LMFDB - Embedded newform 2900.2.c.f.349.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2900,2,Mod(349,2900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2900, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2900.349");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2900 = 2^{2} \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2900.c (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:	$23.1566165862$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.5089536.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 16x^{2} - 24x + 18 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 16x^2 - 24*x + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 580)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			349.4
Root			$0.675970 - 0.675970i$ of defining polynomial
Character	$\chi$	$=$	2900.349
Dual form			2900.2.c.f.349.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.35194i q^{3} +0.648061i q^{7} +1.17226 q^{9} +O(q^{10})$	$q+1.35194i q^{3} +0.648061i q^{7} +1.17226 q^{9} +3.35194 q^{11} +4.17226i q^{13} -4.82032i q^{17} -6.82032 q^{19} -0.876139 q^{21} +5.52420i q^{23} +5.64064i q^{27} +1.00000 q^{29} -2.82032 q^{31} +4.53162i q^{33} +10.2281i q^{37} -5.64064 q^{39} +8.17226 q^{41} -5.69646i q^{43} +2.64806i q^{47} +6.58002 q^{49} +6.51678 q^{51} -2.87614i q^{53} -9.22066i q^{57} +13.2207 q^{59} -1.12386 q^{61} +0.759696i q^{63} -1.52420i q^{67} -7.46838 q^{69} -8.87614 q^{71} +9.69646i q^{73} +2.17226i q^{77} -8.99258 q^{79} -4.10902 q^{81} +1.94418i q^{83} +1.35194i q^{87} -17.0484 q^{89} -2.70388 q^{91} -3.81290i q^{93} +13.3519i q^{97} +3.92935 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 22 q^{9}+O(q^{10}) $ 6 * q - 22 * q^9	$ 6 q - 22 q^{9} + 16 q^{11} - 16 q^{19} + 32 q^{21} + 6 q^{29} + 8 q^{31} + 16 q^{39} + 20 q^{41} - 6 q^{49} - 48 q^{51} - 16 q^{59} - 44 q^{61} - 24 q^{69} - 16 q^{71} + 46 q^{81} - 36 q^{89} - 8 q^{91} - 72 q^{99}+O(q^{100}) $ 6 * q - 22 * q^9 + 16 * q^11 - 16 * q^19 + 32 * q^21 + 6 * q^29 + 8 * q^31 + 16 * q^39 + 20 * q^41 - 6 * q^49 - 48 * q^51 - 16 * q^59 - 44 * q^61 - 24 * q^69 - 16 * q^71 + 46 * q^81 - 36 * q^89 - 8 * q^91 - 72 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2900\mathbb{Z}\right)^\times$.

$n$	$901$	$1277$	$1451$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	1.35194i	0.780542i	0.920700	+	0.390271i	$0.127619\pi$
$3$	1.35194i	0.780542i	−0.920700	+	0.390271i	$0.872381\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.648061i	0.244944i	0.992472	+	0.122472i	$0.0390822\pi$
$7$	0.648061i	0.244944i	−0.992472	+	0.122472i	$0.960918\pi$
$8$	0	0
$9$	1.17226	0.390753
$10$	0	0
$11$	3.35194	1.01065	0.505324	−	0.862930i	$-0.331373\pi$
$11$	3.35194	1.01065	0.505324	+	0.862930i	$0.331373\pi$
$12$	0	0
$13$	4.17226i	1.15718i	0.815620	+	0.578588i	$0.196396\pi$
$13$	4.17226i	1.15718i	−0.815620	+	0.578588i	$0.803604\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 4.82032i	− 1.16910i	−0.811358	−	0.584550i	$-0.801271\pi$
$17$	− 4.82032i	− 1.16910i	0.811358	−	0.584550i	$-0.198729\pi$
$18$	0	0
$19$	−6.82032	−1.56469	−0.782344	−	0.622846i	$-0.785976\pi$
$19$	−6.82032	−1.56469	−0.782344	+	0.622846i	$0.785976\pi$
$20$	0	0
$21$	−0.876139	−0.191189
$22$	0	0
$23$	5.52420i	1.15188i	0.817494	+	0.575938i	$0.195363\pi$
$23$	5.52420i	1.15188i	−0.817494	+	0.575938i	$0.804637\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.64064i	1.08554i
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−2.82032	−0.506545	−0.253272	−	0.967395i	$-0.581507\pi$
$31$	−2.82032	−0.506545	−0.253272	+	0.967395i	$0.581507\pi$
$32$	0	0
$33$	4.53162i	0.788853i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	10.2281i	1.68149i	0.541435	+	0.840743i	$0.317881\pi$
$37$	10.2281i	1.68149i	−0.541435	+	0.840743i	$0.682119\pi$
$38$	0	0
$39$	−5.64064	−0.903226
$40$	0	0
$41$	8.17226	1.27629	0.638146	−	0.769915i	$-0.279701\pi$
$41$	8.17226	1.27629	0.638146	+	0.769915i	$0.279701\pi$
$42$	0	0
$43$	− 5.69646i	− 0.868702i	−0.900744	−	0.434351i	$-0.856978\pi$
$43$	− 5.69646i	− 0.868702i	0.900744	−	0.434351i	$-0.143022\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.64806i	0.386259i	0.981173	+	0.193130i	$0.0618638\pi$
$47$	2.64806i	0.386259i	−0.981173	+	0.193130i	$0.938136\pi$
$48$	0	0
$49$	6.58002	0.940002
$50$	0	0
$51$	6.51678	0.912532
$52$	0	0
$53$	− 2.87614i	− 0.395068i	−0.980296	−	0.197534i	$-0.936707\pi$
$53$	− 2.87614i	− 0.395068i	0.980296	−	0.197534i	$-0.0632933\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 9.22066i	− 1.22131i
$58$	0	0
$59$	13.2207	1.72118	0.860592	−	0.509296i	$-0.170094\pi$
$59$	13.2207	1.72118	0.860592	+	0.509296i	$0.170094\pi$
$60$	0	0
$61$	−1.12386	−0.143896	−0.0719478	−	0.997408i	$-0.522922\pi$
$61$	−1.12386	−0.143896	−0.0719478	+	0.997408i	$0.522922\pi$
