LMFDB - Embedded newform 336.6.h.a.239.16

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			239.16
Root			$9.02465 + 9.02465i$ of defining polynomial
Character	$\chi$	$=$	336.239
Dual form			336.6.h.a.239.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(9.02465 + 12.7105i) q^{3} +89.1061i q^{5} +49.0000i q^{7} +(-80.1112 + 229.415i) q^{9} +O(q^{10})$	\(q+(9.02465 + 12.7105i) q^{3} +89.1061i q^{5} +49.0000i q^{7} +(-80.1112 + 229.415i) q^{9} +0.500662 q^{11} -494.561 q^{13} +(-1132.58 + 804.152i) q^{15} +1567.85i q^{17} -1494.94i q^{19} +(-622.812 + 442.208i) q^{21} +38.6621 q^{23} -4814.90 q^{25} +(-3638.94 + 1052.14i) q^{27} -217.146i q^{29} +8367.22i q^{31} +(4.51830 + 6.36364i) q^{33} -4366.20 q^{35} +13263.5 q^{37} +(-4463.24 - 6286.09i) q^{39} -11918.5i q^{41} -10580.1i q^{43} +(-20442.3 - 7138.40i) q^{45} -407.893 q^{47} -2401.00 q^{49} +(-19928.1 + 14149.3i) q^{51} -5835.89i q^{53} +44.6121i q^{55} +(19001.4 - 13491.3i) q^{57} +36055.8 q^{59} -42120.5 q^{61} +(-11241.3 - 3925.45i) q^{63} -44068.4i q^{65} +61188.5i q^{67} +(348.912 + 491.412i) q^{69} +43952.7 q^{71} -1566.88 q^{73} +(-43452.8 - 61199.5i) q^{75} +24.5324i q^{77} -63492.3i q^{79} +(-46213.4 - 36757.4i) q^{81} +84958.5 q^{83} -139705. q^{85} +(2760.02 - 1959.67i) q^{87} -11088.1i q^{89} -24233.5i q^{91} +(-106351. + 75511.3i) q^{93} +133208. q^{95} -92521.6 q^{97} +(-40.1087 + 114.859i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	9.02465	+	12.7105i	0.578932	+	0.815376i
$3$	9.02465	+	12.7105i	0.578932	+	0.815376i
$4$		0			0
$5$			89.1061i			1.59398i	0.603994	+	0.796989i	$0.293575\pi$
$5$			89.1061i			1.59398i	−0.603994	+	0.796989i	$0.706425\pi$
$6$		0			0
$7$			49.0000i			0.377964i
$7$			49.0000i			0.377964i
$8$		0			0
$9$	−80.1112	+	229.415i	−0.329676	+	0.944094i
$10$		0			0
$11$	0.500662			0.00124756			0.000623782	−	1.00000i	$-0.499801\pi$
$11$	0.500662			0.00124756			0.000623782		1.00000i	$0.499801\pi$
$12$		0			0
$13$	−494.561			−0.811636			−0.405818	−	0.913954i	$-0.633013\pi$
$13$	−494.561			−0.811636			−0.405818	+	0.913954i	$0.633013\pi$
$14$		0			0
$15$	−1132.58	+	804.152i	−1.29969	+	0.922805i
$16$		0			0
$17$			1567.85i			1.31577i	0.753116	+	0.657887i	$0.228550\pi$
$17$			1567.85i			1.31577i	−0.753116	+	0.657887i	$0.771450\pi$
$18$		0			0
$19$		−	1494.94i		−	0.950035i	−0.879976	−	0.475018i	$-0.842442\pi$
$19$		−	1494.94i		−	0.950035i	0.879976	−	0.475018i	$-0.157558\pi$
$20$		0			0
$21$	−622.812	+	442.208i	−0.308183	+	0.218816i
$22$		0			0
$23$	38.6621			0.0152393			0.00761966	−	0.999971i	$-0.497575\pi$
$23$	38.6621			0.0152393			0.00761966	+	0.999971i	$0.497575\pi$
$24$		0			0
$25$	−4814.90			−1.54077
$26$		0			0
$27$	−3638.94	+	1052.14i	−0.960652	+	0.277756i
$28$		0			0
$29$		−	217.146i		−	0.0479465i	−0.999713	−	0.0239732i	$-0.992368\pi$
$29$		−	217.146i		−	0.0479465i	0.999713	−	0.0239732i	$-0.00763165\pi$
$30$		0			0
$31$			8367.22i			1.56378i	0.623413	+	0.781892i	$0.285745\pi$
$31$			8367.22i			1.56378i	−0.623413	+	0.781892i	$0.714255\pi$
$32$		0			0
$33$	4.51830	+	6.36364i	0.000722255	+	0.00101723i
$34$		0			0
$35$	−4366.20			−0.602467
$36$		0			0
$37$	13263.5			1.59278			0.796388	−	0.604786i	$-0.206742\pi$
$37$	13263.5			1.59278			0.796388	+	0.604786i	$0.206742\pi$
$38$		0			0
$39$	−4463.24	−	6286.09i	−0.469882	−	0.661788i
$40$		0			0
$41$		−	11918.5i		−	1.10730i	−0.832751	−	0.553648i	$-0.813235\pi$
$41$		−	11918.5i		−	1.10730i	0.832751	−	0.553648i	$-0.186765\pi$
$42$		0			0
$43$		−	10580.1i		−	0.872608i	−0.899799	−	0.436304i	$-0.856287\pi$
$43$		−	10580.1i		−	0.872608i	0.899799	−	0.436304i	$-0.143713\pi$
$44$		0			0
$45$	−20442.3	−	7138.40i	−1.50487	−	0.525496i
$46$		0			0
$47$	−407.893			−0.0269340			−0.0134670	−	0.999909i	$-0.504287\pi$
$47$	−407.893			−0.0269340			−0.0134670	+	0.999909i	$0.504287\pi$
$48$		0			0
$49$	−2401.00			−0.142857
$50$		0			0
$51$	−19928.1	+	14149.3i	−1.07285	+	0.761744i
$52$		0			0
$53$		−	5835.89i		−	0.285376i	−0.989768	−	0.142688i	$-0.954425\pi$
$53$		−	5835.89i		−	0.285376i	0.989768	−	0.142688i	$-0.0455745\pi$
$54$		0			0
$55$			44.6121i			0.00198859i
$56$		0			0
$57$	19001.4	−	13491.3i	0.774636	−	0.550006i
$58$		0			0
$59$	36055.8			1.34848			0.674241	−	0.738512i	$-0.264471\pi$
$59$	36055.8			1.34848			0.674241	+	0.738512i	$0.264471\pi$
$60$		0			0
$61$	−42120.5			−1.44933			−0.724667	−	0.689099i	$-0.758006\pi$
$61$	−42120.5			−1.44933			−0.724667	+	0.689099i	$0.758006\pi$
$62$		0			0
$63$	−11241.3	−	3925.45i	−0.356834	−	0.124606i
$64$		0			0
$65$		−	44068.4i		−	1.29373i
$66$		0			0
$67$			61188.5i			1.66526i	0.553827	+	0.832632i	$0.313167\pi$
$67$			61188.5i			1.66526i	−0.553827	+	0.832632i	$0.686833\pi$
$68$		0			0
$69$	348.912	+	491.412i	0.00882253	+	0.0124258i
$70$		0			0
$71$	43952.7			1.03476			0.517380	−	0.855756i	$-0.326907\pi$
$71$	43952.7			1.03476			0.517380	+	0.855756i	$0.326907\pi$
$72$		0			0
$73$	−1566.88			−0.0344135			−0.0172067	−	0.999852i	$-0.505477\pi$
$73$	−1566.88			−0.0344135			−0.0172067	+	0.999852i	$0.505477\pi$
$74$		0			0
$75$	−43452.8	−	61199.5i	−0.891999	−	1.25630i
$76$		0			0
$77$			24.5324i			0.000471535i
$78$		0			0
$79$		−	63492.3i		−	1.14460i	−0.820045	−	0.572300i	$-0.806051\pi$
$79$		−	63492.3i		−	1.14460i	0.820045	−	0.572300i	$-0.193949\pi$
$80$		0			0
$81$	−46213.4	−	36757.4i	−0.782628	−	0.622490i
$82$		0			0
$83$	84958.5			1.35367			0.676833	−	0.736136i	$-0.263352\pi$
$83$	84958.5			1.35367			0.676833	+	0.736136i	$0.263352\pi$
$84$		0			0
$85$	−139705.			−2.09732
$86$		0			0
$87$	2760.02	−	1959.67i	0.0390944	−	0.0277577i
$88$		0			0
$89$		−	11088.1i		−	0.148382i	−0.997244	−	0.0741910i	$-0.976363\pi$
