LMFDB - Embedded newform 1680.2.t.e.1009.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1680,2,Mod(1009,1680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1680, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1680.1009");

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.t (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:	$13.4148675396$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 210)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1009.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1680.1009
Dual form			1680.2.t.e.1009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(1.00000 - 2.00000i) q^{5} +1.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +(1.00000 - 2.00000i) q^{5} +1.00000i q^{7} -1.00000 q^{9} -2.00000 q^{11} -6.00000i q^{13} +(2.00000 + 1.00000i) q^{15} +4.00000i q^{17} -6.00000 q^{19} -1.00000 q^{21} -8.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} -1.00000i q^{27} -6.00000 q^{29} +2.00000 q^{31} -2.00000i q^{33} +(2.00000 + 1.00000i) q^{35} +4.00000i q^{37} +6.00000 q^{39} +2.00000 q^{41} +4.00000i q^{43} +(-1.00000 + 2.00000i) q^{45} -8.00000i q^{47} -1.00000 q^{49} -4.00000 q^{51} -6.00000i q^{53} +(-2.00000 + 4.00000i) q^{55} -6.00000i q^{57} -8.00000 q^{59} -10.0000 q^{61} -1.00000i q^{63} +(-12.0000 - 6.00000i) q^{65} -8.00000i q^{67} +8.00000 q^{69} +6.00000 q^{71} +14.0000i q^{73} +(4.00000 - 3.00000i) q^{75} -2.00000i q^{77} -12.0000 q^{79} +1.00000 q^{81} -8.00000i q^{83} +(8.00000 + 4.00000i) q^{85} -6.00000i q^{87} +10.0000 q^{89} +6.00000 q^{91} +2.00000i q^{93} +(-6.00000 + 12.0000i) q^{95} -10.0000i q^{97} +2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^5 - 2 * q^9	$ 2 q + 2 q^{5} - 2 q^{9} - 4 q^{11} + 4 q^{15} - 12 q^{19} - 2 q^{21} - 6 q^{25} - 12 q^{29} + 4 q^{31} + 4 q^{35} + 12 q^{39} + 4 q^{41} - 2 q^{45} - 2 q^{49} - 8 q^{51} - 4 q^{55} - 16 q^{59} - 20 q^{61} - 24 q^{65} + 16 q^{69} + 12 q^{71} + 8 q^{75} - 24 q^{79} + 2 q^{81} + 16 q^{85} + 20 q^{89} + 12 q^{91} - 12 q^{95} + 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^5 - 2 * q^9 - 4 * q^11 + 4 * q^15 - 12 * q^19 - 2 * q^21 - 6 * q^25 - 12 * q^29 + 4 * q^31 + 4 * q^35 + 12 * q^39 + 4 * q^41 - 2 * q^45 - 2 * q^49 - 8 * q^51 - 4 * q^55 - 16 * q^59 - 20 * q^61 - 24 * q^65 + 16 * q^69 + 12 * q^71 + 8 * q^75 - 24 * q^79 + 2 * q^81 + 16 * q^85 + 20 * q^89 + 12 * q^91 - 12 * q^95 + 4 * q^99

Display 100 coefficients 10 coefficients

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
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$	1.00000	−	2.00000i	0.447214	−	0.894427i
$5$	1.00000	−	2.00000i	0.447214	−	0.894427i
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−2.00000			−0.603023			−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000			−0.603023			−0.301511	+	0.953463i	$0.597491\pi$
$12$		0			0
$13$		−	6.00000i		−	1.66410i	−0.554700	−	0.832050i	$-0.687167\pi$
$13$		−	6.00000i		−	1.66410i	0.554700	−	0.832050i	$-0.312833\pi$
$14$		0			0
$15$	2.00000	+	1.00000i	0.516398	+	0.258199i
$16$		0			0
$17$			4.00000i			0.970143i	0.874475	+	0.485071i	$0.161206\pi$
$17$			4.00000i			0.970143i	−0.874475	+	0.485071i	$0.838794\pi$
$18$		0			0
$19$	−6.00000			−1.37649			−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000			−1.37649			−0.688247	+	0.725476i	$0.741620\pi$
$20$		0			0
$21$	−1.00000			−0.218218
$22$		0			0
$23$		−	8.00000i		−	1.66812i	−0.551677	−	0.834058i	$-0.686012\pi$
$23$		−	8.00000i		−	1.66812i	0.551677	−	0.834058i	$-0.313988\pi$
$24$		0			0
$25$	−3.00000	−	4.00000i	−0.600000	−	0.800000i
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	−6.00000			−1.11417			−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000			−1.11417			−0.557086	+	0.830455i	$0.688081\pi$
$30$		0			0
$31$	2.00000			0.359211			0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000			0.359211			0.179605	+	0.983739i	$0.442518\pi$
$32$		0			0
$33$		−	2.00000i		−	0.348155i
$34$		0			0
$35$	2.00000	+	1.00000i	0.338062	+	0.169031i
$36$		0			0
$37$			4.00000i			0.657596i	0.944400	+	0.328798i	$0.106644\pi$
$37$			4.00000i			0.657596i	−0.944400	+	0.328798i	$0.893356\pi$
$38$		0			0
$39$	6.00000			0.960769
$40$		0			0
$41$	2.00000			0.312348			0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000			0.312348			0.156174	+	0.987730i	$0.450084\pi$
$42$		0			0
$43$			4.00000i			0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$			4.00000i			0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$		0			0
$45$	−1.00000	+	2.00000i	−0.149071	+	0.298142i
$46$		0			0
$47$		−	8.00000i		−	1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$		−	8.00000i		−	1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−4.00000			−0.560112
$52$		0			0
$53$		−	6.00000i		−	0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$		−	6.00000i		−	0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$		0			0
$55$	−2.00000	+	4.00000i	−0.269680	+	0.539360i
$56$		0			0
$57$		−	6.00000i		−	0.794719i
$58$		0			0
$59$	−8.00000			−1.04151			−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000			−1.04151			−0.520756	+	0.853706i	$0.674350\pi$
$60$		0			0
$61$	−10.0000			−1.28037			−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000			−1.28037			−0.640184	+	0.768221i	$0.721142\pi$
$62$		0			0
$63$		−	1.00000i		−	0.125988i
$64$		0			0
$65$	−12.0000	−	6.00000i	−1.48842	−	0.744208i
$66$		0			0
$67$		−	8.00000i		−	0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$		−	8.00000i		−	0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$		0			0
$69$	8.00000			0.963087
$70$		0			0
$71$	6.00000			0.712069			0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000			0.712069			0.356034	+	0.934473i	$0.384129\pi$
