LMFDB - Embedded newform 52.3.i.a.43.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [52,3,Mod(23,52)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(52, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 5]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("52.23");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 52 = 2^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	52.i (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$1.41689737467$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			43.1
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	52.43
Dual form			52.3.i.a.23.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.73205 - 1.00000i) q^{2} +(2.00000 + 3.46410i) q^{4} -9.19615i q^{5} -8.00000i q^{8} +(-4.50000 - 7.79423i) q^{9} +O(q^{10})$	\(q+(-1.73205 - 1.00000i) q^{2} +(2.00000 + 3.46410i) q^{4} -9.19615i q^{5} -8.00000i q^{8} +(-4.50000 - 7.79423i) q^{9} +(-9.19615 + 15.9282i) q^{10} +(12.8923 + 1.66987i) q^{13} +(-8.00000 + 13.8564i) q^{16} +(14.4282 + 24.9904i) q^{17} +18.0000i q^{18} +(31.8564 - 18.3923i) q^{20} -59.5692 q^{25} +(-20.6603 - 15.7846i) q^{26} +(6.82051 - 11.8135i) q^{29} +(27.7128 - 16.0000i) q^{32} -57.7128i q^{34} +(18.0000 - 31.1769i) q^{36} +(62.8923 + 36.3109i) q^{37} -73.5692 q^{40} +(-21.1410 - 12.2058i) q^{41} +(-71.6769 + 41.3827i) q^{45} +(24.5000 - 42.4352i) q^{49} +(103.177 + 59.5692i) q^{50} +(20.0000 + 48.0000i) q^{52} +3.49742 q^{53} +(-23.6269 + 13.6410i) q^{58} +(46.4615 + 80.4737i) q^{61} -64.0000 q^{64} +(15.3564 - 118.560i) q^{65} +(-57.7128 + 99.9615i) q^{68} +(-62.3538 + 36.0000i) q^{72} -47.2628i q^{73} +(-72.6218 - 125.785i) q^{74} +(127.426 + 73.5692i) q^{80} +(-40.5000 + 70.1481i) q^{81} +(24.4115 + 42.2820i) q^{82} +(229.815 - 132.684i) q^{85} +(-138.564 - 80.0000i) q^{89} +165.531 q^{90} +(-124.708 + 72.0000i) q^{97} +(-84.8705 + 49.0000i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{4} - 18 q^{9}+O(q^{10}) $ 4 * q + 8 * q^4 - 18 * q^9	$ 4 q + 8 q^{4} - 18 q^{9} - 16 q^{10} + 10 q^{13} - 32 q^{16} + 30 q^{17} + 72 q^{20} - 72 q^{25} - 48 q^{26} - 42 q^{29} + 72 q^{36} + 210 q^{37} - 128 q^{40} + 54 q^{41} - 162 q^{45} + 98 q^{49} + 288 q^{50} + 80 q^{52} - 180 q^{53} - 240 q^{58} - 22 q^{61} - 256 q^{64} + 6 q^{65} - 120 q^{68} - 48 q^{74} + 288 q^{80} - 162 q^{81} + 160 q^{82} + 462 q^{85} + 288 q^{90}+O(q^{100}) $ 4 * q + 8 * q^4 - 18 * q^9 - 16 * q^10 + 10 * q^13 - 32 * q^16 + 30 * q^17 + 72 * q^20 - 72 * q^25 - 48 * q^26 - 42 * q^29 + 72 * q^36 + 210 * q^37 - 128 * q^40 + 54 * q^41 - 162 * q^45 + 98 * q^49 + 288 * q^50 + 80 * q^52 - 180 * q^53 - 240 * q^58 - 22 * q^61 - 256 * q^64 + 6 * q^65 - 120 * q^68 - 48 * q^74 + 288 * q^80 - 162 * q^81 + 160 * q^82 + 462 * q^85 + 288 * q^90

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/52\mathbb{Z}\right)^\times$.

$n$	$27$	$41$
$\chi(n)$	$-1$	$e\left(\frac{1}{6}\right)$

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 $

$2$	−1.73205	−	1.00000i	−0.866025	−	0.500000i
$2$	−1.73205	−	1.00000i	−0.866025	−	0.500000i
$3$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$3$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$4$	2.00000	+	3.46410i	0.500000	+	0.866025i
$5$		−	9.19615i		−	1.83923i	−0.392820	−	0.919615i	$-0.628501\pi$
$5$		−	9.19615i		−	1.83923i	0.392820	−	0.919615i	$-0.371499\pi$
$6$		0			0
$7$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$7$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$8$		−	8.00000i		−	1.00000i
$9$	−4.50000	−	7.79423i	−0.500000	−	0.866025i
$10$	−9.19615	+	15.9282i	−0.919615	+	1.59282i
$11$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$11$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$12$		0			0
$13$	12.8923	+	1.66987i	0.991716	+	0.128452i
$13$	12.8923	+	1.66987i	0.991716	+	0.128452i
$14$		0			0
$15$		0			0
$16$	−8.00000	+	13.8564i	−0.500000	+	0.866025i
$17$	14.4282	+	24.9904i	0.848718	+	1.47002i	0.882353	+	0.470588i	$0.155958\pi$
$17$	14.4282	+	24.9904i	0.848718	+	1.47002i	−0.0336351	+	0.999434i	$0.510708\pi$
$18$			18.0000i			1.00000i
$19$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$19$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$20$	31.8564	−	18.3923i	1.59282	−	0.919615i
$21$		0			0
$22$		0			0
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$		0			0
$25$	−59.5692			−2.38277
$26$	−20.6603	−	15.7846i	−0.794625	−	0.607100i
$27$		0			0
$28$		0			0
$29$	6.82051	−	11.8135i	0.235190	−	0.407361i	−0.724138	−	0.689655i	$-0.757762\pi$
$29$	6.82051	−	11.8135i	0.235190	−	0.407361i	0.959328	+	0.282294i	$0.0910955\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$	27.7128	−	16.0000i	0.866025	−	0.500000i
$33$		0			0
$34$		−	57.7128i		−	1.69744i
$35$		0			0
$36$	18.0000	−	31.1769i	0.500000	−	0.866025i
$37$	62.8923	+	36.3109i	1.69979	+	0.981375i	0.945946	+	0.324324i	$0.105137\pi$
$37$	62.8923	+	36.3109i	1.69979	+	0.981375i	0.753846	+	0.657051i	$0.228196\pi$
$38$		0			0
$39$		0			0
$40$	−73.5692			−1.83923
$41$	−21.1410	−	12.2058i	−0.515635	−	0.297702i	0.219512	−	0.975610i	$-0.429553\pi$
$41$	−21.1410	−	12.2058i	−0.515635	−	0.297702i	−0.735147	+	0.677908i	$0.762887\pi$
$42$		0			0
