LMFDB - Embedded newform 6080.2.a.bp.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6080,2,Mod(1,6080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6080, 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("6080.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6080 = 2^{6} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6080.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:	$48.5490444289$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.564.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3040)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.51414$ of defining polynomial
Character	$\chi$	$=$	6080.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.51414 q^{3} +1.00000 q^{5} +1.32088 q^{7} -0.707389 q^{9} +O(q^{10})$	$q+1.51414 q^{3} +1.00000 q^{5} +1.32088 q^{7} -0.707389 q^{9} -3.32088 q^{11} -1.80675 q^{13} +1.51414 q^{15} +2.00000 q^{17} +1.00000 q^{19} +2.00000 q^{21} -5.32088 q^{23} +1.00000 q^{25} -5.61350 q^{27} -5.02827 q^{29} -5.02827 q^{31} -5.02827 q^{33} +1.32088 q^{35} -0.193252 q^{37} -2.73566 q^{39} +0.971726 q^{41} +4.34916 q^{43} -0.707389 q^{45} +1.32088 q^{47} -5.25526 q^{49} +3.02827 q^{51} -3.80675 q^{53} -3.32088 q^{55} +1.51414 q^{57} +8.05655 q^{59} -12.9909 q^{61} -0.934380 q^{63} -1.80675 q^{65} -6.92892 q^{67} -8.05655 q^{69} -13.0283 q^{71} +5.02827 q^{73} +1.51414 q^{75} -4.38650 q^{77} +2.00000 q^{79} -6.37743 q^{81} -2.29261 q^{83} +2.00000 q^{85} -7.61350 q^{87} +13.0283 q^{89} -2.38650 q^{91} -7.61350 q^{93} +1.00000 q^{95} +1.16498 q^{97} +2.34916 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{3} + 3 q^{5} - 4 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 2 * q^3 + 3 * q^5 - 4 * q^7 + 3 * q^9	$ 3 q - 2 q^{3} + 3 q^{5} - 4 q^{7} + 3 q^{9} - 2 q^{11} - 4 q^{13} - 2 q^{15} + 6 q^{17} + 3 q^{19} + 6 q^{21} - 8 q^{23} + 3 q^{25} - 14 q^{27} - 2 q^{29} - 2 q^{31} - 2 q^{33} - 4 q^{35} - 2 q^{37} + 10 q^{39} + 16 q^{41} - 8 q^{43} + 3 q^{45} - 4 q^{47} + 3 q^{49} - 4 q^{51} - 10 q^{53} - 2 q^{55} - 2 q^{57} - 2 q^{59} - 2 q^{61} + 8 q^{63} - 4 q^{65} - 4 q^{67} + 2 q^{69} - 26 q^{71} + 2 q^{73} - 2 q^{75} - 16 q^{77} + 6 q^{79} + 15 q^{81} - 12 q^{83} + 6 q^{85} - 20 q^{87} + 26 q^{89} - 10 q^{91} - 20 q^{93} + 3 q^{95} + 18 q^{97} - 14 q^{99}+O(q^{100}) $ 3 * q - 2 * q^3 + 3 * q^5 - 4 * q^7 + 3 * q^9 - 2 * q^11 - 4 * q^13 - 2 * q^15 + 6 * q^17 + 3 * q^19 + 6 * q^21 - 8 * q^23 + 3 * q^25 - 14 * q^27 - 2 * q^29 - 2 * q^31 - 2 * q^33 - 4 * q^35 - 2 * q^37 + 10 * q^39 + 16 * q^41 - 8 * q^43 + 3 * q^45 - 4 * q^47 + 3 * q^49 - 4 * q^51 - 10 * q^53 - 2 * q^55 - 2 * q^57 - 2 * q^59 - 2 * q^61 + 8 * q^63 - 4 * q^65 - 4 * q^67 + 2 * q^69 - 26 * q^71 + 2 * q^73 - 2 * q^75 - 16 * q^77 + 6 * q^79 + 15 * q^81 - 12 * q^83 + 6 * q^85 - 20 * q^87 + 26 * q^89 - 10 * q^91 - 20 * q^93 + 3 * q^95 + 18 * q^97 - 14 * 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
$3$	1.51414	0.874187	0.437094	−	0.899416i	$-0.356008\pi$
$3$	1.51414	0.874187	0.437094	+	0.899416i	$0.356008\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.32088	0.499247	0.249624	−	0.968343i	$-0.419693\pi$
$7$	1.32088	0.499247	0.249624	+	0.968343i	$0.419693\pi$
$8$	0	0
$9$	−0.707389	−0.235796
$10$	0	0
$11$	−3.32088	−1.00128	−0.500642	−	0.865654i	$-0.666903\pi$
$11$	−3.32088	−1.00128	−0.500642	+	0.865654i	$0.666903\pi$
$12$	0	0
$13$	−1.80675	−0.501102	−0.250551	−	0.968103i	$-0.580612\pi$
$13$	−1.80675	−0.501102	−0.250551	+	0.968103i	$0.580612\pi$
$14$	0	0
$15$	1.51414	0.390948
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	−5.32088	−1.10948	−0.554741	−	0.832023i	$-0.687182\pi$
$23$	−5.32088	−1.10948	−0.554741	+	0.832023i	$0.687182\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.61350	−1.08032
$28$	0	0
$29$	−5.02827	−0.933727	−0.466864	−	0.884329i	$-0.654616\pi$
$29$	−5.02827	−0.933727	−0.466864	+	0.884329i	$0.654616\pi$
$30$	0	0
$31$	−5.02827	−0.903105	−0.451552	−	0.892245i	$-0.649130\pi$
$31$	−5.02827	−0.903105	−0.451552	+	0.892245i	$0.649130\pi$
$32$	0	0
$33$	−5.02827	−0.875310
$34$	0	0
$35$	1.32088	0.223270
$36$	0	0
$37$	−0.193252	−0.0317705	−0.0158853	−	0.999874i	$-0.505057\pi$
$37$	−0.193252	−0.0317705	−0.0158853	+	0.999874i	$0.505057\pi$
$38$	0	0
$39$	−2.73566	−0.438057
$40$	0	0
$41$	0.971726	0.151758	0.0758791	−	0.997117i	$-0.475824\pi$
$41$	0.971726	0.151758	0.0758791	+	0.997117i	$0.475824\pi$
$42$	0	0
$43$	4.34916	0.663240	0.331620	−	0.943413i	$-0.392405\pi$
$43$	4.34916	0.663240	0.331620	+	0.943413i	$0.392405\pi$
$44$	0	0
$45$	−0.707389	−0.105451
$46$	0	0
$47$	1.32088	0.192671	0.0963354	−	0.995349i	$-0.469288\pi$
$47$	1.32088	0.192671	0.0963354	+	0.995349i	$0.469288\pi$
$48$	0	0
$49$	−5.25526	−0.750752
$50$	0	0
$51$	3.02827	0.424043
$52$	0	0
$53$	−3.80675	−0.522897	−0.261448	−	0.965217i	$-0.584200\pi$