$62$	0	0
$63$	0.759696i	0.0957127i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 1.52420i	− 0.186211i	−0.995656	−	0.0931053i	$-0.970321\pi$
$67$	− 1.52420i	− 0.186211i	0.995656	−	0.0931053i	$-0.0296793\pi$
$68$	0	0
$69$	−7.46838	−0.899088
$70$	0	0
$71$	−8.87614	−1.05340	−0.526702	−	0.850050i	$-0.676572\pi$
$71$	−8.87614	−1.05340	−0.526702	+	0.850050i	$0.676572\pi$
$72$	0	0
$73$	9.69646i	1.13488i	0.823413	+	0.567442i	$0.192067\pi$
$73$	9.69646i	1.13488i	−0.823413	+	0.567442i	$0.807933\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.17226i	0.247552i
$78$	0	0
$79$	−8.99258	−1.01174	−0.505872	−	0.862608i	$-0.668829\pi$
$79$	−8.99258	−1.01174	−0.505872	+	0.862608i	$0.668829\pi$
$80$	0	0
$81$	−4.10902	−0.456558
$82$	0	0
$83$	1.94418i	0.213402i	0.994291	+	0.106701i	$0.0340287\pi$
$83$	1.94418i	0.213402i	−0.994291	+	0.106701i	$0.965971\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.35194i	0.144943i
$88$	0	0
$89$	−17.0484	−1.80713	−0.903563	−	0.428455i	$-0.859058\pi$
$89$	−17.0484	−1.80713	−0.903563	+	0.428455i	$0.859058\pi$
$90$	0	0
$91$	−2.70388	−0.283443
$92$	0	0
$93$	− 3.81290i	− 0.395380i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	13.3519i	1.35568i	0.735208	+	0.677842i	$0.237085\pi$
$97$	13.3519i	1.35568i	−0.735208	+	0.677842i	$0.762915\pi$
$98$	0	0
$99$	3.92935	0.394914
$100$	0	0
$101$	18.4562	1.83646	0.918228	−	0.396052i	$-0.129620\pi$
$101$	18.4562	1.83646	0.918228	+	0.396052i	$0.129620\pi$
$102$	0	0
$103$	− 14.8203i	− 1.46029i	−0.683292	−	0.730145i	$-0.739453\pi$
$103$	− 14.8203i	− 1.46029i	0.683292	−	0.730145i	$-0.260547\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.35194i	0.324044i	0.986787	+	0.162022i	$0.0518016\pi$
$107$	3.35194i	0.324044i	−0.986787	+	0.162022i	$0.948198\pi$
$108$	0	0
$109$	−2.76450	−0.264791	−0.132396	−	0.991197i	$-0.542267\pi$
$109$	−2.76450	−0.264791	−0.132396	+	0.991197i	$0.542267\pi$
$110$	0	0
$111$	−13.8277	−1.31247
$112$	0	0
$113$	7.94418i	0.747326i	0.927565	+	0.373663i	$0.121898\pi$
$113$	7.94418i	0.747326i	−0.927565	+	0.373663i	$0.878102\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.89098i	0.452171i
$118$	0	0
$119$	3.12386	0.286364
$120$	0	0
$121$	0.235496	0.0214088
$122$	0	0
$123$	11.0484i	0.996201i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	17.6965i	1.57031i	0.619301	+	0.785153i	$0.287416\pi$
$127$	17.6965i	1.57031i	−0.619301	+	0.785153i	$0.712584\pi$
$128$	0	0
$129$	7.70127	0.678059
$130$	0	0
$131$	5.52420	0.482652	0.241326	−	0.970444i	$-0.422418\pi$
$131$	5.52420	0.482652	0.241326	+	0.970444i	$0.422418\pi$
$132$	0	0
$133$	− 4.41998i	− 0.383261i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.86872i	0.330527i	0.986249	+	0.165264i	$0.0528475\pi$
$137$	3.86872i	0.330527i	−0.986249	+	0.165264i	$0.947153\pi$
$138$	0	0
$139$	−5.64064	−0.478433	−0.239217	−	0.970966i	$-0.576891\pi$
$139$	−5.64064	−0.478433	−0.239217	+	0.970966i	$0.576891\pi$
$140$	0	0
$141$	−3.58002	−0.301492
$142$	0	0
$143$	13.9852i	1.16950i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	8.89578i	0.733712i
$148$	0	0
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	0	0
$151$	−17.2207	−1.40140	−0.700699	−	0.713457i	$-0.747128\pi$
$151$	−17.2207	−1.40140	−0.700699	+	0.713457i	$0.747128\pi$
$152$	0	0
$153$	− 5.65067i	− 0.456830i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	5.69646i	0.454627i	0.973822	+	0.227314i	$0.0729942\pi$
$157$	5.69646i	0.454627i	−0.973822	+	0.227314i	$0.927006\pi$
$158$	0	0
$159$	3.88836	0.308367
$160$	0	0
$161$	−3.58002	−0.282145
$162$	0	0
$163$	16.4003i	1.28457i	0.766464	+	0.642287i	$0.222014\pi$
$163$	16.4003i	1.28457i	−0.766464	+	0.642287i	$0.777986\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	23.2765i	1.80119i	0.434661	+	0.900594i	$0.356868\pi$
$167$	23.2765i	1.80119i	−0.434661	+	0.900594i	$0.643132\pi$
$168$	0	0
$169$	−4.40776	−0.339058
$170$	0	0
$171$	−7.99519	−0.611408
$172$	0	0
$173$	− 13.9245i	− 1.05866i	−0.848415	−	0.529332i	$-0.822443\pi$
$173$	− 13.9245i	− 1.05866i	0.848415	−	0.529332i	$-0.177557\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	17.8735i	1.34346i
$178$	0	0
$179$	4.34452	0.324725	0.162362	−	0.986731i	$-0.448089\pi$
$179$	4.34452	0.324725	0.162362	+	0.986731i	$0.448089\pi$
$180$	0	0
$181$	20.5168	1.52500	0.762500	−	0.646988i	$-0.223972\pi$
$181$	20.5168	1.52500	0.762500	+	0.646988i	$0.223972\pi$
$182$	0	0
$183$	− 1.51939i	− 0.112317i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 16.1574i	− 1.18155i
$188$	0	0
$189$	−3.65548	−0.265897
$190$	0	0
$191$	−9.10422	−0.658758	−0.329379	−	0.944198i	$-0.606839\pi$
$191$	−9.10422	−0.658758	−0.329379	+	0.944198i	$0.606839\pi$
$192$	0	0
$193$	− 25.1648i	− 1.81140i	−0.423914	−	0.905702i	$-0.639344\pi$
