LMFDB - Embedded newform 2300.2.i.f.1793.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2300,2,Mod(1057,2300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2300, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 1, 2]))

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

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

chi := DirichletCharacter("2300.1057");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2300 = 2^{2} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2300.i (of order $4$, degree $2$, 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:	$18.3655924649$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 74x^{12} + 1357x^{8} + 3177x^{4} + 1296 $ x^16 + 74x^12 + 1357x^8 + 3177*x^4 + 1296
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1793.8
Root			$1.84123 + 1.84123i$ of defining polynomial
Character	$\chi$	$=$	2300.1793
Dual form			2300.2.i.f.1057.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.84123 + 1.84123i) q^{3} +(-2.81318 - 2.81318i) q^{7} +3.78026i q^{9} +O(q^{10})$	$q+(1.84123 + 1.84123i) q^{3} +(-2.81318 - 2.81318i) q^{7} +3.78026i q^{9} +0.669645i q^{11} +(1.35199 + 1.35199i) q^{13} +(2.92239 + 2.92239i) q^{17} +5.33036 q^{19} -10.3594i q^{21} +(2.40858 + 4.14713i) q^{23} +(-1.43663 + 1.43663i) q^{27} +7.95997i q^{29} -0.930892 q^{31} +(-1.23297 + 1.23297i) q^{33} +(-6.08282 - 6.08282i) q^{37} +4.97865i q^{39} +3.17972 q^{41} +(6.02274 - 6.02274i) q^{43} +(-0.701125 + 0.701125i) q^{47} +8.82801i q^{49} +10.7616i q^{51} +(-6.13195 + 6.13195i) q^{53} +(9.81441 + 9.81441i) q^{57} +6.43123i q^{59} +14.7189i q^{61} +(10.6346 - 10.6346i) q^{63} +(2.49862 + 2.49862i) q^{67} +(-3.20107 + 12.0706i) q^{69} +0.918472 q^{71} +(-6.06648 - 6.06648i) q^{73} +(1.88383 - 1.88383i) q^{77} +8.35050 q^{79} +6.05044 q^{81} +(-8.60883 + 8.60883i) q^{83} +(-14.6561 + 14.6561i) q^{87} +1.23842 q^{89} -7.60679i q^{91} +(-1.71399 - 1.71399i) q^{93} +(4.61181 + 4.61181i) q^{97} -2.53143 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 68 q^{19} - 4 q^{41} - 56 q^{69} - 8 q^{71} - 28 q^{79} + 64 q^{81} + 120 q^{89} - 28 q^{99}+O(q^{100}) $ 16 * q + 68 * q^19 - 4 * q^41 - 56 * q^69 - 8 * q^71 - 28 * q^79 + 64 * q^81 + 120 * q^89 - 28 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2300\mathbb{Z}\right)^\times$.

$n$	$277$	$1151$	$1201$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$-1$

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
$3$	1.84123	+	1.84123i	1.06303	+	1.06303i	0.997875	+	0.0651597i	$0.0207557\pi$
$3$	1.84123	+	1.84123i	1.06303	+	1.06303i	0.0651597	+	0.997875i	$0.479244\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−2.81318	−	2.81318i	−1.06328	−	1.06328i	−0.997857	−	0.0654262i	$-0.979159\pi$
$7$	−2.81318	−	2.81318i	−1.06328	−	1.06328i	−0.0654262	−	0.997857i	$-0.520841\pi$
$8$		0			0
$9$			3.78026i			1.26009i
$10$		0			0
$11$			0.669645i			0.201906i	0.994891	+	0.100953i	$0.0321891\pi$
$11$			0.669645i			0.201906i	−0.994891	+	0.100953i	$0.967811\pi$
$12$		0			0
$13$	1.35199	+	1.35199i	0.374974	+	0.374974i	0.869285	−	0.494311i	$-0.164580\pi$
$13$	1.35199	+	1.35199i	0.374974	+	0.374974i	−0.494311	+	0.869285i	$0.664580\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.92239	+	2.92239i	0.708784	+	0.708784i	0.966279	−	0.257496i	$-0.0828973\pi$
$17$	2.92239	+	2.92239i	0.708784	+	0.708784i	−0.257496	+	0.966279i	$0.582897\pi$
$18$		0			0
$19$	5.33036			1.22287			0.611434	−	0.791296i	$-0.290593\pi$
$19$	5.33036			1.22287			0.611434	+	0.791296i	$0.290593\pi$
$20$		0			0
$21$		−	10.3594i		−	2.26061i
$22$		0			0
$23$	2.40858	+	4.14713i	0.502224	+	0.864737i
$23$	2.40858	+	4.14713i	0.502224	+	0.864737i
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−1.43663	+	1.43663i	−0.276479	+	0.276479i
$28$		0			0
$29$			7.95997i			1.47813i	0.673634	+	0.739065i	$0.264732\pi$
$29$			7.95997i			1.47813i	−0.673634	+	0.739065i	$0.735268\pi$
$30$		0			0
$31$	−0.930892			−0.167193			−0.0835965	−	0.996500i	$-0.526641\pi$
$31$	−0.930892			−0.167193			−0.0835965	+	0.996500i	$0.526641\pi$
$32$		0			0
$33$	−1.23297	+	1.23297i	−0.214633	+	0.214633i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−6.08282	−	6.08282i	−1.00001	−	1.00001i	−1.00000		9.09363e-6i	$-0.999997\pi$
$37$	−6.08282	−	6.08282i	−1.00001	−	1.00001i	−9.09363e−6	−	1.00000i	$-0.500003\pi$
$38$		0			0
$39$			4.97865i			0.797221i
$40$		0			0
$41$	3.17972			0.496589			0.248294	−	0.968685i	$-0.420130\pi$
$41$	3.17972			0.496589			0.248294	+	0.968685i	$0.420130\pi$
$42$		0			0
$43$	6.02274	−	6.02274i	0.918460	−	0.918460i	−0.0784576	−	0.996917i	$-0.525000\pi$
$43$	6.02274	−	6.02274i	0.918460	−	0.918460i	0.996917	+	0.0784576i	$0.0249995\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−0.701125	+	0.701125i	−0.102270	+	0.102270i	−0.756390	−	0.654121i	$-0.773039\pi$
$47$	−0.701125	+	0.701125i	−0.102270	+	0.102270i	0.654121	+	0.756390i	$0.273039\pi$
$48$		0			0
$49$			8.82801i			1.26114i
$50$		0			0
$51$			10.7616i			1.50692i
$52$		0			0
$53$	−6.13195	+	6.13195i	−0.842288	+	0.842288i	−0.989156	−	0.146868i	$-0.953081\pi$
$53$	−6.13195	+	6.13195i	−0.842288	+	0.842288i	0.146868	+	0.989156i	$0.453081\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	9.81441	+	9.81441i	1.29995	+	1.29995i
$58$		0			0
$59$			6.43123i			0.837275i	0.908154	+	0.418637i	$0.137492\pi$
$59$			6.43123i			0.837275i	−0.908154	+	0.418637i	$0.862508\pi$
$60$		0			0
$61$			14.7189i			1.88456i	0.334828	+	0.942279i	$0.391322\pi$
$61$			14.7189i			1.88456i	−0.334828	+	0.942279i	$0.608678\pi$
$62$		0			0
$63$	10.6346	−	10.6346i	1.33983	−	1.33983i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	2.49862	+	2.49862i	0.305255	+	0.305255i	0.843066	−	0.537811i	$-0.180748\pi$
$67$	2.49862	+	2.49862i	0.305255	+	0.305255i	−0.537811	+	0.843066i	$0.680748\pi$
$68$		0			0
$69$	−3.20107	+	12.0706i	−0.385364	+	1.45313i
