LMFDB - Embedded newform 4788.2.w.a.1261.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4788,2,Mod(505,4788)] 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("4788.505"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4788, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 4])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4788 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4788.w (of order $3$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-3,0,2,0,0,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$38.2323724878$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1596)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1261.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	4788.1261
Dual form			4788.2.w.a.505.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.50000 - 2.59808i) q^{5} +1.00000 q^{7} -3.00000 q^{11} +(2.00000 - 3.46410i) q^{13} +(1.50000 + 2.59808i) q^{17} +(-0.500000 + 4.33013i) q^{19} +(-2.00000 + 3.46410i) q^{25} +8.00000 q^{31} +(-1.50000 - 2.59808i) q^{35} +11.0000 q^{37} +(4.50000 + 7.79423i) q^{41} +(-4.00000 - 6.92820i) q^{43} +(6.00000 - 10.3923i) q^{47} +1.00000 q^{49} +(6.00000 - 10.3923i) q^{53} +(4.50000 + 7.79423i) q^{55} +(-6.00000 - 10.3923i) q^{59} +(-1.00000 + 1.73205i) q^{61} -12.0000 q^{65} +(2.00000 - 3.46410i) q^{67} +(-6.00000 - 10.3923i) q^{71} +(-1.00000 - 1.73205i) q^{73} -3.00000 q^{77} +(5.00000 + 8.66025i) q^{79} -6.00000 q^{83} +(4.50000 - 7.79423i) q^{85} +(-7.50000 + 12.9904i) q^{89} +(2.00000 - 3.46410i) q^{91} +(12.0000 - 5.19615i) q^{95} +(-4.00000 - 6.92820i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{5} + 2 q^{7} - 6 q^{11} + 4 q^{13} + 3 q^{17} - q^{19} - 4 q^{25} + 16 q^{31} - 3 q^{35} + 22 q^{37} + 9 q^{41} - 8 q^{43} + 12 q^{47} + 2 q^{49} + 12 q^{53} + 9 q^{55} - 12 q^{59} - 2 q^{61}+ \cdots - 8 q^{97}+O(q^{100}) $ 2 * q - 3 * q^5 + 2 * q^7 - 6 * q^11 + 4 * q^13 + 3 * q^17 - q^19 - 4 * q^25 + 16 * q^31 - 3 * q^35 + 22 * q^37 + 9 * q^41 - 8 * q^43 + 12 * q^47 + 2 * q^49 + 12 * q^53 + 9 * q^55 - 12 * q^59 - 2 * q^61 - 24 * q^65 + 4 * q^67 - 12 * q^71 - 2 * q^73 - 6 * q^77 + 10 * q^79 - 12 * q^83 + 9 * q^85 - 15 * q^89 + 4 * q^91 + 24 * q^95 - 8 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4788\mathbb{Z}\right)^\times$.

$n$	$533$	$1009$	$2395$	$4105$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.50000	−	2.59808i	−0.670820	−	1.16190i	−0.977672	−	0.210138i	$-0.932609\pi$
$5$	−1.50000	−	2.59808i	−0.670820	−	1.16190i	0.306851	−	0.951757i	$-0.400725\pi$
$6$		0			0
$7$	1.00000			0.377964
$7$	1.00000			0.377964
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.00000			−0.904534			−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000			−0.904534			−0.452267	+	0.891883i	$0.649385\pi$
$12$		0			0
$13$	2.00000	−	3.46410i	0.554700	−	0.960769i	−0.443227	−	0.896410i	$-0.646166\pi$
$13$	2.00000	−	3.46410i	0.554700	−	0.960769i	0.997927	−	0.0643593i	$-0.0205004\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.50000	+	2.59808i	0.363803	+	0.630126i	0.988583	−	0.150675i	$-0.0481447\pi$
$17$	1.50000	+	2.59808i	0.363803	+	0.630126i	−0.624780	+	0.780801i	$0.714811\pi$
$18$		0			0
$19$	−0.500000	+	4.33013i	−0.114708	+	0.993399i
$19$	−0.500000	+	4.33013i	−0.114708	+	0.993399i
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$23$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$24$		0			0
$25$	−2.00000	+	3.46410i	−0.400000	+	0.692820i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$29$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$30$		0			0
$31$	8.00000			1.43684			0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000			1.43684			0.718421	+	0.695608i	$0.244865\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.50000	−	2.59808i	−0.253546	−	0.439155i
$36$		0			0
$37$	11.0000			1.80839			0.904194	−	0.427121i	$-0.140472\pi$
$37$	11.0000			1.80839			0.904194	+	0.427121i	$0.140472\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	4.50000	+	7.79423i	0.702782	+	1.21725i	0.967486	+	0.252924i	$0.0813924\pi$
$41$	4.50000	+	7.79423i	0.702782	+	1.21725i	−0.264704	+	0.964330i	$0.585274\pi$
$42$		0			0
$43$	−4.00000	−	6.92820i	−0.609994	−	1.05654i	−0.991241	−	0.132068i	$-0.957838\pi$
$43$	−4.00000	−	6.92820i	−0.609994	−	1.05654i	0.381246	−	0.924473i	$-0.375495\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	6.00000	−	10.3923i	0.875190	−	1.51587i	0.0186297	−	0.999826i	$-0.494070\pi$
$47$	6.00000	−	10.3923i	0.875190	−	1.51587i	0.856560	−	0.516047i	$-0.172597\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$		0			0
$51$		0			0
$52$		0			0
$53$	6.00000	−	10.3923i	0.824163	−	1.42749i	−0.0783936	−	0.996922i	$-0.524979\pi$
$53$	6.00000	−	10.3923i	0.824163	−	1.42749i	0.902557	−	0.430570i	$-0.141688\pi$
$54$		0			0
$55$	4.50000	+	7.79423i	0.606780	+	1.05097i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−6.00000	−	10.3923i	−0.781133	−	1.35296i	−0.931282	−	0.364299i	$-0.881308\pi$
$59$	−6.00000	−	10.3923i	−0.781133	−	1.35296i	0.150148	−	0.988663i	$-0.452025\pi$
$60$		0			0
$61$	−1.00000	+	1.73205i	−0.128037	+	0.221766i	−0.922916	−	0.385002i	$-0.874201\pi$
$61$	−1.00000	+	1.73205i	−0.128037	+	0.221766i	0.794879	+	0.606768i	$0.207534\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−12.0000			−1.48842
$66$		0			0
$67$	2.00000	−	3.46410i	0.244339	−	0.423207i	−0.717607	−	0.696449i	$-0.754762\pi$
$67$	2.00000	−	3.46410i	0.244339	−	0.423207i	0.961946	+	0.273241i	$0.0880957\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−6.00000	−	10.3923i	−0.712069	−	1.23334i	−0.964079	−	0.265615i	$-0.914425\pi$
