LMFDB - Embedded newform 1428.2.d.c.169.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1428,2,Mod(169,1428)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1428, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("1428.169");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1428 = 2^{2} \cdot 3 \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1428.d (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:	$11.4026374086$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 43x^{10} + 647x^{8} + 4049x^{6} + 10288x^{4} + 9088x^{2} + 2304 $ x^12 + 43x^10 + 647x^8 + 4049x^6 + 10288x^4 + 9088*x^2 + 2304
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			169.12
Root			$4.02527i$ of defining polynomial
Character	$\chi$	$=$	1428.169
Dual form			1428.2.d.c.169.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +4.02527i q^{5} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +4.02527i q^{5} -1.00000i q^{7} -1.00000 q^{9} -3.54643i q^{11} -3.79991 q^{13} -4.02527 q^{15} +(-3.32819 + 2.43375i) q^{17} -1.49247 q^{19} +1.00000 q^{21} +3.97132i q^{23} -11.2028 q^{25} -1.00000i q^{27} -7.29238i q^{29} -4.20770i q^{31} +3.54643 q^{33} +4.02527 q^{35} +11.3606i q^{37} -3.79991i q^{39} +4.45016i q^{41} -12.7356 q^{43} -4.02527i q^{45} +4.20770 q^{47} -1.00000 q^{49} +(-2.43375 - 3.32819i) q^{51} -4.40291 q^{53} +14.2753 q^{55} -1.49247i q^{57} +6.93570 q^{59} -13.8032i q^{61} +1.00000i q^{63} -15.2957i q^{65} -3.17493 q^{67} -3.97132 q^{69} -5.18919i q^{71} +0.225362i q^{73} -11.2028i q^{75} -3.54643 q^{77} -5.64764i q^{79} +1.00000 q^{81} +10.6927 q^{83} +(-9.79650 - 13.3969i) q^{85} +7.29238 q^{87} +3.07861 q^{89} +3.79991i q^{91} +4.20770 q^{93} -6.00761i q^{95} +15.8757i q^{97} +3.54643i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^9	$ 12 q - 12 q^{9} - 6 q^{13} + 2 q^{15} - 2 q^{17} + 6 q^{19} + 12 q^{21} - 26 q^{25} + 10 q^{33} - 2 q^{35} - 18 q^{43} - 20 q^{47} - 12 q^{49} - 2 q^{51} + 16 q^{53} + 22 q^{55} - 12 q^{59} + 32 q^{67} - 6 q^{69} - 10 q^{77} + 12 q^{81} - 16 q^{83} + 14 q^{85} + 24 q^{87} + 4 q^{89} - 20 q^{93}+O(q^{100}) $ 12 * q - 12 * q^9 - 6 * q^13 + 2 * q^15 - 2 * q^17 + 6 * q^19 + 12 * q^21 - 26 * q^25 + 10 * q^33 - 2 * q^35 - 18 * q^43 - 20 * q^47 - 12 * q^49 - 2 * q^51 + 16 * q^53 + 22 * q^55 - 12 * q^59 + 32 * q^67 - 6 * q^69 - 10 * q^77 + 12 * q^81 - 16 * q^83 + 14 * q^85 + 24 * q^87 + 4 * q^89 - 20 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1428\mathbb{Z}\right)^\times$.

$n$	$409$	$715$	$953$	$1261$
$\chi(n)$	$1$	$1$	$1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$			4.02527i			1.80016i	0.435728	+	0.900078i	$0.356491\pi$
$5$			4.02527i			1.80016i	−0.435728	+	0.900078i	$0.643509\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$		−	3.54643i		−	1.06929i	−0.845077	−	0.534644i	$-0.820446\pi$
$11$		−	3.54643i		−	1.06929i	0.845077	−	0.534644i	$-0.179554\pi$
$12$		0			0
$13$	−3.79991			−1.05391			−0.526953	−	0.849895i	$-0.676666\pi$
$13$	−3.79991			−1.05391			−0.526953	+	0.849895i	$0.676666\pi$
$14$		0			0
$15$	−4.02527			−1.03932
$16$		0			0
$17$	−3.32819	+	2.43375i	−0.807205	+	0.590271i
$17$	−3.32819	+	2.43375i	−0.807205	+	0.590271i
$18$		0			0
$19$	−1.49247			−0.342397			−0.171198	−	0.985237i	$-0.554764\pi$
$19$	−1.49247			−0.342397			−0.171198	+	0.985237i	$0.554764\pi$
$20$		0			0
$21$	1.00000			0.218218
$22$		0			0
$23$			3.97132i			0.828077i	0.910259	+	0.414038i	$0.135882\pi$
$23$			3.97132i			0.828077i	−0.910259	+	0.414038i	$0.864118\pi$
$24$		0			0
$25$	−11.2028			−2.24056
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$		−	7.29238i		−	1.35416i	−0.735909	−	0.677081i	$-0.763245\pi$
$29$		−	7.29238i		−	1.35416i	0.735909	−	0.677081i	$-0.236755\pi$
$30$		0			0
$31$		−	4.20770i		−	0.755725i	−0.925862	−	0.377862i	$-0.876659\pi$
$31$		−	4.20770i		−	0.755725i	0.925862	−	0.377862i	$-0.123341\pi$
$32$		0			0
$33$	3.54643			0.617354
$34$		0			0
$35$	4.02527			0.680395
$36$		0			0
$37$			11.3606i			1.86767i	0.357704	+	0.933835i	$0.383560\pi$
$37$			11.3606i			1.86767i	−0.357704	+	0.933835i	$0.616440\pi$
$38$		0			0
$39$		−	3.79991i		−	0.608473i
$40$		0			0
$41$			4.45016i			0.694998i	0.937680	+	0.347499i	$0.112969\pi$
$41$			4.45016i			0.694998i	−0.937680	+	0.347499i	$0.887031\pi$
$42$		0			0
$43$	−12.7356			−1.94216			−0.971082	−	0.238748i	$-0.923263\pi$
$43$	−12.7356			−1.94216			−0.971082	+	0.238748i	$0.923263\pi$
$44$		0			0
$45$		−	4.02527i		−	0.600052i
$46$		0			0
$47$	4.20770			0.613756			0.306878	−	0.951749i	$-0.400716\pi$
$47$	4.20770			0.613756			0.306878	+	0.951749i	$0.400716\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−2.43375	−	3.32819i	−0.340793	−	0.466040i
$52$		0			0
$53$	−4.40291			−0.604786			−0.302393	−	0.953183i	$-0.597785\pi$
$53$	−4.40291			−0.604786			−0.302393	+	0.953183i	$0.597785\pi$
$54$		0			0
$55$	14.2753			1.92489
$56$		0			0
$57$		−	1.49247i		−	0.197683i
$58$		0			0
$59$	6.93570			0.902952			0.451476	−	0.892283i	$-0.350898\pi$
$59$	6.93570			0.902952			0.451476	+	0.892283i	$0.350898\pi$
$60$		0			0
$61$		−	13.8032i		−	1.76732i	−0.468130	−	0.883659i	$-0.655072\pi$
$61$		−	13.8032i		−	1.76732i	0.468130	−	0.883659i	$-0.344928\pi$
$62$		0			0
$63$			1.00000i			0.125988i
$64$		0			0
$65$		−	15.2957i		−	1.89719i
$66$		0			0
$67$	−3.17493			−0.387880			−0.193940	−	0.981013i	$-0.562127\pi$
