LMFDB - Embedded newform 1368.2.e.c.379.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1368,2,Mod(379,1368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1368, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 0, 1]))

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

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

chi := DirichletCharacter("1368.379");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1368 = 2^{3} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1368.e (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$10.9235349965$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.4919453024256.11
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 12x^{6} + 96x^{4} + 248x^{2} + 900 $ x^8 - 12x^6 + 96x^4 + 248*x^2 + 900
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			379.1
Root			$-3.05923 - 1.41421i$ of defining polynomial
Character	$\chi$	$=$	1368.379
Dual form			1368.2.e.c.379.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} -2.00000 q^{4} -4.32641i q^{5} +2.82843i q^{8} +O(q^{10})$	$q-1.41421i q^{2} -2.00000 q^{4} -4.32641i q^{5} +2.82843i q^{8} -6.11846 q^{10} -4.15719 q^{13} +4.00000 q^{16} -4.35890 q^{19} +8.65281i q^{20} -1.55274i q^{23} -13.7178 q^{25} +5.87915i q^{26} +10.2757 q^{31} -5.65685i q^{32} -8.07974 q^{37} +6.16441i q^{38} +12.2369 q^{40} +12.3288i q^{41} -8.71780 q^{43} -2.19591 q^{46} -7.10007i q^{47} +7.00000 q^{49} +19.3999i q^{50} +8.31437 q^{52} -2.82843i q^{59} -14.5320i q^{62} -8.00000 q^{64} +17.9857i q^{65} -8.71780 q^{73} +11.4265i q^{74} +8.71780 q^{76} +14.1982 q^{79} -17.3056i q^{80} +17.4356 q^{82} +12.3288i q^{86} -11.3137i q^{89} +3.10549i q^{92} -10.0410 q^{94} +18.8584i q^{95} -9.89949i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{4}+O(q^{10}) $ 8 * q - 16 * q^4	$ 8 q - 16 q^{4} + 32 q^{16} - 40 q^{25} + 56 q^{49} - 64 q^{64}+O(q^{100}) $ 8 * q - 16 * q^4 + 32 * q^16 - 40 * q^25 + 56 * q^49 - 64 * q^64

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1368\mathbb{Z}\right)^\times$.

$n$	$343$	$685$	$1009$	$1217$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

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 $

$2$	− 1.41421i	− 1.00000i
$2$	− 1.41421i	− 1.00000i
$3$	0	0
$3$	0	0
$4$	−2.00000	−1.00000
$5$	− 4.32641i	− 1.93483i	−0.253200	−	0.967414i	$-0.581483\pi$
$5$	− 4.32641i	− 1.93483i	0.253200	−	0.967414i	$-0.418517\pi$
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	2.82843i	1.00000i
$9$	0	0
$10$	−6.11846	−1.93483
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−4.15719	−1.15300	−0.576498	−	0.817099i	$-0.695581\pi$
$13$	−4.15719	−1.15300	−0.576498	+	0.817099i	$0.695581\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	1.00000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	−4.35890	−1.00000
$19$	−4.35890	−1.00000
$20$	8.65281i	1.93483i
$21$	0	0
$22$	0	0
$23$	− 1.55274i	− 0.323769i	−0.986810	−	0.161885i	$-0.948243\pi$
$23$	− 1.55274i	− 0.323769i	0.986810	−	0.161885i	$-0.0517572\pi$
$24$	0	0
$25$	−13.7178	−2.74356
$26$	5.87915i	1.15300i
$27$	0	0
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	10.2757	1.84556	0.922781	−	0.385326i	$-0.125911\pi$
$31$	10.2757	1.84556	0.922781	+	0.385326i	$0.125911\pi$
$32$	− 5.65685i	− 1.00000i
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−8.07974	−1.32830	−0.664151	−	0.747599i	$-0.731207\pi$
$37$	−8.07974	−1.32830	−0.664151	+	0.747599i	$0.731207\pi$
$38$	6.16441i	1.00000i
$39$	0	0
