LMFDB - Embedded newform 363.3.h.f.269.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [363,3,Mod(245,363)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(363, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([5, 8])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("363.245"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 363 = 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	363.h (of order $10$, degree $4$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,-5,-8,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$9.89103359628$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{10})$
Coefficient field:	8.0.228765625.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 2x^{6} + 5x^{5} + x^{4} + 15x^{3} - 18x^{2} - 27x + 81 $ x^8 - x^7 - 2x^6 + 5x^5 + x^4 + 15x^3 - 18x^2 - 27*x + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{10}]$

Embedding invariants

Embedding label			269.2
Root			$-0.570223 + 1.63550i$ of defining polynomial
Character	$\chi$	$=$	363.269
Dual form			363.3.h.f.251.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.04781 - 2.81107i) q^{3} +(-3.23607 + 2.35114i) q^{4} +(-9.46289 - 3.07468i) q^{5} +(-6.80418 + 5.89093i) q^{9} +(10.0000 + 6.63325i) q^{12} +(1.27219 + 29.8225i) q^{15} +(4.94427 - 15.2169i) q^{16} +(37.8516 - 12.2987i) q^{20} -29.8496i q^{23} +(59.8673 + 43.4961i) q^{25} +(23.6893 + 12.9544i) q^{27} +(11.4336 + 35.1891i) q^{31} +(8.16839 - 35.0611i) q^{36} +(-20.2254 + 14.6946i) q^{37} +(82.5000 - 34.8246i) q^{45} +(46.7871 - 64.3969i) q^{47} +(-47.9564 + 2.04576i) q^{48} +(-15.1418 + 46.6018i) q^{49} +(-75.7031 + 24.5974i) q^{53} +(29.2419 + 40.2481i) q^{59} +(-74.2338 - 93.5165i) q^{60} +(19.7771 + 60.8676i) q^{64} -35.0000 q^{67} +(-83.9093 + 31.2768i) q^{69} +(47.3145 + 15.3734i) q^{71} +(59.5409 - 213.867i) q^{75} +(-93.5742 + 128.794i) q^{80} +(11.5939 - 80.1660i) q^{81} +149.248i q^{89} +(70.1807 + 96.5954i) q^{92} +(86.9386 - 69.0122i) q^{93} +(29.3566 + 90.3504i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 5 q^{3} - 8 q^{4} - 7 q^{9} + 80 q^{12} - 33 q^{15} - 32 q^{16} + 148 q^{25} + 10 q^{27} - 74 q^{31} - 28 q^{36} - 50 q^{37} + 660 q^{45} - 80 q^{48} + 98 q^{49} - 132 q^{60} - 128 q^{64} - 280 q^{67}+ \cdots - 190 q^{97}+O(q^{100}) $ 8 * q - 5 * q^3 - 8 * q^4 - 7 * q^9 + 80 * q^12 - 33 * q^15 - 32 * q^16 + 148 * q^25 + 10 * q^27 - 74 * q^31 - 28 * q^36 - 50 * q^37 + 660 * q^45 - 80 * q^48 + 98 * q^49 - 132 * q^60 - 128 * q^64 - 280 * q^67 - 99 * q^69 + 370 * q^75 + 113 * q^81 - 185 * q^93 - 190 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/363\mathbb{Z}\right)^\times$.

$n$	$122$	$244$
$\chi(n)$	$-1$	$e\left(\frac{2}{5}\right)$

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		−0.309017	−	0.951057i	$-0.600000\pi$
$2$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$3$	−1.04781	−	2.81107i	−0.349270	−	0.937022i
$3$	−1.04781	−	2.81107i	−0.349270	−	0.937022i
$4$	−3.23607	+	2.35114i	−0.809017	+	0.587785i
$5$	−9.46289	−	3.07468i	−1.89258	−	0.614936i	−0.977191	−	0.212362i	$-0.931884\pi$
$5$	−9.46289	−	3.07468i	−1.89258	−	0.614936i	−0.915388	−	0.402574i	$-0.868116\pi$
$6$		0			0
$7$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$7$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$8$		0			0
$9$	−6.80418	+	5.89093i	−0.756021	+	0.654548i
$10$		0			0
$11$		0			0
$11$		0			0
$12$	10.0000	+	6.63325i	0.833333	+	0.552771i
$13$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$13$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$14$		0			0
$15$	1.27219	+	29.8225i	0.0848129	+	1.98817i
$16$	4.94427	−	15.2169i	0.309017	−	0.951057i
$17$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$17$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$18$		0			0
$19$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$19$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$20$	37.8516	−	12.2987i	1.89258	−	0.614936i
$21$		0			0
$22$		0			0
$23$		−	29.8496i		−	1.29781i	−0.760870	−	0.648905i	$-0.775227\pi$
$23$		−	29.8496i		−	1.29781i	0.760870	−	0.648905i	$-0.224773\pi$
$24$		0			0
$25$	59.8673	+	43.4961i	2.39469	+	1.73984i
$26$		0			0
$27$	23.6893	+	12.9544i	0.877381	+	0.479794i
$28$		0			0
$29$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$29$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$30$		0			0
$31$	11.4336	+	35.1891i	0.368827	+	1.13513i	0.947550	+	0.319608i	$0.103551\pi$
$31$	11.4336	+	35.1891i	0.368827	+	1.13513i	−0.578723	+	0.815524i	$0.696449\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	8.16839	−	35.0611i	0.226900	−	0.973918i
$37$	−20.2254	+	14.6946i	−0.546633	+	0.397152i	−0.826543	−	0.562874i	$-0.809696\pi$
$37$	−20.2254	+	14.6946i	−0.546633	+	0.397152i	0.279909	+	0.960026i	$0.409696\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$41$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$		0			0
$45$	82.5000	−	34.8246i	1.83333	−	0.773879i
$46$		0			0
$47$	46.7871	−	64.3969i	0.995471	−	1.37015i	0.0674071	−	0.997726i	$-0.478527\pi$
