LMFDB - Embedded newform 336.6.h.a.239.10

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			239.10
Root			$-1.95479 + 1.95479i$ of defining polynomial
Character	$\chi$	$=$	336.239
Dual form			336.6.h.a.239.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.95479 + 15.4654i) q^{3} +34.2747i q^{5} -49.0000i q^{7} +(-235.358 - 60.4632i) q^{9} +O(q^{10})$	\(q+(-1.95479 + 15.4654i) q^{3} +34.2747i q^{5} -49.0000i q^{7} +(-235.358 - 60.4632i) q^{9} +609.994 q^{11} -164.872 q^{13} +(-530.072 - 66.9998i) q^{15} -866.294i q^{17} -535.078i q^{19} +(757.805 + 95.7847i) q^{21} -4044.74 q^{23} +1950.25 q^{25} +(1395.16 - 3521.71i) q^{27} -7210.68i q^{29} -4423.05i q^{31} +(-1192.41 + 9433.81i) q^{33} +1679.46 q^{35} +7285.26 q^{37} +(322.290 - 2549.81i) q^{39} -3000.98i q^{41} -18694.4i q^{43} +(2072.36 - 8066.81i) q^{45} -931.299 q^{47} -2401.00 q^{49} +(13397.6 + 1693.42i) q^{51} +40298.7i q^{53} +20907.4i q^{55} +(8275.19 + 1045.96i) q^{57} -38098.4 q^{59} +43901.4 q^{61} +(-2962.70 + 11532.5i) q^{63} -5650.93i q^{65} -4505.45i q^{67} +(7906.61 - 62553.5i) q^{69} -2286.84 q^{71} +64424.6 q^{73} +(-3812.32 + 30161.3i) q^{75} -29889.7i q^{77} +47561.7i q^{79} +(51737.4 + 28461.0i) q^{81} +87756.5 q^{83} +29692.0 q^{85} +(111516. + 14095.4i) q^{87} -12809.7i q^{89} +8078.72i q^{91} +(68404.2 + 8646.12i) q^{93} +18339.6 q^{95} +85262.4 q^{97} +(-143567. - 36882.2i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.95479	+	15.4654i	−0.125400	+	0.992106i
$3$	−1.95479	+	15.4654i	−0.125400	+	0.992106i
$4$		0			0
$5$			34.2747i			0.613124i	0.951851	+	0.306562i	$0.0991787\pi$
$5$			34.2747i			0.613124i	−0.951851	+	0.306562i	$0.900821\pi$
$6$		0			0
$7$		−	49.0000i		−	0.377964i
$7$		−	49.0000i		−	0.377964i
$8$		0			0
$9$	−235.358	−	60.4632i	−0.968550	−	0.248820i
$10$		0			0
$11$	609.994			1.52000			0.760001	−	0.649922i	$-0.225198\pi$
$11$	609.994			1.52000			0.760001	+	0.649922i	$0.225198\pi$
$12$		0			0
$13$	−164.872			−0.270575			−0.135288	−	0.990806i	$-0.543196\pi$
$13$	−164.872			−0.270575			−0.135288	+	0.990806i	$0.543196\pi$
$14$		0			0
$15$	−530.072	−	66.9998i	−0.608285	−	0.0768857i
$16$		0			0
$17$		−	866.294i		−	0.727014i	−0.931591	−	0.363507i	$-0.881579\pi$
$17$		−	866.294i		−	0.727014i	0.931591	−	0.363507i	$-0.118421\pi$
$18$		0			0
$19$		−	535.078i		−	0.340042i	−0.985440	−	0.170021i	$-0.945616\pi$
$19$		−	535.078i		−	0.340042i	0.985440	−	0.170021i	$-0.0543836\pi$
$20$		0			0
$21$	757.805	+	95.7847i	0.374981	+	0.0473967i
$22$		0			0
$23$	−4044.74			−1.59430			−0.797152	−	0.603779i	$-0.793661\pi$
$23$	−4044.74			−1.59430			−0.797152	+	0.603779i	$0.793661\pi$
$24$		0			0
$25$	1950.25			0.624079
$26$		0			0
$27$	1395.16	−	3521.71i	0.368312	−	0.929702i
$28$		0			0
$29$		−	7210.68i		−	1.59214i	−0.605205	−	0.796070i	$-0.706909\pi$
$29$		−	7210.68i		−	1.59214i	0.605205	−	0.796070i	$-0.293091\pi$
$30$		0			0
$31$		−	4423.05i		−	0.826642i	−0.910585	−	0.413321i	$-0.864369\pi$
$31$		−	4423.05i		−	0.826642i	0.910585	−	0.413321i	$-0.135631\pi$
$32$		0			0
$33$	−1192.41	+	9433.81i	−0.190608	+	1.50800i
$34$		0			0
$35$	1679.46			0.231739
$36$		0			0
$37$	7285.26			0.874865			0.437432	−	0.899251i	$-0.355888\pi$
$37$	7285.26			0.874865			0.437432	+	0.899251i	$0.355888\pi$
$38$		0			0
$39$	322.290	−	2549.81i	0.0339301	−	0.268439i
$40$		0			0
$41$		−	3000.98i		−	0.278807i	−0.990236	−	0.139403i	$-0.955482\pi$
$41$		−	3000.98i		−	0.278807i	0.990236	−	0.139403i	$-0.0445185\pi$
$42$		0			0
$43$		−	18694.4i		−	1.54185i	−0.636928	−	0.770923i	$-0.719795\pi$
$43$		−	18694.4i		−	1.54185i	0.636928	−	0.770923i	$-0.280205\pi$
$44$		0			0
$45$	2072.36	−	8066.81i	0.152557	−	0.593841i
$46$		0			0
$47$	−931.299			−0.0614957			−0.0307478	−	0.999527i	$-0.509789\pi$
$47$	−931.299			−0.0614957			−0.0307478	+	0.999527i	$0.509789\pi$
$48$		0			0
$49$	−2401.00			−0.142857
$50$		0			0
$51$	13397.6	+	1693.42i	0.721275	+	0.0911674i
$52$		0			0
$53$			40298.7i			1.97061i	0.170801	+	0.985305i	$0.445364\pi$
$53$			40298.7i			1.97061i	−0.170801	+	0.985305i	$0.554636\pi$
$54$		0			0
$55$			20907.4i			0.931950i
$56$		0			0
$57$	8275.19	+	1045.96i	0.337358	+	0.0426412i
$58$		0			0
$59$	−38098.4			−1.42487			−0.712437	−	0.701736i	$-0.752408\pi$
$59$	−38098.4			−1.42487			−0.712437	+	0.701736i	$0.752408\pi$
$60$		0			0
$61$	43901.4			1.51061			0.755307	−	0.655371i	$-0.227488\pi$
$61$	43901.4			1.51061			0.755307	+	0.655371i	$0.227488\pi$
$62$		0			0
$63$	−2962.70	+	11532.5i	−0.0940451	+	0.366077i
$64$		0			0
$65$		−	5650.93i		−	0.165896i
$66$		0			0
$67$		−	4505.45i		−	0.122617i	−0.998119	−	0.0613085i	$-0.980473\pi$
$67$		−	4505.45i		−	0.122617i	0.998119	−	0.0613085i	$-0.0195274\pi$
$68$		0			0
$69$	7906.61	−	62553.5i	0.199925	−	1.58172i
$70$		0			0
$71$	−2286.84			−0.0538382			−0.0269191	−	0.999638i	$-0.508570\pi$
$71$	−2286.84			−0.0538382			−0.0269191	+	0.999638i	$0.508570\pi$
$72$		0			0
$73$	64424.6			1.41496			0.707480	−	0.706733i	$-0.249832\pi$
$73$	64424.6			1.41496			0.707480	+	0.706733i	$0.249832\pi$
$74$		0			0
$75$	−3812.32	+	30161.3i	−0.0782593	+	0.619152i
$76$		0			0
$77$		−	29889.7i		−	0.574507i
$78$		0			0
$79$			47561.7i			0.857412i	0.903444	+	0.428706i	$0.141030\pi$
$79$			47561.7i			0.857412i	−0.903444	+	0.428706i	$0.858970\pi$
$80$		0			0
$81$	51737.4	+	28461.0i	0.876177	+	0.481989i
$82$		0			0
$83$	87756.5			1.39825			0.699124	−	0.715001i	$-0.253574\pi$
$83$	87756.5			1.39825			0.699124	+	0.715001i	$0.253574\pi$
$84$		0			0
$85$	29692.0			0.445750
$86$		0			0
$87$	111516.	+	14095.4i	1.57957	+	0.199654i
$88$		0			0
$89$		−	12809.7i		−	0.171421i	−0.996320	−	0.0857103i	$-0.972684\pi$
$89$		−	12809.7i		−	0.171421i	0.996320	−	0.0857103i	$-0.0273159\pi$
$90$		0			0
