LMFDB - Embedded newform 304.5.e.f.113.19

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [304,5,Mod(113,304)] 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("304.113"); S:= CuspForms(chi, 5); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(304, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 5, names="a")

Level:	$ N $	$=$	$ 304 = 2^{4} \cdot 19 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	304.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [20,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$31.4244687775$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 996 x^{18} + 408854 x^{16} + 89661524 x^{14} + 11414409521 x^{12} + 861580608848 x^{10} + \cdots + 34\!\cdots\!64 $ x^20 + 996x^18 + 408854x^16 + 89661524x^14 + 11414409521x^12 + 861580608848x^10 + 37911701888064x^8 + 915101881952256x^6 + 10365057927217152x^4 + 34336742150504448*x^2 + 34273490123096064
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{50} $
Twist minimal:	no (minimal twist has level 152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			113.19
Root			$14.8343i$ of defining polynomial
Character	$\chi$	$=$	304.113
Dual form			304.5.e.f.113.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+14.8343i q^{3} -45.4605 q^{5} +2.86483 q^{7} -139.057 q^{9} -100.724 q^{11} +53.2006i q^{13} -674.375i q^{15} -145.257 q^{17} +(336.954 + 129.548i) q^{19} +42.4977i q^{21} -828.203 q^{23} +1441.65 q^{25} -861.237i q^{27} +152.947i q^{29} -1072.35i q^{31} -1494.17i q^{33} -130.236 q^{35} -1201.35i q^{37} -789.195 q^{39} +3018.54i q^{41} +2469.37 q^{43} +6321.60 q^{45} +2268.18 q^{47} -2392.79 q^{49} -2154.79i q^{51} +4530.72i q^{53} +4578.96 q^{55} +(-1921.76 + 4998.49i) q^{57} -1580.60i q^{59} +2849.06 q^{61} -398.374 q^{63} -2418.52i q^{65} +6810.56i q^{67} -12285.8i q^{69} -6602.83i q^{71} -4523.72 q^{73} +21386.0i q^{75} -288.557 q^{77} -1691.70i q^{79} +1512.24 q^{81} +6627.75 q^{83} +6603.46 q^{85} -2268.87 q^{87} +1262.43i q^{89} +152.410i q^{91} +15907.6 q^{93} +(-15318.1 - 5889.33i) q^{95} -15824.6i q^{97} +14006.4 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 32 q^{7} - 372 q^{9} + 24 q^{11} + 216 q^{17} + 596 q^{19} - 576 q^{23} + 1412 q^{25} + 144 q^{35} + 520 q^{39} + 1256 q^{43} + 7232 q^{45} + 3768 q^{47} - 2740 q^{49} + 10128 q^{55} - 728 q^{57}+ \cdots - 760 q^{99}+O(q^{100}) $ 20 * q - 32 * q^7 - 372 * q^9 + 24 * q^11 + 216 * q^17 + 596 * q^19 - 576 * q^23 + 1412 * q^25 + 144 * q^35 + 520 * q^39 + 1256 * q^43 + 7232 * q^45 + 3768 * q^47 - 2740 * q^49 + 10128 * q^55 - 728 * q^57 + 352 * q^61 - 6104 * q^63 + 1352 * q^73 + 9288 * q^77 - 4220 * q^81 + 16104 * q^83 + 10232 * q^85 - 2936 * q^87 + 36432 * q^93 - 14232 * q^95 - 760 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/304\mathbb{Z}\right)^\times$.

$n$	$97$	$191$	$229$
$\chi(n)$	$-1$	$1$	$1$

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
$3$			14.8343i			1.64826i	0.566402	+	0.824129i	$0.308335\pi$
$3$			14.8343i			1.64826i	−0.566402	+	0.824129i	$0.691665\pi$
$4$		0			0
$5$	−45.4605			−1.81842			−0.909209	−	0.416339i	$-0.863313\pi$
$5$	−45.4605			−1.81842			−0.909209	+	0.416339i	$0.863313\pi$
$6$		0			0
$7$	2.86483			0.0584658			0.0292329	−	0.999573i	$-0.490694\pi$
$7$	2.86483			0.0584658			0.0292329	+	0.999573i	$0.490694\pi$
$8$		0			0
$9$	−139.057			−1.71675
$10$		0			0
$11$	−100.724			−0.832430			−0.416215	−	0.909266i	$-0.636644\pi$
$11$	−100.724			−0.832430			−0.416215	+	0.909266i	$0.636644\pi$
$12$		0			0
$13$			53.2006i			0.314797i	0.987535	+	0.157398i	$0.0503106\pi$
$13$			53.2006i			0.314797i	−0.987535	+	0.157398i	$0.949689\pi$
$14$		0			0
$15$		−	674.375i		−	2.99722i
$16$		0			0
$17$	−145.257			−0.502620			−0.251310	−	0.967907i	$-0.580861\pi$
$17$	−145.257			−0.502620			−0.251310	+	0.967907i	$0.580861\pi$
$18$		0			0
$19$	336.954	+	129.548i	0.933391	+	0.358860i
$19$	336.954	+	129.548i	0.933391	+	0.358860i
$20$		0			0
$21$			42.4977i			0.0963668i
$22$		0			0
$23$	−828.203			−1.56560			−0.782801	−	0.622272i	$-0.786210\pi$
$23$	−828.203			−1.56560			−0.782801	+	0.622272i	$0.786210\pi$
$24$		0			0
$25$	1441.65			2.30665
$26$		0			0
$27$		−	861.237i		−	1.18139i
$28$		0			0
$29$			152.947i			0.181864i	0.995857	+	0.0909319i	$0.0289846\pi$
$29$			152.947i			0.181864i	−0.995857	+	0.0909319i	$0.971015\pi$
$30$		0			0
$31$		−	1072.35i		−	1.11587i	−0.829885	−	0.557935i	$-0.811594\pi$
$31$		−	1072.35i		−	1.11587i	0.829885	−	0.557935i	$-0.188406\pi$
$32$		0			0
$33$		−	1494.17i		−	1.37206i
$34$		0			0
$35$	−130.236			−0.106315
$36$		0			0
$37$		−	1201.35i		−	0.877537i	−0.898600	−	0.438769i	$-0.855415\pi$
$37$		−	1201.35i		−	0.877537i	0.898600	−	0.438769i	$-0.144585\pi$
$38$		0			0
$39$	−789.195			−0.518866
$40$		0			0
$41$			3018.54i			1.79568i	0.440318	+	0.897842i	$0.354866\pi$
$41$			3018.54i			1.79568i	−0.440318	+	0.897842i	$0.645134\pi$
$42$		0			0
$43$	2469.37			1.33551			0.667757	−	0.744379i	$-0.267255\pi$
$43$	2469.37			1.33551			0.667757	+	0.744379i	$0.267255\pi$
$44$		0			0
$45$	6321.60			3.12178
$46$		0			0
$47$	2268.18			1.02679			0.513395	−	0.858153i	$-0.328388\pi$
$47$	2268.18			1.02679			0.513395	+	0.858153i	$0.328388\pi$
$48$		0			0
$49$	−2392.79			−0.996582
$50$		0			0
$51$		−	2154.79i		−	0.828447i
$52$		0			0
$53$			4530.72i			1.61293i	0.591282	+	0.806465i	$0.298622\pi$
$53$			4530.72i			1.61293i	−0.591282	+	0.806465i	$0.701378\pi$
$54$		0			0
$55$	4578.96			1.51371
