LMFDB - Embedded newform 495.2.c.e.199.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [495,2,Mod(199,495)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(495, 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("495.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 495 = 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	495.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:	$3.95259490005$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 165)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.5
Root			$0.403032 + 0.403032i$ of defining polynomial
Character	$\chi$	$=$	495.199
Dual form			495.2.c.e.199.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.48119i q^{2} -0.193937 q^{4} +(1.48119 + 1.67513i) q^{5} +1.19394i q^{7} +2.67513i q^{8} +O(q^{10})$	\(q+1.48119i q^{2} -0.193937 q^{4} +(1.48119 + 1.67513i) q^{5} +1.19394i q^{7} +2.67513i q^{8} +(-2.48119 + 2.19394i) q^{10} -1.00000 q^{11} -0.806063i q^{13} -1.76845 q^{14} -4.35026 q^{16} -3.76845i q^{17} +5.35026 q^{19} +(-0.287258 - 0.324869i) q^{20} -1.48119i q^{22} +4.00000i q^{23} +(-0.612127 + 4.96239i) q^{25} +1.19394 q^{26} -0.231548i q^{28} -4.31265 q^{29} +0.962389 q^{31} -1.09332i q^{32} +5.58181 q^{34} +(-2.00000 + 1.76845i) q^{35} +1.61213i q^{37} +7.92478i q^{38} +(-4.48119 + 3.96239i) q^{40} -9.08840 q^{41} -4.41819i q^{43} +0.193937 q^{44} -5.92478 q^{46} -12.3127i q^{47} +5.57452 q^{49} +(-7.35026 - 0.906679i) q^{50} +0.156325i q^{52} +1.42548i q^{53} +(-1.48119 - 1.67513i) q^{55} -3.19394 q^{56} -6.38787i q^{58} +13.2750 q^{59} -0.0752228 q^{61} +1.42548i q^{62} -7.08110 q^{64} +(1.35026 - 1.19394i) q^{65} -2.70052i q^{67} +0.730841i q^{68} +(-2.61942 - 2.96239i) q^{70} +14.0508 q^{71} +10.7308i q^{73} -2.38787 q^{74} -1.03761 q^{76} -1.19394i q^{77} -13.9756 q^{79} +(-6.44358 - 7.28726i) q^{80} -13.4617i q^{82} -9.89446i q^{83} +(6.31265 - 5.58181i) q^{85} +6.54420 q^{86} -2.67513i q^{88} +16.8872 q^{89} +0.962389 q^{91} -0.775746i q^{92} +18.2374 q^{94} +(7.92478 + 8.96239i) q^{95} +11.4763i q^{97} +8.25694i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{4} - 2 q^{5}+O(q^{10}) $ 6 * q - 2 * q^4 - 2 * q^5	$ 6 q - 2 q^{4} - 2 q^{5} - 4 q^{10} - 6 q^{11} + 12 q^{14} - 6 q^{16} + 12 q^{19} + 10 q^{20} - 2 q^{25} + 8 q^{26} + 16 q^{29} - 16 q^{31} + 36 q^{34} - 12 q^{35} - 16 q^{40} - 16 q^{41} + 2 q^{44} + 8 q^{46} + 10 q^{49} - 24 q^{50} + 2 q^{55} - 20 q^{56} + 16 q^{59} - 44 q^{61} + 22 q^{64} - 12 q^{65} - 40 q^{70} + 24 q^{71} - 16 q^{74} - 28 q^{76} + 20 q^{79} - 6 q^{80} - 4 q^{85} + 20 q^{86} + 36 q^{89} - 16 q^{91} + 24 q^{94} + 4 q^{95}+O(q^{100}) $ 6 * q - 2 * q^4 - 2 * q^5 - 4 * q^10 - 6 * q^11 + 12 * q^14 - 6 * q^16 + 12 * q^19 + 10 * q^20 - 2 * q^25 + 8 * q^26 + 16 * q^29 - 16 * q^31 + 36 * q^34 - 12 * q^35 - 16 * q^40 - 16 * q^41 + 2 * q^44 + 8 * q^46 + 10 * q^49 - 24 * q^50 + 2 * q^55 - 20 * q^56 + 16 * q^59 - 44 * q^61 + 22 * q^64 - 12 * q^65 - 40 * q^70 + 24 * q^71 - 16 * q^74 - 28 * q^76 + 20 * q^79 - 6 * q^80 - 4 * q^85 + 20 * q^86 + 36 * q^89 - 16 * q^91 + 24 * q^94 + 4 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/495\mathbb{Z}\right)^\times$.

