LMFDB - Embedded newform 5292.2.w.c.1097.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5292,2,Mod(521,5292)]

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, 1, 1]))

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

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

chi := DirichletCharacter("5292.521");

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.w (of order $6$, 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:	$48$
Relative dimension:	$24$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 1764)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1097.5
Character	$\chi$	$=$	5292.1097
Dual form			5292.2.w.c.521.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.31387 - 2.27569i) q^{5} +O(q^{10})$	$q+(-1.31387 - 2.27569i) q^{5} +(5.18523 + 2.99370i) q^{11} +(2.54231 + 1.46780i) q^{13} +(-0.0167322 - 0.0289811i) q^{17} +(-7.07369 - 4.08400i) q^{19} +(7.04605 - 4.06804i) q^{23} +(-0.952507 + 1.64979i) q^{25} +(0.949006 - 0.547909i) q^{29} +8.10282i q^{31} +(-0.105898 + 0.183421i) q^{37} +(-2.23087 + 3.86399i) q^{41} +(3.78001 + 6.54717i) q^{43} +7.07610 q^{47} +(1.59204 - 0.919166i) q^{53} -15.7333i q^{55} -4.73688 q^{59} +11.9827i q^{61} -7.71401i q^{65} +7.62295 q^{67} -3.52461i q^{71} +(-2.24398 + 1.29556i) q^{73} -7.17904 q^{79} +(-0.932743 - 1.61556i) q^{83} +(-0.0439680 + 0.0761548i) q^{85} +(-3.56125 + 6.16826i) q^{89} +21.4634i q^{95} +(13.2520 - 7.65104i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q+O(q^{10}) $ 48 * q	$ 48 q - 24 q^{11} + 48 q^{23} - 24 q^{25} + 48 q^{53} + 48 q^{79} - 24 q^{85}+O(q^{100}) $ 48 * q - 24 * q^11 + 48 * q^23 - 24 * q^25 + 48 * q^53 + 48 * q^79 - 24 * q^85

Display 100 coefficients 10 coefficients

Character values

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

$n$	$785$	$1081$	$2647$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{5}{6}\right)$	$1$

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
$5$	−1.31387	−	2.27569i	−0.587580	−	1.01772i	−0.994548	−	0.104277i	$-0.966747\pi$
$5$	−1.31387	−	2.27569i	−0.587580	−	1.01772i	0.406968	−	0.913442i	$-0.366586\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	5.18523	+	2.99370i	1.56341	+	0.902633i	0.996909	+	0.0785701i	$0.0250355\pi$
$11$	5.18523	+	2.99370i	1.56341	+	0.902633i	0.566498	+	0.824063i	$0.308298\pi$
$12$		0			0
$13$	2.54231	+	1.46780i	0.705110	+	0.407095i	0.809248	−	0.587467i	$-0.199875\pi$
$13$	2.54231	+	1.46780i	0.705110	+	0.407095i	−0.104138	+	0.994563i	$0.533208\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.0167322	−	0.0289811i	−0.00405817	−	0.00702895i	0.863989	−	0.503510i	$-0.167958\pi$
$17$	−0.0167322	−	0.0289811i	−0.00405817	−	0.00702895i	−0.868047	+	0.496481i	$0.834625\pi$
$18$		0			0
$19$	−7.07369	−	4.08400i	−1.62282	−	0.936933i	−0.986162	−	0.165783i	$-0.946985\pi$
$19$	−7.07369	−	4.08400i	−1.62282	−	0.936933i	−0.636653	−	0.771150i	$-0.719682\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	7.04605	−	4.06804i	1.46920	−	0.848244i	0.469799	−	0.882774i	$-0.344327\pi$
$23$	7.04605	−	4.06804i	1.46920	−	0.848244i	0.999404	+	0.0345292i	$0.0109932\pi$
$24$		0			0
$25$	−0.952507	+	1.64979i	−0.190501	+	0.329958i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	0.949006	−	0.547909i	0.176226	−	0.101744i	−0.409292	−	0.912403i	$-0.634224\pi$
$29$	0.949006	−	0.547909i	0.176226	−	0.101744i	0.585518	+	0.810659i	$0.300891\pi$
$30$		0			0
$31$			8.10282i			1.45531i	0.685944	+	0.727654i	$0.259389\pi$
$31$			8.10282i			1.45531i	−0.685944	+	0.727654i	$0.740611\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−0.105898	+	0.183421i	−0.0174095	+	0.0301542i	−0.874599	−	0.484847i	$-0.838875\pi$
$37$	−0.105898	+	0.183421i	−0.0174095	+	0.0301542i	0.857189	+	0.515001i	$0.172209\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2.23087	+	3.86399i	−0.348404	+	0.603453i	−0.985966	−	0.166946i	$-0.946610\pi$
$41$	−2.23087	+	3.86399i	−0.348404	+	0.603453i	0.637562	+	0.770399i	$0.279943\pi$
$42$		0			0
$43$	3.78001	+	6.54717i	0.576446	+	0.998434i	0.995883	+	0.0906496i	$0.0288943\pi$
$43$	3.78001	+	6.54717i	0.576446	+	0.998434i	−0.419437	+	0.907785i	$0.637772\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	7.07610			1.03216			0.516078	−	0.856542i	$-0.327391\pi$
$47$	7.07610			1.03216			0.516078	+	0.856542i	$0.327391\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	1.59204	−	0.919166i	0.218684	−	0.126257i	−0.386657	−	0.922224i	$-0.626370\pi$
$53$	1.59204	−	0.919166i	0.218684	−	0.126257i	0.605341	+	0.795967i	$0.293037\pi$
$54$		0			0
$55$		−	15.7333i		−	2.12148i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.73688			−0.616689			−0.308344	−	0.951275i	$-0.599775\pi$
$59$	−4.73688			−0.616689			−0.308344	+	0.951275i	$0.599775\pi$
$60$		0			0
$61$			11.9827i			1.53423i	0.641509	+	0.767115i	$0.278309\pi$
$61$			11.9827i			1.53423i	−0.641509	+	0.767115i	$0.721691\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	7.71401i		−	0.956805i
$66$		0			0
$67$	7.62295			0.931291			0.465646	−	0.884971i	$-0.345822\pi$
$67$	7.62295			0.931291			0.465646	+	0.884971i	$0.345822\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	3.52461i		−	0.418294i	−0.977884	−	0.209147i	$-0.932931\pi$
$71$		−	3.52461i		−	0.418294i	0.977884	−	0.209147i	$-0.0670687\pi$
$72$		0			0
$73$	−2.24398	+	1.29556i	−0.262638	+	0.151634i	−0.625537	−	0.780194i	$-0.715120\pi$
$73$	−2.24398	+	1.29556i	−0.262638	+	0.151634i	0.362899	+	0.931828i	$0.381787\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−7.17904			−0.807705			−0.403852	−	0.914824i	$-0.632329\pi$
