LMFDB - Embedded newform 480.2.d.a.49.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [480,2,Mod(49,480)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(480, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("480.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 480 = 2^{5} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	480.d (of order $2$, degree $1$, 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:	$3.83281929702$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.839056.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 6x^{4} + 8x^{2} + 1 $ x^6 + 6x^4 + 8x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.3
Root			$-1.32132i$ of defining polynomial
Character	$\chi$	$=$	480.49
Dual form			480.2.d.a.49.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +(-0.254102 - 2.22158i) q^{5} +2.64265i q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +(-0.254102 - 2.22158i) q^{5} +2.64265i q^{7} +1.00000 q^{9} -1.51363i q^{11} +3.87086 q^{13} +(0.254102 + 2.22158i) q^{15} -3.31415i q^{17} -7.08582i q^{19} -2.64265i q^{21} -4.82778i q^{23} +(-4.87086 + 1.12902i) q^{25} -1.00000 q^{27} -2.18513i q^{29} +7.36266 q^{31} +1.51363i q^{33} +(5.87086 - 0.671502i) q^{35} -7.87086 q^{37} -3.87086 q^{39} +8.72532 q^{41} -1.01641 q^{43} +(-0.254102 - 2.22158i) q^{45} -7.08582i q^{47} +0.0164068 q^{49} +3.31415i q^{51} +4.50820 q^{53} +(-3.36266 + 0.384617i) q^{55} +7.08582i q^{57} +6.79893i q^{59} +3.60104i q^{61} +2.64265i q^{63} +(-0.983593 - 8.59945i) q^{65} +1.01641 q^{67} +4.82778i q^{69} +6.72532 q^{71} +15.5146i q^{73} +(4.87086 - 1.12902i) q^{75} +4.00000 q^{77} -7.36266 q^{79} +1.00000 q^{81} -7.74173 q^{83} +(-7.36266 + 0.842131i) q^{85} +2.18513i q^{87} -14.7581 q^{89} +10.2293i q^{91} -7.36266 q^{93} +(-15.7417 + 1.80052i) q^{95} -11.1444i q^{97} -1.51363i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{3} + 6 q^{9}+O(q^{10}) $ 6 * q - 6 * q^3 + 6 * q^9	$ 6 q - 6 q^{3} + 6 q^{9} - 8 q^{13} + 2 q^{25} - 6 q^{27} + 16 q^{31} + 4 q^{35} - 16 q^{37} + 8 q^{39} - 4 q^{41} - 6 q^{49} + 24 q^{53} + 8 q^{55} - 12 q^{65} - 16 q^{71} - 2 q^{75} + 24 q^{77} - 16 q^{79} + 6 q^{81} + 16 q^{83} - 16 q^{85} - 20 q^{89} - 16 q^{93} - 32 q^{95}+O(q^{100}) $ 6 * q - 6 * q^3 + 6 * q^9 - 8 * q^13 + 2 * q^25 - 6 * q^27 + 16 * q^31 + 4 * q^35 - 16 * q^37 + 8 * q^39 - 4 * q^41 - 6 * q^49 + 24 * q^53 + 8 * q^55 - 12 * q^65 - 16 * q^71 - 2 * q^75 + 24 * q^77 - 16 * q^79 + 6 * q^81 + 16 * q^83 - 16 * q^85 - 20 * q^89 - 16 * q^93 - 32 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/480\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$161$	$421$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$		0			0
$5$	−0.254102	−	2.22158i	−0.113638	−	0.993522i
$5$	−0.254102	−	2.22158i	−0.113638	−	0.993522i
$6$		0			0
$7$			2.64265i			0.998827i	0.866364	+	0.499414i	$0.166451\pi$
$7$			2.64265i			0.998827i	−0.866364	+	0.499414i	$0.833549\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		−	1.51363i		−	0.456377i	−0.973617	−	0.228189i	$-0.926720\pi$
$11$		−	1.51363i		−	0.456377i	0.973617	−	0.228189i	$-0.0732803\pi$
$12$		0			0
$13$	3.87086			1.07358			0.536792	−	0.843714i	$-0.319636\pi$
$13$	3.87086			1.07358			0.536792	+	0.843714i	$0.319636\pi$
$14$		0			0
$15$	0.254102	+	2.22158i	0.0656088	+	0.573610i
$16$		0			0
$17$		−	3.31415i		−	0.803800i	−0.915684	−	0.401900i	$-0.868350\pi$
$17$		−	3.31415i		−	0.803800i	0.915684	−	0.401900i	$-0.131650\pi$
$18$		0			0
$19$		−	7.08582i		−	1.62560i	−0.582545	−	0.812799i	$-0.697943\pi$
$19$		−	7.08582i		−	1.62560i	0.582545	−	0.812799i	$-0.302057\pi$
$20$		0			0
$21$		−	2.64265i		−	0.576673i
$22$		0			0
$23$		−	4.82778i		−	1.00666i	−0.864094	−	0.503331i	$-0.832108\pi$
$23$		−	4.82778i		−	1.00666i	0.864094	−	0.503331i	$-0.167892\pi$
$24$		0			0
$25$	−4.87086	+	1.12902i	−0.974173	+	0.225803i
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$		−	2.18513i		−	0.405769i	−0.979203	−	0.202885i	$-0.934968\pi$
$29$		−	2.18513i		−	0.405769i	0.979203	−	0.202885i	$-0.0650316\pi$
$30$		0			0
$31$	7.36266			1.32237			0.661187	−	0.750222i	$-0.270053\pi$
$31$	7.36266			1.32237			0.661187	+	0.750222i	$0.270053\pi$
$32$		0			0
$33$			1.51363i			0.263490i
$34$		0			0
$35$	5.87086	−	0.671502i	0.992357	−	0.113504i
$36$		0			0
$37$	−7.87086			−1.29396			−0.646981	−	0.762506i	$-0.723969\pi$
$37$	−7.87086			−1.29396			−0.646981	+	0.762506i	$0.723969\pi$
$38$		0			0
$39$	−3.87086			−0.619834
$40$		0			0
$41$	8.72532			1.36267			0.681333	−	0.731973i	$-0.261400\pi$
$41$	8.72532			1.36267			0.681333	+	0.731973i	$0.261400\pi$
$42$		0			0
$43$	−1.01641			−0.155001			−0.0775003	−	0.996992i	$-0.524694\pi$
$43$	−1.01641			−0.155001			−0.0775003	+	0.996992i	$0.524694\pi$
$44$		0			0
$45$	−0.254102	−	2.22158i	−0.0378792	−	0.331174i
$46$		0			0
$47$		−	7.08582i		−	1.03357i	−0.856114	−	0.516786i	$-0.827128\pi$
$47$		−	7.08582i		−	1.03357i	0.856114	−	0.516786i	$-0.172872\pi$
$48$		0			0
$49$	0.0164068			0.00234382
$50$		0			0
$51$			3.31415i			0.464074i
$52$		0			0
$53$	4.50820			0.619249			0.309625	−	0.950859i	$-0.399797\pi$
$53$	4.50820			0.619249			0.309625	+	0.950859i	$0.399797\pi$
$54$		0			0
$55$	−3.36266	+	0.384617i	−0.453421	+	0.0518617i
$56$		0			0
$57$			7.08582i			0.938539i
$58$		0			0
$59$			6.79893i			0.885145i	0.896733	+	0.442573i	$0.145934\pi$
$59$			6.79893i			0.885145i	−0.896733	+	0.442573i	$0.854066\pi$
$60$		0			0
$61$			3.60104i			0.461065i	0.973065	+	0.230533i	$0.0740469\pi$
$61$			3.60104i			0.461065i	−0.973065	+	0.230533i	$0.925953\pi$
$62$		0			0
$63$			2.64265i			0.332942i
$64$		0			0
$65$	−0.983593	−	8.59945i	−0.122000	−	1.06663i
$66$		0			0
$67$	1.01641			0.124174			0.0620869	−	0.998071i	$-0.480224\pi$
$67$	1.01641			0.124174			0.0620869	+	0.998071i	$0.480224\pi$
$68$		0			0
$69$			4.82778i			0.581197i
$70$		0			0
$71$	6.72532			0.798149			0.399074	−	0.916919i	$-0.369331\pi$
