LMFDB - Embedded newform 1815.2.c.e.364.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1815,2,Mod(364,1815)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1815, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1815.364");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1815 = 3 \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1815.c (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:	$14.4928479669$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 165)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			364.5
Root			$0.403032 + 0.403032i$ of defining polynomial
Character	$\chi$	$=$	1815.364
Dual form			1815.2.c.e.364.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.48119i q^{2} -1.00000i q^{3} -0.193937 q^{4} +(-1.48119 - 1.67513i) q^{5} +1.48119 q^{6} -1.19394i q^{7} +2.67513i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.48119i q^{2} -1.00000i q^{3} -0.193937 q^{4} +(-1.48119 - 1.67513i) q^{5} +1.48119 q^{6} -1.19394i q^{7} +2.67513i q^{8} -1.00000 q^{9} +(2.48119 - 2.19394i) q^{10} +0.193937i q^{12} +0.806063i q^{13} +1.76845 q^{14} +(-1.67513 + 1.48119i) q^{15} -4.35026 q^{16} -3.76845i q^{17} -1.48119i q^{18} -5.35026 q^{19} +(0.287258 + 0.324869i) q^{20} -1.19394 q^{21} -4.00000i q^{23} +2.67513 q^{24} +(-0.612127 + 4.96239i) q^{25} -1.19394 q^{26} +1.00000i q^{27} +0.231548i q^{28} -4.31265 q^{29} +(-2.19394 - 2.48119i) q^{30} +0.962389 q^{31} -1.09332i q^{32} +5.58181 q^{34} +(-2.00000 + 1.76845i) q^{35} +0.193937 q^{36} +1.61213i q^{37} -7.92478i q^{38} +0.806063 q^{39} +(4.48119 - 3.96239i) q^{40} -9.08840 q^{41} -1.76845i q^{42} +4.41819i q^{43} +(1.48119 + 1.67513i) q^{45} +5.92478 q^{46} +12.3127i q^{47} +4.35026i q^{48} +5.57452 q^{49} +(-7.35026 - 0.906679i) q^{50} -3.76845 q^{51} -0.156325i q^{52} -1.42548i q^{53} -1.48119 q^{54} +3.19394 q^{56} +5.35026i q^{57} -6.38787i q^{58} -13.2750 q^{59} +(0.324869 - 0.287258i) q^{60} +0.0752228 q^{61} +1.42548i q^{62} +1.19394i q^{63} -7.08110 q^{64} +(1.35026 - 1.19394i) q^{65} -2.70052i q^{67} +0.730841i q^{68} -4.00000 q^{69} +(-2.61942 - 2.96239i) q^{70} -14.0508 q^{71} -2.67513i q^{72} -10.7308i q^{73} -2.38787 q^{74} +(4.96239 + 0.612127i) q^{75} +1.03761 q^{76} +1.19394i q^{78} +13.9756 q^{79} +(6.44358 + 7.28726i) q^{80} +1.00000 q^{81} -13.4617i q^{82} -9.89446i q^{83} +0.231548 q^{84} +(-6.31265 + 5.58181i) q^{85} -6.54420 q^{86} +4.31265i q^{87} -16.8872 q^{89} +(-2.48119 + 2.19394i) q^{90} +0.962389 q^{91} +0.775746i q^{92} -0.962389i q^{93} -18.2374 q^{94} +(7.92478 + 8.96239i) q^{95} -1.09332 q^{96} +11.4763i q^{97} +8.25694i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{4} + 2 q^{5} - 2 q^{6} - 6 q^{9}+O(q^{10}) $ 6 * q - 2 * q^4 + 2 * q^5 - 2 * q^6 - 6 * q^9	$ 6 q - 2 q^{4} + 2 q^{5} - 2 q^{6} - 6 q^{9} + 4 q^{10} - 12 q^{14} - 6 q^{16} - 12 q^{19} - 10 q^{20} - 8 q^{21} + 6 q^{24} - 2 q^{25} - 8 q^{26} + 16 q^{29} - 14 q^{30} - 16 q^{31} + 36 q^{34} - 12 q^{35} + 2 q^{36} + 4 q^{39} + 16 q^{40} - 16 q^{41} - 2 q^{45} - 8 q^{46} + 10 q^{49} - 24 q^{50} + 2 q^{54} + 20 q^{56} - 16 q^{59} + 12 q^{60} + 44 q^{61} + 22 q^{64} - 12 q^{65} - 24 q^{69} - 40 q^{70} - 24 q^{71} - 16 q^{74} + 8 q^{75} + 28 q^{76} - 20 q^{79} + 6 q^{80} + 6 q^{81} + 24 q^{84} + 4 q^{85} - 20 q^{86} - 36 q^{89} - 4 q^{90} - 16 q^{91} - 24 q^{94} + 4 q^{95} + 6 q^{96}+O(q^{100}) $ 6 * q - 2 * q^4 + 2 * q^5 - 2 * q^6 - 6 * q^9 + 4 * q^10 - 12 * q^14 - 6 * q^16 - 12 * q^19 - 10 * q^20 - 8 * q^21 + 6 * q^24 - 2 * q^25 - 8 * q^26 + 16 * q^29 - 14 * q^30 - 16 * q^31 + 36 * q^34 - 12 * q^35 + 2 * q^36 + 4 * q^39 + 16 * q^40 - 16 * q^41 - 2 * q^45 - 8 * q^46 + 10 * q^49 - 24 * q^50 + 2 * q^54 + 20 * q^56 - 16 * q^59 + 12 * q^60 + 44 * q^61 + 22 * q^64 - 12 * q^65 - 24 * q^69 - 40 * q^70 - 24 * q^71 - 16 * q^74 + 8 * q^75 + 28 * q^76 - 20 * q^79 + 6 * q^80 + 6 * q^81 + 24 * q^84 + 4 * q^85 - 20 * q^86 - 36 * q^89 - 4 * q^90 - 16 * q^91 - 24 * q^94 + 4 * q^95 + 6 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1815\mathbb{Z}\right)^\times$.

$n$	$727$	$1211$	$1696$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$			1.48119i			1.04736i	0.851914	+	0.523681i	$0.175442\pi$
$2$			1.48119i			1.04736i	−0.851914	+	0.523681i	$0.824558\pi$
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$	−0.193937			−0.0969683
$5$	−1.48119	−	1.67513i	−0.662410	−	0.749141i
$5$	−1.48119	−	1.67513i	−0.662410	−	0.749141i
$6$	1.48119			0.604695
$7$		−	1.19394i		−	0.451266i	−0.974212	−	0.225633i	$-0.927555\pi$
$7$		−	1.19394i		−	0.451266i	0.974212	−	0.225633i	$-0.0724450\pi$
$8$			2.67513i			0.945802i
$9$	−1.00000			−0.333333
$10$	2.48119	−	2.19394i	0.784623	−	0.693784i
$11$		0			0
$11$		0			0
$12$			0.193937i			0.0559847i
$13$			0.806063i			0.223562i	0.993733	+	0.111781i	$0.0356555\pi$
$13$			0.806063i			0.223562i	−0.993733	+	0.111781i	$0.964345\pi$
$14$	1.76845			0.472639
$15$	−1.67513	+	1.48119i	−0.432517	+	0.382443i
$16$	−4.35026			−1.08757
$17$		−	3.76845i		−	0.913984i	−0.889471	−	0.456992i	$-0.848927\pi$
$17$		−	3.76845i		−	0.913984i	0.889471	−	0.456992i	$-0.151073\pi$
$18$		−	1.48119i		−	0.349121i
$19$	−5.35026			−1.22743			−0.613717	−	0.789526i	$-0.710326\pi$
$19$	−5.35026			−1.22743			−0.613717	+	0.789526i	$0.710326\pi$
$20$	0.287258	+	0.324869i	0.0642328	+	0.0726429i
$21$	−1.19394			−0.260538
$22$		0			0
$23$		−	4.00000i		−	0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$		−	4.00000i		−	0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$	2.67513			0.546059
$25$	−0.612127	+	4.96239i	−0.122425	+	0.992478i
$26$	−1.19394			−0.234150
$27$			1.00000i			0.192450i
$28$			0.231548i			0.0437585i
$29$	−4.31265			−0.800839			−0.400420	−	0.916332i	$-0.631136\pi$
