LMFDB - Embedded newform 400.6.n.g.143.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,6,Mod(143,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 0, 3]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("400.143");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	400.n (of order $4$, 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:	$64.1535279252$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 271 x^{18} + 109637 x^{16} + 25993614 x^{14} + 5522961902 x^{12} + 881545050522 x^{10} + \cdots + 57\!\cdots\!24 $ x^20 + 271x^18 + 109637x^16 + 25993614x^14 + 5522961902x^12 + 881545050522x^10 + 133816049059481x^8 + 14779507781220031x^6 + 824105698447750789x^4 + 12044868290803250652*x^2 + 579398322543528055824
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{67}\cdot 3^{4}\cdot 5^{4} $
Twist minimal:	no (minimal twist has level 80)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			143.9
Root			$-3.75557 - 3.81117i$ of defining polynomial
Character	$\chi$	$=$	400.143
Dual form			400.6.n.g.207.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(17.2921 - 17.2921i) q^{3} +(-154.079 - 154.079i) q^{7} -355.037i q^{9} +O(q^{10})$	$q+(17.2921 - 17.2921i) q^{3} +(-154.079 - 154.079i) q^{7} -355.037i q^{9} -127.489i q^{11} +(-335.067 - 335.067i) q^{13} +(1155.01 - 1155.01i) q^{17} +28.2166 q^{19} -5328.70 q^{21} +(2783.04 - 2783.04i) q^{23} +(-1937.35 - 1937.35i) q^{27} +3388.31i q^{29} +5384.41i q^{31} +(-2204.56 - 2204.56i) q^{33} +(-11534.0 + 11534.0i) q^{37} -11588.1 q^{39} -11147.8 q^{41} +(1437.07 - 1437.07i) q^{43} +(219.040 + 219.040i) q^{47} +30673.4i q^{49} -39945.1i q^{51} +(-22745.6 - 22745.6i) q^{53} +(487.926 - 487.926i) q^{57} -22196.6 q^{59} -1431.25 q^{61} +(-54703.5 + 54703.5i) q^{63} +(28948.2 + 28948.2i) q^{67} -96249.3i q^{69} +24188.3i q^{71} +(-28574.8 - 28574.8i) q^{73} +(-19643.4 + 19643.4i) q^{77} +23417.0 q^{79} +19271.9 q^{81} +(18919.5 - 18919.5i) q^{83} +(58591.1 + 58591.1i) q^{87} -8179.53i q^{89} +103253. i q^{91} +(93107.9 + 93107.9i) q^{93} +(76747.2 - 76747.2i) q^{97} -45263.4 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q+O(q^{10}) $ 20 * q	$ 20 q - 804 q^{13} + 2236 q^{17} - 4520 q^{21} + 11096 q^{33} - 44260 q^{37} - 6760 q^{41} - 182452 q^{53} + 34288 q^{57} - 41080 q^{61} - 264372 q^{73} - 399304 q^{77} - 520220 q^{81} - 713496 q^{93} - 374772 q^{97}+O(q^{100}) $ 20 * q - 804 * q^13 + 2236 * q^17 - 4520 * q^21 + 11096 * q^33 - 44260 * q^37 - 6760 * q^41 - 182452 * q^53 + 34288 * q^57 - 41080 * q^61 - 264372 * q^73 - 399304 * q^77 - 520220 * q^81 - 713496 * q^93 - 374772 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$351$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	17.2921	−	17.2921i	1.10929	−	1.10929i	0.116048	−	0.993244i	$-0.462977\pi$
$3$	17.2921	−	17.2921i	1.10929	−	1.10929i	0.993244	−	0.116048i	$-0.0370227\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−154.079	−	154.079i	−1.18849	−	1.18849i	−0.977484	−	0.211011i	$-0.932324\pi$
$7$	−154.079	−	154.079i	−1.18849	−	1.18849i	−0.211011	−	0.977484i	$-0.567676\pi$
$8$		0			0
$9$		−	355.037i		−	1.46106i
$10$		0			0
$11$		−	127.489i		−	0.317682i	−0.987304	−	0.158841i	$-0.949224\pi$
$11$		−	127.489i		−	0.317682i	0.987304	−	0.158841i	$-0.0507757\pi$
$12$		0			0
$13$	−335.067	−	335.067i	−0.549887	−	0.549887i	0.376521	−	0.926408i	$-0.377120\pi$
$13$	−335.067	−	335.067i	−0.549887	−	0.549887i	−0.926408	+	0.376521i	$0.877120\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1155.01	−	1155.01i	0.969310	−	0.969310i	−0.0302331	−	0.999543i	$-0.509625\pi$
$17$	1155.01	−	1155.01i	0.969310	−	0.969310i	0.999543	+	0.0302331i	$0.00962497\pi$
$18$		0			0
$19$	28.2166			0.0179317			0.00896584	−	0.999960i	$-0.497146\pi$
$19$	28.2166			0.0179317			0.00896584	+	0.999960i	$0.497146\pi$
$20$		0			0
$21$	−5328.70			−2.63677
$22$		0			0
$23$	2783.04	−	2783.04i	1.09698	−	1.09698i	0.102219	−	0.994762i	$-0.467406\pi$
$23$	2783.04	−	2783.04i	1.09698	−	1.09698i	0.994762	−	0.102219i	$-0.0325943\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−1937.35	−	1937.35i	−0.511446	−	0.511446i
$28$		0			0
$29$			3388.31i			0.748149i	0.927399	+	0.374074i	$0.122040\pi$
$29$			3388.31i			0.748149i	−0.927399	+	0.374074i	$0.877960\pi$
$30$		0			0
$31$			5384.41i			1.00631i	0.864195	+	0.503157i	$0.167828\pi$
$31$			5384.41i			1.00631i	−0.864195	+	0.503157i	$0.832172\pi$
$32$		0			0
$33$	−2204.56	−	2204.56i	−0.352402	−	0.352402i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−11534.0	+	11534.0i	−1.38508	+	1.38508i	−0.549757	+	0.835324i	$0.685280\pi$
$37$	−11534.0	+	11534.0i	−1.38508	+	1.38508i	−0.835324	+	0.549757i	$0.814720\pi$
$38$		0			0
$39$	−11588.1			−1.21997
$40$		0			0
$41$	−11147.8			−1.03569			−0.517844	−	0.855475i	$-0.673265\pi$
$41$	−11147.8			−1.03569			−0.517844	+	0.855475i	$0.673265\pi$
$42$		0			0
$43$	1437.07	−	1437.07i	0.118524	−	0.118524i	−0.645357	−	0.763881i	$-0.723291\pi$
$43$	1437.07	−	1437.07i	0.118524	−	0.118524i	0.763881	+	0.645357i	$0.223291\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	219.040	+	219.040i	0.0144637	+	0.0144637i	0.714302	−	0.699838i	$-0.246744\pi$
$47$	219.040	+	219.040i	0.0144637	+	0.0144637i	−0.699838	+	0.714302i	$0.746744\pi$
$48$		0			0
$49$			30673.4i			1.82504i
$50$		0			0
$51$		−	39945.1i		−	2.15049i
$52$		0			0
$53$	−22745.6	−	22745.6i	−1.11226	−	1.11226i	−0.992844	−	0.119421i	$-0.961896\pi$
$53$	−22745.6	−	22745.6i	−1.11226	−	1.11226i	−0.119421	−	0.992844i	$-0.538104\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	487.926	−	487.926i	0.0198915	−	0.0198915i
$58$		0			0
$59$	−22196.6			−0.830149			−0.415074	−	0.909787i	$-0.636244\pi$
$59$	−22196.6			−0.830149			−0.415074	+	0.909787i	$0.636244\pi$
$60$		0			0
$61$	−1431.25			−0.0492484			−0.0246242	−	0.999697i	$-0.507839\pi$
$61$	−1431.25			−0.0492484			−0.0246242	+	0.999697i	$0.507839\pi$
$62$		0			0
$63$	−54703.5	+	54703.5i	−1.73646	+	1.73646i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	28948.2	+	28948.2i	0.787835	+	0.787835i	0.981139	−	0.193304i	$-0.0619204\pi$
