LMFDB - Embedded newform 1520.2.q.m.961.1

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:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} + 9x^{6} + 2x^{5} + 65x^{4} - 20x^{3} + 25x^{2} + 6x + 4 $ x^8 - x^7 + 9x^6 + 2x^5 + 65x^4 - 20x^3 + 25x^2 + 6x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			961.1
Root			$-1.26041 + 2.18309i$ of defining polynomial
Character	$\chi$	$=$	1520.961
Dual form			1520.2.q.m.881.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.26041 + 2.18309i) q^{3} +(-0.500000 + 0.866025i) q^{5} -2.72743 q^{7} +(-1.67727 - 2.90511i) q^{9} +O(q^{10})$	\(q+(-1.26041 + 2.18309i) q^{3} +(-0.500000 + 0.866025i) q^{5} -2.72743 q^{7} +(-1.67727 - 2.90511i) q^{9} -3.31421 q^{11} +(-1.62412 - 2.81306i) q^{13} +(-1.26041 - 2.18309i) q^{15} +(1.17727 - 2.03909i) q^{17} +(3.11494 - 3.04912i) q^{19} +(3.43768 - 5.95423i) q^{21} +(1.07396 + 1.86016i) q^{23} +(-0.500000 - 0.866025i) q^{25} +0.893714 q^{27} +(1.96702 + 3.40697i) q^{29} +10.1896 q^{31} +(4.17727 - 7.23524i) q^{33} +(1.36371 - 2.36202i) q^{35} -3.68579 q^{37} +8.18825 q^{39} +(0.363714 - 0.629971i) q^{41} +(-1.18645 + 2.05499i) q^{43} +3.35453 q^{45} +(-5.51164 - 9.54644i) q^{47} +0.438860 q^{49} +(2.96768 + 5.14017i) q^{51} +(4.49148 + 7.77947i) q^{53} +(1.65711 - 2.87019i) q^{55} +(2.73041 + 10.6434i) q^{57} +(-5.48784 + 9.50521i) q^{59} +(4.22743 + 7.32212i) q^{61} +(4.57462 + 7.92348i) q^{63} +3.24825 q^{65} +(-4.87535 - 8.44436i) q^{67} -5.41453 q^{69} +(3.45850 - 5.99029i) q^{71} +(1.24025 - 2.14818i) q^{73} +2.52082 q^{75} +9.03927 q^{77} +(5.99948 - 10.3914i) q^{79} +(3.90535 - 6.76427i) q^{81} -4.68711 q^{83} +(1.17727 + 2.03909i) q^{85} -9.91699 q^{87} +(-4.27205 - 7.39941i) q^{89} +(4.42968 + 7.67243i) q^{91} +(-12.8430 + 22.2448i) q^{93} +(1.08314 + 4.22218i) q^{95} +(-3.61494 + 6.26127i) q^{97} +(5.55882 + 9.62816i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + q^{3} - 4 q^{5} - 5 q^{9}+O(q^{10}) $ 8 * q + q^3 - 4 * q^5 - 5 * q^9	$ 8 q + q^{3} - 4 q^{5} - 5 q^{9} - 4 q^{11} + 9 q^{13} + q^{15} + q^{17} - 3 q^{19} + 8 q^{21} - 4 q^{25} - 20 q^{27} + 5 q^{29} + 20 q^{31} + 25 q^{33} - 52 q^{37} + 54 q^{39} - 8 q^{41} - 7 q^{43} + 10 q^{45} - 16 q^{47} + 20 q^{49} - 12 q^{51} + 5 q^{53} + 2 q^{55} + 27 q^{57} - 11 q^{59} + 12 q^{61} + 3 q^{63} - 18 q^{65} + 6 q^{69} - 14 q^{71} - 4 q^{73} - 2 q^{75} - 44 q^{77} - 13 q^{79} - 24 q^{81} - 10 q^{83} + q^{85} + 4 q^{87} + 5 q^{89} + 46 q^{91} - 28 q^{93} + 6 q^{95} - q^{97} - 24 q^{99}+O(q^{100}) $ 8 * q + q^3 - 4 * q^5 - 5 * q^9 - 4 * q^11 + 9 * q^13 + q^15 + q^17 - 3 * q^19 + 8 * q^21 - 4 * q^25 - 20 * q^27 + 5 * q^29 + 20 * q^31 + 25 * q^33 - 52 * q^37 + 54 * q^39 - 8 * q^41 - 7 * q^43 + 10 * q^45 - 16 * q^47 + 20 * q^49 - 12 * q^51 + 5 * q^53 + 2 * q^55 + 27 * q^57 - 11 * q^59 + 12 * q^61 + 3 * q^63 - 18 * q^65 + 6 * q^69 - 14 * q^71 - 4 * q^73 - 2 * q^75 - 44 * q^77 - 13 * q^79 - 24 * q^81 - 10 * q^83 + q^85 + 4 * q^87 + 5 * q^89 + 46 * q^91 - 28 * q^93 + 6 * q^95 - q^97 - 24 * 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
$3$	−1.26041	+	2.18309i	−0.727698	+	1.26041i	0.230156	+	0.973154i	$0.426076\pi$
$3$	−1.26041	+	2.18309i	−0.727698	+	1.26041i	−0.957854	+	0.287256i	$0.907257\pi$
$4$		0			0
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$6$		0			0
$7$	−2.72743			−1.03087			−0.515435	−	0.856928i	$-0.672370\pi$
$7$	−2.72743			−1.03087			−0.515435	+	0.856928i	$0.672370\pi$
$8$		0			0
$9$	−1.67727	−	2.90511i	−0.559089	−	0.968370i
$10$		0			0
$11$	−3.31421			−0.999273			−0.499636	−	0.866235i	$-0.666533\pi$
$11$	−3.31421			−0.999273			−0.499636	+	0.866235i	$0.666533\pi$
$12$		0			0
$13$	−1.62412	−	2.81306i	−0.450451	−	0.780204i	0.547963	−	0.836502i	$-0.315403\pi$
$13$	−1.62412	−	2.81306i	−0.450451	−	0.780204i	−0.998414	+	0.0562987i	$0.982070\pi$
$14$		0			0
$15$	−1.26041	−	2.18309i	−0.325436	−	0.563672i
$16$		0			0
$17$	1.17727	−	2.03909i	0.285529	−	0.494551i	−0.687208	−	0.726460i	$-0.741164\pi$
$17$	1.17727	−	2.03909i	0.285529	−	0.494551i	0.972737	+	0.231910i	$0.0744974\pi$
$18$		0			0
$19$	3.11494	−	3.04912i	0.714617	−	0.699516i
$19$	3.11494	−	3.04912i	0.714617	−	0.699516i
$20$		0			0
$21$	3.43768	−	5.95423i	0.750163	−	1.29932i
$22$		0			0
$23$	1.07396	+	1.86016i	0.223937	+	0.387870i	0.956000	−	0.293367i	$-0.0947758\pi$
$23$	1.07396	+	1.86016i	0.223937	+	0.387870i	−0.732063	+	0.681237i	$0.761442\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$		0			0
$27$	0.893714			0.171995
$28$		0			0
$29$	1.96702	+	3.40697i	0.365266	+	0.632659i	0.988819	−	0.149122i	$-0.0476447\pi$
$29$	1.96702	+	3.40697i	0.365266	+	0.632659i	−0.623553	+	0.781781i	$0.714311\pi$
$30$		0			0
$31$	10.1896			1.83010			0.915050	−	0.403340i	$-0.132151\pi$
$31$	10.1896			1.83010			0.915050	+	0.403340i	$0.132151\pi$
$32$		0			0
$33$	4.17727	−	7.23524i	0.727169	−	1.25949i
$34$		0			0
$35$	1.36371	−	2.36202i	0.230510	−	0.399254i
$36$		0			0
$37$	−3.68579			−0.605940			−0.302970	−	0.953000i	$-0.597978\pi$
$37$	−3.68579			−0.605940			−0.302970	+	0.953000i	$0.597978\pi$
$38$		0			0
$39$	8.18825			1.31117
$40$		0			0
$41$	0.363714	−	0.629971i	0.0568025	−	0.0983849i	−0.836226	−	0.548385i	$-0.815243\pi$
$41$	0.363714	−	0.629971i	0.0568025	−	0.0983849i	0.893028	+	0.450000i	$0.148576\pi$
$42$		0			0
