LMFDB - Embedded newform 336.6.h.b.239.32

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:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			239.32
Character	$\chi$	$=$	336.239
Dual form			336.6.h.b.239.31

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(11.6605 + 10.3456i) q^{3} +60.0268i q^{5} -49.0000i q^{7} +(28.9356 + 241.271i) q^{9} +O(q^{10})$	\(q+(11.6605 + 10.3456i) q^{3} +60.0268i q^{5} -49.0000i q^{7} +(28.9356 + 241.271i) q^{9} +595.559 q^{11} -1171.32 q^{13} +(-621.016 + 699.944i) q^{15} +546.441i q^{17} +1026.70i q^{19} +(506.936 - 571.366i) q^{21} +132.143 q^{23} -478.218 q^{25} +(-2158.70 + 3112.70i) q^{27} +4610.62i q^{29} -2766.38i q^{31} +(6944.53 + 6161.44i) q^{33} +2941.31 q^{35} -7219.40 q^{37} +(-13658.2 - 12118.1i) q^{39} +6431.81i q^{41} +6852.22i q^{43} +(-14482.7 + 1736.91i) q^{45} -23988.1 q^{47} -2401.00 q^{49} +(-5653.28 + 6371.78i) q^{51} -30034.6i q^{53} +35749.5i q^{55} +(-10621.9 + 11971.9i) q^{57} -4422.95 q^{59} +49598.2 q^{61} +(11822.3 - 1417.84i) q^{63} -70310.7i q^{65} -25936.4i q^{67} +(1540.86 + 1367.11i) q^{69} -20754.8 q^{71} -39745.8 q^{73} +(-5576.27 - 4947.47i) q^{75} -29182.4i q^{77} +51274.0i q^{79} +(-57374.5 + 13962.6i) q^{81} +18429.8 q^{83} -32801.1 q^{85} +(-47699.8 + 53762.3i) q^{87} -46758.1i q^{89} +57394.8i q^{91} +(28619.9 - 32257.4i) q^{93} -61629.7 q^{95} -138257. q^{97} +(17232.9 + 143691. i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 8 q^{9}+O(q^{10}) $ 40 * q + 8 * q^9	$ 40 q + 8 q^{9} - 1048 q^{13} + 980 q^{21} - 43416 q^{25} + 20296 q^{33} - 16192 q^{37} + 56488 q^{45} - 96040 q^{49} + 31088 q^{57} + 173112 q^{61} - 114176 q^{69} - 267488 q^{73} + 64888 q^{81} + 508112 q^{85} - 224544 q^{93} - 276400 q^{97}+O(q^{100}) $ 40 * q + 8 * q^9 - 1048 * q^13 + 980 * q^21 - 43416 * q^25 + 20296 * q^33 - 16192 * q^37 + 56488 * q^45 - 96040 * q^49 + 31088 * q^57 + 173112 * q^61 - 114176 * q^69 - 267488 * q^73 + 64888 * q^81 + 508112 * q^85 - 224544 * q^93 - 276400 * 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
$3$	11.6605	+	10.3456i	0.748023	+	0.663673i
$3$	11.6605	+	10.3456i	0.748023	+	0.663673i
$4$		0			0
$5$			60.0268i			1.07379i	0.843648	+	0.536896i	$0.180403\pi$
$5$			60.0268i			1.07379i	−0.843648	+	0.536896i	$0.819597\pi$
$6$		0			0
$7$		−	49.0000i		−	0.377964i
$7$		−	49.0000i		−	0.377964i
$8$		0			0
$9$	28.9356	+	241.271i	0.119077	+	0.992885i
$10$		0			0
$11$	595.559			1.48403			0.742016	−	0.670382i	$-0.233870\pi$
$11$	595.559			1.48403			0.742016	+	0.670382i	$0.233870\pi$
$12$		0			0
$13$	−1171.32			−1.92229			−0.961143	−	0.276051i	$-0.910974\pi$
$13$	−1171.32			−1.92229			−0.961143	+	0.276051i	$0.910974\pi$
$14$		0			0
$15$	−621.016	+	699.944i	−0.712647	+	0.803221i
$16$		0			0
$17$			546.441i			0.458586i	0.973357	+	0.229293i	$0.0736414\pi$
$17$			546.441i			0.458586i	−0.973357	+	0.229293i	$0.926359\pi$
$18$		0			0
$19$			1026.70i			0.652471i	0.945289	+	0.326235i	$0.105780\pi$
$19$			1026.70i			0.652471i	−0.945289	+	0.326235i	$0.894220\pi$
$20$		0			0
$21$	506.936	−	571.366i	0.250845	−	0.282726i
$22$		0			0
$23$	132.143			0.0520866			0.0260433	−	0.999661i	$-0.491709\pi$
$23$	132.143			0.0520866			0.0260433	+	0.999661i	$0.491709\pi$
$24$		0			0
$25$	−478.218			−0.153030
$26$		0			0
$27$	−2158.70	+	3112.70i	−0.569879	+	0.821729i
$28$		0			0
$29$			4610.62i			1.01804i	0.860755	+	0.509020i	$0.169992\pi$
$29$			4610.62i			1.01804i	−0.860755	+	0.509020i	$0.830008\pi$
$30$		0			0
$31$		−	2766.38i		−	0.517019i	−0.966009	−	0.258510i	$-0.916769\pi$
$31$		−	2766.38i		−	0.517019i	0.966009	−	0.258510i	$-0.0832314\pi$
$32$		0			0
$33$	6944.53	+	6161.44i	1.11009	+	0.984912i
$34$		0			0
$35$	2941.31			0.405855
$36$		0			0
$37$	−7219.40			−0.866955			−0.433478	−	0.901164i	$-0.642714\pi$
$37$	−7219.40			−0.866955			−0.433478	+	0.901164i	$0.642714\pi$
$38$		0			0
$39$	−13658.2	−	12118.1i	−1.43791	−	1.27577i
$40$		0			0
$41$			6431.81i			0.597549i	0.954324	+	0.298774i	$0.0965778\pi$
$41$			6431.81i			0.597549i	−0.954324	+	0.298774i	$0.903422\pi$
$42$		0			0
$43$			6852.22i			0.565146i	0.959246	+	0.282573i	$0.0911879\pi$
$43$			6852.22i			0.565146i	−0.959246	+	0.282573i	$0.908812\pi$
$44$		0			0
$45$	−14482.7	+	1736.91i	−1.06615	+	0.127863i
$46$		0			0
$47$	−23988.1			−1.58398			−0.791992	−	0.610532i	$-0.790956\pi$
$47$	−23988.1			−1.58398			−0.791992	+	0.610532i	$0.790956\pi$
$48$		0			0
$49$	−2401.00			−0.142857
$50$		0			0
$51$	−5653.28	+	6371.78i	−0.304351	+	0.343033i
$52$		0			0
$53$		−	30034.6i		−	1.46870i	−0.678772	−	0.734349i	$-0.737487\pi$
$53$		−	30034.6i		−	1.46870i	0.678772	−	0.734349i	$-0.262513\pi$
$54$		0			0
$55$			35749.5i			1.59354i
$56$		0			0
$57$	−10621.9	+	11971.9i	−0.433027	+	0.488063i
$58$		0			0
$59$	−4422.95			−0.165418			−0.0827088	−	0.996574i	$-0.526357\pi$
$59$	−4422.95			−0.165418			−0.0827088	+	0.996574i	$0.526357\pi$
$60$		0			0
$61$	49598.2			1.70664			0.853318	−	0.521391i	$-0.174587\pi$
$61$	49598.2			1.70664			0.853318	+	0.521391i	$0.174587\pi$
$62$		0			0
$63$	11822.3	−	1417.84i	0.375275	−	0.0450067i
$64$		0			0
$65$		−	70310.7i		−	2.06414i
$66$		0			0
$67$		−	25936.4i		−	0.705865i	−0.935649	−	0.352933i	$-0.885184\pi$
$67$		−	25936.4i		−	0.705865i	0.935649	−	0.352933i	$-0.114816\pi$
$68$		0			0
$69$	1540.86	+	1367.11i	0.0389620	+	0.0345685i
$70$		0			0
$71$	−20754.8			−0.488621			−0.244310	−	0.969697i	$-0.578562\pi$
$71$	−20754.8			−0.488621			−0.244310	+	0.969697i	$0.578562\pi$
$72$		0			0
$73$	−39745.8			−0.872939			−0.436470	−	0.899719i	$-0.643771\pi$
$73$	−39745.8			−0.872939			−0.436470	+	0.899719i	$0.643771\pi$
$74$		0			0
$75$	−5576.27	−	4947.47i	−0.114470	−	0.101562i
$76$		0			0
$77$		−	29182.4i		−	0.560911i
$78$		0			0
$79$			51274.0i			0.924336i	0.886792	+	0.462168i	$0.152928\pi$