$29$	−4.31265			−0.800839			−0.400420	+	0.916332i	$0.631136\pi$
$30$	−2.19394	−	2.48119i	−0.400556	−	0.453002i
$31$	0.962389			0.172850			0.0864250	−	0.996258i	$-0.472456\pi$
$31$	0.962389			0.172850			0.0864250	+	0.996258i	$0.472456\pi$
$32$		−	1.09332i		−	0.193274i
$33$		0			0
$34$	5.58181			0.957272
$35$	−2.00000	+	1.76845i	−0.338062	+	0.298923i
$36$	0.193937			0.0323228
$37$			1.61213i			0.265032i	0.991181	+	0.132516i	$0.0423056\pi$
$37$			1.61213i			0.265032i	−0.991181	+	0.132516i	$0.957694\pi$
$38$		−	7.92478i		−	1.28557i
$39$	0.806063			0.129073
$40$	4.48119	−	3.96239i	0.708539	−	0.626509i
$41$	−9.08840			−1.41937			−0.709685	−	0.704520i	$-0.751163\pi$
$41$	−9.08840			−1.41937			−0.709685	+	0.704520i	$0.751163\pi$
$42$		−	1.76845i		−	0.272878i
$43$			4.41819i			0.673768i	0.941546	+	0.336884i	$0.109373\pi$
$43$			4.41819i			0.673768i	−0.941546	+	0.336884i	$0.890627\pi$
$44$		0			0
$45$	1.48119	+	1.67513i	0.220803	+	0.249714i
$46$	5.92478			0.873561
$47$			12.3127i			1.79598i	0.440011	+	0.897992i	$0.354974\pi$
$47$			12.3127i			1.79598i	−0.440011	+	0.897992i	$0.645026\pi$
$48$			4.35026i			0.627906i
$49$	5.57452			0.796359
$50$	−7.35026	−	0.906679i	−1.03948	−	0.128224i
$51$	−3.76845			−0.527689
$52$		−	0.156325i		−	0.0216784i
$53$		−	1.42548i		−	0.195805i	−0.995196	−	0.0979027i	$-0.968787\pi$
$53$		−	1.42548i		−	0.195805i	0.995196	−	0.0979027i	$-0.0312134\pi$
$54$	−1.48119			−0.201565
$55$		0			0
$56$	3.19394			0.426808
$57$			5.35026i			0.708659i
$58$		−	6.38787i		−	0.838769i
$59$	−13.2750			−1.72826			−0.864131	−	0.503266i	$-0.832132\pi$
$59$	−13.2750			−1.72826			−0.864131	+	0.503266i	$0.832132\pi$
$60$	0.324869	−	0.287258i	0.0419404	−	0.0370848i
$61$	0.0752228			0.00963129			0.00481565	−	0.999988i	$-0.498467\pi$
$61$	0.0752228			0.00963129			0.00481565	+	0.999988i	$0.498467\pi$
$62$			1.42548i			0.181037i
$63$			1.19394i			0.150422i
$64$	−7.08110			−0.885138
$65$	1.35026	−	1.19394i	0.167479	−	0.148090i
$66$		0			0
$67$		−	2.70052i		−	0.329921i	−0.986300	−	0.164961i	$-0.947250\pi$
$67$		−	2.70052i		−	0.329921i	0.986300	−	0.164961i	$-0.0527497\pi$
$68$			0.730841i			0.0886274i
$69$	−4.00000			−0.481543
$70$	−2.61942	−	2.96239i	−0.313081	−	0.354073i
$71$	−14.0508			−1.66752			−0.833761	−	0.552126i	$-0.813817\pi$
$71$	−14.0508			−1.66752			−0.833761	+	0.552126i	$0.813817\pi$
$72$		−	2.67513i		−	0.315267i
$73$		−	10.7308i		−	1.25595i	−0.778234	−	0.627975i	$-0.783884\pi$
$73$		−	10.7308i		−	1.25595i	0.778234	−	0.627975i	$-0.216116\pi$
$74$	−2.38787			−0.277585
$75$	4.96239	+	0.612127i	0.573007	+	0.0706823i
$76$	1.03761			0.119022
$77$		0			0
$78$			1.19394i			0.135187i
$79$	13.9756			1.57237			0.786187	−	0.617989i	$-0.212052\pi$
$79$	13.9756			1.57237			0.786187	+	0.617989i	$0.212052\pi$
$80$	6.44358	+	7.28726i	0.720414	+	0.814740i
$81$	1.00000			0.111111
$82$		−	13.4617i		−	1.48659i
$83$		−	9.89446i		−	1.08606i	−0.839714	−	0.543029i	$-0.817277\pi$
$83$		−	9.89446i		−	1.08606i	0.839714	−	0.543029i	$-0.182723\pi$
$84$	0.231548			0.0252640
$85$	−6.31265	+	5.58181i	−0.684703	+	0.605432i
$86$	−6.54420			−0.705679
$87$			4.31265i			0.462365i
$88$		0			0
$89$	−16.8872			−1.79004			−0.895018	−	0.446030i	$-0.852837\pi$
$89$	−16.8872			−1.79004			−0.895018	+	0.446030i	$0.852837\pi$
$90$	−2.48119	+	2.19394i	−0.261541	+	0.231261i
$91$	0.962389			0.100886
$92$			0.775746i			0.0808771i
$93$		−	0.962389i		−	0.0997950i
$94$	−18.2374			−1.88105
$95$	7.92478	+	8.96239i	0.813065	+	0.919522i
$96$	−1.09332			−0.111587
$97$			11.4763i			1.16524i	0.812745	+	0.582619i	$0.197972\pi$
$97$			11.4763i			1.16524i	−0.812745	+	0.582619i	$0.802028\pi$
$98$			8.25694i			0.834077i
$99$		0			0
$100$	0.118714	−	0.962389i	0.0118714	−	0.0962389i
$101$	−10.7612			−1.07078			−0.535388	−	0.844606i	$-0.679834\pi$
$101$	−10.7612			−1.07078			−0.535388	+	0.844606i	$0.679834\pi$
$102$		−	5.58181i		−	0.552682i
$103$		−	16.9380i		−	1.66895i	−0.551049	−	0.834473i	$-0.685772\pi$
$103$		−	16.9380i		−	1.66895i	0.551049	−	0.834473i	$-0.314228\pi$
$104$	−2.15633			−0.211445
$105$	1.76845	+	2.00000i	0.172583	+	0.195180i
$106$	2.11142			0.205079
$107$		−	8.28233i		−	0.800683i	−0.916366	−	0.400342i	$-0.868891\pi$
$107$		−	8.28233i		−	0.800683i	0.916366	−	0.400342i	$-0.131109\pi$
$108$		−	0.193937i		−	0.0186616i
$109$	−10.0000			−0.957826			−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000			−0.957826			−0.478913	+	0.877862i	$0.658969\pi$
$110$		0			0
$111$	1.61213			0.153016
$112$			5.19394i			0.490781i
$113$		−	2.26187i		−	0.212778i	−0.994325	−	0.106389i	$-0.966071\pi$
$113$		−	2.26187i		−	0.212778i	0.994325	−	0.106389i	$-0.0339289\pi$
$114$	−7.92478			−0.742223
$115$	−6.70052	+	5.92478i	−0.624827	+	0.552488i
$116$	0.836381			0.0776560
$117$		−	0.806063i		−	0.0745206i
$118$		−	19.6629i		−	1.81012i
$119$	−4.49929			−0.412449
$120$	−3.96239	−	4.48119i	−0.361715	−	0.409075i
$121$		0			0
$122$			0.111420i			0.0100875i
$123$			9.08840i			0.819473i
$124$	−0.186642			−0.0167610
$125$	9.21933	−	6.32487i	0.824602	−	0.565713i
$126$	−1.76845			−0.157546
$127$			13.8192i			1.22626i	0.789982	+	0.613130i	$0.210090\pi$
$127$			13.8192i			1.22626i	−0.789982	+	0.613130i	$0.789910\pi$
$128$		−	12.6751i		−	1.12033i
$129$	4.41819			0.389000
$130$	1.76845	+	2.00000i	0.155104	+	0.175412i
$131$	5.92478			0.517650			0.258825	−	0.965924i	$-0.416665\pi$
$131$	5.92478			0.517650			0.258825	+	0.965924i	$0.416665\pi$
$132$		0			0
$133$			6.38787i			0.553899i
$134$	4.00000			0.345547
$135$	1.67513	−	1.48119i	0.144172	−	0.127481i
$136$	10.0811			0.864447
$137$			3.35026i			0.286232i	0.989706	+	0.143116i	$0.0457122\pi$
$137$			3.35026i			0.286232i	−0.989706	+	0.143116i	$0.954288\pi$