$40$	12.2369	1.93483
$41$	12.3288i	1.92544i	0.270501	+	0.962720i	$0.412811\pi$
$41$	12.3288i	1.92544i	−0.270501	+	0.962720i	$0.587189\pi$
$42$	0	0
$43$	−8.71780	−1.32945	−0.664726	−	0.747087i	$-0.731452\pi$
$43$	−8.71780	−1.32945	−0.664726	+	0.747087i	$0.731452\pi$
$44$	0	0
$45$	0	0
$46$	−2.19591	−0.323769
$47$	− 7.10007i	− 1.03565i	−0.855486	−	0.517826i	$-0.826741\pi$
$47$	− 7.10007i	− 1.03565i	0.855486	−	0.517826i	$-0.173259\pi$
$48$	0	0
$49$	7.00000	1.00000
$50$	19.3999i	2.74356i
$51$	0	0
$52$	8.31437	1.15300
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 2.82843i	− 0.368230i	−0.982905	−	0.184115i	$-0.941058\pi$
$59$	− 2.82843i	− 0.368230i	0.982905	−	0.184115i	$-0.0589419\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	− 14.5320i	− 1.84556i
$63$	0	0
$64$	−8.00000	−1.00000
$65$	17.9857i	2.23085i
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−8.71780	−1.02034	−0.510171	−	0.860073i	$-0.670418\pi$
$73$	−8.71780	−1.02034	−0.510171	+	0.860073i	$0.670418\pi$
$74$	11.4265i	1.32830i
$75$	0	0
$76$	8.71780	1.00000
$77$	0	0
$78$	0	0
$79$	14.1982	1.59742	0.798711	−	0.601714i	$-0.205515\pi$
$79$	14.1982	1.59742	0.798711	+	0.601714i	$0.205515\pi$
$80$	− 17.3056i	− 1.93483i
$81$	0	0
$82$	17.4356	1.92544
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	12.3288i	1.32945i
$87$	0	0
$88$	0	0
$89$	− 11.3137i	− 1.19925i	−0.800281	−	0.599625i	$-0.795316\pi$
$89$	− 11.3137i	− 1.19925i	0.800281	−	0.599625i	$-0.204684\pi$
$90$	0	0
$91$	0	0
$92$	3.10549i	0.323769i
$93$	0	0
$94$	−10.0410	−1.03565
$95$	18.8584i	1.93483i
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	− 9.89949i	− 1.00000i
$99$	0	0
$100$	27.4356	2.74356
$101$	1.22092i	0.121486i	0.998153	+	0.0607431i	$0.0193470\pi$
$101$	1.22092i	0.121486i	−0.998153	+	0.0607431i	$0.980653\pi$
$102$	0	0
$103$	6.35310	0.625989	0.312995	−	0.949755i	$-0.398668\pi$
$103$	6.35310	0.625989	0.312995	+	0.949755i	$0.398668\pi$
$104$	− 11.7583i	− 1.15300i
$105$	0	0
$106$	0	0
$107$	14.1421i	1.36717i	0.729870	+	0.683586i	$0.239581\pi$
$107$	14.1421i	1.36717i	−0.729870	+	0.683586i	$0.760419\pi$
$108$	0	0
$109$	−0.234633	−0.0224738	−0.0112369	−	0.999937i	$-0.503577\pi$
$109$	−0.234633	−0.0224738	−0.0112369	+	0.999937i	$0.503577\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	12.3288i	1.15980i	0.814688	+	0.579899i	$0.196908\pi$
$113$	12.3288i	1.15980i	−0.814688	+	0.579899i	$0.803092\pi$
$114$	0	0
$115$	−6.71780	−0.626438
$116$	0	0
$117$	0	0
$118$	−4.00000	−0.368230
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	0	0
$124$	−20.5513	−1.84556
$125$	37.7167i	3.37349i
$126$	0	0
$127$	−22.5126	−1.99767	−0.998834	−	0.0482746i	$-0.984628\pi$
$127$	−22.5126	−1.99767	−0.998834	+	0.0482746i	$0.984628\pi$
$128$	11.3137i	1.00000i
$129$	0	0
$130$	25.4356	2.23085
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	−10.0000	−0.848189	−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000	−0.848189	−0.424094	+	0.905618i	$0.639408\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	12.3288i	1.02034i
$147$	0	0
$148$	16.1595	1.32830
$149$	− 9.87374i	− 0.808888i	−0.914563	−	0.404444i	$-0.867465\pi$
$149$	− 9.87374i	− 0.808888i	0.914563	−	0.404444i	$-0.132535\pi$
$150$	0	0
$151$	−18.5900	−1.51283	−0.756417	−	0.654089i	$-0.773052\pi$
$151$	−18.5900	−1.51283	−0.756417	+	0.654089i	$0.773052\pi$
$152$	− 12.3288i	− 1.00000i
$153$	0	0
$154$	0	0
$155$	− 44.4566i	− 3.57084i
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	− 20.0793i	− 1.59742i