$79$			51274.0i			0.924336i	−0.886792	+	0.462168i	$0.847072\pi$
$80$		0			0
$81$	−57374.5	+	13962.6i	−0.971642	+	0.236459i
$82$		0			0
$83$	18429.8			0.293646			0.146823	−	0.989163i	$-0.453095\pi$
$83$	18429.8			0.293646			0.146823	+	0.989163i	$0.453095\pi$
$84$		0			0
$85$	−32801.1			−0.492426
$86$		0			0
$87$	−47699.8	+	53762.3i	−0.675645	+	0.761517i
$88$		0			0
$89$		−	46758.1i		−	0.625723i	−0.949799	−	0.312862i	$-0.898712\pi$
$89$		−	46758.1i		−	0.625723i	0.949799	−	0.312862i	$-0.101288\pi$
$90$		0			0
$91$			57394.8i			0.726556i
$92$		0			0
$93$	28619.9	−	32257.4i	0.343132	−	0.386742i
$94$		0			0
$95$	−61629.7			−0.700618
$96$		0			0
$97$	−138257.			−1.49196			−0.745980	−	0.665968i	$-0.768019\pi$
$97$	−138257.			−1.49196			−0.745980	+	0.665968i	$0.768019\pi$
$98$		0			0
$99$	17232.9	+	143691.i	0.176713	+	1.47347i
$100$		0			0
$101$		−	84300.2i		−	0.822291i	−0.911570	−	0.411145i	$-0.865129\pi$
$101$		−	84300.2i		−	0.822291i	0.911570	−	0.411145i	$-0.134871\pi$
$102$		0			0
$103$			199299.i			1.85102i	0.378718	+	0.925512i	$0.376365\pi$
$103$			199299.i			1.85102i	−0.378718	+	0.925512i	$0.623635\pi$
$104$		0			0
$105$	34297.3	+	30429.8i	0.303589	+	0.269355i
$106$		0			0
$107$	−34443.1			−0.290832			−0.145416	−	0.989371i	$-0.546452\pi$
$107$	−34443.1			−0.290832			−0.145416	+	0.989371i	$0.546452\pi$
$108$		0			0
$109$	−51.7146			−0.000416914			−0.000208457	−	1.00000i	$-0.500066\pi$
$109$	−51.7146			−0.000416914			−0.000208457		1.00000i	$0.500066\pi$
$110$		0			0
$111$	−84181.9	−	74689.3i	−0.648502	−	0.575375i
$112$		0			0
$113$		−	19769.6i		−	0.145647i	−0.997345	−	0.0728234i	$-0.976799\pi$
$113$		−	19769.6i		−	0.145647i	0.997345	−	0.0728234i	$-0.0232010\pi$
$114$		0			0
$115$			7932.15i			0.0559302i
$116$		0			0
$117$	−33892.9	−	282606.i	−0.228899	−	1.90861i
$118$		0			0
$119$	26775.6			0.173329
$120$		0			0
$121$	193640.			1.20235
$122$		0			0
$123$	−66541.2	+	74998.3i	−0.396577	+	0.446980i
$124$		0			0
$125$			158878.i			0.909470i
$126$		0			0
$127$		−	217222.i		−	1.19508i	−0.801841	−	0.597538i	$-0.796146\pi$
$127$		−	217222.i		−	1.19508i	0.801841	−	0.597538i	$-0.203854\pi$
$128$		0			0
$129$	−70890.6	+	79900.5i	−0.375072	+	0.422742i
$130$		0			0
$131$	−120831.			−0.615178			−0.307589	−	0.951519i	$-0.599522\pi$
$131$	−120831.			−0.615178			−0.307589	+	0.951519i	$0.599522\pi$
$132$		0			0
$133$	50308.5			0.246611
$134$		0			0
$135$	−186846.	−	129580.i	−0.882366	−	0.611932i
$136$		0			0
$137$			314274.i			1.43056i	0.698836	+	0.715282i	$0.253702\pi$
$137$			314274.i			1.43056i	−0.698836	+	0.715282i	$0.746298\pi$
$138$		0			0
$139$		−	87314.8i		−	0.383310i	−0.981462	−	0.191655i	$-0.938614\pi$
$139$		−	87314.8i		−	0.383310i	0.981462	−	0.191655i	$-0.0613855\pi$
$140$		0			0
$141$	−279713.	−	248172.i	−1.18486	−	1.05125i
$142$		0			0
$143$	−697592.			−2.85273
$144$		0			0
$145$	−276761.			−1.09316
$146$		0			0
$147$	−27996.9	−	24839.9i	−0.106860	−	0.0948104i
$148$		0			0
$149$		−	134418.i		−	0.496013i	−0.968758	−	0.248006i	$-0.920225\pi$
$149$		−	134418.i		−	0.496013i	0.968758	−	0.248006i	$-0.0797754\pi$
$150$		0			0
$151$			377795.i			1.34839i	0.738555	+	0.674193i	$0.235508\pi$
$151$			377795.i			1.34839i	−0.738555	+	0.674193i	$0.764492\pi$
$152$		0			0
$153$	−131840.	+	15811.6i	−0.455323	+	0.0546068i
$154$		0			0
$155$	166057.			0.555171
$156$		0			0
$157$	241186.			0.780912			0.390456	−	0.920622i	$-0.372317\pi$
$157$	241186.			0.780912			0.390456	+	0.920622i	$0.372317\pi$
$158$		0			0
$159$	310727.	−	350220.i	0.974735	−	1.09862i
$160$		0			0
$161$		−	6475.03i		−	0.0196869i
$162$		0			0
$163$			530157.i			1.56292i	0.623958	+	0.781458i	$0.285523\pi$
$163$			530157.i			1.56292i	−0.623958	+	0.781458i	$0.714477\pi$
$164$		0			0
$165$	−369852.	+	416858.i	−1.05759	+	1.19201i
$166$		0			0
$167$	−598192.			−1.65977			−0.829887	−	0.557931i	$-0.811595\pi$
$167$	−598192.			−1.65977			−0.829887	+	0.557931i	$0.811595\pi$
$168$		0			0
$169$	1.00070e6			2.69518
$170$		0			0
$171$	−247714.	+	29708.3i	−0.647829	+	0.0776940i
$172$		0			0
$173$			433196.i			1.10045i	0.835018	+	0.550223i	$0.185457\pi$
$173$			433196.i			1.10045i	−0.835018	+	0.550223i	$0.814543\pi$
$174$		0			0
$175$			23432.7i			0.0578398i
$176$		0			0
$177$	−51573.9	−	45758.2i	−0.123736	−	0.109783i
$178$		0			0
$179$	852576.			1.98884			0.994422	−	0.105478i	$-0.0336372\pi$
$179$	852576.			1.98884			0.994422	+	0.105478i	$0.0336372\pi$
$180$		0			0
$181$	−532828.			−1.20890			−0.604450	−	0.796643i	$-0.706607\pi$
$181$	−532828.			−1.20890			−0.604450	+	0.796643i	$0.706607\pi$
$182$		0			0
$183$	578340.	+	513124.i	1.27660	+	1.13265i
$184$		0			0
$185$		−	433357.i		−	0.930930i
$186$		0			0
$187$			325438.i			0.680556i
$188$		0			0
$189$	152523.	+	105776.i	0.310584	+	0.215394i
$190$		0			0
$191$	440900.			0.874494			0.437247	−	0.899341i	$-0.355953\pi$
$191$	440900.			0.874494			0.437247	+	0.899341i	$0.355953\pi$
$192$		0			0
$193$	167849.			0.324358			0.162179	−	0.986761i	$-0.448148\pi$
$193$	167849.			0.324358			0.162179	+	0.986761i	$0.448148\pi$
$194$		0			0
$195$	727409.	−	819860.i	1.36991	−	1.54402i
$196$		0			0
$197$			199591.i			0.366417i	0.983074	+	0.183209i	$0.0586484\pi$
$197$			199591.i			0.366417i	−0.983074	+	0.183209i	$0.941352\pi$
$198$		0			0
$199$		−	717197.i		−	1.28382i	−0.766778	−	0.641912i	$-0.778141\pi$
$199$		−	717197.i		−	1.28382i	0.766778	−	0.641912i	$-0.221859\pi$
$200$		0			0
$201$	268328.	−	302431.i	0.468464	−	0.528004i
$202$		0			0
$203$	225920.			0.384783
$204$		0			0
$205$	−386081.			−0.641643
$206$		0			0
$207$	3823.65	+	31882.4i	0.00620229	+	0.0517160i
$208$		0			0
$209$			611463.i			0.968288i
$210$		0			0
$211$			114469.i			0.177003i	0.996076	+	0.0885017i	$0.0282079\pi$
$211$			114469.i			0.177003i	−0.996076	+	0.0885017i	$0.971792\pi$