$89$		−	11088.1i		−	0.148382i	0.997244	−	0.0741910i	$-0.0236375\pi$
$90$		0			0
$91$		−	24233.5i		−	0.306769i
$92$		0			0
$93$	−106351.	+	75511.3i	−1.27507	+	0.905325i
$94$		0			0
$95$	133208.			1.51434
$96$		0			0
$97$	−92521.6			−0.998421			−0.499211	−	0.866481i	$-0.666377\pi$
$97$	−92521.6			−0.998421			−0.499211	+	0.866481i	$0.666377\pi$
$98$		0			0
$99$	−40.1087	+	114.859i	−0.000411292	+	0.00117782i
$100$		0			0
$101$			85330.2i			0.832337i	0.909287	+	0.416169i	$0.136627\pi$
$101$			85330.2i			0.832337i	−0.909287	+	0.416169i	$0.863373\pi$
$102$		0			0
$103$			12008.2i			0.111528i	0.998444	+	0.0557639i	$0.0177594\pi$
$103$			12008.2i			0.111528i	−0.998444	+	0.0557639i	$0.982241\pi$
$104$		0			0
$105$	−39403.4	−	55496.4i	−0.348787	−	0.491237i
$106$		0			0
$107$	−201352.			−1.70019			−0.850093	−	0.526632i	$-0.823455\pi$
$107$	−201352.			−1.70019			−0.850093	+	0.526632i	$0.823455\pi$
$108$		0			0
$109$	−33378.2			−0.269089			−0.134545	−	0.990908i	$-0.542957\pi$
$109$	−33378.2			−0.269089			−0.134545	+	0.990908i	$0.542957\pi$
$110$		0			0
$111$	119699.	+	168585.i	0.922108	+	1.29871i
$112$		0			0
$113$		−	207105.i		−	1.52579i	−0.646521	−	0.762896i	$-0.723777\pi$
$113$		−	207105.i		−	1.52579i	0.646521	−	0.762896i	$-0.276223\pi$
$114$		0			0
$115$			3445.03i			0.0242911i
$116$		0			0
$117$	39619.9	−	113460.i	0.267577	−	0.766260i
$118$		0			0
$119$	−76824.5			−0.497316
$120$		0			0
$121$	−161051.			−0.999998
$122$		0			0
$123$	151490.	−	107561.i	0.902862	−	0.641049i
$124$		0			0
$125$		−	150580.i		−	0.861971i
$126$		0			0
$127$		−	35857.1i		−	0.197272i	−0.995124	−	0.0986362i	$-0.968552\pi$
$127$		−	35857.1i		−	0.197272i	0.995124	−	0.0986362i	$-0.0314480\pi$
$128$		0			0
$129$	134478.	−	95481.8i	0.711503	−	0.505180i
$130$		0			0
$131$	−317596.			−1.61695			−0.808476	−	0.588529i	$-0.799707\pi$
$131$	−317596.			−1.61695			−0.808476	+	0.588529i	$0.799707\pi$
$132$		0			0
$133$	73252.0			0.359080
$134$		0			0
$135$	−93752.1	−	324252.i	−0.442738	−	1.53126i
$136$		0			0
$137$		−	301491.i		−	1.37238i	−0.727425	−	0.686188i	$-0.759283\pi$
$137$		−	301491.i		−	1.37238i	0.727425	−	0.686188i	$-0.240717\pi$
$138$		0			0
$139$			399212.i			1.75253i	0.481827	+	0.876267i	$0.339973\pi$
$139$			399212.i			1.75253i	−0.481827	+	0.876267i	$0.660027\pi$
$140$		0			0
$141$	−3681.09	−	5184.50i	−0.0155930	−	0.0219614i
$142$		0			0
$143$	−247.608			−0.00101257
$144$		0			0
$145$	19349.0			0.0764256
$146$		0			0
$147$	−21668.2	−	30517.8i	−0.0827045	−	0.116482i
$148$		0			0
$149$		−	86207.8i		−	0.318113i	−0.987270	−	0.159056i	$-0.949155\pi$
$149$		−	86207.8i		−	0.318113i	0.987270	−	0.159056i	$-0.0508452\pi$
$150$		0			0
$151$			275344.i			0.982729i	0.870954	+	0.491365i	$0.163502\pi$
$151$			275344.i			0.982729i	−0.870954	+	0.491365i	$0.836498\pi$
$152$		0			0
$153$	−359688.	−	125602.i	−1.24222	−	0.433779i
$154$		0			0
$155$	−745570.			−2.49264
$156$		0			0
$157$	−174675.			−0.565562			−0.282781	−	0.959184i	$-0.591257\pi$
$157$	−174675.			−0.565562			−0.282781	+	0.959184i	$0.591257\pi$
$158$		0			0
$159$	74176.7	−	52666.8i	0.232689	−	0.165213i
$160$		0			0
$161$			1894.44i			0.00575992i
$162$		0			0
$163$			267973.i			0.789990i	0.918683	+	0.394995i	$0.129254\pi$
$163$			267973.i			0.789990i	−0.918683	+	0.394995i	$0.870746\pi$
$164$		0			0
$165$	−567.039	+	402.608i	−0.00162145	+	0.00115126i
$166$		0			0
$167$	587945.			1.63134			0.815672	−	0.578514i	$-0.196367\pi$
$167$	587945.			1.63134			0.815672	+	0.578514i	$0.196367\pi$
$168$		0			0
$169$	−126703.			−0.341248
$170$		0			0
$171$	342961.	+	119761.i	0.896923	+	0.313204i
$172$		0			0
$173$			297805.i			0.756514i	0.925701	+	0.378257i	$0.123477\pi$
$173$			297805.i			0.756514i	−0.925701	+	0.378257i	$0.876523\pi$
$174$		0			0
$175$		−	235930.i		−	0.582355i
$176$		0			0
$177$	325391.	+	458285.i	0.780679	+	1.09952i
$178$		0			0
$179$	−132693.			−0.309539			−0.154770	−	0.987951i	$-0.549464\pi$
$179$	−132693.			−0.309539			−0.154770	+	0.987951i	$0.549464\pi$
$180$		0			0
$181$	327860.			0.743862			0.371931	−	0.928260i	$-0.378696\pi$
$181$	327860.			0.743862			0.371931	+	0.928260i	$0.378696\pi$
$182$		0			0
$183$	−380123.	−	535370.i	−0.839066	−	1.18175i
$184$		0			0
$185$			1.18186e6i			2.53885i
$186$		0			0
$187$			784.962i			0.00164151i
$188$		0			0
$189$	−51554.9	−	178308.i	−0.104982	−	0.363092i
$190$		0			0
$191$	−246584.			−0.489081			−0.244541	−	0.969639i	$-0.578637\pi$
$191$	−246584.			−0.489081			−0.244541	+	0.969639i	$0.578637\pi$
$192$		0			0
$193$	−463402.			−0.895498			−0.447749	−	0.894159i	$-0.647774\pi$
$193$	−463402.			−0.895498			−0.447749	+	0.894159i	$0.647774\pi$
$194$		0			0
$195$	560129.	−	397702.i	1.05488	−	0.748981i
$196$		0			0
$197$		−	92200.9i		−	0.169266i	−0.996412	−	0.0846330i	$-0.973028\pi$
$197$		−	92200.9i		−	0.169266i	0.996412	−	0.0846330i	$-0.0269718\pi$
$198$		0			0
$199$			138903.i			0.248644i	0.992242	+	0.124322i	$0.0396756\pi$
$199$			138903.i			0.248644i	−0.992242	+	0.124322i	$0.960324\pi$
$200$		0			0
$201$	−777734.	+	552205.i	−1.35782	+	0.964074i
$202$		0			0
$203$	10640.1			0.0181221
$204$		0			0
$205$	1.06201e6			1.76501
$206$		0			0
$207$	−3097.27	+	8869.65i	−0.00502404	+	0.0143874i
$208$		0			0
$209$		−	748.460i		−	0.00118523i
$210$		0			0
$211$		−	158666.i		−	0.245345i	−0.992447	−	0.122673i	$-0.960853\pi$
$211$		−	158666.i		−	0.245345i	0.992447	−	0.122673i	$-0.0391465\pi$
$212$		0			0
$213$	396658.	+	558659.i	0.599056	+	0.843719i
$214$		0			0
$215$	942753.			1.39092
$216$		0			0
$217$	−409994.			−0.591055
$218$		0			0
$219$	−14140.6	−	19915.8i	−0.0199231	−	0.0280599i
$220$		0			0
$221$		−	775396.i		−	1.06793i
$222$		0			0
