LMFDB - Embedded newform 720.6.f.f.289.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [720,6,Mod(289,720)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(720, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("720.289");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 720 = 2^{4} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	720.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$115.476350265$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-11}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 3 $ x^2 - x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 5)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			289.1
Root			$0.500000 - 1.65831i$ of defining polynomial
Character	$\chi$	$=$	720.289
Dual form			720.6.f.f.289.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(45.0000 - 33.1662i) q^{5} +59.6992i q^{7} +O(q^{10})$	$q+(45.0000 - 33.1662i) q^{5} +59.6992i q^{7} +252.000 q^{11} -119.398i q^{13} +689.858i q^{17} +220.000 q^{19} +2434.40i q^{23} +(925.000 - 2984.96i) q^{25} +6930.00 q^{29} -6752.00 q^{31} +(1980.00 + 2686.47i) q^{35} +13969.6i q^{37} +198.000 q^{41} +417.895i q^{43} -10540.2i q^{47} +13243.0 q^{49} +5823.99i q^{53} +(11340.0 - 8357.89i) q^{55} -24660.0 q^{59} -5698.00 q^{61} +(-3960.00 - 5372.93i) q^{65} +43640.1i q^{67} +53352.0 q^{71} +70922.7i q^{73} +15044.2i q^{77} -51920.0 q^{79} -61841.8i q^{83} +(22880.0 + 31043.6i) q^{85} +9990.00 q^{89} +7128.00 q^{91} +(9900.00 - 7296.57i) q^{95} -101250. i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 90 q^{5}+O(q^{10}) $ 2 * q + 90 * q^5	$ 2 q + 90 q^{5} + 504 q^{11} + 440 q^{19} + 1850 q^{25} + 13860 q^{29} - 13504 q^{31} + 3960 q^{35} + 396 q^{41} + 26486 q^{49} + 22680 q^{55} - 49320 q^{59} - 11396 q^{61} - 7920 q^{65} + 106704 q^{71} - 103840 q^{79} + 45760 q^{85} + 19980 q^{89} + 14256 q^{91} + 19800 q^{95}+O(q^{100}) $ 2 * q + 90 * q^5 + 504 * q^11 + 440 * q^19 + 1850 * q^25 + 13860 * q^29 - 13504 * q^31 + 3960 * q^35 + 396 * q^41 + 26486 * q^49 + 22680 * q^55 - 49320 * q^59 - 11396 * q^61 - 7920 * q^65 + 106704 * q^71 - 103840 * q^79 + 45760 * q^85 + 19980 * q^89 + 14256 * q^91 + 19800 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/720\mathbb{Z}\right)^\times$.

$n$	$181$	$271$	$577$	$641$
$\chi(n)$	$1$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	45.0000	−	33.1662i	0.804984	−	0.593296i
$5$	45.0000	−	33.1662i	0.804984	−	0.593296i
$6$		0			0
$7$			59.6992i			0.460494i	0.973132	+	0.230247i	$0.0739534\pi$
$7$			59.6992i			0.460494i	−0.973132	+	0.230247i	$0.926047\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	252.000			0.627941			0.313970	−	0.949433i	$-0.398341\pi$
$11$	252.000			0.627941			0.313970	+	0.949433i	$0.398341\pi$
$12$		0			0
$13$		−	119.398i		−	0.195948i	−0.995189	−	0.0979739i	$-0.968764\pi$
$13$		−	119.398i		−	0.195948i	0.995189	−	0.0979739i	$-0.0312362\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			689.858i			0.578945i	0.957186	+	0.289473i	$0.0934799\pi$
$17$			689.858i			0.578945i	−0.957186	+	0.289473i	$0.906520\pi$
$18$		0			0
$19$	220.000			0.139810			0.0699051	−	0.997554i	$-0.477730\pi$
$19$	220.000			0.139810			0.0699051	+	0.997554i	$0.477730\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			2434.40i			0.959561i	0.877388	+	0.479781i	$0.159284\pi$
$23$			2434.40i			0.959561i	−0.877388	+	0.479781i	$0.840716\pi$
$24$		0			0
$25$	925.000	−	2984.96i	0.296000	−	0.955188i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	6930.00			1.53016			0.765082	−	0.643932i	$-0.222698\pi$
$29$	6930.00			1.53016			0.765082	+	0.643932i	$0.222698\pi$
$30$		0			0
$31$	−6752.00			−1.26191			−0.630955	−	0.775820i	$-0.717337\pi$
$31$	−6752.00			−1.26191			−0.630955	+	0.775820i	$0.717337\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1980.00	+	2686.47i	0.273209	+	0.370690i
$36$		0			0
$37$			13969.6i			1.67757i	0.544464	+	0.838785i	$0.316733\pi$
$37$			13969.6i			1.67757i	−0.544464	+	0.838785i	$0.683267\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	198.000			0.0183952			0.00919762	−	0.999958i	$-0.497072\pi$
$41$	198.000			0.0183952			0.00919762	+	0.999958i	$0.497072\pi$
$42$		0			0
$43$			417.895i			0.0344664i	0.999851	+	0.0172332i	$0.00548577\pi$
$43$			417.895i			0.0344664i	−0.999851	+	0.0172332i	$0.994514\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	10540.2i		−	0.695994i	−0.937496	−	0.347997i	$-0.886862\pi$
$47$		−	10540.2i		−	0.695994i	0.937496	−	0.347997i	$-0.113138\pi$
$48$		0			0
$49$	13243.0			0.787945
$50$		0			0
$51$		0			0
$52$		0			0
$53$			5823.99i			0.284794i	0.989810	+	0.142397i	$0.0454810\pi$
$53$			5823.99i			0.284794i	−0.989810	+	0.142397i	$0.954519\pi$
$54$		0			0
$55$	11340.0	−	8357.89i	0.505483	−	0.372555i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−24660.0			−0.922281			−0.461140	−	0.887327i	$-0.652560\pi$
$59$	−24660.0			−0.922281			−0.461140	+	0.887327i	$0.652560\pi$
$60$		0			0
$61$	−5698.00			−0.196064			−0.0980320	−	0.995183i	$-0.531255\pi$
$61$	−5698.00			−0.196064			−0.0980320	+	0.995183i	$0.531255\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−3960.00	−	5372.93i	−0.116255	−	0.157735i
$66$		0			0
$67$			43640.1i			1.18768i	0.804583	+	0.593840i	$0.202389\pi$
$67$			43640.1i			1.18768i	−0.804583	+	0.593840i	$0.797611\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	53352.0			1.25604			0.628022	−	0.778196i	$-0.283865\pi$