$67$	−3.17493			−0.387880			−0.193940	+	0.981013i	$0.562127\pi$
$68$		0			0
$69$	−3.97132			−0.478090
$70$		0			0
$71$		−	5.18919i		−	0.615843i	−0.951412	−	0.307922i	$-0.900367\pi$
$71$		−	5.18919i		−	0.615843i	0.951412	−	0.307922i	$-0.0996334\pi$
$72$		0			0
$73$			0.225362i			0.0263766i	0.999913	+	0.0131883i	$0.00419809\pi$
$73$			0.225362i			0.0263766i	−0.999913	+	0.0131883i	$0.995802\pi$
$74$		0			0
$75$		−	11.2028i		−	1.29359i
$76$		0			0
$77$	−3.54643			−0.404153
$78$		0			0
$79$		−	5.64764i		−	0.635409i	−0.948190	−	0.317705i	$-0.897088\pi$
$79$		−	5.64764i		−	0.635409i	0.948190	−	0.317705i	$-0.102912\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	10.6927			1.17367			0.586837	−	0.809705i	$-0.300373\pi$
$83$	10.6927			1.17367			0.586837	+	0.809705i	$0.300373\pi$
$84$		0			0
$85$	−9.79650	−	13.3969i	−1.06258	−	1.45310i
$86$		0			0
$87$	7.29238			0.781826
$88$		0			0
$89$	3.07861			0.326332			0.163166	−	0.986599i	$-0.447829\pi$
$89$	3.07861			0.326332			0.163166	+	0.986599i	$0.447829\pi$
$90$		0			0
$91$			3.79991i			0.398339i
$92$		0			0
$93$	4.20770			0.436318
$94$		0			0
$95$		−	6.00761i		−	0.616368i
$96$		0			0
$97$			15.8757i			1.61194i	0.591959	+	0.805968i	$0.298355\pi$
$97$			15.8757i			1.61194i	−0.591959	+	0.805968i	$0.701645\pi$
$98$		0			0
$99$			3.54643i			0.356430i
$100$		0			0
$101$	−11.8218			−1.17631			−0.588155	−	0.808748i	$-0.700146\pi$
$101$	−11.8218			−1.17631			−0.588155	+	0.808748i	$0.700146\pi$
$102$		0			0
$103$	−6.02527			−0.593688			−0.296844	−	0.954926i	$-0.595934\pi$
$103$	−6.02527			−0.593688			−0.296844	+	0.954926i	$0.595934\pi$
$104$		0			0
$105$			4.02527i			0.392826i
$106$		0			0
$107$			11.6790i			1.12906i	0.825414	+	0.564528i	$0.190942\pi$
$107$			11.6790i			1.12906i	−0.825414	+	0.564528i	$0.809058\pi$
$108$		0			0
$109$			8.31828i			0.796747i	0.917223	+	0.398373i	$0.130425\pi$
$109$			8.31828i			0.796747i	−0.917223	+	0.398373i	$0.869575\pi$
$110$		0			0
$111$	−11.3606			−1.07830
$112$		0			0
$113$		−	12.9351i		−	1.21684i	−0.793617	−	0.608418i	$-0.791805\pi$
$113$		−	12.9351i		−	1.21684i	0.793617	−	0.608418i	$-0.208195\pi$
$114$		0			0
$115$	−15.9856			−1.49067
$116$		0			0
$117$	3.79991			0.351302
$118$		0			0
$119$	2.43375	+	3.32819i	0.223101	+	0.305095i
$120$		0			0
$121$	−1.57716			−0.143378
$122$		0			0
$123$	−4.45016			−0.401257
$124$		0			0
$125$		−	24.9680i		−	2.23321i
$126$		0			0
$127$	−13.4748			−1.19570			−0.597848	−	0.801610i	$-0.703977\pi$
$127$	−13.4748			−1.19570			−0.597848	+	0.801610i	$0.703977\pi$
$128$		0			0
$129$		−	12.7356i		−	1.12131i
$130$		0			0
$131$			14.3012i			1.24950i	0.780825	+	0.624750i	$0.214799\pi$
$131$			14.3012i			1.24950i	−0.780825	+	0.624750i	$0.785201\pi$
$132$		0			0
$133$			1.49247i			0.129414i
$134$		0			0
$135$	4.02527			0.346440
$136$		0			0
$137$	−12.9099			−1.10296			−0.551482	−	0.834187i	$-0.685938\pi$
$137$	−12.9099			−1.10296			−0.551482	+	0.834187i	$0.685938\pi$
$138$		0			0
$139$			21.7855i			1.84783i	0.382604	+	0.923913i	$0.375027\pi$
$139$			21.7855i			1.84783i	−0.382604	+	0.923913i	$0.624973\pi$
$140$		0			0
$141$			4.20770i			0.354352i
$142$		0			0
$143$			13.4761i			1.12693i
$144$		0			0
$145$	29.3538			2.43770
$146$		0			0
$147$		−	1.00000i		−	0.0824786i
$148$		0			0
$149$	−1.59982			−0.131062			−0.0655312	−	0.997851i	$-0.520874\pi$
$149$	−1.59982			−0.131062			−0.0655312	+	0.997851i	$0.520874\pi$
$150$		0			0
$151$	−2.10933			−0.171655			−0.0858276	−	0.996310i	$-0.527353\pi$
$151$	−2.10933			−0.171655			−0.0858276	+	0.996310i	$0.527353\pi$
$152$		0			0
$153$	3.32819	−	2.43375i	0.269068	−	0.196757i
$154$		0			0
$155$	16.9371			1.36042
$156$		0			0
$157$	5.58266			0.445545			0.222772	−	0.974870i	$-0.428489\pi$
$157$	5.58266			0.445545			0.222772	+	0.974870i	$0.428489\pi$
$158$		0			0
$159$		−	4.40291i		−	0.349173i
$160$		0			0
$161$	3.97132			0.312984
$162$		0			0
$163$			15.9549i			1.24968i	0.780751	+	0.624842i	$0.214837\pi$
$163$			15.9549i			1.24968i	−0.780751	+	0.624842i	$0.785163\pi$
$164$		0			0
$165$			14.2753i			1.11133i
$166$		0			0
$167$		−	16.4964i		−	1.27653i	−0.769817	−	0.638264i	$-0.779653\pi$
$167$		−	16.4964i		−	1.27653i	0.769817	−	0.638264i	$-0.220347\pi$
$168$		0			0
$169$	1.43932			0.110717
$170$		0			0
$171$	1.49247			0.114132
$172$		0			0
$173$		−	0.329815i		−	0.0250753i	−0.999921	−	0.0125377i	$-0.996009\pi$
$173$		−	0.329815i		−	0.0250753i	0.999921	−	0.0125377i	$-0.00399097\pi$
$174$		0			0
$175$			11.2028i			0.846853i
$176$		0			0
$177$			6.93570i			0.521319i
$178$		0			0
$179$	−7.32254			−0.547312			−0.273656	−	0.961828i	$-0.588233\pi$
$179$	−7.32254			−0.547312			−0.273656	+	0.961828i	$0.588233\pi$
$180$		0			0
$181$			3.37446i			0.250822i	0.992105	+	0.125411i	$0.0400249\pi$
$181$			3.37446i			0.250822i	−0.992105	+	0.125411i	$0.959975\pi$
$182$		0			0
$183$	13.8032			1.02036
$184$		0			0
$185$	−45.7295			−3.36210
$186$		0			0
$187$	8.63111	+	11.8032i	0.631170	+	0.863136i
$188$		0			0
$189$	−1.00000			−0.0727393
$190$		0			0
$191$	−12.1283			−0.877576			−0.438788	−	0.898591i	$-0.644592\pi$
$191$	−12.1283			−0.877576			−0.438788	+	0.898591i	$0.644592\pi$
$192$		0			0
$193$		−	7.72544i		−	0.556089i	−0.960568	−	0.278045i	$-0.910314\pi$
$193$		−	7.72544i		−	0.556089i	0.960568	−	0.278045i	$-0.0896864\pi$
$194$		0			0
$195$	15.2957			1.09535
$196$		0			0
