LMFDB - Embedded newform 3120.2.r.h.2209.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3120,2,Mod(2209,3120)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3120.2209"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3120, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3120.r (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,2,0,4,0,-6,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$24.9133254306$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 390)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2209.1
Root			$0.403032 + 0.403032i$ of defining polynomial
Character	$\chi$	$=$	3120.2209
Dual form			3120.2.r.h.2209.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +(-1.48119 + 1.67513i) q^{5} +4.15633 q^{7} -1.00000 q^{9} -5.35026i q^{11} +(3.28726 + 1.48119i) q^{13} +(1.67513 + 1.48119i) q^{15} -1.19394i q^{17} -2.80606i q^{19} -4.15633i q^{21} -0.806063i q^{23} +(-0.612127 - 4.96239i) q^{25} +1.00000i q^{27} +7.50659 q^{29} +7.92478i q^{31} -5.35026 q^{33} +(-6.15633 + 6.96239i) q^{35} -7.35026 q^{37} +(1.48119 - 3.28726i) q^{39} -3.61213i q^{41} -6.57452i q^{43} +(1.48119 - 1.67513i) q^{45} -12.3127 q^{47} +10.2750 q^{49} -1.19394 q^{51} +5.53690i q^{53} +(8.96239 + 7.92478i) q^{55} -2.80606 q^{57} -12.8872i q^{59} +6.31265 q^{61} -4.15633 q^{63} +(-7.35026 + 3.31265i) q^{65} +4.57452 q^{67} -0.806063 q^{69} +8.96239i q^{71} +4.08110 q^{73} +(-4.96239 + 0.612127i) q^{75} -22.2374i q^{77} +12.4387 q^{79} +1.00000 q^{81} -10.0508 q^{83} +(2.00000 + 1.76845i) q^{85} -7.50659i q^{87} -5.03761i q^{89} +(13.6629 + 6.15633i) q^{91} +7.92478 q^{93} +(4.70052 + 4.15633i) q^{95} +2.93207 q^{97} +5.35026i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{5} + 4 q^{7} - 6 q^{9} + 8 q^{13} - 2 q^{25} + 4 q^{29} - 12 q^{33} - 16 q^{35} - 24 q^{37} - 2 q^{39} - 2 q^{45} - 32 q^{47} - 2 q^{49} - 8 q^{51} + 32 q^{55} - 16 q^{57} - 4 q^{61} - 4 q^{63}+ \cdots - 12 q^{95}+O(q^{100}) $ 6 * q + 2 * q^5 + 4 * q^7 - 6 * q^9 + 8 * q^13 - 2 * q^25 + 4 * q^29 - 12 * q^33 - 16 * q^35 - 24 * q^37 - 2 * q^39 - 2 * q^45 - 32 * q^47 - 2 * q^49 - 8 * q^51 + 32 * q^55 - 16 * q^57 - 4 * q^61 - 4 * q^63 - 24 * q^65 + 4 * q^67 - 4 * q^69 - 40 * q^73 - 8 * q^75 + 16 * q^79 + 6 * q^81 + 12 * q^85 + 20 * q^91 + 4 * q^93 - 12 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3120\mathbb{Z}\right)^\times$.

$n$	$1951$	$2081$	$2341$	$2497$	$2641$
$\chi(n)$	$1$	$1$	$1$	$-1$	$-1$

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
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$	−1.48119	+	1.67513i	−0.662410	+	0.749141i
$5$	−1.48119	+	1.67513i	−0.662410	+	0.749141i
$6$		0			0
$7$	4.15633			1.57094			0.785472	−	0.618898i	$-0.212420\pi$
$7$	4.15633			1.57094			0.785472	+	0.618898i	$0.212420\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$		−	5.35026i		−	1.61316i	−0.591122	−	0.806582i	$-0.701315\pi$
$11$		−	5.35026i		−	1.61316i	0.591122	−	0.806582i	$-0.298685\pi$
$12$		0			0
$13$	3.28726	+	1.48119i	0.911721	+	0.410809i
$13$	3.28726	+	1.48119i	0.911721	+	0.410809i
$14$		0			0
$15$	1.67513	+	1.48119i	0.432517	+	0.382443i
$16$		0			0
$17$		−	1.19394i		−	0.289572i	−0.989463	−	0.144786i	$-0.953751\pi$
$17$		−	1.19394i		−	0.289572i	0.989463	−	0.144786i	$-0.0462494\pi$
$18$		0			0
$19$		−	2.80606i		−	0.643755i	−0.946781	−	0.321878i	$-0.895686\pi$
$19$		−	2.80606i		−	0.643755i	0.946781	−	0.321878i	$-0.104314\pi$
$20$		0			0
$21$		−	4.15633i		−	0.906985i
$22$		0			0
$23$		−	0.806063i		−	0.168076i	−0.996463	−	0.0840379i	$-0.973218\pi$
$23$		−	0.806063i		−	0.168076i	0.996463	−	0.0840379i	$-0.0267817\pi$
$24$		0			0
$25$	−0.612127	−	4.96239i	−0.122425	−	0.992478i
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	7.50659			1.39394			0.696969	−	0.717101i	$-0.254531\pi$
$29$	7.50659			1.39394			0.696969	+	0.717101i	$0.254531\pi$
$30$		0			0
$31$			7.92478i			1.42333i	0.702518	+	0.711666i	$0.252059\pi$
$31$			7.92478i			1.42333i	−0.702518	+	0.711666i	$0.747941\pi$
$32$		0			0
$33$	−5.35026			−0.931361
$34$		0			0
$35$	−6.15633	+	6.96239i	−1.04061	+	1.17686i
$36$		0			0
$37$	−7.35026			−1.20838			−0.604188	−	0.796842i	$-0.706502\pi$
$37$	−7.35026			−1.20838			−0.604188	+	0.796842i	$0.706502\pi$
$38$		0			0
$39$	1.48119	−	3.28726i	0.237181	−	0.526383i
$40$		0			0
$41$		−	3.61213i		−	0.564119i	−0.959397	−	0.282060i	$-0.908982\pi$
$41$		−	3.61213i		−	0.564119i	0.959397	−	0.282060i	$-0.0910176\pi$
$42$		0			0
$43$		−	6.57452i		−	1.00260i	−0.865272	−	0.501302i	$-0.832855\pi$
$43$		−	6.57452i		−	1.00260i	0.865272	−	0.501302i	$-0.167145\pi$
$44$		0			0
$45$	1.48119	−	1.67513i	0.220803	−	0.249714i
$46$		0			0
$47$	−12.3127			−1.79598			−0.897992	−	0.440011i	$-0.854974\pi$
$47$	−12.3127			−1.79598			−0.897992	+	0.440011i	$0.854974\pi$
$48$		0			0
$49$	10.2750			1.46786
$50$		0			0
$51$	−1.19394			−0.167185
$52$		0			0
$53$			5.53690i			0.760552i	0.924873	+	0.380276i	$0.124171\pi$
$53$			5.53690i			0.760552i	−0.924873	+	0.380276i	$0.875829\pi$
$54$		0			0
$55$	8.96239	+	7.92478i	1.20849	+	1.06858i
$56$		0			0
$57$	−2.80606			−0.371672
$58$		0			0
$59$		−	12.8872i		−	1.67777i	−0.544312	−	0.838883i	$-0.683209\pi$
$59$		−	12.8872i		−	1.67777i	0.544312	−	0.838883i	$-0.316791\pi$
$60$		0			0
$61$	6.31265			0.808252			0.404126	−	0.914703i	$-0.367576\pi$
$61$	6.31265			0.808252			0.404126	+	0.914703i	$0.367576\pi$
$62$		0			0
$63$	−4.15633			−0.523648
$64$		0			0
$65$	−7.35026	+	3.31265i	−0.911688	+	0.410884i
$66$		0			0
$67$	4.57452			0.558866			0.279433	−	0.960165i	$-0.409854\pi$
$67$	4.57452			0.558866			0.279433	+	0.960165i	$0.409854\pi$
$68$		0			0
$69$	−0.806063			−0.0970386
$70$		0			0
$71$			8.96239i			1.06364i	0.846857	+	0.531820i	$0.178492\pi$