$53$	−3.80675	−0.522897	−0.261448	+	0.965217i	$0.584200\pi$
$54$	0	0
$55$	−3.32088	−0.447788
$56$	0	0
$57$	1.51414	0.200552
$58$	0	0
$59$	8.05655	1.04887	0.524437	−	0.851450i	$-0.324276\pi$
$59$	8.05655	1.04887	0.524437	+	0.851450i	$0.324276\pi$
$60$	0	0
$61$	−12.9909	−1.66332	−0.831659	−	0.555287i	$-0.812608\pi$
$61$	−12.9909	−1.66332	−0.831659	+	0.555287i	$0.812608\pi$
$62$	0	0
$63$	−0.934380	−0.117721
$64$	0	0
$65$	−1.80675	−0.224099
$66$	0	0
$67$	−6.92892	−0.846502	−0.423251	−	0.906013i	$-0.639111\pi$
$67$	−6.92892	−0.846502	−0.423251	+	0.906013i	$0.639111\pi$
$68$	0	0
$69$	−8.05655	−0.969894
$70$	0	0
$71$	−13.0283	−1.54617	−0.773086	−	0.634301i	$-0.781288\pi$
$71$	−13.0283	−1.54617	−0.773086	+	0.634301i	$0.781288\pi$
$72$	0	0
$73$	5.02827	0.588515	0.294257	−	0.955726i	$-0.404928\pi$
$73$	5.02827	0.588515	0.294257	+	0.955726i	$0.404928\pi$
$74$	0	0
$75$	1.51414	0.174837
$76$	0	0
$77$	−4.38650	−0.499889
$78$	0	0
$79$	2.00000	0.225018	0.112509	−	0.993651i	$-0.464111\pi$
$79$	2.00000	0.225018	0.112509	+	0.993651i	$0.464111\pi$
$80$	0	0
$81$	−6.37743	−0.708604
$82$	0	0
$83$	−2.29261	−0.251647	−0.125823	−	0.992053i	$-0.540157\pi$
$83$	−2.29261	−0.251647	−0.125823	+	0.992053i	$0.540157\pi$
$84$	0	0
$85$	2.00000	0.216930
$86$	0	0
$87$	−7.61350	−0.816252
$88$	0	0
$89$	13.0283	1.38099	0.690497	−	0.723335i	$-0.257392\pi$
$89$	13.0283	1.38099	0.690497	+	0.723335i	$0.257392\pi$
$90$	0	0
$91$	−2.38650	−0.250174
$92$	0	0
$93$	−7.61350	−0.789483
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	1.16498	0.118286	0.0591428	−	0.998250i	$-0.481163\pi$
$97$	1.16498	0.118286	0.0591428	+	0.998250i	$0.481163\pi$
$98$	0	0
$99$	2.34916	0.236099
$100$	0	0
$101$	3.32088	0.330440	0.165220	−	0.986257i	$-0.447167\pi$
$101$	3.32088	0.330440	0.165220	+	0.986257i	$0.447167\pi$
$102$	0	0
$103$	−3.12763	−0.308175	−0.154087	−	0.988057i	$-0.549244\pi$
$103$	−3.12763	−0.308175	−0.154087	+	0.988057i	$0.549244\pi$
$104$	0	0
$105$	2.00000	0.195180
$106$	0	0
$107$	−9.18418	−0.887868	−0.443934	−	0.896059i	$-0.646418\pi$
$107$	−9.18418	−0.887868	−0.443934	+	0.896059i	$0.646418\pi$
$108$	0	0
$109$	−7.41478	−0.710207	−0.355103	−	0.934827i	$-0.615554\pi$
$109$	−7.41478	−0.710207	−0.355103	+	0.934827i	$0.615554\pi$
$110$	0	0
$111$	−0.292611	−0.0277734
$112$	0	0
$113$	6.77847	0.637665	0.318833	−	0.947811i	$-0.396709\pi$
$113$	6.77847	0.637665	0.318833	+	0.947811i	$0.396709\pi$
$114$	0	0
$115$	−5.32088	−0.496175
$116$	0	0
$117$	1.27807	0.118158
$118$	0	0
$119$	2.64177	0.242171
$120$	0	0
$121$	0.0282739	0.00257035
$122$	0	0
$123$	1.47133	0.132665
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	1.12763	0.100061	0.0500306	−	0.998748i	$-0.484068\pi$
$127$	1.12763	0.100061	0.0500306	+	0.998748i	$0.484068\pi$
$128$	0	0
$129$	6.58522	0.579796
$130$	0	0
$131$	−6.44305	−0.562932	−0.281466	−	0.959571i	$-0.590821\pi$
$131$	−6.44305	−0.562932	−0.281466	+	0.959571i	$0.590821\pi$
$132$	0	0
$133$	1.32088	0.114535
$134$	0	0
$135$	−5.61350	−0.483133
$136$	0	0
$137$	21.9253	1.87321	0.936603	−	0.350393i	$-0.113952\pi$
$137$	21.9253	1.87321	0.936603	+	0.350393i	$0.113952\pi$
$138$	0	0
$139$	−6.73566	−0.571311	−0.285656	−	0.958332i	$-0.592211\pi$
$139$	−6.73566	−0.571311	−0.285656	+	0.958332i	$0.592211\pi$
$140$	0	0
$141$	2.00000	0.168430
$142$	0	0
$143$	6.00000	0.501745
$144$	0	0
$145$	−5.02827	−0.417575
$146$	0	0
$147$	−7.95719	−0.656298
$148$	0	0
$149$	−20.9909	−1.71964	−0.859822	−	0.510594i	$-0.829426\pi$
$149$	−20.9909	−1.71964	−0.859822	+	0.510594i	$0.829426\pi$
$150$	0	0
$151$	2.84049	0.231155	0.115578	−	0.993298i	$-0.463128\pi$
$151$	2.84049	0.231155	0.115578	+	0.993298i	$0.463128\pi$
$152$	0	0
$153$	−1.41478	−0.114378
$154$	0	0
$155$	−5.02827	−0.403881
$156$	0	0
$157$	−15.4713	−1.23475	−0.617373	−	0.786670i	$-0.711803\pi$
$157$	−15.4713	−1.23475	−0.617373	+	0.786670i	$0.711803\pi$
$158$	0	0
$159$	−5.76394	−0.457110
$160$	0	0
$161$	−7.02827	−0.553906
$162$	0	0
$163$	−12.3492	−0.967261	−0.483630	−	0.875272i	$-0.660682\pi$
$163$	−12.3492	−0.967261	−0.483630	+	0.875272i	$0.660682\pi$
$164$	0	0
$165$	−5.02827	−0.391451
$166$	0	0
$167$	2.21245	0.171205	0.0856024	−	0.996329i	$-0.472719\pi$
$167$	2.21245	0.171205	0.0856024	+	0.996329i	$0.472719\pi$
$168$	0	0
$169$	−9.73566	−0.748897
$170$	0	0
$171$	−0.707389	−0.0540954
$172$	0	0
$173$	2.50506	0.190457	0.0952283	−	0.995455i	$-0.469642\pi$
$173$	2.50506	0.190457	0.0952283	+	0.995455i	$0.469642\pi$
$174$	0	0
$175$	1.32088	0.0998495
$176$	0	0
$177$	12.1987	0.916912
$178$	0	0
$179$	−6.00000	−0.448461	−0.224231	−	0.974536i	$-0.571987\pi$
$179$	−6.00000	−0.448461	−0.224231	+	0.974536i	$0.571987\pi$
$180$	0	0
$181$	12.8296	0.953613	0.476807	−	0.879008i	$-0.341794\pi$
$181$	12.8296	0.953613	0.476807	+	0.879008i	$0.341794\pi$
$182$	0	0
$183$	−19.6700	−1.45405
$184$	0	0
$185$	−0.193252	−0.0142082