$212$		0			0
$213$	−242011.	−	214721.i	−0.365499	−	0.324284i
$214$		0			0
$215$	−411317.			−0.606849
$216$		0			0
$217$	−135552.			−0.195415
$218$		0			0
$219$	−463457.	−	411196.i	−0.652979	−	0.579346i
$220$		0			0
$221$		−	640058.i		−	0.881533i
$222$		0			0
$223$			646340.i			0.870360i	0.900343	+	0.435180i	$0.143315\pi$
$223$			646340.i			0.870360i	−0.900343	+	0.435180i	$0.856685\pi$
$224$		0			0
$225$	−13837.5	−	115380.i	−0.0182223	−	0.151941i
$226$		0			0
$227$	−9687.01			−0.0124774			−0.00623872	−	0.999981i	$-0.501986\pi$
$227$	−9687.01			−0.0124774			−0.00623872	+	0.999981i	$0.501986\pi$
$228$		0			0
$229$	1.31685e6			1.65938			0.829692	−	0.558221i	$-0.188516\pi$
$229$	1.31685e6			1.65938			0.829692	+	0.558221i	$0.188516\pi$
$230$		0			0
$231$	301911.	−	340282.i	0.372262	−	0.419575i
$232$		0			0
$233$		−	1.49838e6i		−	1.80814i	−0.427388	−	0.904068i	$-0.640566\pi$
$233$		−	1.49838e6i		−	1.80814i	0.427388	−	0.904068i	$-0.359434\pi$
$234$		0			0
$235$		−	1.43993e6i		−	1.70087i
$236$		0			0
$237$	−530463.	+	597882.i	−0.613457	+	0.691424i
$238$		0			0
$239$	1.19197e6			1.34980			0.674902	−	0.737907i	$-0.264186\pi$
$239$	1.19197e6			1.34980			0.674902	+	0.737907i	$0.264186\pi$
$240$		0			0
$241$	1.37822e6			1.52853			0.764266	−	0.644901i	$-0.223101\pi$
$241$	1.37822e6			1.52853			0.764266	+	0.644901i	$0.223101\pi$
$242$		0			0
$243$	−813469.	−	430764.i	−0.883741	−	0.467976i
$244$		0			0
$245$		−	144124.i		−	0.153399i
$246$		0			0
$247$		−	1.20260e6i		−	1.25424i
$248$		0			0
$249$	214901.	+	190668.i	0.219654	+	0.194885i
$250$		0			0
$251$	1.59147e6			1.59446			0.797230	−	0.603676i	$-0.206298\pi$
$251$	1.59147e6			1.59446			0.797230	+	0.603676i	$0.206298\pi$
$252$		0			0
$253$	78699.3			0.0772982
$254$		0			0
$255$	−382478.	−	339348.i	−0.368346	−	0.326810i
$256$		0			0
$257$			1.95434e6i			1.84572i	0.385133	+	0.922861i	$0.374156\pi$
$257$			1.95434e6i			1.84572i	−0.385133	+	0.922861i	$0.625844\pi$
$258$		0			0
$259$			353750.i			0.327678i
$260$		0			0
$261$	−1.11241e6	+	133411.i	−1.01080	+	0.121225i
$262$		0			0
$263$	2.05157e6			1.82893			0.914464	−	0.404667i	$-0.132613\pi$
$263$	2.05157e6			1.82893			0.914464	+	0.404667i	$0.132613\pi$
$264$		0			0
$265$	1.80288e6			1.57708
$266$		0			0
$267$	483743.	−	545224.i	0.415275	−	0.468055i
$268$		0			0
$269$		−	1.55159e6i		−	1.30737i	−0.756769	−	0.653683i	$-0.773223\pi$
$269$		−	1.55159e6i		−	1.30737i	0.756769	−	0.653683i	$-0.226777\pi$
$270$		0			0
$271$			9623.44i			0.00795989i	0.999992	+	0.00397995i	$0.00126686\pi$
$271$			9623.44i			0.00795989i	−0.999992	+	0.00397995i	$0.998733\pi$
$272$		0			0
$273$	−593786.	+	669253.i	−0.482195	+	0.543480i
$274$		0			0
$275$	−284807.			−0.227101
$276$		0			0
$277$	1.18703e6			0.929529			0.464764	−	0.885434i	$-0.346139\pi$
$277$	1.18703e6			0.929529			0.464764	+	0.885434i	$0.346139\pi$
$278$		0			0
$279$	667446.	−	80046.7i	0.513341	−	0.0615649i
$280$		0			0
$281$			1.80723e6i			1.36536i	0.730715	+	0.682682i	$0.239187\pi$
$281$			1.80723e6i			1.36536i	−0.730715	+	0.682682i	$0.760813\pi$
$282$		0			0
$283$			1.42835e6i			1.06015i	0.847950	+	0.530076i	$0.177836\pi$
$283$			1.42835e6i			1.06015i	−0.847950	+	0.530076i	$0.822164\pi$
$284$		0			0
$285$	−718635.	−	637599.i	−0.524078	−	0.464981i
$286$		0			0
$287$	315159.			0.225852
$288$		0			0
$289$	1.12126e6			0.789699
$290$		0			0
$291$	−1.61215e6	−	1.43036e6i	−1.11602	−	0.990174i
$292$		0			0
$293$		−	1.05898e6i		−	0.720640i	−0.932829	−	0.360320i	$-0.882667\pi$
$293$		−	1.05898e6i		−	0.720640i	0.932829	−	0.360320i	$-0.117333\pi$
$294$		0			0
$295$		−	265496.i		−	0.177624i
$296$		0			0
$297$	−1.28563e6	+	1.85380e6i	−0.845718	+	1.21947i
$298$		0			0
$299$	−154783.			−0.100125
$300$		0			0
$301$	335759.			0.213605
$302$		0			0
$303$	872140.	−	982985.i	0.545732	−	0.615092i
$304$		0			0
$305$			2.97722e6i			1.83257i
$306$		0			0
$307$			223504.i			0.135344i	0.997708	+	0.0676719i	$0.0215571\pi$
$307$			223504.i			0.135344i	−0.997708	+	0.0676719i	$0.978443\pi$
$308$		0			0
$309$	−2.06188e6	+	2.32393e6i	−1.22847	+	1.38461i
$310$		0			0
$311$	−2.90060e6			−1.70054			−0.850271	−	0.526346i	$-0.823562\pi$
$311$	−2.90060e6			−1.70054			−0.850271	+	0.526346i	$0.823562\pi$
$312$		0			0
$313$	886239.			0.511317			0.255658	−	0.966767i	$-0.417708\pi$
$313$	886239.			0.511317			0.255658	+	0.966767i	$0.417708\pi$
$314$		0			0
$315$	85108.7	+	709654.i	0.0483279	+	0.402968i
$316$		0			0
$317$		−	238682.i		−	0.133405i	−0.997773	−	0.0667023i	$-0.978752\pi$
$317$		−	238682.i		−	0.133405i	0.997773	−	0.0667023i	$-0.0212478\pi$
$318$		0			0
$319$			2.74590e6i			1.51080i
$320$		0			0
$321$	−401624.	−	356336.i	−0.217549	−	0.193018i
$322$		0			0
$323$	−561033.			−0.299214
$324$		0			0
$325$	560148.			0.294167
$326$		0			0
$327$	−603.019	−	535.020i	−0.000311861	−	0.000276695i
$328$		0			0
$329$			1.17542e6i			0.598689i
$330$		0			0
$331$		−	1.20841e6i		−	0.606241i	−0.952952	−	0.303120i	$-0.901972\pi$
$331$		−	1.20841e6i		−	0.606241i	0.952952	−	0.303120i	$-0.0980284\pi$
$332$		0			0
$333$	−208898.	−	1.74183e6i	−0.103234	−	0.860787i
$334$		0			0
$335$	1.55688e6			0.757953
$336$		0			0
$337$	−1.63067e6			−0.782153			−0.391076	−	0.920358i	$-0.627897\pi$
$337$	−1.63067e6			−0.782153			−0.391076	+	0.920358i	$0.627897\pi$
$338$		0			0
$339$	204529.	−	230523.i	0.0966619	−	0.108947i
$340$		0			0
$341$		−	1.64754e6i		−	0.767273i
$342$		0			0
$343$			117649.i			0.0539949i
$344$		0			0
$345$	−82063.2	+	92493.0i	−0.0371194	+	0.0418371i
$346$		0			0
$347$	−69040.9			−0.0307810			−0.0153905	−	0.999882i	$-0.504899\pi$
$347$	−69040.9			−0.0307810			−0.0153905	+	0.999882i	$0.504899\pi$
$348$		0			0
$349$	998544.			0.438837			0.219419	−	0.975631i	$-0.429584\pi$
$349$	998544.			0.438837			0.219419	+	0.975631i	$0.429584\pi$