$43$	−1.18645	+	2.05499i	−0.180931	+	0.313383i	−0.942198	−	0.335057i	$-0.891245\pi$
$43$	−1.18645	+	2.05499i	−0.180931	+	0.313383i	0.761267	+	0.648439i	$0.224578\pi$
$44$		0			0
$45$	3.35453			0.500064
$46$		0			0
$47$	−5.51164	−	9.54644i	−0.803955	−	1.39249i	−0.916994	−	0.398901i	$-0.869392\pi$
$47$	−5.51164	−	9.54644i	−0.803955	−	1.39249i	0.113039	−	0.993591i	$-0.463942\pi$
$48$		0			0
$49$	0.438860			0.0626942
$50$		0			0
$51$	2.96768	+	5.14017i	0.415558	+	0.719767i
$52$		0			0
$53$	4.49148	+	7.77947i	0.616952	+	1.06859i	0.990039	+	0.140796i	$0.0449661\pi$
$53$	4.49148	+	7.77947i	0.616952	+	1.06859i	−0.373087	+	0.927797i	$0.621701\pi$
$54$		0			0
$55$	1.65711	−	2.87019i	0.223444	−	0.387017i
$56$		0			0
$57$	2.73041	+	10.6434i	0.361652	+	1.40975i
$58$		0			0
$59$	−5.48784	+	9.50521i	−0.714456	+	1.23747i	0.248714	+	0.968577i	$0.419992\pi$
$59$	−5.48784	+	9.50521i	−0.714456	+	1.23747i	−0.963169	+	0.268896i	$0.913341\pi$
$60$		0			0
$61$	4.22743	+	7.32212i	0.541267	+	0.937501i	0.998832	+	0.0483251i	$0.0153884\pi$
$61$	4.22743	+	7.32212i	0.541267	+	0.937501i	−0.457565	+	0.889176i	$0.651278\pi$
$62$		0			0
$63$	4.57462	+	7.92348i	0.576348	+	0.998264i
$64$		0			0
$65$	3.24825			0.402895
$66$		0			0
$67$	−4.87535	−	8.44436i	−0.595619	−	1.03164i	−0.993459	−	0.114188i	$-0.963573\pi$
$67$	−4.87535	−	8.44436i	−0.595619	−	1.03164i	0.397840	−	0.917455i	$-0.369760\pi$
$68$		0			0
$69$	−5.41453			−0.651833
$70$		0			0
$71$	3.45850	−	5.99029i	0.410448	−	0.710917i	−0.584491	−	0.811400i	$-0.698706\pi$
$71$	3.45850	−	5.99029i	0.410448	−	0.710917i	0.994939	+	0.100484i	$0.0320390\pi$
$72$		0			0
$73$	1.24025	−	2.14818i	0.145160	−	0.251425i	−0.784272	−	0.620417i	$-0.786964\pi$
$73$	1.24025	−	2.14818i	0.145160	−	0.251425i	0.929433	+	0.368992i	$0.120297\pi$
$74$		0			0
$75$	2.52082			0.291079
$76$		0			0
$77$	9.03927			1.03012
$78$		0			0
$79$	5.99948	−	10.3914i	0.674994	−	1.16912i	−0.301477	−	0.953474i	$-0.597480\pi$
$79$	5.99948	−	10.3914i	0.674994	−	1.16912i	0.976471	−	0.215650i	$-0.0691871\pi$
$80$		0			0
$81$	3.90535	−	6.76427i	0.433928	−	0.751586i
$82$		0			0
$83$	−4.68711			−0.514477			−0.257238	−	0.966348i	$-0.582813\pi$
$83$	−4.68711			−0.514477			−0.257238	+	0.966348i	$0.582813\pi$
$84$		0			0
$85$	1.17727	+	2.03909i	0.127692	+	0.221170i
$86$		0			0
$87$	−9.91699			−1.06321
$88$		0			0
$89$	−4.27205	−	7.39941i	−0.452836	−	0.784336i	0.545725	−	0.837965i	$-0.316254\pi$
$89$	−4.27205	−	7.39941i	−0.452836	−	0.784336i	−0.998561	+	0.0536291i	$0.982921\pi$
$90$		0			0
$91$	4.42968	+	7.67243i	0.464357	+	0.804289i
$92$		0			0
$93$	−12.8430	+	22.2448i	−1.33176	+	2.30668i
$94$		0			0
$95$	1.08314	+	4.22218i	0.111128	+	0.433186i
$96$		0			0
$97$	−3.61494	+	6.26127i	−0.367042	+	0.635735i	−0.989102	−	0.147235i	$-0.952963\pi$
$97$	−3.61494	+	6.26127i	−0.367042	+	0.635735i	0.622060	+	0.782970i	$0.286296\pi$
$98$		0			0
$99$	5.55882	+	9.62816i	0.558682	+	0.967666i
$100$		0			0
$101$	2.88818	+	5.00247i	0.287384	+	0.497764i	0.973185	−	0.230026i	$-0.0738810\pi$
$101$	2.88818	+	5.00247i	0.287384	+	0.497764i	−0.685800	+	0.727790i	$0.740548\pi$
$102$		0			0
$103$	15.0576			1.48367			0.741836	−	0.670581i	$-0.233955\pi$
$103$	15.0576			1.48367			0.741836	+	0.670581i	$0.233955\pi$
$104$		0			0
$105$	3.43768	+	5.95423i	0.335483	+	0.581073i
$106$		0			0
$107$	15.1896			1.46843			0.734215	−	0.678917i	$-0.237550\pi$
$107$	15.1896			1.46843			0.734215	+	0.678917i	$0.237550\pi$
$108$		0			0
$109$	6.31057	−	10.9302i	0.604443	−	1.04693i	−0.387696	−	0.921787i	$-0.626729\pi$
$109$	6.31057	−	10.9302i	0.604443	−	1.04693i	0.992139	−	0.125139i	$-0.0399376\pi$
$110$		0			0
$111$	4.64560	−	8.04642i	0.440941	−	0.763732i
$112$		0			0
$113$	−6.10761			−0.574555			−0.287278	−	0.957847i	$-0.592750\pi$
$113$	−6.10761			−0.574555			−0.287278	+	0.957847i	$0.592750\pi$
$114$		0			0
$115$	−2.14793			−0.200295
$116$		0			0
$117$	−5.44818	+	9.43652i	−0.503684	+	0.872406i
$118$		0			0
$119$	−3.21091	+	5.56146i	−0.294344	+	0.509818i
$120$		0			0
$121$	−0.0159950			−0.00145409
$122$		0			0
$123$	0.916857	+	1.58804i	0.0826702	+	0.143189i
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	−4.56114	−	7.90013i	−0.404736	−	0.701023i	0.589555	−	0.807728i	$-0.299303\pi$
$127$	−4.56114	−	7.90013i	−0.404736	−	0.701023i	−0.994291	+	0.106705i	$0.965970\pi$
$128$		0			0
$129$	−2.99082	−	5.18025i	−0.263327	−	0.456096i
$130$		0			0
$131$	11.1328	−	19.2825i	0.972676	−	1.68472i	0.285274	−	0.958446i	$-0.407915\pi$
$131$	11.1328	−	19.2825i	0.972676	−	1.68472i	0.687402	−	0.726277i	$-0.258751\pi$
$132$		0			0
$133$	−8.49578	+	8.31625i	−0.736678	+	0.721110i
$134$		0			0
$135$	−0.446857	+	0.773979i	−0.0384593	+	0.0666135i
$136$		0			0
$137$	3.81355	+	6.60527i	0.325814	+	0.564326i	0.981677	−	0.190554i	$-0.0610284\pi$
$137$	3.81355	+	6.60527i	0.325814	+	0.564326i	−0.655863	+	0.754880i	$0.727695\pi$
$138$		0			0
$139$	8.45916	+	14.6517i	0.717496	+	1.24274i	0.961989	+	0.273089i	$0.0880453\pi$
$139$	8.45916	+	14.6517i	0.717496	+	1.24274i	−0.244493	+	0.969651i	$0.578621\pi$
$140$		0			0
$141$	27.7877			2.34015
$142$		0			0
$143$	5.38269	+	9.32309i	0.450123	+	0.779636i
$144$		0			0
$145$	−3.93403			−0.326704
$146$		0			0
$147$	−0.553143	+	0.958072i	−0.0456225	+	0.0790204i
$148$		0			0
$149$	6.19809	−	10.7354i	0.507767	−	0.879478i	−0.492193	−	0.870486i	$-0.663804\pi$
