LMFDB - Embedded newform 804.2.g.c.401.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [804,2,Mod(401,804)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(804, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("804.401");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 804 = 2^{2} \cdot 3 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	804.g (of order $2$, degree $1$, 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:	$6.41997232251$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{14} - x^{12} - 27x^{10} + 88x^{8} - 243x^{6} - 81x^{4} - 729x^{2} + 6561 $ x^16 - x^14 - x^12 - 27x^10 + 88x^8 - 243x^6 - 81x^4 - 729*x^2 + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4}\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			401.9
Root			$0.673707 - 1.59566i$ of defining polynomial
Character	$\chi$	$=$	804.401
Dual form			804.2.g.c.401.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.673707 - 1.59566i) q^{3} -0.796732 q^{5} -3.71051i q^{7} +(-2.09224 - 2.15001i) q^{9} +O(q^{10})$	\(q+(0.673707 - 1.59566i) q^{3} -0.796732 q^{5} -3.71051i q^{7} +(-2.09224 - 2.15001i) q^{9} -4.36304 q^{11} -0.348117i q^{13} +(-0.536763 + 1.27131i) q^{15} +4.57737i q^{17} +0.708308 q^{19} +(-5.92069 - 2.49979i) q^{21} -0.921525i q^{23} -4.36522 q^{25} +(-4.84023 + 1.89002i) q^{27} +2.31211i q^{29} -2.21649i q^{31} +(-2.93941 + 6.96191i) q^{33} +2.95628i q^{35} -2.29169 q^{37} +(-0.555476 - 0.234529i) q^{39} +10.7955 q^{41} -7.03492i q^{43} +(1.66695 + 1.71298i) q^{45} -8.87739i q^{47} -6.76786 q^{49} +(7.30391 + 3.08381i) q^{51} -7.46157 q^{53} +3.47617 q^{55} +(0.477192 - 1.13022i) q^{57} -3.18679i q^{59} -6.17358i q^{61} +(-7.97762 + 7.76327i) q^{63} +0.277356i q^{65} +(-4.18448 - 7.03492i) q^{67} +(-1.47044 - 0.620838i) q^{69} +9.57696i q^{71} +0.110952 q^{73} +(-2.94088 + 6.96539i) q^{75} +16.1891i q^{77} +1.45669i q^{79} +(-0.245071 + 8.99666i) q^{81} -6.66753i q^{83} -3.64694i q^{85} +(3.68933 + 1.55768i) q^{87} -11.8034i q^{89} -1.29169 q^{91} +(-3.53676 - 1.49327i) q^{93} -0.564331 q^{95} -9.35296i q^{97} +(9.12852 + 9.38057i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 2 q^{9}+O(q^{10}) $ 16 * q + 2 * q^9	$ 16 q + 2 q^{9} + 18 q^{15} + 28 q^{19} - 16 q^{21} + 14 q^{33} - 20 q^{37} - 4 q^{49} - 32 q^{55} + 4 q^{67} - 16 q^{73} + 6 q^{81} - 4 q^{91} - 30 q^{93}+O(q^{100}) $ 16 * q + 2 * q^9 + 18 * q^15 + 28 * q^19 - 16 * q^21 + 14 * q^33 - 20 * q^37 - 4 * q^49 - 32 * q^55 + 4 * q^67 - 16 * q^73 + 6 * q^81 - 4 * q^91 - 30 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/804\mathbb{Z}\right)^\times$.

$n$	$269$	$337$	$403$
$\chi(n)$	$-1$	$-1$	$1$

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
$3$	0.673707	−	1.59566i	0.388965	−	0.921253i
$3$	0.673707	−	1.59566i	0.388965	−	0.921253i
$4$		0			0
$5$	−0.796732			−0.356309			−0.178155	−	0.984003i	$-0.557013\pi$
$5$	−0.796732			−0.356309			−0.178155	+	0.984003i	$0.557013\pi$
$6$		0			0
$7$		−	3.71051i		−	1.40244i	−0.712945	−	0.701220i	$-0.752639\pi$
$7$		−	3.71051i		−	1.40244i	0.712945	−	0.701220i	$-0.247361\pi$
$8$		0			0
$9$	−2.09224	−	2.15001i	−0.697413	−	0.716669i
$10$		0			0
$11$	−4.36304			−1.31551			−0.657753	−	0.753234i	$-0.728493\pi$
$11$	−4.36304			−1.31551			−0.657753	+	0.753234i	$0.728493\pi$
$12$		0			0
$13$		−	0.348117i		−	0.0965504i	−0.998834	−	0.0482752i	$-0.984628\pi$
$13$		−	0.348117i		−	0.0965504i	0.998834	−	0.0482752i	$-0.0153724\pi$
$14$		0			0
$15$	−0.536763	+	1.27131i	−0.138592	+	0.328251i
$16$		0			0
$17$			4.57737i			1.11018i	0.831792	+	0.555088i	$0.187315\pi$
$17$			4.57737i			1.11018i	−0.831792	+	0.555088i	$0.812685\pi$
$18$		0			0
$19$	0.708308			0.162497			0.0812485	−	0.996694i	$-0.474109\pi$
$19$	0.708308			0.162497			0.0812485	+	0.996694i	$0.474109\pi$
$20$		0			0
$21$	−5.92069	−	2.49979i	−1.29200	−	0.545499i
$22$		0			0
$23$		−	0.921525i		−	0.192151i	−0.995374	−	0.0960757i	$-0.969371\pi$
$23$		−	0.921525i		−	0.192151i	0.995374	−	0.0960757i	$-0.0306291\pi$
$24$		0			0
$25$	−4.36522			−0.873044
$26$		0			0
$27$	−4.84023	+	1.89002i	−0.931503	+	0.363735i
$28$		0			0
$29$			2.31211i			0.429348i	0.976686	+	0.214674i	$0.0688689\pi$
$29$			2.31211i			0.429348i	−0.976686	+	0.214674i	$0.931131\pi$
$30$		0			0
$31$		−	2.21649i		−	0.398094i	−0.979990	−	0.199047i	$-0.936215\pi$
$31$		−	2.21649i		−	0.398094i	0.979990	−	0.199047i	$-0.0637847\pi$
$32$		0			0
$33$	−2.93941	+	6.96191i	−0.511685	+	1.21191i
$34$		0			0
$35$			2.95628i			0.499702i
$36$		0			0
$37$	−2.29169			−0.376752			−0.188376	−	0.982097i	$-0.560322\pi$
$37$	−2.29169			−0.376752			−0.188376	+	0.982097i	$0.560322\pi$
$38$		0			0
$39$	−0.555476	−	0.234529i	−0.0889473	−	0.0375547i
$40$		0			0
$41$	10.7955			1.68597			0.842985	−	0.537937i	$-0.180796\pi$
$41$	10.7955			1.68597			0.842985	+	0.537937i	$0.180796\pi$
$42$		0			0
$43$		−	7.03492i		−	1.07282i	−0.843959	−	0.536408i	$-0.819781\pi$
$43$		−	7.03492i		−	1.07282i	0.843959	−	0.536408i	$-0.180219\pi$
$44$		0			0
$45$	1.66695	+	1.71298i	0.248495	+	0.255356i
$46$		0			0
$47$		−	8.87739i		−	1.29490i	−0.762108	−	0.647450i	$-0.775835\pi$
$47$		−	8.87739i		−	1.29490i	0.762108	−	0.647450i	$-0.224165\pi$
$48$		0			0
$49$	−6.76786			−0.966838
$50$		0			0
$51$	7.30391	+	3.08381i	1.02275	+	0.431819i
$52$		0			0
$53$	−7.46157			−1.02492			−0.512462	−	0.858710i	$-0.671267\pi$
$53$	−7.46157			−1.02492			−0.512462	+	0.858710i	$0.671267\pi$
$54$		0			0
$55$	3.47617			0.468727
$56$		0			0
$57$	0.477192	−	1.13022i	0.0632056	−	0.149701i
$58$		0			0
$59$		−	3.18679i		−	0.414885i	−0.978247	−	0.207442i	$-0.933486\pi$
$59$		−	3.18679i		−	0.414885i	0.978247	−	0.207442i	$-0.0665139\pi$
$60$		0			0
$61$		−	6.17358i		−	0.790445i	−0.918585	−	0.395223i	$-0.870667\pi$
