LMFDB - Embedded newform 336.6.k.d.209.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,6,Mod(209,336)]

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([0, 0, 1, 1]))

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

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

chi := DirichletCharacter("336.209");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	336.k (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$53.8889634572$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 484x^{10} + 194194x^{8} - 39867800x^{6} + 5398720873x^{4} - 310089434788x^{2} + 9371104623076 $ x^12 - 484x^10 + 194194x^8 - 39867800x^6 + 5398720873x^4 - 310089434788*x^2 + 9371104623076
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{10} $
Twist minimal:	no (minimal twist has level 21)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			209.9
Root			$-6.98672 + 2.99433i$ of defining polynomial
Character	$\chi$	$=$	336.209
Dual form			336.6.k.d.209.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(9.94100 - 12.0074i) q^{3} +61.8023 q^{5} +(-92.4210 - 90.9140i) q^{7} +(-45.3530 - 238.730i) q^{9} +O(q^{10})$	\(q+(9.94100 - 12.0074i) q^{3} +61.8023 q^{5} +(-92.4210 - 90.9140i) q^{7} +(-45.3530 - 238.730i) q^{9} -615.301i q^{11} -322.895i q^{13} +(614.377 - 742.083i) q^{15} -1537.42 q^{17} +2171.63i q^{19} +(-2010.39 + 205.956i) q^{21} +793.329i q^{23} +694.530 q^{25} +(-3317.37 - 1828.65i) q^{27} +513.596i q^{29} +161.215i q^{31} +(-7388.14 - 6116.71i) q^{33} +(-5711.84 - 5618.70i) q^{35} -8593.37 q^{37} +(-3877.11 - 3209.90i) q^{39} +7004.53 q^{41} +7726.50 q^{43} +(-2802.92 - 14754.1i) q^{45} +17604.2 q^{47} +(276.290 + 16804.7i) q^{49} +(-15283.5 + 18460.3i) q^{51} -7345.26i q^{53} -38027.1i q^{55} +(26075.5 + 21588.2i) q^{57} -23416.6 q^{59} -13344.8i q^{61} +(-17512.3 + 26186.9i) q^{63} -19955.7i q^{65} -14471.1 q^{67} +(9525.78 + 7886.48i) q^{69} +7370.42i q^{71} -9422.68i q^{73} +(6904.32 - 8339.47i) q^{75} +(-55939.5 + 56866.8i) q^{77} -27229.8 q^{79} +(-54935.2 + 21654.3i) q^{81} -15577.0 q^{83} -95016.2 q^{85} +(6166.93 + 5105.66i) q^{87} +90024.7 q^{89} +(-29355.7 + 29842.3i) q^{91} +(1935.77 + 1602.64i) q^{93} +134212. i q^{95} -84850.1i q^{97} +(-146891. + 27905.8i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 112 q^{7} - 492 q^{9}+O(q^{10}) $ 12 * q - 112 * q^7 - 492 * q^9	$ 12 q - 112 q^{7} - 492 q^{9} - 1392 q^{15} + 4116 q^{21} + 7812 q^{25} + 27464 q^{37} + 16080 q^{39} + 73840 q^{43} + 6972 q^{49} - 23760 q^{51} + 103968 q^{57} + 120624 q^{63} - 79344 q^{67} + 247104 q^{79} - 248868 q^{81} - 320112 q^{85} - 310128 q^{91} + 397272 q^{93} - 696576 q^{99}+O(q^{100}) $ 12 * q - 112 * q^7 - 492 * q^9 - 1392 * q^15 + 4116 * q^21 + 7812 * q^25 + 27464 * q^37 + 16080 * q^39 + 73840 * q^43 + 6972 * q^49 - 23760 * q^51 + 103968 * q^57 + 120624 * q^63 - 79344 * q^67 + 247104 * q^79 - 248868 * q^81 - 320112 * q^85 - 310128 * q^91 + 397272 * q^93 - 696576 * q^99

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	9.94100	−	12.0074i	0.637716	−	0.770272i
$3$	9.94100	−	12.0074i	0.637716	−	0.770272i
$4$		0			0
$5$	61.8023			1.10555			0.552777	−	0.833329i	$-0.313568\pi$
$5$	61.8023			1.10555			0.552777	+	0.833329i	$0.313568\pi$
$6$		0			0
$7$	−92.4210	−	90.9140i	−0.712895	−	0.701271i
$7$	−92.4210	−	90.9140i	−0.712895	−	0.701271i
$8$		0			0
$9$	−45.3530	−	238.730i	−0.186638	−	0.982429i
$10$		0			0
$11$		−	615.301i		−	1.53323i	−0.642110	−	0.766613i	$-0.721941\pi$
$11$		−	615.301i		−	1.53323i	0.642110	−	0.766613i	$-0.278059\pi$
$12$		0			0
$13$		−	322.895i		−	0.529911i	−0.964261	−	0.264955i	$-0.914643\pi$
$13$		−	322.895i		−	0.529911i	0.964261	−	0.264955i	$-0.0853572\pi$
$14$		0			0
$15$	614.377	−	742.083i	0.705029	−	0.851577i
$16$		0			0
$17$	−1537.42			−1.29024			−0.645120	−	0.764081i	$-0.723193\pi$
$17$	−1537.42			−1.29024			−0.645120	+	0.764081i	$0.723193\pi$
$18$		0			0
$19$			2171.63i			1.38007i	0.723775	+	0.690036i	$0.242405\pi$
$19$			2171.63i			1.38007i	−0.723775	+	0.690036i	$0.757595\pi$
$20$		0			0
$21$	−2010.39	+	205.956i	−0.994793	+	0.101912i
$22$		0			0
$23$			793.329i			0.312704i	0.987701	+	0.156352i	$0.0499735\pi$
$23$			793.329i			0.312704i	−0.987701	+	0.156352i	$0.950027\pi$
$24$		0			0
$25$	694.530			0.222250
$26$		0			0
$27$	−3317.37	−	1828.65i	−0.875759	−	0.482748i
$28$		0			0
$29$			513.596i			0.113404i	0.998391	+	0.0567018i	$0.0180584\pi$
$29$			513.596i			0.113404i	−0.998391	+	0.0567018i	$0.981942\pi$
$30$		0			0
$31$			161.215i			0.0301302i	0.999887	+	0.0150651i	$0.00479555\pi$
$31$			161.215i			0.0301302i	−0.999887	+	0.0150651i	$0.995204\pi$
$32$		0			0
$33$	−7388.14	−	6116.71i	−1.18100	−	0.977762i
$34$		0			0
$35$	−5711.84	−	5618.70i	−0.788144	−	0.775293i
$36$		0			0
$37$	−8593.37			−1.03195			−0.515976	−	0.856603i	$-0.672571\pi$
$37$	−8593.37			−1.03195			−0.515976	+	0.856603i	$0.672571\pi$
$38$		0			0
$39$	−3877.11	−	3209.90i	−0.408175	−	0.337932i
$40$		0			0
$41$	7004.53			0.650758			0.325379	−	0.945584i	$-0.394508\pi$
$41$	7004.53			0.650758			0.325379	+	0.945584i	$0.394508\pi$
$42$		0			0
$43$	7726.50			0.637253			0.318627	−	0.947880i	$-0.396778\pi$
$43$	7726.50			0.637253			0.318627	+	0.947880i	$0.396778\pi$
$44$		0			0
$45$	−2802.92	−	14754.1i	−0.206338	−	1.08613i
$46$		0			0
$47$	17604.2			1.16244			0.581221	−	0.813745i	$-0.302575\pi$
$47$	17604.2			1.16244			0.581221	+	0.813745i	$0.302575\pi$
$48$		0			0
$49$	276.290	+	16804.7i	0.0164390	+	0.999865i
$50$		0			0
$51$	−15283.5	+	18460.3i	−0.822806	+	0.993835i
$52$		0			0
$53$		−	7345.26i		−	0.359184i	−0.983741	−	0.179592i	$-0.942522\pi$
$53$		−	7345.26i		−	0.359184i	0.983741	−	0.179592i	$-0.0574778\pi$
$54$		0			0
$55$		−	38027.1i		−	1.69506i
$56$		0			0
$57$	26075.5	+	21588.2i	1.06303	+	0.880093i
$58$		0			0
$59$	−23416.6			−0.875779			−0.437889	−	0.899029i	$-0.644274\pi$
$59$	−23416.6			−0.875779			−0.437889	+	0.899029i	$0.644274\pi$
$60$		0			0
$61$		−	13344.8i		−	0.459186i	−0.973287	−	0.229593i	$-0.926260\pi$
$61$		−	13344.8i		−	0.459186i	0.973287	−	0.229593i	$-0.0737395\pi$
$62$		0			0