$197$		−	3.10858i		−	0.221477i	−0.993850	−	0.110739i	$-0.964678\pi$
$197$		−	3.10858i		−	0.221477i	0.993850	−	0.110739i	$-0.0353216\pi$
$198$		0			0
$199$			21.4126i			1.51790i	0.651151	+	0.758948i	$0.274286\pi$
$199$			21.4126i			1.51790i	−0.651151	+	0.758948i	$0.725714\pi$
$200$		0			0
$201$		−	3.17493i		−	0.223943i
$202$		0			0
$203$	−7.29238			−0.511825
$204$		0			0
$205$	−17.9131			−1.25111
$206$		0			0
$207$		−	3.97132i		−	0.276026i
$208$		0			0
$209$			5.29295i			0.366121i
$210$		0			0
$211$		−	0.531496i		−	0.0365897i	−0.999833	−	0.0182948i	$-0.994176\pi$
$211$		−	0.531496i		−	0.0365897i	0.999833	−	0.0182948i	$-0.00582375\pi$
$212$		0			0
$213$	5.18919			0.355557
$214$		0			0
$215$		−	51.2643i		−	3.49620i
$216$		0			0
$217$	−4.20770			−0.285637
$218$		0			0
$219$	−0.225362			−0.0152285
$220$		0			0
$221$	12.6468	−	9.24802i	0.850718	−	0.622089i
$222$		0			0
$223$	15.7382			1.05391			0.526955	−	0.849893i	$-0.323334\pi$
$223$	15.7382			1.05391			0.526955	+	0.849893i	$0.323334\pi$
$224$		0			0
$225$	11.2028			0.746854
$226$		0			0
$227$			26.4883i			1.75809i	0.476738	+	0.879046i	$0.341819\pi$
$227$			26.4883i			1.75809i	−0.476738	+	0.879046i	$0.658181\pi$
$228$		0			0
$229$	−10.9835			−0.725812			−0.362906	−	0.931826i	$-0.618215\pi$
$229$	−10.9835			−0.725812			−0.362906	+	0.931826i	$0.618215\pi$
$230$		0			0
$231$		−	3.54643i		−	0.233338i
$232$		0			0
$233$		−	12.4597i		−	0.816263i	−0.912923	−	0.408131i	$-0.866181\pi$
$233$		−	12.4597i		−	0.816263i	0.912923	−	0.408131i	$-0.133819\pi$
$234$		0			0
$235$			16.9371i			1.10486i
$236$		0			0
$237$	5.64764			0.366854
$238$		0			0
$239$	19.5452			1.26427			0.632136	−	0.774857i	$-0.282178\pi$
$239$	19.5452			1.26427			0.632136	+	0.774857i	$0.282178\pi$
$240$		0			0
$241$		−	18.8607i		−	1.21492i	−0.794349	−	0.607461i	$-0.792188\pi$
$241$		−	18.8607i		−	1.21492i	0.794349	−	0.607461i	$-0.207812\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$		−	4.02527i		−	0.257165i
$246$		0			0
$247$	5.67126			0.360854
$248$		0			0
$249$			10.6927i			0.677621i
$250$		0			0
$251$	24.2705			1.53194			0.765971	−	0.642875i	$-0.222259\pi$
$251$	24.2705			1.53194			0.765971	+	0.642875i	$0.222259\pi$
$252$		0			0
$253$	14.0840			0.885453
$254$		0			0
$255$	13.3969	−	9.79650i	0.838945	−	0.613481i
$256$		0			0
$257$	−1.49394			−0.0931895			−0.0465948	−	0.998914i	$-0.514837\pi$
$257$	−1.49394			−0.0931895			−0.0465948	+	0.998914i	$0.514837\pi$
$258$		0			0
$259$	11.3606			0.705913
$260$		0			0
$261$			7.29238i			0.451387i
$262$		0			0
$263$	−16.7786			−1.03461			−0.517305	−	0.855801i	$-0.673065\pi$
$263$	−16.7786			−1.03461			−0.517305	+	0.855801i	$0.673065\pi$
$264$		0			0
$265$		−	17.7229i		−	1.08871i
$266$		0			0
$267$			3.07861i			0.188408i
$268$		0			0
$269$			28.6209i			1.74505i	0.488573	+	0.872523i	$0.337518\pi$
$269$			28.6209i			1.74505i	−0.488573	+	0.872523i	$0.662482\pi$
$270$		0			0
$271$	4.53479			0.275469			0.137734	−	0.990469i	$-0.456018\pi$
$271$	4.53479			0.275469			0.137734	+	0.990469i	$0.456018\pi$
$272$		0			0
$273$	−3.79991			−0.229981
$274$		0			0
$275$			39.7300i			2.39581i
$276$		0			0
$277$			28.3988i			1.70632i	0.521650	+	0.853159i	$0.325317\pi$
$277$			28.3988i			1.70632i	−0.521650	+	0.853159i	$0.674683\pi$
$278$		0			0
$279$			4.20770i			0.251908i
$280$		0			0
$281$	4.69268			0.279942			0.139971	−	0.990156i	$-0.455299\pi$
$281$	4.69268			0.279942			0.139971	+	0.990156i	$0.455299\pi$
$282$		0			0
$283$		−	5.67348i		−	0.337253i	−0.985680	−	0.168627i	$-0.946067\pi$
$283$		−	5.67348i		−	0.337253i	0.985680	−	0.168627i	$-0.0539332\pi$
$284$		0			0
$285$	6.00761			0.355860
$286$		0			0
$287$	4.45016			0.262685
$288$		0			0
$289$	5.15374	−	16.2000i	0.303161	−	0.952939i
$290$		0			0
$291$	−15.8757			−0.930652
$292$		0			0
$293$	6.94865			0.405945			0.202972	−	0.979184i	$-0.434940\pi$
$293$	6.94865			0.405945			0.202972	+	0.979184i	$0.434940\pi$
$294$		0			0
$295$			27.9181i			1.62545i
$296$		0			0
$297$	−3.54643			−0.205785
$298$		0			0
$299$		−	15.0906i		−	0.872714i
$300$		0			0
$301$			12.7356i			0.734069i
$302$		0			0
$303$		−	11.8218i		−	0.679143i
$304$		0			0
$305$	55.5616			3.18145
$306$		0			0
$307$	−6.40450			−0.365524			−0.182762	−	0.983157i	$-0.558504\pi$
$307$	−6.40450			−0.365524			−0.182762	+	0.983157i	$0.558504\pi$
$308$		0			0
$309$		−	6.02527i		−	0.342766i
$310$		0			0
$311$			5.57256i			0.315991i	0.987440	+	0.157996i	$0.0505032\pi$
$311$			5.57256i			0.315991i	−0.987440	+	0.157996i	$0.949497\pi$
$312$		0			0
$313$		−	6.23758i		−	0.352569i	−0.984339	−	0.176284i	$-0.943592\pi$
$313$		−	6.23758i		−	0.352569i	0.984339	−	0.176284i	$-0.0564078\pi$
$314$		0			0
$315$	−4.02527			−0.226798
$316$		0			0
$317$		−	2.11223i		−	0.118635i	−0.998239	−	0.0593173i	$-0.981108\pi$
$317$		−	2.11223i		−	0.118635i	0.998239	−	0.0593173i	$-0.0188924\pi$
$318$		0			0
$319$	−25.8619			−1.44799
$320$		0			0
$321$	−11.6790			−0.651861
$322$		0			0
$323$	4.96724	−	3.63230i	0.276384	−	0.202107i
$324$		0			0
$325$	42.5697			2.36134
$326$		0			0
$327$	−8.31828			−0.460002
$328$		0			0
$329$		−	4.20770i		−	0.231978i
$330$		0			0
$331$	−13.6906			−0.752505			−0.376253	−	0.926517i	$-0.622788\pi$
$331$	−13.6906			−0.752505			−0.376253	+	0.926517i	$0.622788\pi$
$332$		0			0
$333$		−	11.3606i		−	0.622557i
$334$		0			0
$335$		−	12.7800i		−	0.698245i
$336$		0			0
