LMFDB - Embedded newform 336.9.f.d.97.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,9,Mod(97,336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(336, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("336.97");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	336.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$136.879212981$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			97.9
Character	$\chi$	$=$	336.97
Dual form			336.9.f.d.97.24

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-46.7654i q^{3} +36.6522i q^{5} +(1277.06 + 2033.20i) q^{7} -2187.00 q^{9} +O(q^{10})$	$q-46.7654i q^{3} +36.6522i q^{5} +(1277.06 + 2033.20i) q^{7} -2187.00 q^{9} +22682.7 q^{11} -3619.14i q^{13} +1714.05 q^{15} +44158.6i q^{17} -10828.6i q^{19} +(95083.4 - 59722.4i) q^{21} +70487.9 q^{23} +389282. q^{25} +102276. i q^{27} -40854.0 q^{29} +387397. i q^{31} -1.06076e6i q^{33} +(-74521.3 + 46807.2i) q^{35} +775814. q^{37} -169250. q^{39} -1.44536e6i q^{41} -3.35382e6 q^{43} -80158.4i q^{45} -3.28548e6i q^{47} +(-2.50302e6 + 5.19306e6i) q^{49} +2.06509e6 q^{51} +3.39129e6 q^{53} +831371. i q^{55} -506402. q^{57} +1.77644e7i q^{59} +2.03993e7i q^{61} +(-2.79294e6 - 4.44661e6i) q^{63} +132649. q^{65} -1.12148e7 q^{67} -3.29639e6i q^{69} -1.27220e7 q^{71} -1.88221e6i q^{73} -1.82049e7i q^{75} +(2.89672e7 + 4.61185e7i) q^{77} -4.08976e7 q^{79} +4.78297e6 q^{81} -3.35353e7i q^{83} -1.61851e6 q^{85} +1.91055e6i q^{87} +2.39480e7i q^{89} +(7.35843e6 - 4.62187e6i) q^{91} +1.81168e7 q^{93} +396891. q^{95} -4.47828e7i q^{97} -4.96071e7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 1424 q^{7} - 69984 q^{9}+O(q^{10}) $ 32 * q - 1424 * q^7 - 69984 * q^9	$ 32 q - 1424 q^{7} - 69984 q^{9} - 23616 q^{11} + 59616 q^{15} + 60912 q^{21} - 455040 q^{23} - 3404928 q^{25} + 632064 q^{29} + 1543200 q^{35} + 4150496 q^{37} + 4162752 q^{39} - 6028000 q^{43} + 15115072 q^{49} + 1340064 q^{51} - 37728576 q^{53} + 7286112 q^{57} + 3114288 q^{63} + 29977536 q^{65} - 68431648 q^{67} + 19788096 q^{71} + 87499392 q^{77} + 86954656 q^{79} + 153055008 q^{81} - 31326496 q^{85} + 24540288 q^{91} - 38833344 q^{93} - 329546688 q^{95} + 51648192 q^{99}+O(q^{100}) $ 32 * q - 1424 * q^7 - 69984 * q^9 - 23616 * q^11 + 59616 * q^15 + 60912 * q^21 - 455040 * q^23 - 3404928 * q^25 + 632064 * q^29 + 1543200 * q^35 + 4150496 * q^37 + 4162752 * q^39 - 6028000 * q^43 + 15115072 * q^49 + 1340064 * q^51 - 37728576 * q^53 + 7286112 * q^57 + 3114288 * q^63 + 29977536 * q^65 - 68431648 * q^67 + 19788096 * q^71 + 87499392 * q^77 + 86954656 * q^79 + 153055008 * q^81 - 31326496 * q^85 + 24540288 * q^91 - 38833344 * q^93 - 329546688 * q^95 + 51648192 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$.

$n$	$85$	$113$	$127$	$241$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$		−	46.7654i		−	0.577350i
$3$		−	46.7654i		−	0.577350i
$4$		0			0
$5$			36.6522i			0.0586435i	0.999570	+	0.0293218i	$0.00933474\pi$
$5$			36.6522i			0.0586435i	−0.999570	+	0.0293218i	$0.990665\pi$
$6$		0			0
$7$	1277.06	+	2033.20i	0.531888	+	0.846815i
$7$	1277.06	+	2033.20i	0.531888	+	0.846815i
$8$		0			0
$9$	−2187.00			−0.333333
$10$		0			0
$11$	22682.7			1.54926			0.774629	−	0.632416i	$-0.217936\pi$
$11$	22682.7			1.54926			0.774629	+	0.632416i	$0.217936\pi$
$12$		0			0
$13$		−	3619.14i		−	0.126716i	−0.997991	−	0.0633580i	$-0.979819\pi$
$13$		−	3619.14i		−	0.126716i	0.997991	−	0.0633580i	$-0.0201810\pi$
$14$		0			0
$15$	1714.05			0.0338578
$16$		0			0
$17$			44158.6i			0.528712i	0.964425	+	0.264356i	$0.0851594\pi$
$17$			44158.6i			0.528712i	−0.964425	+	0.264356i	$0.914841\pi$
$18$		0			0
$19$		−	10828.6i		−	0.0830915i	−0.999137	−	0.0415457i	$-0.986772\pi$
$19$		−	10828.6i		−	0.0830915i	0.999137	−	0.0415457i	$-0.0132282\pi$
$20$		0			0
$21$	95083.4	−	59722.4i	0.488909	−	0.307086i
$22$		0			0
$23$	70487.9			0.251886			0.125943	−	0.992038i	$-0.459804\pi$
$23$	70487.9			0.251886			0.125943	+	0.992038i	$0.459804\pi$
$24$		0			0
$25$	389282.			0.996561
$26$		0			0
$27$			102276.i			0.192450i
$28$		0			0
$29$	−40854.0			−0.0577621			−0.0288810	−	0.999583i	$-0.509194\pi$
$29$	−40854.0			−0.0577621			−0.0288810	+	0.999583i	$0.509194\pi$
$30$		0			0
$31$			387397.i			0.419479i	0.977757	+	0.209739i	$0.0672616\pi$
$31$			387397.i			0.419479i	−0.977757	+	0.209739i	$0.932738\pi$
$32$		0			0
$33$		−	1.06076e6i		−	0.894465i
$34$		0			0
$35$	−74521.3	+	46807.2i	−0.0496602	+	0.0311918i
$36$		0			0
$37$	775814.			0.413953			0.206976	−	0.978346i	$-0.433638\pi$
$37$	775814.			0.413953			0.206976	+	0.978346i	$0.433638\pi$
$38$		0			0
$39$	−169250.			−0.0731595
$40$		0			0
$41$		−	1.44536e6i		−	0.511493i	−0.966744	−	0.255747i	$-0.917679\pi$
$41$		−	1.44536e6i		−	0.511493i	0.966744	−	0.255747i	$-0.0823212\pi$
$42$		0			0
$43$	−3.35382e6			−0.980992			−0.490496	−	0.871443i	$-0.663184\pi$
$43$	−3.35382e6			−0.980992			−0.490496	+	0.871443i	$0.663184\pi$
$44$		0			0
$45$		−	80158.4i		−	0.0195478i
$46$		0			0
$47$		−	3.28548e6i		−	0.673298i	−0.941630	−	0.336649i	$-0.890706\pi$
$47$		−	3.28548e6i		−	0.673298i	0.941630	−	0.336649i	$-0.109294\pi$
$48$		0			0
$49$	−2.50302e6	+	5.19306e6i	−0.434190	+	0.900821i
$50$		0			0
$51$	2.06509e6			0.305252
$52$		0			0
$53$	3.39129e6			0.429795			0.214898	−	0.976637i	$-0.431058\pi$
$53$	3.39129e6			0.429795			0.214898	+	0.976637i	$0.431058\pi$
$54$		0			0
$55$			831371.i			0.0908540i
$56$		0			0
$57$	−506402.			−0.0479729
$58$		0			0
$59$			1.77644e7i			1.46603i	0.680211	+	0.733016i	$0.261888\pi$
$59$			1.77644e7i			1.46603i	−0.680211	+	0.733016i	$0.738112\pi$
$60$		0			0
$61$			2.03993e7i			1.47331i	0.676266	+	0.736657i	$0.263597\pi$
$61$			2.03993e7i			1.47331i	−0.676266	+	0.736657i	$0.736403\pi$
$62$		0			0
$63$	−2.79294e6	−	4.44661e6i	−0.177296	−	0.282272i
$64$		0			0
$65$	132649.			0.00743107
