LMFDB - Embedded newform 5700.2.f.m.3649.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5700,2,Mod(3649,5700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5700, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5700.3649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5700 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5700.f (of order $2$, degree $1$, not 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:	$45.5147291521$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{33})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 17x^{2} + 64 $ x^4 + 17*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 228)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3649.2
Root			$3.37228i$ of defining polynomial
Character	$\chi$	$=$	5700.3649
Dual form			5700.2.f.m.3649.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +3.37228i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +3.37228i q^{7} -1.00000 q^{9} +1.37228 q^{11} -2.00000i q^{13} +1.37228i q^{17} -1.00000 q^{19} +3.37228 q^{21} +8.74456i q^{23} +1.00000i q^{27} -2.74456 q^{29} -6.74456 q^{31} -1.37228i q^{33} +4.74456i q^{37} -2.00000 q^{39} -3.37228i q^{43} -13.3723i q^{47} -4.37228 q^{49} +1.37228 q^{51} +2.74456i q^{53} +1.00000i q^{57} -2.62772 q^{61} -3.37228i q^{63} -9.48913i q^{67} +8.74456 q^{69} -12.0000 q^{71} +5.37228i q^{73} +4.62772i q^{77} -8.00000 q^{79} +1.00000 q^{81} -8.74456i q^{83} +2.74456i q^{87} -14.7446 q^{89} +6.74456 q^{91} +6.74456i q^{93} +14.0000i q^{97} -1.37228 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^9	$ 4 q - 4 q^{9} - 6 q^{11} - 4 q^{19} + 2 q^{21} + 12 q^{29} - 4 q^{31} - 8 q^{39} - 6 q^{49} - 6 q^{51} - 22 q^{61} + 12 q^{69} - 48 q^{71} - 32 q^{79} + 4 q^{81} - 36 q^{89} + 4 q^{91} + 6 q^{99}+O(q^{100}) $ 4 * q - 4 * q^9 - 6 * q^11 - 4 * q^19 + 2 * q^21 + 12 * q^29 - 4 * q^31 - 8 * q^39 - 6 * q^49 - 6 * q^51 - 22 * q^61 + 12 * q^69 - 48 * q^71 - 32 * q^79 + 4 * q^81 - 36 * q^89 + 4 * q^91 + 6 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5700\mathbb{Z}\right)^\times$.

$n$	$1901$	$2851$	$3877$	$4201$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.37228i	1.27460i	0.770615	+	0.637301i	$0.219949\pi$
$7$	3.37228i	1.27460i	−0.770615	+	0.637301i	$0.780051\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	1.37228	0.413758	0.206879	−	0.978366i	$-0.433669\pi$
$11$	1.37228	0.413758	0.206879	+	0.978366i	$0.433669\pi$
$12$	0	0
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.37228i	0.332827i	0.986056	+	0.166414i	$0.0532187\pi$
$17$	1.37228i	0.332827i	−0.986056	+	0.166414i	$0.946781\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	3.37228	0.735892
$22$	0	0
$23$	8.74456i	1.82337i	0.410893	+	0.911684i	$0.365217\pi$
$23$	8.74456i	1.82337i	−0.410893	+	0.911684i	$0.634783\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	−2.74456	−0.509652	−0.254826	−	0.966987i	$-0.582018\pi$
$29$	−2.74456	−0.509652	−0.254826	+	0.966987i	$0.582018\pi$
$30$	0	0
$31$	−6.74456	−1.21136	−0.605680	−	0.795709i	$-0.707099\pi$
$31$	−6.74456	−1.21136	−0.605680	+	0.795709i	$0.707099\pi$
$32$	0	0
$33$	− 1.37228i	− 0.238884i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.74456i	0.780001i	0.920815	+	0.390001i	$0.127525\pi$
$37$	4.74456i	0.780001i	−0.920815	+	0.390001i	$0.872475\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	− 3.37228i	− 0.514268i	−0.966376	−	0.257134i	$-0.917222\pi$
$43$	− 3.37228i	− 0.514268i	0.966376	−	0.257134i	$-0.0827782\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 13.3723i	− 1.95055i	−0.221000	−	0.975274i	$-0.570932\pi$
$47$	− 13.3723i	− 1.95055i	0.221000	−	0.975274i	$-0.429068\pi$
$48$	0	0
$49$	−4.37228	−0.624612
$50$	0	0
$51$	1.37228	0.192158
$52$	0	0
$53$	2.74456i	0.376995i	0.982074	+	0.188497i	$0.0603617\pi$
$53$	2.74456i	0.376995i	−0.982074	+	0.188497i	$0.939638\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.00000i	0.132453i
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−2.62772	−0.336445	−0.168222	−	0.985749i	$-0.553803\pi$
$61$	−2.62772	−0.336445	−0.168222	+	0.985749i	$0.553803\pi$
$62$	0	0
$63$	− 3.37228i	− 0.424868i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 9.48913i	− 1.15928i	−0.814872	−	0.579641i	$-0.803193\pi$
$67$	− 9.48913i	− 1.15928i	0.814872	−	0.579641i	$-0.196807\pi$
$68$	0	0
$69$	8.74456	1.05272
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	0	0
$73$	5.37228i	0.628778i	0.949294	+	0.314389i	$0.101800\pi$
$73$	5.37228i	0.628778i	−0.949294	+	0.314389i	$0.898200\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.62772i	0.527377i