$n$	$46$	$56$	$397$
$\chi(n)$	$1$	$1$	$-1$

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	$0.824558\pi$
$3$		0			0
$3$		0			0
$4$	−0.193937			−0.0969683
$5$	1.48119	+	1.67513i	0.662410	+	0.749141i
$5$	1.48119	+	1.67513i	0.662410	+	0.749141i
$6$		0			0
$7$			1.19394i			0.451266i	0.974212	+	0.225633i	$0.0724450\pi$
$7$			1.19394i			0.451266i	−0.974212	+	0.225633i	$0.927555\pi$
$8$			2.67513i			0.945802i
$9$		0			0
$10$	−2.48119	+	2.19394i	−0.784623	+	0.693784i
$11$	−1.00000			−0.301511
$11$	−1.00000			−0.301511
$12$		0			0
$13$		−	0.806063i		−	0.223562i	−0.993733	−	0.111781i	$-0.964345\pi$
$13$		−	0.806063i		−	0.223562i	0.993733	−	0.111781i	$-0.0356555\pi$
$14$	−1.76845			−0.472639
$15$		0			0
$16$	−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	$-0.151073\pi$
$18$		0			0
$19$	5.35026			1.22743			0.613717	−	0.789526i	$-0.289674\pi$
$19$	5.35026			1.22743			0.613717	+	0.789526i	$0.289674\pi$
$20$	−0.287258	−	0.324869i	−0.0642328	−	0.0726429i
$21$		0			0
$22$		−	1.48119i		−	0.315792i
$23$			4.00000i			0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$			4.00000i			0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$		0			0
$25$	−0.612127	+	4.96239i	−0.122425	+	0.992478i
$26$	1.19394			0.234150
$27$		0			0
$28$		−	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	$0.631136\pi$
$30$		0			0
$31$	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			0
$37$			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			0
$40$	−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	$0.751163\pi$
$42$		0			0
$43$		−	4.41819i		−	0.673768i	−0.941546	−	0.336884i	$-0.890627\pi$
$43$		−	4.41819i		−	0.673768i	0.941546	−	0.336884i	$-0.109373\pi$
$44$	0.193937			0.0292370
$45$		0			0
$46$	−5.92478			−0.873561
$47$		−	12.3127i		−	1.79598i	−0.440011	−	0.897992i	$-0.645026\pi$
$47$		−	12.3127i		−	1.79598i	0.440011	−	0.897992i	$-0.354974\pi$
$48$		0			0
$49$	5.57452			0.796359
$50$	−7.35026	−	0.906679i	−1.03948	−	0.128224i
$51$		0			0
$52$			0.156325i			0.0216784i
$53$			1.42548i			0.195805i	0.995196	+	0.0979027i	$0.0312134\pi$
$53$			1.42548i			0.195805i	−0.995196	+	0.0979027i	$0.968787\pi$
$54$		0			0
$55$	−1.48119	−	1.67513i	−0.199724	−	0.225875i
$56$	−3.19394			−0.426808
$57$		0			0
$58$		−	6.38787i		−	0.838769i
$59$	13.2750			1.72826			0.864131	−	0.503266i	$-0.167868\pi$
$59$	13.2750			1.72826			0.864131	+	0.503266i	$0.167868\pi$
$60$		0			0
$61$	−0.0752228			−0.00963129			−0.00481565	−	0.999988i	$-0.501533\pi$
$61$	−0.0752228			−0.00963129			−0.00481565	+	0.999988i	$0.501533\pi$
$62$			1.42548i			0.181037i
$63$		0			0
$64$	−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
$69$		0			0
$70$	−2.61942	−	2.96239i	−0.313081	−	0.354073i
$71$	14.0508			1.66752			0.833761	−	0.552126i	$-0.186183\pi$
$71$	14.0508			1.66752			0.833761	+	0.552126i	$0.186183\pi$
$72$		0			0
$73$			10.7308i			1.25595i	0.778234	+	0.627975i	$0.216116\pi$
$73$			10.7308i			1.25595i	−0.778234	+	0.627975i	$0.783884\pi$
$74$	−2.38787			−0.277585
$75$		0			0
$76$	−1.03761			−0.119022
$77$		−	1.19394i		−	0.136062i
$78$		0			0
$79$	−13.9756			−1.57237			−0.786187	−	0.617989i	$-0.787948\pi$
$79$	−13.9756			−1.57237			−0.786187	+	0.617989i	$0.787948\pi$
$80$	−6.44358	−	7.28726i	−0.720414	−	0.814740i
$81$		0			0
$82$		−	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			0
$85$	6.31265	−	5.58181i	0.684703	−	0.605432i
$86$	6.54420			0.705679
$87$		0			0
$88$		−	2.67513i		−	0.285170i
$89$	16.8872			1.79004			0.895018	−	0.446030i	$-0.147163\pi$
$89$	16.8872			1.79004			0.895018	+	0.446030i	$0.147163\pi$
$90$		0			0
$91$	0.962389			0.100886
$92$		−	0.775746i		−	0.0808771i
$93$		0			0
$94$	18.2374			1.88105
$95$	7.92478	+	8.96239i	0.813065	+	0.919522i
$96$		0			0
$97$			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	$0.679834\pi$
$102$		0			0
$103$		−	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
$105$		0			0
$106$	−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			0
$109$	10.0000			0.957826			0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000			0.957826			0.478913	+	0.877862i	$0.341031\pi$
$110$	2.48119	−	2.19394i	0.236573	−	0.209184i
