$	0	0
$349$	15.5140	0.830443	0.415222	−	0.909720i	$-0.363704\pi$
$349$	15.5140	0.830443	0.415222	+	0.909720i	$0.363704\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	10.6770	0.568277	0.284139	−	0.958783i	$-0.408292\pi$
$353$	10.6770	0.568277	0.284139	+	0.958783i	$0.408292\pi$
$354$	0	0
$355$	8.60719	0.456822
$356$	0	0
$357$	−4.00000	−0.211702
$358$	0	0
$359$	−8.31207	−0.438694	−0.219347	−	0.975647i	$-0.570393\pi$
$359$	−8.31207	−0.438694	−0.219347	+	0.975647i	$0.570393\pi$
$360$	0	0
$361$	−1.10866	−0.0583508
$362$	0	0
$363$	9.35097	0.490798
$364$	0	0
$365$	−0.945668	−0.0494985
$366$	0	0
$367$	28.0000	1.46159	0.730794	−	0.682598i	$-0.239150\pi$
$367$	28.0000	1.46159	0.730794	+	0.682598i	$0.239150\pi$
$368$	0	0
$369$	−10.4596	−0.544507
$370$	0	0
$371$	27.3230	1.41854
$372$	0	0
$373$	−31.1616	−1.61348	−0.806742	−	0.590903i	$-0.798771\pi$
$373$	−31.1616	−1.61348	−0.806742	+	0.590903i	$0.798771\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	4.94567	0.254715
$378$	0	0
$379$	10.5808	0.543498	0.271749	−	0.962368i	$-0.412398\pi$
$379$	10.5808	0.543498	0.271749	+	0.962368i	$0.412398\pi$
$380$	0	0
$381$	1.43171	0.0733485
$382$	0	0
$383$	14.7159	0.751945	0.375972	−	0.926631i	$-0.377309\pi$
$383$	14.7159	0.751945	0.375972	+	0.926631i	$0.377309\pi$
$384$	0	0
$385$	−5.43171	−0.276825
$386$	0	0
$387$	−8.00000	−0.406663
$388$	0	0
$389$	−19.7563	−1.00168	−0.500841	−	0.865539i	$-0.666976\pi$
$389$	−19.7563	−1.00168	−0.500841	+	0.865539i	$0.666976\pi$
$390$	0	0
$391$	−6.56829	−0.332173
$392$	0	0
$393$	4.37737	0.220809
$394$	0	0
$395$	−14.3510	−0.722075
$396$	0	0
$397$	−1.02792	−0.0515901	−0.0257950	−	0.999667i	$-0.508212\pi$
$397$	−1.02792	−0.0515901	−0.0257950	+	0.999667i	$0.508212\pi$
$398$	0	0
$399$	−17.8913	−0.895687
$400$	0	0
$401$	18.2423	0.910977	0.455489	−	0.890242i	$-0.349465\pi$
$401$	18.2423	0.910977	0.455489	+	0.890242i	$0.349465\pi$
$402$	0	0
$403$	7.17548	0.357436
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−10.1336	−0.502306
$408$	0	0
$409$	36.5933	1.80942	0.904710	−	0.426027i	$-0.140087\pi$
$409$	36.5933	1.80942	0.904710	+	0.426027i	$0.140087\pi$
$410$	0	0
$411$	−22.4596	−1.10785
$412$	0	0
$413$	39.2702	1.93236
$414$	0	0
$415$	4.60719	0.226158
$416$	0	0
$417$	9.89134	0.484381
$418$	0	0
$419$	24.3246	1.18833	0.594166	−	0.804342i	$-0.297482\pi$
$419$	24.3246	1.18833	0.594166	+	0.804342i	$0.297482\pi$
$420$	0	0
$421$	−14.3774	−0.700710	−0.350355	−	0.936617i	$-0.613939\pi$
$421$	−14.3774	−0.700710	−0.350355	+	0.936617i	$0.613939\pi$
$422$	0	0
$423$	−9.74378	−0.473759
$424$	0	0
$425$	−0.945668	−0.0458716
$426$	0	0
$427$	−25.3789	−1.22817
$428$	0	0
$429$	−1.28415	−0.0619992
$430$	0	0
$431$	25.7438	1.24003	0.620017	−	0.784588i	$-0.287126\pi$
$431$	25.7438	1.24003	0.620017	+	0.784588i	$0.287126\pi$
$432$	0	0
$433$	−9.02792	−0.433854	−0.216927	−	0.976188i	$-0.569603\pi$
$433$	−9.02792	−0.433854	−0.216927	+	0.976188i	$0.569603\pi$