$67$	28948.2	+	28948.2i	0.787835	+	0.787835i	−0.193304	+	0.981139i	$0.561920\pi$
$68$		0			0
$69$		−	96249.3i		−	2.43374i
$70$		0			0
$71$			24188.3i			0.569456i	0.958608	+	0.284728i	$0.0919032\pi$
$71$			24188.3i			0.569456i	−0.958608	+	0.284728i	$0.908097\pi$
$72$		0			0
$73$	−28574.8	−	28574.8i	−0.627591	−	0.627591i	0.319871	−	0.947461i	$-0.396361\pi$
$73$	−28574.8	−	28574.8i	−0.627591	−	0.627591i	−0.947461	+	0.319871i	$0.896361\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−19643.4	+	19643.4i	−0.377563	+	0.377563i
$78$		0			0
$79$	23417.0			0.422147			0.211073	−	0.977470i	$-0.432304\pi$
$79$	23417.0			0.422147			0.211073	+	0.977470i	$0.432304\pi$
$80$		0			0
$81$	19271.9			0.326371
$82$		0			0
$83$	18919.5	−	18919.5i	0.301450	−	0.301450i	−0.540131	−	0.841581i	$-0.681625\pi$
$83$	18919.5	−	18919.5i	0.301450	−	0.301450i	0.841581	+	0.540131i	$0.181625\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	58591.1	+	58591.1i	0.829915	+	0.829915i
$88$		0			0
$89$		−	8179.53i		−	0.109459i	−0.998501	−	0.0547297i	$-0.982570\pi$
$89$		−	8179.53i		−	0.109459i	0.998501	−	0.0547297i	$-0.0174297\pi$
$90$		0			0
$91$			103253.i			1.30708i
$92$		0			0
$93$	93107.9	+	93107.9i	1.11630	+	1.11630i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	76747.2	−	76747.2i	0.828196	−	0.828196i	−0.159071	−	0.987267i	$-0.550850\pi$
$97$	76747.2	−	76747.2i	0.828196	−	0.828196i	0.987267	+	0.159071i	$0.0508499\pi$
$98$		0			0
$99$	−45263.4			−0.464151
$100$		0			0
$101$	−1379.60			−0.0134570			−0.00672851	−	0.999977i	$-0.502142\pi$
$101$	−1379.60			−0.0134570			−0.00672851	+	0.999977i	$0.502142\pi$
$102$		0			0
$103$	−127370.	+	127370.i	−1.18297	+	1.18297i	−0.203994	+	0.978972i	$0.565392\pi$
$103$	−127370.	+	127370.i	−1.18297	+	1.18297i	−0.978972	+	0.203994i	$0.934608\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	17327.3	+	17327.3i	0.146309	+	0.146309i	0.776467	−	0.630158i	$-0.217010\pi$
$107$	17327.3	+	17327.3i	0.146309	+	0.146309i	−0.630158	+	0.776467i	$0.717010\pi$
$108$		0			0
$109$		−	105777.i		−	0.852759i	−0.904544	−	0.426379i	$-0.859789\pi$
$109$		−	105777.i		−	0.852759i	0.904544	−	0.426379i	$-0.140211\pi$
$110$		0			0
$111$			398895.i			3.07292i
$112$		0			0
$113$	−63449.7	−	63449.7i	−0.467449	−	0.467449i	0.433638	−	0.901087i	$-0.357230\pi$
$113$	−63449.7	−	63449.7i	−0.467449	−	0.467449i	−0.901087	+	0.433638i	$0.857230\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−118961.	+	118961.i	−0.803416	+	0.803416i
$118$		0			0
$119$	−355924.			−2.30404
$120$		0			0
$121$	144797.			0.899078
$122$		0			0
$123$	−192769.	+	192769.i	−1.14888	+	1.14888i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−150685.	−	150685.i	−0.829009	−	0.829009i	0.158370	−	0.987380i	$-0.449376\pi$
$127$	−150685.	−	150685.i	−0.829009	−	0.829009i	−0.987380	+	0.158370i	$0.949376\pi$
$128$		0			0
$129$		−	49700.1i		−	0.262956i
$130$		0			0
$131$		−	218406.i		−	1.11195i	−0.831199	−	0.555976i	$-0.812345\pi$
$131$		−	218406.i		−	1.11195i	0.831199	−	0.555976i	$-0.187655\pi$
$132$		0			0
$133$	−4347.58	−	4347.58i	−0.0213117	−	0.0213117i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−48858.6	+	48858.6i	−0.222403	+	0.222403i	−0.809509	−	0.587107i	$-0.800267\pi$
$137$	−48858.6	+	48858.6i	−0.222403	+	0.222403i	0.587107	+	0.809509i	$0.300267\pi$
$138$		0			0
$139$	356608.			1.56550			0.782752	−	0.622334i	$-0.213815\pi$
$139$	356608.			1.56550			0.782752	+	0.622334i	$0.213815\pi$
$140$		0			0
$141$	7575.36			0.0320889
$142$		0			0
$143$	−42717.5	+	42717.5i	−0.174689	+	0.174689i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	530409.	+	530409.i	2.02450	+	2.02450i
$148$		0			0
$149$		−	39079.0i		−	0.144204i	−0.997397	−	0.0721022i	$-0.977029\pi$
$149$		−	39079.0i		−	0.144204i	0.997397	−	0.0721022i	$-0.0229708\pi$
$150$		0			0
$151$		−	355671.i		−	1.26942i	−0.772750	−	0.634711i	$-0.781119\pi$
$151$		−	355671.i		−	1.26942i	0.772750	−	0.634711i	$-0.218881\pi$
$152$		0			0
$153$	−410070.	−	410070.i	−1.41622	−	1.41622i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	37305.0	−	37305.0i	0.120786	−	0.120786i	−0.644130	−	0.764916i	$-0.722780\pi$
$157$	37305.0	−	37305.0i	0.120786	−	0.120786i	0.764916	+	0.644130i	$0.222780\pi$
$158$		0			0
$159$	−786641.			−2.46765
$160$		0			0
$161$	−857612.			−2.60751
$162$		0			0
$163$	−180954.	+	180954.i	−0.533458	+	0.533458i	−0.921600	−	0.388142i	$-0.873117\pi$
$163$	−180954.	+	180954.i	−0.533458	+	0.533458i	0.388142	+	0.921600i	$0.373117\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	174882.	+	174882.i	0.485236	+	0.485236i	0.906799	−	0.421563i	$-0.138518\pi$
$167$	174882.	+	174882.i	0.485236	+	0.485236i	−0.421563	+	0.906799i	$0.638518\pi$
$168$		0			0
$169$		−	146753.i		−	0.395249i
$170$		0			0
$171$		−	10017.9i		−	0.0261992i
$172$		0			0
$173$	−376164.	−	376164.i	−0.955568	−	0.955568i	0.0434856	−	0.999054i	$-0.486154\pi$
$173$	−376164.	−	376164.i	−0.955568	−	0.955568i	−0.999054	+	0.0434856i	$0.986154\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−383826.	+	383826.i	−0.920877	+	0.920877i
$178$		0			0
$179$	384984.			0.898070			0.449035	−	0.893514i	$-0.351768\pi$
$179$	384984.			0.898070			0.449035	+	0.893514i	$0.351768\pi$
$180$		0			0
$181$	−332337.			−0.754020			−0.377010	−	0.926209i	$-0.623048\pi$
$181$	−332337.			−0.754020			−0.377010	+	0.926209i	$0.623048\pi$
$182$		0			0
$183$	−24749.5	+	24749.5i	−0.0546308	+	0.0546308i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−147251.	−	147251.i	−0.307932	−	0.307932i
$188$		0			0
$189$			597010.i			1.21570i
$190$		0			0
$191$		−	164553.i		−	0.326378i	−0.986595	−	0.163189i	$-0.947822\pi$
$191$		−	164553.i		−	0.326378i	0.986595	−	0.163189i	$-0.0521781\pi$
$192$		0			0
$193$	58078.0	+	58078.0i	0.112232	+	0.112232i	0.760993	−	0.648760i	$-0.224712\pi$
$193$	58078.0	+	58078.0i	0.112232	+	0.112232i	−0.648760	+	0.760993i	$0.724712\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−386719.	+	386719.i	−0.709954	+	0.709954i	−0.966525	−	0.256571i	$-0.917407\pi$