$79$	−7.17904			−0.807705			−0.403852	+	0.914824i	$0.632329\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−0.932743	−	1.61556i	−0.102382	−	0.177331i	0.810284	−	0.586038i	$-0.199313\pi$
$83$	−0.932743	−	1.61556i	−0.102382	−	0.177331i	−0.912666	+	0.408707i	$0.865980\pi$
$84$		0			0
$85$	−0.0439680	+	0.0761548i	−0.00476900	+	0.00826014i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−3.56125	+	6.16826i	−0.377492	+	0.653835i	−0.990697	−	0.136089i	$-0.956547\pi$
$89$	−3.56125	+	6.16826i	−0.377492	+	0.653835i	0.613205	+	0.789924i	$0.289880\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$			21.4634i			2.20209i
$96$		0			0
$97$	13.2520	−	7.65104i	1.34554	−	0.776845i	0.357922	−	0.933752i	$-0.383486\pi$
$97$	13.2520	−	7.65104i	1.34554	−	0.776845i	0.987613	+	0.156907i	$0.0501522\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	3.38758	−	5.86746i	0.337077	−	0.583834i	−0.646805	−	0.762655i	$-0.723895\pi$
$101$	3.38758	−	5.86746i	0.337077	−	0.583834i	0.983881	+	0.178822i	$0.0572286\pi$
$102$		0			0
$103$	4.22811	−	2.44110i	0.416608	−	0.240529i	−0.277017	−	0.960865i	$-0.589346\pi$
$103$	4.22811	−	2.44110i	0.416608	−	0.240529i	0.693625	+	0.720336i	$0.256012\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−12.8238	−	7.40383i	−1.23972	−	0.715756i	−0.270687	−	0.962667i	$-0.587251\pi$
$107$	−12.8238	−	7.40383i	−1.23972	−	0.715756i	−0.969038	+	0.246912i	$0.920584\pi$
$108$		0			0
$109$	5.82144	+	10.0830i	0.557593	+	0.965780i	0.997697	+	0.0678327i	$0.0216084\pi$
$109$	5.82144	+	10.0830i	0.557593	+	0.965780i	−0.440104	+	0.897947i	$0.645058\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.24947	+	0.721383i	0.117540	+	0.0678620i	0.557618	−	0.830098i	$-0.311716\pi$
$113$	1.24947	+	0.721383i	0.117540	+	0.0678620i	−0.440077	+	0.897960i	$0.645049\pi$
$114$		0			0
$115$	−18.5152	−	10.6897i	−1.72655	−	0.996824i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	12.4244	+	21.5197i	1.12949	+	1.95634i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−8.13282			−0.727421
$126$		0			0
$127$	8.25511			0.732523			0.366261	−	0.930512i	$-0.380638\pi$
$127$	8.25511			0.732523			0.366261	+	0.930512i	$0.380638\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	6.64453	+	11.5087i	0.580536	+	1.00552i	0.995416	+	0.0956411i	$0.0304901\pi$
$131$	6.64453	+	11.5087i	0.580536	+	1.00552i	−0.414880	+	0.909876i	$0.636177\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	5.79362	+	3.34495i	0.494982	+	0.285778i	0.726639	−	0.687020i	$-0.241081\pi$
$137$	5.79362	+	3.34495i	0.494982	+	0.285778i	−0.231657	+	0.972798i	$0.574415\pi$
$138$		0			0
$139$	−5.65156	−	3.26293i	−0.479359	−	0.276758i	0.240790	−	0.970577i	$-0.422593\pi$
$139$	−5.65156	−	3.26293i	−0.479359	−	0.276758i	−0.720149	+	0.693819i	$0.755927\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	8.78831	+	15.2218i	0.734916	+	1.27291i
$144$		0			0
$145$	−2.49374	−	1.43976i	−0.207094	−	0.119566i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	4.68597	−	2.70544i	0.383889	−	0.221639i	−0.295620	−	0.955306i	$-0.595526\pi$
$149$	4.68597	−	2.70544i	0.383889	−	0.221639i	0.679509	+	0.733667i	$0.262193\pi$
$150$		0			0
$151$	−10.0358	+	17.3825i	−0.816699	+	1.41456i	0.0914023	+	0.995814i	$0.470865\pi$
$151$	−10.0358	+	17.3825i	−0.816699	+	1.41456i	−0.908101	+	0.418750i	$0.862468\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	18.4395	−	10.6460i	1.48110	−	0.855111i
$156$		0			0
$157$		−	1.31400i		−	0.104869i	−0.998624	−	0.0524343i	$-0.983302\pi$
$157$		−	1.31400i		−	0.104869i	0.998624	−	0.0524343i	$-0.0166980\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	8.30283	−	14.3809i	0.650328	−	1.12640i	−0.332716	−	0.943027i	$-0.607965\pi$
$163$	8.30283	−	14.3809i	0.650328	−	1.12640i	0.983043	−	0.183374i	$-0.0587018\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	6.18145	−	10.7066i	0.478335	−	0.828501i	−0.521356	−	0.853339i	$-0.674574\pi$
$167$	6.18145	−	10.7066i	0.478335	−	0.828501i	0.999691	+	0.0248384i	$0.00790711\pi$
$168$		0			0
$169$	−2.19111	−	3.79511i	−0.168547	−	0.291931i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	10.3691			0.788349			0.394174	−	0.919036i	$-0.371031\pi$
$173$	10.3691			0.788349			0.394174	+	0.919036i	$0.371031\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	19.5936	−	11.3124i	1.46450	−	0.845528i	0.465284	−	0.885162i	$-0.345952\pi$
$179$	19.5936	−	11.3124i	1.46450	−	0.845528i	0.999214	+	0.0396332i	$0.0126189\pi$
$180$		0			0
$181$		−	6.15765i		−	0.457695i	−0.973462	−	0.228847i	$-0.926504\pi$
$181$		−	6.15765i		−	0.457695i	0.973462	−	0.228847i	$-0.0734956\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.556545			0.0409180
$186$		0			0
$187$		−	0.200365i		−	0.0146521i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	22.9913i		−	1.66359i	−0.555084	−	0.831794i	$-0.687314\pi$
$191$		−	22.9913i		−	1.66359i	0.555084	−	0.831794i	$-0.312686\pi$
$192$		0			0
$193$	−17.2302			−1.24026			−0.620130	−	0.784499i	$-0.712920\pi$
$193$	−17.2302			−1.24026			−0.620130	+	0.784499i	$0.712920\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			4.31094i			0.307141i	0.988138	+	0.153571i	$0.0490773\pi$
$197$			4.31094i			0.307141i	−0.988138	+	0.153571i	$0.950923\pi$
$198$		0			0
$199$	6.35934	−	3.67156i	0.450801	−	0.260270i	−0.257367	−	0.966314i	$-0.582855\pi$
$199$	6.35934	−	3.67156i	0.450801	−	0.260270i	0.708169	+	0.706043i	$0.249522\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	11.7243			0.818861
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−24.4525	−	42.3529i	−1.69141	−	2.92961i
$210$		0			0
$211$	−6.79668	+	11.7722i	−0.467903	+	0.810431i	−0.999327	−	0.0366744i	$-0.988324\pi$