$71$	−6.00000	−	10.3923i	−0.712069	−	1.23334i	0.252010	−	0.967725i	$-0.418908\pi$
$72$		0			0
$73$	−1.00000	−	1.73205i	−0.117041	−	0.202721i	0.801553	−	0.597924i	$-0.204008\pi$
$73$	−1.00000	−	1.73205i	−0.117041	−	0.202721i	−0.918594	+	0.395203i	$0.870674\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−3.00000			−0.341882
$78$		0			0
$79$	5.00000	+	8.66025i	0.562544	+	0.974355i	0.997274	+	0.0737937i	$0.0235106\pi$
$79$	5.00000	+	8.66025i	0.562544	+	0.974355i	−0.434730	+	0.900561i	$0.643156\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−6.00000			−0.658586			−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000			−0.658586			−0.329293	+	0.944228i	$0.606810\pi$
$84$		0			0
$85$	4.50000	−	7.79423i	0.488094	−	0.845403i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−7.50000	+	12.9904i	−0.794998	+	1.37698i	0.127842	+	0.991795i	$0.459195\pi$
$89$	−7.50000	+	12.9904i	−0.794998	+	1.37698i	−0.922840	+	0.385183i	$0.874138\pi$
$90$		0			0
$91$	2.00000	−	3.46410i	0.209657	−	0.363137i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	12.0000	−	5.19615i	1.23117	−	0.533114i
$96$		0			0
$97$	−4.00000	−	6.92820i	−0.406138	−	0.703452i	0.588315	−	0.808632i	$-0.299792\pi$
$97$	−4.00000	−	6.92820i	−0.406138	−	0.703452i	−0.994453	+	0.105180i	$0.966458\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−4.50000	+	7.79423i	−0.447767	+	0.775555i	−0.998240	−	0.0592978i	$-0.981114\pi$
$101$	−4.50000	+	7.79423i	−0.447767	+	0.775555i	0.550474	+	0.834853i	$0.314447\pi$
$102$		0			0
$103$	−13.0000			−1.28093			−0.640464	−	0.767988i	$-0.721258\pi$
$103$	−13.0000			−1.28093			−0.640464	+	0.767988i	$0.721258\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−12.0000			−1.16008			−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000			−1.16008			−0.580042	+	0.814587i	$0.696964\pi$
$108$		0			0
$109$	3.50000	+	6.06218i	0.335239	+	0.580651i	0.983531	−	0.180741i	$-0.0578495\pi$
$109$	3.50000	+	6.06218i	0.335239	+	0.580651i	−0.648292	+	0.761392i	$0.724516\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	1.50000	+	2.59808i	0.137505	+	0.238165i
$120$		0			0
$121$	−2.00000			−0.181818
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−3.00000			−0.268328
$126$		0			0
$127$	8.00000	−	13.8564i	0.709885	−	1.22956i	−0.255014	−	0.966937i	$-0.582080\pi$
$127$	8.00000	−	13.8564i	0.709885	−	1.22956i	0.964899	−	0.262620i	$-0.0845865\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−6.00000	−	10.3923i	−0.524222	−	0.907980i	−0.999602	−	0.0281993i	$-0.991023\pi$
$131$	−6.00000	−	10.3923i	−0.524222	−	0.907980i	0.475380	−	0.879781i	$-0.342311\pi$
$132$		0			0
$133$	−0.500000	+	4.33013i	−0.0433555	+	0.375470i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−6.00000	+	10.3923i	−0.512615	+	0.887875i	0.487278	+	0.873247i	$0.337990\pi$
$137$	−6.00000	+	10.3923i	−0.512615	+	0.887875i	−0.999893	+	0.0146279i	$0.995344\pi$
$138$		0			0
$139$	6.50000	−	11.2583i	0.551323	−	0.954919i	−0.446857	−	0.894606i	$-0.647457\pi$
$139$	6.50000	−	11.2583i	0.551323	−	0.954919i	0.998179	−	0.0603135i	$-0.0192101\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−6.00000	+	10.3923i	−0.501745	+	0.869048i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	3.00000	+	5.19615i	0.245770	+	0.425685i	0.962348	−	0.271821i	$-0.0876260\pi$
$149$	3.00000	+	5.19615i	0.245770	+	0.425685i	−0.716578	+	0.697507i	$0.754293\pi$
$150$		0			0
$151$	8.00000			0.651031			0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000			0.651031			0.325515	+	0.945537i	$0.394462\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−12.0000	−	20.7846i	−0.963863	−	1.66946i
$156$		0			0
$157$	−10.0000	−	17.3205i	−0.798087	−	1.38233i	−0.920860	−	0.389892i	$-0.872512\pi$
$157$	−10.0000	−	17.3205i	−0.798087	−	1.38233i	0.122774	−	0.992435i	$-0.460821\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−22.0000			−1.72317			−0.861586	−	0.507611i	$-0.830529\pi$
$163$	−22.0000			−1.72317			−0.861586	+	0.507611i	$0.830529\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	9.00000	−	15.5885i	0.696441	−	1.20627i	−0.273252	−	0.961943i	$-0.588099\pi$
$167$	9.00000	−	15.5885i	0.696441	−	1.20627i	0.969693	−	0.244328i	$-0.0785675\pi$
$168$		0			0
$169$	−1.50000	−	2.59808i	−0.115385	−	0.199852i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−3.00000	−	5.19615i	−0.228086	−	0.395056i	0.729155	−	0.684349i	$-0.239913\pi$
$173$	−3.00000	−	5.19615i	−0.228086	−	0.395056i	−0.957241	+	0.289292i	$0.906580\pi$
$174$		0			0
$175$	−2.00000	+	3.46410i	−0.151186	+	0.261861i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	21.0000			1.56961			0.784807	−	0.619740i	$-0.212762\pi$
$179$	21.0000			1.56961			0.784807	+	0.619740i	$0.212762\pi$
$180$		0			0
$181$	−4.00000	+	6.92820i	−0.297318	+	0.514969i	−0.975521	−	0.219905i	$-0.929425\pi$
$181$	−4.00000	+	6.92820i	−0.297318	+	0.514969i	0.678204	+	0.734874i	$0.262759\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−16.5000	−	28.5788i	−1.21310	−	2.10116i
$186$		0			0
$187$	−4.50000	−	7.79423i	−0.329073	−	0.569970i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$	6.50000	+	11.2583i	0.467880	+	0.810392i	0.999326	−	0.0366998i	$-0.0116845\pi$
$193$	6.50000	+	11.2583i	0.467880	+	0.810392i	−0.531446	+	0.847092i	$0.678351\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−6.00000			−0.427482			−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000			−0.427482			−0.213741	+	0.976890i	$0.568565\pi$
$198$		0			0
$199$	−10.0000	+	17.3205i	−0.708881	+	1.22782i	0.256391	+	0.966573i	$0.417466\pi$
