LMFDB - Embedded newform 855.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [855,2,Mod(1,855)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(855, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("855.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 855 = 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	855.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:	$6.82720937282$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.48119$ of defining polynomial
Character	$\chi$	$=$	855.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.48119 q^{2} +0.193937 q^{4} +1.00000 q^{5} -1.67513 q^{7} +2.67513 q^{8} +O(q^{10})$	\(q-1.48119 q^{2} +0.193937 q^{4} +1.00000 q^{5} -1.67513 q^{7} +2.67513 q^{8} -1.48119 q^{10} -0.481194 q^{11} -1.28726 q^{13} +2.48119 q^{14} -4.35026 q^{16} +6.15633 q^{17} -1.00000 q^{19} +0.193937 q^{20} +0.712742 q^{22} -2.54420 q^{23} +1.00000 q^{25} +1.90668 q^{26} -0.324869 q^{28} +9.83146 q^{29} -9.50659 q^{31} +1.09332 q^{32} -9.11871 q^{34} -1.67513 q^{35} +11.5999 q^{37} +1.48119 q^{38} +2.67513 q^{40} -5.18172 q^{41} +2.06300 q^{43} -0.0933212 q^{44} +3.76845 q^{46} +5.76845 q^{47} -4.19394 q^{49} -1.48119 q^{50} -0.249646 q^{52} +1.84367 q^{53} -0.481194 q^{55} -4.48119 q^{56} -14.5623 q^{58} +14.4993 q^{59} +10.0811 q^{61} +14.0811 q^{62} +7.08110 q^{64} -1.28726 q^{65} +4.57452 q^{67} +1.19394 q^{68} +2.48119 q^{70} -9.66291 q^{71} +9.27504 q^{73} -17.1817 q^{74} -0.193937 q^{76} +0.806063 q^{77} +6.88717 q^{79} -4.35026 q^{80} +7.67513 q^{82} +2.54420 q^{83} +6.15633 q^{85} -3.05571 q^{86} -1.28726 q^{88} +4.48119 q^{89} +2.15633 q^{91} -0.493413 q^{92} -8.54420 q^{94} -1.00000 q^{95} +3.54912 q^{97} +6.21203 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + q^{4} + 3 q^{5} + 3 q^{8}+O(q^{10}) $ 3 * q + q^2 + q^4 + 3 * q^5 + 3 * q^8	$ 3 q + q^{2} + q^{4} + 3 q^{5} + 3 q^{8} + q^{10} + 4 q^{11} + 2 q^{13} + 2 q^{14} - 3 q^{16} + 8 q^{17} - 3 q^{19} + q^{20} + 8 q^{22} + 2 q^{23} + 3 q^{25} + 12 q^{26} - 6 q^{28} + 14 q^{29} - 8 q^{31} - 3 q^{32} - 6 q^{34} + 8 q^{37} - q^{38} + 3 q^{40} + 10 q^{41} + 2 q^{43} + 6 q^{44} + 6 q^{47} - 13 q^{49} + q^{50} + 16 q^{52} + 16 q^{53} + 4 q^{55} - 8 q^{56} - 6 q^{58} + 10 q^{59} - 2 q^{61} + 10 q^{62} - 11 q^{64} + 2 q^{65} + 2 q^{67} + 4 q^{68} + 2 q^{70} + 2 q^{71} - 4 q^{73} - 26 q^{74} - q^{76} + 2 q^{77} - 12 q^{79} - 3 q^{80} + 18 q^{82} - 2 q^{83} + 8 q^{85} + 8 q^{86} + 2 q^{88} + 8 q^{89} - 4 q^{91} - 22 q^{92} - 16 q^{94} - 3 q^{95} + 14 q^{97} - 9 q^{98}+O(q^{100}) $ 3 * q + q^2 + q^4 + 3 * q^5 + 3 * q^8 + q^10 + 4 * q^11 + 2 * q^13 + 2 * q^14 - 3 * q^16 + 8 * q^17 - 3 * q^19 + q^20 + 8 * q^22 + 2 * q^23 + 3 * q^25 + 12 * q^26 - 6 * q^28 + 14 * q^29 - 8 * q^31 - 3 * q^32 - 6 * q^34 + 8 * q^37 - q^38 + 3 * q^40 + 10 * q^41 + 2 * q^43 + 6 * q^44 + 6 * q^47 - 13 * q^49 + q^50 + 16 * q^52 + 16 * q^53 + 4 * q^55 - 8 * q^56 - 6 * q^58 + 10 * q^59 - 2 * q^61 + 10 * q^62 - 11 * q^64 + 2 * q^65 + 2 * q^67 + 4 * q^68 + 2 * q^70 + 2 * q^71 - 4 * q^73 - 26 * q^74 - q^76 + 2 * q^77 - 12 * q^79 - 3 * q^80 + 18 * q^82 - 2 * q^83 + 8 * q^85 + 8 * q^86 + 2 * q^88 + 8 * q^89 - 4 * q^91 - 22 * q^92 - 16 * q^94 - 3 * q^95 + 14 * q^97 - 9 * q^98

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 $

$2$	−1.48119	−1.04736	−0.523681	−	0.851914i	$-0.675442\pi$
$2$	−1.48119	−1.04736	−0.523681	+	0.851914i	$0.675442\pi$
$3$	0	0
$3$	0	0
$4$	0.193937	0.0969683
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.67513	−0.633140	−0.316570	−	0.948569i	$-0.602531\pi$
$7$	−1.67513	−0.633140	−0.316570	+	0.948569i	$0.602531\pi$
$8$	2.67513	0.945802
$9$	0	0
$10$	−1.48119	−0.468395
$11$	−0.481194	−0.145086	−0.0725428	−	0.997365i	$-0.523111\pi$
$11$	−0.481194	−0.145086	−0.0725428	+	0.997365i	$0.523111\pi$
$12$	0	0
$13$	−1.28726	−0.357021	−0.178511	−	0.983938i	$-0.557128\pi$
$13$	−1.28726	−0.357021	−0.178511	+	0.983938i	$0.557128\pi$
$14$	2.48119	0.663127
$15$	0	0
$16$	−4.35026	−1.08757
$17$	6.15633	1.49313	0.746564	−	0.665314i	$-0.231702\pi$
$17$	6.15633	1.49313	0.746564	+	0.665314i	$0.231702\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0.193937	0.0433655
$21$	0	0
$22$	0.712742	0.151957
$23$	−2.54420	−0.530502	−0.265251	−	0.964179i	$-0.585455\pi$
$23$	−2.54420	−0.530502	−0.265251	+	0.964179i	$0.585455\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	1.90668	0.373930
$27$	0	0
$28$	−0.324869	−0.0613945
$29$	9.83146	1.82566	0.912828	−	0.408345i	$-0.133894\pi$
$29$	9.83146	1.82566	0.912828	+	0.408345i	$0.133894\pi$
$30$	0	0
$31$	−9.50659	−1.70743	−0.853717	−	0.520738i	$-0.825657\pi$
$31$	−9.50659	−1.70743	−0.853717	+	0.520738i	$0.825657\pi$
$32$	1.09332	0.193274
$33$	0	0
$34$	−9.11871	−1.56385
$35$	−1.67513	−0.283149
$36$	0	0
$37$	11.5999	1.90701	0.953507	−	0.301372i	$-0.0974447\pi$
$37$	11.5999	1.90701	0.953507	+	0.301372i	$0.0974447\pi$
$38$	1.48119	0.240281
$39$	0	0
$40$	2.67513	0.422975
$41$	−5.18172	−0.809248	−0.404624	−	0.914483i	$-0.632598\pi$
$41$	−5.18172	−0.809248	−0.404624	+	0.914483i	$0.632598\pi$
$42$	0	0
$43$	2.06300	0.314605	0.157303	−	0.987550i	$-0.449720\pi$
$43$	2.06300	0.314605	0.157303	+	0.987550i	$0.449720\pi$
$44$	−0.0933212	−0.0140687
$45$	0	0
$46$	3.76845	0.555628