$186$	0	0
$187$	−6.64177	−0.485694
$188$	0	0
$189$	−7.41478	−0.539346
$190$	0	0
$191$	11.9253	0.862885	0.431442	−	0.902140i	$-0.358005\pi$
$191$	11.9253	0.862885	0.431442	+	0.902140i	$0.358005\pi$
$192$	0	0
$193$	−2.06201	−0.148427	−0.0742134	−	0.997242i	$-0.523645\pi$
$193$	−2.06201	−0.148427	−0.0742134	+	0.997242i	$0.523645\pi$
$194$	0	0
$195$	−2.73566	−0.195905
$196$	0	0
$197$	2.31181	0.164710	0.0823549	−	0.996603i	$-0.473756\pi$
$197$	2.31181	0.164710	0.0823549	+	0.996603i	$0.473756\pi$
$198$	0	0
$199$	−0.773010	−0.0547972	−0.0273986	−	0.999625i	$-0.508722\pi$
$199$	−0.773010	−0.0547972	−0.0273986	+	0.999625i	$0.508722\pi$
$200$	0	0
$201$	−10.4913	−0.740001
$202$	0	0
$203$	−6.64177	−0.466161
$204$	0	0
$205$	0.971726	0.0678683
$206$	0	0
$207$	3.76394	0.261612
$208$	0	0
$209$	−3.32088	−0.229710
$210$	0	0
$211$	0.386505	0.0266081	0.0133040	−	0.999911i	$-0.495765\pi$
$211$	0.386505	0.0266081	0.0133040	+	0.999911i	$0.495765\pi$
$212$	0	0
$213$	−19.7266	−1.35164
$214$	0	0
$215$	4.34916	0.296610
$216$	0	0
$217$	−6.64177	−0.450873
$218$	0	0
$219$	7.61350	0.514472
$220$	0	0
$221$	−3.61350	−0.243070
$222$	0	0
$223$	−5.71285	−0.382561	−0.191280	−	0.981535i	$-0.561264\pi$
$223$	−5.71285	−0.382561	−0.191280	+	0.981535i	$0.561264\pi$
$224$	0	0
$225$	−0.707389	−0.0471593
$226$	0	0
$227$	−17.3720	−1.15302	−0.576509	−	0.817091i	$-0.695585\pi$
$227$	−17.3720	−1.15302	−0.576509	+	0.817091i	$0.695585\pi$
$228$	0	0
$229$	8.79221	0.581006	0.290503	−	0.956874i	$-0.406177\pi$
$229$	8.79221	0.581006	0.290503	+	0.956874i	$0.406177\pi$
$230$	0	0
$231$	−6.64177	−0.436996
$232$	0	0
$233$	20.3684	1.33438	0.667188	−	0.744890i	$-0.267498\pi$
$233$	20.3684	1.33438	0.667188	+	0.744890i	$0.267498\pi$
$234$	0	0
$235$	1.32088	0.0861650
$236$	0	0
$237$	3.02827	0.196708
$238$	0	0
$239$	−27.5388	−1.78134	−0.890669	−	0.454653i	$-0.849763\pi$
$239$	−27.5388	−1.78134	−0.890669	+	0.454653i	$0.849763\pi$
$240$	0	0
$241$	−3.61350	−0.232766	−0.116383	−	0.993204i	$-0.537130\pi$
$241$	−3.61350	−0.232766	−0.116383	+	0.993204i	$0.537130\pi$
$242$	0	0
$243$	7.18418	0.460865
$244$	0	0
$245$	−5.25526	−0.335747
$246$	0	0
$247$	−1.80675	−0.114961
$248$	0	0
$249$	−3.47133	−0.219986
$250$	0	0
$251$	−15.9253	−1.00520	−0.502598	−	0.864520i	$-0.667622\pi$
$251$	−15.9253	−1.00520	−0.502598	+	0.864520i	$0.667622\pi$
$252$	0	0
$253$	17.6700	1.11091
$254$	0	0
$255$	3.02827	0.189638
$256$	0	0
$257$	23.5333	1.46797	0.733985	−	0.679166i	$-0.237658\pi$
$257$	23.5333	1.46797	0.733985	+	0.679166i	$0.237658\pi$
$258$	0	0
$259$	−0.255264	−0.0158613
$260$	0	0
$261$	3.55695	0.220170
$262$	0	0
$263$	20.6610	1.27401	0.637005	−	0.770860i	$-0.280173\pi$
$263$	20.6610	1.27401	0.637005	+	0.770860i	$0.280173\pi$
$264$	0	0
$265$	−3.80675	−0.233847
$266$	0	0
$267$	19.7266	1.20725
$268$	0	0
$269$	18.5671	1.13205	0.566027	−	0.824386i	$-0.308480\pi$
$269$	18.5671	1.13205	0.566027	+	0.824386i	$0.308480\pi$
$270$	0	0
$271$	14.1504	0.859578	0.429789	−	0.902929i	$-0.358588\pi$
$271$	14.1504	0.859578	0.429789	+	0.902929i	$0.358588\pi$
$272$	0	0
$273$	−3.61350	−0.218699
$274$	0	0
$275$	−3.32088	−0.200257
$276$	0	0
$277$	−10.5852	−0.636004	−0.318002	−	0.948090i	$-0.603012\pi$
$277$	−10.5852	−0.636004	−0.318002	+	0.948090i	$0.603012\pi$
$278$	0	0
$279$	3.55695	0.212949
$280$	0	0
$281$	19.1414	1.14188	0.570939	−	0.820992i	$-0.306579\pi$
$281$	19.1414	1.14188	0.570939	+	0.820992i	$0.306579\pi$
$282$	0	0
$283$	14.2926	0.849608	0.424804	−	0.905285i	$-0.360343\pi$
$283$	14.2926	0.849608	0.424804	+	0.905285i	$0.360343\pi$
$284$	0	0
$285$	1.51414	0.0896897
$286$	0	0
$287$	1.28354	0.0757649
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	1.76394	0.103404
$292$	0	0
$293$	14.1751	0.828119	0.414059	−	0.910250i	$-0.364111\pi$
$293$	14.1751	0.828119	0.414059	+	0.910250i	$0.364111\pi$
$294$	0	0
$295$	8.05655	0.469070
$296$	0	0
$297$	18.6418	1.08171
$298$	0	0
$299$	9.61350	0.555963
$300$	0	0
$301$	5.74474	0.331121
$302$	0	0
$303$	5.02827	0.288867
$304$	0	0
$305$	−12.9909	−0.743858
$306$	0	0
$307$	−17.9572	−1.02487	−0.512435	−	0.858726i	$-0.671257\pi$
$307$	−17.9572	−1.02487	−0.512435	+	0.858726i	$0.671257\pi$
$308$	0	0
$309$	−4.73566	−0.269402
$310$	0	0
$311$	0.0938942	0.00532425	0.00266213	−	0.999996i	$-0.499153\pi$
$311$	0.0938942	0.00532425	0.00266213	+	0.999996i	$0.499153\pi$
$312$	0	0
$313$	26.4249	1.49362	0.746812	−	0.665035i	$-0.231583\pi$
$313$	26.4249	1.49362	0.746812	+	0.665035i	$0.231583\pi$
$314$	0	0
$315$	−0.934380	−0.0526463
$316$	0	0
$317$	1.23245	0.0692215	0.0346108	−	0.999401i	$-0.488981\pi$
$317$	1.23245	0.0692215	0.0346108	+	0.999401i	$0.488981\pi$
$318$	0	0
$319$	16.6983	0.934926
$320$	0	0
$321$	−13.9061	−0.776163
$322$	0	0
$323$	2.00000	0.111283
$324$	0	0
$325$	−1.80675	−0.100220
$326$	0	0
$327$	−11.2270	−0.620854
$328$	0	0
$329$	1.74474	0.0961904
$330$	0	0