$70$		0			0
$71$	0.918472			0.109003			0.0545013	−	0.998514i	$-0.482643\pi$
$71$	0.918472			0.109003			0.0545013	+	0.998514i	$0.482643\pi$
$72$		0			0
$73$	−6.06648	−	6.06648i	−0.710027	−	0.710027i	0.256513	−	0.966541i	$-0.417426\pi$
$73$	−6.06648	−	6.06648i	−0.710027	−	0.710027i	−0.966541	+	0.256513i	$0.917426\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.88383	−	1.88383i	0.214683	−	0.214683i
$78$		0			0
$79$	8.35050			0.939505			0.469752	−	0.882798i	$-0.344343\pi$
$79$	8.35050			0.939505			0.469752	+	0.882798i	$0.344343\pi$
$80$		0			0
$81$	6.05044			0.672271
$82$		0			0
$83$	−8.60883	+	8.60883i	−0.944942	+	0.944942i	−0.998561	−	0.0536194i	$-0.982924\pi$
$83$	−8.60883	+	8.60883i	−0.944942	+	0.944942i	0.0536194	+	0.998561i	$0.482924\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−14.6561	+	14.6561i	−1.57130	+	1.57130i
$88$		0			0
$89$	1.23842			0.131272			0.0656360	−	0.997844i	$-0.479092\pi$
$89$	1.23842			0.131272			0.0656360	+	0.997844i	$0.479092\pi$
$90$		0			0
$91$		−	7.60679i		−	0.797408i
$92$		0			0
$93$	−1.71399	−	1.71399i	−0.177732	−	0.177732i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	4.61181	+	4.61181i	0.468258	+	0.468258i	0.901350	−	0.433092i	$-0.142577\pi$
$97$	4.61181	+	4.61181i	0.468258	+	0.468258i	−0.433092	+	0.901350i	$0.642577\pi$
$98$		0			0
$99$	−2.53143			−0.254418
$100$		0			0
$101$	5.89087			0.586163			0.293082	−	0.956087i	$-0.405319\pi$
$101$	5.89087			0.586163			0.293082	+	0.956087i	$0.405319\pi$
$102$		0			0
$103$	2.78050	−	2.78050i	0.273971	−	0.273971i	−0.556725	−	0.830697i	$-0.687942\pi$
$103$	2.78050	−	2.78050i	0.273971	−	0.273971i	0.830697	+	0.556725i	$0.187942\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	10.2709	+	10.2709i	0.992921	+	0.992921i	0.999975	−	0.00705369i	$-0.00224528\pi$
$107$	10.2709	+	10.2709i	0.992921	+	0.992921i	−0.00705369	+	0.999975i	$0.502245\pi$
$108$		0			0
$109$	−2.89980			−0.277750			−0.138875	−	0.990310i	$-0.544349\pi$
$109$	−2.89980			−0.277750			−0.138875	+	0.990310i	$0.544349\pi$
$110$		0			0
$111$		−	22.3997i		−	2.12609i
$112$		0			0
$113$	2.88971	−	2.88971i	0.271841	−	0.271841i	−0.558000	−	0.829841i	$-0.688431\pi$
$113$	2.88971	−	2.88971i	0.271841	−	0.271841i	0.829841	+	0.558000i	$0.188431\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−5.11086	+	5.11086i	−0.472500	+	0.472500i
$118$		0			0
$119$		−	16.4424i		−	1.50728i
$120$		0			0
$121$	10.5516			0.959234
$122$		0			0
$123$	5.85459	+	5.85459i	0.527891	+	0.527891i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−4.69162	+	4.69162i	−0.416314	+	0.416314i	−0.883931	−	0.467617i	$-0.845113\pi$
$127$	−4.69162	+	4.69162i	−0.416314	+	0.416314i	0.467617	+	0.883931i	$0.345113\pi$
$128$		0			0
$129$	22.1785			1.95271
$130$		0			0
$131$	17.0077			1.48597			0.742986	−	0.669307i	$-0.233409\pi$
$131$	17.0077			1.48597			0.742986	+	0.669307i	$0.233409\pi$
$132$		0			0
$133$	−14.9953	−	14.9953i	−1.30025	−	1.30025i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−13.8224	−	13.8224i	−1.18093	−	1.18093i	−0.979505	−	0.201420i	$-0.935444\pi$
$137$	−13.8224	−	13.8224i	−1.18093	−	1.18093i	−0.201420	−	0.979505i	$-0.564556\pi$
$138$		0			0
$139$			4.91847i			0.417179i	0.978003	+	0.208590i	$0.0668873\pi$
$139$			4.91847i			0.417179i	−0.978003	+	0.208590i	$0.933113\pi$
$140$		0			0
$141$	−2.58186			−0.217432
$142$		0			0
$143$	−0.905352	+	0.905352i	−0.0757094	+	0.0757094i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−16.2544	+	16.2544i	−1.34064	+	1.34064i
$148$		0			0
$149$	−15.4223			−1.26344			−0.631722	−	0.775195i	$-0.717651\pi$
$149$	−15.4223			−1.26344			−0.631722	+	0.775195i	$0.717651\pi$
$150$		0			0
$151$	−2.23989			−0.182280			−0.0911399	−	0.995838i	$-0.529051\pi$
$151$	−2.23989			−0.182280			−0.0911399	+	0.995838i	$0.529051\pi$
$152$		0			0
$153$	−11.0474	+	11.0474i	−0.893128	+	0.893128i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−10.5745	−	10.5745i	−0.843935	−	0.843935i	0.145433	−	0.989368i	$-0.453543\pi$
$157$	−10.5745	−	10.5745i	−0.843935	−	0.843935i	−0.989368	+	0.145433i	$0.953543\pi$
$158$		0			0
$159$	−22.5807			−1.79076
$160$		0			0
$161$	4.89087	−	18.4424i	0.385454	−	1.45347i
$162$		0			0
$163$	6.46285	+	6.46285i	0.506210	+	0.506210i	0.913361	−	0.407151i	$-0.133478\pi$
$163$	6.46285	+	6.46285i	0.506210	+	0.506210i	−0.407151	+	0.913361i	$0.633478\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−8.19329	+	8.19329i	−0.634016	+	0.634016i	−0.949073	−	0.315057i	$-0.897976\pi$
$167$	−8.19329	+	8.19329i	−0.634016	+	0.634016i	0.315057	+	0.949073i	$0.397976\pi$
$168$		0			0
$169$		−	9.34425i		−	0.718789i
$170$		0			0
$171$			20.1501i			1.54092i
$172$		0			0
$173$	−15.6379	−	15.6379i	−1.18893	−	1.18893i	−0.977364	−	0.211564i	$-0.932144\pi$
$173$	−15.6379	−	15.6379i	−1.18893	−	1.18893i	−0.211564	−	0.977364i	$-0.567856\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−11.8414	+	11.8414i	−0.890052	+	0.890052i
$178$		0			0
$179$			1.18946i			0.0889041i	0.999012	+	0.0444521i	$0.0141542\pi$
$179$			1.18946i			0.0889041i	−0.999012	+	0.0444521i	$0.985846\pi$
$180$		0			0
$181$		−	5.16373i		−	0.383817i	−0.981413	−	0.191908i	$-0.938532\pi$
$181$		−	5.16373i		−	0.383817i	0.981413	−	0.191908i	$-0.0614677\pi$
$182$		0			0
$183$	−27.1008	+	27.1008i	−2.00335	+	2.00335i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−1.95696	+	1.95696i	−0.143107	+	0.143107i
$188$		0			0
$189$	8.08300			0.587952
$190$		0			0
$191$		−	5.45070i		−	0.394399i	−0.980363	−	0.197200i	$-0.936815\pi$
$191$		−	5.45070i		−	0.394399i	0.980363	−	0.197200i	$-0.0631847\pi$
$192$		0			0
$193$	−14.2287	−	14.2287i	−1.02420	−	1.02420i	−0.999700	−	0.0245025i	$-0.992200\pi$
$193$	−14.2287	−	14.2287i	−1.02420	−	1.02420i	−0.0245025	−	0.999700i	$-0.507800\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	17.1367	−	17.1367i	1.22094	−	1.22094i	0.253643	−	0.967298i	$-0.418371\pi$
