LMFDB - Embedded newform 400.3.p.a.193.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,3,Mod(193,400)]

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([0, 0, 3]))

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

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

chi := DirichletCharacter("400.193");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	400.p (of order $4$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$10.8992105744$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 50)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			193.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	400.193
Dual form			400.3.p.a.257.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.00000 - 3.00000i) q^{3} +(3.00000 - 3.00000i) q^{7} +9.00000i q^{9} +O(q^{10})$	$q+(-3.00000 - 3.00000i) q^{3} +(3.00000 - 3.00000i) q^{7} +9.00000i q^{9} -12.0000 q^{11} +(-12.0000 - 12.0000i) q^{13} +(12.0000 - 12.0000i) q^{17} +20.0000i q^{19} -18.0000 q^{21} +(-3.00000 - 3.00000i) q^{23} +30.0000i q^{29} +8.00000 q^{31} +(36.0000 + 36.0000i) q^{33} +(-48.0000 + 48.0000i) q^{37} +72.0000i q^{39} -48.0000 q^{41} +(27.0000 + 27.0000i) q^{43} +(-27.0000 + 27.0000i) q^{47} +31.0000i q^{49} -72.0000 q^{51} +(-12.0000 - 12.0000i) q^{53} +(60.0000 - 60.0000i) q^{57} -60.0000i q^{59} +32.0000 q^{61} +(27.0000 + 27.0000i) q^{63} +(3.00000 - 3.00000i) q^{67} +18.0000i q^{69} +48.0000 q^{71} +(-12.0000 - 12.0000i) q^{73} +(-36.0000 + 36.0000i) q^{77} +40.0000i q^{79} +81.0000 q^{81} +(-93.0000 - 93.0000i) q^{83} +(90.0000 - 90.0000i) q^{87} -30.0000i q^{89} -72.0000 q^{91} +(-24.0000 - 24.0000i) q^{93} +(12.0000 - 12.0000i) q^{97} -108.000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} + 6 q^{7}+O(q^{10}) $ 2 * q - 6 * q^3 + 6 * q^7	$ 2 q - 6 q^{3} + 6 q^{7} - 24 q^{11} - 24 q^{13} + 24 q^{17} - 36 q^{21} - 6 q^{23} + 16 q^{31} + 72 q^{33} - 96 q^{37} - 96 q^{41} + 54 q^{43} - 54 q^{47} - 144 q^{51} - 24 q^{53} + 120 q^{57} + 64 q^{61} + 54 q^{63} + 6 q^{67} + 96 q^{71} - 24 q^{73} - 72 q^{77} + 162 q^{81} - 186 q^{83} + 180 q^{87} - 144 q^{91} - 48 q^{93} + 24 q^{97}+O(q^{100}) $ 2 * q - 6 * q^3 + 6 * q^7 - 24 * q^11 - 24 * q^13 + 24 * q^17 - 36 * q^21 - 6 * q^23 + 16 * q^31 + 72 * q^33 - 96 * q^37 - 96 * q^41 + 54 * q^43 - 54 * q^47 - 144 * q^51 - 24 * q^53 + 120 * q^57 + 64 * q^61 + 54 * q^63 + 6 * q^67 + 96 * q^71 - 24 * q^73 - 72 * q^77 + 162 * q^81 - 186 * q^83 + 180 * q^87 - 144 * q^91 - 48 * q^93 + 24 * 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
$3$	−3.00000	−	3.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$3$	−3.00000	−	3.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	3.00000	−	3.00000i	0.428571	−	0.428571i	−0.459570	−	0.888142i	$-0.651996\pi$
$7$	3.00000	−	3.00000i	0.428571	−	0.428571i	0.888142	+	0.459570i	$0.151996\pi$
$8$		0			0
$9$			9.00000i			1.00000i
$10$		0			0
$11$	−12.0000			−1.09091			−0.545455	−	0.838140i	$-0.683643\pi$
$11$	−12.0000			−1.09091			−0.545455	+	0.838140i	$0.683643\pi$
$12$		0			0
$13$	−12.0000	−	12.0000i	−0.923077	−	0.923077i	0.0741688	−	0.997246i	$-0.476370\pi$
$13$	−12.0000	−	12.0000i	−0.923077	−	0.923077i	−0.997246	+	0.0741688i	$0.976370\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	12.0000	−	12.0000i	0.705882	−	0.705882i	−0.259784	−	0.965667i	$-0.583651\pi$
$17$	12.0000	−	12.0000i	0.705882	−	0.705882i	0.965667	+	0.259784i	$0.0836515\pi$
$18$		0			0
$19$			20.0000i			1.05263i	0.850289	+	0.526316i	$0.176427\pi$
$19$			20.0000i			1.05263i	−0.850289	+	0.526316i	$0.823573\pi$
$20$		0			0
$21$	−18.0000			−0.857143
$22$		0			0
$23$	−3.00000	−	3.00000i	−0.130435	−	0.130435i	0.638875	−	0.769310i	$-0.279400\pi$
$23$	−3.00000	−	3.00000i	−0.130435	−	0.130435i	−0.769310	+	0.638875i	$0.779400\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$			30.0000i			1.03448i	0.855840	+	0.517241i	$0.173041\pi$
$29$			30.0000i			1.03448i	−0.855840	+	0.517241i	$0.826959\pi$
$30$		0			0
$31$	8.00000			0.258065			0.129032	−	0.991640i	$-0.458813\pi$
$31$	8.00000			0.258065			0.129032	+	0.991640i	$0.458813\pi$
$32$		0			0
$33$	36.0000	+	36.0000i	1.09091	+	1.09091i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−48.0000	+	48.0000i	−1.29730	+	1.29730i	−0.367126	+	0.930171i	$0.619658\pi$
$37$	−48.0000	+	48.0000i	−1.29730	+	1.29730i	−0.930171	+	0.367126i	$0.880342\pi$
$38$		0			0
$39$			72.0000i			1.84615i
$40$		0			0
$41$	−48.0000			−1.17073			−0.585366	−	0.810769i	$-0.699049\pi$
$41$	−48.0000			−1.17073			−0.585366	+	0.810769i	$0.699049\pi$
$42$		0			0
$43$	27.0000	+	27.0000i	0.627907	+	0.627907i	0.947541	−	0.319634i	$-0.103560\pi$
$43$	27.0000	+	27.0000i	0.627907	+	0.627907i	−0.319634	+	0.947541i	$0.603560\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−27.0000	+	27.0000i	−0.574468	+	0.574468i	−0.933374	−	0.358906i	$-0.883150\pi$
$47$	−27.0000	+	27.0000i	−0.574468	+	0.574468i	0.358906	+	0.933374i	$0.383150\pi$
$48$		0			0
$49$			31.0000i			0.632653i
$50$		0			0
$51$	−72.0000			−1.41176
$52$		0			0
$53$	−12.0000	−	12.0000i	−0.226415	−	0.226415i	0.584778	−	0.811193i	$-0.301182\pi$
$53$	−12.0000	−	12.0000i	−0.226415	−	0.226415i	−0.811193	+	0.584778i	$0.801182\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	60.0000	−	60.0000i	1.05263	−	1.05263i
$58$		0			0
$59$		−	60.0000i		−	1.01695i	−0.861077	−	0.508475i	$-0.830210\pi$
$59$		−	60.0000i		−	1.01695i	0.861077	−	0.508475i	$-0.169790\pi$
$60$		0			0
$61$	32.0000			0.524590			0.262295	−	0.964988i	$-0.415521\pi$
$61$	32.0000			0.524590			0.262295	+	0.964988i	$0.415521\pi$
$62$		0			0
$63$	27.0000	+	27.0000i	0.428571	+	0.428571i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	3.00000	−	3.00000i	0.0447761	−	0.0447761i	−0.684364	−	0.729140i	$-0.739920\pi$
$67$	3.00000	−	3.00000i	0.0447761	−	0.0447761i	0.729140	+	0.684364i	$0.239920\pi$
$68$		0			0
$69$			18.0000i			0.260870i