$61$		−	6.17358i		−	0.790445i	0.918585	−	0.395223i	$-0.129333\pi$
$62$		0			0
$63$	−7.97762	+	7.76327i	−1.00509	+	0.978080i
$64$		0			0
$65$			0.277356i			0.0344018i
$66$		0			0
$67$	−4.18448	−	7.03492i	−0.511215	−	0.859453i
$67$	−4.18448	−	7.03492i	−0.511215	−	0.859453i
$68$		0			0
$69$	−1.47044	−	0.620838i	−0.177020	−	0.0747401i
$70$		0			0
$71$			9.57696i			1.13658i	0.822830	+	0.568288i	$0.192394\pi$
$71$			9.57696i			1.13658i	−0.822830	+	0.568288i	$0.807606\pi$
$72$		0			0
$73$	0.110952			0.0129859			0.00649296	−	0.999979i	$-0.497933\pi$
$73$	0.110952			0.0129859			0.00649296	+	0.999979i	$0.497933\pi$
$74$		0			0
$75$	−2.94088	+	6.96539i	−0.339583	+	0.804294i
$76$		0			0
$77$			16.1891i			1.84492i
$78$		0			0
$79$			1.45669i			0.163890i	0.996637	+	0.0819452i	$0.0261132\pi$
$79$			1.45669i			0.163890i	−0.996637	+	0.0819452i	$0.973887\pi$
$80$		0			0
$81$	−0.245071	+	8.99666i	−0.0272301	+	0.999629i
$82$		0			0
$83$		−	6.66753i		−	0.731856i	−0.930643	−	0.365928i	$-0.880752\pi$
$83$		−	6.66753i		−	0.731856i	0.930643	−	0.365928i	$-0.119248\pi$
$84$		0			0
$85$		−	3.64694i		−	0.395566i
$86$		0			0
$87$	3.68933	+	1.55768i	0.395538	+	0.167001i
$88$		0			0
$89$		−	11.8034i		−	1.25116i	−0.780160	−	0.625580i	$-0.784863\pi$
$89$		−	11.8034i		−	1.25116i	0.780160	−	0.625580i	$-0.215137\pi$
$90$		0			0
$91$	−1.29169			−0.135406
$92$		0			0
$93$	−3.53676	−	1.49327i	−0.366745	−	0.154845i
$94$		0			0
$95$	−0.564331			−0.0578992
$96$		0			0
$97$		−	9.35296i		−	0.949649i	−0.880080	−	0.474825i	$-0.842511\pi$
$97$		−	9.35296i		−	0.949649i	0.880080	−	0.474825i	$-0.157489\pi$
$98$		0			0
$99$	9.12852	+	9.38057i	0.917451	+	0.942783i
$100$		0			0
$101$	9.29041			0.924430			0.462215	−	0.886768i	$-0.347055\pi$
$101$	9.29041			0.924430			0.462215	+	0.886768i	$0.347055\pi$
$102$		0			0
$103$	9.13308			0.899909			0.449955	−	0.893051i	$-0.351440\pi$
$103$	9.13308			0.899909			0.449955	+	0.893051i	$0.351440\pi$
$104$		0			0
$105$	4.71720	+	1.99166i	0.460352	+	0.194366i
$106$		0			0
$107$		−	1.80424i		−	0.174423i	−0.996190	−	0.0872113i	$-0.972204\pi$
$107$		−	1.80424i		−	0.174423i	0.996190	−	0.0872113i	$-0.0277955\pi$
$108$		0			0
$109$			6.62324i			0.634391i	0.948360	+	0.317195i	$0.102741\pi$
$109$			6.62324i			0.634391i	−0.948360	+	0.317195i	$0.897259\pi$
$110$		0			0
$111$	−1.54393	+	3.65675i	−0.146543	+	0.347084i
$112$		0			0
$113$	−18.7248			−1.76148			−0.880741	−	0.473598i	$-0.842955\pi$
$113$	−18.7248			−1.76148			−0.880741	+	0.473598i	$0.842955\pi$
$114$		0			0
$115$			0.734209i			0.0684653i
$116$		0			0
$117$	−0.748455	+	0.728345i	−0.0691947	+	0.0673355i
$118$		0			0
$119$	16.9844			1.55695
$120$		0			0
$121$	8.03610			0.730555
$122$		0			0
$123$	7.27298	−	17.2259i	0.655782	−	1.55320i
$124$		0			0
$125$	7.46157			0.667383
$126$		0			0
$127$	17.3550			1.54001			0.770003	−	0.638040i	$-0.220255\pi$
$127$	17.3550			1.54001			0.770003	+	0.638040i	$0.220255\pi$
$128$		0			0
$129$	−11.2253	−	4.73947i	−0.988334	−	0.417287i
$130$		0			0
$131$			4.99103i			0.436068i	0.975941	+	0.218034i	$0.0699644\pi$
$131$			4.99103i			0.436068i	−0.975941	+	0.218034i	$0.930036\pi$
$132$		0			0
$133$		−	2.62818i		−	0.227892i
$134$		0			0
$135$	3.85636	−	1.50584i	0.331903	−	0.129602i
$136$		0			0
$137$	−17.7811			−1.51914			−0.759571	−	0.650424i	$-0.774591\pi$
$137$	−17.7811			−1.51914			−0.759571	+	0.650424i	$0.774591\pi$
$138$		0			0
$139$		−	0.861348i		−	0.0730585i	−0.999333	−	0.0365293i	$-0.988370\pi$
$139$		−	0.861348i		−	0.0730585i	0.999333	−	0.0365293i	$-0.0116302\pi$
$140$		0			0
$141$	−14.1653	−	5.98075i	−1.19293	−	0.503670i
$142$		0			0
$143$			1.51885i			0.127013i
$144$		0			0
$145$		−	1.84213i		−	0.152981i
$146$		0			0
$147$	−4.55955	+	10.7992i	−0.376066	+	0.890702i
$148$		0			0
$149$		−	11.6974i		−	0.958286i	−0.877737	−	0.479143i	$-0.840948\pi$
$149$		−	11.6974i		−	0.958286i	0.877737	−	0.479143i	$-0.159052\pi$
$150$		0			0
$151$	9.84513			0.801185			0.400593	−	0.916256i	$-0.368804\pi$
$151$	9.84513			0.801185			0.400593	+	0.916256i	$0.368804\pi$
$152$		0			0
$153$	9.84139	−	9.57696i	0.795629	−	0.774251i
$154$		0			0
$155$			1.76595i			0.141845i
$156$		0			0
$157$	−4.65691			−0.371662			−0.185831	−	0.982582i	$-0.559498\pi$
$157$	−4.65691			−0.371662			−0.185831	+	0.982582i	$0.559498\pi$
$158$		0			0
$159$	−5.02691	+	11.9061i	−0.398660	+	0.944215i
$160$		0			0
$161$	−3.41933			−0.269481
$162$		0			0
$163$	22.9782			1.79979			0.899896	−	0.436104i	$-0.143642\pi$
$163$	22.9782			1.79979			0.899896	+	0.436104i	$0.143642\pi$
$164$		0			0
$165$	2.34192	−	5.54677i	0.182318	−	0.431816i
$166$		0			0
$167$		−	12.3884i		−	0.958642i	−0.877640	−	0.479321i	$-0.840883\pi$
$167$		−	12.3884i		−	0.958642i	0.877640	−	0.479321i	$-0.159117\pi$
$168$		0			0
$169$	12.8788			0.990678
$170$		0			0
$171$	−1.48195	−	1.52287i	−0.113327	−	0.116457i
$172$		0			0
$173$		−	19.7895i		−	1.50457i	−0.658838	−	0.752285i	$-0.728952\pi$
$173$		−	19.7895i		−	1.50457i	0.658838	−	0.752285i	$-0.271048\pi$
$174$		0			0
$175$			16.1972i			1.22439i
$176$		0			0
$177$	−5.08502	−	2.14696i	−0.382214	−	0.161375i
$178$		0			0
$179$	−5.07137			−0.379052			−0.189526	−	0.981876i	$-0.560695\pi$
$179$	−5.07137			−0.379052			−0.189526	+	0.981876i	$0.560695\pi$
$180$		0			0
$181$	4.74441			0.352649			0.176325	−	0.984332i	$-0.443579\pi$
$181$	4.74441			0.352649			0.176325	+	0.984332i	$0.443579\pi$
$182$		0			0
$183$	−9.85091	−	4.15918i	−0.728200	−	0.307455i
$184$		0			0
$185$	1.82586			0.134240
$186$		0			0
$187$		−	19.9713i		−	1.46044i
$188$		0			0
$189$	7.01293	+	17.9597i	0.510116	+	1.30638i
$190$		0			0
$191$	1.34993			0.0976776			0.0488388	−	0.998807i	$-0.484448\pi$
$191$	1.34993			0.0976776			0.0488388	+	0.998807i	$0.484448\pi$
$192$		0			0
$193$	11.9898			0.863042			0.431521	−	0.902103i	$-0.357977\pi$