$199$	−10.0000	+	17.3205i	−0.708881	+	1.22782i	−0.965272	+	0.261245i	$0.915867\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	13.5000	−	23.3827i	0.942881	−	1.63312i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1.50000	−	12.9904i	0.103757	−	0.898563i
$210$		0			0
$211$	2.00000	+	3.46410i	0.137686	+	0.238479i	0.926620	−	0.375999i	$-0.122700\pi$
$211$	2.00000	+	3.46410i	0.137686	+	0.238479i	−0.788935	+	0.614477i	$0.789367\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−12.0000	+	20.7846i	−0.818393	+	1.41750i
$216$		0			0
$217$	8.00000			0.543075
$218$		0			0
$219$		0			0
$220$		0			0
$221$	12.0000			0.807207
$222$		0			0
$223$	−11.5000	−	19.9186i	−0.770097	−	1.33385i	−0.937509	−	0.347960i	$-0.886874\pi$
$223$	−11.5000	−	19.9186i	−0.770097	−	1.33385i	0.167412	−	0.985887i	$-0.446459\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	18.0000			1.19470			0.597351	−	0.801980i	$-0.296220\pi$
$227$	18.0000			1.19470			0.597351	+	0.801980i	$0.296220\pi$
$228$		0			0
$229$	8.00000			0.528655			0.264327	−	0.964433i	$-0.414850\pi$
$229$	8.00000			0.528655			0.264327	+	0.964433i	$0.414850\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	3.00000	+	5.19615i	0.196537	+	0.340411i	0.947403	−	0.320043i	$-0.103697\pi$
$233$	3.00000	+	5.19615i	0.196537	+	0.340411i	−0.750867	+	0.660454i	$0.770364\pi$
$234$		0			0
$235$	−36.0000			−2.34838
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−9.00000			−0.582162			−0.291081	−	0.956698i	$-0.594015\pi$
$239$	−9.00000			−0.582162			−0.291081	+	0.956698i	$0.594015\pi$
$240$		0			0
$241$	−4.00000	+	6.92820i	−0.257663	+	0.446285i	−0.965615	−	0.259975i	$-0.916286\pi$
$241$	−4.00000	+	6.92820i	−0.257663	+	0.446285i	0.707953	+	0.706260i	$0.249619\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.50000	−	2.59808i	−0.0958315	−	0.165985i
$246$		0			0
$247$	14.0000	+	10.3923i	0.890799	+	0.661247i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	6.00000	−	10.3923i	0.378717	−	0.655956i	−0.612159	−	0.790735i	$-0.709699\pi$
$251$	6.00000	−	10.3923i	0.378717	−	0.655956i	0.990876	+	0.134778i	$0.0430322\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	10.5000	−	18.1865i	0.654972	−	1.13444i	−0.326929	−	0.945049i	$-0.606014\pi$
$257$	10.5000	−	18.1865i	0.654972	−	1.13444i	0.981901	−	0.189396i	$-0.0606529\pi$
$258$		0			0
$259$	11.0000			0.683507
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−4.50000	−	7.79423i	−0.277482	−	0.480613i	0.693276	−	0.720672i	$-0.256167\pi$
$263$	−4.50000	−	7.79423i	−0.277482	−	0.480613i	−0.970758	+	0.240059i	$0.922833\pi$
$264$		0			0
$265$	−36.0000			−2.21146
$266$		0			0
$267$		0			0
$268$		0			0
$269$	4.50000	+	7.79423i	0.274370	+	0.475223i	0.969976	−	0.243201i	$-0.0781974\pi$
$269$	4.50000	+	7.79423i	0.274370	+	0.475223i	−0.695606	+	0.718423i	$0.744864\pi$
$270$		0			0
$271$	−5.50000	−	9.52628i	−0.334101	−	0.578680i	0.649211	−	0.760609i	$-0.275099\pi$
$271$	−5.50000	−	9.52628i	−0.334101	−	0.578680i	−0.983312	+	0.181928i	$0.941766\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	6.00000	−	10.3923i	0.361814	−	0.626680i
$276$		0			0
$277$	−1.00000			−0.0600842			−0.0300421	−	0.999549i	$-0.509564\pi$
$277$	−1.00000			−0.0600842			−0.0300421	+	0.999549i	$0.509564\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	3.00000	−	5.19615i	0.178965	−	0.309976i	−0.762561	−	0.646916i	$-0.776058\pi$
$281$	3.00000	−	5.19615i	0.178965	−	0.309976i	0.941526	+	0.336939i	$0.109392\pi$
$282$		0			0
$283$	−11.5000	−	19.9186i	−0.683604	−	1.18404i	−0.973873	−	0.227092i	$-0.927078\pi$
$283$	−11.5000	−	19.9186i	−0.683604	−	1.18404i	0.290269	−	0.956945i	$-0.406255\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	4.50000	+	7.79423i	0.265627	+	0.460079i
$288$		0			0
$289$	4.00000	−	6.92820i	0.235294	−	0.407541i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	9.00000			0.525786			0.262893	−	0.964825i	$-0.415323\pi$
$293$	9.00000			0.525786			0.262893	+	0.964825i	$0.415323\pi$
$294$		0			0
$295$	−18.0000	+	31.1769i	−1.04800	+	1.81519i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−4.00000	−	6.92820i	−0.230556	−	0.399335i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	6.00000			0.343559
$306$		0			0
$307$	−10.0000	−	17.3205i	−0.570730	−	0.988534i	−0.996491	−	0.0836980i	$-0.973327\pi$
$307$	−10.0000	−	17.3205i	−0.570730	−	0.988534i	0.425761	−	0.904836i	$-0.360006\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	24.0000			1.36092			0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000			1.36092			0.680458	+	0.732787i	$0.261781\pi$
$312$		0			0
$313$	−16.0000	+	27.7128i	−0.904373	+	1.56642i	−0.0826174	+	0.996581i	$0.526328\pi$
$313$	−16.0000	+	27.7128i	−0.904373	+	1.56642i	−0.821756	+	0.569839i	$0.807005\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	3.00000	−	5.19615i	0.168497	−	0.291845i	−0.769395	−	0.638774i	$-0.779442\pi$
$317$	3.00000	−	5.19615i	0.168497	−	0.291845i	0.937892	+	0.346929i	$0.112775\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−12.0000	+	5.19615i	−0.667698	+	0.289122i
$324$		0			0
$325$	8.00000	+	13.8564i	0.443760	+	0.768615i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	6.00000	−	10.3923i	0.330791	−	0.572946i
$330$		0			0
$331$	32.0000			1.75888			0.879440	−	0.476011i	$-0.157918\pi$
$331$	32.0000			1.75888			0.879440	+	0.476011i	$0.157918\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−12.0000			−0.655630
$336$		0			0
$337$	−7.00000	−	12.1244i	−0.381314	−	0.660456i	0.609936	−	0.792451i	$-0.291195\pi$
