LMFDB - Embedded newform 5292.2.j.i.3529.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5292,2,Mod(1765,5292)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5292, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("5292.1765");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5292.j (of order $3$, degree $2$, 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:	$42.2568327497$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(\zeta_{3})$
Twist minimal:	no (minimal twist has level 1764)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			3529.3
Character	$\chi$	$=$	5292.3529
Dual form			5292.2.j.i.1765.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.19243 - 2.06535i) q^{5} +(-1.12978 + 1.95684i) q^{11} +(2.37884 + 4.12027i) q^{13} +4.30404 q^{17} -8.59629 q^{19} +(0.664550 + 1.15103i) q^{23} +(-0.343781 + 0.595446i) q^{25} +(3.87886 - 6.71839i) q^{29} +(-0.405320 - 0.702036i) q^{31} -4.63226 q^{37} +(-5.00426 - 8.66764i) q^{41} +(-1.74292 + 3.01883i) q^{43} +(-2.18338 + 3.78173i) q^{47} +11.6787 q^{53} +5.38874 q^{55} +(-2.40463 - 4.16495i) q^{59} +(0.575967 - 0.997604i) q^{61} +(5.67319 - 9.82626i) q^{65} +(2.06381 + 3.57463i) q^{67} +4.41593 q^{71} -12.1118 q^{73} +(4.23312 - 7.33198i) q^{79} +(-0.817808 + 1.41648i) q^{83} +(-5.13226 - 8.88934i) q^{85} +6.34310 q^{89} +(10.2505 + 17.7543i) q^{95} +(5.98278 - 10.3625i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 4 q^{11} + 8 q^{23} - 12 q^{25} + 32 q^{29} + 24 q^{37} - 32 q^{53} + 36 q^{65} + 12 q^{67} - 48 q^{71} + 12 q^{79} + 12 q^{85} - 32 q^{95}+O(q^{100}) $ 24 * q + 4 * q^11 + 8 * q^23 - 12 * q^25 + 32 * q^29 + 24 * q^37 - 32 * q^53 + 36 * q^65 + 12 * q^67 - 48 * q^71 + 12 * q^79 + 12 * q^85 - 32 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5292\mathbb{Z}\right)^\times$.

$n$	$785$	$1081$	$2647$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.19243	−	2.06535i	−0.533271	−	0.923653i	−0.999245	−	0.0388541i	$-0.987629\pi$
$5$	−1.19243	−	2.06535i	−0.533271	−	0.923653i	0.465974	−	0.884799i	$-0.345704\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.12978	+	1.95684i	−0.340642	+	0.590009i	−0.984552	−	0.175092i	$-0.943978\pi$
$11$	−1.12978	+	1.95684i	−0.340642	+	0.590009i	0.643910	+	0.765101i	$0.277311\pi$
$12$		0			0
$13$	2.37884	+	4.12027i	0.659770	+	1.14276i	0.980675	+	0.195644i	$0.0626798\pi$
$13$	2.37884	+	4.12027i	0.659770	+	1.14276i	−0.320904	+	0.947112i	$0.603987\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.30404			1.04388			0.521941	−	0.852982i	$-0.325208\pi$
$17$	4.30404			1.04388			0.521941	+	0.852982i	$0.325208\pi$
$18$		0			0
$19$	−8.59629			−1.97212			−0.986062	−	0.166378i	$-0.946793\pi$
$19$	−8.59629			−1.97212			−0.986062	+	0.166378i	$0.946793\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.664550	+	1.15103i	0.138568	+	0.240007i	0.926955	−	0.375173i	$-0.122417\pi$
$23$	0.664550	+	1.15103i	0.138568	+	0.240007i	−0.788387	+	0.615180i	$0.789083\pi$
$24$		0			0
$25$	−0.343781	+	0.595446i	−0.0687562	+	0.119089i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.87886	−	6.71839i	0.720287	−	1.24757i	−0.240598	−	0.970625i	$-0.577344\pi$
$29$	3.87886	−	6.71839i	0.720287	−	1.24757i	0.960885	−	0.276948i	$-0.0893231\pi$
$30$		0			0
$31$	−0.405320	−	0.702036i	−0.0727977	−	0.126089i	0.827329	−	0.561718i	$-0.189859\pi$
$31$	−0.405320	−	0.702036i	−0.0727977	−	0.126089i	−0.900126	+	0.435629i	$0.856526\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−4.63226			−0.761539			−0.380770	−	0.924670i	$-0.624341\pi$
$37$	−4.63226			−0.761539			−0.380770	+	0.924670i	$0.624341\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−5.00426	−	8.66764i	−0.781534	−	1.35366i	−0.931048	−	0.364898i	$-0.881104\pi$
$41$	−5.00426	−	8.66764i	−0.781534	−	1.35366i	0.149513	−	0.988760i	$-0.452229\pi$
$42$		0			0
$43$	−1.74292	+	3.01883i	−0.265793	+	0.460367i	−0.967771	−	0.251831i	$-0.918967\pi$
$43$	−1.74292	+	3.01883i	−0.265793	+	0.460367i	0.701978	+	0.712199i	$0.252300\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−2.18338	+	3.78173i	−0.318479	+	0.551622i	−0.980171	−	0.198154i	$-0.936505\pi$
$47$	−2.18338	+	3.78173i	−0.318479	+	0.551622i	0.661692	+	0.749776i	$0.269839\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	11.6787			1.60419			0.802094	−	0.597197i	$-0.203719\pi$
$53$	11.6787			1.60419			0.802094	+	0.597197i	$0.203719\pi$
$54$		0			0
$55$	5.38874			0.726617
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−2.40463	−	4.16495i	−0.313056	−	0.542230i	0.665966	−	0.745982i	$-0.268020\pi$
$59$	−2.40463	−	4.16495i	−0.313056	−	0.542230i	−0.979022	+	0.203752i	$0.934686\pi$
$60$		0			0
$61$	0.575967	−	0.997604i	0.0737450	−	0.127730i	−0.826795	−	0.562504i	$-0.809838\pi$
$61$	0.575967	−	0.997604i	0.0737450	−	0.127730i	0.900540	+	0.434774i	$0.143172\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	5.67319	−	9.82626i	0.703673	−	1.21880i
$66$		0			0
$67$	2.06381	+	3.57463i	0.252135	+	0.436710i	0.964113	−	0.265491i	$-0.0855341\pi$
$67$	2.06381	+	3.57463i	0.252135	+	0.436710i	−0.711979	+	0.702201i	$0.752201\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	4.41593			0.524074			0.262037	−	0.965058i	$-0.415606\pi$
$71$	4.41593			0.524074			0.262037	+	0.965058i	$0.415606\pi$
$72$		0			0
$73$	−12.1118			−1.41758			−0.708790	−	0.705420i	$-0.750759\pi$
$73$	−12.1118			−1.41758			−0.708790	+	0.705420i	$0.750759\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	4.23312	−	7.33198i	0.476263	−	0.824913i	−0.523367	−	0.852108i	$-0.675324\pi$
