LMFDB - Embedded newform 1305.2.f.c.289.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1305,2,Mod(289,1305)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1305, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1305.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1305 = 3^{2} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1305.f (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:	$10.4204774638$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 435)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			289.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1305.289
Dual form			1305.2.f.c.289.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{4} +(1.00000 + 2.00000i) q^{5} -2.00000i q^{7} -3.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{4} +(1.00000 + 2.00000i) q^{5} -2.00000i q^{7} -3.00000 q^{8} +(1.00000 + 2.00000i) q^{10} -2.00000i q^{11} -4.00000i q^{13} -2.00000i q^{14} -1.00000 q^{16} +6.00000 q^{17} +2.00000i q^{19} +(-1.00000 - 2.00000i) q^{20} -2.00000i q^{22} -6.00000i q^{23} +(-3.00000 + 4.00000i) q^{25} -4.00000i q^{26} +2.00000i q^{28} +(5.00000 - 2.00000i) q^{29} -2.00000i q^{31} +5.00000 q^{32} +6.00000 q^{34} +(4.00000 - 2.00000i) q^{35} -2.00000 q^{37} +2.00000i q^{38} +(-3.00000 - 6.00000i) q^{40} +4.00000 q^{43} +2.00000i q^{44} -6.00000i q^{46} +8.00000 q^{47} +3.00000 q^{49} +(-3.00000 + 4.00000i) q^{50} +4.00000i q^{52} -12.0000i q^{53} +(4.00000 - 2.00000i) q^{55} +6.00000i q^{56} +(5.00000 - 2.00000i) q^{58} +4.00000 q^{59} +12.0000i q^{61} -2.00000i q^{62} +7.00000 q^{64} +(8.00000 - 4.00000i) q^{65} -6.00000i q^{67} -6.00000 q^{68} +(4.00000 - 2.00000i) q^{70} -8.00000 q^{71} -6.00000 q^{73} -2.00000 q^{74} -2.00000i q^{76} -4.00000 q^{77} -2.00000i q^{79} +(-1.00000 - 2.00000i) q^{80} -2.00000i q^{83} +(6.00000 + 12.0000i) q^{85} +4.00000 q^{86} +6.00000i q^{88} +16.0000i q^{89} -8.00000 q^{91} +6.00000i q^{92} +8.00000 q^{94} +(-4.00000 + 2.00000i) q^{95} -14.0000 q^{97} +3.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 2 q^{4} + 2 q^{5} - 6 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 - 2 * q^4 + 2 * q^5 - 6 * q^8	$ 2 q + 2 q^{2} - 2 q^{4} + 2 q^{5} - 6 q^{8} + 2 q^{10} - 2 q^{16} + 12 q^{17} - 2 q^{20} - 6 q^{25} + 10 q^{29} + 10 q^{32} + 12 q^{34} + 8 q^{35} - 4 q^{37} - 6 q^{40} + 8 q^{43} + 16 q^{47} + 6 q^{49} - 6 q^{50} + 8 q^{55} + 10 q^{58} + 8 q^{59} + 14 q^{64} + 16 q^{65} - 12 q^{68} + 8 q^{70} - 16 q^{71} - 12 q^{73} - 4 q^{74} - 8 q^{77} - 2 q^{80} + 12 q^{85} + 8 q^{86} - 16 q^{91} + 16 q^{94} - 8 q^{95} - 28 q^{97} + 6 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 - 2 * q^4 + 2 * q^5 - 6 * q^8 + 2 * q^10 - 2 * q^16 + 12 * q^17 - 2 * q^20 - 6 * q^25 + 10 * q^29 + 10 * q^32 + 12 * q^34 + 8 * q^35 - 4 * q^37 - 6 * q^40 + 8 * q^43 + 16 * q^47 + 6 * q^49 - 6 * q^50 + 8 * q^55 + 10 * q^58 + 8 * q^59 + 14 * q^64 + 16 * q^65 - 12 * q^68 + 8 * q^70 - 16 * q^71 - 12 * q^73 - 4 * q^74 - 8 * q^77 - 2 * q^80 + 12 * q^85 + 8 * q^86 - 16 * q^91 + 16 * q^94 - 8 * q^95 - 28 * q^97 + 6 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1305\mathbb{Z}\right)^\times$.

$n$	$146$	$262$	$901$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$	1.00000			0.707107			0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000			0.707107			0.353553	+	0.935414i	$0.384973\pi$
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$	1.00000	+	2.00000i	0.447214	+	0.894427i
$5$	1.00000	+	2.00000i	0.447214	+	0.894427i
$6$		0			0
$7$		−	2.00000i		−	0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$		−	2.00000i		−	0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	−3.00000			−1.06066
$9$		0			0
$10$	1.00000	+	2.00000i	0.316228	+	0.632456i
$11$		−	2.00000i		−	0.603023i	−0.953463	−	0.301511i	$-0.902509\pi$
$11$		−	2.00000i		−	0.603023i	0.953463	−	0.301511i	$-0.0974911\pi$
$12$		0			0
$13$		−	4.00000i		−	1.10940i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$		−	4.00000i		−	1.10940i	0.832050	−	0.554700i	$-0.187167\pi$
$14$		−	2.00000i		−	0.534522i
$15$		0			0
$16$	−1.00000			−0.250000
$17$	6.00000			1.45521			0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000			1.45521			0.727607	+	0.685994i	$0.240633\pi$
$18$		0			0
$19$			2.00000i			0.458831i	0.973329	+	0.229416i	$0.0736815\pi$
$19$			2.00000i			0.458831i	−0.973329	+	0.229416i	$0.926318\pi$
$20$	−1.00000	−	2.00000i	−0.223607	−	0.447214i
$21$		0			0
$22$		−	2.00000i		−	0.426401i
$23$		−	6.00000i		−	1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$		−	6.00000i		−	1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$		0			0
$25$	−3.00000	+	4.00000i	−0.600000	+	0.800000i
$26$		−	4.00000i		−	0.784465i
$27$		0			0
$28$			2.00000i			0.377964i
$29$	5.00000	−	2.00000i	0.928477	−	0.371391i
$29$	5.00000	−	2.00000i	0.928477	−	0.371391i
$30$		0			0
$31$		−	2.00000i		−	0.359211i	−0.983739	−	0.179605i	$-0.942518\pi$
$31$		−	2.00000i		−	0.359211i	0.983739	−	0.179605i	$-0.0574821\pi$
$32$	5.00000			0.883883
$33$		0			0
