LMFDB - Embedded newform 240.3.c.d.209.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [240,3,Mod(209,240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(240, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("240.209");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 240 = 2^{4} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	240.c (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:	$6.53952634465$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 60)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			209.1
Root			$-1.61803i$ of defining polynomial
Character	$\chi$	$=$	240.209
Dual form			240.3.c.d.209.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.23607 - 2.00000i) q^{3} +(2.23607 - 4.47214i) q^{5} -8.00000i q^{7} +(1.00000 + 8.94427i) q^{9} +O(q^{10})$	\(q+(-2.23607 - 2.00000i) q^{3} +(2.23607 - 4.47214i) q^{5} -8.00000i q^{7} +(1.00000 + 8.94427i) q^{9} +8.94427i q^{11} -12.0000i q^{13} +(-13.9443 + 5.52786i) q^{15} -31.3050 q^{17} +6.00000 q^{19} +(-16.0000 + 17.8885i) q^{21} -4.47214 q^{23} +(-15.0000 - 20.0000i) q^{25} +(15.6525 - 22.0000i) q^{27} -26.8328i q^{29} -34.0000 q^{31} +(17.8885 - 20.0000i) q^{33} +(-35.7771 - 17.8885i) q^{35} +44.0000i q^{37} +(-24.0000 + 26.8328i) q^{39} +17.8885i q^{41} -28.0000i q^{43} +(42.2361 + 15.5279i) q^{45} -4.47214 q^{47} -15.0000 q^{49} +(70.0000 + 62.6099i) q^{51} +40.2492 q^{53} +(40.0000 + 20.0000i) q^{55} +(-13.4164 - 12.0000i) q^{57} -98.3870i q^{59} +74.0000 q^{61} +(71.5542 - 8.00000i) q^{63} +(-53.6656 - 26.8328i) q^{65} +92.0000i q^{67} +(10.0000 + 8.94427i) q^{69} -53.6656i q^{71} -56.0000i q^{73} +(-6.45898 + 74.7214i) q^{75} +71.5542 q^{77} +78.0000 q^{79} +(-79.0000 + 17.8885i) q^{81} +102.859 q^{83} +(-70.0000 + 140.000i) q^{85} +(-53.6656 + 60.0000i) q^{87} -17.8885i q^{89} -96.0000 q^{91} +(76.0263 + 68.0000i) q^{93} +(13.4164 - 26.8328i) q^{95} -32.0000i q^{97} +(-80.0000 + 8.94427i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^9	$ 4 q + 4 q^{9} - 20 q^{15} + 24 q^{19} - 64 q^{21} - 60 q^{25} - 136 q^{31} - 96 q^{39} + 160 q^{45} - 60 q^{49} + 280 q^{51} + 160 q^{55} + 296 q^{61} + 40 q^{69} - 160 q^{75} + 312 q^{79} - 316 q^{81} - 280 q^{85} - 384 q^{91} - 320 q^{99}+O(q^{100}) $ 4 * q + 4 * q^9 - 20 * q^15 + 24 * q^19 - 64 * q^21 - 60 * q^25 - 136 * q^31 - 96 * q^39 + 160 * q^45 - 60 * q^49 + 280 * q^51 + 160 * q^55 + 296 * q^61 + 40 * q^69 - 160 * q^75 + 312 * q^79 - 316 * q^81 - 280 * q^85 - 384 * q^91 - 320 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/240\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$161$	$181$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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
$3$	−2.23607	−	2.00000i	−0.745356	−	0.666667i
$3$	−2.23607	−	2.00000i	−0.745356	−	0.666667i
$4$		0			0
$5$	2.23607	−	4.47214i	0.447214	−	0.894427i
$5$	2.23607	−	4.47214i	0.447214	−	0.894427i
$6$		0			0
$7$		−	8.00000i		−	1.14286i	−0.820652	−	0.571429i	$-0.806389\pi$
$7$		−	8.00000i		−	1.14286i	0.820652	−	0.571429i	$-0.193611\pi$
$8$		0			0
$9$	1.00000	+	8.94427i	0.111111	+	0.993808i
$10$		0			0
$11$			8.94427i			0.813116i	0.913625	+	0.406558i	$0.133271\pi$
$11$			8.94427i			0.813116i	−0.913625	+	0.406558i	$0.866729\pi$
$12$		0			0
$13$		−	12.0000i		−	0.923077i	−0.887120	−	0.461538i	$-0.847298\pi$
$13$		−	12.0000i		−	0.923077i	0.887120	−	0.461538i	$-0.152702\pi$
$14$		0			0
$15$	−13.9443	+	5.52786i	−0.929618	+	0.368524i
$16$		0			0
$17$	−31.3050			−1.84147			−0.920734	−	0.390191i	$-0.872409\pi$
$17$	−31.3050			−1.84147			−0.920734	+	0.390191i	$0.872409\pi$
$18$		0			0
$19$	6.00000			0.315789			0.157895	−	0.987456i	$-0.449529\pi$
$19$	6.00000			0.315789			0.157895	+	0.987456i	$0.449529\pi$
$20$		0			0
$21$	−16.0000	+	17.8885i	−0.761905	+	0.851835i
$22$		0			0
$23$	−4.47214			−0.194441			−0.0972203	−	0.995263i	$-0.530995\pi$
$23$	−4.47214			−0.194441			−0.0972203	+	0.995263i	$0.530995\pi$
$24$		0			0
$25$	−15.0000	−	20.0000i	−0.600000	−	0.800000i
$26$		0			0
$27$	15.6525	−	22.0000i	0.579721	−	0.814815i
$28$		0			0
$29$		−	26.8328i		−	0.925270i	−0.886549	−	0.462635i	$-0.846904\pi$
$29$		−	26.8328i		−	0.925270i	0.886549	−	0.462635i	$-0.153096\pi$
$30$		0			0
$31$	−34.0000			−1.09677			−0.548387	−	0.836225i	$-0.684758\pi$
$31$	−34.0000			−1.09677			−0.548387	+	0.836225i	$0.684758\pi$
$32$		0			0
$33$	17.8885	−	20.0000i	0.542077	−	0.606061i
$34$		0			0
$35$	−35.7771	−	17.8885i	−1.02220	−	0.511101i
$36$		0			0
$37$			44.0000i			1.18919i	0.804026	+	0.594595i	$0.202687\pi$
$37$			44.0000i			1.18919i	−0.804026	+	0.594595i	$0.797313\pi$
$38$		0			0
$39$	−24.0000	+	26.8328i	−0.615385	+	0.688021i
$40$		0			0
$41$			17.8885i			0.436306i	0.975915	+	0.218153i	$0.0700032\pi$
$41$			17.8885i			0.436306i	−0.975915	+	0.218153i	$0.929997\pi$
$42$		0			0
$43$		−	28.0000i		−	0.651163i	−0.945514	−	0.325581i	$-0.894440\pi$
$43$		−	28.0000i		−	0.651163i	0.945514	−	0.325581i	$-0.105560\pi$
$44$		0			0
$45$	42.2361	+	15.5279i	0.938579	+	0.345064i
$46$		0			0
$47$	−4.47214			−0.0951518			−0.0475759	−	0.998868i	$-0.515150\pi$
$47$	−4.47214			−0.0951518			−0.0475759	+	0.998868i	$0.515150\pi$
$48$		0			0
$49$	−15.0000			−0.306122
$50$		0			0
$51$	70.0000	+	62.6099i	1.37255	+	1.22765i
$52$		0			0
$53$	40.2492			0.759419			0.379710	−	0.925106i	$-0.376024\pi$
$53$	40.2492			0.759419			0.379710	+	0.925106i	$0.376024\pi$
$54$		0			0
$55$	40.0000	+	20.0000i	0.727273	+	0.363636i
$56$		0			0
$57$	−13.4164	−	12.0000i	−0.235376	−	0.210526i
$58$		0			0
$59$		−	98.3870i		−	1.66758i	−0.552085	−	0.833788i	$-0.686167\pi$
$59$		−	98.3870i		−	1.66758i	0.552085	−	0.833788i	$-0.313833\pi$
$60$		0			0
$61$	74.0000			1.21311			0.606557	−	0.795040i	$-0.292550\pi$
$61$	74.0000			1.21311			0.606557	+	0.795040i	$0.292550\pi$
$62$		0			0
$63$	71.5542	−	8.00000i	1.13578	−	0.126984i
$64$		0			0
$65$	−53.6656	−	26.8328i	−0.825625	−	0.412813i
$66$		0			0
$67$			92.0000i			1.37313i	0.727066	+	0.686567i	$0.240883\pi$
$67$			92.0000i			1.37313i	−0.727066	+	0.686567i	$0.759117\pi$
$68$		0			0
$69$	10.0000	+	8.94427i	0.144928	+	0.129627i
$70$		0			0