$350$		0			0
$351$	2.52853e6	−	3.64598e6i	1.09547	−	1.57960i
$352$		0			0
$353$			2.75932e6i			1.17860i	0.807915	+	0.589299i	$0.200596\pi$
$353$			2.75932e6i			1.17860i	−0.807915	+	0.589299i	$0.799404\pi$
$354$		0			0
$355$		−	1.24584e6i		−	0.524677i
$356$		0			0
$357$	312217.	+	277010.i	0.129654	+	0.115034i
$358$		0			0
$359$	−1.44244e6			−0.590693			−0.295346	−	0.955390i	$-0.595435\pi$
$359$	−1.44244e6			−0.590693			−0.295346	+	0.955390i	$0.595435\pi$
$360$		0			0
$361$	1.42198e6			0.574282
$362$		0			0
$363$	2.25794e6	+	2.00333e6i	0.899386	+	0.797967i
$364$		0			0
$365$		−	2.38581e6i		−	0.937356i
$366$		0			0
$367$			2.51843e6i			0.976033i	0.872834	+	0.488016i	$0.162279\pi$
$367$			2.51843e6i			0.976033i	−0.872834	+	0.488016i	$0.837721\pi$
$368$		0			0
$369$	−1.55181e6	+	186108.i	−0.593297	+	0.0711541i
$370$		0			0
$371$	−1.47170e6			−0.555116
$372$		0			0
$373$	−2.66959e6			−0.993509			−0.496754	−	0.867891i	$-0.665475\pi$
$373$	−2.66959e6			−0.993509			−0.496754	+	0.867891i	$0.665475\pi$
$374$		0			0
$375$	−1.64369e6	+	1.85260e6i	−0.603591	+	0.680304i
$376$		0			0
$377$		−	5.40053e6i		−	1.95696i
$378$		0			0
$379$			1.60130e6i			0.572632i	0.958135	+	0.286316i	$0.0924307\pi$
$379$			1.60130e6i			0.572632i	−0.958135	+	0.286316i	$0.907569\pi$
$380$		0			0
$381$	2.24730e6	−	2.53293e6i	0.793139	−	0.893944i
$382$		0			0
$383$	1.50678e6			0.524870			0.262435	−	0.964950i	$-0.415474\pi$
$383$	1.50678e6			0.524870			0.262435	+	0.964950i	$0.415474\pi$
$384$		0			0
$385$	1.75173e6			0.602302
$386$		0			0
$387$	−1.65324e6	+	198273.i	−0.561125	+	0.0672956i
$388$		0			0
$389$			1.54703e6i			0.518352i	0.965830	+	0.259176i	$0.0834510\pi$
$389$			1.54703e6i			0.518352i	−0.965830	+	0.259176i	$0.916549\pi$
$390$		0			0
$391$			72208.5i			0.0238862i
$392$		0			0
$393$	−1.40895e6	−	1.25008e6i	−0.460167	−	0.408277i
$394$		0			0
$395$	−3.07782e6			−0.992545
$396$		0			0
$397$	127542.			0.0406141			0.0203071	−	0.999794i	$-0.493536\pi$
$397$	127542.			0.0406141			0.0203071	+	0.999794i	$0.493536\pi$
$398$		0			0
$399$	586623.	+	520473.i	0.184471	+	0.163669i
$400$		0			0
$401$			1.37151e6i			0.425929i	0.977060	+	0.212965i	$0.0683119\pi$
$401$			1.37151e6i			0.425929i	−0.977060	+	0.212965i	$0.931688\pi$
$402$		0			0
$403$			3.24032e6i			0.993859i
$404$		0			0
$405$	−838133.	−	3.44401e6i	−0.253907	−	1.04334i
$406$		0			0
$407$	−4.29958e6			−1.28659
$408$		0			0
$409$	3.00447e6			0.888094			0.444047	−	0.896003i	$-0.353542\pi$
$409$	3.00447e6			0.888094			0.444047	+	0.896003i	$0.353542\pi$
$410$		0			0
$411$	−3.25137e6	+	3.66460e6i	−0.949427	+	1.07009i
$412$		0			0
$413$			216725.i			0.0625220i
$414$		0			0
$415$			1.10628e6i			0.315315i
$416$		0			0
$417$	903327.	−	1.01814e6i	0.254393	−	0.286725i
$418$		0			0
$419$	−6.67092e6			−1.85631			−0.928155	−	0.372195i	$-0.878605\pi$
$419$	−6.67092e6			−1.85631			−0.928155	+	0.372195i	$0.878605\pi$
$420$		0			0
$421$	−4.76638e6			−1.31064			−0.655320	−	0.755352i	$-0.727466\pi$
$421$	−4.76638e6			−1.31064			−0.655320	+	0.755352i	$0.727466\pi$
$422$		0			0
$423$	−694109.	−	5.78763e6i	−0.188615	−	1.57271i
$424$		0			0
$425$		−	261318.i		−	0.0701773i
$426$		0			0
$427$		−	2.43031e6i		−	0.645048i
$428$		0			0
$429$	−8.13429e6	−	7.21703e6i	−2.13391	−	1.89328i
$430$		0			0
$431$	5.90821e6			1.53201			0.766007	−	0.642832i	$-0.222241\pi$
$431$	5.90821e6			1.53201			0.766007	+	0.642832i	$0.222241\pi$
$432$		0			0
$433$	4.29870e6			1.10184			0.550918	−	0.834559i	$-0.314278\pi$
$433$	4.29870e6			1.10184			0.550918	+	0.834559i	$0.314278\pi$
$434$		0			0
$435$	−3.22718e6	−	2.86327e6i	−0.817711	−	0.725503i
$436$		0			0
$437$			135672.i			0.0339850i
$438$		0			0
$439$			2.97884e6i			0.737711i	0.929487	+	0.368855i	$0.120250\pi$
$439$			2.97884e6i			0.737711i	−0.929487	+	0.368855i	$0.879750\pi$
$440$		0			0
$441$	−69474.4	−	579292.i	−0.0170109	−	0.141841i
$442$		0			0
$443$	5.93519e6			1.43689			0.718447	−	0.695581i	$-0.244853\pi$
$443$	5.93519e6			1.43689			0.718447	+	0.695581i	$0.244853\pi$
$444$		0			0
$445$	2.80674e6			0.671897
$446$		0			0
$447$	1.39064e6	−	1.56739e6i	0.329190	−	0.371029i
$448$		0			0
$449$		−	2.89503e6i		−	0.677699i	−0.940841	−	0.338850i	$-0.889962\pi$
$449$		−	2.89503e6i		−	0.677699i	0.940841	−	0.338850i	$-0.110038\pi$
$450$		0			0
$451$			3.83052e6i			0.886782i
$452$		0			0
$453$	−3.90853e6	+	4.40529e6i	−0.894887	+	1.00862i
$454$		0			0
$455$	−3.44523e6			−0.780170
$456$		0			0
$457$	−4.14857e6			−0.929198			−0.464599	−	0.885521i	$-0.653801\pi$
$457$	−4.14857e6			−0.929198			−0.464599	+	0.885521i	$0.653801\pi$
$458$		0			0
$459$	−1.70091e6	−	1.17960e6i	−0.376833	−	0.261338i
$460$		0			0
$461$		−	4.49539e6i		−	0.985179i	−0.870262	−	0.492589i	$-0.836050\pi$
$461$		−	4.49539e6i		−	0.985179i	0.870262	−	0.492589i	$-0.163950\pi$
$462$		0			0
$463$			7.08677e6i			1.53637i	0.640228	+	0.768185i	$0.278840\pi$
$463$			7.08677e6i			1.53637i	−0.640228	+	0.768185i	$0.721160\pi$
$464$		0			0
$465$	1.93631e6	+	1.71796e6i	0.415281	+	0.368452i
$466$		0			0
$467$	1.96616e6			0.417182			0.208591	−	0.978003i	$-0.433112\pi$
$467$	1.96616e6			0.417182			0.208591	+	0.978003i	$0.433112\pi$
$468$		0			0
$469$	−1.27088e6			−0.266792
$470$		0			0
$471$	2.81235e6	+	2.49522e6i	0.584140	+	0.518270i
$472$		0			0
$473$			4.08091e6i			0.838694i
$474$		0			0
$475$		−	490988.i		−	0.0998475i
$476$		0			0
$477$	7.24649e6	−	869070.i	1.45825	−	0.174888i
$478$		0			0
$479$	−3.56699e6			−0.710335			−0.355167	−	0.934803i	$-0.615576\pi$
$479$	−3.56699e6			−0.710335			−0.355167	+	0.934803i	$0.615576\pi$
$480$		0			0
$481$	8.45624e6			1.66654
$482$		0			0
$483$	66988.3	−	75502.2i	0.0130657	−	0.0147262i
$484$		0			0
$485$		−	8.29912e6i		−	1.60206i
$486$		0			0
$487$			5.42395e6i			1.03632i	0.855284	+	0.518159i	$0.173383\pi$