$138$		−	5.92478i		−	0.504351i
$139$	−21.1998			−1.79814			−0.899072	−	0.437800i	$-0.855758\pi$
$139$	−21.1998			−1.79814			−0.899072	+	0.437800i	$0.855758\pi$
$140$	0.387873	−	0.342968i	0.0327813	−	0.0289860i
$141$	12.3127			1.03691
$142$		−	20.8119i		−	1.74650i
$143$		0			0
$144$	4.35026			0.362522
$145$	6.38787	+	7.22425i	0.530484	+	0.599942i
$146$	15.8945			1.31543
$147$		−	5.57452i		−	0.459778i
$148$		−	0.312650i		−	0.0256997i
$149$	6.38787			0.523315			0.261657	−	0.965161i	$-0.415731\pi$
$149$	6.38787			0.523315			0.261657	+	0.965161i	$0.415731\pi$
$150$	−0.906679	+	7.35026i	−0.0740300	+	0.600146i
$151$	2.64974			0.215633			0.107816	−	0.994171i	$-0.465614\pi$
$151$	2.64974			0.215633			0.107816	+	0.994171i	$0.465614\pi$
$152$		−	14.3127i		−	1.16091i
$153$			3.76845i			0.304661i
$154$		0			0
$155$	−1.42548	−	1.61213i	−0.114498	−	0.129489i
$156$	−0.156325			−0.0125160
$157$			1.61213i			0.128662i	0.997929	+	0.0643309i	$0.0204913\pi$
$157$			1.61213i			0.128662i	−0.997929	+	0.0643309i	$0.979509\pi$
$158$			20.7005i			1.64685i
$159$	−1.42548			−0.113048
$160$	−1.83146	+	1.61942i	−0.144789	+	0.128026i
$161$	−4.77575			−0.376382
$162$			1.48119i			0.116374i
$163$			0.312650i			0.0244887i	0.999925	+	0.0122443i	$0.00389759\pi$
$163$			0.312650i			0.0244887i	−0.999925	+	0.0122443i	$0.996102\pi$
$164$	1.76257			0.137634
$165$		0			0
$166$	14.6556			1.13750
$167$		−	0.493413i		−	0.0381815i	−0.999818	−	0.0190907i	$-0.993923\pi$
$167$		−	0.493413i		−	0.0381815i	0.999818	−	0.0190907i	$-0.00607714\pi$
$168$		−	3.19394i		−	0.246418i
$169$	12.3503			0.950020
$170$	−8.26774	−	9.35026i	−0.634107	−	0.717132i
$171$	5.35026			0.409145
$172$		−	0.856849i		−	0.0653341i
$173$		−	23.3054i		−	1.77187i	−0.463806	−	0.885937i	$-0.653517\pi$
$173$		−	23.3054i		−	1.77187i	0.463806	−	0.885937i	$-0.346483\pi$
$174$	−6.38787			−0.484263
$175$	5.92478	+	0.730841i	0.447871	+	0.0552464i
$176$		0			0
$177$			13.2750i			0.997813i
$178$		−	25.0132i		−	1.87482i
$179$	10.7005			0.799795			0.399897	−	0.916560i	$-0.369046\pi$
$179$	10.7005			0.799795			0.399897	+	0.916560i	$0.369046\pi$
$180$	−0.287258	−	0.324869i	−0.0214109	−	0.0242143i
$181$	−7.79877			−0.579678			−0.289839	−	0.957075i	$-0.593602\pi$
$181$	−7.79877			−0.579678			−0.289839	+	0.957075i	$0.593602\pi$
$182$			1.42548i			0.105664i
$183$		−	0.0752228i		−	0.00556063i
$184$	10.7005			0.788853
$185$	2.70052	−	2.38787i	0.198546	−	0.175560i
$186$	1.42548			0.104522
$187$		0			0
$188$		−	2.38787i		−	0.174154i
$189$	1.19394			0.0868461
$190$	−13.2750	+	11.7381i	−0.963073	+	0.851574i
$191$	1.29948			0.0940268			0.0470134	−	0.998894i	$-0.485030\pi$
$191$	1.29948			0.0940268			0.0470134	+	0.998894i	$0.485030\pi$
$192$			7.08110i			0.511035i
$193$			8.59498i			0.618680i	0.950951	+	0.309340i	$0.100108\pi$
$193$			8.59498i			0.618680i	−0.950951	+	0.309340i	$0.899892\pi$
$194$	−16.9986			−1.22043
$195$	−1.19394	−	1.35026i	−0.0854996	−	0.0966943i
$196$	−1.08110			−0.0772216
$197$			20.7064i			1.47527i	0.675200	+	0.737635i	$0.264057\pi$
$197$			20.7064i			1.47527i	−0.675200	+	0.737635i	$0.735943\pi$
$198$		0			0
$199$	−5.55149			−0.393535			−0.196767	−	0.980450i	$-0.563044\pi$
$199$	−5.55149			−0.393535			−0.196767	+	0.980450i	$0.563044\pi$
$200$	−13.2750	−	1.63752i	−0.938687	−	0.115790i
$201$	−2.70052			−0.190480
$202$		−	15.9394i		−	1.12149i
$203$			5.14903i			0.361391i
$204$	0.730841			0.0511691
$205$	13.4617	+	15.2243i	0.940205	+	1.06331i
$206$	25.0884			1.74799
$207$			4.00000i			0.278019i
$208$		−	3.50659i		−	0.243138i
$209$		0			0
$210$	−2.96239	+	2.61942i	−0.204424	+	0.180757i
$211$	18.4993			1.27354			0.636772	−	0.771052i	$-0.280269\pi$
$211$	18.4993			1.27354			0.636772	+	0.771052i	$0.280269\pi$
$212$			0.276454i			0.0189869i
$213$			14.0508i			0.962744i
$214$	12.2677			0.838606
$215$	7.40105	−	6.54420i	0.504747	−	0.446311i
$216$	−2.67513			−0.182020
$217$		−	1.14903i		−	0.0780013i
$218$		−	14.8119i		−	1.00319i
$219$	−10.7308			−0.725123
$220$		0			0
$221$	3.03761			0.204332
$222$			2.38787i			0.160264i
$223$		−	17.6121i		−	1.17940i	−0.807624	−	0.589698i	$-0.799247\pi$
$223$		−	17.6121i		−	1.17940i	0.807624	−	0.589698i	$-0.200753\pi$
$224$	−1.30536			−0.0872178
$225$	0.612127	−	4.96239i	0.0408085	−	0.330826i
$226$	3.35026			0.222856
$227$			17.4314i			1.15696i	0.815696	+	0.578480i	$0.196354\pi$
$227$			17.4314i			1.15696i	−0.815696	+	0.578480i	$0.803646\pi$
$228$		−	1.03761i		−	0.0687175i
$229$	−13.0738			−0.863942			−0.431971	−	0.901888i	$-0.642182\pi$
$229$	−13.0738			−0.863942			−0.431971	+	0.901888i	$0.642182\pi$
$230$	−8.77575	−	9.92478i	−0.578656	−	0.654420i
$231$		0			0
$232$		−	11.5369i		−	0.757435i
$233$		−	13.8437i		−	0.906929i	−0.891274	−	0.453465i	$-0.850188\pi$
$233$		−	13.8437i		−	0.906929i	0.891274	−	0.453465i	$-0.149812\pi$
$234$	1.19394			0.0780501
$235$	20.6253	−	18.2374i	1.34545	−	1.18968i
$236$	2.57452			0.167587
$237$		−	13.9756i		−	0.907810i
$238$		−	6.66433i		−	0.431984i
$239$	−12.3733			−0.800361			−0.400181	−	0.916436i	$-0.631053\pi$
$239$	−12.3733			−0.800361			−0.400181	+	0.916436i	$0.631053\pi$
$240$	7.28726	−	6.44358i	0.470390	−	0.415931i
$241$	24.5501			1.58141			0.790705	−	0.612198i	$-0.209714\pi$
$241$	24.5501			1.58141			0.790705	+	0.612198i	$0.209714\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$	−0.0145884			−0.000933930
$245$	−8.25694	−	9.33804i	−0.527517	−	0.596586i
$246$	−13.4617			−0.858285
$247$		−	4.31265i		−	0.274407i
$248$			2.57452i			0.163482i
$249$	−9.89446			−0.627036
$250$	9.36836	+	13.6556i	0.592507	+	0.863657i
$251$	13.9003			0.877382			0.438691	−	0.898638i	$-0.355442\pi$
$251$	13.9003			0.877382			0.438691	+	0.898638i	$0.355442\pi$