$79$	4.23312	−	7.33198i	0.476263	−	0.824913i	0.999630	+	0.0271950i	$0.00865752\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−0.817808	+	1.41648i	−0.0897661	+	0.155479i	−0.907412	−	0.420242i	$-0.861945\pi$
$83$	−0.817808	+	1.41648i	−0.0897661	+	0.155479i	0.817646	+	0.575721i	$0.195279\pi$
$84$		0			0
$85$	−5.13226	−	8.88934i	−0.556672	−	0.964184i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	6.34310			0.672367			0.336184	−	0.941796i	$-0.390864\pi$
$89$	6.34310			0.672367			0.336184	+	0.941796i	$0.390864\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	10.2505	+	17.7543i	1.05168	+	1.82156i
$96$		0			0
$97$	5.98278	−	10.3625i	0.607459	−	1.05215i	−0.384198	−	0.923251i	$-0.625522\pi$
$97$	5.98278	−	10.3625i	0.607459	−	1.05215i	0.991658	−	0.128900i	$-0.0411445\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	5.42348	−	9.39374i	0.539656	−	0.934712i	−0.459266	−	0.888299i	$-0.651888\pi$
$101$	5.42348	−	9.39374i	0.539656	−	0.934712i	0.998922	−	0.0464132i	$-0.0147791\pi$
$102$		0			0
$103$	−9.17226	−	15.8868i	−0.903769	−	1.56537i	−0.822561	−	0.568677i	$-0.807455\pi$
$103$	−9.17226	−	15.8868i	−0.903769	−	1.56537i	−0.0812086	−	0.996697i	$-0.525878\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−3.83412			−0.370658			−0.185329	−	0.982676i	$-0.559335\pi$
$107$	−3.83412			−0.370658			−0.185329	+	0.982676i	$0.559335\pi$
$108$		0			0
$109$	2.64496			0.253341			0.126671	−	0.991945i	$-0.459571\pi$
$109$	2.64496			0.253341			0.126671	+	0.991945i	$0.459571\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	2.64275	+	4.57738i	0.248609	+	0.430603i	0.963140	−	0.269000i	$-0.0866933\pi$
$113$	2.64275	+	4.57738i	0.248609	+	0.430603i	−0.714531	+	0.699604i	$0.753360\pi$
$114$		0			0
$115$	1.58486	−	2.74506i	0.147789	−	0.255978i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	2.94719	+	5.10469i	0.267927	+	0.464062i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−10.2846			−0.919880
$126$		0			0
$127$	−7.67115			−0.680704			−0.340352	−	0.940298i	$-0.610546\pi$
$127$	−7.67115			−0.680704			−0.340352	+	0.940298i	$0.610546\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−9.69007	−	16.7837i	−0.846625	−	1.46640i	−0.884202	−	0.467104i	$-0.845297\pi$
$131$	−9.69007	−	16.7837i	−0.846625	−	1.46640i	0.0375770	−	0.999294i	$-0.488036\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−4.84498	+	8.39176i	−0.413935	+	0.716956i	−0.995316	−	0.0966750i	$-0.969179\pi$
$137$	−4.84498	+	8.39176i	−0.413935	+	0.716956i	0.581381	+	0.813631i	$0.302513\pi$
$138$		0			0
$139$	3.81197	+	6.60252i	0.323327	+	0.560018i	0.981172	−	0.193135i	$-0.0618654\pi$
$139$	3.81197	+	6.60252i	0.323327	+	0.560018i	−0.657846	+	0.753153i	$0.728532\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−10.7503			−0.898981
$144$		0			0
$145$	−18.5011			−1.53643
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−4.54590	−	7.87372i	−0.372414	−	0.645041i	0.617522	−	0.786554i	$-0.288137\pi$
$149$	−4.54590	−	7.87372i	−0.372414	−	0.645041i	−0.989936	+	0.141513i	$0.954803\pi$
$150$		0			0
$151$	10.1291	−	17.5441i	0.824294	−	1.42772i	−0.0781631	−	0.996941i	$-0.524905\pi$
$151$	10.1291	−	17.5441i	0.824294	−	1.42772i	0.902457	−	0.430779i	$-0.141761\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−0.966633	+	1.67426i	−0.0776418	+	0.134480i
$156$		0			0
$157$	−4.18075	−	7.24127i	−0.333660	−	0.577917i	0.649566	−	0.760305i	$-0.274951\pi$
$157$	−4.18075	−	7.24127i	−0.333660	−	0.577917i	−0.983227	+	0.182388i	$0.941617\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−19.3839			−1.51826			−0.759132	−	0.650937i	$-0.774376\pi$
$163$	−19.3839			−1.51826			−0.759132	+	0.650937i	$0.774376\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	8.84158	+	15.3141i	0.684182	+	1.18504i	0.973693	+	0.227864i	$0.0731740\pi$
$167$	8.84158	+	15.3141i	0.684182	+	1.18504i	−0.289511	+	0.957175i	$0.593493\pi$
$168$		0			0
$169$	−4.81772	+	8.34454i	−0.370594	+	0.641888i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	10.5928	−	18.3473i	0.805356	−	1.39492i	−0.110695	−	0.993854i	$-0.535308\pi$
$173$	10.5928	−	18.3473i	0.805356	−	1.39492i	0.916051	−	0.401063i	$-0.131359\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−12.9878			−0.970753			−0.485376	−	0.874305i	$-0.661317\pi$
$179$	−12.9878			−0.970753			−0.485376	+	0.874305i	$0.661317\pi$
$180$		0			0
$181$	−16.8238			−1.25050			−0.625251	−	0.780423i	$-0.715004\pi$
$181$	−16.8238			−1.25050			−0.625251	+	0.780423i	$0.715004\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	5.52365	+	9.56725i	0.406107	+	0.703398i
$186$		0			0
$187$	−4.86262	+	8.42230i	−0.355590	+	0.615899i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	4.58561	−	7.94251i	0.331803	−	0.574700i	−0.651062	−	0.759024i	$-0.725676\pi$
$191$	4.58561	−	7.94251i	0.331803	−	0.574700i	0.982865	+	0.184324i	$0.0590096\pi$
$192$		0			0
$193$	12.8153	+	22.1968i	0.922466	+	1.59776i	0.795586	+	0.605840i	$0.207163\pi$
$193$	12.8153	+	22.1968i	0.922466	+	1.59776i	0.126880	+	0.991918i	$0.459504\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−16.1036			−1.14734			−0.573668	−	0.819088i	$-0.694480\pi$
$197$	−16.1036			−1.14734			−0.573668	+	0.819088i	$0.694480\pi$
$198$		0			0
$199$	−19.2282			−1.36305			−0.681526	−	0.731794i	$-0.738683\pi$
$199$	−19.2282			−1.36305			−0.681526	+	0.731794i	$0.738683\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−11.9345	+	20.6711i	−0.833540	+	1.44373i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	9.71192	−	16.8215i	0.671788	−	1.16357i
$210$		0			0
$211$	−12.3251	−	21.3477i	−0.848496	−	1.46964i	−0.882551	−	0.470218i	$-0.844175\pi$
