LMFDB - Embedded newform 304.5.e.c.113.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [304,5,Mod(113,304)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

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

Self dual:	no
Analytic conductor:	$31.4244687775$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.12107488.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 35x^{2} + 142 $ x^4 + 35*x^2 + 142
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 19)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			113.4
Root			$-2.16425i$ of defining polynomial
Character	$\chi$	$=$	304.113
Dual form			304.5.e.c.113.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+16.8206i q^{3} -14.7720 q^{5} -59.6320 q^{7} -201.932 q^{9} +O(q^{10})$	$q+16.8206i q^{3} -14.7720 q^{5} -59.6320 q^{7} -201.932 q^{9} +17.1240 q^{11} +163.384i q^{13} -248.474i q^{15} +177.528 q^{17} +(-272.004 - 237.349i) q^{19} -1003.05i q^{21} +502.236 q^{23} -406.788 q^{25} -2034.15i q^{27} -500.627i q^{29} +1465.70i q^{31} +288.036i q^{33} +880.884 q^{35} -600.763i q^{37} -2748.21 q^{39} +325.088i q^{41} +752.973 q^{43} +2982.94 q^{45} +2257.62 q^{47} +1154.98 q^{49} +2986.13i q^{51} +1501.20i q^{53} -252.956 q^{55} +(3992.36 - 4575.27i) q^{57} -4555.57i q^{59} +770.012 q^{61} +12041.6 q^{63} -2413.51i q^{65} -1613.61i q^{67} +8447.91i q^{69} +7676.78i q^{71} -2330.98 q^{73} -6842.41i q^{75} -1021.14 q^{77} -53.3968i q^{79} +17859.1 q^{81} -9438.26 q^{83} -2622.44 q^{85} +8420.83 q^{87} -9238.06i q^{89} -9742.91i q^{91} -24654.0 q^{93} +(4018.05 + 3506.13i) q^{95} -2562.56i q^{97} -3457.89 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 42 q^{5} - 136 q^{7} - 278 q^{9}+O(q^{10}) $ 4 * q - 42 * q^5 - 136 * q^7 - 278 * q^9	$ 4 q - 42 q^{5} - 136 q^{7} - 278 q^{9} - 222 q^{11} + 300 q^{17} - 114 q^{19} + 78 q^{23} - 1986 q^{25} + 1866 q^{35} - 6362 q^{39} - 2986 q^{43} + 5182 q^{45} + 7578 q^{47} - 2352 q^{49} + 1090 q^{55} + 10450 q^{57} + 158 q^{61} + 23030 q^{63} - 15168 q^{73} + 102 q^{77} + 40712 q^{81} - 33276 q^{83} - 4902 q^{85} + 7214 q^{87} - 40004 q^{93} + 5358 q^{95} - 23042 q^{99}+O(q^{100}) $ 4 * q - 42 * q^5 - 136 * q^7 - 278 * q^9 - 222 * q^11 + 300 * q^17 - 114 * q^19 + 78 * q^23 - 1986 * q^25 + 1866 * q^35 - 6362 * q^39 - 2986 * q^43 + 5182 * q^45 + 7578 * q^47 - 2352 * q^49 + 1090 * q^55 + 10450 * q^57 + 158 * q^61 + 23030 * q^63 - 15168 * q^73 + 102 * q^77 + 40712 * q^81 - 33276 * q^83 - 4902 * q^85 + 7214 * q^87 - 40004 * q^93 + 5358 * q^95 - 23042 * q^99

Display 100 coefficients 10 coefficients

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
$3$			16.8206i			1.86895i	0.356024	+	0.934477i	$0.384132\pi$
$3$			16.8206i			1.86895i	−0.356024	+	0.934477i	$0.615868\pi$
$4$		0			0
$5$	−14.7720			−0.590880			−0.295440	−	0.955361i	$-0.595466\pi$
$5$	−14.7720			−0.590880			−0.295440	+	0.955361i	$0.595466\pi$
$6$		0			0
$7$	−59.6320			−1.21698			−0.608490	−	0.793562i	$-0.708224\pi$
$7$	−59.6320			−1.21698			−0.608490	+	0.793562i	$0.708224\pi$
$8$		0			0
$9$	−201.932			−2.49299
$10$		0			0
$11$	17.1240			0.141521			0.0707605	−	0.997493i	$-0.477457\pi$
$11$	17.1240			0.141521			0.0707605	+	0.997493i	$0.477457\pi$
$12$		0			0
$13$			163.384i			0.966769i	0.875408	+	0.483384i	$0.160593\pi$
$13$			163.384i			0.966769i	−0.875408	+	0.483384i	$0.839407\pi$
$14$		0			0
$15$		−	248.474i		−	1.10433i
$16$		0			0
$17$	177.528			0.614284			0.307142	−	0.951664i	$-0.400627\pi$
$17$	177.528			0.614284			0.307142	+	0.951664i	$0.400627\pi$
$18$		0			0
$19$	−272.004	−	237.349i	−0.753474	−	0.657478i
$19$	−272.004	−	237.349i	−0.753474	−	0.657478i
$20$		0			0
$21$		−	1003.05i		−	2.27448i
$22$		0			0
$23$	502.236			0.949407			0.474703	−	0.880146i	$-0.342555\pi$
$23$	502.236			0.949407			0.474703	+	0.880146i	$0.342555\pi$
$24$		0			0
$25$	−406.788			−0.650861
$26$		0			0
$27$		−	2034.15i		−	2.79033i
$28$		0			0
$29$		−	500.627i		−	0.595275i	−0.954679	−	0.297638i	$-0.903801\pi$
$29$		−	500.627i		−	0.595275i	0.954679	−	0.297638i	$-0.0961987\pi$
$30$		0			0
$31$			1465.70i			1.52518i	0.646880	+	0.762592i	$0.276074\pi$
$31$			1465.70i			1.52518i	−0.646880	+	0.762592i	$0.723926\pi$
$32$		0			0
$33$			288.036i			0.264496i
$34$		0			0
$35$	880.884			0.719089
$36$		0			0
$37$		−	600.763i		−	0.438833i	−0.975631	−	0.219417i	$-0.929585\pi$
$37$		−	600.763i		−	0.438833i	0.975631	−	0.219417i	$-0.0704154\pi$
$38$		0			0
$39$	−2748.21			−1.80685
$40$		0			0
$41$			325.088i			0.193390i	0.995314	+	0.0966948i	$0.0308271\pi$
$41$			325.088i			0.193390i	−0.995314	+	0.0966948i	$0.969173\pi$
$42$		0			0
$43$	752.973			0.407232			0.203616	−	0.979051i	$-0.434731\pi$
$43$	752.973			0.407232			0.203616	+	0.979051i	$0.434731\pi$
$44$		0			0
$45$	2982.94			1.47306
$46$		0			0
$47$	2257.62			1.02201			0.511005	−	0.859578i	$-0.329273\pi$
$47$	2257.62			1.02201			0.511005	+	0.859578i	$0.329273\pi$
$48$		0			0
$49$	1154.98			0.481040
$50$		0			0
$51$			2986.13i			1.14807i
$52$		0			0
$53$			1501.20i			0.534427i	0.963637	+	0.267213i	$0.0861028\pi$
$53$			1501.20i			0.534427i	−0.963637	+	0.267213i	$0.913897\pi$
$54$		0			0
$55$	−252.956			−0.0836219
$56$		0			0
$57$	3992.36	−	4575.27i	1.22880	−	1.40821i
$58$		0			0
$59$		−	4555.57i		−	1.30870i	−0.756193	−	0.654348i	$-0.772943\pi$
$59$		−	4555.57i		−	1.30870i	0.756193	−	0.654348i	$-0.227057\pi$
$60$		0			0
$61$	770.012			0.206937			0.103468	−	0.994633i	$-0.467006\pi$