$66$		0			0
$67$	−1.12148e7			−0.556534			−0.278267	−	0.960504i	$-0.589760\pi$
$67$	−1.12148e7			−0.556534			−0.278267	+	0.960504i	$0.589760\pi$
$68$		0			0
$69$		−	3.29639e6i		−	0.145426i
$70$		0			0
$71$	−1.27220e7			−0.500637			−0.250318	−	0.968164i	$-0.580535\pi$
$71$	−1.27220e7			−0.500637			−0.250318	+	0.968164i	$0.580535\pi$
$72$		0			0
$73$		−	1.88221e6i		−	0.0662792i	−0.999451	−	0.0331396i	$-0.989449\pi$
$73$		−	1.88221e6i		−	0.0662792i	0.999451	−	0.0331396i	$-0.0105506\pi$
$74$		0			0
$75$		−	1.82049e7i		−	0.575365i
$76$		0			0
$77$	2.89672e7	+	4.61185e7i	0.824032	+	1.31193i
$78$		0			0
$79$	−4.08976e7			−1.05000			−0.525000	−	0.851102i	$-0.675935\pi$
$79$	−4.08976e7			−1.05000			−0.525000	+	0.851102i	$0.675935\pi$
$80$		0			0
$81$	4.78297e6			0.111111
$82$		0			0
$83$		−	3.35353e7i		−	0.706626i	−0.935505	−	0.353313i	$-0.885055\pi$
$83$		−	3.35353e7i		−	0.706626i	0.935505	−	0.353313i	$-0.114945\pi$
$84$		0			0
$85$	−1.61851e6			−0.0310055
$86$		0			0
$87$			1.91055e6i			0.0333490i
$88$		0			0
$89$			2.39480e7i			0.381689i	0.981620	+	0.190844i	$0.0611226\pi$
$89$			2.39480e7i			0.381689i	−0.981620	+	0.190844i	$0.938877\pi$
$90$		0			0
$91$	7.35843e6	−	4.62187e6i	0.107305	−	0.0673988i
$92$		0			0
$93$	1.81168e7			0.242186
$94$		0			0
$95$	396891.			0.00487277
$96$		0			0
$97$		−	4.47828e7i		−	0.505853i	−0.967486	−	0.252926i	$-0.918607\pi$
$97$		−	4.47828e7i		−	0.505853i	0.967486	−	0.252926i	$-0.0813930\pi$
$98$		0			0
$99$	−4.96071e7			−0.516420
$100$		0			0
$101$			1.38718e8i			1.33305i	0.745483	+	0.666525i	$0.232219\pi$
$101$			1.38718e8i			1.33305i	−0.745483	+	0.666525i	$0.767781\pi$
$102$		0			0
$103$		−	2.38466e7i		−	0.211874i	−0.994373	−	0.105937i	$-0.966216\pi$
$103$		−	2.38466e7i		−	0.211874i	0.994373	−	0.105937i	$-0.0337842\pi$
$104$		0			0
$105$	2.18896e6	+	3.48502e6i	0.0180086	+	0.0286713i
$106$		0			0
$107$	−1.45057e7			−0.110663			−0.0553316	−	0.998468i	$-0.517622\pi$
$107$	−1.45057e7			−0.110663			−0.0553316	+	0.998468i	$0.517622\pi$
$108$		0			0
$109$	1.77198e8			1.25531			0.627656	−	0.778491i	$-0.284014\pi$
$109$	1.77198e8			1.25531			0.627656	+	0.778491i	$0.284014\pi$
$110$		0			0
$111$		−	3.62812e7i		−	0.238996i
$112$		0			0
$113$	2.73960e8			1.68025			0.840125	−	0.542392i	$-0.182481\pi$
$113$	2.73960e8			1.68025			0.840125	+	0.542392i	$0.182481\pi$
$114$		0			0
$115$			2.58354e6i			0.0147715i
$116$		0			0
$117$			7.91505e6i			0.0422387i
$118$		0			0
$119$	−8.97833e7	+	5.63933e7i	−0.447721	+	0.281216i
$120$		0			0
$121$	3.00146e8			1.40020
$122$		0			0
$123$	−6.75927e7			−0.295311
$124$		0			0
$125$			2.85853e7i			0.117085i
$126$		0			0
$127$	−9.71441e6			−0.0373424			−0.0186712	−	0.999826i	$-0.505944\pi$
$127$	−9.71441e6			−0.0373424			−0.0186712	+	0.999826i	$0.505944\pi$
$128$		0			0
$129$			1.56842e8i			0.566376i
$130$		0			0
$131$			1.57678e8i			0.535408i	0.963501	+	0.267704i	$0.0862650\pi$
$131$			1.57678e8i			0.535408i	−0.963501	+	0.267704i	$0.913735\pi$
$132$		0			0
$133$	2.20167e7	−	1.38288e7i	0.0703631	−	0.0441954i
$134$		0			0
$135$	−3.74864e6			−0.0112859
$136$		0			0
$137$	3.30455e8			0.938060			0.469030	−	0.883182i	$-0.344604\pi$
$137$	3.30455e8			0.938060			0.469030	+	0.883182i	$0.344604\pi$
$138$		0			0
$139$		−	3.64206e8i		−	0.975637i	−0.872945	−	0.487818i	$-0.837793\pi$
$139$		−	3.64206e8i		−	0.975637i	0.872945	−	0.487818i	$-0.162207\pi$
$140$		0			0
$141$	−1.53647e8			−0.388729
$142$		0			0
$143$		−	8.20918e7i		−	0.196316i
$144$		0			0
$145$		−	1.49739e6i		−	0.00338737i
$146$		0			0
$147$	2.42855e8	+	1.17055e8i	0.520090	+	0.250680i
$148$		0			0
$149$	−4.08404e8			−0.828600			−0.414300	−	0.910140i	$-0.635974\pi$
$149$	−4.08404e8			−0.828600			−0.414300	+	0.910140i	$0.635974\pi$
$150$		0			0
$151$	−1.66682e8			−0.320613			−0.160306	−	0.987067i	$-0.551248\pi$
$151$	−1.66682e8			−0.320613			−0.160306	+	0.987067i	$0.551248\pi$
$152$		0			0
$153$		−	9.65748e7i		−	0.176237i
$154$		0			0
$155$	−1.41990e7			−0.0245997
$156$		0			0
$157$			6.31955e8i			1.04013i	0.854127	+	0.520065i	$0.174092\pi$
$157$			6.31955e8i			1.04013i	−0.854127	+	0.520065i	$0.825908\pi$
$158$		0			0
$159$		−	1.58595e8i		−	0.248142i
$160$		0			0
$161$	9.00176e7	+	1.43316e8i	0.133975	+	0.213300i
$162$		0			0
$163$	2.58717e8			0.366501			0.183251	−	0.983066i	$-0.441338\pi$
$163$	2.58717e8			0.366501			0.183251	+	0.983066i	$0.441338\pi$
$164$		0			0
$165$	3.88794e7			0.0524546
$166$		0			0
$167$		−	6.23385e8i		−	0.801476i	−0.916193	−	0.400738i	$-0.868754\pi$
$167$		−	6.23385e8i		−	0.801476i	0.916193	−	0.400738i	$-0.131246\pi$
$168$		0			0
$169$	8.02633e8			0.983943
$170$		0			0
$171$			2.36821e7i			0.0276972i
$172$		0			0
$173$			1.83280e8i			0.204612i	0.994753	+	0.102306i	$0.0326220\pi$
$173$			1.83280e8i			0.204612i	−0.994753	+	0.102306i	$0.967378\pi$
$174$		0			0
$175$	4.97137e8	+	7.91488e8i	0.530059	+	0.843902i
$176$		0			0
$177$	8.30761e8			0.846414
$178$		0			0
$179$	8.36466e8			0.814772			0.407386	−	0.913256i	$-0.366440\pi$
$179$	8.36466e8			0.814772			0.407386	+	0.913256i	$0.366440\pi$
$180$		0			0
$181$		−	6.79300e8i		−	0.632918i	−0.948606	−	0.316459i	$-0.897506\pi$
$181$		−	6.79300e8i		−	0.632918i	0.948606	−	0.316459i	$-0.102494\pi$
$182$		0			0
$183$	9.53980e8			0.850619
$184$		0			0
$185$			2.84353e7i			0.0242756i
$186$		0			0
$187$			1.00164e9i			0.819112i
$188$		0			0
$189$	−2.07947e8	+	1.30613e8i	−0.162970	+	0.102362i
$190$		0			0
$191$	2.62916e9			1.97553			0.987766	−	0.155943i	$-0.0498416\pi$
$191$	2.62916e9			1.97553			0.987766	+	0.155943i	$0.0498416\pi$
$192$		0			0
$193$	5.98531e8			0.431377			0.215689	−	0.976462i	$-0.430800\pi$
$193$	5.98531e8			0.431377			0.215689	+	0.976462i	$0.430800\pi$
$194$		0			0
$195$		−	6.20339e6i		−	0.00429033i
$196$		0			0
$197$	8.95001e8			0.594236			0.297118	−	0.954841i	$-0.403975\pi$
