LMFDB - Embedded newform 1680.2.t.f.1009.1

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 105)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +(1.00000 - 2.00000i) q^{5} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +(1.00000 - 2.00000i) q^{5} -1.00000i q^{7} -1.00000 q^{9} +6.00000 q^{11} -2.00000i q^{13} +(-2.00000 - 1.00000i) q^{15} -4.00000i q^{17} -6.00000 q^{19} -1.00000 q^{21} +(-3.00000 - 4.00000i) q^{25} +1.00000i q^{27} +2.00000 q^{29} +10.0000 q^{31} -6.00000i q^{33} +(-2.00000 - 1.00000i) q^{35} +4.00000i q^{37} -2.00000 q^{39} +2.00000 q^{41} +4.00000i q^{43} +(-1.00000 + 2.00000i) q^{45} -1.00000 q^{49} -4.00000 q^{51} +6.00000i q^{53} +(6.00000 - 12.0000i) q^{55} +6.00000i q^{57} -8.00000 q^{59} -2.00000 q^{61} +1.00000i q^{63} +(-4.00000 - 2.00000i) q^{65} -16.0000i q^{67} -10.0000 q^{71} -6.00000i q^{73} +(-4.00000 + 3.00000i) q^{75} -6.00000i q^{77} +4.00000 q^{79} +1.00000 q^{81} -8.00000i q^{83} +(-8.00000 - 4.00000i) q^{85} -2.00000i q^{87} -6.00000 q^{89} -2.00000 q^{91} -10.0000i q^{93} +(-6.00000 + 12.0000i) q^{95} +2.00000i q^{97} -6.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 12 q^{11} - 4 q^{15} - 12 q^{19} - 2 q^{21} - 6 q^{25} + 4 q^{29} + 20 q^{31} - 4 q^{35} - 4 q^{39} + 4 q^{41} - 2 q^{45} - 2 q^{49} - 8 q^{51} + 12 q^{55} - 16 q^{59} - 4 q^{61} - 8 q^{65} - 20 q^{71} - 8 q^{75} + 8 q^{79} + 2 q^{81} - 16 q^{85} - 12 q^{89} - 4 q^{91} - 12 q^{95} - 12 q^{99}+O(q^{100}) $ 2 * q + 2 * q^5 - 2 * q^9 + 12 * q^11 - 4 * q^15 - 12 * q^19 - 2 * q^21 - 6 * q^25 + 4 * q^29 + 20 * q^31 - 4 * q^35 - 4 * q^39 + 4 * q^41 - 2 * q^45 - 2 * q^49 - 8 * q^51 + 12 * q^55 - 16 * q^59 - 4 * q^61 - 8 * q^65 - 20 * q^71 - 8 * q^75 + 8 * q^79 + 2 * q^81 - 16 * q^85 - 12 * q^89 - 4 * q^91 - 12 * q^95 - 12 * 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
$11$	6.00000			1.80907			0.904534	−	0.426401i	$-0.140219\pi$
$11$	6.00000			1.80907			0.904534	+	0.426401i	$0.140219\pi$
$12$		0			0
$13$		−	2.00000i		−	0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$		−	2.00000i		−	0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$		0			0
$15$	−2.00000	−	1.00000i	−0.516398	−	0.258199i
$16$		0			0
$17$		−	4.00000i		−	0.970143i	−0.874475	−	0.485071i	$-0.838794\pi$
$17$		−	4.00000i		−	0.970143i	0.874475	−	0.485071i	$-0.161206\pi$
$18$		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
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	−3.00000	−	4.00000i	−0.600000	−	0.800000i
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	2.00000			0.371391			0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000			0.371391			0.185695	+	0.982607i	$0.440546\pi$
$30$		0			0
$31$	10.0000			1.79605			0.898027	−	0.439941i	$-0.145001\pi$
$31$	10.0000			1.79605			0.898027	+	0.439941i	$0.145001\pi$
$32$		0			0
$33$		−	6.00000i		−	1.04447i
$34$		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
$39$	−2.00000			−0.320256
$40$		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
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		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	$0.135198\pi$
$53$			6.00000i			0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$		0			0
$55$	6.00000	−	12.0000i	0.809040	−	1.61808i
$56$		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
$61$	−2.00000			−0.256074			−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000			−0.256074			−0.128037	+	0.991769i	$0.540868\pi$
$62$		0			0
$63$			1.00000i			0.125988i
$64$		0			0
$65$	−4.00000	−	2.00000i	−0.496139	−	0.248069i
$66$		0			0
$67$		−	16.0000i		−	1.95471i	−0.211604	−	0.977356i	$-0.567869\pi$
$67$		−	16.0000i		−	1.95471i	0.211604	−	0.977356i	$-0.432131\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−10.0000			−1.18678			−0.593391	−	0.804914i	$-0.702211\pi$
$71$	−10.0000			−1.18678			−0.593391	+	0.804914i	$0.702211\pi$
$72$		0			0
$73$		−	6.00000i		−	0.702247i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$		−	6.00000i		−	0.702247i	0.936329	−	0.351123i	$-0.114200\pi$
$74$		0			0
$75$	−4.00000	+	3.00000i	−0.461880	+	0.346410i
$76$		0			0
$77$		−	6.00000i		−	0.683763i
$78$		0			0
$79$	4.00000			0.450035			0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000			0.450035			0.225018	+	0.974355i	$0.427756\pi$
$80$		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
$87$		−	2.00000i		−	0.214423i
$88$		0			0
$89$	−6.00000			−0.635999			−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000			−0.635999			−0.317999	+	0.948091i	$0.603011\pi$
$90$		0			0
$91$	−2.00000			−0.209657
$92$		0			0
$93$		−	10.0000i		−	1.03695i
$94$		0			0
$95$	−6.00000	+	12.0000i	−0.615587	+	1.23117i
$96$		0			0
$97$			2.00000i			0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$			2.00000i			0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$		0			0
$99$	−6.00000			−0.603023
$100$		0			0
$101$	−6.00000			−0.597022			−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000			−0.597022			−0.298511	+	0.954406i	$0.596490\pi$
$102$		0			0
$103$		−	8.00000i		−	0.788263i	−0.919054	−	0.394132i	$-0.871045\pi$
$103$		−	8.00000i		−	0.788263i	0.919054	−	0.394132i	$-0.128955\pi$
