LMFDB - Embedded newform 432.6.i.b.289.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [432,6,Mod(145,432)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(432, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

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

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

chi := DirichletCharacter("432.145");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 432 = 2^{4} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	432.i (of order $3$, degree $2$, 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:	$69.2858101592$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.47347183152.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} + 118x^{4} - 231x^{3} + 3700x^{2} - 3585x + 32331 $ x^6 - 3x^5 + 118x^4 - 231x^3 + 3700x^2 - 3585*x + 32331
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{9} $
Twist minimal:	no (minimal twist has level 18)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			289.2
Root			$0.500000 + 8.40123i$ of defining polynomial
Character	$\chi$	$=$	432.289
Dual form			432.6.i.b.145.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(20.8014 + 36.0292i) q^{5} +(-101.661 + 176.082i) q^{7} +O(q^{10})$	$q+(20.8014 + 36.0292i) q^{5} +(-101.661 + 176.082i) q^{7} +(-235.168 + 407.323i) q^{11} +(-241.506 - 418.300i) q^{13} -1259.86 q^{17} -1978.94 q^{19} +(-239.119 - 414.166i) q^{23} +(697.100 - 1207.41i) q^{25} +(-580.249 + 1005.02i) q^{29} +(-1186.50 - 2055.07i) q^{31} -8458.76 q^{35} +8185.10 q^{37} +(8758.87 + 15170.8i) q^{41} +(11435.0 - 19806.1i) q^{43} +(8685.43 - 15043.6i) q^{47} +(-12266.3 - 21245.9i) q^{49} +5390.58 q^{53} -19567.3 q^{55} +(22281.8 + 38593.2i) q^{59} +(-2084.17 + 3609.88i) q^{61} +(10047.3 - 17402.5i) q^{65} +(1228.46 + 2127.76i) q^{67} +2184.37 q^{71} +3037.34 q^{73} +(-47814.7 - 82817.5i) q^{77} +(-25266.2 + 43762.3i) q^{79} +(25913.9 - 44884.2i) q^{83} +(-26206.9 - 45391.7i) q^{85} +20154.7 q^{89} +98206.5 q^{91} +(-41164.9 - 71299.6i) q^{95} +(-40214.4 + 69653.3i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 54 q^{5} + 132 q^{7}+O(q^{10}) $ 6 * q + 54 * q^5 + 132 * q^7	$ 6 q + 54 q^{5} + 132 q^{7} - 315 q^{11} - 744 q^{13} - 2898 q^{17} - 2262 q^{19} - 3168 q^{23} - 2883 q^{25} + 5148 q^{29} + 8610 q^{31} + 2700 q^{35} + 39936 q^{37} - 5049 q^{41} + 31389 q^{43} + 12924 q^{47} - 52857 q^{49} + 96048 q^{53} - 126252 q^{55} + 62955 q^{59} - 75966 q^{61} - 108702 q^{65} + 32991 q^{67} - 129672 q^{71} - 8466 q^{73} - 88740 q^{77} - 89202 q^{79} + 32634 q^{83} + 71388 q^{85} - 66132 q^{89} + 301836 q^{91} - 82944 q^{95} + 46245 q^{97}+O(q^{100}) $ 6 * q + 54 * q^5 + 132 * q^7 - 315 * q^11 - 744 * q^13 - 2898 * q^17 - 2262 * q^19 - 3168 * q^23 - 2883 * q^25 + 5148 * q^29 + 8610 * q^31 + 2700 * q^35 + 39936 * q^37 - 5049 * q^41 + 31389 * q^43 + 12924 * q^47 - 52857 * q^49 + 96048 * q^53 - 126252 * q^55 + 62955 * q^59 - 75966 * q^61 - 108702 * q^65 + 32991 * q^67 - 129672 * q^71 - 8466 * q^73 - 88740 * q^77 - 89202 * q^79 + 32634 * q^83 + 71388 * q^85 - 66132 * q^89 + 301836 * q^91 - 82944 * q^95 + 46245 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/432\mathbb{Z}\right)^\times$.

$n$	$271$	$325$	$353$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	20.8014	+	36.0292i	0.372108	+	0.644509i	0.989890	−	0.141840i	$-0.0453019\pi$
$5$	20.8014	+	36.0292i	0.372108	+	0.644509i	−0.617782	+	0.786349i	$0.711969\pi$
$6$		0			0
$7$	−101.661	+	176.082i	−0.784166	+	1.35822i	0.145330	+	0.989383i	$0.453576\pi$
$7$	−101.661	+	176.082i	−0.784166	+	1.35822i	−0.929496	+	0.368832i	$0.879758\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−235.168	+	407.323i	−0.585998	+	1.01498i	0.408752	+	0.912646i	$0.365964\pi$
$11$	−235.168	+	407.323i	−0.585998	+	1.01498i	−0.994750	+	0.102333i	$0.967369\pi$
$12$		0			0
$13$	−241.506	−	418.300i	−0.396341	−	0.686482i	0.596931	−	0.802293i	$-0.296387\pi$
$13$	−241.506	−	418.300i	−0.396341	−	0.686482i	−0.993271	+	0.115811i	$0.963053\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1259.86			−1.05730			−0.528652	−	0.848839i	$-0.677302\pi$
$17$	−1259.86			−1.05730			−0.528652	+	0.848839i	$0.677302\pi$
$18$		0			0
$19$	−1978.94			−1.25762			−0.628810	−	0.777559i	$-0.716458\pi$
$19$	−1978.94			−1.25762			−0.628810	+	0.777559i	$0.716458\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−239.119	−	414.166i	−0.0942529	−	0.163251i	0.815044	−	0.579400i	$-0.196713\pi$
$23$	−239.119	−	414.166i	−0.0942529	−	0.163251i	−0.909297	+	0.416149i	$0.863380\pi$
$24$		0			0
$25$	697.100	−	1207.41i	0.223072	−	0.386372i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−580.249	+	1005.02i	−0.128121	+	0.221912i	−0.922949	−	0.384923i	$-0.874228\pi$
$29$	−580.249	+	1005.02i	−0.128121	+	0.221912i	0.794828	+	0.606835i	$0.207561\pi$
$30$		0			0
$31$	−1186.50	−	2055.07i	−0.221749	−	0.384081i	0.733590	−	0.679592i	$-0.237843\pi$
$31$	−1186.50	−	2055.07i	−0.221749	−	0.384081i	−0.955339	+	0.295511i	$0.904510\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−8458.76			−1.16718
$36$		0			0
$37$	8185.10			0.982923			0.491462	−	0.870899i	$-0.336463\pi$
$37$	8185.10			0.982923			0.491462	+	0.870899i	$0.336463\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	8758.87	+	15170.8i	0.813745	+	1.40945i	0.910226	+	0.414113i	$0.135908\pi$
$41$	8758.87	+	15170.8i	0.813745	+	1.40945i	−0.0964809	+	0.995335i	$0.530759\pi$
$42$		0			0
$43$	11435.0	−	19806.1i	0.943119	−	1.63353i	0.183644	−	0.982993i	$-0.441211\pi$
$43$	11435.0	−	19806.1i	0.943119	−	1.63353i	0.759475	−	0.650537i	$-0.225456\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	8685.43	−	15043.6i	0.573518	−	0.993362i	−0.422683	−	0.906277i	$-0.638912\pi$
$47$	8685.43	−	15043.6i	0.573518	−	0.993362i	0.996201	−	0.0870843i	$-0.0277550\pi$
$48$		0			0
$49$	−12266.3	−	21245.9i	−0.729833	−	1.26411i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	5390.58			0.263600			0.131800	−	0.991276i	$-0.457924\pi$
$53$	5390.58			0.263600			0.131800	+	0.991276i	$0.457924\pi$
$54$		0			0
$55$	−19567.3			−0.872218
$56$		0			0
$57$		0			0
$58$		0			0
$59$	22281.8	+	38593.2i	0.833335	+	1.44338i	0.895379	+	0.445306i	$0.146905\pi$