$149$	6.19809	−	10.7354i	0.507767	−	0.879478i	0.999960	−	0.00899193i	$-0.00286226\pi$
$150$		0			0
$151$	14.4549			1.17632			0.588160	−	0.808745i	$-0.299853\pi$
$151$	14.4549			1.17632			0.588160	+	0.808745i	$0.299853\pi$
$152$		0			0
$153$	−7.89836			−0.638544
$154$		0			0
$155$	−5.09478	+	8.82442i	−0.409223	+	0.708795i
$156$		0			0
$157$	9.10642	−	15.7728i	0.726772	−	1.25881i	−0.231469	−	0.972842i	$-0.574353\pi$
$157$	9.10642	−	15.7728i	0.726772	−	1.25881i	0.958241	−	0.285963i	$-0.0923135\pi$
$158$		0			0
$159$	−22.6444			−1.79582
$160$		0			0
$161$	−2.92916	−	5.07345i	−0.230850	−	0.399844i
$162$		0			0
$163$	−19.2642			−1.50889			−0.754446	−	0.656362i	$-0.772094\pi$
$163$	−19.2642			−1.50889			−0.754446	+	0.656362i	$0.772094\pi$
$164$		0			0
$165$	4.17727	+	7.23524i	0.325200	+	0.563263i
$166$		0			0
$167$	2.84289	+	4.92404i	0.219990	+	0.381033i	0.954805	−	0.297234i	$-0.0960643\pi$
$167$	2.84289	+	4.92404i	0.219990	+	0.381033i	−0.734815	+	0.678268i	$0.762731\pi$
$168$		0			0
$169$	1.22445	−	2.12080i	0.0941881	−	0.163139i
$170$		0			0
$171$	−14.0826	−	3.93507i	−1.07692	−	0.300922i
$172$		0			0
$173$	4.57760	−	7.92864i	0.348029	−	0.602804i	−0.637870	−	0.770144i	$-0.720184\pi$
$173$	4.57760	−	7.92864i	0.348029	−	0.602804i	0.985899	+	0.167340i	$0.0535178\pi$
$174$		0			0
$175$	1.36371	+	2.36202i	0.103087	+	0.178552i
$176$		0			0
$177$	−13.8338	−	23.9609i	−1.03982	−	1.80101i
$178$		0			0
$179$	−12.1810			−0.910448			−0.455224	−	0.890377i	$-0.650441\pi$
$179$	−12.1810			−0.910448			−0.455224	+	0.890377i	$0.650441\pi$
$180$		0			0
$181$	−7.70608	−	13.3473i	−0.572789	−	0.992099i	−0.996278	−	0.0861980i	$-0.972528\pi$
$181$	−7.70608	−	13.3473i	−0.572789	−	0.992099i	0.423489	−	0.905901i	$-0.360805\pi$
$182$		0			0
$183$	−21.3132			−1.57551
$184$		0			0
$185$	1.84289	−	3.19199i	0.135492	−	0.234679i
$186$		0			0
$187$	−3.90171	+	6.75796i	−0.285321	+	0.494191i
$188$		0			0
$189$	−2.43754			−0.177305
$190$		0			0
$191$	16.2482			1.17568			0.587841	−	0.808977i	$-0.299978\pi$
$191$	16.2482			1.17568			0.587841	+	0.808977i	$0.299978\pi$
$192$		0			0
$193$	−0.425514	+	0.737011i	−0.0306292	+	0.0530512i	−0.880934	−	0.473240i	$-0.843084\pi$
$193$	−0.425514	+	0.737011i	−0.0306292	+	0.0530512i	0.850305	+	0.526291i	$0.176418\pi$
$194$		0			0
$195$	−4.09412	+	7.09123i	−0.293186	+	0.507813i
$196$		0			0
$197$	19.9843			1.42382			0.711910	−	0.702270i	$-0.247830\pi$
$197$	19.9843			1.42382			0.711910	+	0.702270i	$0.247830\pi$
$198$		0			0
$199$	−3.22861	−	5.59212i	−0.228870	−	0.396415i	0.728603	−	0.684936i	$-0.240170\pi$
$199$	−3.22861	−	5.59212i	−0.228870	−	0.396415i	−0.957474	+	0.288521i	$0.906836\pi$
$200$		0			0
$201$	24.5798			1.73372
$202$		0			0
$203$	−5.36490	−	9.29227i	−0.376542	−	0.652190i
$204$		0			0
$205$	0.363714	+	0.629971i	0.0254029	+	0.0439990i
$206$		0			0
$207$	3.60264	−	6.23996i	0.250401	−	0.433707i
$208$		0			0
$209$	−10.3236	+	10.1054i	−0.714097	+	0.699007i
$210$		0			0
$211$	5.27205	−	9.13146i	0.362943	−	0.628635i	−0.625501	−	0.780223i	$-0.715105\pi$
$211$	5.27205	−	9.13146i	0.362943	−	0.628635i	0.988444	+	0.151588i	$0.0484387\pi$
$212$		0			0
$213$	8.71825	+	15.1004i	0.597364	+	1.03467i
$214$		0			0
$215$	−1.18645	−	2.05499i	−0.0809150	−	0.140149i
$216$		0			0
$217$	−27.7913			−1.88660
$218$		0			0
$219$	3.12645	+	5.41516i	0.211266	+	0.365923i
$220$		0			0
$221$	−7.64811			−0.514467
$222$		0			0
$223$	4.86319	−	8.42329i	0.325663	−	0.564065i	−0.655983	−	0.754776i	$-0.727746\pi$
$223$	4.86319	−	8.42329i	0.325663	−	0.564065i	0.981646	+	0.190710i	$0.0610791\pi$
$224$		0			0
$225$	−1.67727	+	2.90511i	−0.111818	+	0.193674i
$226$		0			0
$227$	−22.5798			−1.49867			−0.749336	−	0.662190i	$-0.769627\pi$
$227$	−22.5798			−1.49867			−0.749336	+	0.662190i	$0.769627\pi$
$228$		0			0
$229$	−16.2239			−1.07211			−0.536053	−	0.844184i	$-0.680085\pi$
$229$	−16.2239			−1.07211			−0.536053	+	0.844184i	$0.680085\pi$
$230$		0			0
$231$	−11.3932	+	19.7336i	−0.749617	+	1.29837i
$232$		0			0
$233$	9.33139	−	16.1624i	0.611320	−	1.05884i	−0.379699	−	0.925110i	$-0.623972\pi$
$233$	9.33139	−	16.1624i	0.611320	−	1.05884i	0.991018	−	0.133727i	$-0.0426944\pi$
$234$		0			0
$235$	11.0233			0.719079
$236$		0			0
$237$	15.1236	+	26.1948i	0.982383	+	1.70154i
$238$		0			0
$239$	−0.602780			−0.0389906			−0.0194953	−	0.999810i	$-0.506206\pi$
$239$	−0.602780			−0.0389906			−0.0194953	+	0.999810i	$0.506206\pi$
$240$		0			0
$241$	−10.9408	−	18.9500i	−0.704759	−	1.22068i	−0.966779	−	0.255615i	$-0.917722\pi$
$241$	−10.9408	−	18.9500i	−0.704759	−	1.22068i	0.262020	−	0.965062i	$-0.415611\pi$
$242$		0			0
$243$	11.1853	+	19.3734i	0.717535	+	1.24281i
$244$		0			0
$245$	−0.219430	+	0.380064i	−0.0140189	+	0.0242814i
$246$		0			0
$247$	−13.6364	−	3.81039i	−0.867665	−	0.242449i
$248$		0			0
$249$	5.90768	−	10.2324i	0.374384	−	0.648452i
$250$		0			0
$251$	4.13629	+	7.16426i	0.261080	+	0.452204i	0.966529	−	0.256557i	$-0.0825880\pi$
$251$	4.13629	+	7.16426i	0.261080	+	0.452204i	−0.705449	+	0.708761i	$0.749255\pi$
$252$		0			0
$253$	−3.55934	−	6.16496i	−0.223774	−	0.387588i
$254$		0			0
$255$	−5.93535			−0.371686
$256$		0			0
$257$	−7.09544	−	12.2897i	−0.442602	−	0.766608i	0.555280	−	0.831663i	$-0.312611\pi$
$257$	−7.09544	−	12.2897i	−0.442602	−	0.766608i	−0.997882	+	0.0650550i	$0.979278\pi$
$258$		0			0