$197$	8.95001e8			0.594236			0.297118	+	0.954841i	$0.403975\pi$
$198$		0			0
$199$			2.71031e9i			1.72825i	0.503276	+	0.864126i	$0.332128\pi$
$199$			2.71031e9i			1.72825i	−0.503276	+	0.864126i	$0.667872\pi$
$200$		0			0
$201$			5.24464e8i			0.321315i
$202$		0			0
$203$	−5.21732e7	−	8.30645e7i	−0.0307230	−	0.0489138i
$204$		0			0
$205$	5.29755e7			0.0299957
$206$		0			0
$207$	−1.54157e8			−0.0839619
$208$		0			0
$209$		−	2.45621e8i		−	0.128730i
$210$		0			0
$211$	2.50037e9			1.26146			0.630732	−	0.776001i	$-0.282755\pi$
$211$	2.50037e9			1.26146			0.630732	+	0.776001i	$0.282755\pi$
$212$		0			0
$213$			5.94950e8i			0.289043i
$214$		0			0
$215$		−	1.22925e8i		−	0.0575288i
$216$		0			0
$217$	−7.87657e8	+	4.94731e8i	−0.355221	+	0.223116i
$218$		0			0
$219$	−8.80224e7			−0.0382663
$220$		0			0
$221$	1.59816e8			0.0669963
$222$		0			0
$223$			3.78547e9i			1.53074i	0.643592	+	0.765369i	$0.277443\pi$
$223$			3.78547e9i			1.53074i	−0.643592	+	0.765369i	$0.722557\pi$
$224$		0			0
$225$	−8.51359e8			−0.332187
$226$		0			0
$227$			2.23992e9i			0.843587i	0.906692	+	0.421793i	$0.138599\pi$
$227$			2.23992e9i			0.843587i	−0.906692	+	0.421793i	$0.861401\pi$
$228$		0			0
$229$			4.30162e8i			0.156419i	0.996937	+	0.0782095i	$0.0249203\pi$
$229$			4.30162e8i			0.156419i	−0.996937	+	0.0782095i	$0.975080\pi$
$230$		0			0
$231$	2.15675e9	−	1.35466e9i	0.757446	−	0.475755i
$232$		0			0
$233$	−3.41665e9			−1.15925			−0.579624	−	0.814884i	$-0.696801\pi$
$233$	−3.41665e9			−1.15925			−0.579624	+	0.814884i	$0.696801\pi$
$234$		0			0
$235$	1.20420e8			0.0394846
$236$		0			0
$237$			1.91259e9i			0.606218i
$238$		0			0
$239$	1.43804e8			0.0440738			0.0220369	−	0.999757i	$-0.492985\pi$
$239$	1.43804e8			0.0440738			0.0220369	+	0.999757i	$0.492985\pi$
$240$		0			0
$241$		−	6.12384e9i		−	1.81533i	−0.419693	−	0.907666i	$-0.637862\pi$
$241$		−	6.12384e9i		−	1.81533i	0.419693	−	0.907666i	$-0.362138\pi$
$242$		0			0
$243$		−	2.23677e8i		−	0.0641500i
$244$		0			0
$245$	−1.90337e8	−	9.17411e7i	−0.0528273	−	0.0254624i
$246$		0			0
$247$	−3.91900e7			−0.0105290
$248$		0			0
$249$	−1.56829e9			−0.407971
$250$		0			0
$251$			5.19465e9i			1.30876i	0.756164	+	0.654382i	$0.227071\pi$
$251$			5.19465e9i			1.30876i	−0.756164	+	0.654382i	$0.772929\pi$
$252$		0			0
$253$	1.59886e9			0.390236
$254$		0			0
$255$			7.56902e7i			0.0179011i
$256$		0			0
$257$			4.72089e9i			1.08216i	0.840972	+	0.541079i	$0.181984\pi$
$257$			4.72089e9i			1.08216i	−0.840972	+	0.541079i	$0.818016\pi$
$258$		0			0
$259$	9.90764e8	+	1.57739e9i	0.220177	+	0.350541i
$260$		0			0
$261$	8.93478e7			0.0192540
$262$		0			0
$263$	6.48185e8			0.135480			0.0677401	−	0.997703i	$-0.478421\pi$
$263$	6.48185e8			0.135480			0.0677401	+	0.997703i	$0.478421\pi$
$264$		0			0
$265$			1.24298e8i			0.0252047i
$266$		0			0
$267$	1.11994e9			0.220368
$268$		0			0
$269$			5.01618e9i			0.957997i	0.877816	+	0.478999i	$0.159000\pi$
$269$			5.01618e9i			0.957997i	−0.877816	+	0.478999i	$0.841000\pi$
$270$		0			0
$271$			5.25230e9i			0.973806i	0.873456	+	0.486903i	$0.161874\pi$
$271$			5.25230e9i			0.973806i	−0.873456	+	0.486903i	$0.838126\pi$
$272$		0			0
$273$	−2.16143e8	−	3.44120e8i	−0.0389127	−	0.0619526i
$274$		0			0
$275$	8.82996e9			1.54393
$276$		0			0
$277$	1.11359e10			1.89149			0.945747	−	0.324903i	$-0.105332\pi$
$277$	1.11359e10			1.89149			0.945747	+	0.324903i	$0.105332\pi$
$278$		0			0
$279$		−	8.47238e8i		−	0.139826i
$280$		0			0
$281$	1.63482e9			0.262207			0.131103	−	0.991369i	$-0.458148\pi$
$281$	1.63482e9			0.262207			0.131103	+	0.991369i	$0.458148\pi$
$282$		0			0
$283$		−	9.11053e9i		−	1.42036i	−0.704021	−	0.710179i	$-0.748614\pi$
$283$		−	9.11053e9i		−	1.42036i	0.704021	−	0.710179i	$-0.251386\pi$
$284$		0			0
$285$		−	1.85607e7i		−	0.00281330i
$286$		0			0
$287$	2.93870e9	−	1.84581e9i	0.433140	−	0.272057i
$288$		0			0
$289$	5.02578e9			0.720463
$290$		0			0
$291$	−2.09428e9			−0.292054
$292$		0			0
$293$			1.02026e10i			1.38433i	0.721738	+	0.692166i	$0.243344\pi$
$293$			1.02026e10i			1.38433i	−0.721738	+	0.692166i	$0.756656\pi$
$294$		0			0
$295$	−6.51106e8			−0.0859733
$296$		0			0
$297$			2.31989e9i			0.298155i
$298$		0			0
$299$		−	2.55105e8i		−	0.0319179i
$300$		0			0
$301$	−4.28304e9	−	6.81898e9i	−0.521778	−	0.830718i
$302$		0			0
$303$	6.48719e9			0.769637
$304$		0			0
$305$	−7.47678e8			−0.0864003
$306$		0			0
$307$			5.54857e9i			0.624637i	0.949977	+	0.312319i	$0.101106\pi$
$307$			5.54857e9i			0.624637i	−0.949977	+	0.312319i	$0.898894\pi$
$308$		0			0
$309$	−1.11520e9			−0.122326
$310$		0			0
$311$		−	7.92937e9i		−	0.847612i	−0.905753	−	0.423806i	$-0.860694\pi$
$311$		−	7.92937e9i		−	0.847612i	0.905753	−	0.423806i	$-0.139306\pi$
$312$		0			0
$313$		−	7.55770e9i		−	0.787431i	−0.919232	−	0.393715i	$-0.871190\pi$
$313$		−	7.55770e9i		−	0.787431i	0.919232	−	0.393715i	$-0.128810\pi$
$314$		0			0
$315$	1.62978e8	−	1.02367e8i	0.0165534	−	0.0103973i
$316$		0			0
$317$	2.62992e9			0.260439			0.130219	−	0.991485i	$-0.458432\pi$
$317$	2.62992e9			0.260439			0.130219	+	0.991485i	$0.458432\pi$
$318$		0			0
$319$	−9.26679e8			−0.0894884
$320$		0			0
$321$			6.78364e8i			0.0638914i
$322$		0			0
$323$	4.78174e8			0.0439315
$324$		0			0
$325$		−	1.40886e9i		−	0.126280i
$326$		0			0
$327$		−	8.28671e9i		−	0.724755i
$328$		0			0
$329$	6.68004e9	−	4.19577e9i	0.570159	−	0.358119i
$330$		0			0
$331$	−1.56319e10			−1.30227			−0.651135	−	0.758962i	$-0.725707\pi$
$331$	−1.56319e10			−1.30227			−0.651135	+	0.758962i	$0.725707\pi$
$332$		0			0
$333$	−1.69670e9			−0.137984
$334$		0			0
$335$		−	4.11047e8i		−	0.0326371i
$336$		0			0
$337$	1.65290e10			1.28153			0.640763	−	0.767739i	$-0.278618\pi$
$337$	1.65290e10			1.28153			0.640763	+	0.767739i	$0.278618\pi$
$338$		0			0
$339$		−	1.28119e10i		−	0.970093i
$340$		0			0
$341$			8.78722e9i			0.649881i