$59$	22281.8	+	38593.2i	0.833335	+	1.44338i	−0.0620431	+	0.998073i	$0.519762\pi$
$60$		0			0
$61$	−2084.17	+	3609.88i	−0.0717147	+	0.124213i	−0.899653	−	0.436606i	$-0.856180\pi$
$61$	−2084.17	+	3609.88i	−0.0717147	+	0.124213i	0.827938	+	0.560819i	$0.189514\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	10047.3	−	17402.5i	0.294963	−	0.510891i
$66$		0			0
$67$	1228.46	+	2127.76i	0.0334330	+	0.0579076i	0.882258	−	0.470767i	$-0.156023\pi$
$67$	1228.46	+	2127.76i	0.0334330	+	0.0579076i	−0.848825	+	0.528674i	$0.822689\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	2184.37			0.0514256			0.0257128	−	0.999669i	$-0.491814\pi$
$71$	2184.37			0.0514256			0.0257128	+	0.999669i	$0.491814\pi$
$72$		0			0
$73$	3037.34			0.0667093			0.0333546	−	0.999444i	$-0.489381\pi$
$73$	3037.34			0.0667093			0.0333546	+	0.999444i	$0.489381\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−47814.7	−	82817.5i	−0.919040	−	1.59182i
$78$		0			0
$79$	−25266.2	+	43762.3i	−0.455482	+	0.788918i	−0.998716	−	0.0506633i	$-0.983866\pi$
$79$	−25266.2	+	43762.3i	−0.455482	+	0.788918i	0.543234	+	0.839582i	$0.317200\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	25913.9	−	44884.2i	0.412893	−	0.715152i	−0.582311	−	0.812966i	$-0.697852\pi$
$83$	25913.9	−	44884.2i	0.412893	−	0.715152i	0.995205	+	0.0978135i	$0.0311849\pi$
$84$		0			0
$85$	−26206.9	−	45391.7i	−0.393431	−	0.681442i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	20154.7			0.269713			0.134857	−	0.990865i	$-0.456943\pi$
$89$	20154.7			0.269713			0.134857	+	0.990865i	$0.456943\pi$
$90$		0			0
$91$	98206.5			1.24319
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−41164.9	−	71299.6i	−0.467970	−	0.810547i
$96$		0			0
$97$	−40214.4	+	69653.3i	−0.433962	+	0.751644i	−0.997210	−	0.0746433i	$-0.976218\pi$
$97$	−40214.4	+	69653.3i	−0.433962	+	0.751644i	0.563248	+	0.826288i	$0.309552\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	28450.2	−	49277.2i	0.277512	−	0.480665i	−0.693254	−	0.720694i	$-0.743823\pi$
$101$	28450.2	−	49277.2i	0.277512	−	0.480665i	0.970766	+	0.240028i	$0.0771568\pi$
$102$		0			0
$103$	−98505.0	−	170616.i	−0.914882	−	1.58462i	−0.807074	−	0.590451i	$-0.798950\pi$
$103$	−98505.0	−	170616.i	−0.914882	−	1.58462i	−0.107808	−	0.994172i	$-0.534383\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−23482.4			−0.198282			−0.0991411	−	0.995073i	$-0.531610\pi$
$107$	−23482.4			−0.198282			−0.0991411	+	0.995073i	$0.531610\pi$
$108$		0			0
$109$	−118550.			−0.955727			−0.477863	−	0.878434i	$-0.658589\pi$
$109$	−118550.			−0.955727			−0.477863	+	0.878434i	$0.658589\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−223.401	−	386.942i	−0.00164585	−	0.00285069i	0.865201	−	0.501425i	$-0.167191\pi$
$113$	−223.401	−	386.942i	−0.00164585	−	0.00285069i	−0.866847	+	0.498574i	$0.833857\pi$
$114$		0			0
$115$	9948.04	−	17230.5i	0.0701444	−	0.121494i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	128078.	−	221838.i	0.829102	−	1.43605i
$120$		0			0
$121$	−30082.4	−	52104.3i	−0.186788	−	0.323527i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	188012.			1.07624
$126$		0			0
$127$	−193803.			−1.06623			−0.533116	−	0.846042i	$-0.678979\pi$
$127$	−193803.			−1.06623			−0.533116	+	0.846042i	$0.678979\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−48037.8	−	83203.8i	−0.244571	−	0.423609i	0.717440	−	0.696620i	$-0.245314\pi$
$131$	−48037.8	−	83203.8i	−0.244571	−	0.423609i	−0.962011	+	0.273011i	$0.911980\pi$
$132$		0			0
$133$	201181.	−	348455.i	0.986183	−	1.70812i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	116089.	−	201071.i	0.528431	−	0.915269i	−0.471020	−	0.882123i	$-0.656114\pi$
$137$	116089.	−	201071.i	0.528431	−	0.915269i	0.999451	−	0.0331465i	$-0.0105528\pi$
$138$		0			0
$139$	−80284.0	−	139056.i	−0.352445	−	0.610453i	0.634232	−	0.773143i	$-0.281316\pi$
$139$	−80284.0	−	139056.i	−0.352445	−	0.610453i	−0.986677	+	0.162689i	$0.947983\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	227177.			0.929020
$144$		0			0
$145$	−48280.1			−0.190699
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−139592.	−	241781.i	−0.515105	−	0.892189i	−0.999846	−	0.0175309i	$-0.994419\pi$
$149$	−139592.	−	241781.i	−0.515105	−	0.892189i	0.484741	−	0.874658i	$-0.338914\pi$
$150$		0			0
$151$	−14288.8	+	24749.0i	−0.0509981	+	0.0883314i	−0.890398	−	0.455184i	$-0.849574\pi$
$151$	−14288.8	+	24749.0i	−0.0509981	+	0.0883314i	0.839399	+	0.543515i	$0.182907\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	49361.6	−	85496.9i	0.165029	−	0.285839i
$156$		0			0
$157$	62127.0	+	107607.i	0.201155	+	0.348411i	0.948901	−	0.315574i	$-0.102197\pi$
$157$	62127.0	+	107607.i	0.201155	+	0.348411i	−0.747746	+	0.663985i	$0.768864\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	97236.1			0.295640
$162$		0			0
$163$	−163892.			−0.483159			−0.241579	−	0.970381i	$-0.577665\pi$
$163$	−163892.			−0.483159			−0.241579	+	0.970381i	$0.577665\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	81164.7	+	140581.i	0.225204	+	0.390065i	0.956381	−	0.292123i	$-0.0943618\pi$
$167$	81164.7	+	140581.i	0.225204	+	0.390065i	−0.731177	+	0.682188i	$0.761028\pi$
$168$		0			0
$169$	68996.6	−	119506.i	0.185828	−	0.321863i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−192179.	+	332863.i	−0.488191	+	0.845572i	−0.999908	−	0.0135821i	$-0.995677\pi$
$173$	−192179.	+	332863.i	−0.488191	+	0.845572i	0.511716	+	0.859154i	$0.329010\pi$
$174$		0			0
$175$	141735.	+	245493.i	0.349851	+	0.605960i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−511991.			−1.19434			−0.597172	−	0.802113i	$-0.703709\pi$
$179$	−511991.			−1.19434			−0.597172	+	0.802113i	$0.703709\pi$
$180$		0			0
$181$	−285832.			−0.648507			−0.324254	−	0.945970i	$-0.605113\pi$
$181$	−285832.			−0.648507			−0.324254	+	0.945970i	$0.605113\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	170262.	+	294902.i	0.365753	+	0.633503i
$186$		0			0
$187$	296279.	−	513170.i	0.619578	−	1.07314i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−18626.1	+	32261.4i	−0.0369436	+	0.0639882i	−0.883906	−	0.467665i	$-0.845096\pi$
$191$	−18626.1	+	32261.4i	−0.0369436	+	0.0639882i	0.846962	+	0.531653i	$0.178429\pi$