$71$	53352.0			1.25604			0.628022	+	0.778196i	$0.283865\pi$
$72$		0			0
$73$			70922.7i			1.55768i	0.627223	+	0.778840i	$0.284192\pi$
$73$			70922.7i			1.55768i	−0.627223	+	0.778840i	$0.715808\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			15044.2i			0.289163i
$78$		0			0
$79$	−51920.0			−0.935981			−0.467990	−	0.883734i	$-0.655022\pi$
$79$	−51920.0			−0.935981			−0.467990	+	0.883734i	$0.655022\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	61841.8i		−	0.985342i	−0.870216	−	0.492671i	$-0.836021\pi$
$83$		−	61841.8i		−	0.985342i	0.870216	−	0.492671i	$-0.163979\pi$
$84$		0			0
$85$	22880.0	+	31043.6i	0.343486	+	0.466042i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	9990.00			0.133687			0.0668437	−	0.997763i	$-0.478707\pi$
$89$	9990.00			0.133687			0.0668437	+	0.997763i	$0.478707\pi$
$90$		0			0
$91$	7128.00			0.0902328
$92$		0			0
$93$		0			0
$94$		0			0
$95$	9900.00	−	7296.57i	0.112545	−	0.0829488i
$96$		0			0
$97$		−	101250.i		−	1.09261i	−0.837586	−	0.546305i	$-0.816034\pi$
$97$		−	101250.i		−	1.09261i	0.837586	−	0.546305i	$-0.183966\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	109098.			1.06418			0.532088	−	0.846689i	$-0.321408\pi$
$101$	109098.			1.06418			0.532088	+	0.846689i	$0.321408\pi$
$102$		0			0
$103$		−	70624.2i		−	0.655935i	−0.944689	−	0.327967i	$-0.893636\pi$
$103$		−	70624.2i		−	0.655935i	0.944689	−	0.327967i	$-0.106364\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			97117.4i			0.820045i	0.912075	+	0.410022i	$0.134479\pi$
$107$			97117.4i			0.820045i	−0.912075	+	0.410022i	$0.865521\pi$
$108$		0			0
$109$	−21010.0			−0.169379			−0.0846895	−	0.996407i	$-0.526990\pi$
$109$	−21010.0			−0.169379			−0.0846895	+	0.996407i	$0.526990\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	105018.i		−	0.773688i	−0.922145	−	0.386844i	$-0.873565\pi$
$113$		−	105018.i		−	0.773688i	0.922145	−	0.386844i	$-0.126435\pi$
$114$		0			0
$115$	80740.0	+	109548.i	0.569304	+	0.772432i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−41184.0			−0.266601
$120$		0			0
$121$	−97547.0			−0.605690
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−57375.0	−	165002.i	−0.328434	−	0.944527i
$126$		0			0
$127$			87220.6i			0.479855i	0.970791	+	0.239927i	$0.0771236\pi$
$127$			87220.6i			0.479855i	−0.970791	+	0.239927i	$0.922876\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	192852.			0.981852			0.490926	−	0.871201i	$-0.336659\pi$
$131$	192852.			0.981852			0.490926	+	0.871201i	$0.336659\pi$
$132$		0			0
$133$			13133.8i			0.0643817i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			143570.i			0.653525i	0.945106	+	0.326763i	$0.105958\pi$
$137$			143570.i			0.653525i	−0.945106	+	0.326763i	$0.894042\pi$
$138$		0			0
$139$	318340.			1.39751			0.698754	−	0.715362i	$-0.253738\pi$
$139$	318340.			1.39751			0.698754	+	0.715362i	$0.253738\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	30088.4i		−	0.123044i
$144$		0			0
$145$	311850.	−	229842.i	1.23176	−	0.907841i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−84150.0			−0.310519			−0.155260	−	0.987874i	$-0.549621\pi$
$149$	−84150.0			−0.310519			−0.155260	+	0.987874i	$0.549621\pi$
$150$		0			0
$151$	155848.			0.556236			0.278118	−	0.960547i	$-0.410289\pi$
$151$	155848.			0.556236			0.278118	+	0.960547i	$0.410289\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−303840.	+	223939.i	−1.01582	+	0.748686i
$156$		0			0
$157$			356643.i			1.15474i	0.816482	+	0.577371i	$0.195921\pi$
$157$			356643.i			1.15474i	−0.816482	+	0.577371i	$0.804079\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−145332.			−0.441872
$162$		0			0
$163$			144890.i			0.427139i	0.976928	+	0.213570i	$0.0685090\pi$
$163$			144890.i			0.427139i	−0.976928	+	0.213570i	$0.931491\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	18102.1i		−	0.0502272i	−0.999685	−	0.0251136i	$-0.992005\pi$
$167$		−	18102.1i		−	0.0502272i	0.999685	−	0.0251136i	$-0.00799474\pi$
$168$		0			0
$169$	357037.			0.961604
$170$		0			0
$171$		0			0
$172$		0			0
$173$			492572.i			1.25128i	0.780112	+	0.625640i	$0.215162\pi$
$173$			492572.i			1.25128i	−0.780112	+	0.625640i	$0.784838\pi$
$174$		0			0
$175$	178200.	+	55221.8i	0.439858	+	0.136306i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	444420.			1.03672			0.518359	−	0.855163i	$-0.326543\pi$
$179$	444420.			1.03672			0.518359	+	0.855163i	$0.326543\pi$
$180$		0			0
$181$	156902.			0.355985			0.177993	−	0.984032i	$-0.443040\pi$
$181$	156902.			0.355985			0.177993	+	0.984032i	$0.443040\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	463320.	+	628633.i	0.995295	+	1.35042i
$186$		0			0
$187$			173844.i			0.363543i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	332352.			0.659196			0.329598	−	0.944121i	$-0.393087\pi$
$191$	332352.			0.659196			0.329598	+	0.944121i	$0.393087\pi$
$192$		0			0
$193$			786120.i			1.51913i	0.650430	+	0.759566i	$0.274589\pi$
$193$			786120.i			1.51913i	−0.650430	+	0.759566i	$0.725411\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	59606.4i		−	0.109428i	−0.998502	−	0.0547138i	$-0.982575\pi$
$197$		−	59606.4i		−	0.109428i	0.998502	−	0.0547138i	$-0.0174247\pi$
$198$		0			0
$199$	395800.			0.708505			0.354253	−	0.935150i	$-0.384735\pi$
$199$	395800.			0.708505			0.354253	+	0.935150i	$0.384735\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			413716.i			0.704631i