$56$		0			0
$57$	−1921.76	+	4998.49i	−0.591494	+	1.53847i
$58$		0			0
$59$		−	1580.60i		−	0.454064i	−0.973887	−	0.227032i	$-0.927098\pi$
$59$		−	1580.60i		−	0.454064i	0.973887	−	0.227032i	$-0.0729022\pi$
$60$		0			0
$61$	2849.06			0.765671			0.382836	−	0.923816i	$-0.374948\pi$
$61$	2849.06			0.765671			0.382836	+	0.923816i	$0.374948\pi$
$62$		0			0
$63$	−398.374			−0.100371
$64$		0			0
$65$		−	2418.52i		−	0.572432i
$66$		0			0
$67$			6810.56i			1.51717i	0.651576	+	0.758584i	$0.274108\pi$
$67$			6810.56i			1.51717i	−0.651576	+	0.758584i	$0.725892\pi$
$68$		0			0
$69$		−	12285.8i		−	2.58052i
$70$		0			0
$71$		−	6602.83i		−	1.30982i	−0.755705	−	0.654912i	$-0.772706\pi$
$71$		−	6602.83i		−	1.30982i	0.755705	−	0.654912i	$-0.227294\pi$
$72$		0			0
$73$	−4523.72			−0.848888			−0.424444	−	0.905454i	$-0.639530\pi$
$73$	−4523.72			−0.848888			−0.424444	+	0.905454i	$0.639530\pi$
$74$		0			0
$75$			21386.0i			3.80195i
$76$		0			0
$77$	−288.557			−0.0486687
$78$		0			0
$79$		−	1691.70i		−	0.271063i	−0.990773	−	0.135531i	$-0.956726\pi$
$79$		−	1691.70i		−	0.271063i	0.990773	−	0.135531i	$-0.0432742\pi$
$80$		0			0
$81$	1512.24			0.230489
$82$		0			0
$83$	6627.75			0.962077			0.481039	−	0.876699i	$-0.340260\pi$
$83$	6627.75			0.962077			0.481039	+	0.876699i	$0.340260\pi$
$84$		0			0
$85$	6603.46			0.913973
$86$		0			0
$87$	−2268.87			−0.299758
$88$		0			0
$89$			1262.43i			0.159377i	0.996820	+	0.0796886i	$0.0253926\pi$
$89$			1262.43i			0.159377i	−0.996820	+	0.0796886i	$0.974607\pi$
$90$		0			0
$91$			152.410i			0.0184048i
$92$		0			0
$93$	15907.6			1.83924
$94$		0			0
$95$	−15318.1	−	5889.33i	−1.69730	−	0.652558i
$96$		0			0
$97$		−	15824.6i		−	1.68185i	−0.541149	−	0.840927i	$-0.682010\pi$
$97$		−	15824.6i		−	1.68185i	0.541149	−	0.840927i	$-0.317990\pi$
$98$		0			0
$99$	14006.4			1.42908
$100$		0			0
$101$	−1728.80			−0.169474			−0.0847370	−	0.996403i	$-0.527005\pi$
$101$	−1728.80			−0.169474			−0.0847370	+	0.996403i	$0.527005\pi$
$102$		0			0
$103$		−	18702.4i		−	1.76288i	−0.472299	−	0.881439i	$-0.656576\pi$
$103$		−	18702.4i		−	1.76288i	0.472299	−	0.881439i	$-0.343424\pi$
$104$		0			0
$105$		−	1931.97i		−	0.175235i
$106$		0			0
$107$		−	5713.61i		−	0.499049i	−0.968368	−	0.249524i	$-0.919726\pi$
$107$		−	5713.61i		−	0.499049i	0.968368	−	0.249524i	$-0.0802743\pi$
$108$		0			0
$109$		−	13528.3i		−	1.13865i	−0.822112	−	0.569326i	$-0.807204\pi$
$109$		−	13528.3i		−	1.13865i	0.822112	−	0.569326i	$-0.192796\pi$
$110$		0			0
$111$	17821.2			1.44641
$112$		0			0
$113$		−	14367.9i		−	1.12522i	−0.826723	−	0.562609i	$-0.809798\pi$
$113$		−	14367.9i		−	1.12522i	0.826723	−	0.562609i	$-0.190202\pi$
$114$		0			0
$115$	37650.5			2.84692
$116$		0			0
$117$		−	7397.92i		−	0.540428i
$118$		0			0
$119$	−416.136			−0.0293861
$120$		0			0
$121$	−4495.66			−0.307060
$122$		0			0
$123$	−44778.1			−2.95975
$124$		0			0
$125$	−37125.5			−2.37603
$126$		0			0
$127$		−	3564.00i		−	0.220968i	−0.993878	−	0.110484i	$-0.964760\pi$
$127$		−	3564.00i		−	0.220968i	0.993878	−	0.110484i	$-0.0352401\pi$
$128$		0			0
$129$			36631.4i			2.20127i
$130$		0			0
$131$	−7223.20			−0.420908			−0.210454	−	0.977604i	$-0.567494\pi$
$131$	−7223.20			−0.420908			−0.210454	+	0.977604i	$0.567494\pi$
$132$		0			0
$133$	965.315	+	371.134i	0.0545715	+	0.0209810i
$134$		0			0
$135$			39152.2i			2.14827i
$136$		0			0
$137$	−7261.51			−0.386888			−0.193444	−	0.981111i	$-0.561966\pi$
$137$	−7261.51			−0.386888			−0.193444	+	0.981111i	$0.561966\pi$
$138$		0			0
$139$	−7455.27			−0.385864			−0.192932	−	0.981212i	$-0.561800\pi$
$139$	−7455.27			−0.385864			−0.192932	+	0.981212i	$0.561800\pi$
$140$		0			0
$141$			33646.9i			1.69241i
$142$		0			0
$143$		−	5358.58i		−	0.262046i
$144$		0			0
$145$		−	6953.06i		−	0.330705i
$146$		0			0
$147$		−	35495.5i		−	1.64262i
$148$		0			0
$149$	−15653.0			−0.705058			−0.352529	−	0.935801i	$-0.614678\pi$
$149$	−15653.0			−0.705058			−0.352529	+	0.935801i	$0.614678\pi$
$150$		0			0
$151$		−	15060.9i		−	0.660536i	−0.943887	−	0.330268i	$-0.892861\pi$
$151$		−	15060.9i		−	0.660536i	0.943887	−	0.330268i	$-0.107139\pi$
$152$		0			0
$153$	20199.0			0.862874
$154$		0			0
$155$			48749.6i			2.02912i
$156$		0			0
$157$	−9569.26			−0.388221			−0.194110	−	0.980980i	$-0.562182\pi$
$157$	−9569.26			−0.388221			−0.194110	+	0.980980i	$0.562182\pi$
$158$		0			0
$159$	−67210.1			−2.65852
$160$		0			0
$161$	−2372.66			−0.0915342
$162$		0			0
$163$	−11904.1			−0.448045			−0.224022	−	0.974584i	$-0.571919\pi$
$163$	−11904.1			−0.448045			−0.224022	+	0.974584i	$0.571919\pi$
$164$		0			0
$165$			67925.8i			2.49498i
$166$		0			0
$167$		−	19033.2i		−	0.682461i	−0.939980	−	0.341231i	$-0.889156\pi$
$167$		−	19033.2i		−	0.682461i	0.939980	−	0.341231i	$-0.110844\pi$
$168$		0			0
$169$	25730.7			0.900903
$170$		0			0
$171$	−46855.9	−	18014.6i	−1.60240	−	0.616074i
$172$		0			0
$173$		−	27328.5i		−	0.913112i	−0.889695	−	0.456556i	$-0.849083\pi$
$173$		−	27328.5i		−	0.913112i	0.889695	−	0.456556i	$-0.150917\pi$
$174$		0			0
$175$	4130.09			0.134860
$176$		0			0
$177$	23447.1			0.748414
$178$		0			0
$179$			3468.70i			0.108258i	0.998534	+	0.0541291i	$0.0172383\pi$
$179$			3468.70i			0.108258i	−0.998534	+	0.0541291i	$0.982762\pi$
$180$		0			0
$181$			42357.6i			1.29293i	0.762945	+	0.646464i	$0.223753\pi$
$181$			42357.6i			1.29293i	−0.762945	+	0.646464i	$0.776247\pi$
$182$		0			0
$183$			42263.9i			1.26202i
$184$		0			0
$185$			54613.9i			1.59573i