$252$		−	0.231548i		−	0.0145862i
$253$		0			0
$254$	−20.4690			−1.28434
$255$	5.58181	+	6.31265i	0.349546	+	0.395313i
$256$	4.61213			0.288258
$257$		−	18.8872i		−	1.17815i	−0.808079	−	0.589075i	$-0.799492\pi$
$257$		−	18.8872i		−	1.17815i	0.808079	−	0.589075i	$-0.200508\pi$
$258$			6.54420i			0.407424i
$259$	1.92478			0.119600
$260$	−0.261865	+	0.231548i	−0.0162402	+	0.0143600i
$261$	4.31265			0.266946
$262$			8.77575i			0.542167i
$263$			20.8061i			1.28296i	0.767141	+	0.641478i	$0.221679\pi$
$263$			20.8061i			1.28296i	−0.767141	+	0.641478i	$0.778321\pi$
$264$		0			0
$265$	−2.38787	+	2.11142i	−0.146686	+	0.129703i
$266$	−9.46168			−0.580133
$267$			16.8872i			1.03348i
$268$			0.523730i			0.0319919i
$269$	32.3996			1.97544			0.987720	−	0.156233i	$-0.0499351\pi$
$269$	32.3996			1.97544			0.987720	+	0.156233i	$0.0499351\pi$
$270$	2.19394	+	2.48119i	0.133519	+	0.151001i
$271$	16.8265			1.02214			0.511069	−	0.859539i	$-0.329249\pi$
$271$	16.8265			1.02214			0.511069	+	0.859539i	$0.329249\pi$
$272$			16.3938i			0.994017i
$273$		−	0.962389i		−	0.0582464i
$274$	−4.96239			−0.299789
$275$		0			0
$276$	0.775746			0.0466944
$277$		−	16.9076i		−	1.01588i	−0.861392	−	0.507941i	$-0.830407\pi$
$277$		−	16.9076i		−	1.01588i	0.861392	−	0.507941i	$-0.169593\pi$
$278$		−	31.4010i		−	1.88331i
$279$	−0.962389			−0.0576167
$280$	−4.73084	−	5.35026i	−0.282722	−	0.319739i
$281$	−5.61213			−0.334791			−0.167396	−	0.985890i	$-0.553536\pi$
$281$	−5.61213			−0.334791			−0.167396	+	0.985890i	$0.553536\pi$
$282$			18.2374i			1.08602i
$283$			5.81924i			0.345918i	0.984929	+	0.172959i	$0.0553328\pi$
$283$			5.81924i			0.345918i	−0.984929	+	0.172959i	$0.944667\pi$
$284$	2.72496			0.161697
$285$	8.96239	−	7.92478i	0.530886	−	0.469423i
$286$		0			0
$287$			10.8510i			0.640512i
$288$			1.09332i			0.0644246i
$289$	2.79877			0.164633
$290$	−10.7005	+	9.46168i	−0.628356	+	0.555609i
$291$	11.4763			0.672751
$292$			2.08110i			0.121787i
$293$		−	8.29218i		−	0.484434i	−0.970222	−	0.242217i	$-0.922125\pi$
$293$		−	8.29218i		−	0.484434i	0.970222	−	0.242217i	$-0.0778747\pi$
$294$	8.25694			0.481555
$295$	19.6629	+	22.2374i	1.14482	+	1.29471i
$296$	−4.31265			−0.250668
$297$		0			0
$298$			9.46168i			0.548100i
$299$	3.22425			0.186463
$300$	−0.962389	−	0.118714i	−0.0555635	−	0.00685394i
$301$	5.27504			0.304048
$302$			3.92478i			0.225846i
$303$			10.7612i			0.618212i
$304$	23.2750			1.33492
$305$	−0.111420	−	0.126008i	−0.00637987	−	0.00721520i
$306$	−5.58181			−0.319091
$307$		−	25.6688i		−	1.46500i	−0.680770	−	0.732498i	$-0.738354\pi$
$307$		−	25.6688i		−	1.46500i	0.680770	−	0.732498i	$-0.261646\pi$
$308$		0			0
$309$	−16.9380			−0.963566
$310$	2.38787	−	2.11142i	0.135622	−	0.119921i
$311$	−15.7235			−0.891601			−0.445800	−	0.895132i	$-0.647081\pi$
$311$	−15.7235			−0.891601			−0.445800	+	0.895132i	$0.647081\pi$
$312$			2.15633i			0.122078i
$313$			26.8627i			1.51837i	0.650874	+	0.759186i	$0.274403\pi$
$313$			26.8627i			1.51837i	−0.650874	+	0.759186i	$0.725597\pi$
$314$	−2.38787			−0.134755
$315$	2.00000	−	1.76845i	0.112687	−	0.0996410i
$316$	−2.71037			−0.152470
$317$			0.710373i			0.0398985i	0.999801	+	0.0199492i	$0.00635046\pi$
$317$			0.710373i			0.0398985i	−0.999801	+	0.0199492i	$0.993650\pi$
$318$		−	2.11142i		−	0.118403i
$319$		0			0
$320$	10.4885	+	11.8618i	0.586324	+	0.663093i
$321$	−8.28233			−0.462275
$322$		−	7.07381i		−	0.394208i
$323$			20.1622i			1.12186i
$324$	−0.193937			−0.0107743
$325$	−4.00000	−	0.493413i	−0.221880	−	0.0273696i
$326$	−0.463096			−0.0256485
$327$			10.0000i			0.553001i
$328$		−	24.3127i		−	1.34244i
$329$	14.7005			0.810466
$330$		0			0
$331$	0.962389			0.0528977			0.0264488	−	0.999650i	$-0.491580\pi$
$331$	0.962389			0.0528977			0.0264488	+	0.999650i	$0.491580\pi$
$332$			1.91890i			0.105313i
$333$		−	1.61213i		−	0.0883440i
$334$	0.730841			0.0399898
$335$	−4.52373	+	4.00000i	−0.247158	+	0.218543i
$336$	5.19394			0.283352
$337$		−	19.8192i		−	1.07962i	−0.841786	−	0.539811i	$-0.818496\pi$
$337$		−	19.8192i		−	1.07962i	0.841786	−	0.539811i	$-0.181504\pi$
$338$			18.2931i			0.995015i
$339$	−2.26187			−0.122848
$340$	1.22425	−	1.08252i	0.0663945	−	0.0587077i
$341$		0			0
$342$			7.92478i			0.428523i
$343$		−	15.0132i		−	0.810635i
$344$	−11.8192			−0.637251
$345$	5.92478	+	6.70052i	0.318979	+	0.360744i
$346$	34.5198			1.85579
$347$		−	6.20711i		−	0.333215i	−0.986023	−	0.166608i	$-0.946719\pi$
$347$		−	6.20711i		−	0.333215i	0.986023	−	0.166608i	$-0.0532813\pi$
$348$		−	0.836381i		−	0.0448347i
$349$	4.44851			0.238123			0.119062	−	0.992887i	$-0.462011\pi$
$349$	4.44851			0.238123			0.119062	+	0.992887i	$0.462011\pi$
$350$	−1.08252	+	8.77575i	−0.0578630	+	0.469083i
$351$	−0.806063			−0.0430245
$352$		0			0
$353$		−	6.57452i		−	0.349926i	−0.984575	−	0.174963i	$-0.944019\pi$
$353$		−	6.57452i		−	0.349926i	0.984575	−	0.174963i	$-0.0559806\pi$
$354$	−19.6629			−1.04507
$355$	20.8119	+	23.5369i	1.10458	+	1.24921i
$356$	3.27504			0.173577
$357$			4.49929i			0.238128i
$358$			15.8496i			0.837675i
$359$	8.62530			0.455226			0.227613	−	0.973752i	$-0.426908\pi$
$359$	8.62530			0.455226			0.227613	+	0.973752i	$0.426908\pi$
$360$	−4.48119	+	3.96239i	−0.236180	+	0.208836i
$361$	9.62530			0.506595
$362$		−	11.5515i		−	0.607133i
$363$		0			0
$364$	−0.186642			−0.00978272
$365$	−17.9756	+	15.8945i	−0.940884	+	0.831954i
$366$	0.111420			0.00582399
$367$			23.0132i			1.20128i	0.799520	+	0.600639i	$0.205087\pi$
$367$			23.0132i			1.20128i	−0.799520	+	0.600639i	$0.794913\pi$
$368$			17.4010i			0.907092i
$369$	9.08840			0.473123
$370$	3.53690	+	4.00000i	0.183875	+	0.207950i
$371$	−1.70194			−0.0883602
$372$			0.186642i			0.00967695i