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 8.74456i	− 0.959840i	−0.877312	−	0.479920i	$-0.840666\pi$
$83$	− 8.74456i	− 0.959840i	0.877312	−	0.479920i	$-0.159334\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	2.74456i	0.294248i
$88$	0	0
$89$	−14.7446	−1.56292	−0.781460	−	0.623955i	$-0.785525\pi$
$89$	−14.7446	−1.56292	−0.781460	+	0.623955i	$0.785525\pi$
$90$	0	0
$91$	6.74456	0.707022
$92$	0	0
$93$	6.74456i	0.699379i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	14.0000i	1.42148i	0.703452	+	0.710742i	$0.251641\pi$
$97$	14.0000i	1.42148i	−0.703452	+	0.710742i	$0.748359\pi$
$98$	0	0
$99$	−1.37228	−0.137919
$100$	0	0
$101$	11.4891	1.14321	0.571605	−	0.820529i	$-0.306321\pi$
$101$	11.4891	1.14321	0.571605	+	0.820529i	$0.306321\pi$
$102$	0	0
$103$	1.25544i	0.123702i	0.998085	+	0.0618510i	$0.0197003\pi$
$103$	1.25544i	0.123702i	−0.998085	+	0.0618510i	$0.980300\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.7446i	1.42541i	0.701464	+	0.712705i	$0.252530\pi$
$107$	14.7446i	1.42541i	−0.701464	+	0.712705i	$0.747470\pi$
$108$	0	0
$109$	18.2337	1.74647	0.873235	−	0.487299i	$-0.162018\pi$
$109$	18.2337	1.74647	0.873235	+	0.487299i	$0.162018\pi$
$110$	0	0
$111$	4.74456	0.450334
$112$	0	0
$113$	− 14.7446i	− 1.38705i	−0.720432	−	0.693526i	$-0.756056\pi$
$113$	− 14.7446i	− 1.38705i	0.720432	−	0.693526i	$-0.243944\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.00000i	0.184900i
$118$	0	0
$119$	−4.62772	−0.424222
$120$	0	0
$121$	−9.11684	−0.828804
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 12.2337i	− 1.08556i	−0.839874	−	0.542782i	$-0.817371\pi$
$127$	− 12.2337i	− 1.08556i	0.839874	−	0.542782i	$-0.182629\pi$
$128$	0	0
$129$	−3.37228	−0.296913
$130$	0	0
$131$	−18.8614	−1.64793	−0.823964	−	0.566642i	$-0.808242\pi$
$131$	−18.8614	−1.64793	−0.823964	+	0.566642i	$0.808242\pi$
$132$	0	0
$133$	− 3.37228i	− 0.292414i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1.37228i	0.117242i	0.998280	+	0.0586210i	$0.0186703\pi$
$137$	1.37228i	0.117242i	−0.998280	+	0.0586210i	$0.981330\pi$
$138$	0	0
$139$	−15.3723	−1.30386	−0.651930	−	0.758279i	$-0.726040\pi$
$139$	−15.3723	−1.30386	−0.651930	+	0.758279i	$0.726040\pi$
$140$	0	0
$141$	−13.3723	−1.12615
$142$	0	0
$143$	− 2.74456i	− 0.229512i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	4.37228i	0.360620i
$148$	0	0
$149$	−1.37228	−0.112422	−0.0562108	−	0.998419i	$-0.517902\pi$
$149$	−1.37228	−0.112422	−0.0562108	+	0.998419i	$0.517902\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	0	0
$153$	− 1.37228i	− 0.110942i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	7.48913i	0.597697i	0.954301	+	0.298849i	$0.0966026\pi$
$157$	7.48913i	0.597697i	−0.954301	+	0.298849i	$0.903397\pi$
$158$	0	0
$159$	2.74456	0.217658
$160$	0	0
$161$	−29.4891	−2.32407
$162$	0	0
$163$	9.48913i	0.743246i	0.928384	+	0.371623i	$0.121199\pi$
$163$	9.48913i	0.743246i	−0.928384	+	0.371623i	$0.878801\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	20.2337i	1.56573i	0.622192	+	0.782865i	$0.286242\pi$
$167$	20.2337i	1.56573i	−0.622192	+	0.782865i	$0.713758\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	− 9.25544i	− 0.703678i	−0.936061	−	0.351839i	$-0.885556\pi$
$173$	− 9.25544i	− 0.703678i	0.936061	−	0.351839i	$-0.114444\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−14.7446	−1.10206	−0.551030	−	0.834485i	$-0.685765\pi$
$179$	−14.7446	−1.10206	−0.551030	+	0.834485i	$0.685765\pi$
$180$	0	0
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	0	0
$183$	2.62772i	0.194247i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1.88316i	0.137710i
$188$	0	0
$189$	−3.37228	−0.245297
$190$	0	0
$191$	−10.6277	−0.768995	−0.384497	−	0.923126i	$-0.625625\pi$
$191$	−10.6277	−0.768995	−0.384497	+	0.923126i	$0.625625\pi$
$192$	0	0
$193$	− 16.7446i	− 1.20530i	−0.798006	−	0.602650i	$-0.794112\pi$
$193$	− 16.7446i	− 1.20530i	0.798006	−	0.602650i	$-0.205888\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 11.4891i	− 0.818566i	−0.912407	−	0.409283i	$-0.865779\pi$
$197$	− 11.4891i	− 0.818566i	0.912407	−	0.409283i	$-0.134221\pi$
$198$	0	0
$199$	14.1168	1.00072	0.500358	−	0.865818i	$-0.333202\pi$
$199$	14.1168	1.00072	0.500358	+	0.865818i	$0.333202\pi$
$200$	0	0
$201$	−9.48913	−0.669311
$202$	0	0
$203$	− 9.25544i	− 0.649604i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 8.74456i	− 0.607789i
$208$	0	0
$209$	−1.37228	−0.0949227
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	0	0
$213$	12.0000i	0.822226i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 22.7446i	− 1.54400i