$47$	5.76845	0.841415	0.420708	−	0.907196i	$-0.361782\pi$
$47$	5.76845	0.841415	0.420708	+	0.907196i	$0.361782\pi$
$48$	0	0
$49$	−4.19394	−0.599134
$50$	−1.48119	−0.209473
$51$	0	0
$52$	−0.249646	−0.0346197
$53$	1.84367	0.253248	0.126624	−	0.991951i	$-0.459586\pi$
$53$	1.84367	0.253248	0.126624	+	0.991951i	$0.459586\pi$
$54$	0	0
$55$	−0.481194	−0.0648842
$56$	−4.48119	−0.598825
$57$	0	0
$58$	−14.5623	−1.91212
$59$	14.4993	1.88765	0.943824	−	0.330450i	$-0.107200\pi$
$59$	14.4993	1.88765	0.943824	+	0.330450i	$0.107200\pi$
$60$	0	0
$61$	10.0811	1.29075	0.645376	−	0.763865i	$-0.276701\pi$
$61$	10.0811	1.29075	0.645376	+	0.763865i	$0.276701\pi$
$62$	14.0811	1.78830
$63$	0	0
$64$	7.08110	0.885138
$65$	−1.28726	−0.159665
$66$	0	0
$67$	4.57452	0.558866	0.279433	−	0.960165i	$-0.409854\pi$
$67$	4.57452	0.558866	0.279433	+	0.960165i	$0.409854\pi$
$68$	1.19394	0.144786
$69$	0	0
$70$	2.48119	0.296559
$71$	−9.66291	−1.14678	−0.573388	−	0.819284i	$-0.694371\pi$
$71$	−9.66291	−1.14678	−0.573388	+	0.819284i	$0.694371\pi$
$72$	0	0
$73$	9.27504	1.08556	0.542781	−	0.839875i	$-0.317372\pi$
$73$	9.27504	1.08556	0.542781	+	0.839875i	$0.317372\pi$
$74$	−17.1817	−1.99733
$75$	0	0
$76$	−0.193937	−0.0222460
$77$	0.806063	0.0918595
$78$	0	0
$79$	6.88717	0.774867	0.387433	−	0.921898i	$-0.373362\pi$
$79$	6.88717	0.774867	0.387433	+	0.921898i	$0.373362\pi$
$80$	−4.35026	−0.486374
$81$	0	0
$82$	7.67513	0.847576
$83$	2.54420	0.279262	0.139631	−	0.990204i	$-0.455408\pi$
$83$	2.54420	0.279262	0.139631	+	0.990204i	$0.455408\pi$
$84$	0	0
$85$	6.15633	0.667747
$86$	−3.05571	−0.329506
$87$	0	0
$88$	−1.28726	−0.137222
$89$	4.48119	0.475006	0.237503	−	0.971387i	$-0.423671\pi$
$89$	4.48119	0.475006	0.237503	+	0.971387i	$0.423671\pi$
$90$	0	0
$91$	2.15633	0.226044
$92$	−0.493413	−0.0514419
$93$	0	0
$94$	−8.54420	−0.881267
$95$	−1.00000	−0.102598
$96$	0	0
$97$	3.54912	0.360359	0.180179	−	0.983634i	$-0.442332\pi$
$97$	3.54912	0.360359	0.180179	+	0.983634i	$0.442332\pi$
$98$	6.21203	0.627510
$99$	0	0
$100$	0.193937	0.0193937
$101$	4.51388	0.449148	0.224574	−	0.974457i	$-0.427901\pi$
$101$	4.51388	0.449148	0.224574	+	0.974457i	$0.427901\pi$
$102$	0	0
$103$	16.6253	1.63814	0.819070	−	0.573694i	$-0.194490\pi$
$103$	16.6253	1.63814	0.819070	+	0.573694i	$0.194490\pi$
$104$	−3.44358	−0.337671
$105$	0	0
$106$	−2.73084	−0.265243
$107$	12.4387	1.20249	0.601245	−	0.799065i	$-0.294671\pi$
$107$	12.4387	1.20249	0.601245	+	0.799065i	$0.294671\pi$
$108$	0	0
$109$	−17.6629	−1.69180	−0.845900	−	0.533341i	$-0.820936\pi$
$109$	−17.6629	−1.69180	−0.845900	+	0.533341i	$0.820936\pi$
$110$	0.712742	0.0679573
$111$	0	0
$112$	7.28726	0.688581
$113$	9.31994	0.876747	0.438373	−	0.898793i	$-0.355555\pi$
$113$	9.31994	0.876747	0.438373	+	0.898793i	$0.355555\pi$
$114$	0	0
$115$	−2.54420	−0.237248
$116$	1.90668	0.177031
$117$	0	0
$118$	−21.4763	−1.97705
$119$	−10.3127	−0.945359
$120$	0	0
$121$	−10.7685	−0.978950
$122$	−14.9321	−1.35189
$123$	0	0
$124$	−1.84367	−0.165567
$125$	1.00000	0.0894427
$126$	0	0
$127$	6.96239	0.617812	0.308906	−	0.951093i	$-0.400037\pi$
$127$	6.96239	0.617812	0.308906	+	0.951093i	$0.400037\pi$
$128$	−12.6751	−1.12033
$129$	0	0
$130$	1.90668	0.167227
$131$	−1.70545	−0.149006	−0.0745028	−	0.997221i	$-0.523737\pi$
$131$	−1.70545	−0.149006	−0.0745028	+	0.997221i	$0.523737\pi$
$132$	0	0
$133$	1.67513	0.145252
$134$	−6.77575	−0.585335
$135$	0	0
$136$	16.4690	1.41220
$137$	−1.81336	−0.154926	−0.0774628	−	0.996995i	$-0.524682\pi$
$137$	−1.81336	−0.154926	−0.0774628	+	0.996995i	$0.524682\pi$
$138$	0	0
$139$	2.51388	0.213225	0.106612	−	0.994301i	$-0.466000\pi$
$139$	2.51388	0.213225	0.106612	+	0.994301i	$0.466000\pi$
$140$	−0.324869	−0.0274565
$141$	0	0
$142$	14.3127	1.20109
$143$	0.619421	0.0517986
$144$	0	0
$145$	9.83146	0.816458
$146$	−13.7381	−1.13698
$147$	0	0
$148$	2.24965	0.184920
$149$	8.62530	0.706612	0.353306	−	0.935508i	$-0.385057\pi$
$149$	8.62530	0.706612	0.353306	+	0.935508i	$0.385057\pi$
$150$	0	0
$151$	4.85685	0.395245	0.197622	−	0.980278i	$-0.436678\pi$
$151$	4.85685	0.395245	0.197622	+	0.980278i	$0.436678\pi$
$152$	−2.67513	−0.216982
$153$	0	0
$154$	−1.19394	−0.0962102
$155$	−9.50659	−0.763587
$156$	0	0
$157$	−14.5501	−1.16122	−0.580611	−	0.814181i	$-0.697186\pi$
$157$	−14.5501	−1.16122	−0.580611	+	0.814181i	$0.697186\pi$
$158$	−10.2012	−0.811566
$159$	0	0
$160$	1.09332	0.0864346
$161$	4.26187	0.335882
$162$	0	0
$163$	−5.67513	−0.444511	−0.222255	−	0.974989i	$-0.571342\pi$
$163$	−5.67513	−0.444511	−0.222255	+	0.974989i	$0.571342\pi$
$164$	−1.00492	−0.0784714
$165$	0	0
$166$	−3.76845	−0.292489
$167$	2.41819	0.187125	0.0935626	−	0.995613i	$-0.470174\pi$
$167$	2.41819	0.187125	0.0935626	+	0.995613i	$0.470174\pi$
$168$	0	0
$169$	−11.3430	−0.872536
$170$	−9.11871	−0.699373
$171$	0	0
$172$	0.400092	0.0305067
$173$	−4.85685	−0.369259	−0.184630	−	0.982808i	$-0.559109\pi$
$173$	−4.85685	−0.369259	−0.184630	+	0.982808i	$0.559109\pi$
$174$	0	0
$175$	−1.67513	−0.126628
$176$	2.09332	0.157790
$177$	0	0
$178$	−6.63752	−0.497503
$179$	−1.03761	−0.0775547	−0.0387774	−	0.999248i	$-0.512346\pi$
$179$	−1.03761	−0.0775547	−0.0387774	+	0.999248i	$0.512346\pi$
$180$	0	0
$181$	7.27504	0.540749	0.270375	−	0.962755i	$-0.412852\pi$
$181$	7.27504	0.540749	0.270375	+	0.962755i	$0.412852\pi$