$91$			8078.72i			0.102268i
$92$		0			0
$93$	68404.2	+	8646.12i	0.820116	+	0.103661i
$94$		0			0
$95$	18339.6			0.208488
$96$		0			0
$97$	85262.4			0.920086			0.460043	−	0.887897i	$-0.347834\pi$
$97$	85262.4			0.920086			0.460043	+	0.887897i	$0.347834\pi$
$98$		0			0
$99$	−143567.	−	36882.2i	−1.47220	−	0.378207i
$100$		0			0
$101$		−	101150.i		−	0.986647i	−0.869846	−	0.493323i	$-0.835782\pi$
$101$		−	101150.i		−	0.986647i	0.869846	−	0.493323i	$-0.164218\pi$
$102$		0			0
$103$			44497.4i			0.413277i	0.978417	+	0.206639i	$0.0662524\pi$
$103$			44497.4i			0.413277i	−0.978417	+	0.206639i	$0.933748\pi$
$104$		0			0
$105$	−3282.99	+	25973.5i	−0.0290600	+	0.229910i
$106$		0			0
$107$	−140485.			−1.18624			−0.593118	−	0.805115i	$-0.702103\pi$
$107$	−140485.			−1.18624			−0.593118	+	0.805115i	$0.702103\pi$
$108$		0			0
$109$	23941.9			0.193015			0.0965076	−	0.995332i	$-0.469233\pi$
$109$	23941.9			0.193015			0.0965076	+	0.995332i	$0.469233\pi$
$110$		0			0
$111$	−14241.2	+	112670.i	−0.109708	+	0.867959i
$112$		0			0
$113$			4392.80i			0.0323627i	0.999869	+	0.0161814i	$0.00515092\pi$
$113$			4392.80i			0.0323627i	−0.999869	+	0.0161814i	$0.994849\pi$
$114$		0			0
$115$		−	138632.i		−	0.977506i
$116$		0			0
$117$	38803.8	+	9968.68i	0.262066	+	0.0673245i
$118$		0			0
$119$	−42448.4			−0.274786
$120$		0			0
$121$	211042.			1.31041
$122$		0			0
$123$	46411.4	+	5866.28i	0.276606	+	0.0349623i
$124$		0			0
$125$			173952.i			0.995762i
$126$		0			0
$127$		−	56313.3i		−	0.309814i	−0.987929	−	0.154907i	$-0.950492\pi$
$127$		−	56313.3i		−	0.309814i	0.987929	−	0.154907i	$-0.0495079\pi$
$128$		0			0
$129$	289117.	+	36543.7i	1.52968	+	0.193347i
$130$		0			0
$131$	166069.			0.845496			0.422748	−	0.906247i	$-0.361066\pi$
$131$	166069.			0.845496			0.422748	+	0.906247i	$0.361066\pi$
$132$		0			0
$133$	−26218.8			−0.128524
$134$		0			0
$135$	120705.	+	47818.8i	0.570023	+	0.225821i
$136$		0			0
$137$		−	323471.i		−	1.47243i	−0.676748	−	0.736215i	$-0.736611\pi$
$137$		−	323471.i		−	1.47243i	0.676748	−	0.736215i	$-0.263389\pi$
$138$		0			0
$139$		−	248432.i		−	1.09061i	−0.838237	−	0.545306i	$-0.816413\pi$
$139$		−	248432.i		−	1.09061i	0.838237	−	0.545306i	$-0.183587\pi$
$140$		0			0
$141$	1820.49	−	14402.9i	0.00771154	−	0.0610102i
$142$		0			0
$143$	−100571.			−0.411275
$144$		0			0
$145$	247144.			0.976180
$146$		0			0
$147$	4693.45	−	37132.4i	0.0179143	−	0.141729i
$148$		0			0
$149$			152342.i			0.562154i	0.959685	+	0.281077i	$0.0906916\pi$
$149$			152342.i			0.562154i	−0.959685	+	0.281077i	$0.909308\pi$
$150$		0			0
$151$		−	161537.i		−	0.576541i	−0.957549	−	0.288271i	$-0.906920\pi$
$151$		−	161537.i		−	0.576541i	0.957549	−	0.288271i	$-0.0930802\pi$
$152$		0			0
$153$	−52378.9	+	203889.i	−0.180896	+	0.704149i
$154$		0			0
$155$	151599.			0.506834
$156$		0			0
$157$	417516.			1.35184			0.675918	−	0.736977i	$-0.263747\pi$
$157$	417516.			1.35184			0.675918	+	0.736977i	$0.263747\pi$
$158$		0			0
$159$	−623235.	−	78775.4i	−1.95506	−	0.247114i
$160$		0			0
$161$			198192.i			0.602590i
$162$		0			0
$163$		−	247821.i		−	0.730582i	−0.930893	−	0.365291i	$-0.880969\pi$
$163$		−	247821.i		−	0.730582i	0.930893	−	0.365291i	$-0.119031\pi$
$164$		0			0
$165$	−323341.	−	40869.5i	−0.924593	−	0.116866i
$166$		0			0
$167$	446665.			1.23934			0.619671	−	0.784862i	$-0.287266\pi$
$167$	446665.			1.23934			0.619671	+	0.784862i	$0.287266\pi$
$168$		0			0
$169$	−344110.			−0.926789
$170$		0			0
$171$	−32352.5	+	125935.i	−0.0846092	+	0.329348i
$172$		0			0
$173$		−	444308.i		−	1.12868i	−0.825544	−	0.564338i	$-0.809132\pi$
$173$		−	444308.i		−	1.12868i	0.825544	−	0.564338i	$-0.190868\pi$
$174$		0			0
$175$		−	95562.0i		−	0.235880i
$176$		0			0
$177$	74474.3	−	589207.i	0.178679	−	1.41363i
$178$		0			0
$179$	−136247.			−0.317830			−0.158915	−	0.987292i	$-0.550800\pi$
$179$	−136247.			−0.317830			−0.158915	+	0.987292i	$0.550800\pi$
$180$		0			0
$181$	−333288.			−0.756177			−0.378089	−	0.925769i	$-0.623419\pi$
$181$	−333288.			−0.756177			−0.378089	+	0.925769i	$0.623419\pi$
$182$		0			0
$183$	−85817.9	+	678952.i	−0.189431	+	1.49869i
$184$		0			0
$185$			249700.i			0.536401i
$186$		0			0
$187$		−	528434.i		−	1.10506i
$188$		0			0
$189$	−172564.	−	68363.0i	−0.351394	−	0.139209i
$190$		0			0
$191$	−525292.			−1.04188			−0.520940	−	0.853593i	$-0.674418\pi$
$191$	−525292.			−1.04188			−0.520940	+	0.853593i	$0.674418\pi$
$192$		0			0
$193$	−579015.			−1.11891			−0.559457	−	0.828859i	$-0.688990\pi$
$193$	−579015.			−1.11891			−0.559457	+	0.828859i	$0.688990\pi$
$194$		0			0
$195$	87393.9	+	11046.4i	0.164587	+	0.0208034i
$196$		0			0
$197$			779918.i			1.43180i	0.698201	+	0.715902i	$0.253984\pi$
$197$			779918.i			1.43180i	−0.698201	+	0.715902i	$0.746016\pi$
$198$		0			0
$199$			146631.i			0.262477i	0.991351	+	0.131239i	$0.0418954\pi$
$199$			146631.i			0.262477i	−0.991351	+	0.131239i	$0.958105\pi$
$200$		0			0
$201$	69678.6	+	8807.20i	0.121649	+	0.0153762i
$202$		0			0
$203$	−353323.			−0.601772
$204$		0			0
$205$	102858.			0.170943
$206$		0			0
$207$	951960.	+	244558.i	1.54416	+	0.396694i
$208$		0			0
$209$		−	326394.i		−	0.516865i
$210$		0			0
$211$			256593.i			0.396769i	0.980124	+	0.198385i	$0.0635695\pi$
$211$			256593.i			0.396769i	−0.980124	+	0.198385i	$0.936431\pi$
$212$		0			0
$213$	4470.30	−	35367.0i	0.00675130	−	0.0534132i
$214$		0			0
$215$	640746.			0.945344
$216$		0			0
$217$	−216729.			−0.312441
$218$		0			0
$219$	−125936.	+	996352.i	−0.177436	+	1.40379i
$220$		0			0
$221$			142827.i			0.196712i
$222$		0			0
$223$			54363.1i			0.0732052i	0.999330	+	0.0366026i	$0.0116536\pi$
$223$			54363.1i			0.0732052i	−0.999330	+	0.0366026i	$0.988346\pi$
$224$		0			0
