LMFDB - Embedded newform 1305.2.f.d.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:	$ 1 $
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.d.289.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} +2.00000 q^{4} +(-2.00000 + 1.00000i) q^{5} -4.00000i q^{7} +O(q^{10})$	\(q+2.00000 q^{2} +2.00000 q^{4} +(-2.00000 + 1.00000i) q^{5} -4.00000i q^{7} +(-4.00000 + 2.00000i) q^{10} +1.00000i q^{11} -2.00000i q^{13} -8.00000i q^{14} -4.00000 q^{16} +6.00000 q^{17} -4.00000i q^{19} +(-4.00000 + 2.00000i) q^{20} +2.00000i q^{22} -9.00000i q^{23} +(3.00000 - 4.00000i) q^{25} -4.00000i q^{26} -8.00000i q^{28} +(2.00000 - 5.00000i) q^{29} -2.00000i q^{31} -8.00000 q^{32} +12.0000 q^{34} +(4.00000 + 8.00000i) q^{35} -1.00000 q^{37} -8.00000i q^{38} +9.00000i q^{41} -1.00000 q^{43} +2.00000i q^{44} -18.0000i q^{46} -8.00000 q^{47} -9.00000 q^{49} +(6.00000 - 8.00000i) q^{50} -4.00000i q^{52} +9.00000i q^{53} +(-1.00000 - 2.00000i) q^{55} +(4.00000 - 10.0000i) q^{58} -8.00000 q^{59} -6.00000i q^{61} -4.00000i q^{62} -8.00000 q^{64} +(2.00000 + 4.00000i) q^{65} +12.0000i q^{67} +12.0000 q^{68} +(8.00000 + 16.0000i) q^{70} -2.00000 q^{71} +15.0000 q^{73} -2.00000 q^{74} -8.00000i q^{76} +4.00000 q^{77} +4.00000i q^{79} +(8.00000 - 4.00000i) q^{80} +18.0000i q^{82} -7.00000i q^{83} +(-12.0000 + 6.00000i) q^{85} -2.00000 q^{86} -2.00000i q^{89} -8.00000 q^{91} -18.0000i q^{92} -16.0000 q^{94} +(4.00000 + 8.00000i) q^{95} +11.0000 q^{97} -18.0000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} + 4 q^{4} - 4 q^{5}+O(q^{10}) $ 2 * q + 4 * q^2 + 4 * q^4 - 4 * q^5	$ 2 q + 4 q^{2} + 4 q^{4} - 4 q^{5} - 8 q^{10} - 8 q^{16} + 12 q^{17} - 8 q^{20} + 6 q^{25} + 4 q^{29} - 16 q^{32} + 24 q^{34} + 8 q^{35} - 2 q^{37} - 2 q^{43} - 16 q^{47} - 18 q^{49} + 12 q^{50} - 2 q^{55} + 8 q^{58} - 16 q^{59} - 16 q^{64} + 4 q^{65} + 24 q^{68} + 16 q^{70} - 4 q^{71} + 30 q^{73} - 4 q^{74} + 8 q^{77} + 16 q^{80} - 24 q^{85} - 4 q^{86} - 16 q^{91} - 32 q^{94} + 8 q^{95} + 22 q^{97} - 36 q^{98}+O(q^{100}) $ 2 * q + 4 * q^2 + 4 * q^4 - 4 * q^5 - 8 * q^10 - 8 * q^16 + 12 * q^17 - 8 * q^20 + 6 * q^25 + 4 * q^29 - 16 * q^32 + 24 * q^34 + 8 * q^35 - 2 * q^37 - 2 * q^43 - 16 * q^47 - 18 * q^49 + 12 * q^50 - 2 * q^55 + 8 * q^58 - 16 * q^59 - 16 * q^64 + 4 * q^65 + 24 * q^68 + 16 * q^70 - 4 * q^71 + 30 * q^73 - 4 * q^74 + 8 * q^77 + 16 * q^80 - 24 * q^85 - 4 * q^86 - 16 * q^91 - 32 * q^94 + 8 * q^95 + 22 * q^97 - 36 * 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 $

$2$	2.00000			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$2$	2.00000			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$3$		0			0
$3$		0			0
$4$	2.00000			1.00000
$5$	−2.00000	+	1.00000i	−0.894427	+	0.447214i
$5$	−2.00000	+	1.00000i	−0.894427	+	0.447214i
$6$		0			0
$7$		−	4.00000i		−	1.51186i	−0.654654	−	0.755929i	$-0.727186\pi$
$7$		−	4.00000i		−	1.51186i	0.654654	−	0.755929i	$-0.272814\pi$
$8$		0			0
$9$		0			0
$10$	−4.00000	+	2.00000i	−1.26491	+	0.632456i
$11$			1.00000i			0.301511i	0.988571	+	0.150756i	$0.0481707\pi$
$11$			1.00000i			0.301511i	−0.988571	+	0.150756i	$0.951829\pi$
$12$		0			0
$13$		−	2.00000i		−	0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$		−	2.00000i		−	0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$		−	8.00000i		−	2.13809i
$15$		0			0
$16$	−4.00000			−1.00000
$17$	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
$19$		−	4.00000i		−	0.917663i	−0.888523	−	0.458831i	$-0.848268\pi$
$19$		−	4.00000i		−	0.917663i	0.888523	−	0.458831i	$-0.151732\pi$
$20$	−4.00000	+	2.00000i	−0.894427	+	0.447214i
$21$		0			0
$22$			2.00000i			0.426401i
$23$		−	9.00000i		−	1.87663i	−0.345782	−	0.938315i	$-0.612386\pi$
$23$		−	9.00000i		−	1.87663i	0.345782	−	0.938315i	$-0.387614\pi$
$24$		0			0
$25$	3.00000	−	4.00000i	0.600000	−	0.800000i
$26$		−	4.00000i		−	0.784465i
$27$		0			0
$28$		−	8.00000i		−	1.51186i
$29$	2.00000	−	5.00000i	0.371391	−	0.928477i
$29$	2.00000	−	5.00000i	0.371391	−	0.928477i
$30$		0			0
$31$		−	2.00000i		−	0.359211i	−0.983739	−	0.179605i	$-0.942518\pi$
$31$		−	2.00000i		−	0.359211i	0.983739	−	0.179605i	$-0.0574821\pi$
$32$	−8.00000			−1.41421
$33$		0			0
$34$	12.0000			2.05798
$35$	4.00000	+	8.00000i	0.676123	+	1.35225i
$36$		0			0
$37$	−1.00000			−0.164399			−0.0821995	−	0.996616i	$-0.526194\pi$
$37$	−1.00000			−0.164399			−0.0821995	+	0.996616i	$0.526194\pi$
$38$		−	8.00000i		−	1.29777i
$39$		0			0
$40$		0			0
$41$			9.00000i			1.40556i	0.711405	+	0.702782i	$0.248059\pi$
$41$			9.00000i			1.40556i	−0.711405	+	0.702782i	$0.751941\pi$
$42$		0			0
$43$	−1.00000			−0.152499			−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000			−0.152499			−0.0762493	+	0.997089i	$0.524294\pi$
$44$			2.00000i			0.301511i