$43$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$43$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$44$		0			0
$45$	−71.6769	+	41.3827i	−1.59282	+	0.919615i
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	24.5000	−	42.4352i	0.500000	−	0.866025i
$50$	103.177	+	59.5692i	2.06354	+	1.19138i
$51$		0			0
$52$	20.0000	+	48.0000i	0.384615	+	0.923077i
$53$	3.49742			0.0659891			0.0329946	−	0.999456i	$-0.489496\pi$
$53$	3.49742			0.0659891			0.0329946	+	0.999456i	$0.489496\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−23.6269	+	13.6410i	−0.407361	+	0.235190i
$59$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$59$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$60$		0			0
$61$	46.4615	+	80.4737i	0.761664	+	1.31924i	0.941992	+	0.335635i	$0.108951\pi$
$61$	46.4615	+	80.4737i	0.761664	+	1.31924i	−0.180328	+	0.983607i	$0.557716\pi$
$62$		0			0
$63$		0			0
$64$	−64.0000			−1.00000
$65$	15.3564	−	118.560i	0.236252	−	1.82399i
$66$		0			0
$67$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$67$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$68$	−57.7128	+	99.9615i	−0.848718	+	1.47002i
$69$		0			0
$70$		0			0
$71$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$71$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$72$	−62.3538	+	36.0000i	−0.866025	+	0.500000i
$73$		−	47.2628i		−	0.647436i	−0.946154	−	0.323718i	$-0.895067\pi$
$73$		−	47.2628i		−	0.647436i	0.946154	−	0.323718i	$-0.104933\pi$
$74$	−72.6218	−	125.785i	−0.981375	−	1.69979i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$	127.426	+	73.5692i	1.59282	+	0.919615i
$81$	−40.5000	+	70.1481i	−0.500000	+	0.866025i
$82$	24.4115	+	42.2820i	0.297702	+	0.515635i
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$	229.815	−	132.684i	2.70371	−	1.56099i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−138.564	−	80.0000i	−1.55690	−	0.898876i	−0.997551	−	0.0699439i	$-0.977718\pi$
$89$	−138.564	−	80.0000i	−1.55690	−	0.898876i	−0.559349	−	0.828932i	$-0.688949\pi$
$90$	165.531			1.83923
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−124.708	+	72.0000i	−1.28565	+	0.742268i	−0.977875	−	0.209192i	$-0.932917\pi$
$97$	−124.708	+	72.0000i	−1.28565	+	0.742268i	−0.307771	+	0.951460i	$0.599583\pi$
$98$	−84.8705	+	49.0000i	−0.866025	+	0.500000i
$99$		0			0
$100$	−119.138	−	206.354i	−1.19138	−	2.06354i
$101$	−32.1795	+	55.7365i	−0.318609	+	0.551847i	−0.980198	−	0.198020i	$-0.936549\pi$
$101$	−32.1795	+	55.7365i	−0.318609	+	0.551847i	0.661589	+	0.749866i	$0.269882\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$	13.3590	−	103.138i	0.128452	−	0.991716i
$105$		0			0
$106$	−6.05771	−	3.49742i	−0.0571482	−	0.0329946i
$107$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$107$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$108$		0			0
$109$		−	120.000i		−	1.10092i	−0.834862	−	0.550459i	$-0.814453\pi$
$109$		−	120.000i		−	1.10092i	0.834862	−	0.550459i	$-0.185547\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	104.495	+	180.990i	0.924733	+	1.60168i	0.791990	+	0.610534i	$0.209045\pi$
$113$	104.495	+	180.990i	0.924733	+	1.60168i	0.132743	+	0.991150i	$0.457621\pi$
$114$		0			0
$115$		0			0
$116$	54.5641			0.470380
$117$	−45.0000	−	108.000i	−0.384615	−	0.923077i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	60.5000	+	104.789i	0.500000	+	0.866025i
$122$		−	185.846i		−	1.52333i
$123$		0			0
$124$		0			0
$125$			317.904i			2.54323i
$126$		0			0
$127$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$127$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$128$	110.851	+	64.0000i	0.866025	+	0.500000i
$129$		0			0
$130$	−145.158	+	189.995i	−1.11660	+	1.46150i
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	199.923	−	115.426i	1.47002	−	0.848718i
$137$	−233.710	+	134.933i	−1.70591	+	0.984910i	−0.766423	+	0.642336i	$0.777965\pi$
$137$	−233.710	+	134.933i	−1.70591	+	0.984910i	−0.939491	+	0.342574i	$0.888701\pi$
$138$		0			0
$139$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$139$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	144.000			1.00000
$145$	−108.638	−	62.7224i	−0.749231	−	0.432568i
$146$	−47.2628	+	81.8616i	−0.323718	+	0.560696i
$147$		0			0
$148$			290.487i			1.96275i
$149$	−44.7436	+	25.8327i	−0.300292	+	0.173374i	−0.642574	−	0.766223i	$-0.722134\pi$
$149$	−44.7436	+	25.8327i	−0.300292	+	0.173374i	0.342282	+	0.939597i	$0.388800\pi$
$150$		0			0
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		0			0
$153$	129.854	−	224.913i	0.848718	−	1.47002i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	313.631			1.99765			0.998824	−	0.0484851i	$-0.0154393\pi$
$157$	313.631			1.99765			0.998824	+	0.0484851i	$0.0154393\pi$
$158$		0			0
$159$		0			0
$160$	−147.138	−	254.851i	−0.919615	−	1.59282i
$161$		0			0
$162$	140.296	−	81.0000i	0.866025	−	0.500000i
$163$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$163$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$164$		−	97.6462i		−	0.595403i
$165$		0			0
$166$		0			0
$167$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$167$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$168$		0			0
$169$	163.423	+	43.0570i	0.967000	+	0.254775i
$170$	−530.736			−3.12198
$171$		0			0
$172$		0			0