$211$	−12.3251	−	21.3477i	−0.848496	−	1.46964i	0.0340549	−	0.999420i	$-0.489158\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	8.31325			0.566959
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	10.2386	+	17.7338i	0.688723	+	1.19290i
$222$		0			0
$223$	−7.41074	+	12.8358i	−0.496260	+	0.859547i	−0.999991	−	0.00431335i	$-0.998627\pi$
$223$	−7.41074	+	12.8358i	−0.496260	+	0.859547i	0.503731	+	0.863861i	$0.331960\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	9.07752	−	15.7227i	0.602496	−	1.04355i	−0.389946	−	0.920838i	$-0.627506\pi$
$227$	9.07752	−	15.7227i	0.602496	−	1.04355i	0.992442	−	0.122716i	$-0.0391604\pi$
$228$		0			0
$229$	4.75346	+	8.23323i	0.314117	+	0.544067i	0.979249	−	0.202659i	$-0.0649581\pi$
$229$	4.75346	+	8.23323i	0.314117	+	0.544067i	−0.665132	+	0.746726i	$0.731625\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−20.3431			−1.33272			−0.666360	−	0.745630i	$-0.732149\pi$
$233$	−20.3431			−1.33272			−0.666360	+	0.745630i	$0.732149\pi$
$234$		0			0
$235$	10.4141			0.679343
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−8.90544	−	15.4247i	−0.576045	−	0.997739i	−0.995927	−	0.0901607i	$-0.971262\pi$
$239$	−8.90544	−	15.4247i	−0.576045	−	0.997739i	0.419882	−	0.907579i	$-0.362071\pi$
$240$		0			0
$241$	3.14437	−	5.44620i	0.202546	−	0.350821i	−0.746802	−	0.665047i	$-0.768412\pi$
$241$	3.14437	−	5.44620i	0.202546	−	0.350821i	0.949348	+	0.314226i	$0.101745\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−20.4492	−	35.4190i	−1.30115	−	2.25366i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−25.1868			−1.58978			−0.794889	−	0.606755i	$-0.792471\pi$
$251$	−25.1868			−1.58978			−0.794889	+	0.606755i	$0.792471\pi$
$252$		0			0
$253$	−3.00318			−0.188809
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−14.3401	−	24.8379i	−0.894514	−	1.54934i	−0.834405	−	0.551152i	$-0.814189\pi$
$257$	−14.3401	−	24.8379i	−0.894514	−	1.54934i	−0.0601087	−	0.998192i	$-0.519145\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−7.20840	+	12.4853i	−0.444489	+	0.769877i	−0.998016	−	0.0629537i	$-0.979948\pi$
$263$	−7.20840	+	12.4853i	−0.444489	+	0.769877i	0.553528	+	0.832831i	$0.313281\pi$
$264$		0			0
$265$	−13.9260	−	24.1205i	−0.855468	−	1.48171i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−12.8999			−0.786524			−0.393262	−	0.919426i	$-0.628653\pi$
$269$	−12.8999			−0.786524			−0.393262	+	0.919426i	$0.628653\pi$
$270$		0			0
$271$	19.4622			1.18224			0.591122	−	0.806582i	$-0.298685\pi$
$271$	19.4622			1.18224			0.591122	+	0.806582i	$0.298685\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−0.776794	−	1.34545i	−0.0468424	−	0.0811335i
$276$		0			0
$277$	−7.95620	+	13.7805i	−0.478042	+	0.827993i	−0.999683	−	0.0251721i	$-0.991987\pi$
$277$	−7.95620	+	13.7805i	−0.478042	+	0.827993i	0.521641	+	0.853165i	$0.325320\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	13.9998	−	24.2484i	0.835158	−	1.44654i	−0.0587432	−	0.998273i	$-0.518709\pi$
$281$	13.9998	−	24.2484i	0.835158	−	1.44654i	0.893901	−	0.448263i	$-0.147957\pi$
$282$		0			0
$283$	2.62345	+	4.54394i	0.155948	+	0.270109i	0.933404	−	0.358828i	$-0.116824\pi$
$283$	2.62345	+	4.54394i	0.155948	+	0.270109i	−0.777456	+	0.628937i	$0.783490\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	1.52473			0.0896898
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1.65939	+	2.87415i	0.0969428	+	0.167910i	0.910418	−	0.413690i	$-0.135760\pi$
$293$	1.65939	+	2.87415i	0.0969428	+	0.167910i	−0.813475	+	0.581600i	$0.802427\pi$
$294$		0			0
$295$	−5.73471	+	9.93282i	−0.333888	+	0.578311i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−3.16171	+	5.47625i	−0.182847	+	0.316700i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−2.74720			−0.157304
$306$		0			0
$307$	−12.4703			−0.711715			−0.355857	−	0.934540i	$-0.615811\pi$
$307$	−12.4703			−0.711715			−0.355857	+	0.934540i	$0.615811\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−11.3383	−	19.6385i	−0.642936	−	1.11360i	−0.984774	−	0.173840i	$-0.944383\pi$
$311$	−11.3383	−	19.6385i	−0.642936	−	1.11360i	0.341838	−	0.939759i	$-0.388951\pi$
$312$		0			0
$313$	3.16108	−	5.47515i	0.178675	−	0.309474i	−0.762752	−	0.646691i	$-0.776152\pi$
$313$	3.16108	−	5.47515i	0.178675	−	0.309474i	0.941427	+	0.337217i	$0.109486\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−7.79041	+	13.4934i	−0.437553	+	0.757864i	−0.997500	−	0.0706643i	$-0.977488\pi$
$317$	−7.79041	+	13.4934i	−0.437553	+	0.757864i	0.559947	+	0.828528i	$0.310821\pi$
$318$		0			0
$319$	8.76453	+	15.1806i	0.490719	+	0.849951i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−36.9987			−2.05867
$324$		0			0
$325$	−3.27119			−0.181453
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	10.8634	−	18.8159i	0.597104	−	1.03422i	−0.396142	−	0.918189i	$-0.629651\pi$
$331$	10.8634	−	18.8159i	0.597104	−	1.03422i	0.993246	−	0.116026i	$-0.0370155\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	4.92190	−	8.52499i	0.268912	−	0.465770i
$336$		0			0
$337$	4.04329	+	7.00319i	0.220252	+	0.381488i	0.954884	−	0.296977i	$-0.0959786\pi$
$337$	4.04329	+	7.00319i	0.220252	+	0.381488i	−0.734632	+	0.678465i	$0.762645\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	1.83169			0.0991917
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−0.466877	−	0.808654i	−0.0250632	−	0.0434108i	0.853222	−	0.521548i	$-0.174645\pi$
$347$	−0.466877	−	0.808654i	−0.0250632	−	0.0434108i	−0.878285	+	0.478138i	$0.841312\pi$
$348$		0			0
$349$	−1.90264	+	3.29548i	−0.101846	+	0.176403i	−0.912445	−	0.409199i	$-0.865808\pi$