$225$	−459005.	−	117918.i	−0.604451	−	0.155283i
$226$		0			0
$227$	557039.			0.717499			0.358749	−	0.933434i	$-0.383203\pi$
$227$	557039.			0.717499			0.358749	+	0.933434i	$0.383203\pi$
$228$		0			0
$229$	−712802.			−0.898215			−0.449107	−	0.893478i	$-0.648258\pi$
$229$	−712802.			−0.898215			−0.449107	+	0.893478i	$0.648258\pi$
$230$		0			0
$231$	462257.	+	58428.1i	0.569972	+	0.0720430i
$232$		0			0
$233$		−	660799.i		−	0.797406i	−0.917080	−	0.398703i	$-0.869461\pi$
$233$		−	660799.i		−	0.797406i	0.917080	−	0.398703i	$-0.130539\pi$
$234$		0			0
$235$		−	31920.0i		−	0.0377045i
$236$		0			0
$237$	−735561.	−	92973.1i	−0.850644	−	0.107519i
$238$		0			0
$239$	91882.9			0.104049			0.0520247	−	0.998646i	$-0.483433\pi$
$239$	91882.9			0.104049			0.0520247	+	0.998646i	$0.483433\pi$
$240$		0			0
$241$	1.31284e6			1.45603			0.728015	−	0.685561i	$-0.240443\pi$
$241$	1.31284e6			1.45603			0.728015	+	0.685561i	$0.240443\pi$
$242$		0			0
$243$	−541296.	+	744505.i	−0.588057	+	0.808820i
$244$		0			0
$245$		−	82293.5i		−	0.0875892i
$246$		0			0
$247$			88219.2i			0.0920070i
$248$		0			0
$249$	−171545.	+	1.35719e6i	−0.175340	+	1.38721i
$250$		0			0
$251$	−817907.			−0.819444			−0.409722	−	0.912210i	$-0.634374\pi$
$251$	−817907.			−0.819444			−0.409722	+	0.912210i	$0.634374\pi$
$252$		0			0
$253$	−2.46727e6			−2.42334
$254$		0			0
$255$	−58041.5	+	459198.i	−0.0558970	+	0.442231i
$256$		0			0
$257$			1.36984e6i			1.29371i	0.762615	+	0.646853i	$0.223915\pi$
$257$			1.36984e6i			1.29371i	−0.762615	+	0.646853i	$0.776085\pi$
$258$		0			0
$259$		−	356978.i		−	0.330668i
$260$		0			0
$261$	−435981.	+	1.69709e6i	−0.396156	+	1.54207i
$262$		0			0
$263$	−1.66364e6			−1.48310			−0.741551	−	0.670897i	$-0.765909\pi$
$263$	−1.66364e6			−1.48310			−0.741551	+	0.670897i	$0.765909\pi$
$264$		0			0
$265$	−1.38122e6			−1.20823
$266$		0			0
$267$	198107.	+	25040.2i	0.170067	+	0.0214961i
$268$		0			0
$269$		−	1.34836e6i		−	1.13612i	−0.822986	−	0.568062i	$-0.807693\pi$
$269$		−	1.34836e6i		−	1.13612i	0.822986	−	0.568062i	$-0.192307\pi$
$270$		0			0
$271$		−	1.70813e6i		−	1.41285i	−0.707786	−	0.706427i	$-0.750306\pi$
$271$		−	1.70813e6i		−	1.41285i	0.707786	−	0.706427i	$-0.249694\pi$
$272$		0			0
$273$	−124941.	−	15792.2i	−0.101461	−	0.0128244i
$274$		0			0
$275$	1.18964e6			0.948600
$276$		0			0
$277$	−1.98772e6			−1.55653			−0.778264	−	0.627938i	$-0.783899\pi$
$277$	−1.98772e6			−1.55653			−0.778264	+	0.627938i	$0.783899\pi$
$278$		0			0
$279$	−267432.	+	1.04100e6i	−0.205685	+	0.800644i
$280$		0			0
$281$		−	1.40701e6i		−	1.06300i	−0.847060	−	0.531498i	$-0.821629\pi$
$281$		−	1.40701e6i		−	1.06300i	0.847060	−	0.531498i	$-0.178371\pi$
$282$		0			0
$283$		−	2.43397e6i		−	1.80654i	−0.429070	−	0.903271i	$-0.641158\pi$
$283$		−	2.43397e6i		−	1.80654i	0.429070	−	0.903271i	$-0.358842\pi$
$284$		0			0
$285$	−35850.1	+	283630.i	−0.0261444	+	0.206842i
$286$		0			0
$287$	−147048.			−0.105379
$288$		0			0
$289$	669392.			0.471450
$290$		0			0
$291$	−166670.	+	1.31862e6i	−0.115379	+	0.912823i
$292$		0			0
$293$			1.26082e6i			0.857997i	0.903305	+	0.428998i	$0.141133\pi$
$293$			1.26082e6i			0.857997i	−0.903305	+	0.428998i	$0.858867\pi$
$294$		0			0
$295$		−	1.30581e6i		−	0.873624i
$296$		0			0
$297$	851041.	−	2.14822e6i	0.559834	−	1.41315i
$298$		0			0
$299$	666863.			0.431379
$300$		0			0
$301$	−916027.			−0.582763
$302$		0			0
$303$	1.56432e6	+	197727.i	0.978859	+	0.123725i
$304$		0			0
$305$			1.50471e6i			0.926194i
$306$		0			0
$307$		−	534646.i		−	0.323758i	−0.986811	−	0.161879i	$-0.948245\pi$
$307$		−	534646.i		−	0.323758i	0.986811	−	0.161879i	$-0.0517554\pi$
$308$		0			0
$309$	−688170.	−	86983.0i	−0.410015	−	0.0518249i
$310$		0			0
$311$	−1.32628e6			−0.777563			−0.388781	−	0.921330i	$-0.627104\pi$
$311$	−1.32628e6			−0.777563			−0.388781	+	0.921330i	$0.627104\pi$
$312$		0			0
$313$	1.09046e6			0.629140			0.314570	−	0.949234i	$-0.398140\pi$
$313$	1.09046e6			0.629140			0.314570	+	0.949234i	$0.398140\pi$
$314$		0			0
$315$	−395274.	−	101546.i	−0.224451	−	0.0576613i
$316$		0			0
$317$		−	41654.7i		−	0.0232818i	−0.999932	−	0.0116409i	$-0.996295\pi$
$317$		−	41654.7i		−	0.0232818i	0.999932	−	0.0116409i	$-0.00370549\pi$
$318$		0			0
$319$		−	4.39847e6i		−	2.42006i
$320$		0			0
$321$	274619.	−	2.17266e6i	0.148754	−	1.17687i
$322$		0			0
$323$	−463534.			−0.247216
$324$		0			0
$325$	−321540.			−0.168860
$326$		0			0
$327$	−46801.3	+	370271.i	−0.0242041	+	0.191492i
$328$		0			0
$329$			45633.6i			0.0232432i
$330$		0			0
$331$			2.21425e6i			1.11085i	0.831565	+	0.555427i	$0.187445\pi$
$331$			2.21425e6i			1.11085i	−0.831565	+	0.555427i	$0.812555\pi$
$332$		0			0
$333$	−1.71464e6	−	440491.i	−0.847350	−	0.217684i
$334$		0			0
$335$	154423.			0.0751795
$336$		0			0
$337$	1.26033e6			0.604518			0.302259	−	0.953226i	$-0.402259\pi$
$337$	1.26033e6			0.604518			0.302259	+	0.953226i	$0.402259\pi$
$338$		0			0
$339$	−67936.4	−	8587.00i	−0.0321073	−	0.00405828i
$340$		0			0
$341$		−	2.69803e6i		−	1.25650i
$342$		0			0
$343$			117649.i			0.0539949i
$344$		0			0
$345$	2.14400e6	+	270997.i	0.969790	+	0.122579i
$346$		0			0
$347$	1.32678e6			0.591530			0.295765	−	0.955261i	$-0.404426\pi$
$347$	1.32678e6			0.591530			0.295765	+	0.955261i	$0.404426\pi$
$348$		0			0
$349$	−236837.			−0.104085			−0.0520423	−	0.998645i	$-0.516573\pi$
$349$	−236837.			−0.104085			−0.0520423	+	0.998645i	$0.516573\pi$
$350$		0			0
$351$	−230023.	+	580630.i	−0.0996560	+	0.251554i
$352$		0			0
$353$			2.95509e6i			1.26221i	0.775696	+	0.631107i	$0.217399\pi$
$353$			2.95509e6i			1.26221i	−0.775696	+	0.631107i	$0.782601\pi$
$354$		0			0
$355$		−	78380.8i		−	0.0330095i
$356$		0			0
$357$	82977.7	−	656482.i	0.0344581	−	0.272616i
$358$		0			0