$173$	−165.000	−	285.788i	−0.953757	−	1.65196i	−0.737187	−	0.675689i	$-0.763846\pi$
$173$	−165.000	−	285.788i	−0.953757	−	1.65196i	−0.216570	−	0.976267i	$-0.569487\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	160.000	+	277.128i	0.898876	+	1.55690i
$179$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$179$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$180$	−286.708	−	165.531i	−1.59282	−	0.919615i
$181$	−330.769			−1.82745			−0.913727	−	0.406329i	$-0.866809\pi$
$181$	−330.769			−1.82745			−0.913727	+	0.406329i	$0.866809\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	333.920	−	578.367i	1.80498	−	3.12631i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$		0			0
$193$	2.99227	+	1.72759i	0.0155040	+	0.00895123i	0.507732	−	0.861515i	$-0.330484\pi$
$193$	2.99227	+	1.72759i	0.0155040	+	0.00895123i	−0.492228	+	0.870466i	$0.663817\pi$
$194$	288.000			1.48454
$195$		0			0
$196$	196.000			1.00000
$197$	−48.4974	−	28.0000i	−0.246180	−	0.142132i	0.371834	−	0.928299i	$-0.378729\pi$
$197$	−48.4974	−	28.0000i	−0.246180	−	0.142132i	−0.618014	+	0.786167i	$0.712062\pi$
$198$		0			0
$199$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$199$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$200$			476.554i			2.38277i
$201$		0			0
$202$	111.473	−	64.3590i	0.551847	−	0.318609i
$203$		0			0
$204$		0			0
$205$	−112.246	+	194.416i	−0.547542	+	0.948371i
$206$		0			0
$207$		0			0
$208$	−126.277	+	165.282i	−0.607100	+	0.794625i
$209$		0			0
$210$		0			0
$211$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$211$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$212$	6.99485	+	12.1154i	0.0329946	+	0.0571482i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−120.000	+	207.846i	−0.550459	+	0.953422i
$219$		0			0
$220$		0			0
$221$	144.282	+	346.277i	0.652860	+	1.56686i
$222$		0			0
$223$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$223$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$224$		0			0
$225$	268.061	+	464.296i	1.19138	+	2.06354i
$226$		−	417.979i		−	1.84947i
$227$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$227$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$228$		0			0
$229$		−	120.000i		−	0.524017i	−0.965066	−	0.262009i	$-0.915615\pi$
$229$		−	120.000i		−	0.524017i	0.965066	−	0.262009i	$-0.0843849\pi$
$230$		0			0
$231$		0			0
$232$	−94.5077	−	54.5641i	−0.407361	−	0.235190i
$233$	210.000			0.901288			0.450644	−	0.892704i	$-0.351194\pi$
$233$	210.000			0.901288			0.450644	+	0.892704i	$0.351194\pi$
$234$	−30.0577	+	232.061i	−0.128452	+	0.991716i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−209.577	+	120.999i	−0.869614	+	0.502072i	−0.867220	−	0.497925i	$-0.834095\pi$
$241$	−209.577	+	120.999i	−0.869614	+	0.502072i	−0.00239399	+	0.999997i	$0.500762\pi$
$242$		−	242.000i		−	1.00000i
$243$		0			0
$244$	−185.846	+	321.895i	−0.761664	+	1.31924i
$245$	−390.241	−	225.306i	−1.59282	−	0.919615i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	317.904	−	550.626i	1.27162	−	2.20250i
$251$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$251$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−128.000	−	221.703i	−0.500000	−	0.866025i
$257$	−155.213	+	268.836i	−0.603941	+	1.04606i	0.388277	+	0.921543i	$0.373070\pi$
$257$	−155.213	+	268.836i	−0.603941	+	1.04606i	−0.992218	+	0.124514i	$0.960263\pi$
$258$		0			0
$259$		0			0
$260$	441.415	−	183.923i	1.69775	−	0.707396i
$261$	−122.769			−0.470380
$262$		0			0
$263$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$263$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$264$		0			0
$265$		−	32.1628i		−	0.121369i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	69.0000	+	119.512i	0.256506	+	0.444281i	0.965303	−	0.261131i	$-0.0840955\pi$
$269$	69.0000	+	119.512i	0.256506	+	0.444281i	−0.708798	+	0.705412i	$0.750762\pi$
$270$		0			0
$271$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$271$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$272$	−461.703			−1.69744
$273$		0			0
$274$	539.731			1.96982
$275$		0			0
$276$		0			0
$277$	−275.738	−	477.593i	−0.995445	−	1.72416i	−0.580283	−	0.814415i	$-0.697058\pi$
$277$	−275.738	−	477.593i	−0.995445	−	1.72416i	−0.415162	−	0.909747i	$-0.636275\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			240.104i			0.854462i	0.904143	+	0.427231i	$0.140511\pi$
$281$			240.104i			0.854462i	−0.904143	+	0.427231i	$0.859489\pi$
$282$		0			0
$283$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$283$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−249.415	−	144.000i	−0.866025	−	0.500000i
$289$	−271.846	+	470.851i	−0.940644	+	1.62924i
$290$	125.445	+	217.277i	0.432568	+	0.749231i
$291$		0			0
$292$	163.723	−	94.5256i	0.560696	−	0.323718i
$293$	486.390	−	280.817i	1.66003	−	0.958421i	0.687337	−	0.726339i	$-0.258780\pi$
$293$	486.390	−	280.817i	1.66003	−	0.958421i	0.972696	−	0.232082i	$-0.0745537\pi$
$294$		0			0
$295$		0			0
$296$	290.487	−	503.138i	0.981375	−	1.69979i
$297$		0			0
$298$	103.331			0.346748
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	740.049	−	427.267i	2.42639	−	1.40088i
$306$	−449.827	+	259.708i	−1.47002	+	0.848718i
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$	−50.0000			−0.159744			−0.0798722	−	0.996805i	$-0.525451\pi$
