LMFDB - Embedded newform 1680.2.k.e.209.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1680,2,Mod(209,1680)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1680, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1680.209"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1680.k (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

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

Self dual:	no
Analytic conductor:	$13.4148675396$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.10070523904.11
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 10x^{4} + 81 $ x^8 - 10*x^4 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 210)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			209.4
Root			$0.420861 + 1.68014i$ of defining polynomial
Character	$\chi$	$=$	1680.209
Dual form			1680.2.k.e.209.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.420861 + 1.68014i) q^{3} +(-1.95522 - 1.08495i) q^{5} +(2.37608 + 1.16372i) q^{7} +(-2.64575 - 1.41421i) q^{9} -2.82843i q^{11} -0.841723 q^{13} +(2.64575 - 2.82843i) q^{15} -1.19038i q^{17} +4.55066i q^{19} +(-2.95522 + 3.50238i) q^{21} +3.29150 q^{23} +(2.64575 + 4.24264i) q^{25} +(3.48957 - 3.85005i) q^{27} -7.98430i q^{29} +5.53019i q^{31} +(4.75216 + 1.19038i) q^{33} +(-3.38317 - 4.85326i) q^{35} -10.8127i q^{37} +(0.354249 - 1.41421i) q^{39} +7.82087 q^{41} -4.65489i q^{43} +(3.63866 + 5.63561i) q^{45} -4.33981i q^{47} +(4.29150 + 5.53019i) q^{49} +(2.00000 + 0.500983i) q^{51} +12.5830 q^{53} +(-3.06871 + 5.53019i) q^{55} +(-7.64575 - 1.91520i) q^{57} +3.91044 q^{59} +10.0808i q^{61} +(-4.64076 - 6.43920i) q^{63} +(1.64575 + 0.913230i) q^{65} +4.65489i q^{67} +(-1.38527 + 5.53019i) q^{69} +12.6392i q^{71} +3.06871 q^{73} +(-8.24173 + 2.65967i) q^{75} +(3.29150 - 6.72057i) q^{77} +7.29150 q^{79} +(5.00000 + 7.48331i) q^{81} +7.70010i q^{83} +(-1.29150 + 2.32744i) q^{85} +(13.4148 + 3.36028i) q^{87} -12.8712 q^{89} +(-2.00000 - 0.979531i) q^{91} +(-9.29150 - 2.32744i) q^{93} +(4.93725 - 8.89753i) q^{95} +8.11905 q^{97} +(-4.00000 + 7.48331i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{21} - 16 q^{23} - 16 q^{35} + 24 q^{39} - 8 q^{49} + 16 q^{51} + 16 q^{53} - 40 q^{57} - 8 q^{63} - 8 q^{65} - 16 q^{77} + 16 q^{79} + 40 q^{81} + 32 q^{85} - 16 q^{91} - 32 q^{93} - 24 q^{95}+ \cdots - 32 q^{99}+O(q^{100}) $ 8 * q - 8 * q^21 - 16 * q^23 - 16 * q^35 + 24 * q^39 - 8 * q^49 + 16 * q^51 + 16 * q^53 - 40 * q^57 - 8 * q^63 - 8 * q^65 - 16 * q^77 + 16 * q^79 + 40 * q^81 + 32 * q^85 - 16 * q^91 - 32 * q^93 - 24 * q^95 - 32 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$421$	$1121$	$1471$
$\chi(n)$	$-1$	$-1$	$1$	$-1$	$1$

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
$3$	−0.420861	+	1.68014i	−0.242984	+	0.970030i
$3$	−0.420861	+	1.68014i	−0.242984	+	0.970030i
$4$		0			0
$5$	−1.95522	−	1.08495i	−0.874400	−	0.485206i
$5$	−1.95522	−	1.08495i	−0.874400	−	0.485206i
$6$		0			0
$7$	2.37608	+	1.16372i	0.898073	+	0.439846i
$7$	2.37608	+	1.16372i	0.898073	+	0.439846i
$8$		0			0
$9$	−2.64575	−	1.41421i	−0.881917	−	0.471405i
$10$		0			0
$11$		−	2.82843i		−	0.852803i	−0.904534	−	0.426401i	$-0.859781\pi$
$11$		−	2.82843i		−	0.852803i	0.904534	−	0.426401i	$-0.140219\pi$
$12$		0			0
$13$	−0.841723			−0.233452			−0.116726	−	0.993164i	$-0.537240\pi$
$13$	−0.841723			−0.233452			−0.116726	+	0.993164i	$0.537240\pi$
$14$		0			0
$15$	2.64575	−	2.82843i	0.683130	−	0.730297i
$16$		0			0
$17$		−	1.19038i		−	0.288709i	−0.989526	−	0.144354i	$-0.953890\pi$
$17$		−	1.19038i		−	0.288709i	0.989526	−	0.144354i	$-0.0461105\pi$
$18$		0			0
$19$			4.55066i			1.04399i	0.852948	+	0.521996i	$0.174813\pi$
$19$			4.55066i			1.04399i	−0.852948	+	0.521996i	$0.825187\pi$
$20$		0			0
$21$	−2.95522	+	3.50238i	−0.644881	+	0.764283i
$22$		0			0
$23$	3.29150			0.686326			0.343163	−	0.939276i	$-0.388502\pi$
$23$	3.29150			0.686326			0.343163	+	0.939276i	$0.388502\pi$
$24$		0			0
$25$	2.64575	+	4.24264i	0.529150	+	0.848528i
$26$		0			0
$27$	3.48957	−	3.85005i	0.671569	−	0.740942i
$28$		0			0
$29$		−	7.98430i		−	1.48265i	−0.671148	−	0.741323i	$-0.734198\pi$
$29$		−	7.98430i		−	1.48265i	0.671148	−	0.741323i	$-0.265802\pi$
$30$		0			0
$31$			5.53019i			0.993252i	0.867965	+	0.496626i	$0.165428\pi$
$31$			5.53019i			0.993252i	−0.867965	+	0.496626i	$0.834572\pi$
$32$		0			0
$33$	4.75216	+	1.19038i	0.827245	+	0.207218i
$34$		0			0
$35$	−3.38317	−	4.85326i	−0.571860	−	0.820352i
$36$		0			0
$37$		−	10.8127i		−	1.77760i	−0.458294	−	0.888801i	$-0.651539\pi$
$37$		−	10.8127i		−	1.77760i	0.458294	−	0.888801i	$-0.348461\pi$
$38$		0			0
$39$	0.354249	−	1.41421i	0.0567252	−	0.226455i
$40$		0			0
$41$	7.82087			1.22141			0.610707	−	0.791856i	$-0.290885\pi$
$41$	7.82087			1.22141			0.610707	+	0.791856i	$0.290885\pi$
$42$		0			0
$43$		−	4.65489i		−	0.709864i	−0.934892	−	0.354932i	$-0.884504\pi$
$43$		−	4.65489i		−	0.709864i	0.934892	−	0.354932i	$-0.115496\pi$
$44$		0			0
$45$	3.63866	+	5.63561i	0.542420	+	0.840108i
$46$		0			0
$47$		−	4.33981i		−	0.633027i	−0.948588	−	0.316513i	$-0.897488\pi$
$47$		−	4.33981i		−	0.633027i	0.948588	−	0.316513i	$-0.102512\pi$
$48$		0			0
$49$	4.29150	+	5.53019i	0.613072	+	0.790027i
$50$		0			0
$51$	2.00000	+	0.500983i	0.280056	+	0.0701517i
$52$		0			0
$53$	12.5830			1.72841			0.864204	−	0.503141i	$-0.167822\pi$
$53$	12.5830			1.72841			0.864204	+	0.503141i	$0.167822\pi$
$54$		0			0
$55$	−3.06871	+	5.53019i	−0.413785	+	0.745691i
$56$		0			0
$57$	−7.64575	−	1.91520i	−1.01270	−	0.253674i
$58$		0			0
$59$	3.91044			0.509095			0.254548	−	0.967060i	$-0.418073\pi$
$59$	3.91044			0.509095			0.254548	+	0.967060i	$0.418073\pi$
$60$		0			0
$61$			10.0808i			1.29072i	0.763878	+	0.645360i	$0.223293\pi$
$61$			10.0808i			1.29072i	−0.763878	+	0.645360i	$0.776707\pi$
$62$		0			0
$63$	−4.64076	−	6.43920i	−0.584681	−	0.811263i
$64$		0			0
$65$	1.64575	+	0.913230i	0.204130	+	0.113272i
$66$		0			0
$67$			4.65489i			0.568685i	0.958723	+	0.284343i	$0.0917753\pi$
$67$			4.65489i			0.568685i	−0.958723	+	0.284343i	$0.908225\pi$
$68$		0			0
$69$	−1.38527	+	5.53019i	−0.166766	+	0.665757i
