LMFDB - Embedded newform 900.3.c.p.451.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [900,3,Mod(451,900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(900, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("900.451");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	900.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$24.5232237924$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.12239922073600.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{6} + 4x^{4} - 32x^{2} + 256 $ x^8 - 2x^6 + 4x^4 - 32*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			451.5
Root			$0.950636 + 1.75963i$ of defining polynomial
Character	$\chi$	$=$	900.451
Dual form			900.3.c.p.451.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.950636 - 1.75963i) q^{2} +(-2.19258 - 3.34553i) q^{4} +3.98982i q^{7} +(-7.97124 + 0.677747i) q^{8} +O(q^{10})$	\(q+(0.950636 - 1.75963i) q^{2} +(-2.19258 - 3.34553i) q^{4} +3.98982i q^{7} +(-7.97124 + 0.677747i) q^{8} +8.39401i q^{11} +9.77033 q^{13} +(7.02060 + 3.79287i) q^{14} +(-6.38516 + 14.6707i) q^{16} +12.1399 q^{17} -17.3719i q^{19} +(14.7703 + 7.97964i) q^{22} +2.97203i q^{23} +(9.28803 - 17.1921i) q^{26} +(13.3481 - 8.74801i) q^{28} +51.6300 q^{29} +30.7541i q^{31} +(19.7450 + 25.1820i) q^{32} +(11.5407 - 21.3618i) q^{34} +19.5407 q^{37} +(-30.5682 - 16.5144i) q^{38} +42.5603 q^{41} -62.9208i q^{43} +(28.0824 - 18.4046i) q^{44} +(5.22967 + 2.82532i) q^{46} -39.5201i q^{47} +33.0813 q^{49} +(-21.4223 - 32.6869i) q^{52} -21.2096 q^{53} +(-2.70409 - 31.8038i) q^{56} +(49.0813 - 90.8496i) q^{58} +50.3640i q^{59} -70.3923 q^{61} +(54.1157 + 29.2359i) q^{62} +(63.0813 - 10.8050i) q^{64} -94.8394i q^{67} +(-26.6178 - 40.6145i) q^{68} +132.376i q^{71} +77.7033 q^{73} +(18.5761 - 34.3843i) q^{74} +(-58.1184 + 38.0894i) q^{76} -33.4906 q^{77} -48.3742i q^{79} +(40.4593 - 74.8903i) q^{82} +140.248i q^{83} +(-110.717 - 59.8148i) q^{86} +(-5.68901 - 66.9106i) q^{88} -18.1394 q^{89} +38.9819i q^{91} +(9.94303 - 6.51642i) q^{92} +(-69.5407 - 37.5692i) q^{94} -15.0000 q^{97} +(31.4483 - 58.2108i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{4}+O(q^{10}) $ 8 * q + 4 * q^4	$ 8 q + 4 q^{4} - 8 q^{13} - 8 q^{16} + 32 q^{22} - 44 q^{28} - 80 q^{34} - 16 q^{37} + 128 q^{46} - 80 q^{49} - 236 q^{52} + 48 q^{58} + 40 q^{61} + 160 q^{64} - 240 q^{73} - 228 q^{76} + 496 q^{82} - 304 q^{88} - 384 q^{94} - 120 q^{97}+O(q^{100}) $ 8 * q + 4 * q^4 - 8 * q^13 - 8 * q^16 + 32 * q^22 - 44 * q^28 - 80 * q^34 - 16 * q^37 + 128 * q^46 - 80 * q^49 - 236 * q^52 + 48 * q^58 + 40 * q^61 + 160 * q^64 - 240 * q^73 - 228 * q^76 + 496 * q^82 - 304 * q^88 - 384 * q^94 - 120 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/900\mathbb{Z}\right)^\times$.