$186$		0			0
$187$	14630.9			0.418396
$188$		0			0
$189$		−	2467.29i		−	0.0690712i
$190$		0			0
$191$	23795.4			0.652270			0.326135	−	0.945323i	$-0.394254\pi$
$191$	23795.4			0.652270			0.326135	+	0.945323i	$0.394254\pi$
$192$		0			0
$193$			36775.1i			0.987277i	0.869667	+	0.493639i	$0.164333\pi$
$193$			36775.1i			0.987277i	−0.869667	+	0.493639i	$0.835667\pi$
$194$		0			0
$195$	35877.2			0.943515
$196$		0			0
$197$	67966.3			1.75130			0.875651	−	0.482944i	$-0.160433\pi$
$197$	67966.3			1.75130			0.875651	+	0.482944i	$0.160433\pi$
$198$		0			0
$199$	9109.02			0.230020			0.115010	−	0.993364i	$-0.463310\pi$
$199$	9109.02			0.230020			0.115010	+	0.993364i	$0.463310\pi$
$200$		0			0
$201$	−101030.			−2.50068
$202$		0			0
$203$			438.168i			0.0106328i
$204$		0			0
$205$		−	137224.i		−	3.26531i
$206$		0			0
$207$	115168.			2.68775
$208$		0			0
$209$	−33939.4	−	13048.6i	−0.776983	−	0.298726i
$210$		0			0
$211$			71608.1i			1.60841i	0.594351	+	0.804206i	$0.297409\pi$
$211$			71608.1i			1.60841i	−0.594351	+	0.804206i	$0.702591\pi$
$212$		0			0
$213$	97948.4			2.15893
$214$		0			0
$215$	−112259.			−2.42852
$216$		0			0
$217$		−	3072.10i		−	0.0652403i
$218$		0			0
$219$		−	67106.4i		−	1.39919i
$220$		0			0
$221$		−	7727.77i		−	0.158223i
$222$		0			0
$223$			30553.2i			0.614393i	0.951646	+	0.307197i	$0.0993909\pi$
$223$			30553.2i			0.614393i	−0.951646	+	0.307197i	$0.900609\pi$
$224$		0			0
$225$	−200472.			−3.95994
$226$		0			0
$227$		−	95469.6i		−	1.85274i	−0.376620	−	0.926368i	$-0.622914\pi$
$227$		−	95469.6i		−	1.85274i	0.376620	−	0.926368i	$-0.377086\pi$
$228$		0			0
$229$	−84302.1			−1.60756			−0.803781	−	0.594926i	$-0.797181\pi$
$229$	−84302.1			−1.60756			−0.803781	+	0.594926i	$0.797181\pi$
$230$		0			0
$231$		−	4280.55i		−	0.0802186i
$232$		0			0
$233$	−55650.0			−1.02507			−0.512534	−	0.858667i	$-0.671293\pi$
$233$	−55650.0			−1.02507			−0.512534	+	0.858667i	$0.671293\pi$
$234$		0			0
$235$	−103112.			−1.86713
$236$		0			0
$237$	25095.3			0.446782
$238$		0			0
$239$	36259.9			0.634791			0.317395	−	0.948293i	$-0.397192\pi$
$239$	36259.9			0.634791			0.317395	+	0.948293i	$0.397192\pi$
$240$		0			0
$241$		−	37725.4i		−	0.649531i	−0.945795	−	0.324766i	$-0.894715\pi$
$241$		−	37725.4i		−	0.649531i	0.945795	−	0.324766i	$-0.105285\pi$
$242$		0			0
$243$		−	47327.1i		−	0.801489i
$244$		0			0
$245$	108777.			1.81220
$246$		0			0
$247$	−6892.06	+	17926.2i	−0.112968	+	0.293828i
$248$		0			0
$249$			98318.2i			1.58575i
$250$		0			0
$251$	47956.5			0.761203			0.380601	−	0.924739i	$-0.375717\pi$
$251$	47956.5			0.761203			0.380601	+	0.924739i	$0.375717\pi$
$252$		0			0
$253$	83420.0			1.30325
$254$		0			0
$255$			97957.8i			1.50646i
$256$		0			0
$257$			2340.34i			0.0354334i	0.999843	+	0.0177167i	$0.00563969\pi$
$257$			2340.34i			0.0354334i	−0.999843	+	0.0177167i	$0.994360\pi$
$258$		0			0
$259$		−	3441.65i		−	0.0513060i
$260$		0			0
$261$		−	21268.4i		−	0.312215i
$262$		0			0
$263$	−96120.0			−1.38964			−0.694820	−	0.719183i	$-0.744516\pi$
$263$	−96120.0			−1.38964			−0.694820	+	0.719183i	$0.744516\pi$
$264$		0			0
$265$		−	205969.i		−	2.93298i
$266$		0			0
$267$	−18727.2			−0.262695
$268$		0			0
$269$		−	44228.9i		−	0.611226i	−0.952156	−	0.305613i	$-0.901139\pi$
$269$		−	44228.9i		−	0.611226i	0.952156	−	0.305613i	$-0.0988614\pi$
$270$		0			0
$271$	121267.			1.65122			0.825610	−	0.564241i	$-0.190831\pi$
$271$	121267.			1.65122			0.825610	+	0.564241i	$0.190831\pi$
$272$		0			0
$273$	−2260.91			−0.0303359
$274$		0			0
$275$	−145209.			−1.92012
$276$		0			0
$277$	−53311.7			−0.694805			−0.347402	−	0.937716i	$-0.612936\pi$
$277$	−53311.7			−0.694805			−0.347402	+	0.937716i	$0.612936\pi$
$278$		0			0
$279$			149118.i			1.91567i
$280$		0			0
$281$			91196.0i			1.15495i	0.816408	+	0.577475i	$0.195962\pi$
$281$			91196.0i			1.15495i	−0.816408	+	0.577475i	$0.804038\pi$
$282$		0			0
$283$	−19894.3			−0.248403			−0.124201	−	0.992257i	$-0.539637\pi$
$283$	−19894.3			−0.248403			−0.124201	+	0.992257i	$0.539637\pi$
$284$		0			0
$285$	87364.3	−	227234.i	1.07558	−	2.79758i
$286$		0			0
$287$			8647.60i			0.104986i
$288$		0			0
$289$	−62421.4			−0.747373
$290$		0			0
$291$	234747.			2.77213
$292$		0			0
$293$		−	55158.5i		−	0.642505i	−0.946993	−	0.321253i	$-0.895896\pi$
$293$		−	55158.5i		−	0.642505i	0.946993	−	0.321253i	$-0.104104\pi$
$294$		0			0
$295$			71854.7i			0.825678i
$296$		0			0
$297$			86747.3i			0.983429i
$298$		0			0
$299$		−	44060.9i		−	0.492846i
$300$		0			0
$301$	7074.30			0.0780819
$302$		0			0
$303$		−	25645.6i		−	0.279337i
$304$		0			0
$305$	−129520.			−1.39231
$306$		0			0
$307$		−	124990.i		−	1.32617i	−0.748545	−	0.663084i	$-0.769247\pi$
$307$		−	124990.i		−	1.32617i	0.748545	−	0.663084i	$-0.230753\pi$
$308$		0			0
$309$	277437.			2.90568
$310$		0			0
$311$	40835.5			0.422199			0.211099	−	0.977465i	$-0.432296\pi$
$311$	40835.5			0.422199			0.211099	+	0.977465i	$0.432296\pi$
$312$		0			0
$313$	99556.2			1.01620			0.508100	−	0.861298i	$-0.330348\pi$
$313$	99556.2			1.01620			0.508100	+	0.861298i	$0.330348\pi$
$314$		0			0
$315$	18110.3			0.182517
$316$		0			0
$317$			95876.3i			0.954097i	0.878877	+	0.477049i	$0.158293\pi$
$317$			95876.3i			0.954097i	−0.878877	+	0.477049i	$0.841707\pi$
$318$		0			0
$319$		−	15405.5i		−	0.151389i
$320$		0			0
$321$	84757.5			0.822561
$322$		0			0
$323$	−48945.0	−	18817.8i	−0.469141	−	0.180370i
$324$		0			0
$325$			76696.9i			0.726124i
