LMFDB - Embedded newform 1404.2.j.a.289.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1404,2,Mod(289,1404)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1404.289"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1404, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4, 2])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1404 = 2^{2} \cdot 3^{3} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1404.j (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$11.2109964438$
Analytic rank:	$0$
Dimension:	$28$
Relative dimension:	$14$ over $\Q(\zeta_{3})$
Twist minimal:	no (minimal twist has level 468)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			289.8
Character	$\chi$	$=$	1404.289
Dual form			1404.2.j.a.685.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.216950 + 0.375768i) q^{5} +(-0.791636 - 1.37115i) q^{7} +3.28568 q^{11} +(-1.50098 - 3.27827i) q^{13} +(-0.820732 + 1.42155i) q^{17} +(0.0150406 - 0.0260511i) q^{19} +(-0.0282011 + 0.0488457i) q^{23} +(2.40587 - 4.16708i) q^{25} +3.37346 q^{29} +(-2.73925 - 4.74451i) q^{31} +(0.343491 - 0.594943i) q^{35} +(-1.81655 - 3.14636i) q^{37} +(-1.81506 + 3.14378i) q^{41} +(-1.33998 - 2.32092i) q^{43} +(3.78962 - 6.56382i) q^{47} +(2.24662 - 3.89127i) q^{49} +4.05057 q^{53} +(0.712828 + 1.23465i) q^{55} +0.362821 q^{59} +(1.72774 + 2.99253i) q^{61} +(0.906231 - 1.27524i) q^{65} +(-1.28103 + 2.21882i) q^{67} +(7.29808 - 12.6407i) q^{71} -0.162888 q^{73} +(-2.60107 - 4.50518i) q^{77} +(5.70128 - 9.87490i) q^{79} +(5.52019 - 9.56125i) q^{83} -0.712231 q^{85} +(6.13248 + 10.6218i) q^{89} +(-3.30678 + 4.65327i) q^{91} +0.0130522 q^{95} +(6.46904 + 11.2047i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 2 q^{7} - 8 q^{11} + q^{13} + 8 q^{17} - q^{19} + 4 q^{23} - 14 q^{25} - 26 q^{29} + 2 q^{31} - 3 q^{35} - q^{37} - 4 q^{41} + 2 q^{43} - 11 q^{47} - 12 q^{49} - 52 q^{53} - 16 q^{59} - 7 q^{61}+ \cdots - 25 q^{97}+O(q^{100}) $ 28 * q + 2 * q^7 - 8 * q^11 + q^13 + 8 * q^17 - q^19 + 4 * q^23 - 14 * q^25 - 26 * q^29 + 2 * q^31 - 3 * q^35 - q^37 - 4 * q^41 + 2 * q^43 - 11 * q^47 - 12 * q^49 - 52 * q^53 - 16 * q^59 - 7 * q^61 + 31 * q^65 - 7 * q^67 + 12 * q^71 + 14 * q^73 + 28 * q^77 + 5 * q^79 - 9 * q^83 - 36 * q^85 + 11 * q^89 - q^91 + 56 * q^95 - 25 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1404\mathbb{Z}\right)^\times$.

$n$	$677$	$703$	$1081$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$e\left(\frac{1}{3}\right)$

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
$5$	0.216950	+	0.375768i	0.0970229	+	0.168049i	0.910451	−	0.413617i	$-0.135735\pi$
$5$	0.216950	+	0.375768i	0.0970229	+	0.168049i	−0.813428	+	0.581665i	$0.802401\pi$
$6$		0			0
$7$	−0.791636	−	1.37115i	−0.299210	−	0.518248i	0.676745	−	0.736217i	$-0.263390\pi$
$7$	−0.791636	−	1.37115i	−0.299210	−	0.518248i	−0.975956	+	0.217970i	$0.930057\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.28568			0.990671			0.495335	−	0.868702i	$-0.335045\pi$
$11$	3.28568			0.990671			0.495335	+	0.868702i	$0.335045\pi$
$12$		0			0
$13$	−1.50098	−	3.27827i	−0.416298	−	0.909228i
$13$	−1.50098	−	3.27827i	−0.416298	−	0.909228i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.820732	+	1.42155i	−0.199057	+	0.344777i	−0.948223	−	0.317606i	$-0.897121\pi$
$17$	−0.820732	+	1.42155i	−0.199057	+	0.344777i	0.749166	+	0.662382i	$0.230455\pi$
$18$		0			0
$19$	0.0150406	−	0.0260511i	0.00345055	−	0.00597653i	−0.864295	−	0.502985i	$-0.832235\pi$
$19$	0.0150406	−	0.0260511i	0.00345055	−	0.00597653i	0.867746	+	0.497009i	$0.165568\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.0282011	+	0.0488457i	−0.00588033	+	0.0101850i	−0.868951	−	0.494899i	$-0.835205\pi$
$23$	−0.0282011	+	0.0488457i	−0.00588033	+	0.0101850i	0.863070	+	0.505084i	$0.168538\pi$
$24$		0			0
$25$	2.40587	−	4.16708i	0.481173	−	0.833416i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.37346			0.626436			0.313218	−	0.949681i	$-0.398593\pi$
$29$	3.37346			0.626436			0.313218	+	0.949681i	$0.398593\pi$
$30$		0			0
$31$	−2.73925	−	4.74451i	−0.491983	−	0.852140i	0.507974	−	0.861372i	$-0.330394\pi$
$31$	−2.73925	−	4.74451i	−0.491983	−	0.852140i	−0.999957	+	0.00923265i	$0.997061\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.343491	−	0.594943i	0.0580605	−	0.100564i
$36$		0			0
$37$	−1.81655	−	3.14636i	−0.298639	−	0.517258i	0.677186	−	0.735812i	$-0.263199\pi$
$37$	−1.81655	−	3.14636i	−0.298639	−	0.517258i	−0.975825	+	0.218554i	$0.929866\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−1.81506	+	3.14378i	−0.283465	+	0.490976i	−0.972236	−	0.234004i	$-0.924817\pi$
$41$	−1.81506	+	3.14378i	−0.283465	+	0.490976i	0.688771	+	0.724979i	$0.258151\pi$
$42$		0			0
$43$	−1.33998	−	2.32092i	−0.204345	−	0.353937i	0.745579	−	0.666418i	$-0.232173\pi$
$43$	−1.33998	−	2.32092i	−0.204345	−	0.353937i	−0.949924	+	0.312481i	$0.898840\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.78962	−	6.56382i	0.552773	−	0.957432i	−0.445300	−	0.895382i	$-0.646903\pi$
$47$	3.78962	−	6.56382i	0.552773	−	0.957432i	0.998073	−	0.0620500i	$-0.0197638\pi$
$48$		0			0
$49$	2.24662	−	3.89127i	0.320946	−	0.555895i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	4.05057			0.556388			0.278194	−	0.960525i	$-0.410264\pi$
$53$	4.05057			0.556388			0.278194	+	0.960525i	$0.410264\pi$
$54$		0			0
$55$	0.712828	+	1.23465i	0.0961178	+	0.166481i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.362821			0.0472353			0.0236176	−	0.999721i	$-0.492482\pi$
$59$	0.362821			0.0472353			0.0236176	+	0.999721i	$0.492482\pi$
$60$		0			0
$61$	1.72774	+	2.99253i	0.221214	+	0.383154i	0.955177	−	0.296035i	$-0.0956646\pi$
$61$	1.72774	+	2.99253i	0.221214	+	0.383154i	−0.733963	+	0.679190i	$0.762331\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	0.906231	−	1.27524i	0.112404	−	0.158174i
$66$		0			0
$67$	−1.28103	+	2.21882i	−0.156503	+	0.271071i	−0.933605	−	0.358303i	$-0.883355\pi$