$34$	6.00000			1.02899
$35$	4.00000	−	2.00000i	0.676123	−	0.338062i
$36$		0			0
$37$	−2.00000			−0.328798			−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000			−0.328798			−0.164399	+	0.986394i	$0.552568\pi$
$38$			2.00000i			0.324443i
$39$		0			0
$40$	−3.00000	−	6.00000i	−0.474342	−	0.948683i
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	4.00000			0.609994			0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000			0.609994			0.304997	+	0.952353i	$0.401344\pi$
$44$			2.00000i			0.301511i
$45$		0			0
$46$		−	6.00000i		−	0.884652i
$47$	8.00000			1.16692			0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000			1.16692			0.583460	+	0.812142i	$0.301699\pi$
$48$		0			0
$49$	3.00000			0.428571
$50$	−3.00000	+	4.00000i	−0.424264	+	0.565685i
$51$		0			0
$52$			4.00000i			0.554700i
$53$		−	12.0000i		−	1.64833i	−0.566352	−	0.824163i	$-0.691646\pi$
$53$		−	12.0000i		−	1.64833i	0.566352	−	0.824163i	$-0.308354\pi$
$54$		0			0
$55$	4.00000	−	2.00000i	0.539360	−	0.269680i
$56$			6.00000i			0.801784i
$57$		0			0
$58$	5.00000	−	2.00000i	0.656532	−	0.262613i
$59$	4.00000			0.520756			0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000			0.520756			0.260378	+	0.965507i	$0.416153\pi$
$60$		0			0
$61$			12.0000i			1.53644i	0.640184	+	0.768221i	$0.278858\pi$
$61$			12.0000i			1.53644i	−0.640184	+	0.768221i	$0.721142\pi$
$62$		−	2.00000i		−	0.254000i
$63$		0			0
$64$	7.00000			0.875000
$65$	8.00000	−	4.00000i	0.992278	−	0.496139i
$66$		0			0
$67$		−	6.00000i		−	0.733017i	−0.930415	−	0.366508i	$-0.880553\pi$
$67$		−	6.00000i		−	0.733017i	0.930415	−	0.366508i	$-0.119447\pi$
$68$	−6.00000			−0.727607
$69$		0			0
$70$	4.00000	−	2.00000i	0.478091	−	0.239046i
$71$	−8.00000			−0.949425			−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000			−0.949425			−0.474713	+	0.880141i	$0.657448\pi$
$72$		0			0
$73$	−6.00000			−0.702247			−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000			−0.702247			−0.351123	+	0.936329i	$0.614200\pi$
$74$	−2.00000			−0.232495
$75$		0			0
$76$		−	2.00000i		−	0.229416i
$77$	−4.00000			−0.455842
$78$		0			0
$79$		−	2.00000i		−	0.225018i	−0.993651	−	0.112509i	$-0.964111\pi$
$79$		−	2.00000i		−	0.225018i	0.993651	−	0.112509i	$-0.0358886\pi$
$80$	−1.00000	−	2.00000i	−0.111803	−	0.223607i
$81$		0			0
$82$		0			0
$83$		−	2.00000i		−	0.219529i	−0.993958	−	0.109764i	$-0.964990\pi$
$83$		−	2.00000i		−	0.219529i	0.993958	−	0.109764i	$-0.0350096\pi$
$84$		0			0
$85$	6.00000	+	12.0000i	0.650791	+	1.30158i
$86$	4.00000			0.431331
$87$		0			0
$88$			6.00000i			0.639602i
$89$			16.0000i			1.69600i	0.529999	+	0.847998i	$0.322192\pi$
$89$			16.0000i			1.69600i	−0.529999	+	0.847998i	$0.677808\pi$
$90$		0			0
$91$	−8.00000			−0.838628
$92$			6.00000i			0.625543i
$93$		0			0
$94$	8.00000			0.825137
$95$	−4.00000	+	2.00000i	−0.410391	+	0.205196i
$96$		0			0
$97$	−14.0000			−1.42148			−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000			−1.42148			−0.710742	+	0.703452i	$0.751641\pi$
$98$	3.00000			0.303046
$99$		0			0
$100$	3.00000	−	4.00000i	0.300000	−	0.400000i
$101$		−	12.0000i		−	1.19404i	−0.802225	−	0.597022i	$-0.796350\pi$
$101$		−	12.0000i		−	1.19404i	0.802225	−	0.597022i	$-0.203650\pi$
$102$		0			0
$103$		−	18.0000i		−	1.77359i	−0.462160	−	0.886796i	$-0.652926\pi$
$103$		−	18.0000i		−	1.77359i	0.462160	−	0.886796i	$-0.347074\pi$
$104$			12.0000i			1.17670i
$105$		0			0
$106$		−	12.0000i		−	1.16554i
$107$		−	10.0000i		−	0.966736i	−0.875417	−	0.483368i	$-0.839413\pi$
$107$		−	10.0000i		−	0.966736i	0.875417	−	0.483368i	$-0.160587\pi$
$108$		0			0
$109$	−10.0000			−0.957826			−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000			−0.957826			−0.478913	+	0.877862i	$0.658969\pi$
$110$	4.00000	−	2.00000i	0.381385	−	0.190693i
$111$		0			0
$112$			2.00000i			0.188982i
$113$	−2.00000			−0.188144			−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000			−0.188144			−0.0940721	+	0.995565i	$0.529988\pi$
$114$		0			0
$115$	12.0000	−	6.00000i	1.11901	−	0.559503i
$116$	−5.00000	+	2.00000i	−0.464238	+	0.185695i
$117$		0			0
$118$	4.00000			0.368230
$119$		−	12.0000i		−	1.10004i
$120$		0			0
$121$	7.00000			0.636364
$122$			12.0000i			1.08643i
$123$		0			0
$124$			2.00000i			0.179605i
$125$	−11.0000	−	2.00000i	−0.983870	−	0.178885i
$126$		0			0
$127$	8.00000			0.709885			0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000			0.709885			0.354943	+	0.934888i	$0.384500\pi$
$128$	−3.00000			−0.265165
$129$		0			0
$130$	8.00000	−	4.00000i	0.701646	−	0.350823i
$131$			22.0000i			1.92215i	0.276289	+	0.961074i	$0.410895\pi$
$131$			22.0000i			1.92215i	−0.276289	+	0.961074i	$0.589105\pi$
$132$		0			0
$133$	4.00000			0.346844
$134$		−	6.00000i		−	0.518321i
$135$		0			0
$136$	−18.0000			−1.54349
$137$	14.0000			1.19610			0.598050	−	0.801459i	$-0.295942\pi$
$137$	14.0000			1.19610			0.598050	+	0.801459i	$0.295942\pi$
$138$		0			0
$139$	−4.00000			−0.339276			−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000			−0.339276			−0.169638	+	0.985506i	$0.554260\pi$