$111$		0			0
$112$		−	5.19394i		−	0.490781i
$113$			2.26187i			0.212778i	0.994325	+	0.106389i	$0.0339289\pi$
$113$			2.26187i			0.212778i	−0.994325	+	0.106389i	$0.966071\pi$
$114$		0			0
$115$	−6.70052	+	5.92478i	−0.624827	+	0.552488i
$116$	0.836381			0.0776560
$117$		0			0
$118$			19.6629i			1.81012i
$119$	4.49929			0.412449
$120$		0			0
$121$	1.00000			0.0909091
$122$		−	0.111420i		−	0.0100875i
$123$		0			0
$124$	−0.186642			−0.0167610
$125$	−9.21933	+	6.32487i	−0.824602	+	0.565713i
$126$		0			0
$127$		−	13.8192i		−	1.22626i	−0.789982	−	0.613130i	$-0.789910\pi$
$127$		−	13.8192i		−	1.22626i	0.789982	−	0.613130i	$-0.210090\pi$
$128$		−	12.6751i		−	1.12033i
$129$		0			0
$130$	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
$135$		0			0
$136$	10.0811			0.864447
$137$		−	3.35026i		−	0.286232i	−0.989706	−	0.143116i	$-0.954288\pi$
$137$		−	3.35026i		−	0.286232i	0.989706	−	0.143116i	$-0.0457122\pi$
$138$		0			0
$139$	21.1998			1.79814			0.899072	−	0.437800i	$-0.144242\pi$
$139$	21.1998			1.79814			0.899072	+	0.437800i	$0.144242\pi$
$140$	0.387873	−	0.342968i	0.0327813	−	0.0289860i
$141$		0			0
$142$			20.8119i			1.74650i
$143$			0.806063i			0.0674064i
$144$		0			0
$145$	−6.38787	−	7.22425i	−0.530484	−	0.599942i
$146$	−15.8945			−1.31543
$147$		0			0
$148$		−	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			0
$151$	−2.64974			−0.215633			−0.107816	−	0.994171i	$-0.534386\pi$
$151$	−2.64974			−0.215633			−0.107816	+	0.994171i	$0.534386\pi$
$152$			14.3127i			1.16091i
$153$		0			0
$154$	1.76845			0.142506
$155$	1.42548	+	1.61213i	0.114498	+	0.129489i
$156$		0			0
$157$			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
$159$		0			0
$160$	1.83146	−	1.61942i	0.144789	−	0.128026i
$161$	−4.77575			−0.376382
$162$		0			0
$163$			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	$-0.00607714\pi$
$168$		0			0
$169$	12.3503			0.950020
$170$	8.26774	+	9.35026i	0.634107	+	0.717132i
$171$		0			0
$172$			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	$-0.346483\pi$
$174$		0			0
$175$	−5.92478	−	0.730841i	−0.447871	−	0.0552464i
$176$	4.35026			0.327913
$177$		0			0
$178$			25.0132i			1.87482i
$179$	−10.7005			−0.799795			−0.399897	−	0.916560i	$-0.630954\pi$
$179$	−10.7005			−0.799795			−0.399897	+	0.916560i	$0.630954\pi$
$180$		0			0
$181$	−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			0
$184$	−10.7005			−0.788853
$185$	−2.70052	+	2.38787i	−0.198546	+	0.175560i
$186$		0			0
$187$			3.76845i			0.275577i
$188$			2.38787i			0.174154i
$189$		0			0
$190$	−13.2750	+	11.7381i	−0.963073	+	0.851574i
$191$	−1.29948			−0.0940268			−0.0470134	−	0.998894i	$-0.514970\pi$
$191$	−1.29948			−0.0940268			−0.0470134	+	0.998894i	$0.514970\pi$
$192$		0			0
$193$		−	8.59498i		−	0.618680i	−0.950951	−	0.309340i	$-0.899892\pi$
$193$		−	8.59498i		−	0.618680i	0.950951	−	0.309340i	$-0.100108\pi$
$194$	−16.9986			−1.22043
$195$		0			0
$196$	−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
$201$		0			0
$202$		−	15.9394i		−	1.12149i
$203$		−	5.14903i		−	0.361391i
$204$		0			0
$205$	−13.4617	−	15.2243i	−0.940205	−	1.06331i
$206$	25.0884			1.74799
$207$		0			0
$208$			3.50659i			0.243138i
$209$	−5.35026			−0.370085
$210$		0			0
$211$	−18.4993			−1.27354			−0.636772	−	0.771052i	$-0.719731\pi$
$211$	−18.4993			−1.27354			−0.636772	+	0.771052i	$0.719731\pi$
$212$		−	0.276454i		−	0.0189869i
$213$		0			0
$214$	12.2677			0.838606
$215$	7.40105	−	6.54420i	0.504747	−	0.446311i
$216$		0			0
$217$			1.14903i			0.0780013i
$218$			14.8119i			1.00319i
$219$		0			0
$220$	0.287258	+	0.324869i	0.0193669	+	0.0219027i
$221$	−3.03761			−0.204332
$222$		0			0
$223$		−	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			0
$226$	−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	$0.803646\pi$
$228$		0			0
$229$	−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	$-0.149812\pi$
$234$		0			0
$235$	20.6253	−	18.2374i	1.34545	−	1.18968i
$236$	−2.57452			−0.167587
$237$		0			0
$238$			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	$0.631053\pi$
$240$		0			0
$241$	−24.5501			−1.58141			−0.790705	−	0.612198i	$-0.790286\pi$