$63$	−17512.3	+	26186.9i	−0.555895	+	0.831252i
$64$		0			0
$65$		−	19955.7i		−	0.585845i
$66$		0			0
$67$	−14471.1			−0.393835			−0.196918	−	0.980420i	$-0.563093\pi$
$67$	−14471.1			−0.393835			−0.196918	+	0.980420i	$0.563093\pi$
$68$		0			0
$69$	9525.78	+	7886.48i	0.240867	+	0.199416i
$70$		0			0
$71$			7370.42i			0.173519i	0.996229	+	0.0867593i	$0.0276511\pi$
$71$			7370.42i			0.173519i	−0.996229	+	0.0867593i	$0.972349\pi$
$72$		0			0
$73$		−	9422.68i		−	0.206951i	−0.994632	−	0.103475i	$-0.967004\pi$
$73$		−	9422.68i		−	0.206951i	0.994632	−	0.103475i	$-0.0329963\pi$
$74$		0			0
$75$	6904.32	−	8339.47i	0.141732	−	0.171193i
$76$		0			0
$77$	−55939.5	+	56866.8i	−1.07521	+	1.09303i
$78$		0			0
$79$	−27229.8			−0.490882			−0.245441	−	0.969412i	$-0.578933\pi$
$79$	−27229.8			−0.490882			−0.245441	+	0.969412i	$0.578933\pi$
$80$		0			0
$81$	−54935.2	+	21654.3i	−0.930333	+	0.366717i
$82$		0			0
$83$	−15577.0			−0.248192			−0.124096	−	0.992270i	$-0.539603\pi$
$83$	−15577.0			−0.248192			−0.124096	+	0.992270i	$0.539603\pi$
$84$		0			0
$85$	−95016.2			−1.42643
$86$		0			0
$87$	6166.93	+	5105.66i	0.0873516	+	0.0723192i
$88$		0			0
$89$	90024.7			1.20472			0.602361	−	0.798224i	$-0.294227\pi$
$89$	90024.7			1.20472			0.602361	+	0.798224i	$0.294227\pi$
$90$		0			0
$91$	−29355.7	+	29842.3i	−0.371611	+	0.377771i
$92$		0			0
$93$	1935.77	+	1602.64i	0.0232085	+	0.0192145i
$94$		0			0
$95$			134212.i			1.52574i
$96$		0			0
$97$		−	84850.1i		−	0.915637i	−0.889046	−	0.457818i	$-0.848631\pi$
$97$		−	84850.1i		−	0.915637i	0.889046	−	0.457818i	$-0.151369\pi$
$98$		0			0
$99$	−146891.	+	27905.8i	−1.50628	+	0.286158i
$100$		0			0
$101$	−80893.8			−0.789063			−0.394532	−	0.918882i	$-0.629093\pi$
$101$	−80893.8			−0.789063			−0.394532	+	0.918882i	$0.629093\pi$
$102$		0			0
$103$		−	214822.i		−	1.99520i	−0.0692543	−	0.997599i	$-0.522062\pi$
$103$		−	214822.i		−	1.99520i	0.0692543	−	0.997599i	$-0.477938\pi$
$104$		0			0
$105$	−124247.	+	12728.5i	−1.09980	+	0.112669i
$106$		0			0
$107$		−	120007.i		−	1.01332i	−0.862146	−	0.506660i	$-0.830880\pi$
$107$		−	120007.i		−	1.01332i	0.862146	−	0.506660i	$-0.169120\pi$
$108$		0			0
$109$	29204.1			0.235439			0.117719	−	0.993047i	$-0.462442\pi$
$109$	29204.1			0.235439			0.117719	+	0.993047i	$0.462442\pi$
$110$		0			0
$111$	−85426.7	+	103184.i	−0.658091	+	0.794883i
$112$		0			0
$113$			64313.7i			0.473813i	0.971532	+	0.236907i	$0.0761335\pi$
$113$			64313.7i			0.473813i	−0.971532	+	0.236907i	$0.923866\pi$
$114$		0			0
$115$			49029.6i			0.345711i
$116$		0			0
$117$	−77084.8	+	14644.3i	−0.520600	+	0.0989014i
$118$		0			0
$119$	142090.	+	139773.i	0.919805	+	0.904807i
$120$		0			0
$121$	−217545.			−1.35078
$122$		0			0
$123$	69632.0	−	84105.8i	0.414998	−	0.501260i
$124$		0			0
$125$	−150209.			−0.859845
$126$		0			0
$127$	293422.			1.61430			0.807149	−	0.590348i	$-0.201010\pi$
$127$	293422.			1.61430			0.807149	+	0.590348i	$0.201010\pi$
$128$		0			0
$129$	76809.2	−	92774.9i	0.406386	−	0.490858i
$130$		0			0
$131$	−159323.			−0.811147			−0.405573	−	0.914063i	$-0.632928\pi$
$131$	−159323.			−0.811147			−0.405573	+	0.914063i	$0.632928\pi$
$132$		0			0
$133$	197432.	−	200704.i	0.967804	−	0.983847i
$134$		0			0
$135$	−205021.	−	113015.i	−0.968199	−	0.533704i
$136$		0			0
$137$			387950.i			1.76593i	0.469436	+	0.882967i	$0.344457\pi$
$137$			387950.i			1.76593i	−0.469436	+	0.882967i	$0.655543\pi$
$138$		0			0
$139$		−	324317.i		−	1.42375i	−0.702308	−	0.711874i	$-0.747847\pi$
$139$		−	324317.i		−	1.42375i	0.702308	−	0.711874i	$-0.252153\pi$
$140$		0			0
$141$	175003.	−	211380.i	0.741308	−	0.895397i
$142$		0			0
$143$	−198678.			−0.812473
$144$		0			0
$145$			31741.4i			0.125374i
$146$		0			0
$147$	204527.	+	163738.i	0.780651	+	0.624967i
$148$		0			0
$149$		−	318945.i		−	1.17693i	−0.808523	−	0.588464i	$-0.799733\pi$
$149$		−	318945.i		−	1.17693i	0.808523	−	0.588464i	$-0.200267\pi$
$150$		0			0
$151$	−479410.			−1.71106			−0.855529	−	0.517754i	$-0.826768\pi$
$151$	−479410.			−1.71106			−0.855529	+	0.517754i	$0.826768\pi$
$152$		0			0
$153$	69726.6	+	367029.i	0.240808	+	1.26757i
$154$		0			0
$155$			9963.49i			0.0333106i
$156$		0			0
$157$		−	323801.i		−	1.04840i	−0.851594	−	0.524202i	$-0.824364\pi$
$157$		−	323801.i		−	1.04840i	0.851594	−	0.524202i	$-0.175636\pi$
$158$		0			0
$159$	−88197.1	−	73019.2i	−0.276670	−	0.229057i
$160$		0			0
$161$	72124.7	−	73320.3i	0.219290	−	0.222925i
$162$		0			0
$163$	21423.8			0.0631579			0.0315789	−	0.999501i	$-0.489946\pi$
$163$	21423.8			0.0631579			0.0315789	+	0.999501i	$0.489946\pi$
$164$		0			0
$165$	−456604.	−	378027.i	−1.30566	−	1.08097i
$166$		0			0
$167$	−436305.			−1.21059			−0.605297	−	0.796000i	$-0.706946\pi$
$167$	−436305.			−1.21059			−0.605297	+	0.796000i	$0.706946\pi$
$168$		0			0
$169$	267032.			0.719195
$170$		0			0
$171$	518434.	−	98489.9i	1.35582	−	0.257574i
$172$		0			0
$173$	−185581.			−0.471431			−0.235715	−	0.971822i	$-0.575743\pi$
$173$	−185581.			−0.471431			−0.235715	+	0.971822i	$0.575743\pi$
$174$		0			0
$175$	−64189.2	−	63142.5i	−0.158441	−	0.155857i
$176$		0			0
$177$	−232785.	+	281172.i	−0.558498	+	0.674588i
$178$		0			0
$179$			170761.i			0.398341i	0.979965	+	0.199171i	$0.0638248\pi$
$179$			170761.i			0.398341i	−0.979965	+	0.199171i	$0.936175\pi$
$180$		0			0
$181$			326285.i			0.740289i	0.928974	+	0.370144i	$0.120692\pi$
$181$			326285.i			0.740289i	−0.928974	+	0.370144i	$0.879308\pi$
$182$		0			0
$183$	−160236.	−	132661.i	−0.353698	−	0.292830i
$184$		0			0
$185$	−531090.			−1.14088
$186$		0			0
$187$			945977.i			1.97823i
$188$		0			0
$189$	140345.	+	470601.i	0.285787	+	0.958293i
$190$		0			0
$191$		−	296995.i		−	0.589069i	−0.955641	−	0.294534i	$-0.904835\pi$
$191$		−	296995.i		−	0.589069i	0.955641	−	0.294534i	$-0.0951646\pi$
$192$		0			0
$193$	414092.			0.800208			0.400104	−	0.916470i	$-0.368974\pi$
$193$	414092.			0.800208			0.400104	+	0.916470i	$0.368974\pi$
$194$		0			0
$195$	−239615.	−	198379.i	−0.451260	−	0.373602i
$196$		0			0
$197$		−	216937.i		−	0.398262i	−0.979973	−	0.199131i	$-0.936188\pi$