$313$	−50.0000			−0.159744			−0.0798722	+	0.996805i	$0.525451\pi$
$314$	−543.224	−	313.631i	−1.73001	−	0.998824i
$315$		0			0
$316$		0			0
$317$			437.904i			1.38140i	0.723141	+	0.690700i	$0.242697\pi$
$317$			437.904i			1.38140i	−0.723141	+	0.690700i	$0.757303\pi$
$318$		0			0
$319$		0			0
$320$			588.554i			1.83923i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−324.000			−1.00000
$325$	−767.985	−	99.4730i	−2.36303	−	0.306071i
$326$		0			0
$327$		0			0
$328$	−97.6462	+	169.128i	−0.297702	+	0.515635i
$329$		0			0
$330$		0			0
$331$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$331$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$332$		0			0
$333$		−	653.596i		−	1.96275i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	323.831			0.960922			0.480461	−	0.877016i	$-0.340469\pi$
$337$	323.831			0.960922			0.480461	+	0.877016i	$0.340469\pi$
$338$	−240.000	−	238.000i	−0.710059	−	0.704142i
$339$		0			0
$340$	919.261	+	530.736i	2.70371	+	1.56099i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$			660.000i			1.90751i
$347$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$347$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$348$		0			0
$349$	311.769	+	180.000i	0.893321	+	0.515759i	0.875027	−	0.484073i	$-0.160843\pi$
$349$	311.769	+	180.000i	0.893321	+	0.515759i	0.0182939	+	0.999833i	$0.494177\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−101.941	−	58.8557i	−0.288785	−	0.166730i	0.348609	−	0.937268i	$-0.386654\pi$
$353$	−101.941	−	58.8557i	−0.288785	−	0.166730i	−0.637394	+	0.770538i	$0.719988\pi$
$354$		0			0
$355$		0			0
$356$		−	640.000i		−	1.79775i
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$	331.061	+	573.415i	0.919615	+	1.59282i
$361$	180.500	−	312.635i	0.500000	−	0.866025i
$362$	572.909	+	330.769i	1.58262	+	0.913727i
$363$		0			0
$364$		0			0
$365$	−434.636			−1.19078
$366$		0			0
$367$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$367$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$368$		0			0
$369$			219.704i			0.595403i
$370$	−1156.73	+	667.841i	−3.12631	+	1.80498i
$371$		0			0
$372$		0			0
$373$	−80.7384	−	139.843i	−0.216457	−	0.374914i	0.737265	−	0.675603i	$-0.236117\pi$
$373$	−80.7384	−	139.843i	−0.216457	−	0.374914i	−0.953722	+	0.300689i	$0.902783\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	107.659	−	140.913i	0.285568	−	0.373776i
$378$		0			0
$379$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$379$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$383$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$384$		0			0
$385$		0			0
$386$	−3.45517	−	5.98454i	−0.00895123	−	0.0155040i
$387$		0			0
$388$	−498.831	−	288.000i	−1.28565	−	0.742268i
$389$	777.897			1.99974			0.999868	−	0.0162499i	$-0.00517272\pi$
$389$	777.897			1.99974			0.999868	+	0.0162499i	$0.00517272\pi$
$390$		0			0
$391$		0			0
$392$	−339.482	−	196.000i	−0.866025	−	0.500000i
$393$		0			0
$394$	56.0000	+	96.9948i	0.142132	+	0.246180i
$395$		0			0
$396$		0			0
$397$	−394.908	+	228.000i	−0.994729	+	0.574307i	−0.906685	−	0.421809i	$-0.861395\pi$
$397$	−394.908	+	228.000i	−0.994729	+	0.574307i	−0.0880448	+	0.996117i	$0.528062\pi$
$398$		0			0
$399$		0			0
$400$	476.554	−	825.415i	1.19138	−	2.06354i
$401$	563.859	+	325.544i	1.40613	+	0.811831i	0.995012	−	0.0997506i	$-0.0318045\pi$
$401$	563.859	+	325.544i	1.40613	+	0.811831i	0.411120	+	0.911581i	$0.365138\pi$
$402$		0			0
$403$		0			0
$404$	−257.436			−0.637218
$405$	645.092	+	372.444i	1.59282	+	0.919615i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−482.577	+	278.616i	−1.17989	+	0.681213i	−0.955990	−	0.293399i	$-0.905214\pi$
$409$	−482.577	+	278.616i	−1.17989	+	0.681213i	−0.223905	+	0.974611i	$0.571880\pi$
$410$	388.832	−	224.492i	0.948371	−	0.547542i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	384.000	−	160.000i	0.923077	−	0.384615i
$417$		0			0
$418$		0			0
$419$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$419$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$420$		0			0
$421$		−	470.229i		−	1.11693i	−0.829527	−	0.558467i	$-0.811390\pi$
$421$		−	470.229i		−	1.11693i	0.829527	−	0.558467i	$-0.188610\pi$
$422$		0			0
$423$		0			0
$424$		−	27.9794i		−	0.0659891i
$425$	−859.477	−	1488.66i	−2.02230	−	3.50272i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$431$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$432$		0			0
$433$	−280.838	−	486.426i	−0.648587	−	1.12339i	−0.983460	−	0.181123i	$-0.942027\pi$
$433$	−280.838	−	486.426i	−0.648587	−	1.12339i	0.334873	−	0.942263i	$-0.391307\pi$
$434$		0			0
$435$		0			0
$436$	415.692	−	240.000i	0.953422	−	0.550459i
$437$		0			0
$438$		0			0
$439$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$439$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$440$		0			0
$441$	−441.000			−1.00000
$442$	96.3731	−	744.051i	0.218039	−	1.68337i
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$	−735.692	+	1274.26i	−1.65324	+	2.86350i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−484.974	+	280.000i	−1.08012	+	0.623608i	−0.930929	−	0.365200i	$-0.881001\pi$
$449$	−484.974	+	280.000i	−1.08012	+	0.623608i	−0.149192	+	0.988808i	$0.547667\pi$
$450$		−	1072.25i		−	2.38277i