$45$		0			0
$46$		−	18.0000i		−	2.65396i
$47$	−8.00000			−1.16692			−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000			−1.16692			−0.583460	+	0.812142i	$0.698301\pi$
$48$		0			0
$49$	−9.00000			−1.28571
$50$	6.00000	−	8.00000i	0.848528	−	1.13137i
$51$		0			0
$52$		−	4.00000i		−	0.554700i
$53$			9.00000i			1.23625i	0.786082	+	0.618123i	$0.212106\pi$
$53$			9.00000i			1.23625i	−0.786082	+	0.618123i	$0.787894\pi$
$54$		0			0
$55$	−1.00000	−	2.00000i	−0.134840	−	0.269680i
$56$		0			0
$57$		0			0
$58$	4.00000	−	10.0000i	0.525226	−	1.31306i
$59$	−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
$61$		−	6.00000i		−	0.768221i	−0.923287	−	0.384111i	$-0.874508\pi$
$61$		−	6.00000i		−	0.768221i	0.923287	−	0.384111i	$-0.125492\pi$
$62$		−	4.00000i		−	0.508001i
$63$		0			0
$64$	−8.00000			−1.00000
$65$	2.00000	+	4.00000i	0.248069	+	0.496139i
$66$		0			0
$67$			12.0000i			1.46603i	0.680211	+	0.733017i	$0.261888\pi$
$67$			12.0000i			1.46603i	−0.680211	+	0.733017i	$0.738112\pi$
$68$	12.0000			1.45521
$69$		0			0
$70$	8.00000	+	16.0000i	0.956183	+	1.91237i
$71$	−2.00000			−0.237356			−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000			−0.237356			−0.118678	+	0.992933i	$0.537866\pi$
$72$		0			0
$73$	15.0000			1.75562			0.877809	−	0.479012i	$-0.159005\pi$
$73$	15.0000			1.75562			0.877809	+	0.479012i	$0.159005\pi$
$74$	−2.00000			−0.232495
$75$		0			0
$76$		−	8.00000i		−	0.917663i
$77$	4.00000			0.455842
$78$		0			0
$79$			4.00000i			0.450035i	0.974355	+	0.225018i	$0.0722440\pi$
$79$			4.00000i			0.450035i	−0.974355	+	0.225018i	$0.927756\pi$
$80$	8.00000	−	4.00000i	0.894427	−	0.447214i
$81$		0			0
$82$			18.0000i			1.98777i
$83$		−	7.00000i		−	0.768350i	−0.923260	−	0.384175i	$-0.874486\pi$
$83$		−	7.00000i		−	0.768350i	0.923260	−	0.384175i	$-0.125514\pi$
$84$		0			0
$85$	−12.0000	+	6.00000i	−1.30158	+	0.650791i
$86$	−2.00000			−0.215666
$87$		0			0
$88$		0			0
$89$		−	2.00000i		−	0.212000i	−0.994366	−	0.106000i	$-0.966196\pi$
$89$		−	2.00000i		−	0.212000i	0.994366	−	0.106000i	$-0.0338043\pi$
$90$		0			0
$91$	−8.00000			−0.838628
$92$		−	18.0000i		−	1.87663i
$93$		0			0
$94$	−16.0000			−1.65027
$95$	4.00000	+	8.00000i	0.410391	+	0.820783i
$96$		0			0
$97$	11.0000			1.11688			0.558440	−	0.829545i	$-0.311400\pi$
$97$	11.0000			1.11688			0.558440	+	0.829545i	$0.311400\pi$
$98$	−18.0000			−1.81827
$99$		0			0
$100$	6.00000	−	8.00000i	0.600000	−	0.800000i
$101$		−	3.00000i		−	0.298511i	−0.988799	−	0.149256i	$-0.952312\pi$
$101$		−	3.00000i		−	0.298511i	0.988799	−	0.149256i	$-0.0476877\pi$
$102$		0			0
$103$			6.00000i			0.591198i	0.955312	+	0.295599i	$0.0955191\pi$
$103$			6.00000i			0.591198i	−0.955312	+	0.295599i	$0.904481\pi$
$104$		0			0
$105$		0			0
$106$			18.0000i			1.74831i
$107$			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
$109$	−1.00000			−0.0957826			−0.0478913	−	0.998853i	$-0.515250\pi$
$109$	−1.00000			−0.0957826			−0.0478913	+	0.998853i	$0.515250\pi$
$110$	−2.00000	−	4.00000i	−0.190693	−	0.381385i
$111$		0			0
$112$			16.0000i			1.51186i
$113$	8.00000			0.752577			0.376288	−	0.926503i	$-0.377200\pi$
$113$	8.00000			0.752577			0.376288	+	0.926503i	$0.377200\pi$
$114$		0			0
$115$	9.00000	+	18.0000i	0.839254	+	1.67851i
$116$	4.00000	−	10.0000i	0.371391	−	0.928477i
$117$		0			0
$118$	−16.0000			−1.47292
$119$		−	24.0000i		−	2.20008i
$120$		0			0
$121$	10.0000			0.909091
$122$		−	12.0000i		−	1.08643i
$123$		0			0
$124$		−	4.00000i		−	0.359211i
$125$	−2.00000	+	11.0000i	−0.178885	+	0.983870i
$126$		0			0
$127$	7.00000			0.621150			0.310575	−	0.950549i	$-0.399478\pi$
$127$	7.00000			0.621150			0.310575	+	0.950549i	$0.399478\pi$
$128$		0			0
$129$		0			0
$130$	4.00000	+	8.00000i	0.350823	+	0.701646i
$131$		−	20.0000i		−	1.74741i	−0.486458	−	0.873704i	$-0.661711\pi$
$131$		−	20.0000i		−	1.74741i	0.486458	−	0.873704i	$-0.338289\pi$
$132$		0			0
$133$	−16.0000			−1.38738
$134$			24.0000i			2.07328i
$135$		0			0
$136$		0			0
$137$	16.0000			1.36697			0.683486	−	0.729964i	$-0.260463\pi$
$137$	16.0000			1.36697			0.683486	+	0.729964i	$0.260463\pi$
$138$		0			0
$139$	11.0000			0.933008			0.466504	−	0.884519i	$-0.345513\pi$
$139$	11.0000			0.933008			0.466504	+	0.884519i	$0.345513\pi$
$140$	8.00000	+	16.0000i	0.676123	+	1.35225i
$141$		0			0
$142$	−4.00000			−0.335673
$143$	2.00000			0.167248
$144$		0			0
$145$	1.00000	+	12.0000i	0.0830455	+	0.996546i
$146$	30.0000			2.48282
$147$		0			0
$148$	−2.00000			−0.164399
$149$	−14.0000			−1.14692			−0.573462	−	0.819232i	$-0.694400\pi$
$149$	−14.0000			−1.14692			−0.573462	+	0.819232i	$0.694400\pi$
$150$		0			0
$151$	−17.0000			−1.38344			−0.691720	−	0.722166i	$-0.743147\pi$
$151$	−17.0000			−1.38344			−0.691720	+	0.722166i	$0.743147\pi$