$204$		0			0
$205$	8910.00	−	6566.92i	0.0148079	−	0.0109138i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	55440.0			0.0877925
$210$		0			0
$211$	251548.			0.388969			0.194484	−	0.980906i	$-0.437697\pi$
$211$	251548.			0.388969			0.194484	+	0.980906i	$0.437697\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	13860.0	+	18805.3i	0.0204488	+	0.0277449i
$216$		0			0
$217$		−	403089.i		−	0.581101i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	82368.0			0.113443
$222$		0			0
$223$			288765.i			0.388851i	0.980917	+	0.194425i	$0.0622842\pi$
$223$			288765.i			0.388851i	−0.980917	+	0.194425i	$0.937716\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			1.16414e6i			1.49948i	0.661731	+	0.749741i	$0.269822\pi$
$227$			1.16414e6i			1.49948i	−0.661731	+	0.749741i	$0.730178\pi$
$228$		0			0
$229$	547670.			0.690129			0.345064	−	0.938579i	$-0.387857\pi$
$229$	547670.			0.690129			0.345064	+	0.938579i	$0.387857\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	48104.3i		−	0.0580489i	−0.999579	−	0.0290245i	$-0.990760\pi$
$233$		−	48104.3i		−	0.0580489i	0.999579	−	0.0290245i	$-0.00924007\pi$
$234$		0			0
$235$	−349580.	−	474311.i	−0.412930	−	0.560264i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1.00584e6			−1.13903			−0.569514	−	0.821982i	$-0.692868\pi$
$239$	−1.00584e6			−1.13903			−0.569514	+	0.821982i	$0.692868\pi$
$240$		0			0
$241$	895202.			0.992838			0.496419	−	0.868083i	$-0.334648\pi$
$241$	895202.			0.992838			0.496419	+	0.868083i	$0.334648\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	595935.	−	439221.i	0.634284	−	0.467485i
$246$		0			0
$247$		−	26267.7i		−	0.0273955i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	558252.			0.559301			0.279651	−	0.960102i	$-0.409781\pi$
$251$	558252.			0.559301			0.279651	+	0.960102i	$0.409781\pi$
$252$		0			0
$253$			613469.i			0.602548i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			787924.i			0.744135i	0.928206	+	0.372067i	$0.121351\pi$
$257$			787924.i			0.744135i	−0.928206	+	0.372067i	$0.878649\pi$
$258$		0			0
$259$	−833976.			−0.772510
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	1.63173e6i		−	1.45465i	−0.686291	−	0.727327i	$-0.740762\pi$
$263$		−	1.63173e6i		−	1.45465i	0.686291	−	0.727327i	$-0.259238\pi$
$264$		0			0
$265$	193160.	+	262080.i	0.168967	+	0.229255i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	1.73637e6			1.46306			0.731529	−	0.681810i	$-0.238807\pi$
$269$	1.73637e6			1.46306			0.731529	+	0.681810i	$0.238807\pi$
$270$		0			0
$271$	1.72005e6			1.42271			0.711357	−	0.702831i	$-0.248081\pi$
$271$	1.72005e6			1.42271			0.711357	+	0.702831i	$0.248081\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	233100.	−	752211.i	0.185871	−	0.599802i
$276$		0			0
$277$			1.27243e6i			0.996402i	0.867062	+	0.498201i	$0.166006\pi$
$277$			1.27243e6i			0.996402i	−0.867062	+	0.498201i	$0.833994\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1.46500e6			−1.10681			−0.553404	−	0.832913i	$-0.686671\pi$
$281$	−1.46500e6			−1.10681			−0.553404	+	0.832913i	$0.686671\pi$
$282$		0			0
$283$			1.65051e6i			1.22504i	0.790455	+	0.612521i	$0.209844\pi$
$283$			1.65051e6i			1.22504i	−0.790455	+	0.612521i	$0.790156\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			11820.5i			0.00847089i
$288$		0			0
$289$	943953.			0.664823
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	2.38772e6i		−	1.62485i	−0.583064	−	0.812426i	$-0.698146\pi$
$293$		−	2.38772e6i		−	1.62485i	0.583064	−	0.812426i	$-0.301854\pi$
$294$		0			0
$295$	−1.10970e6	+	817880.i	−0.742422	+	0.547185i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	290664.			0.188024
$300$		0			0
$301$	−24948.0			−0.0158716
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−256410.	+	188981.i	−0.157828	+	0.116324i
$306$		0			0
$307$		−	928264.i		−	0.562115i	−0.959691	−	0.281058i	$-0.909315\pi$
$307$		−	928264.i		−	0.562115i	0.959691	−	0.281058i	$-0.0906852\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	568152.			0.333092			0.166546	−	0.986034i	$-0.446739\pi$
$311$	568152.			0.333092			0.166546	+	0.986034i	$0.446739\pi$
$312$		0			0
$313$		−	1.72244e6i		−	0.993766i	−0.867818	−	0.496883i	$-0.834478\pi$
$313$		−	1.72244e6i		−	0.993766i	0.867818	−	0.496883i	$-0.165522\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	131643.i		−	0.0735785i	−0.999323	−	0.0367893i	$-0.988287\pi$
$317$		−	131643.i		−	0.0735785i	0.999323	−	0.0367893i	$-0.0117130\pi$
$318$		0			0
$319$	1.74636e6			0.960853
$320$		0			0
$321$		0			0
$322$		0			0
$323$			151769.i			0.0809424i
$324$		0			0
$325$	−356400.	−	110444.i	−0.187167	−	0.0580006i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	629244.			0.320501
$330$		0			0
$331$	1.58055e6			0.792935			0.396468	−	0.918049i	$-0.370236\pi$
$331$	1.58055e6			0.792935			0.396468	+	0.918049i	$0.370236\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	1.44738e6	+	1.96381e6i	0.704645	+	0.956063i
$336$		0			0
$337$			1.22885e6i			0.589419i	0.955587	+	0.294709i	$0.0952228\pi$
$337$			1.22885e6i			0.589419i	−0.955587	+	0.294709i	$0.904777\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1.70150e6			−0.792405
$342$		0			0
$343$			1.79396e6i			0.823338i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	3.84224e6i		−	1.71301i	−0.516137	−	0.856506i	$-0.672630\pi$
$347$		−	3.84224e6i		−	1.71301i	0.516137	−	0.856506i	$-0.327370\pi$
$348$		0			0