$241$	−24.5501			−1.58141			−0.790705	+	0.612198i	$0.790286\pi$
$242$			1.48119i			0.0952148i
$243$		0			0
$244$	0.0145884			0.000933930
$245$	8.25694	+	9.33804i	0.527517	+	0.596586i
$246$		0			0
$247$		−	4.31265i		−	0.274407i
$248$			2.57452i			0.163482i
$249$		0			0
$250$	−9.36836	−	13.6556i	−0.592507	−	0.863657i
$251$	−13.9003			−0.877382			−0.438691	−	0.898638i	$-0.644558\pi$
$251$	−13.9003			−0.877382			−0.438691	+	0.898638i	$0.644558\pi$
$252$		0			0
$253$		−	4.00000i		−	0.251478i
$254$	20.4690			1.28434
$255$		0			0
$256$	4.61213			0.288258
$257$			18.8872i			1.17815i	0.808079	+	0.589075i	$0.200508\pi$
$257$			18.8872i			1.17815i	−0.808079	+	0.589075i	$0.799492\pi$
$258$		0			0
$259$	−1.92478			−0.119600
$260$	−0.261865	+	0.231548i	−0.0162402	+	0.0143600i
$261$		0			0
$262$			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
$267$		0			0
$268$			0.523730i			0.0319919i
$269$	−32.3996			−1.97544			−0.987720	−	0.156233i	$-0.950065\pi$
$269$	−32.3996			−1.97544			−0.987720	+	0.156233i	$0.950065\pi$
$270$		0			0
$271$	−16.8265			−1.02214			−0.511069	−	0.859539i	$-0.670751\pi$
$271$	−16.8265			−1.02214			−0.511069	+	0.859539i	$0.670751\pi$
$272$			16.3938i			0.994017i
$273$		0			0
$274$	4.96239			0.299789
$275$	0.612127	−	4.96239i	0.0369126	−	0.299243i
$276$		0			0
$277$			16.9076i			1.01588i	0.861392	+	0.507941i	$0.169593\pi$
$277$			16.9076i			1.01588i	−0.861392	+	0.507941i	$0.830407\pi$
$278$			31.4010i			1.88331i
$279$		0			0
$280$	−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	$0.553536\pi$
$282$		0			0
$283$		−	5.81924i		−	0.345918i	−0.984929	−	0.172959i	$-0.944667\pi$
$283$		−	5.81924i		−	0.345918i	0.984929	−	0.172959i	$-0.0553328\pi$
$284$	−2.72496			−0.161697
$285$		0			0
$286$	−1.19394			−0.0705989
$287$		−	10.8510i		−	0.640512i
$288$		0			0
$289$	2.79877			0.164633
$290$	10.7005	−	9.46168i	0.628356	−	0.555609i
$291$		0			0
$292$		−	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	$-0.0778747\pi$
$294$		0			0
$295$	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			0
$301$	5.27504			0.304048
$302$		−	3.92478i		−	0.225846i
$303$		0			0
$304$	−23.2750			−1.33492
$305$	−0.111420	−	0.126008i	−0.00637987	−	0.00721520i
$306$		0			0
$307$			25.6688i			1.46500i	0.680770	+	0.732498i	$0.261646\pi$
$307$			25.6688i			1.46500i	−0.680770	+	0.732498i	$0.738354\pi$
$308$			0.231548i			0.0131937i
$309$		0			0
$310$	−2.38787	+	2.11142i	−0.135622	+	0.119921i
$311$	15.7235			0.891601			0.445800	−	0.895132i	$-0.352919\pi$
$311$	15.7235			0.891601			0.445800	+	0.895132i	$0.352919\pi$
$312$		0			0
$313$			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
$315$		0			0
$316$	2.71037			0.152470
$317$		−	0.710373i		−	0.0398985i	−0.999801	−	0.0199492i	$-0.993650\pi$
$317$		−	0.710373i		−	0.0398985i	0.999801	−	0.0199492i	$-0.00635046\pi$
$318$		0			0
$319$	4.31265			0.241462
$320$	−10.4885	−	11.8618i	−0.586324	−	0.663093i
$321$		0			0
$322$		−	7.07381i		−	0.394208i
$323$		−	20.1622i		−	1.12186i
$324$		0			0
$325$	4.00000	+	0.493413i	0.221880	+	0.0273696i
$326$	−0.463096			−0.0256485
$327$		0			0
$328$		−	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
$333$		0			0
$334$	0.730841			0.0399898
$335$	4.52373	−	4.00000i	0.247158	−	0.218543i
$336$		0			0
$337$			19.8192i			1.07962i	0.841786	+	0.539811i	$0.181504\pi$
$337$			19.8192i			1.07962i	−0.841786	+	0.539811i	$0.818496\pi$
$338$			18.2931i			0.995015i
$339$		0			0
$340$	−1.22425	+	1.08252i	−0.0663945	+	0.0587077i
$341$	−0.962389			−0.0521163
$342$		0			0
$343$			15.0132i			0.810635i
$344$	11.8192			0.637251
$345$		0			0
$346$	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			0
$349$	−4.44851			−0.238123			−0.119062	−	0.992887i	$-0.537989\pi$
$349$	−4.44851			−0.238123			−0.119062	+	0.992887i	$0.537989\pi$
$350$	1.08252	−	8.77575i	0.0578630	−	0.469083i
$351$		0			0
$352$			1.09332i			0.0582742i
$353$			6.57452i			0.349926i	0.984575	+	0.174963i	$0.0559806\pi$
$353$			6.57452i			0.349926i	−0.984575	+	0.174963i	$0.944019\pi$
$354$		0			0
$355$	20.8119	+	23.5369i	1.10458	+	1.24921i
$356$	−3.27504			−0.173577
$357$		0			0
