LMFDB - Embedded newform 1620.3.t.b.1349.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,3,Mod(269,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 1, 3]))

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

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

chi := DirichletCharacter("1620.269");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1620.t (of order $6$, 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:	$44.1418028264$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.12960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 $ x^8 - 3x^6 + 8x^4 - 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 60)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1349.2
Root			$1.40126 + 0.809017i$ of defining polynomial
Character	$\chi$	$=$	1620.1349
Dual form			1620.3.t.b.269.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.75495 - 4.17256i) q^{5} +(6.92820 - 4.00000i) q^{7} +O(q^{10})$	$q+(-2.75495 - 4.17256i) q^{5} +(6.92820 - 4.00000i) q^{7} +(7.74597 - 4.47214i) q^{11} +(10.3923 + 6.00000i) q^{13} +31.3050 q^{17} -6.00000 q^{19} +(2.23607 - 3.87298i) q^{23} +(-9.82051 + 22.9904i) q^{25} +(23.2379 - 13.4164i) q^{29} +(-17.0000 + 29.4449i) q^{31} +(-35.7771 - 17.8885i) q^{35} +44.0000i q^{37} +(15.4919 + 8.94427i) q^{41} +(24.2487 - 14.0000i) q^{43} +(2.23607 + 3.87298i) q^{47} +(7.50000 - 12.9904i) q^{49} -40.2492 q^{53} +(-40.0000 - 20.0000i) q^{55} +(85.2056 + 49.1935i) q^{59} +(-37.0000 - 64.0859i) q^{61} +(-3.59492 - 59.8922i) q^{65} +(79.6743 + 46.0000i) q^{67} -53.6656i q^{71} -56.0000i q^{73} +(35.7771 - 61.9677i) q^{77} +(39.0000 + 67.5500i) q^{79} +(-51.4296 - 89.0786i) q^{83} +(-86.2436 - 130.622i) q^{85} +17.8885i q^{89} +96.0000 q^{91} +(16.5297 + 25.0354i) q^{95} +(-27.7128 + 16.0000i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 48 q^{19} + 60 q^{25} - 136 q^{31} + 60 q^{49} - 320 q^{55} - 296 q^{61} + 312 q^{79} + 280 q^{85} + 768 q^{91}+O(q^{100}) $ 8 * q - 48 * q^19 + 60 * q^25 - 136 * q^31 + 60 * q^49 - 320 * q^55 - 296 * q^61 + 312 * q^79 + 280 * q^85 + 768 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$.

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$-1$	$e\left(\frac{5}{6}\right)$

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
$5$	−2.75495	−	4.17256i	−0.550990	−	0.834512i
$5$	−2.75495	−	4.17256i	−0.550990	−	0.834512i
$6$		0			0
$7$	6.92820	−	4.00000i	0.989743	−	0.571429i	0.0845458	−	0.996420i	$-0.473056\pi$
$7$	6.92820	−	4.00000i	0.989743	−	0.571429i	0.905198	+	0.424991i	$0.139723\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	7.74597	−	4.47214i	0.704179	−	0.406558i	−0.104723	−	0.994501i	$-0.533396\pi$
$11$	7.74597	−	4.47214i	0.704179	−	0.406558i	0.808902	+	0.587944i	$0.200062\pi$
$12$		0			0
$13$	10.3923	+	6.00000i	0.799408	+	0.461538i	0.843264	−	0.537499i	$-0.180631\pi$
$13$	10.3923	+	6.00000i	0.799408	+	0.461538i	−0.0438561	+	0.999038i	$0.513964\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	31.3050			1.84147			0.920734	−	0.390191i	$-0.127591\pi$
$17$	31.3050			1.84147			0.920734	+	0.390191i	$0.127591\pi$
$18$		0			0
$19$	−6.00000			−0.315789			−0.157895	−	0.987456i	$-0.550471\pi$
$19$	−6.00000			−0.315789			−0.157895	+	0.987456i	$0.550471\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.23607	−	3.87298i	0.0972203	−	0.168391i	−0.813313	−	0.581827i	$-0.802338\pi$
$23$	2.23607	−	3.87298i	0.0972203	−	0.168391i	0.910533	+	0.413436i	$0.135672\pi$
$24$		0			0
$25$	−9.82051	+	22.9904i	−0.392820	+	0.919615i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	23.2379	−	13.4164i	0.801307	−	0.462635i	−0.0426210	−	0.999091i	$-0.513571\pi$
$29$	23.2379	−	13.4164i	0.801307	−	0.462635i	0.843928	+	0.536457i	$0.180237\pi$
$30$		0			0
$31$	−17.0000	+	29.4449i	−0.548387	+	0.949834i	0.449998	+	0.893029i	$0.351425\pi$
$31$	−17.0000	+	29.4449i	−0.548387	+	0.949834i	−0.998385	+	0.0568049i	$0.981909\pi$
$32$		0			0
$33$		0			0
$34$		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
$39$		0			0
$40$		0			0
$41$	15.4919	+	8.94427i	0.377852	+	0.218153i	0.676883	−	0.736090i	$-0.263330\pi$
$41$	15.4919	+	8.94427i	0.377852	+	0.218153i	−0.299031	+	0.954243i	$0.596663\pi$
$42$		0			0
$43$	24.2487	−	14.0000i	0.563924	−	0.325581i	−0.190795	−	0.981630i	$-0.561107\pi$
$43$	24.2487	−	14.0000i	0.563924	−	0.325581i	0.754719	+	0.656048i	$0.227773\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	2.23607	+	3.87298i	0.0475759	+	0.0824039i	0.888833	−	0.458232i	$-0.151517\pi$
$47$	2.23607	+	3.87298i	0.0475759	+	0.0824039i	−0.841257	+	0.540636i	$0.818184\pi$
$48$		0			0
$49$	7.50000	−	12.9904i	0.153061	−	0.265110i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−40.2492			−0.759419			−0.379710	−	0.925106i	$-0.623976\pi$
$53$	−40.2492			−0.759419			−0.379710	+	0.925106i	$0.623976\pi$
$54$		0			0
$55$	−40.0000	−	20.0000i	−0.727273	−	0.363636i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	85.2056	+	49.1935i	1.44416	+	0.833788i	0.998124	−	0.0612253i	$-0.0195008\pi$
$59$	85.2056	+	49.1935i	1.44416	+	0.833788i	0.446039	+	0.895013i	$0.352834\pi$
$60$		0			0
$61$	−37.0000	−	64.0859i	−0.606557	−	1.05059i	−0.991803	−	0.127774i	$-0.959217\pi$
$61$	−37.0000	−	64.0859i	−0.606557	−	1.05059i	0.385246	−	0.922814i	$-0.374117\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−3.59492	−	59.8922i	−0.0553064	−	0.921419i
$66$		0			0
$67$	79.6743	+	46.0000i	1.18917	+	0.686567i	0.958118	−	0.286374i	$-0.0924501\pi$
$67$	79.6743	+	46.0000i	1.18917	+	0.686567i	0.231051	+	0.972942i	$0.425783\pi$
$68$		0			0
$69$		0			0
$70$		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