$159$	0	0
$160$	−24.4739	−1.93483
$161$	0	0
$162$	0	0
$163$	14.0000	1.09656	0.548282	−	0.836293i	$-0.315282\pi$
$163$	14.0000	1.09656	0.548282	+	0.836293i	$0.315282\pi$
$164$	− 24.6577i	− 1.92544i
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	4.28220	0.329400
$170$	0	0
$171$	0	0
$172$	17.4356	1.32945
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	−16.0000	−1.19925
$179$	− 24.6577i	− 1.84300i	−0.388379	−	0.921500i	$-0.626965\pi$
$179$	− 24.6577i	− 1.84300i	0.388379	−	0.921500i	$-0.373035\pi$
$180$	0	0
$181$	−12.0023	−0.892123	−0.446062	−	0.895002i	$-0.647174\pi$
$181$	−12.0023	−0.892123	−0.446062	+	0.895002i	$0.647174\pi$
$182$	0	0
$183$	0	0
$184$	4.39182	0.323769
$185$	34.9562i	2.57003i
$186$	0	0
$187$	0	0
$188$	14.2001i	1.03565i
$189$	0	0
$190$	26.6698	1.93483
$191$	− 27.5112i	− 1.99064i	−0.0966364	−	0.995320i	$-0.530808\pi$
$191$	− 27.5112i	− 1.99064i	0.0966364	−	0.995320i	$-0.469192\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	−14.0000	−1.00000
$197$	− 24.7375i	− 1.76248i	−0.472673	−	0.881238i	$-0.656711\pi$
$197$	− 24.7375i	− 1.76248i	0.472673	−	0.881238i	$-0.343289\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	− 38.7998i	− 2.74356i
$201$	0	0
$202$	1.72664	0.121486
$203$	0	0
$204$	0	0
$205$	53.3395	3.72539
$206$	− 8.98464i	− 0.625989i
$207$	0	0
$208$	−16.6287	−1.15300
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	20.0000	1.36717
$215$	37.7167i	2.57226i
$216$	0	0
$217$	0	0
$218$	0.331822i	0.0224738i
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−26.4351	−1.77023	−0.885114	−	0.465375i	$-0.845919\pi$
$223$	−26.4351	−1.77023	−0.885114	+	0.465375i	$0.845919\pi$
$224$	0	0
$225$	0	0
$226$	17.4356	1.15980
$227$	− 24.6577i	− 1.63659i	−0.574801	−	0.818293i	$-0.694921\pi$
$227$	− 24.6577i	− 1.63659i	0.574801	−	0.818293i	$-0.305079\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	9.50040i	0.626438i
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	−30.7178	−2.00381
$236$	5.65685i	0.368230i
$237$	0	0
$238$	0	0
$239$	3.99459i	0.258388i	0.991619	+	0.129194i	$0.0412390\pi$
$239$	3.99459i	0.258388i	−0.991619	+	0.129194i	$0.958761\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	15.5563i	1.00000i
$243$	0	0
$244$	0	0
$245$	− 30.2848i	− 1.93483i
$246$	0	0
$247$	18.1208	1.15300
$248$	29.0639i	1.84556i
$249$	0	0
$250$	53.3395	3.37349
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	31.8376i	1.99767i
$255$	0	0
$256$	16.0000	1.00000
$257$	− 28.2843i	− 1.76432i	−0.470946	−	0.882162i	$-0.656087\pi$
$257$	− 28.2843i	− 1.76432i	0.470946	−	0.882162i	$-0.343913\pi$
$258$	0	0
$259$	0	0
$260$	− 35.9714i	− 2.23085i
$261$	0	0
$262$	0	0
$263$	− 21.9639i	− 1.35435i	−0.735822	−	0.677175i	$-0.763204\pi$
$263$	− 21.9639i	− 1.35435i	0.735822	−	0.677175i	$-0.236796\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	14.1421i	0.848189i
$279$	0	0
$280$	0	0
$281$	12.3288i	0.735476i	0.929929	+	0.367738i	$0.119868\pi$
$281$	12.3288i	0.735476i	−0.929929	+	0.367738i	$0.880132\pi$
$282$	0	0
$283$	26.0000	1.54554	0.772770	−	0.634686i	$-0.218871\pi$
$283$	26.0000	1.54554	0.772770	+	0.634686i	$0.218871\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	17.4356	1.02034
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	−12.2369	−0.712461
$296$	− 22.8530i	− 1.32830i
$297$	0	0
$298$	−13.9636	−0.808888
$299$	6.45504i	0.373305i
$300$	0	0
$301$	0	0
$302$	26.2903i	1.51283i
$303$	0	0
$304$	−17.4356	−1.00000