$193$	11.9898			0.863042			0.431521	+	0.902103i	$0.357977\pi$
$194$		0			0
$195$	0.442565	+	0.186857i	0.0316927	+	0.0133811i
$196$		0			0
$197$	7.46157			0.531615			0.265807	−	0.964026i	$-0.414362\pi$
$197$	7.46157			0.531615			0.265807	+	0.964026i	$0.414362\pi$
$198$		0			0
$199$	20.7563			1.47138			0.735688	−	0.677321i	$-0.236859\pi$
$199$	20.7563			1.47138			0.735688	+	0.677321i	$0.236859\pi$
$200$		0			0
$201$	−14.0444	+	1.93752i	−0.990618	+	0.136662i
$202$		0			0
$203$	8.57910			0.602135
$204$		0			0
$205$	−8.60109			−0.600726
$206$		0			0
$207$	−1.98129	+	1.92805i	−0.137709	+	0.134009i
$208$		0			0
$209$	−3.09037			−0.213766
$210$		0			0
$211$	−22.9149			−1.57753			−0.788764	−	0.614696i	$-0.789279\pi$
$211$	−22.9149			−1.57753			−0.788764	+	0.614696i	$0.789279\pi$
$212$		0			0
$213$	15.2815	+	6.45206i	1.04707	+	0.442088i
$214$		0			0
$215$			5.60495i			0.382254i
$216$		0			0
$217$	−8.22432			−0.558303
$218$		0			0
$219$	0.0747489	−	0.177041i	0.00505106	−	0.0119633i
$220$		0			0
$221$	1.59346			0.107188
$222$		0			0
$223$	25.2464			1.69063			0.845314	−	0.534271i	$-0.179414\pi$
$223$	25.2464			1.69063			0.845314	+	0.534271i	$0.179414\pi$
$224$		0			0
$225$	9.13308	+	9.38526i	0.608872	+	0.625684i
$226$		0			0
$227$			26.7344i			1.77443i	0.461361	+	0.887213i	$0.347361\pi$
$227$			26.7344i			1.77443i	−0.461361	+	0.887213i	$0.652639\pi$
$228$		0			0
$229$			22.9475i			1.51642i	0.652012	+	0.758208i	$0.273925\pi$
$229$			22.9475i			1.51642i	−0.652012	+	0.758208i	$0.726075\pi$
$230$		0			0
$231$	25.8322	+	10.9067i	1.69964	+	0.717608i
$232$		0			0
$233$	20.5507			1.34632			0.673160	−	0.739497i	$-0.264937\pi$
$233$	20.5507			1.34632			0.673160	+	0.739497i	$0.264937\pi$
$234$		0			0
$235$			7.07290i			0.461385i
$236$		0			0
$237$	2.32438	+	0.981381i	0.150984	+	0.0637476i
$238$		0			0
$239$	−3.33391			−0.215652			−0.107826	−	0.994170i	$-0.534389\pi$
$239$	−3.33391			−0.215652			−0.107826	+	0.994170i	$0.534389\pi$
$240$		0			0
$241$	−21.6055			−1.39173			−0.695867	−	0.718171i	$-0.744980\pi$
$241$	−21.6055			−1.39173			−0.695867	+	0.718171i	$0.744980\pi$
$242$		0			0
$243$	14.1905	+	6.45216i	0.910320	+	0.413906i
$244$		0			0
$245$	5.39217			0.344493
$246$		0			0
$247$		−	0.246574i		−	0.0156891i
$248$		0			0
$249$	−10.6391	−	4.49195i	−0.674224	−	0.284666i
$250$		0			0
$251$	−14.9231			−0.941940			−0.470970	−	0.882149i	$-0.656096\pi$
$251$	−14.9231			−0.941940			−0.470970	+	0.882149i	$0.656096\pi$
$252$		0			0
$253$			4.02065i			0.252776i
$254$		0			0
$255$	−5.81926	−	2.45697i	−0.364416	−	0.153861i
$256$		0			0
$257$		−	8.24177i		−	0.514108i	−0.966397	−	0.257054i	$-0.917248\pi$
$257$		−	8.24177i		−	0.514108i	0.966397	−	0.257054i	$-0.0827518\pi$
$258$		0			0
$259$			8.50334i			0.528372i
$260$		0			0
$261$	4.97105	−	4.83748i	0.307700	−	0.299433i
$262$		0			0
$263$		−	6.66753i		−	0.411137i	−0.978643	−	0.205569i	$-0.934096\pi$
$263$		−	6.66753i		−	0.411137i	0.978643	−	0.205569i	$-0.0659044\pi$
$264$		0			0
$265$	5.94487			0.365190
$266$		0			0
$267$	−18.8342	−	7.95204i	−1.15263	−	0.486657i
$268$		0			0
$269$			8.41637i			0.513155i	0.966524	+	0.256577i	$0.0825949\pi$
$269$			8.41637i			0.513155i	−0.966524	+	0.256577i	$0.917405\pi$
$270$		0			0
$271$			19.7247i			1.19819i	0.800678	+	0.599094i	$0.204473\pi$
$271$			19.7247i			1.19819i	−0.800678	+	0.599094i	$0.795527\pi$
$272$		0			0
$273$	−0.870221	+	2.06110i	−0.0526682	+	0.124743i
$274$		0			0
$275$	19.0456			1.14849
$276$		0			0
$277$	−6.91624			−0.415557			−0.207778	−	0.978176i	$-0.566623\pi$
$277$	−6.91624			−0.415557			−0.207778	+	0.978176i	$0.566623\pi$
$278$		0			0
$279$	−4.76548	+	4.63744i	−0.285302	+	0.277636i
$280$		0			0
$281$	13.9444			0.831855			0.415927	−	0.909398i	$-0.363457\pi$
$281$	13.9444			0.831855			0.415927	+	0.909398i	$0.363457\pi$
$282$		0			0
$283$	−5.25427			−0.312334			−0.156167	−	0.987731i	$-0.549914\pi$
$283$	−5.25427			−0.312334			−0.156167	+	0.987731i	$0.549914\pi$
$284$		0			0
$285$	−0.380194	+	0.900479i	−0.0225207	+	0.0533398i
$286$		0			0
$287$		−	40.0567i		−	2.36447i
$288$		0			0
$289$	−3.95234			−0.232491
$290$		0			0
$291$	−14.9241	−	6.30115i	−0.874867	−	0.369380i
$292$		0			0
$293$			27.6683i			1.61640i	0.588909	+	0.808200i	$0.299558\pi$
$293$			27.6683i			1.61640i	−0.588909	+	0.808200i	$0.700442\pi$
$294$		0			0
$295$			2.53902i			0.147827i
$296$		0			0
$297$	21.1181	−	8.24623i	1.22540	−	0.478495i
$298$		0			0
$299$	−0.320799			−0.0185523
$300$		0			0
$301$	−26.1031			−1.50456
$302$		0			0
$303$	6.25901	−	14.8243i	0.359571	−	0.851634i
$304$		0			0
$305$			4.91868i			0.281643i
$306$		0			0
$307$	−16.1191			−0.919966			−0.459983	−	0.887928i	$-0.652145\pi$
$307$	−16.1191			−0.919966			−0.459983	+	0.887928i	$0.652145\pi$
$308$		0			0
$309$	6.15302	−	14.5733i	0.350033	−	0.829044i
$310$		0			0
$311$	−10.6105			−0.601667			−0.300834	−	0.953677i	$-0.597265\pi$
$311$	−10.6105			−0.601667			−0.300834	+	0.953677i	$0.597265\pi$
$312$		0			0
$313$		−	21.0156i		−	1.18787i	−0.804512	−	0.593936i	$-0.797573\pi$
$313$		−	21.0156i		−	1.18787i	0.804512	−	0.593936i	$-0.202427\pi$
$314$		0			0
$315$	6.35602	−	6.18524i	0.358121	−	0.348499i
$316$		0			0
$317$			25.1053i			1.41005i	0.709181	+	0.705027i	$0.249065\pi$
$317$			25.1053i			1.41005i	−0.709181	+	0.705027i	$0.750935\pi$
$318$		0			0
$319$		−	10.0878i		−	0.564809i
$320$		0			0
$321$	−2.87895	−	1.21553i	−0.160687	−	0.0678442i
$322$		0			0
$323$			3.24219i			0.180400i
$324$		0			0
$325$			1.51961i			0.0842927i
$326$		0			0
$327$	10.5684	+	4.46212i	0.584434	+	0.246756i
$328$		0			0
$329$	−32.9396			−1.81602
$330$		0			0
$331$		−	17.3051i		−	0.951174i	−0.879669	−	0.475587i	$-0.842236\pi$
$331$		−	17.3051i		−	0.951174i	0.879669	−	0.475587i	$-0.157764\pi$
$332$		0			0