$72$		0			0
$73$			14.0000i			1.63858i	0.573382	+	0.819288i	$0.305631\pi$
$73$			14.0000i			1.63858i	−0.573382	+	0.819288i	$0.694369\pi$
$74$		0			0
$75$	4.00000	−	3.00000i	0.461880	−	0.346410i
$76$		0			0
$77$		−	2.00000i		−	0.227921i
$78$		0			0
$79$	−12.0000			−1.35011			−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000			−1.35011			−0.675053	+	0.737769i	$0.735879\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	8.00000i		−	0.878114i	−0.898459	−	0.439057i	$-0.855313\pi$
$83$		−	8.00000i		−	0.878114i	0.898459	−	0.439057i	$-0.144687\pi$
$84$		0			0
$85$	8.00000	+	4.00000i	0.867722	+	0.433861i
$86$		0			0
$87$		−	6.00000i		−	0.643268i
$88$		0			0
$89$	10.0000			1.06000			0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000			1.06000			0.529999	+	0.847998i	$0.322192\pi$
$90$		0			0
$91$	6.00000			0.628971
$92$		0			0
$93$			2.00000i			0.207390i
$94$		0			0
$95$	−6.00000	+	12.0000i	−0.615587	+	1.23117i
$96$		0			0
$97$		−	10.0000i		−	1.01535i	−0.861550	−	0.507673i	$-0.830506\pi$
$97$		−	10.0000i		−	1.01535i	0.861550	−	0.507673i	$-0.169494\pi$
$98$		0			0
$99$	2.00000			0.201008
$100$		0			0
$101$	10.0000			0.995037			0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000			0.995037			0.497519	+	0.867453i	$0.334245\pi$
$102$		0			0
$103$			8.00000i			0.788263i	0.919054	+	0.394132i	$0.128955\pi$
$103$			8.00000i			0.788263i	−0.919054	+	0.394132i	$0.871045\pi$
$104$		0			0
$105$	−1.00000	+	2.00000i	−0.0975900	+	0.195180i
$106$		0			0
$107$		−	12.0000i		−	1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$		−	12.0000i		−	1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$		0			0
$109$	14.0000			1.34096			0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000			1.34096			0.670478	+	0.741929i	$0.266089\pi$
$110$		0			0
$111$	−4.00000			−0.379663
$112$		0			0
$113$		−	6.00000i		−	0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$		−	6.00000i		−	0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$		0			0
$115$	−16.0000	−	8.00000i	−1.49201	−	0.746004i
$116$		0			0
$117$			6.00000i			0.554700i
$118$		0			0
$119$	−4.00000			−0.366679
$120$		0			0
$121$	−7.00000			−0.636364
$122$		0			0
$123$			2.00000i			0.180334i
$124$		0			0
$125$	−11.0000	+	2.00000i	−0.983870	+	0.178885i
$126$		0			0
$127$		−	4.00000i		−	0.354943i	−0.984126	−	0.177471i	$-0.943208\pi$
$127$		−	4.00000i		−	0.354943i	0.984126	−	0.177471i	$-0.0567917\pi$
$128$		0			0
$129$	−4.00000			−0.352180
$130$		0			0
$131$	12.0000			1.04844			0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000			1.04844			0.524222	+	0.851581i	$0.324356\pi$
$132$		0			0
$133$		−	6.00000i		−	0.520266i
$134$		0			0
$135$	−2.00000	−	1.00000i	−0.172133	−	0.0860663i
$136$		0			0
$137$		−	6.00000i		−	0.512615i	−0.966595	−	0.256307i	$-0.917494\pi$
$137$		−	6.00000i		−	0.512615i	0.966595	−	0.256307i	$-0.0825059\pi$
$138$		0			0
$139$	−14.0000			−1.18746			−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000			−1.18746			−0.593732	+	0.804663i	$0.702346\pi$
$140$		0			0
$141$	8.00000			0.673722
$142$		0			0
$143$			12.0000i			1.00349i
$144$		0			0
$145$	−6.00000	+	12.0000i	−0.498273	+	0.996546i
$146$		0			0
$147$		−	1.00000i		−	0.0824786i
$148$		0			0
$149$	−10.0000			−0.819232			−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000			−0.819232			−0.409616	+	0.912258i	$0.634337\pi$
$150$		0			0
$151$	8.00000			0.651031			0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000			0.651031			0.325515	+	0.945537i	$0.394462\pi$
$152$		0			0
$153$		−	4.00000i		−	0.323381i
$154$		0			0
$155$	2.00000	−	4.00000i	0.160644	−	0.321288i
$156$		0			0
$157$		−	22.0000i		−	1.75579i	−0.478852	−	0.877896i	$-0.658947\pi$
$157$		−	22.0000i		−	1.75579i	0.478852	−	0.877896i	$-0.341053\pi$
$158$		0			0
$159$	6.00000			0.475831
$160$		0			0
$161$	8.00000			0.630488
$162$		0			0
$163$		−	4.00000i		−	0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$		−	4.00000i		−	0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$		0			0
$165$	−4.00000	−	2.00000i	−0.311400	−	0.155700i
$166$		0			0
$167$			12.0000i			0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$			12.0000i			0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$		0			0
$169$	−23.0000			−1.76923
$170$		0			0
$171$	6.00000			0.458831
$172$		0			0
$173$		−	8.00000i		−	0.608229i	−0.952636	−	0.304114i	$-0.901639\pi$
$173$		−	8.00000i		−	0.608229i	0.952636	−	0.304114i	$-0.0983605\pi$
$174$		0			0
$175$	4.00000	−	3.00000i	0.302372	−	0.226779i
$176$		0			0
$177$		−	8.00000i		−	0.601317i
$178$		0			0
$179$	10.0000			0.747435			0.373718	−	0.927543i	$-0.378083\pi$
$179$	10.0000			0.747435			0.373718	+	0.927543i	$0.378083\pi$
$180$		0			0
$181$	2.00000			0.148659			0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000			0.148659			0.0743294	+	0.997234i	$0.476318\pi$
$182$		0			0
$183$		−	10.0000i		−	0.739221i
$184$		0			0
$185$	8.00000	+	4.00000i	0.588172	+	0.294086i
$186$		0			0
$187$		−	8.00000i		−	0.585018i
$188$		0			0
$189$	1.00000			0.0727393
$190$		0			0
$191$	18.0000			1.30243			0.651217	−	0.758891i	$-0.274259\pi$
$191$	18.0000			1.30243			0.651217	+	0.758891i	$0.274259\pi$
$192$		0			0
$193$			8.00000i			0.575853i	0.957653	+	0.287926i	$0.0929658\pi$
$193$			8.00000i			0.575853i	−0.957653	+	0.287926i	$0.907034\pi$
$194$		0			0
$195$	6.00000	−	12.0000i	0.429669	−	0.859338i
$196$		0			0
$197$			2.00000i			0.142494i	0.997459	+	0.0712470i	$0.0226979\pi$
$197$			2.00000i			0.142494i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$		0			0
$199$	6.00000			0.425329			0.212664	−	0.977125i	$-0.431786\pi$