$211$	−6.79668	+	11.7722i	−0.467903	+	0.810431i	0.531425	+	0.847106i	$0.321657\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	9.93288	−	17.2043i	0.677417	−	1.17332i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	0.0982386i		−	0.00660824i
$222$		0			0
$223$	22.6248	−	13.0624i	1.51507	−	0.874725i	0.515225	−	0.857055i	$-0.327708\pi$
$223$	22.6248	−	13.0624i	1.51507	−	0.874725i	0.999844	−	0.0176705i	$-0.00562499\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1.58102	−	2.73840i	0.104936	−	0.181754i	−0.808776	−	0.588117i	$-0.799870\pi$
$227$	1.58102	−	2.73840i	0.104936	−	0.181754i	0.913712	+	0.406362i	$0.133203\pi$
$228$		0			0
$229$	11.7934	−	6.80892i	0.779330	−	0.449946i	−0.0568630	−	0.998382i	$-0.518110\pi$
$229$	11.7934	−	6.80892i	0.779330	−	0.449946i	0.836193	+	0.548436i	$0.184777\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	23.3765	+	13.4964i	1.53145	+	0.884182i	0.999295	+	0.0375332i	$0.0119500\pi$
$233$	23.3765	+	13.4964i	1.53145	+	0.884182i	0.532152	+	0.846649i	$0.321383\pi$
$234$		0			0
$235$	−9.29708	−	16.1030i	−0.606475	−	1.05044i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	7.46107	+	4.30765i	0.482617	+	0.278639i	0.721506	−	0.692408i	$-0.243450\pi$
$239$	7.46107	+	4.30765i	0.482617	+	0.278639i	−0.238890	+	0.971047i	$0.576783\pi$
$240$		0			0
$241$	20.1317	+	11.6230i	1.29680	+	0.748706i	0.979849	−	0.199737i	$-0.0640089\pi$
$241$	20.1317	+	11.6230i	1.29680	+	0.748706i	0.316947	+	0.948443i	$0.397342\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−11.9890	−	20.7656i	−0.762842	−	1.32128i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	9.09095			0.573816			0.286908	−	0.957958i	$-0.407373\pi$
$251$	9.09095			0.573816			0.286908	+	0.957958i	$0.407373\pi$
$252$		0			0
$253$	48.7139			3.06261
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−7.36958	−	12.7645i	−0.459702	−	0.796226i	0.539243	−	0.842150i	$-0.318710\pi$
$257$	−7.36958	−	12.7645i	−0.459702	−	0.796226i	−0.998945	+	0.0459235i	$0.985377\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−24.1622	−	13.9501i	−1.48991	−	0.860198i	−0.489974	−	0.871737i	$-0.662994\pi$
$263$	−24.1622	−	13.9501i	−1.48991	−	0.860198i	−0.999933	+	0.0115386i	$0.996327\pi$
$264$		0			0
$265$	−4.18347	−	2.41533i	−0.256989	−	0.148372i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−6.57488	−	11.3880i	−0.400878	−	0.694341i	0.592954	−	0.805236i	$-0.297961\pi$
$269$	−6.57488	−	11.3880i	−0.400878	−	0.694341i	−0.993832	+	0.110896i	$0.964628\pi$
$270$		0			0
$271$	15.4231	+	8.90451i	0.936885	+	0.540911i	0.888982	−	0.457941i	$-0.151413\pi$
$271$	15.4231	+	8.90451i	0.936885	+	0.540911i	0.0479022	+	0.998852i	$0.484746\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−9.87794	+	5.70303i	−0.595662	+	0.343906i
$276$		0			0
$277$	−8.89435	+	15.4055i	−0.534410	+	0.925625i	0.464782	+	0.885425i	$0.346133\pi$
$277$	−8.89435	+	15.4055i	−0.534410	+	0.925625i	−0.999192	+	0.0401999i	$0.987201\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	18.3680	−	10.6048i	1.09574	−	0.632628i	0.160644	−	0.987012i	$-0.448643\pi$
$281$	18.3680	−	10.6048i	1.09574	−	0.632628i	0.935100	+	0.354384i	$0.115309\pi$
$282$		0			0
$283$		−	18.7643i		−	1.11542i	−0.830035	−	0.557711i	$-0.811680\pi$
$283$		−	18.7643i		−	1.11542i	0.830035	−	0.557711i	$-0.188320\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	8.49944	−	14.7215i	0.499967	−	0.865968i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	6.40500	−	11.0938i	0.374184	−	0.648106i	−0.616020	−	0.787730i	$-0.711256\pi$
$293$	6.40500	−	11.0938i	0.374184	−	0.648106i	0.990205	+	0.139624i	$0.0445894\pi$
$294$		0			0
$295$	6.22364	+	10.7797i	0.362354	+	0.627616i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	23.8843			1.38127
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	27.2689	−	15.7437i	1.56142	−	0.901484i
$306$		0			0
$307$		−	3.97281i		−	0.226740i	−0.993553	−	0.113370i	$-0.963835\pi$
$307$		−	3.97281i		−	0.226740i	0.993553	−	0.113370i	$-0.0361646\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	0.586407			0.0332521			0.0166260	−	0.999862i	$-0.494708\pi$
$311$	0.586407			0.0332521			0.0166260	+	0.999862i	$0.494708\pi$
$312$		0			0
$313$			0.993112i			0.0561340i	0.999606	+	0.0280670i	$0.00893518\pi$
$313$			0.993112i			0.0561340i	−0.999606	+	0.0280670i	$0.991065\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			23.7894i			1.33615i	0.744096	+	0.668073i	$0.232881\pi$
$317$			23.7894i			1.33615i	−0.744096	+	0.668073i	$0.767119\pi$
$318$		0			0
$319$	6.56109			0.367350
$320$		0			0
$321$		0			0
$322$		0			0
$323$			0.273338i			0.0152089i
$324$		0			0
$325$	−4.84314	+	2.79619i	−0.268649	+	0.155105i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	27.0971			1.48939			0.744697	−	0.667403i	$-0.232594\pi$
$331$	27.0971			1.48939			0.744697	+	0.667403i	$0.232594\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−10.0156	−	17.3475i	−0.547208	−	0.947793i
$336$		0			0
$337$	0.618503	−	1.07128i	0.0336920	−	0.0583562i	−0.848688	−	0.528894i	$-0.822607\pi$
$337$	0.618503	−	1.07128i	0.0336920	−	0.0583562i	0.882380	+	0.470538i	$0.155940\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−24.2574	+	42.0150i	−1.31361	+	2.27524i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$			13.4486i			0.721958i	0.932574	+	0.360979i	$0.117557\pi$
$347$			13.4486i			0.721958i	−0.932574	+	0.360979i	$0.882443\pi$
$348$		0			0
$349$	−12.4728	+	7.20115i	−0.667652	+	0.385469i	−0.795186	−	0.606365i	$-0.792627\pi$