$47$	46.7871	−	64.3969i	0.995471	−	1.37015i	0.928063	−	0.372422i	$-0.121473\pi$
$48$	−47.9564	+	2.04576i	−0.999091	+	0.0426201i
$49$	−15.1418	+	46.6018i	−0.309017	+	0.951057i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−75.7031	+	24.5974i	−1.42836	+	0.464103i	−0.918248	−	0.396005i	$-0.870396\pi$
$53$	−75.7031	+	24.5974i	−1.42836	+	0.464103i	−0.510113	+	0.860108i	$0.670396\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	29.2419	+	40.2481i	0.495626	+	0.682171i	0.981413	−	0.191906i	$-0.0614669\pi$
$59$	29.2419	+	40.2481i	0.495626	+	0.682171i	−0.485787	+	0.874077i	$0.661467\pi$
$60$	−74.2338	−	93.5165i	−1.23723	−	1.55861i
$61$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$61$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$62$		0			0
$63$		0			0
$64$	19.7771	+	60.8676i	0.309017	+	0.951057i
$65$		0			0
$66$		0			0
$67$	−35.0000			−0.522388			−0.261194	−	0.965286i	$-0.584116\pi$
$67$	−35.0000			−0.522388			−0.261194	+	0.965286i	$0.584116\pi$
$68$		0			0
$69$	−83.9093	+	31.2768i	−1.21608	+	0.453286i
$70$		0			0
$71$	47.3145	+	15.3734i	0.666401	+	0.216527i	0.622632	−	0.782515i	$-0.286063\pi$
$71$	47.3145	+	15.3734i	0.666401	+	0.216527i	0.0437690	+	0.999042i	$0.486063\pi$
$72$		0			0
$73$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$73$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$74$		0			0
$75$	59.5409	−	213.867i	0.793878	−	2.85155i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$79$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$80$	−93.5742	+	128.794i	−1.16968	+	1.60992i
$81$	11.5939	−	80.1660i	0.143134	−	0.989703i
$82$		0			0
$83$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$83$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$			149.248i			1.67695i	0.544944	+	0.838473i	$0.316551\pi$
$89$			149.248i			1.67695i	−0.544944	+	0.838473i	$0.683449\pi$
$90$		0			0
$91$		0			0
$92$	70.1807	+	96.5954i	0.762833	+	1.04995i
$93$	86.9386	−	69.0122i	0.934823	−	0.742067i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	29.3566	+	90.3504i	0.302646	+	0.931447i	0.980545	+	0.196293i	$0.0628903\pi$
$97$	29.3566	+	90.3504i	0.302646	+	0.931447i	−0.677900	+	0.735154i	$0.737110\pi$
$98$		0			0
$99$		0			0
$100$	−296.000			−2.96000
$101$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$101$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$102$		0			0
$103$	153.713	−	111.679i	1.49236	−	1.08426i	0.519059	−	0.854739i	$-0.326283\pi$
$103$	153.713	−	111.679i	1.49236	−	1.08426i	0.973303	−	0.229526i	$-0.0737174\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$107$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$108$	−107.118	+	13.7755i	−0.991832	+	0.127551i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$	62.5000	+	41.4578i	0.563063	+	0.373494i
$112$		0			0
$113$	40.9387	−	56.3473i	0.362290	−	0.498649i	−0.588495	−	0.808501i	$-0.700279\pi$
$113$	40.9387	−	56.3473i	0.362290	−	0.498649i	0.950785	+	0.309852i	$0.100279\pi$
$114$		0			0
$115$	−91.7780	+	282.464i	−0.798070	+	2.45621i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$		0			0
$122$		0			0
$123$		0			0
$124$	−119.735	−	86.9922i	−0.965601	−	0.701550i
$125$	−286.571	−	394.431i	−2.29257	−	3.15545i
$126$		0			0
$127$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$127$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−184.339	−	195.423i	−1.36547	−	1.44758i
$136$		0			0
$137$	−66.2402	−	21.5228i	−0.483505	−	0.157100i	0.0571141	−	0.998368i	$-0.481810\pi$
$137$	−66.2402	−	21.5228i	−0.483505	−	0.157100i	−0.540620	+	0.841267i	$0.681810\pi$
$138$		0			0
$139$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$139$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$140$		0			0
$141$	−230.048	−	64.0459i	−1.63155	−	0.454226i
$142$		0			0
$143$		0			0
$144$	56.0000	+	132.665i	0.388889	+	0.921285i
$145$		0			0
$146$		0			0
$147$	146.866	−	6.26515i	0.999091	−	0.0426201i
$148$	30.9017	−	95.1057i	0.208795	−	0.642606i
$149$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$149$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$150$		0			0
$151$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$151$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	368.145i		−	2.37513i
$156$		0			0
$157$	173.939	+	126.374i	1.10789	+	0.804929i	0.982330	−	0.187157i	$-0.0599274\pi$
$157$	173.939	+	126.374i	1.10789	+	0.804929i	0.125560	+	0.992086i	$0.459927\pi$
$158$		0			0
$159$	148.468	+	187.033i	0.933759	+	1.17631i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−21.6312	−	66.5740i	−0.132707	−	0.408429i	0.862520	−	0.506024i	$-0.168885\pi$
$163$	−21.6312	−	66.5740i	−0.132707	−	0.408429i	−0.995226	+	0.0975945i	$0.968885\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$167$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$168$		0			0
$169$	136.724	−	99.3357i	0.809017	−	0.587785i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$173$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	82.5000	−	124.373i	0.466102	−	0.702675i
$178$		0			0
$179$	−204.694	+	281.737i	−1.14354	+	1.57395i	−0.384205	+	0.923248i	$0.625524\pi$