$152$		0			0
$153$		0			0
$154$	8.00000			0.644658
$155$	2.00000	+	4.00000i	0.160644	+	0.321288i
$156$		0			0
$157$	14.0000			1.11732			0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000			1.11732			0.558661	+	0.829396i	$0.311315\pi$
$158$			8.00000i			0.636446i
$159$		0			0
$160$	16.0000	−	8.00000i	1.26491	−	0.632456i
$161$	−36.0000			−2.83720
$162$		0			0
$163$	−13.0000			−1.01824			−0.509119	−	0.860696i	$-0.670029\pi$
$163$	−13.0000			−1.01824			−0.509119	+	0.860696i	$0.670029\pi$
$164$			18.0000i			1.40556i
$165$		0			0
$166$		−	14.0000i		−	1.08661i
$167$			24.0000i			1.85718i	0.371113	+	0.928588i	$0.378976\pi$
$167$			24.0000i			1.85718i	−0.371113	+	0.928588i	$0.621024\pi$
$168$		0			0
$169$	9.00000			0.692308
$170$	−24.0000	+	12.0000i	−1.84072	+	0.920358i
$171$		0			0
$172$	−2.00000			−0.152499
$173$		−	21.0000i		−	1.59660i	−0.602260	−	0.798300i	$-0.705733\pi$
$173$		−	21.0000i		−	1.59660i	0.602260	−	0.798300i	$-0.294267\pi$
$174$		0			0
$175$	−16.0000	−	12.0000i	−1.20949	−	0.907115i
$176$		−	4.00000i		−	0.301511i
$177$		0			0
$178$		−	4.00000i		−	0.299813i
$179$	−6.00000			−0.448461			−0.224231	−	0.974536i	$-0.571987\pi$
$179$	−6.00000			−0.448461			−0.224231	+	0.974536i	$0.571987\pi$
$180$		0			0
$181$	−7.00000			−0.520306			−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000			−0.520306			−0.260153	+	0.965567i	$0.583773\pi$
$182$	−16.0000			−1.18600
$183$		0			0
$184$		0			0
$185$	2.00000	−	1.00000i	0.147043	−	0.0735215i
$186$		0			0
$187$			6.00000i			0.438763i
$188$	−16.0000			−1.16692
$189$		0			0
$190$	8.00000	+	16.0000i	0.580381	+	1.16076i
$191$		−	3.00000i		−	0.217072i	−0.994092	−	0.108536i	$-0.965384\pi$
$191$		−	3.00000i		−	0.217072i	0.994092	−	0.108536i	$-0.0346163\pi$
$192$		0			0
$193$	6.00000			0.431889			0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000			0.431889			0.215945	+	0.976406i	$0.430717\pi$
$194$	22.0000			1.57951
$195$		0			0
$196$	−18.0000			−1.28571
$197$			23.0000i			1.63868i	0.573306	+	0.819341i	$0.305660\pi$
$197$			23.0000i			1.63868i	−0.573306	+	0.819341i	$0.694340\pi$
$198$		0			0
$199$	−19.0000			−1.34687			−0.673437	−	0.739244i	$-0.735183\pi$
$199$	−19.0000			−1.34687			−0.673437	+	0.739244i	$0.735183\pi$
$200$		0			0
$201$		0			0
$202$		−	6.00000i		−	0.422159i
$203$	−20.0000	−	8.00000i	−1.40372	−	0.561490i
$204$		0			0
$205$	−9.00000	−	18.0000i	−0.628587	−	1.25717i
$206$			12.0000i			0.836080i
$207$		0			0
$208$			8.00000i			0.554700i
$209$	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	$0.978069\pi$
$212$			18.0000i			1.23625i
$213$		0			0
$214$			8.00000i			0.546869i
$215$	2.00000	−	1.00000i	0.136399	−	0.0681994i
$216$		0			0
$217$	−8.00000			−0.543075
$218$	−2.00000			−0.135457
$219$		0			0
$220$	−2.00000	−	4.00000i	−0.134840	−	0.269680i
$221$		−	12.0000i		−	0.807207i
$222$		0			0
$223$		−	4.00000i		−	0.267860i	−0.990991	−	0.133930i	$-0.957240\pi$
$223$		−	4.00000i		−	0.267860i	0.990991	−	0.133930i	$-0.0427597\pi$
$224$			32.0000i			2.13809i
$225$		0			0
$226$	16.0000			1.06430
$227$		−	7.00000i		−	0.464606i	−0.972643	−	0.232303i	$-0.925374\pi$
$227$		−	7.00000i		−	0.464606i	0.972643	−	0.232303i	$-0.0746261\pi$
$228$		0			0
$229$		−	4.00000i		−	0.264327i	−0.991228	−	0.132164i	$-0.957808\pi$
$229$		−	4.00000i		−	0.264327i	0.991228	−	0.132164i	$-0.0421925\pi$
$230$	18.0000	+	36.0000i	1.18688	+	2.37377i
$231$		0			0
$232$		0			0
$233$		−	1.00000i		−	0.0655122i	−0.999463	−	0.0327561i	$-0.989572\pi$
$233$		−	1.00000i		−	0.0655122i	0.999463	−	0.0327561i	$-0.0104285\pi$
$234$		0			0
$235$	16.0000	−	8.00000i	1.04372	−	0.521862i
$236$	−16.0000			−1.04151
$237$		0			0
$238$		−	48.0000i		−	3.11138i
$239$	26.0000			1.68180			0.840900	−	0.541190i	$-0.182026\pi$
$239$	26.0000			1.68180			0.840900	+	0.541190i	$0.182026\pi$
$240$		0			0
$241$	11.0000			0.708572			0.354286	−	0.935137i	$-0.384724\pi$
$241$	11.0000			0.708572			0.354286	+	0.935137i	$0.384724\pi$
$242$	20.0000			1.28565
$243$		0			0
$244$		−	12.0000i		−	0.768221i
$245$	18.0000	−	9.00000i	1.14998	−	0.574989i
$246$		0			0
$247$	−8.00000			−0.509028
$248$		0			0
$249$		0			0
$250$	−4.00000	+	22.0000i	−0.252982	+	1.39140i
$251$			12.0000i			0.757433i	0.925513	+	0.378717i	$0.123635\pi$
$251$			12.0000i			0.757433i	−0.925513	+	0.378717i	$0.876365\pi$
$252$		0			0
$253$	9.00000			0.565825
$254$	14.0000			0.878438
$255$		0			0
$256$	16.0000			1.00000
$257$		−	15.0000i		−	0.935674i	−0.883815	−	0.467837i	$-0.845033\pi$
$257$		−	15.0000i		−	0.935674i	0.883815	−	0.467837i	$-0.154967\pi$
$258$		0			0
$259$			4.00000i			0.248548i
$260$	4.00000	+	8.00000i	0.248069	+	0.496139i
$261$		0			0
$262$		−	40.0000i		−	2.47121i
$263$	−26.0000			−1.60323			−0.801614	−	0.597841i	$-0.796025\pi$