$349$	−12.4728	+	7.20115i	−0.667652	+	0.385469i	0.127535	+	0.991834i	$0.459294\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	3.77665	−	6.54134i	0.201011	−	0.348161i	−0.747844	−	0.663875i	$-0.768911\pi$
$353$	3.77665	−	6.54134i	0.201011	−	0.348161i	0.948854	+	0.315714i	$0.102244\pi$
$354$		0			0
$355$	−8.02091	+	4.63087i	−0.425706	+	0.245781i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−20.8285	−	12.0253i	−1.09928	−	0.634672i	−0.163251	−	0.986585i	$-0.552198\pi$
$359$	−20.8285	−	12.0253i	−1.09928	−	0.634672i	−0.936033	+	0.351913i	$0.885531\pi$
$360$		0			0
$361$	23.8580	+	41.3233i	1.25569	+	2.17491i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	5.89659	+	3.40440i	0.308642	+	0.178194i
$366$		0			0
$367$	6.28058	+	3.62610i	0.327844	+	0.189281i	0.654883	−	0.755730i	$-0.272718\pi$
$367$	6.28058	+	3.62610i	0.327844	+	0.189281i	−0.327040	+	0.945011i	$0.606051\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−6.15315	−	10.6576i	−0.318598	−	0.551828i	0.661598	−	0.749859i	$-0.269879\pi$
$373$	−6.15315	−	10.6576i	−0.318598	−	0.551828i	−0.980196	+	0.198031i	$0.936545\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	3.21689			0.165678
$378$		0			0
$379$	−20.4289			−1.04936			−0.524680	−	0.851299i	$-0.675815\pi$
$379$	−20.4289			−1.04936			−0.524680	+	0.851299i	$0.675815\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−17.6460	−	30.5638i	−0.901668	−	1.56174i	−0.825328	−	0.564653i	$-0.809010\pi$
$383$	−17.6460	−	30.5638i	−0.901668	−	1.56174i	−0.0763399	−	0.997082i	$-0.524323\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	19.2835	+	11.1334i	0.977714	+	0.564484i	0.901579	−	0.432614i	$-0.142409\pi$
$389$	19.2835	+	11.1334i	0.977714	+	0.564484i	0.0761350	+	0.997098i	$0.475742\pi$
$390$		0			0
$391$	−0.235792	−	0.136135i	−0.0119245	−	0.00688463i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	9.43232	+	16.3373i	0.474592	+	0.822017i
$396$		0			0
$397$	16.8558	+	9.73170i	0.845969	+	0.488420i	0.859289	−	0.511491i	$-0.170907\pi$
$397$	16.8558	+	9.73170i	0.845969	+	0.488420i	−0.0133200	+	0.999911i	$0.504240\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	7.45051	−	4.30155i	0.372061	−	0.214809i	−0.302298	−	0.953214i	$-0.597754\pi$
$401$	7.45051	−	4.30155i	0.372061	−	0.214809i	0.674358	+	0.738404i	$0.264420\pi$
$402$		0			0
$403$	−11.8933	+	20.5999i	−0.592449	+	1.02615i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−1.09821	+	0.634053i	−0.0544364	+	0.0314289i
$408$		0			0
$409$		−	19.3494i		−	0.956768i	−0.878151	−	0.478384i	$-0.841223\pi$
$409$		−	19.3494i		−	0.956768i	0.878151	−	0.478384i	$-0.158777\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−2.45101	+	4.24527i	−0.120315	+	0.208392i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−8.81222	+	15.2632i	−0.430505	+	0.745657i	−0.996917	−	0.0784657i	$-0.974998\pi$
$419$	−8.81222	+	15.2632i	−0.430505	+	0.745657i	0.566412	+	0.824122i	$0.308331\pi$
$420$		0			0
$421$	−5.77040	−	9.99463i	−0.281232	−	0.487109i	0.690456	−	0.723374i	$-0.257410\pi$
$421$	−5.77040	−	9.99463i	−0.281232	−	0.487109i	−0.971689	+	0.236266i	$0.924076\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	0.0637503			0.00309235
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−9.90849	+	5.72067i	−0.477275	+	0.275555i	−0.719280	−	0.694720i	$-0.755528\pi$
$431$	−9.90849	+	5.72067i	−0.477275	+	0.275555i	0.242005	+	0.970275i	$0.422195\pi$
$432$		0			0
$433$			6.25525i			0.300608i	0.988640	+	0.150304i	$0.0480253\pi$
$433$			6.25525i			0.300608i	−0.988640	+	0.150304i	$0.951975\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−66.4554			−3.17899
$438$		0			0
$439$			15.3048i			0.730457i	0.930918	+	0.365229i	$0.119009\pi$
$439$			15.3048i			0.730457i	−0.930918	+	0.365229i	$0.880991\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			22.4980i			1.06891i	0.845196	+	0.534456i	$0.179483\pi$
$443$			22.4980i			1.06891i	−0.845196	+	0.534456i	$0.820517\pi$
$444$		0			0
$445$	18.7161			0.887227
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	25.9709i		−	1.22564i	−0.790222	−	0.612821i	$-0.790035\pi$
$449$		−	25.9709i		−	1.22564i	0.790222	−	0.612821i	$-0.209965\pi$
$450$		0			0
$451$	−23.1352	+	13.3571i	−1.08939	+	0.628962i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.0444037			0.00207712			0.00103856	−	0.999999i	$-0.499669\pi$
$457$	0.0444037			0.00207712			0.00103856	+	0.999999i	$0.499669\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−1.46783	−	2.54236i	−0.0683636	−	0.118409i	0.829817	−	0.558035i	$-0.188444\pi$
$461$	−1.46783	−	2.54236i	−0.0683636	−	0.118409i	−0.898181	+	0.439626i	$0.855111\pi$
$462$		0			0
$463$	−19.2017	+	33.2583i	−0.892378	+	1.54564i	−0.0553609	+	0.998466i	$0.517631\pi$
$463$	−19.2017	+	33.2583i	−0.892378	+	1.54564i	−0.837017	+	0.547177i	$0.815702\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−8.63913	+	14.9634i	−0.399771	+	0.692424i	−0.993697	−	0.112095i	$-0.964244\pi$
$467$	−8.63913	+	14.9634i	−0.399771	+	0.692424i	0.593926	+	0.804520i	$0.297577\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$			45.2648i			2.08128i
$474$		0			0
$475$	13.4755	−	7.78007i	0.618297	−	0.356974i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−18.4571	+	31.9686i	−0.843326	+	1.46068i	0.0437419	+	0.999043i	$0.486072\pi$
$479$	−18.4571	+	31.9686i	−0.843326	+	1.46068i	−0.887067	+	0.461640i	$0.847261\pi$
$480$		0			0
$481$	−0.538451	+	0.310875i	−0.0245513	+	0.0141747i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−34.8228	−	20.1049i	−1.58122	−	0.912918i
$486$		0			0
$487$	−2.03199	−	3.51951i	−0.0920783	−	0.159484i	0.816307	−	0.577618i	$-0.196018\pi$