$70$		0			0
$71$			12.6392i			1.50000i	0.661440	+	0.749998i	$0.269945\pi$
$71$			12.6392i			1.50000i	−0.661440	+	0.749998i	$0.730055\pi$
$72$		0			0
$73$	3.06871			0.359166			0.179583	−	0.983743i	$-0.442525\pi$
$73$	3.06871			0.359166			0.179583	+	0.983743i	$0.442525\pi$
$74$		0			0
$75$	−8.24173	+	2.65967i	−0.951673	+	0.307113i
$76$		0			0
$77$	3.29150	−	6.72057i	0.375102	−	0.765880i
$78$		0			0
$79$	7.29150			0.820358			0.410179	−	0.912005i	$-0.365466\pi$
$79$	7.29150			0.820358			0.410179	+	0.912005i	$0.365466\pi$
$80$		0			0
$81$	5.00000	+	7.48331i	0.555556	+	0.831479i
$82$		0			0
$83$			7.70010i			0.845196i	0.906317	+	0.422598i	$0.138882\pi$
$83$			7.70010i			0.845196i	−0.906317	+	0.422598i	$0.861118\pi$
$84$		0			0
$85$	−1.29150	+	2.32744i	−0.140083	+	0.252447i
$86$		0			0
$87$	13.4148	+	3.36028i	1.43821	+	0.360260i
$88$		0			0
$89$	−12.8712			−1.36435			−0.682173	−	0.731191i	$-0.738965\pi$
$89$	−12.8712			−1.36435			−0.682173	+	0.731191i	$0.738965\pi$
$90$		0			0
$91$	−2.00000	−	0.979531i	−0.209657	−	0.102683i
$92$		0			0
$93$	−9.29150	−	2.32744i	−0.963484	−	0.241345i
$94$		0			0
$95$	4.93725	−	8.89753i	0.506552	−	0.912867i
$96$		0			0
$97$	8.11905			0.824365			0.412182	−	0.911101i	$-0.364767\pi$
$97$	8.11905			0.824365			0.412182	+	0.911101i	$0.364767\pi$
$98$		0			0
$99$	−4.00000	+	7.48331i	−0.402015	+	0.752101i
$100$		0			0
$101$	3.91044			0.389103			0.194551	−	0.980892i	$-0.437675\pi$
$101$	3.91044			0.389103			0.194551	+	0.980892i	$0.437675\pi$
$102$		0			0
$103$	12.5730			1.23886			0.619429	−	0.785053i	$-0.287364\pi$
$103$	12.5730			1.23886			0.619429	+	0.785053i	$0.287364\pi$
$104$		0			0
$105$	9.57802	−	3.64165i	0.934719	−	0.355388i
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		0			0
$109$	2.00000			0.191565			0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000			0.191565			0.0957826	+	0.995402i	$0.469465\pi$
$110$		0			0
$111$	18.1669	+	4.55066i	1.72433	+	0.431929i
$112$		0			0
$113$	−6.00000			−0.564433			−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000			−0.564433			−0.282216	+	0.959351i	$0.591070\pi$
$114$		0			0
$115$	−6.43560	−	3.57113i	−0.600123	−	0.333009i
$116$		0			0
$117$	2.22699	+	1.19038i	0.205885	+	0.110050i
$118$		0			0
$119$	1.38527	−	2.82843i	0.126987	−	0.259281i
$120$		0			0
$121$	3.00000			0.272727
$122$		0			0
$123$	−3.29150	+	13.1402i	−0.296785	+	1.18481i
$124$		0			0
$125$	−0.569951	−	11.1658i	−0.0509780	−	0.998700i
$126$		0			0
$127$		−	10.8127i		−	0.959474i	−0.877412	−	0.479737i	$-0.840732\pi$
$127$		−	10.8127i		−	0.959474i	0.877412	−	0.479737i	$-0.159268\pi$
$128$		0			0
$129$	7.82087	+	1.95906i	0.688589	+	0.172486i
$130$		0			0
$131$	−8.96077			−0.782906			−0.391453	−	0.920198i	$-0.628027\pi$
$131$	−8.96077			−0.782906			−0.391453	+	0.920198i	$0.628027\pi$
$132$		0			0
$133$	−5.29570	+	10.8127i	−0.459196	+	0.937582i
$134$		0			0
$135$	−11.0000	+	3.74166i	−0.946729	+	0.322031i
$136$		0			0
$137$	−9.29150			−0.793827			−0.396913	−	0.917856i	$-0.629919\pi$
$137$	−9.29150			−0.793827			−0.396913	+	0.917856i	$0.629919\pi$
$138$		0			0
$139$		−	15.6110i		−	1.32411i	−0.749455	−	0.662056i	$-0.769684\pi$
$139$		−	15.6110i		−	1.32411i	0.749455	−	0.662056i	$-0.230316\pi$
$140$		0			0
$141$	7.29150	+	1.82646i	0.614055	+	0.153816i
$142$		0			0
$143$			2.38075i			0.199088i
$144$		0			0
$145$	−8.66259	+	15.6110i	−0.719389	+	1.29643i
$146$		0			0
$147$	−11.0976	+	4.88289i	−0.915317	+	0.402734i
$148$		0			0
$149$		−	3.32941i		−	0.272756i	−0.990657	−	0.136378i	$-0.956454\pi$
$149$		−	3.32941i		−	0.272756i	0.990657	−	0.136378i	$-0.0435462\pi$
$150$		0			0
$151$	22.5830			1.83778			0.918889	−	0.394515i	$-0.129087\pi$
$151$	22.5830			1.83778			0.918889	+	0.394515i	$0.129087\pi$
$152$		0			0
$153$	−1.68345	+	3.14944i	−0.136099	+	0.254617i
$154$		0			0
$155$	6.00000	−	10.8127i	0.481932	−	0.868499i
$156$		0			0
$157$	22.6209			1.80534			0.902672	−	0.430330i	$-0.141603\pi$
$157$	22.6209			1.80534			0.902672	+	0.430330i	$0.141603\pi$
$158$		0			0
$159$	−5.29570	+	21.1412i	−0.419976	+	1.67661i
$160$		0			0
$161$	7.82087	+	3.83039i	0.616371	+	0.301877i
$162$		0			0
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$	−8.00000	−	7.48331i	−0.622799	−	0.582575i
$166$		0			0
$167$		−	11.4821i		−	0.888509i	−0.895901	−	0.444255i	$-0.853469\pi$
$167$		−	11.4821i		−	0.888509i	0.895901	−	0.444255i	$-0.146531\pi$
$168$		0			0
$169$	−12.2915			−0.945500
$170$		0			0
$171$	6.43560	−	12.0399i	0.492143	−	0.920715i
$172$		0			0
$173$			8.89047i			0.675930i	0.941159	+	0.337965i	$0.109739\pi$
$173$			8.89047i			0.675930i	−0.941159	+	0.337965i	$0.890261\pi$
$174$		0			0
$175$	1.34926	+	13.1598i	0.101994	+	0.994785i
$176$		0			0
$177$	−1.64575	+	6.57008i	−0.123702	+	0.493838i
$178$		0			0
$179$			9.48725i			0.709110i	0.935035	+	0.354555i	$0.115368\pi$
$179$			9.48725i			0.709110i	−0.935035	+	0.354555i	$0.884632\pi$
$180$		0			0
$181$		−	12.0399i		−	0.894920i	−0.894304	−	0.447460i	$-0.852329\pi$
$181$		−	12.0399i		−	0.894920i	0.894304	−	0.447460i	$-0.147671\pi$
$182$		0			0
$183$	−16.9373	−	4.24264i	−1.25204	−	0.313625i
$184$		0			0
$185$	−11.7313	+	21.1412i	−0.862503	+	1.55433i
$186$		0			0
$187$	−3.36689			−0.246211
$188$		0			0
$189$	12.7719	−	5.08713i	0.929018	−	0.370034i
$190$		0			0
$191$			7.98430i			0.577724i	0.957371	+	0.288862i	$0.0932768\pi$
$191$			7.98430i			0.577724i	−0.957371	+	0.288862i	$0.906723\pi$
$192$		0			0
$193$		−	8.48528i		−	0.610784i	−0.952227	−	0.305392i	$-0.901213\pi$
$193$		−	8.48528i		−	0.610784i	0.952227	−	0.305392i	$-0.0987875\pi$
$194$		0			0
$195$	−2.22699	+	2.38075i	−0.159478	+	0.170489i
$196$		0			0
$197$	18.0000			1.28245			0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000			1.28245			0.641223	+	0.767354i	$0.278427\pi$
$198$		0			0
$199$			16.5906i			1.17607i	0.808834	+	0.588037i	$0.200099\pi$
$199$			16.5906i			1.17607i	−0.808834	+	0.588037i	$0.799901\pi$
$200$		0			0
$201$	−7.82087	−	1.95906i	−0.551642	−	0.138182i
$202$		0			0
$203$	9.29150	−	18.9713i	0.652136	−	1.33153i
$204$		0			0
$205$	−15.2915	−	8.48528i	−1.06800	−	0.592638i
$206$		0			0