$259$	10.0527			0.624645
$260$		0			0
$261$	6.59842	−	11.4288i	0.408432	−	0.707425i
$262$		0			0
$263$	−9.27153	+	16.0588i	−0.571707	+	0.990225i	0.424684	+	0.905342i	$0.360385\pi$
$263$	−9.27153	+	16.0588i	−0.571707	+	0.990225i	−0.996391	+	0.0848836i	$0.972948\pi$
$264$		0			0
$265$	−8.98296			−0.551819
$266$		0			0
$267$	21.5381			1.31811
$268$		0			0
$269$	−3.33371	+	5.77416i	−0.203260	+	0.352057i	−0.949577	−	0.313534i	$-0.898487\pi$
$269$	−3.33371	+	5.77416i	−0.203260	+	0.352057i	0.746317	+	0.665591i	$0.231820\pi$
$270$		0			0
$271$	11.0331	−	19.1099i	0.670214	−	1.16085i	−0.307629	−	0.951506i	$-0.599536\pi$
$271$	11.0331	−	19.1099i	0.670214	−	1.16085i	0.977843	−	0.209339i	$-0.0671311\pi$
$272$		0			0
$273$	−22.3328			−1.35165
$274$		0			0
$275$	1.65711	+	2.87019i	0.0999273	+	0.173079i
$276$		0			0
$277$	−24.4624			−1.46980			−0.734902	−	0.678173i	$-0.762772\pi$
$277$	−24.4624			−1.46980			−0.734902	+	0.678173i	$0.762772\pi$
$278$		0			0
$279$	−17.0906	−	29.6018i	−1.02319	−	1.77221i
$280$		0			0
$281$	−9.33139	−	16.1624i	−0.556664	−	0.964170i	−0.997772	−	0.0667164i	$-0.978748\pi$
$281$	−9.33139	−	16.1624i	−0.556664	−	0.964170i	0.441108	−	0.897454i	$-0.354586\pi$
$282$		0			0
$283$	2.24811	−	3.89384i	0.133636	−	0.231465i	−0.791439	−	0.611248i	$-0.790668\pi$
$283$	2.24811	−	3.89384i	0.133636	−	0.231465i	0.925076	+	0.379783i	$0.124001\pi$
$284$		0			0
$285$	−10.5826	−	2.95707i	−0.626860	−	0.175162i
$286$		0			0
$287$	−0.992002	+	1.71820i	−0.0585561	+	0.101422i
$288$		0			0
$289$	5.72809	+	9.92134i	0.336946	+	0.583608i
$290$		0			0
$291$	−9.11262	−	15.7835i	−0.534191	−	0.925246i
$292$		0			0
$293$	1.27021			0.0742063			0.0371031	−	0.999311i	$-0.488187\pi$
$293$	1.27021			0.0742063			0.0371031	+	0.999311i	$0.488187\pi$
$294$		0			0
$295$	−5.48784	−	9.50521i	−0.319514	−	0.553415i
$296$		0			0
$297$	−2.96196			−0.171870
$298$		0			0
$299$	3.48850	−	6.04225i	0.201745	−	0.349433i
$300$		0			0
$301$	3.23595	−	5.60483i	0.186517	−	0.323057i
$302$		0			0
$303$	−14.5611			−0.836516
$304$		0			0
$305$	−8.45485			−0.484124
$306$		0			0
$307$	−4.32638	+	7.49350i	−0.246919	+	0.427677i	−0.962669	−	0.270680i	$-0.912752\pi$
$307$	−4.32638	+	7.49350i	−0.246919	+	0.427677i	0.715750	+	0.698356i	$0.246085\pi$
$308$		0			0
$309$	−18.9788	+	32.8722i	−1.07967	+	1.87004i
$310$		0			0
$311$	−23.8497			−1.35239			−0.676196	−	0.736721i	$-0.736373\pi$
$311$	−23.8497			−1.35239			−0.676196	+	0.736721i	$0.736373\pi$
$312$		0			0
$313$	0.388175	+	0.672340i	0.0219410	+	0.0380029i	0.876787	−	0.480878i	$-0.159682\pi$
$313$	0.388175	+	0.672340i	0.0219410	+	0.0380029i	−0.854846	+	0.518881i	$0.826349\pi$
$314$		0			0
$315$	−9.14925			−0.515502
$316$		0			0
$317$	6.92116	+	11.9878i	0.388731	+	0.673302i	0.992279	−	0.124025i	$-0.0395802\pi$
$317$	6.92116	+	11.9878i	0.388731	+	0.673302i	−0.603548	+	0.797327i	$0.706247\pi$
$318$		0			0
$319$	−6.51911	−	11.2914i	−0.365000	−	0.632199i
$320$		0			0
$321$	−19.1451	+	33.1603i	−1.06857	+	1.85082i
$322$		0			0
$323$	−2.55030	−	9.94126i	−0.141902	−	0.553147i
$324$		0			0
$325$	−1.62412	+	2.81306i	−0.0900902	+	0.156041i
$326$		0			0
$327$	15.9078	+	27.5531i	0.879704	+	1.52369i
$328$		0			0
$329$	15.0326	+	26.0372i	0.828774	+	1.43548i
$330$		0			0
$331$	−28.6993			−1.57746			−0.788728	−	0.614742i	$-0.789260\pi$
$331$	−28.6993			−1.57746			−0.788728	+	0.614742i	$0.789260\pi$
$332$		0			0
$333$	6.18205	+	10.7076i	0.338774	+	0.586774i
$334$		0			0
$335$	9.75071			0.532738
$336$		0			0
$337$	17.6628	−	30.5928i	0.962153	−	1.66650i	0.245076	−	0.969504i	$-0.421187\pi$
$337$	17.6628	−	30.5928i	0.962153	−	1.66650i	0.717077	−	0.696994i	$-0.245480\pi$
$338$		0			0
$339$	7.69809	−	13.3335i	0.418103	−	0.724175i
$340$		0			0
$341$	−33.7704			−1.82877
$342$		0			0
$343$	17.8950			0.966241
$344$		0			0
$345$	2.70727	−	4.68912i	0.145754	−	0.252454i
$346$		0			0
$347$	13.0031	−	22.5221i	0.698044	−	1.20905i	−0.271100	−	0.962551i	$-0.587387\pi$
$347$	13.0031	−	22.5221i	0.698044	−	1.20905i	0.969144	−	0.246496i	$-0.0792793\pi$
$348$		0			0
$349$	2.44757			0.131015			0.0655077	−	0.997852i	$-0.479133\pi$
$349$	2.44757			0.131015			0.0655077	+	0.997852i	$0.479133\pi$
$350$		0			0
$351$	−1.45150	−	2.51408i	−0.0774755	−	0.134191i
$352$		0			0
$353$	−4.85103			−0.258194			−0.129097	−	0.991632i	$-0.541208\pi$
$353$	−4.85103			−0.258194			−0.129097	+	0.991632i	$0.541208\pi$
$354$		0			0
$355$	3.45850	+	5.99029i	0.183558	+	0.317932i
$356$		0			0
$357$	−8.09412	−	14.0194i	−0.428386	−	0.741987i
$358$		0			0
$359$	−3.67376	+	6.36314i	−0.193894	+	0.335834i	−0.946537	−	0.322594i	$-0.895445\pi$
$359$	−3.67376	+	6.36314i	−0.193894	+	0.335834i	0.752644	+	0.658428i	$0.228778\pi$
$360$		0			0
$361$	0.405741	−	18.9957i	0.0213548	−	0.999772i
$362$		0			0
$363$	0.0201603	−	0.0349186i	0.00105814	−	0.00183275i
$364$		0			0
$365$	1.24025	+	2.14818i	0.0649176	+	0.112441i
$366$		0			0
$367$	6.51784	+	11.2892i	0.340228	+	0.589293i	0.984475	−	0.175525i	$-0.0561623\pi$
$367$	6.51784	+	11.2892i	0.340228	+	0.589293i	−0.644247	+	0.764818i	$0.722829\pi$
$368$		0			0
$369$	−2.44018			−0.127031
$370$		0			0
$371$	−12.2502	−	21.2179i	−0.635998	−	1.10158i
$372$		0			0
$373$	−28.5514			−1.47833			−0.739167	−	0.673522i	$-0.764781\pi$
$373$	−28.5514			−1.47833			−0.739167	+	0.673522i	$0.764781\pi$
$374$		0			0