$487$	−2.03199	−	3.51951i	−0.0920783	−	0.159484i	−0.908385	+	0.418134i	$0.862684\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−4.30460	−	2.48526i	−0.194264	−	0.112158i	0.399713	−	0.916640i	$-0.369110\pi$
$491$	−4.30460	−	2.48526i	−0.194264	−	0.112158i	−0.593977	+	0.804482i	$0.702443\pi$
$492$		0			0
$493$	−0.0317580	−	0.0183355i	−0.00143031	−	0.000825789i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−4.60335	−	7.97323i	−0.206074	−	0.356931i	0.744400	−	0.667734i	$-0.232735\pi$
$499$	−4.60335	−	7.97323i	−0.206074	−	0.356931i	−0.950474	+	0.310803i	$0.899402\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−1.16946			−0.0521436			−0.0260718	−	0.999660i	$-0.508300\pi$
$503$	−1.16946			−0.0521436			−0.0260718	+	0.999660i	$0.508300\pi$
$504$		0			0
$505$	−17.8033			−0.792238
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−4.86206	−	8.42133i	−0.215507	−	0.373269i	0.737922	−	0.674886i	$-0.235807\pi$
$509$	−4.86206	−	8.42133i	−0.215507	−	0.373269i	−0.953429	+	0.301617i	$0.902474\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−11.1104	−	6.41458i	−0.489582	−	0.282660i
$516$		0			0
$517$	36.6912	+	21.1837i	1.61368	+	0.931658i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	6.23317	+	10.7962i	0.273080	+	0.472989i	0.969649	−	0.244501i	$-0.0786243\pi$
$521$	6.23317	+	10.7962i	0.273080	+	0.472989i	−0.696569	+	0.717490i	$0.745291\pi$
$522$		0			0
$523$	15.6222	+	9.01951i	0.683113	+	0.394395i	0.801027	−	0.598628i	$-0.204287\pi$
$523$	15.6222	+	9.01951i	0.683113	+	0.394395i	−0.117914	+	0.993024i	$0.537621\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	0.234828	−	0.135578i	0.0102293	−	0.00590588i
$528$		0			0
$529$	21.5979	−	37.4086i	0.939037	−	1.62646i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−11.3431	+	6.54897i	−0.491326	+	0.283667i
$534$		0			0
$535$			38.9107i			1.68226i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−1.87785	+	3.25253i	−0.0807349	+	0.139837i	−0.903566	−	0.428449i	$-0.859060\pi$
$541$	−1.87785	+	3.25253i	−0.0807349	+	0.139837i	0.822831	+	0.568286i	$0.192393\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	15.2972	−	26.4956i	0.655262	−	1.13495i
$546$		0			0
$547$	5.05062	+	8.74793i	0.215949	+	0.374034i	0.953566	−	0.301185i	$-0.0973822\pi$
$547$	5.05062	+	8.74793i	0.215949	+	0.374034i	−0.737617	+	0.675219i	$0.764049\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−8.95063			−0.381310
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	15.7817	−	9.11158i	0.668693	−	0.386070i	−0.126888	−	0.991917i	$-0.540499\pi$
$557$	15.7817	−	9.11158i	0.668693	−	0.386070i	0.795581	+	0.605847i	$0.207166\pi$
$558$		0			0
$559$			22.1933i			0.938674i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	16.6938			0.703560			0.351780	−	0.936083i	$-0.385576\pi$
$563$	16.6938			0.703560			0.351780	+	0.936083i	$0.385576\pi$
$564$		0			0
$565$		−	3.79121i		−	0.159497i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	25.2546i		−	1.05873i	−0.848394	−	0.529365i	$-0.822430\pi$
$569$		−	25.2546i		−	1.05873i	0.848394	−	0.529365i	$-0.177570\pi$
$570$		0			0
$571$	−5.77962			−0.241869			−0.120935	−	0.992660i	$-0.538589\pi$
$571$	−5.77962			−0.241869			−0.120935	+	0.992660i	$0.538589\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			15.4993i			0.646367i
$576$		0			0
$577$	28.6539	−	16.5433i	1.19288	−	0.688708i	0.233919	−	0.972256i	$-0.424845\pi$
$577$	28.6539	−	16.5433i	1.19288	−	0.688708i	0.958958	+	0.283548i	$0.0915115\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	11.0068			0.455855
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−14.1186	−	24.4541i	−0.582737	−	1.00933i	−0.995153	−	0.0983341i	$-0.968649\pi$
$587$	−14.1186	−	24.4541i	−0.582737	−	1.00933i	0.412417	−	0.910995i	$-0.364685\pi$
$588$		0			0
$589$	33.0919	−	57.3168i	1.36353	−	2.36170i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	9.29769	−	16.1041i	0.381810	−	0.661315i	−0.609511	−	0.792778i	$-0.708634\pi$
$593$	9.29769	−	16.1041i	0.381810	−	0.661315i	0.991321	+	0.131463i	$0.0419674\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$			8.97521i			0.366717i	0.983046	+	0.183358i	$0.0586969\pi$
$599$			8.97521i			0.366717i	−0.983046	+	0.183358i	$0.941303\pi$
$600$		0			0
$601$	4.71245	−	2.72073i	0.192225	−	0.110981i	−0.400799	−	0.916166i	$-0.631267\pi$
$601$	4.71245	−	2.72073i	0.192225	−	0.110981i	0.593024	+	0.805185i	$0.297934\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	32.6482	−	56.5483i	1.32734	−	2.29901i
$606$		0			0
$607$	−32.0240	+	18.4890i	−1.29981	+	0.750447i	−0.980372	−	0.197159i	$-0.936829\pi$
$607$	−32.0240	+	18.4890i	−1.29981	+	0.750447i	−0.319441	+	0.947606i	$0.603495\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	17.9897	+	10.3863i	0.727783	+	0.420186i
$612$		0			0
$613$	5.93439	+	10.2787i	0.239688	+	0.415151i	0.960625	−	0.277849i	$-0.0896216\pi$
$613$	5.93439	+	10.2787i	0.239688	+	0.415151i	−0.720937	+	0.693001i	$0.756288\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−16.2845	−	9.40188i	−0.655591	−	0.378506i	0.135004	−	0.990845i	$-0.456895\pi$
$617$	−16.2845	−	9.40188i	−0.655591	−	0.378506i	−0.790595	+	0.612339i	$0.790229\pi$
$618$		0			0
$619$	−10.2892	−	5.94048i	−0.413558	−	0.238768i	0.278759	−	0.960361i	$-0.410077\pi$
$619$	−10.2892	−	5.94048i	−0.413558	−	0.238768i	−0.692318	+	0.721593i	$0.743410\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	15.4480	+	26.7567i	0.617920	+	1.07027i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	0.00708765			0.000282603
$630$		0			0
$631$	−28.3350			−1.12800			−0.563998	−	0.825776i	$-0.690738\pi$