$193$	− 25.1648i	− 1.81140i	0.423914	−	0.905702i	$-0.360656\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0.703878i	0.0501493i	0.999686	+	0.0250746i	$0.00798234\pi$
$197$	0.703878i	0.0501493i	−0.999686	+	0.0250746i	$0.992018\pi$
$198$	0	0
$199$	5.22066	0.370083	0.185041	−	0.982731i	$-0.440758\pi$
$199$	5.22066	0.370083	0.185041	+	0.982731i	$0.440758\pi$
$200$	0	0
$201$	2.06063	0.145345
$202$	0	0
$203$	0.648061i	0.0454850i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	6.47580i	0.450099i
$208$	0	0
$209$	−22.8613	−1.58135
$210$	0	0
$211$	−13.1042	−0.902131	−0.451066	−	0.892491i	$-0.648956\pi$
$211$	−13.1042	−0.902131	−0.451066	+	0.892491i	$0.648956\pi$
$212$	0	0
$213$	− 12.0000i	− 0.822226i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 1.82774i	− 0.124075i
$218$	0	0
$219$	−13.1090	−0.885826
$220$	0	0
$221$	20.1116	1.35285
$222$	0	0
$223$	− 20.5726i	− 1.37764i	−0.724931	−	0.688822i	$-0.758128\pi$
$223$	− 20.5726i	− 1.37764i	0.724931	−	0.688822i	$-0.241872\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	9.41256i	0.624734i	0.949962	+	0.312367i	$0.101122\pi$
$227$	9.41256i	0.624734i	−0.949962	+	0.312367i	$0.898878\pi$
$228$	0	0
$229$	−2.00000	−0.132164	−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000	−0.132164	−0.0660819	+	0.997814i	$0.521050\pi$
$230$	0	0
$231$	−2.93676	−0.193225
$232$	0	0
$233$	10.3445i	0.677692i	0.940842	+	0.338846i	$0.110037\pi$
$233$	10.3445i	0.677692i	−0.940842	+	0.338846i	$0.889963\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 12.1574i	− 0.789710i
$238$	0	0
$239$	−20.8761	−1.35037	−0.675183	−	0.737651i	$-0.735935\pi$
$239$	−20.8761	−1.35037	−0.675183	+	0.737651i	$0.735935\pi$
$240$	0	0
$241$	21.1600	1.36304	0.681519	−	0.731801i	$-0.261320\pi$
$241$	21.1600	1.36304	0.681519	+	0.731801i	$0.261320\pi$
$242$	0	0
$243$	11.3668i	0.729179i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 28.4562i	− 1.81062i
$248$	0	0
$249$	−2.62842	−0.166569
$250$	0	0
$251$	−8.64806	−0.545861	−0.272930	−	0.962034i	$-0.587993\pi$
$251$	−8.64806	−0.545861	−0.272930	+	0.962034i	$0.587993\pi$
$252$	0	0
$253$	18.5168i	1.16414i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 27.4535i	− 1.71251i	−0.516557	−	0.856253i	$-0.672787\pi$
$257$	− 27.4535i	− 1.71251i	0.516557	−	0.856253i	$-0.327213\pi$
$258$	0	0
$259$	−6.62842	−0.411870
$260$	0	0
$261$	1.17226	0.0725611
$262$	0	0
$263$	− 11.3371i	− 0.699076i	−0.936922	−	0.349538i	$-0.886339\pi$
$263$	− 11.3371i	− 0.699076i	0.936922	−	0.349538i	$-0.113661\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 23.0484i	− 1.41054i
$268$	0	0
$269$	20.1723	1.22992	0.614962	−	0.788557i	$-0.289171\pi$
$269$	20.1723	1.22992	0.614962	+	0.788557i	$0.289171\pi$
$270$	0	0
$271$	−10.7449	−0.652704	−0.326352	−	0.945248i	$-0.605819\pi$
$271$	−10.7449	−0.652704	−0.326352	+	0.945248i	$0.605819\pi$
$272$	0	0
$273$	− 3.65548i	− 0.221240i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	12.6284i	0.758768i	0.925239	+	0.379384i	$0.123864\pi$
$277$	12.6284i	0.758768i	−0.925239	+	0.379384i	$0.876136\pi$
$278$	0	0
$279$	−3.30615	−0.197934
$280$	0	0
$281$	6.76450	0.403536	0.201768	−	0.979433i	$-0.435331\pi$
$281$	6.76450	0.403536	0.201768	+	0.979433i	$0.435331\pi$
$282$	0	0
$283$	1.06804i	0.0634886i	0.999496	+	0.0317443i	$0.0101062\pi$
$283$	1.06804i	0.0634886i	−0.999496	+	0.0317443i	$0.989894\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.29612i	0.312620i
$288$	0	0
$289$	−6.23550	−0.366794
$290$	0	0
$291$	−18.0510	−1.05817
$292$	0	0
$293$	5.46357i	0.319185i	0.987183	+	0.159593i	$0.0510181\pi$
$293$	5.46357i	0.319185i	−0.987183	+	0.159593i	$0.948982\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	18.9071i	1.09710i
$298$	0	0
$299$	−23.0484	−1.33292
$300$	0	0
$301$	3.69165	0.212783
$302$	0	0
$303$	24.9516i	1.43343i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 6.30354i	− 0.359762i	−0.983688	−	0.179881i	$-0.942429\pi$
$307$	− 6.30354i	− 0.359762i	0.983688	−	0.179881i	$-0.0575713\pi$
$308$	0	0
$309$	20.0362	1.13982
$310$	0	0
$311$	24.6842	1.39971	0.699857	−	0.714283i	$-0.253247\pi$
$311$	24.6842	1.39971	0.699857	+	0.714283i	$0.253247\pi$
$312$	0	0
$313$	− 24.1723i	− 1.36630i	−0.730280	−	0.683148i	$-0.760610\pi$
$313$	− 24.1723i	− 1.36630i	0.730280	−	0.683148i	$-0.239390\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 1.42740i	− 0.0801708i	−0.999196	−	0.0400854i	$-0.987237\pi$
$317$	− 1.42740i	− 0.0801708i	0.999196	−	0.0400854i	$-0.0127630\pi$
$318$	0	0
$319$	3.35194	0.187673
$320$	0	0
$321$	−4.53162	−0.252930
$322$	0	0
$323$	32.8761i	1.82928i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 3.73744i	− 0.206681i
$328$	0	0
$329$	−1.71610	−0.0946119
$330$	0	0
$331$	34.2132	1.88053	0.940265	−	0.340444i	$-0.110577\pi$
$331$	34.2132	1.88053	0.940265	+	0.340444i	$0.110577\pi$
$332$	0	0