$207$	−8.70850	−	4.65489i	−0.605282	−	0.323537i
$208$		0			0
$209$	12.8712			0.890320
$210$		0			0
$211$	−21.1660			−1.45713			−0.728564	−	0.684978i	$-0.759812\pi$
$211$	−21.1660			−1.45713			−0.728564	+	0.684978i	$0.759812\pi$
$212$		0			0
$213$	−21.2356	−	5.31935i	−1.45504	−	0.364476i
$214$		0			0
$215$	−5.05034	+	9.10132i	−0.344430	+	0.620705i
$216$		0			0
$217$	−6.43560	+	13.1402i	−0.436877	+	0.892013i
$218$		0			0
$219$	−1.29150	+	5.15587i	−0.0872717	+	0.348401i
$220$		0			0
$221$			1.00197i			0.0673996i
$222$		0			0
$223$	4.75216			0.318228			0.159114	−	0.987260i	$-0.449136\pi$
$223$	4.75216			0.318228			0.159114	+	0.987260i	$0.449136\pi$
$224$		0			0
$225$	−1.00000	−	14.9666i	−0.0666667	−	0.997775i
$226$		0			0
$227$		−	1.40122i		−	0.0930023i	−0.998918	−	0.0465011i	$-0.985193\pi$
$227$		−	1.40122i		−	0.0930023i	0.998918	−	0.0465011i	$-0.0148071\pi$
$228$		0			0
$229$		−	28.2835i		−	1.86903i	−0.355930	−	0.934513i	$-0.615836\pi$
$229$		−	28.2835i		−	1.86903i	0.355930	−	0.934513i	$-0.384164\pi$
$230$		0			0
$231$	9.90624	+	8.35862i	0.651782	+	0.549957i
$232$		0			0
$233$	2.70850			0.177440			0.0887198	−	0.996057i	$-0.471722\pi$
$233$	2.70850			0.177440			0.0887198	+	0.996057i	$0.471722\pi$
$234$		0			0
$235$	−4.70850	+	8.48528i	−0.307149	+	0.553519i
$236$		0			0
$237$	−3.06871	+	12.2508i	−0.199334	+	0.795772i
$238$		0			0
$239$		−	22.1264i		−	1.43124i	−0.698490	−	0.715620i	$-0.746144\pi$
$239$		−	22.1264i		−	1.43124i	0.698490	−	0.715620i	$-0.253856\pi$
$240$		0			0
$241$		−	1.95906i		−	0.126194i	−0.998007	−	0.0630972i	$-0.979902\pi$
$241$		−	1.95906i		−	0.126194i	0.998007	−	0.0630972i	$-0.0200978\pi$
$242$		0			0
$243$	−14.6773	+	5.25127i	−0.941551	+	0.336869i
$244$		0			0
$245$	−2.39082	−	15.4688i	−0.152744	−	0.988266i
$246$		0			0
$247$		−	3.83039i		−	0.243722i
$248$		0			0
$249$	−12.9373	−	3.24067i	−0.819865	−	0.205369i
$250$		0			0
$251$	−11.7313			−0.740473			−0.370237	−	0.928937i	$-0.620723\pi$
$251$	−11.7313			−0.740473			−0.370237	+	0.928937i	$0.620723\pi$
$252$		0			0
$253$		−	9.30978i		−	0.585301i
$254$		0			0
$255$	−3.36689	−	3.14944i	−0.210843	−	0.197225i
$256$		0			0
$257$		−	14.2098i		−	0.886384i	−0.896427	−	0.443192i	$-0.853846\pi$
$257$		−	14.2098i		−	0.886384i	0.896427	−	0.443192i	$-0.146154\pi$
$258$		0			0
$259$	12.5830	−	25.6919i	0.781870	−	1.59642i
$260$		0			0
$261$	−11.2915	+	21.1245i	−0.698926	+	1.30757i
$262$		0			0
$263$	6.58301			0.405925			0.202963	−	0.979186i	$-0.434943\pi$
$263$	6.58301			0.405925			0.202963	+	0.979186i	$0.434943\pi$
$264$		0			0
$265$	−24.6025	−	13.6520i	−1.51132	−	0.838634i
$266$		0			0
$267$	5.41699	−	21.6255i	0.331515	−	1.32346i
$268$		0			0
$269$	−24.6025			−1.50004			−0.750021	−	0.661414i	$-0.769957\pi$
$269$	−24.6025			−1.50004			−0.750021	+	0.661414i	$0.769957\pi$
$270$		0			0
$271$		−	7.48925i		−	0.454940i	−0.973785	−	0.227470i	$-0.926955\pi$
$271$		−	7.48925i		−	0.454940i	0.973785	−	0.227470i	$-0.0730453\pi$
$272$		0			0
$273$	2.48747	−	2.94804i	0.150549	−	0.178423i
$274$		0			0
$275$	12.0000	−	7.48331i	0.723627	−	0.451261i
$276$		0			0
$277$			15.4676i			0.929359i	0.885479	+	0.464679i	$0.153830\pi$
$277$			15.4676i			0.929359i	−0.885479	+	0.464679i	$0.846170\pi$
$278$		0			0
$279$	7.82087	−	14.6315i	0.468223	−	0.875965i
$280$		0			0
$281$			20.6235i			1.23029i	0.788412	+	0.615147i	$0.210903\pi$
$281$			20.6235i			1.23029i	−0.788412	+	0.615147i	$0.789097\pi$
$282$		0			0
$283$	−9.74968			−0.579558			−0.289779	−	0.957094i	$-0.593582\pi$
$283$	−9.74968			−0.579558			−0.289779	+	0.957094i	$0.593582\pi$
$284$		0			0
$285$	12.8712	+	12.0399i	0.762425	+	0.713183i
$286$		0			0
$287$	18.5830	+	9.10132i	1.09692	+	0.537234i
$288$		0			0
$289$	15.5830			0.916647
$290$		0			0
$291$	−3.41699	+	13.6412i	−0.200308	+	0.799659i
$292$		0			0
$293$		−	11.2712i		−	0.658472i	−0.944248	−	0.329236i	$-0.893209\pi$
$293$		−	11.2712i		−	0.658472i	0.944248	−	0.329236i	$-0.106791\pi$
$294$		0			0
$295$	−7.64575	−	4.24264i	−0.445153	−	0.247016i
$296$		0			0
$297$	−10.8896	−	9.87000i	−0.631878	−	0.572716i
$298$		0			0
$299$	−2.77053			−0.160224
$300$		0			0
$301$	5.41699	−	11.0604i	0.312230	−	0.637510i
$302$		0			0
$303$	−1.64575	+	6.57008i	−0.0945459	+	0.377441i
$304$		0			0
$305$	10.9373	−	19.7103i	0.626265	−	1.12861i
$306$		0			0
$307$	−14.8000			−0.844682			−0.422341	−	0.906437i	$-0.638791\pi$
$307$	−14.8000			−0.844682			−0.422341	+	0.906437i	$0.638791\pi$
$308$		0			0
$309$	−5.29150	+	21.1245i	−0.301023	+	1.20173i
$310$		0			0
$311$	2.77053			0.157103			0.0785513	−	0.996910i	$-0.474971\pi$
$311$	2.77053			0.157103			0.0785513	+	0.996910i	$0.474971\pi$
$312$		0			0
$313$	8.11905			0.458916			0.229458	−	0.973319i	$-0.426305\pi$
$313$	8.11905			0.458916			0.229458	+	0.973319i	$0.426305\pi$
$314$		0			0
$315$	2.08747	+	17.6251i	0.117615	+	0.993059i
$316$		0			0
$317$	6.00000			0.336994			0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000			0.336994			0.168497	+	0.985702i	$0.446109\pi$
$318$		0			0
$319$	−22.5830			−1.26441
$320$		0			0
$321$		0			0
$322$		0			0
$323$	5.41699			0.301410
$324$		0			0
$325$	−2.22699	−	3.57113i	−0.123531	−	0.198091i
$326$		0			0
$327$	−0.841723	+	3.36028i	−0.0465474	+	0.185824i
$328$		0			0
$329$	5.05034	−	10.3117i	0.278434	−	0.568505i
$330$		0			0
$331$	−8.00000			−0.439720			−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000			−0.439720			−0.219860	+	0.975531i	$0.570560\pi$
$332$		0			0
$333$	−15.2915	+	28.6078i	−0.837969	+	1.56770i
$334$		0			0
$335$	5.05034	−	9.10132i	0.275929	−	0.497258i
$336$		0			0
$337$		−	22.4499i		−	1.22293i	−0.791273	−	0.611463i	$-0.790581\pi$
$337$		−	22.4499i		−	1.22293i	0.791273	−	0.611463i	$-0.209419\pi$
$338$		0			0
$339$	2.52517	−	10.0808i	0.137148	−	0.547517i
$340$		0			0
$341$	15.6417			0.847048
$342$		0			0
$343$	3.76135	+	18.1343i	0.203094	+	0.979159i
$344$		0			0
$345$	8.70850	−	9.30978i	0.468850	−	0.501221i
$346$		0			0
$347$	24.0000			1.28839			0.644194	−	0.764862i	$-0.277193\pi$
$347$	24.0000			1.28839			0.644194	+	0.764862i	$0.277193\pi$
$348$		0			0
$349$			19.1822i			1.02680i	0.858150	+	0.513399i	$0.171614\pi$