$67$	−1.28103	+	2.21882i	−0.156503	+	0.271071i	0.777102	+	0.629374i	$0.216689\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	7.29808	−	12.6407i	0.866123	−	1.50017i	0.000195676	−	1.00000i	$-0.499938\pi$
$71$	7.29808	−	12.6407i	0.866123	−	1.50017i	0.865928	−	0.500169i	$-0.166729\pi$
$72$		0			0
$73$	−0.162888			−0.0190646			−0.00953230	−	0.999955i	$-0.503034\pi$
$73$	−0.162888			−0.0190646			−0.00953230	+	0.999955i	$0.503034\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−2.60107	−	4.50518i	−0.296419	−	0.513413i
$78$		0			0
$79$	5.70128	−	9.87490i	0.641444	−	1.11101i	−0.343667	−	0.939092i	$-0.611669\pi$
$79$	5.70128	−	9.87490i	0.641444	−	1.11101i	0.985111	−	0.171922i	$-0.0549976\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	5.52019	−	9.56125i	0.605920	−	1.04948i	−0.385986	−	0.922505i	$-0.626139\pi$
$83$	5.52019	−	9.56125i	0.605920	−	1.04948i	0.991905	−	0.126979i	$-0.0405281\pi$
$84$		0			0
$85$	−0.712231			−0.0772523
$86$		0			0
$87$		0			0
$88$		0			0
$89$	6.13248	+	10.6218i	0.650042	+	1.12591i	0.983112	+	0.183004i	$0.0585820\pi$
$89$	6.13248	+	10.6218i	0.650042	+	1.12591i	−0.333070	+	0.942902i	$0.608085\pi$
$90$		0			0
$91$	−3.30678	+	4.65327i	−0.346645	+	0.487796i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	0.0130522			0.00133913
$96$		0			0
$97$	6.46904	+	11.2047i	0.656831	+	1.13767i	0.981431	+	0.191814i	$0.0614371\pi$
$97$	6.46904	+	11.2047i	0.656831	+	1.13767i	−0.324600	+	0.945851i	$0.605230\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	7.63565			0.759775			0.379888	−	0.925033i	$-0.375963\pi$
$101$	7.63565			0.759775			0.379888	+	0.925033i	$0.375963\pi$
$102$		0			0
$103$	−3.58661	−	6.21219i	−0.353399	−	0.612105i	0.633444	−	0.773789i	$-0.281641\pi$
$103$	−3.58661	−	6.21219i	−0.353399	−	0.612105i	−0.986843	+	0.161684i	$0.948307\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−7.29326	−	12.6323i	−0.705066	−	1.22121i	−0.966668	−	0.256035i	$-0.917584\pi$
$107$	−7.29326	−	12.6323i	−0.705066	−	1.22121i	0.261601	−	0.965176i	$-0.415750\pi$
$108$		0			0
$109$	−16.2665			−1.55805			−0.779026	−	0.626992i	$-0.784286\pi$
$109$	−16.2665			−1.55805			−0.779026	+	0.626992i	$0.784286\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−19.2845			−1.81413			−0.907066	−	0.420988i	$-0.861683\pi$
$113$	−19.2845			−1.81413			−0.907066	+	0.420988i	$0.861683\pi$
$114$		0			0
$115$	−0.0244729			−0.00228211
$116$		0			0
$117$		0			0
$118$		0			0
$119$	2.59889			0.238239
$120$		0			0
$121$	−0.204282			−0.0185711
$122$		0			0
$123$		0			0
$124$		0			0
$125$	4.25731			0.380785
$126$		0			0
$127$	4.39676	+	7.61542i	0.390150	+	0.675759i	0.992469	−	0.122496i	$-0.0390899\pi$
$127$	4.39676	+	7.61542i	0.390150	+	0.675759i	−0.602319	+	0.798255i	$0.705757\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	9.05299	+	15.6802i	0.790964	+	1.36999i	0.925371	+	0.379063i	$0.123754\pi$
$131$	9.05299	+	15.6802i	0.790964	+	1.36999i	−0.134407	+	0.990926i	$0.542913\pi$
$132$		0			0
$133$	−0.0476268			−0.00412976
$134$		0			0
$135$		0			0
$136$		0			0
$137$	2.05714	+	3.56307i	0.175753	+	0.304414i	0.940422	−	0.340010i	$-0.110431\pi$
$137$	2.05714	+	3.56307i	0.175753	+	0.304414i	−0.764668	+	0.644424i	$0.777097\pi$
$138$		0			0
$139$	12.1691			1.03217			0.516084	−	0.856538i	$-0.327389\pi$
$139$	12.1691			1.03217			0.516084	+	0.856538i	$0.327389\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−4.93175	−	10.7714i	−0.412414	−	0.900746i
$144$		0			0
$145$	0.731872	+	1.26764i	0.0607786	+	0.105272i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−9.45044			−0.774210			−0.387105	−	0.922036i	$-0.626525\pi$
$149$	−9.45044			−0.774210			−0.387105	+	0.922036i	$0.626525\pi$
$150$		0			0
$151$	5.31651	−	9.20847i	0.432651	−	0.749374i	−0.564449	−	0.825468i	$-0.690911\pi$
$151$	5.31651	−	9.20847i	0.432651	−	0.749374i	0.997101	+	0.0760934i	$0.0242447\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1.18856	−	2.05864i	0.0954672	−	0.165354i
$156$		0			0
$157$	−0.146130	−	0.253105i	−0.0116624	−	0.0202000i	0.860135	−	0.510066i	$-0.170379\pi$
$157$	−0.146130	−	0.253105i	−0.0116624	−	0.0202000i	−0.871798	+	0.489866i	$0.837046\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	0.0893000			0.00703782
$162$		0			0
$163$	7.12118	−	12.3342i	0.557774	−	0.966092i	−0.439908	−	0.898043i	$-0.644989\pi$
$163$	7.12118	−	12.3342i	0.557774	−	0.966092i	0.997682	−	0.0680496i	$-0.0216776\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−7.62713	+	13.2106i	−0.590205	+	1.02227i	0.403999	+	0.914759i	$0.367620\pi$
$167$	−7.62713	+	13.2106i	−0.590205	+	1.02227i	−0.994204	+	0.107506i	$0.965714\pi$
$168$		0			0
$169$	−8.49411	+	9.84125i	−0.653393	+	0.757019i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	3.88027	+	6.72083i	0.295012	+	0.510975i	0.974988	−	0.222259i	$-0.0713431\pi$
$173$	3.88027	+	6.72083i	0.295012	+	0.510975i	−0.679976	+	0.733235i	$0.738010\pi$
$174$		0			0
$175$	−7.61828			−0.575888
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−4.40510	−	7.62986i	−0.329253	−	0.570282i	0.653111	−	0.757262i	$-0.273463\pi$
$179$	−4.40510	−	7.62986i	−0.329253	−	0.570282i	−0.982364	+	0.186980i	$0.940130\pi$
$180$		0			0
$181$	9.31165			0.692129			0.346065	−	0.938211i	$-0.387518\pi$
$181$	9.31165			0.692129			0.346065	+	0.938211i	$0.387518\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.788200	−	1.36520i	0.0579496	−	0.100372i
$186$		0			0
$187$	−2.69667	+	4.67076i	−0.197200	+	0.341560i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	9.04730	+	15.6704i	0.654639	+	1.13387i	0.981984	+	0.188964i	$0.0605128\pi$
$191$	9.04730	+	15.6704i	0.654639	+	1.13387i	−0.327345	+	0.944905i	$0.606154\pi$
$192$		0			0
$193$	2.08885	−	3.61799i	0.150359	−	0.260429i	−0.781001	−	0.624530i	$-0.785290\pi$
$193$	2.08885	−	3.61799i	0.150359	−	0.260429i	0.931359	+	0.364101i	$0.118624\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−10.8080	−	18.7200i	−0.770038	−	1.33374i	−0.937542	−	0.347873i	$-0.886904\pi$