$373$			28.1925i			1.45975i	0.683579	+	0.729877i	$0.260423\pi$
$373$			28.1925i			1.45975i	−0.683579	+	0.729877i	$0.739577\pi$
$374$		0			0
$375$	−6.32487	−	9.21933i	−0.326615	−	0.476084i
$376$	−32.9380			−1.69865
$377$		−	3.47627i		−	0.179037i
$378$			1.76845i			0.0909594i
$379$	−3.74798			−0.192521			−0.0962605	−	0.995356i	$-0.530688\pi$
$379$	−3.74798			−0.192521			−0.0962605	+	0.995356i	$0.530688\pi$
$380$	−1.53690	−	1.73813i	−0.0788415	−	0.0891644i
$381$	13.8192			0.707981
$382$			1.92478i			0.0984802i
$383$		−	1.76257i		−	0.0900632i	−0.998986	−	0.0450316i	$-0.985661\pi$
$383$		−	1.76257i		−	0.0900632i	0.998986	−	0.0450316i	$-0.0143389\pi$
$384$	−12.6751			−0.646825
$385$		0			0
$386$	−12.7308			−0.647983
$387$		−	4.41819i		−	0.224589i
$388$		−	2.22567i		−	0.112991i
$389$	−6.52373			−0.330766			−0.165383	−	0.986229i	$-0.552886\pi$
$389$	−6.52373			−0.330766			−0.165383	+	0.986229i	$0.552886\pi$
$390$	2.00000	−	1.76845i	0.101274	−	0.0895491i
$391$	−15.0738			−0.762315
$392$			14.9126i			0.753198i
$393$		−	5.92478i		−	0.298865i
$394$	−30.6702			−1.54514
$395$	−20.7005	−	23.4109i	−1.04156	−	1.17793i
$396$		0			0
$397$		−	23.6991i		−	1.18942i	−0.803939	−	0.594712i	$-0.797266\pi$
$397$		−	23.6991i		−	1.18942i	0.803939	−	0.594712i	$-0.202734\pi$
$398$		−	8.22284i		−	0.412174i
$399$	6.38787			0.319794
$400$	2.66291	−	21.5877i	0.133146	−	1.07938i
$401$	8.88717			0.443804			0.221902	−	0.975069i	$-0.428774\pi$
$401$	8.88717			0.443804			0.221902	+	0.975069i	$0.428774\pi$
$402$		−	4.00000i		−	0.199502i
$403$			0.775746i			0.0386427i
$404$	2.08698			0.103831
$405$	−1.48119	−	1.67513i	−0.0736011	−	0.0832379i
$406$	−7.62672			−0.378508
$407$		0			0
$408$		−	10.0811i		−	0.499089i
$409$	−4.85097			−0.239865			−0.119932	−	0.992782i	$-0.538268\pi$
$409$	−4.85097			−0.239865			−0.119932	+	0.992782i	$0.538268\pi$
$410$	−22.5501	+	19.9394i	−1.11367	+	0.984735i
$411$	3.35026			0.165256
$412$			3.28489i			0.161835i
$413$			15.8496i			0.779906i
$414$	−5.92478			−0.291187
$415$	−16.5745	+	14.6556i	−0.813611	+	0.719416i
$416$	0.881286			0.0432086
$417$			21.1998i			1.03816i
$418$		0			0
$419$	10.7005			0.522755			0.261377	−	0.965237i	$-0.415823\pi$
$419$	10.7005			0.522755			0.261377	+	0.965237i	$0.415823\pi$
$420$	−0.342968	−	0.387873i	−0.0167351	−	0.0189263i
$421$	−30.6009			−1.49139			−0.745697	−	0.666285i	$-0.767884\pi$
$421$	−30.6009			−1.49139			−0.745697	+	0.666285i	$0.767884\pi$
$422$			27.4010i			1.33386i
$423$		−	12.3127i		−	0.598662i
$424$	3.81336			0.185193
$425$	18.7005	+	2.30677i	0.907109	+	0.111895i
$426$	−20.8119			−1.00834
$427$		−	0.0898112i		−	0.00434627i
$428$			1.60625i			0.0776409i
$429$		0			0
$430$	9.69323	+	10.9624i	0.467449	+	0.528653i
$431$	5.92478			0.285386			0.142693	−	0.989767i	$-0.454424\pi$
$431$	5.92478			0.285386			0.142693	+	0.989767i	$0.454424\pi$
$432$		−	4.35026i		−	0.209302i
$433$			16.0000i			0.768911i	0.923144	+	0.384455i	$0.125611\pi$
$433$			16.0000i			0.768911i	−0.923144	+	0.384455i	$0.874389\pi$
$434$	1.70194			0.0816956
$435$	7.22425	−	6.38787i	0.346376	−	0.306275i
$436$	1.93937			0.0928788
$437$			21.4010i			1.02375i
$438$		−	15.8945i		−	0.759467i
$439$	5.35026			0.255354			0.127677	−	0.991816i	$-0.459248\pi$
$439$	5.35026			0.255354			0.127677	+	0.991816i	$0.459248\pi$
$440$		0			0
$441$	−5.57452			−0.265453
$442$			4.49929i			0.214010i
$443$		−	19.6873i		−	0.935374i	−0.883894	−	0.467687i	$-0.845087\pi$
$443$		−	19.6873i		−	0.935374i	0.883894	−	0.467687i	$-0.154913\pi$
$444$	−0.312650			−0.0148377
$445$	25.0132	+	28.2882i	1.18574	+	1.34099i
$446$	26.0870			1.23525
$447$		−	6.38787i		−	0.302136i
$448$			8.45439i			0.399432i
$449$	31.3357			1.47882			0.739411	−	0.673254i	$-0.235104\pi$
$449$	31.3357			1.47882			0.739411	+	0.673254i	$0.235104\pi$
$450$	7.35026	+	0.906679i	0.346495	+	0.0427412i
$451$		0			0
$452$			0.438658i			0.0206328i
$453$		−	2.64974i		−	0.124496i
$454$	−25.8192			−1.21176
$455$	−1.42548	−	1.61213i	−0.0668277	−	0.0755777i
$456$	−14.3127			−0.670251
$457$			37.5936i			1.75855i	0.476312	+	0.879276i	$0.341973\pi$
$457$			37.5936i			1.75855i	−0.476312	+	0.879276i	$0.658027\pi$
$458$		−	19.3649i		−	0.904860i
$459$	3.76845			0.175896
$460$	1.29948	−	1.14903i	0.0605884	−	0.0535738i
$461$	−17.9854			−0.837664			−0.418832	−	0.908064i	$-0.637560\pi$
$461$	−17.9854			−0.837664			−0.418832	+	0.908064i	$0.637560\pi$
$462$		0			0
$463$		−	39.0132i		−	1.81310i	−0.422103	−	0.906548i	$-0.638708\pi$
$463$		−	39.0132i		−	1.81310i	0.422103	−	0.906548i	$-0.361292\pi$
$464$	18.7612			0.870965
$465$	−1.61213	+	1.42548i	−0.0747606	+	0.0661053i
$466$	20.5052			0.949884
$467$			14.5501i			0.673297i	0.941630	+	0.336649i	$0.109293\pi$
$467$			14.5501i			0.673297i	−0.941630	+	0.336649i	$0.890707\pi$
$468$			0.156325i			0.00722613i
$469$	−3.22425			−0.148882
$470$	27.0132	+	30.5501i	1.24602	+	1.40917i
$471$	1.61213			0.0742829
$472$		−	35.5125i		−	1.63459i
$473$		0			0
$474$	20.7005			0.950807
$475$	3.27504	−	26.5501i	0.150269	−	1.21820i
$476$	0.872577			0.0399945
$477$			1.42548i			0.0652685i
$478$		−	18.3272i		−	0.838268i
$479$	28.6253			1.30792			0.653962	−	0.756528i	$-0.273106\pi$
$479$	28.6253			1.30792			0.653962	+	0.756528i	$0.273106\pi$
$480$	1.61942	+	1.83146i	0.0739161	+	0.0835941i
$481$	−1.29948			−0.0592510
$482$			36.3634i			1.65631i
$483$			4.77575i			0.217304i
$484$		0			0
$485$	19.2243	−	16.9986i	0.872928	−	0.771866i
$486$	1.48119			0.0671883
$487$			1.44992i			0.0657022i	0.999460	+	0.0328511i	$0.0104587\pi$
$487$			1.44992i			0.0657022i	−0.999460	+	0.0328511i	$0.989541\pi$
$488$			0.201231i			0.00910929i
$489$	0.312650			0.0141385
$490$	13.8315	−	12.2301i	0.624841	−	0.552501i