$349$			19.1822i			1.02680i	−0.858150	+	0.513399i	$0.828386\pi$
$350$		0			0
$351$	−2.93725	+	3.24067i	−0.156779	+	0.172974i
$352$		0			0
$353$		−	21.3521i		−	1.13646i	−0.822871	−	0.568228i	$-0.807629\pi$
$353$		−	21.3521i		−	1.13646i	0.822871	−	0.568228i	$-0.192371\pi$
$354$		0			0
$355$	13.7129	−	24.7124i	0.727807	−	1.31160i
$356$		0			0
$357$	4.16915	+	3.51782i	0.220655	+	0.186183i
$358$		0			0
$359$		−	26.7813i		−	1.41346i	−0.707481	−	0.706732i	$-0.750169\pi$
$359$		−	26.7813i		−	1.41346i	0.707481	−	0.706732i	$-0.249831\pi$
$360$		0			0
$361$	−1.70850			−0.0899209
$362$		0			0
$363$	−1.26258	+	5.04042i	−0.0662685	+	0.264554i
$364$		0			0
$365$	−6.00000	−	3.32941i	−0.314054	−	0.174269i
$366$		0			0
$367$	−8.11905			−0.423811			−0.211905	−	0.977290i	$-0.567967\pi$
$367$	−8.11905			−0.423811			−0.211905	+	0.977290i	$0.567967\pi$
$368$		0			0
$369$	−20.6921	−	11.0604i	−1.07719	−	0.575780i
$370$		0			0
$371$	29.8982	+	14.6431i	1.55224	+	0.760233i
$372$		0			0
$373$		−	15.4676i		−	0.800883i	−0.916322	−	0.400441i	$-0.868857\pi$
$373$		−	15.4676i		−	0.800883i	0.916322	−	0.400441i	$-0.131143\pi$
$374$		0			0
$375$	19.0000	+	3.74166i	0.981156	+	0.193218i
$376$		0			0
$377$			6.72057i			0.346127i
$378$		0			0
$379$	−14.5830			−0.749079			−0.374539	−	0.927211i	$-0.622199\pi$
$379$	−14.5830			−0.749079			−0.374539	+	0.927211i	$0.622199\pi$
$380$		0			0
$381$	18.1669	+	4.55066i	0.930719	+	0.233137i
$382$		0			0
$383$		−	11.4821i		−	0.586706i	−0.956004	−	0.293353i	$-0.905229\pi$
$383$		−	11.4821i		−	0.586706i	0.956004	−	0.293353i	$-0.0947712\pi$
$384$		0			0
$385$	−13.7271	+	9.56904i	−0.699598	+	0.487684i
$386$		0			0
$387$	−6.58301	+	12.3157i	−0.334633	+	0.626041i
$388$		0			0
$389$		−	0.323511i		−	0.0164026i	−0.999966	−	0.00820132i	$-0.997389\pi$
$389$		−	0.323511i		−	0.0164026i	0.999966	−	0.00820132i	$-0.00261059\pi$
$390$		0			0
$391$		−	3.91813i		−	0.198148i
$392$		0			0
$393$	3.77124	−	15.0554i	0.190234	−	0.759443i
$394$		0			0
$395$	−14.2565	−	7.91094i	−0.717321	−	0.398043i
$396$		0			0
$397$	−21.5338			−1.08075			−0.540375	−	0.841424i	$-0.681718\pi$
$397$	−21.5338			−1.08075			−0.540375	+	0.841424i	$0.681718\pi$
$398$		0			0
$399$	−15.9382	−	13.4482i	−0.797906	−	0.673251i
$400$		0			0
$401$			7.48331i			0.373699i	0.982389	+	0.186849i	$0.0598277\pi$
$401$			7.48331i			0.373699i	−0.982389	+	0.186849i	$0.940172\pi$
$402$		0			0
$403$		−	4.65489i		−	0.231876i
$404$		0			0
$405$	−1.65704	−	20.0563i	−0.0823389	−	0.996604i
$406$		0			0
$407$	−30.5830			−1.51594
$408$		0			0
$409$		−	13.0194i		−	0.643770i	−0.946779	−	0.321885i	$-0.895684\pi$
$409$		−	13.0194i		−	0.643770i	0.946779	−	0.321885i	$-0.104316\pi$
$410$		0			0
$411$	3.91044	−	15.6110i	0.192888	−	0.770036i
$412$		0			0
$413$	9.29150	+	4.55066i	0.457205	+	0.223923i
$414$		0			0
$415$	8.35425	−	15.0554i	0.410094	−	0.739039i
$416$		0			0
$417$	26.2288	+	6.57008i	1.28443	+	0.321738i
$418$		0			0
$419$	−21.8320			−1.06656			−0.533281	−	0.845938i	$-0.679041\pi$
$419$	−21.8320			−1.06656			−0.533281	+	0.845938i	$0.679041\pi$
$420$		0			0
$421$	−10.0000			−0.487370			−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000			−0.487370			−0.243685	+	0.969854i	$0.578356\pi$
$422$		0			0
$423$	−6.13742	+	11.4821i	−0.298412	+	0.558277i
$424$		0			0
$425$	5.05034	−	3.14944i	0.244977	−	0.152770i
$426$		0			0
$427$	−11.7313	+	23.9529i	−0.567718	+	1.15916i
$428$		0			0
$429$	−4.00000	−	1.00197i	−0.193122	−	0.0483754i
$430$		0			0
$431$			16.4696i			0.793312i	0.917967	+	0.396656i	$0.129829\pi$
$431$			16.4696i			0.793312i	−0.917967	+	0.396656i	$0.870171\pi$
$432$		0			0
$433$	26.5313			1.27501			0.637507	−	0.770445i	$-0.279966\pi$
$433$	26.5313			1.27501			0.637507	+	0.770445i	$0.279966\pi$
$434$		0			0
$435$	−22.5830	−	21.1245i	−1.08277	−	1.01284i
$436$		0			0
$437$			14.9785i			0.716519i
$438$		0			0
$439$			18.5496i			0.885326i	0.896688	+	0.442663i	$0.145966\pi$
$439$			18.5496i			0.885326i	−0.896688	+	0.442663i	$0.854034\pi$
$440$		0			0
$441$	−3.53338	−	20.7006i	−0.168256	−	0.985743i
$442$		0			0
$443$	−13.1660			−0.625536			−0.312768	−	0.949830i	$-0.601256\pi$
$443$	−13.1660			−0.625536			−0.312768	+	0.949830i	$0.601256\pi$
$444$		0			0
$445$	25.1660	+	13.9647i	1.19298	+	0.661989i
$446$		0			0
$447$	5.59388	+	1.40122i	0.264581	+	0.0662755i
$448$		0			0
$449$			8.30781i			0.392070i	0.980597	+	0.196035i	$0.0628066\pi$
$449$			8.30781i			0.392070i	−0.980597	+	0.196035i	$0.937193\pi$
$450$		0			0
$451$		−	22.1208i		−	1.04163i
$452$		0			0
$453$	−9.50432	+	37.9426i	−0.446552	+	1.78270i
$454$		0			0
$455$	2.84769	+	4.08510i	0.133502	+	0.191513i
$456$		0			0
$457$			8.48528i			0.396925i	0.980109	+	0.198462i	$0.0635948\pi$
$457$			8.48528i			0.396925i	−0.980109	+	0.198462i	$0.936405\pi$
$458$		0			0
$459$	−4.58301	−	4.15390i	−0.213916	−	0.193888i
$460$		0			0
$461$	−8.96077			−0.417345			−0.208672	−	0.977986i	$-0.566914\pi$
$461$	−8.96077			−0.417345			−0.208672	+	0.977986i	$0.566914\pi$
$462$		0			0
$463$		−	23.9529i		−	1.11319i	−0.830786	−	0.556593i	$-0.812108\pi$
$463$		−	23.9529i		−	1.11319i	0.830786	−	0.556593i	$-0.187892\pi$
$464$		0			0
$465$	15.6417	+	14.6315i	0.725368	+	0.678520i
$466$		0			0
$467$		−	3.36028i		−	0.155495i	−0.996973	−	0.0777477i	$-0.975227\pi$
$467$		−	3.36028i		−	0.155495i	0.996973	−	0.0777477i	$-0.0247729\pi$
$468$		0			0
$469$	−5.41699	+	11.0604i	−0.250134	+	0.510721i
$470$		0			0
$471$	−9.52026	+	38.0063i	−0.438670	+	1.75124i
$472$		0			0
$473$	−13.1660			−0.605374
$474$		0			0
$475$	−19.3068	+	12.0399i	−0.885857	+	0.552429i
$476$		0			0
$477$	−33.2915	−	17.7951i	−1.52431	−	0.814780i
$478$		0			0
$479$	−39.1044			−1.78672			−0.893362	−	0.449338i	$-0.851660\pi$
$479$	−39.1044			−1.78672			−0.893362	+	0.449338i	$0.851660\pi$
$480$		0			0
$481$			9.10132i			0.414984i
$482$		0			0
$483$	−9.72711	+	11.5281i	−0.442599	+	0.524547i
$484$		0			0
$485$	−15.8745	−	8.80879i	−0.720824	−	0.399987i
$486$		0			0
$487$			32.4382i			1.46991i	0.678114	+	0.734957i	$0.262798\pi$