$487$			5.42395e6i			1.03632i	−0.855284	+	0.518159i	$0.826617\pi$
$488$		0			0
$489$	−5.48481e6	+	6.18191e6i	−1.03726	+	1.16910i
$490$		0			0
$491$	2.22352e6			0.416234			0.208117	−	0.978104i	$-0.433267\pi$
$491$	2.22352e6			0.416234			0.208117	+	0.978104i	$0.433267\pi$
$492$		0			0
$493$	−2.51943e6			−0.466858
$494$		0			0
$495$	−8.62533e6	+	1.03443e6i	−1.58220	+	0.189753i
$496$		0			0
$497$			1.01698e6i			0.184681i
$498$		0			0
$499$		−	2.07264e6i		−	0.372626i	−0.982490	−	0.186313i	$-0.940346\pi$
$499$		−	2.07264e6i		−	0.372626i	0.982490	−	0.186313i	$-0.0596538\pi$
$500$		0			0
$501$	−6.97523e6	−	6.18867e6i	−1.24155	−	1.10155i
$502$		0			0
$503$	757775.			0.133543			0.0667714	−	0.997768i	$-0.478730\pi$
$503$	757775.			0.133543			0.0667714	+	0.997768i	$0.478730\pi$
$504$		0			0
$505$	5.06028e6			0.882969
$506$		0			0
$507$	1.16687e7	+	1.03529e7i	2.01606	+	1.78872i
$508$		0			0
$509$			1.15618e7i			1.97801i	0.147872	+	0.989007i	$0.452758\pi$
$509$			1.15618e7i			1.97801i	−0.147872	+	0.989007i	$0.547242\pi$
$510$		0			0
$511$			1.94754e6i			0.329940i
$512$		0			0
$513$	−3.19582e6	−	2.21634e6i	−0.536154	−	0.371829i
$514$		0			0
$515$	−1.19633e7			−1.98762
$516$		0			0
$517$	−1.42863e7			−2.35068
$518$		0			0
$519$	−4.48168e6	+	5.05129e6i	−0.730336	+	0.823159i
$520$		0			0
$521$			4.64184e6i			0.749197i	0.927187	+	0.374599i	$0.122220\pi$
$521$			4.64184e6i			0.749197i	−0.927187	+	0.374599i	$0.877780\pi$
$522$		0			0
$523$			1.20127e6i			0.192038i	0.995379	+	0.0960190i	$0.0306110\pi$
$523$			1.20127e6i			0.192038i	−0.995379	+	0.0960190i	$0.969389\pi$
$524$		0			0
$525$	−242426.	+	273237.i	−0.0383867	+	0.0432655i
$526$		0			0
$527$	1.51166e6			0.237098
$528$		0			0
$529$	−6.41888e6			−0.997287
$530$		0			0
$531$	−127981.	−	1.06713e6i	−0.0196974	−	0.164241i
$532$		0			0
$533$		−	7.53372e6i		−	1.14866i
$534$		0			0
$535$		−	2.06751e6i		−	0.312294i
$536$		0			0
$537$	9.94148e6	+	8.82044e6i	1.48770	+	1.31994i
$538$		0			0
$539$	−1.42994e6			−0.212005
$540$		0			0
$541$	−4.17422e6			−0.613172			−0.306586	−	0.951843i	$-0.599187\pi$
$541$	−4.17422e6			−0.613172			−0.306586	+	0.951843i	$0.599187\pi$
$542$		0			0
$543$	−6.21305e6	−	5.51244e6i	−0.904285	−	0.802314i
$544$		0			0
$545$		−	3104.26i		−	0.000447679i
$546$		0			0
$547$		−	7.21542e6i		−	1.03108i	−0.856865	−	0.515541i	$-0.827591\pi$
$547$		−	7.21542e6i		−	1.03108i	0.856865	−	0.515541i	$-0.172409\pi$
$548$		0			0
$549$	1.43515e6	+	1.19666e7i	0.203220	+	1.69449i
$550$		0			0
$551$	−4.73374e6			−0.664241
$552$		0			0
$553$	2.51243e6			0.349366
$554$		0			0
$555$	4.48336e6	−	5.05317e6i	0.617833	−	0.696357i
$556$		0			0
$557$			419591.i			0.0573044i	0.999589	+	0.0286522i	$0.00912153\pi$
$557$			419591.i			0.0573044i	−0.999589	+	0.0286522i	$0.990878\pi$
$558$		0			0
$559$		−	8.02616e6i		−	1.08637i
$560$		0			0
$561$	−3.36686e6	+	3.79477e6i	−0.451667	+	0.509071i
$562$		0			0
$563$	7.08302e6			0.941775			0.470888	−	0.882193i	$-0.343934\pi$
$563$	7.08302e6			0.941775			0.470888	+	0.882193i	$0.343934\pi$
$564$		0			0
$565$	1.18670e6			0.156394
$566$		0			0
$567$	684170.	+	2.81135e6i	0.0893730	+	0.367246i
$568$		0			0
$569$			32234.0i			0.00417382i	0.999998	+	0.00208691i	$0.000664285\pi$
$569$			32234.0i			0.00417382i	−0.999998	+	0.00208691i	$0.999336\pi$
$570$		0			0
$571$		−	8.76445e6i		−	1.12495i	−0.826813	−	0.562476i	$-0.809849\pi$
$571$		−	8.76445e6i		−	1.12495i	0.826813	−	0.562476i	$-0.190151\pi$
$572$		0			0
$573$	5.14113e6	+	4.56140e6i	0.654142	+	0.580378i
$574$		0			0
$575$	−63193.4			−0.00797080
$576$		0			0
$577$	8.67226e6			1.08441			0.542204	−	0.840247i	$-0.317590\pi$
$577$	8.67226e6			1.08441			0.542204	+	0.840247i	$0.317590\pi$
$578$		0			0
$579$	1.95720e6	+	1.73650e6i	0.242627	+	0.215267i
$580$		0			0
$581$		−	903059.i		−	0.110988i
$582$		0			0
$583$		−	1.78874e7i		−	2.17960i
$584$		0			0
$585$	1.69639e7	−	2.03448e6i	2.04945	−	0.245790i
$586$		0			0
$587$	−9.91907e6			−1.18816			−0.594081	−	0.804405i	$-0.702484\pi$
$587$	−9.91907e6			−1.18816			−0.594081	+	0.804405i	$0.702484\pi$
$588$		0			0
$589$	2.84025e6			0.337340
$590$		0			0
$591$	−2.06490e6	+	2.32734e6i	−0.243181	+	0.274089i
$592$		0			0
$593$		−	1.05207e7i		−	1.22859i	−0.789075	−	0.614297i	$-0.789440\pi$
$593$		−	1.05207e7i		−	1.22859i	0.789075	−	0.614297i	$-0.210560\pi$
$594$		0			0
$595$			1.60725e6i			0.186119i
$596$		0			0
$597$	7.41986e6	−	8.36289e6i	0.852039	−	0.960330i
$598$		0			0
$599$	−3.88901e6			−0.442865			−0.221433	−	0.975176i	$-0.571073\pi$
$599$	−3.88901e6			−0.442865			−0.221433	+	0.975176i	$0.571073\pi$
$600$		0			0
$601$	−1.15374e7			−1.30293			−0.651464	−	0.758679i	$-0.725845\pi$
$601$	−1.15374e7			−1.30293			−0.651464	+	0.758679i	$0.725845\pi$
$602$		0			0
$603$	6.25769e6	−	750484.i	0.700843	−	0.0840520i
$604$		0			0
$605$			1.16236e7i			1.29107i
$606$		0			0
$607$			981043.i			0.108073i	0.998539	+	0.0540364i	$0.0172087\pi$
$607$			981043.i			0.108073i	−0.998539	+	0.0540364i	$0.982791\pi$
$608$		0			0
$609$	2.63435e6	+	2.33729e6i	0.287826	+	0.255370i
$610$		0			0
$611$	2.80978e7			3.04487
$612$		0			0
$613$	1.43521e7			1.54264			0.771321	−	0.636446i	$-0.219596\pi$
$613$	1.43521e7			1.54264			0.771321	+	0.636446i	$0.219596\pi$
$614$		0			0
$615$	−4.50191e6	−	3.99425e6i	−0.479964	−	0.425841i
$616$		0			0
$617$		−	1.52490e7i		−	1.61261i	−0.591502	−	0.806304i	$-0.701465\pi$
$617$		−	1.52490e7i		−	1.61261i	0.591502	−	0.806304i	$-0.298535\pi$
$618$		0			0
$619$			9.80690e6i			1.02874i	0.857569	+	0.514369i	$0.171974\pi$
$619$			9.80690e6i			1.02874i	−0.857569	+	0.514369i	$0.828026\pi$
$620$		0			0
$621$	−285258.	+	411324.i	−0.0296831	+	0.0428011i
$622$		0			0
$623$	−2.29115e6			−0.236501
$624$		0			0
$625$	−1.10314e7			−1.12961
$626$		0			0
$627$	−6.32597e6	+	7.12998e6i	−0.642626	+	0.724301i
