LMFDB - Embedded newform 1520.2.q.k.961.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1520,2,Mod(881,1520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1520, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))

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

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

chi := DirichletCharacter("1520.881");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1520 = 2^{4} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1520.q (of order $3$, degree $2$, not 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:	$12.1372611072$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	8.0.4601315889.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} + 6x^{6} - 3x^{5} + 26x^{4} - 14x^{3} + 31x^{2} + 12x + 9 $ x^8 - x^7 + 6x^6 - 3x^5 + 26x^4 - 14x^3 + 31x^2 + 12x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 760)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			961.3
Root			$-1.02359 + 1.77290i$ of defining polynomial
Character	$\chi$	$=$	1520.961
Dual form			1520.2.q.k.881.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.595455 - 1.03136i) q^{3} +(-0.500000 + 0.866025i) q^{5} +3.62891 q^{7} +(0.790867 + 1.36982i) q^{9} +O(q^{10})$	\(q+(0.595455 - 1.03136i) q^{3} +(-0.500000 + 0.866025i) q^{5} +3.62891 q^{7} +(0.790867 + 1.36982i) q^{9} +1.46544 q^{11} +(-0.828173 - 1.43444i) q^{13} +(0.595455 + 1.03136i) q^{15} +(0.709133 - 1.22825i) q^{17} +(-2.75180 + 3.38047i) q^{19} +(2.16085 - 3.74270i) q^{21} +(3.55704 + 6.16097i) q^{23} +(-0.500000 - 0.866025i) q^{25} +5.45643 q^{27} +(-0.704630 - 1.22046i) q^{29} +2.45643 q^{31} +(0.872601 - 1.51139i) q^{33} +(-1.81445 + 3.14272i) q^{35} +0.865267 q^{37} -1.97256 q^{39} +(-1.62354 + 2.81206i) q^{41} +(0.276499 - 0.478911i) q^{43} -1.58173 q^{45} +(1.67071 + 2.89376i) q^{47} +6.16895 q^{49} +(-0.844513 - 1.46274i) q^{51} +(-3.53795 - 6.12791i) q^{53} +(-0.732718 + 1.26911i) q^{55} +(1.84790 + 4.85101i) q^{57} +(-2.67719 + 4.63703i) q^{59} +(-3.34726 - 5.79762i) q^{61} +(2.86998 + 4.97095i) q^{63} +1.65635 q^{65} +(1.39082 + 2.40898i) q^{67} +8.47222 q^{69} +(4.50811 - 7.80827i) q^{71} +(-2.53731 + 4.39474i) q^{73} -1.19091 q^{75} +5.31793 q^{77} +(2.50065 - 4.33125i) q^{79} +(0.876457 - 1.51807i) q^{81} +11.4946 q^{83} +(0.709133 + 1.22825i) q^{85} -1.67830 q^{87} +(5.84078 + 10.1165i) q^{89} +(-3.00536 - 5.20544i) q^{91} +(1.46269 - 2.53346i) q^{93} +(-1.55167 - 4.07337i) q^{95} +(5.44271 - 9.42705i) q^{97} +(1.15897 + 2.00739i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{3} - 4 q^{5} - 4 q^{7} - q^{9}+O(q^{10}) $ 8 * q - q^3 - 4 * q^5 - 4 * q^7 - q^9	$ 8 q - q^{3} - 4 q^{5} - 4 q^{7} - q^{9} + 8 q^{11} + q^{13} - q^{15} + 13 q^{17} - q^{19} + 12 q^{21} - 8 q^{23} - 4 q^{25} + 20 q^{27} - 3 q^{29} - 4 q^{31} - 15 q^{33} + 2 q^{35} + 20 q^{37} + 2 q^{39} - 8 q^{41} + 3 q^{43} + 2 q^{45} - 10 q^{47} + 12 q^{49} + 16 q^{51} - 11 q^{53} - 4 q^{55} - 29 q^{57} - q^{59} + 25 q^{63} - 2 q^{65} + 8 q^{67} + 22 q^{69} + 4 q^{71} - 20 q^{73} + 2 q^{75} - 36 q^{77} + 3 q^{79} + 12 q^{81} + 30 q^{83} + 13 q^{85} - 24 q^{87} + 17 q^{89} + 4 q^{91} + 12 q^{93} - 4 q^{95} + 11 q^{97} - 30 q^{99}+O(q^{100}) $ 8 * q - q^3 - 4 * q^5 - 4 * q^7 - q^9 + 8 * q^11 + q^13 - q^15 + 13 * q^17 - q^19 + 12 * q^21 - 8 * q^23 - 4 * q^25 + 20 * q^27 - 3 * q^29 - 4 * q^31 - 15 * q^33 + 2 * q^35 + 20 * q^37 + 2 * q^39 - 8 * q^41 + 3 * q^43 + 2 * q^45 - 10 * q^47 + 12 * q^49 + 16 * q^51 - 11 * q^53 - 4 * q^55 - 29 * q^57 - q^59 + 25 * q^63 - 2 * q^65 + 8 * q^67 + 22 * q^69 + 4 * q^71 - 20 * q^73 + 2 * q^75 - 36 * q^77 + 3 * q^79 + 12 * q^81 + 30 * q^83 + 13 * q^85 - 24 * q^87 + 17 * q^89 + 4 * q^91 + 12 * q^93 - 4 * q^95 + 11 * q^97 - 30 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times$.

$n$	$191$	$401$	$1141$	$1217$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	0.595455	−	1.03136i	0.343786	−	0.595455i	−0.641346	−	0.767251i	$-0.721624\pi$
$3$	0.595455	−	1.03136i	0.343786	−	0.595455i	0.985132	+	0.171797i	$0.0549572\pi$
$4$		0			0
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$6$		0			0
$7$	3.62891			1.37160			0.685799	−	0.727791i	$-0.259453\pi$
$7$	3.62891			1.37160			0.685799	+	0.727791i	$0.259453\pi$
$8$		0			0
$9$	0.790867	+	1.36982i	0.263622	+	0.456607i
$10$		0			0
$11$	1.46544			0.441846			0.220923	−	0.975291i	$-0.429093\pi$
$11$	1.46544			0.441846			0.220923	+	0.975291i	$0.429093\pi$
$12$		0			0
$13$	−0.828173	−	1.43444i	−0.229694	−	0.397842i	0.728023	−	0.685552i	$-0.240439\pi$
$13$	−0.828173	−	1.43444i	−0.229694	−	0.397842i	−0.957717	+	0.287711i	$0.907106\pi$
$14$		0			0
$15$	0.595455	+	1.03136i	0.153746	+	0.266295i
$16$		0			0
$17$	0.709133	−	1.22825i	0.171990	−	0.297895i	−0.767126	−	0.641497i	$-0.778314\pi$
$17$	0.709133	−	1.22825i	0.171990	−	0.297895i	0.939115	+	0.343602i	$0.111647\pi$
$18$		0			0
$19$	−2.75180	+	3.38047i	−0.631307	+	0.775533i
$19$	−2.75180	+	3.38047i	−0.631307	+	0.775533i
$20$		0			0
$21$	2.16085	−	3.74270i	0.471536	−	0.816724i
$22$		0			0
$23$	3.55704	+	6.16097i	0.741693	+	1.28465i	0.951724	+	0.306956i	$0.0993103\pi$
$23$	3.55704	+	6.16097i	0.741693	+	1.28465i	−0.210031	+	0.977695i	$0.567356\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$		0			0
$27$	5.45643			1.05009
$28$		0			0
$29$	−0.704630	−	1.22046i	−0.130847	−	0.226633i	0.793157	−	0.609018i	$-0.208436\pi$
$29$	−0.704630	−	1.22046i	−0.130847	−	0.226633i	−0.924003	+	0.382385i	$0.875103\pi$