$179$	−204.694	+	281.737i	−1.14354	+	1.57395i	−0.759335	+	0.650700i	$0.774476\pi$
$180$	−185.098	+	306.664i	−1.02832	+	1.70369i
$181$	81.2715	−	250.128i	0.449014	−	1.38192i	−0.429008	−	0.903301i	$-0.641137\pi$
$181$	81.2715	−	250.128i	0.449014	−	1.38192i	0.878022	−	0.478621i	$-0.158863\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	236.572	−	76.8670i	1.27877	−	0.415497i
$186$		0			0
$187$		0			0
$188$			318.396i			1.69360i
$189$		0			0
$190$		0			0
$191$	204.694	+	281.737i	1.07169	+	1.47506i	0.868349	+	0.495954i	$0.165181\pi$
$191$	204.694	+	281.737i	1.07169	+	1.47506i	0.203345	+	0.979107i	$0.434819\pi$
$192$	150.380	−	119.372i	0.783230	−	0.621732i
$193$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$193$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$194$		0			0
$195$		0			0
$196$	−60.5673	−	186.407i	−0.309017	−	0.951057i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	−2.00000			−0.0100503			−0.00502513	−	0.999987i	$-0.501600\pi$
$199$	−2.00000			−0.0100503			−0.00502513	+	0.999987i	$0.501600\pi$
$200$		0			0
$201$	36.6734	+	98.3873i	0.182455	+	0.489489i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	175.842	+	203.102i	0.849479	+	0.981171i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$211$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$212$	187.148	−	257.588i	0.882776	−	1.21504i
$213$	−6.36097	−	149.113i	−0.0298637	−	0.700059i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	360.013	+	261.564i	1.61441	+	1.17293i	0.846542	+	0.532322i	$0.178680\pi$
$223$	360.013	+	261.564i	1.61441	+	1.17293i	0.767864	+	0.640613i	$0.221320\pi$
$224$		0			0
$225$	−663.580	+	56.7184i	−2.94925	+	0.252082i
$226$		0			0
$227$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$227$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$228$		0			0
$229$	133.804	+	411.807i	0.584299	+	1.79829i	0.602069	+	0.798444i	$0.294343\pi$
$229$	133.804	+	411.807i	0.584299	+	1.79829i	−0.0177708	+	0.999842i	$0.505657\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$233$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$234$		0			0
$235$	−640.741	+	465.526i	−2.72656	+	1.98096i
$236$	−189.258	−	61.4936i	−0.801940	−	0.260566i
$237$		0			0
$238$		0			0
$239$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$239$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$240$	460.096	+	128.092i	1.91707	+	0.533715i
$241$		0			0			−	1.00000i	$-0.5\pi$
$241$		0			0				1.00000i	$0.5\pi$
$242$		0			0
$243$	−237.500	+	51.4077i	−0.977366	+	0.211554i
$244$		0			0
$245$	286.571	−	394.431i	1.16968	−	1.60992i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−425.830	+	138.361i	−1.69653	+	0.551237i	−0.988002	−	0.154441i	$-0.950642\pi$
$251$	−425.830	+	138.361i	−1.69653	+	0.551237i	−0.708533	+	0.705678i	$0.750642\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−207.108	−	150.473i	−0.809017	−	0.587785i
$257$	280.723	+	386.382i	1.09231	+	1.50343i	0.845206	+	0.534441i	$0.179478\pi$
$257$	280.723	+	386.382i	1.09231	+	1.50343i	0.247100	+	0.968990i	$0.420522\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$	792.000			2.98868
$266$		0			0
$267$	419.546	−	156.384i	1.57133	−	0.585707i
$268$	113.262	−	82.2899i	0.422621	−	0.307052i
$269$	378.516	+	122.987i	1.40712	+	0.457202i	0.911487	−	0.411329i	$-0.134935\pi$
$269$	378.516	+	122.987i	1.40712	+	0.457202i	0.495635	+	0.868531i	$0.334935\pi$
$270$		0			0
$271$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$271$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$	198.000	−	298.496i	0.717391	−	1.08151i
$277$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$277$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$278$		0			0
$279$	−285.093	−	172.078i	−1.02184	−	0.616768i
$280$		0			0
$281$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$281$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$282$		0			0
$283$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$283$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$284$	−189.258	+	61.4936i	−0.666401	+	0.216527i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−233.806	−	169.870i	−0.809017	−	0.587785i
$290$		0			0
$291$	223.221	−	177.193i	0.767081	−	0.608912i
$292$		0			0
$293$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$293$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$294$		0			0
$295$	−152.963	−	470.773i	−0.518520	−	1.59584i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	310.152	+	832.076i	1.03384	+	2.77359i
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$		0			0
$309$	−475.000	−	315.079i	−1.53722	−	1.01967i
$310$		0			0
$311$	−233.936	+	321.985i	−0.752204	+	1.03532i	0.245618	+	0.969367i	$0.421009\pi$
$311$	−233.936	+	321.985i	−0.752204	+	1.03532i	−0.997823	+	0.0659539i	$0.978991\pi$
$312$		0			0
$313$	81.8895	−	252.030i	0.261628	−	0.805208i	−0.730823	−	0.682567i	$-0.760864\pi$
$313$	81.8895	−	252.030i	0.261628	−	0.805208i	0.992451	−	0.122641i	$-0.0391363\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	596.162	−	193.705i	1.88064	−	0.611056i	0.894015	−	0.448037i	$-0.147877\pi$