$342$		0			0
$343$	−1.37550e10	+	1.54272e9i	−0.993769	+	0.111458i
$344$		0			0
$345$	1.20820e8			0.00852830
$346$		0			0
$347$	−2.21707e10			−1.52919			−0.764596	−	0.644510i	$-0.777061\pi$
$347$	−2.21707e10			−1.52919			−0.764596	+	0.644510i	$0.777061\pi$
$348$		0			0
$349$		−	1.24747e10i		−	0.840869i	−0.907323	−	0.420435i	$-0.861878\pi$
$349$		−	1.24747e10i		−	0.840869i	0.907323	−	0.420435i	$-0.138122\pi$
$350$		0			0
$351$	3.70150e8			0.0243865
$352$		0			0
$353$			6.63372e9i			0.427227i	0.976918	+	0.213613i	$0.0685233\pi$
$353$			6.63372e9i			0.427227i	−0.976918	+	0.213613i	$0.931477\pi$
$354$		0			0
$355$		−	4.66290e8i		−	0.0293591i
$356$		0			0
$357$	2.63725e9	+	4.19875e9i	0.162360	+	0.258492i
$358$		0			0
$359$	−1.80545e10			−1.08694			−0.543472	−	0.839427i	$-0.682891\pi$
$359$	−1.80545e10			−1.08694			−0.543472	+	0.839427i	$0.682891\pi$
$360$		0			0
$361$	1.68663e10			0.993096
$362$		0			0
$363$		−	1.40364e10i		−	0.808407i
$364$		0			0
$365$	6.89873e7			0.00388685
$366$		0			0
$367$			2.36769e10i			1.30515i	0.757723	+	0.652576i	$0.226312\pi$
$367$			2.36769e10i			1.30515i	−0.757723	+	0.652576i	$0.773688\pi$
$368$		0			0
$369$			3.16100e9i			0.170498i
$370$		0			0
$371$	4.33090e9	+	6.89518e9i	0.228603	+	0.363957i
$372$		0			0
$373$	−1.73066e10			−0.894078			−0.447039	−	0.894514i	$-0.647521\pi$
$373$	−1.73066e10			−0.894078			−0.447039	+	0.894514i	$0.647521\pi$
$374$		0			0
$375$	1.33680e9			0.0675993
$376$		0			0
$377$			1.47856e8i			0.00731938i
$378$		0			0
$379$	−1.35880e10			−0.658563			−0.329281	−	0.944232i	$-0.606806\pi$
$379$	−1.35880e10			−0.658563			−0.329281	+	0.944232i	$0.606806\pi$
$380$		0			0
$381$			4.54298e8i			0.0215596i
$382$		0			0
$383$		−	2.76701e10i		−	1.28593i	−0.765897	−	0.642963i	$-0.777705\pi$
$383$		−	2.76701e10i		−	1.28593i	0.765897	−	0.642963i	$-0.222295\pi$
$384$		0			0
$385$	−1.69034e9	+	1.06171e9i	−0.0769365	+	0.0483242i
$386$		0			0
$387$	7.33480e9			0.326997
$388$		0			0
$389$	−3.42151e10			−1.49423			−0.747117	−	0.664692i	$-0.768563\pi$
$389$	−3.42151e10			−1.49423			−0.747117	+	0.664692i	$0.768563\pi$
$390$		0			0
$391$			3.11265e9i			0.133175i
$392$		0			0
$393$	7.37386e9			0.309118
$394$		0			0
$395$		−	1.49899e9i		−	0.0615757i
$396$		0			0
$397$			2.96013e9i			0.119165i	0.998223	+	0.0595824i	$0.0189769\pi$
$397$			2.96013e9i			0.119165i	−0.998223	+	0.0595824i	$0.981023\pi$
$398$		0			0
$399$	−6.46707e8	−	1.02962e9i	−0.0255162	−	0.0406241i
$400$		0			0
$401$	2.08599e10			0.806743			0.403372	−	0.915036i	$-0.367838\pi$
$401$	2.08599e10			0.806743			0.403372	+	0.915036i	$0.367838\pi$
$402$		0			0
$403$	1.40204e9			0.0531547
$404$		0			0
$405$			1.75306e8i			0.00651595i
$406$		0			0
$407$	1.75975e10			0.641320
$408$		0			0
$409$		−	1.86838e10i		−	0.667685i	−0.942629	−	0.333842i	$-0.891655\pi$
$409$		−	1.86838e10i		−	0.667685i	0.942629	−	0.333842i	$-0.108345\pi$
$410$		0			0
$411$		−	1.54539e10i		−	0.541589i
$412$		0			0
$413$	−3.61187e10	+	2.26863e10i	−1.24146	+	0.779766i
$414$		0			0
$415$	1.22914e9			0.0414390
$416$		0			0
$417$	−1.70322e10			−0.563284
$418$		0			0
$419$		−	2.19687e10i		−	0.712767i	−0.934340	−	0.356383i	$-0.884010\pi$
$419$		−	2.19687e10i		−	0.712767i	0.934340	−	0.356383i	$-0.115990\pi$
$420$		0			0
$421$	2.04109e10			0.649731			0.324866	−	0.945760i	$-0.394681\pi$
$421$	2.04109e10			0.649731			0.324866	+	0.945760i	$0.394681\pi$
$422$		0			0
$423$			7.18534e9i			0.224433i
$424$		0			0
$425$			1.71901e10i			0.526894i
$426$		0			0
$427$	−4.14759e10	+	2.60512e10i	−1.24762	+	0.783639i
$428$		0			0
$429$	−3.83905e9			−0.113343
$430$		0			0
$431$	−4.57148e10			−1.32479			−0.662396	−	0.749154i	$-0.730460\pi$
$431$	−4.57148e10			−1.32479			−0.662396	+	0.749154i	$0.730460\pi$
$432$		0			0
$433$		−	4.67336e10i		−	1.32947i	−0.747080	−	0.664734i	$-0.768545\pi$
$433$		−	4.67336e10i		−	1.32947i	0.747080	−	0.664734i	$-0.231455\pi$
$434$		0			0
$435$	−7.00260e7			−0.00195570
$436$		0			0
$437$		−	7.63283e8i		−	0.0209295i
$438$		0			0
$439$		−	4.75010e10i		−	1.27892i	−0.768823	−	0.639462i	$-0.779157\pi$
$439$		−	4.75010e10i		−	1.27892i	0.768823	−	0.639462i	$-0.220843\pi$
$440$		0			0
$441$	5.47410e9	−	1.13572e10i	0.144730	−	0.300274i
$442$		0			0
$443$	−1.39740e10			−0.362833			−0.181416	−	0.983406i	$-0.558068\pi$
$443$	−1.39740e10			−0.362833			−0.181416	+	0.983406i	$0.558068\pi$
$444$		0			0
$445$	−8.77748e8			−0.0223836
$446$		0			0
$447$			1.90992e10i			0.478393i
$448$		0			0
$449$	−6.66742e10			−1.64049			−0.820243	−	0.572015i	$-0.806162\pi$
$449$	−6.66742e10			−1.64049			−0.820243	+	0.572015i	$0.806162\pi$
$450$		0			0
$451$		−	3.27846e10i		−	0.792435i
$452$		0			0
$453$			7.79495e9i			0.185106i
$454$		0			0
$455$	1.69402e8	+	2.69703e8i	0.00395250	+	0.00629274i
$456$		0			0
$457$	−3.69304e10			−0.846679			−0.423340	−	0.905971i	$-0.639142\pi$
$457$	−3.69304e10			−0.846679			−0.423340	+	0.905971i	$0.639142\pi$
$458$		0			0
$459$	−4.51636e9			−0.101751
$460$		0			0
$461$		−	4.75584e10i		−	1.05299i	−0.850179	−	0.526494i	$-0.823506\pi$
$461$		−	4.75584e10i		−	1.05299i	0.850179	−	0.526494i	$-0.176494\pi$
$462$		0			0
$463$	2.33733e10			0.508622			0.254311	−	0.967122i	$-0.418151\pi$
$463$	2.33733e10			0.508622			0.254311	+	0.967122i	$0.418151\pi$
$464$		0			0
$465$			6.64020e8i			0.0142026i
$466$		0			0
$467$			2.85291e10i			0.599820i	0.953968	+	0.299910i	$0.0969566\pi$
$467$			2.85291e10i			0.599820i	−0.953968	+	0.299910i	$0.903043\pi$
$468$		0			0
$469$	−1.43220e10	−	2.28019e10i	−0.296014	−	0.471281i
$470$		0			0
$471$	2.95536e10			0.600519
$472$		0			0
$473$	−7.60736e10			−1.51981
$474$		0			0
$475$		−	4.21536e9i		−	0.0828057i
$476$		0			0
$477$	−7.41675e9			−0.143265
$478$		0			0
$479$			6.23637e10i			1.18465i	0.805699	+	0.592325i	$0.201790\pi$
$479$			6.23637e10i			1.18465i	−0.805699	+	0.592325i	$0.798210\pi$