$n$	$101$	$451$	$577$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$	0.950636	−	1.75963i	0.475318	−	0.879814i
$2$	0.950636	−	1.75963i	0.475318	−	0.879814i
$3$		0			0
$3$		0			0
$4$	−2.19258	−	3.34553i	−0.548146	−	0.836383i
$5$		0			0
$5$		0			0
$6$		0			0
$7$			3.98982i			0.569975i	0.958531	+	0.284987i	$0.0919894\pi$
$7$			3.98982i			0.569975i	−0.958531	+	0.284987i	$0.908011\pi$
$8$	−7.97124	+	0.677747i	−0.996405	+	0.0847184i
$9$		0			0
$10$		0			0
$11$			8.39401i			0.763091i	0.924350	+	0.381546i	$0.124608\pi$
$11$			8.39401i			0.763091i	−0.924350	+	0.381546i	$0.875392\pi$
$12$		0			0
$13$	9.77033			0.751564			0.375782	−	0.926708i	$-0.377374\pi$
$13$	9.77033			0.751564			0.375782	+	0.926708i	$0.377374\pi$
$14$	7.02060	+	3.79287i	0.501472	+	0.270919i
$15$		0			0
$16$	−6.38516	+	14.6707i	−0.399073	+	0.916919i
$17$	12.1399			0.714114			0.357057	−	0.934083i	$-0.383780\pi$
$17$	12.1399			0.714114			0.357057	+	0.934083i	$0.383780\pi$
$18$		0			0
$19$		−	17.3719i		−	0.914313i	−0.889386	−	0.457157i	$-0.848868\pi$
$19$		−	17.3719i		−	0.914313i	0.889386	−	0.457157i	$-0.151132\pi$
$20$		0			0
$21$		0			0
$22$	14.7703	+	7.97964i	0.671379	+	0.362711i
$23$			2.97203i			0.129219i	0.997911	+	0.0646094i	$0.0205801\pi$
$23$			2.97203i			0.129219i	−0.997911	+	0.0646094i	$0.979420\pi$
$24$		0			0
$25$		0			0
$26$	9.28803	−	17.1921i	0.357232	−	0.661236i
$27$		0			0
$28$	13.3481	−	8.74801i	0.476717	−	0.312429i
$29$	51.6300			1.78034			0.890172	−	0.455624i	$-0.150584\pi$
$29$	51.6300			1.78034			0.890172	+	0.455624i	$0.150584\pi$
$30$		0			0
$31$			30.7541i			0.992067i	0.868303	+	0.496033i	$0.165211\pi$
$31$			30.7541i			0.992067i	−0.868303	+	0.496033i	$0.834789\pi$
$32$	19.7450	+	25.1820i	0.617032	+	0.786938i
$33$		0			0
$34$	11.5407	−	21.3618i	0.339431	−	0.628287i
$35$		0			0
$36$		0			0
$37$	19.5407			0.528126			0.264063	−	0.964505i	$-0.414937\pi$
$37$	19.5407			0.528126			0.264063	+	0.964505i	$0.414937\pi$
$38$	−30.5682	−	16.5144i	−0.804426	−	0.434589i
$39$		0			0
$40$		0			0
$41$	42.5603			1.03806			0.519028	−	0.854757i	$-0.326294\pi$
$41$	42.5603			1.03806			0.519028	+	0.854757i	$0.326294\pi$
$42$		0			0
$43$		−	62.9208i		−	1.46327i	−0.681694	−	0.731637i	$-0.738756\pi$
$43$		−	62.9208i		−	1.46327i	0.681694	−	0.731637i	$-0.261244\pi$
$44$	28.0824	−	18.4046i	0.638237	−	0.418285i
$45$		0			0
$46$	5.22967	+	2.82532i	0.113688	+	0.0614200i
$47$		−	39.5201i		−	0.840853i	−0.907327	−	0.420426i	$-0.861880\pi$
$47$		−	39.5201i		−	0.840853i	0.907327	−	0.420426i	$-0.138120\pi$
$48$		0			0
$49$	33.0813			0.675129
$50$		0			0
$51$		0			0
$52$	−21.4223	−	32.6869i	−0.411966	−	0.628595i
$53$	−21.2096			−0.400182			−0.200091	−	0.979777i	$-0.564124\pi$
$53$	−21.2096			−0.400182			−0.200091	+	0.979777i	$0.564124\pi$
$54$		0			0
$55$		0			0
$56$	−2.70409	−	31.8038i	−0.0482873	−	0.567926i
$57$		0			0