$70$		0			0
$71$	48.0000			0.676056			0.338028	−	0.941136i	$-0.390240\pi$
$71$	48.0000			0.676056			0.338028	+	0.941136i	$0.390240\pi$
$72$		0			0
$73$	−12.0000	−	12.0000i	−0.164384	−	0.164384i	0.620122	−	0.784505i	$-0.287083\pi$
$73$	−12.0000	−	12.0000i	−0.164384	−	0.164384i	−0.784505	+	0.620122i	$0.787083\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−36.0000	+	36.0000i	−0.467532	+	0.467532i
$78$		0			0
$79$			40.0000i			0.506329i	0.967423	+	0.253165i	$0.0814714\pi$
$79$			40.0000i			0.506329i	−0.967423	+	0.253165i	$0.918529\pi$
$80$		0			0
$81$	81.0000			1.00000
$82$		0			0
$83$	−93.0000	−	93.0000i	−1.12048	−	1.12048i	−0.991669	−	0.128813i	$-0.958883\pi$
$83$	−93.0000	−	93.0000i	−1.12048	−	1.12048i	−0.128813	−	0.991669i	$-0.541117\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	90.0000	−	90.0000i	1.03448	−	1.03448i
$88$		0			0
$89$		−	30.0000i		−	0.337079i	−0.985695	−	0.168539i	$-0.946095\pi$
$89$		−	30.0000i		−	0.337079i	0.985695	−	0.168539i	$-0.0539050\pi$
$90$		0			0
$91$	−72.0000			−0.791209
$92$		0			0
$93$	−24.0000	−	24.0000i	−0.258065	−	0.258065i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	12.0000	−	12.0000i	0.123711	−	0.123711i	−0.642540	−	0.766252i	$-0.722120\pi$
$97$	12.0000	−	12.0000i	0.123711	−	0.123711i	0.766252	+	0.642540i	$0.222120\pi$
$98$		0			0
$99$		−	108.000i		−	1.09091i
$100$		0			0
$101$	−78.0000			−0.772277			−0.386139	−	0.922441i	$-0.626191\pi$
$101$	−78.0000			−0.772277			−0.386139	+	0.922441i	$0.626191\pi$
$102$		0			0
$103$	−93.0000	−	93.0000i	−0.902913	−	0.902913i	0.0927745	−	0.995687i	$-0.470426\pi$
$103$	−93.0000	−	93.0000i	−0.902913	−	0.902913i	−0.995687	+	0.0927745i	$0.970426\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−27.0000	+	27.0000i	−0.252336	+	0.252336i	−0.821928	−	0.569591i	$-0.807101\pi$
$107$	−27.0000	+	27.0000i	−0.252336	+	0.252336i	0.569591	+	0.821928i	$0.307101\pi$
$108$		0			0
$109$		−	160.000i		−	1.46789i	−0.679209	−	0.733945i	$-0.737677\pi$
$109$		−	160.000i		−	1.46789i	0.679209	−	0.733945i	$-0.262323\pi$
$110$		0			0
$111$	288.000			2.59459
$112$		0			0
$113$	−72.0000	−	72.0000i	−0.637168	−	0.637168i	0.312688	−	0.949856i	$-0.398771\pi$
$113$	−72.0000	−	72.0000i	−0.637168	−	0.637168i	−0.949856	+	0.312688i	$0.898771\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	108.000	−	108.000i	0.923077	−	0.923077i
$118$		0			0
$119$		−	72.0000i		−	0.605042i
$120$		0			0
$121$	23.0000			0.190083
$122$		0			0
$123$	144.000	+	144.000i	1.17073	+	1.17073i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−117.000	+	117.000i	−0.921260	+	0.921260i	−0.997119	−	0.0758587i	$-0.975830\pi$
$127$	−117.000	+	117.000i	−0.921260	+	0.921260i	0.0758587	+	0.997119i	$0.475830\pi$
$128$		0			0
$129$		−	162.000i		−	1.25581i
$130$		0			0
$131$	−132.000			−1.00763			−0.503817	−	0.863811i	$-0.668071\pi$
$131$	−132.000			−1.00763			−0.503817	+	0.863811i	$0.668071\pi$
$132$		0			0
$133$	60.0000	+	60.0000i	0.451128	+	0.451128i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−168.000	+	168.000i	−1.22628	+	1.22628i	−0.260916	+	0.965362i	$0.584025\pi$
$137$	−168.000	+	168.000i	−1.22628	+	1.22628i	−0.965362	+	0.260916i	$0.915975\pi$
$138$		0			0
$139$		−	100.000i		−	0.719424i	−0.933063	−	0.359712i	$-0.882875\pi$
$139$		−	100.000i		−	0.719424i	0.933063	−	0.359712i	$-0.117125\pi$
$140$		0			0
$141$	162.000			1.14894
$142$		0			0
$143$	144.000	+	144.000i	1.00699	+	1.00699i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	93.0000	−	93.0000i	0.632653	−	0.632653i
$148$		0			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$		0			0
$151$	248.000			1.64238			0.821192	−	0.570652i	$-0.193309\pi$
$151$	248.000			1.64238			0.821192	+	0.570652i	$0.193309\pi$
$152$		0			0
$153$	108.000	+	108.000i	0.705882	+	0.705882i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	72.0000	−	72.0000i	0.458599	−	0.458599i	−0.439597	−	0.898195i	$-0.644879\pi$
$157$	72.0000	−	72.0000i	0.458599	−	0.458599i	0.898195	+	0.439597i	$0.144879\pi$
$158$		0			0
$159$			72.0000i			0.452830i
$160$		0			0
$161$	−18.0000			−0.111801
$162$		0			0
$163$	−93.0000	−	93.0000i	−0.570552	−	0.570552i	0.361731	−	0.932283i	$-0.382186\pi$
$163$	−93.0000	−	93.0000i	−0.570552	−	0.570552i	−0.932283	+	0.361731i	$0.882186\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	3.00000	−	3.00000i	0.0179641	−	0.0179641i	−0.698068	−	0.716032i	$-0.745957\pi$
$167$	3.00000	−	3.00000i	0.0179641	−	0.0179641i	0.716032	+	0.698068i	$0.245957\pi$
$168$		0			0
$169$			119.000i			0.704142i
$170$		0			0
$171$	−180.000			−1.05263
$172$		0			0
$173$	168.000	+	168.000i	0.971098	+	0.971098i	0.999594	−	0.0284957i	$-0.00907168\pi$
$173$	168.000	+	168.000i	0.971098	+	0.971098i	−0.0284957	+	0.999594i	$0.509072\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−180.000	+	180.000i	−1.01695	+	1.01695i
$178$		0			0
$179$		−	300.000i		−	1.67598i	−0.545687	−	0.837989i	$-0.683731\pi$
$179$		−	300.000i		−	1.67598i	0.545687	−	0.837989i	$-0.316269\pi$
$180$		0			0
$181$	142.000			0.784530			0.392265	−	0.919852i	$-0.371692\pi$
$181$	142.000			0.784530			0.392265	+	0.919852i	$0.371692\pi$
$182$		0			0
$183$	−96.0000	−	96.0000i	−0.524590	−	0.524590i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−144.000	+	144.000i	−0.770053	+	0.770053i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−192.000			−1.00524			−0.502618	−	0.864509i	$-0.667630\pi$
$191$	−192.000			−1.00524			−0.502618	+	0.864509i	$0.667630\pi$
$192$		0			0
$193$	−132.000	−	132.000i	−0.683938	−	0.683938i	0.276947	−	0.960885i	$-0.410677\pi$
$193$	−132.000	−	132.000i	−0.683938	−	0.683938i	−0.960885	+	0.276947i	$0.910677\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	132.000	−	132.000i	0.670051	−	0.670051i	−0.287677	−	0.957728i	$-0.592883\pi$
$197$	132.000	−	132.000i	0.670051	−	0.670051i	0.957728	+	0.287677i	$0.0928829\pi$
$198$		0			0