$199$	6.00000			0.425329			0.212664	+	0.977125i	$0.431786\pi$
$200$		0			0
$201$	8.00000			0.564276
$202$		0			0
$203$		−	6.00000i		−	0.421117i
$204$		0			0
$205$	2.00000	−	4.00000i	0.139686	−	0.279372i
$206$		0			0
$207$			8.00000i			0.556038i
$208$		0			0
$209$	12.0000			0.830057
$210$		0			0
$211$		0			0			−	1.00000i	$-0.5\pi$
$211$		0			0				1.00000i	$0.5\pi$
$212$		0			0
$213$			6.00000i			0.411113i
$214$		0			0
$215$	8.00000	+	4.00000i	0.545595	+	0.272798i
$216$		0			0
$217$			2.00000i			0.135769i
$218$		0			0
$219$	−14.0000			−0.946032
$220$		0			0
$221$	24.0000			1.61441
$222$		0			0
$223$		−	8.00000i		−	0.535720i	−0.963458	−	0.267860i	$-0.913684\pi$
$223$		−	8.00000i		−	0.535720i	0.963458	−	0.267860i	$-0.0863164\pi$
$224$		0			0
$225$	3.00000	+	4.00000i	0.200000	+	0.266667i
$226$		0			0
$227$			8.00000i			0.530979i	0.964114	+	0.265489i	$0.0855335\pi$
$227$			8.00000i			0.530979i	−0.964114	+	0.265489i	$0.914466\pi$
$228$		0			0
$229$	−14.0000			−0.925146			−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000			−0.925146			−0.462573	+	0.886581i	$0.653074\pi$
$230$		0			0
$231$	2.00000			0.131590
$232$		0			0
$233$			10.0000i			0.655122i	0.944830	+	0.327561i	$0.106227\pi$
$233$			10.0000i			0.655122i	−0.944830	+	0.327561i	$0.893773\pi$
$234$		0			0
$235$	−16.0000	−	8.00000i	−1.04372	−	0.521862i
$236$		0			0
$237$		−	12.0000i		−	0.779484i
$238$		0			0
$239$	26.0000			1.68180			0.840900	−	0.541190i	$-0.182026\pi$
$239$	26.0000			1.68180			0.840900	+	0.541190i	$0.182026\pi$
$240$		0			0
$241$	−26.0000			−1.67481			−0.837404	−	0.546585i	$-0.815928\pi$
$241$	−26.0000			−1.67481			−0.837404	+	0.546585i	$0.815928\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	−1.00000	+	2.00000i	−0.0638877	+	0.127775i
$246$		0			0
$247$			36.0000i			2.29063i
$248$		0			0
$249$	8.00000			0.506979
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$			16.0000i			1.00591i
$254$		0			0
$255$	−4.00000	+	8.00000i	−0.250490	+	0.500979i
$256$		0			0
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$	−4.00000			−0.248548
$260$		0			0
$261$	6.00000			0.371391
$262$		0			0
$263$			16.0000i			0.986602i	0.869859	+	0.493301i	$0.164210\pi$
$263$			16.0000i			0.986602i	−0.869859	+	0.493301i	$0.835790\pi$
$264$		0			0
$265$	−12.0000	−	6.00000i	−0.737154	−	0.368577i
$266$		0			0
$267$			10.0000i			0.611990i
$268$		0			0
$269$	−18.0000			−1.09748			−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000			−1.09748			−0.548740	+	0.835993i	$0.684892\pi$
$270$		0			0
$271$	−10.0000			−0.607457			−0.303728	−	0.952759i	$-0.598232\pi$
$271$	−10.0000			−0.607457			−0.303728	+	0.952759i	$0.598232\pi$
$272$		0			0
$273$			6.00000i			0.363137i
$274$		0			0
$275$	6.00000	+	8.00000i	0.361814	+	0.482418i
$276$		0			0
$277$			28.0000i			1.68236i	0.540758	+	0.841178i	$0.318138\pi$
$277$			28.0000i			1.68236i	−0.540758	+	0.841178i	$0.681862\pi$
$278$		0			0
$279$	−2.00000			−0.119737
$280$		0			0
$281$	30.0000			1.78965			0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000			1.78965			0.894825	+	0.446417i	$0.147300\pi$
$282$		0			0
$283$			20.0000i			1.18888i	0.804141	+	0.594438i	$0.202626\pi$
$283$			20.0000i			1.18888i	−0.804141	+	0.594438i	$0.797374\pi$
$284$		0			0
$285$	−12.0000	−	6.00000i	−0.710819	−	0.355409i
$286$		0			0
$287$			2.00000i			0.118056i
$288$		0			0
$289$	1.00000			0.0588235
$290$		0			0
$291$	10.0000			0.586210
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$	−8.00000	+	16.0000i	−0.465778	+	0.931556i
$296$		0			0
$297$			2.00000i			0.116052i
$298$		0			0
$299$	−48.0000			−2.77591
$300$		0			0
$301$	−4.00000			−0.230556
$302$		0			0
$303$			10.0000i			0.574485i
$304$		0			0
$305$	−10.0000	+	20.0000i	−0.572598	+	1.14520i
$306$		0			0
$307$			12.0000i			0.684876i	0.939540	+	0.342438i	$0.111253\pi$
$307$			12.0000i			0.684876i	−0.939540	+	0.342438i	$0.888747\pi$
$308$		0			0
$309$	−8.00000			−0.455104
$310$		0			0
$311$	−24.0000			−1.36092			−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000			−1.36092			−0.680458	+	0.732787i	$0.738219\pi$
$312$		0			0
$313$			2.00000i			0.113047i	0.998401	+	0.0565233i	$0.0180015\pi$
$313$			2.00000i			0.113047i	−0.998401	+	0.0565233i	$0.981998\pi$
$314$		0			0
$315$	−2.00000	−	1.00000i	−0.112687	−	0.0563436i
$316$		0			0
$317$		−	2.00000i		−	0.112331i	−0.998421	−	0.0561656i	$-0.982113\pi$
$317$		−	2.00000i		−	0.112331i	0.998421	−	0.0561656i	$-0.0178875\pi$
$318$		0			0
$319$	12.0000			0.671871
$320$		0			0
$321$	12.0000			0.669775
$322$		0			0
$323$		−	24.0000i		−	1.33540i
$324$		0			0
$325$	−24.0000	+	18.0000i	−1.33128	+	0.998460i
$326$		0			0
$327$			14.0000i			0.774202i
$328$		0			0
$329$	8.00000			0.441054
$330$		0			0
$331$	8.00000			0.439720			0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000			0.439720			0.219860	+	0.975531i	$0.429440\pi$
$332$		0			0
$333$		−	4.00000i		−	0.219199i
$334$		0			0
$335$	−16.0000	−	8.00000i	−0.874173	−	0.437087i
$336$		0			0
$337$		−	8.00000i		−	0.435788i	−0.975972	−	0.217894i	$-0.930081\pi$
$337$		−	8.00000i		−	0.435788i	0.975972	−	0.217894i	$-0.0699187\pi$
$338$		0			0
$339$	6.00000			0.325875
$340$		0			0
$341$	−4.00000			−0.216612
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$	8.00000	−	16.0000i	0.430706	−	0.861411i
$346$		0			0
$347$		−	36.0000i		−	1.93258i	−0.257454	−	0.966291i	$-0.582883\pi$
$347$		−	36.0000i		−	1.93258i	0.257454	−	0.966291i	$-0.417117\pi$
$348$		0			0
$349$	26.0000			1.39175			0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000			1.39175			0.695874	+	0.718164i	$0.255017\pi$
$350$		0			0
$351$	−6.00000			−0.320256
$352$		0			0
$353$		−	20.0000i		−	1.06449i	−0.846590	−	0.532246i	$-0.821348\pi$