$337$	−7.00000	−	12.1244i	−0.381314	−	0.660456i	−0.991250	+	0.131995i	$0.957862\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−24.0000			−1.29967
$342$		0			0
$343$	1.00000			0.0539949
$344$		0			0
$345$		0			0
$346$		0			0
$347$	6.00000	+	10.3923i	0.322097	+	0.557888i	0.980921	−	0.194409i	$-0.0622790\pi$
$347$	6.00000	+	10.3923i	0.322097	+	0.557888i	−0.658824	+	0.752297i	$0.728946\pi$
$348$		0			0
$349$	−34.0000			−1.81998			−0.909989	−	0.414632i	$-0.863910\pi$
$349$	−34.0000			−1.81998			−0.909989	+	0.414632i	$0.863910\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	9.00000			0.479022			0.239511	−	0.970894i	$-0.423013\pi$
$353$	9.00000			0.479022			0.239511	+	0.970894i	$0.423013\pi$
$354$		0			0
$355$	−18.0000	+	31.1769i	−0.955341	+	1.65470i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$359$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$360$		0			0
$361$	−18.5000	−	4.33013i	−0.973684	−	0.227901i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−3.00000	+	5.19615i	−0.157027	+	0.271979i
$366$		0			0
$367$	6.50000	−	11.2583i	0.339297	−	0.587680i	−0.645003	−	0.764180i	$-0.723144\pi$
$367$	6.50000	−	11.2583i	0.339297	−	0.587680i	0.984301	+	0.176500i	$0.0564774\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	6.00000	−	10.3923i	0.311504	−	0.539542i
$372$		0			0
$373$	29.0000			1.50156			0.750782	−	0.660551i	$-0.229677\pi$
$373$	29.0000			1.50156			0.750782	+	0.660551i	$0.229677\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−16.0000			−0.821865			−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000			−0.821865			−0.410932	+	0.911666i	$0.634797\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	3.00000	+	5.19615i	0.153293	+	0.265511i	0.932436	−	0.361335i	$-0.117679\pi$
$383$	3.00000	+	5.19615i	0.153293	+	0.265511i	−0.779143	+	0.626846i	$0.784346\pi$
$384$		0			0
$385$	4.50000	+	7.79423i	0.229341	+	0.397231i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	3.00000	−	5.19615i	0.152106	−	0.263455i	−0.779895	−	0.625910i	$-0.784728\pi$
$389$	3.00000	−	5.19615i	0.152106	−	0.263455i	0.932002	+	0.362454i	$0.118061\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$	15.0000	−	25.9808i	0.754732	−	1.30723i
$396$		0			0
$397$	−7.00000	−	12.1244i	−0.351320	−	0.608504i	0.635161	−	0.772380i	$-0.280934\pi$
$397$	−7.00000	−	12.1244i	−0.351320	−	0.608504i	−0.986481	+	0.163876i	$0.947600\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−6.00000	−	10.3923i	−0.299626	−	0.518967i	0.676425	−	0.736512i	$-0.263528\pi$
$401$	−6.00000	−	10.3923i	−0.299626	−	0.518967i	−0.976050	+	0.217545i	$0.930195\pi$
$402$		0			0
$403$	16.0000	−	27.7128i	0.797017	−	1.38047i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−33.0000			−1.63575
$408$		0			0
$409$	5.00000	−	8.66025i	0.247234	−	0.428222i	−0.715523	−	0.698589i	$-0.753812\pi$
$409$	5.00000	−	8.66025i	0.247234	−	0.428222i	0.962757	+	0.270367i	$0.0871450\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−6.00000	−	10.3923i	−0.295241	−	0.511372i
$414$		0			0
$415$	9.00000	+	15.5885i	0.441793	+	0.765207i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	18.0000			0.879358			0.439679	−	0.898155i	$-0.355092\pi$
$419$	18.0000			0.879358			0.439679	+	0.898155i	$0.355092\pi$
$420$		0			0
$421$	−11.5000	−	19.9186i	−0.560476	−	0.970772i	−0.997455	−	0.0713008i	$-0.977285\pi$
$421$	−11.5000	−	19.9186i	−0.560476	−	0.970772i	0.436979	−	0.899472i	$-0.356048\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−12.0000			−0.582086
$426$		0			0
$427$	−1.00000	+	1.73205i	−0.0483934	+	0.0838198i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−16.5000	+	28.5788i	−0.794777	+	1.37659i	0.128204	+	0.991748i	$0.459079\pi$
$431$	−16.5000	+	28.5788i	−0.794777	+	1.37659i	−0.922981	+	0.384846i	$0.874254\pi$
$432$		0			0
$433$	−1.00000	+	1.73205i	−0.0480569	+	0.0832370i	−0.889053	−	0.457804i	$-0.848636\pi$
$433$	−1.00000	+	1.73205i	−0.0480569	+	0.0832370i	0.840996	+	0.541041i	$0.181970\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−11.5000	−	19.9186i	−0.548865	−	0.950662i	−0.998353	−	0.0573756i	$-0.981727\pi$
$439$	−11.5000	−	19.9186i	−0.548865	−	0.950662i	0.449488	−	0.893287i	$-0.351607\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$443$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$444$		0			0
$445$	45.0000			2.13320
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−6.00000			−0.283158			−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000			−0.283158			−0.141579	+	0.989927i	$0.545218\pi$
$450$		0			0
$451$	−13.5000	−	23.3827i	−0.635690	−	1.10105i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−12.0000			−0.562569
$456$		0			0
$457$	−7.00000			−0.327446			−0.163723	−	0.986506i	$-0.552350\pi$
$457$	−7.00000			−0.327446			−0.163723	+	0.986506i	$0.552350\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	19.5000	+	33.7750i	0.908206	+	1.57306i	0.816556	+	0.577267i	$0.195881\pi$
$461$	19.5000	+	33.7750i	0.908206	+	1.57306i	0.0916500	+	0.995791i	$0.470786\pi$
$462$		0			0
$463$	20.0000			0.929479			0.464739	−	0.885448i	$-0.346148\pi$
$463$	20.0000			0.929479			0.464739	+	0.885448i	$0.346148\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−6.00000			−0.277647			−0.138823	−	0.990317i	$-0.544332\pi$
$467$	−6.00000			−0.277647			−0.138823	+	0.990317i	$0.544332\pi$
$468$		0			0
$469$	2.00000	−	3.46410i	0.0923514	−	0.159957i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	12.0000	+	20.7846i	0.551761	+	0.955677i
$474$		0			0