$223$			1.13680e6i			1.53081i	0.643552	+	0.765403i	$0.277460\pi$
$223$			1.13680e6i			1.53081i	−0.643552	+	0.765403i	$0.722540\pi$
$224$		0			0
$225$	385727.	−	1.10461e6i	0.507954	−	1.45463i
$226$		0			0
$227$	447447.			0.576338			0.288169	−	0.957580i	$-0.406954\pi$
$227$	447447.			0.576338			0.288169	+	0.957580i	$0.406954\pi$
$228$		0			0
$229$	1.47206e6			1.85497			0.927485	−	0.373860i	$-0.121966\pi$
$229$	1.47206e6			1.85497			0.927485	+	0.373860i	$0.121966\pi$
$230$		0			0
$231$	−311.818	+	221.397i	−0.000384478	+	0.000272987i
$232$		0			0
$233$			678540.i			0.818814i	0.912352	+	0.409407i	$0.134264\pi$
$233$			678540.i			0.818814i	−0.912352	+	0.409407i	$0.865736\pi$
$234$		0			0
$235$		−	36345.7i		−	0.0429323i
$236$		0			0
$237$	807016.	−	572996.i	0.933279	−	0.662645i
$238$		0			0
$239$	475048.			0.537951			0.268975	−	0.963147i	$-0.413315\pi$
$239$	475048.			0.537951			0.268975	+	0.963147i	$0.413315\pi$
$240$		0			0
$241$	1.09469e6			1.21408			0.607040	−	0.794672i	$-0.292357\pi$
$241$	1.09469e6			1.21408			0.607040	+	0.794672i	$0.292357\pi$
$242$		0			0
$243$	50143.7	−	919116.i	0.0544755	−	0.998515i
$244$		0			0
$245$		−	213944.i		−	0.227711i
$246$		0			0
$247$			739338.i			0.771082i
$248$		0			0
$249$	766721.	+	1.07986e6i	0.783681	+	1.10375i
$250$		0			0
$251$	−8624.98			−0.00864119			−0.00432060	−	0.999991i	$-0.501375\pi$
$251$	−8624.98			−0.00864119			−0.00432060	+	0.999991i	$0.501375\pi$
$252$		0			0
$253$	19.3566			1.90120e−5
$254$		0			0
$255$	−1.26079e6	−	1.77571e6i	−1.21420	−	1.71010i
$256$		0			0
$257$		−	750309.i		−	0.708610i	−0.935130	−	0.354305i	$-0.884717\pi$
$257$		−	750309.i		−	0.708610i	0.935130	−	0.354305i	$-0.115283\pi$
$258$		0			0
$259$			649913.i			0.602013i
$260$		0			0
$261$	49816.5	+	17395.8i	0.0452660	+	0.0158068i
$262$		0			0
$263$	1.24009e6			1.10551			0.552755	−	0.833344i	$-0.313577\pi$
$263$	1.24009e6			1.10551			0.552755	+	0.833344i	$0.313577\pi$
$264$		0			0
$265$	520013.			0.454883
$266$		0			0
$267$	140935.	−	100066.i	0.120987	−	0.0859031i
$268$		0			0
$269$			2.33127e6i			1.96432i	0.188056	+	0.982158i	$0.439781\pi$
$269$			2.33127e6i			1.96432i	−0.188056	+	0.982158i	$0.560219\pi$
$270$		0			0
$271$		−	335146.i		−	0.277211i	−0.990348	−	0.138606i	$-0.955738\pi$
$271$		−	335146.i		−	0.277211i	0.990348	−	0.138606i	$-0.0442621\pi$
$272$		0			0
$273$	308018.	−	218699.i	0.250132	−	0.177599i
$274$		0			0
$275$	−2410.64			−0.00192221
$276$		0			0
$277$	651685.			0.510315			0.255157	−	0.966900i	$-0.417873\pi$
$277$	651685.			0.510315			0.255157	+	0.966900i	$0.417873\pi$
$278$		0			0
$279$	−1.91957e6	−	670308.i	−1.47636	−	0.515542i
$280$		0			0
$281$			205434.i			0.155205i	0.996984	+	0.0776025i	$0.0247265\pi$
$281$			205434.i			0.155205i	−0.996984	+	0.0776025i	$0.975273\pi$
$282$		0			0
$283$			1.19611e6i			0.887780i	0.896081	+	0.443890i	$0.146402\pi$
$283$			1.19611e6i			0.887780i	−0.896081	+	0.443890i	$0.853598\pi$
$284$		0			0
$285$	1.20216e6	+	1.69314e6i	0.876697	+	1.23475i
$286$		0			0
$287$	584009.			0.418518
$288$		0			0
$289$	−1.03829e6			−0.731264
$290$		0			0
$291$	−834976.	−	1.17599e6i	−0.578018	−	0.814089i
$292$		0			0
$293$			2.18573e6i			1.48740i	0.668513	+	0.743701i	$0.266931\pi$
$293$			2.18573e6i			1.48740i	−0.668513	+	0.743701i	$0.733069\pi$
$294$		0			0
$295$			3.21279e6i			2.14945i
$296$		0			0
$297$	−1821.88	+	526.767i	−0.00119847	+	0.000346519i
$298$		0			0
$299$	−19120.7			−0.0123688
$300$		0			0
$301$	518425.			0.329815
$302$		0			0
$303$	−1.08459e6	+	770076.i	−0.678668	+	0.481867i
$304$		0			0
$305$		−	3.75319e6i		−	2.31021i
$306$		0			0
$307$		−	1.01159e6i		−	0.612576i	−0.951939	−	0.306288i	$-0.900913\pi$
$307$		−	1.01159e6i		−	0.612576i	0.951939	−	0.306288i	$-0.0990870\pi$
$308$		0			0
$309$	−152629.	+	108369.i	−0.0909371	+	0.0645670i
$310$		0			0
$311$	−1.06060e6			−0.621800			−0.310900	−	0.950443i	$-0.600630\pi$
$311$	−1.06060e6			−0.621800			−0.310900	+	0.950443i	$0.600630\pi$
$312$		0			0
$313$	−1.14675e6			−0.661617			−0.330808	−	0.943698i	$-0.607321\pi$
$313$	−1.14675e6			−0.661617			−0.330808	+	0.943698i	$0.607321\pi$
$314$		0			0
$315$	349782.	−	1.00167e6i	0.198619	−	0.568786i
$316$		0			0
$317$			3.10580e6i			1.73590i	0.496651	+	0.867950i	$0.334563\pi$
$317$			3.10580e6i			1.73590i	−0.496651	+	0.867950i	$0.665437\pi$
$318$		0			0
$319$		−	108.717i		−	5.98163e-5i
$320$		0			0
$321$	−1.81713e6	−	2.55928e6i	−0.984292	−	1.38629i
$322$		0			0
$323$	2.34384e6			1.25003
$324$		0			0
$325$	2.38126e6			1.25054
$326$		0			0
$327$	−301227.	−	424252.i	−0.155784	−	0.219409i
$328$		0			0
$329$		−	19986.7i		−	0.0101801i
$330$		0			0
$331$			3.12207e6i			1.56629i	0.621838	+	0.783146i	$0.286386\pi$
$331$			3.12207e6i			1.56629i	−0.621838	+	0.783146i	$0.713614\pi$
$332$		0			0
$333$	−1.06256e6	+	3.04285e6i	−0.525100	+	1.50373i
$334$		0			0
$335$	−5.45227e6			−2.65439
$336$		0			0
$337$	−583792.			−0.280016			−0.140008	−	0.990150i	$-0.544713\pi$
$337$	−583792.			−0.280016			−0.140008	+	0.990150i	$0.544713\pi$
$338$		0			0
$339$	2.63240e6	−	1.86906e6i	1.24409	−	0.883330i
$340$		0			0
$341$			4189.15i			0.00195092i
$342$		0			0
$343$		−	117649.i		−	0.0539949i
$344$		0			0
$345$	−43787.8	+	31090.2i	−0.0198064	+	0.0140629i
$346$		0			0
$347$	−3.79948e6			−1.69395			−0.846975	−	0.531632i	$-0.821579\pi$
$347$	−3.79948e6			−1.69395			−0.846975	+	0.531632i	$0.821579\pi$
$348$		0			0
$349$	−2.70346e6			−1.18811			−0.594055	−	0.804425i	$-0.702474\pi$
$349$	−2.70346e6			−1.18811			−0.594055	+	0.804425i	$0.702474\pi$
$350$		0			0
$351$	1.79968e6	−	520347.i	0.779699	−	0.225437i
$352$		0			0
$353$			744026.i			0.317798i	0.987295	+	0.158899i	$0.0507944\pi$