$317$	596.162	−	193.705i	1.88064	−	0.611056i	0.986623	−	0.163020i	$-0.0521234\pi$
$318$		0			0
$319$		0			0
$320$		−	636.792i		−	1.98997i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	150.963	+	286.681i	0.465935	+	0.884819i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	563.000			1.70091			0.850453	−	0.526051i	$-0.176328\pi$
$331$	563.000			1.70091			0.850453	+	0.526051i	$0.176328\pi$
$332$		0			0
$333$	51.0525	−	219.132i	0.153311	−	0.658053i
$334$		0			0
$335$	331.201	+	107.614i	0.988660	+	0.321235i
$336$		0			0
$337$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$337$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$338$		0			0
$339$	−201.292	−	56.0401i	−0.593782	−	0.165310i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$	890.190	−	37.9745i	2.58026	−	0.110071i
$346$		0			0
$347$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$347$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$348$		0			0
$349$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$349$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			328.346i			0.930158i	0.885269	+	0.465079i	$0.153974\pi$
$353$			328.346i			0.930158i	−0.885269	+	0.465079i	$0.846026\pi$
$354$		0			0
$355$	−400.463	−	290.954i	−1.12807	−	0.819588i
$356$	−350.903	−	482.977i	−0.985684	−	1.35668i
$357$		0			0
$358$		0			0
$359$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$359$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$360$		0			0
$361$	−111.555	−	343.331i	−0.309017	−	0.951057i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−513.726	+	373.244i	−1.39980	+	1.01701i	−0.405089	+	0.914277i	$0.632759\pi$
$367$	−513.726	+	373.244i	−1.39980	+	1.01701i	−0.994709	+	0.102735i	$0.967241\pi$
$368$	−454.219	−	147.585i	−1.23429	−	0.401045i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−119.082	+	427.733i	−0.320112	+	1.14982i
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$	−808.500	+	1218.86i	−2.15600	+	3.25029i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−41.0993	+	126.491i	−0.108441	+	0.333748i	−0.990523	−	0.137349i	$-0.956142\pi$
$379$	−41.0993	+	126.491i	−0.108441	+	0.333748i	0.882081	+	0.471097i	$0.156142\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−123.018	+	39.9708i	−0.321195	+	0.104363i	−0.465177	−	0.885218i	$-0.654009\pi$
$383$	−123.018	+	39.9708i	−0.321195	+	0.104363i	0.143982	+	0.989580i	$0.454009\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−307.426	−	223.358i	−0.792336	−	0.575666i
$389$	−321.661	−	442.729i	−0.826893	−	1.13812i	−0.988493	−	0.151264i	$-0.951666\pi$
$389$	−321.661	−	442.729i	−0.826893	−	1.13812i	0.161600	−	0.986856i	$-0.448334\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−790.000			−1.98992			−0.994962	−	0.100251i	$-0.968036\pi$
$397$	−790.000			−1.98992			−0.994962	+	0.100251i	$0.968036\pi$
$398$		0			0
$399$		0			0
$400$	957.876	−	695.938i	2.39469	−	1.73984i
$401$	−757.031	−	245.974i	−1.88786	−	0.613403i	−0.981690	−	0.190487i	$-0.938993\pi$
$401$	−757.031	−	245.974i	−1.88786	−	0.613403i	−0.906169	−	0.422915i	$-0.861007\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−356.196	+	722.955i	−0.879497	+	1.78507i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$409$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$410$		0			0
$411$	8.90535	+	208.758i	0.0216675	+	0.507926i
$412$	−234.853	+	722.803i	−0.570031	+	1.75438i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$			795.990i			1.89974i	0.312649	+	0.949869i	$0.398784\pi$
$419$			795.990i			1.89974i	−0.312649	+	0.949869i	$0.601216\pi$
$420$		0			0
$421$	−600.291	−	436.137i	−1.42587	−	1.03595i	−0.990767	−	0.135572i	$-0.956713\pi$
$421$	−600.291	−	436.137i	−1.42587	−	1.03595i	−0.435101	−	0.900382i	$-0.643287\pi$
$422$		0			0
$423$	61.0098	+	713.788i	0.144231	+	1.68744i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$431$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$432$	314.253	−	296.428i	0.727437	−	0.686175i
$433$	20.2254	−	14.6946i	0.0467100	−	0.0339368i	−0.564185	−	0.825648i	$-0.690810\pi$
$433$	20.2254	−	14.6946i	0.0467100	−	0.0339368i	0.610895	+	0.791711i	$0.290810\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	−171.500	−	406.287i	−0.388889	−	0.921285i
$442$		0			0
$443$	−450.326	+	619.821i	−1.01654	+	1.39914i	−0.101934	+	0.994791i	$0.532503\pi$
$443$	−450.326	+	619.821i	−1.01654	+	1.39914i	−0.914603	+	0.404352i	$0.867497\pi$
$444$	−299.727	+	12.7860i	−0.675062	+	0.0287973i
$445$	458.890	−	1412.32i	1.03121	−	3.17375i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	615.088	−	199.854i	1.36991	−	0.445110i	0.470568	−	0.882364i	$-0.344049\pi$
$449$	615.088	−	199.854i	1.36991	−	0.445110i	0.899338	+	0.437254i	$0.144049\pi$
$450$		0			0
$451$		0			0
$452$			278.596i			0.616364i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$457$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$458$		0			0
$459$		0			0
$460$	−367.112	−	1129.86i	−0.798070	−	2.45621i
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	35.0000			0.0755940			0.0377970	−	0.999285i	$-0.487966\pi$
