LMFDB - Embedded newform 7248.2.a.bm.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7248,2,Mod(1,7248)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7248, 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("7248.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7248 = 2^{4} \cdot 3 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7248.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:	$57.8755713850$
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} - 27x^{8} + 45x^{7} + 258x^{6} - 289x^{5} - 1133x^{4} + 510x^{3} + 2070x^{2} + 341x - 500 $ x^10 - 2x^9 - 27x^8 + 45x^7 + 258x^6 - 289x^5 - 1133x^4 + 510x^3 + 2070x^2 + 341*x - 500
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3624)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.840171$ of defining polynomial
Character	$\chi$	$=$	7248.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -0.840171 q^{5} -5.06275 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -0.840171 q^{5} -5.06275 q^{7} +1.00000 q^{9} -5.28116 q^{11} +2.84960 q^{13} +0.840171 q^{15} +6.04127 q^{17} +2.17709 q^{19} +5.06275 q^{21} +3.88550 q^{23} -4.29411 q^{25} -1.00000 q^{27} +3.42565 q^{29} -6.90163 q^{31} +5.28116 q^{33} +4.25357 q^{35} -9.10148 q^{37} -2.84960 q^{39} +8.55092 q^{41} +6.84802 q^{43} -0.840171 q^{45} -4.60862 q^{47} +18.6314 q^{49} -6.04127 q^{51} -9.69715 q^{53} +4.43708 q^{55} -2.17709 q^{57} +10.2604 q^{59} +4.38199 q^{61} -5.06275 q^{63} -2.39415 q^{65} +15.5362 q^{67} -3.88550 q^{69} -1.61922 q^{71} -11.6675 q^{73} +4.29411 q^{75} +26.7372 q^{77} +5.79273 q^{79} +1.00000 q^{81} +2.59569 q^{83} -5.07570 q^{85} -3.42565 q^{87} -7.38593 q^{89} -14.4268 q^{91} +6.90163 q^{93} -1.82913 q^{95} -17.7507 q^{97} -5.28116 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 10 q^{3} + 2 q^{5} - 8 q^{7} + 10 q^{9}+O(q^{10}) $ 10 * q - 10 * q^3 + 2 * q^5 - 8 * q^7 + 10 * q^9	$ 10 q - 10 q^{3} + 2 q^{5} - 8 q^{7} + 10 q^{9} - 7 q^{11} + 6 q^{13} - 2 q^{15} + 7 q^{17} + 8 q^{21} - 25 q^{23} + 8 q^{25} - 10 q^{27} + 12 q^{29} - 11 q^{31} + 7 q^{33} - 9 q^{35} - 3 q^{37} - 6 q^{39} + 12 q^{41} + 2 q^{45} - 31 q^{47} + 14 q^{49} - 7 q^{51} + q^{53} - 9 q^{55} - 19 q^{59} + 24 q^{61} - 8 q^{63} + 20 q^{65} + q^{67} + 25 q^{69} - 34 q^{71} - 18 q^{73} - 8 q^{75} + 27 q^{77} - 25 q^{79} + 10 q^{81} - 14 q^{83} - 3 q^{85} - 12 q^{87} + 20 q^{89} + 12 q^{91} + 11 q^{93} - 48 q^{95} - 15 q^{97} - 7 q^{99}+O(q^{100}) $ 10 * q - 10 * q^3 + 2 * q^5 - 8 * q^7 + 10 * q^9 - 7 * q^11 + 6 * q^13 - 2 * q^15 + 7 * q^17 + 8 * q^21 - 25 * q^23 + 8 * q^25 - 10 * q^27 + 12 * q^29 - 11 * q^31 + 7 * q^33 - 9 * q^35 - 3 * q^37 - 6 * q^39 + 12 * q^41 + 2 * q^45 - 31 * q^47 + 14 * q^49 - 7 * q^51 + q^53 - 9 * q^55 - 19 * q^59 + 24 * q^61 - 8 * q^63 + 20 * q^65 + q^67 + 25 * q^69 - 34 * q^71 - 18 * q^73 - 8 * q^75 + 27 * q^77 - 25 * q^79 + 10 * q^81 - 14 * q^83 - 3 * q^85 - 12 * q^87 + 20 * q^89 + 12 * q^91 + 11 * q^93 - 48 * q^95 - 15 * q^97 - 7 * 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
$5$	−0.840171	−0.375736	−0.187868	−	0.982194i	$-0.560158\pi$
$5$	−0.840171	−0.375736	−0.187868	+	0.982194i	$0.560158\pi$
$6$	0	0
$7$	−5.06275	−1.91354	−0.956769	−	0.290848i	$-0.906062\pi$
$7$	−5.06275	−1.91354	−0.956769	+	0.290848i	$0.906062\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.28116	−1.59233	−0.796164	−	0.605080i	$-0.793141\pi$
$11$	−5.28116	−1.59233	−0.796164	+	0.605080i	$0.793141\pi$
$12$	0	0
$13$	2.84960	0.790337	0.395168	−	0.918609i	$-0.370686\pi$
$13$	2.84960	0.790337	0.395168	+	0.918609i	$0.370686\pi$
$14$	0	0
$15$	0.840171	0.216931
$16$	0	0
$17$	6.04127	1.46522	0.732612	−	0.680647i	$-0.238301\pi$
$17$	6.04127	1.46522	0.732612	+	0.680647i	$0.238301\pi$
$18$	0	0
$19$	2.17709	0.499458	0.249729	−	0.968316i	$-0.419658\pi$
$19$	2.17709	0.499458	0.249729	+	0.968316i	$0.419658\pi$
$20$	0	0
$21$	5.06275	1.10478
$22$	0	0
$23$	3.88550	0.810182	0.405091	−	0.914276i	$-0.367240\pi$
$23$	3.88550	0.810182	0.405091	+	0.914276i	$0.367240\pi$
$24$	0	0
$25$	−4.29411	−0.858822
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	3.42565	0.636127	0.318064	−	0.948069i	$-0.396967\pi$
$29$	3.42565	0.636127	0.318064	+	0.948069i	$0.396967\pi$
$30$	0	0
$31$	−6.90163	−1.23957	−0.619784	−	0.784772i	$-0.712780\pi$
$31$	−6.90163	−1.23957	−0.619784	+	0.784772i	$0.712780\pi$
$32$	0	0
$33$	5.28116	0.919332
$34$	0	0
$35$	4.25357	0.718985
$36$	0	0
$37$	−9.10148	−1.49627	−0.748137	−	0.663544i	$-0.769052\pi$
$37$	−9.10148	−1.49627	−0.748137	+	0.663544i	$0.769052\pi$
$38$	0	0
$39$	−2.84960	−0.456301
$40$	0	0
$41$	8.55092	1.33543	0.667715	−	0.744417i	$-0.267273\pi$
$41$	8.55092	1.33543	0.667715	+	0.744417i	$0.267273\pi$
$42$	0	0
$43$	6.84802	1.04431	0.522157	−	0.852849i	$-0.325128\pi$
$43$	6.84802	1.04431	0.522157	+	0.852849i	$0.325128\pi$
$44$	0	0