$631$	−28.3350			−1.12800			−0.563998	+	0.825776i	$0.690738\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−10.8461	−	18.7861i	−0.430416	−	0.745502i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−24.5634	−	14.1817i	−0.970195	−	0.560142i	−0.0708994	−	0.997483i	$-0.522587\pi$
$641$	−24.5634	−	14.1817i	−0.970195	−	0.560142i	−0.899296	+	0.437341i	$0.855920\pi$
$642$		0			0
$643$	−16.0912	−	9.29024i	−0.634573	−	0.366371i	0.147948	−	0.988995i	$-0.452733\pi$
$643$	−16.0912	−	9.29024i	−0.634573	−	0.366371i	−0.782521	+	0.622624i	$0.786067\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	21.4482	+	37.1494i	0.843216	+	1.46049i	0.887161	+	0.461460i	$0.152674\pi$
$647$	21.4482	+	37.1494i	0.843216	+	1.46049i	−0.0439448	+	0.999034i	$0.513993\pi$
$648$		0			0
$649$	−24.5618	−	14.1808i	−0.964135	−	0.556644i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−3.18721	+	1.84014i	−0.124725	+	0.0720101i	−0.561065	−	0.827772i	$-0.689608\pi$
$653$	−3.18721	+	1.84014i	−0.124725	+	0.0720101i	0.436339	+	0.899782i	$0.356275\pi$
$654$		0			0
$655$	17.4601	−	30.2418i	0.682223	−	1.18164i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	13.7955	−	7.96483i	0.537396	−	0.310266i	−0.206627	−	0.978420i	$-0.566249\pi$
$659$	13.7955	−	7.96483i	0.537396	−	0.310266i	0.744023	+	0.668154i	$0.232915\pi$
$660$		0			0
$661$			1.32072i			0.0513700i	0.999670	+	0.0256850i	$0.00817669\pi$
$661$			1.32072i			0.0513700i	−0.999670	+	0.0256850i	$0.991823\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	4.45783	−	7.72118i	0.172608	−	0.298965i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−35.8726	+	62.1332i	−1.38485	+	2.39863i
$672$		0			0
$673$	21.7987	+	37.7565i	0.840280	+	1.45541i	0.889658	+	0.456627i	$0.150943\pi$
$673$	21.7987	+	37.7565i	0.840280	+	1.45541i	−0.0493788	+	0.998780i	$0.515724\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	28.9354			1.11208			0.556039	−	0.831156i	$-0.312320\pi$
$677$	28.9354			1.11208			0.556039	+	0.831156i	$0.312320\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1.90755	+	1.10132i	−0.0729903	+	0.0421410i	−0.536051	−	0.844186i	$-0.680085\pi$
$683$	−1.90755	+	1.10132i	−0.0729903	+	0.0421410i	0.463061	+	0.886327i	$0.346751\pi$
$684$		0			0
$685$		−	17.5793i		−	0.671670i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	5.39662			0.205595
$690$		0			0
$691$		−	29.8948i		−	1.13725i	−0.822597	−	0.568625i	$-0.807475\pi$
$691$		−	29.8948i		−	1.13725i	0.822597	−	0.568625i	$-0.192525\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			17.1483i			0.650471i
$696$		0			0
$697$	0.149310			0.00565552
$698$		0			0
$699$		0			0
$700$		0			0
$701$			45.4119i			1.71518i	0.514332	+	0.857591i	$0.328040\pi$
$701$			45.4119i			1.71518i	−0.514332	+	0.857591i	$0.671960\pi$
$702$		0			0
$703$	1.49818	−	0.864975i	0.0565049	−	0.0326231i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−15.8518			−0.595328			−0.297664	−	0.954671i	$-0.596207\pi$
$709$	−15.8518			−0.595328			−0.297664	+	0.954671i	$0.596207\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	32.9626	+	57.0928i	1.23446	+	2.13814i
$714$		0			0
$715$	23.0934	−	39.9989i	0.863644	−	1.49588i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−3.30154	+	5.71844i	−0.123127	+	0.213262i	−0.920999	−	0.389565i	$-0.872625\pi$
$719$	−3.30154	+	5.71844i	−0.123127	+	0.213262i	0.797872	+	0.602826i	$0.205959\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$			2.08755i			0.0775296i
$726$		0			0
$727$	−31.2086	+	18.0183i	−1.15746	+	0.668261i	−0.950694	−	0.310129i	$-0.899628\pi$
$727$	−31.2086	+	18.0183i	−1.15746	+	0.668261i	−0.206767	+	0.978390i	$0.566294\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	0.126496	−	0.219098i	0.00467863	−	0.00810362i
$732$		0			0
$733$	−26.0312	+	15.0291i	−0.961486	+	0.555114i	−0.896630	−	0.442781i	$-0.853992\pi$
$733$	−26.0312	+	15.0291i	−0.961486	+	0.555114i	−0.0648557	+	0.997895i	$0.520659\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	39.5268	+	22.8208i	1.45599	+	0.840614i
$738$		0			0
$739$	−4.83576	−	8.37577i	−0.177886	−	0.308108i	0.763270	−	0.646079i	$-0.223593\pi$
$739$	−4.83576	−	8.37577i	−0.177886	−	0.308108i	−0.941156	+	0.337972i	$0.890259\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−11.0205	−	6.36269i	−0.404303	−	0.233424i	0.284036	−	0.958814i	$-0.408326\pi$
$743$	−11.0205	−	6.36269i	−0.404303	−	0.233424i	−0.688339	+	0.725389i	$0.741660\pi$
$744$		0			0
$745$	−12.3135	−	7.10920i	−0.451132	−	0.260461i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−10.6870	−	18.5105i	−0.389975	−	0.675457i	0.602471	−	0.798141i	$-0.294183\pi$
$751$	−10.6870	−	18.5105i	−0.389975	−	0.675457i	−0.992446	+	0.122684i	$0.960850\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	52.7428			1.91951
$756$		0			0
$757$	−7.21065			−0.262076			−0.131038	−	0.991377i	$-0.541831\pi$
$757$	−7.21065			−0.262076			−0.131038	+	0.991377i	$0.541831\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−11.1620	−	19.3332i	−0.404623	−	0.700827i	0.589655	−	0.807655i	$-0.299264\pi$
$761$	−11.1620	−	19.3332i	−0.404623	−	0.700827i	−0.994277	+	0.106829i	$0.965930\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−12.0426	−	6.95280i	−0.434833	−	0.251051i
$768$		0			0
$769$	13.3202	+	7.69042i	0.480338	+	0.277324i	0.720558	−	0.693395i	$-0.243886\pi$
$769$	13.3202	+	7.69042i	0.480338	+	0.277324i	−0.240219	+	0.970719i	$0.577219\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−6.17527	−	10.6959i	−0.222109	−	0.384704i	0.733339	−	0.679863i	$-0.237961\pi$
$773$	−6.17527	−	10.6959i	−0.222109	−	0.384704i	−0.955448	+	0.295159i	$0.904627\pi$
$774$		0			0