$463$	35.0000			0.0755940			0.0377970	+	0.999285i	$0.487966\pi$
$464$		0			0
$465$	−1034.88	+	385.747i	−2.22555	+	0.829563i
$466$		0			0
$467$	−123.018	−	39.9708i	−0.263421	−	0.0855907i	0.174329	−	0.984687i	$-0.444224\pi$
$467$	−123.018	−	39.9708i	−0.263421	−	0.0855907i	−0.437750	+	0.899097i	$0.644224\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	172.990	−	621.369i	0.367283	−	1.31925i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	370.196	−	613.328i	0.776093	−	1.28580i
$478$		0			0
$479$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$479$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	945.238i		−	1.94894i
$486$		0			0
$487$	707.890	+	514.312i	1.45357	+	1.05608i	0.984980	+	0.172668i	$0.0552387\pi$
$487$	707.890	+	514.312i	1.45357	+	1.05608i	0.468593	+	0.883414i	$0.344761\pi$
$488$		0			0
$489$	−164.478	+	130.564i	−0.336357	+	0.267001i
$490$		0			0
$491$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$491$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	592.000			1.19355
$497$		0			0
$498$		0			0
$499$	487.028	−	353.847i	0.976008	−	0.709112i	0.0191955	−	0.999816i	$-0.493890\pi$
$499$	487.028	−	353.847i	0.976008	−	0.709112i	0.956813	+	0.290704i	$0.0938895\pi$
$500$	1854.73	+	602.637i	3.70945	+	1.20527i
$501$		0			0
$502$		0			0
$503$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$503$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−422.500	−	280.255i	−0.833333	−	0.552771i
$508$		0			0
$509$	−87.7258	+	120.744i	−0.172349	+	0.237219i	−0.886450	−	0.462825i	$-0.846836\pi$
$509$	−87.7258	+	120.744i	−0.172349	+	0.237219i	0.714100	+	0.700043i	$0.246836\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1797.95	+	584.189i	−3.49116	+	1.13435i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	380.145	+	523.225i	0.729646	+	1.00427i	0.999148	+	0.0412712i	$0.0131408\pi$
$521$	380.145	+	523.225i	0.729646	+	1.00427i	−0.269502	+	0.963000i	$0.586859\pi$
$522$		0			0
$523$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$523$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−362.000			−0.684310
$530$		0			0
$531$	−436.066	−	101.593i	−0.821217	−	0.191324i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	1006.46	+	280.201i	1.87423	+	0.521789i
$538$		0			0
$539$		0			0
$540$	1056.00	+	198.997i	1.95556	+	0.368514i
$541$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$541$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$542$		0			0
$543$	−788.283	+	33.6272i	−1.45172	+	0.0619286i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$547$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$548$	264.961	−	86.0910i	0.483505	−	0.157100i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−463.961	−	584.478i	−0.835966	−	1.05311i
$556$		0			0
$557$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$557$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$563$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$564$	895.032	−	333.619i	1.58694	−	0.591523i
$565$	−560.649	+	407.335i	−0.992299	+	0.720947i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$569$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$570$		0			0
$571$		0			0			−	1.00000i	$-0.5\pi$
$571$		0			0				1.00000i	$0.5\pi$
$572$		0			0
$573$	577.500	−	870.614i	1.00785	−	1.51940i
$574$		0			0
$575$	1298.34	−	1787.02i	2.25799	−	3.10785i
$576$	−493.134	−	297.649i	−0.856135	−	0.516752i
$577$	−326.013	+	1003.36i	−0.565014	+	1.73893i	0.102897	+	0.994692i	$0.467189\pi$
$577$	−326.013	+	1003.36i	−0.565014	+	1.73893i	−0.667911	+	0.744241i	$0.732811\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−187.148	−	257.588i	−0.318822	−	0.438821i	0.619285	−	0.785166i	$-0.287423\pi$
$587$	−187.148	−	257.588i	−0.318822	−	0.438821i	−0.938107	+	0.346345i	$0.887423\pi$
$588$	−460.539	+	365.578i	−0.783230	+	0.621732i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	123.607	+	380.423i	0.208795	+	0.642606i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	2.09562	+	5.62213i	0.00351025	+	0.00941731i
$598$		0			0
$599$	−1135.55	−	368.962i	−1.89574	−	0.615963i	−0.973148	−	0.230182i	$-0.926068\pi$
$599$	−1135.55	−	368.962i	−1.89574	−	0.615963i	−0.922591	−	0.385781i	$-0.873932\pi$
$600$		0			0
$601$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$601$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$602$		0			0
$603$	238.146	−	206.183i	0.394936	−	0.341928i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$607$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$613$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			1114.39i			1.80614i	0.429498	+	0.903068i	$0.358691\pi$
$617$			1114.39i			1.80614i	−0.429498	+	0.903068i	$0.641309\pi$
$618$		0			0
$619$	1000.75	+	727.090i	1.61673	+	1.17462i	0.831984	+	0.554800i	$0.187205\pi$
$619$	1000.75	+	727.090i	1.61673	+	1.17462i	0.784743	+	0.619821i	$0.212795\pi$
$620$	865.562	+	1191.34i	1.39607	+	1.92152i
$621$	386.685	−	707.117i	0.622681	−	1.13867i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	927.360	+	2854.12i	1.48378	+	4.56659i
$626$		0			0
