LMFDB - Embedded newform 336.6.h.a.239.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,6,Mod(239,336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(336, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("336.239");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	336.h (of order $2$, degree $1$, 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:	$53.8889634572$
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} + 444940 x^{16} + 56262171366 x^{12} + \cdots + 77\!\cdots\!16 $ x^20 + 444940x^16 + 56262171366x^12 + 1699160102026524x^8 + 13319972960343660513x^4 + 772185348835324025616
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{35}\cdot 3^{14}\cdot 7^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			239.2
Root			$-15.5877 - 15.5877i$ of defining polynomial
Character	$\chi$	$=$	336.239
Dual form			336.6.h.a.239.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-15.5877 + 0.152257i) q^{3} +4.09696i q^{5} +49.0000i q^{7} +(242.954 - 4.74667i) q^{9} +O(q^{10})$	\(q+(-15.5877 + 0.152257i) q^{3} +4.09696i q^{5} +49.0000i q^{7} +(242.954 - 4.74667i) q^{9} +272.131 q^{11} -663.953 q^{13} +(-0.623790 - 63.8623i) q^{15} +1677.60i q^{17} -2038.56i q^{19} +(-7.46058 - 763.798i) q^{21} +1364.54 q^{23} +3108.21 q^{25} +(-3786.37 + 110.981i) q^{27} -4497.04i q^{29} -10289.0i q^{31} +(-4241.90 + 41.4338i) q^{33} -200.751 q^{35} -505.640 q^{37} +(10349.5 - 101.091i) q^{39} +16134.6i q^{41} +6006.38i q^{43} +(19.4469 + 995.372i) q^{45} -23040.8 q^{47} -2401.00 q^{49} +(-255.426 - 26150.0i) q^{51} +23596.2i q^{53} +1114.91i q^{55} +(310.385 + 31776.5i) q^{57} +31448.1 q^{59} +30296.6 q^{61} +(232.587 + 11904.7i) q^{63} -2720.19i q^{65} +18489.1i q^{67} +(-21270.1 + 207.760i) q^{69} +38569.6 q^{71} -58093.8 q^{73} +(-48450.0 + 473.247i) q^{75} +13334.4i q^{77} +79280.5i q^{79} +(59003.9 - 2306.44i) q^{81} -49464.7 q^{83} -6873.08 q^{85} +(684.705 + 70098.6i) q^{87} +114757. i q^{89} -32533.7i q^{91} +(1566.57 + 160382. i) q^{93} +8351.92 q^{95} +56497.0 q^{97} +(66115.3 - 1291.72i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 140 q^{9}+O(q^{10}) $ 20 * q - 140 * q^9	$ 20 q - 140 q^{9} + 1048 q^{13} - 980 q^{21} + 5916 q^{25} - 26056 q^{33} + 61360 q^{37} - 92512 q^{45} - 48020 q^{49} - 20720 q^{57} + 46680 q^{61} - 28360 q^{69} - 54280 q^{73} + 152660 q^{81} - 150536 q^{85} + 41688 q^{93} - 421352 q^{97}+O(q^{100}) $ 20 * q - 140 * q^9 + 1048 * q^13 - 980 * q^21 + 5916 * q^25 - 26056 * q^33 + 61360 * q^37 - 92512 * q^45 - 48020 * q^49 - 20720 * q^57 + 46680 * q^61 - 28360 * q^69 - 54280 * q^73 + 152660 * q^81 - 150536 * q^85 + 41688 * q^93 - 421352 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$.

$n$	$85$	$113$	$127$	$241$
$\chi(n)$	$1$	$-1$	$-1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−15.5877	+	0.152257i	−0.999952	+	0.00976727i
$3$	−15.5877	+	0.152257i	−0.999952	+	0.00976727i
$4$		0			0
$5$			4.09696i			0.0732887i	0.999328	+	0.0366444i	$0.0116669\pi$
$5$			4.09696i			0.0732887i	−0.999328	+	0.0366444i	$0.988333\pi$
$6$		0			0
$7$			49.0000i			0.377964i
$7$			49.0000i			0.377964i
$8$		0			0
$9$	242.954	−	4.74667i	0.999809	−	0.0195336i
$10$		0			0
$11$	272.131			0.678105			0.339052	−	0.940768i	$-0.389894\pi$
$11$	272.131			0.678105			0.339052	+	0.940768i	$0.389894\pi$
$12$		0			0
$13$	−663.953			−1.08963			−0.544814	−	0.838557i	$-0.683400\pi$
$13$	−663.953			−1.08963			−0.544814	+	0.838557i	$0.683400\pi$
$14$		0			0
$15$	−0.623790	−	63.8623i	−0.000715831	−	0.0732852i
$16$		0			0
$17$			1677.60i			1.40788i	0.710258	+	0.703942i	$0.248578\pi$
$17$			1677.60i			1.40788i	−0.710258	+	0.703942i	$0.751422\pi$
$18$		0			0
$19$		−	2038.56i		−	1.29551i	−0.761849	−	0.647754i	$-0.775708\pi$
$19$		−	2038.56i		−	1.29551i	0.761849	−	0.647754i	$-0.224292\pi$
$20$		0			0
$21$	−7.46058	−	763.798i	−0.00369168	−	0.377946i
$22$		0			0
$23$	1364.54			0.537857			0.268928	−	0.963160i	$-0.413330\pi$
$23$	1364.54			0.537857			0.268928	+	0.963160i	$0.413330\pi$
$24$		0			0
$25$	3108.21			0.994629
$26$		0			0
$27$	−3786.37	+	110.981i	−0.999571	+	0.0292981i
$28$		0			0
$29$		−	4497.04i		−	0.992961i	−0.868048	−	0.496480i	$-0.834625\pi$
$29$		−	4497.04i		−	0.992961i	0.868048	−	0.496480i	$-0.165375\pi$
$30$		0			0
$31$		−	10289.0i		−	1.92296i	−0.274883	−	0.961478i	$-0.588639\pi$
$31$		−	10289.0i		−	1.92296i	0.274883	−	0.961478i	$-0.411361\pi$
$32$		0			0
$33$	−4241.90	+	41.4338i	−0.678072	+	0.00662323i
$34$		0			0
$35$	−200.751			−0.0277005
$36$		0			0
$37$	−505.640			−0.0607207			−0.0303604	−	0.999539i	$-0.509665\pi$
$37$	−505.640			−0.0607207			−0.0303604	+	0.999539i	$0.509665\pi$
$38$		0			0
$39$	10349.5	−	101.091i	1.08958	−	0.0106427i
$40$		0			0
$41$			16134.6i			1.49899i	0.662013	+	0.749493i	$0.269702\pi$
$41$			16134.6i			1.49899i	−0.662013	+	0.749493i	$0.730298\pi$
$42$		0			0
$43$			6006.38i			0.495383i	0.968839	+	0.247692i	$0.0796720\pi$
$43$			6006.38i			0.495383i	−0.968839	+	0.247692i	$0.920328\pi$
$44$		0			0
$45$	19.4469	+	995.372i	0.00143159	+	0.0732747i
$46$		0			0
$47$	−23040.8			−1.52143			−0.760717	−	0.649083i	$-0.775153\pi$
$47$	−23040.8			−1.52143			−0.760717	+	0.649083i	$0.775153\pi$
$48$		0			0
$49$	−2401.00			−0.142857
$50$		0			0
$51$	−255.426	−	26150.0i	−0.0137512	−	1.40782i
$52$		0			0
$53$			23596.2i			1.15386i	0.816795	+	0.576928i	$0.195749\pi$
$53$			23596.2i			1.15386i	−0.816795	+	0.576928i	$0.804251\pi$
$54$		0			0
$55$			1114.91i			0.0496974i
$56$		0			0
$57$	310.385	+	31776.5i	0.0126536	+	1.29545i
$58$		0			0
$59$	31448.1			1.17615			0.588077	−	0.808805i	$-0.299885\pi$
$59$	31448.1			1.17615			0.588077	+	0.808805i	$0.299885\pi$
$60$		0			0
$61$	30296.6			1.04248			0.521242	−	0.853409i	$-0.325469\pi$
$61$	30296.6			1.04248			0.521242	+	0.853409i	$0.325469\pi$
$62$		0			0
$63$	232.587	+	11904.7i	0.00738301	+	0.377892i
$64$		0			0
$65$		−	2720.19i		−	0.0798575i