$197$	−10.8080	−	18.7200i	−0.770038	−	1.33374i	0.167504	−	0.985871i	$-0.446429\pi$
$198$		0			0
$199$	7.59202	−	13.1498i	0.538184	−	0.932162i	−0.460818	−	0.887495i	$-0.652444\pi$
$199$	7.59202	−	13.1498i	0.538184	−	0.932162i	0.999002	−	0.0446676i	$-0.0142229\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−2.67055	−	4.62553i	−0.187436	−	0.324649i
$204$		0			0
$205$	−1.57511			−0.110010
$206$		0			0
$207$		0			0
$208$		0			0
$209$	0.0494187	−	0.0855957i	0.00341836	−	0.00592078i
$210$		0			0
$211$	−9.96737	+	17.2640i	−0.686182	+	1.18850i	0.286881	+	0.957966i	$0.407381\pi$
$211$	−9.96737	+	17.2640i	−0.686182	+	1.18850i	−0.973064	+	0.230537i	$0.925952\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	0.581418	−	1.00705i	0.0396524	−	0.0686799i
$216$		0			0
$217$	−4.33697	+	7.51185i	−0.294413	+	0.509938i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	5.89213	+	0.556861i	0.396348	+	0.0374585i
$222$		0			0
$223$	−19.1053			−1.27938			−0.639692	−	0.768631i	$-0.720938\pi$
$223$	−19.1053			−1.27938			−0.639692	+	0.768631i	$0.720938\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	6.44350	+	11.1605i	0.427670	+	0.740747i	0.996666	−	0.0815941i	$-0.0260011\pi$
$227$	6.44350	+	11.1605i	0.427670	+	0.740747i	−0.568995	+	0.822341i	$0.692668\pi$
$228$		0			0
$229$	5.06389	+	8.77091i	0.334631	+	0.579598i	0.983414	−	0.181376i	$-0.0580550\pi$
$229$	5.06389	+	8.77091i	0.334631	+	0.579598i	−0.648783	+	0.760974i	$0.724722\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−19.4552			−1.27455			−0.637275	−	0.770637i	$-0.719938\pi$
$233$	−19.4552			−1.27455			−0.637275	+	0.770637i	$0.719938\pi$
$234$		0			0
$235$	3.28863			0.214527
$236$		0			0
$237$		0			0
$238$		0			0
$239$	6.33263	+	10.9684i	0.409623	+	0.709489i	0.994847	−	0.101383i	$-0.0323268\pi$
$239$	6.33263	+	10.9684i	0.409623	+	0.709489i	−0.585224	+	0.810872i	$0.698993\pi$
$240$		0			0
$241$	−7.50431	−	12.9979i	−0.483395	−	0.837265i	0.516423	−	0.856334i	$-0.327263\pi$
$241$	−7.50431	−	12.9979i	−0.483395	−	0.837265i	−0.999818	+	0.0190684i	$0.993930\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.94962			0.124557
$246$		0			0
$247$	−0.107978	−	0.0102049i	−0.00687049	−	0.000649325i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−3.90158	+	6.75773i	−0.246265	+	0.426544i	−0.962487	−	0.271329i	$-0.912537\pi$
$251$	−3.90158	+	6.75773i	−0.246265	+	0.426544i	0.716221	+	0.697873i	$0.245870\pi$
$252$		0			0
$253$	−0.0926598	+	0.160492i	−0.00582547	+	0.0100900i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−3.72220	+	6.44703i	−0.232184	+	0.402155i	−0.958451	−	0.285259i	$-0.907921\pi$
$257$	−3.72220	+	6.44703i	−0.232184	+	0.402155i	0.726266	+	0.687413i	$0.241254\pi$
$258$		0			0
$259$	−2.87609	+	4.98154i	−0.178712	+	0.309538i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−7.13493			−0.439958			−0.219979	−	0.975505i	$-0.570599\pi$
$263$	−7.13493			−0.439958			−0.219979	+	0.975505i	$0.570599\pi$
$264$		0			0
$265$	0.878769	+	1.52207i	0.0539824	+	0.0935002i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−14.1183	+	24.4536i	−0.860807	+	1.49096i	0.0103450	+	0.999946i	$0.496707\pi$
$269$	−14.1183	+	24.4536i	−0.860807	+	1.49096i	−0.871152	+	0.491014i	$0.836626\pi$
$270$		0			0
$271$	−3.82382	−	6.62305i	−0.232281	−	0.402322i	0.726198	−	0.687485i	$-0.241285\pi$
$271$	−3.82382	−	6.62305i	−0.232281	−	0.402322i	−0.958479	+	0.285164i	$0.907952\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	7.90491	−	13.6917i	0.476684	−	0.825641i
$276$		0			0
$277$	9.47813	+	16.4166i	0.569486	+	0.986378i	0.996617	+	0.0821882i	$0.0261909\pi$
$277$	9.47813	+	16.4166i	0.569486	+	0.986378i	−0.427131	+	0.904190i	$0.640476\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−12.7457	+	22.0763i	−0.760347	+	1.31696i	0.182325	+	0.983238i	$0.441638\pi$
$281$	−12.7457	+	22.0763i	−0.760347	+	1.31696i	−0.942672	+	0.333721i	$0.891696\pi$
$282$		0			0
$283$	−4.26197	+	7.38195i	−0.253348	+	0.438811i	−0.964445	−	0.264282i	$-0.914865\pi$
$283$	−4.26197	+	7.38195i	−0.253348	+	0.438811i	0.711098	+	0.703093i	$0.248198\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	5.74747			0.339262
$288$		0			0
$289$	7.15280	+	12.3890i	0.420753	+	0.728765i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	2.30817			0.134845			0.0674223	−	0.997725i	$-0.478523\pi$
$293$	2.30817			0.134845			0.0674223	+	0.997725i	$0.478523\pi$
$294$		0			0
$295$	0.0787140	+	0.136337i	0.00458290	+	0.00793782i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	0.202459	+	0.0191342i	0.0117085	+	0.00110656i
$300$		0			0
$301$	−2.12156	+	3.67464i	−0.122285	+	0.211803i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−0.749665	+	1.29846i	−0.0429257	+	0.0743495i
$306$		0			0
$307$	25.9276			1.47976			0.739882	−	0.672736i	$-0.234881\pi$
$307$	25.9276			1.47976			0.739882	+	0.672736i	$0.234881\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	16.1747	+	28.0154i	0.917181	+	1.58860i	0.803676	+	0.595068i	$0.202875\pi$
$311$	16.1747	+	28.0154i	0.917181	+	1.58860i	0.113506	+	0.993537i	$0.463792\pi$
$312$		0			0
$313$	−1.84638	+	3.19803i	−0.104364	+	0.180763i	−0.913478	−	0.406888i	$-0.866614\pi$
$313$	−1.84638	+	3.19803i	−0.104364	+	0.180763i	0.809114	+	0.587651i	$0.199947\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−6.38892	+	11.0659i	−0.358837	+	0.621524i	−0.987767	−	0.155938i	$-0.950160\pi$
$317$	−6.38892	+	11.0659i	−0.358837	+	0.621524i	0.628930	+	0.777462i	$0.283493\pi$
$318$		0			0
$319$	11.0841			0.620592
$320$		0			0
$321$		0			0
$322$		0			0
$323$	0.0246886	+	0.0427620i	0.00137371	+	0.00237934i
$324$		0			0
$325$	−17.2720	−	1.63236i	−0.958077	−	0.0905471i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−12.0000			−0.661582
$330$		0			0
$331$	−4.05904	−	7.03047i	−0.223105	−	0.386430i	0.732644	−	0.680612i	$-0.238286\pi$
$331$	−4.05904	−	7.03047i	−0.223105	−	0.386430i	−0.955749	+	0.294182i	$0.904953\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1.11168			−0.0607376