$197$		−	216937.i		−	0.398262i	0.979973	−	0.199131i	$-0.0638119\pi$
$198$		0			0
$199$		−	36716.3i		−	0.0657244i	−0.999460	−	0.0328622i	$-0.989538\pi$
$199$		−	36716.3i		−	0.0657244i	0.999460	−	0.0328622i	$-0.0104623\pi$
$200$		0			0
$201$	−143857.	+	173760.i	−0.251155	+	0.303360i
$202$		0			0
$203$	46693.1	−	47467.1i	0.0795266	−	0.0808448i
$204$		0			0
$205$	432896.			0.719448
$206$		0			0
$207$	189392.	−	35979.8i	0.307210	−	0.0583624i
$208$		0			0
$209$	1.33621e6			2.11596
$210$		0			0
$211$	587416.			0.908321			0.454161	−	0.890920i	$-0.349939\pi$
$211$	587416.			0.908321			0.454161	+	0.890920i	$0.349939\pi$
$212$		0			0
$213$	88499.2	+	73269.3i	0.133657	+	0.110656i
$214$		0			0
$215$	477516.			0.704518
$216$		0			0
$217$	14656.7	−	14899.7i	0.0211294	−	0.0214797i
$218$		0			0
$219$	−113141.	−	93670.8i	−0.159408	−	0.131976i
$220$		0			0
$221$			496425.i			0.683712i
$222$		0			0
$223$		−	349812.i		−	0.471056i	−0.971868	−	0.235528i	$-0.924318\pi$
$223$		−	349812.i		−	0.471056i	0.971868	−	0.235528i	$-0.0756819\pi$
$224$		0			0
$225$	−31499.0	−	165805.i	−0.0414802	−	0.218344i
$226$		0			0
$227$	925060.			1.19153			0.595766	−	0.803158i	$-0.296849\pi$
$227$	925060.			1.19153			0.595766	+	0.803158i	$0.296849\pi$
$228$		0			0
$229$			136041.i			0.171428i	0.996320	+	0.0857142i	$0.0273172\pi$
$229$			136041.i			0.171428i	−0.996320	+	0.0857142i	$0.972683\pi$
$230$		0			0
$231$	126725.	+	1.23700e6i	0.156254	+	1.52524i
$232$		0			0
$233$		−	1.13764e6i		−	1.37283i	−0.727212	−	0.686413i	$-0.759184\pi$
$233$		−	1.13764e6i		−	1.37283i	0.727212	−	0.686413i	$-0.240816\pi$
$234$		0			0
$235$	1.08798e6			1.28514
$236$		0			0
$237$	−270692.	+	326958.i	−0.313043	+	0.378113i
$238$		0			0
$239$		−	1.04427e6i		−	1.18254i	−0.806473	−	0.591271i	$-0.798626\pi$
$239$		−	1.04427e6i		−	1.18254i	0.806473	−	0.591271i	$-0.201374\pi$
$240$		0			0
$241$		−	503492.i		−	0.558406i	−0.960232	−	0.279203i	$-0.909930\pi$
$241$		−	503492.i		−	0.558406i	0.960232	−	0.279203i	$-0.0900702\pi$
$242$		0			0
$243$	−286101.	+	874891.i	−0.310816	+	0.950470i
$244$		0			0
$245$	17075.4	+	1.03857e6i	0.0181742	+	1.10540i
$246$		0			0
$247$	701208.			0.731315
$248$		0			0
$249$	−154851.	+	187038.i	−0.158276	+	0.191175i
$250$		0			0
$251$	1.68197e6			1.68514			0.842568	−	0.538590i	$-0.181043\pi$
$251$	1.68197e6			1.68514			0.842568	+	0.538590i	$0.181043\pi$
$252$		0			0
$253$	488136.			0.479446
$254$		0			0
$255$	−944556.	+	1.14089e6i	−0.909656	+	1.09874i
$256$		0			0
$257$	953741.			0.900736			0.450368	−	0.892843i	$-0.351293\pi$
$257$	953741.			0.900736			0.450368	+	0.892843i	$0.351293\pi$
$258$		0			0
$259$	794208.	+	781258.i	0.735673	+	0.723677i
$260$		0			0
$261$	122611.	−	23293.1i	0.111411	−	0.0211654i
$262$		0			0
$263$			1.64469e6i			1.46621i	0.680117	+	0.733104i	$0.261929\pi$
$263$			1.64469e6i			1.46621i	−0.680117	+	0.733104i	$0.738071\pi$
$264$		0			0
$265$		−	453954.i		−	0.397098i
$266$		0			0
$267$	894935.	−	1.08096e6i	0.768269	−	0.927963i
$268$		0			0
$269$	76973.2			0.0648573			0.0324286	−	0.999474i	$-0.489676\pi$
$269$	76973.2			0.0648573			0.0324286	+	0.999474i	$0.489676\pi$
$270$		0			0
$271$		−	1.79795e6i		−	1.48715i	−0.668652	−	0.743576i	$-0.733128\pi$
$271$		−	1.79795e6i		−	1.48715i	0.668652	−	0.743576i	$-0.266872\pi$
$272$		0			0
$273$	66502.0	+	649146.i	0.0540043	+	0.527152i
$274$		0			0
$275$		−	427345.i		−	0.340759i
$276$		0			0
$277$	1.16102e6			0.909160			0.454580	−	0.890706i	$-0.349789\pi$
$277$	1.16102e6			0.909160			0.454580	+	0.890706i	$0.349789\pi$
$278$		0			0
$279$	38487.0	−	7311.60i	0.0296008	−	0.00562344i
$280$		0			0
$281$		−	1.50031e6i		−	1.13348i	−0.823896	−	0.566741i	$-0.808204\pi$
$281$		−	1.50031e6i		−	1.13348i	0.823896	−	0.566741i	$-0.191796\pi$
$282$		0			0
$283$		−	775321.i		−	0.575460i	−0.957712	−	0.287730i	$-0.907099\pi$
$283$		−	775321.i		−	0.575460i	0.957712	−	0.287730i	$-0.0929006\pi$
$284$		0			0
$285$	1.61153e6	+	1.33420e6i	1.17524	+	0.972991i
$286$		0			0
$287$	−647366.	−	636810.i	−0.463922	−	0.456357i
$288$		0			0
$289$	943804.			0.664718
$290$		0			0
$291$	−1.01883e6	−	843495.i	−0.705289	−	0.583916i
$292$		0			0
$293$	194034.			0.132041			0.0660206	−	0.997818i	$-0.478970\pi$
$293$	194034.			0.132041			0.0660206	+	0.997818i	$0.478970\pi$
$294$		0			0
$295$	−1.44720e6			−0.968221
$296$		0			0
$297$	−1.12517e6	+	2.04118e6i	−0.740162	+	1.34274i
$298$		0			0
$299$	256162.			0.165705
$300$		0			0
$301$	−714091.	−	702447.i	−0.454295	−	0.446887i
$302$		0			0
$303$	−804165.	+	971320.i	−0.503198	+	0.607793i
$304$		0			0
$305$		−	824742.i		−	0.507655i
$306$		0			0
$307$			394659.i			0.238988i	0.992835	+	0.119494i	$0.0381273\pi$
$307$			394659.i			0.238988i	−0.992835	+	0.119494i	$0.961873\pi$
$308$		0			0
$309$	−2.57945e6	−	2.13555e6i	−1.53685	−	1.27237i
$310$		0			0
$311$	−2.29757e6			−1.34700			−0.673501	−	0.739186i	$-0.735210\pi$
$311$	−2.29757e6			−1.34700			−0.673501	+	0.739186i	$0.735210\pi$
$312$		0			0
$313$			634859.i			0.366283i	0.983087	+	0.183141i	$0.0586266\pi$
$313$			634859.i			0.366283i	−0.983087	+	0.183141i	$0.941373\pi$
$314$		0			0
$315$	−1.08230e6	+	1.61841e6i	−0.614572	+	0.918994i
$316$		0			0
$317$			192873.i			0.107801i	0.998546	+	0.0539005i	$0.0171654\pi$
$317$			192873.i			0.107801i	−0.998546	+	0.0539005i	$0.982835\pi$
$318$		0			0
$319$	316016.			0.173873
$320$		0			0
$321$	−1.44096e6	−	1.19299e6i	−0.780532	−	0.646210i
$322$		0			0
$323$		−	3.33871e6i		−	1.78062i
$324$		0			0
$325$		−	224260.i		−	0.117772i
$326$		0			0
$327$	290318.	−	350664.i	0.150143	−	0.181352i
$328$		0			0
$329$	−1.62700e6	−	1.60047e6i	−0.828700	−	0.815187i
$330$		0			0
$331$	2.85474e6			1.43218			0.716088	−	0.698010i	$-0.245931\pi$
$331$	2.85474e6			1.43218			0.716088	+	0.698010i	$0.245931\pi$
$332$		0			0
$333$	389735.	+	2.05150e6i	0.192601	+	1.01382i
$334$		0			0
$335$	−894348.			−0.435406
$336$		0			0
$337$	3.37722e6			1.61989			0.809944	−	0.586507i	$-0.199497\pi$
$337$	3.37722e6			1.61989			0.809944	+	0.586507i	$0.199497\pi$
$338$		0			0
$339$	772237.	+	639342.i	0.364965	+	0.302158i
$340$		0			0
$341$	99196.0			0.0461964
$342$		0			0