$197$	17.1367	−	17.1367i	1.22094	−	1.22094i	0.967298	−	0.253643i	$-0.0816289\pi$
$198$		0			0
$199$	10.8482			0.769005			0.384503	−	0.923124i	$-0.374373\pi$
$199$	10.8482			0.769005			0.384503	+	0.923124i	$0.374373\pi$
$200$		0			0
$201$			9.20107i			0.648994i
$202$		0			0
$203$	22.3929	−	22.3929i	1.57167	−	1.57167i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−15.6772	+	9.10506i	−1.08964	+	0.632845i
$208$		0			0
$209$			3.56944i			0.246904i
$210$		0			0
$211$	−11.3505			−0.781401			−0.390700	−	0.920518i	$-0.627767\pi$
$211$	−11.3505			−0.781401			−0.390700	+	0.920518i	$0.627767\pi$
$212$		0			0
$213$	1.69112	+	1.69112i	0.115873	+	0.115873i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	2.61877	+	2.61877i	0.177774	+	0.177774i
$218$		0			0
$219$		−	22.3396i		−	1.50957i
$220$		0			0
$221$			7.90208i			0.531551i
$222$		0			0
$223$	−15.5418	−	15.5418i	−1.04075	−	1.04075i	−0.999134	−	0.0416196i	$-0.986748\pi$
$223$	−15.5418	−	15.5418i	−1.04075	−	1.04075i	−0.0416196	−	0.999134i	$-0.513252\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	3.69891	+	3.69891i	0.245505	+	0.245505i	0.819123	−	0.573618i	$-0.194461\pi$
$227$	3.69891	+	3.69891i	0.245505	+	0.245505i	−0.573618	+	0.819123i	$0.694461\pi$
$228$		0			0
$229$	24.1412			1.59529			0.797647	−	0.603125i	$-0.206078\pi$
$229$	24.1412			1.59529			0.797647	+	0.603125i	$0.206078\pi$
$230$		0			0
$231$	6.93714			0.456431
$232$		0			0
$233$	−3.30895	−	3.30895i	−0.216777	−	0.216777i	0.590362	−	0.807139i	$-0.298985\pi$
$233$	−3.30895	−	3.30895i	−0.216777	−	0.216777i	−0.807139	+	0.590362i	$0.798985\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	15.3752	+	15.3752i	0.998726	+	0.998726i
$238$		0			0
$239$		−	12.7492i		−	0.824675i	−0.911031	−	0.412337i	$-0.864712\pi$
$239$		−	12.7492i		−	0.824675i	0.911031	−	0.412337i	$-0.135288\pi$
$240$		0			0
$241$		−	19.8826i		−	1.28075i	−0.768062	−	0.640376i	$-0.778779\pi$
$241$		−	19.8826i		−	1.28075i	0.768062	−	0.640376i	$-0.221221\pi$
$242$		0			0
$243$	15.4501	+	15.4501i	0.991126	+	0.991126i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	7.20658	+	7.20658i	0.458544	+	0.458544i
$248$		0			0
$249$	−31.7017			−2.00901
$250$		0			0
$251$		−	2.49765i		−	0.157651i	−0.996888	−	0.0788253i	$-0.974883\pi$
$251$		−	2.49765i		−	0.157651i	0.996888	−	0.0788253i	$-0.0251169\pi$
$252$		0			0
$253$	−2.77711	+	1.61290i	−0.174595	+	0.101402i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	14.1216	−	14.1216i	0.880880	−	0.880880i	−0.112744	−	0.993624i	$-0.535964\pi$
$257$	14.1216	−	14.1216i	0.880880	−	0.880880i	0.993624	+	0.112744i	$0.0359640\pi$
$258$		0			0
$259$			34.2242i			2.12659i
$260$		0			0
$261$	−30.0907			−1.86257
$262$		0			0
$263$	−13.1966	+	13.1966i	−0.813740	+	0.813740i	−0.985192	−	0.171452i	$-0.945154\pi$
$263$	−13.1966	+	13.1966i	−0.813740	+	0.813740i	0.171452	+	0.985192i	$0.445154\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	2.28021	+	2.28021i	0.139547	+	0.139547i
$268$		0			0
$269$			11.0299i			0.672504i	0.941772	+	0.336252i	$0.109159\pi$
$269$			11.0299i			0.672504i	−0.941772	+	0.336252i	$0.890841\pi$
$270$		0			0
$271$	17.1322			1.04071			0.520355	−	0.853950i	$-0.325800\pi$
$271$	17.1322			1.04071			0.520355	+	0.853950i	$0.325800\pi$
$272$		0			0
$273$	14.0058	−	14.0058i	0.847672	−	0.847672i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	20.5025	−	20.5025i	1.23188	−	1.23188i	0.268634	−	0.963242i	$-0.413428\pi$
$277$	20.5025	−	20.5025i	1.23188	−	1.23188i	0.963242	−	0.268634i	$-0.0865722\pi$
$278$		0			0
$279$		−	3.51901i		−	0.210678i
$280$		0			0
$281$			1.78173i			0.106289i	0.998587	+	0.0531446i	$0.0169244\pi$
$281$			1.78173i			0.106289i	−0.998587	+	0.0531446i	$0.983076\pi$
$282$		0			0
$283$	8.95608	−	8.95608i	0.532383	−	0.532383i	−0.388898	−	0.921281i	$-0.627144\pi$
$283$	8.95608	−	8.95608i	0.532383	−	0.532383i	0.921281	+	0.388898i	$0.127144\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−8.94513	−	8.94513i	−0.528015	−	0.528015i
$288$		0			0
$289$			0.0807255i			0.00474856i
$290$		0			0
$291$			16.9828i			0.995550i
$292$		0			0
$293$	−0.456450	+	0.456450i	−0.0266661	+	0.0266661i	−0.720314	−	0.693648i	$-0.756002\pi$
$293$	−0.456450	+	0.456450i	−0.0266661	+	0.0266661i	0.693648	+	0.720314i	$0.256002\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−0.962031	−	0.962031i	−0.0558227	−	0.0558227i
$298$		0			0
$299$	−2.35050	+	8.86326i	−0.135933	+	0.512576i
$300$		0			0
$301$	−33.8862			−1.95317
$302$		0			0
$303$	10.8464	+	10.8464i	0.623112	+	0.623112i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	11.4876	−	11.4876i	0.655632	−	0.655632i	−0.298712	−	0.954343i	$-0.596557\pi$
$307$	11.4876	−	11.4876i	0.655632	−	0.655632i	0.954343	+	0.298712i	$0.0965569\pi$
$308$		0			0
$309$	10.2391			0.582481
$310$		0			0
$311$	10.2702			0.582369			0.291184	−	0.956667i	$-0.405951\pi$
$311$	10.2702			0.582369			0.291184	+	0.956667i	$0.405951\pi$
$312$		0			0
$313$	2.40036	−	2.40036i	0.135676	−	0.135676i	−0.636007	−	0.771683i	$-0.719415\pi$
$313$	2.40036	−	2.40036i	0.135676	−	0.135676i	0.771683	+	0.636007i	$0.219415\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−19.0467	+	19.0467i	−1.06977	+	1.06977i	−0.0723938	+	0.997376i	$0.523064\pi$
$317$	−19.0467	+	19.0467i	−1.06977	+	1.06977i	−0.997376	+	0.0723938i	$0.976936\pi$
$318$		0			0
$319$	−5.33036			−0.298443
$320$		0			0
$321$			37.8220i			2.11102i
$322$		0			0
$323$	15.5774	+	15.5774i	0.866748	+	0.866748i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−5.33920	−	5.33920i	−0.295258	−	0.295258i
$328$		0			0
$329$	3.94479			0.217483
$330$		0			0
$331$	−1.88671			−0.103703			−0.0518514	−	0.998655i	$-0.516512\pi$
$331$	−1.88671			−0.103703			−0.0518514	+	0.998655i	$0.516512\pi$
$332$		0			0
$333$	22.9946	−	22.9946i	1.26010	−	1.26010i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−18.8523	−	18.8523i	−1.02695	−	1.02695i	−0.999627	−	0.0273235i	$-0.991302\pi$
