LMFDB - Embedded newform 2394.2.b.h.1709.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2394,2,Mod(1709,2394)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2394, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2394.1709");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2394.b (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:	$19.1161862439$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.2803712.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 6x^{4} + 8x^{2} + 2 $ x^6 + 6x^4 + 8x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1709.2
Root			$2.05288i$ of defining polynomial
Character	$\chi$	$=$	2394.1709
Dual form			2394.2.b.h.1709.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -2.69155i q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -2.69155i q^{5} +1.00000 q^{7} +1.00000 q^{8} -2.69155i q^{10} -4.10576i q^{11} -2.15728i q^{13} +1.00000 q^{14} +1.00000 q^{16} +6.93419i q^{17} +(3.42864 - 2.69155i) q^{19} -2.69155i q^{20} -4.10576i q^{22} +6.39992i q^{23} -2.24443 q^{25} -2.15728i q^{26} +1.00000 q^{28} +4.00000 q^{29} -8.55720i q^{31} +1.00000 q^{32} +6.93419i q^{34} -2.69155i q^{35} -10.5057i q^{37} +(3.42864 - 2.69155i) q^{38} -2.69155i q^{40} -7.05086 q^{41} +3.05086 q^{43} -4.10576i q^{44} +6.39992i q^{46} -6.26304i q^{47} +1.00000 q^{49} -2.24443 q^{50} -2.15728i q^{52} -10.8573 q^{53} -11.0509 q^{55} +1.00000 q^{56} +4.00000 q^{58} +2.75557 q^{59} -5.80642 q^{61} -8.55720i q^{62} +1.00000 q^{64} -5.80642 q^{65} +0.136879i q^{67} +6.93419i q^{68} -2.69155i q^{70} -10.3684 q^{71} +2.56199 q^{73} -10.5057i q^{74} +(3.42864 - 2.69155i) q^{76} -4.10576i q^{77} +7.54038i q^{79} -2.69155i q^{80} -7.05086 q^{82} +4.84883i q^{83} +18.6637 q^{85} +3.05086 q^{86} -4.10576i q^{88} -3.05086 q^{89} -2.15728i q^{91} +6.39992i q^{92} -6.26304i q^{94} +(-7.24443 - 9.22835i) q^{95} -7.19470i q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{2} + 6 q^{4} + 6 q^{7} + 6 q^{8}+O(q^{10}) $ 6 * q + 6 * q^2 + 6 * q^4 + 6 * q^7 + 6 * q^8	$ 6 q + 6 q^{2} + 6 q^{4} + 6 q^{7} + 6 q^{8} + 6 q^{14} + 6 q^{16} - 6 q^{19} - 14 q^{25} + 6 q^{28} + 24 q^{29} + 6 q^{32} - 6 q^{38} - 16 q^{41} - 8 q^{43} + 6 q^{49} - 14 q^{50} - 12 q^{53} - 40 q^{55} + 6 q^{56} + 24 q^{58} + 16 q^{59} - 8 q^{61} + 6 q^{64} - 8 q^{65} - 8 q^{71} - 12 q^{73} - 6 q^{76} - 16 q^{82} + 32 q^{85} - 8 q^{86} + 8 q^{89} - 44 q^{95} + 6 q^{98}+O(q^{100}) $ 6 * q + 6 * q^2 + 6 * q^4 + 6 * q^7 + 6 * q^8 + 6 * q^14 + 6 * q^16 - 6 * q^19 - 14 * q^25 + 6 * q^28 + 24 * q^29 + 6 * q^32 - 6 * q^38 - 16 * q^41 - 8 * q^43 + 6 * q^49 - 14 * q^50 - 12 * q^53 - 40 * q^55 + 6 * q^56 + 24 * q^58 + 16 * q^59 - 8 * q^61 + 6 * q^64 - 8 * q^65 - 8 * q^71 - 12 * q^73 - 6 * q^76 - 16 * q^82 + 32 * q^85 - 8 * q^86 + 8 * q^89 - 44 * q^95 + 6 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2394\mathbb{Z}\right)^\times$.

$n$	$533$	$1009$	$1711$
$\chi(n)$	$-1$	$-1$	$1$

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 $

$2$	1.00000			0.707107
$2$	1.00000			0.707107
$3$		0			0
$3$		0			0
$4$	1.00000			0.500000
$5$		−	2.69155i		−	1.20370i	−0.798610	−	0.601848i	$-0.794431\pi$
$5$		−	2.69155i		−	1.20370i	0.798610	−	0.601848i	$-0.205569\pi$
$6$		0			0
$7$	1.00000			0.377964
$7$	1.00000			0.377964
$8$	1.00000			0.353553
$9$		0			0
$10$		−	2.69155i		−	0.851142i
$11$		−	4.10576i		−	1.23793i	−0.785417	−	0.618967i	$-0.787552\pi$
$11$		−	4.10576i		−	1.23793i	0.785417	−	0.618967i	$-0.212448\pi$
$12$		0			0
$13$		−	2.15728i		−	0.598322i	−0.954203	−	0.299161i	$-0.903293\pi$
$13$		−	2.15728i		−	0.598322i	0.954203	−	0.299161i	$-0.0967068\pi$
$14$	1.00000			0.267261
$15$		0			0
$16$	1.00000			0.250000
$17$			6.93419i			1.68179i	0.541200	+	0.840894i	$0.317970\pi$
$17$			6.93419i			1.68179i	−0.541200	+	0.840894i	$0.682030\pi$
$18$		0			0
$19$	3.42864	−	2.69155i	0.786584	−	0.617483i
$19$	3.42864	−	2.69155i	0.786584	−	0.617483i
$20$		−	2.69155i		−	0.601848i
$21$		0			0
$22$		−	4.10576i		−	0.875351i
$23$			6.39992i			1.33448i	0.744845	+	0.667238i	$0.232524\pi$
$23$			6.39992i			1.33448i	−0.744845	+	0.667238i	$0.767476\pi$
$24$		0			0
$25$	−2.24443			−0.448886
$26$		−	2.15728i		−	0.423077i
$27$		0			0
$28$	1.00000			0.188982
$29$	4.00000			0.742781			0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000			0.742781			0.371391	+	0.928477i	$0.378881\pi$
$30$		0			0
$31$		−	8.55720i		−	1.53692i	−0.639899	−	0.768459i	$-0.721024\pi$
$31$		−	8.55720i		−	1.53692i	0.639899	−	0.768459i	$-0.278976\pi$
$32$	1.00000			0.176777
$33$		0			0
$34$			6.93419i			1.18920i
$35$		−	2.69155i		−	0.454955i
$36$		0			0
$37$		−	10.5057i		−	1.72712i	−0.504243	−	0.863562i	$-0.668228\pi$
$37$		−	10.5057i		−	1.72712i	0.504243	−	0.863562i	$-0.331772\pi$
$38$	3.42864	−	2.69155i	0.556199	−	0.436627i
$39$		0			0
$40$		−	2.69155i		−	0.425571i
$41$	−7.05086			−1.10116			−0.550579	−	0.834783i	$-0.685593\pi$
$41$	−7.05086			−1.10116			−0.550579	+	0.834783i	$0.685593\pi$
$42$		0			0
$43$	3.05086			0.465251			0.232626	−	0.972566i	$-0.425268\pi$
$43$	3.05086			0.465251			0.232626	+	0.972566i	$0.425268\pi$
$44$		−	4.10576i		−	0.618967i
$45$		0			0
$46$			6.39992i			0.943617i
$47$		−	6.26304i		−	0.913559i	−0.889580	−	0.456779i	$-0.849003\pi$
$47$		−	6.26304i		−	0.913559i	0.889580	−	0.456779i	$-0.150997\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$	−2.24443			−0.317411
$51$		0			0
$52$		−	2.15728i		−	0.299161i
$53$	−10.8573			−1.49136			−0.745681	−	0.666303i	$-0.767876\pi$
$53$	−10.8573			−1.49136			−0.745681	+	0.666303i	$0.767876\pi$
$54$		0			0
$55$	−11.0509			−1.49010
$56$	1.00000			0.133631
$57$		0			0
$58$	4.00000			0.525226
$59$	2.75557			0.358744			0.179372	−	0.983781i	$-0.442593\pi$
$59$	2.75557			0.358744			0.179372	+	0.983781i	$0.442593\pi$
$60$		0			0
$61$	−5.80642			−0.743436			−0.371718	−	0.928346i	$-0.621231\pi$