$71$	6.72532			0.798149			0.399074	+	0.916919i	$0.369331\pi$
$72$		0			0
$73$			15.5146i			1.81585i	0.419132	+	0.907925i	$0.362334\pi$
$73$			15.5146i			1.81585i	−0.419132	+	0.907925i	$0.637666\pi$
$74$		0			0
$75$	4.87086	−	1.12902i	0.562439	−	0.130368i
$76$		0			0
$77$	4.00000			0.455842
$78$		0			0
$79$	−7.36266			−0.828364			−0.414182	−	0.910194i	$-0.635932\pi$
$79$	−7.36266			−0.828364			−0.414182	+	0.910194i	$0.635932\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	−7.74173			−0.849765			−0.424883	−	0.905248i	$-0.639685\pi$
$83$	−7.74173			−0.849765			−0.424883	+	0.905248i	$0.639685\pi$
$84$		0			0
$85$	−7.36266	+	0.842131i	−0.798593	+	0.0913420i
$86$		0			0
$87$			2.18513i			0.234271i
$88$		0			0
$89$	−14.7581			−1.56436			−0.782180	−	0.623053i	$-0.785892\pi$
$89$	−14.7581			−1.56436			−0.782180	+	0.623053i	$0.785892\pi$
$90$		0			0
$91$			10.2293i			1.07233i
$92$		0			0
$93$	−7.36266			−0.763472
$94$		0			0
$95$	−15.7417	+	1.80052i	−1.61507	+	0.184729i
$96$		0			0
$97$		−	11.1444i		−	1.13154i	−0.824563	−	0.565769i	$-0.808579\pi$
$97$		−	11.1444i		−	1.13154i	0.824563	−	0.565769i	$-0.191421\pi$
$98$		0			0
$99$		−	1.51363i		−	0.152126i
$100$		0			0
$101$			13.3295i			1.32633i	0.748471	+	0.663167i	$0.230788\pi$
$101$			13.3295i			1.32633i	−0.748471	+	0.663167i	$0.769212\pi$
$102$		0			0
$103$			0.958386i			0.0944326i	0.998885	+	0.0472163i	$0.0150350\pi$
$103$			0.958386i			0.0944326i	−0.998885	+	0.0472163i	$0.984965\pi$
$104$		0			0
$105$	−5.87086	+	0.671502i	−0.572938	+	0.0655318i
$106$		0			0
$107$	4.00000			0.386695			0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000			0.386695			0.193347	+	0.981130i	$0.438066\pi$
$108$		0			0
$109$			0.769233i			0.0736792i	0.999321	+	0.0368396i	$0.0117291\pi$
$109$			0.769233i			0.0736792i	−0.999321	+	0.0368396i	$0.988271\pi$
$110$		0			0
$111$	7.87086			0.747069
$112$		0			0
$113$		−	14.4585i		−	1.36014i	−0.733146	−	0.680071i	$-0.761949\pi$
$113$		−	14.4585i		−	1.36014i	0.733146	−	0.680071i	$-0.238051\pi$
$114$		0			0
$115$	−10.7253	+	1.22675i	−1.00014	+	0.114395i
$116$		0			0
$117$	3.87086			0.357862
$118$		0			0
$119$	8.75814			0.802857
$120$		0			0
$121$	8.70892			0.791720
$122$		0			0
$123$	−8.72532			−0.786736
$124$		0			0
$125$	3.74590	+	10.5341i	0.335043	+	0.942203i
$126$		0			0
$127$			11.5290i			1.02303i	0.859274	+	0.511516i	$0.170916\pi$
$127$			11.5290i			1.02303i	−0.859274	+	0.511516i	$0.829084\pi$
$128$		0			0
$129$	1.01641			0.0894896
$130$		0			0
$131$			7.37270i			0.644156i	0.946713	+	0.322078i	$0.104381\pi$
$131$			7.37270i			0.644156i	−0.946713	+	0.322078i	$0.895619\pi$
$132$		0			0
$133$	18.7253			1.62369
$134$		0			0
$135$	0.254102	+	2.22158i	0.0218696	+	0.191203i
$136$		0			0
$137$			3.88792i			0.332167i	0.986112	+	0.166084i	$0.0531122\pi$
$137$			3.88792i			0.332167i	−0.986112	+	0.166084i	$0.946888\pi$
$138$		0			0
$139$			14.6291i			1.24083i	0.784275	+	0.620414i	$0.213035\pi$
$139$			14.6291i			1.24083i	−0.784275	+	0.620414i	$0.786965\pi$
$140$		0			0
$141$			7.08582i			0.596733i
$142$		0			0
$143$		−	5.85907i		−	0.489960i
$144$		0			0
$145$	−4.85446	+	0.555246i	−0.403141	+	0.0461107i
$146$		0			0
$147$	−0.0164068			−0.00135321
$148$		0			0
$149$		−	11.0715i		−	0.907010i	−0.891254	−	0.453505i	$-0.850173\pi$
$149$		−	11.0715i		−	0.907010i	0.891254	−	0.453505i	$-0.149827\pi$
$150$		0			0
$151$	−0.637339			−0.0518659			−0.0259329	−	0.999664i	$-0.508256\pi$
$151$	−0.637339			−0.0518659			−0.0259329	+	0.999664i	$0.508256\pi$
$152$		0			0
$153$		−	3.31415i		−	0.267933i
$154$		0			0
$155$	−1.87086	−	16.3568i	−0.150271	−	1.31381i
$156$		0			0
$157$	0.129135			0.0103061			0.00515306	−	0.999987i	$-0.498360\pi$
$157$	0.129135			0.0103061			0.00515306	+	0.999987i	$0.498360\pi$
$158$		0			0
$159$	−4.50820			−0.357524
$160$		0			0
$161$	12.7581			1.00548
$162$		0			0
$163$	19.4835			1.52606			0.763031	−	0.646362i	$-0.223710\pi$
$163$	19.4835			1.52606			0.763031	+	0.646362i	$0.223710\pi$
$164$		0			0
$165$	3.36266	−	0.384617i	0.261783	−	0.0299424i
$166$		0			0
$167$			1.80052i			0.139328i	0.997571	+	0.0696641i	$0.0221928\pi$
$167$			1.80052i			0.139328i	−0.997571	+	0.0696641i	$0.977807\pi$
$168$		0			0
$169$	1.98359			0.152584
$170$		0			0
$171$		−	7.08582i		−	0.541866i
$172$		0			0
$173$	−23.2335			−1.76641			−0.883206	−	0.468985i	$-0.844620\pi$
$173$	−23.2335			−1.76641			−0.883206	+	0.468985i	$0.844620\pi$
$174$		0			0
$175$	−2.98359	−	12.8720i	−0.225538	−	0.973031i
$176$		0			0
$177$		−	6.79893i		−	0.511039i
$178$		0			0
$179$			2.85664i			0.213515i	0.994285	+	0.106757i	$0.0340468\pi$
$179$			2.85664i			0.213515i	−0.994285	+	0.106757i	$0.965953\pi$
$180$		0			0
$181$		−	5.28530i		−	0.392853i	−0.980519	−	0.196427i	$-0.937066\pi$
$181$		−	5.28530i		−	0.392853i	0.980519	−	0.196427i	$-0.0629337\pi$
$182$		0			0
$183$		−	3.60104i		−	0.266196i
$184$		0			0
$185$	2.00000	+	17.4858i	0.147043	+	1.28558i
$186$		0			0
$187$	−5.01641			−0.366836
$188$		0			0
$189$		−	2.64265i		−	0.192224i
$190$		0			0
$191$	5.96719			0.431770			0.215885	−	0.976419i	$-0.430736\pi$
$191$	5.96719			0.431770			0.215885	+	0.976419i	$0.430736\pi$
$192$		0			0
$193$		−	14.9409i		−	1.07547i	−0.843115	−	0.537733i	$-0.819281\pi$
$193$		−	14.9409i		−	1.07547i	0.843115	−	0.537733i	$-0.180719\pi$
$194$		0			0
$195$	0.983593	+	8.59945i	0.0704366	+	0.615819i
$196$		0			0
$197$	3.23353			0.230379			0.115190	−	0.993344i	$-0.463252\pi$
$197$	3.23353			0.230379			0.115190	+	0.993344i	$0.463252\pi$
$198$		0			0
$199$	−8.12080			−0.575668			−0.287834	−	0.957680i	$-0.592935\pi$
$199$	−8.12080			−0.575668			−0.287834	+	0.957680i	$0.592935\pi$
$200$		0			0
$201$	−1.01641			−0.0716918
$202$		0			0
$203$	5.77454			0.405293
$204$		0			0
$205$	−2.21712	−	19.3840i	−0.154850	−	1.35384i
$206$		0			0
$207$		−	4.82778i		−	0.335554i
$208$		0			0
$209$	−10.7253			−0.741886
$210$		0			0