$61$	770.012			0.206937			0.103468	+	0.994633i	$0.467006\pi$
$62$		0			0
$63$	12041.6			3.03392
$64$		0			0
$65$		−	2413.51i		−	0.571244i
$66$		0			0
$67$		−	1613.61i		−	0.359458i	−0.983716	−	0.179729i	$-0.942478\pi$
$67$		−	1613.61i		−	0.359458i	0.983716	−	0.179729i	$-0.0575221\pi$
$68$		0			0
$69$			8447.91i			1.77440i
$70$		0			0
$71$			7676.78i			1.52287i	0.648242	+	0.761435i	$0.275505\pi$
$71$			7676.78i			1.52287i	−0.648242	+	0.761435i	$0.724495\pi$
$72$		0			0
$73$	−2330.98			−0.437413			−0.218707	−	0.975791i	$-0.570184\pi$
$73$	−2330.98			−0.437413			−0.218707	+	0.975791i	$0.570184\pi$
$74$		0			0
$75$		−	6842.41i		−	1.21643i
$76$		0			0
$77$	−1021.14			−0.172228
$78$		0			0
$79$		−	53.3968i		−	0.00855580i	−0.999991	−	0.00427790i	$-0.998638\pi$
$79$		−	53.3968i		−	0.00855580i	0.999991	−	0.00427790i	$-0.00136170\pi$
$80$		0			0
$81$	17859.1			2.72200
$82$		0			0
$83$	−9438.26			−1.37005			−0.685024	−	0.728520i	$-0.740208\pi$
$83$	−9438.26			−1.37005			−0.685024	+	0.728520i	$0.740208\pi$
$84$		0			0
$85$	−2622.44			−0.362968
$86$		0			0
$87$	8420.83			1.11254
$88$		0			0
$89$		−	9238.06i		−	1.16627i	−0.812374	−	0.583137i	$-0.801825\pi$
$89$		−	9238.06i		−	1.16627i	0.812374	−	0.583137i	$-0.198175\pi$
$90$		0			0
$91$		−	9742.91i		−	1.17654i
$92$		0			0
$93$	−24654.0			−2.85050
$94$		0			0
$95$	4018.05	+	3506.13i	0.445213	+	0.388490i
$96$		0			0
$97$		−	2562.56i		−	0.272352i	−0.990685	−	0.136176i	$-0.956519\pi$
$97$		−	2562.56i		−	0.272352i	0.990685	−	0.136176i	$-0.0434813\pi$
$98$		0			0
$99$	−3457.89			−0.352810
$100$		0			0
$101$	7050.24			0.691133			0.345566	−	0.938394i	$-0.387687\pi$
$101$	7050.24			0.691133			0.345566	+	0.938394i	$0.387687\pi$
$102$		0			0
$103$		−	8275.94i		−	0.780086i	−0.920797	−	0.390043i	$-0.872460\pi$
$103$		−	8275.94i		−	0.780086i	0.920797	−	0.390043i	$-0.127540\pi$
$104$		0			0
$105$			14817.0i			1.34394i
$106$		0			0
$107$		−	6473.72i		−	0.565440i	−0.959202	−	0.282720i	$-0.908763\pi$
$107$		−	6473.72i		−	0.565440i	0.959202	−	0.282720i	$-0.0912367\pi$
$108$		0			0
$109$		−	5991.18i		−	0.504266i	−0.967693	−	0.252133i	$-0.918868\pi$
$109$		−	5991.18i		−	0.504266i	0.967693	−	0.252133i	$-0.0811320\pi$
$110$		0			0
$111$	10105.2			0.820159
$112$		0			0
$113$		−	20337.5i		−	1.59273i	−0.604817	−	0.796364i	$-0.706754\pi$
$113$		−	20337.5i		−	1.59273i	0.604817	−	0.796364i	$-0.293246\pi$
$114$		0			0
$115$	−7419.03			−0.560986
$116$		0			0
$117$		−	32992.4i		−	2.41014i
$118$		0			0
$119$	−10586.4			−0.747571
$120$		0			0
$121$	−14347.8			−0.979972
$122$		0			0
$123$	−5468.17			−0.361436
$124$		0			0
$125$	15241.6			0.975461
$126$		0			0
$127$		−	15199.0i		−	0.942342i	−0.882042	−	0.471171i	$-0.843831\pi$
$127$		−	15199.0i		−	0.942342i	0.882042	−	0.471171i	$-0.156169\pi$
$128$		0			0
$129$			12665.4i			0.761098i
$130$		0			0
$131$	−29704.4			−1.73092			−0.865461	−	0.500976i	$-0.832975\pi$
$131$	−29704.4			−1.73092			−0.865461	+	0.500976i	$0.832975\pi$
$132$		0			0
$133$	16220.2	+	14153.6i	0.916963	+	0.800137i
$134$		0			0
$135$			30048.4i			1.64875i
$136$		0			0
$137$	7080.63			0.377251			0.188626	−	0.982049i	$-0.439597\pi$
$137$	7080.63			0.377251			0.188626	+	0.982049i	$0.439597\pi$
$138$		0			0
$139$	16337.3			0.845574			0.422787	−	0.906229i	$-0.361052\pi$
$139$	16337.3			0.845574			0.422787	+	0.906229i	$0.361052\pi$
$140$		0			0
$141$			37974.5i			1.91009i
$142$		0			0
$143$			2797.79i			0.136818i
$144$		0			0
$145$			7395.26i			0.351736i
$146$		0			0
$147$			19427.4i			0.899041i
$148$		0			0
$149$	−12128.3			−0.546294			−0.273147	−	0.961972i	$-0.588065\pi$
$149$	−12128.3			−0.546294			−0.273147	+	0.961972i	$0.588065\pi$
$150$		0			0
$151$			26542.8i			1.16411i	0.813151	+	0.582053i	$0.197750\pi$
$151$			26542.8i			1.16411i	−0.813151	+	0.582053i	$0.802250\pi$
$152$		0			0
$153$	−35848.6			−1.53140
$154$		0			0
$155$		−	21651.4i		−	0.901201i
$156$		0			0
$157$	−8292.18			−0.336411			−0.168205	−	0.985752i	$-0.553797\pi$
$157$	−8292.18			−0.336411			−0.168205	+	0.985752i	$0.553797\pi$
$158$		0			0
$159$	−25251.1			−0.998819
$160$		0			0
$161$	−29949.4			−1.15541
$162$		0			0
$163$	−16703.6			−0.628688			−0.314344	−	0.949309i	$-0.601785\pi$
$163$	−16703.6			−0.628688			−0.314344	+	0.949309i	$0.601785\pi$
$164$		0			0
$165$		−	4254.87i		−	0.156285i
$166$		0			0
$167$		−	20319.3i		−	0.728576i	−0.931286	−	0.364288i	$-0.881312\pi$
$167$		−	20319.3i		−	0.728576i	0.931286	−	0.364288i	$-0.118688\pi$
$168$		0			0
$169$	1866.70			0.0653585
$170$		0			0
$171$	54926.3	+	47928.5i	1.87840	+	1.63908i
$172$		0			0
$173$			19570.3i			0.653893i	0.945043	+	0.326946i	$0.106020\pi$
$173$			19570.3i			0.653893i	−0.945043	+	0.326946i	$0.893980\pi$
$174$		0			0
$175$	24257.6			0.792084
$176$		0			0
$177$	76627.4			2.44589
$178$		0			0
$179$			23233.3i			0.725110i	0.931962	+	0.362555i	$0.118096\pi$
$179$			23233.3i			0.725110i	−0.931962	+	0.362555i	$0.881904\pi$
$180$		0			0
$181$		−	53293.1i		−	1.62672i	−0.581759	−	0.813361i	$-0.697635\pi$
$181$		−	53293.1i		−	1.62672i	0.581759	−	0.813361i	$-0.302365\pi$
$182$		0			0
$183$			12952.1i			0.386756i
$184$		0			0
$185$			8874.46i			0.259298i
$186$		0			0
$187$	3040.00			0.0869340
$188$		0			0
$189$			121300.i			3.39577i
$190$		0			0
$191$	−5420.24			−0.148577			−0.0742886	−	0.997237i	$-0.523669\pi$
$191$	−5420.24			−0.148577			−0.0742886	+	0.997237i	$0.523669\pi$
$192$		0			0
$193$			38918.6i			1.04482i	0.852694	+	0.522411i	$0.174967\pi$