$331$	−21.8578	−1.20142	−0.600708	−	0.799469i	$-0.705114\pi$
$331$	−21.8578	−1.20142	−0.600708	+	0.799469i	$0.705114\pi$
$332$	0	0
$333$	0.136705	0.00749137
$334$	0	0
$335$	−6.92892	−0.378567
$336$	0	0
$337$	4.64723	0.253151	0.126575	−	0.991957i	$-0.459601\pi$
$337$	4.64723	0.253151	0.126575	+	0.991957i	$0.459601\pi$
$338$	0	0
$339$	10.2635	0.557439
$340$	0	0
$341$	16.6983	0.904265
$342$	0	0
$343$	−16.1878	−0.874058
$344$	0	0
$345$	−8.05655	−0.433750
$346$	0	0
$347$	−13.9061	−0.746519	−0.373259	−	0.927727i	$-0.621760\pi$
$347$	−13.9061	−0.746519	−0.373259	+	0.927727i	$0.621760\pi$
$348$	0	0
$349$	−18.4249	−0.986263	−0.493131	−	0.869955i	$-0.664148\pi$
$349$	−18.4249	−0.986263	−0.493131	+	0.869955i	$0.664148\pi$
$350$	0	0
$351$	10.1422	0.541349
$352$	0	0
$353$	36.1696	1.92512	0.962558	−	0.271076i	$-0.0873795\pi$
$353$	36.1696	1.92512	0.962558	+	0.271076i	$0.0873795\pi$
$354$	0	0
$355$	−13.0283	−0.691469
$356$	0	0
$357$	4.00000	0.211702
$358$	0	0
$359$	−21.8880	−1.15520	−0.577601	−	0.816319i	$-0.696011\pi$
$359$	−21.8880	−1.15520	−0.577601	+	0.816319i	$0.696011\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0.0428105	0.00224697
$364$	0	0
$365$	5.02827	0.263192
$366$	0	0
$367$	−2.10482	−0.109871	−0.0549354	−	0.998490i	$-0.517495\pi$
$367$	−2.10482	−0.109871	−0.0549354	+	0.998490i	$0.517495\pi$
$368$	0	0
$369$	−0.687389	−0.0357840
$370$	0	0
$371$	−5.02827	−0.261055
$372$	0	0
$373$	34.1751	1.76952	0.884760	−	0.466047i	$-0.154322\pi$
$373$	34.1751	1.76952	0.884760	+	0.466047i	$0.154322\pi$
$374$	0	0
$375$	1.51414	0.0781897
$376$	0	0
$377$	9.08482	0.467892
$378$	0	0
$379$	−2.71646	−0.139535	−0.0697676	−	0.997563i	$-0.522226\pi$
$379$	−2.71646	−0.139535	−0.0697676	+	0.997563i	$0.522226\pi$
$380$	0	0
$381$	1.70739	0.0874722
$382$	0	0
$383$	−6.15591	−0.314552	−0.157276	−	0.987555i	$-0.550271\pi$
$383$	−6.15591	−0.314552	−0.157276	+	0.987555i	$0.550271\pi$
$384$	0	0
$385$	−4.38650	−0.223557
$386$	0	0
$387$	−3.07655	−0.156390
$388$	0	0
$389$	−11.4823	−0.582173	−0.291087	−	0.956697i	$-0.594017\pi$
$389$	−11.4823	−0.582173	−0.291087	+	0.956697i	$0.594017\pi$
$390$	0	0
$391$	−10.6418	−0.538177
$392$	0	0
$393$	−9.75566	−0.492108
$394$	0	0
$395$	2.00000	0.100631
$396$	0	0
$397$	−8.06748	−0.404895	−0.202447	−	0.979293i	$-0.564890\pi$
$397$	−8.06748	−0.404895	−0.202447	+	0.979293i	$0.564890\pi$
$398$	0	0
$399$	2.00000	0.100125
$400$	0	0
$401$	−34.8114	−1.73840	−0.869199	−	0.494462i	$-0.835365\pi$
$401$	−34.8114	−1.73840	−0.869199	+	0.494462i	$0.835365\pi$
$402$	0	0
$403$	9.08482	0.452547
$404$	0	0
$405$	−6.37743	−0.316897
$406$	0	0
$407$	0.641769	0.0318113
$408$	0	0
$409$	0.773010	0.0382229	0.0191114	−	0.999817i	$-0.493916\pi$
$409$	0.773010	0.0382229	0.0191114	+	0.999817i	$0.493916\pi$
$410$	0	0
$411$	33.1979	1.63753
$412$	0	0
$413$	10.6418	0.523647
$414$	0	0
$415$	−2.29261	−0.112540
$416$	0	0
$417$	−10.1987	−0.499433
$418$	0	0
$419$	21.0848	1.03006	0.515030	−	0.857172i	$-0.327781\pi$
$419$	21.0848	1.03006	0.515030	+	0.857172i	$0.327781\pi$
$420$	0	0
$421$	37.0101	1.80376	0.901882	−	0.431983i	$-0.142186\pi$
$421$	37.0101	1.80376	0.901882	+	0.431983i	$0.142186\pi$
$422$	0	0
$423$	−0.934380	−0.0454311
$424$	0	0
$425$	2.00000	0.0970143
$426$	0	0
$427$	−17.1595	−0.830407
$428$	0	0
$429$	9.08482	0.438619
$430$	0	0
$431$	−22.3684	−1.07745	−0.538723	−	0.842483i	$-0.681093\pi$
$431$	−22.3684	−1.07745	−0.538723	+	0.842483i	$0.681093\pi$
$432$	0	0
$433$	−11.6755	−0.561089	−0.280545	−	0.959841i	$-0.590515\pi$
$433$	−11.6755	−0.561089	−0.280545	+	0.959841i	$0.590515\pi$
$434$	0	0
$435$	−7.61350	−0.365039
$436$	0	0
$437$	−5.32088	−0.254532
$438$	0	0
$439$	−8.52867	−0.407051	−0.203526	−	0.979070i	$-0.565240\pi$
$439$	−8.52867	−0.407051	−0.203526	+	0.979070i	$0.565240\pi$
$440$	0	0
$441$	3.71752	0.177025
$442$	0	0
$443$	−15.9627	−0.758409	−0.379204	−	0.925313i	$-0.623802\pi$
$443$	−15.9627	−0.758409	−0.379204	+	0.925313i	$0.623802\pi$
$444$	0	0
$445$	13.0283	0.617599
$446$	0	0
$447$	−31.7831	−1.50329
$448$	0	0
$449$	23.9253	1.12911	0.564553	−	0.825397i	$-0.309049\pi$
$449$	23.9253	1.12911	0.564553	+	0.825397i	$0.309049\pi$
$450$	0	0
$451$	−3.22699	−0.151953
$452$	0	0
$453$	4.30088	0.202073
$454$	0	0
$455$	−2.38650	−0.111881
$456$	0	0
$457$	−17.7375	−0.829726	−0.414863	−	0.909884i	$-0.636171\pi$
$457$	−17.7375	−0.829726	−0.414863	+	0.909884i	$0.636171\pi$
$458$	0	0
$459$	−11.2270	−0.524031
$460$	0	0
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	0	0
$463$	−18.2070	−0.846151	−0.423075	−	0.906095i	$-0.639049\pi$
$463$	−18.2070	−0.846151	−0.423075	+	0.906095i	$0.639049\pi$
$464$	0	0
$465$	−7.61350	−0.353067
$466$	0	0
$467$	−22.7922	−1.05470	−0.527349	−	0.849649i	$-0.676814\pi$
$467$	−22.7922	−1.05470	−0.527349	+	0.849649i	$0.676814\pi$
$468$	0	0
$469$	−9.15230	−0.422614
$470$	0	0