$71$			8.96239i			1.06364i	−0.846857	+	0.531820i	$0.821508\pi$
$72$		0			0
$73$	4.08110			0.477657			0.238828	−	0.971062i	$-0.423237\pi$
$73$	4.08110			0.477657			0.238828	+	0.971062i	$0.423237\pi$
$74$		0			0
$75$	−4.96239	+	0.612127i	−0.573007	+	0.0706823i
$76$		0			0
$77$		−	22.2374i		−	2.53419i
$78$		0			0
$79$	12.4387			1.39946			0.699729	−	0.714408i	$-0.253304\pi$
$79$	12.4387			1.39946			0.699729	+	0.714408i	$0.253304\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	−10.0508			−1.10322			−0.551609	−	0.834103i	$-0.685986\pi$
$83$	−10.0508			−1.10322			−0.551609	+	0.834103i	$0.685986\pi$
$84$		0			0
$85$	2.00000	+	1.76845i	0.216930	+	0.191816i
$86$		0			0
$87$		−	7.50659i		−	0.804791i
$88$		0			0
$89$		−	5.03761i		−	0.533986i	−0.963699	−	0.266993i	$-0.913970\pi$
$89$		−	5.03761i		−	0.533986i	0.963699	−	0.266993i	$-0.0860300\pi$
$90$		0			0
$91$	13.6629	+	6.15633i	1.43226	+	0.645358i
$92$		0			0
$93$	7.92478			0.821761
$94$		0			0
$95$	4.70052	+	4.15633i	0.482264	+	0.426430i
$96$		0			0
$97$	2.93207			0.297707			0.148853	−	0.988859i	$-0.452442\pi$
$97$	2.93207			0.297707			0.148853	+	0.988859i	$0.452442\pi$
$98$		0			0
$99$			5.35026i			0.537722i
$100$		0			0
$101$	5.89446			0.586521			0.293260	−	0.956033i	$-0.405260\pi$
$101$	5.89446			0.586521			0.293260	+	0.956033i	$0.405260\pi$
$102$		0			0
$103$		−	3.29948i		−	0.325107i	−0.986700	−	0.162554i	$-0.948027\pi$
$103$		−	3.29948i		−	0.325107i	0.986700	−	0.162554i	$-0.0519730\pi$
$104$		0			0
$105$	6.96239	+	6.15633i	0.679460	+	0.600796i
$106$		0			0
$107$		−	19.7889i		−	1.91307i	−0.291622	−	0.956534i	$-0.594195\pi$
$107$		−	19.7889i		−	1.91307i	0.291622	−	0.956534i	$-0.405805\pi$
$108$		0			0
$109$		−	9.43136i		−	0.903361i	−0.892180	−	0.451680i	$-0.850825\pi$
$109$		−	9.43136i		−	0.903361i	0.892180	−	0.451680i	$-0.149175\pi$
$110$		0			0
$111$			7.35026i			0.697656i
$112$		0			0
$113$			4.41819i			0.415628i	0.978168	+	0.207814i	$0.0666349\pi$
$113$			4.41819i			0.415628i	−0.978168	+	0.207814i	$0.933365\pi$
$114$		0			0
$115$	1.35026	+	1.19394i	0.125913	+	0.111335i
$116$		0			0
$117$	−3.28726	−	1.48119i	−0.303907	−	0.136936i
$118$		0			0
$119$		−	4.96239i		−	0.454901i
$120$		0			0
$121$	−17.6253			−1.60230
$122$		0			0
$123$	−3.61213			−0.325695
$124$		0			0
$125$	9.21933	+	6.32487i	0.824602	+	0.565713i
$126$		0			0
$127$			18.1622i			1.61164i	0.592164	+	0.805818i	$0.298274\pi$
$127$			18.1622i			1.61164i	−0.592164	+	0.805818i	$0.701726\pi$
$128$		0			0
$129$	−6.57452			−0.578854
$130$		0			0
$131$	9.81924			0.857911			0.428955	−	0.903326i	$-0.358882\pi$
$131$	9.81924			0.857911			0.428955	+	0.903326i	$0.358882\pi$
$132$		0			0
$133$		−	11.6629i		−	1.01130i
$134$		0			0
$135$	−1.67513	−	1.48119i	−0.144172	−	0.127481i
$136$		0			0
$137$	0.911603			0.0778835			0.0389418	−	0.999241i	$-0.487601\pi$
$137$	0.911603			0.0778835			0.0389418	+	0.999241i	$0.487601\pi$
$138$		0			0
$139$	−0.836381			−0.0709409			−0.0354704	−	0.999371i	$-0.511293\pi$
$139$	−0.836381			−0.0709409			−0.0354704	+	0.999371i	$0.511293\pi$
$140$		0			0
$141$			12.3127i			1.03691i
$142$		0			0
$143$	7.92478	−	17.5877i	0.662703	−	1.47076i
$144$		0			0
$145$	−11.1187	+	12.5745i	−0.923359	+	1.04426i
$146$		0			0
$147$		−	10.2750i		−	0.847471i
$148$		0			0
$149$		−	10.8872i		−	0.891911i	−0.895055	−	0.445956i	$-0.852864\pi$
$149$		−	10.8872i		−	0.891911i	0.895055	−	0.445956i	$-0.147136\pi$
$150$		0			0
$151$		−	17.7889i		−	1.44764i	−0.689988	−	0.723821i	$-0.742384\pi$
$151$		−	17.7889i		−	1.44764i	0.689988	−	0.723821i	$-0.257616\pi$
$152$		0			0
$153$			1.19394i			0.0965241i
$154$		0			0
$155$	−13.2750	−	11.7381i	−1.06628	−	0.942830i
$156$		0			0
$157$		−	10.7757i		−	0.859998i	−0.902829	−	0.429999i	$-0.858514\pi$
$157$		−	10.7757i		−	0.859998i	0.902829	−	0.429999i	$-0.141486\pi$
$158$		0			0
$159$	5.53690			0.439105
$160$		0			0
$161$		−	3.35026i		−	0.264038i
$162$		0			0
$163$	12.0508			0.943890			0.471945	−	0.881628i	$-0.343552\pi$
$163$	12.0508			0.943890			0.471945	+	0.881628i	$0.343552\pi$
$164$		0			0
$165$	7.92478	−	8.96239i	0.616943	−	0.697721i
$166$		0			0
$167$	−1.61213			−0.124750			−0.0623751	−	0.998053i	$-0.519867\pi$
$167$	−1.61213			−0.124750			−0.0623751	+	0.998053i	$0.519867\pi$
$168$		0			0
$169$	8.61213	+	9.73813i	0.662471	+	0.749087i
$170$		0			0
$171$			2.80606i			0.214585i
$172$		0			0
$173$			1.22425i			0.0930783i	0.998916	+	0.0465391i	$0.0148192\pi$
$173$			1.22425i			0.0930783i	−0.998916	+	0.0465391i	$0.985181\pi$
$174$		0			0
$175$	−2.54420	−	20.6253i	−0.192323	−	1.55913i
$176$		0			0
$177$	−12.8872			−0.968659
$178$		0			0
$179$	1.25457			0.0937710			0.0468855	−	0.998900i	$-0.485070\pi$
$179$	1.25457			0.0937710			0.0468855	+	0.998900i	$0.485070\pi$
$180$		0			0
$181$	5.47627			0.407048			0.203524	−	0.979070i	$-0.434761\pi$
$181$	5.47627			0.407048			0.203524	+	0.979070i	$0.434761\pi$
$182$		0			0
$183$		−	6.31265i		−	0.466645i
$184$		0			0
$185$	10.8872	−	12.3127i	0.800440	−	0.905244i
$186$		0			0
$187$	−6.38787			−0.467128
$188$		0			0
$189$			4.15633i			0.302328i
$190$		0			0
$191$	10.2374			0.740754			0.370377	−	0.928881i	$-0.379228\pi$
$191$	10.2374			0.740754			0.370377	+	0.928881i	$0.379228\pi$
$192$		0			0
$193$	9.06793			0.652724			0.326362	−	0.945245i	$-0.394177\pi$
$193$	9.06793			0.652724			0.326362	+	0.945245i	$0.394177\pi$
$194$		0			0
$195$	3.31265	+	7.35026i	0.237224	+	0.526363i
$196$		0			0
$197$	3.14903			0.224359			0.112180	−	0.993688i	$-0.464217\pi$
$197$	3.14903			0.224359			0.112180	+	0.993688i	$0.464217\pi$
$198$		0			0
$199$	−14.1114			−1.00033			−0.500166	−	0.865930i	$-0.666728\pi$
$199$	−14.1114			−1.00033			−0.500166	+	0.865930i	$0.666728\pi$
