LMFDB - Embedded newform 435.2.f.d.289.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [435,2,Mod(289,435)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(435, 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("435.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 435 = 3 \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	435.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:	$3.47349248793$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			289.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	435.289
Dual form			435.2.f.d.289.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} +(2.00000 - 1.00000i) q^{5} -2.00000 q^{6} -4.00000i q^{7} +1.00000 q^{9} +O(q^{10})$	\(q+2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} +(2.00000 - 1.00000i) q^{5} -2.00000 q^{6} -4.00000i q^{7} +1.00000 q^{9} +(4.00000 - 2.00000i) q^{10} +1.00000i q^{11} -2.00000 q^{12} -2.00000i q^{13} -8.00000i q^{14} +(-2.00000 + 1.00000i) q^{15} -4.00000 q^{16} +6.00000 q^{17} +2.00000 q^{18} +4.00000i q^{19} +(4.00000 - 2.00000i) q^{20} +4.00000i q^{21} +2.00000i q^{22} +9.00000i q^{23} +(3.00000 - 4.00000i) q^{25} -4.00000i q^{26} -1.00000 q^{27} -8.00000i q^{28} +(-2.00000 - 5.00000i) q^{29} +(-4.00000 + 2.00000i) q^{30} +2.00000i q^{31} -8.00000 q^{32} -1.00000i q^{33} +12.0000 q^{34} +(-4.00000 - 8.00000i) q^{35} +2.00000 q^{36} +1.00000 q^{37} +8.00000i q^{38} +2.00000i q^{39} +9.00000i q^{41} +8.00000i q^{42} +1.00000 q^{43} +2.00000i q^{44} +(2.00000 - 1.00000i) q^{45} +18.0000i q^{46} -8.00000 q^{47} +4.00000 q^{48} -9.00000 q^{49} +(6.00000 - 8.00000i) q^{50} -6.00000 q^{51} -4.00000i q^{52} -9.00000i q^{53} -2.00000 q^{54} +(1.00000 + 2.00000i) q^{55} -4.00000i q^{57} +(-4.00000 - 10.0000i) q^{58} +8.00000 q^{59} +(-4.00000 + 2.00000i) q^{60} +6.00000i q^{61} +4.00000i q^{62} -4.00000i q^{63} -8.00000 q^{64} +(-2.00000 - 4.00000i) q^{65} -2.00000i q^{66} +12.0000i q^{67} +12.0000 q^{68} -9.00000i q^{69} +(-8.00000 - 16.0000i) q^{70} +2.00000 q^{71} -15.0000 q^{73} +2.00000 q^{74} +(-3.00000 + 4.00000i) q^{75} +8.00000i q^{76} +4.00000 q^{77} +4.00000i q^{78} -4.00000i q^{79} +(-8.00000 + 4.00000i) q^{80} +1.00000 q^{81} +18.0000i q^{82} +7.00000i q^{83} +8.00000i q^{84} +(12.0000 - 6.00000i) q^{85} +2.00000 q^{86} +(2.00000 + 5.00000i) q^{87} -2.00000i q^{89} +(4.00000 - 2.00000i) q^{90} -8.00000 q^{91} +18.0000i q^{92} -2.00000i q^{93} -16.0000 q^{94} +(4.00000 + 8.00000i) q^{95} +8.00000 q^{96} -11.0000 q^{97} -18.0000 q^{98} +1.00000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} - 2 q^{3} + 4 q^{4} + 4 q^{5} - 4 q^{6} + 2 q^{9}+O(q^{10}) $ 2 * q + 4 * q^2 - 2 * q^3 + 4 * q^4 + 4 * q^5 - 4 * q^6 + 2 * q^9	$ 2 q + 4 q^{2} - 2 q^{3} + 4 q^{4} + 4 q^{5} - 4 q^{6} + 2 q^{9} + 8 q^{10} - 4 q^{12} - 4 q^{15} - 8 q^{16} + 12 q^{17} + 4 q^{18} + 8 q^{20} + 6 q^{25} - 2 q^{27} - 4 q^{29} - 8 q^{30} - 16 q^{32} + 24 q^{34} - 8 q^{35} + 4 q^{36} + 2 q^{37} + 2 q^{43} + 4 q^{45} - 16 q^{47} + 8 q^{48} - 18 q^{49} + 12 q^{50} - 12 q^{51} - 4 q^{54} + 2 q^{55} - 8 q^{58} + 16 q^{59} - 8 q^{60} - 16 q^{64} - 4 q^{65} + 24 q^{68} - 16 q^{70} + 4 q^{71} - 30 q^{73} + 4 q^{74} - 6 q^{75} + 8 q^{77} - 16 q^{80} + 2 q^{81} + 24 q^{85} + 4 q^{86} + 4 q^{87} + 8 q^{90} - 16 q^{91} - 32 q^{94} + 8 q^{95} + 16 q^{96} - 22 q^{97} - 36 q^{98}+O(q^{100}) $ 2 * q + 4 * q^2 - 2 * q^3 + 4 * q^4 + 4 * q^5 - 4 * q^6 + 2 * q^9 + 8 * q^10 - 4 * q^12 - 4 * q^15 - 8 * q^16 + 12 * q^17 + 4 * q^18 + 8 * q^20 + 6 * q^25 - 2 * q^27 - 4 * q^29 - 8 * q^30 - 16 * q^32 + 24 * q^34 - 8 * q^35 + 4 * q^36 + 2 * q^37 + 2 * q^43 + 4 * q^45 - 16 * q^47 + 8 * q^48 - 18 * q^49 + 12 * q^50 - 12 * q^51 - 4 * q^54 + 2 * q^55 - 8 * q^58 + 16 * q^59 - 8 * q^60 - 16 * q^64 - 4 * q^65 + 24 * q^68 - 16 * q^70 + 4 * q^71 - 30 * q^73 + 4 * q^74 - 6 * q^75 + 8 * q^77 - 16 * q^80 + 2 * q^81 + 24 * q^85 + 4 * q^86 + 4 * q^87 + 8 * q^90 - 16 * q^91 - 32 * q^94 + 8 * q^95 + 16 * q^96 - 22 * q^97 - 36 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/435\mathbb{Z}\right)^\times$.