$193$			38918.6i			1.04482i	−0.852694	+	0.522411i	$0.825033\pi$
$194$		0			0
$195$	40596.6			1.06763
$196$		0			0
$197$	−41122.0			−1.05960			−0.529800	−	0.848122i	$-0.677733\pi$
$197$	−41122.0			−1.05960			−0.529800	+	0.848122i	$0.677733\pi$
$198$		0			0
$199$	−22778.9			−0.575211			−0.287606	−	0.957749i	$-0.592859\pi$
$199$	−22778.9			−0.575211			−0.287606	+	0.957749i	$0.592859\pi$
$200$		0			0
$201$	27141.8			0.671811
$202$		0			0
$203$			29853.4i			0.724438i
$204$		0			0
$205$		−	4802.20i		−	0.114270i
$206$		0			0
$207$	−101418.			−2.36686
$208$		0			0
$209$	−4657.81	−	4064.38i	−0.106632	−	0.0930469i
$210$		0			0
$211$		−	80470.1i		−	1.80746i	−0.428099	−	0.903732i	$-0.640817\pi$
$211$		−	80470.1i		−	1.80746i	0.428099	−	0.903732i	$-0.359183\pi$
$212$		0			0
$213$	−129128.			−2.84617
$214$		0			0
$215$	−11122.9			−0.240625
$216$		0			0
$217$		−	87402.8i		−	1.85612i
$218$		0			0
$219$		−	39208.4i		−	0.817505i
$220$		0			0
$221$			29005.2i			0.593870i
$222$		0			0
$223$		−	8564.16i		−	0.172217i	−0.996286	−	0.0861083i	$-0.972557\pi$
$223$		−	8564.16i		−	0.172217i	0.996286	−	0.0861083i	$-0.0274431\pi$
$224$		0			0
$225$	82143.5			1.62259
$226$		0			0
$227$			37739.2i			0.732387i	0.930539	+	0.366193i	$0.119339\pi$
$227$			37739.2i			0.732387i	−0.930539	+	0.366193i	$0.880661\pi$
$228$		0			0
$229$	−44028.8			−0.839587			−0.419793	−	0.907620i	$-0.637897\pi$
$229$	−44028.8			−0.839587			−0.419793	+	0.907620i	$0.637897\pi$
$230$		0			0
$231$		−	17176.2i		−	0.321886i
$232$		0			0
$233$	−5412.82			−0.0997038			−0.0498519	−	0.998757i	$-0.515875\pi$
$233$	−5412.82			−0.0997038			−0.0498519	+	0.998757i	$0.515875\pi$
$234$		0			0
$235$	−33349.6			−0.603885
$236$		0			0
$237$	898.165			0.0159904
$238$		0			0
$239$	24069.2			0.421372			0.210686	−	0.977554i	$-0.432430\pi$
$239$	24069.2			0.421372			0.210686	+	0.977554i	$0.432430\pi$
$240$		0			0
$241$		−	24651.4i		−	0.424432i	−0.977223	−	0.212216i	$-0.931932\pi$
$241$		−	24651.4i		−	0.424432i	0.977223	−	0.212216i	$-0.0680680\pi$
$242$		0			0
$243$			135634.i			2.29697i
$244$		0			0
$245$	−17061.3			−0.284237
$246$		0			0
$247$	38779.1	−	44441.1i	0.635629	−	0.728435i
$248$		0			0
$249$		−	158757.i		−	2.56056i
$250$		0			0
$251$	−91859.2			−1.45806			−0.729030	−	0.684482i	$-0.760028\pi$
$251$	−91859.2			−1.45806			−0.729030	+	0.684482i	$0.760028\pi$
$252$		0			0
$253$	8600.31			0.134361
$254$		0			0
$255$		−	44111.1i		−	0.678371i
$256$		0			0
$257$		−	63651.3i		−	0.963698i	−0.876254	−	0.481849i	$-0.839965\pi$
$257$		−	63651.3i		−	0.963698i	0.876254	−	0.481849i	$-0.160035\pi$
$258$		0			0
$259$			35824.7i			0.534051i
$260$		0			0
$261$			101093.i			1.48401i
$262$		0			0
$263$	−113072.			−1.63472			−0.817360	−	0.576127i	$-0.804563\pi$
$263$	−113072.			−1.63472			−0.817360	+	0.576127i	$0.804563\pi$
$264$		0			0
$265$		−	22175.8i		−	0.315782i
$266$		0			0
$267$	155390.			2.17971
$268$		0			0
$269$		−	54854.5i		−	0.758067i	−0.925383	−	0.379034i	$-0.876257\pi$
$269$		−	54854.5i		−	0.758067i	0.925383	−	0.379034i	$-0.123743\pi$
$270$		0			0
$271$	−61064.9			−0.831482			−0.415741	−	0.909483i	$-0.636478\pi$
$271$	−61064.9			−0.831482			−0.415741	+	0.909483i	$0.636478\pi$
$272$		0			0
$273$	163881.			2.19889
$274$		0			0
$275$	−6965.85			−0.0921104
$276$		0			0
$277$	−31384.3			−0.409027			−0.204514	−	0.978864i	$-0.565561\pi$
$277$	−31384.3			−0.409027			−0.204514	+	0.978864i	$0.565561\pi$
$278$		0			0
$279$		−	295972.i		−	3.80227i
$280$		0			0
$281$			79910.3i			1.01202i	0.862527	+	0.506011i	$0.168880\pi$
$281$			79910.3i			1.01202i	−0.862527	+	0.506011i	$0.831120\pi$
$282$		0			0
$283$	99909.8			1.24749			0.623743	−	0.781630i	$-0.285611\pi$
$283$	99909.8			1.24749			0.623743	+	0.781630i	$0.285611\pi$
$284$		0			0
$285$	−58975.1	+	67585.9i	−0.726071	+	0.832082i
$286$		0			0
$287$		−	19385.7i		−	0.235351i
$288$		0			0
$289$	−52004.8			−0.622655
$290$		0			0
$291$	43103.8			0.509014
$292$		0			0
$293$		−	1641.27i		−	0.0191182i	−0.999954	−	0.00955908i	$-0.996957\pi$
$293$		−	1641.27i		−	0.0191182i	0.999954	−	0.00955908i	$-0.00304279\pi$
$294$		0			0
$295$			67294.9i			0.773283i
$296$		0			0
$297$		−	34832.8i		−	0.394890i
$298$		0			0
$299$			82057.3i			0.917857i
$300$		0			0
$301$	−44901.3			−0.495594
$302$		0			0
$303$			118589.i			1.29170i
$304$		0			0
$305$	−11374.6			−0.122275
$306$		0			0
$307$		−	72395.1i		−	0.768126i	−0.923307	−	0.384063i	$-0.874524\pi$
$307$		−	72395.1i		−	0.768126i	0.923307	−	0.384063i	$-0.125476\pi$
$308$		0			0
$309$	139206.			1.45795
$310$		0			0
$311$	124301.			1.28515			0.642573	−	0.766225i	$-0.277867\pi$
$311$	124301.			1.28515			0.642573	+	0.766225i	$0.277867\pi$
$312$		0			0
$313$	87747.3			0.895664			0.447832	−	0.894118i	$-0.352196\pi$
$313$	87747.3			0.895664			0.447832	+	0.894118i	$0.352196\pi$
$314$		0			0
$315$	−177879.			−1.79268
$316$		0			0
$317$		−	40640.3i		−	0.404426i	−0.979342	−	0.202213i	$-0.935187\pi$
$317$		−	40640.3i		−	0.404426i	0.979342	−	0.202213i	$-0.0648133\pi$
$318$		0			0
$319$		−	8572.74i		−	0.0842439i
$320$		0			0
$321$	108892.			1.05678
$322$		0			0
$323$	−48288.4	−	42136.2i	−0.462847	−	0.403878i
$324$		0			0
$325$		−	66462.6i		−	0.629232i
$326$		0			0
$327$	100775.			0.942450
$328$		0			0
$329$	−134626.			−1.24377
$330$		0			0
$331$			146431.i			1.33652i	0.743927	+	0.668260i	$0.232961\pi$
$331$			146431.i			1.33652i	−0.743927	+	0.668260i	$0.767039\pi$