$211$		−	13.7141i		−	0.944119i	−0.881567	−	0.472059i	$-0.843511\pi$
$211$		−	13.7141i		−	0.944119i	0.881567	−	0.472059i	$-0.156489\pi$
$212$		0			0
$213$	−6.72532			−0.460812
$214$		0			0
$215$	0.258271	+	2.25803i	0.0176139	+	0.153997i
$216$		0			0
$217$			19.4569i			1.32082i
$218$		0			0
$219$		−	15.5146i		−	1.04838i
$220$		0			0
$221$		−	12.8286i		−	0.862947i
$222$		0			0
$223$			9.84472i			0.659251i	0.944112	+	0.329626i	$0.106923\pi$
$223$			9.84472i			0.659251i	−0.944112	+	0.329626i	$0.893077\pi$
$224$		0			0
$225$	−4.87086	+	1.12902i	−0.324724	+	0.0752677i
$226$		0			0
$227$	5.70892			0.378914			0.189457	−	0.981889i	$-0.439327\pi$
$227$	5.70892			0.378914			0.189457	+	0.981889i	$0.439327\pi$
$228$		0			0
$229$		−	0.769233i		−	0.0508324i	−0.999677	−	0.0254162i	$-0.991909\pi$
$229$		−	0.769233i		−	0.0508324i	0.999677	−	0.0254162i	$-0.00809109\pi$
$230$		0			0
$231$	−4.00000			−0.263181
$232$		0			0
$233$			18.4008i			1.20548i	0.797939	+	0.602739i	$0.205924\pi$
$233$			18.4008i			1.20548i	−0.797939	+	0.602739i	$0.794076\pi$
$234$		0			0
$235$	−15.7417	+	1.80052i	−1.02688	+	0.117453i
$236$		0			0
$237$	7.36266			0.478256
$238$		0			0
$239$	10.0328			0.648969			0.324484	−	0.945891i	$-0.394809\pi$
$239$	10.0328			0.648969			0.324484	+	0.945891i	$0.394809\pi$
$240$		0			0
$241$	10.7581			0.692992			0.346496	−	0.938051i	$-0.387371\pi$
$241$	10.7581			0.692992			0.346496	+	0.938051i	$0.387371\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$	−0.00416898	−	0.0364490i	−0.000266347	−	0.00232864i
$246$		0			0
$247$		−	27.4282i		−	1.74522i
$248$		0			0
$249$	7.74173			0.490612
$250$		0			0
$251$		−	12.6580i		−	0.798966i	−0.916741	−	0.399483i	$-0.869190\pi$
$251$		−	12.6580i		−	0.798966i	0.916741	−	0.399483i	$-0.130810\pi$
$252$		0			0
$253$	−7.30749			−0.459418
$254$		0			0
$255$	7.36266	−	0.842131i	0.461068	−	0.0527363i
$256$		0			0
$257$			13.3110i			0.830316i	0.909749	+	0.415158i	$0.136274\pi$
$257$			13.3110i			0.830316i	−0.909749	+	0.415158i	$0.863726\pi$
$258$		0			0
$259$		−	20.7999i		−	1.29244i
$260$		0			0
$261$		−	2.18513i		−	0.135256i
$262$		0			0
$263$			18.4256i			1.13617i	0.822969	+	0.568087i	$0.192316\pi$
$263$			18.4256i			1.13617i	−0.822969	+	0.568087i	$0.807684\pi$
$264$		0			0
$265$	−1.14554	−	10.0153i	−0.0703701	−	0.615238i
$266$		0			0
$267$	14.7581			0.903183
$268$		0			0
$269$		−	3.86940i		−	0.235921i	−0.993018	−	0.117961i	$-0.962364\pi$
$269$		−	3.86940i		−	0.235921i	0.993018	−	0.117961i	$-0.0376357\pi$
$270$		0			0
$271$	17.3955			1.05670			0.528350	−	0.849027i	$-0.322811\pi$
$271$	17.3955			1.05670			0.528350	+	0.849027i	$0.322811\pi$
$272$		0			0
$273$		−	10.2293i		−	0.619108i
$274$		0			0
$275$	1.70892	+	7.37270i	0.103052	+	0.444591i
$276$		0			0
$277$	0.887271			0.0533110			0.0266555	−	0.999645i	$-0.491514\pi$
$277$	0.887271			0.0533110			0.0266555	+	0.999645i	$0.491514\pi$
$278$		0			0
$279$	7.36266			0.440791
$280$		0			0
$281$	−13.4835			−0.804356			−0.402178	−	0.915562i	$-0.631747\pi$
$281$	−13.4835			−0.804356			−0.402178	+	0.915562i	$0.631747\pi$
$282$		0			0
$283$	28.4342			1.69024			0.845120	−	0.534577i	$-0.179529\pi$
$283$	28.4342			1.69024			0.845120	+	0.534577i	$0.179529\pi$
$284$		0			0
$285$	15.7417	−	1.80052i	0.932460	−	0.106653i
$286$		0			0
$287$			23.0580i			1.36107i
$288$		0			0
$289$	6.01641			0.353906
$290$		0			0
$291$			11.1444i			0.653294i
$292$		0			0
$293$	−7.99166			−0.466878			−0.233439	−	0.972371i	$-0.574998\pi$
$293$	−7.99166			−0.466878			−0.233439	+	0.972371i	$0.574998\pi$
$294$		0			0
$295$	15.1044	−	1.72762i	0.879412	−	0.100586i
$296$		0			0
$297$			1.51363i			0.0878299i
$298$		0			0
$299$		−	18.6877i		−	1.08074i
$300$		0			0
$301$		−	2.68601i		−	0.154819i
$302$		0			0
$303$		−	13.3295i		−	0.765760i
$304$		0			0
$305$	8.00000	−	0.915029i	0.458079	−	0.0523944i
$306$		0			0
$307$	−17.4506			−0.995961			−0.497980	−	0.867188i	$-0.665925\pi$
$307$	−17.4506			−0.995961			−0.497980	+	0.867188i	$0.665925\pi$
$308$		0			0
$309$		−	0.958386i		−	0.0545207i
$310$		0			0
$311$	−21.4506			−1.21635			−0.608177	−	0.793801i	$-0.708099\pi$
$311$	−21.4506			−1.21635			−0.608177	+	0.793801i	$0.708099\pi$
$312$		0			0
$313$		−	7.73879i		−	0.437422i	−0.975790	−	0.218711i	$-0.929815\pi$
$313$		−	7.73879i		−	0.437422i	0.975790	−	0.218711i	$-0.0701853\pi$
$314$		0			0
$315$	5.87086	−	0.671502i	0.330786	−	0.0378348i
$316$		0			0
$317$	11.2335			0.630938			0.315469	−	0.948936i	$-0.397838\pi$
$317$	11.2335			0.630938			0.315469	+	0.948936i	$0.397838\pi$
$318$		0			0
$319$	−3.30749			−0.185184
$320$		0			0
$321$	−4.00000			−0.223258
$322$		0			0
$323$	−23.4835			−1.30665
$324$		0			0
$325$	−18.8545	+	4.37027i	−1.04586	+	0.242419i
$326$		0			0
$327$		−	0.769233i		−	0.0425387i
$328$		0			0
$329$	18.7253			1.03236
$330$		0			0
$331$			8.00084i			0.439766i	0.975526	+	0.219883i	$0.0705676\pi$
$331$			8.00084i			0.439766i	−0.975526	+	0.219883i	$0.929432\pi$
$332$		0			0
$333$	−7.87086			−0.431321
$334$		0			0
$335$	−0.258271	−	2.25803i	−0.0141108	−	0.123369i
$336$		0			0
$337$		−	21.5692i		−	1.17495i	−0.809243	−	0.587474i	$-0.800123\pi$
$337$		−	21.5692i		−	1.17495i	0.809243	−	0.587474i	$-0.199877\pi$
$338$		0			0
$339$			14.4585i			0.785279i
$340$		0			0
$341$		−	11.1444i		−	0.603501i
$342$		0			0
$343$			18.5419i			1.00117i
$344$		0			0
$345$	10.7253	−	1.22675i	0.577432	−	0.0660459i
$346$		0			0
$347$	−21.7089			−1.16540			−0.582698	−	0.812689i	$-0.698003\pi$
$347$	−21.7089			−1.16540			−0.582698	+	0.812689i	$0.698003\pi$
$348$		0			0
$349$			24.7422i			1.32442i	0.749318	+	0.662211i	$0.230382\pi$
$349$			24.7422i			1.32442i	−0.749318	+	0.662211i	$0.769618\pi$
$350$		0			0
$351$	−3.87086			−0.206611
$352$		0			0
$353$		−	3.31415i		−	0.176394i	−0.996103	−	0.0881972i	$-0.971889\pi$
$353$		−	3.31415i		−	0.176394i	0.996103	−	0.0881972i	$-0.0281106\pi$
$354$		0			0
$355$	−1.70892	−	14.9409i	−0.0906998	−	0.792979i