$628$		0			0
$629$		−	3.94497e6i		−	0.397573i
$630$		0			0
$631$		−	2.01361e6i		−	0.201327i	−0.994921	−	0.100663i	$-0.967903\pi$
$631$		−	2.01361e6i		−	0.201327i	0.994921	−	0.100663i	$-0.0320965\pi$
$632$		0			0
$633$	−1.18425e6	+	1.33477e6i	−0.117472	+	0.132403i
$634$		0			0
$635$	1.30392e7			1.28326
$636$		0			0
$637$	2.81234e6			0.274612
$638$		0			0
$639$	−600551.	−	5.00752e6i	−0.0581833	−	0.485144i
$640$		0			0
$641$			1.50699e7i			1.44866i	0.689456	+	0.724328i	$0.257850\pi$
$641$			1.50699e7i			1.44866i	−0.689456	+	0.724328i	$0.742150\pi$
$642$		0			0
$643$			431918.i			0.0411978i	0.999788	+	0.0205989i	$0.00655730\pi$
$643$			431918.i			0.0411978i	−0.999788	+	0.0205989i	$0.993443\pi$
$644$		0			0
$645$	−4.79617e6	−	4.25534e6i	−0.453937	−	0.402749i
$646$		0			0
$647$	1.03026e7			0.967576			0.483788	−	0.875185i	$-0.339261\pi$
$647$	1.03026e7			0.967576			0.483788	+	0.875185i	$0.339261\pi$
$648$		0			0
$649$	−2.63413e6			−0.245485
$650$		0			0
$651$	−1.58061e6	−	1.40238e6i	−0.146175	−	0.129692i
$652$		0			0
$653$		−	1.55582e7i		−	1.42783i	−0.700231	−	0.713917i	$-0.746920\pi$
$653$		−	1.55582e7i		−	1.42783i	0.700231	−	0.713917i	$-0.253080\pi$
$654$		0			0
$655$		−	7.25311e6i		−	0.660573i
$656$		0			0
$657$	−1.15007e6	−	9.58951e6i	−0.103947	−	0.866728i
$658$		0			0
$659$	−1.68578e6			−0.151212			−0.0756060	−	0.997138i	$-0.524089\pi$
$659$	−1.68578e6			−0.151212			−0.0756060	+	0.997138i	$0.524089\pi$
$660$		0			0
$661$	−872809.			−0.0776990			−0.0388495	−	0.999245i	$-0.512369\pi$
$661$	−872809.			−0.0776990			−0.0388495	+	0.999245i	$0.512369\pi$
$662$		0			0
$663$	6.62181e6	−	7.46341e6i	0.585050	−	0.659407i
$664$		0			0
$665$			3.01986e6i			0.264809i
$666$		0			0
$667$			609264.i			0.0530262i
$668$		0			0
$669$	−6.68680e6	+	7.53666e6i	−0.577634	+	0.651049i
$670$		0			0
$671$	2.95386e7			2.53270
$672$		0			0
$673$	578719.			0.0492527			0.0246264	−	0.999697i	$-0.492160\pi$
$673$	578719.			0.0492527			0.0246264	+	0.999697i	$0.492160\pi$
$674$		0			0
$675$	1.03233e6	−	1.48855e6i	0.0872085	−	0.125749i
$676$		0			0
$677$			9.63738e6i			0.808141i	0.914728	+	0.404070i	$0.132405\pi$
$677$			9.63738e6i			0.808141i	−0.914728	+	0.404070i	$0.867595\pi$
$678$		0			0
$679$			6.77459e6i			0.563908i
$680$		0			0
$681$	−112956.	−	100218.i	−0.00933341	−	0.00828094i
$682$		0			0
$683$	1.81197e6			0.148628			0.0743139	−	0.997235i	$-0.476323\pi$
$683$	1.81197e6			0.148628			0.0743139	+	0.997235i	$0.476323\pi$
$684$		0			0
$685$	−1.88649e7			−1.53613
$686$		0			0
$687$	1.53551e7	+	1.36236e7i	1.24126	+	1.10129i
$688$		0			0
$689$			3.51802e7i			2.82326i
$690$		0			0
$691$			3.50492e6i			0.279244i	0.990205	+	0.139622i	$0.0445887\pi$
$691$			3.50492e6i			0.279244i	−0.990205	+	0.139622i	$0.955411\pi$
$692$		0			0
$693$	7.04087e6	−	844410.i	0.556920	−	0.0667914i
$694$		0			0
$695$	5.24123e6			0.411596
$696$		0			0
$697$	−3.51460e6			−0.274027
$698$		0			0
$699$	1.55017e7	−	1.74719e7i	1.20001	−	1.35253i
$700$		0			0
$701$			3.50366e6i			0.269294i	0.990894	+	0.134647i	$0.0429901\pi$
$701$			3.50366e6i			0.269294i	−0.990894	+	0.134647i	$0.957010\pi$
$702$		0			0
$703$		−	7.41218e6i		−	0.565663i
$704$		0			0
$705$	1.48970e7	−	1.67903e7i	1.12882	−	1.27229i
$706$		0			0
$707$	−4.13071e6			−0.310797
$708$		0			0
$709$	2.66641e6			0.199210			0.0996051	−	0.995027i	$-0.468242\pi$
$709$	2.66641e6			0.199210			0.0996051	+	0.995027i	$0.468242\pi$
$710$		0			0
$711$	−1.23709e7	+	1.48365e6i	−0.917759	+	0.110067i
$712$		0			0
$713$		−	365558.i		−	0.0269298i
$714$		0			0
$715$		−	4.18742e7i		−	3.06324i
$716$		0			0
$717$	1.38990e7	+	1.23317e7i	1.00968	+	0.895829i
$718$		0			0
$719$	−539114.			−0.0388918			−0.0194459	−	0.999811i	$-0.506190\pi$
$719$	−539114.			−0.0388918			−0.0194459	+	0.999811i	$0.506190\pi$
$720$		0			0
$721$	9.76565e6			0.699622
$722$		0			0
$723$	1.60707e7	+	1.42585e7i	1.14338	+	1.01445i
$724$		0			0
$725$		−	2.20488e6i		−	0.155790i
$726$		0			0
$727$			6.14080e6i			0.430913i	0.976513	+	0.215456i	$0.0691239\pi$
$727$			6.14080e6i			0.430913i	−0.976513	+	0.215456i	$0.930876\pi$
$728$		0			0
$729$	−5.02895e6	−	1.34388e7i	−0.350476	−	0.936572i
$730$		0			0
$731$	−3.74433e6			−0.259168
$732$		0			0
$733$	9.13605e6			0.628056			0.314028	−	0.949414i	$-0.398321\pi$
$733$	9.13605e6			0.628056			0.314028	+	0.949414i	$0.398321\pi$
$734$		0			0
$735$	1.49106e6	−	1.68057e6i	0.101807	−	0.114746i
$736$		0			0
$737$		−	1.54466e7i		−	1.04753i
$738$		0			0
$739$		−	4.29195e6i		−	0.289097i	−0.989498	−	0.144548i	$-0.953827\pi$
$739$		−	4.29195e6i		−	0.289097i	0.989498	−	0.144548i	$-0.0461730\pi$
$740$		0			0
$741$	1.24417e7	−	1.40230e7i	0.832402	−	0.938197i
$742$		0			0
$743$	−3.68696e6			−0.245017			−0.122509	−	0.992467i	$-0.539094\pi$
$743$	−3.68696e6			−0.245017			−0.122509	+	0.992467i	$0.539094\pi$
$744$		0			0
$745$	8.06870e6			0.532615
$746$		0			0
$747$	533276.	+	4.44657e6i	0.0349664	+	0.291557i
$748$		0			0
$749$			1.68771e6i			0.109924i
$750$		0			0
$751$			1.16822e7i			0.755831i	0.925840	+	0.377916i	$0.123359\pi$
$751$			1.16822e7i			0.755831i	−0.925840	+	0.377916i	$0.876641\pi$
$752$		0			0
$753$	1.85573e7	+	1.64647e7i	1.19269	+	1.05820i
$754$		0			0
$755$	−2.26779e7			−1.44789
$756$		0			0
$757$	−1.76630e7			−1.12028			−0.560138	−	0.828399i	$-0.689252\pi$
$757$	−1.76630e7			−1.12028			−0.560138	+	0.828399i	$0.689252\pi$
$758$		0			0
$759$	917674.	+	814194.i	0.0578208	+	0.0513007i
$760$		0			0
$761$			1.60065e7i			1.00193i	0.865469	+	0.500963i	$0.167021\pi$
$761$			1.60065e7i			1.00193i	−0.865469	+	0.500963i	$0.832979\pi$
$762$		0			0
$763$			2534.01i			0.000157579i
$764$		0			0
$765$	−949119.	−	7.91395e6i	−0.0586364	−	0.488922i
$766$		0			0
$767$	5.18070e6			0.317980
$768$		0			0
$769$	454748.			0.0277303			0.0138652	−	0.999904i	$-0.495586\pi$