$336$		0			0
$337$	17.4314	+	30.1921i	0.949551	+	1.64467i	0.746373	+	0.665528i	$0.231794\pi$
$337$	17.4314	+	30.1921i	0.949551	+	1.64467i	0.203178	+	0.979142i	$0.434873\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−9.00029	−	15.5890i	−0.487393	−	0.844190i
$342$		0			0
$343$	−18.1969			−0.982543
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−22.1469			−1.18891			−0.594453	−	0.804130i	$-0.702631\pi$
$347$	−22.1469			−1.18891			−0.594453	+	0.804130i	$0.702631\pi$
$348$		0			0
$349$	17.8426			0.955094			0.477547	−	0.878606i	$-0.341526\pi$
$349$	17.8426			0.955094			0.477547	+	0.878606i	$0.341526\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−14.9858			−0.797616			−0.398808	−	0.917034i	$-0.630576\pi$
$353$	−14.9858			−0.797616			−0.398808	+	0.917034i	$0.630576\pi$
$354$		0			0
$355$	6.33327			0.336135
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−2.49334			−0.131594			−0.0657968	−	0.997833i	$-0.520959\pi$
$359$	−2.49334			−0.131594			−0.0657968	+	0.997833i	$0.520959\pi$
$360$		0			0
$361$	9.49955	+	16.4537i	0.499976	+	0.865984i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−0.0353385	−	0.0612081i	−0.00184970	−	0.00320378i
$366$		0			0
$367$	−4.38816			−0.229060			−0.114530	−	0.993420i	$-0.536536\pi$
$367$	−4.38816			−0.229060			−0.114530	+	0.993420i	$0.536536\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−3.20657	−	5.55395i	−0.166477	−	0.288347i
$372$		0			0
$373$	15.0702			0.780304			0.390152	−	0.920750i	$-0.372422\pi$
$373$	15.0702			0.780304			0.390152	+	0.920750i	$0.372422\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−5.06350	−	11.0591i	−0.260784	−	0.569573i
$378$		0			0
$379$	−13.2405	−	22.9331i	−0.680117	−	1.17800i	−0.974945	−	0.222446i	$-0.928596\pi$
$379$	−13.2405	−	22.9331i	−0.680117	−	1.17800i	0.294828	−	0.955550i	$-0.404738\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	3.05939			0.156327			0.0781637	−	0.996941i	$-0.475094\pi$
$383$	3.05939			0.156327			0.0781637	+	0.996941i	$0.475094\pi$
$384$		0			0
$385$	1.12860	−	1.95480i	0.0575189	−	0.0996256i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	7.58306	−	13.1342i	0.384476	−	0.665932i	−0.607220	−	0.794534i	$-0.707715\pi$
$389$	7.58306	−	13.1342i	0.384476	−	0.665932i	0.991696	+	0.128601i	$0.0410487\pi$
$390$		0			0
$391$	−0.0462911	−	0.0801785i	−0.00234104	−	0.00405480i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	4.94756			0.248939
$396$		0			0
$397$	6.92576	−	11.9958i	0.347594	−	0.602050i	−0.638228	−	0.769848i	$-0.720332\pi$
$397$	6.92576	−	11.9958i	0.347594	−	0.602050i	0.985821	+	0.167798i	$0.0536655\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1.31025	+	2.26942i	−0.0654309	+	0.113330i	−0.896885	−	0.442264i	$-0.854176\pi$
$401$	−1.31025	+	2.26942i	−0.0654309	+	0.113330i	0.831454	+	0.555593i	$0.187509\pi$
$402$		0			0
$403$	−11.4422	+	16.1014i	−0.569978	+	0.802068i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−5.96861	−	10.3379i	−0.295853	−	0.512432i
$408$		0			0
$409$	−15.7656			−0.779558			−0.389779	−	0.920909i	$-0.627449\pi$
$409$	−15.7656			−0.779558			−0.389779	+	0.920909i	$0.627449\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−0.287222	−	0.497484i	−0.0141333	−	0.0244796i
$414$		0			0
$415$	4.79042			0.235152
$416$		0			0
$417$		0			0
$418$		0			0
$419$	4.26344	−	7.38449i	0.208282	−	0.360756i	−0.742891	−	0.669412i	$-0.766546\pi$
$419$	4.26344	−	7.38449i	0.208282	−	0.360756i	0.951174	+	0.308656i	$0.0998793\pi$
$420$		0			0
$421$	−7.83552	+	13.5715i	−0.381880	+	0.661436i	−0.991331	−	0.131388i	$-0.958057\pi$
$421$	−7.83552	+	13.5715i	−0.381880	+	0.661436i	0.609451	+	0.792824i	$0.291390\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	3.94914	+	6.84012i	0.191562	+	0.331794i
$426$		0			0
$427$	2.73548	−	4.73799i	0.132379	−	0.229288i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−12.8332	−	22.2278i	−0.618155	−	1.07068i	−0.989822	−	0.142310i	$-0.954547\pi$
$431$	−12.8332	−	22.2278i	−0.618155	−	1.07068i	0.371667	−	0.928366i	$-0.378786\pi$
$432$		0			0
$433$	−18.3374	+	31.7614i	−0.881241	+	1.52635i	−0.0312791	+	0.999511i	$0.509958\pi$
$433$	−18.3374	+	31.7614i	−0.881241	+	1.52635i	−0.849962	+	0.526844i	$0.823375\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	0.000848323		0.00146934i	4.05808e−5		7.02880e-5i
$438$		0			0
$439$	4.99493			0.238395			0.119197	−	0.992871i	$-0.461968\pi$
$439$	4.99493			0.238395			0.119197	+	0.992871i	$0.461968\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	15.1200	−	26.1886i	0.718373	−	1.24426i	−0.243271	−	0.969958i	$-0.578220\pi$
$443$	15.1200	−	26.1886i	0.718373	−	1.24426i	0.961644	−	0.274300i	$-0.0884462\pi$
$444$		0			0
$445$	−2.66088	+	4.60878i	−0.126138	+	0.218477i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	9.70002	−	16.8009i	0.457772	−	0.792885i	−0.541071	−	0.840977i	$-0.681981\pi$
$449$	9.70002	−	16.8009i	0.457772	−	0.792885i	0.998843	+	0.0480922i	$0.0153141\pi$
$450$		0			0
$451$	−5.96372	+	10.3295i	−0.280820	+	0.486395i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−2.46596	−	0.233056i	−0.115606	−	0.0109258i
$456$		0			0
$457$	−9.45223			−0.442156			−0.221078	−	0.975256i	$-0.570958\pi$
$457$	−9.45223			−0.442156			−0.221078	+	0.975256i	$0.570958\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	11.5227	+	19.9580i	0.536668	+	0.929536i	0.999081	+	0.0428711i	$0.0136505\pi$
$461$	11.5227	+	19.9580i	0.536668	+	0.929536i	−0.462413	+	0.886665i	$0.653016\pi$
$462$		0			0
$463$	−10.3312	−	17.8942i	−0.480134	−	0.831616i	0.519607	−	0.854406i	$-0.326078\pi$
$463$	−10.3312	−	17.8942i	−0.480134	−	0.831616i	−0.999740	+	0.0227898i	$0.992745\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	5.84410			0.270433			0.135216	−	0.990816i	$-0.456827\pi$
$467$	5.84410			0.270433			0.135216	+	0.990816i	$0.456827\pi$
$468$		0			0
$469$	4.05645			0.187309