$337$			5.03276i			0.274152i	0.990561	+	0.137076i	$0.0437705\pi$
$337$			5.03276i			0.274152i	−0.990561	+	0.137076i	$0.956230\pi$
$338$		0			0
$339$	12.9351			0.702540
$340$		0			0
$341$	−14.9223			−0.808088
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$		−	15.9856i		−	0.860637i
$346$		0			0
$347$			0.545806i			0.0293004i	0.999893	+	0.0146502i	$0.00466347\pi$
$347$			0.545806i			0.0293004i	−0.999893	+	0.0146502i	$0.995337\pi$
$348$		0			0
$349$	−27.3682			−1.46499			−0.732493	−	0.680775i	$-0.761643\pi$
$349$	−27.3682			−1.46499			−0.732493	+	0.680775i	$0.761643\pi$
$350$		0			0
$351$			3.79991i			0.202824i
$352$		0			0
$353$	34.2575			1.82334			0.911672	−	0.410919i	$-0.134792\pi$
$353$	34.2575			1.82334			0.911672	+	0.410919i	$0.134792\pi$
$354$		0			0
$355$	20.8879			1.10861
$356$		0			0
$357$	−3.32819	+	2.43375i	−0.176147	+	0.128808i
$358$		0			0
$359$	22.9314			1.21027			0.605137	−	0.796121i	$-0.293118\pi$
$359$	22.9314			1.21027			0.605137	+	0.796121i	$0.293118\pi$
$360$		0			0
$361$	−16.7725			−0.882765
$362$		0			0
$363$		−	1.57716i		−	0.0827793i
$364$		0			0
$365$	−0.907142			−0.0474820
$366$		0			0
$367$		−	8.00000i		−	0.417597i	−0.977959	−	0.208798i	$-0.933045\pi$
$367$		−	8.00000i		−	0.417597i	0.977959	−	0.208798i	$-0.0669552\pi$
$368$		0			0
$369$		−	4.45016i		−	0.231666i
$370$		0			0
$371$			4.40291i			0.228587i
$372$		0			0
$373$	−22.0342			−1.14089			−0.570445	−	0.821336i	$-0.693229\pi$
$373$	−22.0342			−1.14089			−0.570445	+	0.821336i	$0.693229\pi$
$374$		0			0
$375$	24.9680			1.28934
$376$		0			0
$377$			27.7104i			1.42716i
$378$		0			0
$379$			6.73095i			0.345746i	0.984944	+	0.172873i	$0.0553049\pi$
$379$			6.73095i			0.345746i	−0.984944	+	0.172873i	$0.944695\pi$
$380$		0			0
$381$		−	13.4748i		−	0.690335i
$382$		0			0
$383$	−14.5135			−0.741604			−0.370802	−	0.928712i	$-0.620917\pi$
$383$	−14.5135			−0.741604			−0.370802	+	0.928712i	$0.620917\pi$
$384$		0			0
$385$		−	14.2753i		−	0.727539i
$386$		0			0
$387$	12.7356			0.647388
$388$		0			0
$389$	−3.22145			−0.163334			−0.0816670	−	0.996660i	$-0.526024\pi$
$389$	−3.22145			−0.163334			−0.0816670	+	0.996660i	$0.526024\pi$
$390$		0			0
$391$	−9.66518	−	13.2173i	−0.488789	−	0.668428i
$392$		0			0
$393$	−14.3012			−0.721399
$394$		0			0
$395$	22.7333			1.14384
$396$		0			0
$397$			29.1530i			1.46315i	0.681763	+	0.731573i	$0.261214\pi$
$397$			29.1530i			1.46315i	−0.681763	+	0.731573i	$0.738786\pi$
$398$		0			0
$399$	−1.49247			−0.0747171
$400$		0			0
$401$			31.9706i			1.59654i	0.602302	+	0.798269i	$0.294250\pi$
$401$			31.9706i			1.59654i	−0.602302	+	0.798269i	$0.705750\pi$
$402$		0			0
$403$			15.9889i			0.796462i
$404$		0			0
$405$			4.02527i			0.200017i
$406$		0			0
$407$	40.2895			1.99708
$408$		0			0
$409$	−25.7658			−1.27404			−0.637019	−	0.770848i	$-0.719833\pi$
$409$	−25.7658			−1.27404			−0.637019	+	0.770848i	$0.719833\pi$
$410$		0			0
$411$		−	12.9099i		−	0.636797i
$412$		0			0
$413$		−	6.93570i		−	0.341284i
$414$		0			0
$415$			43.0409i			2.11280i
$416$		0			0
$417$	−21.7855			−1.06684
$418$		0			0
$419$		−	5.32633i		−	0.260208i	−0.991500	−	0.130104i	$-0.958469\pi$
$419$		−	5.32633i		−	0.260208i	0.991500	−	0.130104i	$-0.0415312\pi$
$420$		0			0
$421$	−7.24637			−0.353167			−0.176583	−	0.984286i	$-0.556505\pi$
$421$	−7.24637			−0.353167			−0.176583	+	0.984286i	$0.556505\pi$
$422$		0			0
$423$	−4.20770			−0.204585
$424$		0			0
$425$	37.2851	−	27.2648i	1.80859	−	1.32254i
$426$		0			0
$427$	−13.8032			−0.667984
$428$		0			0
$429$	−13.4761			−0.650633
$430$		0			0
$431$			21.1577i			1.01913i	0.860433	+	0.509564i	$0.170193\pi$
$431$			21.1577i			1.01913i	−0.860433	+	0.509564i	$0.829807\pi$
$432$		0			0
$433$	−9.59901			−0.461299			−0.230649	−	0.973037i	$-0.574085\pi$
$433$	−9.59901			−0.461299			−0.230649	+	0.973037i	$0.574085\pi$
$434$		0			0
$435$			29.3538i			1.40741i
$436$		0			0
$437$		−	5.92708i		−	0.283531i
$438$		0			0
$439$		−	30.3634i		−	1.44917i	−0.689187	−	0.724583i	$-0.742032\pi$
$439$		−	30.3634i		−	1.44917i	0.689187	−	0.724583i	$-0.257968\pi$
$440$		0			0
$441$	1.00000			0.0476190
$442$		0			0
$443$	−34.7910			−1.65297			−0.826486	−	0.562957i	$-0.809664\pi$
$443$	−34.7910			−1.65297			−0.826486	+	0.562957i	$0.809664\pi$
$444$		0			0
$445$			12.3922i			0.587448i
$446$		0			0
$447$		−	1.59982i		−	0.0756689i
$448$		0			0
$449$			23.7010i			1.11852i	0.828993	+	0.559259i	$0.188914\pi$
$449$			23.7010i			1.11852i	−0.828993	+	0.559259i	$0.811086\pi$
$450$		0			0
$451$	15.7822			0.743153
$452$		0			0
$453$		−	2.10933i		−	0.0991052i
$454$		0			0
$455$	−15.2957			−0.717072
$456$		0			0
$457$	31.4240			1.46995			0.734977	−	0.678092i	$-0.237193\pi$
$457$	31.4240			1.46995			0.734977	+	0.678092i	$0.237193\pi$
$458$		0			0
$459$	2.43375	+	3.32819i	0.113598	+	0.155347i
$460$		0			0
$461$	−34.9554			−1.62804			−0.814018	−	0.580840i	$-0.802724\pi$
$461$	−34.9554			−1.62804			−0.814018	+	0.580840i	$0.802724\pi$
$462$		0			0
$463$	18.3061			0.850755			0.425377	−	0.905016i	$-0.360141\pi$
$463$	18.3061			0.850755			0.425377	+	0.905016i	$0.360141\pi$
$464$		0			0
$465$			16.9371i			0.785440i
$466$		0			0
$467$	11.0478			0.511230			0.255615	−	0.966779i	$-0.417722\pi$
$467$	11.0478			0.511230			0.255615	+	0.966779i	$0.417722\pi$
$468$		0			0
$469$			3.17493i			0.146605i
$470$		0			0
$471$			5.58266i			0.257236i
$472$		0			0