$197$	−386719.	+	386719.i	−0.709954	+	0.709954i	0.256571	+	0.966525i	$0.417407\pi$
$198$		0			0
$199$	14055.2			0.0251596			0.0125798	−	0.999921i	$-0.495996\pi$
$199$	14055.2			0.0251596			0.0125798	+	0.999921i	$0.495996\pi$
$200$		0			0
$201$	1.00115e6			1.74788
$202$		0			0
$203$	522066.	−	522066.i	0.889171	−	0.889171i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−988080.	−	988080.i	−1.60275	−	1.60275i
$208$		0			0
$209$		−	3597.32i		−	0.00569657i
$210$		0			0
$211$			438963.i			0.678768i	0.940648	+	0.339384i	$0.110219\pi$
$211$			438963.i			0.678768i	−0.940648	+	0.339384i	$0.889781\pi$
$212$		0			0
$213$	418268.	+	418268.i	0.631692	+	0.631692i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	829622.	−	829622.i	1.19600	−	1.19600i
$218$		0			0
$219$	−988240.			−1.39236
$220$		0			0
$221$	−774010.			−1.06602
$222$		0			0
$223$	−413890.	+	413890.i	−0.557343	+	0.557343i	−0.928550	−	0.371207i	$-0.878944\pi$
$223$	−413890.	+	413890.i	−0.557343	+	0.557343i	0.371207	+	0.928550i	$0.378944\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	165742.	+	165742.i	0.213485	+	0.213485i	0.805746	−	0.592261i	$-0.201765\pi$
$227$	165742.	+	165742.i	0.213485	+	0.213485i	−0.592261	+	0.805746i	$0.701765\pi$
$228$		0			0
$229$			590717.i			0.744373i	0.928158	+	0.372187i	$0.121392\pi$
$229$			590717.i			0.744373i	−0.928158	+	0.372187i	$0.878608\pi$
$230$		0			0
$231$			679352.i			0.837655i
$232$		0			0
$233$	849795.	+	849795.i	1.02547	+	1.02547i	0.999667	+	0.0258063i	$0.00821531\pi$
$233$	849795.	+	849795.i	1.02547	+	1.02547i	0.0258063	+	0.999667i	$0.491785\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	404930.	−	404930.i	0.468284	−	0.468284i
$238$		0			0
$239$	72025.5			0.0815627			0.0407814	−	0.999168i	$-0.487015\pi$
$239$	72025.5			0.0815627			0.0407814	+	0.999168i	$0.487015\pi$
$240$		0			0
$241$	358847.			0.397985			0.198992	−	0.980001i	$-0.436233\pi$
$241$	358847.			0.397985			0.198992	+	0.980001i	$0.436233\pi$
$242$		0			0
$243$	804029.	−	804029.i	0.873486	−	0.873486i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−9454.46	−	9454.46i	−0.00986040	−	0.00986040i
$248$		0			0
$249$		−	654318.i		−	0.668791i
$250$		0			0
$251$		−	209249.i		−	0.209642i	−0.994491	−	0.104821i	$-0.966573\pi$
$251$		−	209249.i		−	0.209642i	0.994491	−	0.104821i	$-0.0334270\pi$
$252$		0			0
$253$	−354807.	−	354807.i	−0.348491	−	0.348491i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	228218.	−	228218.i	0.215535	−	0.215535i	−0.591079	−	0.806614i	$-0.701298\pi$
$257$	228218.	−	228218.i	0.215535	−	0.215535i	0.806614	+	0.591079i	$0.201298\pi$
$258$		0			0
$259$	3.55428e6			3.29232
$260$		0			0
$261$	1.20297e6			1.09309
$262$		0			0
$263$	945242.	−	945242.i	0.842663	−	0.842663i	−0.146542	−	0.989204i	$-0.546814\pi$
$263$	945242.	−	945242.i	0.842663	−	0.842663i	0.989204	+	0.146542i	$0.0468143\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−141442.	−	141442.i	−0.121422	−	0.121422i
$268$		0			0
$269$		−	1.56559e6i		−	1.31916i	−0.751635	−	0.659579i	$-0.770735\pi$
$269$		−	1.56559e6i		−	1.31916i	0.751635	−	0.659579i	$-0.229265\pi$
$270$		0			0
$271$			990821.i			0.819543i	0.912188	+	0.409772i	$0.134392\pi$
$271$			990821.i			0.819543i	−0.912188	+	0.409772i	$0.865608\pi$
$272$		0			0
$273$	1.78547e6	+	1.78547e6i	1.44993	+	1.44993i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	174283.	−	174283.i	0.136476	−	0.136476i	−0.635569	−	0.772044i	$-0.719234\pi$
$277$	174283.	−	174283.i	0.136476	−	0.136476i	0.772044	+	0.635569i	$0.219234\pi$
$278$		0			0
$279$	1.91166e6			1.47028
$280$		0			0
$281$	2.17766e6			1.64522			0.822609	−	0.568607i	$-0.192517\pi$
$281$	2.17766e6			1.64522			0.822609	+	0.568607i	$0.192517\pi$
$282$		0			0
$283$	−502126.	+	502126.i	−0.372689	+	0.372689i	−0.868456	−	0.495767i	$-0.834887\pi$
$283$	−502126.	+	502126.i	−0.372689	+	0.372689i	0.495767	+	0.868456i	$0.334887\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	1.71763e6	+	1.71763e6i	1.23091	+	1.23091i
$288$		0			0
$289$		−	1.24823e6i		−	0.879123i
$290$		0			0
$291$		−	2.65425e6i		−	1.83742i
$292$		0			0
$293$	−45772.5	−	45772.5i	−0.0311484	−	0.0311484i	0.691361	−	0.722509i	$-0.257011\pi$
$293$	−45772.5	−	45772.5i	−0.0311484	−	0.0311484i	−0.722509	+	0.691361i	$0.757011\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−246992.	+	246992.i	−0.162477	+	0.162477i
$298$		0			0
$299$	−1.86501e6			−1.20643
$300$		0			0
$301$	−442844.			−0.281731
$302$		0			0
$303$	−23856.2	+	23856.2i	−0.0149277	+	0.0149277i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−1.39565e6	−	1.39565e6i	−0.845144	−	0.845144i	0.144379	−	0.989523i	$-0.453882\pi$
$307$	−1.39565e6	−	1.39565e6i	−0.845144	−	0.845144i	−0.989523	+	0.144379i	$0.953882\pi$
$308$		0			0
$309$			4.40498e6i			2.62451i
$310$		0			0
$311$		−	1.84115e6i		−	1.07942i	−0.841852	−	0.539709i	$-0.818534\pi$
$311$		−	1.84115e6i		−	1.07942i	0.841852	−	0.539709i	$-0.181466\pi$
$312$		0			0
$313$	−2.22340e6	−	2.22340e6i	−1.28279	−	1.28279i	−0.939073	−	0.343718i	$-0.888314\pi$
$313$	−2.22340e6	−	2.22340e6i	−1.28279	−	1.28279i	−0.343718	−	0.939073i	$-0.611686\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1.35501e6	−	1.35501e6i	0.757344	−	0.757344i	−0.218495	−	0.975838i	$-0.570115\pi$
$317$	1.35501e6	−	1.35501e6i	0.757344	−	0.757344i	0.975838	+	0.218495i	$0.0701146\pi$
$318$		0			0
$319$	431973.			0.237673
$320$		0			0
$321$	599254.			0.324600
$322$		0			0
$323$	32590.4	−	32590.4i	0.0173814	−	0.0173814i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−1.82912e6	−	1.82912e6i	−0.945958	−	0.945958i
$328$		0			0
$329$		−	67498.9i		−	0.0343801i
$330$		0			0
$331$		−	2.40493e6i		−	1.20652i	−0.797546	−	0.603258i	$-0.793869\pi$
$331$		−	2.40493e6i		−	1.20652i	0.797546	−	0.603258i	$-0.206131\pi$
$332$		0			0
$333$	4.09499e6	+	4.09499e6i	2.02368	+	2.02368i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	1.72476e6	−	1.72476e6i	0.827284	−	0.827284i	−0.159856	−	0.987140i	$-0.551103\pi$
$337$	1.72476e6	−	1.72476e6i	0.827284	−	0.827284i	0.987140	+	0.159856i	$0.0511031\pi$
$338$		0			0
$339$	−2.19436e6			−1.03707