$104$		0			0
$105$	−1.00000	+	2.00000i	−0.0975900	+	0.195180i
$106$		0			0
$107$			4.00000i			0.386695i	0.981130	+	0.193347i	$0.0619344\pi$
$107$			4.00000i			0.386695i	−0.981130	+	0.193347i	$0.938066\pi$
$108$		0			0
$109$	−2.00000			−0.191565			−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000			−0.191565			−0.0957826	+	0.995402i	$0.530535\pi$
$110$		0			0
$111$	4.00000			0.379663
$112$		0			0
$113$			6.00000i			0.564433i	0.959351	+	0.282216i	$0.0910696\pi$
$113$			6.00000i			0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$			2.00000i			0.184900i
$118$		0			0
$119$	−4.00000			−0.366679
$120$		0			0
$121$	25.0000			2.27273
$122$		0			0
$123$		−	2.00000i		−	0.180334i
$124$		0			0
$125$	−11.0000	+	2.00000i	−0.983870	+	0.178885i
$126$		0			0
$127$			20.0000i			1.77471i	0.461084	+	0.887357i	$0.347461\pi$
$127$			20.0000i			1.77471i	−0.461084	+	0.887357i	$0.652539\pi$
$128$		0			0
$129$	4.00000			0.352180
$130$		0			0
$131$	−4.00000			−0.349482			−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000			−0.349482			−0.174741	+	0.984614i	$0.555909\pi$
$132$		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	$0.0825059\pi$
$137$			6.00000i			0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$		0			0
$139$	2.00000			0.169638			0.0848189	−	0.996396i	$-0.472969\pi$
$139$	2.00000			0.169638			0.0848189	+	0.996396i	$0.472969\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	12.0000i		−	1.00349i
$144$		0			0
$145$	2.00000	−	4.00000i	0.166091	−	0.332182i
$146$		0			0
$147$			1.00000i			0.0824786i
$148$		0			0
$149$	14.0000			1.14692			0.573462	−	0.819232i	$-0.305600\pi$
$149$	14.0000			1.14692			0.573462	+	0.819232i	$0.305600\pi$
$150$		0			0
$151$	−8.00000			−0.651031			−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000			−0.651031			−0.325515	+	0.945537i	$0.605538\pi$
$152$		0			0
$153$			4.00000i			0.323381i
$154$		0			0
$155$	10.0000	−	20.0000i	0.803219	−	1.60644i
$156$		0			0
$157$		−	18.0000i		−	1.43656i	−0.695756	−	0.718278i	$-0.744931\pi$
$157$		−	18.0000i		−	1.43656i	0.695756	−	0.718278i	$-0.255069\pi$
$158$		0			0
$159$	6.00000			0.475831
$160$		0			0
$161$		0			0
$162$		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
$165$	−12.0000	−	6.00000i	−0.934199	−	0.467099i
$166$		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
$169$	9.00000			0.692308
$170$		0			0
$171$	6.00000			0.458831
$172$		0			0
$173$		0			0		1.00000			$0$
$173$		0			0		−1.00000			$\pi$
$174$		0			0
$175$	−4.00000	+	3.00000i	−0.302372	+	0.226779i
$176$		0			0
$177$			8.00000i			0.601317i
$178$		0			0
$179$	−14.0000			−1.04641			−0.523205	−	0.852207i	$-0.675264\pi$
$179$	−14.0000			−1.04641			−0.523205	+	0.852207i	$0.675264\pi$
$180$		0			0
$181$	−6.00000			−0.445976			−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000			−0.445976			−0.222988	+	0.974821i	$0.571581\pi$
$182$		0			0
$183$			2.00000i			0.147844i
$184$		0			0
$185$	8.00000	+	4.00000i	0.588172	+	0.294086i
$186$		0			0
$187$		−	24.0000i		−	1.75505i
$188$		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	$-0.907034\pi$
$193$		−	8.00000i		−	0.575853i	0.957653	−	0.287926i	$-0.0929658\pi$
$194$		0			0
$195$	−2.00000	+	4.00000i	−0.143223	+	0.286446i
$196$		0			0
$197$		−	2.00000i		−	0.142494i	−0.997459	−	0.0712470i	$-0.977302\pi$
$197$		−	2.00000i		−	0.142494i	0.997459	−	0.0712470i	$-0.0226979\pi$
$198$		0			0
$199$	14.0000			0.992434			0.496217	−	0.868199i	$-0.334722\pi$
$199$	14.0000			0.992434			0.496217	+	0.868199i	$0.334722\pi$
$200$		0			0
$201$	−16.0000			−1.12855
$202$		0			0
$203$		−	2.00000i		−	0.140372i
$204$		0			0
$205$	2.00000	−	4.00000i	0.139686	−	0.279372i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−36.0000			−2.49017
$210$		0			0
$211$	16.0000			1.10149			0.550743	−	0.834675i	$-0.314345\pi$
$211$	16.0000			1.10149			0.550743	+	0.834675i	$0.314345\pi$
$212$		0			0
$213$			10.0000i			0.685189i
$214$		0			0
$215$	8.00000	+	4.00000i	0.545595	+	0.272798i
$216$		0			0
$217$		−	10.0000i		−	0.678844i
$218$		0			0
$219$	−6.00000			−0.405442
$220$		0			0
$221$	−8.00000			−0.538138
$222$		0			0
$223$			24.0000i			1.60716i	0.595198	+	0.803579i	$0.297074\pi$
$223$			24.0000i			1.60716i	−0.595198	+	0.803579i	$0.702926\pi$
$224$		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
$229$	10.0000			0.660819			0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000			0.660819			0.330409	+	0.943838i	$0.392813\pi$
$230$		0			0
$231$	−6.00000			−0.394771
$232$		0			0
$233$		−	26.0000i		−	1.70332i	−0.524097	−	0.851658i	$-0.675597\pi$
$233$		−	26.0000i		−	1.70332i	0.524097	−	0.851658i	$-0.324403\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		−	4.00000i		−	0.259828i
$238$		0			0
$239$	−6.00000			−0.388108			−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000			−0.388108			−0.194054	+	0.980991i	$0.562164\pi$
$240$		0			0
$241$	22.0000			1.41714			0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000			1.41714			0.708572	+	0.705638i	$0.249340\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	−1.00000	+	2.00000i	−0.0638877	+	0.127775i
$246$		0			0
$247$			12.0000i			0.763542i