$353$		−	20.0000i		−	1.06449i	0.846590	−	0.532246i	$-0.178652\pi$
$354$		0			0
$355$	6.00000	−	12.0000i	0.318447	−	0.636894i
$356$		0			0
$357$		−	4.00000i		−	0.211702i
$358$		0			0
$359$	6.00000			0.316668			0.158334	−	0.987386i	$-0.449388\pi$
$359$	6.00000			0.316668			0.158334	+	0.987386i	$0.449388\pi$
$360$		0			0
$361$	17.0000			0.894737
$362$		0			0
$363$		−	7.00000i		−	0.367405i
$364$		0			0
$365$	28.0000	+	14.0000i	1.46559	+	0.732793i
$366$		0			0
$367$			32.0000i			1.67039i	0.549957	+	0.835193i	$0.314644\pi$
$367$			32.0000i			1.67039i	−0.549957	+	0.835193i	$0.685356\pi$
$368$		0			0
$369$	−2.00000			−0.104116
$370$		0			0
$371$	6.00000			0.311504
$372$		0			0
$373$			24.0000i			1.24267i	0.783544	+	0.621336i	$0.213410\pi$
$373$			24.0000i			1.24267i	−0.783544	+	0.621336i	$0.786590\pi$
$374$		0			0
$375$	−2.00000	−	11.0000i	−0.103280	−	0.568038i
$376$		0			0
$377$			36.0000i			1.85409i
$378$		0			0
$379$	20.0000			1.02733			0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000			1.02733			0.513665	+	0.857991i	$0.328287\pi$
$380$		0			0
$381$	4.00000			0.204926
$382$		0			0
$383$			20.0000i			1.02195i	0.859595	+	0.510976i	$0.170716\pi$
$383$			20.0000i			1.02195i	−0.859595	+	0.510976i	$0.829284\pi$
$384$		0			0
$385$	−4.00000	−	2.00000i	−0.203859	−	0.101929i
$386$		0			0
$387$		−	4.00000i		−	0.203331i
$388$		0			0
$389$	−2.00000			−0.101404			−0.0507020	−	0.998714i	$-0.516146\pi$
$389$	−2.00000			−0.101404			−0.0507020	+	0.998714i	$0.516146\pi$
$390$		0			0
$391$	32.0000			1.61831
$392$		0			0
$393$			12.0000i			0.605320i
$394$		0			0
$395$	−12.0000	+	24.0000i	−0.603786	+	1.20757i
$396$		0			0
$397$		−	2.00000i		−	0.100377i	−0.998740	−	0.0501886i	$-0.984018\pi$
$397$		−	2.00000i		−	0.100377i	0.998740	−	0.0501886i	$-0.0159822\pi$
$398$		0			0
$399$	6.00000			0.300376
$400$		0			0
$401$	−14.0000			−0.699127			−0.349563	−	0.936913i	$-0.613670\pi$
$401$	−14.0000			−0.699127			−0.349563	+	0.936913i	$0.613670\pi$
$402$		0			0
$403$		−	12.0000i		−	0.597763i
$404$		0			0
$405$	1.00000	−	2.00000i	0.0496904	−	0.0993808i
$406$		0			0
$407$		−	8.00000i		−	0.396545i
$408$		0			0
$409$	−26.0000			−1.28562			−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000			−1.28562			−0.642809	+	0.766027i	$0.722231\pi$
$410$		0			0
$411$	6.00000			0.295958
$412$		0			0
$413$		−	8.00000i		−	0.393654i
$414$		0			0
$415$	−16.0000	−	8.00000i	−0.785409	−	0.392705i
$416$		0			0
$417$		−	14.0000i		−	0.685583i
$418$		0			0
$419$	20.0000			0.977064			0.488532	−	0.872546i	$-0.337533\pi$
$419$	20.0000			0.977064			0.488532	+	0.872546i	$0.337533\pi$
$420$		0			0
$421$	34.0000			1.65706			0.828529	−	0.559946i	$-0.189178\pi$
$421$	34.0000			1.65706			0.828529	+	0.559946i	$0.189178\pi$
$422$		0			0
$423$			8.00000i			0.388973i
$424$		0			0
$425$	16.0000	−	12.0000i	0.776114	−	0.582086i
$426$		0			0
$427$		−	10.0000i		−	0.483934i
$428$		0			0
$429$	−12.0000			−0.579365
$430$		0			0
$431$	−2.00000			−0.0963366			−0.0481683	−	0.998839i	$-0.515338\pi$
$431$	−2.00000			−0.0963366			−0.0481683	+	0.998839i	$0.515338\pi$
$432$		0			0
$433$			26.0000i			1.24948i	0.780833	+	0.624740i	$0.214795\pi$
$433$			26.0000i			1.24948i	−0.780833	+	0.624740i	$0.785205\pi$
$434$		0			0
$435$	−12.0000	−	6.00000i	−0.575356	−	0.287678i
$436$		0			0
$437$			48.0000i			2.29615i
$438$		0			0
$439$	−18.0000			−0.859093			−0.429547	−	0.903045i	$-0.641327\pi$
$439$	−18.0000			−0.859093			−0.429547	+	0.903045i	$0.641327\pi$
$440$		0			0
$441$	1.00000			0.0476190
$442$		0			0
$443$		−	12.0000i		−	0.570137i	−0.958507	−	0.285069i	$-0.907984\pi$
$443$		−	12.0000i		−	0.570137i	0.958507	−	0.285069i	$-0.0920164\pi$
$444$		0			0
$445$	10.0000	−	20.0000i	0.474045	−	0.948091i
$446$		0			0
$447$		−	10.0000i		−	0.472984i
$448$		0			0
$449$	6.00000			0.283158			0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000			0.283158			0.141579	+	0.989927i	$0.454782\pi$
$450$		0			0
$451$	−4.00000			−0.188353
$452$		0			0
$453$			8.00000i			0.375873i
$454$		0			0
$455$	6.00000	−	12.0000i	0.281284	−	0.562569i
$456$		0			0
$457$			28.0000i			1.30978i	0.755722	+	0.654892i	$0.227286\pi$
$457$			28.0000i			1.30978i	−0.755722	+	0.654892i	$0.772714\pi$
$458$		0			0
$459$	4.00000			0.186704
$460$		0			0
$461$	−18.0000			−0.838344			−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000			−0.838344			−0.419172	+	0.907907i	$0.637680\pi$
$462$		0			0
$463$			36.0000i			1.67306i	0.547920	+	0.836531i	$0.315420\pi$
$463$			36.0000i			1.67306i	−0.547920	+	0.836531i	$0.684580\pi$
$464$		0			0
$465$	4.00000	+	2.00000i	0.185496	+	0.0927478i
$466$		0			0
$467$			24.0000i			1.11059i	0.831654	+	0.555294i	$0.187394\pi$
$467$			24.0000i			1.11059i	−0.831654	+	0.555294i	$0.812606\pi$
$468$		0			0
$469$	8.00000			0.369406
$470$		0			0
$471$	22.0000			1.01371
$472$		0			0
$473$		−	8.00000i		−	0.367840i
$474$		0			0
$475$	18.0000	+	24.0000i	0.825897	+	1.10120i
$476$		0			0
$477$			6.00000i			0.274721i
$478$		0			0
$479$	8.00000			0.365529			0.182765	−	0.983157i	$-0.441495\pi$
$479$	8.00000			0.365529			0.182765	+	0.983157i	$0.441495\pi$
$480$		0			0
$481$	24.0000			1.09431
$482$		0			0
$483$			8.00000i			0.364013i
$484$		0			0
$485$	−20.0000	−	10.0000i	−0.908153	−	0.454077i
$486$		0			0
$487$		−	12.0000i		−	0.543772i	−0.962329	−	0.271886i	$-0.912353\pi$
$487$		−	12.0000i		−	0.543772i	0.962329	−	0.271886i	$-0.0876473\pi$
$488$		0			0
$489$	4.00000			0.180886
$490$		0			0
$491$	30.0000			1.35388			0.676941	−	0.736038i	$-0.263305\pi$
$491$	30.0000			1.35388			0.676941	+	0.736038i	$0.263305\pi$
$492$		0			0
$493$		−	24.0000i		−	1.08091i
$494$		0			0
$495$	2.00000	−	4.00000i	0.0898933	−	0.179787i