$45$	−0.840171	−0.125245
$46$	0	0
$47$	−4.60862	−0.672237	−0.336118	−	0.941820i	$-0.609114\pi$
$47$	−4.60862	−0.672237	−0.336118	+	0.941820i	$0.609114\pi$
$48$	0	0
$49$	18.6314	2.66163
$50$	0	0
$51$	−6.04127	−0.845947
$52$	0	0
$53$	−9.69715	−1.33201	−0.666003	−	0.745949i	$-0.731996\pi$
$53$	−9.69715	−1.33201	−0.666003	+	0.745949i	$0.731996\pi$
$54$	0	0
$55$	4.43708	0.598295
$56$	0	0
$57$	−2.17709	−0.288362
$58$	0	0
$59$	10.2604	1.33579	0.667893	−	0.744257i	$-0.267196\pi$
$59$	10.2604	1.33579	0.667893	+	0.744257i	$0.267196\pi$
$60$	0	0
$61$	4.38199	0.561056	0.280528	−	0.959846i	$-0.409490\pi$
$61$	4.38199	0.561056	0.280528	+	0.959846i	$0.409490\pi$
$62$	0	0
$63$	−5.06275	−0.637846
$64$	0	0
$65$	−2.39415	−0.296958
$66$	0	0
$67$	15.5362	1.89805	0.949026	−	0.315198i	$-0.102071\pi$
$67$	15.5362	1.89805	0.949026	+	0.315198i	$0.102071\pi$
$68$	0	0
$69$	−3.88550	−0.467759
$70$	0	0
$71$	−1.61922	−0.192166	−0.0960832	−	0.995373i	$-0.530631\pi$
$71$	−1.61922	−0.192166	−0.0960832	+	0.995373i	$0.530631\pi$
$72$	0	0
$73$	−11.6675	−1.36558	−0.682789	−	0.730615i	$-0.739233\pi$
$73$	−11.6675	−1.36558	−0.682789	+	0.730615i	$0.739233\pi$
$74$	0	0
$75$	4.29411	0.495841
$76$	0	0
$77$	26.7372	3.04698
$78$	0	0
$79$	5.79273	0.651733	0.325867	−	0.945416i	$-0.394344\pi$
$79$	5.79273	0.651733	0.325867	+	0.945416i	$0.394344\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	2.59569	0.284914	0.142457	−	0.989801i	$-0.454500\pi$
$83$	2.59569	0.284914	0.142457	+	0.989801i	$0.454500\pi$
$84$	0	0
$85$	−5.07570	−0.550537
$86$	0	0
$87$	−3.42565	−0.367268
$88$	0	0
$89$	−7.38593	−0.782907	−0.391454	−	0.920198i	$-0.628028\pi$
$89$	−7.38593	−0.782907	−0.391454	+	0.920198i	$0.628028\pi$
$90$	0	0
$91$	−14.4268	−1.51234
$92$	0	0
$93$	6.90163	0.715665
$94$	0	0
$95$	−1.82913	−0.187665
$96$	0	0
$97$	−17.7507	−1.80231	−0.901155	−	0.433498i	$-0.857279\pi$
$97$	−17.7507	−1.80231	−0.901155	+	0.433498i	$0.857279\pi$
$98$	0	0
$99$	−5.28116	−0.530776
$100$	0	0
$101$	7.46889	0.743183	0.371591	−	0.928396i	$-0.378812\pi$
$101$	7.46889	0.743183	0.371591	+	0.928396i	$0.378812\pi$
$102$	0	0
$103$	18.7415	1.84666	0.923328	−	0.384013i	$-0.125458\pi$
$103$	18.7415	1.84666	0.923328	+	0.384013i	$0.125458\pi$
$104$	0	0
$105$	−4.25357	−0.415106
$106$	0	0
$107$	8.73112	0.844070	0.422035	−	0.906580i	$-0.361316\pi$
$107$	8.73112	0.844070	0.422035	+	0.906580i	$0.361316\pi$
$108$	0	0
$109$	9.95458	0.953476	0.476738	−	0.879046i	$-0.341819\pi$
$109$	9.95458	0.953476	0.476738	+	0.879046i	$0.341819\pi$
$110$	0	0
$111$	9.10148	0.863875
$112$	0	0
$113$	2.32267	0.218498	0.109249	−	0.994014i	$-0.465155\pi$
$113$	2.32267	0.218498	0.109249	+	0.994014i	$0.465155\pi$
$114$	0	0
$115$	−3.26448	−0.304415
$116$	0	0
$117$	2.84960	0.263446
$118$	0	0
$119$	−30.5854	−2.80376
$120$	0	0
$121$	16.8906	1.53551
$122$	0	0
$123$	−8.55092	−0.771010
$124$	0	0
$125$	7.80865	0.698426
$126$	0	0
$127$	15.9688	1.41700	0.708502	−	0.705709i	$-0.249371\pi$
$127$	15.9688	1.41700	0.708502	+	0.705709i	$0.249371\pi$
$128$	0	0
$129$	−6.84802	−0.602935
$130$	0	0
$131$	−3.62082	−0.316353	−0.158176	−	0.987411i	$-0.550561\pi$
$131$	−3.62082	−0.316353	−0.158176	+	0.987411i	$0.550561\pi$
$132$	0	0
$133$	−11.0220	−0.955733
$134$	0	0
$135$	0.840171	0.0723104
$136$	0	0
$137$	20.1501	1.72154	0.860768	−	0.508997i	$-0.169984\pi$
$137$	20.1501	1.72154	0.860768	+	0.508997i	$0.169984\pi$
$138$	0	0
$139$	−10.4495	−0.886317	−0.443158	−	0.896443i	$-0.646142\pi$
$139$	−10.4495	−0.886317	−0.443158	+	0.896443i	$0.646142\pi$
$140$	0	0
$141$	4.60862	0.388116
$142$	0	0
$143$	−15.0492	−1.25848
$144$	0	0
$145$	−2.87813	−0.239016
$146$	0	0
$147$	−18.6314	−1.53669
$148$	0	0
$149$	−18.9620	−1.55343	−0.776714	−	0.629853i	$-0.783115\pi$
$149$	−18.9620	−1.55343	−0.776714	+	0.629853i	$0.783115\pi$
$150$	0	0
$151$	−1.00000	−0.0813788
$151$	−1.00000	−0.0813788
$152$	0	0
$153$	6.04127	0.488408
$154$	0	0
$155$	5.79855	0.465751
$156$	0	0
$157$	−0.973646	−0.0777054	−0.0388527	−	0.999245i	$-0.512370\pi$
$157$	−0.973646	−0.0777054	−0.0388527	+	0.999245i	$0.512370\pi$
$158$	0	0
$159$	9.69715	0.769034
$160$	0	0
$161$	−19.6713	−1.55032
$162$	0	0
$163$	−13.0115	−1.01914	−0.509570	−	0.860429i	$-0.670196\pi$
$163$	−13.0115	−1.01914	−0.509570	+	0.860429i	$0.670196\pi$
$164$	0	0
$165$	−4.43708	−0.345426
$166$	0	0
$167$	−22.9324	−1.77456	−0.887281	−	0.461230i	$-0.847408\pi$
$167$	−22.9324	−1.77456	−0.887281	+	0.461230i	$0.847408\pi$
$168$	0	0
$169$	−4.87978	−0.375368
$170$	0	0
$171$	2.17709	0.166486
$172$	0	0
$173$	−3.35409	−0.255007	−0.127503	−	0.991838i	$-0.540696\pi$
$173$	−3.35409	−0.255007	−0.127503	+	0.991838i	$0.540696\pi$
$174$	0	0
$175$	21.7400	1.64339
$176$	0	0
$177$	−10.2604	−0.771216
$178$	0	0
$179$	−6.84046	−0.511280	−0.255640	−	0.966772i	$-0.582286\pi$
$179$	−6.84046	−0.511280	−0.255640	+	0.966772i	$0.582286\pi$
$180$	0	0