$359$	−1.11486e6			−0.456544			−0.228272	−	0.973597i	$-0.573308\pi$
$359$	−1.11486e6			−0.456544			−0.228272	+	0.973597i	$0.573308\pi$
$360$		0			0
$361$	2.18979e6			0.884371
$362$		0			0
$363$	−412543.	+	3.26385e6i	−0.164325	+	1.30006i
$364$		0			0
$365$			2.20813e6i			0.867547i
$366$		0			0
$367$		−	3.40638e6i		−	1.32016i	−0.751194	−	0.660081i	$-0.770522\pi$
$367$		−	3.40638e6i		−	1.32016i	0.751194	−	0.660081i	$-0.229478\pi$
$368$		0			0
$369$	−181449.	+	706303.i	−0.0693727	+	0.270038i
$370$		0			0
$371$	1.97463e6			0.744821
$372$		0			0
$373$	−1.16530e6			−0.433677			−0.216838	−	0.976207i	$-0.569574\pi$
$373$	−1.16530e6			−0.433677			−0.216838	+	0.976207i	$0.569574\pi$
$374$		0			0
$375$	−2.69025e6	−	340040.i	−0.987902	−	0.124868i
$376$		0			0
$377$			1.18884e6i			0.430794i
$378$		0			0
$379$		−	2.23745e6i		−	0.800121i	−0.916489	−	0.400060i	$-0.868989\pi$
$379$		−	2.23745e6i		−	0.800121i	0.916489	−	0.400060i	$-0.131011\pi$
$380$		0			0
$381$	870908.	+	110081.i	0.307369	+	0.0388507i
$382$		0			0
$383$	−915237.			−0.318813			−0.159407	−	0.987213i	$-0.550958\pi$
$383$	−915237.			−0.318813			−0.159407	+	0.987213i	$0.550958\pi$
$384$		0			0
$385$	1.02446e6			0.352244
$386$		0			0
$387$	−1.13033e6	+	4.39988e6i	−0.383642	+	1.49336i
$388$		0			0
$389$		−	3.79183e6i		−	1.27050i	−0.772307	−	0.635249i	$-0.780897\pi$
$389$		−	3.79183e6i		−	1.27050i	0.772307	−	0.635249i	$-0.219103\pi$
$390$		0			0
$391$			3.50393e6i			1.15908i
$392$		0			0
$393$	−324631.	+	2.56833e6i	−0.106025	+	0.838822i
$394$		0			0
$395$	−1.63016e6			−0.525700
$396$		0			0
$397$	990144.			0.315299			0.157649	−	0.987495i	$-0.449608\pi$
$397$	990144.			0.315299			0.157649	+	0.987495i	$0.449608\pi$
$398$		0			0
$399$	51252.2	−	405484.i	0.0161169	−	0.127509i
$400$		0			0
$401$		−	3.36375e6i		−	1.04463i	−0.852752	−	0.522315i	$-0.825068\pi$
$401$		−	3.36375e6i		−	1.04463i	0.852752	−	0.522315i	$-0.174932\pi$
$402$		0			0
$403$			729236.i			0.223669i
$404$		0			0
$405$	−975491.	+	1.77328e6i	−0.295519	+	0.537206i
$406$		0			0
$407$	4.44397e6			1.32980
$408$		0			0
$409$	4.10276e6			1.21274			0.606370	−	0.795182i	$-0.292625\pi$
$409$	4.10276e6			1.21274			0.606370	+	0.795182i	$0.292625\pi$
$410$		0			0
$411$	5.00262e6	+	632318.i	1.46081	+	0.184642i
$412$		0			0
$413$			1.86682e6i			0.538551i
$414$		0			0
$415$			3.00783e6i			0.857299i
$416$		0			0
$417$	3.84210e6	+	485632.i	1.08200	+	0.136763i
$418$		0			0
$419$	−2.10585e6			−0.585993			−0.292996	−	0.956114i	$-0.594652\pi$
$419$	−2.10585e6			−0.585993			−0.292996	+	0.956114i	$0.594652\pi$
$420$		0			0
$421$	3.43459e6			0.944430			0.472215	−	0.881483i	$-0.343455\pi$
$421$	3.43459e6			0.944430			0.472215	+	0.881483i	$0.343455\pi$
$422$		0			0
$423$	219188.	+	56309.3i	0.0595616	+	0.0153013i
$424$		0			0
$425$		−	1.68949e6i		−	0.453714i
$426$		0			0
$427$		−	2.15117e6i		−	0.570958i
$428$		0			0
$429$	196595.	−	1.55537e6i	0.0515738	−	0.408028i
$430$		0			0
$431$	3.16234e6			0.820003			0.410002	−	0.912085i	$-0.365528\pi$
$431$	3.16234e6			0.820003			0.410002	+	0.912085i	$0.365528\pi$
$432$		0			0
$433$	6.76256e6			1.73337			0.866685	−	0.498856i	$-0.166246\pi$
$433$	6.76256e6			1.73337			0.866685	+	0.498856i	$0.166246\pi$
$434$		0			0
$435$	−483114.	+	3.82218e6i	−0.122413	+	0.968474i
$436$		0			0
$437$			2.16425e6i			0.542130i
$438$		0			0
$439$		−	4.95019e6i		−	1.22592i	−0.790115	−	0.612958i	$-0.789979\pi$
$439$		−	4.95019e6i		−	1.22592i	0.790115	−	0.612958i	$-0.210021\pi$
$440$		0			0
$441$	565094.	+	145172.i	0.138364	+	0.0355457i
$442$		0			0
$443$	7.71755e6			1.86840			0.934200	−	0.356749i	$-0.116115\pi$
$443$	7.71755e6			1.86840			0.934200	+	0.356749i	$0.116115\pi$
$444$		0			0
$445$	439047.			0.105102
$446$		0			0
$447$	−2.35604e6	−	297797.i	−0.557717	−	0.0704940i
$448$		0			0
$449$			760145.i			0.177943i	0.996034	+	0.0889715i	$0.0283580\pi$
$449$			760145.i			0.177943i	−0.996034	+	0.0889715i	$0.971642\pi$
$450$		0			0
$451$		−	1.83058e6i		−	0.423787i
$452$		0			0
$453$	2.49824e6	+	315771.i	0.571990	+	0.0722981i
$454$		0			0
$455$	−276896.			−0.0627029
$456$		0			0
$457$	4.11280e6			0.921186			0.460593	−	0.887612i	$-0.347637\pi$
$457$	4.11280e6			0.921186			0.460593	+	0.887612i	$0.347637\pi$
$458$		0			0
$459$	−3.05083e6	−	1.20862e6i	−0.675907	−	0.267768i
$460$		0			0
$461$		−	237892.i		−	0.0521347i	−0.999660	−	0.0260674i	$-0.991702\pi$
$461$		−	237892.i		−	0.0521347i	0.999660	−	0.0260674i	$-0.00829844\pi$
$462$		0			0
$463$			5.40815e6i			1.17245i	0.810147	+	0.586227i	$0.199387\pi$
$463$			5.40815e6i			1.17245i	−0.810147	+	0.586227i	$0.800613\pi$
$464$		0			0
$465$	−296343.	+	2.34453e6i	−0.0635569	+	0.502833i
$466$		0			0
$467$	−8.00905e6			−1.69937			−0.849687	−	0.527287i	$-0.823209\pi$
$467$	−8.00905e6			−1.69937			−0.849687	+	0.527287i	$0.823209\pi$
$468$		0			0
$469$	−220767.			−0.0463449
$470$		0			0
$471$	−816156.	+	6.45705e6i	−0.169520	+	1.34116i
$472$		0			0
$473$		−	1.14035e7i		−	2.34361i
$474$		0			0
$475$		−	1.04353e6i		−	0.212213i
$476$		0			0
$477$	2.43659e6	−	9.48460e6i	0.490327	−	1.90863i
$478$		0			0
$479$	−690366.			−0.137480			−0.0687401	−	0.997635i	$-0.521898\pi$
$479$	−690366.			−0.137480			−0.0687401	+	0.997635i	$0.521898\pi$
$480$		0			0
$481$	−1.20113e6			−0.236717
$482$		0			0
$483$	−3.06512e6	−	387424.i	−0.597833	−	0.0755646i
$484$		0			0
$485$			2.92234e6i			0.564127i
$486$		0			0
$487$			305940.i			0.0584539i	0.999573	+	0.0292270i	$0.00930456\pi$
$487$			305940.i			0.0584539i	−0.999573	+	0.0292270i	$0.990695\pi$
$488$		0			0
$489$	3.83265e6	+	484438.i	0.724815	+	0.0916148i
$490$		0			0
$491$	−5.87843e6			−1.10042			−0.550209	−	0.835027i	$-0.685452\pi$
$491$	−5.87843e6			−1.10042			−0.550209	+	0.835027i	$0.685452\pi$