$496$		0			0
$497$			6.00000i			0.269137i
$498$		0			0
$499$	−12.0000			−0.537194			−0.268597	−	0.963253i	$-0.586560\pi$
$499$	−12.0000			−0.537194			−0.268597	+	0.963253i	$0.586560\pi$
$500$		0			0
$501$	−12.0000			−0.536120
$502$		0			0
$503$		−	12.0000i		−	0.535054i	−0.963550	−	0.267527i	$-0.913794\pi$
$503$		−	12.0000i		−	0.535054i	0.963550	−	0.267527i	$-0.0862064\pi$
$504$		0			0
$505$	10.0000	−	20.0000i	0.444994	−	0.889988i
$506$		0			0
$507$		−	23.0000i		−	1.02147i
$508$		0			0
$509$	−30.0000			−1.32973			−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000			−1.32973			−0.664863	+	0.746965i	$0.731510\pi$
$510$		0			0
$511$	−14.0000			−0.619324
$512$		0			0
$513$			6.00000i			0.264906i
$514$		0			0
$515$	16.0000	+	8.00000i	0.705044	+	0.352522i
$516$		0			0
$517$			16.0000i			0.703679i
$518$		0			0
$519$	8.00000			0.351161
$520$		0			0
$521$	−26.0000			−1.13908			−0.569540	−	0.821963i	$-0.692879\pi$
$521$	−26.0000			−1.13908			−0.569540	+	0.821963i	$0.692879\pi$
$522$		0			0
$523$		−	28.0000i		−	1.22435i	−0.790721	−	0.612177i	$-0.790294\pi$
$523$		−	28.0000i		−	1.22435i	0.790721	−	0.612177i	$-0.209706\pi$
$524$		0			0
$525$	3.00000	+	4.00000i	0.130931	+	0.174574i
$526$		0			0
$527$			8.00000i			0.348485i
$528$		0			0
$529$	−41.0000			−1.78261
$530$		0			0
$531$	8.00000			0.347170
$532$		0			0
$533$		−	12.0000i		−	0.519778i
$534$		0			0
$535$	−24.0000	−	12.0000i	−1.03761	−	0.518805i
$536$		0			0
$537$			10.0000i			0.431532i
$538$		0			0
$539$	2.00000			0.0861461
$540$		0			0
$541$	22.0000			0.945854			0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000			0.945854			0.472927	+	0.881102i	$0.343197\pi$
$542$		0			0
$543$			2.00000i			0.0858282i
$544$		0			0
$545$	14.0000	−	28.0000i	0.599694	−	1.19939i
$546$		0			0
$547$		−	40.0000i		−	1.71028i	−0.518400	−	0.855138i	$-0.673472\pi$
$547$		−	40.0000i		−	1.71028i	0.518400	−	0.855138i	$-0.326528\pi$
$548$		0			0
$549$	10.0000			0.426790
$550$		0			0
$551$	36.0000			1.53365
$552$		0			0
$553$		−	12.0000i		−	0.510292i
$554$		0			0
$555$	−4.00000	+	8.00000i	−0.169791	+	0.339581i
$556$		0			0
$557$		−	42.0000i		−	1.77960i	−0.456354	−	0.889799i	$-0.650845\pi$
$557$		−	42.0000i		−	1.77960i	0.456354	−	0.889799i	$-0.349155\pi$
$558$		0			0
$559$	24.0000			1.01509
$560$		0			0
$561$	8.00000			0.337760
$562$		0			0
$563$		−	4.00000i		−	0.168580i	−0.996441	−	0.0842900i	$-0.973138\pi$
$563$		−	4.00000i		−	0.168580i	0.996441	−	0.0842900i	$-0.0268622\pi$
$564$		0			0
$565$	−12.0000	−	6.00000i	−0.504844	−	0.252422i
$566$		0			0
$567$			1.00000i			0.0419961i
$568$		0			0
$569$	46.0000			1.92842			0.964210	−	0.265139i	$-0.0854179\pi$
$569$	46.0000			1.92842			0.964210	+	0.265139i	$0.0854179\pi$
$570$		0			0
$571$	−4.00000			−0.167395			−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000			−0.167395			−0.0836974	+	0.996491i	$0.526673\pi$
$572$		0			0
$573$			18.0000i			0.751961i
$574$		0			0
$575$	−32.0000	+	24.0000i	−1.33449	+	1.00087i
$576$		0			0
$577$		−	26.0000i		−	1.08239i	−0.840896	−	0.541197i	$-0.817971\pi$
$577$		−	26.0000i		−	1.08239i	0.840896	−	0.541197i	$-0.182029\pi$
$578$		0			0
$579$	−8.00000			−0.332469
$580$		0			0
$581$	8.00000			0.331896
$582$		0			0
$583$			12.0000i			0.496989i
$584$		0			0
$585$	12.0000	+	6.00000i	0.496139	+	0.248069i
$586$		0			0
$587$			12.0000i			0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$			12.0000i			0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$		0			0
$589$	−12.0000			−0.494451
$590$		0			0
$591$	−2.00000			−0.0822690
$592$		0			0
$593$		−	12.0000i		−	0.492781i	−0.969171	−	0.246390i	$-0.920755\pi$
$593$		−	12.0000i		−	0.492781i	0.969171	−	0.246390i	$-0.0792446\pi$
$594$		0			0
$595$	−4.00000	+	8.00000i	−0.163984	+	0.327968i
$596$		0			0
$597$			6.00000i			0.245564i
$598$		0			0
$599$	30.0000			1.22577			0.612883	−	0.790173i	$-0.290010\pi$
$599$	30.0000			1.22577			0.612883	+	0.790173i	$0.290010\pi$
$600$		0			0
$601$	38.0000			1.55005			0.775026	−	0.631929i	$-0.217737\pi$
$601$	38.0000			1.55005			0.775026	+	0.631929i	$0.217737\pi$
$602$		0			0
$603$			8.00000i			0.325785i
$604$		0			0
$605$	−7.00000	+	14.0000i	−0.284590	+	0.569181i
$606$		0			0
$607$		−	8.00000i		−	0.324710i	−0.986732	−	0.162355i	$-0.948091\pi$
$607$		−	8.00000i		−	0.324710i	0.986732	−	0.162355i	$-0.0519090\pi$
$608$		0			0
$609$	6.00000			0.243132
$610$		0			0
$611$	−48.0000			−1.94187
$612$		0			0
$613$		−	28.0000i		−	1.13091i	−0.824779	−	0.565455i	$-0.808701\pi$
$613$		−	28.0000i		−	1.13091i	0.824779	−	0.565455i	$-0.191299\pi$
$614$		0			0
$615$	4.00000	+	2.00000i	0.161296	+	0.0806478i
$616$		0			0
$617$		−	26.0000i		−	1.04672i	−0.852111	−	0.523360i	$-0.824678\pi$
$617$		−	26.0000i		−	1.04672i	0.852111	−	0.523360i	$-0.175322\pi$
$618$		0			0
$619$	−22.0000			−0.884255			−0.442127	−	0.896952i	$-0.645776\pi$
$619$	−22.0000			−0.884255			−0.442127	+	0.896952i	$0.645776\pi$
$620$		0			0
$621$	−8.00000			−0.321029
$622$		0			0
$623$			10.0000i			0.400642i
$624$		0			0
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$		0			0
$627$			12.0000i			0.479234i
$628$		0			0
$629$	−16.0000			−0.637962
$630$		0			0
$631$	4.00000			0.159237			0.0796187	−	0.996825i	$-0.474630\pi$
$631$	4.00000			0.159237			0.0796187	+	0.996825i	$0.474630\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−8.00000	−	4.00000i	−0.317470	−	0.158735i
$636$		0			0
$637$			6.00000i			0.237729i
$638$		0			0
$639$	−6.00000			−0.237356
$640$		0			0
$641$	14.0000			0.552967			0.276483	−	0.961019i	$-0.410831\pi$
$641$	14.0000			0.552967			0.276483	+	0.961019i	$0.410831\pi$
$642$		0			0
$643$			12.0000i			0.473234i	0.971603	+	0.236617i	$0.0760386\pi$
