LMFDB - Embedded newform 405.3.h.a.269.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,3,Mod(134,405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([5, 3]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("405.134");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	405.h (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$11.0354507066$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 15)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			269.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	405.269
Dual form			405.3.h.a.134.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{2} +(1.50000 + 2.59808i) q^{4} +(-2.50000 - 4.33013i) q^{5} -7.00000 q^{8} +O(q^{10})$	$q+(-0.500000 + 0.866025i) q^{2} +(1.50000 + 2.59808i) q^{4} +(-2.50000 - 4.33013i) q^{5} -7.00000 q^{8} +5.00000 q^{10} +(-2.50000 + 4.33013i) q^{16} -14.0000 q^{17} -22.0000 q^{19} +(7.50000 - 12.9904i) q^{20} +(-17.0000 - 29.4449i) q^{23} +(-12.5000 + 21.6506i) q^{25} +(-1.00000 - 1.73205i) q^{31} +(-16.5000 - 28.5788i) q^{32} +(7.00000 - 12.1244i) q^{34} +(11.0000 - 19.0526i) q^{38} +(17.5000 + 30.3109i) q^{40} +34.0000 q^{46} +(7.00000 - 12.1244i) q^{47} +(-24.5000 - 42.4352i) q^{49} +(-12.5000 - 21.6506i) q^{50} -86.0000 q^{53} +(59.0000 - 102.191i) q^{61} +2.00000 q^{62} +13.0000 q^{64} +(-21.0000 - 36.3731i) q^{68} +(-33.0000 - 57.1577i) q^{76} +(-49.0000 + 84.8705i) q^{79} +25.0000 q^{80} +(-77.0000 + 133.368i) q^{83} +(35.0000 + 60.6218i) q^{85} +(51.0000 - 88.3346i) q^{92} +(7.00000 + 12.1244i) q^{94} +(55.0000 + 95.2628i) q^{95} +49.0000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} + 3 q^{4} - 5 q^{5} - 14 q^{8}+O(q^{10}) $ 2 * q - q^2 + 3 * q^4 - 5 * q^5 - 14 * q^8	$ 2 q - q^{2} + 3 q^{4} - 5 q^{5} - 14 q^{8} + 10 q^{10} - 5 q^{16} - 28 q^{17} - 44 q^{19} + 15 q^{20} - 34 q^{23} - 25 q^{25} - 2 q^{31} - 33 q^{32} + 14 q^{34} + 22 q^{38} + 35 q^{40} + 68 q^{46} + 14 q^{47} - 49 q^{49} - 25 q^{50} - 172 q^{53} + 118 q^{61} + 4 q^{62} + 26 q^{64} - 42 q^{68} - 66 q^{76} - 98 q^{79} + 50 q^{80} - 154 q^{83} + 70 q^{85} + 102 q^{92} + 14 q^{94} + 110 q^{95} + 98 q^{98}+O(q^{100}) $ 2 * q - q^2 + 3 * q^4 - 5 * q^5 - 14 * q^8 + 10 * q^10 - 5 * q^16 - 28 * q^17 - 44 * q^19 + 15 * q^20 - 34 * q^23 - 25 * q^25 - 2 * q^31 - 33 * q^32 + 14 * q^34 + 22 * q^38 + 35 * q^40 + 68 * q^46 + 14 * q^47 - 49 * q^49 - 25 * q^50 - 172 * q^53 + 118 * q^61 + 4 * q^62 + 26 * q^64 - 42 * q^68 - 66 * q^76 - 98 * q^79 + 50 * q^80 - 154 * q^83 + 70 * q^85 + 102 * q^92 + 14 * q^94 + 110 * q^95 + 98 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/405\mathbb{Z}\right)^\times$.

$n$	$82$	$326$
$\chi(n)$	$-1$	$e\left(\frac{1}{6}\right)$

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 $

$2$	−0.500000	+	0.866025i	−0.250000	+	0.433013i	−0.963525	−	0.267617i	$-0.913764\pi$
$2$	−0.500000	+	0.866025i	−0.250000	+	0.433013i	0.713525	+	0.700629i	$0.247097\pi$
$3$		0			0
$3$		0			0
$4$	1.50000	+	2.59808i	0.375000	+	0.649519i
$5$	−2.50000	−	4.33013i	−0.500000	−	0.866025i
$5$	−2.50000	−	4.33013i	−0.500000	−	0.866025i
$6$		0			0
$7$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$7$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$8$	−7.00000			−0.875000
$9$		0			0
$10$	5.00000			0.500000
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$		0			0
$13$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$13$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$14$		0			0
$15$		0			0
$16$	−2.50000	+	4.33013i	−0.156250	+	0.270633i
$17$	−14.0000			−0.823529			−0.411765	−	0.911290i	$-0.635087\pi$
$17$	−14.0000			−0.823529			−0.411765	+	0.911290i	$0.635087\pi$
$18$		0			0
$19$	−22.0000			−1.15789			−0.578947	−	0.815365i	$-0.696536\pi$
$19$	−22.0000			−1.15789			−0.578947	+	0.815365i	$0.696536\pi$
$20$	7.50000	−	12.9904i	0.375000	−	0.649519i
$21$		0			0
$22$		0			0
$23$	−17.0000	−	29.4449i	−0.739130	−	1.28021i	−0.952887	−	0.303325i	$-0.901903\pi$
$23$	−17.0000	−	29.4449i	−0.739130	−	1.28021i	0.213757	−	0.976887i	$-0.431430\pi$
$24$		0			0
$25$	−12.5000	+	21.6506i	−0.500000	+	0.866025i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$29$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$30$		0			0
$31$	−1.00000	−	1.73205i	−0.0322581	−	0.0558726i	0.849446	−	0.527676i	$-0.176937\pi$
$31$	−1.00000	−	1.73205i	−0.0322581	−	0.0558726i	−0.881704	+	0.471803i	$0.843603\pi$
$32$	−16.5000	−	28.5788i	−0.515625	−	0.893089i
$33$		0			0
$34$	7.00000	−	12.1244i	0.205882	−	0.356599i
$35$		0			0
$36$		0			0
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	11.0000	−	19.0526i	0.289474	−	0.501383i
$39$		0			0
$40$	17.5000	+	30.3109i	0.437500	+	0.757772i
$41$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$41$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$42$		0			0
$43$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$43$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$44$		0			0
$45$		0			0
$46$	34.0000			0.739130