$333$	4.79477	+	4.92716i	0.262752	+	0.270007i
$334$		0			0
$335$	3.33391	+	5.60495i	0.182151	+	0.306231i
$336$		0			0
$337$		−	21.5412i		−	1.17342i	−0.809795	−	0.586712i	$-0.800422\pi$
$337$		−	21.5412i		−	1.17342i	0.809795	−	0.586712i	$-0.199578\pi$
$338$		0			0
$339$	−12.6150	+	29.8784i	−0.685154	+	1.62277i
$340$		0			0
$341$			9.67065i			0.523695i
$342$		0			0
$343$		−	0.861348i		−	0.0465084i
$344$		0			0
$345$	1.17154	+	0.494641i	0.0630738	+	0.0266306i
$346$		0			0
$347$	−23.5825			−1.26597			−0.632987	−	0.774163i	$-0.718171\pi$
$347$	−23.5825			−1.26597			−0.632987	+	0.774163i	$0.718171\pi$
$348$		0			0
$349$	−14.7703			−0.790635			−0.395317	−	0.918545i	$-0.629365\pi$
$349$	−14.7703			−0.790635			−0.395317	+	0.918545i	$0.629365\pi$
$350$		0			0
$351$	0.657949	+	1.68497i	0.0351187	+	0.0899370i
$352$		0			0
$353$	−21.4060			−1.13933			−0.569663	−	0.821878i	$-0.692926\pi$
$353$	−21.4060			−1.13933			−0.569663	+	0.821878i	$0.692926\pi$
$354$		0			0
$355$		−	7.63027i		−	0.404972i
$356$		0			0
$357$	11.4425	−	27.1012i	0.605600	−	1.43435i
$358$		0			0
$359$		−	6.85923i		−	0.362016i	−0.983482	−	0.181008i	$-0.942064\pi$
$359$		−	6.85923i		−	0.362016i	0.983482	−	0.181008i	$-0.0579360\pi$
$360$		0			0
$361$	−18.4983			−0.973595
$362$		0			0
$363$	5.41397	−	12.8229i	0.284160	−	0.673025i
$364$		0			0
$365$	−0.0883987			−0.00462700
$366$		0			0
$367$			2.91338i			0.152077i	0.997105	+	0.0760386i	$0.0242272\pi$
$367$			2.91338i			0.152077i	−0.997105	+	0.0760386i	$0.975773\pi$
$368$		0			0
$369$	−22.5867	−	23.2104i	−1.17582	−	1.20828i
$370$		0			0
$371$			27.6862i			1.43740i
$372$		0			0
$373$			30.9750i			1.60382i	0.597442	+	0.801912i	$0.296184\pi$
$373$			30.9750i			1.60382i	−0.597442	+	0.801912i	$0.703816\pi$
$374$		0			0
$375$	5.02691	−	11.9061i	0.259588	−	0.614828i
$376$		0			0
$377$	0.804885			0.0414537
$378$		0			0
$379$		−	26.8534i		−	1.37937i	−0.724110	−	0.689684i	$-0.757749\pi$
$379$		−	26.8534i		−	1.37937i	0.724110	−	0.689684i	$-0.242251\pi$
$380$		0			0
$381$	11.6922	−	27.6926i	0.599008	−	1.41873i
$382$		0			0
$383$	24.9248			1.27360			0.636800	−	0.771029i	$-0.280258\pi$
$383$	24.9248			1.27360			0.636800	+	0.771029i	$0.280258\pi$
$384$		0			0
$385$		−	12.8984i		−	0.657361i
$386$		0			0
$387$	−15.1251	+	14.7187i	−0.768854	+	0.748196i
$388$		0			0
$389$		−	12.2725i		−	0.622239i	−0.950371	−	0.311119i	$-0.899296\pi$
$389$		−	12.2725i		−	0.622239i	0.950371	−	0.311119i	$-0.100704\pi$
$390$		0			0
$391$	4.21817			0.213322
$392$		0			0
$393$	7.96397	+	3.36249i	0.401729	+	0.169615i
$394$		0			0
$395$		−	1.16059i		−	0.0583957i
$396$		0			0
$397$	−25.6528			−1.28748			−0.643739	−	0.765245i	$-0.722618\pi$
$397$	−25.6528			−1.28748			−0.643739	+	0.765245i	$0.722618\pi$
$398$		0			0
$399$	−4.19367	−	1.77062i	−0.209946	−	0.0886420i
$400$		0			0
$401$	−5.96466			−0.297861			−0.148930	−	0.988848i	$-0.547583\pi$
$401$	−5.96466			−0.297861			−0.148930	+	0.988848i	$0.547583\pi$
$402$		0			0
$403$	−0.771600			−0.0384361
$404$		0			0
$405$	0.195256	−	7.16793i	0.00970234	−	0.356177i
$406$		0			0
$407$	9.99874			0.495619
$408$		0			0
$409$			1.72270i			0.0851818i	0.999093	+	0.0425909i	$0.0135612\pi$
$409$			1.72270i			0.0851818i	−0.999093	+	0.0425909i	$0.986439\pi$
$410$		0			0
$411$	−11.9792	+	28.3725i	−0.590893	+	1.39951i
$412$		0			0
$413$	−11.8246			−0.581851
$414$		0			0
$415$			5.31223i			0.260767i
$416$		0			0
$417$	−1.37441	−	0.580295i	−0.0673054	−	0.0284172i
$418$		0			0
$419$		−	21.7220i		−	1.06119i	−0.847625	−	0.530596i	$-0.821968\pi$
$419$		−	21.7220i		−	1.06119i	0.847625	−	0.530596i	$-0.178032\pi$
$420$		0			0
$421$	28.3587			1.38212			0.691060	−	0.722798i	$-0.257144\pi$
$421$	28.3587			1.38212			0.691060	+	0.722798i	$0.257144\pi$
$422$		0			0
$423$	−19.0865	+	18.5736i	−0.928015	+	0.903080i
$424$		0			0
$425$		−	19.9812i		−	0.969232i
$426$		0			0
$427$	−22.9071			−1.10855
$428$		0			0
$429$	2.42356	+	1.02326i	0.117011	+	0.0494034i
$430$		0			0
$431$		−	8.01507i		−	0.386072i	−0.981192	−	0.193036i	$-0.938167\pi$
$431$		−	8.01507i		−	0.386072i	0.981192	−	0.193036i	$-0.0618334\pi$
$432$		0			0
$433$			21.8017i			1.04772i	0.851804	+	0.523860i	$0.175509\pi$
$433$			21.8017i			1.04772i	−0.851804	+	0.523860i	$0.824491\pi$
$434$		0			0
$435$	−2.93941	−	1.24106i	−0.140934	−	0.0595040i
$436$		0			0
$437$		−	0.652724i		−	0.0312240i
$438$		0			0
$439$	7.16476			0.341955			0.170978	−	0.985275i	$-0.445307\pi$
$439$	7.16476			0.341955			0.170978	+	0.985275i	$0.445307\pi$
$440$		0			0
$441$	14.1600	+	14.5510i	0.674285	+	0.692903i
$442$		0			0
$443$	−21.0564			−1.00042			−0.500211	−	0.865904i	$-0.666744\pi$
$443$	−21.0564			−1.00042			−0.500211	+	0.865904i	$0.666744\pi$
$444$		0			0
$445$			9.40415i			0.445800i
$446$		0			0
$447$	−18.6650	−	7.88059i	−0.882823	−	0.372739i
$448$		0			0
$449$		−	30.2326i		−	1.42677i	−0.700775	−	0.713383i	$-0.747162\pi$
$449$		−	30.2326i		−	1.42677i	0.700775	−	0.713383i	$-0.252838\pi$
$450$		0			0
$451$	−47.1011			−2.21790
$452$		0			0
$453$	6.63273	−	15.7094i	0.311633	−	0.738094i
$454$		0			0
$455$	1.02913			0.0482465
$456$		0			0
$457$	12.5718			0.588085			0.294043	−	0.955792i	$-0.404999\pi$
$457$	12.5718			0.588085			0.294043	+	0.955792i	$0.404999\pi$
$458$		0			0
$459$	−8.65133	−	22.1555i	−0.403809	−	1.03413i
$460$		0			0
$461$		−	25.7580i		−	1.19967i	−0.800124	−	0.599835i	$-0.795233\pi$
$461$		−	25.7580i		−	1.19967i	0.800124	−	0.599835i	$-0.204767\pi$
$462$		0			0
$463$			14.6652i			0.681549i	0.940145	+	0.340775i	$0.110689\pi$
$463$			14.6652i			0.681549i	−0.940145	+	0.340775i	$0.889311\pi$
$464$		0			0
$465$	2.81785	+	1.18973i	0.130675	+	0.0551725i
$466$		0			0
$467$			0.687212i			0.0318004i	0.999874	+	0.0159002i	$0.00506140\pi$
$467$			0.687212i			0.0318004i	−0.999874	+	0.0159002i	$0.994939\pi$