$340$		0			0
$341$	686454.			0.319687
$342$		0			0
$343$	2.13652e6	−	2.13652e6i	0.980554	−	0.980554i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−2.06948e6	−	2.06948e6i	−0.922653	−	0.922653i	0.0745635	−	0.997216i	$-0.476244\pi$
$347$	−2.06948e6	−	2.06948e6i	−0.922653	−	0.922653i	−0.997216	+	0.0745635i	$0.976244\pi$
$348$		0			0
$349$			1.88338e6i			0.827702i	0.910345	+	0.413851i	$0.135817\pi$
$349$			1.88338e6i			0.827702i	−0.910345	+	0.413851i	$0.864183\pi$
$350$		0			0
$351$			1.29829e6i			0.562475i
$352$		0			0
$353$	−1.77943e6	−	1.77943e6i	−0.760054	−	0.760054i	0.216278	−	0.976332i	$-0.430608\pi$
$353$	−1.77943e6	−	1.77943e6i	−0.760054	−	0.760054i	−0.976332	+	0.216278i	$0.930608\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−6.15469e6	+	6.15469e6i	−2.55585	+	2.55585i
$358$		0			0
$359$	4.50238e6			1.84377			0.921884	−	0.387466i	$-0.126650\pi$
$359$	4.50238e6			1.84377			0.921884	+	0.387466i	$0.126650\pi$
$360$		0			0
$361$	−2.47530e6			−0.999678
$362$		0			0
$363$	2.50386e6	−	2.50386e6i	0.997340	−	0.997340i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	604369.	+	604369.i	0.234227	+	0.234227i	0.814455	−	0.580227i	$-0.197036\pi$
$367$	604369.	+	604369.i	0.234227	+	0.234227i	−0.580227	+	0.814455i	$0.697036\pi$
$368$		0			0
$369$			3.95787e6i			1.51320i
$370$		0			0
$371$			7.00922e6i			2.64384i
$372$		0			0
$373$	−87601.5	−	87601.5i	−0.0326016	−	0.0326016i	0.690618	−	0.723220i	$-0.257339\pi$
$373$	−87601.5	−	87601.5i	−0.0326016	−	0.0326016i	−0.723220	+	0.690618i	$0.757339\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	1.13531e6	−	1.13531e6i	0.411397	−	0.411397i
$378$		0			0
$379$	−2.48550e6			−0.888824			−0.444412	−	0.895823i	$-0.646587\pi$
$379$	−2.48550e6			−0.888824			−0.444412	+	0.895823i	$0.646587\pi$
$380$		0			0
$381$	−5.21132e6			−1.83923
$382$		0			0
$383$	601337.	−	601337.i	0.209470	−	0.209470i	−0.594572	−	0.804042i	$-0.702679\pi$
$383$	601337.	−	601337.i	0.209470	−	0.209470i	0.804042	+	0.594572i	$0.202679\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−510213.	−	510213.i	−0.173171	−	0.173171i
$388$		0			0
$389$		−	3.73874e6i		−	1.25271i	−0.779537	−	0.626356i	$-0.784546\pi$
$389$		−	3.73874e6i		−	1.25271i	0.779537	−	0.626356i	$-0.215454\pi$
$390$		0			0
$391$		−	6.42886e6i		−	2.12663i
$392$		0			0
$393$	−3.77670e6	−	3.77670e6i	−1.23348	−	1.23348i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	209001.	−	209001.i	0.0665538	−	0.0665538i	−0.673046	−	0.739600i	$-0.735015\pi$
$397$	209001.	−	209001.i	0.0665538	−	0.0665538i	0.739600	+	0.673046i	$0.235015\pi$
$398$		0			0
$399$	−150358.			−0.0472818
$400$		0			0
$401$	−3.88404e6			−1.20621			−0.603104	−	0.797662i	$-0.706070\pi$
$401$	−3.88404e6			−1.20621			−0.603104	+	0.797662i	$0.706070\pi$
$402$		0			0
$403$	1.80414e6	−	1.80414e6i	0.553359	−	0.553359i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1.47046e6	+	1.47046e6i	0.440015	+	0.440015i
$408$		0			0
$409$		−	1.17358e6i		−	0.346899i	−0.984843	−	0.173450i	$-0.944509\pi$
$409$		−	1.17358e6i		−	0.346899i	0.984843	−	0.173450i	$-0.0554913\pi$
$410$		0			0
$411$			1.68974e6i			0.493419i
$412$		0			0
$413$	3.42002e6	+	3.42002e6i	0.986627	+	0.986627i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	6.16652e6	−	6.16652e6i	1.73660	−	1.73660i
$418$		0			0
$419$	1.60340e6			0.446177			0.223089	−	0.974798i	$-0.428386\pi$
$419$	1.60340e6			0.446177			0.223089	+	0.974798i	$0.428386\pi$
$420$		0			0
$421$	−3.71020e6			−1.02022			−0.510108	−	0.860111i	$-0.670394\pi$
$421$	−3.71020e6			−1.02022			−0.510108	+	0.860111i	$0.670394\pi$
$422$		0			0
$423$	77767.4	−	77767.4i	0.0211323	−	0.0211323i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	220526.	+	220526.i	0.0585315	+	0.0585315i
$428$		0			0
$429$			1.47735e6i			0.387562i
$430$		0			0
$431$		−	2.29101e6i		−	0.594065i	−0.954867	−	0.297033i	$-0.904003\pi$
$431$		−	2.29101e6i		−	0.594065i	0.954867	−	0.297033i	$-0.0959971\pi$
$432$		0			0
$433$	972567.	+	972567.i	0.249287	+	0.249287i	0.820678	−	0.571391i	$-0.193596\pi$
$433$	972567.	+	972567.i	0.249287	+	0.249287i	−0.571391	+	0.820678i	$0.693596\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	78527.9	−	78527.9i	0.0196707	−	0.0196707i
$438$		0			0
$439$	−3.49284e6			−0.865002			−0.432501	−	0.901633i	$-0.642369\pi$
$439$	−3.49284e6			−0.865002			−0.432501	+	0.901633i	$0.642369\pi$
$440$		0			0
$441$	1.08902e7			2.66648
$442$		0			0
$443$	−4.91464e6	+	4.91464e6i	−1.18982	+	1.18982i	−0.212707	+	0.977116i	$0.568228\pi$
$443$	−4.91464e6	+	4.91464e6i	−1.18982	+	1.18982i	−0.977116	+	0.212707i	$0.931772\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−675761.	−	675761.i	−0.159965	−	0.159965i
$448$		0			0
$449$		−	3.79845e6i		−	0.889182i	−0.895734	−	0.444591i	$-0.853349\pi$
$449$		−	3.79845e6i		−	0.889182i	0.895734	−	0.444591i	$-0.146651\pi$
$450$		0			0
$451$			1.42122e6i			0.329019i
$452$		0			0
$453$	−6.15031e6	−	6.15031e6i	−1.40816	−	1.40816i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−4.14446e6	+	4.14446e6i	−0.928276	+	0.928276i	−0.997595	−	0.0693189i	$-0.977917\pi$
$457$	−4.14446e6	+	4.14446e6i	−0.928276	+	0.928276i	0.0693189	+	0.997595i	$0.477917\pi$
$458$		0			0
$459$	−4.47532e6			−0.991499
$460$		0			0
$461$	1.33612e6			0.292815			0.146407	−	0.989224i	$-0.453229\pi$
$461$	1.33612e6			0.292815			0.146407	+	0.989224i	$0.453229\pi$
$462$		0			0
$463$	−3.66263e6	+	3.66263e6i	−0.794036	+	0.794036i	−0.982148	−	0.188112i	$-0.939763\pi$
$463$	−3.66263e6	+	3.66263e6i	−0.794036	+	0.794036i	0.188112	+	0.982148i	$0.439763\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	4.58287e6	+	4.58287e6i	0.972401	+	0.972401i	0.999629	−	0.0272279i	$-0.00866797\pi$
$467$	4.58287e6	+	4.58287e6i	0.972401	+	0.972401i	−0.0272279	+	0.999629i	$0.508668\pi$
$468$		0			0
$469$		−	8.92061e6i		−	1.87267i
$470$		0			0
$471$		−	1.29017e6i		−	0.267975i
$472$		0			0
$473$	−183211.	−	183211.i	−0.0376530	−	0.0376530i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−8.07553e6	+	8.07553e6i	−1.62508	+	1.62508i
$478$		0			0