$61$	−5.80642			−0.743436			−0.371718	+	0.928346i	$0.621231\pi$
$62$		−	8.55720i		−	1.08677i
$63$		0			0
$64$	1.00000			0.125000
$65$	−5.80642			−0.720198
$66$		0			0
$67$			0.136879i			0.0167224i	0.999965	+	0.00836122i	$0.00266149\pi$
$67$			0.136879i			0.0167224i	−0.999965	+	0.00836122i	$0.997339\pi$
$68$			6.93419i			0.840894i
$69$		0			0
$70$		−	2.69155i		−	0.321702i
$71$	−10.3684			−1.23050			−0.615252	−	0.788330i	$-0.710946\pi$
$71$	−10.3684			−1.23050			−0.615252	+	0.788330i	$0.710946\pi$
$72$		0			0
$73$	2.56199			0.299859			0.149929	−	0.988697i	$-0.452095\pi$
$73$	2.56199			0.299859			0.149929	+	0.988697i	$0.452095\pi$
$74$		−	10.5057i		−	1.22126i
$75$		0			0
$76$	3.42864	−	2.69155i	0.393292	−	0.308742i
$77$		−	4.10576i		−	0.467895i
$78$		0			0
$79$			7.54038i			0.848359i	0.905578	+	0.424179i	$0.139437\pi$
$79$			7.54038i			0.848359i	−0.905578	+	0.424179i	$0.860563\pi$
$80$		−	2.69155i		−	0.300924i
$81$		0			0
$82$	−7.05086			−0.778637
$83$			4.84883i			0.532228i	0.963942	+	0.266114i	$0.0857398\pi$
$83$			4.84883i			0.532228i	−0.963942	+	0.266114i	$0.914260\pi$
$84$		0			0
$85$	18.6637			2.02436
$86$	3.05086			0.328982
$87$		0			0
$88$		−	4.10576i		−	0.437676i
$89$	−3.05086			−0.323390			−0.161695	−	0.986841i	$-0.551696\pi$
$89$	−3.05086			−0.323390			−0.161695	+	0.986841i	$0.551696\pi$
$90$		0			0
$91$		−	2.15728i		−	0.226144i
$92$			6.39992i			0.667238i
$93$		0			0
$94$		−	6.26304i		−	0.645983i
$95$	−7.24443	−	9.22835i	−0.743263	−	0.946809i
$96$		0			0
$97$		−	7.19470i		−	0.730511i	−0.930907	−	0.365255i	$-0.880982\pi$
$97$		−	7.19470i		−	0.730511i	0.930907	−	0.365255i	$-0.119018\pi$
$98$	1.00000			0.101015
$99$		0			0
$100$	−2.24443			−0.224443
$101$		−	9.41694i		−	0.937020i	−0.883458	−	0.468510i	$-0.844791\pi$
$101$		−	9.41694i		−	0.937020i	0.883458	−	0.468510i	$-0.155209\pi$
$102$		0			0
$103$		−	5.72877i		−	0.564473i	−0.959345	−	0.282236i	$-0.908924\pi$
$103$		−	5.72877i		−	0.564473i	0.959345	−	0.282236i	$-0.0910763\pi$
$104$		−	2.15728i		−	0.211539i
$105$		0			0
$106$	−10.8573			−1.05455
$107$	15.4193			1.49064			0.745319	−	0.666708i	$-0.232297\pi$
$107$	15.4193			1.49064			0.745319	+	0.666708i	$0.232297\pi$
$108$		0			0
$109$			16.1625i			1.54809i	0.633130	+	0.774045i	$0.281770\pi$
$109$			16.1625i			1.54809i	−0.633130	+	0.774045i	$0.718230\pi$
$110$	−11.0509			−1.05366
$111$		0			0
$112$	1.00000			0.0944911
$113$	18.8573			1.77394			0.886972	−	0.461824i	$-0.152805\pi$
$113$	18.8573			1.77394			0.886972	+	0.461824i	$0.152805\pi$
$114$		0			0
$115$	17.2257			1.60630
$116$	4.00000			0.371391
$117$		0			0
$118$	2.75557			0.253671
$119$			6.93419i			0.635656i
$120$		0			0
$121$	−5.85728			−0.532480
$122$	−5.80642			−0.525689
$123$		0			0
$124$		−	8.55720i		−	0.768459i
$125$		−	7.41675i		−	0.663374i
$126$		0			0
$127$			11.7111i			1.03919i	0.854412	+	0.519596i	$0.173918\pi$
$127$			11.7111i			1.03919i	−0.854412	+	0.519596i	$0.826082\pi$
$128$	1.00000			0.0883883
$129$		0			0
$130$	−5.80642			−0.509257
$131$			5.12259i			0.447562i	0.974639	+	0.223781i	$0.0718401\pi$
$131$			5.12259i			0.447562i	−0.974639	+	0.223781i	$0.928160\pi$
$132$		0			0
$133$	3.42864	−	2.69155i	0.297301	−	0.233387i
$134$			0.136879i			0.0118245i
$135$		0			0
$136$			6.93419i			0.594602i
$137$			2.20899i			0.188727i	0.995538	+	0.0943634i	$0.0300816\pi$
$137$			2.20899i			0.188727i	−0.995538	+	0.0943634i	$0.969918\pi$
$138$		0			0
$139$	7.61285			0.645713			0.322857	−	0.946448i	$-0.395357\pi$
$139$	7.61285			0.645713			0.322857	+	0.946448i	$0.395357\pi$
$140$		−	2.69155i		−	0.227477i
$141$		0			0
$142$	−10.3684			−0.870098
$143$	−8.85728			−0.740683
$144$		0			0
$145$		−	10.7662i		−	0.894084i
$146$	2.56199			0.212032
$147$		0			0
$148$		−	10.5057i		−	0.863562i
$149$		−	3.43461i		−	0.281375i	−0.990054	−	0.140687i	$-0.955069\pi$
$149$		−	3.43461i		−	0.281375i	0.990054	−	0.140687i	$-0.0449312\pi$
$150$		0			0
$151$		−	2.43104i		−	0.197835i	−0.995096	−	0.0989175i	$-0.968462\pi$
$151$		−	2.43104i		−	0.197835i	0.995096	−	0.0989175i	$-0.0315380\pi$
$152$	3.42864	−	2.69155i	0.278099	−	0.218313i
$153$		0			0
$154$		−	4.10576i		−	0.330852i
$155$	−23.0321			−1.84998
$156$		0			0
$157$	8.10171			0.646587			0.323293	−	0.946299i	$-0.395210\pi$
$157$	8.10171			0.646587			0.323293	+	0.946299i	$0.395210\pi$
$158$			7.54038i			0.599880i
$159$		0			0
$160$		−	2.69155i		−	0.212786i
$161$			6.39992i			0.504384i
$162$		0			0
$163$	4.29529			0.336433			0.168216	−	0.985750i	$-0.446199\pi$
$163$	4.29529			0.336433			0.168216	+	0.985750i	$0.446199\pi$
$164$	−7.05086			−0.550579
$165$		0			0
$166$			4.84883i			0.376342i
$167$	−14.3684			−1.11186			−0.555931	−	0.831229i	$-0.687638\pi$
$167$	−14.3684			−1.11186			−0.555931	+	0.831229i	$0.687638\pi$
$168$		0			0
$169$	8.34614			0.642011
$170$	18.6637			1.43144
$171$		0			0
$172$	3.05086			0.232626
$173$	21.0509			1.60047			0.800233	−	0.599689i	$-0.204709\pi$
$173$	21.0509			1.60047			0.800233	+	0.599689i	$0.204709\pi$
$174$		0			0
$175$	−2.24443			−0.169663
$176$		−	4.10576i		−	0.309483i
$177$		0			0
$178$	−3.05086			−0.228671
$179$	−7.61285			−0.569011			−0.284505	−	0.958674i	$-0.591829\pi$
$179$	−7.61285			−0.569011			−0.284505	+	0.958674i	$0.591829\pi$
$180$		0			0
$181$			15.2309i			1.13210i	0.824370	+	0.566051i	$0.191530\pi$
$181$			15.2309i			1.13210i	−0.824370	+	0.566051i	$0.808470\pi$
$182$		−	2.15728i		−	0.159908i
$183$		0			0
$184$			6.39992i			0.471808i
$185$	−28.2766			−2.07893
$186$		0			0
$187$	28.4701			2.08194
$188$		−	6.26304i		−	0.456779i
$189$		0			0
$190$	−7.24443	−	9.22835i	−0.525566	−	0.669495i
$191$			8.95459i			0.647931i	0.946069	+	0.323966i	$0.105016\pi$
$191$			8.95459i			0.647931i	−0.946069	+	0.323966i	$0.894984\pi$
$192$		0			0
$193$		−	12.7998i		−	0.921353i	−0.887568	−	0.460676i	$-0.847607\pi$
