LMFDB - Embedded newform 256.5.c.c.255.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,5,Mod(255,256)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(256, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("256.255");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	256.c (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:	$26.4627105495$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 6 $ x^2 + 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 128)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			255.1
Root			$-2.44949i$ of defining polynomial
Character	$\chi$	$=$	256.255
Dual form			256.5.c.c.255.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.79796i q^{3} -8.00000 q^{5} -78.3837i q^{7} -15.0000 q^{9} +O(q^{10})$	$q-9.79796i q^{3} -8.00000 q^{5} -78.3837i q^{7} -15.0000 q^{9} -107.778i q^{11} +216.000 q^{13} +78.3837i q^{15} -162.000 q^{17} -440.908i q^{19} -768.000 q^{21} +705.453i q^{23} -561.000 q^{25} -646.665i q^{27} +1304.00 q^{29} -627.069i q^{31} -1056.00 q^{33} +627.069i q^{35} -1512.00 q^{37} -2116.36i q^{39} -1890.00 q^{41} +2909.99i q^{43} +120.000 q^{45} +1410.91i q^{47} -3743.00 q^{49} +1587.27i q^{51} +1976.00 q^{53} +862.220i q^{55} -4320.00 q^{57} -2263.33i q^{59} -2376.00 q^{61} +1175.76i q^{63} -1728.00 q^{65} +1675.45i q^{67} +6912.00 q^{69} +7759.98i q^{71} -2750.00 q^{73} +5496.65i q^{75} -8448.00 q^{77} -7995.13i q^{79} -7551.00 q^{81} -9337.45i q^{83} +1296.00 q^{85} -12776.5i q^{87} -2430.00 q^{89} -16930.9i q^{91} -6144.00 q^{93} +3527.27i q^{95} +7454.00 q^{97} +1616.66i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 16 q^{5} - 30 q^{9}+O(q^{10}) $ 2 * q - 16 * q^5 - 30 * q^9	$ 2 q - 16 q^{5} - 30 q^{9} + 432 q^{13} - 324 q^{17} - 1536 q^{21} - 1122 q^{25} + 2608 q^{29} - 2112 q^{33} - 3024 q^{37} - 3780 q^{41} + 240 q^{45} - 7486 q^{49} + 3952 q^{53} - 8640 q^{57} - 4752 q^{61} - 3456 q^{65} + 13824 q^{69} - 5500 q^{73} - 16896 q^{77} - 15102 q^{81} + 2592 q^{85} - 4860 q^{89} - 12288 q^{93} + 14908 q^{97}+O(q^{100}) $ 2 * q - 16 * q^5 - 30 * q^9 + 432 * q^13 - 324 * q^17 - 1536 * q^21 - 1122 * q^25 + 2608 * q^29 - 2112 * q^33 - 3024 * q^37 - 3780 * q^41 + 240 * q^45 - 7486 * q^49 + 3952 * q^53 - 8640 * q^57 - 4752 * q^61 - 3456 * q^65 + 13824 * q^69 - 5500 * q^73 - 16896 * q^77 - 15102 * q^81 + 2592 * q^85 - 4860 * q^89 - 12288 * q^93 + 14908 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/256\mathbb{Z}\right)^\times$.

$n$	$5$	$255$
$\chi(n)$	$1$	$-1$

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
$3$	− 9.79796i	− 1.08866i	−0.838870	−	0.544331i	$-0.816784\pi$
$3$	− 9.79796i	− 1.08866i	0.838870	−	0.544331i	$-0.183216\pi$
$4$	0	0
$5$	−8.00000	−0.320000	−0.160000	−	0.987117i	$-0.551149\pi$
$5$	−8.00000	−0.320000	−0.160000	+	0.987117i	$0.551149\pi$
$6$	0	0
$7$	− 78.3837i	− 1.59967i	−0.600222	−	0.799833i	$-0.704921\pi$
$7$	− 78.3837i	− 1.59967i	0.600222	−	0.799833i	$-0.295079\pi$
$8$	0	0
$9$	−15.0000	−0.185185
$10$	0	0
$11$	− 107.778i	− 0.890724i	−0.895351	−	0.445362i	$-0.853075\pi$
$11$	− 107.778i	− 0.890724i	0.895351	−	0.445362i	$-0.146925\pi$
$12$	0	0
$13$	216.000	1.27811	0.639053	−	0.769162i	$-0.279326\pi$
$13$	216.000	1.27811	0.639053	+	0.769162i	$0.279326\pi$
$14$	0	0
$15$	78.3837i	0.348372i
$16$	0	0
$17$	−162.000	−0.560554	−0.280277	−	0.959919i	$-0.590426\pi$
$17$	−162.000	−0.560554	−0.280277	+	0.959919i	$0.590426\pi$
$18$	0	0
$19$	− 440.908i	− 1.22135i	−0.791880	−	0.610676i	$-0.790898\pi$
$19$	− 440.908i	− 1.22135i	0.791880	−	0.610676i	$-0.209102\pi$
$20$	0	0
$21$	−768.000	−1.74150
$22$	0	0
$23$	705.453i	1.33356i	0.745255	+	0.666780i	$0.232328\pi$
$23$	705.453i	1.33356i	−0.745255	+	0.666780i	$0.767672\pi$
$24$	0	0
$25$	−561.000	−0.897600
$26$	0	0
$27$	− 646.665i	− 0.887058i
$28$	0	0
$29$	1304.00	1.55054	0.775268	−	0.631633i	$-0.217615\pi$
$29$	1304.00	1.55054	0.775268	+	0.631633i	$0.217615\pi$
$30$	0	0
$31$	− 627.069i	− 0.652518i	−0.945280	−	0.326259i	$-0.894212\pi$
$31$	− 627.069i	− 0.652518i	0.945280	−	0.326259i	$-0.105788\pi$
$32$	0	0
$33$	−1056.00	−0.969697
$34$	0	0
$35$	627.069i	0.511893i
$36$	0	0
$37$	−1512.00	−1.10446	−0.552228	−	0.833693i	$-0.686222\pi$
$37$	−1512.00	−1.10446	−0.552228	+	0.833693i	$0.686222\pi$
$38$	0	0
$39$	− 2116.36i	− 1.39143i
$40$	0	0
$41$	−1890.00	−1.12433	−0.562165	−	0.827025i	$-0.690032\pi$
$41$	−1890.00	−1.12433	−0.562165	+	0.827025i	$0.690032\pi$
$42$	0	0
$43$	2909.99i	1.57382i	0.617068	+	0.786910i	$0.288321\pi$
$43$	2909.99i	1.57382i	−0.617068	+	0.786910i	$0.711679\pi$
$44$	0	0
$45$	120.000	0.0592593
$46$	0	0
$47$	1410.91i	0.638708i	0.947635	+	0.319354i	$0.103466\pi$
$47$	1410.91i	0.638708i	−0.947635	+	0.319354i	$0.896534\pi$
$48$	0	0
$49$	−3743.00	−1.55893
$50$	0	0
$51$	1587.27i	0.610253i
$52$	0	0
$53$	1976.00	0.703453	0.351727	−	0.936103i	$-0.385595\pi$
$53$	1976.00	0.703453	0.351727	+	0.936103i	$0.385595\pi$
$54$	0	0
$55$	862.220i	0.285032i
$56$	0	0
$57$	−4320.00	−1.32964