$491$	8.00000			0.361035			0.180517	−	0.983572i	$-0.442223\pi$
$491$	8.00000			0.361035			0.180517	+	0.983572i	$0.442223\pi$
$492$		−	1.76257i		−	0.0794629i
$493$			16.2520i			0.731954i
$494$	6.38787			0.287404
$495$		0			0
$496$	−4.18664			−0.187986
$497$			16.7757i			0.752495i
$498$		−	14.6556i		−	0.656734i
$499$	−30.7005			−1.37434			−0.687172	−	0.726495i	$-0.741148\pi$
$499$	−30.7005			−1.37434			−0.687172	+	0.726495i	$0.741148\pi$
$500$	−1.78797	+	1.22662i	−0.0799602	+	0.0548563i
$501$	−0.493413			−0.0220441
$502$			20.5891i			0.918937i
$503$		−	19.7586i		−	0.880993i	−0.897754	−	0.440496i	$-0.854802\pi$
$503$		−	19.7586i		−	0.880993i	0.897754	−	0.440496i	$-0.145198\pi$
$504$	−3.19394			−0.142269
$505$	15.9394	+	18.0263i	0.709292	+	0.802162i
$506$		0			0
$507$		−	12.3503i		−	0.548494i
$508$		−	2.68006i		−	0.118908i
$509$	−22.1016			−0.979635			−0.489817	−	0.871825i	$-0.662937\pi$
$509$	−22.1016			−0.979635			−0.489817	+	0.871825i	$0.662937\pi$
$510$	−9.35026	+	8.26774i	−0.414037	+	0.366102i
$511$	−12.8119			−0.566767
$512$		−	18.5188i		−	0.818423i
$513$		−	5.35026i		−	0.236220i
$514$	27.9756			1.23395
$515$	−28.3733	+	25.0884i	−1.25028	+	1.10553i
$516$	−0.856849			−0.0377207
$517$		0			0
$518$			2.85097i			0.125264i
$519$	−23.3054			−1.02299
$520$	3.19394	+	3.61213i	0.140063	+	0.158402i
$521$	−22.8119			−0.999409			−0.499705	−	0.866196i	$-0.666558\pi$
$521$	−22.8119			−0.999409			−0.499705	+	0.866196i	$0.666558\pi$
$522$			6.38787i			0.279590i
$523$		−	12.2677i		−	0.536431i	−0.963359	−	0.268216i	$-0.913566\pi$
$523$		−	12.2677i		−	0.536431i	0.963359	−	0.268216i	$-0.0864339\pi$
$524$	−1.14903			−0.0501957
$525$	0.730841	−	5.92478i	0.0318965	−	0.258578i
$526$	−30.8178			−1.34372
$527$		−	3.62672i		−	0.157982i
$528$		0			0
$529$	7.00000			0.304348
$530$	−3.12742	−	3.53690i	−0.135847	−	0.153633i
$531$	13.2750			0.576088
$532$		−	1.23884i		−	0.0537106i
$533$		−	7.32582i		−	0.317317i
$534$	−25.0132			−1.08243
$535$	−13.8740	+	12.2677i	−0.599825	+	0.530381i
$536$	7.22425			0.312040
$537$		−	10.7005i		−	0.461762i
$538$			47.9902i			2.06900i
$539$		0			0
$540$	−0.324869	+	0.287258i	−0.0139801	+	0.0123616i
$541$	5.22425			0.224608			0.112304	−	0.993674i	$-0.464177\pi$
$541$	5.22425			0.224608			0.112304	+	0.993674i	$0.464177\pi$
$542$			24.9234i			1.07055i
$543$			7.79877i			0.334677i
$544$	−4.12013			−0.176649
$545$	14.8119	+	16.7513i	0.634474	+	0.717547i
$546$	1.42548			0.0610051
$547$			17.9697i			0.768328i	0.923265	+	0.384164i	$0.125510\pi$
$547$			17.9697i			0.768328i	−0.923265	+	0.384164i	$0.874490\pi$
$548$		−	0.649738i		−	0.0277554i
$549$	−0.0752228			−0.00321043
$550$		0			0
$551$	23.0738			0.982977
$552$		−	10.7005i		−	0.455445i
$553$		−	16.6859i		−	0.709558i
$554$	25.0435			1.06400
$555$	−2.38787	−	2.70052i	−0.101360	−	0.114631i
$556$	4.11142			0.174363
$557$		−	15.8700i		−	0.672434i	−0.941784	−	0.336217i	$-0.890852\pi$
$557$		−	15.8700i		−	0.672434i	0.941784	−	0.336217i	$-0.109148\pi$
$558$		−	1.42548i		−	0.0603456i
$559$	−3.56134			−0.150629
$560$	8.70052	−	7.69323i	0.367664	−	0.325098i
$561$		0			0
$562$		−	8.31265i		−	0.350648i
$563$			31.6688i			1.33468i	0.744753	+	0.667340i	$0.232567\pi$
$563$			31.6688i			1.33468i	−0.744753	+	0.667340i	$0.767433\pi$
$564$	−2.38787			−0.100548
$565$	−3.78892	+	3.35026i	−0.159401	+	0.140947i
$566$	−8.61942			−0.362301
$567$		−	1.19394i		−	0.0501406i
$568$		−	37.5877i		−	1.57714i
$569$	−24.3127			−1.01924			−0.509620	−	0.860400i	$-0.670214\pi$
$569$	−24.3127			−1.01924			−0.509620	+	0.860400i	$0.670214\pi$
$570$	11.7381	+	13.2750i	0.491656	+	0.556030i
$571$	−8.05079			−0.336915			−0.168457	−	0.985709i	$-0.553879\pi$
$571$	−8.05079			−0.336915			−0.168457	+	0.985709i	$0.553879\pi$
$572$		0			0
$573$		−	1.29948i		−	0.0542864i
$574$	−16.0724			−0.670849
$575$	19.8496	+	2.44851i	0.827784	+	0.102110i
$576$	7.08110			0.295046
$577$		−	44.5355i		−	1.85404i	−0.375016	−	0.927018i	$-0.622363\pi$
$577$		−	44.5355i		−	1.85404i	0.375016	−	0.927018i	$-0.377637\pi$
$578$			4.14552i			0.172431i
$579$	8.59498			0.357195
$580$	−1.23884	−	1.40105i	−0.0514401	−	0.0581753i
$581$	−11.8134			−0.490101
$582$			16.9986i			0.704614i
$583$		0			0
$584$	28.7064			1.18788
$585$	−1.35026	+	1.19394i	−0.0558265	+	0.0493632i
$586$	12.2823			0.507379
$587$		−	15.4763i		−	0.638774i	−0.947624	−	0.319387i	$-0.896523\pi$
$587$		−	15.4763i		−	0.638774i	0.947624	−	0.319387i	$-0.103477\pi$
$588$			1.08110i			0.0445839i
$589$	−5.14903			−0.212162
$590$	−32.9380	+	29.1246i	−1.35603	+	1.19904i
$591$	20.7064			0.851748
$592$		−	7.01317i		−	0.288240i
$593$		−	9.53102i		−	0.391392i	−0.980665	−	0.195696i	$-0.937303\pi$
$593$		−	9.53102i		−	0.391392i	0.980665	−	0.195696i	$-0.0626966\pi$
$594$		0			0
$595$	6.66433	+	7.53690i	0.273211	+	0.308983i
$596$	−1.23884			−0.0507450
$597$			5.55149i			0.227207i
$598$			4.77575i			0.195295i
$599$	25.5515			1.04401			0.522003	−	0.852944i	$-0.325185\pi$
$599$	25.5515			1.04401			0.522003	+	0.852944i	$0.325185\pi$
$600$	−1.63752	+	13.2750i	−0.0668515	+	0.541951i
$601$	−12.0263			−0.490565			−0.245282	−	0.969452i	$-0.578881\pi$
$601$	−12.0263			−0.490565			−0.245282	+	0.969452i	$0.578881\pi$
$602$			7.81336i			0.318449i
$603$			2.70052i			0.109974i
$604$	−0.513881			−0.0209095
$605$		0			0
$606$	−15.9394			−0.647492
$607$			6.86670i			0.278711i	0.990242	+	0.139355i	$0.0445030\pi$
$607$			6.86670i			0.278711i	−0.990242	+	0.139355i	$0.955497\pi$
$608$			5.84955i			0.237231i
$609$	5.14903			0.208649
$610$	0.186642	−	0.165034i	0.00755693	−	0.00668203i
$611$	−9.92478			−0.401514
$612$		−	0.730841i		−	0.0295425i
$613$		−	7.25457i		−	0.293009i	−0.989210	−	0.146505i	$-0.953198\pi$