$218$	0	0
$219$	5.37228	0.363025
$220$	0	0
$221$	2.74456	0.184619
$222$	0	0
$223$	− 13.4891i	− 0.903299i	−0.892196	−	0.451649i	$-0.850836\pi$
$223$	− 13.4891i	− 0.903299i	0.892196	−	0.451649i	$-0.149164\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 8.23369i	− 0.546489i	−0.961945	−	0.273245i	$-0.911903\pi$
$227$	− 8.23369i	− 0.546489i	0.961945	−	0.273245i	$-0.0880968\pi$
$228$	0	0
$229$	5.37228	0.355010	0.177505	−	0.984120i	$-0.443197\pi$
$229$	5.37228	0.355010	0.177505	+	0.984120i	$0.443197\pi$
$230$	0	0
$231$	4.62772	0.304482
$232$	0	0
$233$	1.37228i	0.0899011i	0.998989	+	0.0449506i	$0.0143130\pi$
$233$	1.37228i	0.0899011i	−0.998989	+	0.0449506i	$0.985687\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	8.00000i	0.519656i
$238$	0	0
$239$	−18.8614	−1.22004	−0.610021	−	0.792385i	$-0.708839\pi$
$239$	−18.8614	−1.22004	−0.610021	+	0.792385i	$0.708839\pi$
$240$	0	0
$241$	−0.744563	−0.0479615	−0.0239807	−	0.999712i	$-0.507634\pi$
$241$	−0.744563	−0.0479615	−0.0239807	+	0.999712i	$0.507634\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.00000i	0.127257i
$248$	0	0
$249$	−8.74456	−0.554164
$250$	0	0
$251$	−1.37228	−0.0866176	−0.0433088	−	0.999062i	$-0.513790\pi$
$251$	−1.37228	−0.0866176	−0.0433088	+	0.999062i	$0.513790\pi$
$252$	0	0
$253$	12.0000i	0.754434i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 17.4891i	− 1.09094i	−0.838130	−	0.545471i	$-0.816351\pi$
$257$	− 17.4891i	− 1.09094i	0.838130	−	0.545471i	$-0.183649\pi$
$258$	0	0
$259$	−16.0000	−0.994192
$260$	0	0
$261$	2.74456	0.169884
$262$	0	0
$263$	30.8614i	1.90300i	0.307655	+	0.951498i	$0.400456\pi$
$263$	30.8614i	1.90300i	−0.307655	+	0.951498i	$0.599544\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	14.7446i	0.902353i
$268$	0	0
$269$	−14.7446	−0.898992	−0.449496	−	0.893282i	$-0.648396\pi$
$269$	−14.7446	−0.898992	−0.449496	+	0.893282i	$0.648396\pi$
$270$	0	0
$271$	−4.00000	−0.242983	−0.121491	−	0.992592i	$-0.538768\pi$
$271$	−4.00000	−0.242983	−0.121491	+	0.992592i	$0.538768\pi$
$272$	0	0
$273$	− 6.74456i	− 0.408200i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	24.1168i	1.44904i	0.689253	+	0.724520i	$0.257939\pi$
$277$	24.1168i	1.44904i	−0.689253	+	0.724520i	$0.742061\pi$
$278$	0	0
$279$	6.74456	0.403786
$280$	0	0
$281$	−26.7446	−1.59545	−0.797723	−	0.603023i	$-0.793963\pi$
$281$	−26.7446	−1.59545	−0.797723	+	0.603023i	$0.793963\pi$
$282$	0	0
$283$	26.1168i	1.55249i	0.630434	+	0.776243i	$0.282877\pi$
$283$	26.1168i	1.55249i	−0.630434	+	0.776243i	$0.717123\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	15.1168	0.889226
$290$	0	0
$291$	14.0000	0.820695
$292$	0	0
$293$	− 29.4891i	− 1.72277i	−0.507950	−	0.861387i	$-0.669597\pi$
$293$	− 29.4891i	− 1.72277i	0.507950	−	0.861387i	$-0.330403\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.37228i	0.0796278i
$298$	0	0
$299$	17.4891	1.01142
$300$	0	0
$301$	11.3723	0.655487
$302$	0	0
$303$	− 11.4891i	− 0.660033i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	20.0000i	1.14146i	0.821138	+	0.570730i	$0.193340\pi$
$307$	20.0000i	1.14146i	−0.821138	+	0.570730i	$0.806660\pi$
$308$	0	0
$309$	1.25544	0.0714193
$310$	0	0
$311$	−18.8614	−1.06953	−0.534766	−	0.845000i	$-0.679600\pi$
$311$	−18.8614	−1.06953	−0.534766	+	0.845000i	$0.679600\pi$
$312$	0	0
$313$	− 14.0000i	− 0.791327i	−0.918396	−	0.395663i	$-0.870515\pi$
$313$	− 14.0000i	− 0.791327i	0.918396	−	0.395663i	$-0.129485\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	29.4891i	1.65627i	0.560526	+	0.828137i	$0.310599\pi$
$317$	29.4891i	1.65627i	−0.560526	+	0.828137i	$0.689401\pi$
$318$	0	0
$319$	−3.76631	−0.210873
$320$	0	0
$321$	14.7446	0.822961
$322$	0	0
$323$	− 1.37228i	− 0.0763558i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 18.2337i	− 1.00833i
$328$	0	0
$329$	45.0951	2.48617
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	0	0
$333$	− 4.74456i	− 0.260000i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	19.4891i	1.06164i	0.847484	+	0.530820i	$0.178116\pi$
$337$	19.4891i	1.06164i	−0.847484	+	0.530820i	$0.821884\pi$
$338$	0	0
$339$	−14.7446	−0.800815
$340$	0	0
$341$	−9.25544	−0.501210
$342$	0	0
$343$	8.86141i	0.478471i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	16.1168i	0.865198i	0.901587	+	0.432599i	$0.142403\pi$
$347$	16.1168i	0.865198i	−0.901587	+	0.432599i	$0.857597\pi$
$348$	0	0
$349$	−18.6277	−0.997119	−0.498559	−	0.866856i	$-0.666137\pi$
$349$	−18.6277	−0.997119	−0.498559	+	0.866856i	$0.666137\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	0	0
$353$	− 30.0000i	− 1.59674i	−0.602168	−	0.798369i	$-0.705696\pi$