$349$	−1.59445e6			−0.700725			−0.350362	−	0.936614i	$-0.613942\pi$
$349$	−1.59445e6			−0.700725			−0.350362	+	0.936614i	$0.613942\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			295365.i			0.126160i	0.998008	+	0.0630802i	$0.0200924\pi$
$353$			295365.i			0.126160i	−0.998008	+	0.0630802i	$0.979908\pi$
$354$		0			0
$355$	2.40084e6	−	1.76949e6i	1.01110	−	0.745206i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1.10484e6			0.452442			0.226221	−	0.974076i	$-0.427363\pi$
$359$	1.10484e6			0.452442			0.226221	+	0.974076i	$0.427363\pi$
$360$		0			0
$361$	−2.42770e6			−0.980453
$362$		0			0
$363$		0			0
$364$		0			0
$365$	2.35224e6	+	3.19152e6i	0.924165	+	1.25391i
$366$		0			0
$367$			1.83760e6i			0.712174i	0.934453	+	0.356087i	$0.115889\pi$
$367$			1.83760e6i			0.712174i	−0.934453	+	0.356087i	$0.884111\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−347688.			−0.131146
$372$		0			0
$373$			2.93350e6i			1.09173i	0.837874	+	0.545864i	$0.183798\pi$
$373$			2.93350e6i			1.09173i	−0.837874	+	0.545864i	$0.816202\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	827432.i		−	0.299832i
$378$		0			0
$379$	−5.09342e6			−1.82143			−0.910713	−	0.413040i	$-0.864467\pi$
$379$	−5.09342e6			−1.82143			−0.910713	+	0.413040i	$0.864467\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			3.17485e6i			1.10593i	0.833205	+	0.552964i	$0.186503\pi$
$383$			3.17485e6i			1.10593i	−0.833205	+	0.552964i	$0.813497\pi$
$384$		0			0
$385$	498960.	+	676989.i	0.171559	+	0.232772i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−1.79991e6			−0.603083			−0.301541	−	0.953453i	$-0.597501\pi$
$389$	−1.79991e6			−0.603083			−0.301541	+	0.953453i	$0.597501\pi$
$390$		0			0
$391$	−1.67939e6			−0.555533
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−2.33640e6	+	1.72199e6i	−0.753450	+	0.555314i
$396$		0			0
$397$		−	4.90405e6i		−	1.56163i	−0.624760	−	0.780817i	$-0.714803\pi$
$397$		−	4.90405e6i		−	1.56163i	0.624760	−	0.780817i	$-0.285197\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	642798.			0.199624			0.0998122	−	0.995006i	$-0.468176\pi$
$401$	642798.			0.199624			0.0998122	+	0.995006i	$0.468176\pi$
$402$		0			0
$403$			806179.i			0.247268i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			3.52035e6i			1.05341i
$408$		0			0
$409$	−2.05711e6			−0.608064			−0.304032	−	0.952662i	$-0.598333\pi$
$409$	−2.05711e6			−0.608064			−0.304032	+	0.952662i	$0.598333\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	1.47218e6i		−	0.424704i
$414$		0			0
$415$	−2.05106e6	−	2.78288e6i	−0.584599	−	0.793185i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−2.93742e6			−0.817393			−0.408697	−	0.912670i	$-0.634017\pi$
$419$	−2.93742e6			−0.817393			−0.408697	+	0.912670i	$0.634017\pi$
$420$		0			0
$421$	2.71770e6			0.747303			0.373651	−	0.927569i	$-0.378106\pi$
$421$	2.71770e6			0.747303			0.373651	+	0.927569i	$0.378106\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	2.05920e6	+	638119.i	0.553001	+	0.171368i
$426$		0			0
$427$		−	340166.i		−	0.0902862i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	4.99435e6			1.29505			0.647524	−	0.762045i	$-0.275804\pi$
$431$	4.99435e6			1.29505			0.647524	+	0.762045i	$0.275804\pi$
$432$		0			0
$433$		−	2.08183e6i		−	0.533612i	−0.963750	−	0.266806i	$-0.914032\pi$
$433$		−	2.08183e6i		−	0.533612i	0.963750	−	0.266806i	$-0.0859684\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			535569.i			0.134156i
$438$		0			0
$439$	4.70404e6			1.16496			0.582478	−	0.812846i	$-0.302083\pi$
$439$	4.70404e6			1.16496			0.582478	+	0.812846i	$0.302083\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	5.70103e6i		−	1.38021i	−0.723711	−	0.690103i	$-0.757565\pi$
$443$		−	5.70103e6i		−	1.38021i	0.723711	−	0.690103i	$-0.242435\pi$
$444$		0			0
$445$	449550.	−	331331.i	0.107616	−	0.0793162i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−6.20325e6			−1.45212			−0.726062	−	0.687630i	$-0.758651\pi$
$449$	−6.20325e6			−1.45212			−0.726062	+	0.687630i	$0.758651\pi$
$450$		0			0
$451$	49896.0			0.0115511
$452$		0			0
$453$		0			0
$454$		0			0
$455$	320760.	−	236409.i	0.0726360	−	0.0535347i
$456$		0			0
$457$		−	2.15371e6i		−	0.482388i	−0.970477	−	0.241194i	$-0.922461\pi$
$457$		−	2.15371e6i		−	0.482388i	0.970477	−	0.241194i	$-0.0775391\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	3.85130e6			0.844024			0.422012	−	0.906590i	$-0.361324\pi$
$461$	3.85130e6			0.844024			0.422012	+	0.906590i	$0.361324\pi$
$462$		0			0
$463$			2.08213e6i			0.451394i	0.974198	+	0.225697i	$0.0724659\pi$
$463$			2.08213e6i			0.451394i	−0.974198	+	0.225697i	$0.927534\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			1.30822e6i			0.277579i	0.990322	+	0.138790i	$0.0443212\pi$
$467$			1.30822e6i			0.277579i	−0.990322	+	0.138790i	$0.955679\pi$
$468$		0			0
$469$	−2.60528e6			−0.546919
$470$		0			0
$471$		0			0
$472$		0			0
$473$			105309.i			0.0216429i
$474$		0			0
$475$	203500.	−	656692.i	0.0413838	−	0.133545i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−6.76368e6			−1.34693			−0.673464	−	0.739220i	$-0.735194\pi$
$479$	−6.76368e6			−1.34693			−0.673464	+	0.739220i	$0.735194\pi$
$480$		0			0
$481$	1.66795e6			0.328716
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−3.35808e6	−	4.55625e6i	−0.648241	−	0.879534i
$486$		0			0
$487$		−	6.67193e6i		−	1.27476i	−0.770549	−	0.637381i	$-0.780018\pi$