$434$	0	0
$435$	−4.94567	−0.237127
$436$	0	0
$437$	−29.3789	−1.40538
$438$	0	0
$439$	−10.3510	−0.494025	−0.247012	−	0.969012i	$-0.579449\pi$
$439$	−10.3510	−0.494025	−0.247012	+	0.969012i	$0.579449\pi$
$440$	0	0
$441$	10.8913	0.518635
$442$	0	0
$443$	−23.9472	−1.13777	−0.568883	−	0.822419i	$-0.692624\pi$
$443$	−23.9472	−1.13777	−0.568883	+	0.822419i	$0.692624\pi$
$444$	0	0
$445$	−2.00000	−0.0948091
$446$	0	0
$447$	−15.8913	−0.751634
$448$	0	0
$449$	−34.7019	−1.63769	−0.818843	−	0.574018i	$-0.805384\pi$
$449$	−34.7019	−1.63769	−0.818843	+	0.574018i	$0.805384\pi$
$450$	0	0
$451$	13.4317	0.632474
$452$	0	0
$453$	−15.1755	−0.713006
$454$	0	0
$455$	−4.22982	−0.198297
$456$	0	0
$457$	8.94567	0.418461	0.209230	−	0.977866i	$-0.432904\pi$
$457$	8.94567	0.418461	0.209230	+	0.977866i	$0.432904\pi$
$458$	0	0
$459$	0.945668	0.0441400
$460$	0	0
$461$	18.9193	0.881158	0.440579	−	0.897714i	$-0.354773\pi$
$461$	18.9193	0.881158	0.440579	+	0.897714i	$0.354773\pi$
$462$	0	0
$463$	−32.7672	−1.52282	−0.761411	−	0.648269i	$-0.775493\pi$
$463$	−32.7672	−1.52282	−0.761411	+	0.648269i	$0.775493\pi$
$464$	0	0
$465$	−7.17548	−0.332755
$466$	0	0
$467$	9.43171	0.436447	0.218224	−	0.975899i	$-0.429974\pi$
$467$	9.43171	0.436447	0.218224	+	0.975899i	$0.429974\pi$
$468$	0	0
$469$	47.2702	2.18274
$470$	0	0
$471$	5.32304	0.245273
$472$	0	0
$473$	10.2732	0.472361
$474$	0	0
$475$	−4.22982	−0.194077
$476$	0	0
$477$	−6.45963	−0.295766
$478$	0	0
$479$	−21.5264	−0.983569	−0.491784	−	0.870717i	$-0.663655\pi$
$479$	−21.5264	−0.983569	−0.491784	+	0.870717i	$0.663655\pi$
$480$	0	0
$481$	−7.89134	−0.359814
$482$	0	0
$483$	29.3789	1.33679
$484$	0	0
$485$	10.8370	0.492083
$486$	0	0
$487$	43.0404	1.95035	0.975174	−	0.221442i	$-0.0710763\pi$
$487$	43.0404	1.95035	0.975174	+	0.221442i	$0.0710763\pi$
$488$	0	0
$489$	7.85244	0.355100
$490$	0	0
$491$	13.7563	0.620811	0.310406	−	0.950604i	$-0.399535\pi$
$491$	13.7563	0.620811	0.310406	+	0.950604i	$0.399535\pi$
$492$	0	0
$493$	4.67696	0.210640
$494$	0	0
$495$	1.28415	0.0577181
$496$	0	0
$497$	36.4068	1.63307
$498$	0	0
$499$	−26.1212	−1.16934	−0.584672	−	0.811270i	$-0.698777\pi$
$499$	−26.1212	−1.16934	−0.584672	+	0.811270i	$0.698777\pi$
$500$	0	0
$501$	−4.14756	−0.185299
$502$	0	0
$503$	19.6226	0.874930	0.437465	−	0.899236i	$-0.355876\pi$
$503$	19.6226	0.874930	0.437465	+	0.899236i	$0.355876\pi$
$504$	0	0
$505$	−3.51396	−0.156369
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	21.7827	0.965500	0.482750	−	0.875758i	$-0.339638\pi$
$509$	21.7827	0.965500	0.482750	+	0.875758i	$0.339638\pi$
$510$	0	0
$511$	−4.00000	−0.176950
$512$	0	0
$513$	4.22982	0.186751
$514$	0	0
$515$	−18.3510	−0.808640
$516$	0	0
$517$	12.5124	0.550297
$518$	0	0
$519$	−15.5962	−0.684598
$520$	0	0
$521$	−39.3789	−1.72522	−0.862610	−	0.505869i	$-0.831172\pi$
$521$	−39.3789	−1.72522	−0.862610	+	0.505869i	$0.831172\pi$
$522$	0	0