$199$			160.000i			0.804020i	0.915635	+	0.402010i	$0.131688\pi$
$199$			160.000i			0.804020i	−0.915635	+	0.402010i	$0.868312\pi$
$200$		0			0
$201$	−18.0000			−0.0895522
$202$		0			0
$203$	90.0000	+	90.0000i	0.443350	+	0.443350i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	27.0000	−	27.0000i	0.130435	−	0.130435i
$208$		0			0
$209$		−	240.000i		−	1.14833i
$210$		0			0
$211$	28.0000			0.132701			0.0663507	−	0.997796i	$-0.478864\pi$
$211$	28.0000			0.132701			0.0663507	+	0.997796i	$0.478864\pi$
$212$		0			0
$213$	−144.000	−	144.000i	−0.676056	−	0.676056i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	24.0000	−	24.0000i	0.110599	−	0.110599i
$218$		0			0
$219$			72.0000i			0.328767i
$220$		0			0
$221$	−288.000			−1.30317
$222$		0			0
$223$	117.000	+	117.000i	0.524664	+	0.524664i	0.918976	−	0.394313i	$-0.129017\pi$
$223$	117.000	+	117.000i	0.524664	+	0.524664i	−0.394313	+	0.918976i	$0.629017\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	93.0000	−	93.0000i	0.409692	−	0.409692i	−0.471939	−	0.881631i	$-0.656446\pi$
$227$	93.0000	−	93.0000i	0.409692	−	0.409692i	0.881631	+	0.471939i	$0.156446\pi$
$228$		0			0
$229$			370.000i			1.61572i	0.589374	+	0.807860i	$0.299374\pi$
$229$			370.000i			1.61572i	−0.589374	+	0.807860i	$0.700626\pi$
$230$		0			0
$231$	216.000			0.935065
$232$		0			0
$233$	−252.000	−	252.000i	−1.08155	−	1.08155i	−0.996366	−	0.0851794i	$-0.972854\pi$
$233$	−252.000	−	252.000i	−1.08155	−	1.08155i	−0.0851794	−	0.996366i	$-0.527146\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	120.000	−	120.000i	0.506329	−	0.506329i
$238$		0			0
$239$			360.000i			1.50628i	0.657862	+	0.753138i	$0.271461\pi$
$239$			360.000i			1.50628i	−0.657862	+	0.753138i	$0.728539\pi$
$240$		0			0
$241$	32.0000			0.132780			0.0663900	−	0.997794i	$-0.478852\pi$
$241$	32.0000			0.132780			0.0663900	+	0.997794i	$0.478852\pi$
$242$		0			0
$243$	−243.000	−	243.000i	−1.00000	−	1.00000i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	240.000	−	240.000i	0.971660	−	0.971660i
$248$		0			0
$249$			558.000i			2.24096i
$250$		0			0
$251$	−252.000			−1.00398			−0.501992	−	0.864872i	$-0.667399\pi$
$251$	−252.000			−1.00398			−0.501992	+	0.864872i	$0.667399\pi$
$252$		0			0
$253$	36.0000	+	36.0000i	0.142292	+	0.142292i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	192.000	−	192.000i	0.747082	−	0.747082i	−0.226848	−	0.973930i	$-0.572842\pi$
$257$	192.000	−	192.000i	0.747082	−	0.747082i	0.973930	+	0.226848i	$0.0728422\pi$
$258$		0			0
$259$			288.000i			1.11197i
$260$		0			0
$261$	−270.000			−1.03448
$262$		0			0
$263$	−333.000	−	333.000i	−1.26616	−	1.26616i	−0.948056	−	0.318104i	$-0.896954\pi$
$263$	−333.000	−	333.000i	−1.26616	−	1.26616i	−0.318104	−	0.948056i	$-0.603046\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−90.0000	+	90.0000i	−0.337079	+	0.337079i
$268$		0			0
$269$		−	480.000i		−	1.78439i	−0.451654	−	0.892193i	$-0.649166\pi$
$269$		−	480.000i		−	1.78439i	0.451654	−	0.892193i	$-0.350834\pi$
$270$		0			0
$271$	88.0000			0.324723			0.162362	−	0.986731i	$-0.448089\pi$
$271$	88.0000			0.324723			0.162362	+	0.986731i	$0.448089\pi$
$272$		0			0
$273$	216.000	+	216.000i	0.791209	+	0.791209i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−288.000	+	288.000i	−1.03971	+	1.03971i	−0.0405330	+	0.999178i	$0.512906\pi$
$277$	−288.000	+	288.000i	−1.03971	+	1.03971i	−0.999178	+	0.0405330i	$0.987094\pi$
$278$		0			0
$279$			72.0000i			0.258065i
$280$		0			0
$281$	−288.000			−1.02491			−0.512456	−	0.858714i	$-0.671264\pi$
$281$	−288.000			−1.02491			−0.512456	+	0.858714i	$0.671264\pi$
$282$		0			0
$283$	117.000	+	117.000i	0.413428	+	0.413428i	0.882931	−	0.469503i	$-0.155567\pi$
$283$	117.000	+	117.000i	0.413428	+	0.413428i	−0.469503	+	0.882931i	$0.655567\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−144.000	+	144.000i	−0.501742	+	0.501742i
$288$		0			0
$289$			1.00000i			0.00346021i
$290$		0			0
$291$	−72.0000			−0.247423
$292$		0			0
$293$	168.000	+	168.000i	0.573379	+	0.573379i	0.933071	−	0.359692i	$-0.117118\pi$
$293$	168.000	+	168.000i	0.573379	+	0.573379i	−0.359692	+	0.933071i	$0.617118\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$			72.0000i			0.240803i
$300$		0			0
$301$	162.000			0.538206
$302$		0			0
$303$	234.000	+	234.000i	0.772277	+	0.772277i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	243.000	−	243.000i	0.791531	−	0.791531i	−0.190212	−	0.981743i	$-0.560918\pi$
$307$	243.000	−	243.000i	0.791531	−	0.791531i	0.981743	+	0.190212i	$0.0609176\pi$
$308$		0			0
$309$			558.000i			1.80583i
$310$		0			0
$311$	−552.000			−1.77492			−0.887460	−	0.460885i	$-0.847532\pi$
$311$	−552.000			−1.77492			−0.887460	+	0.460885i	$0.847532\pi$
$312$		0			0
$313$	48.0000	+	48.0000i	0.153355	+	0.153355i	0.779614	−	0.626260i	$-0.215415\pi$
$313$	48.0000	+	48.0000i	0.153355	+	0.153355i	−0.626260	+	0.779614i	$0.715415\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−228.000	+	228.000i	−0.719243	+	0.719243i	−0.968450	−	0.249207i	$-0.919830\pi$
$317$	−228.000	+	228.000i	−0.719243	+	0.719243i	0.249207	+	0.968450i	$0.419830\pi$
$318$		0			0
$319$		−	360.000i		−	1.12853i
$320$		0			0
$321$	162.000			0.504673
$322$		0			0
$323$	240.000	+	240.000i	0.743034	+	0.743034i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−480.000	+	480.000i	−1.46789	+	1.46789i
$328$		0			0
$329$			162.000i			0.492401i
$330$		0			0
$331$	148.000			0.447130			0.223565	−	0.974689i	$-0.428231\pi$
$331$	148.000			0.447130			0.223565	+	0.974689i	$0.428231\pi$
$332$		0			0
$333$	−432.000	−	432.000i	−1.29730	−	1.29730i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	192.000	−	192.000i	0.569733	−	0.569733i	−0.362321	−	0.932054i	$-0.618015\pi$
$337$	192.000	−	192.000i	0.569733	−	0.569733i	0.932054	+	0.362321i	$0.118015\pi$
$338$		0			0
$339$			432.000i			1.27434i
$340$		0			0
$341$	−96.0000			−0.281525