$451$		0			0
$452$	−417.979	+	723.962i	−0.924733	+	1.60168i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	782.992	+	452.061i	1.71333	+	0.989192i	0.929978	+	0.367615i	$0.119826\pi$
$457$	782.992	+	452.061i	1.71333	+	0.989192i	0.783353	+	0.621577i	$0.213508\pi$
$458$	−120.000	+	207.846i	−0.262009	+	0.453812i
$459$		0			0
$460$		0			0
$461$	−62.4103	+	36.0326i	−0.135380	+	0.0781619i	−0.566161	−	0.824295i	$-0.691572\pi$
$461$	−62.4103	+	36.0326i	−0.135380	+	0.0781619i	0.430780	+	0.902457i	$0.358238\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$	109.128	+	189.015i	0.235190	+	0.407361i
$465$		0			0
$466$	−363.731	−	210.000i	−0.780538	−	0.450644i
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$	284.123	−	371.885i	0.607100	−	0.794625i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−15.7384	−	27.2597i	−0.0329946	−	0.0571482i
$478$		0			0
$479$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$479$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$480$		0			0
$481$	750.192	+	573.153i	1.55965	+	1.19159i
$482$	483.997			1.00414
$483$		0			0
$484$	−242.000	+	419.156i	−0.500000	+	0.866025i
$485$	662.123	+	1146.83i	1.36520	+	2.36460i
$486$		0			0
$487$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$487$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$488$	643.790	−	371.692i	1.31924	−	0.761664i
$489$		0			0
$490$	450.611	+	780.482i	0.919615	+	1.59282i
$491$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$491$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$492$		0			0
$493$	393.631			0.798440
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$	−1101.25	+	635.808i	−2.20250	+	1.27162i
$501$		0			0
$502$		0			0
$503$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$503$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$504$		0			0
$505$	512.561	+	295.928i	1.01497	+	0.585995i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−497.974	−	287.506i	−0.978339	−	0.564844i	−0.0765706	−	0.997064i	$-0.524397\pi$
$509$	−497.974	−	287.506i	−0.978339	−	0.564844i	−0.901768	+	0.432220i	$0.857730\pi$
$510$		0			0
$511$		0			0
$512$			512.000i			1.00000i
$513$		0			0
$514$	537.673	−	310.426i	1.04606	−	0.603941i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	−948.477	−	122.851i	−1.82399	−	0.236252i
$521$	−1041.10			−1.99828			−0.999139	−	0.0414992i	$-0.986787\pi$
$521$	−1041.10			−1.99828			−0.999139	+	0.0414992i	$0.986787\pi$
$522$	212.642	+	122.769i	0.407361	+	0.235190i
$523$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$523$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−264.500	−	458.127i	−0.500000	−	0.866025i
$530$	−32.1628	+	55.7077i	−0.0606846	+	0.105109i
$531$		0			0
$532$		0			0
$533$	−252.174	−	192.663i	−0.473123	−	0.361470i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		−	276.000i		−	0.513011i
$539$		0			0
$540$		0			0
$541$		−	1010.63i		−	1.86808i	−0.357173	−	0.934038i	$-0.616259\pi$
$541$		−	1010.63i		−	1.86808i	0.357173	−	0.934038i	$-0.383741\pi$
$542$		0			0
$543$		0			0
$544$	799.692	+	461.703i	1.47002	+	0.848718i
$545$	−1103.54			−2.02484
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$	−934.841	−	539.731i	−1.70591	−	0.984910i
$549$	418.154	−	724.263i	0.761664	−	1.31924i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$			1102.95i			1.99089i
$555$		0			0
$556$		0			0
$557$	708.226	+	408.894i	1.27150	+	0.734101i	0.975270	−	0.221016i	$-0.0709372\pi$
$557$	708.226	+	408.894i	1.27150	+	0.734101i	0.296230	+	0.955117i	$0.404271\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	240.104	−	415.872i	0.427231	−	0.739986i
$563$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$563$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$564$		0			0
$565$	1664.42	−	960.951i	2.94587	−	1.70080i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	231.000	−	400.104i	0.405975	−	0.703170i	−0.588459	−	0.808527i	$-0.700265\pi$
$569$	231.000	−	400.104i	0.405975	−	0.703170i	0.994434	+	0.105357i	$0.0335985\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	288.000	+	498.831i	0.500000	+	0.866025i
$577$		−	947.929i		−	1.64286i	−0.570311	−	0.821429i	$-0.693177\pi$
$577$		−	947.929i		−	1.64286i	0.570311	−	0.821429i	$-0.306823\pi$
$578$	941.703	−	543.692i	1.62924	−	0.940644i
$579$		0			0
$580$		−	501.779i		−	0.865137i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−378.102			−0.647436
$585$	−993.184	+	413.827i	−1.69775	+	0.707396i
$586$	−1123.27			−1.91684
$587$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$587$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−1006.28	+	580.974i	−1.69979	+	0.981375i
$593$			437.404i			0.737612i	0.929506	+	0.368806i	$0.120233\pi$
$593$			437.404i			0.737612i	−0.929506	+	0.368806i	$0.879767\pi$
$594$		0			0
$595$		0			0
$596$	−178.974	−	103.331i	−0.300292	−	0.173374i
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$	67.6539	−	117.180i	0.112569	−	0.194975i	−0.804236	−	0.594309i	$-0.797425\pi$
$601$	67.6539	−	117.180i	0.112569	−	0.194975i	0.916805	+	0.399334i	$0.130759\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	963.656	−	556.367i	1.59282	−	0.919615i
$606$		0			0
$607$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$607$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$608$		0			0