$332$		0			0
$333$			121313.i			1.09401i
$334$		0			0
$335$			23836.2i			0.212397i
$336$		0			0
$337$			191079.i			1.68249i	0.540654	+	0.841245i	$0.318177\pi$
$337$			191079.i			1.68249i	−0.540654	+	0.841245i	$0.681823\pi$
$338$		0			0
$339$	342089.			2.97674
$340$		0			0
$341$			25098.7i			0.215845i
$342$		0			0
$343$	74302.9			0.631564
$344$		0			0
$345$		−	124792.i		−	1.04846i
$346$		0			0
$347$	−75833.7			−0.629801			−0.314900	−	0.949125i	$-0.601971\pi$
$347$	−75833.7			−0.629801			−0.314900	+	0.949125i	$0.601971\pi$
$348$		0			0
$349$	167272.			1.37332			0.686659	−	0.726980i	$-0.259077\pi$
$349$	167272.			1.37332			0.686659	+	0.726980i	$0.259077\pi$
$350$		0			0
$351$	332347.			2.69760
$352$		0			0
$353$	71568.7			0.574346			0.287173	−	0.957879i	$-0.407285\pi$
$353$	71568.7			0.574346			0.287173	+	0.957879i	$0.407285\pi$
$354$		0			0
$355$		−	113401.i		−	0.899833i
$356$		0			0
$357$		−	178069.i		−	1.39718i
$358$		0			0
$359$	121681.			0.944132			0.472066	−	0.881563i	$-0.343508\pi$
$359$	121681.			0.944132			0.472066	+	0.881563i	$0.343508\pi$
$360$		0			0
$361$	17651.5	+	129120.i	0.135446	+	0.990785i
$362$		0			0
$363$		−	241338.i		−	1.83152i
$364$		0			0
$365$	34433.2			0.258459
$366$		0			0
$367$	−233608.			−1.73443			−0.867214	−	0.497935i	$-0.834092\pi$
$367$	−233608.			−1.73443			−0.867214	+	0.497935i	$0.834092\pi$
$368$		0			0
$369$		−	65645.7i		−	0.482118i
$370$		0			0
$371$		−	89519.8i		−	0.650387i
$372$		0			0
$373$			42576.8i			0.306024i	0.988224	+	0.153012i	$0.0488973\pi$
$373$			42576.8i			0.306024i	−0.988224	+	0.153012i	$0.951103\pi$
$374$		0			0
$375$			256372.i			1.82309i
$376$		0			0
$377$	81794.3			0.575493
$378$		0			0
$379$		−	99133.0i		−	0.690144i	−0.938576	−	0.345072i	$-0.887854\pi$
$379$		−	99133.0i		−	0.690144i	0.938576	−	0.345072i	$-0.112146\pi$
$380$		0			0
$381$	255657.			1.76119
$382$		0			0
$383$		−	198667.i		−	1.35434i	−0.735826	−	0.677171i	$-0.763206\pi$
$383$		−	198667.i		−	1.35434i	0.735826	−	0.677171i	$-0.236794\pi$
$384$		0			0
$385$	15084.3			0.101766
$386$		0			0
$387$	−152049.			−1.01523
$388$		0			0
$389$	−252659.			−1.66969			−0.834845	−	0.550485i	$-0.814443\pi$
$389$	−252659.			−1.66969			−0.834845	+	0.550485i	$0.814443\pi$
$390$		0			0
$391$	89161.0			0.583205
$392$		0			0
$393$		−	499645.i		−	3.23501i
$394$		0			0
$395$			788.777i			0.00505545i
$396$		0			0
$397$	37586.2			0.238478			0.119239	−	0.992866i	$-0.461955\pi$
$397$	37586.2			0.238478			0.119239	+	0.992866i	$0.461955\pi$
$398$		0			0
$399$	−238072.	+	272832.i	−1.49542	+	1.71376i
$400$		0			0
$401$			175688.i			1.09258i	0.837596	+	0.546290i	$0.183960\pi$
$401$			175688.i			1.09258i	−0.837596	+	0.546290i	$0.816040\pi$
$402$		0			0
$403$	−239472.			−1.47450
$404$		0			0
$405$	−263814.			−1.60838
$406$		0			0
$407$		−	10287.5i		−	0.0621041i
$408$		0			0
$409$		−	276594.i		−	1.65347i	−0.562591	−	0.826735i	$-0.690195\pi$
$409$		−	276594.i		−	1.65347i	0.562591	−	0.826735i	$-0.309805\pi$
$410$		0			0
$411$			119100.i			0.705066i
$412$		0			0
$413$			271658.i			1.59266i
$414$		0			0
$415$	139422.			0.809534
$416$		0			0
$417$			274804.i			1.58034i
$418$		0			0
$419$	91901.2			0.523471			0.261736	−	0.965140i	$-0.415705\pi$
$419$	91901.2			0.523471			0.261736	+	0.965140i	$0.415705\pi$
$420$		0			0
$421$		−	154865.i		−	0.873753i	−0.899521	−	0.436877i	$-0.856085\pi$
$421$		−	154865.i		−	0.873753i	0.899521	−	0.436877i	$-0.143915\pi$
$422$		0			0
$423$	−455886.			−2.54786
$424$		0			0
$425$	−72216.3			−0.399813
$426$		0			0
$427$	−45917.4			−0.251838
$428$		0			0
$429$	−47060.5			−0.255706
$430$		0			0
$431$			110424.i			0.594443i	0.954809	+	0.297221i	$0.0960600\pi$
$431$			110424.i			0.594443i	−0.954809	+	0.297221i	$0.903940\pi$
$432$		0			0
$433$		−	34749.0i		−	0.185339i	−0.995697	−	0.0926694i	$-0.970460\pi$
$433$		−	34749.0i		−	0.185339i	0.995697	−	0.0926694i	$-0.0295400\pi$
$434$		0			0
$435$	−124393.			−0.657379
$436$		0			0
$437$	−136610.	−	119205.i	−0.715353	−	0.624214i
$438$		0			0
$439$			82718.3i			0.429213i	0.976701	+	0.214606i	$0.0688468\pi$
$439$			82718.3i			0.429213i	−0.976701	+	0.214606i	$0.931153\pi$
$440$		0			0
$441$	−233227.			−1.19923
$442$		0			0
$443$	214703.			1.09404			0.547018	−	0.837121i	$-0.315763\pi$
$443$	214703.			1.09404			0.547018	+	0.837121i	$0.315763\pi$
$444$		0			0
$445$			136465.i			0.689128i
$446$		0			0
$447$		−	204005.i		−	1.02100i
$448$		0			0
$449$			200750.i			0.995781i	0.867240	+	0.497891i	$0.165892\pi$
$449$			200750.i			0.995781i	−0.867240	+	0.497891i	$0.834108\pi$
$450$		0			0
$451$			5566.82i			0.0273687i
$452$		0			0
$453$	−446465.			−2.17566
$454$		0			0
$455$			143922.i			0.695193i
$456$		0			0
$457$	130986.			0.627183			0.313591	−	0.949558i	$-0.398468\pi$
$457$	130986.			0.627183			0.313591	+	0.949558i	$0.398468\pi$
$458$		0			0
$459$		−	361118.i		−	1.71405i
$460$		0			0
$461$	35097.5			0.165149			0.0825743	−	0.996585i	$-0.473686\pi$
$461$	35097.5			0.165149			0.0825743	+	0.996585i	$0.473686\pi$
$462$		0			0
$463$	28532.2			0.133099			0.0665493	−	0.997783i	$-0.478801\pi$
$463$	28532.2			0.133099			0.0665493	+	0.997783i	$0.478801\pi$
$464$		0			0
$465$	364188.			1.68430
$466$		0			0
$467$	−402160.			−1.84402			−0.922008	−	0.387170i	$-0.873453\pi$
$467$	−402160.			−1.84402			−0.922008	+	0.387170i	$0.873453\pi$
$468$		0			0
$469$			96222.7i			0.437453i
$470$		0			0
$471$		−	139479.i		−	0.628736i
$472$		0			0
$473$	12893.9			0.0576319
