LMFDB - Embedded newform 320.6.c.f.129.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [320,6,Mod(129,320)] 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("320.129"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(320, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	320.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,90,0,0,0,-306,0,-504] 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:	$51.3228223402$
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:	$\Z[a_1, \ldots, a_{5}]$
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			129.1
Root			$0.500000 + 1.65831i$ of defining polynomial
Character	$\chi$	$=$	320.129
Dual form			320.6.c.f.129.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-19.8997i q^{3} +(45.0000 + 33.1662i) q^{5} +59.6992i q^{7} -153.000 q^{9} -252.000 q^{11} -119.398i q^{13} +(660.000 - 895.489i) q^{15} +689.858i q^{17} +220.000 q^{19} +1188.00 q^{21} -2434.40i q^{23} +(925.000 + 2984.96i) q^{25} -1790.98i q^{27} +6930.00 q^{29} +6752.00 q^{31} +5014.74i q^{33} +(-1980.00 + 2686.47i) q^{35} +13969.6i q^{37} -2376.00 q^{39} -198.000 q^{41} -417.895i q^{43} +(-6885.00 - 5074.44i) q^{45} +10540.2i q^{47} +13243.0 q^{49} +13728.0 q^{51} -5823.99i q^{53} +(-11340.0 - 8357.89i) q^{55} -4377.94i q^{57} +24660.0 q^{59} +5698.00 q^{61} -9133.98i q^{63} +(3960.00 - 5372.93i) q^{65} -43640.1i q^{67} -48444.0 q^{69} +53352.0 q^{71} -70922.7i q^{73} +(59400.0 - 18407.3i) q^{75} -15044.2i q^{77} +51920.0 q^{79} -72819.0 q^{81} -61841.8i q^{83} +(-22880.0 + 31043.6i) q^{85} -137905. i q^{87} -9990.00 q^{89} +7128.00 q^{91} -134363. i q^{93} +(9900.00 + 7296.57i) q^{95} +101250. i q^{97} +38556.0 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 90 q^{5} - 306 q^{9} - 504 q^{11} + 1320 q^{15} + 440 q^{19} + 2376 q^{21} + 1850 q^{25} + 13860 q^{29} + 13504 q^{31} - 3960 q^{35} - 4752 q^{39} - 396 q^{41} - 13770 q^{45} + 26486 q^{49} + 27456 q^{51}+ \cdots + 77112 q^{99}+O(q^{100}) $ 2 * q + 90 * q^5 - 306 * q^9 - 504 * q^11 + 1320 * q^15 + 440 * q^19 + 2376 * q^21 + 1850 * q^25 + 13860 * q^29 + 13504 * q^31 - 3960 * q^35 - 4752 * q^39 - 396 * q^41 - 13770 * q^45 + 26486 * q^49 + 27456 * q^51 - 22680 * q^55 + 49320 * q^59 + 11396 * q^61 + 7920 * q^65 - 96888 * q^69 + 106704 * q^71 + 118800 * q^75 + 103840 * q^79 - 145638 * q^81 - 45760 * q^85 - 19980 * q^89 + 14256 * q^91 + 19800 * q^95 + 77112 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/320\mathbb{Z}\right)^\times$.

$n$	$191$	$257$	$261$
$\chi(n)$	$1$	$-1$	$1$

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
$3$		−	19.8997i		−	1.27657i	−0.769800	−	0.638285i	$-0.779644\pi$
$3$		−	19.8997i		−	1.27657i	0.769800	−	0.638285i	$-0.220356\pi$
$4$		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
$9$	−153.000			−0.629630
$10$		0			0
$11$	−252.000			−0.627941			−0.313970	−	0.949433i	$-0.601659\pi$
$11$	−252.000			−0.627941			−0.313970	+	0.949433i	$0.601659\pi$
$12$		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
$15$	660.000	−	895.489i	0.757383	−	1.02762i
$16$		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
$21$	1188.00			0.587852
$22$		0			0
$23$		−	2434.40i		−	0.959561i	−0.877388	−	0.479781i	$-0.840716\pi$
$23$		−	2434.40i		−	0.959561i	0.877388	−	0.479781i	$-0.159284\pi$
$24$		0			0
$25$	925.000	+	2984.96i	0.296000	+	0.955188i
$26$		0			0
$27$		−	1790.98i		−	0.472804i
$28$		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	$-0.282663\pi$
$31$	6752.00			1.26191			0.630955	+	0.775820i	$0.282663\pi$
$32$		0			0
$33$			5014.74i			0.801610i
$34$		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
$39$	−2376.00			−0.250141
$40$		0			0
$41$	−198.000			−0.0183952			−0.00919762	−	0.999958i	$-0.502928\pi$
$41$	−198.000			−0.0183952			−0.00919762	+	0.999958i	$0.502928\pi$
$42$		0			0
$43$		−	417.895i		−	0.0344664i	−0.999851	−	0.0172332i	$-0.994514\pi$
$43$		−	417.895i		−	0.0344664i	0.999851	−	0.0172332i	$-0.00548577\pi$
$44$		0			0
$45$	−6885.00	−	5074.44i	−0.506842	−	0.373557i
$46$		0			0
$47$			10540.2i			0.695994i	0.937496	+	0.347997i	$0.113138\pi$
$47$			10540.2i			0.695994i	−0.937496	+	0.347997i	$0.886862\pi$
$48$		0			0
$49$	13243.0			0.787945
$50$		0			0
$51$	13728.0			0.739064
$52$		0			0
$53$		−	5823.99i		−	0.284794i	−0.989810	−	0.142397i	$-0.954519\pi$