$627$		0			0
$628$	−860.000			−1.36943
$629$		0			0
$630$		0			0
$631$	−981.338	+	712.984i	−1.55521	+	1.12993i	−0.615407	+	0.788209i	$0.711008\pi$
$631$	−981.338	+	712.984i	−1.55521	+	1.12993i	−0.939803	+	0.341717i	$0.888992\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$	−920.192	−	256.183i	−1.44684	−	0.402804i
$637$		0			0
$638$		0			0
$639$	−412.500	+	174.123i	−0.645540	+	0.272493i
$640$		0			0
$641$	−614.081	+	845.210i	−0.958005	+	1.31858i	−0.0101258	+	0.999949i	$0.503223\pi$
$641$	−614.081	+	845.210i	−0.958005	+	1.31858i	−0.947879	+	0.318631i	$0.896777\pi$
$642$		0			0
$643$	122.062	−	375.667i	0.189832	−	0.584242i	−0.810166	−	0.586200i	$-0.800623\pi$
$643$	122.062	−	375.667i	0.189832	−	0.584242i	0.999998	+	0.00195828i	$0.000623341\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1107.16	−	359.738i	1.71122	−	0.556009i	0.720683	−	0.693265i	$-0.243828\pi$
$647$	1107.16	−	359.738i	1.71122	−	0.556009i	0.990536	+	0.137256i	$0.0438283\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	226.525	+	164.580i	0.347431	+	0.252423i
$653$	−99.4226	−	136.843i	−0.152255	−	0.209561i	0.726075	−	0.687615i	$-0.241342\pi$
$653$	−99.4226	−	136.843i	−0.152255	−	0.209561i	−0.878330	+	0.478054i	$0.841342\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	1153.00			1.74433			0.872163	−	0.489215i	$-0.162717\pi$
$661$	1153.00			1.74433			0.872163	+	0.489215i	$0.162717\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	358.050	−	1286.09i	0.535202	−	1.92241i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$673$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$674$		0			0
$675$	854.746	+	1805.94i	1.26629	+	2.67546i
$676$	−208.895	+	642.914i	−0.309017	+	0.951057i
$677$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$677$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	1114.39i		−	1.63160i	−0.578331	−	0.815802i	$-0.696296\pi$
$683$		−	1114.39i		−	1.63160i	0.578331	−	0.815802i	$-0.303704\pi$
$684$		0			0
$685$	560.649	+	407.335i	0.818465	+	0.594650i
$686$		0			0
$687$	1017.42	−	807.629i	1.48096	−	1.17559i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−337.756	−	1039.50i	−0.488792	−	1.50435i	−0.826412	−	0.563066i	$-0.809622\pi$
$691$	−337.756	−	1039.50i	−0.488792	−	1.50435i	0.337620	−	0.941283i	$-0.390378\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$701$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	1980.00	+	1313.38i	2.80851	+	1.86296i
$706$		0			0
$707$		0			0
$708$	25.4439	+	596.450i	0.0359377	+	0.842444i
$709$	326.631	−	1005.27i	0.460692	−	1.41787i	−0.403628	−	0.914923i	$-0.632251\pi$
$709$	326.631	−	1005.27i	0.460692	−	1.41787i	0.864320	−	0.502942i	$-0.167749\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1050.38	−	341.290i	1.47319	−	0.478667i
$714$		0			0
$715$		0			0
$716$		−	1392.98i		−	1.94551i
$717$		0			0
$718$		0			0
$719$	497.113	+	684.217i	0.691395	+	0.951624i	1.00000		0.000433404i	$0.000137957\pi$
$719$	497.113	+	684.217i	0.691395	+	0.951624i	−0.308605	+	0.951190i	$0.599862\pi$
$720$	−122.020	−	1427.58i	−0.169472	−	1.98275i
$721$		0			0
$722$		0			0
$723$		0			0
$724$	325.086	+	1000.51i	0.449014	+	1.38192i
$725$		0			0
$726$		0			0
$727$	−1355.00			−1.86382			−0.931912	−	0.362685i	$-0.881860\pi$
$727$	−1355.00			−1.86382			−0.931912	+	0.362685i	$0.881860\pi$
$728$		0			0
$729$	393.365	+	613.763i	0.539596	+	0.841924i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$733$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$734$		0			0
$735$	−1409.04	−	392.281i	−1.91707	−	0.533715i
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$739$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$740$	−584.839	+	804.962i	−0.790323	+	1.08779i
$741$		0			0
$742$		0			0
$743$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$743$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−787.174	−	571.915i	−1.04817	−	0.761538i	−0.0763041	−	0.997085i	$-0.524312\pi$
$751$	−787.174	−	571.915i	−1.04817	−	0.761538i	−0.971863	+	0.235547i	$0.924312\pi$
$752$	−748.594	−	1030.35i	−0.995471	−	1.37015i
$753$	835.130	+	1052.06i	1.10907	+	1.39716i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	21.6312	+	66.5740i	0.0285749	+	0.0879445i	0.964327	−	0.264714i	$-0.0852776\pi$
$757$	21.6312	+	66.5740i	0.0285749	+	0.0879445i	−0.935752	+	0.352659i	$0.885278\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$761$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$762$		0			0
$763$		0			0
$764$	−1324.80	−	430.455i	−1.73404	−	0.563423i
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−205.979	+	739.863i	−0.268202	+	0.963363i
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$	792.000	−	1193.98i	1.02724	−	1.54862i
$772$		0			0
$773$	421.084	−	579.572i	0.544740	−	0.749770i	−0.444547	−	0.895756i	$-0.646635\pi$
$773$	421.084	−	579.572i	0.544740	−	0.749770i	0.989287	+	0.145985i	$0.0466352\pi$
$774$		0			0
$775$	−846.089	+	2603.99i	−1.09173	+	3.35999i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	634.269	+	460.824i	0.809017	+	0.587785i
$785$	−1257.40	−	1730.67i	−1.60179	−	2.20467i