$475$	−14.0000	−	10.3923i	−0.642364	−	0.476832i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	21.0000	−	36.3731i	0.959514	−	1.66193i	0.235833	−	0.971794i	$-0.424218\pi$
$479$	21.0000	−	36.3731i	0.959514	−	1.66193i	0.723681	−	0.690134i	$-0.242449\pi$
$480$		0			0
$481$	22.0000	−	38.1051i	1.00311	−	1.73744i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−12.0000	+	20.7846i	−0.544892	+	0.943781i
$486$		0			0
$487$	8.00000			0.362515			0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000			0.362515			0.181257	+	0.983436i	$0.441983\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	4.50000	+	7.79423i	0.203082	+	0.351749i	0.949520	−	0.313707i	$-0.101571\pi$
$491$	4.50000	+	7.79423i	0.203082	+	0.351749i	−0.746438	+	0.665455i	$0.768237\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−6.00000	−	10.3923i	−0.269137	−	0.466159i
$498$		0			0
$499$	5.00000	+	8.66025i	0.223831	+	0.387686i	0.955968	−	0.293471i	$-0.0948104\pi$
$499$	5.00000	+	8.66025i	0.223831	+	0.387686i	−0.732137	+	0.681157i	$0.761477\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	21.0000	−	36.3731i	0.936344	−	1.62179i	0.164124	−	0.986440i	$-0.447520\pi$
$503$	21.0000	−	36.3731i	0.936344	−	1.62179i	0.772220	−	0.635355i	$-0.219146\pi$
$504$		0			0
$505$	27.0000			1.20148
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−1.50000	+	2.59808i	−0.0664863	+	0.115158i	−0.897352	−	0.441315i	$-0.854512\pi$
$509$	−1.50000	+	2.59808i	−0.0664863	+	0.115158i	0.830866	+	0.556473i	$0.187846\pi$
$510$		0			0
$511$	−1.00000	−	1.73205i	−0.0442374	−	0.0766214i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	19.5000	+	33.7750i	0.859273	+	1.48830i
$516$		0			0
$517$	−18.0000	+	31.1769i	−0.791639	+	1.37116i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	18.0000			0.788594			0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000			0.788594			0.394297	+	0.918983i	$0.370988\pi$
$522$		0			0
$523$	12.5000	−	21.6506i	0.546587	−	0.946716i	−0.451918	−	0.892059i	$-0.649260\pi$
$523$	12.5000	−	21.6506i	0.546587	−	0.946716i	0.998505	−	0.0546569i	$-0.0174065\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	12.0000	+	20.7846i	0.522728	+	0.905392i
$528$		0			0
$529$	11.5000	+	19.9186i	0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	36.0000			1.55933
$534$		0			0
$535$	18.0000	+	31.1769i	0.778208	+	1.34790i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−3.00000			−0.129219
$540$		0			0
$541$	18.5000	−	32.0429i	0.795377	−	1.37763i	−0.127222	−	0.991874i	$-0.540606\pi$
$541$	18.5000	−	32.0429i	0.795377	−	1.37763i	0.922599	−	0.385759i	$-0.126061\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	10.5000	−	18.1865i	0.449771	−	0.779026i
$546$		0			0
$547$	−10.0000	+	17.3205i	−0.427569	+	0.740571i	−0.996657	−	0.0817056i	$-0.973963\pi$
$547$	−10.0000	+	17.3205i	−0.427569	+	0.740571i	0.569087	+	0.822277i	$0.307297\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	5.00000	+	8.66025i	0.212622	+	0.368271i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−9.00000	+	15.5885i	−0.381342	+	0.660504i	−0.991254	−	0.131965i	$-0.957871\pi$
$557$	−9.00000	+	15.5885i	−0.381342	+	0.660504i	0.609912	+	0.792469i	$0.291205\pi$
$558$		0			0
$559$	−32.0000			−1.35346
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−24.0000			−1.01148			−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000			−1.01148			−0.505740	+	0.862686i	$0.668780\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	24.0000			1.00613			0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000			1.00613			0.503066	+	0.864248i	$0.332205\pi$
$570$		0			0
$571$	20.0000			0.836974			0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000			0.836974			0.418487	+	0.908223i	$0.362561\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	32.0000			1.33218			0.666089	−	0.745873i	$-0.267967\pi$
$577$	32.0000			1.33218			0.666089	+	0.745873i	$0.267967\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−6.00000			−0.248922
$582$		0			0
$583$	−18.0000	+	31.1769i	−0.745484	+	1.29122i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$587$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$588$		0			0
$589$	−4.00000	+	34.6410i	−0.164817	+	1.42736i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−10.5000	+	18.1865i	−0.431183	+	0.746831i	−0.996976	−	0.0777165i	$-0.975237\pi$
$593$	−10.5000	+	18.1865i	−0.431183	+	0.746831i	0.565792	+	0.824548i	$0.308570\pi$
$594$		0			0
$595$	4.50000	−	7.79423i	0.184482	−	0.319532i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−19.5000	+	33.7750i	−0.796748	+	1.38001i	0.124975	+	0.992160i	$0.460115\pi$
$599$	−19.5000	+	33.7750i	−0.796748	+	1.38001i	−0.921723	+	0.387849i	$0.873218\pi$
$600$		0			0
$601$	2.00000			0.0815817			0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000			0.0815817			0.0407909	+	0.999168i	$0.487012\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	3.00000	+	5.19615i	0.121967	+	0.211254i
$606$		0			0
$607$	32.0000			1.29884			0.649420	−	0.760430i	$-0.275012\pi$
$607$	32.0000			1.29884			0.649420	+	0.760430i	$0.275012\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−24.0000	−	41.5692i	−0.970936	−	1.68171i
$612$		0			0
$613$	−23.5000	−	40.7032i	−0.949156	−	1.64399i	−0.747208	−	0.664590i	$-0.768606\pi$
$613$	−23.5000	−	40.7032i	−0.949156	−	1.64399i	−0.201948	−	0.979396i	$-0.564727\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	9.00000	−	15.5885i	0.362326	−	0.627568i	−0.626017	−	0.779809i	$-0.715316\pi$
$617$	9.00000	−	15.5885i	0.362326	−	0.627568i	0.988343	+	0.152242i	$0.0486493\pi$
$618$		0			0