$53$		−	5823.99i		−	0.284794i	0.989810	−	0.142397i	$-0.0454810\pi$
$54$		0			0
$55$	−11340.0	−	8357.89i	−0.505483	−	0.372555i
$56$		0			0
$57$		−	4377.94i		−	0.178477i
$58$		0			0
$59$	24660.0			0.922281			0.461140	−	0.887327i	$-0.347440\pi$
$59$	24660.0			0.922281			0.461140	+	0.887327i	$0.347440\pi$
$60$		0			0
$61$	5698.00			0.196064			0.0980320	−	0.995183i	$-0.468745\pi$
$61$	5698.00			0.196064			0.0980320	+	0.995183i	$0.468745\pi$
$62$		0			0
$63$		−	9133.98i		−	0.289941i
$64$		0			0
$65$	3960.00	−	5372.93i	0.116255	−	0.157735i
$66$		0			0
$67$		−	43640.1i		−	1.18768i	−0.804583	−	0.593840i	$-0.797611\pi$
$67$		−	43640.1i		−	1.18768i	0.804583	−	0.593840i	$-0.202389\pi$
$68$		0			0
$69$	−48444.0			−1.22495
$70$		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	$-0.715808\pi$
$73$		−	70922.7i		−	1.55768i	0.627223	−	0.778840i	$-0.284192\pi$
$74$		0			0
$75$	59400.0	−	18407.3i	1.21936	−	0.377865i
$76$		0			0
$77$		−	15044.2i		−	0.289163i
$78$		0			0
$79$	51920.0			0.935981			0.467990	−	0.883734i	$-0.344978\pi$
$79$	51920.0			0.935981			0.467990	+	0.883734i	$0.344978\pi$
$80$		0			0
$81$	−72819.0			−1.23320
$82$		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
$87$		−	137905.i		−	1.95336i
$88$		0			0
$89$	−9990.00			−0.133687			−0.0668437	−	0.997763i	$-0.521293\pi$
$89$	−9990.00			−0.133687			−0.0668437	+	0.997763i	$0.521293\pi$
$90$		0			0
$91$	7128.00			0.0902328
$92$		0			0
$93$		−	134363.i		−	1.61092i
$94$		0			0
$95$	9900.00	+	7296.57i	0.112545	+	0.0829488i
$96$		0			0
$97$			101250.i			1.09261i	0.837586	+	0.546305i	$0.183966\pi$
$97$			101250.i			1.09261i	−0.837586	+	0.546305i	$0.816034\pi$
$98$		0			0
$99$	38556.0			0.395370
$100$		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
$105$	53460.0	+	39401.5i	0.473212	+	0.348770i
$106$		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	$-0.473010\pi$
$109$	21010.0			0.169379			0.0846895	+	0.996407i	$0.473010\pi$
$110$		0			0
$111$	277992.			2.14153
$112$		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
$117$			18268.0i			0.123375i
$118$		0			0
$119$	−41184.0			−0.266601
$120$		0			0
$121$	−97547.0			−0.605690
$122$		0			0
$123$			3940.15i			0.0234828i
$124$		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
$129$	−8316.00			−0.0439987
$130$		0			0
$131$	−192852.			−0.981852			−0.490926	−	0.871201i	$-0.663341\pi$
$131$	−192852.			−0.981852			−0.490926	+	0.871201i	$0.663341\pi$
$132$		0			0
$133$			13133.8i			0.0643817i
$134$		0			0
$135$	59400.0	−	80594.0i	0.280512	−	0.380599i
$136$		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
$141$	209748.			0.888485
$142$		0			0
$143$			30088.4i			0.123044i
$144$		0			0
$145$	311850.	+	229842.i	1.23176	+	0.907841i
$146$		0			0
$147$		−	263532.i		−	1.00587i
$148$		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	$-0.589711\pi$
$151$	−155848.			−0.556236			−0.278118	+	0.960547i	$0.589711\pi$
$152$		0			0
$153$		−	105548.i		−	0.364521i
$154$		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
$159$	−115896.			−0.363560
$160$		0			0
$161$	145332.			0.441872
$162$		0			0
$163$		−	144890.i		−	0.427139i	−0.976928	−	0.213570i	$-0.931491\pi$
$163$		−	144890.i		−	0.427139i	0.976928	−	0.213570i	$-0.0685090\pi$
$164$		0			0
$165$	−166320.	+	225663.i	−0.475592	+	0.645284i
$166$		0			0
$167$			18102.1i			0.0502272i	0.999685	+	0.0251136i	$0.00799474\pi$
$167$			18102.1i			0.0502272i	−0.999685	+	0.0251136i	$0.992005\pi$
$168$		0			0
$169$	357037.			0.961604
$170$		0			0
$171$	−33660.0			−0.0880286
$172$		0			0
$173$		−	492572.i		−	1.25128i	−0.780112	−	0.625640i	$-0.784838\pi$
$173$		−	492572.i		−	1.25128i	0.780112	−	0.625640i	$-0.215162\pi$
$174$		0			0
$175$	−178200.	+	55221.8i	−0.439858	+	0.136306i
$176$		0			0
$177$		−	490728.i		−	1.17736i
$178$		0			0
$179$	−444420.			−1.03672			−0.518359	−	0.855163i	$-0.673457\pi$
$179$	−444420.			−1.03672			−0.518359	+	0.855163i	$0.673457\pi$
$180$		0			0
$181$	−156902.			−0.355985			−0.177993	−	0.984032i	$-0.556960\pi$
$181$	−156902.			−0.355985			−0.177993	+	0.984032i	$0.556960\pi$
$182$		0			0
$183$		−	113389.i		−	0.250289i
$184$		0			0
$185$	−463320.	+	628633.i	−0.995295	+	1.35042i
$186$		0			0
$187$		−	173844.i		−	0.363543i
$188$		0			0
$189$	106920.			0.217723
$190$		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	$-0.725411\pi$
$193$		−	786120.i		−	1.51913i	0.650430	−	0.759566i	$-0.274589\pi$