$609$		0			0
$610$	−1709.07			−2.80175
$611$		0			0
$612$	1038.83			1.69744
$613$	−477.508	−	275.689i	−0.778968	−	0.449738i	0.0570962	−	0.998369i	$-0.481816\pi$
$613$	−477.508	−	275.689i	−0.778968	−	0.449738i	−0.836065	+	0.548631i	$0.815149\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−684.043	+	394.933i	−1.10866	+	0.640085i	−0.938482	−	0.345328i	$-0.887768\pi$
$617$	−684.043	+	394.933i	−1.10866	+	0.640085i	−0.170178	+	0.985413i	$0.554434\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	1434.26			2.29482
$626$	86.6025	+	50.0000i	0.138343	+	0.0798722i
$627$		0			0
$628$	627.261	+	1086.45i	0.998824	+	1.73001i
$629$			2095.60i			3.33164i
$630$		0			0
$631$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$631$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$632$		0			0
$633$		0			0
$634$	437.904	−	758.472i	0.690700	−	1.19633i
$635$		0			0
$636$		0			0
$637$	386.723	−	506.176i	0.607100	−	0.794625i
$638$		0			0
$639$		0			0
$640$	588.554	−	1019.41i	0.919615	−	1.59282i
$641$	−477.705	−	827.409i	−0.745250	−	1.29081i	−0.950078	−	0.312012i	$-0.898997\pi$
$641$	−477.705	−	827.409i	−0.745250	−	1.29081i	0.204828	−	0.978798i	$-0.434336\pi$
$642$		0			0
$643$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$643$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$647$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$648$	561.184	+	324.000i	0.866025	+	0.500000i
$649$		0			0
$650$	1230.72	+	940.277i	1.89341	+	1.44658i
$651$		0			0
$652$		0			0
$653$	−315.000	+	545.596i	−0.482389	+	0.835522i	−0.999796	−	0.0202175i	$-0.993564\pi$
$653$	−315.000	+	545.596i	−0.482389	+	0.835522i	0.517407	+	0.855740i	$0.326897\pi$
$654$		0			0
$655$		0			0
$656$	338.256	−	195.292i	0.515635	−	0.297702i
$657$	−368.377	+	212.683i	−0.560696	+	0.323718i
$658$		0			0
$659$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$659$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$660$		0			0
$661$	−1143.31	−	660.089i	−1.72966	−	0.998622i	−0.891074	−	0.453858i	$-0.850047\pi$
$661$	−1143.31	−	660.089i	−1.72966	−	0.998622i	−0.838589	−	0.544764i	$-0.816619\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−653.596	+	1132.06i	−0.981375	+	1.69979i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−670.546	+	1161.42i	−0.996354	+	1.72574i	−0.424288	+	0.905527i	$0.639476\pi$
$673$	−670.546	+	1161.42i	−0.996354	+	1.72574i	−0.572065	+	0.820208i	$0.693858\pi$
$674$	−560.891	−	323.831i	−0.832183	−	0.480461i
$675$		0			0
$676$	177.692	+	652.228i	0.262858	+	0.964834i
$677$	−1350.00			−1.99409			−0.997046	−	0.0768095i	$-0.975527\pi$
$677$	−1350.00			−1.99409			−0.997046	+	0.0768095i	$0.975527\pi$
$678$		0			0
$679$		0			0
$680$	−1061.47	−	1838.52i	−1.56099	−	2.70371i
$681$		0			0
$682$		0			0
$683$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$683$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$684$		0			0
$685$	1240.86	+	2149.23i	1.81148	+	3.13757i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	45.0898	+	5.84025i	0.0654424	+	0.00847642i
$690$		0			0
$691$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$691$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$692$	660.000	−	1143.15i	0.953757	−	1.65196i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	704.429i		−	1.01066i
$698$	−360.000	−	623.538i	−0.515759	−	0.893321i
$699$		0			0
$700$		0			0
$701$	1302.00			1.85735			0.928673	−	0.370899i	$-0.120950\pi$
$701$	1302.00			1.85735			0.928673	+	0.370899i	$0.120950\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	117.711	+	203.882i	0.166730	+	0.288785i
$707$		0			0
$708$		0			0
$709$	−183.077	+	105.699i	−0.258218	+	0.149082i	−0.623522	−	0.781806i	$-0.714299\pi$
$709$	−183.077	+	105.699i	−0.258218	+	0.149082i	0.365303	+	0.930889i	$0.380965\pi$
$710$		0			0
$711$		0			0
$712$	−640.000	+	1108.51i	−0.898876	+	1.55690i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$719$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$720$		−	1324.25i		−	1.83923i
$721$		0			0
$722$	−625.270	+	361.000i	−0.866025	+	0.500000i
$723$		0			0
$724$	−661.538	−	1145.82i	−0.913727	−	1.58262i
$725$	−406.292	+	703.719i	−0.560403	+	0.970647i
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	729.000			1.00000
$730$	752.811	+	434.636i	1.03125	+	0.595392i
$731$		0			0
$732$		0			0
$733$			1147.74i			1.56581i	0.622143	+	0.782904i	$0.286262\pi$
$733$			1147.74i			1.56581i	−0.622143	+	0.782904i	$0.713738\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	219.704	−	380.538i	0.297702	−	0.515635i
$739$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$739$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$740$	2671.36			3.60995
$741$		0			0
$742$		0			0
$743$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$743$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$744$		0			0
$745$	237.561	+	411.469i	0.318874	+	0.552307i
$746$			322.954i			0.432914i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$751$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$752$		0			0
$753$		0			0
$754$	−327.384	+	136.410i	−0.434197	+	0.180915i
$755$		0			0
$756$		0			0
$757$	595.000	−	1030.57i	0.785997	−	1.36139i	−0.142404	−	0.989809i	$-0.545483\pi$