$47$	7.00000	−	12.1244i	0.148936	−	0.257965i	−0.781898	−	0.623406i	$-0.785748\pi$
$47$	7.00000	−	12.1244i	0.148936	−	0.257965i	0.930835	+	0.365441i	$0.119082\pi$
$48$		0			0
$49$	−24.5000	−	42.4352i	−0.500000	−	0.866025i
$50$	−12.5000	−	21.6506i	−0.250000	−	0.433013i
$51$		0			0
$52$		0			0
$53$	−86.0000			−1.62264			−0.811321	−	0.584601i	$-0.801251\pi$
$53$	−86.0000			−1.62264			−0.811321	+	0.584601i	$0.801251\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$59$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$60$		0			0
$61$	59.0000	−	102.191i	0.967213	−	1.67526i	0.263665	−	0.964614i	$-0.415069\pi$
$61$	59.0000	−	102.191i	0.967213	−	1.67526i	0.703548	−	0.710648i	$-0.251598\pi$
$62$	2.00000			0.0322581
$63$		0			0
$64$	13.0000			0.203125
$65$		0			0
$66$		0			0
$67$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$67$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$68$	−21.0000	−	36.3731i	−0.308824	−	0.534898i
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$		0			0
$76$	−33.0000	−	57.1577i	−0.434211	−	0.752075i
$77$		0			0
$78$		0			0
$79$	−49.0000	+	84.8705i	−0.620253	+	1.07431i	0.369185	+	0.929356i	$0.379637\pi$
$79$	−49.0000	+	84.8705i	−0.620253	+	1.07431i	−0.989438	+	0.144954i	$0.953697\pi$
$80$	25.0000			0.312500
$81$		0			0
$82$		0			0
$83$	−77.0000	+	133.368i	−0.927711	+	1.60684i	−0.140568	+	0.990071i	$0.544893\pi$
$83$	−77.0000	+	133.368i	−0.927711	+	1.60684i	−0.787142	+	0.616771i	$0.788440\pi$
$84$		0			0
$85$	35.0000	+	60.6218i	0.411765	+	0.713197i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$	51.0000	−	88.3346i	0.554348	−	0.960159i
$93$		0			0
$94$	7.00000	+	12.1244i	0.0744681	+	0.128983i
$95$	55.0000	+	95.2628i	0.578947	+	1.00277i
$96$		0			0
$97$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$97$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$98$	49.0000			0.500000
$99$		0			0
$100$	−75.0000			−0.750000
$101$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$101$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$103$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$104$		0			0
$105$		0			0
$106$	43.0000	−	74.4782i	0.405660	−	0.702624i
$107$	106.000			0.990654			0.495327	−	0.868707i	$-0.335048\pi$
$107$	106.000			0.990654			0.495327	+	0.868707i	$0.335048\pi$
$108$		0			0
$109$	−22.0000			−0.201835			−0.100917	−	0.994895i	$-0.532178\pi$
$109$	−22.0000			−0.201835			−0.100917	+	0.994895i	$0.532178\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	103.000	+	178.401i	0.911504	+	1.57877i	0.811940	+	0.583741i	$0.198412\pi$
$113$	103.000	+	178.401i	0.911504	+	1.57877i	0.0995644	+	0.995031i	$0.468255\pi$
$114$		0			0
$115$	−85.0000	+	147.224i	−0.739130	+	1.28021i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−60.5000	−	104.789i	−0.500000	−	0.866025i
$122$	59.0000	+	102.191i	0.483607	+	0.837631i
$123$		0			0
$124$	3.00000	−	5.19615i	0.0241935	−	0.0419045i
$125$	125.000			1.00000
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$	59.5000	−	103.057i	0.464844	−	0.805133i
$129$		0			0
$130$		0			0
$131$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$131$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	98.0000			0.720588
$137$	−113.000	+	195.722i	−0.824818	+	1.42863i	0.0772412	+	0.997012i	$0.475389\pi$
$137$	−113.000	+	195.722i	−0.824818	+	1.42863i	−0.902059	+	0.431613i	$0.857944\pi$
$138$		0			0
$139$	131.000	+	226.899i	0.942446	+	1.63236i	0.760786	+	0.649003i	$0.224814\pi$
$139$	131.000	+	226.899i	0.942446	+	1.63236i	0.181660	+	0.983361i	$0.441853\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$		0			0
$151$	119.000	−	206.114i	0.788079	−	1.36499i	−0.139063	−	0.990284i	$-0.544409\pi$
$151$	119.000	−	206.114i	0.788079	−	1.36499i	0.927142	−	0.374710i	$-0.122258\pi$
$152$	154.000			1.01316
$153$		0			0
$154$		0			0
$155$	−5.00000	+	8.66025i	−0.0322581	+	0.0558726i
$156$		0			0
$157$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$157$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$158$	−49.0000	−	84.8705i	−0.310127	−	0.537155i
$159$		0			0
$160$	−82.5000	+	142.894i	−0.515625	+	0.893089i
$161$		0			0
$162$		0			0
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$		0			0
$166$	−77.0000	−	133.368i	−0.463855	−	0.803421i
$167$	127.000	+	219.970i	0.760479	+	1.31719i	0.942604	+	0.333913i	$0.108369\pi$
$167$	127.000	+	219.970i	0.760479	+	1.31719i	−0.182125	+	0.983275i	$0.558298\pi$
$168$		0			0
$169$	−84.5000	+	146.358i	−0.500000	+	0.866025i
$170$	−70.0000			−0.411765
$171$		0			0
$172$		0			0
$173$	−77.0000	+	133.368i	−0.445087	+	0.770913i	−0.998058	−	0.0622873i	$-0.980160\pi$
$173$	−77.0000	+	133.368i	−0.445087	+	0.770913i	0.552972	+	0.833200i	$0.313494\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$	122.000			0.674033			0.337017	−	0.941499i	$-0.390582\pi$
$181$	122.000			0.674033			0.337017	+	0.941499i	$0.390582\pi$
$182$		0			0
$183$		0			0
$184$	119.000	+	206.114i	0.646739	+	1.12019i
$185$		0			0
$186$		0			0