$58$	0	0
$59$	− 2263.33i	− 0.650195i	−0.945681	−	0.325097i	$-0.894603\pi$
$59$	− 2263.33i	− 0.650195i	0.945681	−	0.325097i	$-0.105397\pi$
$60$	0	0
$61$	−2376.00	−0.638538	−0.319269	−	0.947664i	$-0.603437\pi$
$61$	−2376.00	−0.638538	−0.319269	+	0.947664i	$0.603437\pi$
$62$	0	0
$63$	1175.76i	0.296235i
$64$	0	0
$65$	−1728.00	−0.408994
$66$	0	0
$67$	1675.45i	0.373235i	0.982433	+	0.186617i	$0.0597525\pi$
$67$	1675.45i	0.373235i	−0.982433	+	0.186617i	$0.940248\pi$
$68$	0	0
$69$	6912.00	1.45180
$70$	0	0
$71$	7759.98i	1.53937i	0.638422	+	0.769687i	$0.279588\pi$
$71$	7759.98i	1.53937i	−0.638422	+	0.769687i	$0.720412\pi$
$72$	0	0
$73$	−2750.00	−0.516044	−0.258022	−	0.966139i	$-0.583071\pi$
$73$	−2750.00	−0.516044	−0.258022	+	0.966139i	$0.583071\pi$
$74$	0	0
$75$	5496.65i	0.977183i
$76$	0	0
$77$	−8448.00	−1.42486
$78$	0	0
$79$	− 7995.13i	− 1.28107i	−0.767931	−	0.640533i	$-0.778713\pi$
$79$	− 7995.13i	− 1.28107i	0.767931	−	0.640533i	$-0.221287\pi$
$80$	0	0
$81$	−7551.00	−1.15089
$82$	0	0
$83$	− 9337.45i	− 1.35542i	−0.735332	−	0.677708i	$-0.762974\pi$
$83$	− 9337.45i	− 1.35542i	0.735332	−	0.677708i	$-0.237026\pi$
$84$	0	0
$85$	1296.00	0.179377
$86$	0	0
$87$	− 12776.5i	− 1.68801i
$88$	0	0
$89$	−2430.00	−0.306779	−0.153390	−	0.988166i	$-0.549019\pi$
$89$	−2430.00	−0.306779	−0.153390	+	0.988166i	$0.549019\pi$
$90$	0	0
$91$	− 16930.9i	− 2.04454i
$92$	0	0
$93$	−6144.00	−0.710371
$94$	0	0
$95$	3527.27i	0.390833i
$96$	0	0
$97$	7454.00	0.792220	0.396110	−	0.918203i	$-0.370360\pi$
$97$	7454.00	0.792220	0.396110	+	0.918203i	$0.370360\pi$
$98$	0	0
$99$	1616.66i	0.164949i
$100$	0	0
$101$	1496.00	0.146652	0.0733261	−	0.997308i	$-0.476639\pi$
$101$	1496.00	0.146652	0.0733261	+	0.997308i	$0.476639\pi$
$102$	0	0
$103$	− 2586.66i	− 0.243818i	−0.992541	−	0.121909i	$-0.961098\pi$
$103$	− 2586.66i	− 0.243818i	0.992541	−	0.121909i	$-0.0389015\pi$
$104$	0	0
$105$	6144.00	0.557279
$106$	0	0
$107$	1067.98i	0.0932813i	0.998912	+	0.0466406i	$0.0148516\pi$
$107$	1067.98i	0.0932813i	−0.998912	+	0.0466406i	$0.985148\pi$
$108$	0	0
$109$	14904.0	1.25444	0.627220	−	0.778842i	$-0.284193\pi$
$109$	14904.0	1.25444	0.627220	+	0.778842i	$0.284193\pi$
$110$	0	0
$111$	14814.5i	1.20238i
$112$	0	0
$113$	−702.000	−0.0549769	−0.0274884	−	0.999622i	$-0.508751\pi$
$113$	−702.000	−0.0549769	−0.0274884	+	0.999622i	$0.508751\pi$
$114$	0	0
$115$	− 5643.62i	− 0.426739i
$116$	0	0
$117$	−3240.00	−0.236686
$118$	0	0
$119$	12698.2i	0.896699i
$120$	0	0
$121$	3025.00	0.206612
$122$	0	0
$123$	18518.1i	1.22402i
$124$	0	0
$125$	9488.00	0.607232
$126$	0	0
$127$	− 3448.88i	− 0.213831i	−0.994268	−	0.106916i	$-0.965903\pi$
$127$	− 3448.88i	− 0.213831i	0.994268	−	0.106916i	$-0.0340974\pi$
$128$	0	0
$129$	28512.0	1.71336
$130$	0	0
$131$	9592.20i	0.558954i	0.960152	+	0.279477i	$0.0901610\pi$
$131$	9592.20i	0.558954i	−0.960152	+	0.279477i	$0.909839\pi$
$132$	0	0
$133$	−34560.0	−1.95376
$134$	0	0
$135$	5173.32i	0.283859i
$136$	0	0
$137$	32670.0	1.74064	0.870318	−	0.492490i	$-0.163913\pi$
$137$	32670.0	1.74064	0.870318	+	0.492490i	$0.163913\pi$
$138$	0	0
$139$	− 1322.72i	− 0.0684605i	−0.999414	−	0.0342302i	$-0.989102\pi$
$139$	− 1322.72i	− 0.0684605i	0.999414	−	0.0342302i	$-0.0108979\pi$
$140$	0	0
$141$	13824.0	0.695337
$142$	0	0
$143$	− 23280.0i	− 1.13844i
$144$	0	0
$145$	−10432.0	−0.496171
$146$	0	0
$147$	36673.8i	1.69715i
$148$	0	0
$149$	−872.000	−0.0392775	−0.0196388	−	0.999807i	$-0.506252\pi$
$149$	−872.000	−0.0392775	−0.0196388	+	0.999807i	$0.506252\pi$
$150$	0	0
$151$	9484.42i	0.415965i	0.978133	+	0.207983i	$0.0666898\pi$
$151$	9484.42i	0.415965i	−0.978133	+	0.207983i	$0.933310\pi$
$152$	0	0
$153$	2430.00	0.103806
$154$	0	0
$155$	5016.55i	0.208806i
$156$	0	0
$157$	28728.0	1.16548	0.582742	−	0.812657i	$-0.301980\pi$
$157$	28728.0	1.16548	0.582742	+	0.812657i	$0.301980\pi$
$158$	0	0
$159$	− 19360.8i	− 0.765823i
$160$	0	0
$161$	55296.0	2.13325
$162$	0	0
$163$	− 31128.1i	− 1.17160i	−0.810457	−	0.585798i	$-0.800781\pi$
$163$	− 31128.1i	− 1.17160i	0.810457	−	0.585798i	$-0.199219\pi$
$164$	0	0
$165$	8448.00	0.310303
$166$	0	0
$167$	− 54319.9i	− 1.94772i	−0.227155	−	0.973859i	$-0.572942\pi$
$167$	− 54319.9i	− 1.94772i	0.227155	−	0.973859i	$-0.427058\pi$
$168$	0	0
$169$	18095.0	0.633556
$170$	0	0
$171$	6613.62i	0.226176i
$172$	0	0
$173$	−25256.0	−0.843864	−0.421932	−	0.906628i	$-0.638648\pi$
$173$	−25256.0	−0.843864	−0.421932	+	0.906628i	$0.638648\pi$
$174$	0	0
$175$	43973.2i	1.43586i
$176$	0	0
$177$	−22176.0	−0.707843
$178$	0	0
$179$	− 8945.54i	− 0.279190i	−0.990209	−	0.139595i	$-0.955420\pi$
$179$	− 8945.54i	− 0.279190i	0.990209	−	0.139595i	$-0.0445801\pi$
$180$	0	0
$181$	−36072.0	−1.10107	−0.550533	−	0.834814i	$-0.685575\pi$
$181$	−36072.0	−1.10107	−0.550533	+	0.834814i	$0.685575\pi$
$182$	0	0
$183$	23280.0i	0.695152i
$184$	0	0
$185$	12096.0	0.353426
$186$	0	0
$187$	17460.0i	0.499298i
$188$	0	0
$189$	−50688.0	−1.41900
$190$	0	0
$191$	31039.9i	0.850852i	0.904993	+	0.425426i	$0.139876\pi$