$613$		−	7.25457i		−	0.293009i	0.989210	−	0.146505i	$-0.0468023\pi$
$614$	38.0205			1.53438
$615$	15.2243	−	13.4617i	0.613901	−	0.542827i
$616$		0			0
$617$			38.3634i			1.54445i	0.635347	+	0.772227i	$0.280857\pi$
$617$			38.3634i			1.54445i	−0.635347	+	0.772227i	$0.719143\pi$
$618$		−	25.0884i		−	1.00920i
$619$	29.6893			1.19331			0.596656	−	0.802497i	$-0.296496\pi$
$619$	29.6893			1.19331			0.596656	+	0.802497i	$0.296496\pi$
$620$	0.276454	+	0.312650i	0.0111026	+	0.0125563i
$621$	4.00000			0.160514
$622$		−	23.2896i		−	0.933829i
$623$			20.1622i			0.807782i
$624$	−3.50659			−0.140376
$625$	−24.2506	−	6.07522i	−0.970024	−	0.243009i
$626$	−39.7889			−1.59029
$627$		0			0
$628$		−	0.312650i		−	0.0124761i
$629$	6.07522			0.242235
$630$	2.61942	+	2.96239i	0.104360	+	0.118024i
$631$	−19.6991			−0.784209			−0.392105	−	0.919921i	$-0.628253\pi$
$631$	−19.6991			−0.784209			−0.392105	+	0.919921i	$0.628253\pi$
$632$			37.3865i			1.48715i
$633$		−	18.4993i		−	0.735281i
$634$	−1.05220			−0.0417882
$635$	23.1490	−	20.4690i	0.918641	−	0.812287i
$636$	0.276454			0.0109621
$637$			4.49341i			0.178036i
$638$		0			0
$639$	14.0508			0.555840
$640$	−21.2325	+	18.7743i	−0.839288	+	0.742121i
$641$	−31.4372			−1.24170			−0.620848	−	0.783931i	$-0.713212\pi$
$641$	−31.4372			−1.24170			−0.620848	+	0.783931i	$0.713212\pi$
$642$		−	12.2677i		−	0.484169i
$643$			34.4894i			1.36013i	0.733151	+	0.680065i	$0.238049\pi$
$643$			34.4894i			1.36013i	−0.733151	+	0.680065i	$0.761951\pi$
$644$	0.926192			0.0364971
$645$	−6.54420	−	7.40105i	−0.257678	−	0.291416i
$646$	−29.8641			−1.17499
$647$		−	5.61213i		−	0.220635i	−0.993896	−	0.110318i	$-0.964813\pi$
$647$		−	5.61213i		−	0.220635i	0.993896	−	0.110318i	$-0.0351868\pi$
$648$			2.67513i			0.105089i
$649$		0			0
$650$	0.730841	−	5.92478i	0.0286659	−	0.232389i
$651$	−1.14903			−0.0450341
$652$		−	0.0606343i		−	0.00237462i
$653$		−	4.06537i		−	0.159090i	−0.996831	−	0.0795452i	$-0.974653\pi$
$653$		−	4.06537i		−	0.159090i	0.996831	−	0.0795452i	$-0.0253468\pi$
$654$	−14.8119			−0.579193
$655$	−8.77575	−	9.92478i	−0.342897	−	0.387793i
$656$	39.5369			1.54366
$657$			10.7308i			0.418650i
$658$			21.7743i			0.848852i
$659$	11.3747			0.443095			0.221548	−	0.975150i	$-0.428889\pi$
$659$	11.3747			0.443095			0.221548	+	0.975150i	$0.428889\pi$
$660$		0			0
$661$	−42.4749			−1.65208			−0.826040	−	0.563611i	$-0.809412\pi$
$661$	−42.4749			−1.65208			−0.826040	+	0.563611i	$0.809412\pi$
$662$			1.42548i			0.0554030i
$663$		−	3.03761i		−	0.117971i
$664$	26.4690			1.02720
$665$	10.7005	−	9.46168i	0.414949	−	0.366908i
$666$	2.38787			0.0925282
$667$			17.2506i			0.667946i
$668$			0.0956908i			0.00370239i
$669$	−17.6121			−0.680924
$670$	−5.92478	−	6.70052i	−0.228894	−	0.258864i
$671$		0			0
$672$			1.30536i			0.0503552i
$673$		−	14.8813i		−	0.573631i	−0.957986	−	0.286816i	$-0.907403\pi$
$673$		−	14.8813i		−	0.573631i	0.957986	−	0.286816i	$-0.0925967\pi$
$674$	29.3561			1.13076
$675$	−4.96239	−	0.612127i	−0.191002	−	0.0235608i
$676$	−2.39517			−0.0921218
$677$			27.7685i			1.06723i	0.845728	+	0.533614i	$0.179167\pi$
$677$			27.7685i			1.06723i	−0.845728	+	0.533614i	$0.820833\pi$
$678$		−	3.35026i		−	0.128666i
$679$	13.7019			0.525832
$680$	−14.9321	−	16.8872i	−0.572619	−	0.647593i
$681$	17.4314			0.667971
$682$		0			0
$683$			25.4617i			0.974264i	0.873328	+	0.487132i	$0.161957\pi$
$683$			25.4617i			0.974264i	−0.873328	+	0.487132i	$0.838043\pi$
$684$	−1.03761			−0.0396741
$685$	5.61213	−	4.96239i	0.214428	−	0.189603i
$686$	22.2374			0.849029
$687$			13.0738i			0.498797i
$688$		−	19.2203i		−	0.732766i
$689$	1.14903			0.0437746
$690$	−9.92478	+	8.77575i	−0.377830	+	0.334087i
$691$	43.6991			1.66239			0.831196	−	0.555979i	$-0.187657\pi$
$691$	43.6991			1.66239			0.831196	+	0.555979i	$0.187657\pi$
$692$			4.51976i			0.171816i
$693$		0			0
$694$	9.19394			0.348997
$695$	31.4010	+	35.5125i	1.19111	+	1.34706i
$696$	−11.5369			−0.437305
$697$			34.2492i			1.29728i
$698$			6.58910i			0.249401i
$699$	−13.8437			−0.523616
$700$	−1.14903	−	0.141737i	−0.0434293	−	0.00535714i
$701$	−7.01317			−0.264884			−0.132442	−	0.991191i	$-0.542282\pi$
$701$	−7.01317			−0.264884			−0.132442	+	0.991191i	$0.542282\pi$
$702$		−	1.19394i		−	0.0450622i
$703$		−	8.62530i		−	0.325309i
$704$		0			0
$705$	−18.2374	−	20.6253i	−0.686861	−	0.776794i
$706$	9.73813			0.366500
$707$			12.8481i			0.483204i
$708$		−	2.57452i		−	0.0967562i
$709$	−45.6747			−1.71535			−0.857674	−	0.514194i	$-0.828091\pi$
$709$	−45.6747			−1.71535			−0.857674	+	0.514194i	$0.828091\pi$
$710$	−34.8627	+	30.8265i	−1.30837	+	1.15690i
$711$	−13.9756			−0.524125
$712$		−	45.1754i		−	1.69302i
$713$		−	3.84955i		−	0.144167i
$714$	−6.66433			−0.249406
$715$		0			0
$716$	−2.07522			−0.0775547
$717$			12.3733i			0.462089i
$718$			12.7757i			0.476787i
$719$	−16.2520			−0.606098			−0.303049	−	0.952975i	$-0.598005\pi$
$719$	−16.2520			−0.606098			−0.303049	+	0.952975i	$0.598005\pi$
$720$	−6.44358	−	7.28726i	−0.240138	−	0.271580i
$721$	−20.2228			−0.753138
$722$			14.2569i			0.530588i
$723$		−	24.5501i		−	0.913027i
$724$	1.51247			0.0562104
$725$	2.63989	−	21.4010i	0.0980430	−	0.794815i
$726$		0			0
$727$			15.2243i			0.564636i	0.959321	+	0.282318i	$0.0911034\pi$
$727$			15.2243i			0.564636i	−0.959321	+	0.282318i	$0.908897\pi$
$728$			2.57452i			0.0954179i
$729$	−1.00000			−0.0370370
$730$	−23.5428	−	26.6253i	−0.871358	−	0.985447i
$731$	16.6497			0.615813
$732$			0.0145884i			0.000539205i
$733$		−	43.5066i		−	1.60695i	−0.595337	−	0.803476i	$-0.702981\pi$
$733$		−	43.5066i		−	1.60695i	0.595337	−	0.803476i	$-0.297019\pi$
$734$	−34.0870			−1.25817
$735$	−9.33804	+	8.25694i	−0.344439	+	0.304562i
$736$	−4.37328			−0.161201
$737$		0			0