$181$	5.62937	0.418428	0.209214	−	0.977870i	$-0.432910\pi$
$181$	5.62937	0.418428	0.209214	+	0.977870i	$0.432910\pi$
$182$	0	0
$183$	−4.38199	−0.323926
$184$	0	0
$185$	7.64680	0.562204
$186$	0	0
$187$	−31.9049	−2.33312
$188$	0	0
$189$	5.06275	0.368261
$190$	0	0
$191$	−22.3799	−1.61935	−0.809676	−	0.586877i	$-0.800357\pi$
$191$	−22.3799	−1.61935	−0.809676	+	0.586877i	$0.800357\pi$
$192$	0	0
$193$	−7.14401	−0.514237	−0.257118	−	0.966380i	$-0.582773\pi$
$193$	−7.14401	−0.514237	−0.257118	+	0.966380i	$0.582773\pi$
$194$	0	0
$195$	2.39415	0.171449
$196$	0	0
$197$	−10.0110	−0.713256	−0.356628	−	0.934247i	$-0.616074\pi$
$197$	−10.0110	−0.713256	−0.356628	+	0.934247i	$0.616074\pi$
$198$	0	0
$199$	0.288851	0.0204761	0.0102380	−	0.999948i	$-0.496741\pi$
$199$	0.288851	0.0204761	0.0102380	+	0.999948i	$0.496741\pi$
$200$	0	0
$201$	−15.5362	−1.09584
$202$	0	0
$203$	−17.3432	−1.21725
$204$	0	0
$205$	−7.18424	−0.501769
$206$	0	0
$207$	3.88550	0.270061
$208$	0	0
$209$	−11.4975	−0.795302
$210$	0	0
$211$	−22.7364	−1.56524	−0.782619	−	0.622501i	$-0.786117\pi$
$211$	−22.7364	−1.56524	−0.782619	+	0.622501i	$0.786117\pi$
$212$	0	0
$213$	1.61922	0.110947
$214$	0	0
$215$	−5.75351	−0.392386
$216$	0	0
$217$	34.9412	2.37196
$218$	0	0
$219$	11.6675	0.788417
$220$	0	0
$221$	17.2152	1.15802
$222$	0	0
$223$	−4.00024	−0.267876	−0.133938	−	0.990990i	$-0.542762\pi$
$223$	−4.00024	−0.267876	−0.133938	+	0.990990i	$0.542762\pi$
$224$	0	0
$225$	−4.29411	−0.286274
$226$	0	0
$227$	−17.4400	−1.15753	−0.578766	−	0.815494i	$-0.696465\pi$
$227$	−17.4400	−1.15753	−0.578766	+	0.815494i	$0.696465\pi$
$228$	0	0
$229$	11.9220	0.787826	0.393913	−	0.919148i	$-0.371121\pi$
$229$	11.9220	0.787826	0.393913	+	0.919148i	$0.371121\pi$
$230$	0	0
$231$	−26.7372	−1.75918
$232$	0	0
$233$	−20.2911	−1.32931	−0.664657	−	0.747149i	$-0.731422\pi$
$233$	−20.2911	−1.32931	−0.664657	+	0.747149i	$0.731422\pi$
$234$	0	0
$235$	3.87203	0.252583
$236$	0	0
$237$	−5.79273	−0.376278
$238$	0	0
$239$	−5.39330	−0.348864	−0.174432	−	0.984669i	$-0.555809\pi$
$239$	−5.39330	−0.348864	−0.174432	+	0.984669i	$0.555809\pi$
$240$	0	0
$241$	−5.28935	−0.340717	−0.170358	−	0.985382i	$-0.554493\pi$
$241$	−5.28935	−0.340717	−0.170358	+	0.985382i	$0.554493\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−15.6536	−1.00007
$246$	0	0
$247$	6.20383	0.394740
$248$	0	0
$249$	−2.59569	−0.164495
$250$	0	0
$251$	−7.87886	−0.497309	−0.248654	−	0.968592i	$-0.579988\pi$
$251$	−7.87886	−0.497309	−0.248654	+	0.968592i	$0.579988\pi$
$252$	0	0
$253$	−20.5199	−1.29008
$254$	0	0
$255$	5.07570	0.317853
$256$	0	0
$257$	3.30095	0.205907	0.102954	−	0.994686i	$-0.467171\pi$
$257$	3.30095	0.205907	0.102954	+	0.994686i	$0.467171\pi$
$258$	0	0
$259$	46.0785	2.86318
$260$	0	0
$261$	3.42565	0.212042
$262$	0	0
$263$	−4.22907	−0.260776	−0.130388	−	0.991463i	$-0.541622\pi$
$263$	−4.22907	−0.260776	−0.130388	+	0.991463i	$0.541622\pi$
$264$	0	0
$265$	8.14727	0.500483
$266$	0	0
$267$	7.38593	0.452012
$268$	0	0
$269$	−20.1585	−1.22908	−0.614542	−	0.788884i	$-0.710659\pi$
$269$	−20.1585	−1.22908	−0.614542	+	0.788884i	$0.710659\pi$
$270$	0	0
$271$	−23.4935	−1.42713	−0.713566	−	0.700588i	$-0.752921\pi$
$271$	−23.4935	−1.42713	−0.713566	+	0.700588i	$0.752921\pi$
$272$	0	0
$273$	14.4268	0.873150
$274$	0	0
$275$	22.6779	1.36753
$276$	0	0
$277$	−29.4348	−1.76857	−0.884284	−	0.466950i	$-0.845353\pi$
$277$	−29.4348	−1.76857	−0.884284	+	0.466950i	$0.845353\pi$
$278$	0	0
$279$	−6.90163	−0.413190
$280$	0	0
$281$	−11.3527	−0.677247	−0.338623	−	0.940922i	$-0.609961\pi$
$281$	−11.3527	−0.677247	−0.338623	+	0.940922i	$0.609961\pi$
$282$	0	0
$283$	−24.1900	−1.43795	−0.718973	−	0.695038i	$-0.755387\pi$
$283$	−24.1900	−1.43795	−0.718973	+	0.695038i	$0.755387\pi$
$284$	0	0
$285$	1.82913	0.108348
$286$	0	0
$287$	−43.2911	−2.55539
$288$	0	0
$289$	19.4969	1.14688
$290$	0	0
$291$	17.7507	1.04056
$292$	0	0
$293$	2.64953	0.154787	0.0773936	−	0.997001i	$-0.475340\pi$
$293$	2.64953	0.154787	0.0773936	+	0.997001i	$0.475340\pi$
$294$	0	0
$295$	−8.62046	−0.501903
$296$	0	0
$297$	5.28116	0.306444
$298$	0	0
$299$	11.0721	0.640317
$300$	0	0
$301$	−34.6698	−1.99833
$302$	0	0
$303$	−7.46889	−0.429077
$304$	0	0
$305$	−3.68162	−0.210809
$306$	0	0
$307$	23.5848	1.34606	0.673028	−	0.739617i	$-0.264994\pi$
$307$	23.5848	1.34606	0.673028	+	0.739617i	$0.264994\pi$
$308$	0	0
$309$	−18.7415	−1.06617
$310$	0	0
$311$	−6.65183	−0.377191	−0.188595	−	0.982055i	$-0.560393\pi$
$311$	−6.65183	−0.377191	−0.188595	+	0.982055i	$0.560393\pi$
$312$	0	0
$313$	0.147479	0.00833603	0.00416802	−	0.999991i	$-0.498673\pi$
$313$	0.147479	0.00833603	0.00416802	+	0.999991i	$0.498673\pi$
$314$	0	0
$315$	4.25357	0.239662
$316$	0	0
$317$	21.1626	1.18861	0.594304	−	0.804240i	$-0.297428\pi$
$317$	21.1626	1.18861	0.594304	+	0.804240i	$0.297428\pi$
$318$	0	0