$337$	−18.8523	−	18.8523i	−1.02695	−	1.02695i	−0.0273235	−	0.999627i	$-0.508698\pi$
$338$		0			0
$339$	10.6412			0.577953
$340$		0			0
$341$		−	0.623367i		−	0.0337572i
$342$		0			0
$343$	5.14253	−	5.14253i	0.277670	−	0.277670i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	21.8137	−	21.8137i	1.17102	−	1.17102i	0.189054	−	0.981967i	$-0.439458\pi$
$347$	21.8137	−	21.8137i	1.17102	−	1.17102i	0.981967	−	0.189054i	$-0.0605422\pi$
$348$		0			0
$349$		−	33.3093i		−	1.78300i	−0.453016	−	0.891502i	$-0.649652\pi$
$349$		−	33.3093i		−	1.78300i	0.453016	−	0.891502i	$-0.350348\pi$
$350$		0			0
$351$	−3.88461			−0.207345
$352$		0			0
$353$	5.24181	+	5.24181i	0.278993	+	0.278993i	0.832707	−	0.553714i	$-0.186790\pi$
$353$	5.24181	+	5.24181i	0.278993	+	0.278993i	−0.553714	+	0.832707i	$0.686790\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	30.2743	−	30.2743i	1.60229	−	1.60229i
$358$		0			0
$359$	−31.4662			−1.66072			−0.830360	−	0.557227i	$-0.811865\pi$
$359$	−31.4662			−1.66072			−0.830360	+	0.557227i	$0.811865\pi$
$360$		0			0
$361$	9.41269			0.495405
$362$		0			0
$363$	19.4279	+	19.4279i	1.01970	+	1.01970i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	12.7642	+	12.7642i	0.666285	+	0.666285i	0.956854	−	0.290569i	$-0.0938445\pi$
$367$	12.7642	+	12.7642i	0.666285	+	0.666285i	−0.290569	+	0.956854i	$0.593844\pi$
$368$		0			0
$369$			12.0201i			0.625744i
$370$		0			0
$371$	34.5006			1.79118
$372$		0			0
$373$	−24.4624	+	24.4624i	−1.26662	+	1.26662i	−0.318792	+	0.947825i	$0.603277\pi$
$373$	−24.4624	+	24.4624i	−1.26662	+	1.26662i	−0.947825	+	0.318792i	$0.896723\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−10.7618	+	10.7618i	−0.554261	+	0.554261i
$378$		0			0
$379$	−28.3197			−1.45468			−0.727342	−	0.686275i	$-0.759245\pi$
$379$	−28.3197			−1.45468			−0.727342	+	0.686275i	$0.759245\pi$
$380$		0			0
$381$	−17.2767			−0.885112
$382$		0			0
$383$	−0.992819	+	0.992819i	−0.0507307	+	0.0507307i	−0.732017	−	0.681286i	$-0.761421\pi$
$383$	−0.992819	+	0.992819i	−0.0507307	+	0.0507307i	0.681286	+	0.732017i	$0.261421\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	22.7675	+	22.7675i	1.15734	+	1.15734i
$388$		0			0
$389$	31.2592			1.58491			0.792453	−	0.609933i	$-0.208804\pi$
$389$	31.2592			1.58491			0.792453	+	0.609933i	$0.208804\pi$
$390$		0			0
$391$	−5.08073	+	19.1584i	−0.256943	+	0.968880i
$392$		0			0
$393$	31.3151	+	31.3151i	1.57964	+	1.57964i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	8.95068	−	8.95068i	0.449222	−	0.449222i	−0.445874	−	0.895096i	$-0.647107\pi$
$397$	8.95068	−	8.95068i	0.449222	−	0.449222i	0.895096	+	0.445874i	$0.147107\pi$
$398$		0			0
$399$		−	55.2195i		−	2.76443i
$400$		0			0
$401$			27.6435i			1.38045i	0.723594	+	0.690226i	$0.242489\pi$
$401$			27.6435i			1.38045i	−0.723594	+	0.690226i	$0.757511\pi$
$402$		0			0
$403$	−1.25856	−	1.25856i	−0.0626931	−	0.0626931i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	4.07333	−	4.07333i	0.201907	−	0.201907i
$408$		0			0
$409$			35.2603i			1.74351i	0.489942	+	0.871755i	$0.337018\pi$
$409$			35.2603i			1.74351i	−0.489942	+	0.871755i	$0.662982\pi$
$410$		0			0
$411$		−	50.9003i		−	2.51073i
$412$		0			0
$413$	18.0922	−	18.0922i	0.890260	−	0.890260i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−9.05604	+	9.05604i	−0.443476	+	0.443476i
$418$		0			0
$419$	20.5522			1.00404			0.502022	−	0.864855i	$-0.332590\pi$
$419$	20.5522			1.00404			0.502022	+	0.864855i	$0.332590\pi$
$420$		0			0
$421$			17.7182i			0.863532i	0.901986	+	0.431766i	$0.142109\pi$
$421$			17.7182i			0.863532i	−0.901986	+	0.431766i	$0.857891\pi$
$422$		0			0
$423$	−2.65043	−	2.65043i	−0.128868	−	0.128868i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	41.4069	−	41.4069i	2.00382	−	2.00382i
$428$		0			0
$429$	−3.33392			−0.160963
$430$		0			0
$431$		−	33.6435i		−	1.62055i	−0.586049	−	0.810276i	$-0.699317\pi$
$431$		−	33.6435i		−	1.62055i	0.586049	−	0.810276i	$-0.300683\pi$
$432$		0			0
$433$	−3.60805	+	3.60805i	−0.173392	+	0.173392i	−0.788468	−	0.615076i	$-0.789125\pi$
$433$	−3.60805	+	3.60805i	−0.173392	+	0.173392i	0.615076	+	0.788468i	$0.289125\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	12.8386	+	22.1057i	0.614154	+	1.05746i
$438$		0			0
$439$			2.11329i			0.100862i	0.998728	+	0.0504310i	$0.0160595\pi$
$439$			2.11329i			0.100862i	−0.998728	+	0.0504310i	$0.983941\pi$
$440$		0			0
$441$	−33.3721			−1.58915
$442$		0			0
$443$	20.8608	+	20.8608i	0.991127	+	0.991127i	0.999961	−	0.00883420i	$-0.00281205\pi$
$443$	20.8608	+	20.8608i	0.991127	+	0.991127i	−0.00883420	+	0.999961i	$0.502812\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−28.3960	−	28.3960i	−1.34308	−	1.34308i
$448$		0			0
$449$		−	24.8683i		−	1.17361i	−0.809729	−	0.586804i	$-0.800386\pi$
$449$		−	24.8683i		−	1.17361i	0.809729	−	0.586804i	$-0.199614\pi$
$450$		0			0
$451$			2.12928i			0.100264i
$452$		0			0
$453$	−4.12416	−	4.12416i	−0.193770	−	0.193770i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−12.1625	−	12.1625i	−0.568936	−	0.568936i	0.362894	−	0.931830i	$-0.381789\pi$
$457$	−12.1625	−	12.1625i	−0.568936	−	0.568936i	−0.931830	+	0.362894i	$0.881789\pi$
$458$		0			0
$459$	−8.39678			−0.391928
$460$		0			0
$461$	−7.76663			−0.361728			−0.180864	−	0.983508i	$-0.557889\pi$
$461$	−7.76663			−0.361728			−0.180864	+	0.983508i	$0.557889\pi$
$462$		0			0
$463$	−4.57959	−	4.57959i	−0.212831	−	0.212831i	0.592638	−	0.805469i	$-0.298087\pi$
$463$	−4.57959	−	4.57959i	−0.212831	−	0.212831i	−0.805469	+	0.592638i	$0.798087\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−23.9292	−	23.9292i	−1.10731	−	1.10731i	−0.993502	−	0.113811i	$-0.963694\pi$
$467$	−23.9292	−	23.9292i	−1.10731	−	1.10731i	−0.113811	−	0.993502i	$-0.536306\pi$
$468$		0			0
$469$		−	14.0582i		−	0.649146i
$470$		0			0
$471$		−	38.9401i		−	1.79426i
