LMFDB - Embedded newform 315.2.d.d.64.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [315,2,Mod(64,315)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(315, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("315.64");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	315.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$2.51528766367$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			64.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	315.64
Dual form			315.2.d.d.64.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000 q^{4} +(2.00000 - 1.00000i) q^{5} +1.00000i q^{7} -3.00000i q^{8} +(-1.00000 - 2.00000i) q^{10} +4.00000i q^{13} +1.00000 q^{14} -1.00000 q^{16} -2.00000i q^{17} +(2.00000 - 1.00000i) q^{20} +(3.00000 - 4.00000i) q^{25} +4.00000 q^{26} +1.00000i q^{28} -8.00000 q^{29} -4.00000 q^{31} -5.00000i q^{32} -2.00000 q^{34} +(1.00000 + 2.00000i) q^{35} -8.00000i q^{37} +(-3.00000 - 6.00000i) q^{40} +4.00000 q^{41} +8.00000i q^{43} +12.0000i q^{47} -1.00000 q^{49} +(-4.00000 - 3.00000i) q^{50} +4.00000i q^{52} +6.00000i q^{53} +3.00000 q^{56} +8.00000i q^{58} -8.00000 q^{59} +10.0000 q^{61} +4.00000i q^{62} -7.00000 q^{64} +(4.00000 + 8.00000i) q^{65} -8.00000i q^{67} -2.00000i q^{68} +(2.00000 - 1.00000i) q^{70} -16.0000 q^{71} +12.0000i q^{73} -8.00000 q^{74} +8.00000 q^{79} +(-2.00000 + 1.00000i) q^{80} -4.00000i q^{82} +16.0000i q^{83} +(-2.00000 - 4.00000i) q^{85} +8.00000 q^{86} +12.0000 q^{89} -4.00000 q^{91} +12.0000 q^{94} -4.00000i q^{97} +1.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} + 4 q^{5} - 2 q^{10} + 2 q^{14} - 2 q^{16} + 4 q^{20} + 6 q^{25} + 8 q^{26} - 16 q^{29} - 8 q^{31} - 4 q^{34} + 2 q^{35} - 6 q^{40} + 8 q^{41} - 2 q^{49} - 8 q^{50} + 6 q^{56} - 16 q^{59}+ \cdots + 24 q^{94}+O(q^{100}) $ 2 * q + 2 * q^4 + 4 * q^5 - 2 * q^10 + 2 * q^14 - 2 * q^16 + 4 * q^20 + 6 * q^25 + 8 * q^26 - 16 * q^29 - 8 * q^31 - 4 * q^34 + 2 * q^35 - 6 * q^40 + 8 * q^41 - 2 * q^49 - 8 * q^50 + 6 * q^56 - 16 * q^59 + 20 * q^61 - 14 * q^64 + 8 * q^65 + 4 * q^70 - 32 * q^71 - 16 * q^74 + 16 * q^79 - 4 * q^80 - 4 * q^85 + 16 * q^86 + 24 * q^89 - 8 * q^91 + 24 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/315\mathbb{Z}\right)^\times$.

$n$	$127$	$136$	$281$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		−	1.00000i		−	0.707107i	−0.935414	−	0.353553i	$-0.884973\pi$
$2$		−	1.00000i		−	0.707107i	0.935414	−	0.353553i	$-0.115027\pi$
$3$		0			0
$3$		0			0
$4$	1.00000			0.500000
$5$	2.00000	−	1.00000i	0.894427	−	0.447214i
$5$	2.00000	−	1.00000i	0.894427	−	0.447214i
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		−	3.00000i		−	1.06066i
$9$		0			0
$10$	−1.00000	−	2.00000i	−0.316228	−	0.632456i
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$		0			0
$13$			4.00000i			1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$			4.00000i			1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$	1.00000			0.267261
$15$		0			0
$16$	−1.00000			−0.250000
$17$		−	2.00000i		−	0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$		−	2.00000i		−	0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$	2.00000	−	1.00000i	0.447214	−	0.223607i
$21$		0			0
$22$		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
$26$	4.00000			0.784465
$27$		0			0
$28$			1.00000i			0.188982i
$29$	−8.00000			−1.48556			−0.742781	−	0.669534i	$-0.766494\pi$
$29$	−8.00000			−1.48556			−0.742781	+	0.669534i	$0.766494\pi$
$30$		0			0
$31$	−4.00000			−0.718421			−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000			−0.718421			−0.359211	+	0.933257i	$0.616954\pi$
$32$		−	5.00000i		−	0.883883i
$33$		0			0
$34$	−2.00000			−0.342997
$35$	1.00000	+	2.00000i	0.169031	+	0.338062i
$36$		0			0
$37$		−	8.00000i		−	1.31519i	−0.753371	−	0.657596i	$-0.771573\pi$
$37$		−	8.00000i		−	1.31519i	0.753371	−	0.657596i	$-0.228427\pi$
$38$		0			0
$39$		0			0
$40$	−3.00000	−	6.00000i	−0.474342	−	0.948683i
$41$	4.00000			0.624695			0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000			0.624695			0.312348	+	0.949968i	$0.398885\pi$
$42$		0			0
$43$			8.00000i			1.21999i	0.792406	+	0.609994i	$0.208828\pi$
$43$			8.00000i			1.21999i	−0.792406	+	0.609994i	$0.791172\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			12.0000i			1.75038i	0.483779	+	0.875190i	$0.339264\pi$
$47$			12.0000i			1.75038i	−0.483779	+	0.875190i	$0.660736\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$	−4.00000	−	3.00000i	−0.565685	−	0.424264i
$51$		0			0
$52$			4.00000i			0.554700i
$53$			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
$55$		0			0
$56$	3.00000			0.400892
$57$		0			0
$58$			8.00000i			1.05045i
$59$	−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	$-0.278858\pi$
$61$	10.0000			1.28037			0.640184	+	0.768221i	$0.278858\pi$
$62$			4.00000i			0.508001i
$63$		0			0
$64$	−7.00000			−0.875000
$65$	4.00000	+	8.00000i	0.496139	+	0.992278i
$66$		0			0
$67$		−	8.00000i		−	0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$		−	8.00000i		−	0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$		−	2.00000i		−	0.242536i
$69$		0			0
$70$	2.00000	−	1.00000i	0.239046	−	0.119523i
$71$	−16.0000			−1.89885			−0.949425	−	0.313993i	$-0.898333\pi$
$71$	−16.0000			−1.89885			−0.949425	+	0.313993i	$0.898333\pi$
$72$		0			0
$73$			12.0000i			1.40449i	0.711934	+	0.702247i	$0.247820\pi$
$73$			12.0000i			1.40449i	−0.711934	+	0.702247i	$0.752180\pi$