$353$			744026.i			0.317798i	−0.987295	+	0.158899i	$0.949206\pi$
$354$		0			0
$355$			3.91645e6i			1.64939i
$356$		0			0
$357$	−693315.	−	976475.i	−0.287912	−	0.405500i
$358$		0			0
$359$	380221.			0.155704			0.0778521	−	0.996965i	$-0.475194\pi$
$359$	380221.			0.155704			0.0778521	+	0.996965i	$0.475194\pi$
$360$		0			0
$361$	241255.			0.0974333
$362$		0			0
$363$	−1.45343e6	−	2.04703e6i	−0.578931	−	0.815375i
$364$		0			0
$365$		−	139619.i		−	0.0548544i
$366$		0			0
$367$		−	2.71708e6i		−	1.05302i	−0.850168	−	0.526511i	$-0.823500\pi$
$367$		−	2.71708e6i		−	1.05302i	0.850168	−	0.526511i	$-0.176500\pi$
$368$		0			0
$369$	2.73429e6	+	954809.i	1.04539	+	0.365049i
$370$		0			0
$371$	285958.			0.107862
$372$		0			0
$373$	1.04911e6			0.390436			0.195218	−	0.980760i	$-0.437459\pi$
$373$	1.04911e6			0.390436			0.195218	+	0.980760i	$0.437459\pi$
$374$		0			0
$375$	1.91394e6	−	1.35893e6i	0.702831	−	0.499023i
$376$		0			0
$377$			107392.i			0.0389151i
$378$		0			0
$379$			2.10101e6i			0.751329i	0.926756	+	0.375665i	$0.122586\pi$
$379$			2.10101e6i			0.751329i	−0.926756	+	0.375665i	$0.877414\pi$
$380$		0			0
$381$	455760.	−	323598.i	0.160851	−	0.114207i
$382$		0			0
$383$	−6955.85			−0.00242300			−0.00121150	−	0.999999i	$-0.500386\pi$
$383$	−6955.85			−0.00242300			−0.00121150	+	0.999999i	$0.500386\pi$
$384$		0			0
$385$	−2185.99			−0.000751617
$386$		0			0
$387$	2.42724e6	+	847586.i	0.823824	+	0.287678i
$388$		0			0
$389$		−	5.31694e6i		−	1.78151i	−0.454485	−	0.890754i	$-0.650177\pi$
$389$		−	5.31694e6i		−	1.78151i	0.454485	−	0.890754i	$-0.349823\pi$
$390$		0			0
$391$			60616.2i			0.0200515i
$392$		0			0
$393$	−2.86620e6	−	4.03679e6i	−0.936105	−	1.31842i
$394$		0			0
$395$	5.65755e6			1.82447
$396$		0			0
$397$	−3.38197e6			−1.07695			−0.538473	−	0.842643i	$-0.680999\pi$
$397$	−3.38197e6			−1.07695			−0.538473	+	0.842643i	$0.680999\pi$
$398$		0			0
$399$	661074.	+	931067.i	0.207883	+	0.292785i
$400$		0			0
$401$		−	5.07917e6i		−	1.57737i	−0.614801	−	0.788683i	$-0.710764\pi$
$401$		−	5.07917e6i		−	1.57737i	0.614801	−	0.788683i	$-0.289236\pi$
$402$		0			0
$403$		−	4.13810e6i		−	1.26922i
$404$		0			0
$405$	3.27531e6	−	4.11789e6i	0.992236	−	1.24749i
$406$		0			0
$407$	6640.54			0.00198709
$408$		0			0
$409$	−5.89940e6			−1.74381			−0.871907	−	0.489672i	$-0.837116\pi$
$409$	−5.89940e6			−1.74381			−0.871907	+	0.489672i	$0.837116\pi$
$410$		0			0
$411$	3.83209e6	−	2.72085e6i	1.11900	−	0.794512i
$412$		0			0
$413$			1.76673e6i			0.509678i
$414$		0			0
$415$			7.57032e6i			2.15772i
$416$		0			0
$417$	−5.07416e6	+	3.60275e6i	−1.42897	+	1.01460i
$418$		0			0
$419$	−5.66949e6			−1.57764			−0.788822	−	0.614622i	$-0.789309\pi$
$419$	−5.66949e6			−1.57764			−0.788822	+	0.614622i	$0.789309\pi$
$420$		0			0
$421$	3.19589e6			0.878794			0.439397	−	0.898293i	$-0.355192\pi$
$421$	3.19589e6			0.878794			0.439397	+	0.898293i	$0.355192\pi$
$422$		0			0
$423$	32676.8	−	93576.7i	0.00887950	−	0.0254283i
$424$		0			0
$425$		−	7.54903e6i		−	2.02730i
$426$		0			0
$427$		−	2.06390e6i		−	0.547797i
$428$		0			0
$429$	−2234.57	−	3147.21i	−0.000586208	−	0.000825624i
$430$		0			0
$431$	4.34244e6			1.12601			0.563003	−	0.826455i	$-0.309646\pi$
$431$	4.34244e6			1.12601			0.563003	+	0.826455i	$0.309646\pi$
$432$		0			0
$433$	1.69513e6			0.434494			0.217247	−	0.976117i	$-0.430292\pi$
$433$	1.69513e6			0.434494			0.217247	+	0.976117i	$0.430292\pi$
$434$		0			0
$435$	174618.	+	245935.i	0.0442452	+	0.0623156i
$436$		0			0
$437$		−	57797.5i		−	0.0144779i
$438$		0			0
$439$		−	6.05622e6i		−	1.49982i	−0.661538	−	0.749912i	$-0.730096\pi$
$439$		−	6.05622e6i		−	1.49982i	0.661538	−	0.749912i	$-0.269904\pi$
$440$		0			0
$441$	192347.	−	550825.i	0.0470966	−	0.134871i
$442$		0			0
$443$	−3.17377e6			−0.768363			−0.384182	−	0.923258i	$-0.625516\pi$
$443$	−3.17377e6			−0.768363			−0.384182	+	0.923258i	$0.625516\pi$
$444$		0			0
$445$	988016.			0.236518
$446$		0			0
$447$	1.09574e6	−	777996.i	0.259381	−	0.184166i
$448$		0			0
$449$			3.16370e6i			0.740594i	0.928913	+	0.370297i	$0.120744\pi$
$449$			3.16370e6i			0.740594i	−0.928913	+	0.370297i	$0.879256\pi$
$450$		0			0
$451$		−	5967.16i		−	0.00138142i
$452$		0			0
$453$	−3.49975e6	+	2.48489e6i	−0.801294	+	0.568933i
$454$		0			0
$455$	2.15935e6			0.488984
$456$		0			0
$457$	−2.66857e6			−0.597707			−0.298854	−	0.954299i	$-0.596604\pi$
$457$	−2.66857e6			−0.597707			−0.298854	+	0.954299i	$0.596604\pi$
$458$		0			0
$459$	−1.64960e6	−	5.70531e6i	−0.365465	−	1.26400i
$460$		0			0
$461$			5.45772e6i			1.19608i	0.801468	+	0.598038i	$0.204053\pi$
$461$			5.45772e6i			1.19608i	−0.801468	+	0.598038i	$0.795947\pi$
$462$		0			0
$463$			2.05681e6i			0.445905i	0.974829	+	0.222953i	$0.0715695\pi$
$463$			2.05681e6i			0.445905i	−0.974829	+	0.222953i	$0.928430\pi$
$464$		0			0
$465$	−6.72852e6	−	9.47654e6i	−1.44307	−	2.03244i
$466$		0			0
$467$	7.67403e6			1.62829			0.814144	−	0.580663i	$-0.197206\pi$
$467$	7.67403e6			1.62829			0.814144	+	0.580663i	$0.197206\pi$
$468$		0			0
$469$	−2.99824e6			−0.629410
$470$		0			0
$471$	−1.57638e6	−	2.22019e6i	−0.327422	−	0.461146i
$472$		0			0
$473$		−	5297.06i		−	0.00108863i
$474$		0			0
$475$			7.19798e6i			1.46378i
$476$		0			0
$477$	1.33884e6	+	467520.i	0.269422	+	0.0940815i
$478$		0			0
$479$	−7.77488e6			−1.54830			−0.774149	−	0.633003i	$-0.781822\pi$
$479$	−7.77488e6			−1.54830			−0.774149	+	0.633003i	$0.781822\pi$
$480$		0			0
$481$	−6.55961e6			−1.29275
$482$		0			0
$483$	−24079.2	+	17096.7i	−0.00469650	+	0.00333460i
$484$		0			0
$485$		−	8.24424e6i		−	1.59146i
$486$		0			0
$487$			5.65842e6i			1.08112i	0.841306	+	0.540559i	$0.181787\pi$
$487$			5.65842e6i			1.08112i	−0.841306	+	0.540559i	$0.818213\pi$