$30$		0			0
$31$	2.45643			0.441188			0.220594	−	0.975366i	$-0.429200\pi$
$31$	2.45643			0.441188			0.220594	+	0.975366i	$0.429200\pi$
$32$		0			0
$33$	0.872601	−	1.51139i	0.151900	−	0.263099i
$34$		0			0
$35$	−1.81445	+	3.14272i	−0.306698	+	0.531217i
$36$		0			0
$37$	0.865267			0.142249			0.0711245	−	0.997467i	$-0.477341\pi$
$37$	0.865267			0.142249			0.0711245	+	0.997467i	$0.477341\pi$
$38$		0			0
$39$	−1.97256			−0.315862
$40$		0			0
$41$	−1.62354	+	2.81206i	−0.253555	+	0.439170i	−0.964502	−	0.264076i	$-0.914933\pi$
$41$	−1.62354	+	2.81206i	−0.253555	+	0.439170i	0.710947	+	0.703245i	$0.248266\pi$
$42$		0			0
$43$	0.276499	−	0.478911i	0.0421658	−	0.0730332i	−0.844172	−	0.536072i	$-0.819908\pi$
$43$	0.276499	−	0.478911i	0.0421658	−	0.0730332i	0.886338	+	0.463039i	$0.153241\pi$
$44$		0			0
$45$	−1.58173			−0.235791
$46$		0			0
$47$	1.67071	+	2.89376i	0.243699	+	0.422099i	0.961765	−	0.273876i	$-0.0883059\pi$
$47$	1.67071	+	2.89376i	0.243699	+	0.422099i	−0.718066	+	0.695975i	$0.754973\pi$
$48$		0			0
$49$	6.16895			0.881279
$50$		0			0
$51$	−0.844513	−	1.46274i	−0.118255	−	0.204825i
$52$		0			0
$53$	−3.53795	−	6.12791i	−0.485975	−	0.841734i	0.513895	−	0.857853i	$-0.328202\pi$
$53$	−3.53795	−	6.12791i	−0.485975	−	0.841734i	−0.999870	+	0.0161194i	$0.994869\pi$
$54$		0			0
$55$	−0.732718	+	1.26911i	−0.0987997	+	0.171126i
$56$		0			0
$57$	1.84790	+	4.85101i	0.244761	+	0.642532i
$58$		0			0
$59$	−2.67719	+	4.63703i	−0.348540	+	0.603690i	−0.985990	−	0.166802i	$-0.946656\pi$
$59$	−2.67719	+	4.63703i	−0.348540	+	0.603690i	0.637450	+	0.770492i	$0.279989\pi$
$60$		0			0
$61$	−3.34726	−	5.79762i	−0.428572	−	0.742309i	0.568174	−	0.822908i	$-0.307650\pi$
$61$	−3.34726	−	5.79762i	−0.428572	−	0.742309i	−0.996747	+	0.0805993i	$0.974317\pi$
$62$		0			0
$63$	2.86998	+	4.97095i	0.361584	+	0.626281i
$64$		0			0
$65$	1.65635			0.205444
$66$		0			0
$67$	1.39082	+	2.40898i	0.169916	+	0.294304i	0.938390	−	0.345577i	$-0.112317\pi$
$67$	1.39082	+	2.40898i	0.169916	+	0.294304i	−0.768474	+	0.639881i	$0.778984\pi$
$68$		0			0
$69$	8.47222			1.01994
$70$		0			0
$71$	4.50811	−	7.80827i	0.535014	−	0.926671i	−0.464149	−	0.885757i	$-0.653640\pi$
$71$	4.50811	−	7.80827i	0.535014	−	0.926671i	0.999163	−	0.0409137i	$-0.0130269\pi$
$72$		0			0
$73$	−2.53731	+	4.39474i	−0.296969	+	0.514366i	−0.975441	−	0.220261i	$-0.929309\pi$
$73$	−2.53731	+	4.39474i	−0.296969	+	0.514366i	0.678472	+	0.734626i	$0.262643\pi$
$74$		0			0
$75$	−1.19091			−0.137514
$76$		0			0
$77$	5.31793			0.606035
$78$		0			0
$79$	2.50065	−	4.33125i	0.281345	−	0.487303i	−0.690371	−	0.723455i	$-0.742553\pi$
$79$	2.50065	−	4.33125i	0.281345	−	0.487303i	0.971716	+	0.236152i	$0.0758862\pi$
$80$		0			0
$81$	0.876457	−	1.51807i	0.0973841	−	0.168674i
$82$		0			0
$83$	11.4946			1.26170			0.630848	−	0.775906i	$-0.282707\pi$
$83$	11.4946			1.26170			0.630848	+	0.775906i	$0.282707\pi$
$84$		0			0
$85$	0.709133	+	1.22825i	0.0769163	+	0.133223i
$86$		0			0
$87$	−1.67830			−0.179933
$88$		0			0
$89$	5.84078	+	10.1165i	0.619122	+	1.07235i	0.989646	+	0.143527i	$0.0458445\pi$
$89$	5.84078	+	10.1165i	0.619122	+	1.07235i	−0.370525	+	0.928823i	$0.620822\pi$
$90$		0			0
$91$	−3.00536	−	5.20544i	−0.315048	−	0.545678i
$92$		0			0
$93$	1.46269	−	2.53346i	0.151674	−	0.262708i
$94$		0			0
$95$	−1.55167	−	4.07337i	−0.159198	−	0.417919i
$96$		0			0
$97$	5.44271	−	9.42705i	0.552624	−	0.957172i	−0.445461	−	0.895302i	$-0.646960\pi$
$97$	5.44271	−	9.42705i	0.552624	−	0.957172i	0.998084	−	0.0618706i	$-0.0197066\pi$
$98$		0			0
$99$	1.15897	+	2.00739i	0.116480	+	0.201750i
$100$		0			0
$101$	0.324783	+	0.562541i	0.0323171	+	0.0559749i	0.881732	−	0.471751i	$-0.156378\pi$
$101$	0.324783	+	0.562541i	0.0323171	+	0.0559749i	−0.849414	+	0.527726i	$0.823045\pi$
$102$		0			0
$103$	13.1141			1.29217			0.646084	−	0.763266i	$-0.276406\pi$
$103$	13.1141			1.29217			0.646084	+	0.763266i	$0.276406\pi$
$104$		0			0
$105$	2.16085	+	3.74270i	0.210877	+	0.365250i
$106$		0			0
$107$	4.73225			0.457484			0.228742	−	0.973487i	$-0.426539\pi$
$107$	4.73225			0.457484			0.228742	+	0.973487i	$0.426539\pi$
$108$		0			0
$109$	6.47166	−	11.2092i	0.619872	−	1.07365i	−0.369636	−	0.929177i	$-0.620518\pi$
$109$	6.47166	−	11.2092i	0.619872	−	1.07365i	0.989509	−	0.144474i	$-0.0461490\pi$
$110$		0			0
$111$	0.515227	−	0.892400i	0.0489032	−	0.0847029i
$112$		0			0
$113$	−7.34272			−0.690745			−0.345373	−	0.938466i	$-0.612247\pi$
$113$	−7.34272			−0.690745			−0.345373	+	0.938466i	$0.612247\pi$
$114$		0			0
$115$	−7.11407			−0.663391
$116$		0			0
$117$	1.30995	−	2.26890i	0.121105	−	0.209760i
$118$		0			0
$119$	2.57338	−	4.45722i	0.235901	−	0.408593i
$120$		0			0
$121$	−8.85249			−0.804772
$122$		0			0
$123$	1.93349	+	3.34891i	0.174337	+	0.301961i
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	−9.95060	−	17.2350i	−0.882973	−	1.52935i	−0.848019	−	0.529966i	$-0.822205\pi$
$127$	−9.95060	−	17.2350i	−0.882973	−	1.52935i	−0.0349542	−	0.999389i	$-0.511129\pi$
$128$		0			0
$129$	−0.329286	−	0.570340i	−0.0289920	−	0.0502156i
$130$		0			0
$131$	2.74533	−	4.75504i	0.239860	−	0.415450i	−0.720814	−	0.693129i	$-0.756232\pi$
$131$	2.74533	−	4.75504i	0.239860	−	0.415450i	0.960674	+	0.277679i	$0.0895650\pi$
$132$		0			0
$133$	−9.98603	+	12.2674i	−0.865898	+	1.06372i
$134$		0			0
$135$	−2.72822	+	4.72541i	−0.234807	+	0.406698i
$136$		0			0
$137$	5.80884	+	10.0612i	0.496282	+	0.859586i	0.999991	−	0.00428732i	$-0.00136470\pi$