$75$		0			0
$76$		0			0
$77$	35.7771	−	61.9677i	0.464638	−	0.804776i
$78$		0			0
$79$	39.0000	+	67.5500i	0.493671	+	0.855063i	0.999973	−	0.00729288i	$-0.00232142\pi$
$79$	39.0000	+	67.5500i	0.493671	+	0.855063i	−0.506303	+	0.862356i	$0.668988\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−51.4296	−	89.0786i	−0.619633	−	1.07324i	−0.989553	−	0.144172i	$-0.953948\pi$
$83$	−51.4296	−	89.0786i	−0.619633	−	1.07324i	0.369919	−	0.929064i	$-0.379385\pi$
$84$		0			0
$85$	−86.2436	−	130.622i	−1.01463	−	1.53673i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			17.8885i			0.200995i	0.994937	+	0.100497i	$0.0320434\pi$
$89$			17.8885i			0.200995i	−0.994937	+	0.100497i	$0.967957\pi$
$90$		0			0
$91$	96.0000			1.05495
$92$		0			0
$93$		0			0
$94$		0			0
$95$	16.5297	+	25.0354i	0.173997	+	0.263530i
$96$		0			0
$97$	−27.7128	+	16.0000i	−0.285699	+	0.164948i	−0.636001	−	0.771689i	$-0.719412\pi$
$97$	−27.7128	+	16.0000i	−0.285699	+	0.164948i	0.350302	+	0.936637i	$0.386079\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	131.681	−	76.0263i	1.30378	−	0.752736i	0.322727	−	0.946492i	$-0.395400\pi$
$101$	131.681	−	76.0263i	1.30378	−	0.752736i	0.981050	+	0.193756i	$0.0620671\pi$
$102$		0			0
$103$	−90.0666	−	52.0000i	−0.874433	−	0.504854i	−0.00561444	−	0.999984i	$-0.501787\pi$
$103$	−90.0666	−	52.0000i	−0.874433	−	0.504854i	−0.868819	+	0.495130i	$0.835120\pi$
$104$		0			0
$105$		0			0
$106$		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
$111$		0			0
$112$		0			0
$113$	20.1246	−	34.8569i	0.178094	−	0.308468i	−0.763134	−	0.646241i	$-0.776340\pi$
$113$	20.1246	−	34.8569i	0.178094	−	0.308468i	0.941228	+	0.337773i	$0.109674\pi$
$114$		0			0
$115$	−22.3205	+	1.33975i	−0.194091	+	0.0116500i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	216.887	−	125.220i	1.82258	−	1.05227i
$120$		0			0
$121$	−20.5000	+	35.5070i	−0.169421	+	0.293447i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	122.984	−	22.3607i	0.983870	−	0.178885i
$126$		0			0
$127$		−	16.0000i		−	0.125984i	−0.998014	−	0.0629921i	$-0.979936\pi$
$127$		−	16.0000i		−	0.125984i	0.998014	−	0.0629921i	$-0.0200643\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	69.7137	+	40.2492i	0.532166	+	0.307246i	0.741898	−	0.670513i	$-0.233926\pi$
$131$	69.7137	+	40.2492i	0.532166	+	0.307246i	−0.209732	+	0.977759i	$0.567259\pi$
$132$		0			0
$133$	−41.5692	+	24.0000i	−0.312551	+	0.180451i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−87.2067	−	151.046i	−0.636545	−	1.10253i	−0.986186	−	0.165644i	$-0.947030\pi$
$137$	−87.2067	−	151.046i	−0.636545	−	1.10253i	0.349641	−	0.936884i	$-0.386304\pi$
$138$		0			0
$139$	59.0000	−	102.191i	0.424460	−	0.735187i	−0.571909	−	0.820317i	$-0.693797\pi$
$139$	59.0000	−	102.191i	0.424460	−	0.735187i	0.996370	+	0.0851297i	$0.0271305\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	107.331			0.750568
$144$		0			0
$145$	−120.000	−	60.0000i	−0.827586	−	0.413793i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	85.2056	+	49.1935i	0.571850	+	0.330158i	0.757888	−	0.652385i	$-0.226231\pi$
$149$	85.2056	+	49.1935i	0.571850	+	0.330158i	−0.186038	+	0.982543i	$0.559565\pi$
$150$		0			0
$151$	−17.0000	−	29.4449i	−0.112583	−	0.194999i	0.804228	−	0.594321i	$-0.202579\pi$
$151$	−17.0000	−	29.4449i	−0.112583	−	0.194999i	−0.916811	+	0.399322i	$0.869246\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	169.695	−	10.1856i	1.09480	−	0.0657135i
$156$		0			0
$157$	79.6743	+	46.0000i	0.507480	+	0.292994i	0.731797	−	0.681523i	$-0.238682\pi$
$157$	79.6743	+	46.0000i	0.507480	+	0.292994i	−0.224317	+	0.974516i	$0.572015\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	35.7771i		−	0.222218i
$162$		0			0
$163$			68.0000i			0.417178i	0.978003	+	0.208589i	$0.0668871\pi$
$163$			68.0000i			0.417178i	−0.978003	+	0.208589i	$0.933113\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−33.5410	+	58.0948i	−0.200844	+	0.347873i	−0.948801	−	0.315875i	$-0.897702\pi$
$167$	−33.5410	+	58.0948i	−0.200844	+	0.347873i	0.747956	+	0.663748i	$0.231035\pi$
$168$		0			0
$169$	−12.5000	−	21.6506i	−0.0739645	−	0.128110i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	38.0132	+	65.8407i	0.219729	+	0.380582i	0.954725	−	0.297489i	$-0.0961493\pi$
$173$	38.0132	+	65.8407i	0.219729	+	0.380582i	−0.734996	+	0.678072i	$0.762816\pi$
$174$		0			0
$175$	23.9230	+	198.564i	0.136703	+	1.13465i
$176$		0			0
$177$		0			0
$178$		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
$183$		0			0
$184$		0			0
$185$	183.593	−	121.218i	0.992393	−	0.655231i
$186$		0			0
$187$	242.487	−	140.000i	1.29672	−	0.748663i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	185.903	−	107.331i	0.973315	−	0.561944i	0.0730699	−	0.997327i	$-0.476720\pi$
$191$	185.903	−	107.331i	0.973315	−	0.561944i	0.900245	+	0.435383i	$0.143387\pi$
$192$		0			0
$193$	27.7128	+	16.0000i	0.143590	+	0.0829016i	0.570074	−	0.821594i	$-0.306915\pi$
$193$	27.7128	+	16.0000i	0.143590	+	0.0829016i	−0.426484	+	0.904495i	$0.640248\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−4.47214			−0.0227012			−0.0113506	−	0.999936i	$-0.503613\pi$
$197$	−4.47214			−0.0227012			−0.0113506	+	0.999936i	$0.503613\pi$
$198$		0			0
$199$	114.000			0.572864			0.286432	−	0.958101i	$-0.407531\pi$
$199$	114.000			0.572864			0.286432	+	0.958101i	$0.407531\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	107.331	−	185.903i	0.528725	−	0.915779i
$204$		0			0
$205$	−5.35898	−	89.2820i	−0.0261414	−	0.435522i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−46.4758	+	26.8328i	−0.222372	+	0.128387i
$210$		0			0