$74$	−8.00000			−0.929981
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	8.00000			0.900070			0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000			0.900070			0.450035	+	0.893011i	$0.351411\pi$
$80$	−2.00000	+	1.00000i	−0.223607	+	0.111803i
$81$		0			0
$82$		−	4.00000i		−	0.441726i
$83$			16.0000i			1.75623i	0.478451	+	0.878114i	$0.341198\pi$
$83$			16.0000i			1.75623i	−0.478451	+	0.878114i	$0.658802\pi$
$84$		0			0
$85$	−2.00000	−	4.00000i	−0.216930	−	0.433861i
$86$	8.00000			0.862662
$87$		0			0
$88$		0			0
$89$	12.0000			1.27200			0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000			1.27200			0.635999	+	0.771690i	$0.280588\pi$
$90$		0			0
$91$	−4.00000			−0.419314
$92$		0			0
$93$		0			0
$94$	12.0000			1.23771
$95$		0			0
$96$		0			0
$97$		−	4.00000i		−	0.406138i	−0.979164	−	0.203069i	$-0.934908\pi$
$97$		−	4.00000i		−	0.406138i	0.979164	−	0.203069i	$-0.0650917\pi$
$98$			1.00000i			0.101015i
$99$		0			0
$100$	3.00000	−	4.00000i	0.300000	−	0.400000i
$101$	−12.0000			−1.19404			−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000			−1.19404			−0.597022	+	0.802225i	$0.703650\pi$
$102$		0			0
$103$			8.00000i			0.788263i	0.919054	+	0.394132i	$0.128955\pi$
$103$			8.00000i			0.788263i	−0.919054	+	0.394132i	$0.871045\pi$
$104$	12.0000			1.17670
$105$		0			0
$106$	6.00000			0.582772
$107$			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
$111$		0			0
$112$		−	1.00000i		−	0.0944911i
$113$		−	18.0000i		−	1.69330i	−0.532152	−	0.846649i	$-0.678617\pi$
$113$		−	18.0000i		−	1.69330i	0.532152	−	0.846649i	$-0.321383\pi$
$114$		0			0
$115$		0			0
$116$	−8.00000			−0.742781
$117$		0			0
$118$			8.00000i			0.736460i
$119$	2.00000			0.183340
$120$		0			0
$121$	−11.0000			−1.00000
$122$		−	10.0000i		−	0.905357i
$123$		0			0
$124$	−4.00000			−0.359211
$125$	2.00000	−	11.0000i	0.178885	−	0.983870i
$126$		0			0
$127$		−	8.00000i		−	0.709885i	−0.934888	−	0.354943i	$-0.884500\pi$
$127$		−	8.00000i		−	0.709885i	0.934888	−	0.354943i	$-0.115500\pi$
$128$		−	3.00000i		−	0.265165i
$129$		0			0
$130$	8.00000	−	4.00000i	0.701646	−	0.350823i
$131$	8.00000			0.698963			0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000			0.698963			0.349482	+	0.936943i	$0.386358\pi$
$132$		0			0
$133$		0			0
$134$	−8.00000			−0.691095
$135$		0			0
$136$	−6.00000			−0.514496
$137$		−	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
$139$	−8.00000			−0.678551			−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000			−0.678551			−0.339276	+	0.940687i	$0.610182\pi$
$140$	1.00000	+	2.00000i	0.0845154	+	0.169031i
$141$		0			0
$142$			16.0000i			1.34269i
$143$		0			0
$144$		0			0
$145$	−16.0000	+	8.00000i	−1.32873	+	0.664364i
$146$	12.0000			0.993127
$147$		0			0
$148$		−	8.00000i		−	0.657596i
$149$	16.0000			1.31077			0.655386	−	0.755295i	$-0.272506\pi$
$149$	16.0000			1.31077			0.655386	+	0.755295i	$0.272506\pi$
$150$		0			0
$151$	−16.0000			−1.30206			−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000			−1.30206			−0.651031	+	0.759051i	$0.725663\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−8.00000	+	4.00000i	−0.642575	+	0.321288i
$156$		0			0
$157$		−	12.0000i		−	0.957704i	−0.877896	−	0.478852i	$-0.841053\pi$
$157$		−	12.0000i		−	0.957704i	0.877896	−	0.478852i	$-0.158947\pi$
$158$		−	8.00000i		−	0.636446i
$159$		0			0
$160$	−5.00000	−	10.0000i	−0.395285	−	0.790569i
$161$		0			0
$162$		0			0
$163$		−	8.00000i		−	0.626608i	−0.949653	−	0.313304i	$-0.898564\pi$
$163$		−	8.00000i		−	0.626608i	0.949653	−	0.313304i	$-0.101436\pi$
$164$	4.00000			0.312348
$165$		0			0
$166$	16.0000			1.24184
$167$		−	12.0000i		−	0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$		−	12.0000i		−	0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$		0			0
$169$	−3.00000			−0.230769
$170$	−4.00000	+	2.00000i	−0.306786	+	0.153393i
$171$		0			0
$172$			8.00000i			0.609994i
$173$		−	6.00000i		−	0.456172i	−0.973641	−	0.228086i	$-0.926753\pi$
$173$		−	6.00000i		−	0.456172i	0.973641	−	0.228086i	$-0.0732467\pi$
$174$		0			0
$175$	4.00000	+	3.00000i	0.302372	+	0.226779i
$176$		0			0
$177$		0			0
$178$		−	12.0000i		−	0.899438i
$179$	16.0000			1.19590			0.597948	−	0.801535i	$-0.295983\pi$
$179$	16.0000			1.19590			0.597948	+	0.801535i	$0.295983\pi$
$180$		0			0
$181$	18.0000			1.33793			0.668965	−	0.743294i	$-0.266738\pi$
$181$	18.0000			1.33793			0.668965	+	0.743294i	$0.266738\pi$
$182$			4.00000i			0.296500i
$183$		0			0
$184$		0			0
$185$	−8.00000	−	16.0000i	−0.588172	−	1.17634i
$186$		0			0
$187$		0			0
$188$			12.0000i			0.875190i
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$			16.0000i			1.15171i	0.817554	+	0.575853i	$0.195330\pi$
$193$			16.0000i			1.15171i	−0.817554	+	0.575853i	$0.804670\pi$
$194$	−4.00000			−0.287183
$195$		0			0
$196$	−1.00000			−0.0714286
$197$		−	10.0000i		−	0.712470i	−0.934396	−	0.356235i	$-0.884060\pi$
$197$		−	10.0000i		−	0.712470i	0.934396	−	0.356235i	$-0.115940\pi$
$198$		0			0
$199$	4.00000			0.283552			0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000			0.283552			0.141776	+	0.989899i	$0.454719\pi$
$200$	−12.0000	−	9.00000i	−0.848528	−	0.636396i
$201$		0			0
$202$			12.0000i			0.844317i
$203$		−	8.00000i		−	0.561490i
$204$		0			0
$205$	8.00000	−	4.00000i	0.558744	−	0.279372i
$206$	8.00000			0.557386
$207$		0			0
$208$		−	4.00000i		−	0.277350i
$209$		0			0