$66$		0			0
$67$			18489.1i			0.503187i	0.967833	+	0.251593i	$0.0809546\pi$
$67$			18489.1i			0.503187i	−0.967833	+	0.251593i	$0.919045\pi$
$68$		0			0
$69$	−21270.1	+	207.760i	−0.537831	+	0.00525339i
$70$		0			0
$71$	38569.6			0.908028			0.454014	−	0.890995i	$-0.349992\pi$
$71$	38569.6			0.908028			0.454014	+	0.890995i	$0.349992\pi$
$72$		0			0
$73$	−58093.8			−1.27592			−0.637959	−	0.770071i	$-0.720221\pi$
$73$	−58093.8			−1.27592			−0.637959	+	0.770071i	$0.720221\pi$
$74$		0			0
$75$	−48450.0	+	473.247i	−0.994581	+	0.00971481i
$76$		0			0
$77$			13334.4i			0.256299i
$78$		0			0
$79$			79280.5i			1.42922i	0.699524	+	0.714609i	$0.253395\pi$
$79$			79280.5i			1.42922i	−0.699524	+	0.714609i	$0.746605\pi$
$80$		0			0
$81$	59003.9	−	2306.44i	0.999237	−	0.0390598i
$82$		0			0
$83$	−49464.7			−0.788134			−0.394067	−	0.919082i	$-0.628932\pi$
$83$	−49464.7			−0.788134			−0.394067	+	0.919082i	$0.628932\pi$
$84$		0			0
$85$	−6873.08			−0.103182
$86$		0			0
$87$	684.705	+	70098.6i	0.00969852	+	0.992913i
$88$		0			0
$89$			114757.i			1.53569i	0.640634	+	0.767847i	$0.278672\pi$
$89$			114757.i			1.53569i	−0.640634	+	0.767847i	$0.721328\pi$
$90$		0			0
$91$		−	32533.7i		−	0.411841i
$92$		0			0
$93$	1566.57	+	160382.i	0.0187820	+	1.92286i
$94$		0			0
$95$	8351.92			0.0949461
$96$		0			0
$97$	56497.0			0.609672			0.304836	−	0.952405i	$-0.401398\pi$
$97$	56497.0			0.609672			0.304836	+	0.952405i	$0.401398\pi$
$98$		0			0
$99$	66115.3	−	1291.72i	0.677975	−	0.0132458i
$100$		0			0
$101$			171252.i			1.67045i	0.549911	+	0.835223i	$0.314662\pi$
$101$			171252.i			1.67045i	−0.549911	+	0.835223i	$0.685338\pi$
$102$		0			0
$103$			58072.4i			0.539357i	0.962950	+	0.269679i	$0.0869175\pi$
$103$			58072.4i			0.539357i	−0.962950	+	0.269679i	$0.913082\pi$
$104$		0			0
$105$	3129.25	−	30.5657i	0.0276992	−	0.000270559i
$106$		0			0
$107$	−45320.6			−0.382680			−0.191340	−	0.981524i	$-0.561283\pi$
$107$	−45320.6			−0.382680			−0.191340	+	0.981524i	$0.561283\pi$
$108$		0			0
$109$	4731.46			0.0381442			0.0190721	−	0.999818i	$-0.493929\pi$
$109$	4731.46			0.0381442			0.0190721	+	0.999818i	$0.493929\pi$
$110$		0			0
$111$	7881.77	−	76.9870i	0.0607178	−	0.000593076i
$112$		0			0
$113$			142120.i			1.04703i	0.852016	+	0.523516i	$0.175380\pi$
$113$			142120.i			1.04703i	−0.852016	+	0.523516i	$0.824620\pi$
$114$		0			0
$115$			5590.47i			0.0394188i
$116$		0			0
$117$	−161310.	+	3151.56i	−1.08942	+	0.0212844i
$118$		0			0
$119$	−82202.5			−0.532130
$120$		0			0
$121$	−86995.6			−0.540174
$122$		0			0
$123$	−2456.59	−	251501.i	−0.0146410	−	1.49891i
$124$		0			0
$125$			25537.3i			0.146184i
$126$		0			0
$127$		−	62469.6i		−	0.343684i	−0.985124	−	0.171842i	$-0.945028\pi$
$127$		−	62469.6i		−	0.343684i	0.985124	−	0.171842i	$-0.0549719\pi$
$128$		0			0
$129$	−914.511	−	93625.7i	−0.00483854	−	0.495360i
$130$		0			0
$131$	227591.			1.15872			0.579359	−	0.815072i	$-0.303303\pi$
$131$	227591.			1.15872			0.579359	+	0.815072i	$0.303303\pi$
$132$		0			0
$133$	99889.6			0.489656
$134$		0			0
$135$	−454.685	−	15512.6i	−0.00214722	−	0.0732573i
$136$		0			0
$137$			268762.i			1.22339i	0.791093	+	0.611696i	$0.209513\pi$
$137$			268762.i			1.22339i	−0.791093	+	0.611696i	$0.790487\pi$
$138$		0			0
$139$		−	124803.i		−	0.547885i	−0.961746	−	0.273943i	$-0.911672\pi$
$139$		−	124803.i		−	0.547885i	0.961746	−	0.273943i	$-0.0883278\pi$
$140$		0			0
$141$	359154.	−	3508.12i	1.52136	−	0.0148603i
$142$		0			0
$143$	−180682.			−0.738882
$144$		0			0
$145$	18424.2			0.0727728
$146$		0			0
$147$	37426.1	−	365.568i	0.142850	−	0.00139532i
$148$		0			0
$149$		−	188083.i		−	0.694038i	−0.937858	−	0.347019i	$-0.887194\pi$
$149$		−	188083.i		−	0.694038i	0.937858	−	0.347019i	$-0.112806\pi$
$150$		0			0
$151$			351527.i			1.25463i	0.778765	+	0.627315i	$0.215846\pi$
$151$			351527.i			1.25463i	−0.778765	+	0.627315i	$0.784154\pi$
$152$		0			0
$153$	7963.02	+	407580.i	0.0275010	+	1.40761i
$154$		0			0
$155$	42153.7			0.140931
$156$		0			0
$157$	−509930.			−1.65106			−0.825528	−	0.564361i	$-0.809123\pi$
$157$	−509930.			−1.65106			−0.825528	+	0.564361i	$0.809123\pi$
$158$		0			0
$159$	−3592.67	−	367810.i	−0.0112700	−	1.15380i
$160$		0			0
$161$			66862.5i			0.203291i
$162$		0			0
$163$		−	228751.i		−	0.674362i	−0.941440	−	0.337181i	$-0.890526\pi$
$163$		−	228751.i		−	0.674362i	0.941440	−	0.337181i	$-0.109474\pi$
$164$		0			0
$165$	−169.753	−	17378.9i	−0.000485408	−	0.0496950i
$166$		0			0
$167$	375939.			1.04310			0.521550	−	0.853220i	$-0.325354\pi$
$167$	375939.			1.04310			0.521550	+	0.853220i	$0.325354\pi$
$168$		0			0
$169$	69539.9			0.187291
$170$		0			0
$171$	−9676.38	−	495276.i	−0.0253060	−	1.29526i
$172$		0			0
$173$		−	4113.40i		−	0.0104493i	−0.999986	−	0.00522463i	$-0.998337\pi$
$173$		−	4113.40i		−	0.0104493i	0.999986	−	0.00522463i	$-0.00166306\pi$
$174$		0			0
$175$			152303.i			0.375934i
$176$		0			0
$177$	−490204.	+	4788.18i	−1.17610	+	0.0114878i
$178$		0			0
$179$	−586416.			−1.36796			−0.683980	−	0.729501i	$-0.739752\pi$
$179$	−586416.			−1.36796			−0.683980	+	0.729501i	$0.739752\pi$
$180$		0			0
$181$	628904.			1.42688			0.713441	−	0.700716i	$-0.247136\pi$
$181$	628904.			1.42688			0.713441	+	0.700716i	$0.247136\pi$
$182$		0			0
$183$	−472254.	+	4612.86i	−1.04243	+	0.0101822i
$184$		0			0
$185$		−	2071.59i		−	0.00445014i
$186$		0			0
$187$			456528.i			0.954692i
$188$		0			0
$189$	−5438.07	−	185532.i	−0.0110736	−	0.377802i
$190$		0			0
$191$	214154.			0.424759			0.212380	−	0.977187i	$-0.431879\pi$
$191$	214154.			0.424759			0.212380	+	0.977187i	$0.431879\pi$
$192$		0			0
$193$	303938.			0.587344			0.293672	−	0.955906i	$-0.405123\pi$
$193$	303938.			0.587344			0.293672	+	0.955906i	$0.405123\pi$
$194$		0			0
$195$	414.167	+	42401.5i	0.000779990	+	0.0798537i
$196$		0			0
$197$		−	460739.i		−	0.845841i	−0.906167	−	0.422921i	$-0.861005\pi$