$263$	−26.0000			−1.60323			−0.801614	+	0.597841i	$0.796025\pi$
$264$		0			0
$265$	−9.00000	−	18.0000i	−0.552866	−	1.10573i
$266$	−32.0000			−1.96205
$267$		0			0
$268$			24.0000i			1.46603i
$269$		−	10.0000i		−	0.609711i	−0.952399	−	0.304855i	$-0.901392\pi$
$269$		−	10.0000i		−	0.609711i	0.952399	−	0.304855i	$-0.0986081\pi$
$270$		0			0
$271$		−	2.00000i		−	0.121491i	−0.998153	−	0.0607457i	$-0.980652\pi$
$271$		−	2.00000i		−	0.121491i	0.998153	−	0.0607457i	$-0.0193479\pi$
$272$	−24.0000			−1.45521
$273$		0			0
$274$	32.0000			1.93319
$275$	4.00000	+	3.00000i	0.241209	+	0.180907i
$276$		0			0
$277$			2.00000i			0.120168i	0.998193	+	0.0600842i	$0.0191369\pi$
$277$			2.00000i			0.120168i	−0.998193	+	0.0600842i	$0.980863\pi$
$278$	22.0000			1.31947
$279$		0			0
$280$		0			0
$281$	24.0000			1.43172			0.715860	−	0.698244i	$-0.246035\pi$
$281$	24.0000			1.43172			0.715860	+	0.698244i	$0.246035\pi$
$282$		0			0
$283$			22.0000i			1.30776i	0.756596	+	0.653882i	$0.226861\pi$
$283$			22.0000i			1.30776i	−0.756596	+	0.653882i	$0.773139\pi$
$284$	−4.00000			−0.237356
$285$		0			0
$286$	4.00000			0.236525
$287$	36.0000			2.12501
$288$		0			0
$289$	19.0000			1.11765
$290$	2.00000	+	24.0000i	0.117444	+	1.40933i
$291$		0			0
$292$	30.0000			1.75562
$293$	−2.00000			−0.116841			−0.0584206	−	0.998292i	$-0.518606\pi$
$293$	−2.00000			−0.116841			−0.0584206	+	0.998292i	$0.518606\pi$
$294$		0			0
$295$	16.0000	−	8.00000i	0.931556	−	0.465778i
$296$		0			0
$297$		0			0
$298$	−28.0000			−1.62200
$299$	−18.0000			−1.04097
$300$		0			0
$301$			4.00000i			0.230556i
$302$	−34.0000			−1.95648
$303$		0			0
$304$			16.0000i			0.917663i
$305$	6.00000	+	12.0000i	0.343559	+	0.687118i
$306$		0			0
$307$	15.0000			0.856095			0.428048	−	0.903756i	$-0.359202\pi$
$307$	15.0000			0.856095			0.428048	+	0.903756i	$0.359202\pi$
$308$	8.00000			0.455842
$309$		0			0
$310$	4.00000	+	8.00000i	0.227185	+	0.454369i
$311$		−	23.0000i		−	1.30421i	−0.758129	−	0.652105i	$-0.773886\pi$
$311$		−	23.0000i		−	1.30421i	0.758129	−	0.652105i	$-0.226114\pi$
$312$		0			0
$313$		−	8.00000i		−	0.452187i	−0.974106	−	0.226093i	$-0.927405\pi$
$313$		−	8.00000i		−	0.452187i	0.974106	−	0.226093i	$-0.0725954\pi$
$314$	28.0000			1.58013
$315$		0			0
$316$			8.00000i			0.450035i
$317$	12.0000			0.673987			0.336994	−	0.941507i	$-0.390590\pi$
$317$	12.0000			0.673987			0.336994	+	0.941507i	$0.390590\pi$
$318$		0			0
$319$	5.00000	+	2.00000i	0.279946	+	0.111979i
$320$	16.0000	−	8.00000i	0.894427	−	0.447214i
$321$		0			0
$322$	−72.0000			−4.01240
$323$		−	24.0000i		−	1.33540i
$324$		0			0
$325$	−8.00000	−	6.00000i	−0.443760	−	0.332820i
$326$	−26.0000			−1.44001
$327$		0			0
$328$		0			0
$329$			32.0000i			1.76422i
$330$		0			0
$331$			32.0000i			1.75888i	0.476011	+	0.879440i	$0.342082\pi$
$331$			32.0000i			1.75888i	−0.476011	+	0.879440i	$0.657918\pi$
$332$		−	14.0000i		−	0.768350i
$333$		0			0
$334$			48.0000i			2.62644i
$335$	−12.0000	−	24.0000i	−0.655630	−	1.31126i
$336$		0			0
$337$	−10.0000			−0.544735			−0.272367	−	0.962193i	$-0.587807\pi$
$337$	−10.0000			−0.544735			−0.272367	+	0.962193i	$0.587807\pi$
$338$	18.0000			0.979071
$339$		0			0
$340$	−24.0000	+	12.0000i	−1.30158	+	0.650791i
$341$	2.00000			0.108306
$342$		0			0
$343$			8.00000i			0.431959i
$344$		0			0
$345$		0			0
$346$		−	42.0000i		−	2.25793i
$347$			19.0000i			1.01997i	0.860182	+	0.509987i	$0.170350\pi$
$347$			19.0000i			1.01997i	−0.860182	+	0.509987i	$0.829650\pi$
$348$		0			0
$349$	−11.0000			−0.588817			−0.294408	−	0.955680i	$-0.595123\pi$
$349$	−11.0000			−0.588817			−0.294408	+	0.955680i	$0.595123\pi$
$350$	−32.0000	−	24.0000i	−1.71047	−	1.28285i
$351$		0			0
$352$		−	8.00000i		−	0.426401i
$353$		−	18.0000i		−	0.958043i	−0.877803	−	0.479022i	$-0.840992\pi$
$353$		−	18.0000i		−	0.958043i	0.877803	−	0.479022i	$-0.159008\pi$
$354$		0			0
$355$	4.00000	−	2.00000i	0.212298	−	0.106149i
$356$		−	4.00000i		−	0.212000i
$357$		0			0
$358$	−12.0000			−0.634220
$359$		−	21.0000i		−	1.10834i	−0.832404	−	0.554169i	$-0.813036\pi$
$359$		−	21.0000i		−	1.10834i	0.832404	−	0.554169i	$-0.186964\pi$
$360$		0			0
$361$	3.00000			0.157895
$362$	−14.0000			−0.735824
$363$		0			0
$364$	−16.0000			−0.838628
$365$	−30.0000	+	15.0000i	−1.57027	+	0.785136i
$366$		0			0
$367$	27.0000			1.40939			0.704694	−	0.709511i	$-0.251084\pi$
$367$	27.0000			1.40939			0.704694	+	0.709511i	$0.251084\pi$
$368$			36.0000i			1.87663i
$369$		0			0
$370$	4.00000	−	2.00000i	0.207950	−	0.103975i
$371$	36.0000			1.86903
$372$		0			0
$373$			2.00000i			0.103556i	0.998659	+	0.0517780i	$0.0164888\pi$
$373$			2.00000i			0.103556i	−0.998659	+	0.0517780i	$0.983511\pi$
$374$			12.0000i			0.620505i
$375$		0			0
$376$		0			0