$769$	454748.			0.0277303			0.0138652	+	0.999904i	$0.495586\pi$
$770$		0			0
$771$	−2.02188e7	+	2.27886e7i	−1.22496	+	1.38064i
$772$		0			0
$773$		−	1.02478e7i		−	0.616854i	−0.951248	−	0.308427i	$-0.900198\pi$
$773$		−	1.02478e7i		−	0.616854i	0.951248	−	0.308427i	$-0.0998025\pi$
$774$		0			0
$775$			1.32293e6i			0.0791194i
$776$		0			0
$777$	−3.65977e6	+	4.12491e6i	−0.217471	+	0.245111i
$778$		0			0
$779$	−6.60356e6			−0.389883
$780$		0			0
$781$	−1.23607e7			−0.725129
$782$		0			0
$783$	−1.43515e7	−	9.95294e6i	−0.836552	−	0.580159i
$784$		0			0
$785$			1.44776e7i			0.838538i
$786$		0			0
$787$		−	2.48007e7i		−	1.42734i	−0.700484	−	0.713669i	$-0.747032\pi$
$787$		−	2.48007e7i		−	1.42734i	0.700484	−	0.713669i	$-0.252968\pi$
$788$		0			0
$789$	2.39224e7	+	2.12248e7i	1.36808	+	1.21381i
$790$		0			0
$791$	−968709.			−0.0550493
$792$		0			0
$793$	−5.80954e7			−3.28064
$794$		0			0
$795$	2.10226e7	+	1.86520e7i	1.17969	+	1.04666i
$796$		0			0
$797$			1.38588e6i			0.0772824i	0.999253	+	0.0386412i	$0.0123029\pi$
$797$			1.38588e6i			0.0772824i	−0.999253	+	0.0386412i	$0.987697\pi$
$798$		0			0
$799$		−	1.31081e7i		−	0.726392i
$800$		0			0
$801$	1.12814e7	−	1.35297e6i	0.621271	−	0.0745089i
$802$		0			0
$803$	−2.36710e7			−1.29547
$804$		0			0
$805$	388675.			0.0211396
$806$		0			0
$807$	1.60522e7	−	1.80924e7i	0.867663	−	0.977939i
$808$		0			0
$809$			6.94399e6i			0.373025i	0.982453	+	0.186513i	$0.0597185\pi$
$809$			6.94399e6i			0.373025i	−0.982453	+	0.186513i	$0.940281\pi$
$810$		0			0
$811$		−	1.42362e7i		−	0.760049i	−0.924976	−	0.380025i	$-0.875916\pi$
$811$		−	1.42362e7i		−	0.760049i	0.924976	−	0.380025i	$-0.124084\pi$
$812$		0			0
$813$	−99560.6	+	112214.i	−0.00528276	+	0.00595418i
$814$		0			0
$815$	−3.18236e7			−1.67825
$816$		0			0
$817$	−7.03520e6			−0.368741
$818$		0			0
$819$	−1.38477e7	+	1.66075e6i	−0.721387	+	0.0865158i
$820$		0			0
$821$		−	3.21481e7i		−	1.66455i	−0.554360	−	0.832277i	$-0.687037\pi$
$821$		−	3.21481e7i		−	1.66455i	0.554360	−	0.832277i	$-0.312963\pi$
$822$		0			0
$823$			2.16556e7i			1.11447i	0.830353	+	0.557237i	$0.188139\pi$
$823$			2.16556e7i			1.11447i	−0.830353	+	0.557237i	$0.811861\pi$
$824$		0			0
$825$	−3.32100e6	−	2.94651e6i	−0.169877	−	0.150721i
$826$		0			0
$827$	−8.32946e6			−0.423500			−0.211750	−	0.977324i	$-0.567916\pi$
$827$	−8.32946e6			−0.423500			−0.211750	+	0.977324i	$0.567916\pi$
$828$		0			0
$829$	8.33090e6			0.421023			0.210511	−	0.977591i	$-0.432487\pi$
$829$	8.33090e6			0.421023			0.210511	+	0.977591i	$0.432487\pi$
$830$		0			0
$831$	1.38414e7	+	1.22806e7i	0.695309	+	0.616903i
$832$		0			0
$833$		−	1.31200e6i		−	0.0655123i
$834$		0			0
$835$		−	3.59075e7i		−	1.78225i
$836$		0			0
$837$	8.61091e6	+	5.97177e6i	0.424850	+	0.294639i
$838$		0			0
$839$	4.28482e6			0.210149			0.105075	−	0.994464i	$-0.466492\pi$
$839$	4.28482e6			0.210149			0.105075	+	0.994464i	$0.466492\pi$
$840$		0			0
$841$	−746690.			−0.0364041
$842$		0			0
$843$	−1.86970e7	+	2.10733e7i	−0.906156	+	1.02132i
$844$		0			0
$845$			6.00690e7i			2.89407i
$846$		0			0
$847$		−	9.48835e6i		−	0.454446i
$848$		0			0
$849$	−1.47772e7	+	1.66553e7i	−0.703594	+	0.793018i
$850$		0			0
$851$	−953996.			−0.0451567
$852$		0			0
$853$	1.37087e7			0.645093			0.322547	−	0.946554i	$-0.395461\pi$
$853$	1.37087e7			0.645093			0.322547	+	0.946554i	$0.395461\pi$
$854$		0			0
$855$	−1.78329e6	−	1.48695e7i	−0.0834272	−	0.695633i
$856$		0			0
$857$		−	1.64750e7i		−	0.766253i	−0.923696	−	0.383127i	$-0.874847\pi$
$857$		−	1.64750e7i		−	0.766253i	0.923696	−	0.383127i	$-0.125153\pi$
$858$		0			0
$859$			1.74418e7i			0.806506i	0.915088	+	0.403253i	$0.132121\pi$
$859$			1.74418e7i			0.806506i	−0.915088	+	0.403253i	$0.867879\pi$
$860$		0			0
$861$	3.67491e6	+	3.26052e6i	0.168943	+	0.149892i
$862$		0			0
$863$	1.72754e7			0.789591			0.394795	−	0.918769i	$-0.370815\pi$
$863$	1.72754e7			0.789591			0.394795	+	0.918769i	$0.370815\pi$
$864$		0			0
$865$	−2.60034e7			−1.18165
$866$		0			0
$867$	1.30745e7	+	1.16001e7i	0.590713	+	0.524102i
$868$		0			0
$869$			3.05367e7i			1.37174i
$870$		0			0
$871$			3.03798e7i			1.35688i
$872$		0			0
$873$	−4.00054e6	−	3.33574e7i	−0.177657	−	1.48135i
$874$		0			0
$875$	7.78502e6			0.343747
$876$		0			0
$877$	−1.76889e7			−0.776609			−0.388305	−	0.921531i	$-0.626939\pi$
$877$	−1.76889e7			−0.776609			−0.388305	+	0.921531i	$0.626939\pi$
$878$		0			0
$879$	1.09558e7	−	1.23483e7i	0.478270	−	0.539056i
$880$		0			0
$881$		−	4.38768e7i		−	1.90456i	−0.305225	−	0.952280i	$-0.598732\pi$
$881$		−	4.38768e7i		−	1.90456i	0.305225	−	0.952280i	$-0.401268\pi$
$882$		0			0
$883$			9.44221e6i			0.407541i	0.979019	+	0.203771i	$0.0653197\pi$
$883$			9.44221e6i			0.407541i	−0.979019	+	0.203771i	$0.934680\pi$
$884$		0			0
$885$	2.74672e6	−	3.09582e6i	0.117884	−	0.132867i
$886$		0			0
$887$	−3.78161e7			−1.61387			−0.806934	−	0.590642i	$-0.798875\pi$
$887$	−3.78161e7			−1.61387			−0.806934	+	0.590642i	$0.798875\pi$
$888$		0			0
$889$	−1.06439e7			−0.451696
$890$		0			0
$891$	−3.41699e7	+	8.31558e6i	−1.44195	+	0.350912i
$892$		0			0
$893$		−	2.46286e7i		−	1.03350i
$894$		0			0
$895$			5.11774e7i			2.13560i
$896$		0			0
$897$	−1.80485e6	−	1.60132e6i	−0.0748961	−	0.0664505i
$898$		0			0
$899$	1.27547e7			0.526346
$900$		0			0
$901$	1.64121e7			0.673524
$902$		0			0
$903$	3.91513e6	+	3.47364e6i	0.159781	+	0.141764i
$904$		0			0
$905$		−	3.19839e7i		−	1.29811i
$906$		0			0
$907$			2.36126e7i			0.953070i	0.879155	+	0.476535i	$0.158108\pi$
$907$			2.36126e7i			0.953070i	−0.879155	+	0.476535i	$0.841892\pi$
$908$		0			0
$909$	2.03392e7	−	2.43928e6i	0.816440	−	0.0979155i
$910$		0			0
$911$	2.17739e7			0.869242			0.434621	−	0.900613i	$-0.356882\pi$
$911$	2.17739e7			0.869242			0.434621	+	0.900613i	$0.356882\pi$