$471$	−23.4257	−1.07940
$472$	0	0
$473$	−14.4431	−0.664092
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	2.69285	0.123297
$478$	0	0
$479$	26.7357	1.22158	0.610792	−	0.791791i	$-0.290851\pi$
$479$	26.7357	1.22158	0.610792	+	0.791791i	$0.290851\pi$
$480$	0	0
$481$	0.349158	0.0159203
$482$	0	0
$483$	−10.6418	−0.484217
$484$	0	0
$485$	1.16498	0.0528990
$486$	0	0
$487$	−25.7694	−1.16772	−0.583862	−	0.811853i	$-0.698459\pi$
$487$	−25.7694	−1.16772	−0.583862	+	0.811853i	$0.698459\pi$
$488$	0	0
$489$	−18.6983	−0.845567
$490$	0	0
$491$	26.3793	1.19048	0.595240	−	0.803548i	$-0.297057\pi$
$491$	26.3793	1.19048	0.595240	+	0.803548i	$0.297057\pi$
$492$	0	0
$493$	−10.0565	−0.452924
$494$	0	0
$495$	2.34916	0.105587
$496$	0	0
$497$	−17.2088	−0.771922
$498$	0	0
$499$	5.96265	0.266925	0.133463	−	0.991054i	$-0.457390\pi$
$499$	5.96265	0.266925	0.133463	+	0.991054i	$0.457390\pi$
$500$	0	0
$501$	3.34996	0.149665
$502$	0	0
$503$	19.1896	0.855624	0.427812	−	0.903868i	$-0.359285\pi$
$503$	19.1896	0.855624	0.427812	+	0.903868i	$0.359285\pi$
$504$	0	0
$505$	3.32088	0.147777
$506$	0	0
$507$	−14.7411	−0.654676
$508$	0	0
$509$	−23.6700	−1.04916	−0.524578	−	0.851362i	$-0.675777\pi$
$509$	−23.6700	−1.04916	−0.524578	+	0.851362i	$0.675777\pi$
$510$	0	0
$511$	6.64177	0.293815
$512$	0	0
$513$	−5.61350	−0.247842
$514$	0	0
$515$	−3.12763	−0.137820
$516$	0	0
$517$	−4.38650	−0.192918
$518$	0	0
$519$	3.79301	0.166495
$520$	0	0
$521$	−7.35823	−0.322370	−0.161185	−	0.986924i	$-0.551532\pi$
$521$	−7.35823	−0.322370	−0.161185	+	0.986924i	$0.551532\pi$
$522$	0	0
$523$	6.48586	0.283607	0.141803	−	0.989895i	$-0.454710\pi$
$523$	6.48586	0.283607	0.141803	+	0.989895i	$0.454710\pi$
$524$	0	0
$525$	2.00000	0.0872872
$526$	0	0
$527$	−10.0565	−0.438070
$528$	0	0
$529$	5.31181	0.230948
$530$	0	0
$531$	−5.69912	−0.247321
$532$	0	0
$533$	−1.75566	−0.0760462
$534$	0	0
$535$	−9.18418	−0.397067
$536$	0	0
$537$	−9.08482	−0.392039
$538$	0	0
$539$	17.4521	0.751716
$540$	0	0
$541$	2.22513	0.0956660	0.0478330	−	0.998855i	$-0.484768\pi$
$541$	2.22513	0.0956660	0.0478330	+	0.998855i	$0.484768\pi$
$542$	0	0
$543$	19.4257	0.833637
$544$	0	0
$545$	−7.41478	−0.317614
$546$	0	0
$547$	22.1378	0.946542	0.473271	−	0.880917i	$-0.343073\pi$
$547$	22.1378	0.946542	0.473271	+	0.880917i	$0.343073\pi$
$548$	0	0
$549$	9.18964	0.392204
$550$	0	0
$551$	−5.02827	−0.214212
$552$	0	0
$553$	2.64177	0.112339
$554$	0	0
$555$	−0.292611	−0.0124206
$556$	0	0
$557$	−31.8688	−1.35032	−0.675161	−	0.737670i	$-0.735926\pi$
$557$	−31.8688	−1.35032	−0.675161	+	0.737670i	$0.735926\pi$
$558$	0	0
$559$	−7.85783	−0.332351
$560$	0	0
$561$	−10.0565	−0.424588
$562$	0	0
$563$	−23.9681	−1.01014	−0.505068	−	0.863080i	$-0.668533\pi$
$563$	−23.9681	−1.01014	−0.505068	+	0.863080i	$0.668533\pi$
$564$	0	0
$565$	6.77847	0.285173
$566$	0	0
$567$	−8.42385	−0.353769
$568$	0	0
$569$	0.659914	0.0276650	0.0138325	−	0.999904i	$-0.495597\pi$
$569$	0.659914	0.0276650	0.0138325	+	0.999904i	$0.495597\pi$
$570$	0	0
$571$	35.2462	1.47501	0.737504	−	0.675343i	$-0.236004\pi$
$571$	35.2462	1.47501	0.737504	+	0.675343i	$0.236004\pi$
$572$	0	0
$573$	18.0565	0.754323
$574$	0	0
$575$	−5.32088	−0.221896
$576$	0	0
$577$	33.1414	1.37969	0.689847	−	0.723956i	$-0.257678\pi$
$577$	33.1414	1.37969	0.689847	+	0.723956i	$0.257678\pi$
$578$	0	0
$579$	−3.12217	−0.129753
$580$	0	0
$581$	−3.02827	−0.125634
$582$	0	0
$583$	12.6418	0.523569
$584$	0	0
$585$	1.27807	0.0528419
$586$	0	0
$587$	−11.7639	−0.485550	−0.242775	−	0.970083i	$-0.578058\pi$
$587$	−11.7639	−0.485550	−0.242775	+	0.970083i	$0.578058\pi$
$588$	0	0
$589$	−5.02827	−0.207186
$590$	0	0
$591$	3.50040	0.143987
$592$	0	0
$593$	5.34009	0.219291	0.109646	−	0.993971i	$-0.465028\pi$
$593$	5.34009	0.219291	0.109646	+	0.993971i	$0.465028\pi$
$594$	0	0
$595$	2.64177	0.108302
$596$	0	0
$597$	−1.17044	−0.0479030
$598$	0	0
$599$	−4.64177	−0.189658	−0.0948288	−	0.995494i	$-0.530230\pi$
$599$	−4.64177	−0.189658	−0.0948288	+	0.995494i	$0.530230\pi$
$600$	0	0
$601$	−18.4431	−0.752308	−0.376154	−	0.926557i	$-0.622754\pi$
$601$	−18.4431	−0.752308	−0.376154	+	0.926557i	$0.622754\pi$
$602$	0	0
$603$	4.90144	0.199602
$604$	0	0
$605$	0.0282739	0.00114950
$606$	0	0
$607$	2.92892	0.118881	0.0594405	−	0.998232i	$-0.481068\pi$
$607$	2.92892	0.118881	0.0594405	+	0.998232i	$0.481068\pi$
$608$	0	0
$609$	−10.0565	−0.407512
$610$	0	0
$611$	−2.38650	−0.0965477
$612$	0	0
$613$	7.48225	0.302205	0.151103	−	0.988518i	$-0.451718\pi$
$613$	7.48225	0.302205	0.151103	+	0.988518i	$0.451718\pi$
$614$	0	0
$615$	1.47133	0.0593296
$616$	0	0
$617$	−5.53880	−0.222984	−0.111492	−	0.993765i	$-0.535563\pi$
$617$	−5.53880	−0.222984	−0.111492	+	0.993765i	$0.535563\pi$
$618$	0	0
$619$	−13.3027	−0.534682	−0.267341	−	0.963602i	$-0.586145\pi$
$619$	−13.3027	−0.534682	−0.267341	+	0.963602i	$0.586145\pi$
$620$	0	0