$375$	−1.26041	+	2.18309i	−0.0650873	+	0.112734i
$376$		0			0
$377$	6.38936	−	11.0667i	0.329069	−	0.569964i
$378$		0			0
$379$	−13.2128			−0.678698			−0.339349	−	0.940661i	$-0.610207\pi$
$379$	−13.2128			−0.678698			−0.339349	+	0.940661i	$0.610207\pi$
$380$		0			0
$381$	22.9956			1.17810
$382$		0			0
$383$	−18.8309	+	32.6160i	−0.962212	+	1.66660i	−0.245287	+	0.969450i	$0.578882\pi$
$383$	−18.8309	+	32.6160i	−0.962212	+	1.66660i	−0.716925	+	0.697150i	$0.754451\pi$
$384$		0			0
$385$	−4.51964	+	7.82824i	−0.230342	+	0.398964i
$386$		0			0
$387$	7.95995			0.404627
$388$		0			0
$389$	12.8223	+	22.2090i	0.650119	+	1.12604i	0.983094	+	0.183103i	$0.0586142\pi$
$389$	12.8223	+	22.2090i	0.650119	+	1.12604i	−0.332975	+	0.942936i	$0.608052\pi$
$390$		0			0
$391$	5.05736			0.255762
$392$		0			0
$393$	28.0637	+	48.6078i	1.41563	+	2.45194i
$394$		0			0
$395$	5.99948	+	10.3914i	0.301866	+	0.522848i
$396$		0			0
$397$	16.6366	−	28.8154i	0.834965	−	1.44620i	−0.0590940	−	0.998252i	$-0.518821\pi$
$397$	16.6366	−	28.8154i	0.834965	−	1.44620i	0.894059	−	0.447949i	$-0.147845\pi$
$398$		0			0
$399$	−7.44699	−	29.0290i	−0.372816	−	1.45327i
$400$		0			0
$401$	−4.70674	+	8.15232i	−0.235044	+	0.407107i	−0.959285	−	0.282439i	$-0.908857\pi$
$401$	−4.70674	+	8.15232i	−0.235044	+	0.407107i	0.724242	+	0.689546i	$0.242190\pi$
$402$		0			0
$403$	−16.5491	−	28.6639i	−0.824370	−	1.42785i
$404$		0			0
$405$	3.90535	+	6.76427i	0.194059	+	0.336119i
$406$		0			0
$407$	12.2155			0.605499
$408$		0			0
$409$	1.25545	+	2.17450i	0.0620779	+	0.107522i	0.895394	−	0.445275i	$-0.146894\pi$
$409$	1.25545	+	2.17450i	0.0620779	+	0.107522i	−0.833316	+	0.552797i	$0.813561\pi$
$410$		0			0
$411$	−19.2266			−0.948376
$412$		0			0
$413$	14.9677	−	25.9248i	0.736511	−	1.27567i
$414$		0			0
$415$	2.34355	−	4.05915i	0.115041	−	0.199256i
$416$		0			0
$417$	−42.6480			−2.08848
$418$		0			0
$419$	39.5638			1.93282			0.966409	−	0.257011i	$-0.0827376\pi$
$419$	39.5638			1.93282			0.966409	+	0.257011i	$0.0827376\pi$
$420$		0			0
$421$	7.07892	−	12.2611i	0.345006	−	0.597567i	−0.640349	−	0.768084i	$-0.721210\pi$
$421$	7.07892	−	12.2611i	0.345006	−	0.597567i	0.985355	+	0.170517i	$0.0545437\pi$
$422$		0			0
$423$	−18.4890	+	32.0238i	−0.898965	+	1.55705i
$424$		0			0
$425$	−2.35453			−0.114212
$426$		0			0
$427$	−11.5300	−	19.9705i	−0.557976	−	0.966442i
$428$		0			0
$429$	−27.1376			−1.31022
$430$		0			0
$431$	−3.26287	−	5.65145i	−0.157167	−	0.272221i	0.776679	−	0.629897i	$-0.216903\pi$
$431$	−3.26287	−	5.65145i	−0.157167	−	0.272221i	−0.933846	+	0.357676i	$0.883569\pi$
$432$		0			0
$433$	4.30073	+	7.44908i	0.206680	+	0.357980i	0.950667	−	0.310214i	$-0.100401\pi$
$433$	4.30073	+	7.44908i	0.206680	+	0.357980i	−0.743987	+	0.668194i	$0.767067\pi$
$434$		0			0
$435$	4.95850	−	8.58837i	0.237742	−	0.411781i
$436$		0			0
$437$	9.01718	+	2.51965i	0.431350	+	0.120531i
$438$		0			0
$439$	−5.77939	+	10.0102i	−0.275835	+	0.477760i	−0.970345	−	0.241722i	$-0.922288\pi$
$439$	−5.77939	+	10.0102i	−0.275835	+	0.477760i	0.694510	+	0.719483i	$0.255621\pi$
$440$		0			0
$441$	−0.736084	−	1.27494i	−0.0350516	−	0.0607112i
$442$		0			0
$443$	−5.69629	−	9.86626i	−0.270639	−	0.468760i	0.698387	−	0.715721i	$-0.253902\pi$
$443$	−5.69629	−	9.86626i	−0.270639	−	0.468760i	−0.969026	+	0.246960i	$0.920568\pi$
$444$		0			0
$445$	8.54410			0.405029
$446$		0			0
$447$	15.6243	+	27.0620i	0.739002	+	1.27999i
$448$		0			0
$449$	17.2252			0.812909			0.406455	−	0.913671i	$-0.366765\pi$
$449$	17.2252			0.812909			0.406455	+	0.913671i	$0.366765\pi$
$450$		0			0
$451$	−1.20542	+	2.08786i	−0.0567612	+	0.0983133i
$452$		0			0
$453$	−18.2190	+	31.5563i	−0.856005	+	1.48264i
$454$		0			0
$455$	−8.85936			−0.415333
$456$		0			0
$457$	−26.1885			−1.22505			−0.612524	−	0.790452i	$-0.709846\pi$
$457$	−26.1885			−1.22505			−0.612524	+	0.790452i	$0.709846\pi$
$458$		0			0
$459$	1.05214	−	1.82236i	0.0491097	−	0.0850605i
$460$		0			0
$461$	−20.2166	+	35.0162i	−0.941580	+	1.63086i	−0.179122	+	0.983827i	$0.557326\pi$
$461$	−20.2166	+	35.0162i	−0.941580	+	1.63086i	−0.762458	+	0.647038i	$0.776008\pi$
$462$		0			0
$463$	−9.48542			−0.440825			−0.220412	−	0.975407i	$-0.570740\pi$
$463$	−9.48542			−0.440825			−0.220412	+	0.975407i	$0.570740\pi$
$464$		0			0
$465$	−12.8430	−	22.2448i	−0.595581	−	1.03158i
$466$		0			0
$467$	39.2462			1.81610			0.908048	−	0.418867i	$-0.137573\pi$
$467$	39.2462			1.81610			0.908048	+	0.418867i	$0.137573\pi$
$468$		0			0
$469$	13.2972	+	23.0314i	0.614006	+	1.06349i
$470$		0			0
$471$	22.9557	+	39.7604i	1.05774	+	1.83206i
$472$		0			0
$473$	3.93214	−	6.81066i	0.180800	−	0.313155i
$474$		0			0
$475$	−4.19809	−	1.17306i	−0.192621	−	0.0538237i
$476$		0			0
$477$	15.0668	−	26.0965i	0.689862	−	1.19488i
$478$		0			0
$479$	2.72677	+	4.72290i	0.124589	+	0.215795i	0.921572	−	0.388207i	$-0.126905\pi$
$479$	2.72677	+	4.72290i	0.124589	+	0.215795i	−0.796983	+	0.604002i	$0.793572\pi$
$480$		0			0
$481$	5.98617	+	10.3684i	0.272946	+	0.472756i
$482$		0			0
$483$	14.7677			0.671956
$484$		0			0
$485$	−3.61494	−	6.26127i	−0.164146	−	0.284309i
$486$		0			0
$487$	−25.0662			−1.13586			−0.567930	−	0.823077i	$-0.692256\pi$
$487$	−25.0662			−1.13586			−0.567930	+	0.823077i	$0.692256\pi$
$488$		0			0
$489$	24.2808	−	42.0557i	1.09802	−	1.90182i
$490$		0			0
$491$	14.8412	−	25.7057i	0.669773	−	1.16008i	−0.308194	−	0.951324i	$-0.599725\pi$