$191$	31039.9i	0.850852i	−0.904993	+	0.425426i	$0.860124\pi$
$192$	0	0
$193$	41374.0	1.11074	0.555371	−	0.831603i	$-0.312576\pi$
$193$	41374.0	1.11074	0.555371	+	0.831603i	$0.312576\pi$
$194$	0	0
$195$	16930.9i	0.445256i
$196$	0	0
$197$	67640.0	1.74289	0.871447	−	0.490489i	$-0.163182\pi$
$197$	67640.0	1.74289	0.871447	+	0.490489i	$0.163182\pi$
$198$	0	0
$199$	25004.4i	0.631408i	0.948858	+	0.315704i	$0.102241\pi$
$199$	25004.4i	0.631408i	−0.948858	+	0.315704i	$0.897759\pi$
$200$	0	0
$201$	16416.0	0.406327
$202$	0	0
$203$	− 102212.i	− 2.48034i
$204$	0	0
$205$	15120.0	0.359786
$206$	0	0
$207$	− 10581.8i	− 0.246955i
$208$	0	0
$209$	−47520.0	−1.08789
$210$	0	0
$211$	− 26189.9i	− 0.588260i	−0.955765	−	0.294130i	$-0.904970\pi$
$211$	− 26189.9i	− 0.588260i	0.955765	−	0.294130i	$-0.0950299\pi$
$212$	0	0
$213$	76032.0	1.67586
$214$	0	0
$215$	− 23280.0i	− 0.503623i
$216$	0	0
$217$	−49152.0	−1.04381
$218$	0	0
$219$	26944.4i	0.561798i
$220$	0	0
$221$	−34992.0	−0.716447
$222$	0	0
$223$	− 20693.3i	− 0.416121i	−0.978116	−	0.208061i	$-0.933285\pi$
$223$	− 20693.3i	− 0.416121i	0.978116	−	0.208061i	$-0.0667151\pi$
$224$	0	0
$225$	8415.00	0.166222
$226$	0	0
$227$	− 77923.2i	− 1.51222i	−0.654445	−	0.756110i	$-0.727098\pi$
$227$	− 77923.2i	− 1.51222i	0.654445	−	0.756110i	$-0.272902\pi$
$228$	0	0
$229$	−14472.0	−0.275967	−0.137984	−	0.990435i	$-0.544062\pi$
$229$	−14472.0	−0.275967	−0.137984	+	0.990435i	$0.544062\pi$
$230$	0	0
$231$	82773.2i	1.55119i
$232$	0	0
$233$	2754.00	0.0507285	0.0253643	−	0.999678i	$-0.491925\pi$
$233$	2754.00	0.0507285	0.0253643	+	0.999678i	$0.491925\pi$
$234$	0	0
$235$	− 11287.2i	− 0.204387i
$236$	0	0
$237$	−78336.0	−1.39465
$238$	0	0
$239$	− 91708.9i	− 1.60552i	−0.596302	−	0.802760i	$-0.703364\pi$
$239$	− 91708.9i	− 1.60552i	0.596302	−	0.802760i	$-0.296636\pi$
$240$	0	0
$241$	−97570.0	−1.67990	−0.839948	−	0.542667i	$-0.817414\pi$
$241$	−97570.0	−1.67990	−0.839948	+	0.542667i	$0.817414\pi$
$242$	0	0
$243$	21604.5i	0.365874i
$244$	0	0
$245$	29944.0	0.498859
$246$	0	0
$247$	− 95236.2i	− 1.56102i
$248$	0	0
$249$	−91488.0	−1.47559
$250$	0	0
$251$	67811.7i	1.07636i	0.842830	+	0.538179i	$0.180888\pi$
$251$	67811.7i	1.07636i	−0.842830	+	0.538179i	$0.819112\pi$
$252$	0	0
$253$	76032.0	1.18783
$254$	0	0
$255$	− 12698.2i	− 0.195281i
$256$	0	0
$257$	80514.0	1.21900	0.609502	−	0.792785i	$-0.291369\pi$
$257$	80514.0	1.21900	0.609502	+	0.792785i	$0.291369\pi$
$258$	0	0
$259$	118516.i	1.76676i
$260$	0	0
$261$	−19560.0	−0.287136
$262$	0	0
$263$	− 14814.5i	− 0.214179i	−0.994249	−	0.107089i	$-0.965847\pi$
$263$	− 14814.5i	− 0.214179i	0.994249	−	0.107089i	$-0.0341531\pi$
$264$	0	0
$265$	−15808.0	−0.225105
$266$	0	0
$267$	23809.0i	0.333979i
$268$	0	0
$269$	−44872.0	−0.620113	−0.310057	−	0.950718i	$-0.600348\pi$
$269$	−44872.0	−0.620113	−0.310057	+	0.950718i	$0.600348\pi$
$270$	0	0
$271$	91395.4i	1.24447i	0.782829	+	0.622237i	$0.213776\pi$
$271$	91395.4i	1.24447i	−0.782829	+	0.622237i	$0.786224\pi$
$272$	0	0
$273$	−165888.	−2.22582
$274$	0	0
$275$	60463.2i	0.799513i
$276$	0	0
$277$	−90504.0	−1.17953	−0.589764	−	0.807576i	$-0.700779\pi$
$277$	−90504.0	−1.17953	−0.589764	+	0.807576i	$0.700779\pi$
$278$	0	0
$279$	9406.04i	0.120837i
$280$	0	0
$281$	−23166.0	−0.293385	−0.146693	−	0.989182i	$-0.546863\pi$
$281$	−23166.0	−0.293385	−0.146693	+	0.989182i	$0.546863\pi$
$282$	0	0
$283$	− 70809.8i	− 0.884140i	−0.896981	−	0.442070i	$-0.854244\pi$
$283$	− 70809.8i	− 0.884140i	0.896981	−	0.442070i	$-0.145756\pi$
$284$	0	0
$285$	34560.0	0.425485
$286$	0	0
$287$	148145.i	1.79855i
$288$	0	0
$289$	−57277.0	−0.685780
$290$	0	0
$291$	− 73034.0i	− 0.862460i
$292$	0	0
$293$	−37768.0	−0.439935	−0.219968	−	0.975507i	$-0.570595\pi$
$293$	−37768.0	−0.439935	−0.219968	+	0.975507i	$0.570595\pi$
$294$	0	0
$295$	18106.6i	0.208062i
$296$	0	0
$297$	−69696.0	−0.790123
$298$	0	0
$299$	152378.i	1.70443i
$300$	0	0
$301$	228096.	2.51759
$302$	0	0
$303$	− 14657.7i	− 0.159655i
$304$	0	0
$305$	19008.0	0.204332
$306$	0	0
$307$	− 38535.4i	− 0.408868i	−0.978880	−	0.204434i	$-0.934465\pi$
$307$	− 38535.4i	− 0.408868i	0.978880	−	0.204434i	$-0.0655353\pi$
$308$	0	0
$309$	−25344.0	−0.265435
$310$	0	0
$311$	85359.8i	0.882537i	0.897375	+	0.441268i	$0.145471\pi$
$311$	85359.8i	0.882537i	−0.897375	+	0.441268i	$0.854529\pi$
$312$	0	0
$313$	−93602.0	−0.955425	−0.477712	−	0.878516i	$-0.658534\pi$
$313$	−93602.0	−0.955425	−0.477712	+	0.878516i	$0.658534\pi$
$314$	0	0
$315$	− 9406.04i	− 0.0947951i
$316$	0	0
$317$	−49192.0	−0.489526	−0.244763	−	0.969583i	$-0.578710\pi$
$317$	−49192.0	−0.489526	−0.244763	+	0.969583i	$0.578710\pi$
$318$	0	0
$319$	− 140542.i	− 1.38110i
$320$	0	0
$321$	10464.0	0.101552
$322$	0	0
$323$	71427.1i	0.684633i
$324$	0	0
$325$	−121176.	−1.14723
$326$	0	0
$327$	− 146029.i	− 1.36566i
$328$	0	0
$329$	110592.	1.02172
$330$	0	0
$331$	− 16490.0i	− 0.150509i	−0.997164	−	0.0752547i	$-0.976023\pi$
$331$	− 16490.0i	− 0.150509i	0.997164	−	0.0752547i	$-0.0239770\pi$