$492$		0			0
$493$	−6.24657e6			−1.15751
$494$		0			0
$495$	1.26413e6	−	4.92071e6i	0.231888	−	0.902640i
$496$		0			0
$497$			112055.i			0.0203489i
$498$		0			0
$499$			3.48586e6i			0.626699i	0.949638	+	0.313349i	$0.101451\pi$
$499$			3.48586e6i			0.626699i	−0.949638	+	0.313349i	$0.898549\pi$
$500$		0			0
$501$	−873136.	+	6.90786e6i	−0.155413	+	1.22956i
$502$		0			0
$503$	4.79530e6			0.845077			0.422538	−	0.906345i	$-0.361139\pi$
$503$	4.79530e6			0.845077			0.422538	+	0.906345i	$0.361139\pi$
$504$		0			0
$505$	3.46688e6			0.604937
$506$		0			0
$507$	672663.	−	5.32181e6i	0.116219	−	0.919473i
$508$		0			0
$509$			4.61364e6i			0.789313i	0.918829	+	0.394657i	$0.129136\pi$
$509$			4.61364e6i			0.789313i	−0.918829	+	0.394657i	$0.870864\pi$
$510$		0			0
$511$		−	3.15680e6i		−	0.534805i
$512$		0			0
$513$	−1.88439e6	−	746520.i	−0.316138	−	0.125242i
$514$		0			0
$515$	−1.52513e6			−0.253390
$516$		0			0
$517$	−568087.			−0.0934735
$518$		0			0
$519$	6.87140e6	+	868529.i	1.11977	+	0.141536i
$520$		0			0
$521$		−	4.20268e6i		−	0.678316i	−0.940730	−	0.339158i	$-0.889858\pi$
$521$		−	4.20268e6i		−	0.678316i	0.940730	−	0.339158i	$-0.110142\pi$
$522$		0			0
$523$		−	1.05219e7i		−	1.68205i	−0.540994	−	0.841026i	$-0.681952\pi$
$523$		−	1.05219e7i		−	1.68205i	0.540994	−	0.841026i	$-0.318048\pi$
$524$		0			0
$525$	1.47791e6	+	186804.i	0.234018	+	0.0295792i
$526$		0			0
$527$	−3.83166e6			−0.600980
$528$		0			0
$529$	9.92357e6			1.54180
$530$		0			0
$531$	8.96674e6	+	2.30355e6i	1.38006	+	0.354537i
$532$		0			0
$533$			494777.i			0.0754382i
$534$		0			0
$535$		−	4.81509e6i		−	0.727310i
$536$		0			0
$537$	266334.	−	2.10711e6i	0.0398558	−	0.315321i
$538$		0			0
$539$	−1.46460e6			−0.217143
$540$		0			0
$541$	−1.03899e7			−1.52623			−0.763114	−	0.646264i	$-0.776330\pi$
$541$	−1.03899e7			−1.52623			−0.763114	+	0.646264i	$0.776330\pi$
$542$		0			0
$543$	651508.	−	5.15444e6i	0.0948244	−	0.750208i
$544$		0			0
$545$			820600.i			0.118342i
$546$		0			0
$547$			2.99120e6i			0.427441i	0.976895	+	0.213721i	$0.0685582\pi$
$547$			2.99120e6i			0.427441i	−0.976895	+	0.213721i	$0.931442\pi$
$548$		0			0
$549$	−1.03325e7	−	2.65442e6i	−1.46310	−	0.375871i
$550$		0			0
$551$	−3.85827e6			−0.541395
$552$		0			0
$553$	2.33052e6			0.324071
$554$		0			0
$555$	−3.86171e6	−	488111.i	−0.532167	−	0.0672646i
$556$		0			0
$557$			9.46974e6i			1.29330i	0.762786	+	0.646651i	$0.223831\pi$
$557$			9.46974e6i			1.29330i	−0.762786	+	0.646651i	$0.776169\pi$
$558$		0			0
$559$			3.08219e6i			0.417185i
$560$		0			0
$561$	8.17245e6	+	1.03298e6i	1.09634	+	0.138575i
$562$		0			0
$563$	1.42689e7			1.89723			0.948613	−	0.316439i	$-0.102487\pi$
$563$	1.42689e7			1.89723			0.948613	+	0.316439i	$0.102487\pi$
$564$		0			0
$565$	−150562.			−0.0198424
$566$		0			0
$567$	1.39459e6	−	2.53513e6i	0.182175	−	0.331164i
$568$		0			0
$569$			1.12931e7i			1.46228i	0.682225	+	0.731142i	$0.261012\pi$
$569$			1.12931e7i			1.46228i	−0.682225	+	0.731142i	$0.738988\pi$
$570$		0			0
$571$		−	1.15403e7i		−	1.48124i	−0.671922	−	0.740621i	$-0.734531\pi$
$571$		−	1.15403e7i		−	1.48124i	0.671922	−	0.740621i	$-0.265469\pi$
$572$		0			0
$573$	1.02684e6	−	8.12386e6i	0.130651	−	1.03366i
$574$		0			0
$575$	−7.88823e6			−0.994970
$576$		0			0
$577$	−8.17134e6			−1.02177			−0.510886	−	0.859649i	$-0.670683\pi$
$577$	−8.17134e6			−1.02177			−0.510886	+	0.859649i	$0.670683\pi$
$578$		0			0
$579$	1.13185e6	−	8.95471e6i	0.140312	−	1.11008i
$580$		0			0
$581$		−	4.30007e6i		−	0.528488i
$582$		0			0
$583$			2.45820e7i			2.99533i
$584$		0			0
$585$	−341673.	+	1.32999e6i	−0.0412783	+	0.160679i
$586$		0			0
$587$	−1.14971e7			−1.37719			−0.688593	−	0.725148i	$-0.741771\pi$
$587$	−1.14971e7			−1.37719			−0.688593	+	0.725148i	$0.741771\pi$
$588$		0			0
$589$	−2.36667e6			−0.281093
$590$		0			0
$591$	−1.20617e7	−	1.52458e6i	−1.42050	−	0.179548i
$592$		0			0
$593$		−	5.38468e6i		−	0.628815i	−0.949288	−	0.314408i	$-0.898194\pi$
$593$		−	5.38468e6i		−	0.628815i	0.949288	−	0.314408i	$-0.101806\pi$
$594$		0			0
$595$		−	1.45491e6i		−	0.168478i
$596$		0			0
$597$	−2.26770e6	−	286632.i	−0.260406	−	0.0329146i
$598$		0			0
$599$	6.14219e6			0.699449			0.349724	−	0.936853i	$-0.386275\pi$
$599$	6.14219e6			0.699449			0.349724	+	0.936853i	$0.386275\pi$
$600$		0			0
$601$	−1.79061e6			−0.202216			−0.101108	−	0.994875i	$-0.532239\pi$
$601$	−1.79061e6			−0.202216			−0.101108	+	0.994875i	$0.532239\pi$
$602$		0			0
$603$	−272414.	+	1.06039e6i	−0.0305096	+	0.118761i
$604$		0			0
$605$			7.23340e6i			0.803441i
$606$		0			0
$607$		−	121847.i		−	0.0134228i	−0.999977	−	0.00671138i	$-0.997864\pi$
$607$		−	121847.i		−	0.0134228i	0.999977	−	0.00671138i	$-0.00213631\pi$
$608$		0			0
$609$	690673.	−	5.46429e6i	0.0754621	−	0.597022i
$610$		0			0
$611$	153545.			0.0166392
$612$		0			0
$613$	−1.07351e6			−0.115386			−0.0576932	−	0.998334i	$-0.518375\pi$
$613$	−1.07351e6			−0.115386			−0.0576932	+	0.998334i	$0.518375\pi$
$614$		0			0
$615$	−201065.	+	1.59074e6i	−0.0214362	+	0.169594i
$616$		0			0
$617$		−	44550.3i		−	0.00471126i	−0.999997	−	0.00235563i	$-0.999250\pi$
$617$		−	44550.3i		−	0.00471126i	0.999997	−	0.00235563i	$-0.000749821\pi$
$618$		0			0
$619$		−	1.34292e7i		−	1.40871i	−0.709847	−	0.704356i	$-0.751236\pi$
$619$		−	1.34292e7i		−	1.40871i	0.709847	−	0.704356i	$-0.248764\pi$
$620$		0			0
$621$	−5.64307e6	+	1.42444e7i	−0.587200	+	1.48223i
$622$		0			0
$623$	−627674.			−0.0647909
$624$		0			0
$625$	132350.			0.0135526
$626$		0			0
$627$	5.04782e6	+	638032.i	0.512785	+	0.0648147i
$628$		0			0
$629$		−	6.31118e6i		−	0.636039i
$630$		0			0
$631$			1.55136e7i			1.55110i	0.631286	+	0.775551i	$0.282528\pi$
$631$			1.55136e7i			1.55110i	−0.631286	+	0.775551i	$0.717472\pi$