$523$	−18.6461	−0.815336	−0.407668	−	0.913130i	$-0.633658\pi$
$523$	−18.6461	−0.815336	−0.407668	+	0.913130i	$0.633658\pi$
$524$	0	0
$525$	4.22982	0.184604
$526$	0	0
$527$	6.78562	0.295586
$528$	0	0
$529$	25.2423	1.09749
$530$	0	0
$531$	−9.28415	−0.402898
$532$	0	0
$533$	10.4596	0.453057
$534$	0	0
$535$	−4.00000	−0.172935
$536$	0	0
$537$	−23.8649	−1.02985
$538$	0	0
$539$	−13.9861	−0.602423
$540$	0	0
$541$	−41.5698	−1.78723	−0.893613	−	0.448838i	$-0.851838\pi$
$541$	−41.5698	−1.78723	−0.893613	+	0.448838i	$0.851838\pi$
$542$	0	0
$543$	10.4596	0.448866
$544$	0	0
$545$	0.486038	0.0208196
$546$	0	0
$547$	9.13659	0.390652	0.195326	−	0.980738i	$-0.437423\pi$
$547$	9.13659	0.390652	0.195326	+	0.980738i	$0.437423\pi$
$548$	0	0
$549$	6.00000	0.256074
$550$	0	0
$551$	20.9193	0.891190
$552$	0	0
$553$	−60.7019	−2.58131
$554$	0	0
$555$	7.89134	0.334969
$556$	0	0
$557$	−3.13659	−0.132902	−0.0664508	−	0.997790i	$-0.521168\pi$
$557$	−3.13659	−0.132902	−0.0664508	+	0.997790i	$0.521168\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	0	0
$561$	−1.21438	−0.0512710
$562$	0	0
$563$	−9.59622	−0.404432	−0.202216	−	0.979341i	$-0.564814\pi$
$563$	−9.59622	−0.404432	−0.202216	+	0.979341i	$0.564814\pi$
$564$	0	0
$565$	−4.94567	−0.208066
$566$	0	0
$567$	−4.22982	−0.177636
$568$	0	0
$569$	9.24525	0.387581	0.193791	−	0.981043i	$-0.437922\pi$
$569$	9.24525	0.387581	0.193791	+	0.981043i	$0.437922\pi$
$570$	0	0
$571$	3.54037	0.148160	0.0740799	−	0.997252i	$-0.476398\pi$
$571$	3.54037	0.148160	0.0740799	+	0.997252i	$0.476398\pi$
$572$	0	0
$573$	−13.4317	−0.561118
$574$	0	0
$575$	6.94567	0.289654
$576$	0	0
$577$	−12.9457	−0.538935	−0.269468	−	0.963009i	$-0.586848\pi$
$577$	−12.9457	−0.538935	−0.269468	+	0.963009i	$0.586848\pi$
$578$	0	0
$579$	10.0823	0.419004
$580$	0	0
$581$	19.4876	0.808480
$582$	0	0
$583$	8.29512	0.343549
$584$	0	0
$585$	1.00000	0.0413449
$586$	0	0
$587$	14.6630	0.605208	0.302604	−	0.953116i	$-0.402144\pi$
$587$	14.6630	0.605208	0.302604	+	0.953116i	$0.402144\pi$
$588$	0	0
$589$	30.3510	1.25059
$590$	0	0
$591$	−5.54037	−0.227900
$592$	0	0
$593$	34.4068	1.41292	0.706459	−	0.707754i	$-0.250291\pi$
$593$	34.4068	1.41292	0.706459	+	0.707754i	$0.250291\pi$
$594$	0	0
$595$	−4.00000	−0.163984
$596$	0	0
$597$	19.7827	0.809651
$598$	0	0
$599$	24.6241	1.00612	0.503058	−	0.864253i	$-0.332208\pi$
$599$	24.6241	1.00612	0.503058	+	0.864253i	$0.332208\pi$
$600$	0	0
$601$	−37.0279	−1.51040	−0.755200	−	0.655494i	$-0.772460\pi$
$601$	−37.0279	−1.51040	−0.755200	+	0.655494i	$0.772460\pi$
$602$	0	0
$603$	−11.1755	−0.455101
$604$	0	0
$605$	9.35097	0.380171
$606$	0	0
$607$	−35.5653	−1.44355	−0.721776	−	0.692126i	$-0.756674\pi$
$607$	−35.5653	−1.44355	−0.721776	+	0.692126i	$0.756674\pi$
$608$	0	0
$609$	−20.9193	−0.847691
$610$	0	0
$611$	9.74378	0.394191
$612$	0	0
$613$	−21.7299	−0.877661	−0.438830	−	0.898570i	$-0.644607\pi$