$487$			32.4382i			1.46991i	−0.678114	+	0.734957i	$0.737202\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			14.1421i			0.638226i	0.947717	+	0.319113i	$0.103385\pi$
$491$			14.1421i			0.638226i	−0.947717	+	0.319113i	$0.896615\pi$
$492$		0			0
$493$	−9.50432			−0.428053
$494$		0			0
$495$	15.9399	−	10.2917i	0.716446	−	0.462577i
$496$		0			0
$497$	−14.7085	+	30.0317i	−0.659766	+	1.34711i
$498$		0			0
$499$	−1.41699			−0.0634334			−0.0317167	−	0.999497i	$-0.510097\pi$
$499$	−1.41699			−0.0634334			−0.0317167	+	0.999497i	$0.510097\pi$
$500$		0			0
$501$	19.2915	+	4.83236i	0.861881	+	0.215894i
$502$		0			0
$503$			8.67963i			0.387006i	0.981100	+	0.193503i	$0.0619848\pi$
$503$			8.67963i			0.387006i	−0.981100	+	0.193503i	$0.938015\pi$
$504$		0			0
$505$	−7.64575	−	4.24264i	−0.340231	−	0.188795i
$506$		0			0
$507$	5.17302	−	20.6515i	0.229742	−	0.917164i
$508$		0			0
$509$	30.1436			1.33609			0.668045	−	0.744121i	$-0.267131\pi$
$509$	30.1436			1.33609			0.668045	+	0.744121i	$0.267131\pi$
$510$		0			0
$511$	7.29150	+	3.57113i	0.322557	+	0.157977i
$512$		0			0
$513$	17.5203	+	15.8799i	0.773538	+	0.701113i
$514$		0			0
$515$	−24.5830	−	13.6412i	−1.08326	−	0.601101i
$516$		0			0
$517$	−12.2748			−0.539847
$518$		0			0
$519$	−14.9373	−	3.74166i	−0.655673	−	0.164241i
$520$		0			0
$521$	−23.4626			−1.02792			−0.513958	−	0.857815i	$-0.671821\pi$
$521$	−23.4626			−1.02792			−0.513958	+	0.857815i	$0.671821\pi$
$522$		0			0
$523$	−4.20861			−0.184030			−0.0920149	−	0.995758i	$-0.529331\pi$
$523$	−4.20861			−0.184030			−0.0920149	+	0.995758i	$0.529331\pi$
$524$		0			0
$525$	−22.6781	−	3.27149i	−0.989755	−	0.142780i
$526$		0			0
$527$	6.58301			0.286760
$528$		0			0
$529$	−12.1660			−0.528957
$530$		0			0
$531$	−10.3460	−	5.53019i	−0.448980	−	0.239990i
$532$		0			0
$533$	−6.58301			−0.285142
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−15.9399	−	3.99282i	−0.687858	−	0.172303i
$538$		0			0
$539$	15.6417	−	12.1382i	0.673737	−	0.522829i
$540$		0			0
$541$	2.00000			0.0859867			0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000			0.0859867			0.0429934	+	0.999075i	$0.486311\pi$
$542$		0			0
$543$	20.2288	+	5.06713i	0.868099	+	0.217452i
$544$		0			0
$545$	−3.91044	−	2.16991i	−0.167505	−	0.0929486i
$546$		0			0
$547$			16.9706i			0.725609i	0.931865	+	0.362804i	$0.118181\pi$
$547$			16.9706i			0.725609i	−0.931865	+	0.362804i	$0.881819\pi$
$548$		0			0
$549$	14.2565	−	26.6714i	0.608451	−	1.13831i
$550$		0			0
$551$	36.3338			1.54787
$552$		0			0
$553$	17.3252	+	8.48528i	0.736742	+	0.360831i
$554$		0			0
$555$	−30.5830	−	28.6078i	−1.29818	−	1.21433i
$556$		0			0
$557$	7.16601			0.303634			0.151817	−	0.988409i	$-0.451488\pi$
$557$	7.16601			0.303634			0.151817	+	0.988409i	$0.451488\pi$
$558$		0			0
$559$			3.91813i			0.165719i
$560$		0			0
$561$	1.41699	−	5.65685i	0.0598256	−	0.238833i
$562$		0			0
$563$			18.7605i			0.790660i	0.918539	+	0.395330i	$0.129370\pi$
$563$			18.7605i			0.790660i	−0.918539	+	0.395330i	$0.870630\pi$
$564$		0			0
$565$	11.7313	+	6.50972i	0.493540	+	0.273866i
$566$		0			0
$567$	3.17190	+	23.5996i	0.133207	+	0.991088i
$568$		0			0
$569$		−	11.1362i		−	0.466855i	−0.972374	−	0.233428i	$-0.925006\pi$
$569$		−	11.1362i		−	0.466855i	0.972374	−	0.233428i	$-0.0749942\pi$
$570$		0			0
$571$	34.5830			1.44725			0.723627	−	0.690191i	$-0.242474\pi$
$571$	34.5830			1.44725			0.723627	+	0.690191i	$0.242474\pi$
$572$		0			0
$573$	−13.4148	−	3.36028i	−0.560409	−	0.140378i
$574$		0			0
$575$	8.70850	+	13.9647i	0.363169	+	0.582367i
$576$		0			0
$577$	−30.9853			−1.28993			−0.644967	−	0.764210i	$-0.723129\pi$
$577$	−30.9853			−1.28993			−0.644967	+	0.764210i	$0.723129\pi$
$578$		0			0
$579$	14.2565	+	3.57113i	0.592479	+	0.148411i
$580$		0			0
$581$	−8.96077	+	18.2960i	−0.371755	+	0.759048i
$582$		0			0
$583$		−	35.5901i		−	1.47399i
$584$		0			0
$585$	−3.06275	−	4.74362i	−0.126629	−	0.196125i
$586$		0			0
$587$			44.1054i			1.82042i	0.414143	+	0.910212i	$0.364081\pi$
$587$			44.1054i			1.82042i	−0.414143	+	0.910212i	$0.635919\pi$
$588$		0			0
$589$	−25.1660			−1.03695
$590$		0			0
$591$	−7.57551	+	30.2425i	−0.311615	+	1.24401i
$592$		0			0
$593$			7.91094i			0.324863i	0.986720	+	0.162432i	$0.0519337\pi$
$593$			7.91094i			0.324863i	−0.986720	+	0.162432i	$0.948066\pi$
$594$		0			0
$595$	−5.77721	+	4.02724i	−0.236843	+	0.165101i
$596$		0			0
$597$	−27.8745	−	6.98233i	−1.14083	−	0.285768i
$598$		0			0
$599$		−	18.2960i		−	0.747556i	−0.927518	−	0.373778i	$-0.878062\pi$
$599$		−	18.2960i		−	0.747556i	0.927518	−	0.373778i	$-0.121938\pi$
$600$		0			0
$601$		−	11.0604i		−	0.451162i	−0.974224	−	0.225581i	$-0.927572\pi$
$601$		−	11.0604i		−	0.451162i	0.974224	−	0.225581i	$-0.0724281\pi$
$602$		0			0
$603$	6.58301	−	12.3157i	0.268081	−	0.501533i
$604$		0			0
$605$	−5.86565	−	3.25486i	−0.238473	−	0.132329i
$606$		0			0
$607$	−23.7608			−0.964421			−0.482210	−	0.876055i	$-0.660166\pi$
$607$	−23.7608			−0.964421			−0.482210	+	0.876055i	$0.660166\pi$
$608$		0			0
$609$	27.9641	+	23.5953i	1.13316	+	0.956131i
$610$		0			0
$611$			3.65292i			0.147781i
$612$		0			0
$613$			40.0990i			1.61958i	0.586719	+	0.809791i	$0.300419\pi$
$613$			40.0990i			1.61958i	−0.586719	+	0.809791i	$0.699581\pi$
$614$		0			0
$615$	20.6921	−	22.1208i	0.834385	−	0.891995i
$616$		0			0
$617$	4.83399			0.194609			0.0973045	−	0.995255i	$-0.468978\pi$
$617$	4.83399			0.194609			0.0973045	+	0.995255i	$0.468978\pi$
$618$		0			0
$619$			0.632534i			0.0254237i	0.999919	+	0.0127118i	$0.00404641\pi$
$619$			0.632534i			0.0254237i	−0.999919	+	0.0127118i	$0.995954\pi$
$620$		0			0
$621$	11.4859	−	12.6724i	0.460915	−	0.508528i
$622$		0			0
$623$	−30.5830	−	14.9785i	−1.22528	−	0.600101i
$624$		0			0
$625$	−11.0000	+	22.4499i	−0.440000	+	0.897998i
$626$		0			0
$627$	−5.41699	+	21.6255i	−0.216334	+	0.863637i
$628$		0			0
$629$	−12.8712			−0.513209
$630$		0			0
$631$	20.4575			0.814401			0.407200	−	0.913339i	$-0.366505\pi$
$631$	20.4575			0.814401			0.407200	+	0.913339i	$0.366505\pi$
$632$		0			0
$633$	8.90796	−	35.5619i	0.354060	−	1.41346i
$634$		0			0
$635$	−11.7313	+	21.1412i	−0.465543	+	0.838964i
$636$		0			0
$637$	−3.61226	−	4.65489i	−0.143123	−	0.184433i