$632$		0			0
$633$	−3.96831e6	−	501584.i	−0.393637	−	0.0497548i
$634$		0			0
$635$	1.93012e6			0.189955
$636$		0			0
$637$	395857.			0.0386536
$638$		0			0
$639$	538226.	+	138270.i	0.0521450	+	0.0133960i
$640$		0			0
$641$			6.58501e6i			0.633011i	0.948591	+	0.316505i	$0.102510\pi$
$641$			6.58501e6i			0.633011i	−0.948591	+	0.316505i	$0.897490\pi$
$642$		0			0
$643$			1.87288e7i			1.78642i	0.449644	+	0.893208i	$0.351551\pi$
$643$			1.87288e7i			1.78642i	−0.449644	+	0.893208i	$0.648449\pi$
$644$		0			0
$645$	−1.25252e6	+	9.90940e6i	−0.118546	+	0.937882i
$646$		0			0
$647$	−6.56023e6			−0.616110			−0.308055	−	0.951369i	$-0.599678\pi$
$647$	−6.56023e6			−0.616110			−0.308055	+	0.951369i	$0.599678\pi$
$648$		0			0
$649$	−2.32398e7			−2.16581
$650$		0			0
$651$	423660.	−	3.35181e6i	0.0391801	−	0.309975i
$652$		0			0
$653$		−	1.02071e7i		−	0.936739i	−0.883533	−	0.468369i	$-0.844842\pi$
$653$		−	1.02071e7i		−	0.936739i	0.883533	−	0.468369i	$-0.155158\pi$
$654$		0			0
$655$			5.69198e6i			0.518394i
$656$		0			0
$657$	−1.51628e7	−	3.89532e6i	−1.37046	−	0.352070i
$658$		0			0
$659$	7.37359e6			0.661402			0.330701	−	0.943736i	$-0.392715\pi$
$659$	7.37359e6			0.661402			0.330701	+	0.943736i	$0.392715\pi$
$660$		0			0
$661$	−1.86568e7			−1.66086			−0.830430	−	0.557124i	$-0.811905\pi$
$661$	−1.86568e7			−1.66086			−0.830430	+	0.557124i	$0.811905\pi$
$662$		0			0
$663$	−2.20888e6	−	279198.i	−0.195159	−	0.0246676i
$664$		0			0
$665$		−	898642.i		−	0.0788011i
$666$		0			0
$667$			2.91653e7i			2.53835i
$668$		0			0
$669$	−840747.	−	106268.i	−0.0726274	−	0.00917992i
$670$		0			0
$671$	2.67796e7			2.29614
$672$		0			0
$673$	−4.10422e6			−0.349295			−0.174648	−	0.984631i	$-0.555879\pi$
$673$	−4.10422e6			−0.349295			−0.174648	+	0.984631i	$0.555879\pi$
$674$		0			0
$675$	2.72091e6	−	6.86820e6i	0.229855	−	0.580207i
$676$		0			0
$677$		−	6.59602e6i		−	0.553109i	−0.960998	−	0.276554i	$-0.910807\pi$
$677$		−	6.59602e6i		−	0.553109i	0.960998	−	0.276554i	$-0.0891926\pi$
$678$		0			0
$679$		−	4.17786e6i		−	0.347760i
$680$		0			0
$681$	−1.08889e6	+	8.61484e6i	−0.0899742	+	0.711835i
$682$		0			0
$683$	−1.20505e7			−0.988449			−0.494225	−	0.869334i	$-0.664548\pi$
$683$	−1.20505e7			−0.988449			−0.494225	+	0.869334i	$0.664548\pi$
$684$		0			0
$685$	1.10869e7			0.902782
$686$		0			0
$687$	1.39338e6	−	1.10238e7i	0.112636	−	0.891124i
$688$		0			0
$689$		−	6.64411e6i		−	0.533198i
$690$		0			0
$691$			1.13116e6i			0.0901216i	0.998984	+	0.0450608i	$0.0143482\pi$
$691$			1.13116e6i			0.0901216i	−0.998984	+	0.0450608i	$0.985652\pi$
$692$		0			0
$693$	−1.80723e6	+	7.03477e6i	−0.142949	+	0.556438i
$694$		0			0
$695$	8.51492e6			0.668681
$696$		0			0
$697$	−2.59973e6			−0.202697
$698$		0			0
$699$	1.02195e7	+	1.29172e6i	0.791111	+	0.0999945i
$700$		0			0
$701$		−	1.77396e7i		−	1.36348i	−0.731596	−	0.681739i	$-0.761224\pi$
$701$		−	1.77396e7i		−	1.36348i	0.731596	−	0.681739i	$-0.238776\pi$
$702$		0			0
$703$		−	3.89818e6i		−	0.297491i
$704$		0			0
$705$	493656.	+	62396.8i	0.0374069	+	0.00472813i
$706$		0			0
$707$	−4.95634e6			−0.372917
$708$		0			0
$709$	−2.08646e7			−1.55882			−0.779409	−	0.626516i	$-0.784480\pi$
$709$	−2.08646e7			−1.55882			−0.779409	+	0.626516i	$0.784480\pi$
$710$		0			0
$711$	2.87573e6	−	1.11940e7i	0.213341	−	0.830446i
$712$		0			0
$713$			1.78901e7i			1.31792i
$714$		0			0
$715$		−	3.44704e6i		−	0.252163i
$716$		0			0
$717$	−179612.	+	1.42101e6i	−0.0130478	+	0.103228i
$718$		0			0
$719$	−3.18080e6			−0.229464			−0.114732	−	0.993396i	$-0.536601\pi$
$719$	−3.18080e6			−0.229464			−0.114732	+	0.993396i	$0.536601\pi$
$720$		0			0
$721$	2.18037e6			0.156204
$722$		0			0
$723$	−2.56633e6	+	2.03037e7i	−0.182586	+	1.44454i
$724$		0			0
$725$		−	1.40626e7i		−	0.993620i
$726$		0			0
$727$			1.95894e7i			1.37462i	0.726362	+	0.687312i	$0.241210\pi$
$727$			1.95894e7i			1.37462i	−0.726362	+	0.687312i	$0.758790\pi$
$728$		0			0
$729$	−1.04559e7	−	9.82671e6i	−0.728693	−	0.684840i
$730$		0			0
$731$	−1.61949e7			−1.12094
$732$		0			0
$733$	1.75977e7			1.20975			0.604874	−	0.796321i	$-0.293223\pi$
$733$	1.75977e7			1.20975			0.604874	+	0.796321i	$0.293223\pi$
$734$		0			0
$735$	1.27270e6	+	160867.i	0.0868978	+	0.0109837i
$736$		0			0
$737$		−	2.74830e6i		−	0.186378i
$738$		0			0
$739$		−	1.68232e7i		−	1.13317i	−0.824002	−	0.566587i	$-0.808263\pi$
$739$		−	1.68232e7i		−	1.13317i	0.824002	−	0.566587i	$-0.191737\pi$
$740$		0			0
$741$	−1.36435e6	−	172450.i	−0.0912807	−	0.0115377i
$742$		0			0
$743$	−9.00427e6			−0.598379			−0.299190	−	0.954194i	$-0.596716\pi$
$743$	−9.00427e6			−0.598379			−0.299190	+	0.954194i	$0.596716\pi$
$744$		0			0
$745$	−5.22149e6			−0.344670
$746$		0			0
$747$	−2.06542e7	−	5.30604e6i	−1.35427	−	0.347912i
$748$		0			0
$749$			6.88378e6i			0.448355i
$750$		0			0
$751$			2.44872e7i			1.58431i	0.610321	+	0.792154i	$0.291040\pi$
$751$			2.44872e7i			1.58431i	−0.610321	+	0.792154i	$0.708960\pi$
$752$		0			0
$753$	1.59884e6	−	1.26493e7i	0.102758	−	0.812976i
$754$		0			0
$755$	5.53664e6			0.353491
$756$		0			0
$757$	1.05546e7			0.669427			0.334713	−	0.942320i	$-0.391361\pi$
$757$	1.05546e7			0.669427			0.334713	+	0.942320i	$0.391361\pi$
$758$		0			0
$759$	4.82299e6	−	3.81573e7i	0.303887	−	2.40421i
$760$		0			0
$761$			2.29673e7i			1.43763i	0.695200	+	0.718816i	$0.255316\pi$
$761$			2.29673e7i			1.43763i	−0.695200	+	0.718816i	$0.744684\pi$
$762$		0			0
$763$		−	1.17315e6i		−	0.0729529i
$764$		0			0
$765$	−6.98823e6	−	1.79527e6i	−0.431731	−	0.110911i
$766$		0			0
$767$	6.28134e6			0.385535
$768$		0			0
$769$	4.94280e6			0.301410			0.150705	−	0.988579i	$-0.451846\pi$
$769$	4.94280e6			0.301410			0.150705	+	0.988579i	$0.451846\pi$