$193$		−	12.7998i		−	0.921353i	0.887568	−	0.460676i	$-0.152393\pi$
$194$		−	7.19470i		−	0.516549i
$195$		0			0
$196$	1.00000			0.0714286
$197$			4.12597i			0.293963i	0.989139	+	0.146982i	$0.0469558\pi$
$197$			4.12597i			0.293963i	−0.989139	+	0.146982i	$0.953044\pi$
$198$		0			0
$199$	−19.0509			−1.35048			−0.675240	−	0.737598i	$-0.735960\pi$
$199$	−19.0509			−1.35048			−0.675240	+	0.737598i	$0.735960\pi$
$200$	−2.24443			−0.158705
$201$		0			0
$202$		−	9.41694i		−	0.662573i
$203$	4.00000			0.280745
$204$		0			0
$205$			18.9777i			1.32546i
$206$		−	5.72877i		−	0.399143i
$207$		0			0
$208$		−	2.15728i		−	0.149580i
$209$	−11.0509	−	14.0772i	−0.764404	−	0.973739i
$210$		0			0
$211$			2.69155i			0.185294i	0.995699	+	0.0926469i	$0.0295328\pi$
$211$			2.69155i			0.185294i	−0.995699	+	0.0926469i	$0.970467\pi$
$212$	−10.8573			−0.745681
$213$		0			0
$214$	15.4193			1.05404
$215$		−	8.21152i		−	0.560021i
$216$		0			0
$217$		−	8.55720i		−	0.580901i
$218$			16.1625i			1.09467i
$219$		0			0
$220$	−11.0509			−0.745049
$221$	14.9590			1.00625
$222$		0			0
$223$		−	7.76242i		−	0.519810i	−0.965634	−	0.259905i	$-0.916309\pi$
$223$		−	7.76242i		−	0.519810i	0.965634	−	0.259905i	$-0.0836913\pi$
$224$	1.00000			0.0668153
$225$		0			0
$226$	18.8573			1.25437
$227$	−24.3783			−1.61804			−0.809021	−	0.587780i	$-0.800002\pi$
$227$	−24.3783			−1.61804			−0.809021	+	0.587780i	$0.800002\pi$
$228$		0			0
$229$	−15.7146			−1.03845			−0.519224	−	0.854638i	$-0.673779\pi$
$229$	−15.7146			−1.03845			−0.519224	+	0.854638i	$0.673779\pi$
$230$	17.2257			1.13583
$231$		0			0
$232$	4.00000			0.262613
$233$			17.0425i			1.11649i	0.829676	+	0.558245i	$0.188525\pi$
$233$			17.0425i			1.11649i	−0.829676	+	0.558245i	$0.811475\pi$
$234$		0			0
$235$	−16.8573			−1.09965
$236$	2.75557			0.179372
$237$		0			0
$238$			6.93419i			0.449477i
$239$			25.7952i			1.66856i	0.551345	+	0.834278i	$0.314115\pi$
$239$			25.7952i			1.66856i	−0.551345	+	0.834278i	$0.685885\pi$
$240$		0			0
$241$		−	14.3377i		−	0.923572i	−0.886991	−	0.461786i	$-0.847209\pi$
$241$		−	14.3377i		−	0.923572i	0.886991	−	0.461786i	$-0.152791\pi$
$242$	−5.85728			−0.376520
$243$		0			0
$244$	−5.80642			−0.371718
$245$		−	2.69155i		−	0.171957i
$246$		0			0
$247$	−5.80642	−	7.39654i	−0.369454	−	0.470630i
$248$		−	8.55720i		−	0.543383i
$249$		0			0
$250$		−	7.41675i		−	0.469076i
$251$		−	2.15032i		−	0.135727i	−0.997695	−	0.0678635i	$-0.978382\pi$
$251$		−	2.15032i		−	0.135727i	0.997695	−	0.0678635i	$-0.0216182\pi$
$252$		0			0
$253$	26.2766			1.65199
$254$			11.7111i			0.734820i
$255$		0			0
$256$	1.00000			0.0625000
$257$	14.6637			0.914697			0.457348	−	0.889288i	$-0.348799\pi$
$257$	14.6637			0.914697			0.457348	+	0.889288i	$0.348799\pi$
$258$		0			0
$259$		−	10.5057i		−	0.652791i
$260$	−5.80642			−0.360099
$261$		0			0
$262$			5.12259i			0.316474i
$263$			20.9597i			1.29243i	0.763156	+	0.646214i	$0.223649\pi$
$263$			20.9597i			1.29243i	−0.763156	+	0.646214i	$0.776351\pi$
$264$		0			0
$265$			29.2229i			1.79515i
$266$	3.42864	−	2.69155i	0.210223	−	0.165029i
$267$		0			0
$268$			0.136879i			0.00836122i
$269$	−11.4193			−0.696245			−0.348123	−	0.937449i	$-0.613181\pi$
$269$	−11.4193			−0.696245			−0.348123	+	0.937449i	$0.613181\pi$
$270$		0			0
$271$	6.75557			0.410372			0.205186	−	0.978723i	$-0.434220\pi$
$271$	6.75557			0.410372			0.205186	+	0.978723i	$0.434220\pi$
$272$			6.93419i			0.420447i
$273$		0			0
$274$			2.20899i			0.133450i
$275$			9.21510i			0.555691i
$276$		0			0
$277$	16.9590			1.01897			0.509483	−	0.860480i	$-0.329837\pi$
$277$	16.9590			1.01897			0.509483	+	0.860480i	$0.329837\pi$
$278$	7.61285			0.456588
$279$		0			0
$280$		−	2.69155i		−	0.160851i
$281$	12.7556			0.760933			0.380467	−	0.924795i	$-0.375763\pi$
$281$	12.7556			0.760933			0.380467	+	0.924795i	$0.375763\pi$
$282$		0			0
$283$	24.9590			1.48366			0.741829	−	0.670589i	$-0.233959\pi$
$283$	24.9590			1.48366			0.741829	+	0.670589i	$0.233959\pi$
$284$	−10.3684			−0.615252
$285$		0			0
$286$	−8.85728			−0.523742
$287$	−7.05086			−0.416199
$288$		0			0
$289$	−31.0830			−1.82841
$290$		−	10.7662i		−	0.632213i
$291$		0			0
$292$	2.56199			0.149929
$293$	10.2953			0.601457			0.300729	−	0.953710i	$-0.402770\pi$
$293$	10.2953			0.601457			0.300729	+	0.953710i	$0.402770\pi$
$294$		0			0
$295$		−	7.41675i		−	0.431820i
$296$		−	10.5057i		−	0.610630i
$297$		0			0
$298$		−	3.43461i		−	0.198962i
$299$	13.8064			0.798446
$300$		0			0
$301$	3.05086			0.175848
$302$		−	2.43104i		−	0.139891i
$303$		0			0
$304$	3.42864	−	2.69155i	0.196646	−	0.154371i
$305$			15.6283i			0.894872i
$306$		0			0
$307$			7.66401i			0.437408i	0.975791	+	0.218704i	$0.0701829\pi$
$307$			7.66401i			0.437408i	−0.975791	+	0.218704i	$0.929817\pi$
$308$		−	4.10576i		−	0.233947i
$309$		0			0
$310$	−23.0321			−1.30814
$311$			28.0692i			1.59166i	0.605522	+	0.795829i	$0.292964\pi$
$311$			28.0692i			1.59166i	−0.605522	+	0.795829i	$0.707036\pi$
$312$		0			0
$313$	32.9590			1.86295			0.931476	−	0.363803i	$-0.118522\pi$
$313$	32.9590			1.86295			0.931476	+	0.363803i	$0.118522\pi$
$314$	8.10171			0.457206
$315$		0			0
$316$			7.54038i			0.424179i
$317$	25.4479			1.42929			0.714647	−	0.699485i	$-0.246587\pi$
$317$	25.4479			1.42929			0.714647	+	0.699485i	$0.246587\pi$
$318$		0			0
$319$		−	16.4230i		−	0.919514i
$320$		−	2.69155i		−	0.150462i
$321$		0			0
$322$			6.39992i			0.356654i
$323$	18.6637	+	23.7748i	1.03848	+	1.32287i
$324$		0			0
$325$			4.84187i			0.268578i
$326$	4.29529			0.237894
$327$		0			0
$328$	−7.05086			−0.389318
$329$		−	6.26304i		−	0.345293i
$330$		0			0
$331$			32.8531i			1.80577i	0.429885	+	0.902884i	$0.358554\pi$
$331$			32.8531i			1.80577i	−0.429885	+	0.902884i	$0.641446\pi$
$332$			4.84883i			0.266114i
$333$		0			0
$334$	−14.3684			−0.786205