$137$	5.80884	+	10.0612i	0.496282	+	0.859586i	−0.503708	+	0.863874i	$0.668031\pi$
$138$		0			0
$139$	−2.68443	−	4.64958i	−0.227691	−	0.394372i	0.729433	−	0.684053i	$-0.239784\pi$
$139$	−2.68443	−	4.64958i	−0.227691	−	0.394372i	−0.957123	+	0.289681i	$0.906451\pi$
$140$		0			0
$141$	3.97934			0.335121
$142$		0			0
$143$	−1.21364	−	2.10208i	−0.101489	−	0.175785i
$144$		0			0
$145$	1.40926			0.117033
$146$		0			0
$147$	3.67333	−	6.36240i	0.302971	−	0.524762i
$148$		0			0
$149$	−7.63942	+	13.2319i	−0.625845	+	1.08400i	0.362531	+	0.931972i	$0.381913\pi$
$149$	−7.63942	+	13.2319i	−0.625845	+	1.08400i	−0.988377	+	0.152025i	$0.951421\pi$
$150$		0			0
$151$	11.2067			0.911988			0.455994	−	0.889983i	$-0.349284\pi$
$151$	11.2067			0.911988			0.455994	+	0.889983i	$0.349284\pi$
$152$		0			0
$153$	2.24332			0.181362
$154$		0			0
$155$	−1.22822	+	2.12733i	−0.0986527	+	0.170871i
$156$		0			0
$157$	6.98967	−	12.1065i	0.557836	−	0.966201i	−0.439840	−	0.898076i	$-0.644965\pi$
$157$	6.98967	−	12.1065i	0.557836	−	0.966201i	0.997677	−	0.0681250i	$-0.0217017\pi$
$158$		0			0
$159$	−8.42677			−0.668286
$160$		0			0
$161$	12.9081	+	22.3576i	1.01730	+	1.76202i
$162$		0			0
$163$	−19.4397			−1.52264			−0.761318	−	0.648379i	$-0.775447\pi$
$163$	−19.4397			−1.52264			−0.761318	+	0.648379i	$0.775447\pi$
$164$		0			0
$165$	0.872601	+	1.51139i	0.0679319	+	0.117662i
$166$		0			0
$167$	−4.60553	−	7.97702i	−0.356387	−	0.617280i	0.630967	−	0.775809i	$-0.282658\pi$
$167$	−4.60553	−	7.97702i	−0.356387	−	0.617280i	−0.987354	+	0.158529i	$0.949325\pi$
$168$		0			0
$169$	5.12826	−	8.88240i	0.394481	−	0.683262i
$170$		0			0
$171$	−6.80695	−	1.09597i	−0.520541	−	0.0838112i
$172$		0			0
$173$	−7.36523	+	12.7569i	−0.559968	+	0.969893i	0.437531	+	0.899204i	$0.355853\pi$
$173$	−7.36523	+	12.7569i	−0.559968	+	0.969893i	−0.997498	+	0.0706891i	$0.977480\pi$
$174$		0			0
$175$	−1.81445	−	3.14272i	−0.137160	−	0.237568i
$176$		0			0
$177$	3.18829	+	5.52228i	0.239647	+	0.415080i
$178$		0			0
$179$	0.174699			0.0130576			0.00652881	−	0.999979i	$-0.497922\pi$
$179$	0.174699			0.0130576			0.00652881	+	0.999979i	$0.497922\pi$
$180$		0			0
$181$	−11.3515	−	19.6614i	−0.843751	−	1.46142i	−0.886702	−	0.462342i	$-0.847009\pi$
$181$	−11.3515	−	19.6614i	−0.843751	−	1.46142i	0.0429509	−	0.999077i	$-0.486324\pi$
$182$		0			0
$183$	−7.97256			−0.589349
$184$		0			0
$185$	−0.432633	+	0.749343i	−0.0318078	+	0.0550928i
$186$		0			0
$187$	1.03919	−	1.79993i	0.0759931	−	0.131624i
$188$		0			0
$189$	19.8009			1.44030
$190$		0			0
$191$	10.5466			0.763124			0.381562	−	0.924343i	$-0.375386\pi$
$191$	10.5466			0.763124			0.381562	+	0.924343i	$0.375386\pi$
$192$		0			0
$193$	−7.62329	+	13.2039i	−0.548736	+	0.950439i	0.449625	+	0.893217i	$0.351558\pi$
$193$	−7.62329	+	13.2039i	−0.548736	+	0.950439i	−0.998361	+	0.0572220i	$0.981776\pi$
$194$		0			0
$195$	0.986279	−	1.70829i	0.0706289	−	0.122333i
$196$		0			0
$197$	−22.9630			−1.63605			−0.818025	−	0.575183i	$-0.804931\pi$
$197$	−22.9630			−1.63605			−0.818025	+	0.575183i	$0.804931\pi$
$198$		0			0
$199$	−11.1932	−	19.3872i	−0.793464	−	1.37432i	−0.923810	−	0.382851i	$-0.874942\pi$
$199$	−11.1932	−	19.3872i	−0.793464	−	1.37432i	0.130346	−	0.991469i	$-0.458391\pi$
$200$		0			0
$201$	3.31269			0.233659
$202$		0			0
$203$	−2.55704	−	4.42892i	−0.179469	−	0.310849i
$204$		0			0
$205$	−1.62354	−	2.81206i	−0.113393	−	0.196403i
$206$		0			0
$207$	−5.62629	+	9.74501i	−0.391054	+	0.677325i
$208$		0			0
$209$	−4.03259	+	4.95387i	−0.278940	+	0.342666i
$210$		0			0
$211$	−11.0734	+	19.1796i	−0.762323	+	1.32038i	0.179328	+	0.983789i	$0.442608\pi$
$211$	−11.0734	+	19.1796i	−0.762323	+	1.32038i	−0.941651	+	0.336592i	$0.890726\pi$
$212$		0			0
$213$	−5.36875	−	9.29894i	−0.367860	−	0.637153i
$214$		0			0
$215$	0.276499	+	0.478911i	0.0188571	+	0.0326615i
$216$		0			0
$217$	8.91416			0.605132
$218$		0			0
$219$	3.02170	+	5.23374i	0.204188	+	0.353663i
$220$		0			0
$221$	−2.34914			−0.158020
$222$		0			0
$223$	−2.54606	+	4.40990i	−0.170497	+	0.295309i	−0.938594	−	0.345025i	$-0.887871\pi$
$223$	−2.54606	+	4.40990i	−0.170497	+	0.295309i	0.768097	+	0.640334i	$0.221204\pi$
$224$		0			0
$225$	0.790867	−	1.36982i	0.0527245	−	0.0913215i
$226$		0			0
$227$	21.3127			1.41457			0.707287	−	0.706927i	$-0.249919\pi$
$227$	21.3127			1.41457			0.707287	+	0.706927i	$0.249919\pi$
$228$		0			0
$229$	5.55978			0.367401			0.183700	−	0.982982i	$-0.441192\pi$
$229$	5.55978			0.367401			0.183700	+	0.982982i	$0.441192\pi$
$230$		0			0
$231$	3.16659	−	5.48469i	0.208346	−	0.360866i
$232$		0			0
$233$	−13.3603	+	23.1406i	−0.875260	+	1.51599i	−0.0187737	+	0.999824i	$0.505976\pi$
$233$	−13.3603	+	23.1406i	−0.875260	+	1.51599i	−0.856486	+	0.516170i	$0.827357\pi$
$234$		0			0
$235$	−3.34143			−0.217971
$236$		0			0
$237$	−2.97804	−	5.15812i	−0.193445	−	0.335056i
$238$		0			0
$239$	−27.6233			−1.78681			−0.893403	−	0.449257i	$-0.851689\pi$
$239$	−27.6233			−1.78681			−0.893403	+	0.449257i	$0.851689\pi$
$240$		0			0
$241$	−12.7078	−	22.0105i	−0.818579	−	1.41782i	−0.906729	−	0.421714i	$-0.861429\pi$
$241$	−12.7078	−	22.0105i	−0.818579	−	1.41782i	0.0881498	−	0.996107i	$-0.471905\pi$
$242$		0			0
$243$	7.14087	+	12.3683i	0.458087	+	0.793430i
$244$		0			0
$245$	−3.08448	+	5.34247i	−0.197060	+	0.341318i
$246$		0			0
$247$	7.12805	+	1.14767i	0.453547	+	0.0730246i
$248$		0			0
$249$	6.84451	−	11.8550i	0.433753	−	0.751283i
$250$		0			0