$326$		0			0
$327$	200683.			1.87679
$328$		0			0
$329$	6497.94			0.0600321
$330$		0			0
$331$			143098.i			1.30610i	0.757315	+	0.653050i	$0.226511\pi$
$331$			143098.i			1.30610i	−0.757315	+	0.653050i	$0.773489\pi$
$332$		0			0
$333$			167056.i			1.50652i
$334$		0			0
$335$		−	309611.i		−	2.75885i
$336$		0			0
$337$		−	218865.i		−	1.92715i	−0.267433	−	0.963576i	$-0.586175\pi$
$337$		−	218865.i		−	1.92715i	0.267433	−	0.963576i	$-0.413825\pi$
$338$		0			0
$339$	213138.			1.85465
$340$		0			0
$341$			108012.i			0.928884i
$342$		0			0
$343$	−13733.4			−0.116732
$344$		0			0
$345$			558520.i			4.69246i
$346$		0			0
$347$	121057.			1.00538			0.502692	−	0.864466i	$-0.332343\pi$
$347$	121057.			1.00538			0.502692	+	0.864466i	$0.332343\pi$
$348$		0			0
$349$	−232555.			−1.90931			−0.954653	−	0.297722i	$-0.903773\pi$
$349$	−232555.			−1.90931			−0.954653	+	0.297722i	$0.903773\pi$
$350$		0			0
$351$	45818.3			0.371899
$352$		0			0
$353$	142311.			1.14206			0.571030	−	0.820929i	$-0.306544\pi$
$353$	142311.			1.14206			0.571030	+	0.820929i	$0.306544\pi$
$354$		0			0
$355$			300168.i			2.38181i
$356$		0			0
$357$		−	6173.10i		−	0.0484358i
$358$		0			0
$359$	−74343.0			−0.576834			−0.288417	−	0.957505i	$-0.593129\pi$
$359$	−74343.0			−0.576834			−0.288417	+	0.957505i	$0.593129\pi$
$360$		0			0
$361$	96755.4	+	87303.8i	0.742439	+	0.669914i
$362$		0			0
$363$		−	66690.1i		−	0.506114i
$364$		0			0
$365$	205651.			1.54363
$366$		0			0
$367$	−84813.8			−0.629701			−0.314850	−	0.949141i	$-0.601954\pi$
$367$	−84813.8			−0.629701			−0.314850	+	0.949141i	$0.601954\pi$
$368$		0			0
$369$		−	419750.i		−	3.08275i
$370$		0			0
$371$			12979.7i			0.0943012i
$372$		0			0
$373$		−	120963.i		−	0.869433i	−0.900567	−	0.434717i	$-0.856849\pi$
$373$		−	120963.i		−	0.869433i	0.900567	−	0.434717i	$-0.143151\pi$
$374$		0			0
$375$		−	550731.i		−	3.91631i
$376$		0			0
$377$	−8136.90			−0.0572501
$378$		0			0
$379$		−	25901.6i		−	0.180322i	−0.995927	−	0.0901608i	$-0.971262\pi$
$379$		−	25901.6i		−	0.180322i	0.995927	−	0.0901608i	$-0.0287381\pi$
$380$		0			0
$381$	52869.4			0.364212
$382$		0			0
$383$			118419.i			0.807280i	0.914918	+	0.403640i	$0.132255\pi$
$383$			118419.i			0.807280i	−0.914918	+	0.403640i	$0.867745\pi$
$384$		0			0
$385$	13117.9			0.0885001
$386$		0			0
$387$	−343383.			−2.29275
$388$		0			0
$389$	45823.7			0.302824			0.151412	−	0.988471i	$-0.451618\pi$
$389$	45823.7			0.302824			0.151412	+	0.988471i	$0.451618\pi$
$390$		0			0
$391$	120302.			0.786902
$392$		0			0
$393$		−	107151.i		−	0.693765i
$394$		0			0
$395$			76905.6i			0.492906i
$396$		0			0
$397$	−112423.			−0.713304			−0.356652	−	0.934237i	$-0.616082\pi$
$397$	−112423.			−0.713304			−0.356652	+	0.934237i	$0.616082\pi$
$398$		0			0
$399$	−5505.52	+	14319.8i	−0.0345822	+	0.0899479i
$400$		0			0
$401$			155180.i			0.965043i	0.875884	+	0.482522i	$0.160279\pi$
$401$			155180.i			0.965043i	−0.875884	+	0.482522i	$0.839721\pi$
$402$		0			0
$403$	57049.7			0.351272
$404$		0			0
$405$	−68747.2			−0.419126
$406$		0			0
$407$			121005.i			0.730489i
$408$		0			0
$409$			2674.57i			0.0159885i	0.999968	+	0.00799424i	$0.00254467\pi$
$409$			2674.57i			0.0159885i	−0.999968	+	0.00799424i	$0.997455\pi$
$410$		0			0
$411$		−	107720.i		−	0.637692i
$412$		0			0
$413$		−	4528.13i		−	0.0265472i
$414$		0			0
$415$	−301301.			−1.74946
$416$		0			0
$417$		−	110594.i		−	0.636003i
$418$		0			0
$419$	−47400.1			−0.269992			−0.134996	−	0.990846i	$-0.543102\pi$
$419$	−47400.1			−0.269992			−0.134996	+	0.990846i	$0.543102\pi$
$420$		0			0
$421$			101396.i			0.572081i	0.958218	+	0.286040i	$0.0923391\pi$
$421$			101396.i			0.572081i	−0.958218	+	0.286040i	$0.907661\pi$
$422$		0			0
$423$	−315406.			−1.76274
$424$		0			0
$425$	−209411.			−1.15937
$426$		0			0
$427$	8162.07			0.0447656
$428$		0			0
$429$	79490.9			0.431920
$430$		0			0
$431$			99089.3i			0.533424i	0.963776	+	0.266712i	$0.0859372\pi$
$431$			99089.3i			0.533424i	−0.963776	+	0.266712i	$0.914063\pi$
$432$		0			0
$433$			256640.i			1.36883i	0.729094	+	0.684414i	$0.239942\pi$
$433$			256640.i			1.36883i	−0.729094	+	0.684414i	$0.760058\pi$
$434$		0			0
$435$	103144.			0.545086
$436$		0			0
$437$	−279067.	−	107292.i	−1.46132	−	0.561832i
$438$		0			0
$439$			135549.i			0.703345i	0.936123	+	0.351673i	$0.114387\pi$
$439$			135549.i			0.703345i	−0.936123	+	0.351673i	$0.885613\pi$
$440$		0			0
$441$	332735.			1.71089
$442$		0			0
$443$	−211491.			−1.07767			−0.538834	−	0.842412i	$-0.681135\pi$
$443$	−211491.			−1.07767			−0.538834	+	0.842412i	$0.681135\pi$
$444$		0			0
$445$		−	57390.5i		−	0.289814i
$446$		0			0
$447$		−	232201.i		−	1.16212i
$448$		0			0
$449$		−	242635.i		−	1.20354i	−0.798669	−	0.601770i	$-0.794462\pi$
$449$		−	242635.i		−	1.20354i	0.798669	−	0.601770i	$-0.205538\pi$
$450$		0			0
$451$		−	304040.i		−	1.49478i
$452$		0			0
$453$	223418.			1.08873
$454$		0			0
$455$		−	6928.65i		−	0.0334677i
$456$		0			0
$457$	84466.2			0.404437			0.202218	−	0.979340i	$-0.435185\pi$
$457$	84466.2			0.404437			0.202218	+	0.979340i	$0.435185\pi$
$458$		0			0
$459$			125101.i			0.593792i
$460$		0			0
$461$	−314530.			−1.48000			−0.739998	−	0.672609i	$-0.765174\pi$
$461$	−314530.			−1.48000			−0.739998	+	0.672609i	$0.765174\pi$
$462$		0			0
$463$	230478.			1.07514			0.537572	−	0.843218i	$-0.319342\pi$
$463$	230478.			1.07514			0.537572	+	0.843218i	$0.319342\pi$
$464$		0			0
$465$	−723167.			−3.34451
$466$		0			0