$182$	−3.19394	−0.236750
$183$	0	0
$184$	−6.80606	−0.501750
$185$	11.5999	0.852842
$186$	0	0
$187$	−2.96239	−0.216631
$188$	1.11871	0.0815906
$189$	0	0
$190$	1.48119	0.107457
$191$	−17.4436	−1.26217	−0.631087	−	0.775712i	$-0.717391\pi$
$191$	−17.4436	−1.26217	−0.631087	+	0.775712i	$0.717391\pi$
$192$	0	0
$193$	−12.1138	−0.871970	−0.435985	−	0.899954i	$-0.643600\pi$
$193$	−12.1138	−0.871970	−0.435985	+	0.899954i	$0.643600\pi$
$194$	−5.25694	−0.377426
$195$	0	0
$196$	−0.813358	−0.0580970
$197$	−12.8568	−0.916013	−0.458006	−	0.888949i	$-0.651436\pi$
$197$	−12.8568	−0.916013	−0.458006	+	0.888949i	$0.651436\pi$
$198$	0	0
$199$	−3.81336	−0.270322	−0.135161	−	0.990824i	$-0.543155\pi$
$199$	−3.81336	−0.270322	−0.135161	+	0.990824i	$0.543155\pi$
$200$	2.67513	0.189160
$201$	0	0
$202$	−6.68594	−0.470421
$203$	−16.4690	−1.15590
$204$	0	0
$205$	−5.18172	−0.361907
$206$	−24.6253	−1.71573
$207$	0	0
$208$	5.59991	0.388284
$209$	0.481194	0.0332849
$210$	0	0
$211$	−10.8872	−0.749503	−0.374752	−	0.927125i	$-0.622272\pi$
$211$	−10.8872	−0.749503	−0.374752	+	0.927125i	$0.622272\pi$
$212$	0.357556	0.0245570
$213$	0	0
$214$	−18.4241	−1.25944
$215$	2.06300	0.140696
$216$	0	0
$217$	15.9248	1.08104
$218$	26.1622	1.77193
$219$	0	0
$220$	−0.0933212	−0.00629171
$221$	−7.92478	−0.533078
$222$	0	0
$223$	−22.5501	−1.51006	−0.755032	−	0.655687i	$-0.772379\pi$
$223$	−22.5501	−1.51006	−0.755032	+	0.655687i	$0.772379\pi$
$224$	−1.83146	−0.122369
$225$	0	0
$226$	−13.8046	−0.918272
$227$	−12.0303	−0.798480	−0.399240	−	0.916846i	$-0.630726\pi$
$227$	−12.0303	−0.798480	−0.399240	+	0.916846i	$0.630726\pi$
$228$	0	0
$229$	−14.2677	−0.942839	−0.471420	−	0.881909i	$-0.656258\pi$
$229$	−14.2677	−0.942839	−0.471420	+	0.881909i	$0.656258\pi$
$230$	3.76845	0.248484
$231$	0	0
$232$	26.3004	1.72671
$233$	23.4617	1.53703	0.768513	−	0.639834i	$-0.220997\pi$
$233$	23.4617	1.53703	0.768513	+	0.639834i	$0.220997\pi$
$234$	0	0
$235$	5.76845	0.376292
$236$	2.81194	0.183042
$237$	0	0
$238$	15.2750	0.990134
$239$	7.63023	0.493558	0.246779	−	0.969072i	$-0.420628\pi$
$239$	7.63023	0.493558	0.246779	+	0.969072i	$0.420628\pi$
$240$	0	0
$241$	−7.14903	−0.460510	−0.230255	−	0.973130i	$-0.573956\pi$
$241$	−7.14903	−0.460510	−0.230255	+	0.973130i	$0.573956\pi$
$242$	15.9502	1.02532
$243$	0	0
$244$	1.95509	0.125162
$245$	−4.19394	−0.267941
$246$	0	0
$247$	1.28726	0.0819062
$248$	−25.4314	−1.61489
$249$	0	0
$250$	−1.48119	−0.0936790
$251$	−10.6072	−0.669521	−0.334760	−	0.942303i	$-0.608655\pi$
$251$	−10.6072	−0.669521	−0.334760	+	0.942303i	$0.608655\pi$
$252$	0	0
$253$	1.22425	0.0769682
$254$	−10.3127	−0.647073
$255$	0	0
$256$	4.61213	0.288258
$257$	27.8945	1.74001	0.870004	−	0.493044i	$-0.164116\pi$
$257$	27.8945	1.74001	0.870004	+	0.493044i	$0.164116\pi$
$258$	0	0
$259$	−19.4314	−1.20741
$260$	−0.249646	−0.0154824
$261$	0	0
$262$	2.52610	0.156063
$263$	−28.3430	−1.74770	−0.873851	−	0.486194i	$-0.838385\pi$
$263$	−28.3430	−1.74770	−0.873851	+	0.486194i	$0.838385\pi$
$264$	0	0
$265$	1.84367	0.113256
$266$	−2.48119	−0.152132
$267$	0	0
$268$	0.887166	0.0541923
$269$	13.2569	0.808290	0.404145	−	0.914695i	$-0.367569\pi$
$269$	13.2569	0.808290	0.404145	+	0.914695i	$0.367569\pi$
$270$	0	0
$271$	−25.9003	−1.57333	−0.786667	−	0.617378i	$-0.788195\pi$
$271$	−25.9003	−1.57333	−0.786667	+	0.617378i	$0.788195\pi$
$272$	−26.7816	−1.62387
$273$	0	0
$274$	2.68594	0.162263
$275$	−0.481194	−0.0290171
$276$	0	0
$277$	14.3127	0.859964	0.429982	−	0.902838i	$-0.358520\pi$
$277$	14.3127	0.859964	0.429982	+	0.902838i	$0.358520\pi$
$278$	−3.72355	−0.223323
$279$	0	0
$280$	−4.48119	−0.267803
$281$	19.8315	1.18305	0.591523	−	0.806288i	$-0.298527\pi$
$281$	19.8315	1.18305	0.591523	+	0.806288i	$0.298527\pi$
$282$	0	0
$283$	−20.5379	−1.22085	−0.610425	−	0.792074i	$-0.709001\pi$
$283$	−20.5379	−1.22085	−0.610425	+	0.792074i	$0.709001\pi$
$284$	−1.87399	−0.111201
$285$	0	0
$286$	−0.917483	−0.0542519
$287$	8.68006	0.512367
$288$	0	0
$289$	20.9003	1.22943
$290$	−14.5623	−0.855128
$291$	0	0
$292$	1.79877	0.105265
$293$	−13.0435	−0.762009	−0.381004	−	0.924573i	$-0.624422\pi$
$293$	−13.0435	−0.762009	−0.381004	+	0.924573i	$0.624422\pi$
$294$	0	0
$295$	14.4993	0.844181
$296$	31.0313	1.80366
$297$	0	0
$298$	−12.7757	−0.740079
$299$	3.27504	0.189400
$300$	0	0
$301$	−3.45580	−0.199189
$302$	−7.19394	−0.413965
$303$	0	0
$304$	4.35026	0.249505
$305$	10.0811	0.577242
$306$	0	0
$307$	33.8397	1.93133	0.965667	−	0.259783i	$-0.0836511\pi$
$307$	33.8397	1.93133	0.965667	+	0.259783i	$0.0836511\pi$
$308$	0.156325	0.00890745
$309$	0	0
$310$	14.0811	0.799753
$311$	24.0082	1.36138	0.680691	−	0.732570i	$-0.261679\pi$
$311$	24.0082	1.36138	0.680691	+	0.732570i	$0.261679\pi$
$312$	0	0
$313$	11.0132	0.622501	0.311251	−	0.950328i	$-0.399252\pi$
$313$	11.0132	0.622501	0.311251	+	0.950328i	$0.399252\pi$
$314$	21.5515	1.21622
$315$	0	0
$316$	1.33567	0.0751375
$317$	−7.64244	−0.429242	−0.214621	−	0.976697i	$-0.568852\pi$
$317$	−7.64244	−0.429242	−0.214621	+	0.976697i	$0.568852\pi$
$318$	0	0
$319$	−4.73084	−0.264876
$320$	7.08110	0.395846
$321$	0	0
$322$	−6.31265	−0.351790
$323$	−6.15633	−0.342547
$324$	0	0
$325$	−1.28726	−0.0714042
$326$	8.40597	0.465564
$327$	0	0
$328$	−13.8618	−0.765388
$329$	−9.66291	−0.532734
$330$	0	0