$643$			12.0000i			0.473234i	−0.971603	+	0.236617i	$0.923961\pi$
$644$		0			0
$645$	−4.00000	+	8.00000i	−0.157500	+	0.315000i
$646$		0			0
$647$		−	36.0000i		−	1.41531i	−0.706560	−	0.707653i	$-0.749754\pi$
$647$		−	36.0000i		−	1.41531i	0.706560	−	0.707653i	$-0.250246\pi$
$648$		0			0
$649$	16.0000			0.628055
$650$		0			0
$651$	−2.00000			−0.0783862
$652$		0			0
$653$		−	34.0000i		−	1.33052i	−0.746611	−	0.665261i	$-0.768320\pi$
$653$		−	34.0000i		−	1.33052i	0.746611	−	0.665261i	$-0.231680\pi$
$654$		0			0
$655$	12.0000	−	24.0000i	0.468879	−	0.937758i
$656$		0			0
$657$		−	14.0000i		−	0.546192i
$658$		0			0
$659$	−26.0000			−1.01282			−0.506408	−	0.862294i	$-0.669027\pi$
$659$	−26.0000			−1.01282			−0.506408	+	0.862294i	$0.669027\pi$
$660$		0			0
$661$	38.0000			1.47803			0.739014	−	0.673690i	$-0.235292\pi$
$661$	38.0000			1.47803			0.739014	+	0.673690i	$0.235292\pi$
$662$		0			0
$663$			24.0000i			0.932083i
$664$		0			0
$665$	−12.0000	−	6.00000i	−0.465340	−	0.232670i
$666$		0			0
$667$			48.0000i			1.85857i
$668$		0			0
$669$	8.00000			0.309298
$670$		0			0
$671$	20.0000			0.772091
$672$		0			0
$673$		−	12.0000i		−	0.462566i	−0.972887	−	0.231283i	$-0.925708\pi$
$673$		−	12.0000i		−	0.462566i	0.972887	−	0.231283i	$-0.0742923\pi$
$674$		0			0
$675$	−4.00000	+	3.00000i	−0.153960	+	0.115470i
$676$		0			0
$677$			40.0000i			1.53732i	0.639655	+	0.768662i	$0.279077\pi$
$677$			40.0000i			1.53732i	−0.639655	+	0.768662i	$0.720923\pi$
$678$		0			0
$679$	10.0000			0.383765
$680$		0			0
$681$	−8.00000			−0.306561
$682$		0			0
$683$			4.00000i			0.153056i	0.997067	+	0.0765279i	$0.0243834\pi$
$683$			4.00000i			0.153056i	−0.997067	+	0.0765279i	$0.975617\pi$
$684$		0			0
$685$	−12.0000	−	6.00000i	−0.458496	−	0.229248i
$686$		0			0
$687$		−	14.0000i		−	0.534133i
$688$		0			0
$689$	−36.0000			−1.37149
$690$		0			0
$691$	2.00000			0.0760836			0.0380418	−	0.999276i	$-0.487888\pi$
$691$	2.00000			0.0760836			0.0380418	+	0.999276i	$0.487888\pi$
$692$		0			0
$693$			2.00000i			0.0759737i
$694$		0			0
$695$	−14.0000	+	28.0000i	−0.531050	+	1.06210i
$696$		0			0
$697$			8.00000i			0.303022i
$698$		0			0
$699$	−10.0000			−0.378235
$700$		0			0
$701$	−34.0000			−1.28416			−0.642081	−	0.766637i	$-0.721929\pi$
$701$	−34.0000			−1.28416			−0.642081	+	0.766637i	$0.721929\pi$
$702$		0			0
$703$		−	24.0000i		−	0.905177i
$704$		0			0
$705$	8.00000	−	16.0000i	0.301297	−	0.602595i
$706$		0			0
$707$			10.0000i			0.376089i
$708$		0			0
$709$	6.00000			0.225335			0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000			0.225335			0.112667	+	0.993633i	$0.464061\pi$
$710$		0			0
$711$	12.0000			0.450035
$712$		0			0
$713$		−	16.0000i		−	0.599205i
$714$		0			0
$715$	24.0000	+	12.0000i	0.897549	+	0.448775i
$716$		0			0
$717$			26.0000i			0.970988i
$718$		0			0
$719$	−4.00000			−0.149175			−0.0745874	−	0.997214i	$-0.523764\pi$
$719$	−4.00000			−0.149175			−0.0745874	+	0.997214i	$0.523764\pi$
$720$		0			0
$721$	−8.00000			−0.297936
$722$		0			0
$723$		−	26.0000i		−	0.966950i
$724$		0			0
$725$	18.0000	+	24.0000i	0.668503	+	0.891338i
$726$		0			0
$727$		−	24.0000i		−	0.890111i	−0.895503	−	0.445055i	$-0.853184\pi$
$727$		−	24.0000i		−	0.890111i	0.895503	−	0.445055i	$-0.146816\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−16.0000			−0.591781
$732$		0			0
$733$		−	14.0000i		−	0.517102i	−0.965998	−	0.258551i	$-0.916755\pi$
$733$		−	14.0000i		−	0.517102i	0.965998	−	0.258551i	$-0.0832450\pi$
$734$		0			0
$735$	−2.00000	−	1.00000i	−0.0737711	−	0.0368856i
$736$		0			0
$737$			16.0000i			0.589368i
$738$		0			0
$739$	−44.0000			−1.61857			−0.809283	−	0.587419i	$-0.800144\pi$
$739$	−44.0000			−1.61857			−0.809283	+	0.587419i	$0.800144\pi$
$740$		0			0
$741$	−36.0000			−1.32249
$742$		0			0
$743$		−	48.0000i		−	1.76095i	−0.474093	−	0.880475i	$-0.657224\pi$
$743$		−	48.0000i		−	1.76095i	0.474093	−	0.880475i	$-0.342776\pi$
$744$		0			0
$745$	−10.0000	+	20.0000i	−0.366372	+	0.732743i
$746$		0			0
$747$			8.00000i			0.292705i
$748$		0			0
$749$	12.0000			0.438470
$750$		0			0
$751$	−20.0000			−0.729810			−0.364905	−	0.931045i	$-0.618899\pi$
$751$	−20.0000			−0.729810			−0.364905	+	0.931045i	$0.618899\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	8.00000	−	16.0000i	0.291150	−	0.582300i
$756$		0			0
$757$		−	32.0000i		−	1.16306i	−0.813525	−	0.581530i	$-0.802454\pi$
$757$		−	32.0000i		−	1.16306i	0.813525	−	0.581530i	$-0.197546\pi$
$758$		0			0
$759$	−16.0000			−0.580763
$760$		0			0
$761$	−6.00000			−0.217500			−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000			−0.217500			−0.108750	+	0.994069i	$0.534685\pi$
$762$		0			0
$763$			14.0000i			0.506834i
$764$		0			0
$765$	−8.00000	−	4.00000i	−0.289241	−	0.144620i
$766$		0			0
$767$			48.0000i			1.73318i
$768$		0			0
$769$	−30.0000			−1.08183			−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000			−1.08183			−0.540914	+	0.841078i	$0.681921\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$	−6.00000	−	8.00000i	−0.215526	−	0.287368i
$776$		0			0
$777$		−	4.00000i		−	0.143499i
$778$		0			0
$779$	−12.0000			−0.429945
$780$		0			0
$781$	−12.0000			−0.429394
$782$		0			0
$783$			6.00000i			0.214423i
$784$		0			0
$785$	−44.0000	−	22.0000i	−1.57043	−	0.785214i
$786$		0			0
$787$		−	28.0000i		−	0.998092i	−0.866575	−	0.499046i	$-0.833684\pi$
$787$		−	28.0000i		−	0.998092i	0.866575	−	0.499046i	$-0.166316\pi$
$788$		0			0
$789$	−16.0000			−0.569615
$790$		0			0
$791$	6.00000			0.213335
$792$		0			0
$793$			60.0000i			2.13066i
$794$		0			0
$795$	6.00000	−	12.0000i	0.212798	−	0.425596i
$796$		0			0
$797$		−	8.00000i		−	0.283375i	−0.989911	−	0.141687i	$-0.954747\pi$