$619$	−28.0000			−1.12542			−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000			−1.12542			−0.562708	+	0.826656i	$0.690240\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−7.50000	+	12.9904i	−0.300481	+	0.520449i
$624$		0			0
$625$	14.5000	+	25.1147i	0.580000	+	1.00459i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	16.5000	+	28.5788i	0.657898	+	1.13951i
$630$		0			0
$631$	2.00000	−	3.46410i	0.0796187	−	0.137904i	−0.823467	−	0.567365i	$-0.807963\pi$
$631$	2.00000	−	3.46410i	0.0796187	−	0.137904i	0.903085	+	0.429461i	$0.141296\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−48.0000			−1.90482
$636$		0			0
$637$	2.00000	−	3.46410i	0.0792429	−	0.137253i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	21.0000	+	36.3731i	0.829450	+	1.43665i	0.898470	+	0.439034i	$0.144679\pi$
$641$	21.0000	+	36.3731i	0.829450	+	1.43665i	−0.0690201	+	0.997615i	$0.521987\pi$
$642$		0			0
$643$	9.50000	+	16.4545i	0.374643	+	0.648901i	0.990274	−	0.139134i	$-0.0444318\pi$
$643$	9.50000	+	16.4545i	0.374643	+	0.648901i	−0.615630	+	0.788035i	$0.711098\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−30.0000			−1.17942			−0.589711	−	0.807614i	$-0.700758\pi$
$647$	−30.0000			−1.17942			−0.589711	+	0.807614i	$0.700758\pi$
$648$		0			0
$649$	18.0000	+	31.1769i	0.706562	+	1.22380i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	24.0000			0.939193			0.469596	−	0.882881i	$-0.344399\pi$
$653$	24.0000			0.939193			0.469596	+	0.882881i	$0.344399\pi$
$654$		0			0
$655$	−18.0000	+	31.1769i	−0.703318	+	1.21818i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	16.5000	−	28.5788i	0.642749	−	1.11327i	−0.342068	−	0.939675i	$-0.611127\pi$
$659$	16.5000	−	28.5788i	0.642749	−	1.11327i	0.984817	−	0.173598i	$-0.0555394\pi$
$660$		0			0
$661$	17.0000	−	29.4449i	0.661223	−	1.14527i	−0.319071	−	0.947731i	$-0.603371\pi$
$661$	17.0000	−	29.4449i	0.661223	−	1.14527i	0.980294	−	0.197542i	$-0.0632958\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	12.0000	−	5.19615i	0.465340	−	0.201498i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$	3.00000	−	5.19615i	0.115814	−	0.200595i
$672$		0			0
$673$	17.0000			0.655302			0.327651	−	0.944799i	$-0.393743\pi$
$673$	17.0000			0.655302			0.327651	+	0.944799i	$0.393743\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	33.0000			1.26829			0.634147	−	0.773213i	$-0.281352\pi$
$677$	33.0000			1.26829			0.634147	+	0.773213i	$0.281352\pi$
$678$		0			0
$679$	−4.00000	−	6.92820i	−0.153506	−	0.265880i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	21.0000			0.803543			0.401771	−	0.915740i	$-0.368395\pi$
$683$	21.0000			0.803543			0.401771	+	0.915740i	$0.368395\pi$
$684$		0			0
$685$	36.0000			1.37549
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−24.0000	−	41.5692i	−0.914327	−	1.58366i
$690$		0			0
$691$	−37.0000			−1.40755			−0.703773	−	0.710425i	$-0.748503\pi$
$691$	−37.0000			−1.40755			−0.703773	+	0.710425i	$0.748503\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−39.0000			−1.47935
$696$		0			0
$697$	−13.5000	+	23.3827i	−0.511349	+	0.885682i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−3.00000	−	5.19615i	−0.113308	−	0.196256i	0.803794	−	0.594908i	$-0.202811\pi$
$701$	−3.00000	−	5.19615i	−0.113308	−	0.196256i	−0.917102	+	0.398652i	$0.869478\pi$
$702$		0			0
$703$	−5.50000	+	47.6314i	−0.207436	+	1.79645i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−4.50000	+	7.79423i	−0.169240	+	0.293132i
$708$		0			0
$709$	9.50000	−	16.4545i	0.356780	−	0.617961i	−0.630641	−	0.776075i	$-0.717208\pi$
$709$	9.50000	−	16.4545i	0.356780	−	0.617961i	0.987421	+	0.158114i	$0.0505412\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	36.0000			1.34632
$716$		0			0
$717$		0			0
$718$		0			0
$719$	15.0000	+	25.9808i	0.559406	+	0.968919i	0.997546	+	0.0700124i	$0.0223039\pi$
$719$	15.0000	+	25.9808i	0.559406	+	0.968919i	−0.438141	+	0.898906i	$0.644363\pi$
$720$		0			0
$721$	−13.0000			−0.484145
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−17.5000	−	30.3109i	−0.649039	−	1.12417i	−0.983353	−	0.181707i	$-0.941838\pi$
$727$	−17.5000	−	30.3109i	−0.649039	−	1.12417i	0.334314	−	0.942462i	$-0.391496\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	12.0000	−	20.7846i	0.443836	−	0.768747i
$732$		0			0
$733$	14.0000			0.517102			0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000			0.517102			0.258551	+	0.965998i	$0.416755\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−6.00000	+	10.3923i	−0.221013	+	0.382805i
$738$		0			0
$739$	17.0000	+	29.4449i	0.625355	+	1.08315i	0.988472	+	0.151403i	$0.0483792\pi$
$739$	17.0000	+	29.4449i	0.625355	+	1.08315i	−0.363117	+	0.931744i	$0.618287\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−19.5000	−	33.7750i	−0.715386	−	1.23908i	−0.962811	−	0.270177i	$-0.912918\pi$
$743$	−19.5000	−	33.7750i	−0.715386	−	1.23908i	0.247425	−	0.968907i	$-0.420416\pi$
$744$		0			0
$745$	9.00000	−	15.5885i	0.329734	−	0.571117i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−12.0000			−0.438470
$750$		0			0
$751$	−10.0000	+	17.3205i	−0.364905	+	0.632034i	−0.988761	−	0.149505i	$-0.952232\pi$
$751$	−10.0000	+	17.3205i	−0.364905	+	0.632034i	0.623856	+	0.781540i	$0.285565\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−12.0000	−	20.7846i	−0.436725	−	0.756429i
$756$		0			0
$757$	−20.5000	−	35.5070i	−0.745085	−	1.29053i	−0.950155	−	0.311778i	$-0.899075\pi$
$757$	−20.5000	−	35.5070i	−0.745085	−	1.29053i	0.205070	−	0.978747i	$-0.434258\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	30.0000			1.08750			0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000			1.08750			0.543750	+	0.839248i	$0.317004\pi$