$474$		0			0
$475$	110648.	+	96550.9i	0.490407	+	0.427926i
$476$		0			0
$477$		−	303141.i		−	1.33232i
$478$		0			0
$479$	272020.			1.18558			0.592788	−	0.805359i	$-0.298027\pi$
$479$	272020.			1.18558			0.592788	+	0.805359i	$0.298027\pi$
$480$		0			0
$481$	98154.9			0.424250
$482$		0			0
$483$		−	503766.i		−	2.15941i
$484$		0			0
$485$			37854.2i			0.160928i
$486$		0			0
$487$			412092.i			1.73754i	0.495212	+	0.868772i	$0.335090\pi$
$487$			412092.i			1.73754i	−0.495212	+	0.868772i	$0.664910\pi$
$488$		0			0
$489$		−	280965.i		−	1.17499i
$490$		0			0
$491$	−343889.			−1.42645			−0.713223	−	0.700938i	$-0.752765\pi$
$491$	−343889.			−1.42645			−0.713223	+	0.700938i	$0.752765\pi$
$492$		0			0
$493$		−	88875.3i		−	0.365668i
$494$		0			0
$495$	51080.0			0.208468
$496$		0			0
$497$		−	457782.i		−	1.85330i
$498$		0			0
$499$	166687.			0.669425			0.334712	−	0.942320i	$-0.391361\pi$
$499$	166687.			0.669425			0.334712	+	0.942320i	$0.391361\pi$
$500$		0			0
$501$	341782.			1.36168
$502$		0			0
$503$	341395.			1.34934			0.674669	−	0.738121i	$-0.264286\pi$
$503$	341395.			1.34934			0.674669	+	0.738121i	$0.264286\pi$
$504$		0			0
$505$	−104146.			−0.408377
$506$		0			0
$507$			31399.1i			0.122152i
$508$		0			0
$509$			390009.i			1.50535i	0.658390	+	0.752677i	$0.271238\pi$
$509$			390009.i			1.50535i	−0.658390	+	0.752677i	$0.728762\pi$
$510$		0			0
$511$	139001.			0.532323
$512$		0			0
$513$	−482804.	+	553297.i	−1.83458	+	2.10244i
$514$		0			0
$515$			122252.i			0.460938i
$516$		0			0
$517$	38659.6			0.144636
$518$		0			0
$519$	−329185.			−1.22209
$520$		0			0
$521$			385501.i			1.42020i	0.704099	+	0.710102i	$0.251351\pi$
$521$			385501.i			1.42020i	−0.704099	+	0.710102i	$0.748649\pi$
$522$		0			0
$523$		−	480249.i		−	1.75575i	−0.478889	−	0.877875i	$-0.658961\pi$
$523$		−	480249.i		−	1.75575i	0.478889	−	0.877875i	$-0.341039\pi$
$524$		0			0
$525$			408027.i			1.48037i
$526$		0			0
$527$			260203.i			0.936896i
$528$		0			0
$529$	−27599.8			−0.0986267
$530$		0			0
$531$			919916.i			3.26257i
$532$		0			0
$533$	−53114.2			−0.186963
$534$		0			0
$535$			95629.9i			0.334107i
$536$		0			0
$537$	−390797.			−1.35520
$538$		0			0
$539$	19777.9			0.0680772
$540$		0			0
$541$	398980.			1.36319			0.681595	−	0.731730i	$-0.261287\pi$
$541$	398980.			1.36319			0.681595	+	0.731730i	$0.261287\pi$
$542$		0			0
$543$	896420.			3.04027
$544$		0			0
$545$			88501.8i			0.297961i
$546$		0			0
$547$			4309.05i			0.0144015i	0.999974	+	0.00720073i	$0.00229208\pi$
$547$			4309.05i			0.0144015i	−0.999974	+	0.00720073i	$0.997708\pi$
$548$		0			0
$549$	−155490.			−0.515891
$550$		0			0
$551$	−118823.	+	136172.i	−0.391380	+	0.448524i
$552$		0			0
$553$			3184.16i			0.0104122i
$554$		0			0
$555$	−149274.			−0.484615
$556$		0			0
$557$	−208935.			−0.673444			−0.336722	−	0.941604i	$-0.609318\pi$
$557$	−208935.			−0.673444			−0.336722	+	0.941604i	$0.609318\pi$
$558$		0			0
$559$			123024.i			0.393699i
$560$		0			0
$561$			51134.5i			0.162476i
$562$		0			0
$563$			552442.i			1.74289i	0.490493	+	0.871445i	$0.336817\pi$
$563$			552442.i			1.74289i	−0.490493	+	0.871445i	$0.663183\pi$
$564$		0			0
$565$			300426.i			0.941111i
$566$		0			0
$567$	−1.06497e6			−3.31262
$568$		0			0
$569$			376174.i			1.16189i	0.813943	+	0.580944i	$0.197317\pi$
$569$			376174.i			1.16189i	−0.813943	+	0.580944i	$0.802683\pi$
$570$		0			0
$571$	390679.			1.19825			0.599125	−	0.800656i	$-0.295515\pi$
$571$	390679.			1.19825			0.599125	+	0.800656i	$0.295515\pi$
$572$		0			0
$573$		−	91171.7i		−	0.277684i
$574$		0			0
$575$	−204304.			−0.617932
$576$		0			0
$577$	−374980.			−1.12631			−0.563153	−	0.826352i	$-0.690412\pi$
$577$	−374980.			−1.12631			−0.563153	+	0.826352i	$0.690412\pi$
$578$		0			0
$579$	−654633.			−1.95272
$580$		0			0
$581$	562823.			1.66732
$582$		0			0
$583$			25706.7i			0.0756326i
$584$		0			0
$585$			487364.i			1.42411i
$586$		0			0
$587$	−269098.			−0.780970			−0.390485	−	0.920609i	$-0.627693\pi$
$587$	−269098.			−0.780970			−0.390485	+	0.920609i	$0.627693\pi$
$588$		0			0
$589$	347884.	−	398677.i	1.00277	−	1.14919i
$590$		0			0
$591$		−	691697.i		−	1.98035i
$592$		0			0
$593$	56198.5			0.159814			0.0799071	−	0.996802i	$-0.474538\pi$
$593$	56198.5			0.159814			0.0799071	+	0.996802i	$0.474538\pi$
$594$		0			0
$595$	156382.			0.441725
$596$		0			0
$597$		−	383155.i		−	1.07504i
$598$		0			0
$599$			149473.i			0.416592i	0.978066	+	0.208296i	$0.0667917\pi$
$599$			149473.i			0.416592i	−0.978066	+	0.208296i	$0.933208\pi$
$600$		0			0
$601$			111386.i			0.308377i	0.988041	+	0.154188i	$0.0492763\pi$
$601$			111386.i			0.308377i	−0.988041	+	0.154188i	$0.950724\pi$
$602$		0			0
$603$			325839.i			0.896125i
$604$		0			0
$605$	211945.			0.579046
$606$		0			0
$607$		−	496593.i		−	1.34779i	−0.738826	−	0.673896i	$-0.764619\pi$
$607$		−	496593.i		−	1.34779i	0.738826	−	0.673896i	$-0.235381\pi$
$608$		0			0
$609$	−502151.			−1.35394
$610$		0			0
$611$			368859.i			0.988047i
$612$		0			0
$613$	21019.3			0.0559369			0.0279684	−	0.999609i	$-0.491096\pi$
$613$	21019.3			0.0559369			0.0279684	+	0.999609i	$0.491096\pi$
$614$		0			0
$615$	80775.8			0.213566
$616$		0			0
$617$	−415170.			−1.09058			−0.545288	−	0.838249i	$-0.683580\pi$
$617$	−415170.			−1.09058			−0.545288	+	0.838249i	$0.683580\pi$
$618$		0			0
$619$	40823.3			0.106543			0.0532717	−	0.998580i	$-0.483035\pi$
$619$	40823.3			0.106543			0.0532717	+	0.998580i	$0.483035\pi$