$71$		−	53.6656i		−	0.755854i	−0.925835	−	0.377927i	$-0.876637\pi$
$71$		−	53.6656i		−	0.755854i	0.925835	−	0.377927i	$-0.123363\pi$
$72$		0			0
$73$		−	56.0000i		−	0.767123i	−0.923515	−	0.383562i	$-0.874697\pi$
$73$		−	56.0000i		−	0.767123i	0.923515	−	0.383562i	$-0.125303\pi$
$74$		0			0
$75$	−6.45898	+	74.7214i	−0.0861197	+	0.996285i
$76$		0			0
$77$	71.5542			0.929275
$78$		0			0
$79$	78.0000			0.987342			0.493671	−	0.869649i	$-0.335655\pi$
$79$	78.0000			0.987342			0.493671	+	0.869649i	$0.335655\pi$
$80$		0			0
$81$	−79.0000	+	17.8885i	−0.975309	+	0.220846i
$82$		0			0
$83$	102.859			1.23927			0.619633	−	0.784891i	$-0.287281\pi$
$83$	102.859			1.23927			0.619633	+	0.784891i	$0.287281\pi$
$84$		0			0
$85$	−70.0000	+	140.000i	−0.823529	+	1.64706i
$86$		0			0
$87$	−53.6656	+	60.0000i	−0.616846	+	0.689655i
$88$		0			0
$89$		−	17.8885i		−	0.200995i	−0.994937	−	0.100497i	$-0.967957\pi$
$89$		−	17.8885i		−	0.200995i	0.994937	−	0.100497i	$-0.0320434\pi$
$90$		0			0
$91$	−96.0000			−1.05495
$92$		0			0
$93$	76.0263	+	68.0000i	0.817487	+	0.731183i
$94$		0			0
$95$	13.4164	−	26.8328i	0.141225	−	0.282451i
$96$		0			0
$97$		−	32.0000i		−	0.329897i	−0.986302	−	0.164948i	$-0.947254\pi$
$97$		−	32.0000i		−	0.329897i	0.986302	−	0.164948i	$-0.0527458\pi$
$98$		0			0
$99$	−80.0000	+	8.94427i	−0.808081	+	0.0903462i
$100$		0			0
$101$		−	152.053i		−	1.50547i	−0.658323	−	0.752736i	$-0.728734\pi$
$101$		−	152.053i		−	1.50547i	0.658323	−	0.752736i	$-0.271266\pi$
$102$		0			0
$103$		−	104.000i		−	1.00971i	−0.863205	−	0.504854i	$-0.831546\pi$
$103$		−	104.000i		−	1.00971i	0.863205	−	0.504854i	$-0.168454\pi$
$104$		0			0
$105$	44.2229	+	111.554i	0.421171	+	1.06242i
$106$		0			0
$107$	−147.580			−1.37926			−0.689628	−	0.724163i	$-0.742226\pi$
$107$	−147.580			−1.37926			−0.689628	+	0.724163i	$0.742226\pi$
$108$		0			0
$109$	74.0000			0.678899			0.339450	−	0.940624i	$-0.389759\pi$
$109$	74.0000			0.678899			0.339450	+	0.940624i	$0.389759\pi$
$110$		0			0
$111$	88.0000	−	98.3870i	0.792793	−	0.886369i
$112$		0			0
$113$	40.2492			0.356188			0.178094	−	0.984013i	$-0.443007\pi$
$113$	40.2492			0.356188			0.178094	+	0.984013i	$0.443007\pi$
$114$		0			0
$115$	−10.0000	+	20.0000i	−0.0869565	+	0.173913i
$116$		0			0
$117$	107.331	−	12.0000i	0.917361	−	0.102564i
$118$		0			0
$119$			250.440i			2.10453i
$120$		0			0
$121$	41.0000			0.338843
$122$		0			0
$123$	35.7771	−	40.0000i	0.290871	−	0.325203i
$124$		0			0
$125$	−122.984	+	22.3607i	−0.983870	+	0.178885i
$126$		0			0
$127$			16.0000i			0.125984i	0.998014	+	0.0629921i	$0.0200643\pi$
$127$			16.0000i			0.125984i	−0.998014	+	0.0629921i	$0.979936\pi$
$128$		0			0
$129$	−56.0000	+	62.6099i	−0.434109	+	0.485348i
$130$		0			0
$131$		−	80.4984i		−	0.614492i	−0.951630	−	0.307246i	$-0.900593\pi$
$131$		−	80.4984i		−	0.614492i	0.951630	−	0.307246i	$-0.0994074\pi$
$132$		0			0
$133$		−	48.0000i		−	0.360902i
$134$		0			0
$135$	−63.3870	−	119.193i	−0.469533	−	0.882915i
$136$		0			0
$137$	−174.413			−1.27309			−0.636545	−	0.771240i	$-0.719637\pi$
$137$	−174.413			−1.27309			−0.636545	+	0.771240i	$0.719637\pi$
$138$		0			0
$139$	118.000			0.848921			0.424460	−	0.905446i	$-0.360464\pi$
$139$	118.000			0.848921			0.424460	+	0.905446i	$0.360464\pi$
$140$		0			0
$141$	10.0000	+	8.94427i	0.0709220	+	0.0634346i
$142$		0			0
$143$	107.331			0.750568
$144$		0			0
$145$	−120.000	−	60.0000i	−0.827586	−	0.413793i
$146$		0			0
$147$	33.5410	+	30.0000i	0.228170	+	0.204082i
$148$		0			0
$149$			98.3870i			0.660315i	0.943926	+	0.330158i	$0.107102\pi$
$149$			98.3870i			0.660315i	−0.943926	+	0.330158i	$0.892898\pi$
$150$		0			0
$151$	−34.0000			−0.225166			−0.112583	−	0.993642i	$-0.535912\pi$
$151$	−34.0000			−0.225166			−0.112583	+	0.993642i	$0.535912\pi$
$152$		0			0
$153$	−31.3050	−	280.000i	−0.204608	−	1.83007i
$154$		0			0
$155$	−76.0263	+	152.053i	−0.490492	+	0.980985i
$156$		0			0
$157$		−	92.0000i		−	0.585987i	−0.956114	−	0.292994i	$-0.905349\pi$
$157$		−	92.0000i		−	0.585987i	0.956114	−	0.292994i	$-0.0946515\pi$
$158$		0			0
$159$	−90.0000	−	80.4984i	−0.566038	−	0.506280i
$160$		0			0
$161$			35.7771i			0.222218i
$162$		0			0
$163$		−	68.0000i		−	0.417178i	−0.978003	−	0.208589i	$-0.933113\pi$
$163$		−	68.0000i		−	0.417178i	0.978003	−	0.208589i	$-0.0668871\pi$
$164$		0			0
$165$	−49.4427	−	124.721i	−0.299653	−	0.755887i
$166$		0			0
$167$	67.0820			0.401689			0.200844	−	0.979623i	$-0.435631\pi$
$167$	67.0820			0.401689			0.200844	+	0.979623i	$0.435631\pi$
$168$		0			0
$169$	25.0000			0.147929
$170$		0			0
$171$	6.00000	+	53.6656i	0.0350877	+	0.313834i
$172$		0			0
$173$	76.0263			0.439458			0.219729	−	0.975561i	$-0.429483\pi$
$173$	76.0263			0.439458			0.219729	+	0.975561i	$0.429483\pi$
$174$		0			0
$175$	−160.000	+	120.000i	−0.914286	+	0.685714i
$176$		0			0
$177$	−196.774	+	220.000i	−1.11172	+	1.24294i
$178$		0			0
$179$		−	259.384i		−	1.44907i	−0.689237	−	0.724536i	$-0.742054\pi$
$179$		−	259.384i		−	1.44907i	0.689237	−	0.724536i	$-0.257946\pi$
$180$		0			0
$181$	−166.000			−0.917127			−0.458564	−	0.888662i	$-0.651636\pi$
$181$	−166.000			−0.917127			−0.458564	+	0.888662i	$0.651636\pi$
$182$		0			0
$183$	−165.469	−	148.000i	−0.904202	−	0.808743i
$184$		0			0
$185$	196.774	+	98.3870i	1.06364	+	0.531822i
$186$		0			0
$187$		−	280.000i		−	1.49733i
$188$		0			0
$189$	−176.000	−	125.220i	−0.931217	−	0.662539i
$190$		0			0
$191$			214.663i			1.12389i	0.827175	+	0.561944i	$0.189946\pi$
$191$			214.663i			1.12389i	−0.827175	+	0.561944i	$0.810054\pi$
$192$		0			0
$193$		−	32.0000i		−	0.165803i	−0.996558	−	0.0829016i	$-0.973581\pi$
$193$		−	32.0000i		−	0.165803i	0.996558	−	0.0829016i	$-0.0264187\pi$
$194$		0			0
$195$	66.3344	+	167.331i	0.340176	+	0.858109i
$196$		0			0
$197$	4.47214			0.0227012			0.0113506	−	0.999936i	$-0.496387\pi$
$197$	4.47214			0.0227012			0.0113506	+	0.999936i	$0.496387\pi$
$198$		0			0
$199$	−114.000			−0.572864			−0.286432	−	0.958101i	$-0.592469\pi$
$199$	−114.000			−0.572864			−0.286432	+	0.958101i	$0.592469\pi$
$200$		0			0
$201$	184.000	−	205.718i	0.915423	−	1.02347i
$202$		0			0
$203$	−214.663			−1.05745
$204$		0			0
$205$	80.0000	+	40.0000i	0.390244	+	0.195122i