$613$	−21.7299	−0.877661	−0.438830	+	0.898570i	$0.644607\pi$
$614$	0	0
$615$	−10.4596	−0.421773
$616$	0	0
$617$	18.6770	0.751906	0.375953	−	0.926639i	$-0.377315\pi$
$617$	18.6770	0.751906	0.375953	+	0.926639i	$0.377315\pi$
$618$	0	0
$619$	−24.4721	−0.983617	−0.491809	−	0.870703i	$-0.663664\pi$
$619$	−24.4721	−0.983617	−0.491809	+	0.870703i	$0.663664\pi$
$620$	0	0
$621$	−6.94567	−0.278720
$622$	0	0
$623$	−8.45963	−0.338928
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−5.43171	−0.216921
$628$	0	0
$629$	−7.46258	−0.297553
$630$	0	0
$631$	40.8495	1.62619	0.813096	−	0.582129i	$-0.197780\pi$
$631$	40.8495	1.62619	0.813096	+	0.582129i	$0.197780\pi$
$632$	0	0
$633$	12.0000	0.476957
$634$	0	0
$635$	1.43171	0.0568155
$636$	0	0
$637$	−10.8913	−0.431530
$638$	0	0
$639$	−8.60719	−0.340495
$640$	0	0
$641$	−7.64903	−0.302119	−0.151059	−	0.988525i	$-0.548268\pi$
$641$	−7.64903	−0.302119	−0.151059	+	0.988525i	$0.548268\pi$
$642$	0	0
$643$	−16.3121	−0.643285	−0.321643	−	0.946861i	$-0.604235\pi$
$643$	−16.3121	−0.643285	−0.321643	+	0.946861i	$0.604235\pi$
$644$	0	0
$645$	−8.00000	−0.315000
$646$	0	0
$647$	25.9736	1.02113	0.510564	−	0.859840i	$-0.329437\pi$
$647$	25.9736	1.02113	0.510564	+	0.859840i	$0.329437\pi$
$648$	0	0
$649$	11.9222	0.467988
$650$	0	0
$651$	−30.3510	−1.18955
$652$	0	0
$653$	−5.08074	−0.198825	−0.0994124	−	0.995046i	$-0.531696\pi$
$653$	−5.08074	−0.198825	−0.0994124	+	0.995046i	$0.531696\pi$
$654$	0	0
$655$	4.37737	0.171038
$656$	0	0
$657$	0.945668	0.0368940
$658$	0	0
$659$	23.4053	0.911741	0.455870	−	0.890046i	$-0.349328\pi$
$659$	23.4053	0.911741	0.455870	+	0.890046i	$0.349328\pi$
$660$	0	0
$661$	−18.1600	−0.706344	−0.353172	−	0.935558i	$-0.614897\pi$
$661$	−18.1600	−0.706344	−0.353172	+	0.935558i	$0.614897\pi$
$662$	0	0
$663$	−0.945668	−0.0367267
$664$	0	0
$665$	−17.8913	−0.693796
$666$	0	0
$667$	−34.3510	−1.33007
$668$	0	0
$669$	−13.1491	−0.508373
$670$	0	0
$671$	−7.70488	−0.297444
$672$	0	0
$673$	−38.6241	−1.48885	−0.744426	−	0.667705i	$-0.767277\pi$
$673$	−38.6241	−1.48885	−0.744426	+	0.667705i	$0.767277\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	1.70488	0.0655240	0.0327620	−	0.999463i	$-0.489570\pi$
$677$	1.70488	0.0655240	0.0327620	+	0.999463i	$0.489570\pi$
$678$	0	0
$679$	45.8385	1.75912
$680$	0	0
$681$	−21.7438	−0.833223
$682$	0	0
$683$	−21.2313	−0.812394	−0.406197	−	0.913785i	$-0.633145\pi$
$683$	−21.2313	−0.812394	−0.406197	+	0.913785i	$0.633145\pi$
$684$	0	0
$685$	−22.4596	−0.858138
$686$	0	0
$687$	1.40530	0.0536155
$688$	0	0
$689$	6.45963	0.246092
$690$	0	0
$691$	−10.8759	−0.413739	−0.206869	−	0.978369i	$-0.566327\pi$
$691$	−10.8759	−0.413739	−0.206869	+	0.978369i	$0.566327\pi$
$692$	0	0
$693$	5.43171	0.206333
$694$	0	0
$695$	9.89134	0.375200
$696$	0	0
$697$	9.89134	0.374661
$698$	0	0
$699$	−21.4053	−0.809623
$700$	0	0
$701$	−24.8929	−0.940190	−0.470095	−	0.882616i	$-0.655780\pi$