$468$		0			0
$469$	−26.1031	+	15.5265i	−1.20533	+	0.716949i
$470$		0			0
$471$	−3.13739	+	7.43083i	−0.144563	+	0.342395i
$472$		0			0
$473$			30.6936i			1.41129i
$474$		0			0
$475$	−3.09192			−0.141867
$476$		0			0
$477$	15.6114	+	16.0424i	0.714796	+	0.734532i
$478$		0			0
$479$			16.4281i			0.750621i	0.926899	+	0.375311i	$0.122464\pi$
$479$			16.4281i			0.750621i	−0.926899	+	0.375311i	$0.877536\pi$
$480$		0			0
$481$			0.797778i			0.0363755i
$482$		0			0
$483$	−2.30362	+	5.45607i	−0.104818	+	0.248260i
$484$		0			0
$485$			7.45180i			0.338369i
$486$		0			0
$487$		−	21.4915i		−	0.973873i	−0.873437	−	0.486937i	$-0.838114\pi$
$487$		−	21.4915i		−	0.973873i	0.873437	−	0.486937i	$-0.161886\pi$
$488$		0			0
$489$	15.4806	−	36.6653i	0.700055	−	1.65806i
$490$		0			0
$491$			13.9200i			0.628202i	0.949390	+	0.314101i	$0.101703\pi$
$491$			13.9200i			0.628202i	−0.949390	+	0.314101i	$0.898297\pi$
$492$		0			0
$493$	−10.5834			−0.476652
$494$		0			0
$495$	−7.27298	−	7.47380i	−0.326896	−	0.335922i
$496$		0			0
$497$	35.5354			1.59398
$498$		0			0
$499$			23.3855i			1.04688i	0.852063	+	0.523439i	$0.175351\pi$
$499$			23.3855i			1.04688i	−0.852063	+	0.523439i	$0.824649\pi$
$500$		0			0
$501$	−19.7676	−	8.34613i	−0.883151	−	0.372878i
$502$		0			0
$503$	−0.711311			−0.0317158			−0.0158579	−	0.999874i	$-0.505048\pi$
$503$	−0.711311			−0.0317158			−0.0158579	+	0.999874i	$0.505048\pi$
$504$		0			0
$505$	−7.40196			−0.329383
$506$		0			0
$507$	8.67654	−	20.5502i	0.385339	−	0.912665i
$508$		0			0
$509$			31.5977i			1.40054i	0.713877	+	0.700272i	$0.246938\pi$
$509$			31.5977i			1.40054i	−0.713877	+	0.700272i	$0.753062\pi$
$510$		0			0
$511$		−	0.411687i		−	0.0182120i
$512$		0			0
$513$	−3.42837	+	1.33872i	−0.151366	+	0.0591058i
$514$		0			0
$515$	−7.27661			−0.320646
$516$		0			0
$517$			38.7324i			1.70345i
$518$		0			0
$519$	−31.5773	−	13.3323i	−1.38609	−	0.585225i
$520$		0			0
$521$	2.06124			0.0903046			0.0451523	−	0.998980i	$-0.485623\pi$
$521$	2.06124			0.0903046			0.0451523	+	0.998980i	$0.485623\pi$
$522$		0			0
$523$	12.9112			0.564566			0.282283	−	0.959331i	$-0.408908\pi$
$523$	12.9112			0.564566			0.282283	+	0.959331i	$0.408908\pi$
$524$		0			0
$525$	25.8451	+	10.9121i	1.12797	+	0.476245i
$526$		0			0
$527$	10.1457			0.441955
$528$		0			0
$529$	22.1508			0.963078
$530$		0			0
$531$	−6.85162	+	6.66753i	−0.297335	+	0.289346i
$532$		0			0
$533$		−	3.75809i		−	0.162781i
$534$		0			0
$535$			1.43750i			0.0621484i
$536$		0			0
$537$	−3.41662	+	8.09217i	−0.147438	+	0.349203i
$538$		0			0
$539$	29.5284			1.27188
$540$		0			0
$541$			23.0864i			0.992563i	0.868162	+	0.496281i	$0.165302\pi$
$541$			23.0864i			0.992563i	−0.868162	+	0.496281i	$0.834698\pi$
$542$		0			0
$543$	3.19634	−	7.57045i	0.137168	−	0.324879i
$544$		0			0
$545$		−	5.27694i		−	0.226039i
$546$		0			0
$547$		−	39.4666i		−	1.68747i	−0.536760	−	0.843735i	$-0.680352\pi$
$547$		−	39.4666i		−	1.68747i	0.536760	−	0.843735i	$-0.319648\pi$
$548$		0			0
$549$	−13.2732	+	12.9166i	−0.566488	+	0.551267i
$550$		0			0
$551$			1.63768i			0.0697677i
$552$		0			0
$553$	5.40506			0.229846
$554$		0			0
$555$	1.23010	−	2.91345i	0.0522147	−	0.123669i
$556$		0			0
$557$		−	24.8748i		−	1.05398i	−0.849872	−	0.526989i	$-0.823321\pi$
$557$		−	24.8748i		−	1.05398i	0.849872	−	0.526989i	$-0.176679\pi$
$558$		0			0
$559$	−2.44898			−0.103581
$560$		0			0
$561$	−31.8673	−	13.4548i	−1.34544	−	0.568060i
$562$		0			0
$563$	32.2015			1.35713			0.678565	−	0.734540i	$-0.262602\pi$
$563$	32.2015			1.35713			0.678565	+	0.734540i	$0.262602\pi$
$564$		0			0
$565$	14.9187			0.627632
$566$		0			0
$567$	33.3822	+	0.909338i	1.40192	+	0.0381886i
$568$		0			0
$569$			40.9314i			1.71593i	0.513707	+	0.857966i	$0.328272\pi$
$569$			40.9314i			1.71593i	−0.513707	+	0.857966i	$0.671728\pi$
$570$		0			0
$571$	−12.7600			−0.533991			−0.266996	−	0.963698i	$-0.586031\pi$
$571$	−12.7600			−0.533991			−0.266996	+	0.963698i	$0.586031\pi$
$572$		0			0
$573$	0.909457	−	2.15403i	0.0379931	−	0.0899857i
$574$		0			0
$575$			4.02266i			0.167757i
$576$		0			0
$577$			24.8014i			1.03250i	0.856439	+	0.516248i	$0.172672\pi$
$577$			24.8014i			1.03250i	−0.856439	+	0.516248i	$0.827328\pi$
$578$		0			0
$579$	8.07758	−	19.1315i	0.335693	−	0.795080i
$580$		0			0
$581$	−24.7399			−1.02638
$582$		0			0
$583$	32.5551			1.34829
$584$		0			0
$585$	0.596318	−	0.580295i	0.0246547	−	0.0239923i
$586$		0			0
$587$	−39.8480			−1.64470			−0.822351	−	0.568980i	$-0.807338\pi$
$587$	−39.8480			−1.64470			−0.822351	+	0.568980i	$0.807338\pi$
$588$		0			0
$589$		−	1.56996i		−	0.0646891i
$590$		0			0
$591$	5.02691	−	11.9061i	0.206779	−	0.489751i
$592$		0			0
$593$	37.7119			1.54864			0.774320	−	0.632794i	$-0.218092\pi$
$593$	37.7119			1.54864			0.774320	+	0.632794i	$0.218092\pi$
$594$		0			0
$595$	−13.5320			−0.554757
$596$		0			0
$597$	13.9837	−	33.1199i	0.572313	−	1.35551i
$598$		0			0
$599$	−0.948887			−0.0387705			−0.0193852	−	0.999812i	$-0.506171\pi$
$599$	−0.948887			−0.0387705			−0.0193852	+	0.999812i	$0.506171\pi$
$600$		0			0
$601$	7.56367			0.308528			0.154264	−	0.988030i	$-0.450699\pi$
$601$	7.56367			0.308528			0.154264	+	0.988030i	$0.450699\pi$
$602$		0			0
$603$	−6.37021	+	23.7154i	−0.259415	+	0.965766i
$604$		0			0
$605$	−6.40262			−0.260303
$606$		0			0
$607$	−24.8349			−1.00802			−0.504009	−	0.863699i	$-0.668142\pi$
$607$	−24.8349			−1.00802			−0.504009	+	0.863699i	$0.668142\pi$
$608$		0			0
$609$	5.77979	−	13.6893i	0.234209	−	0.554718i
$610$		0			0
$611$	−3.09037			−0.125023
$612$		0			0
$613$	−14.4241			−0.582584			−0.291292	−	0.956634i	$-0.594085\pi$
$613$	−14.4241			−0.582584			−0.291292	+	0.956634i	$0.594085\pi$
$614$		0			0
$615$	−5.79461	+	13.7244i	−0.233661	+	0.553421i