$331$	23.9307	1.31535	0.657674	−	0.753303i	$-0.271540\pi$
$331$	23.9307	1.31535	0.657674	+	0.753303i	$0.271540\pi$
$332$	0.493413	0.0270796
$333$	0	0
$334$	−3.58181	−0.195988
$335$	4.57452	0.249932
$336$	0	0
$337$	4.65211	0.253416	0.126708	−	0.991940i	$-0.459559\pi$
$337$	4.65211	0.253416	0.126708	+	0.991940i	$0.459559\pi$
$338$	16.8011	0.913861
$339$	0	0
$340$	1.19394	0.0647503
$341$	4.57452	0.247724
$342$	0	0
$343$	18.7513	1.01248
$344$	5.51881	0.297554
$345$	0	0
$346$	7.19394	0.386748
$347$	−5.61801	−0.301590	−0.150795	−	0.988565i	$-0.548183\pi$
$347$	−5.61801	−0.301590	−0.150795	+	0.988565i	$0.548183\pi$
$348$	0	0
$349$	−17.7235	−0.948720	−0.474360	−	0.880331i	$-0.657320\pi$
$349$	−17.7235	−0.948720	−0.474360	+	0.880331i	$0.657320\pi$
$350$	2.48119	0.132625
$351$	0	0
$352$	−0.526100	−0.0280412
$353$	−7.40105	−0.393918	−0.196959	−	0.980412i	$-0.563107\pi$
$353$	−7.40105	−0.393918	−0.196959	+	0.980412i	$0.563107\pi$
$354$	0	0
$355$	−9.66291	−0.512854
$356$	0.869067	0.0460605
$357$	0	0
$358$	1.53690	0.0812279
$359$	−18.5926	−0.981281	−0.490640	−	0.871362i	$-0.663237\pi$
$359$	−18.5926	−0.981281	−0.490640	+	0.871362i	$0.663237\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−10.7757	−0.566361
$363$	0	0
$364$	0.418190	0.0219191
$365$	9.27504	0.485478
$366$	0	0
$367$	−24.8895	−1.29922	−0.649612	−	0.760266i	$-0.725068\pi$
$367$	−24.8895	−1.29922	−0.649612	+	0.760266i	$0.725068\pi$
$368$	11.0679	0.576956
$369$	0	0
$370$	−17.1817	−0.893235
$371$	−3.08840	−0.160342
$372$	0	0
$373$	13.6243	0.705442	0.352721	−	0.935729i	$-0.385256\pi$
$373$	13.6243	0.705442	0.352721	+	0.935729i	$0.385256\pi$
$374$	4.38787	0.226892
$375$	0	0
$376$	15.4314	0.795812
$377$	−12.6556	−0.651797
$378$	0	0
$379$	3.82909	0.196687	0.0983435	−	0.995153i	$-0.468646\pi$
$379$	3.82909	0.196687	0.0983435	+	0.995153i	$0.468646\pi$
$380$	−0.193937	−0.00994874
$381$	0	0
$382$	25.8373	1.32195
$383$	20.3127	1.03793	0.518964	−	0.854796i	$-0.326318\pi$
$383$	20.3127	1.03793	0.518964	+	0.854796i	$0.326318\pi$
$384$	0	0
$385$	0.806063	0.0410808
$386$	17.9429	0.913268
$387$	0	0
$388$	0.688305	0.0349434
$389$	−15.2506	−0.773236	−0.386618	−	0.922240i	$-0.626357\pi$
$389$	−15.2506	−0.773236	−0.386618	+	0.922240i	$0.626357\pi$
$390$	0	0
$391$	−15.6629	−0.792108
$392$	−11.2193	−0.566662
$393$	0	0
$394$	19.0435	0.959397
$395$	6.88717	0.346531
$396$	0	0
$397$	−23.7137	−1.19016	−0.595078	−	0.803668i	$-0.702879\pi$
$397$	−23.7137	−1.19016	−0.595078	+	0.803668i	$0.702879\pi$
$398$	5.64832	0.283125
$399$	0	0
$400$	−4.35026	−0.217513
$401$	21.3928	1.06831	0.534153	−	0.845388i	$-0.320631\pi$
$401$	21.3928	1.06831	0.534153	+	0.845388i	$0.320631\pi$
$402$	0	0
$403$	12.2374	0.609590
$404$	0.875407	0.0435531
$405$	0	0
$406$	24.3938	1.21064
$407$	−5.58181	−0.276680
$408$	0	0
$409$	19.2506	0.951881	0.475940	−	0.879477i	$-0.342108\pi$
$409$	19.2506	0.951881	0.475940	+	0.879477i	$0.342108\pi$
$410$	7.67513	0.379048
$411$	0	0
$412$	3.22425	0.158848
$413$	−24.2882	−1.19514
$414$	0	0
$415$	2.54420	0.124890
$416$	−1.40739	−0.0690028
$417$	0	0
$418$	−0.712742	−0.0348614
$419$	33.0821	1.61616	0.808082	−	0.589070i	$-0.200506\pi$
$419$	33.0821	1.61616	0.808082	+	0.589070i	$0.200506\pi$
$420$	0	0
$421$	−16.5139	−0.804837	−0.402419	−	0.915456i	$-0.631830\pi$
$421$	−16.5139	−0.804837	−0.402419	+	0.915456i	$0.631830\pi$
$422$	16.1260	0.785002
$423$	0	0
$424$	4.93207	0.239523
$425$	6.15633	0.298626
$426$	0	0
$427$	−16.8872	−0.817227
$428$	2.41231	0.116603
$429$	0	0
$430$	−3.05571	−0.147359
$431$	7.19982	0.346803	0.173401	−	0.984851i	$-0.444524\pi$
$431$	7.19982	0.346803	0.173401	+	0.984851i	$0.444524\pi$
$432$	0	0
$433$	−28.0240	−1.34675	−0.673373	−	0.739303i	$-0.735155\pi$
$433$	−28.0240	−1.34675	−0.673373	+	0.739303i	$0.735155\pi$
$434$	−23.5877	−1.13225
$435$	0	0
$436$	−3.42548	−0.164051
$437$	2.54420	0.121706
$438$	0	0
$439$	24.8773	1.18733	0.593665	−	0.804712i	$-0.297681\pi$
$439$	24.8773	1.18733	0.593665	+	0.804712i	$0.297681\pi$
$440$	−1.28726	−0.0613676
$441$	0	0
$442$	11.7381	0.558326
$443$	27.1451	1.28970	0.644850	−	0.764309i	$-0.276920\pi$
$443$	27.1451	1.28970	0.644850	+	0.764309i	$0.276920\pi$
$444$	0	0
$445$	4.48119	0.212429
$446$	33.4010	1.58159
$447$	0	0
$448$	−11.8618	−0.560416
$449$	9.11634	0.430227	0.215113	−	0.976589i	$-0.430988\pi$
$449$	9.11634	0.430227	0.215113	+	0.976589i	$0.430988\pi$
$450$	0	0
$451$	2.49341	0.117410
$452$	1.80748	0.0850166
$453$	0	0
$454$	17.8192	0.836298
$455$	2.15633	0.101090
$456$	0	0
$457$	−21.3404	−0.998262	−0.499131	−	0.866526i	$-0.666347\pi$
$457$	−21.3404	−0.998262	−0.499131	+	0.866526i	$0.666347\pi$
$458$	21.1333	0.987494
$459$	0	0
$460$	−0.493413	−0.0230055
$461$	18.2981	0.852226	0.426113	−	0.904670i	$-0.359883\pi$
$461$	18.2981	0.852226	0.426113	+	0.904670i	$0.359883\pi$
$462$	0	0
$463$	33.5999	1.56152	0.780760	−	0.624831i	$-0.214832\pi$
$463$	33.5999	1.56152	0.780760	+	0.624831i	$0.214832\pi$
$464$	−42.7694	−1.98552
$465$	0	0
$466$	−34.7513	−1.60982
$467$	−22.5198	−1.04209	−0.521045	−	0.853529i	$-0.674458\pi$
$467$	−22.5198	−1.04209	−0.521045	+	0.853529i	$0.674458\pi$
$468$	0	0
$469$	−7.66291	−0.353840
$470$	−8.54420	−0.394114
$471$	0	0
$472$	38.7875	1.78534
$473$	−0.992706	−0.0456447
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	−2.00000	−0.0916698