$701$	−24.8929	−0.940190	−0.470095	+	0.882616i	$0.655780\pi$
$702$	0	0
$703$	−33.3789	−1.25891
$704$	0	0
$705$	−9.74378	−0.366972
$706$	0	0
$707$	−14.8634	−0.558996
$708$	0	0
$709$	45.1352	1.69509	0.847543	−	0.530726i	$-0.178081\pi$
$709$	45.1352	1.69509	0.847543	+	0.530726i	$0.178081\pi$
$710$	0	0
$711$	14.3510	0.538203
$712$	0	0
$713$	−49.8385	−1.86647
$714$	0	0
$715$	−1.28415	−0.0480244
$716$	0	0
$717$	12.3121	0.459803
$718$	0	0
$719$	14.8106	0.552342	0.276171	−	0.961109i	$-0.410934\pi$
$719$	14.8106	0.552342	0.276171	+	0.961109i	$0.410934\pi$
$720$	0	0
$721$	−77.6212	−2.89076
$722$	0	0
$723$	−2.00000	−0.0743808
$724$	0	0
$725$	−4.94567	−0.183677
$726$	0	0
$727$	9.59622	0.355904	0.177952	−	0.984039i	$-0.443053\pi$
$727$	9.59622	0.355904	0.177952	+	0.984039i	$0.443053\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	7.56534	0.279814
$732$	0	0
$733$	13.0279	0.481197	0.240599	−	0.970625i	$-0.422656\pi$
$733$	13.0279	0.481197	0.240599	+	0.970625i	$0.422656\pi$
$734$	0	0
$735$	10.8913	0.401733
$736$	0	0
$737$	14.3510	0.528625
$738$	0	0
$739$	26.8759	0.988646	0.494323	−	0.869278i	$-0.335416\pi$
$739$	26.8759	0.988646	0.494323	+	0.869278i	$0.335416\pi$
$740$	0	0
$741$	−4.22982	−0.155386
$742$	0	0
$743$	46.2812	1.69789	0.848946	−	0.528479i	$-0.177238\pi$
$743$	46.2812	1.69789	0.848946	+	0.528479i	$0.177238\pi$
$744$	0	0
$745$	−15.8913	−0.582213
$746$	0	0
$747$	−4.60719	−0.168568
$748$	0	0
$749$	−16.9193	−0.618216
$750$	0	0
$751$	34.3510	1.25348	0.626742	−	0.779227i	$-0.284388\pi$
$751$	34.3510	1.25348	0.626742	+	0.779227i	$0.284388\pi$
$752$	0	0
$753$	29.7563	1.08438
$754$	0	0
$755$	−15.1755	−0.552292
$756$	0	0
$757$	−13.2702	−0.482315	−0.241157	−	0.970486i	$-0.577527\pi$
$757$	−13.2702	−0.482315	−0.241157	+	0.970486i	$0.577527\pi$
$758$	0	0
$759$	8.91926	0.323749
$760$	0	0
$761$	−47.5962	−1.72536	−0.862681	−	0.505749i	$-0.831216\pi$
$761$	−47.5962	−1.72536	−0.862681	+	0.505749i	$0.831216\pi$
$762$	0	0
$763$	2.05585	0.0744267
$764$	0	0
$765$	0.945668	0.0341907
$766$	0	0
$767$	9.28415	0.335231
$768$	0	0
$769$	16.6461	0.600273	0.300137	−	0.953896i	$-0.402968\pi$
$769$	16.6461	0.600273	0.300137	+	0.953896i	$0.402968\pi$
$770$	0	0
$771$	−2.37737	−0.0856190
$772$	0	0
$773$	19.1616	0.689193	0.344597	−	0.938751i	$-0.388016\pi$
$773$	19.1616	0.689193	0.344597	+	0.938751i	$0.388016\pi$
$774$	0	0
$775$	−7.17548	−0.257751
$776$	0	0
$777$	33.3789	1.19746
$778$	0	0
$779$	44.2423	1.58514
$780$	0	0
$781$	11.0529	0.395504
$782$	0	0
$783$	4.94567	0.176744
$784$	0	0
$785$	5.32304	0.189987
$786$	0	0
$787$	9.98608	0.355965	0.177983	−	0.984034i	$-0.443043\pi$
$787$	9.98608	0.355965	0.177983	+	0.984034i	$0.443043\pi$
$788$	0	0
$789$	−20.5419	−0.731311
$790$	0	0
$791$	−20.9193	−0.743803
$792$	0	0
$793$	−6.00000	−0.213066
$794$	0	0
$795$	−6.45963	−0.229099
$796$	0	0
$797$	51.0838	1.80948	0.904740	−	0.425964i	$-0.140065\pi$
$797$	51.0838	1.80948	0.904740	+	0.425964i	$0.140065\pi$