$616$		0			0
$617$		−	16.0102i		−	0.644545i	−0.946647	−	0.322273i	$-0.895553\pi$
$617$		−	16.0102i		−	0.644545i	0.946647	−	0.322273i	$-0.104447\pi$
$618$		0			0
$619$	−3.42035			−0.137476			−0.0687378	−	0.997635i	$-0.521897\pi$
$619$	−3.42035			−0.137476			−0.0687378	+	0.997635i	$0.521897\pi$
$620$		0			0
$621$	1.74170	+	4.46040i	0.0698921	+	0.178989i
$622$		0			0
$623$	−43.7966			−1.75468
$624$		0			0
$625$	15.8812			0.635249
$626$		0			0
$627$	−2.08201	+	4.93118i	−0.0831473	+	0.196932i
$628$		0			0
$629$		−	10.4899i		−	0.418261i
$630$		0			0
$631$			12.1883i			0.485207i	0.970126	+	0.242603i	$0.0780013\pi$
$631$			12.1883i			0.485207i	−0.970126	+	0.242603i	$0.921999\pi$
$632$		0			0
$633$	−15.4379	+	36.5643i	−0.613603	+	1.45330i
$634$		0			0
$635$	−13.8273			−0.548718
$636$		0			0
$637$			2.35601i			0.0933486i
$638$		0			0
$639$	20.5905	−	20.0373i	0.814549	−	0.792663i
$640$		0			0
$641$	40.0329			1.58121			0.790603	−	0.612329i	$-0.209767\pi$
$641$	40.0329			1.58121			0.790603	+	0.612329i	$0.209767\pi$
$642$		0			0
$643$	14.5381			0.573328			0.286664	−	0.958031i	$-0.407454\pi$
$643$	14.5381			0.573328			0.286664	+	0.958031i	$0.407454\pi$
$644$		0			0
$645$	8.94357	+	3.77609i	0.352153	+	0.148683i
$646$		0			0
$647$	−36.5141			−1.43552			−0.717758	−	0.696292i	$-0.754832\pi$
$647$	−36.5141			−1.43552			−0.717758	+	0.696292i	$0.754832\pi$
$648$		0			0
$649$			13.9041i			0.545783i
$650$		0			0
$651$	−5.54078	+	13.1232i	−0.217160	+	0.514338i
$652$		0			0
$653$	41.6546			1.63007			0.815034	−	0.579413i	$-0.196718\pi$
$653$	41.6546			1.63007			0.815034	+	0.579413i	$0.196718\pi$
$654$		0			0
$655$		−	3.97651i		−	0.155375i
$656$		0			0
$657$	−0.232137	−	0.238547i	−0.00905654	−	0.00930660i
$658$		0			0
$659$		−	26.4490i		−	1.03031i	−0.857098	−	0.515154i	$-0.827735\pi$
$659$		−	26.4490i		−	1.03031i	0.857098	−	0.515154i	$-0.172265\pi$
$660$		0			0
$661$			43.2296i			1.68144i	0.541473	+	0.840718i	$0.317867\pi$
$661$			43.2296i			1.68144i	−0.541473	+	0.840718i	$0.682133\pi$
$662$		0			0
$663$	1.07353	−	2.54262i	0.0416923	−	0.0987472i
$664$		0			0
$665$			2.09395i			0.0812001i
$666$		0			0
$667$	2.13067			0.0824998
$668$		0			0
$669$	17.0087	−	40.2847i	0.657594	−	1.55749i
$670$		0			0
$671$			26.9355i			1.03984i
$672$		0			0
$673$			32.4447i			1.25065i	0.780364	+	0.625326i	$0.215034\pi$
$673$			32.4447i			1.25065i	−0.780364	+	0.625326i	$0.784966\pi$
$674$		0			0
$675$	21.1287	−	8.25035i	0.813243	−	0.317556i
$676$		0			0
$677$	−27.2852			−1.04866			−0.524328	−	0.851516i	$-0.675684\pi$
$677$	−27.2852			−1.04866			−0.524328	+	0.851516i	$0.675684\pi$
$678$		0			0
$679$	−34.7042			−1.33183
$680$		0			0
$681$	42.6589	+	18.0111i	1.63469	+	0.690189i
$682$		0			0
$683$	28.8676			1.10459			0.552293	−	0.833650i	$-0.313753\pi$
$683$	28.8676			1.10459			0.552293	+	0.833650i	$0.313753\pi$
$684$		0			0
$685$	14.1668			0.541284
$686$		0			0
$687$	36.6164	+	15.4599i	1.39700	+	0.589832i
$688$		0			0
$689$			2.59750i			0.0989569i
$690$		0			0
$691$	−16.0443			−0.610352			−0.305176	−	0.952296i	$-0.598715\pi$
$691$	−16.0443			−0.610352			−0.305176	+	0.952296i	$0.598715\pi$
$692$		0			0
$693$	34.8067	−	33.8714i	1.32220	−	1.28667i
$694$		0			0
$695$			0.686263i			0.0260314i
$696$		0			0
$697$			49.4149i			1.87172i
$698$		0			0
$699$	13.8451	−	32.7918i	0.523671	−	1.24030i
$700$		0			0
$701$	−40.7250			−1.53816			−0.769080	−	0.639152i	$-0.779285\pi$
$701$	−40.7250			−1.53816			−0.769080	+	0.639152i	$0.779285\pi$
$702$		0			0
$703$	−1.62322			−0.0612210
$704$		0			0
$705$	11.2859	+	4.76506i	0.425052	+	0.179462i
$706$		0			0
$707$		−	34.4721i		−	1.29646i
$708$		0			0
$709$	31.6984			1.19046			0.595229	−	0.803556i	$-0.297061\pi$
$709$	31.6984			1.19046			0.595229	+	0.803556i	$0.297061\pi$
$710$		0			0
$711$	3.13190	−	3.04774i	0.117455	−	0.114299i
$712$		0			0
$713$	−2.04256			−0.0764943
$714$		0			0
$715$		−	1.21012i		−	0.0452558i
$716$		0			0
$717$	−2.24607	+	5.31977i	−0.0838812	+	0.198670i
$718$		0			0
$719$			26.7856i			0.998934i	0.866333	+	0.499467i	$0.166471\pi$
$719$			26.7856i			0.998934i	−0.866333	+	0.499467i	$0.833529\pi$
$720$		0			0
$721$		−	33.8884i		−	1.26207i
$722$		0			0
$723$	−14.5558	+	34.4750i	−0.541335	+	1.28214i
$724$		0			0
$725$		−	10.0929i		−	0.374839i
$726$		0			0
$727$		−	11.4050i		−	0.422988i	−0.977379	−	0.211494i	$-0.932167\pi$
$727$		−	11.4050i		−	0.422988i	0.977379	−	0.211494i	$-0.0678328\pi$
$728$		0			0
$729$	19.8556	−	18.2963i	0.735394	−	0.677639i
$730$		0			0
$731$	32.2015			1.19101
$732$		0			0
$733$		−	1.10241i		−	0.0407183i	−0.999793	−	0.0203592i	$-0.993519\pi$
$733$		−	1.10241i		−	0.0407183i	0.999793	−	0.0203592i	$-0.00648097\pi$
$734$		0			0
$735$	3.63274	−	8.60405i	0.133996	−	0.317365i
$736$		0			0
$737$	18.2570	+	30.6936i	0.672507	+	1.13061i
$738$		0			0
$739$		−	33.0270i		−	1.21492i	−0.794351	−	0.607459i	$-0.792189\pi$
$739$		−	33.0270i		−	1.21492i	0.794351	−	0.607459i	$-0.207811\pi$
$740$		0			0
$741$	−0.393448	−	0.166119i	−0.0144537	−	0.00610252i
$742$		0			0
$743$		−	45.6795i		−	1.67582i	−0.545809	−	0.837909i	$-0.683778\pi$
$743$		−	45.6795i		−	1.67582i	0.545809	−	0.837909i	$-0.316222\pi$
$744$		0			0
$745$			9.31966i			0.341446i
$746$		0			0
$747$	−14.3352	+	13.9501i	−0.524499	+	0.510406i
$748$		0			0
$749$	−6.69465			−0.244617
$750$		0			0
$751$	−15.1184			−0.551679			−0.275840	−	0.961204i	$-0.588956\pi$
$751$	−15.1184			−0.551679			−0.275840	+	0.961204i	$0.588956\pi$
$752$		0			0
$753$	−10.0538	+	23.8122i	−0.366381	+	0.867765i
$754$		0			0
$755$	−7.84392			−0.285470
$756$		0			0
$757$			16.7354i			0.608257i	0.952631	+	0.304128i	$0.0983652\pi$
$757$			16.7354i			0.608257i	−0.952631	+	0.304128i	$0.901635\pi$
$758$		0			0
$759$	6.41558	+	2.70874i	0.232871	+	0.0983210i
$760$		0			0