$335$	0.368416			0.0201287
$336$		0			0
$337$			22.0799i			1.20277i	0.798960	+	0.601384i	$0.205384\pi$
$337$			22.0799i			1.20277i	−0.798960	+	0.601384i	$0.794616\pi$
$338$	8.34614			0.453970
$339$		0			0
$340$	18.6637			1.01218
$341$	−35.1338			−1.90260
$342$		0			0
$343$	1.00000			0.0539949
$344$	3.05086			0.164491
$345$		0			0
$346$	21.0509			1.13170
$347$			11.2487i			0.603865i	0.953329	+	0.301932i	$0.0976316\pi$
$347$			11.2487i			0.603865i	−0.953329	+	0.301932i	$0.902368\pi$
$348$		0			0
$349$	31.9081			1.70800			0.854002	−	0.520270i	$-0.174169\pi$
$349$	31.9081			1.70800			0.854002	+	0.520270i	$0.174169\pi$
$350$	−2.24443			−0.119970
$351$		0			0
$352$		−	4.10576i		−	0.218838i
$353$		−	33.1848i		−	1.76625i	−0.469138	−	0.883125i	$-0.655435\pi$
$353$		−	33.1848i		−	1.76625i	0.469138	−	0.883125i	$-0.344565\pi$
$354$		0			0
$355$			27.9071i			1.48115i
$356$	−3.05086			−0.161695
$357$		0			0
$358$	−7.61285			−0.402352
$359$			13.1253i			0.692727i	0.938100	+	0.346364i	$0.112584\pi$
$359$			13.1253i			0.692727i	−0.938100	+	0.346364i	$0.887416\pi$
$360$		0			0
$361$	4.51114	−	18.4567i	0.237428	−	0.971405i
$362$			15.2309i			0.800517i
$363$		0			0
$364$		−	2.15728i		−	0.113072i
$365$		−	6.89573i		−	0.360939i
$366$		0			0
$367$	14.5718			0.760644			0.380322	−	0.924854i	$-0.375813\pi$
$367$	14.5718			0.760644			0.380322	+	0.924854i	$0.375813\pi$
$368$			6.39992i			0.333619i
$369$		0			0
$370$	−28.2766			−1.47003
$371$	−10.8573			−0.563682
$372$		0			0
$373$		−	18.5734i		−	0.961692i	−0.876805	−	0.480846i	$-0.840330\pi$
$373$		−	18.5734i		−	0.961692i	0.876805	−	0.480846i	$-0.159670\pi$
$374$	28.4701			1.47216
$375$		0			0
$376$		−	6.26304i		−	0.322992i
$377$		−	8.62912i		−	0.444422i
$378$		0			0
$379$			2.04061i			0.104819i	0.998626	+	0.0524096i	$0.0166901\pi$
$379$			2.04061i			0.104819i	−0.998626	+	0.0524096i	$0.983310\pi$
$380$	−7.24443	−	9.22835i	−0.371631	−	0.473404i
$381$		0			0
$382$			8.95459i			0.458157i
$383$	−23.1240			−1.18158			−0.590790	−	0.806825i	$-0.701184\pi$
$383$	−23.1240			−1.18158			−0.590790	+	0.806825i	$0.701184\pi$
$384$		0			0
$385$	−11.0509			−0.563204
$386$		−	12.7998i		−	0.651495i
$387$		0			0
$388$		−	7.19470i		−	0.365255i
$389$		−	23.1037i		−	1.17140i	−0.810527	−	0.585702i	$-0.800819\pi$
$389$		−	23.1037i		−	1.17140i	0.810527	−	0.585702i	$-0.199181\pi$
$390$		0			0
$391$	−44.3783			−2.24431
$392$	1.00000			0.0505076
$393$		0			0
$394$			4.12597i			0.207863i
$395$	20.2953			1.02117
$396$		0			0
$397$	−10.5906			−0.531526			−0.265763	−	0.964038i	$-0.585624\pi$
$397$	−10.5906			−0.531526			−0.265763	+	0.964038i	$0.585624\pi$
$398$	−19.0509			−0.954933
$399$		0			0
$400$	−2.24443			−0.112222
$401$	17.1338			0.855623			0.427811	−	0.903868i	$-0.359285\pi$
$401$	17.1338			0.855623			0.427811	+	0.903868i	$0.359285\pi$
$402$		0			0
$403$	−18.4603			−0.919572
$404$		−	9.41694i		−	0.468510i
$405$		0			0
$406$	4.00000			0.198517
$407$	−43.1338			−2.13806
$408$		0			0
$409$			7.46846i			0.369291i	0.982805	+	0.184646i	$0.0591138\pi$
$409$			7.46846i			0.369291i	−0.982805	+	0.184646i	$0.940886\pi$
$410$			18.9777i			0.937243i
$411$		0			0
$412$		−	5.72877i		−	0.282236i
$413$	2.75557			0.135593
$414$		0			0
$415$	13.0509			0.640641
$416$		−	2.15728i		−	0.105769i
$417$		0			0
$418$	−11.0509	−	14.0772i	−0.540515	−	0.688537i
$419$			16.9573i			0.828419i	0.910182	+	0.414210i	$0.135942\pi$
$419$			16.9573i			0.828419i	−0.910182	+	0.414210i	$0.864058\pi$
$420$		0			0
$421$			23.9704i			1.16825i	0.811665	+	0.584123i	$0.198561\pi$
$421$			23.9704i			1.16825i	−0.811665	+	0.584123i	$0.801439\pi$
$422$			2.69155i			0.131023i
$423$		0			0
$424$	−10.8573			−0.527276
$425$		−	15.5633i		−	0.754931i
$426$		0			0
$427$	−5.80642			−0.280993
$428$	15.4193			0.745319
$429$		0			0
$430$		−	8.21152i		−	0.395995i
$431$	21.6860			1.04458			0.522288	−	0.852769i	$-0.325078\pi$
$431$	21.6860			1.04458			0.522288	+	0.852769i	$0.325078\pi$
$432$		0			0
$433$			28.1026i			1.35053i	0.737577	+	0.675263i	$0.235970\pi$
$433$			28.1026i			1.35053i	−0.737577	+	0.675263i	$0.764030\pi$
$434$		−	8.55720i		−	0.410759i
$435$		0			0
$436$			16.1625i			0.774045i
$437$	17.2257	+	21.9430i	0.824017	+	1.04968i
$438$		0			0
$439$		−	1.83181i		−	0.0874276i	−0.999044	−	0.0437138i	$-0.986081\pi$
$439$		−	1.83181i		−	0.0874276i	0.999044	−	0.0437138i	$-0.0139190\pi$
$440$	−11.0509			−0.526829
$441$		0			0
$442$	14.9590			0.711527
$443$		−	14.6247i		−	0.694840i	−0.937710	−	0.347420i	$-0.887058\pi$
$443$		−	14.6247i		−	0.694840i	0.937710	−	0.347420i	$-0.112942\pi$
$444$		0			0
$445$			8.21152i			0.389264i
$446$		−	7.76242i		−	0.367561i
$447$		0			0
$448$	1.00000			0.0472456
$449$	−30.0000			−1.41579			−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000			−1.41579			−0.707894	+	0.706319i	$0.750354\pi$
$450$		0			0
$451$			28.9491i			1.36316i
$452$	18.8573			0.886972
$453$		0			0
$454$	−24.3783			−1.14413
$455$	−5.80642			−0.272209
$456$		0			0
$457$	−39.6958			−1.85689			−0.928446	−	0.371467i	$-0.878855\pi$
$457$	−39.6958			−1.85689			−0.928446	+	0.371467i	$0.878855\pi$
$458$	−15.7146			−0.734293
$459$		0			0
$460$	17.2257			0.803152
$461$			1.20541i			0.0561418i	0.999606	+	0.0280709i	$0.00893641\pi$
$461$			1.20541i			0.0561418i	−0.999606	+	0.0280709i	$0.991064\pi$
$462$		0			0
$463$	−20.6637			−0.960324			−0.480162	−	0.877180i	$-0.659422\pi$
$463$	−20.6637			−0.960324			−0.480162	+	0.877180i	$0.659422\pi$
$464$	4.00000			0.185695
$465$		0			0
$466$			17.0425i			0.789478i
$467$		−	41.7622i		−	1.93253i	−0.257559	−	0.966263i	$-0.582918\pi$
$467$		−	41.7622i		−	1.93253i	0.257559	−	0.966263i	$-0.417082\pi$
$468$		0			0
$469$			0.136879i			0.00632049i
$470$	−16.8573			−0.777568
$471$		0			0
$472$	2.75557			0.126835
$473$		−	12.5261i		−	0.575950i
$474$		0			0
$475$	−7.69535	+	6.04099i	−0.353087	+	0.277180i