$194$		0			0
$195$	−106920.	−	78803.0i	−0.201360	−	0.148408i
$196$		0			0
$197$			59606.4i			0.109428i	0.998502	+	0.0547138i	$0.0174247\pi$
$197$			59606.4i			0.109428i	−0.998502	+	0.0547138i	$0.982575\pi$
$198$		0			0
$199$	−395800.			−0.708505			−0.354253	−	0.935150i	$-0.615265\pi$
$199$	−395800.			−0.708505			−0.354253	+	0.935150i	$0.615265\pi$
$200$		0			0
$201$	−868428.			−1.51616
$202$		0			0
$203$			413716.i			0.704631i
$204$		0			0
$205$	−8910.00	−	6566.92i	−0.0148079	−	0.0109138i
$206$		0			0
$207$			372464.i			0.604168i
$208$		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
$213$		−	1.06169e6i		−	1.60343i
$214$		0			0
$215$	13860.0	−	18805.3i	0.0204488	−	0.0277449i
$216$		0			0
$217$			403089.i			0.581101i
$218$		0			0
$219$	−1.41134e6			−1.98849
$220$		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
$225$	−141525.	−	456699.i	−0.186370	−	0.601415i
$226$		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	$-0.612143\pi$
$229$	−547670.			−0.690129			−0.345064	+	0.938579i	$0.612143\pi$
$230$		0			0
$231$	−299376.			−0.369137
$232$		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
$237$		−	1.03319e6i		−	1.19484i
$238$		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
$243$			1.01387e6i			1.10146i
$244$		0			0
$245$	595935.	+	439221.i	0.634284	+	0.467485i
$246$		0			0
$247$		−	26267.7i		−	0.0273955i
$248$		0			0
$249$	−1.23064e6			−1.25786
$250$		0			0
$251$	−558252.			−0.559301			−0.279651	−	0.960102i	$-0.590219\pi$
$251$	−558252.			−0.559301			−0.279651	+	0.960102i	$0.590219\pi$
$252$		0			0
$253$			613469.i			0.602548i
$254$		0			0
$255$	617760.	+	455306.i	0.594935	+	0.438483i
$256$		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
$261$	−1.06029e6			−0.963437
$262$		0			0
$263$			1.63173e6i			1.45465i	0.686291	+	0.727327i	$0.259238\pi$
$263$			1.63173e6i			1.45465i	−0.686291	+	0.727327i	$0.740762\pi$
$264$		0			0
$265$	193160.	−	262080.i	0.168967	−	0.229255i
$266$		0			0
$267$			198798.i			0.170661i
$268$		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	$-0.751919\pi$
$271$	−1.72005e6			−1.42271			−0.711357	+	0.702831i	$0.751919\pi$
$272$		0			0
$273$		−	141845.i		−	0.115188i
$274$		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
$279$	−1.03306e6			−0.794536
$280$		0			0
$281$	1.46500e6			1.10681			0.553404	−	0.832913i	$-0.313329\pi$
$281$	1.46500e6			1.10681			0.553404	+	0.832913i	$0.313329\pi$
$282$		0			0
$283$		−	1.65051e6i		−	1.22504i	−0.790455	−	0.612521i	$-0.790156\pi$
$283$		−	1.65051e6i		−	1.22504i	0.790455	−	0.612521i	$-0.209844\pi$
$284$		0			0
$285$	145200.	−	197008.i	0.105890	−	0.143672i
$286$		0			0
$287$		−	11820.5i		−	0.00847089i
$288$		0			0
$289$	943953.			0.664823
$290$		0			0
$291$	2.01485e6			1.39479
$292$		0			0
$293$			2.38772e6i			1.62485i	0.583064	+	0.812426i	$0.301854\pi$
$293$			2.38772e6i			1.62485i	−0.583064	+	0.812426i	$0.698146\pi$
$294$		0			0
$295$	1.10970e6	+	817880.i	0.742422	+	0.547185i
$296$		0			0
$297$			451326.i			0.296893i
$298$		0			0
$299$	−290664.			−0.188024
$300$		0			0
$301$	24948.0			0.0158716
$302$		0			0
$303$		−	2.17102e6i		−	1.35849i
$304$		0			0
$305$	256410.	+	188981.i	0.157828	+	0.116324i
$306$		0			0
$307$			928264.i			0.562115i	0.959691	+	0.281058i	$0.0906852\pi$
$307$			928264.i			0.562115i	−0.959691	+	0.281058i	$0.909315\pi$
$308$		0			0
$309$	−1.40540e6			−0.837346
$310$		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	$0.165522\pi$
$313$			1.72244e6i			0.993766i	−0.867818	+	0.496883i	$0.834478\pi$
$314$		0			0
$315$	302940.	−	411029.i	0.172021	−	0.233398i
$316$		0			0
$317$			131643.i			0.0735785i	0.999323	+	0.0367893i	$0.0117130\pi$
$317$			131643.i			0.0735785i	−0.999323	+	0.0367893i	$0.988287\pi$
$318$		0			0
$319$	−1.74636e6			−0.960853
$320$		0			0
$321$	1.93261e6			1.04684
$322$		0			0
$323$			151769.i			0.0809424i
$324$		0			0
$325$	356400.	−	110444.i	0.187167	−	0.0580006i
$326$		0			0
$327$		−	418094.i		−	0.216224i
$328$		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
$333$		−	2.13735e6i		−	1.05625i
$334$		0			0
$335$	1.44738e6	−	1.96381e6i	0.704645	−	0.956063i
$336$		0			0
$337$		−	1.22885e6i		−	0.589419i	−0.955587	−	0.294709i	$-0.904777\pi$
$337$		−	1.22885e6i		−	0.589419i	0.955587	−	0.294709i	$-0.0952228\pi$
$338$		0			0
$339$	−2.08982e6			−0.987667
$340$		0			0