$479$	7.00075e6			1.39414			0.697069	−	0.717004i	$-0.254487\pi$
$479$	7.00075e6			1.39414			0.697069	+	0.717004i	$0.254487\pi$
$480$		0			0
$481$	7.72932e6			1.52328
$482$		0			0
$483$	−1.48300e7	+	1.48300e7i	−2.89249	+	2.89249i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	3.98892e6	+	3.98892e6i	0.762137	+	0.762137i	0.976708	−	0.214571i	$-0.0688355\pi$
$487$	3.98892e6	+	3.98892e6i	0.762137	+	0.762137i	−0.214571	+	0.976708i	$0.568835\pi$
$488$		0			0
$489$			6.25818e6i			1.18352i
$490$		0			0
$491$			2.30379e6i			0.431260i	0.976475	+	0.215630i	$0.0691805\pi$
$491$			2.30379e6i			0.431260i	−0.976475	+	0.215630i	$0.930819\pi$
$492$		0			0
$493$	3.91352e6	+	3.91352e6i	0.725188	+	0.725188i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	3.72690e6	−	3.72690e6i	0.676795	−	0.676795i
$498$		0			0
$499$	−2.47982e6			−0.445829			−0.222914	−	0.974838i	$-0.571557\pi$
$499$	−2.47982e6			−0.445829			−0.222914	+	0.974838i	$0.571557\pi$
$500$		0			0
$501$	6.04816e6			1.07654
$502$		0			0
$503$	4.08077e6	−	4.08077e6i	0.719154	−	0.719154i	−0.249278	−	0.968432i	$-0.580193\pi$
$503$	4.08077e6	−	4.08077e6i	0.719154	−	0.719154i	0.968432	+	0.249278i	$0.0801934\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−2.53768e6	−	2.53768e6i	−0.438446	−	0.438446i
$508$		0			0
$509$			2.80205e6i			0.479382i	0.970849	+	0.239691i	$0.0770461\pi$
$509$			2.80205e6i			0.479382i	−0.970849	+	0.239691i	$0.922954\pi$
$510$		0			0
$511$			8.80554e6i			1.49178i
$512$		0			0
$513$	−54665.6	−	54665.6i	−0.00917109	−	0.00917109i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	27925.3	−	27925.3i	0.00459485	−	0.00459485i
$518$		0			0
$519$	−1.30094e7			−2.12001
$520$		0			0
$521$	144765.			0.0233652			0.0116826	−	0.999932i	$-0.496281\pi$
$521$	144765.			0.0233652			0.0116826	+	0.999932i	$0.496281\pi$
$522$		0			0
$523$	3.79906e6	−	3.79906e6i	0.607327	−	0.607327i	−0.334920	−	0.942247i	$-0.608709\pi$
$523$	3.79906e6	−	3.79906e6i	0.607327	−	0.607327i	0.942247	+	0.334920i	$0.108709\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	6.21903e6	+	6.21903e6i	0.975430	+	0.975430i
$528$		0			0
$529$		−	9.05423e6i		−	1.40674i
$530$		0			0
$531$			7.88060e6i			1.21289i
$532$		0			0
$533$	3.73526e6	+	3.73526e6i	0.569511	+	0.569511i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	6.65720e6	−	6.65720e6i	0.996221	−	0.996221i
$538$		0			0
$539$	3.91053e6			0.579781
$540$		0			0
$541$	−1.17321e7			−1.72339			−0.861695	−	0.507426i	$-0.830597\pi$
$541$	−1.17321e7			−1.72339			−0.861695	+	0.507426i	$0.830597\pi$
$542$		0			0
$543$	−5.74683e6	+	5.74683e6i	−0.836428	+	0.836428i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	5.24186e6	+	5.24186e6i	0.749060	+	0.749060i	0.974303	−	0.225243i	$-0.0723174\pi$
$547$	5.24186e6	+	5.24186e6i	0.749060	+	0.749060i	−0.225243	+	0.974303i	$0.572317\pi$
$548$		0			0
$549$			508148.i			0.0719547i
$550$		0			0
$551$			95606.6i			0.0134156i
$552$		0			0
$553$	−3.60806e6	−	3.60806e6i	−0.501719	−	0.501719i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−7.61782e6	+	7.61782e6i	−1.04038	+	1.04038i	−0.0412325	+	0.999150i	$0.513128\pi$
$557$	−7.61782e6	+	7.61782e6i	−1.04038	+	1.04038i	−0.999150	+	0.0412325i	$0.986872\pi$
$558$		0			0
$559$	−963030.			−0.130350
$560$		0			0
$561$	−5.09258e6			−0.683173
$562$		0			0
$563$	86277.5	−	86277.5i	0.0114717	−	0.0114717i	−0.701348	−	0.712819i	$-0.747418\pi$
$563$	86277.5	−	86277.5i	0.0114717	−	0.0114717i	0.712819	+	0.701348i	$0.247418\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−2.96938e6	−	2.96938e6i	−0.387890	−	0.387890i
$568$		0			0
$569$		−	5.02097e6i		−	0.650140i	−0.945690	−	0.325070i	$-0.894612\pi$
$569$		−	5.02097e6i		−	0.650140i	0.945690	−	0.325070i	$-0.105388\pi$
$570$		0			0
$571$		−	8.08321e6i		−	1.03751i	−0.854922	−	0.518756i	$-0.826395\pi$
$571$		−	8.08321e6i		−	1.03751i	0.854922	−	0.518756i	$-0.173605\pi$
$572$		0			0
$573$	−2.84547e6	−	2.84547e6i	−0.362049	−	0.362049i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	3.25025e6	−	3.25025e6i	0.406422	−	0.406422i	−0.474067	−	0.880489i	$-0.657214\pi$
$577$	3.25025e6	−	3.25025e6i	0.406422	−	0.406422i	0.880489	+	0.474067i	$0.157214\pi$
$578$		0			0
$579$	2.00859e6			0.248997
$580$		0			0
$581$	−5.83018e6			−0.716542
$582$		0			0
$583$	−2.89982e6	+	2.89982e6i	−0.353346	+	0.353346i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−3.79350e6	−	3.79350e6i	−0.454407	−	0.454407i	0.442407	−	0.896814i	$-0.354125\pi$
$587$	−3.79350e6	−	3.79350e6i	−0.454407	−	0.454407i	−0.896814	+	0.442407i	$0.854125\pi$
$588$		0			0
$589$			151930.i			0.0180449i
$590$		0			0
$591$			1.33744e7i			1.57509i
$592$		0			0
$593$	−1.01322e7	−	1.01322e7i	−1.18323	−	1.18323i	−0.978904	−	0.204321i	$-0.934501\pi$
$593$	−1.01322e7	−	1.01322e7i	−1.18323	−	1.18323i	−0.204321	−	0.978904i	$-0.565499\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	243044.	−	243044.i	0.0279093	−	0.0279093i
$598$		0			0
$599$	−1.84016e6			−0.209550			−0.104775	−	0.994496i	$-0.533412\pi$
$599$	−1.84016e6			−0.209550			−0.104775	+	0.994496i	$0.533412\pi$
$600$		0			0
$601$	2.07660e6			0.234513			0.117257	−	0.993102i	$-0.462590\pi$
$601$	2.07660e6			0.234513			0.117257	+	0.993102i	$0.462590\pi$
$602$		0			0
$603$	1.02777e7	−	1.02777e7i	1.15107	−	1.15107i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−7.47671e6	−	7.47671e6i	−0.823643	−	0.823643i	0.162986	−	0.986628i	$-0.447888\pi$
$607$	−7.47671e6	−	7.47671e6i	−0.823643	−	0.823643i	−0.986628	+	0.162986i	$0.947888\pi$
$608$		0			0
$609$		−	1.80553e7i		−	1.97270i
$610$		0			0
$611$		−	146786.i		−	0.0159068i
$612$		0			0
$613$	−421185.	−	421185.i	−0.0452712	−	0.0452712i	0.684109	−	0.729380i	$-0.260191\pi$
$613$	−421185.	−	421185.i	−0.0452712	−	0.0452712i	−0.729380	+	0.684109i	$0.760191\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−4.79146e6	+	4.79146e6i	−0.506705	+	0.506705i	−0.913514	−	0.406808i	$-0.866642\pi$
$617$	−4.79146e6	+	4.79146e6i	−0.506705	+	0.506705i	0.406808	+	0.913514i	$0.366642\pi$
$618$		0			0
$619$	−8.81805e6			−0.925009			−0.462505	−	0.886617i	$-0.653049\pi$