$476$			6.93419i			0.317828i
$477$		0			0
$478$			25.7952i			1.17985i
$479$			15.1659i			0.692949i	0.938059	+	0.346474i	$0.112621\pi$
$479$			15.1659i			0.692949i	−0.938059	+	0.346474i	$0.887379\pi$
$480$		0			0
$481$	−22.6637			−1.03338
$482$		−	14.3377i		−	0.653064i
$483$		0			0
$484$	−5.85728			−0.266240
$485$	−19.3649			−0.879314
$486$		0			0
$487$		−	10.6426i		−	0.482261i	−0.970493	−	0.241130i	$-0.922482\pi$
$487$		−	10.6426i		−	0.482261i	0.970493	−	0.241130i	$-0.0775181\pi$
$488$	−5.80642			−0.262844
$489$		0			0
$490$		−	2.69155i		−	0.121592i
$491$			21.7412i			0.981166i	0.871394	+	0.490583i	$0.163216\pi$
$491$			21.7412i			0.981166i	−0.871394	+	0.490583i	$0.836784\pi$
$492$		0			0
$493$			27.7368i			1.24920i
$494$	−5.80642	−	7.39654i	−0.261243	−	0.332786i
$495$		0			0
$496$		−	8.55720i		−	0.384230i
$497$	−10.3684			−0.465087
$498$		0			0
$499$	−12.0830			−0.540908			−0.270454	−	0.962733i	$-0.587174\pi$
$499$	−12.0830			−0.540908			−0.270454	+	0.962733i	$0.587174\pi$
$500$		−	7.41675i		−	0.331687i
$501$		0			0
$502$		−	2.15032i		−	0.0959734i
$503$			13.6798i			0.609952i	0.952360	+	0.304976i	$0.0986484\pi$
$503$			13.6798i			0.609952i	−0.952360	+	0.304976i	$0.901352\pi$
$504$		0			0
$505$	−25.3461			−1.12789
$506$	26.2766			1.16814
$507$		0			0
$508$			11.7111i			0.519596i
$509$	5.87955			0.260607			0.130303	−	0.991474i	$-0.458405\pi$
$509$	5.87955			0.260607			0.130303	+	0.991474i	$0.458405\pi$
$510$		0			0
$511$	2.56199			0.113336
$512$	1.00000			0.0441942
$513$		0			0
$514$	14.6637			0.646788
$515$	−15.4193			−0.679454
$516$		0			0
$517$	−25.7146			−1.13092
$518$		−	10.5057i		−	0.461593i
$519$		0			0
$520$	−5.80642			−0.254629
$521$	−3.51114			−0.153826			−0.0769129	−	0.997038i	$-0.524506\pi$
$521$	−3.51114			−0.153826			−0.0769129	+	0.997038i	$0.524506\pi$
$522$		0			0
$523$		−	26.1207i		−	1.14218i	−0.820888	−	0.571089i	$-0.806521\pi$
$523$		−	26.1207i		−	1.14218i	0.820888	−	0.571089i	$-0.193479\pi$
$524$			5.12259i			0.223781i
$525$		0			0
$526$			20.9597i			0.913884i
$527$	59.3372			2.58477
$528$		0			0
$529$	−17.9590			−0.780826
$530$			29.2229i			1.26936i
$531$		0			0
$532$	3.42864	−	2.69155i	0.148650	−	0.116693i
$533$			15.2107i			0.658847i
$534$		0			0
$535$		−	41.5017i		−	1.79428i
$536$			0.136879i			0.00591227i
$537$		0			0
$538$	−11.4193			−0.492320
$539$		−	4.10576i		−	0.176848i
$540$		0			0
$541$	−17.5210			−0.753286			−0.376643	−	0.926359i	$-0.622922\pi$
$541$	−17.5210			−0.753286			−0.376643	+	0.926359i	$0.622922\pi$
$542$	6.75557			0.290177
$543$		0			0
$544$			6.93419i			0.297301i
$545$	43.5022			1.86343
$546$		0			0
$547$			30.1545i			1.28932i	0.764471	+	0.644658i	$0.223000\pi$
$547$			30.1545i			1.28932i	−0.764471	+	0.644658i	$0.777000\pi$
$548$			2.20899i			0.0943634i
$549$		0			0
$550$			9.21510i			0.392933i
$551$	13.7146	−	10.7662i	0.584260	−	0.458655i
$552$		0			0
$553$			7.54038i			0.320649i
$554$	16.9590			0.720518
$555$		0			0
$556$	7.61285			0.322857
$557$		−	12.4409i		−	0.527139i	−0.964640	−	0.263569i	$-0.915100\pi$
$557$		−	12.4409i		−	0.527139i	0.964640	−	0.263569i	$-0.0848998\pi$
$558$		0			0
$559$		−	6.58155i		−	0.278370i
$560$		−	2.69155i		−	0.113739i
$561$		0			0
$562$	12.7556			0.538061
$563$	−27.8796			−1.17498			−0.587492	−	0.809230i	$-0.699884\pi$
$563$	−27.8796			−1.17498			−0.587492	+	0.809230i	$0.699884\pi$
$564$		0			0
$565$		−	50.7553i		−	2.13529i
$566$	24.9590			1.04910
$567$		0			0
$568$	−10.3684			−0.435049
$569$	29.9911			1.25729			0.628646	−	0.777691i	$-0.283609\pi$
$569$	29.9911			1.25729			0.628646	+	0.777691i	$0.283609\pi$
$570$		0			0
$571$	34.6637			1.45063			0.725315	−	0.688417i	$-0.241694\pi$
$571$	34.6637			1.45063			0.725315	+	0.688417i	$0.241694\pi$
$572$	−8.85728			−0.370341
$573$		0			0
$574$	−7.05086			−0.294297
$575$		−	14.3642i		−	0.599028i
$576$		0			0
$577$	19.8064			0.824552			0.412276	−	0.911059i	$-0.364734\pi$
$577$	19.8064			0.824552			0.412276	+	0.911059i	$0.364734\pi$
$578$	−31.0830			−1.29288
$579$		0			0
$580$		−	10.7662i		−	0.447042i
$581$			4.84883i			0.201163i
$582$		0			0
$583$			44.5774i			1.84621i
$584$	2.56199			0.106016
$585$		0			0
$586$	10.2953			0.425294
$587$			38.9338i			1.60697i	0.595325	+	0.803485i	$0.297023\pi$
$587$			38.9338i			1.60697i	−0.595325	+	0.803485i	$0.702977\pi$
$588$		0			0
$589$	−23.0321	−	29.3396i	−0.949022	−	1.20892i
$590$		−	7.41675i		−	0.305343i
$591$		0			0
$592$		−	10.5057i		−	0.431781i
$593$			8.55024i			0.351116i	0.984469	+	0.175558i	$0.0561730\pi$
$593$			8.55024i			0.351116i	−0.984469	+	0.175558i	$0.943827\pi$
$594$		0			0
$595$	18.6637			0.765137
$596$		−	3.43461i		−	0.140687i
$597$		0			0
$598$	13.8064			0.564587
$599$	26.8385			1.09659			0.548297	−	0.836284i	$-0.315277\pi$
$599$	26.8385			1.09659			0.548297	+	0.836284i	$0.315277\pi$
$600$		0			0
$601$		−	10.1530i		−	0.414151i	−0.978325	−	0.207076i	$-0.933605\pi$
$601$		−	10.1530i		−	0.414151i	0.978325	−	0.207076i	$-0.0663946\pi$
$602$	3.05086			0.124344
$603$		0			0
$604$		−	2.43104i		−	0.0989175i
$605$			15.7651i			0.640944i
$606$		0			0
$607$			45.5740i			1.84979i	0.380220	+	0.924896i	$0.375848\pi$
$607$			45.5740i			1.84979i	−0.380220	+	0.924896i	$0.624152\pi$
$608$	3.42864	−	2.69155i	0.139050	−	0.109157i
$609$		0			0
$610$			15.6283i			0.632770i
$611$	−13.5111			−0.546602
$612$		0			0
$613$	−45.8992			−1.85385			−0.926926	−	0.375243i	$-0.877559\pi$
$613$	−45.8992			−1.85385			−0.926926	+	0.375243i	$0.877559\pi$
$614$			7.66401i			0.309294i
$615$		0			0
$616$		−	4.10576i		−	0.165426i
$617$		−	8.93438i		−	0.359685i	−0.983695	−	0.179842i	$-0.942441\pi$
$617$		−	8.93438i		−	0.359685i	0.983695	−	0.179842i	$-0.0575588\pi$
$618$		0			0
$619$	19.4924			0.783466			0.391733	−	0.920079i	$-0.371876\pi$
$619$	19.4924			0.783466			0.391733	+	0.920079i	$0.371876\pi$