$638$		0			0
$639$	17.8745	−	33.4401i	0.707105	−	1.32287i
$640$		0			0
$641$			16.7931i			0.663287i	0.943405	+	0.331644i	$0.107603\pi$
$641$			16.7931i			0.663287i	−0.943405	+	0.331644i	$0.892397\pi$
$642$		0			0
$643$	47.7669			1.88374			0.941872	−	0.335971i	$-0.109065\pi$
$643$	47.7669			1.88374			0.941872	+	0.335971i	$0.109065\pi$
$644$		0			0
$645$	−13.1660	−	12.3157i	−0.518411	−	0.484929i
$646$		0			0
$647$		−	15.4002i		−	0.605444i	−0.953079	−	0.302722i	$-0.902105\pi$
$647$		−	15.4002i		−	0.605444i	0.953079	−	0.302722i	$-0.0978954\pi$
$648$		0			0
$649$		−	11.0604i		−	0.434158i
$650$		0			0
$651$	−19.3688	−	16.3429i	−0.759125	−	0.640529i
$652$		0			0
$653$	19.1660			0.750024			0.375012	−	0.927020i	$-0.377639\pi$
$653$	19.1660			0.750024			0.375012	+	0.927020i	$0.377639\pi$
$654$		0			0
$655$	17.5203	+	9.72202i	0.684573	+	0.379871i
$656$		0			0
$657$	−8.11905	−	4.33981i	−0.316754	−	0.169312i
$658$		0			0
$659$			32.7617i			1.27621i	0.769948	+	0.638107i	$0.220282\pi$
$659$			32.7617i			1.27621i	−0.769948	+	0.638107i	$0.779718\pi$
$660$		0			0
$661$		−	0.979531i		−	0.0380994i	−0.999819	−	0.0190497i	$-0.993936\pi$
$661$		−	0.979531i		−	0.0380994i	0.999819	−	0.0190497i	$-0.00606407\pi$
$662$		0			0
$663$	−1.68345	−	0.421689i	−0.0653796	−	0.0163770i
$664$		0			0
$665$	22.0856	−	15.3956i	0.856441	−	0.597017i
$666$		0			0
$667$		−	26.2803i		−	1.01758i
$668$		0			0
$669$	−2.00000	+	7.98430i	−0.0773245	+	0.308691i
$670$		0			0
$671$	28.5129			1.10073
$672$		0			0
$673$		−	38.5960i		−	1.48777i	−0.668309	−	0.743883i	$-0.732982\pi$
$673$		−	38.5960i		−	1.48777i	0.668309	−	0.743883i	$-0.267018\pi$
$674$		0			0
$675$	25.5669	+	4.61874i	0.984071	+	0.177775i
$676$		0			0
$677$		−	42.4933i		−	1.63315i	−0.577239	−	0.816575i	$-0.695870\pi$
$677$		−	42.4933i		−	1.63315i	0.577239	−	0.816575i	$-0.304130\pi$
$678$		0			0
$679$	19.2915	+	9.44832i	0.740340	+	0.362593i
$680$		0			0
$681$	2.35425	+	0.589720i	0.0902150	+	0.0225981i
$682$		0			0
$683$	−5.41699			−0.207276			−0.103638	−	0.994615i	$-0.533048\pi$
$683$	−5.41699			−0.207276			−0.103638	+	0.994615i	$0.533048\pi$
$684$		0			0
$685$	18.1669	+	10.0808i	0.694122	+	0.385169i
$686$		0			0
$687$	47.5203	+	11.9034i	1.81301	+	0.454144i
$688$		0			0
$689$	−10.5914			−0.403500
$690$		0			0
$691$			6.50972i			0.247641i	0.992305	+	0.123821i	$0.0395148\pi$
$691$			6.50972i			0.247641i	−0.992305	+	0.123821i	$0.960485\pi$
$692$		0			0
$693$	−18.2128	+	13.1261i	−0.691848	+	0.498618i
$694$		0			0
$695$	−16.9373	+	30.5230i	−0.642467	+	1.15780i
$696$		0			0
$697$		−	9.30978i		−	0.352633i
$698$		0			0
$699$	−1.13990	+	4.55066i	−0.0431151	+	0.172122i
$700$		0			0
$701$			1.32548i			0.0500626i	0.999687	+	0.0250313i	$0.00796854\pi$
$701$			1.32548i			0.0500626i	−0.999687	+	0.0250313i	$0.992031\pi$
$702$		0			0
$703$	49.2050			1.85580
$704$		0			0
$705$	−12.2748	−	11.4821i	−0.462298	−	0.432440i
$706$		0			0
$707$	9.29150	+	4.55066i	0.349443	+	0.171145i
$708$		0			0
$709$	20.5830			0.773011			0.386505	−	0.922287i	$-0.373682\pi$
$709$	20.5830			0.773011			0.386505	+	0.922287i	$0.373682\pi$
$710$		0			0
$711$	−19.2915	−	10.3117i	−0.723488	−	0.386721i
$712$		0			0
$713$			18.2026i			0.681694i
$714$		0			0
$715$	2.58301	−	4.65489i	0.0965989	−	0.174083i
$716$		0			0
$717$	37.1755	+	9.31216i	1.38835	+	0.347769i
$718$		0			0
$719$	36.3338			1.35502			0.677511	−	0.735512i	$-0.263058\pi$
$719$	36.3338			1.35502			0.677511	+	0.735512i	$0.263058\pi$
$720$		0			0
$721$	29.8745	+	14.6315i	1.11258	+	0.544906i
$722$		0			0
$723$	3.29150	+	0.824494i	0.122412	+	0.0306633i
$724$		0			0
$725$	33.8745	−	21.1245i	1.25807	−	0.784543i
$726$		0			0
$727$	7.03196			0.260801			0.130401	−	0.991461i	$-0.458374\pi$
$727$	7.03196			0.260801			0.130401	+	0.991461i	$0.458374\pi$
$728$		0			0
$729$	−2.64575	−	26.8701i	−0.0979908	−	0.995187i
$730$		0			0
$731$	−5.54107			−0.204944
$732$		0			0
$733$	24.9007			0.919728			0.459864	−	0.887989i	$-0.347898\pi$
$733$	24.9007			0.919728			0.459864	+	0.887989i	$0.347898\pi$
$734$		0			0
$735$	26.9960	+	2.49331i	0.995762	+	0.0919670i
$736$		0			0
$737$	13.1660			0.484976
$738$		0			0
$739$	−9.16601			−0.337177			−0.168589	−	0.985687i	$-0.553921\pi$
$739$	−9.16601			−0.337177			−0.168589	+	0.985687i	$0.553921\pi$
$740$		0			0
$741$	6.43560	+	1.61206i	0.236418	+	0.0592207i
$742$		0			0
$743$	33.8745			1.24274			0.621368	−	0.783519i	$-0.286577\pi$
$743$	33.8745			1.24274			0.621368	+	0.783519i	$0.286577\pi$
$744$		0			0
$745$	−3.61226	+	6.50972i	−0.132343	+	0.238498i
$746$		0			0
$747$	10.8896	−	20.3725i	0.398429	−	0.745392i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−21.1660			−0.772359			−0.386179	−	0.922424i	$-0.626205\pi$
$751$	−21.1660			−0.772359			−0.386179	+	0.922424i	$0.626205\pi$
$752$		0			0
$753$	4.93725	−	19.7103i	0.179924	−	0.718282i
$754$		0			0
$755$	−44.1547	−	24.5015i	−1.60695	−	0.891701i
$756$		0			0
$757$			3.15194i			0.114559i	0.998358	+	0.0572796i	$0.0182426\pi$
$757$			3.15194i			0.114559i	−0.998358	+	0.0572796i	$0.981757\pi$
$758$		0			0
$759$	15.6417	+	3.91813i	0.567759	+	0.142219i
$760$		0			0
$761$	20.6921			0.750087			0.375044	−	0.927007i	$-0.377628\pi$
$761$	20.6921			0.750087			0.375044	+	0.927007i	$0.377628\pi$
$762$		0			0
$763$	4.75216	+	2.32744i	0.172040	+	0.0842591i
$764$		0			0
$765$	6.70850	−	4.33138i	0.242546	−	0.156601i
$766$		0			0
$767$	−3.29150			−0.118849
$768$		0			0
$769$			35.1402i			1.26719i	0.773666	+	0.633594i	$0.218421\pi$
$769$			35.1402i			1.26719i	−0.773666	+	0.633594i	$0.781579\pi$
$770$		0			0
$771$	23.8745	+	5.98036i	0.859819	+	0.215378i
$772$		0			0
$773$		−	7.35310i		−	0.264473i	−0.991218	−	0.132236i	$-0.957784\pi$
$773$		−	7.35310i		−	0.264473i	0.991218	−	0.132236i	$-0.0422158\pi$
$774$		0			0
$775$	−23.4626	+	14.6315i	−0.842802	+	0.525579i
$776$		0			0
$777$	37.8703	+	31.9540i	1.35859	+	1.14634i
$778$		0			0
$779$			35.5901i			1.27515i
$780$		0			0
$781$	35.7490			1.27920
$782$		0			0
$783$	−30.7399	−	27.8618i	−1.09856	−	0.995699i
$784$		0			0
$785$	−44.2288	−	24.5426i	−1.57859	−	0.875963i
$786$		0			0