$n$	$31$	$146$	$262$
$\chi(n)$	$-1$	$1$	$-1$

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	$0.250000\pi$
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$	2.00000			1.00000
$5$	2.00000	−	1.00000i	0.894427	−	0.447214i
$5$	2.00000	−	1.00000i	0.894427	−	0.447214i
$6$	−2.00000			−0.816497
$7$		−	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
$9$	1.00000			0.333333
$10$	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	$0.951829\pi$
$12$	−2.00000			−0.577350
$13$		−	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
$15$	−2.00000	+	1.00000i	−0.516398	+	0.258199i
$16$	−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	$0.240633\pi$
$18$	2.00000			0.471405
$19$			4.00000i			0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$			4.00000i			0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$	4.00000	−	2.00000i	0.894427	−	0.447214i
$21$			4.00000i			0.872872i
$22$			2.00000i			0.426401i
$23$			9.00000i			1.87663i	0.345782	+	0.938315i	$0.387614\pi$
$23$			9.00000i			1.87663i	−0.345782	+	0.938315i	$0.612386\pi$
$24$		0			0
$25$	3.00000	−	4.00000i	0.600000	−	0.800000i
$26$		−	4.00000i		−	0.784465i
$27$	−1.00000			−0.192450
$28$		−	8.00000i		−	1.51186i
$29$	−2.00000	−	5.00000i	−0.371391	−	0.928477i
$29$	−2.00000	−	5.00000i	−0.371391	−	0.928477i
$30$	−4.00000	+	2.00000i	−0.730297	+	0.365148i
$31$			2.00000i			0.359211i	0.983739	+	0.179605i	$0.0574821\pi$
$31$			2.00000i			0.359211i	−0.983739	+	0.179605i	$0.942518\pi$
$32$	−8.00000			−1.41421
$33$		−	1.00000i		−	0.174078i
$34$	12.0000			2.05798
$35$	−4.00000	−	8.00000i	−0.676123	−	1.35225i
$36$	2.00000			0.333333
$37$	1.00000			0.164399			0.0821995	−	0.996616i	$-0.473806\pi$
$37$	1.00000			0.164399			0.0821995	+	0.996616i	$0.473806\pi$
$38$			8.00000i			1.29777i
$39$			2.00000i			0.320256i
$40$		0			0
$41$			9.00000i			1.40556i	0.711405	+	0.702782i	$0.248059\pi$
$41$			9.00000i			1.40556i	−0.711405	+	0.702782i	$0.751941\pi$
$42$			8.00000i			1.23443i
$43$	1.00000			0.152499			0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000			0.152499			0.0762493	+	0.997089i	$0.475706\pi$
$44$			2.00000i			0.301511i
$45$	2.00000	−	1.00000i	0.298142	−	0.149071i
$46$			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	$0.698301\pi$
$48$	4.00000			0.577350
$49$	−9.00000			−1.28571
$50$	6.00000	−	8.00000i	0.848528	−	1.13137i
$51$	−6.00000			−0.840168
$52$		−	4.00000i		−	0.554700i
$53$		−	9.00000i		−	1.23625i	−0.786082	−	0.618123i	$-0.787894\pi$
$53$		−	9.00000i		−	1.23625i	0.786082	−	0.618123i	$-0.212106\pi$
$54$	−2.00000			−0.272166
$55$	1.00000	+	2.00000i	0.134840	+	0.269680i
$56$		0			0
$57$		−	4.00000i		−	0.529813i
$58$	−4.00000	−	10.0000i	−0.525226	−	1.31306i
$59$	8.00000			1.04151			0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000			1.04151			0.520756	+	0.853706i	$0.325650\pi$
$60$	−4.00000	+	2.00000i	−0.516398	+	0.258199i
$61$			6.00000i			0.768221i	0.923287	+	0.384111i	$0.125492\pi$
$61$			6.00000i			0.768221i	−0.923287	+	0.384111i	$0.874508\pi$
$62$			4.00000i			0.508001i
$63$		−	4.00000i		−	0.503953i
$64$	−8.00000			−1.00000
$65$	−2.00000	−	4.00000i	−0.248069	−	0.496139i
$66$		−	2.00000i		−	0.246183i
$67$			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
$69$		−	9.00000i		−	1.08347i
$70$	−8.00000	−	16.0000i	−0.956183	−	1.91237i
$71$	2.00000			0.237356			0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000			0.237356			0.118678	+	0.992933i	$0.462134\pi$
$72$		0			0
$73$	−15.0000			−1.75562			−0.877809	−	0.479012i	$-0.840995\pi$
$73$	−15.0000			−1.75562			−0.877809	+	0.479012i	$0.840995\pi$
$74$	2.00000			0.232495
$75$	−3.00000	+	4.00000i	−0.346410	+	0.461880i
$76$			8.00000i			0.917663i
$77$	4.00000			0.455842
$78$			4.00000i			0.452911i
$79$		−	4.00000i		−	0.450035i	−0.974355	−	0.225018i	$-0.927756\pi$
$79$		−	4.00000i		−	0.450035i	0.974355	−	0.225018i	$-0.0722440\pi$
$80$	−8.00000	+	4.00000i	−0.894427	+	0.447214i
$81$	1.00000			0.111111
$82$			18.0000i			1.98777i
$83$			7.00000i			0.768350i	0.923260	+	0.384175i	$0.125514\pi$
$83$			7.00000i			0.768350i	−0.923260	+	0.384175i	$0.874486\pi$
$84$			8.00000i			0.872872i
$85$	12.0000	−	6.00000i	1.30158	−	0.650791i
$86$	2.00000			0.215666
$87$	2.00000	+	5.00000i	0.214423	+	0.536056i
$88$		0			0
$89$		−	2.00000i		−	0.212000i	−0.994366	−	0.106000i	$-0.966196\pi$
$89$		−	2.00000i		−	0.212000i	0.994366	−	0.106000i	$-0.0338043\pi$
$90$	4.00000	−	2.00000i	0.421637	−	0.210819i
$91$	−8.00000			−0.838628
$92$			18.0000i			1.87663i
$93$		−	2.00000i		−	0.207390i
$94$	−16.0000			−1.65027
$95$	4.00000	+	8.00000i	0.410391	+	0.820783i
$96$	8.00000			0.816497
$97$	−11.0000			−1.11688			−0.558440	−	0.829545i	$-0.688600\pi$
$97$	−11.0000			−1.11688			−0.558440	+	0.829545i	$0.688600\pi$
$98$	−18.0000			−1.81827
$99$			1.00000i			0.100504i
$100$	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	$-0.0476877\pi$
$102$	−12.0000			−1.18818
$103$			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