$775$	−13.3680	−	7.71799i	−0.480191	−	0.277238i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	31.5610	−	18.2218i	1.13079	−	0.652862i
$780$		0			0
$781$	10.5516	−	18.2759i	0.377566	−	0.653963i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−2.99025	+	1.72642i	−0.106727	+	0.0616187i
$786$		0			0
$787$			14.5290i			0.517904i	0.965890	+	0.258952i	$0.0833771\pi$
$787$			14.5290i			0.517904i	−0.965890	+	0.258952i	$0.916623\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−17.5883	+	30.4638i	−0.624578	+	1.08180i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−22.0040	+	38.1120i	−0.779420	+	1.35000i	0.152856	+	0.988248i	$0.451153\pi$
$797$	−22.0040	+	38.1120i	−0.779420	+	1.35000i	−0.932276	+	0.361747i	$0.882181\pi$
$798$		0			0
$799$	−0.118399	−	0.205073i	−0.00418866	−	0.00725497i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−15.5141			−0.547480
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	41.4554	−	23.9343i	1.45749	−	0.841484i	0.458606	−	0.888640i	$-0.348349\pi$
$809$	41.4554	−	23.9343i	1.45749	−	0.841484i	0.998888	+	0.0471551i	$0.0150155\pi$
$810$		0			0
$811$		−	17.4775i		−	0.613720i	−0.951755	−	0.306860i	$-0.900722\pi$
$811$		−	17.4775i		−	0.613720i	0.951755	−	0.306860i	$-0.0992783\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−43.6353			−1.52848
$816$		0			0
$817$		−	61.7502i		−	2.16037i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			7.39671i			0.258147i	0.991635	+	0.129074i	$0.0412003\pi$
$821$			7.39671i			0.258147i	−0.991635	+	0.129074i	$0.958800\pi$
$822$		0			0
$823$	−42.9436			−1.49692			−0.748460	−	0.663179i	$-0.769207\pi$
$823$	−42.9436			−1.49692			−0.748460	+	0.663179i	$0.769207\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			43.0042i			1.49540i	0.664035	+	0.747701i	$0.268843\pi$
$827$			43.0042i			1.49540i	−0.664035	+	0.747701i	$0.731157\pi$
$828$		0			0
$829$	10.0780	−	5.81851i	0.350022	−	0.202085i	−0.314673	−	0.949200i	$-0.601895\pi$
$829$	10.0780	−	5.81851i	0.350022	−	0.202085i	0.664695	+	0.747115i	$0.268561\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−32.4865			−1.12424
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−0.936892	−	1.62274i	−0.0323451	−	0.0560234i	0.849400	−	0.527750i	$-0.176964\pi$
$839$	−0.936892	−	1.62274i	−0.0323451	−	0.0560234i	−0.881745	+	0.471727i	$0.843631\pi$
$840$		0			0
$841$	−13.8996	+	24.0748i	−0.479296	+	0.830166i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−5.75766	+	9.97255i	−0.198069	+	0.343066i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$			1.72319i			0.0590702i
$852$		0			0
$853$	−34.9301	+	20.1669i	−1.19598	+	0.690501i	−0.959657	−	0.281173i	$-0.909276\pi$
$853$	−34.9301	+	20.1669i	−1.19598	+	0.690501i	−0.236325	+	0.971674i	$0.575943\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	16.1341	−	27.9450i	0.551129	−	0.954583i	−0.447065	−	0.894502i	$-0.647531\pi$
$857$	16.1341	−	27.9450i	0.551129	−	0.954583i	0.998193	−	0.0600814i	$-0.0191360\pi$
$858$		0			0
$859$	33.6905	−	19.4512i	1.14951	−	0.663668i	0.200740	−	0.979645i	$-0.435665\pi$
$859$	33.6905	−	19.4512i	1.14951	−	0.663668i	0.948767	+	0.315977i	$0.102332\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−33.1319	−	19.1287i	−1.12782	−	0.651148i	−0.184435	−	0.982845i	$-0.559046\pi$
$863$	−33.1319	−	19.1287i	−1.12782	−	0.651148i	−0.943386	+	0.331697i	$0.892379\pi$
$864$		0			0
$865$	−13.6237	−	23.5969i	−0.463218	−	0.802318i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−37.2250	−	21.4919i	−1.26277	−	0.729061i
$870$		0			0
$871$	19.3799	+	11.1890i	0.656663	+	0.379124i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−4.76152	−	8.24719i	−0.160785	−	0.278488i	0.774365	−	0.632739i	$-0.218069\pi$
$877$	−4.76152	−	8.24719i	−0.160785	−	0.278488i	−0.935150	+	0.354251i	$0.884736\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	16.5770			0.558493			0.279247	−	0.960219i	$-0.409915\pi$
$881$	16.5770			0.558493			0.279247	+	0.960219i	$0.409915\pi$
$882$		0			0
$883$	−34.9830			−1.17727			−0.588636	−	0.808398i	$-0.700335\pi$
$883$	−34.9830			−1.17727			−0.588636	+	0.808398i	$0.700335\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	17.3795	+	30.1022i	0.583547	+	1.01073i	0.995055	+	0.0993271i	$0.0316690\pi$
$887$	17.3795	+	30.1022i	0.583547	+	1.01073i	−0.411508	+	0.911406i	$0.634998\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−50.0542	−	28.8988i	−1.67500	−	0.967061i
$894$		0			0
$895$	−51.4870	−	29.7260i	−1.72102	−	0.993632i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	4.43960	+	7.68962i	0.148069	+	0.256463i
$900$		0			0
$901$	−0.0532769	−	0.0307594i	−0.00177491	−	0.00102474i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−14.0129	+	8.09035i	−0.465805	+	0.268932i
$906$		0			0
$907$	−15.9116	+	27.5597i	−0.528336	+	0.915105i	0.471118	+	0.882070i	$0.343851\pi$
$907$	−15.9116	+	27.5597i	−0.528336	+	0.915105i	−0.999454	+	0.0330347i	$0.989483\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−7.51591	+	4.33931i	−0.249013	+	0.143768i	−0.619312	−	0.785145i	$-0.712589\pi$
$911$	−7.51591	+	4.33931i	−0.249013	+	0.143768i	0.370299	+	0.928913i	$0.379255\pi$
$912$		0			0
$913$		−	11.1694i		−	0.369653i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−10.3922	+	17.9999i	−0.342808	+	0.593760i	−0.984953	−	0.172823i	$-0.944711\pi$
$919$	−10.3922	+	17.9999i	−0.342808	+	0.593760i	0.642145	+	0.766583i	$0.278045\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	5.17343	−	8.96064i	0.170286	−	0.294943i
$924$		0			0
$925$	−0.201737	−	0.349419i	−0.00663308	−	0.0114888i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	41.0722			1.34753			0.673767	−	0.738944i	$-0.264675\pi$