$332$	0	0
$333$	22680.0	0.204529
$334$	0	0
$335$	− 13403.6i	− 0.119435i
$336$	0	0
$337$	−93758.0	−0.825560	−0.412780	−	0.910831i	$-0.635442\pi$
$337$	−93758.0	−0.825560	−0.412780	+	0.910831i	$0.635442\pi$
$338$	0	0
$339$	6878.17i	0.0598513i
$340$	0	0
$341$	−67584.0	−0.581213
$342$	0	0
$343$	105191.i	0.894108i
$344$	0	0
$345$	−55296.0	−0.464575
$346$	0	0
$347$	− 220405.i	− 1.83047i	−0.402920	−	0.915235i	$-0.632005\pi$
$347$	− 220405.i	− 1.83047i	0.402920	−	0.915235i	$-0.367995\pi$
$348$	0	0
$349$	48600.0	0.399012	0.199506	−	0.979897i	$-0.436066\pi$
$349$	48600.0	0.399012	0.199506	+	0.979897i	$0.436066\pi$
$350$	0	0
$351$	− 139680.i	− 1.13375i
$352$	0	0
$353$	108162.	0.868011	0.434006	−	0.900910i	$-0.357100\pi$
$353$	108162.	0.868011	0.434006	+	0.900910i	$0.357100\pi$
$354$	0	0
$355$	− 62079.9i	− 0.492600i
$356$	0	0
$357$	124416.	0.976202
$358$	0	0
$359$	50087.2i	0.388631i	0.980939	+	0.194316i	$0.0622486\pi$
$359$	50087.2i	0.388631i	−0.980939	+	0.194316i	$0.937751\pi$
$360$	0	0
$361$	−64079.0	−0.491701
$362$	0	0
$363$	− 29638.8i	− 0.224930i
$364$	0	0
$365$	22000.0	0.165134
$366$	0	0
$367$	− 61296.0i	− 0.455093i	−0.973767	−	0.227547i	$-0.926930\pi$
$367$	− 61296.0i	− 0.455093i	0.973767	−	0.227547i	$-0.0730704\pi$
$368$	0	0
$369$	28350.0	0.208209
$370$	0	0
$371$	− 154886.i	− 1.12529i
$372$	0	0
$373$	54648.0	0.392787	0.196393	−	0.980525i	$-0.437077\pi$
$373$	54648.0	0.392787	0.196393	+	0.980525i	$0.437077\pi$
$374$	0	0
$375$	− 92963.0i	− 0.661070i
$376$	0	0
$377$	281664.	1.98175
$378$	0	0
$379$	103790.i	0.722564i	0.932457	+	0.361282i	$0.117661\pi$
$379$	103790.i	0.722564i	−0.932457	+	0.361282i	$0.882339\pi$
$380$	0	0
$381$	−33792.0	−0.232790
$382$	0	0
$383$	124160.i	0.846415i	0.906033	+	0.423207i	$0.139096\pi$
$383$	124160.i	0.846415i	−0.906033	+	0.423207i	$0.860904\pi$
$384$	0	0
$385$	67584.0	0.455955
$386$	0	0
$387$	− 43649.9i	− 0.291448i
$388$	0	0
$389$	129112.	0.853233	0.426616	−	0.904433i	$-0.359705\pi$
$389$	129112.	0.853233	0.426616	+	0.904433i	$0.359705\pi$
$390$	0	0
$391$	− 114283.i	− 0.747532i
$392$	0	0
$393$	93984.0	0.608512
$394$	0	0
$395$	63961.1i	0.409941i
$396$	0	0
$397$	−232200.	−1.47327	−0.736633	−	0.676293i	$-0.763585\pi$
$397$	−232200.	−1.47327	−0.736633	+	0.676293i	$0.763585\pi$
$398$	0	0
$399$	338617.i	2.12698i
$400$	0	0
$401$	151902.	0.944658	0.472329	−	0.881422i	$-0.343413\pi$
$401$	151902.	0.944658	0.472329	+	0.881422i	$0.343413\pi$
$402$	0	0
$403$	− 135447.i	− 0.833987i
$404$	0	0
$405$	60408.0	0.368285
$406$	0	0
$407$	162960.i	0.983765i
$408$	0	0
$409$	−28450.0	−0.170073	−0.0850366	−	0.996378i	$-0.527101\pi$
$409$	−28450.0	−0.170073	−0.0850366	+	0.996378i	$0.527101\pi$
$410$	0	0
$411$	− 320099.i	− 1.89496i
$412$	0	0
$413$	−177408.	−1.04010
$414$	0	0
$415$	74699.6i	0.433733i
$416$	0	0
$417$	−12960.0	−0.0745303
$418$	0	0
$419$	291382.i	1.65972i	0.557974	+	0.829858i	$0.311579\pi$
$419$	291382.i	1.65972i	−0.557974	+	0.829858i	$0.688421\pi$
$420$	0	0
$421$	−119880.	−0.676367	−0.338184	−	0.941080i	$-0.609812\pi$
$421$	−119880.	−0.676367	−0.338184	+	0.941080i	$0.609812\pi$
$422$	0	0
$423$	− 21163.6i	− 0.118279i
$424$	0	0
$425$	90882.0	0.503153
$426$	0	0
$427$	186240.i	1.02145i
$428$	0	0
$429$	−228096.	−1.23938
$430$	0	0
$431$	− 170720.i	− 0.919028i	−0.888170	−	0.459514i	$-0.848024\pi$
$431$	− 170720.i	− 0.919028i	0.888170	−	0.459514i	$-0.151976\pi$
$432$	0	0
$433$	215518.	1.14950	0.574748	−	0.818330i	$-0.305100\pi$
$433$	215518.	1.14950	0.574748	+	0.818330i	$0.305100\pi$
$434$	0	0
$435$	102212.i	0.540163i
$436$	0	0
$437$	311040.	1.62875
$438$	0	0
$439$	− 75169.9i	− 0.390045i	−0.980799	−	0.195023i	$-0.937522\pi$
$439$	− 75169.9i	− 0.390045i	0.980799	−	0.195023i	$-0.0624781\pi$
$440$	0	0
$441$	56145.0	0.288691
$442$	0	0
$443$	185270.i	0.944054i	0.881584	+	0.472027i	$0.156477\pi$
$443$	185270.i	0.944054i	−0.881584	+	0.472027i	$0.843523\pi$
$444$	0	0
$445$	19440.0	0.0981694
$446$	0	0
$447$	8543.82i	0.0427599i
$448$	0	0
$449$	177822.	0.882049	0.441025	−	0.897495i	$-0.354615\pi$
$449$	177822.	0.882049	0.441025	+	0.897495i	$0.354615\pi$
$450$	0	0
$451$	203700.i	1.00147i
$452$	0	0
$453$	92928.0	0.452846
$454$	0	0
$455$	135447.i	0.654254i
$456$	0	0
$457$	203294.	0.973402	0.486701	−	0.873569i	$-0.338200\pi$
$457$	203294.	0.973402	0.486701	+	0.873569i	$0.338200\pi$
$458$	0	0
$459$	104760.i	0.497244i
$460$	0	0
$461$	286648.	1.34880	0.674399	−	0.738367i	$-0.264403\pi$
$461$	286648.	1.34880	0.674399	+	0.738367i	$0.264403\pi$
$462$	0	0
$463$	− 332817.i	− 1.55254i	−0.630399	−	0.776271i	$-0.717109\pi$
$463$	− 332817.i	− 1.55254i	0.630399	−	0.776271i	$-0.282891\pi$
$464$	0	0
$465$	49152.0	0.227319
$466$	0	0
$467$	58738.8i	0.269334i	0.990891	+	0.134667i	$0.0429965\pi$
$467$	58738.8i	0.269334i	−0.990891	+	0.134667i	$0.957004\pi$
$468$	0	0
$469$	131328.	0.597051
$470$	0	0
$471$	− 281476.i	− 1.26882i
$472$	0	0
$473$	313632.	1.40184
$474$	0	0
$475$	247349.i	1.09629i
$476$	0	0
$477$	−29640.0	−0.130269
$478$	0	0