$140$	−4.00000	+	2.00000i	−0.338062	+	0.169031i
$141$		0			0
$142$	−8.00000			−0.671345
$143$	−8.00000			−0.668994
$144$		0			0
$145$	9.00000	+	8.00000i	0.747409	+	0.664364i
$146$	−6.00000			−0.496564
$147$		0			0
$148$	2.00000			0.164399
$149$	10.0000			0.819232			0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000			0.819232			0.409616	+	0.912258i	$0.365663\pi$
$150$		0			0
$151$	−8.00000			−0.651031			−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000			−0.651031			−0.325515	+	0.945537i	$0.605538\pi$
$152$		−	6.00000i		−	0.486664i
$153$		0			0
$154$	−4.00000			−0.322329
$155$	4.00000	−	2.00000i	0.321288	−	0.160644i
$156$		0			0
$157$	−2.00000			−0.159617			−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000			−0.159617			−0.0798087	+	0.996810i	$0.525431\pi$
$158$		−	2.00000i		−	0.159111i
$159$		0			0
$160$	5.00000	+	10.0000i	0.395285	+	0.790569i
$161$	−12.0000			−0.945732
$162$		0			0
$163$	−20.0000			−1.56652			−0.783260	−	0.621694i	$-0.786445\pi$
$163$	−20.0000			−1.56652			−0.783260	+	0.621694i	$0.786445\pi$
$164$		0			0
$165$		0			0
$166$		−	2.00000i		−	0.155230i
$167$		−	6.00000i		−	0.464294i	−0.972681	−	0.232147i	$-0.925425\pi$
$167$		−	6.00000i		−	0.464294i	0.972681	−	0.232147i	$-0.0745750\pi$
$168$		0			0
$169$	−3.00000			−0.230769
$170$	6.00000	+	12.0000i	0.460179	+	0.920358i
$171$		0			0
$172$	−4.00000			−0.304997
$173$			12.0000i			0.912343i	0.889892	+	0.456172i	$0.150780\pi$
$173$			12.0000i			0.912343i	−0.889892	+	0.456172i	$0.849220\pi$
$174$		0			0
$175$	8.00000	+	6.00000i	0.604743	+	0.453557i
$176$			2.00000i			0.150756i
$177$		0			0
$178$			16.0000i			1.19925i
$179$	12.0000			0.896922			0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000			0.896922			0.448461	+	0.893802i	$0.351972\pi$
$180$		0			0
$181$	−10.0000			−0.743294			−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000			−0.743294			−0.371647	+	0.928374i	$0.621207\pi$
$182$	−8.00000			−0.592999
$183$		0			0
$184$			18.0000i			1.32698i
$185$	−2.00000	−	4.00000i	−0.147043	−	0.294086i
$186$		0			0
$187$		−	12.0000i		−	0.877527i
$188$	−8.00000			−0.583460
$189$		0			0
$190$	−4.00000	+	2.00000i	−0.290191	+	0.145095i
$191$		−	6.00000i		−	0.434145i	−0.976156	−	0.217072i	$-0.930349\pi$
$191$		−	6.00000i		−	0.434145i	0.976156	−	0.217072i	$-0.0696508\pi$
$192$		0			0
$193$	−6.00000			−0.431889			−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000			−0.431889			−0.215945	+	0.976406i	$0.569283\pi$
$194$	−14.0000			−1.00514
$195$		0			0
$196$	−3.00000			−0.214286
$197$			4.00000i			0.284988i	0.989796	+	0.142494i	$0.0455122\pi$
$197$			4.00000i			0.284988i	−0.989796	+	0.142494i	$0.954488\pi$
$198$		0			0
$199$	8.00000			0.567105			0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000			0.567105			0.283552	+	0.958957i	$0.408487\pi$
$200$	9.00000	−	12.0000i	0.636396	−	0.848528i
$201$		0			0
$202$		−	12.0000i		−	0.844317i
$203$	−4.00000	−	10.0000i	−0.280745	−	0.701862i
$204$		0			0
$205$		0			0
$206$		−	18.0000i		−	1.25412i
$207$		0			0
$208$			4.00000i			0.277350i
$209$	4.00000			0.276686
$210$		0			0
$211$			2.00000i			0.137686i	0.997628	+	0.0688428i	$0.0219307\pi$
$211$			2.00000i			0.137686i	−0.997628	+	0.0688428i	$0.978069\pi$
$212$			12.0000i			0.824163i
$213$		0			0
$214$		−	10.0000i		−	0.683586i
$215$	4.00000	+	8.00000i	0.272798	+	0.545595i
$216$		0			0
$217$	−4.00000			−0.271538
$218$	−10.0000			−0.677285
$219$		0			0
$220$	−4.00000	+	2.00000i	−0.269680	+	0.134840i
$221$		−	24.0000i		−	1.61441i
$222$		0			0
$223$			22.0000i			1.47323i	0.676313	+	0.736614i	$0.263577\pi$
$223$			22.0000i			1.47323i	−0.676313	+	0.736614i	$0.736423\pi$
$224$		−	10.0000i		−	0.668153i
$225$		0			0
$226$	−2.00000			−0.133038
$227$		−	2.00000i		−	0.132745i	−0.997795	−	0.0663723i	$-0.978857\pi$
$227$		−	2.00000i		−	0.132745i	0.997795	−	0.0663723i	$-0.0211425\pi$
$228$		0			0
$229$			20.0000i			1.32164i	0.750546	+	0.660819i	$0.229791\pi$
$229$			20.0000i			1.32164i	−0.750546	+	0.660819i	$0.770209\pi$
$230$	12.0000	−	6.00000i	0.791257	−	0.395628i
$231$		0			0
$232$	−15.0000	+	6.00000i	−0.984798	+	0.393919i
$233$			16.0000i			1.04819i	0.851658	+	0.524097i	$0.175597\pi$
$233$			16.0000i			1.04819i	−0.851658	+	0.524097i	$0.824403\pi$
$234$		0			0
$235$	8.00000	+	16.0000i	0.521862	+	1.04372i
$236$	−4.00000			−0.260378
$237$		0			0
$238$		−	12.0000i		−	0.777844i
$239$	−16.0000			−1.03495			−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000			−1.03495			−0.517477	+	0.855697i	$0.673129\pi$
$240$		0			0
$241$	−22.0000			−1.41714			−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000			−1.41714			−0.708572	+	0.705638i	$0.750660\pi$
$242$	7.00000			0.449977
$243$		0			0
$244$		−	12.0000i		−	0.768221i
$245$	3.00000	+	6.00000i	0.191663	+	0.383326i
$246$		0			0
$247$	8.00000			0.509028
$248$			6.00000i			0.381000i
$249$		0			0
$250$	−11.0000	−	2.00000i	−0.695701	−	0.126491i
$251$			30.0000i			1.89358i	0.321847	+	0.946792i	$0.395696\pi$