$487$		−	6.67193e6i		−	1.27476i	0.770549	−	0.637381i	$-0.219982\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−6.87575e6			−1.28711			−0.643556	−	0.765399i	$-0.722542\pi$
$491$	−6.87575e6			−1.28711			−0.643556	+	0.765399i	$0.722542\pi$
$492$		0			0
$493$			4.78072e6i			0.885881i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			3.18507e6i			0.578400i
$498$		0			0
$499$	−6.94010e6			−1.24771			−0.623856	−	0.781539i	$-0.714435\pi$
$499$	−6.94010e6			−1.24771			−0.623856	+	0.781539i	$0.714435\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			921007.i			0.162309i	0.996702	+	0.0811546i	$0.0258607\pi$
$503$			921007.i			0.162309i	−0.996702	+	0.0811546i	$0.974139\pi$
$504$		0			0
$505$	4.90941e6	−	3.61837e6i	0.856645	−	0.631371i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−4.97979e6			−0.851955			−0.425977	−	0.904734i	$-0.640070\pi$
$509$	−4.97979e6			−0.851955			−0.425977	+	0.904734i	$0.640070\pi$
$510$		0			0
$511$	−4.23403e6			−0.717302
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−2.34234e6	−	3.17809e6i	−0.389163	−	0.528017i
$516$		0			0
$517$		−	2.65614e6i		−	0.437043i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	147798.			0.0238547			0.0119274	−	0.999929i	$-0.496203\pi$
$521$	147798.			0.0238547			0.0119274	+	0.999929i	$0.496203\pi$
$522$		0			0
$523$		−	1.23884e7i		−	1.98043i	−0.139543	−	0.990216i	$-0.544563\pi$
$523$		−	1.23884e7i		−	1.98043i	0.139543	−	0.990216i	$-0.455437\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	4.65792e6i		−	0.730576i
$528$		0			0
$529$	510027.			0.0792417
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	23640.9i		−	0.00360451i
$534$		0			0
$535$	3.22102e6	+	4.37028e6i	0.486529	+	0.660123i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	3.33724e6			0.494783
$540$		0			0
$541$	−9.99810e6			−1.46867			−0.734335	−	0.678787i	$-0.762506\pi$
$541$	−9.99810e6			−1.46867			−0.734335	+	0.678787i	$0.762506\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−945450.	+	696823.i	−0.136348	+	0.100492i
$546$		0			0
$547$		−	1.18580e7i		−	1.69451i	−0.531189	−	0.847253i	$-0.678255\pi$
$547$		−	1.18580e7i		−	1.69451i	0.531189	−	0.847253i	$-0.321745\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	1.52460e6			0.213933
$552$		0			0
$553$		−	3.09958e6i		−	0.431013i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	904550.i		−	0.123536i	−0.998091	−	0.0617681i	$-0.980326\pi$
$557$		−	904550.i		−	0.123536i	0.998091	−	0.0617681i	$-0.0196739\pi$
$558$		0			0
$559$	49896.0			0.00675361
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	8.68719e6i		−	1.15507i	−0.816366	−	0.577535i	$-0.804015\pi$
$563$		−	8.68719e6i		−	1.15507i	0.816366	−	0.577535i	$-0.195985\pi$
$564$		0			0
$565$	−3.48304e6	−	4.72579e6i	−0.459026	−	0.622807i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	2.27007e6			0.293940			0.146970	−	0.989141i	$-0.453048\pi$
$569$	2.27007e6			0.293940			0.146970	+	0.989141i	$0.453048\pi$
$570$		0			0
$571$	−1.43807e7			−1.84582			−0.922908	−	0.385021i	$-0.874194\pi$
$571$	−1.43807e7			−1.84582			−0.922908	+	0.385021i	$0.874194\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	7.26660e6	+	2.25182e6i	0.916562	+	0.284030i
$576$		0			0
$577$			5.63943e6i			0.705173i	0.935779	+	0.352586i	$0.114698\pi$
$577$			5.63943e6i			0.705173i	−0.935779	+	0.352586i	$0.885302\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	3.69191e6			0.453744
$582$		0			0
$583$			1.46765e6i			0.178834i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			1.28473e6i			0.153893i	0.997035	+	0.0769464i	$0.0245170\pi$
$587$			1.28473e6i			0.153893i	−0.997035	+	0.0769464i	$0.975483\pi$
$588$		0			0
$589$	−1.48544e6			−0.176428
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	7.00943e6i		−	0.818552i	−0.912411	−	0.409276i	$-0.865781\pi$
$593$		−	7.00943e6i		−	0.818552i	0.912411	−	0.409276i	$-0.134219\pi$
$594$		0			0
$595$	−1.85328e6	+	1.36592e6i	−0.214609	+	0.158173i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−8.80020e6			−1.00213			−0.501067	−	0.865409i	$-0.667059\pi$
$599$	−8.80020e6			−1.00213			−0.501067	+	0.865409i	$0.667059\pi$
$600$		0			0
$601$	−1.07670e7			−1.21593			−0.607965	−	0.793964i	$-0.708014\pi$
$601$	−1.07670e7			−1.21593			−0.607965	+	0.793964i	$0.708014\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−4.38962e6	+	3.23527e6i	−0.487571	+	0.359353i
$606$		0			0
$607$			1.51219e7i			1.66584i	0.553391	+	0.832921i	$0.313333\pi$
$607$			1.51219e7i			1.66584i	−0.553391	+	0.832921i	$0.686667\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−1.25849e6			−0.136379
$612$		0			0
$613$		−	8.31622e6i		−	0.893871i	−0.894566	−	0.446936i	$-0.852515\pi$
$613$		−	8.31622e6i		−	0.893871i	0.894566	−	0.446936i	$-0.147485\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	1.21083e7i		−	1.28047i	−0.768178	−	0.640237i	$-0.778836\pi$
$617$		−	1.21083e7i		−	1.28047i	0.768178	−	0.640237i	$-0.221164\pi$
$618$		0			0
$619$	−9.73238e6			−1.02092			−0.510461	−	0.859901i	$-0.670525\pi$
$619$	−9.73238e6			−1.02092			−0.510461	+	0.859901i	$0.670525\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			596395.i			0.0615622i
$624$		0			0
$625$	−8.05437e6	−	5.52218e6i	−0.824768	−	0.565471i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−9.63706e6			−0.971220
$630$		0			0
$631$	8.60145e6			0.859999			0.430000	−	0.902829i	$-0.358514\pi$