$358$		−	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	$0.426908\pi$
$360$		0			0
$361$	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			0
$367$			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
$369$		0			0
$370$	−3.53690	−	4.00000i	−0.183875	−	0.207950i
$371$	−1.70194			−0.0883602
$372$		0			0
$373$		−	28.1925i		−	1.45975i	−0.683579	−	0.729877i	$-0.739577\pi$
$373$		−	28.1925i		−	1.45975i	0.683579	−	0.729877i	$-0.260423\pi$
$374$	−5.58181			−0.288629
$375$		0			0
$376$	32.9380			1.69865
$377$			3.47627i			0.179037i
$378$		0			0
$379$	−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
$381$		0			0
$382$		−	1.92478i		−	0.0984802i
$383$			1.76257i			0.0900632i	0.998986	+	0.0450316i	$0.0143389\pi$
$383$			1.76257i			0.0900632i	−0.998986	+	0.0450316i	$0.985661\pi$
$384$		0			0
$385$	2.00000	−	1.76845i	0.101929	−	0.0901287i
$386$	12.7308			0.647983
$387$		0			0
$388$		−	2.22567i		−	0.112991i
$389$	6.52373			0.330766			0.165383	−	0.986229i	$-0.447114\pi$
$389$	6.52373			0.330766			0.165383	+	0.986229i	$0.447114\pi$
$390$		0			0
$391$	15.0738			0.762315
$392$			14.9126i			0.753198i
$393$		0			0
$394$	−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
$399$		0			0
$400$	2.66291	−	21.5877i	0.133146	−	1.07938i
$401$	−8.88717			−0.443804			−0.221902	−	0.975069i	$-0.571226\pi$
$401$	−8.88717			−0.443804			−0.221902	+	0.975069i	$0.571226\pi$
$402$		0			0
$403$		−	0.775746i		−	0.0386427i
$404$	2.08698			0.103831
$405$		0			0
$406$	7.62672			0.378508
$407$		−	1.61213i		−	0.0799102i
$408$		0			0
$409$	4.85097			0.239865			0.119932	−	0.992782i	$-0.461732\pi$
$409$	4.85097			0.239865			0.119932	+	0.992782i	$0.461732\pi$
$410$	22.5501	−	19.9394i	1.11367	−	0.984735i
$411$		0			0
$412$			3.28489i			0.161835i
$413$			15.8496i			0.779906i
$414$		0			0
$415$	16.5745	−	14.6556i	0.813611	−	0.719416i
$416$	−0.881286			−0.0432086
$417$		0			0
$418$		−	7.92478i		−	0.387614i
$419$	−10.7005			−0.522755			−0.261377	−	0.965237i	$-0.584177\pi$
$419$	−10.7005			−0.522755			−0.261377	+	0.965237i	$0.584177\pi$
$420$		0			0
$421$	−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
$423$		0			0
$424$	−3.81336			−0.185193
$425$	18.7005	+	2.30677i	0.907109	+	0.111895i
$426$		0			0
$427$		−	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	$0.454424\pi$
$432$		0			0
$433$			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
$435$		0			0
$436$	−1.93937			−0.0928788
$437$			21.4010i			1.02375i
$438$		0			0
$439$	−5.35026			−0.255354			−0.127677	−	0.991816i	$-0.540752\pi$
$439$	−5.35026			−0.255354			−0.127677	+	0.991816i	$0.540752\pi$
$440$	4.48119	−	3.96239i	0.213633	−	0.188899i
$441$		0			0
$442$		−	4.49929i		−	0.214010i
$443$			19.6873i			0.935374i	0.883894	+	0.467687i	$0.154913\pi$
$443$			19.6873i			0.935374i	−0.883894	+	0.467687i	$0.845087\pi$
$444$		0			0
$445$	25.0132	+	28.2882i	1.18574	+	1.34099i
$446$	26.0870			1.23525
$447$		0			0
$448$		−	8.45439i		−	0.399432i
$449$	−31.3357			−1.47882			−0.739411	−	0.673254i	$-0.764896\pi$
$449$	−31.3357			−1.47882			−0.739411	+	0.673254i	$0.764896\pi$
$450$		0			0
$451$	9.08840			0.427956
$452$		−	0.438658i		−	0.0206328i
$453$		0			0
$454$	−25.8192			−1.21176
$455$	1.42548	+	1.61213i	0.0668277	+	0.0755777i
$456$		0			0
$457$		−	37.5936i		−	1.75855i	−0.476312	−	0.879276i	$-0.658027\pi$
$457$		−	37.5936i		−	1.75855i	0.476312	−	0.879276i	$-0.341973\pi$
$458$		−	19.3649i		−	0.904860i
$459$		0			0
$460$	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
$465$		0			0
$466$	20.5052			0.949884
$467$		−	14.5501i		−	0.673297i	−0.941630	−	0.336649i	$-0.890707\pi$
$467$		−	14.5501i		−	0.673297i	0.941630	−	0.336649i	$-0.109293\pi$
$468$		0			0
$469$	3.22425			0.148882
$470$	27.0132	+	30.5501i	1.24602	+	1.40917i
$471$		0			0
$472$			35.5125i			1.63459i
$473$			4.41819i			0.203149i
$474$		0			0
$475$	−3.27504	+	26.5501i	−0.150269	+	1.21820i
$476$	−0.872577			−0.0399945
$477$		0			0
$478$		−	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	$0.273106\pi$
$480$		0			0
$481$	1.29948			0.0592510
$482$		−	36.3634i		−	1.65631i
$483$		0			0
$484$	−0.193937			−0.00881530