$251$			30.0000i			1.89358i	−0.321847	+	0.946792i	$0.604304\pi$
$252$		0			0
$253$	−12.0000			−0.754434
$254$	8.00000			0.501965
$255$		0			0
$256$	−17.0000			−1.06250
$257$			24.0000i			1.49708i	0.663090	+	0.748539i	$0.269245\pi$
$257$			24.0000i			1.49708i	−0.663090	+	0.748539i	$0.730755\pi$
$258$		0			0
$259$			4.00000i			0.248548i
$260$	−8.00000	+	4.00000i	−0.496139	+	0.248069i
$261$		0			0
$262$			22.0000i			1.35916i
$263$	−16.0000			−0.986602			−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000			−0.986602			−0.493301	+	0.869859i	$0.664210\pi$
$264$		0			0
$265$	24.0000	−	12.0000i	1.47431	−	0.737154i
$266$	4.00000			0.245256
$267$		0			0
$268$			6.00000i			0.366508i
$269$		−	4.00000i		−	0.243884i	−0.992537	−	0.121942i	$-0.961088\pi$
$269$		−	4.00000i		−	0.243884i	0.992537	−	0.121942i	$-0.0389122\pi$
$270$		0			0
$271$		−	2.00000i		−	0.121491i	−0.998153	−	0.0607457i	$-0.980652\pi$
$271$		−	2.00000i		−	0.121491i	0.998153	−	0.0607457i	$-0.0193479\pi$
$272$	−6.00000			−0.363803
$273$		0			0
$274$	14.0000			0.845771
$275$	8.00000	+	6.00000i	0.482418	+	0.361814i
$276$		0			0
$277$			4.00000i			0.240337i	0.992754	+	0.120168i	$0.0383434\pi$
$277$			4.00000i			0.240337i	−0.992754	+	0.120168i	$0.961657\pi$
$278$	−4.00000			−0.239904
$279$		0			0
$280$	−12.0000	+	6.00000i	−0.717137	+	0.358569i
$281$	30.0000			1.78965			0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000			1.78965			0.894825	+	0.446417i	$0.147300\pi$
$282$		0			0
$283$			2.00000i			0.118888i	0.998232	+	0.0594438i	$0.0189327\pi$
$283$			2.00000i			0.118888i	−0.998232	+	0.0594438i	$0.981067\pi$
$284$	8.00000			0.474713
$285$		0			0
$286$	−8.00000			−0.473050
$287$		0			0
$288$		0			0
$289$	19.0000			1.11765
$290$	9.00000	+	8.00000i	0.528498	+	0.469776i
$291$		0			0
$292$	6.00000			0.351123
$293$	26.0000			1.51894			0.759468	−	0.650545i	$-0.225459\pi$
$293$	26.0000			1.51894			0.759468	+	0.650545i	$0.225459\pi$
$294$		0			0
$295$	4.00000	+	8.00000i	0.232889	+	0.465778i
$296$	6.00000			0.348743
$297$		0			0
$298$	10.0000			0.579284
$299$	−24.0000			−1.38796
$300$		0			0
$301$		−	8.00000i		−	0.461112i
$302$	−8.00000			−0.460348
$303$		0			0
$304$		−	2.00000i		−	0.114708i
$305$	−24.0000	+	12.0000i	−1.37424	+	0.687118i
$306$		0			0
$307$	12.0000			0.684876			0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000			0.684876			0.342438	+	0.939540i	$0.388747\pi$
$308$	4.00000			0.227921
$309$		0			0
$310$	4.00000	−	2.00000i	0.227185	−	0.113592i
$311$		−	14.0000i		−	0.793867i	−0.917847	−	0.396934i	$-0.870074\pi$
$311$		−	14.0000i		−	0.793867i	0.917847	−	0.396934i	$-0.129926\pi$
$312$		0			0
$313$			8.00000i			0.452187i	0.974106	+	0.226093i	$0.0725954\pi$
$313$			8.00000i			0.452187i	−0.974106	+	0.226093i	$0.927405\pi$
$314$	−2.00000			−0.112867
$315$		0			0
$316$			2.00000i			0.112509i
$317$	−6.00000			−0.336994			−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000			−0.336994			−0.168497	+	0.985702i	$0.553891\pi$
$318$		0			0
$319$	−4.00000	−	10.0000i	−0.223957	−	0.559893i
$320$	7.00000	+	14.0000i	0.391312	+	0.782624i
$321$		0			0
$322$	−12.0000			−0.668734
$323$			12.0000i			0.667698i
$324$		0			0
$325$	16.0000	+	12.0000i	0.887520	+	0.665640i
$326$	−20.0000			−1.10770
$327$		0			0
$328$		0			0
$329$		−	16.0000i		−	0.882109i
$330$		0			0
$331$		−	22.0000i		−	1.20923i	−0.796518	−	0.604615i	$-0.793327\pi$
$331$		−	22.0000i		−	1.20923i	0.796518	−	0.604615i	$-0.206673\pi$
$332$			2.00000i			0.109764i
$333$		0			0
$334$		−	6.00000i		−	0.328305i
$335$	12.0000	−	6.00000i	0.655630	−	0.327815i
$336$		0			0
$337$	34.0000			1.85210			0.926049	−	0.377403i	$-0.123183\pi$
$337$	34.0000			1.85210			0.926049	+	0.377403i	$0.123183\pi$
$338$	−3.00000			−0.163178
$339$		0			0
$340$	−6.00000	−	12.0000i	−0.325396	−	0.650791i
$341$	−4.00000			−0.216612
$342$		0			0
$343$		−	20.0000i		−	1.07990i
$344$	−12.0000			−0.646997
$345$		0			0
$346$			12.0000i			0.645124i
$347$		−	10.0000i		−	0.536828i	−0.963304	−	0.268414i	$-0.913500\pi$
$347$		−	10.0000i		−	0.536828i	0.963304	−	0.268414i	$-0.0864995\pi$
$348$		0			0
$349$	−26.0000			−1.39175			−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000			−1.39175			−0.695874	+	0.718164i	$0.744983\pi$
$350$	8.00000	+	6.00000i	0.427618	+	0.320713i
$351$		0			0
$352$		−	10.0000i		−	0.533002i
$353$			24.0000i			1.27739i	0.769460	+	0.638696i	$0.220526\pi$
$353$			24.0000i			1.27739i	−0.769460	+	0.638696i	$0.779474\pi$
$354$		0			0
$355$	−8.00000	−	16.0000i	−0.424596	−	0.849192i
$356$		−	16.0000i		−	0.847998i
$357$		0			0
$358$	12.0000			0.634220
$359$			18.0000i			0.950004i	0.879985	+	0.475002i	$0.157553\pi$
$359$			18.0000i			0.950004i	−0.879985	+	0.475002i	$0.842447\pi$
$360$		0			0
$361$	15.0000			0.789474
$362$	−10.0000			−0.525588
$363$		0			0
$364$	8.00000			0.419314
$365$	−6.00000	−	12.0000i	−0.314054	−	0.628109i
$366$		0			0
$367$	24.0000			1.25279			0.626395	−	0.779506i	$-0.284530\pi$
$367$	24.0000			1.25279			0.626395	+	0.779506i	$0.284530\pi$
$368$			6.00000i			0.312772i
$369$		0			0