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	−62.8712	−3.57084
$311$	− 12.6474i	− 0.717168i	−0.933497	−	0.358584i	$-0.883260\pi$
$311$	− 12.6474i	− 0.717168i	0.933497	−	0.358584i	$-0.116740\pi$
$312$	0	0
$313$	−8.71780	−0.492759	−0.246380	−	0.969173i	$-0.579241\pi$
$313$	−8.71780	−0.492759	−0.246380	+	0.969173i	$0.579241\pi$
$314$	0	0
$315$	0	0
$316$	−28.3964	−1.59742
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	0	0
$320$	34.6113i	1.93483i
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	57.0274	3.16331
$326$	− 19.7990i	− 1.09656i
$327$	0	0
$328$	−34.8712	−1.92544
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	− 6.05595i	− 0.329400i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	− 24.6577i	− 1.32945i
$345$	0	0
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	0	0
$356$	22.6274i	1.19925i
$357$	0	0
$358$	−34.8712	−1.84300
$359$	− 33.0585i	− 1.74476i	−0.488827	−	0.872381i	$-0.662575\pi$
$359$	− 33.0585i	− 1.74476i	0.488827	−	0.872381i	$-0.337425\pi$
$360$	0	0
$361$	19.0000	1.00000
$362$	16.9738i	0.892123i
$363$	0	0
$364$	0	0
$365$	37.7167i	1.97418i
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	− 6.21097i	− 0.323769i
$369$	0	0
$370$	49.4356	2.57003
$371$	0	0
$372$	0	0
$373$	−36.9454	−1.91296	−0.956481	−	0.291796i	$-0.905747\pi$
$373$	−36.9454	−1.91296	−0.956481	+	0.291796i	$0.905747\pi$
$374$	0	0
$375$	0	0
$376$	20.0820	1.03565
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	− 37.7167i	− 1.93483i
$381$	0	0
$382$	−38.9067	−1.99064
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 19.1902i	− 0.972981i	−0.873686	−	0.486491i	$-0.838277\pi$
$389$	− 19.1902i	− 0.972981i	0.873686	−	0.486491i	$-0.161723\pi$
$390$	0	0
$391$	0	0
$392$	19.7990i	1.00000i
$393$	0	0
$394$	−34.9841	−1.76248
$395$	− 61.4272i	− 3.09074i
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	−54.8712	−2.74356
$401$	12.3288i	0.615672i	0.951439	+	0.307836i	$0.0996049\pi$
$401$	12.3288i	0.615672i	−0.951439	+	0.307836i	$0.900395\pi$
$402$	0	0
$403$	−42.7178	−2.12793
$404$	− 2.44184i	− 0.121486i
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	− 75.4335i	− 3.72539i
$411$	0	0
$412$	−12.7062	−0.625989
$413$	0	0
$414$	0	0
$415$	0	0
$416$	23.5166i	1.15300i
$417$	0	0
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−40.8680	−1.99178	−0.995891	−	0.0905552i	$-0.971136\pi$
$421$	−40.8680	−1.99178	−0.995891	+	0.0905552i	$0.971136\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	− 28.2843i	− 1.36717i
$429$	0	0
$430$	53.3395	2.57226
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0.469267	0.0224738
$437$	6.76825i	0.323769i
$438$	0	0
$439$	−30.3577	−1.44889	−0.724447	−	0.689331i	$-0.757905\pi$
$439$	−30.3577	−1.44889	−0.724447	+	0.689331i	$0.757905\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	−48.9477	−2.32034
$446$	37.3849i	1.77023i
$447$	0	0
$448$	0	0
$449$	39.5980i	1.86874i	0.356299	+	0.934372i	$0.384039\pi$
$449$	39.5980i	1.86874i	−0.356299	+	0.934372i	$0.615961\pi$
$450$	0	0
$451$	0	0
$452$	− 24.6577i	− 1.15980i
$453$	0	0
$454$	−34.8712	−1.63659
$455$	0	0
$456$	0	0
$457$	2.00000	0.0935561	0.0467780	−	0.998905i	$-0.485105\pi$
$457$	2.00000	0.0935561	0.0467780	+	0.998905i	$0.485105\pi$
$458$	0	0
$459$	0	0
$460$	13.4356	0.626438
$461$	42.0431i	1.95814i	0.203513	+	0.979072i	$0.434764\pi$
$461$	42.0431i	1.95814i	−0.203513	+	0.979072i	$0.565236\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	0	0
$470$	43.4415i	2.00381i
$471$	0	0
$472$	8.00000	0.368230