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−4.40276	−	7.62580i	−0.202439	−	0.350635i
$474$		0			0
$475$	−0.0723714	−	0.125351i	−0.00332063	−	0.00575149i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−9.41643			−0.430248			−0.215124	−	0.976587i	$-0.569015\pi$
$479$	−9.41643			−0.430248			−0.215124	+	0.976587i	$0.569015\pi$
$480$		0			0
$481$	−7.58800	+	10.6778i	−0.345983	+	0.486864i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−2.80691	+	4.86172i	−0.127455	+	0.220759i
$486$		0			0
$487$	1.94398	−	3.36708i	0.0880903	−	0.152577i	−0.818614	−	0.574345i	$-0.805257\pi$
$487$	1.94398	−	3.36708i	0.0880903	−	0.152577i	0.906704	+	0.421768i	$0.138590\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−2.62024	+	4.53840i	−0.118250	+	0.204815i	−0.919074	−	0.394084i	$-0.871062\pi$
$491$	−2.62024	+	4.53840i	−0.118250	+	0.204815i	0.800824	+	0.598899i	$0.204395\pi$
$492$		0			0
$493$	−2.76871	+	4.79554i	−0.124696	+	0.215980i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−23.1097			−1.03661
$498$		0			0
$499$	10.8182	+	18.7377i	0.484289	+	0.838813i	0.999837	−	0.0180474i	$-0.00574497\pi$
$499$	10.8182	+	18.7377i	0.484289	+	0.838813i	−0.515548	+	0.856861i	$0.672412\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−14.9763	+	25.9397i	−0.667760	+	1.15659i	0.310769	+	0.950485i	$0.399413\pi$
$503$	−14.9763	+	25.9397i	−0.667760	+	1.15659i	−0.978529	+	0.206108i	$0.933920\pi$
$504$		0			0
$505$	1.65655	+	2.86923i	0.0737156	+	0.127679i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	12.5253	−	21.6945i	0.555176	−	0.961594i	−0.442713	−	0.896663i	$-0.645984\pi$
$509$	12.5253	−	21.6945i	0.555176	−	0.961594i	0.997890	−	0.0649305i	$-0.0206826\pi$
$510$		0			0
$511$	0.128948	+	0.223345i	0.00570433	+	0.00988018i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	1.55623	−	2.69546i	0.0685756	−	0.118776i
$516$		0			0
$517$	12.4515	−	21.5666i	0.547617	−	0.948500i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	21.4586			0.940116			0.470058	−	0.882635i	$-0.344233\pi$
$521$	21.4586			0.940116			0.470058	+	0.882635i	$0.344233\pi$
$522$		0			0
$523$	−0.790297	−	1.36883i	−0.0345573	−	0.0598550i	0.848230	−	0.529629i	$-0.177669\pi$
$523$	−0.790297	−	1.36883i	−0.0345573	−	0.0598550i	−0.882787	+	0.469774i	$0.844335\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	8.99275			0.391730
$528$		0			0
$529$	11.4984	+	19.9158i	0.499931	+	0.865906i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	13.0305	+	1.23151i	0.564415	+	0.0533424i
$534$		0			0
$535$	3.16454	−	5.48115i	0.136815	−	0.236971i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	7.38170	−	12.7855i	0.317952	−	0.550709i
$540$		0			0
$541$	26.1156			1.12280			0.561398	−	0.827546i	$-0.310264\pi$
$541$	26.1156			1.12280			0.561398	+	0.827546i	$0.310264\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−3.52902	−	6.11244i	−0.151167	−	0.261828i
$546$		0			0
$547$	10.4157	−	18.0406i	0.445344	−	0.771359i	−0.552732	−	0.833359i	$-0.686415\pi$
$547$	10.4157	−	18.0406i	0.445344	−	0.771359i	0.998076	+	0.0620002i	$0.0197479\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	0.0507389	−	0.0878823i	0.00216155	−	0.00374391i
$552$		0			0
$553$	−18.0533			−0.767706
$554$		0			0
$555$		0			0
$556$		0			0
$557$	19.4069	+	33.6138i	0.822299	+	1.42426i	0.903967	+	0.427603i	$0.140642\pi$
$557$	19.4069	+	33.6138i	0.822299	+	1.42426i	−0.0816680	+	0.996660i	$0.526025\pi$
$558$		0			0
$559$	−5.59730	+	7.87648i	−0.236741	+	0.333140i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	0.804224			0.0338940			0.0169470	−	0.999856i	$-0.494605\pi$
$563$	0.804224			0.0338940			0.0169470	+	0.999856i	$0.494605\pi$
$564$		0			0
$565$	−4.18377	−	7.24650i	−0.176012	−	0.304862i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−19.8108			−0.830514			−0.415257	−	0.909704i	$-0.636308\pi$
$569$	−19.8108			−0.830514			−0.415257	+	0.909704i	$0.636308\pi$
$570$		0			0
$571$	−18.8485	−	32.6466i	−0.788786	−	1.36622i	−0.926711	−	0.375775i	$-0.877376\pi$
$571$	−18.8485	−	32.6466i	−0.788786	−	1.36622i	0.137924	−	0.990443i	$-0.455957\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	0.135696	+	0.235032i	0.00565891	+	0.00980153i
$576$		0			0
$577$	1.66499			0.0693145			0.0346573	−	0.999399i	$-0.488966\pi$
$577$	1.66499			0.0693145			0.0346573	+	0.999399i	$0.488966\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−17.4799			−0.725190
$582$		0			0
$583$	13.3089			0.551197
$584$		0			0
$585$		0			0
$586$		0			0
$587$	13.7504			0.567541			0.283770	−	0.958892i	$-0.408415\pi$
$587$	13.7504			0.567541			0.283770	+	0.958892i	$0.408415\pi$
$588$		0			0
$589$	−0.164800			−0.00679045
$590$		0			0
$591$		0			0
$592$		0			0
$593$	48.0032			1.97125			0.985627	−	0.168937i	$-0.0540334\pi$
$593$	48.0032			1.97125			0.985627	+	0.168937i	$0.0540334\pi$
$594$		0			0
$595$	0.563828	+	0.976578i	0.0231147	+	0.0400358i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	18.5595	+	32.1459i	0.758319	+	1.31345i	0.943707	+	0.330782i	$0.107312\pi$
$599$	18.5595	+	32.1459i	0.758319	+	1.31345i	−0.185388	+	0.982665i	$0.559354\pi$
$600$		0			0
$601$	7.04826			0.287505			0.143752	−	0.989614i	$-0.454083\pi$
$601$	7.04826			0.287505			0.143752	+	0.989614i	$0.454083\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−0.0443189	−	0.0767627i	−0.00180182	−	0.00312085i
$606$		0			0
$607$	10.3740			0.421068			0.210534	−	0.977587i	$-0.432480\pi$
$607$	10.3740			0.421068			0.210534	+	0.977587i	$0.432480\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−27.2061	−	2.57123i	−1.10064	−	0.104021i
$612$		0			0
$613$	−16.7614	−	29.0316i	−0.676986	−	1.17257i	−0.975884	−	0.218289i	$-0.929952\pi$
$613$	−16.7614	−	29.0316i	−0.676986	−	1.17257i	0.298898	−	0.954285i	$-0.403381\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−35.1206			−1.41390			−0.706952	−	0.707262i	$-0.749930\pi$
$617$	−35.1206			−1.41390			−0.706952	+	0.707262i	$0.749930\pi$
$618$		0			0
$619$	19.5529	−	33.8667i	0.785899	−	1.36122i	−0.142562	−	0.989786i	$-0.545534\pi$