$472$		0			0
$473$	4.03310	+	4.03310i	0.185442	+	0.185442i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−23.1803	−	23.1803i	−1.06135	−	1.06135i
$478$		0			0
$479$	24.7308			1.12998			0.564989	−	0.825099i	$-0.308881\pi$
$479$	24.7308			1.12998			0.564989	+	0.825099i	$0.308881\pi$
$480$		0			0
$481$		−	16.4478i		−	0.749955i
$482$		0			0
$483$	42.9620	−	24.9516i	1.95484	−	1.13534i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	22.7959	−	22.7959i	1.03298	−	1.03298i	0.0335428	−	0.999437i	$-0.489321\pi$
$487$	22.7959	−	22.7959i	1.03298	−	1.03298i	0.999437	−	0.0335428i	$-0.0106790\pi$
$488$		0			0
$489$			23.7992i			1.07624i
$490$		0			0
$491$	−30.4611			−1.37469			−0.687345	−	0.726331i	$-0.741224\pi$
$491$	−30.4611			−1.37469			−0.687345	+	0.726331i	$0.741224\pi$
$492$		0			0
$493$	−23.2621	+	23.2621i	−1.04767	+	1.04767i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−2.58383	−	2.58383i	−0.115901	−	0.115901i
$498$		0			0
$499$			35.0302i			1.56817i	0.620657	+	0.784083i	$0.286866\pi$
$499$			35.0302i			1.56817i	−0.620657	+	0.784083i	$0.713134\pi$
$500$		0			0
$501$	−30.1715			−1.34796
$502$		0			0
$503$	−4.87159	+	4.87159i	−0.217213	+	0.217213i	−0.807323	−	0.590110i	$-0.799084\pi$
$503$	−4.87159	+	4.87159i	−0.217213	+	0.217213i	0.590110	+	0.807323i	$0.299084\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	17.2049	−	17.2049i	0.764097	−	0.764097i
$508$		0			0
$509$			9.95184i			0.441108i	0.975375	+	0.220554i	$0.0707865\pi$
$509$			9.95184i			0.441108i	−0.975375	+	0.220554i	$0.929214\pi$
$510$		0			0
$511$			34.1322i			1.50992i
$512$		0			0
$513$	−7.65774	+	7.65774i	−0.338098	+	0.338098i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−0.469505	−	0.469505i	−0.0206488	−	0.0206488i
$518$		0			0
$519$		−	57.5860i		−	2.52774i
$520$		0			0
$521$		−	28.2794i		−	1.23894i	−0.785019	−	0.619471i	$-0.787347\pi$
$521$		−	28.2794i		−	1.23894i	0.785019	−	0.619471i	$-0.212653\pi$
$522$		0			0
$523$	−30.2142	+	30.2142i	−1.32118	+	1.32118i	−0.408352	+	0.912825i	$0.633896\pi$
$523$	−30.2142	+	30.2142i	−1.32118	+	1.32118i	−0.912825	+	0.408352i	$0.866104\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−2.72043	−	2.72043i	−0.118504	−	0.118504i
$528$		0			0
$529$	−11.3975	+	19.9774i	−0.495541	+	0.868584i
$530$		0			0
$531$	−24.3117			−1.05504
$532$		0			0
$533$	4.29894	+	4.29894i	0.186208	+	0.186208i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−2.19006	+	2.19006i	−0.0945081	+	0.0945081i
$538$		0			0
$539$	−5.91163			−0.254632
$540$		0			0
$541$	−31.7385			−1.36454			−0.682272	−	0.731098i	$-0.739008\pi$
$541$	−31.7385			−1.36454			−0.682272	+	0.731098i	$0.739008\pi$
$542$		0			0
$543$	9.50761	−	9.50761i	0.408011	−	0.408011i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−7.77322	+	7.77322i	−0.332359	+	0.332359i	−0.853482	−	0.521123i	$-0.825513\pi$
$547$	−7.77322	+	7.77322i	−0.332359	+	0.332359i	0.521123	+	0.853482i	$0.325513\pi$
$548$		0			0
$549$	−55.6411			−2.37470
$550$		0			0
$551$			42.4295i			1.80756i
$552$		0			0
$553$	−23.4915	−	23.4915i	−0.998960	−	0.998960i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−21.9221	−	21.9221i	−0.928868	−	0.928868i	0.0687644	−	0.997633i	$-0.478094\pi$
$557$	−21.9221	−	21.9221i	−0.928868	−	0.928868i	−0.997633	+	0.0687644i	$0.978094\pi$
$558$		0			0
$559$	16.2854			0.688798
$560$		0			0
$561$	−7.20644			−0.304256
$562$		0			0
$563$	1.55833	−	1.55833i	0.0656757	−	0.0656757i	−0.673506	−	0.739182i	$-0.735212\pi$
$563$	1.55833	−	1.55833i	0.0656757	−	0.0656757i	0.739182	+	0.673506i	$0.235212\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−17.0210	−	17.0210i	−0.714814	−	0.714814i
$568$		0			0
$569$	−26.4852			−1.11032			−0.555158	−	0.831745i	$-0.687342\pi$
$569$	−26.4852			−1.11032			−0.555158	+	0.831745i	$0.687342\pi$
$570$		0			0
$571$			27.0729i			1.13297i	0.824073	+	0.566484i	$0.191697\pi$
$571$			27.0729i			1.13297i	−0.824073	+	0.566484i	$0.808303\pi$
$572$		0			0
$573$	10.0360	−	10.0360i	0.419260	−	0.419260i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−15.1469	+	15.1469i	−0.630572	+	0.630572i	−0.948212	−	0.317639i	$-0.897110\pi$
$577$	−15.1469	+	15.1469i	−0.630572	+	0.630572i	0.317639	+	0.948212i	$0.397110\pi$
$578$		0			0
$579$		−	52.3965i		−	2.17752i
$580$		0			0
$581$	48.4365			2.00948
$582$		0			0
$583$	−4.10623	−	4.10623i	−0.170063	−	0.170063i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	28.7257	−	28.7257i	1.18563	−	1.18563i	0.207373	−	0.978262i	$-0.433509\pi$
$587$	28.7257	−	28.7257i	1.18563	−	1.18563i	0.978262	−	0.207373i	$-0.0664912\pi$
$588$		0			0
$589$	−4.96198			−0.204455
$590$		0			0
$591$	63.1053			2.59581
$592$		0			0
$593$	−12.5074	−	12.5074i	−0.513619	−	0.513619i	0.402014	−	0.915633i	$-0.368310\pi$
$593$	−12.5074	−	12.5074i	−0.513619	−	0.513619i	−0.915633	+	0.402014i	$0.868310\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	19.9740	+	19.9740i	0.817479	+	0.817479i
$598$		0			0
$599$		−	25.0065i		−	1.02174i	−0.859658	−	0.510869i	$-0.829324\pi$
$599$		−	25.0065i		−	1.02174i	0.859658	−	0.510869i	$-0.170676\pi$
$600$		0			0
$601$	−48.4268			−1.97537			−0.987685	−	0.156454i	$-0.949994\pi$
$601$	−48.4268			−1.97537			−0.987685	+	0.156454i	$0.949994\pi$
$602$		0			0
$603$	−9.44543	+	9.44543i	−0.384648	+	0.384648i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−1.09298	+	1.09298i	−0.0443625	+	0.0443625i	−0.728940	−	0.684578i	$-0.759987\pi$
$607$	−1.09298	+	1.09298i	−0.0443625	+	0.0443625i	0.684578	+	0.728940i	$0.259987\pi$
$608$		0			0
$609$	82.4608			3.34148
$610$		0			0
$611$	−1.89583			−0.0766969
$612$		0			0
$613$	−6.24984	+	6.24984i	−0.252429	+	0.252429i	−0.821966	−	0.569537i	$-0.807123\pi$
$613$	−6.24984	+	6.24984i	−0.252429	+	0.252429i	0.569537	+	0.821966i	$0.307123\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	31.7070	+	31.7070i	1.27648	+	1.27648i	0.942627	+	0.333849i	$0.108347\pi$