$200$		0			0
$201$		−	4.57452i		−	0.322661i
$202$		0			0
$203$	31.1998			2.18980
$204$		0			0
$205$	6.05079	+	5.35026i	0.422605	+	0.373678i
$206$		0			0
$207$			0.806063i			0.0560253i
$208$		0			0
$209$	−15.0132			−1.03848
$210$		0			0
$211$	17.4617			1.20211			0.601056	−	0.799207i	$-0.294747\pi$
$211$	17.4617			1.20211			0.601056	+	0.799207i	$0.294747\pi$
$212$		0			0
$213$	8.96239			0.614093
$214$		0			0
$215$	11.0132	+	9.73813i	0.751092	+	0.664135i
$216$		0			0
$217$			32.9380i			2.23597i
$218$		0			0
$219$		−	4.08110i		−	0.275775i
$220$		0			0
$221$	1.76845	−	3.92478i	0.118959	−	0.264009i
$222$		0			0
$223$	−19.6326			−1.31470			−0.657348	−	0.753587i	$-0.728322\pi$
$223$	−19.6326			−1.31470			−0.657348	+	0.753587i	$0.728322\pi$
$224$		0			0
$225$	0.612127	+	4.96239i	0.0408085	+	0.330826i
$226$		0			0
$227$	4.77575			0.316977			0.158489	−	0.987361i	$-0.449338\pi$
$227$	4.77575			0.316977			0.158489	+	0.987361i	$0.449338\pi$
$228$		0			0
$229$		−	19.5672i		−	1.29304i	−0.762898	−	0.646519i	$-0.776224\pi$
$229$		−	19.5672i		−	1.29304i	0.762898	−	0.646519i	$-0.223776\pi$
$230$		0			0
$231$	−22.2374			−1.46312
$232$		0			0
$233$			2.74543i			0.179859i	0.995948	+	0.0899295i	$0.0286642\pi$
$233$			2.74543i			0.179859i	−0.995948	+	0.0899295i	$0.971336\pi$
$234$		0			0
$235$	18.2374	−	20.6253i	1.18968	−	1.34545i
$236$		0			0
$237$		−	12.4387i		−	0.807978i
$238$		0			0
$239$		−	5.29948i		−	0.342795i	−0.985202	−	0.171397i	$-0.945172\pi$
$239$		−	5.29948i		−	0.342795i	0.985202	−	0.171397i	$-0.0548282\pi$
$240$		0			0
$241$			12.9380i			0.833407i	0.909043	+	0.416703i	$0.136815\pi$
$241$			12.9380i			0.833407i	−0.909043	+	0.416703i	$0.863185\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	−15.2193	+	17.2120i	−0.972327	+	1.09964i
$246$		0			0
$247$	4.15633	−	9.22425i	0.264461	−	0.586925i
$248$		0			0
$249$			10.0508i			0.636943i
$250$		0			0
$251$	−8.10554			−0.511617			−0.255809	−	0.966727i	$-0.582342\pi$
$251$	−8.10554			−0.511617			−0.255809	+	0.966727i	$0.582342\pi$
$252$		0			0
$253$	−4.31265			−0.271134
$254$		0			0
$255$	1.76845	−	2.00000i	0.110745	−	0.125245i
$256$		0			0
$257$		−	12.0567i		−	0.752074i	−0.926605	−	0.376037i	$-0.877287\pi$
$257$		−	12.0567i		−	0.752074i	0.926605	−	0.376037i	$-0.122713\pi$
$258$		0			0
$259$	−30.5501			−1.89829
$260$		0			0
$261$	−7.50659			−0.464646
$262$		0			0
$263$		−	4.28233i		−	0.264060i	−0.991246	−	0.132030i	$-0.957850\pi$
$263$		−	4.28233i		−	0.264060i	0.991246	−	0.132030i	$-0.0421495\pi$
$264$		0			0
$265$	−9.27504	−	8.20123i	−0.569761	−	0.503798i
$266$		0			0
$267$	−5.03761			−0.308297
$268$		0			0
$269$	−19.8799			−1.21210			−0.606049	−	0.795428i	$-0.707246\pi$
$269$	−19.8799			−1.21210			−0.606049	+	0.795428i	$0.707246\pi$
$270$		0			0
$271$			5.01317i			0.304529i	0.988340	+	0.152264i	$0.0486565\pi$
$271$			5.01317i			0.304529i	−0.988340	+	0.152264i	$0.951344\pi$
$272$		0			0
$273$	6.15633	−	13.6629i	0.372598	−	0.826917i
$274$		0			0
$275$	−26.5501	+	3.27504i	−1.60103	+	0.197492i
$276$		0			0
$277$			8.55008i			0.513724i	0.966448	+	0.256862i	$0.0826887\pi$
$277$			8.55008i			0.513724i	−0.966448	+	0.256862i	$0.917311\pi$
$278$		0			0
$279$		−	7.92478i		−	0.474444i
$280$		0			0
$281$		−	25.5125i		−	1.52195i	−0.648783	−	0.760973i	$-0.724722\pi$
$281$		−	25.5125i		−	1.52195i	0.648783	−	0.760973i	$-0.275278\pi$
$282$		0			0
$283$			23.7235i			1.41022i	0.709099	+	0.705109i	$0.249102\pi$
$283$			23.7235i			1.41022i	−0.709099	+	0.705109i	$0.750898\pi$
$284$		0			0
$285$	4.15633	−	4.70052i	0.246199	−	0.278435i
$286$		0			0
$287$		−	15.0132i		−	0.886200i
$288$		0			0
$289$	15.5745			0.916148
$290$		0			0
$291$		−	2.93207i		−	0.171881i
$292$		0			0
$293$	−23.9248			−1.39770			−0.698850	−	0.715268i	$-0.746305\pi$
$293$	−23.9248			−1.39770			−0.698850	+	0.715268i	$0.746305\pi$
$294$		0			0
$295$	21.5877	+	19.0884i	1.25688	+	1.11137i
$296$		0			0
$297$	5.35026			0.310454
$298$		0			0
$299$	1.19394	−	2.64974i	0.0690471	−	0.153238i
$300$		0			0
$301$		−	27.3258i		−	1.57503i
$302$		0			0
$303$		−	5.89446i		−	0.338628i
$304$		0			0
$305$	−9.35026	+	10.5745i	−0.535394	+	0.605495i
$306$		0			0
$307$	−33.5125			−1.91266			−0.956329	−	0.292293i	$-0.905582\pi$
$307$	−33.5125			−1.91266			−0.956329	+	0.292293i	$0.905582\pi$
$308$		0			0
$309$	−3.29948			−0.187701
$310$		0			0
$311$	5.76257			0.326766			0.163383	−	0.986563i	$-0.447759\pi$
$311$	5.76257			0.326766			0.163383	+	0.986563i	$0.447759\pi$
$312$		0			0
$313$			12.6253i			0.713624i	0.934176	+	0.356812i	$0.116136\pi$
$313$			12.6253i			0.713624i	−0.934176	+	0.356812i	$0.883864\pi$
$314$		0			0
$315$	6.15633	−	6.96239i	0.346870	−	0.392286i
$316$		0			0
$317$	−3.73813			−0.209955			−0.104977	−	0.994475i	$-0.533477\pi$
$317$	−3.73813			−0.209955			−0.104977	+	0.994475i	$0.533477\pi$
$318$		0			0
$319$		−	40.1622i		−	2.24865i
$320$		0			0
$321$	−19.7889			−1.10451
$322$		0			0
$323$	−3.35026			−0.186414
$324$		0			0
$325$	5.33804	−	17.2193i	0.296101	−	0.955157i
$326$		0			0
$327$	−9.43136			−0.521556
$328$		0			0
$329$	−51.1754			−2.82139
$330$		0			0
$331$		−	1.81924i		−	0.0999943i	−0.998749	−	0.0499972i	$-0.984079\pi$
$331$		−	1.81924i		−	0.0999943i	0.998749	−	0.0499972i	$-0.0159212\pi$
$332$		0			0
$333$	7.35026			0.402792
$334$		0			0
$335$	−6.77575	+	7.66291i	−0.370199	+	0.418670i
$336$		0			0
$337$		−	8.83638i		−	0.481348i	−0.970606	−	0.240674i	$-0.922631\pi$
$337$		−	8.83638i		−	0.481348i	0.970606	−	0.240674i	$-0.0773685\pi$
$338$		0			0
$339$	4.41819			0.239963
$340$		0			0
$341$	42.3996			2.29607
$342$		0			0
$343$	13.6121			0.734986
$344$		0			0
$345$	1.19394	−	1.35026i	0.0642794	−	0.0726956i
$346$		0			0