$197$		−	460739.i		−	0.845841i	0.906167	−	0.422921i	$-0.138995\pi$
$198$		0			0
$199$			248864.i			0.445480i	0.974878	+	0.222740i	$0.0715002\pi$
$199$			248864.i			0.445480i	−0.974878	+	0.222740i	$0.928500\pi$
$200$		0			0
$201$	−2815.09	−	288203.i	−0.00491476	−	0.503163i
$202$		0			0
$203$	220355.			0.375304
$204$		0			0
$205$	−66102.7			−0.109859
$206$		0			0
$207$	331520.	−	6477.02i	0.537754	−	0.0105063i
$208$		0			0
$209$		−	554757.i		−	0.878490i
$210$		0			0
$211$			764476.i			1.18211i	0.806632	+	0.591055i	$0.201288\pi$
$211$			764476.i			1.18211i	−0.806632	+	0.591055i	$0.798712\pi$
$212$		0			0
$213$	−601212.	+	5872.48i	−0.907985	+	0.00886896i
$214$		0			0
$215$	−24607.9			−0.0363060
$216$		0			0
$217$	504161.			0.726809
$218$		0			0
$219$	905549.	−	8845.17i	1.27586	−	0.0124622i
$220$		0			0
$221$		−	1.11385e6i		−	1.53407i
$222$		0			0
$223$			773325.i			1.04136i	0.853753	+	0.520679i	$0.174321\pi$
$223$			773325.i			1.04136i	−0.853753	+	0.520679i	$0.825679\pi$
$224$		0			0
$225$	755152.	−	14753.7i	0.994439	−	0.0194287i
$226$		0			0
$227$	−488037.			−0.628620			−0.314310	−	0.949320i	$-0.601773\pi$
$227$	−488037.			−0.628620			−0.314310	+	0.949320i	$0.601773\pi$
$228$		0			0
$229$	−665638.			−0.838782			−0.419391	−	0.907806i	$-0.637756\pi$
$229$	−665638.			−0.838782			−0.419391	+	0.907806i	$0.637756\pi$
$230$		0			0
$231$	−2030.26	−	207853.i	−0.00250335	−	0.256287i
$232$		0			0
$233$			219625.i			0.265029i	0.991181	+	0.132514i	$0.0423050\pi$
$233$			219625.i			0.265029i	−0.991181	+	0.132514i	$0.957695\pi$
$234$		0			0
$235$		−	94397.4i		−	0.111504i
$236$		0			0
$237$	−12071.0	−	1.23580e6i	−0.0139596	−	1.42915i
$238$		0			0
$239$	−543608.			−0.615589			−0.307794	−	0.951453i	$-0.599591\pi$
$239$	−543608.			−0.615589			−0.307794	+	0.951453i	$0.599591\pi$
$240$		0			0
$241$	−1.49771e6			−1.66106			−0.830530	−	0.556973i	$-0.811963\pi$
$241$	−1.49771e6			−1.66106			−0.830530	+	0.556973i	$0.811963\pi$
$242$		0			0
$243$	−919385.	+	44935.9i	−0.998808	+	0.0488177i
$244$		0			0
$245$		−	9836.81i		−	0.0104698i
$246$		0			0
$247$			1.35351e6i			1.41162i
$248$		0			0
$249$	771042.	−	7531.33i	0.788097	−	0.00769792i
$250$		0			0
$251$	250520.			0.250991			0.125496	−	0.992094i	$-0.459948\pi$
$251$	250520.			0.250991			0.125496	+	0.992094i	$0.459948\pi$
$252$		0			0
$253$	371334.			0.364723
$254$		0			0
$255$	107136.	−	1046.47i	0.103177	−	0.00100781i
$256$		0			0
$257$		−	522647.i		−	0.493600i	−0.969066	−	0.246800i	$-0.920621\pi$
$257$		−	522647.i		−	0.493600i	0.969066	−	0.246800i	$-0.0793791\pi$
$258$		0			0
$259$		−	24776.3i		−	0.0229503i
$260$		0			0
$261$	−21346.0	−	1.09257e6i	−0.0193961	−	0.992771i
$262$		0			0
$263$	1.11981e6			0.998289			0.499145	−	0.866519i	$-0.333648\pi$
$263$	1.11981e6			0.998289			0.499145	+	0.866519i	$0.333648\pi$
$264$		0			0
$265$	−96672.6			−0.0845646
$266$		0			0
$267$	−17472.5	−	1.78880e6i	−0.0149995	−	1.53562i
$268$		0			0
$269$		−	2.20216e6i		−	1.85553i	−0.373168	−	0.927764i	$-0.621729\pi$
$269$		−	2.20216e6i		−	1.85553i	0.373168	−	0.927764i	$-0.378271\pi$
$270$		0			0
$271$		−	224228.i		−	0.185467i	−0.995691	−	0.0927333i	$-0.970440\pi$
$271$		−	224228.i		−	0.185467i	0.995691	−	0.0927333i	$-0.0295604\pi$
$272$		0			0
$273$	4953.47	+	507126.i	0.00402256	+	0.411821i
$274$		0			0
$275$	845842.			0.674462
$276$		0			0
$277$	2.14354e6			1.67854			0.839271	−	0.543714i	$-0.182982\pi$
$277$	2.14354e6			1.67854			0.839271	+	0.543714i	$0.182982\pi$
$278$		0			0
$279$	−48838.5	−	2.49975e6i	−0.0375623	−	1.92259i
$280$		0			0
$281$			63081.3i			0.0476579i	0.999716	+	0.0238290i	$0.00758571\pi$
$281$			63081.3i			0.0476579i	−0.999716	+	0.0238290i	$0.992414\pi$
$282$		0			0
$283$			1.92441e6i			1.42834i	0.699973	+	0.714169i	$0.253195\pi$
$283$			1.92441e6i			1.42834i	−0.699973	+	0.714169i	$0.746805\pi$
$284$		0			0
$285$	−130187.	+	1271.64i	−0.0949416	+	0.000927365i
$286$		0			0
$287$	−790593.			−0.566563
$288$		0			0
$289$	−1.39449e6			−0.982136
$290$		0			0
$291$	−880659.	+	8602.05i	−0.609643	+	0.00595483i
$292$		0			0
$293$			509387.i			0.346640i	0.984866	+	0.173320i	$0.0554495\pi$
$293$			509387.i			0.346640i	−0.984866	+	0.173320i	$0.944550\pi$
$294$		0			0
$295$			128842.i			0.0861988i
$296$		0			0
$297$	−1.03039e6	+	30201.4i	−0.677814	+	0.0198672i
$298$		0			0
$299$	−905990.			−0.586064
$300$		0			0
$301$	−294312.			−0.187237
$302$		0			0
$303$	−26074.3	−	2.66943e6i	−0.0163157	−	1.67037i
$304$		0			0
$305$			124124.i			0.0764022i
$306$		0			0
$307$			1.50324e6i			0.910298i	0.890415	+	0.455149i	$0.150414\pi$
$307$			1.50324e6i			0.910298i	−0.890415	+	0.455149i	$0.849586\pi$
$308$		0			0
$309$	−8841.91	−	905216.i	−0.00526805	−	0.539332i
$310$		0			0
$311$	−987354.			−0.578858			−0.289429	−	0.957199i	$-0.593465\pi$
$311$	−987354.			−0.578858			−0.289429	+	0.957199i	$0.593465\pi$
$312$		0			0
$313$	718492.			0.414535			0.207267	−	0.978284i	$-0.433543\pi$
$313$	718492.			0.414535			0.207267	+	0.978284i	$0.433543\pi$
$314$		0			0
$315$	−48773.2	+	952.899i	−0.0276952	+	0.000541091i
$316$		0			0
$317$		−	722877.i		−	0.404032i	−0.979382	−	0.202016i	$-0.935251\pi$
$317$		−	722877.i		−	0.404032i	0.979382	−	0.202016i	$-0.0647494\pi$
$318$		0			0
$319$		−	1.22379e6i		−	0.673331i
$320$		0			0
$321$	706444.	−	6900.36i	0.382662	−	0.00373774i
$322$		0			0
$323$	3.41990e6			1.82392
$324$		0			0
$325$	−2.06371e6			−1.08378
$326$		0			0
$327$	−73752.7	+	720.397i	−0.0381424	+	0.000372565i
$328$		0			0
$329$		−	1.12900e6i		−	0.575048i
$330$		0			0
$331$			677831.i			0.340057i	0.985439	+	0.170028i	$0.0543859\pi$
$331$			677831.i			0.340057i	−0.985439	+	0.170028i	$0.945614\pi$
$332$		0			0
$333$	−122847.	+	2400.10i	−0.0607091	+	0.00118609i
$334$		0			0
$335$	−75749.3			−0.0368779
$336$		0			0
$337$	2.76203e6			1.32481			0.662406	−	0.749145i	$-0.269535\pi$
$337$	2.76203e6			1.32481			0.662406	+	0.749145i	$0.269535\pi$