$333$	11.9900i	0.657046i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 4.40034i	− 0.239702i	−0.992792	−	0.119851i	$-0.961758\pi$
$337$	− 4.40034i	− 0.239702i	0.992792	−	0.119851i	$-0.0382416\pi$
$338$	0	0
$339$	−10.7401	−0.583320
$340$	0	0
$341$	−9.45355	−0.511938
$342$	0	0
$343$	8.80068i	0.475192i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 12.9926i	− 0.697478i	−0.937220	−	0.348739i	$-0.886610\pi$
$347$	− 12.9926i	− 0.697478i	0.937220	−	0.348739i	$-0.113390\pi$
$348$	0	0
$349$	20.0968	1.07576	0.537878	−	0.843022i	$-0.319226\pi$
$349$	20.0968	1.07576	0.537878	+	0.843022i	$0.319226\pi$
$350$	0	0
$351$	−23.5342	−1.25616
$352$	0	0
$353$	− 17.8129i	− 0.948085i	−0.880502	−	0.474043i	$-0.842794\pi$
$353$	− 17.8129i	− 0.948085i	0.880502	−	0.474043i	$-0.157206\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	4.22327i	0.223519i
$358$	0	0
$359$	−5.52420	−0.291556	−0.145778	−	0.989317i	$-0.546569\pi$
$359$	−5.52420	−0.291556	−0.145778	+	0.989317i	$0.546569\pi$
$360$	0	0
$361$	27.5168	1.44825
$362$	0	0
$363$	0.318377i	0.0167104i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	20.0558i	1.04691i	0.852055	+	0.523453i	$0.175356\pi$
$367$	20.0558i	1.04691i	−0.852055	+	0.523453i	$0.824644\pi$
$368$	0	0
$369$	9.58002	0.498716
$370$	0	0
$371$	1.86391	0.0967695
$372$	0	0
$373$	− 1.88836i	− 0.0977758i	−0.998804	−	0.0488879i	$-0.984432\pi$
$373$	− 1.88836i	− 0.0977758i	0.998804	−	0.0488879i	$-0.0155677\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.17226i	0.214882i
$378$	0	0
$379$	15.0894	0.775089	0.387545	−	0.921851i	$-0.373323\pi$
$379$	15.0894	0.775089	0.387545	+	0.921851i	$0.373323\pi$
$380$	0	0
$381$	−23.9245	−1.22569
$382$	0	0
$383$	17.3371i	0.885885i	0.896550	+	0.442942i	$0.146065\pi$
$383$	17.3371i	0.885885i	−0.896550	+	0.442942i	$0.853935\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 6.67773i	− 0.339448i
$388$	0	0
$389$	0.890976	0.0451743	0.0225871	−	0.999745i	$-0.492810\pi$
$389$	0.890976	0.0451743	0.0225871	+	0.999745i	$0.492810\pi$
$390$	0	0
$391$	26.6284	1.34666
$392$	0	0
$393$	7.46838i	0.376730i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 5.39292i	− 0.270663i	−0.990800	−	0.135331i	$-0.956790\pi$
$397$	− 5.39292i	− 0.270663i	0.990800	−	0.135331i	$-0.0432099\pi$
$398$	0	0
$399$	5.97555	0.299152
$400$	0	0
$401$	5.23550	0.261448	0.130724	−	0.991419i	$-0.458270\pi$
$401$	5.23550	0.261448	0.130724	+	0.991419i	$0.458270\pi$
$402$	0	0
$403$	− 11.7671i	− 0.586162i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	34.2839i	1.69939i
$408$	0	0
$409$	16.0606	0.794147	0.397073	−	0.917787i	$-0.370026\pi$
$409$	16.0606	0.794147	0.397073	+	0.917787i	$0.370026\pi$
$410$	0	0
$411$	−5.23027	−0.257990
$412$	0	0
$413$	8.56779i	0.421593i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 7.62581i	− 0.373437i
$418$	0	0
$419$	19.9245	0.973377	0.486689	−	0.873575i	$-0.338205\pi$
$419$	19.9245	0.973377	0.486689	+	0.873575i	$0.338205\pi$
$420$	0	0
$421$	−15.5652	−0.758600	−0.379300	−	0.925274i	$-0.623835\pi$
$421$	−15.5652	−0.758600	−0.379300	+	0.925274i	$0.623835\pi$
$422$	0	0
$423$	3.10422i	0.150932i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 0.728330i	− 0.0352464i
$428$	0	0
$429$	−18.9071	−0.912843
$430$	0	0
$431$	−13.4078	−0.645829	−0.322914	−	0.946428i	$-0.604663\pi$
$431$	−13.4078	−0.645829	−0.322914	+	0.946428i	$0.604663\pi$
$432$	0	0
$433$	− 34.6842i	− 1.66682i	−0.552657	−	0.833409i	$-0.686386\pi$
$433$	− 34.6842i	− 1.66682i	0.552657	−	0.833409i	$-0.313614\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 37.6768i	− 1.80233i
$438$	0	0
$439$	15.2355	0.727151	0.363575	−	0.931565i	$-0.381556\pi$
$439$	15.2355	0.727151	0.363575	+	0.931565i	$0.381556\pi$
$440$	0	0
$441$	7.71349	0.367309
$442$	0	0
$443$	5.00742i	0.237910i	0.992900	+	0.118955i	$0.0379544\pi$
$443$	5.00742i	0.237910i	−0.992900	+	0.118955i	$0.962046\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 24.3349i	− 1.15100i
$448$	0	0
$449$	17.8129	0.840643	0.420321	−	0.907375i	$-0.361917\pi$
$449$	17.8129	0.840643	0.420321	+	0.907375i	$0.361917\pi$
$450$	0	0
$451$	27.3929	1.28988
$452$	0	0
$453$	− 23.2813i	− 1.09385i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 24.4051i	− 1.14162i	−0.821081	−	0.570812i	$-0.806628\pi$
$457$	− 24.4051i	− 1.14162i	0.821081	−	0.570812i	$-0.193372\pi$
$458$	0	0
$459$	27.1897	1.26911
$460$	0	0
$461$	−7.98516	−0.371906	−0.185953	−	0.982559i	$-0.559537\pi$
$461$	−7.98516	−0.371906	−0.185953	+	0.982559i	$0.559537\pi$
$462$	0	0
$463$	11.6210i	0.540074i	0.962850	+	0.270037i	$0.0870359\pi$
$463$	11.6210i	0.540074i	−0.962850	+	0.270037i	$0.912964\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 1.24030i	− 0.0573944i	−0.999588	−	0.0286972i	$-0.990864\pi$
$467$	− 1.24030i	− 0.0573944i	0.999588	−	0.0286972i	$-0.00913586\pi$
$468$	0	0
$469$	0.987774	0.0456112
$470$	0	0