$248$		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
$253$		0			0
$254$		0			0
$255$	−4.00000	+	8.00000i	−0.250490	+	0.500979i
$256$		0			0
$257$		−	16.0000i		−	0.998053i	−0.866587	−	0.499026i	$-0.833691\pi$
$257$		−	16.0000i		−	0.998053i	0.866587	−	0.499026i	$-0.166309\pi$
$258$		0			0
$259$	4.00000			0.248548
$260$		0			0
$261$	−2.00000			−0.123797
$262$		0			0
$263$			24.0000i			1.47990i	0.672660	+	0.739952i	$0.265152\pi$
$263$			24.0000i			1.47990i	−0.672660	+	0.739952i	$0.734848\pi$
$264$		0			0
$265$	12.0000	+	6.00000i	0.737154	+	0.368577i
$266$		0			0
$267$			6.00000i			0.367194i
$268$		0			0
$269$	14.0000			0.853595			0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000			0.853595			0.426798	+	0.904347i	$0.359642\pi$
$270$		0			0
$271$	14.0000			0.850439			0.425220	−	0.905090i	$-0.360197\pi$
$271$	14.0000			0.850439			0.425220	+	0.905090i	$0.360197\pi$
$272$		0			0
$273$			2.00000i			0.121046i
$274$		0			0
$275$	−18.0000	−	24.0000i	−1.08544	−	1.44725i
$276$		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
$279$	−10.0000			−0.598684
$280$		0			0
$281$	−2.00000			−0.119310			−0.0596550	−	0.998219i	$-0.519000\pi$
$281$	−2.00000			−0.119310			−0.0596550	+	0.998219i	$0.519000\pi$
$282$		0			0
$283$		−	20.0000i		−	1.18888i	−0.804141	−	0.594438i	$-0.797374\pi$
$283$		−	20.0000i		−	1.18888i	0.804141	−	0.594438i	$-0.202626\pi$
$284$		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
$291$	2.00000			0.117242
$292$		0			0
$293$			24.0000i			1.40209i	0.713115	+	0.701047i	$0.247284\pi$
$293$			24.0000i			1.40209i	−0.713115	+	0.701047i	$0.752716\pi$
$294$		0			0
$295$	−8.00000	+	16.0000i	−0.465778	+	0.931556i
$296$		0			0
$297$			6.00000i			0.348155i
$298$		0			0
$299$		0			0
$300$		0			0
$301$	4.00000			0.230556
$302$		0			0
$303$			6.00000i			0.344691i
$304$		0			0
$305$	−2.00000	+	4.00000i	−0.114520	+	0.229039i
$306$		0			0
$307$			4.00000i			0.228292i	0.993464	+	0.114146i	$0.0364132\pi$
$307$			4.00000i			0.228292i	−0.993464	+	0.114146i	$0.963587\pi$
$308$		0			0
$309$	−8.00000			−0.455104
$310$		0			0
$311$	24.0000			1.36092			0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000			1.36092			0.680458	+	0.732787i	$0.261781\pi$
$312$		0			0
$313$			6.00000i			0.339140i	0.985518	+	0.169570i	$0.0542379\pi$
$313$			6.00000i			0.339140i	−0.985518	+	0.169570i	$0.945762\pi$
$314$		0			0
$315$	2.00000	+	1.00000i	0.112687	+	0.0563436i
$316$		0			0
$317$			18.0000i			1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$			18.0000i			1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$		0			0
$319$	12.0000			0.671871
$320$		0			0
$321$	4.00000			0.223258
$322$		0			0
$323$			24.0000i			1.33540i
$324$		0			0
$325$	−8.00000	+	6.00000i	−0.443760	+	0.332820i
$326$		0			0
$327$			2.00000i			0.110600i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	24.0000			1.31916			0.659580	−	0.751635i	$-0.270734\pi$
$331$	24.0000			1.31916			0.659580	+	0.751635i	$0.270734\pi$
$332$		0			0
$333$		−	4.00000i		−	0.219199i
$334$		0			0
$335$	−32.0000	−	16.0000i	−1.74835	−	0.874173i
$336$		0			0
$337$			24.0000i			1.30736i	0.756770	+	0.653682i	$0.226776\pi$
$337$			24.0000i			1.30736i	−0.756770	+	0.653682i	$0.773224\pi$
$338$		0			0
$339$	6.00000			0.325875
$340$		0			0
$341$	60.0000			3.24918
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			12.0000i			0.644194i	0.946707	+	0.322097i	$0.104388\pi$
$347$			12.0000i			0.644194i	−0.946707	+	0.322097i	$0.895612\pi$
$348$		0			0
$349$	2.00000			0.107058			0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000			0.107058			0.0535288	+	0.998566i	$0.482953\pi$
$350$		0			0
$351$	2.00000			0.106752
$352$		0			0
$353$			20.0000i			1.06449i	0.846590	+	0.532246i	$0.178652\pi$
$353$			20.0000i			1.06449i	−0.846590	+	0.532246i	$0.821348\pi$
$354$		0			0
$355$	−10.0000	+	20.0000i	−0.530745	+	1.06149i
$356$		0			0
$357$			4.00000i			0.211702i
$358$		0			0
$359$	22.0000			1.16112			0.580558	−	0.814219i	$-0.302835\pi$
$359$	22.0000			1.16112			0.580558	+	0.814219i	$0.302835\pi$
$360$		0			0
$361$	17.0000			0.894737
$362$		0			0
$363$		−	25.0000i		−	1.31216i
$364$		0			0
$365$	−12.0000	−	6.00000i	−0.628109	−	0.314054i
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		0			0
$369$	−2.00000			−0.104116
$370$		0			0
$371$	6.00000			0.311504
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$	2.00000	+	11.0000i	0.103280	+	0.568038i
$376$		0			0
$377$		−	4.00000i		−	0.206010i
$378$		0			0
$379$	−28.0000			−1.43826			−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000			−1.43826			−0.719132	+	0.694874i	$0.755460\pi$
$380$		0			0
$381$	20.0000			1.02463
$382$		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
$385$	−12.0000	−	6.00000i	−0.611577	−	0.305788i
$386$		0			0
$387$		−	4.00000i		−	0.203331i
$388$		0			0
$389$	−26.0000			−1.31825			−0.659126	−	0.752032i	$-0.729074\pi$
$389$	−26.0000			−1.31825			−0.659126	+	0.752032i	$0.729074\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$			4.00000i			0.201773i
$394$		0			0
$395$	4.00000	−	8.00000i	0.201262	−	0.402524i
$396$		0			0