$353$	− 30.0000i	− 1.59674i	0.602168	−	0.798369i	$-0.294304\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	4.62772i	0.244925i
$358$	0	0
$359$	5.13859	0.271205	0.135602	−	0.990763i	$-0.456703\pi$
$359$	5.13859	0.271205	0.135602	+	0.990763i	$0.456703\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	9.11684i	0.478510i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	25.4891i	1.33052i	0.746611	+	0.665261i	$0.231680\pi$
$367$	25.4891i	1.33052i	−0.746611	+	0.665261i	$0.768320\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−9.25544	−0.480518
$372$	0	0
$373$	30.2337i	1.56544i	0.622373	+	0.782721i	$0.286169\pi$
$373$	30.2337i	1.56544i	−0.622373	+	0.782721i	$0.713831\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5.48913i	0.282704i
$378$	0	0
$379$	−21.7228	−1.11583	−0.557913	−	0.829899i	$-0.688398\pi$
$379$	−21.7228	−1.11583	−0.557913	+	0.829899i	$0.688398\pi$
$380$	0	0
$381$	−12.2337	−0.626751
$382$	0	0
$383$	− 8.23369i	− 0.420722i	−0.977624	−	0.210361i	$-0.932536\pi$
$383$	− 8.23369i	− 0.420722i	0.977624	−	0.210361i	$-0.0674639\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.37228i	0.171423i
$388$	0	0
$389$	25.3723	1.28643	0.643213	−	0.765687i	$-0.277601\pi$
$389$	25.3723	1.28643	0.643213	+	0.765687i	$0.277601\pi$
$390$	0	0
$391$	−12.0000	−0.606866
$392$	0	0
$393$	18.8614i	0.951432i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	29.6060i	1.48588i	0.669357	+	0.742941i	$0.266569\pi$
$397$	29.6060i	1.48588i	−0.669357	+	0.742941i	$0.733431\pi$
$398$	0	0
$399$	−3.37228	−0.168825
$400$	0	0
$401$	5.48913	0.274114	0.137057	−	0.990563i	$-0.456236\pi$
$401$	5.48913	0.274114	0.137057	+	0.990563i	$0.456236\pi$
$402$	0	0
$403$	13.4891i	0.671941i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6.51087i	0.322732i
$408$	0	0
$409$	−30.4674	−1.50651	−0.753257	−	0.657726i	$-0.771519\pi$
$409$	−30.4674	−1.50651	−0.753257	+	0.657726i	$0.771519\pi$
$410$	0	0
$411$	1.37228	0.0676896
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	15.3723i	0.752784i
$418$	0	0
$419$	19.7228	0.963522	0.481761	−	0.876303i	$-0.339997\pi$
$419$	19.7228	0.963522	0.481761	+	0.876303i	$0.339997\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	0	0
$423$	13.3723i	0.650183i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 8.86141i	− 0.428834i
$428$	0	0
$429$	−2.74456	−0.132509
$430$	0	0
$431$	−2.74456	−0.132201	−0.0661005	−	0.997813i	$-0.521056\pi$
$431$	−2.74456	−0.132201	−0.0661005	+	0.997813i	$0.521056\pi$
$432$	0	0
$433$	− 7.48913i	− 0.359904i	−0.983675	−	0.179952i	$-0.942406\pi$
$433$	− 7.48913i	− 0.359904i	0.983675	−	0.179952i	$-0.0575943\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 8.74456i	− 0.418309i
$438$	0	0
$439$	−4.23369	−0.202063	−0.101031	−	0.994883i	$-0.532214\pi$
$439$	−4.23369	−0.202063	−0.101031	+	0.994883i	$0.532214\pi$
$440$	0	0
$441$	4.37228	0.208204
$442$	0	0
$443$	13.3723i	0.635336i	0.948202	+	0.317668i	$0.102900\pi$
$443$	13.3723i	0.635336i	−0.948202	+	0.317668i	$0.897100\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	1.37228i	0.0649067i
$448$	0	0
$449$	22.9783	1.08441	0.542205	−	0.840246i	$-0.317589\pi$
$449$	22.9783	1.08441	0.542205	+	0.840246i	$0.317589\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	− 8.00000i	− 0.375873i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	14.8614i	0.695187i	0.937645	+	0.347594i	$0.113001\pi$
$457$	14.8614i	0.695187i	−0.937645	+	0.347594i	$0.886999\pi$
$458$	0	0
$459$	−1.37228	−0.0640526
$460$	0	0
$461$	12.3505	0.575222	0.287611	−	0.957747i	$-0.407139\pi$
$461$	12.3505	0.575222	0.287611	+	0.957747i	$0.407139\pi$
$462$	0	0
$463$	34.3505i	1.59640i	0.602389	+	0.798202i	$0.294215\pi$
$463$	34.3505i	1.59640i	−0.602389	+	0.798202i	$0.705785\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4.11684i	0.190505i	0.995453	+	0.0952524i	$0.0303658\pi$
$467$	4.11684i	0.190505i	−0.995453	+	0.0952524i	$0.969634\pi$
$468$	0	0
$469$	32.0000	1.47762
$470$	0	0
$471$	7.48913	0.345081
$472$	0	0
$473$	− 4.62772i	− 0.212783i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 2.74456i	− 0.125665i
$478$	0	0
$479$	−38.2337	−1.74694	−0.873471	−	0.486876i	$-0.838136\pi$
$479$	−38.2337	−1.74694	−0.873471	+	0.486876i	$0.838136\pi$
$480$	0	0
$481$	9.48913	0.432667
$482$	0	0
$483$	29.4891i	1.34180i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	8.00000i	0.362515i	0.983436	+	0.181257i	$0.0580167\pi$
$487$	8.00000i	0.362515i	−0.983436	+	0.181257i	$0.941983\pi$
$488$	0	0
$489$	9.48913	0.429113
$490$	0	0
$491$	−27.2554	−1.23002	−0.615010	−	0.788519i	$-0.710848\pi$