$617$	31.7070	+	31.7070i	1.27648	+	1.27648i	0.333849	+	0.942627i	$0.391653\pi$
$618$		0			0
$619$	−22.1590			−0.890647			−0.445323	−	0.895370i	$-0.646911\pi$
$619$	−22.1590			−0.890647			−0.445323	+	0.895370i	$0.646911\pi$
$620$		0			0
$621$	−9.41814	−	2.49765i	−0.377937	−	0.100227i
$622$		0			0
$623$	−3.48389	−	3.48389i	−0.139579	−	0.139579i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−6.57217	+	6.57217i	−0.262467	+	0.262467i
$628$		0			0
$629$		−	35.5527i		−	1.41758i
$630$		0			0
$631$		−	27.5694i		−	1.09752i	−0.835979	−	0.548761i	$-0.815100\pi$
$631$		−	27.5694i		−	1.09752i	0.835979	−	0.548761i	$-0.184900\pi$
$632$		0			0
$633$	−20.8989	−	20.8989i	−0.830656	−	0.830656i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−11.9354	+	11.9354i	−0.472897	+	0.472897i
$638$		0			0
$639$			3.47206i			0.137352i
$640$		0			0
$641$			0.623367i			0.0246215i	0.999924	+	0.0123108i	$0.00391873\pi$
$641$			0.623367i			0.0246215i	−0.999924	+	0.0123108i	$0.996081\pi$
$642$		0			0
$643$	0.347244	−	0.347244i	0.0136940	−	0.0136940i	−0.700227	−	0.713921i	$-0.746918\pi$
$643$	0.347244	−	0.347244i	0.0136940	−	0.0136940i	0.713921	+	0.700227i	$0.246918\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1.90243	−	1.90243i	0.0747924	−	0.0747924i	−0.668721	−	0.743513i	$-0.733158\pi$
$647$	1.90243	−	1.90243i	0.0747924	−	0.0747924i	0.743513	+	0.668721i	$0.233158\pi$
$648$		0			0
$649$	−4.30664			−0.169050
$650$		0			0
$651$			9.64351i			0.377959i
$652$		0			0
$653$	19.1231	+	19.1231i	0.748346	+	0.748346i	0.974168	−	0.225823i	$-0.0725070\pi$
$653$	19.1231	+	19.1231i	0.748346	+	0.748346i	−0.225823	+	0.974168i	$0.572507\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	22.9328	−	22.9328i	0.894695	−	0.894695i
$658$		0			0
$659$	13.0718			0.509205			0.254602	−	0.967046i	$-0.418055\pi$
$659$	13.0718			0.509205			0.254602	+	0.967046i	$0.418055\pi$
$660$		0			0
$661$		−	7.31982i		−	0.284708i	−0.989816	−	0.142354i	$-0.954533\pi$
$661$		−	7.31982i		−	0.284708i	0.989816	−	0.142354i	$-0.0454671\pi$
$662$		0			0
$663$	−14.5495	+	14.5495i	−0.565057	+	0.565057i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−33.0111	+	19.1723i	−1.27819	+	0.742353i
$668$		0			0
$669$		−	57.2319i		−	2.21271i
$670$		0			0
$671$	−9.85642			−0.380503
$672$		0			0
$673$	−18.4603	−	18.4603i	−0.711594	−	0.711594i	0.255275	−	0.966869i	$-0.417834\pi$
$673$	−18.4603	−	18.4603i	−0.711594	−	0.711594i	−0.966869	+	0.255275i	$0.917834\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	35.4059	+	35.4059i	1.36076	+	1.36076i	0.872951	+	0.487809i	$0.162204\pi$
$677$	35.4059	+	35.4059i	1.36076	+	1.36076i	0.487809	+	0.872951i	$0.337796\pi$
$678$		0			0
$679$		−	25.9477i		−	0.995783i
$680$		0			0
$681$			13.6211i			0.521961i
$682$		0			0
$683$	−24.4514	−	24.4514i	−0.935606	−	0.935606i	0.0624424	−	0.998049i	$-0.480111\pi$
$683$	−24.4514	−	24.4514i	−0.935606	−	0.935606i	−0.998049	+	0.0624424i	$0.980111\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	44.4494	+	44.4494i	1.69585	+	1.69585i
$688$		0			0
$689$	−16.5807			−0.631673
$690$		0			0
$691$	28.0112			1.06560			0.532798	−	0.846242i	$-0.321140\pi$
$691$	28.0112			1.06560			0.532798	+	0.846242i	$0.321140\pi$
$692$		0			0
$693$	7.12137	+	7.12137i	0.270519	+	0.270519i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	9.29238	+	9.29238i	0.351974	+	0.351974i
$698$		0			0
$699$		−	12.1851i		−	0.460882i
$700$		0			0
$701$			9.05991i			0.342188i	0.985255	+	0.171094i	$0.0547302\pi$
$701$			9.05991i			0.342188i	−0.985255	+	0.171094i	$0.945270\pi$
$702$		0			0
$703$	−32.4236	−	32.4236i	−1.22288	−	1.22288i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−16.5721	−	16.5721i	−0.623258	−	0.623258i
$708$		0			0
$709$	−39.5029			−1.48356			−0.741781	−	0.670642i	$-0.766019\pi$
$709$	−39.5029			−1.48356			−0.741781	+	0.670642i	$0.766019\pi$
$710$		0			0
$711$			31.5670i			1.18386i
$712$		0			0
$713$	−2.24213	−	3.86053i	−0.0839684	−	0.144578i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	23.4741	−	23.4741i	0.876658	−	0.876658i
$718$		0			0
$719$			0.0568181i			0.00211896i	0.999999	+	0.00105948i	$0.000337243\pi$
$719$			0.0568181i			0.00211896i	−0.999999	+	0.00105948i	$0.999663\pi$
$720$		0			0
$721$	−15.6441			−0.582618
$722$		0			0
$723$	36.6084	−	36.6084i	1.36148	−	1.36148i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	36.2043	+	36.2043i	1.34274	+	1.34274i	0.893318	+	0.449425i	$0.148371\pi$
$727$	36.2043	+	36.2043i	1.34274	+	1.34274i	0.449425	+	0.893318i	$0.351629\pi$
$728$		0			0
$729$			38.7432i			1.43493i
$730$		0			0
$731$	35.2016			1.30198
$732$		0			0
$733$	17.1850	−	17.1850i	0.634742	−	0.634742i	−0.314512	−	0.949254i	$-0.601841\pi$
$733$	17.1850	−	17.1850i	0.634742	−	0.634742i	0.949254	+	0.314512i	$0.101841\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1.67319	+	1.67319i	−0.0616327	+	0.0616327i
$738$		0			0
$739$			26.6326i			0.979695i	0.871808	+	0.489847i	$0.162948\pi$
$739$			26.6326i			0.979695i	−0.871808	+	0.489847i	$0.837052\pi$
$740$		0			0
$741$			26.5379i			0.974896i
$742$		0			0
$743$	23.5571	−	23.5571i	0.864226	−	0.864226i	−0.127600	−	0.991826i	$-0.540727\pi$
$743$	23.5571	−	23.5571i	0.864226	−	0.864226i	0.991826	+	0.127600i	$0.0407274\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−32.5436	−	32.5436i	−1.19071	−	1.19071i
$748$		0			0
$749$		−	57.7876i		−	2.11151i
$750$		0			0
$751$		−	29.1828i		−	1.06489i	−0.846463	−	0.532447i	$-0.821273\pi$
$751$		−	29.1828i		−	1.06489i	0.846463	−	0.532447i	$-0.178727\pi$
$752$		0			0
$753$	4.59875	−	4.59875i	0.167588	−	0.167588i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−4.57008	−	4.57008i	−0.166102	−	0.166102i	0.619161	−	0.785264i	$-0.287473\pi$
$757$	−4.57008	−	4.57008i	−0.166102	−	0.166102i	−0.785264	+	0.619161i	$0.787473\pi$
$758$		0			0
$759$	−8.08300	−	2.14358i	−0.293394	−	0.0778071i
$760$		0			0