$491$	14.8412	−	25.7057i	0.669773	−	1.16008i	0.977967	−	0.208758i	$-0.0669421\pi$
$492$		0			0
$493$	9.26281			0.417176
$494$		0			0
$495$	−11.1176			−0.499701
$496$		0			0
$497$	−9.43280	+	16.3381i	−0.423119	+	0.732863i
$498$		0			0
$499$	−6.85519	+	11.8735i	−0.306881	+	0.531533i	−0.977678	−	0.210108i	$-0.932619\pi$
$499$	−6.85519	+	11.8735i	−0.306881	+	0.531533i	0.670798	+	0.741640i	$0.265952\pi$
$500$		0			0
$501$	−14.3328			−0.640344
$502$		0			0
$503$	18.3253	+	31.7404i	0.817086	+	1.41523i	0.907820	+	0.419359i	$0.137745\pi$
$503$	18.3253	+	31.7404i	0.817086	+	1.41523i	−0.0907343	+	0.995875i	$0.528921\pi$
$504$		0			0
$505$	−5.77635			−0.257044
$506$		0			0
$507$	3.08661	+	5.34616i	0.137081	+	0.237431i
$508$		0			0
$509$	6.32519	+	10.9556i	0.280359	+	0.485596i	0.971473	−	0.237149i	$-0.0762131\pi$
$509$	6.32519	+	10.9556i	0.280359	+	0.485596i	−0.691114	+	0.722746i	$0.742880\pi$
$510$		0			0
$511$	−3.38269	+	5.85899i	−0.149641	+	0.259187i
$512$		0			0
$513$	2.78387	−	2.72504i	0.122911	−	0.120314i
$514$		0			0
$515$	−7.52882	+	13.0403i	−0.331759	+	0.574624i
$516$		0			0
$517$	18.2667	+	31.6389i	0.803371	+	1.39148i
$518$		0			0
$519$	11.5393	+	19.9867i	0.506520	+	0.877318i
$520$		0			0
$521$	16.3339			0.715601			0.357800	−	0.933798i	$-0.383527\pi$
$521$	16.3339			0.715601			0.357800	+	0.933798i	$0.383527\pi$
$522$		0			0
$523$	−4.56734	−	7.91086i	−0.199716	−	0.345918i	0.748720	−	0.662886i	$-0.230669\pi$
$523$	−4.56734	−	7.91086i	−0.199716	−	0.345918i	−0.948436	+	0.316968i	$0.897335\pi$
$524$		0			0
$525$	−6.87535			−0.300065
$526$		0			0
$527$	11.9958	−	20.7774i	0.522547	−	0.905078i
$528$		0			0
$529$	9.19321	−	15.9231i	0.399705	−	0.692309i
$530$		0			0
$531$	36.8183			1.59778
$532$		0			0
$533$	−2.36286			−0.102347
$534$		0			0
$535$	−7.59478	+	13.1545i	−0.328351	+	0.568721i
$536$		0			0
$537$	15.3530	−	26.5922i	0.662531	−	1.14754i
$538$		0			0
$539$	−1.45447			−0.0626486
$540$		0			0
$541$	1.45954	+	2.52800i	0.0627507	+	0.108687i	0.895694	−	0.444671i	$-0.146679\pi$
$541$	1.45954	+	2.52800i	0.0627507	+	0.108687i	−0.832943	+	0.553358i	$0.813346\pi$
$542$		0			0
$543$	38.8513			1.66727
$544$		0			0
$545$	6.31057	+	10.9302i	0.270315	+	0.468200i
$546$		0			0
$547$	−1.73973	−	3.01329i	−0.0743853	−	0.128839i	0.826433	−	0.563034i	$-0.190366\pi$
$547$	−1.73973	−	3.01329i	−0.0743853	−	0.128839i	−0.900819	+	0.434195i	$0.857033\pi$
$548$		0			0
$549$	14.1810	−	24.5623i	0.605232	−	1.04829i
$550$		0			0
$551$	16.5154	+	4.61486i	0.703580	+	0.196600i
$552$		0			0
$553$	−16.3631	+	28.3418i	−0.695831	+	1.20522i
$554$		0			0
$555$	4.64560	+	8.04642i	0.197195	+	0.341552i
$556$		0			0
$557$	−7.96280	−	13.7920i	−0.337395	−	0.584385i	0.646547	−	0.762874i	$-0.276212\pi$
$557$	−7.96280	−	13.7920i	−0.337395	−	0.584385i	−0.983942	+	0.178489i	$0.942879\pi$
$558$		0			0
$559$	7.70775			0.326003
$560$		0			0
$561$	−9.83551	−	17.0356i	−0.415256	−	0.719244i
$562$		0			0
$563$	−19.6274			−0.827195			−0.413598	−	0.910460i	$-0.635728\pi$
$563$	−19.6274			−0.827195			−0.413598	+	0.910460i	$0.635728\pi$
$564$		0			0
$565$	3.05380	−	5.28934i	0.128474	−	0.222524i
$566$		0			0
$567$	−10.6516	+	18.4491i	−0.447324	+	0.774788i
$568$		0			0
$569$	−11.7544			−0.492770			−0.246385	−	0.969172i	$-0.579243\pi$
$569$	−11.7544			−0.492770			−0.246385	+	0.969172i	$0.579243\pi$
$570$		0			0
$571$	−32.4868			−1.35953			−0.679766	−	0.733429i	$-0.737919\pi$
$571$	−32.4868			−1.35953			−0.679766	+	0.733429i	$0.737919\pi$
$572$		0			0
$573$	−20.4795	+	35.4715i	−0.855541	+	1.48184i
$574$		0			0
$575$	1.07396	−	1.86016i	0.0447873	−	0.0775740i
$576$		0			0
$577$	25.7287			1.07110			0.535551	−	0.844503i	$-0.320104\pi$
$577$	25.7287			1.07110			0.535551	+	0.844503i	$0.320104\pi$
$578$		0			0
$579$	−1.07264	−	1.85787i	−0.0445775	−	0.0772106i
$580$		0			0
$581$	12.7837			0.530359
$582$		0			0
$583$	−14.8857	−	25.7828i	−0.616503	−	1.06782i
$584$		0			0
$585$	−5.44818	−	9.43652i	−0.225254	−	0.390152i
$586$		0			0
$587$	−20.8279	+	36.0750i	−0.859659	+	1.48897i	0.0125958	+	0.999921i	$0.495991\pi$
$587$	−20.8279	+	36.0750i	−0.859659	+	1.48897i	−0.872255	+	0.489052i	$0.837343\pi$
$588$		0			0
$589$	31.7399	−	31.0692i	1.30782	−	1.28018i
$590$		0			0
$591$	−25.1884	+	43.6276i	−1.03611	+	1.79460i
$592$		0			0
$593$	−8.83267	−	15.2986i	−0.362714	−	0.628239i	0.625692	−	0.780070i	$-0.284817\pi$
$593$	−8.83267	−	15.2986i	−0.362714	−	0.628239i	−0.988407	+	0.151831i	$0.951483\pi$
$594$		0			0
$595$	−3.21091	−	5.56146i	−0.131634	−	0.227998i
$596$		0			0
$597$	16.2775			0.666193
$598$		0			0
$599$	−13.1925	−	22.8502i	−0.539033	−	0.933632i	−0.998956	−	0.0456738i	$-0.985457\pi$
$599$	−13.1925	−	22.8502i	−0.539033	−	0.933632i	0.459924	−	0.887959i	$-0.347877\pi$
$600$		0			0
$601$	9.46838			0.386223			0.193112	−	0.981177i	$-0.438142\pi$
$601$	9.46838			0.386223			0.193112	+	0.981177i	$0.438142\pi$
$602$		0			0
$603$	−16.3545	+	28.3269i	−0.666008	+	1.15356i
$604$		0			0
$605$	0.00799750	−	0.0138521i	0.000325145	−	0.000563167i
$606$		0			0
$607$	8.26529			0.335478			0.167739	−	0.985831i	$-0.446353\pi$
$607$	8.26529			0.335478			0.167739	+	0.985831i	$0.446353\pi$
$608$		0			0
$609$	27.0479			1.09604
$610$		0			0
$611$	−17.9032	+	31.0092i	−0.724285	+	1.25450i
$612$		0			0