$210$		0			0
$211$	20.0000			1.37686			0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000			1.37686			0.688428	+	0.725304i	$0.258301\pi$
$212$			6.00000i			0.412082i
$213$		0			0
$214$	4.00000			0.273434
$215$	8.00000	+	16.0000i	0.545595	+	1.09119i
$216$		0			0
$217$		−	4.00000i		−	0.271538i
$218$			2.00000i			0.135457i
$219$		0			0
$220$		0			0
$221$	8.00000			0.538138
$222$		0			0
$223$		−	24.0000i		−	1.60716i	−0.595198	−	0.803579i	$-0.702926\pi$
$223$		−	24.0000i		−	1.60716i	0.595198	−	0.803579i	$-0.297074\pi$
$224$	5.00000			0.334077
$225$		0			0
$226$	−18.0000			−1.19734
$227$			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
$231$		0			0
$232$			24.0000i			1.57568i
$233$		−	22.0000i		−	1.44127i	−0.693316	−	0.720634i	$-0.743851\pi$
$233$		−	22.0000i		−	1.44127i	0.693316	−	0.720634i	$-0.256149\pi$
$234$		0			0
$235$	12.0000	+	24.0000i	0.782794	+	1.56559i
$236$	−8.00000			−0.520756
$237$		0			0
$238$		−	2.00000i		−	0.129641i
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−14.0000			−0.901819			−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000			−0.901819			−0.450910	+	0.892570i	$0.648900\pi$
$242$			11.0000i			0.707107i
$243$		0			0
$244$	10.0000			0.640184
$245$	−2.00000	+	1.00000i	−0.127775	+	0.0638877i
$246$		0			0
$247$		0			0
$248$			12.0000i			0.762001i
$249$		0			0
$250$	−11.0000	−	2.00000i	−0.695701	−	0.126491i
$251$	−24.0000			−1.51487			−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000			−1.51487			−0.757433	+	0.652913i	$0.773547\pi$
$252$		0			0
$253$		0			0
$254$	−8.00000			−0.501965
$255$		0			0
$256$	−17.0000			−1.06250
$257$		−	14.0000i		−	0.873296i	−0.899632	−	0.436648i	$-0.856166\pi$
$257$		−	14.0000i		−	0.873296i	0.899632	−	0.436648i	$-0.143834\pi$
$258$		0			0
$259$	8.00000			0.497096
$260$	4.00000	+	8.00000i	0.248069	+	0.496139i
$261$		0			0
$262$		−	8.00000i		−	0.494242i
$263$			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
$265$	6.00000	+	12.0000i	0.368577	+	0.737154i
$266$		0			0
$267$		0			0
$268$		−	8.00000i		−	0.488678i
$269$	4.00000			0.243884			0.121942	−	0.992537i	$-0.461088\pi$
$269$	4.00000			0.243884			0.121942	+	0.992537i	$0.461088\pi$
$270$		0			0
$271$	28.0000			1.70088			0.850439	−	0.526073i	$-0.176336\pi$
$271$	28.0000			1.70088			0.850439	+	0.526073i	$0.176336\pi$
$272$			2.00000i			0.121268i
$273$		0			0
$274$	−6.00000			−0.362473
$275$		0			0
$276$		0			0
$277$			16.0000i			0.961347i	0.876900	+	0.480673i	$0.159608\pi$
$277$			16.0000i			0.961347i	−0.876900	+	0.480673i	$0.840392\pi$
$278$			8.00000i			0.479808i
$279$		0			0
$280$	6.00000	−	3.00000i	0.358569	−	0.179284i
$281$	8.00000			0.477240			0.238620	−	0.971113i	$-0.423305\pi$
$281$	8.00000			0.477240			0.238620	+	0.971113i	$0.423305\pi$
$282$		0			0
$283$		−	16.0000i		−	0.951101i	−0.879688	−	0.475551i	$-0.842249\pi$
$283$		−	16.0000i		−	0.951101i	0.879688	−	0.475551i	$-0.157751\pi$
$284$	−16.0000			−0.949425
$285$		0			0
$286$		0			0
$287$			4.00000i			0.236113i
$288$		0			0
$289$	13.0000			0.764706
$290$	8.00000	+	16.0000i	0.469776	+	0.939552i
$291$		0			0
$292$			12.0000i			0.702247i
$293$			18.0000i			1.05157i	0.850617	+	0.525786i	$0.176229\pi$
$293$			18.0000i			1.05157i	−0.850617	+	0.525786i	$0.823771\pi$
$294$		0			0
$295$	−16.0000	+	8.00000i	−0.931556	+	0.465778i
$296$	−24.0000			−1.39497
$297$		0			0
$298$		−	16.0000i		−	0.926855i
$299$		0			0
$300$		0			0
$301$	−8.00000			−0.461112
$302$			16.0000i			0.920697i
$303$		0			0
$304$		0			0
$305$	20.0000	−	10.0000i	1.14520	−	0.572598i
$306$		0			0
$307$		−	16.0000i		−	0.913168i	−0.889680	−	0.456584i	$-0.849073\pi$
$307$		−	16.0000i		−	0.913168i	0.889680	−	0.456584i	$-0.150927\pi$
$308$		0			0
$309$		0			0
$310$	4.00000	+	8.00000i	0.227185	+	0.454369i
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$		−	12.0000i		−	0.678280i	−0.940736	−	0.339140i	$-0.889864\pi$
$313$		−	12.0000i		−	0.678280i	0.940736	−	0.339140i	$-0.110136\pi$
$314$	−12.0000			−0.677199
$315$		0			0
$316$	8.00000			0.450035
$317$			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
$319$		0			0
$320$	−14.0000	+	7.00000i	−0.782624	+	0.391312i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	16.0000	+	12.0000i	0.887520	+	0.665640i
$326$	−8.00000			−0.443079
$327$		0			0
$328$		−	12.0000i		−	0.662589i
$329$	−12.0000			−0.661581
$330$		0			0
$331$	−12.0000			−0.659580			−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000			−0.659580			−0.329790	+	0.944054i	$0.606978\pi$
$332$			16.0000i			0.878114i
$333$		0			0
$334$	−12.0000			−0.656611
$335$	−8.00000	−	16.0000i	−0.437087	−	0.874173i
$336$		0			0
$337$		−	24.0000i		−	1.30736i	−0.756770	−	0.653682i	$-0.773224\pi$
$337$		−	24.0000i		−	1.30736i	0.756770	−	0.653682i	$-0.226776\pi$
$338$			3.00000i			0.163178i
$339$		0			0
$340$	−2.00000	−	4.00000i	−0.108465	−	0.216930i
$341$		0			0
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$	24.0000			1.29399
$345$		0			0
$346$	−6.00000			−0.322562
$347$			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
$349$	−10.0000			−0.535288			−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000			−0.535288			−0.267644	+	0.963518i	$0.586245\pi$
$350$	3.00000	−	4.00000i	0.160357	−	0.213809i
$351$		0			0
$352$		0			0
$353$		−	14.0000i		−	0.745145i	−0.928003	−	0.372572i	$-0.878476\pi$
$353$		−	14.0000i		−	0.745145i	0.928003	−	0.372572i	$-0.121524\pi$
$354$		0			0