$480$		0			0
$481$		−	2.80778e9i		−	0.0524544i
$482$		0			0
$483$	6.70223e9	−	4.20970e9i	0.123149	−	0.0773505i
$484$		0			0
$485$	1.64139e9			0.0296650
$486$		0			0
$487$	8.70224e10			1.54709			0.773544	−	0.633743i	$-0.218482\pi$
$487$	8.70224e10			1.54709			0.773544	+	0.633743i	$0.218482\pi$
$488$		0			0
$489$		−	1.20990e10i		−	0.211600i
$490$		0			0
$491$	−1.64844e10			−0.283626			−0.141813	−	0.989893i	$-0.545293\pi$
$491$	−1.64844e10			−0.283626			−0.141813	+	0.989893i	$0.545293\pi$
$492$		0			0
$493$		−	1.80406e9i		−	0.0305395i
$494$		0			0
$495$		−	1.81821e9i		−	0.0302847i
$496$		0			0
$497$	−1.62468e10	−	2.58664e10i	−0.266283	−	0.423946i
$498$		0			0
$499$	−5.07024e10			−0.817761			−0.408881	−	0.912588i	$-0.634081\pi$
$499$	−5.07024e10			−0.817761			−0.408881	+	0.912588i	$0.634081\pi$
$500$		0			0
$501$	−2.91528e10			−0.462732
$502$		0			0
$503$			7.43592e10i			1.16162i	0.814041	+	0.580808i	$0.197263\pi$
$503$			7.43592e10i			1.16162i	−0.814041	+	0.580808i	$0.802737\pi$
$504$		0			0
$505$	−5.08431e9			−0.0781748
$506$		0			0
$507$		−	3.75354e10i		−	0.568080i
$508$		0			0
$509$			4.93156e10i			0.734706i	0.930082	+	0.367353i	$0.119736\pi$
$509$			4.93156e10i			0.734706i	−0.930082	+	0.367353i	$0.880264\pi$
$510$		0			0
$511$	3.82692e9	−	2.40371e9i	0.0561262	−	0.0352532i
$512$		0			0
$513$	1.10750e9			0.0159910
$514$		0			0
$515$	8.74032e8			0.0124251
$516$		0			0
$517$		−	7.45235e10i		−	1.04311i
$518$		0			0
$519$	8.57115e9			0.118133
$520$		0			0
$521$			4.65738e10i			0.632107i	0.948741	+	0.316054i	$0.102358\pi$
$521$			4.65738e10i			0.632107i	−0.948741	+	0.316054i	$0.897642\pi$
$522$		0			0
$523$		−	1.71628e10i		−	0.229394i	−0.993401	−	0.114697i	$-0.963410\pi$
$523$		−	1.71628e10i		−	0.229394i	0.993401	−	0.114697i	$-0.0365897\pi$
$524$		0			0
$525$	3.70142e10	−	2.32488e10i	0.487227	−	0.306030i
$526$		0			0
$527$	−1.71069e10			−0.221784
$528$		0			0
$529$	−7.33424e10			−0.936554
$530$		0			0
$531$		−	3.88508e10i		−	0.488678i
$532$		0			0
$533$	−5.23094e9			−0.0648144
$534$		0			0
$535$		−	5.31665e8i		−	0.00648968i
$536$		0			0
$537$		−	3.91177e10i		−	0.470409i
$538$		0			0
$539$	−5.67752e10	+	1.17793e11i	−0.672672	+	1.39561i
$540$		0			0
$541$	1.44426e10			0.168599			0.0842995	−	0.996440i	$-0.473135\pi$
$541$	1.44426e10			0.168599			0.0842995	+	0.996440i	$0.473135\pi$
$542$		0			0
$543$	−3.17677e10			−0.365415
$544$		0			0
$545$			6.49468e9i			0.0736159i
$546$		0			0
$547$	6.82236e10			0.762054			0.381027	−	0.924564i	$-0.375571\pi$
$547$	6.82236e10			0.762054			0.381027	+	0.924564i	$0.375571\pi$
$548$		0			0
$549$		−	4.46132e10i		−	0.491105i
$550$		0			0
$551$			4.42390e8i			0.00479954i
$552$		0			0
$553$	−5.22288e10	−	8.31531e10i	−0.558483	−	0.889156i
$554$		0			0
$555$	1.32979e9			0.0140155
$556$		0			0
$557$	−1.42350e10			−0.147889			−0.0739447	−	0.997262i	$-0.523559\pi$
$557$	−1.42350e10			−0.147889			−0.0739447	+	0.997262i	$0.523559\pi$
$558$		0			0
$559$			1.21379e10i			0.124307i
$560$		0			0
$561$	4.68419e10			0.472914
$562$		0			0
$563$		−	1.50168e11i		−	1.49467i	−0.664449	−	0.747334i	$-0.731334\pi$
$563$		−	1.50168e11i		−	1.49467i	0.664449	−	0.747334i	$-0.268666\pi$
$564$		0			0
$565$			1.00413e10i			0.0985358i
$566$		0			0
$567$	6.10816e9	+	9.72474e9i	0.0590987	+	0.0940905i
$568$		0			0
$569$	−5.21291e10			−0.497314			−0.248657	−	0.968592i	$-0.579989\pi$
$569$	−5.21291e10			−0.497314			−0.248657	+	0.968592i	$0.579989\pi$
$570$		0			0
$571$	−2.68533e10			−0.252612			−0.126306	−	0.991991i	$-0.540312\pi$
$571$	−2.68533e10			−0.252612			−0.126306	+	0.991991i	$0.540312\pi$
$572$		0			0
$573$		−	1.22954e11i		−	1.14057i
$574$		0			0
$575$	2.74396e10			0.251019
$576$		0			0
$577$			7.56506e10i			0.682510i	0.939971	+	0.341255i	$0.110852\pi$
$577$			7.56506e10i			0.682510i	−0.939971	+	0.341255i	$0.889148\pi$
$578$		0			0
$579$		−	2.79905e10i		−	0.249056i
$580$		0			0
$581$	6.81840e10	−	4.28267e10i	0.598381	−	0.375846i
$582$		0			0
$583$	7.69236e10			0.665864
$584$		0			0
$585$	−2.90104e8			−0.00247702
$586$		0			0
$587$			1.26488e11i			1.06537i	0.846315	+	0.532683i	$0.178816\pi$
$587$			1.26488e11i			1.06537i	−0.846315	+	0.532683i	$0.821184\pi$
$588$		0			0
$589$	4.19496e9			0.0348551
$590$		0			0
$591$		−	4.18551e10i		−	0.343082i
$592$		0			0
$593$			2.15913e11i			1.74607i	0.487660	+	0.873033i	$0.337850\pi$
$593$			2.15913e11i			1.74607i	−0.487660	+	0.873033i	$0.662150\pi$
$594$		0			0
$595$	−2.06694e9	−	3.29075e9i	−0.0164915	−	0.0262559i
$596$		0			0
$597$	1.26749e11			0.997807
$598$		0			0
$599$	9.15933e10			0.711469			0.355735	−	0.934587i	$-0.384231\pi$
$599$	9.15933e10			0.711469			0.355735	+	0.934587i	$0.384231\pi$
$600$		0			0
$601$			1.03906e11i			0.796422i	0.917294	+	0.398211i	$0.130369\pi$
$601$			1.03906e11i			0.796422i	−0.917294	+	0.398211i	$0.869631\pi$
$602$		0			0
$603$	2.45267e10			0.185511
$604$		0			0
$605$			1.10010e10i			0.0821128i
$606$		0			0
$607$		−	1.95438e11i		−	1.43964i	−0.694160	−	0.719820i	$-0.744224\pi$
$607$		−	1.95438e11i		−	1.43964i	0.694160	−	0.719820i	$-0.255776\pi$
$608$		0			0
$609$	−3.88454e9	+	2.43990e9i	−0.0282404	+	0.0177379i
$610$		0			0
$611$	−1.18906e10			−0.0853177
$612$		0			0
$613$	4.24653e10			0.300741			0.150370	−	0.988630i	$-0.451953\pi$
$613$	4.24653e10			0.300741			0.150370	+	0.988630i	$0.451953\pi$
$614$		0			0
$615$		−	2.47742e9i		−	0.0173181i
$616$		0			0
$617$	1.72007e11			1.18688			0.593439	−	0.804879i	$-0.297770\pi$
$617$	1.72007e11			1.18688			0.593439	+	0.804879i	$0.297770\pi$
$618$		0			0
$619$		−	2.47382e11i		−	1.68502i	−0.538679	−	0.842511i	$-0.681076\pi$
$619$		−	2.47382e11i		−	1.68502i	0.538679	−	0.842511i	$-0.318924\pi$
$620$		0			0
$621$			7.20921e9i			0.0484754i
$622$		0			0
$623$	−4.86912e10	+	3.05831e10i	−0.323220	+	0.203016i
$624$		0			0