$491$	−27.2554	−1.23002	−0.615010	+	0.788519i	$0.710848\pi$
$492$	0	0
$493$	− 3.76631i	− 0.169626i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 40.4674i	− 1.81521i
$498$	0	0
$499$	34.3505	1.53774	0.768870	−	0.639405i	$-0.220819\pi$
$499$	34.3505	1.53774	0.768870	+	0.639405i	$0.220819\pi$
$500$	0	0
$501$	20.2337	0.903975
$502$	0	0
$503$	− 3.25544i	− 0.145153i	−0.997363	−	0.0725764i	$-0.976878\pi$
$503$	− 3.25544i	− 0.145153i	0.997363	−	0.0725764i	$-0.0231221\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 9.00000i	− 0.399704i
$508$	0	0
$509$	14.7446	0.653541	0.326771	−	0.945104i	$-0.394040\pi$
$509$	14.7446	0.653541	0.326771	+	0.945104i	$0.394040\pi$
$510$	0	0
$511$	−18.1168	−0.801442
$512$	0	0
$513$	− 1.00000i	− 0.0441511i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 18.3505i	− 0.807055i
$518$	0	0
$519$	−9.25544	−0.406269
$520$	0	0
$521$	5.48913	0.240483	0.120241	−	0.992745i	$-0.461633\pi$
$521$	5.48913	0.240483	0.120241	+	0.992745i	$0.461633\pi$
$522$	0	0
$523$	18.7446i	0.819642i	0.912166	+	0.409821i	$0.134409\pi$
$523$	18.7446i	0.819642i	−0.912166	+	0.409821i	$0.865591\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 9.25544i	− 0.403173i
$528$	0	0
$529$	−53.4674	−2.32467
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	14.7446i	0.636275i
$538$	0	0
$539$	−6.00000	−0.258438
$540$	0	0
$541$	26.8614	1.15486	0.577431	−	0.816439i	$-0.304055\pi$
$541$	26.8614	1.15486	0.577431	+	0.816439i	$0.304055\pi$
$542$	0	0
$543$	22.0000i	0.944110i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 18.7446i	− 0.801460i	−0.916196	−	0.400730i	$-0.868757\pi$
$547$	− 18.7446i	− 0.801460i	0.916196	−	0.400730i	$-0.131243\pi$
$548$	0	0
$549$	2.62772	0.112148
$550$	0	0
$551$	2.74456	0.116922
$552$	0	0
$553$	− 26.9783i	− 1.14723i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	6.86141i	0.290727i	0.989378	+	0.145364i	$0.0464352\pi$
$557$	6.86141i	0.290727i	−0.989378	+	0.145364i	$0.953565\pi$
$558$	0	0
$559$	−6.74456	−0.285265
$560$	0	0
$561$	1.88316	0.0795069
$562$	0	0
$563$	− 46.9783i	− 1.97990i	−0.141428	−	0.989949i	$-0.545169\pi$
$563$	− 46.9783i	− 1.97990i	0.141428	−	0.989949i	$-0.454831\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	3.37228i	0.141623i
$568$	0	0
$569$	21.2554	0.891074	0.445537	−	0.895263i	$-0.353013\pi$
$569$	21.2554	0.891074	0.445537	+	0.895263i	$0.353013\pi$
$570$	0	0
$571$	18.9783	0.794215	0.397108	−	0.917772i	$-0.370014\pi$
$571$	18.9783	0.794215	0.397108	+	0.917772i	$0.370014\pi$
$572$	0	0
$573$	10.6277i	0.443979i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	3.88316i	0.161658i	0.996728	+	0.0808290i	$0.0257568\pi$
$577$	3.88316i	0.161658i	−0.996728	+	0.0808290i	$0.974243\pi$
$578$	0	0
$579$	−16.7446	−0.695880
$580$	0	0
$581$	29.4891	1.22342
$582$	0	0
$583$	3.76631i	0.155985i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	33.6060i	1.38707i	0.720424	+	0.693533i	$0.243947\pi$
$587$	33.6060i	1.38707i	−0.720424	+	0.693533i	$0.756053\pi$
$588$	0	0
$589$	6.74456	0.277905
$590$	0	0
$591$	−11.4891	−0.472599
$592$	0	0
$593$	− 28.9783i	− 1.18999i	−0.803728	−	0.594997i	$-0.797153\pi$
$593$	− 28.9783i	− 1.18999i	0.803728	−	0.594997i	$-0.202847\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 14.1168i	− 0.577764i
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	−12.7446	−0.519862	−0.259931	−	0.965627i	$-0.583700\pi$
$601$	−12.7446	−0.519862	−0.259931	+	0.965627i	$0.583700\pi$
$602$	0	0
$603$	9.48913i	0.386427i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 30.7446i	− 1.24788i	−0.781471	−	0.623942i	$-0.785530\pi$
$607$	− 30.7446i	− 1.24788i	0.781471	−	0.623942i	$-0.214470\pi$
$608$	0	0
$609$	−9.25544	−0.375049
$610$	0	0
$611$	−26.7446	−1.08197
$612$	0	0
$613$	11.8832i	0.479956i	0.970778	+	0.239978i	$0.0771403\pi$
$613$	11.8832i	0.479956i	−0.970778	+	0.239978i	$0.922860\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 28.1168i	− 1.13194i	−0.824425	−	0.565971i	$-0.808502\pi$
$617$	− 28.1168i	− 1.13194i	0.824425	−	0.565971i	$-0.191498\pi$
$618$	0	0
$619$	38.9783	1.56667	0.783334	−	0.621601i	$-0.213517\pi$
$619$	38.9783	1.56667	0.783334	+	0.621601i	$0.213517\pi$
$620$	0	0
$621$	−8.74456	−0.350907
$622$	0	0
$623$	− 49.7228i	− 1.99210i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1.37228i	0.0548036i
$628$	0	0
$629$	−6.51087	−0.259606
$630$	0	0
$631$	32.8614	1.30819	0.654096	−	0.756412i	$-0.273049\pi$
$631$	32.8614	1.30819	0.654096	+	0.756412i	$0.273049\pi$
$632$	0	0
$633$	4.00000i	0.158986i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	8.74456i	0.346472i