$343$	1.50225e6	−	1.57823e6i	0.689457	−	0.724327i
$344$		0			0
$345$	588716.	+	487403.i	0.266292	+	0.220466i
$346$		0			0
$347$		−	547961.i		−	0.244301i	−0.992512	−	0.122151i	$-0.961021\pi$
$347$		−	547961.i		−	0.244301i	0.992512	−	0.122151i	$-0.0389791\pi$
$348$		0			0
$349$		−	1206.49i		−	0.000530223i	−1.00000		0.000265112i	$-0.999916\pi$
$349$		−	1206.49i		−	0.000530223i	1.00000		0.000265112i	$-8.43876e-5\pi$
$350$		0			0
$351$	−590461.	+	1.07116e6i	−0.255813	+	0.464074i
$352$		0			0
$353$	−1.15208e6			−0.492090			−0.246045	−	0.969258i	$-0.579131\pi$
$353$	−1.15208e6			−0.492090			−0.246045	+	0.969258i	$0.579131\pi$
$354$		0			0
$355$			455509.i			0.191834i
$356$		0			0
$357$	3.09082e6	−	316640.i	1.28352	−	0.131491i
$358$		0			0
$359$			2.10870e6i			0.863531i	0.901986	+	0.431766i	$0.142109\pi$
$359$			2.10870e6i			0.863531i	−0.901986	+	0.431766i	$0.857891\pi$
$360$		0			0
$361$	−2.23988e6			−0.904599
$362$		0			0
$363$	−2.16261e6	+	2.61213e6i	−0.861414	+	1.04047i
$364$		0			0
$365$		−	582344.i		−	0.228795i
$366$		0			0
$367$		−	3.37130e6i		−	1.30657i	−0.757113	−	0.653284i	$-0.773391\pi$
$367$		−	3.37130e6i		−	1.30657i	0.757113	−	0.653284i	$-0.226609\pi$
$368$		0			0
$369$	−317676.	−	1.67219e6i	−0.121456	−	0.639323i
$370$		0			0
$371$	−667787.	+	678856.i	−0.251885	+	0.256061i
$372$		0			0
$373$	−65054.8			−0.0242107			−0.0121054	−	0.999927i	$-0.503853\pi$
$373$	−65054.8			−0.0242107			−0.0121054	+	0.999927i	$0.503853\pi$
$374$		0			0
$375$	−1.49323e6	+	1.80361e6i	−0.548337	+	0.662315i
$376$		0			0
$377$	165838.			0.0600938
$378$		0			0
$379$	690875.			0.247059			0.123530	−	0.992341i	$-0.460579\pi$
$379$	690875.			0.247059			0.123530	+	0.992341i	$0.460579\pi$
$380$		0			0
$381$	2.91691e6	−	3.52322e6i	1.02946	−	1.24345i
$382$		0			0
$383$	2.44872e6			0.852986			0.426493	−	0.904491i	$-0.359749\pi$
$383$	2.44872e6			0.852986			0.426493	+	0.904491i	$0.359749\pi$
$384$		0			0
$385$	−3.45719e6	+	3.51450e6i	−1.18870	+	1.20840i
$386$		0			0
$387$	−350420.	−	1.84455e6i	−0.118936	−	0.626056i
$388$		0			0
$389$			3.72849e6i			1.24928i	0.780913	+	0.624640i	$0.214754\pi$
$389$			3.72849e6i			1.24928i	−0.780913	+	0.624640i	$0.785246\pi$
$390$		0			0
$391$		−	1.21968e6i		−	0.403463i
$392$		0			0
$393$	−1.58383e6	+	1.91304e6i	−0.517281	+	0.624803i
$394$		0			0
$395$	−1.68287e6			−0.542697
$396$		0			0
$397$		−	3.19740e6i		−	1.01817i	−0.860716	−	0.509086i	$-0.829984\pi$
$397$		−	3.19740e6i		−	1.01817i	0.860716	−	0.509086i	$-0.170016\pi$
$398$		0			0
$399$	−447259.	−	4.36583e6i	−0.140646	−	1.37289i
$400$		0			0
$401$			6.01416e6i			1.86773i	0.357627	+	0.933865i	$0.383586\pi$
$401$			6.01416e6i			1.86773i	−0.357627	+	0.933865i	$0.616414\pi$
$402$		0			0
$403$	52055.6			0.0159663
$404$		0			0
$405$	−3.39512e6	+	1.33828e6i	−1.02853	+	0.405425i
$406$		0			0
$407$			5.28751e6i			1.58221i
$408$		0			0
$409$		−	5.69952e6i		−	1.68473i	−0.538909	−	0.842364i	$-0.681163\pi$
$409$		−	5.69952e6i		−	1.68473i	0.538909	−	0.842364i	$-0.318837\pi$
$410$		0			0
$411$	4.65825e6	+	3.85661e6i	1.36025	+	1.12616i
$412$		0			0
$413$	2.16419e6	+	2.12890e6i	0.624338	+	0.614158i
$414$		0			0
$415$	−962694.			−0.274390
$416$		0			0
$417$	−3.89419e6	−	3.22404e6i	−1.09667	−	0.907946i
$418$		0			0
$419$	2.40292e6			0.668658			0.334329	−	0.942456i	$-0.391490\pi$
$419$	2.40292e6			0.668658			0.334329	+	0.942456i	$0.391490\pi$
$420$		0			0
$421$	−1.84926e6			−0.508501			−0.254251	−	0.967138i	$-0.581829\pi$
$421$	−1.84926e6			−0.508501			−0.254251	+	0.967138i	$0.581829\pi$
$422$		0			0
$423$	−798403.	−	4.20265e6i	−0.216956	−	1.14202i
$424$		0			0
$425$	−1.06778e6			−0.286755
$426$		0			0
$427$	−1.21323e6	+	1.23334e6i	−0.322014	+	0.327351i
$428$		0			0
$429$	−1.97505e6	+	2.38559e6i	−0.518127	+	0.625825i
$430$		0			0
$431$			670021.i			0.173738i	0.996220	+	0.0868691i	$0.0276862\pi$
$431$			670021.i			0.173738i	−0.996220	+	0.0868691i	$0.972314\pi$
$432$		0			0
$433$		−	3.18314e6i		−	0.815898i	−0.913005	−	0.407949i	$-0.866244\pi$
$433$		−	3.18314e6i		−	0.815898i	0.913005	−	0.407949i	$-0.133756\pi$
$434$		0			0
$435$	381131.	+	315542.i	0.0965719	+	0.0799528i
$436$		0			0
$437$	−1.72282e6			−0.431554
$438$		0			0
$439$			3.04218e6i			0.753396i	0.926336	+	0.376698i	$0.122941\pi$
$439$			3.04218e6i			0.753396i	−0.926336	+	0.376698i	$0.877059\pi$
$440$		0			0
$441$	3.99927e6	−	828104.i	0.979228	−	0.202763i
$442$		0			0
$443$			4.06656e6i			0.984505i	0.870452	+	0.492253i	$0.163826\pi$
$443$			4.06656e6i			0.984505i	−0.870452	+	0.492253i	$0.836174\pi$
$444$		0			0
$445$	5.56374e6			1.33188
$446$		0			0
$447$	−3.82969e6	−	3.17063e6i	−0.906555	−	0.750546i
$448$		0			0
$449$			3.30712e6i			0.774166i	0.922045	+	0.387083i	$0.126517\pi$
$449$			3.30712e6i			0.774166i	−0.922045	+	0.387083i	$0.873483\pi$
$450$		0			0
$451$		−	4.30990e6i		−	0.997758i
$452$		0			0
$453$	−4.76582e6	+	5.75645e6i	−1.09117	+	1.31798i
$454$		0			0
$455$	−1.81425e6	+	1.84432e6i	−0.410836	+	0.417646i
$456$		0			0
$457$	797135.			0.178542			0.0892712	−	0.996007i	$-0.471546\pi$
$457$	797135.			0.178542			0.0892712	+	0.996007i	$0.471546\pi$
$458$		0			0
$459$	5.10019e6	+	2.81140e6i	1.12994	+	0.622861i
$460$		0			0
$461$	−4.32869e6			−0.948646			−0.474323	−	0.880351i	$-0.657307\pi$
$461$	−4.32869e6			−0.948646			−0.474323	+	0.880351i	$0.657307\pi$
$462$		0			0
$463$	−5.38489e6			−1.16741			−0.583706	−	0.811965i	$-0.698398\pi$
$463$	−5.38489e6			−1.16741			−0.583706	+	0.811965i	$0.698398\pi$
$464$		0			0
$465$	119635.	+	99047.1i	0.0256582	+	0.0212427i
$466$		0			0
$467$	−677568.			−0.143767			−0.0718837	−	0.997413i	$-0.522901\pi$
$467$	−677568.			−0.143767			−0.0718837	+	0.997413i	$0.522901\pi$
$468$		0			0
$469$	1.33743e6	+	1.31563e6i	0.280763	+	0.276185i
$470$		0			0
$471$	−3.88799e6	−	3.21891e6i	−0.807557	−	0.668584i
$472$		0			0
$473$		−	4.75413e6i		−	0.977053i
$474$		0			0
$475$			1.50826e6i			0.306720i
$476$		0			0
$477$	−1.75353e6	+	333130.i	−0.352873	+	0.0670374i
$478$		0			0
$479$	−7.87592e6			−1.56842			−0.784210	−	0.620495i	$-0.786932\pi$
$479$	−7.87592e6			−1.56842			−0.784210	+	0.620495i	$0.786932\pi$