$757$	595.000	−	1030.57i	0.785997	−	1.36139i	0.928401	−	0.371579i	$-0.121183\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1316.36	−	760.000i	1.72977	−	0.998686i	0.839263	−	0.543725i	$-0.182987\pi$
$761$	1316.36	−	760.000i	1.72977	−	0.998686i	0.890512	−	0.454961i	$-0.150347\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−2068.34	−	1194.16i	−2.70371	−	1.56099i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−1039.23	−	600.000i	−1.35141	−	0.780234i	−0.362959	−	0.931805i	$-0.618233\pi$
$769$	−1039.23	−	600.000i	−1.35141	−	0.780234i	−0.988446	+	0.151571i	$0.951567\pi$
$770$		0			0
$771$		0			0
$772$			13.8207i			0.0179025i
$773$	−1295.57	+	748.000i	−1.67603	+	0.967658i	−0.711885	+	0.702296i	$0.752158\pi$
$773$	−1295.57	+	748.000i	−1.67603	+	0.967658i	−0.964149	+	0.265362i	$0.914508\pi$
$774$		0			0
$775$		0			0
$776$	576.000	+	997.661i	0.742268	+	1.28565i
$777$		0			0
$778$	−1347.36	−	777.897i	−1.73182	−	0.999868i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	392.000	+	678.964i	0.500000	+	0.866025i
$785$		−	2884.20i		−	3.67413i
$786$		0			0
$787$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$787$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$788$		−	224.000i		−	0.284264i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	464.615	+	1115.08i	0.585896	+	1.40615i
$794$	912.000			1.14861
$795$		0			0
$796$		0			0
$797$	−555.000	−	961.288i	−0.696361	−	1.20613i	−0.969720	−	0.244221i	$-0.921468\pi$
$797$	−555.000	−	961.288i	−0.696361	−	1.20613i	0.273358	−	0.961912i	$-0.411866\pi$
$798$		0			0
$799$		0			0
$800$	−1650.83	+	953.108i	−2.06354	+	1.19138i
$801$			1440.00i			1.79775i
$802$	−651.088	−	1127.72i	−0.811831	−	1.40613i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	445.892	+	257.436i	0.551847	+	0.318609i
$809$	621.987	−	1077.31i	0.768835	−	1.33166i	−0.169361	−	0.985554i	$-0.554170\pi$
$809$	621.987	−	1077.31i	0.768835	−	1.33166i	0.938195	−	0.346106i	$-0.112496\pi$
$810$	−744.888	−	1290.18i	−0.919615	−	1.59282i
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	1114.46			1.36243
$819$		0			0
$820$	−897.969			−1.09508
$821$	1212.44	+	700.000i	1.47678	+	0.852619i	0.999656	−	0.0262179i	$-0.00834637\pi$
$821$	1212.44	+	700.000i	1.47678	+	0.852619i	0.477123	+	0.878837i	$0.341680\pi$
$822$		0			0
$823$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$823$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$		0			0
$829$	782.154	−	1354.73i	0.943491	−	1.63417i	0.184745	−	0.982786i	$-0.440854\pi$
$829$	782.154	−	1354.73i	0.943491	−	1.63417i	0.758745	−	0.651387i	$-0.225813\pi$
$830$		0			0
$831$		0			0
$832$	−825.108	−	106.872i	−0.991716	−	0.128452i
$833$	1413.96			1.69744
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$839$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$840$		0			0
$841$	327.461	+	567.180i	0.389371	+	0.674411i
$842$	−470.229	+	814.461i	−0.558467	+	0.967294i
$843$		0			0
$844$		0			0
$845$	395.959	−	1502.86i	0.468590	−	1.77854i
$846$		0			0
$847$		0			0
$848$	−27.9794	+	48.4617i	−0.0329946	+	0.0571482i
$849$		0			0
$850$			3437.91i			4.04460i
$851$		0			0
$852$		0			0
$853$			1183.07i			1.38695i	0.720480	+	0.693476i	$0.243922\pi$
$853$			1183.07i			1.38695i	−0.720480	+	0.693476i	$0.756078\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1226.84			−1.43155			−0.715774	−	0.698333i	$-0.753926\pi$
$857$	−1226.84			−1.43155			−0.715774	+	0.698333i	$0.753926\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$	−2628.15	+	1517.37i	−3.03833	+	1.75418i
$866$			1123.35i			1.29717i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−960.000			−1.10092
$873$	1122.37	+	648.000i	1.28565	+	0.742268i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	906.123	−	523.150i	1.03321	−	0.596523i	0.115306	−	0.993330i	$-0.463215\pi$
$877$	906.123	−	523.150i	1.03321	−	0.596523i	0.917902	+	0.396807i	$0.129882\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	877.320	−	1519.56i	0.995823	−	1.72482i	0.418842	−	0.908059i	$-0.362436\pi$
$881$	877.320	−	1519.56i	0.995823	−	1.72482i	0.576981	−	0.816758i	$-0.304231\pi$
$882$	763.834	+	441.000i	0.866025	+	0.500000i
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$	−910.974	+	1192.36i	−1.03051	+	1.34883i
$885$		0			0
$886$		0			0
$887$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$887$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$888$		0			0
$889$		0			0
$890$	2548.51	−	1471.38i	2.86350	−	1.65324i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	1120.00			1.24722
$899$		0			0
$900$	−1072.25	+	1857.18i	−1.19138	+	2.06354i
$901$	50.4615	+	87.4019i	0.0560061	+	0.0970055i
$902$		0			0
$903$		0			0
$904$	1447.92	−	835.959i	1.60168	−	0.924733i
$905$			3041.80i			3.36111i
$906$		0			0
$907$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$907$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$908$		0			0
$909$	579.231			0.637218
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$	−904.122	−	1565.98i	−0.989192	−	1.71333i
$915$		0			0
$916$	415.692	−	240.000i	0.453812	−	0.262009i
$917$		0			0
$918$		0			0
$919$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$919$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$920$		0			0