$477$	0	0
$478$	−11.3018	−0.516934
$479$	−3.46802	−0.158458	−0.0792289	−	0.996856i	$-0.525246\pi$
$479$	−3.46802	−0.158458	−0.0792289	+	0.996856i	$0.525246\pi$
$480$	0	0
$481$	−14.9321	−0.680844
$482$	10.5891	0.482320
$483$	0	0
$484$	−2.08840	−0.0949271
$485$	3.54912	0.161157
$486$	0	0
$487$	−19.5223	−0.884641	−0.442320	−	0.896857i	$-0.645845\pi$
$487$	−19.5223	−0.884641	−0.442320	+	0.896857i	$0.645845\pi$
$488$	26.9683	1.22080
$489$	0	0
$490$	6.21203	0.280631
$491$	−22.2193	−1.00274	−0.501372	−	0.865232i	$-0.667171\pi$
$491$	−22.2193	−1.00274	−0.501372	+	0.865232i	$0.667171\pi$
$492$	0	0
$493$	60.5256	2.72594
$494$	−1.90668	−0.0857855
$495$	0	0
$496$	41.3561	1.85695
$497$	16.1866	0.726070
$498$	0	0
$499$	−13.5877	−0.608269	−0.304134	−	0.952629i	$-0.598367\pi$
$499$	−13.5877	−0.608269	−0.304134	+	0.952629i	$0.598367\pi$
$500$	0.193937	0.00867311
$501$	0	0
$502$	15.7113	0.701231
$503$	3.25457	0.145114	0.0725571	−	0.997364i	$-0.476884\pi$
$503$	3.25457	0.145114	0.0725571	+	0.997364i	$0.476884\pi$
$504$	0	0
$505$	4.51388	0.200865
$506$	−1.81336	−0.0806136
$507$	0	0
$508$	1.35026	0.0599082
$509$	40.2555	1.78429	0.892147	−	0.451744i	$-0.149198\pi$
$509$	40.2555	1.78429	0.892147	+	0.451744i	$0.149198\pi$
$510$	0	0
$511$	−15.5369	−0.687312
$512$	18.5188	0.818423
$513$	0	0
$514$	−41.3171	−1.82242
$515$	16.6253	0.732598
$516$	0	0
$517$	−2.77575	−0.122077
$518$	28.7816	1.26459
$519$	0	0
$520$	−3.44358	−0.151011
$521$	14.4713	0.634001	0.317001	−	0.948425i	$-0.397324\pi$
$521$	14.4713	0.634001	0.317001	+	0.948425i	$0.397324\pi$
$522$	0	0
$523$	24.7005	1.08008	0.540039	−	0.841640i	$-0.318409\pi$
$523$	24.7005	1.08008	0.540039	+	0.841640i	$0.318409\pi$
$524$	−0.330749	−0.0144488
$525$	0	0
$526$	41.9814	1.83048
$527$	−58.5256	−2.54942
$528$	0	0
$529$	−16.5271	−0.718568
$530$	−2.73084	−0.118620
$531$	0	0
$532$	0.324869	0.0140849
$533$	6.67021	0.288919
$534$	0	0
$535$	12.4387	0.537770
$536$	12.2374	0.528576
$537$	0	0
$538$	−19.6361	−0.846573
$539$	2.01810	0.0869257
$540$	0	0
$541$	13.8496	0.595439	0.297719	−	0.954653i	$-0.403774\pi$
$541$	13.8496	0.595439	0.297719	+	0.954653i	$0.403774\pi$
$542$	38.3634	1.64785
$543$	0	0
$544$	6.73084	0.288582
$545$	−17.6629	−0.756596
$546$	0	0
$547$	−0.649738	−0.0277808	−0.0138904	−	0.999904i	$-0.504422\pi$
$547$	−0.649738	−0.0277808	−0.0138904	+	0.999904i	$0.504422\pi$
$548$	−0.351676	−0.0150229
$549$	0	0
$550$	0.712742	0.0303914
$551$	−9.83146	−0.418834
$552$	0	0
$553$	−11.5369	−0.490599
$554$	−21.1998	−0.900694
$555$	0	0
$556$	0.487533	0.0206760
$557$	−13.3503	−0.565669	−0.282834	−	0.959169i	$-0.591275\pi$
$557$	−13.3503	−0.565669	−0.282834	+	0.959169i	$0.591275\pi$
$558$	0	0
$559$	−2.65562	−0.112321
$560$	7.28726	0.307943
$561$	0	0
$562$	−29.3742	−1.23908
$563$	−25.2809	−1.06546	−0.532732	−	0.846284i	$-0.678835\pi$
$563$	−25.2809	−1.06546	−0.532732	+	0.846284i	$0.678835\pi$
$564$	0	0
$565$	9.31994	0.392093
$566$	30.4206	1.27867
$567$	0	0
$568$	−25.8496	−1.08462
$569$	−46.4323	−1.94654	−0.973272	−	0.229655i	$-0.926240\pi$
$569$	−46.4323	−1.94654	−0.973272	+	0.229655i	$0.926240\pi$
$570$	0	0
$571$	−24.3127	−1.01745	−0.508726	−	0.860928i	$-0.669884\pi$
$571$	−24.3127	−1.01745	−0.508726	+	0.860928i	$0.669884\pi$
$572$	0.120128	0.00502282
$573$	0	0
$574$	−12.8568	−0.536634
$575$	−2.54420	−0.106100
$576$	0	0
$577$	21.7743	0.906477	0.453239	−	0.891389i	$-0.350269\pi$
$577$	21.7743	0.906477	0.453239	+	0.891389i	$0.350269\pi$
$578$	−30.9575	−1.28766
$579$	0	0
$580$	1.90668	0.0791705
$581$	−4.26187	−0.176812
$582$	0	0
$583$	−0.887166	−0.0367427
$584$	24.8119	1.02673
$585$	0	0
$586$	19.3199	0.798100
$587$	−2.56864	−0.106019	−0.0530095	−	0.998594i	$-0.516881\pi$
$587$	−2.56864	−0.106019	−0.0530095	+	0.998594i	$0.516881\pi$
$588$	0	0
$589$	9.50659	0.391712
$590$	−21.4763	−0.884164
$591$	0	0
$592$	−50.4626	−2.07400
$593$	−8.94780	−0.367442	−0.183721	−	0.982978i	$-0.558814\pi$
$593$	−8.94780	−0.367442	−0.183721	+	0.982978i	$0.558814\pi$
$594$	0	0
$595$	−10.3127	−0.422777
$596$	1.67276	0.0685190
$597$	0	0
$598$	−4.85097	−0.198371
$599$	25.4471	1.03974	0.519870	−	0.854245i	$-0.325980\pi$
$599$	25.4471	1.03974	0.519870	+	0.854245i	$0.325980\pi$
$600$	0	0
$601$	−26.8726	−1.09616	−0.548078	−	0.836427i	$-0.684640\pi$
$601$	−26.8726	−1.09616	−0.548078	+	0.836427i	$0.684640\pi$
$602$	5.11871	0.208623
$603$	0	0
$604$	0.941921	0.0383262
$605$	−10.7685	−0.437800
$606$	0	0
$607$	−11.4499	−0.464738	−0.232369	−	0.972628i	$-0.574648\pi$
$607$	−11.4499	−0.464738	−0.232369	+	0.972628i	$0.574648\pi$
$608$	−1.09332	−0.0443400
$609$	0	0
$610$	−14.9321	−0.604582
$611$	−7.42548	−0.300403
$612$	0	0
$613$	−39.2360	−1.58473	−0.792364	−	0.610049i	$-0.791150\pi$
$613$	−39.2360	−1.58473	−0.792364	+	0.610049i	$0.791150\pi$
$614$	−50.1232	−2.02281
$615$	0	0
$616$	2.15633	0.0868808
$617$	40.8568	1.64483	0.822417	−	0.568885i	$-0.192625\pi$
$617$	40.8568	1.64483	0.822417	+	0.568885i	$0.192625\pi$
$618$	0	0
$619$	18.7612	0.754075	0.377037	−	0.926198i	$-0.376943\pi$
$619$	18.7612	0.754075	0.377037	+	0.926198i	$0.376943\pi$
$620$	−1.84367	−0.0740438
$621$	0	0
$622$	−35.5609	−1.42586
$623$	−7.50659	−0.300745
$624$	0	0
$625$	1.00000	0.0400000
$626$	−16.3127	−0.651985
$627$	0	0
$628$	−2.82179	−0.112602