$762$		0			0
$763$	3.50000	+	6.06218i	0.126709	+	0.219466i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−48.0000			−1.73318
$768$		0			0
$769$	−19.0000	+	32.9090i	−0.685158	+	1.18673i	0.288230	+	0.957561i	$0.406933\pi$
$769$	−19.0000	+	32.9090i	−0.685158	+	1.18673i	−0.973387	+	0.229166i	$0.926400\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−16.5000	+	28.5788i	−0.593464	+	1.02791i	0.400298	+	0.916385i	$0.368907\pi$
$773$	−16.5000	+	28.5788i	−0.593464	+	1.02791i	−0.993762	+	0.111524i	$0.964427\pi$
$774$		0			0
$775$	−16.0000	+	27.7128i	−0.574737	+	0.995474i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−36.0000	+	15.5885i	−1.28983	+	0.558514i
$780$		0			0
$781$	18.0000	+	31.1769i	0.644091	+	1.11560i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−30.0000	+	51.9615i	−1.07075	+	1.85459i
$786$		0			0
$787$	−19.0000			−0.677277			−0.338638	−	0.940917i	$-0.609966\pi$
$787$	−19.0000			−0.677277			−0.338638	+	0.940917i	$0.609966\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	4.00000	+	6.92820i	0.142044	+	0.246028i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	9.00000			0.318796			0.159398	−	0.987214i	$-0.449045\pi$
$797$	9.00000			0.318796			0.159398	+	0.987214i	$0.449045\pi$
$798$		0			0
$799$	36.0000			1.27359
$800$		0			0
$801$		0			0
$802$		0			0
$803$	3.00000	+	5.19615i	0.105868	+	0.183368i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	36.0000			1.26569			0.632846	−	0.774277i	$-0.281886\pi$
$809$	36.0000			1.26569			0.632846	+	0.774277i	$0.281886\pi$
$810$		0			0
$811$	−5.50000	+	9.52628i	−0.193131	+	0.334513i	−0.946286	−	0.323330i	$-0.895198\pi$
$811$	−5.50000	+	9.52628i	−0.193131	+	0.334513i	0.753155	+	0.657843i	$0.228531\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	33.0000	+	57.1577i	1.15594	+	2.00215i
$816$		0			0
$817$	32.0000	−	13.8564i	1.11954	−	0.484774i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	12.0000	−	20.7846i	0.418803	−	0.725388i	−0.577016	−	0.816733i	$-0.695783\pi$
$821$	12.0000	−	20.7846i	0.418803	−	0.725388i	0.995819	+	0.0913446i	$0.0291165\pi$
$822$		0			0
$823$	−7.00000	+	12.1244i	−0.244005	+	0.422628i	−0.961851	−	0.273573i	$-0.911795\pi$
$823$	−7.00000	+	12.1244i	−0.244005	+	0.422628i	0.717847	+	0.696201i	$0.245128\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−16.5000	+	28.5788i	−0.573761	+	0.993784i	0.422414	+	0.906403i	$0.361183\pi$
$827$	−16.5000	+	28.5788i	−0.573761	+	0.993784i	−0.996175	+	0.0873805i	$0.972150\pi$
$828$		0			0
$829$	−34.0000			−1.18087			−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000			−1.18087			−0.590434	+	0.807086i	$0.701044\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1.50000	+	2.59808i	0.0519719	+	0.0900180i
$834$		0			0
$835$	−54.0000			−1.86875
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−12.0000	−	20.7846i	−0.414286	−	0.717564i	0.581067	−	0.813856i	$-0.302635\pi$
$839$	−12.0000	−	20.7846i	−0.414286	−	0.717564i	−0.995353	+	0.0962912i	$0.969302\pi$
$840$		0			0
$841$	14.5000	+	25.1147i	0.500000	+	0.866025i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−4.50000	+	7.79423i	−0.154805	+	0.268130i
$846$		0			0
$847$	−2.00000			−0.0687208
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−22.0000	−	38.1051i	−0.753266	−	1.30469i	−0.946232	−	0.323489i	$-0.895144\pi$
$853$	−22.0000	−	38.1051i	−0.753266	−	1.30469i	0.192966	−	0.981205i	$-0.438189\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1.50000	−	2.59808i	−0.0512390	−	0.0887486i	0.839268	−	0.543718i	$-0.182984\pi$
$857$	−1.50000	−	2.59808i	−0.0512390	−	0.0887486i	−0.890507	+	0.454969i	$0.849650\pi$
$858$		0			0
$859$	0.500000	−	0.866025i	0.0170598	−	0.0295484i	−0.857369	−	0.514701i	$-0.827903\pi$
$859$	0.500000	−	0.866025i	0.0170598	−	0.0295484i	0.874429	+	0.485153i	$0.161236\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−3.00000			−0.102121			−0.0510606	−	0.998696i	$-0.516260\pi$
$863$	−3.00000			−0.102121			−0.0510606	+	0.998696i	$0.516260\pi$
$864$		0			0
$865$	−9.00000	+	15.5885i	−0.306009	+	0.530023i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−15.0000	−	25.9808i	−0.508840	−	0.881337i
$870$		0			0
$871$	−8.00000	−	13.8564i	−0.271070	−	0.469506i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−3.00000			−0.101419
$876$		0			0
$877$	−13.0000	−	22.5167i	−0.438979	−	0.760334i	0.558632	−	0.829416i	$-0.311326\pi$
$877$	−13.0000	−	22.5167i	−0.438979	−	0.760334i	−0.997611	+	0.0690819i	$0.977993\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−51.0000			−1.71823			−0.859117	−	0.511780i	$-0.828986\pi$
$881$	−51.0000			−1.71823			−0.859117	+	0.511780i	$0.828986\pi$
$882$		0			0
$883$	−4.00000	+	6.92820i	−0.134611	+	0.233153i	−0.925449	−	0.378873i	$-0.876312\pi$
$883$	−4.00000	+	6.92820i	−0.134611	+	0.233153i	0.790838	+	0.612026i	$0.209645\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−12.0000	+	20.7846i	−0.402921	+	0.697879i	−0.994077	−	0.108678i	$-0.965338\pi$
$887$	−12.0000	+	20.7846i	−0.402921	+	0.697879i	0.591156	+	0.806557i	$0.298672\pi$
$888$		0			0
$889$	8.00000	−	13.8564i	0.268311	−	0.464729i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	42.0000	+	31.1769i	1.40548	+	1.04330i
$894$		0			0
$895$	−31.5000	−	54.5596i	−1.05293	−	1.82373i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	36.0000			1.19933
$902$		0			0
$903$		0			0
$904$		0			0
$905$	24.0000			0.797787
$906$		0			0
$907$	−19.0000	−	32.9090i	−0.630885	−	1.09272i	−0.987371	−	0.158424i	$-0.949359\pi$