$347$		−	2.13586i		−	0.114659i	−0.998355	−	0.0573294i	$-0.981741\pi$
$347$		−	2.13586i		−	0.114659i	0.998355	−	0.0573294i	$-0.0182585\pi$
$348$		0			0
$349$		−	16.4934i		−	0.882872i	−0.897293	−	0.441436i	$-0.854469\pi$
$349$		−	16.4934i		−	0.882872i	0.897293	−	0.441436i	$-0.145531\pi$
$350$		0			0
$351$	−1.48119	+	3.28726i	−0.0790603	+	0.175461i
$352$		0			0
$353$	3.86414			0.205668			0.102834	−	0.994699i	$-0.467209\pi$
$353$	3.86414			0.205668			0.102834	+	0.994699i	$0.467209\pi$
$354$		0			0
$355$	−15.0132	−	13.2750i	−0.796817	−	0.704566i
$356$		0			0
$357$	−4.96239			−0.262637
$358$		0			0
$359$			18.5139i			0.977125i	0.872529	+	0.488563i	$0.162479\pi$
$359$			18.5139i			0.977125i	−0.872529	+	0.488563i	$0.837521\pi$
$360$		0			0
$361$	11.1260			0.585579
$362$		0			0
$363$			17.6253i			0.925088i
$364$		0			0
$365$	−6.04491	+	6.83638i	−0.316405	+	0.357833i
$366$		0			0
$367$			23.7137i			1.23784i	0.785452	+	0.618922i	$0.212430\pi$
$367$			23.7137i			1.23784i	−0.785452	+	0.618922i	$0.787570\pi$
$368$		0			0
$369$			3.61213i			0.188040i
$370$		0			0
$371$			23.0132i			1.19478i
$372$		0			0
$373$		−	11.1490i		−	0.577275i	−0.957438	−	0.288637i	$-0.906798\pi$
$373$		−	11.1490i		−	0.577275i	0.957438	−	0.288637i	$-0.0932022\pi$
$374$		0			0
$375$	6.32487	−	9.21933i	0.326615	−	0.476084i
$376$		0			0
$377$	24.6761	+	11.1187i	1.27088	+	0.572643i
$378$		0			0
$379$			38.7572i			1.99082i	0.0956865	+	0.995412i	$0.469495\pi$
$379$			38.7572i			1.99082i	−0.0956865	+	0.995412i	$0.530505\pi$
$380$		0			0
$381$	18.1622			0.930478
$382$		0			0
$383$	−1.76257			−0.0900632			−0.0450316	−	0.998986i	$-0.514339\pi$
$383$	−1.76257			−0.0900632			−0.0450316	+	0.998986i	$0.514339\pi$
$384$		0			0
$385$	37.2506	+	32.9380i	1.89847	+	1.67867i
$386$		0			0
$387$			6.57452i			0.334201i
$388$		0			0
$389$	34.0567			1.72674			0.863371	−	0.504570i	$-0.168349\pi$
$389$	34.0567			1.72674			0.863371	+	0.504570i	$0.168349\pi$
$390$		0			0
$391$	−0.962389			−0.0486701
$392$		0			0
$393$		−	9.81924i		−	0.495315i
$394$		0			0
$395$	−18.4241	+	20.8364i	−0.927016	+	1.04839i
$396$		0			0
$397$	−15.8134			−0.793650			−0.396825	−	0.917894i	$-0.629888\pi$
$397$	−15.8134			−0.793650			−0.396825	+	0.917894i	$0.629888\pi$
$398$		0			0
$399$	−11.6629			−0.583876
$400$		0			0
$401$			25.0132i			1.24910i	0.780986	+	0.624549i	$0.214717\pi$
$401$			25.0132i			1.24910i	−0.780986	+	0.624549i	$0.785283\pi$
$402$		0			0
$403$	−11.7381	+	26.0508i	−0.584718	+	1.29768i
$404$		0			0
$405$	−1.48119	+	1.67513i	−0.0736011	+	0.0832379i
$406$		0			0
$407$			39.3258i			1.94931i
$408$		0			0
$409$		−	34.0263i		−	1.68249i	−0.540651	−	0.841247i	$-0.681822\pi$
$409$		−	34.0263i		−	1.68249i	0.540651	−	0.841247i	$-0.318178\pi$
$410$		0			0
$411$		−	0.911603i		−	0.0449661i
$412$		0			0
$413$		−	53.5633i		−	2.63568i
$414$		0			0
$415$	14.8872	−	16.8364i	0.730782	−	0.826465i
$416$		0			0
$417$			0.836381i			0.0409577i
$418$		0			0
$419$	3.73226			0.182333			0.0911663	−	0.995836i	$-0.470941\pi$
$419$	3.73226			0.182333			0.0911663	+	0.995836i	$0.470941\pi$
$420$		0			0
$421$			16.9683i			0.826983i	0.910508	+	0.413491i	$0.135691\pi$
$421$			16.9683i			0.826983i	−0.910508	+	0.413491i	$0.864309\pi$
$422$		0			0
$423$	12.3127			0.598662
$424$		0			0
$425$	−5.92478	+	0.730841i	−0.287394	+	0.0354510i
$426$		0			0
$427$	26.2374			1.26972
$428$		0			0
$429$	−17.5877	−	7.92478i	−0.849142	−	0.382612i
$430$		0			0
$431$			20.9986i			1.01147i	0.862690	+	0.505733i	$0.168778\pi$
$431$			20.9986i			1.01147i	−0.862690	+	0.505733i	$0.831222\pi$
$432$		0			0
$433$			17.3404i			0.833327i	0.909061	+	0.416664i	$0.136801\pi$
$433$			17.3404i			0.833327i	−0.909061	+	0.416664i	$0.863199\pi$
$434$		0			0
$435$	12.5745	+	11.1187i	0.602902	+	0.533102i
$436$		0			0
$437$	−2.26187			−0.108200
$438$		0			0
$439$	−13.7743			−0.657413			−0.328706	−	0.944432i	$-0.606613\pi$
$439$	−13.7743			−0.657413			−0.328706	+	0.944432i	$0.606613\pi$
$440$		0			0
$441$	−10.2750			−0.489288
$442$		0			0
$443$			9.61213i			0.456686i	0.973581	+	0.228343i	$0.0733307\pi$
$443$			9.61213i			0.456686i	−0.973581	+	0.228343i	$0.926669\pi$
$444$		0			0
$445$	8.43866	+	7.46168i	0.400031	+	0.353718i
$446$		0			0
$447$	−10.8872			−0.514945
$448$		0			0
$449$			25.6629i			1.21111i	0.795804	+	0.605554i	$0.207048\pi$
$449$			25.6629i			1.21111i	−0.795804	+	0.605554i	$0.792952\pi$
$450$		0			0
$451$	−19.3258			−0.910018
$452$		0			0
$453$	−17.7889			−0.835796
$454$		0			0
$455$	−30.5501	+	13.7685i	−1.43221	+	0.645475i
$456$		0			0
$457$	22.9321			1.07272			0.536359	−	0.843990i	$-0.319799\pi$
$457$	22.9321			1.07272			0.536359	+	0.843990i	$0.319799\pi$
$458$		0			0
$459$	1.19394			0.0557282
$460$		0			0
$461$			22.7367i			1.05895i	0.848324	+	0.529477i	$0.177612\pi$
$461$			22.7367i			1.05895i	−0.848324	+	0.529477i	$0.822388\pi$
$462$		0			0
$463$	5.08252			0.236205			0.118102	−	0.993001i	$-0.462319\pi$
$463$	5.08252			0.236205			0.118102	+	0.993001i	$0.462319\pi$
$464$		0			0
$465$	−11.7381	+	13.2750i	−0.544343	+	0.615615i
$466$		0			0
$467$			16.4631i			0.761821i	0.924612	+	0.380911i	$0.124389\pi$
$467$			16.4631i			0.761821i	−0.924612	+	0.380911i	$0.875611\pi$
$468$		0			0
$469$	19.0132			0.877947
$470$		0			0
$471$	−10.7757			−0.496520
$472$		0			0
$473$	−35.1754			−1.61737
$474$		0			0
$475$	−13.9248	+	1.71767i	−0.638913	+	0.0788120i
$476$		0			0
$477$		−	5.53690i		−	0.253517i
$478$		0			0
$479$			8.15045i			0.372403i	0.982512	+	0.186202i	$0.0596178\pi$
$479$			8.15045i			0.372403i	−0.982512	+	0.186202i	$0.940382\pi$
$480$		0			0
$481$	−24.1622	−	10.8872i	−1.10170	−	0.496412i
$482$		0			0
$483$	−3.35026			−0.152442
$484$		0			0
$485$	−4.34297	+	4.91160i	−0.197204	+	0.223024i