$206$		0			0
$207$	−4.47214	−	40.0000i	−0.0216045	−	0.193237i
$208$		0			0
$209$			53.6656i			0.256773i
$210$		0			0
$211$	6.00000			0.0284360			0.0142180	−	0.999899i	$-0.495474\pi$
$211$	6.00000			0.0284360			0.0142180	+	0.999899i	$0.495474\pi$
$212$		0			0
$213$	−107.331	+	120.000i	−0.503903	+	0.563380i
$214$		0			0
$215$	−125.220	−	62.6099i	−0.582418	−	0.291209i
$216$		0			0
$217$			272.000i			1.25346i
$218$		0			0
$219$	−112.000	+	125.220i	−0.511416	+	0.571780i
$220$		0			0
$221$			375.659i			1.69982i
$222$		0			0
$223$			272.000i			1.21973i	0.792505	+	0.609865i	$0.208777\pi$
$223$			272.000i			1.21973i	−0.792505	+	0.609865i	$0.791223\pi$
$224$		0			0
$225$	163.885	−	154.164i	0.728380	−	0.685174i
$226$		0			0
$227$	245.967			1.08356			0.541779	−	0.840521i	$-0.317751\pi$
$227$	245.967			1.08356			0.541779	+	0.840521i	$0.317751\pi$
$228$		0			0
$229$	154.000			0.672489			0.336245	−	0.941775i	$-0.390843\pi$
$229$	154.000			0.672489			0.336245	+	0.941775i	$0.390843\pi$
$230$		0			0
$231$	−160.000	−	143.108i	−0.692641	−	0.619517i
$232$		0			0
$233$	183.358			0.786942			0.393471	−	0.919337i	$-0.371274\pi$
$233$	183.358			0.786942			0.393471	+	0.919337i	$0.371274\pi$
$234$		0			0
$235$	−10.0000	+	20.0000i	−0.0425532	+	0.0851064i
$236$		0			0
$237$	−174.413	−	156.000i	−0.735921	−	0.658228i
$238$		0			0
$239$			178.885i			0.748475i	0.927333	+	0.374237i	$0.122095\pi$
$239$			178.885i			0.748475i	−0.927333	+	0.374237i	$0.877905\pi$
$240$		0			0
$241$	−206.000			−0.854772			−0.427386	−	0.904069i	$-0.640565\pi$
$241$	−206.000			−0.854772			−0.427386	+	0.904069i	$0.640565\pi$
$242$		0			0
$243$	212.426	+	118.000i	0.874183	+	0.485597i
$244$		0			0
$245$	−33.5410	+	67.0820i	−0.136902	+	0.273804i
$246$		0			0
$247$		−	72.0000i		−	0.291498i
$248$		0			0
$249$	−230.000	−	205.718i	−0.923695	−	0.826178i
$250$		0			0
$251$		−	26.8328i		−	0.106904i	−0.998570	−	0.0534518i	$-0.982978\pi$
$251$		−	26.8328i		−	0.106904i	0.998570	−	0.0534518i	$-0.0170224\pi$
$252$		0			0
$253$		−	40.0000i		−	0.158103i
$254$		0			0
$255$	436.525	−	173.050i	1.71186	−	0.678626i
$256$		0			0
$257$	40.2492			0.156612			0.0783059	−	0.996929i	$-0.475049\pi$
$257$	40.2492			0.156612			0.0783059	+	0.996929i	$0.475049\pi$
$258$		0			0
$259$	352.000			1.35907
$260$		0			0
$261$	240.000	−	26.8328i	0.919540	−	0.102808i
$262$		0			0
$263$	210.190			0.799203			0.399602	−	0.916689i	$-0.369149\pi$
$263$	210.190			0.799203			0.399602	+	0.916689i	$0.369149\pi$
$264$		0			0
$265$	90.0000	−	180.000i	0.339623	−	0.679245i
$266$		0			0
$267$	−35.7771	+	40.0000i	−0.133997	+	0.149813i
$268$		0			0
$269$		−	134.164i		−	0.498751i	−0.968407	−	0.249376i	$-0.919775\pi$
$269$		−	134.164i		−	0.498751i	0.968407	−	0.249376i	$-0.0802254\pi$
$270$		0			0
$271$	398.000			1.46863			0.734317	−	0.678806i	$-0.237502\pi$
$271$	398.000			1.46863			0.734317	+	0.678806i	$0.237502\pi$
$272$		0			0
$273$	214.663	+	192.000i	0.786310	+	0.703297i
$274$		0			0
$275$	178.885	−	134.164i	0.650493	−	0.487869i
$276$		0			0
$277$		−	292.000i		−	1.05415i	−0.849818	−	0.527076i	$-0.823288\pi$
$277$		−	292.000i		−	1.05415i	0.849818	−	0.527076i	$-0.176712\pi$
$278$		0			0
$279$	−34.0000	−	304.105i	−0.121864	−	1.08998i
$280$		0			0
$281$			53.6656i			0.190981i	0.995430	+	0.0954904i	$0.0304419\pi$
$281$			53.6656i			0.190981i	−0.995430	+	0.0954904i	$0.969558\pi$
$282$		0			0
$283$			52.0000i			0.183746i	0.995771	+	0.0918728i	$0.0292853\pi$
$283$			52.0000i			0.183746i	−0.995771	+	0.0918728i	$0.970715\pi$
$284$		0			0
$285$	−83.6656	+	33.1672i	−0.293564	+	0.116376i
$286$		0			0
$287$	143.108			0.498635
$288$		0			0
$289$	691.000			2.39100
$290$		0			0
$291$	−64.0000	+	71.5542i	−0.219931	+	0.245891i
$292$		0			0
$293$	−389.076			−1.32790			−0.663952	−	0.747775i	$-0.731122\pi$
$293$	−389.076			−1.32790			−0.663952	+	0.747775i	$0.731122\pi$
$294$		0			0
$295$	−440.000	−	220.000i	−1.49153	−	0.745763i
$296$		0			0
$297$	196.774	+	140.000i	0.662539	+	0.471380i
$298$		0			0
$299$			53.6656i			0.179484i
$300$		0			0
$301$	−224.000			−0.744186
$302$		0			0
$303$	−304.105	+	340.000i	−1.00365	+	1.12211i
$304$		0			0
$305$	165.469	−	330.938i	0.542521	−	1.08504i
$306$		0			0
$307$			492.000i			1.60261i	0.598259	+	0.801303i	$0.295859\pi$
$307$			492.000i			1.60261i	−0.598259	+	0.801303i	$0.704141\pi$
$308$		0			0
$309$	−208.000	+	232.551i	−0.673139	+	0.752592i
$310$		0			0
$311$			482.991i			1.55302i	0.630102	+	0.776512i	$0.283013\pi$
$311$			482.991i			1.55302i	−0.630102	+	0.776512i	$0.716987\pi$
$312$		0			0
$313$			568.000i			1.81470i	0.420380	+	0.907348i	$0.361897\pi$
$313$			568.000i			1.81470i	−0.420380	+	0.907348i	$0.638103\pi$
$314$		0			0
$315$	124.223	−	337.889i	0.394358	−	1.07266i
$316$		0			0
$317$	612.683			1.93275			0.966376	−	0.257132i	$-0.0827774\pi$
$317$	612.683			1.93275			0.966376	+	0.257132i	$0.0827774\pi$
$318$		0			0
$319$	240.000			0.752351
$320$		0			0
$321$	330.000	+	295.161i	1.02804	+	0.919505i
$322$		0			0
$323$	−187.830			−0.581516
$324$		0			0
$325$	−240.000	+	180.000i	−0.738462	+	0.553846i
$326$		0			0
$327$	−165.469	−	148.000i	−0.506021	−	0.452599i
$328$		0			0
$329$			35.7771i			0.108745i
$330$		0			0
$331$	−202.000			−0.610272			−0.305136	−	0.952309i	$-0.598702\pi$
$331$	−202.000			−0.610272			−0.305136	+	0.952309i	$0.598702\pi$
$332$		0			0
$333$	−393.548	+	44.0000i	−1.18183	+	0.132132i
$334$		0			0
$335$	411.437	+	205.718i	1.22817	+	0.614084i
$336$		0			0
$337$			368.000i			1.09199i	0.837789	+	0.545994i	$0.183848\pi$
$337$			368.000i			1.09199i	−0.837789	+	0.545994i	$0.816152\pi$
$338$		0			0
$339$	−90.0000	−	80.4984i	−0.265487	−	0.237459i
$340$		0			0
$341$		−	304.105i		−	0.891804i
$342$		0			0
$343$		−	272.000i		−	0.793003i
$344$		0			0
$345$	62.3607	−	24.7214i	0.180756	−	0.0716561i
$346$		0			0
$347$	−254.912			−0.734616			−0.367308	−	0.930099i	$-0.619720\pi$
$347$	−254.912			−0.734616			−0.367308	+	0.930099i	$0.619720\pi$
$348$		0			0
$349$	−118.000			−0.338109			−0.169054	−	0.985607i	$-0.554071\pi$
$349$	−118.000			−0.338109			−0.169054	+	0.985607i	$0.554071\pi$
$350$		0			0
$351$	−264.000	−	187.830i	−0.752137	−	0.535127i