$377$	−10.0000	−	4.00000i	−0.515026	−	0.206010i
$378$		0			0
$379$			22.0000i			1.13006i	0.825069	+	0.565032i	$0.191136\pi$
$379$			22.0000i			1.13006i	−0.825069	+	0.565032i	$0.808864\pi$
$380$	8.00000	+	16.0000i	0.410391	+	0.820783i
$381$		0			0
$382$		−	6.00000i		−	0.306987i
$383$		−	27.0000i		−	1.37964i	−0.723983	−	0.689818i	$-0.757691\pi$
$383$		−	27.0000i		−	1.37964i	0.723983	−	0.689818i	$-0.242309\pi$
$384$		0			0
$385$	−8.00000	+	4.00000i	−0.407718	+	0.203859i
$386$	12.0000			0.610784
$387$		0			0
$388$	22.0000			1.11688
$389$			3.00000i			0.152106i	0.997104	+	0.0760530i	$0.0242318\pi$
$389$			3.00000i			0.152106i	−0.997104	+	0.0760530i	$0.975768\pi$
$390$		0			0
$391$		−	54.0000i		−	2.73090i
$392$		0			0
$393$		0			0
$394$			46.0000i			2.31745i
$395$	−4.00000	−	8.00000i	−0.201262	−	0.402524i
$396$		0			0
$397$		−	16.0000i		−	0.803017i	−0.915855	−	0.401508i	$-0.868486\pi$
$397$		−	16.0000i		−	0.803017i	0.915855	−	0.401508i	$-0.131514\pi$
$398$	−38.0000			−1.90477
$399$		0			0
$400$	−12.0000	+	16.0000i	−0.600000	+	0.800000i
$401$	12.0000			0.599251			0.299626	−	0.954057i	$-0.403138\pi$
$401$	12.0000			0.599251			0.299626	+	0.954057i	$0.403138\pi$
$402$		0			0
$403$	−4.00000			−0.199254
$404$		−	6.00000i		−	0.298511i
$405$		0			0
$406$	−40.0000	−	16.0000i	−1.98517	−	0.794067i
$407$		−	1.00000i		−	0.0495682i
$408$		0			0
$409$		−	14.0000i		−	0.692255i	−0.938187	−	0.346128i	$-0.887496\pi$
$409$		−	14.0000i		−	0.692255i	0.938187	−	0.346128i	$-0.112504\pi$
$410$	−18.0000	−	36.0000i	−0.888957	−	1.77791i
$411$		0			0
$412$			12.0000i			0.591198i
$413$			32.0000i			1.57462i
$414$		0			0
$415$	7.00000	+	14.0000i	0.343616	+	0.687233i
$416$			16.0000i			0.784465i
$417$		0			0
$418$	8.00000			0.391293
$419$	−30.0000			−1.46560			−0.732798	−	0.680446i	$-0.761786\pi$
$419$	−30.0000			−1.46560			−0.732798	+	0.680446i	$0.761786\pi$
$420$		0			0
$421$		−	22.0000i		−	1.07221i	−0.844150	−	0.536107i	$-0.819894\pi$
$421$		−	22.0000i		−	1.07221i	0.844150	−	0.536107i	$-0.180106\pi$
$422$			4.00000i			0.194717i
$423$		0			0
$424$		0			0
$425$	18.0000	−	24.0000i	0.873128	−	1.16417i
$426$		0			0
$427$	−24.0000			−1.16144
$428$			8.00000i			0.386695i
$429$		0			0
$430$	4.00000	−	2.00000i	0.192897	−	0.0964486i
$431$	6.00000			0.289010			0.144505	−	0.989504i	$-0.453841\pi$
$431$	6.00000			0.289010			0.144505	+	0.989504i	$0.453841\pi$
$432$		0			0
$433$	7.00000			0.336399			0.168199	−	0.985753i	$-0.446205\pi$
$433$	7.00000			0.336399			0.168199	+	0.985753i	$0.446205\pi$
$434$	−16.0000			−0.768025
$435$		0			0
$436$	−2.00000			−0.0957826
$437$	−36.0000			−1.72211
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$		0			0
$442$		−	24.0000i		−	1.14156i
$443$	−4.00000			−0.190046			−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000			−0.190046			−0.0950229	+	0.995475i	$0.530292\pi$
$444$		0			0
$445$	2.00000	+	4.00000i	0.0948091	+	0.189618i
$446$		−	8.00000i		−	0.378811i
$447$		0			0
$448$			32.0000i			1.51186i
$449$			15.0000i			0.707894i	0.935266	+	0.353947i	$0.115161\pi$
$449$			15.0000i			0.707894i	−0.935266	+	0.353947i	$0.884839\pi$
$450$		0			0
$451$	−9.00000			−0.423793
$452$	16.0000			0.752577
$453$		0			0
$454$		−	14.0000i		−	0.657053i
$455$	16.0000	−	8.00000i	0.750092	−	0.375046i
$456$		0			0
$457$			10.0000i			0.467780i	0.972263	+	0.233890i	$0.0751456\pi$
$457$			10.0000i			0.467780i	−0.972263	+	0.233890i	$0.924854\pi$
$458$		−	8.00000i		−	0.373815i
$459$		0			0
$460$	18.0000	+	36.0000i	0.839254	+	1.67851i
$461$			33.0000i			1.53696i	0.639872	+	0.768482i	$0.278987\pi$
$461$			33.0000i			1.53696i	−0.639872	+	0.768482i	$0.721013\pi$
$462$		0			0
$463$			2.00000i			0.0929479i	0.998920	+	0.0464739i	$0.0147984\pi$
$463$			2.00000i			0.0929479i	−0.998920	+	0.0464739i	$0.985202\pi$
$464$	−8.00000	+	20.0000i	−0.371391	+	0.928477i
$465$		0			0
$466$		−	2.00000i		−	0.0926482i
$467$	18.0000			0.832941			0.416470	−	0.909149i	$-0.363267\pi$
$467$	18.0000			0.832941			0.416470	+	0.909149i	$0.363267\pi$
$468$		0			0
$469$	48.0000			2.21643
$470$	32.0000	−	16.0000i	1.47605	−	0.738025i
$471$		0			0
$472$		0			0
$473$		−	1.00000i		−	0.0459800i
$474$		0			0
$475$	−16.0000	−	12.0000i	−0.734130	−	0.550598i
$476$		−	48.0000i		−	2.20008i
$477$		0			0
$478$	52.0000			2.37842
$479$		0			0		1.00000			$0$
$479$		0			0		−1.00000			$\pi$
$480$		0			0
$481$			2.00000i			0.0911922i
$482$	22.0000			1.00207
$483$		0			0
$484$	20.0000			0.909091
$485$	−22.0000	+	11.0000i	−0.998969	+	0.499484i
$486$		0			0
$487$			4.00000i			0.181257i	0.995885	+	0.0906287i	$0.0288876\pi$
$487$			4.00000i			0.181257i	−0.995885	+	0.0906287i	$0.971112\pi$
$488$		0			0
$489$		0			0
$490$	36.0000	−	18.0000i	1.62631	−	0.813157i