$187$		0			0
$188$	42.0000			0.223404
$189$		0			0
$190$	−110.000			−0.578947
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$		0			0
$193$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$193$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$194$		0			0
$195$		0			0
$196$	73.5000	−	127.306i	0.375000	−	0.649519i
$197$	−374.000			−1.89848			−0.949239	−	0.314557i	$-0.898144\pi$
$197$	−374.000			−1.89848			−0.949239	+	0.314557i	$0.898144\pi$
$198$		0			0
$199$	−142.000			−0.713568			−0.356784	−	0.934187i	$-0.616127\pi$
$199$	−142.000			−0.713568			−0.356784	+	0.934187i	$0.616127\pi$
$200$	87.5000	−	151.554i	0.437500	−	0.757772i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−181.000	−	313.501i	−0.857820	−	1.48579i	−0.874004	−	0.485919i	$-0.838485\pi$
$211$	−181.000	−	313.501i	−0.857820	−	1.48579i	0.0161841	−	0.999869i	$-0.494848\pi$
$212$	−129.000	−	223.435i	−0.608491	−	1.05394i
$213$		0			0
$214$	−53.0000	+	91.7987i	−0.247664	+	0.428966i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	11.0000	−	19.0526i	0.0504587	−	0.0873971i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$223$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$224$		0			0
$225$		0			0
$226$	−206.000			−0.911504
$227$	67.0000	−	116.047i	0.295154	−	0.511222i	−0.679867	−	0.733336i	$-0.737962\pi$
$227$	67.0000	−	116.047i	0.295154	−	0.511222i	0.975021	+	0.222114i	$0.0712956\pi$
$228$		0			0
$229$	−109.000	−	188.794i	−0.475983	−	0.824426i	0.523639	−	0.851940i	$-0.324574\pi$
$229$	−109.000	−	188.794i	−0.475983	−	0.824426i	−0.999621	+	0.0275144i	$0.991241\pi$
$230$	−85.0000	−	147.224i	−0.369565	−	0.640106i
$231$		0			0
$232$		0			0
$233$	34.0000			0.145923			0.0729614	−	0.997335i	$-0.476755\pi$
$233$	34.0000			0.145923			0.0729614	+	0.997335i	$0.476755\pi$
$234$		0			0
$235$	−70.0000			−0.297872
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$239$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$240$		0			0
$241$	239.000	−	413.960i	0.991701	−	1.71768i	0.384511	−	0.923120i	$-0.374370\pi$
$241$	239.000	−	413.960i	0.991701	−	1.71768i	0.607190	−	0.794557i	$-0.292297\pi$
$242$	121.000			0.500000
$243$		0			0
$244$	354.000			1.45082
$245$	−122.500	+	212.176i	−0.500000	+	0.866025i
$246$		0			0
$247$		0			0
$248$	7.00000	+	12.1244i	0.0282258	+	0.0488885i
$249$		0			0
$250$	−62.5000	+	108.253i	−0.250000	+	0.433013i
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	85.5000	+	148.090i	0.333984	+	0.578478i
$257$	−233.000	−	403.568i	−0.906615	−	1.57030i	−0.818735	−	0.574172i	$-0.805324\pi$
$257$	−233.000	−	403.568i	−0.906615	−	1.57030i	−0.0878799	−	0.996131i	$-0.528009\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	223.000	−	386.247i	0.847909	−	1.46862i	−0.0351622	−	0.999382i	$-0.511195\pi$
$263$	223.000	−	386.247i	0.847909	−	1.46862i	0.883071	−	0.469239i	$-0.155472\pi$
$264$		0			0
$265$	215.000	+	372.391i	0.811321	+	1.40525i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	482.000			1.77860			0.889299	−	0.457326i	$-0.151193\pi$
$271$	482.000			1.77860			0.889299	+	0.457326i	$0.151193\pi$
$272$	35.0000	−	60.6218i	0.128676	−	0.222874i
$273$		0			0
$274$	−113.000	−	195.722i	−0.412409	−	0.714313i
$275$		0			0
$276$		0			0
$277$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$277$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$278$	−262.000			−0.942446
$279$		0			0
$280$		0			0
$281$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$281$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$282$		0			0
$283$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$283$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−93.0000			−0.321799
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−197.000	−	341.214i	−0.672355	−	1.16455i	−0.977235	−	0.212162i	$-0.931950\pi$
$293$	−197.000	−	341.214i	−0.672355	−	1.16455i	0.304880	−	0.952391i	$-0.401384\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$	119.000	+	206.114i	0.394040	+	0.682497i
$303$		0			0
$304$	55.0000	−	95.2628i	0.180921	−	0.313364i
$305$	−590.000			−1.93443
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$	−5.00000	−	8.66025i	−0.0161290	−	0.0279363i
$311$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$311$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$312$		0			0
$313$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$313$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$314$		0			0
$315$		0			0
$316$	−294.000			−0.930380
$317$	67.0000	−	116.047i	0.211356	−	0.366080i	−0.740783	−	0.671745i	$-0.765545\pi$
$317$	67.0000	−	116.047i	0.211356	−	0.366080i	0.952139	+	0.305664i	$0.0988786\pi$
$318$		0			0
$319$		0			0
$320$	−32.5000	−	56.2917i	−0.101562	−	0.175911i
$321$		0			0
$322$		0			0
$323$	308.000			0.953560
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−61.0000	+	105.655i	−0.184290	+	0.319200i	−0.943337	−	0.331836i	$-0.892332\pi$
$331$	−61.0000	+	105.655i	−0.184290	+	0.319200i	0.759047	+	0.651036i	$0.225665\pi$