$486$		0			0
$487$	35.4215			1.60510			0.802551	−	0.596583i	$-0.203476\pi$
$487$	35.4215			1.60510			0.802551	+	0.596583i	$0.203476\pi$
$488$		0			0
$489$		−	12.0508i		−	0.544955i
$490$		0			0
$491$	35.8046			1.61584			0.807921	−	0.589291i	$-0.200593\pi$
$491$	35.8046			1.61584			0.807921	+	0.589291i	$0.200593\pi$
$492$		0			0
$493$		−	8.96239i		−	0.403646i
$494$		0			0
$495$	−8.96239	−	7.92478i	−0.402829	−	0.356192i
$496$		0			0
$497$			37.2506i			1.67092i
$498$		0			0
$499$		−	29.1939i		−	1.30690i	−0.756970	−	0.653450i	$-0.773321\pi$
$499$		−	29.1939i		−	1.30690i	0.756970	−	0.653450i	$-0.226679\pi$
$500$		0			0
$501$			1.61213i			0.0720245i
$502$		0			0
$503$		−	0.806063i		−	0.0359406i	−0.999839	−	0.0179703i	$-0.994280\pi$
$503$		−	0.806063i		−	0.0359406i	0.999839	−	0.0179703i	$-0.00572043\pi$
$504$		0			0
$505$	−8.73084	+	9.87399i	−0.388517	+	0.439387i
$506$		0			0
$507$	9.73813	−	8.61213i	0.432486	−	0.382478i
$508$		0			0
$509$			11.9149i			0.528120i	0.964506	+	0.264060i	$0.0850617\pi$
$509$			11.9149i			0.528120i	−0.964506	+	0.264060i	$0.914938\pi$
$510$		0			0
$511$	16.9624			0.750372
$512$		0			0
$513$	2.80606			0.123891
$514$		0			0
$515$	5.52705	+	4.88717i	0.243551	+	0.215354i
$516$		0			0
$517$			65.8759i			2.89722i
$518$		0			0
$519$	1.22425			0.0537388
$520$		0			0
$521$	16.4485			0.720622			0.360311	−	0.932832i	$-0.382671\pi$
$521$	16.4485			0.720622			0.360311	+	0.932832i	$0.382671\pi$
$522$		0			0
$523$		−	6.07522i		−	0.265651i	−0.991139	−	0.132825i	$-0.957595\pi$
$523$		−	6.07522i		−	0.265651i	0.991139	−	0.132825i	$-0.0424050\pi$
$524$		0			0
$525$	−20.6253	+	2.54420i	−0.900162	+	0.111038i
$526$		0			0
$527$	9.46168			0.412157
$528$		0			0
$529$	22.3503			0.971751
$530$		0			0
$531$			12.8872i			0.559255i
$532$		0			0
$533$	5.35026	−	11.8740i	0.231746	−	0.514320i
$534$		0			0
$535$	33.1490	+	29.3112i	1.43316	+	1.26724i
$536$		0			0
$537$		−	1.25457i		−	0.0541387i
$538$		0			0
$539$		−	54.9741i		−	2.36790i
$540$		0			0
$541$			6.10554i			0.262498i	0.991349	+	0.131249i	$0.0418987\pi$
$541$			6.10554i			0.262498i	−0.991349	+	0.131249i	$0.958101\pi$
$542$		0			0
$543$		−	5.47627i		−	0.235009i
$544$		0			0
$545$	15.7988	+	13.9697i	0.676745	+	0.598395i
$546$		0			0
$547$			0.775746i			0.0331685i	0.999862	+	0.0165843i	$0.00527918\pi$
$547$			0.775746i			0.0331685i	−0.999862	+	0.0165843i	$0.994721\pi$
$548$		0			0
$549$	−6.31265			−0.269417
$550$		0			0
$551$		−	21.0640i		−	0.897355i
$552$		0			0
$553$	51.6991			2.19847
$554$		0			0
$555$	−12.3127	−	10.8872i	−0.522643	−	0.462134i
$556$		0			0
$557$	−5.13918			−0.217754			−0.108877	−	0.994055i	$-0.534725\pi$
$557$	−5.13918			−0.217754			−0.108877	+	0.994055i	$0.534725\pi$
$558$		0			0
$559$	9.73813	−	21.6121i	0.411879	−	0.914096i
$560$		0			0
$561$			6.38787i			0.269696i
$562$		0			0
$563$			30.3390i			1.27864i	0.768942	+	0.639318i	$0.220783\pi$
$563$			30.3390i			1.27864i	−0.768942	+	0.639318i	$0.779217\pi$
$564$		0			0
$565$	−7.40105	−	6.54420i	−0.311364	−	0.275316i
$566$		0			0
$567$	4.15633			0.174549
$568$		0			0
$569$	−7.25060			−0.303961			−0.151981	−	0.988383i	$-0.548565\pi$
$569$	−7.25060			−0.303961			−0.151981	+	0.988383i	$0.548565\pi$
$570$		0			0
$571$	−16.3127			−0.682663			−0.341332	−	0.939943i	$-0.610878\pi$
$571$	−16.3127			−0.682663			−0.341332	+	0.939943i	$0.610878\pi$
$572$		0			0
$573$		−	10.2374i		−	0.427675i
$574$		0			0
$575$	−4.00000	+	0.493413i	−0.166812	+	0.0205767i
$576$		0			0
$577$	−4.85685			−0.202193			−0.101097	−	0.994877i	$-0.532235\pi$
$577$	−4.85685			−0.202193			−0.101097	+	0.994877i	$0.532235\pi$
$578$		0			0
$579$		−	9.06793i		−	0.376850i
$580$		0			0
$581$	−41.7743			−1.73309
$582$		0			0
$583$	29.6239			1.22690
$584$		0			0
$585$	7.35026	−	3.31265i	0.303896	−	0.136961i
$586$		0			0
$587$	18.8218			0.776859			0.388429	−	0.921479i	$-0.373018\pi$
$587$	18.8218			0.776859			0.388429	+	0.921479i	$0.373018\pi$
$588$		0			0
$589$	22.2374			0.916277
$590$		0			0
$591$		−	3.14903i		−	0.129534i
$592$		0			0
$593$	−5.59754			−0.229863			−0.114932	−	0.993373i	$-0.536665\pi$
$593$	−5.59754			−0.229863			−0.114932	+	0.993373i	$0.536665\pi$
$594$		0			0
$595$	8.31265	+	7.35026i	0.340785	+	0.301331i
$596$		0			0
$597$			14.1114i			0.577542i
$598$		0			0
$599$	−47.9511			−1.95923			−0.979615	−	0.200885i	$-0.935618\pi$
$599$	−47.9511			−1.95923			−0.979615	+	0.200885i	$0.935618\pi$
$600$		0			0
$601$	−1.97556			−0.0805849			−0.0402924	−	0.999188i	$-0.512829\pi$
$601$	−1.97556			−0.0805849			−0.0402924	+	0.999188i	$0.512829\pi$
$602$		0			0
$603$	−4.57452			−0.186289
$604$		0			0
$605$	26.1065	−	29.5247i	1.06138	−	1.20035i
$606$		0			0
$607$			18.1622i			0.737181i	0.929592	+	0.368591i	$0.120160\pi$
$607$			18.1622i			0.737181i	−0.929592	+	0.368591i	$0.879840\pi$
$608$		0			0
$609$		−	31.1998i		−	1.26428i
$610$		0			0
$611$	−40.4749	−	18.2374i	−1.63744	−	0.737807i
$612$		0			0
$613$	−30.6761			−1.23900			−0.619498	−	0.784998i	$-0.712664\pi$
$613$	−30.6761			−1.23900			−0.619498	+	0.784998i	$0.712664\pi$
$614$		0			0
$615$	5.35026	−	6.05079i	0.215743	−	0.243991i
$616$		0			0
$617$	37.3258			1.50268			0.751341	−	0.659915i	$-0.229408\pi$
$617$	37.3258			1.50268			0.751341	+	0.659915i	$0.229408\pi$
$618$		0			0
$619$		−	17.9814i		−	0.722735i	−0.932423	−	0.361368i	$-0.882310\pi$
$619$		−	17.9814i		−	0.722735i	0.932423	−	0.361368i	$-0.117690\pi$
$620$		0			0
$621$	0.806063			0.0323462
$622$		0			0
$623$		−	20.9380i		−	0.838861i
$624$		0			0
$625$	−24.2506	+	6.07522i	−0.970024	+	0.243009i
$626$		0			0
$627$			15.0132i			0.599568i
$628$		0			0
$629$			8.77575i			0.349912i
$630$		0			0
$631$			32.2374i			1.28335i	0.766976	+	0.641676i	$0.221761\pi$