$638$	0	0
$639$	12.0000	0.474713
$640$	0	0
$641$	−20.2337	−0.799183	−0.399591	−	0.916693i	$-0.630848\pi$
$641$	−20.2337	−0.799183	−0.399591	+	0.916693i	$0.630848\pi$
$642$	0	0
$643$	− 14.3505i	− 0.565930i	−0.959130	−	0.282965i	$-0.908682\pi$
$643$	− 14.3505i	− 0.565930i	0.959130	−	0.282965i	$-0.0913180\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	6.86141i	0.269750i	0.990863	+	0.134875i	$0.0430632\pi$
$647$	6.86141i	0.269750i	−0.990863	+	0.134875i	$0.956937\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−22.7446	−0.891430
$652$	0	0
$653$	9.60597i	0.375911i	0.982178	+	0.187955i	$0.0601860\pi$
$653$	9.60597i	0.375911i	−0.982178	+	0.187955i	$0.939814\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 5.37228i	− 0.209593i
$658$	0	0
$659$	−2.74456	−0.106913	−0.0534565	−	0.998570i	$-0.517024\pi$
$659$	−2.74456	−0.106913	−0.0534565	+	0.998570i	$0.517024\pi$
$660$	0	0
$661$	10.2337	0.398044	0.199022	−	0.979995i	$-0.436223\pi$
$661$	10.2337	0.398044	0.199022	+	0.979995i	$0.436223\pi$
$662$	0	0
$663$	− 2.74456i	− 0.106590i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 24.0000i	− 0.929284i
$668$	0	0
$669$	−13.4891	−0.521520
$670$	0	0
$671$	−3.60597	−0.139207
$672$	0	0
$673$	1.76631i	0.0680863i	0.999420	+	0.0340432i	$0.0108384\pi$
$673$	1.76631i	0.0680863i	−0.999420	+	0.0340432i	$0.989162\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 2.74456i	− 0.105482i	−0.998608	−	0.0527411i	$-0.983204\pi$
$677$	− 2.74456i	− 0.105482i	0.998608	−	0.0527411i	$-0.0167958\pi$
$678$	0	0
$679$	−47.2119	−1.81183
$680$	0	0
$681$	−8.23369	−0.315516
$682$	0	0
$683$	− 9.25544i	− 0.354149i	−0.984197	−	0.177075i	$-0.943337\pi$
$683$	− 9.25544i	− 0.354149i	0.984197	−	0.177075i	$-0.0566634\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 5.37228i	− 0.204965i
$688$	0	0
$689$	5.48913	0.209119
$690$	0	0
$691$	−22.3505	−0.850254	−0.425127	−	0.905134i	$-0.639771\pi$
$691$	−22.3505	−0.850254	−0.425127	+	0.905134i	$0.639771\pi$
$692$	0	0
$693$	− 4.62772i	− 0.175792i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	1.37228	0.0519044
$700$	0	0
$701$	35.4891	1.34041	0.670203	−	0.742178i	$-0.266207\pi$
$701$	35.4891	1.34041	0.670203	+	0.742178i	$0.266207\pi$
$702$	0	0
$703$	− 4.74456i	− 0.178945i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	38.7446i	1.45714i
$708$	0	0
$709$	−18.4674	−0.693557	−0.346778	−	0.937947i	$-0.612724\pi$
$709$	−18.4674	−0.693557	−0.346778	+	0.937947i	$0.612724\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	0	0
$713$	− 58.9783i	− 2.20875i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	18.8614i	0.704392i
$718$	0	0
$719$	28.1168	1.04858	0.524291	−	0.851539i	$-0.324331\pi$
$719$	28.1168	1.04858	0.524291	+	0.851539i	$0.324331\pi$
$720$	0	0
$721$	−4.23369	−0.157671
$722$	0	0
$723$	0.744563i	0.0276906i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	46.5842i	1.72771i	0.503738	+	0.863857i	$0.331958\pi$
$727$	46.5842i	1.72771i	−0.503738	+	0.863857i	$0.668042\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	4.62772	0.171162
$732$	0	0
$733$	− 2.00000i	− 0.0738717i	−0.999318	−	0.0369358i	$-0.988240\pi$
$733$	− 2.00000i	− 0.0738717i	0.999318	−	0.0369358i	$-0.0117597\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 13.0217i	− 0.479662i
$738$	0	0
$739$	−4.39403	−0.161637	−0.0808185	−	0.996729i	$-0.525753\pi$
$739$	−4.39403	−0.161637	−0.0808185	+	0.996729i	$0.525753\pi$
$740$	0	0
$741$	2.00000	0.0734718
$742$	0	0
$743$	28.4674i	1.04437i	0.852833	+	0.522183i	$0.174882\pi$
$743$	28.4674i	1.04437i	−0.852833	+	0.522183i	$0.825118\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	8.74456i	0.319947i
$748$	0	0
$749$	−49.7228	−1.81683
$750$	0	0
$751$	42.9783	1.56830	0.784149	−	0.620572i	$-0.213100\pi$
$751$	42.9783	1.56830	0.784149	+	0.620572i	$0.213100\pi$
$752$	0	0
$753$	1.37228i	0.0500087i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 31.0951i	− 1.13017i	−0.825033	−	0.565085i	$-0.808843\pi$
$757$	− 31.0951i	− 1.13017i	0.825033	−	0.565085i	$-0.191157\pi$
$758$	0	0
$759$	12.0000	0.435572
$760$	0	0
$761$	−44.5842	−1.61618	−0.808088	−	0.589061i	$-0.799498\pi$
$761$	−44.5842	−1.61618	−0.808088	+	0.589061i	$0.799498\pi$
$762$	0	0
$763$	61.4891i	2.22606i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−3.88316	−0.140030	−0.0700151	−	0.997546i	$-0.522305\pi$
$769$	−3.88316	−0.140030	−0.0700151	+	0.997546i	$0.522305\pi$
$770$	0	0
$771$	−17.4891	−0.629855
$772$	0	0
$773$	− 18.5109i	− 0.665790i	−0.942964	−	0.332895i	$-0.891975\pi$