$58$	49.0813	−	90.8496i	0.846230	−	1.56637i
$59$			50.3640i			0.853628i	0.904339	+	0.426814i	$0.140364\pi$
$59$			50.3640i			0.853628i	−0.904339	+	0.426814i	$0.859636\pi$
$60$		0			0
$61$	−70.3923			−1.15397			−0.576986	−	0.816754i	$-0.695771\pi$
$61$	−70.3923			−1.15397			−0.576986	+	0.816754i	$0.695771\pi$
$62$	54.1157	+	29.2359i	0.872834	+	0.471547i
$63$		0			0
$64$	63.0813	−	10.8050i	0.985646	−	0.168828i
$65$		0			0
$66$		0			0
$67$		−	94.8394i		−	1.41551i	−0.706456	−	0.707757i	$-0.749707\pi$
$67$		−	94.8394i		−	1.41551i	0.706456	−	0.707757i	$-0.250293\pi$
$68$	−26.6178	−	40.6145i	−0.391438	−	0.597273i
$69$		0			0
$70$		0			0
$71$			132.376i			1.86445i	0.361874	+	0.932227i	$0.382137\pi$
$71$			132.376i			1.86445i	−0.361874	+	0.932227i	$0.617863\pi$
$72$		0			0
$73$	77.7033			1.06443			0.532214	−	0.846610i	$-0.321360\pi$
$73$	77.7033			1.06443			0.532214	+	0.846610i	$0.321360\pi$
$74$	18.5761	−	34.3843i	0.251028	−	0.464653i
$75$		0			0
$76$	−58.1184	+	38.0894i	−0.764716	+	0.501177i
$77$	−33.4906			−0.434943
$78$		0			0
$79$		−	48.3742i		−	0.612331i	−0.951978	−	0.306166i	$-0.900954\pi$
$79$		−	48.3742i		−	0.612331i	0.951978	−	0.306166i	$-0.0990462\pi$
$80$		0			0
$81$		0			0
$82$	40.4593	−	74.8903i	0.493407	−	0.913296i
$83$			140.248i			1.68974i	0.534974	+	0.844868i	$0.320321\pi$
$83$			140.248i			1.68974i	−0.534974	+	0.844868i	$0.679679\pi$
$84$		0			0
$85$		0			0
$86$	−110.717	−	59.8148i	−1.28741	−	0.695521i
$87$		0			0
$88$	−5.68901	−	66.9106i	−0.0646479	−	0.760348i
$89$	−18.1394			−0.203813			−0.101907	−	0.994794i	$-0.532494\pi$
$89$	−18.1394			−0.203813			−0.101907	+	0.994794i	$0.532494\pi$
$90$		0			0
$91$			38.9819i			0.428372i
$92$	9.94303	−	6.51642i	0.108076	−	0.0708307i
$93$		0			0
$94$	−69.5407	−	37.5692i	−0.739794	−	0.399673i
$95$		0			0
$96$		0			0
$97$	−15.0000			−0.154639			−0.0773196	−	0.997006i	$-0.524636\pi$
$97$	−15.0000			−0.154639			−0.0773196	+	0.997006i	$0.524636\pi$
$98$	31.4483	−	58.2108i	0.320901	−	0.593988i
$99$		0			0
$100$		0			0
$101$	118.611			1.17437			0.587184	−	0.809453i	$-0.300237\pi$
$101$	118.611			1.17437			0.587184	+	0.809453i	$0.300237\pi$
$102$		0			0
$103$		−	165.740i		−	1.60912i	−0.593868	−	0.804562i	$-0.702400\pi$
$103$		−	165.740i		−	1.60912i	0.593868	−	0.804562i	$-0.297600\pi$
$104$	−77.8816	+	6.62181i	−0.748862	+	0.0636712i
$105$		0			0
$106$	−20.1626	+	37.3211i	−0.190214	+	0.352085i
$107$		−	104.222i		−	0.974039i	−0.873391	−	0.487020i	$-0.838084\pi$
$107$		−	104.222i		−	0.974039i	0.873391	−	0.487020i	$-0.161916\pi$
$108$		0			0
$109$	60.8516			0.558272			0.279136	−	0.960252i	$-0.409952\pi$
$109$	60.8516			0.558272			0.279136	+	0.960252i	$0.409952\pi$
$110$		0			0
$111$		0			0
$112$	−58.5335	−	25.4757i	−0.522621	−	0.227461i
$113$	139.821			1.23735			0.618676	−	0.785646i	$-0.287669\pi$
$113$	139.821			1.23735			0.618676	+	0.785646i	$0.287669\pi$