$397$		−	22.0000i		−	1.10415i	−0.833795	−	0.552074i	$-0.813837\pi$
$397$		−	22.0000i		−	1.10415i	0.833795	−	0.552074i	$-0.186163\pi$
$398$		0			0
$399$	6.00000			0.300376
$400$		0			0
$401$	−30.0000			−1.49813			−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000			−1.49813			−0.749064	+	0.662497i	$0.769497\pi$
$402$		0			0
$403$		−	20.0000i		−	0.996271i
$404$		0			0
$405$	1.00000	−	2.00000i	0.0496904	−	0.0993808i
$406$		0			0
$407$			24.0000i			1.18964i
$408$		0			0
$409$	22.0000			1.08783			0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000			1.08783			0.543915	+	0.839140i	$0.316941\pi$
$410$		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
$417$		−	2.00000i		−	0.0979404i
$418$		0			0
$419$	−12.0000			−0.586238			−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000			−0.586238			−0.293119	+	0.956076i	$0.594693\pi$
$420$		0			0
$421$	18.0000			0.877266			0.438633	−	0.898666i	$-0.355463\pi$
$421$	18.0000			0.877266			0.438633	+	0.898666i	$0.355463\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−16.0000	+	12.0000i	−0.776114	+	0.582086i
$426$		0			0
$427$			2.00000i			0.0967868i
$428$		0			0
$429$	−12.0000			−0.579365
$430$		0			0
$431$	14.0000			0.674356			0.337178	−	0.941441i	$-0.390528\pi$
$431$	14.0000			0.674356			0.337178	+	0.941441i	$0.390528\pi$
$432$		0			0
$433$		−	34.0000i		−	1.63394i	−0.576683	−	0.816968i	$-0.695653\pi$
$433$		−	34.0000i		−	1.63394i	0.576683	−	0.816968i	$-0.304347\pi$
$434$		0			0
$435$	−4.00000	−	2.00000i	−0.191785	−	0.0958927i
$436$		0			0
$437$		0			0
$438$		0			0
$439$	6.00000			0.286364			0.143182	−	0.989696i	$-0.454267\pi$
$439$	6.00000			0.286364			0.143182	+	0.989696i	$0.454267\pi$
$440$		0			0
$441$	1.00000			0.0476190
$442$		0			0
$443$			4.00000i			0.190046i	0.995475	+	0.0950229i	$0.0302924\pi$
$443$			4.00000i			0.190046i	−0.995475	+	0.0950229i	$0.969708\pi$
$444$		0			0
$445$	−6.00000	+	12.0000i	−0.284427	+	0.568855i
$446$		0			0
$447$		−	14.0000i		−	0.662177i
$448$		0			0
$449$	−10.0000			−0.471929			−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000			−0.471929			−0.235965	+	0.971762i	$0.575825\pi$
$450$		0			0
$451$	12.0000			0.565058
$452$		0			0
$453$			8.00000i			0.375873i
$454$		0			0
$455$	−2.00000	+	4.00000i	−0.0937614	+	0.187523i
$456$		0			0
$457$			20.0000i			0.935561i	0.883845	+	0.467780i	$0.154946\pi$
$457$			20.0000i			0.935561i	−0.883845	+	0.467780i	$0.845054\pi$
$458$		0			0
$459$	4.00000			0.186704
$460$		0			0
$461$	30.0000			1.39724			0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000			1.39724			0.698620	+	0.715493i	$0.253798\pi$
$462$		0			0
$463$		−	36.0000i		−	1.67306i	−0.547920	−	0.836531i	$-0.684580\pi$
$463$		−	36.0000i		−	1.67306i	0.547920	−	0.836531i	$-0.315420\pi$
$464$		0			0
$465$	−20.0000	−	10.0000i	−0.927478	−	0.463739i
$466$		0			0
$467$		−	24.0000i		−	1.11059i	−0.831654	−	0.555294i	$-0.812606\pi$
$467$		−	24.0000i		−	1.11059i	0.831654	−	0.555294i	$-0.187394\pi$
$468$		0			0
$469$	−16.0000			−0.738811
$470$		0			0
$471$	−18.0000			−0.829396
$472$		0			0
$473$			24.0000i			1.10352i
$474$		0			0
$475$	18.0000	+	24.0000i	0.825897	+	1.10120i
$476$		0			0
$477$		−	6.00000i		−	0.274721i
$478$		0			0
$479$	24.0000			1.09659			0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000			1.09659			0.548294	+	0.836286i	$0.315277\pi$
$480$		0			0
$481$	8.00000			0.364769
$482$		0			0
$483$		0			0
$484$		0			0
$485$	4.00000	+	2.00000i	0.181631	+	0.0908153i
$486$		0			0
$487$			12.0000i			0.543772i	0.962329	+	0.271886i	$0.0876473\pi$
$487$			12.0000i			0.543772i	−0.962329	+	0.271886i	$0.912353\pi$
$488$		0			0
$489$	−4.00000			−0.180886
$490$		0			0
$491$	−10.0000			−0.451294			−0.225647	−	0.974209i	$-0.572450\pi$
$491$	−10.0000			−0.451294			−0.225647	+	0.974209i	$0.572450\pi$
$492$		0			0
$493$		−	8.00000i		−	0.360302i
$494$		0			0
$495$	−6.00000	+	12.0000i	−0.269680	+	0.539360i
$496$		0			0
$497$			10.0000i			0.448561i
$498$		0			0
$499$	4.00000			0.179065			0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000			0.179065			0.0895323	+	0.995984i	$0.471463\pi$
$500$		0			0
$501$	12.0000			0.536120
$502$		0			0
$503$			36.0000i			1.60516i	0.596544	+	0.802580i	$0.296540\pi$
$503$			36.0000i			1.60516i	−0.596544	+	0.802580i	$0.703460\pi$
$504$		0			0
$505$	−6.00000	+	12.0000i	−0.266996	+	0.533993i
$506$		0			0
$507$		−	9.00000i		−	0.399704i
$508$		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
$511$	−6.00000			−0.265424
$512$		0			0
$513$		−	6.00000i		−	0.264906i
$514$		0			0
$515$	−16.0000	−	8.00000i	−0.705044	−	0.352522i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	38.0000			1.66481			0.832405	−	0.554168i	$-0.186963\pi$
$521$	38.0000			1.66481			0.832405	+	0.554168i	$0.186963\pi$
$522$		0			0
$523$		−	20.0000i		−	0.874539i	−0.899331	−	0.437269i	$-0.855946\pi$
$523$		−	20.0000i		−	0.874539i	0.899331	−	0.437269i	$-0.144054\pi$
$524$		0			0
$525$	3.00000	+	4.00000i	0.130931	+	0.174574i
$526$		0			0
$527$		−	40.0000i		−	1.74243i
$528$		0			0
$529$	23.0000			1.00000
$530$		0			0
$531$	8.00000			0.347170
$532$		0			0
$533$		−	4.00000i		−	0.173259i
$534$		0			0
$535$	8.00000	+	4.00000i	0.345870	+	0.172935i