$491$		−	8.00000i		−	0.361035i	−0.983572	−	0.180517i	$-0.942223\pi$
$491$		−	8.00000i		−	0.361035i	0.983572	−	0.180517i	$-0.0577772\pi$
$492$		0			0
$493$	12.0000	−	30.0000i	0.540453	−	1.35113i
$494$	−16.0000			−0.719874
$495$		0			0
$496$			8.00000i			0.359211i
$497$			8.00000i			0.358849i
$498$		0			0
$499$	−12.0000			−0.537194			−0.268597	−	0.963253i	$-0.586560\pi$
$499$	−12.0000			−0.537194			−0.268597	+	0.963253i	$0.586560\pi$
$500$	−4.00000	+	22.0000i	−0.178885	+	0.983870i
$501$		0			0
$502$			24.0000i			1.07117i
$503$	8.00000			0.356702			0.178351	−	0.983967i	$-0.442924\pi$
$503$	8.00000			0.356702			0.178351	+	0.983967i	$0.442924\pi$
$504$		0			0
$505$	3.00000	+	6.00000i	0.133498	+	0.266996i
$506$	18.0000			0.800198
$507$		0			0
$508$	14.0000			0.621150
$509$	28.0000			1.24108			0.620539	−	0.784176i	$-0.286914\pi$
$509$	28.0000			1.24108			0.620539	+	0.784176i	$0.286914\pi$
$510$		0			0
$511$		−	60.0000i		−	2.65424i
$512$	32.0000			1.41421
$513$		0			0
$514$		−	30.0000i		−	1.32324i
$515$	−6.00000	−	12.0000i	−0.264392	−	0.528783i
$516$		0			0
$517$		−	8.00000i		−	0.351840i
$518$			8.00000i			0.351500i
$519$		0			0
$520$		0			0
$521$	8.00000			0.350486			0.175243	−	0.984525i	$-0.443929\pi$
$521$	8.00000			0.350486			0.175243	+	0.984525i	$0.443929\pi$
$522$		0			0
$523$			2.00000i			0.0874539i	0.999044	+	0.0437269i	$0.0139232\pi$
$523$			2.00000i			0.0874539i	−0.999044	+	0.0437269i	$0.986077\pi$
$524$		−	40.0000i		−	1.74741i
$525$		0			0
$526$	−52.0000			−2.26731
$527$		−	12.0000i		−	0.522728i
$528$		0			0
$529$	−58.0000			−2.52174
$530$	−18.0000	−	36.0000i	−0.781870	−	1.56374i
$531$		0			0
$532$	−32.0000			−1.38738
$533$	18.0000			0.779667
$534$		0			0
$535$	−4.00000	−	8.00000i	−0.172935	−	0.345870i
$536$		0			0
$537$		0			0
$538$		−	20.0000i		−	0.862261i
$539$		−	9.00000i		−	0.387657i
$540$		0			0
$541$			32.0000i			1.37579i	0.725811	+	0.687894i	$0.241464\pi$
$541$			32.0000i			1.37579i	−0.725811	+	0.687894i	$0.758536\pi$
$542$		−	4.00000i		−	0.171815i
$543$		0			0
$544$	−48.0000			−2.05798
$545$	2.00000	−	1.00000i	0.0856706	−	0.0428353i
$546$		0			0
$547$		−	26.0000i		−	1.11168i	−0.831289	−	0.555840i	$-0.812397\pi$
$547$		−	26.0000i		−	1.11168i	0.831289	−	0.555840i	$-0.187603\pi$
$548$	32.0000			1.36697
$549$		0			0
$550$	8.00000	+	6.00000i	0.341121	+	0.255841i
$551$	−20.0000	−	8.00000i	−0.852029	−	0.340811i
$552$		0			0
$553$	16.0000			0.680389
$554$			4.00000i			0.169944i
$555$		0			0
$556$	22.0000			0.933008
$557$			45.0000i			1.90671i	0.301849	+	0.953356i	$0.402396\pi$
$557$			45.0000i			1.90671i	−0.301849	+	0.953356i	$0.597604\pi$
$558$		0			0
$559$			2.00000i			0.0845910i
$560$	−16.0000	−	32.0000i	−0.676123	−	1.35225i
$561$		0			0
$562$	48.0000			2.02476
$563$	36.0000			1.51722			0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000			1.51722			0.758610	+	0.651546i	$0.225879\pi$
$564$		0			0
$565$	−16.0000	+	8.00000i	−0.673125	+	0.336563i
$566$			44.0000i			1.84946i
$567$		0			0
$568$		0			0
$569$		−	38.0000i		−	1.59304i	−0.604610	−	0.796521i	$-0.706671\pi$
$569$		−	38.0000i		−	1.59304i	0.604610	−	0.796521i	$-0.293329\pi$
$570$		0			0
$571$	−23.0000			−0.962520			−0.481260	−	0.876578i	$-0.659821\pi$
$571$	−23.0000			−0.962520			−0.481260	+	0.876578i	$0.659821\pi$
$572$	4.00000			0.167248
$573$		0			0
$574$	72.0000			3.00522
$575$	−36.0000	−	27.0000i	−1.50130	−	1.12598i
$576$		0			0
$577$	−34.0000			−1.41544			−0.707719	−	0.706494i	$-0.750276\pi$
$577$	−34.0000			−1.41544			−0.707719	+	0.706494i	$0.750276\pi$
$578$	38.0000			1.58059
$579$		0			0
$580$	2.00000	+	24.0000i	0.0830455	+	0.996546i
$581$	−28.0000			−1.16164
$582$		0			0
$583$	−9.00000			−0.372742
$584$		0			0
$585$		0			0
$586$	−4.00000			−0.165238
$587$		−	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
$589$	−8.00000			−0.329634
$590$	32.0000	−	16.0000i	1.31742	−	0.658710i
$591$		0			0
$592$	4.00000			0.164399
$593$			14.0000i			0.574911i	0.957794	+	0.287456i	$0.0928094\pi$
$593$			14.0000i			0.574911i	−0.957794	+	0.287456i	$0.907191\pi$
$594$		0			0
$595$	24.0000	+	48.0000i	0.983904	+	1.96781i
$596$	−28.0000			−1.14692
$597$		0			0
$598$	−36.0000			−1.47215
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$			26.0000i			1.06056i	0.847822	+	0.530281i	$0.177914\pi$
$601$			26.0000i			1.06056i	−0.847822	+	0.530281i	$0.822086\pi$
$602$			8.00000i			0.326056i
$603$		0			0
$604$	−34.0000			−1.38344
$605$	−20.0000	+	10.0000i	−0.813116	+	0.406558i
$606$		0			0
$607$	40.0000			1.62355			0.811775	−	0.583970i	$-0.198502\pi$
$607$	40.0000			1.62355			0.811775	+	0.583970i	$0.198502\pi$
$608$			32.0000i			1.29777i
$609$		0			0
$610$	12.0000	+	24.0000i	0.485866	+	0.971732i
$611$			16.0000i			0.647291i
$612$		0			0