$471$	−7.70127	−0.354856
$472$	0	0
$473$	− 19.0942i	− 0.877952i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 3.37158i	− 0.154374i
$478$	0	0
$479$	−19.6965	−0.899954	−0.449977	−	0.893040i	$-0.648568\pi$
$479$	−19.6965	−0.899954	−0.449977	+	0.893040i	$0.648568\pi$
$480$	0	0
$481$	−42.6742	−1.94578
$482$	0	0
$483$	− 4.83997i	− 0.220226i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	15.0894i	0.683765i	0.939743	+	0.341883i	$0.111064\pi$
$487$	15.0894i	0.683765i	−0.939743	+	0.341883i	$0.888936\pi$
$488$	0	0
$489$	−22.1723	−1.00266
$490$	0	0
$491$	−12.2281	−0.551845	−0.275923	−	0.961180i	$-0.588983\pi$
$491$	−12.2281	−0.551845	−0.275923	+	0.961180i	$0.588983\pi$
$492$	0	0
$493$	− 4.82032i	− 0.217096i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 5.75228i	− 0.258025i
$498$	0	0
$499$	5.52901	0.247512	0.123756	−	0.992313i	$-0.460506\pi$
$499$	5.52901	0.247512	0.123756	+	0.992313i	$0.460506\pi$
$500$	0	0
$501$	−31.4684	−1.40590
$502$	0	0
$503$	24.2887i	1.08298i	0.840707	+	0.541490i	$0.182140\pi$
$503$	24.2887i	1.08298i	−0.840707	+	0.541490i	$0.817860\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 5.95902i	− 0.264649i
$508$	0	0
$509$	9.58002	0.424627	0.212313	−	0.977202i	$-0.431900\pi$
$509$	9.58002	0.424627	0.212313	+	0.977202i	$0.431900\pi$
$510$	0	0
$511$	−6.28390	−0.277983
$512$	0	0
$513$	− 38.4710i	− 1.69854i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	8.87614i	0.390372i
$518$	0	0
$519$	18.8251	0.826331
$520$	0	0
$521$	−21.5800	−0.945438	−0.472719	−	0.881213i	$-0.656727\pi$
$521$	−21.5800	−0.945438	−0.472719	+	0.881213i	$0.656727\pi$
$522$	0	0
$523$	− 42.2132i	− 1.84586i	−0.384972	−	0.922928i	$-0.625789\pi$
$523$	− 42.2132i	− 1.84586i	0.384972	−	0.922928i	$-0.374211\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	13.5949i	0.592201i
$528$	0	0
$529$	−7.51678	−0.326817
$530$	0	0
$531$	15.4981	0.672558
$532$	0	0
$533$	34.0968i	1.47690i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	5.87353i	0.253461i
$538$	0	0
$539$	22.0558	0.950011
$540$	0	0
$541$	36.5530	1.57153	0.785767	−	0.618522i	$-0.212268\pi$
$541$	36.5530	1.57153	0.785767	+	0.618522i	$0.212268\pi$
$542$	0	0
$543$	27.7374i	1.19033i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 26.6236i	− 1.13834i	−0.822219	−	0.569172i	$-0.807264\pi$
$547$	− 26.6236i	− 1.13834i	0.822219	−	0.569172i	$-0.192736\pi$
$548$	0	0
$549$	−1.31746	−0.0562277
$550$	0	0
$551$	−6.82032	−0.290555
$552$	0	0
$553$	− 5.82774i	− 0.247821i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 22.2233i	− 0.941630i	−0.882232	−	0.470815i	$-0.843960\pi$
$557$	− 22.2233i	− 0.941630i	0.882232	−	0.470815i	$-0.156040\pi$
$558$	0	0
$559$	23.7671	1.00524
$560$	0	0
$561$	21.8439	0.922248
$562$	0	0
$563$	1.11905i	0.0471625i	0.999722	+	0.0235812i	$0.00750684\pi$
$563$	1.11905i	0.0471625i	−0.999722	+	0.0235812i	$0.992493\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 2.66290i	− 0.111831i
$568$	0	0
$569$	10.5316	0.441508	0.220754	−	0.975329i	$-0.429148\pi$
$569$	10.5316	0.441508	0.220754	+	0.975329i	$0.429148\pi$
$570$	0	0
$571$	−1.18449	−0.0495692	−0.0247846	−	0.999693i	$-0.507890\pi$
$571$	−1.18449	−0.0495692	−0.0247846	+	0.999693i	$0.507890\pi$
$572$	0	0
$573$	− 12.3083i	− 0.514189i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 11.8687i	− 0.494101i	−0.969003	−	0.247051i	$-0.920539\pi$
$577$	− 11.8687i	− 0.494101i	0.969003	−	0.247051i	$-0.0794614\pi$
$578$	0	0
$579$	34.0213	1.41388
$580$	0	0
$581$	−1.25995	−0.0522715
$582$	0	0
$583$	− 9.64064i	− 0.399275i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 32.4610i	− 1.33981i	−0.742448	−	0.669904i	$-0.766335\pi$
$587$	− 32.4610i	− 1.33981i	0.742448	−	0.669904i	$-0.233665\pi$
$588$	0	0
$589$	19.2355	0.792585
$590$	0	0
$591$	−0.951601	−0.0391436
$592$	0	0
$593$	9.12386i	0.374672i	0.982296	+	0.187336i	$0.0599853\pi$
$593$	9.12386i	0.374672i	−0.982296	+	0.187336i	$0.940015\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	7.05801i	0.288865i
$598$	0	0
$599$	30.2887	1.23756	0.618781	−	0.785563i	$-0.287627\pi$
$599$	30.2887	1.23756	0.618781	+	0.785563i	$0.287627\pi$
$600$	0	0
$601$	20.5168	0.836897	0.418448	−	0.908241i	$-0.362574\pi$
$601$	20.5168	0.836897	0.418448	+	0.908241i	$0.362574\pi$
$602$	0	0
$603$	− 1.78676i	− 0.0727624i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	13.9293i	0.565375i	0.959212	+	0.282687i	$0.0912259\pi$
$607$	13.9293i	0.565375i	−0.959212	+	0.282687i	$0.908774\pi$
$608$	0	0
$609$	−0.876139	−0.0355029
$610$	0	0
$611$	−11.0484	−0.446970
$612$	0	0
$613$	19.7523i	0.797787i	0.916997	+	0.398893i	$0.130606\pi$
$613$	19.7523i	0.797787i	−0.916997	+	0.398893i	$0.869394\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	37.2010i	1.49766i	0.662764	+	0.748828i	$0.269383\pi$
$617$	37.2010i	1.49766i	−0.662764	+	0.748828i	$0.730617\pi$