$349$	−1.90264	+	3.29548i	−0.101846	+	0.176403i	0.810599	+	0.585601i	$0.199142\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−6.55974	+	11.3618i	−0.349140	+	0.604728i	−0.986097	−	0.166171i	$-0.946859\pi$
$353$	−6.55974	+	11.3618i	−0.349140	+	0.604728i	0.636957	+	0.770899i	$0.280193\pi$
$354$		0			0
$355$	−5.26569	−	9.12043i	−0.279474	−	0.484062i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	21.5658			1.13820			0.569100	−	0.822268i	$-0.307292\pi$
$359$	21.5658			1.13820			0.569100	+	0.822268i	$0.307292\pi$
$360$		0			0
$361$	54.8962			2.88927
$362$		0			0
$363$		0			0
$364$		0			0
$365$	14.4425	+	25.0151i	0.755954	+	1.30935i
$366$		0			0
$367$	−9.33095	+	16.1617i	−0.487072	+	0.843633i	−0.999890	−	0.0148645i	$-0.995268\pi$
$367$	−9.33095	+	16.1617i	−0.487072	+	0.843633i	0.512818	+	0.858497i	$0.328602\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	10.0854	+	17.4684i	0.522201	+	0.904478i	0.999666	+	0.0258279i	$0.00822218\pi$
$373$	10.0854	+	17.4684i	0.522201	+	0.904478i	−0.477466	+	0.878650i	$0.658444\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	36.9087			1.90090
$378$		0			0
$379$	18.2436			0.937110			0.468555	−	0.883434i	$-0.344775\pi$
$379$	18.2436			0.937110			0.468555	+	0.883434i	$0.344775\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−11.4282	−	19.7942i	−0.583952	−	1.01143i	−0.995005	−	0.0998233i	$-0.968172\pi$
$383$	−11.4282	−	19.7942i	−0.583952	−	1.01143i	0.411053	−	0.911611i	$-0.365161\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	6.17146	−	10.6893i	0.312905	−	0.541968i	−0.666085	−	0.745876i	$-0.732031\pi$
$389$	6.17146	−	10.6893i	0.312905	−	0.541968i	0.978990	+	0.203908i	$0.0653644\pi$
$390$		0			0
$391$	2.86025	+	4.95410i	0.144649	+	0.250539i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−20.1908			−1.01591
$396$		0			0
$397$	−26.2032			−1.31510			−0.657551	−	0.753410i	$-0.728408\pi$
$397$	−26.2032			−1.31510			−0.657551	+	0.753410i	$0.728408\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	1.95559	+	3.38718i	0.0976575	+	0.169148i	0.910715	−	0.413036i	$-0.135532\pi$
$401$	1.95559	+	3.38718i	0.0976575	+	0.169148i	−0.813057	+	0.582184i	$0.802198\pi$
$402$		0			0
$403$	1.92838	−	3.34006i	0.0960596	−	0.166380i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	5.23344	−	9.06459i	0.259412	−	0.449315i
$408$		0			0
$409$	−1.05065	−	1.81978i	−0.0519513	−	0.0899823i	0.838880	−	0.544316i	$-0.183211\pi$
$409$	−1.05065	−	1.81978i	−0.0519513	−	0.0899823i	−0.890832	+	0.454334i	$0.849877\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	3.90072			0.191479
$416$		0			0
$417$		0			0
$418$		0			0
$419$	11.7017	+	20.2679i	0.571664	+	0.990150i	0.996395	+	0.0848311i	$0.0270351\pi$
$419$	11.7017	+	20.2679i	0.571664	+	0.990150i	−0.424732	+	0.905319i	$0.639632\pi$
$420$		0			0
$421$	9.78341	−	16.9454i	0.476814	−	0.825866i	−0.522833	−	0.852435i	$-0.675125\pi$
$421$	9.78341	−	16.9454i	0.476814	−	0.825866i	0.999647	+	0.0265688i	$0.00845812\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.47965	+	2.56282i	−0.0717733	+	0.124315i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−6.70877			−0.323150			−0.161575	−	0.986860i	$-0.551657\pi$
$431$	−6.70877			−0.323150			−0.161575	+	0.986860i	$0.551657\pi$
$432$		0			0
$433$	9.46607			0.454910			0.227455	−	0.973789i	$-0.426959\pi$
$433$	9.46607			0.454910			0.227455	+	0.973789i	$0.426959\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−5.71267	−	9.89463i	−0.273274	−	0.473324i
$438$		0			0
$439$	−3.82386	+	6.62312i	−0.182503	+	0.316104i	−0.942732	−	0.333550i	$-0.891753\pi$
$439$	−3.82386	+	6.62312i	−0.182503	+	0.316104i	0.760229	+	0.649655i	$0.225087\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	18.0763	−	31.3091i	0.858833	−	1.48754i	−0.0142102	−	0.999899i	$-0.504523\pi$
$443$	18.0763	−	31.3091i	0.858833	−	1.48754i	0.873043	−	0.487643i	$-0.162143\pi$
$444$		0			0
$445$	−7.56371	−	13.1007i	−0.358554	−	0.621034i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−13.7337			−0.648132			−0.324066	−	0.946034i	$-0.605050\pi$
$449$	−13.7337			−0.648132			−0.324066	+	0.946034i	$0.605050\pi$
$450$		0			0
$451$	22.6149			1.06489
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.213109	−	0.369115i	0.00996881	−	0.0172665i	−0.860998	−	0.508608i	$-0.830160\pi$
$457$	0.213109	−	0.369115i	0.00996881	−	0.0172665i	0.870967	+	0.491342i	$0.163493\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	2.26453	−	3.92228i	0.105470	−	0.182679i	−0.808460	−	0.588551i	$-0.799699\pi$
$461$	2.26453	−	3.92228i	0.105470	−	0.182679i	0.913930	+	0.405872i	$0.133032\pi$
$462$		0			0
$463$	16.5598	+	28.6825i	0.769600	+	1.33299i	0.937780	+	0.347230i	$0.112878\pi$
$463$	16.5598	+	28.6825i	0.769600	+	1.33299i	−0.168179	+	0.985756i	$0.553789\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	1.51332			0.0700281			0.0350140	−	0.999387i	$-0.488852\pi$
$467$	1.51332			0.0700281			0.0350140	+	0.999387i	$0.488852\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−3.93824	−	6.82123i	−0.181080	−	0.313640i
$474$		0			0
$475$	2.95524	−	5.11863i	0.135596	−	0.234859i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	11.1463	−	19.3060i	0.509288	−	0.882112i	−0.490654	−	0.871354i	$-0.663242\pi$
$479$	11.1463	−	19.3060i	0.509288	−	0.882112i	0.999942	−	0.0107580i	$-0.00342444\pi$
$480$		0			0
$481$	−11.0194	−	19.0862i	−0.502441	−	0.870254i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−28.5362			−1.29576
$486$		0			0
$487$	4.63353			0.209965			0.104983	−	0.994474i	$-0.466521\pi$