$319$	−18.0914	−1.01292
$320$	0	0
$321$	−8.73112	−0.487324
$322$	0	0
$323$	13.1524	0.731818
$324$	0	0
$325$	−12.2365	−0.678759
$326$	0	0
$327$	−9.95458	−0.550489
$328$	0	0
$329$	23.3323	1.28635
$330$	0	0
$331$	26.7904	1.47253	0.736265	−	0.676693i	$-0.236587\pi$
$331$	26.7904	1.47253	0.736265	+	0.676693i	$0.236587\pi$
$332$	0	0
$333$	−9.10148	−0.498758
$334$	0	0
$335$	−13.0531	−0.713166
$336$	0	0
$337$	3.83343	0.208820	0.104410	−	0.994534i	$-0.466705\pi$
$337$	3.83343	0.208820	0.104410	+	0.994534i	$0.466705\pi$
$338$	0	0
$339$	−2.32267	−0.126150
$340$	0	0
$341$	36.4486	1.97380
$342$	0	0
$343$	−58.8868	−3.17959
$344$	0	0
$345$	3.26448	0.175754
$346$	0	0
$347$	22.2895	1.19656	0.598281	−	0.801286i	$-0.295851\pi$
$347$	22.2895	1.19656	0.598281	+	0.801286i	$0.295851\pi$
$348$	0	0
$349$	12.2971	0.658247	0.329123	−	0.944287i	$-0.393247\pi$
$349$	12.2971	0.658247	0.329123	+	0.944287i	$0.393247\pi$
$350$	0	0
$351$	−2.84960	−0.152100
$352$	0	0
$353$	−9.20341	−0.489848	−0.244924	−	0.969542i	$-0.578763\pi$
$353$	−9.20341	−0.489848	−0.244924	+	0.969542i	$0.578763\pi$
$354$	0	0
$355$	1.36042	0.0722038
$356$	0	0
$357$	30.5854	1.61875
$358$	0	0
$359$	29.2305	1.54273	0.771364	−	0.636394i	$-0.219575\pi$
$359$	29.2305	1.54273	0.771364	+	0.636394i	$0.219575\pi$
$360$	0	0
$361$	−14.2603	−0.750541
$362$	0	0
$363$	−16.8906	−0.886528
$364$	0	0
$365$	9.80271	0.513097
$366$	0	0
$367$	−1.20631	−0.0629688	−0.0314844	−	0.999504i	$-0.510023\pi$
$367$	−1.20631	−0.0629688	−0.0314844	+	0.999504i	$0.510023\pi$
$368$	0	0
$369$	8.55092	0.445143
$370$	0	0
$371$	49.0942	2.54884
$372$	0	0
$373$	−0.315179	−0.0163194	−0.00815969	−	0.999967i	$-0.502597\pi$
$373$	−0.315179	−0.0163194	−0.00815969	+	0.999967i	$0.502597\pi$
$374$	0	0
$375$	−7.80865	−0.403237
$376$	0	0
$377$	9.76173	0.502755
$378$	0	0
$379$	9.71795	0.499178	0.249589	−	0.968352i	$-0.419705\pi$
$379$	9.71795	0.499178	0.249589	+	0.968352i	$0.419705\pi$
$380$	0	0
$381$	−15.9688	−0.818108
$382$	0	0
$383$	−1.79806	−0.0918764	−0.0459382	−	0.998944i	$-0.514628\pi$
$383$	−1.79806	−0.0918764	−0.0459382	+	0.998944i	$0.514628\pi$
$384$	0	0
$385$	−22.4638	−1.14486
$386$	0	0
$387$	6.84802	0.348105
$388$	0	0
$389$	2.41147	0.122266	0.0611332	−	0.998130i	$-0.480529\pi$
$389$	2.41147	0.122266	0.0611332	+	0.998130i	$0.480529\pi$
$390$	0	0
$391$	23.4733	1.18710
$392$	0	0
$393$	3.62082	0.182646
$394$	0	0
$395$	−4.86689	−0.244880
$396$	0	0
$397$	13.9868	0.701979	0.350989	−	0.936379i	$-0.385845\pi$
$397$	13.9868	0.701979	0.350989	+	0.936379i	$0.385845\pi$
$398$	0	0
$399$	11.0220	0.551793
$400$	0	0
$401$	5.13135	0.256247	0.128124	−	0.991758i	$-0.459105\pi$
$401$	5.13135	0.256247	0.128124	+	0.991758i	$0.459105\pi$
$402$	0	0
$403$	−19.6669	−0.979677
$404$	0	0
$405$	−0.840171	−0.0417484
$406$	0	0
$407$	48.0664	2.38256
$408$	0	0
$409$	9.83539	0.486329	0.243164	−	0.969985i	$-0.421815\pi$
$409$	9.83539	0.486329	0.243164	+	0.969985i	$0.421815\pi$
$410$	0	0
$411$	−20.1501	−0.993930
$412$	0	0
$413$	−51.9456	−2.55608
$414$	0	0
$415$	−2.18083	−0.107052
$416$	0	0
$417$	10.4495	0.511715
$418$	0	0
$419$	9.70549	0.474144	0.237072	−	0.971492i	$-0.423812\pi$
$419$	9.70549	0.474144	0.237072	+	0.971492i	$0.423812\pi$
$420$	0	0
$421$	14.0905	0.686729	0.343365	−	0.939202i	$-0.388433\pi$
$421$	14.0905	0.686729	0.343365	+	0.939202i	$0.388433\pi$
$422$	0	0
$423$	−4.60862	−0.224079
$424$	0	0
$425$	−25.9419	−1.25837
$426$	0	0
$427$	−22.1849	−1.07360
$428$	0	0
$429$	15.0492	0.726582
$430$	0	0
$431$	−19.5772	−0.943002	−0.471501	−	0.881866i	$-0.656288\pi$
$431$	−19.5772	−0.943002	−0.471501	+	0.881866i	$0.656288\pi$
$432$	0	0
$433$	19.6191	0.942834	0.471417	−	0.881911i	$-0.343743\pi$
$433$	19.6191	0.942834	0.471417	+	0.881911i	$0.343743\pi$
$434$	0	0
$435$	2.87813	0.137996
$436$	0	0
$437$	8.45908	0.404652
$438$	0	0
$439$	37.0490	1.76825	0.884127	−	0.467247i	$-0.154754\pi$
$439$	37.0490	1.76825	0.884127	+	0.467247i	$0.154754\pi$
$440$	0	0
$441$	18.6314	0.887210
$442$	0	0
$443$	32.2348	1.53152	0.765761	−	0.643125i	$-0.222363\pi$
$443$	32.2348	1.53152	0.765761	+	0.643125i	$0.222363\pi$
$444$	0	0
$445$	6.20545	0.294166
$446$	0	0
$447$	18.9620	0.896872
$448$	0	0
$449$	−3.86193	−0.182256	−0.0911278	−	0.995839i	$-0.529047\pi$
$449$	−3.86193	−0.182256	−0.0911278	+	0.995839i	$0.529047\pi$
$450$	0	0
$451$	−45.1587	−2.12644
$452$	0	0
$453$	1.00000	0.0469841
$454$	0	0
$455$	12.1210	0.568241
$456$	0	0
$457$	−41.1197	−1.92350	−0.961750	−	0.273928i	$-0.911677\pi$
$457$	−41.1197	−1.92350	−0.961750	+	0.273928i	$0.911677\pi$
$458$	0	0
$459$	−6.04127	−0.281982
$460$	0	0
$461$	−38.4267	−1.78971	−0.894856	−	0.446355i	$-0.852722\pi$
$461$	−38.4267	−1.78971	−0.894856	+	0.446355i	$0.852722\pi$
$462$	0	0
$463$	−12.2378	−0.568738	−0.284369	−	0.958715i	$-0.591784\pi$
$463$	−12.2378	−0.568738	−0.284369	+	0.958715i	$0.591784\pi$
$464$	0	0