$625$	1.51015e11			0.989695
$626$		0			0
$627$	−1.14866e10			−0.0743224
$628$		0			0
$629$			3.42588e10i			0.218862i
$630$		0			0
$631$	1.49618e11			0.943772			0.471886	−	0.881659i	$-0.343573\pi$
$631$	1.49618e11			0.943772			0.471886	+	0.881659i	$0.343573\pi$
$632$		0			0
$633$		−	1.16931e11i		−	0.728306i
$634$		0			0
$635$		−	3.56055e8i		−	0.00218989i
$636$		0			0
$637$	1.87944e10	+	9.05876e9i	0.114149	+	0.0550188i
$638$		0			0
$639$	2.78231e10			0.166879
$640$		0			0
$641$	6.63756e10			0.393167			0.196583	−	0.980487i	$-0.437015\pi$
$641$	6.63756e10			0.393167			0.196583	+	0.980487i	$0.437015\pi$
$642$		0			0
$643$		−	4.22748e10i		−	0.247308i	−0.992325	−	0.123654i	$-0.960539\pi$
$643$		−	4.22748e10i		−	0.247308i	0.992325	−	0.123654i	$-0.0394613\pi$
$644$		0			0
$645$	−5.74862e9			−0.0332143
$646$		0			0
$647$			3.08165e11i			1.75860i	0.476271	+	0.879299i	$0.341988\pi$
$647$			3.08165e11i			1.75860i	−0.476271	+	0.879299i	$0.658012\pi$
$648$		0			0
$649$			4.02945e11i			2.27126i
$650$		0			0
$651$	2.31363e10	+	3.68351e10i	0.128816	+	0.205087i
$652$		0			0
$653$	−6.98857e8			−0.00384358			−0.00192179	−	0.999998i	$-0.500612\pi$
$653$	−6.98857e8			−0.00384358			−0.00192179	+	0.999998i	$0.500612\pi$
$654$		0			0
$655$	−5.77924e9			−0.0313982
$656$		0			0
$657$			4.11640e9i			0.0220931i
$658$		0			0
$659$	−2.46713e11			−1.30813			−0.654064	−	0.756440i	$-0.726937\pi$
$659$	−2.46713e11			−1.30813			−0.654064	+	0.756440i	$0.726937\pi$
$660$		0			0
$661$		−	1.98544e10i		−	0.104004i	−0.998647	−	0.0520022i	$-0.983440\pi$
$661$		−	1.98544e10i		−	0.104004i	0.998647	−	0.0520022i	$-0.0165603\pi$
$662$		0			0
$663$		−	7.47385e9i		−	0.0386803i
$664$		0			0
$665$	5.06855e8	+	8.06959e8i	0.00259177	+	0.00412634i
$666$		0			0
$667$	−2.87971e9			−0.0145494
$668$		0			0
$669$	1.77029e11			0.883772
$670$		0			0
$671$			4.62711e11i			2.28255i
$672$		0			0
$673$	1.21344e11			0.591506			0.295753	−	0.955264i	$-0.404429\pi$
$673$	1.21344e11			0.591506			0.295753	+	0.955264i	$0.404429\pi$
$674$		0			0
$675$			3.98141e10i			0.191788i
$676$		0			0
$677$			3.15922e11i			1.50392i	0.659208	+	0.751961i	$0.270892\pi$
$677$			3.15922e11i			1.50392i	−0.659208	+	0.751961i	$0.729108\pi$
$678$		0			0
$679$	9.10524e10	−	5.71905e10i	0.428363	−	0.269057i
$680$		0			0
$681$	1.04751e11			0.487045
$682$		0			0
$683$	−3.01173e11			−1.38399			−0.691995	−	0.721903i	$-0.743268\pi$
$683$	−3.01173e11			−1.38399			−0.691995	+	0.721903i	$0.743268\pi$
$684$		0			0
$685$			1.21119e10i			0.0550111i
$686$		0			0
$687$	2.01167e10			0.0903086
$688$		0			0
$689$		−	1.22735e10i		−	0.0544619i
$690$		0			0
$691$		−	3.71500e11i		−	1.62947i	−0.579833	−	0.814735i	$-0.696882\pi$
$691$		−	3.71500e11i		−	1.62947i	0.579833	−	0.814735i	$-0.303118\pi$
$692$		0			0
$693$	−6.33514e10	−	1.00861e11i	−0.274677	−	0.437312i
$694$		0			0
$695$	1.33490e10			0.0572148
$696$		0			0
$697$	6.38249e10			0.270433
$698$		0			0
$699$			1.59781e11i			0.669293i
$700$		0			0
$701$	1.23838e11			0.512841			0.256420	−	0.966565i	$-0.417457\pi$
$701$	1.23838e11			0.512841			0.256420	+	0.966565i	$0.417457\pi$
$702$		0			0
$703$		−	8.40095e9i		−	0.0343959i
$704$		0			0
$705$		−	5.63149e9i		−	0.0227964i
$706$		0			0
$707$	−2.82041e11	+	1.77151e11i	−1.12885	+	0.709034i
$708$		0			0
$709$	1.29658e11			0.513114			0.256557	−	0.966529i	$-0.417412\pi$
$709$	1.29658e11			0.513114			0.256557	+	0.966529i	$0.417412\pi$
$710$		0			0
$711$	8.94430e10			0.350000
$712$		0			0
$713$			2.73068e10i			0.105661i
$714$		0			0
$715$	3.00884e9			0.0115127
$716$		0			0
$717$		−	6.72507e9i		−	0.0254460i
$718$		0			0
$719$		−	1.22880e11i		−	0.459795i	−0.973215	−	0.229898i	$-0.926161\pi$
$719$		−	1.22880e11i		−	0.459795i	0.973215	−	0.229898i	$-0.0738392\pi$
$720$		0			0
$721$	4.84850e10	−	3.04537e10i	0.179418	−	0.112693i
$722$		0			0
$723$	−2.86384e11			−1.04808
$724$		0			0
$725$	−1.59037e10			−0.0575634
$726$		0			0
$727$			1.68766e11i			0.604155i	0.953283	+	0.302078i	$0.0976802\pi$
$727$			1.68766e11i			0.604155i	−0.953283	+	0.302078i	$0.902320\pi$
$728$		0			0
$729$	−1.04604e10			−0.0370370
$730$		0			0
$731$		−	1.48100e11i		−	0.518662i
$732$		0			0
$733$			3.87008e11i			1.34061i	0.742084	+	0.670307i	$0.233838\pi$
$733$			3.87008e11i			1.34061i	−0.742084	+	0.670307i	$0.766162\pi$
$734$		0			0
$735$	−4.29031e9	+	8.90118e9i	−0.0147007	+	0.0304999i
$736$		0			0
$737$	−2.54382e11			−0.862215
$738$		0			0
$739$	1.03147e11			0.345844			0.172922	−	0.984936i	$-0.444679\pi$
$739$	1.03147e11			0.345844			0.172922	+	0.984936i	$0.444679\pi$
$740$		0			0
$741$			1.83274e9i			0.00607893i
$742$		0			0
$743$	2.92881e11			0.961029			0.480515	−	0.876987i	$-0.340450\pi$
$743$	2.92881e11			0.961029			0.480515	+	0.876987i	$0.340450\pi$
$744$		0			0
$745$		−	1.49689e10i		−	0.0485920i
$746$		0			0
$747$			7.33417e10i			0.235542i
$748$		0			0
$749$	−1.85247e10	−	2.94930e10i	−0.0588604	−	0.0937112i
$750$		0			0
$751$	5.26110e11			1.65393			0.826964	−	0.562255i	$-0.190066\pi$
$751$	5.26110e11			1.65393			0.826964	+	0.562255i	$0.190066\pi$
$752$		0			0
$753$	2.42930e11			0.755615
$754$		0			0
$755$		−	6.10926e9i		−	0.0188019i
$756$		0			0
$757$	1.45621e11			0.443444			0.221722	−	0.975110i	$-0.428832\pi$
$757$	1.45621e11			0.443444			0.221722	+	0.975110i	$0.428832\pi$
$758$		0			0
$759$		−	7.47711e10i		−	0.225303i
$760$		0			0
$761$			1.83220e11i			0.546303i	0.961971	+	0.273152i	$0.0880660\pi$
$761$			1.83220e11i			0.546303i	−0.961971	+	0.273152i	$0.911934\pi$
$762$		0			0
$763$	2.26293e11	+	3.60278e11i	0.667686	+	1.06302i
$764$		0			0
$765$	3.53968e9			0.0103352
$766$		0			0
$767$	6.42920e10			0.185770
$768$		0			0
$769$		−	3.70029e11i		−	1.05811i	−0.848588	−	0.529055i	$-0.822547\pi$
$769$		−	3.70029e11i		−	1.05811i	0.848588	−	0.529055i	$-0.177453\pi$
$770$		0			0
$771$	2.20774e11			0.624785
$772$		0			0