$356$		0			0
$357$	−8.75814			−0.463530
$358$		0			0
$359$	−16.7581			−0.884461			−0.442230	−	0.896902i	$-0.645813\pi$
$359$	−16.7581			−0.884461			−0.442230	+	0.896902i	$0.645813\pi$
$360$		0			0
$361$	−31.2088			−1.64257
$362$		0			0
$363$	−8.70892			−0.457100
$364$		0			0
$365$	34.4671	−	3.94229i	1.80409	−	0.206349i
$366$		0			0
$367$		−	28.5324i		−	1.48938i	−0.667411	−	0.744690i	$-0.732597\pi$
$367$		−	28.5324i		−	1.48938i	0.667411	−	0.744690i	$-0.267403\pi$
$368$		0			0
$369$	8.72532			0.454222
$370$		0			0
$371$			11.9136i			0.618523i
$372$		0			0
$373$	37.5798			1.94581			0.972904	−	0.231211i	$-0.0742688\pi$
$373$	37.5798			1.94581			0.972904	+	0.231211i	$0.0742688\pi$
$374$		0			0
$375$	−3.74590	−	10.5341i	−0.193437	−	0.543981i
$376$		0			0
$377$		−	8.45836i		−	0.435628i
$378$		0			0
$379$			6.74456i			0.346445i	0.984883	+	0.173222i	$0.0554179\pi$
$379$			6.74456i			0.346445i	−0.984883	+	0.173222i	$0.944582\pi$
$380$		0			0
$381$		−	11.5290i		−	0.590648i
$382$		0			0
$383$			21.8312i			1.11552i	0.830001	+	0.557762i	$0.188340\pi$
$383$			21.8312i			1.11552i	−0.830001	+	0.557762i	$0.811660\pi$
$384$		0			0
$385$	−1.01641	−	8.88633i	−0.0518009	−	0.452889i
$386$		0			0
$387$	−1.01641			−0.0516669
$388$		0			0
$389$			8.81344i			0.446859i	0.974720	+	0.223429i	$0.0717252\pi$
$389$			8.81344i			0.446859i	−0.974720	+	0.223429i	$0.928275\pi$
$390$		0			0
$391$	−16.0000			−0.809155
$392$		0			0
$393$		−	7.37270i		−	0.371904i
$394$		0			0
$395$	1.87086	+	16.3568i	0.0941334	+	0.822998i
$396$		0			0
$397$	−0.821644			−0.0412372			−0.0206186	−	0.999787i	$-0.506564\pi$
$397$	−0.821644			−0.0412372			−0.0206186	+	0.999787i	$0.506564\pi$
$398$		0			0
$399$	−18.7253			−0.937439
$400$		0			0
$401$	−12.7253			−0.635472			−0.317736	−	0.948179i	$-0.602923\pi$
$401$	−12.7253			−0.635472			−0.317736	+	0.948179i	$0.602923\pi$
$402$		0			0
$403$	28.4999			1.41968
$404$		0			0
$405$	−0.254102	−	2.22158i	−0.0126264	−	0.110391i
$406$		0			0
$407$			11.9136i			0.590535i
$408$		0			0
$409$	−2.25827			−0.111664			−0.0558321	−	0.998440i	$-0.517781\pi$
$409$	−2.25827			−0.111664			−0.0558321	+	0.998440i	$0.517781\pi$
$410$		0			0
$411$		−	3.88792i		−	0.191777i
$412$		0			0
$413$	−17.9672			−0.884107
$414$		0			0
$415$	1.96719	+	17.1989i	0.0965654	+	0.844261i
$416$		0			0
$417$		−	14.6291i		−	0.716392i
$418$		0			0
$419$			33.4579i			1.63453i	0.576264	+	0.817263i	$0.304510\pi$
$419$			33.4579i			1.63453i	−0.576264	+	0.817263i	$0.695490\pi$
$420$		0			0
$421$			11.3398i			0.552669i	0.961061	+	0.276335i	$0.0891198\pi$
$421$			11.3398i			0.552669i	−0.961061	+	0.276335i	$0.910880\pi$
$422$		0			0
$423$		−	7.08582i		−	0.344524i
$424$		0			0
$425$	3.74173	+	16.1428i	0.181501	+	0.783040i
$426$		0			0
$427$	−9.51627			−0.460525
$428$		0			0
$429$			5.85907i			0.282878i
$430$		0			0
$431$	−10.6597			−0.513459			−0.256730	−	0.966483i	$-0.582645\pi$
$431$	−10.6597			−0.513459			−0.256730	+	0.966483i	$0.582645\pi$
$432$		0			0
$433$			26.5132i			1.27414i	0.770805	+	0.637072i	$0.219854\pi$
$433$			26.5132i			1.27414i	−0.770805	+	0.637072i	$0.780146\pi$
$434$		0			0
$435$	4.85446	−	0.555246i	0.232753	−	0.0266220i
$436$		0			0
$437$	−34.2088			−1.63643
$438$		0			0
$439$	32.8789			1.56923			0.784613	−	0.619986i	$-0.212862\pi$
$439$	32.8789			1.56923			0.784613	+	0.619986i	$0.212862\pi$
$440$		0			0
$441$	0.0164068			0.000781274
$442$		0			0
$443$	−5.70892			−0.271239			−0.135619	−	0.990761i	$-0.543302\pi$
$443$	−5.70892			−0.271239			−0.135619	+	0.990761i	$0.543302\pi$
$444$		0			0
$445$	3.75007	+	32.7864i	0.177770	+	1.55423i
$446$		0			0
$447$			11.0715i			0.523662i
$448$		0			0
$449$	2.00000			0.0943858			0.0471929	−	0.998886i	$-0.484972\pi$
$449$	2.00000			0.0943858			0.0471929	+	0.998886i	$0.484972\pi$
$450$		0			0
$451$		−	13.2069i		−	0.621890i
$452$		0			0
$453$	0.637339			0.0299448
$454$		0			0
$455$	22.7253	−	2.59929i	1.06538	−	0.121857i
$456$		0			0
$457$			3.94229i			0.184413i	0.995740	+	0.0922064i	$0.0293920\pi$
$457$			3.94229i			0.184413i	−0.995740	+	0.0922064i	$0.970608\pi$
$458$		0			0
$459$			3.31415i			0.154691i
$460$		0			0
$461$			33.8969i			1.57874i	0.613920	+	0.789369i	$0.289592\pi$
$461$			33.8969i			1.57874i	−0.613920	+	0.789369i	$0.710408\pi$
$462$		0			0
$463$		−	22.8688i		−	1.06280i	−0.847120	−	0.531402i	$-0.821665\pi$
$463$		−	22.8688i		−	1.06280i	0.847120	−	0.531402i	$-0.178335\pi$
$464$		0			0
$465$	1.87086	+	16.3568i	0.0867593	+	0.758527i
$466$		0			0
$467$	15.7417			0.728440			0.364220	−	0.931313i	$-0.381336\pi$
$467$	15.7417			0.728440			0.364220	+	0.931313i	$0.381336\pi$
$468$		0			0
$469$			2.68601i			0.124028i
$470$		0			0
$471$	−0.129135			−0.00595024
$472$		0			0
$473$			1.53847i			0.0707388i
$474$		0			0
$475$	8.00000	+	34.5140i	0.367065	+	1.58361i
$476$		0			0
$477$	4.50820			0.206416
$478$		0			0
$479$	20.6925			0.945465			0.472732	−	0.881206i	$-0.343268\pi$
$479$	20.6925			0.945465			0.472732	+	0.881206i	$0.343268\pi$
$480$		0			0
$481$	−30.4671			−1.38918
$482$		0			0
$483$	−12.7581			−0.580515
$484$		0			0
$485$	−24.7581	+	2.83180i	−1.12421	+	0.128586i
$486$		0			0
$487$			30.8401i			1.39750i	0.715366	+	0.698750i	$0.246260\pi$
$487$			30.8401i			1.39750i	−0.715366	+	0.698750i	$0.753740\pi$
$488$		0			0
$489$	−19.4835			−0.881072
$490$		0			0
$491$		−	10.9737i		−	0.495238i	−0.968858	−	0.247619i	$-0.920352\pi$
$491$		−	10.9737i		−	0.495238i	0.968858	−	0.247619i	$-0.0796481\pi$
$492$		0			0
$493$	−7.24186			−0.326157
$494$		0			0
$495$	−3.36266	+	0.384617i	−0.151140	+	0.0172872i
$496$		0			0
$497$			17.7727i			0.797213i
$498$		0			0
$499$		−	3.71729i		−	0.166409i	−0.996533	−	0.0832044i	$-0.973485\pi$
$499$		−	3.71729i		−	0.166409i	0.996533	−	0.0832044i	$-0.0265154\pi$
$500$		0			0
$501$		−	1.80052i		−	0.0804412i
$502$		0			0
$503$		−	39.9451i		−	1.78107i	−0.454919	−	0.890533i	$-0.650332\pi$