$370$	−2.00000	−	4.00000i	−0.103975	−	0.207950i
$371$	−24.0000			−1.24602
$372$		0			0
$373$			4.00000i			0.207112i	0.994624	+	0.103556i	$0.0330221\pi$
$373$			4.00000i			0.207112i	−0.994624	+	0.103556i	$0.966978\pi$
$374$		−	12.0000i		−	0.620505i
$375$		0			0
$376$	−24.0000			−1.23771
$377$	−8.00000	−	20.0000i	−0.412021	−	1.03005i
$378$		0			0
$379$		−	38.0000i		−	1.95193i	−0.217930	−	0.975964i	$-0.569930\pi$
$379$		−	38.0000i		−	1.95193i	0.217930	−	0.975964i	$-0.430070\pi$
$380$	4.00000	−	2.00000i	0.205196	−	0.102598i
$381$		0			0
$382$		−	6.00000i		−	0.306987i
$383$		−	30.0000i		−	1.53293i	−0.642287	−	0.766464i	$-0.722014\pi$
$383$		−	30.0000i		−	1.53293i	0.642287	−	0.766464i	$-0.277986\pi$
$384$		0			0
$385$	−4.00000	−	8.00000i	−0.203859	−	0.407718i
$386$	−6.00000			−0.305392
$387$		0			0
$388$	14.0000			0.710742
$389$		−	12.0000i		−	0.608424i	−0.952604	−	0.304212i	$-0.901607\pi$
$389$		−	12.0000i		−	0.608424i	0.952604	−	0.304212i	$-0.0983931\pi$
$390$		0			0
$391$		−	36.0000i		−	1.82060i
$392$	−9.00000			−0.454569
$393$		0			0
$394$			4.00000i			0.201517i
$395$	4.00000	−	2.00000i	0.201262	−	0.100631i
$396$		0			0
$397$			28.0000i			1.40528i	0.711546	+	0.702640i	$0.247995\pi$
$397$			28.0000i			1.40528i	−0.711546	+	0.702640i	$0.752005\pi$
$398$	8.00000			0.401004
$399$		0			0
$400$	3.00000	−	4.00000i	0.150000	−	0.200000i
$401$	−18.0000			−0.898877			−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000			−0.898877			−0.449439	+	0.893311i	$0.648376\pi$
$402$		0			0
$403$	−8.00000			−0.398508
$404$			12.0000i			0.597022i
$405$		0			0
$406$	−4.00000	−	10.0000i	−0.198517	−	0.496292i
$407$			4.00000i			0.198273i
$408$		0			0
$409$		−	8.00000i		−	0.395575i	−0.980245	−	0.197787i	$-0.936624\pi$
$409$		−	8.00000i		−	0.395575i	0.980245	−	0.197787i	$-0.0633755\pi$
$410$		0			0
$411$		0			0
$412$			18.0000i			0.886796i
$413$		−	8.00000i		−	0.393654i
$414$		0			0
$415$	4.00000	−	2.00000i	0.196352	−	0.0981761i
$416$		−	20.0000i		−	0.980581i
$417$		0			0
$418$	4.00000			0.195646
$419$	12.0000			0.586238			0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000			0.586238			0.293119	+	0.956076i	$0.405307\pi$
$420$		0			0
$421$		−	28.0000i		−	1.36464i	−0.731055	−	0.682318i	$-0.760972\pi$
$421$		−	28.0000i		−	1.36464i	0.731055	−	0.682318i	$-0.239028\pi$
$422$			2.00000i			0.0973585i
$423$		0			0
$424$			36.0000i			1.74831i
$425$	−18.0000	+	24.0000i	−0.873128	+	1.16417i
$426$		0			0
$427$	24.0000			1.16144
$428$			10.0000i			0.483368i
$429$		0			0
$430$	4.00000	+	8.00000i	0.192897	+	0.385794i
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$	−22.0000			−1.05725			−0.528626	−	0.848855i	$-0.677293\pi$
$433$	−22.0000			−1.05725			−0.528626	+	0.848855i	$0.677293\pi$
$434$	−4.00000			−0.192006
$435$		0			0
$436$	10.0000			0.478913
$437$	12.0000			0.574038
$438$		0			0
$439$	24.0000			1.14546			0.572729	−	0.819745i	$-0.305885\pi$
$439$	24.0000			1.14546			0.572729	+	0.819745i	$0.305885\pi$
$440$	−12.0000	+	6.00000i	−0.572078	+	0.286039i
$441$		0			0
$442$		−	24.0000i		−	1.14156i
$443$	28.0000			1.33032			0.665160	−	0.746701i	$-0.268363\pi$
$443$	28.0000			1.33032			0.665160	+	0.746701i	$0.268363\pi$
$444$		0			0
$445$	−32.0000	+	16.0000i	−1.51695	+	0.758473i
$446$			22.0000i			1.04173i
$447$		0			0
$448$		−	14.0000i		−	0.661438i
$449$			24.0000i			1.13263i	0.824189	+	0.566315i	$0.191631\pi$
$449$			24.0000i			1.13263i	−0.824189	+	0.566315i	$0.808369\pi$
$450$		0			0
$451$		0			0
$452$	2.00000			0.0940721
$453$		0			0
$454$		−	2.00000i		−	0.0938647i
$455$	−8.00000	−	16.0000i	−0.375046	−	0.750092i
$456$		0			0
$457$			8.00000i			0.374224i	0.982339	+	0.187112i	$0.0599128\pi$
$457$			8.00000i			0.374224i	−0.982339	+	0.187112i	$0.940087\pi$
$458$			20.0000i			0.934539i
$459$		0			0
$460$	−12.0000	+	6.00000i	−0.559503	+	0.279751i
$461$			12.0000i			0.558896i	0.960161	+	0.279448i	$0.0901514\pi$
$461$			12.0000i			0.558896i	−0.960161	+	0.279448i	$0.909849\pi$
$462$		0			0
$463$		−	26.0000i		−	1.20832i	−0.796862	−	0.604161i	$-0.793508\pi$
$463$		−	26.0000i		−	1.20832i	0.796862	−	0.604161i	$-0.206492\pi$
$464$	−5.00000	+	2.00000i	−0.232119	+	0.0928477i
$465$		0			0
$466$			16.0000i			0.741186i
$467$	−12.0000			−0.555294			−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000			−0.555294			−0.277647	+	0.960683i	$0.589555\pi$
$468$		0			0
$469$	−12.0000			−0.554109
$470$	8.00000	+	16.0000i	0.369012	+	0.738025i
$471$		0			0
$472$	−12.0000			−0.552345
$473$		−	8.00000i		−	0.367840i
$474$		0			0
$475$	−8.00000	−	6.00000i	−0.367065	−	0.275299i
$476$			12.0000i			0.550019i
$477$		0			0
$478$	−16.0000			−0.731823
$479$			42.0000i			1.91903i	0.281659	+	0.959514i	$0.409115\pi$
$479$			42.0000i			1.91903i	−0.281659	+	0.959514i	$0.590885\pi$
$480$		0			0
$481$			8.00000i			0.364769i
$482$	−22.0000			−1.00207
$483$		0			0
$484$	−7.00000			−0.318182
$485$	−14.0000	−	28.0000i	−0.635707	−	1.27141i
$486$		0			0
$487$			14.0000i			0.634401i	0.948359	+	0.317200i	$0.102743\pi$