$912$		0			0
$913$	1.09760e7			0.435781
$914$		0			0
$915$	−3.08012e7	+	3.47159e7i	−1.21623	+	1.37081i
$916$		0			0
$917$			5.92073e6i			0.232515i
$918$		0			0
$919$		−	4.56575e7i		−	1.78330i	−0.452729	−	0.891648i	$-0.649550\pi$
$919$		−	4.56575e7i		−	1.78330i	0.452729	−	0.891648i	$-0.350450\pi$
$920$		0			0
$921$	−2.31229e6	+	2.60617e6i	−0.0898241	+	0.101240i
$922$		0			0
$923$	2.43105e7			0.939269
$924$		0			0
$925$	3.45245e6			0.132670
$926$		0			0
$927$	−4.80851e7	+	5.76684e6i	−1.83785	+	0.220414i
$928$		0			0
$929$		−	4.31259e6i		−	0.163945i	−0.996635	−	0.0819727i	$-0.973878\pi$
$929$		−	4.31259e6i		−	0.163945i	0.996635	−	0.0819727i	$-0.0261220\pi$
$930$		0			0
$931$		−	2.46512e6i		−	0.0932101i
$932$		0			0
$933$	−3.38225e7	−	3.00086e7i	−1.27204	−	1.12860i
$934$		0			0
$935$	−1.95350e7			−0.730776
$936$		0			0
$937$	−4.92937e7			−1.83418			−0.917091	−	0.398678i	$-0.869469\pi$
$937$	−4.92937e7			−1.83418			−0.917091	+	0.398678i	$0.869469\pi$
$938$		0			0
$939$	1.03340e7	+	9.16871e6i	0.382477	+	0.339347i
$940$		0			0
$941$		−	5.41248e6i		−	0.199261i	−0.995025	−	0.0996305i	$-0.968234\pi$
$941$		−	5.41248e6i		−	0.199261i	0.995025	−	0.0996305i	$-0.0317661\pi$
$942$		0			0
$943$			849922.i			0.0311243i
$944$		0			0
$945$	−6.34941e6	+	9.15544e6i	−0.231288	+	0.333503i
$946$		0			0
$947$	−2.12447e6			−0.0769796			−0.0384898	−	0.999259i	$-0.512255\pi$
$947$	−2.12447e6			−0.0769796			−0.0384898	+	0.999259i	$0.512255\pi$
$948$		0			0
$949$	4.65551e7			1.67804
$950$		0			0
$951$	2.46931e6	−	2.78315e6i	0.0885370	−	0.0997897i
$952$		0			0
$953$		−	5.03502e7i		−	1.79584i	−0.440154	−	0.897922i	$-0.645076\pi$
$953$		−	5.03502e7i		−	1.79584i	0.440154	−	0.897922i	$-0.354924\pi$
$954$		0			0
$955$			2.64658e7i			0.939025i
$956$		0			0
$957$	−2.84081e7	+	3.20186e7i	−1.00268	+	1.13012i
$958$		0			0
$959$	1.53994e7			0.540702
$960$		0			0
$961$	2.09763e7			0.732691
$962$		0			0
$963$	−996632.	−	8.31012e6i	−0.0346313	−	0.288763i
$964$		0			0
$965$			1.00754e7i			0.348293i
$966$		0			0
$967$			4.94644e7i			1.70109i	0.525903	+	0.850544i	$0.323727\pi$
$967$			4.94644e7i			1.70109i	−0.525903	+	0.850544i	$0.676273\pi$
$968$		0			0
$969$	−6.54193e6	−	5.80424e6i	−0.223819	−	0.198580i
$970$		0			0
$971$	−3.44238e7			−1.17168			−0.585842	−	0.810426i	$-0.699236\pi$
$971$	−3.44238e7			−1.17168			−0.585842	+	0.810426i	$0.699236\pi$
$972$		0			0
$973$	−4.27842e6			−0.144878
$974$		0			0
$975$	6.53161e6	+	5.79508e6i	0.220044	+	0.195231i
$976$		0			0
$977$		−	3.72790e7i		−	1.24948i	−0.780834	−	0.624738i	$-0.785206\pi$
$977$		−	3.72790e7i		−	1.24948i	0.780834	−	0.624738i	$-0.214794\pi$
$978$		0			0
$979$		−	2.78472e7i		−	0.928593i
$980$		0			0
$981$	−1496.39	−	12477.2i	−4.96447e−5	−	0.000413948i
$982$		0			0
$983$	4.27871e6			0.141231			0.0706153	−	0.997504i	$-0.477504\pi$
$983$	4.27871e6			0.141231			0.0706153	+	0.997504i	$0.477504\pi$
$984$		0			0
$985$	−1.19808e7			−0.393456
$986$		0			0
$987$	−1.21604e7	+	1.37060e7i	−0.397334	+	0.447833i
$988$		0			0
$989$			905477.i			0.0294365i
$990$		0			0
$991$			1.08165e7i			0.349868i	0.984580	+	0.174934i	$0.0559711\pi$
$991$			1.08165e7i			0.349868i	−0.984580	+	0.174934i	$0.944029\pi$
$992$		0			0
$993$	1.25018e7	−	1.40907e7i	0.402346	−	0.453482i
$994$		0			0
$995$	4.30510e7			1.37856
$996$		0			0
$997$	−2.37974e7			−0.758213			−0.379107	−	0.925353i	$-0.623769\pi$
$997$	−2.37974e7			−0.758213			−0.379107	+	0.925353i	$0.623769\pi$
$998$		0			0
$999$	1.55845e7	−	2.24718e7i	0.494059	−	0.712402i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.6.h.b.239.32	yes	40
3.2	odd	2	inner	336.6.h.b.239.10	yes	40
4.3	odd	2	inner	336.6.h.b.239.9	✓	40
12.11	even	2	inner	336.6.h.b.239.31	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.6.h.b.239.9	✓	40	4.3	odd	2	inner
336.6.h.b.239.10	yes	40	3.2	odd	2	inner
336.6.h.b.239.31	yes	40	12.11	even	2	inner
336.6.h.b.239.32	yes	40	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)

\(f(q)\)	\(=\)	\(q+(11.6605 + 10.3456i) q^{3} +60.0268i q^{5} -49.0000i q^{7} +(28.9356 + 241.271i) q^{9} +O(q^{10})\)	\(q+(11.6605 + 10.3456i) q^{3} +60.0268i q^{5} -49.0000i q^{7} +(28.9356 + 241.271i) q^{9} +595.559 q^{11} -1171.32 q^{13} +(-621.016 + 699.944i) q^{15} +546.441i q^{17} +1026.70i q^{19} +(506.936 - 571.366i) q^{21} +132.143 q^{23} -478.218 q^{25} +(-2158.70 + 3112.70i) q^{27} +4610.62i q^{29} -2766.38i q^{31} +(6944.53 + 6161.44i) q^{33} +2941.31 q^{35} -7219.40 q^{37} +(-13658.2 - 12118.1i) q^{39} +6431.81i q^{41} +6852.22i q^{43} +(-14482.7 + 1736.91i) q^{45} -23988.1 q^{47} -2401.00 q^{49} +(-5653.28 + 6371.78i) q^{51} -30034.6i q^{53} +35749.5i q^{55} +(-10621.9 + 11971.9i) q^{57} -4422.95 q^{59} +49598.2 q^{61} +(11822.3 - 1417.84i) q^{63} -70310.7i q^{65} -25936.4i q^{67} +(1540.86 + 1367.11i) q^{69} -20754.8 q^{71} -39745.8 q^{73} +(-5576.27 - 4947.47i) q^{75} -29182.4i q^{77} +51274.0i q^{79} +(-57374.5 + 13962.6i) q^{81} +18429.8 q^{83} -32801.1 q^{85} +(-47699.8 + 53762.3i) q^{87} -46758.1i q^{89} +57394.8i q^{91} +(28619.9 - 32257.4i) q^{93} -61629.7 q^{95} -138257. q^{97} +(17232.9 + 143691. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 8 q^{9}+O(q^{10}) \) 40 * q + 8 * q^9	\( 40 q + 8 q^{9} - 1048 q^{13} + 980 q^{21} - 43416 q^{25} + 20296 q^{33} - 16192 q^{37} + 56488 q^{45} - 96040 q^{49} + 31088 q^{57} + 173112 q^{61} - 114176 q^{69} - 267488 q^{73} + 64888 q^{81} + 508112 q^{85} - 224544 q^{93} - 276400 q^{97}+O(q^{100}) \) 40 * q + 8 * q^9 - 1048 * q^13 + 980 * q^21 - 43416 * q^25 + 20296 * q^33 - 16192 * q^37 + 56488 * q^45 - 96040 * q^49 + 31088 * q^57 + 173112 * q^61 - 114176 * q^69 - 267488 * q^73 + 64888 * q^81 + 508112 * q^85 - 224544 * q^93 - 276400 * q^97

Introduction

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