$341$	−1.70150e6			−0.792405
$342$		0			0
$343$			1.79396e6i			0.823338i
$344$		0			0
$345$	−2.17998e6	−	1.60671e6i	−0.986063	−	0.726756i
$346$		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	$-0.386058\pi$
$349$	1.59445e6			0.700725			0.350362	+	0.936614i	$0.386058\pi$
$350$		0			0
$351$	−213840.			−0.0926448
$352$		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
$357$			819551.i			0.340334i
$358$		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
$363$			1.94116e6i			0.773206i
$364$		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
$369$	30294.0			0.0115822
$370$		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
$375$	3.28350e6	+	1.14175e6i	1.20575	+	0.419268i
$376$		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
$381$	1.73567e6			0.612568
$382$		0			0
$383$		−	3.17485e6i		−	1.10593i	−0.833205	−	0.552964i	$-0.813497\pi$
$383$		−	3.17485e6i		−	1.10593i	0.833205	−	0.552964i	$-0.186503\pi$
$384$		0			0
$385$	498960.	−	676989.i	0.171559	−	0.232772i
$386$		0			0
$387$			63937.9i			0.0217011i
$388$		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
$393$			3.83771e6i			1.25340i
$394$		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
$399$	261360.			0.0821877
$400$		0			0
$401$	−642798.			−0.199624			−0.0998122	−	0.995006i	$-0.531824\pi$
$401$	−642798.			−0.199624			−0.0998122	+	0.995006i	$0.531824\pi$
$402$		0			0
$403$		−	806179.i		−	0.247268i
$404$		0			0
$405$	−3.27686e6	−	2.41513e6i	−0.992704	−	0.731650i
$406$		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
$411$	2.85701e6			0.834271
$412$		0			0
$413$			1.47218e6i			0.424704i
$414$		0			0
$415$	2.05106e6	−	2.78288e6i	0.584599	−	0.793185i
$416$		0			0
$417$		−	6.33489e6i		−	1.78402i
$418$		0			0
$419$	2.93742e6			0.817393			0.408697	−	0.912670i	$-0.365983\pi$
$419$	2.93742e6			0.817393			0.408697	+	0.912670i	$0.365983\pi$
$420$		0			0
$421$	−2.71770e6			−0.747303			−0.373651	−	0.927569i	$-0.621894\pi$
$421$	−2.71770e6			−0.747303			−0.373651	+	0.927569i	$0.621894\pi$
$422$		0			0
$423$		−	1.61266e6i		−	0.438219i
$424$		0			0
$425$	−2.05920e6	+	638119.i	−0.553001	+	0.171368i
$426$		0			0
$427$			340166.i			0.0902862i
$428$		0			0
$429$	598752.			0.157074
$430$		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	$0.0859684\pi$
$433$			2.08183e6i			0.533612i	−0.963750	+	0.266806i	$0.914032\pi$
$434$		0			0
$435$	4.57380e6	−	6.20574e6i	1.15892	−	1.57243i
$436$		0			0
$437$		−	535569.i		−	0.134156i
$438$		0			0
$439$	−4.70404e6			−1.16496			−0.582478	−	0.812846i	$-0.697917\pi$
$439$	−4.70404e6			−1.16496			−0.582478	+	0.812846i	$0.697917\pi$
$440$		0			0
$441$	−2.02618e6			−0.496114
$442$		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
$447$			1.67456e6i			0.396399i
$448$		0			0
$449$	6.20325e6			1.45212			0.726062	−	0.687630i	$-0.241349\pi$
$449$	6.20325e6			1.45212			0.726062	+	0.687630i	$0.241349\pi$
$450$		0			0
$451$	49896.0			0.0115511
$452$		0			0
$453$			3.10134e6i			0.710074i
$454$		0			0
$455$	320760.	+	236409.i	0.0726360	+	0.0535347i
$456$		0			0
$457$			2.15371e6i			0.482388i	0.970477	+	0.241194i	$0.0775391\pi$
$457$			2.15371e6i			0.482388i	−0.970477	+	0.241194i	$0.922461\pi$
$458$		0			0
$459$	1.23552e6			0.273727
$460$		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
$465$	4.45632e6	−	6.04634e6i	0.955749	−	1.29676i
$466$		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
$471$	7.09711e6			1.47411
$472$		0			0
$473$			105309.i			0.0216429i
$474$		0			0
$475$	203500.	+	656692.i	0.0413838	+	0.133545i
$476$		0			0
$477$			891071.i			0.179315i
$478$		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
$483$		−	2.89207e6i		−	0.564080i
$484$		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
$489$	−2.88328e6			−0.545273
$490$		0			0
$491$	6.87575e6			1.28711			0.643556	−	0.765399i	$-0.277458\pi$
$491$	6.87575e6			1.28711			0.643556	+	0.765399i	$0.277458\pi$
$492$		0			0
$493$			4.78072e6i			0.885881i
$494$		0			0
$495$	1.73502e6	+	1.27876e6i	0.318267	+	0.234572i
$496$		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
$501$	360228.			0.0641185
$502$		0			0
$503$		−	921007.i		−	0.162309i	−0.996702	−	0.0811546i	$-0.974139\pi$
$503$		−	921007.i		−	0.162309i	0.996702	−	0.0811546i	$-0.0258607\pi$
$504$		0			0
$505$	4.90941e6	+	3.61837e6i	0.856645	+	0.631371i
$506$		0			0
$507$		−	7.10495e6i		−	1.22755i