$332$	−462.000			−1.39157
$333$		0			0
$334$	−254.000			−0.760479
$335$		0			0
$336$		0			0
$337$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$337$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$338$	−84.5000	−	146.358i	−0.250000	−	0.433013i
$339$		0			0
$340$	−105.000	+	181.865i	−0.308824	+	0.534898i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−77.0000	−	133.368i	−0.222543	−	0.385456i
$347$	−293.000	−	507.491i	−0.844380	−	1.46251i	−0.886158	−	0.463383i	$-0.846635\pi$
$347$	−293.000	−	507.491i	−0.844380	−	1.46251i	0.0417778	−	0.999127i	$-0.486698\pi$
$348$		0			0
$349$	−229.000	+	396.640i	−0.656160	+	1.13650i	0.325441	+	0.945562i	$0.394487\pi$
$349$	−229.000	+	396.640i	−0.656160	+	1.13650i	−0.981602	+	0.190941i	$0.938846\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−137.000	+	237.291i	−0.388102	+	0.672212i	−0.992194	−	0.124702i	$-0.960203\pi$
$353$	−137.000	+	237.291i	−0.388102	+	0.672212i	0.604092	+	0.796914i	$0.293536\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$	123.000			0.340720
$362$	−61.0000	+	105.655i	−0.168508	+	0.291865i
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$367$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$368$	170.000			0.461957
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$373$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$374$		0			0
$375$		0			0
$376$	−49.0000	+	84.8705i	−0.130319	+	0.225719i
$377$		0			0
$378$		0			0
$379$	−742.000			−1.95778			−0.978892	−	0.204379i	$-0.934482\pi$
$379$	−742.000			−1.95778			−0.978892	+	0.204379i	$0.934482\pi$
$380$	−165.000	+	285.788i	−0.434211	+	0.752075i
$381$		0			0
$382$		0			0
$383$	343.000	+	594.093i	0.895561	+	1.55116i	0.833108	+	0.553110i	$0.186559\pi$
$383$	343.000	+	594.093i	0.895561	+	1.55116i	0.0624530	+	0.998048i	$0.480108\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$389$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$390$		0			0
$391$	238.000	+	412.228i	0.608696	+	1.05429i
$392$	171.500	+	297.047i	0.437500	+	0.757772i
$393$		0			0
$394$	187.000	−	323.894i	0.474619	−	0.822065i
$395$	490.000			1.24051
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$	71.0000	−	122.976i	0.178392	−	0.308984i
$399$		0			0
$400$	−62.5000	−	108.253i	−0.156250	−	0.270633i
$401$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$401$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	71.0000	+	122.976i	0.173594	+	0.300674i	0.939674	−	0.342072i	$-0.111129\pi$
$409$	71.0000	+	122.976i	0.173594	+	0.300674i	−0.766080	+	0.642746i	$0.777795\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	770.000			1.85542
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$419$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$420$		0			0
$421$	−301.000	+	521.347i	−0.714964	+	1.23835i	0.248009	+	0.968758i	$0.420224\pi$
$421$	−301.000	+	521.347i	−0.714964	+	1.23835i	−0.962973	+	0.269597i	$0.913110\pi$
$422$	362.000			0.857820
$423$		0			0
$424$	602.000			1.41981
$425$	175.000	−	303.109i	0.411765	−	0.713197i
$426$		0			0
$427$		0			0
$428$	159.000	+	275.396i	0.371495	+	0.643449i
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$	−33.0000	−	57.1577i	−0.0756881	−	0.131096i
$437$	374.000	+	647.787i	0.855835	+	1.48235i
$438$		0			0
$439$	311.000	−	538.668i	0.708428	−	1.22703i	−0.257012	−	0.966408i	$-0.582738\pi$
$439$	311.000	−	538.668i	0.708428	−	1.22703i	0.965440	−	0.260625i	$-0.0839288\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	283.000	−	490.170i	0.638826	−	1.10648i	−0.346864	−	0.937915i	$-0.612754\pi$
$443$	283.000	−	490.170i	0.638826	−	1.10648i	0.985691	−	0.168564i	$-0.0539130\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$	−309.000	+	535.204i	−0.683628	+	1.18408i
$453$		0			0
$454$	67.0000	+	116.047i	0.147577	+	0.255611i
$455$		0			0
$456$		0			0
$457$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$457$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$458$	218.000			0.475983
$459$		0			0
$460$	−510.000			−1.10870
$461$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$461$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$462$		0			0
$463$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$463$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$464$		0			0
$465$		0			0
$466$	−17.0000	+	29.4449i	−0.0364807	+	0.0631864i
$467$	346.000			0.740899			0.370450	−	0.928853i	$-0.379204\pi$
$467$	346.000			0.740899			0.370450	+	0.928853i	$0.379204\pi$
$468$		0			0
$469$		0			0
$470$	35.0000	−	60.6218i	0.0744681	−	0.128983i
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	275.000	−	476.314i	0.578947	−	1.00277i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$479$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$480$		0			0
$481$		0			0
$482$	239.000	+	413.960i	0.495851	+	0.858838i
$483$		0			0
$484$	181.500	−	314.367i	0.375000	−	0.649519i
$485$		0			0
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$	−413.000	+	715.337i	−0.846311	+	1.46585i