$761$	−18.4243			−0.667880			−0.333940	−	0.942594i	$-0.608378\pi$
$761$	−18.4243			−0.667880			−0.333940	+	0.942594i	$0.608378\pi$
$762$		0			0
$763$	8.15767	+	8.15767i	0.295328	+	0.295328i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−8.69495	+	8.69495i	−0.313956	+	0.313956i
$768$		0			0
$769$	−12.9052			−0.465372			−0.232686	−	0.972552i	$-0.574751\pi$
$769$	−12.9052			−0.465372			−0.232686	+	0.972552i	$0.574751\pi$
$770$		0			0
$771$	52.0021			1.87281
$772$		0			0
$773$	−10.0851	+	10.0851i	−0.362737	+	0.362737i	−0.864820	−	0.502083i	$-0.832567\pi$
$773$	−10.0851	+	10.0851i	−0.362737	+	0.362737i	0.502083	+	0.864820i	$0.332567\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−63.0146	+	63.0146i	−2.26064	+	2.26064i
$778$		0			0
$779$	16.9490			0.607262
$780$		0			0
$781$			0.615050i			0.0220082i
$782$		0			0
$783$	−11.4355	−	11.4355i	−0.408672	−	0.408672i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−4.30608	−	4.30608i	−0.153495	−	0.153495i	0.626182	−	0.779677i	$-0.284617\pi$
$787$	−4.30608	−	4.30608i	−0.153495	−	0.153495i	−0.779677	+	0.626182i	$0.784617\pi$
$788$		0			0
$789$	−48.5961			−1.73007
$790$		0			0
$791$	−16.2586			−0.578088
$792$		0			0
$793$	−19.8998	+	19.8998i	−0.706661	+	0.706661i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	8.43179	+	8.43179i	0.298669	+	0.298669i	0.840492	−	0.541823i	$-0.182266\pi$
$797$	8.43179	+	8.43179i	0.298669	+	0.298669i	−0.541823	+	0.840492i	$0.682266\pi$
$798$		0			0
$799$	−4.09792			−0.144974
$800$		0			0
$801$			4.68153i			0.165414i
$802$		0			0
$803$	4.06239	−	4.06239i	0.143358	−	0.143358i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−20.3086	+	20.3086i	−0.714895	+	0.714895i
$808$		0			0
$809$		−	29.6831i		−	1.04360i	−0.853067	−	0.521802i	$-0.825260\pi$
$809$		−	29.6831i		−	1.04360i	0.853067	−	0.521802i	$-0.174740\pi$
$810$		0			0
$811$	−21.5627			−0.757167			−0.378584	−	0.925567i	$-0.623589\pi$
$811$	−21.5627			−0.757167			−0.378584	+	0.925567i	$0.623589\pi$
$812$		0			0
$813$	31.5444	+	31.5444i	1.10631	+	1.10631i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	32.1034	−	32.1034i	1.12315	−	1.12315i
$818$		0			0
$819$	28.7556			1.00480
$820$		0			0
$821$	41.8339			1.46001			0.730006	−	0.683441i	$-0.239517\pi$
$821$	41.8339			1.46001			0.730006	+	0.683441i	$0.239517\pi$
$822$		0			0
$823$	4.72846	+	4.72846i	0.164824	+	0.164824i	0.784700	−	0.619876i	$-0.212817\pi$
$823$	4.72846	+	4.72846i	0.164824	+	0.164824i	−0.619876	+	0.784700i	$0.712817\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−26.1279	−	26.1279i	−0.908555	−	0.908555i	0.0876010	−	0.996156i	$-0.472080\pi$
$827$	−26.1279	−	26.1279i	−0.908555	−	0.908555i	−0.996156	+	0.0876010i	$0.972080\pi$
$828$		0			0
$829$			3.40215i			0.118161i	0.998253	+	0.0590807i	$0.0188169\pi$
$829$			3.40215i			0.118161i	−0.998253	+	0.0590807i	$0.981183\pi$
$830$		0			0
$831$	75.4997			2.61905
$832$		0			0
$833$	−25.7989	+	25.7989i	−0.893878	+	0.893878i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	1.33735	−	1.33735i	0.0462254	−	0.0462254i
$838$		0			0
$839$	−52.9698			−1.82872			−0.914361	−	0.404900i	$-0.867306\pi$
$839$	−52.9698			−1.82872			−0.914361	+	0.404900i	$0.867306\pi$
$840$		0			0
$841$	−34.3612			−1.18487
$842$		0			0
$843$	−3.28058	+	3.28058i	−0.112989	+	0.112989i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−29.6835	−	29.6835i	−1.01994	−	1.01994i
$848$		0			0
$849$	32.9804			1.13188
$850$		0			0
$851$	10.5753	−	39.8772i	0.362516	−	1.36697i
$852$		0			0
$853$	−10.4349	−	10.4349i	−0.357282	−	0.357282i	0.505528	−	0.862810i	$-0.331298\pi$
$853$	−10.4349	−	10.4349i	−0.357282	−	0.357282i	−0.862810	+	0.505528i	$0.831298\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−17.4579	+	17.4579i	−0.596351	+	0.596351i	−0.939340	−	0.342988i	$-0.888561\pi$
$857$	−17.4579	+	17.4579i	−0.596351	+	0.596351i	0.342988	+	0.939340i	$0.388561\pi$
$858$		0			0
$859$			9.59295i			0.327307i	0.986518	+	0.163654i	$0.0523279\pi$
$859$			9.59295i			0.327307i	−0.986518	+	0.163654i	$0.947672\pi$
$860$		0			0
$861$		−	32.9401i		−	1.12260i
$862$		0			0
$863$	24.4145	+	24.4145i	0.831081	+	0.831081i	0.987665	−	0.156584i	$-0.0500482\pi$
$863$	24.4145	+	24.4145i	0.831081	+	0.831081i	−0.156584	+	0.987665i	$0.550048\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−0.148634	+	0.148634i	−0.00504788	+	0.00504788i
$868$		0			0
$869$			5.59187i			0.189691i
$870$		0			0
$871$			6.75622i			0.228926i
$872$		0			0
$873$	−17.4338	+	17.4338i	−0.590045	+	0.590045i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−31.4588	+	31.4588i	−1.06229	+	1.06229i	−0.0643627	+	0.997927i	$0.520501\pi$
$877$	−31.4588	+	31.4588i	−1.06229	+	1.06229i	−0.997927	+	0.0643627i	$0.979499\pi$
$878$		0			0
$879$	−1.68086			−0.0566939
$880$		0			0
$881$		−	15.0201i		−	0.506042i	−0.967461	−	0.253021i	$-0.918576\pi$
$881$		−	15.0201i		−	0.506042i	0.967461	−	0.253021i	$-0.0814241\pi$
$882$		0			0
$883$	−7.68431	−	7.68431i	−0.258598	−	0.258598i	0.565886	−	0.824484i	$-0.308534\pi$
$883$	−7.68431	−	7.68431i	−0.258598	−	0.258598i	−0.824484	+	0.565886i	$0.808534\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	24.7287	−	24.7287i	0.830310	−	0.830310i	−0.157249	−	0.987559i	$-0.550263\pi$
$887$	24.7287	−	24.7287i	0.830310	−	0.830310i	0.987559	+	0.157249i	$0.0502626\pi$
$888$		0			0
$889$	26.3968			0.885320
$890$		0			0
$891$			4.05164i			0.135735i
$892$		0			0
$893$	−3.73725	+	3.73725i	−0.125062	+	0.125062i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−20.6471	+	11.9915i	−0.689387	+	0.400384i
$898$		0			0
$899$		−	7.40987i		−	0.247133i
$900$		0			0
$901$	−35.8399			−1.19400
$902$		0			0
$903$	−62.3922	−	62.3922i	−2.07628	−	2.07628i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	21.0258	+	21.0258i	0.698150	+	0.698150i	0.964011	−	0.265862i	$-0.0856564\pi$
$907$	21.0258	+	21.0258i	0.698150	+	0.698150i	−0.265862	+	0.964011i	$0.585656\pi$