$619$	19.5529	−	33.8667i	0.785899	−	1.36122i	0.928461	−	0.371431i	$-0.121133\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	9.70939	−	16.8172i	0.388999	−	0.673765i
$624$		0			0
$625$	−11.1057	−	19.2356i	−0.444228	−	0.769426i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	5.96361			0.237785
$630$		0			0
$631$	11.2857	−	19.5474i	0.449277	−	0.778170i	−0.549062	−	0.835781i	$-0.685015\pi$
$631$	11.2857	−	19.5474i	0.449277	−	0.778170i	0.998339	+	0.0576114i	$0.0183484\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−1.90775	+	3.30433i	−0.0757069	+	0.131128i
$636$		0			0
$637$	−16.1288	−	1.52432i	−0.639045	−	0.0603957i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−5.60461	−	9.70746i	−0.221369	−	0.383422i	0.733855	−	0.679306i	$-0.237719\pi$
$641$	−5.60461	−	9.70746i	−0.221369	−	0.383422i	−0.955224	+	0.295884i	$0.904386\pi$
$642$		0			0
$643$	2.98337			0.117653			0.0588264	−	0.998268i	$-0.481264\pi$
$643$	2.98337			0.117653			0.0588264	+	0.998268i	$0.481264\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	19.2748	+	33.3850i	0.757772	+	1.31250i	0.943984	+	0.329990i	$0.107045\pi$
$647$	19.2748	+	33.3850i	0.757772	+	1.31250i	−0.186212	+	0.982510i	$0.559621\pi$
$648$		0			0
$649$	1.19212			0.0467946
$650$		0			0
$651$		0			0
$652$		0			0
$653$	25.0269	−	43.3478i	0.979377	−	1.69633i	0.314716	−	0.949186i	$-0.398091\pi$
$653$	25.0269	−	43.3478i	0.979377	−	1.69633i	0.664661	−	0.747145i	$-0.268576\pi$
$654$		0			0
$655$	−3.92809	+	6.80365i	−0.153483	+	0.265841i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	1.65831	+	2.87228i	0.0645985	+	0.111888i	0.896516	−	0.443012i	$-0.146090\pi$
$659$	1.65831	+	2.87228i	0.0645985	+	0.111888i	−0.831917	+	0.554900i	$0.812757\pi$
$660$		0			0
$661$	5.56081	−	9.63160i	0.216290	−	0.374626i	−0.737381	−	0.675478i	$-0.763938\pi$
$661$	5.56081	−	9.63160i	0.216290	−	0.374626i	0.953671	+	0.300852i	$0.0972709\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−0.0103326	−	0.0178966i	−0.000400682	−	0.000694001i
$666$		0			0
$667$	−0.0951352	+	0.164779i	−0.00368365	+	0.00638027i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	5.67680	+	9.83251i	0.219151	+	0.379580i
$672$		0			0
$673$	32.4363			1.25033			0.625163	−	0.780494i	$-0.285032\pi$
$673$	32.4363			1.25033			0.625163	+	0.780494i	$0.285032\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	12.6356	−	21.8854i	0.485624	−	0.841126i	−0.514239	−	0.857647i	$-0.671926\pi$
$677$	12.6356	−	21.8854i	0.485624	−	0.841126i	0.999864	+	0.0165208i	$0.00525897\pi$
$678$		0			0
$679$	10.2423	−	17.7401i	0.393062	−	0.680803i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−0.343395	+	0.594778i	−0.0131397	+	0.0227585i	−0.872520	−	0.488578i	$-0.837516\pi$
$683$	−0.343395	+	0.594778i	−0.0131397	+	0.0227585i	0.859381	+	0.511336i	$0.170849\pi$
$684$		0			0
$685$	−0.892592	+	1.54601i	−0.0341042	+	0.0590702i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−6.07983	−	13.2788i	−0.231623	−	0.505884i
$690$		0			0
$691$	−44.4463			−1.69082			−0.845409	−	0.534119i	$-0.820643\pi$
$691$	−44.4463			−1.69082			−0.845409	+	0.534119i	$0.820643\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	2.64008	+	4.57275i	0.100144	+	0.173454i
$696$		0			0
$697$	−2.97936	−	5.16040i	−0.112851	−	0.195464i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	5.00391			0.188995			0.0944975	−	0.995525i	$-0.469876\pi$
$701$	5.00391			0.188995			0.0944975	+	0.995525i	$0.469876\pi$
$702$		0			0
$703$	−0.109288			−0.00412188
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−6.04466	−	10.4697i	−0.227333	−	0.393752i
$708$		0			0
$709$	−22.0166	−	38.1339i	−0.826852	−	1.43215i	−0.900495	−	0.434866i	$-0.856796\pi$
$709$	−22.0166	−	38.1339i	−0.826852	−	1.43215i	0.0736428	−	0.997285i	$-0.476538\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	0.308999			0.0115721
$714$		0			0
$715$	2.97759	−	4.19004i	0.111356	−	0.156699i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−16.8809	+	29.2385i	−0.629551	+	1.09041i	0.358091	+	0.933687i	$0.383428\pi$
$719$	−16.8809	+	29.2385i	−0.629551	+	1.09041i	−0.987642	+	0.156727i	$0.949906\pi$
$720$		0			0
$721$	−5.67858	+	9.83558i	−0.211481	+	0.366296i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	8.11609	−	14.0575i	0.301424	−	0.522082i
$726$		0			0
$727$	19.1552	−	33.1778i	0.710427	−	1.23050i	−0.254270	−	0.967133i	$-0.581835\pi$
$727$	19.1552	−	33.1778i	0.710427	−	1.23050i	0.964697	−	0.263362i	$-0.0848313\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	4.39907			0.162705
$732$		0			0
$733$	−24.2010	−	41.9174i	−0.893885	−	1.54825i	−0.835179	−	0.549978i	$-0.814636\pi$
$733$	−24.2010	−	41.9174i	−0.893885	−	1.54825i	−0.0587059	−	0.998275i	$-0.518697\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−4.20907	+	7.29033i	−0.155043	+	0.268543i
$738$		0			0
$739$	20.1732	+	34.9409i	0.742082	+	1.28532i	0.951546	+	0.307507i	$0.0994946\pi$
$739$	20.1732	+	34.9409i	0.742082	+	1.28532i	−0.209464	+	0.977816i	$0.567172\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	22.8084	−	39.5053i	0.836758	−	1.44931i	−0.0558331	−	0.998440i	$-0.517781\pi$
$743$	22.8084	−	39.5053i	0.836758	−	1.44931i	0.892591	−	0.450867i	$-0.148885\pi$
$744$		0			0
$745$	−2.05027	−	3.55117i	−0.0751161	−	0.130105i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−11.5472	+	20.0004i	−0.421926	+	0.730798i
$750$		0			0
$751$	−21.0467	+	36.4540i	−0.768007	+	1.33023i	0.170636	+	0.985334i	$0.445418\pi$
$751$	−21.0467	+	36.4540i	−0.768007	+	1.33023i	−0.938642	+	0.344892i	$0.887915\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	4.61366			0.167908
$756$		0			0
$757$	−15.6559	−	27.1168i	−0.569024	−	0.985578i	−0.996663	−	0.0816283i	$-0.973988\pi$
$757$	−15.6559	−	27.1168i	−0.569024	−	0.985578i	0.427639	−	0.903949i	$-0.359345\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	51.3570			1.86169			0.930845	−	0.365414i	$-0.119073\pi$