$618$	0	0
$619$	−7.46357	−0.299986	−0.149993	−	0.988687i	$-0.547925\pi$
$619$	−7.46357	−0.299986	−0.149993	+	0.988687i	$0.547925\pi$
$620$	0	0
$621$	−31.1600	−1.25041
$622$	0	0
$623$	− 11.0484i	− 0.442645i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 30.9071i	− 1.23431i
$628$	0	0
$629$	49.3026	1.96582
$630$	0	0
$631$	44.4562	1.76977	0.884886	−	0.465808i	$-0.154236\pi$
$631$	44.4562	1.76977	0.884886	+	0.465808i	$0.154236\pi$
$632$	0	0
$633$	− 17.7161i	− 0.704152i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	27.4535i	1.08775i
$638$	0	0
$639$	−10.4051	−0.411621
$640$	0	0
$641$	−40.7401	−1.60914	−0.804568	−	0.593861i	$-0.797603\pi$
$641$	−40.7401	−1.60914	−0.804568	+	0.593861i	$0.797603\pi$
$642$	0	0
$643$	20.3493i	0.802499i	0.915969	+	0.401250i	$0.131424\pi$
$643$	20.3493i	0.802499i	−0.915969	+	0.401250i	$0.868576\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 41.7933i	− 1.64306i	−0.570163	−	0.821531i	$-0.693120\pi$
$647$	− 41.7933i	− 1.64306i	0.570163	−	0.821531i	$-0.306880\pi$
$648$	0	0
$649$	44.3148	1.73951
$650$	0	0
$651$	2.47099	0.0968458
$652$	0	0
$653$	23.7933i	0.931102i	0.885021	+	0.465551i	$0.154144\pi$
$653$	23.7933i	0.931102i	−0.885021	+	0.465551i	$0.845856\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	11.3668i	0.443460i
$658$	0	0
$659$	−7.50936	−0.292523	−0.146262	−	0.989246i	$-0.546724\pi$
$659$	−7.50936	−0.292523	−0.146262	+	0.989246i	$0.546724\pi$
$660$	0	0
$661$	45.9245	1.78626	0.893129	−	0.449801i	$-0.148505\pi$
$661$	45.9245	1.78626	0.893129	+	0.449801i	$0.148505\pi$
$662$	0	0
$663$	27.1897i	1.05596i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	5.52420i	0.213898i
$668$	0	0
$669$	27.8129	1.07531
$670$	0	0
$671$	−3.76711	−0.145428
$672$	0	0
$673$	7.29612i	0.281245i	0.990063	+	0.140622i	$0.0449104\pi$
$673$	7.29612i	0.281245i	−0.990063	+	0.140622i	$0.955090\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 36.2494i	− 1.39318i	−0.717470	−	0.696589i	$-0.754700\pi$
$677$	− 36.2494i	− 1.39318i	0.717470	−	0.696589i	$-0.245300\pi$
$678$	0	0
$679$	−8.65287	−0.332067
$680$	0	0
$681$	−12.7252	−0.487631
$682$	0	0
$683$	− 5.40295i	− 0.206738i	−0.994643	−	0.103369i	$-0.967038\pi$
$683$	− 5.40295i	− 0.206738i	0.994643	−	0.103369i	$-0.0329623\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 2.70388i	− 0.103159i
$688$	0	0
$689$	12.0000	0.457164
$690$	0	0
$691$	−36.0458	−1.37125	−0.685623	−	0.727957i	$-0.740470\pi$
$691$	−36.0458	−1.37125	−0.685623	+	0.727957i	$0.740470\pi$
$692$	0	0
$693$	2.54645i	0.0967318i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 39.3929i	− 1.49211i
$698$	0	0
$699$	−13.9852	−0.528967
$700$	0	0
$701$	3.33229	0.125859	0.0629295	−	0.998018i	$-0.479956\pi$
$701$	3.33229	0.125859	0.0629295	+	0.998018i	$0.479956\pi$
$702$	0	0
$703$	− 69.7588i	− 2.63100i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	11.9607i	0.449829i
$708$	0	0
$709$	−7.48322	−0.281038	−0.140519	−	0.990078i	$-0.544877\pi$
$709$	−7.48322	−0.281038	−0.140519	+	0.990078i	$0.544877\pi$
$710$	0	0
$711$	−10.5416	−0.395343
$712$	0	0
$713$	− 15.5800i	− 0.583476i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 28.2233i	− 1.05402i
$718$	0	0
$719$	39.6620	1.47914	0.739571	−	0.673078i	$-0.235028\pi$
$719$	39.6620	1.47914	0.739571	+	0.673078i	$0.235028\pi$
$720$	0	0
$721$	9.60447	0.357689
$722$	0	0
$723$	28.6071i	1.06391i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	8.05582i	0.298774i	0.988779	+	0.149387i	$0.0477300\pi$
$727$	8.05582i	0.298774i	−0.988779	+	0.149387i	$0.952270\pi$
$728$	0	0
$729$	−27.6943	−1.02571
$730$	0	0
$731$	−27.4588	−1.01560
$732$	0	0
$733$	− 28.9320i	− 1.06863i	−0.845287	−	0.534313i	$-0.820570\pi$
$733$	− 28.9320i	− 1.06863i	0.845287	−	0.534313i	$-0.179430\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 5.10902i	− 0.188193i
$738$	0	0
$739$	−36.0772	−1.32712	−0.663560	−	0.748123i	$-0.730955\pi$
$739$	−36.0772	−1.32712	−0.663560	+	0.748123i	$0.730955\pi$
$740$	0	0
$741$	38.4710	1.41327
$742$	0	0
$743$	− 40.9681i	− 1.50297i	−0.659747	−	0.751487i	$-0.729337\pi$
$743$	− 40.9681i	− 1.50297i	0.659747	−	0.751487i	$-0.270663\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	2.27909i	0.0833875i
$748$	0	0
$749$	−2.17226	−0.0793727
$750$	0	0
$751$	23.2861	0.849722	0.424861	−	0.905259i	$-0.360323\pi$
$751$	23.2861	0.849722	0.424861	+	0.905259i	$0.360323\pi$
$752$	0	0
$753$	− 11.6917i	− 0.426067i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 27.3275i	− 0.993234i	−0.867970	−	0.496617i	$-0.834575\pi$
$757$	− 27.3275i	− 0.993234i	0.867970	−	0.496617i	$-0.165425\pi$
$758$	0	0
$759$	−25.0336	−0.908661
$760$	0	0
$761$	44.5626	1.61539	0.807696	−	0.589599i	$-0.200714\pi$
$761$	44.5626	1.61539	0.807696	+	0.589599i	$0.200714\pi$
$762$	0	0