$761$			7.42249i			0.269065i	0.990909	+	0.134533i	$0.0429533\pi$
$761$			7.42249i			0.269065i	−0.990909	+	0.134533i	$0.957047\pi$
$762$		0			0
$763$	24.5756			0.889695
$764$		0			0
$765$	−7.84095	+	7.63027i	−0.283490	+	0.275873i
$766$		0			0
$767$	−1.10938			−0.0400573
$768$		0			0
$769$			18.7867i			0.677467i	0.940882	+	0.338733i	$0.109998\pi$
$769$			18.7867i			0.677467i	−0.940882	+	0.338733i	$0.890002\pi$
$770$		0			0
$771$	−13.1510	−	5.55254i	−0.473623	−	0.199970i
$772$		0			0
$773$		−	5.26458i		−	0.189354i	−0.995508	−	0.0946770i	$-0.969818\pi$
$773$		−	5.26458i		−	0.189354i	0.995508	−	0.0946770i	$-0.0301818\pi$
$774$		0			0
$775$			9.67548i			0.347554i
$776$		0			0
$777$	13.5684	+	5.72876i	0.486764	+	0.205518i
$778$		0			0
$779$	7.64652			0.273965
$780$		0			0
$781$		−	41.7846i		−	1.49517i
$782$		0			0
$783$	−4.36993	−	11.1911i	−0.156169	−	0.399939i
$784$		0			0
$785$	3.71031			0.132427
$786$		0			0
$787$		−	31.5703i		−	1.12536i	−0.826675	−	0.562680i	$-0.809770\pi$
$787$		−	31.5703i		−	1.12536i	0.826675	−	0.562680i	$-0.190230\pi$
$788$		0			0
$789$	−10.6391	−	4.49195i	−0.378761	−	0.159918i
$790$		0			0
$791$			69.4786i			2.47037i
$792$		0			0
$793$	−2.14913			−0.0763178
$794$		0			0
$795$	4.00509	−	9.48596i	0.142046	−	0.336432i
$796$		0			0
$797$		−	47.5264i		−	1.68347i	−0.539891	−	0.841735i	$-0.681534\pi$
$797$		−	47.5264i		−	1.68347i	0.539891	−	0.841735i	$-0.318466\pi$
$798$		0			0
$799$	40.6351			1.43757
$800$		0			0
$801$	−25.3774	+	24.6956i	−0.896668	+	0.872575i
$802$		0			0
$803$	−0.484086			−0.0170830
$804$		0			0
$805$	2.72429			0.0960185
$806$		0			0
$807$	13.4296	+	5.67016i	0.472745	+	0.199599i
$808$		0			0
$809$	6.47988			0.227821			0.113910	−	0.993491i	$-0.463662\pi$
$809$	6.47988			0.227821			0.113910	+	0.993491i	$0.463662\pi$
$810$		0			0
$811$		−	19.2232i		−	0.675017i	−0.941322	−	0.337508i	$-0.890416\pi$
$811$		−	19.2232i		−	0.675017i	0.941322	−	0.337508i	$-0.109584\pi$
$812$		0			0
$813$	31.4738	+	13.2886i	1.10383	+	0.466053i
$814$		0			0
$815$	−18.3075			−0.641283
$816$		0			0
$817$		−	4.98289i		−	0.174329i
$818$		0			0
$819$	2.70253	+	2.77715i	0.0944340	+	0.0970414i
$820$		0			0
$821$		−	36.8059i		−	1.28454i	−0.766480	−	0.642268i	$-0.777994\pi$
$821$		−	36.8059i		−	1.28454i	0.766480	−	0.642268i	$-0.222006\pi$
$822$		0			0
$823$	34.7600			1.21166			0.605829	−	0.795595i	$-0.292841\pi$
$823$	34.7600			1.21166			0.605829	+	0.795595i	$0.292841\pi$
$824$		0			0
$825$	12.8312	−	30.3903i	0.446723	−	1.05805i
$826$		0			0
$827$			28.2779i			0.983319i	0.870788	+	0.491659i	$0.163609\pi$
$827$			28.2779i			0.983319i	−0.870788	+	0.491659i	$0.836391\pi$
$828$		0			0
$829$	−18.6542			−0.647885			−0.323943	−	0.946077i	$-0.605009\pi$
$829$	−18.6542			−0.647885			−0.323943	+	0.946077i	$0.605009\pi$
$830$		0			0
$831$	−4.65952	+	11.0359i	−0.161637	+	0.382833i
$832$		0			0
$833$		−	30.9790i		−	1.07336i
$834$		0			0
$835$			9.87021i			0.341573i
$836$		0			0
$837$	4.18922	+	10.7283i	0.144801	+	0.370826i
$838$		0			0
$839$		−	1.61729i		−	0.0558351i	−0.999610	−	0.0279176i	$-0.991112\pi$
$839$		−	1.61729i		−	0.0558351i	0.999610	−	0.0279176i	$-0.00888759\pi$
$840$		0			0
$841$	23.6542			0.815660
$842$		0			0
$843$	9.39445	−	22.2505i	0.323562	−	0.766348i
$844$		0			0
$845$	−10.2610			−0.352988
$846$		0			0
$847$		−	29.8180i		−	1.02456i
$848$		0			0
$849$	−3.53983	+	8.38401i	−0.121487	+	0.287738i
$850$		0			0
$851$			2.11185i			0.0723934i
$852$		0			0
$853$	41.0973			1.40715			0.703573	−	0.710623i	$-0.251587\pi$
$853$	41.0973			1.40715			0.703573	+	0.710623i	$0.251587\pi$
$854$		0			0
$855$	1.18072	+	1.21332i	0.0403796	+	0.0414946i
$856$		0			0
$857$	−3.34206			−0.114163			−0.0570813	−	0.998370i	$-0.518179\pi$
$857$	−3.34206			−0.114163			−0.0570813	+	0.998370i	$0.518179\pi$
$858$		0			0
$859$	−20.4659			−0.698289			−0.349144	−	0.937069i	$-0.613528\pi$
$859$	−20.4659			−0.698289			−0.349144	+	0.937069i	$0.613528\pi$
$860$		0			0
$861$	−63.9167	−	26.9864i	−2.17828	−	0.919695i
$862$		0			0
$863$		−	35.3173i		−	1.20222i	−0.799168	−	0.601108i	$-0.794726\pi$
$863$		−	35.3173i		−	1.20222i	0.799168	−	0.601108i	$-0.205274\pi$
$864$		0			0
$865$			15.7669i			0.536092i
$866$		0			0
$867$	−2.66272	+	6.30658i	−0.0904306	+	0.214183i
$868$		0			0
$869$		−	6.35559i		−	0.215599i
$870$		0			0
$871$	−2.44898	+	1.45669i	−0.0829805	+	0.0493581i
$872$		0			0
$873$	−20.1089	+	19.5686i	−0.680585	+	0.662298i
$874$		0			0
$875$		−	27.6862i		−	0.935964i
$876$		0			0
$877$	41.3023			1.39468			0.697339	−	0.716741i	$-0.254367\pi$
$877$	41.3023			1.39468			0.697339	+	0.716741i	$0.254367\pi$
$878$		0			0
$879$	44.1491	+	18.6403i	1.48911	+	0.628722i
$880$		0			0
$881$			3.58334i			0.120726i	0.998176	+	0.0603629i	$0.0192258\pi$
$881$			3.58334i			0.120726i	−0.998176	+	0.0603629i	$0.980774\pi$
$882$		0			0
$883$			34.4837i			1.16047i	0.814449	+	0.580235i	$0.197039\pi$
$883$			34.4837i			1.16047i	−0.814449	+	0.580235i	$0.802961\pi$
$884$		0			0
$885$	4.05140	+	1.71055i	0.136186	+	0.0574996i
$886$		0			0
$887$			8.83478i			0.296643i	0.988939	+	0.148321i	$0.0473870\pi$
$887$			8.83478i			0.296643i	−0.988939	+	0.148321i	$0.952613\pi$
$888$		0			0
$889$		−	64.3958i		−	2.15977i
$890$		0			0
$891$	1.06925	−	39.2528i	0.0358214	−	1.31502i
$892$		0			0
$893$		−	6.28792i		−	0.210417i
$894$		0			0
$895$	4.04052			0.135060
$896$		0			0
$897$	−0.216124	+	0.511885i	−0.00721618	+	0.0170913i
$898$		0			0
$899$	5.12478			0.170921
$900$		0			0
$901$		−	34.1544i		−	1.13785i
$902$		0			0
$903$	−17.5858	+	41.6516i	−0.585220	+	1.38608i
$904$		0			0
$905$	−3.78002			−0.125652
$906$		0			0
$907$	−41.3117			−1.37173			−0.685867	−	0.727727i	$-0.740577\pi$
$907$	−41.3117			−1.37173			−0.685867	+	0.727727i	$0.740577\pi$
$908$		0			0