$479$	− 186240.i	− 0.811710i	−0.913937	−	0.405855i	$-0.866974\pi$
$479$	− 186240.i	− 0.811710i	0.913937	−	0.405855i	$-0.133026\pi$
$480$	0	0
$481$	−326592.	−1.41161
$482$	0	0
$483$	− 541788.i	− 2.32239i
$484$	0	0
$485$	−59632.0	−0.253510
$486$	0	0
$487$	363308.i	1.53185i	0.642927	+	0.765927i	$0.277720\pi$
$487$	363308.i	1.53185i	−0.642927	+	0.765927i	$0.722280\pi$
$488$	0	0
$489$	−304992.	−1.27547
$490$	0	0
$491$	− 107533.i	− 0.446043i	−0.974813	−	0.223022i	$-0.928408\pi$
$491$	− 107533.i	− 0.446043i	0.974813	−	0.223022i	$-0.0715921\pi$
$492$	0	0
$493$	−211248.	−0.869158
$494$	0	0
$495$	− 12933.3i	− 0.0527836i
$496$	0	0
$497$	608256.	2.46249
$498$	0	0
$499$	385089.i	1.54654i	0.634079	+	0.773268i	$0.281379\pi$
$499$	385089.i	1.54654i	−0.634079	+	0.773268i	$0.718621\pi$
$500$	0	0
$501$	−532224.	−2.12041
$502$	0	0
$503$	− 210930.i	− 0.833688i	−0.908978	−	0.416844i	$-0.863136\pi$
$503$	− 210930.i	− 0.833688i	0.908978	−	0.416844i	$-0.136864\pi$
$504$	0	0
$505$	−11968.0	−0.0469287
$506$	0	0
$507$	− 177294.i	− 0.689729i
$508$	0	0
$509$	171512.	0.662001	0.331001	−	0.943630i	$-0.392614\pi$
$509$	171512.	0.662001	0.331001	+	0.943630i	$0.392614\pi$
$510$	0	0
$511$	215555.i	0.825499i
$512$	0	0
$513$	−285120.	−1.08341
$514$	0	0
$515$	20693.3i	0.0780216i
$516$	0	0
$517$	152064.	0.568912
$518$	0	0
$519$	247457.i	0.918683i
$520$	0	0
$521$	184734.	0.680568	0.340284	−	0.940323i	$-0.389477\pi$
$521$	184734.	0.680568	0.340284	+	0.940323i	$0.389477\pi$
$522$	0	0
$523$	− 420009.i	− 1.53552i	−0.640738	−	0.767760i	$-0.721371\pi$
$523$	− 420009.i	− 1.53552i	0.640738	−	0.767760i	$-0.278629\pi$
$524$	0	0
$525$	430848.	1.56317
$526$	0	0
$527$	101585.i	0.365771i
$528$	0	0
$529$	−217823.	−0.778381
$530$	0	0
$531$	33949.9i	0.120406i
$532$	0	0
$533$	−408240.	−1.43701
$534$	0	0
$535$	− 8543.82i	− 0.0298500i
$536$	0	0
$537$	−87648.0	−0.303944
$538$	0	0
$539$	403411.i	1.38858i
$540$	0	0
$541$	230904.	0.788927	0.394464	−	0.918912i	$-0.370930\pi$
$541$	230904.	0.788927	0.394464	+	0.918912i	$0.370930\pi$
$542$	0	0
$543$	353432.i	1.19869i
$544$	0	0
$545$	−119232.	−0.401421
$546$	0	0
$547$	39769.9i	0.132917i	0.997789	+	0.0664584i	$0.0211700\pi$
$547$	39769.9i	0.132917i	−0.997789	+	0.0664584i	$0.978830\pi$
$548$	0	0
$549$	35640.0	0.118248
$550$	0	0
$551$	− 574944.i	− 1.89375i
$552$	0	0
$553$	−626688.	−2.04928
$554$	0	0
$555$	− 118516.i	− 0.384761i
$556$	0	0
$557$	−23144.0	−0.0745981	−0.0372991	−	0.999304i	$-0.511875\pi$
$557$	−23144.0	−0.0745981	−0.0372991	+	0.999304i	$0.511875\pi$
$558$	0	0
$559$	628559.i	2.01151i
$560$	0	0
$561$	171072.	0.543567
$562$	0	0
$563$	− 491123.i	− 1.54943i	−0.632308	−	0.774717i	$-0.717892\pi$
$563$	− 491123.i	− 1.54943i	0.632308	−	0.774717i	$-0.282108\pi$
$564$	0	0
$565$	5616.00	0.0175926
$566$	0	0
$567$	591875.i	1.84104i
$568$	0	0
$569$	127710.	0.394458	0.197229	−	0.980357i	$-0.436806\pi$
$569$	127710.	0.394458	0.197229	+	0.980357i	$0.436806\pi$
$570$	0	0
$571$	− 39064.5i	− 0.119815i	−0.998204	−	0.0599073i	$-0.980919\pi$
$571$	− 39064.5i	− 0.119815i	0.998204	−	0.0599073i	$-0.0190805\pi$
$572$	0	0
$573$	304128.	0.926290
$574$	0	0
$575$	− 395759.i	− 1.19700i
$576$	0	0
$577$	−270718.	−0.813140	−0.406570	−	0.913620i	$-0.633275\pi$
$577$	−270718.	−0.813140	−0.406570	+	0.913620i	$0.633275\pi$
$578$	0	0
$579$	− 405381.i	− 1.20922i
$580$	0	0
$581$	−731904.	−2.16821
$582$	0	0
$583$	− 212968.i	− 0.626582i
$584$	0	0
$585$	25920.0	0.0757396
$586$	0	0
$587$	− 273353.i	− 0.793319i	−0.917966	−	0.396660i	$-0.870169\pi$
$587$	− 273353.i	− 0.793319i	0.917966	−	0.396660i	$-0.129831\pi$
$588$	0	0
$589$	−276480.	−0.796954
$590$	0	0
$591$	− 662734.i	− 1.89742i
$592$	0	0
$593$	−62910.0	−0.178900	−0.0894500	−	0.995991i	$-0.528511\pi$
$593$	−62910.0	−0.178900	−0.0894500	+	0.995991i	$0.528511\pi$
$594$	0	0
$595$	− 101585.i	− 0.286944i
$596$	0	0
$597$	244992.	0.687390
$598$	0	0
$599$	− 261723.i	− 0.729438i	−0.931118	−	0.364719i	$-0.881165\pi$
$599$	− 261723.i	− 0.729438i	0.931118	−	0.364719i	$-0.118835\pi$
$600$	0	0
$601$	−203902.	−0.564511	−0.282256	−	0.959339i	$-0.591083\pi$
$601$	−203902.	−0.564511	−0.282256	+	0.959339i	$0.591083\pi$
$602$	0	0
$603$	− 25131.8i	− 0.0691176i
$604$	0	0
$605$	−24200.0	−0.0661157
$606$	0	0
$607$	− 369657.i	− 1.00328i	−0.865077	−	0.501640i	$-0.832730\pi$
$607$	− 369657.i	− 1.00328i	0.865077	−	0.501640i	$-0.167270\pi$
$608$	0	0
$609$	−1.00147e6	−2.70025
$610$	0	0
$611$	304756.i	0.816337i
$612$	0	0
$613$	222264.	0.591491	0.295746	−	0.955267i	$-0.404432\pi$
$613$	222264.	0.591491	0.295746	+	0.955267i	$0.404432\pi$
$614$	0	0
$615$	− 148145.i	− 0.391685i
$616$	0	0
$617$	−363582.	−0.955063	−0.477532	−	0.878615i	$-0.658468\pi$
$617$	−363582.	−0.955063	−0.477532	+	0.878615i	$0.658468\pi$
$618$	0	0
$619$	− 317718.i	− 0.829203i	−0.910003	−	0.414602i	$-0.863921\pi$
$619$	− 317718.i	− 0.829203i	0.910003	−	0.414602i	$-0.136079\pi$
$620$	0	0
$621$	456192.	1.18294
$622$	0	0
$623$	190472.i	0.490745i
$624$	0	0
$625$	274721.	0.703286