$487$	4.63353			0.209965			0.104983	+	0.994474i	$0.466521\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	12.5858	+	21.7993i	0.567990	+	0.983788i	0.996765	+	0.0803766i	$0.0256123\pi$
$491$	12.5858	+	21.7993i	0.567990	+	0.983788i	−0.428774	+	0.903412i	$0.641054\pi$
$492$		0			0
$493$	16.6948	−	28.9162i	0.751894	−	1.30232i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	1.13727	+	1.96982i	0.0509114	+	0.0881811i	0.890358	−	0.455261i	$-0.150454\pi$
$499$	1.13727	+	1.96982i	0.0509114	+	0.0881811i	−0.839447	+	0.543442i	$0.817121\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	8.94793			0.398968			0.199484	−	0.979901i	$-0.436073\pi$
$503$	8.94793			0.398968			0.199484	+	0.979901i	$0.436073\pi$
$504$		0			0
$505$	−25.8685			−1.15113
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−16.6343	−	28.8115i	−0.737304	−	1.27705i	−0.953705	−	0.300743i	$-0.902765\pi$
$509$	−16.6343	−	28.8115i	−0.737304	−	1.27705i	0.216402	−	0.976304i	$-0.430568\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−21.8746	+	37.8878i	−0.963908	+	1.66954i
$516$		0			0
$517$	−4.93349	−	8.54505i	−0.216975	−	0.375811i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	15.1479			0.663642			0.331821	−	0.943342i	$-0.392337\pi$
$521$	15.1479			0.663642			0.331821	+	0.943342i	$0.392337\pi$
$522$		0			0
$523$	−10.4790			−0.458217			−0.229108	−	0.973401i	$-0.573581\pi$
$523$	−10.4790			−0.458217			−0.229108	+	0.973401i	$0.573581\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1.74451	−	3.02159i	−0.0759922	−	0.131622i
$528$		0			0
$529$	10.6167	−	18.3887i	0.461598	−	0.799511i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	23.8086	−	41.2378i	1.03127	−	1.78621i
$534$		0			0
$535$	4.57192	+	7.91880i	0.197661	+	0.342360i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	4.87748			0.209699			0.104850	−	0.994488i	$-0.466564\pi$
$541$	4.87748			0.209699			0.104850	+	0.994488i	$0.466564\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−3.15393	−	5.46277i	−0.135100	−	0.234000i
$546$		0			0
$547$	9.62179	−	16.6654i	0.411398	−	0.712562i	−0.583645	−	0.812009i	$-0.698374\pi$
$547$	9.62179	−	16.6654i	0.411398	−	0.712562i	0.995043	+	0.0994468i	$0.0317073\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−33.3438	+	57.7532i	−1.42049	+	2.46037i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−14.6478			−0.620646			−0.310323	−	0.950631i	$-0.600437\pi$
$557$	−14.6478			−0.620646			−0.310323	+	0.950631i	$0.600437\pi$
$558$		0			0
$559$	−16.5845			−0.701450
$560$		0			0
$561$		0			0
$562$		0			0
$563$	1.85335	+	3.21010i	0.0781095	+	0.135290i	0.902434	−	0.430828i	$-0.141778\pi$
$563$	1.85335	+	3.21010i	0.0781095	+	0.135290i	−0.824325	+	0.566117i	$0.808445\pi$
$564$		0			0
$565$	6.30259	−	10.9164i	0.265152	−	0.459257i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	15.1768	−	26.2871i	0.636246	−	1.10201i	−0.350004	−	0.936748i	$-0.613820\pi$
$569$	15.1768	−	26.2871i	0.636246	−	1.10201i	0.986250	−	0.165262i	$-0.0528470\pi$
$570$		0			0
$571$	5.88458	+	10.1924i	0.246262	+	0.426539i	0.962486	−	0.271332i	$-0.0874642\pi$
$571$	5.88458	+	10.1924i	0.246262	+	0.426539i	−0.716224	+	0.697871i	$0.754131\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.913839			−0.0381097
$576$		0			0
$577$	−3.59281			−0.149570			−0.0747852	−	0.997200i	$-0.523827\pi$
$577$	−3.59281			−0.149570			−0.0747852	+	0.997200i	$0.523827\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−13.1943	+	22.8533i	−0.546454	+	0.946485i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	4.62298	−	8.00724i	0.190811	−	0.330494i	−0.754708	−	0.656060i	$-0.772222\pi$
$587$	4.62298	−	8.00724i	0.190811	−	0.330494i	0.945519	+	0.325566i	$0.105555\pi$
$588$		0			0
$589$	3.48425	+	6.03490i	0.143566	+	0.248664i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	34.0793			1.39947			0.699735	−	0.714403i	$-0.253302\pi$
$593$	34.0793			1.39947			0.699735	+	0.714403i	$0.253302\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−3.51340	−	6.08539i	−0.143554	−	0.248642i	0.785279	−	0.619142i	$-0.212520\pi$
$599$	−3.51340	−	6.08539i	−0.143554	−	0.248642i	−0.928832	+	0.370500i	$0.879186\pi$
$600$		0			0
$601$	2.31218	−	4.00481i	0.0943158	−	0.163360i	−0.815007	−	0.579451i	$-0.803267\pi$
$601$	2.31218	−	4.00481i	0.0943158	−	0.163360i	0.909323	+	0.416091i	$0.136600\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	7.02864	−	12.1740i	0.285755	−	0.494942i
$606$		0			0
$607$	10.1041	+	17.5007i	0.410111	+	0.710333i	0.994901	−	0.100852i	$-0.0321567\pi$
$607$	10.1041	+	17.5007i	0.410111	+	0.710333i	−0.584791	+	0.811184i	$0.698823\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−20.7756			−0.840493
$612$		0			0
$613$	−17.8207			−0.719773			−0.359887	−	0.932996i	$-0.617185\pi$
$613$	−17.8207			−0.719773			−0.359887	+	0.932996i	$0.617185\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−15.0671	−	26.0969i	−0.606577	−	1.05062i	−0.991800	−	0.127799i	$-0.959209\pi$
$617$	−15.0671	−	26.0969i	−0.606577	−	1.05062i	0.385223	−	0.922824i	$-0.374125\pi$
$618$		0			0
$619$	−15.9558	+	27.6363i	−0.641318	+	1.11080i	0.343821	+	0.939035i	$0.388279\pi$
$619$	−15.9558	+	27.6363i	−0.641318	+	1.11080i	−0.985139	+	0.171760i	$0.945055\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	13.9825	+	24.2185i	0.559301	+	0.968738i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−19.9374			−0.794957
$630$		0			0
$631$	−10.1430			−0.403788			−0.201894	−	0.979407i	$-0.564710\pi$
$631$	−10.1430			−0.403788			−0.201894	+	0.979407i	$0.564710\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	9.14731	+	15.8436i	0.363000	+	0.628734i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	21.9256	−	37.9762i	0.866009	−	1.49997i	−3.28428e−5	−	1.00000i	$-0.500010\pi$