$798$	0	0
$799$	9.21438	0.325981
$800$	0	0
$801$	2.00000	0.0706665
$802$	0	0
$803$	−1.21438	−0.0428544
$804$	0	0
$805$	29.3789	1.03547
$806$	0	0
$807$	18.8370	0.663094
$808$	0	0
$809$	52.1336	1.83292	0.916461	−	0.400125i	$-0.131033\pi$
$809$	52.1336	1.83292	0.916461	+	0.400125i	$0.131033\pi$
$810$	0	0
$811$	33.3914	1.17253	0.586265	−	0.810119i	$-0.300598\pi$
$811$	33.3914	1.17253	0.586265	+	0.810119i	$0.300598\pi$
$812$	0	0
$813$	9.74378	0.341729
$814$	0	0
$815$	7.85244	0.275059
$816$	0	0
$817$	33.8385	1.18386
$818$	0	0
$819$	4.22982	0.147802
$820$	0	0
$821$	−3.89134	−0.135809	−0.0679043	−	0.997692i	$-0.521631\pi$
$821$	−3.89134	−0.135809	−0.0679043	+	0.997692i	$0.521631\pi$
$822$	0	0
$823$	−44.6241	−1.55550	−0.777750	−	0.628574i	$-0.783639\pi$
$823$	−44.6241	−1.55550	−0.777750	+	0.628574i	$0.783639\pi$
$824$	0	0
$825$	1.28415	0.0447083
$826$	0	0
$827$	−23.1755	−0.805891	−0.402945	−	0.915224i	$-0.632013\pi$
$827$	−23.1755	−0.805891	−0.402945	+	0.915224i	$0.632013\pi$
$828$	0	0
$829$	−35.8385	−1.24472	−0.622362	−	0.782730i	$-0.713827\pi$
$829$	−35.8385	−1.24472	−0.622362	+	0.782730i	$0.713827\pi$
$830$	0	0
$831$	−19.8913	−0.690023
$832$	0	0
$833$	−10.2996	−0.356859
$834$	0	0
$835$	−4.14756	−0.143532
$836$	0	0
$837$	7.17548	0.248021
$838$	0	0
$839$	40.0947	1.38422	0.692112	−	0.721790i	$-0.256680\pi$
$839$	40.0947	1.38422	0.692112	+	0.721790i	$0.256680\pi$
$840$	0	0
$841$	−4.54037	−0.156564
$842$	0	0
$843$	−18.9193	−0.651614
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	39.5529	1.35905
$848$	0	0
$849$	−0.919260	−0.0315489
$850$	0	0
$851$	54.8106	1.87888
$852$	0	0
$853$	5.61816	0.192362	0.0961810	−	0.995364i	$-0.469337\pi$
$853$	5.61816	0.192362	0.0961810	+	0.995364i	$0.469337\pi$
$854$	0	0
$855$	4.22982	0.144657
$856$	0	0
$857$	1.45811	0.0498082	0.0249041	−	0.999690i	$-0.492072\pi$
$857$	1.45811	0.0498082	0.0249041	+	0.999690i	$0.492072\pi$
$858$	0	0
$859$	50.5405	1.72442	0.862209	−	0.506553i	$-0.169080\pi$
$859$	50.5405	1.72442	0.862209	+	0.506553i	$0.169080\pi$
$860$	0	0
$861$	−44.2423	−1.50777
$862$	0	0
$863$	44.3370	1.50925	0.754625	−	0.656156i	$-0.227819\pi$
$863$	44.3370	1.50925	0.754625	+	0.656156i	$0.227819\pi$
$864$	0	0
$865$	−15.5962	−0.530287
$866$	0	0
$867$	16.1057	0.546979
$868$	0	0
$869$	−18.4288	−0.625153
$870$	0	0
$871$	11.1755	0.378667
$872$	0	0
$873$	−10.8370	−0.366777
$874$	0	0
$875$	4.22982	0.142994
$876$	0	0
$877$	−19.1366	−0.646197	−0.323098	−	0.946365i	$-0.604724\pi$
$877$	−19.1366	−0.646197	−0.323098	+	0.946365i	$0.604724\pi$
$878$	0	0
$879$	14.2423	0.480381
$880$	0	0
$881$	−14.7547	−0.497100	−0.248550	−	0.968619i	$-0.579954\pi$
$881$	−14.7547	−0.497100	−0.248550	+	0.968619i	$0.579954\pi$
$882$	0	0
$883$	−13.2144	−0.444699	−0.222350	−	0.974967i	$-0.571373\pi$
$883$	−13.2144	−0.444699	−0.222350	+	0.974967i	$0.571373\pi$
$884$	0	0
$885$	−9.28415	−0.312083
$886$	0	0