$192$		0			0
$193$	−289378.	−	501217.i	−0.559206	−	0.968573i	−0.997563	−	0.0697722i	$-0.977773\pi$
$193$	−289378.	−	501217.i	−0.559206	−	0.968573i	0.438357	−	0.898801i	$-0.355561\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−234386.			−0.430294			−0.215147	−	0.976582i	$-0.569023\pi$
$197$	−234386.			−0.430294			−0.215147	+	0.976582i	$0.569023\pi$
$198$		0			0
$199$	−200551.			−0.358997			−0.179499	−	0.983758i	$-0.557448\pi$
$199$	−200551.			−0.358997			−0.179499	+	0.983758i	$0.557448\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−117977.	−	204342.i	−0.200936	−	0.348031i
$204$		0			0
$205$	−364394.	+	631149.i	−0.605601	+	1.04893i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	465384.	−	806069.i	0.736963	−	1.27646i
$210$		0			0
$211$	−269826.	−	467352.i	−0.417232	−	0.722667i	0.578428	−	0.815733i	$-0.303666\pi$
$211$	−269826.	−	467352.i	−0.417232	−	0.722667i	−0.995660	+	0.0930667i	$0.970333\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	951461.			1.40377
$216$		0			0
$217$	482480.			0.695553
$218$		0			0
$219$		0			0
$220$		0			0
$221$	304263.	+	526999.i	0.419053	+	0.725821i
$222$		0			0
$223$	442584.	−	766578.i	0.595983	−	1.03227i	−0.397424	−	0.917635i	$-0.630096\pi$
$223$	442584.	−	766578.i	0.595983	−	1.03227i	0.993407	−	0.114638i	$-0.0365708\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−159067.	+	275512.i	−0.204887	+	0.354875i	−0.950097	−	0.311955i	$-0.899016\pi$
$227$	−159067.	+	275512.i	−0.204887	+	0.354875i	0.745210	+	0.666830i	$0.232349\pi$
$228$		0			0
$229$	−246934.	−	427703.i	−0.311166	−	0.538956i	0.667449	−	0.744656i	$-0.267386\pi$
$229$	−246934.	−	427703.i	−0.311166	−	0.538956i	−0.978615	+	0.205700i	$0.934053\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−1.16189e6			−1.40208			−0.701041	−	0.713121i	$-0.747281\pi$
$233$	−1.16189e6			−1.40208			−0.701041	+	0.713121i	$0.747281\pi$
$234$		0			0
$235$	722678.			0.853641
$236$		0			0
$237$		0			0
$238$		0			0
$239$	579263.	+	1.00331e6i	0.655965	+	1.13617i	0.981651	+	0.190687i	$0.0610716\pi$
$239$	579263.	+	1.00331e6i	0.655965	+	1.13617i	−0.325685	+	0.945478i	$0.605595\pi$
$240$		0			0
$241$	410557.	−	711106.i	0.455335	−	0.788663i	−0.543372	−	0.839492i	$-0.682853\pi$
$241$	410557.	−	711106.i	0.455335	−	0.788663i	0.998707	+	0.0508285i	$0.0161862\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	510314.	−	883889.i	0.543153	−	0.940768i
$246$		0			0
$247$	477926.	+	827792.i	0.498446	+	0.863334i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−852357.			−0.853959			−0.426980	−	0.904261i	$-0.640422\pi$
$251$	−852357.			−0.853959			−0.426980	+	0.904261i	$0.640422\pi$
$252$		0			0
$253$	224933.			0.220928
$254$		0			0
$255$		0			0
$256$		0			0
$257$	650171.	+	1.12613e6i	0.614037	+	1.06354i	0.990553	+	0.137133i	$0.0437889\pi$
$257$	650171.	+	1.12613e6i	0.614037	+	1.06354i	−0.376515	+	0.926410i	$0.622878\pi$
$258$		0			0
$259$	−832103.	+	1.44124e6i	−0.770775	+	1.33502i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	572288.	−	991232.i	0.510182	−	0.883662i	−0.489748	−	0.871864i	$-0.662911\pi$
$263$	572288.	−	991232.i	0.510182	−	0.883662i	0.999930	−	0.0117977i	$-0.00375542\pi$
$264$		0			0
$265$	112132.	+	194218.i	0.0980876	+	0.169893i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.54095e6			−1.29840			−0.649199	−	0.760619i	$-0.724896\pi$
$269$	−1.54095e6			−1.29840			−0.649199	+	0.760619i	$0.724896\pi$
$270$		0			0
$271$	−1.42397e6			−1.17782			−0.588908	−	0.808200i	$-0.700442\pi$
$271$	−1.42397e6			−1.17782			−0.588908	+	0.808200i	$0.700442\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	327871.	+	567889.i	0.261440	+	0.452827i
$276$		0			0
$277$	−798009.	+	1.38219e6i	−0.624897	+	1.08235i	0.363664	+	0.931530i	$0.381526\pi$
$277$	−798009.	+	1.38219e6i	−0.624897	+	1.08235i	−0.988561	+	0.150823i	$0.951808\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	200767.	−	347738.i	0.151679	−	0.262716i	−0.780166	−	0.625573i	$-0.784865\pi$
$281$	200767.	−	347738.i	0.151679	−	0.262716i	0.931845	+	0.362857i	$0.118199\pi$
$282$		0			0
$283$	535582.	+	927656.i	0.397521	+	0.688527i	0.993419	−	0.114533i	$-0.0365372\pi$
$283$	535582.	+	927656.i	0.397521	+	0.688527i	−0.595898	+	0.803060i	$0.703204\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−3.56173e6			−2.55244
$288$		0			0
$289$	167390.			0.117892
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−358830.	−	621511.i	−0.244185	−	0.422941i	0.717717	−	0.696335i	$-0.245187\pi$
$293$	−358830.	−	621511.i	−0.244185	−	0.422941i	−0.961902	+	0.273394i	$0.911854\pi$
$294$		0			0
$295$	−926986.	+	1.60559e6i	−0.620181	+	1.07418i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−115497.	+	200047.i	−0.0747125	+	0.129406i
$300$		0			0
$301$	2.32499e6	+	4.02700e6i	1.47912	+	2.56192i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−173415.			−0.106742
$306$		0			0
$307$	2.25621e6			1.36626			0.683131	−	0.730296i	$-0.260618\pi$
$307$	2.25621e6			1.36626			0.683131	+	0.730296i	$0.260618\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−897190.	−	1.55398e6i	−0.525997	−	0.911054i	−0.999541	−	0.0302837i	$-0.990359\pi$
$311$	−897190.	−	1.55398e6i	−0.525997	−	0.911054i	0.473544	−	0.880770i	$-0.342974\pi$
$312$		0			0
$313$	281895.	−	488257.i	0.162640	−	0.281701i	−0.773175	−	0.634193i	$-0.781332\pi$
$313$	281895.	−	488257.i	0.162640	−	0.281701i	0.935815	+	0.352492i	$0.114666\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1.59153e6	+	2.75662e6i	−0.889544	+	1.54074i	−0.0491288	+	0.998792i	$0.515644\pi$
$317$	−1.59153e6	+	2.75662e6i	−0.889544	+	1.54074i	−0.840415	+	0.541943i	$0.817689\pi$
$318$		0			0
$319$	−272912.	−	472698.i	−0.150157	−	0.260080i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	2.49319e6			1.32969
$324$		0			0
$325$	−673414.			−0.353650
$326$		0			0
$327$		0			0
$328$		0			0
$329$	1.76593e6	+	3.05869e6i	0.899466	+	1.55792i
$330$		0			0
$331$	−3762.47	+	6516.79i	−0.00188757	+	0.00326937i	−0.866968	−	0.498364i	$-0.833934\pi$
$331$	−3762.47	+	6516.79i	−0.00188757	+	0.00326937i	0.865080	+	0.501634i	$0.167267\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−51107.6	+	88520.9i	−0.0248813	+	0.0430957i