$641$	21.9256	−	37.9762i	0.866009	−	1.49997i	0.866042	−	0.499972i	$-0.166656\pi$
$642$		0			0
$643$	−1.10737	−	1.91802i	−0.0436704	−	0.0756394i	0.843364	−	0.537343i	$-0.180572\pi$
$643$	−1.10737	−	1.91802i	−0.0436704	−	0.0756394i	−0.887034	+	0.461703i	$0.847238\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	20.9266			0.822710			0.411355	−	0.911475i	$-0.365056\pi$
$647$	20.9266			0.822710			0.411355	+	0.911475i	$0.365056\pi$
$648$		0			0
$649$	10.8668			0.426560
$650$		0			0
$651$		0			0
$652$		0			0
$653$	4.01442	+	6.95317i	0.157096	+	0.272099i	0.933820	−	0.357742i	$-0.116453\pi$
$653$	4.01442	+	6.95317i	0.157096	+	0.272099i	−0.776724	+	0.629841i	$0.783120\pi$
$654$		0			0
$655$	−23.1095	+	40.0268i	−0.902962	+	1.56398i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−9.25793	+	16.0352i	−0.360638	+	0.624643i	−0.988066	−	0.154031i	$-0.950774\pi$
$659$	−9.25793	+	16.0352i	−0.360638	+	0.624643i	0.627428	+	0.778675i	$0.284108\pi$
$660$		0			0
$661$	−10.4273	−	18.0606i	−0.405574	−	0.702474i	0.588814	−	0.808268i	$-0.299595\pi$
$661$	−10.4273	−	18.0606i	−0.405574	−	0.702474i	−0.994388	+	0.105794i	$0.966262\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	10.3108			0.399236
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1.30143	+	2.25415i	0.0502412	+	0.0870204i
$672$		0			0
$673$	11.4484	−	19.8292i	0.441303	−	0.764360i	−0.556483	−	0.830859i	$-0.687850\pi$
$673$	11.4484	−	19.8292i	0.441303	−	0.764360i	0.997786	+	0.0664992i	$0.0211830\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−23.3119	+	40.3774i	−0.895948	+	1.55183i	−0.0633218	+	0.997993i	$0.520169\pi$
$677$	−23.3119	+	40.3774i	−0.895948	+	1.55183i	−0.832627	+	0.553835i	$0.813164\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	36.6203			1.40124			0.700618	−	0.713536i	$-0.252908\pi$
$683$	36.6203			1.40124			0.700618	+	0.713536i	$0.252908\pi$
$684$		0			0
$685$	23.1092			0.882958
$686$		0			0
$687$		0			0
$688$		0			0
$689$	27.7816	+	48.1192i	1.05840	+	1.83320i
$690$		0			0
$691$	−13.2586	+	22.9645i	−0.504380	+	0.873611i	0.495607	+	0.868547i	$0.334945\pi$
$691$	−13.2586	+	22.9645i	−0.504380	+	0.873611i	−0.999987	+	0.00506472i	$0.998388\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	9.09101	−	15.7461i	0.344842	−	0.597283i
$696$		0			0
$697$	−21.5385	−	37.3058i	−0.815830	−	1.41306i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	29.9931			1.13282			0.566411	−	0.824123i	$-0.308331\pi$
$701$	29.9931			1.13282			0.566411	+	0.824123i	$0.308331\pi$
$702$		0			0
$703$	39.8203			1.50185
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−5.48805	+	9.50558i	−0.206108	+	0.356990i	−0.950485	−	0.310770i	$-0.899413\pi$
$709$	−5.48805	+	9.50558i	−0.206108	+	0.356990i	0.744377	+	0.667759i	$0.232746\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	0.538712	−	0.933076i	0.0201749	−	0.0349440i
$714$		0			0
$715$	12.8189	+	22.2030i	0.479401	+	0.830346i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	13.1042			0.488705			0.244352	−	0.969687i	$-0.421425\pi$
$719$	13.1042			0.488705			0.244352	+	0.969687i	$0.421425\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	2.66696	+	4.61931i	0.0990483	+	0.171557i
$726$		0			0
$727$	−8.96026	+	15.5196i	−0.332318	+	0.575591i	−0.982966	−	0.183788i	$-0.941164\pi$
$727$	−8.96026	+	15.5196i	−0.332318	+	0.575591i	0.650648	+	0.759379i	$0.274497\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−7.50160	+	12.9932i	−0.277457	+	0.480569i
$732$		0			0
$733$	−2.95660	−	5.12098i	−0.109204	−	0.189148i	0.806244	−	0.591583i	$-0.201497\pi$
$733$	−2.95660	−	5.12098i	−0.109204	−	0.189148i	−0.915448	+	0.402436i	$0.868164\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−9.32662			−0.343550
$738$		0			0
$739$	−29.1774			−1.07331			−0.536653	−	0.843803i	$-0.680312\pi$
$739$	−29.1774			−1.07331			−0.536653	+	0.843803i	$0.680312\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	18.8512	+	32.6513i	0.691584	+	1.19786i	0.971319	+	0.237781i	$0.0764201\pi$
$743$	18.8512	+	32.6513i	0.691584	+	1.19786i	−0.279735	+	0.960077i	$0.590247\pi$
$744$		0			0
$745$	−10.8413	+	18.7777i	−0.397196	+	0.687963i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−20.3277	−	35.2085i	−0.741767	−	1.28478i	−0.951690	−	0.307060i	$-0.900655\pi$
$751$	−20.3277	−	35.2085i	−0.741767	−	1.28478i	0.209923	−	0.977718i	$-0.432679\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−48.3130			−1.75829
$756$		0			0
$757$	17.6704			0.642241			0.321120	−	0.947038i	$-0.395941\pi$
$757$	17.6704			0.642241			0.321120	+	0.947038i	$0.395941\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−0.644183	−	1.11576i	−0.0233516	−	0.0404462i	0.854113	−	0.520087i	$-0.174100\pi$
$761$	−0.644183	−	1.11576i	−0.0233516	−	0.0404462i	−0.877465	+	0.479641i	$0.840767\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	11.4405	−	19.8154i	0.413091	−	0.715494i
$768$		0			0
$769$	4.19275	+	7.26205i	0.151194	+	0.261876i	0.931667	−	0.363314i	$-0.118355\pi$
$769$	4.19275	+	7.26205i	0.151194	+	0.261876i	−0.780472	+	0.625190i	$0.785021\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−50.3060			−1.80938			−0.904691	−	0.426068i	$-0.859898\pi$
$773$	−50.3060			−1.80938			−0.904691	+	0.426068i	$0.859898\pi$
$774$		0			0
$775$	0.557366			0.0200212
$776$		0			0
$777$		0			0
$778$		0			0
$779$	43.0181	+	74.5095i	1.54128	+	2.66958i
$780$		0			0
$781$	−4.98903	+	8.64125i	−0.178521	+	0.309208i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−9.97051	+	17.2694i	−0.355863	+	0.616373i