$508$		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
$513$		−	394015.i		−	0.0661027i
$514$		0			0
$515$	2.34234e6	−	3.17809e6i	0.389163	−	0.528017i
$516$		0			0
$517$		−	2.65614e6i		−	0.437043i
$518$		0			0
$519$	−9.80206e6			−1.59735
$520$		0			0
$521$	−147798.			−0.0238547			−0.0119274	−	0.999929i	$-0.503797\pi$
$521$	−147798.			−0.0238547			−0.0119274	+	0.999929i	$0.503797\pi$
$522$		0			0
$523$			1.23884e7i			1.98043i	0.139543	+	0.990216i	$0.455437\pi$
$523$			1.23884e7i			1.98043i	−0.139543	+	0.990216i	$0.544563\pi$
$524$		0			0
$525$	1.09890e6	+	3.54614e6i	0.174004	+	0.561509i
$526$		0			0
$527$			4.65792e6i			0.730576i
$528$		0			0
$529$	510027.			0.0792417
$530$		0			0
$531$	−3.77298e6			−0.580695
$532$		0			0
$533$			23640.9i			0.00360451i
$534$		0			0
$535$	−3.22102e6	+	4.37028e6i	−0.486529	+	0.660123i
$536$		0			0
$537$			8.84385e6i			1.32344i
$538$		0			0
$539$	−3.33724e6			−0.494783
$540$		0			0
$541$	9.99810e6			1.46867			0.734335	−	0.678787i	$-0.237494\pi$
$541$	9.99810e6			1.46867			0.734335	+	0.678787i	$0.237494\pi$
$542$		0			0
$543$			3.12231e6i			0.454440i
$544$		0			0
$545$	945450.	+	696823.i	0.136348	+	0.100492i
$546$		0			0
$547$			1.18580e7i			1.69451i	0.531189	+	0.847253i	$0.321745\pi$
$547$			1.18580e7i			1.69451i	−0.531189	+	0.847253i	$0.678255\pi$
$548$		0			0
$549$	−871794.			−0.123448
$550$		0			0
$551$	1.52460e6			0.213933
$552$		0			0
$553$			3.09958e6i			0.431013i
$554$		0			0
$555$	1.25096e7	+	9.21995e6i	1.72390	+	1.27056i
$556$		0			0
$557$			904550.i			0.123536i	0.998091	+	0.0617681i	$0.0196739\pi$
$557$			904550.i			0.123536i	−0.998091	+	0.0617681i	$0.980326\pi$
$558$		0			0
$559$	−49896.0			−0.00675361
$560$		0			0
$561$	−3.45946e6			−0.464088
$562$		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
$567$		−	4.34724e6i		−	0.567879i
$568$		0			0
$569$	−2.27007e6			−0.293940			−0.146970	−	0.989141i	$-0.546952\pi$
$569$	−2.27007e6			−0.293940			−0.146970	+	0.989141i	$0.546952\pi$
$570$		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
$573$		−	6.61372e6i		−	0.841510i
$574$		0			0
$575$	7.26660e6	−	2.25182e6i	0.916562	−	0.284030i
$576$		0			0
$577$		−	5.63943e6i		−	0.705173i	−0.935779	−	0.352586i	$-0.885302\pi$
$577$		−	5.63943e6i		−	0.705173i	0.935779	−	0.352586i	$-0.114698\pi$
$578$		0			0
$579$	−1.56436e7			−1.93928
$580$		0			0
$581$	3.69191e6			0.453744
$582$		0			0
$583$			1.46765e6i			0.178834i
$584$		0			0
$585$	−605880.	+	822059.i	−0.0731976	+	0.0993146i
$586$		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
$591$	1.18615e6			0.139692
$592$		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
$597$			7.87632e6i			0.904456i
$598$		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
$603$			6.67694e6i			0.747798i
$604$		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
$609$	8.23284e6			0.899511
$610$		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
$615$	−130680.	+	177307.i	−0.0139323	+	0.0189033i
$616$		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
$621$	−4.35996e6			−0.453684
$622$		0			0
$623$		−	596395.i		−	0.0615622i
$624$		0			0
$625$	−8.05437e6	+	5.52218e6i	−0.824768	+	0.565471i
$626$		0			0
$627$			1.10324e6i			0.112073i
$628$		0			0
$629$	−9.63706e6			−0.971220
$630$		0			0
$631$	−8.60145e6			−0.859999			−0.430000	−	0.902829i	$-0.641486\pi$
$631$	−8.60145e6			−0.859999			−0.430000	+	0.902829i	$0.641486\pi$
$632$		0			0
$633$		−	5.00574e6i		−	0.496546i
$634$		0			0
$635$	−2.89278e6	+	3.92493e6i	−0.284696	+	0.386276i
$636$		0			0
$637$		−	1.58119e6i		−	0.154396i
$638$		0			0
$639$	−8.16286e6			−0.790842
$640$		0			0
$641$	−6.42440e6			−0.617572			−0.308786	−	0.951132i	$-0.599923\pi$
$641$	−6.42440e6			−0.617572			−0.308786	+	0.951132i	$0.599923\pi$
$642$		0			0
$643$			3.64721e6i			0.347883i	0.984756	+	0.173941i	$0.0556503\pi$
$643$			3.64721e6i			0.347883i	−0.984756	+	0.173941i	$0.944350\pi$
$644$		0			0
$645$	−374220.	−	275811.i	−0.0354183	−	0.0261043i
$646$		0			0
$647$		−	3.78036e6i		−	0.355036i	−0.984118	−	0.177518i	$-0.943193\pi$
$647$		−	3.78036e6i		−	0.355036i	0.984118	−	0.177518i	$-0.0568068\pi$
$648$		0			0
$649$	−6.21432e6			−0.579138
$650$		0			0
$651$	8.02138e6			0.741816
$652$		0			0
$653$		−	1.66957e7i		−	1.53223i	−0.642706	−	0.766113i	$-0.722188\pi$
$653$		−	1.66957e7i		−	1.53223i	0.642706	−	0.766113i	$-0.277812\pi$