$631$	8.60145e6			0.859999			0.430000	+	0.902829i	$0.358514\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	2.89278e6	+	3.92493e6i	0.284696	+	0.386276i
$636$		0			0
$637$		−	1.58119e6i		−	0.154396i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	6.42440e6			0.617572			0.308786	−	0.951132i	$-0.400077\pi$
$641$	6.42440e6			0.617572			0.308786	+	0.951132i	$0.400077\pi$
$642$		0			0
$643$		−	3.64721e6i		−	0.347883i	−0.984756	−	0.173941i	$-0.944350\pi$
$643$		−	3.64721e6i		−	0.347883i	0.984756	−	0.173941i	$-0.0556503\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			3.78036e6i			0.355036i	0.984118	+	0.177518i	$0.0568068\pi$
$647$			3.78036e6i			0.355036i	−0.984118	+	0.177518i	$0.943193\pi$
$648$		0			0
$649$	−6.21432e6			−0.579138
$650$		0			0
$651$		0			0
$652$		0			0
$653$			1.66957e7i			1.53223i	0.642706	+	0.766113i	$0.277812\pi$
$653$			1.66957e7i			1.53223i	−0.642706	+	0.766113i	$0.722188\pi$
$654$		0			0
$655$	8.67834e6	−	6.39618e6i	0.790375	−	0.582529i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−1.22166e6			−0.109581			−0.0547907	−	0.998498i	$-0.517449\pi$
$659$	−1.22166e6			−0.109581			−0.0547907	+	0.998498i	$0.517449\pi$
$660$		0			0
$661$	1.62789e7			1.44918			0.724589	−	0.689182i	$-0.242030\pi$
$661$	1.62789e7			1.44918			0.724589	+	0.689182i	$0.242030\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	435600.	+	591023.i	0.0381974	+	0.0518263i
$666$		0			0
$667$			1.68704e7i			1.46829i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−1.43590e6			−0.123117
$672$		0			0
$673$			1.43928e7i			1.22492i	0.790503	+	0.612459i	$0.209819\pi$
$673$			1.43928e7i			1.22492i	−0.790503	+	0.612459i	$0.790181\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	2.62429e6i		−	0.220059i	−0.993928	−	0.110030i	$-0.964905\pi$
$677$		−	2.62429e6i		−	0.220059i	0.993928	−	0.110030i	$-0.0350946\pi$
$678$		0			0
$679$	6.04454e6			0.503140
$680$		0			0
$681$		0			0
$682$		0			0
$683$			1.03039e7i			0.845184i	0.906320	+	0.422592i	$0.138880\pi$
$683$			1.03039e7i			0.845184i	−0.906320	+	0.422592i	$0.861120\pi$
$684$		0			0
$685$	4.76168e6	+	6.46065e6i	0.387734	+	0.526078i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	695376.			0.0558048
$690$		0			0
$691$	−4.50285e6			−0.358751			−0.179375	−	0.983781i	$-0.557408\pi$
$691$	−4.50285e6			−0.358751			−0.179375	+	0.983781i	$0.557408\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1.43253e7	−	1.05581e7i	1.12497	−	0.829136i
$696$		0			0
$697$			136592.i			0.0106498i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	4.88090e6			0.375150			0.187575	−	0.982250i	$-0.439937\pi$
$701$	4.88090e6			0.375150			0.187575	+	0.982250i	$0.439937\pi$
$702$		0			0
$703$			3.07332e6i			0.234541i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			6.51307e6i			0.490046i
$708$		0			0
$709$	−9.96961e6			−0.744839			−0.372420	−	0.928064i	$-0.621472\pi$
$709$	−9.96961e6			−0.744839			−0.372420	+	0.928064i	$0.621472\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	1.64371e7i		−	1.21088i
$714$		0			0
$715$	−997920.	−	1.35398e6i	−0.0730013	−	0.0990482i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−1.19167e7			−0.859675			−0.429838	−	0.902906i	$-0.641429\pi$
$719$	−1.19167e7			−0.859675			−0.429838	+	0.902906i	$0.641429\pi$
$720$		0			0
$721$	4.21621e6			0.302054
$722$		0			0
$723$		0			0
$724$		0			0
$725$	6.41025e6	−	2.06858e7i	0.452929	−	1.46160i
$726$		0			0
$727$			1.38269e6i			0.0970264i	0.998823	+	0.0485132i	$0.0154483\pi$
$727$			1.38269e6i			0.0970264i	−0.998823	+	0.0485132i	$0.984552\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−288288.			−0.0199541
$732$		0			0
$733$		−	6.09661e6i		−	0.419110i	−0.977797	−	0.209555i	$-0.932798\pi$
$733$		−	6.09661e6i		−	0.419110i	0.977797	−	0.209555i	$-0.0672016\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			1.09973e7i			0.745793i
$738$		0			0
$739$	−6.16946e6			−0.415562			−0.207781	−	0.978175i	$-0.566624\pi$
$739$	−6.16946e6			−0.415562			−0.207781	+	0.978175i	$0.566624\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	1.57574e7i		−	1.04716i	−0.851978	−	0.523578i	$-0.824597\pi$
$743$		−	1.57574e7i		−	1.04716i	0.851978	−	0.523578i	$-0.175403\pi$
$744$		0			0
$745$	−3.78675e6	+	2.79094e6i	−0.249963	+	0.184230i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−5.79784e6			−0.377626
$750$		0			0
$751$	1.51816e7			0.982243			0.491122	−	0.871091i	$-0.336587\pi$
$751$	1.51816e7			0.982243			0.491122	+	0.871091i	$0.336587\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	7.01316e6	−	5.16889e6i	0.447761	−	0.330012i
$756$		0			0
$757$		−	652274.i		−	0.0413705i	−0.999786	−	0.0206852i	$-0.993415\pi$
$757$		−	652274.i		−	0.0413705i	0.999786	−	0.0206852i	$-0.00658478\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−4.51420e6			−0.282566			−0.141283	−	0.989969i	$-0.545123\pi$
$761$	−4.51420e6			−0.282566			−0.141283	+	0.989969i	$0.545123\pi$
$762$		0			0
$763$		−	1.25428e6i		−	0.0779980i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			2.94437e6i			0.180719i
$768$		0			0
$769$	−1.20799e7			−0.736625			−0.368312	−	0.929702i	$-0.620064\pi$
$769$	−1.20799e7			−0.736625			−0.368312	+	0.929702i	$0.620064\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			1.04245e7i			0.627492i	0.949507	+	0.313746i	$0.101584\pi$
$773$			1.04245e7i			0.627492i	−0.949507	+	0.313746i	$0.898416\pi$
$774$		0			0