$473$			45.1660i			2.07673i
$474$		0			0
$475$	16.7199			0.767161
$476$		0			0
$477$	4.40291			0.201595
$478$		0			0
$479$			11.4968i			0.525302i	0.964891	+	0.262651i	$0.0845967\pi$
$479$			11.4968i			0.525302i	−0.964891	+	0.262651i	$0.915403\pi$
$480$		0			0
$481$		−	43.1692i		−	1.96835i
$482$		0			0
$483$			3.97132i			0.180701i
$484$		0			0
$485$	−63.9041			−2.90174
$486$		0			0
$487$		−	30.5791i		−	1.38567i	−0.721096	−	0.692835i	$-0.756361\pi$
$487$		−	30.5791i		−	1.38567i	0.721096	−	0.692835i	$-0.243639\pi$
$488$		0			0
$489$	−15.9549			−0.721506
$490$		0			0
$491$	36.4795			1.64630			0.823148	−	0.567828i	$-0.192216\pi$
$491$	36.4795			1.64630			0.823148	+	0.567828i	$0.192216\pi$
$492$		0			0
$493$	17.7478	+	24.2705i	0.799322	+	1.09309i
$494$		0			0
$495$	−14.2753			−0.641629
$496$		0			0
$497$	−5.18919			−0.232767
$498$		0			0
$499$		−	16.7116i		−	0.748115i	−0.927405	−	0.374058i	$-0.877966\pi$
$499$		−	16.7116i		−	0.748115i	0.927405	−	0.374058i	$-0.122034\pi$
$500$		0			0
$501$	16.4964			0.737004
$502$		0			0
$503$		−	14.2686i		−	0.636206i	−0.948056	−	0.318103i	$-0.896954\pi$
$503$		−	14.2686i		−	0.636206i	0.948056	−	0.318103i	$-0.103046\pi$
$504$		0			0
$505$		−	47.5858i		−	2.11754i
$506$		0			0
$507$			1.43932i			0.0639224i
$508$		0			0
$509$	27.6361			1.22495			0.612475	−	0.790490i	$-0.290174\pi$
$509$	27.6361			1.22495			0.612475	+	0.790490i	$0.290174\pi$
$510$		0			0
$511$	0.225362			0.00996941
$512$		0			0
$513$			1.49247i			0.0658943i
$514$		0			0
$515$		−	24.2534i		−	1.06873i
$516$		0			0
$517$		−	14.9223i		−	0.656282i
$518$		0			0
$519$	0.329815			0.0144773
$520$		0			0
$521$			39.5867i			1.73433i	0.498024	+	0.867163i	$0.334059\pi$
$521$			39.5867i			1.73433i	−0.498024	+	0.867163i	$0.665941\pi$
$522$		0			0
$523$	−7.41386			−0.324185			−0.162093	−	0.986776i	$-0.551824\pi$
$523$	−7.41386			−0.324185			−0.162093	+	0.986776i	$0.551824\pi$
$524$		0			0
$525$	−11.2028			−0.488931
$526$		0			0
$527$	10.2405	+	14.0040i	0.446082	+	0.610025i
$528$		0			0
$529$	7.22865			0.314289
$530$		0			0
$531$	−6.93570			−0.300984
$532$		0			0
$533$		−	16.9102i		−	0.732462i
$534$		0			0
$535$	−47.0114			−2.03248
$536$		0			0
$537$		−	7.32254i		−	0.315991i
$538$		0			0
$539$			3.54643i			0.152756i
$540$		0			0
$541$			5.00936i			0.215369i	0.994185	+	0.107685i	$0.0343437\pi$
$541$			5.00936i			0.215369i	−0.994185	+	0.107685i	$0.965656\pi$
$542$		0			0
$543$	−3.37446			−0.144812
$544$		0			0
$545$	−33.4833			−1.43427
$546$		0			0
$547$		−	34.5766i		−	1.47839i	−0.673493	−	0.739194i	$-0.735207\pi$
$547$		−	34.5766i		−	1.47839i	0.673493	−	0.739194i	$-0.264793\pi$
$548$		0			0
$549$			13.8032i			0.589106i
$550$		0			0
$551$			10.8837i			0.463660i
$552$		0			0
$553$	−5.64764			−0.240162
$554$		0			0
$555$		−	45.7295i		−	1.94111i
$556$		0			0
$557$	−16.1327			−0.683564			−0.341782	−	0.939779i	$-0.611030\pi$
$557$	−16.1327			−0.683564			−0.341782	+	0.939779i	$0.611030\pi$
$558$		0			0
$559$	48.3942			2.04686
$560$		0			0
$561$	−11.8032	+	8.63111i	−0.498332	+	0.364406i
$562$		0			0
$563$	−18.4329			−0.776854			−0.388427	−	0.921480i	$-0.626981\pi$
$563$	−18.4329			−0.776854			−0.388427	+	0.921480i	$0.626981\pi$
$564$		0			0
$565$	52.0675			2.19049
$566$		0			0
$567$		−	1.00000i		−	0.0419961i
$568$		0			0
$569$	37.0096			1.55152			0.775761	−	0.631026i	$-0.217366\pi$
$569$	37.0096			1.55152			0.775761	+	0.631026i	$0.217366\pi$
$570$		0			0
$571$			15.4862i			0.648078i	0.946044	+	0.324039i	$0.105041\pi$
$571$			15.4862i			0.648078i	−0.946044	+	0.324039i	$0.894959\pi$
$572$		0			0
$573$		−	12.1283i		−	0.506669i
$574$		0			0
$575$		−	44.4899i		−	1.85536i
$576$		0			0
$577$	−39.2314			−1.63322			−0.816611	−	0.577188i	$-0.804150\pi$
$577$	−39.2314			−1.63322			−0.816611	+	0.577188i	$0.804150\pi$
$578$		0			0
$579$	7.72544			0.321058
$580$		0			0
$581$		−	10.6927i		−	0.443607i
$582$		0			0
$583$			15.6146i			0.646690i
$584$		0			0
$585$			15.2957i			0.632398i
$586$		0			0
$587$	36.8605			1.52140			0.760698	−	0.649106i	$-0.224857\pi$
$587$	36.8605			1.52140			0.760698	+	0.649106i	$0.224857\pi$
$588$		0			0
$589$			6.27987i			0.258758i
$590$		0			0
$591$	3.10858			0.127870
$592$		0			0
$593$	−39.7165			−1.63096			−0.815480	−	0.578785i	$-0.803527\pi$
$593$	−39.7165			−1.63096			−0.815480	+	0.578785i	$0.803527\pi$
$594$		0			0
$595$	−13.3969	+	9.79650i	−0.549219	+	0.401617i
$596$		0			0
$597$	−21.4126			−0.876358
$598$		0			0
$599$	37.9426			1.55029			0.775147	−	0.631781i	$-0.217676\pi$
$599$	37.9426			1.55029			0.775147	+	0.631781i	$0.217676\pi$
$600$		0			0
$601$		−	2.81826i		−	0.114959i	−0.998347	−	0.0574795i	$-0.981694\pi$
$601$		−	2.81826i		−	0.114959i	0.998347	−	0.0574795i	$-0.0183064\pi$
$602$		0			0
$603$	3.17493			0.129293
$604$		0			0
$605$		−	6.34849i		−	0.258103i
$606$		0			0
$607$			5.32123i			0.215982i	0.994152	+	0.107991i	$0.0344418\pi$
$607$			5.32123i			0.215982i	−0.994152	+	0.107991i	$0.965558\pi$
$608$		0			0
$609$		−	7.29238i		−	0.295502i
$610$		0			0
$611$	−15.9889			−0.646841
$612$		0			0
$613$	22.7793			0.920049			0.460024	−	0.887906i	$-0.347841\pi$
$613$	22.7793			0.920049			0.460024	+	0.887906i	$0.347841\pi$
$614$		0			0
$615$		−	17.9131i		−	0.722326i
$616$		0			0
$617$			24.2908i			0.977913i	0.872308	+	0.488956i	$0.162622\pi$
$617$			24.2908i			0.977913i	−0.872308	+	0.488956i	$0.837378\pi$