$336$		0			0
$337$	734831.	+	1.27277e6i	0.352463	+	0.610483i	0.986680	−	0.162671i	$-0.0520111\pi$
$337$	734831.	+	1.27277e6i	0.352463	+	0.610483i	−0.634218	+	0.773154i	$0.718678\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	1.11610e6			0.519779
$342$		0			0
$343$	1.57078e6			0.720909
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1.31149e6	+	2.27156e6i	0.584709	+	1.01275i	0.994912	+	0.100751i	$0.0321246\pi$
$347$	1.31149e6	+	2.27156e6i	0.584709	+	1.01275i	−0.410203	+	0.911994i	$0.634542\pi$
$348$		0			0
$349$	−582327.	+	1.00862e6i	−0.255919	+	0.443265i	−0.965145	−	0.261716i	$-0.915711\pi$
$349$	−582327.	+	1.00862e6i	−0.255919	+	0.443265i	0.709225	+	0.704982i	$0.249045\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1.57339e6	−	2.72518e6i	0.672045	−	1.16402i	−0.305278	−	0.952263i	$-0.598749\pi$
$353$	1.57339e6	−	2.72518e6i	0.672045	−	1.16402i	0.977323	−	0.211753i	$-0.0679172\pi$
$354$		0			0
$355$	45438.0	+	78700.9i	0.0191359	+	0.0331443i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−720847.			−0.295194			−0.147597	−	0.989048i	$-0.547154\pi$
$359$	−720847.			−0.295194			−0.147597	+	0.989048i	$0.547154\pi$
$360$		0			0
$361$	1.44012e6			0.581607
$362$		0			0
$363$		0			0
$364$		0			0
$365$	63181.1	+	109433.i	0.0248230	+	0.0429947i
$366$		0			0
$367$	41258.2	−	71461.4i	0.0159899	−	0.0276953i	−0.857920	−	0.513784i	$-0.828243\pi$
$367$	41258.2	−	71461.4i	0.0159899	−	0.0276953i	0.873910	+	0.486088i	$0.161577\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−548010.	+	949181.i	−0.206706	+	0.358026i
$372$		0			0
$373$	−2.65152e6	−	4.59256e6i	−0.986784	−	1.70916i	−0.633724	−	0.773560i	$-0.718474\pi$
$373$	−2.65152e6	−	4.59256e6i	−0.986784	−	1.70916i	−0.353060	−	0.935601i	$-0.614859\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	560534.			0.203118
$378$		0			0
$379$	49534.8			0.0177138			0.00885691	−	0.999961i	$-0.497181\pi$
$379$	49534.8			0.0177138			0.00885691	+	0.999961i	$0.497181\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	1.92179e6	+	3.32863e6i	0.669434	+	1.15949i	0.978063	+	0.208311i	$0.0667968\pi$
$383$	1.92179e6	+	3.32863e6i	0.669434	+	1.15949i	−0.308628	+	0.951183i	$0.599870\pi$
$384$		0			0
$385$	1.98923e6	−	3.44545e6i	0.683963	−	1.18466i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−1.96659e6	+	3.40623e6i	−0.658930	+	1.14130i	0.321963	+	0.946752i	$0.395657\pi$
$389$	−1.96659e6	+	3.40623e6i	−0.658930	+	1.14130i	−0.980893	+	0.194548i	$0.937676\pi$
$390$		0			0
$391$	301257.	+	521792.i	0.0996540	+	0.172606i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−2.10229e6			−0.677953
$396$		0			0
$397$	1.48840e6			0.473960			0.236980	−	0.971514i	$-0.423842\pi$
$397$	1.48840e6			0.473960			0.236980	+	0.971514i	$0.423842\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1.84020e6	−	3.18731e6i	−0.571483	−	0.989837i	−0.996414	−	0.0846115i	$-0.973035\pi$
$401$	−1.84020e6	−	3.18731e6i	−0.571483	−	0.989837i	0.424931	−	0.905226i	$-0.360298\pi$
$402$		0			0
$403$	−573091.	+	992622.i	−0.175776	+	0.304454i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−1.92487e6	+	3.33398e6i	−0.575991	+	0.997646i
$408$		0			0
$409$	−290013.	−	502318.i	−0.0857254	−	0.148481i	0.819975	−	0.572400i	$-0.193987\pi$
$409$	−290013.	−	502318.i	−0.0857254	−	0.148481i	−0.905700	+	0.423919i	$0.860654\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−9.06073e6			−2.61389
$414$		0			0
$415$	2.15619e6			0.614563
$416$		0			0
$417$		0			0
$418$		0			0
$419$	759450.	+	1.31541e6i	0.211331	+	0.366037i	0.952131	−	0.305689i	$-0.0988868\pi$
$419$	759450.	+	1.31541e6i	0.211331	+	0.366037i	−0.740800	+	0.671726i	$0.765553\pi$
$420$		0			0
$421$	−2.70128e6	+	4.67875e6i	−0.742787	+	1.28654i	0.208435	+	0.978036i	$0.433163\pi$
$421$	−2.70128e6	+	4.67875e6i	−0.742787	+	1.28654i	−0.951222	+	0.308508i	$0.900170\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−878248.	+	1.52117e6i	−0.235855	+	0.408513i
$426$		0			0
$427$	−423756.	−	733967.i	−0.112472	−	0.194808i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−3.04614e6			−0.789871			−0.394935	−	0.918709i	$-0.629233\pi$
$431$	−3.04614e6			−0.789871			−0.394935	+	0.918709i	$0.629233\pi$
$432$		0			0
$433$	−1.07293e6			−0.275011			−0.137505	−	0.990501i	$-0.543908\pi$
$433$	−1.07293e6			−0.275011			−0.137505	+	0.990501i	$0.543908\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	473203.	+	819612.i	0.118534	+	0.205307i
$438$		0			0
$439$	3.81738e6	−	6.61190e6i	0.945376	−	1.63744i	0.190380	−	0.981710i	$-0.439028\pi$
$439$	3.81738e6	−	6.61190e6i	0.945376	−	1.63744i	0.754996	−	0.655729i	$-0.227639\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	3.30067e6	−	5.71693e6i	0.799085	−	1.38406i	−0.121127	−	0.992637i	$-0.538651\pi$
$443$	3.30067e6	−	5.71693e6i	0.799085	−	1.38406i	0.920213	−	0.391419i	$-0.128016\pi$
$444$		0			0
$445$	419248.	+	726159.i	0.100362	+	0.173833i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	5.52311e6			1.29291			0.646454	−	0.762953i	$-0.276251\pi$
$449$	5.52311e6			1.29291			0.646454	+	0.762953i	$0.276251\pi$
$450$		0			0
$451$	−8.23922e6			−1.90741
$452$		0			0
$453$		0			0
$454$		0			0
$455$	2.04284e6	+	3.53830e6i	0.462600	+	0.801246i
$456$		0			0
$457$	−706717.	+	1.22407e6i	−0.158291	+	0.274167i	−0.934252	−	0.356613i	$-0.883932\pi$
$457$	−706717.	+	1.22407e6i	−0.158291	+	0.274167i	0.775962	+	0.630780i	$0.217265\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	4.25416e6	−	7.36841e6i	0.932312	−	1.61481i	0.152953	−	0.988234i	$-0.451122\pi$
$461$	4.25416e6	−	7.36841e6i	0.932312	−	1.61481i	0.779359	−	0.626578i	$-0.215545\pi$
$462$		0			0
$463$	2.04077e6	+	3.53472e6i	0.442427	+	0.766307i	0.997869	−	0.0652489i	$-0.0207841\pi$
$463$	2.04077e6	+	3.53472e6i	0.442427	+	0.766307i	−0.555442	+	0.831555i	$0.687451\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−5.38239e6			−1.14204			−0.571022	−	0.820935i	$-0.693453\pi$
$467$	−5.38239e6			−1.14204			−0.571022	+	0.820935i	$0.693453\pi$
$468$		0			0
$469$	−499545.			−0.104868
$470$		0			0
$471$		0			0
$472$		0			0
$473$	5.37831e6	+	9.31550e6i	1.10533	+	1.91449i
$474$		0			0