$909$	−19.4378	−	19.9745i	−0.644710	−	0.662511i
$910$		0			0
$911$		−	49.6762i		−	1.64585i	−0.568152	−	0.822924i	$-0.692341\pi$
$911$		−	49.6762i		−	1.64585i	0.568152	−	0.822924i	$-0.307659\pi$
$912$		0			0
$913$			29.0907i			0.962761i
$914$		0			0
$915$	7.84853	+	3.31375i	0.259464	+	0.109549i
$916$		0			0
$917$	18.5193			0.611560
$918$		0			0
$919$			38.9926i			1.28625i	0.765762	+	0.643124i	$0.222362\pi$
$919$			38.9926i			1.28625i	−0.765762	+	0.643124i	$0.777638\pi$
$920$		0			0
$921$	−10.8595	+	25.7206i	−0.357834	+	0.847521i
$922$		0			0
$923$	3.33391			0.109737
$924$		0			0
$925$	10.0037			0.328921
$926$		0			0
$927$	−19.1086	−	19.6362i	−0.627608	−	0.644937i
$928$		0			0
$929$	−27.0604			−0.887822			−0.443911	−	0.896071i	$-0.646409\pi$
$929$	−27.0604			−0.887822			−0.443911	+	0.896071i	$0.646409\pi$
$930$		0			0
$931$	−4.79373			−0.157108
$932$		0			0
$933$	−7.14838	+	16.9307i	−0.234027	+	0.554288i
$934$		0			0
$935$			15.9117i			0.520369i
$936$		0			0
$937$			11.4417i			0.373783i	0.982381	+	0.186891i	$0.0598413\pi$
$937$			11.4417i			0.373783i	−0.982381	+	0.186891i	$0.940159\pi$
$938$		0			0
$939$	−33.5337	−	14.1583i	−1.09433	−	0.462040i
$940$		0			0
$941$	40.5295			1.32122			0.660612	−	0.750728i	$-0.270297\pi$
$941$	40.5295			1.32122			0.660612	+	0.750728i	$0.270297\pi$
$942$		0			0
$943$		−	9.94830i		−	0.323961i
$944$		0			0
$945$	−5.58743	−	14.3091i	−0.181759	−	0.465474i
$946$		0			0
$947$			49.7397i			1.61632i	0.588961	+	0.808161i	$0.299537\pi$
$947$			49.7397i			1.61632i	−0.588961	+	0.808161i	$0.700463\pi$
$948$		0			0
$949$		−	0.0386242i		−	0.00125379i
$950$		0			0
$951$	40.0594	+	16.9136i	1.29902	+	0.548461i
$952$		0			0
$953$		−	36.0894i		−	1.16905i	−0.811375	−	0.584525i	$-0.801281\pi$
$953$		−	36.0894i		−	1.16905i	0.811375	−	0.584525i	$-0.198719\pi$
$954$		0			0
$955$	−1.07553			−0.0348034
$956$		0			0
$957$	−16.0967	−	6.79623i	−0.520332	−	0.219691i
$958$		0			0
$959$			65.9769i			2.13051i
$960$		0			0
$961$	26.0872			0.841521
$962$		0			0
$963$	−3.87913	+	3.77491i	−0.125003	+	0.121645i
$964$		0			0
$965$	−9.55263			−0.307510
$966$		0			0
$967$	−20.3931			−0.655798			−0.327899	−	0.944713i	$-0.606341\pi$
$967$	−20.3931			−0.655798			−0.327899	+	0.944713i	$0.606341\pi$
$968$		0			0
$969$	5.17342	+	2.18428i	0.166194	+	0.0701693i
$970$		0			0
$971$		−	20.9753i		−	0.673128i	−0.941661	−	0.336564i	$-0.890735\pi$
$971$		−	20.9753i		−	0.673128i	0.941661	−	0.336564i	$-0.109265\pi$
$972$		0			0
$973$	−3.19604			−0.102460
$974$		0			0
$975$	2.42477	+	1.02377i	0.0776549	+	0.0327869i
$976$		0			0
$977$			30.3311i			0.970376i	0.874410	+	0.485188i	$0.161249\pi$
$977$			30.3311i			0.970376i	−0.874410	+	0.485188i	$0.838751\pi$
$978$		0			0
$979$			51.4987i			1.64591i
$980$		0			0
$981$	14.2400	−	13.8574i	0.454649	−	0.442433i
$982$		0			0
$983$	60.0965			1.91678			0.958391	−	0.285458i	$-0.0921458\pi$
$983$	60.0965			1.91678			0.958391	+	0.285458i	$0.0921458\pi$
$984$		0			0
$985$	−5.94487			−0.189419
$986$		0			0
$987$	−22.1916	+	52.5603i	−0.706367	+	1.67301i
$988$		0			0
$989$	−6.48286			−0.206143
$990$		0			0
$991$		−	50.0041i		−	1.58843i	−0.607636	−	0.794216i	$-0.707882\pi$
$991$		−	50.0041i		−	1.58843i	0.607636	−	0.794216i	$-0.292118\pi$
$992$		0			0
$993$	−27.6130	−	11.6586i	−0.876272	−	0.369973i
$994$		0			0
$995$	−16.5372			−0.524265
$996$		0			0
$997$	24.3250			0.770382			0.385191	−	0.922837i	$-0.374136\pi$
$997$	24.3250			0.770382			0.385191	+	0.922837i	$0.374136\pi$
$998$		0			0
$999$	11.0923	−	4.33134i	0.350945	−	0.137038i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.g.c.401.9	yes	16
3.2	odd	2	inner	804.2.g.c.401.7	✓	16
67.66	odd	2	inner	804.2.g.c.401.8	yes	16
201.200	even	2	inner	804.2.g.c.401.10	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.g.c.401.7	✓	16	3.2	odd	2	inner
804.2.g.c.401.8	yes	16	67.66	odd	2	inner
804.2.g.c.401.9	yes	16	1.1	even	1	trivial
804.2.g.c.401.10	yes	16	201.200	even	2	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:	\( N \)	\(=\)	\( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	804.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.673707 - 1.59566i) q^{3} -0.796732 q^{5} -3.71051i q^{7} +(-2.09224 - 2.15001i) q^{9} +O(q^{10})\)	\(q+(0.673707 - 1.59566i) q^{3} -0.796732 q^{5} -3.71051i q^{7} +(-2.09224 - 2.15001i) q^{9} -4.36304 q^{11} -0.348117i q^{13} +(-0.536763 + 1.27131i) q^{15} +4.57737i q^{17} +0.708308 q^{19} +(-5.92069 - 2.49979i) q^{21} -0.921525i q^{23} -4.36522 q^{25} +(-4.84023 + 1.89002i) q^{27} +2.31211i q^{29} -2.21649i q^{31} +(-2.93941 + 6.96191i) q^{33} +2.95628i q^{35} -2.29169 q^{37} +(-0.555476 - 0.234529i) q^{39} +10.7955 q^{41} -7.03492i q^{43} +(1.66695 + 1.71298i) q^{45} -8.87739i q^{47} -6.76786 q^{49} +(7.30391 + 3.08381i) q^{51} -7.46157 q^{53} +3.47617 q^{55} +(0.477192 - 1.13022i) q^{57} -3.18679i q^{59} -6.17358i q^{61} +(-7.97762 + 7.76327i) q^{63} +0.277356i q^{65} +(-4.18448 - 7.03492i) q^{67} +(-1.47044 - 0.620838i) q^{69} +9.57696i q^{71} +0.110952 q^{73} +(-2.94088 + 6.96539i) q^{75} +16.1891i q^{77} +1.45669i q^{79} +(-0.245071 + 8.99666i) q^{81} -6.66753i q^{83} -3.64694i q^{85} +(3.68933 + 1.55768i) q^{87} -11.8034i q^{89} -1.29169 q^{91} +(-3.53676 - 1.49327i) q^{93} -0.564331 q^{95} -9.35296i q^{97} +(9.12852 + 9.38057i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 2 q^{9}+O(q^{10}) \) 16 * q + 2 * q^9	\( 16 q + 2 q^{9} + 18 q^{15} + 28 q^{19} - 16 q^{21} + 14 q^{33} - 20 q^{37} - 4 q^{49} - 32 q^{55} + 4 q^{67} - 16 q^{73} + 6 q^{81} - 4 q^{91} - 30 q^{93}+O(q^{100}) \) 16 * q + 2 * q^9 + 18 * q^15 + 28 * q^19 - 16 * q^21 + 14 * q^33 - 20 * q^37 - 4 * q^49 - 32 * q^55 + 4 * q^67 - 16 * q^73 + 6 * q^81 - 4 * q^91 - 30 * q^93

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