$654$		0			0
$655$	−8.67834e6	−	6.39618e6i	−0.790375	−	0.582529i
$656$		0			0
$657$			1.08512e7i			0.980761i
$658$		0			0
$659$	1.22166e6			0.109581			0.0547907	−	0.998498i	$-0.482551\pi$
$659$	1.22166e6			0.109581			0.0547907	+	0.998498i	$0.482551\pi$
$660$		0			0
$661$	−1.62789e7			−1.44918			−0.724589	−	0.689182i	$-0.757970\pi$
$661$	−1.62789e7			−1.44918			−0.724589	+	0.689182i	$0.757970\pi$
$662$		0			0
$663$		−	1.63910e6i		−	0.144818i
$664$		0			0
$665$	−435600.	+	591023.i	−0.0381974	+	0.0518263i
$666$		0			0
$667$		−	1.68704e7i		−	1.46829i
$668$		0			0
$669$	5.74636e6			0.496395
$670$		0			0
$671$	−1.43590e6			−0.123117
$672$		0			0
$673$		−	1.43928e7i		−	1.22492i	−0.790503	−	0.612459i	$-0.790181\pi$
$673$		−	1.43928e7i		−	1.22492i	0.790503	−	0.612459i	$-0.209819\pi$
$674$		0			0
$675$	5.34600e6	−	1.65665e6i	0.451616	−	0.139950i
$676$		0			0
$677$			2.62429e6i			0.220059i	0.993928	+	0.110030i	$0.0350946\pi$
$677$			2.62429e6i			0.220059i	−0.993928	+	0.110030i	$0.964905\pi$
$678$		0			0
$679$	−6.04454e6			−0.503140
$680$		0			0
$681$	2.31661e7			1.91419
$682$		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
$687$			1.08985e7i			0.880998i
$688$		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
$693$			2.30176e6i			0.182066i
$694$		0			0
$695$	1.43253e7	+	1.05581e7i	1.12497	+	0.829136i
$696$		0			0
$697$		−	136592.i		−	0.0106498i
$698$		0			0
$699$	−957264.			−0.0741035
$700$		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
$705$	9.43866e6	+	6.95655e6i	0.715217	+	0.527134i
$706$		0			0
$707$			6.51307e6i			0.490046i
$708$		0			0
$709$	9.96961e6			0.744839			0.372420	−	0.928064i	$-0.378528\pi$
$709$	9.96961e6			0.744839			0.372420	+	0.928064i	$0.378528\pi$
$710$		0			0
$711$	−7.94376e6			−0.589321
$712$		0			0
$713$		−	1.64371e7i		−	1.21088i
$714$		0			0
$715$	−997920.	+	1.35398e6i	−0.0730013	+	0.0990482i
$716$		0			0
$717$			2.00160e7i			1.45405i
$718$		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
$723$		−	1.78143e7i		−	1.26743i
$724$		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
$729$	2.48079e6			0.172890
$730$		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
$735$	8.74038e6	−	1.18590e7i	0.596777	−	0.809707i
$736$		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
$741$	−522720.			−0.0349723
$742$		0			0
$743$			1.57574e7i			1.04716i	0.851978	+	0.523578i	$0.175403\pi$
$743$			1.57574e7i			1.04716i	−0.851978	+	0.523578i	$0.824597\pi$
$744$		0			0
$745$	−3.78675e6	−	2.79094e6i	−0.249963	−	0.184230i
$746$		0			0
$747$			9.46179e6i			0.620400i
$748$		0			0
$749$	−5.79784e6			−0.377626
$750$		0			0
$751$	−1.51816e7			−0.982243			−0.491122	−	0.871091i	$-0.663413\pi$
$751$	−1.51816e7			−0.982243			−0.491122	+	0.871091i	$0.663413\pi$
$752$		0			0
$753$			1.11091e7i			0.713987i
$754$		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
$759$	1.22079e7			0.769194
$760$		0			0
$761$	4.51420e6			0.282566			0.141283	−	0.989969i	$-0.454877\pi$
$761$	4.51420e6			0.282566			0.141283	+	0.989969i	$0.454877\pi$
$762$		0			0
$763$			1.25428e6i			0.0779980i
$764$		0			0
$765$	3.50064e6	−	4.74967e6i	0.216269	−	0.293434i
$766$		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
$771$	1.56795e7			0.949939
$772$		0			0
$773$		−	1.04245e7i		−	0.627492i	−0.949507	−	0.313746i	$-0.898416\pi$
$773$		−	1.04245e7i		−	0.627492i	0.949507	−	0.313746i	$-0.101584\pi$
$774$		0			0
$775$	6.24560e6	+	2.01545e7i	0.373525	+	1.20536i
$776$		0			0
$777$			1.65959e7i			0.986163i
$778$		0			0
$779$	−43560.0			−0.00257184
$780$		0			0
$781$	−1.34447e7			−0.788721
$782$		0			0
$783$		−	1.24115e7i		−	0.723467i
$784$		0			0
$785$	−1.18285e7	+	1.60489e7i	−0.685104	+	0.929549i
$786$		0			0
$787$			3.45366e7i			1.98766i	0.110913	+	0.993830i	$0.464622\pi$
$787$			3.45366e7i			1.98766i	−0.110913	+	0.993830i	$0.535378\pi$
$788$		0			0
$789$	3.24711e7			1.85697
$790$		0			0
$791$	6.26947e6			0.356279
$792$		0			0
$793$		−	680333.i		−	0.0384183i
$794$		0			0
$795$	−5.21532e6	−	3.84384e6i	−0.292660	−	0.215698i
$796$		0			0
$797$		−	2.09287e7i		−	1.16707i	−0.812089	−	0.583533i	$-0.801670\pi$
$797$		−	2.09287e7i		−	1.16707i	0.812089	−	0.583533i	$-0.198330\pi$
$798$		0			0
$799$	−7.27126e6			−0.402942
$800$		0			0
$801$	1.52847e6			0.0841735
$802$		0			0
$803$			1.78725e7i			0.978131i
$804$		0			0