$338$		0			0
$339$	−21638.7	−	2.21533e6i	−0.0102266	−	1.04698i
$340$		0			0
$341$		−	2.79996e6i		−	1.30396i
$342$		0			0
$343$		−	117649.i		−	0.0539949i
$344$		0			0
$345$	−851.187	−	87142.7i	−0.000385014	−	0.0394170i
$346$		0			0
$347$	−3.48005e6			−1.55154			−0.775768	−	0.631018i	$-0.782637\pi$
$347$	−3.48005e6			−1.55154			−0.775768	+	0.631018i	$0.782637\pi$
$348$		0			0
$349$	565820.			0.248665			0.124333	−	0.992241i	$-0.460321\pi$
$349$	565820.			0.248665			0.124333	+	0.992241i	$0.460321\pi$
$350$		0			0
$351$	2.51397e6	−	73686.1i	1.08916	−	0.0319240i
$352$		0			0
$353$			3.55759e6i			1.51956i	0.650178	+	0.759782i	$0.274694\pi$
$353$			3.55759e6i			1.51956i	−0.650178	+	0.759782i	$0.725306\pi$
$354$		0			0
$355$			158018.i			0.0665482i
$356$		0			0
$357$	1.28135e6	−	12515.9i	0.532105	−	0.00519746i
$358$		0			0
$359$	2.01060e6			0.823358			0.411679	−	0.911329i	$-0.364942\pi$
$359$	2.01060e6			0.823358			0.411679	+	0.911329i	$0.364942\pi$
$360$		0			0
$361$	−1.67964e6			−0.678342
$362$		0			0
$363$	1.35606e6	−	13245.7i	0.540148	−	0.00527603i
$364$		0			0
$365$		−	238008.i		−	0.0935103i
$366$		0			0
$367$		−	2.01074e6i		−	0.779276i	−0.920968	−	0.389638i	$-0.872600\pi$
$367$		−	2.01074e6i		−	0.779276i	0.920968	−	0.389638i	$-0.127400\pi$
$368$		0			0
$369$	76585.4	+	3.91995e6i	0.0292806	+	1.49870i
$370$		0			0
$371$	−1.15621e6			−0.436117
$372$		0			0
$373$	−392550.			−0.146091			−0.0730453	−	0.997329i	$-0.523272\pi$
$373$	−392550.			−0.146091			−0.0730453	+	0.997329i	$0.523272\pi$
$374$		0			0
$375$	−3888.22	−	398067.i	−0.00142782	−	0.146177i
$376$		0			0
$377$			2.98582e6i			1.08196i
$378$		0			0
$379$			3.18669e6i			1.13957i	0.821793	+	0.569787i	$0.192974\pi$
$379$			3.18669e6i			1.13957i	−0.821793	+	0.569787i	$0.807026\pi$
$380$		0			0
$381$	9511.42	+	973759.i	0.00335686	+	0.343668i
$382$		0			0
$383$	−1.91541e6			−0.667215			−0.333607	−	0.942712i	$-0.608266\pi$
$383$	−1.91541e6			−0.667215			−0.333607	+	0.942712i	$0.608266\pi$
$384$		0			0
$385$	−54630.7			−0.0187839
$386$		0			0
$387$	28510.3	+	1.45927e6i	0.00967662	+	0.495289i
$388$		0			0
$389$			1.11824e6i			0.374680i	0.982295	+	0.187340i	$0.0599866\pi$
$389$			1.11824e6i			0.374680i	−0.982295	+	0.187340i	$0.940013\pi$
$390$		0			0
$391$			2.28916e6i			0.757240i
$392$		0			0
$393$	−3.54763e6	+	34652.3i	−1.15866	+	0.0113175i
$394$		0			0
$395$	−324809.			−0.104746
$396$		0			0
$397$	−2.25436e6			−0.717873			−0.358937	−	0.933362i	$-0.616861\pi$
$397$	−2.25436e6			−0.717873			−0.358937	+	0.933362i	$0.616861\pi$
$398$		0			0
$399$	−1.55705e6	+	15208.9i	−0.489633	+	0.00478260i
$400$		0			0
$401$			4.17593e6i			1.29686i	0.761275	+	0.648429i	$0.224574\pi$
$401$			4.17593e6i			1.29686i	−0.761275	+	0.648429i	$0.775426\pi$
$402$		0			0
$403$			6.83141e6i			2.09531i
$404$		0			0
$405$	9449.40	+	241737.i	0.00286264	+	0.0732328i
$406$		0			0
$407$	−137600.			−0.0411750
$408$		0			0
$409$	3.93559e6			1.16333			0.581663	−	0.813430i	$-0.302402\pi$
$409$	3.93559e6			1.16333			0.581663	+	0.813430i	$0.302402\pi$
$410$		0			0
$411$	−40920.8	−	4.18938e6i	−0.0119492	−	1.22333i
$412$		0			0
$413$			1.54096e6i			0.444544i
$414$		0			0
$415$		−	202655.i		−	0.0577614i
$416$		0			0
$417$	19002.2	+	1.94540e6i	0.00535134	+	0.547859i
$418$		0			0
$419$	3.19998e6			0.890456			0.445228	−	0.895417i	$-0.353123\pi$
$419$	3.19998e6			0.890456			0.445228	+	0.895417i	$0.353123\pi$
$420$		0			0
$421$	390819.			0.107466			0.0537329	−	0.998555i	$-0.482888\pi$
$421$	390819.			0.107466			0.0537329	+	0.998555i	$0.482888\pi$
$422$		0			0
$423$	−5.59785e6	+	109367.i	−1.52114	+	0.0297191i
$424$		0			0
$425$			5.21435e6i			1.40032i
$426$		0			0
$427$			1.48453e6i			0.394022i
$428$		0			0
$429$	2.81642e6	−	27510.1i	0.738847	−	0.00721687i
$430$		0			0
$431$	242947.			0.0629967			0.0314984	−	0.999504i	$-0.489972\pi$
$431$	242947.			0.0629967			0.0314984	+	0.999504i	$0.489972\pi$
$432$		0			0
$433$	502840.			0.128887			0.0644436	−	0.997921i	$-0.479473\pi$
$433$	502840.			0.128887			0.0644436	+	0.997921i	$0.479473\pi$
$434$		0			0
$435$	−287192.	+	2805.21i	−0.0727693	+	0.000710792i
$436$		0			0
$437$		−	2.78170e6i		−	0.696798i
$438$		0			0
$439$			5.25188e6i			1.30063i	0.759665	+	0.650314i	$0.225363\pi$
$439$			5.25188e6i			1.30063i	−0.759665	+	0.650314i	$0.774637\pi$
$440$		0			0
$441$	−583332.	+	11396.7i	−0.142830	+	0.00279052i
$442$		0			0
$443$	7.14397e6			1.72954			0.864770	−	0.502169i	$-0.167464\pi$
$443$	7.14397e6			1.72954			0.864770	+	0.502169i	$0.167464\pi$
$444$		0			0
$445$	−470156.			−0.112549
$446$		0			0
$447$	28636.8	+	2.93178e6i	0.00677886	+	0.694005i
$448$		0			0
$449$			7.05471e6i			1.65144i	0.564080	+	0.825720i	$0.309231\pi$
$449$			7.05471e6i			1.65144i	−0.564080	+	0.825720i	$0.690769\pi$
$450$		0			0
$451$			4.39072e6i			1.01647i
$452$		0			0
$453$	−53522.3	−	5.47950e6i	−0.0122543	−	1.25457i
$454$		0			0
$455$	133289.			0.0301833
$456$		0			0
$457$	−1.85068e6			−0.414515			−0.207258	−	0.978286i	$-0.566454\pi$
$457$	−1.85068e6			−0.414515			−0.207258	+	0.978286i	$0.566454\pi$
$458$		0			0
$459$	−186182.	−	6.35202e6i	−0.0412483	−	1.40728i
$460$		0			0
$461$		−	3.20400e6i		−	0.702166i	−0.936344	−	0.351083i	$-0.885813\pi$
$461$		−	3.20400e6i		−	0.702166i	0.936344	−	0.351083i	$-0.114187\pi$
$462$		0			0
$463$		−	2.10485e6i		−	0.456320i	−0.973624	−	0.228160i	$-0.926729\pi$
$463$		−	2.10485e6i		−	0.456320i	0.973624	−	0.228160i	$-0.0732710\pi$
$464$		0			0
$465$	−657079.	+	6418.18i	−0.140924	+	0.00137651i
$466$		0			0
$467$	−6.60453e6			−1.40136			−0.700680	−	0.713475i	$-0.747120\pi$
$467$	−6.60453e6			−1.40136			−0.700680	+	0.713475i	$0.747120\pi$
$468$		0			0
$469$	−905967.			−0.190187
$470$		0			0
$471$	7.94865e6	−	77640.3i	1.65098	−	0.0161263i
$472$		0			0
$473$			1.63452e6i			0.335922i
$474$		0			0
$475$		−	6.33629e6i		−	1.28855i
$476$		0			0
$477$	112003.	+	5.73277e6i	0.0225390	+	1.15364i