$629$	71.4128	2.84742
$630$	0	0
$631$	32.5599	1.29619	0.648095	−	0.761559i	$-0.275566\pi$
$631$	32.5599	1.29619	0.648095	+	0.761559i	$0.275566\pi$
$632$	18.4241	0.732870
$633$	0	0
$634$	11.3199	0.449572
$635$	6.96239	0.276294
$636$	0	0
$637$	5.39868	0.213903
$638$	7.00729	0.277421
$639$	0	0
$640$	−12.6751	−0.501029
$641$	25.6712	1.01395	0.506975	−	0.861961i	$-0.330764\pi$
$641$	25.6712	1.01395	0.506975	+	0.861961i	$0.330764\pi$
$642$	0	0
$643$	−6.22521	−0.245498	−0.122749	−	0.992438i	$-0.539171\pi$
$643$	−6.22521	−0.245498	−0.122749	+	0.992438i	$0.539171\pi$
$644$	0.826531	0.0325699
$645$	0	0
$646$	9.11871	0.358771
$647$	−37.8653	−1.48864	−0.744319	−	0.667824i	$-0.767226\pi$
$647$	−37.8653	−1.48864	−0.744319	+	0.667824i	$0.767226\pi$
$648$	0	0
$649$	−6.97698	−0.273870
$650$	1.90668	0.0747861
$651$	0	0
$652$	−1.10062	−0.0431034
$653$	−41.5428	−1.62569	−0.812847	−	0.582477i	$-0.802084\pi$
$653$	−41.5428	−1.62569	−0.812847	+	0.582477i	$0.802084\pi$
$654$	0	0
$655$	−1.70545	−0.0666374
$656$	22.5418	0.880111
$657$	0	0
$658$	14.3127	0.557965
$659$	−39.3719	−1.53371	−0.766855	−	0.641820i	$-0.778180\pi$
$659$	−39.3719	−1.53371	−0.766855	+	0.641820i	$0.778180\pi$
$660$	0	0
$661$	−21.5369	−0.837688	−0.418844	−	0.908058i	$-0.637565\pi$
$661$	−21.5369	−0.837688	−0.418844	+	0.908058i	$0.637565\pi$
$662$	−35.4460	−1.37765
$663$	0	0
$664$	6.80606	0.264126
$665$	1.67513	0.0649588
$666$	0	0
$667$	−25.0132	−0.968514
$668$	0.468976	0.0181452
$669$	0	0
$670$	−6.77575	−0.261770
$671$	−4.85097	−0.187270
$672$	0	0
$673$	−10.4509	−0.402852	−0.201426	−	0.979504i	$-0.564558\pi$
$673$	−10.4509	−0.402852	−0.201426	+	0.979504i	$0.564558\pi$
$674$	−6.89068	−0.265419
$675$	0	0
$676$	−2.19982	−0.0846083
$677$	−35.5329	−1.36564	−0.682821	−	0.730586i	$-0.739247\pi$
$677$	−35.5329	−1.36564	−0.682821	+	0.730586i	$0.739247\pi$
$678$	0	0
$679$	−5.94525	−0.228158
$680$	16.4690	0.631556
$681$	0	0
$682$	−6.77575	−0.259457
$683$	−20.5891	−0.787820	−0.393910	−	0.919149i	$-0.628878\pi$
$683$	−20.5891	−0.787820	−0.393910	+	0.919149i	$0.628878\pi$
$684$	0	0
$685$	−1.81336	−0.0692848
$686$	−27.7743	−1.06043
$687$	0	0
$688$	−8.97461	−0.342154
$689$	−2.37328	−0.0904149
$690$	0	0
$691$	−29.0884	−1.10657	−0.553287	−	0.832990i	$-0.686627\pi$
$691$	−29.0884	−1.10657	−0.553287	+	0.832990i	$0.686627\pi$
$692$	−0.941921	−0.0358064
$693$	0	0
$694$	8.32136	0.315874
$695$	2.51388	0.0953569
$696$	0	0
$697$	−31.9003	−1.20831
$698$	26.2520	0.993653
$699$	0	0
$700$	−0.324869	−0.0122789
$701$	7.38646	0.278983	0.139491	−	0.990223i	$-0.455453\pi$
$701$	7.38646	0.278983	0.139491	+	0.990223i	$0.455453\pi$
$702$	0	0
$703$	−11.5999	−0.437499
$704$	−3.40739	−0.128421
$705$	0	0
$706$	10.9624	0.412575
$707$	−7.56134	−0.284374
$708$	0	0
$709$	39.5837	1.48660	0.743299	−	0.668959i	$-0.233260\pi$
$709$	39.5837	1.48660	0.743299	+	0.668959i	$0.233260\pi$
$710$	14.3127	0.537144
$711$	0	0
$712$	11.9878	0.449261
$713$	24.1866	0.905797
$714$	0	0
$715$	0.619421	0.0231650
$716$	−0.201231	−0.00752035
$717$	0	0
$718$	27.5393	1.02776
$719$	−31.5940	−1.17826	−0.589129	−	0.808039i	$-0.700529\pi$
$719$	−31.5940	−1.17826	−0.589129	+	0.808039i	$0.700529\pi$
$720$	0	0
$721$	−27.8496	−1.03717
$722$	−1.48119	−0.0551243
$723$	0	0
$724$	1.41090	0.0524355
$725$	9.83146	0.365131
$726$	0	0
$727$	−13.8011	−0.511856	−0.255928	−	0.966696i	$-0.582381\pi$
$727$	−13.8011	−0.511856	−0.255928	+	0.966696i	$0.582381\pi$
$728$	5.76845	0.213793
$729$	0	0
$730$	−13.7381	−0.508471
$731$	12.7005	0.469746
$732$	0	0
$733$	35.5731	1.31392	0.656961	−	0.753924i	$-0.271842\pi$
$733$	35.5731	1.31392	0.656961	+	0.753924i	$0.271842\pi$
$734$	36.8662	1.36076
$735$	0	0
$736$	−2.78163	−0.102532
$737$	−2.20123	−0.0810834
$738$	0	0
$739$	21.2144	0.780384	0.390192	−	0.920733i	$-0.372409\pi$
$739$	21.2144	0.780384	0.390192	+	0.920733i	$0.372409\pi$
$740$	2.24965	0.0826986
$741$	0	0
$742$	4.57452	0.167936
$743$	−40.1133	−1.47162	−0.735808	−	0.677191i	$-0.763197\pi$
$743$	−40.1133	−1.47162	−0.735808	+	0.677191i	$0.763197\pi$
$744$	0	0
$745$	8.62530	0.316007
$746$	−20.1803	−0.738853
$747$	0	0
$748$	−0.574515	−0.0210064
$749$	−20.8364	−0.761345
$750$	0	0
$751$	3.87002	0.141219	0.0706096	−	0.997504i	$-0.477506\pi$
$751$	3.87002	0.141219	0.0706096	+	0.997504i	$0.477506\pi$
$752$	−25.0943	−0.915094
$753$	0	0
$754$	18.7454	0.682668
$755$	4.85685	0.176759
$756$	0	0
$757$	−15.3963	−0.559588	−0.279794	−	0.960060i	$-0.590266\pi$
$757$	−15.3963	−0.559588	−0.279794	+	0.960060i	$0.590266\pi$
$758$	−5.67162	−0.206003
$759$	0	0
$760$	−2.67513	−0.0970372
$761$	−29.6385	−1.07439	−0.537197	−	0.843457i	$-0.680517\pi$
$761$	−29.6385	−1.07439	−0.537197	+	0.843457i	$0.680517\pi$
$762$	0	0
$763$	29.5877	1.07115
$764$	−3.38295	−0.122391
$765$	0	0
$766$	−30.0870	−1.08709
$767$	−18.6643	−0.673930
$768$	0	0
$769$	−29.1754	−1.05209	−0.526046	−	0.850456i	$-0.676326\pi$
$769$	−29.1754	−1.05209	−0.526046	+	0.850456i	$0.676326\pi$
$770$	−1.19394	−0.0430265
$771$	0	0
$772$	−2.34931	−0.0845534
$773$	17.3317	0.623378	0.311689	−	0.950184i	$-0.399105\pi$
$773$	17.3317	0.623378	0.311689	+	0.950184i	$0.399105\pi$
$774$	0	0
$775$	−9.50659	−0.341487
$776$	9.49437	0.340828
$777$	0	0
$778$	22.5891	0.809859
$779$	5.18172	0.185654
$780$	0	0
$781$	4.64974	0.166381