$355$	−32.0000	+	16.0000i	−1.69838	+	0.849192i
$356$	12.0000			0.635999
$357$		0			0
$358$		−	16.0000i		−	0.845626i
$359$	−32.0000			−1.68890			−0.844448	−	0.535638i	$-0.820071\pi$
$359$	−32.0000			−1.68890			−0.844448	+	0.535638i	$0.820071\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		−	18.0000i		−	0.946059i
$363$		0			0
$364$	−4.00000			−0.209657
$365$	12.0000	+	24.0000i	0.628109	+	1.25622i
$366$		0			0
$367$			24.0000i			1.25279i	0.779506	+	0.626395i	$0.215470\pi$
$367$			24.0000i			1.25279i	−0.779506	+	0.626395i	$0.784530\pi$
$368$		0			0
$369$		0			0
$370$	−16.0000	+	8.00000i	−0.831800	+	0.415900i
$371$	−6.00000			−0.311504
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$	36.0000			1.85656
$377$		−	32.0000i		−	1.64808i
$378$		0			0
$379$	4.00000			0.205466			0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000			0.205466			0.102733	+	0.994709i	$0.467241\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	4.00000i		−	0.204390i	−0.994764	−	0.102195i	$-0.967413\pi$
$383$		−	4.00000i		−	0.204390i	0.994764	−	0.102195i	$-0.0325866\pi$
$384$		0			0
$385$		0			0
$386$	16.0000			0.814379
$387$		0			0
$388$		−	4.00000i		−	0.203069i
$389$	8.00000			0.405616			0.202808	−	0.979219i	$-0.434993\pi$
$389$	8.00000			0.405616			0.202808	+	0.979219i	$0.434993\pi$
$390$		0			0
$391$		0			0
$392$			3.00000i			0.151523i
$393$		0			0
$394$	−10.0000			−0.503793
$395$	16.0000	−	8.00000i	0.805047	−	0.402524i
$396$		0			0
$397$		−	4.00000i		−	0.200754i	−0.994949	−	0.100377i	$-0.967995\pi$
$397$		−	4.00000i		−	0.200754i	0.994949	−	0.100377i	$-0.0320049\pi$
$398$		−	4.00000i		−	0.200502i
$399$		0			0
$400$	−3.00000	+	4.00000i	−0.150000	+	0.200000i
$401$	24.0000			1.19850			0.599251	−	0.800561i	$-0.295465\pi$
$401$	24.0000			1.19850			0.599251	+	0.800561i	$0.295465\pi$
$402$		0			0
$403$		−	16.0000i		−	0.797017i
$404$	−12.0000			−0.597022
$405$		0			0
$406$	−8.00000			−0.397033
$407$		0			0
$408$		0			0
$409$	10.0000			0.494468			0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000			0.494468			0.247234	+	0.968956i	$0.420478\pi$
$410$	−4.00000	−	8.00000i	−0.197546	−	0.395092i
$411$		0			0
$412$			8.00000i			0.394132i
$413$		−	8.00000i		−	0.393654i
$414$		0			0
$415$	16.0000	+	32.0000i	0.785409	+	1.57082i
$416$	20.0000			0.980581
$417$		0			0
$418$		0			0
$419$	−24.0000			−1.17248			−0.586238	−	0.810139i	$-0.699392\pi$
$419$	−24.0000			−1.17248			−0.586238	+	0.810139i	$0.699392\pi$
$420$		0			0
$421$	6.00000			0.292422			0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000			0.292422			0.146211	+	0.989253i	$0.453292\pi$
$422$		−	20.0000i		−	0.973585i
$423$		0			0
$424$	18.0000			0.874157
$425$	−8.00000	−	6.00000i	−0.388057	−	0.291043i
$426$		0			0
$427$			10.0000i			0.483934i
$428$			4.00000i			0.193347i
$429$		0			0
$430$	16.0000	−	8.00000i	0.771589	−	0.385794i
$431$	32.0000			1.54139			0.770693	−	0.637207i	$-0.219910\pi$
$431$	32.0000			1.54139			0.770693	+	0.637207i	$0.219910\pi$
$432$		0			0
$433$			20.0000i			0.961139i	0.876957	+	0.480569i	$0.159570\pi$
$433$			20.0000i			0.961139i	−0.876957	+	0.480569i	$0.840430\pi$
$434$	−4.00000			−0.192006
$435$		0			0
$436$	−2.00000			−0.0957826
$437$		0			0
$438$		0			0
$439$	−12.0000			−0.572729			−0.286364	−	0.958121i	$-0.592447\pi$
$439$	−12.0000			−0.572729			−0.286364	+	0.958121i	$0.592447\pi$
$440$		0			0
$441$		0			0
$442$		−	8.00000i		−	0.380521i
$443$		−	20.0000i		−	0.950229i	−0.879924	−	0.475114i	$-0.842407\pi$
$443$		−	20.0000i		−	0.950229i	0.879924	−	0.475114i	$-0.157593\pi$
$444$		0			0
$445$	24.0000	−	12.0000i	1.13771	−	0.568855i
$446$	−24.0000			−1.13643
$447$		0			0
$448$		−	7.00000i		−	0.330719i
$449$	16.0000			0.755087			0.377543	−	0.925992i	$-0.376769\pi$
$449$	16.0000			0.755087			0.377543	+	0.925992i	$0.376769\pi$
$450$		0			0
$451$		0			0
$452$		−	18.0000i		−	0.846649i
$453$		0			0
$454$	8.00000			0.375459
$455$	−8.00000	+	4.00000i	−0.375046	+	0.187523i
$456$		0			0
$457$		−	16.0000i		−	0.748448i	−0.927338	−	0.374224i	$-0.877909\pi$
$457$		−	16.0000i		−	0.748448i	0.927338	−	0.374224i	$-0.122091\pi$
$458$			14.0000i			0.654177i
$459$		0			0
$460$		0			0
$461$	−12.0000			−0.558896			−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000			−0.558896			−0.279448	+	0.960161i	$0.590151\pi$
$462$		0			0
$463$		−	24.0000i		−	1.11537i	−0.830051	−	0.557687i	$-0.811689\pi$
$463$		−	24.0000i		−	1.11537i	0.830051	−	0.557687i	$-0.188311\pi$
$464$	8.00000			0.371391
$465$		0			0
$466$	−22.0000			−1.01913
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$	8.00000			0.369406
$470$	24.0000	−	12.0000i	1.10704	−	0.553519i
$471$		0			0
$472$			24.0000i			1.10469i
$473$		0			0
$474$		0			0
$475$		0			0
$476$	2.00000			0.0916698
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$	32.0000			1.45907
$482$			14.0000i			0.637683i
$483$		0			0
$484$	−11.0000			−0.500000
$485$	−4.00000	−	8.00000i	−0.181631	−	0.363261i
$486$		0			0
$487$		−	24.0000i		−	1.08754i	−0.839233	−	0.543772i	$-0.816996\pi$
$487$		−	24.0000i		−	1.08754i	0.839233	−	0.543772i	$-0.183004\pi$
$488$		−	30.0000i		−	1.35804i
$489$		0			0
$490$	1.00000	+	2.00000i	0.0451754	+	0.0903508i
$491$	32.0000			1.44414			0.722070	−	0.691820i	$-0.243191\pi$
$491$	32.0000			1.44414			0.722070	+	0.691820i	$0.243191\pi$
$492$		0			0
$493$			16.0000i			0.720604i
$494$		0			0
$495$		0			0
$496$	4.00000			0.179605