$631$			32.2374i			1.28335i	−0.766976	+	0.641676i	$0.778239\pi$
$632$		0			0
$633$		−	17.4617i		−	0.694040i
$634$		0			0
$635$	−30.4241	−	26.9018i	−1.20734	−	1.06756i
$636$		0			0
$637$	33.7767	+	15.2193i	1.33828	+	0.603012i
$638$		0			0
$639$		−	8.96239i		−	0.354547i
$640$		0			0
$641$	−12.6107			−0.498093			−0.249047	−	0.968492i	$-0.580117\pi$
$641$	−12.6107			−0.498093			−0.249047	+	0.968492i	$0.580117\pi$
$642$		0			0
$643$	−11.0230			−0.434706			−0.217353	−	0.976093i	$-0.569742\pi$
$643$	−11.0230			−0.434706			−0.217353	+	0.976093i	$0.569742\pi$
$644$		0			0
$645$	9.73813	−	11.0132i	0.383439	−	0.433643i
$646$		0			0
$647$		−	29.8046i		−	1.17174i	−0.810404	−	0.585871i	$-0.800753\pi$
$647$		−	29.8046i		−	1.17174i	0.810404	−	0.585871i	$-0.199247\pi$
$648$		0			0
$649$	−68.9497			−2.70651
$650$		0			0
$651$	32.9380			1.29094
$652$		0			0
$653$			44.8627i			1.75561i	0.479014	+	0.877807i	$0.340994\pi$
$653$			44.8627i			1.75561i	−0.479014	+	0.877807i	$0.659006\pi$
$654$		0			0
$655$	−14.5442	+	16.4485i	−0.568289	+	0.642696i
$656$		0			0
$657$	−4.08110			−0.159219
$658$		0			0
$659$	−26.9683			−1.05053			−0.525267	−	0.850937i	$-0.676035\pi$
$659$	−26.9683			−1.05053			−0.525267	+	0.850937i	$0.676035\pi$
$660$		0			0
$661$			42.9537i			1.67070i	0.549715	+	0.835352i	$0.314736\pi$
$661$			42.9537i			1.67070i	−0.549715	+	0.835352i	$0.685264\pi$
$662$		0			0
$663$	−3.92478	−	1.76845i	−0.152426	−	0.0686810i
$664$		0			0
$665$	19.5369	+	17.2750i	0.757609	+	0.669897i
$666$		0			0
$667$		−	6.05079i		−	0.234287i
$668$		0			0
$669$			19.6326i			0.759040i
$670$		0			0
$671$		−	33.7743i		−	1.30384i
$672$		0			0
$673$			14.8627i			0.572916i	0.958093	+	0.286458i	$0.0924779\pi$
$673$			14.8627i			0.572916i	−0.958093	+	0.286458i	$0.907522\pi$
$674$		0			0
$675$	4.96239	−	0.612127i	0.191002	−	0.0235608i
$676$		0			0
$677$		−	0.911603i		−	0.0350358i	−0.999847	−	0.0175179i	$-0.994424\pi$
$677$		−	0.911603i		−	0.0350358i	0.999847	−	0.0175179i	$-0.00557640\pi$
$678$		0			0
$679$	12.1866			0.467680
$680$		0			0
$681$		−	4.77575i		−	0.183007i
$682$		0			0
$683$	27.1998			1.04077			0.520386	−	0.853931i	$-0.325788\pi$
$683$	27.1998			1.04077			0.520386	+	0.853931i	$0.325788\pi$
$684$		0			0
$685$	−1.35026	+	1.52705i	−0.0515908	+	0.0583458i
$686$		0			0
$687$	−19.5672			−0.746536
$688$		0			0
$689$	−8.20123	+	18.2012i	−0.312442	+	0.693412i
$690$		0			0
$691$		−	25.9697i		−	0.987933i	−0.869481	−	0.493967i	$-0.835546\pi$
$691$		−	25.9697i		−	0.987933i	0.869481	−	0.493967i	$-0.164454\pi$
$692$		0			0
$693$			22.2374i			0.844730i
$694$		0			0
$695$	1.23884	−	1.40105i	0.0469920	−	0.0531447i
$696$		0			0
$697$	−4.31265			−0.163353
$698$		0			0
$699$	2.74543			0.103842
$700$		0			0
$701$	−5.20853			−0.196723			−0.0983616	−	0.995151i	$-0.531360\pi$
$701$	−5.20853			−0.196723			−0.0983616	+	0.995151i	$0.531360\pi$
$702$		0			0
$703$			20.6253i			0.777898i
$704$		0			0
$705$	−20.6253	−	18.2374i	−0.776794	−	0.686861i
$706$		0			0
$707$	24.4993			0.921391
$708$		0			0
$709$		−	6.79147i		−	0.255059i	−0.991835	−	0.127530i	$-0.959295\pi$
$709$		−	6.79147i		−	0.255059i	0.991835	−	0.127530i	$-0.0407048\pi$
$710$		0			0
$711$	−12.4387			−0.466486
$712$		0			0
$713$	6.38787			0.239228
$714$		0			0
$715$	17.7235	+	39.3258i	0.662823	+	1.47070i
$716$		0			0
$717$	−5.29948			−0.197913
$718$		0			0
$719$	20.9986			0.783115			0.391558	−	0.920154i	$-0.371936\pi$
$719$	20.9986			0.783115			0.391558	+	0.920154i	$0.371936\pi$
$720$		0			0
$721$		−	13.7137i		−	0.510725i
$722$		0			0
$723$	12.9380			0.481168
$724$		0			0
$725$	−4.59498	−	37.2506i	−0.170653	−	1.38345i
$726$		0			0
$727$			21.4763i			0.796511i	0.917275	+	0.398255i	$0.130384\pi$
$727$			21.4763i			0.796511i	−0.917275	+	0.398255i	$0.869616\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−7.84955			−0.290326
$732$		0			0
$733$	−10.5139			−0.388339			−0.194170	−	0.980968i	$-0.562201\pi$
$733$	−10.5139			−0.388339			−0.194170	+	0.980968i	$0.562201\pi$
$734$		0			0
$735$	17.2120	+	15.2193i	0.634875	+	0.561373i
$736$		0			0
$737$		−	24.4749i		−	0.901543i
$738$		0			0
$739$			22.7454i			0.836704i	0.908285	+	0.418352i	$0.137392\pi$
$739$			22.7454i			0.836704i	−0.908285	+	0.418352i	$0.862608\pi$
$740$		0			0
$741$	−9.22425	−	4.15633i	−0.338861	−	0.152686i
$742$		0			0
$743$	−31.9511			−1.17217			−0.586087	−	0.810248i	$-0.699332\pi$
$743$	−31.9511			−1.17217			−0.586087	+	0.810248i	$0.699332\pi$
$744$		0			0
$745$	18.2374	+	16.1260i	0.668168	+	0.590811i
$746$		0			0
$747$	10.0508			0.367739
$748$		0			0
$749$		−	82.2492i		−	3.00532i
$750$		0			0
$751$	−11.5975			−0.423200			−0.211600	−	0.977356i	$-0.567867\pi$
$751$	−11.5975			−0.423200			−0.211600	+	0.977356i	$0.567867\pi$
$752$		0			0
$753$			8.10554i			0.295382i
$754$		0			0
$755$	29.7988	+	26.3488i	1.08449	+	0.958933i
$756$		0			0
$757$		−	33.5125i		−	1.21803i	−0.793158	−	0.609016i	$-0.791565\pi$
$757$		−	33.5125i		−	1.21803i	0.793158	−	0.609016i	$-0.208435\pi$
$758$		0			0
$759$			4.31265i			0.156539i
$760$		0			0
$761$			36.7123i			1.33082i	0.746478	+	0.665410i	$0.231743\pi$
$761$			36.7123i			1.33082i	−0.746478	+	0.665410i	$0.768257\pi$
$762$		0			0
$763$		−	39.1998i		−	1.41913i
$764$		0			0
$765$	−2.00000	−	1.76845i	−0.0723102	−	0.0639385i
$766$		0			0
$767$	19.0884	−	42.3634i	0.689242	−	1.52966i
$768$		0			0
$769$		−	45.4010i		−	1.63720i	−0.574361	−	0.818602i	$-0.694750\pi$
$769$		−	45.4010i		−	1.63720i	0.574361	−	0.818602i	$-0.305250\pi$
$770$		0			0
$771$	−12.0567			−0.434210
$772$		0			0
$773$	−2.52373			−0.0907723			−0.0453861	−	0.998970i	$-0.514452\pi$
$773$	−2.52373			−0.0907723			−0.0453861	+	0.998970i	$0.514452\pi$
$774$		0			0