$907$	−19.0000	−	32.9090i	−0.630885	−	1.09272i	0.356487	−	0.934300i	$-0.383975\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−45.0000			−1.49092			−0.745458	−	0.666552i	$-0.767769\pi$
$911$	−45.0000			−1.49092			−0.745458	+	0.666552i	$0.767769\pi$
$912$		0			0
$913$	18.0000			0.595713
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−6.00000	−	10.3923i	−0.198137	−	0.343184i
$918$		0			0
$919$	−4.00000			−0.131948			−0.0659739	−	0.997821i	$-0.521015\pi$
$919$	−4.00000			−0.131948			−0.0659739	+	0.997821i	$0.521015\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−48.0000			−1.57994
$924$		0			0
$925$	−22.0000	+	38.1051i	−0.723356	+	1.25289i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	22.5000	+	38.9711i	0.738201	+	1.27860i	0.953305	+	0.302010i	$0.0976578\pi$
$929$	22.5000	+	38.9711i	0.738201	+	1.27860i	−0.215104	+	0.976591i	$0.569009\pi$
$930$		0			0
$931$	−0.500000	+	4.33013i	−0.0163868	+	0.141914i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−13.5000	+	23.3827i	−0.441497	+	0.764696i
$936$		0			0
$937$	8.00000	−	13.8564i	0.261349	−	0.452669i	−0.705252	−	0.708957i	$-0.749166\pi$
$937$	8.00000	−	13.8564i	0.261349	−	0.452669i	0.966601	+	0.256288i	$0.0824995\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−7.50000	+	12.9904i	−0.244493	+	0.423474i	−0.961989	−	0.273088i	$-0.911955\pi$
$941$	−7.50000	+	12.9904i	−0.244493	+	0.423474i	0.717496	+	0.696563i	$0.245288\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−4.50000	−	7.79423i	−0.146230	−	0.253278i	0.783601	−	0.621264i	$-0.213381\pi$
$947$	−4.50000	−	7.79423i	−0.146230	−	0.253278i	−0.929831	+	0.367986i	$0.880047\pi$
$948$		0			0
$949$	−8.00000			−0.259691
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−12.0000	−	20.7846i	−0.388718	−	0.673280i	0.603559	−	0.797318i	$-0.293749\pi$
$953$	−12.0000	−	20.7846i	−0.388718	−	0.673280i	−0.992277	+	0.124039i	$0.960415\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−6.00000	+	10.3923i	−0.193750	+	0.335585i
$960$		0			0
$961$	33.0000			1.06452
$962$		0			0
$963$		0			0
$964$		0			0
$965$	19.5000	−	33.7750i	0.627727	−	1.08726i
$966$		0			0
$967$	11.0000	+	19.0526i	0.353736	+	0.612689i	0.986901	−	0.161328i	$-0.0515777\pi$
$967$	11.0000	+	19.0526i	0.353736	+	0.612689i	−0.633165	+	0.774017i	$0.718244\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−6.00000	−	10.3923i	−0.192549	−	0.333505i	0.753545	−	0.657396i	$-0.228342\pi$
$971$	−6.00000	−	10.3923i	−0.192549	−	0.333505i	−0.946094	+	0.323891i	$0.895009\pi$
$972$		0			0
$973$	6.50000	−	11.2583i	0.208380	−	0.360925i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−30.0000			−0.959785			−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000			−0.959785			−0.479893	+	0.877327i	$0.659324\pi$
$978$		0			0
$979$	22.5000	−	38.9711i	0.719103	−	1.24552i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−12.0000	−	20.7846i	−0.382741	−	0.662926i	0.608712	−	0.793391i	$-0.291686\pi$
$983$	−12.0000	−	20.7846i	−0.382741	−	0.662926i	−0.991453	+	0.130465i	$0.958353\pi$
$984$		0			0
$985$	9.00000	+	15.5885i	0.286764	+	0.496690i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	17.0000	+	29.4449i	0.540023	+	0.935347i	0.998902	+	0.0468483i	$0.0149177\pi$
$991$	17.0000	+	29.4449i	0.540023	+	0.935347i	−0.458879	+	0.888499i	$0.651749\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	60.0000			1.90213
$996$		0			0
$997$	−4.00000	+	6.92820i	−0.126681	+	0.219418i	−0.922389	−	0.386263i	$-0.873766\pi$
$997$	−4.00000	+	6.92820i	−0.126681	+	0.219418i	0.795708	+	0.605681i	$0.207099\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4788.2.w.a.1261.1		2
3.2	odd	2		1596.2.s.b.1261.1	yes	2
19.11	even	3	inner	4788.2.w.a.505.1		2
57.11	odd	6		1596.2.s.b.505.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1596.2.s.b.505.1	✓	2	57.11	odd	6
1596.2.s.b.1261.1	yes	2	3.2	odd	2
4788.2.w.a.505.1		2	19.11	even	3	inner
4788.2.w.a.1261.1		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4788 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4788.w (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.50000 - 2.59808i) q^{5} +1.00000 q^{7} -3.00000 q^{11} +(2.00000 - 3.46410i) q^{13} +(1.50000 + 2.59808i) q^{17} +(-0.500000 + 4.33013i) q^{19} +(-2.00000 + 3.46410i) q^{25} +8.00000 q^{31} +(-1.50000 - 2.59808i) q^{35} +11.0000 q^{37} +(4.50000 + 7.79423i) q^{41} +(-4.00000 - 6.92820i) q^{43} +(6.00000 - 10.3923i) q^{47} +1.00000 q^{49} +(6.00000 - 10.3923i) q^{53} +(4.50000 + 7.79423i) q^{55} +(-6.00000 - 10.3923i) q^{59} +(-1.00000 + 1.73205i) q^{61} -12.0000 q^{65} +(2.00000 - 3.46410i) q^{67} +(-6.00000 - 10.3923i) q^{71} +(-1.00000 - 1.73205i) q^{73} -3.00000 q^{77} +(5.00000 + 8.66025i) q^{79} -6.00000 q^{83} +(4.50000 - 7.79423i) q^{85} +(-7.50000 + 12.9904i) q^{89} +(2.00000 - 3.46410i) q^{91} +(12.0000 - 5.19615i) q^{95} +(-4.00000 - 6.92820i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{5} + 2 q^{7} - 6 q^{11} + 4 q^{13} + 3 q^{17} - q^{19} - 4 q^{25} + 16 q^{31} - 3 q^{35} + 22 q^{37} + 9 q^{41} - 8 q^{43} + 12 q^{47} + 2 q^{49} + 12 q^{53} + 9 q^{55} - 12 q^{59} - 2 q^{61}+ \cdots - 8 q^{97}+O(q^{100}) \) 2 * q - 3 * q^5 + 2 * q^7 - 6 * q^11 + 4 * q^13 + 3 * q^17 - q^19 - 4 * q^25 + 16 * q^31 - 3 * q^35 + 22 * q^37 + 9 * q^41 - 8 * q^43 + 12 * q^47 + 2 * q^49 + 12 * q^53 + 9 * q^55 - 12 * q^59 - 2 * q^61 - 24 * q^65 + 4 * q^67 - 12 * q^71 - 2 * q^73 - 6 * q^77 + 10 * q^79 - 12 * q^83 + 9 * q^85 - 15 * q^89 + 4 * q^91 + 24 * q^95 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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