$465$	−5.79855	−0.268901
$466$	0	0
$467$	22.5546	1.04370	0.521850	−	0.853037i	$-0.325242\pi$
$467$	22.5546	1.04370	0.521850	+	0.853037i	$0.325242\pi$
$468$	0	0
$469$	−78.6560	−3.63199
$470$	0	0
$471$	0.973646	0.0448633
$472$	0	0
$473$	−36.1655	−1.66289
$474$	0	0
$475$	−9.34866	−0.428946
$476$	0	0
$477$	−9.69715	−0.444002
$478$	0	0
$479$	−26.4527	−1.20865	−0.604327	−	0.796736i	$-0.706558\pi$
$479$	−26.4527	−1.20865	−0.604327	+	0.796736i	$0.706558\pi$
$480$	0	0
$481$	−25.9356	−1.18256
$482$	0	0
$483$	19.6713	0.895075
$484$	0	0
$485$	14.9136	0.677192
$486$	0	0
$487$	−13.8872	−0.629287	−0.314644	−	0.949210i	$-0.601885\pi$
$487$	−13.8872	−0.629287	−0.314644	+	0.949210i	$0.601885\pi$
$488$	0	0
$489$	13.0115	0.588401
$490$	0	0
$491$	4.82759	0.217866	0.108933	−	0.994049i	$-0.465257\pi$
$491$	4.82759	0.217866	0.108933	+	0.994049i	$0.465257\pi$
$492$	0	0
$493$	20.6953	0.932068
$494$	0	0
$495$	4.43708	0.199432
$496$	0	0
$497$	8.19771	0.367718
$498$	0	0
$499$	33.9233	1.51862	0.759308	−	0.650731i	$-0.225537\pi$
$499$	33.9233	1.51862	0.759308	+	0.650731i	$0.225537\pi$
$500$	0	0
$501$	22.9324	1.02454
$502$	0	0
$503$	−13.7619	−0.613615	−0.306807	−	0.951772i	$-0.599261\pi$
$503$	−13.7619	−0.613615	−0.306807	+	0.951772i	$0.599261\pi$
$504$	0	0
$505$	−6.27515	−0.279240
$506$	0	0
$507$	4.87978	0.216719
$508$	0	0
$509$	−42.7862	−1.89646	−0.948232	−	0.317579i	$-0.897130\pi$
$509$	−42.7862	−1.89646	−0.948232	+	0.317579i	$0.897130\pi$
$510$	0	0
$511$	59.0697	2.61309
$512$	0	0
$513$	−2.17709	−0.0961208
$514$	0	0
$515$	−15.7461	−0.693855
$516$	0	0
$517$	24.3389	1.07042
$518$	0	0
$519$	3.35409	0.147228
$520$	0	0
$521$	5.48520	0.240311	0.120155	−	0.992755i	$-0.461661\pi$
$521$	5.48520	0.240311	0.120155	+	0.992755i	$0.461661\pi$
$522$	0	0
$523$	−3.51584	−0.153737	−0.0768685	−	0.997041i	$-0.524492\pi$
$523$	−3.51584	−0.153737	−0.0768685	+	0.997041i	$0.524492\pi$
$524$	0	0
$525$	−21.7400	−0.948811
$526$	0	0
$527$	−41.6946	−1.81625
$528$	0	0
$529$	−7.90290	−0.343604
$530$	0	0
$531$	10.2604	0.445262
$532$	0	0
$533$	24.3667	1.05544
$534$	0	0
$535$	−7.33564	−0.317147
$536$	0	0
$537$	6.84046	0.295188
$538$	0	0
$539$	−98.3954	−4.23819
$540$	0	0
$541$	20.8765	0.897550	0.448775	−	0.893645i	$-0.351860\pi$
$541$	20.8765	0.897550	0.448775	+	0.893645i	$0.351860\pi$
$542$	0	0
$543$	−5.62937	−0.241579
$544$	0	0
$545$	−8.36355	−0.358255
$546$	0	0
$547$	22.4602	0.960330	0.480165	−	0.877178i	$-0.340577\pi$
$547$	22.4602	0.960330	0.480165	+	0.877178i	$0.340577\pi$
$548$	0	0
$549$	4.38199	0.187019
$550$	0	0
$551$	7.45794	0.317719
$552$	0	0
$553$	−29.3271	−1.24712
$554$	0	0
$555$	−7.64680	−0.324589
$556$	0	0
$557$	28.9328	1.22592	0.612961	−	0.790113i	$-0.289978\pi$
$557$	28.9328	1.22592	0.612961	+	0.790113i	$0.289978\pi$
$558$	0	0
$559$	19.5141	0.825360
$560$	0	0
$561$	31.9049	1.34703
$562$	0	0
$563$	−38.5214	−1.62348	−0.811741	−	0.584018i	$-0.801480\pi$
$563$	−38.5214	−1.62348	−0.811741	+	0.584018i	$0.801480\pi$
$564$	0	0
$565$	−1.95144	−0.0820976
$566$	0	0
$567$	−5.06275	−0.212615
$568$	0	0
$569$	−0.0270422	−0.00113367	−0.000566835	−	1.00000i	$-0.500180\pi$
$569$	−0.0270422	−0.00113367	−0.000566835		1.00000i	$0.500180\pi$
$570$	0	0
$571$	−15.3854	−0.643861	−0.321930	−	0.946763i	$-0.604332\pi$
$571$	−15.3854	−0.643861	−0.321930	+	0.946763i	$0.604332\pi$
$572$	0	0
$573$	22.3799	0.934934
$574$	0	0
$575$	−16.6848	−0.695803
$576$	0	0
$577$	17.8931	0.744901	0.372451	−	0.928052i	$-0.378518\pi$
$577$	17.8931	0.744901	0.372451	+	0.928052i	$0.378518\pi$
$578$	0	0
$579$	7.14401	0.296895
$580$	0	0
$581$	−13.1413	−0.545194
$582$	0	0
$583$	51.2122	2.12099
$584$	0	0
$585$	−2.39415	−0.0989860
$586$	0	0
$587$	−14.7573	−0.609099	−0.304549	−	0.952497i	$-0.598506\pi$
$587$	−14.7573	−0.609099	−0.304549	+	0.952497i	$0.598506\pi$
$588$	0	0
$589$	−15.0255	−0.619113
$590$	0	0
$591$	10.0110	0.411798
$592$	0	0
$593$	−13.4118	−0.550755	−0.275378	−	0.961336i	$-0.588803\pi$
$593$	−13.4118	−0.550755	−0.275378	+	0.961336i	$0.588803\pi$
$594$	0	0
$595$	25.6970	1.05347
$596$	0	0
$597$	−0.288851	−0.0118219
$598$	0	0
$599$	−29.8532	−1.21977	−0.609885	−	0.792490i	$-0.708784\pi$
$599$	−29.8532	−1.21977	−0.609885	+	0.792490i	$0.708784\pi$
$600$	0	0
$601$	37.1115	1.51381	0.756906	−	0.653524i	$-0.226710\pi$
$601$	37.1115	1.51381	0.756906	+	0.653524i	$0.226710\pi$
$602$	0	0
$603$	15.5362	0.632684
$604$	0	0
$605$	−14.1910	−0.576947
$606$	0	0
$607$	15.2079	0.617268	0.308634	−	0.951181i	$-0.400128\pi$
$607$	15.2079	0.617268	0.308634	+	0.951181i	$0.400128\pi$
$608$	0	0
$609$	17.3432	0.702782
$610$	0	0
$611$	−13.1327	−0.531293
$612$	0	0
$613$	−37.1964	−1.50235	−0.751174	−	0.660104i	$-0.770512\pi$
$613$	−37.1964	−1.50235	−0.751174	+	0.660104i	$0.770512\pi$
$614$	0	0
$615$	7.18424	0.289696
$616$	0	0
$617$	−20.4083	−0.821609	−0.410804	−	0.911724i	$-0.634752\pi$