$352$		0			0
$353$	−31.3050			−0.0886826			−0.0443413	−	0.999016i	$-0.514119\pi$
$353$	−31.3050			−0.0886826			−0.0443413	+	0.999016i	$0.514119\pi$
$354$		0			0
$355$	−240.000	−	120.000i	−0.676056	−	0.338028i
$356$		0			0
$357$	500.879	−	560.000i	1.40302	−	1.56863i
$358$		0			0
$359$		−	53.6656i		−	0.149486i	−0.997203	−	0.0747432i	$-0.976186\pi$
$359$		−	53.6656i		−	0.149486i	0.997203	−	0.0747432i	$-0.0238137\pi$
$360$		0			0
$361$	−325.000			−0.900277
$362$		0			0
$363$	−91.6788	−	82.0000i	−0.252559	−	0.225895i
$364$		0			0
$365$	−250.440	−	125.220i	−0.686136	−	0.343068i
$366$		0			0
$367$			352.000i			0.959128i	0.877507	+	0.479564i	$0.159205\pi$
$367$			352.000i			0.959128i	−0.877507	+	0.479564i	$0.840795\pi$
$368$		0			0
$369$	−160.000	+	17.8885i	−0.433604	+	0.0484784i
$370$		0			0
$371$		−	321.994i		−	0.867908i
$372$		0			0
$373$		−	132.000i		−	0.353887i	−0.984221	−	0.176944i	$-0.943379\pi$
$373$		−	132.000i		−	0.353887i	0.984221	−	0.176944i	$-0.0566211\pi$
$374$		0			0
$375$	319.721	+	195.967i	0.852590	+	0.522580i
$376$		0			0
$377$	−321.994			−0.854095
$378$		0			0
$379$	−394.000			−1.03958			−0.519789	−	0.854295i	$-0.673989\pi$
$379$	−394.000			−1.03958			−0.519789	+	0.854295i	$0.673989\pi$
$380$		0			0
$381$	32.0000	−	35.7771i	0.0839895	−	0.0939031i
$382$		0			0
$383$	−76.0263			−0.198502			−0.0992511	−	0.995062i	$-0.531645\pi$
$383$	−76.0263			−0.198502			−0.0992511	+	0.995062i	$0.531645\pi$
$384$		0			0
$385$	160.000	−	320.000i	0.415584	−	0.831169i
$386$		0			0
$387$	250.440	−	28.0000i	0.647131	−	0.0723514i
$388$		0			0
$389$			277.272i			0.712783i	0.934337	+	0.356391i	$0.115993\pi$
$389$			277.272i			0.712783i	−0.934337	+	0.356391i	$0.884007\pi$
$390$		0			0
$391$	140.000			0.358056
$392$		0			0
$393$	−160.997	+	180.000i	−0.409661	+	0.458015i
$394$		0			0
$395$	174.413	−	348.827i	0.441553	−	0.883105i
$396$		0			0
$397$		−	652.000i		−	1.64232i	−0.570700	−	0.821159i	$-0.693328\pi$
$397$		−	652.000i		−	1.64232i	0.570700	−	0.821159i	$-0.306672\pi$
$398$		0			0
$399$	−96.0000	+	107.331i	−0.240602	+	0.269001i
$400$		0			0
$401$		−	178.885i		−	0.446098i	−0.974807	−	0.223049i	$-0.928399\pi$
$401$		−	178.885i		−	0.446098i	0.974807	−	0.223049i	$-0.0716011\pi$
$402$		0			0
$403$			408.000i			1.01241i
$404$		0			0
$405$	−96.6494	+	393.299i	−0.238640	+	0.971108i
$406$		0			0
$407$	−393.548			−0.966948
$408$		0			0
$409$	−206.000			−0.503667			−0.251834	−	0.967771i	$-0.581034\pi$
$409$	−206.000			−0.503667			−0.251834	+	0.967771i	$0.581034\pi$
$410$		0			0
$411$	390.000	+	348.827i	0.948905	+	0.848727i
$412$		0			0
$413$	−787.096			−1.90580
$414$		0			0
$415$	230.000	−	460.000i	0.554217	−	1.10843i
$416$		0			0
$417$	−263.856	−	236.000i	−0.632748	−	0.565947i
$418$		0			0
$419$			205.718i			0.490974i	0.969400	+	0.245487i	$0.0789479\pi$
$419$			205.718i			0.490974i	−0.969400	+	0.245487i	$0.921052\pi$
$420$		0			0
$421$	−38.0000			−0.0902613			−0.0451306	−	0.998981i	$-0.514370\pi$
$421$	−38.0000			−0.0902613			−0.0451306	+	0.998981i	$0.514370\pi$
$422$		0			0
$423$	−4.47214	−	40.0000i	−0.0105724	−	0.0945626i
$424$		0			0
$425$	469.574	+	626.099i	1.10488	+	1.47317i
$426$		0			0
$427$		−	592.000i		−	1.38642i
$428$		0			0
$429$	−240.000	−	214.663i	−0.559441	−	0.500379i
$430$		0			0
$431$		−	608.210i		−	1.41116i	−0.708630	−	0.705581i	$-0.750686\pi$
$431$		−	608.210i		−	1.41116i	0.708630	−	0.705581i	$-0.249314\pi$
$432$		0			0
$433$		−	272.000i		−	0.628176i	−0.949394	−	0.314088i	$-0.898301\pi$
$433$		−	272.000i		−	0.628176i	0.949394	−	0.314088i	$-0.101699\pi$
$434$		0			0
$435$	148.328	+	374.164i	0.340984	+	0.860147i
$436$		0			0
$437$	−26.8328			−0.0614023
$438$		0			0
$439$	366.000			0.833713			0.416856	−	0.908972i	$-0.363132\pi$
$439$	366.000			0.833713			0.416856	+	0.908972i	$0.363132\pi$
$440$		0			0
$441$	−15.0000	−	134.164i	−0.0340136	−	0.304227i
$442$		0			0
$443$	−576.906			−1.30227			−0.651135	−	0.758962i	$-0.725707\pi$
$443$	−576.906			−1.30227			−0.651135	+	0.758962i	$0.725707\pi$
$444$		0			0
$445$	−80.0000	−	40.0000i	−0.179775	−	0.0898876i
$446$		0			0
$447$	196.774	−	220.000i	0.440210	−	0.492170i
$448$		0			0
$449$		−	429.325i		−	0.956181i	−0.878311	−	0.478090i	$-0.841329\pi$
$449$		−	429.325i		−	0.956181i	0.878311	−	0.478090i	$-0.158671\pi$
$450$		0			0
$451$	−160.000			−0.354767
$452$		0			0
$453$	76.0263	+	68.0000i	0.167829	+	0.150110i
$454$		0			0
$455$	−214.663	+	429.325i	−0.471786	+	0.943572i
$456$		0			0
$457$			104.000i			0.227571i	0.993505	+	0.113786i	$0.0362977\pi$
$457$			104.000i			0.227571i	−0.993505	+	0.113786i	$0.963702\pi$
$458$		0			0
$459$	−490.000	+	688.709i	−1.06754	+	1.50046i
$460$		0			0
$461$			509.823i			1.10591i	0.833212	+	0.552954i	$0.186499\pi$
$461$			509.823i			1.10591i	−0.833212	+	0.552954i	$0.813501\pi$
$462$		0			0
$463$			96.0000i			0.207343i	0.994612	+	0.103672i	$0.0330591\pi$
$463$			96.0000i			0.207343i	−0.994612	+	0.103672i	$0.966941\pi$
$464$		0			0
$465$	474.105	−	187.947i	1.01958	−	0.404188i
$466$		0			0
$467$	−147.580			−0.316018			−0.158009	−	0.987438i	$-0.550508\pi$
$467$	−147.580			−0.316018			−0.158009	+	0.987438i	$0.550508\pi$
$468$		0			0
$469$	736.000			1.56930
$470$		0			0
$471$	−184.000	+	205.718i	−0.390658	+	0.436769i
$472$		0			0
$473$	250.440			0.529471
$474$		0			0
$475$	−90.0000	−	120.000i	−0.189474	−	0.252632i
$476$		0			0
$477$	40.2492	+	360.000i	0.0843799	+	0.754717i
$478$		0			0
$479$		−	572.433i		−	1.19506i	−0.801847	−	0.597530i	$-0.796149\pi$
$479$		−	572.433i		−	1.19506i	0.801847	−	0.597530i	$-0.203851\pi$
$480$		0			0
$481$	528.000			1.09771
$482$		0			0
$483$	71.5542	−	80.0000i	0.148145	−	0.165631i
$484$		0			0
$485$	−143.108	−	71.5542i	−0.295069	−	0.147534i
$486$		0			0
$487$		−	648.000i		−	1.33060i	−0.746578	−	0.665298i	$-0.768305\pi$
$487$		−	648.000i		−	1.33060i	0.746578	−	0.665298i	$-0.231695\pi$
$488$		0			0
$489$	−136.000	+	152.053i	−0.278119	+	0.310946i
$490$		0			0
$491$		−	134.164i		−	0.273247i	−0.990623	−	0.136623i	$-0.956375\pi$
$491$		−	134.164i		−	0.273247i	0.990623	−	0.136623i	$-0.0436250\pi$
$492$		0			0
$493$			840.000i			1.70385i
$494$		0			0