$620$	−23.0321			−0.924992
$621$		0			0
$622$			28.0692i			1.12547i
$623$	−3.05086			−0.122230
$624$		0			0
$625$	−31.1847			−1.24739
$626$	32.9590			1.31731
$627$		0			0
$628$	8.10171			0.323293
$629$	72.8484			2.90466
$630$		0			0
$631$	−6.63512			−0.264140			−0.132070	−	0.991240i	$-0.542162\pi$
$631$	−6.63512			−0.264140			−0.132070	+	0.991240i	$0.542162\pi$
$632$			7.54038i			0.299940i
$633$		0			0
$634$	25.4479			1.01066
$635$	31.5210			1.25087
$636$		0			0
$637$		−	2.15728i		−	0.0854746i
$638$		−	16.4230i		−	0.650195i
$639$		0			0
$640$		−	2.69155i		−	0.106393i
$641$	−0.488863			−0.0193089			−0.00965445	−	0.999953i	$-0.503073\pi$
$641$	−0.488863			−0.0193089			−0.00965445	+	0.999953i	$0.503073\pi$
$642$		0			0
$643$	26.9590			1.06316			0.531579	−	0.847008i	$-0.321599\pi$
$643$	26.9590			1.06316			0.531579	+	0.847008i	$0.321599\pi$
$644$			6.39992i			0.252192i
$645$		0			0
$646$	18.6637	+	23.7748i	0.734314	+	0.935408i
$647$		−	29.1377i		−	1.14552i	−0.819722	−	0.572761i	$-0.805872\pi$
$647$		−	29.1377i		−	1.14552i	0.819722	−	0.572761i	$-0.194128\pi$
$648$		0			0
$649$		−	11.3137i		−	0.444102i
$650$			4.84187i			0.189914i
$651$		0			0
$652$	4.29529			0.168216
$653$		−	8.44053i		−	0.330304i	−0.986268	−	0.165152i	$-0.947189\pi$
$653$		−	8.44053i		−	0.330304i	0.986268	−	0.165152i	$-0.0528114\pi$
$654$		0			0
$655$	13.7877			0.538730
$656$	−7.05086			−0.275290
$657$		0			0
$658$		−	6.26304i		−	0.244159i
$659$	−0.930409			−0.0362436			−0.0181218	−	0.999836i	$-0.505769\pi$
$659$	−0.930409			−0.0362436			−0.0181218	+	0.999836i	$0.505769\pi$
$660$		0			0
$661$		−	46.3171i		−	1.80153i	−0.434311	−	0.900763i	$-0.643008\pi$
$661$		−	46.3171i		−	1.80153i	0.434311	−	0.900763i	$-0.356992\pi$
$662$			32.8531i			1.27687i
$663$		0			0
$664$			4.84883i			0.188171i
$665$	−7.24443	−	9.22835i	−0.280927	−	0.357860i
$666$		0			0
$667$			25.5997i			0.991224i
$668$	−14.3684			−0.555931
$669$		0			0
$670$	0.368416			0.0142332
$671$			23.8398i			0.920325i
$672$		0			0
$673$			5.90412i			0.227587i	0.993504	+	0.113793i	$0.0363002\pi$
$673$			5.90412i			0.227587i	−0.993504	+	0.113793i	$0.963700\pi$
$674$			22.0799i			0.850486i
$675$		0			0
$676$	8.34614			0.321005
$677$	−14.0286			−0.539162			−0.269581	−	0.962978i	$-0.586885\pi$
$677$	−14.0286			−0.539162			−0.269581	+	0.962978i	$0.586885\pi$
$678$		0			0
$679$		−	7.19470i		−	0.276107i
$680$	18.6637			0.715720
$681$		0			0
$682$	−35.1338			−1.34534
$683$	−20.7467			−0.793850			−0.396925	−	0.917851i	$-0.629923\pi$
$683$	−20.7467			−0.793850			−0.396925	+	0.917851i	$0.629923\pi$
$684$		0			0
$685$	5.94561			0.227170
$686$	1.00000			0.0381802
$687$		0			0
$688$	3.05086			0.116313
$689$			23.4222i			0.892315i
$690$		0			0
$691$	−22.2480			−0.846353			−0.423176	−	0.906047i	$-0.639085\pi$
$691$	−22.2480			−0.846353			−0.423176	+	0.906047i	$0.639085\pi$
$692$	21.0509			0.800233
$693$		0			0
$694$			11.2487i			0.426997i
$695$		−	20.4903i		−	0.777243i
$696$		0			0
$697$		−	48.8920i		−	1.85192i
$698$	31.9081			1.20774
$699$		0			0
$700$	−2.24443			−0.0848315
$701$			40.7252i			1.53817i	0.639147	+	0.769085i	$0.279288\pi$
$701$			40.7252i			1.53817i	−0.639147	+	0.769085i	$0.720712\pi$
$702$		0			0
$703$	−28.2766	−	36.0202i	−1.06647	−	1.35853i
$704$		−	4.10576i		−	0.154742i
$705$		0			0
$706$		−	33.1848i		−	1.24893i
$707$		−	9.41694i		−	0.354160i
$708$		0			0
$709$	−40.0098			−1.50260			−0.751301	−	0.659960i	$-0.770573\pi$
$709$	−40.0098			−1.50260			−0.751301	+	0.659960i	$0.770573\pi$
$710$			27.9071i			1.04733i
$711$		0			0
$712$	−3.05086			−0.114336
$713$	54.7654			2.05098
$714$		0			0
$715$			23.8398i			0.891558i
$716$	−7.61285			−0.284505
$717$		0			0
$718$			13.1253i			0.489832i
$719$		−	46.1222i		−	1.72007i	−0.510237	−	0.860034i	$-0.670442\pi$
$719$		−	46.1222i		−	1.72007i	0.510237	−	0.860034i	$-0.329558\pi$
$720$		0			0
$721$		−	5.72877i		−	0.213351i
$722$	4.51114	−	18.4567i	0.167887	−	0.686887i
$723$		0			0
$724$			15.2309i			0.566051i
$725$	−8.97773			−0.333424
$726$		0			0
$727$	−0.266706			−0.00989159			−0.00494579	−	0.999988i	$-0.501574\pi$
$727$	−0.266706			−0.00989159			−0.00494579	+	0.999988i	$0.501574\pi$
$728$		−	2.15728i		−	0.0799541i
$729$		0			0
$730$		−	6.89573i		−	0.255222i
$731$			21.1552i			0.782454i
$732$		0			0
$733$	7.13383			0.263494			0.131747	−	0.991283i	$-0.457941\pi$
$733$	7.13383			0.263494			0.131747	+	0.991283i	$0.457941\pi$
$734$	14.5718			0.537856
$735$		0			0
$736$			6.39992i			0.235904i
$737$	0.561993			0.0207013
$738$		0			0
$739$	26.1936			0.963546			0.481773	−	0.876296i	$-0.339993\pi$
$739$	26.1936			0.963546			0.481773	+	0.876296i	$0.339993\pi$
$740$	−28.2766			−1.03947
$741$		0			0
$742$	−10.8573			−0.398583
$743$	−27.3176			−1.00218			−0.501092	−	0.865394i	$-0.667068\pi$
$743$	−27.3176			−1.00218			−0.501092	+	0.865394i	$0.667068\pi$
$744$		0			0
$745$	−9.24443			−0.338690
$746$		−	18.5734i		−	0.680019i
$747$		0			0
$748$	28.4701			1.04097
$749$	15.4193			0.563408
$750$		0			0
$751$			34.0787i			1.24355i	0.783196	+	0.621774i	$0.213588\pi$
$751$			34.0787i			1.24355i	−0.783196	+	0.621774i	$0.786412\pi$
$752$		−	6.26304i		−	0.228390i
$753$		0			0
$754$		−	8.62912i		−	0.314254i
$755$	−6.54326			−0.238133
$756$		0			0
$757$	−51.7975			−1.88261			−0.941307	−	0.337553i	$-0.890401\pi$
$757$	−51.7975			−1.88261			−0.941307	+	0.337553i	$0.890401\pi$
$758$			2.04061i			0.0741183i
$759$		0			0
$760$	−7.24443	−	9.22835i	−0.262783	−	0.334747i
$761$			27.5280i			0.997888i	0.866634	+	0.498944i	$0.166279\pi$
$761$			27.5280i			0.997888i	−0.866634	+	0.498944i	$0.833721\pi$
$762$		0			0
$763$			16.1625i			0.585123i
$764$			8.95459i			0.323966i
$765$		0			0
$766$	−23.1240			−0.835504
$767$		−	5.94453i		−	0.214645i
$768$		0			0
$769$	17.5299			0.632143			0.316072	−	0.948735i	$-0.397636\pi$