$485$	−19.2243	+	16.9986i	−0.872928	+	0.771866i
$486$		0			0
$487$			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			0
$490$	−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	$0.442223\pi$
$492$		0			0
$493$			16.2520i			0.731954i
$494$	6.38787			0.287404
$495$		0			0
$496$	−4.18664			−0.187986
$497$			16.7757i			0.752495i
$498$		0			0
$499$	−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			0
$502$		−	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	$-0.145198\pi$
$504$		0			0
$505$	−15.9394	−	18.0263i	−0.709292	−	0.802162i
$506$	5.92478			0.263388
$507$		0			0
$508$			2.68006i			0.118908i
$509$	22.1016			0.979635			0.489817	−	0.871825i	$-0.337063\pi$
$509$	22.1016			0.979635			0.489817	+	0.871825i	$0.337063\pi$
$510$		0			0
$511$	−12.8119			−0.566767
$512$		−	18.5188i		−	0.818423i
$513$		0			0
$514$	−27.9756			−1.23395
$515$	28.3733	−	25.0884i	1.25028	−	1.10553i
$516$		0			0
$517$			12.3127i			0.541510i
$518$		−	2.85097i		−	0.125264i
$519$		0			0
$520$	3.19394	+	3.61213i	0.140063	+	0.158402i
$521$	22.8119			0.999409			0.499705	−	0.866196i	$-0.333442\pi$
$521$	22.8119			0.999409			0.499705	+	0.866196i	$0.333442\pi$
$522$		0			0
$523$			12.2677i			0.536431i	0.963359	+	0.268216i	$0.0864339\pi$
$523$			12.2677i			0.536431i	−0.963359	+	0.268216i	$0.913566\pi$
$524$	−1.14903			−0.0501957
$525$		0			0
$526$	−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
$531$		0			0
$532$		−	1.23884i		−	0.0537106i
$533$			7.32582i			0.317317i
$534$		0			0
$535$	13.8740	−	12.2677i	0.599825	−	0.530381i
$536$	7.22425			0.312040
$537$		0			0
$538$		−	47.9902i		−	2.06900i
$539$	−5.57452			−0.240111
$540$		0			0
$541$	−5.22425			−0.224608			−0.112304	−	0.993674i	$-0.535823\pi$
$541$	−5.22425			−0.224608			−0.112304	+	0.993674i	$0.535823\pi$
$542$		−	24.9234i		−	1.07055i
$543$		0			0
$544$	−4.12013			−0.176649
$545$	14.8119	+	16.7513i	0.634474	+	0.717547i
$546$		0			0
$547$		−	17.9697i		−	0.768328i	−0.923265	−	0.384164i	$-0.874490\pi$
$547$		−	17.9697i		−	0.768328i	0.923265	−	0.384164i	$-0.125510\pi$
$548$			0.649738i			0.0277554i
$549$		0			0
$550$	7.35026	+	0.906679i	0.313416	+	0.0386609i
$551$	−23.0738			−0.982977
$552$		0			0
$553$		−	16.6859i		−	0.709558i
$554$	−25.0435			−1.06400
$555$		0			0
$556$	−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	$-0.109148\pi$
$558$		0			0
$559$	−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	$0.767433\pi$
$564$		0			0
$565$	−3.78892	+	3.35026i	−0.159401	+	0.140947i
$566$	8.61942			0.362301
$567$		0			0
$568$			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	$0.670214\pi$
$570$		0			0
$571$	8.05079			0.336915			0.168457	−	0.985709i	$-0.446121\pi$
$571$	8.05079			0.336915			0.168457	+	0.985709i	$0.446121\pi$
$572$		−	0.156325i		−	0.00653628i
$573$		0			0
$574$	16.0724			0.670849
$575$	−19.8496	−	2.44851i	−0.827784	−	0.102110i
$576$		0			0
$577$		−	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
$579$		0			0
$580$	1.23884	+	1.40105i	0.0514401	+	0.0581753i
$581$	11.8134			0.490101
$582$		0			0
$583$		−	1.42548i		−	0.0590375i
$584$	−28.7064			−1.18788
$585$		0			0
$586$	12.2823			0.507379
$587$			15.4763i			0.638774i	0.947624	+	0.319387i	$0.103477\pi$
$587$			15.4763i			0.638774i	−0.947624	+	0.319387i	$0.896523\pi$
$588$		0			0
$589$	5.14903			0.212162
$590$	−32.9380	+	29.1246i	−1.35603	+	1.19904i
$591$		0			0
$592$		−	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
$597$		0			0
$598$			4.77575i			0.195295i
$599$	−25.5515			−1.04401			−0.522003	−	0.852944i	$-0.674815\pi$
$599$	−25.5515			−1.04401			−0.522003	+	0.852944i	$0.674815\pi$
$600$		0			0
$601$	12.0263			0.490565			0.245282	−	0.969452i	$-0.421119\pi$
$601$	12.0263			0.490565			0.245282	+	0.969452i	$0.421119\pi$
$602$			7.81336i			0.318449i
$603$		0			0
$604$	0.513881			0.0209095
$605$	1.48119	+	1.67513i	0.0602191	+	0.0681038i
$606$		0			0
$607$		−	6.86670i		−	0.278711i	−0.990242	−	0.139355i	$-0.955497\pi$
$607$		−	6.86670i		−	0.278711i	0.990242	−	0.139355i	$-0.0445030\pi$
$608$		−	5.84955i		−	0.237231i
$609$		0			0
$610$	0.186642	−	0.165034i	0.00755693	−	0.00668203i
$611$	−9.92478			−0.401514
$612$		0			0