$211$	3.00000	−	5.19615i	0.0142180	−	0.0246263i	−0.858829	−	0.512263i	$-0.828807\pi$
$211$	3.00000	−	5.19615i	0.0142180	−	0.0246263i	0.873047	+	0.487636i	$0.162141\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−125.220	−	62.6099i	−0.582418	−	0.291209i
$216$		0			0
$217$			272.000i			1.25346i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	325.331	+	187.830i	1.47208	+	0.849908i
$222$		0			0
$223$	−235.559	+	136.000i	−1.05632	+	0.609865i	−0.924411	−	0.381397i	$-0.875443\pi$
$223$	−235.559	+	136.000i	−1.05632	+	0.609865i	−0.131907	+	0.991262i	$0.542110\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−122.984	−	213.014i	−0.541779	−	0.938388i	−0.998802	−	0.0489334i	$-0.984418\pi$
$227$	−122.984	−	213.014i	−0.541779	−	0.938388i	0.457023	−	0.889455i	$-0.348916\pi$
$228$		0			0
$229$	−77.0000	+	133.368i	−0.336245	+	0.582393i	−0.983723	−	0.179691i	$-0.942490\pi$
$229$	−77.0000	+	133.368i	−0.336245	+	0.582393i	0.647479	+	0.762084i	$0.275824\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−183.358			−0.786942			−0.393471	−	0.919337i	$-0.628726\pi$
$233$	−183.358			−0.786942			−0.393471	+	0.919337i	$0.628726\pi$
$234$		0			0
$235$	10.0000	−	20.0000i	0.0425532	−	0.0851064i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−154.919	−	89.4427i	−0.648198	−	0.374237i	0.139567	−	0.990213i	$-0.455429\pi$
$239$	−154.919	−	89.4427i	−0.648198	−	0.374237i	−0.787766	+	0.615975i	$0.788762\pi$
$240$		0			0
$241$	103.000	+	178.401i	0.427386	+	0.740254i	0.996640	−	0.0819076i	$-0.0261012\pi$
$241$	103.000	+	178.401i	0.427386	+	0.740254i	−0.569254	+	0.822162i	$0.692768\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−74.8653	+	4.49364i	−0.305572	+	0.0183414i
$246$		0			0
$247$	−62.3538	−	36.0000i	−0.252445	−	0.145749i
$248$		0			0
$249$		0			0
$250$		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
$255$		0			0
$256$		0			0
$257$	20.1246	−	34.8569i	0.0783059	−	0.135630i	−0.824213	−	0.566280i	$-0.808382\pi$
$257$	20.1246	−	34.8569i	0.0783059	−	0.135630i	0.902519	+	0.430650i	$0.141716\pi$
$258$		0			0
$259$	176.000	+	304.841i	0.679537	+	1.17699i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−105.095	−	182.030i	−0.399602	−	0.692130i	0.594075	−	0.804410i	$-0.297518\pi$
$263$	−105.095	−	182.030i	−0.399602	−	0.692130i	−0.993677	+	0.112279i	$0.964185\pi$
$264$		0			0
$265$	110.885	+	167.942i	0.418432	+	0.633744i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			134.164i			0.498751i	0.968407	+	0.249376i	$0.0802254\pi$
$269$			134.164i			0.498751i	−0.968407	+	0.249376i	$0.919775\pi$
$270$		0			0
$271$	−398.000			−1.46863			−0.734317	−	0.678806i	$-0.762498\pi$
$271$	−398.000			−1.46863			−0.734317	+	0.678806i	$0.762498\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	26.7468	+	222.001i	0.0972610	+	0.807278i
$276$		0			0
$277$	−252.879	+	146.000i	−0.912922	+	0.527076i	−0.881370	−	0.472426i	$-0.843378\pi$
$277$	−252.879	+	146.000i	−0.912922	+	0.527076i	−0.0315519	+	0.999502i	$0.510045\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−46.4758	+	26.8328i	−0.165394	+	0.0954904i	−0.580412	−	0.814323i	$-0.697109\pi$
$281$	−46.4758	+	26.8328i	−0.165394	+	0.0954904i	0.415018	+	0.909813i	$0.363775\pi$
$282$		0			0
$283$	45.0333	+	26.0000i	0.159128	+	0.0918728i	0.577450	−	0.816426i	$-0.304048\pi$
$283$	45.0333	+	26.0000i	0.159128	+	0.0918728i	−0.418321	+	0.908299i	$0.637381\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	143.108			0.498635
$288$		0			0
$289$	691.000			2.39100
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−194.538	+	336.950i	−0.663952	+	1.15000i	0.315616	+	0.948887i	$0.397789\pi$
$293$	−194.538	+	336.950i	−0.663952	+	1.15000i	−0.979568	+	0.201112i	$0.935545\pi$
$294$		0			0
$295$	−29.4744	−	491.051i	−0.0999133	−	1.66458i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	46.4758	−	26.8328i	0.155437	−	0.0897419i
$300$		0			0
$301$	112.000	−	193.990i	0.372093	−	0.644484i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−165.469	+	330.938i	−0.542521	+	1.08504i
$306$		0			0
$307$		−	492.000i		−	1.60261i	−0.598259	−	0.801303i	$-0.704141\pi$
$307$		−	492.000i		−	1.60261i	0.598259	−	0.801303i	$-0.295859\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−418.282	−	241.495i	−1.34496	−	0.776512i	−0.357428	−	0.933941i	$-0.616346\pi$
$311$	−418.282	−	241.495i	−1.34496	−	0.776512i	−0.987530	+	0.157428i	$0.949680\pi$
$312$		0			0
$313$	491.902	−	284.000i	1.57157	−	0.907348i	0.575597	−	0.817734i	$-0.304770\pi$
$313$	491.902	−	284.000i	1.57157	−	0.907348i	0.995977	−	0.0896145i	$-0.0285635\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	306.341	+	530.599i	0.966376	+	1.67381i	0.705871	+	0.708341i	$0.250556\pi$
$317$	306.341	+	530.599i	0.966376	+	1.67381i	0.260506	+	0.965472i	$0.416111\pi$
$318$		0			0
$319$	120.000	−	207.846i	0.376176	−	0.651555i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−187.830			−0.581516
$324$		0			0
$325$	−240.000	+	180.000i	−0.738462	+	0.553846i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	30.9839	+	17.8885i	0.0941759	+	0.0543725i
$330$		0			0
$331$	−101.000	−	174.937i	−0.305136	−	0.528511i	0.672156	−	0.740410i	$-0.265369\pi$
$331$	−101.000	−	174.937i	−0.305136	−	0.528511i	−0.977292	+	0.211899i	$0.932035\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−27.5610	−	459.174i	−0.0822717	−	1.37067i
$336$		0			0
$337$	−318.697	−	184.000i	−0.945689	−	0.545994i	−0.0539502	−	0.998544i	$-0.517181\pi$
$337$	−318.697	−	184.000i	−0.945689	−	0.545994i	−0.891739	+	0.452550i	$0.850515\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			304.105i			0.891804i
$342$		0			0