$738$			13.4617i			0.495531i
$739$	−7.02302			−0.258346			−0.129173	−	0.991622i	$-0.541232\pi$
$739$	−7.02302			−0.258346			−0.129173	+	0.991622i	$0.541232\pi$
$740$	−0.523730	+	0.463096i	−0.0192527	+	0.0170237i
$741$	−4.31265			−0.158429
$742$		−	2.52090i		−	0.0925452i
$743$		−	2.94192i		−	0.107929i	−0.998543	−	0.0539643i	$-0.982814\pi$
$743$		−	2.94192i		−	0.107929i	0.998543	−	0.0539643i	$-0.0171857\pi$
$744$	2.57452			0.0943863
$745$	−9.46168	−	10.7005i	−0.346649	−	0.392037i
$746$	−41.7586			−1.52889
$747$			9.89446i			0.362019i
$748$		0			0
$749$	−9.88858			−0.361321
$750$	13.6556	−	9.36836i	0.498633	−	0.342084i
$751$	24.1016			0.879479			0.439739	−	0.898125i	$-0.355071\pi$
$751$	24.1016			0.879479			0.439739	+	0.898125i	$0.355071\pi$
$752$		−	53.5633i		−	1.95325i
$753$		−	13.9003i		−	0.506557i
$754$	5.14903			0.187517
$755$	−3.92478	−	4.43866i	−0.142837	−	0.161539i
$756$	−0.231548			−0.00842132
$757$		−	16.3127i		−	0.592893i	−0.955049	−	0.296447i	$-0.904198\pi$
$757$		−	16.3127i		−	0.592893i	0.955049	−	0.296447i	$-0.0958017\pi$
$758$		−	5.55149i		−	0.201639i
$759$		0			0
$760$	−23.9756	+	21.1998i	−0.869685	+	0.768998i
$761$	5.08840			0.184454			0.0922271	−	0.995738i	$-0.470601\pi$
$761$	5.08840			0.184454			0.0922271	+	0.995738i	$0.470601\pi$
$762$			20.4690i			0.741513i
$763$			11.9394i			0.432234i
$764$	−0.252016			−0.00911762
$765$	6.31265	−	5.58181i	0.228234	−	0.201811i
$766$	2.61071			0.0943289
$767$		−	10.7005i		−	0.386374i
$768$		−	4.61213i		−	0.166426i
$769$	2.10157			0.0757846			0.0378923	−	0.999282i	$-0.487936\pi$
$769$	2.10157			0.0757846			0.0378923	+	0.999282i	$0.487936\pi$
$770$		0			0
$771$	−18.8872			−0.680205
$772$		−	1.66688i		−	0.0599924i
$773$		−	46.0625i		−	1.65675i	−0.560171	−	0.828377i	$-0.689264\pi$
$773$		−	46.0625i		−	1.65675i	0.560171	−	0.828377i	$-0.310736\pi$
$774$	6.54420			0.235226
$775$	−0.589104	+	4.77575i	−0.0211612	+	0.171550i
$776$	−30.7005			−1.10208
$777$		−	1.92478i		−	0.0690510i
$778$		−	9.66291i		−	0.346432i
$779$	48.6253			1.74218
$780$	0.231548	+	0.261865i	0.00829075	+	0.00937628i
$781$		0			0
$782$		−	22.3272i		−	0.798420i
$783$		−	4.31265i		−	0.154122i
$784$	−24.2506			−0.866093

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:	\( N \)	\(=\)	\( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1815.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.48119i q^{2} -1.00000i q^{3} -0.193937 q^{4} +(-1.48119 - 1.67513i) q^{5} +1.48119 q^{6} -1.19394i q^{7} +2.67513i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.48119i q^{2} -1.00000i q^{3} -0.193937 q^{4} +(-1.48119 - 1.67513i) q^{5} +1.48119 q^{6} -1.19394i q^{7} +2.67513i q^{8} -1.00000 q^{9} +(2.48119 - 2.19394i) q^{10} +0.193937i q^{12} +0.806063i q^{13} +1.76845 q^{14} +(-1.67513 + 1.48119i) q^{15} -4.35026 q^{16} -3.76845i q^{17} -1.48119i q^{18} -5.35026 q^{19} +(0.287258 + 0.324869i) q^{20} -1.19394 q^{21} -4.00000i q^{23} +2.67513 q^{24} +(-0.612127 + 4.96239i) q^{25} -1.19394 q^{26} +1.00000i q^{27} +0.231548i q^{28} -4.31265 q^{29} +(-2.19394 - 2.48119i) q^{30} +0.962389 q^{31} -1.09332i q^{32} +5.58181 q^{34} +(-2.00000 + 1.76845i) q^{35} +0.193937 q^{36} +1.61213i q^{37} -7.92478i q^{38} +0.806063 q^{39} +(4.48119 - 3.96239i) q^{40} -9.08840 q^{41} -1.76845i q^{42} +4.41819i q^{43} +(1.48119 + 1.67513i) q^{45} +5.92478 q^{46} +12.3127i q^{47} +4.35026i q^{48} +5.57452 q^{49} +(-7.35026 - 0.906679i) q^{50} -3.76845 q^{51} -0.156325i q^{52} -1.42548i q^{53} -1.48119 q^{54} +3.19394 q^{56} +5.35026i q^{57} -6.38787i q^{58} -13.2750 q^{59} +(0.324869 - 0.287258i) q^{60} +0.0752228 q^{61} +1.42548i q^{62} +1.19394i q^{63} -7.08110 q^{64} +(1.35026 - 1.19394i) q^{65} -2.70052i q^{67} +0.730841i q^{68} -4.00000 q^{69} +(-2.61942 - 2.96239i) q^{70} -14.0508 q^{71} -2.67513i q^{72} -10.7308i q^{73} -2.38787 q^{74} +(4.96239 + 0.612127i) q^{75} +1.03761 q^{76} +1.19394i q^{78} +13.9756 q^{79} +(6.44358 + 7.28726i) q^{80} +1.00000 q^{81} -13.4617i q^{82} -9.89446i q^{83} +0.231548 q^{84} +(-6.31265 + 5.58181i) q^{85} -6.54420 q^{86} +4.31265i q^{87} -16.8872 q^{89} +(-2.48119 + 2.19394i) q^{90} +0.962389 q^{91} +0.775746i q^{92} -0.962389i q^{93} -18.2374 q^{94} +(7.92478 + 8.96239i) q^{95} -1.09332 q^{96} +11.4763i q^{97} +8.25694i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{4} + 2 q^{5} - 2 q^{6} - 6 q^{9}+O(q^{10}) \) 6 * q - 2 * q^4 + 2 * q^5 - 2 * q^6 - 6 * q^9	\( 6 q - 2 q^{4} + 2 q^{5} - 2 q^{6} - 6 q^{9} + 4 q^{10} - 12 q^{14} - 6 q^{16} - 12 q^{19} - 10 q^{20} - 8 q^{21} + 6 q^{24} - 2 q^{25} - 8 q^{26} + 16 q^{29} - 14 q^{30} - 16 q^{31} + 36 q^{34} - 12 q^{35} + 2 q^{36} + 4 q^{39} + 16 q^{40} - 16 q^{41} - 2 q^{45} - 8 q^{46} + 10 q^{49} - 24 q^{50} + 2 q^{54} + 20 q^{56} - 16 q^{59} + 12 q^{60} + 44 q^{61} + 22 q^{64} - 12 q^{65} - 24 q^{69} - 40 q^{70} - 24 q^{71} - 16 q^{74} + 8 q^{75} + 28 q^{76} - 20 q^{79} + 6 q^{80} + 6 q^{81} + 24 q^{84} + 4 q^{85} - 20 q^{86} - 36 q^{89} - 4 q^{90} - 16 q^{91} - 24 q^{94} + 4 q^{95} + 6 q^{96}+O(q^{100}) \) 6 * q - 2 * q^4 + 2 * q^5 - 2 * q^6 - 6 * q^9 + 4 * q^10 - 12 * q^14 - 6 * q^16 - 12 * q^19 - 10 * q^20 - 8 * q^21 + 6 * q^24 - 2 * q^25 - 8 * q^26 + 16 * q^29 - 14 * q^30 - 16 * q^31 + 36 * q^34 - 12 * q^35 + 2 * q^36 + 4 * q^39 + 16 * q^40 - 16 * q^41 - 2 * q^45 - 8 * q^46 + 10 * q^49 - 24 * q^50 + 2 * q^54 + 20 * q^56 - 16 * q^59 + 12 * q^60 + 44 * q^61 + 22 * q^64 - 12 * q^65 - 24 * q^69 - 40 * q^70 - 24 * q^71 - 16 * q^74 + 8 * q^75 + 28 * q^76 - 20 * q^79 + 6 * q^80 + 6 * q^81 + 24 * q^84 + 4 * q^85 - 20 * q^86 - 36 * q^89 - 4 * q^90 - 16 * q^91 - 24 * q^94 + 4 * q^95 + 6 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more