$497$		−	16.0000i		−	0.717698i
$498$		0			0
$499$	20.0000			0.895323			0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000			0.895323			0.447661	+	0.894203i	$0.352257\pi$
$500$	2.00000	−	11.0000i	0.0894427	−	0.491935i
$501$		0			0
$502$			24.0000i			1.07117i
$503$		−	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
$505$	−24.0000	+	12.0000i	−1.06799	+	0.533993i
$506$		0			0
$507$		0			0
$508$		−	8.00000i		−	0.354943i
$509$	−36.0000			−1.59567			−0.797836	−	0.602875i	$-0.794022\pi$
$509$	−36.0000			−1.59567			−0.797836	+	0.602875i	$0.794022\pi$
$510$		0			0
$511$	−12.0000			−0.530849
$512$			11.0000i			0.486136i
$513$		0			0
$514$	−14.0000			−0.617514
$515$	8.00000	+	16.0000i	0.352522	+	0.705044i
$516$		0			0
$517$		0			0
$518$		−	8.00000i		−	0.351500i
$519$		0			0
$520$	24.0000	−	12.0000i	1.05247	−	0.526235i
$521$	4.00000			0.175243			0.0876216	−	0.996154i	$-0.472073\pi$
$521$	4.00000			0.175243			0.0876216	+	0.996154i	$0.472073\pi$
$522$		0			0
$523$		−	16.0000i		−	0.699631i	−0.936819	−	0.349816i	$-0.886244\pi$
$523$		−	16.0000i		−	0.699631i	0.936819	−	0.349816i	$-0.113756\pi$
$524$	8.00000			0.349482
$525$		0			0
$526$	24.0000			1.04645
$527$			8.00000i			0.348485i
$528$		0			0
$529$	23.0000			1.00000
$530$	12.0000	−	6.00000i	0.521247	−	0.260623i
$531$		0			0
$532$		0			0
$533$			16.0000i			0.693037i
$534$		0			0
$535$	4.00000	+	8.00000i	0.172935	+	0.345870i
$536$	−24.0000			−1.03664
$537$		0			0
$538$		−	4.00000i		−	0.172452i
$539$		0			0
$540$		0			0
$541$	−30.0000			−1.28980			−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000			−1.28980			−0.644900	+	0.764267i	$0.723101\pi$
$542$		−	28.0000i		−	1.20270i
$543$		0			0
$544$	−10.0000			−0.428746
$545$	−4.00000	+	2.00000i	−0.171341	+	0.0856706i
$546$		0			0
$547$			40.0000i			1.71028i	0.518400	+	0.855138i	$0.326528\pi$
$547$			40.0000i			1.71028i	−0.518400	+	0.855138i	$0.673472\pi$
$548$		−	6.00000i		−	0.256307i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$			8.00000i			0.340195i
$554$	16.0000			0.679775
$555$		0			0
$556$	−8.00000			−0.339276
$557$			2.00000i			0.0847427i	0.999102	+	0.0423714i	$0.0134913\pi$
$557$			2.00000i			0.0847427i	−0.999102	+	0.0423714i	$0.986509\pi$
$558$		0			0
$559$	−32.0000			−1.35346
$560$	−1.00000	−	2.00000i	−0.0422577	−	0.0845154i
$561$		0			0
$562$		−	8.00000i		−	0.337460i
$563$			24.0000i			1.01148i	0.862686	+	0.505740i	$0.168780\pi$
$563$			24.0000i			1.01148i	−0.862686	+	0.505740i	$0.831220\pi$
$564$		0			0
$565$	−18.0000	−	36.0000i	−0.757266	−	1.51453i
$566$	−16.0000			−0.672530
$567$		0			0
$568$			48.0000i			2.01404i
$569$	24.0000			1.00613			0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000			1.00613			0.503066	+	0.864248i	$0.332205\pi$
$570$		0			0
$571$	−12.0000			−0.502184			−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000			−0.502184			−0.251092	+	0.967963i	$0.580790\pi$
$572$		0			0
$573$		0			0
$574$	4.00000			0.166957
$575$		0			0
$576$		0			0
$577$		−	20.0000i		−	0.832611i	−0.909225	−	0.416305i	$-0.863325\pi$
$577$		−	20.0000i		−	0.832611i	0.909225	−	0.416305i	$-0.136675\pi$
$578$		−	13.0000i		−	0.540729i
$579$		0			0
$580$	−16.0000	+	8.00000i	−0.664364	+	0.332182i
$581$	−16.0000			−0.663792
$582$		0			0
$583$		0			0
$584$	36.0000			1.48969
$585$		0			0
$586$	18.0000			0.743573
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$		0			0
$589$		0			0
$590$	8.00000	+	16.0000i	0.329355	+	0.658710i
$591$		0			0
$592$			8.00000i			0.328798i
$593$		−	2.00000i		−	0.0821302i	−0.999156	−	0.0410651i	$-0.986925\pi$
$593$		−	2.00000i		−	0.0821302i	0.999156	−	0.0410651i	$-0.0130751\pi$
$594$		0			0
$595$	4.00000	−	2.00000i	0.163984	−	0.0819920i
$596$	16.0000			0.655386
$597$		0			0
$598$		0			0
$599$	16.0000			0.653742			0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000			0.653742			0.326871	+	0.945069i	$0.394006\pi$
$600$		0			0
$601$	22.0000			0.897399			0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000			0.897399			0.448699	+	0.893683i	$0.351887\pi$
$602$			8.00000i			0.326056i
$603$		0			0
$604$	−16.0000			−0.651031
$605$	−22.0000	+	11.0000i	−0.894427	+	0.447214i
$606$		0			0
$607$		−	24.0000i		−	0.974130i	−0.873366	−	0.487065i	$-0.838067\pi$
$607$		−	24.0000i		−	0.974130i	0.873366	−	0.487065i	$-0.161933\pi$
$608$		0			0
$609$		0			0
$610$	−10.0000	−	20.0000i	−0.404888	−	0.809776i
$611$	−48.0000			−1.94187
$612$		0			0
$613$		−	8.00000i		−	0.323117i	−0.986863	−	0.161558i	$-0.948348\pi$
$613$		−	8.00000i		−	0.323117i	0.986863	−	0.161558i	$-0.0516520\pi$
$614$	−16.0000			−0.645707
$615$		0			0
$616$		0			0
$617$			22.0000i			0.885687i	0.896599	+	0.442843i	$0.146030\pi$
$617$			22.0000i			0.885687i	−0.896599	+	0.442843i	$0.853970\pi$
$618$		0			0
$619$	−40.0000			−1.60774			−0.803868	−	0.594808i	$-0.797228\pi$
$619$	−40.0000			−1.60774			−0.803868	+	0.594808i	$0.797228\pi$
$620$	−8.00000	+	4.00000i	−0.321288	+	0.160644i
$621$		0			0
$622$		0			0
$623$			12.0000i			0.480770i
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$	−12.0000			−0.479616
$627$		0			0
$628$		−	12.0000i		−	0.478852i
$629$	−16.0000			−0.637962
$630$		0			0
$631$	−8.00000			−0.318475			−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000			−0.318475			−0.159237	+	0.987240i	$0.550904\pi$
$632$		−	24.0000i		−	0.954669i
$633$		0			0
$634$	18.0000			0.714871
$635$	−8.00000	−	16.0000i	−0.317470	−	0.634941i