$489$		0			0
$490$	−122.500	−	212.176i	−0.250000	−	0.433013i
$491$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$491$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	10.0000			0.0201613
$497$		0			0
$498$		0			0
$499$	−469.000	−	812.332i	−0.939880	−	1.62792i	−0.765692	−	0.643207i	$-0.777603\pi$
$499$	−469.000	−	812.332i	−0.939880	−	1.62792i	−0.174187	−	0.984713i	$-0.555730\pi$
$500$	187.500	+	324.760i	0.375000	+	0.649519i
$501$		0			0
$502$		0			0
$503$	994.000			1.97614			0.988072	−	0.153995i	$-0.0492141\pi$
$503$	994.000			1.97614			0.988072	+	0.153995i	$0.0492141\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$509$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$510$		0			0
$511$		0			0
$512$	305.000			0.595703
$513$		0			0
$514$	466.000			0.906615
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$		0			0
$525$		0			0
$526$	223.000	+	386.247i	0.423954	+	0.734311i
$527$	14.0000	+	24.2487i	0.0265655	+	0.0460127i
$528$		0			0
$529$	−313.500	+	542.998i	−0.592628	+	1.02646i
$530$	−430.000			−0.811321
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−265.000	−	458.993i	−0.495327	−	0.857932i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−1078.00			−1.99261			−0.996303	−	0.0859072i	$-0.972621\pi$
$541$	−1078.00			−1.99261			−0.996303	+	0.0859072i	$0.972621\pi$
$542$	−241.000	+	417.424i	−0.444649	+	0.770155i
$543$		0			0
$544$	231.000	+	400.104i	0.424632	+	0.735485i
$545$	55.0000	+	95.2628i	0.100917	+	0.174794i
$546$		0			0
$547$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$547$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$548$	−678.000			−1.23723
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−393.000	+	680.696i	−0.706835	+	1.22427i
$557$	−614.000			−1.10233			−0.551167	−	0.834395i	$-0.685817\pi$
$557$	−614.000			−1.10233			−0.551167	+	0.834395i	$0.685817\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−77.0000	−	133.368i	−0.136767	−	0.236888i	0.789504	−	0.613746i	$-0.210338\pi$
$563$	−77.0000	−	133.368i	−0.136767	−	0.236888i	−0.926271	+	0.376858i	$0.877005\pi$
$564$		0			0
$565$	515.000	−	892.006i	0.911504	−	1.57877i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$569$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$570$		0			0
$571$	179.000	+	310.037i	0.313485	+	0.542972i	0.979114	−	0.203310i	$-0.0651701\pi$
$571$	179.000	+	310.037i	0.313485	+	0.542972i	−0.665629	+	0.746283i	$0.731837\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	850.000			1.47826
$576$		0			0
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	46.5000	−	80.5404i	0.0804498	−	0.139343i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	394.000			0.672355
$587$	427.000	−	739.586i	0.727428	−	1.25994i	−0.230539	−	0.973063i	$-0.574049\pi$
$587$	427.000	−	739.586i	0.727428	−	1.25994i	0.957967	−	0.286879i	$-0.0926176\pi$
$588$		0			0
$589$	22.0000	+	38.1051i	0.0373514	+	0.0646946i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−1166.00			−1.96627			−0.983137	−	0.182873i	$-0.941460\pi$
$593$	−1166.00			−1.96627			−0.983137	+	0.182873i	$0.941460\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$599$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$600$		0			0
$601$	−121.000	+	209.578i	−0.201331	+	0.348716i	−0.948958	−	0.315404i	$-0.897860\pi$
$601$	−121.000	+	209.578i	−0.201331	+	0.348716i	0.747626	+	0.664119i	$0.231193\pi$
$602$		0			0
$603$		0			0
$604$	714.000			1.18212
$605$	−302.500	+	523.945i	−0.500000	+	0.866025i
$606$		0			0
$607$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$607$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$608$	363.000	+	628.734i	0.597039	+	1.03410i
$609$		0			0
$610$	295.000	−	510.955i	0.483607	−	0.837631i
$611$		0			0
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−593.000	−	1027.11i	−0.961102	−	1.66468i	−0.719741	−	0.694242i	$-0.755740\pi$
$617$	−593.000	−	1027.11i	−0.961102	−	1.66468i	−0.241361	−	0.970435i	$-0.577594\pi$
$618$		0			0
$619$	−349.000	+	604.486i	−0.563813	+	0.976552i	0.433346	+	0.901227i	$0.357333\pi$
$619$	−349.000	+	604.486i	−0.563813	+	0.976552i	−0.997159	+	0.0753247i	$0.976001\pi$
$620$	−30.0000			−0.0483871
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−312.500	−	541.266i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−238.000			−0.377179			−0.188590	−	0.982056i	$-0.560392\pi$
$631$	−238.000			−0.377179			−0.188590	+	0.982056i	$0.560392\pi$
$632$	343.000	−	594.093i	0.542722	−	0.940021i
$633$		0			0
$634$	67.0000	+	116.047i	0.105678	+	0.183040i
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	−595.000			−0.929688
$641$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$641$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$642$		0			0
$643$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$643$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$644$		0			0
$645$		0			0
$646$	−154.000	+	266.736i	−0.238390	+	0.412904i
$647$	706.000			1.09119			0.545595	−	0.838049i	$-0.316304\pi$