$786$		0			0
$787$	−11.1922	−	19.3855i	−0.398960	−	0.691020i	0.594638	−	0.803994i	$-0.297296\pi$
$787$	−11.1922	−	19.3855i	−0.398960	−	0.691020i	−0.993598	+	0.112974i	$0.963962\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	5.48052			0.194619
$794$		0			0
$795$		0			0
$796$		0			0
$797$	0.706182	+	1.22314i	0.0250143	+	0.0433260i	0.878262	−	0.478181i	$-0.158704\pi$
$797$	0.706182	+	1.22314i	0.0250143	+	0.0433260i	−0.853247	+	0.521507i	$0.825370\pi$
$798$		0			0
$799$	−9.39736	+	16.2767i	−0.332455	+	0.575828i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	13.6837	−	23.7008i	0.482887	−	0.836384i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−20.0848			−0.706142			−0.353071	−	0.935596i	$-0.614863\pi$
$809$	−20.0848			−0.706142			−0.353071	+	0.935596i	$0.614863\pi$
$810$		0			0
$811$	55.7821			1.95878			0.979388	−	0.201988i	$-0.0647401\pi$
$811$	55.7821			1.95878			0.979388	+	0.201988i	$0.0647401\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	23.1140	+	40.0345i	0.809646	+	1.40235i
$816$		0			0
$817$	14.9827	−	25.9507i	0.524177	−	0.907901i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−24.4346	+	42.3219i	−0.852772	+	1.47704i	0.0259249	+	0.999664i	$0.491747\pi$
$821$	−24.4346	+	42.3219i	−0.852772	+	1.47704i	−0.878697	+	0.477380i	$0.841586\pi$
$822$		0			0
$823$	0.266319	+	0.461277i	0.00928328	+	0.0160791i	0.870630	−	0.491939i	$-0.163712\pi$
$823$	0.266319	+	0.461277i	0.00928328	+	0.0160791i	−0.861346	+	0.508018i	$0.830378\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−10.1951			−0.354518			−0.177259	−	0.984164i	$-0.556723\pi$
$827$	−10.1951			−0.354518			−0.177259	+	0.984164i	$0.556723\pi$
$828$		0			0
$829$	21.2692			0.738711			0.369355	−	0.929288i	$-0.379579\pi$
$829$	21.2692			0.738711			0.369355	+	0.929288i	$0.379579\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	21.0859	−	36.5219i	0.729709	−	1.26389i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−9.47717	+	16.4149i	−0.327188	+	0.566707i	−0.981953	−	0.189126i	$-0.939434\pi$
$839$	−9.47717	+	16.4149i	−0.327188	+	0.566707i	0.654765	+	0.755833i	$0.272768\pi$
$840$		0			0
$841$	−15.5912	−	27.0047i	−0.537626	−	0.931196i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	22.9792			0.790509
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−3.07837	−	5.33190i	−0.105525	−	0.182775i
$852$		0			0
$853$	−2.75811	+	4.77718i	−0.0944358	+	0.163568i	−0.909373	−	0.415982i	$-0.863438\pi$
$853$	−2.75811	+	4.77718i	−0.0944358	+	0.163568i	0.814937	+	0.579549i	$0.196771\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	7.92471	−	13.7260i	0.270703	−	0.468871i	−0.698339	−	0.715767i	$-0.746077\pi$
$857$	7.92471	−	13.7260i	0.270703	−	0.468871i	0.969042	+	0.246896i	$0.0794106\pi$
$858$		0			0
$859$	10.7004	+	18.5337i	0.365095	+	0.632362i	0.988791	−	0.149304i	$-0.0477033\pi$
$859$	10.7004	+	18.5337i	0.365095	+	0.632362i	−0.623697	+	0.781666i	$0.714370\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	4.03720			0.137428			0.0687140	−	0.997636i	$-0.478110\pi$
$863$	4.03720			0.137428			0.0687140	+	0.997636i	$0.478110\pi$
$864$		0			0
$865$	−50.5247			−1.71789
$866$		0			0
$867$		0			0
$868$		0			0
$869$	9.56500	+	16.5671i	0.324470	+	0.561999i
$870$		0			0
$871$	−9.81894	+	17.0069i	−0.332702	+	0.576257i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	12.6905	+	21.9806i	0.428529	+	0.742233i	0.996743	−	0.0806475i	$-0.0256988\pi$
$877$	12.6905	+	21.9806i	0.428529	+	0.742233i	−0.568214	+	0.822881i	$0.692365\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	18.7755			0.632561			0.316281	−	0.948666i	$-0.397566\pi$
$881$	18.7755			0.632561			0.316281	+	0.948666i	$0.397566\pi$
$882$		0			0
$883$	8.52167			0.286777			0.143388	−	0.989666i	$-0.454200\pi$
$883$	8.52167			0.286777			0.143388	+	0.989666i	$0.454200\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−5.05040	−	8.74755i	−0.169576	−	0.293714i	0.768695	−	0.639616i	$-0.220906\pi$
$887$	−5.05040	−	8.74755i	−0.169576	−	0.293714i	−0.938271	+	0.345902i	$0.887573\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	18.7690	−	32.5089i	0.628080	−	1.08787i
$894$		0			0
$895$	15.4870	+	26.8243i	0.517674	+	0.896638i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−6.28873			−0.209741
$900$		0			0
$901$	50.2654			1.67458
$902$		0			0
$903$		0			0
$904$		0			0
$905$	20.0612	+	34.7470i	0.666857	+	1.15503i
$906$		0			0
$907$	10.6088	−	18.3750i	0.352260	−	0.610131i	−0.634385	−	0.773017i	$-0.718747\pi$
$907$	10.6088	−	18.3750i	0.352260	−	0.610131i	0.986645	+	0.162885i	$0.0520801\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	4.46609	−	7.73550i	0.147968	−	0.256289i	−0.782508	−	0.622640i	$-0.786060\pi$
$911$	4.46609	−	7.73550i	0.147968	−	0.256289i	0.930476	+	0.366352i	$0.119393\pi$
$912$		0			0
$913$	−1.84789	−	3.20063i	−0.0611561	−	0.105925i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	34.9129			1.15167			0.575834	−	0.817566i	$-0.304677\pi$
$919$	34.9129			1.15167			0.575834	+	0.817566i	$0.304677\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	10.5048	+	18.1948i	0.345769	+	0.598889i
$924$		0			0
$925$	1.59248	−	2.75826i	0.0523605	−	0.0906911i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	5.51692	−	9.55559i	0.181004	−	0.313509i	−0.761218	−	0.648496i	$-0.775399\pi$
$929$	5.51692	−	9.55559i	0.181004	−	0.313509i	0.942223	+	0.334987i	$0.108732\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	23.1933			0.758503
$936$		0			0
$937$	13.6426			0.445686			0.222843	−	0.974854i	$-0.428466\pi$