$636$		0			0
$637$		−	4.00000i		−	0.158486i
$638$		0			0
$639$		0			0
$640$	−3.00000	−	6.00000i	−0.118585	−	0.237171i
$641$	8.00000			0.315981			0.157991	−	0.987441i	$-0.449498\pi$
$641$	8.00000			0.315981			0.157991	+	0.987441i	$0.449498\pi$
$642$		0			0
$643$			16.0000i			0.630978i	0.948929	+	0.315489i	$0.102169\pi$
$643$			16.0000i			0.630978i	−0.948929	+	0.315489i	$0.897831\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			28.0000i			1.10079i	0.834903	+	0.550397i	$0.185524\pi$
$647$			28.0000i			1.10079i	−0.834903	+	0.550397i	$0.814476\pi$
$648$		0			0
$649$		0			0
$650$	12.0000	−	16.0000i	0.470679	−	0.627572i
$651$		0			0
$652$		−	8.00000i		−	0.313304i
$653$			46.0000i			1.80012i	0.435767	+	0.900060i	$0.356477\pi$
$653$			46.0000i			1.80012i	−0.435767	+	0.900060i	$0.643523\pi$
$654$		0			0
$655$	16.0000	−	8.00000i	0.625172	−	0.312586i
$656$	−4.00000			−0.156174
$657$		0			0
$658$			12.0000i			0.467809i
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	−30.0000			−1.16686			−0.583432	−	0.812162i	$-0.698291\pi$
$661$	−30.0000			−1.16686			−0.583432	+	0.812162i	$0.698291\pi$
$662$			12.0000i			0.466393i
$663$		0			0
$664$	48.0000			1.86276
$665$		0			0
$666$		0			0
$667$		0			0
$668$		−	12.0000i		−	0.464294i
$669$		0			0
$670$	−16.0000	+	8.00000i	−0.618134	+	0.309067i
$671$		0			0
$672$		0			0
$673$			24.0000i			0.925132i	0.886585	+	0.462566i	$0.153071\pi$
$673$			24.0000i			0.925132i	−0.886585	+	0.462566i	$0.846929\pi$
$674$	−24.0000			−0.924445
$675$		0			0
$676$	−3.00000			−0.115385
$677$			50.0000i			1.92166i	0.277145	+	0.960828i	$0.410612\pi$
$677$			50.0000i			1.92166i	−0.277145	+	0.960828i	$0.589388\pi$
$678$		0			0
$679$	4.00000			0.153506
$680$	−12.0000	+	6.00000i	−0.460179	+	0.230089i
$681$		0			0
$682$		0			0
$683$			20.0000i			0.765279i	0.923898	+	0.382639i	$0.124985\pi$
$683$			20.0000i			0.765279i	−0.923898	+	0.382639i	$0.875015\pi$
$684$		0			0
$685$	−6.00000	−	12.0000i	−0.229248	−	0.458496i
$686$	−1.00000			−0.0381802
$687$		0			0
$688$		−	8.00000i		−	0.304997i
$689$	−24.0000			−0.914327
$690$		0			0
$691$	40.0000			1.52167			0.760836	−	0.648944i	$-0.224789\pi$
$691$	40.0000			1.52167			0.760836	+	0.648944i	$0.224789\pi$
$692$		−	6.00000i		−	0.228086i
$693$		0			0
$694$	12.0000			0.455514
$695$	−16.0000	+	8.00000i	−0.606915	+	0.303457i
$696$		0			0
$697$		−	8.00000i		−	0.303022i
$698$			10.0000i			0.378506i
$699$		0			0
$700$	4.00000	+	3.00000i	0.151186	+	0.113389i
$701$	−8.00000			−0.302156			−0.151078	−	0.988522i	$-0.548274\pi$
$701$	−8.00000			−0.302156			−0.151078	+	0.988522i	$0.548274\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−14.0000			−0.526897
$707$		−	12.0000i		−	0.451306i
$708$		0			0
$709$	10.0000			0.375558			0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000			0.375558			0.187779	+	0.982211i	$0.439871\pi$
$710$	16.0000	+	32.0000i	0.600469	+	1.20094i
$711$		0			0
$712$		−	36.0000i		−	1.34916i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	16.0000			0.597948
$717$		0			0
$718$			32.0000i			1.19423i
$719$	48.0000			1.79010			0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000			1.79010			0.895049	+	0.445968i	$0.147140\pi$
$720$		0			0
$721$	−8.00000			−0.297936
$722$			19.0000i			0.707107i
$723$		0			0
$724$	18.0000			0.668965
$725$	−24.0000	+	32.0000i	−0.891338	+	1.18845i
$726$		0			0
$727$			40.0000i			1.48352i	0.670667	+	0.741759i	$0.266008\pi$
$727$			40.0000i			1.48352i	−0.670667	+	0.741759i	$0.733992\pi$
$728$			12.0000i			0.444750i
$729$		0			0
$730$	24.0000	−	12.0000i	0.888280	−	0.444140i
$731$	16.0000			0.591781
$732$		0			0
$733$			52.0000i			1.92066i	0.278859	+	0.960332i	$0.410044\pi$
$733$			52.0000i			1.92066i	−0.278859	+	0.960332i	$0.589956\pi$
$734$	24.0000			0.885856
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−4.00000			−0.147142			−0.0735712	−	0.997290i	$-0.523440\pi$
$739$	−4.00000			−0.147142			−0.0735712	+	0.997290i	$0.523440\pi$
$740$	−8.00000	−	16.0000i	−0.294086	−	0.588172i
$741$		0			0
$742$			6.00000i			0.220267i
$743$			8.00000i			0.293492i	0.989174	+	0.146746i	$0.0468799\pi$
$743$			8.00000i			0.293492i	−0.989174	+	0.146746i	$0.953120\pi$
$744$		0			0
$745$	32.0000	−	16.0000i	1.17239	−	0.586195i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−4.00000			−0.146157
$750$		0			0
$751$	−24.0000			−0.875772			−0.437886	−	0.899030i	$-0.644273\pi$
$751$	−24.0000			−0.875772			−0.437886	+	0.899030i	$0.644273\pi$
$752$		−	12.0000i		−	0.437595i
$753$		0			0
$754$	−32.0000			−1.16537
$755$	−32.0000	+	16.0000i	−1.16460	+	0.582300i
$756$		0			0
$757$		−	8.00000i		−	0.290765i	−0.989376	−	0.145382i	$-0.953559\pi$
$757$		−	8.00000i		−	0.290765i	0.989376	−	0.145382i	$-0.0464413\pi$
$758$		−	4.00000i		−	0.145287i
$759$		0			0
$760$		0			0
$761$	−20.0000			−0.724999			−0.362500	−	0.931984i	$-0.618077\pi$
$761$	−20.0000			−0.724999			−0.362500	+	0.931984i	$0.618077\pi$
$762$		0			0
$763$		−	2.00000i		−	0.0724049i
$764$		0			0
$765$		0			0
$766$	−4.00000			−0.144526
$767$		−	32.0000i		−	1.15545i
$768$		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
$772$			16.0000i			0.575853i
$773$		−	18.0000i		−	0.647415i	−0.946157	−	0.323708i	$-0.895071\pi$
$773$		−	18.0000i		−	0.647415i	0.946157	−	0.323708i	$-0.104929\pi$
$774$		0			0
$775$	−12.0000	+	16.0000i	−0.431053	+	0.574737i
$776$	−12.0000			−0.430775
$777$		0			0