$342$		0			0
$343$	240.000	+	240.000i	0.699708	+	0.699708i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−117.000	+	117.000i	−0.337176	+	0.337176i	−0.855303	−	0.518128i	$-0.826629\pi$
$347$	−117.000	+	117.000i	−0.337176	+	0.337176i	0.518128	+	0.855303i	$0.326629\pi$
$348$		0			0
$349$			130.000i			0.372493i	0.982503	+	0.186246i	$0.0596323\pi$
$349$			130.000i			0.372493i	−0.982503	+	0.186246i	$0.940368\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	288.000	+	288.000i	0.815864	+	0.815864i	0.985506	−	0.169642i	$-0.0542611\pi$
$353$	288.000	+	288.000i	0.815864	+	0.815864i	−0.169642	+	0.985506i	$0.554261\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−216.000	+	216.000i	−0.605042	+	0.605042i
$358$		0			0
$359$		−	120.000i		−	0.334262i	−0.985935	−	0.167131i	$-0.946550\pi$
$359$		−	120.000i		−	0.334262i	0.985935	−	0.167131i	$-0.0534503\pi$
$360$		0			0
$361$	−39.0000			−0.108033
$362$		0			0
$363$	−69.0000	−	69.0000i	−0.190083	−	0.190083i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	213.000	−	213.000i	0.580381	−	0.580381i	−0.354627	−	0.935008i	$-0.615392\pi$
$367$	213.000	−	213.000i	0.580381	−	0.580381i	0.935008	+	0.354627i	$0.115392\pi$
$368$		0			0
$369$		−	432.000i		−	1.17073i
$370$		0			0
$371$	−72.0000			−0.194070
$372$		0			0
$373$	168.000	+	168.000i	0.450402	+	0.450402i	0.895488	−	0.445086i	$-0.146827\pi$
$373$	168.000	+	168.000i	0.450402	+	0.450402i	−0.445086	+	0.895488i	$0.646827\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	360.000	−	360.000i	0.954907	−	0.954907i
$378$		0			0
$379$			20.0000i			0.0527704i	0.999652	+	0.0263852i	$0.00839965\pi$
$379$			20.0000i			0.0527704i	−0.999652	+	0.0263852i	$0.991600\pi$
$380$		0			0
$381$	702.000			1.84252
$382$		0			0
$383$	−123.000	−	123.000i	−0.321149	−	0.321149i	0.528059	−	0.849208i	$-0.322920\pi$
$383$	−123.000	−	123.000i	−0.321149	−	0.321149i	−0.849208	+	0.528059i	$0.822920\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−243.000	+	243.000i	−0.627907	+	0.627907i
$388$		0			0
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$	−72.0000			−0.184143
$392$		0			0
$393$	396.000	+	396.000i	1.00763	+	1.00763i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−108.000	+	108.000i	−0.272040	+	0.272040i	−0.829921	−	0.557881i	$-0.811615\pi$
$397$	−108.000	+	108.000i	−0.272040	+	0.272040i	0.557881	+	0.829921i	$0.311615\pi$
$398$		0			0
$399$		−	360.000i		−	0.902256i
$400$		0			0
$401$	−18.0000			−0.0448878			−0.0224439	−	0.999748i	$-0.507145\pi$
$401$	−18.0000			−0.0448878			−0.0224439	+	0.999748i	$0.507145\pi$
$402$		0			0
$403$	−96.0000	−	96.0000i	−0.238213	−	0.238213i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	576.000	−	576.000i	1.41523	−	1.41523i
$408$		0			0
$409$		−	80.0000i		−	0.195599i	−0.995206	−	0.0977995i	$-0.968820\pi$
$409$		−	80.0000i		−	0.195599i	0.995206	−	0.0977995i	$-0.0311804\pi$
$410$		0			0
$411$	1008.00			2.45255
$412$		0			0
$413$	−180.000	−	180.000i	−0.435835	−	0.435835i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−300.000	+	300.000i	−0.719424	+	0.719424i
$418$		0			0
$419$		−	540.000i		−	1.28878i	−0.764696	−	0.644391i	$-0.777111\pi$
$419$		−	540.000i		−	1.28878i	0.764696	−	0.644391i	$-0.222889\pi$
$420$		0			0
$421$	−608.000			−1.44418			−0.722090	−	0.691799i	$-0.756818\pi$
$421$	−608.000			−1.44418			−0.722090	+	0.691799i	$0.756818\pi$
$422$		0			0
$423$	−243.000	−	243.000i	−0.574468	−	0.574468i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	96.0000	−	96.0000i	0.224824	−	0.224824i
$428$		0			0
$429$		−	864.000i		−	2.01399i
$430$		0			0
$431$	−312.000			−0.723898			−0.361949	−	0.932198i	$-0.617889\pi$
$431$	−312.000			−0.723898			−0.361949	+	0.932198i	$0.617889\pi$
$432$		0			0
$433$	−252.000	−	252.000i	−0.581986	−	0.581986i	0.353463	−	0.935449i	$-0.385004\pi$
$433$	−252.000	−	252.000i	−0.581986	−	0.581986i	−0.935449	+	0.353463i	$0.885004\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	60.0000	−	60.0000i	0.137300	−	0.137300i
$438$		0			0
$439$			40.0000i			0.0911162i	0.998962	+	0.0455581i	$0.0145066\pi$
$439$			40.0000i			0.0911162i	−0.998962	+	0.0455581i	$0.985493\pi$
$440$		0			0
$441$	−279.000			−0.632653
$442$		0			0
$443$	−213.000	−	213.000i	−0.480813	−	0.480813i	0.424578	−	0.905391i	$-0.360422\pi$
$443$	−213.000	−	213.000i	−0.480813	−	0.480813i	−0.905391	+	0.424578i	$0.860422\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	480.000i		−	1.06904i	−0.845155	−	0.534521i	$-0.820492\pi$
$449$		−	480.000i		−	1.06904i	0.845155	−	0.534521i	$-0.179508\pi$
$450$		0			0
$451$	576.000			1.27716
$452$		0			0
$453$	−744.000	−	744.000i	−1.64238	−	1.64238i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	432.000	−	432.000i	0.945295	−	0.945295i	−0.0532840	−	0.998579i	$-0.516969\pi$
$457$	432.000	−	432.000i	0.945295	−	0.945295i	0.998579	+	0.0532840i	$0.0169689\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	222.000			0.481562			0.240781	−	0.970579i	$-0.422596\pi$
$461$	222.000			0.481562			0.240781	+	0.970579i	$0.422596\pi$
$462$		0			0
$463$	−213.000	−	213.000i	−0.460043	−	0.460043i	0.438626	−	0.898670i	$-0.355465\pi$
$463$	−213.000	−	213.000i	−0.460043	−	0.460043i	−0.898670	+	0.438626i	$0.855465\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	3.00000	−	3.00000i	0.00642398	−	0.00642398i	−0.703887	−	0.710311i	$-0.748554\pi$
$467$	3.00000	−	3.00000i	0.00642398	−	0.00642398i	0.710311	+	0.703887i	$0.248554\pi$
$468$		0			0
$469$		−	18.0000i		−	0.0383795i
$470$		0			0
$471$	−432.000			−0.917197
$472$		0			0
$473$	−324.000	−	324.000i	−0.684989	−	0.684989i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	108.000	−	108.000i	0.226415	−	0.226415i
$478$		0			0
$479$		−	240.000i		−	0.501044i	−0.968111	−	0.250522i	$-0.919398\pi$
$479$		−	240.000i		−	0.501044i	0.968111	−	0.250522i	$-0.0806022\pi$
$480$		0			0
$481$	1152.00			2.39501
$482$		0			0