$467$	−117986.			−0.541000			−0.270500	−	0.962720i	$-0.587189\pi$
$467$	−117986.			−0.541000			−0.270500	+	0.962720i	$0.587189\pi$
$468$		0			0
$469$			19511.1i			0.0887025i
$470$		0			0
$471$		−	141953.i		−	0.639888i
$472$		0			0
$473$	−248725.			−1.11172
$474$		0			0
$475$	485772.	+	186764.i	2.15300	+	0.827763i
$476$		0			0
$477$		−	630028.i		−	2.76900i
$478$		0			0
$479$	190455.			0.830081			0.415040	−	0.909803i	$-0.363767\pi$
$479$	190455.			0.830081			0.415040	+	0.909803i	$0.363767\pi$
$480$		0			0
$481$	63912.5			0.276246
$482$		0			0
$483$		−	35196.8i		−	0.150872i
$484$		0			0
$485$			719392.i			3.05831i
$486$		0			0
$487$		−	176874.i		−	0.745774i	−0.927877	−	0.372887i	$-0.878368\pi$
$487$		−	176874.i		−	0.745774i	0.927877	−	0.372887i	$-0.121632\pi$
$488$		0			0
$489$		−	176589.i		−	0.738493i
$490$		0			0
$491$	−85636.3			−0.355218			−0.177609	−	0.984101i	$-0.556836\pi$
$491$	−85636.3			−0.355218			−0.177609	+	0.984101i	$0.556836\pi$
$492$		0			0
$493$		−	22216.7i		−	0.0914083i
$494$		0			0
$495$	−636737.			−2.59866
$496$		0			0
$497$		−	18915.9i		−	0.0765800i
$498$		0			0
$499$	399577.			1.60472			0.802361	−	0.596839i	$-0.203577\pi$
$499$	399577.			1.60472			0.802361	+	0.596839i	$0.203577\pi$
$500$		0			0
$501$	282344.			1.12487
$502$		0			0
$503$	377058.			1.49029			0.745147	−	0.666900i	$-0.232379\pi$
$503$	377058.			1.49029			0.745147	+	0.666900i	$0.232379\pi$
$504$		0			0
$505$	78592.3			0.308175
$506$		0			0
$507$			381697.i			1.48492i
$508$		0			0
$509$		−	439558.i		−	1.69661i	−0.529511	−	0.848303i	$-0.677625\pi$
$509$		−	439558.i		−	1.69661i	0.529511	−	0.848303i	$-0.322375\pi$
$510$		0			0
$511$	−12959.7			−0.0496310
$512$		0			0
$513$	111572.	−	290197.i	0.423955	−	1.10270i
$514$		0			0
$515$			850218.i			3.20565i
$516$		0			0
$517$	−228460.			−0.854731
$518$		0			0
$519$	405400.			1.50504
$520$		0			0
$521$			188374.i			0.693977i	0.937869	+	0.346988i	$0.112796\pi$
$521$			188374.i			0.693977i	−0.937869	+	0.346988i	$0.887204\pi$
$522$		0			0
$523$			120335.i			0.439936i	0.975507	+	0.219968i	$0.0705952\pi$
$523$			120335.i			0.439936i	−0.975507	+	0.219968i	$0.929405\pi$
$524$		0			0
$525$			61267.0i			0.222284i
$526$		0			0
$527$			155767.i			0.560858i
$528$		0			0
$529$	406080.			1.45111
$530$		0			0
$531$			219793.i			0.779516i
$532$		0			0
$533$	−160588.			−0.565275
$534$		0			0
$535$			259743.i			0.907480i
$536$		0			0
$537$	−51455.8			−0.178437
$538$		0			0
$539$	241012.			0.829585
$540$		0			0
$541$	−73485.7			−0.251078			−0.125539	−	0.992089i	$-0.540066\pi$
$541$	−73485.7			−0.251078			−0.125539	+	0.992089i	$0.540066\pi$
$542$		0			0
$543$	−628346.			−2.13108
$544$		0			0
$545$			615004.i			2.07054i
$546$		0			0
$547$		−	192601.i		−	0.643700i	−0.946791	−	0.321850i	$-0.895695\pi$
$547$		−	192601.i		−	0.643700i	0.946791	−	0.321850i	$-0.104305\pi$
$548$		0			0
$549$	−396182.			−1.31447
$550$		0			0
$551$	−19814.1	+	51536.3i	−0.0652637	+	0.169750i
$552$		0			0
$553$		−	4846.44i		−	0.0158479i
$554$		0			0
$555$	−810160.			−2.63018
$556$		0			0
$557$	−234775.			−0.756730			−0.378365	−	0.925656i	$-0.623514\pi$
$557$	−234775.			−0.756730			−0.378365	+	0.925656i	$0.623514\pi$
$558$		0			0
$559$			131372.i			0.420415i
$560$		0			0
$561$			217039.i			0.689624i
$562$		0			0
$563$		−	43274.1i		−	0.136525i	−0.997667	−	0.0682623i	$-0.978255\pi$
$563$		−	43274.1i		−	0.136525i	0.997667	−	0.0682623i	$-0.0217455\pi$
$564$		0			0
$565$			653172.i			2.04612i
$566$		0			0
$567$	4332.31			0.0134758
$568$		0			0
$569$		−	369293.i		−	1.14064i	−0.821424	−	0.570318i	$-0.806820\pi$
$569$		−	369293.i		−	1.14064i	0.821424	−	0.570318i	$-0.193180\pi$
$570$		0			0
$571$	56108.0			0.172089			0.0860444	−	0.996291i	$-0.472577\pi$
$571$	56108.0			0.172089			0.0860444	+	0.996291i	$0.472577\pi$
$572$		0			0
$573$			352989.i			1.07511i
$574$		0			0
$575$	−1.19398e6			−3.61129
$576$		0			0
$577$	82362.1			0.247386			0.123693	−	0.992321i	$-0.460526\pi$
$577$	82362.1			0.247386			0.123693	+	0.992321i	$0.460526\pi$
$578$		0			0
$579$	−545534.			−1.62729
$580$		0			0
$581$	18987.4			0.0562486
$582$		0			0
$583$		−	456352.i		−	1.34265i
$584$		0			0
$585$			336313.i			0.982724i
$586$		0			0
$587$	−172458.			−0.500502			−0.250251	−	0.968181i	$-0.580513\pi$
$587$	−172458.			−0.500502			−0.250251	+	0.968181i	$0.580513\pi$
$588$		0			0
$589$	138921.	−	361333.i	0.400441	−	1.04154i
$590$		0			0
$591$			1.00823e6i			2.88660i
$592$		0			0
$593$	−379651.			−1.07963			−0.539815	−	0.841784i	$-0.681506\pi$
$593$	−379651.			−1.07963			−0.539815	+	0.841784i	$0.681506\pi$
$594$		0			0
$595$	18917.8			0.0534362
$596$		0			0
$597$			135126.i			0.379132i
$598$		0			0
$599$		−	425819.i		−	1.18678i	−0.804914	−	0.593391i	$-0.797789\pi$
$599$		−	425819.i		−	1.18678i	0.804914	−	0.593391i	$-0.202211\pi$
$600$		0			0
$601$		−	571193.i		−	1.58137i	−0.612222	−	0.790686i	$-0.709724\pi$
$601$		−	571193.i		−	1.58137i	0.612222	−	0.790686i	$-0.290276\pi$
$602$		0			0
$603$		−	947057.i		−	2.60460i
$604$		0			0
$605$	204375.			0.558363
$606$		0			0
$607$		−	287559.i		−	0.780458i	−0.920718	−	0.390229i	$-0.872396\pi$
$607$		−	287559.i		−	0.780458i	0.920718	−	0.390229i	$-0.127604\pi$
$608$		0			0
$609$	−6499.92			−0.0175256
$610$		0			0
$611$			120668.i			0.323230i
$612$		0			0
$613$	338076.			0.899692			0.449846	−	0.893106i	$-0.351479\pi$
$613$	338076.			0.899692			0.449846	+	0.893106i	$0.351479\pi$
$614$		0			0
$615$	2.03563e6			5.38206