$613$	−3.40417	+	5.89620i	−0.137493	+	0.238145i	−0.926547	−	0.376179i	$-0.877238\pi$
$613$	−3.40417	+	5.89620i	−0.137493	+	0.238145i	0.789054	+	0.614324i	$0.210571\pi$
$614$		0			0
$615$	−1.83371			−0.0739425
$616$		0			0
$617$	−3.00668	−	5.20772i	−0.121044	−	0.209655i	0.799135	−	0.601151i	$-0.205291\pi$
$617$	−3.00668	−	5.20772i	−0.121044	−	0.209655i	−0.920180	+	0.391496i	$0.871958\pi$
$618$		0			0
$619$	−13.2299			−0.531754			−0.265877	−	0.964007i	$-0.585662\pi$
$619$	−13.2299			−0.531754			−0.265877	+	0.964007i	$0.585662\pi$
$620$		0			0
$621$	0.959816	+	1.66245i	0.0385161	+	0.0667118i
$622$		0			0
$623$	11.6517	+	20.1813i	0.466816	+	0.808548i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$		0			0
$627$	−9.04916	−	35.2743i	−0.361389	−	1.40872i
$628$		0			0
$629$	−4.33915	+	7.51564i	−0.173013	+	0.299668i
$630$		0			0
$631$	−11.8932	−	20.5996i	−0.473460	−	0.820058i	0.526078	−	0.850436i	$-0.323662\pi$
$631$	−11.8932	−	20.5996i	−0.473460	−	0.820058i	−0.999538	+	0.0303788i	$0.990329\pi$
$632$		0			0
$633$	13.2899	+	23.0188i	0.528226	+	0.914914i
$634$		0			0
$635$	9.12228			0.362007
$636$		0			0
$637$	−0.712762	−	1.23454i	−0.0282407	−	0.0489143i
$638$		0			0
$639$	−23.2033			−0.917908
$640$		0			0
$641$	−6.14239	+	10.6389i	−0.242610	+	0.420212i	−0.961457	−	0.274956i	$-0.911337\pi$
$641$	−6.14239	+	10.6389i	−0.242610	+	0.420212i	0.718847	+	0.695168i	$0.244670\pi$
$642$		0			0
$643$	14.5776	−	25.2492i	0.574885	−	0.995729i	−0.421170	−	0.906982i	$-0.638380\pi$
$643$	14.5776	−	25.2492i	0.574885	−	0.995729i	0.996054	−	0.0887474i	$-0.0282864\pi$
$644$		0			0
$645$	5.98164			0.235527
$646$		0			0
$647$	36.7499			1.44479			0.722394	−	0.691481i	$-0.243042\pi$
$647$	36.7499			1.44479			0.722394	+	0.691481i	$0.243042\pi$
$648$		0			0
$649$	18.1879	−	31.5023i	0.713936	−	1.23657i
$650$		0			0
$651$	35.0284	−	60.6710i	1.37287	−	2.37788i
$652$		0			0
$653$	24.7707			0.969351			0.484675	−	0.874694i	$-0.338938\pi$
$653$	24.7707			0.969351			0.484675	+	0.874694i	$0.338938\pi$
$654$		0			0
$655$	11.1328	+	19.2825i	0.434994	+	0.753431i
$656$		0			0
$657$	−8.32092			−0.324630
$658$		0			0
$659$	11.1156	+	19.2528i	0.433002	+	0.749982i	0.997130	−	0.0757053i	$-0.0241208\pi$
$659$	11.1156	+	19.2528i	0.433002	+	0.749982i	−0.564128	+	0.825687i	$0.690787\pi$
$660$		0			0
$661$	−20.4684	−	35.4524i	−0.796130	−	1.37894i	−0.922119	−	0.386905i	$-0.873544\pi$
$661$	−20.4684	−	35.4524i	−0.796130	−	1.37894i	0.125990	−	0.992032i	$-0.459789\pi$
$662$		0			0
$663$	9.63975	−	16.6965i	0.374377	−	0.648440i
$664$		0			0
$665$	−2.95419	−	11.5157i	−0.114559	−	0.446559i
$666$		0			0
$667$	−4.22501	+	7.31793i	−0.163593	+	0.283351i
$668$		0			0
$669$	12.2592	+	21.2336i	0.473969	+	0.820939i
$670$		0			0
$671$	−14.0106	−	24.2671i	−0.540873	−	0.936819i
$672$		0			0
$673$	−33.5992			−1.29515			−0.647577	−	0.762000i	$-0.724217\pi$
$673$	−33.5992			−1.29515			−0.647577	+	0.762000i	$0.724217\pi$
$674$		0			0
$675$	−0.446857	−	0.773979i	−0.0171995	−	0.0297905i
$676$		0			0
$677$	29.5883			1.13717			0.568585	−	0.822624i	$-0.307491\pi$
$677$	29.5883			1.13717			0.568585	+	0.822624i	$0.307491\pi$
$678$		0			0
$679$	9.85949	−	17.0771i	0.378373	−	0.655361i
$680$		0			0
$681$	28.4598	−	49.2938i	1.09058	−	1.88894i
$682$		0			0
$683$	−41.0567			−1.57099			−0.785495	−	0.618868i	$-0.787592\pi$
$683$	−41.0567			−1.57099			−0.785495	+	0.618868i	$0.787592\pi$
$684$		0			0
$685$	−7.62711			−0.291417
$686$		0			0
$687$	20.4488	−	35.4183i	0.780170	−	1.35129i
$688$		0			0
$689$	14.5894	−	25.2696i	0.555813	−	0.962697i
$690$		0			0
$691$	−1.22497			−0.0466000			−0.0233000	−	0.999729i	$-0.507417\pi$
$691$	−1.22497			−0.0466000			−0.0233000	+	0.999729i	$0.507417\pi$
$692$		0			0
$693$	−15.1613	−	26.2601i	−0.575929	−	0.997538i
$694$		0			0
$695$	−16.9183			−0.641748
$696$		0			0
$697$	−0.856376	−	1.48329i	−0.0324375	−	0.0561835i
$698$		0			0
$699$	23.5228	+	40.7426i	0.889712	+	1.54103i
$700$		0			0
$701$	−22.0101	+	38.1226i	−0.831309	+	1.43987i	0.0656918	+	0.997840i	$0.479075\pi$
$701$	−22.0101	+	38.1226i	−0.831309	+	1.43987i	−0.897001	+	0.442029i	$0.854259\pi$
$702$		0			0
$703$	−11.4810	+	11.2384i	−0.433015	+	0.423865i
$704$		0			0
$705$	−13.8939	+	24.0649i	−0.523273	+	0.906335i
$706$		0			0
$707$	−7.87729	−	13.6439i	−0.296256	−	0.513130i
$708$		0			0
$709$	−8.16255	−	14.1379i	−0.306551	−	0.530962i	0.671055	−	0.741408i	$-0.265842\pi$
$709$	−8.16255	−	14.1379i	−0.306551	−	0.530962i	−0.977605	+	0.210446i	$0.932508\pi$
$710$		0			0
$711$	−40.2509			−1.50953
$712$		0			0
$713$	10.9432	+	18.9542i	0.409827	+	0.709841i
$714$		0			0
$715$	−10.7654			−0.402602
$716$		0			0
$717$	0.759750	−	1.31593i	0.0283734	−	0.0491442i
$718$		0			0
$719$	5.08574	−	8.80876i	0.189666	−	0.328511i	−0.755473	−	0.655180i	$-0.772593\pi$
$719$	5.08574	−	8.80876i	0.189666	−	0.328511i	0.945139	+	0.326669i	$0.105926\pi$
$720$		0			0
$721$	−41.0686			−1.52947
$722$		0			0
$723$	55.1595			2.05141
$724$		0			0
$725$	1.96702	−	3.40697i	0.0730532	−	0.126532i
$726$		0			0
$727$	4.02148	−	6.96541i	0.149148	−	0.258333i	−0.781765	−	0.623574i	$-0.785680\pi$
$727$	4.02148	−	6.96541i	0.149148	−	0.258333i	0.930913	+	0.365241i	$0.119013\pi$
$728$		0			0
$729$	−32.9600			−1.22074
$730$		0			0
$731$	2.79353	+	4.83853i	0.103322	+	0.178960i
$732$		0			0
$733$	41.2325			1.52296			0.761479	−	0.648190i	$-0.224474\pi$