$773$			3.68722e11i			1.03272i	0.856372	+	0.516359i	$0.172713\pi$
$773$			3.68722e11i			1.03272i	−0.856372	+	0.516359i	$0.827287\pi$
$774$		0			0
$775$			1.50807e11i			0.418036i
$776$		0			0
$777$	7.37670e10	−	4.63334e10i	0.202385	−	0.127119i
$778$		0			0
$779$	−1.56511e10			−0.0425007
$780$		0			0
$781$	−2.88570e11			−0.775616
$782$		0			0
$783$		−	4.17838e9i		−	0.0111163i
$784$		0			0
$785$	−2.31625e10			−0.0609968
$786$		0			0
$787$		−	5.27998e11i		−	1.37636i	−0.725539	−	0.688182i	$-0.758409\pi$
$787$		−	5.27998e11i		−	1.37636i	0.725539	−	0.688182i	$-0.241591\pi$
$788$		0			0
$789$		−	3.03126e10i		−	0.0782195i
$790$		0			0
$791$	3.49865e11	+	5.57017e11i	0.893706	+	1.42286i
$792$		0			0
$793$	7.38278e10			0.186693
$794$		0			0
$795$	5.81285e9			0.0145519
$796$		0			0
$797$		−	1.99982e11i		−	0.495631i	−0.968807	−	0.247815i	$-0.920287\pi$
$797$		−	1.99982e11i		−	0.495631i	0.968807	−	0.247815i	$-0.0797126\pi$
$798$		0			0
$799$	1.45082e11			0.355981
$800$		0			0
$801$		−	5.23743e10i		−	0.127230i
$802$		0			0
$803$		−	4.26937e10i		−	0.102684i
$804$		0			0
$805$	−5.25285e9	+	3.29934e9i	−0.0125087	+	0.00785676i
$806$		0			0
$807$	2.34584e11			0.553100
$808$		0			0
$809$	2.78539e11			0.650268			0.325134	−	0.945668i	$-0.394591\pi$
$809$	2.78539e11			0.650268			0.325134	+	0.945668i	$0.394591\pi$
$810$		0			0
$811$		−	9.36819e10i		−	0.216557i	−0.994121	−	0.108279i	$-0.965466\pi$
$811$		−	9.36819e10i		−	0.216557i	0.994121	−	0.108279i	$-0.0345338\pi$
$812$		0			0
$813$	2.45626e11			0.562227
$814$		0			0
$815$			9.48256e9i			0.0214929i
$816$		0			0
$817$			3.63170e10i			0.0815120i
$818$		0			0
$819$	−1.60929e10	+	1.01080e10i	−0.0357683	+	0.0224663i
$820$		0			0
$821$	5.20403e9			0.0114543			0.00572713	−	0.999984i	$-0.498177\pi$
$821$	5.20403e9			0.0114543			0.00572713	+	0.999984i	$0.498177\pi$
$822$		0			0
$823$	−2.20643e9			−0.00480940			−0.00240470	−	0.999997i	$-0.500765\pi$
$823$	−2.20643e9			−0.00480940			−0.00240470	+	0.999997i	$0.500765\pi$
$824$		0			0
$825$		−	4.12936e11i		−	0.891389i
$826$		0			0
$827$	−4.65423e11			−0.995006			−0.497503	−	0.867462i	$-0.665750\pi$
$827$	−4.65423e11			−0.995006			−0.497503	+	0.867462i	$0.665750\pi$
$828$		0			0
$829$			3.39398e11i			0.718607i	0.933221	+	0.359303i	$0.116986\pi$
$829$			3.39398e11i			0.718607i	−0.933221	+	0.359303i	$0.883014\pi$
$830$		0			0
$831$		−	5.20773e11i		−	1.09206i
$832$		0			0
$833$	−2.29318e11	−	1.10530e11i	−0.476275	−	0.229561i
$834$		0			0
$835$	2.28484e10			0.0470014
$836$		0			0
$837$	−3.96214e10			−0.0807287
$838$		0			0
$839$			3.15712e11i			0.637153i	0.947897	+	0.318577i	$0.103205\pi$
$839$			3.15712e11i			0.637153i	−0.947897	+	0.318577i	$0.896795\pi$
$840$		0			0
$841$	−4.98577e11			−0.996664
$842$		0			0
$843$		−	7.64528e10i		−	0.151385i
$844$		0			0
$845$			2.94182e10i			0.0577019i
$846$		0			0
$847$	3.83305e11	+	6.10257e11i	0.744751	+	1.18571i
$848$		0			0
$849$	−4.26057e11			−0.820044
$850$		0			0
$851$	5.46855e10			0.104269
$852$		0			0
$853$		−	8.44380e11i		−	1.59493i	−0.603365	−	0.797465i	$-0.706174\pi$
$853$		−	8.44380e11i		−	1.59493i	0.603365	−	0.797465i	$-0.293826\pi$
$854$		0			0
$855$	−8.68000e8			−0.00162426
$856$		0			0
$857$		−	9.61672e10i		−	0.178280i	−0.996019	−	0.0891402i	$-0.971588\pi$
$857$		−	9.61672e10i		−	0.178280i	0.996019	−	0.0891402i	$-0.0284119\pi$
$858$		0			0
$859$		−	6.32033e11i		−	1.16083i	−0.814322	−	0.580413i	$-0.802891\pi$
$859$		−	6.32033e11i		−	1.16083i	0.814322	−	0.580413i	$-0.197109\pi$
$860$		0			0
$861$	−8.63201e10	−	1.37430e11i	−0.157072	−	0.250073i
$862$		0			0
$863$	4.46614e11			0.805173			0.402587	−	0.915382i	$-0.368111\pi$
$863$	4.46614e11			0.805173			0.402587	+	0.915382i	$0.368111\pi$
$864$		0			0
$865$	−6.71761e9			−0.0119991
$866$		0			0
$867$		−	2.35032e11i		−	0.415960i
$868$		0			0
$869$	−9.27668e11			−1.62672
$870$		0			0
$871$			4.05878e10i			0.0705218i
$872$		0			0
$873$			9.79399e10i			0.168618i
$874$		0			0
$875$	−5.81197e10	+	3.65052e10i	−0.0991496	+	0.0622763i
$876$		0			0
$877$	−6.35009e11			−1.07345			−0.536724	−	0.843758i	$-0.680339\pi$
$877$	−6.35009e11			−1.07345			−0.536724	+	0.843758i	$0.680339\pi$
$878$		0			0
$879$	4.77128e11			0.799245
$880$		0			0
$881$		−	6.13202e11i		−	1.01789i	−0.860800	−	0.508944i	$-0.830036\pi$
$881$		−	6.13202e11i		−	1.01789i	0.860800	−	0.508944i	$-0.169964\pi$
$882$		0			0
$883$	−4.81745e11			−0.792454			−0.396227	−	0.918153i	$-0.629681\pi$
$883$	−4.81745e11			−0.792454			−0.396227	+	0.918153i	$0.629681\pi$
$884$		0			0
$885$			3.04492e10i			0.0496367i
$886$		0			0
$887$			3.85645e11i			0.623008i	0.950245	+	0.311504i	$0.100833\pi$
$887$			3.85645e11i			0.623008i	−0.950245	+	0.311504i	$0.899167\pi$
$888$		0			0
$889$	−1.24059e10	−	1.97514e10i	−0.0198620	−	0.0316221i
$890$		0			0
$891$	1.08491e11			0.172140
$892$		0			0
$893$	−3.55770e10			−0.0559453
$894$		0			0
$895$			3.06583e10i			0.0477811i
$896$		0			0
$897$	−1.19301e10			−0.0184278
$898$		0			0
$899$		−	1.58267e10i		−	0.0242300i
$900$		0			0
$901$			1.49755e11i			0.227238i
$902$		0			0
$903$	−3.18892e11	+	2.00298e11i	−0.479615	+	0.301249i
$904$		0			0
$905$	2.48978e10			0.0371165
$906$		0			0
$907$	−9.06408e11			−1.33935			−0.669676	−	0.742654i	$-0.733567\pi$
$907$	−9.06408e11			−1.33935			−0.669676	+	0.742654i	$0.733567\pi$
$908$		0			0
$909$		−	3.03376e11i		−	0.444350i
$910$		0			0
$911$	4.63945e11			0.673586			0.336793	−	0.941579i	$-0.390658\pi$
$911$	4.63945e11			0.673586			0.336793	+	0.941579i	$0.390658\pi$
$912$		0			0
$913$		−	7.60671e11i		−	1.09475i
$914$		0			0
$915$			3.49655e10i			0.0498833i
$916$		0			0
$917$	−3.20591e11	+	2.01365e11i	−0.453392	+	0.284777i
$918$		0			0
$919$	3.17911e10			0.0445701			0.0222850	−	0.999752i	$-0.492906\pi$
$919$	3.17911e10			0.0445701			0.0222850	+	0.999752i	$0.492906\pi$
$920$		0			0