$613$			6.00000i			0.242338i	0.992632	+	0.121169i	$0.0386643\pi$
$613$			6.00000i			0.242338i	−0.992632	+	0.121169i	$0.961336\pi$
$614$	30.0000			1.21070
$615$		0			0
$616$		0			0
$617$	−36.0000			−1.44931			−0.724653	−	0.689114i	$-0.758000\pi$
$617$	−36.0000			−1.44931			−0.724653	+	0.689114i	$0.758000\pi$
$618$		0			0
$619$		−	10.0000i		−	0.401934i	−0.979598	−	0.200967i	$-0.935592\pi$
$619$		−	10.0000i		−	0.401934i	0.979598	−	0.200967i	$-0.0644084\pi$
$620$	4.00000	+	8.00000i	0.160644	+	0.321288i
$621$		0			0
$622$		−	46.0000i		−	1.84443i
$623$	−8.00000			−0.320513
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$		−	16.0000i		−	0.639489i
$627$		0			0
$628$	28.0000			1.11732
$629$	−6.00000			−0.239236
$630$		0			0
$631$	32.0000			1.27390			0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000			1.27390			0.636950	+	0.770905i	$0.280196\pi$
$632$		0			0
$633$		0			0
$634$	24.0000			0.953162
$635$	−14.0000	+	7.00000i	−0.555573	+	0.277787i
$636$		0			0
$637$			18.0000i			0.713186i
$638$	10.0000	+	4.00000i	0.395904	+	0.158362i
$639$		0			0
$640$		0			0
$641$			33.0000i			1.30342i	0.758468	+	0.651711i	$0.225948\pi$
$641$			33.0000i			1.30342i	−0.758468	+	0.651711i	$0.774052\pi$
$642$		0			0
$643$			24.0000i			0.946468i	0.880937	+	0.473234i	$0.156913\pi$
$643$			24.0000i			0.946468i	−0.880937	+	0.473234i	$0.843087\pi$
$644$	−72.0000			−2.83720
$645$		0			0
$646$		−	48.0000i		−	1.88853i
$647$			17.0000i			0.668339i	0.942513	+	0.334169i	$0.108456\pi$
$647$			17.0000i			0.668339i	−0.942513	+	0.334169i	$0.891544\pi$
$648$		0			0
$649$		−	8.00000i		−	0.314027i
$650$	−16.0000	−	12.0000i	−0.627572	−	0.470679i
$651$		0			0
$652$	−26.0000			−1.01824
$653$	24.0000			0.939193			0.469596	−	0.882881i	$-0.344399\pi$
$653$	24.0000			0.939193			0.469596	+	0.882881i	$0.344399\pi$
$654$		0			0
$655$	20.0000	+	40.0000i	0.781465	+	1.56293i
$656$		−	36.0000i		−	1.40556i
$657$		0			0
$658$			64.0000i			2.49498i
$659$		−	29.0000i		−	1.12968i	−0.825201	−	0.564840i	$-0.808938\pi$
$659$		−	29.0000i		−	1.12968i	0.825201	−	0.564840i	$-0.191062\pi$
$660$		0			0
$661$	−5.00000			−0.194477			−0.0972387	−	0.995261i	$-0.531001\pi$
$661$	−5.00000			−0.194477			−0.0972387	+	0.995261i	$0.531001\pi$
$662$			64.0000i			2.48743i
$663$		0			0
$664$		0			0
$665$	32.0000	−	16.0000i	1.24091	−	0.620453i
$666$		0			0
$667$	−45.0000	−	18.0000i	−1.74241	−	0.696963i
$668$			48.0000i			1.85718i
$669$		0			0
$670$	−24.0000	−	48.0000i	−0.927201	−	1.85440i
$671$	6.00000			0.231627
$672$		0			0
$673$			32.0000i			1.23351i	0.787155	+	0.616755i	$0.211553\pi$
$673$			32.0000i			1.23351i	−0.787155	+	0.616755i	$0.788447\pi$
$674$	−20.0000			−0.770371
$675$		0			0
$676$	18.0000			0.692308
$677$	18.0000			0.691796			0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000			0.691796			0.345898	+	0.938272i	$0.387574\pi$
$678$		0			0
$679$		−	44.0000i		−	1.68857i
$680$		0			0
$681$		0			0
$682$	4.00000			0.153168
$683$		−	9.00000i		−	0.344375i	−0.985064	−	0.172188i	$-0.944916\pi$
$683$		−	9.00000i		−	0.344375i	0.985064	−	0.172188i	$-0.0550836\pi$
$684$		0			0
$685$	−32.0000	+	16.0000i	−1.22266	+	0.611329i
$686$			16.0000i			0.610883i
$687$		0			0
$688$	4.00000			0.152499
$689$	18.0000			0.685745
$690$		0			0
$691$	20.0000			0.760836			0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000			0.760836			0.380418	+	0.924815i	$0.375780\pi$
$692$		−	42.0000i		−	1.59660i
$693$		0			0
$694$			38.0000i			1.44246i
$695$	−22.0000	+	11.0000i	−0.834508	+	0.417254i
$696$		0			0
$697$			54.0000i			2.04540i
$698$	−22.0000			−0.832712
$699$		0			0
$700$	−32.0000	−	24.0000i	−1.20949	−	0.907115i
$701$	−6.00000			−0.226617			−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000			−0.226617			−0.113308	+	0.993560i	$0.536145\pi$
$702$		0			0
$703$			4.00000i			0.150863i
$704$		−	8.00000i		−	0.301511i
$705$		0			0
$706$		−	36.0000i		−	1.35488i
$707$	−12.0000			−0.451306
$708$		0			0
$709$	19.0000			0.713560			0.356780	−	0.934188i	$-0.383875\pi$
$709$	19.0000			0.713560			0.356780	+	0.934188i	$0.383875\pi$
$710$	8.00000	−	4.00000i	0.300235	−	0.150117i
$711$		0			0
$712$		0			0
$713$	−18.0000			−0.674105
$714$		0			0
$715$	−4.00000	+	2.00000i	−0.149592	+	0.0747958i
$716$	−12.0000			−0.448461
$717$		0			0
$718$		−	42.0000i		−	1.56743i
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	24.0000			0.893807
$722$	6.00000			0.223297
$723$		0			0
$724$	−14.0000			−0.520306
$725$	−14.0000	−	23.0000i	−0.519947	−	0.854199i
$726$		0			0
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$		0			0
$730$	−60.0000	+	30.0000i	−2.22070	+	1.11035i
$731$	−6.00000			−0.221918
$732$		0			0
$733$	2.00000			0.0738717			0.0369358	−	0.999318i	$-0.488240\pi$
$733$	2.00000			0.0738717			0.0369358	+	0.999318i	$0.488240\pi$