$887$	−24.3246	−0.816739	−0.408369	−	0.912817i	$-0.633902\pi$
$887$	−24.3246	−0.816739	−0.408369	+	0.912817i	$0.633902\pi$
$888$	0	0
$889$	6.05585	0.203107
$890$	0	0
$891$	−1.28415	−0.0430206
$892$	0	0
$893$	41.2144	1.37919
$894$	0	0
$895$	−23.8649	−0.797717
$896$	0	0
$897$	6.94567	0.231909
$898$	0	0
$899$	35.4876	1.18358
$900$	0	0
$901$	6.10866	0.203509
$902$	0	0
$903$	−33.8385	−1.12608
$904$	0	0
$905$	10.4596	0.347690
$906$	0	0
$907$	37.6212	1.24919	0.624596	−	0.780948i	$-0.285264\pi$
$907$	37.6212	1.24919	0.624596	+	0.780948i	$0.285264\pi$
$908$	0	0
$909$	3.51396	0.116551
$910$	0	0
$911$	−27.3230	−0.905253	−0.452626	−	0.891700i	$-0.649513\pi$
$911$	−27.3230	−0.905253	−0.452626	+	0.891700i	$0.649513\pi$
$912$	0	0
$913$	5.91631	0.195801
$914$	0	0
$915$	6.00000	0.198354
$916$	0	0
$917$	18.5155	0.611435
$918$	0	0
$919$	−4.00000	−0.131948	−0.0659739	−	0.997821i	$-0.521015\pi$
$919$	−4.00000	−0.131948	−0.0659739	+	0.997821i	$0.521015\pi$
$920$	0	0
$921$	−2.25622	−0.0743451
$922$	0	0
$923$	8.60719	0.283309
$924$	0	0
$925$	7.89134	0.259466
$926$	0	0
$927$	18.3510	0.602725
$928$	0	0
$929$	−46.4846	−1.52511	−0.762555	−	0.646923i	$-0.776055\pi$
$929$	−46.4846	−1.52511	−0.762555	+	0.646923i	$0.776055\pi$
$930$	0	0
$931$	−46.0683	−1.50983
$932$	0	0
$933$	16.9193	0.553912
$934$	0	0
$935$	−1.21438	−0.0397144
$936$	0	0
$937$	−28.0558	−0.916545	−0.458272	−	0.888812i	$-0.651532\pi$
$937$	−28.0558	−0.916545	−0.458272	+	0.888812i	$0.651532\pi$
$938$	0	0
$939$	−26.4596	−0.863477
$940$	0	0
$941$	−18.9721	−0.618472	−0.309236	−	0.950985i	$-0.600073\pi$
$941$	−18.9721	−0.618472	−0.309236	+	0.950985i	$0.600073\pi$
$942$	0	0
$943$	−72.6491	−2.36578
$944$	0	0
$945$	−4.22982	−0.137596
$946$	0	0
$947$	−8.09474	−0.263044	−0.131522	−	0.991313i	$-0.541986\pi$
$947$	−8.09474	−0.263044	−0.131522	+	0.991313i	$0.541986\pi$
$948$	0	0
$949$	−0.945668	−0.0306977
$950$	0	0
$951$	3.13659	0.101711
$952$	0	0
$953$	28.4860	0.922753	0.461377	−	0.887204i	$-0.347356\pi$
$953$	28.4860	0.922753	0.461377	+	0.887204i	$0.347356\pi$
$954$	0	0
$955$	−13.4317	−0.434640
$956$	0	0
$957$	−6.35097	−0.205298
$958$	0	0
$959$	−95.0001	−3.06771
$960$	0	0
$961$	20.4876	0.660889
$962$	0	0
$963$	4.00000	0.128898
$964$	0	0
$965$	10.0823	0.324559
$966$	0	0
$967$	31.2577	1.00518	0.502591	−	0.864525i	$-0.332380\pi$
$967$	31.2577	1.00518	0.502591	+	0.864525i	$0.332380\pi$
$968$	0	0
$969$	−4.00000	−0.128499
$970$	0	0
$971$	4.37737	0.140477	0.0702383	−	0.997530i	$-0.477624\pi$
$971$	4.37737	0.140477	0.0702383	+	0.997530i	$0.477624\pi$
$972$	0	0
$973$	41.8385	1.34128
$974$	0	0
$975$	1.00000	0.0320256
$976$	0	0
$977$	−17.2702	−0.552523	−0.276262	−	0.961082i	$-0.589096\pi$
$977$	−17.2702	−0.552523	−0.276262	+	0.961082i	$0.589096\pi$
$978$	0	0
$979$	−2.56829	−0.0820830
$980$	0	0
$981$	−0.486038	−0.0155180
$982$	0	0
$983$	−5.36193	−0.171019	−0.0855096	−	0.996337i	$-0.527252\pi$