$782$	23.1998	0.829624
$783$	0	0
$784$	18.2447	0.651597
$785$	−14.5501	−0.519315
$786$	0	0
$787$	−13.3747	−0.476757	−0.238378	−	0.971172i	$-0.576616\pi$
$787$	−13.3747	−0.476757	−0.238378	+	0.971172i	$0.576616\pi$
$788$	−2.49341	−0.0888242
$789$	0	0
$790$	−10.2012	−0.362944
$791$	−15.6121	−0.555103
$792$	0	0
$793$	−12.9770	−0.460826
$794$	35.1246	1.24653
$795$	0	0
$796$	−0.739549	−0.0262126
$797$	4.26774	0.151171	0.0755856	−	0.997139i	$-0.475917\pi$
$797$	4.26774	0.151171	0.0755856	+	0.997139i	$0.475917\pi$
$798$	0	0
$799$	35.5125	1.25634
$800$	1.09332	0.0386547
$801$	0	0
$802$	−31.6869	−1.11890
$803$	−4.46310	−0.157499
$804$	0	0
$805$	4.26187	0.150211
$806$	−18.1260	−0.638461
$807$	0	0
$808$	12.0752	0.424805
$809$	−11.2243	−0.394624	−0.197312	−	0.980341i	$-0.563221\pi$
$809$	−11.2243	−0.394624	−0.197312	+	0.980341i	$0.563221\pi$
$810$	0	0
$811$	−15.1754	−0.532880	−0.266440	−	0.963852i	$-0.585847\pi$
$811$	−15.1754	−0.532880	−0.266440	+	0.963852i	$0.585847\pi$
$812$	−3.19394	−0.112085
$813$	0	0
$814$	8.26774	0.289784
$815$	−5.67513	−0.198791
$816$	0	0
$817$	−2.06300	−0.0721754
$818$	−28.5139	−0.996964
$819$	0	0
$820$	−1.00492	−0.0350935
$821$	−47.1852	−1.64678	−0.823388	−	0.567479i	$-0.807919\pi$
$821$	−47.1852	−1.64678	−0.823388	+	0.567479i	$0.807919\pi$
$822$	0	0
$823$	50.4363	1.75810	0.879049	−	0.476731i	$-0.158178\pi$
$823$	50.4363	1.75810	0.879049	+	0.476731i	$0.158178\pi$
$824$	44.4749	1.54935
$825$	0	0
$826$	35.9756	1.25175
$827$	8.03032	0.279241	0.139621	−	0.990205i	$-0.455412\pi$
$827$	8.03032	0.279241	0.139621	+	0.990205i	$0.455412\pi$
$828$	0	0
$829$	10.4339	0.362385	0.181192	−	0.983448i	$-0.442004\pi$
$829$	10.4339	0.362385	0.181192	+	0.983448i	$0.442004\pi$
$830$	−3.76845	−0.130805
$831$	0	0
$832$	−9.11520	−0.316013
$833$	−25.8192	−0.894584
$834$	0	0
$835$	2.41819	0.0836849
$836$	0.0933212	0.00322758
$837$	0	0
$838$	−49.0010	−1.69271
$839$	−20.2130	−0.697830	−0.348915	−	0.937154i	$-0.613450\pi$
$839$	−20.2130	−0.697830	−0.348915	+	0.937154i	$0.613450\pi$
$840$	0	0
$841$	67.6575	2.33302
$842$	24.4603	0.842956
$843$	0	0
$844$	−2.11142	−0.0726781
$845$	−11.3430	−0.390210
$846$	0	0
$847$	18.0386	0.619812
$848$	−8.02047	−0.275424
$849$	0	0
$850$	−9.11871	−0.312769
$851$	−29.5125	−1.01167
$852$	0	0
$853$	33.1998	1.13674	0.568370	−	0.822773i	$-0.307574\pi$
$853$	33.1998	1.13674	0.568370	+	0.822773i	$0.307574\pi$
$854$	25.0132	0.855933
$855$	0	0
$856$	33.2750	1.13732
$857$	20.4836	0.699705	0.349853	−	0.936805i	$-0.386232\pi$
$857$	20.4836	0.699705	0.349853	+	0.936805i	$0.386232\pi$
$858$	0	0
$859$	41.1246	1.40315	0.701577	−	0.712594i	$-0.252480\pi$
$859$	41.1246	1.40315	0.701577	+	0.712594i	$0.252480\pi$
$860$	0.400092	0.0136430
$861$	0	0
$862$	−10.6643	−0.363228
$863$	7.51247	0.255727	0.127864	−	0.991792i	$-0.459188\pi$
$863$	7.51247	0.255727	0.127864	+	0.991792i	$0.459188\pi$
$864$	0	0
$865$	−4.85685	−0.165138
$866$	41.5090	1.41053
$867$	0	0
$868$	3.08840	0.104827
$869$	−3.31406	−0.112422
$870$	0	0
$871$	−5.88858	−0.199527
$872$	−47.2506	−1.60011
$873$	0	0
$874$	−3.76845	−0.127470
$875$	−1.67513	−0.0566298
$876$	0	0
$877$	27.4542	0.927063	0.463531	−	0.886081i	$-0.346582\pi$
$877$	27.4542	0.927063	0.463531	+	0.886081i	$0.346582\pi$
$878$	−36.8481	−1.24356
$879$	0	0
$880$	2.09332	0.0705658
$881$	−25.3766	−0.854960	−0.427480	−	0.904025i	$-0.640599\pi$
$881$	−25.3766	−0.854960	−0.427480	+	0.904025i	$0.640599\pi$
$882$	0	0
$883$	−18.9140	−0.636506	−0.318253	−	0.948006i	$-0.603096\pi$
$883$	−18.9140	−0.636506	−0.318253	+	0.948006i	$0.603096\pi$
$884$	−1.53690	−0.0516917
$885$	0	0
$886$	−40.2071	−1.35078
$887$	33.6239	1.12898	0.564490	−	0.825440i	$-0.309073\pi$
$887$	33.6239	1.12898	0.564490	+	0.825440i	$0.309073\pi$
$888$	0	0
$889$	−11.6629	−0.391162
$890$	−6.63752	−0.222490
$891$	0	0
$892$	−4.37328	−0.146428
$893$	−5.76845	−0.193034
$894$	0	0
$895$	−1.03761	−0.0346835
$896$	21.2325	0.709328
$897$	0	0
$898$	−13.5031	−0.450604
$899$	−93.4636	−3.11719
$900$	0	0
$901$	11.3503	0.378132
$902$	−3.69323	−0.122971
$903$	0	0
$904$	24.9321	0.829228
$905$	7.27504	0.241830
$906$	0	0
$907$	22.7269	0.754633	0.377317	−	0.926084i	$-0.376847\pi$
$907$	22.7269	0.754633	0.377317	+	0.926084i	$0.376847\pi$
$908$	−2.33312	−0.0774273
$909$	0	0
$910$	−3.19394	−0.105878
$911$	−15.2243	−0.504402	−0.252201	−	0.967675i	$-0.581154\pi$
$911$	−15.2243	−0.504402	−0.252201	+	0.967675i	$0.581154\pi$
$912$	0	0
$913$	−1.22425	−0.0405169
$914$	31.6093	1.04554
$915$	0	0
$916$	−2.76704	−0.0914255
$917$	2.85685	0.0943415
$918$	0	0
$919$	37.5271	1.23790	0.618952	−	0.785429i	$-0.287558\pi$
$919$	37.5271	1.23790	0.618952	+	0.785429i	$0.287558\pi$
$920$	−6.80606	−0.224389
$921$	0	0
$922$	−27.1030	−0.892589
$923$	12.4387	0.409423
$924$	0	0
$925$	11.5999	0.381403
$926$	−49.7680	−1.63548
$927$	0	0
$928$	10.7489	0.352851
$929$	25.8740	0.848898	0.424449	−	0.905452i	$-0.360468\pi$
$929$	25.8740	0.848898	0.424449	+	0.905452i	$0.360468\pi$
$930$	0	0
$931$	4.19394	0.137451
$932$	4.55008	0.149043
$933$	0	0
$934$	33.3561	1.09145
$935$	−2.96239	−0.0968805
$936$	0	0
$937$	−9.81336	−0.320588	−0.160294	−	0.987069i	$-0.551244\pi$
$937$	−9.81336	−0.320588	−0.160294	+	0.987069i	$0.551244\pi$
$938$	11.3503	0.370599
$939$	0	0
$940$	1.11871	0.0364884