$734$	54.0000			1.99318
$735$		0			0
$736$			72.0000i			2.65396i
$737$	−12.0000			−0.442026
$738$		0			0
$739$			10.0000i			0.367856i	0.982940	+	0.183928i	$0.0588813\pi$
$739$			10.0000i			0.367856i	−0.982940	+	0.183928i	$0.941119\pi$
$740$	4.00000	−	2.00000i	0.147043	−	0.0735215i
$741$		0			0
$742$	72.0000			2.64320
$743$	−6.00000			−0.220119			−0.110059	−	0.993925i	$-0.535104\pi$
$743$	−6.00000			−0.220119			−0.110059	+	0.993925i	$0.535104\pi$
$744$		0			0
$745$	28.0000	−	14.0000i	1.02584	−	0.512920i
$746$			4.00000i			0.146450i
$747$		0			0
$748$			12.0000i			0.438763i
$749$	16.0000			0.584627
$750$		0			0
$751$			28.0000i			1.02173i	0.859660	+	0.510867i	$0.170676\pi$
$751$			28.0000i			1.02173i	−0.859660	+	0.510867i	$0.829324\pi$
$752$	32.0000			1.16692
$753$		0			0
$754$	−20.0000	−	8.00000i	−0.728357	−	0.291343i
$755$	34.0000	−	17.0000i	1.23739	−	0.618693i
$756$		0			0
$757$	−33.0000			−1.19941			−0.599703	−	0.800223i	$-0.704714\pi$
$757$	−33.0000			−1.19941			−0.599703	+	0.800223i	$0.704714\pi$
$758$			44.0000i			1.59815i
$759$		0			0
$760$		0			0
$761$	34.0000			1.23250			0.616250	−	0.787551i	$-0.288651\pi$
$761$	34.0000			1.23250			0.616250	+	0.787551i	$0.288651\pi$
$762$		0			0
$763$			4.00000i			0.144810i
$764$		−	6.00000i		−	0.217072i
$765$		0			0
$766$		−	54.0000i		−	1.95110i
$767$			16.0000i			0.577727i
$768$		0			0
$769$		−	14.0000i		−	0.504853i	−0.967616	−	0.252426i	$-0.918771\pi$
$769$		−	14.0000i		−	0.504853i	0.967616	−	0.252426i	$-0.0812286\pi$
$770$	−16.0000	+	8.00000i	−0.576600	+	0.288300i
$771$		0			0
$772$	12.0000			0.431889
$773$	−24.0000			−0.863220			−0.431610	−	0.902060i	$-0.642054\pi$
$773$	−24.0000			−0.863220			−0.431610	+	0.902060i	$0.642054\pi$
$774$		0			0
$775$	−8.00000	−	6.00000i	−0.287368	−	0.215526i
$776$		0			0
$777$		0			0
$778$			6.00000i			0.215110i
$779$	36.0000			1.28983
$780$		0			0
$781$		−	2.00000i		−	0.0715656i
$782$		−	108.000i		−	3.86207i
$783$		0			0
$784$	36.0000			1.28571
$785$	−28.0000	+	14.0000i	−0.999363	+	0.499681i
$786$		0			0
$787$		−	42.0000i		−	1.49714i	−0.663057	−	0.748569i	$-0.730741\pi$
$787$		−	42.0000i		−	1.49714i	0.663057	−	0.748569i	$-0.269259\pi$
$788$			46.0000i			1.63868i
$789$		0			0
$790$	−8.00000	−	16.0000i	−0.284627	−	0.569254i

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)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +2.00000 q^{4} +(-2.00000 + 1.00000i) q^{5} -4.00000i q^{7} +O(q^{10})\)	\(q+2.00000 q^{2} +2.00000 q^{4} +(-2.00000 + 1.00000i) q^{5} -4.00000i q^{7} +(-4.00000 + 2.00000i) q^{10} +1.00000i q^{11} -2.00000i q^{13} -8.00000i q^{14} -4.00000 q^{16} +6.00000 q^{17} -4.00000i q^{19} +(-4.00000 + 2.00000i) q^{20} +2.00000i q^{22} -9.00000i q^{23} +(3.00000 - 4.00000i) q^{25} -4.00000i q^{26} -8.00000i q^{28} +(2.00000 - 5.00000i) q^{29} -2.00000i q^{31} -8.00000 q^{32} +12.0000 q^{34} +(4.00000 + 8.00000i) q^{35} -1.00000 q^{37} -8.00000i q^{38} +9.00000i q^{41} -1.00000 q^{43} +2.00000i q^{44} -18.0000i q^{46} -8.00000 q^{47} -9.00000 q^{49} +(6.00000 - 8.00000i) q^{50} -4.00000i q^{52} +9.00000i q^{53} +(-1.00000 - 2.00000i) q^{55} +(4.00000 - 10.0000i) q^{58} -8.00000 q^{59} -6.00000i q^{61} -4.00000i q^{62} -8.00000 q^{64} +(2.00000 + 4.00000i) q^{65} +12.0000i q^{67} +12.0000 q^{68} +(8.00000 + 16.0000i) q^{70} -2.00000 q^{71} +15.0000 q^{73} -2.00000 q^{74} -8.00000i q^{76} +4.00000 q^{77} +4.00000i q^{79} +(8.00000 - 4.00000i) q^{80} +18.0000i q^{82} -7.00000i q^{83} +(-12.0000 + 6.00000i) q^{85} -2.00000 q^{86} -2.00000i q^{89} -8.00000 q^{91} -18.0000i q^{92} -16.0000 q^{94} +(4.00000 + 8.00000i) q^{95} +11.0000 q^{97} -18.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 4 q^{4} - 4 q^{5}+O(q^{10}) \) 2 * q + 4 * q^2 + 4 * q^4 - 4 * q^5	\( 2 q + 4 q^{2} + 4 q^{4} - 4 q^{5} - 8 q^{10} - 8 q^{16} + 12 q^{17} - 8 q^{20} + 6 q^{25} + 4 q^{29} - 16 q^{32} + 24 q^{34} + 8 q^{35} - 2 q^{37} - 2 q^{43} - 16 q^{47} - 18 q^{49} + 12 q^{50} - 2 q^{55} + 8 q^{58} - 16 q^{59} - 16 q^{64} + 4 q^{65} + 24 q^{68} + 16 q^{70} - 4 q^{71} + 30 q^{73} - 4 q^{74} + 8 q^{77} + 16 q^{80} - 24 q^{85} - 4 q^{86} - 16 q^{91} - 32 q^{94} + 8 q^{95} + 22 q^{97} - 36 q^{98}+O(q^{100}) \) 2 * q + 4 * q^2 + 4 * q^4 - 4 * q^5 - 8 * q^10 - 8 * q^16 + 12 * q^17 - 8 * q^20 + 6 * q^25 + 4 * q^29 - 16 * q^32 + 24 * q^34 + 8 * q^35 - 2 * q^37 - 2 * q^43 - 16 * q^47 - 18 * q^49 + 12 * q^50 - 2 * q^55 + 8 * q^58 - 16 * q^59 - 16 * q^64 + 4 * q^65 + 24 * q^68 + 16 * q^70 - 4 * q^71 + 30 * q^73 - 4 * q^74 + 8 * q^77 + 16 * q^80 - 24 * q^85 - 4 * q^86 - 16 * q^91 - 32 * q^94 + 8 * q^95 + 22 * q^97 - 36 * q^98

Introduction

L-functions

Modular forms

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