$105$	4.00000	+	8.00000i	0.390360	+	0.780720i
$106$		−	18.0000i		−	1.74831i
$107$		−	4.00000i		−	0.386695i	−0.981130	−	0.193347i	$-0.938066\pi$
$107$		−	4.00000i		−	0.386695i	0.981130	−	0.193347i	$-0.0619344\pi$
$108$	−2.00000			−0.192450
$109$	−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
$111$	−1.00000			−0.0949158
$112$			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	$0.377200\pi$
$114$		−	8.00000i		−	0.749269i
$115$	9.00000	+	18.0000i	0.839254	+	1.67851i
$116$	−4.00000	−	10.0000i	−0.371391	−	0.928477i
$117$		−	2.00000i		−	0.184900i
$118$	16.0000			1.47292
$119$		−	24.0000i		−	2.20008i
$120$		0			0
$121$	10.0000			0.909091
$122$			12.0000i			1.08643i
$123$		−	9.00000i		−	0.811503i
$124$			4.00000i			0.359211i
$125$	2.00000	−	11.0000i	0.178885	−	0.983870i
$126$		−	8.00000i		−	0.712697i
$127$	−7.00000			−0.621150			−0.310575	−	0.950549i	$-0.600522\pi$
$127$	−7.00000			−0.621150			−0.310575	+	0.950549i	$0.600522\pi$
$128$		0			0
$129$	−1.00000			−0.0880451
$130$	−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	$-0.338289\pi$
$132$		−	2.00000i		−	0.174078i
$133$	16.0000			1.38738
$134$			24.0000i			2.07328i
$135$	−2.00000	+	1.00000i	−0.172133	+	0.0860663i
$136$		0			0
$137$	16.0000			1.36697			0.683486	−	0.729964i	$-0.260463\pi$
$137$	16.0000			1.36697			0.683486	+	0.729964i	$0.260463\pi$
$138$		−	18.0000i		−	1.53226i
$139$	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
$141$	8.00000			0.673722
$142$	4.00000			0.335673
$143$	2.00000			0.167248
$144$	−4.00000			−0.333333
$145$	−9.00000	−	8.00000i	−0.747409	−	0.664364i
$146$	−30.0000			−2.48282
$147$	9.00000			0.742307
$148$	2.00000			0.164399
$149$	14.0000			1.14692			0.573462	−	0.819232i	$-0.305600\pi$
$149$	14.0000			1.14692			0.573462	+	0.819232i	$0.305600\pi$
$150$	−6.00000	+	8.00000i	−0.489898	+	0.653197i
$151$	−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
$153$	6.00000			0.485071
$154$	8.00000			0.644658
$155$	2.00000	+	4.00000i	0.160644	+	0.321288i
$156$			4.00000i			0.320256i
$157$	−14.0000			−1.11732			−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000			−1.11732			−0.558661	+	0.829396i	$0.688685\pi$
$158$		−	8.00000i		−	0.636446i
$159$			9.00000i			0.713746i
$160$	−16.0000	+	8.00000i	−1.26491	+	0.632456i
$161$	36.0000			2.83720
$162$	2.00000			0.157135
$163$	13.0000			1.01824			0.509119	−	0.860696i	$-0.329971\pi$
$163$	13.0000			1.01824			0.509119	+	0.860696i	$0.329971\pi$
$164$			18.0000i			1.40556i
$165$	−1.00000	−	2.00000i	−0.0778499	−	0.155700i
$166$			14.0000i			1.08661i
$167$		−	24.0000i		−	1.85718i	−0.371113	−	0.928588i	$-0.621024\pi$
$167$		−	24.0000i		−	1.85718i	0.371113	−	0.928588i	$-0.378976\pi$
$168$		0			0
$169$	9.00000			0.692308
$170$	24.0000	−	12.0000i	1.84072	−	0.920358i
$171$			4.00000i			0.305888i
$172$	2.00000			0.152499
$173$			21.0000i			1.59660i	0.602260	+	0.798300i	$0.294267\pi$
$173$			21.0000i			1.59660i	−0.602260	+	0.798300i	$0.705733\pi$
$174$	4.00000	+	10.0000i	0.303239	+	0.758098i
$175$	−16.0000	−	12.0000i	−1.20949	−	0.907115i
$176$		−	4.00000i		−	0.301511i
$177$	−8.00000			−0.601317
$178$		−	4.00000i		−	0.299813i
$179$	6.00000			0.448461			0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000			0.448461			0.224231	+	0.974536i	$0.428013\pi$
$180$	4.00000	−	2.00000i	0.298142	−	0.149071i
$181$	−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
$183$		−	6.00000i		−	0.443533i
$184$		0			0
$185$	2.00000	−	1.00000i	0.147043	−	0.0735215i
$186$		−	4.00000i		−	0.293294i
$187$			6.00000i			0.438763i
$188$	−16.0000			−1.16692
$189$			4.00000i			0.290957i
$190$	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	$-0.0346163\pi$
$192$	8.00000			0.577350
$193$	−6.00000			−0.431889			−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000			−0.431889			−0.215945	+	0.976406i	$0.569283\pi$
$194$	−22.0000			−1.57951
$195$	2.00000	+	4.00000i	0.143223	+	0.286446i
$196$	−18.0000			−1.28571
$197$		−	23.0000i		−	1.63868i	−0.573306	−	0.819341i	$-0.694340\pi$
$197$		−	23.0000i		−	1.63868i	0.573306	−	0.819341i	$-0.305660\pi$
$198$			2.00000i			0.142134i
$199$	−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
$201$		−	12.0000i		−	0.846415i
$202$		−	6.00000i		−	0.422159i
$203$	−20.0000	+	8.00000i	−1.40372	+	0.561490i
$204$	−12.0000			−0.840168
$205$	9.00000	+	18.0000i	0.628587	+	1.25717i
$206$			12.0000i			0.836080i
$207$			9.00000i			0.625543i
$208$			8.00000i			0.554700i
$209$	−4.00000			−0.276686
$210$	8.00000	+	16.0000i	0.552052	+	1.10410i
$211$		−	2.00000i		−	0.137686i	−0.997628	−	0.0688428i	$-0.978069\pi$
$211$		−	2.00000i		−	0.137686i	0.997628	−	0.0688428i	$-0.0219307\pi$
$212$		−	18.0000i		−	1.23625i
$213$	−2.00000			−0.137038
$214$		−	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
$219$	15.0000			1.01361
$220$	2.00000	+	4.00000i	0.134840	+	0.269680i
$221$		−	12.0000i		−	0.807207i
$222$	−2.00000			−0.134231
$223$		−	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
$225$	3.00000	−	4.00000i	0.200000	−	0.266667i