$616$		0			0
$617$	−91953.9			−0.241546			−0.120773	−	0.992680i	$-0.538537\pi$
$617$	−91953.9			−0.241546			−0.120773	+	0.992680i	$0.538537\pi$
$618$		0			0
$619$	−309424.			−0.807557			−0.403778	−	0.914857i	$-0.632303\pi$
$619$	−309424.			−0.807557			−0.403778	+	0.914857i	$0.632303\pi$
$620$		0			0
$621$			713279.i			1.84959i
$622$		0			0
$623$			3616.63i			0.00931812i
$624$		0			0
$625$	786708.			2.01397
$626$		0			0
$627$	193568.	−	503468.i	0.492377	−	1.28067i
$628$		0			0
$629$			174504.i			0.441068i
$630$		0			0
$631$	347970.			0.873944			0.436972	−	0.899475i	$-0.356051\pi$
$631$	347970.			0.873944			0.436972	+	0.899475i	$0.356051\pi$
$632$		0			0
$633$	−1.06226e6			−2.65108
$634$		0			0
$635$			162021.i			0.401813i
$636$		0			0
$637$		−	127298.i		−	0.313720i
$638$		0			0
$639$			918170.i			2.24865i
$640$		0			0
$641$			196909.i			0.479237i	0.970867	+	0.239619i	$0.0770224\pi$
$641$			196909.i			0.479237i	−0.970867	+	0.239619i	$0.922978\pi$
$642$		0			0
$643$	458651.			1.10933			0.554665	−	0.832074i	$-0.312847\pi$
$643$	458651.			1.10933			0.554665	+	0.832074i	$0.312847\pi$
$644$		0			0
$645$		−	1.66528e6i		−	4.00283i
$646$		0			0
$647$	−455186.			−1.08738			−0.543689	−	0.839287i	$-0.682973\pi$
$647$	−455186.			−1.08738			−0.543689	+	0.839287i	$0.682973\pi$
$648$		0			0
$649$			159204.i			0.377977i
$650$		0			0
$651$	45572.5			0.107533
$652$		0			0
$653$	−196893.			−0.461747			−0.230873	−	0.972984i	$-0.574158\pi$
$653$	−196893.			−0.461747			−0.230873	+	0.972984i	$0.574158\pi$
$654$		0			0
$655$	328370.			0.765387
$656$		0			0
$657$	629056.			1.45733
$658$		0			0
$659$		−	697104.i		−	1.60519i	−0.596524	−	0.802595i	$-0.703452\pi$
$659$		−	697104.i		−	1.60519i	0.596524	−	0.802595i	$-0.296548\pi$
$660$		0			0
$661$			326279.i			0.746770i	0.927676	+	0.373385i	$0.121803\pi$
$661$			326279.i			0.746770i	−0.927676	+	0.373385i	$0.878197\pi$
$662$		0			0
$663$	114636.			0.260792
$664$		0			0
$665$	−43883.7	−	16871.9i	−0.0992338	−	0.0381523i
$666$		0			0
$667$		−	126672.i		−	0.284726i
$668$		0			0
$669$	−453235.			−1.01268
$670$		0			0
$671$	−286969.			−0.637368
$672$		0			0
$673$			257251.i			0.567972i	0.958828	+	0.283986i	$0.0916569\pi$
$673$			257251.i			0.567972i	−0.958828	+	0.283986i	$0.908343\pi$
$674$		0			0
$675$		−	1.24161e6i		−	2.72506i
$676$		0			0
$677$			365636.i			0.797759i	0.917003	+	0.398879i	$0.130601\pi$
$677$			365636.i			0.797759i	−0.917003	+	0.398879i	$0.869399\pi$
$678$		0			0
$679$		−	45334.6i		−	0.0983310i
$680$		0			0
$681$	1.41623e6			3.05379
$682$		0			0
$683$		−	135949.i		−	0.291429i	−0.989327	−	0.145715i	$-0.953452\pi$
$683$		−	135949.i		−	0.291429i	0.989327	−	0.145715i	$-0.0465481\pi$
$684$		0			0
$685$	330112.			0.703525
$686$		0			0
$687$		−	1.25056e6i		−	2.64967i
$688$		0			0
$689$	−241037.			−0.507744
$690$		0			0
$691$	−166089.			−0.347844			−0.173922	−	0.984759i	$-0.555644\pi$
$691$	−166089.			−0.347844			−0.173922	+	0.984759i	$0.555644\pi$
$692$		0			0
$693$	40125.9			0.0835522
$694$		0			0
$695$	338920.			0.701662
$696$		0			0
$697$		−	438465.i		−	0.902546i
$698$		0			0
$699$		−	825529.i		−	1.68958i
$700$		0			0
$701$	72123.3			0.146771			0.0733854	−	0.997304i	$-0.476620\pi$
$701$	72123.3			0.146771			0.0733854	+	0.997304i	$0.476620\pi$
$702$		0			0
$703$	155633.	−	404800.i	0.314913	−	0.819086i
$704$		0			0
$705$		−	1.52960e6i		−	3.07752i
$706$		0			0
$707$	−4952.73			−0.00990844
$708$		0			0
$709$	−566782.			−1.12752			−0.563759	−	0.825939i	$-0.690645\pi$
$709$	−566782.			−1.12752			−0.563759	+	0.825939i	$0.690645\pi$
$710$		0			0
$711$			235243.i			0.465348i
$712$		0			0
$713$			888125.i			1.74701i
$714$		0			0
$715$			243604.i			0.476510i
$716$		0			0
$717$			537891.i			1.04630i
$718$		0			0
$719$	−425756.			−0.823574			−0.411787	−	0.911280i	$-0.635095\pi$
$719$	−425756.			−0.823574			−0.411787	+	0.911280i	$0.635095\pi$
$720$		0			0
$721$		−	53579.0i		−	0.103068i
$722$		0			0
$723$	559631.			1.07059
$724$		0			0
$725$			220497.i			0.419496i
$726$		0			0
$727$	−363837.			−0.688395			−0.344197	−	0.938897i	$-0.611849\pi$
$727$	−363837.			−0.688395			−0.344197	+	0.938897i	$0.611849\pi$
$728$		0			0
$729$	824557.			1.55155
$730$		0			0
$731$	−358693.			−0.671256
$732$		0			0
$733$	117252.			0.218229			0.109115	−	0.994029i	$-0.465198\pi$
$733$	117252.			0.218229			0.109115	+	0.994029i	$0.465198\pi$
$734$		0			0
$735$			1.61364e6i			2.98698i
$736$		0			0
$737$		−	685988.i		−	1.26294i
$738$		0			0
$739$	326297.			0.597481			0.298740	−	0.954334i	$-0.403434\pi$
$739$	326297.			0.597481			0.298740	+	0.954334i	$0.403434\pi$
$740$		0			0
$741$	−265923.	−	102239.i	−0.484305	−	0.186200i
$742$		0			0
$743$			179042.i			0.324323i	0.986764	+	0.162161i	$0.0518465\pi$
$743$			179042.i			0.324323i	−0.986764	+	0.162161i	$0.948154\pi$
$744$		0			0
$745$	711592.			1.28209
$746$		0			0
$747$	−921635.			−1.65165
$748$		0			0
$749$		−	16368.5i		−	0.0291773i
$750$		0			0
$751$			407455.i			0.722436i	0.932481	+	0.361218i	$0.117639\pi$
$751$			407455.i			0.722436i	−0.932481	+	0.361218i	$0.882361\pi$
$752$		0			0
$753$			711402.i			1.25466i
$754$		0			0
$755$			684674.i			1.20113i
$756$		0			0
$757$	714736.			1.24725			0.623626	−	0.781723i	$-0.285659\pi$
$757$	714736.			1.24725			0.623626	+	0.781723i	$0.285659\pi$
$758$		0			0
$759$			1.23748e6i			2.14810i
$760$		0			0
$761$	419311.			0.724047			0.362024	−	0.932169i	$-0.382086\pi$