$620$		0			0
$621$		−	1.02162e6i		−	2.64915i
$622$		0			0
$623$			550884.i			1.41933i
$624$		0			0
$625$	29093.9			0.0744804
$626$		0			0
$627$	68365.2	−	78347.0i	0.173900	−	0.199291i
$628$		0			0
$629$		−	106652.i		−	0.269568i
$630$		0			0
$631$	515651.			1.29508			0.647541	−	0.762030i	$-0.275797\pi$
$631$	515651.			1.29508			0.647541	+	0.762030i	$0.275797\pi$
$632$		0			0
$633$	1.35355e6			3.37807
$634$		0			0
$635$			224520.i			0.556811i
$636$		0			0
$637$			188705.i			0.465054i
$638$		0			0
$639$		−	1.55019e6i		−	3.79649i
$640$		0			0
$641$		−	100947.i		−	0.245685i	−0.992426	−	0.122843i	$-0.960799\pi$
$641$		−	100947.i		−	0.245685i	0.992426	−	0.122843i	$-0.0392010\pi$
$642$		0			0
$643$	256498.			0.620386			0.310193	−	0.950674i	$-0.399606\pi$
$643$	256498.			0.620386			0.310193	+	0.950674i	$0.399606\pi$
$644$		0			0
$645$		−	187094.i		−	0.449718i
$646$		0			0
$647$	−752626.			−1.79792			−0.898961	−	0.438029i	$-0.855677\pi$
$647$	−752626.			−1.79792			−0.898961	+	0.438029i	$0.855677\pi$
$648$		0			0
$649$		−	78009.8i		−	0.185208i
$650$		0			0
$651$	1.47017e6			3.46900
$652$		0			0
$653$	−703535.			−1.64991			−0.824953	−	0.565201i	$-0.808798\pi$
$653$	−703535.			−1.64991			−0.824953	+	0.565201i	$0.808798\pi$
$654$		0			0
$655$	438793.			1.02277
$656$		0			0
$657$	470699.			1.09047
$658$		0			0
$659$		−	813907.i		−	1.87415i	−0.349130	−	0.937074i	$-0.613523\pi$
$659$		−	813907.i		−	1.87415i	0.349130	−	0.937074i	$-0.386477\pi$
$660$		0			0
$661$		−	426236.i		−	0.975545i	−0.872971	−	0.487772i	$-0.837810\pi$
$661$		−	426236.i		−	0.975545i	0.872971	−	0.487772i	$-0.162190\pi$
$662$		0			0
$663$	−487885.			−1.10992
$664$		0			0
$665$	−239604.	−	209077.i	−0.541815	−	0.472785i
$666$		0			0
$667$		−	251433.i		−	0.565158i
$668$		0			0
$669$	144054.			0.321865
$670$		0			0
$671$	13185.7			0.0292859
$672$		0			0
$673$		−	513214.i		−	1.13310i	−0.824027	−	0.566550i	$-0.808278\pi$
$673$		−	513214.i		−	1.13310i	0.824027	−	0.566550i	$-0.191722\pi$
$674$		0			0
$675$			827467.i			1.81611i
$676$		0			0
$677$		−	139181.i		−	0.303671i	−0.988406	−	0.151836i	$-0.951482\pi$
$677$		−	139181.i		−	0.303671i	0.988406	−	0.151836i	$-0.0485184\pi$
$678$		0			0
$679$			152811.i			0.331447i
$680$		0			0
$681$	−634795.			−1.36880
$682$		0			0
$683$		−	340635.i		−	0.730211i	−0.930966	−	0.365105i	$-0.881033\pi$
$683$		−	340635.i		−	0.730211i	0.930966	−	0.365105i	$-0.118967\pi$
$684$		0			0
$685$	−104595.			−0.222910
$686$		0			0
$687$		−	740590.i		−	1.56915i
$688$		0			0
$689$	−245273.			−0.516667
$690$		0			0
$691$	−665671.			−1.39413			−0.697065	−	0.717008i	$-0.745511\pi$
$691$	−665671.			−1.39413			−0.697065	+	0.717008i	$0.745511\pi$
$692$		0			0
$693$	206201.			0.429363
$694$		0			0
$695$	−241335.			−0.499633
$696$		0			0
$697$			57712.3i			0.118796i
$698$		0			0
$699$		−	91046.8i		−	0.186342i
$700$		0			0
$701$	−287944.			−0.585965			−0.292982	−	0.956118i	$-0.594648\pi$
$701$	−287944.			−0.585965			−0.292982	+	0.956118i	$0.594648\pi$
$702$		0			0
$703$	−142591.	+	163410.i	−0.288523	+	0.330649i
$704$		0			0
$705$		−	560959.i		−	1.12863i
$706$		0			0
$707$	−420420.			−0.841095
$708$		0			0
$709$	461125.			0.917331			0.458665	−	0.888609i	$-0.348328\pi$
$709$	461125.			0.917331			0.458665	+	0.888609i	$0.348328\pi$
$710$		0			0
$711$			10782.5i			0.0213295i
$712$		0			0
$713$			736129.i			1.44802i
$714$		0			0
$715$		−	41329.0i		−	0.0808430i
$716$		0			0
$717$			404858.i			0.787525i
$718$		0			0
$719$	−680993.			−1.31730			−0.658650	−	0.752449i	$-0.728872\pi$
$719$	−680993.			−1.31730			−0.658650	+	0.752449i	$0.728872\pi$
$720$		0			0
$721$			493511.i			0.949349i
$722$		0			0
$723$	414651.			0.793243
$724$		0			0
$725$			203649.i			0.387441i
$726$		0			0
$727$	−11537.8			−0.0218301			−0.0109150	−	0.999940i	$-0.503474\pi$
$727$	−11537.8			−0.0218301			−0.0109150	+	0.999940i	$0.503474\pi$
$728$		0			0
$729$	−834857.			−1.57093
$730$		0			0
$731$	133674.			0.250156
$732$		0			0
$733$	27216.9			0.0506560			0.0253280	−	0.999679i	$-0.491937\pi$
$733$	27216.9			0.0506560			0.0253280	+	0.999679i	$0.491937\pi$
$734$		0			0
$735$		−	286981.i		−	0.531226i
$736$		0			0
$737$		−	27631.5i		−	0.0508708i
$738$		0			0
$739$	−488729.			−0.894909			−0.447455	−	0.894307i	$-0.647669\pi$
$739$	−488729.			−0.894909			−0.447455	+	0.894307i	$0.647669\pi$
$740$		0			0
$741$	747525.	+	652287.i	1.36141	+	1.18796i
$742$		0			0
$743$			4601.59i			0.00833548i	0.999991	+	0.00416774i	$0.00132664\pi$
$743$			4601.59i			0.00833548i	−0.999991	+	0.00416774i	$0.998673\pi$
$744$		0			0
$745$	179159.			0.322794
$746$		0			0
$747$	1.90589e6			3.41552
$748$		0			0
$749$			386041.i			0.688129i
$750$		0			0
$751$			62141.6i			0.110180i	0.998481	+	0.0550900i	$0.0175446\pi$
$751$			62141.6i			0.110180i	−0.998481	+	0.0550900i	$0.982455\pi$
$752$		0			0
$753$		−	1.54513e6i		−	2.72505i
$754$		0			0
$755$		−	392090.i		−	0.687847i
$756$		0			0
$757$	−295159.			−0.515067			−0.257533	−	0.966269i	$-0.582910\pi$
$757$	−295159.			−0.515067			−0.257533	+	0.966269i	$0.582910\pi$
$758$		0			0
$759$			144662.i			0.251114i
$760$		0			0
$761$	327534.			0.565571			0.282785	−	0.959183i	$-0.408742\pi$
$761$	327534.			0.565571			0.282785	+	0.959183i	$0.408742\pi$
$762$		0			0
$763$			357266.i			0.613681i
$764$		0			0
$765$	529556.			0.904875
$766$		0			0
$767$	744307.			1.26521
$768$		0			0
$769$	805330.			1.36182			0.680912	−	0.732365i	$-0.261583\pi$