$226$	16.0000			1.06430
$227$			7.00000i			0.464606i	0.972643	+	0.232303i	$0.0746261\pi$
$227$			7.00000i			0.464606i	−0.972643	+	0.232303i	$0.925374\pi$
$228$		−	8.00000i		−	0.529813i
$229$			4.00000i			0.264327i	0.991228	+	0.132164i	$0.0421925\pi$
$229$			4.00000i			0.264327i	−0.991228	+	0.132164i	$0.957808\pi$
$230$	18.0000	+	36.0000i	1.18688	+	2.37377i
$231$	−4.00000			−0.263181
$232$		0			0
$233$			1.00000i			0.0655122i	0.999463	+	0.0327561i	$0.0104285\pi$
$233$			1.00000i			0.0655122i	−0.999463	+	0.0327561i	$0.989572\pi$
$234$		−	4.00000i		−	0.261488i
$235$	−16.0000	+	8.00000i	−1.04372	+	0.521862i
$236$	16.0000			1.04151
$237$			4.00000i			0.259828i
$238$		−	48.0000i		−	3.11138i
$239$	−26.0000			−1.68180			−0.840900	−	0.541190i	$-0.817974\pi$
$239$	−26.0000			−1.68180			−0.840900	+	0.541190i	$0.817974\pi$
$240$	8.00000	−	4.00000i	0.516398	−	0.258199i
$241$	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
$243$	−1.00000			−0.0641500
$244$			12.0000i			0.768221i
$245$	−18.0000	+	9.00000i	−1.14998	+	0.574989i
$246$		−	18.0000i		−	1.14764i
$247$	8.00000			0.509028
$248$		0			0
$249$		−	7.00000i		−	0.443607i
$250$	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	$0.876365\pi$
$252$		−	8.00000i		−	0.503953i
$253$	−9.00000			−0.565825
$254$	−14.0000			−0.878438
$255$	−12.0000	+	6.00000i	−0.751469	+	0.375735i
$256$	16.0000			1.00000
$257$			15.0000i			0.935674i	0.883815	+	0.467837i	$0.154967\pi$
$257$			15.0000i			0.935674i	−0.883815	+	0.467837i	$0.845033\pi$
$258$	−2.00000			−0.124515
$259$		−	4.00000i		−	0.248548i
$260$	−4.00000	−	8.00000i	−0.248069	−	0.496139i
$261$	−2.00000	−	5.00000i	−0.123797	−	0.309492i
$262$		−	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
$267$			2.00000i			0.122398i
$268$			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	$-0.0986081\pi$
$270$	−4.00000	+	2.00000i	−0.243432	+	0.121716i
$271$			2.00000i			0.121491i	0.998153	+	0.0607457i	$0.0193479\pi$
$271$			2.00000i			0.121491i	−0.998153	+	0.0607457i	$0.980652\pi$
$272$	−24.0000			−1.45521
$273$	8.00000			0.484182
$274$	32.0000			1.93319
$275$	4.00000	+	3.00000i	0.241209	+	0.180907i
$276$		−	18.0000i		−	1.08347i
$277$			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
$279$			2.00000i			0.119737i
$280$		0			0
$281$	−24.0000			−1.43172			−0.715860	−	0.698244i	$-0.753965\pi$
$281$	−24.0000			−1.43172			−0.715860	+	0.698244i	$0.753965\pi$
$282$	16.0000			0.952786
$283$			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
$285$	−4.00000	−	8.00000i	−0.236940	−	0.473879i
$286$	4.00000			0.236525
$287$	36.0000			2.12501
$288$	−8.00000			−0.471405
$289$	19.0000			1.11765
$290$	−18.0000	−	16.0000i	−1.05700	−	0.939552i
$291$	11.0000			0.644831
$292$	−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	$0.518606\pi$
$294$	18.0000			1.04978
$295$	16.0000	−	8.00000i	0.931556	−	0.465778i
$296$		0			0
$297$		−	1.00000i		−	0.0580259i
$298$	28.0000			1.62200
$299$	18.0000			1.04097
$300$	−6.00000	+	8.00000i	−0.346410	+	0.461880i
$301$		−	4.00000i		−	0.230556i
$302$	−34.0000			−1.95648
$303$			3.00000i			0.172345i
$304$		−	16.0000i		−	0.917663i
$305$	6.00000	+	12.0000i	0.343559	+	0.687118i
$306$	12.0000			0.685994
$307$	−15.0000			−0.856095			−0.428048	−	0.903756i	$-0.640798\pi$
$307$	−15.0000			−0.856095			−0.428048	+	0.903756i	$0.640798\pi$
$308$	8.00000			0.455842
$309$		−	6.00000i		−	0.341328i
$310$	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
$315$	−4.00000	−	8.00000i	−0.225374	−	0.450749i
$316$		−	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	$0.390590\pi$
$318$			18.0000i			1.00939i
$319$	5.00000	−	2.00000i	0.279946	−	0.111979i
$320$	−16.0000	+	8.00000i	−0.894427	+	0.447214i
$321$			4.00000i			0.223258i
$322$	72.0000			4.01240
$323$			24.0000i			1.33540i
$324$	2.00000			0.111111
$325$	−8.00000	−	6.00000i	−0.443760	−	0.332820i
$326$	26.0000			1.44001
$327$	1.00000			0.0553001
$328$		0			0
$329$			32.0000i			1.76422i
$330$	−2.00000	−	4.00000i	−0.110096	−	0.220193i
$331$		−	32.0000i		−	1.75888i	−0.476011	−	0.879440i	$-0.657918\pi$
$331$		−	32.0000i		−	1.75888i	0.476011	−	0.879440i	$-0.342082\pi$
$332$			14.0000i			0.768350i
$333$	1.00000			0.0547997
$334$		−	48.0000i		−	2.62644i
$335$	12.0000	+	24.0000i	0.655630	+	1.31126i
$336$		−	16.0000i		−	0.872872i
$337$	10.0000			0.544735			0.272367	−	0.962193i	$-0.412193\pi$
$337$	10.0000			0.544735			0.272367	+	0.962193i	$0.412193\pi$
$338$	18.0000			0.979071
$339$	−8.00000			−0.434500
$340$	24.0000	−	12.0000i	1.30158	−	0.650791i
$341$	−2.00000			−0.108306
$342$			8.00000i			0.432590i
$343$			8.00000i			0.431959i
$344$		0			0
$345$	−9.00000	−	18.0000i	−0.484544	−	0.969087i
$346$			42.0000i			2.25793i
$347$		−	19.0000i		−	1.01997i	−0.860182	−	0.509987i	$-0.829650\pi$
$347$		−	19.0000i		−	1.01997i	0.860182	−	0.509987i	$-0.170350\pi$
$348$	4.00000	+	10.0000i	0.214423	+	0.536056i
$349$	−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
$351$			2.00000i			0.106752i
$352$		−	8.00000i		−	0.426401i