$343$			272.000i			0.793003i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	127.456	−	220.760i	0.367308	−	0.636196i	−0.621836	−	0.783148i	$-0.713613\pi$
$347$	127.456	−	220.760i	0.367308	−	0.636196i	0.989144	+	0.146952i	$0.0469462\pi$
$348$		0			0
$349$	59.0000	+	102.191i	0.169054	+	0.292811i	0.938088	−	0.346398i	$-0.112595\pi$
$349$	59.0000	+	102.191i	0.169054	+	0.292811i	−0.769033	+	0.639209i	$0.779262\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−15.6525	−	27.1109i	−0.0443413	−	0.0768014i	0.843003	−	0.537909i	$-0.180786\pi$
$353$	−15.6525	−	27.1109i	−0.0443413	−	0.0768014i	−0.887344	+	0.461108i	$0.847452\pi$
$354$		0			0
$355$	−223.923	+	147.846i	−0.630769	+	0.416468i
$356$		0			0
$357$		0			0
$358$		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
$363$		0			0
$364$		0			0
$365$	−233.663	+	154.277i	−0.640174	+	0.422677i
$366$		0			0
$367$	−304.841	+	176.000i	−0.830629	+	0.479564i	−0.854068	−	0.520161i	$-0.825872\pi$
$367$	−304.841	+	176.000i	−0.830629	+	0.479564i	0.0234388	+	0.999725i	$0.492539\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−278.855	+	160.997i	−0.751630	+	0.433954i
$372$		0			0
$373$	114.315	+	66.0000i	0.306475	+	0.176944i	0.645348	−	0.763889i	$-0.276712\pi$
$373$	114.315	+	66.0000i	0.306475	+	0.176944i	−0.338873	+	0.940832i	$0.610046\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	321.994			0.854095
$378$		0			0
$379$	394.000			1.03958			0.519789	−	0.854295i	$-0.326011\pi$
$379$	394.000			1.03958			0.519789	+	0.854295i	$0.326011\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	38.0132	−	65.8407i	0.0992511	−	0.171908i	−0.812124	−	0.583485i	$-0.801689\pi$
$383$	38.0132	−	65.8407i	0.0992511	−	0.171908i	0.911375	+	0.411577i	$0.135022\pi$
$384$		0			0
$385$	−357.128	+	21.4359i	−0.927606	+	0.0556778i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−240.125	+	138.636i	−0.617288	+	0.356391i	−0.775812	−	0.630964i	$-0.782660\pi$
$389$	−240.125	+	138.636i	−0.617288	+	0.356391i	0.158524	+	0.987355i	$0.449326\pi$
$390$		0			0
$391$	70.0000	−	121.244i	0.179028	−	0.310086i
$392$		0			0
$393$		0			0
$394$		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
$399$		0			0
$400$		0			0
$401$	−154.919	−	89.4427i	−0.386333	−	0.223049i	0.294237	−	0.955732i	$-0.404934\pi$
$401$	−154.919	−	89.4427i	−0.386333	−	0.223049i	−0.680570	+	0.732683i	$0.738268\pi$
$402$		0			0
$403$	−353.338	+	204.000i	−0.876770	+	0.506203i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	196.774	+	340.823i	0.483474	+	0.837402i
$408$		0			0
$409$	103.000	−	178.401i	0.251834	−	0.436189i	−0.712197	−	0.701980i	$-0.752300\pi$
$409$	103.000	−	178.401i	0.251834	−	0.436189i	0.964031	+	0.265791i	$0.0856331\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	787.096			1.90580
$414$		0			0
$415$	−230.000	+	460.000i	−0.554217	+	1.10843i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−178.157	−	102.859i	−0.425196	−	0.245487i	0.272102	−	0.962268i	$-0.412281\pi$
$419$	−178.157	−	102.859i	−0.425196	−	0.245487i	−0.697298	+	0.716781i	$0.745615\pi$
$420$		0			0
$421$	19.0000	+	32.9090i	0.0451306	+	0.0781686i	0.887708	−	0.460406i	$-0.152296\pi$
$421$	19.0000	+	32.9090i	0.0451306	+	0.0781686i	−0.842578	+	0.538575i	$0.818963\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−307.431	+	719.713i	−0.723366	+	1.69344i
$426$		0			0
$427$	−512.687	−	296.000i	−1.20067	−	0.693208i
$428$		0			0
$429$		0			0
$430$		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
$435$		0			0
$436$		0			0
$437$	−13.4164	+	23.2379i	−0.0307012	+	0.0531760i
$438$		0			0
$439$	183.000	+	316.965i	0.416856	+	0.722017i	0.995621	−	0.0934778i	$-0.0297984\pi$
$439$	183.000	+	316.965i	0.416856	+	0.722017i	−0.578765	+	0.815494i	$0.696465\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	288.453	+	499.615i	0.651135	+	1.12780i	0.982848	+	0.184418i	$0.0590401\pi$
$443$	288.453	+	499.615i	0.651135	+	1.12780i	−0.331713	+	0.943380i	$0.607627\pi$
$444$		0			0
$445$	74.6410	−	49.2820i	0.167733	−	0.110746i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			429.325i			0.956181i	0.878311	+	0.478090i	$0.158671\pi$
$449$			429.325i			0.956181i	−0.878311	+	0.478090i	$0.841329\pi$
$450$		0			0
$451$	160.000			0.354767
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−264.475	−	400.566i	−0.581264	−	0.880364i
$456$		0			0
$457$	90.0666	−	52.0000i	0.197082	−	0.113786i	−0.398211	−	0.917294i	$-0.630369\pi$
$457$	90.0666	−	52.0000i	0.197082	−	0.113786i	0.595294	+	0.803508i	$0.297036\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−441.520	+	254.912i	−0.957744	+	0.552954i	−0.895478	−	0.445106i	$-0.853166\pi$
$461$	−441.520	+	254.912i	−0.957744	+	0.552954i	−0.0622662	+	0.998060i	$0.519833\pi$
$462$		0			0
$463$	83.1384	+	48.0000i	0.179565	+	0.103672i	0.587088	−	0.809523i	$-0.300274\pi$
$463$	83.1384	+	48.0000i	0.179565	+	0.103672i	−0.407523	+	0.913195i	$0.633608\pi$
$464$		0			0
$465$		0			0
$466$		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
$471$		0			0
$472$		0			0
$473$	125.220	−	216.887i	0.264735	−	0.458535i
$474$		0			0
$475$	58.9230	−	137.942i	0.124049	−	0.290405i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−495.742	+	286.217i	−1.03495	+	0.597530i	−0.918399	−	0.395655i	$-0.870518\pi$
$479$	−495.742	+	286.217i	−1.03495	+	0.597530i	−0.116552	+	0.993185i	$0.537184\pi$
$480$		0			0
$481$	−264.000	+	457.261i	−0.548857	+	0.950647i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	143.108	+	71.5542i	0.295069	+	0.147534i
$486$		0			0
$487$			648.000i			1.33060i	0.746578	+	0.665298i	$0.231695\pi$