$495$	−138.885	+	377.771i	−0.280577	+	0.763173i
$496$		0			0
$497$	−429.325			−0.863833
$498$		0			0
$499$	486.000			0.973948			0.486974	−	0.873416i	$-0.338101\pi$
$499$	486.000			0.973948			0.486974	+	0.873416i	$0.338101\pi$
$500$		0			0
$501$	−150.000	−	134.164i	−0.299401	−	0.267793i
$502$		0			0
$503$	−791.568			−1.57369			−0.786847	−	0.617148i	$-0.788288\pi$
$503$	−791.568			−1.57369			−0.786847	+	0.617148i	$0.788288\pi$
$504$		0			0
$505$	−680.000	−	340.000i	−1.34653	−	0.673267i
$506$		0			0
$507$	−55.9017	−	50.0000i	−0.110260	−	0.0986193i
$508$		0			0
$509$		−	26.8328i		−	0.0527167i	−0.999653	−	0.0263584i	$-0.991609\pi$
$509$		−	26.8328i		−	0.0527167i	0.999653	−	0.0263584i	$-0.00839110\pi$
$510$		0			0
$511$	−448.000			−0.876712
$512$		0			0
$513$	93.9149	−	132.000i	0.183070	−	0.257310i
$514$		0			0
$515$	−465.102	−	232.551i	−0.903111	−	0.451555i
$516$		0			0
$517$		−	40.0000i		−	0.0773694i
$518$		0			0
$519$	−170.000	−	152.053i	−0.327553	−	0.292972i
$520$		0			0
$521$		−	983.870i		−	1.88843i	−0.329336	−	0.944213i	$-0.606825\pi$
$521$		−	983.870i		−	1.88843i	0.329336	−	0.944213i	$-0.393175\pi$
$522$		0			0
$523$			292.000i			0.558317i	0.960245	+	0.279159i	$0.0900556\pi$
$523$			292.000i			0.558317i	−0.960245	+	0.279159i	$0.909944\pi$
$524$		0			0
$525$	597.771	+	51.6718i	1.13861	+	0.0984226i
$526$		0			0
$527$	1064.37			2.01967
$528$		0			0
$529$	−509.000			−0.962193
$530$		0			0
$531$	880.000	−	98.3870i	1.65725	−	0.185286i
$532$		0			0
$533$	214.663			0.402744
$534$		0			0
$535$	−330.000	+	660.000i	−0.616822	+	1.23364i
$536$		0			0
$537$	−518.768	+	580.000i	−0.966048	+	1.08007i
$538$		0			0
$539$		−	134.164i		−	0.248913i
$540$		0			0
$541$	−86.0000			−0.158965			−0.0794824	−	0.996836i	$-0.525327\pi$
$541$	−86.0000			−0.158965			−0.0794824	+	0.996836i	$0.525327\pi$
$542$		0			0
$543$	371.187	+	332.000i	0.683586	+	0.611418i
$544$		0			0
$545$	165.469	−	330.938i	0.303613	−	0.607226i
$546$		0			0
$547$		−	164.000i		−	0.299817i	−0.988700	−	0.149909i	$-0.952102\pi$
$547$		−	164.000i		−	0.299817i	0.988700	−	0.149909i	$-0.0478979\pi$
$548$		0			0
$549$	74.0000	+	661.876i	0.134791	+	1.20560i
$550$		0			0
$551$		−	160.997i		−	0.292190i
$552$		0			0
$553$		−	624.000i		−	1.12839i
$554$		0			0
$555$	−243.226	−	613.548i	−0.438245	−	1.10549i
$556$		0			0
$557$	362.243			0.650347			0.325173	−	0.945654i	$-0.394577\pi$
$557$	362.243			0.650347			0.325173	+	0.945654i	$0.394577\pi$
$558$		0			0
$559$	−336.000			−0.601073
$560$		0			0
$561$	−560.000	+	626.099i	−0.998217	+	1.11604i
$562$		0			0
$563$	997.286			1.77138			0.885689	−	0.464278i	$-0.153686\pi$
$563$	997.286			1.77138			0.885689	+	0.464278i	$0.153686\pi$
$564$		0			0
$565$	90.0000	−	180.000i	0.159292	−	0.318584i
$566$		0			0
$567$	143.108	+	632.000i	0.252396	+	1.11464i
$568$		0			0
$569$			626.099i			1.10035i	0.835049	+	0.550175i	$0.185439\pi$
$569$			626.099i			1.10035i	−0.835049	+	0.550175i	$0.814561\pi$
$570$		0			0
$571$	−394.000			−0.690018			−0.345009	−	0.938599i	$-0.612124\pi$
$571$	−394.000			−0.690018			−0.345009	+	0.938599i	$0.612124\pi$
$572$		0			0
$573$	429.325	−	480.000i	0.749258	−	0.837696i
$574$		0			0
$575$	67.0820	+	89.4427i	0.116664	+	0.155553i
$576$		0			0
$577$			608.000i			1.05373i	0.849950	+	0.526863i	$0.176632\pi$
$577$			608.000i			1.05373i	−0.849950	+	0.526863i	$0.823368\pi$
$578$		0			0
$579$	−64.0000	+	71.5542i	−0.110535	+	0.123582i
$580$		0			0
$581$		−	822.873i		−	1.41630i
$582$		0			0
$583$			360.000i			0.617496i
$584$		0			0
$585$	186.334	−	506.833i	0.318520	−	0.866381i
$586$		0			0
$587$	711.070			1.21136			0.605681	−	0.795707i	$-0.292901\pi$
$587$	711.070			1.21136			0.605681	+	0.795707i	$0.292901\pi$
$588$		0			0
$589$	−204.000			−0.346350
$590$		0			0
$591$	−10.0000	−	8.94427i	−0.0169205	−	0.0151341i
$592$		0			0
$593$	−603.738			−1.01811			−0.509054	−	0.860734i	$-0.670005\pi$
$593$	−603.738			−1.01811			−0.509054	+	0.860734i	$0.670005\pi$
$594$		0			0
$595$	1120.00	+	560.000i	1.88235	+	0.941176i
$596$		0			0
$597$	254.912	+	228.000i	0.426988	+	0.381910i
$598$		0			0
$599$			53.6656i			0.0895920i	0.998996	+	0.0447960i	$0.0142638\pi$
$599$			53.6656i			0.0895920i	−0.998996	+	0.0447960i	$0.985736\pi$
$600$		0			0
$601$	434.000			0.722130			0.361065	−	0.932541i	$-0.382413\pi$
$601$	434.000			0.722130			0.361065	+	0.932541i	$0.382413\pi$
$602$		0			0
$603$	−822.873	+	92.0000i	−1.36463	+	0.152570i
$604$		0			0
$605$	91.6788	−	183.358i	0.151535	−	0.303070i
$606$		0			0
$607$			656.000i			1.08072i	0.841432	+	0.540362i	$0.181713\pi$
$607$			656.000i			1.08072i	−0.841432	+	0.540362i	$0.818287\pi$
$608$		0			0
$609$	480.000	+	429.325i	0.788177	+	0.704967i
$610$		0			0
$611$			53.6656i			0.0878325i
$612$		0			0
$613$			844.000i			1.37684i	0.725315	+	0.688418i	$0.241694\pi$
$613$			844.000i			1.37684i	−0.725315	+	0.688418i	$0.758306\pi$
$614$		0			0
$615$	−98.8854	−	249.443i	−0.160789	−	0.405598i
$616$		0			0
$617$	−460.630			−0.746564			−0.373282	−	0.927718i	$-0.621768\pi$
$617$	−460.630			−0.746564			−0.373282	+	0.927718i	$0.621768\pi$
$618$		0			0
$619$	1046.00			1.68982			0.844911	−	0.534907i	$-0.179653\pi$
$619$	1046.00			1.68982			0.844911	+	0.534907i	$0.179653\pi$
$620$		0			0
$621$	−70.0000	+	98.3870i	−0.112721	+	0.158433i
$622$		0			0
$623$	−143.108			−0.229708
$624$		0			0
$625$	−175.000	+	600.000i	−0.280000	+	0.960000i
$626$		0			0
$627$	107.331	−	120.000i	0.171182	−	0.191388i
$628$		0			0
$629$		−	1377.42i		−	2.18985i
$630$		0			0
$631$	46.0000			0.0729002			0.0364501	−	0.999335i	$-0.488395\pi$
$631$	46.0000			0.0729002			0.0364501	+	0.999335i	$0.488395\pi$
$632$		0			0
$633$	−13.4164	−	12.0000i	−0.0211950	−	0.0189573i
$634$		0			0
$635$	71.5542	+	35.7771i	0.112684	+	0.0563419i
$636$		0			0
$637$			180.000i			0.282575i
$638$		0			0
$639$	480.000	−	53.6656i	0.751174	−	0.0839838i
$640$		0			0
$641$			1144.87i			1.78606i	0.449994	+	0.893032i	$0.351426\pi$
$641$			1144.87i			1.78606i	−0.449994	+	0.893032i	$0.648574\pi$
$642$		0			0
$643$		−	804.000i		−	1.25039i	−0.780469	−	0.625194i	$-0.785020\pi$
$643$		−	804.000i		−	1.25039i	0.780469	−	0.625194i	$-0.214980\pi$
$644$		0			0