$775$	−6.24560e6	+	2.01545e7i	−0.373525	+	1.20536i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	43560.0			0.00257184
$780$		0			0
$781$	1.34447e7			0.788721
$782$		0			0
$783$		0			0
$784$		0			0
$785$	1.18285e7	+	1.60489e7i	0.685104	+	0.929549i
$786$		0			0
$787$		−	3.45366e7i		−	1.98766i	−0.110913	−	0.993830i	$-0.535378\pi$
$787$		−	3.45366e7i		−	1.98766i	0.110913	−	0.993830i	$-0.464622\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	6.26947e6			0.356279
$792$		0			0
$793$			680333.i			0.0384183i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			2.09287e7i			1.16707i	0.812089	+	0.583533i	$0.198330\pi$
$797$			2.09287e7i			1.16707i	−0.812089	+	0.583533i	$0.801670\pi$
$798$		0			0
$799$	7.27126e6			0.402942
$800$		0			0
$801$		0			0
$802$		0			0
$803$			1.78725e7i			0.978131i
$804$		0			0
$805$	−6.53994e6	+	4.82012e6i	−0.355700	+	0.262161i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−2.48797e7			−1.33651			−0.668257	−	0.743930i	$-0.732959\pi$
$809$	−2.48797e7			−1.33651			−0.668257	+	0.743930i	$0.732959\pi$
$810$		0			0
$811$	3.95415e6			0.211106			0.105553	−	0.994414i	$-0.466339\pi$
$811$	3.95415e6			0.211106			0.105553	+	0.994414i	$0.466339\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	4.80546e6	+	6.52005e6i	0.253420	+	0.343841i
$816$		0			0
$817$			91936.8i			0.00481875i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	3.43550e6			0.177882			0.0889410	−	0.996037i	$-0.471652\pi$
$821$	3.43550e6			0.177882			0.0889410	+	0.996037i	$0.471652\pi$
$822$		0			0
$823$			3.94833e6i			0.203195i	0.994826	+	0.101598i	$0.0323954\pi$
$823$			3.94833e6i			0.203195i	−0.994826	+	0.101598i	$0.967605\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	3.38176e7i		−	1.71941i	−0.510791	−	0.859705i	$-0.670647\pi$
$827$		−	3.38176e7i		−	1.71941i	0.510791	−	0.859705i	$-0.329353\pi$
$828$		0			0
$829$	1.52015e7			0.768244			0.384122	−	0.923282i	$-0.374504\pi$
$829$	1.52015e7			0.768244			0.384122	+	0.923282i	$0.374504\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			9.13579e6i			0.456177i
$834$		0			0
$835$	−600380.	−	814596.i	−0.0297996	−	0.0404321i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	2.89012e7			1.41746			0.708729	−	0.705481i	$-0.249269\pi$
$839$	2.89012e7			1.41746			0.708729	+	0.705481i	$0.249269\pi$
$840$		0			0
$841$	2.75138e7			1.34140
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1.60667e7	−	1.18416e7i	0.774077	−	0.570516i
$846$		0			0
$847$		−	5.82348e6i		−	0.278917i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−3.40077e7			−1.60973
$852$		0			0
$853$		−	2.02107e7i		−	0.951062i	−0.879699	−	0.475531i	$-0.842256\pi$
$853$		−	2.02107e7i		−	0.951062i	0.879699	−	0.475531i	$-0.157744\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	1.70522e7i		−	0.793101i	−0.918013	−	0.396550i	$-0.870207\pi$
$857$		−	1.70522e7i		−	0.793101i	0.918013	−	0.396550i	$-0.129793\pi$
$858$		0			0
$859$	−1.95505e7			−0.904015			−0.452008	−	0.892014i	$-0.649292\pi$
$859$	−1.95505e7			−0.904015			−0.452008	+	0.892014i	$0.649292\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			2.70896e7i			1.23816i	0.785330	+	0.619078i	$0.212493\pi$
$863$			2.70896e7i			1.23816i	−0.785330	+	0.619078i	$0.787507\pi$
$864$		0			0
$865$	1.63368e7	+	2.21657e7i	0.742379	+	1.00726i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1.30838e7			−0.587741
$870$		0			0
$871$	5.21057e6			0.232723
$872$		0			0
$873$		0			0
$874$		0			0
$875$	9.85050e6	−	3.42524e6i	0.434949	−	0.151242i
$876$		0			0
$877$			1.98285e6i			0.0870545i	0.999052	+	0.0435272i	$0.0138595\pi$
$877$			1.98285e6i			0.0870545i	−0.999052	+	0.0435272i	$0.986140\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	4.22840e7			1.83542			0.917712	−	0.397247i	$-0.130034\pi$
$881$	4.22840e7			1.83542			0.917712	+	0.397247i	$0.130034\pi$
$882$		0			0
$883$		−	134502.i		−	0.00580535i	−0.999996	−	0.00290267i	$-0.999076\pi$
$883$		−	134502.i		−	0.00580535i	0.999996	−	0.00290267i	$-0.000923951\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			3.87668e6i			0.165444i	0.996573	+	0.0827219i	$0.0263613\pi$
$887$			3.87668e6i			0.165444i	−0.996573	+	0.0827219i	$0.973639\pi$
$888$		0			0
$889$	−5.20700e6			−0.220970
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	2.31885e6i		−	0.0973070i
$894$		0			0
$895$	1.99989e7	−	1.47397e7i	0.834543	−	0.615081i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−4.67914e7			−1.93093
$900$		0			0
$901$	−4.01773e6			−0.164880
$902$		0			0
$903$		0			0
$904$		0			0
$905$	7.06059e6	−	5.20385e6i	0.286563	−	0.211205i
$906$		0			0
$907$			2.87363e7i			1.15988i	0.814660	+	0.579939i	$0.196924\pi$
$907$			2.87363e7i			1.15988i	−0.814660	+	0.579939i	$0.803076\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1.87675e6			0.0749223			0.0374611	−	0.999298i	$-0.488073\pi$
$911$	1.87675e6			0.0749223			0.0374611	+	0.999298i	$0.488073\pi$
$912$		0			0
$913$		−	1.55841e7i		−	0.618736i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			1.15131e7i			0.452137i
$918$		0			0
$919$	6.76852e6			0.264366			0.132183	−	0.991225i	$-0.457801\pi$
$919$	6.76852e6			0.264366			0.132183	+	0.991225i	$0.457801\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	6.37015e6i		−	0.246119i
$924$		0			0
$925$	4.16988e7	+	1.29219e7i	1.60239	+	0.496560i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1.15356e7			−0.438530			−0.219265	−	0.975665i	$-0.570366\pi$