$761$	51.3570			1.86169			0.930845	+	0.365414i	$0.119073\pi$
$762$		0			0
$763$	12.8772	+	22.3039i	0.466185	+	0.807456i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−0.544588	−	1.18943i	−0.0196639	−	0.0429477i
$768$		0			0
$769$	−10.8919	+	18.8653i	−0.392772	+	0.680301i	−0.992814	−	0.119667i	$-0.961817\pi$
$769$	−10.8919	+	18.8653i	−0.392772	+	0.680301i	0.600042	+	0.799969i	$0.295151\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	9.64820	−	16.7112i	0.347022	−	0.601059i	−0.638697	−	0.769458i	$-0.720526\pi$
$773$	9.64820	−	16.7112i	0.347022	−	0.601059i	0.985719	+	0.168399i	$0.0538597\pi$
$774$		0			0
$775$	−26.3610			−0.946916
$776$		0			0
$777$		0			0
$778$		0			0
$779$	0.0545992	+	0.0945687i	0.00195622	+	0.00338827i
$780$		0			0
$781$	23.9792	−	41.5332i	0.858043	−	1.48617i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	0.0634058	−	0.109822i	0.00226305	−	0.00391972i
$786$		0			0
$787$	−10.4299			−0.371785			−0.185893	−	0.982570i	$-0.559518\pi$
$787$	−10.4299			−0.371785			−0.185893	+	0.982570i	$0.559518\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	15.2663	+	26.4420i	0.542807	+	0.940170i
$792$		0			0
$793$	7.21702	−	10.1557i	0.256284	−	0.360641i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	0.852516			0.0301977			0.0150988	−	0.999886i	$-0.495194\pi$
$797$	0.852516			0.0301977			0.0150988	+	0.999886i	$0.495194\pi$
$798$		0			0
$799$	6.22053	+	10.7743i	0.220067	+	0.381167i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−0.535199			−0.0188867
$804$		0			0
$805$	0.0193736	+	0.0335561i	0.000682830	+	0.00118270i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	22.8597	+	39.5942i	0.803704	+	1.39206i	0.917162	+	0.398514i	$0.130474\pi$
$809$	22.8597	+	39.5942i	0.803704	+	1.39206i	−0.113458	+	0.993543i	$0.536193\pi$
$810$		0			0
$811$	14.7191			0.516857			0.258429	−	0.966030i	$-0.416795\pi$
$811$	14.7191			0.516857			0.258429	+	0.966030i	$0.416795\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	6.17975			0.216467
$816$		0			0
$817$	−0.0806166			−0.00282042
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−12.6029			−0.439843			−0.219922	−	0.975518i	$-0.570580\pi$
$821$	−12.6029			−0.439843			−0.219922	+	0.975518i	$0.570580\pi$
$822$		0			0
$823$	−2.12581			−0.0741012			−0.0370506	−	0.999313i	$-0.511796\pi$
$823$	−2.12581			−0.0741012			−0.0370506	+	0.999313i	$0.511796\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	30.0089			1.04351			0.521756	−	0.853095i	$-0.325277\pi$
$827$	30.0089			1.04351			0.521756	+	0.853095i	$0.325277\pi$
$828$		0			0
$829$	−19.6435	−	34.0235i	−0.682246	−	1.18168i	−0.974294	−	0.225281i	$-0.927670\pi$
$829$	−19.6435	−	34.0235i	−0.682246	−	1.18168i	0.292048	−	0.956404i	$-0.405663\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	3.68775	+	6.38738i	0.127773	+	0.221310i
$834$		0			0
$835$	−6.61882			−0.229054
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−26.4687	−	45.8452i	−0.913802	−	1.58275i	−0.808646	−	0.588296i	$-0.799799\pi$
$839$	−26.4687	−	45.8452i	−0.913802	−	1.58275i	−0.105156	−	0.994456i	$-0.533534\pi$
$840$		0			0
$841$	−17.6198			−0.607578
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−5.54082	−	1.05676i	−0.190610	−	0.0363535i
$846$		0			0
$847$	0.161717	+	0.280102i	0.00555666	+	0.00962442i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	0.204915			0.00702438
$852$		0			0
$853$	−2.51731	+	4.36011i	−0.0861911	+	0.149287i	−0.905898	−	0.423496i	$-0.860803\pi$
$853$	−2.51731	+	4.36011i	−0.0861911	+	0.149287i	0.819707	+	0.572783i	$0.194136\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1.95321	+	3.38305i	−0.0667203	+	0.115563i	−0.897456	−	0.441104i	$-0.854587\pi$
$857$	−1.95321	+	3.38305i	−0.0667203	+	0.115563i	0.830736	+	0.556667i	$0.187920\pi$
$858$		0			0
$859$	−0.493148	−	0.854157i	−0.0168260	−	0.0291435i	0.857490	−	0.514501i	$-0.172023\pi$
$859$	−0.493148	−	0.854157i	−0.0168260	−	0.0291435i	−0.874316	+	0.485357i	$0.838689\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−44.2030			−1.50469			−0.752345	−	0.658770i	$-0.771077\pi$
$863$	−44.2030			−1.50469			−0.752345	+	0.658770i	$0.771077\pi$
$864$		0			0
$865$	−1.68365	+	2.91617i	−0.0572458	+	0.0991526i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	18.7326	−	32.4458i	0.635460	−	1.10065i
$870$		0			0
$871$	9.19669	+	0.869172i	0.311618	+	0.0294508i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−3.37024	−	5.83742i	−0.113935	−	0.197341i
$876$		0			0
$877$	39.7236			1.34137			0.670685	−	0.741743i	$-0.266000\pi$
$877$	39.7236			1.34137			0.670685	+	0.741743i	$0.266000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	21.1534	+	36.6387i	0.712675	+	1.23439i	0.963849	+	0.266448i	$0.0858501\pi$
$881$	21.1534	+	36.6387i	0.712675	+	1.23439i	−0.251174	+	0.967942i	$0.580817\pi$
$882$		0			0
$883$	−32.9536			−1.10898			−0.554488	−	0.832192i	$-0.687086\pi$
$883$	−32.9536			−1.10898			−0.554488	+	0.832192i	$0.687086\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	5.45287	−	9.44465i	0.183090	−	0.317120i	−0.759842	−	0.650108i	$-0.774724\pi$
$887$	5.45287	−	9.44465i	0.183090	−	0.317120i	0.942931	+	0.332988i	$0.108057\pi$
$888$		0			0
$889$	6.96127	−	12.0573i	0.233474	−	0.404388i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−0.113996	−	0.197448i	−0.00381475	−	0.00660733i
$894$		0			0
$895$	1.91137	−	3.31059i	0.0638901	−	0.110661i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−9.24074	−	16.0054i	−0.308196	−	0.533811i
$900$		0			0
$901$	−3.32443	+	5.75808i	−0.110753	+	0.191830i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	2.02016	+	3.49902i	0.0671524	+	0.116311i
$906$		0			0
$907$	18.7810			0.623612			0.311806	−	0.950146i	$-0.399066\pi$
$907$	18.7810			0.623612			0.311806	+	0.950146i	$0.399066\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	3.88256	−	6.72478i	0.128635	−	0.222802i	−0.794513	−	0.607247i	$-0.792274\pi$
$911$	3.88256	−	6.72478i	0.128635	−	0.222802i	0.923148	+	0.384445i	$0.125607\pi$