$487$			648.000i			1.33060i	−0.746578	+	0.665298i	$0.768305\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	116.190	+	67.0820i	0.236638	+	0.136623i	0.613631	−	0.789593i	$-0.289708\pi$
$491$	116.190	+	67.0820i	0.236638	+	0.136623i	−0.376992	+	0.926216i	$0.623042\pi$
$492$		0			0
$493$	727.461	−	420.000i	1.47558	−	0.851927i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−214.663	−	371.806i	−0.431917	−	0.748101i
$498$		0			0
$499$	243.000	−	420.888i	0.486974	−	0.843464i	−0.512914	−	0.858440i	$-0.671434\pi$
$499$	243.000	−	420.888i	0.486974	−	0.843464i	0.999888	+	0.0149764i	$0.00476733\pi$
$500$		0			0
$501$		0			0
$502$		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
$507$		0			0
$508$		0			0
$509$	−23.2379	−	13.4164i	−0.0456540	−	0.0263584i	0.476999	−	0.878904i	$-0.341724\pi$
$509$	−23.2379	−	13.4164i	−0.0456540	−	0.0263584i	−0.522653	+	0.852545i	$0.675058\pi$
$510$		0			0
$511$	−224.000	−	387.979i	−0.438356	−	0.759255i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	31.1559	+	519.066i	0.0604970	+	1.00789i
$516$		0			0
$517$	34.6410	+	20.0000i	0.0670039	+	0.0386847i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			983.870i			1.88843i	0.329336	+	0.944213i	$0.393175\pi$
$521$			983.870i			1.88843i	−0.329336	+	0.944213i	$0.606825\pi$
$522$		0			0
$523$		−	292.000i		−	0.558317i	−0.960245	−	0.279159i	$-0.909944\pi$
$523$		−	292.000i		−	0.558317i	0.960245	−	0.279159i	$-0.0900556\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−532.184	+	921.770i	−1.00984	+	1.74909i
$528$		0			0
$529$	254.500	+	440.807i	0.481096	+	0.833283i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	107.331	+	185.903i	0.201372	+	0.348786i
$534$		0			0
$535$	406.577	+	615.788i	0.759957	+	1.15101i
$536$		0			0
$537$		0			0
$538$		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
$543$		0			0
$544$		0			0
$545$	−203.866	−	308.769i	−0.374067	−	0.566549i
$546$		0			0
$547$	142.028	−	82.0000i	0.259649	−	0.149909i	−0.364525	−	0.931193i	$-0.618769\pi$
$547$	142.028	−	82.0000i	0.259649	−	0.149909i	0.624175	+	0.781285i	$0.285435\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−139.427	+	80.4984i	−0.253044	+	0.146095i
$552$		0			0
$553$	540.400	+	312.000i	0.977215	+	0.564195i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−362.243			−0.650347			−0.325173	−	0.945654i	$-0.605423\pi$
$557$	−362.243			−0.650347			−0.325173	+	0.945654i	$0.605423\pi$
$558$		0			0
$559$	336.000			0.601073
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−498.643	+	863.675i	−0.885689	+	1.53406i	−0.0407680	+	0.999169i	$0.512980\pi$
$563$	−498.643	+	863.675i	−0.885689	+	1.53406i	−0.844921	+	0.534890i	$0.820353\pi$
$564$		0			0
$565$	−200.885	+	12.0577i	−0.355548	+	0.0213411i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−542.218	+	313.050i	−0.952931	+	0.550175i	−0.893990	−	0.448087i	$-0.852106\pi$
$569$	−542.218	+	313.050i	−0.952931	+	0.550175i	−0.0589407	+	0.998261i	$0.518772\pi$
$570$		0			0
$571$	−197.000	+	341.214i	−0.345009	+	0.597573i	−0.985355	−	0.170513i	$-0.945457\pi$
$571$	−197.000	+	341.214i	−0.345009	+	0.597573i	0.640347	+	0.768086i	$0.278791\pi$
$572$		0			0
$573$		0			0
$574$		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
$579$		0			0
$580$		0			0
$581$	−712.629	−	411.437i	−1.22656	−	0.708152i
$582$		0			0
$583$	−311.769	+	180.000i	−0.534767	+	0.308748i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−355.535	−	615.804i	−0.605681	−	1.04907i	−0.991943	−	0.126681i	$-0.959567\pi$
$587$	−355.535	−	615.804i	−0.605681	−	1.04907i	0.386262	−	0.922389i	$-0.373766\pi$
$588$		0			0
$589$	102.000	−	176.669i	0.173175	−	0.299948i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	603.738			1.01811			0.509054	−	0.860734i	$-0.329995\pi$
$593$	603.738			1.01811			0.509054	+	0.860734i	$0.329995\pi$
$594$		0			0
$595$	−1120.00	−	560.000i	−1.88235	−	0.941176i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−46.4758	−	26.8328i	−0.0775890	−	0.0447960i	0.460704	−	0.887554i	$-0.347597\pi$
$599$	−46.4758	−	26.8328i	−0.0775890	−	0.0447960i	−0.538293	+	0.842758i	$0.680930\pi$
$600$		0			0
$601$	−217.000	−	375.855i	−0.361065	−	0.625383i	0.627071	−	0.778962i	$-0.284253\pi$
$601$	−217.000	−	375.855i	−0.361065	−	0.625383i	−0.988136	+	0.153579i	$0.950920\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	204.632	−	12.2826i	0.338234	−	0.0203019i
$606$		0			0
$607$	568.113	+	328.000i	0.935935	+	0.540362i	0.888684	−	0.458521i	$-0.151620\pi$
$607$	568.113	+	328.000i	0.935935	+	0.540362i	0.0472514	+	0.998883i	$0.484954\pi$
$608$		0			0
$609$		0			0
$610$		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
$615$		0			0
$616$		0			0
$617$	−230.315	+	398.917i	−0.373282	+	0.646543i	−0.990068	−	0.140587i	$-0.955101\pi$
$617$	−230.315	+	398.917i	−0.373282	+	0.646543i	0.616786	+	0.787131i	$0.288434\pi$
$618$		0			0
$619$	523.000	+	905.863i	0.844911	+	1.46343i	0.885698	+	0.464261i	$0.153680\pi$
$619$	523.000	+	905.863i	0.844911	+	1.46343i	−0.0407872	+	0.999168i	$0.512987\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	71.5542	+	123.935i	0.114854	+	0.198933i
$624$		0			0
$625$	−432.115	−	451.554i	−0.691384	−	0.722487i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			1377.42i			2.18985i
$630$		0			0
$631$	−46.0000			−0.0729002			−0.0364501	−	0.999335i	$-0.511605\pi$
$631$	−46.0000			−0.0729002			−0.0364501	+	0.999335i	$0.511605\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−66.7610	+	44.0792i	−0.105135	+	0.0694160i
$636$		0			0
$637$	155.885	−	90.0000i	0.244717	−	0.141287i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−991.484	+	572.433i	−1.54678	+	0.893032i	−0.548391	+	0.836222i	$0.684760\pi$