$805$	6.53994e6	+	4.82012e6i	0.355700	+	0.262161i
$806$		0			0
$807$		−	3.45533e7i		−	1.86770i
$808$		0			0
$809$	2.48797e7			1.33651			0.668257	−	0.743930i	$-0.267041\pi$
$809$	2.48797e7			1.33651			0.668257	+	0.743930i	$0.267041\pi$
$810$		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
$813$			3.42285e7i			1.81619i
$814$		0			0
$815$	4.80546e6	−	6.52005e6i	0.253420	−	0.343841i
$816$		0			0
$817$		−	91936.8i		−	0.00481875i
$818$		0			0
$819$	−1.09058e6			−0.0568132
$820$		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
$825$	−1.49688e7	+	4.63863e6i	−0.765688	+	0.237277i
$826$		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	$-0.625496\pi$
$829$	−1.52015e7			−0.768244			−0.384122	+	0.923282i	$0.625496\pi$
$830$		0			0
$831$	2.53210e7			1.27198
$832$		0			0
$833$			9.13579e6i			0.456177i
$834$		0			0
$835$	−600380.	+	814596.i	−0.0297996	+	0.0404321i
$836$		0			0
$837$		−	1.20927e7i		−	0.596635i
$838$		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
$843$		−	2.91532e7i		−	1.41292i
$844$		0			0
$845$	1.60667e7	+	1.18416e7i	0.774077	+	0.570516i
$846$		0			0
$847$		−	5.82348e6i		−	0.278917i
$848$		0			0
$849$	−3.28446e7			−1.56385
$850$		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
$855$	−1.51470e6	−	1.11638e6i	−0.0708617	−	0.0522270i
$856$		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
$861$	−235224.			−0.0108137
$862$		0			0
$863$		−	2.70896e7i		−	1.23816i	−0.785330	−	0.619078i	$-0.787507\pi$
$863$		−	2.70896e7i		−	1.23816i	0.785330	−	0.619078i	$-0.212493\pi$
$864$		0			0
$865$	1.63368e7	−	2.21657e7i	0.742379	−	1.00726i
$866$		0			0
$867$		−	1.87844e7i		−	0.848692i
$868$		0			0
$869$	−1.30838e7			−0.587741
$870$		0			0
$871$	−5.21057e6			−0.232723
$872$		0			0
$873$		−	1.54912e7i		−	0.687940i
$874$		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
$879$	4.75150e7			2.07424
$880$		0			0
$881$	−4.22840e7			−1.83542			−0.917712	−	0.397247i	$-0.869966\pi$
$881$	−4.22840e7			−1.83542			−0.917712	+	0.397247i	$0.869966\pi$
$882$		0			0
$883$			134502.i			0.00580535i	0.999996	+	0.00290267i	$0.000923951\pi$
$883$			134502.i			0.00580535i	−0.999996	+	0.00290267i	$0.999076\pi$
$884$		0			0
$885$	1.62756e7	−	2.20828e7i	0.698520	−	0.947753i
$886$		0			0
$887$		−	3.87668e6i		−	0.165444i	−0.996573	−	0.0827219i	$-0.973639\pi$
$887$		−	3.87668e6i		−	0.165444i	0.996573	−	0.0827219i	$-0.0263613\pi$
$888$		0			0
$889$	−5.20700e6			−0.220970
$890$		0			0
$891$	1.83504e7			0.774374
$892$		0			0
$893$			2.31885e6i			0.0973070i
$894$		0			0
$895$	−1.99989e7	−	1.47397e7i	−0.834543	−	0.615081i
$896$		0			0
$897$			5.78414e6i			0.240026i
$898$		0			0
$899$	4.67914e7			1.93093
$900$		0			0
$901$	4.01773e6			0.164880
$902$		0			0
$903$		−	496459.i		−	0.0202611i
$904$		0			0
$905$	−7.06059e6	−	5.20385e6i	−0.286563	−	0.211205i
$906$		0			0
$907$		−	2.87363e7i		−	1.15988i	−0.814660	−	0.579939i	$-0.803076\pi$
$907$		−	2.87363e7i		−	1.15988i	0.814660	−	0.579939i	$-0.196924\pi$
$908$		0			0
$909$	−1.66920e7			−0.670037
$910$		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
$915$	3.76068e6	−	5.10249e6i	0.148496	−	0.201479i
$916$		0			0
$917$		−	1.15131e7i		−	0.452137i
$918$		0			0
$919$	−6.76852e6			−0.264366			−0.132183	−	0.991225i	$-0.542199\pi$
$919$	−6.76852e6			−0.264366			−0.132183	+	0.991225i	$0.542199\pi$
$920$		0			0
$921$	1.84722e7			0.717579
$922$		0			0
$923$		−	6.37015e6i		−	0.246119i
$924$		0			0
$925$	−4.16988e7	+	1.29219e7i	−1.60239	+	0.496560i
$926$		0			0
$927$			1.08055e7i			0.412996i
$928$		0			0
$929$	1.15356e7			0.438530			0.219265	−	0.975665i	$-0.429634\pi$
$929$	1.15356e7			0.438530			0.219265	+	0.975665i	$0.429634\pi$
$930$		0			0
$931$	2.91346e6			0.110163
$932$		0			0
$933$		−	1.13061e7i		−	0.425214i
$934$		0			0
$935$	5.76576e6	−	7.82299e6i	0.215689	−	0.292647i
$936$		0			0
$937$		−	3.92632e7i		−	1.46096i	−0.682936	−	0.730478i	$-0.739297\pi$
$937$		−	3.92632e7i		−	1.46096i	0.682936	−	0.730478i	$-0.260703\pi$
$938$		0			0
$939$	3.42762e7			1.26861
$940$		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
$945$	4.81140e6	+	3.54614e6i	0.175264	+	0.129174i
$946$		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
$951$	2.61967e6			0.0939281
$952$		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
$957$			3.47521e7i			1.22660i
$958$		0			0
$959$	−8.57102e6			−0.300944
$960$		0			0