$473$	0	0
$474$	0	0
$475$	59.7945	2.74356
$476$	0	0
$477$	0	0
$478$	5.64920	0.258388
$479$	39.2695i	1.79427i	0.441758	+	0.897134i	$0.354355\pi$
$479$	39.2695i	1.79427i	−0.441758	+	0.897134i	$0.645645\pi$
$480$	0	0
$481$	33.5890	1.53153
$482$	0	0
$483$	0	0
$484$	22.0000	1.00000
$485$	0	0
$486$	0	0
$487$	43.0639	1.95141	0.975705	−	0.219087i	$-0.0703079\pi$
$487$	43.0639	1.95141	0.975705	+	0.219087i	$0.0703079\pi$
$488$	0	0
$489$	0	0
$490$	−42.8292	−1.93483
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	0	0
$494$	− 25.6266i	− 1.15300i
$495$	0	0
$496$	41.1026	1.84556
$497$	0	0
$498$	0	0
$499$	43.5890	1.95131	0.975656	−	0.219308i	$-0.0703801\pi$
$499$	43.5890	1.95131	0.975656	+	0.219308i	$0.0703801\pi$
$500$	− 75.4335i	− 3.37349i
$501$	0	0
$502$	0	0
$503$	44.8168i	1.99828i	0.0414285	+	0.999141i	$0.486809\pi$
$503$	44.8168i	1.99828i	−0.0414285	+	0.999141i	$0.513191\pi$
$504$	0	0
$505$	5.28220	0.235055
$506$	0	0
$507$	0	0
$508$	45.0252	1.99767
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	− 22.6274i	− 1.00000i
$513$	0	0
$514$	−40.0000	−1.76432
$515$	− 27.4861i	− 1.21118i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	−50.8712	−2.23085
$521$	5.65685i	0.247831i	0.992293	+	0.123916i	$0.0395452\pi$
$521$	5.65685i	0.247831i	−0.992293	+	0.123916i	$0.960455\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	0	0
$526$	−31.0616	−1.35435
$527$	0	0
$528$	0	0
$529$	20.5890	0.895173
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 51.2532i	− 2.22002i
$534$	0	0
$535$	61.1846	2.64524
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	1.01512i	0.0434829i
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	20.0000	0.848189
$557$	36.4958i	1.54638i	0.634176	+	0.773189i	$0.281339\pi$
$557$	36.4958i	1.54638i	−0.634176	+	0.773189i	$0.718661\pi$
$558$	0	0
$559$	36.2415	1.53285
$560$	0	0
$561$	0	0
$562$	17.4356	0.735476
$563$	14.1421i	0.596020i	0.954563	+	0.298010i	$0.0963229\pi$
$563$	14.1421i	0.596020i	−0.954563	+	0.298010i	$0.903677\pi$
$564$	0	0
$565$	53.3395	2.24401
$566$	− 36.7696i	− 1.54554i
$567$	0	0
$568$	0	0
$569$	− 45.2548i	− 1.89718i	−0.316506	−	0.948591i	$-0.602510\pi$
$569$	− 45.2548i	− 1.89718i	0.316506	−	0.948591i	$-0.397490\pi$
$570$	0	0
$571$	43.5890	1.82414	0.912071	−	0.410032i	$-0.134482\pi$
$571$	43.5890	1.82414	0.912071	+	0.410032i	$0.134482\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	21.3002i	0.888280i
$576$	0	0
$577$	−22.0000	−0.915872	−0.457936	−	0.888985i	$-0.651411\pi$
$577$	−22.0000	−0.915872	−0.457936	+	0.888985i	$0.651411\pi$
$578$	24.0416i	1.00000i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	− 24.6577i	− 1.02034i
$585$	0	0
$586$	0	0
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	−44.7905	−1.84556
$590$	17.3056i	0.712461i
$591$	0	0
$592$	−32.3190	−1.32830
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	19.7475i	0.808888i
$597$	0	0
$598$	9.12881	0.373305
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	37.1800	1.51283
$605$	47.5905i	1.93483i
$606$	0	0
$607$	−1.49201	−0.0605588	−0.0302794	−	0.999541i	$-0.509640\pi$
$607$	−1.49201	−0.0605588	−0.0302794	+	0.999541i	$0.509640\pi$
$608$	24.6577i	1.00000i
$609$	0	0
$610$	0	0
$611$	29.5163i	1.19410i
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0	0
$619$	−8.71780	−0.350398	−0.175199	−	0.984533i	$-0.556057\pi$
$619$	−8.71780	−0.350398	−0.175199	+	0.984533i	$0.556057\pi$
$620$	88.9133i	3.57084i
$621$	0	0
$622$	−17.8861	−0.717168
$623$	0	0