$769$	17.5299			0.632143			0.316072	+	0.948735i	$0.397636\pi$
$770$	−11.0509			−0.398245
$771$		0			0
$772$		−	12.7998i		−	0.460676i
$773$	−0.222156			−0.00799041			−0.00399520	−	0.999992i	$-0.501272\pi$
$773$	−0.222156			−0.00799041			−0.00399520	+	0.999992i	$0.501272\pi$
$774$		0			0
$775$			19.2061i			0.689902i
$776$		−	7.19470i		−	0.258275i
$777$		0			0
$778$		−	23.1037i		−	0.828307i
$779$	−24.1748	+	18.9777i	−0.866154	+	0.679947i
$780$		0			0
$781$			42.5702i			1.52328i
$782$	−44.3783			−1.58696
$783$		0			0
$784$	1.00000			0.0357143
$785$		−	21.8061i		−	0.778295i
$786$		0			0
$787$			20.8675i			0.743847i	0.928263	+	0.371923i	$0.121302\pi$
$787$			20.8675i			0.743847i	−0.928263	+	0.371923i	$0.878698\pi$
$788$			4.12597i			0.146982i
$789$		0			0
$790$	20.2953			0.722074
$791$	18.8573			0.670488
$792$		0			0
$793$			12.5261i			0.444814i
$794$	−10.5906			−0.375845
$795$		0			0
$796$	−19.0509			−0.675240
$797$	−21.0223			−0.744647			−0.372324	−	0.928103i	$-0.621439\pi$
$797$	−21.0223			−0.744647			−0.372324	+	0.928103i	$0.621439\pi$
$798$		0			0
$799$	43.4291			1.53641
$800$	−2.24443			−0.0793526
$801$		0			0
$802$	17.1338			0.605017
$803$		−	10.5189i		−	0.371205i
$804$		0			0
$805$	17.2257			0.607126
$806$	−18.4603			−0.650236
$807$		0			0
$808$		−	9.41694i		−	0.331287i
$809$		−	39.5135i		−	1.38922i	−0.719386	−	0.694610i	$-0.755577\pi$
$809$		−	39.5135i		−	1.38922i	0.719386	−	0.694610i	$-0.244423\pi$
$810$		0			0
$811$		−	27.4895i		−	0.965287i	−0.875817	−	0.482643i	$-0.839677\pi$
$811$		−	27.4895i		−	0.965287i	0.875817	−	0.482643i	$-0.160323\pi$
$812$	4.00000			0.140372
$813$		0			0
$814$	−43.1338			−1.51184
$815$		−	11.5610i		−	0.404963i
$816$		0			0
$817$	10.4603	−	8.21152i	0.365959	−	0.287285i
$818$			7.46846i			0.261128i
$819$		0			0
$820$			18.9777i			0.662731i
$821$		−	23.6512i		−	0.825433i	−0.910860	−	0.412716i	$-0.864580\pi$
$821$		−	23.6512i		−	0.825433i	0.910860	−	0.412716i	$-0.135420\pi$
$822$		0			0
$823$	5.97142			0.208151			0.104075	−	0.994569i	$-0.466812\pi$
$823$	5.97142			0.208151			0.104075	+	0.994569i	$0.466812\pi$
$824$		−	5.72877i		−	0.199571i
$825$		0			0
$826$	2.75557			0.0958785
$827$	43.8163			1.52364			0.761820	−	0.647788i	$-0.224306\pi$
$827$	43.8163			1.52364			0.761820	+	0.647788i	$0.224306\pi$
$828$		0			0
$829$		−	14.1623i		−	0.491879i	−0.969285	−	0.245939i	$-0.920904\pi$
$829$		−	14.1623i		−	0.491879i	0.969285	−	0.245939i	$-0.0790964\pi$
$830$	13.0509			0.453002
$831$		0			0
$832$		−	2.15728i		−	0.0747902i
$833$			6.93419i			0.240255i
$834$		0			0
$835$			38.6733i			1.33834i
$836$	−11.0509	−	14.0772i	−0.382202	−	0.486869i
$837$		0			0
$838$			16.9573i			0.585781i
$839$	43.7975			1.51206			0.756029	−	0.654538i	$-0.227137\pi$
$839$	43.7975			1.51206			0.756029	+	0.654538i	$0.227137\pi$
$840$		0			0
$841$	−13.0000			−0.448276
$842$			23.9704i			0.826074i
$843$		0			0
$844$			2.69155i			0.0926469i
$845$		−	22.4640i		−	0.772787i
$846$		0			0
$847$	−5.85728			−0.201258
$848$	−10.8573			−0.372840
$849$		0			0
$850$		−	15.5633i		−	0.533817i
$851$	67.2355			2.30480
$852$		0			0
$853$	32.8385			1.12437			0.562185	−	0.827011i	$-0.309961\pi$
$853$	32.8385			1.12437			0.562185	+	0.827011i	$0.309961\pi$
$854$	−5.80642			−0.198692
$855$		0			0
$856$	15.4193			0.527020
$857$	−27.6040			−0.942933			−0.471467	−	0.881884i	$-0.656275\pi$
$857$	−27.6040			−0.942933			−0.471467	+	0.881884i	$0.656275\pi$
$858$		0			0
$859$	17.5941			0.600303			0.300152	−	0.953891i	$-0.402963\pi$
$859$	17.5941			0.600303			0.300152	+	0.953891i	$0.402963\pi$
$860$		−	8.21152i		−	0.280011i
$861$		0			0
$862$	21.6860			0.738627
$863$	−48.8484			−1.66282			−0.831409	−	0.555661i	$-0.812465\pi$
$863$	−48.8484			−1.66282			−0.831409	+	0.555661i	$0.812465\pi$
$864$		0			0
$865$		−	56.6594i		−	1.92648i
$866$			28.1026i			0.954967i
$867$		0			0
$868$		−	8.55720i		−	0.290450i
$869$	30.9590			1.05021
$870$		0			0
$871$	0.295286			0.0100054
$872$			16.1625i			0.547333i
$873$		0			0
$874$	17.2257	+	21.9430i	0.582668	+	0.742234i
$875$		−	7.41675i		−	0.250732i
$876$		0			0
$877$		−	39.1671i		−	1.32258i	−0.750130	−	0.661290i	$-0.770009\pi$
$877$		−	39.1671i		−	1.32258i	0.750130	−	0.661290i	$-0.229991\pi$
$878$		−	1.83181i		−	0.0618206i
$879$		0			0
$880$	−11.0509			−0.372524
$881$		−	4.10576i		−	0.138327i	−0.997605	−	0.0691633i	$-0.977967\pi$
$881$		−	4.10576i		−	0.138327i	0.997605	−	0.0691633i	$-0.0220330\pi$
$882$		0			0
$883$	27.1052			0.912164			0.456082	−	0.889938i	$-0.349252\pi$
$883$	27.1052			0.912164			0.456082	+	0.889938i	$0.349252\pi$
$884$	14.9590			0.503125
$885$		0			0
$886$		−	14.6247i		−	0.491326i
$887$	−24.6923			−0.829086			−0.414543	−	0.910030i	$-0.636059\pi$
$887$	−24.6923			−0.829086			−0.414543	+	0.910030i	$0.636059\pi$
$888$		0			0
$889$			11.7111i			0.392778i
$890$			8.21152i			0.275251i
$891$		0			0
$892$		−	7.76242i		−	0.259905i
$893$	−16.8573	−	21.4737i	−0.564107	−	0.718590i
$894$		0			0
$895$			20.4903i			0.684917i
$896$	1.00000			0.0334077
$897$		0			0
$898$	−30.0000			−1.00111
$899$		−	34.2288i		−	1.14159i
$900$		0			0
$901$		−	75.2864i		−	2.50815i
$902$			28.9491i			0.963901i
$903$		0			0
$904$	18.8573			0.627184
$905$	40.9946			1.36271
$906$		0			0
$907$		−	13.3278i		−	0.442543i	−0.975212	−	0.221272i	$-0.928979\pi$
$907$		−	13.3278i		−	0.442543i	0.975212	−	0.221272i	$-0.0710207\pi$
$908$	−24.3783			−0.809021
$909$		0			0
$910$	−5.80642			−0.192481
$911$	44.3497			1.46937			0.734685	−	0.678408i	$-0.237330\pi$
$911$	44.3497			1.46937			0.734685	+	0.678408i	$0.237330\pi$
$912$		0			0
$913$	19.9081			0.658863
$914$	−39.6958			−1.31302
$915$		0			0
$916$	−15.7146			−0.519224
$917$			5.12259i			0.169163i
$918$		0			0
$919$	10.0286			0.330812			0.165406	−	0.986226i	$-0.447106\pi$
$919$	10.0286			0.330812			0.165406	+	0.986226i	$0.447106\pi$