$763$	− 1.79157i	− 0.0648591i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	55.1600i	1.99171i
$768$	0	0
$769$	−51.6110	−1.86114	−0.930570	−	0.366115i	$-0.880688\pi$
$769$	−51.6110	−1.86114	−0.930570	+	0.366115i	$0.880688\pi$
$770$	0	0
$771$	37.1155	1.33668
$772$	0	0
$773$	− 19.6358i	− 0.706252i	−0.935576	−	0.353126i	$-0.885119\pi$
$773$	− 19.6358i	− 0.706252i	0.935576	−	0.353126i	$-0.114881\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 8.96122i	− 0.321482i
$778$	0	0
$779$	−55.7374	−1.99700
$780$	0	0
$781$	−29.7523	−1.06462
$782$	0	0
$783$	5.64064i	0.201580i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 37.3371i	− 1.33092i	−0.746432	−	0.665462i	$-0.768235\pi$
$787$	− 37.3371i	− 1.33092i	0.746432	−	0.665462i	$-0.231765\pi$
$788$	0	0
$789$	15.3271	0.545658
$790$	0	0
$791$	−5.14831	−0.183053
$792$	0	0
$793$	− 4.68904i	− 0.166513i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	11.2159i	0.397286i	0.980072	+	0.198643i	$0.0636534\pi$
$797$	11.2159i	0.397286i	−0.980072	+	0.198643i	$0.936347\pi$
$798$	0	0
$799$	12.7645	0.451576
$800$	0	0
$801$	−19.9852	−0.706141
$802$	0	0
$803$	32.5019i	1.14697i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	27.2717i	0.960008i
$808$	0	0
$809$	−11.1749	−0.392888	−0.196444	−	0.980515i	$-0.562939\pi$
$809$	−11.1749	−0.392888	−0.196444	+	0.980515i	$0.562939\pi$
$810$	0	0
$811$	17.6406	0.619447	0.309723	−	0.950827i	$-0.399764\pi$
$811$	17.6406	0.619447	0.309723	+	0.950827i	$0.399764\pi$
$812$	0	0
$813$	− 14.5264i	− 0.509463i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	38.8517i	1.35925i
$818$	0	0
$819$	−3.16965	−0.110757
$820$	0	0
$821$	24.2477	0.846251	0.423126	−	0.906071i	$-0.360933\pi$
$821$	24.2477	0.846251	0.423126	+	0.906071i	$0.360933\pi$
$822$	0	0
$823$	16.9777i	0.591807i	0.955218	+	0.295903i	$0.0956207\pi$
$823$	16.9777i	0.591807i	−0.955218	+	0.295903i	$0.904379\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	5.58482i	0.194203i	0.995274	+	0.0971017i	$0.0309572\pi$
$827$	5.58482i	0.194203i	−0.995274	+	0.0971017i	$0.969043\pi$
$828$	0	0
$829$	−18.6136	−0.646476	−0.323238	−	0.946318i	$-0.604771\pi$
$829$	−18.6136	−0.646476	−0.323238	+	0.946318i	$0.604771\pi$
$830$	0	0
$831$	−17.0729	−0.592251
$832$	0	0
$833$	− 31.7178i	− 1.09896i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 15.9084i	− 0.549876i
$838$	0	0
$839$	15.9293	0.549942	0.274971	−	0.961453i	$-0.411332\pi$
$839$	15.9293	0.549942	0.274971	+	0.961453i	$0.411332\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	9.14520i	0.314977i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0.152616i	0.00524395i
$848$	0	0
$849$	−1.44393	−0.0495555
$850$	0	0
$851$	−56.5019	−1.93686
$852$	0	0
$853$	− 27.0288i	− 0.925447i	−0.886503	−	0.462723i	$-0.846872\pi$
$853$	− 27.0288i	− 0.925447i	0.886503	−	0.462723i	$-0.153128\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 30.2691i	− 1.03397i	−0.855994	−	0.516986i	$-0.827054\pi$
$857$	− 30.2691i	− 1.03397i	0.855994	−	0.516986i	$-0.172946\pi$
$858$	0	0
$859$	41.4487	1.41421	0.707106	−	0.707107i	$-0.250000\pi$
$859$	41.4487	1.41421	0.707106	+	0.707107i	$0.250000\pi$
$860$	0	0
$861$	−7.16003	−0.244013
$862$	0	0
$863$	42.2494i	1.43819i	0.694913	+	0.719093i	$0.255443\pi$
$863$	42.2494i	1.43819i	−0.694913	+	0.719093i	$0.744557\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 8.43001i	− 0.286298i
$868$	0	0
$869$	−30.1426	−1.02252
$870$	0	0
$871$	6.35936	0.215479
$872$	0	0
$873$	15.6519i	0.529738i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 37.8129i	− 1.27685i	−0.769684	−	0.638426i	$-0.779586\pi$
$877$	− 37.8129i	− 1.27685i	0.769684	−	0.638426i	$-0.220414\pi$
$878$	0	0
$879$	−7.38642	−0.249138
$880$	0	0
$881$	−5.35675	−0.180473	−0.0902367	−	0.995920i	$-0.528762\pi$
$881$	−5.35675	−0.180473	−0.0902367	+	0.995920i	$0.528762\pi$
$882$	0	0
$883$	8.53643i	0.287274i	0.989630	+	0.143637i	$0.0458797\pi$
$883$	8.53643i	0.287274i	−0.989630	+	0.143637i	$0.954120\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	22.7597i	0.764196i	0.924122	+	0.382098i	$0.124798\pi$
$887$	22.7597i	0.764196i	−0.924122	+	0.382098i	$0.875202\pi$
$888$	0	0
$889$	−11.4684	−0.384637
$890$	0	0
$891$	−13.7732	−0.461420
$892$	0	0
$893$	− 18.0606i	− 0.604376i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 31.1600i	− 1.04040i
$898$	0	0
$899$	−2.82032	−0.0940630
$900$	0	0
$901$	−13.8639	−0.461874
$902$	0	0
$903$	4.99089i	0.166086i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 3.94418i	− 0.130964i	−0.997854	−	0.0654822i	$-0.979141\pi$
$907$	− 3.94418i	− 0.130964i	0.997854	−	0.0654822i	$-0.0208586\pi$
$908$	0	0
$909$	21.6354	0.717602
$910$	0	0
$911$	−40.6546	−1.34695	−0.673473	−	0.739212i	$-0.735198\pi$
$911$	−40.6546	−1.34695	−0.673473	+	0.739212i	$0.735198\pi$
$912$	0	0
$913$	6.51678i	0.215674i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3.58002i	0.118223i