$478$		0			0
$479$	−4.40171e6			−0.876562			−0.438281	−	0.898838i	$-0.644413\pi$
$479$	−4.40171e6			−0.876562			−0.438281	+	0.898838i	$0.644413\pi$
$480$		0			0
$481$	335721.			0.0661631
$482$		0			0
$483$	−10180.3	−	1.04223e6i	−0.00198560	−	0.203281i
$484$		0			0
$485$			231466.i			0.0446821i
$486$		0			0
$487$		−	943866.i		−	0.180338i	−0.995926	−	0.0901691i	$-0.971259\pi$
$487$		−	943866.i		−	0.180338i	0.995926	−	0.0901691i	$-0.0287408\pi$
$488$		0			0
$489$	34828.8	+	3.56570e6i	0.00658668	+	0.674330i
$490$		0			0
$491$	−5.91042e6			−1.10641			−0.553203	−	0.833046i	$-0.686595\pi$
$491$	−5.91042e6			−1.10641			−0.553203	+	0.833046i	$0.686595\pi$
$492$		0			0
$493$	7.54425e6			1.39797
$494$		0			0
$495$	5292.12	+	270872.i	0.000970770	+	0.0496879i
$496$		0			0
$497$			1.88991e6i			0.343202i
$498$		0			0
$499$		−	1.65549e6i		−	0.297629i	−0.988865	−	0.148814i	$-0.952454\pi$
$499$		−	1.65549e6i		−	0.297629i	0.988865	−	0.148814i	$-0.0475457\pi$
$500$		0			0
$501$	−5.86003e6	+	57239.2i	−1.04305	+	0.0101882i
$502$		0			0
$503$	3.06947e6			0.540933			0.270466	−	0.962729i	$-0.412822\pi$
$503$	3.06947e6			0.540933			0.270466	+	0.962729i	$0.412822\pi$
$504$		0			0
$505$	−701614.			−0.122425
$506$		0			0
$507$	−1.08397e6	+	10587.9i	−0.187282	+	0.00182932i
$508$		0			0
$509$			8.15245e6i			1.39474i	0.716711	+	0.697370i	$0.245647\pi$
$509$			8.15245e6i			1.39474i	−0.716711	+	0.697370i	$0.754353\pi$
$510$		0			0
$511$		−	2.84660e6i		−	0.482251i
$512$		0			0
$513$	226242.	+	7.71875e6i	0.0379559	+	1.29495i
$514$		0			0
$515$	−237921.			−0.0395288
$516$		0			0
$517$	−6.27013e6			−1.03169
$518$		0			0
$519$	626.292	+	64118.5i	0.000102061	+	0.0104488i
$520$		0			0
$521$			7.71608e6i			1.24538i	0.782468	+	0.622691i	$0.213961\pi$
$521$			7.71608e6i			1.24538i	−0.782468	+	0.622691i	$0.786039\pi$
$522$		0			0
$523$			432211.i			0.0690942i	0.999403	+	0.0345471i	$0.0109989\pi$
$523$			432211.i			0.0690942i	−0.999403	+	0.0345471i	$0.989001\pi$
$524$		0			0
$525$	−23189.1	−	2.37405e6i	−0.00367185	−	0.375916i
$526$		0			0
$527$	1.72609e7			2.70730
$528$		0			0
$529$	−4.57437e6			−0.710710
$530$		0			0
$531$	7.64042e6	−	149274.i	1.17593	−	0.0229745i
$532$		0			0
$533$		−	1.07126e7i		−	1.63334i
$534$		0			0
$535$		−	185677.i		−	0.0280461i
$536$		0			0
$537$	9.14089e6	−	89285.8i	1.36789	−	0.0133612i
$538$		0			0
$539$	−653387.			−0.0968721
$540$		0			0
$541$	6.84802e6			1.00594			0.502970	−	0.864304i	$-0.332241\pi$
$541$	6.84802e6			1.00594			0.502970	+	0.864304i	$0.332241\pi$
$542$		0			0
$543$	−9.80317e6	+	95754.8i	−1.42681	+	0.0139367i
$544$		0			0
$545$			19384.6i			0.00279554i
$546$		0			0
$547$		−	9.38397e6i		−	1.34097i	−0.741924	−	0.670484i	$-0.766087\pi$
$547$		−	9.38397e6i		−	1.34097i	0.741924	−	0.670484i	$-0.233913\pi$
$548$		0			0
$549$	7.36066e6	−	143808.i	1.04228	−	0.0203635i
$550$		0			0
$551$	−9.16751e6			−1.28639
$552$		0			0
$553$	−3.88474e6			−0.540194
$554$		0			0
$555$	315.413	+	32291.3i	4.34658e−5	+	0.00444993i
$556$		0			0
$557$		−	4.22272e6i		−	0.576706i	−0.957524	−	0.288353i	$-0.906892\pi$
$557$		−	4.22272e6i		−	0.576706i	0.957524	−	0.288353i	$-0.0931077\pi$
$558$		0			0
$559$		−	3.98795e6i		−	0.539784i
$560$		0			0
$561$	−69509.4	−	7.11623e6i	−0.00932474	−	0.954647i
$562$		0			0
$563$	−1.30349e7			−1.73316			−0.866579	−	0.499039i	$-0.833686\pi$
$563$	−1.30349e7			−1.73316			−0.866579	+	0.499039i	$0.833686\pi$
$564$		0			0
$565$	−582261.			−0.0767356
$566$		0			0
$567$	113016.	+	2.89119e6i	0.0147632	+	0.377676i
$568$		0			0
$569$		−	1.14986e7i		−	1.48889i	−0.667682	−	0.744446i	$-0.732713\pi$
$569$		−	1.14986e7i		−	1.48889i	0.667682	−	0.744446i	$-0.267287\pi$
$570$		0			0
$571$		−	4.68419e6i		−	0.601234i	−0.953745	−	0.300617i	$-0.902807\pi$
$571$		−	4.68419e6i		−	0.601234i	0.953745	−	0.300617i	$-0.0971926\pi$
$572$		0			0
$573$	−3.33817e6	+	32606.4i	−0.424739	+	0.00414874i
$574$		0			0
$575$	4.24128e6			0.534968
$576$		0			0
$577$	1.92861e6			0.241159			0.120580	−	0.992704i	$-0.461525\pi$
$577$	1.92861e6			0.241159			0.120580	+	0.992704i	$0.461525\pi$
$578$		0			0
$579$	−4.73770e6	+	46276.7i	−0.587316	+	0.00573675i
$580$		0			0
$581$		−	2.42377e6i		−	0.297887i
$582$		0			0
$583$			6.42125e6i			0.782435i
$584$		0			0
$585$	−12911.8	−	660880.i	−0.00155991	−	0.0798423i
$586$		0			0
$587$	1.01345e7			1.21397			0.606986	−	0.794712i	$-0.292378\pi$
$587$	1.01345e7			1.21397			0.606986	+	0.794712i	$0.292378\pi$
$588$		0			0
$589$	−2.09748e7			−2.49120
$590$		0			0
$591$	70150.5	+	7.18186e6i	0.00826156	+	0.845801i
$592$		0			0
$593$		−	1.43086e7i		−	1.67093i	−0.549540	−	0.835467i	$-0.685197\pi$
$593$		−	1.43086e7i		−	1.67093i	0.549540	−	0.835467i	$-0.314803\pi$
$594$		0			0
$595$		−	336781.i		−	0.0389991i
$596$		0			0
$597$	−37891.1	−	3.87921e6i	−0.00435113	−	0.445459i
$598$		0			0
$599$	−7.47422e6			−0.851136			−0.425568	−	0.904927i	$-0.639926\pi$
$599$	−7.47422e6			−0.851136			−0.425568	+	0.904927i	$0.639926\pi$
$600$		0			0
$601$	2.52157e6			0.284764			0.142382	−	0.989812i	$-0.454524\pi$
$601$	2.52157e6			0.284764			0.142382	+	0.989812i	$0.454524\pi$
$602$		0			0
$603$	87761.7	+	4.49200e6i	0.00982906	+	0.503091i
$604$		0			0
$605$		−	356418.i		−	0.0395887i
$606$		0			0
$607$		−	1.41126e7i		−	1.55466i	−0.629094	−	0.777330i	$-0.716574\pi$
$607$		−	1.41126e7i		−	1.55466i	0.629094	−	0.777330i	$-0.283426\pi$
$608$		0			0
$609$	−3.43483e6	+	33550.5i	−0.375286	+	0.00366569i
$610$		0			0
$611$	1.52980e7			1.65780
$612$		0			0
$613$	1.13520e7			1.22018			0.610088	−	0.792333i	$-0.291134\pi$
$613$	1.13520e7			1.22018			0.610088	+	0.792333i	$0.291134\pi$
$614$		0			0
$615$	1.03039e6	−	10064.6i	0.109853	−	0.00107302i
$616$		0			0
$617$			4.25681e6i			0.450164i	0.974340	+	0.225082i	$0.0722650\pi$
$617$			4.25681e6i			0.450164i	−0.974340	+	0.225082i	$0.927735\pi$
$618$		0			0
$619$		−	1.65329e7i		−	1.73429i	−0.498054	−	0.867146i	$-0.665952\pi$