$761$	419311.			0.724047			0.362024	+	0.932169i	$0.382086\pi$
$762$		0			0
$763$		−	38756.3i		−	0.0665722i
$764$		0			0
$765$	−918257.			−1.56907
$766$		0			0
$767$	84088.7			0.142938
$768$		0			0
$769$	−460066.			−0.777978			−0.388989	−	0.921242i	$-0.627176\pi$
$769$	−460066.			−0.777978			−0.388989	+	0.921242i	$0.627176\pi$
$770$		0			0
$771$	−34717.3			−0.0584034
$772$		0			0
$773$		−	757267.i		−	1.26733i	−0.773608	−	0.633665i	$-0.781550\pi$
$773$		−	757267.i		−	1.26733i	0.773608	−	0.633665i	$-0.218450\pi$
$774$		0			0
$775$		−	1.54596e6i		−	2.57392i
$776$		0			0
$777$	51054.6			0.0845654
$778$		0			0
$779$	−391048.	+	1.01711e6i	−0.644399	+	1.67608i
$780$		0			0
$781$			665064.i			1.09034i
$782$		0			0
$783$	131724.			0.214853
$784$		0			0
$785$	435023.			0.705948
$786$		0			0
$787$			332574.i			0.536957i	0.963286	+	0.268478i	$0.0865208\pi$
$787$			332574.i			0.536957i	−0.963286	+	0.268478i	$0.913479\pi$
$788$		0			0
$789$		−	1.42588e6i		−	2.29049i
$790$		0			0
$791$		−	41161.6i		−	0.0657868i
$792$		0			0
$793$			151572.i			0.241031i
$794$		0			0
$795$	3.05540e6			4.83431
$796$		0			0
$797$		−	388000.i		−	0.610823i	−0.952220	−	0.305412i	$-0.901206\pi$
$797$		−	388000.i		−	0.610823i	0.952220	−	0.305412i	$-0.0987940\pi$
$798$		0			0
$799$	−329469.			−0.516085
$800$		0			0
$801$		−	175549.i		−	0.273611i
$802$		0			0
$803$	455648.			0.706640
$804$		0			0
$805$	107862.			0.166448
$806$		0			0
$807$	656106.			1.00746
$808$		0			0
$809$	−864221.			−1.32047			−0.660234	−	0.751060i	$-0.729543\pi$
$809$	−864221.			−1.32047			−0.660234	+	0.751060i	$0.729543\pi$
$810$		0			0
$811$		−	61653.6i		−	0.0937382i	−0.998901	−	0.0468691i	$-0.985076\pi$
$811$		−	61653.6i		−	0.0937382i	0.998901	−	0.0468691i	$-0.0149244\pi$
$812$		0			0
$813$			1.79892e6i			2.72164i
$814$		0			0
$815$	541166.			0.814733
$816$		0			0
$817$	832063.	+	319902.i	1.24656	+	0.479263i
$818$		0			0
$819$		−	21193.8i		−	0.0315966i
$820$		0			0
$821$	84273.1			0.125027			0.0625134	−	0.998044i	$-0.480088\pi$
$821$	84273.1			0.125027			0.0625134	+	0.998044i	$0.480088\pi$
$822$		0			0
$823$	−983646.			−1.45224			−0.726121	−	0.687567i	$-0.758679\pi$
$823$	−983646.			−1.45224			−0.726121	+	0.687567i	$0.758679\pi$
$824$		0			0
$825$		−	2.15408e6i		−	3.16486i
$826$		0			0
$827$			361582.i			0.528684i	0.964429	+	0.264342i	$0.0851548\pi$
$827$			361582.i			0.528684i	−0.964429	+	0.264342i	$0.914845\pi$
$828$		0			0
$829$			411606.i			0.598925i	0.954108	+	0.299462i	$0.0968073\pi$
$829$			411606.i			0.598925i	−0.954108	+	0.299462i	$0.903193\pi$
$830$		0			0
$831$		−	790843.i		−	1.14522i
$832$		0			0
$833$	347570.			0.500902
$834$		0			0
$835$			865256.i			1.24100i
$836$		0			0
$837$	−923548.			−1.31828
$838$		0			0
$839$			626506.i			0.890023i	0.895525	+	0.445012i	$0.146800\pi$
$839$			626506.i			0.890023i	−0.895525	+	0.445012i	$0.853200\pi$
$840$		0			0
$841$	683888.			0.966926
$842$		0			0
$843$	−1.35283e6			−1.90366
$844$		0			0
$845$	−1.16973e6			−1.63822
$846$		0			0
$847$	−12879.3			−0.0179525
$848$		0			0
$849$		−	295119.i		−	0.409432i
$850$		0			0
$851$			994961.i			1.37387i
$852$		0			0
$853$	−1.43364e6			−1.97034			−0.985172	−	0.171570i	$-0.945116\pi$
$853$	−1.43364e6			−1.97034			−0.985172	+	0.171570i	$0.945116\pi$
$854$		0			0
$855$	2.13009e6	+	818953.i	2.91384	+	1.12028i
$856$		0			0
$857$		−	1.13615e6i		−	1.54695i	−0.633828	−	0.773474i	$-0.718517\pi$
$857$		−	1.13615e6i		−	1.54695i	0.633828	−	0.773474i	$-0.281483\pi$
$858$		0			0
$859$	−270468.			−0.366548			−0.183274	−	0.983062i	$-0.558669\pi$
$859$	−270468.			−0.366548			−0.183274	+	0.983062i	$0.558669\pi$
$860$		0			0
$861$	−128281.			−0.173044
$862$		0			0
$863$		−	104658.i		−	0.140525i	−0.997529	−	0.0702623i	$-0.977616\pi$
$863$		−	104658.i		−	0.140525i	0.997529	−	0.0702623i	$-0.0223836\pi$
$864$		0			0
$865$			1.24237e6i			1.66042i
$866$		0			0
$867$		−	925979.i		−	1.23186i
$868$		0			0
$869$			170395.i			0.225641i
$870$		0			0
$871$	−362326.			−0.477599
$872$		0			0
$873$			2.20052e6i			2.88733i
$874$		0			0
$875$	−106358.			−0.138917
$876$		0			0
$877$			1.18104e6i			1.53556i	0.640717	+	0.767778i	$0.278637\pi$
$877$			1.18104e6i			1.53556i	−0.640717	+	0.767778i	$0.721363\pi$
$878$		0			0
$879$	818238.			1.05901
$880$		0			0
$881$	648093.			0.834998			0.417499	−	0.908677i	$-0.362907\pi$
$881$	648093.			0.834998			0.417499	+	0.908677i	$0.362907\pi$
$882$		0			0
$883$	−617809.			−0.792379			−0.396190	−	0.918169i	$-0.629668\pi$
$883$	−617809.			−0.792379			−0.396190	+	0.918169i	$0.629668\pi$
$884$		0			0
$885$	−1.06591e6			−1.36093
$886$		0			0
$887$		−	895467.i		−	1.13816i	−0.822283	−	0.569079i	$-0.807300\pi$
$887$		−	895467.i		−	1.13816i	0.822283	−	0.569079i	$-0.192700\pi$
$888$		0			0
$889$		−	10210.2i		−	0.0129191i
$890$		0			0
$891$	−152319.			−0.191866
$892$		0			0
$893$	764272.	+	293839.i	0.958397	+	0.368474i
$894$		0			0
$895$		−	157689.i		−	0.196859i
$896$		0			0
$897$	653614.			0.812337
$898$		0			0
$899$	164013.			0.202936
$900$		0			0
$901$		−	658119.i		−	0.810690i
$902$		0			0
$903$			104942.i			0.128699i
$904$		0			0
$905$		−	1.92560e6i		−	2.35108i
$906$		0			0
$907$		−	356161.i		−	0.432944i	−0.976289	−	0.216472i	$-0.930545\pi$
$907$		−	356161.i		−	0.432944i	0.976289	−	0.216472i	$-0.0694551\pi$
$908$		0			0
$909$	240403.			0.290945
$910$		0			0
$911$		−	551857.i		−	0.664952i	−0.943112	−	0.332476i	$-0.892116\pi$
$911$		−	551857.i		−	0.664952i	0.943112	−	0.332476i	$-0.107884\pi$