$733$	41.2325			1.52296			0.761479	+	0.648190i	$0.224474\pi$
$734$		0			0
$735$	−0.553143	−	0.958072i	−0.0204030	−	0.0353390i
$736$		0			0
$737$	16.1580	+	27.9864i	0.595186	+	1.03089i
$738$		0			0
$739$	16.5253	−	28.6227i	0.607893	−	1.05290i	−0.383694	−	0.923460i	$-0.625348\pi$
$739$	16.5253	−	28.6227i	0.607893	−	1.05290i	0.991587	−	0.129442i	$-0.0413186\pi$
$740$		0			0
$741$	25.5059	−	24.9669i	0.936983	−	0.917184i
$742$		0			0
$743$	17.3933	−	30.1261i	0.638099	−	1.10522i	−0.347750	−	0.937587i	$-0.613054\pi$
$743$	17.3933	−	30.1261i	0.638099	−	1.10522i	0.985849	−	0.167633i	$-0.0536124\pi$
$744$		0			0
$745$	6.19809	+	10.7354i	0.227080	+	0.393315i
$746$		0			0
$747$	7.86153	+	13.6166i	0.287638	+	0.498204i
$748$		0			0
$749$	−41.4284			−1.51376
$750$		0			0
$751$	−21.7350	−	37.6462i	−0.793123	−	1.37373i	−0.924024	−	0.382334i	$-0.875121\pi$
$751$	−21.7350	−	37.6462i	−0.793123	−	1.37373i	0.130902	−	0.991395i	$-0.458213\pi$
$752$		0			0
$753$	−20.8537			−0.759950
$754$		0			0
$755$	−7.22743	+	12.5183i	−0.263033	+	0.455587i
$756$		0			0
$757$	3.08494	−	5.34328i	0.112124	−	0.194205i	−0.804502	−	0.593949i	$-0.797568\pi$
$757$	3.08494	−	5.34328i	0.112124	−	0.194205i	0.916626	+	0.399745i	$0.130901\pi$
$758$		0			0
$759$	17.9449			0.651359
$760$		0			0
$761$	−9.81318			−0.355728			−0.177864	−	0.984055i	$-0.556919\pi$
$761$	−9.81318			−0.355728			−0.177864	+	0.984055i	$0.556919\pi$
$762$		0			0
$763$	−17.2116	+	29.8114i	−0.623103	+	1.07925i
$764$		0			0
$765$	3.94918	−	6.84018i	0.142783	−	0.247307i
$766$		0			0
$767$	35.6517			1.28731
$768$		0			0
$769$	20.7452	+	35.9317i	0.748090	+	1.29573i	0.948737	+	0.316066i	$0.102362\pi$
$769$	20.7452	+	35.9317i	0.748090	+	1.29573i	−0.200647	+	0.979664i	$0.564305\pi$
$770$		0			0
$771$	35.7727			1.28832
$772$		0			0
$773$	−20.8584	−	36.1279i	−0.750226	−	1.29943i	−0.947713	−	0.319124i	$-0.896611\pi$
$773$	−20.8584	−	36.1279i	−0.750226	−	1.29943i	0.197487	−	0.980306i	$-0.436722\pi$
$774$		0			0
$775$	−5.09478	−	8.82442i	−0.183010	−	0.316983i
$776$		0			0
$777$	−12.6705	+	21.9460i	−0.454553	+	0.787309i
$778$		0			0
$779$	−0.787908	−	3.07133i	−0.0282297	−	0.110042i
$780$		0			0
$781$	−11.4622	+	19.8531i	−0.410149	+	0.710400i
$782$		0			0
$783$	1.75795	+	3.04486i	0.0628240	+	0.108814i
$784$		0			0
$785$	9.10642	+	15.7728i	0.325022	+	0.562955i
$786$		0			0
$787$	13.0342			0.464621			0.232310	−	0.972642i	$-0.425371\pi$
$787$	13.0342			0.464621			0.232310	+	0.972642i	$0.425371\pi$
$788$		0			0
$789$	−23.3718	−	40.4812i	−0.832060	−	1.44117i
$790$		0			0
$791$	16.6580			0.592292

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)

\(f(q)\)	\(=\)	\(q+(-1.26041 + 2.18309i) q^{3} +(-0.500000 + 0.866025i) q^{5} -2.72743 q^{7} +(-1.67727 - 2.90511i) q^{9} +O(q^{10})\)	\(q+(-1.26041 + 2.18309i) q^{3} +(-0.500000 + 0.866025i) q^{5} -2.72743 q^{7} +(-1.67727 - 2.90511i) q^{9} -3.31421 q^{11} +(-1.62412 - 2.81306i) q^{13} +(-1.26041 - 2.18309i) q^{15} +(1.17727 - 2.03909i) q^{17} +(3.11494 - 3.04912i) q^{19} +(3.43768 - 5.95423i) q^{21} +(1.07396 + 1.86016i) q^{23} +(-0.500000 - 0.866025i) q^{25} +0.893714 q^{27} +(1.96702 + 3.40697i) q^{29} +10.1896 q^{31} +(4.17727 - 7.23524i) q^{33} +(1.36371 - 2.36202i) q^{35} -3.68579 q^{37} +8.18825 q^{39} +(0.363714 - 0.629971i) q^{41} +(-1.18645 + 2.05499i) q^{43} +3.35453 q^{45} +(-5.51164 - 9.54644i) q^{47} +0.438860 q^{49} +(2.96768 + 5.14017i) q^{51} +(4.49148 + 7.77947i) q^{53} +(1.65711 - 2.87019i) q^{55} +(2.73041 + 10.6434i) q^{57} +(-5.48784 + 9.50521i) q^{59} +(4.22743 + 7.32212i) q^{61} +(4.57462 + 7.92348i) q^{63} +3.24825 q^{65} +(-4.87535 - 8.44436i) q^{67} -5.41453 q^{69} +(3.45850 - 5.99029i) q^{71} +(1.24025 - 2.14818i) q^{73} +2.52082 q^{75} +9.03927 q^{77} +(5.99948 - 10.3914i) q^{79} +(3.90535 - 6.76427i) q^{81} -4.68711 q^{83} +(1.17727 + 2.03909i) q^{85} -9.91699 q^{87} +(-4.27205 - 7.39941i) q^{89} +(4.42968 + 7.67243i) q^{91} +(-12.8430 + 22.2448i) q^{93} +(1.08314 + 4.22218i) q^{95} +(-3.61494 + 6.26127i) q^{97} +(5.55882 + 9.62816i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{3} - 4 q^{5} - 5 q^{9}+O(q^{10}) \) 8 * q + q^3 - 4 * q^5 - 5 * q^9	\( 8 q + q^{3} - 4 q^{5} - 5 q^{9} - 4 q^{11} + 9 q^{13} + q^{15} + q^{17} - 3 q^{19} + 8 q^{21} - 4 q^{25} - 20 q^{27} + 5 q^{29} + 20 q^{31} + 25 q^{33} - 52 q^{37} + 54 q^{39} - 8 q^{41} - 7 q^{43} + 10 q^{45} - 16 q^{47} + 20 q^{49} - 12 q^{51} + 5 q^{53} + 2 q^{55} + 27 q^{57} - 11 q^{59} + 12 q^{61} + 3 q^{63} - 18 q^{65} + 6 q^{69} - 14 q^{71} - 4 q^{73} - 2 q^{75} - 44 q^{77} - 13 q^{79} - 24 q^{81} - 10 q^{83} + q^{85} + 4 q^{87} + 5 q^{89} + 46 q^{91} - 28 q^{93} + 6 q^{95} - q^{97} - 24 q^{99}+O(q^{100}) \) 8 * q + q^3 - 4 * q^5 - 5 * q^9 - 4 * q^11 + 9 * q^13 + q^15 + q^17 - 3 * q^19 + 8 * q^21 - 4 * q^25 - 20 * q^27 + 5 * q^29 + 20 * q^31 + 25 * q^33 - 52 * q^37 + 54 * q^39 - 8 * q^41 - 7 * q^43 + 10 * q^45 - 16 * q^47 + 20 * q^49 - 12 * q^51 + 5 * q^53 + 2 * q^55 + 27 * q^57 - 11 * q^59 + 12 * q^61 + 3 * q^63 - 18 * q^65 + 6 * q^69 - 14 * q^71 - 4 * q^73 - 2 * q^75 - 44 * q^77 - 13 * q^79 - 24 * q^81 - 10 * q^83 + q^85 + 4 * q^87 + 5 * q^89 + 46 * q^91 - 28 * q^93 + 6 * q^95 - q^97 - 24 * q^99

Introduction

L-functions

Modular forms

Varieties

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