$487$			14.0000i			0.634401i	−0.948359	+	0.317200i	$0.897257\pi$
$488$		−	36.0000i		−	1.62964i
$489$		0			0
$490$	3.00000	+	6.00000i	0.135526	+	0.271052i
$491$		−	2.00000i		−	0.0902587i	−0.998981	−	0.0451294i	$-0.985630\pi$
$491$		−	2.00000i		−	0.0902587i	0.998981	−	0.0451294i	$-0.0143700\pi$
$492$		0			0
$493$	30.0000	−	12.0000i	1.35113	−	0.540453i
$494$	8.00000			0.359937
$495$		0			0
$496$			2.00000i			0.0898027i
$497$			16.0000i			0.717698i
$498$		0			0
$499$	−12.0000			−0.537194			−0.268597	−	0.963253i	$-0.586560\pi$
$499$	−12.0000			−0.537194			−0.268597	+	0.963253i	$0.586560\pi$
$500$	11.0000	+	2.00000i	0.491935	+	0.0894427i
$501$		0			0
$502$			30.0000i			1.33897i
$503$	16.0000			0.713405			0.356702	−	0.934218i	$-0.383901\pi$
$503$	16.0000			0.713405			0.356702	+	0.934218i	$0.383901\pi$
$504$		0			0
$505$	24.0000	−	12.0000i	1.06799	−	0.533993i
$506$	−12.0000			−0.533465
$507$		0			0
$508$	−8.00000			−0.354943
$509$	10.0000			0.443242			0.221621	−	0.975133i	$-0.428865\pi$
$509$	10.0000			0.443242			0.221621	+	0.975133i	$0.428865\pi$
$510$		0			0
$511$			12.0000i			0.530849i
$512$	−11.0000			−0.486136
$513$		0			0
$514$			24.0000i			1.05859i
$515$	36.0000	−	18.0000i	1.58635	−	0.793175i
$516$		0			0
$517$		−	16.0000i		−	0.703679i
$518$			4.00000i			0.175750i
$519$		0			0
$520$	−24.0000	+	12.0000i	−1.05247	+	0.526235i
$521$	14.0000			0.613351			0.306676	−	0.951814i	$-0.400783\pi$
$521$	14.0000			0.613351			0.306676	+	0.951814i	$0.400783\pi$
$522$		0			0
$523$			34.0000i			1.48672i	0.668894	+	0.743358i	$0.266768\pi$
$523$			34.0000i			1.48672i	−0.668894	+	0.743358i	$0.733232\pi$
$524$		−	22.0000i		−	0.961074i
$525$		0			0
$526$	−16.0000			−0.697633
$527$		−	12.0000i		−	0.522728i
$528$		0			0
$529$	−13.0000			−0.565217
$530$	24.0000	−	12.0000i	1.04249	−	0.521247i
$531$		0			0
$532$	−4.00000			−0.173422
$533$		0			0
$534$		0			0
$535$	20.0000	−	10.0000i	0.864675	−	0.432338i
$536$			18.0000i			0.777482i
$537$		0			0
$538$		−	4.00000i		−	0.172452i
$539$		−	6.00000i		−	0.258438i
$540$		0			0
$541$			44.0000i			1.89171i	0.324593	+	0.945854i	$0.394773\pi$
$541$			44.0000i			1.89171i	−0.324593	+	0.945854i	$0.605227\pi$
$542$		−	2.00000i		−	0.0859074i
$543$		0			0
$544$	30.0000			1.28624
$545$	−10.0000	−	20.0000i	−0.428353	−	0.856706i
$546$		0			0
$547$			26.0000i			1.11168i	0.831289	+	0.555840i	$0.187603\pi$
$547$			26.0000i			1.11168i	−0.831289	+	0.555840i	$0.812397\pi$
$548$	−14.0000			−0.598050
$549$		0			0
$550$	8.00000	+	6.00000i	0.341121	+	0.255841i
$551$	4.00000	+	10.0000i	0.170406	+	0.426014i
$552$		0			0
$553$	−4.00000			−0.170097
$554$			4.00000i			0.169944i
$555$		0			0
$556$	4.00000			0.169638
$557$			12.0000i			0.508456i	0.967144	+	0.254228i	$0.0818214\pi$
$557$			12.0000i			0.508456i	−0.967144	+	0.254228i	$0.918179\pi$
$558$		0			0
$559$		−	16.0000i		−	0.676728i
$560$	−4.00000	+	2.00000i	−0.169031	+	0.0845154i
$561$		0			0
$562$	30.0000			1.26547
$563$	−12.0000			−0.505740			−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000			−0.505740			−0.252870	+	0.967500i	$0.581374\pi$
$564$		0			0
$565$	−2.00000	−	4.00000i	−0.0841406	−	0.168281i
$566$			2.00000i			0.0840663i
$567$		0			0
$568$	24.0000			1.00702
$569$		−	32.0000i		−	1.34151i	−0.741679	−	0.670755i	$-0.765970\pi$
$569$		−	32.0000i		−	1.34151i	0.741679	−	0.670755i	$-0.234030\pi$
$570$		0			0
$571$	−20.0000			−0.836974			−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000			−0.836974			−0.418487	+	0.908223i	$0.637439\pi$
$572$	8.00000			0.334497
$573$		0			0
$574$		0			0
$575$	24.0000	+	18.0000i	1.00087	+	0.750652i
$576$		0			0
$577$	−38.0000			−1.58196			−0.790980	−	0.611842i	$-0.790429\pi$
$577$	−38.0000			−1.58196			−0.790980	+	0.611842i	$0.790429\pi$
$578$	19.0000			0.790296
$579$		0			0
$580$	−9.00000	−	8.00000i	−0.373705	−	0.332182i
$581$	−4.00000			−0.165948
$582$		0			0
$583$	−24.0000			−0.993978
$584$	18.0000			0.744845
$585$		0			0
$586$	26.0000			1.07405
$587$		−	42.0000i		−	1.73353i	−0.498721	−	0.866763i	$-0.666197\pi$
$587$		−	42.0000i		−	1.73353i	0.498721	−	0.866763i	$-0.333803\pi$
$588$		0			0
$589$	4.00000			0.164817
$590$	4.00000	+	8.00000i	0.164677	+	0.329355i
$591$		0			0
$592$	2.00000			0.0821995
$593$		−	8.00000i		−	0.328521i	−0.986417	−	0.164260i	$-0.947476\pi$
$593$		−	8.00000i		−	0.328521i	0.986417	−	0.164260i	$-0.0525237\pi$
$594$		0			0
$595$	24.0000	−	12.0000i	0.983904	−	0.491952i
$596$	−10.0000			−0.409616
$597$		0			0
$598$	−24.0000			−0.981433
$599$			18.0000i			0.735460i	0.929933	+	0.367730i	$0.119865\pi$
$599$			18.0000i			0.735460i	−0.929933	+	0.367730i	$0.880135\pi$
$600$		0			0
$601$			8.00000i			0.326327i	0.986599	+	0.163163i	$0.0521698\pi$
$601$			8.00000i			0.326327i	−0.986599	+	0.163163i	$0.947830\pi$
$602$		−	8.00000i		−	0.326056i
$603$		0			0
$604$	8.00000			0.325515
$605$	7.00000	+	14.0000i	0.284590	+	0.569181i
$606$		0			0
$607$	8.00000			0.324710			0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000			0.324710			0.162355	+	0.986732i	$0.448091\pi$