$619$	−8.81805e6			−0.925009			−0.462505	+	0.886617i	$0.653049\pi$
$620$		0			0
$621$	−1.07835e7			−1.12209
$622$		0			0
$623$	−1.26029e6	+	1.26029e6i	−0.130092	+	0.130092i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−62205.4	−	62205.4i	−0.00631915	−	0.00631915i
$628$		0			0
$629$			2.66437e7i			2.68515i
$630$		0			0
$631$		−	4.57807e6i		−	0.457730i	−0.973458	−	0.228865i	$-0.926499\pi$
$631$		−	4.57807e6i		−	0.457730i	0.973458	−	0.228865i	$-0.0735014\pi$
$632$		0			0
$633$	7.59061e6	+	7.59061e6i	0.752952	+	0.752952i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	1.02777e7	−	1.02777e7i	1.00356	−	1.00356i
$638$		0			0
$639$	8.58774e6			0.832007
$640$		0			0
$641$	−1.62758e7			−1.56458			−0.782288	−	0.622917i	$-0.785947\pi$
$641$	−1.62758e7			−1.56458			−0.782288	+	0.622917i	$0.785947\pi$
$642$		0			0
$643$	2.99416e6	−	2.99416e6i	0.285593	−	0.285593i	−0.549742	−	0.835335i	$-0.685274\pi$
$643$	2.99416e6	−	2.99416e6i	0.285593	−	0.285593i	0.835335	+	0.549742i	$0.185274\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−4.92387e6	−	4.92387e6i	−0.462430	−	0.462430i	0.437021	−	0.899451i	$-0.356033\pi$
$647$	−4.92387e6	−	4.92387e6i	−0.462430	−	0.462430i	−0.899451	+	0.437021i	$0.856033\pi$
$648$		0			0
$649$			2.82983e6i			0.263723i
$650$		0			0
$651$		−	2.86919e7i		−	2.65342i
$652$		0			0
$653$	−1.05307e7	−	1.05307e7i	−0.966441	−	0.966441i	0.0330138	−	0.999455i	$-0.489489\pi$
$653$	−1.05307e7	−	1.05307e7i	−0.966441	−	0.966441i	−0.999455	+	0.0330138i	$0.989489\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−1.01451e7	+	1.01451e7i	−0.916945	+	0.916945i
$658$		0			0
$659$	−1.29023e7			−1.15732			−0.578659	−	0.815570i	$-0.696424\pi$
$659$	−1.29023e7			−1.15732			−0.578659	+	0.815570i	$0.696424\pi$
$660$		0			0
$661$	6.60437e6			0.587933			0.293967	−	0.955816i	$-0.405025\pi$
$661$	6.60437e6			0.587933			0.293967	+	0.955816i	$0.405025\pi$
$662$		0			0
$663$	−1.33843e7	+	1.33843e7i	−1.18253	+	1.18253i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	9.42978e6	+	9.42978e6i	0.820705	+	0.820705i
$668$		0			0
$669$			1.43141e7i			1.23651i
$670$		0			0
$671$			182470.i			0.0156453i
$672$		0			0
$673$	−1.01077e7	−	1.01077e7i	−0.860230	−	0.860230i	0.131135	−	0.991365i	$-0.458138\pi$
$673$	−1.01077e7	−	1.01077e7i	−0.860230	−	0.860230i	−0.991365	+	0.131135i	$0.958138\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−5.24850e6	+	5.24850e6i	−0.440112	+	0.440112i	−0.892050	−	0.451938i	$-0.850733\pi$
$677$	−5.24850e6	+	5.24850e6i	−0.440112	+	0.440112i	0.451938	+	0.892050i	$0.350733\pi$
$678$		0			0
$679$	−2.36502e7			−1.96861
$680$		0			0
$681$	5.73207e6			0.473635
$682$		0			0
$683$	1.03798e7	−	1.03798e7i	0.851407	−	0.851407i	−0.138900	−	0.990306i	$-0.544357\pi$
$683$	1.03798e7	−	1.03798e7i	0.851407	−	0.851407i	0.990306	+	0.138900i	$0.0443565\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	1.02148e7	+	1.02148e7i	0.825727	+	0.825727i
$688$		0			0
$689$			1.52426e7i			1.22324i
$690$		0			0
$691$			6.06155e6i			0.482935i	0.970409	+	0.241468i	$0.0776288\pi$
$691$			6.06155e6i			0.482935i	−0.970409	+	0.241468i	$0.922371\pi$
$692$		0			0
$693$	6.97412e6	+	6.97412e6i	0.551641	+	0.551641i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−1.28758e7	+	1.28758e7i	−1.00390	+	1.00390i
$698$		0			0
$699$	2.93896e7			2.27510
$700$		0			0
$701$	2.87503e6			0.220977			0.110489	−	0.993877i	$-0.464758\pi$
$701$	2.87503e6			0.220977			0.110489	+	0.993877i	$0.464758\pi$
$702$		0			0
$703$	−325450.	+	325450.i	−0.0248368	+	0.0248368i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	212566.	+	212566.i	0.0159936	+	0.0159936i
$708$		0			0
$709$			528547.i			0.0394883i	0.999805	+	0.0197441i	$0.00628516\pi$
$709$			528547.i			0.0394883i	−0.999805	+	0.0197441i	$0.993715\pi$
$710$		0			0
$711$		−	8.31390e6i		−	0.616780i
$712$		0			0
$713$	1.49850e7	+	1.49850e7i	1.10391	+	1.10391i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	1.24548e6	−	1.24548e6i	0.0904769	−	0.0904769i
$718$		0			0
$719$	1.66055e7			1.19793			0.598963	−	0.800777i	$-0.295580\pi$
$719$	1.66055e7			1.19793			0.598963	+	0.800777i	$0.295580\pi$
$720$		0			0
$721$	3.92498e7			2.81190
$722$		0			0
$723$	6.20523e6	−	6.20523e6i	0.441481	−	0.441481i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−1.54075e7	−	1.54075e7i	−1.08118	−	1.08118i	−0.996400	−	0.0847783i	$-0.972982\pi$
$727$	−1.54075e7	−	1.54075e7i	−1.08118	−	1.08118i	−0.0847783	−	0.996400i	$-0.527018\pi$
$728$		0			0
$729$		−	2.31237e7i		−	1.61153i
$730$		0			0
$731$		−	3.31966e6i		−	0.229773i
$732$		0			0
$733$	1.69113e7	+	1.69113e7i	1.16256	+	1.16256i	0.983913	+	0.178650i	$0.0571731\pi$
$733$	1.69113e7	+	1.69113e7i	1.16256	+	1.16256i	0.178650	+	0.983913i	$0.442827\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	3.69059e6	−	3.69059e6i	0.250281	−	0.250281i
$738$		0			0
$739$	2.76180e7			1.86029			0.930147	−	0.367187i	$-0.119679\pi$
$739$	2.76180e7			1.86029			0.930147	+	0.367187i	$0.119679\pi$
$740$		0			0
$741$	−326976.			−0.0218761
$742$		0			0
$743$	1.71922e7	−	1.71922e7i	1.14251	−	1.14251i	0.154521	−	0.987990i	$-0.450617\pi$
$743$	1.71922e7	−	1.71922e7i	1.14251	−	1.14251i	0.987990	−	0.154521i	$-0.0493833\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−6.71712e6	−	6.71712e6i	−0.440435	−	0.440435i
$748$		0			0
$749$		−	5.33954e6i		−	0.347776i
$750$		0			0
$751$			1.11896e7i			0.723961i	0.932186	+	0.361981i	$0.117899\pi$
$751$			1.11896e7i			0.723961i	−0.932186	+	0.361981i	$0.882101\pi$
$752$		0			0
$753$	−3.61836e6	−	3.61836e6i	−0.232554	−	0.232554i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	1.17247e6	−	1.17247e6i	0.0743637	−	0.0743637i	−0.668947	−	0.743310i	$-0.733255\pi$
$757$	1.17247e6	−	1.17247e6i	0.0743637	−	0.0743637i	0.743310	+	0.668947i	$0.233255\pi$
$758$		0			0
$759$	−1.22708e7			−0.773156
$760$		0			0
$761$	−9.79138e6			−0.612890			−0.306445	−	0.951888i	$-0.599140\pi$
$761$	−9.79138e6			−0.612890			−0.306445	+	0.951888i	$0.599140\pi$
$762$		0			0