$908$		0			0
$909$			22.2690i			0.738615i
$910$		0			0
$911$			5.49181i			0.181952i	0.995853	+	0.0909758i	$0.0289986\pi$
$911$			5.49181i			0.181952i	−0.995853	+	0.0909758i	$0.971001\pi$
$912$		0			0
$913$	−5.76486	−	5.76486i	−0.190789	−	0.190789i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−47.8459	−	47.8459i	−1.58001	−	1.58001i
$918$		0			0
$919$	42.6156			1.40576			0.702879	−	0.711309i	$-0.251897\pi$
$919$	42.6156			1.40576			0.702879	+	0.711309i	$0.251897\pi$
$920$		0			0
$921$	42.3026			1.39392
$922$		0			0
$923$	1.24176	+	1.24176i	0.0408732	+	0.0408732i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	10.5110	+	10.5110i	0.345227	+	0.345227i
$928$		0			0
$929$			52.0763i			1.70857i	0.519807	+	0.854284i	$0.326004\pi$
$929$			52.0763i			1.70857i	−0.519807	+	0.854284i	$0.673996\pi$
$930$		0			0
$931$			47.0564i			1.54221i
$932$		0			0
$933$	18.9098	+	18.9098i	0.619078	+	0.619078i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	19.7161	+	19.7161i	0.644096	+	0.644096i	0.951560	−	0.307464i	$-0.0994803\pi$
$937$	19.7161	+	19.7161i	0.644096	+	0.644096i	−0.307464	+	0.951560i	$0.599480\pi$
$938$		0			0
$939$	8.83922			0.288457
$940$		0			0
$941$			36.2420i			1.18146i	0.806870	+	0.590728i	$0.201160\pi$
$941$			36.2420i			1.18146i	−0.806870	+	0.590728i	$0.798840\pi$
$942$		0			0
$943$	7.65862	+	13.1867i	0.249399	+	0.429419i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	6.11772	−	6.11772i	0.198799	−	0.198799i	−0.600686	−	0.799485i	$-0.705106\pi$
$947$	6.11772	−	6.11772i	0.198799	−	0.198799i	0.799485	+	0.600686i	$0.205106\pi$
$948$		0			0
$949$		−	16.4036i		−	0.532484i
$950$		0			0
$951$	−70.1388			−2.27440
$952$		0			0
$953$	4.88253	−	4.88253i	0.158161	−	0.158161i	−0.623591	−	0.781751i	$-0.714327\pi$
$953$	4.88253	−	4.88253i	0.158161	−	0.158161i	0.781751	+	0.623591i	$0.214327\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−9.81441	−	9.81441i	−0.317255	−	0.317255i
$958$		0			0
$959$			77.7698i			2.51132i
$960$		0			0
$961$	−30.1334			−0.972046
$962$		0			0
$963$	−38.8265	+	38.8265i	−1.25117	+	1.25117i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−3.73083	+	3.73083i	−0.119975	+	0.119975i	−0.764545	−	0.644570i	$-0.777036\pi$
$967$	−3.73083	+	3.73083i	−0.119975	+	0.119975i	0.644570	+	0.764545i	$0.277036\pi$
$968$		0			0
$969$			57.3631i			1.84277i
$970$		0			0
$971$		−	54.6174i		−	1.75276i	−0.481625	−	0.876378i	$-0.659953\pi$
$971$		−	54.6174i		−	1.75276i	0.481625	−	0.876378i	$-0.340047\pi$
$972$		0			0
$973$	13.8366	−	13.8366i	0.443580	−	0.443580i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	30.6893	+	30.6893i	0.981836	+	0.981836i	0.999838	−	0.0180018i	$-0.00573045\pi$
$977$	30.6893	+	30.6893i	0.981836	+	0.981836i	−0.0180018	+	0.999838i	$0.505730\pi$
$978$		0			0
$979$			0.829299i			0.0265045i
$980$		0			0
$981$		−	10.9620i		−	0.349989i
$982$		0			0
$983$	1.68391	−	1.68391i	0.0537084	−	0.0537084i	−0.679742	−	0.733451i	$-0.737908\pi$
$983$	1.68391	−	1.68391i	0.0537084	−	0.0537084i	0.733451	+	0.679742i	$0.237908\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	7.26326	+	7.26326i	0.231192	+	0.231192i
$988$		0			0
$989$	39.4834	+	10.4709i	1.25550	+	0.332954i
$990$		0			0
$991$	−43.4655			−1.38073			−0.690363	−	0.723463i	$-0.742549\pi$
$991$	−43.4655			−1.38073			−0.690363	+	0.723463i	$0.742549\pi$
$992$		0			0
$993$	−3.47386	−	3.47386i	−0.110240	−	0.110240i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	32.3959	−	32.3959i	1.02599	−	1.02599i	0.0263368	−	0.999653i	$-0.491616\pi$
$997$	32.3959	−	32.3959i	1.02599	−	1.02599i	0.999653	−	0.0263368i	$-0.00838424\pi$
$998$		0			0
$999$	17.4775			0.552964

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2300.2.i.f.1793.8	yes	16
5.2	odd	4		2300.2.i.e.1057.8	yes	16
5.3	odd	4		2300.2.i.e.1057.1	✓	16
5.4	even	2	inner	2300.2.i.f.1793.1	yes	16
23.22	odd	2		2300.2.i.e.1793.8	yes	16
115.22	even	4	inner	2300.2.i.f.1057.8	yes	16
115.68	even	4	inner	2300.2.i.f.1057.1	yes	16
115.114	odd	2		2300.2.i.e.1793.1	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2300.2.i.e.1057.1	✓	16	5.3	odd	4
2300.2.i.e.1057.8	yes	16	5.2	odd	4
2300.2.i.e.1793.1	yes	16	115.114	odd	2
2300.2.i.e.1793.8	yes	16	23.22	odd	2
2300.2.i.f.1057.1	yes	16	115.68	even	4	inner
2300.2.i.f.1057.8	yes	16	115.22	even	4	inner
2300.2.i.f.1793.1	yes	16	5.4	even	2	inner
2300.2.i.f.1793.8	yes	16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2300.i (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.84123 + 1.84123i) q^{3} +(-2.81318 - 2.81318i) q^{7} +3.78026i q^{9} +O(q^{10})\)	\(q+(1.84123 + 1.84123i) q^{3} +(-2.81318 - 2.81318i) q^{7} +3.78026i q^{9} +0.669645i q^{11} +(1.35199 + 1.35199i) q^{13} +(2.92239 + 2.92239i) q^{17} +5.33036 q^{19} -10.3594i q^{21} +(2.40858 + 4.14713i) q^{23} +(-1.43663 + 1.43663i) q^{27} +7.95997i q^{29} -0.930892 q^{31} +(-1.23297 + 1.23297i) q^{33} +(-6.08282 - 6.08282i) q^{37} +4.97865i q^{39} +3.17972 q^{41} +(6.02274 - 6.02274i) q^{43} +(-0.701125 + 0.701125i) q^{47} +8.82801i q^{49} +10.7616i q^{51} +(-6.13195 + 6.13195i) q^{53} +(9.81441 + 9.81441i) q^{57} +6.43123i q^{59} +14.7189i q^{61} +(10.6346 - 10.6346i) q^{63} +(2.49862 + 2.49862i) q^{67} +(-3.20107 + 12.0706i) q^{69} +0.918472 q^{71} +(-6.06648 - 6.06648i) q^{73} +(1.88383 - 1.88383i) q^{77} +8.35050 q^{79} +6.05044 q^{81} +(-8.60883 + 8.60883i) q^{83} +(-14.6561 + 14.6561i) q^{87} +1.23842 q^{89} -7.60679i q^{91} +(-1.71399 - 1.71399i) q^{93} +(4.61181 + 4.61181i) q^{97} -2.53143 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 68 q^{19} - 4 q^{41} - 56 q^{69} - 8 q^{71} - 28 q^{79} + 64 q^{81} + 120 q^{89} - 28 q^{99}+O(q^{100}) \) 16 * q + 68 * q^19 - 4 * q^41 - 56 * q^69 - 8 * q^71 - 28 * q^79 + 64 * q^81 + 120 * q^89 - 28 * q^99

Introduction

L-functions

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