$647$	706.000			1.09119			0.545595	+	0.838049i	$0.316304\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−557.000	−	964.752i	−0.852986	−	1.47742i	−0.878501	−	0.477741i	$-0.841456\pi$
$653$	−557.000	−	964.752i	−0.852986	−	1.47742i	0.0255145	−	0.999674i	$-0.491878\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$659$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$660$		0			0
$661$	419.000	+	725.729i	0.633888	+	1.09793i	0.986750	+	0.162251i	$0.0518753\pi$
$661$	419.000	+	725.729i	0.633888	+	1.09793i	−0.352862	+	0.935676i	$0.614791\pi$
$662$	−61.0000	−	105.655i	−0.0921450	−	0.159600i
$663$		0			0
$664$	539.000	−	933.575i	0.811747	−	1.40599i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−381.000	+	659.911i	−0.570359	+	0.987891i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$673$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$674$		0			0
$675$		0			0
$676$	−507.000			−0.750000
$677$	187.000	−	323.894i	0.276219	−	0.478425i	−0.694223	−	0.719760i	$-0.744252\pi$
$677$	187.000	−	323.894i	0.276219	−	0.478425i	0.970442	+	0.241335i	$0.0775853\pi$
$678$		0			0
$679$		0			0
$680$	−245.000	−	424.352i	−0.360294	−	0.624048i
$681$		0			0
$682$		0			0
$683$	−86.0000			−0.125915			−0.0629575	−	0.998016i	$-0.520053\pi$
$683$	−86.0000			−0.125915			−0.0629575	+	0.998016i	$0.520053\pi$
$684$		0			0
$685$	1130.00			1.64964
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−661.000	+	1144.89i	−0.956585	+	1.65685i	−0.225885	+	0.974154i	$0.572527\pi$
$691$	−661.000	+	1144.89i	−0.956585	+	1.65685i	−0.730699	+	0.682699i	$0.760806\pi$
$692$	−462.000			−0.667630
$693$		0			0
$694$	586.000			0.844380
$695$	655.000	−	1134.49i	0.942446	−	1.63236i
$696$		0			0
$697$		0			0
$698$	−229.000	−	396.640i	−0.328080	−	0.568252i
$699$		0			0
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−137.000	−	237.291i	−0.194051	−	0.336106i
$707$		0			0
$708$		0			0
$709$	371.000	−	642.591i	0.523272	−	0.906334i	−0.476361	−	0.879250i	$-0.658044\pi$
$709$	371.000	−	642.591i	0.523272	−	0.906334i	0.999633	−	0.0270842i	$-0.00862223\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−34.0000	+	58.8897i	−0.0476858	+	0.0825943i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		0			0
$721$		0			0
$722$	−61.5000	+	106.521i	−0.0851801	+	0.147536i
$723$		0			0
$724$	183.000	+	316.965i	0.252762	+	0.437797i
$725$		0			0
$726$		0			0
$727$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$727$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$733$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$734$		0			0
$735$		0			0
$736$	−561.000	+	971.681i	−0.762228	+	1.32022i
$737$		0			0
$738$		0			0
$739$	−1462.00			−1.97835			−0.989175	−	0.146744i	$-0.953121\pi$
$739$	−1462.00			−1.97835			−0.989175	+	0.146744i	$0.953121\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−257.000	−	445.137i	−0.345895	−	0.599108i	0.639621	−	0.768690i	$-0.279091\pi$
$743$	−257.000	−	445.137i	−0.345895	−	0.599108i	−0.985516	+	0.169583i	$0.945758\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	719.000	+	1245.34i	0.957390	+	1.65825i	0.728801	+	0.684725i	$0.240078\pi$
$751$	719.000	+	1245.34i	0.957390	+	1.65825i	0.228589	+	0.973523i	$0.426589\pi$
$752$	35.0000	+	60.6218i	0.0465426	+	0.0806141i
$753$		0			0
$754$		0			0
$755$	−1190.00			−1.57616
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$	371.000	−	642.591i	0.489446	−	0.847745i
$759$		0			0
$760$	−385.000	−	666.840i	−0.506579	−	0.877420i
$761$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$761$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	−686.000			−0.895561
$767$		0			0
$768$		0			0
$769$	−289.000	−	500.563i	−0.375813	−	0.650927i	0.614636	−	0.788811i	$-0.289303\pi$
$769$	−289.000	−	500.563i	−0.375813	−	0.650927i	−0.990448	+	0.137884i	$0.955970\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−1526.00			−1.97413			−0.987063	−	0.160330i	$-0.948744\pi$
$773$	−1526.00			−1.97413			−0.987063	+	0.160330i	$0.948744\pi$
$774$		0			0
$775$	50.0000			0.0645161
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$	−476.000			−0.608696
$783$		0			0
$784$	245.000			0.312500
$785$		0			0
$786$		0			0
$787$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$787$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$788$	−561.000	−	971.681i	−0.711929	−	1.23310i
$789$		0			0
$790$	−245.000	+	424.352i	−0.310127	+	0.537155i
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	−213.000	−	368.927i	−0.267588	−	0.463476i
$797$	−413.000	−	715.337i	−0.518193	−	0.897537i	−0.999777	−	0.0211367i	$-0.993271\pi$
$797$	−413.000	−	715.337i	−0.518193	−	0.897537i	0.481583	−	0.876400i	$-0.340062\pi$
$798$		0			0
$799$	−98.0000	+	169.741i	−0.122653	+	0.212442i
$800$	825.000			1.03125
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$		0			0
$811$	1082.00			1.33416			0.667078	−	0.744988i	$-0.267545\pi$
$811$	1082.00			1.33416			0.667078	+	0.744988i	$0.267545\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−142.000			−0.173594