$114$		0			0
$115$		0			0
$116$	−113.203	−	172.730i	−0.975888	−	1.48905i
$117$		0			0
$118$	88.6220	+	47.8779i	0.751034	+	0.405745i
$119$			48.4362i			0.407027i
$120$		0			0
$121$	50.5407			0.417691
$122$	−66.9175	+	123.864i	−0.548504	+	1.01528i
$123$		0			0
$124$	102.889	−	67.4308i	0.829748	−	0.543797i
$125$		0			0
$126$		0			0
$127$			63.8372i			0.502655i	0.967902	+	0.251327i	$0.0808671\pi$
$127$			63.8372i			0.502655i	−0.967902	+	0.251327i	$0.919133\pi$
$128$	40.9547	−	121.271i	0.319958	−	0.947432i
$129$		0			0
$130$		0			0
$131$			205.994i			1.57248i	0.617923	+	0.786238i	$0.287974\pi$
$131$			205.994i			1.57248i	−0.617923	+	0.786238i	$0.712026\pi$
$132$		0			0
$133$	69.3110			0.521135
$134$	−166.882	−	90.1577i	−1.24539	−	0.672819i
$135$		0			0
$136$	−96.7703	+	8.22780i	−0.711547	+	0.0604985i
$137$	−243.081			−1.77431			−0.887156	−	0.461470i	$-0.847322\pi$
$137$	−243.081			−1.77431			−0.887156	+	0.461470i	$0.847322\pi$
$138$		0			0
$139$		−	122.520i		−	0.881439i	−0.897645	−	0.440719i	$-0.854723\pi$
$139$		−	122.520i		−	0.881439i	0.897645	−	0.440719i	$-0.145277\pi$
$140$		0			0
$141$		0			0
$142$	232.933	+	125.842i	1.64037	+	0.886209i
$143$			82.0122i			0.573512i
$144$		0			0
$145$		0			0
$146$	73.8676	−	136.729i	0.505942	−	0.936499i
$147$		0			0
$148$	−42.8445	−	65.3739i	−0.289490	−	0.441716i
$149$	−121.399			−0.814761			−0.407380	−	0.913259i	$-0.633558\pi$
$149$	−121.399			−0.814761			−0.407380	+	0.913259i	$0.633558\pi$
$150$		0			0
$151$			113.376i			0.750833i	0.926856	+	0.375417i	$0.122500\pi$
$151$			113.376i			0.750833i	−0.926856	+	0.375417i	$0.877500\pi$
$152$	11.7738	+	138.476i	0.0774591	+	0.911026i
$153$		0			0
$154$	−31.8374	+	58.9310i	−0.206736	+	0.382669i
$155$		0			0
$156$		0			0
$157$	117.933			0.751165			0.375583	−	0.926789i	$-0.377443\pi$
$157$	117.933			0.751165			0.375583	+	0.926789i	$0.377443\pi$
$158$	−85.1206	−	45.9862i	−0.538738	−	0.291052i
$159$		0			0
$160$		0			0
$161$	−11.8579			−0.0736514
$162$		0			0
$163$			158.677i			0.973476i	0.873548	+	0.486738i	$0.161813\pi$
$163$			158.677i			0.973476i	−0.873548	+	0.486738i	$0.838187\pi$
$164$	−93.3169	−	142.387i	−0.569006	−	0.868212i
$165$		0			0
$166$	246.785	+	133.325i	1.48665	+	0.803162i
$167$		−	260.736i		−	1.56130i	−0.624972	−	0.780648i	$-0.714889\pi$
$167$		−	260.736i		−	1.56130i	0.624972	−	0.780648i	$-0.285111\pi$
$168$		0			0
$169$	−73.5407			−0.435152
$170$		0			0
$171$		0			0
$172$	−210.504	+	137.959i	−1.22386	+	0.802088i
$173$	294.711			1.70353			0.851765	−	0.523924i	$-0.175532\pi$
$173$	294.711			1.70353			0.851765	+	0.523924i	$0.175532\pi$
$174$		0			0
$175$		0			0
$176$	−123.146	−	53.5971i	−0.699693	−	0.304529i
$177$		0			0
$178$	−17.2440	+	31.9186i	−0.0968762	+	0.179318i
$179$			147.236i			0.822550i	0.911511	+	0.411275i	$0.134916\pi$
$179$			147.236i			0.822550i	−0.911511	+	0.411275i	$0.865084\pi$