$797$		−	8.00000i		−	0.283375i	0.989911	−	0.141687i	$-0.0452527\pi$
$798$		0			0
$799$	32.0000			1.13208
$800$		0			0
$801$	−10.0000			−0.353333
$802$		0			0
$803$		−	28.0000i		−	0.988099i
$804$		0			0
$805$	8.00000	−	16.0000i	0.281963	−	0.563926i
$806$		0			0
$807$		−	18.0000i		−	0.633630i
$808$		0			0
$809$	−30.0000			−1.05474			−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000			−1.05474			−0.527372	+	0.849635i	$0.676823\pi$
$810$		0			0
$811$	14.0000			0.491606			0.245803	−	0.969320i	$-0.420948\pi$
$811$	14.0000			0.491606			0.245803	+	0.969320i	$0.420948\pi$
$812$		0			0
$813$		−	10.0000i		−	0.350715i
$814$		0			0
$815$	−8.00000	−	4.00000i	−0.280228	−	0.140114i
$816$		0			0
$817$		−	24.0000i		−	0.839654i
$818$		0			0
$819$	−6.00000			−0.209657
$820$		0			0
$821$	22.0000			0.767805			0.383903	−	0.923374i	$-0.374580\pi$
$821$	22.0000			0.767805			0.383903	+	0.923374i	$0.374580\pi$
$822$		0			0
$823$			12.0000i			0.418294i	0.977884	+	0.209147i	$0.0670687\pi$
$823$			12.0000i			0.418294i	−0.977884	+	0.209147i	$0.932931\pi$
$824$		0			0
$825$	−8.00000	+	6.00000i	−0.278524	+	0.208893i
$826$		0			0
$827$			20.0000i			0.695468i	0.937593	+	0.347734i	$0.113049\pi$
$827$			20.0000i			0.695468i	−0.937593	+	0.347734i	$0.886951\pi$
$828$		0			0
$829$	−18.0000			−0.625166			−0.312583	−	0.949890i	$-0.601194\pi$
$829$	−18.0000			−0.625166			−0.312583	+	0.949890i	$0.601194\pi$
$830$		0			0
$831$	−28.0000			−0.971309
$832$		0			0
$833$		−	4.00000i		−	0.138592i
$834$		0			0
$835$	24.0000	+	12.0000i	0.830554	+	0.415277i
$836$		0			0
$837$		−	2.00000i		−	0.0691301i
$838$		0			0
$839$	−20.0000			−0.690477			−0.345238	−	0.938515i	$-0.612202\pi$
$839$	−20.0000			−0.690477			−0.345238	+	0.938515i	$0.612202\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$			30.0000i			1.03325i
$844$		0			0
$845$	−23.0000	+	46.0000i	−0.791224	+	1.58245i
$846$		0			0
$847$		−	7.00000i		−	0.240523i
$848$		0			0
$849$	−20.0000			−0.686398
$850$		0			0
$851$	32.0000			1.09695
$852$		0			0
$853$		−	38.0000i		−	1.30110i	−0.759465	−	0.650548i	$-0.774539\pi$
$853$		−	38.0000i		−	1.30110i	0.759465	−	0.650548i	$-0.225461\pi$
$854$		0			0
$855$	6.00000	−	12.0000i	0.205196	−	0.410391i
$856$		0			0
$857$			24.0000i			0.819824i	0.912125	+	0.409912i	$0.134441\pi$
$857$			24.0000i			0.819824i	−0.912125	+	0.409912i	$0.865559\pi$
$858$		0			0
$859$	−10.0000			−0.341196			−0.170598	−	0.985341i	$-0.554570\pi$
$859$	−10.0000			−0.341196			−0.170598	+	0.985341i	$0.554570\pi$
$860$		0			0
$861$	−2.00000			−0.0681598
$862$		0			0
$863$		−	48.0000i		−	1.63394i	−0.576681	−	0.816970i	$-0.695652\pi$
$863$		−	48.0000i		−	1.63394i	0.576681	−	0.816970i	$-0.304348\pi$
$864$		0			0
$865$	−16.0000	−	8.00000i	−0.544016	−	0.272008i
$866$		0			0
$867$			1.00000i			0.0339618i
$868$		0			0
$869$	24.0000			0.814144
$870$		0			0
$871$	−48.0000			−1.62642
$872$		0			0
$873$			10.0000i			0.338449i
$874$		0			0
$875$	−2.00000	−	11.0000i	−0.0676123	−	0.371868i
$876$		0			0
$877$		−	28.0000i		−	0.945493i	−0.881199	−	0.472746i	$-0.843263\pi$
$877$		−	28.0000i		−	0.945493i	0.881199	−	0.472746i	$-0.156737\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	30.0000			1.01073			0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000			1.01073			0.505363	+	0.862907i	$0.331359\pi$
$882$		0			0
$883$			24.0000i			0.807664i	0.914833	+	0.403832i	$0.132322\pi$
$883$			24.0000i			0.807664i	−0.914833	+	0.403832i	$0.867678\pi$
$884$		0			0
$885$	−16.0000	−	8.00000i	−0.537834	−	0.268917i
$886$		0			0
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$	4.00000			0.134156
$890$		0			0
$891$	−2.00000			−0.0670025
$892$		0			0
$893$			48.0000i			1.60626i
$894$		0			0
$895$	10.0000	−	20.0000i	0.334263	−	0.668526i
$896$		0			0
$897$		−	48.0000i		−	1.60267i
$898$		0			0
$899$	−12.0000			−0.400222
$900$		0			0
$901$	24.0000			0.799556
$902$		0			0
$903$		−	4.00000i		−	0.133112i
$904$		0			0
$905$	2.00000	−	4.00000i	0.0664822	−	0.132964i
$906$		0			0
$907$		−	24.0000i		−	0.796907i	−0.917189	−	0.398453i	$-0.869547\pi$
$907$		−	24.0000i		−	0.796907i	0.917189	−	0.398453i	$-0.130453\pi$
$908$		0			0
$909$	−10.0000			−0.331679
$910$		0			0
$911$	−6.00000			−0.198789			−0.0993944	−	0.995048i	$-0.531691\pi$
$911$	−6.00000			−0.198789			−0.0993944	+	0.995048i	$0.531691\pi$
$912$		0			0
$913$			16.0000i			0.529523i
$914$		0			0
$915$	−20.0000	−	10.0000i	−0.661180	−	0.330590i
$916$		0			0
$917$			12.0000i			0.396275i
$918$		0			0
$919$	60.0000			1.97922			0.989609	−	0.143787i	$-0.0459280\pi$
$919$	60.0000			1.97922			0.989609	+	0.143787i	$0.0459280\pi$
$920$		0			0
$921$	−12.0000			−0.395413
$922$		0			0
$923$		−	36.0000i		−	1.18495i
$924$		0			0
$925$	16.0000	−	12.0000i	0.526077	−	0.394558i
$926$		0			0
$927$		−	8.00000i		−	0.262754i
$928$		0			0
$929$	30.0000			0.984268			0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000			0.984268			0.492134	+	0.870519i	$0.336217\pi$
$930$		0			0
$931$	6.00000			0.196642
$932$		0			0
$933$		−	24.0000i		−	0.785725i
$934$		0			0
$935$	−16.0000	−	8.00000i	−0.523256	−	0.261628i
$936$		0			0
$937$			26.0000i			0.849383i	0.905338	+	0.424691i	$0.139617\pi$
$937$			26.0000i			0.849383i	−0.905338	+	0.424691i	$0.860383\pi$
$938$		0			0
$939$	−2.00000			−0.0652675
$940$		0			0
$941$	−10.0000			−0.325991			−0.162995	−	0.986627i	$-0.552116\pi$
$941$	−10.0000			−0.325991			−0.162995	+	0.986627i	$0.552116\pi$
$942$		0			0
$943$		−	16.0000i		−	0.521032i
$944$		0			0
$945$	1.00000	−	2.00000i	0.0325300	−	0.0650600i
$946$		0			0
$947$			12.0000i			0.389948i	0.980808	+	0.194974i	$0.0624622\pi$
$947$			12.0000i			0.389948i	−0.980808	+	0.194974i	$0.937538\pi$