$483$	54.0000	+	54.0000i	0.111801	+	0.111801i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−627.000	+	627.000i	−1.28747	+	1.28747i	−0.351158	+	0.936316i	$0.614212\pi$
$487$	−627.000	+	627.000i	−1.28747	+	1.28747i	−0.936316	+	0.351158i	$0.885788\pi$
$488$		0			0
$489$			558.000i			1.14110i
$490$		0			0
$491$	588.000			1.19756			0.598778	−	0.800915i	$-0.295653\pi$
$491$	588.000			1.19756			0.598778	+	0.800915i	$0.295653\pi$
$492$		0			0
$493$	360.000	+	360.000i	0.730223	+	0.730223i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	144.000	−	144.000i	0.289738	−	0.289738i
$498$		0			0
$499$			460.000i			0.921844i	0.887441	+	0.460922i	$0.152481\pi$
$499$			460.000i			0.921844i	−0.887441	+	0.460922i	$0.847519\pi$
$500$		0			0
$501$	−18.0000			−0.0359281
$502$		0			0
$503$	627.000	+	627.000i	1.24652	+	1.24652i	0.957246	+	0.289275i	$0.0934141\pi$
$503$	627.000	+	627.000i	1.24652	+	1.24652i	0.289275	+	0.957246i	$0.406586\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	357.000	−	357.000i	0.704142	−	0.704142i
$508$		0			0
$509$		−	450.000i		−	0.884086i	−0.896994	−	0.442043i	$-0.854254\pi$
$509$		−	450.000i		−	0.884086i	0.896994	−	0.442043i	$-0.145746\pi$
$510$		0			0
$511$	−72.0000			−0.140900
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	324.000	−	324.000i	0.626692	−	0.626692i
$518$		0			0
$519$		−	1008.00i		−	1.94220i
$520$		0			0
$521$	−558.000			−1.07102			−0.535509	−	0.844530i	$-0.679880\pi$
$521$	−558.000			−1.07102			−0.535509	+	0.844530i	$0.679880\pi$
$522$		0			0
$523$	−123.000	−	123.000i	−0.235182	−	0.235182i	0.579670	−	0.814851i	$-0.303182\pi$
$523$	−123.000	−	123.000i	−0.235182	−	0.235182i	−0.814851	+	0.579670i	$0.803182\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	96.0000	−	96.0000i	0.182163	−	0.182163i
$528$		0			0
$529$		−	511.000i		−	0.965974i
$530$		0			0
$531$	540.000			1.01695
$532$		0			0
$533$	576.000	+	576.000i	1.08068	+	1.08068i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−900.000	+	900.000i	−1.67598	+	1.67598i
$538$		0			0
$539$		−	372.000i		−	0.690167i
$540$		0			0
$541$	542.000			1.00185			0.500924	−	0.865491i	$-0.332994\pi$
$541$	542.000			1.00185			0.500924	+	0.865491i	$0.332994\pi$
$542$		0			0
$543$	−426.000	−	426.000i	−0.784530	−	0.784530i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−147.000	+	147.000i	−0.268739	+	0.268739i	−0.828592	−	0.559853i	$-0.810858\pi$
$547$	−147.000	+	147.000i	−0.268739	+	0.268739i	0.559853	+	0.828592i	$0.310858\pi$
$548$		0			0
$549$			288.000i			0.524590i
$550$		0			0
$551$	−600.000			−1.08893
$552$		0			0
$553$	120.000	+	120.000i	0.216998	+	0.216998i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−288.000	+	288.000i	−0.517056	+	0.517056i	−0.916679	−	0.399624i	$-0.869141\pi$
$557$	−288.000	+	288.000i	−0.517056	+	0.517056i	0.399624	+	0.916679i	$0.369141\pi$
$558$		0			0
$559$		−	648.000i		−	1.15921i
$560$		0			0
$561$	864.000			1.54011
$562$		0			0
$563$	477.000	+	477.000i	0.847247	+	0.847247i	0.989789	−	0.142542i	$-0.0455276\pi$
$563$	477.000	+	477.000i	0.847247	+	0.847247i	−0.142542	+	0.989789i	$0.545528\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	243.000	−	243.000i	0.428571	−	0.428571i
$568$		0			0
$569$			240.000i			0.421793i	0.977508	+	0.210896i	$0.0676382\pi$
$569$			240.000i			0.421793i	−0.977508	+	0.210896i	$0.932362\pi$
$570$		0			0
$571$	−692.000			−1.21191			−0.605954	−	0.795499i	$-0.707209\pi$
$571$	−692.000			−1.21191			−0.605954	+	0.795499i	$0.707209\pi$
$572$		0			0
$573$	576.000	+	576.000i	1.00524	+	1.00524i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−168.000	+	168.000i	−0.291161	+	0.291161i	−0.837539	−	0.546378i	$-0.816006\pi$
$577$	−168.000	+	168.000i	−0.291161	+	0.291161i	0.546378	+	0.837539i	$0.316006\pi$
$578$		0			0
$579$			792.000i			1.36788i
$580$		0			0
$581$	−558.000			−0.960413
$582$		0			0
$583$	144.000	+	144.000i	0.246998	+	0.246998i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	213.000	−	213.000i	0.362862	−	0.362862i	−0.502004	−	0.864866i	$-0.667404\pi$
$587$	213.000	−	213.000i	0.362862	−	0.362862i	0.864866	+	0.502004i	$0.167404\pi$
$588$		0			0
$589$			160.000i			0.271647i
$590$		0			0
$591$	−792.000			−1.34010
$592$		0			0
$593$	−312.000	−	312.000i	−0.526138	−	0.526138i	0.393280	−	0.919419i	$-0.371340\pi$
$593$	−312.000	−	312.000i	−0.526138	−	0.526138i	−0.919419	+	0.393280i	$0.871340\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	480.000	−	480.000i	0.804020	−	0.804020i
$598$		0			0
$599$			240.000i			0.400668i	0.979728	+	0.200334i	$0.0642027\pi$
$599$			240.000i			0.400668i	−0.979728	+	0.200334i	$0.935797\pi$
$600$		0			0
$601$	−608.000			−1.01165			−0.505824	−	0.862637i	$-0.668811\pi$
$601$	−608.000			−1.01165			−0.505824	+	0.862637i	$0.668811\pi$
$602$		0			0
$603$	27.0000	+	27.0000i	0.0447761	+	0.0447761i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−267.000	+	267.000i	−0.439868	+	0.439868i	−0.891968	−	0.452099i	$-0.850675\pi$
$607$	−267.000	+	267.000i	−0.439868	+	0.439868i	0.452099	+	0.891968i	$0.350675\pi$
$608$		0			0
$609$		−	540.000i		−	0.886700i
$610$		0			0
$611$	648.000			1.06056
$612$		0			0
$613$	228.000	+	228.000i	0.371941	+	0.371941i	0.868184	−	0.496243i	$-0.165287\pi$
$613$	228.000	+	228.000i	0.371941	+	0.371941i	−0.496243	+	0.868184i	$0.665287\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−348.000	+	348.000i	−0.564019	+	0.564019i	−0.930447	−	0.366427i	$-0.880581\pi$
$617$	−348.000	+	348.000i	−0.564019	+	0.564019i	0.366427	+	0.930447i	$0.380581\pi$
$618$		0			0
$619$			940.000i			1.51858i	0.650753	+	0.759289i	$0.274453\pi$
$619$			940.000i			1.51858i	−0.650753	+	0.759289i	$0.725547\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−90.0000	−	90.0000i	−0.144462	−	0.144462i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−720.000	+	720.000i	−1.14833	+	1.14833i
$628$		0			0