$251$	−4.92363	−	8.52797i	−0.310777	−	0.538281i	0.667754	−	0.744382i	$-0.267256\pi$
$251$	−4.92363	−	8.52797i	−0.310777	−	0.538281i	−0.978531	+	0.206101i	$0.933922\pi$
$252$		0			0
$253$	5.21261	+	9.02851i	0.327714	+	0.567617i
$254$		0			0
$255$	1.68903			0.105771
$256$		0			0
$257$	1.62968	+	2.82268i	0.101656	+	0.176074i	0.912367	−	0.409373i	$-0.134252\pi$
$257$	1.62968	+	2.82268i	0.101656	+	0.176074i	−0.810711	+	0.585447i	$0.800919\pi$
$258$		0			0
$259$	3.13997			0.195108
$260$		0			0
$261$	1.11454	−	1.93044i	0.0689881	−	0.119491i
$262$		0			0
$263$	−6.68508	+	11.5789i	−0.412220	+	0.713986i	−0.995132	−	0.0985494i	$-0.968580\pi$
$263$	−6.68508	+	11.5789i	−0.412220	+	0.713986i	0.582912	+	0.812535i	$0.301913\pi$
$264$		0			0
$265$	7.07591			0.434669
$266$		0			0
$267$	13.9117			0.851381
$268$		0			0
$269$	6.76682	−	11.7205i	0.412580	−	0.714610i	−0.582591	−	0.812765i	$-0.697961\pi$
$269$	6.76682	−	11.7205i	0.412580	−	0.714610i	0.995171	+	0.0981559i	$0.0312944\pi$
$270$		0			0
$271$	−5.14130	+	8.90499i	−0.312312	+	0.540940i	−0.978862	−	0.204520i	$-0.934437\pi$
$271$	−5.14130	+	8.90499i	−0.312312	+	0.540940i	0.666551	+	0.745460i	$0.267770\pi$
$272$		0			0
$273$	−7.15823			−0.433236
$274$		0			0
$275$	−0.732718	−	1.26911i	−0.0441846	−	0.0765299i
$276$		0			0
$277$	−23.2902			−1.39937			−0.699687	−	0.714449i	$-0.746677\pi$
$277$	−23.2902			−1.39937			−0.699687	+	0.714449i	$0.746677\pi$
$278$		0			0
$279$	1.94271	+	3.36487i	0.116307	+	0.201450i
$280$		0			0
$281$	−0.828426	−	1.43488i	−0.0494198	−	0.0855975i	0.840257	−	0.542188i	$-0.182404\pi$
$281$	−0.828426	−	1.43488i	−0.0494198	−	0.0855975i	−0.889677	+	0.456590i	$0.849071\pi$
$282$		0			0
$283$	−8.71385	+	15.0928i	−0.517984	+	0.897175i	0.481797	+	0.876283i	$0.339984\pi$
$283$	−8.71385	+	15.0928i	−0.517984	+	0.897175i	−0.999782	+	0.0208927i	$0.993349\pi$
$284$		0			0
$285$	−5.12505	−	0.825174i	−0.303582	−	0.0488791i
$286$		0			0
$287$	−5.89168	+	10.2047i	−0.347775	+	0.602364i
$288$		0			0
$289$	7.49426	+	12.9804i	0.440839	+	0.763555i
$290$		0			0
$291$	−6.48178	−	11.2268i	−0.379968	−	0.658125i
$292$		0			0
$293$	−13.7575			−0.803720			−0.401860	−	0.915701i	$-0.631636\pi$
$293$	−13.7575			−0.803720			−0.401860	+	0.915701i	$0.631636\pi$
$294$		0			0
$295$	−2.67719	−	4.63703i	−0.155872	−	0.269978i
$296$		0			0
$297$	7.99606			0.463978
$298$		0			0
$299$	5.89168	−	10.2047i	0.340725	−	0.590153i
$300$		0			0
$301$	1.00339	−	1.73792i	0.0578344	−	0.100172i
$302$		0			0
$303$	0.773575			0.0444407
$304$		0			0
$305$	6.69451			0.383327
$306$		0			0
$307$	−8.28997	+	14.3586i	−0.473133	+	0.819491i	−0.999527	−	0.0307498i	$-0.990210\pi$
$307$	−8.28997	+	14.3586i	−0.473133	+	0.819491i	0.526394	+	0.850241i	$0.323544\pi$
$308$		0			0
$309$	7.80884	−	13.5253i	0.444229	−	0.769428i
$310$		0			0
$311$	28.1051			1.59369			0.796846	−	0.604182i	$-0.206500\pi$
$311$	28.1051			1.59369			0.796846	+	0.604182i	$0.206500\pi$
$312$		0			0
$313$	11.4644	+	19.8569i	0.648007	+	1.12238i	0.983598	+	0.180373i	$0.0577306\pi$
$313$	11.4644	+	19.8569i	0.648007	+	1.12238i	−0.335591	+	0.942008i	$0.608936\pi$
$314$		0			0
$315$	−5.73996			−0.323410
$316$		0			0
$317$	−7.44675	−	12.8981i	−0.418251	−	0.724432i	0.577513	−	0.816382i	$-0.304023\pi$
$317$	−7.44675	−	12.8981i	−0.418251	−	0.724432i	−0.995764	+	0.0919497i	$0.970690\pi$
$318$		0			0
$319$	−1.03259	−	1.78850i	−0.0578140	−	0.100137i
$320$		0			0
$321$	2.81784	−	4.88065i	0.157277	−	0.272411i
$322$		0			0
$323$	2.20069	+	5.77711i	0.122449	+	0.321447i
$324$		0			0
$325$	−0.828173	+	1.43444i	−0.0459388	+	0.0795683i
$326$		0			0
$327$	−7.70716	−	13.3492i	−0.426207	−	0.738212i
$328$		0			0
$329$	6.06286	+	10.5012i	0.334257	+	0.578949i
$330$		0			0
$331$	8.06690			0.443397			0.221698	−	0.975115i	$-0.428840\pi$
$331$	8.06690			0.443397			0.221698	+	0.975115i	$0.428840\pi$
$332$		0			0
$333$	0.684311	+	1.18526i	0.0375000	+	0.0649519i
$334$		0			0
$335$	−2.78165			−0.151978
$336$		0			0
$337$	−4.29648	+	7.44173i	−0.234044	+	0.405377i	−0.958995	−	0.283425i	$-0.908529\pi$
$337$	−4.29648	+	7.44173i	−0.234044	+	0.405377i	0.724950	+	0.688801i	$0.241863\pi$
$338$		0			0
$339$	−4.37226	+	7.57298i	−0.237469	+	0.411308i
$340$		0			0
$341$	3.59975			0.194937
$342$		0			0
$343$	−3.01579			−0.162837
$344$		0			0
$345$	−4.23611	+	7.33716i	−0.228064	+	0.395019i
$346$		0			0
$347$	2.73293	−	4.73358i	0.146711	−	0.254112i	−0.783299	−	0.621645i	$-0.786464\pi$
$347$	2.73293	−	4.73358i	0.146711	−	0.254112i	0.930010	+	0.367534i	$0.119798\pi$
$348$		0			0
$349$	−15.1617			−0.811585			−0.405793	−	0.913965i	$-0.633005\pi$
$349$	−15.1617			−0.811585			−0.405793	+	0.913965i	$0.633005\pi$
$350$		0			0
$351$	−4.51887	−	7.82691i	−0.241199	−	0.417770i
$352$		0			0
$353$	−0.156005			−0.00830329			−0.00415165	−	0.999991i	$-0.501322\pi$
$353$	−0.156005			−0.00830329			−0.00415165	+	0.999991i	$0.501322\pi$
$354$		0			0
$355$	4.50811	+	7.80827i	0.239265	+	0.414420i
$356$		0			0
$357$	−3.06466	−	5.30814i	−0.162199	−	0.280937i
$358$		0			0
$359$	−5.46205	+	9.46054i	−0.288276	+	0.499308i	−0.973398	−	0.229120i	$-0.926415\pi$
$359$	−5.46205	+	9.46054i	−0.288276	+	0.499308i	0.685122	+	0.728428i	$0.259749\pi$
$360$		0			0
$361$	−3.85518	−	18.6048i	−0.202904	−	0.979199i
$362$		0			0
$363$	−5.27126	+	9.13009i	−0.276669	+	0.479206i
$364$		0			0
$365$	−2.53731	−	4.39474i	−0.132809	−	0.230031i
$366$		0			0
$367$	8.66094	+	15.0012i	0.452097	+	0.783055i	0.998516	−	0.0544566i	$-0.0173426\pi$