$353$			18.0000i			0.958043i	0.877803	+	0.479022i	$0.159008\pi$
$353$			18.0000i			0.958043i	−0.877803	+	0.479022i	$0.840992\pi$
$354$	−16.0000			−0.850390
$355$	4.00000	−	2.00000i	0.212298	−	0.106149i
$356$		−	4.00000i		−	0.212000i
$357$			24.0000i			1.27021i
$358$	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
$363$	−10.0000			−0.524864
$364$	−16.0000			−0.838628
$365$	−30.0000	+	15.0000i	−1.57027	+	0.785136i
$366$		−	12.0000i		−	0.627250i
$367$	−27.0000			−1.40939			−0.704694	−	0.709511i	$-0.748916\pi$
$367$	−27.0000			−1.40939			−0.704694	+	0.709511i	$0.748916\pi$
$368$		−	36.0000i		−	1.87663i
$369$			9.00000i			0.468521i
$370$	4.00000	−	2.00000i	0.207950	−	0.103975i
$371$	−36.0000			−1.86903
$372$		−	4.00000i		−	0.207390i
$373$			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
$375$	−2.00000	+	11.0000i	−0.103280	+	0.568038i
$376$		0			0
$377$	−10.0000	+	4.00000i	−0.515026	+	0.206010i
$378$			8.00000i			0.411476i
$379$		−	22.0000i		−	1.13006i	−0.825069	−	0.565032i	$-0.808864\pi$
$379$		−	22.0000i		−	1.13006i	0.825069	−	0.565032i	$-0.191136\pi$
$380$	8.00000	+	16.0000i	0.410391	+	0.820783i
$381$	7.00000			0.358621
$382$		−	6.00000i		−	0.306987i
$383$			27.0000i			1.37964i	0.723983	+	0.689818i	$0.242309\pi$
$383$			27.0000i			1.37964i	−0.723983	+	0.689818i	$0.757691\pi$
$384$		0			0
$385$	8.00000	−	4.00000i	0.407718	−	0.203859i
$386$	−12.0000			−0.610784
$387$	1.00000			0.0508329
$388$	−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	$0.975768\pi$
$390$	4.00000	+	8.00000i	0.202548	+	0.405096i
$391$			54.0000i			2.73090i
$392$		0			0
$393$			20.0000i			1.00887i
$394$		−	46.0000i		−	2.31745i
$395$	−4.00000	−	8.00000i	−0.201262	−	0.402524i
$396$			2.00000i			0.100504i
$397$		−	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
$399$	−16.0000			−0.801002
$400$	−12.0000	+	16.0000i	−0.600000	+	0.800000i
$401$	−12.0000			−0.599251			−0.299626	−	0.954057i	$-0.596862\pi$
$401$	−12.0000			−0.599251			−0.299626	+	0.954057i	$0.596862\pi$
$402$		−	24.0000i		−	1.19701i
$403$	4.00000			0.199254
$404$		−	6.00000i		−	0.298511i
$405$	2.00000	−	1.00000i	0.0993808	−	0.0496904i
$406$	−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	$0.112504\pi$
$409$			14.0000i			0.692255i	−0.938187	+	0.346128i	$0.887496\pi$
$410$	18.0000	+	36.0000i	0.888957	+	1.77791i
$411$	−16.0000			−0.789222
$412$			12.0000i			0.591198i
$413$		−	32.0000i		−	1.57462i
$414$			18.0000i			0.884652i
$415$	7.00000	+	14.0000i	0.343616	+	0.687233i
$416$			16.0000i			0.784465i
$417$	−11.0000			−0.538672
$418$	−8.00000			−0.391293
$419$	30.0000			1.46560			0.732798	−	0.680446i	$-0.238214\pi$
$419$	30.0000			1.46560			0.732798	+	0.680446i	$0.238214\pi$
$420$	8.00000	+	16.0000i	0.390360	+	0.780720i
$421$			22.0000i			1.07221i	0.844150	+	0.536107i	$0.180106\pi$
$421$			22.0000i			1.07221i	−0.844150	+	0.536107i	$0.819894\pi$
$422$		−	4.00000i		−	0.194717i
$423$	−8.00000			−0.388973
$424$		0			0
$425$	18.0000	−	24.0000i	0.873128	−	1.16417i
$426$	−4.00000			−0.193801
$427$	24.0000			1.16144
$428$		−	8.00000i		−	0.386695i
$429$	−2.00000			−0.0965609
$430$	4.00000	−	2.00000i	0.192897	−	0.0964486i
$431$	−6.00000			−0.289010			−0.144505	−	0.989504i	$-0.546159\pi$
$431$	−6.00000			−0.289010			−0.144505	+	0.989504i	$0.546159\pi$
$432$	4.00000			0.192450
$433$	−7.00000			−0.336399			−0.168199	−	0.985753i	$-0.553795\pi$
$433$	−7.00000			−0.336399			−0.168199	+	0.985753i	$0.553795\pi$
$434$	16.0000			0.768025
$435$	9.00000	+	8.00000i	0.431517	+	0.383571i
$436$	−2.00000			−0.0957826
$437$	−36.0000			−1.72211
$438$	30.0000			1.43346
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	−9.00000			−0.428571
$442$		−	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	$0.530292\pi$
$444$	−2.00000			−0.0949158
$445$	−2.00000	−	4.00000i	−0.0948091	−	0.189618i
$446$		−	8.00000i		−	0.378811i
$447$	−14.0000			−0.662177
$448$			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	$0.884839\pi$
$450$	6.00000	−	8.00000i	0.282843	−	0.377124i
$451$	−9.00000			−0.423793
$452$	16.0000			0.752577
$453$	17.0000			0.798730
$454$			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
$459$	−6.00000			−0.280056
$460$	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	$0.721013\pi$
$462$	−8.00000			−0.372194
$463$			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
$465$	−2.00000	−	4.00000i	−0.0927478	−	0.185496i
$466$			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	$0.363267\pi$
$468$		−	4.00000i		−	0.184900i
$469$	48.0000			2.21643
$470$	−32.0000	+	16.0000i	−1.47605	+	0.738025i
$471$	14.0000			0.645086
$472$		0			0
$473$			1.00000i			0.0459800i
$474$			8.00000i			0.367452i
$475$	16.0000	+	12.0000i	0.734130	+	0.550598i
$476$		−	48.0000i		−	2.20008i
$477$		−	9.00000i		−	0.412082i
$478$	−52.0000			−2.37842
$479$		0			0		1.00000			$0$
$479$		0			0		−1.00000			$\pi$
$480$	16.0000	−	8.00000i	0.730297	−	0.365148i
$481$		−	2.00000i		−	0.0911922i
$482$	22.0000			1.00207
$483$	−36.0000			−1.63806