$787$	−24.9007			−0.887614			−0.443807	−	0.896122i	$-0.646372\pi$
$787$	−24.9007			−0.887614			−0.443807	+	0.896122i	$0.646372\pi$
$788$		0			0
$789$	−2.77053	+	11.0604i	−0.0986336	+	0.393760i
$790$		0			0
$791$	−14.2565	−	6.98233i	−0.506902	−	0.248263i
$792$		0			0
$793$		−	8.48528i		−	0.301321i
$794$		0			0
$795$	33.2915	−	35.5901i	1.18073	−	1.26225i
$796$		0			0
$797$			6.93141i			0.245523i	0.992436	+	0.122762i	$0.0391750\pi$
$797$			6.93141i			0.245523i	−0.992436	+	0.122762i	$0.960825\pi$
$798$		0			0
$799$	−5.16601			−0.182760
$800$		0			0
$801$	34.0540	+	18.2026i	1.20324	+	0.643159i
$802$		0			0
$803$		−	8.67963i		−	0.306297i
$804$		0			0
$805$	−11.1357	−	15.9745i	−0.392482	−	0.563028i
$806$		0			0
$807$	10.3542	−	41.3357i	0.364487	−	1.45509i
$808$		0			0
$809$			33.7637i			1.18707i	0.804809	+	0.593533i	$0.202268\pi$
$809$			33.7637i			1.18707i	−0.804809	+	0.593533i	$0.797732\pi$
$810$		0			0
$811$			28.6305i			1.00535i	0.864475	+	0.502676i	$0.167651\pi$
$811$			28.6305i			1.00535i	−0.864475	+	0.502676i	$0.832349\pi$
$812$		0			0
$813$	12.5830	+	3.15194i	0.441305	+	0.110543i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	21.1828			0.741093
$818$		0			0
$819$	3.90624	+	5.42002i	0.136495	+	0.189391i
$820$		0			0
$821$			5.98036i			0.208716i	0.994540	+	0.104358i	$0.0332788\pi$
$821$			5.98036i			0.208716i	−0.994540	+	0.104358i	$0.966721\pi$
$822$		0			0
$823$			19.2980i			0.672686i	0.941740	+	0.336343i	$0.109190\pi$
$823$			19.2980i			0.672686i	−0.941740	+	0.336343i	$0.890810\pi$
$824$		0			0
$825$	7.52269	+	23.3111i	0.261906	+	0.811590i
$826$		0			0
$827$	−36.0000			−1.25184			−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000			−1.25184			−0.625921	+	0.779886i	$0.715277\pi$
$828$		0			0
$829$			45.2211i			1.57059i	0.619120	+	0.785296i	$0.287489\pi$
$829$			45.2211i			1.57059i	−0.619120	+	0.785296i	$0.712511\pi$
$830$		0			0
$831$	−25.9878	−	6.50972i	−0.901506	−	0.225820i
$832$		0			0
$833$	6.58301	−	5.10850i	0.228088	−	0.176999i
$834$		0			0
$835$	−12.4575	+	22.4499i	−0.431110	+	0.776912i
$836$		0			0
$837$	21.2915	+	19.2980i	0.735942	+	0.667037i
$838$		0			0
$839$	−26.2331			−0.905669			−0.452834	−	0.891595i	$-0.649587\pi$
$839$	−26.2331			−0.905669			−0.452834	+	0.891595i	$0.649587\pi$
$840$		0			0
$841$	−34.7490			−1.19824
$842$		0			0
$843$	−34.6504	−	8.67963i	−1.19342	−	0.298942i
$844$		0			0
$845$	24.0326	+	13.3357i	0.826745	+	0.458762i
$846$		0			0
$847$	7.12824	+	3.49117i	0.244929	+	0.119958i
$848$		0			0
$849$	4.10326	−	16.3808i	0.140824	−	0.562189i
$850$		0			0
$851$		−	35.5901i		−	1.22001i
$852$		0			0
$853$	−5.89206			−0.201740			−0.100870	−	0.994900i	$-0.532163\pi$
$853$	−5.89206			−0.201740			−0.100870	+	0.994900i	$0.532163\pi$
$854$		0			0
$855$	−25.6458	+	16.5583i	−0.877066	+	0.566282i
$856$		0			0
$857$			24.1545i			0.825103i	0.910934	+	0.412551i	$0.135362\pi$
$857$			24.1545i			0.825103i	−0.910934	+	0.412551i	$0.864638\pi$
$858$		0			0
$859$			2.59160i			0.0884241i	0.999022	+	0.0442121i	$0.0140777\pi$
$859$			2.59160i			0.0884241i	−0.999022	+	0.0442121i	$0.985922\pi$
$860$		0			0
$861$	−23.1124	+	27.3917i	−0.787668	+	0.933506i
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$	9.64575	−	17.3828i	0.327965	−	0.591033i
$866$		0			0
$867$	−6.55829	+	26.1817i	−0.222731	+	0.889176i
$868$		0			0
$869$		−	20.6235i		−	0.699604i
$870$		0			0
$871$		−	3.91813i		−	0.132761i
$872$		0			0
$873$	−21.4810	−	11.4821i	−0.727021	−	0.388609i
$874$		0			0
$875$	11.6396	−	27.1941i	0.393492	−	0.919328i
$876$		0			0
$877$		−	1.50295i		−	0.0507510i	−0.999678	−	0.0253755i	$-0.991922\pi$
$877$		−	1.50295i		−	0.0507510i	0.999678	−	0.0253755i	$-0.00807814\pi$
$878$		0			0
$879$	18.9373	+	4.74362i	0.638738	+	0.159998i
$880$		0			0
$881$	12.8712			0.433642			0.216821	−	0.976211i	$-0.430431\pi$
$881$	12.8712			0.433642			0.216821	+	0.976211i	$0.430431\pi$
$882$		0			0
$883$		−	13.9647i		−	0.469948i	−0.972002	−	0.234974i	$-0.924499\pi$
$883$		−	13.9647i		−	0.469948i	0.972002	−	0.234974i	$-0.0755006\pi$
$884$		0			0
$885$	10.3460	−	11.0604i	0.347778	−	0.371791i
$886$		0			0
$887$		−	44.6632i		−	1.49964i	−0.661640	−	0.749822i	$-0.730139\pi$
$887$		−	44.6632i		−	1.49964i	0.661640	−	0.749822i	$-0.269861\pi$
$888$		0			0
$889$	12.5830	−	25.6919i	0.422020	−	0.861678i
$890$		0			0
$891$	21.1660	−	14.1421i	0.709088	−	0.473779i
$892$		0			0
$893$	19.7490			0.660876
$894$		0			0
$895$	10.2932	−	18.5496i	0.344065	−	0.620046i
$896$		0			0
$897$	1.16601	−	4.65489i	0.0389320	−	0.155422i
$898$		0			0
$899$	44.1547			1.47264
$900$		0			0
$901$		−	14.9785i		−	0.499006i
$902$		0			0
$903$	16.3032	+	13.7562i	0.542537	+	0.457778i
$904$		0			0
$905$	−13.0627	+	23.5406i	−0.434220	+	0.782518i
$906$		0			0
$907$		−	33.9411i		−	1.12700i	−0.826117	−	0.563498i	$-0.809455\pi$
$907$		−	33.9411i		−	1.12700i	0.826117	−	0.563498i	$-0.190545\pi$
$908$		0			0
$909$	−10.3460	−	5.53019i	−0.343156	−	0.183425i
$910$		0			0
$911$			4.15390i			0.137625i	0.997630	+	0.0688125i	$0.0219210\pi$
$911$			4.15390i			0.137625i	−0.997630	+	0.0688125i	$0.978079\pi$
$912$		0			0
$913$	21.7792			0.720785
$914$		0			0
$915$	28.5129	+	26.6714i	0.942609	+	0.881730i
$916$		0			0
$917$	−21.2915	−	10.4278i	−0.703107	−	0.344358i
$918$		0			0
$919$	−10.1255			−0.334009			−0.167005	−	0.985956i	$-0.553409\pi$
$919$	−10.1255			−0.334009			−0.167005	+	0.985956i	$0.553409\pi$
$920$		0			0
$921$	6.22876	−	24.8661i	0.205245	−	0.819367i
$922$		0			0
$923$		−	10.6387i		−	0.350177i
$924$		0			0
$925$	45.8745	−	28.6078i	1.50834	−	0.940618i
$926$		0			0
$927$	−33.2651	−	17.7809i	−1.09257	−	0.584003i
$928$		0			0
$929$	30.7928			1.01028			0.505139	−	0.863038i	$-0.331441\pi$
$929$	30.7928			1.01028			0.505139	+	0.863038i	$0.331441\pi$
$930$		0			0
$931$	−25.1660	+	19.5292i	−0.824783	+	0.640043i
$932$		0			0
$933$	−1.16601	+	4.65489i	−0.0381735	+	0.152394i
$934$		0			0
$935$	6.58301	+	3.65292i	0.215287	+	0.119463i
$936$		0			0
$937$	3.06871			0.100250			0.0501252	−	0.998743i	$-0.484038\pi$
$937$	3.06871			0.100250			0.0501252	+	0.998743i	$0.484038\pi$
$938$		0			0