$613$			7.25457i			0.293009i	0.989210	+	0.146505i	$0.0468023\pi$
$613$			7.25457i			0.293009i	−0.989210	+	0.146505i	$0.953198\pi$
$614$	−38.0205			−1.53438
$615$		0			0
$616$	3.19394			0.128687
$617$		−	38.3634i		−	1.54445i	−0.635347	−	0.772227i	$-0.719143\pi$
$617$		−	38.3634i		−	1.54445i	0.635347	−	0.772227i	$-0.280857\pi$
$618$		0			0
$619$	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
$621$		0			0
$622$			23.2896i			0.933829i
$623$			20.1622i			0.807782i
$624$		0			0
$625$	−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
$630$		0			0
$631$	−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
$633$		0			0
$634$	1.05220			0.0417882
$635$	23.1490	−	20.4690i	0.918641	−	0.812287i
$636$		0			0
$637$		−	4.49341i		−	0.178036i
$638$			6.38787i			0.252898i
$639$		0			0
$640$	21.2325	−	18.7743i	0.839288	−	0.742121i
$641$	31.4372			1.24170			0.620848	−	0.783931i	$-0.286788\pi$
$641$	31.4372			1.24170			0.620848	+	0.783931i	$0.286788\pi$
$642$		0			0
$643$			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
$645$		0			0
$646$	29.8641			1.17499
$647$			5.61213i			0.220635i	0.993896	+	0.110318i	$0.0351868\pi$
$647$			5.61213i			0.220635i	−0.993896	+	0.110318i	$0.964813\pi$
$648$		0			0
$649$	−13.2750			−0.521091
$650$	−0.730841	+	5.92478i	−0.0286659	+	0.232389i
$651$		0			0
$652$		−	0.0606343i		−	0.00237462i
$653$			4.06537i			0.159090i	0.996831	+	0.0795452i	$0.0253468\pi$
$653$			4.06537i			0.159090i	−0.996831	+	0.0795452i	$0.974653\pi$
$654$		0			0
$655$	8.77575	+	9.92478i	0.342897	+	0.387793i
$656$	39.5369			1.54366
$657$		0			0
$658$			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
$663$		0			0
$664$	26.4690			1.02720
$665$	−10.7005	+	9.46168i	−0.414949	+	0.366908i
$666$		0			0
$667$		−	17.2506i		−	0.667946i
$668$			0.0956908i			0.00370239i
$669$		0			0
$670$	5.92478	+	6.70052i	0.228894	+	0.258864i
$671$	0.0752228			0.00290394
$672$		0			0
$673$			14.8813i			0.573631i	0.957986	+	0.286816i	$0.0925967\pi$
$673$			14.8813i			0.573631i	−0.957986	+	0.286816i	$0.907403\pi$
$674$	−29.3561			−1.13076
$675$		0			0
$676$	−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	$0.820833\pi$
$678$		0			0
$679$	−13.7019			−0.525832
$680$	14.9321	+	16.8872i	0.572619	+	0.647593i
$681$		0			0
$682$		−	1.42548i		−	0.0545846i
$683$		−	25.4617i		−	0.974264i	−0.873328	−	0.487132i	$-0.838043\pi$
$683$		−	25.4617i		−	0.974264i	0.873328	−	0.487132i	$-0.161957\pi$
$684$		0			0
$685$	5.61213	−	4.96239i	0.214428	−	0.189603i
$686$	−22.2374			−0.849029
$687$		0			0
$688$			19.2203i			0.732766i
$689$	1.14903			0.0437746
$690$		0			0
$691$	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
$696$		0			0
$697$			34.2492i			1.29728i
$698$		−	6.58910i		−	0.249401i
$699$		0			0
$700$	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	$0.542282\pi$
$702$		0			0
$703$			8.62530i			0.325309i
$704$	7.08110			0.266879
$705$		0			0
$706$	−9.73813			−0.366500
$707$		−	12.8481i		−	0.483204i
$708$		0			0
$709$	−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
$711$		0			0
$712$			45.1754i			1.69302i
$713$			3.84955i			0.144167i
$714$		0			0
$715$	−1.35026	+	1.19394i	−0.0504969	+	0.0446507i
$716$	2.07522			0.0775547
$717$		0			0
$718$			12.7757i			0.476787i
$719$	16.2520			0.606098			0.303049	−	0.952975i	$-0.401995\pi$
$719$	16.2520			0.606098			0.303049	+	0.952975i	$0.401995\pi$
$720$		0			0
$721$	20.2228			0.753138
$722$			14.2569i			0.530588i
$723$		0			0
$724$	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
$729$		0			0
$730$	−23.5428	−	26.6253i	−0.871358	−	0.985447i
$731$	−16.6497			−0.615813
$732$		0			0
$733$			43.5066i			1.60695i	0.595337	+	0.803476i	$0.297019\pi$
$733$			43.5066i			1.60695i	−0.595337	+	0.803476i	$0.702981\pi$
$734$	−34.0870			−1.25817
$735$		0			0
$736$	4.37328			0.161201
$737$			2.70052i			0.0994751i
$738$		0			0
$739$	7.02302			0.258346			0.129173	−	0.991622i	$-0.458768\pi$
$739$	7.02302			0.258346			0.129173	+	0.991622i	$0.458768\pi$
$740$	0.523730	−	0.463096i	0.0192527	−	0.0170237i
$741$		0			0
$742$		−	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	$-0.0171857\pi$
$744$		0			0