$367$	8.66094	+	15.0012i	0.452097	+	0.783055i	−0.546419	+	0.837512i	$0.684009\pi$
$368$		0			0
$369$	−5.13603			−0.267371
$370$		0			0
$371$	−12.8389	−	22.2376i	−0.666562	−	1.15452i
$372$		0			0
$373$	−14.3633			−0.743704			−0.371852	−	0.928292i	$-0.621277\pi$
$373$	−14.3633			−0.743704			−0.371852	+	0.928292i	$0.621277\pi$
$374$		0			0
$375$	0.595455	−	1.03136i	0.0307492	−	0.0532591i
$376$		0			0
$377$	−1.16711	+	2.02150i	−0.0601093	+	0.104112i
$378$		0			0
$379$	−4.83096			−0.248150			−0.124075	−	0.992273i	$-0.539596\pi$
$379$	−4.83096			−0.248150			−0.124075	+	0.992273i	$0.539596\pi$
$380$		0			0
$381$	−23.7005			−1.21422
$382$		0			0
$383$	−5.72522	+	9.91637i	−0.292545	+	0.506703i	−0.974411	−	0.224774i	$-0.927835\pi$
$383$	−5.72522	+	9.91637i	−0.292545	+	0.506703i	0.681866	+	0.731477i	$0.261169\pi$
$384$		0			0
$385$	−2.65897	+	4.60546i	−0.135513	+	0.234716i
$386$		0			0
$387$	0.874697			0.0444634
$388$		0			0
$389$	0.391290	+	0.677734i	0.0198392	+	0.0343625i	0.875775	−	0.482720i	$-0.160351\pi$
$389$	0.391290	+	0.677734i	0.0198392	+	0.0343625i	−0.855935	+	0.517083i	$0.827018\pi$
$390$		0			0
$391$	10.0896			0.510255
$392$		0			0
$393$	−3.26944	−	5.66283i	−0.164921	−	0.285652i
$394$		0			0
$395$	2.50065	+	4.33125i	0.125821	+	0.217929i
$396$		0			0
$397$	4.79602	−	8.30695i	0.240705	−	0.416914i	−0.720210	−	0.693756i	$-0.755955\pi$
$397$	4.79602	−	8.30695i	0.240705	−	0.416914i	0.960915	+	0.276842i	$0.0892880\pi$
$398$		0			0
$399$	6.70587	+	17.6039i	0.335713	+	0.881295i
$400$		0			0
$401$	−2.07140	+	3.58778i	−0.103441	+	0.179165i	−0.913100	−	0.407735i	$-0.866319\pi$
$401$	−2.07140	+	3.58778i	−0.103441	+	0.179165i	0.809659	+	0.586900i	$0.199652\pi$
$402$		0			0
$403$	−2.03435	−	3.52360i	−0.101338	−	0.175523i
$404$		0			0
$405$	0.876457	+	1.51807i	0.0435515	+	0.0754334i
$406$		0			0
$407$	1.26799			0.0628521
$408$		0			0
$409$	−19.3052	−	33.4376i	−0.954580	−	1.65338i	−0.735325	−	0.677714i	$-0.762971\pi$
$409$	−19.3052	−	33.4376i	−0.954580	−	1.65338i	−0.219255	−	0.975668i	$-0.570363\pi$
$410$		0			0
$411$	13.8356			0.682460
$412$		0			0
$413$	−9.71527	+	16.8273i	−0.478057	+	0.828019i
$414$		0			0
$415$	−5.74730	+	9.95461i	−0.282124	+	0.488653i
$416$		0			0
$417$	−6.39384			−0.313108
$418$		0			0
$419$	6.24529			0.305102			0.152551	−	0.988296i	$-0.451251\pi$
$419$	6.24529			0.305102			0.152551	+	0.988296i	$0.451251\pi$
$420$		0			0
$421$	9.46497	−	16.3938i	0.461294	−	0.798985i	−0.537731	−	0.843116i	$-0.680719\pi$
$421$	9.46497	−	16.3938i	0.461294	−	0.798985i	0.999026	+	0.0441308i	$0.0140518\pi$
$422$		0			0
$423$	−2.64263	+	4.57716i	−0.128489	+	0.222549i
$424$		0			0
$425$	−1.41827			−0.0687960
$426$		0			0
$427$	−12.1469	−	21.0390i	−0.587829	−	1.01815i
$428$		0			0
$429$	−2.89066			−0.139562
$430$		0			0
$431$	13.7166	+	23.7579i	0.660708	+	1.14438i	0.980430	+	0.196868i	$0.0630771\pi$
$431$	13.7166	+	23.7579i	0.660708	+	1.14438i	−0.319722	+	0.947511i	$0.603590\pi$
$432$		0			0
$433$	−15.6349	−	27.0805i	−0.751366	−	1.30140i	−0.947161	−	0.320759i	$-0.896062\pi$
$433$	−15.6349	−	27.0805i	−0.751366	−	1.30140i	0.195795	−	0.980645i	$-0.437271\pi$
$434$		0			0
$435$	0.839151	−	1.45345i	0.0402342	−	0.0696877i
$436$		0			0
$437$	−30.6152	−	4.92930i	−1.46453	−	0.235800i
$438$		0			0
$439$	−0.572769	+	0.992066i	−0.0273368	+	0.0473487i	−0.879370	−	0.476139i	$-0.842036\pi$
$439$	−0.572769	+	0.992066i	−0.0273368	+	0.0473487i	0.852033	+	0.523487i	$0.175369\pi$
$440$		0			0
$441$	4.87882	+	8.45037i	0.232325	+	0.402399i
$442$		0			0
$443$	−18.3193	−	31.7300i	−0.870379	−	1.50754i	−0.861606	−	0.507578i	$-0.830541\pi$
$443$	−18.3193	−	31.7300i	−0.870379	−	1.50754i	−0.00877299	−	0.999962i	$-0.502793\pi$
$444$		0			0
$445$	−11.6816			−0.553759
$446$		0			0
$447$	9.09786	+	15.7579i	0.430314	+	0.745325i
$448$		0			0
$449$	−24.4353			−1.15317			−0.576586	−	0.817037i	$-0.695615\pi$
$449$	−24.4353			−1.15317			−0.576586	+	0.817037i	$0.695615\pi$
$450$		0			0
$451$	−2.37920	+	4.12090i	−0.112032	+	0.194045i
$452$		0			0
$453$	6.67308	−	11.5581i	0.313529	−	0.543048i
$454$		0			0
$455$	6.01072			0.281787
$456$		0			0
$457$	15.0004			0.701690			0.350845	−	0.936433i	$-0.385894\pi$
$457$	15.0004			0.701690			0.350845	+	0.936433i	$0.385894\pi$
$458$		0			0
$459$	3.86933	−	6.70188i	0.180605	−	0.312817i
$460$		0			0
$461$	19.4476	−	33.6842i	0.905765	−	1.56883i	0.0858789	−	0.996306i	$-0.472630\pi$
$461$	19.4476	−	33.6842i	0.905765	−	1.56883i	0.819886	−	0.572526i	$-0.194036\pi$
$462$		0			0
$463$	−6.74322			−0.313384			−0.156692	−	0.987647i	$-0.550083\pi$
$463$	−6.74322			−0.313384			−0.156692	+	0.987647i	$0.550083\pi$
$464$		0			0
$465$	1.46269	+	2.53346i	0.0678308	+	0.117486i
$466$		0			0
$467$	−35.2591			−1.63160			−0.815798	−	0.578336i	$-0.803702\pi$
$467$	−35.2591			−1.63160			−0.815798	+	0.578336i	$0.803702\pi$
$468$		0			0
$469$	5.04717	+	8.74196i	0.233057	+	0.403666i
$470$		0			0
$471$	−8.32406	−	14.4177i	−0.383553	−	0.664333i
$472$		0			0
$473$	0.405192	−	0.701814i	0.0186308	−	0.0322694i
$474$		0			0
$475$	4.30348	+	0.692894i	0.197457	+	0.0317922i
$476$		0			0
$477$	5.59610	−	9.69273i	0.256228	−	0.443800i
$478$		0			0
$479$	−10.4067	−	18.0249i	−0.475494	−	0.823580i	0.524112	−	0.851649i	$-0.324397\pi$
$479$	−10.4067	−	18.0249i	−0.475494	−	0.823580i	−0.999606	+	0.0280693i	$0.991064\pi$
$480$		0			0
$481$	−0.716591	−	1.24117i	−0.0326737	−	0.0565926i
$482$		0			0
$483$	30.7449			1.39894
$484$		0			0
$485$	5.44271	+	9.42705i	0.247141	+	0.428060i
$486$		0			0