$626$	0	0
$627$	465599.i	1.18434i
$628$	0	0
$629$	244944.	0.619107
$630$	0	0
$631$	348415.i	0.875062i	0.899203	+	0.437531i	$0.144147\pi$
$631$	348415.i	0.875062i	−0.899203	+	0.437531i	$0.855853\pi$
$632$	0	0
$633$	−256608.	−0.640417
$634$	0	0
$635$	27591.1i	0.0684259i
$636$	0	0
$637$	−808488.	−1.99248
$638$	0	0
$639$	− 116400.i	− 0.285069i
$640$	0	0
$641$	67230.0	0.163624	0.0818120	−	0.996648i	$-0.473929\pi$
$641$	67230.0	0.163624	0.0818120	+	0.996648i	$0.473929\pi$
$642$	0	0
$643$	− 74513.5i	− 0.180224i	−0.995932	−	0.0901121i	$-0.971277\pi$
$643$	− 74513.5i	− 0.180224i	0.995932	−	0.0901121i	$-0.0287225\pi$
$644$	0	0
$645$	−228096.	−0.548275
$646$	0	0
$647$	380239.i	0.908340i	0.890915	+	0.454170i	$0.150064\pi$
$647$	380239.i	0.908340i	−0.890915	+	0.454170i	$0.849936\pi$
$648$	0	0
$649$	−243936.	−0.579144
$650$	0	0
$651$	481589.i	1.13636i
$652$	0	0
$653$	−89288.0	−0.209395	−0.104698	−	0.994504i	$-0.533387\pi$
$653$	−89288.0	−0.209395	−0.104698	+	0.994504i	$0.533387\pi$
$654$	0	0
$655$	− 76737.6i	− 0.178865i
$656$	0	0
$657$	41250.0	0.0955638
$658$	0	0
$659$	223149.i	0.513834i	0.966433	+	0.256917i	$0.0827068\pi$
$659$	223149.i	0.513834i	−0.966433	+	0.256917i	$0.917293\pi$
$660$	0	0
$661$	560088.	1.28190	0.640949	−	0.767584i	$-0.278541\pi$
$661$	560088.	1.28190	0.640949	+	0.767584i	$0.278541\pi$
$662$	0	0
$663$	342850.i	0.779969i
$664$	0	0
$665$	276480.	0.625202
$666$	0	0
$667$	919911.i	2.06773i
$668$	0	0
$669$	−202752.	−0.453015
$670$	0	0
$671$	256079.i	0.568761i
$672$	0	0
$673$	−810722.	−1.78995	−0.894977	−	0.446112i	$-0.852808\pi$
$673$	−810722.	−1.78995	−0.894977	+	0.446112i	$0.852808\pi$
$674$	0	0
$675$	362779.i	0.796223i
$676$	0	0
$677$	655768.	1.43078	0.715390	−	0.698725i	$-0.246249\pi$
$677$	655768.	1.43078	0.715390	+	0.698725i	$0.246249\pi$
$678$	0	0
$679$	− 584272.i	− 1.26729i
$680$	0	0
$681$	−763488.	−1.64630
$682$	0	0
$683$	− 189208.i	− 0.405601i	−0.979220	−	0.202800i	$-0.934996\pi$
$683$	− 189208.i	− 0.405601i	0.979220	−	0.202800i	$-0.0650043\pi$
$684$	0	0
$685$	−261360.	−0.557004
$686$	0	0
$687$	141796.i	0.300435i
$688$	0	0
$689$	426816.	0.899088
$690$	0	0
$691$	− 569389.i	− 1.19248i	−0.802804	−	0.596242i	$-0.796660\pi$
$691$	− 569389.i	− 1.19248i	0.802804	−	0.596242i	$-0.203340\pi$
$692$	0	0
$693$	126720.	0.263863
$694$	0	0
$695$	10581.8i	0.0219073i
$696$	0	0
$697$	306180.	0.630248
$698$	0	0
$699$	− 26983.6i	− 0.0552262i
$700$	0	0
$701$	519224.	1.05662	0.528310	−	0.849052i	$-0.322826\pi$
$701$	519224.	1.05662	0.528310	+	0.849052i	$0.322826\pi$
$702$	0	0
$703$	666653.i	1.34893i
$704$	0	0
$705$	−110592.	−0.222508
$706$	0	0
$707$	− 117262.i	− 0.234595i
$708$	0	0
$709$	919512.	1.82922	0.914608	−	0.404342i	$-0.132499\pi$
$709$	919512.	1.82922	0.914608	+	0.404342i	$0.132499\pi$
$710$	0	0
$711$	119927.i	0.237234i
$712$	0	0
$713$	442368.	0.870171
$714$	0	0
$715$	186240.i	0.364301i
$716$	0	0
$717$	−898560.	−1.74787
$718$	0	0
$719$	656071.i	1.26909i	0.772885	+	0.634546i	$0.218813\pi$
$719$	656071.i	1.26909i	−0.772885	+	0.634546i	$0.781187\pi$
$720$	0	0
$721$	−202752.	−0.390027
$722$	0	0
$723$	955987.i	1.82884i
$724$	0	0
$725$	−731544.	−1.39176
$726$	0	0
$727$	− 488722.i	− 0.924684i	−0.886702	−	0.462342i	$-0.847009\pi$
$727$	− 488722.i	− 0.924684i	0.886702	−	0.462342i	$-0.152991\pi$
$728$	0	0
$729$	−399951.	−0.752578
$730$	0	0
$731$	− 471419.i	− 0.882211i
$732$	0	0
$733$	−706536.	−1.31500	−0.657501	−	0.753454i	$-0.728386\pi$
$733$	−706536.	−1.31500	−0.657501	+	0.753454i	$0.728386\pi$
$734$	0	0
$735$	− 293390.i	− 0.543089i
$736$	0	0
$737$	180576.	0.332449
$738$	0	0
$739$	172748.i	0.316318i	0.987414	+	0.158159i	$0.0505558\pi$
$739$	172748.i	0.316318i	−0.987414	+	0.158159i	$0.949444\pi$
$740$	0	0
$741$	−933120.	−1.69942
$742$	0	0
$743$	250436.i	0.453648i	0.973936	+	0.226824i	$0.0728342\pi$
$743$	250436.i	0.453648i	−0.973936	+	0.226824i	$0.927166\pi$
$744$	0	0
$745$	6976.00	0.0125688
$746$	0	0
$747$	140062.i	0.251003i
$748$	0	0
$749$	83712.0	0.149219
$750$	0	0
$751$	− 536301.i	− 0.950887i	−0.879746	−	0.475443i	$-0.842288\pi$
$751$	− 536301.i	− 0.950887i	0.879746	−	0.475443i	$-0.157712\pi$
$752$	0	0
$753$	664416.	1.17179
$754$	0	0
$755$	− 75875.4i	− 0.133109i
$756$	0	0
$757$	470232.	0.820579	0.410290	−	0.911955i	$-0.365428\pi$
$757$	470232.	0.820579	0.410290	+	0.911955i	$0.365428\pi$
$758$	0	0
$759$	− 744958.i	− 1.29315i
$760$	0	0
$761$	691038.	1.19325	0.596627	−	0.802519i	$-0.296507\pi$
$761$	691038.	1.19325	0.596627	+	0.802519i	$0.296507\pi$
$762$	0	0
$763$	− 1.16823e6i	− 2.00669i
$764$	0	0
$765$	−19440.0	−0.0332180
$766$	0	0
$767$	− 488879.i	− 0.831018i
$768$	0	0
$769$	304030.	0.514119	0.257060	−	0.966396i	$-0.417246\pi$
$769$	304030.	0.514119	0.257060	+	0.966396i	$0.417246\pi$
$770$	0	0
$771$	− 788873.i	− 1.32708i
$772$	0	0
$773$	533720.	0.893212	0.446606	−	0.894731i	$-0.352633\pi$
$773$	533720.	0.893212	0.446606	+	0.894731i	$0.352633\pi$
$774$	0	0
$775$	351786.i	0.585700i
$776$	0	0
$777$	1.16122e6	1.92341