$480$		0			0
$481$			2.77476e6i			0.546842i
$482$		0			0
$483$	−163391.	−	1.59490e6i	−0.0318683	−	0.311076i
$484$		0			0
$485$		−	5.24394e6i		−	1.01229i
$486$		0			0
$487$	−6.38336e6			−1.21963			−0.609813	−	0.792545i	$-0.708756\pi$
$487$	−6.38336e6			−1.21963			−0.609813	+	0.792545i	$0.708756\pi$
$488$		0			0
$489$	212974.	−	257243.i	0.0402768	−	0.0486488i
$490$		0			0
$491$		−	837734.i		−	0.156820i	−0.996921	−	0.0784102i	$-0.975016\pi$
$491$		−	837734.i		−	0.156820i	0.996921	−	0.0784102i	$-0.0249844\pi$
$492$		0			0
$493$		−	789613.i		−	0.146318i
$494$		0			0
$495$	−9.07821e6	+	1.72464e6i	−1.66528	+	0.316363i
$496$		0			0
$497$	670074.	−	681182.i	0.121684	−	0.123701i
$498$		0			0
$499$	4.80919e6			0.864611			0.432306	−	0.901727i	$-0.357700\pi$
$499$	4.80919e6			0.864611			0.432306	+	0.901727i	$0.357700\pi$
$500$		0			0
$501$	−4.33731e6	+	5.23886e6i	−0.772015	+	0.932487i
$502$		0			0
$503$	2.58231e6			0.455081			0.227540	−	0.973769i	$-0.426932\pi$
$503$	2.58231e6			0.455081			0.227540	+	0.973769i	$0.426932\pi$
$504$		0			0
$505$	−4.99943e6			−0.872352
$506$		0			0
$507$	2.65456e6	−	3.20635e6i	0.458642	−	0.553975i
$508$		0			0
$509$	5.81775e6			0.995315			0.497658	−	0.867374i	$-0.334194\pi$
$509$	5.81775e6			0.995315			0.497658	+	0.867374i	$0.334194\pi$
$510$		0			0
$511$	−856653.	+	870853.i	−0.145129	+	0.147534i
$512$		0			0
$513$	3.97115e6	−	7.20410e6i	0.666227	−	1.20861i
$514$		0			0
$515$		−	1.32765e7i		−	2.20580i
$516$		0			0
$517$		−	1.08319e7i		−	1.78229i
$518$		0			0
$519$	−1.84486e6	+	2.22833e6i	−0.300639	+	0.363130i
$520$		0			0
$521$	−7.44578e6			−1.20175			−0.600877	−	0.799341i	$-0.705182\pi$
$521$	−7.44578e6			−1.20175			−0.600877	+	0.799341i	$0.705182\pi$
$522$		0			0
$523$		−	2.68003e6i		−	0.428435i	−0.976786	−	0.214218i	$-0.931280\pi$
$523$		−	2.68003e6i		−	0.428435i	0.976786	−	0.214218i	$-0.0687202\pi$
$524$		0			0
$525$	−1.39628e6	+	143042.i	−0.221092	+	0.0226499i
$526$		0			0
$527$		−	247856.i		−	0.0388752i
$528$		0			0
$529$	5.80697e6			0.902216
$530$		0			0
$531$	1.06201e6	+	5.59026e6i	0.163453	+	0.860390i
$532$		0			0
$533$		−	2.26173e6i		−	0.344844i
$534$		0			0
$535$		−	7.41670e6i		−	1.12028i
$536$		0			0
$537$	2.05038e6	+	1.69753e6i	0.306831	+	0.254028i
$538$		0			0
$539$	1.03400e7	−	170002.i	1.53302	−	0.0252047i
$540$		0			0
$541$	−7.33897e6			−1.07806			−0.539029	−	0.842288i	$-0.681209\pi$
$541$	−7.33897e6			−1.07806			−0.539029	+	0.842288i	$0.681209\pi$
$542$		0			0
$543$	3.91782e6	+	3.24360e6i	0.570224	+	0.472094i
$544$		0			0
$545$	1.80488e6			0.260290
$546$		0			0
$547$	3.63983e6			0.520131			0.260066	−	0.965591i	$-0.416256\pi$
$547$	3.63983e6			0.520131			0.260066	+	0.965591i	$0.416256\pi$
$548$		0			0
$549$	−3.18582e6	+	605228.i	−0.451118	+	0.0857015i
$550$		0			0
$551$	−1.11534e6			−0.156505
$552$		0			0
$553$	2.51661e6	+	2.47557e6i	0.349947	+	0.344241i
$554$		0			0
$555$	−5.27957e6	+	6.37699e6i	−0.727556	+	0.878786i
$556$		0			0
$557$		−	1.12162e7i		−	1.53183i	−0.642944	−	0.765913i	$-0.722287\pi$
$557$		−	1.12162e7i		−	1.53183i	0.642944	−	0.765913i	$-0.277713\pi$
$558$		0			0
$559$		−	2.49485e6i		−	0.337687i
$560$		0			0
$561$	1.13587e7	+	9.40395e6i	1.52377	+	1.26155i
$562$		0			0
$563$	−1.46722e7			−1.95085			−0.975426	−	0.220326i	$-0.929288\pi$
$563$	−1.46722e7			−1.95085			−0.975426	+	0.220326i	$0.929288\pi$
$564$		0			0
$565$			3.97474e6i			0.523826i
$566$		0			0
$567$	7.04584e6	+	2.99307e6i	0.920397	+	0.390984i
$568$		0			0
$569$			5.12118e6i			0.663116i	0.943435	+	0.331558i	$0.107574\pi$
$569$			5.12118e6i			0.663116i	−0.943435	+	0.331558i	$0.892426\pi$
$570$		0			0
$571$	−4.08710e6			−0.524596			−0.262298	−	0.964987i	$-0.584480\pi$
$571$	−4.08710e6			−0.524596			−0.262298	+	0.964987i	$0.584480\pi$
$572$		0			0
$573$	−3.56613e6	−	2.95243e6i	−0.453743	−	0.375658i
$574$		0			0
$575$			550991.i			0.0694984i
$576$		0			0
$577$			2.58345e6i			0.323043i	0.986869	+	0.161521i	$0.0516401\pi$
$577$			2.58345e6i			0.323043i	−0.986869	+	0.161521i	$0.948360\pi$
$578$		0			0
$579$	4.11648e6	−	4.97214e6i	0.510305	−	0.616378i
$580$		0			0
$581$	1.43964e6	+	1.41617e6i	0.176935	+	0.174050i
$582$		0			0
$583$	−4.51955e6			−0.550711
$584$		0			0
$585$	−4.76402e6	+	905049.i	−0.575551	+	0.109341i
$586$		0			0
$587$	9.72773e6			1.16524			0.582621	−	0.812744i	$-0.302027\pi$
$587$	9.72773e6			1.16524			0.582621	+	0.812744i	$0.302027\pi$
$588$		0			0
$589$	−350100.			−0.0415819
$590$		0			0
$591$	−2.60484e6	−	2.15657e6i	−0.306770	−	0.253978i
$592$		0			0
$593$	9.57397e6			1.11803			0.559017	−	0.829156i	$-0.311179\pi$
$593$	9.57397e6			1.11803			0.559017	+	0.829156i	$0.311179\pi$
$594$		0			0
$595$	8.78149e6	+	8.63830e6i	1.01689	+	1.00031i
$596$		0			0
$597$	−440866.	−	364997.i	−0.0506257	−	0.0419135i
$598$		0			0
$599$			5.71115e6i			0.650364i	0.945651	+	0.325182i	$0.105426\pi$
$599$			5.71115e6i			0.650364i	−0.945651	+	0.325182i	$0.894574\pi$
$600$		0			0
$601$			1.49010e6i			0.168279i	0.996454	+	0.0841395i	$0.0268141\pi$
$601$			1.49010e6i			0.168279i	−0.996454	+	0.0841395i	$0.973186\pi$
$602$		0			0
$603$	656308.	+	3.45469e6i	0.0735046	+	0.386915i
$604$		0			0
$605$	−1.34448e7			−1.49336
$606$		0			0
$607$			1.13768e7i			1.25328i	0.779308	+	0.626641i	$0.215571\pi$
$607$			1.13768e7i			1.25328i	−0.779308	+	0.626641i	$0.784429\pi$
$608$		0			0
$609$	−105778.	−	1.03253e6i	−0.0115572	−	0.112813i
$610$		0			0
$611$		−	5.68431e6i		−	0.615991i
$612$		0			0
$613$	8.50093e6			0.913724			0.456862	−	0.889537i	$-0.348973\pi$
$613$	8.50093e6			0.913724			0.456862	+	0.889537i	$0.348973\pi$
$614$		0			0
$615$	4.30342e6	−	5.19794e6i	0.458803	−	0.554170i
$616$		0			0
$617$			1.82563e7i			1.93064i	0.261073	+	0.965319i	$0.415924\pi$
$617$			1.82563e7i			1.93064i	−0.261073	+	0.965319i	$0.584076\pi$
$618$		0			0
$619$			1.24791e7i			1.30905i	0.756041	+	0.654524i	$0.227131\pi$
$619$			1.24791e7i			1.30905i	−0.756041	+	0.654524i	$0.772869\pi$
$620$		0			0
$621$	1.45072e6	−	2.63177e6i	0.150957	−	0.273854i
$622$		0			0
$623$	−8.32017e6	−	8.18450e6i	−0.858840	−	0.844836i
$624$		0			0