$624$	0	0
$625$	94.5890	3.78356
$626$	12.3288i	0.492759i
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	40.1586i	1.59742i
$633$	0	0
$634$	0	0
$635$	97.3986i	3.86514i
$636$	0	0
$637$	−29.1003	−1.15300
$638$	0	0
$639$	0	0
$640$	48.9477	1.93483
$641$	22.6274i	0.893729i	0.894602	+	0.446865i	$0.147459\pi$
$641$	22.6274i	0.893729i	−0.894602	+	0.446865i	$0.852541\pi$
$642$	0	0
$643$	−46.0000	−1.81406	−0.907031	−	0.421063i	$-0.861657\pi$
$643$	−46.0000	−1.81406	−0.907031	+	0.421063i	$0.861657\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 47.9223i	− 1.88402i	−0.335585	−	0.942010i	$-0.608934\pi$
$647$	− 47.9223i	− 1.88402i	0.335585	−	0.942010i	$-0.391066\pi$
$648$	0	0
$649$	0	0
$650$	− 80.6490i	− 3.16331i
$651$	0	0
$652$	−28.0000	−1.09656
$653$	− 50.6960i	− 1.98389i	−0.126684	−	0.991943i	$-0.540433\pi$
$653$	− 50.6960i	− 1.98389i	0.126684	−	0.991943i	$-0.459567\pi$
$654$	0	0
$655$	0	0
$656$	49.3153i	1.92544i
$657$	0	0
$658$	0	0
$659$	− 24.6577i	− 0.960526i	−0.877125	−	0.480263i	$-0.840541\pi$
$659$	− 24.6577i	− 0.960526i	0.877125	−	0.480263i	$-0.159459\pi$
$660$	0	0
$661$	7.61047	0.296013	0.148007	−	0.988986i	$-0.452714\pi$
$661$	7.61047	0.296013	0.148007	+	0.988986i	$0.452714\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	0	0
$676$	−8.56440	−0.329400
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	49.3153i	1.88700i	0.331375	+	0.943499i	$0.392487\pi$
$683$	49.3153i	1.88700i	−0.331375	+	0.943499i	$0.607513\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	−34.8712	−1.32945
$689$	0	0
$690$	0	0
$691$	43.5890	1.65820	0.829102	−	0.559098i	$-0.188852\pi$
$691$	43.5890	1.65820	0.829102	+	0.559098i	$0.188852\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	43.2641i	1.64110i
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 45.1486i	− 1.70524i	−0.522531	−	0.852620i	$-0.675012\pi$
$701$	− 45.1486i	− 1.70524i	0.522531	−	0.852620i	$-0.324988\pi$
$702$	0	0
$703$	35.2188	1.32830
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	32.0000	1.19925
$713$	− 15.9554i	− 0.597536i
$714$	0	0
$715$	0	0
$716$	49.3153i	1.84300i
$717$	0	0
$718$	−46.7518	−1.74476
$719$	− 53.4696i	− 1.99408i	−0.0768806	−	0.997040i	$-0.524496\pi$
$719$	− 53.4696i	− 1.99408i	0.0768806	−	0.997040i	$-0.475504\pi$
$720$	0	0
$721$	0	0
$722$	− 26.8701i	− 1.00000i
$723$	0	0
$724$	24.0046	0.892123
$725$	0	0
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	0	0
$730$	53.3395	1.97418
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	−8.78364	−0.323769
$737$	0	0
$738$	0	0
$739$	50.0000	1.83928	0.919640	−	0.392763i	$-0.128481\pi$
$739$	50.0000	1.83928	0.919640	+	0.392763i	$0.128481\pi$
$740$	− 69.9125i	− 2.57003i
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	−42.7178	−1.56506
$746$	52.2487i	1.91296i
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	50.9090	1.85770	0.928848	−	0.370462i	$-0.120801\pi$
$751$	50.9090	1.85770	0.928848	+	0.370462i	$0.120801\pi$
$752$	− 28.4003i	− 1.03565i
$753$	0	0
$754$	0	0
$755$	80.4280i	2.92707i
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	−53.3395	−1.93483
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	0	0
$764$	55.0224i	1.99064i
$765$	0	0
$766$	0	0
$767$	11.7583i	0.424568i
$768$	0	0
$769$	43.5890	1.57186	0.785930	−	0.618316i	$-0.212185\pi$
$769$	43.5890	1.57186	0.785930	+	0.618316i	$0.212185\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	−140.959	−5.06341
$776$	0	0
$777$	0	0
$778$	−27.1390	−0.972981