$983$	−5.36193	−0.171019	−0.0855096	+	0.996337i	$0.527252\pi$
$984$	0	0
$985$	−5.54037	−0.176531
$986$	0	0
$987$	−41.2144	−1.31187
$988$	0	0
$989$	−55.5653	−1.76688
$990$	0	0
$991$	−43.0529	−1.36762	−0.683810	−	0.729660i	$-0.739678\pi$
$991$	−43.0529	−1.36762	−0.683810	+	0.729660i	$0.739678\pi$
$992$	0	0
$993$	24.0125	0.762013
$994$	0	0
$995$	19.7827	0.627153
$996$	0	0
$997$	47.0838	1.49116	0.745579	−	0.666417i	$-0.232173\pi$
$997$	47.0838	1.49116	0.745579	+	0.666417i	$0.232173\pi$
$998$	0	0
$999$	−7.89134	−0.249671

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6240.2.a.bx.1.1	✓	3
4.3	odd	2		6240.2.a.ca.1.3	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6240.2.a.bx.1.1	✓	3	1.1	even	1	trivial
6240.2.a.ca.1.3	yes	3	4.3	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:	\( N \)	\(=\)	\( 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6240.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} -4.22982 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.00000 q^{5} -4.22982 q^{7} +1.00000 q^{9} -1.28415 q^{11} -1.00000 q^{13} +1.00000 q^{15} -0.945668 q^{17} -4.22982 q^{19} +4.22982 q^{21} +6.94567 q^{23} +1.00000 q^{25} -1.00000 q^{27} -4.94567 q^{29} -7.17548 q^{31} +1.28415 q^{33} +4.22982 q^{35} +7.89134 q^{37} +1.00000 q^{39} -10.4596 q^{41} -8.00000 q^{43} -1.00000 q^{45} -9.74378 q^{47} +10.8913 q^{49} +0.945668 q^{51} -6.45963 q^{53} +1.28415 q^{55} +4.22982 q^{57} -9.28415 q^{59} +6.00000 q^{61} -4.22982 q^{63} +1.00000 q^{65} -11.1755 q^{67} -6.94567 q^{69} -8.60719 q^{71} +0.945668 q^{73} -1.00000 q^{75} +5.43171 q^{77} +14.3510 q^{79} +1.00000 q^{81} -4.60719 q^{83} +0.945668 q^{85} +4.94567 q^{87} +2.00000 q^{89} +4.22982 q^{91} +7.17548 q^{93} +4.22982 q^{95} -10.8370 q^{97} -1.28415 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 3 q^{5} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 - 3 * q^5 + 3 * q^9	\( 3 q - 3 q^{3} - 3 q^{5} + 3 q^{9} - 2 q^{11} - 3 q^{13} + 3 q^{15} + 8 q^{17} + 10 q^{23} + 3 q^{25} - 3 q^{27} - 4 q^{29} + 2 q^{31} + 2 q^{33} + 2 q^{37} + 3 q^{39} - 6 q^{41} - 24 q^{43} - 3 q^{45} - 2 q^{47} + 11 q^{49} - 8 q^{51} + 6 q^{53} + 2 q^{55} - 26 q^{59} + 18 q^{61} + 3 q^{65} - 10 q^{67} - 10 q^{69} - 6 q^{71} - 8 q^{73} - 3 q^{75} + 20 q^{77} - 4 q^{79} + 3 q^{81} + 6 q^{83} - 8 q^{85} + 4 q^{87} + 6 q^{89} - 2 q^{93} - 2 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - 3 * q^5 + 3 * q^9 - 2 * q^11 - 3 * q^13 + 3 * q^15 + 8 * q^17 + 10 * q^23 + 3 * q^25 - 3 * q^27 - 4 * q^29 + 2 * q^31 + 2 * q^33 + 2 * q^37 + 3 * q^39 - 6 * q^41 - 24 * q^43 - 3 * q^45 - 2 * q^47 + 11 * q^49 - 8 * q^51 + 6 * q^53 + 2 * q^55 - 26 * q^59 + 18 * q^61 + 3 * q^65 - 10 * q^67 - 10 * q^69 - 6 * q^71 - 8 * q^73 - 3 * q^75 + 20 * q^77 - 4 * q^79 + 3 * q^81 + 6 * q^83 - 8 * q^85 + 4 * q^87 + 6 * q^89 - 2 * q^93 - 2 * q^99

Introduction

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