$778$		−	8.00000i		−	0.286814i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	1.00000			0.0357143
$785$	−12.0000	−	24.0000i	−0.428298	−	0.856597i
$786$		0			0
$787$		−	32.0000i		−	1.14068i	−0.821410	−	0.570338i	$-0.806812\pi$
$787$		−	32.0000i		−	1.14068i	0.821410	−	0.570338i	$-0.193188\pi$
$788$		−	10.0000i		−	0.356235i
$789$		0			0
$790$	−8.00000	−	16.0000i	−0.284627	−	0.569254i
$791$	18.0000			0.640006
$792$		0			0
$793$			40.0000i			1.42044i
$794$	−4.00000			−0.141955
$795$		0			0
$796$	4.00000			0.141776
$797$			38.0000i			1.34603i	0.739629	+	0.673015i	$0.235001\pi$
$797$			38.0000i			1.34603i	−0.739629	+	0.673015i	$0.764999\pi$
$798$		0			0
$799$	24.0000			0.849059
$800$	−20.0000	−	15.0000i	−0.707107	−	0.530330i
$801$		0			0
$802$		−	24.0000i		−	0.847469i
$803$		0			0
$804$		0			0
$805$		0			0
$806$	−16.0000			−0.563576
$807$		0			0
$808$			36.0000i			1.26648i
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$	16.0000			0.561836			0.280918	−	0.959732i	$-0.409361\pi$
$811$	16.0000			0.561836			0.280918	+	0.959732i	$0.409361\pi$
$812$		−	8.00000i		−	0.280745i
$813$		0			0
$814$		0			0
$815$	−8.00000	−	16.0000i	−0.280228	−	0.560456i
$816$		0			0
$817$		0			0
$818$		−	10.0000i		−	0.349642i
$819$		0			0
$820$	8.00000	−	4.00000i	0.279372	−	0.139686i
$821$	32.0000			1.11681			0.558404	−	0.829569i	$-0.311414\pi$
$821$	32.0000			1.11681			0.558404	+	0.829569i	$0.311414\pi$
$822$		0			0
$823$			8.00000i			0.278862i	0.990232	+	0.139431i	$0.0445274\pi$
$823$			8.00000i			0.278862i	−0.990232	+	0.139431i	$0.955473\pi$
$824$	24.0000			0.836080
$825$		0			0
$826$	−8.00000			−0.278356
$827$		−	36.0000i		−	1.25184i	−0.779886	−	0.625921i	$-0.784723\pi$
$827$		−	36.0000i		−	1.25184i	0.779886	−	0.625921i	$-0.215277\pi$
$828$		0			0
$829$	6.00000			0.208389			0.104194	−	0.994557i	$-0.466774\pi$
$829$	6.00000			0.208389			0.104194	+	0.994557i	$0.466774\pi$
$830$	32.0000	−	16.0000i	1.11074	−	0.555368i
$831$		0			0
$832$		−	28.0000i		−	0.970725i
$833$			2.00000i			0.0692959i
$834$		0			0
$835$	−12.0000	−	24.0000i	−0.415277	−	0.830554i
$836$		0			0
$837$		0			0
$838$			24.0000i			0.829066i
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	35.0000			1.20690
$842$		−	6.00000i		−	0.206774i
$843$		0			0
$844$	20.0000			0.688428
$845$	−6.00000	+	3.00000i	−0.206406	+	0.103203i
$846$		0			0
$847$		−	11.0000i		−	0.377964i
$848$		−	6.00000i		−	0.206041i
$849$		0			0
$850$	−6.00000	+	8.00000i	−0.205798	+	0.274398i
$851$		0			0
$852$		0			0
$853$			36.0000i			1.23262i	0.787505	+	0.616308i	$0.211372\pi$
$853$			36.0000i			1.23262i	−0.787505	+	0.616308i	$0.788628\pi$
$854$	10.0000			0.342193
$855$		0			0
$856$	12.0000			0.410152
$857$			6.00000i			0.204956i	0.994735	+	0.102478i	$0.0326771\pi$
$857$			6.00000i			0.204956i	−0.994735	+	0.102478i	$0.967323\pi$
$858$		0			0
$859$	8.00000			0.272956			0.136478	−	0.990643i	$-0.456422\pi$
$859$	8.00000			0.272956			0.136478	+	0.990643i	$0.456422\pi$
$860$	8.00000	+	16.0000i	0.272798	+	0.545595i
$861$		0			0
$862$		−	32.0000i		−	1.08992i
$863$		−	24.0000i		−	0.816970i	−0.912765	−	0.408485i	$-0.866057\pi$
$863$		−	24.0000i		−	0.816970i	0.912765	−	0.408485i	$-0.133943\pi$
$864$		0			0
$865$	−6.00000	−	12.0000i	−0.204006	−	0.408012i
$866$	20.0000			0.679628
$867$		0			0
$868$		−	4.00000i		−	0.135769i
$869$		0			0
$870$		0			0
$871$	32.0000			1.08428
$872$			6.00000i			0.203186i
$873$		0			0
$874$		0			0
$875$	11.0000	+	2.00000i	0.371868	+	0.0676123i
$876$		0			0
$877$			40.0000i			1.35070i	0.737496	+	0.675352i	$0.236008\pi$
$877$			40.0000i			1.35070i	−0.737496	+	0.675352i	$0.763992\pi$
$878$			12.0000i			0.404980i
$879$		0			0
$880$		0			0
$881$	28.0000			0.943344			0.471672	−	0.881774i	$-0.343651\pi$
$881$	28.0000			0.943344			0.471672	+	0.881774i	$0.343651\pi$
$882$		0			0
$883$		−	24.0000i		−	0.807664i	−0.914833	−	0.403832i	$-0.867678\pi$
$883$		−	24.0000i		−	0.807664i	0.914833	−	0.403832i	$-0.132322\pi$
$884$	8.00000			0.269069
$885$		0			0
$886$	−20.0000			−0.671913
$887$		−	4.00000i		−	0.134307i	−0.997743	−	0.0671534i	$-0.978608\pi$
$887$		−	4.00000i		−	0.134307i	0.997743	−	0.0671534i	$-0.0213917\pi$
$888$		0			0
$889$	8.00000			0.268311
$890$	−12.0000	−	24.0000i	−0.402241	−	0.804482i
$891$		0			0
$892$		−	24.0000i		−	0.803579i
$893$		0			0
$894$		0			0
$895$	32.0000	−	16.0000i	1.06964	−	0.534821i
$896$	3.00000			0.100223
$897$		0			0
$898$		−	16.0000i		−	0.533927i
$899$	32.0000			1.06726
$900$		0			0
$901$	12.0000			0.399778
$902$		0			0
$903$		0			0
$904$	−54.0000			−1.79601
$905$	36.0000	−	18.0000i	1.19668	−	0.598340i
$906$		0			0
$907$			8.00000i			0.265636i	0.991140	+	0.132818i	$0.0424025\pi$
$907$			8.00000i			0.265636i	−0.991140	+	0.132818i	$0.957597\pi$
$908$			8.00000i			0.265489i
$909$		0			0
$910$	4.00000	+	8.00000i	0.132599	+	0.265197i
$911$	−48.0000			−1.59031			−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000			−1.59031			−0.795155	+	0.606406i	$0.792611\pi$
$912$		0			0
$913$		0			0
$914$	−16.0000			−0.529233
$915$		0			0
$916$	−14.0000			−0.462573
$917$			8.00000i			0.264183i
$918$		0			0
$919$	8.00000			0.263896			0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000			0.263896			0.131948	+	0.991257i	$0.457877\pi$
$920$		0			0
$921$		0			0
$922$			12.0000i			0.395199i
$923$		−	64.0000i		−	2.10659i
$924$		0			0
$925$	−32.0000	−	24.0000i	−1.05215	−	0.789115i
$926$	−24.0000			−0.788689