$608$			10.0000i			0.405554i
$609$		0			0
$610$	−24.0000	+	12.0000i	−0.971732	+	0.485866i
$611$		−	32.0000i		−	1.29458i
$612$		0			0
$613$		−	12.0000i		−	0.484675i	−0.970192	−	0.242338i	$-0.922086\pi$
$613$		−	12.0000i		−	0.484675i	0.970192	−	0.242338i	$-0.0779142\pi$
$614$	12.0000			0.484281
$615$		0			0
$616$	12.0000			0.483494
$617$	6.00000			0.241551			0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000			0.241551			0.120775	+	0.992680i	$0.461462\pi$
$618$		0			0
$619$			26.0000i			1.04503i	0.852631	+	0.522514i	$0.175006\pi$
$619$			26.0000i			1.04503i	−0.852631	+	0.522514i	$0.824994\pi$
$620$	−4.00000	+	2.00000i	−0.160644	+	0.0803219i
$621$		0			0
$622$		−	14.0000i		−	0.561349i
$623$	32.0000			1.28205
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$			8.00000i			0.319744i
$627$		0			0
$628$	2.00000			0.0798087
$629$	−12.0000			−0.478471
$630$		0			0
$631$	8.00000			0.318475			0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000			0.318475			0.159237	+	0.987240i	$0.449096\pi$
$632$			6.00000i			0.238667i
$633$		0			0
$634$	−6.00000			−0.238290
$635$	8.00000	+	16.0000i	0.317470	+	0.634941i
$636$		0			0
$637$		−	12.0000i		−	0.475457i
$638$	−4.00000	−	10.0000i	−0.158362	−	0.395904i
$639$		0			0
$640$	−3.00000	−	6.00000i	−0.118585	−	0.237171i
$641$		−	24.0000i		−	0.947943i	−0.880540	−	0.473972i	$-0.842820\pi$
$641$		−	24.0000i		−	0.947943i	0.880540	−	0.473972i	$-0.157180\pi$
$642$		0			0
$643$		−	6.00000i		−	0.236617i	−0.992977	−	0.118308i	$-0.962253\pi$
$643$		−	6.00000i		−	0.236617i	0.992977	−	0.118308i	$-0.0377472\pi$
$644$	12.0000			0.472866
$645$		0			0
$646$			12.0000i			0.472134i
$647$			10.0000i			0.393141i	0.980490	+	0.196570i	$0.0629804\pi$
$647$			10.0000i			0.393141i	−0.980490	+	0.196570i	$0.937020\pi$
$648$		0			0
$649$		−	8.00000i		−	0.314027i
$650$	16.0000	+	12.0000i	0.627572	+	0.470679i
$651$		0			0
$652$	20.0000			0.783260
$653$	18.0000			0.704394			0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000			0.704394			0.352197	+	0.935926i	$0.385435\pi$
$654$		0			0
$655$	−44.0000	+	22.0000i	−1.71922	+	0.859611i
$656$		0			0
$657$		0			0
$658$		−	16.0000i		−	0.623745i
$659$		−	26.0000i		−	1.01282i	−0.862294	−	0.506408i	$-0.830973\pi$
$659$		−	26.0000i		−	1.01282i	0.862294	−	0.506408i	$-0.169027\pi$
$660$		0			0
$661$	22.0000			0.855701			0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000			0.855701			0.427850	+	0.903850i	$0.359271\pi$
$662$		−	22.0000i		−	0.855054i
$663$		0			0
$664$			6.00000i			0.232845i
$665$	4.00000	+	8.00000i	0.155113	+	0.310227i
$666$		0			0
$667$	−12.0000	−	30.0000i	−0.464642	−	1.16160i
$668$			6.00000i			0.232147i
$669$		0			0
$670$	12.0000	−	6.00000i	0.463600	−	0.231800i
$671$	24.0000			0.926510
$672$		0			0
$673$		−	32.0000i		−	1.23351i	−0.787155	−	0.616755i	$-0.788447\pi$
$673$		−	32.0000i		−	1.23351i	0.787155	−	0.616755i	$-0.211553\pi$
$674$	34.0000			1.30963
$675$		0			0
$676$	3.00000			0.115385
$677$	−6.00000			−0.230599			−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000			−0.230599			−0.115299	+	0.993331i	$0.536783\pi$
$678$		0			0
$679$			28.0000i			1.07454i
$680$	−18.0000	−	36.0000i	−0.690268	−	1.38054i
$681$		0			0
$682$	−4.00000			−0.153168
$683$			6.00000i			0.229584i	0.993390	+	0.114792i	$0.0366201\pi$
$683$			6.00000i			0.229584i	−0.993390	+	0.114792i	$0.963380\pi$
$684$		0			0
$685$	14.0000	+	28.0000i	0.534913	+	1.06983i
$686$		−	20.0000i		−	0.763604i
$687$		0			0
$688$	−4.00000			−0.152499
$689$	−48.0000			−1.82865
$690$		0			0
$691$	20.0000			0.760836			0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000			0.760836			0.380418	+	0.924815i	$0.375780\pi$
$692$		−	12.0000i		−	0.456172i
$693$		0			0
$694$		−	10.0000i		−	0.379595i
$695$	−4.00000	−	8.00000i	−0.151729	−	0.303457i
$696$		0			0
$697$		0			0
$698$	−26.0000			−0.984115
$699$		0			0
$700$	−8.00000	−	6.00000i	−0.302372	−	0.226779i
$701$	−30.0000			−1.13308			−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000			−1.13308			−0.566542	+	0.824033i	$0.691719\pi$
$702$		0			0
$703$		−	4.00000i		−	0.150863i
$704$		−	14.0000i		−	0.527645i
$705$		0			0
$706$			24.0000i			0.903252i
$707$	−24.0000			−0.902613
$708$		0			0
$709$	−50.0000			−1.87779			−0.938895	−	0.344204i	$-0.888149\pi$
$709$	−50.0000			−1.87779			−0.938895	+	0.344204i	$0.888149\pi$
$710$	−8.00000	−	16.0000i	−0.300235	−	0.600469i
$711$		0			0
$712$		−	48.0000i		−	1.79888i
$713$	−12.0000			−0.449404
$714$		0			0
$715$	−8.00000	−	16.0000i	−0.299183	−	0.598366i
$716$	−12.0000			−0.448461
$717$		0			0
$718$			18.0000i			0.671754i
$719$	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
$721$	−36.0000			−1.34071
$722$	15.0000			0.558242
$723$		0			0
$724$	10.0000			0.371647
$725$	−7.00000	+	26.0000i	−0.259973	+	0.965616i
$726$		0			0
$727$	48.0000			1.78022			0.890111	−	0.455744i	$-0.150627\pi$
$727$	48.0000			1.78022			0.890111	+	0.455744i	$0.150627\pi$
$728$	24.0000			0.889499
$729$		0			0
$730$	−6.00000	−	12.0000i	−0.222070	−	0.444140i
$731$	24.0000			0.887672