$488$		0			0
$489$	−3.40606e6	+	2.41836e6i	−0.644139	+	0.457351i
$490$		0			0
$491$	−7.61103e6			−1.42475			−0.712377	−	0.701797i	$-0.752381\pi$
$491$	−7.61103e6			−1.42475			−0.712377	+	0.701797i	$0.752381\pi$
$492$		0			0
$493$	340452.			0.0630868
$494$		0			0
$495$	−10234.7	−	3573.93i	−0.00187742	−	0.000655591i
$496$		0			0
$497$			2.15368e6i			0.391103i
$498$		0			0
$499$			9.64253e6i			1.73356i	0.498687	+	0.866782i	$0.333816\pi$
$499$			9.64253e6i			1.73356i	−0.498687	+	0.866782i	$0.666184\pi$
$500$		0			0
$501$	5.30600e6	+	7.47305e6i	0.944437	+	1.33016i
$502$		0			0
$503$	1.06371e7			1.87458			0.937289	−	0.348552i	$-0.113327\pi$
$503$	1.06371e7			1.87458			0.937289	+	0.348552i	$0.113327\pi$
$504$		0			0
$505$	−7.60344e6			−1.32673
$506$		0			0
$507$	−1.14345e6	−	1.61045e6i	−0.197559	−	0.278245i
$508$		0			0
$509$			3.27060e6i			0.559542i	0.960067	+	0.279771i	$0.0902586\pi$
$509$			3.27060e6i			0.559542i	−0.960067	+	0.279771i	$0.909741\pi$
$510$		0			0
$511$		−	76777.1i		−	0.0130071i
$512$		0			0
$513$	1.57289e6	+	5.44000e6i	0.263878	+	0.912653i
$514$		0			0
$515$	−1.07000e6			−0.177773
$516$		0			0
$517$	−204.216			−3.36019e−5
$518$		0			0
$519$	−3.78524e6	+	2.68759e6i	−0.616844	+	0.437970i
$520$		0			0
$521$			3.22903e6i			0.521168i	0.965451	+	0.260584i	$0.0839152\pi$
$521$			3.22903e6i			0.521168i	−0.965451	+	0.260584i	$0.916085\pi$
$522$		0			0
$523$		−	6.39933e6i		−	1.02301i	−0.859280	−	0.511506i	$-0.829088\pi$
$523$		−	6.39933e6i		−	1.02301i	0.859280	−	0.511506i	$-0.170912\pi$
$524$		0			0
$525$	2.99878e6	−	2.12919e6i	0.474839	−	0.337144i
$526$		0			0
$527$	−1.31185e7			−2.05759
$528$		0			0
$529$	−6.43485e6			−0.999768
$530$		0			0
$531$	−2.88847e6	+	8.27173e6i	−0.444562	+	1.27309i
$532$		0			0
$533$			5.89444e6i			0.898720i
$534$		0			0
$535$		−	1.79417e7i		−	2.71006i
$536$		0			0
$537$	−1.19751e6	−	1.68659e6i	−0.179202	−	0.252391i
$538$		0			0
$539$	−1202.09			−0.000178224
$540$		0			0
$541$	−5.64318e6			−0.828954			−0.414477	−	0.910060i	$-0.636036\pi$
$541$	−5.64318e6			−0.828954			−0.414477	+	0.910060i	$0.636036\pi$
$542$		0			0
$543$	2.95883e6	+	4.16725e6i	0.430645	+	0.606527i
$544$		0			0
$545$		−	2.97420e6i		−	0.428923i
$546$		0			0
$547$		−	7.56523e6i		−	1.08107i	−0.841322	−	0.540535i	$-0.818222\pi$
$547$		−	7.56523e6i		−	1.08107i	0.841322	−	0.540535i	$-0.181778\pi$
$548$		0			0
$549$	3.37432e6	−	9.66306e6i	0.477811	−	1.36831i
$550$		0			0
$551$	−324620.			−0.0455508
$552$		0			0
$553$	3.11112e6			0.432618
$554$		0			0
$555$	−1.50220e7	+	1.06659e7i	−2.07012	+	1.46982i
$556$		0			0
$557$		−	7.45725e6i		−	1.01845i	−0.860633	−	0.509226i	$-0.829932\pi$
$557$		−	7.45725e6i		−	1.01845i	0.860633	−	0.509226i	$-0.170068\pi$
$558$		0			0
$559$			5.23251e6i			0.708240i
$560$		0			0
$561$	−9977.22	+	7084.01i	−0.00133845	+	0.000950325i
$562$		0			0
$563$	1.37333e7			1.82601			0.913005	−	0.407949i	$-0.133756\pi$
$563$	1.37333e7			1.82601			0.913005	+	0.407949i	$0.133756\pi$
$564$		0			0
$565$	1.84544e7			2.43208
$566$		0			0
$567$	1.80111e6	−	2.26446e6i	0.235279	−	0.295805i
$568$		0			0
$569$		−	3.40447e6i		−	0.440827i	−0.975407	−	0.220414i	$-0.929259\pi$
$569$		−	3.40447e6i		−	0.440827i	0.975407	−	0.220414i	$-0.0707407\pi$
$570$		0			0
$571$			1.88581e6i			0.242051i	0.992649	+	0.121025i	$0.0386183\pi$
$571$			1.88581e6i			0.242051i	−0.992649	+	0.121025i	$0.961382\pi$
$572$		0			0
$573$	−2.22533e6	−	3.13419e6i	−0.283145	−	0.398785i
$574$		0			0
$575$	−186154.			−0.0234802
$576$		0			0
$577$	8.66113e6			1.08302			0.541508	−	0.840695i	$-0.317854\pi$
$577$	8.66113e6			1.08302			0.541508	+	0.840695i	$0.317854\pi$
$578$		0			0
$579$	−4.18204e6	−	5.89005e6i	−0.518433	−	0.730168i
$580$		0			0
$581$			4.16297e6i			0.511638i
$582$		0			0
$583$		−	2921.81i		−	0.000356025i
$584$		0			0
$585$	1.01099e7	+	3.53037e6i	1.22140	+	0.426511i
$586$		0			0
$587$	4.71177e6			0.564402			0.282201	−	0.959355i	$-0.408936\pi$
$587$	4.71177e6			0.564402			0.282201	+	0.959355i	$0.408936\pi$
$588$		0			0
$589$	1.25085e7			1.48565
$590$		0			0
$591$	1.17192e6	−	832081.i	0.138015	−	0.0979935i
$592$		0			0
$593$		−	9.98387e6i		−	1.16590i	−0.812507	−	0.582951i	$-0.801898\pi$
$593$		−	9.98387e6i		−	1.16590i	0.812507	−	0.582951i	$-0.198102\pi$
$594$		0			0
$595$		−	6.84554e6i		−	0.792711i
$596$		0			0
$597$	−1.76552e6	+	1.25355e6i	−0.202739	+	0.143948i
$598$		0			0
$599$	1.14206e7			1.30053			0.650267	−	0.759706i	$-0.274657\pi$
$599$	1.14206e7			1.30053			0.650267	+	0.759706i	$0.274657\pi$
$600$		0			0
$601$	−819682.			−0.0925677			−0.0462839	−	0.998928i	$-0.514738\pi$
$601$	−819682.			−0.0925677			−0.0462839	+	0.998928i	$0.514738\pi$
$602$		0			0
$603$	−1.40376e7	−	4.90189e6i	−1.57217	−	0.548997i
$604$		0			0
$605$		−	1.43506e7i		−	1.59398i
$606$		0			0
$607$		−	3.74981e6i		−	0.413084i	−0.978438	−	0.206542i	$-0.933779\pi$
$607$		−	3.74981e6i		−	0.413084i	0.978438	−	0.206542i	$-0.0662210\pi$
$608$		0			0
$609$	96023.6	+	135241.i	0.0104914	+	0.0147763i
$610$		0			0
$611$	201728.			0.0218606
$612$		0			0
$613$	7.69400e6			0.826991			0.413496	−	0.910506i	$-0.364308\pi$
$613$	7.69400e6			0.826991			0.413496	+	0.910506i	$0.364308\pi$
$614$		0			0
$615$	9.58432e6	+	1.34987e7i	1.02182	+	1.43914i
$616$		0			0
$617$			7.02931e6i			0.743361i	0.928361	+	0.371681i	$0.121218\pi$
$617$			7.02931e6i			0.743361i	−0.928361	+	0.371681i	$0.878782\pi$
$618$		0			0
$619$			1.11190e7i			1.16638i	0.812336	+	0.583189i	$0.198195\pi$
$619$			1.11190e7i			1.16638i	−0.812336	+	0.583189i	$0.801805\pi$
$620$		0			0
$621$	−140689.	+	40677.9i	−0.0146397	+	0.00423282i
$622$		0			0
$623$	543316.			0.0560832
$624$		0			0
$625$	−1.62894e6			−0.166803