$641$	−991.484	+	572.433i	−1.54678	+	0.893032i	−0.998385	+	0.0568101i	$0.981907\pi$
$642$		0			0
$643$	−696.284	−	402.000i	−1.08287	−	0.625194i	−0.151200	−	0.988503i	$-0.548314\pi$
$643$	−696.284	−	402.000i	−1.08287	−	0.625194i	−0.931669	+	0.363309i	$0.881647\pi$
$644$		0			0
$645$		0			0
$646$		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
$651$		0			0
$652$		0			0
$653$	538.892	−	933.389i	0.825256	−	1.42939i	−0.0764669	−	0.997072i	$-0.524364\pi$
$653$	538.892	−	933.389i	0.825256	−	1.42939i	0.901723	−	0.432314i	$-0.142303\pi$
$654$		0			0
$655$	−24.1154	−	401.769i	−0.0368174	−	0.613388i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	704.883	−	406.964i	1.06963	−	0.617548i	0.141546	−	0.989932i	$-0.454793\pi$
$659$	704.883	−	406.964i	1.06963	−	0.617548i	0.928079	+	0.372383i	$0.121459\pi$
$660$		0			0
$661$	−541.000	+	937.039i	−0.818457	+	1.41761i	0.0883620	+	0.996088i	$0.471837\pi$
$661$	−541.000	+	937.039i	−0.818457	+	1.41761i	−0.906819	+	0.421520i	$0.861497\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	214.663	+	107.331i	0.322801	+	0.161400i
$666$		0			0
$667$		−	120.000i		−	0.179910i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−573.202	−	330.938i	−0.854250	−	0.493201i
$672$		0			0
$673$	−914.523	+	528.000i	−1.35887	+	0.784547i	−0.989472	−	0.144723i	$-0.953771\pi$
$673$	−914.523	+	528.000i	−1.35887	+	0.784547i	−0.369403	+	0.929269i	$0.620438\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	306.341	+	530.599i	0.452498	+	0.783750i	0.998541	−	0.0540077i	$-0.0171995\pi$
$677$	306.341	+	530.599i	0.452498	+	0.783750i	−0.546042	+	0.837758i	$0.683866\pi$
$678$		0			0
$679$	−128.000	+	221.703i	−0.188513	+	0.326513i
$680$		0			0
$681$		0			0
$682$		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
$687$		0			0
$688$		0			0
$689$	−418.282	−	241.495i	−0.607086	−	0.350501i
$690$		0			0
$691$	−461.000	−	798.475i	−0.667149	−	1.15554i	−0.978698	−	0.205306i	$-0.934181\pi$
$691$	−461.000	−	798.475i	−0.667149	−	1.15554i	0.311549	−	0.950230i	$-0.399152\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−588.940	+	35.3500i	−0.847396	+	0.0508633i
$696$		0			0
$697$	484.974	+	280.000i	0.695802	+	0.401722i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	474.046i		−	0.676243i	−0.941102	−	0.338122i	$-0.890209\pi$
$701$		−	474.046i		−	0.676243i	0.941102	−	0.338122i	$-0.109791\pi$
$702$		0			0
$703$		−	264.000i		−	0.375533i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	608.210	−	1053.45i	0.860269	−	1.49003i
$708$		0			0
$709$	483.000	+	836.581i	0.681241	+	1.17994i	0.974602	+	0.223943i	$0.0718928\pi$
$709$	483.000	+	836.581i	0.681241	+	1.17994i	−0.293361	+	0.956002i	$0.594774\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	76.0263	+	131.681i	0.106629	+	0.184686i
$714$		0			0
$715$	−295.692	−	447.846i	−0.413556	−	0.626358i
$716$		0			0
$717$		0			0
$718$		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
$723$		0			0
$724$		0			0
$725$	80.2403	+	666.004i	0.110676	+	0.918626i
$726$		0			0
$727$	353.338	−	204.000i	0.486023	−	0.280605i	−0.236900	−	0.971534i	$-0.576132\pi$
$727$	353.338	−	204.000i	0.486023	−	0.280605i	0.722923	+	0.690929i	$0.242798\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	759.105	−	438.269i	1.03845	−	0.599548i
$732$		0			0
$733$	−142.028	−	82.0000i	−0.193763	−	0.111869i	0.399980	−	0.916524i	$-0.369017\pi$
$733$	−142.028	−	82.0000i	−0.193763	−	0.111869i	−0.593743	+	0.804655i	$0.702350\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	822.873			1.11652
$738$		0			0
$739$	1082.00			1.46414			0.732070	−	0.681229i	$-0.238554\pi$
$739$	1082.00			1.46414			0.732070	+	0.681229i	$0.238554\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−570.197	+	987.611i	−0.767426	+	1.32922i	0.171529	+	0.985179i	$0.445129\pi$
$743$	−570.197	+	987.611i	−0.767426	+	1.32922i	−0.938955	+	0.344041i	$0.888204\pi$
$744$		0			0
$745$	−29.4744	−	491.051i	−0.0395630	−	0.659129i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1022.47	+	590.322i	−1.36511	+	0.788147i
$750$		0			0
$751$	479.000	−	829.652i	0.637816	−	1.10473i	−0.348095	−	0.937459i	$-0.613171\pi$
$751$	479.000	−	829.652i	0.637816	−	1.10473i	0.985911	−	0.167271i	$-0.0534954\pi$
$752$		0			0
$753$		0			0
$754$		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
$759$		0			0
$760$		0			0
$761$	975.992	+	563.489i	1.28251	+	0.740459i	0.977307	−	0.211828i	$-0.0679416\pi$
$761$	975.992	+	563.489i	1.28251	+	0.740459i	0.305205	+	0.952287i	$0.401275\pi$
$762$		0			0
$763$	512.687	−	296.000i	0.671936	−	0.387942i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	590.322	+	1022.47i	0.769651	+	1.33307i
$768$		0			0
$769$	663.000	−	1148.35i	0.862159	−	1.49330i	−0.00768240	−	0.999970i	$-0.502445\pi$
$769$	663.000	−	1148.35i	0.862159	−	1.49330i	0.869841	−	0.493332i	$-0.164221\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−147.580			−0.190919			−0.0954596	−	0.995433i	$-0.530432\pi$
$773$	−147.580			−0.190919			−0.0954596	+	0.995433i	$0.530432\pi$
$774$		0			0
$775$	−510.000	−	680.000i	−0.658065	−	0.877419i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−92.9516	−	53.6656i	−0.119322	−	0.0688904i
$780$		0			0
$781$	−240.000	−	415.692i	−0.307298	−	0.532256i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−27.5610	−	459.174i	−0.0351096	−	0.584935i
$786$		0			0
$787$	980.341	+	566.000i	1.24567	+	0.719187i	0.970242	−	0.242136i	$-0.0778479\pi$
$787$	980.341	+	566.000i	1.24567	+	0.719187i	0.275426	+	0.961322i	$0.411181\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	321.994i		−	0.407072i
$792$		0			0