$918$	0	0
$919$	−27.3471	−0.902099	−0.451049	−	0.892499i	$-0.648950\pi$
$919$	−27.3471	−0.902099	−0.451049	+	0.892499i	$0.648950\pi$
$920$	0	0
$921$	8.52200	0.280810
$922$	0	0
$923$	− 37.0336i	− 1.21897i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 17.3733i	− 0.570613i
$928$	0	0
$929$	−20.9368	−0.686913	−0.343456	−	0.939169i	$-0.611598\pi$
$929$	−20.9368	−0.686913	−0.343456	+	0.939169i	$0.611598\pi$
$930$	0	0
$931$	−44.8778	−1.47081
$932$	0	0
$933$	33.3716i	1.09254i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 36.4051i	− 1.18930i	−0.803983	−	0.594652i	$-0.797290\pi$
$937$	− 36.4051i	− 1.18930i	0.803983	−	0.594652i	$-0.202710\pi$
$938$	0	0
$939$	32.6794	1.06645
$940$	0	0
$941$	28.0968	0.915929	0.457965	−	0.888970i	$-0.348579\pi$
$941$	28.0968	0.915929	0.457965	+	0.888970i	$0.348579\pi$
$942$	0	0
$943$	45.1452i	1.47013i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	10.8809i	0.353583i	0.984248	+	0.176792i	$0.0565719\pi$
$947$	10.8809i	0.353583i	−0.984248	+	0.176792i	$0.943428\pi$
$948$	0	0
$949$	−40.4562	−1.31326
$950$	0	0
$951$	1.92976	0.0625767
$952$	0	0
$953$	38.8103i	1.25719i	0.777733	+	0.628594i	$0.216369\pi$
$953$	38.8103i	1.25719i	−0.777733	+	0.628594i	$0.783631\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	4.53162i	0.146486i
$958$	0	0
$959$	−2.50717	−0.0809606
$960$	0	0
$961$	−23.0458	−0.743413
$962$	0	0
$963$	3.92935i	0.126621i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 47.1862i	− 1.51741i	−0.651437	−	0.758703i	$-0.725834\pi$
$967$	− 47.1862i	− 1.51741i	0.651437	−	0.758703i	$-0.274166\pi$
$968$	0	0
$969$	−44.4465	−1.42783
$970$	0	0
$971$	40.0772	1.28614	0.643069	−	0.765809i	$-0.277661\pi$
$971$	40.0772	1.28614	0.643069	+	0.765809i	$0.277661\pi$
$972$	0	0
$973$	− 3.65548i	− 0.117189i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 2.29873i	− 0.0735430i	−0.999324	−	0.0367715i	$-0.988293\pi$
$977$	− 2.29873i	− 0.0735430i	0.999324	−	0.0367715i	$-0.0117074\pi$
$978$	0	0
$979$	−57.1452	−1.82637
$980$	0	0
$981$	−3.24072	−0.103468
$982$	0	0
$983$	− 8.89578i	− 0.283731i	−0.989886	−	0.141866i	$-0.954690\pi$
$983$	− 8.89578i	− 0.283731i	0.989886	−	0.141866i	$-0.0453101\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 2.32007i	− 0.0738486i
$988$	0	0
$989$	31.4684	1.00064
$990$	0	0
$991$	37.5652	1.19330	0.596649	−	0.802503i	$-0.296499\pi$
$991$	37.5652	1.19330	0.596649	+	0.802503i	$0.296499\pi$
$992$	0	0
$993$	46.2542i	1.46783i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	25.8081i	0.817351i	0.912680	+	0.408675i	$0.134009\pi$
$997$	25.8081i	0.817351i	−0.912680	+	0.408675i	$0.865991\pi$
$998$	0	0
$999$	−57.6929	−1.82532

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2900.2.c.f.349.4		6
5.2	odd	4		580.2.a.c.1.2	✓	3
5.3	odd	4		2900.2.a.g.1.2		3
5.4	even	2	inner	2900.2.c.f.349.3		6
15.2	even	4		5220.2.a.x.1.2		3
20.7	even	4		2320.2.a.m.1.2		3
40.27	even	4		9280.2.a.bw.1.2		3
40.37	odd	4		9280.2.a.bk.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
580.2.a.c.1.2	✓	3	5.2	odd	4
2320.2.a.m.1.2		3	20.7	even	4
2900.2.a.g.1.2		3	5.3	odd	4
2900.2.c.f.349.3		6	5.4	even	2	inner
2900.2.c.f.349.4		6	1.1	even	1	trivial
5220.2.a.x.1.2		3	15.2	even	4
9280.2.a.bk.1.2		3	40.37	odd	4
9280.2.a.bw.1.2		3	40.27	even	4

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:	\( N \)	\(=\)	\( 2900 = 2^{2} \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2900.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.35194i q^{3} +0.648061i q^{7} +1.17226 q^{9} +O(q^{10})\)	\(q+1.35194i q^{3} +0.648061i q^{7} +1.17226 q^{9} +3.35194 q^{11} +4.17226i q^{13} -4.82032i q^{17} -6.82032 q^{19} -0.876139 q^{21} +5.52420i q^{23} +5.64064i q^{27} +1.00000 q^{29} -2.82032 q^{31} +4.53162i q^{33} +10.2281i q^{37} -5.64064 q^{39} +8.17226 q^{41} -5.69646i q^{43} +2.64806i q^{47} +6.58002 q^{49} +6.51678 q^{51} -2.87614i q^{53} -9.22066i q^{57} +13.2207 q^{59} -1.12386 q^{61} +0.759696i q^{63} -1.52420i q^{67} -7.46838 q^{69} -8.87614 q^{71} +9.69646i q^{73} +2.17226i q^{77} -8.99258 q^{79} -4.10902 q^{81} +1.94418i q^{83} +1.35194i q^{87} -17.0484 q^{89} -2.70388 q^{91} -3.81290i q^{93} +13.3519i q^{97} +3.92935 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 22 q^{9}+O(q^{10}) \) 6 * q - 22 * q^9	\( 6 q - 22 q^{9} + 16 q^{11} - 16 q^{19} + 32 q^{21} + 6 q^{29} + 8 q^{31} + 16 q^{39} + 20 q^{41} - 6 q^{49} - 48 q^{51} - 16 q^{59} - 44 q^{61} - 24 q^{69} - 16 q^{71} + 46 q^{81} - 36 q^{89} - 8 q^{91} - 72 q^{99}+O(q^{100}) \) 6 * q - 22 * q^9 + 16 * q^11 - 16 * q^19 + 32 * q^21 + 6 * q^29 + 8 * q^31 + 16 * q^39 + 20 * q^41 - 6 * q^49 - 48 * q^51 - 16 * q^59 - 44 * q^61 - 24 * q^69 - 16 * q^71 + 46 * q^81 - 36 * q^89 - 8 * q^91 - 72 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more