$961$	1.69604e7			0.592416
$962$		0			0
$963$		−	1.48590e7i		−	0.516325i
$964$		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
$969$	3.02016e6			0.103329
$970$		0			0
$971$	−3.84771e7			−1.30965			−0.654823	−	0.755783i	$-0.727257\pi$
$971$	−3.84771e7			−1.30965			−0.654823	+	0.755783i	$0.727257\pi$
$972$		0			0
$973$			1.90047e7i			0.643544i
$974$		0			0
$975$	−2.19780e6	−	7.09227e6i	−0.0740417	−	0.238932i
$976$		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
$981$	−3.21453e6			−0.106646
$982$		0			0
$983$		−	2.88475e7i		−	0.952192i	−0.879393	−	0.476096i	$-0.842051\pi$
$983$		−	2.88475e7i		−	0.952192i	0.879393	−	0.476096i	$-0.157949\pi$
$984$		0			0
$985$	−1.97692e6	+	2.68229e6i	−0.0649230	+	0.0880876i
$986$		0			0
$987$			1.25218e7i			0.409142i
$988$		0			0
$989$	−1.01732e6			−0.0330726
$990$		0			0
$991$	−5.21596e7			−1.68714			−0.843569	−	0.537021i	$-0.819550\pi$
$991$	−5.21596e7			−1.68714			−0.843569	+	0.537021i	$0.819550\pi$
$992$		0			0
$993$		−	3.14525e7i		−	1.01224i
$994$		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
$999$	2.50193e7			0.793161

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.6.c.f.129.1		2
4.3	odd	2		320.6.c.g.129.2		2
5.4	even	2	inner	320.6.c.f.129.2		2
8.3	odd	2		80.6.c.a.49.1		2
8.5	even	2		5.6.b.a.4.1	✓	2
20.19	odd	2		320.6.c.g.129.1		2
24.5	odd	2		45.6.b.b.19.2		2
24.11	even	2		720.6.f.f.289.2		2
40.3	even	4		400.6.a.t.1.2		2
40.13	odd	4		25.6.a.c.1.1		2
40.19	odd	2		80.6.c.a.49.2		2
40.27	even	4		400.6.a.t.1.1		2
40.29	even	2		5.6.b.a.4.2	yes	2
40.37	odd	4		25.6.a.c.1.2		2
56.13	odd	2		245.6.b.a.99.1		2
120.29	odd	2		45.6.b.b.19.1		2
120.53	even	4		225.6.a.n.1.2		2
120.59	even	2		720.6.f.f.289.1		2
120.77	even	4		225.6.a.n.1.1		2
280.69	odd	2		245.6.b.a.99.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.6.b.a.4.1	✓	2	8.5	even	2
5.6.b.a.4.2	yes	2	40.29	even	2
25.6.a.c.1.1		2	40.13	odd	4
25.6.a.c.1.2		2	40.37	odd	4
45.6.b.b.19.1		2	120.29	odd	2
45.6.b.b.19.2		2	24.5	odd	2
80.6.c.a.49.1		2	8.3	odd	2
80.6.c.a.49.2		2	40.19	odd	2
225.6.a.n.1.1		2	120.77	even	4
225.6.a.n.1.2		2	120.53	even	4
245.6.b.a.99.1		2	56.13	odd	2
245.6.b.a.99.2		2	280.69	odd	2
320.6.c.f.129.1		2	1.1	even	1	trivial
320.6.c.f.129.2		2	5.4	even	2	inner
320.6.c.g.129.1		2	20.19	odd	2
320.6.c.g.129.2		2	4.3	odd	2
400.6.a.t.1.1		2	40.27	even	4
400.6.a.t.1.2		2	40.3	even	4
720.6.f.f.289.1		2	120.59	even	2
720.6.f.f.289.2		2	24.11	even	2

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:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	320.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-19.8997i q^{3} +(45.0000 + 33.1662i) q^{5} +59.6992i q^{7} -153.000 q^{9} -252.000 q^{11} -119.398i q^{13} +(660.000 - 895.489i) q^{15} +689.858i q^{17} +220.000 q^{19} +1188.00 q^{21} -2434.40i q^{23} +(925.000 + 2984.96i) q^{25} -1790.98i q^{27} +6930.00 q^{29} +6752.00 q^{31} +5014.74i q^{33} +(-1980.00 + 2686.47i) q^{35} +13969.6i q^{37} -2376.00 q^{39} -198.000 q^{41} -417.895i q^{43} +(-6885.00 - 5074.44i) q^{45} +10540.2i q^{47} +13243.0 q^{49} +13728.0 q^{51} -5823.99i q^{53} +(-11340.0 - 8357.89i) q^{55} -4377.94i q^{57} +24660.0 q^{59} +5698.00 q^{61} -9133.98i q^{63} +(3960.00 - 5372.93i) q^{65} -43640.1i q^{67} -48444.0 q^{69} +53352.0 q^{71} -70922.7i q^{73} +(59400.0 - 18407.3i) q^{75} -15044.2i q^{77} +51920.0 q^{79} -72819.0 q^{81} -61841.8i q^{83} +(-22880.0 + 31043.6i) q^{85} -137905. i q^{87} -9990.00 q^{89} +7128.00 q^{91} -134363. i q^{93} +(9900.00 + 7296.57i) q^{95} +101250. i q^{97} +38556.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 90 q^{5} - 306 q^{9} - 504 q^{11} + 1320 q^{15} + 440 q^{19} + 2376 q^{21} + 1850 q^{25} + 13860 q^{29} + 13504 q^{31} - 3960 q^{35} - 4752 q^{39} - 396 q^{41} - 13770 q^{45} + 26486 q^{49} + 27456 q^{51}+ \cdots + 77112 q^{99}+O(q^{100}) \) 2 * q + 90 * q^5 - 306 * q^9 - 504 * q^11 + 1320 * q^15 + 440 * q^19 + 2376 * q^21 + 1850 * q^25 + 13860 * q^29 + 13504 * q^31 - 3960 * q^35 - 4752 * q^39 - 396 * q^41 - 13770 * q^45 + 26486 * q^49 + 27456 * q^51 - 22680 * q^55 + 49320 * q^59 + 11396 * q^61 + 7920 * q^65 - 96888 * q^69 + 106704 * q^71 + 118800 * q^75 + 103840 * q^79 - 145638 * q^81 - 45760 * q^85 - 19980 * q^89 + 14256 * q^91 + 19800 * q^95 + 77112 * q^99

Introduction

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