$625$	−1.14537e7			−1.17285
$626$		0			0
$627$	1.32832e7	−	1.60443e7i	1.34938	−	1.62987i
$628$		0			0
$629$	1.32116e7			1.33146
$630$		0			0
$631$	1.52412e7			1.52387			0.761933	−	0.647656i	$-0.224251\pi$
$631$	1.52412e7			1.52387			0.761933	+	0.647656i	$0.224251\pi$
$632$		0			0
$633$	5.83950e6	−	7.05331e6i	0.579250	−	0.699654i
$634$		0			0
$635$	1.81342e7			1.78469
$636$		0			0
$637$	5.42616e6	−	89212.6i	0.529839	−	0.00871119i
$638$		0			0
$639$	1.75954e6	−	334271.i	0.170470	−	0.0323852i
$640$		0			0
$641$		−	3.06269e6i		−	0.294414i	−0.989106	−	0.147207i	$-0.952972\pi$
$641$		−	3.06269e6i		−	0.294414i	0.989106	−	0.147207i	$-0.0470283\pi$
$642$		0			0
$643$		−	4.16015e6i		−	0.396809i	−0.980120	−	0.198404i	$-0.936424\pi$
$643$		−	4.16015e6i		−	0.396809i	0.980120	−	0.198404i	$-0.0635760\pi$
$644$		0			0
$645$	4.74699e6	−	5.73370e6i	0.449282	−	0.542670i
$646$		0			0
$647$	1.44997e7			1.36175			0.680877	−	0.732398i	$-0.261599\pi$
$647$	1.44997e7			1.36175			0.680877	+	0.732398i	$0.261599\pi$
$648$		0			0
$649$			1.44083e7i			1.34277i
$650$		0			0
$651$	−33203.2	−	324106.i	−0.00307063	−	0.0299733i
$652$		0			0
$653$		−	5.47090e6i		−	0.502084i	−0.967976	−	0.251042i	$-0.919227\pi$
$653$		−	5.47090e6i		−	0.502084i	0.967976	−	0.251042i	$-0.0807732\pi$
$654$		0			0
$655$	−9.84652e6			−0.896766
$656$		0			0
$657$	−2.24948e6	+	427347.i	−0.203314	+	0.0386249i
$658$		0			0
$659$			1.88834e7i			1.69381i	0.531741	+	0.846907i	$0.321538\pi$
$659$			1.88834e7i			1.69381i	−0.531741	+	0.846907i	$0.678462\pi$
$660$		0			0
$661$		−	4.64223e6i		−	0.413260i	−0.978419	−	0.206630i	$-0.933750\pi$
$661$		−	4.64223e6i		−	0.413260i	0.978419	−	0.206630i	$-0.0662496\pi$
$662$		0			0
$663$	5.96075e6	+	4.93496e6i	0.526644	+	0.436014i
$664$		0			0
$665$	1.22017e7	−	1.24040e7i	1.06996	−	1.08770i
$666$		0			0
$667$	−407451.			−0.0354618
$668$		0			0
$669$	−4.20031e6	−	3.47748e6i	−0.362841	−	0.300400i
$670$		0			0
$671$	−8.21109e6			−0.704036
$672$		0			0
$673$	629529.			0.0535770			0.0267885	−	0.999641i	$-0.491472\pi$
$673$	629529.			0.0535770			0.0267885	+	0.999641i	$0.491472\pi$
$674$		0			0
$675$	−2.30401e6	−	1.27005e6i	−0.194637	−	0.107291i
$676$		0			0
$677$	6.49528e6			0.544661			0.272331	−	0.962204i	$-0.412206\pi$
$677$	6.49528e6			0.544661			0.272331	+	0.962204i	$0.412206\pi$
$678$		0			0
$679$	−7.71406e6	+	7.84193e6i	−0.642109	+	0.652753i
$680$		0			0
$681$	9.19603e6	−	1.11075e7i	0.759858	−	0.917803i
$682$		0			0
$683$		−	8.25581e6i		−	0.677186i	−0.940933	−	0.338593i	$-0.890049\pi$
$683$		−	8.25581e6i		−	0.677186i	0.940933	−	0.338593i	$-0.109951\pi$
$684$		0			0
$685$			2.39762e7i			1.95233i
$686$		0			0
$687$	1.63350e6	+	1.35239e6i	0.132046	+	0.109322i
$688$		0			0
$689$	−2.37175e6			−0.190336
$690$		0			0
$691$			2.21202e7i			1.76235i	0.472787	+	0.881177i	$0.343248\pi$
$691$			2.21202e7i			1.76235i	−0.472787	+	0.881177i	$0.656752\pi$
$692$		0			0
$693$	1.61128e7	+	1.07754e7i	1.27450	+	0.852313i
$694$		0			0
$695$		−	2.00436e7i		−	1.57403i
$696$		0			0
$697$	−1.07689e7			−0.839633
$698$		0			0
$699$	−1.36600e7	−	1.13093e7i	−1.05745	−	0.875472i
$700$		0			0
$701$		−	4.20151e6i		−	0.322932i	−0.986878	−	0.161466i	$-0.948378\pi$
$701$		−	4.20151e6i		−	0.322932i	0.986878	−	0.161466i	$-0.0516222\pi$
$702$		0			0
$703$		−	1.86616e7i		−	1.42417i
$704$		0			0
$705$	1.08156e7	−	1.30638e7i	0.819556	−	0.989910i
$706$		0			0
$707$	7.47629e6	+	7.35438e6i	0.562519	+	0.553347i
$708$		0			0
$709$	1.27013e7			0.948928			0.474464	−	0.880275i	$-0.342642\pi$
$709$	1.27013e7			0.948928			0.474464	+	0.880275i	$0.342642\pi$
$710$		0			0
$711$	1.23495e6	+	6.50058e6i	0.0916172	+	0.482257i
$712$		0			0
$713$	−127897.			−0.00942185
$714$		0			0
$715$	−1.22787e7			−0.898233
$716$		0			0
$717$	−1.25389e7	−	1.03810e7i	−0.910879	−	0.754125i
$718$		0			0
$719$	1.21200e7			0.874337			0.437169	−	0.899380i	$-0.355981\pi$
$719$	1.21200e7			0.874337			0.437169	+	0.899380i	$0.355981\pi$
$720$		0			0
$721$	−1.95303e7	+	1.98541e7i	−1.39917	+	1.42237i
$722$		0			0
$723$	−6.04560e6	−	5.00521e6i	−0.430124	−	0.356104i
$724$		0			0
$725$			356708.i			0.0252039i
$726$		0			0
$727$			3.77140e6i			0.264647i	0.991207	+	0.132323i	$0.0422437\pi$
$727$			3.77140e6i			0.264647i	−0.991207	+	0.132323i	$0.957756\pi$
$728$		0			0
$729$	7.66100e6	+	1.21326e7i	0.533908	+	0.845542i
$730$		0			0
$731$	−1.18789e7			−0.822209
$732$		0			0
$733$			1.16976e7i			0.804148i	0.915607	+	0.402074i	$0.131711\pi$
$733$			1.16976e7i			0.804148i	−0.915607	+	0.402074i	$0.868289\pi$
$734$		0			0
$735$	1.26402e7	+	1.01194e7i	0.863052	+	0.690935i
$736$		0			0
$737$			8.90409e6i			0.603838i
$738$		0			0
$739$	1.02243e6			0.0688689			0.0344344	−	0.999407i	$-0.489037\pi$
$739$	1.02243e6			0.0688689			0.0344344	+	0.999407i	$0.489037\pi$
$740$		0			0
$741$	6.97071e6	−	8.41965e6i	0.466371	−	0.563312i
$742$		0			0
$743$			1.29463e7i			0.860345i	0.902747	+	0.430173i	$0.141547\pi$
$743$			1.29463e7i			0.860345i	−0.902747	+	0.430173i	$0.858453\pi$
$744$		0			0
$745$		−	1.97116e7i		−	1.30116i
$746$		0			0
$747$	706463.	+	3.71870e6i	0.0463221	+	0.243831i
$748$		0			0
$749$	−1.09103e7	+	1.10912e7i	−0.710612	+	0.722391i
$750$		0			0
$751$	2.90253e7			1.87792			0.938959	−	0.344028i	$-0.111792\pi$
$751$	2.90253e7			1.87792			0.938959	+	0.344028i	$0.111792\pi$
$752$		0			0
$753$	1.67205e7	−	2.01960e7i	1.07464	−	1.29801i
$754$		0			0
$755$	−2.96287e7			−1.89167
$756$		0			0
$757$	−1.29941e7			−0.824148			−0.412074	−	0.911150i	$-0.635195\pi$
$757$	−1.29941e7			−0.824148			−0.412074	+	0.911150i	$0.635195\pi$
$758$		0			0
$759$	4.85256e6	−	5.86122e6i	0.305750	−	0.369304i
$760$		0			0
$761$	−1.68382e7			−1.05398			−0.526991	−	0.849871i	$-0.676680\pi$
$761$	−1.68382e7			−1.05398			−0.526991	+	0.849871i	$0.676680\pi$
$762$		0			0
$763$	−2.69908e6	−	2.65506e6i	−0.167843	−	0.165106i
$764$		0			0
$765$	4.30927e6	+	2.26832e7i	0.266226	+	1.40137i
$766$		0			0
$767$			7.56111e6i			0.464085i
$768$		0			0
$769$			2.53765e7i			1.54745i	0.633525	+	0.773723i	$0.281608\pi$
$769$			2.53765e7i			1.54745i	−0.633525	+	0.773723i	$0.718392\pi$
$770$		0			0