$619$		−	1.65329e7i		−	1.73429i	0.498054	−	0.867146i	$-0.334048\pi$
$620$		0			0
$621$	−5.16665e6	+	151438.i	−0.537626	+	0.0157582i
$622$		0			0
$623$	−5.62310e6			−0.580437
$624$		0			0
$625$	9.60855e6			0.983915
$626$		0			0
$627$	84465.4	+	8.64739e6i	0.00858045	+	0.878448i
$628$		0			0
$629$		−	848262.i		−	0.0854877i
$630$		0			0
$631$			1.45941e7i			1.45917i	0.683892	+	0.729583i	$0.260286\pi$
$631$			1.45941e7i			1.45917i	−0.683892	+	0.729583i	$0.739714\pi$
$632$		0			0
$633$	−116397.	−	1.19164e7i	−0.0115460	−	1.18205i
$634$		0			0
$635$	255936.			0.0251882
$636$		0			0
$637$	1.59415e6			0.155661
$638$		0			0
$639$	9.37062e6	−	183077.i	0.907855	−	0.0177371i
$640$		0			0
$641$			8.29112e6i			0.797018i	0.917164	+	0.398509i	$0.130472\pi$
$641$			8.29112e6i			0.797018i	−0.917164	+	0.398509i	$0.869528\pi$
$642$		0			0
$643$			1.67207e7i			1.59488i	0.603398	+	0.797440i	$0.293813\pi$
$643$			1.67207e7i			1.59488i	−0.603398	+	0.797440i	$0.706187\pi$
$644$		0			0
$645$	383581.	−	3746.72i	0.0363043	−	0.000354611i
$646$		0			0
$647$	−1.07577e7			−1.01032			−0.505159	−	0.863027i	$-0.668566\pi$
$647$	−1.07577e7			−1.01032			−0.505159	+	0.863027i	$0.668566\pi$
$648$		0			0
$649$	8.55800e6			0.797555
$650$		0			0
$651$	−7.85872e6	+	76761.9i	−0.726774	+	0.00709894i
$652$		0			0
$653$		−	6.65436e6i		−	0.610693i	−0.952241	−	0.305347i	$-0.901228\pi$
$653$		−	6.65436e6i		−	0.610693i	0.952241	−	0.305347i	$-0.0987723\pi$
$654$		0			0
$655$			932434.i			0.0849210i
$656$		0			0
$657$	−1.41141e7	+	275752.i	−1.27567	+	0.0249233i
$658$		0			0
$659$	1.29230e7			1.15917			0.579587	−	0.814910i	$-0.303214\pi$
$659$	1.29230e7			1.15917			0.579587	+	0.814910i	$0.303214\pi$
$660$		0			0
$661$	1.45547e7			1.29569			0.647843	−	0.761774i	$-0.275671\pi$
$661$	1.45547e7			1.29569			0.647843	+	0.761774i	$0.275671\pi$
$662$		0			0
$663$	169591.	+	1.73623e7i	0.0149837	+	1.53400i
$664$		0			0
$665$			409244.i			0.0358863i
$666$		0			0
$667$		−	6.13640e6i		−	0.534071i
$668$		0			0
$669$	−117744.	−	1.20544e7i	−0.0101712	−	1.04131i
$670$		0			0
$671$	8.24465e6			0.706913
$672$		0			0
$673$	−4.21161e6			−0.358435			−0.179218	−	0.983809i	$-0.557357\pi$
$673$	−4.21161e6			−0.358435			−0.179218	+	0.983809i	$0.557357\pi$
$674$		0			0
$675$	−1.17688e7	+	344953.i	−0.994202	+	0.0291407i
$676$		0			0
$677$		−	7.02130e6i		−	0.588771i	−0.955687	−	0.294385i	$-0.904885\pi$
$677$		−	7.02130e6i		−	0.588771i	0.955687	−	0.294385i	$-0.0951149\pi$
$678$		0			0
$679$			2.76835e6i			0.230434i
$680$		0			0
$681$	7.60738e6	−	74306.9i	0.628590	−	0.00613990i
$682$		0			0
$683$	−6.98754e6			−0.573156			−0.286578	−	0.958057i	$-0.592518\pi$
$683$	−6.98754e6			−0.573156			−0.286578	+	0.958057i	$0.592518\pi$
$684$		0			0
$685$	−1.10111e6			−0.0896609
$686$		0			0
$687$	1.03758e7	−	101348.i	0.838742	−	0.00819261i
$688$		0			0
$689$		−	1.56667e7i		−	1.25727i
$690$		0			0
$691$			8.56530e6i			0.682413i	0.939988	+	0.341206i	$0.110836\pi$
$691$			8.56530e6i			0.682413i	−0.939988	+	0.341206i	$0.889164\pi$
$692$		0			0
$693$	63294.1	+	3.23965e6i	0.00500645	+	0.256251i
$694$		0			0
$695$	511315.			0.0401538
$696$		0			0
$697$	−2.70674e7			−2.11040
$698$		0			0
$699$	−33439.4	−	3.42346e6i	−0.00258861	−	0.265016i
$700$		0			0
$701$		−	8.32268e6i		−	0.639688i	−0.947470	−	0.319844i	$-0.896370\pi$
$701$		−	8.32268e6i		−	0.639688i	0.947470	−	0.319844i	$-0.103630\pi$
$702$		0			0
$703$			1.03078e6i			0.0786642i
$704$		0			0
$705$	14372.6	+	1.47144e6i	0.00108909	+	0.111499i
$706$		0			0
$707$	−8.39136e6			−0.631370
$708$		0			0
$709$	−5.62710e6			−0.420406			−0.210203	−	0.977658i	$-0.567412\pi$
$709$	−5.62710e6			−0.420406			−0.210203	+	0.977658i	$0.567412\pi$
$710$		0			0
$711$	376318.	+	1.92615e7i	0.0279178	+	1.42895i
$712$		0			0
$713$		−	1.40398e7i		−	1.03427i
$714$		0			0
$715$		−	740248.i		−	0.0541517i
$716$		0			0
$717$	8.47360e6	−	82767.9i	0.615560	−	0.00601262i
$718$		0			0
$719$	−1.71285e7			−1.23566			−0.617828	−	0.786313i	$-0.711987\pi$
$719$	−1.71285e7			−1.23566			−0.617828	+	0.786313i	$0.711987\pi$
$720$		0			0
$721$	−2.84555e6			−0.203858
$722$		0			0
$723$	2.33459e7	−	228037.i	1.66098	−	0.0162240i
$724$		0			0
$725$		−	1.39778e7i		−	0.987627i
$726$		0			0
$727$		−	1.50550e7i		−	1.05644i	−0.849107	−	0.528221i	$-0.822859\pi$
$727$		−	1.50550e7i		−	1.05644i	0.849107	−	0.528221i	$-0.177141\pi$
$728$		0			0
$729$	1.43243e7	−	840430.i	0.998283	−	0.0585710i
$730$		0			0
$731$	−1.00763e7			−0.697442
$732$		0			0
$733$	1.20830e6			0.0830647			0.0415323	−	0.999137i	$-0.486776\pi$
$733$	1.20830e6			0.0830647			0.0415323	+	0.999137i	$0.486776\pi$
$734$		0			0
$735$	1497.72	+	153333.i	0.000102262	+	0.0104693i
$736$		0			0
$737$			5.03147e6i			0.341213i
$738$		0			0
$739$		−	1.96203e7i		−	1.32158i	−0.750570	−	0.660791i	$-0.770221\pi$
$739$		−	1.96203e7i		−	1.32158i	0.750570	−	0.660791i	$-0.229779\pi$
$740$		0			0
$741$	−206081.	−	2.10981e7i	−0.0137877	−	1.41156i
$742$		0			0
$743$	2.43935e7			1.62107			0.810536	−	0.585689i	$-0.199176\pi$
$743$	2.43935e7			1.62107			0.810536	+	0.585689i	$0.199176\pi$
$744$		0			0
$745$	770568.			0.0508651
$746$		0			0
$747$	−1.20176e7	+	234793.i	−0.787984	+	0.0153951i
$748$		0			0
$749$		−	2.22071e6i		−	0.144639i
$750$		0			0
$751$		−	1.35217e7i		−	0.874844i	−0.899256	−	0.437422i	$-0.855892\pi$
$751$		−	1.35217e7i		−	0.874844i	0.899256	−	0.437422i	$-0.144108\pi$
$752$		0			0
$753$	−3.90504e6	+	38143.4i	−0.250979	+	0.00245150i
$754$		0			0
$755$	−1.44019e6			−0.0919503
$756$		0			0
$757$	−1.60829e6			−0.102006			−0.0510028	−	0.998699i	$-0.516242\pi$
$757$	−1.60829e6			−0.102006			−0.0510028	+	0.998699i	$0.516242\pi$
$758$		0			0
$759$	−5.78825e6	+	56538.1i	−0.364706	+	0.00356235i
$760$		0			0
$761$			7.89048e6i			0.493903i	0.969028	+	0.246952i	$0.0794288\pi$
$761$			7.89048e6i			0.493903i	−0.969028	+	0.246952i	$0.920571\pi$