$939$	−3.41699	+	13.6412i	−0.111509	+	0.445162i
$940$		0			0
$941$	−40.2443			−1.31193			−0.655963	−	0.754793i	$-0.727737\pi$
$941$	−40.2443			−1.31193			−0.655963	+	0.754793i	$0.727737\pi$
$942$		0			0
$943$	25.7424			0.838288
$944$		0			0
$945$	−30.4911	−	3.91047i	−0.991876	−	0.127208i
$946$		0			0
$947$	−24.0000			−0.779895			−0.389948	−	0.920837i	$-0.627507\pi$
$947$	−24.0000			−0.779895			−0.389948	+	0.920837i	$0.627507\pi$
$948$		0			0
$949$	−2.58301			−0.0838479
$950$		0			0
$951$	−2.52517	+	10.0808i	−0.0818842	+	0.326894i
$952$		0			0
$953$	−27.8745			−0.902944			−0.451472	−	0.892285i	$-0.649101\pi$
$953$	−27.8745			−0.902944			−0.451472	+	0.892285i	$0.649101\pi$
$954$		0			0
$955$	8.66259	−	15.6110i	0.280315	−	0.505161i
$956$		0			0
$957$	9.50432	−	37.9426i	0.307231	−	1.22651i
$958$		0			0
$959$	−22.0773	−	10.8127i	−0.712915	−	0.349161i
$960$		0			0
$961$	0.416995			0.0134514
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−9.20614	+	16.5906i	−0.296356	+	0.534069i
$966$		0			0
$967$		−	49.4087i		−	1.58888i	−0.607344	−	0.794439i	$-0.707765\pi$
$967$		−	49.4087i		−	1.58888i	0.607344	−	0.794439i	$-0.292235\pi$
$968$		0			0
$969$	−2.27980	+	9.10132i	−0.0732379	+	0.292376i
$970$		0			0
$971$	24.6025			0.789532			0.394766	−	0.918782i	$-0.370826\pi$
$971$	24.6025			0.789532			0.394766	+	0.918782i	$0.370826\pi$
$972$		0			0
$973$	18.1669	−	37.0931i	0.582404	−	1.18915i
$974$		0			0
$975$	6.93725	−	2.23871i	0.222170	−	0.0716960i
$976$		0			0
$977$	−51.8745			−1.65961			−0.829806	−	0.558052i	$-0.811549\pi$
$977$	−51.8745			−1.65961			−0.829806	+	0.558052i	$0.811549\pi$
$978$		0			0
$979$			36.4053i			1.16352i
$980$		0			0
$981$	−5.29150	−	2.82843i	−0.168945	−	0.0903047i
$982$		0			0
$983$			62.0225i			1.97821i	0.147213	+	0.989105i	$0.452970\pi$
$983$			62.0225i			1.97821i	−0.147213	+	0.989105i	$0.547030\pi$
$984$		0			0
$985$	−35.1939	−	19.5292i	−1.12137	−	0.622251i
$986$		0			0
$987$	15.1997	+	12.8251i	0.483812	+	0.408227i
$988$		0			0
$989$		−	15.3216i		−	0.487198i
$990$		0			0
$991$	0.708497			0.0225062			0.0112531	−	0.999937i	$-0.496418\pi$
$991$	0.708497			0.0225062			0.0112531	+	0.999937i	$0.496418\pi$
$992$		0			0
$993$	3.36689	−	13.4411i	0.106845	−	0.426541i
$994$		0			0
$995$	18.0000	−	32.4382i	0.570638	−	1.02836i
$996$		0			0
$997$	−42.2259			−1.33731			−0.668653	−	0.743574i	$-0.733129\pi$
$997$	−42.2259			−1.33731			−0.668653	+	0.743574i	$0.733129\pi$
$998$		0			0
$999$	−41.6295	−	37.7318i	−1.31710	−	1.19378i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.2.k.e.209.4		8
3.2	odd	2		1680.2.k.f.209.3		8
4.3	odd	2		210.2.d.a.209.5	yes	8
5.4	even	2		1680.2.k.f.209.5		8
7.6	odd	2	inner	1680.2.k.e.209.5		8
12.11	even	2		210.2.d.b.209.6	yes	8
15.14	odd	2	inner	1680.2.k.e.209.6		8
20.3	even	4		1050.2.b.f.251.16		16
20.7	even	4		1050.2.b.f.251.1		16
20.19	odd	2		210.2.d.b.209.4	yes	8
21.20	even	2		1680.2.k.f.209.6		8
28.27	even	2		210.2.d.a.209.4	yes	8
35.34	odd	2		1680.2.k.f.209.4		8
60.23	odd	4		1050.2.b.f.251.2		16
60.47	odd	4		1050.2.b.f.251.15		16
60.59	even	2		210.2.d.a.209.3	✓	8
84.83	odd	2		210.2.d.b.209.3	yes	8
105.104	even	2	inner	1680.2.k.e.209.3		8
140.27	odd	4		1050.2.b.f.251.8		16
140.83	odd	4		1050.2.b.f.251.9		16
140.139	even	2		210.2.d.b.209.5	yes	8
420.83	even	4		1050.2.b.f.251.7		16
420.167	even	4		1050.2.b.f.251.10		16
420.419	odd	2		210.2.d.a.209.6	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.2.d.a.209.3	✓	8	60.59	even	2
210.2.d.a.209.4	yes	8	28.27	even	2
210.2.d.a.209.5	yes	8	4.3	odd	2
210.2.d.a.209.6	yes	8	420.419	odd	2
210.2.d.b.209.3	yes	8	84.83	odd	2
210.2.d.b.209.4	yes	8	20.19	odd	2
210.2.d.b.209.5	yes	8	140.139	even	2
210.2.d.b.209.6	yes	8	12.11	even	2
1050.2.b.f.251.1		16	20.7	even	4
1050.2.b.f.251.2		16	60.23	odd	4
1050.2.b.f.251.7		16	420.83	even	4
1050.2.b.f.251.8		16	140.27	odd	4
1050.2.b.f.251.9		16	140.83	odd	4
1050.2.b.f.251.10		16	420.167	even	4
1050.2.b.f.251.15		16	60.47	odd	4
1050.2.b.f.251.16		16	20.3	even	4
1680.2.k.e.209.3		8	105.104	even	2	inner
1680.2.k.e.209.4		8	1.1	even	1	trivial
1680.2.k.e.209.5		8	7.6	odd	2	inner
1680.2.k.e.209.6		8	15.14	odd	2	inner
1680.2.k.f.209.3		8	3.2	odd	2
1680.2.k.f.209.4		8	35.34	odd	2
1680.2.k.f.209.5		8	5.4	even	2
1680.2.k.f.209.6		8	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1680.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.420861 + 1.68014i) q^{3} +(-1.95522 - 1.08495i) q^{5} +(2.37608 + 1.16372i) q^{7} +(-2.64575 - 1.41421i) q^{9} -2.82843i q^{11} -0.841723 q^{13} +(2.64575 - 2.82843i) q^{15} -1.19038i q^{17} +4.55066i q^{19} +(-2.95522 + 3.50238i) q^{21} +3.29150 q^{23} +(2.64575 + 4.24264i) q^{25} +(3.48957 - 3.85005i) q^{27} -7.98430i q^{29} +5.53019i q^{31} +(4.75216 + 1.19038i) q^{33} +(-3.38317 - 4.85326i) q^{35} -10.8127i q^{37} +(0.354249 - 1.41421i) q^{39} +7.82087 q^{41} -4.65489i q^{43} +(3.63866 + 5.63561i) q^{45} -4.33981i q^{47} +(4.29150 + 5.53019i) q^{49} +(2.00000 + 0.500983i) q^{51} +12.5830 q^{53} +(-3.06871 + 5.53019i) q^{55} +(-7.64575 - 1.91520i) q^{57} +3.91044 q^{59} +10.0808i q^{61} +(-4.64076 - 6.43920i) q^{63} +(1.64575 + 0.913230i) q^{65} +4.65489i q^{67} +(-1.38527 + 5.53019i) q^{69} +12.6392i q^{71} +3.06871 q^{73} +(-8.24173 + 2.65967i) q^{75} +(3.29150 - 6.72057i) q^{77} +7.29150 q^{79} +(5.00000 + 7.48331i) q^{81} +7.70010i q^{83} +(-1.29150 + 2.32744i) q^{85} +(13.4148 + 3.36028i) q^{87} -12.8712 q^{89} +(-2.00000 - 0.979531i) q^{91} +(-9.29150 - 2.32744i) q^{93} +(4.93725 - 8.89753i) q^{95} +8.11905 q^{97} +(-4.00000 + 7.48331i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{21} - 16 q^{23} - 16 q^{35} + 24 q^{39} - 8 q^{49} + 16 q^{51} + 16 q^{53} - 40 q^{57} - 8 q^{63} - 8 q^{65} - 16 q^{77} + 16 q^{79} + 40 q^{81} + 32 q^{85} - 16 q^{91} - 32 q^{93} - 24 q^{95}+ \cdots - 32 q^{99}+O(q^{100}) \) 8 * q - 8 * q^21 - 16 * q^23 - 16 * q^35 + 24 * q^39 - 8 * q^49 + 16 * q^51 + 16 * q^53 - 40 * q^57 - 8 * q^63 - 8 * q^65 - 16 * q^77 + 16 * q^79 + 40 * q^81 + 32 * q^85 - 16 * q^91 - 32 * q^93 - 24 * q^95 - 32 * q^99

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