$621$	29.8688	1.19859
$622$	0	0
$623$	17.2088	0.689458
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−5.02827	−0.200810
$628$	0	0
$629$	−0.386505	−0.0154110
$630$	0	0
$631$	−31.2462	−1.24389	−0.621946	−	0.783060i	$-0.713658\pi$
$631$	−31.2462	−1.24389	−0.621946	+	0.783060i	$0.713658\pi$
$632$	0	0
$633$	0.585221	0.0232605
$634$	0	0
$635$	1.12763	0.0447487
$636$	0	0
$637$	9.49494	0.376203
$638$	0	0
$639$	9.21606	0.364582
$640$	0	0
$641$	17.0101	0.671860	0.335930	−	0.941887i	$-0.390949\pi$
$641$	17.0101	0.671860	0.335930	+	0.941887i	$0.390949\pi$
$642$	0	0
$643$	20.0757	0.791710	0.395855	−	0.918313i	$-0.370448\pi$
$643$	20.0757	0.791710	0.395855	+	0.918313i	$0.370448\pi$
$644$	0	0
$645$	6.58522	0.259293
$646$	0	0
$647$	−23.6508	−0.929811	−0.464905	−	0.885360i	$-0.653912\pi$
$647$	−23.6508	−0.929811	−0.464905	+	0.885360i	$0.653912\pi$
$648$	0	0
$649$	−26.7549	−1.05022
$650$	0	0
$651$	−10.0565	−0.394147
$652$	0	0
$653$	−12.3684	−0.484011	−0.242006	−	0.970275i	$-0.577805\pi$
$653$	−12.3684	−0.484011	−0.242006	+	0.970275i	$0.577805\pi$
$654$	0	0
$655$	−6.44305	−0.251751
$656$	0	0
$657$	−3.55695	−0.138770
$658$	0	0
$659$	−14.7730	−0.575475	−0.287737	−	0.957709i	$-0.592903\pi$
$659$	−14.7730	−0.575475	−0.287737	+	0.957709i	$0.592903\pi$
$660$	0	0
$661$	41.7831	1.62518	0.812588	−	0.582839i	$-0.198058\pi$
$661$	41.7831	1.62518	0.812588	+	0.582839i	$0.198058\pi$
$662$	0	0
$663$	−5.47133	−0.212489
$664$	0	0
$665$	1.32088	0.0512217
$666$	0	0
$667$	26.7549	1.03595
$668$	0	0
$669$	−8.65004	−0.334430
$670$	0	0
$671$	43.1414	1.66545
$672$	0	0
$673$	−29.1359	−1.12311	−0.561553	−	0.827441i	$-0.689796\pi$
$673$	−29.1359	−1.12311	−0.561553	+	0.827441i	$0.689796\pi$
$674$	0	0
$675$	−5.61350	−0.216064
$676$	0	0
$677$	10.9481	0.420770	0.210385	−	0.977619i	$-0.432528\pi$
$677$	10.9481	0.420770	0.210385	+	0.977619i	$0.432528\pi$
$678$	0	0
$679$	1.53880	0.0590538
$680$	0	0
$681$	−26.3035	−1.00795
$682$	0	0
$683$	36.2799	1.38821	0.694106	−	0.719872i	$-0.255800\pi$
$683$	36.2799	1.38821	0.694106	+	0.719872i	$0.255800\pi$
$684$	0	0
$685$	21.9253	0.837723
$686$	0	0
$687$	13.3126	0.507908
$688$	0	0
$689$	6.87783	0.262025
$690$	0	0
$691$	−31.5471	−1.20011	−0.600054	−	0.799960i	$-0.704854\pi$
$691$	−31.5471	−1.20011	−0.600054	+	0.799960i	$0.704854\pi$
$692$	0	0
$693$	3.10297	0.117872
$694$	0	0
$695$	−6.73566	−0.255498
$696$	0	0
$697$	1.94345	0.0736135
$698$	0	0
$699$	30.8405	1.16649
$700$	0	0
$701$	51.9336	1.96150	0.980752	−	0.195257i	$-0.0625541\pi$
$701$	51.9336	1.96150	0.980752	+	0.195257i	$0.0625541\pi$
$702$	0	0
$703$	−0.193252	−0.00728865
$704$	0	0
$705$	2.00000	0.0753244
$706$	0	0
$707$	4.38650	0.164971
$708$	0	0
$709$	35.5953	1.33681	0.668406	−	0.743797i	$-0.266977\pi$
$709$	35.5953	1.33681	0.668406	+	0.743797i	$0.266977\pi$
$710$	0	0
$711$	−1.41478	−0.0530583
$712$	0	0
$713$	26.7549	1.00198
$714$	0	0
$715$	6.00000	0.224387
$716$	0	0
$717$	−41.6975	−1.55722
$718$	0	0
$719$	26.2635	0.979465	0.489732	−	0.871873i	$-0.337094\pi$
$719$	26.2635	0.979465	0.489732	+	0.871873i	$0.337094\pi$
$720$	0	0
$721$	−4.13124	−0.153855
$722$	0	0
$723$	−5.47133	−0.203481
$724$	0	0
$725$	−5.02827	−0.186745
$726$	0	0
$727$	−22.4804	−0.833752	−0.416876	−	0.908963i	$-0.636875\pi$
$727$	−22.4804	−0.833752	−0.416876	+	0.908963i	$0.636875\pi$
$728$	0	0
$729$	30.0101	1.11149
$730$	0	0
$731$	8.69832	0.321719
$732$	0	0
$733$	−28.1806	−1.04087	−0.520437	−	0.853900i	$-0.674231\pi$
$733$	−28.1806	−1.04087	−0.520437	+	0.853900i	$0.674231\pi$
$734$	0	0
$735$	−7.95719	−0.293505
$736$	0	0
$737$	23.0101	0.847589
$738$	0	0
$739$	−35.8506	−1.31879	−0.659393	−	0.751798i	$-0.729187\pi$
$739$	−35.8506	−1.31879	−0.659393	+	0.751798i	$0.729187\pi$
$740$	0	0
$741$	−2.73566	−0.100497
$742$	0	0
$743$	−33.5525	−1.23092	−0.615462	−	0.788167i	$-0.711030\pi$
$743$	−33.5525	−1.23092	−0.615462	+	0.788167i	$0.711030\pi$
$744$	0	0
$745$	−20.9909	−0.769048
$746$	0	0
$747$	1.62177	0.0593374
$748$	0	0
$749$	−12.1312	−0.443266
$750$	0	0
$751$	−11.4257	−0.416930	−0.208465	−	0.978030i	$-0.566847\pi$
$751$	−11.4257	−0.416930	−0.208465	+	0.978030i	$0.566847\pi$
$752$	0	0
$753$	−24.1131	−0.878730
$754$	0	0
$755$	2.84049	0.103376
$756$	0	0
$757$	−13.9144	−0.505727	−0.252863	−	0.967502i	$-0.581372\pi$
$757$	−13.9144	−0.505727	−0.252863	+	0.967502i	$0.581372\pi$
$758$	0	0
$759$	26.7549	0.971140
$760$	0	0
$761$	1.30274	0.0472243	0.0236121	−	0.999721i	$-0.492483\pi$
$761$	1.30274	0.0472243	0.0236121	+	0.999721i	$0.492483\pi$
$762$	0	0
$763$	−9.79407	−0.354569
$764$	0	0
$765$	−1.41478	−0.0511514
$766$	0	0
$767$	−14.5561	−0.525592
$768$	0	0
$769$	16.5297	0.596077	0.298039	−	0.954554i	$-0.403668\pi$
$769$	16.5297	0.596077	0.298039	+	0.954554i	$0.403668\pi$
$770$	0	0
$771$	35.6327	1.28328
$772$	0	0
$773$	−17.1084	−0.615347	−0.307674	−	0.951492i	$-0.599551\pi$