$617$	−20.4083	−0.821609	−0.410804	+	0.911724i	$0.634752\pi$
$618$	0	0
$619$	−21.7350	−0.873605	−0.436802	−	0.899558i	$-0.643889\pi$
$619$	−21.7350	−0.873605	−0.436802	+	0.899558i	$0.643889\pi$
$620$	0	0
$621$	−3.88550	−0.155920
$622$	0	0
$623$	37.3931	1.49812
$624$	0	0
$625$	14.9100	0.596399
$626$	0	0
$627$	11.4975	0.459168
$628$	0	0
$629$	−54.9845	−2.19238
$630$	0	0
$631$	−28.3135	−1.12714	−0.563571	−	0.826068i	$-0.690573\pi$
$631$	−28.3135	−1.12714	−0.563571	+	0.826068i	$0.690573\pi$
$632$	0	0
$633$	22.7364	0.903690
$634$	0	0
$635$	−13.4165	−0.532419
$636$	0	0
$637$	53.0920	2.10358
$638$	0	0
$639$	−1.61922	−0.0640554
$640$	0	0
$641$	20.9033	0.825632	0.412816	−	0.910814i	$-0.364545\pi$
$641$	20.9033	0.825632	0.412816	+	0.910814i	$0.364545\pi$
$642$	0	0
$643$	21.8460	0.861520	0.430760	−	0.902466i	$-0.358246\pi$
$643$	21.8460	0.861520	0.430760	+	0.902466i	$0.358246\pi$
$644$	0	0
$645$	5.75351	0.226544
$646$	0	0
$647$	−34.0888	−1.34017	−0.670085	−	0.742284i	$-0.733742\pi$
$647$	−34.0888	−1.34017	−0.670085	+	0.742284i	$0.733742\pi$
$648$	0	0
$649$	−54.1866	−2.12701
$650$	0	0
$651$	−34.9412	−1.36945
$652$	0	0
$653$	11.2431	0.439978	0.219989	−	0.975502i	$-0.429398\pi$
$653$	11.2431	0.439978	0.219989	+	0.975502i	$0.429398\pi$
$654$	0	0
$655$	3.04211	0.118865
$656$	0	0
$657$	−11.6675	−0.455193
$658$	0	0
$659$	28.1601	1.09696	0.548480	−	0.836164i	$-0.315207\pi$
$659$	28.1601	1.09696	0.548480	+	0.836164i	$0.315207\pi$
$660$	0	0
$661$	−15.5861	−0.606228	−0.303114	−	0.952954i	$-0.598026\pi$
$661$	−15.5861	−0.606228	−0.303114	+	0.952954i	$0.598026\pi$
$662$	0	0
$663$	−17.2152	−0.668583
$664$	0	0
$665$	9.26041	0.359103
$666$	0	0
$667$	13.3104	0.515379
$668$	0	0
$669$	4.00024	0.154658
$670$	0	0
$671$	−23.1420	−0.893386
$672$	0	0
$673$	−28.2598	−1.08934	−0.544669	−	0.838651i	$-0.683345\pi$
$673$	−28.2598	−1.08934	−0.544669	+	0.838651i	$0.683345\pi$
$674$	0	0
$675$	4.29411	0.165280
$676$	0	0
$677$	−30.9869	−1.19092	−0.595461	−	0.803384i	$-0.703031\pi$
$677$	−30.9869	−1.19092	−0.595461	+	0.803384i	$0.703031\pi$
$678$	0	0
$679$	89.8672	3.44879
$680$	0	0
$681$	17.4400	0.668301
$682$	0	0
$683$	−5.16899	−0.197786	−0.0988929	−	0.995098i	$-0.531530\pi$
$683$	−5.16899	−0.197786	−0.0988929	+	0.995098i	$0.531530\pi$
$684$	0	0
$685$	−16.9295	−0.646843
$686$	0	0
$687$	−11.9220	−0.454852
$688$	0	0
$689$	−27.6330	−1.05273
$690$	0	0
$691$	24.6842	0.939031	0.469515	−	0.882924i	$-0.344429\pi$
$691$	24.6842	0.939031	0.469515	+	0.882924i	$0.344429\pi$
$692$	0	0
$693$	26.7372	1.01566
$694$	0	0
$695$	8.77939	0.333021
$696$	0	0
$697$	51.6584	1.95670
$698$	0	0
$699$	20.2911	0.767480
$700$	0	0
$701$	47.7367	1.80299	0.901495	−	0.432790i	$-0.142471\pi$
$701$	47.7367	1.80299	0.901495	+	0.432790i	$0.142471\pi$
$702$	0	0
$703$	−19.8147	−0.747327
$704$	0	0
$705$	−3.87203	−0.145829
$706$	0	0
$707$	−37.8131	−1.42211
$708$	0	0
$709$	20.0737	0.753882	0.376941	−	0.926237i	$-0.376976\pi$
$709$	20.0737	0.753882	0.376941	+	0.926237i	$0.376976\pi$
$710$	0	0
$711$	5.79273	0.217244
$712$	0	0
$713$	−26.8163	−1.00428
$714$	0	0
$715$	12.6439	0.472855
$716$	0	0
$717$	5.39330	0.201416
$718$	0	0
$719$	−25.2695	−0.942393	−0.471197	−	0.882028i	$-0.656178\pi$
$719$	−25.2695	−0.942393	−0.471197	+	0.882028i	$0.656178\pi$
$720$	0	0
$721$	−94.8835	−3.53365
$722$	0	0
$723$	5.28935	0.196713
$724$	0	0
$725$	−14.7101	−0.546320
$726$	0	0
$727$	−32.3463	−1.19966	−0.599830	−	0.800128i	$-0.704765\pi$
$727$	−32.3463	−1.19966	−0.599830	+	0.800128i	$0.704765\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	41.3708	1.53015
$732$	0	0
$733$	−14.5569	−0.537673	−0.268836	−	0.963186i	$-0.586639\pi$
$733$	−14.5569	−0.537673	−0.268836	+	0.963186i	$0.586639\pi$
$734$	0	0
$735$	15.6536	0.577391
$736$	0	0
$737$	−82.0492	−3.02232
$738$	0	0
$739$	−9.62805	−0.354173	−0.177087	−	0.984195i	$-0.556667\pi$
$739$	−9.62805	−0.354173	−0.177087	+	0.984195i	$0.556667\pi$
$740$	0	0
$741$	−6.20383	−0.227903
$742$	0	0
$743$	−47.2029	−1.73171	−0.865854	−	0.500297i	$-0.833224\pi$
$743$	−47.2029	−1.73171	−0.865854	+	0.500297i	$0.833224\pi$
$744$	0	0
$745$	15.9313	0.583679
$746$	0	0
$747$	2.59569	0.0949714
$748$	0	0
$749$	−44.2035	−1.61516
$750$	0	0
$751$	9.53587	0.347969	0.173984	−	0.984748i	$-0.444336\pi$
$751$	9.53587	0.347969	0.173984	+	0.984748i	$0.444336\pi$
$752$	0	0
$753$	7.87886	0.287121
$754$	0	0
$755$	0.840171	0.0305770
$756$	0	0
$757$	10.2654	0.373103	0.186551	−	0.982445i	$-0.440269\pi$
$757$	10.2654	0.373103	0.186551	+	0.982445i	$0.440269\pi$
$758$	0	0
$759$	20.5199	0.744826
$760$	0	0
$761$	5.86797	0.212714	0.106357	−	0.994328i	$-0.466081\pi$
$761$	5.86797	0.212714	0.106357	+	0.994328i	$0.466081\pi$
$762$	0	0
$763$	−50.3975	−1.82451
$764$	0	0
$765$	−5.07570	−0.183512
$766$	0	0
$767$	29.2379	1.05572
$768$	0	0
$769$	35.7436	1.28895	0.644474	−	0.764626i	$-0.277076\pi$