$475$	−1.37952e6	+	2.38940e6i	−0.280540	+	0.485909i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	4.29153e6	−	7.43315e6i	0.854621	−	1.48025i	−0.0223760	−	0.999750i	$-0.507123\pi$
$479$	4.29153e6	−	7.43315e6i	0.854621	−	1.48025i	0.876997	−	0.480497i	$-0.159544\pi$
$480$		0			0
$481$	−1.97675e6	−	3.42383e6i	−0.389573	−	0.674760i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−3.34607e6			−0.645922
$486$		0			0
$487$	3.96670e6			0.757891			0.378946	−	0.925419i	$-0.376287\pi$
$487$	3.96670e6			0.757891			0.378946	+	0.925419i	$0.376287\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−2.88205e6	−	4.99185e6i	−0.539507	−	0.934454i	−0.998931	−	0.0462362i	$-0.985277\pi$
$491$	−2.88205e6	−	4.99185e6i	−0.539507	−	0.934454i	0.459424	−	0.888217i	$-0.348056\pi$
$492$		0			0
$493$	731033.	−	1.26619e6i	0.135463	−	0.234628i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−222064.	+	384627.i	−0.0403262	+	0.0698471i
$498$		0			0
$499$	−2.32003e6	−	4.01841e6i	−0.417102	−	0.722442i	0.578544	−	0.815651i	$-0.303621\pi$
$499$	−2.32003e6	−	4.01841e6i	−0.417102	−	0.722442i	−0.995647	+	0.0932088i	$0.970288\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−2.73360e6			−0.481743			−0.240872	−	0.970557i	$-0.577433\pi$
$503$	−2.73360e6			−0.481743			−0.240872	+	0.970557i	$0.577433\pi$
$504$		0			0
$505$	2.36722e6			0.413057
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−2.97841e6	−	5.15877e6i	−0.509555	−	0.882574i	−0.999939	−	0.0110680i	$-0.996477\pi$
$509$	−2.97841e6	−	5.15877e6i	−0.509555	−	0.882574i	0.490384	−	0.871506i	$-0.336856\pi$
$510$		0			0
$511$	−308778.	+	534820.i	−0.0523112	+	0.0906056i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	4.09809e6	−	7.09810e6i	0.680869	−	1.17930i
$516$		0			0
$517$	4.08507e6	+	7.07555e6i	0.672161	+	1.16422i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	1.11813e7			1.80468			0.902339	−	0.431027i	$-0.141849\pi$
$521$	1.11813e7			1.80468			0.902339	+	0.431027i	$0.141849\pi$
$522$		0			0
$523$	−4.60970e6			−0.736917			−0.368458	−	0.929644i	$-0.620114\pi$
$523$	−4.60970e6			−0.736917			−0.368458	+	0.929644i	$0.620114\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1.49482e6	+	2.58910e6i	0.234456	+	0.406090i
$528$		0			0
$529$	3.10382e6	−	5.37597e6i	0.482233	−	0.835252i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	4.23063e6	−	7.32767e6i	0.645041	−	1.11724i
$534$		0			0
$535$	−488468.	−	846052.i	−0.0737823	−	0.127795i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	1.15386e7			1.71072
$540$		0			0
$541$	−8.65572e6			−1.27148			−0.635741	−	0.771902i	$-0.719305\pi$
$541$	−8.65572e6			−1.27148			−0.635741	+	0.771902i	$0.719305\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−2.46600e6	−	4.27124e6i	−0.355633	−	0.615975i
$546$		0			0
$547$	581626.	−	1.00741e6i	0.0831143	−	0.143958i	−0.821472	−	0.570249i	$-0.806847\pi$
$547$	581626.	−	1.00741e6i	0.0831143	−	0.143958i	0.904586	+	0.426291i	$0.140180\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	1.14828e6	−	1.98888e6i	0.161127	−	0.279081i
$552$		0			0
$553$	−5.13715e6	−	8.89781e6i	−0.714347	−	1.23729i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−1.23037e7			−1.68034			−0.840168	−	0.542326i	$-0.817544\pi$
$557$	−1.23037e7			−1.68034			−0.840168	+	0.542326i	$0.817544\pi$
$558$		0			0
$559$	−1.10465e7			−1.49519
$560$		0			0
$561$		0			0
$562$		0			0
$563$	2.76949e6	+	4.79690e6i	0.368239	+	0.637808i	0.989290	−	0.145962i	$-0.0466276\pi$
$563$	2.76949e6	+	4.79690e6i	0.368239	+	0.637808i	−0.621052	+	0.783770i	$0.713294\pi$
$564$		0			0
$565$	9294.13	−	16097.9i	0.00122486	−	0.00212152i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	6.91495e6	−	1.19770e7i	0.895382	−	1.55085i	0.0620514	−	0.998073i	$-0.480236\pi$
$569$	6.91495e6	−	1.19770e7i	0.895382	−	1.55085i	0.833331	−	0.552775i	$-0.186431\pi$
$570$		0			0
$571$	990947.	+	1.71637e6i	0.127192	+	0.220303i	0.922588	−	0.385787i	$-0.126070\pi$
$571$	990947.	+	1.71637e6i	0.127192	+	0.220303i	−0.795396	+	0.606091i	$0.792737\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−666760.			−0.0841007
$576$		0			0
$577$	9.36673e6			1.17125			0.585623	−	0.810583i	$-0.300850\pi$
$577$	9.36673e6			1.17125			0.585623	+	0.810583i	$0.300850\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	5.26886e6	+	9.12593e6i	0.647554	+	1.12160i
$582$		0			0
$583$	−1.26769e6	+	2.19570e6i	−0.154469	+	0.267549i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	929269.	−	1.60954e6i	0.111313	−	0.192800i	−0.804987	−	0.593293i	$-0.797828\pi$
$587$	929269.	−	1.60954e6i	0.111313	−	0.192800i	0.916300	+	0.400493i	$0.131161\pi$
$588$		0			0
$589$	2.34801e6	+	4.06687e6i	0.278876	+	0.483028i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−1.50844e6			−0.176154			−0.0880770	−	0.996114i	$-0.528072\pi$
$593$	−1.50844e6			−0.176154			−0.0880770	+	0.996114i	$0.528072\pi$
$594$		0			0
$595$	1.06568e7			1.23406
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−6.50791e6	−	1.12720e7i	−0.741096	−	1.28362i	−0.951997	−	0.306108i	$-0.900973\pi$
$599$	−6.50791e6	−	1.12720e7i	−0.741096	−	1.28362i	0.210901	−	0.977507i	$-0.432360\pi$
$600$		0			0
$601$	6.92629e6	−	1.19967e7i	0.782194	−	1.35480i	−0.148467	−	0.988917i	$-0.547434\pi$
$601$	6.92629e6	−	1.19967e7i	0.782194	−	1.35480i	0.930661	−	0.365882i	$-0.119233\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.25152e6	−	2.16769e6i	0.139011	−	0.240773i
$606$		0			0
$607$	−4.75435e6	−	8.23478e6i	−0.523745	−	0.907153i	−0.999618	−	0.0276388i	$-0.991201\pi$
$607$	−4.75435e6	−	8.23478e6i	−0.523745	−	0.907153i	0.475873	−	0.879514i	$-0.342132\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−8.39032e6			−0.909234
$612$		0			0
$613$	−1.65462e7			−1.77848			−0.889238	−	0.457445i	$-0.848765\pi$
$613$	−1.65462e7			−1.77848			−0.889238	+	0.457445i	$0.848765\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−2.75765e6	−	4.77638e6i	−0.291626	−	0.505111i	0.682569	−	0.730822i	$-0.260863\pi$
$617$	−2.75765e6	−	4.77638e6i	−0.291626	−	0.505111i	−0.974194	+	0.225711i	$0.927529\pi$
$618$		0			0
$619$	1.24124e6	−	2.14989e6i	0.130205	−	0.225522i	−0.793550	−	0.608505i	$-0.791770\pi$