$921$	2.59481e11			0.360634
$922$		0			0
$923$			4.60427e10i			0.0634387i
$924$		0			0
$925$	3.02010e11			0.412529
$926$		0			0
$927$			5.21526e10i			0.0706248i
$928$		0			0
$929$			1.15735e12i			1.55382i	0.629609	+	0.776912i	$0.283215\pi$
$929$			1.15735e12i			1.55382i	−0.629609	+	0.776912i	$0.716785\pi$
$930$		0			0
$931$	5.62333e10	+	2.71041e10i	0.0748506	+	0.0360775i
$932$		0			0
$933$	−3.70820e11			−0.489369
$934$		0			0
$935$	−3.67121e10			−0.0480356
$936$		0			0
$937$		−	4.45116e11i		−	0.577450i	−0.957412	−	0.288725i	$-0.906769\pi$
$937$		−	4.45116e11i		−	0.577450i	0.957412	−	0.288725i	$-0.0932314\pi$
$938$		0			0
$939$	−3.53439e11			−0.454623
$940$		0			0
$941$		−	2.70422e11i		−	0.344892i	−0.985019	−	0.172446i	$-0.944833\pi$
$941$		−	2.70422e11i		−	0.344892i	0.985019	−	0.172446i	$-0.0551670\pi$
$942$		0			0
$943$		−	1.01880e11i		−	0.128838i
$944$		0			0
$945$	−4.78725e9	−	7.62173e9i	−0.00600286	−	0.00955711i
$946$		0			0
$947$	−1.15537e12			−1.43655			−0.718273	−	0.695762i	$-0.755067\pi$
$947$	−1.15537e12			−1.43655			−0.718273	+	0.695762i	$0.755067\pi$
$948$		0			0
$949$	−6.81199e9			−0.00839864
$950$		0			0
$951$		−	1.22989e11i		−	0.150364i
$952$		0			0
$953$	−1.78259e11			−0.216112			−0.108056	−	0.994145i	$-0.534463\pi$
$953$	−1.78259e11			−0.216112			−0.108056	+	0.994145i	$0.534463\pi$
$954$		0			0
$955$			9.63646e10i			0.115852i
$956$		0			0
$957$			4.33365e10i			0.0516662i
$958$		0			0
$959$	4.22013e11	+	6.71882e11i	0.498943	+	0.794363i
$960$		0			0
$961$	7.02814e11			0.824038
$962$		0			0
$963$	3.17239e10			0.0368877
$964$		0			0
$965$			2.19375e10i			0.0252975i
$966$		0			0
$967$	−1.01164e12			−1.15696			−0.578481	−	0.815696i	$-0.696354\pi$
$967$	−1.01164e12			−1.15696			−0.578481	+	0.815696i	$0.696354\pi$
$968$		0			0
$969$		−	2.23620e10i		−	0.0253638i
$970$		0			0
$971$		−	1.33837e12i		−	1.50556i	−0.658270	−	0.752782i	$-0.728711\pi$
$971$		−	1.33837e12i		−	1.50556i	0.658270	−	0.752782i	$-0.271289\pi$
$972$		0			0
$973$	7.40505e11	−	4.65115e11i	0.826183	−	0.518930i
$974$		0			0
$975$	−6.58860e10			−0.0729079
$976$		0			0
$977$	−3.69147e11			−0.405155			−0.202577	−	0.979266i	$-0.564932\pi$
$977$	−3.69147e11			−0.405155			−0.202577	+	0.979266i	$0.564932\pi$
$978$		0			0
$979$			5.43206e11i			0.591335i
$980$		0			0
$981$	−3.87531e11			−0.418437
$982$		0			0
$983$		−	1.14406e12i		−	1.22527i	−0.790364	−	0.612637i	$-0.790109\pi$
$983$		−	1.14406e12i		−	1.22527i	0.790364	−	0.612637i	$-0.209891\pi$
$984$		0			0
$985$			3.28038e10i			0.0348481i
$986$		0			0
$987$	−1.96217e11	−	3.12395e11i	−0.206760	−	0.329181i
$988$		0			0
$989$	−2.36403e11			−0.247098
$990$		0			0
$991$	7.90661e11			0.819777			0.409888	−	0.912136i	$-0.365568\pi$
$991$	7.90661e11			0.819777			0.409888	+	0.912136i	$0.365568\pi$
$992$		0			0
$993$			7.31033e11i			0.751865i
$994$		0			0
$995$	−9.93389e10			−0.101351
$996$		0			0
$997$		−	1.71492e12i		−	1.73566i	−0.496862	−	0.867830i	$-0.665514\pi$
$997$		−	1.71492e12i		−	1.73566i	0.496862	−	0.867830i	$-0.334486\pi$
$998$		0			0
$999$			7.93470e10i			0.0796652i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.9.f.d.97.9		32
4.3	odd	2		168.9.f.a.97.25	yes	32
7.6	odd	2	inner	336.9.f.d.97.24		32
28.27	even	2		168.9.f.a.97.8	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.9.f.a.97.8	✓	32	28.27	even	2
168.9.f.a.97.25	yes	32	4.3	odd	2
336.9.f.d.97.9		32	1.1	even	1	trivial
336.9.f.d.97.24		32	7.6	odd	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:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	336.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-46.7654i q^{3} +36.6522i q^{5} +(1277.06 + 2033.20i) q^{7} -2187.00 q^{9} +O(q^{10})\)	\(q-46.7654i q^{3} +36.6522i q^{5} +(1277.06 + 2033.20i) q^{7} -2187.00 q^{9} +22682.7 q^{11} -3619.14i q^{13} +1714.05 q^{15} +44158.6i q^{17} -10828.6i q^{19} +(95083.4 - 59722.4i) q^{21} +70487.9 q^{23} +389282. q^{25} +102276. i q^{27} -40854.0 q^{29} +387397. i q^{31} -1.06076e6i q^{33} +(-74521.3 + 46807.2i) q^{35} +775814. q^{37} -169250. q^{39} -1.44536e6i q^{41} -3.35382e6 q^{43} -80158.4i q^{45} -3.28548e6i q^{47} +(-2.50302e6 + 5.19306e6i) q^{49} +2.06509e6 q^{51} +3.39129e6 q^{53} +831371. i q^{55} -506402. q^{57} +1.77644e7i q^{59} +2.03993e7i q^{61} +(-2.79294e6 - 4.44661e6i) q^{63} +132649. q^{65} -1.12148e7 q^{67} -3.29639e6i q^{69} -1.27220e7 q^{71} -1.88221e6i q^{73} -1.82049e7i q^{75} +(2.89672e7 + 4.61185e7i) q^{77} -4.08976e7 q^{79} +4.78297e6 q^{81} -3.35353e7i q^{83} -1.61851e6 q^{85} +1.91055e6i q^{87} +2.39480e7i q^{89} +(7.35843e6 - 4.62187e6i) q^{91} +1.81168e7 q^{93} +396891. q^{95} -4.47828e7i q^{97} -4.96071e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 1424 q^{7} - 69984 q^{9}+O(q^{10}) \) 32 * q - 1424 * q^7 - 69984 * q^9	\( 32 q - 1424 q^{7} - 69984 q^{9} - 23616 q^{11} + 59616 q^{15} + 60912 q^{21} - 455040 q^{23} - 3404928 q^{25} + 632064 q^{29} + 1543200 q^{35} + 4150496 q^{37} + 4162752 q^{39} - 6028000 q^{43} + 15115072 q^{49} + 1340064 q^{51} - 37728576 q^{53} + 7286112 q^{57} + 3114288 q^{63} + 29977536 q^{65} - 68431648 q^{67} + 19788096 q^{71} + 87499392 q^{77} + 86954656 q^{79} + 153055008 q^{81} - 31326496 q^{85} + 24540288 q^{91} - 38833344 q^{93} - 329546688 q^{95} + 51648192 q^{99}+O(q^{100}) \) 32 * q - 1424 * q^7 - 69984 * q^9 - 23616 * q^11 + 59616 * q^15 + 60912 * q^21 - 455040 * q^23 - 3404928 * q^25 + 632064 * q^29 + 1543200 * q^35 + 4150496 * q^37 + 4162752 * q^39 - 6028000 * q^43 + 15115072 * q^49 + 1340064 * q^51 - 37728576 * q^53 + 7286112 * q^57 + 3114288 * q^63 + 29977536 * q^65 - 68431648 * q^67 + 19788096 * q^71 + 87499392 * q^77 + 86954656 * q^79 + 153055008 * q^81 - 31326496 * q^85 + 24540288 * q^91 - 38833344 * q^93 - 329546688 * q^95 + 51648192 * q^99

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