$629$			1152.00i			1.83148i
$630$		0			0
$631$	808.000			1.28051			0.640254	−	0.768164i	$-0.278829\pi$
$631$	808.000			1.28051			0.640254	+	0.768164i	$0.278829\pi$
$632$		0			0
$633$	−84.0000	−	84.0000i	−0.132701	−	0.132701i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	372.000	−	372.000i	0.583987	−	0.583987i
$638$		0			0
$639$			432.000i			0.676056i
$640$		0			0
$641$	−768.000			−1.19813			−0.599064	−	0.800701i	$-0.704460\pi$
$641$	−768.000			−1.19813			−0.599064	+	0.800701i	$0.704460\pi$
$642$		0			0
$643$	477.000	+	477.000i	0.741835	+	0.741835i	0.972931	−	0.231096i	$-0.0742311\pi$
$643$	477.000	+	477.000i	0.741835	+	0.741835i	−0.231096	+	0.972931i	$0.574231\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−627.000	+	627.000i	−0.969088	+	0.969088i	−0.999536	−	0.0304482i	$-0.990307\pi$
$647$	−627.000	+	627.000i	−0.969088	+	0.969088i	0.0304482	+	0.999536i	$0.490307\pi$
$648$		0			0
$649$			720.000i			1.10940i
$650$		0			0
$651$	−144.000			−0.221198
$652$		0			0
$653$	−12.0000	−	12.0000i	−0.0183767	−	0.0183767i	0.697859	−	0.716235i	$-0.254136\pi$
$653$	−12.0000	−	12.0000i	−0.0183767	−	0.0183767i	−0.716235	+	0.697859i	$0.754136\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	108.000	−	108.000i	0.164384	−	0.164384i
$658$		0			0
$659$		−	540.000i		−	0.819423i	−0.912215	−	0.409712i	$-0.865629\pi$
$659$		−	540.000i		−	0.819423i	0.912215	−	0.409712i	$-0.134371\pi$
$660$		0			0
$661$	352.000			0.532526			0.266263	−	0.963900i	$-0.414211\pi$
$661$	352.000			0.532526			0.266263	+	0.963900i	$0.414211\pi$
$662$		0			0
$663$	864.000	+	864.000i	1.30317	+	1.30317i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	90.0000	−	90.0000i	0.134933	−	0.134933i
$668$		0			0
$669$		−	702.000i		−	1.04933i
$670$		0			0
$671$	−384.000			−0.572280
$672$		0			0
$673$	−732.000	−	732.000i	−1.08767	−	1.08767i	−0.995768	−	0.0918988i	$-0.970706\pi$
$673$	−732.000	−	732.000i	−1.08767	−	1.08767i	−0.0918988	−	0.995768i	$-0.529294\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−108.000	+	108.000i	−0.159527	+	0.159527i	−0.782357	−	0.622830i	$-0.785983\pi$
$677$	−108.000	+	108.000i	−0.159527	+	0.159527i	0.622830	+	0.782357i	$0.285983\pi$
$678$		0			0
$679$		−	72.0000i		−	0.106038i
$680$		0			0
$681$	−558.000			−0.819383
$682$		0			0
$683$	−933.000	−	933.000i	−1.36603	−	1.36603i	−0.866016	−	0.500016i	$-0.833327\pi$
$683$	−933.000	−	933.000i	−1.36603	−	1.36603i	−0.500016	−	0.866016i	$-0.666673\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	1110.00	−	1110.00i	1.61572	−	1.61572i
$688$		0			0
$689$			288.000i			0.417997i
$690$		0			0
$691$	68.0000			0.0984081			0.0492041	−	0.998789i	$-0.484332\pi$
$691$	68.0000			0.0984081			0.0492041	+	0.998789i	$0.484332\pi$
$692$		0			0
$693$	−324.000	−	324.000i	−0.467532	−	0.467532i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−576.000	+	576.000i	−0.826399	+	0.826399i
$698$		0			0
$699$			1512.00i			2.16309i
$700$		0			0
$701$	192.000			0.273894			0.136947	−	0.990578i	$-0.456271\pi$
$701$	192.000			0.273894			0.136947	+	0.990578i	$0.456271\pi$
$702$		0			0
$703$	−960.000	−	960.000i	−1.36558	−	1.36558i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−234.000	+	234.000i	−0.330976	+	0.330976i
$708$		0			0
$709$		−	50.0000i		−	0.0705219i	−0.999378	−	0.0352609i	$-0.988774\pi$
$709$		−	50.0000i		−	0.0705219i	0.999378	−	0.0352609i	$-0.0112262\pi$
$710$		0			0
$711$	−360.000			−0.506329
$712$		0			0
$713$	−24.0000	−	24.0000i	−0.0336606	−	0.0336606i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	1080.00	−	1080.00i	1.50628	−	1.50628i
$718$		0			0
$719$			840.000i			1.16829i	0.811650	+	0.584145i	$0.198570\pi$
$719$			840.000i			1.16829i	−0.811650	+	0.584145i	$0.801430\pi$
$720$		0			0
$721$	−558.000			−0.773925
$722$		0			0
$723$	−96.0000	−	96.0000i	−0.132780	−	0.132780i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	963.000	−	963.000i	1.32462	−	1.32462i	0.414633	−	0.909989i	$-0.363910\pi$
$727$	963.000	−	963.000i	1.32462	−	1.32462i	0.909989	−	0.414633i	$-0.136090\pi$
$728$		0			0
$729$			729.000i			1.00000i
$730$		0			0
$731$	648.000			0.886457
$732$		0			0
$733$	−72.0000	−	72.0000i	−0.0982265	−	0.0982265i	0.656286	−	0.754512i	$-0.272127\pi$
$733$	−72.0000	−	72.0000i	−0.0982265	−	0.0982265i	−0.754512	+	0.656286i	$0.772127\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−36.0000	+	36.0000i	−0.0488467	+	0.0488467i
$738$		0			0
$739$			20.0000i			0.0270636i	0.999908	+	0.0135318i	$0.00430744\pi$
$739$			20.0000i			0.0270636i	−0.999908	+	0.0135318i	$0.995693\pi$
$740$		0			0
$741$	−1440.00			−1.94332
$742$		0			0
$743$	−243.000	−	243.000i	−0.327052	−	0.327052i	0.524412	−	0.851465i	$-0.324285\pi$
$743$	−243.000	−	243.000i	−0.327052	−	0.327052i	−0.851465	+	0.524412i	$0.824285\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	837.000	−	837.000i	1.12048	−	1.12048i
$748$		0			0
$749$			162.000i			0.216288i
$750$		0			0
$751$	−1072.00			−1.42743			−0.713715	−	0.700436i	$-0.752989\pi$
$751$	−1072.00			−1.42743			−0.713715	+	0.700436i	$0.752989\pi$
$752$		0			0
$753$	756.000	+	756.000i	1.00398	+	1.00398i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−408.000	+	408.000i	−0.538970	+	0.538970i	−0.923226	−	0.384257i	$-0.874458\pi$
$757$	−408.000	+	408.000i	−0.538970	+	0.538970i	0.384257	+	0.923226i	$0.374458\pi$
$758$		0			0
$759$		−	216.000i		−	0.284585i
$760$		0			0
$761$	1362.00			1.78975			0.894875	−	0.446317i	$-0.147264\pi$
$761$	1362.00			1.78975			0.894875	+	0.446317i	$0.147264\pi$
$762$		0			0
$763$	−480.000	−	480.000i	−0.629096	−	0.629096i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−720.000	+	720.000i	−0.938722	+	0.938722i
$768$		0			0
$769$		−	370.000i		−	0.481144i	−0.970631	−	0.240572i	$-0.922665\pi$
$769$		−	370.000i		−	0.481144i	0.970631	−	0.240572i	$-0.0773351\pi$
$770$		0			0
$771$	−1152.00			−1.49416
$772$		0			0