$773$	−17.1084	−0.615347	−0.307674	+	0.951492i	$0.599551\pi$
$774$	0	0
$775$	−5.02827	−0.180621
$776$	0	0
$777$	−0.386505	−0.0138658
$778$	0	0
$779$	0.971726	0.0348157
$780$	0	0
$781$	43.2654	1.54816
$782$	0	0
$783$	28.2262	1.00872
$784$	0	0
$785$	−15.4713	−0.552195
$786$	0	0
$787$	−20.4293	−0.728226	−0.364113	−	0.931355i	$-0.618628\pi$
$787$	−20.4293	−0.728226	−0.364113	+	0.931355i	$0.618628\pi$
$788$	0	0
$789$	31.2835	1.11372
$790$	0	0
$791$	8.95358	0.318353
$792$	0	0
$793$	23.4713	0.833491
$794$	0	0
$795$	−5.76394	−0.204426
$796$	0	0
$797$	−4.69285	−0.166229	−0.0831147	−	0.996540i	$-0.526487\pi$
$797$	−4.69285	−0.166229	−0.0831147	+	0.996540i	$0.526487\pi$
$798$	0	0
$799$	2.64177	0.0934591
$800$	0	0
$801$	−9.21606	−0.325634
$802$	0	0
$803$	−16.6983	−0.589271
$804$	0	0
$805$	−7.02827	−0.247714
$806$	0	0
$807$	28.1131	0.989628
$808$	0	0
$809$	−24.7658	−0.870719	−0.435359	−	0.900257i	$-0.643379\pi$
$809$	−24.7658	−0.870719	−0.435359	+	0.900257i	$0.643379\pi$
$810$	0	0
$811$	−48.8789	−1.71637	−0.858185	−	0.513341i	$-0.828408\pi$
$811$	−48.8789	−1.71637	−0.858185	+	0.513341i	$0.828408\pi$
$812$	0	0
$813$	21.4257	0.751432
$814$	0	0
$815$	−12.3492	−0.432572
$816$	0	0
$817$	4.34916	0.152158
$818$	0	0
$819$	1.68819	0.0589901
$820$	0	0
$821$	−17.0283	−0.594291	−0.297145	−	0.954832i	$-0.596035\pi$
$821$	−17.0283	−0.594291	−0.297145	+	0.954832i	$0.596035\pi$
$822$	0	0
$823$	53.8205	1.87606	0.938032	−	0.346548i	$-0.112646\pi$
$823$	53.8205	1.87606	0.938032	+	0.346548i	$0.112646\pi$
$824$	0	0
$825$	−5.02827	−0.175062
$826$	0	0
$827$	−23.1951	−0.806573	−0.403286	−	0.915074i	$-0.632132\pi$
$827$	−23.1951	−0.806573	−0.403286	+	0.915074i	$0.632132\pi$
$828$	0	0
$829$	17.3582	0.602876	0.301438	−	0.953486i	$-0.402533\pi$
$829$	17.3582	0.602876	0.301438	+	0.953486i	$0.402533\pi$
$830$	0	0
$831$	−16.0275	−0.555987
$832$	0	0
$833$	−10.5105	−0.364168
$834$	0	0
$835$	2.21245	0.0765651
$836$	0	0
$837$	28.2262	0.975640
$838$	0	0
$839$	45.4148	1.56789	0.783946	−	0.620829i	$-0.213204\pi$
$839$	45.4148	1.56789	0.783946	+	0.620829i	$0.213204\pi$
$840$	0	0
$841$	−3.71646	−0.128154
$842$	0	0
$843$	28.9827	0.998216
$844$	0	0
$845$	−9.73566	−0.334917
$846$	0	0
$847$	0.0373465	0.00128324
$848$	0	0
$849$	21.6410	0.742716
$850$	0	0
$851$	1.02827	0.0352488
$852$	0	0
$853$	28.0950	0.961953	0.480976	−	0.876734i	$-0.340282\pi$
$853$	28.0950	0.961953	0.480976	+	0.876734i	$0.340282\pi$
$854$	0	0
$855$	−0.707389	−0.0241922
$856$	0	0
$857$	−34.6000	−1.18191	−0.590957	−	0.806703i	$-0.701250\pi$
$857$	−34.6000	−1.18191	−0.590957	+	0.806703i	$0.701250\pi$
$858$	0	0
$859$	5.74474	0.196008	0.0980039	−	0.995186i	$-0.468754\pi$
$859$	5.74474	0.196008	0.0980039	+	0.995186i	$0.468754\pi$
$860$	0	0
$861$	1.94345	0.0662327
$862$	0	0
$863$	4.03188	0.137247	0.0686234	−	0.997643i	$-0.478139\pi$
$863$	4.03188	0.137247	0.0686234	+	0.997643i	$0.478139\pi$
$864$	0	0
$865$	2.50506	0.0851747
$866$	0	0
$867$	−19.6838	−0.668496
$868$	0	0
$869$	−6.64177	−0.225307
$870$	0	0
$871$	12.5188	0.424183
$872$	0	0
$873$	−0.824093	−0.0278913
$874$	0	0
$875$	1.32088	0.0446540
$876$	0	0
$877$	−38.9590	−1.31555	−0.657777	−	0.753213i	$-0.728503\pi$
$877$	−38.9590	−1.31555	−0.657777	+	0.753213i	$0.728503\pi$
$878$	0	0
$879$	21.4631	0.723931
$880$	0	0
$881$	−11.7074	−0.394432	−0.197216	−	0.980360i	$-0.563190\pi$
$881$	−11.7074	−0.394432	−0.197216	+	0.980360i	$0.563190\pi$
$882$	0	0
$883$	−28.3382	−0.953657	−0.476829	−	0.878996i	$-0.658214\pi$
$883$	−28.3382	−0.953657	−0.476829	+	0.878996i	$0.658214\pi$
$884$	0	0
$885$	12.1987	0.410055
$886$	0	0
$887$	−3.13856	−0.105383	−0.0526913	−	0.998611i	$-0.516780\pi$
$887$	−3.13856	−0.105383	−0.0526913	+	0.998611i	$0.516780\pi$
$888$	0	0
$889$	1.48947	0.0499553
$890$	0	0
$891$	21.1787	0.709514
$892$	0	0
$893$	1.32088	0.0442017
$894$	0	0
$895$	−6.00000	−0.200558
$896$	0	0
$897$	14.5561	0.486016
$898$	0	0
$899$	25.2835	0.843253
$900$	0	0
$901$	−7.61350	−0.253642
$902$	0	0
$903$	8.69832	0.289462
$904$	0	0
$905$	12.8296	0.426469
$906$	0	0
$907$	−13.1842	−0.437774	−0.218887	−	0.975750i	$-0.570243\pi$
$907$	−13.1842	−0.437774	−0.218887	+	0.975750i	$0.570243\pi$
$908$	0	0
$909$	−2.34916	−0.0779167
$910$	0	0
$911$	43.6519	1.44625	0.723126	−	0.690716i	$-0.242705\pi$
$911$	43.6519	1.44625	0.723126	+	0.690716i	$0.242705\pi$
$912$	0	0
$913$	7.61350	0.251970
$914$	0	0
$915$	−19.6700	−0.650272
$916$	0	0
$917$	−8.51053	−0.281042
$918$	0	0
$919$	−19.8013	−0.653184	−0.326592	−	0.945165i	$-0.605900\pi$
$919$	−19.8013	−0.653184	−0.326592	+	0.945165i	$0.605900\pi$
$920$	0	0
$921$	−27.1896	−0.895929
$922$	0	0
$923$	23.5388	0.774789
$924$	0	0
$925$	−0.193252	−0.00635410
$926$	0	0
$927$	2.21245	0.0726665
$928$	0	0
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	−5.25526	−0.172234
$932$	0	0
$933$	0.142169	0.00465439
$934$	0	0