$948$		0			0
$949$	84.0000			2.72676
$950$		0			0
$951$	2.00000			0.0648544
$952$		0			0
$953$			10.0000i			0.323932i	0.986796	+	0.161966i	$0.0517835\pi$
$953$			10.0000i			0.323932i	−0.986796	+	0.161966i	$0.948217\pi$
$954$		0			0
$955$	18.0000	−	36.0000i	0.582466	−	1.16493i
$956$		0			0
$957$			12.0000i			0.387905i
$958$		0			0
$959$	6.00000			0.193750
$960$		0			0
$961$	−27.0000			−0.870968
$962$		0			0
$963$			12.0000i			0.386695i
$964$		0			0
$965$	16.0000	+	8.00000i	0.515058	+	0.257529i
$966$		0			0
$967$		−	56.0000i		−	1.80084i	−0.435023	−	0.900419i	$-0.643260\pi$
$967$		−	56.0000i		−	1.80084i	0.435023	−	0.900419i	$-0.356740\pi$
$968$		0			0
$969$	24.0000			0.770991
$970$		0			0
$971$	24.0000			0.770197			0.385098	−	0.922876i	$-0.374168\pi$
$971$	24.0000			0.770197			0.385098	+	0.922876i	$0.374168\pi$
$972$		0			0
$973$		−	14.0000i		−	0.448819i
$974$		0			0
$975$	−18.0000	−	24.0000i	−0.576461	−	0.768615i
$976$		0			0
$977$		−	18.0000i		−	0.575871i	−0.957650	−	0.287936i	$-0.907031\pi$
$977$		−	18.0000i		−	0.575871i	0.957650	−	0.287936i	$-0.0929689\pi$
$978$		0			0
$979$	−20.0000			−0.639203
$980$		0			0
$981$	−14.0000			−0.446986
$982$		0			0
$983$		−	16.0000i		−	0.510321i	−0.966899	−	0.255160i	$-0.917872\pi$
$983$		−	16.0000i		−	0.510321i	0.966899	−	0.255160i	$-0.0821283\pi$
$984$		0			0
$985$	4.00000	+	2.00000i	0.127451	+	0.0637253i
$986$		0			0
$987$			8.00000i			0.254643i
$988$		0			0
$989$	32.0000			1.01754
$990$		0			0
$991$	−4.00000			−0.127064			−0.0635321	−	0.997980i	$-0.520237\pi$
$991$	−4.00000			−0.127064			−0.0635321	+	0.997980i	$0.520237\pi$
$992$		0			0
$993$			8.00000i			0.253872i
$994$		0			0
$995$	6.00000	−	12.0000i	0.190213	−	0.380426i
$996$		0			0
$997$		−	42.0000i		−	1.33015i	−0.746775	−	0.665077i	$-0.768399\pi$
$997$		−	42.0000i		−	1.33015i	0.746775	−	0.665077i	$-0.231601\pi$
$998$		0			0
$999$	4.00000			0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.2.t.e.1009.2		2
3.2	odd	2		5040.2.t.h.1009.2		2
4.3	odd	2		210.2.g.b.169.2	yes	2
5.2	odd	4		8400.2.a.bp.1.1		1
5.3	odd	4		8400.2.a.w.1.1		1
5.4	even	2	inner	1680.2.t.e.1009.1		2
12.11	even	2		630.2.g.c.379.1		2
15.14	odd	2		5040.2.t.h.1009.1		2
20.3	even	4		1050.2.a.p.1.1		1
20.7	even	4		1050.2.a.d.1.1		1
20.19	odd	2		210.2.g.b.169.1	✓	2
28.3	even	6		1470.2.n.f.79.1		4
28.11	odd	6		1470.2.n.b.79.1		4
28.19	even	6		1470.2.n.f.949.2		4
28.23	odd	6		1470.2.n.b.949.2		4
28.27	even	2		1470.2.g.b.589.2		2
60.23	odd	4		3150.2.a.d.1.1		1
60.47	odd	4		3150.2.a.bk.1.1		1
60.59	even	2		630.2.g.c.379.2		2
140.19	even	6		1470.2.n.f.949.1		4
140.27	odd	4		7350.2.a.bk.1.1		1
140.39	odd	6		1470.2.n.b.79.2		4
140.59	even	6		1470.2.n.f.79.2		4
140.79	odd	6		1470.2.n.b.949.1		4
140.83	odd	4		7350.2.a.bz.1.1		1
140.139	even	2		1470.2.g.b.589.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.2.g.b.169.1	✓	2	20.19	odd	2
210.2.g.b.169.2	yes	2	4.3	odd	2
630.2.g.c.379.1		2	12.11	even	2
630.2.g.c.379.2		2	60.59	even	2
1050.2.a.d.1.1		1	20.7	even	4
1050.2.a.p.1.1		1	20.3	even	4
1470.2.g.b.589.1		2	140.139	even	2
1470.2.g.b.589.2		2	28.27	even	2
1470.2.n.b.79.1		4	28.11	odd	6
1470.2.n.b.79.2		4	140.39	odd	6
1470.2.n.b.949.1		4	140.79	odd	6
1470.2.n.b.949.2		4	28.23	odd	6
1470.2.n.f.79.1		4	28.3	even	6
1470.2.n.f.79.2		4	140.59	even	6
1470.2.n.f.949.1		4	140.19	even	6
1470.2.n.f.949.2		4	28.19	even	6
1680.2.t.e.1009.1		2	5.4	even	2	inner
1680.2.t.e.1009.2		2	1.1	even	1	trivial
3150.2.a.d.1.1		1	60.23	odd	4
3150.2.a.bk.1.1		1	60.47	odd	4
5040.2.t.h.1009.1		2	15.14	odd	2
5040.2.t.h.1009.2		2	3.2	odd	2
7350.2.a.bk.1.1		1	140.27	odd	4
7350.2.a.bz.1.1		1	140.83	odd	4
8400.2.a.w.1.1		1	5.3	odd	4
8400.2.a.bp.1.1		1	5.2	odd	4

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

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(1.00000 - 2.00000i) q^{5} +1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +(1.00000 - 2.00000i) q^{5} +1.00000i q^{7} -1.00000 q^{9} -2.00000 q^{11} -6.00000i q^{13} +(2.00000 + 1.00000i) q^{15} +4.00000i q^{17} -6.00000 q^{19} -1.00000 q^{21} -8.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} -1.00000i q^{27} -6.00000 q^{29} +2.00000 q^{31} -2.00000i q^{33} +(2.00000 + 1.00000i) q^{35} +4.00000i q^{37} +6.00000 q^{39} +2.00000 q^{41} +4.00000i q^{43} +(-1.00000 + 2.00000i) q^{45} -8.00000i q^{47} -1.00000 q^{49} -4.00000 q^{51} -6.00000i q^{53} +(-2.00000 + 4.00000i) q^{55} -6.00000i q^{57} -8.00000 q^{59} -10.0000 q^{61} -1.00000i q^{63} +(-12.0000 - 6.00000i) q^{65} -8.00000i q^{67} +8.00000 q^{69} +6.00000 q^{71} +14.0000i q^{73} +(4.00000 - 3.00000i) q^{75} -2.00000i q^{77} -12.0000 q^{79} +1.00000 q^{81} -8.00000i q^{83} +(8.00000 + 4.00000i) q^{85} -6.00000i q^{87} +10.0000 q^{89} +6.00000 q^{91} +2.00000i q^{93} +(-6.00000 + 12.0000i) q^{95} -10.0000i q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^5 - 2 * q^9	\( 2 q + 2 q^{5} - 2 q^{9} - 4 q^{11} + 4 q^{15} - 12 q^{19} - 2 q^{21} - 6 q^{25} - 12 q^{29} + 4 q^{31} + 4 q^{35} + 12 q^{39} + 4 q^{41} - 2 q^{45} - 2 q^{49} - 8 q^{51} - 4 q^{55} - 16 q^{59} - 20 q^{61} - 24 q^{65} + 16 q^{69} + 12 q^{71} + 8 q^{75} - 24 q^{79} + 2 q^{81} + 16 q^{85} + 20 q^{89} + 12 q^{91} - 12 q^{95} + 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^5 - 2 * q^9 - 4 * q^11 + 4 * q^15 - 12 * q^19 - 2 * q^21 - 6 * q^25 - 12 * q^29 + 4 * q^31 + 4 * q^35 + 12 * q^39 + 4 * q^41 - 2 * q^45 - 2 * q^49 - 8 * q^51 - 4 * q^55 - 16 * q^59 - 20 * q^61 - 24 * q^65 + 16 * q^69 + 12 * q^71 + 8 * q^75 - 24 * q^79 + 2 * q^81 + 16 * q^85 + 20 * q^89 + 12 * q^91 - 12 * q^95 + 4 * q^99

Introduction

L-functions

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