$769$	805330.			1.36182			0.680912	+	0.732365i	$0.261583\pi$
$770$		0			0
$771$	1.07065e6			1.80111
$772$		0			0
$773$			585854.i			0.980462i	0.871593	+	0.490231i	$0.163088\pi$
$773$			585854.i			0.980462i	−0.871593	+	0.490231i	$0.836912\pi$
$774$		0			0
$775$		−	596230.i		−	0.992683i
$776$		0			0
$777$	−602592.			−0.998117
$778$		0			0
$779$	77159.5	−	88425.3i	0.127149	−	0.145714i
$780$		0			0
$781$			131457.i			0.215518i
$782$		0			0
$783$	−1.01835e6			−1.66101
$784$		0			0
$785$	122492.			0.198778
$786$		0			0
$787$		−	762631.i		−	1.23130i	−0.788018	−	0.615652i	$-0.788893\pi$
$787$		−	762631.i		−	1.23130i	0.788018	−	0.615652i	$-0.211107\pi$
$788$		0			0
$789$		−	1.90194e6i		−	3.05522i
$790$		0			0
$791$			1.21277e6i			1.93832i
$792$		0			0
$793$			125808.i			0.200060i
$794$		0			0
$795$	373010.			0.590182
$796$		0			0
$797$		−	739146.i		−	1.16363i	−0.813322	−	0.581813i	$-0.802343\pi$
$797$		−	739146.i		−	1.16363i	0.813322	−	0.581813i	$-0.197657\pi$
$798$		0			0
$799$	400791.			0.627804
$800$		0			0
$801$			1.86546e6i			2.90751i
$802$		0			0
$803$	−39915.7			−0.0619031
$804$		0			0
$805$	442412.			0.682708
$806$		0			0
$807$	922685.			1.41679
$808$		0			0
$809$	−283400.			−0.433015			−0.216508	−	0.976281i	$-0.569467\pi$
$809$	−283400.			−0.433015			−0.216508	+	0.976281i	$0.569467\pi$
$810$		0			0
$811$			14690.1i			0.0223348i	0.999938	+	0.0111674i	$0.00355477\pi$
$811$			14690.1i			0.0223348i	−0.999938	+	0.0111674i	$0.996445\pi$
$812$		0			0
$813$		−	1.02715e6i		−	1.55400i
$814$		0			0
$815$	246746.			0.371479
$816$		0			0
$817$	−204812.	−	178718.i	−0.306839	−	0.267746i
$818$		0			0
$819$			1.96741e6i			2.93310i
$820$		0			0
$821$	668244.			0.991399			0.495699	−	0.868494i	$-0.334912\pi$
$821$	668244.			0.991399			0.495699	+	0.868494i	$0.334912\pi$
$822$		0			0
$823$	−6475.45			−0.00956028			−0.00478014	−	0.999989i	$-0.501522\pi$
$823$	−6475.45			−0.00956028			−0.00478014	+	0.999989i	$0.501522\pi$
$824$		0			0
$825$		−	117170.i		−	0.172150i
$826$		0			0
$827$		−	55503.2i		−	0.0811534i	−0.999176	−	0.0405767i	$-0.987080\pi$
$827$		−	55503.2i		−	0.0811534i	0.999176	−	0.0405767i	$-0.0129195\pi$
$828$		0			0
$829$			475443.i			0.691813i	0.938269	+	0.345907i	$0.112429\pi$
$829$			475443.i			0.691813i	−0.938269	+	0.345907i	$0.887571\pi$
$830$		0			0
$831$		−	527902.i		−	0.764453i
$832$		0			0
$833$	205041.			0.295495
$834$		0			0
$835$			300156.i			0.430501i
$836$		0			0
$837$	2.98145e6			4.25576
$838$		0			0
$839$			389106.i			0.552770i	0.961047	+	0.276385i	$0.0891364\pi$
$839$			389106.i			0.552770i	−0.961047	+	0.276385i	$0.910864\pi$
$840$		0			0
$841$	456654.			0.645647
$842$		0			0
$843$	−1.34414e6			−1.89142
$844$		0			0
$845$	−27575.0			−0.0386190
$846$		0			0
$847$	855586.			1.19261
$848$		0			0
$849$			1.68054e6i			2.33149i
$850$		0			0
$851$		−	301725.i		−	0.416631i
$852$		0			0
$853$	−1.17337e6			−1.61264			−0.806322	−	0.591476i	$-0.798545\pi$
$853$	−1.17337e6			−1.61264			−0.806322	+	0.591476i	$0.798545\pi$
$854$		0			0
$855$	−811372.	−	707999.i	−1.10991	−	0.968502i
$856$		0			0
$857$		−	170943.i		−	0.232750i	−0.993205	−	0.116375i	$-0.962873\pi$
$857$		−	170943.i		−	0.232750i	0.993205	−	0.116375i	$-0.0371274\pi$
$858$		0			0
$859$	−494381.			−0.670002			−0.335001	−	0.942218i	$-0.608737\pi$
$859$	−494381.			−0.670002			−0.335001	+	0.942218i	$0.608737\pi$
$860$		0			0
$861$	326078.			0.439861
$862$		0			0
$863$			1.38223e6i			1.85591i	0.372691	+	0.927956i	$0.378435\pi$
$863$			1.38223e6i			1.85591i	−0.372691	+	0.927956i	$0.621565\pi$
$864$		0			0
$865$		−	289093.i		−	0.386372i
$866$		0			0
$867$		−	874751.i		−	1.16371i
$868$		0			0
$869$		−	914.368i		−	0.00121083i
$870$		0			0
$871$	263637.			0.347513
$872$		0			0
$873$			517464.i			0.678971i
$874$		0			0
$875$	−908886.			−1.18712
$876$		0			0
$877$		−	1.30994e6i		−	1.70315i	−0.524235	−	0.851574i	$-0.675649\pi$
$877$		−	1.30994e6i		−	1.70315i	0.524235	−	0.851574i	$-0.324351\pi$
$878$		0			0
$879$	27607.2			0.0357309
$880$		0			0
$881$	−1.16822e6			−1.50513			−0.752565	−	0.658517i	$-0.771184\pi$
$881$	−1.16822e6			−1.50513			−0.752565	+	0.658517i	$0.771184\pi$
$882$		0			0
$883$	−761544.			−0.976728			−0.488364	−	0.872640i	$-0.662406\pi$
$883$	−761544.			−0.976728			−0.488364	+	0.872640i	$0.662406\pi$
$884$		0			0
$885$	−1.13194e6			−1.44523
$886$		0			0
$887$			347725.i			0.441966i	0.975278	+	0.220983i	$0.0709265\pi$
$887$			347725.i			0.441966i	−0.975278	+	0.220983i	$0.929074\pi$
$888$		0			0
$889$			906349.i			1.14681i
$890$		0			0
$891$	305819.			0.385220
$892$		0			0
$893$	−614082.	−	535845.i	−0.770058	−	0.671949i
$894$		0			0
$895$		−	343202.i		−	0.428453i
$896$		0			0
$897$	−1.38025e6			−1.71543
$898$		0			0
$899$	733769.			0.907905
$900$		0			0
$901$			266506.i			0.328290i
$902$		0			0
$903$		−	755266.i		−	0.926242i
$904$		0			0
$905$			787245.i			0.961198i
$906$		0			0
$907$		−	687599.i		−	0.835835i	−0.908485	−	0.417917i	$-0.862760\pi$
$907$		−	687599.i		−	0.835835i	0.908485	−	0.417917i	$-0.137240\pi$
$908$		0			0
$909$	−1.42367e6			−1.72299
$910$		0			0
$911$		−	449372.i		−	0.541464i	−0.962655	−	0.270732i	$-0.912734\pi$
$911$		−	449372.i		−	0.541464i	0.962655	−	0.270732i	$-0.0872657\pi$
$912$		0			0
$913$	−161621.			−0.193891
$914$		0			0
$915$		−	191328.i		−	0.228526i
$916$		0			0
$917$	1.77133e6			2.10650
$918$		0			0
$919$	−74612.4			−0.0883446			−0.0441723	−	0.999024i	$-0.514065\pi$