$762$		0			0
$763$			231842.i			0.0144172i
$764$		0			0
$765$	−1.66984e6	+	32624.2i	−0.103162	+	0.00201552i
$766$		0			0
$767$	−2.08800e7			−1.28157
$768$		0			0
$769$	2.87428e7			1.75272			0.876361	−	0.481655i	$-0.159964\pi$
$769$	2.87428e7			1.75272			0.876361	+	0.481655i	$0.159964\pi$
$770$		0			0
$771$	79576.5	+	8.14687e6i	0.00482113	+	0.493577i
$772$		0			0
$773$			9.18854e6i			0.553093i	0.961001	+	0.276546i	$0.0891899\pi$
$773$			9.18854e6i			0.553093i	−0.961001	+	0.276546i	$0.910810\pi$
$774$		0			0
$775$		−	3.19804e7i		−	1.91263i
$776$		0			0
$777$	3772.37	+	386207.i	0.000224162	+	0.0229492i
$778$		0			0
$779$	3.28913e7			1.94195
$780$		0			0
$781$	1.04960e7			0.615738
$782$		0			0
$783$	499086.	+	1.70275e7i	0.0290919	+	0.992535i
$784$		0			0
$785$		−	2.08917e6i		−	0.121004i
$786$		0			0
$787$			1.61932e7i			0.931959i	0.884796	+	0.465979i	$0.154298\pi$
$787$			1.61932e7i			0.931959i	−0.884796	+	0.465979i	$0.845702\pi$
$788$		0			0
$789$	−1.74553e7	+	170499.i	−0.998242	+	0.00975056i
$790$		0			0
$791$	−6.96389e6			−0.395741
$792$		0			0
$793$	−2.01155e7			−1.13592
$794$		0			0
$795$	1.50690e6	−	14719.0i	0.0845606	−	0.000825966i
$796$		0			0
$797$			1.60903e7i			0.897263i	0.893717	+	0.448631i	$0.148088\pi$
$797$			1.60903e7i			0.897263i	−0.893717	+	0.448631i	$0.851912\pi$
$798$		0			0
$799$		−	3.86533e7i		−	2.14200i
$800$		0			0
$801$	544714.	+	2.78806e7i	0.0299976	+	1.53540i
$802$		0			0
$803$	−1.58091e7			−0.865206
$804$		0			0
$805$	−273933.			−0.0148989
$806$		0			0
$807$	335293.	+	3.43266e7i	0.0181234	+	1.85544i
$808$		0			0
$809$		−	2.33051e7i		−	1.25193i	−0.779851	−	0.625965i	$-0.784705\pi$
$809$		−	2.33051e7i		−	1.25193i	0.779851	−	0.625965i	$-0.215295\pi$
$810$		0			0
$811$		−	1.85228e7i		−	0.988905i	−0.869205	−	0.494453i	$-0.835369\pi$
$811$		−	1.85228e7i		−	0.988905i	0.869205	−	0.494453i	$-0.164631\pi$
$812$		0			0
$813$	34140.1	+	3.49519e6i	0.00181150	+	0.185458i
$814$		0			0
$815$	937183.			0.0494232
$816$		0			0
$817$	1.22444e7			0.641773
$818$		0			0
$819$	−154427.	−	7.90417e6i	−0.00804474	−	0.411762i
$820$		0			0
$821$			3.04818e7i			1.57827i	0.614217	+	0.789137i	$0.289472\pi$
$821$			3.04818e7i			1.57827i	−0.614217	+	0.789137i	$0.710528\pi$
$822$		0			0
$823$		−	2.57951e7i		−	1.32751i	−0.747951	−	0.663754i	$-0.768962\pi$
$823$		−	2.57951e7i		−	1.32751i	0.747951	−	0.663754i	$-0.231038\pi$
$824$		0			0
$825$	−1.31847e7	+	128785.i	−0.674430	+	0.00658766i
$826$		0			0
$827$	−3.97713e6			−0.202211			−0.101106	−	0.994876i	$-0.532238\pi$
$827$	−3.97713e6			−0.202211			−0.101106	+	0.994876i	$0.532238\pi$
$828$		0			0
$829$	−2.08838e7			−1.05542			−0.527708	−	0.849426i	$-0.676948\pi$
$829$	−2.08838e7			−1.05542			−0.527708	+	0.849426i	$0.676948\pi$
$830$		0			0
$831$	−3.34129e7	+	326368.i	−1.67846	+	0.0163948i
$832$		0			0
$833$		−	4.02792e6i		−	0.201126i
$834$		0			0
$835$			1.54021e6i			0.0764475i
$836$		0			0
$837$	1.14188e6	+	3.89580e7i	0.0563389	+	1.92213i
$838$		0			0
$839$	3.72217e7			1.82554			0.912770	−	0.408475i	$-0.133939\pi$
$839$	3.72217e7			1.82554			0.912770	+	0.408475i	$0.133939\pi$
$840$		0			0
$841$	287749.			0.0140289
$842$		0			0
$843$	−9604.55	−	983294.i	−0.000465488	−	0.0476556i
$844$		0			0
$845$			284903.i			0.0137263i
$846$		0			0
$847$		−	4.26278e6i		−	0.204167i
$848$		0			0
$849$	−293004.	−	2.99971e7i	−0.0139510	−	1.42827i
$850$		0			0
$851$	−689966.			−0.0326591
$852$		0			0
$853$	−4.71096e6			−0.221685			−0.110843	−	0.993838i	$-0.535355\pi$
$853$	−4.71096e6			−0.221685			−0.110843	+	0.993838i	$0.535355\pi$
$854$		0			0
$855$	2.02913e6	−	39643.8i	0.0949280	−	0.00185464i
$856$		0			0
$857$		−	2.40844e7i		−	1.12017i	−0.828436	−	0.560084i	$-0.810769\pi$
$857$		−	2.40844e7i		−	1.12017i	0.828436	−	0.560084i	$-0.189231\pi$
$858$		0			0
$859$			1.90435e7i			0.880570i	0.897858	+	0.440285i	$0.145123\pi$
$859$			1.90435e7i			0.880570i	−0.897858	+	0.440285i	$0.854877\pi$
$860$		0			0
$861$	1.23235e7	−	120373.i	0.566536	−	0.00553378i
$862$		0			0
$863$	3.96564e7			1.81254			0.906268	−	0.422704i	$-0.138919\pi$
$863$	3.96564e7			1.81254			0.906268	+	0.422704i	$0.138919\pi$
$864$		0			0
$865$	16852.4			0.000765813
$866$		0			0
$867$	2.17369e7	−	212321.i	0.982089	−	0.00959279i
$868$		0			0
$869$			2.15747e7i			0.969160i
$870$		0			0
$871$		−	1.22759e7i		−	0.548287i
$872$		0			0
$873$	1.37262e7	−	268173.i	0.609556	−	0.0119091i
$874$		0			0
$875$	−1.25133e6			−0.0552523
$876$		0			0
$877$	2.58895e7			1.13664			0.568322	−	0.822806i	$-0.307593\pi$
$877$	2.58895e7			1.13664			0.568322	+	0.822806i	$0.307593\pi$
$878$		0			0
$879$	−77557.6	−	7.94018e6i	−0.00338573	−	0.346624i
$880$		0			0
$881$			2.52007e7i			1.09389i	0.837170	+	0.546943i	$0.184209\pi$
$881$			2.52007e7i			1.09389i	−0.837170	+	0.546943i	$0.815791\pi$
$882$		0			0
$883$			1.82716e7i			0.788632i	0.918975	+	0.394316i	$0.129019\pi$
$883$			1.82716e7i			0.788632i	−0.918975	+	0.394316i	$0.870981\pi$
$884$		0			0
$885$	−19617.0	−	2.00835e6i	−0.000841927	−	0.0861947i
$886$		0			0
$887$	−1.64555e7			−0.702267			−0.351133	−	0.936325i	$-0.614204\pi$
$887$	−1.64555e7			−0.702267			−0.351133	+	0.936325i	$0.614204\pi$
$888$		0			0
$889$	3.06101e6			0.129900
$890$		0			0
$891$	1.60568e7	−	627654.i	0.677587	−	0.0264866i
$892$		0			0
$893$			4.69702e7i			1.97103i
$894$		0			0
$895$		−	2.40253e6i		−	0.100256i
$896$		0			0
$897$	1.41223e7	−	137943.i	0.586036	−	0.00572425i
$898$		0			0
$899$	−4.62701e7			−1.90942
$900$		0			0
$901$	−3.95850e7			−1.62449
$902$		0			0
$903$	4.58766e6	−	44811.0i	0.187228	−	0.00182880i
$904$		0			0
$905$			2.57660e6i			0.104574i
$906$		0			0
$907$			3.25491e7i			1.31377i	0.753989	+	0.656887i	$0.228127\pi$
$907$			3.25491e7i			1.31377i	−0.753989	+	0.656887i	$0.771873\pi$
$908$		0			0
$909$	812877.	+	4.16063e7i	0.0326299	+	1.67013i
$910$		0			0
$911$	−2.19704e7			−0.877085			−0.438543	−	0.898710i	$-0.644505\pi$