$770$		0			0
$771$	−2.11851e7	−	2.67774e6i	−1.28349	−	0.162230i
$772$		0			0
$773$			2.76210e7i			1.66261i	0.555814	+	0.831307i	$0.312407\pi$
$773$			2.76210e7i			1.66261i	−0.555814	+	0.831307i	$0.687593\pi$
$774$		0			0
$775$		−	8.62603e6i		−	0.515889i
$776$		0			0
$777$	5.52081e6	+	697817.i	0.328058	+	0.0414657i
$778$		0			0
$779$	−1.60576e6			−0.0948061
$780$		0			0
$781$	−1.39496e6			−0.0818342
$782$		0			0
$783$	−2.53939e7	−	1.00601e7i	−1.48022	−	0.586404i
$784$		0			0
$785$			1.43102e7i			0.828843i
$786$		0			0
$787$		−	1.31437e7i		−	0.756448i	−0.925714	−	0.378224i	$-0.876535\pi$
$787$		−	1.31437e7i		−	0.756448i	0.925714	−	0.378224i	$-0.123465\pi$
$788$		0			0
$789$	3.25207e6	−	2.57289e7i	0.185981	−	1.47139i
$790$		0			0
$791$	215247.			0.0122320
$792$		0			0
$793$	−7.23810e6			−0.408735
$794$		0			0
$795$	2.70000e6	−	2.13612e7i	0.151512	−	1.19869i
$796$		0			0
$797$			4.21837e6i			0.235233i	0.993059	+	0.117617i	$0.0375254\pi$
$797$			4.21837e6i			0.235233i	−0.993059	+	0.117617i	$0.962475\pi$
$798$		0			0
$799$			806778.i			0.0447082i
$800$		0			0
$801$	−774514.	+	3.01485e6i	−0.0426528	+	0.166029i
$802$		0			0
$803$	3.92986e7			2.15074
$804$		0			0
$805$	−6.79298e6			−0.369463
$806$		0			0
$807$	2.08530e7	+	2.63576e6i	1.12716	+	0.142470i
$808$		0			0
$809$		−	4.42530e6i		−	0.237723i	−0.992911	−	0.118862i	$-0.962076\pi$
$809$		−	4.42530e6i		−	0.237723i	0.992911	−	0.118862i	$-0.0379245\pi$
$810$		0			0
$811$			2.95635e7i			1.57835i	0.614168	+	0.789175i	$0.289492\pi$
$811$			2.95635e7i			1.57835i	−0.614168	+	0.789175i	$0.710508\pi$
$812$		0			0
$813$	2.64169e7	+	3.33903e6i	1.40170	+	0.177172i
$814$		0			0
$815$	8.49399e6			0.447938
$816$		0			0
$817$	−1.00030e7			−0.524293
$818$		0			0
$819$	488465.	−	1.90139e6i	0.0254463	−	0.0990515i
$820$		0			0
$821$		−	1.06331e7i		−	0.550557i	−0.961365	−	0.275278i	$-0.911230\pi$
$821$		−	1.06331e7i		−	0.550557i	0.961365	−	0.275278i	$-0.0887700\pi$
$822$		0			0
$823$			4.52511e6i			0.232879i	0.993198	+	0.116439i	$0.0371481\pi$
$823$			4.52511e6i			0.232879i	−0.993198	+	0.116439i	$0.962852\pi$
$824$		0			0
$825$	−2.32549e6	+	1.83982e7i	−0.118954	+	0.941112i
$826$		0			0
$827$	2.52748e7			1.28506			0.642531	−	0.766260i	$-0.277884\pi$
$827$	2.52748e7			1.28506			0.642531	+	0.766260i	$0.277884\pi$
$828$		0			0
$829$	1.80586e7			0.912639			0.456319	−	0.889816i	$-0.349167\pi$
$829$	1.80586e7			0.912639			0.456319	+	0.889816i	$0.349167\pi$
$830$		0			0
$831$	3.88558e6	−	3.07410e7i	0.195188	−	1.54424i
$832$		0			0
$833$			2.07997e6i			0.103859i
$834$		0			0
$835$			1.53093e7i			0.759870i
$836$		0			0
$837$	−1.55767e7	−	6.17087e6i	−0.768531	−	0.304462i
$838$		0			0
$839$	−2.55450e7			−1.25285			−0.626427	−	0.779480i	$-0.715484\pi$
$839$	−2.55450e7			−1.25285			−0.626427	+	0.779480i	$0.715484\pi$
$840$		0			0
$841$	−3.14828e7			−1.53491
$842$		0			0
$843$	2.17600e7	+	2.75041e6i	1.05460	+	0.133299i
$844$		0			0
$845$		−	1.17943e7i		−	0.568237i
$846$		0			0
$847$		−	1.03411e7i		−	0.495287i
$848$		0			0
$849$	3.76423e7	+	4.75789e6i	1.79228	+	0.226540i
$850$		0			0
$851$	−2.94670e7			−1.39480
$852$		0			0
$853$	7.27960e6			0.342559			0.171279	−	0.985223i	$-0.445210\pi$
$853$	7.27960e6			0.342559			0.171279	+	0.985223i	$0.445210\pi$
$854$		0			0
$855$	−4.31637e6	−	1.10887e6i	−0.201931	−	0.0518760i
$856$		0			0
$857$		−	1.03285e7i		−	0.480381i	−0.970726	−	0.240191i	$-0.922790\pi$
$857$		−	1.03285e7i		−	0.480381i	0.970726	−	0.240191i	$-0.0772099\pi$
$858$		0			0
$859$		−	3.05579e7i		−	1.41300i	−0.707715	−	0.706498i	$-0.750274\pi$
$859$		−	3.05579e7i		−	1.41300i	0.707715	−	0.706498i	$-0.249726\pi$
$860$		0			0
$861$	287448.	−	2.27416e6i	0.0132145	−	0.104547i
$862$		0			0
$863$	−2.89837e7			−1.32473			−0.662363	−	0.749183i	$-0.730446\pi$
$863$	−2.89837e7			−1.32473			−0.662363	+	0.749183i	$0.730446\pi$
$864$		0			0
$865$	1.52285e7			0.692018
$866$		0			0
$867$	−1.30852e6	+	1.03524e7i	−0.0591198	+	0.467729i
$868$		0			0
$869$			2.90124e7i			1.30327i
$870$		0			0
$871$			742821.i			0.0331771i
$872$		0			0
$873$	−2.00672e7	−	5.15524e6i	−0.891149	−	0.228936i
$874$		0			0
$875$	8.52367e6			0.376363
$876$		0			0
$877$	3.55930e7			1.56267			0.781333	−	0.624114i	$-0.214540\pi$
$877$	3.55930e7			1.56267			0.781333	+	0.624114i	$0.214540\pi$
$878$		0			0
$879$	−1.94992e7	−	2.46465e6i	−0.851224	−	0.107593i
$880$		0			0
$881$		−	3.76355e7i		−	1.63365i	−0.576888	−	0.816823i	$-0.695733\pi$
$881$		−	3.76355e7i		−	1.63365i	0.576888	−	0.816823i	$-0.304267\pi$
$882$		0			0
$883$			2.66728e7i			1.15124i	0.817717	+	0.575621i	$0.195240\pi$
$883$			2.66728e7i			1.15124i	−0.817717	+	0.575621i	$0.804760\pi$
$884$		0			0
$885$	2.01949e7	+	2.55258e6i	0.866728	+	0.109552i
$886$		0			0
$887$	−1.30365e7			−0.556357			−0.278178	−	0.960529i	$-0.589731\pi$
$887$	−1.30365e7			−0.556357			−0.278178	+	0.960529i	$0.589731\pi$
$888$		0			0
$889$	−2.75935e6			−0.117099
$890$		0			0
$891$	3.15595e7	+	1.73610e7i	1.33179	+	0.732624i
$892$		0			0
$893$			498317.i			0.0209111i
$894$		0			0
$895$		−	4.66982e6i		−	0.194869i
$896$		0			0
$897$	−1.30358e6	+	1.03133e7i	−0.0540948	+	0.427974i
$898$		0			0
$899$	−3.18932e7			−1.31613
$900$		0			0
$901$	3.49105e7			1.43266
$902$		0			0
$903$	1.79064e6	−	1.41667e7i	0.0730784	−	0.578163i
$904$		0			0
$905$		−	1.14234e7i		−	0.463631i
$906$		0			0
$907$			59787.1i			0.00241318i	0.999999	+	0.00120659i	$0.000384069\pi$
$907$			59787.1i			0.00241318i	−0.999999	+	0.00120659i	$0.999616\pi$
$908$		0			0
$909$	−6.11584e6	+	2.38064e7i	−0.245497	+	0.955617i
$910$		0			0
$911$	3.07153e7			1.22619			0.613097	−	0.790008i	$-0.289924\pi$
$911$	3.07153e7			1.22619			0.613097	+	0.790008i	$0.289924\pi$
$912$		0			0