$819$		0			0
$820$		0			0
$821$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$821$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$822$		0			0
$823$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$823$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−374.000			−0.452237			−0.226119	−	0.974100i	$-0.572604\pi$
$827$	−374.000			−0.452237			−0.226119	+	0.974100i	$0.572604\pi$
$828$		0			0
$829$	−502.000			−0.605549			−0.302774	−	0.953062i	$-0.597913\pi$
$829$	−502.000			−0.605549			−0.302774	+	0.953062i	$0.597913\pi$
$830$	−385.000	+	666.840i	−0.463855	+	0.803421i
$831$		0			0
$832$		0			0
$833$	343.000	+	594.093i	0.411765	+	0.713197i
$834$		0			0
$835$	635.000	−	1099.85i	0.760479	−	1.31719i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$839$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$840$		0			0
$841$	−420.500	−	728.327i	−0.500000	−	0.866025i
$842$	−301.000	−	521.347i	−0.357482	−	0.619177i
$843$		0			0
$844$	543.000	−	940.504i	0.643365	−	1.11434i
$845$	845.000			1.00000
$846$		0			0
$847$		0			0
$848$	215.000	−	372.391i	0.253538	−	0.439140i
$849$		0			0
$850$	175.000	+	303.109i	0.205882	+	0.356599i
$851$		0			0
$852$		0			0
$853$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$853$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$854$		0			0
$855$		0			0
$856$	−742.000			−0.866822
$857$	−833.000	+	1442.80i	−0.971995	+	1.68355i	−0.282482	+	0.959273i	$0.591158\pi$
$857$	−833.000	+	1442.80i	−0.971995	+	1.68355i	−0.689514	+	0.724273i	$0.742176\pi$
$858$		0			0
$859$	−109.000	−	188.794i	−0.126892	−	0.219783i	0.795579	−	0.605850i	$-0.207167\pi$
$859$	−109.000	−	188.794i	−0.126892	−	0.219783i	−0.922471	+	0.386067i	$0.873833\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	274.000			0.317497			0.158749	−	0.987319i	$-0.449254\pi$
$863$	274.000			0.317497			0.158749	+	0.987319i	$0.449254\pi$
$864$		0			0
$865$	770.000			0.890173
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	154.000			0.176606
$873$		0			0
$874$	−748.000			−0.855835
$875$		0			0
$876$		0			0
$877$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$877$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$878$	311.000	+	538.668i	0.354214	+	0.613517i
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$	283.000	+	490.170i	0.319413	+	0.553240i
$887$	847.000	+	1467.05i	0.954904	+	1.65394i	0.734587	+	0.678514i	$0.237376\pi$
$887$	847.000	+	1467.05i	0.954904	+	1.65394i	0.220317	+	0.975428i	$0.429291\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−154.000	+	266.736i	−0.172452	+	0.298696i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	1204.00			1.33629
$902$		0			0
$903$		0			0
$904$	−721.000	−	1248.81i	−0.797566	−	1.38143i
$905$	−305.000	−	528.275i	−0.337017	−	0.583730i
$906$		0			0
$907$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$907$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$908$	402.000			0.442731
$909$		0			0
$910$		0			0
$911$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$911$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	327.000	−	566.381i	0.356987	−	0.618319i
$917$		0			0
$918$		0			0
$919$	1298.00			1.41240			0.706202	−	0.708010i	$-0.250407\pi$
$919$	1298.00			1.41240			0.706202	+	0.708010i	$0.250407\pi$
$920$	595.000	−	1030.57i	0.646739	−	1.12019i
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$929$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$930$		0			0
$931$	539.000	+	933.575i	0.578947	+	1.00277i
$932$	51.0000	+	88.3346i	0.0547210	+	0.0947796i
$933$		0			0
$934$	−173.000	+	299.645i	−0.185225	+	0.320819i
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$	−105.000	−	181.865i	−0.111702	−	0.193474i
$941$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$941$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	787.000	−	1363.12i	0.831045	−	1.43941i	−0.0661646	−	0.997809i	$-0.521076\pi$
$947$	787.000	−	1363.12i	0.831045	−	1.43941i	0.897210	−	0.441604i	$-0.145590\pi$
$948$		0			0
$949$		0			0
$950$	275.000	+	476.314i	0.289474	+	0.501383i
$951$		0			0
$952$		0			0
$953$	1474.00			1.54669			0.773347	−	0.633983i	$-0.218581\pi$
$953$	1474.00			1.54669			0.773347	+	0.633983i	$0.218581\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	478.500	−	828.786i	0.497919	−	0.862421i
$962$		0			0
$963$		0			0
$964$	1434.00			1.48755
$965$		0			0
$966$		0			0
$967$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$967$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$968$	423.500	+	733.524i	0.437500	+	0.757772i
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	295.000	+	510.955i	0.302254	+	0.523519i
$977$	967.000	+	1674.89i	0.989765	+	1.71432i	0.618473	+	0.785806i	$0.287752\pi$
$977$	967.000	+	1674.89i	0.989765	+	1.71432i	0.371292	+	0.928516i	$0.378915\pi$
$978$		0			0
$979$		0			0
$980$	−735.000			−0.750000
$981$		0			0
$982$		0			0