$779$	− 53.7401i	− 1.92544i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	28.0000	1.00000
$785$	0	0
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	49.4750i	1.76248i
$789$	0	0
$790$	−86.8712	−3.09074
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	77.5996i	2.74356i
$801$	0	0
$802$	17.4356	0.615672
$803$	0	0
$804$	0	0
$805$	0	0
$806$	60.4121i	2.12793i
$807$	0	0
$808$	−3.45329	−0.121486
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 60.5697i	− 2.12166i
$816$	0	0
$817$	38.0000	1.32945
$818$	0	0
$819$	0	0
$820$	−106.679	−3.72539
$821$	− 13.6429i	− 0.476139i	−0.971248	−	0.238070i	$-0.923485\pi$
$821$	− 13.6429i	− 0.476139i	0.971248	−	0.238070i	$-0.0765146\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	17.9693i	0.625989i
$825$	0	0
$826$	0	0
$827$	− 24.6577i	− 0.857431i	−0.903440	−	0.428715i	$-0.858966\pi$
$827$	− 24.6577i	− 0.857431i	0.903440	−	0.428715i	$-0.141034\pi$
$828$	0	0
$829$	−19.8474	−0.689329	−0.344664	−	0.938726i	$-0.612007\pi$
$829$	−19.8474	−0.689329	−0.344664	+	0.938726i	$0.612007\pi$
$830$	0	0
$831$	0	0
$832$	33.2575	1.15300
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	57.7960i	1.99178i
$843$	0	0
$844$	0	0
$845$	− 18.5265i	− 0.637333i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	12.5458i	0.430063i
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	−40.0000	−1.36717
$857$	56.5685i	1.93234i	0.257897	+	0.966172i	$0.416970\pi$
$857$	56.5685i	1.93234i	−0.257897	+	0.966172i	$0.583030\pi$
$858$	0	0
$859$	43.5890	1.48724	0.743619	−	0.668604i	$-0.233108\pi$
$859$	43.5890	1.48724	0.743619	+	0.668604i	$0.233108\pi$
$860$	− 75.4335i	− 2.57226i
$861$	0	0
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	− 0.663643i	− 0.0224738i
$873$	0	0
$874$	9.57175	0.323769
$875$	0	0
$876$	0	0
$877$	−48.7131	−1.64492	−0.822462	−	0.568820i	$-0.807400\pi$
$877$	−48.7131	−1.64492	−0.822462	+	0.568820i	$0.807400\pi$
$878$	42.9322i	1.44889i
$879$	0	0
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	−8.71780	−0.293377	−0.146689	−	0.989183i	$-0.546862\pi$
$883$	−8.71780	−0.293377	−0.146689	+	0.989183i	$0.546862\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	0	0
$890$	69.2225i	2.32034i
$891$	0	0
$892$	52.8703	1.77023
$893$	30.9485i	1.03565i
$894$	0	0
$895$	−106.679	−3.56589
$896$	0	0
$897$	0	0
$898$	56.0000	1.86874
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	−34.8712	−1.15980
$905$	51.9268i	1.72611i
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	49.3153i	1.63659i
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	0	0
$914$	− 2.82843i	− 0.0935561i
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	− 19.0008i	− 0.626438i
$921$	0	0
$922$	59.4580	1.95814
$923$	0	0
$924$	0	0
$925$	110.836	3.64427
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	0	0
$931$	−30.5123	−1.00000
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−61.0246	−1.99359	−0.996793	−	0.0800213i	$-0.974501\pi$
$937$	−61.0246	−1.99359	−0.996793	+	0.0800213i	$0.974501\pi$
$938$	0	0
$939$	0	0
$940$	61.4356	2.00381
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	19.1435	0.623398
$944$	− 11.3137i	− 0.368230i
$945$	0	0
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	36.2415	1.17645
$950$	− 84.5622i	− 2.74356i
$951$	0	0
$952$	0	0
$953$	− 61.6441i	− 1.99685i	−0.0561066	−	0.998425i	$-0.517869\pi$
$953$	− 61.6441i	− 1.99685i	0.0561066	−	0.998425i	$-0.482131\pi$
$954$	0	0
$955$	−119.025	−3.85155
$956$	− 7.98917i	− 0.258388i
$957$	0	0