$929$	41.0722			1.34753			0.673767	+	0.738944i	$0.264675\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−0.455968	+	0.263254i	−0.0149118	+	0.00860931i
$936$		0			0
$937$		−	8.38277i		−	0.273853i	−0.990581	−	0.136927i	$-0.956278\pi$
$937$		−	8.38277i		−	0.273853i	0.990581	−	0.136927i	$-0.0437225\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−9.99340			−0.325775			−0.162888	−	0.986645i	$-0.552081\pi$
$941$	−9.99340			−0.325775			−0.162888	+	0.986645i	$0.552081\pi$
$942$		0			0
$943$			36.3011i			1.18213i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	23.5165i		−	0.764182i	−0.924125	−	0.382091i	$-0.875204\pi$
$947$		−	23.5165i		−	0.764182i	0.924125	−	0.382091i	$-0.124796\pi$
$948$		0			0
$949$	−7.60652			−0.246918
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	34.7300i		−	1.12501i	−0.826793	−	0.562507i	$-0.809837\pi$
$953$		−	34.7300i		−	1.12501i	0.826793	−	0.562507i	$-0.190163\pi$
$954$		0			0
$955$	−52.3210	+	30.2075i	−1.69307	+	0.977492i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−34.6556			−1.11792
$962$		0			0
$963$		0			0
$964$		0			0
$965$	22.6383	+	39.2107i	0.728753	+	1.26224i
$966$		0			0
$967$	8.98645	−	15.5650i	0.288985	−	0.500536i	−0.684583	−	0.728935i	$-0.740016\pi$
$967$	8.98645	−	15.5650i	0.288985	−	0.500536i	0.973568	+	0.228399i	$0.0733489\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−20.9001	+	36.2001i	−0.670717	+	1.16172i	0.306984	+	0.951715i	$0.400680\pi$
$971$	−20.9001	+	36.2001i	−0.670717	+	1.16172i	−0.977701	+	0.210001i	$0.932653\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	44.6291i		−	1.42781i	−0.700242	−	0.713905i	$-0.746925\pi$
$977$		−	44.6291i		−	1.42781i	0.700242	−	0.713905i	$-0.253075\pi$
$978$		0			0
$979$	−36.9318	+	21.3226i	−1.18035	+	0.681473i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	28.5601	−	49.4675i	0.910925	−	1.57777i	0.0981655	−	0.995170i	$-0.468703\pi$
$983$	28.5601	−	49.4675i	0.910925	−	1.57777i	0.812760	−	0.582599i	$-0.197964\pi$
$984$		0			0
$985$	9.81035	−	5.66401i	0.312584	−	0.180470i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	53.2683	+	30.7544i	1.69383	+	0.977935i
$990$		0			0
$991$	−18.6791	−	32.3532i	−0.593362	−	1.02773i	−0.993776	−	0.111399i	$-0.964467\pi$
$991$	−18.6791	−	32.3532i	−0.593362	−	1.02773i	0.400413	−	0.916335i	$-0.368867\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−16.7107	−	9.64792i	−0.529764	−	0.305859i
$996$		0			0
$997$	−34.0530	−	19.6605i	−1.07847	−	0.622654i	−0.147986	−	0.988990i	$-0.547279\pi$
$997$	−34.0530	−	19.6605i	−1.07847	−	0.622654i	−0.930483	+	0.366335i	$0.880612\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5292.2.w.c.1097.5		48
3.2	odd	2		1764.2.w.c.509.16		48
7.2	even	3		5292.2.x.c.881.5		48
7.3	odd	6		5292.2.bm.c.2285.5		48
7.4	even	3		5292.2.bm.c.2285.20		48
7.5	odd	6		5292.2.x.c.881.20		48
7.6	odd	2	inner	5292.2.w.c.1097.20		48
9.2	odd	6		5292.2.bm.c.4625.5		48
9.7	even	3		1764.2.bm.c.1685.7		48
21.2	odd	6		1764.2.x.c.293.2	✓	48
21.5	even	6		1764.2.x.c.293.23	yes	48
21.11	odd	6		1764.2.bm.c.1697.18		48
21.17	even	6		1764.2.bm.c.1697.7		48
21.20	even	2		1764.2.w.c.509.9		48
63.2	odd	6		5292.2.x.c.4409.20		48
63.11	odd	6	inner	5292.2.w.c.521.20		48
63.16	even	3		1764.2.x.c.1469.23	yes	48
63.20	even	6		5292.2.bm.c.4625.20		48
63.25	even	3		1764.2.w.c.1109.9		48
63.34	odd	6		1764.2.bm.c.1685.18		48
63.38	even	6	inner	5292.2.w.c.521.5		48
63.47	even	6		5292.2.x.c.4409.5		48
63.52	odd	6		1764.2.w.c.1109.16		48
63.61	odd	6		1764.2.x.c.1469.2	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1764.2.w.c.509.9		48	21.20	even	2
1764.2.w.c.509.16		48	3.2	odd	2
1764.2.w.c.1109.9		48	63.25	even	3
1764.2.w.c.1109.16		48	63.52	odd	6
1764.2.x.c.293.2	✓	48	21.2	odd	6
1764.2.x.c.293.23	yes	48	21.5	even	6
1764.2.x.c.1469.2	yes	48	63.61	odd	6
1764.2.x.c.1469.23	yes	48	63.16	even	3
1764.2.bm.c.1685.7		48	9.7	even	3
1764.2.bm.c.1685.18		48	63.34	odd	6
1764.2.bm.c.1697.7		48	21.17	even	6
1764.2.bm.c.1697.18		48	21.11	odd	6
5292.2.w.c.521.5		48	63.38	even	6	inner
5292.2.w.c.521.20		48	63.11	odd	6	inner
5292.2.w.c.1097.5		48	1.1	even	1	trivial
5292.2.w.c.1097.20		48	7.6	odd	2	inner
5292.2.x.c.881.5		48	7.2	even	3
5292.2.x.c.881.20		48	7.5	odd	6
5292.2.x.c.4409.5		48	63.47	even	6
5292.2.x.c.4409.20		48	63.2	odd	6
5292.2.bm.c.2285.5		48	7.3	odd	6
5292.2.bm.c.2285.20		48	7.4	even	3
5292.2.bm.c.4625.5		48	9.2	odd	6
5292.2.bm.c.4625.20		48	63.20	even	6

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.w (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.31387 - 2.27569i) q^{5} +O(q^{10})\)	\(q+(-1.31387 - 2.27569i) q^{5} +(5.18523 + 2.99370i) q^{11} +(2.54231 + 1.46780i) q^{13} +(-0.0167322 - 0.0289811i) q^{17} +(-7.07369 - 4.08400i) q^{19} +(7.04605 - 4.06804i) q^{23} +(-0.952507 + 1.64979i) q^{25} +(0.949006 - 0.547909i) q^{29} +8.10282i q^{31} +(-0.105898 + 0.183421i) q^{37} +(-2.23087 + 3.86399i) q^{41} +(3.78001 + 6.54717i) q^{43} +7.07610 q^{47} +(1.59204 - 0.919166i) q^{53} -15.7333i q^{55} -4.73688 q^{59} +11.9827i q^{61} -7.71401i q^{65} +7.62295 q^{67} -3.52461i q^{71} +(-2.24398 + 1.29556i) q^{73} -7.17904 q^{79} +(-0.932743 - 1.61556i) q^{83} +(-0.0439680 + 0.0761548i) q^{85} +(-3.56125 + 6.16826i) q^{89} +21.4634i q^{95} +(13.2520 - 7.65104i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 48 * q	\( 48 q - 24 q^{11} + 48 q^{23} - 24 q^{25} + 48 q^{53} + 48 q^{79} - 24 q^{85}+O(q^{100}) \) 48 * q - 24 * q^11 + 48 * q^23 - 24 * q^25 + 48 * q^53 + 48 * q^79 - 24 * q^85

Introduction

L-functions

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