$626$		0			0
$627$	9513.26	−	6754.59i	0.000966408	−	0.000686167i
$628$		0			0
$629$			2.07952e7i			2.09573i
$630$		0			0
$631$			2.58766e6i			0.258723i	0.991598	+	0.129361i	$0.0412927\pi$
$631$			2.58766e6i			0.258723i	−0.991598	+	0.129361i	$0.958707\pi$
$632$		0			0
$633$	2.01672e6	−	1.43191e6i	0.200049	−	0.142038i
$634$		0			0
$635$	3.19509e6			0.314448
$636$		0			0
$637$	1.18744e6			0.115948
$638$		0			0
$639$	−3.52111e6	+	1.00834e7i	−0.341136	+	0.976911i
$640$		0			0
$641$			1.53208e7i			1.47277i	0.676561	+	0.736387i	$0.263470\pi$
$641$			1.53208e7i			1.47277i	−0.676561	+	0.736387i	$0.736530\pi$
$642$		0			0
$643$			5.71046e6i			0.544683i	0.962201	+	0.272341i	$0.0877980\pi$
$643$			5.71046e6i			0.544683i	−0.962201	+	0.272341i	$0.912202\pi$
$644$		0			0
$645$	8.50802e6	+	1.19828e7i	0.805247	+	1.13412i
$646$		0			0
$647$	1.31655e7			1.23645			0.618225	−	0.786001i	$-0.287852\pi$
$647$	1.31655e7			1.23645			0.618225	+	0.786001i	$0.287852\pi$
$648$		0			0
$649$	18051.8			0.00168232
$650$		0			0
$651$	−3.70005e6	−	5.21121e6i	−0.342181	−	0.481932i
$652$		0			0
$653$		−	607604.i		−	0.0557619i	−0.999611	−	0.0278810i	$-0.991124\pi$
$653$		−	607604.i		−	0.0557619i	0.999611	−	0.0278810i	$-0.00887593\pi$
$654$		0			0
$655$		−	2.82998e7i		−	2.57739i
$656$		0			0
$657$	125525.	−	359466.i	0.0113453	−	0.0324896i
$658$		0			0
$659$	−1.50697e7			−1.35174			−0.675868	−	0.737023i	$-0.736231\pi$
$659$	−1.50697e7			−1.35174			−0.675868	+	0.737023i	$0.736231\pi$
$660$		0			0
$661$	−2.02480e7			−1.80251			−0.901255	−	0.433289i	$-0.857353\pi$
$661$	−2.02480e7			−1.80251			−0.901255	+	0.433289i	$0.857353\pi$
$662$		0			0
$663$	9.85563e6	−	6.99768e6i	0.870764	−	0.618259i
$664$		0			0
$665$			6.52720e6i			0.572365i
$666$		0			0
$667$		−	8395.31i		−	0.000730671i
$668$		0			0
$669$	−1.44492e7	+	1.02592e7i	−1.24818	+	0.886232i
$670$		0			0
$671$	−21088.1			−0.00180814
$672$		0			0
$673$	769492.			0.0654887			0.0327443	−	0.999464i	$-0.489575\pi$
$673$	769492.			0.0654887			0.0327443	+	0.999464i	$0.489575\pi$
$674$		0			0
$675$	1.75211e7	−	5.06595e6i	1.48014	−	0.427958i
$676$		0			0
$677$			1.36511e7i			1.14471i	0.820006	+	0.572356i	$0.193970\pi$
$677$			1.36511e7i			1.14471i	−0.820006	+	0.572356i	$0.806030\pi$
$678$		0			0
$679$		−	4.53356e6i		−	0.377368i
$680$		0			0
$681$	4.03806e6	+	5.68726e6i	0.333660	+	0.469932i
$682$		0			0
$683$	1.85411e7			1.52084			0.760421	−	0.649430i	$-0.224992\pi$
$683$	1.85411e7			1.52084			0.760421	+	0.649430i	$0.224992\pi$
$684$		0			0
$685$	2.68647e7			2.18754
$686$		0			0
$687$	1.32848e7	+	1.87106e7i	1.07390	+	1.51250i
$688$		0			0
$689$			2.88620e6i			0.231621i
$690$		0			0
$691$		−	1.44856e7i		−	1.15410i	−0.816710	−	0.577049i	$-0.804204\pi$
$691$		−	1.44856e7i		−	1.15410i	0.816710	−	0.577049i	$-0.195796\pi$
$692$		0			0
$693$	−5628.11	−	1965.32i	−0.000445174	−	0.000155454i
$694$		0			0
$695$	−3.55722e7			−2.79350
$696$		0			0
$697$	1.86865e7			1.45695
$698$		0			0
$699$	−8.62455e6	+	6.12359e6i	−0.667642	+	0.474038i
$700$		0			0
$701$			1.11306e7i			0.855509i	0.903895	+	0.427755i	$0.140695\pi$
$701$			1.11306e7i			0.855509i	−0.903895	+	0.427755i	$0.859305\pi$
$702$		0			0
$703$		−	1.98282e7i		−	1.51319i
$704$		0			0
$705$	461971.	−	328008.i	0.0350059	−	0.0248549i
$706$		0			0
$707$	−4.18118e6			−0.314594
$708$		0			0
$709$	8.73823e6			0.652842			0.326421	−	0.945225i	$-0.394157\pi$
$709$	8.73823e6			0.652842			0.326421	+	0.945225i	$0.394157\pi$
$710$		0			0
$711$	1.45661e7	+	5.08645e6i	1.08061	+	0.377347i
$712$		0			0
$713$			323494.i			0.0238310i
$714$		0			0
$715$		−	22063.4i		−	0.00161401i
$716$		0			0
$717$	4.28714e6	+	6.03807e6i	0.311437	+	0.438632i
$718$		0			0
$719$	−4.87877e6			−0.351956			−0.175978	−	0.984394i	$-0.556309\pi$
$719$	−4.87877e6			−0.351956			−0.175978	+	0.984394i	$0.556309\pi$
$720$		0			0
$721$	−588400.			−0.0421536
$722$		0			0
$723$	9.87916e6	+	1.39140e7i	0.702869	+	0.989931i
$724$		0			0
$725$			1.04554e6i			0.0738743i
$726$		0			0
$727$			1.96724e7i			1.38045i	0.723595	+	0.690225i	$0.242488\pi$
$727$			1.96724e7i			1.38045i	−0.723595	+	0.690225i	$0.757512\pi$
$728$		0			0
$729$	1.21349e7	−	7.65735e6i	0.845703	−	0.533654i
$730$		0			0
$731$	1.65880e7			1.14816
$732$		0			0
$733$	−1.39786e7			−0.960955			−0.480477	−	0.877007i	$-0.659537\pi$
$733$	−1.39786e7			−0.960955			−0.480477	+	0.877007i	$0.659537\pi$
$734$		0			0
$735$	2.71932e6	−	1.93077e6i	0.185670	−	0.131829i
$736$		0			0
$737$			30634.8i			0.00207752i
$738$		0			0
$739$		−	5.21685e6i		−	0.351396i	−0.984444	−	0.175698i	$-0.943782\pi$
$739$		−	5.21685e6i		−	0.351396i	0.984444	−	0.175698i	$-0.0562183\pi$
$740$		0			0
$741$	−9.39732e6	+	6.67227e6i	−0.628722	+	0.446404i
$742$		0			0
$743$	5.71859e6			0.380029			0.190015	−	0.981781i	$-0.439146\pi$
$743$	5.71859e6			0.380029			0.190015	+	0.981781i	$0.439146\pi$
$744$		0			0
$745$	7.68164e6			0.507065
$746$		0			0
$747$	−6.80613e6	+	1.94907e7i	−0.446271	+	1.27799i
$748$		0			0
$749$		−	9.86625e6i		−	0.642610i
$750$		0			0
$751$			1.23839e7i			0.801234i	0.916246	+	0.400617i	$0.131204\pi$
$751$			1.23839e7i			0.801234i	−0.916246	+	0.400617i	$0.868796\pi$
$752$		0			0
$753$	−77837.4	−	109627.i	−0.00500266	−	0.00704582i
$754$		0			0
$755$	−2.45349e7			−1.56645
$756$		0			0
$757$	−2.68220e7			−1.70118			−0.850591	−	0.525827i	$-0.823756\pi$
$757$	−2.68220e7			−1.70118			−0.850591	+	0.525827i	$0.823756\pi$
$758$		0			0
$759$	174.687	+	246.032i	1.10067e−5	+	1.55020e-5i
$760$		0			0
$761$		−	7.50832e6i		−	0.469982i	−0.971998	−	0.234991i	$-0.924494\pi$
$761$		−	7.50832e6i		−	0.469982i	0.971998	−	0.234991i	$-0.0755060\pi$
$762$		0			0
$763$		−	1.63553e6i		−	0.101706i