$645$	154.780	+	390.440i	0.239969	+	0.605333i
$646$		0			0
$647$	−576.906			−0.891662			−0.445831	−	0.895117i	$-0.647092\pi$
$647$	−576.906			−0.891662			−0.445831	+	0.895117i	$0.647092\pi$
$648$		0			0
$649$	880.000			1.35593
$650$		0			0
$651$	544.000	−	608.210i	0.835637	−	0.934271i
$652$		0			0
$653$	1077.78			1.65051			0.825256	−	0.564758i	$-0.191031\pi$
$653$	1077.78			1.65051			0.825256	+	0.564758i	$0.191031\pi$
$654$		0			0
$655$	−360.000	−	180.000i	−0.549618	−	0.274809i
$656$		0			0
$657$	500.879	−	56.0000i	0.762373	−	0.0852359i
$658$		0			0
$659$			813.929i			1.23510i	0.786533	+	0.617548i	$0.211874\pi$
$659$			813.929i			1.23510i	−0.786533	+	0.617548i	$0.788126\pi$
$660$		0			0
$661$	1082.00			1.63691			0.818457	−	0.574568i	$-0.194830\pi$
$661$	1082.00			1.63691			0.818457	+	0.574568i	$0.194830\pi$
$662$		0			0
$663$	751.319	−	840.000i	1.13321	−	1.26697i
$664$		0			0
$665$	−214.663	−	107.331i	−0.322801	−	0.161400i
$666$		0			0
$667$			120.000i			0.179910i
$668$		0			0
$669$	544.000	−	608.210i	0.813154	−	0.909134i
$670$		0			0
$671$			661.876i			0.986403i
$672$		0			0
$673$		−	1056.00i		−	1.56909i	−0.620070	−	0.784547i	$-0.712896\pi$
$673$		−	1056.00i		−	1.56909i	0.620070	−	0.784547i	$-0.287104\pi$
$674$		0			0
$675$	−674.787	+	16.9505i	−0.999685	+	0.0251118i
$676$		0			0
$677$	612.683			0.904996			0.452498	−	0.891765i	$-0.350533\pi$
$677$	612.683			0.904996			0.452498	+	0.891765i	$0.350533\pi$
$678$		0			0
$679$	−256.000			−0.377025
$680$		0			0
$681$	−550.000	−	491.935i	−0.807636	−	0.722371i
$682$		0			0
$683$	317.522			0.464893			0.232446	−	0.972609i	$-0.425327\pi$
$683$	317.522			0.464893			0.232446	+	0.972609i	$0.425327\pi$
$684$		0			0
$685$	−390.000	+	780.000i	−0.569343	+	1.13869i
$686$		0			0
$687$	−344.354	−	308.000i	−0.501244	−	0.448326i
$688$		0			0
$689$		−	482.991i		−	0.701002i
$690$		0			0
$691$	−922.000			−1.33430			−0.667149	−	0.744924i	$-0.732486\pi$
$691$	−922.000			−1.33430			−0.667149	+	0.744924i	$0.732486\pi$
$692$		0			0
$693$	71.5542	+	640.000i	0.103253	+	0.923521i
$694$		0			0
$695$	263.856	−	527.712i	0.379649	−	0.759298i
$696$		0			0
$697$		−	560.000i		−	0.803443i
$698$		0			0
$699$	−410.000	−	366.715i	−0.586552	−	0.524628i
$700$		0			0
$701$			474.046i			0.676243i	0.941102	+	0.338122i	$0.109791\pi$
$701$			474.046i			0.676243i	−0.941102	+	0.338122i	$0.890209\pi$
$702$		0			0
$703$			264.000i			0.375533i
$704$		0			0
$705$	62.3607	−	24.7214i	0.0884549	−	0.0350658i
$706$		0			0
$707$	−1216.42			−1.72054
$708$		0			0
$709$	−966.000			−1.36248			−0.681241	−	0.732059i	$-0.738559\pi$
$709$	−966.000			−1.36248			−0.681241	+	0.732059i	$0.738559\pi$
$710$		0			0
$711$	78.0000	+	697.653i	0.109705	+	0.981228i
$712$		0			0
$713$	152.053			0.213258
$714$		0			0
$715$	240.000	−	480.000i	0.335664	−	0.671329i
$716$		0			0
$717$	357.771	−	400.000i	0.498983	−	0.557880i
$718$		0			0
$719$		−	1109.09i		−	1.54254i	−0.636505	−	0.771272i	$-0.719621\pi$
$719$		−	1109.09i		−	1.54254i	0.636505	−	0.771272i	$-0.280379\pi$
$720$		0			0
$721$	−832.000			−1.15395
$722$		0			0
$723$	460.630	+	412.000i	0.637109	+	0.569848i
$724$		0			0
$725$	−536.656	+	402.492i	−0.740216	+	0.555162i
$726$		0			0
$727$		−	408.000i		−	0.561210i	−0.959823	−	0.280605i	$-0.909465\pi$
$727$		−	408.000i		−	0.561210i	0.959823	−	0.280605i	$-0.0905352\pi$
$728$		0			0
$729$	−239.000	−	688.709i	−0.327846	−	0.944731i
$730$		0			0
$731$			876.539i			1.19910i
$732$		0			0
$733$			164.000i			0.223738i	0.993723	+	0.111869i	$0.0356837\pi$
$733$			164.000i			0.223738i	−0.993723	+	0.111869i	$0.964316\pi$
$734$		0			0
$735$	209.164	−	82.9180i	0.284577	−	0.112814i
$736$		0			0
$737$	−822.873			−1.11652
$738$		0			0
$739$	−1082.00			−1.46414			−0.732070	−	0.681229i	$-0.761446\pi$
$739$	−1082.00			−1.46414			−0.732070	+	0.681229i	$0.761446\pi$
$740$		0			0
$741$	−144.000	+	160.997i	−0.194332	+	0.217270i
$742$		0			0
$743$	1140.39			1.53485			0.767426	−	0.641138i	$-0.221537\pi$
$743$	1140.39			1.53485			0.767426	+	0.641138i	$0.221537\pi$
$744$		0			0
$745$	440.000	+	220.000i	0.590604	+	0.295302i
$746$		0			0
$747$	102.859	+	920.000i	0.137696	+	1.23159i
$748$		0			0
$749$			1180.64i			1.57629i
$750$		0			0
$751$	958.000			1.27563			0.637816	−	0.770189i	$-0.279838\pi$
$751$	958.000			1.27563			0.637816	+	0.770189i	$0.279838\pi$
$752$		0			0
$753$	−53.6656	+	60.0000i	−0.0712691	+	0.0796813i
$754$		0			0
$755$	−76.0263	+	152.053i	−0.100697	+	0.201394i
$756$		0			0
$757$		−	772.000i		−	1.01982i	−0.860229	−	0.509908i	$-0.829680\pi$
$757$		−	772.000i		−	1.01982i	0.860229	−	0.509908i	$-0.170320\pi$
$758$		0			0
$759$	−80.0000	+	89.4427i	−0.105402	+	0.117843i
$760$		0			0
$761$			1126.98i			1.48092i	0.672102	+	0.740459i	$0.265392\pi$
$761$			1126.98i			1.48092i	−0.672102	+	0.740459i	$0.734608\pi$
$762$		0			0
$763$		−	592.000i		−	0.775885i
$764$		0			0
$765$	−1322.20	−	486.099i	−1.72836	−	0.635424i
$766$		0			0
$767$	−1180.64			−1.53930
$768$		0			0
$769$	−1326.00			−1.72432			−0.862159	−	0.506638i	$-0.830888\pi$
$769$	−1326.00			−1.72432			−0.862159	+	0.506638i	$0.830888\pi$
$770$		0			0
$771$	−90.0000	−	80.4984i	−0.116732	−	0.104408i
$772$		0			0
$773$	147.580			0.190919			0.0954596	−	0.995433i	$-0.469568\pi$
$773$	147.580			0.190919			0.0954596	+	0.995433i	$0.469568\pi$
$774$		0			0
$775$	510.000	+	680.000i	0.658065	+	0.877419i
$776$		0			0
$777$	−787.096	−	704.000i	−1.01299	−	0.906049i
$778$		0			0
$779$			107.331i			0.137781i
$780$		0			0
$781$	480.000			0.614597
$782$		0			0
$783$	−590.322	−	420.000i	−0.753923	−	0.536398i
$784$		0			0
$785$	−411.437	−	205.718i	−0.524123	−	0.262061i
$786$		0			0
$787$			1132.00i			1.43837i	0.694817	+	0.719187i	$0.255485\pi$
$787$			1132.00i			1.43837i	−0.694817	+	0.719187i	$0.744515\pi$
$788$		0			0
$789$	−470.000	−	420.381i	−0.595691	−	0.532802i
$790$		0			0
$791$		−	321.994i		−	0.407072i
$792$		0			0
$793$		−	888.000i		−	1.11980i
$794$		0			0
$795$	−561.246	+	222.492i	−0.705970	+	0.279864i
$796$		0			0
$797$	612.683			0.768736			0.384368	−	0.923180i	$-0.374419\pi$
$797$	612.683			0.768736			0.384368	+	0.923180i	$0.374419\pi$
$798$		0			0
$799$	140.000			0.175219