$912$		0			0
$913$	−667574.			−0.800862
$914$		0			0
$915$		−	1.92134e6i		−	2.29489i
$916$		0			0
$917$	−20693.2			−0.0246087
$918$		0			0
$919$	−276649.			−0.327566			−0.163783	−	0.986496i	$-0.552370\pi$
$919$	−276649.			−0.327566			−0.163783	+	0.986496i	$0.552370\pi$
$920$		0			0
$921$	1.85414e6			2.18587
$922$		0			0
$923$	351274.			0.412328
$924$		0			0
$925$		−	1.73193e6i		−	2.02417i
$926$		0			0
$927$			2.60070e6i			3.02643i
$928$		0			0
$929$	277351.			0.321365			0.160682	−	0.987006i	$-0.448630\pi$
$929$	277351.			0.321365			0.160682	+	0.987006i	$0.448630\pi$
$930$		0			0
$931$	−806262.	−	309983.i	−0.930201	−	0.357633i
$932$		0			0
$933$			605767.i			0.695892i
$934$		0			0
$935$	−665127.			−0.760819
$936$		0			0
$937$	−130928.			−0.149126			−0.0745631	−	0.997216i	$-0.523756\pi$
$937$	−130928.			−0.149126			−0.0745631	+	0.997216i	$0.523756\pi$
$938$		0			0
$939$			1.47685e6i			1.67496i
$940$		0			0
$941$			271511.i			0.306626i	0.988178	+	0.153313i	$0.0489943\pi$
$941$			271511.i			0.306626i	−0.988178	+	0.153313i	$0.951006\pi$
$942$		0			0
$943$		−	2.49997e6i		−	2.81133i
$944$		0			0
$945$			112164.i			0.125600i
$946$		0			0
$947$	939290.			1.04737			0.523684	−	0.851912i	$-0.324557\pi$
$947$	939290.			1.04737			0.523684	+	0.851912i	$0.324557\pi$
$948$		0			0
$949$		−	240665.i		−	0.267227i
$950$		0			0
$951$	−1.42226e6			−1.57260
$952$		0			0
$953$		−	1.37941e6i		−	1.51882i	−0.650611	−	0.759411i	$-0.725487\pi$
$953$		−	1.37941e6i		−	1.51882i	0.650611	−	0.759411i	$-0.274513\pi$
$954$		0			0
$955$	−1.08175e6			−1.18610
$956$		0			0
$957$	228530.			0.249528
$958$		0			0
$959$	−20803.0			−0.0226197
$960$		0			0
$961$	−226415.			−0.245165
$962$		0			0
$963$			794518.i			0.856744i
$964$		0			0
$965$		−	1.67181e6i		−	1.79528i
$966$		0			0
$967$	−849703.			−0.908687			−0.454343	−	0.890827i	$-0.650126\pi$
$967$	−849703.			−0.908687			−0.454343	+	0.890827i	$0.650126\pi$
$968$		0			0
$969$	279150.	−	726066.i	0.297296	−	0.773265i
$970$		0			0
$971$		−	267402.i		−	0.283613i	−0.989894	−	0.141806i	$-0.954709\pi$
$971$		−	267402.i		−	0.283613i	0.989894	−	0.141806i	$-0.0452910\pi$
$972$		0			0
$973$	−21358.1			−0.0225598
$974$		0			0
$975$	−1.13775e6			−1.19684
$976$		0			0
$977$		−	169611.i		−	0.177690i	−0.996045	−	0.0888451i	$-0.971682\pi$
$977$		−	169611.i		−	0.177690i	0.996045	−	0.0888451i	$-0.0283176\pi$
$978$		0			0
$979$		−	127157.i		−	0.132670i
$980$		0			0
$981$			1.88121e6i			1.95478i
$982$		0			0
$983$		−	1.42103e6i		−	1.47060i	−0.677740	−	0.735302i	$-0.737040\pi$
$983$		−	1.42103e6i		−	1.47060i	0.677740	−	0.735302i	$-0.262960\pi$
$984$		0			0
$985$	−3.08978e6			−3.18460
$986$		0			0
$987$			96392.5i			0.0989484i
$988$		0			0
$989$	−2.04514e6			−2.09088
$990$		0			0
$991$		−	1.43454e6i		−	1.46071i	−0.683067	−	0.730356i	$-0.739354\pi$
$991$		−	1.43454e6i		−	1.46071i	0.683067	−	0.730356i	$-0.260646\pi$
$992$		0			0
$993$	−2.12276e6			−2.15279
$994$		0			0
$995$	−414100.			−0.418272
$996$		0			0
$997$	425661.			0.428226			0.214113	−	0.976809i	$-0.431314\pi$
$997$	425661.			0.428226			0.214113	+	0.976809i	$0.431314\pi$
$998$		0			0
$999$	−1.03465e6			−1.03672

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.5.e.f.113.19		20
4.3	odd	2		152.5.e.a.113.2	✓	20
12.11	even	2		1368.5.o.a.721.20		20
19.18	odd	2	inner	304.5.e.f.113.2		20
76.75	even	2		152.5.e.a.113.19	yes	20
228.227	odd	2		1368.5.o.a.721.19		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.5.e.a.113.2	✓	20	4.3	odd	2
152.5.e.a.113.19	yes	20	76.75	even	2
304.5.e.f.113.2		20	19.18	odd	2	inner
304.5.e.f.113.19		20	1.1	even	1	trivial
1368.5.o.a.721.19		20	228.227	odd	2
1368.5.o.a.721.20		20	12.11	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:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	304.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+14.8343i q^{3} -45.4605 q^{5} +2.86483 q^{7} -139.057 q^{9} -100.724 q^{11} +53.2006i q^{13} -674.375i q^{15} -145.257 q^{17} +(336.954 + 129.548i) q^{19} +42.4977i q^{21} -828.203 q^{23} +1441.65 q^{25} -861.237i q^{27} +152.947i q^{29} -1072.35i q^{31} -1494.17i q^{33} -130.236 q^{35} -1201.35i q^{37} -789.195 q^{39} +3018.54i q^{41} +2469.37 q^{43} +6321.60 q^{45} +2268.18 q^{47} -2392.79 q^{49} -2154.79i q^{51} +4530.72i q^{53} +4578.96 q^{55} +(-1921.76 + 4998.49i) q^{57} -1580.60i q^{59} +2849.06 q^{61} -398.374 q^{63} -2418.52i q^{65} +6810.56i q^{67} -12285.8i q^{69} -6602.83i q^{71} -4523.72 q^{73} +21386.0i q^{75} -288.557 q^{77} -1691.70i q^{79} +1512.24 q^{81} +6627.75 q^{83} +6603.46 q^{85} -2268.87 q^{87} +1262.43i q^{89} +152.410i q^{91} +15907.6 q^{93} +(-15318.1 - 5889.33i) q^{95} -15824.6i q^{97} +14006.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 32 q^{7} - 372 q^{9} + 24 q^{11} + 216 q^{17} + 596 q^{19} - 576 q^{23} + 1412 q^{25} + 144 q^{35} + 520 q^{39} + 1256 q^{43} + 7232 q^{45} + 3768 q^{47} - 2740 q^{49} + 10128 q^{55} - 728 q^{57}+ \cdots - 760 q^{99}+O(q^{100}) \) 20 * q - 32 * q^7 - 372 * q^9 + 24 * q^11 + 216 * q^17 + 596 * q^19 - 576 * q^23 + 1412 * q^25 + 144 * q^35 + 520 * q^39 + 1256 * q^43 + 7232 * q^45 + 3768 * q^47 - 2740 * q^49 + 10128 * q^55 - 728 * q^57 + 352 * q^61 - 6104 * q^63 + 1352 * q^73 + 9288 * q^77 - 4220 * q^81 + 16104 * q^83 + 10232 * q^85 - 2936 * q^87 + 36432 * q^93 - 14232 * q^95 - 760 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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