$773$	−132.000	−	132.000i	−0.170763	−	0.170763i	0.616551	−	0.787315i	$-0.288529\pi$
$773$	−132.000	−	132.000i	−0.170763	−	0.170763i	−0.787315	+	0.616551i	$0.788529\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	864.000	−	864.000i	1.11197	−	1.11197i
$778$		0			0
$779$		−	960.000i		−	1.23235i
$780$		0			0
$781$	−576.000			−0.737516
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	93.0000	−	93.0000i	0.118170	−	0.118170i	−0.645549	−	0.763719i	$-0.723371\pi$
$787$	93.0000	−	93.0000i	0.118170	−	0.118170i	0.763719	+	0.645549i	$0.223371\pi$
$788$		0			0
$789$			1998.00i			2.53232i
$790$		0			0
$791$	−432.000			−0.546144
$792$		0			0
$793$	−384.000	−	384.000i	−0.484237	−	0.484237i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−228.000	+	228.000i	−0.286073	+	0.286073i	−0.835525	−	0.549452i	$-0.814836\pi$
$797$	−228.000	+	228.000i	−0.286073	+	0.286073i	0.549452	+	0.835525i	$0.314836\pi$
$798$		0			0
$799$			648.000i			0.811014i
$800$		0			0
$801$	270.000			0.337079
$802$		0			0
$803$	144.000	+	144.000i	0.179328	+	0.179328i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−1440.00	+	1440.00i	−1.78439	+	1.78439i
$808$		0			0
$809$			750.000i			0.927070i	0.886078	+	0.463535i	$0.153419\pi$
$809$			750.000i			0.927070i	−0.886078	+	0.463535i	$0.846581\pi$
$810$		0			0
$811$	−412.000			−0.508015			−0.254007	−	0.967202i	$-0.581749\pi$
$811$	−412.000			−0.508015			−0.254007	+	0.967202i	$0.581749\pi$
$812$		0			0
$813$	−264.000	−	264.000i	−0.324723	−	0.324723i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−540.000	+	540.000i	−0.660955	+	0.660955i
$818$		0			0
$819$		−	648.000i		−	0.791209i
$820$		0			0
$821$	672.000			0.818514			0.409257	−	0.912419i	$-0.365788\pi$
$821$	672.000			0.818514			0.409257	+	0.912419i	$0.365788\pi$
$822$		0			0
$823$	717.000	+	717.000i	0.871203	+	0.871203i	0.992604	−	0.121401i	$-0.0387386\pi$
$823$	717.000	+	717.000i	0.871203	+	0.871203i	−0.121401	+	0.992604i	$0.538739\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	123.000	−	123.000i	0.148730	−	0.148730i	−0.628820	−	0.777551i	$-0.716462\pi$
$827$	123.000	−	123.000i	0.148730	−	0.148730i	0.777551	+	0.628820i	$0.216462\pi$
$828$		0			0
$829$		−	1280.00i		−	1.54403i	−0.635605	−	0.772014i	$-0.719249\pi$
$829$		−	1280.00i		−	1.54403i	0.635605	−	0.772014i	$-0.280751\pi$
$830$		0			0
$831$	1728.00			2.07942
$832$		0			0
$833$	372.000	+	372.000i	0.446579	+	0.446579i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$			1560.00i			1.85936i	0.368373	+	0.929678i	$0.379915\pi$
$839$			1560.00i			1.85936i	−0.368373	+	0.929678i	$0.620085\pi$
$840$		0			0
$841$	−59.0000			−0.0701546
$842$		0			0
$843$	864.000	+	864.000i	1.02491	+	1.02491i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	69.0000	−	69.0000i	0.0814640	−	0.0814640i
$848$		0			0
$849$		−	702.000i		−	0.826855i
$850$		0			0
$851$	288.000			0.338425
$852$		0			0
$853$	−372.000	−	372.000i	−0.436108	−	0.436108i	0.454592	−	0.890700i	$-0.349785\pi$
$853$	−372.000	−	372.000i	−0.436108	−	0.436108i	−0.890700	+	0.454592i	$0.849785\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	552.000	−	552.000i	0.644107	−	0.644107i	−0.307455	−	0.951563i	$-0.599477\pi$
$857$	552.000	−	552.000i	0.644107	−	0.644107i	0.951563	+	0.307455i	$0.0994774\pi$
$858$		0			0
$859$		−	620.000i		−	0.721769i	−0.932610	−	0.360885i	$-0.882475\pi$
$859$		−	620.000i		−	0.721769i	0.932610	−	0.360885i	$-0.117525\pi$
$860$		0			0
$861$	864.000			1.00348
$862$		0			0
$863$	−123.000	−	123.000i	−0.142526	−	0.142526i	0.632244	−	0.774770i	$-0.282134\pi$
$863$	−123.000	−	123.000i	−0.142526	−	0.142526i	−0.774770	+	0.632244i	$0.782134\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	3.00000	−	3.00000i	0.00346021	−	0.00346021i
$868$		0			0
$869$		−	480.000i		−	0.552359i
$870$		0			0
$871$	−72.0000			−0.0826636
$872$		0			0
$873$	108.000	+	108.000i	0.123711	+	0.123711i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−1128.00	+	1128.00i	−1.28620	+	1.28620i	−0.349128	+	0.937075i	$0.613522\pi$
$877$	−1128.00	+	1128.00i	−1.28620	+	1.28620i	−0.937075	+	0.349128i	$0.886478\pi$
$878$		0			0
$879$		−	1008.00i		−	1.14676i
$880$		0			0
$881$	912.000			1.03519			0.517594	−	0.855627i	$-0.326828\pi$
$881$	912.000			1.03519			0.517594	+	0.855627i	$0.326828\pi$
$882$		0			0
$883$	957.000	+	957.000i	1.08381	+	1.08381i	0.996151	+	0.0876543i	$0.0279371\pi$
$883$	957.000	+	957.000i	1.08381	+	1.08381i	0.0876543	+	0.996151i	$0.472063\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	483.000	−	483.000i	0.544532	−	0.544532i	−0.380322	−	0.924854i	$-0.624187\pi$
$887$	483.000	−	483.000i	0.544532	−	0.544532i	0.924854	+	0.380322i	$0.124187\pi$
$888$		0			0
$889$			702.000i			0.789651i
$890$		0			0
$891$	−972.000			−1.09091
$892$		0			0
$893$	−540.000	−	540.000i	−0.604703	−	0.604703i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	216.000	−	216.000i	0.240803	−	0.240803i
$898$		0			0
$899$			240.000i			0.266963i
$900$		0			0
$901$	−288.000			−0.319645
$902$		0			0
$903$	−486.000	−	486.000i	−0.538206	−	0.538206i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−1077.00	+	1077.00i	−1.18743	+	1.18743i	−0.209656	+	0.977775i	$0.567234\pi$
$907$	−1077.00	+	1077.00i	−1.18743	+	1.18743i	−0.977775	+	0.209656i	$0.932766\pi$
$908$		0			0
$909$		−	702.000i		−	0.772277i
$910$		0			0
$911$	1128.00			1.23820			0.619100	−	0.785312i	$-0.287498\pi$
$911$	1128.00			1.23820			0.619100	+	0.785312i	$0.287498\pi$
$912$		0			0
$913$	1116.00	+	1116.00i	1.22234	+	1.22234i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−396.000	+	396.000i	−0.431843	+	0.431843i
$918$		0			0
$919$		−	1600.00i		−	1.74102i	−0.492148	−	0.870511i	$-0.663788\pi$
$919$		−	1600.00i		−	1.74102i	0.492148	−	0.870511i	$-0.336212\pi$
$920$		0			0
$921$	−1458.00			−1.58306
$922$		0			0
$923$	−576.000	−	576.000i	−0.624052	−	0.624052i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	837.000	−	837.000i	0.902913	−	0.902913i