$775$	39.3258	−	4.85097i	1.41263	−	0.174252i
$776$		0			0
$777$			30.5501i			1.09598i
$778$		0			0
$779$	−10.1359			−0.363155
$780$		0			0
$781$	47.9511			1.71583
$782$		0			0
$783$			7.50659i			0.268264i
$784$		0			0
$785$	18.0508	+	15.9610i	0.644260	+	0.569672i
$786$		0			0
$787$	−7.12459			−0.253964			−0.126982	−	0.991905i	$-0.540529\pi$
$787$	−7.12459			−0.253964			−0.126982	+	0.991905i	$0.540529\pi$
$788$		0			0
$789$	−4.28233			−0.152455
$790$		0			0
$791$			18.3634i			0.652929i
$792$		0			0
$793$	20.7513	+	9.35026i	0.736901	+	0.332038i
$794$		0			0
$795$	−8.20123	+	9.27504i	−0.290868	+	0.328952i
$796$		0			0
$797$		−	42.2031i		−	1.49491i	−0.664311	−	0.747456i	$-0.731275\pi$
$797$		−	42.2031i		−	1.49491i	0.664311	−	0.747456i	$-0.268725\pi$
$798$		0			0
$799$			14.7005i			0.520067i
$800$		0			0
$801$			5.03761i			0.177995i
$802$		0			0
$803$		−	21.8350i		−	0.770539i
$804$		0			0
$805$	5.61213	+	4.96239i	0.197801	+	0.174901i
$806$		0			0
$807$			19.8799i			0.699805i
$808$		0			0
$809$	−32.2374			−1.13341			−0.566704	−	0.823922i	$-0.691782\pi$
$809$	−32.2374			−1.13341			−0.566704	+	0.823922i	$0.691782\pi$
$810$		0			0
$811$			38.9076i			1.36623i	0.730310	+	0.683116i	$0.239376\pi$
$811$			38.9076i			1.36623i	−0.730310	+	0.683116i	$0.760624\pi$
$812$		0			0
$813$	5.01317			0.175820
$814$		0			0
$815$	−17.8496	+	20.1866i	−0.625243	+	0.707107i
$816$		0			0
$817$	−18.4485			−0.645432
$818$		0			0
$819$	−13.6629	−	6.15633i	−0.477421	−	0.215119i
$820$		0			0
$821$			44.4095i			1.54990i	0.632022	+	0.774951i	$0.282225\pi$
$821$			44.4095i			1.54990i	−0.632022	+	0.774951i	$0.717775\pi$
$822$		0			0
$823$		−	14.0000i		−	0.488009i	−0.969774	−	0.244005i	$-0.921539\pi$
$823$		−	14.0000i		−	0.488009i	0.969774	−	0.244005i	$-0.0784612\pi$
$824$		0			0
$825$	3.27504	+	26.5501i	0.114022	+	0.924355i
$826$		0			0
$827$	9.02302			0.313761			0.156881	−	0.987618i	$-0.449856\pi$
$827$	9.02302			0.313761			0.156881	+	0.987618i	$0.449856\pi$
$828$		0			0
$829$	−43.1002			−1.49693			−0.748465	−	0.663174i	$-0.769209\pi$
$829$	−43.1002			−1.49693			−0.748465	+	0.663174i	$0.769209\pi$
$830$		0			0
$831$	8.55008			0.296599
$832$		0			0
$833$		−	12.2677i		−	0.425052i
$834$		0			0
$835$	2.38787	−	2.70052i	0.0826358	−	0.0934555i
$836$		0			0
$837$	−7.92478			−0.273920
$838$		0			0
$839$		−	12.5237i		−	0.432367i	−0.976353	−	0.216184i	$-0.930639\pi$
$839$		−	12.5237i		−	0.432367i	0.976353	−	0.216184i	$-0.0693610\pi$
$840$		0			0
$841$	27.3488			0.943064
$842$		0			0
$843$	−25.5125			−0.878696
$844$		0			0
$845$	−29.0689	+	0.00236967i	−1.00000	+	8.15191e-5i
$846$		0			0
$847$	−73.2565			−2.51712
$848$		0			0
$849$	23.7235			0.814190
$850$		0			0
$851$			5.92478i			0.203099i
$852$		0			0
$853$	42.3146			1.44882			0.724411	−	0.689368i	$-0.242112\pi$
$853$	42.3146			1.44882			0.724411	+	0.689368i	$0.242112\pi$
$854$		0			0
$855$	−4.70052	−	4.15633i	−0.160755	−	0.142143i
$856$		0			0
$857$			27.9960i			0.956326i	0.878271	+	0.478163i	$0.158697\pi$
$857$			27.9960i			0.956326i	−0.878271	+	0.478163i	$0.841303\pi$
$858$		0			0
$859$	27.3865			0.934414			0.467207	−	0.884148i	$-0.345260\pi$
$859$	27.3865			0.934414			0.467207	+	0.884148i	$0.345260\pi$
$860$		0			0
$861$	−15.0132			−0.511648
$862$		0			0
$863$	−40.0118			−1.36202			−0.681008	−	0.732276i	$-0.738458\pi$
$863$	−40.0118			−1.36202			−0.681008	+	0.732276i	$0.738458\pi$
$864$		0			0
$865$	−2.05079	−	1.81336i	−0.0697288	−	0.0616560i
$866$		0			0
$867$		−	15.5745i		−	0.528938i
$868$		0			0
$869$		−	66.5501i		−	2.25756i
$870$		0			0
$871$	15.0376	+	6.77575i	0.509530	+	0.229587i
$872$		0			0
$873$	−2.93207			−0.0992356
$874$		0			0
$875$	38.3185	+	26.2882i	1.29540	+	0.888704i
$876$		0			0
$877$	−21.9610			−0.741569			−0.370785	−	0.928719i	$-0.620911\pi$
$877$	−21.9610			−0.741569			−0.370785	+	0.928719i	$0.620911\pi$
$878$		0			0
$879$			23.9248i			0.806963i
$880$		0			0
$881$	−39.9657			−1.34648			−0.673240	−	0.739424i	$-0.735098\pi$
$881$	−39.9657			−1.34648			−0.673240	+	0.739424i	$0.735098\pi$
$882$		0			0
$883$			19.4763i			0.655429i	0.944777	+	0.327714i	$0.106278\pi$
$883$			19.4763i			0.655429i	−0.944777	+	0.327714i	$0.893722\pi$
$884$		0			0
$885$	19.0884	−	21.5877i	0.641649	−	0.725662i
$886$		0			0
$887$			43.5672i			1.46284i	0.681925	+	0.731422i	$0.261143\pi$
$887$			43.5672i			1.46284i	−0.681925	+	0.731422i	$0.738857\pi$
$888$		0			0
$889$			75.4880i			2.53179i
$890$		0			0
$891$		−	5.35026i		−	0.179241i
$892$		0			0
$893$			34.5501i			1.15617i
$894$		0			0
$895$	−1.85826	+	2.10157i	−0.0621149	+	0.0702478i
$896$		0			0
$897$	−2.64974	−	1.19394i	−0.0884722	−	0.0398644i
$898$		0			0
$899$			59.4880i			1.98404i
$900$		0			0
$901$	6.61071			0.220235
$902$		0			0
$903$	−27.3258			−0.909346
$904$		0			0
$905$	−8.11142	+	9.17347i	−0.269633	+	0.304936i
$906$		0			0
$907$		−	35.3014i		−	1.17216i	−0.810252	−	0.586082i	$-0.800670\pi$
$907$		−	35.3014i		−	1.17216i	0.810252	−	0.586082i	$-0.199330\pi$
$908$		0			0
$909$	−5.89446			−0.195507
$910$		0			0
$911$	−36.8773			−1.22180			−0.610900	−	0.791708i	$-0.709192\pi$
$911$	−36.8773			−1.22180			−0.610900	+	0.791708i	$0.709192\pi$
$912$		0			0
$913$			53.7743i			1.77967i
$914$		0			0
$915$	10.5745	+	9.35026i	0.349583	+	0.309110i
$916$		0			0
$917$	40.8119			1.34773
$918$		0			0
$919$	45.1490			1.48933			0.744665	−	0.667439i	$-0.232609\pi$
$919$	45.1490			1.48933			0.744665	+	0.667439i	$0.232609\pi$
$920$		0			0
$921$			33.5125i			1.10427i
$922$		0			0
$923$	−13.2750	+	29.4617i	−0.436953	+	0.969743i
$924$		0			0
$925$	4.49929	+	36.4749i	0.147936	+	1.19929i
$926$		0			0
$927$			3.29948i			0.108369i
$928$		0			0
$929$		−	11.8594i		−	0.389094i	−0.980893	−	0.194547i	$-0.937676\pi$