$983$	−977.000	+	1692.21i	−0.993896	+	1.72148i	−0.401409	+	0.915899i	$0.631479\pi$
$983$	−977.000	+	1692.21i	−0.993896	+	1.72148i	−0.592487	+	0.805580i	$0.701854\pi$
$984$		0			0
$985$	935.000	+	1619.47i	0.949239	+	1.64413i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−958.000			−0.966700			−0.483350	−	0.875427i	$-0.660580\pi$
$991$	−958.000			−0.966700			−0.483350	+	0.875427i	$0.660580\pi$
$992$	−33.0000	+	57.1577i	−0.0332661	+	0.0576186i
$993$		0			0
$994$		0			0
$995$	355.000	+	614.878i	0.356784	+	0.617968i
$996$		0			0
$997$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$997$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$998$	938.000			0.939880
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.3.h.a.269.1		2
3.2	odd	2		405.3.h.b.269.1		2
5.4	even	2		405.3.h.b.269.1		2
9.2	odd	6		15.3.d.a.14.1	✓	1
9.4	even	3	inner	405.3.h.a.134.1		2
9.5	odd	6		405.3.h.b.134.1		2
9.7	even	3		15.3.d.b.14.1	yes	1
15.14	odd	2	CM	405.3.h.a.269.1		2
36.7	odd	6		240.3.c.b.209.1		1
36.11	even	6		240.3.c.a.209.1		1
45.2	even	12		75.3.c.d.26.1		2
45.4	even	6		405.3.h.b.134.1		2
45.7	odd	12		75.3.c.d.26.2		2
45.14	odd	6	inner	405.3.h.a.134.1		2
45.29	odd	6		15.3.d.b.14.1	yes	1
45.34	even	6		15.3.d.a.14.1	✓	1
45.38	even	12		75.3.c.d.26.2		2
45.43	odd	12		75.3.c.d.26.1		2
72.11	even	6		960.3.c.d.449.1		1
72.29	odd	6		960.3.c.b.449.1		1
72.43	odd	6		960.3.c.a.449.1		1
72.61	even	6		960.3.c.c.449.1		1
180.7	even	12		1200.3.l.l.401.1		2
180.43	even	12		1200.3.l.l.401.2		2
180.47	odd	12		1200.3.l.l.401.2		2
180.79	odd	6		240.3.c.a.209.1		1
180.83	odd	12		1200.3.l.l.401.1		2
180.119	even	6		240.3.c.b.209.1		1
360.29	odd	6		960.3.c.c.449.1		1
360.259	odd	6		960.3.c.d.449.1		1
360.299	even	6		960.3.c.a.449.1		1
360.349	even	6		960.3.c.b.449.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.3.d.a.14.1	✓	1	9.2	odd	6
15.3.d.a.14.1	✓	1	45.34	even	6
15.3.d.b.14.1	yes	1	9.7	even	3
15.3.d.b.14.1	yes	1	45.29	odd	6
75.3.c.d.26.1		2	45.2	even	12
75.3.c.d.26.1		2	45.43	odd	12
75.3.c.d.26.2		2	45.7	odd	12
75.3.c.d.26.2		2	45.38	even	12
240.3.c.a.209.1		1	36.11	even	6
240.3.c.a.209.1		1	180.79	odd	6
240.3.c.b.209.1		1	36.7	odd	6
240.3.c.b.209.1		1	180.119	even	6
405.3.h.a.134.1		2	9.4	even	3	inner
405.3.h.a.134.1		2	45.14	odd	6	inner
405.3.h.a.269.1		2	1.1	even	1	trivial
405.3.h.a.269.1		2	15.14	odd	2	CM
405.3.h.b.134.1		2	9.5	odd	6
405.3.h.b.134.1		2	45.4	even	6
405.3.h.b.269.1		2	3.2	odd	2
405.3.h.b.269.1		2	5.4	even	2
960.3.c.a.449.1		1	72.43	odd	6
960.3.c.a.449.1		1	360.299	even	6
960.3.c.b.449.1		1	72.29	odd	6
960.3.c.b.449.1		1	360.349	even	6
960.3.c.c.449.1		1	72.61	even	6
960.3.c.c.449.1		1	360.29	odd	6
960.3.c.d.449.1		1	72.11	even	6
960.3.c.d.449.1		1	360.259	odd	6
1200.3.l.l.401.1		2	180.7	even	12
1200.3.l.l.401.1		2	180.83	odd	12
1200.3.l.l.401.2		2	180.43	even	12
1200.3.l.l.401.2		2	180.47	odd	12

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	405.h (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{2} +(1.50000 + 2.59808i) q^{4} +(-2.50000 - 4.33013i) q^{5} -7.00000 q^{8} +O(q^{10})\)	\(q+(-0.500000 + 0.866025i) q^{2} +(1.50000 + 2.59808i) q^{4} +(-2.50000 - 4.33013i) q^{5} -7.00000 q^{8} +5.00000 q^{10} +(-2.50000 + 4.33013i) q^{16} -14.0000 q^{17} -22.0000 q^{19} +(7.50000 - 12.9904i) q^{20} +(-17.0000 - 29.4449i) q^{23} +(-12.5000 + 21.6506i) q^{25} +(-1.00000 - 1.73205i) q^{31} +(-16.5000 - 28.5788i) q^{32} +(7.00000 - 12.1244i) q^{34} +(11.0000 - 19.0526i) q^{38} +(17.5000 + 30.3109i) q^{40} +34.0000 q^{46} +(7.00000 - 12.1244i) q^{47} +(-24.5000 - 42.4352i) q^{49} +(-12.5000 - 21.6506i) q^{50} -86.0000 q^{53} +(59.0000 - 102.191i) q^{61} +2.00000 q^{62} +13.0000 q^{64} +(-21.0000 - 36.3731i) q^{68} +(-33.0000 - 57.1577i) q^{76} +(-49.0000 + 84.8705i) q^{79} +25.0000 q^{80} +(-77.0000 + 133.368i) q^{83} +(35.0000 + 60.6218i) q^{85} +(51.0000 - 88.3346i) q^{92} +(7.00000 + 12.1244i) q^{94} +(55.0000 + 95.2628i) q^{95} +49.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 3 q^{4} - 5 q^{5} - 14 q^{8}+O(q^{10}) \) 2 * q - q^2 + 3 * q^4 - 5 * q^5 - 14 * q^8	\( 2 q - q^{2} + 3 q^{4} - 5 q^{5} - 14 q^{8} + 10 q^{10} - 5 q^{16} - 28 q^{17} - 44 q^{19} + 15 q^{20} - 34 q^{23} - 25 q^{25} - 2 q^{31} - 33 q^{32} + 14 q^{34} + 22 q^{38} + 35 q^{40} + 68 q^{46} + 14 q^{47} - 49 q^{49} - 25 q^{50} - 172 q^{53} + 118 q^{61} + 4 q^{62} + 26 q^{64} - 42 q^{68} - 66 q^{76} - 98 q^{79} + 50 q^{80} - 154 q^{83} + 70 q^{85} + 102 q^{92} + 14 q^{94} + 110 q^{95} + 98 q^{98}+O(q^{100}) \) 2 * q - q^2 + 3 * q^4 - 5 * q^5 - 14 * q^8 + 10 * q^10 - 5 * q^16 - 28 * q^17 - 44 * q^19 + 15 * q^20 - 34 * q^23 - 25 * q^25 - 2 * q^31 - 33 * q^32 + 14 * q^34 + 22 * q^38 + 35 * q^40 + 68 * q^46 + 14 * q^47 - 49 * q^49 - 25 * q^50 - 172 * q^53 + 118 * q^61 + 4 * q^62 + 26 * q^64 - 42 * q^68 - 66 * q^76 - 98 * q^79 + 50 * q^80 - 154 * q^83 + 70 * q^85 + 102 * q^92 + 14 * q^94 + 110 * q^95 + 98 * q^98

Introduction

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