$619$	1.24124e6	−	2.14989e6i	0.130205	−	0.225522i	0.923756	+	0.382982i	$0.125103\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−2.04895e6	+	3.54888e6i	−0.211500	+	0.366329i
$624$		0			0
$625$	1.73248e6	+	3.00074e6i	0.177406	+	0.307276i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−1.03121e7			−1.03925
$630$		0			0
$631$	9.16852e6			0.916697			0.458349	−	0.888772i	$-0.348441\pi$
$631$	9.16852e6			0.916697			0.458349	+	0.888772i	$0.348441\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−4.03138e6	−	6.98256e6i	−0.396753	−	0.687196i
$636$		0			0
$637$	−5.92476e6	+	1.02620e7i	−0.578525	+	1.00204i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−994072.	+	1.72178e6i	−0.0955593	+	0.165514i	−0.909842	−	0.414955i	$-0.863797\pi$
$641$	−994072.	+	1.72178e6i	−0.0955593	+	0.165514i	0.814283	+	0.580469i	$0.197131\pi$
$642$		0			0
$643$	−2.62237e6	−	4.54208e6i	−0.250130	−	0.433239i	0.713431	−	0.700725i	$-0.247140\pi$
$643$	−2.62237e6	−	4.54208e6i	−0.250130	−	0.433239i	−0.963562	+	0.267487i	$0.913807\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	9.94212e6			0.933723			0.466862	−	0.884330i	$-0.345385\pi$
$647$	9.94212e6			0.933723			0.466862	+	0.884330i	$0.345385\pi$
$648$		0			0
$649$	−2.09598e7			−1.95333
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1.21132e6	−	2.09806e6i	−0.111167	−	0.192547i	0.805074	−	0.593174i	$-0.202125\pi$
$653$	−1.21132e6	−	2.09806e6i	−0.111167	−	0.192547i	−0.916241	+	0.400628i	$0.868792\pi$
$654$		0			0
$655$	1.99851e6	−	3.46152e6i	0.182013	−	0.315256i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−7.70304e6	+	1.33421e7i	−0.690953	+	1.19677i	0.280573	+	0.959833i	$0.409476\pi$
$659$	−7.70304e6	+	1.33421e7i	−0.690953	+	1.19677i	−0.971526	+	0.236933i	$0.923858\pi$
$660$		0			0
$661$	−3.40975e6	−	5.90586e6i	−0.303542	−	0.525751i	0.673393	−	0.739284i	$-0.264836\pi$
$661$	−3.40975e6	−	5.90586e6i	−0.303542	−	0.525751i	−0.976936	+	0.213534i	$0.931503\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	1.67394e7			1.46786
$666$		0			0
$667$	554995.			0.0483030
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−980259.	−	1.69786e6i	−0.0840494	−	0.145578i
$672$		0			0
$673$	114441.	−	198218.i	0.00973969	−	0.0168696i	−0.861114	−	0.508411i	$-0.830233\pi$
$673$	114441.	−	198218.i	0.00973969	−	0.0168696i	0.870854	+	0.491541i	$0.163566\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1.00487e7	+	1.74048e7i	−0.842629	+	1.45948i	0.0450356	+	0.998985i	$0.485660\pi$
$677$	−1.00487e7	+	1.74048e7i	−0.842629	+	1.45948i	−0.887665	+	0.460491i	$0.847673\pi$
$678$		0			0
$679$	−8.17644e6	−	1.41620e7i	−0.680597	−	1.17883i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1.34916e7			1.10665			0.553325	−	0.832965i	$-0.313359\pi$
$683$	1.34916e7			1.10665			0.553325	+	0.832965i	$0.313359\pi$
$684$		0			0
$685$	9.65924e6			0.786533
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−1.30185e6	−	2.25488e6i	−0.104475	−	0.180957i
$690$		0			0
$691$	2.16664e6	−	3.75273e6i	0.172620	−	0.298987i	−0.766715	−	0.641988i	$-0.778110\pi$
$691$	2.16664e6	−	3.75273e6i	0.172620	−	0.298987i	0.939335	+	0.343001i	$0.111443\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	3.34005e6	−	5.78513e6i	0.262295	−	0.454309i
$696$		0			0
$697$	−1.10349e7	−	1.91131e7i	−0.860376	−	1.49021i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	4.00999e6			0.308211			0.154106	−	0.988054i	$-0.450750\pi$
$701$	4.00999e6			0.308211			0.154106	+	0.988054i	$0.450750\pi$
$702$		0			0
$703$	−1.61978e7			−1.23614
$704$		0			0
$705$		0			0
$706$		0			0
$707$	5.78454e6	+	1.00191e7i	0.435231	+	0.753843i
$708$		0			0
$709$	−2.57127e6	+	4.45358e6i	−0.192102	+	0.332731i	−0.945947	−	0.324322i	$-0.894864\pi$
$709$	−2.57127e6	+	4.45358e6i	−0.192102	+	0.332731i	0.753844	+	0.657053i	$0.228197\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−567428.	+	982813.i	−0.0418010	+	0.0724014i
$714$		0			0
$715$	4.72562e6	+	8.18501e6i	0.345695	+	0.598762i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−1.15318e6			−0.0831906			−0.0415953	−	0.999135i	$-0.513244\pi$
$719$	−1.15318e6			−0.0831906			−0.0415953	+	0.999135i	$0.513244\pi$
$720$		0			0
$721$	4.00564e7			2.86968
$722$		0			0
$723$		0			0
$724$		0			0
$725$	808984.	+	1.40120e6i	0.0571603	+	0.0990046i
$726$		0			0
$727$	−9.27449e6	+	1.60639e7i	−0.650810	+	1.12724i	0.332117	+	0.943238i	$0.392237\pi$
$727$	−9.27449e6	+	1.60639e7i	−0.650810	+	1.12724i	−0.982927	+	0.183997i	$0.941096\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1.44065e7	+	2.49529e7i	−0.997163	+	1.72714i
$732$		0			0
$733$	2.05765e6	+	3.56395e6i	0.141453	+	0.245003i	0.928044	−	0.372471i	$-0.121489\pi$
$733$	2.05765e6	+	3.56395e6i	0.141453	+	0.245003i	−0.786591	+	0.617474i	$0.788156\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1.15558e6			−0.0783666
$738$		0			0
$739$	2.27852e7			1.53477			0.767384	−	0.641188i	$-0.221558\pi$
$739$	2.27852e7			1.53477			0.767384	+	0.641188i	$0.221558\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	1.05194e7	+	1.82202e7i	0.699068	+	1.21082i	0.968790	+	0.247884i	$0.0797351\pi$
$743$	1.05194e7	+	1.82202e7i	0.699068	+	1.21082i	−0.269721	+	0.962938i	$0.586932\pi$
$744$		0			0
$745$	5.80745e6	−	1.00588e7i	0.383349	−	0.663980i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	2.38724e6	−	4.13482e6i	0.155486	−	0.269310i
$750$		0			0
$751$	1.03484e7	+	1.79240e7i	0.669537	+	1.15967i	0.978034	+	0.208446i	$0.0668406\pi$
$751$	1.03484e7	+	1.79240e7i	0.669537	+	1.15967i	−0.308497	+	0.951225i	$0.599826\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−1.18891e6			−0.0759072
$756$		0			0
$757$	1.22154e7			0.774761			0.387380	−	0.921920i	$-0.373380\pi$
$757$	1.22154e7			0.774761			0.387380	+	0.921920i	$0.373380\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	642893.	+	1.11352e6i	0.0402417	+	0.0697007i	0.885445	−	0.464745i	$-0.153854\pi$
$761$	642893.	+	1.11352e6i	0.0402417	+	0.0697007i	−0.845203	+	0.534445i	$0.820521\pi$
$762$		0			0
$763$	1.20518e7	−	2.08744e7i	0.749449	−	1.29808i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1.07623e7	−	1.86409e7i	0.660570	−	1.14414i
$768$		0			0