$487$	−26.5233			−1.20188			−0.600942	−	0.799293i	$-0.705208\pi$
$487$	−26.5233			−1.20188			−0.600942	+	0.799293i	$0.705208\pi$
$488$		0			0
$489$	−11.5755	+	20.0493i	−0.523461	+	0.906661i
$490$		0			0
$491$	16.2099	−	28.0764i	0.731544	−	1.26707i	−0.224680	−	0.974433i	$-0.572134\pi$
$491$	16.2099	−	28.0764i	0.731544	−	1.26707i	0.956223	−	0.292638i	$-0.0945330\pi$
$492$		0			0
$493$	−1.99871			−0.0900172
$494$		0			0
$495$	−2.31793			−0.104183
$496$		0			0
$497$	16.3595	−	28.3355i	0.733823	−	1.27102i
$498$		0			0
$499$	4.43602	−	7.68342i	0.198584	−	0.343957i	−0.749486	−	0.662020i	$-0.769699\pi$
$499$	4.43602	−	7.68342i	0.198584	−	0.343957i	0.948069	+	0.318063i	$0.103033\pi$
$500$		0			0
$501$	−10.9695			−0.490083
$502$		0			0
$503$	12.2221	+	21.1694i	0.544958	+	0.943895i	0.998610	+	0.0527159i	$0.0167878\pi$
$503$	12.2221	+	21.1694i	0.544958	+	0.943895i	−0.453651	+	0.891179i	$0.649879\pi$
$504$		0			0
$505$	−0.649566			−0.0289053
$506$		0			0
$507$	−6.10729	−	10.5781i	−0.271234	−	0.469792i
$508$		0			0
$509$	−15.1982	−	26.3241i	−0.673650	−	1.16680i	−0.976861	−	0.213873i	$-0.931392\pi$
$509$	−15.1982	−	26.3241i	−0.673650	−	1.16680i	0.303211	−	0.952923i	$-0.401941\pi$
$510$		0			0
$511$	−9.20764	+	15.9481i	−0.407322	+	0.705503i
$512$		0			0
$513$	−15.0150	+	18.4453i	−0.662929	+	0.814380i
$514$		0			0
$515$	−6.55704	+	11.3571i	−0.288938	+	0.500454i
$516$		0			0
$517$	2.44833	+	4.24063i	0.107677	+	0.186502i
$518$		0			0
$519$	8.77132	+	15.1924i	0.385018	+	0.666871i
$520$		0			0
$521$	37.8421			1.65789			0.828945	−	0.559330i	$-0.188941\pi$
$521$	37.8421			1.65789			0.828945	+	0.559330i	$0.188941\pi$
$522$		0			0
$523$	−7.73697	−	13.4008i	−0.338314	−	0.585977i	0.645802	−	0.763505i	$-0.276523\pi$
$523$	−7.73697	−	13.4008i	−0.338314	−	0.585977i	−0.984116	+	0.177528i	$0.943190\pi$
$524$		0			0
$525$	−4.32170			−0.188614
$526$		0			0
$527$	1.74194	−	3.01712i	0.0758799	−	0.131428i
$528$		0			0
$529$	−13.8050	+	23.9110i	−0.600218	+	1.03961i
$530$		0			0
$531$	−8.46920			−0.367532
$532$		0			0
$533$	5.37830			0.232960
$534$		0			0
$535$	−2.36613	+	4.09825i	−0.102297	+	0.177183i
$536$		0			0
$537$	0.104025	−	0.180177i	0.00448903	−	0.00777522i
$538$		0			0
$539$	9.04021			0.389390
$540$		0			0
$541$	14.1405	+	24.4921i	0.607949	+	1.05300i	0.991578	+	0.129511i	$0.0413408\pi$
$541$	14.1405	+	24.4921i	0.607949	+	1.05300i	−0.383629	+	0.923487i	$0.625326\pi$
$542$		0			0
$543$	−27.0372			−1.16028
$544$		0			0
$545$	6.47166	+	11.2092i	0.277215	+	0.480151i
$546$		0			0
$547$	−12.6628	−	21.9325i	−0.541420	−	0.937767i	−0.998823	−	0.0485075i	$-0.984554\pi$
$547$	−12.6628	−	21.9325i	−0.541420	−	0.937767i	0.457403	−	0.889260i	$-0.348780\pi$
$548$		0			0
$549$	5.29447	−	9.17029i	0.225962	−	0.391378i
$550$		0			0
$551$	6.06472	+	0.976468i	0.258366	+	0.0415989i
$552$		0			0
$553$	9.07461	−	15.7177i	0.385892	−	0.668384i
$554$		0			0
$555$	0.515227	+	0.892400i	0.0218702	+	0.0378803i
$556$		0			0
$557$	4.42153	+	7.65832i	0.187346	+	0.324493i	0.944365	−	0.328900i	$-0.106678\pi$
$557$	4.42153	+	7.65832i	0.187346	+	0.324493i	−0.757018	+	0.653394i	$0.773345\pi$
$558$		0			0
$559$	−0.915958			−0.0387409
$560$		0			0
$561$	−1.23758	−	2.14355i	−0.0522507	−	0.0905009i
$562$		0			0
$563$	−18.3607			−0.773809			−0.386905	−	0.922120i	$-0.626456\pi$
$563$	−18.3607			−0.773809			−0.386905	+	0.922120i	$0.626456\pi$
$564$		0			0
$565$	3.67136	−	6.35898i	0.154455	−	0.267524i
$566$		0			0
$567$	3.18058	−	5.50892i	0.133572	−	0.231353i
$568$		0			0
$569$	−32.4768			−1.36150			−0.680749	−	0.732517i	$-0.738345\pi$
$569$	−32.4768			−1.36150			−0.680749	+	0.732517i	$0.738345\pi$
$570$		0			0
$571$	29.3199			1.22700			0.613500	−	0.789695i	$-0.289761\pi$
$571$	29.3199			1.22700			0.613500	+	0.789695i	$0.289761\pi$
$572$		0			0
$573$	6.28001	−	10.8773i	0.262351	−	0.454406i
$574$		0			0
$575$	3.55704	−	6.16097i	0.148339	−	0.256930i
$576$		0			0
$577$	34.1001			1.41961			0.709803	−	0.704400i	$-0.248784\pi$
$577$	34.1001			1.41961			0.709803	+	0.704400i	$0.248784\pi$
$578$		0			0
$579$	9.07865	+	15.7247i	0.377296	+	0.653495i
$580$		0			0
$581$	41.7128			1.73054
$582$		0			0
$583$	−5.18465	−	8.98007i	−0.214726	−	0.371917i
$584$		0			0
$585$	1.30995	+	2.26890i	0.0541598	+	0.0938075i
$586$		0			0
$587$	−2.80999	+	4.86705i	−0.115981	+	0.200885i	−0.918171	−	0.396183i	$-0.870334\pi$
$587$	−2.80999	+	4.86705i	−0.115981	+	0.200885i	0.802191	+	0.597068i	$0.203668\pi$
$588$		0			0
$589$	−6.75961	+	8.30390i	−0.278525	+	0.342156i
$590$		0			0
$591$	−13.6735	+	23.6831i	−0.562451	+	0.974193i
$592$		0			0
$593$	−4.89795	−	8.48349i	−0.201135	−	0.348375i	0.747760	−	0.663969i	$-0.231130\pi$
$593$	−4.89795	−	8.48349i	−0.201135	−	0.348375i	−0.948894	+	0.315594i	$0.897796\pi$
$594$		0			0
$595$	2.57338	+	4.45722i	0.105498	+	0.182728i
$596$		0			0
$597$	−26.6601			−1.09113
$598$		0			0
$599$	−8.65570	−	14.9921i	−0.353662	−	0.612561i	0.633226	−	0.773967i	$-0.281730\pi$
$599$	−8.65570	−	14.9921i	−0.353662	−	0.612561i	−0.986888	+	0.161406i	$0.948397\pi$
$600$		0			0
$601$	−0.675207			−0.0275423			−0.0137711	−	0.999905i	$-0.504384\pi$
$601$	−0.675207			−0.0275423			−0.0137711	+	0.999905i	$0.504384\pi$
$602$		0			0
$603$	−2.19992	+	3.81036i	−0.0895875	+	0.155170i
$604$		0			0
$605$	4.42625	−	7.66649i	0.179953	−	0.311687i
$606$		0			0
$607$	36.5256			1.48253			0.741263	−	0.671214i	$-0.234227\pi$
$607$	36.5256			1.48253			0.741263	+	0.671214i	$0.234227\pi$
$608$		0			0