$536$		0			0
$537$			14.0000i			0.604145i
$538$		0			0
$539$	−6.00000			−0.258438
$540$		0			0
$541$	6.00000			0.257960			0.128980	−	0.991647i	$-0.458830\pi$
$541$	6.00000			0.257960			0.128980	+	0.991647i	$0.458830\pi$
$542$		0			0
$543$			6.00000i			0.257485i
$544$		0			0
$545$	−2.00000	+	4.00000i	−0.0856706	+	0.171341i
$546$		0			0
$547$		−	16.0000i		−	0.684111i	−0.939680	−	0.342055i	$-0.888877\pi$
$547$		−	16.0000i		−	0.684111i	0.939680	−	0.342055i	$-0.111123\pi$
$548$		0			0
$549$	2.00000			0.0853579
$550$		0			0
$551$	−12.0000			−0.511217
$552$		0			0
$553$		−	4.00000i		−	0.170097i
$554$		0			0
$555$	4.00000	−	8.00000i	0.169791	−	0.339581i
$556$		0			0
$557$		−	38.0000i		−	1.61011i	−0.593199	−	0.805056i	$-0.702135\pi$
$557$		−	38.0000i		−	1.61011i	0.593199	−	0.805056i	$-0.297865\pi$
$558$		0			0
$559$	8.00000			0.338364
$560$		0			0
$561$	−24.0000			−1.01328
$562$		0			0
$563$			36.0000i			1.51722i	0.651546	+	0.758610i	$0.274121\pi$
$563$			36.0000i			1.51722i	−0.651546	+	0.758610i	$0.725879\pi$
$564$		0			0
$565$	12.0000	+	6.00000i	0.504844	+	0.252422i
$566$		0			0
$567$		−	1.00000i		−	0.0419961i
$568$		0			0
$569$	30.0000			1.25767			0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000			1.25767			0.628833	+	0.777541i	$0.283533\pi$
$570$		0			0
$571$	−36.0000			−1.50655			−0.753277	−	0.657704i	$-0.771528\pi$
$571$	−36.0000			−1.50655			−0.753277	+	0.657704i	$0.771528\pi$
$572$		0			0
$573$		−	18.0000i		−	0.751961i
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	14.0000i		−	0.582828i	−0.956597	−	0.291414i	$-0.905874\pi$
$577$		−	14.0000i		−	0.582828i	0.956597	−	0.291414i	$-0.0941257\pi$
$578$		0			0
$579$	−8.00000			−0.332469
$580$		0			0
$581$	−8.00000			−0.331896
$582$		0			0
$583$			36.0000i			1.49097i
$584$		0			0
$585$	4.00000	+	2.00000i	0.165380	+	0.0826898i
$586$		0			0
$587$		−	12.0000i		−	0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$		−	12.0000i		−	0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$		0			0
$589$	−60.0000			−2.47226
$590$		0			0
$591$	−2.00000			−0.0822690
$592$		0			0
$593$			44.0000i			1.80686i	0.428732	+	0.903432i	$0.358960\pi$
$593$			44.0000i			1.80686i	−0.428732	+	0.903432i	$0.641040\pi$
$594$		0			0
$595$	−4.00000	+	8.00000i	−0.163984	+	0.327968i
$596$		0			0
$597$		−	14.0000i		−	0.572982i
$598$		0			0
$599$	−2.00000			−0.0817178			−0.0408589	−	0.999165i	$-0.513009\pi$
$599$	−2.00000			−0.0817178			−0.0408589	+	0.999165i	$0.513009\pi$
$600$		0			0
$601$	−26.0000			−1.06056			−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000			−1.06056			−0.530281	+	0.847822i	$0.677914\pi$
$602$		0			0
$603$			16.0000i			0.651570i
$604$		0			0
$605$	25.0000	−	50.0000i	1.01639	−	2.03279i
$606$		0			0
$607$			24.0000i			0.974130i	0.873366	+	0.487065i	$0.161933\pi$
$607$			24.0000i			0.974130i	−0.873366	+	0.487065i	$0.838067\pi$
$608$		0			0
$609$	−2.00000			−0.0810441
$610$		0			0
$611$		0			0
$612$		0			0
$613$			4.00000i			0.161558i	0.996732	+	0.0807792i	$0.0257409\pi$
$613$			4.00000i			0.161558i	−0.996732	+	0.0807792i	$0.974259\pi$
$614$		0			0
$615$	−4.00000	−	2.00000i	−0.161296	−	0.0806478i
$616$		0			0
$617$			26.0000i			1.04672i	0.852111	+	0.523360i	$0.175322\pi$
$617$			26.0000i			1.04672i	−0.852111	+	0.523360i	$0.824678\pi$
$618$		0			0
$619$	10.0000			0.401934			0.200967	−	0.979598i	$-0.435592\pi$
$619$	10.0000			0.401934			0.200967	+	0.979598i	$0.435592\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			6.00000i			0.240385i
$624$		0			0
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$		0			0
$627$			36.0000i			1.43770i
$628$		0			0
$629$	16.0000			0.637962
$630$		0			0
$631$	20.0000			0.796187			0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000			0.796187			0.398094	+	0.917345i	$0.369672\pi$
$632$		0			0
$633$		−	16.0000i		−	0.635943i
$634$		0			0
$635$	40.0000	+	20.0000i	1.58735	+	0.793676i
$636$		0			0
$637$			2.00000i			0.0792429i
$638$		0			0
$639$	10.0000			0.395594
$640$		0			0
$641$	−2.00000			−0.0789953			−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000			−0.0789953			−0.0394976	+	0.999220i	$0.512576\pi$
$642$		0			0
$643$			20.0000i			0.788723i	0.918955	+	0.394362i	$0.129034\pi$
$643$			20.0000i			0.788723i	−0.918955	+	0.394362i	$0.870966\pi$
$644$		0			0
$645$	4.00000	−	8.00000i	0.157500	−	0.315000i
$646$		0			0
$647$		−	20.0000i		−	0.786281i	−0.919478	−	0.393141i	$-0.871389\pi$
$647$		−	20.0000i		−	0.786281i	0.919478	−	0.393141i	$-0.128611\pi$
$648$		0			0
$649$	−48.0000			−1.88416
$650$		0			0
$651$	−10.0000			−0.391931
$652$		0			0
$653$			2.00000i			0.0782660i	0.999234	+	0.0391330i	$0.0124596\pi$
$653$			2.00000i			0.0782660i	−0.999234	+	0.0391330i	$0.987540\pi$
$654$		0			0
$655$	−4.00000	+	8.00000i	−0.156293	+	0.312586i
$656$		0			0
$657$			6.00000i			0.234082i
$658$		0			0
$659$	−18.0000			−0.701180			−0.350590	−	0.936529i	$-0.614019\pi$
$659$	−18.0000			−0.701180			−0.350590	+	0.936529i	$0.614019\pi$
$660$		0			0
$661$	−18.0000			−0.700119			−0.350059	−	0.936727i	$-0.613839\pi$
$661$	−18.0000			−0.700119			−0.350059	+	0.936727i	$0.613839\pi$
$662$		0			0