$618$		0			0
$619$		−	24.7786i		−	0.995936i	−0.867195	−	0.497968i	$-0.834080\pi$
$619$		−	24.7786i		−	0.995936i	0.867195	−	0.497968i	$-0.165920\pi$
$620$		0			0
$621$	3.97132			0.159363
$622$		0			0
$623$		−	3.07861i		−	0.123342i
$624$		0			0
$625$	44.4890			1.77956
$626$		0			0
$627$	−5.29295			−0.211380
$628$		0			0
$629$	−27.6488	−	37.8102i	−1.10243	−	1.50759i
$630$		0			0
$631$	9.76043			0.388557			0.194278	−	0.980946i	$-0.437763\pi$
$631$	9.76043			0.388557			0.194278	+	0.980946i	$0.437763\pi$
$632$		0			0
$633$	0.531496			0.0211251
$634$		0			0
$635$		−	54.2398i		−	2.15244i
$636$		0			0
$637$	3.79991			0.150558
$638$		0			0
$639$			5.18919i			0.205281i
$640$		0			0
$641$		−	34.7916i		−	1.37419i	−0.726570	−	0.687093i	$-0.758887\pi$
$641$		−	34.7916i		−	1.37419i	0.726570	−	0.687093i	$-0.241113\pi$
$642$		0			0
$643$			45.8855i			1.80955i	0.425893	+	0.904774i	$0.359960\pi$
$643$			45.8855i			1.80955i	−0.425893	+	0.904774i	$0.640040\pi$
$644$		0			0
$645$	51.2643			2.01853
$646$		0			0
$647$	−15.4357			−0.606839			−0.303419	−	0.952857i	$-0.598128\pi$
$647$	−15.4357			−0.606839			−0.303419	+	0.952857i	$0.598128\pi$
$648$		0			0
$649$		−	24.5970i		−	0.965516i
$650$		0			0
$651$		−	4.20770i		−	0.164913i
$652$		0			0
$653$		−	9.66864i		−	0.378363i	−0.981942	−	0.189182i	$-0.939417\pi$
$653$		−	9.66864i		−	0.378363i	0.981942	−	0.189182i	$-0.0605835\pi$
$654$		0			0
$655$	−57.5661			−2.24929
$656$		0			0
$657$		−	0.225362i		−	0.00879220i
$658$		0			0
$659$	−10.3756			−0.404178			−0.202089	−	0.979367i	$-0.564773\pi$
$659$	−10.3756			−0.404178			−0.202089	+	0.979367i	$0.564773\pi$
$660$		0			0
$661$	−48.5894			−1.88991			−0.944954	−	0.327202i	$-0.893894\pi$
$661$	−48.5894			−1.88991			−0.944954	+	0.327202i	$0.893894\pi$
$662$		0			0
$663$	9.24802	+	12.6468i	0.359164	+	0.491162i
$664$		0			0
$665$	−6.00761			−0.232965
$666$		0			0
$667$	28.9604			1.12135
$668$		0			0
$669$			15.7382i			0.608475i
$670$		0			0
$671$	−48.9521			−1.88977
$672$		0			0
$673$		−	25.3194i		−	0.975989i	−0.872847	−	0.487995i	$-0.837729\pi$
$673$		−	25.3194i		−	0.975989i	0.872847	−	0.487995i	$-0.162271\pi$
$674$		0			0
$675$			11.2028i			0.431197i
$676$		0			0
$677$			2.64564i			0.101680i	0.998707	+	0.0508401i	$0.0161899\pi$
$677$			2.64564i			0.101680i	−0.998707	+	0.0508401i	$0.983810\pi$
$678$		0			0
$679$	15.8757			0.609254
$680$		0			0
$681$	−26.4883			−1.01503
$682$		0			0
$683$		−	15.9903i		−	0.611853i	−0.952055	−	0.305926i	$-0.901034\pi$
$683$		−	15.9903i		−	0.611853i	0.952055	−	0.305926i	$-0.0989662\pi$
$684$		0			0
$685$		−	51.9657i		−	1.98551i
$686$		0			0
$687$		−	10.9835i		−	0.419048i
$688$		0			0
$689$	16.7306			0.637387
$690$		0			0
$691$		−	13.4728i		−	0.512529i	−0.966607	−	0.256264i	$-0.917508\pi$
$691$		−	13.4728i		−	0.512529i	0.966607	−	0.256264i	$-0.0824917\pi$
$692$		0			0
$693$	3.54643			0.134718
$694$		0			0
$695$	−87.6927			−3.32637
$696$		0			0
$697$	−10.8306	−	14.8110i	−0.410237	−	0.561006i
$698$		0			0
$699$	12.4597			0.471269
$700$		0			0
$701$	−15.8140			−0.597288			−0.298644	−	0.954365i	$-0.596534\pi$
$701$	−15.8140			−0.597288			−0.298644	+	0.954365i	$0.596534\pi$
$702$		0			0
$703$		−	16.9554i		−	0.639484i
$704$		0			0
$705$	−16.9371			−0.637889
$706$		0			0
$707$			11.8218i			0.444603i
$708$		0			0
$709$			8.00000i			0.300446i	0.988652	+	0.150223i	$0.0479992\pi$
$709$			8.00000i			0.300446i	−0.988652	+	0.150223i	$0.952001\pi$
$710$		0			0
$711$			5.64764i			0.211803i
$712$		0			0
$713$	16.7101			0.625798
$714$		0			0
$715$	−54.2450			−2.02865
$716$		0			0
$717$			19.5452i			0.729928i
$718$		0			0
$719$		−	3.78606i		−	0.141196i	−0.997505	−	0.0705981i	$-0.977509\pi$
$719$		−	3.78606i		−	0.141196i	0.997505	−	0.0705981i	$-0.0224908\pi$
$720$		0			0
$721$			6.02527i			0.224393i
$722$		0			0
$723$	18.8607			0.701436
$724$		0			0
$725$			81.6952i			3.03408i
$726$		0			0
$727$	−33.7337			−1.25111			−0.625557	−	0.780178i	$-0.715128\pi$
$727$	−33.7337			−1.25111			−0.625557	+	0.780178i	$0.715128\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	42.3866	−	30.9953i	1.56772	−	1.14640i
$732$		0			0
$733$	38.3129			1.41512			0.707560	−	0.706653i	$-0.249796\pi$
$733$	38.3129			1.41512			0.707560	+	0.706653i	$0.249796\pi$
$734$		0			0
$735$	4.02527			0.148474
$736$		0			0
$737$			11.2597i			0.414756i
$738$		0			0
$739$	30.3912			1.11796			0.558979	−	0.829182i	$-0.311193\pi$
$739$	30.3912			1.11796			0.558979	+	0.829182i	$0.311193\pi$
$740$		0			0
$741$			5.67126i			0.208339i
$742$		0			0
$743$		−	37.2427i		−	1.36630i	−0.730278	−	0.683151i	$-0.760609\pi$
$743$		−	37.2427i		−	1.36630i	0.730278	−	0.683151i	$-0.239391\pi$
$744$		0			0
$745$		−	6.43971i		−	0.235933i
$746$		0			0
$747$	−10.6927			−0.391225
$748$		0			0
$749$	11.6790			0.426743
$750$		0			0
$751$		−	29.0450i		−	1.05987i	−0.848039	−	0.529934i	$-0.822217\pi$
$751$		−	29.0450i		−	1.05987i	0.848039	−	0.529934i	$-0.177783\pi$
$752$		0			0
$753$			24.2705i			0.884467i
$754$		0			0
$755$		−	8.49065i		−	0.309006i
$756$		0			0
$757$	−33.4692			−1.21646			−0.608230	−	0.793761i	$-0.708120\pi$
$757$	−33.4692			−1.21646			−0.608230	+	0.793761i	$0.708120\pi$
$758$		0			0
$759$			14.0840i			0.511216i
$760$		0			0
$761$	−30.9761			−1.12288			−0.561442	−	0.827516i	$-0.689753\pi$
$761$	−30.9761			−1.12288			−0.561442	+	0.827516i	$0.689753\pi$
$762$		0			0