$609$	−6.09040			−0.246795
$610$		0			0
$611$	2.76728	−	4.79307i	0.111952	−	0.193907i
$612$		0			0
$613$	14.4464	−	25.0219i	0.583485	−	1.01063i	−0.411578	−	0.911375i	$-0.635022\pi$
$613$	14.4464	−	25.0219i	0.583485	−	1.01063i	0.995062	−	0.0992505i	$-0.0316445\pi$
$614$		0			0
$615$	−3.86699			−0.155932
$616$		0			0
$617$	18.8491	+	32.6475i	0.758835	+	1.31434i	0.943445	+	0.331529i	$0.107564\pi$
$617$	18.8491	+	32.6475i	0.758835	+	1.31434i	−0.184610	+	0.982812i	$0.559102\pi$
$618$		0			0
$619$	−15.1225			−0.607823			−0.303912	−	0.952700i	$-0.598293\pi$
$619$	−15.1225			−0.607823			−0.303912	+	0.952700i	$0.598293\pi$
$620$		0			0
$621$	19.4087	+	33.6169i	0.778845	+	1.34900i
$622$		0			0
$623$	21.1956	+	36.7119i	0.849185	+	1.47083i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$		0			0
$627$	2.70799	+	7.10885i	0.108147	+	0.283900i
$628$		0			0
$629$	0.613589	−	1.06277i	0.0244654	−	0.0423753i
$630$		0			0
$631$	−21.4332	−	37.1235i	−0.853244	−	1.47786i	−0.878265	−	0.478175i	$-0.841299\pi$
$631$	−21.4332	−	37.1235i	−0.853244	−	1.47786i	0.0250209	−	0.999687i	$-0.492035\pi$
$632$		0			0
$633$	13.1874	+	22.8412i	0.524152	+	0.907857i
$634$		0			0
$635$	19.9012			0.789755
$636$		0			0
$637$	−5.10896	−	8.84898i	−0.202424	−	0.350609i
$638$		0			0
$639$	14.2612			0.564166
$640$		0			0
$641$	−13.7344	+	23.7887i	−0.542476	+	0.939597i	0.456285	+	0.889834i	$0.349180\pi$
$641$	−13.7344	+	23.7887i	−0.542476	+	0.939597i	−0.998761	+	0.0497629i	$0.984153\pi$
$642$		0			0
$643$	−6.11943	+	10.5992i	−0.241327	+	0.417991i	−0.961093	−	0.276227i	$-0.910916\pi$
$643$	−6.11943	+	10.5992i	−0.241327	+	0.417991i	0.719766	+	0.694217i	$0.244249\pi$
$644$		0			0
$645$	0.658572			0.0259312
$646$		0			0
$647$	9.34562			0.367414			0.183707	−	0.982981i	$-0.441190\pi$
$647$	9.34562			0.367414			0.183707	+	0.982981i	$0.441190\pi$
$648$		0			0
$649$	−3.92325	+	6.79527i	−0.154001	+	0.266738i
$650$		0			0
$651$	5.30798	−	9.19369i	0.208036	−	0.360329i
$652$		0			0
$653$	31.6815			1.23979			0.619896	−	0.784684i	$-0.287175\pi$
$653$	31.6815			1.23979			0.619896	+	0.784684i	$0.287175\pi$
$654$		0			0
$655$	2.74533	+	4.75504i	0.107269	+	0.185795i
$656$		0			0
$657$	−8.02669			−0.313151
$658$		0			0
$659$	18.0843	+	31.3229i	0.704465	+	1.22017i	0.966884	+	0.255215i	$0.0821462\pi$
$659$	18.0843	+	31.3229i	0.704465	+	1.22017i	−0.262420	+	0.964954i	$0.584520\pi$
$660$		0			0
$661$	−11.1139	−	19.2498i	−0.432280	−	0.748731i	0.564789	−	0.825235i	$-0.308958\pi$
$661$	−11.1139	−	19.2498i	−0.432280	−	0.748731i	−0.997069	+	0.0765040i	$0.975624\pi$
$662$		0			0
$663$	−1.39881	+	2.42280i	−0.0543251	+	0.0940939i
$664$		0			0
$665$	−5.63088	−	14.7819i	−0.218356	−	0.573216i
$666$		0			0
$667$	5.01279	−	8.68241i	0.194096	−	0.336184i
$668$		0			0
$669$	3.03213	+	5.25180i	0.117229	+	0.203046i
$670$		0			0
$671$	−4.90519	−	8.49604i	−0.189363	−	0.327986i
$672$		0			0
$673$	5.97256			0.230225			0.115113	−	0.993352i	$-0.463277\pi$
$673$	5.97256			0.230225			0.115113	+	0.993352i	$0.463277\pi$
$674$		0			0
$675$	−2.72822	−	4.72541i	−0.105009	−	0.181881i
$676$		0			0
$677$	48.4002			1.86017			0.930085	−	0.367345i	$-0.119733\pi$
$677$	48.4002			1.86017			0.930085	+	0.367345i	$0.119733\pi$
$678$		0			0
$679$	19.7511	−	34.2099i	0.757977	−	1.31285i
$680$		0			0
$681$	12.6907	−	21.9810i	0.486310	−	0.842314i
$682$		0			0
$683$	13.0777			0.500405			0.250202	−	0.968194i	$-0.419503\pi$
$683$	13.0777			0.500405			0.250202	+	0.968194i	$0.419503\pi$
$684$		0			0
$685$	−11.6177			−0.443889
$686$		0			0
$687$	3.31060	−	5.73412i	0.126307	−	0.218770i
$688$		0			0
$689$	−5.86008	+	10.1500i	−0.223251	+	0.386682i
$690$		0			0
$691$	28.5178			1.08487			0.542435	−	0.840098i	$-0.317502\pi$
$691$	28.5178			1.08487			0.542435	+	0.840098i	$0.317502\pi$
$692$		0			0
$693$	4.20578	+	7.28462i	0.159764	+	0.276720i
$694$		0			0
$695$	5.36887			0.203653
$696$		0			0
$697$	2.30262	+	3.98825i	0.0872178	+	0.151066i
$698$		0			0
$699$	15.9109	+	27.5584i	0.601804	+	1.04236i
$700$		0			0
$701$	−18.5868	+	32.1933i	−0.702014	+	1.21592i	0.265744	+	0.964044i	$0.414382\pi$
$701$	−18.5868	+	32.1933i	−0.702014	+	1.21592i	−0.967758	+	0.251881i	$0.918951\pi$
$702$		0			0
$703$	−2.38104	+	2.92501i	−0.0898027	+	0.110319i
$704$		0			0
$705$	−1.98967	+	3.44621i	−0.0749353	+	0.129792i
$706$		0			0
$707$	1.17861	+	2.04141i	0.0443261	+	0.0767750i
$708$		0			0
$709$	16.7445	+	29.0024i	0.628853	+	1.08921i	0.987782	+	0.155841i	$0.0498088\pi$
$709$	16.7445	+	29.0024i	0.628853	+	1.08921i	−0.358929	+	0.933365i	$0.616858\pi$
$710$		0			0
$711$	7.91072			0.296675
$712$		0			0
$713$	8.73762	+	15.1340i	0.327226	+	0.566773i
$714$		0			0
$715$	2.42727			0.0907748
$716$		0			0
$717$	−16.4484	+	28.4896i	−0.614279	+	1.06396i
$718$		0			0
$719$	9.15807	−	15.8622i	0.341538	−	0.591562i	−0.643180	−	0.765715i	$-0.722385\pi$
$719$	9.15807	−	15.8622i	0.341538	−	0.591562i	0.984719	+	0.174153i	$0.0557188\pi$
$720$		0			0
$721$	47.5897			1.77233
$722$		0			0
$723$	−30.2676			−1.12566
$724$		0			0
$725$	−0.704630	+	1.22046i	−0.0261693	+	0.0453266i
$726$		0			0
$727$	15.6629	−	27.1289i	0.580904	−	1.00615i	−0.414469	−	0.910064i	$-0.636033\pi$
$727$	15.6629	−	27.1289i	0.580904	−	1.00615i	0.995373	−	0.0960914i	$-0.0306341\pi$
$728$		0			0
$729$	22.2670			0.824703
$730$		0			0
$731$	−0.392150	−	0.679223i	−0.0145042	−	0.0251220i
$732$		0			0
$733$	40.8010			1.50702			0.753510	−	0.657436i	$-0.228359\pi$
$733$	40.8010			1.50702			0.753510	+	0.657436i	$0.228359\pi$