$503$		−	39.9451i		−	1.78107i	0.454919	−	0.890533i	$-0.349668\pi$
$504$		0			0
$505$	29.6126	−	3.38705i	1.31774	−	0.150722i
$506$		0			0
$507$	−1.98359			−0.0880945
$508$		0			0
$509$		−	0.0728979i		−	0.00323114i	−0.999999	−	0.00161557i	$-0.999486\pi$
$509$		−	0.0728979i		−	0.00323114i	0.999999	−	0.00161557i	$-0.000514253\pi$
$510$		0			0
$511$	−40.9997			−1.81372
$512$		0			0
$513$			7.08582i			0.312846i
$514$		0			0
$515$	2.12914	−	0.243528i	0.0938209	−	0.0107311i
$516$		0			0
$517$	−10.7253			−0.471699
$518$		0			0
$519$	23.2335			1.01984
$520$		0			0
$521$	−11.9672			−0.524292			−0.262146	−	0.965028i	$-0.584430\pi$
$521$	−11.9672			−0.524292			−0.262146	+	0.965028i	$0.584430\pi$
$522$		0			0
$523$	−16.0656			−0.702501			−0.351250	−	0.936282i	$-0.614243\pi$
$523$	−16.0656			−0.702501			−0.351250	+	0.936282i	$0.614243\pi$
$524$		0			0
$525$	2.98359	+	12.8720i	0.130215	+	0.561779i
$526$		0			0
$527$		−	24.4010i		−	1.06292i
$528$		0			0
$529$	−0.307491			−0.0133692
$530$		0			0
$531$			6.79893i			0.295048i
$532$		0			0
$533$	33.7745			1.46294
$534$		0			0
$535$	−1.01641	−	8.88633i	−0.0439431	−	0.384190i
$536$		0			0
$537$		−	2.85664i		−	0.123273i
$538$		0			0
$539$		−	0.0248338i		−	0.00106967i
$540$		0			0
$541$		−	15.8559i		−	0.681698i	−0.940118	−	0.340849i	$-0.889285\pi$
$541$		−	15.8559i		−	0.681698i	0.940118	−	0.340849i	$-0.110715\pi$
$542$		0			0
$543$			5.28530i			0.226814i
$544$		0			0
$545$	1.70892	−	0.195463i	0.0732019	−	0.00837274i
$546$		0			0
$547$	−4.95078			−0.211680			−0.105840	−	0.994383i	$-0.533753\pi$
$547$	−4.95078			−0.211680			−0.105840	+	0.994383i	$0.533753\pi$
$548$		0			0
$549$			3.60104i			0.153688i
$550$		0			0
$551$	−15.4835			−0.659618
$552$		0			0
$553$		−	19.4569i		−	0.827393i
$554$		0			0
$555$	−2.00000	−	17.4858i	−0.0848953	−	0.742230i
$556$		0			0
$557$	−1.26634			−0.0536565			−0.0268283	−	0.999640i	$-0.508541\pi$
$557$	−1.26634			−0.0536565			−0.0268283	+	0.999640i	$0.508541\pi$
$558$		0			0
$559$	−3.93437			−0.166406
$560$		0			0
$561$	5.01641			0.211793
$562$		0			0
$563$	5.70892			0.240602			0.120301	−	0.992737i	$-0.461614\pi$
$563$	5.70892			0.240602			0.120301	+	0.992737i	$0.461614\pi$
$564$		0			0
$565$	−32.1208	+	3.67393i	−1.35133	+	0.154564i
$566$		0			0
$567$			2.64265i			0.110981i
$568$		0			0
$569$	2.75814			0.115627			0.0578135	−	0.998327i	$-0.481587\pi$
$569$	2.75814			0.115627			0.0578135	+	0.998327i	$0.481587\pi$
$570$		0			0
$571$			25.7735i			1.07859i	0.842118	+	0.539294i	$0.181309\pi$
$571$			25.7735i			1.07859i	−0.842118	+	0.539294i	$0.818691\pi$
$572$		0			0
$573$	−5.96719			−0.249283
$574$		0			0
$575$	5.45065	+	23.5155i	0.227308	+	0.980663i
$576$		0			0
$577$			32.7135i			1.36188i	0.732338	+	0.680941i	$0.238429\pi$
$577$			32.7135i			1.36188i	−0.732338	+	0.680941i	$0.761571\pi$
$578$		0			0
$579$			14.9409i			0.620921i
$580$		0			0
$581$		−	20.4587i		−	0.848769i
$582$		0			0
$583$		−	6.82376i		−	0.282611i
$584$		0			0
$585$	−0.983593	−	8.59945i	−0.0406666	−	0.355543i
$586$		0			0
$587$	43.4835			1.79475			0.897377	−	0.441264i	$-0.145470\pi$
$587$	43.4835			1.79475			0.897377	+	0.441264i	$0.145470\pi$
$588$		0			0
$589$		−	52.1705i		−	2.14965i
$590$		0			0
$591$	−3.23353			−0.133009
$592$		0			0
$593$			7.83021i			0.321548i	0.986991	+	0.160774i	$0.0513991\pi$
$593$			7.83021i			0.321548i	−0.986991	+	0.160774i	$0.948601\pi$
$594$		0			0
$595$	−2.22546	−	19.4569i	−0.0912348	−	0.797656i
$596$		0			0
$597$	8.12080			0.332362
$598$		0			0
$599$	−32.7581			−1.33846			−0.669231	−	0.743055i	$-0.733376\pi$
$599$	−32.7581			−1.33846			−0.669231	+	0.743055i	$0.733376\pi$
$600$		0			0
$601$	17.8074			0.726377			0.363189	−	0.931716i	$-0.381688\pi$
$601$	17.8074			0.726377			0.363189	+	0.931716i	$0.381688\pi$
$602$		0			0
$603$	1.01641			0.0413913
$604$		0			0
$605$	−2.21295	−	19.3476i	−0.0899692	−	0.786591i
$606$		0			0
$607$		−	3.41188i		−	0.138484i	−0.997600	−	0.0692420i	$-0.977942\pi$
$607$		−	3.41188i		−	0.138484i	0.997600	−	0.0692420i	$-0.0220581\pi$
$608$		0			0
$609$	−5.77454			−0.233996
$610$		0			0
$611$		−	27.4282i		−	1.10963i
$612$		0			0
$613$	36.6290			1.47943			0.739716	−	0.672920i	$-0.234960\pi$
$613$	36.6290			1.47943			0.739716	+	0.672920i	$0.234960\pi$
$614$		0			0
$615$	2.21712	+	19.3840i	0.0894029	+	0.781640i
$616$		0			0
$617$		−	40.3979i		−	1.62636i	−0.582012	−	0.813180i	$-0.697734\pi$
$617$		−	40.3979i		−	1.62636i	0.582012	−	0.813180i	$-0.302266\pi$
$618$		0			0
$619$			24.5172i			0.985430i	0.870191	+	0.492715i	$0.163996\pi$
$619$			24.5172i			0.985430i	−0.870191	+	0.492715i	$0.836004\pi$
$620$		0			0
$621$			4.82778i			0.193732i
$622$		0			0
$623$		−	39.0006i		−	1.56252i
$624$		0			0
$625$	22.4506	−	10.9986i	0.898026	−	0.439943i
$626$		0			0
$627$	10.7253			0.428328
$628$		0			0
$629$			26.0852i			1.04009i
$630$		0			0
$631$	−18.7805			−0.747640			−0.373820	−	0.927501i	$-0.621952\pi$
$631$	−18.7805			−0.747640			−0.373820	+	0.927501i	$0.621952\pi$
$632$		0			0
$633$			13.7141i			0.545087i
$634$		0			0
$635$	25.6126	−	2.92953i	1.01640	−	0.116255i
$636$		0			0
$637$	0.0635083			0.00251629
$638$		0			0
$639$	6.72532			0.266050
$640$		0			0
$641$	−15.5163			−0.612856			−0.306428	−	0.951894i	$-0.599134\pi$
$641$	−15.5163			−0.612856			−0.306428	+	0.951894i	$0.599134\pi$
$642$		0			0
$643$	−17.4506			−0.688186			−0.344093	−	0.938936i	$-0.611814\pi$
$643$	−17.4506			−0.688186			−0.344093	+	0.938936i	$0.611814\pi$
$644$		0			0
$645$	−0.258271	−	2.25803i	−0.0101694	−	0.0889099i
$646$		0			0
$647$		−	13.1403i		−	0.516600i	−0.966065	−	0.258300i	$-0.916838\pi$
$647$		−	13.1403i		−	0.516600i	0.966065	−	0.258300i	$-0.0831624\pi$
$648$		0			0
$649$	10.2911			0.403960
$650$		0			0
$651$		−	19.4569i		−	0.762577i
$652$		0			0
$653$	−14.7993			−0.579141			−0.289570	−	0.957157i	$-0.593512\pi$
$653$	−14.7993			−0.579141			−0.289570	+	0.957157i	$0.593512\pi$
$654$		0			0