$773$	− 18.5109i	− 0.665790i	0.942964	−	0.332895i	$-0.108025\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	16.0000i	0.573997i
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−16.4674	−0.589249
$782$	0	0
$783$	− 2.74456i	− 0.0980827i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1.48913i	0.0530816i	0.999648	+	0.0265408i	$0.00844919\pi$
$787$	1.48913i	0.0530816i	−0.999648	+	0.0265408i	$0.991551\pi$
$788$	0	0
$789$	30.8614	1.09870
$790$	0	0
$791$	49.7228	1.76794
$792$	0	0
$793$	5.25544i	0.186626i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 29.4891i	− 1.04456i	−0.852775	−	0.522279i	$-0.825082\pi$
$797$	− 29.4891i	− 1.04456i	0.852775	−	0.522279i	$-0.174918\pi$
$798$	0	0
$799$	18.3505	0.649195
$800$	0	0
$801$	14.7446	0.520974
$802$	0	0
$803$	7.37228i	0.260162i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	14.7446i	0.519033i
$808$	0	0
$809$	25.3723	0.892042	0.446021	−	0.895023i	$-0.352841\pi$
$809$	25.3723	0.892042	0.446021	+	0.895023i	$0.352841\pi$
$810$	0	0
$811$	−0.233688	−0.00820589	−0.00410295	−	0.999992i	$-0.501306\pi$
$811$	−0.233688	−0.00820589	−0.00410295	+	0.999992i	$0.501306\pi$
$812$	0	0
$813$	4.00000i	0.140286i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	3.37228i	0.117981i
$818$	0	0
$819$	−6.74456	−0.235674
$820$	0	0
$821$	−44.5842	−1.55600	−0.778000	−	0.628264i	$-0.783766\pi$
$821$	−44.5842	−1.55600	−0.778000	+	0.628264i	$0.783766\pi$
$822$	0	0
$823$	− 14.3505i	− 0.500228i	−0.968216	−	0.250114i	$-0.919532\pi$
$823$	− 14.3505i	− 0.500228i	0.968216	−	0.250114i	$-0.0804681\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	5.48913i	0.190876i	0.995435	+	0.0954378i	$0.0304251\pi$
$827$	5.48913i	0.190876i	−0.995435	+	0.0954378i	$0.969575\pi$
$828$	0	0
$829$	42.2337	1.46684	0.733418	−	0.679778i	$-0.237924\pi$
$829$	42.2337	1.46684	0.733418	+	0.679778i	$0.237924\pi$
$830$	0	0
$831$	24.1168	0.836604
$832$	0	0
$833$	− 6.00000i	− 0.207888i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 6.74456i	− 0.233126i
$838$	0	0
$839$	2.74456	0.0947528	0.0473764	−	0.998877i	$-0.484914\pi$
$839$	2.74456	0.0947528	0.0473764	+	0.998877i	$0.484914\pi$
$840$	0	0
$841$	−21.4674	−0.740254
$842$	0	0
$843$	26.7446i	0.921132i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 30.7446i	− 1.05640i
$848$	0	0
$849$	26.1168	0.896328
$850$	0	0
$851$	−41.4891	−1.42223
$852$	0	0
$853$	− 8.51087i	− 0.291407i	−0.989328	−	0.145703i	$-0.953455\pi$
$853$	− 8.51087i	− 0.291407i	0.989328	−	0.145703i	$-0.0465445\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 34.9783i	− 1.19483i	−0.801931	−	0.597417i	$-0.796194\pi$
$857$	− 34.9783i	− 1.19483i	0.801931	−	0.597417i	$-0.203806\pi$
$858$	0	0
$859$	−4.39403	−0.149922	−0.0749612	−	0.997186i	$-0.523883\pi$
$859$	−4.39403	−0.149922	−0.0749612	+	0.997186i	$0.523883\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 46.9783i	− 1.59916i	−0.600561	−	0.799579i	$-0.705056\pi$
$863$	− 46.9783i	− 1.59916i	0.600561	−	0.799579i	$-0.294944\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 15.1168i	− 0.513395i
$868$	0	0
$869$	−10.9783	−0.372412
$870$	0	0
$871$	−18.9783	−0.643053
$872$	0	0
$873$	− 14.0000i	− 0.473828i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 39.4891i	− 1.33345i	−0.745302	−	0.666727i	$-0.767695\pi$
$877$	− 39.4891i	− 1.33345i	0.745302	−	0.666727i	$-0.232305\pi$
$878$	0	0
$879$	−29.4891	−0.994644
$880$	0	0
$881$	17.8397	0.601033	0.300517	−	0.953777i	$-0.402841\pi$
$881$	17.8397	0.601033	0.300517	+	0.953777i	$0.402841\pi$
$882$	0	0
$883$	− 50.3505i	− 1.69443i	−0.531250	−	0.847215i	$-0.678277\pi$
$883$	− 50.3505i	− 1.69443i	0.531250	−	0.847215i	$-0.321723\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 18.5109i	− 0.621534i	−0.950486	−	0.310767i	$-0.899414\pi$
$887$	− 18.5109i	− 0.621534i	0.950486	−	0.310767i	$-0.100586\pi$
$888$	0	0
$889$	41.2554	1.38366
$890$	0	0
$891$	1.37228	0.0459732
$892$	0	0
$893$	13.3723i	0.447486i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 17.4891i	− 0.583945i
$898$	0	0
$899$	18.5109	0.617372
$900$	0	0
$901$	−3.76631	−0.125474
$902$	0	0
$903$	− 11.3723i	− 0.378446i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	40.2337i	1.33594i	0.744189	+	0.667969i	$0.232836\pi$
$907$	40.2337i	1.33594i	−0.744189	+	0.667969i	$0.767164\pi$
$908$	0	0
$909$	−11.4891	−0.381070
$910$	0	0
$911$	−3.76631	−0.124783	−0.0623917	−	0.998052i	$-0.519873\pi$
$911$	−3.76631	−0.124783	−0.0623917	+	0.998052i	$0.519873\pi$
$912$	0	0
$913$	− 12.0000i	− 0.397142i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 63.6060i	− 2.10045i
$918$	0	0
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	0	0