$764$		0			0
$765$	1.11919e7	−	3.20504e7i	0.691435	−	1.98006i
$766$		0			0
$767$	−1.78318e7			−1.09448
$768$		0			0
$769$	1.34168e7			0.818149			0.409074	−	0.912501i	$-0.365852\pi$
$769$	1.34168e7			0.818149			0.409074	+	0.912501i	$0.365852\pi$
$770$		0			0
$771$	9.53677e6	−	6.77128e6i	0.577784	−	0.410237i
$772$		0			0
$773$		−	2.64320e6i		−	0.159104i	−0.996831	−	0.0795521i	$-0.974651\pi$
$773$		−	2.64320e6i		−	0.159104i	0.996831	−	0.0795521i	$-0.0253490\pi$
$774$		0			0
$775$		−	4.02873e7i		−	2.40943i
$776$		0			0
$777$	−8.26068e6	+	5.86524e6i	−0.490867	+	0.348524i
$778$		0			0
$779$	−1.78175e7			−1.05197
$780$		0			0
$781$	22005.5			0.00129093
$782$		0			0
$783$	228468.	+	790181.i	0.0133174	+	0.0460598i
$784$		0			0
$785$		−	1.55646e7i		−	0.901494i
$786$		0			0
$787$		−	2.36773e7i		−	1.36268i	−0.731966	−	0.681342i	$-0.761397\pi$
$787$		−	2.36773e7i		−	1.36268i	0.731966	−	0.681342i	$-0.238603\pi$
$788$		0			0
$789$	1.11913e7	+	1.57621e7i	0.640015	+	0.901406i
$790$		0			0
$791$	1.01482e7			0.576695
$792$		0			0
$793$	2.08311e7			1.17633
$794$		0			0
$795$	4.69294e6	+	6.60960e6i	0.263346	+	0.370900i
$796$		0			0
$797$		−	1.54278e7i		−	0.860316i	−0.902754	−	0.430158i	$-0.858458\pi$
$797$		−	1.54278e7i		−	0.860316i	0.902754	−	0.430158i	$-0.141542\pi$
$798$		0			0
$799$		−	639514.i		−	0.0354391i
$800$		0			0
$801$	2.54377e6	+	888280.i	0.140087	+	0.0489180i
$802$		0			0
$803$	−784.478			−4.29331e−5
$804$		0			0
$805$	−168806.			−0.00918119
$806$		0			0
$807$	−2.96315e7	+	2.10389e7i	−1.60166	+	1.13721i
$808$		0			0
$809$		−	1.63302e6i		−	0.0877242i	−0.999038	−	0.0438621i	$-0.986034\pi$
$809$		−	1.63302e6i		−	0.0877242i	0.999038	−	0.0438621i	$-0.0139662\pi$
$810$		0			0
$811$		−	1.15435e7i		−	0.616289i	−0.951340	−	0.308145i	$-0.900292\pi$
$811$		−	1.15435e7i		−	0.616289i	0.951340	−	0.308145i	$-0.0997081\pi$
$812$		0			0
$813$	4.25986e6	−	3.02458e6i	0.226032	−	0.160487i
$814$		0			0
$815$	−2.38780e7			−1.25923
$816$		0			0
$817$	−1.58166e7			−0.829008
$818$		0			0
$819$	5.55952e6	+	1.94137e6i	0.289619	+	0.101134i
$820$		0			0
$821$			1.63497e7i			0.846551i	0.906001	+	0.423275i	$0.139120\pi$
$821$			1.63497e7i			0.846551i	−0.906001	+	0.423275i	$0.860880\pi$
$822$		0			0
$823$		−	2.63280e6i		−	0.135494i	−0.997703	−	0.0677468i	$-0.978419\pi$
$823$		−	2.63280e6i		−	0.135494i	0.997703	−	0.0677468i	$-0.0215810\pi$
$824$		0			0
$825$	−21755.2	−	30640.3i	−0.00111283	−	0.00156732i
$826$		0			0
$827$	1.25025e7			0.635672			0.317836	−	0.948146i	$-0.397044\pi$
$827$	1.25025e7			0.635672			0.317836	+	0.948146i	$0.397044\pi$
$828$		0			0
$829$	9.99769e6			0.505258			0.252629	−	0.967563i	$-0.418705\pi$
$829$	9.99769e6			0.505258			0.252629	+	0.967563i	$0.418705\pi$
$830$		0			0
$831$	5.88123e6	+	8.28321e6i	0.295438	+	0.416099i
$832$		0			0
$833$		−	3.76440e6i		−	0.187968i
$834$		0			0
$835$			5.23895e7i			2.60033i
$836$		0			0
$837$	−8.80349e6	−	3.04478e7i	−0.434351	−	1.50225i
$838$		0			0
$839$	−1.71523e7			−0.841233			−0.420617	−	0.907238i	$-0.638186\pi$
$839$	−1.71523e7			−0.841233			−0.420617	+	0.907238i	$0.638186\pi$
$840$		0			0
$841$	2.04640e7			0.997701
$842$		0			0
$843$	−2.61116e6	+	1.85397e6i	−0.126550	+	0.0898532i
$844$		0			0
$845$		−	1.12900e7i		−	0.543941i
$846$		0			0
$847$		−	7.89149e6i		−	0.377964i
$848$		0			0
$849$	−1.52031e7	+	1.07945e7i	−0.723875	+	0.513964i
$850$		0			0
$851$	512795.			0.0242728
$852$		0			0
$853$	−2.24058e7			−1.05436			−0.527178	−	0.849755i	$-0.676750\pi$
$853$	−2.24058e7			−1.05436			−0.527178	+	0.849755i	$0.676750\pi$
$854$		0			0
$855$	−1.06715e7	+	3.05600e7i	−0.499240	+	1.42968i
$856$		0			0
$857$		−	763420.i		−	0.0355068i	−0.999842	−	0.0177534i	$-0.994349\pi$
$857$		−	763420.i		−	0.0355068i	0.999842	−	0.0177534i	$-0.00565138\pi$
$858$		0			0
$859$		−	4.85268e6i		−	0.224388i	−0.993686	−	0.112194i	$-0.964212\pi$
$859$		−	4.85268e6i		−	0.224388i	0.993686	−	0.112194i	$-0.0357878\pi$
$860$		0			0
$861$	5.27048e6	+	7.42301e6i	0.242294	+	0.341250i
$862$		0			0
$863$	6.68390e6			0.305494			0.152747	−	0.988265i	$-0.451188\pi$
$863$	6.68390e6			0.305494			0.152747	+	0.988265i	$0.451188\pi$
$864$		0			0
$865$	−2.65363e7			−1.20587
$866$		0			0
$867$	−9.37021e6	−	1.31971e7i	−0.423352	−	0.596255i
$868$		0			0
$869$		−	31788.2i		−	0.00142796i
$870$		0			0
$871$		−	3.02614e7i		−	1.35159i
$872$		0			0
$873$	7.41202e6	−	2.12258e7i	0.329155	−	0.942604i
$874$		0			0
$875$	7.37843e6			0.325795
$876$		0			0
$877$	−3.78652e7			−1.66242			−0.831211	−	0.555957i	$-0.812352\pi$
$877$	−3.78652e7			−1.66242			−0.831211	+	0.555957i	$0.812352\pi$
$878$		0			0
$879$	−2.77817e7	+	1.97255e7i	−1.21279	+	0.861104i
$880$		0			0
$881$		−	9.23058e6i		−	0.400672i	−0.979727	−	0.200336i	$-0.935797\pi$
$881$		−	9.23058e6i		−	0.400672i	0.979727	−	0.200336i	$-0.0642034\pi$
$882$		0			0
$883$			2.81375e7i			1.21446i	0.794526	+	0.607230i	$0.207719\pi$
$883$			2.81375e7i			1.21446i	−0.794526	+	0.607230i	$0.792281\pi$
$884$		0			0
$885$	−4.08360e7	+	2.89943e7i	−1.75261	+	1.24439i
$886$		0			0
$887$	3.66169e7			1.56269			0.781344	−	0.624100i	$-0.214534\pi$
$887$	3.66169e7			1.56269			0.781344	+	0.624100i	$0.214534\pi$
$888$		0			0
$889$	1.75700e6			0.0745619
$890$		0			0
$891$	−23137.3	−	18403.0i	−0.000976379	−	0.000776597i
$892$		0			0
$893$			609775.i			0.0255883i
$894$		0			0
$895$		−	1.18238e7i		−	0.493399i
$896$		0			0
$897$	−172558.	−	243033.i	−0.00716068	−	0.0100852i
$898$		0			0
$899$	1.81691e6			0.0749780
$900$		0			0
$901$	9.14978e6			0.375490
$902$		0			0
$903$	4.67861e6	+	6.58942e6i	0.190940	+	0.268923i
$904$		0			0
$905$			2.92144e7i			1.18570i
$906$		0			0
$907$		−	4.29527e7i		−	1.73369i	−0.498576	−	0.866846i	$-0.666144\pi$