$484$	20.0000			0.909091
$485$	−22.0000	+	11.0000i	−0.998969	+	0.499484i
$486$	−2.00000			−0.0907218
$487$			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
$489$	−13.0000			−0.587880
$490$	−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	$-0.0577772\pi$
$492$		−	18.0000i		−	0.811503i
$493$	−12.0000	−	30.0000i	−0.540453	−	1.35113i
$494$	16.0000			0.719874
$495$	1.00000	+	2.00000i	0.0449467	+	0.0898933i
$496$		−	8.00000i		−	0.359211i
$497$		−	8.00000i		−	0.358849i
$498$		−	14.0000i		−	0.627355i
$499$	−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
$501$			24.0000i			1.07224i
$502$			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
$507$	−9.00000			−0.399704
$508$	−14.0000			−0.621150
$509$	−28.0000			−1.24108			−0.620539	−	0.784176i	$-0.713086\pi$
$509$	−28.0000			−1.24108			−0.620539	+	0.784176i	$0.713086\pi$
$510$	−24.0000	+	12.0000i	−1.06274	+	0.531369i
$511$			60.0000i			2.65424i
$512$	32.0000			1.41421
$513$		−	4.00000i		−	0.176604i
$514$			30.0000i			1.32324i
$515$	6.00000	+	12.0000i	0.264392	+	0.528783i
$516$	−2.00000			−0.0880451
$517$		−	8.00000i		−	0.351840i
$518$		−	8.00000i		−	0.351500i
$519$		−	21.0000i		−	0.921798i
$520$		0			0
$521$	−8.00000			−0.350486			−0.175243	−	0.984525i	$-0.556071\pi$
$521$	−8.00000			−0.350486			−0.175243	+	0.984525i	$0.556071\pi$
$522$	−4.00000	−	10.0000i	−0.175075	−	0.437688i
$523$			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
$525$	16.0000	+	12.0000i	0.698297	+	0.523723i
$526$	−52.0000			−2.26731
$527$			12.0000i			0.522728i
$528$			4.00000i			0.174078i
$529$	−58.0000			−2.52174
$530$	−18.0000	−	36.0000i	−0.781870	−	1.56374i
$531$	8.00000			0.347170
$532$	32.0000			1.38738
$533$	18.0000			0.779667
$534$			4.00000i			0.173097i
$535$	−4.00000	−	8.00000i	−0.172935	−	0.345870i
$536$		0			0
$537$	−6.00000			−0.258919
$538$		−	20.0000i		−	0.862261i
$539$		−	9.00000i		−	0.387657i
$540$	−4.00000	+	2.00000i	−0.172133	+	0.0860663i
$541$		−	32.0000i		−	1.37579i	−0.725811	−	0.687894i	$-0.758536\pi$
$541$		−	32.0000i		−	1.37579i	0.725811	−	0.687894i	$-0.241464\pi$
$542$			4.00000i			0.171815i
$543$	7.00000			0.300399
$544$	−48.0000			−2.05798
$545$	−2.00000	+	1.00000i	−0.0856706	+	0.0428353i
$546$	16.0000			0.684737
$547$		−	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
$549$			6.00000i			0.256074i
$550$	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
$555$	−2.00000	+	1.00000i	−0.0848953	+	0.0424476i
$556$	22.0000			0.933008
$557$		−	45.0000i		−	1.90671i	−0.301849	−	0.953356i	$-0.597604\pi$
$557$		−	45.0000i		−	1.90671i	0.301849	−	0.953356i	$-0.402396\pi$
$558$			4.00000i			0.169334i
$559$		−	2.00000i		−	0.0845910i
$560$	16.0000	+	32.0000i	0.676123	+	1.35225i
$561$		−	6.00000i		−	0.253320i
$562$	−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	$0.225879\pi$
$564$	16.0000			0.673722
$565$	16.0000	−	8.00000i	0.673125	−	0.336563i
$566$			44.0000i			1.84946i
$567$		−	4.00000i		−	0.167984i
$568$		0			0
$569$		−	38.0000i		−	1.59304i	−0.604610	−	0.796521i	$-0.706671\pi$
$569$		−	38.0000i		−	1.59304i	0.604610	−	0.796521i	$-0.293329\pi$
$570$	−8.00000	−	16.0000i	−0.335083	−	0.670166i
$571$	−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
$573$			3.00000i			0.125327i
$574$	72.0000			3.00522
$575$	36.0000	+	27.0000i	1.50130	+	1.12598i
$576$	−8.00000			−0.333333
$577$	34.0000			1.41544			0.707719	−	0.706494i	$-0.249724\pi$
$577$	34.0000			1.41544			0.707719	+	0.706494i	$0.249724\pi$
$578$	38.0000			1.58059
$579$	6.00000			0.249351
$580$	−18.0000	−	16.0000i	−0.747409	−	0.664364i
$581$	28.0000			1.16164
$582$	22.0000			0.911929
$583$	9.00000			0.372742
$584$		0			0
$585$	−2.00000	−	4.00000i	−0.0826898	−	0.165380i
$586$	−4.00000			−0.165238
$587$			12.0000i			0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$			12.0000i			0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$	18.0000			0.742307
$589$	−8.00000			−0.329634
$590$	32.0000	−	16.0000i	1.31742	−	0.658710i
$591$			23.0000i			0.946094i
$592$	−4.00000			−0.164399
$593$		−	14.0000i		−	0.574911i	−0.957794	−	0.287456i	$-0.907191\pi$
$593$		−	14.0000i		−	0.574911i	0.957794	−	0.287456i	$-0.0928094\pi$
$594$		−	2.00000i		−	0.0820610i
$595$	−24.0000	−	48.0000i	−0.983904	−	1.96781i
$596$	28.0000			1.14692
$597$	19.0000			0.777618
$598$	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	$-0.822086\pi$
$601$		−	26.0000i		−	1.06056i	0.847822	−	0.530281i	$-0.177914\pi$
$602$		−	8.00000i		−	0.326056i
$603$			12.0000i			0.488678i
$604$	−34.0000			−1.38344
$605$	20.0000	−	10.0000i	0.813116	−	0.406558i
$606$			6.00000i			0.243733i
$607$	−40.0000			−1.62355			−0.811775	−	0.583970i	$-0.801498\pi$
$607$	−40.0000			−1.62355			−0.811775	+	0.583970i	$0.801498\pi$
$608$		−	32.0000i		−	1.29777i
$609$	20.0000	−	8.00000i	0.810441	−	0.324176i
$610$	12.0000	+	24.0000i	0.485866	+	0.971732i
$611$			16.0000i			0.647291i
$612$	12.0000			0.485071
$613$			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
$615$	−9.00000	−	18.0000i	−0.362915	−	0.725830i