$927$		0			0
$928$			40.0000i			1.31306i
$929$	4.00000			0.131236			0.0656179	−	0.997845i	$-0.479098\pi$
$929$	4.00000			0.131236			0.0656179	+	0.997845i	$0.479098\pi$
$930$		0			0
$931$		0			0
$932$		−	22.0000i		−	0.720634i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	44.0000i		−	1.43742i	−0.695311	−	0.718709i	$-0.744734\pi$
$937$		−	44.0000i		−	1.43742i	0.695311	−	0.718709i	$-0.255266\pi$
$938$		−	8.00000i		−	0.261209i
$939$		0			0
$940$	12.0000	+	24.0000i	0.391397	+	0.782794i
$941$	−36.0000			−1.17357			−0.586783	−	0.809744i	$-0.699606\pi$
$941$	−36.0000			−1.17357			−0.586783	+	0.809744i	$0.699606\pi$
$942$		0			0
$943$		0			0
$944$	8.00000			0.260378
$945$		0			0
$946$		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
$949$	−48.0000			−1.55815
$950$		0			0
$951$		0			0
$952$		−	6.00000i		−	0.194461i
$953$			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
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	6.00000			0.193750
$960$		0			0
$961$	−15.0000			−0.483871
$962$		−	32.0000i		−	1.03172i
$963$		0			0
$964$	−14.0000			−0.450910
$965$	16.0000	+	32.0000i	0.515058	+	1.03012i
$966$		0			0
$967$			8.00000i			0.257263i	0.991692	+	0.128631i	$0.0410584\pi$
$967$			8.00000i			0.257263i	−0.991692	+	0.128631i	$0.958942\pi$
$968$			33.0000i			1.06066i
$969$		0			0
$970$	−8.00000	+	4.00000i	−0.256865	+	0.128432i
$971$	56.0000			1.79713			0.898563	−	0.438845i	$-0.144612\pi$
$971$	56.0000			1.79713			0.898563	+	0.438845i	$0.144612\pi$
$972$		0			0
$973$		−	8.00000i		−	0.256468i
$974$	−24.0000			−0.769010
$975$		0			0
$976$	−10.0000			−0.320092
$977$		−	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
$979$		0			0
$980$	−2.00000	+	1.00000i	−0.0638877	+	0.0319438i
$981$		0			0
$982$		−	32.0000i		−	1.02116i
$983$		−	36.0000i		−	1.14822i	−0.818778	−	0.574111i	$-0.805348\pi$
$983$		−	36.0000i		−	1.14822i	0.818778	−	0.574111i	$-0.194652\pi$
$984$		0			0
$985$	−10.0000	−	20.0000i	−0.318626	−	0.637253i
$986$	16.0000			0.509544
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−32.0000			−1.01651			−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000			−1.01651			−0.508257	+	0.861206i	$0.669710\pi$
$992$			20.0000i			0.635001i
$993$		0			0
$994$	−16.0000			−0.507489
$995$	8.00000	−	4.00000i	0.253617	−	0.126809i
$996$		0			0
$997$			28.0000i			0.886769i	0.896332	+	0.443384i	$0.146222\pi$
$997$			28.0000i			0.886769i	−0.896332	+	0.443384i	$0.853778\pi$
$998$		−	20.0000i		−	0.633089i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.d.d.64.1	yes	2
3.2	odd	2		315.2.d.b.64.2	yes	2
4.3	odd	2		5040.2.t.r.1009.1		2
5.2	odd	4		1575.2.a.g.1.1		1
5.3	odd	4		1575.2.a.d.1.1		1
5.4	even	2	inner	315.2.d.d.64.2	yes	2
7.6	odd	2		2205.2.d.c.1324.1		2
12.11	even	2		5040.2.t.c.1009.2		2
15.2	even	4		1575.2.a.b.1.1		1
15.8	even	4		1575.2.a.j.1.1		1
15.14	odd	2		315.2.d.b.64.1	✓	2
20.19	odd	2		5040.2.t.r.1009.2		2
21.20	even	2		2205.2.d.g.1324.2		2
35.34	odd	2		2205.2.d.c.1324.2		2
60.59	even	2		5040.2.t.c.1009.1		2
105.104	even	2		2205.2.d.g.1324.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.d.b.64.1	✓	2	15.14	odd	2
315.2.d.b.64.2	yes	2	3.2	odd	2
315.2.d.d.64.1	yes	2	1.1	even	1	trivial
315.2.d.d.64.2	yes	2	5.4	even	2	inner
1575.2.a.b.1.1		1	15.2	even	4
1575.2.a.d.1.1		1	5.3	odd	4
1575.2.a.g.1.1		1	5.2	odd	4
1575.2.a.j.1.1		1	15.8	even	4
2205.2.d.c.1324.1		2	7.6	odd	2
2205.2.d.c.1324.2		2	35.34	odd	2
2205.2.d.g.1324.1		2	105.104	even	2
2205.2.d.g.1324.2		2	21.20	even	2
5040.2.t.c.1009.1		2	60.59	even	2
5040.2.t.c.1009.2		2	12.11	even	2
5040.2.t.r.1009.1		2	4.3	odd	2
5040.2.t.r.1009.2		2	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	315.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000 q^{4} +(2.00000 - 1.00000i) q^{5} +1.00000i q^{7} -3.00000i q^{8} +(-1.00000 - 2.00000i) q^{10} +4.00000i q^{13} +1.00000 q^{14} -1.00000 q^{16} -2.00000i q^{17} +(2.00000 - 1.00000i) q^{20} +(3.00000 - 4.00000i) q^{25} +4.00000 q^{26} +1.00000i q^{28} -8.00000 q^{29} -4.00000 q^{31} -5.00000i q^{32} -2.00000 q^{34} +(1.00000 + 2.00000i) q^{35} -8.00000i q^{37} +(-3.00000 - 6.00000i) q^{40} +4.00000 q^{41} +8.00000i q^{43} +12.0000i q^{47} -1.00000 q^{49} +(-4.00000 - 3.00000i) q^{50} +4.00000i q^{52} +6.00000i q^{53} +3.00000 q^{56} +8.00000i q^{58} -8.00000 q^{59} +10.0000 q^{61} +4.00000i q^{62} -7.00000 q^{64} +(4.00000 + 8.00000i) q^{65} -8.00000i q^{67} -2.00000i q^{68} +(2.00000 - 1.00000i) q^{70} -16.0000 q^{71} +12.0000i q^{73} -8.00000 q^{74} +8.00000 q^{79} +(-2.00000 + 1.00000i) q^{80} -4.00000i q^{82} +16.0000i q^{83} +(-2.00000 - 4.00000i) q^{85} +8.00000 q^{86} +12.0000 q^{89} -4.00000 q^{91} +12.0000 q^{94} -4.00000i q^{97} +1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} + 4 q^{5} - 2 q^{10} + 2 q^{14} - 2 q^{16} + 4 q^{20} + 6 q^{25} + 8 q^{26} - 16 q^{29} - 8 q^{31} - 4 q^{34} + 2 q^{35} - 6 q^{40} + 8 q^{41} - 2 q^{49} - 8 q^{50} + 6 q^{56} - 16 q^{59}+ \cdots + 24 q^{94}+O(q^{100}) \) 2 * q + 2 * q^4 + 4 * q^5 - 2 * q^10 + 2 * q^14 - 2 * q^16 + 4 * q^20 + 6 * q^25 + 8 * q^26 - 16 * q^29 - 8 * q^31 - 4 * q^34 + 2 * q^35 - 6 * q^40 + 8 * q^41 - 2 * q^49 - 8 * q^50 + 6 * q^56 - 16 * q^59 + 20 * q^61 - 14 * q^64 + 8 * q^65 + 4 * q^70 - 32 * q^71 - 16 * q^74 + 16 * q^79 - 4 * q^80 - 4 * q^85 + 16 * q^86 + 24 * q^89 - 8 * q^91 + 24 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more