$778$	0	0
$779$	833316.i	1.37320i
$780$	0	0
$781$	836352.	1.37116
$782$	0	0
$783$	− 843252.i	− 1.37541i
$784$	0	0
$785$	−229824.	−0.372955
$786$	0	0
$787$	− 1.17467e6i	− 1.89656i	−0.317442	−	0.948278i	$-0.602824\pi$
$787$	− 1.17467e6i	− 1.89656i	0.317442	−	0.948278i	$-0.397176\pi$
$788$	0	0
$789$	−145152.	−0.233168
$790$	0	0
$791$	55025.3i	0.0879447i
$792$	0	0
$793$	−513216.	−0.816120
$794$	0	0
$795$	154886.i	0.245063i
$796$	0	0
$797$	552376.	0.869597	0.434799	−	0.900528i	$-0.356820\pi$
$797$	552376.	0.869597	0.434799	+	0.900528i	$0.356820\pi$
$798$	0	0
$799$	− 228567.i	− 0.358030i
$800$	0	0
$801$	36450.0	0.0568110
$802$	0	0
$803$	296388.i	0.459653i
$804$	0	0
$805$	−442368.	−0.682640
$806$	0	0
$807$	439654.i	0.675094i
$808$	0	0
$809$	−803682.	−1.22797	−0.613984	−	0.789318i	$-0.710434\pi$
$809$	−803682.	−1.22797	−0.613984	+	0.789318i	$0.710434\pi$
$810$	0	0
$811$	68869.9i	0.104710i	0.998629	+	0.0523549i	$0.0166727\pi$
$811$	68869.9i	0.104710i	−0.998629	+	0.0523549i	$0.983327\pi$
$812$	0	0
$813$	895488.	1.35481
$814$	0	0
$815$	249025.i	0.374910i
$816$	0	0
$817$	1.28304e6	1.92219
$818$	0	0
$819$	253963.i	0.378619i
$820$	0	0
$821$	−867592.	−1.28715	−0.643575	−	0.765383i	$-0.722550\pi$
$821$	−867592.	−1.28715	−0.643575	+	0.765383i	$0.722550\pi$
$822$	0	0
$823$	30177.7i	0.0445540i	0.999752	+	0.0222770i	$0.00709158\pi$
$823$	30177.7i	0.0445540i	−0.999752	+	0.0222770i	$0.992908\pi$
$824$	0	0
$825$	592416.	0.870400
$826$	0	0
$827$	− 864680.i	− 1.26428i	−0.774853	−	0.632141i	$-0.782176\pi$
$827$	− 864680.i	− 1.26428i	0.774853	−	0.632141i	$-0.217824\pi$
$828$	0	0
$829$	−449928.	−0.654687	−0.327344	−	0.944905i	$-0.606153\pi$
$829$	−449928.	−0.654687	−0.327344	+	0.944905i	$0.606153\pi$
$830$	0	0
$831$	886754.i	1.28411i
$832$	0	0
$833$	606366.	0.873866
$834$	0	0
$835$	434559.i	0.623269i
$836$	0	0
$837$	−405504.	−0.578821
$838$	0	0
$839$	800689.i	1.13747i	0.822521	+	0.568735i	$0.192567\pi$
$839$	800689.i	1.13747i	−0.822521	+	0.568735i	$0.807433\pi$
$840$	0	0
$841$	993135.	1.40416
$842$	0	0
$843$	226980.i	0.319398i
$844$	0	0
$845$	−144760.	−0.202738
$846$	0	0
$847$	− 237111.i	− 0.330510i
$848$	0	0
$849$	−693792.	−0.962529
$850$	0	0
$851$	− 1.06665e6i	− 1.47286i
$852$	0	0
$853$	−121608.	−0.167134	−0.0835669	−	0.996502i	$-0.526631\pi$
$853$	−121608.	−0.167134	−0.0835669	+	0.996502i	$0.526631\pi$
$854$	0	0
$855$	− 52909.0i	− 0.0723764i
$856$	0	0
$857$	1.26473e6	1.72202	0.861009	−	0.508590i	$-0.169833\pi$
$857$	1.26473e6	1.72202	0.861009	+	0.508590i	$0.169833\pi$
$858$	0	0
$859$	− 967088.i	− 1.31063i	−0.755356	−	0.655314i	$-0.772536\pi$
$859$	− 967088.i	− 1.31063i	0.755356	−	0.655314i	$-0.227464\pi$
$860$	0	0
$861$	1.45152e6	1.95802
$862$	0	0
$863$	428915.i	0.575904i	0.957645	+	0.287952i	$0.0929744\pi$
$863$	428915.i	0.575904i	−0.957645	+	0.287952i	$0.907026\pi$
$864$	0	0
$865$	202048.	0.270036
$866$	0	0
$867$	561198.i	0.746582i
$868$	0	0
$869$	−861696.	−1.14108
$870$	0	0
$871$	361897.i	0.477034i
$872$	0	0
$873$	−111810.	−0.146707
$874$	0	0
$875$	− 743704.i	− 0.971369i
$876$	0	0
$877$	−519048.	−0.674852	−0.337426	−	0.941352i	$-0.609556\pi$
$877$	−519048.	−0.674852	−0.337426	+	0.941352i	$0.609556\pi$
$878$	0	0
$879$	370049.i	0.478941i
$880$	0	0
$881$	434754.	0.560134	0.280067	−	0.959980i	$-0.409643\pi$
$881$	434754.	0.560134	0.280067	+	0.959980i	$0.409643\pi$
$882$	0	0
$883$	474329.i	0.608357i	0.952615	+	0.304178i	$0.0983819\pi$
$883$	474329.i	0.608357i	−0.952615	+	0.304178i	$0.901618\pi$
$884$	0	0
$885$	177408.	0.226510
$886$	0	0
$887$	− 298407.i	− 0.379281i	−0.981854	−	0.189641i	$-0.939268\pi$
$887$	− 298407.i	− 0.379281i	0.981854	−	0.189641i	$-0.0607323\pi$
$888$	0	0
$889$	−270336.	−0.342058
$890$	0	0
$891$	813828.i	1.02513i
$892$	0	0
$893$	622080.	0.780088
$894$	0	0
$895$	71564.3i	0.0893409i
$896$	0	0
$897$	1.49299e6	1.85555
$898$	0	0
$899$	− 817698.i	− 1.01175i
$900$	0	0
$901$	−320112.	−0.394323
$902$	0	0
$903$	− 2.23488e6i	− 2.74080i
$904$	0	0
$905$	288576.	0.352341
$906$	0	0
$907$	693549.i	0.843067i	0.906813	+	0.421534i	$0.138508\pi$
$907$	693549.i	0.843067i	−0.906813	+	0.421534i	$0.861492\pi$
$908$	0	0
$909$	−22440.0	−0.0271578
$910$	0	0
$911$	1.13296e6i	1.36514i	0.730821	+	0.682570i	$0.239138\pi$
$911$	1.13296e6i	1.36514i	−0.730821	+	0.682570i	$0.760862\pi$
$912$	0	0
$913$	−1.00637e6	−1.20730
$914$	0	0
$915$	− 186240.i	− 0.222449i
$916$	0	0
$917$	751872.	0.894139
$918$	0	0
$919$	526817.i	0.623776i	0.950119	+	0.311888i	$0.100961\pi$
$919$	526817.i	0.623776i	−0.950119	+	0.311888i	$0.899039\pi$
$920$	0	0
$921$	−377568.	−0.445119
$922$	0	0
$923$	1.67616e6i	1.96748i
$924$	0	0
$925$	848232.	0.991360
$926$	0	0
$927$	38799.9i	0.0451514i
$928$	0	0
$929$	−371682.	−0.430666	−0.215333	−	0.976541i	$-0.569084\pi$
$929$	−371682.	−0.430666	−0.215333	+	0.976541i	$0.569084\pi$
$930$	0	0
$931$	1.65032e6i	1.90401i
$932$	0	0
$933$	836352.	0.960784
$934$	0	0
$935$	− 139680.i	− 0.159775i
$936$	0	0
$937$	532418.	0.606420	0.303210	−	0.952924i	$-0.401942\pi$