$732$		0			0
$733$	22.0000			0.812589			0.406294	−	0.913742i	$-0.366821\pi$
$733$	22.0000			0.812589			0.406294	+	0.913742i	$0.366821\pi$
$734$	24.0000			0.885856
$735$		0			0
$736$		−	30.0000i		−	1.10581i
$737$	−12.0000			−0.442026
$738$		0			0
$739$			34.0000i			1.25071i	0.780340	+	0.625355i	$0.215046\pi$
$739$			34.0000i			1.25071i	−0.780340	+	0.625355i	$0.784954\pi$
$740$	2.00000	+	4.00000i	0.0735215	+	0.147043i
$741$		0			0
$742$	−24.0000			−0.881068
$743$	48.0000			1.76095			0.880475	−	0.474093i	$-0.157224\pi$
$743$	48.0000			1.76095			0.880475	+	0.474093i	$0.157224\pi$
$744$		0			0
$745$	10.0000	+	20.0000i	0.366372	+	0.732743i
$746$			4.00000i			0.146450i
$747$		0			0
$748$			12.0000i			0.438763i
$749$	−20.0000			−0.730784
$750$		0			0
$751$			46.0000i			1.67856i	0.543696	+	0.839282i	$0.317024\pi$
$751$			46.0000i			1.67856i	−0.543696	+	0.839282i	$0.682976\pi$
$752$	−8.00000			−0.291730
$753$		0			0
$754$	−8.00000	−	20.0000i	−0.291343	−	0.728357i
$755$	−8.00000	−	16.0000i	−0.291150	−	0.582300i
$756$		0			0
$757$	−42.0000			−1.52652			−0.763258	−	0.646094i	$-0.776401\pi$
$757$	−42.0000			−1.52652			−0.763258	+	0.646094i	$0.776401\pi$
$758$		−	38.0000i		−	1.38022i
$759$		0			0
$760$	12.0000	−	6.00000i	0.435286	−	0.217643i
$761$	−2.00000			−0.0724999			−0.0362500	−	0.999343i	$-0.511541\pi$
$761$	−2.00000			−0.0724999			−0.0362500	+	0.999343i	$0.511541\pi$
$762$		0			0
$763$			20.0000i			0.724049i
$764$			6.00000i			0.217072i
$765$		0			0
$766$		−	30.0000i		−	1.08394i
$767$		−	16.0000i		−	0.577727i
$768$		0			0
$769$		−	32.0000i		−	1.15395i	−0.816762	−	0.576975i	$-0.804233\pi$
$769$		−	32.0000i		−	1.15395i	0.816762	−	0.576975i	$-0.195767\pi$
$770$	−4.00000	−	8.00000i	−0.144150	−	0.288300i
$771$		0			0
$772$	6.00000			0.215945
$773$	18.0000			0.647415			0.323708	−	0.946157i	$-0.395071\pi$
$773$	18.0000			0.647415			0.323708	+	0.946157i	$0.395071\pi$
$774$		0			0
$775$	8.00000	+	6.00000i	0.287368	+	0.215526i
$776$	42.0000			1.50771
$777$		0			0
$778$		−	12.0000i		−	0.430221i
$779$		0			0
$780$		0			0
$781$			16.0000i			0.572525i
$782$		−	36.0000i		−	1.28736i
$783$		0			0
$784$	−3.00000			−0.107143
$785$	−2.00000	−	4.00000i	−0.0713831	−	0.142766i
$786$		0			0
$787$		−	6.00000i		−	0.213877i	−0.994266	−	0.106938i	$-0.965895\pi$
$787$		−	6.00000i		−	0.213877i	0.994266	−	0.106938i	$-0.0341048\pi$
$788$		−	4.00000i		−	0.142494i
$789$		0			0

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:	\( N \)	\(=\)	\( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1305.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{4} +(1.00000 + 2.00000i) q^{5} -2.00000i q^{7} -3.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{4} +(1.00000 + 2.00000i) q^{5} -2.00000i q^{7} -3.00000 q^{8} +(1.00000 + 2.00000i) q^{10} -2.00000i q^{11} -4.00000i q^{13} -2.00000i q^{14} -1.00000 q^{16} +6.00000 q^{17} +2.00000i q^{19} +(-1.00000 - 2.00000i) q^{20} -2.00000i q^{22} -6.00000i q^{23} +(-3.00000 + 4.00000i) q^{25} -4.00000i q^{26} +2.00000i q^{28} +(5.00000 - 2.00000i) q^{29} -2.00000i q^{31} +5.00000 q^{32} +6.00000 q^{34} +(4.00000 - 2.00000i) q^{35} -2.00000 q^{37} +2.00000i q^{38} +(-3.00000 - 6.00000i) q^{40} +4.00000 q^{43} +2.00000i q^{44} -6.00000i q^{46} +8.00000 q^{47} +3.00000 q^{49} +(-3.00000 + 4.00000i) q^{50} +4.00000i q^{52} -12.0000i q^{53} +(4.00000 - 2.00000i) q^{55} +6.00000i q^{56} +(5.00000 - 2.00000i) q^{58} +4.00000 q^{59} +12.0000i q^{61} -2.00000i q^{62} +7.00000 q^{64} +(8.00000 - 4.00000i) q^{65} -6.00000i q^{67} -6.00000 q^{68} +(4.00000 - 2.00000i) q^{70} -8.00000 q^{71} -6.00000 q^{73} -2.00000 q^{74} -2.00000i q^{76} -4.00000 q^{77} -2.00000i q^{79} +(-1.00000 - 2.00000i) q^{80} -2.00000i q^{83} +(6.00000 + 12.0000i) q^{85} +4.00000 q^{86} +6.00000i q^{88} +16.0000i q^{89} -8.00000 q^{91} +6.00000i q^{92} +8.00000 q^{94} +(-4.00000 + 2.00000i) q^{95} -14.0000 q^{97} +3.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{4} + 2 q^{5} - 6 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 - 2 * q^4 + 2 * q^5 - 6 * q^8	\( 2 q + 2 q^{2} - 2 q^{4} + 2 q^{5} - 6 q^{8} + 2 q^{10} - 2 q^{16} + 12 q^{17} - 2 q^{20} - 6 q^{25} + 10 q^{29} + 10 q^{32} + 12 q^{34} + 8 q^{35} - 4 q^{37} - 6 q^{40} + 8 q^{43} + 16 q^{47} + 6 q^{49} - 6 q^{50} + 8 q^{55} + 10 q^{58} + 8 q^{59} + 14 q^{64} + 16 q^{65} - 12 q^{68} + 8 q^{70} - 16 q^{71} - 12 q^{73} - 4 q^{74} - 8 q^{77} - 2 q^{80} + 12 q^{85} + 8 q^{86} - 16 q^{91} + 16 q^{94} - 8 q^{95} - 28 q^{97} + 6 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^4 + 2 * q^5 - 6 * q^8 + 2 * q^10 - 2 * q^16 + 12 * q^17 - 2 * q^20 - 6 * q^25 + 10 * q^29 + 10 * q^32 + 12 * q^34 + 8 * q^35 - 4 * q^37 - 6 * q^40 + 8 * q^43 + 16 * q^47 + 6 * q^49 - 6 * q^50 + 8 * q^55 + 10 * q^58 + 8 * q^59 + 14 * q^64 + 16 * q^65 - 12 * q^68 + 8 * q^70 - 16 * q^71 - 12 * q^73 - 4 * q^74 - 8 * q^77 - 2 * q^80 + 12 * q^85 + 8 * q^86 - 16 * q^91 + 16 * q^94 - 8 * q^95 - 28 * q^97 + 6 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more