$958$	55.5354	1.79427
$959$	0	0
$960$	0	0
$961$	74.5890	2.40610
$962$	− 47.5020i	− 1.53153i
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	− 31.1127i	− 1.00000i
$969$	0	0
$970$	0	0
$971$	49.3153i	1.58260i	0.611426	+	0.791302i	$0.290596\pi$
$971$	49.3153i	1.58260i	−0.611426	+	0.791302i	$0.709404\pi$
$972$	0	0
$973$	0	0
$974$	− 60.9015i	− 1.95141i
$975$	0	0
$976$	0	0
$977$	− 61.6441i	− 1.97217i	−0.166240	−	0.986085i	$-0.553163\pi$
$977$	− 61.6441i	− 1.97217i	0.166240	−	0.986085i	$-0.446837\pi$
$978$	0	0
$979$	0	0
$980$	60.5697i	1.93483i
$981$	0	0
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	−107.025	−3.41009
$986$	0	0
$987$	0	0
$988$	−36.2415	−1.15300
$989$	13.5365i	0.430436i
$990$	0	0
$991$	31.2962	0.994157	0.497079	−	0.867706i	$-0.334406\pi$
$991$	31.2962	0.994157	0.497079	+	0.867706i	$0.334406\pi$
$992$	− 58.1279i	− 1.84556i
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	− 61.6441i	− 1.95131i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1368.2.e.c.379.1	✓	8
3.2	odd	2	inner	1368.2.e.c.379.8	yes	8
4.3	odd	2		5472.2.e.c.5167.1		8
8.3	odd	2	inner	1368.2.e.c.379.4	yes	8
8.5	even	2		5472.2.e.c.5167.8		8
12.11	even	2		5472.2.e.c.5167.7		8
19.18	odd	2	inner	1368.2.e.c.379.5	yes	8
24.5	odd	2		5472.2.e.c.5167.2		8
24.11	even	2	inner	1368.2.e.c.379.5	yes	8
57.56	even	2	inner	1368.2.e.c.379.4	yes	8
76.75	even	2		5472.2.e.c.5167.2		8
152.37	odd	2		5472.2.e.c.5167.7		8
152.75	even	2	inner	1368.2.e.c.379.8	yes	8
228.227	odd	2		5472.2.e.c.5167.8		8
456.227	odd	2	CM	1368.2.e.c.379.1	✓	8
456.341	even	2		5472.2.e.c.5167.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1368.2.e.c.379.1	✓	8	1.1	even	1	trivial
1368.2.e.c.379.1	✓	8	456.227	odd	2	CM
1368.2.e.c.379.4	yes	8	8.3	odd	2	inner
1368.2.e.c.379.4	yes	8	57.56	even	2	inner
1368.2.e.c.379.5	yes	8	19.18	odd	2	inner
1368.2.e.c.379.5	yes	8	24.11	even	2	inner
1368.2.e.c.379.8	yes	8	3.2	odd	2	inner
1368.2.e.c.379.8	yes	8	152.75	even	2	inner
5472.2.e.c.5167.1		8	4.3	odd	2
5472.2.e.c.5167.1		8	456.341	even	2
5472.2.e.c.5167.2		8	24.5	odd	2
5472.2.e.c.5167.2		8	76.75	even	2
5472.2.e.c.5167.7		8	12.11	even	2
5472.2.e.c.5167.7		8	152.37	odd	2
5472.2.e.c.5167.8		8	8.5	even	2
5472.2.e.c.5167.8		8	228.227	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:	\( N \)	\(=\)	\( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1368.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -2.00000 q^{4} -4.32641i q^{5} +2.82843i q^{8} +O(q^{10})\)	\(q-1.41421i q^{2} -2.00000 q^{4} -4.32641i q^{5} +2.82843i q^{8} -6.11846 q^{10} -4.15719 q^{13} +4.00000 q^{16} -4.35890 q^{19} +8.65281i q^{20} -1.55274i q^{23} -13.7178 q^{25} +5.87915i q^{26} +10.2757 q^{31} -5.65685i q^{32} -8.07974 q^{37} +6.16441i q^{38} +12.2369 q^{40} +12.3288i q^{41} -8.71780 q^{43} -2.19591 q^{46} -7.10007i q^{47} +7.00000 q^{49} +19.3999i q^{50} +8.31437 q^{52} -2.82843i q^{59} -14.5320i q^{62} -8.00000 q^{64} +17.9857i q^{65} -8.71780 q^{73} +11.4265i q^{74} +8.71780 q^{76} +14.1982 q^{79} -17.3056i q^{80} +17.4356 q^{82} +12.3288i q^{86} -11.3137i q^{89} +3.10549i q^{92} -10.0410 q^{94} +18.8584i q^{95} -9.89949i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{4}+O(q^{10}) \) 8 * q - 16 * q^4	\( 8 q - 16 q^{4} + 32 q^{16} - 40 q^{25} + 56 q^{49} - 64 q^{64}+O(q^{100}) \) 8 * q - 16 * q^4 + 32 * q^16 - 40 * q^25 + 56 * q^49 - 64 * q^64

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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