$763$	8.31828			0.301142
$764$		0			0
$765$	9.79650	+	13.3969i	0.354193	+	0.484365i
$766$		0			0
$767$	−26.3551			−0.951626
$768$		0			0
$769$	24.0089			0.865783			0.432892	−	0.901446i	$-0.357493\pi$
$769$	24.0089			0.865783			0.432892	+	0.901446i	$0.357493\pi$
$770$		0			0
$771$		−	1.49394i		−	0.0538030i
$772$		0			0
$773$	2.37904			0.0855683			0.0427841	−	0.999084i	$-0.486377\pi$
$773$	2.37904			0.0855683			0.0427841	+	0.999084i	$0.486377\pi$
$774$		0			0
$775$			47.1381i			1.69325i
$776$		0			0
$777$			11.3606i			0.407559i
$778$		0			0
$779$		−	6.64174i		−	0.237965i
$780$		0			0
$781$	−18.4031			−0.658514
$782$		0			0
$783$	−7.29238			−0.260609
$784$		0			0
$785$			22.4717i			0.802051i
$786$		0			0
$787$			18.8238i			0.670997i	0.942041	+	0.335498i	$0.108905\pi$
$787$			18.8238i			0.670997i	−0.942041	+	0.335498i	$0.891095\pi$
$788$		0			0
$789$		−	16.7786i		−	0.597332i
$790$		0			0
$791$	−12.9351			−0.459921
$792$		0			0
$793$			52.4509i			1.86259i
$794$		0			0
$795$	17.7229			0.628566
$796$		0			0
$797$	10.1455			0.359370			0.179685	−	0.983724i	$-0.442492\pi$
$797$	10.1455			0.359370			0.179685	+	0.983724i	$0.442492\pi$
$798$		0			0
$799$	−14.0040	+	10.2405i	−0.495427	+	0.362282i
$800$		0			0
$801$	−3.07861			−0.108777
$802$		0			0
$803$	0.799229			0.0282042
$804$		0			0
$805$			15.9856i			0.563419i
$806$		0			0
$807$	−28.6209			−1.00750
$808$		0			0
$809$			27.0198i			0.949964i	0.879995	+	0.474982i	$0.157546\pi$
$809$			27.0198i			0.949964i	−0.879995	+	0.474982i	$0.842454\pi$
$810$		0			0
$811$		−	6.32100i		−	0.221960i	−0.993823	−	0.110980i	$-0.964601\pi$
$811$		−	6.32100i		−	0.221960i	0.993823	−	0.110980i	$-0.0353990\pi$
$812$		0			0
$813$			4.53479i			0.159042i
$814$		0			0
$815$	−64.2228			−2.24963
$816$		0			0
$817$	19.0076			0.664990
$818$		0			0
$819$		−	3.79991i		−	0.132780i
$820$		0			0
$821$		−	17.4679i		−	0.609635i	−0.952411	−	0.304818i	$-0.901404\pi$
$821$		−	17.4679i		−	0.609635i	0.952411	−	0.304818i	$-0.0985955\pi$
$822$		0			0
$823$		−	43.9467i		−	1.53189i	−0.642908	−	0.765943i	$-0.722272\pi$
$823$		−	43.9467i		−	1.53189i	0.642908	−	0.765943i	$-0.277728\pi$
$824$		0			0
$825$	−39.7300			−1.38322
$826$		0			0
$827$			56.6175i			1.96878i	0.175993	+	0.984391i	$0.443686\pi$
$827$			56.6175i			1.96878i	−0.175993	+	0.984391i	$0.556314\pi$
$828$		0			0
$829$	−26.9631			−0.936468			−0.468234	−	0.883604i	$-0.655110\pi$
$829$	−26.9631			−0.936468			−0.468234	+	0.883604i	$0.655110\pi$
$830$		0			0
$831$	−28.3988			−0.985144
$832$		0			0
$833$	3.32819	−	2.43375i	0.115315	−	0.0843244i
$834$		0			0
$835$	66.4024			2.29795
$836$		0			0
$837$	−4.20770			−0.145439
$838$		0			0
$839$		−	1.88596i		−	0.0651106i	−0.999470	−	0.0325553i	$-0.989636\pi$
$839$		−	1.88596i		−	0.0651106i	0.999470	−	0.0325553i	$-0.0103645\pi$
$840$		0			0
$841$	−24.1788			−0.833753
$842$		0			0
$843$			4.69268i			0.161624i
$844$		0			0
$845$			5.79365i			0.199308i
$846$		0			0
$847$			1.57716i			0.0541918i
$848$		0			0
$849$	5.67348			0.194713
$850$		0			0
$851$	−45.1165			−1.54657
$852$		0			0
$853$			51.6040i			1.76689i	0.468539	+	0.883443i	$0.344781\pi$
$853$			51.6040i			1.76689i	−0.468539	+	0.883443i	$0.655219\pi$
$854$		0			0
$855$			6.00761i			0.205456i
$856$		0			0
$857$		−	48.5395i		−	1.65808i	−0.559190	−	0.829040i	$-0.688888\pi$
$857$		−	48.5395i		−	1.65808i	0.559190	−	0.829040i	$-0.311112\pi$
$858$		0			0
$859$	7.36582			0.251319			0.125659	−	0.992073i	$-0.459895\pi$
$859$	7.36582			0.251319			0.125659	+	0.992073i	$0.459895\pi$
$860$		0			0
$861$			4.45016i			0.151661i
$862$		0			0
$863$	43.7936			1.49075			0.745375	−	0.666645i	$-0.232270\pi$
$863$	43.7936			1.49075			0.745375	+	0.666645i	$0.232270\pi$
$864$		0			0
$865$	1.32759			0.0451395
$866$		0			0
$867$	16.2000	+	5.15374i	0.550180	+	0.175030i
$868$		0			0
$869$	−20.0290			−0.679436
$870$		0			0
$871$	12.0645			0.408789
$872$		0			0
$873$		−	15.8757i		−	0.537312i
$874$		0			0
$875$	−24.9680			−0.844073
$876$		0			0
$877$		−	49.5900i		−	1.67454i	−0.546793	−	0.837268i	$-0.684151\pi$
$877$		−	49.5900i		−	1.67454i	0.546793	−	0.837268i	$-0.315849\pi$
$878$		0			0
$879$			6.94865i			0.234372i
$880$		0			0
$881$			23.7745i			0.800984i	0.916300	+	0.400492i	$0.131161\pi$
$881$			23.7745i			0.800984i	−0.916300	+	0.400492i	$0.868839\pi$
$882$		0			0
$883$	2.98568			0.100476			0.0502381	−	0.998737i	$-0.484002\pi$
$883$	2.98568			0.100476			0.0502381	+	0.998737i	$0.484002\pi$
$884$		0			0
$885$	−27.9181			−0.938457
$886$		0			0
$887$			15.0154i			0.504168i	0.967705	+	0.252084i	$0.0811158\pi$
$887$			15.0154i			0.504168i	−0.967705	+	0.252084i	$0.918884\pi$
$888$		0			0
$889$			13.4748i			0.451931i
$890$		0			0
$891$		−	3.54643i		−	0.118810i
$892$		0			0
$893$	−6.27987			−0.210148
$894$		0			0
$895$		−	29.4752i		−	0.985247i
$896$		0			0
$897$	15.0906			0.503862
$898$		0			0
$899$	−30.6841			−1.02337
$900$		0			0
$901$	14.6537	−	10.7156i	0.488186	−	0.356987i
$902$		0			0
$903$	−12.7356			−0.423815
$904$		0			0
$905$	−13.5831			−0.451518
$906$		0			0
$907$			39.1586i			1.30024i	0.759831	+	0.650120i	$0.225282\pi$
$907$			39.1586i			1.30024i	−0.759831	+	0.650120i	$0.774718\pi$
$908$		0			0
$909$	11.8218			0.392103
$910$		0			0
$911$			17.4113i			0.576861i	0.957501	+	0.288430i	$0.0931334\pi$
$911$			17.4113i			0.576861i	−0.957501	+	0.288430i	$0.906867\pi$
$912$		0			0