$786$		0			0
$787$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$787$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	−829.866	−	2226.36i	−1.04386	−	2.80046i
$796$	6.47214	−	4.70228i	0.00813082	−	0.00590739i
$797$	160.869	+	52.2696i	0.201843	+	0.0655829i	0.408194	−	0.912895i	$-0.366159\pi$
$797$	160.869	+	52.2696i	0.201843	+	0.0655829i	−0.206351	+	0.978478i	$0.566159\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−879.210	−	1015.51i	−1.09764	−	1.26780i
$802$		0			0
$803$		0			0
$804$	−350.000	−	232.164i	−0.435323	−	0.288761i
$805$		0			0
$806$		0			0
$807$	−50.8877	−	1192.90i	−0.0630579	−	1.47819i
$808$		0			0
$809$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$809$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$810$		0			0
$811$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$811$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			696.491i			0.854590i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$821$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$822$		0			0
$823$	233.308	+	718.048i	0.283485	+	0.872476i	0.986849	+	0.161646i	$0.0516803\pi$
$823$	233.308	+	718.048i	0.283485	+	0.872476i	−0.703364	+	0.710830i	$0.748320\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$827$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$828$	−1046.56	−	243.823i	−1.26396	−	0.294473i
$829$	660.967	−	480.221i	0.797306	−	0.579277i	−0.112816	−	0.993616i	$-0.535987\pi$
$829$	660.967	−	480.221i	0.797306	−	0.579277i	0.910123	+	0.414339i	$0.135987\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−185.000	+	981.721i	−0.221027	+	1.17290i
$838$		0			0
$839$	964.984	−	1328.19i	1.15016	−	1.58306i	0.407780	−	0.913080i	$-0.366303\pi$
$839$	964.984	−	1328.19i	1.15016	−	1.58306i	0.742380	−	0.669979i	$-0.233697\pi$
$840$		0			0
$841$	259.883	−	799.839i	0.309017	−	0.951057i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1599.23	+	519.621i	−1.89258	+	0.614936i
$846$		0			0
$847$		0			0
$848$			1273.58i			1.50187i
$849$		0			0
$850$		0			0
$851$	438.629	+	603.721i	0.515428	+	0.709426i
$852$	371.169	+	467.583i	0.435644	+	0.548806i
$853$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$853$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$	−757.000			−0.881257			−0.440629	−	0.897689i	$-0.645244\pi$
$859$	−757.000			−0.881257			−0.440629	+	0.897689i	$0.645244\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1589.77	+	516.546i	1.84214	+	0.598547i	0.998055	+	0.0623360i	$0.0198550\pi$
$863$	1589.77	+	516.546i	1.84214	+	0.598547i	0.844084	+	0.536211i	$0.180145\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−232.531	+	835.235i	−0.268202	+	0.963363i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−731.996	−	441.823i	−0.838483	−	0.506097i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$877$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	945.238i		−	1.07291i	−0.843927	−	0.536457i	$-0.819762\pi$
$881$		−	945.238i		−	1.07291i	0.843927	−	0.536457i	$-0.180238\pi$
$882$		0			0
$883$	1108.35	+	805.266i	1.25521	+	0.911966i	0.998512	−	0.0545251i	$-0.0173645\pi$
$883$	1108.35	+	805.266i	1.25521	+	0.911966i	0.256701	+	0.966491i	$0.417364\pi$
$884$		0			0
$885$	−1163.10	+	923.271i	−1.31423	+	1.04324i
$886$		0			0
$887$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$887$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$	−1780.00			−1.99552
$893$		0			0
$894$		0			0
$895$	2803.24	−	2036.68i	3.13212	−	2.27562i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	2014.04	−	1743.72i	2.23782	−	1.93746i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−1538.13	+	2117.05i	−1.69959	+	2.33928i
$906$		0			0
$907$	−540.780	+	1664.35i	−0.596229	+	1.83500i	−0.0477182	+	0.998861i	$0.515195\pi$
$907$	−540.780	+	1664.35i	−0.596229	+	1.83500i	−0.548511	+	0.836143i	$0.684805\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−378.516	+	122.987i	−0.415495	+	0.135002i	−0.509302	−	0.860588i	$-0.670096\pi$
$911$	−378.516	+	122.987i	−0.415495	+	0.135002i	0.0938071	+	0.995590i	$0.470096\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−1401.22	−	1018.04i	−1.52971	−	1.11140i
$917$		0			0
$918$		0			0
$919$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$919$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−1850.00			−2.00000
$926$		0			0
$927$	−387.999	+	1665.40i	−0.418553	+	1.79655i
$928$		0			0
$929$	1514.06	+	491.949i	1.62978	+	0.529547i	0.974220	−	0.225599i	$-0.0724340\pi$
$929$	1514.06	+	491.949i	1.62978	+	0.529547i	0.655557	+	0.755146i	$0.272434\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	1150.24	+	320.229i	1.23284	+	0.343225i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$937$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$938$		0			0
$939$	−794.278	+	33.8830i	−0.845876	+	0.0360841i
$940$	978.966	−	3012.95i	1.04145	−	3.20526i
$941$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$941$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$942$		0			0
$943$		0			0
$944$	757.031	−	245.974i	0.801940	−	0.260566i
$945$		0			0