$769$	35.7436	1.28895	0.644474	+	0.764626i	$0.277076\pi$
$770$	0	0
$771$	−3.30095	−0.118881
$772$	0	0
$773$	29.7074	1.06850	0.534251	−	0.845326i	$-0.320594\pi$
$773$	29.7074	1.06850	0.534251	+	0.845326i	$0.320594\pi$
$774$	0	0
$775$	29.6364	1.06457
$776$	0	0
$777$	−46.0785	−1.65306
$778$	0	0
$779$	18.6161	0.666991
$780$	0	0
$781$	8.55137	0.305992
$782$	0	0
$783$	−3.42565	−0.122423
$784$	0	0
$785$	0.818030	0.0291967
$786$	0	0
$787$	7.49847	0.267292	0.133646	−	0.991029i	$-0.457332\pi$
$787$	7.49847	0.267292	0.133646	+	0.991029i	$0.457332\pi$
$788$	0	0
$789$	4.22907	0.150559
$790$	0	0
$791$	−11.7591	−0.418104
$792$	0	0
$793$	12.4869	0.443423
$794$	0	0
$795$	−8.14727	−0.288954
$796$	0	0
$797$	−9.77743	−0.346334	−0.173167	−	0.984892i	$-0.555400\pi$
$797$	−9.77743	−0.346334	−0.173167	+	0.984892i	$0.555400\pi$
$798$	0	0
$799$	−27.8419	−0.984977
$800$	0	0
$801$	−7.38593	−0.260969
$802$	0	0
$803$	61.6180	2.17445
$804$	0	0
$805$	16.5273	0.582509
$806$	0	0
$807$	20.1585	0.709612
$808$	0	0
$809$	27.8357	0.978653	0.489326	−	0.872101i	$-0.337243\pi$
$809$	27.8357	0.978653	0.489326	+	0.872101i	$0.337243\pi$
$810$	0	0
$811$	42.9749	1.50905	0.754527	−	0.656269i	$-0.227866\pi$
$811$	42.9749	1.50905	0.754527	+	0.656269i	$0.227866\pi$
$812$	0	0
$813$	23.4935	0.823955
$814$	0	0
$815$	10.9319	0.382928
$816$	0	0
$817$	14.9088	0.521591
$818$	0	0
$819$	−14.4268	−0.504113
$820$	0	0
$821$	38.7511	1.35242	0.676212	−	0.736707i	$-0.263621\pi$
$821$	38.7511	1.35242	0.676212	+	0.736707i	$0.263621\pi$
$822$	0	0
$823$	−16.1237	−0.562036	−0.281018	−	0.959702i	$-0.590672\pi$
$823$	−16.1237	−0.562036	−0.281018	+	0.959702i	$0.590672\pi$
$824$	0	0
$825$	−22.6779	−0.789543
$826$	0	0
$827$	25.5961	0.890064	0.445032	−	0.895515i	$-0.353192\pi$
$827$	25.5961	0.890064	0.445032	+	0.895515i	$0.353192\pi$
$828$	0	0
$829$	24.8634	0.863542	0.431771	−	0.901983i	$-0.357889\pi$
$829$	24.8634	0.863542	0.431771	+	0.901983i	$0.357889\pi$
$830$	0	0
$831$	29.4348	1.02108
$832$	0	0
$833$	112.557	3.89988
$834$	0	0
$835$	19.2671	0.666767
$836$	0	0
$837$	6.90163	0.238555
$838$	0	0
$839$	−17.6682	−0.609975	−0.304987	−	0.952356i	$-0.598652\pi$
$839$	−17.6682	−0.609975	−0.304987	+	0.952356i	$0.598652\pi$
$840$	0	0
$841$	−17.2649	−0.595342
$842$	0	0
$843$	11.3527	0.391009
$844$	0	0
$845$	4.09985	0.141039
$846$	0	0
$847$	−85.5130	−2.93826
$848$	0	0
$849$	24.1900	0.830198
$850$	0	0
$851$	−35.3638	−1.21226
$852$	0	0
$853$	22.0559	0.755180	0.377590	−	0.925973i	$-0.376753\pi$
$853$	22.0559	0.755180	0.377590	+	0.925973i	$0.376753\pi$
$854$	0	0
$855$	−1.82913	−0.0625548
$856$	0	0
$857$	−4.32370	−0.147695	−0.0738474	−	0.997270i	$-0.523528\pi$
$857$	−4.32370	−0.147695	−0.0738474	+	0.997270i	$0.523528\pi$
$858$	0	0
$859$	0.182458	0.00622539	0.00311269	−	0.999995i	$-0.499009\pi$
$859$	0.182458	0.00622539	0.00311269	+	0.999995i	$0.499009\pi$
$860$	0	0
$861$	43.2911	1.47536
$862$	0	0
$863$	−37.2310	−1.26736	−0.633679	−	0.773596i	$-0.718456\pi$
$863$	−37.2310	−1.26736	−0.633679	+	0.773596i	$0.718456\pi$
$864$	0	0
$865$	2.81801	0.0958152
$866$	0	0
$867$	−19.4969	−0.662151
$868$	0	0
$869$	−30.5923	−1.03777
$870$	0	0
$871$	44.2720	1.50010
$872$	0	0
$873$	−17.7507	−0.600770
$874$	0	0
$875$	−39.5332	−1.33647
$876$	0	0
$877$	28.7126	0.969555	0.484777	−	0.874638i	$-0.338901\pi$
$877$	28.7126	0.969555	0.484777	+	0.874638i	$0.338901\pi$
$878$	0	0
$879$	−2.64953	−0.0893664
$880$	0	0
$881$	−25.0450	−0.843788	−0.421894	−	0.906645i	$-0.638635\pi$
$881$	−25.0450	−0.843788	−0.421894	+	0.906645i	$0.638635\pi$
$882$	0	0
$883$	17.6861	0.595183	0.297591	−	0.954693i	$-0.403817\pi$
$883$	17.6861	0.595183	0.297591	+	0.954693i	$0.403817\pi$
$884$	0	0
$885$	8.62046	0.289774
$886$	0	0
$887$	4.49503	0.150928	0.0754641	−	0.997149i	$-0.475956\pi$
$887$	4.49503	0.150928	0.0754641	+	0.997149i	$0.475956\pi$
$888$	0	0
$889$	−80.8461	−2.71149
$890$	0	0
$891$	−5.28116	−0.176925
$892$	0	0
$893$	−10.0334	−0.335754
$894$	0	0
$895$	5.74716	0.192106
$896$	0	0
$897$	−11.0721	−0.369687
$898$	0	0
$899$	−23.6426	−0.788523
$900$	0	0
$901$	−58.5831	−1.95169
$902$	0	0
$903$	34.6698	1.15374
$904$	0	0
$905$	−4.72963	−0.157218
$906$	0	0
$907$	51.2646	1.70221	0.851107	−	0.524992i	$-0.175932\pi$
$907$	51.2646	1.70221	0.851107	+	0.524992i	$0.175932\pi$
$908$	0	0
$909$	7.46889	0.247728
$910$	0	0
$911$	5.58145	0.184922	0.0924608	−	0.995716i	$-0.470527\pi$
$911$	5.58145	0.184922	0.0924608	+	0.995716i	$0.470527\pi$
$912$	0	0
$913$	−13.7083	−0.453677
$914$	0	0
$915$	3.68162	0.121711
$916$	0	0
$917$	18.3313	0.605353
$918$	0	0
$919$	−26.4389	−0.872140	−0.436070	−	0.899913i	$-0.643630\pi$
$919$	−26.4389	−0.872140	−0.436070	+	0.899913i	$0.643630\pi$
$920$	0	0
$921$	−23.5848	−0.777145
$922$	0	0
$923$	−4.61414	−0.151876
$924$	0	0
$925$	39.0828	1.28503
$926$	0	0
$927$	18.7415	0.615552
$928$	0	0
$929$	−17.6003	−0.577447	−0.288723	−	0.957413i	$-0.593231\pi$