$771$	9.48114e6	−	1.14519e7i	0.574413	−	0.693812i
$772$		0			0
$773$	3.00362e6			0.180799			0.0903996	−	0.995906i	$-0.471186\pi$
$773$	3.00362e6			0.180799			0.0903996	+	0.995906i	$0.471186\pi$
$774$		0			0
$775$			111969.i			0.00669643i
$776$		0			0
$777$	1.72761e7	−	1.76985e6i	1.02658	−	0.105168i
$778$		0			0
$779$			1.52112e7i			0.898093i
$780$		0			0
$781$	4.53503e6			0.266043
$782$		0			0
$783$	939186.	−	1.70379e6i	0.0547454	−	0.0993142i
$784$		0			0
$785$		−	2.00117e7i		−	1.15907i
$786$		0			0
$787$		−	9.42926e6i		−	0.542676i	−0.962484	−	0.271338i	$-0.912534\pi$
$787$		−	9.42926e6i		−	0.542676i	0.962484	−	0.271338i	$-0.0874662\pi$
$788$		0			0
$789$	1.97484e7	+	1.63499e7i	1.12938	+	0.935023i
$790$		0			0
$791$	5.84701e6	−	5.94393e6i	0.332271	−	0.337779i
$792$		0			0
$793$	−4.30898e6			−0.243328
$794$		0			0
$795$	−5.45079e6	−	4.51276e6i	−0.305873	−	0.253235i
$796$		0			0
$797$	−6.70033e6			−0.373637			−0.186819	−	0.982394i	$-0.559818\pi$
$797$	−6.70033e6			−0.373637			−0.186819	+	0.982394i	$0.559818\pi$
$798$		0			0
$799$	−2.70651e7			−1.49983
$800$		0			0
$801$	−4.08289e6	−	2.14916e7i	−0.224847	−	1.18355i
$802$		0			0
$803$	−5.79778e6			−0.317302
$804$		0			0
$805$	4.45748e6	−	4.53137e6i	0.242437	−	0.246456i
$806$		0			0
$807$	765190.	−	924244.i	0.0413605	−	0.0499577i
$808$		0			0
$809$			2.51823e7i			1.35277i	0.736547	+	0.676386i	$0.236455\pi$
$809$			2.51823e7i			1.35277i	−0.736547	+	0.676386i	$0.763545\pi$
$810$		0			0
$811$			2.59174e6i			0.138369i	0.997604	+	0.0691846i	$0.0220397\pi$
$811$			2.59174e6i			0.138369i	−0.997604	+	0.0691846i	$0.977960\pi$
$812$		0			0
$813$	−2.15887e7	−	1.78735e7i	−1.14551	−	0.948380i
$814$		0			0
$815$	1.32404e6			0.0698245
$816$		0			0
$817$			1.67791e7i			0.879455i
$818$		0			0
$819$	8.45562e6	+	5.65465e6i	0.440490	+	0.294575i
$820$		0			0
$821$		−	2.39963e7i		−	1.24247i	−0.783623	−	0.621237i	$-0.786630\pi$
$821$		−	2.39963e7i		−	1.24247i	0.783623	−	0.621237i	$-0.213370\pi$
$822$		0			0
$823$	1.47240e7			0.757748			0.378874	−	0.925448i	$-0.376311\pi$
$823$	1.47240e7			0.757748			0.378874	+	0.925448i	$0.376311\pi$
$824$		0			0
$825$	−5.13128e6	−	4.24824e6i	−0.262477	−	0.217307i
$826$		0			0
$827$		−	2.95165e7i		−	1.50072i	−0.661028	−	0.750361i	$-0.729879\pi$
$827$		−	2.95165e7i		−	1.50072i	0.661028	−	0.750361i	$-0.270121\pi$
$828$		0			0
$829$		−	3.36080e7i		−	1.69846i	−0.528019	−	0.849232i	$-0.677065\pi$
$829$		−	3.36080e7i		−	1.69846i	0.528019	−	0.849232i	$-0.322935\pi$
$830$		0			0
$831$	1.15417e7	−	1.39408e7i	0.579785	−	0.700300i
$832$		0			0
$833$	−424774.	−	2.58359e7i	−0.0212102	−	1.29007i
$834$		0			0
$835$	−2.69647e7			−1.33838
$836$		0			0
$837$	294806.	−	534811.i	0.0145453	−	0.0263868i
$838$		0			0
$839$	−6.45947e6			−0.316805			−0.158402	−	0.987375i	$-0.550634\pi$
$839$	−6.45947e6			−0.316805			−0.158402	+	0.987375i	$0.550634\pi$
$840$		0			0
$841$	2.02474e7			0.987140
$842$		0			0
$843$	−1.80147e7	−	1.49146e7i	−0.873089	−	0.722839i
$844$		0			0
$845$	1.65032e7			0.795108
$846$		0			0
$847$	2.01057e7	+	1.97779e7i	0.962965	+	0.947263i
$848$		0			0
$849$	−9.30955e6	−	7.70746e6i	−0.443261	−	0.366980i
$850$		0			0
$851$		−	6.81737e6i		−	0.322696i
$852$		0			0
$853$		−	4.03008e7i		−	1.89645i	−0.317602	−	0.948224i	$-0.602877\pi$
$853$		−	4.03008e7i		−	1.89645i	0.317602	−	0.948224i	$-0.397123\pi$
$854$		0			0
$855$	3.20404e7	−	6.08691e6i	1.49893	−	0.284762i
$856$		0			0
$857$	−9.87862e6			−0.459456			−0.229728	−	0.973255i	$-0.573784\pi$
$857$	−9.87862e6			−0.459456			−0.229728	+	0.973255i	$0.573784\pi$
$858$		0			0
$859$		−	4.71083e6i		−	0.217829i	−0.994051	−	0.108914i	$-0.965263\pi$
$859$		−	4.71083e6i		−	0.217829i	0.994051	−	0.108914i	$-0.0347374\pi$
$860$		0			0
$861$	−1.40819e7	+	1.44262e6i	−0.647369	+	0.0663200i
$862$		0			0
$863$		−	8.17467e6i		−	0.373632i	−0.982395	−	0.186816i	$-0.940183\pi$
$863$		−	8.17467e6i		−	0.373632i	0.982395	−	0.186816i	$-0.0598167\pi$
$864$		0			0
$865$	−1.14693e7			−0.521192
$866$		0			0
$867$	9.38236e6	−	1.13326e7i	0.423901	−	0.512014i
$868$		0			0
$869$			1.67545e7i			0.752633i
$870$		0			0
$871$			4.67264e6i			0.208698i
$872$		0			0
$873$	−2.02563e7	+	3.84821e6i	−0.899548	+	0.170892i
$874$		0			0
$875$	1.38824e7	+	1.36561e7i	0.612979	+	0.602984i
$876$		0			0
$877$	−1.16202e7			−0.510169			−0.255084	−	0.966919i	$-0.582103\pi$
$877$	−1.16202e7			−0.510169			−0.255084	+	0.966919i	$0.582103\pi$
$878$		0			0
$879$	1.92889e6	−	2.32984e6i	0.0842047	−	0.101708i
$880$		0			0
$881$	−4.28179e7			−1.85860			−0.929300	−	0.369327i	$-0.879588\pi$
$881$	−4.28179e7			−1.85860			−0.929300	+	0.369327i	$0.879588\pi$
$882$		0			0
$883$	−2.20297e7			−0.950839			−0.475420	−	0.879759i	$-0.657704\pi$
$883$	−2.20297e7			−0.950839			−0.475420	+	0.879759i	$0.657704\pi$
$884$		0			0
$885$	−1.43866e7	+	1.73771e7i	−0.617449	+	0.745793i
$886$		0			0
$887$	−1.86327e7			−0.795181			−0.397590	−	0.917563i	$-0.630153\pi$
$887$	−1.86327e7			−0.795181			−0.397590	+	0.917563i	$0.630153\pi$
$888$		0			0
$889$	−2.71184e7	−	2.66762e7i	−1.15082	−	1.13206i
$890$		0			0
$891$	1.33239e7	+	3.38017e7i	0.562260	+	1.42641i
$892$		0			0
$893$			3.82298e7i			1.60426i
$894$		0			0
$895$			1.05534e7i			0.440388i
$896$		0			0
$897$	2.54651e6	−	3.07583e6i	0.105673	−	0.127638i
$898$		0			0
$899$	−82799.6			−0.00341687
$900$		0			0
$901$			1.12927e7i			0.463434i
$902$		0			0
$903$	−1.55333e7	+	1.59132e6i	−0.633935	+	0.0649437i
$904$		0			0
$905$			2.01652e7i			0.818429i
$906$		0			0
$907$	−5.81361e6			−0.234654			−0.117327	−	0.993093i	$-0.537433\pi$
$907$	−5.81361e6			−0.234654			−0.117327	+	0.993093i	$0.537433\pi$
$908$		0			0
$909$	3.66878e6	+	1.93118e7i	0.147269	+	0.775198i
$910$		0			0
$911$		−	2.69096e7i		−	1.07426i	−0.843498	−	0.537132i	$-0.819508\pi$
$911$		−	2.69096e7i		−	1.07426i	0.843498	−	0.537132i	$-0.180492\pi$
$912$		0			0
$913$			9.58454e6i			0.380535i
$914$		0			0
$915$	−9.90297e6	−	8.19876e6i	−0.391032	−	0.323739i
$916$		0			0
$917$	1.47248e7	+	1.44847e7i	0.578262	+	0.568833i
$918$		0			0
$919$	−1.86255e7			−0.727476			−0.363738	−	0.931501i	$-0.618500\pi$