$946$		0			0
$947$			1323.33i			1.39740i	0.715417	+	0.698698i	$0.246237\pi$
$947$			1323.33i			1.39740i	−0.715417	+	0.698698i	$0.753763\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	−1169.18	−	1472.89i	−1.22942	−	1.54878i
$952$		0			0
$953$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$953$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$954$		0			0
$955$	−1070.74	−	3295.41i	−1.12120	−	3.45069i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	−1790.06	+	667.238i	−1.86465	+	0.695039i
$961$	−330.079	+	239.816i	−0.343474	+	0.249549i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1140.44	+	1569.68i	−1.17450	+	1.61656i	−0.552671	+	0.833399i	$0.686391\pi$
$971$	−1140.44	+	1569.68i	−1.17450	+	1.61656i	−0.621825	+	0.783156i	$0.713609\pi$
$972$	647.699	−	724.755i	0.666357	−	0.745632i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1447.82	+	470.426i	−1.48191	+	0.481501i	−0.934682	−	0.355485i	$-0.884316\pi$
$977$	−1447.82	+	470.426i	−1.48191	+	0.481501i	−0.547225	+	0.836986i	$0.684316\pi$
$978$		0			0
$979$		0			0
$980$			1950.18i			1.98997i
$981$		0			0
$982$		0			0
$983$	1093.65	+	1505.28i	1.11256	+	1.53131i	0.817586	+	0.575806i	$0.195312\pi$
$983$	1093.65	+	1505.28i	1.11256	+	1.53131i	0.294976	+	0.955505i	$0.404688\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	1582.00			1.59637			0.798184	−	0.602414i	$-0.205794\pi$
$991$	1582.00			1.59637			0.798184	+	0.602414i	$0.205794\pi$
$992$		0			0
$993$	−589.918	−	1582.63i	−0.594076	−	1.59379i
$994$		0			0
$995$	18.9258	+	6.14936i	0.0190209	+	0.00618026i
$996$		0			0
$997$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$997$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$998$		0			0
$999$	−669.487	+	86.0966i	−0.670157	+	0.0861828i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	363.3.h.f.269.2		8
3.2	odd	2	inner	363.3.h.f.269.1		8
11.2	odd	10	inner	363.3.h.f.251.1		8
11.3	even	5		363.3.b.c.122.2	yes	2
11.4	even	5	inner	363.3.h.f.245.1		8
11.5	even	5	inner	363.3.h.f.323.2		8
11.6	odd	10	inner	363.3.h.f.323.2		8
11.7	odd	10	inner	363.3.h.f.245.1		8
11.8	odd	10		363.3.b.c.122.2	yes	2
11.9	even	5	inner	363.3.h.f.251.1		8
11.10	odd	2	CM	363.3.h.f.269.2		8
33.2	even	10	inner	363.3.h.f.251.2		8
33.5	odd	10	inner	363.3.h.f.323.1		8
33.8	even	10		363.3.b.c.122.1	✓	2
33.14	odd	10		363.3.b.c.122.1	✓	2
33.17	even	10	inner	363.3.h.f.323.1		8
33.20	odd	10	inner	363.3.h.f.251.2		8
33.26	odd	10	inner	363.3.h.f.245.2		8
33.29	even	10	inner	363.3.h.f.245.2		8
33.32	even	2	inner	363.3.h.f.269.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
363.3.b.c.122.1	✓	2	33.8	even	10
363.3.b.c.122.1	✓	2	33.14	odd	10
363.3.b.c.122.2	yes	2	11.3	even	5
363.3.b.c.122.2	yes	2	11.8	odd	10
363.3.h.f.245.1		8	11.4	even	5	inner
363.3.h.f.245.1		8	11.7	odd	10	inner
363.3.h.f.245.2		8	33.26	odd	10	inner
363.3.h.f.245.2		8	33.29	even	10	inner
363.3.h.f.251.1		8	11.2	odd	10	inner
363.3.h.f.251.1		8	11.9	even	5	inner
363.3.h.f.251.2		8	33.2	even	10	inner
363.3.h.f.251.2		8	33.20	odd	10	inner
363.3.h.f.269.1		8	3.2	odd	2	inner
363.3.h.f.269.1		8	33.32	even	2	inner
363.3.h.f.269.2		8	1.1	even	1	trivial
363.3.h.f.269.2		8	11.10	odd	2	CM
363.3.h.f.323.1		8	33.5	odd	10	inner
363.3.h.f.323.1		8	33.17	even	10	inner
363.3.h.f.323.2		8	11.5	even	5	inner
363.3.h.f.323.2		8	11.6	odd	10	inner

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:	\( N \)	\(=\)	\( 363 = 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	363.h (of order \(10\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.04781 - 2.81107i) q^{3} +(-3.23607 + 2.35114i) q^{4} +(-9.46289 - 3.07468i) q^{5} +(-6.80418 + 5.89093i) q^{9} +(10.0000 + 6.63325i) q^{12} +(1.27219 + 29.8225i) q^{15} +(4.94427 - 15.2169i) q^{16} +(37.8516 - 12.2987i) q^{20} -29.8496i q^{23} +(59.8673 + 43.4961i) q^{25} +(23.6893 + 12.9544i) q^{27} +(11.4336 + 35.1891i) q^{31} +(8.16839 - 35.0611i) q^{36} +(-20.2254 + 14.6946i) q^{37} +(82.5000 - 34.8246i) q^{45} +(46.7871 - 64.3969i) q^{47} +(-47.9564 + 2.04576i) q^{48} +(-15.1418 + 46.6018i) q^{49} +(-75.7031 + 24.5974i) q^{53} +(29.2419 + 40.2481i) q^{59} +(-74.2338 - 93.5165i) q^{60} +(19.7771 + 60.8676i) q^{64} -35.0000 q^{67} +(-83.9093 + 31.2768i) q^{69} +(47.3145 + 15.3734i) q^{71} +(59.5409 - 213.867i) q^{75} +(-93.5742 + 128.794i) q^{80} +(11.5939 - 80.1660i) q^{81} +149.248i q^{89} +(70.1807 + 96.5954i) q^{92} +(86.9386 - 69.0122i) q^{93} +(29.3566 + 90.3504i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 5 q^{3} - 8 q^{4} - 7 q^{9} + 80 q^{12} - 33 q^{15} - 32 q^{16} + 148 q^{25} + 10 q^{27} - 74 q^{31} - 28 q^{36} - 50 q^{37} + 660 q^{45} - 80 q^{48} + 98 q^{49} - 132 q^{60} - 128 q^{64} - 280 q^{67}+ \cdots - 190 q^{97}+O(q^{100}) \) 8 * q - 5 * q^3 - 8 * q^4 - 7 * q^9 + 80 * q^12 - 33 * q^15 - 32 * q^16 + 148 * q^25 + 10 * q^27 - 74 * q^31 - 28 * q^36 - 50 * q^37 + 660 * q^45 - 80 * q^48 + 98 * q^49 - 132 * q^60 - 128 * q^64 - 280 * q^67 - 99 * q^69 + 370 * q^75 + 113 * q^81 - 185 * q^93 - 190 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more