$763$	−1.62980e7	+	1.62980e7i	−1.01350	+	1.01350i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	7.43734e6	+	7.43734e6i	0.456488	+	0.456488i
$768$		0			0
$769$			1.10059e7i			0.671138i	0.942016	+	0.335569i	$0.108929\pi$
$769$			1.10059e7i			0.671138i	−0.942016	+	0.335569i	$0.891071\pi$
$770$		0			0
$771$		−	7.89277e6i		−	0.478182i
$772$		0			0
$773$	−8.90959e6	−	8.90959e6i	−0.536302	−	0.536302i	0.386139	−	0.922441i	$-0.373809\pi$
$773$	−8.90959e6	−	8.90959e6i	−0.536302	−	0.536302i	−0.922441	+	0.386139i	$0.873809\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	6.14612e7	−	6.14612e7i	3.65215	−	3.65215i
$778$		0			0
$779$	−314553.			−0.0185716
$780$		0			0
$781$	3.08375e6			0.180906
$782$		0			0
$783$	6.56435e6	−	6.56435e6i	0.382638	−	0.382638i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−4.88094e6	−	4.88094e6i	−0.280910	−	0.280910i	0.552562	−	0.833472i	$-0.313650\pi$
$787$	−4.88094e6	−	4.88094e6i	−0.280910	−	0.280910i	−0.833472	+	0.552562i	$0.813650\pi$
$788$		0			0
$789$		−	3.26905e7i		−	1.86952i
$790$		0			0
$791$			1.95525e7i			1.11112i
$792$		0			0
$793$	479566.	+	479566.i	0.0270811	+	0.0270811i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−7.86412e6	+	7.86412e6i	−0.438535	+	0.438535i	−0.891519	−	0.452984i	$-0.850360\pi$
$797$	−7.86412e6	+	7.86412e6i	−0.438535	+	0.438535i	0.452984	+	0.891519i	$0.350360\pi$
$798$		0			0
$799$	505987.			0.0280396
$800$		0			0
$801$	−2.90403e6			−0.159926
$802$		0			0
$803$	−3.64299e6	+	3.64299e6i	−0.199374	+	0.199374i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−2.70724e7	−	2.70724e7i	−1.46333	−	1.46333i
$808$		0			0
$809$			3.59398e6i			0.193065i	0.995330	+	0.0965327i	$0.0307752\pi$
$809$			3.59398e6i			0.193065i	−0.995330	+	0.0965327i	$0.969225\pi$
$810$		0			0
$811$		−	1.96828e7i		−	1.05084i	−0.850844	−	0.525418i	$-0.823909\pi$
$811$		−	1.96828e7i		−	1.05084i	0.850844	−	0.525418i	$-0.176091\pi$
$812$		0			0
$813$	1.71334e7	+	1.71334e7i	0.909113	+	0.909113i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	40549.3	−	40549.3i	0.00212534	−	0.00212534i
$818$		0			0
$819$	3.66587e7			1.90971
$820$		0			0
$821$	−6.87305e6			−0.355870			−0.177935	−	0.984042i	$-0.556942\pi$
$821$	−6.87305e6			−0.355870			−0.177935	+	0.984042i	$0.556942\pi$
$822$		0			0
$823$	2.46199e7	−	2.46199e7i	1.26703	−	1.26703i	0.319415	−	0.947615i	$-0.396514\pi$
$823$	2.46199e7	−	2.46199e7i	1.26703	−	1.26703i	0.947615	−	0.319415i	$-0.103486\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	2.64914e7	+	2.64914e7i	1.34692	+	1.34692i	0.888988	+	0.457931i	$0.151409\pi$
$827$	2.64914e7	+	2.64914e7i	1.34692	+	1.34692i	0.457931	+	0.888988i	$0.348591\pi$
$828$		0			0
$829$		−	4.71839e6i		−	0.238456i	−0.992867	−	0.119228i	$-0.961958\pi$
$829$		−	4.71839e6i		−	0.238456i	0.992867	−	0.119228i	$-0.0380419\pi$
$830$		0			0
$831$		−	6.02745e6i		−	0.302782i
$832$		0			0
$833$	3.54280e7	+	3.54280e7i	1.76903	+	1.76903i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	1.04315e7	−	1.04315e7i	0.514675	−	0.514675i
$838$		0			0
$839$	2.18851e6			0.107336			0.0536678	−	0.998559i	$-0.482909\pi$
$839$	2.18851e6			0.107336			0.0536678	+	0.998559i	$0.482909\pi$
$840$		0			0
$841$	9.03051e6			0.440273
$842$		0			0
$843$	3.76564e7	−	3.76564e7i	1.82503	−	1.82503i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−2.23102e7	−	2.23102e7i	−1.06855	−	1.06855i
$848$		0			0
$849$			1.73657e7i			0.826842i
$850$		0			0
$851$			6.41990e7i			3.03882i
$852$		0			0
$853$	1.82806e7	+	1.82806e7i	0.860238	+	0.860238i	0.991366	−	0.131127i	$-0.0418596\pi$
$853$	1.82806e7	+	1.82806e7i	0.860238	+	0.860238i	−0.131127	+	0.991366i	$0.541860\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−2.39122e7	+	2.39122e7i	−1.11216	+	1.11216i	−0.119303	+	0.992858i	$0.538066\pi$
$857$	−2.39122e7	+	2.39122e7i	−1.11216	+	1.11216i	−0.992858	+	0.119303i	$0.961934\pi$
$858$		0			0
$859$	1.15768e7			0.535309			0.267654	−	0.963515i	$-0.413751\pi$
$859$	1.15768e7			0.535309			0.267654	+	0.963515i	$0.413751\pi$
$860$		0			0
$861$	5.94032e7			2.73088
$862$		0			0
$863$	3.08022e6	−	3.08022e6i	0.140784	−	0.140784i	−0.633202	−	0.773987i	$-0.718260\pi$
$863$	3.08022e6	−	3.08022e6i	0.140784	−	0.140784i	0.773987	+	0.633202i	$0.218260\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−2.15846e7	−	2.15846e7i	−0.975204	−	0.975204i
$868$		0			0
$869$		−	2.98542e6i		−	0.134108i
$870$		0			0
$871$		−	1.93992e7i		−	0.866440i
$872$		0			0
$873$	−2.72481e7	−	2.72481e7i	−1.21004	−	1.21004i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1.24010e7	−	1.24010e7i	0.544449	−	0.544449i	−0.380381	−	0.924830i	$-0.624207\pi$
$877$	1.24010e7	−	1.24010e7i	0.544449	−	0.544449i	0.924830	+	0.380381i	$0.124207\pi$
$878$		0			0
$879$	−1.58301e6			−0.0691053
$880$		0			0
$881$	8.61119e6			0.373786			0.186893	−	0.982380i	$-0.440158\pi$
$881$	8.61119e6			0.373786			0.186893	+	0.982380i	$0.440158\pi$
$882$		0			0
$883$	−2.44433e7	+	2.44433e7i	−1.05502	+	1.05502i	−0.0566197	+	0.998396i	$0.518032\pi$
$883$	−2.44433e7	+	2.44433e7i	−1.05502	+	1.05502i	−0.998396	+	0.0566197i	$0.981968\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1.94644e7	+	1.94644e7i	0.830678	+	0.830678i	0.987609	−	0.156932i	$-0.0501603\pi$
$887$	1.94644e7	+	1.94644e7i	0.830678	+	0.830678i	−0.156932	+	0.987609i	$0.550160\pi$
$888$		0			0
$889$			4.64345e7i			1.97055i
$890$		0			0
$891$		−	2.45696e6i		−	0.103682i
$892$		0			0
$893$	6180.58	+	6180.58i	0.000259359	+	0.000259359i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−3.22500e7	+	3.22500e7i	−1.33828	+	1.33828i
$898$		0			0
$899$	−1.82440e7			−0.752873
$900$		0			0
$901$	−5.25427e7			−2.15626
$902$		0			0
$903$	−7.65772e6	+	7.65772e6i	−0.312522	+	0.312522i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−2.02098e7	−	2.02098e7i	−0.815724	−	0.815724i	0.169761	−	0.985485i	$-0.445700\pi$
$907$	−2.02098e7	−	2.02098e7i	−0.815724	−	0.815724i	−0.985485	+	0.169761i	$0.945700\pi$
$908$		0			0
$909$			489807.i			0.0196614i
$910$		0			0
$911$		−	2.45272e7i		−	0.979155i	−0.871960	−	0.489577i	$-0.837151\pi$
$911$		−	2.45272e7i		−	0.979155i	0.871960	−	0.489577i	$-0.162849\pi$