$734$		0			0
$735$	3.67333	+	6.36240i	0.135493	+	0.234681i
$736$		0			0
$737$	2.03817	+	3.53021i	0.0750768	+	0.130037i
$738$		0			0
$739$	−17.2205	+	29.8268i	−0.633467	+	1.09720i	0.353371	+	0.935483i	$0.385035\pi$
$739$	−17.2205	+	29.8268i	−0.633467	+	1.09720i	−0.986838	+	0.161713i	$0.948298\pi$
$740$		0			0
$741$	5.42809	−	6.66818i	0.199406	−	0.244962i
$742$		0			0
$743$	14.0285	−	24.2980i	0.514655	−	0.891409i	−0.485200	−	0.874403i	$-0.661253\pi$
$743$	14.0285	−	24.2980i	0.514655	−	0.891409i	0.999855	−	0.0170058i	$-0.00541336\pi$
$744$		0			0
$745$	−7.63942	−	13.2319i	−0.279887	−	0.484778i
$746$		0			0
$747$	9.09070	+	15.7456i	0.332611	+	0.576100i
$748$		0			0
$749$	17.1729			0.627484
$750$		0			0
$751$	−27.0985	−	46.9359i	−0.988837	−	1.71272i	−0.623457	−	0.781858i	$-0.714272\pi$
$751$	−27.0985	−	46.9359i	−0.988837	−	1.71272i	−0.365381	−	0.930858i	$-0.619061\pi$
$752$		0			0
$753$	−11.7272			−0.427363
$754$		0			0
$755$	−5.60335	+	9.70528i	−0.203927	+	0.353211i
$756$		0			0
$757$	−19.9109	+	34.4866i	−0.723673	+	1.25344i	0.235846	+	0.971791i	$0.424214\pi$
$757$	−19.9109	+	34.4866i	−0.723673	+	1.25344i	−0.959518	+	0.281647i	$0.909119\pi$
$758$		0			0
$759$	12.4155			0.450654
$760$		0			0
$761$	11.6211			0.421265			0.210633	−	0.977565i	$-0.432448\pi$
$761$	11.6211			0.421265			0.210633	+	0.977565i	$0.432448\pi$
$762$		0			0
$763$	23.4850	−	40.6773i	0.850215	−	1.47262i
$764$		0			0
$765$	−1.12166	+	1.94277i	−0.0405537	+	0.0702411i
$766$		0			0
$767$	8.86871			0.320230
$768$		0			0
$769$	5.19906	+	9.00503i	0.187483	+	0.324730i	0.944410	−	0.328769i	$-0.106634\pi$
$769$	5.19906	+	9.00503i	0.187483	+	0.324730i	−0.756928	+	0.653499i	$0.773300\pi$
$770$		0			0
$771$	3.88159			0.139792
$772$		0			0
$773$	16.8670	+	29.2144i	0.606662	+	1.05077i	0.991786	+	0.127905i	$0.0408253\pi$
$773$	16.8670	+	29.2144i	0.606662	+	1.05077i	−0.385124	+	0.922865i	$0.625841\pi$
$774$		0			0
$775$	−1.22822	−	2.12733i	−0.0441188	−	0.0764160i
$776$		0			0
$777$	1.86971	−	3.23844i	0.0670755	−	0.116178i
$778$		0			0
$779$	−5.03842	−	13.2266i	−0.180520	−	0.473891i
$780$		0			0
$781$	6.60634	−	11.4425i	0.236394	−	0.409446i
$782$		0			0
$783$	−3.84477	−	6.65933i	−0.137401	−	0.237985i
$784$		0			0
$785$	6.98967	+	12.1065i	0.249472	+	0.432098i
$786$		0			0
$787$	−35.5090			−1.26576			−0.632880	−	0.774250i	$-0.718127\pi$
$787$	−35.5090			−1.26576			−0.632880	+	0.774250i	$0.718127\pi$
$788$		0			0
$789$	7.96133	+	13.7894i	0.283431	+	0.490917i
$790$		0			0
$791$	−26.6460			−0.947424
$792$

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:	\( N \)	\(=\)	\( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1520.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.595455 - 1.03136i) q^{3} +(-0.500000 + 0.866025i) q^{5} +3.62891 q^{7} +(0.790867 + 1.36982i) q^{9} +O(q^{10})\)	\(q+(0.595455 - 1.03136i) q^{3} +(-0.500000 + 0.866025i) q^{5} +3.62891 q^{7} +(0.790867 + 1.36982i) q^{9} +1.46544 q^{11} +(-0.828173 - 1.43444i) q^{13} +(0.595455 + 1.03136i) q^{15} +(0.709133 - 1.22825i) q^{17} +(-2.75180 + 3.38047i) q^{19} +(2.16085 - 3.74270i) q^{21} +(3.55704 + 6.16097i) q^{23} +(-0.500000 - 0.866025i) q^{25} +5.45643 q^{27} +(-0.704630 - 1.22046i) q^{29} +2.45643 q^{31} +(0.872601 - 1.51139i) q^{33} +(-1.81445 + 3.14272i) q^{35} +0.865267 q^{37} -1.97256 q^{39} +(-1.62354 + 2.81206i) q^{41} +(0.276499 - 0.478911i) q^{43} -1.58173 q^{45} +(1.67071 + 2.89376i) q^{47} +6.16895 q^{49} +(-0.844513 - 1.46274i) q^{51} +(-3.53795 - 6.12791i) q^{53} +(-0.732718 + 1.26911i) q^{55} +(1.84790 + 4.85101i) q^{57} +(-2.67719 + 4.63703i) q^{59} +(-3.34726 - 5.79762i) q^{61} +(2.86998 + 4.97095i) q^{63} +1.65635 q^{65} +(1.39082 + 2.40898i) q^{67} +8.47222 q^{69} +(4.50811 - 7.80827i) q^{71} +(-2.53731 + 4.39474i) q^{73} -1.19091 q^{75} +5.31793 q^{77} +(2.50065 - 4.33125i) q^{79} +(0.876457 - 1.51807i) q^{81} +11.4946 q^{83} +(0.709133 + 1.22825i) q^{85} -1.67830 q^{87} +(5.84078 + 10.1165i) q^{89} +(-3.00536 - 5.20544i) q^{91} +(1.46269 - 2.53346i) q^{93} +(-1.55167 - 4.07337i) q^{95} +(5.44271 - 9.42705i) q^{97} +(1.15897 + 2.00739i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{3} - 4 q^{5} - 4 q^{7} - q^{9}+O(q^{10}) \) 8 * q - q^3 - 4 * q^5 - 4 * q^7 - q^9	\( 8 q - q^{3} - 4 q^{5} - 4 q^{7} - q^{9} + 8 q^{11} + q^{13} - q^{15} + 13 q^{17} - q^{19} + 12 q^{21} - 8 q^{23} - 4 q^{25} + 20 q^{27} - 3 q^{29} - 4 q^{31} - 15 q^{33} + 2 q^{35} + 20 q^{37} + 2 q^{39} - 8 q^{41} + 3 q^{43} + 2 q^{45} - 10 q^{47} + 12 q^{49} + 16 q^{51} - 11 q^{53} - 4 q^{55} - 29 q^{57} - q^{59} + 25 q^{63} - 2 q^{65} + 8 q^{67} + 22 q^{69} + 4 q^{71} - 20 q^{73} + 2 q^{75} - 36 q^{77} + 3 q^{79} + 12 q^{81} + 30 q^{83} + 13 q^{85} - 24 q^{87} + 17 q^{89} + 4 q^{91} + 12 q^{93} - 4 q^{95} + 11 q^{97} - 30 q^{99}+O(q^{100}) \) 8 * q - q^3 - 4 * q^5 - 4 * q^7 - q^9 + 8 * q^11 + q^13 - q^15 + 13 * q^17 - q^19 + 12 * q^21 - 8 * q^23 - 4 * q^25 + 20 * q^27 - 3 * q^29 - 4 * q^31 - 15 * q^33 + 2 * q^35 + 20 * q^37 + 2 * q^39 - 8 * q^41 + 3 * q^43 + 2 * q^45 - 10 * q^47 + 12 * q^49 + 16 * q^51 - 11 * q^53 - 4 * q^55 - 29 * q^57 - q^59 + 25 * q^63 - 2 * q^65 + 8 * q^67 + 22 * q^69 + 4 * q^71 - 20 * q^73 + 2 * q^75 - 36 * q^77 + 3 * q^79 + 12 * q^81 + 30 * q^83 + 13 * q^85 - 24 * q^87 + 17 * q^89 + 4 * q^91 + 12 * q^93 - 4 * q^95 + 11 * q^97 - 30 * q^99

Introduction

L-functions

Modular forms

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