$937$	532418.	0.606420	0.303210	+	0.952924i	$0.401942\pi$
$938$	0	0
$939$	917109.i	1.04013i
$940$	0	0
$941$	−614600.	−0.694086	−0.347043	−	0.937849i	$-0.612814\pi$
$941$	−614600.	−0.694086	−0.347043	+	0.937849i	$0.612814\pi$
$942$	0	0
$943$	− 1.33331e6i	− 1.49936i
$944$	0	0
$945$	405504.	0.454079
$946$	0	0
$947$	583988.i	0.651184i	0.945510	+	0.325592i	$0.105564\pi$
$947$	583988.i	0.651184i	−0.945510	+	0.325592i	$0.894436\pi$
$948$	0	0
$949$	−594000.	−0.659560
$950$	0	0
$951$	481981.i	0.532929i
$952$	0	0
$953$	−604962.	−0.666104	−0.333052	−	0.942908i	$-0.608079\pi$
$953$	−604962.	−0.666104	−0.333052	+	0.942908i	$0.608079\pi$
$954$	0	0
$955$	− 248319.i	− 0.272273i
$956$	0	0
$957$	−1.37702e6	−1.50355
$958$	0	0
$959$	− 2.56079e6i	− 2.78444i
$960$	0	0
$961$	530305.	0.574221
$962$	0	0
$963$	− 16019.7i	− 0.0172743i
$964$	0	0
$965$	−330992.	−0.355437
$966$	0	0
$967$	− 1.39405e6i	− 1.49082i	−0.666604	−	0.745412i	$-0.732253\pi$
$967$	− 1.39405e6i	− 1.49082i	0.666604	−	0.745412i	$-0.267747\pi$
$968$	0	0
$969$	699840.	0.745334
$970$	0	0
$971$	− 662175.i	− 0.702319i	−0.936316	−	0.351160i	$-0.885787\pi$
$971$	− 662175.i	− 0.702319i	0.936316	−	0.351160i	$-0.114213\pi$
$972$	0	0
$973$	−103680.	−0.109514
$974$	0	0
$975$	1.18728e6i	1.24894i
$976$	0	0
$977$	−815778.	−0.854639	−0.427320	−	0.904101i	$-0.640542\pi$
$977$	−815778.	−0.854639	−0.427320	+	0.904101i	$0.640542\pi$
$978$	0	0
$979$	261899.i	0.273256i
$980$	0	0
$981$	−223560.	−0.232304
$982$	0	0
$983$	1.23384e6i	1.27688i	0.769671	+	0.638441i	$0.220420\pi$
$983$	1.23384e6i	1.27688i	−0.769671	+	0.638441i	$0.779580\pi$
$984$	0	0
$985$	−541120.	−0.557726
$986$	0	0
$987$	− 1.08358e6i	− 1.11231i
$988$	0	0
$989$	−2.05286e6	−2.09878
$990$	0	0
$991$	417315.i	0.424929i	0.977169	+	0.212464i	$0.0681490\pi$
$991$	417315.i	0.424929i	−0.977169	+	0.212464i	$0.931851\pi$
$992$	0	0
$993$	−161568.	−0.163854
$994$	0	0
$995$	− 200035.i	− 0.202051i
$996$	0	0
$997$	1.80252e6	1.81338	0.906692	−	0.421793i	$-0.138599\pi$
$997$	1.80252e6	1.81338	0.906692	+	0.421793i	$0.138599\pi$
$998$	0	0
$999$	977758.i	0.979716i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.5.c.c.255.1		2
4.3	odd	2	inner	256.5.c.c.255.2		2
8.3	odd	2		256.5.c.f.255.1		2
8.5	even	2		256.5.c.f.255.2		2
16.3	odd	4		128.5.d.c.63.2	yes	4
16.5	even	4		128.5.d.c.63.1	✓	4
16.11	odd	4		128.5.d.c.63.3	yes	4
16.13	even	4		128.5.d.c.63.4	yes	4
48.5	odd	4		1152.5.b.i.703.4		4
48.11	even	4		1152.5.b.i.703.3		4
48.29	odd	4		1152.5.b.i.703.2		4
48.35	even	4		1152.5.b.i.703.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.5.d.c.63.1	✓	4	16.5	even	4
128.5.d.c.63.2	yes	4	16.3	odd	4
128.5.d.c.63.3	yes	4	16.11	odd	4
128.5.d.c.63.4	yes	4	16.13	even	4
256.5.c.c.255.1		2	1.1	even	1	trivial
256.5.c.c.255.2		2	4.3	odd	2	inner
256.5.c.f.255.1		2	8.3	odd	2
256.5.c.f.255.2		2	8.5	even	2
1152.5.b.i.703.1		4	48.35	even	4
1152.5.b.i.703.2		4	48.29	odd	4
1152.5.b.i.703.3		4	48.11	even	4
1152.5.b.i.703.4		4	48.5	odd	4

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:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	256.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-9.79796i q^{3} -8.00000 q^{5} -78.3837i q^{7} -15.0000 q^{9} +O(q^{10})\)	\(q-9.79796i q^{3} -8.00000 q^{5} -78.3837i q^{7} -15.0000 q^{9} -107.778i q^{11} +216.000 q^{13} +78.3837i q^{15} -162.000 q^{17} -440.908i q^{19} -768.000 q^{21} +705.453i q^{23} -561.000 q^{25} -646.665i q^{27} +1304.00 q^{29} -627.069i q^{31} -1056.00 q^{33} +627.069i q^{35} -1512.00 q^{37} -2116.36i q^{39} -1890.00 q^{41} +2909.99i q^{43} +120.000 q^{45} +1410.91i q^{47} -3743.00 q^{49} +1587.27i q^{51} +1976.00 q^{53} +862.220i q^{55} -4320.00 q^{57} -2263.33i q^{59} -2376.00 q^{61} +1175.76i q^{63} -1728.00 q^{65} +1675.45i q^{67} +6912.00 q^{69} +7759.98i q^{71} -2750.00 q^{73} +5496.65i q^{75} -8448.00 q^{77} -7995.13i q^{79} -7551.00 q^{81} -9337.45i q^{83} +1296.00 q^{85} -12776.5i q^{87} -2430.00 q^{89} -16930.9i q^{91} -6144.00 q^{93} +3527.27i q^{95} +7454.00 q^{97} +1616.66i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 16 q^{5} - 30 q^{9}+O(q^{10}) \) 2 * q - 16 * q^5 - 30 * q^9	\( 2 q - 16 q^{5} - 30 q^{9} + 432 q^{13} - 324 q^{17} - 1536 q^{21} - 1122 q^{25} + 2608 q^{29} - 2112 q^{33} - 3024 q^{37} - 3780 q^{41} + 240 q^{45} - 7486 q^{49} + 3952 q^{53} - 8640 q^{57} - 4752 q^{61} - 3456 q^{65} + 13824 q^{69} - 5500 q^{73} - 16896 q^{77} - 15102 q^{81} + 2592 q^{85} - 4860 q^{89} - 12288 q^{93} + 14908 q^{97}+O(q^{100}) \) 2 * q - 16 * q^5 - 30 * q^9 + 432 * q^13 - 324 * q^17 - 1536 * q^21 - 1122 * q^25 + 2608 * q^29 - 2112 * q^33 - 3024 * q^37 - 3780 * q^41 + 240 * q^45 - 7486 * q^49 + 3952 * q^53 - 8640 * q^57 - 4752 * q^61 - 3456 * q^65 + 13824 * q^69 - 5500 * q^73 - 16896 * q^77 - 15102 * q^81 + 2592 * q^85 - 4860 * q^89 - 12288 * q^93 + 14908 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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