$655$	16.3791	−	1.87342i	0.639983	−	0.0732004i
$656$		0			0
$657$			15.5146i			0.605284i
$658$		0			0
$659$		−	7.99614i		−	0.311485i	−0.987798	−	0.155743i	$-0.950223\pi$
$659$		−	7.99614i		−	0.311485i	0.987798	−	0.155743i	$-0.0497771\pi$
$660$		0			0
$661$			0.915029i			0.0355905i	0.999842	+	0.0177953i	$0.00566470\pi$
$661$			0.915029i			0.0355905i	−0.999842	+	0.0177953i	$0.994335\pi$
$662$		0			0
$663$			12.8286i			0.498223i
$664$		0			0
$665$	−4.75814	−	41.5999i	−0.184513	−	1.61317i
$666$		0			0
$667$	−10.5494			−0.408473
$668$		0			0
$669$		−	9.84472i		−	0.380619i
$670$		0			0
$671$	5.45065			0.210420
$672$		0			0
$673$			34.3978i			1.32594i	0.748647	+	0.662969i	$0.230704\pi$
$673$			34.3978i			1.32594i	−0.748647	+	0.662969i	$0.769296\pi$
$674$		0			0
$675$	4.87086	−	1.12902i	0.187480	−	0.0434559i
$676$		0			0
$677$	40.1676			1.54377			0.771884	−	0.635764i	$-0.219315\pi$
$677$	40.1676			1.54377			0.771884	+	0.635764i	$0.219315\pi$
$678$		0			0
$679$	29.4506			1.13021
$680$		0			0
$681$	−5.70892			−0.218766
$682$		0			0
$683$	−33.2580			−1.27258			−0.636291	−	0.771449i	$-0.719532\pi$
$683$	−33.2580			−1.27258			−0.636291	+	0.771449i	$0.719532\pi$
$684$		0			0
$685$	8.63734	−	0.987927i	0.330016	−	0.0377468i
$686$		0			0
$687$			0.769233i			0.0293481i
$688$		0			0
$689$	17.4506			0.664817
$690$		0			0
$691$		−	50.2241i		−	1.91062i	−0.295611	−	0.955308i	$-0.595523\pi$
$691$		−	50.2241i		−	1.91062i	0.295611	−	0.955308i	$-0.404477\pi$
$692$		0			0
$693$	4.00000			0.151947
$694$		0			0
$695$	32.4999	−	3.71729i	1.23279	−	0.141005i
$696$		0			0
$697$		−	28.9170i		−	1.09531i
$698$		0			0
$699$		−	18.4008i		−	0.695983i
$700$		0			0
$701$		−	23.7543i		−	0.897188i	−0.893736	−	0.448594i	$-0.851925\pi$
$701$		−	23.7543i		−	0.897188i	0.893736	−	0.448594i	$-0.148075\pi$
$702$		0			0
$703$			55.7715i			2.10346i
$704$		0			0
$705$	15.7417	−	1.80052i	0.592868	−	0.0678114i
$706$		0			0
$707$	−35.2252			−1.32478
$708$		0			0
$709$		−	36.3146i		−	1.36382i	−0.731435	−	0.681911i	$-0.761149\pi$
$709$		−	36.3146i		−	1.36382i	0.731435	−	0.681911i	$-0.238851\pi$
$710$		0			0
$711$	−7.36266			−0.276121
$712$		0			0
$713$		−	35.5453i		−	1.33118i
$714$		0			0
$715$	−13.0164	+	1.48880i	−0.486786	+	0.0556779i
$716$		0			0
$717$	−10.0328			−0.374682
$718$		0			0
$719$	30.7253			1.14586			0.572931	−	0.819604i	$-0.305806\pi$
$719$	30.7253			1.14586			0.572931	+	0.819604i	$0.305806\pi$
$720$		0			0
$721$	−2.53268			−0.0943219
$722$		0			0
$723$	−10.7581			−0.400099
$724$		0			0
$725$	2.46705	+	10.6435i	0.0916240	+	0.395289i
$726$		0			0
$727$			5.47445i			0.203036i	0.994834	+	0.101518i	$0.0323700\pi$
$727$			5.47445i			0.203036i	−0.994834	+	0.101518i	$0.967630\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$			3.36852i			0.124589i
$732$		0			0
$733$	−17.1455			−0.633285			−0.316643	−	0.948545i	$-0.602556\pi$
$733$	−17.1455			−0.633285			−0.316643	+	0.948545i	$0.602556\pi$
$734$		0			0
$735$	0.00416898	+	0.0364490i	0.000153775	+	0.00134444i
$736$		0			0
$737$		−	1.53847i		−	0.0566701i
$738$		0			0
$739$		−	11.6019i		−	0.426782i	−0.976967	−	0.213391i	$-0.931549\pi$
$739$		−	11.6019i		−	0.426782i	0.976967	−	0.213391i	$-0.0684508\pi$
$740$		0			0
$741$			27.4282i			1.00760i
$742$		0			0
$743$			23.6613i			0.868048i	0.900901	+	0.434024i	$0.142907\pi$
$743$			23.6613i			0.868048i	−0.900901	+	0.434024i	$0.857093\pi$
$744$		0			0
$745$	−24.5962	+	2.81328i	−0.901135	+	0.103071i
$746$		0			0
$747$	−7.74173			−0.283255
$748$		0			0
$749$			10.5706i			0.386241i
$750$		0			0
$751$	11.4283			0.417024			0.208512	−	0.978020i	$-0.433138\pi$
$751$	11.4283			0.417024			0.208512	+	0.978020i	$0.433138\pi$
$752$		0			0
$753$			12.6580i			0.461283i
$754$		0			0
$755$	0.161949	+	1.41590i	0.00589392	+	0.0515299i
$756$		0			0
$757$	−19.1784			−0.697049			−0.348525	−	0.937300i	$-0.613317\pi$
$757$	−19.1784			−0.697049			−0.348525	+	0.937300i	$0.613317\pi$
$758$		0			0
$759$	7.30749			0.265245
$760$		0			0
$761$	4.03281			0.146189			0.0730947	−	0.997325i	$-0.476712\pi$
$761$	4.03281			0.146189			0.0730947	+	0.997325i	$0.476712\pi$
$762$		0			0
$763$	−2.03281			−0.0735928
$764$		0			0
$765$	−7.36266	+	0.842131i	−0.266198	+	0.0304473i
$766$		0			0
$767$			26.3177i			0.950279i
$768$		0			0
$769$	2.95078			0.106408			0.0532039	−	0.998584i	$-0.483057\pi$
$769$	2.95078			0.106408			0.0532039	+	0.998584i	$0.483057\pi$
$770$		0			0
$771$		−	13.3110i		−	0.479383i
$772$		0			0
$773$	45.2663			1.62812			0.814059	−	0.580783i	$-0.197253\pi$
$773$	45.2663			1.62812			0.814059	+	0.580783i	$0.197253\pi$
$774$		0			0
$775$	−35.8625	+	8.31256i	−1.28822	+	0.298596i
$776$		0			0
$777$			20.7999i			0.746193i
$778$		0			0
$779$		−	61.8260i		−	2.21515i
$780$		0			0
$781$		−	10.1797i		−	0.364257i
$782$		0			0
$783$			2.18513i			0.0780903i
$784$		0			0
$785$	−0.0328135	−	0.286885i	−0.00117116	−	0.0102394i
$786$		0			0
$787$	52.9997			1.88924			0.944618	−	0.328171i	$-0.106432\pi$
$787$	52.9997			1.88924			0.944618	+	0.328171i	$0.106432\pi$
$788$		0			0
$789$		−	18.4256i		−	0.655970i
$790$		0			0
$791$	38.2088			1.35855
$792$		0			0
$793$			13.9391i			0.494993i
$794$		0			0
$795$	1.14554	+	10.0153i	0.0406282	+	0.355208i
$796$		0			0
$797$	16.5738			0.587075			0.293538	−	0.955948i	$-0.405167\pi$
$797$	16.5738			0.587075			0.293538	+	0.955948i	$0.405167\pi$
$798$		0			0
$799$	−23.4835			−0.830785
$800$		0			0
$801$	−14.7581			−0.521453
$802$		0			0
$803$	23.4835			0.828713
$804$		0			0
$805$	−3.24186	−	28.3433i	−0.114261	−	0.998969i
$806$		0			0
$807$			3.86940i			0.136209i
$808$		0			0
$809$	37.5491			1.32016			0.660078	−	0.751197i	$-0.270523\pi$
$809$	37.5491			1.32016			0.660078	+	0.751197i	$0.270523\pi$
$810$		0			0
$811$			32.1102i			1.12754i	0.825931	+	0.563771i	$0.190650\pi$
$811$			32.1102i			1.12754i	−0.825931	+	0.563771i	$0.809350\pi$
$812$		0			0
$813$	−17.3955			−0.610086
$814$		0			0