$663$			8.00000i			0.310694i
$664$		0			0
$665$	12.0000	+	6.00000i	0.465340	+	0.232670i
$666$		0			0
$667$		0			0
$668$		0			0
$669$	24.0000			0.927894
$670$		0			0
$671$	−12.0000			−0.463255
$672$		0			0
$673$		−	36.0000i		−	1.38770i	−0.720121	−	0.693849i	$-0.755914\pi$
$673$		−	36.0000i		−	1.38770i	0.720121	−	0.693849i	$-0.244086\pi$
$674$		0			0
$675$	4.00000	−	3.00000i	0.153960	−	0.115470i
$676$		0			0
$677$		−	32.0000i		−	1.22986i	−0.788582	−	0.614930i	$-0.789184\pi$
$677$		−	32.0000i		−	1.22986i	0.788582	−	0.614930i	$-0.210816\pi$
$678$		0			0
$679$	2.00000			0.0767530
$680$		0			0
$681$	8.00000			0.306561
$682$		0			0
$683$		−	28.0000i		−	1.07139i	−0.844411	−	0.535695i	$-0.820050\pi$
$683$		−	28.0000i		−	1.07139i	0.844411	−	0.535695i	$-0.179950\pi$
$684$		0			0
$685$	12.0000	+	6.00000i	0.458496	+	0.229248i
$686$		0			0
$687$		−	10.0000i		−	0.381524i
$688$		0			0
$689$	12.0000			0.457164
$690$		0			0
$691$	50.0000			1.90209			0.951045	−	0.309053i	$-0.100012\pi$
$691$	50.0000			1.90209			0.951045	+	0.309053i	$0.100012\pi$
$692$		0			0
$693$			6.00000i			0.227921i
$694$		0			0
$695$	2.00000	−	4.00000i	0.0758643	−	0.151729i
$696$		0			0
$697$		−	8.00000i		−	0.303022i
$698$		0			0
$699$	−26.0000			−0.983410
$700$		0			0
$701$	−10.0000			−0.377695			−0.188847	−	0.982006i	$-0.560475\pi$
$701$	−10.0000			−0.377695			−0.188847	+	0.982006i	$0.560475\pi$
$702$		0			0
$703$		−	24.0000i		−	0.905177i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			6.00000i			0.225653i
$708$		0			0
$709$	−26.0000			−0.976450			−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000			−0.976450			−0.488225	+	0.872718i	$0.662356\pi$
$710$		0			0
$711$	−4.00000			−0.150012
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−24.0000	−	12.0000i	−0.897549	−	0.448775i
$716$		0			0
$717$			6.00000i			0.224074i
$718$		0			0
$719$	12.0000			0.447524			0.223762	−	0.974644i	$-0.428166\pi$
$719$	12.0000			0.447524			0.223762	+	0.974644i	$0.428166\pi$
$720$		0			0
$721$	−8.00000			−0.297936
$722$		0			0
$723$		−	22.0000i		−	0.818189i
$724$		0			0
$725$	−6.00000	−	8.00000i	−0.222834	−	0.297113i
$726$		0			0
$727$		−	40.0000i		−	1.48352i	−0.670667	−	0.741759i	$-0.733992\pi$
$727$		−	40.0000i		−	1.48352i	0.670667	−	0.741759i	$-0.266008\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	16.0000			0.591781
$732$		0			0
$733$			22.0000i			0.812589i	0.913742	+	0.406294i	$0.133179\pi$
$733$			22.0000i			0.812589i	−0.913742	+	0.406294i	$0.866821\pi$
$734$		0			0
$735$	2.00000	+	1.00000i	0.0737711	+	0.0368856i
$736$		0			0
$737$		−	96.0000i		−	3.53621i
$738$		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
$741$	12.0000			0.440831
$742$		0			0
$743$		−	40.0000i		−	1.46746i	−0.679442	−	0.733729i	$-0.737778\pi$
$743$		−	40.0000i		−	1.46746i	0.679442	−	0.733729i	$-0.262222\pi$
$744$		0			0
$745$	14.0000	−	28.0000i	0.512920	−	1.02584i
$746$		0			0
$747$			8.00000i			0.292705i
$748$		0			0
$749$	4.00000			0.146157
$750$		0			0
$751$	12.0000			0.437886			0.218943	−	0.975738i	$-0.429739\pi$
$751$	12.0000			0.437886			0.218943	+	0.975738i	$0.429739\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−8.00000	+	16.0000i	−0.291150	+	0.582300i
$756$		0			0
$757$			40.0000i			1.45382i	0.686730	+	0.726912i	$0.259045\pi$
$757$			40.0000i			1.45382i	−0.686730	+	0.726912i	$0.740955\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−22.0000			−0.797499			−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000			−0.797499			−0.398750	+	0.917060i	$0.630556\pi$
$762$		0			0
$763$			2.00000i			0.0724049i
$764$		0			0
$765$	8.00000	+	4.00000i	0.289241	+	0.144620i
$766$		0			0
$767$			16.0000i			0.577727i
$768$		0			0
$769$	18.0000			0.649097			0.324548	−	0.945869i	$-0.394788\pi$
$769$	18.0000			0.649097			0.324548	+	0.945869i	$0.394788\pi$
$770$		0			0
$771$	−16.0000			−0.576226
$772$		0			0
$773$			24.0000i			0.863220i	0.902060	+	0.431610i	$0.142054\pi$
$773$			24.0000i			0.863220i	−0.902060	+	0.431610i	$0.857946\pi$
$774$		0			0
$775$	−30.0000	−	40.0000i	−1.07763	−	1.43684i
$776$		0			0
$777$		−	4.00000i		−	0.143499i
$778$		0			0
$779$	−12.0000			−0.429945
$780$		0			0
$781$	−60.0000			−2.14697
$782$		0			0
$783$			2.00000i			0.0714742i
$784$		0			0
$785$	−36.0000	−	18.0000i	−1.28490	−	0.642448i
$786$		0			0
$787$		−	4.00000i		−	0.142585i	−0.997455	−	0.0712923i	$-0.977288\pi$
$787$		−	4.00000i		−	0.142585i	0.997455	−	0.0712923i	$-0.0227123\pi$
$788$		0			0
$789$	24.0000			0.854423
$790$		0			0
$791$	6.00000			0.213335
$792$		0			0
$793$			4.00000i			0.142044i
$794$		0			0
$795$	6.00000	−	12.0000i	0.212798	−	0.425596i
$796$		0			0
$797$			16.0000i			0.566749i	0.959009	+	0.283375i	$0.0914540\pi$
$797$			16.0000i			0.566749i	−0.959009	+	0.283375i	$0.908546\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	6.00000			0.212000
$802$		0			0
$803$		−	36.0000i		−	1.27041i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		−	14.0000i		−	0.492823i
$808$		0			0
$809$	18.0000			0.632846			0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000			0.632846			0.316423	+	0.948618i	$0.397518\pi$
$810$		0			0