$769$	5.43009e6	+	9.40520e6i	0.331125	+	0.573525i	0.982733	−	0.185031i	$-0.0592386\pi$
$769$	5.43009e6	+	9.40520e6i	0.331125	+	0.573525i	−0.651608	+	0.758556i	$0.725905\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	7.80047e6			0.469540			0.234770	−	0.972051i	$-0.424566\pi$
$773$	7.80047e6			0.469540			0.234770	+	0.972051i	$0.424566\pi$
$774$		0			0
$775$	−3.30842e6			−0.197864
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1.73333e7	−	3.00222e7i	−1.02338	−	1.77255i
$780$		0			0
$781$	−513693.	+	889742.i	−0.0301353	+	0.0521959i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−2.58466e6	+	4.47676e6i	−0.149703	+	0.259293i
$786$		0			0
$787$	2.36760e6	+	4.10081e6i	0.136261	+	0.236011i	0.926079	−	0.377331i	$-0.123158\pi$
$787$	2.36760e6	+	4.10081e6i	0.136261	+	0.236011i	−0.789817	+	0.613342i	$0.789825\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	90844.4			0.00516246
$792$		0			0
$793$	2.01335e6			0.113694
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−2.59183e6	−	4.48918e6i	−0.144531	−	0.250335i	0.784667	−	0.619918i	$-0.212834\pi$
$797$	−2.59183e6	−	4.48918e6i	−0.144531	−	0.250335i	−0.929198	+	0.369583i	$0.879501\pi$
$798$		0			0
$799$	−1.09424e7	+	1.89528e7i	−0.606383	+	1.05029i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−714285.	+	1.23718e6i	−0.0390915	+	0.0677085i
$804$		0			0
$805$	2.02265e6	+	3.50333e6i	0.110010	+	0.190542i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−7.85790e6			−0.422119			−0.211060	−	0.977473i	$-0.567691\pi$
$809$	−7.85790e6			−0.422119			−0.211060	+	0.977473i	$0.567691\pi$
$810$		0			0
$811$	−2.34495e7			−1.25193			−0.625967	−	0.779849i	$-0.715296\pi$
$811$	−2.34495e7			−1.25193			−0.625967	+	0.779849i	$0.715296\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−3.40920e6	−	5.90491e6i	−0.179787	−	0.311400i
$816$		0			0
$817$	−2.26293e7	+	3.91951e7i	−1.18608	+	2.05436i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1.29037e7	+	2.23499e7i	−0.668125	+	1.15723i	0.310303	+	0.950638i	$0.399570\pi$
$821$	−1.29037e7	+	2.23499e7i	−0.668125	+	1.15723i	−0.978428	+	0.206589i	$0.933764\pi$
$822$		0			0
$823$	5.93578e6	+	1.02811e7i	0.305477	+	0.529101i	0.977367	−	0.211549i	$-0.0678509\pi$
$823$	5.93578e6	+	1.02811e7i	0.305477	+	0.529101i	−0.671891	+	0.740650i	$0.734518\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−4.70487e6			−0.239213			−0.119606	−	0.992821i	$-0.538163\pi$
$827$	−4.70487e6			−0.239213			−0.119606	+	0.992821i	$0.538163\pi$
$828$		0			0
$829$	−1.00951e7			−0.510183			−0.255091	−	0.966917i	$-0.582106\pi$
$829$	−1.00951e7			−0.510183			−0.255091	+	0.966917i	$0.582106\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1.54538e7	+	2.67668e7i	0.771656	+	1.33655i
$834$		0			0
$835$	−3.37669e6	+	5.84859e6i	−0.167600	+	0.290292i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−3.68716e6	+	6.38634e6i	−0.180837	+	0.313218i	−0.942166	−	0.335147i	$-0.891214\pi$
$839$	−3.68716e6	+	6.38634e6i	−0.180837	+	0.313218i	0.761329	+	0.648366i	$0.224547\pi$
$840$		0			0
$841$	9.58220e6	+	1.65968e7i	0.467170	+	0.809162i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	5.74092e6			0.276592
$846$		0			0
$847$	1.22328e7			0.585892
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−1.95721e6	−	3.38999e6i	−0.0926434	−	0.160463i
$852$		0			0
$853$	−1.15979e7	+	2.00882e7i	−0.545769	+	0.945299i	0.452790	+	0.891617i	$0.350429\pi$
$853$	−1.15979e7	+	2.00882e7i	−0.545769	+	0.945299i	−0.998558	+	0.0536815i	$0.982904\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	4.71610e6	−	8.16852e6i	0.219346	−	0.379919i	−0.735262	−	0.677783i	$-0.762941\pi$
$857$	4.71610e6	−	8.16852e6i	0.219346	−	0.379919i	0.954608	+	0.297864i	$0.0962742\pi$
$858$		0			0
$859$	7.97110e6	+	1.38064e7i	0.368583	+	0.638405i	0.989344	−	0.145595i	$-0.0465097\pi$
$859$	7.97110e6	+	1.38064e7i	0.368583	+	0.638405i	−0.620761	+	0.784000i	$0.713176\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−2.89248e7			−1.32203			−0.661017	−	0.750371i	$-0.729875\pi$
$863$	−2.89248e7			−1.32203			−0.661017	+	0.750371i	$0.729875\pi$
$864$		0			0
$865$	−1.59904e7			−0.726639
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1.18836e7	−	2.05830e7i	−0.533824	−	0.924610i
$870$		0			0
$871$	593361.	−	1.02773e6i	0.0265017	−	0.0459023i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1.91134e7	+	3.31054e7i	−0.843953	+	1.46177i
$876$		0			0
$877$	8.40397e6	+	1.45561e7i	0.368965	+	0.639066i	0.989404	−	0.145188i	$-0.0463788\pi$
$877$	8.40397e6	+	1.45561e7i	0.368965	+	0.639066i	−0.620439	+	0.784255i	$0.713046\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−2.56183e7			−1.11202			−0.556008	−	0.831177i	$-0.687668\pi$
$881$	−2.56183e7			−1.11202			−0.556008	+	0.831177i	$0.687668\pi$
$882$		0			0
$883$	−1.53830e7			−0.663955			−0.331978	−	0.943287i	$-0.607716\pi$
$883$	−1.53830e7			−0.663955			−0.331978	+	0.943287i	$0.607716\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	8.93554e6	+	1.54768e7i	0.381340	+	0.660500i	0.991254	−	0.131968i	$-0.0421295\pi$
$887$	8.93554e6	+	1.54768e7i	0.381340	+	0.660500i	−0.609914	+	0.792467i	$0.708796\pi$
$888$		0			0
$889$	1.97022e7	−	3.41252e7i	0.836103	−	1.44817i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1.71880e7	+	2.97705e7i	−0.721267	+	1.24927i
$894$		0			0
$895$	−1.06501e7	−	1.84466e7i	−0.444424	−	0.769766i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	2.75385e6			0.113643
$900$		0			0
$901$	−6.79137e6			−0.278705
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−5.94572e6	−	1.02983e7i	−0.241314	−	0.417969i
$906$		0			0
$907$	−2.54251e6	+	4.40375e6i	−0.102623	+	0.177748i	−0.912764	−	0.408486i	$-0.866057\pi$
$907$	−2.54251e6	+	4.40375e6i	−0.102623	+	0.177748i	0.810142	+	0.586234i	$0.199390\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	2.52606e6	−	4.37526e6i	0.100843	−	0.174666i	−0.811189	−	0.584784i	$-0.801179\pi$
$911$	2.52606e6	−	4.37526e6i	0.100843	−	0.174666i	0.912032	+	0.410118i	$0.134513\pi$
$912$		0			0
$913$	1.21882e7	+	2.11107e7i	0.483910	+	0.838156i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1.95342e7			0.767136
$918$		0			0
$919$	3.46731e7			1.35426			0.677132	−	0.735861i	$-0.263223\pi$