$921$		0			0
$922$	144.131			0.156324
$923$		0			0
$924$		0			0
$925$	−3746.45	−	2163.01i	−4.05021	−	2.33839i
$926$		0			0
$927$		0			0
$928$		−	436.513i		−	0.470380i
$929$	−603.243	+	348.283i	−0.649347	+	0.374901i	−0.788206	−	0.615411i	$-0.788990\pi$
$929$	−603.243	+	348.283i	−0.649347	+	0.374901i	0.138859	+	0.990312i	$0.455657\pi$
$930$		0			0
$931$		0			0
$932$	420.000	+	727.461i	0.450644	+	0.780538i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	−864.000	+	360.000i	−0.923077	+	0.384615i
$937$	1794.63			1.91529			0.957647	−	0.287945i	$-0.0929721\pi$
$937$	1794.63			1.91529			0.957647	+	0.287945i	$0.0929721\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	1160.00i		−	1.23273i	−0.787460	−	0.616366i	$-0.788604\pi$
$941$		−	1160.00i		−	1.23273i	0.787460	−	0.616366i	$-0.211396\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$947$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$948$		0			0
$949$	78.9229	−	609.326i	0.0831642	−	0.642072i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	615.000	+	1065.21i	0.645331	+	1.11775i	0.984225	+	0.176921i	$0.0566137\pi$
$953$	615.000	+	1065.21i	0.645331	+	1.11775i	−0.338895	+	0.940824i	$0.610053\pi$
$954$			62.9536i			0.0659891i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−961.000			−1.00000
$962$	−726.218	−	1742.92i	−0.754904	−	1.81177i
$963$		0			0
$964$	−838.308	−	483.997i	−0.869614	−	0.502072i
$965$	15.8872	−	27.5174i	0.0164634	−	0.0285154i
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$	838.313	−	484.000i	0.866025	−	0.500000i
$969$		0			0
$970$		−	2648.49i		−	2.73040i
$971$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$971$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−1486.77			−1.52333
$977$	1202.73	+	694.394i	1.23104	+	0.710741i	0.967247	−	0.253838i	$-0.0816931\pi$
$977$	1202.73	+	694.394i	1.23104	+	0.710741i	0.263793	+	0.964579i	$0.415026\pi$
$978$		0			0
$979$		0			0
$980$		−	1802.45i		−	1.83923i
$981$	−935.307	+	540.000i	−0.953422	+	0.550459i
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$	−257.492	+	445.990i	−0.261413	+	0.452781i
$986$	−681.788	−	393.631i	−0.691469	−	0.399220i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$991$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	784.661	+	1359.07i	0.787023	+	1.36316i	0.927783	+	0.373119i	$0.121712\pi$
$997$	784.661	+	1359.07i	0.787023	+	1.36316i	−0.140761	+	0.990044i	$0.544955\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	52.3.i.a.43.1	yes	4
4.3	odd	2	CM	52.3.i.a.43.1	yes	4
13.10	even	6	inner	52.3.i.a.23.1	✓	4
52.23	odd	6	inner	52.3.i.a.23.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
52.3.i.a.23.1	✓	4	13.10	even	6	inner
52.3.i.a.23.1	✓	4	52.23	odd	6	inner
52.3.i.a.43.1	yes	4	1.1	even	1	trivial
52.3.i.a.43.1	yes	4	4.3	odd	2	CM

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:	\( N \)	\(=\)	\( 52 = 2^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	52.i (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.73205 - 1.00000i) q^{2} +(2.00000 + 3.46410i) q^{4} -9.19615i q^{5} -8.00000i q^{8} +(-4.50000 - 7.79423i) q^{9} +O(q^{10})\)	\(q+(-1.73205 - 1.00000i) q^{2} +(2.00000 + 3.46410i) q^{4} -9.19615i q^{5} -8.00000i q^{8} +(-4.50000 - 7.79423i) q^{9} +(-9.19615 + 15.9282i) q^{10} +(12.8923 + 1.66987i) q^{13} +(-8.00000 + 13.8564i) q^{16} +(14.4282 + 24.9904i) q^{17} +18.0000i q^{18} +(31.8564 - 18.3923i) q^{20} -59.5692 q^{25} +(-20.6603 - 15.7846i) q^{26} +(6.82051 - 11.8135i) q^{29} +(27.7128 - 16.0000i) q^{32} -57.7128i q^{34} +(18.0000 - 31.1769i) q^{36} +(62.8923 + 36.3109i) q^{37} -73.5692 q^{40} +(-21.1410 - 12.2058i) q^{41} +(-71.6769 + 41.3827i) q^{45} +(24.5000 - 42.4352i) q^{49} +(103.177 + 59.5692i) q^{50} +(20.0000 + 48.0000i) q^{52} +3.49742 q^{53} +(-23.6269 + 13.6410i) q^{58} +(46.4615 + 80.4737i) q^{61} -64.0000 q^{64} +(15.3564 - 118.560i) q^{65} +(-57.7128 + 99.9615i) q^{68} +(-62.3538 + 36.0000i) q^{72} -47.2628i q^{73} +(-72.6218 - 125.785i) q^{74} +(127.426 + 73.5692i) q^{80} +(-40.5000 + 70.1481i) q^{81} +(24.4115 + 42.2820i) q^{82} +(229.815 - 132.684i) q^{85} +(-138.564 - 80.0000i) q^{89} +165.531 q^{90} +(-124.708 + 72.0000i) q^{97} +(-84.8705 + 49.0000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{4} - 18 q^{9}+O(q^{10}) \) 4 * q + 8 * q^4 - 18 * q^9	\( 4 q + 8 q^{4} - 18 q^{9} - 16 q^{10} + 10 q^{13} - 32 q^{16} + 30 q^{17} + 72 q^{20} - 72 q^{25} - 48 q^{26} - 42 q^{29} + 72 q^{36} + 210 q^{37} - 128 q^{40} + 54 q^{41} - 162 q^{45} + 98 q^{49} + 288 q^{50} + 80 q^{52} - 180 q^{53} - 240 q^{58} - 22 q^{61} - 256 q^{64} + 6 q^{65} - 120 q^{68} - 48 q^{74} + 288 q^{80} - 162 q^{81} + 160 q^{82} + 462 q^{85} + 288 q^{90}+O(q^{100}) \) 4 * q + 8 * q^4 - 18 * q^9 - 16 * q^10 + 10 * q^13 - 32 * q^16 + 30 * q^17 + 72 * q^20 - 72 * q^25 - 48 * q^26 - 42 * q^29 + 72 * q^36 + 210 * q^37 - 128 * q^40 + 54 * q^41 - 162 * q^45 + 98 * q^49 + 288 * q^50 + 80 * q^52 - 180 * q^53 - 240 * q^58 - 22 * q^61 - 256 * q^64 + 6 * q^65 - 120 * q^68 - 48 * q^74 + 288 * q^80 - 162 * q^81 + 160 * q^82 + 462 * q^85 + 288 * q^90

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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