$913$		−	37.9208i		−	1.25500i
$914$		0			0
$915$			55.5616i			1.83681i
$916$		0			0
$917$	14.3012			0.472267
$918$		0			0
$919$	−57.1727			−1.88595			−0.942977	−	0.332859i	$-0.891987\pi$
$919$	−57.1727			−1.88595			−0.942977	+	0.332859i	$0.891987\pi$
$920$		0			0
$921$		−	6.40450i		−	0.211035i
$922$		0			0
$923$			19.7184i			0.649040i
$924$		0			0
$925$		−	127.271i		−	4.18463i
$926$		0			0
$927$	6.02527			0.197896
$928$		0			0
$929$			6.09643i			0.200018i	0.994987	+	0.100009i	$0.0318871\pi$
$929$			6.09643i			0.200018i	−0.994987	+	0.100009i	$0.968113\pi$
$930$		0			0
$931$	1.49247			0.0489138
$932$		0			0
$933$	−5.57256			−0.182437
$934$		0			0
$935$	−47.5111	+	34.7426i	−1.55378	+	1.13620i
$936$		0			0
$937$	50.3510			1.64490			0.822448	−	0.568840i	$-0.192607\pi$
$937$	50.3510			1.64490			0.822448	+	0.568840i	$0.192607\pi$
$938$		0			0
$939$	6.23758			0.203556
$940$		0			0
$941$		−	12.9192i		−	0.421153i	−0.977577	−	0.210576i	$-0.932466\pi$
$941$		−	12.9192i		−	0.421153i	0.977577	−	0.210576i	$-0.0675341\pi$
$942$		0			0
$943$	−17.6730			−0.575512
$944$		0			0
$945$		−	4.02527i		−	0.130942i
$946$		0			0
$947$		−	4.01283i		−	0.130399i	−0.997872	−	0.0651997i	$-0.979232\pi$
$947$		−	4.01283i		−	0.130399i	0.997872	−	0.0651997i	$-0.0207685\pi$
$948$		0			0
$949$		−	0.856354i		−	0.0277984i
$950$		0			0
$951$	2.11223			0.0684938
$952$		0			0
$953$	3.65422			0.118372			0.0591859	−	0.998247i	$-0.481150\pi$
$953$	3.65422			0.118372			0.0591859	+	0.998247i	$0.481150\pi$
$954$		0			0
$955$		−	48.8199i		−	1.57977i
$956$		0			0
$957$		−	25.8619i		−	0.835997i
$958$		0			0
$959$			12.9099i			0.416881i
$960$		0			0
$961$	13.2953			0.428880
$962$		0			0
$963$		−	11.6790i		−	0.376352i
$964$		0			0
$965$	31.0970			1.00105
$966$		0			0
$967$	9.57044			0.307765			0.153882	−	0.988089i	$-0.450822\pi$
$967$	9.57044			0.307765			0.153882	+	0.988089i	$0.450822\pi$
$968$		0			0
$969$	3.63230	+	4.96724i	0.116686	+	0.159571i
$970$		0			0
$971$	50.4539			1.61914			0.809571	−	0.587022i	$-0.199700\pi$
$971$	50.4539			1.61914			0.809571	+	0.587022i	$0.199700\pi$
$972$		0			0
$973$	21.7855			0.698412
$974$		0			0
$975$			42.5697i			1.36332i
$976$		0			0
$977$	14.7281			0.471194			0.235597	−	0.971851i	$-0.424295\pi$
$977$	14.7281			0.471194			0.235597	+	0.971851i	$0.424295\pi$
$978$		0			0
$979$		−	10.9181i		−	0.348943i
$980$		0			0
$981$		−	8.31828i		−	0.265582i
$982$		0			0
$983$		−	34.7957i		−	1.10981i	−0.831913	−	0.554905i	$-0.812754\pi$
$983$		−	34.7957i		−	1.10981i	0.831913	−	0.554905i	$-0.187246\pi$
$984$		0			0
$985$	12.5129			0.398693
$986$		0			0
$987$	4.20770			0.133932
$988$		0			0
$989$		−	50.5771i		−	1.60826i
$990$		0			0
$991$		−	45.7910i		−	1.45460i	−0.686320	−	0.727299i	$-0.740775\pi$
$991$		−	45.7910i		−	1.45460i	0.686320	−	0.727299i	$-0.259225\pi$
$992$		0			0
$993$		−	13.6906i		−	0.434459i
$994$		0			0
$995$	−86.1914			−2.73245
$996$		0			0
$997$			27.7393i			0.878513i	0.898362	+	0.439256i	$0.144758\pi$
$997$			27.7393i			0.878513i	−0.898362	+	0.439256i	$0.855242\pi$
$998$		0			0
$999$	11.3606			0.359433

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1428.2.d.c.169.12	yes	12
3.2	odd	2		4284.2.d.f.3025.1		12
17.16	even	2	inner	1428.2.d.c.169.1	✓	12
51.50	odd	2		4284.2.d.f.3025.12		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1428.2.d.c.169.1	✓	12	17.16	even	2	inner
1428.2.d.c.169.12	yes	12	1.1	even	1	trivial
4284.2.d.f.3025.1		12	3.2	odd	2
4284.2.d.f.3025.12		12	51.50	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:	\( N \)	\(=\)	\( 1428 = 2^{2} \cdot 3 \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1428.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +4.02527i q^{5} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +4.02527i q^{5} -1.00000i q^{7} -1.00000 q^{9} -3.54643i q^{11} -3.79991 q^{13} -4.02527 q^{15} +(-3.32819 + 2.43375i) q^{17} -1.49247 q^{19} +1.00000 q^{21} +3.97132i q^{23} -11.2028 q^{25} -1.00000i q^{27} -7.29238i q^{29} -4.20770i q^{31} +3.54643 q^{33} +4.02527 q^{35} +11.3606i q^{37} -3.79991i q^{39} +4.45016i q^{41} -12.7356 q^{43} -4.02527i q^{45} +4.20770 q^{47} -1.00000 q^{49} +(-2.43375 - 3.32819i) q^{51} -4.40291 q^{53} +14.2753 q^{55} -1.49247i q^{57} +6.93570 q^{59} -13.8032i q^{61} +1.00000i q^{63} -15.2957i q^{65} -3.17493 q^{67} -3.97132 q^{69} -5.18919i q^{71} +0.225362i q^{73} -11.2028i q^{75} -3.54643 q^{77} -5.64764i q^{79} +1.00000 q^{81} +10.6927 q^{83} +(-9.79650 - 13.3969i) q^{85} +7.29238 q^{87} +3.07861 q^{89} +3.79991i q^{91} +4.20770 q^{93} -6.00761i q^{95} +15.8757i q^{97} +3.54643i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^9	\( 12 q - 12 q^{9} - 6 q^{13} + 2 q^{15} - 2 q^{17} + 6 q^{19} + 12 q^{21} - 26 q^{25} + 10 q^{33} - 2 q^{35} - 18 q^{43} - 20 q^{47} - 12 q^{49} - 2 q^{51} + 16 q^{53} + 22 q^{55} - 12 q^{59} + 32 q^{67} - 6 q^{69} - 10 q^{77} + 12 q^{81} - 16 q^{83} + 14 q^{85} + 24 q^{87} + 4 q^{89} - 20 q^{93}+O(q^{100}) \) 12 * q - 12 * q^9 - 6 * q^13 + 2 * q^15 - 2 * q^17 + 6 * q^19 + 12 * q^21 - 26 * q^25 + 10 * q^33 - 2 * q^35 - 18 * q^43 - 20 * q^47 - 12 * q^49 - 2 * q^51 + 16 * q^53 + 22 * q^55 - 12 * q^59 + 32 * q^67 - 6 * q^69 - 10 * q^77 + 12 * q^81 - 16 * q^83 + 14 * q^85 + 24 * q^87 + 4 * q^89 - 20 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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