$811$	−34.0000			−1.19390			−0.596951	−	0.802278i	$-0.703621\pi$
$811$	−34.0000			−1.19390			−0.596951	+	0.802278i	$0.703621\pi$
$812$		0			0
$813$		−	14.0000i		−	0.491001i
$814$		0			0
$815$	−8.00000	−	4.00000i	−0.280228	−	0.140114i
$816$		0			0
$817$		−	24.0000i		−	0.839654i
$818$		0			0
$819$	2.00000			0.0698857
$820$		0			0
$821$	−2.00000			−0.0698005			−0.0349002	−	0.999391i	$-0.511111\pi$
$821$	−2.00000			−0.0698005			−0.0349002	+	0.999391i	$0.511111\pi$
$822$		0			0
$823$		−	44.0000i		−	1.53374i	−0.641800	−	0.766872i	$-0.721812\pi$
$823$		−	44.0000i		−	1.53374i	0.641800	−	0.766872i	$-0.278188\pi$
$824$		0			0
$825$	−24.0000	+	18.0000i	−0.835573	+	0.626680i
$826$		0			0
$827$		−	12.0000i		−	0.417281i	−0.977992	−	0.208640i	$-0.933096\pi$
$827$		−	12.0000i		−	0.417281i	0.977992	−	0.208640i	$-0.0669038\pi$
$828$		0			0
$829$	6.00000			0.208389			0.104194	−	0.994557i	$-0.466774\pi$
$829$	6.00000			0.208389			0.104194	+	0.994557i	$0.466774\pi$
$830$		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
$837$			10.0000i			0.345651i
$838$		0			0
$839$	12.0000			0.414286			0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000			0.414286			0.207143	+	0.978311i	$0.433583\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$		0			0
$843$			2.00000i			0.0688837i
$844$		0			0
$845$	9.00000	−	18.0000i	0.309609	−	0.619219i
$846$		0			0
$847$		−	25.0000i		−	0.859010i
$848$		0			0
$849$	−20.0000			−0.686398
$850$		0			0
$851$		0			0
$852$		0			0
$853$			30.0000i			1.02718i	0.858036	+	0.513590i	$0.171685\pi$
$853$			30.0000i			1.02718i	−0.858036	+	0.513590i	$0.828315\pi$
$854$		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
$859$	−26.0000			−0.887109			−0.443554	−	0.896248i	$-0.646283\pi$
$859$	−26.0000			−0.887109			−0.443554	+	0.896248i	$0.646283\pi$
$860$		0			0
$861$	−2.00000			−0.0681598
$862$		0			0
$863$			24.0000i			0.816970i	0.912765	+	0.408485i	$0.133943\pi$
$863$			24.0000i			0.816970i	−0.912765	+	0.408485i	$0.866057\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		−	1.00000i		−	0.0339618i
$868$		0			0
$869$	24.0000			0.814144
$870$		0			0
$871$	−32.0000			−1.08428
$872$		0			0
$873$		−	2.00000i		−	0.0676897i
$874$		0			0
$875$	2.00000	+	11.0000i	0.0676123	+	0.371868i
$876$		0			0
$877$			4.00000i			0.135070i	0.997717	+	0.0675352i	$0.0215135\pi$
$877$			4.00000i			0.135070i	−0.997717	+	0.0675352i	$0.978487\pi$
$878$		0			0
$879$	24.0000			0.809500
$880$		0			0
$881$	14.0000			0.471672			0.235836	−	0.971793i	$-0.424217\pi$
$881$	14.0000			0.471672			0.235836	+	0.971793i	$0.424217\pi$
$882$		0			0
$883$			48.0000i			1.61533i	0.589643	+	0.807664i	$0.299269\pi$
$883$			48.0000i			1.61533i	−0.589643	+	0.807664i	$0.700731\pi$
$884$		0			0
$885$	16.0000	+	8.00000i	0.537834	+	0.268917i
$886$		0			0
$887$			8.00000i			0.268614i	0.990940	+	0.134307i	$0.0428808\pi$
$887$			8.00000i			0.268614i	−0.990940	+	0.134307i	$0.957119\pi$
$888$		0			0
$889$	20.0000			0.670778
$890$		0			0
$891$	6.00000			0.201008
$892$		0			0
$893$		0			0
$894$		0			0
$895$	−14.0000	+	28.0000i	−0.467968	+	0.935937i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	20.0000			0.667037
$900$		0			0
$901$	24.0000			0.799556
$902$		0			0
$903$		−	4.00000i		−	0.133112i
$904$		0			0
$905$	−6.00000	+	12.0000i	−0.199447	+	0.398893i
$906$		0			0
$907$			16.0000i			0.531271i	0.964073	+	0.265636i	$0.0855818\pi$
$907$			16.0000i			0.531271i	−0.964073	+	0.265636i	$0.914418\pi$
$908$		0			0
$909$	6.00000			0.199007
$910$		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
$913$		−	48.0000i		−	1.58857i
$914$		0			0
$915$	4.00000	+	2.00000i	0.132236	+	0.0661180i
$916$		0			0
$917$			4.00000i			0.132092i
$918$		0			0
$919$	28.0000			0.923635			0.461817	−	0.886975i	$-0.347198\pi$
$919$	28.0000			0.923635			0.461817	+	0.886975i	$0.347198\pi$
$920$		0			0
$921$	4.00000			0.131804
$922$		0			0
$923$			20.0000i			0.658308i
$924$		0			0
$925$	16.0000	−	12.0000i	0.526077	−	0.394558i
$926$		0			0
$927$			8.00000i			0.262754i
$928$		0			0
$929$	14.0000			0.459325			0.229663	−	0.973270i	$-0.426238\pi$
$929$	14.0000			0.459325			0.229663	+	0.973270i	$0.426238\pi$
$930$		0			0
$931$	6.00000			0.196642
$932$		0			0
$933$		−	24.0000i		−	0.785725i
$934$		0			0
$935$	−48.0000	−	24.0000i	−1.56977	−	0.784884i
$936$		0			0
$937$		−	2.00000i		−	0.0653372i	−0.999466	−	0.0326686i	$-0.989599\pi$
$937$		−	2.00000i		−	0.0653372i	0.999466	−	0.0326686i	$-0.0104006\pi$
$938$		0			0
$939$	6.00000			0.195803
$940$		0			0
$941$	−42.0000			−1.36916			−0.684580	−	0.728937i	$-0.740015\pi$
$941$	−42.0000			−1.36916			−0.684580	+	0.728937i	$0.740015\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	1.00000	−	2.00000i	0.0325300	−	0.0650600i
$946$		0			0
$947$		−	36.0000i		−	1.16984i	−0.811090	−	0.584921i	$-0.801125\pi$
$947$		−	36.0000i		−	1.16984i	0.811090	−	0.584921i	$-0.198875\pi$
$948$		0			0
$949$	−12.0000			−0.389536
$950$		0			0
$951$	18.0000			0.583690
$952$		0			0
$953$			6.00000i			0.194359i	0.995267	+	0.0971795i	$0.0309821\pi$
$953$			6.00000i			0.194359i	−0.995267	+	0.0971795i	$0.969018\pi$
$954$		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