$912$		0			0
$913$	18.1376	−	31.4153i	0.600267	−	1.03969i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	14.3334	−	24.8261i	0.473329	−	0.819830i
$918$		0			0
$919$	−20.9504	+	36.2872i	−0.691090	+	1.19700i	0.280391	+	0.959886i	$0.409536\pi$
$919$	−20.9504	+	36.2872i	−0.691090	+	1.19700i	−0.971481	+	0.237118i	$0.923797\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−52.3938	−	4.95170i	−1.72456	−	0.162987i
$924$		0			0
$925$	−17.4815			−0.574788
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−20.8258	−	36.0714i	−0.683273	−	1.18346i	−0.973976	−	0.226651i	$-0.927222\pi$
$929$	−20.8258	−	36.0714i	−0.683273	−	1.18346i	0.290703	−	0.956813i	$-0.406111\pi$
$930$		0			0
$931$	−0.0675812	−	0.117054i	−0.00221488	−	0.00383629i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−2.34017			−0.0765316
$936$		0			0
$937$	46.9101			1.53249			0.766244	−	0.642550i	$-0.222124\pi$
$937$	46.9101			1.53249			0.766244	+	0.642550i	$0.222124\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1.71346	−	2.96780i	−0.0558571	−	0.0967474i	0.836745	−	0.547593i	$-0.184456\pi$
$941$	−1.71346	−	2.96780i	−0.0558571	−	0.0967474i	−0.892602	+	0.450846i	$0.851122\pi$
$942$		0			0
$943$	−0.102373	−	0.177316i	−0.00333373	−	0.00577420i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−21.0542			−0.684170			−0.342085	−	0.939669i	$-0.611133\pi$
$947$	−21.0542			−0.684170			−0.342085	+	0.939669i	$0.611133\pi$
$948$		0			0
$949$	0.244492	+	0.533991i	0.00793655	+	0.0173341i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−5.77633	+	10.0049i	−0.187114	+	0.324090i	−0.944287	−	0.329124i	$-0.893247\pi$
$953$	−5.77633	+	10.0049i	−0.187114	+	0.324090i	0.757173	+	0.653214i	$0.226580\pi$
$954$		0			0
$955$	−3.92562	+	6.79937i	−0.127030	+	0.220022i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	3.25701	−	5.64131i	0.105174	−	0.182167i
$960$		0			0
$961$	0.493070	−	0.854023i	0.0159055	−	0.0275491i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1.81270			0.0583529
$966$		0			0
$967$	9.28441	+	16.0811i	0.298567	+	0.517133i	0.975808	−	0.218628i	$-0.0701582\pi$
$967$	9.28441	+	16.0811i	0.298567	+	0.517133i	−0.677242	+	0.735761i	$0.736825\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	9.28362	−	16.0797i	0.297926	−	0.516022i	−0.677736	−	0.735306i	$-0.737039\pi$
$971$	9.28362	−	16.0797i	0.297926	−	0.516022i	0.975661	+	0.219283i	$0.0703719\pi$
$972$		0			0
$973$	−9.63348	−	16.6857i	−0.308835	−	0.534918i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−23.9560	+	41.4930i	−0.766421	+	1.32748i	0.173071	+	0.984909i	$0.444631\pi$
$977$	−23.9560	+	41.4930i	−0.766421	+	1.32748i	−0.939492	+	0.342571i	$0.888702\pi$
$978$		0			0
$979$	20.1494	+	34.8998i	0.643978	+	1.11540i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	28.1105	−	48.6888i	0.896586	−	1.55293i	0.0647561	−	0.997901i	$-0.479373\pi$
$983$	28.1105	−	48.6888i	0.896586	−	1.55293i	0.831830	−	0.555031i	$-0.187294\pi$
$984$		0			0
$985$	4.68958	−	8.12260i	0.149423	−	0.258808i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	0.151156			0.00480647
$990$		0			0
$991$	0.830293	+	1.43811i	0.0263751	+	0.0456831i	0.878912	−	0.476985i	$-0.158270\pi$
$991$	0.830293	+	1.43811i	0.0263751	+	0.0456831i	−0.852537	+	0.522668i	$0.824937\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	6.58835			0.208865
$996$		0			0
$997$	27.4394	+	47.5265i	0.869016	+	1.50518i	0.863004	+	0.505197i	$0.168580\pi$
$997$	27.4394	+	47.5265i	0.869016	+	1.50518i	0.00601173	+	0.999982i	$0.498086\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1404.2.j.a.289.8		28
3.2	odd	2		468.2.j.a.133.5	✓	28
9.4	even	3		1404.2.k.a.1225.8		28
9.5	odd	6		468.2.k.a.445.14	yes	28
13.9	even	3		1404.2.k.a.1153.8		28
39.35	odd	6		468.2.k.a.61.14	yes	28
117.22	even	3	inner	1404.2.j.a.685.8		28
117.113	odd	6		468.2.j.a.373.5	yes	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
468.2.j.a.133.5	✓	28	3.2	odd	2
468.2.j.a.373.5	yes	28	117.113	odd	6
468.2.k.a.61.14	yes	28	39.35	odd	6
468.2.k.a.445.14	yes	28	9.5	odd	6
1404.2.j.a.289.8		28	1.1	even	1	trivial
1404.2.j.a.685.8		28	117.22	even	3	inner
1404.2.k.a.1153.8		28	13.9	even	3
1404.2.k.a.1225.8		28	9.4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1404 = 2^{2} \cdot 3^{3} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1404.j (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.216950 + 0.375768i) q^{5} +(-0.791636 - 1.37115i) q^{7} +3.28568 q^{11} +(-1.50098 - 3.27827i) q^{13} +(-0.820732 + 1.42155i) q^{17} +(0.0150406 - 0.0260511i) q^{19} +(-0.0282011 + 0.0488457i) q^{23} +(2.40587 - 4.16708i) q^{25} +3.37346 q^{29} +(-2.73925 - 4.74451i) q^{31} +(0.343491 - 0.594943i) q^{35} +(-1.81655 - 3.14636i) q^{37} +(-1.81506 + 3.14378i) q^{41} +(-1.33998 - 2.32092i) q^{43} +(3.78962 - 6.56382i) q^{47} +(2.24662 - 3.89127i) q^{49} +4.05057 q^{53} +(0.712828 + 1.23465i) q^{55} +0.362821 q^{59} +(1.72774 + 2.99253i) q^{61} +(0.906231 - 1.27524i) q^{65} +(-1.28103 + 2.21882i) q^{67} +(7.29808 - 12.6407i) q^{71} -0.162888 q^{73} +(-2.60107 - 4.50518i) q^{77} +(5.70128 - 9.87490i) q^{79} +(5.52019 - 9.56125i) q^{83} -0.712231 q^{85} +(6.13248 + 10.6218i) q^{89} +(-3.30678 + 4.65327i) q^{91} +0.0130522 q^{95} +(6.46904 + 11.2047i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 2 q^{7} - 8 q^{11} + q^{13} + 8 q^{17} - q^{19} + 4 q^{23} - 14 q^{25} - 26 q^{29} + 2 q^{31} - 3 q^{35} - q^{37} - 4 q^{41} + 2 q^{43} - 11 q^{47} - 12 q^{49} - 52 q^{53} - 16 q^{59} - 7 q^{61}+ \cdots - 25 q^{97}+O(q^{100}) \) 28 * q + 2 * q^7 - 8 * q^11 + q^13 + 8 * q^17 - q^19 + 4 * q^23 - 14 * q^25 - 26 * q^29 + 2 * q^31 - 3 * q^35 - q^37 - 4 * q^41 + 2 * q^43 - 11 * q^47 - 12 * q^49 - 52 * q^53 - 16 * q^59 - 7 * q^61 + 31 * q^65 - 7 * q^67 + 12 * q^71 + 14 * q^73 + 28 * q^77 + 5 * q^79 - 9 * q^83 - 36 * q^85 + 11 * q^89 - q^91 + 56 * q^95 - 25 * q^97

Introduction

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