$919$	−1.86255e7			−0.727476			−0.363738	+	0.931501i	$0.618500\pi$
$920$		0			0
$921$	4.73881e6	+	3.92331e6i	0.184086	+	0.152406i
$922$		0			0
$923$	2.37987e6			0.0919494
$924$		0			0
$925$	−5.96835e6			−0.229351
$926$		0			0
$927$	−5.12845e7	+	9.74283e6i	−1.96014	+	0.372380i
$928$		0			0
$929$	2.43516e7			0.925740			0.462870	−	0.886426i	$-0.346820\pi$
$929$	2.43516e7			0.925740			0.462870	+	0.886426i	$0.346820\pi$
$930$		0			0
$931$	−3.64936e7	+	600000.i	−1.37989	+	0.0226870i
$932$		0			0
$933$	−2.28402e7	+	2.75878e7i	−0.859004	+	1.03756i
$934$		0			0
$935$			5.84636e7i			2.18704i
$936$		0			0
$937$			6.06143e6i			0.225541i	0.993621	+	0.112771i	$0.0359725\pi$
$937$			6.06143e6i			0.225541i	−0.993621	+	0.112771i	$0.964027\pi$
$938$		0			0
$939$	7.62298e6	+	6.31114e6i	0.282137	+	0.233584i
$940$		0			0
$941$	−1.76788e6			−0.0650847			−0.0325424	−	0.999470i	$-0.510360\pi$
$941$	−1.76788e6			−0.0650847			−0.0325424	+	0.999470i	$0.510360\pi$
$942$		0			0
$943$			5.55690e6i			0.203495i
$944$		0			0
$945$	8.67366e6	+	2.90842e7i	0.315953	+	1.05944i
$946$		0			0
$947$			4.74165e7i			1.71812i	0.511872	+	0.859061i	$0.328952\pi$
$947$			4.74165e7i			1.71812i	−0.511872	+	0.859061i	$0.671048\pi$
$948$		0			0
$949$	−3.04253e6			−0.109665
$950$		0			0
$951$	2.31589e6	+	1.91735e6i	0.0830362	+	0.0687464i
$952$		0			0
$953$		−	3.22073e7i		−	1.14874i	−0.818596	−	0.574370i	$-0.805247\pi$
$953$		−	3.22073e7i		−	1.14874i	0.818596	−	0.574370i	$-0.194753\pi$
$954$		0			0
$955$		−	1.83550e7i		−	0.651247i
$956$		0			0
$957$	3.14152e6	−	3.79452e6i	0.110882	−	0.133930i
$958$		0			0
$959$	3.52701e7	−	3.58547e7i	1.23840	−	1.25893i
$960$		0			0
$961$	2.86032e7			0.999092
$962$		0			0
$963$	−2.86493e7	+	5.44267e6i	−0.995515	+	0.189124i
$964$		0			0
$965$	2.55918e7			0.884674
$966$		0			0
$967$	−2.97124e7			−1.02181			−0.510906	−	0.859637i	$-0.670690\pi$
$967$	−2.97124e7			−1.02181			−0.510906	+	0.859637i	$0.670690\pi$
$968$		0			0
$969$	−4.00890e7	−	3.31901e7i	−1.37156	−	1.13553i
$970$		0			0
$971$	−1.54497e7			−0.525861			−0.262931	−	0.964815i	$-0.584689\pi$
$971$	−1.54497e7			−0.525861			−0.262931	+	0.964815i	$0.584689\pi$
$972$		0			0
$973$	−2.94850e7	+	2.99737e7i	−0.998432	+	1.01498i
$974$		0			0
$975$	−2.69277e6	−	2.22937e6i	−0.0907168	−	0.0751053i
$976$		0			0
$977$		−	2.78966e7i		−	0.935008i	−0.883991	−	0.467504i	$-0.845153\pi$
$977$		−	2.78966e7i		−	0.935008i	0.883991	−	0.467504i	$-0.154847\pi$
$978$		0			0
$979$		−	5.53923e7i		−	1.84711i
$980$		0			0
$981$	−1.32449e6	−	6.97191e6i	−0.0439418	−	0.231302i
$982$		0			0
$983$	−3.66294e6			−0.120905			−0.0604527	−	0.998171i	$-0.519254\pi$
$983$	−3.66294e6			−0.120905			−0.0604527	+	0.998171i	$0.519254\pi$
$984$		0			0
$985$		−	1.34072e7i		−	0.440300i
$986$		0			0
$987$	−3.53914e7	+	3.62568e6i	−1.15639	+	0.118467i
$988$		0			0
$989$			6.12966e6i			0.199272i
$990$		0			0
$991$	−5.89627e6			−0.190719			−0.0953594	−	0.995443i	$-0.530400\pi$
$991$	−5.89627e6			−0.190719			−0.0953594	+	0.995443i	$0.530400\pi$
$992$		0			0
$993$	2.83789e7	−	3.42778e7i	0.913320	−	1.10316i
$994$		0			0
$995$		−	2.26916e6i		−	0.0726619i
$996$		0			0
$997$			1.33192e7i			0.424366i	0.977230	+	0.212183i	$0.0680572\pi$
$997$			1.33192e7i			0.424366i	−0.977230	+	0.212183i	$0.931943\pi$
$998$		0			0
$999$	2.85074e7	+	1.57142e7i	0.903741	+	0.498173i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.6.k.d.209.9		12
3.2	odd	2	inner	336.6.k.d.209.3		12
4.3	odd	2		21.6.c.a.20.5	✓	12
7.6	odd	2	inner	336.6.k.d.209.4		12
12.11	even	2		21.6.c.a.20.8	yes	12
21.20	even	2	inner	336.6.k.d.209.10		12
28.27	even	2		21.6.c.a.20.6	yes	12
84.83	odd	2		21.6.c.a.20.7	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.6.c.a.20.5	✓	12	4.3	odd	2
21.6.c.a.20.6	yes	12	28.27	even	2
21.6.c.a.20.7	yes	12	84.83	odd	2
21.6.c.a.20.8	yes	12	12.11	even	2
336.6.k.d.209.3		12	3.2	odd	2	inner
336.6.k.d.209.4		12	7.6	odd	2	inner
336.6.k.d.209.9		12	1.1	even	1	trivial
336.6.k.d.209.10		12	21.20	even	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.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(9.94100 - 12.0074i) q^{3} +61.8023 q^{5} +(-92.4210 - 90.9140i) q^{7} +(-45.3530 - 238.730i) q^{9} +O(q^{10})\)	\(q+(9.94100 - 12.0074i) q^{3} +61.8023 q^{5} +(-92.4210 - 90.9140i) q^{7} +(-45.3530 - 238.730i) q^{9} -615.301i q^{11} -322.895i q^{13} +(614.377 - 742.083i) q^{15} -1537.42 q^{17} +2171.63i q^{19} +(-2010.39 + 205.956i) q^{21} +793.329i q^{23} +694.530 q^{25} +(-3317.37 - 1828.65i) q^{27} +513.596i q^{29} +161.215i q^{31} +(-7388.14 - 6116.71i) q^{33} +(-5711.84 - 5618.70i) q^{35} -8593.37 q^{37} +(-3877.11 - 3209.90i) q^{39} +7004.53 q^{41} +7726.50 q^{43} +(-2802.92 - 14754.1i) q^{45} +17604.2 q^{47} +(276.290 + 16804.7i) q^{49} +(-15283.5 + 18460.3i) q^{51} -7345.26i q^{53} -38027.1i q^{55} +(26075.5 + 21588.2i) q^{57} -23416.6 q^{59} -13344.8i q^{61} +(-17512.3 + 26186.9i) q^{63} -19955.7i q^{65} -14471.1 q^{67} +(9525.78 + 7886.48i) q^{69} +7370.42i q^{71} -9422.68i q^{73} +(6904.32 - 8339.47i) q^{75} +(-55939.5 + 56866.8i) q^{77} -27229.8 q^{79} +(-54935.2 + 21654.3i) q^{81} -15577.0 q^{83} -95016.2 q^{85} +(6166.93 + 5105.66i) q^{87} +90024.7 q^{89} +(-29355.7 + 29842.3i) q^{91} +(1935.77 + 1602.64i) q^{93} +134212. i q^{95} -84850.1i q^{97} +(-146891. + 27905.8i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 112 q^{7} - 492 q^{9}+O(q^{10}) \) 12 * q - 112 * q^7 - 492 * q^9	\( 12 q - 112 q^{7} - 492 q^{9} - 1392 q^{15} + 4116 q^{21} + 7812 q^{25} + 27464 q^{37} + 16080 q^{39} + 73840 q^{43} + 6972 q^{49} - 23760 q^{51} + 103968 q^{57} + 120624 q^{63} - 79344 q^{67} + 247104 q^{79} - 248868 q^{81} - 320112 q^{85} - 310128 q^{91} + 397272 q^{93} - 696576 q^{99}+O(q^{100}) \) 12 * q - 112 * q^7 - 492 * q^9 - 1392 * q^15 + 4116 * q^21 + 7812 * q^25 + 27464 * q^37 + 16080 * q^39 + 73840 * q^43 + 6972 * q^49 - 23760 * q^51 + 103968 * q^57 + 120624 * q^63 - 79344 * q^67 + 247104 * q^79 - 248868 * q^81 - 320112 * q^85 - 310128 * q^91 + 397272 * q^93 - 696576 * q^99

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