$745$	9.46168	+	10.7005i	0.346649	+	0.392037i
$746$	41.7586			1.52889
$747$		0			0
$748$		−	0.730841i		−	0.0267222i
$749$	9.88858			0.361321
$750$		0			0
$751$	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
$753$		0			0
$754$	−5.14903			−0.187517
$755$	−3.92478	−	4.43866i	−0.142837	−	0.161539i
$756$		0			0
$757$		−	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	$0.470601\pi$
$762$		0			0
$763$			11.9394i			0.432234i
$764$	0.252016			0.00911762
$765$		0			0
$766$	−2.61071			−0.0943289
$767$		−	10.7005i		−	0.386374i
$768$		0			0
$769$	−2.10157			−0.0757846			−0.0378923	−	0.999282i	$-0.512064\pi$
$769$	−2.10157			−0.0757846			−0.0378923	+	0.999282i	$0.512064\pi$
$770$	2.61942	+	2.96239i	0.0943974	+	0.106757i
$771$		0			0
$772$			1.66688i			0.0599924i
$773$			46.0625i			1.65675i	0.560171	+	0.828377i	$0.310736\pi$
$773$			46.0625i			1.65675i	−0.560171	+	0.828377i	$0.689264\pi$
$774$		0			0
$775$	−0.589104	+	4.77575i	−0.0211612	+	0.171550i
$776$	−30.7005			−1.10208
$777$		0			0
$778$			9.66291i			0.346432i
$779$	−48.6253			−1.74218
$780$		0			0
$781$	−14.0508			−0.502777
$782$			22.3272i			0.798420i
$783$		0			0
$784$	−24.2506			−0.866093
$785$	−2.70052	+	2.38787i	−0.0963858	+	0.0852268i
$786$		0			0
$787$		−	26.0303i		−	0.927881i	−0.885866	−	0.463940i	$-0.846435\pi$
$787$		−	26.0303i		−	0.927881i	0.885866	−	0.463940i	$-0.153565\pi$
$788$		−	4.01573i		−	0.143054i
$789$		0			0
$790$	34.6761	−	30.6615i	1.23372	−	1.09089i
$791$	−2.70052

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:	\( N \)	\(=\)	\( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	495.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.48119i q^{2} -0.193937 q^{4} +(1.48119 + 1.67513i) q^{5} +1.19394i q^{7} +2.67513i q^{8} +O(q^{10})\)	\(q+1.48119i q^{2} -0.193937 q^{4} +(1.48119 + 1.67513i) q^{5} +1.19394i q^{7} +2.67513i q^{8} +(-2.48119 + 2.19394i) q^{10} -1.00000 q^{11} -0.806063i q^{13} -1.76845 q^{14} -4.35026 q^{16} -3.76845i q^{17} +5.35026 q^{19} +(-0.287258 - 0.324869i) q^{20} -1.48119i q^{22} +4.00000i q^{23} +(-0.612127 + 4.96239i) q^{25} +1.19394 q^{26} -0.231548i q^{28} -4.31265 q^{29} +0.962389 q^{31} -1.09332i q^{32} +5.58181 q^{34} +(-2.00000 + 1.76845i) q^{35} +1.61213i q^{37} +7.92478i q^{38} +(-4.48119 + 3.96239i) q^{40} -9.08840 q^{41} -4.41819i q^{43} +0.193937 q^{44} -5.92478 q^{46} -12.3127i q^{47} +5.57452 q^{49} +(-7.35026 - 0.906679i) q^{50} +0.156325i q^{52} +1.42548i q^{53} +(-1.48119 - 1.67513i) q^{55} -3.19394 q^{56} -6.38787i q^{58} +13.2750 q^{59} -0.0752228 q^{61} +1.42548i q^{62} -7.08110 q^{64} +(1.35026 - 1.19394i) q^{65} -2.70052i q^{67} +0.730841i q^{68} +(-2.61942 - 2.96239i) q^{70} +14.0508 q^{71} +10.7308i q^{73} -2.38787 q^{74} -1.03761 q^{76} -1.19394i q^{77} -13.9756 q^{79} +(-6.44358 - 7.28726i) q^{80} -13.4617i q^{82} -9.89446i q^{83} +(6.31265 - 5.58181i) q^{85} +6.54420 q^{86} -2.67513i q^{88} +16.8872 q^{89} +0.962389 q^{91} -0.775746i q^{92} +18.2374 q^{94} +(7.92478 + 8.96239i) q^{95} +11.4763i q^{97} +8.25694i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{4} - 2 q^{5}+O(q^{10}) \) 6 * q - 2 * q^4 - 2 * q^5	\( 6 q - 2 q^{4} - 2 q^{5} - 4 q^{10} - 6 q^{11} + 12 q^{14} - 6 q^{16} + 12 q^{19} + 10 q^{20} - 2 q^{25} + 8 q^{26} + 16 q^{29} - 16 q^{31} + 36 q^{34} - 12 q^{35} - 16 q^{40} - 16 q^{41} + 2 q^{44} + 8 q^{46} + 10 q^{49} - 24 q^{50} + 2 q^{55} - 20 q^{56} + 16 q^{59} - 44 q^{61} + 22 q^{64} - 12 q^{65} - 40 q^{70} + 24 q^{71} - 16 q^{74} - 28 q^{76} + 20 q^{79} - 6 q^{80} - 4 q^{85} + 20 q^{86} + 36 q^{89} - 16 q^{91} + 24 q^{94} + 4 q^{95}+O(q^{100}) \) 6 * q - 2 * q^4 - 2 * q^5 - 4 * q^10 - 6 * q^11 + 12 * q^14 - 6 * q^16 + 12 * q^19 + 10 * q^20 - 2 * q^25 + 8 * q^26 + 16 * q^29 - 16 * q^31 + 36 * q^34 - 12 * q^35 - 16 * q^40 - 16 * q^41 + 2 * q^44 + 8 * q^46 + 10 * q^49 - 24 * q^50 + 2 * q^55 - 20 * q^56 + 16 * q^59 - 44 * q^61 + 22 * q^64 - 12 * q^65 - 40 * q^70 + 24 * q^71 - 16 * q^74 - 28 * q^76 + 20 * q^79 - 6 * q^80 - 4 * q^85 + 20 * q^86 + 36 * q^89 - 16 * q^91 + 24 * q^94 + 4 * q^95

Introduction

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