$961$	69.0000			2.22581
$962$		0			0
$963$		−	4.00000i		−	0.128898i
$964$		0			0
$965$	−16.0000	−	8.00000i	−0.515058	−	0.257529i
$966$		0			0
$967$			40.0000i			1.28631i	0.765735	+	0.643157i	$0.222376\pi$
$967$			40.0000i			1.28631i	−0.765735	+	0.643157i	$0.777624\pi$
$968$		0			0
$969$	24.0000			0.770991
$970$		0			0
$971$	8.00000			0.256732			0.128366	−	0.991727i	$-0.459027\pi$
$971$	8.00000			0.256732			0.128366	+	0.991727i	$0.459027\pi$
$972$		0			0
$973$		−	2.00000i		−	0.0641171i
$974$		0			0
$975$	6.00000	+	8.00000i	0.192154	+	0.256205i
$976$		0			0
$977$		−	30.0000i		−	0.959785i	−0.877327	−	0.479893i	$-0.840676\pi$
$977$		−	30.0000i		−	0.959785i	0.877327	−	0.479893i	$-0.159324\pi$
$978$		0			0
$979$	−36.0000			−1.15056
$980$		0			0
$981$	2.00000			0.0638551
$982$		0			0
$983$		−	24.0000i		−	0.765481i	−0.923856	−	0.382741i	$-0.874980\pi$
$983$		−	24.0000i		−	0.765481i	0.923856	−	0.382741i	$-0.125020\pi$
$984$		0			0
$985$	−4.00000	−	2.00000i	−0.127451	−	0.0637253i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		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
$993$		−	24.0000i		−	0.761617i
$994$		0			0
$995$	14.0000	−	28.0000i	0.443830	−	0.887660i
$996$		0			0
$997$		−	14.0000i		−	0.443384i	−0.975117	−	0.221692i	$-0.928842\pi$
$997$		−	14.0000i		−	0.443384i	0.975117	−	0.221692i	$-0.0711580\pi$
$998$		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.f.1009.1		2
3.2	odd	2		5040.2.t.e.1009.2		2
4.3	odd	2		105.2.d.a.64.2	yes	2
5.2	odd	4		8400.2.a.bj.1.1		1
5.3	odd	4		8400.2.a.ch.1.1		1
5.4	even	2	inner	1680.2.t.f.1009.2		2
12.11	even	2		315.2.d.c.64.1		2
15.14	odd	2		5040.2.t.e.1009.1		2
20.3	even	4		525.2.a.c.1.1		1
20.7	even	4		525.2.a.b.1.1		1
20.19	odd	2		105.2.d.a.64.1	✓	2
28.3	even	6		735.2.q.b.79.1		4
28.11	odd	6		735.2.q.a.79.1		4
28.19	even	6		735.2.q.b.214.2		4
28.23	odd	6		735.2.q.a.214.2		4
28.27	even	2		735.2.d.a.589.2		2
60.23	odd	4		1575.2.a.e.1.1		1
60.47	odd	4		1575.2.a.i.1.1		1
60.59	even	2		315.2.d.c.64.2		2
84.83	odd	2		2205.2.d.f.1324.1		2
140.19	even	6		735.2.q.b.214.1		4
140.27	odd	4		3675.2.a.d.1.1		1
140.39	odd	6		735.2.q.a.79.2		4
140.59	even	6		735.2.q.b.79.2		4
140.79	odd	6		735.2.q.a.214.1		4
140.83	odd	4		3675.2.a.l.1.1		1
140.139	even	2		735.2.d.a.589.1		2
420.419	odd	2		2205.2.d.f.1324.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.d.a.64.1	✓	2	20.19	odd	2
105.2.d.a.64.2	yes	2	4.3	odd	2
315.2.d.c.64.1		2	12.11	even	2
315.2.d.c.64.2		2	60.59	even	2
525.2.a.b.1.1		1	20.7	even	4
525.2.a.c.1.1		1	20.3	even	4
735.2.d.a.589.1		2	140.139	even	2
735.2.d.a.589.2		2	28.27	even	2
735.2.q.a.79.1		4	28.11	odd	6
735.2.q.a.79.2		4	140.39	odd	6
735.2.q.a.214.1		4	140.79	odd	6
735.2.q.a.214.2		4	28.23	odd	6
735.2.q.b.79.1		4	28.3	even	6
735.2.q.b.79.2		4	140.59	even	6
735.2.q.b.214.1		4	140.19	even	6
735.2.q.b.214.2		4	28.19	even	6
1575.2.a.e.1.1		1	60.23	odd	4
1575.2.a.i.1.1		1	60.47	odd	4
1680.2.t.f.1009.1		2	1.1	even	1	trivial
1680.2.t.f.1009.2		2	5.4	even	2	inner
2205.2.d.f.1324.1		2	84.83	odd	2
2205.2.d.f.1324.2		2	420.419	odd	2
3675.2.a.d.1.1		1	140.27	odd	4
3675.2.a.l.1.1		1	140.83	odd	4
5040.2.t.e.1009.1		2	15.14	odd	2
5040.2.t.e.1009.2		2	3.2	odd	2
8400.2.a.bj.1.1		1	5.2	odd	4
8400.2.a.ch.1.1		1	5.3	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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +(1.00000 - 2.00000i) q^{5} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +(1.00000 - 2.00000i) q^{5} -1.00000i q^{7} -1.00000 q^{9} +6.00000 q^{11} -2.00000i q^{13} +(-2.00000 - 1.00000i) q^{15} -4.00000i q^{17} -6.00000 q^{19} -1.00000 q^{21} +(-3.00000 - 4.00000i) q^{25} +1.00000i q^{27} +2.00000 q^{29} +10.0000 q^{31} -6.00000i q^{33} +(-2.00000 - 1.00000i) q^{35} +4.00000i q^{37} -2.00000 q^{39} +2.00000 q^{41} +4.00000i q^{43} +(-1.00000 + 2.00000i) q^{45} -1.00000 q^{49} -4.00000 q^{51} +6.00000i q^{53} +(6.00000 - 12.0000i) q^{55} +6.00000i q^{57} -8.00000 q^{59} -2.00000 q^{61} +1.00000i q^{63} +(-4.00000 - 2.00000i) q^{65} -16.0000i q^{67} -10.0000 q^{71} -6.00000i q^{73} +(-4.00000 + 3.00000i) q^{75} -6.00000i q^{77} +4.00000 q^{79} +1.00000 q^{81} -8.00000i q^{83} +(-8.00000 - 4.00000i) q^{85} -2.00000i q^{87} -6.00000 q^{89} -2.00000 q^{91} -10.0000i q^{93} +(-6.00000 + 12.0000i) q^{95} +2.00000i q^{97} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^5 - 2 * q^9	\( 2 q + 2 q^{5} - 2 q^{9} + 12 q^{11} - 4 q^{15} - 12 q^{19} - 2 q^{21} - 6 q^{25} + 4 q^{29} + 20 q^{31} - 4 q^{35} - 4 q^{39} + 4 q^{41} - 2 q^{45} - 2 q^{49} - 8 q^{51} + 12 q^{55} - 16 q^{59} - 4 q^{61} - 8 q^{65} - 20 q^{71} - 8 q^{75} + 8 q^{79} + 2 q^{81} - 16 q^{85} - 12 q^{89} - 4 q^{91} - 12 q^{95} - 12 q^{99}+O(q^{100}) \) 2 * q + 2 * q^5 - 2 * q^9 + 12 * q^11 - 4 * q^15 - 12 * q^19 - 2 * q^21 - 6 * q^25 + 4 * q^29 + 20 * q^31 - 4 * q^35 - 4 * q^39 + 4 * q^41 - 2 * q^45 - 2 * q^49 - 8 * q^51 + 12 * q^55 - 16 * q^59 - 4 * q^61 - 8 * q^65 - 20 * q^71 - 8 * q^75 + 8 * q^79 + 2 * q^81 - 16 * q^85 - 12 * q^89 - 4 * q^91 - 12 * q^95 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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