$919$	3.46731e7			1.35426			0.677132	+	0.735861i	$0.263223\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−527537.	−	913720.i	−0.0203821	−	0.0353028i
$924$		0			0
$925$	5.70583e6	−	9.88279e6i	0.219263	−	0.379774i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−8.00390e6	+	1.38632e7i	−0.304272	+	0.527015i	−0.977099	−	0.212785i	$-0.931747\pi$
$929$	−8.00390e6	+	1.38632e7i	−0.304272	+	0.527015i	0.672827	+	0.739800i	$0.265080\pi$
$930$		0			0
$931$	2.42743e7	+	4.20444e7i	0.917852	+	1.58977i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	2.46521e7			0.922199
$936$		0			0
$937$	−1.90843e7			−0.710112			−0.355056	−	0.934845i	$-0.615538\pi$
$937$	−1.90843e7			−0.710112			−0.355056	+	0.934845i	$0.615538\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−2.57109e7	−	4.45326e7i	−0.946550	−	1.63947i	−0.752617	−	0.658458i	$-0.771209\pi$
$941$	−2.57109e7	−	4.45326e7i	−0.946550	−	1.63947i	−0.193933	−	0.981015i	$-0.562124\pi$
$942$		0			0
$943$	4.18882e6	−	7.25526e6i	0.153396	−	0.265689i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	2.11813e7	−	3.66871e7i	0.767498	−	1.32935i	−0.171418	−	0.985198i	$-0.554835\pi$
$947$	2.11813e7	−	3.66871e7i	0.767498	−	1.32935i	0.938916	−	0.344147i	$-0.111832\pi$
$948$		0			0
$949$	−733535.	−	1.27052e6i	−0.0264396	−	0.0457948i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	3.48928e7			1.24452			0.622262	−	0.782809i	$-0.286214\pi$
$953$	3.48928e7			1.24452			0.622262	+	0.782809i	$0.286214\pi$
$954$		0			0
$955$	−1.54980e6			−0.0549880
$956$		0			0
$957$		0			0
$958$		0			0
$959$	2.36033e7	+	4.08821e7i	0.828755	+	1.43545i
$960$		0			0
$961$	1.14990e7	−	1.99169e7i	0.401655	−	0.695686i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1.20389e7	−	2.08521e7i	0.416170	−	0.720827i
$966$		0			0
$967$	3.66580e6	+	6.34935e6i	0.126067	+	0.218355i	0.922150	−	0.386833i	$-0.126431\pi$
$967$	3.66580e6	+	6.34935e6i	0.126067	+	0.218355i	−0.796082	+	0.605188i	$0.793098\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	3.45114e6			0.117467			0.0587334	−	0.998274i	$-0.481294\pi$
$971$	3.45114e6			0.117467			0.0587334	+	0.998274i	$0.481294\pi$
$972$		0			0
$973$	3.26469e7			1.10550
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1.14779e7	−	1.98803e7i	−0.384704	−	0.666326i	0.607024	−	0.794683i	$-0.292363\pi$
$977$	−1.14779e7	−	1.98803e7i	−0.384704	−	0.666326i	−0.991728	+	0.128357i	$0.959030\pi$
$978$		0			0
$979$	−4.73975e6	+	8.20949e6i	−0.158052	+	0.273753i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−2.52879e7	+	4.38000e7i	−0.834698	+	1.44574i	0.0595772	+	0.998224i	$0.481025\pi$
$983$	−2.52879e7	+	4.38000e7i	−0.834698	+	1.44574i	−0.894276	+	0.447516i	$0.852309\pi$
$984$		0			0
$985$	−4.87556e6	−	8.44471e6i	−0.160116	−	0.277328i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1.09373e7			−0.355567
$990$		0			0
$991$	3.58604e7			1.15993			0.579964	−	0.814642i	$-0.303067\pi$
$991$	3.58604e7			1.15993			0.579964	+	0.814642i	$0.303067\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−4.17174e6	−	7.22567e6i	−0.133586	−	0.231377i
$996$		0			0
$997$	1.75536e7	−	3.04037e7i	0.559279	−	0.968699i	−0.438278	−	0.898839i	$-0.644411\pi$
$997$	1.75536e7	−	3.04037e7i	0.559279	−	0.968699i	0.997557	−	0.0698599i	$-0.0222552\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.6.i.b.289.2		6
3.2	odd	2		144.6.i.b.97.2		6
4.3	odd	2		54.6.c.b.19.2		6
9.4	even	3	inner	432.6.i.b.145.2		6
9.5	odd	6		144.6.i.b.49.2		6
12.11	even	2		18.6.c.b.7.2	✓	6
36.7	odd	6		162.6.a.i.1.2		3
36.11	even	6		162.6.a.j.1.2		3
36.23	even	6		18.6.c.b.13.2	yes	6
36.31	odd	6		54.6.c.b.37.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.6.c.b.7.2	✓	6	12.11	even	2
18.6.c.b.13.2	yes	6	36.23	even	6
54.6.c.b.19.2		6	4.3	odd	2
54.6.c.b.37.2		6	36.31	odd	6
144.6.i.b.49.2		6	9.5	odd	6
144.6.i.b.97.2		6	3.2	odd	2
162.6.a.i.1.2		3	36.7	odd	6
162.6.a.j.1.2		3	36.11	even	6
432.6.i.b.145.2		6	9.4	even	3	inner
432.6.i.b.289.2		6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 432 = 2^{4} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	432.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(20.8014 + 36.0292i) q^{5} +(-101.661 + 176.082i) q^{7} +O(q^{10})\)	\(q+(20.8014 + 36.0292i) q^{5} +(-101.661 + 176.082i) q^{7} +(-235.168 + 407.323i) q^{11} +(-241.506 - 418.300i) q^{13} -1259.86 q^{17} -1978.94 q^{19} +(-239.119 - 414.166i) q^{23} +(697.100 - 1207.41i) q^{25} +(-580.249 + 1005.02i) q^{29} +(-1186.50 - 2055.07i) q^{31} -8458.76 q^{35} +8185.10 q^{37} +(8758.87 + 15170.8i) q^{41} +(11435.0 - 19806.1i) q^{43} +(8685.43 - 15043.6i) q^{47} +(-12266.3 - 21245.9i) q^{49} +5390.58 q^{53} -19567.3 q^{55} +(22281.8 + 38593.2i) q^{59} +(-2084.17 + 3609.88i) q^{61} +(10047.3 - 17402.5i) q^{65} +(1228.46 + 2127.76i) q^{67} +2184.37 q^{71} +3037.34 q^{73} +(-47814.7 - 82817.5i) q^{77} +(-25266.2 + 43762.3i) q^{79} +(25913.9 - 44884.2i) q^{83} +(-26206.9 - 45391.7i) q^{85} +20154.7 q^{89} +98206.5 q^{91} +(-41164.9 - 71299.6i) q^{95} +(-40214.4 + 69653.3i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 54 q^{5} + 132 q^{7}+O(q^{10}) \) 6 * q + 54 * q^5 + 132 * q^7	\( 6 q + 54 q^{5} + 132 q^{7} - 315 q^{11} - 744 q^{13} - 2898 q^{17} - 2262 q^{19} - 3168 q^{23} - 2883 q^{25} + 5148 q^{29} + 8610 q^{31} + 2700 q^{35} + 39936 q^{37} - 5049 q^{41} + 31389 q^{43} + 12924 q^{47} - 52857 q^{49} + 96048 q^{53} - 126252 q^{55} + 62955 q^{59} - 75966 q^{61} - 108702 q^{65} + 32991 q^{67} - 129672 q^{71} - 8466 q^{73} - 88740 q^{77} - 89202 q^{79} + 32634 q^{83} + 71388 q^{85} - 66132 q^{89} + 301836 q^{91} - 82944 q^{95} + 46245 q^{97}+O(q^{100}) \) 6 * q + 54 * q^5 + 132 * q^7 - 315 * q^11 - 744 * q^13 - 2898 * q^17 - 2262 * q^19 - 3168 * q^23 - 2883 * q^25 + 5148 * q^29 + 8610 * q^31 + 2700 * q^35 + 39936 * q^37 - 5049 * q^41 + 31389 * q^43 + 12924 * q^47 - 52857 * q^49 + 96048 * q^53 - 126252 * q^55 + 62955 * q^59 - 75966 * q^61 - 108702 * q^65 + 32991 * q^67 - 129672 * q^71 - 8466 * q^73 - 88740 * q^77 - 89202 * q^79 + 32634 * q^83 + 71388 * q^85 - 66132 * q^89 + 301836 * q^91 - 82944 * q^95 + 46245 * q^97

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