$920$	17.2257			0.567914
$921$		0			0
$922$			1.20541i			0.0396982i
$923$			22.3676i			0.736238i
$924$		0			0
$925$			23.5793i			0.775282i
$926$	−20.6637			−0.679051
$927$		0			0
$928$	4.00000			0.131306
$929$		−	52.0326i		−	1.70713i	−0.520983	−	0.853567i	$-0.674434\pi$
$929$		−	52.0326i		−	1.70713i	0.520983	−	0.853567i	$-0.325566\pi$
$930$		0			0
$931$	3.42864	−	2.69155i	0.112369	−	0.0882119i
$932$			17.0425i			0.558245i
$933$		0			0
$934$		−	41.7622i		−	1.36650i
$935$		−	76.6287i		−	2.50603i
$936$		0			0
$937$	−13.1971			−0.431131			−0.215565	−	0.976489i	$-0.569159\pi$
$937$	−13.1971			−0.431131			−0.215565	+	0.976489i	$0.569159\pi$
$938$			0.136879i			0.00446926i
$939$		0			0
$940$	−16.8573			−0.549824
$941$	−6.65386			−0.216910			−0.108455	−	0.994101i	$-0.534590\pi$
$941$	−6.65386			−0.216910			−0.108455	+	0.994101i	$0.534590\pi$
$942$		0			0
$943$		−	45.1249i		−	1.46947i
$944$	2.75557			0.0896861
$945$		0			0
$946$		−	12.5261i		−	0.407258i
$947$			24.4662i			0.795044i	0.917593	+	0.397522i	$0.130130\pi$
$947$			24.4662i			0.795044i	−0.917593	+	0.397522i	$0.869870\pi$
$948$		0			0
$949$		−	5.52694i		−	0.179412i
$950$	−7.69535	+	6.04099i	−0.249670	+	0.195996i
$951$		0			0
$952$			6.93419i			0.224738i
$953$	−33.9367			−1.09932			−0.549659	−	0.835389i	$-0.685242\pi$
$953$	−33.9367			−1.09932			−0.549659	+	0.835389i	$0.685242\pi$
$954$		0			0
$955$	24.1017			0.779913
$956$			25.7952i			0.834278i
$957$		0			0
$958$			15.1659i			0.489989i
$959$			2.20899i			0.0713320i
$960$		0			0
$961$	−42.2257			−1.36212
$962$	−22.6637			−0.730707
$963$		0			0
$964$		−	14.3377i		−	0.461786i
$965$	−34.4514			−1.10903
$966$		0			0
$967$	32.0731			1.03140			0.515701	−	0.856769i	$-0.327531\pi$
$967$	32.0731			1.03140			0.515701	+	0.856769i	$0.327531\pi$
$968$	−5.85728			−0.188260
$969$		0			0
$970$	−19.3649			−0.621769
$971$	−43.2168			−1.38689			−0.693447	−	0.720508i	$-0.743909\pi$
$971$	−43.2168			−1.38689			−0.693447	+	0.720508i	$0.743909\pi$
$972$		0			0
$973$	7.61285			0.244057
$974$		−	10.6426i		−	0.341010i
$975$		0			0
$976$	−5.80642			−0.185859
$977$	5.90813			0.189018			0.0945090	−	0.995524i	$-0.469872\pi$
$977$	5.90813			0.189018			0.0945090	+	0.995524i	$0.469872\pi$
$978$		0			0
$979$			12.5261i			0.400335i
$980$		−	2.69155i		−	0.0859784i
$981$		0			0
$982$			21.7412i			0.693789i
$983$	−23.0607			−0.735522			−0.367761	−	0.929920i	$-0.619876\pi$
$983$	−23.0607			−0.735522			−0.367761	+	0.929920i	$0.619876\pi$
$984$		0			0
$985$	11.1052			0.353843
$986$			27.7368i			0.883318i
$987$		0			0
$988$	−5.80642	−	7.39654i	−0.184727	−	0.235315i
$989$			19.5252i			0.620866i
$990$		0			0
$991$		−	8.88267i		−	0.282167i	−0.989998	−	0.141084i	$-0.954941\pi$
$991$		−	8.88267i		−	0.282167i	0.989998	−	0.141084i	$-0.0450587\pi$
$992$		−	8.55720i		−	0.271691i
$993$		0			0
$994$	−10.3684			−0.328866
$995$			51.2763i			1.62557i
$996$		0			0
$997$	−42.5906			−1.34886			−0.674428	−	0.738340i	$-0.735610\pi$
$997$	−42.5906			−1.34886			−0.674428	+	0.738340i	$0.735610\pi$
$998$	−12.0830			−0.382480
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2394.2.b.h.1709.2	yes	6
3.2	odd	2		2394.2.b.g.1709.5	yes	6
19.18	odd	2		2394.2.b.g.1709.2	✓	6
57.56	even	2	inner	2394.2.b.h.1709.5	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2394.2.b.g.1709.2	✓	6	19.18	odd	2
2394.2.b.g.1709.5	yes	6	3.2	odd	2
2394.2.b.h.1709.2	yes	6	1.1	even	1	trivial
2394.2.b.h.1709.5	yes	6	57.56	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:	\( N \)	\(=\)	\( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2394.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -2.69155i q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -2.69155i q^{5} +1.00000 q^{7} +1.00000 q^{8} -2.69155i q^{10} -4.10576i q^{11} -2.15728i q^{13} +1.00000 q^{14} +1.00000 q^{16} +6.93419i q^{17} +(3.42864 - 2.69155i) q^{19} -2.69155i q^{20} -4.10576i q^{22} +6.39992i q^{23} -2.24443 q^{25} -2.15728i q^{26} +1.00000 q^{28} +4.00000 q^{29} -8.55720i q^{31} +1.00000 q^{32} +6.93419i q^{34} -2.69155i q^{35} -10.5057i q^{37} +(3.42864 - 2.69155i) q^{38} -2.69155i q^{40} -7.05086 q^{41} +3.05086 q^{43} -4.10576i q^{44} +6.39992i q^{46} -6.26304i q^{47} +1.00000 q^{49} -2.24443 q^{50} -2.15728i q^{52} -10.8573 q^{53} -11.0509 q^{55} +1.00000 q^{56} +4.00000 q^{58} +2.75557 q^{59} -5.80642 q^{61} -8.55720i q^{62} +1.00000 q^{64} -5.80642 q^{65} +0.136879i q^{67} +6.93419i q^{68} -2.69155i q^{70} -10.3684 q^{71} +2.56199 q^{73} -10.5057i q^{74} +(3.42864 - 2.69155i) q^{76} -4.10576i q^{77} +7.54038i q^{79} -2.69155i q^{80} -7.05086 q^{82} +4.84883i q^{83} +18.6637 q^{85} +3.05086 q^{86} -4.10576i q^{88} -3.05086 q^{89} -2.15728i q^{91} +6.39992i q^{92} -6.26304i q^{94} +(-7.24443 - 9.22835i) q^{95} -7.19470i q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} + 6 q^{4} + 6 q^{7} + 6 q^{8}+O(q^{10}) \) 6 * q + 6 * q^2 + 6 * q^4 + 6 * q^7 + 6 * q^8	\( 6 q + 6 q^{2} + 6 q^{4} + 6 q^{7} + 6 q^{8} + 6 q^{14} + 6 q^{16} - 6 q^{19} - 14 q^{25} + 6 q^{28} + 24 q^{29} + 6 q^{32} - 6 q^{38} - 16 q^{41} - 8 q^{43} + 6 q^{49} - 14 q^{50} - 12 q^{53} - 40 q^{55} + 6 q^{56} + 24 q^{58} + 16 q^{59} - 8 q^{61} + 6 q^{64} - 8 q^{65} - 8 q^{71} - 12 q^{73} - 6 q^{76} - 16 q^{82} + 32 q^{85} - 8 q^{86} + 8 q^{89} - 44 q^{95} + 6 q^{98}+O(q^{100}) \) 6 * q + 6 * q^2 + 6 * q^4 + 6 * q^7 + 6 * q^8 + 6 * q^14 + 6 * q^16 - 6 * q^19 - 14 * q^25 + 6 * q^28 + 24 * q^29 + 6 * q^32 - 6 * q^38 - 16 * q^41 - 8 * q^43 + 6 * q^49 - 14 * q^50 - 12 * q^53 - 40 * q^55 + 6 * q^56 + 24 * q^58 + 16 * q^59 - 8 * q^61 + 6 * q^64 - 8 * q^65 - 8 * q^71 - 12 * q^73 - 6 * q^76 - 16 * q^82 + 32 * q^85 - 8 * q^86 + 8 * q^89 - 44 * q^95 + 6 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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