$180$		0			0
$181$	−71.7703			−0.396521			−0.198261	−	0.980149i	$-0.563529\pi$
$181$	−71.7703			−0.396521			−0.198261	+	0.980149i	$0.563529\pi$
$182$	68.5936	+	37.0576i	0.376888	+	0.203613i
$183$		0			0
$184$	−2.01428	−	23.6908i	−0.0109472	−	0.128754i
$185$		0			0
$186$		0			0
$187$			101.903i			0.544934i
$188$	−132.216	+	86.6510i	−0.703275	+	0.460910i
$189$		0			0
$190$		0			0
$191$			333.832i			1.74781i	0.486094	+	0.873907i	$0.338421\pi$
$191$			333.832i			1.74781i	−0.486094	+	0.873907i	$0.661579\pi$
$192$		0			0
$193$	−192.703			−0.998463			−0.499231	−	0.866469i	$-0.666384\pi$
$193$	−192.703			−0.998463			−0.499231	+	0.866469i	$0.666384\pi$
$194$	−14.2595	+	26.3944i	−0.0735028	+	0.136054i
$195$		0			0
$196$	−72.5335	−	110.675i	−0.370069	−	0.564666i
$197$	−36.5608			−0.185588			−0.0927940	−	0.995685i	$-0.529580\pi$
$197$	−36.5608			−0.185588			−0.0927940	+	0.995685i	$0.529580\pi$
$198$		0			0
$199$			230.742i			1.15951i	0.814793	+	0.579753i	$0.196851\pi$
$199$			230.742i			1.15951i	−0.814793	+	0.579753i	$0.803149\pi$
$200$		0			0
$201$		0			0
$202$	112.756	−	208.712i	0.558198	−	1.03323i
$203$			205.994i			1.01475i
$204$		0			0
$205$		0			0
$206$	−291.640	−	157.558i	−1.41573	−	0.764846i
$207$		0			0
$208$	−62.3852	+	143.338i	−0.299929	+	0.689123i
$209$	145.820			0.697705
$210$		0			0
$211$		−	177.957i		−	0.843400i	−0.906735	−	0.421700i	$-0.861433\pi$
$211$		−	177.957i		−	0.843400i	0.906735	−	0.421700i	$-0.138567\pi$
$212$	46.5039	+	70.9575i	0.219358	+	0.334705i
$213$		0			0
$214$	−183.392	−	99.0774i	−0.856973	−	0.462978i
$215$		0			0
$216$		0			0
$217$	−122.703			−0.565453
$218$	57.8478	−	107.076i	0.265357	−	0.491176i
$219$		0			0
$220$		0			0
$221$	118.611			0.536702
$222$		0			0
$223$			175.877i			0.788684i	0.918964	+	0.394342i	$0.129028\pi$
$223$			175.877i			0.788684i	−0.918964	+	0.394342i	$0.870972\pi$
$224$	−100.472	+	78.7791i	−0.448535	+	0.351693i
$225$		0			0
$226$	132.919	−	246.033i	0.588136	−	1.08864i
$227$		−	76.5902i		−	0.337402i	−0.985667	−	0.168701i	$-0.946043\pi$
$227$		−	76.5902i		−	0.337402i	0.985667	−	0.168701i	$-0.0539572\pi$
$228$		0			0
$229$	−77.6363			−0.339023			−0.169511	−	0.985528i	$-0.554219\pi$
$229$	−77.6363			−0.339023			−0.169511	+	0.985528i	$0.554219\pi$
$230$		0			0
$231$		0			0
$232$	−411.555	+	34.9921i	−1.77394	+	0.150828i
$233$	−133.539			−0.573130			−0.286565	−	0.958061i	$-0.592513\pi$
$233$	−133.539			−0.573130			−0.286565	+	0.958061i	$0.592513\pi$
$234$		0			0
$235$		0			0
$236$	168.494	−	110.427i	0.713960	−	0.467912i
$237$		0			0
$238$	85.2297	+	46.0452i	0.358108	+	0.193467i
$239$		−	14.8602i		−	0.0621764i	−0.999517	−	0.0310882i	$-0.990103\pi$
$239$		−	14.8602i		−	0.0621764i	0.999517	−	0.0310882i	$-0.00989727\pi$
$240$		0			0
$241$	125.622			0.521253			0.260627	−	0.965440i	$-0.416071\pi$