$929$	−1.15356e7			−0.438530			−0.219265	+	0.975665i	$0.570366\pi$
$930$		0			0
$931$	2.91346e6			0.110163
$932$		0			0
$933$		0			0
$934$		0			0
$935$	5.76576e6	+	7.82299e6i	0.215689	+	0.292647i
$936$		0			0
$937$			3.92632e7i			1.46096i	0.682936	+	0.730478i	$0.260703\pi$
$937$			3.92632e7i			1.46096i	−0.682936	+	0.730478i	$0.739297\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−2.94919e7			−1.08575			−0.542874	−	0.839814i	$-0.682664\pi$
$941$	−2.94919e7			−1.08575			−0.542874	+	0.839814i	$0.682664\pi$
$942$		0			0
$943$			482012.i			0.0176514i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			2.09628e7i			0.759581i	0.925072	+	0.379791i	$0.124004\pi$
$947$			2.09628e7i			0.759581i	−0.925072	+	0.379791i	$0.875996\pi$
$948$		0			0
$949$	8.46806e6			0.305224
$950$		0			0
$951$		0			0
$952$		0			0
$953$			1.64122e7i			0.585375i	0.956208	+	0.292687i	$0.0945495\pi$
$953$			1.64122e7i			0.585375i	−0.956208	+	0.292687i	$0.905451\pi$
$954$		0			0
$955$	1.49558e7	−	1.10229e7i	0.530643	−	0.391099i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−8.57102e6			−0.300944
$960$		0			0
$961$	1.69604e7			0.592416
$962$		0			0
$963$		0			0
$964$		0			0
$965$	2.60726e7	+	3.53754e7i	0.901295	+	1.22288i
$966$		0			0
$967$			4.71911e7i			1.62291i	0.584416	+	0.811454i	$0.301324\pi$
$967$			4.71911e7i			1.62291i	−0.584416	+	0.811454i	$0.698676\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	3.84771e7			1.30965			0.654823	−	0.755783i	$-0.272743\pi$
$971$	3.84771e7			1.30965			0.654823	+	0.755783i	$0.272743\pi$
$972$		0			0
$973$			1.90047e7i			0.643544i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	2.70184e7i		−	0.905572i	−0.891619	−	0.452786i	$-0.850430\pi$
$977$		−	2.70184e7i		−	0.905572i	0.891619	−	0.452786i	$-0.149570\pi$
$978$		0			0
$979$	2.51748e6			0.0839478
$980$		0			0
$981$		0			0
$982$		0			0
$983$			2.88475e7i			0.952192i	0.879393	+	0.476096i	$0.157949\pi$
$983$			2.88475e7i			0.952192i	−0.879393	+	0.476096i	$0.842051\pi$
$984$		0			0
$985$	−1.97692e6	−	2.68229e6i	−0.0649230	−	0.0880876i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1.01732e6			−0.0330726
$990$		0			0
$991$	5.21596e7			1.68714			0.843569	−	0.537021i	$-0.180450\pi$
$991$	5.21596e7			1.68714			0.843569	+	0.537021i	$0.180450\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	1.78110e7	−	1.31272e7i	0.570336	−	0.420353i
$996$		0			0
$997$		−	9.78148e6i		−	0.311650i	−0.987785	−	0.155825i	$-0.950196\pi$
$997$		−	9.78148e6i		−	0.311650i	0.987785	−	0.155825i	$-0.0498036\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	720.6.f.f.289.1		2
3.2	odd	2		80.6.c.a.49.2		2
4.3	odd	2		45.6.b.b.19.1		2
5.4	even	2	inner	720.6.f.f.289.2		2
12.11	even	2		5.6.b.a.4.2	yes	2
15.2	even	4		400.6.a.t.1.2		2
15.8	even	4		400.6.a.t.1.1		2
15.14	odd	2		80.6.c.a.49.1		2
20.3	even	4		225.6.a.n.1.1		2
20.7	even	4		225.6.a.n.1.2		2
20.19	odd	2		45.6.b.b.19.2		2
24.5	odd	2		320.6.c.g.129.1		2
24.11	even	2		320.6.c.f.129.2		2
60.23	odd	4		25.6.a.c.1.2		2
60.47	odd	4		25.6.a.c.1.1		2
60.59	even	2		5.6.b.a.4.1	✓	2
84.83	odd	2		245.6.b.a.99.2		2
120.29	odd	2		320.6.c.g.129.2		2
120.59	even	2		320.6.c.f.129.1		2
420.419	odd	2		245.6.b.a.99.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.6.b.a.4.1	✓	2	60.59	even	2
5.6.b.a.4.2	yes	2	12.11	even	2
25.6.a.c.1.1		2	60.47	odd	4
25.6.a.c.1.2		2	60.23	odd	4
45.6.b.b.19.1		2	4.3	odd	2
45.6.b.b.19.2		2	20.19	odd	2
80.6.c.a.49.1		2	15.14	odd	2
80.6.c.a.49.2		2	3.2	odd	2
225.6.a.n.1.1		2	20.3	even	4
225.6.a.n.1.2		2	20.7	even	4
245.6.b.a.99.1		2	420.419	odd	2
245.6.b.a.99.2		2	84.83	odd	2
320.6.c.f.129.1		2	120.59	even	2
320.6.c.f.129.2		2	24.11	even	2
320.6.c.g.129.1		2	24.5	odd	2
320.6.c.g.129.2		2	120.29	odd	2
400.6.a.t.1.1		2	15.8	even	4
400.6.a.t.1.2		2	15.2	even	4
720.6.f.f.289.1		2	1.1	even	1	trivial
720.6.f.f.289.2		2	5.4	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	720.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(45.0000 - 33.1662i) q^{5} +59.6992i q^{7} +O(q^{10})\)	\(q+(45.0000 - 33.1662i) q^{5} +59.6992i q^{7} +252.000 q^{11} -119.398i q^{13} +689.858i q^{17} +220.000 q^{19} +2434.40i q^{23} +(925.000 - 2984.96i) q^{25} +6930.00 q^{29} -6752.00 q^{31} +(1980.00 + 2686.47i) q^{35} +13969.6i q^{37} +198.000 q^{41} +417.895i q^{43} -10540.2i q^{47} +13243.0 q^{49} +5823.99i q^{53} +(11340.0 - 8357.89i) q^{55} -24660.0 q^{59} -5698.00 q^{61} +(-3960.00 - 5372.93i) q^{65} +43640.1i q^{67} +53352.0 q^{71} +70922.7i q^{73} +15044.2i q^{77} -51920.0 q^{79} -61841.8i q^{83} +(22880.0 + 31043.6i) q^{85} +9990.00 q^{89} +7128.00 q^{91} +(9900.00 - 7296.57i) q^{95} -101250. i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 90 q^{5}+O(q^{10}) \) 2 * q + 90 * q^5	\( 2 q + 90 q^{5} + 504 q^{11} + 440 q^{19} + 1850 q^{25} + 13860 q^{29} - 13504 q^{31} + 3960 q^{35} + 396 q^{41} + 26486 q^{49} + 22680 q^{55} - 49320 q^{59} - 11396 q^{61} - 7920 q^{65} + 106704 q^{71} - 103840 q^{79} + 45760 q^{85} + 19980 q^{89} + 14256 q^{91} + 19800 q^{95}+O(q^{100}) \) 2 * q + 90 * q^5 + 504 * q^11 + 440 * q^19 + 1850 * q^25 + 13860 * q^29 - 13504 * q^31 + 3960 * q^35 + 396 * q^41 + 26486 * q^49 + 22680 * q^55 - 49320 * q^59 - 11396 * q^61 - 7920 * q^65 + 106704 * q^71 - 103840 * q^79 + 45760 * q^85 + 19980 * q^89 + 14256 * q^91 + 19800 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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