$793$		−	888.000i		−	1.11980i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	306.341	−	530.599i	0.384368	−	0.665745i	−0.607313	−	0.794462i	$-0.707753\pi$
$797$	306.341	−	530.599i	0.384368	−	0.665745i	0.991681	+	0.128718i	$0.0410860\pi$
$798$		0			0
$799$	70.0000	+	121.244i	0.0876095	+	0.151744i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−250.440	−	433.774i	−0.311880	−	0.540192i
$804$		0			0
$805$	−149.282	+	98.5641i	−0.185444	+	0.122440i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	1234.31i		−	1.52572i	−0.646562	−	0.762861i	$-0.723794\pi$
$809$		−	1234.31i		−	1.52572i	0.646562	−	0.762861i	$-0.276206\pi$
$810$		0			0
$811$	874.000			1.07768			0.538841	−	0.842408i	$-0.318862\pi$
$811$	874.000			1.07768			0.538841	+	0.842408i	$0.318862\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	283.734	−	187.337i	0.348140	−	0.229861i
$816$		0			0
$817$	−145.492	+	84.0000i	−0.178081	+	0.102815i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	224.633	−	129.692i	0.273609	−	0.157968i	−0.356918	−	0.934136i	$-0.616172\pi$
$821$	224.633	−	129.692i	0.273609	−	0.157968i	0.630527	+	0.776168i	$0.282839\pi$
$822$		0			0
$823$	−1184.72	−	684.000i	−1.43952	−	0.831106i	−0.441701	−	0.897162i	$-0.645625\pi$
$823$	−1184.72	−	684.000i	−1.43952	−	0.831106i	−0.997816	+	0.0660564i	$0.978958\pi$
$824$		0			0
$825$		0			0
$826$		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
$831$		0			0
$832$		0			0
$833$	234.787	−	406.663i	0.281857	−	0.488191i
$834$		0			0
$835$	334.808	−	20.0962i	0.400967	−	0.0240673i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	263.363	−	152.053i	0.313901	−	0.181231i	−0.334770	−	0.942300i	$-0.608659\pi$
$839$	263.363	−	152.053i	0.313901	−	0.181231i	0.648671	+	0.761069i	$0.275325\pi$
$840$		0			0
$841$	−60.5000	+	104.789i	−0.0719382	+	0.124601i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−55.9017	+	111.803i	−0.0661559	+	0.132312i
$846$		0			0
$847$			328.000i			0.387249i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	170.411	+	98.3870i	0.200248	+	0.115613i
$852$		0			0
$853$	−668.572	+	386.000i	−0.783789	+	0.452521i	−0.837771	−	0.546021i	$-0.816142\pi$
$853$	−668.572	+	386.000i	−0.783789	+	0.452521i	0.0539827	+	0.998542i	$0.482808\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	449.450	+	778.470i	0.524445	+	0.908366i	0.999595	+	0.0284609i	$0.00906062\pi$
$857$	449.450	+	778.470i	0.524445	+	0.908366i	−0.475150	+	0.879905i	$0.657606\pi$
$858$		0			0
$859$	139.000	−	240.755i	0.161816	−	0.280274i	−0.773704	−	0.633547i	$-0.781598\pi$
$859$	139.000	−	240.755i	0.161816	−	0.280274i	0.935520	+	0.353274i	$0.114932\pi$
$860$		0			0
$861$		0			0
$862$		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
$867$		0			0
$868$		0			0
$869$	604.185	+	348.827i	0.695265	+	0.401412i
$870$		0			0
$871$	552.000	+	956.092i	0.633754	+	1.09769i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	762.614	−	646.854i	0.871558	−	0.739262i
$876$		0			0
$877$	−543.864	−	314.000i	−0.620141	−	0.358039i	0.156783	−	0.987633i	$-0.449888\pi$
$877$	−543.864	−	314.000i	−0.620141	−	0.358039i	−0.776924	+	0.629594i	$0.783221\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			536.656i			0.609145i	0.952489	+	0.304572i	$0.0985135\pi$
$881$			536.656i			0.609145i	−0.952489	+	0.304572i	$0.901486\pi$
$882$		0			0
$883$			948.000i			1.07361i	0.843705	+	0.536806i	$0.180369\pi$
$883$			948.000i			1.07361i	−0.843705	+	0.536806i	$0.819631\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−140.872	+	243.998i	−0.158819	+	0.275082i	−0.934443	−	0.356113i	$-0.884102\pi$
$887$	−140.872	+	243.998i	−0.158819	+	0.275082i	0.775624	+	0.631195i	$0.217435\pi$
$888$		0			0
$889$	−64.0000	−	110.851i	−0.0719910	−	0.124692i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−13.4164	−	23.2379i	−0.0150240	−	0.0260223i
$894$		0			0
$895$	−1082.29	+	714.589i	−1.20927	+	0.798424i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			912.316i			1.01481i
$900$		0			0
$901$	−1260.00			−1.39845
$902$		0			0
$903$		0			0
$904$		0			0
$905$	457.322	+	692.645i	0.505328	+	0.765353i
$906$		0			0
$907$	−446.869	+	258.000i	−0.492689	+	0.284454i	−0.725689	−	0.688022i	$-0.758479\pi$
$907$	−446.869	+	258.000i	−0.492689	+	0.284454i	0.233000	+	0.972477i	$0.425146\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	278.855	−	160.997i	0.306097	−	0.176725i	−0.339081	−	0.940757i	$-0.610116\pi$
$911$	278.855	−	160.997i	0.306097	−	0.176725i	0.645179	+	0.764032i	$0.276783\pi$
$912$		0			0
$913$	−796.743	−	460.000i	−0.872665	−	0.503834i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	643.988			0.702277
$918$		0			0
$919$	1314.00			1.42982			0.714908	−	0.699219i	$-0.246469\pi$
$919$	1314.00			1.42982			0.714908	+	0.699219i	$0.246469\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	321.994	−	557.710i	0.348856	−	0.604236i
$924$		0			0
$925$	−1011.58	−	432.102i	−1.09360	−	0.467138i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	1301.32	−	751.319i	1.40078	−	0.808739i	0.406305	−	0.913737i	$-0.366817\pi$
$929$	1301.32	−	751.319i	1.40078	−	0.808739i	0.994472	+	0.104998i	$0.0334836\pi$
$930$		0			0
$931$	−45.0000	+	77.9423i	−0.0483351	+	0.0837189i
$932$		0			0
$933$		0			0
$934$		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
$939$		0			0
$940$		0			0
$941$	317.585	+	183.358i	0.337497	+	0.194854i	0.659165	−	0.751999i	$-0.270910\pi$
$941$	317.585	+	183.358i	0.337497	+	0.194854i	−0.321668	+	0.946853i	$0.604243\pi$
$942$		0			0
$943$	69.2820	−	40.0000i	0.0734698	−	0.0424178i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−838.525	−	1452.37i	−0.885455	−	1.53365i	−0.845192	−	0.534463i	$-0.820514\pi$