$616$		0			0
$617$	−36.0000			−1.44931			−0.724653	−	0.689114i	$-0.758000\pi$
$617$	−36.0000			−1.44931			−0.724653	+	0.689114i	$0.758000\pi$
$618$		−	12.0000i		−	0.482711i
$619$			10.0000i			0.401934i	0.979598	+	0.200967i	$0.0644084\pi$
$619$			10.0000i			0.401934i	−0.979598	+	0.200967i	$0.935592\pi$
$620$	4.00000	+	8.00000i	0.160644	+	0.321288i
$621$		−	9.00000i		−	0.361158i
$622$		−	46.0000i		−	1.84443i
$623$	−8.00000			−0.320513
$624$		−	8.00000i		−	0.320256i
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$		−	16.0000i		−	0.639489i
$627$	4.00000			0.159745
$628$	−28.0000			−1.11732
$629$	6.00000			0.239236
$630$	−8.00000	−	16.0000i	−0.318728	−	0.637455i
$631$	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
$633$			2.00000i			0.0794929i
$634$	24.0000			0.953162
$635$	−14.0000	+	7.00000i	−0.555573	+	0.277787i
$636$			18.0000i			0.713746i
$637$			18.0000i			0.713186i
$638$	10.0000	−	4.00000i	0.395904	−	0.158362i
$639$	2.00000			0.0791188
$640$		0			0
$641$			33.0000i			1.30342i	0.758468	+	0.651711i	$0.225948\pi$
$641$			33.0000i			1.30342i	−0.758468	+	0.651711i	$0.774052\pi$
$642$			8.00000i			0.315735i
$643$			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
$645$	−2.00000	+	1.00000i	−0.0787499	+	0.0393750i
$646$			48.0000i			1.88853i
$647$		−	17.0000i		−	0.668339i	−0.942513	−	0.334169i	$-0.891544\pi$
$647$		−	17.0000i		−	0.668339i	0.942513	−	0.334169i	$-0.108456\pi$
$648$		0			0
$649$			8.00000i			0.314027i
$650$	−16.0000	−	12.0000i	−0.627572	−	0.470679i
$651$	−8.00000			−0.313545
$652$	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	$0.344399\pi$
$654$	2.00000			0.0782062
$655$	−20.0000	−	40.0000i	−0.781465	−	1.56293i
$656$		−	36.0000i		−	1.40556i
$657$	−15.0000			−0.585206
$658$			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	$-0.191062\pi$
$660$	−2.00000	−	4.00000i	−0.0778499	−	0.155700i
$661$	−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
$663$			12.0000i			0.466041i
$664$		0			0
$665$	32.0000	−	16.0000i	1.24091	−	0.620453i
$666$	2.00000			0.0774984
$667$	45.0000	−	18.0000i	1.74241	−	0.696963i
$668$		−	48.0000i		−	1.85718i
$669$			4.00000i			0.154649i
$670$	24.0000	+	48.0000i	0.927201	+	1.85440i
$671$	−6.00000			−0.231627
$672$		−	32.0000i		−	1.23443i
$673$			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
$675$	−3.00000	+	4.00000i	−0.115470	+	0.153960i
$676$	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	$0.387574\pi$
$678$	−16.0000			−0.614476
$679$			44.0000i			1.68857i
$680$		0			0
$681$		−	7.00000i		−	0.268241i
$682$	−4.00000			−0.153168
$683$			9.00000i			0.344375i	0.985064	+	0.172188i	$0.0550836\pi$
$683$			9.00000i			0.344375i	−0.985064	+	0.172188i	$0.944916\pi$
$684$			8.00000i			0.305888i
$685$	32.0000	−	16.0000i	1.22266	−	0.611329i
$686$			16.0000i			0.610883i
$687$		−	4.00000i		−	0.152610i
$688$	−4.00000			−0.152499
$689$	−18.0000			−0.685745
$690$	−18.0000	−	36.0000i	−0.685248	−	1.37050i
$691$	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
$693$	4.00000			0.151947
$694$		−	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
$699$		−	1.00000i		−	0.0378235i
$700$	−32.0000	−	24.0000i	−1.20949	−	0.907115i
$701$	6.00000			0.226617			0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000			0.226617			0.113308	+	0.993560i	$0.463855\pi$
$702$			4.00000i			0.150970i
$703$			4.00000i			0.150863i
$704$		−	8.00000i		−	0.301511i
$705$	16.0000	−	8.00000i	0.602595	−	0.301297i
$706$			36.0000i			1.35488i
$707$	−12.0000			−0.451306
$708$	−16.0000			−0.601317
$709$	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
$711$		−	4.00000i		−	0.150012i
$712$		0			0
$713$	−18.0000			−0.674105
$714$			48.0000i			1.79635i
$715$	4.00000	−	2.00000i	0.149592	−	0.0747958i
$716$	12.0000			0.448461
$717$	26.0000			0.970988
$718$		−	42.0000i		−	1.56743i
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$	−8.00000	+	4.00000i	−0.298142	+	0.149071i
$721$	24.0000			0.893807
$722$	6.00000			0.223297
$723$	−11.0000			−0.409094
$724$	−14.0000			−0.520306
$725$	−26.0000	−	7.00000i	−0.965616	−	0.259973i
$726$	−20.0000			−0.742270
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$	−60.0000	+	30.0000i	−2.22070	+	1.11035i
$731$	6.00000			0.221918
$732$		−	12.0000i		−	0.443533i
$733$	−2.00000			−0.0738717			−0.0369358	−	0.999318i	$-0.511760\pi$
$733$	−2.00000			−0.0738717			−0.0369358	+	0.999318i	$0.511760\pi$
$734$	−54.0000			−1.99318
$735$	18.0000	−	9.00000i	0.663940	−	0.331970i
$736$		−	72.0000i		−	2.65396i
$737$	−12.0000			−0.442026
$738$			18.0000i			0.662589i
$739$		−	10.0000i		−	0.367856i	−0.982940	−	0.183928i	$-0.941119\pi$
$739$		−	10.0000i		−	0.367856i	0.982940	−	0.183928i	$-0.0588813\pi$
$740$	4.00000	−	2.00000i	0.147043	−	0.0735215i
$741$	−8.00000			−0.293887
$742$	−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
$747$			7.00000i			0.256117i
$748$			12.0000i			0.438763i
$749$	−16.0000			−0.584627
$750$	−4.00000	+	22.0000i	−0.146059	+	0.803326i