$928$		0			0
$929$		−	960.000i		−	1.03337i	−0.856176	−	0.516685i	$-0.827166\pi$
$929$		−	960.000i		−	1.03337i	0.856176	−	0.516685i	$-0.172834\pi$
$930$		0			0
$931$	−620.000			−0.665951
$932$		0			0
$933$	1656.00	+	1656.00i	1.77492	+	1.77492i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	492.000	−	492.000i	0.525080	−	0.525080i	−0.394021	−	0.919101i	$-0.628916\pi$
$937$	492.000	−	492.000i	0.525080	−	0.525080i	0.919101	+	0.394021i	$0.128916\pi$
$938$		0			0
$939$		−	288.000i		−	0.306709i
$940$		0			0
$941$	−738.000			−0.784272			−0.392136	−	0.919907i	$-0.628264\pi$
$941$	−738.000			−0.784272			−0.392136	+	0.919907i	$0.628264\pi$
$942$		0			0
$943$	144.000	+	144.000i	0.152704	+	0.152704i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−237.000	+	237.000i	−0.250264	+	0.250264i	−0.821079	−	0.570815i	$-0.806627\pi$
$947$	−237.000	+	237.000i	−0.250264	+	0.250264i	0.570815	+	0.821079i	$0.306627\pi$
$948$		0			0
$949$			288.000i			0.303477i
$950$		0			0
$951$	1368.00			1.43849
$952$		0			0
$953$	648.000	+	648.000i	0.679958	+	0.679958i	0.959991	−	0.280032i	$-0.0903452\pi$
$953$	648.000	+	648.000i	0.679958	+	0.679958i	−0.280032	+	0.959991i	$0.590345\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−1080.00	+	1080.00i	−1.12853	+	1.12853i
$958$		0			0
$959$			1008.00i			1.05109i
$960$		0			0
$961$	−897.000			−0.933403
$962$		0			0
$963$	−243.000	−	243.000i	−0.252336	−	0.252336i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−627.000	+	627.000i	−0.648397	+	0.648397i	−0.952606	−	0.304208i	$-0.901608\pi$
$967$	−627.000	+	627.000i	−0.648397	+	0.648397i	0.304208	+	0.952606i	$0.401608\pi$
$968$		0			0
$969$		−	1440.00i		−	1.48607i
$970$		0			0
$971$	708.000			0.729145			0.364573	−	0.931175i	$-0.381215\pi$
$971$	708.000			0.729145			0.364573	+	0.931175i	$0.381215\pi$
$972$		0			0
$973$	−300.000	−	300.000i	−0.308325	−	0.308325i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	612.000	−	612.000i	0.626407	−	0.626407i	−0.320755	−	0.947162i	$-0.603937\pi$
$977$	612.000	−	612.000i	0.626407	−	0.626407i	0.947162	+	0.320755i	$0.103937\pi$
$978$		0			0
$979$			360.000i			0.367722i
$980$		0			0
$981$	1440.00			1.46789
$982$		0			0
$983$	627.000	+	627.000i	0.637843	+	0.637843i	0.950023	−	0.312180i	$-0.101059\pi$
$983$	627.000	+	627.000i	0.637843	+	0.637843i	−0.312180	+	0.950023i	$0.601059\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	486.000	−	486.000i	0.492401	−	0.492401i
$988$		0			0
$989$		−	162.000i		−	0.163802i
$990$		0			0
$991$	1168.00			1.17861			0.589304	−	0.807912i	$-0.299402\pi$
$991$	1168.00			1.17861			0.589304	+	0.807912i	$0.299402\pi$
$992$		0			0
$993$	−444.000	−	444.000i	−0.447130	−	0.447130i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−108.000	+	108.000i	−0.108325	+	0.108325i	−0.759192	−	0.650867i	$-0.774406\pi$
$997$	−108.000	+	108.000i	−0.108325	+	0.108325i	0.650867	+	0.759192i	$0.274406\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.3.p.a.193.1		2
4.3	odd	2		50.3.c.a.43.1	yes	2
5.2	odd	4	inner	400.3.p.a.257.1		2
5.3	odd	4		400.3.p.g.257.1		2
5.4	even	2		400.3.p.g.193.1		2
12.11	even	2		450.3.g.e.343.1		2
20.3	even	4		50.3.c.b.7.1	yes	2
20.7	even	4		50.3.c.a.7.1	✓	2
20.19	odd	2		50.3.c.b.43.1	yes	2
60.23	odd	4		450.3.g.c.307.1		2
60.47	odd	4		450.3.g.e.307.1		2
60.59	even	2		450.3.g.c.343.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.3.c.a.7.1	✓	2	20.7	even	4
50.3.c.a.43.1	yes	2	4.3	odd	2
50.3.c.b.7.1	yes	2	20.3	even	4
50.3.c.b.43.1	yes	2	20.19	odd	2
400.3.p.a.193.1		2	1.1	even	1	trivial
400.3.p.a.257.1		2	5.2	odd	4	inner
400.3.p.g.193.1		2	5.4	even	2
400.3.p.g.257.1		2	5.3	odd	4
450.3.g.c.307.1		2	60.23	odd	4
450.3.g.c.343.1		2	60.59	even	2
450.3.g.e.307.1		2	60.47	odd	4
450.3.g.e.343.1		2	12.11	even	2

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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	400.p (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-3.00000 - 3.00000i) q^{3} +(3.00000 - 3.00000i) q^{7} +9.00000i q^{9} +O(q^{10})\)	\(q+(-3.00000 - 3.00000i) q^{3} +(3.00000 - 3.00000i) q^{7} +9.00000i q^{9} -12.0000 q^{11} +(-12.0000 - 12.0000i) q^{13} +(12.0000 - 12.0000i) q^{17} +20.0000i q^{19} -18.0000 q^{21} +(-3.00000 - 3.00000i) q^{23} +30.0000i q^{29} +8.00000 q^{31} +(36.0000 + 36.0000i) q^{33} +(-48.0000 + 48.0000i) q^{37} +72.0000i q^{39} -48.0000 q^{41} +(27.0000 + 27.0000i) q^{43} +(-27.0000 + 27.0000i) q^{47} +31.0000i q^{49} -72.0000 q^{51} +(-12.0000 - 12.0000i) q^{53} +(60.0000 - 60.0000i) q^{57} -60.0000i q^{59} +32.0000 q^{61} +(27.0000 + 27.0000i) q^{63} +(3.00000 - 3.00000i) q^{67} +18.0000i q^{69} +48.0000 q^{71} +(-12.0000 - 12.0000i) q^{73} +(-36.0000 + 36.0000i) q^{77} +40.0000i q^{79} +81.0000 q^{81} +(-93.0000 - 93.0000i) q^{83} +(90.0000 - 90.0000i) q^{87} -30.0000i q^{89} -72.0000 q^{91} +(-24.0000 - 24.0000i) q^{93} +(12.0000 - 12.0000i) q^{97} -108.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} + 6 q^{7}+O(q^{10}) \) 2 * q - 6 * q^3 + 6 * q^7	\( 2 q - 6 q^{3} + 6 q^{7} - 24 q^{11} - 24 q^{13} + 24 q^{17} - 36 q^{21} - 6 q^{23} + 16 q^{31} + 72 q^{33} - 96 q^{37} - 96 q^{41} + 54 q^{43} - 54 q^{47} - 144 q^{51} - 24 q^{53} + 120 q^{57} + 64 q^{61} + 54 q^{63} + 6 q^{67} + 96 q^{71} - 24 q^{73} - 72 q^{77} + 162 q^{81} - 186 q^{83} + 180 q^{87} - 144 q^{91} - 48 q^{93} + 24 q^{97}+O(q^{100}) \) 2 * q - 6 * q^3 + 6 * q^7 - 24 * q^11 - 24 * q^13 + 24 * q^17 - 36 * q^21 - 6 * q^23 + 16 * q^31 + 72 * q^33 - 96 * q^37 - 96 * q^41 + 54 * q^43 - 54 * q^47 - 144 * q^51 - 24 * q^53 + 120 * q^57 + 64 * q^61 + 54 * q^63 + 6 * q^67 + 96 * q^71 - 24 * q^73 - 72 * q^77 + 162 * q^81 - 186 * q^83 + 180 * q^87 - 144 * q^91 - 48 * q^93 + 24 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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