$947$	−838.525	−	1452.37i	−0.885455	−	1.53365i	−0.0402628	−	0.999189i	$-0.512820\pi$
$948$		0			0
$949$	336.000	−	581.969i	0.354057	−	0.613245i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−827.345			−0.868148			−0.434074	−	0.900877i	$-0.642924\pi$
$953$	−827.345			−0.868148			−0.434074	+	0.900877i	$0.642924\pi$
$954$		0			0
$955$	−960.000	−	480.000i	−1.00524	−	0.502618i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1208.37	−	697.653i	−1.26003	−	0.727480i
$960$		0			0
$961$	−97.5000	−	168.875i	−0.101457	−	0.175728i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−9.58644	−	159.713i	−0.00993414	−	0.165505i
$966$		0			0
$967$	464.190	+	268.000i	0.480031	+	0.277146i	0.720429	−	0.693528i	$-0.243945\pi$
$967$	464.190	+	268.000i	0.480031	+	0.277146i	−0.240399	+	0.970674i	$0.577278\pi$
$968$		0			0
$969$		0			0
$970$		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
$975$		0			0
$976$		0			0
$977$	−87.2067	+	151.046i	−0.0892596	+	0.154602i	−0.907198	−	0.420703i	$-0.861783\pi$
$977$	−87.2067	+	151.046i	−0.0892596	+	0.154602i	0.817939	+	0.575305i	$0.195117\pi$
$978$		0			0
$979$	80.0000	+	138.564i	0.0817160	+	0.141536i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−927.968	−	1607.29i	−0.944016	−	1.63508i	−0.757709	−	0.652593i	$-0.773681\pi$
$983$	−927.968	−	1607.29i	−0.944016	−	1.63508i	−0.186308	−	0.982491i	$-0.559652\pi$
$984$		0			0
$985$	12.3205	+	18.6603i	0.0125081	+	0.0189444i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	125.220i		−	0.126613i
$990$		0			0
$991$	−1646.00			−1.66095			−0.830474	−	0.557057i	$-0.811930\pi$
$991$	−1646.00			−1.66095			−0.830474	+	0.557057i	$0.811930\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−314.064	−	475.672i	−0.315642	−	0.478062i
$996$		0			0
$997$	1008.05	−	582.000i	1.01109	−	0.583751i	0.0995771	−	0.995030i	$-0.468251\pi$
$997$	1008.05	−	582.000i	1.01109	−	0.583751i	0.911510	+	0.411279i	$0.134918\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.3.t.b.1349.2		8
3.2	odd	2	inner	1620.3.t.b.1349.3		8
5.4	even	2	inner	1620.3.t.b.1349.1		8
9.2	odd	6	inner	1620.3.t.b.269.1		8
9.4	even	3		60.3.b.a.29.2	yes	4
9.5	odd	6		60.3.b.a.29.4	yes	4
9.7	even	3	inner	1620.3.t.b.269.4		8
15.14	odd	2	inner	1620.3.t.b.1349.4		8
36.23	even	6		240.3.c.d.209.1		4
36.31	odd	6		240.3.c.d.209.3		4
45.4	even	6		60.3.b.a.29.3	yes	4
45.13	odd	12		300.3.g.e.101.1		2
45.14	odd	6		60.3.b.a.29.1	✓	4
45.22	odd	12		300.3.g.h.101.2		2
45.23	even	12		300.3.g.e.101.2		2
45.29	odd	6	inner	1620.3.t.b.269.2		8
45.32	even	12		300.3.g.h.101.1		2
45.34	even	6	inner	1620.3.t.b.269.3		8
72.5	odd	6		960.3.c.h.449.1		4
72.13	even	6		960.3.c.h.449.3		4
72.59	even	6		960.3.c.g.449.4		4
72.67	odd	6		960.3.c.g.449.2		4
180.23	odd	12		1200.3.l.q.401.1		2
180.59	even	6		240.3.c.d.209.4		4
180.67	even	12		1200.3.l.h.401.1		2
180.103	even	12		1200.3.l.q.401.2		2
180.139	odd	6		240.3.c.d.209.2		4
180.167	odd	12		1200.3.l.h.401.2		2
360.59	even	6		960.3.c.g.449.1		4
360.139	odd	6		960.3.c.g.449.3		4
360.149	odd	6		960.3.c.h.449.4		4
360.229	even	6		960.3.c.h.449.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.3.b.a.29.1	✓	4	45.14	odd	6
60.3.b.a.29.2	yes	4	9.4	even	3
60.3.b.a.29.3	yes	4	45.4	even	6
60.3.b.a.29.4	yes	4	9.5	odd	6
240.3.c.d.209.1		4	36.23	even	6
240.3.c.d.209.2		4	180.139	odd	6
240.3.c.d.209.3		4	36.31	odd	6
240.3.c.d.209.4		4	180.59	even	6
300.3.g.e.101.1		2	45.13	odd	12
300.3.g.e.101.2		2	45.23	even	12
300.3.g.h.101.1		2	45.32	even	12
300.3.g.h.101.2		2	45.22	odd	12
960.3.c.g.449.1		4	360.59	even	6
960.3.c.g.449.2		4	72.67	odd	6
960.3.c.g.449.3		4	360.139	odd	6
960.3.c.g.449.4		4	72.59	even	6
960.3.c.h.449.1		4	72.5	odd	6
960.3.c.h.449.2		4	360.229	even	6
960.3.c.h.449.3		4	72.13	even	6
960.3.c.h.449.4		4	360.149	odd	6
1200.3.l.h.401.1		2	180.67	even	12
1200.3.l.h.401.2		2	180.167	odd	12
1200.3.l.q.401.1		2	180.23	odd	12
1200.3.l.q.401.2		2	180.103	even	12
1620.3.t.b.269.1		8	9.2	odd	6	inner
1620.3.t.b.269.2		8	45.29	odd	6	inner
1620.3.t.b.269.3		8	45.34	even	6	inner
1620.3.t.b.269.4		8	9.7	even	3	inner
1620.3.t.b.1349.1		8	5.4	even	2	inner
1620.3.t.b.1349.2		8	1.1	even	1	trivial
1620.3.t.b.1349.3		8	3.2	odd	2	inner
1620.3.t.b.1349.4		8	15.14	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1620.t (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.75495 - 4.17256i) q^{5} +(6.92820 - 4.00000i) q^{7} +O(q^{10})\)	\(q+(-2.75495 - 4.17256i) q^{5} +(6.92820 - 4.00000i) q^{7} +(7.74597 - 4.47214i) q^{11} +(10.3923 + 6.00000i) q^{13} +31.3050 q^{17} -6.00000 q^{19} +(2.23607 - 3.87298i) q^{23} +(-9.82051 + 22.9904i) q^{25} +(23.2379 - 13.4164i) q^{29} +(-17.0000 + 29.4449i) q^{31} +(-35.7771 - 17.8885i) q^{35} +44.0000i q^{37} +(15.4919 + 8.94427i) q^{41} +(24.2487 - 14.0000i) q^{43} +(2.23607 + 3.87298i) q^{47} +(7.50000 - 12.9904i) q^{49} -40.2492 q^{53} +(-40.0000 - 20.0000i) q^{55} +(85.2056 + 49.1935i) q^{59} +(-37.0000 - 64.0859i) q^{61} +(-3.59492 - 59.8922i) q^{65} +(79.6743 + 46.0000i) q^{67} -53.6656i q^{71} -56.0000i q^{73} +(35.7771 - 61.9677i) q^{77} +(39.0000 + 67.5500i) q^{79} +(-51.4296 - 89.0786i) q^{83} +(-86.2436 - 130.622i) q^{85} +17.8885i q^{89} +96.0000 q^{91} +(16.5297 + 25.0354i) q^{95} +(-27.7128 + 16.0000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 48 q^{19} + 60 q^{25} - 136 q^{31} + 60 q^{49} - 320 q^{55} - 296 q^{61} + 312 q^{79} + 280 q^{85} + 768 q^{91}+O(q^{100}) \) 8 * q - 48 * q^19 + 60 * q^25 - 136 * q^31 + 60 * q^49 - 320 * q^55 - 296 * q^61 + 312 * q^79 + 280 * q^85 + 768 * q^91

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