$751$		−	28.0000i		−	1.02173i	−0.859660	−	0.510867i	$-0.829324\pi$
$751$		−	28.0000i		−	1.02173i	0.859660	−	0.510867i	$-0.170676\pi$
$752$	32.0000			1.16692
$753$		−	12.0000i		−	0.437304i
$754$	−20.0000	+	8.00000i	−0.728357	+	0.291343i
$755$	−34.0000	+	17.0000i	−1.23739	+	0.618693i
$756$			8.00000i			0.290957i
$757$	33.0000			1.19941			0.599703	−	0.800223i	$-0.295286\pi$
$757$	33.0000			1.19941			0.599703	+	0.800223i	$0.295286\pi$
$758$		−	44.0000i		−	1.59815i
$759$	9.00000			0.326679
$760$		0			0
$761$	−34.0000			−1.23250			−0.616250	−	0.787551i	$-0.711349\pi$
$761$	−34.0000			−1.23250			−0.616250	+	0.787551i	$0.711349\pi$
$762$	14.0000			0.507166
$763$			4.00000i			0.144810i
$764$		−	6.00000i		−	0.217072i
$765$	12.0000	−	6.00000i	0.433861	−	0.216930i
$766$			54.0000i			1.95110i
$767$		−	16.0000i		−	0.577727i
$768$	−16.0000			−0.577350
$769$			14.0000i			0.504853i	0.967616	+	0.252426i	$0.0812286\pi$
$769$			14.0000i			0.504853i	−0.967616	+	0.252426i	$0.918771\pi$
$770$	16.0000	−	8.00000i	0.576600	−	0.288300i
$771$		−	15.0000i		−	0.540212i
$772$	−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	$0.642054\pi$
$774$	2.00000			0.0718885
$775$	8.00000	+	6.00000i	0.287368	+	0.215526i
$776$		0			0
$777$			4.00000i			0.143499i
$778$			6.00000i			0.215110i
$779$	−36.0000			−1.28983
$780$	4.00000	+	8.00000i	0.143223	+	0.286446i
$781$			2.00000i			0.0715656i

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:	\( N \)	\(=\)	\( 435 = 3 \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	435.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} +(2.00000 - 1.00000i) q^{5} -2.00000 q^{6} -4.00000i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} +(2.00000 - 1.00000i) q^{5} -2.00000 q^{6} -4.00000i q^{7} +1.00000 q^{9} +(4.00000 - 2.00000i) q^{10} +1.00000i q^{11} -2.00000 q^{12} -2.00000i q^{13} -8.00000i q^{14} +(-2.00000 + 1.00000i) q^{15} -4.00000 q^{16} +6.00000 q^{17} +2.00000 q^{18} +4.00000i q^{19} +(4.00000 - 2.00000i) q^{20} +4.00000i q^{21} +2.00000i q^{22} +9.00000i q^{23} +(3.00000 - 4.00000i) q^{25} -4.00000i q^{26} -1.00000 q^{27} -8.00000i q^{28} +(-2.00000 - 5.00000i) q^{29} +(-4.00000 + 2.00000i) q^{30} +2.00000i q^{31} -8.00000 q^{32} -1.00000i q^{33} +12.0000 q^{34} +(-4.00000 - 8.00000i) q^{35} +2.00000 q^{36} +1.00000 q^{37} +8.00000i q^{38} +2.00000i q^{39} +9.00000i q^{41} +8.00000i q^{42} +1.00000 q^{43} +2.00000i q^{44} +(2.00000 - 1.00000i) q^{45} +18.0000i q^{46} -8.00000 q^{47} +4.00000 q^{48} -9.00000 q^{49} +(6.00000 - 8.00000i) q^{50} -6.00000 q^{51} -4.00000i q^{52} -9.00000i q^{53} -2.00000 q^{54} +(1.00000 + 2.00000i) q^{55} -4.00000i q^{57} +(-4.00000 - 10.0000i) q^{58} +8.00000 q^{59} +(-4.00000 + 2.00000i) q^{60} +6.00000i q^{61} +4.00000i q^{62} -4.00000i q^{63} -8.00000 q^{64} +(-2.00000 - 4.00000i) q^{65} -2.00000i q^{66} +12.0000i q^{67} +12.0000 q^{68} -9.00000i q^{69} +(-8.00000 - 16.0000i) q^{70} +2.00000 q^{71} -15.0000 q^{73} +2.00000 q^{74} +(-3.00000 + 4.00000i) q^{75} +8.00000i q^{76} +4.00000 q^{77} +4.00000i q^{78} -4.00000i q^{79} +(-8.00000 + 4.00000i) q^{80} +1.00000 q^{81} +18.0000i q^{82} +7.00000i q^{83} +8.00000i q^{84} +(12.0000 - 6.00000i) q^{85} +2.00000 q^{86} +(2.00000 + 5.00000i) q^{87} -2.00000i q^{89} +(4.00000 - 2.00000i) q^{90} -8.00000 q^{91} +18.0000i q^{92} -2.00000i q^{93} -16.0000 q^{94} +(4.00000 + 8.00000i) q^{95} +8.00000 q^{96} -11.0000 q^{97} -18.0000 q^{98} +1.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} - 2 q^{3} + 4 q^{4} + 4 q^{5} - 4 q^{6} + 2 q^{9}+O(q^{10}) \) 2 * q + 4 * q^2 - 2 * q^3 + 4 * q^4 + 4 * q^5 - 4 * q^6 + 2 * q^9	\( 2 q + 4 q^{2} - 2 q^{3} + 4 q^{4} + 4 q^{5} - 4 q^{6} + 2 q^{9} + 8 q^{10} - 4 q^{12} - 4 q^{15} - 8 q^{16} + 12 q^{17} + 4 q^{18} + 8 q^{20} + 6 q^{25} - 2 q^{27} - 4 q^{29} - 8 q^{30} - 16 q^{32} + 24 q^{34} - 8 q^{35} + 4 q^{36} + 2 q^{37} + 2 q^{43} + 4 q^{45} - 16 q^{47} + 8 q^{48} - 18 q^{49} + 12 q^{50} - 12 q^{51} - 4 q^{54} + 2 q^{55} - 8 q^{58} + 16 q^{59} - 8 q^{60} - 16 q^{64} - 4 q^{65} + 24 q^{68} - 16 q^{70} + 4 q^{71} - 30 q^{73} + 4 q^{74} - 6 q^{75} + 8 q^{77} - 16 q^{80} + 2 q^{81} + 24 q^{85} + 4 q^{86} + 4 q^{87} + 8 q^{90} - 16 q^{91} - 32 q^{94} + 8 q^{95} + 16 q^{96} - 22 q^{97} - 36 q^{98}+O(q^{100}) \) 2 * q + 4 * q^2 - 2 * q^3 + 4 * q^4 + 4 * q^5 - 4 * q^6 + 2 * q^9 + 8 * q^10 - 4 * q^12 - 4 * q^15 - 8 * q^16 + 12 * q^17 + 4 * q^18 + 8 * q^20 + 6 * q^25 - 2 * q^27 - 4 * q^29 - 8 * q^30 - 16 * q^32 + 24 * q^34 - 8 * q^35 + 4 * q^36 + 2 * q^37 + 2 * q^43 + 4 * q^45 - 16 * q^47 + 8 * q^48 - 18 * q^49 + 12 * q^50 - 12 * q^51 - 4 * q^54 + 2 * q^55 - 8 * q^58 + 16 * q^59 - 8 * q^60 - 16 * q^64 - 4 * q^65 + 24 * q^68 - 16 * q^70 + 4 * q^71 - 30 * q^73 + 4 * q^74 - 6 * q^75 + 8 * q^77 - 16 * q^80 + 2 * q^81 + 24 * q^85 + 4 * q^86 + 4 * q^87 + 8 * q^90 - 16 * q^91 - 32 * q^94 + 8 * q^95 + 16 * q^96 - 22 * q^97 - 36 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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