$921$	20.0000	0.659022
$922$	0	0
$923$	24.0000i	0.789970i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 1.25544i	− 0.0412340i
$928$	0	0
$929$	11.4891	0.376946	0.188473	−	0.982078i	$-0.439646\pi$
$929$	11.4891	0.376946	0.188473	+	0.982078i	$0.439646\pi$
$930$	0	0
$931$	4.37228	0.143296
$932$	0	0
$933$	18.8614i	0.617495i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 57.8397i	− 1.88954i	−0.327735	−	0.944770i	$-0.606285\pi$
$937$	− 57.8397i	− 1.88954i	0.327735	−	0.944770i	$-0.393715\pi$
$938$	0	0
$939$	−14.0000	−0.456873
$940$	0	0
$941$	38.7446	1.26304	0.631518	−	0.775361i	$-0.282432\pi$
$941$	38.7446	1.26304	0.631518	+	0.775361i	$0.282432\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 43.7228i	− 1.42080i	−0.703798	−	0.710400i	$-0.748514\pi$
$947$	− 43.7228i	− 1.42080i	0.703798	−	0.710400i	$-0.251486\pi$
$948$	0	0
$949$	10.7446	0.348783
$950$	0	0
$951$	29.4891	0.956250
$952$	0	0
$953$	52.4674i	1.69959i	0.527117	+	0.849793i	$0.323273\pi$
$953$	52.4674i	1.69959i	−0.527117	+	0.849793i	$0.676727\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	3.76631i	0.121748i
$958$	0	0
$959$	−4.62772	−0.149437
$960$	0	0
$961$	14.4891	0.467391
$962$	0	0
$963$	− 14.7446i	− 0.475137i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 5.02175i	− 0.161489i	−0.996735	−	0.0807443i	$-0.974270\pi$
$967$	− 5.02175i	− 0.161489i	0.996735	−	0.0807443i	$-0.0257297\pi$
$968$	0	0
$969$	−1.37228	−0.0440840
$970$	0	0
$971$	−46.9783	−1.50760	−0.753802	−	0.657102i	$-0.771782\pi$
$971$	−46.9783	−1.50760	−0.753802	+	0.657102i	$0.771782\pi$
$972$	0	0
$973$	− 51.8397i	− 1.66190i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 49.7228i	− 1.59077i	−0.606102	−	0.795387i	$-0.707268\pi$
$977$	− 49.7228i	− 1.59077i	0.606102	−	0.795387i	$-0.292732\pi$
$978$	0	0
$979$	−20.2337	−0.646671
$980$	0	0
$981$	−18.2337	−0.582157
$982$	0	0
$983$	29.4891i	0.940557i	0.882518	+	0.470279i	$0.155847\pi$
$983$	29.4891i	0.940557i	−0.882518	+	0.470279i	$0.844153\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 45.0951i	− 1.43539i
$988$	0	0
$989$	29.4891	0.937700
$990$	0	0
$991$	53.9565	1.71398	0.856992	−	0.515329i	$-0.172330\pi$
$991$	53.9565	1.71398	0.856992	+	0.515329i	$0.172330\pi$
$992$	0	0
$993$	4.00000i	0.126936i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	36.1168i	1.14383i	0.820312	+	0.571916i	$0.193800\pi$
$997$	36.1168i	1.14383i	−0.820312	+	0.571916i	$0.806200\pi$
$998$	0	0
$999$	−4.74456	−0.150111

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5700.2.f.m.3649.2		4
5.2	odd	4		5700.2.a.t.1.1		2
5.3	odd	4		228.2.a.c.1.1	✓	2
5.4	even	2	inner	5700.2.f.m.3649.3		4
15.8	even	4		684.2.a.d.1.2		2
20.3	even	4		912.2.a.n.1.1		2
40.3	even	4		3648.2.a.bq.1.2		2
40.13	odd	4		3648.2.a.bk.1.2		2
60.23	odd	4		2736.2.a.y.1.2		2
95.18	even	4		4332.2.a.i.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
228.2.a.c.1.1	✓	2	5.3	odd	4
684.2.a.d.1.2		2	15.8	even	4
912.2.a.n.1.1		2	20.3	even	4
2736.2.a.y.1.2		2	60.23	odd	4
3648.2.a.bk.1.2		2	40.13	odd	4
3648.2.a.bq.1.2		2	40.3	even	4
4332.2.a.i.1.1		2	95.18	even	4
5700.2.a.t.1.1		2	5.2	odd	4
5700.2.f.m.3649.2		4	1.1	even	1	trivial
5700.2.f.m.3649.3		4	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 5700 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5700.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +3.37228i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +3.37228i q^{7} -1.00000 q^{9} +1.37228 q^{11} -2.00000i q^{13} +1.37228i q^{17} -1.00000 q^{19} +3.37228 q^{21} +8.74456i q^{23} +1.00000i q^{27} -2.74456 q^{29} -6.74456 q^{31} -1.37228i q^{33} +4.74456i q^{37} -2.00000 q^{39} -3.37228i q^{43} -13.3723i q^{47} -4.37228 q^{49} +1.37228 q^{51} +2.74456i q^{53} +1.00000i q^{57} -2.62772 q^{61} -3.37228i q^{63} -9.48913i q^{67} +8.74456 q^{69} -12.0000 q^{71} +5.37228i q^{73} +4.62772i q^{77} -8.00000 q^{79} +1.00000 q^{81} -8.74456i q^{83} +2.74456i q^{87} -14.7446 q^{89} +6.74456 q^{91} +6.74456i q^{93} +14.0000i q^{97} -1.37228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^9	\( 4 q - 4 q^{9} - 6 q^{11} - 4 q^{19} + 2 q^{21} + 12 q^{29} - 4 q^{31} - 8 q^{39} - 6 q^{49} - 6 q^{51} - 22 q^{61} + 12 q^{69} - 48 q^{71} - 32 q^{79} + 4 q^{81} - 36 q^{89} + 4 q^{91} + 6 q^{99}+O(q^{100}) \) 4 * q - 4 * q^9 - 6 * q^11 - 4 * q^19 + 2 * q^21 + 12 * q^29 - 4 * q^31 - 8 * q^39 - 6 * q^49 - 6 * q^51 - 22 * q^61 + 12 * q^69 - 48 * q^71 - 32 * q^79 + 4 * q^81 - 36 * q^89 + 4 * q^91 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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