$919$	−74612.4			−0.0883446			−0.0441723	+	0.999024i	$0.514065\pi$
$920$		0			0
$921$	1.21773e6			1.43559
$922$		0			0
$923$	−1.25426e6			−1.47226
$924$		0			0
$925$			244383.i			0.285619i
$926$		0			0
$927$			1.67118e6i			1.94475i
$928$		0			0
$929$	178101.			0.206365			0.103182	−	0.994662i	$-0.467097\pi$
$929$	178101.			0.206365			0.103182	+	0.994662i	$0.467097\pi$
$930$		0			0
$931$	−314158.	−	274133.i	−0.362451	−	0.316273i
$932$		0			0
$933$			2.09081e6i			2.40188i
$934$		0			0
$935$	−44906.8			−0.0513676
$936$		0			0
$937$	−1.24353e6			−1.41637			−0.708186	−	0.706026i	$-0.750486\pi$
$937$	−1.24353e6			−1.41637			−0.708186	+	0.706026i	$0.750486\pi$
$938$		0			0
$939$			1.47596e6i			1.67395i
$940$		0			0
$941$		−	1.31010e6i		−	1.47954i	−0.672861	−	0.739769i	$-0.734935\pi$
$941$		−	1.31010e6i		−	1.47954i	0.672861	−	0.739769i	$-0.265065\pi$
$942$		0			0
$943$			163271.i			0.183606i
$944$		0			0
$945$		−	1.79185e6i		−	2.00649i
$946$		0			0
$947$	−660926.			−0.736975			−0.368488	−	0.929633i	$-0.620124\pi$
$947$	−660926.			−0.736975			−0.368488	+	0.929633i	$0.620124\pi$
$948$		0			0
$949$		−	380844.i		−	0.422877i
$950$		0			0
$951$	683594.			0.755853
$952$		0			0
$953$			61065.5i			0.0672373i	0.999435	+	0.0336187i	$0.0107032\pi$
$953$			61065.5i			0.0672373i	−0.999435	+	0.0336187i	$0.989297\pi$
$954$		0			0
$955$	80067.8			0.0877913
$956$		0			0
$957$	144199.			0.157448
$958$		0			0
$959$	−422232.			−0.459107
$960$		0			0
$961$	−1.22476e6			−1.32619
$962$		0			0
$963$			1.30725e6i			1.40964i
$964$		0			0
$965$		−	574905.i		−	0.617364i
$966$		0			0
$967$	269267.			0.287959			0.143979	−	0.989581i	$-0.454010\pi$
$967$	269267.			0.287959			0.143979	+	0.989581i	$0.454010\pi$
$968$		0			0
$969$	708755.	−	812238.i	0.754829	−	0.865040i
$970$		0			0
$971$		−	1.49753e6i		−	1.58832i	−0.607709	−	0.794160i	$-0.707911\pi$
$971$		−	1.49753e6i		−	1.58832i	0.607709	−	0.794160i	$-0.292089\pi$
$972$		0			0
$973$	−974228.			−1.02905
$974$		0			0
$975$	1.11794e6			1.17601
$976$		0			0
$977$		−	1.59483e6i		−	1.67080i	−0.549639	−	0.835402i	$-0.685235\pi$
$977$		−	1.59483e6i		−	1.67080i	0.549639	−	0.835402i	$-0.314765\pi$
$978$		0			0
$979$		−	158193.i		−	0.165052i
$980$		0			0
$981$			1.20981e6i			1.25713i
$982$		0			0
$983$		−	1.32566e6i		−	1.37191i	−0.727642	−	0.685957i	$-0.759384\pi$
$983$		−	1.32566e6i		−	1.37191i	0.727642	−	0.685957i	$-0.240616\pi$
$984$		0			0
$985$	607455.			0.626097
$986$		0			0
$987$		−	2.26450e6i		−	2.32454i
$988$		0			0
$989$	378170.			0.386629
$990$		0			0
$991$			666235.i			0.678391i	0.940716	+	0.339196i	$0.110155\pi$
$991$			666235.i			0.678391i	−0.940716	+	0.339196i	$0.889845\pi$
$992$		0			0
$993$	−2.46305e6			−2.49790
$994$		0			0
$995$	336490.			0.339881
$996$		0			0
$997$	−823511.			−0.828475			−0.414237	−	0.910169i	$-0.635952\pi$
$997$	−823511.			−0.828475			−0.414237	+	0.910169i	$0.635952\pi$
$998$		0			0
$999$	−1.22204e6			−1.22449

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.5.e.c.113.4		4
4.3	odd	2		19.5.b.b.18.3	yes	4
12.11	even	2		171.5.c.c.37.2		4
19.18	odd	2	inner	304.5.e.c.113.1		4
76.75	even	2		19.5.b.b.18.2	✓	4
228.227	odd	2		171.5.c.c.37.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.5.b.b.18.2	✓	4	76.75	even	2
19.5.b.b.18.3	yes	4	4.3	odd	2
171.5.c.c.37.2		4	12.11	even	2
171.5.c.c.37.3		4	228.227	odd	2
304.5.e.c.113.1		4	19.18	odd	2	inner
304.5.e.c.113.4		4	1.1	even	1	trivial

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

\(f(q)\)	\(=\)	\(q+16.8206i q^{3} -14.7720 q^{5} -59.6320 q^{7} -201.932 q^{9} +O(q^{10})\)	\(q+16.8206i q^{3} -14.7720 q^{5} -59.6320 q^{7} -201.932 q^{9} +17.1240 q^{11} +163.384i q^{13} -248.474i q^{15} +177.528 q^{17} +(-272.004 - 237.349i) q^{19} -1003.05i q^{21} +502.236 q^{23} -406.788 q^{25} -2034.15i q^{27} -500.627i q^{29} +1465.70i q^{31} +288.036i q^{33} +880.884 q^{35} -600.763i q^{37} -2748.21 q^{39} +325.088i q^{41} +752.973 q^{43} +2982.94 q^{45} +2257.62 q^{47} +1154.98 q^{49} +2986.13i q^{51} +1501.20i q^{53} -252.956 q^{55} +(3992.36 - 4575.27i) q^{57} -4555.57i q^{59} +770.012 q^{61} +12041.6 q^{63} -2413.51i q^{65} -1613.61i q^{67} +8447.91i q^{69} +7676.78i q^{71} -2330.98 q^{73} -6842.41i q^{75} -1021.14 q^{77} -53.3968i q^{79} +17859.1 q^{81} -9438.26 q^{83} -2622.44 q^{85} +8420.83 q^{87} -9238.06i q^{89} -9742.91i q^{91} -24654.0 q^{93} +(4018.05 + 3506.13i) q^{95} -2562.56i q^{97} -3457.89 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 42 q^{5} - 136 q^{7} - 278 q^{9}+O(q^{10}) \) 4 * q - 42 * q^5 - 136 * q^7 - 278 * q^9	\( 4 q - 42 q^{5} - 136 q^{7} - 278 q^{9} - 222 q^{11} + 300 q^{17} - 114 q^{19} + 78 q^{23} - 1986 q^{25} + 1866 q^{35} - 6362 q^{39} - 2986 q^{43} + 5182 q^{45} + 7578 q^{47} - 2352 q^{49} + 1090 q^{55} + 10450 q^{57} + 158 q^{61} + 23030 q^{63} - 15168 q^{73} + 102 q^{77} + 40712 q^{81} - 33276 q^{83} - 4902 q^{85} + 7214 q^{87} - 40004 q^{93} + 5358 q^{95} - 23042 q^{99}+O(q^{100}) \) 4 * q - 42 * q^5 - 136 * q^7 - 278 * q^9 - 222 * q^11 + 300 * q^17 - 114 * q^19 + 78 * q^23 - 1986 * q^25 + 1866 * q^35 - 6362 * q^39 - 2986 * q^43 + 5182 * q^45 + 7578 * q^47 - 2352 * q^49 + 1090 * q^55 + 10450 * q^57 + 158 * q^61 + 23030 * q^63 - 15168 * q^73 + 102 * q^77 + 40712 * q^81 - 33276 * q^83 - 4902 * q^85 + 7214 * q^87 - 40004 * q^93 + 5358 * q^95 - 23042 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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