$800$		0			0
$801$	160.000	−	17.8885i	0.199750	−	0.0223328i
$802$		0			0
$803$	500.879			0.623760
$804$		0			0
$805$	160.000	+	80.0000i	0.198758	+	0.0993789i
$806$		0			0
$807$	−268.328	+	300.000i	−0.332501	+	0.371747i
$808$		0			0
$809$			1234.31i			1.52572i	0.646562	+	0.762861i	$0.276206\pi$
$809$			1234.31i			1.52572i	−0.646562	+	0.762861i	$0.723794\pi$
$810$		0			0
$811$	−874.000			−1.07768			−0.538841	−	0.842408i	$-0.681138\pi$
$811$	−874.000			−1.07768			−0.538841	+	0.842408i	$0.681138\pi$
$812$		0			0
$813$	−889.955	−	796.000i	−1.09466	−	0.979090i
$814$		0			0
$815$	−304.105	−	152.053i	−0.373135	−	0.186568i
$816$		0			0
$817$		−	168.000i		−	0.205630i
$818$		0			0
$819$	−96.0000	−	858.650i	−0.117216	−	1.04841i
$820$		0			0
$821$		−	259.384i		−	0.315937i	−0.987444	−	0.157968i	$-0.949506\pi$
$821$		−	259.384i		−	0.315937i	0.987444	−	0.157968i	$-0.0504944\pi$
$822$		0			0
$823$		−	1368.00i		−	1.66221i	−0.556114	−	0.831106i	$-0.687708\pi$
$823$		−	1368.00i		−	1.66221i	0.556114	−	0.831106i	$-0.312292\pi$
$824$		0			0
$825$	−668.328	−	57.7709i	−0.810095	−	0.0700253i
$826$		0			0
$827$	460.630			0.556989			0.278495	−	0.960438i	$-0.410165\pi$
$827$	460.630			0.556989			0.278495	+	0.960438i	$0.410165\pi$
$828$		0			0
$829$	1002.00			1.20869			0.604343	−	0.796725i	$-0.293436\pi$
$829$	1002.00			1.20869			0.604343	+	0.796725i	$0.293436\pi$
$830$		0			0
$831$	−584.000	+	652.932i	−0.702768	+	0.785718i
$832$		0			0
$833$	469.574			0.563715
$834$		0			0
$835$	150.000	−	300.000i	0.179641	−	0.359281i
$836$		0			0
$837$	−532.184	+	748.000i	−0.635823	+	0.893668i
$838$		0			0
$839$			304.105i			0.362462i	0.983441	+	0.181231i	$0.0580081\pi$
$839$			304.105i			0.362462i	−0.983441	+	0.181231i	$0.941992\pi$
$840$		0			0
$841$	121.000			0.143876
$842$		0			0
$843$	107.331	−	120.000i	0.127321	−	0.142349i
$844$		0			0
$845$	55.9017	−	111.803i	0.0661559	−	0.132312i
$846$		0			0
$847$		−	328.000i		−	0.387249i
$848$		0			0
$849$	104.000	−	116.276i	0.122497	−	0.136956i
$850$		0			0
$851$		−	196.774i		−	0.231227i
$852$		0			0
$853$		−	772.000i		−	0.905041i	−0.891754	−	0.452521i	$-0.850525\pi$
$853$		−	772.000i		−	0.905041i	0.891754	−	0.452521i	$-0.149475\pi$
$854$		0			0
$855$	253.416	+	93.1672i	0.296393	+	0.108967i
$856$		0			0
$857$	898.899			1.04889			0.524445	−	0.851444i	$-0.324273\pi$
$857$	898.899			1.04889			0.524445	+	0.851444i	$0.324273\pi$
$858$		0			0
$859$	278.000			0.323632			0.161816	−	0.986821i	$-0.448265\pi$
$859$	278.000			0.323632			0.161816	+	0.986821i	$0.448265\pi$
$860$		0			0
$861$	−320.000	−	286.217i	−0.371661	−	0.332424i
$862$		0			0
$863$	−576.906			−0.668488			−0.334244	−	0.942486i	$-0.608481\pi$
$863$	−576.906			−0.668488			−0.334244	+	0.942486i	$0.608481\pi$
$864$		0			0
$865$	170.000	−	340.000i	0.196532	−	0.393064i
$866$		0			0
$867$	−1545.12	−	1382.00i	−1.78215	−	1.59400i
$868$		0			0
$869$			697.653i			0.802823i
$870$		0			0
$871$	1104.00			1.26751
$872$		0			0
$873$	286.217	−	32.0000i	0.327854	−	0.0366552i
$874$		0			0
$875$	178.885	+	983.870i	0.204441	+	1.12442i
$876$		0			0
$877$			628.000i			0.716078i	0.933707	+	0.358039i	$0.116554\pi$
$877$			628.000i			0.716078i	−0.933707	+	0.358039i	$0.883446\pi$
$878$		0			0
$879$	870.000	+	778.152i	0.989761	+	0.885269i
$880$		0			0
$881$		−	536.656i		−	0.609145i	−0.952489	−	0.304572i	$-0.901486\pi$
$881$		−	536.656i		−	0.609145i	0.952489	−	0.304572i	$-0.0985135\pi$
$882$		0			0
$883$		−	948.000i		−	1.07361i	−0.843705	−	0.536806i	$-0.819631\pi$
$883$		−	948.000i		−	1.07361i	0.843705	−	0.536806i	$-0.180369\pi$
$884$		0			0
$885$	543.870	+	1371.93i	0.614542	+	1.55021i
$886$		0			0
$887$	281.745			0.317638			0.158819	−	0.987308i	$-0.449231\pi$
$887$	281.745			0.317638			0.158819	+	0.987308i	$0.449231\pi$
$888$		0			0
$889$	128.000			0.143982
$890$		0			0
$891$	−160.000	−	706.597i	−0.179574	−	0.793039i
$892$		0			0
$893$	−26.8328			−0.0300479
$894$		0			0
$895$	−1160.00	−	580.000i	−1.29609	−	0.648045i
$896$		0			0
$897$	107.331	−	120.000i	0.119656	−	0.133779i
$898$		0			0
$899$			912.316i			1.01481i
$900$		0			0
$901$	−1260.00			−1.39845
$902$		0			0
$903$	500.879	+	448.000i	0.554684	+	0.496124i
$904$		0			0
$905$	−371.187	+	742.375i	−0.410152	+	0.820303i
$906$		0			0
$907$			516.000i			0.568908i	0.958690	+	0.284454i	$0.0918124\pi$
$907$			516.000i			0.568908i	−0.958690	+	0.284454i	$0.908188\pi$
$908$		0			0
$909$	1360.00	−	152.053i	1.49615	−	0.167275i
$910$		0			0
$911$			321.994i			0.353451i	0.984260	+	0.176725i	$0.0565505\pi$
$911$			321.994i			0.353451i	−0.984260	+	0.176725i	$0.943450\pi$
$912$		0			0
$913$			920.000i			1.00767i
$914$		0			0
$915$	−1031.88	+	409.062i	−1.12773	+	0.447062i
$916$		0			0
$917$	−643.988			−0.702277
$918$		0			0
$919$	−1314.00			−1.42982			−0.714908	−	0.699219i	$-0.753531\pi$
$919$	−1314.00			−1.42982			−0.714908	+	0.699219i	$0.753531\pi$
$920$		0			0
$921$	984.000	−	1100.15i	1.06840	−	1.19451i
$922$		0			0
$923$	−643.988			−0.697711
$924$		0			0
$925$	880.000	−	660.000i	0.951351	−	0.713514i
$926$		0			0
$927$	930.204	−	104.000i	1.00346	−	0.112190i
$928$		0			0
$929$		−	1502.64i		−	1.61748i	−0.588167	−	0.808739i	$-0.700150\pi$
$929$		−	1502.64i		−	1.61748i	0.588167	−	0.808739i	$-0.299850\pi$
$930$		0			0
$931$	−90.0000			−0.0966702
$932$		0			0
$933$	965.981	−	1080.00i	1.03535	−	1.15756i
$934$		0			0
$935$	−1252.20	−	626.099i	−1.33925	−	0.669625i
$936$		0			0
$937$			1288.00i			1.37460i	0.726374	+	0.687300i	$0.241204\pi$
$937$			1288.00i			1.37460i	−0.726374	+	0.687300i	$0.758796\pi$
$938$		0			0
$939$	1136.00	−	1270.09i	1.20980	−	1.35259i
$940$		0			0
$941$			366.715i			0.389708i	0.980832	+	0.194854i	$0.0624233\pi$
$941$			366.715i			0.389708i	−0.980832	+	0.194854i	$0.937577\pi$
$942$		0			0
$943$		−	80.0000i		−	0.0848356i
$944$		0			0
$945$	−953.548	+	507.096i	−1.00905	+	0.536609i
$946$		0			0
$947$	1677.05			1.77091			0.885455	−	0.464726i	$-0.153847\pi$
$947$	1677.05			1.77091			0.885455	+	0.464726i	$0.153847\pi$
$948$		0			0
$949$	−672.000			−0.708114
$950$		0			0
$951$	−1370.00	−	1225.37i	−1.44059	−	1.28850i
$952$		0			0
$953$	827.345			0.868148			0.434074	−	0.900877i	$-0.357076\pi$