$913$	5.35309e7			2.12534
$914$		0			0
$915$	−2.32709e7	−	2.94138e6i	−0.918883	−	0.116145i
$916$		0			0
$917$		−	8.13740e6i		−	0.319567i
$918$		0			0
$919$			3.92950e7i			1.53479i	0.641175	+	0.767395i	$0.278447\pi$
$919$			3.92950e7i			1.53479i	−0.641175	+	0.767395i	$0.721553\pi$
$920$		0			0
$921$	8.26852e6	+	1.04512e6i	0.321202	+	0.0405992i
$922$		0			0
$923$	377036.			0.0145673
$924$		0			0
$925$	1.42081e7			0.545984
$926$		0			0
$927$	2.69046e6	−	1.04728e7i	0.102832	−	0.400280i
$928$		0			0
$929$		−	8.50979e6i		−	0.323504i	−0.986831	−	0.161752i	$-0.948286\pi$
$929$		−	8.50979e6i		−	0.323504i	0.986831	−	0.161752i	$-0.0517145\pi$
$930$		0			0
$931$			1.28472e6i			0.0485775i
$932$		0			0
$933$	2.59260e6	−	2.05115e7i	0.0975062	−	0.771425i
$934$		0			0
$935$	1.81119e7			0.677541
$936$		0			0
$937$	1.87059e6			0.0696031			0.0348016	−	0.999394i	$-0.488920\pi$
$937$	1.87059e6			0.0696031			0.0348016	+	0.999394i	$0.488920\pi$
$938$		0			0
$939$	−2.13161e6	+	1.68643e7i	−0.0788940	+	0.624174i
$940$		0			0
$941$		−	4.51059e7i		−	1.66058i	−0.557332	−	0.830290i	$-0.688175\pi$
$941$		−	4.51059e7i		−	1.66058i	0.557332	−	0.830290i	$-0.311825\pi$
$942$		0			0
$943$			1.21382e7i			0.444503i
$944$		0			0
$945$	2.34312e6	−	5.91457e6i	0.0853523	−	0.215448i
$946$		0			0
$947$	2.45322e7			0.888917			0.444458	−	0.895799i	$-0.353396\pi$
$947$	2.45322e7			0.888917			0.444458	+	0.895799i	$0.353396\pi$
$948$		0			0
$949$	−1.06218e7			−0.382853
$950$		0			0
$951$	644207.	+	81426.2i	0.0230980	+	0.00291953i
$952$		0			0
$953$		−	3.07970e7i		−	1.09844i	−0.835678	−	0.549219i	$-0.814925\pi$
$953$		−	3.07970e7i		−	1.09844i	0.835678	−	0.549219i	$-0.185075\pi$
$954$		0			0
$955$		−	1.80042e7i		−	0.638802i
$956$		0			0
$957$	6.80242e7	+	8.59809e6i	2.40095	+	0.303474i
$958$		0			0
$959$	−1.58501e7			−0.556526
$960$		0			0
$961$	9.06581e6			0.316663
$962$		0			0
$963$	3.30643e7	+	8.49419e6i	1.14893	+	0.295159i
$964$		0			0
$965$		−	1.98456e7i		−	0.686033i
$966$		0			0
$967$			2.99527e7i			1.03008i	0.857167	+	0.515039i	$0.172223\pi$
$967$			2.99527e7i			1.03008i	−0.857167	+	0.515039i	$0.827777\pi$
$968$		0			0
$969$	906112.	−	7.16875e6i	0.0310008	−	0.245264i
$970$		0			0
$971$	5.30760e6			0.180655			0.0903275	−	0.995912i	$-0.471209\pi$
$971$	5.30760e6			0.180655			0.0903275	+	0.995912i	$0.471209\pi$
$972$		0			0
$973$	−1.21732e7			−0.412213
$974$		0			0
$975$	628544.	−	4.97275e6i	0.0211750	−	0.167527i
$976$		0			0
$977$			4.22136e7i			1.41487i	0.706779	+	0.707434i	$0.250147\pi$
$977$			4.22136e7i			1.41487i	−0.706779	+	0.707434i	$0.749853\pi$
$978$		0			0
$979$		−	7.81383e6i		−	0.260560i
$980$		0			0
$981$	−5.63490e6	−	1.44760e6i	−0.186945	−	0.0480260i
$982$		0			0
$983$	−3.29247e7			−1.08677			−0.543385	−	0.839483i	$-0.682858\pi$
$983$	−3.29247e7			−1.08677			−0.543385	+	0.839483i	$0.682858\pi$
$984$		0			0
$985$	−2.67315e7			−0.877874
$986$		0			0
$987$	−705743.	−	89204.2i	−0.0230597	−	0.00291469i
$988$		0			0
$989$			7.56141e7i			2.45817i
$990$		0			0
$991$			8.64486e6i			0.279624i	0.990178	+	0.139812i	$0.0446498\pi$
$991$			8.64486e6i			0.279624i	−0.990178	+	0.139812i	$0.955350\pi$
$992$		0			0
$993$	−3.42443e7	−	4.32839e6i	−1.10208	−	0.139301i
$994$		0			0
$995$	−5.02572e6			−0.160931
$996$		0			0
$997$	4.70071e7			1.49770			0.748851	−	0.662739i	$-0.230606\pi$
$997$	4.70071e7			1.49770			0.748851	+	0.662739i	$0.230606\pi$
$998$		0			0
$999$	1.01641e7	−	2.56566e7i	0.322223	−	0.813364i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.6.h.a.239.10	yes	20
3.2	odd	2	inner	336.6.h.a.239.12	yes	20
4.3	odd	2	inner	336.6.h.a.239.11	yes	20
12.11	even	2	inner	336.6.h.a.239.9	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.6.h.a.239.9	✓	20	12.11	even	2	inner
336.6.h.a.239.10	yes	20	1.1	even	1	trivial
336.6.h.a.239.11	yes	20	4.3	odd	2	inner
336.6.h.a.239.12	yes	20	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-1.95479 + 15.4654i) q^{3} +34.2747i q^{5} -49.0000i q^{7} +(-235.358 - 60.4632i) q^{9} +O(q^{10})\)	\(q+(-1.95479 + 15.4654i) q^{3} +34.2747i q^{5} -49.0000i q^{7} +(-235.358 - 60.4632i) q^{9} +609.994 q^{11} -164.872 q^{13} +(-530.072 - 66.9998i) q^{15} -866.294i q^{17} -535.078i q^{19} +(757.805 + 95.7847i) q^{21} -4044.74 q^{23} +1950.25 q^{25} +(1395.16 - 3521.71i) q^{27} -7210.68i q^{29} -4423.05i q^{31} +(-1192.41 + 9433.81i) q^{33} +1679.46 q^{35} +7285.26 q^{37} +(322.290 - 2549.81i) q^{39} -3000.98i q^{41} -18694.4i q^{43} +(2072.36 - 8066.81i) q^{45} -931.299 q^{47} -2401.00 q^{49} +(13397.6 + 1693.42i) q^{51} +40298.7i q^{53} +20907.4i q^{55} +(8275.19 + 1045.96i) q^{57} -38098.4 q^{59} +43901.4 q^{61} +(-2962.70 + 11532.5i) q^{63} -5650.93i q^{65} -4505.45i q^{67} +(7906.61 - 62553.5i) q^{69} -2286.84 q^{71} +64424.6 q^{73} +(-3812.32 + 30161.3i) q^{75} -29889.7i q^{77} +47561.7i q^{79} +(51737.4 + 28461.0i) q^{81} +87756.5 q^{83} +29692.0 q^{85} +(111516. + 14095.4i) q^{87} -12809.7i q^{89} +8078.72i q^{91} +(68404.2 + 8646.12i) q^{93} +18339.6 q^{95} +85262.4 q^{97} +(-143567. - 36882.2i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 140 q^{9}+O(q^{10}) \) 20 * q - 140 * q^9	\( 20 q - 140 q^{9} + 1048 q^{13} - 980 q^{21} + 5916 q^{25} - 26056 q^{33} + 61360 q^{37} - 92512 q^{45} - 48020 q^{49} - 20720 q^{57} + 46680 q^{61} - 28360 q^{69} - 54280 q^{73} + 152660 q^{81} - 150536 q^{85} + 41688 q^{93} - 421352 q^{97}+O(q^{100}) \) 20 * q - 140 * q^9 + 1048 * q^13 - 980 * q^21 + 5916 * q^25 - 26056 * q^33 + 61360 * q^37 - 92512 * q^45 - 48020 * q^49 - 20720 * q^57 + 46680 * q^61 - 28360 * q^69 - 54280 * q^73 + 152660 * q^81 - 150536 * q^85 + 41688 * q^93 - 421352 * q^97

Introduction

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