$911$	−2.19704e7			−0.877085			−0.438543	+	0.898710i	$0.644505\pi$
$912$		0			0
$913$	−1.34609e7			−0.534438
$914$		0			0
$915$	−18898.7	−	1.93481e6i	−0.000746241	−	0.0763986i
$916$		0			0
$917$			1.11520e7i			0.437954i
$918$		0			0
$919$		−	4.39809e7i		−	1.71781i	−0.512134	−	0.858906i	$-0.671145\pi$
$919$		−	4.39809e7i		−	1.71781i	0.512134	−	0.858906i	$-0.328855\pi$
$920$		0			0
$921$	−228879.	−	2.34321e7i	−0.00889113	−	0.910255i
$922$		0			0
$923$	−2.56084e7			−0.989414
$924$		0			0
$925$	−1.57164e6			−0.0603946
$926$		0			0
$927$	275650.	+	1.41089e7i	0.0105356	+	0.539255i
$928$		0			0
$929$			3.65148e6i			0.138813i	0.997588	+	0.0694063i	$0.0221105\pi$
$929$			3.65148e6i			0.138813i	−0.997588	+	0.0694063i	$0.977889\pi$
$930$		0			0
$931$			4.89459e6i			0.185073i
$932$		0			0
$933$	1.53906e7	−	150331.i	0.578830	−	0.00565386i
$934$		0			0
$935$	−1.87038e6			−0.0699682
$936$		0			0
$937$	3.07140e7			1.14284			0.571422	−	0.820656i	$-0.306392\pi$
$937$	3.07140e7			1.14284			0.571422	+	0.820656i	$0.306392\pi$
$938$		0			0
$939$	−1.11996e7	+	109395.i	−0.414515	+	0.00404887i
$940$		0			0
$941$			2.30090e7i			0.847079i	0.905878	+	0.423539i	$0.139213\pi$
$941$			2.30090e7i			0.847079i	−0.905878	+	0.423539i	$0.860787\pi$
$942$		0			0
$943$			2.20162e7i			0.806239i
$944$		0			0
$945$	760118.	−	22279.6i	0.0276886	−	0.000811573i
$946$		0			0
$947$	4.03740e7			1.46294			0.731471	−	0.681872i	$-0.238834\pi$
$947$	4.03740e7			1.46294			0.731471	+	0.681872i	$0.238834\pi$
$948$		0			0
$949$	3.85715e7			1.39028
$950$		0			0
$951$	110063.	+	1.12680e7i	0.00394629	+	0.404013i
$952$		0			0
$953$		−	4.28451e6i		−	0.152816i	−0.997077	−	0.0764080i	$-0.975655\pi$
$953$		−	4.28451e6i		−	0.152816i	0.997077	−	0.0764080i	$-0.0243452\pi$
$954$		0			0
$955$			877381.i			0.0311301i
$956$		0			0
$957$	186330.	+	1.90760e7i	0.00657661	+	0.673299i
$958$		0			0
$959$	−1.31693e7			−0.462399
$960$		0			0
$961$	−7.72345e7			−2.69776
$962$		0			0
$963$	−1.10108e7	+	215122.i	−0.382607	+	0.00747512i
$964$		0			0
$965$			1.24522e6i			0.0430457i
$966$		0			0
$967$			1.25501e6i			0.0431601i	0.999767	+	0.0215800i	$0.00686968\pi$
$967$			1.25501e6i			0.0431601i	−0.999767	+	0.0215800i	$0.993130\pi$
$968$		0			0
$969$	−5.33084e7	+	520702.i	−1.82384	+	0.0178148i
$970$		0			0
$971$	3.11183e7			1.05918			0.529588	−	0.848255i	$-0.322347\pi$
$971$	3.11183e7			1.05918			0.529588	+	0.848255i	$0.322347\pi$
$972$		0			0
$973$	6.11537e6			0.207081
$974$		0			0
$975$	3.21685e7	−	314213.i	1.08372	−	0.0105855i
$976$		0			0
$977$			2.26471e7i			0.759059i	0.925180	+	0.379530i	$0.123914\pi$
$977$			2.26471e7i			0.759059i	−0.925180	+	0.379530i	$0.876086\pi$
$978$		0			0
$979$			3.12290e7i			1.04136i
$980$		0			0
$981$	1.14953e6	−	22458.7i	0.0381370	−	0.000745095i
$982$		0			0
$983$	−1.95377e7			−0.644896			−0.322448	−	0.946587i	$-0.604506\pi$
$983$	−1.95377e7			−0.644896			−0.322448	+	0.946587i	$0.604506\pi$
$984$		0			0
$985$	1.88763e6			0.0619906
$986$		0			0
$987$	171898.	+	1.75985e7i	0.00561665	+	0.575021i
$988$		0			0
$989$			8.19594e6i			0.266445i
$990$		0			0
$991$			2.44752e7i			0.791665i	0.918323	+	0.395833i	$0.129544\pi$
$991$			2.44752e7i			0.791665i	−0.918323	+	0.395833i	$0.870456\pi$
$992$		0			0
$993$	−103204.	−	1.05658e7i	−0.00332142	−	0.340040i
$994$		0			0
$995$	−1.01958e6			−0.0326487
$996$		0			0
$997$	−9.11549e6			−0.290430			−0.145215	−	0.989400i	$-0.546387\pi$
$997$	−9.11549e6			−0.290430			−0.145215	+	0.989400i	$0.546387\pi$
$998$		0			0
$999$	1.91454e6	−	56116.4i	0.0606947	−	0.00177900i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.6.h.a.239.2	yes	20
3.2	odd	2	inner	336.6.h.a.239.20	yes	20
4.3	odd	2	inner	336.6.h.a.239.19	yes	20
12.11	even	2	inner	336.6.h.a.239.1	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.6.h.a.239.1	✓	20	12.11	even	2	inner
336.6.h.a.239.2	yes	20	1.1	even	1	trivial
336.6.h.a.239.19	yes	20	4.3	odd	2	inner
336.6.h.a.239.20	yes	20	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	336.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-15.5877 + 0.152257i) q^{3} +4.09696i q^{5} +49.0000i q^{7} +(242.954 - 4.74667i) q^{9} +O(q^{10})\)	\(q+(-15.5877 + 0.152257i) q^{3} +4.09696i q^{5} +49.0000i q^{7} +(242.954 - 4.74667i) q^{9} +272.131 q^{11} -663.953 q^{13} +(-0.623790 - 63.8623i) q^{15} +1677.60i q^{17} -2038.56i q^{19} +(-7.46058 - 763.798i) q^{21} +1364.54 q^{23} +3108.21 q^{25} +(-3786.37 + 110.981i) q^{27} -4497.04i q^{29} -10289.0i q^{31} +(-4241.90 + 41.4338i) q^{33} -200.751 q^{35} -505.640 q^{37} +(10349.5 - 101.091i) q^{39} +16134.6i q^{41} +6006.38i q^{43} +(19.4469 + 995.372i) q^{45} -23040.8 q^{47} -2401.00 q^{49} +(-255.426 - 26150.0i) q^{51} +23596.2i q^{53} +1114.91i q^{55} +(310.385 + 31776.5i) q^{57} +31448.1 q^{59} +30296.6 q^{61} +(232.587 + 11904.7i) q^{63} -2720.19i q^{65} +18489.1i q^{67} +(-21270.1 + 207.760i) q^{69} +38569.6 q^{71} -58093.8 q^{73} +(-48450.0 + 473.247i) q^{75} +13334.4i q^{77} +79280.5i q^{79} +(59003.9 - 2306.44i) q^{81} -49464.7 q^{83} -6873.08 q^{85} +(684.705 + 70098.6i) q^{87} +114757. i q^{89} -32533.7i q^{91} +(1566.57 + 160382. i) q^{93} +8351.92 q^{95} +56497.0 q^{97} +(66115.3 - 1291.72i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 140 q^{9}+O(q^{10}) \) 20 * q - 140 * q^9	\( 20 q - 140 q^{9} + 1048 q^{13} - 980 q^{21} + 5916 q^{25} - 26056 q^{33} + 61360 q^{37} - 92512 q^{45} - 48020 q^{49} - 20720 q^{57} + 46680 q^{61} - 28360 q^{69} - 54280 q^{73} + 152660 q^{81} - 150536 q^{85} + 41688 q^{93} - 421352 q^{97}+O(q^{100}) \) 20 * q - 140 * q^9 + 1048 * q^13 - 980 * q^21 + 5916 * q^25 - 26056 * q^33 + 61360 * q^37 - 92512 * q^45 - 48020 * q^49 - 20720 * q^57 + 46680 * q^61 - 28360 * q^69 - 54280 * q^73 + 152660 * q^81 - 150536 * q^85 + 41688 * q^93 - 421352 * q^97

Introduction

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