$907$		−	4.29527e7i		−	1.73369i	0.498576	−	0.866846i	$-0.333856\pi$
$908$		0			0
$909$	−1.95760e7	−	6.83591e6i	−0.785805	−	0.274402i
$910$		0			0
$911$	7.38907e6			0.294981			0.147491	−	0.989063i	$-0.452880\pi$
$911$	7.38907e6			0.294981			0.147491	+	0.989063i	$0.452880\pi$
$912$		0			0
$913$	42535.5			0.00168879
$914$		0			0
$915$	4.77048e7	−	3.38712e7i	1.88369	−	1.33745i
$916$		0			0
$917$		−	1.55622e7i		−	0.611151i
$918$		0			0
$919$			1.21985e7i			0.476449i	0.971210	+	0.238224i	$0.0765653\pi$
$919$			1.21985e7i			0.476449i	−0.971210	+	0.238224i	$0.923435\pi$
$920$		0			0
$921$	1.28578e7	−	9.12927e6i	0.499479	−	0.354639i
$922$		0			0
$923$	−2.17373e7			−0.839848
$924$		0			0
$925$	−6.38625e7			−2.45410
$926$		0			0
$927$	−2.75485e6	−	961989.i	−0.105293	−	0.0367681i
$928$		0			0
$929$			1.67369e7i			0.636262i	0.948047	+	0.318131i	$0.103055\pi$
$929$			1.67369e7i			0.636262i	−0.948047	+	0.318131i	$0.896945\pi$
$930$		0			0
$931$			3.58935e6i			0.135719i
$932$		0			0
$933$	−9.57154e6	−	1.34807e7i	−0.359980	−	0.507000i
$934$		0			0
$935$	−69944.9			−0.00261654
$936$		0			0
$937$	−2.53134e7			−0.941892			−0.470946	−	0.882162i	$-0.656087\pi$
$937$	−2.53134e7			−0.941892			−0.470946	+	0.882162i	$0.656087\pi$
$938$		0			0
$939$	−1.03490e7	−	1.45757e7i	−0.383031	−	0.539466i
$940$		0			0
$941$		−	3.57268e7i		−	1.31529i	−0.753329	−	0.657643i	$-0.771553\pi$
$941$		−	3.57268e7i		−	1.31529i	0.753329	−	0.657643i	$-0.228447\pi$
$942$		0			0
$943$		−	460796.i		−	0.0168744i
$944$		0			0
$945$	1.58884e7	−	4.59385e6i	0.578761	−	0.167339i
$946$		0			0
$947$	2.31475e7			0.838743			0.419371	−	0.907815i	$-0.362251\pi$
$947$	2.31475e7			0.838743			0.419371	+	0.907815i	$0.362251\pi$
$948$		0			0
$949$	774917.			0.0279312
$950$		0			0
$951$	−3.94761e7	+	2.80287e7i	−1.41541	+	1.00497i
$952$		0			0
$953$		−	2.95731e7i		−	1.05479i	−0.849621	−	0.527394i	$-0.823169\pi$
$953$		−	2.95731e7i		−	1.05479i	0.849621	−	0.527394i	$-0.176831\pi$
$954$		0			0
$955$		−	2.19721e7i		−	0.779585i
$956$		0			0
$957$	1381.84	−	981.131i	4.87728e−5	−	3.46296e-5i
$958$		0			0
$959$	1.47731e7			0.518709
$960$		0			0
$961$	−4.13812e7			−1.44542
$962$		0			0
$963$	1.61306e7	−	4.61932e7i	0.560511	−	1.60514i
$964$		0			0
$965$		−	4.12920e7i		−	1.42741i
$966$		0			0
$967$		−	1.29687e7i		−	0.445995i	−0.974819	−	0.222997i	$-0.928416\pi$
$967$		−	1.29687e7i		−	0.445995i	0.974819	−	0.222997i	$-0.0715841\pi$
$968$		0			0
$969$	2.11523e7	+	2.97912e7i	0.723684	+	1.01925i
$970$		0			0
$971$	5.47865e7			1.86477			0.932386	−	0.361464i	$-0.117723\pi$
$971$	5.47865e7			1.86477			0.932386	+	0.361464i	$0.117723\pi$
$972$		0			0
$973$	−1.95614e7			−0.662395
$974$		0			0
$975$	2.14900e7	+	3.02669e7i	0.723978	+	1.01966i
$976$		0			0
$977$			3.19937e7i			1.07233i	0.844113	+	0.536165i	$0.180128\pi$
$977$			3.19937e7i			1.07233i	−0.844113	+	0.536165i	$0.819872\pi$
$978$		0			0
$979$		−	5551.38i		−	0.000185116i
$980$		0			0
$981$	2.67397e6	−	7.65746e6i	0.0887123	−	0.254046i
$982$		0			0
$983$	4.41464e7			1.45717			0.728587	−	0.684954i	$-0.240178\pi$
$983$	4.41464e7			1.45717			0.728587	+	0.684954i	$0.240178\pi$
$984$		0			0
$985$	8.21566e6			0.269806
$986$		0			0
$987$	254041.	−	180373.i	0.00830061	−	0.00589359i
$988$		0			0
$989$		−	409049.i		−	0.0132979i
$990$		0			0
$991$		−	2.36415e7i		−	0.764700i	−0.924018	−	0.382350i	$-0.875115\pi$
$991$		−	2.36415e7i		−	0.764700i	0.924018	−	0.382350i	$-0.124885\pi$
$992$		0			0
$993$	−3.96829e7	+	2.81756e7i	−1.27712	+	0.906776i
$994$		0			0
$995$	−1.23771e7			−0.396334
$996$		0			0
$997$	3.56987e7			1.13740			0.568702	−	0.822544i	$-0.307446\pi$
$997$	3.56987e7			1.13740			0.568702	+	0.822544i	$0.307446\pi$
$998$		0			0
$999$	−4.82652e7	+	1.39551e7i	−1.53010	+	0.442404i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.6.h.a.239.16	yes	20
3.2	odd	2	inner	336.6.h.a.239.6	yes	20
4.3	odd	2	inner	336.6.h.a.239.5	✓	20
12.11	even	2	inner	336.6.h.a.239.15	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.6.h.a.239.5	✓	20	4.3	odd	2	inner
336.6.h.a.239.6	yes	20	3.2	odd	2	inner
336.6.h.a.239.15	yes	20	12.11	even	2	inner
336.6.h.a.239.16	yes	20	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:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	336.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(9.02465 + 12.7105i) q^{3} +89.1061i q^{5} +49.0000i q^{7} +(-80.1112 + 229.415i) q^{9} +O(q^{10})\)	\(q+(9.02465 + 12.7105i) q^{3} +89.1061i q^{5} +49.0000i q^{7} +(-80.1112 + 229.415i) q^{9} +0.500662 q^{11} -494.561 q^{13} +(-1132.58 + 804.152i) q^{15} +1567.85i q^{17} -1494.94i q^{19} +(-622.812 + 442.208i) q^{21} +38.6621 q^{23} -4814.90 q^{25} +(-3638.94 + 1052.14i) q^{27} -217.146i q^{29} +8367.22i q^{31} +(4.51830 + 6.36364i) q^{33} -4366.20 q^{35} +13263.5 q^{37} +(-4463.24 - 6286.09i) q^{39} -11918.5i q^{41} -10580.1i q^{43} +(-20442.3 - 7138.40i) q^{45} -407.893 q^{47} -2401.00 q^{49} +(-19928.1 + 14149.3i) q^{51} -5835.89i q^{53} +44.6121i q^{55} +(19001.4 - 13491.3i) q^{57} +36055.8 q^{59} -42120.5 q^{61} +(-11241.3 - 3925.45i) q^{63} -44068.4i q^{65} +61188.5i q^{67} +(348.912 + 491.412i) q^{69} +43952.7 q^{71} -1566.88 q^{73} +(-43452.8 - 61199.5i) q^{75} +24.5324i q^{77} -63492.3i q^{79} +(-46213.4 - 36757.4i) q^{81} +84958.5 q^{83} -139705. q^{85} +(2760.02 - 1959.67i) q^{87} -11088.1i q^{89} -24233.5i q^{91} +(-106351. + 75511.3i) q^{93} +133208. q^{95} -92521.6 q^{97} +(-40.1087 + 114.859i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 140 q^{9}+O(q^{10}) \) 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

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more