$937$	13.6426			0.445686			0.222843	+	0.974854i	$0.428466\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−12.0278	−	20.8328i	−0.392095	−	0.679129i	0.600630	−	0.799527i	$-0.294916\pi$
$941$	−12.0278	−	20.8328i	−0.392095	−	0.679129i	−0.992726	+	0.120398i	$0.961583\pi$
$942$		0			0
$943$	6.65117	−	11.5202i	0.216592	−	0.375148i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	6.69602	−	11.5978i	0.217591	−	0.376879i	−0.736480	−	0.676460i	$-0.763513\pi$
$947$	6.69602	−	11.5978i	0.217591	−	0.376879i	0.954071	+	0.299580i	$0.0968466\pi$
$948$		0			0
$949$	−28.8120	−	49.9038i	−0.935277	−	1.61995i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−36.3444			−1.17731			−0.588655	−	0.808384i	$-0.700342\pi$
$953$	−36.3444			−1.17731			−0.588655	+	0.808384i	$0.700342\pi$
$954$		0			0
$955$	−21.8721			−0.707764
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	15.1714	−	26.2777i	0.489401	−	0.847667i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	30.5627	−	52.9362i	0.983849	−	1.70408i
$966$		0			0
$967$	4.82455	+	8.35637i	0.155147	+	0.268723i	0.933113	−	0.359584i	$-0.117082\pi$
$967$	4.82455	+	8.35637i	0.155147	+	0.268723i	−0.777965	+	0.628307i	$0.783748\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	43.3775			1.39205			0.696025	−	0.718017i	$-0.254950\pi$
$971$	43.3775			1.39205			0.696025	+	0.718017i	$0.254950\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	2.60004	+	4.50340i	0.0831826	+	0.144076i	0.904615	−	0.426229i	$-0.140158\pi$
$977$	2.60004	+	4.50340i	0.0831826	+	0.144076i	−0.821433	+	0.570305i	$0.806825\pi$
$978$		0			0
$979$	−7.16631	+	12.4124i	−0.229036	+	0.396703i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−11.6349	+	20.1522i	−0.371095	+	0.642755i	−0.989734	−	0.142920i	$-0.954351\pi$
$983$	−11.6349	+	20.1522i	−0.371095	+	0.642755i	0.618639	+	0.785675i	$0.287684\pi$
$984$		0			0
$985$	19.2025	+	33.2596i	0.611841	+	1.05974i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−4.63304			−0.147322
$990$		0			0
$991$	−39.0382			−1.24009			−0.620044	−	0.784567i	$-0.712885\pi$
$991$	−39.0382			−1.24009			−0.620044	+	0.784567i	$0.712885\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	22.9283	+	39.7130i	0.726877	+	1.25899i
$996$		0			0
$997$	−12.0905	+	20.9413i	−0.382909	+	0.663218i	−0.991477	−	0.130284i	$-0.958411\pi$
$997$	−12.0905	+	20.9413i	−0.382909	+	0.663218i	0.608568	+	0.793502i	$0.291744\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5292.2.j.i.3529.3		24
3.2	odd	2		1764.2.j.i.1177.5	yes	24
7.2	even	3		5292.2.i.j.2125.3		24
7.3	odd	6		5292.2.l.j.3313.3		24
7.4	even	3		5292.2.l.j.3313.10		24
7.5	odd	6		5292.2.i.j.2125.10		24
7.6	odd	2	inner	5292.2.j.i.3529.10		24
9.4	even	3	inner	5292.2.j.i.1765.3		24
9.5	odd	6		1764.2.j.i.589.5	✓	24
21.2	odd	6		1764.2.i.j.1537.5		24
21.5	even	6		1764.2.i.j.1537.8		24
21.11	odd	6		1764.2.l.j.961.12		24
21.17	even	6		1764.2.l.j.961.1		24
21.20	even	2		1764.2.j.i.1177.8	yes	24
63.4	even	3		5292.2.i.j.1549.3		24
63.5	even	6		1764.2.l.j.949.1		24
63.13	odd	6	inner	5292.2.j.i.1765.10		24
63.23	odd	6		1764.2.l.j.949.12		24
63.31	odd	6		5292.2.i.j.1549.10		24
63.32	odd	6		1764.2.i.j.373.5		24
63.40	odd	6		5292.2.l.j.361.3		24
63.41	even	6		1764.2.j.i.589.8	yes	24
63.58	even	3		5292.2.l.j.361.10		24
63.59	even	6		1764.2.i.j.373.8		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1764.2.i.j.373.5		24	63.32	odd	6
1764.2.i.j.373.8		24	63.59	even	6
1764.2.i.j.1537.5		24	21.2	odd	6
1764.2.i.j.1537.8		24	21.5	even	6
1764.2.j.i.589.5	✓	24	9.5	odd	6
1764.2.j.i.589.8	yes	24	63.41	even	6
1764.2.j.i.1177.5	yes	24	3.2	odd	2
1764.2.j.i.1177.8	yes	24	21.20	even	2
1764.2.l.j.949.1		24	63.5	even	6
1764.2.l.j.949.12		24	63.23	odd	6
1764.2.l.j.961.1		24	21.17	even	6
1764.2.l.j.961.12		24	21.11	odd	6
5292.2.i.j.1549.3		24	63.4	even	3
5292.2.i.j.1549.10		24	63.31	odd	6
5292.2.i.j.2125.3		24	7.2	even	3
5292.2.i.j.2125.10		24	7.5	odd	6
5292.2.j.i.1765.3		24	9.4	even	3	inner
5292.2.j.i.1765.10		24	63.13	odd	6	inner
5292.2.j.i.3529.3		24	1.1	even	1	trivial
5292.2.j.i.3529.10		24	7.6	odd	2	inner
5292.2.l.j.361.3		24	63.40	odd	6
5292.2.l.j.361.10		24	63.58	even	3
5292.2.l.j.3313.3		24	7.3	odd	6
5292.2.l.j.3313.10		24	7.4	even	3

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:	\( N \)	\(=\)	\( 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5292.j (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.19243 - 2.06535i) q^{5} +(-1.12978 + 1.95684i) q^{11} +(2.37884 + 4.12027i) q^{13} +4.30404 q^{17} -8.59629 q^{19} +(0.664550 + 1.15103i) q^{23} +(-0.343781 + 0.595446i) q^{25} +(3.87886 - 6.71839i) q^{29} +(-0.405320 - 0.702036i) q^{31} -4.63226 q^{37} +(-5.00426 - 8.66764i) q^{41} +(-1.74292 + 3.01883i) q^{43} +(-2.18338 + 3.78173i) q^{47} +11.6787 q^{53} +5.38874 q^{55} +(-2.40463 - 4.16495i) q^{59} +(0.575967 - 0.997604i) q^{61} +(5.67319 - 9.82626i) q^{65} +(2.06381 + 3.57463i) q^{67} +4.41593 q^{71} -12.1118 q^{73} +(4.23312 - 7.33198i) q^{79} +(-0.817808 + 1.41648i) q^{83} +(-5.13226 - 8.88934i) q^{85} +6.34310 q^{89} +(10.2505 + 17.7543i) q^{95} +(5.98278 - 10.3625i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 4 q^{11} + 8 q^{23} - 12 q^{25} + 32 q^{29} + 24 q^{37} - 32 q^{53} + 36 q^{65} + 12 q^{67} - 48 q^{71} + 12 q^{79} + 12 q^{85} - 32 q^{95}+O(q^{100}) \) 24 * q + 4 * q^11 + 8 * q^23 - 12 * q^25 + 32 * q^29 + 24 * q^37 - 32 * q^53 + 36 * q^65 + 12 * q^67 - 48 * q^71 + 12 * q^79 + 12 * q^85 - 32 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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