$929$		−	11.8594i		−	0.389094i	0.980893	−	0.194547i	$-0.0623237\pi$
$930$		0			0
$931$		−	28.8324i		−	0.944944i
$932$		0			0
$933$		−	5.76257i		−	0.188658i
$934$		0			0
$935$	9.46168	−	10.7005i	0.309430	−	0.349945i
$936$		0			0
$937$			14.6399i			0.478264i	0.970987	+	0.239132i	$0.0768629\pi$
$937$			14.6399i			0.478264i	−0.970987	+	0.239132i	$0.923137\pi$
$938$		0			0
$939$	12.6253			0.412011
$940$		0			0
$941$		−	33.0033i		−	1.07588i	−0.842984	−	0.537939i	$-0.819203\pi$
$941$		−	33.0033i		−	1.07588i	0.842984	−	0.537939i	$-0.180797\pi$
$942$		0			0
$943$	−2.91160			−0.0948149
$944$		0			0
$945$	−6.96239	−	6.15633i	−0.226487	−	0.200265i
$946$		0			0
$947$	14.5745			0.473608			0.236804	−	0.971557i	$-0.423900\pi$
$947$	14.5745			0.473608			0.236804	+	0.971557i	$0.423900\pi$
$948$		0			0
$949$	13.4156	+	6.04491i	0.435490	+	0.196226i
$950$		0			0
$951$			3.73813i			0.121217i
$952$		0			0
$953$			26.0910i			0.845169i	0.906324	+	0.422584i	$0.138877\pi$
$953$			26.0910i			0.845169i	−0.906324	+	0.422584i	$0.861123\pi$
$954$		0			0
$955$	−15.1636	+	17.1490i	−0.490683	+	0.554930i
$956$		0			0
$957$	−40.1622			−1.29826
$958$		0			0
$959$	3.78892			0.122351
$960$		0			0
$961$	−31.8021			−1.02587
$962$		0			0
$963$			19.7889i			0.637689i
$964$		0			0
$965$	−13.4314	+	15.1900i	−0.432371	+	0.488982i
$966$		0			0
$967$	−56.7328			−1.82440			−0.912201	−	0.409743i	$-0.865618\pi$
$967$	−56.7328			−1.82440			−0.912201	+	0.409743i	$0.865618\pi$
$968$		0			0
$969$			3.35026i			0.107626i
$970$		0			0
$971$	−33.4168			−1.07240			−0.536198	−	0.844092i	$-0.680140\pi$
$971$	−33.4168			−1.07240			−0.536198	+	0.844092i	$0.680140\pi$
$972$		0			0
$973$	−3.47627			−0.111444
$974$		0			0
$975$	−17.2193	−	5.33804i	−0.551460	−	0.170954i
$976$		0			0
$977$	−37.5778			−1.20222			−0.601111	−	0.799166i	$-0.705275\pi$
$977$	−37.5778			−1.20222			−0.601111	+	0.799166i	$0.705275\pi$
$978$		0			0
$979$	−26.9525			−0.861407
$980$		0			0
$981$			9.43136i			0.301120i
$982$		0			0
$983$	−34.2981			−1.09394			−0.546969	−	0.837153i	$-0.684219\pi$
$983$	−34.2981			−1.09394			−0.546969	+	0.837153i	$0.684219\pi$
$984$		0			0
$985$	−4.66433	+	5.27504i	−0.148618	+	0.168077i
$986$		0			0
$987$			51.1754i			1.62893i
$988$		0			0
$989$	−5.29948			−0.168514
$990$		0			0
$991$	−19.5613			−0.621386			−0.310693	−	0.950510i	$-0.600561\pi$
$991$	−19.5613			−0.621386			−0.310693	+	0.950510i	$0.600561\pi$
$992$		0			0
$993$	−1.81924			−0.0577317
$994$		0			0
$995$	20.9018	−	23.6385i	0.662630	−	0.749390i
$996$		0			0
$997$		−	45.9511i		−	1.45529i	−0.685956	−	0.727643i	$-0.740616\pi$
$997$		−	45.9511i		−	1.45529i	0.685956	−	0.727643i	$-0.259384\pi$
$998$		0			0
$999$		−	7.35026i		−	0.232552i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3120.2.r.h.2209.1		6
4.3	odd	2		390.2.f.b.259.4	yes	6
5.4	even	2		3120.2.r.g.2209.6		6
12.11	even	2		1170.2.f.c.649.5		6
13.12	even	2		3120.2.r.g.2209.3		6
20.3	even	4		1950.2.b.l.1351.3		6
20.7	even	4		1950.2.b.m.1351.4		6
20.19	odd	2		390.2.f.a.259.3	✓	6
52.51	odd	2		390.2.f.a.259.6	yes	6
60.59	even	2		1170.2.f.d.649.1		6
65.64	even	2	inner	3120.2.r.h.2209.4		6
156.155	even	2		1170.2.f.d.649.2		6
260.103	even	4		1950.2.b.l.1351.4		6
260.207	even	4		1950.2.b.m.1351.3		6
260.259	odd	2		390.2.f.b.259.1	yes	6
780.779	even	2		1170.2.f.c.649.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.f.a.259.3	✓	6	20.19	odd	2
390.2.f.a.259.6	yes	6	52.51	odd	2
390.2.f.b.259.1	yes	6	260.259	odd	2
390.2.f.b.259.4	yes	6	4.3	odd	2
1170.2.f.c.649.5		6	12.11	even	2
1170.2.f.c.649.6		6	780.779	even	2
1170.2.f.d.649.1		6	60.59	even	2
1170.2.f.d.649.2		6	156.155	even	2
1950.2.b.l.1351.3		6	20.3	even	4
1950.2.b.l.1351.4		6	260.103	even	4
1950.2.b.m.1351.3		6	260.207	even	4
1950.2.b.m.1351.4		6	20.7	even	4
3120.2.r.g.2209.3		6	13.12	even	2
3120.2.r.g.2209.6		6	5.4	even	2
3120.2.r.h.2209.1		6	1.1	even	1	trivial
3120.2.r.h.2209.4		6	65.64	even	2	inner

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3120.r (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +(-1.48119 + 1.67513i) q^{5} +4.15633 q^{7} -1.00000 q^{9} -5.35026i q^{11} +(3.28726 + 1.48119i) q^{13} +(1.67513 + 1.48119i) q^{15} -1.19394i q^{17} -2.80606i q^{19} -4.15633i q^{21} -0.806063i q^{23} +(-0.612127 - 4.96239i) q^{25} +1.00000i q^{27} +7.50659 q^{29} +7.92478i q^{31} -5.35026 q^{33} +(-6.15633 + 6.96239i) q^{35} -7.35026 q^{37} +(1.48119 - 3.28726i) q^{39} -3.61213i q^{41} -6.57452i q^{43} +(1.48119 - 1.67513i) q^{45} -12.3127 q^{47} +10.2750 q^{49} -1.19394 q^{51} +5.53690i q^{53} +(8.96239 + 7.92478i) q^{55} -2.80606 q^{57} -12.8872i q^{59} +6.31265 q^{61} -4.15633 q^{63} +(-7.35026 + 3.31265i) q^{65} +4.57452 q^{67} -0.806063 q^{69} +8.96239i q^{71} +4.08110 q^{73} +(-4.96239 + 0.612127i) q^{75} -22.2374i q^{77} +12.4387 q^{79} +1.00000 q^{81} -10.0508 q^{83} +(2.00000 + 1.76845i) q^{85} -7.50659i q^{87} -5.03761i q^{89} +(13.6629 + 6.15633i) q^{91} +7.92478 q^{93} +(4.70052 + 4.15633i) q^{95} +2.93207 q^{97} +5.35026i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{5} + 4 q^{7} - 6 q^{9} + 8 q^{13} - 2 q^{25} + 4 q^{29} - 12 q^{33} - 16 q^{35} - 24 q^{37} - 2 q^{39} - 2 q^{45} - 32 q^{47} - 2 q^{49} - 8 q^{51} + 32 q^{55} - 16 q^{57} - 4 q^{61} - 4 q^{63}+ \cdots - 12 q^{95}+O(q^{100}) \) 6 * q + 2 * q^5 + 4 * q^7 - 6 * q^9 + 8 * q^13 - 2 * q^25 + 4 * q^29 - 12 * q^33 - 16 * q^35 - 24 * q^37 - 2 * q^39 - 2 * q^45 - 32 * q^47 - 2 * q^49 - 8 * q^51 + 32 * q^55 - 16 * q^57 - 4 * q^61 - 4 * q^63 - 24 * q^65 + 4 * q^67 - 4 * q^69 - 40 * q^73 - 8 * q^75 + 16 * q^79 + 6 * q^81 + 12 * q^85 + 20 * q^91 + 4 * q^93 - 12 * q^95

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