$815$	−4.95078	−	43.2841i	−0.173418	−	1.51618i
$816$		0			0
$817$			7.20207i			0.251969i
$818$		0			0
$819$			10.2293i			0.357442i
$820$		0			0
$821$		−	29.3809i		−	1.02540i	−0.858568	−	0.512699i	$-0.828646\pi$
$821$		−	29.3809i		−	1.02540i	0.858568	−	0.512699i	$-0.171354\pi$
$822$		0			0
$823$		−	28.3866i		−	0.989495i	−0.869037	−	0.494748i	$-0.835260\pi$
$823$		−	28.3866i		−	0.989495i	0.869037	−	0.494748i	$-0.164740\pi$
$824$		0			0
$825$	−1.70892	−	7.37270i	−0.0594968	−	0.256684i
$826$		0			0
$827$	1.45065			0.0504439			0.0252219	−	0.999682i	$-0.491971\pi$
$827$	1.45065			0.0504439			0.0252219	+	0.999682i	$0.491971\pi$
$828$		0			0
$829$			37.4621i			1.30111i	0.759458	+	0.650556i	$0.225464\pi$
$829$			37.4621i			1.30111i	−0.759458	+	0.650556i	$0.774536\pi$
$830$		0			0
$831$	−0.887271			−0.0307791
$832$		0			0
$833$		−	0.0543744i		−	0.00188396i
$834$		0			0
$835$	4.00000	−	0.457515i	0.138426	−	0.0158329i
$836$		0			0
$837$	−7.36266			−0.254491
$838$		0			0
$839$	−48.7581			−1.68332			−0.841659	−	0.540010i	$-0.818421\pi$
$839$	−48.7581			−1.68332			−0.841659	+	0.540010i	$0.818421\pi$
$840$		0			0
$841$	24.2252			0.835351
$842$		0			0
$843$	13.4835			0.464395
$844$		0			0
$845$	−0.504034	−	4.40672i	−0.0173393	−	0.151596i
$846$		0			0
$847$			23.0146i			0.790791i
$848$		0			0
$849$	−28.4342			−0.975861
$850$		0			0
$851$			37.9988i			1.30258i
$852$		0			0
$853$	−4.37073			−0.149651			−0.0748255	−	0.997197i	$-0.523840\pi$
$853$	−4.37073			−0.149651			−0.0748255	+	0.997197i	$0.523840\pi$
$854$		0			0
$855$	−15.7417	+	1.80052i	−0.538356	+	0.0615764i
$856$		0			0
$857$		−	20.5130i		−	0.700712i	−0.936617	−	0.350356i	$-0.886061\pi$
$857$		−	20.5130i		−	0.700712i	0.936617	−	0.350356i	$-0.113939\pi$
$858$		0			0
$859$			10.1131i			0.345054i	0.985005	+	0.172527i	$0.0551932\pi$
$859$			10.1131i			0.345054i	−0.985005	+	0.172527i	$0.944807\pi$
$860$		0			0
$861$		−	23.0580i		−	0.785813i
$862$		0			0
$863$		−	13.2861i		−	0.452266i	−0.974096	−	0.226133i	$-0.927392\pi$
$863$		−	13.2861i		−	0.452266i	0.974096	−	0.226133i	$-0.0726083\pi$
$864$		0			0
$865$	5.90368	+	51.6152i	0.200731	+	1.75497i
$866$		0			0
$867$	−6.01641			−0.204328
$868$		0			0
$869$			11.1444i			0.378047i
$870$		0			0
$871$	3.93437			0.133311
$872$		0			0
$873$		−	11.1444i		−	0.377180i
$874$		0			0
$875$	−27.8381	+	9.89909i	−0.941098	+	0.334650i
$876$		0			0
$877$	33.6454			1.13612			0.568062	−	0.822986i	$-0.307693\pi$
$877$	33.6454			1.13612			0.568062	+	0.822986i	$0.307693\pi$
$878$		0			0
$879$	7.99166			0.269552
$880$		0			0
$881$	−32.7909			−1.10476			−0.552378	−	0.833594i	$-0.686279\pi$
$881$	−32.7909			−1.10476			−0.552378	+	0.833594i	$0.686279\pi$
$882$		0			0
$883$	−33.4506			−1.12570			−0.562852	−	0.826558i	$-0.690296\pi$
$883$	−33.4506			−1.12570			−0.562852	+	0.826558i	$0.690296\pi$
$884$		0			0
$885$	−15.1044	+	1.72762i	−0.507729	+	0.0580733i
$886$		0			0
$887$		−	34.8924i		−	1.17157i	−0.810466	−	0.585785i	$-0.800786\pi$
$887$		−	34.8924i		−	1.17157i	0.810466	−	0.585785i	$-0.199214\pi$
$888$		0			0
$889$	−30.4671			−1.02183
$890$		0			0
$891$		−	1.51363i		−	0.0507086i
$892$		0			0
$893$	−50.2088			−1.68017
$894$		0			0
$895$	6.34625	−	0.725876i	0.212132	−	0.0242634i
$896$		0			0
$897$			18.6877i			0.623964i
$898$		0			0
$899$		−	16.0884i		−	0.536578i
$900$		0			0
$901$		−	14.9409i		−	0.497752i
$902$		0			0
$903$			2.68601i			0.0893847i
$904$		0			0
$905$	−11.7417	+	1.34300i	−0.390308	+	0.0446429i
$906$		0			0
$907$	30.9836			1.02879			0.514397	−	0.857552i	$-0.328016\pi$
$907$	30.9836			1.02879			0.514397	+	0.857552i	$0.328016\pi$
$908$		0			0
$909$			13.3295i			0.442112i
$910$		0			0
$911$	−16.0000			−0.530104			−0.265052	−	0.964234i	$-0.585389\pi$
$911$	−16.0000			−0.530104			−0.265052	+	0.964234i	$0.585389\pi$
$912$		0			0
$913$			11.7181i			0.387814i
$914$		0			0
$915$	−8.00000	+	0.915029i	−0.264472	+	0.0302499i
$916$		0			0
$917$	−19.4835			−0.643400
$918$		0			0
$919$	−15.6043			−0.514737			−0.257368	−	0.966313i	$-0.582855\pi$
$919$	−15.6043			−0.514737			−0.257368	+	0.966313i	$0.582855\pi$
$920$		0			0
$921$	17.4506			0.575018
$922$		0			0
$923$	26.0328			0.856880
$924$		0			0
$925$	38.3379	−	8.88633i	1.26054	−	0.292181i
$926$		0			0
$927$			0.958386i			0.0314775i
$928$		0			0
$929$	38.9341			1.27739			0.638693	−	0.769461i	$-0.279475\pi$
$929$	38.9341			1.27739			0.638693	+	0.769461i	$0.279475\pi$
$930$		0			0
$931$		−	0.116255i		−	0.00381011i
$932$		0			0
$933$	21.4506			0.702263
$934$		0			0
$935$	1.27468	+	11.1444i	0.0416864	+	0.364460i
$936$		0			0
$937$		−	19.6027i		−	0.640393i	−0.947351	−	0.320197i	$-0.896251\pi$
$937$		−	19.6027i		−	0.640393i	0.947351	−	0.320197i	$-0.103749\pi$
$938$		0			0
$939$			7.73879i			0.252546i
$940$		0			0
$941$			25.0476i			0.816530i	0.912864	+	0.408265i	$0.133866\pi$
$941$			25.0476i			0.816530i	−0.912864	+	0.408265i	$0.866134\pi$
$942$		0			0
$943$		−	42.1240i		−	1.37175i
$944$		0			0
$945$	−5.87086	+	0.671502i	−0.190979	+	0.0218439i
$946$		0			0
$947$	7.93437			0.257832			0.128916	−	0.991655i	$-0.458850\pi$
$947$	7.93437			0.257832			0.128916	+	0.991655i	$0.458850\pi$
$948$		0			0
$949$			60.0550i			1.94947i
$950$		0			0
$951$	−11.2335			−0.364272
$952$		0			0
$953$		−	11.4809i		−	0.371903i	−0.982559	−	0.185952i	$-0.940463\pi$
$953$		−	11.4809i		−	0.371903i	0.982559	−	0.185952i	$-0.0595368\pi$
$954$		0			0
$955$	−1.51627	−	13.2566i	−0.0490654	−	0.428974i
$956$		0			0
$957$	3.30749			0.106916
$958$		0			0
$959$	−10.2744			−0.331778
$960$		0			0
$961$	23.2088			0.748670
$962$		0			0
$963$	4.00000			0.128898
$964$		0			0
$965$	−33.1924	+	3.79650i	−1.06850	+	0.122214i
$966$		0			0
$967$			15.8993i			0.511285i	0.966771	+	0.255643i	$0.0822871\pi$
$967$			15.8993i			0.511285i	−0.966771	+	0.255643i	$0.917713\pi$
$968$		0			0
$969$	23.4835			0.754397
$970$		0			0
$971$			40.6600i			1.30484i	0.757857	+	0.652421i	$0.226246\pi$