$912$		0			0
$913$	−2.41204e6	−	2.41204e6i	−0.0957650	−	0.0957650i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−3.36516e7	+	3.36516e7i	−1.32155	+	1.32155i
$918$		0			0
$919$	1.34572e7			0.525613			0.262806	−	0.964849i	$-0.415352\pi$
$919$	1.34572e7			0.525613			0.262806	+	0.964849i	$0.415352\pi$
$920$		0			0
$921$	−4.82676e7			−1.87502
$922$		0			0
$923$	8.10471e6	−	8.10471e6i	0.313136	−	0.313136i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	4.52208e7	+	4.52208e7i	1.72838	+	1.72838i
$928$		0			0
$929$			1.09539e7i			0.416417i	0.978084	+	0.208209i	$0.0667633\pi$
$929$			1.09539e7i			0.416417i	−0.978084	+	0.208209i	$0.933237\pi$
$930$		0			0
$931$			865500.i			0.0327260i
$932$		0			0
$933$	−3.18375e7	−	3.18375e7i	−1.19739	−	1.19739i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	2.46236e7	−	2.46236e7i	0.916224	−	0.916224i	−0.0805282	−	0.996752i	$-0.525661\pi$
$937$	2.46236e7	−	2.46236e7i	0.916224	−	0.916224i	0.996752	+	0.0805282i	$0.0256607\pi$
$938$		0			0
$939$	−7.68946e7			−2.84598
$940$		0			0
$941$	2.79774e7			1.02999			0.514995	−	0.857193i	$-0.327794\pi$
$941$	2.79774e7			1.02999			0.514995	+	0.857193i	$0.327794\pi$
$942$		0			0
$943$	−3.10247e7	+	3.10247e7i	−1.13613	+	1.13613i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1.34277e7	+	1.34277e7i	0.486550	+	0.486550i	0.907216	−	0.420666i	$-0.138204\pi$
$947$	1.34277e7	+	1.34277e7i	0.486550	+	0.486550i	−0.420666	+	0.907216i	$0.638204\pi$
$948$		0			0
$949$			1.91490e7i			0.690208i
$950$		0			0
$951$		−	4.68619e7i		−	1.68023i
$952$		0			0
$953$	2.47107e7	+	2.47107e7i	0.881359	+	0.881359i	0.993673	−	0.112313i	$-0.0358261\pi$
$953$	2.47107e7	+	2.47107e7i	0.881359	+	0.881359i	−0.112313	+	0.993673i	$0.535826\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	7.46974e6	−	7.46974e6i	0.263649	−	0.263649i
$958$		0			0
$959$	1.50561e7			0.528648
$960$		0			0
$961$	−362678.			−0.0126681
$962$		0			0
$963$	6.15184e6	−	6.15184e6i	0.213766	−	0.213766i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−2.49726e7	−	2.49726e7i	−0.858811	−	0.858811i	0.132387	−	0.991198i	$-0.457736\pi$
$967$	−2.49726e7	−	2.49726e7i	−0.858811	−	0.858811i	−0.991198	+	0.132387i	$0.957736\pi$
$968$		0			0
$969$		−	1.12712e6i		−	0.0385620i
$970$		0			0
$971$		−	1.77496e7i		−	0.604143i	−0.953285	−	0.302071i	$-0.902322\pi$
$971$		−	1.77496e7i		−	0.604143i	0.953285	−	0.302071i	$-0.0976781\pi$
$972$		0			0
$973$	−5.49457e7	−	5.49457e7i	−1.86059	−	1.86059i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	2.06328e7	−	2.06328e7i	0.691548	−	0.691548i	−0.271024	−	0.962572i	$-0.587362\pi$
$977$	2.06328e7	−	2.06328e7i	0.691548	−	0.691548i	0.962572	+	0.271024i	$0.0873624\pi$
$978$		0			0
$979$	−1.04280e6			−0.0347733
$980$		0			0
$981$	−3.75548e7			−1.24593
$982$		0			0
$983$	1.82547e7	−	1.82547e7i	0.602546	−	0.602546i	−0.338442	−	0.940987i	$-0.609900\pi$
$983$	1.82547e7	−	1.82547e7i	0.602546	−	0.602546i	0.940987	+	0.338442i	$0.109900\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−1.16720e6	−	1.16720e6i	−0.0381375	−	0.0381375i
$988$		0			0
$989$		−	7.99884e6i		−	0.260038i
$990$		0			0
$991$		−	3.83326e7i		−	1.23989i	−0.784644	−	0.619947i	$-0.787154\pi$
$991$		−	3.83326e7i		−	1.23989i	0.784644	−	0.619947i	$-0.212846\pi$
$992$		0			0
$993$	−4.15865e7	−	4.15865e7i	−1.33838	−	1.33838i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1.22874e7	+	1.22874e7i	−0.391491	+	0.391491i	−0.875219	−	0.483727i	$-0.839283\pi$
$997$	−1.22874e7	+	1.22874e7i	−0.391491	+	0.391491i	0.483727	+	0.875219i	$0.339283\pi$
$998$		0			0
$999$	4.46909e7			1.41679

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.6.n.g.143.9		20
4.3	odd	2	inner	400.6.n.g.143.2		20
5.2	odd	4	inner	400.6.n.g.207.2		20
5.3	odd	4		80.6.n.d.47.9	yes	20
5.4	even	2		80.6.n.d.63.2	yes	20
20.3	even	4		80.6.n.d.47.2	✓	20
20.7	even	4	inner	400.6.n.g.207.9		20
20.19	odd	2		80.6.n.d.63.9	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.6.n.d.47.2	✓	20	20.3	even	4
80.6.n.d.47.9	yes	20	5.3	odd	4
80.6.n.d.63.2	yes	20	5.4	even	2
80.6.n.d.63.9	yes	20	20.19	odd	2
400.6.n.g.143.2		20	4.3	odd	2	inner
400.6.n.g.143.9		20	1.1	even	1	trivial
400.6.n.g.207.2		20	5.2	odd	4	inner
400.6.n.g.207.9		20	20.7	even	4	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	400.n (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(17.2921 - 17.2921i) q^{3} +(-154.079 - 154.079i) q^{7} -355.037i q^{9} +O(q^{10})\)	\(q+(17.2921 - 17.2921i) q^{3} +(-154.079 - 154.079i) q^{7} -355.037i q^{9} -127.489i q^{11} +(-335.067 - 335.067i) q^{13} +(1155.01 - 1155.01i) q^{17} +28.2166 q^{19} -5328.70 q^{21} +(2783.04 - 2783.04i) q^{23} +(-1937.35 - 1937.35i) q^{27} +3388.31i q^{29} +5384.41i q^{31} +(-2204.56 - 2204.56i) q^{33} +(-11534.0 + 11534.0i) q^{37} -11588.1 q^{39} -11147.8 q^{41} +(1437.07 - 1437.07i) q^{43} +(219.040 + 219.040i) q^{47} +30673.4i q^{49} -39945.1i q^{51} +(-22745.6 - 22745.6i) q^{53} +(487.926 - 487.926i) q^{57} -22196.6 q^{59} -1431.25 q^{61} +(-54703.5 + 54703.5i) q^{63} +(28948.2 + 28948.2i) q^{67} -96249.3i q^{69} +24188.3i q^{71} +(-28574.8 - 28574.8i) q^{73} +(-19643.4 + 19643.4i) q^{77} +23417.0 q^{79} +19271.9 q^{81} +(18919.5 - 18919.5i) q^{83} +(58591.1 + 58591.1i) q^{87} -8179.53i q^{89} +103253. i q^{91} +(93107.9 + 93107.9i) q^{93} +(76747.2 - 76747.2i) q^{97} -45263.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q+O(q^{10}) \) 20 * q	\( 20 q - 804 q^{13} + 2236 q^{17} - 4520 q^{21} + 11096 q^{33} - 44260 q^{37} - 6760 q^{41} - 182452 q^{53} + 34288 q^{57} - 41080 q^{61} - 264372 q^{73} - 399304 q^{77} - 520220 q^{81} - 713496 q^{93} - 374772 q^{97}+O(q^{100}) \) 20 * q - 804 * q^13 + 2236 * q^17 - 4520 * q^21 + 11096 * q^33 - 44260 * q^37 - 6760 * q^41 - 182452 * q^53 + 34288 * q^57 - 41080 * q^61 - 264372 * q^73 - 399304 * q^77 - 520220 * q^81 - 713496 * q^93 - 374772 * q^97

Introduction

L-functions

Modular forms

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Database

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