$953$	827.345			0.868148			0.434074	+	0.900877i	$0.357076\pi$
$954$		0			0
$955$	960.000	+	480.000i	1.00524	+	0.502618i
$956$		0			0
$957$	−536.656	−	480.000i	−0.560769	−	0.501567i
$958$		0			0
$959$			1395.31i			1.45496i
$960$		0			0
$961$	195.000			0.202914
$962$		0			0
$963$	−147.580	−	1320.00i	−0.153251	−	1.37072i
$964$		0			0
$965$	−143.108	−	71.5542i	−0.148299	−	0.0741494i
$966$		0			0
$967$			536.000i			0.554292i	0.960828	+	0.277146i	$0.0893885\pi$
$967$			536.000i			0.554292i	−0.960828	+	0.277146i	$0.910611\pi$
$968$		0			0
$969$	420.000	+	375.659i	0.433437	+	0.387677i
$970$		0			0
$971$		−	134.164i		−	0.138171i	−0.997611	−	0.0690855i	$-0.977992\pi$
$971$		−	134.164i		−	0.138171i	0.997611	−	0.0690855i	$-0.0220081\pi$
$972$		0			0
$973$		−	944.000i		−	0.970195i
$974$		0			0
$975$	896.656	+	77.5078i	0.919648	+	0.0794951i
$976$		0			0
$977$	−174.413			−0.178519			−0.0892596	−	0.996008i	$-0.528450\pi$
$977$	−174.413			−0.178519			−0.0892596	+	0.996008i	$0.528450\pi$
$978$		0			0
$979$	160.000			0.163432
$980$		0			0
$981$	74.0000	+	661.876i	0.0754332	+	0.674695i
$982$		0			0
$983$	1855.94			1.88803			0.944016	−	0.329898i	$-0.107014\pi$
$983$	1855.94			1.88803			0.944016	+	0.329898i	$0.107014\pi$
$984$		0			0
$985$	10.0000	−	20.0000i	0.0101523	−	0.0203046i
$986$		0			0
$987$	71.5542	−	80.0000i	0.0724966	−	0.0810537i
$988$		0			0
$989$			125.220i			0.126613i
$990$		0			0
$991$	1646.00			1.66095			0.830474	−	0.557057i	$-0.188070\pi$
$991$	1646.00			1.66095			0.830474	+	0.557057i	$0.188070\pi$
$992$		0			0
$993$	451.686	+	404.000i	0.454870	+	0.406848i
$994$		0			0
$995$	−254.912	+	509.823i	−0.256193	+	0.512385i
$996$		0			0
$997$			1164.00i			1.16750i	0.811933	+	0.583751i	$0.198416\pi$
$997$			1164.00i			1.16750i	−0.811933	+	0.583751i	$0.801584\pi$
$998$		0			0
$999$	968.000	+	688.709i	0.968969	+	0.689398i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.3.c.d.209.1		4
3.2	odd	2	inner	240.3.c.d.209.3		4
4.3	odd	2		60.3.b.a.29.4	yes	4
5.2	odd	4		1200.3.l.h.401.2		2
5.3	odd	4		1200.3.l.q.401.1		2
5.4	even	2	inner	240.3.c.d.209.4		4
8.3	odd	2		960.3.c.h.449.1		4
8.5	even	2		960.3.c.g.449.4		4
12.11	even	2		60.3.b.a.29.2	yes	4
15.2	even	4		1200.3.l.h.401.1		2
15.8	even	4		1200.3.l.q.401.2		2
15.14	odd	2	inner	240.3.c.d.209.2		4
20.3	even	4		300.3.g.e.101.2		2
20.7	even	4		300.3.g.h.101.1		2
20.19	odd	2		60.3.b.a.29.1	✓	4
24.5	odd	2		960.3.c.g.449.2		4
24.11	even	2		960.3.c.h.449.3		4
36.7	odd	6		1620.3.t.b.1349.3		8
36.11	even	6		1620.3.t.b.1349.2		8
36.23	even	6		1620.3.t.b.269.4		8
36.31	odd	6		1620.3.t.b.269.1		8
40.19	odd	2		960.3.c.h.449.4		4
40.29	even	2		960.3.c.g.449.1		4
60.23	odd	4		300.3.g.e.101.1		2
60.47	odd	4		300.3.g.h.101.2		2
60.59	even	2		60.3.b.a.29.3	yes	4
120.29	odd	2		960.3.c.g.449.3		4
120.59	even	2		960.3.c.h.449.2		4
180.59	even	6		1620.3.t.b.269.3		8
180.79	odd	6		1620.3.t.b.1349.4		8
180.119	even	6		1620.3.t.b.1349.1		8
180.139	odd	6		1620.3.t.b.269.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.3.b.a.29.1	✓	4	20.19	odd	2
60.3.b.a.29.2	yes	4	12.11	even	2
60.3.b.a.29.3	yes	4	60.59	even	2
60.3.b.a.29.4	yes	4	4.3	odd	2
240.3.c.d.209.1		4	1.1	even	1	trivial
240.3.c.d.209.2		4	15.14	odd	2	inner
240.3.c.d.209.3		4	3.2	odd	2	inner
240.3.c.d.209.4		4	5.4	even	2	inner
300.3.g.e.101.1		2	60.23	odd	4
300.3.g.e.101.2		2	20.3	even	4
300.3.g.h.101.1		2	20.7	even	4
300.3.g.h.101.2		2	60.47	odd	4
960.3.c.g.449.1		4	40.29	even	2
960.3.c.g.449.2		4	24.5	odd	2
960.3.c.g.449.3		4	120.29	odd	2
960.3.c.g.449.4		4	8.5	even	2
960.3.c.h.449.1		4	8.3	odd	2
960.3.c.h.449.2		4	120.59	even	2
960.3.c.h.449.3		4	24.11	even	2
960.3.c.h.449.4		4	40.19	odd	2
1200.3.l.h.401.1		2	15.2	even	4
1200.3.l.h.401.2		2	5.2	odd	4
1200.3.l.q.401.1		2	5.3	odd	4
1200.3.l.q.401.2		2	15.8	even	4
1620.3.t.b.269.1		8	36.31	odd	6
1620.3.t.b.269.2		8	180.139	odd	6
1620.3.t.b.269.3		8	180.59	even	6
1620.3.t.b.269.4		8	36.23	even	6
1620.3.t.b.1349.1		8	180.119	even	6
1620.3.t.b.1349.2		8	36.11	even	6
1620.3.t.b.1349.3		8	36.7	odd	6
1620.3.t.b.1349.4		8	180.79	odd	6

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:	\( N \)	\(=\)	\( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	240.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.23607 - 2.00000i) q^{3} +(2.23607 - 4.47214i) q^{5} -8.00000i q^{7} +(1.00000 + 8.94427i) q^{9} +O(q^{10})\)	\(q+(-2.23607 - 2.00000i) q^{3} +(2.23607 - 4.47214i) q^{5} -8.00000i q^{7} +(1.00000 + 8.94427i) q^{9} +8.94427i q^{11} -12.0000i q^{13} +(-13.9443 + 5.52786i) q^{15} -31.3050 q^{17} +6.00000 q^{19} +(-16.0000 + 17.8885i) q^{21} -4.47214 q^{23} +(-15.0000 - 20.0000i) q^{25} +(15.6525 - 22.0000i) q^{27} -26.8328i q^{29} -34.0000 q^{31} +(17.8885 - 20.0000i) q^{33} +(-35.7771 - 17.8885i) q^{35} +44.0000i q^{37} +(-24.0000 + 26.8328i) q^{39} +17.8885i q^{41} -28.0000i q^{43} +(42.2361 + 15.5279i) q^{45} -4.47214 q^{47} -15.0000 q^{49} +(70.0000 + 62.6099i) q^{51} +40.2492 q^{53} +(40.0000 + 20.0000i) q^{55} +(-13.4164 - 12.0000i) q^{57} -98.3870i q^{59} +74.0000 q^{61} +(71.5542 - 8.00000i) q^{63} +(-53.6656 - 26.8328i) q^{65} +92.0000i q^{67} +(10.0000 + 8.94427i) q^{69} -53.6656i q^{71} -56.0000i q^{73} +(-6.45898 + 74.7214i) q^{75} +71.5542 q^{77} +78.0000 q^{79} +(-79.0000 + 17.8885i) q^{81} +102.859 q^{83} +(-70.0000 + 140.000i) q^{85} +(-53.6656 + 60.0000i) q^{87} -17.8885i q^{89} -96.0000 q^{91} +(76.0263 + 68.0000i) q^{93} +(13.4164 - 26.8328i) q^{95} -32.0000i q^{97} +(-80.0000 + 8.94427i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^9	\( 4 q + 4 q^{9} - 20 q^{15} + 24 q^{19} - 64 q^{21} - 60 q^{25} - 136 q^{31} - 96 q^{39} + 160 q^{45} - 60 q^{49} + 280 q^{51} + 160 q^{55} + 296 q^{61} + 40 q^{69} - 160 q^{75} + 312 q^{79} - 316 q^{81} - 280 q^{85} - 384 q^{91} - 320 q^{99}+O(q^{100}) \) 4 * q + 4 * q^9 - 20 * q^15 + 24 * q^19 - 64 * q^21 - 60 * q^25 - 136 * q^31 - 96 * q^39 + 160 * q^45 - 60 * q^49 + 280 * q^51 + 160 * q^55 + 296 * q^61 + 40 * q^69 - 160 * q^75 + 312 * q^79 - 316 * q^81 - 280 * q^85 - 384 * q^91 - 320 * q^99

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