$935$	−6.64177	−0.217209
$936$	0	0
$937$	−31.9709	−1.04444	−0.522222	−	0.852809i	$-0.674897\pi$
$937$	−31.9709	−1.04444	−0.522222	+	0.852809i	$0.674897\pi$
$938$	0	0
$939$	40.0109	1.30571
$940$	0	0
$941$	38.8114	1.26522	0.632608	−	0.774472i	$-0.281984\pi$
$941$	38.8114	1.26522	0.632608	+	0.774472i	$0.281984\pi$
$942$	0	0
$943$	−5.17044	−0.168373
$944$	0	0
$945$	−7.41478	−0.241203
$946$	0	0
$947$	32.9619	1.07112	0.535558	−	0.844498i	$-0.320101\pi$
$947$	32.9619	1.07112	0.535558	+	0.844498i	$0.320101\pi$
$948$	0	0
$949$	−9.08482	−0.294906
$950$	0	0
$951$	1.86610	0.0605126
$952$	0	0
$953$	−6.82409	−0.221054	−0.110527	−	0.993873i	$-0.535254\pi$
$953$	−6.82409	−0.221054	−0.110527	+	0.993873i	$0.535254\pi$
$954$	0	0
$955$	11.9253	0.385894
$956$	0	0
$957$	25.2835	0.817301
$958$	0	0
$959$	28.9608	0.935193
$960$	0	0
$961$	−5.71646	−0.184402
$962$	0	0
$963$	6.49679	0.209356
$964$	0	0
$965$	−2.06201	−0.0663785
$966$	0	0
$967$	−17.7458	−0.570666	−0.285333	−	0.958428i	$-0.592104\pi$
$967$	−17.7458	−0.570666	−0.285333	+	0.958428i	$0.592104\pi$
$968$	0	0
$969$	3.02827	0.0972822
$970$	0	0
$971$	31.1595	0.999956	0.499978	−	0.866038i	$-0.333341\pi$
$971$	31.1595	0.999956	0.499978	+	0.866038i	$0.333341\pi$
$972$	0	0
$973$	−8.89703	−0.285226
$974$	0	0
$975$	−2.73566	−0.0876113
$976$	0	0
$977$	−14.3063	−0.457701	−0.228850	−	0.973462i	$-0.573497\pi$
$977$	−14.3063	−0.457701	−0.228850	+	0.973462i	$0.573497\pi$
$978$	0	0
$979$	−43.2654	−1.38277
$980$	0	0
$981$	5.24514	0.167464
$982$	0	0
$983$	−48.2509	−1.53896	−0.769482	−	0.638669i	$-0.779485\pi$
$983$	−48.2509	−1.53896	−0.769482	+	0.638669i	$0.779485\pi$
$984$	0	0
$985$	2.31181	0.0736605
$986$	0	0
$987$	2.64177	0.0840884
$988$	0	0
$989$	−23.1414	−0.735853
$990$	0	0
$991$	6.04562	0.192045	0.0960227	−	0.995379i	$-0.469388\pi$
$991$	6.04562	0.192045	0.0960227	+	0.995379i	$0.469388\pi$
$992$	0	0
$993$	−33.0957	−1.05026
$994$	0	0
$995$	−0.773010	−0.0245061
$996$	0	0
$997$	23.9072	0.757147	0.378574	−	0.925571i	$-0.376415\pi$
$997$	23.9072	0.757147	0.378574	+	0.925571i	$0.376415\pi$
$998$	0	0
$999$	1.08482	0.0343222

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6080.2.a.bp.1.3		3
4.3	odd	2		6080.2.a.bz.1.1		3
8.3	odd	2		3040.2.a.k.1.3	✓	3
8.5	even	2		3040.2.a.n.1.1	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3040.2.a.k.1.3	✓	3	8.3	odd	2
3040.2.a.n.1.1	yes	3	8.5	even	2
6080.2.a.bp.1.3		3	1.1	even	1	trivial
6080.2.a.bz.1.1		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:	\( N \)	\(=\)	\( 6080 = 2^{6} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6080.a (trivial)

\(f(q)\)	\(=\)	\(q+1.51414 q^{3} +1.00000 q^{5} +1.32088 q^{7} -0.707389 q^{9} +O(q^{10})\)	\(q+1.51414 q^{3} +1.00000 q^{5} +1.32088 q^{7} -0.707389 q^{9} -3.32088 q^{11} -1.80675 q^{13} +1.51414 q^{15} +2.00000 q^{17} +1.00000 q^{19} +2.00000 q^{21} -5.32088 q^{23} +1.00000 q^{25} -5.61350 q^{27} -5.02827 q^{29} -5.02827 q^{31} -5.02827 q^{33} +1.32088 q^{35} -0.193252 q^{37} -2.73566 q^{39} +0.971726 q^{41} +4.34916 q^{43} -0.707389 q^{45} +1.32088 q^{47} -5.25526 q^{49} +3.02827 q^{51} -3.80675 q^{53} -3.32088 q^{55} +1.51414 q^{57} +8.05655 q^{59} -12.9909 q^{61} -0.934380 q^{63} -1.80675 q^{65} -6.92892 q^{67} -8.05655 q^{69} -13.0283 q^{71} +5.02827 q^{73} +1.51414 q^{75} -4.38650 q^{77} +2.00000 q^{79} -6.37743 q^{81} -2.29261 q^{83} +2.00000 q^{85} -7.61350 q^{87} +13.0283 q^{89} -2.38650 q^{91} -7.61350 q^{93} +1.00000 q^{95} +1.16498 q^{97} +2.34916 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{3} + 3 q^{5} - 4 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 2 * q^3 + 3 * q^5 - 4 * q^7 + 3 * q^9	\( 3 q - 2 q^{3} + 3 q^{5} - 4 q^{7} + 3 q^{9} - 2 q^{11} - 4 q^{13} - 2 q^{15} + 6 q^{17} + 3 q^{19} + 6 q^{21} - 8 q^{23} + 3 q^{25} - 14 q^{27} - 2 q^{29} - 2 q^{31} - 2 q^{33} - 4 q^{35} - 2 q^{37} + 10 q^{39} + 16 q^{41} - 8 q^{43} + 3 q^{45} - 4 q^{47} + 3 q^{49} - 4 q^{51} - 10 q^{53} - 2 q^{55} - 2 q^{57} - 2 q^{59} - 2 q^{61} + 8 q^{63} - 4 q^{65} - 4 q^{67} + 2 q^{69} - 26 q^{71} + 2 q^{73} - 2 q^{75} - 16 q^{77} + 6 q^{79} + 15 q^{81} - 12 q^{83} + 6 q^{85} - 20 q^{87} + 26 q^{89} - 10 q^{91} - 20 q^{93} + 3 q^{95} + 18 q^{97} - 14 q^{99}+O(q^{100}) \) 3 * q - 2 * q^3 + 3 * q^5 - 4 * q^7 + 3 * q^9 - 2 * q^11 - 4 * q^13 - 2 * q^15 + 6 * q^17 + 3 * q^19 + 6 * q^21 - 8 * q^23 + 3 * q^25 - 14 * q^27 - 2 * q^29 - 2 * q^31 - 2 * q^33 - 4 * q^35 - 2 * q^37 + 10 * q^39 + 16 * q^41 - 8 * q^43 + 3 * q^45 - 4 * q^47 + 3 * q^49 - 4 * q^51 - 10 * q^53 - 2 * q^55 - 2 * q^57 - 2 * q^59 - 2 * q^61 + 8 * q^63 - 4 * q^65 - 4 * q^67 + 2 * q^69 - 26 * q^71 + 2 * q^73 - 2 * q^75 - 16 * q^77 + 6 * q^79 + 15 * q^81 - 12 * q^83 + 6 * q^85 - 20 * q^87 + 26 * q^89 - 10 * q^91 - 20 * q^93 + 3 * q^95 + 18 * q^97 - 14 * q^99

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