$929$	−17.6003	−0.577447	−0.288723	+	0.957413i	$0.593231\pi$
$930$	0	0
$931$	40.5622	1.32937
$932$	0	0
$933$	6.65183	0.217771
$934$	0	0
$935$	26.8056	0.876636
$936$	0	0
$937$	−19.0182	−0.621298	−0.310649	−	0.950525i	$-0.600546\pi$
$937$	−19.0182	−0.621298	−0.310649	+	0.950525i	$0.600546\pi$
$938$	0	0
$939$	−0.147479	−0.00481281
$940$	0	0
$941$	−57.5170	−1.87500	−0.937501	−	0.347983i	$-0.886867\pi$
$941$	−57.5170	−1.87500	−0.937501	+	0.347983i	$0.886867\pi$
$942$	0	0
$943$	33.2246	1.08194
$944$	0	0
$945$	−4.25357	−0.138369
$946$	0	0
$947$	23.9913	0.779611	0.389806	−	0.920897i	$-0.372542\pi$
$947$	23.9913	0.779611	0.389806	+	0.920897i	$0.372542\pi$
$948$	0	0
$949$	−33.2477	−1.07927
$950$	0	0
$951$	−21.1626	−0.686243
$952$	0	0
$953$	−44.6215	−1.44543	−0.722716	−	0.691145i	$-0.757107\pi$
$953$	−44.6215	−1.44543	−0.722716	+	0.691145i	$0.757107\pi$
$954$	0	0
$955$	18.8029	0.608449
$956$	0	0
$957$	18.0914	0.584812
$958$	0	0
$959$	−102.015	−3.29423
$960$	0	0
$961$	16.6325	0.536531
$962$	0	0
$963$	8.73112	0.281357
$964$	0	0
$965$	6.00219	0.193217
$966$	0	0
$967$	−12.7289	−0.409333	−0.204666	−	0.978832i	$-0.565611\pi$
$967$	−12.7289	−0.409333	−0.204666	+	0.978832i	$0.565611\pi$
$968$	0	0
$969$	−13.1524	−0.422515
$970$	0	0
$971$	−26.0219	−0.835082	−0.417541	−	0.908658i	$-0.637108\pi$
$971$	−26.0219	−0.835082	−0.417541	+	0.908658i	$0.637108\pi$
$972$	0	0
$973$	52.9033	1.69600
$974$	0	0
$975$	12.2365	0.391882
$976$	0	0
$977$	15.4035	0.492802	0.246401	−	0.969168i	$-0.420752\pi$
$977$	15.4035	0.492802	0.246401	+	0.969168i	$0.420752\pi$
$978$	0	0
$979$	39.0063	1.24665
$980$	0	0
$981$	9.95458	0.317825
$982$	0	0
$983$	−35.1072	−1.11975	−0.559873	−	0.828579i	$-0.689150\pi$
$983$	−35.1072	−1.11975	−0.559873	+	0.828579i	$0.689150\pi$
$984$	0	0
$985$	8.41097	0.267996
$986$	0	0
$987$	−23.3323	−0.742675
$988$	0	0
$989$	26.6080	0.846085
$990$	0	0
$991$	−10.0954	−0.320691	−0.160345	−	0.987061i	$-0.551261\pi$
$991$	−10.0954	−0.320691	−0.160345	+	0.987061i	$0.551261\pi$
$992$	0	0
$993$	−26.7904	−0.850166
$994$	0	0
$995$	−0.242684	−0.00769360
$996$	0	0
$997$	−38.7845	−1.22832	−0.614159	−	0.789182i	$-0.710505\pi$
$997$	−38.7845	−1.22832	−0.614159	+	0.789182i	$0.710505\pi$
$998$	0	0
$999$	9.10148	0.287958

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7248.2.a.bm.1.5		10
4.3	odd	2		3624.2.a.l.1.5	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3624.2.a.l.1.5	✓	10	4.3	odd	2
7248.2.a.bm.1.5		10	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7248 = 2^{4} \cdot 3 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7248.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -0.840171 q^{5} -5.06275 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -0.840171 q^{5} -5.06275 q^{7} +1.00000 q^{9} -5.28116 q^{11} +2.84960 q^{13} +0.840171 q^{15} +6.04127 q^{17} +2.17709 q^{19} +5.06275 q^{21} +3.88550 q^{23} -4.29411 q^{25} -1.00000 q^{27} +3.42565 q^{29} -6.90163 q^{31} +5.28116 q^{33} +4.25357 q^{35} -9.10148 q^{37} -2.84960 q^{39} +8.55092 q^{41} +6.84802 q^{43} -0.840171 q^{45} -4.60862 q^{47} +18.6314 q^{49} -6.04127 q^{51} -9.69715 q^{53} +4.43708 q^{55} -2.17709 q^{57} +10.2604 q^{59} +4.38199 q^{61} -5.06275 q^{63} -2.39415 q^{65} +15.5362 q^{67} -3.88550 q^{69} -1.61922 q^{71} -11.6675 q^{73} +4.29411 q^{75} +26.7372 q^{77} +5.79273 q^{79} +1.00000 q^{81} +2.59569 q^{83} -5.07570 q^{85} -3.42565 q^{87} -7.38593 q^{89} -14.4268 q^{91} +6.90163 q^{93} -1.82913 q^{95} -17.7507 q^{97} -5.28116 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{3} + 2 q^{5} - 8 q^{7} + 10 q^{9}+O(q^{10}) \) 10 * q - 10 * q^3 + 2 * q^5 - 8 * q^7 + 10 * q^9	\( 10 q - 10 q^{3} + 2 q^{5} - 8 q^{7} + 10 q^{9} - 7 q^{11} + 6 q^{13} - 2 q^{15} + 7 q^{17} + 8 q^{21} - 25 q^{23} + 8 q^{25} - 10 q^{27} + 12 q^{29} - 11 q^{31} + 7 q^{33} - 9 q^{35} - 3 q^{37} - 6 q^{39} + 12 q^{41} + 2 q^{45} - 31 q^{47} + 14 q^{49} - 7 q^{51} + q^{53} - 9 q^{55} - 19 q^{59} + 24 q^{61} - 8 q^{63} + 20 q^{65} + q^{67} + 25 q^{69} - 34 q^{71} - 18 q^{73} - 8 q^{75} + 27 q^{77} - 25 q^{79} + 10 q^{81} - 14 q^{83} - 3 q^{85} - 12 q^{87} + 20 q^{89} + 12 q^{91} + 11 q^{93} - 48 q^{95} - 15 q^{97} - 7 q^{99}+O(q^{100}) \) 10 * q - 10 * q^3 + 2 * q^5 - 8 * q^7 + 10 * q^9 - 7 * q^11 + 6 * q^13 - 2 * q^15 + 7 * q^17 + 8 * q^21 - 25 * q^23 + 8 * q^25 - 10 * q^27 + 12 * q^29 - 11 * q^31 + 7 * q^33 - 9 * q^35 - 3 * q^37 - 6 * q^39 + 12 * q^41 + 2 * q^45 - 31 * q^47 + 14 * q^49 - 7 * q^51 + q^53 - 9 * q^55 - 19 * q^59 + 24 * q^61 - 8 * q^63 + 20 * q^65 + q^67 + 25 * q^69 - 34 * q^71 - 18 * q^73 - 8 * q^75 + 27 * q^77 - 25 * q^79 + 10 * q^81 - 14 * q^83 - 3 * q^85 - 12 * q^87 + 20 * q^89 + 12 * q^91 + 11 * q^93 - 48 * q^95 - 15 * q^97 - 7 * q^99

Introduction

L-functions

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