$941$	−44.0346	−1.43549	−0.717743	−	0.696308i	$-0.754825\pi$
$941$	−44.0346	−1.43549	−0.717743	+	0.696308i	$0.754825\pi$
$942$	0	0
$943$	13.1833	0.429308
$944$	−63.0757	−2.05294
$945$	0	0
$946$	1.47039	0.0478065
$947$	24.8832	0.808595	0.404298	−	0.914627i	$-0.367516\pi$
$947$	24.8832	0.808595	0.404298	+	0.914627i	$0.367516\pi$
$948$	0	0
$949$	−11.9394	−0.387568
$950$	1.48119	0.0480563
$951$	0	0
$952$	−27.5877	−0.894122
$953$	28.3938	0.919764	0.459882	−	0.887980i	$-0.347892\pi$
$953$	28.3938	0.919764	0.459882	+	0.887980i	$0.347892\pi$
$954$	0	0
$955$	−17.4436	−0.564461
$956$	1.47978	0.0478595
$957$	0	0
$958$	5.13681	0.165963
$959$	3.03761	0.0980896
$960$	0	0
$961$	59.3752	1.91533
$962$	22.1173	0.713090
$963$	0	0
$964$	−1.38646	−0.0446548
$965$	−12.1138	−0.389957
$966$	0	0
$967$	−15.3117	−0.492391	−0.246195	−	0.969220i	$-0.579181\pi$
$967$	−15.3117	−0.492391	−0.246195	+	0.969220i	$0.579181\pi$
$968$	−28.8070	−0.925893
$969$	0	0
$970$	−5.25694	−0.168790
$971$	46.2520	1.48430	0.742149	−	0.670235i	$-0.233807\pi$
$971$	46.2520	1.48430	0.742149	+	0.670235i	$0.233807\pi$
$972$	0	0
$973$	−4.21108	−0.135001
$974$	28.9163	0.926539
$975$	0	0
$976$	−43.8554	−1.40378
$977$	−2.03032	−0.0649556	−0.0324778	−	0.999472i	$-0.510340\pi$
$977$	−2.03032	−0.0649556	−0.0324778	+	0.999472i	$0.510340\pi$
$978$	0	0
$979$	−2.15633	−0.0689165
$980$	−0.813358	−0.0259818
$981$	0	0
$982$	32.9111	1.05024
$983$	−18.1662	−0.579411	−0.289705	−	0.957116i	$-0.593557\pi$
$983$	−18.1662	−0.579411	−0.289705	+	0.957116i	$0.593557\pi$
$984$	0	0
$985$	−12.8568	−0.409653
$986$	−89.6502	−2.85505
$987$	0	0
$988$	0.249646	0.00794231
$989$	−5.24869	−0.166899
$990$	0	0
$991$	−34.0263	−1.08088	−0.540441	−	0.841382i	$-0.681743\pi$
$991$	−34.0263	−1.08088	−0.540441	+	0.841382i	$0.681743\pi$
$992$	−10.3938	−0.330002
$993$	0	0
$994$	−23.9756	−0.760459
$995$	−3.81336	−0.120892
$996$	0	0
$997$	27.7889	0.880084	0.440042	−	0.897977i	$-0.354964\pi$
$997$	27.7889	0.880084	0.440042	+	0.897977i	$0.354964\pi$
$998$	20.1260	0.637078
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	855.2.a.l.1.1	yes	3
3.2	odd	2		855.2.a.h.1.3	✓	3
5.4	even	2		4275.2.a.bb.1.3		3
15.14	odd	2		4275.2.a.bj.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
855.2.a.h.1.3	✓	3	3.2	odd	2
855.2.a.l.1.1	yes	3	1.1	even	1	trivial
4275.2.a.bb.1.3		3	5.4	even	2
4275.2.a.bj.1.1		3	15.14	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:	\( N \)	\(=\)	\( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	855.a (trivial)

\(f(q)\)	\(=\)	\(q-1.48119 q^{2} +0.193937 q^{4} +1.00000 q^{5} -1.67513 q^{7} +2.67513 q^{8} +O(q^{10})\)	\(q-1.48119 q^{2} +0.193937 q^{4} +1.00000 q^{5} -1.67513 q^{7} +2.67513 q^{8} -1.48119 q^{10} -0.481194 q^{11} -1.28726 q^{13} +2.48119 q^{14} -4.35026 q^{16} +6.15633 q^{17} -1.00000 q^{19} +0.193937 q^{20} +0.712742 q^{22} -2.54420 q^{23} +1.00000 q^{25} +1.90668 q^{26} -0.324869 q^{28} +9.83146 q^{29} -9.50659 q^{31} +1.09332 q^{32} -9.11871 q^{34} -1.67513 q^{35} +11.5999 q^{37} +1.48119 q^{38} +2.67513 q^{40} -5.18172 q^{41} +2.06300 q^{43} -0.0933212 q^{44} +3.76845 q^{46} +5.76845 q^{47} -4.19394 q^{49} -1.48119 q^{50} -0.249646 q^{52} +1.84367 q^{53} -0.481194 q^{55} -4.48119 q^{56} -14.5623 q^{58} +14.4993 q^{59} +10.0811 q^{61} +14.0811 q^{62} +7.08110 q^{64} -1.28726 q^{65} +4.57452 q^{67} +1.19394 q^{68} +2.48119 q^{70} -9.66291 q^{71} +9.27504 q^{73} -17.1817 q^{74} -0.193937 q^{76} +0.806063 q^{77} +6.88717 q^{79} -4.35026 q^{80} +7.67513 q^{82} +2.54420 q^{83} +6.15633 q^{85} -3.05571 q^{86} -1.28726 q^{88} +4.48119 q^{89} +2.15633 q^{91} -0.493413 q^{92} -8.54420 q^{94} -1.00000 q^{95} +3.54912 q^{97} +6.21203 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + q^{4} + 3 q^{5} + 3 q^{8}+O(q^{10}) \) 3 * q + q^2 + q^4 + 3 * q^5 + 3 * q^8	\( 3 q + q^{2} + q^{4} + 3 q^{5} + 3 q^{8} + q^{10} + 4 q^{11} + 2 q^{13} + 2 q^{14} - 3 q^{16} + 8 q^{17} - 3 q^{19} + q^{20} + 8 q^{22} + 2 q^{23} + 3 q^{25} + 12 q^{26} - 6 q^{28} + 14 q^{29} - 8 q^{31} - 3 q^{32} - 6 q^{34} + 8 q^{37} - q^{38} + 3 q^{40} + 10 q^{41} + 2 q^{43} + 6 q^{44} + 6 q^{47} - 13 q^{49} + q^{50} + 16 q^{52} + 16 q^{53} + 4 q^{55} - 8 q^{56} - 6 q^{58} + 10 q^{59} - 2 q^{61} + 10 q^{62} - 11 q^{64} + 2 q^{65} + 2 q^{67} + 4 q^{68} + 2 q^{70} + 2 q^{71} - 4 q^{73} - 26 q^{74} - q^{76} + 2 q^{77} - 12 q^{79} - 3 q^{80} + 18 q^{82} - 2 q^{83} + 8 q^{85} + 8 q^{86} + 2 q^{88} + 8 q^{89} - 4 q^{91} - 22 q^{92} - 16 q^{94} - 3 q^{95} + 14 q^{97} - 9 q^{98}+O(q^{100}) \) 3 * q + q^2 + q^4 + 3 * q^5 + 3 * q^8 + q^10 + 4 * q^11 + 2 * q^13 + 2 * q^14 - 3 * q^16 + 8 * q^17 - 3 * q^19 + q^20 + 8 * q^22 + 2 * q^23 + 3 * q^25 + 12 * q^26 - 6 * q^28 + 14 * q^29 - 8 * q^31 - 3 * q^32 - 6 * q^34 + 8 * q^37 - q^38 + 3 * q^40 + 10 * q^41 + 2 * q^43 + 6 * q^44 + 6 * q^47 - 13 * q^49 + q^50 + 16 * q^52 + 16 * q^53 + 4 * q^55 - 8 * q^56 - 6 * q^58 + 10 * q^59 - 2 * q^61 + 10 * q^62 - 11 * q^64 + 2 * q^65 + 2 * q^67 + 4 * q^68 + 2 * q^70 + 2 * q^71 - 4 * q^73 - 26 * q^74 - q^76 + 2 * q^77 - 12 * q^79 - 3 * q^80 + 18 * q^82 - 2 * q^83 + 8 * q^85 + 8 * q^86 + 2 * q^88 + 8 * q^89 - 4 * q^91 - 22 * q^92 - 16 * q^94 - 3 * q^95 + 14 * q^97 - 9 * q^98

Introduction

L-functions

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