$971$			40.6600i			1.30484i	−0.757857	+	0.652421i	$0.773754\pi$
$972$		0			0
$973$	−38.6597			−1.23937
$974$		0			0
$975$	18.8545	−	4.37027i	0.603826	−	0.139961i
$976$		0			0
$977$			26.5676i			0.849972i	0.905200	+	0.424986i	$0.139721\pi$
$977$			26.5676i			0.849972i	−0.905200	+	0.424986i	$0.860279\pi$
$978$		0			0
$979$			22.3384i			0.713938i
$980$		0			0
$981$			0.769233i			0.0245597i
$982$		0			0
$983$		−	9.88057i		−	0.315141i	−0.987508	−	0.157571i	$-0.949634\pi$
$983$		−	9.88057i		−	0.315141i	0.987508	−	0.157571i	$-0.0503662\pi$
$984$		0			0
$985$	−0.821644	−	7.18355i	−0.0261798	−	0.228887i
$986$		0			0
$987$	−18.7253			−0.596034
$988$		0			0
$989$			4.90699i			0.156033i
$990$		0			0
$991$	53.0549			1.68534			0.842672	−	0.538427i	$-0.180981\pi$
$991$	53.0549			1.68534			0.842672	+	0.538427i	$0.180981\pi$
$992$		0			0
$993$		−	8.00084i		−	0.253899i
$994$		0			0
$995$	2.06351	+	18.0410i	0.0654176	+	0.571939i
$996$		0			0
$997$	−32.3051			−1.02311			−0.511556	−	0.859250i	$-0.670931\pi$
$997$	−32.3051			−1.02311			−0.511556	+	0.859250i	$0.670931\pi$
$998$		0			0
$999$	7.87086			0.249023

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	480.2.d.a.49.3		6
3.2	odd	2		1440.2.d.e.1009.4		6
4.3	odd	2		120.2.d.a.109.2	yes	6
5.2	odd	4		2400.2.k.f.1201.8		12
5.3	odd	4		2400.2.k.f.1201.5		12
5.4	even	2		480.2.d.b.49.3		6
8.3	odd	2		120.2.d.b.109.6	yes	6
8.5	even	2		480.2.d.b.49.4		6
12.11	even	2		360.2.d.f.109.5		6
15.2	even	4		7200.2.k.u.3601.4		12
15.8	even	4		7200.2.k.u.3601.10		12
15.14	odd	2		1440.2.d.f.1009.4		6
16.3	odd	4		3840.2.f.l.769.7		12
16.5	even	4		3840.2.f.m.769.12		12
16.11	odd	4		3840.2.f.l.769.6		12
16.13	even	4		3840.2.f.m.769.1		12
20.3	even	4		600.2.k.f.301.8		12
20.7	even	4		600.2.k.f.301.5		12
20.19	odd	2		120.2.d.b.109.5	yes	6
24.5	odd	2		1440.2.d.f.1009.3		6
24.11	even	2		360.2.d.e.109.1		6
40.3	even	4		600.2.k.f.301.7		12
40.13	odd	4		2400.2.k.f.1201.11		12
40.19	odd	2		120.2.d.a.109.1	✓	6
40.27	even	4		600.2.k.f.301.6		12
40.29	even	2	inner	480.2.d.a.49.4		6
40.37	odd	4		2400.2.k.f.1201.2		12
60.23	odd	4		1800.2.k.u.901.5		12
60.47	odd	4		1800.2.k.u.901.8		12
60.59	even	2		360.2.d.e.109.2		6
80.19	odd	4		3840.2.f.l.769.1		12
80.29	even	4		3840.2.f.m.769.7		12
80.59	odd	4		3840.2.f.l.769.12		12
80.69	even	4		3840.2.f.m.769.6		12
120.29	odd	2		1440.2.d.e.1009.3		6
120.53	even	4		7200.2.k.u.3601.9		12
120.59	even	2		360.2.d.f.109.6		6
120.77	even	4		7200.2.k.u.3601.3		12
120.83	odd	4		1800.2.k.u.901.6		12
120.107	odd	4		1800.2.k.u.901.7		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.2.d.a.109.1	✓	6	40.19	odd	2
120.2.d.a.109.2	yes	6	4.3	odd	2
120.2.d.b.109.5	yes	6	20.19	odd	2
120.2.d.b.109.6	yes	6	8.3	odd	2
360.2.d.e.109.1		6	24.11	even	2
360.2.d.e.109.2		6	60.59	even	2
360.2.d.f.109.5		6	12.11	even	2
360.2.d.f.109.6		6	120.59	even	2
480.2.d.a.49.3		6	1.1	even	1	trivial
480.2.d.a.49.4		6	40.29	even	2	inner
480.2.d.b.49.3		6	5.4	even	2
480.2.d.b.49.4		6	8.5	even	2
600.2.k.f.301.5		12	20.7	even	4
600.2.k.f.301.6		12	40.27	even	4
600.2.k.f.301.7		12	40.3	even	4
600.2.k.f.301.8		12	20.3	even	4
1440.2.d.e.1009.3		6	120.29	odd	2
1440.2.d.e.1009.4		6	3.2	odd	2
1440.2.d.f.1009.3		6	24.5	odd	2
1440.2.d.f.1009.4		6	15.14	odd	2
1800.2.k.u.901.5		12	60.23	odd	4
1800.2.k.u.901.6		12	120.83	odd	4
1800.2.k.u.901.7		12	120.107	odd	4
1800.2.k.u.901.8		12	60.47	odd	4
2400.2.k.f.1201.2		12	40.37	odd	4
2400.2.k.f.1201.5		12	5.3	odd	4
2400.2.k.f.1201.8		12	5.2	odd	4
2400.2.k.f.1201.11		12	40.13	odd	4
3840.2.f.l.769.1		12	80.19	odd	4
3840.2.f.l.769.6		12	16.11	odd	4
3840.2.f.l.769.7		12	16.3	odd	4
3840.2.f.l.769.12		12	80.59	odd	4
3840.2.f.m.769.1		12	16.13	even	4
3840.2.f.m.769.6		12	80.69	even	4
3840.2.f.m.769.7		12	80.29	even	4
3840.2.f.m.769.12		12	16.5	even	4
7200.2.k.u.3601.3		12	120.77	even	4
7200.2.k.u.3601.4		12	15.2	even	4
7200.2.k.u.3601.9		12	120.53	even	4
7200.2.k.u.3601.10		12	15.8	even	4

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:	\( N \)	\(=\)	\( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	480.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +(-0.254102 - 2.22158i) q^{5} +2.64265i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +(-0.254102 - 2.22158i) q^{5} +2.64265i q^{7} +1.00000 q^{9} -1.51363i q^{11} +3.87086 q^{13} +(0.254102 + 2.22158i) q^{15} -3.31415i q^{17} -7.08582i q^{19} -2.64265i q^{21} -4.82778i q^{23} +(-4.87086 + 1.12902i) q^{25} -1.00000 q^{27} -2.18513i q^{29} +7.36266 q^{31} +1.51363i q^{33} +(5.87086 - 0.671502i) q^{35} -7.87086 q^{37} -3.87086 q^{39} +8.72532 q^{41} -1.01641 q^{43} +(-0.254102 - 2.22158i) q^{45} -7.08582i q^{47} +0.0164068 q^{49} +3.31415i q^{51} +4.50820 q^{53} +(-3.36266 + 0.384617i) q^{55} +7.08582i q^{57} +6.79893i q^{59} +3.60104i q^{61} +2.64265i q^{63} +(-0.983593 - 8.59945i) q^{65} +1.01641 q^{67} +4.82778i q^{69} +6.72532 q^{71} +15.5146i q^{73} +(4.87086 - 1.12902i) q^{75} +4.00000 q^{77} -7.36266 q^{79} +1.00000 q^{81} -7.74173 q^{83} +(-7.36266 + 0.842131i) q^{85} +2.18513i q^{87} -14.7581 q^{89} +10.2293i q^{91} -7.36266 q^{93} +(-15.7417 + 1.80052i) q^{95} -11.1444i q^{97} -1.51363i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} + 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^3 + 6 * q^9	\( 6 q - 6 q^{3} + 6 q^{9} - 8 q^{13} + 2 q^{25} - 6 q^{27} + 16 q^{31} + 4 q^{35} - 16 q^{37} + 8 q^{39} - 4 q^{41} - 6 q^{49} + 24 q^{53} + 8 q^{55} - 12 q^{65} - 16 q^{71} - 2 q^{75} + 24 q^{77} - 16 q^{79} + 6 q^{81} + 16 q^{83} - 16 q^{85} - 20 q^{89} - 16 q^{93} - 32 q^{95}+O(q^{100}) \) 6 * q - 6 * q^3 + 6 * q^9 - 8 * q^13 + 2 * q^25 - 6 * q^27 + 16 * q^31 + 4 * q^35 - 16 * q^37 + 8 * q^39 - 4 * q^41 - 6 * q^49 + 24 * q^53 + 8 * q^55 - 12 * q^65 - 16 * q^71 - 2 * q^75 + 24 * q^77 - 16 * q^79 + 6 * q^81 + 16 * q^83 - 16 * q^85 - 20 * q^89 - 16 * q^93 - 32 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more