$241$	125.622			0.521253			0.260627	+	0.965440i	$0.416071\pi$
$242$	48.0458	−	88.9328i	0.198536	−	0.367491i
$243$		0			0
$244$	154.341	+	235.500i	0.632545	+	0.965163i
$245$		0			0
$246$		0			0
$247$		−	169.730i		−	0.687165i
$248$	−20.8435	−	245.148i	−0.0840463	−	0.988500i
$249$		0			0
$250$		0			0
$251$			164.024i			0.653484i	0.945114	+	0.326742i	$0.105951\pi$
$251$			164.024i			0.653484i	−0.945114	+	0.326742i	$0.894049\pi$
$252$		0			0
$253$	−24.9473			−0.0986057
$254$	112.330	+	60.6859i	0.442243	+	0.238921i
$255$		0			0
$256$	−174.459	−	187.350i	−0.681482	−	0.731835i
$257$	188.098			0.731901			0.365950	−	0.930634i	$-0.380744\pi$
$257$	188.098			0.731901			0.365950	+	0.930634i	$0.380744\pi$
$258$		0			0
$259$			77.9638i			0.301018i
$260$		0			0
$261$		0			0
$262$	362.474	+	195.826i	1.38349	+	0.747426i
$263$			114.544i			0.435529i	0.976001	+	0.217764i	$0.0698764\pi$
$263$			114.544i			0.435529i	−0.976001	+	0.217764i	$0.930124\pi$
$264$		0			0
$265$		0			0
$266$	65.8895	−	121.962i	0.247705	−	0.458502i
$267$		0			0
$268$	−317.288	+	207.943i	−1.18391	+	0.775907i
$269$	221.871			0.824800			0.412400	−	0.911003i	$-0.364691\pi$
$269$	221.871			0.824800			0.412400	+	0.911003i	$0.364691\pi$
$270$		0			0
$271$		−	390.163i		−	1.43971i	−0.694122	−	0.719857i	$-0.744207\pi$
$271$		−	390.163i		−	1.43971i	0.694122	−	0.719857i	$-0.255793\pi$
$272$	−77.5155	+	178.101i	−0.284983	+	0.654785i
$273$		0			0
$274$	−231.081	+	427.732i	−0.843362	+	1.56106i
$275$		0			0
$276$		0			0
$277$	−272.421			−0.983469			−0.491734	−	0.870745i	$-0.663637\pi$
$277$	−272.421			−0.983469			−0.491734	+	0.870745i	$0.663637\pi$
$278$	−215.590	−	116.472i	−0.775502	−	0.418964i
$279$		0			0
$280$		0			0
$281$	473.740			1.68591			0.842953	−	0.537987i	$-0.180815\pi$
$281$	473.740			1.68591			0.842953	+	0.537987i	$0.180815\pi$
$282$		0			0
$283$			164.824i			0.582415i	0.956660	+	0.291208i	$0.0940570\pi$
$283$			164.824i			0.582415i	−0.956660	+	0.291208i	$0.905943\pi$
$284$	442.869	−	290.246i	1.55940	−	1.02199i
$285$		0			0
$286$	144.311	+	77.9638i	0.504584	+	0.272601i
$287$			169.808i			0.591665i
$288$		0			0
$289$	−141.622			−0.490041
$290$		0			0
$291$		0			0
$292$	−170.371	−	259.959i	−0.583462	−	0.890270i
$293$	−273.219			−0.932488			−0.466244	−	0.884656i	$-0.654393\pi$
$293$	−273.219			−0.932488			−0.466244	+	0.884656i	$0.654393\pi$
$294$		0			0
$295$		0			0
$296$	−155.763	+	13.2436i	−0.526227	+	0.0447420i
$297$		0			0
$298$	−115.407	+	213.618i	−0.387270	+	0.716838i
$299$			29.0377i			0.0971161i
$300$		0			0
$301$	251.043			0.834029
$302$	199.499	+	107.779i	0.660594	+	0.356885i
$303$		0			0
$304$	254.859	+	110.923i	0.838351	+	0.364877i
$305$		0			0
$306$		0			0
$307$			132.905i			0.432915i	0.976292	+	0.216458i	$0.0694503\pi$
$307$			132.905i			0.432915i	−0.976292	+	0.216458i	$0.930550\pi$
$308$	73.4309	+	112.044i	0.238412	+	0.363779i
$309$		0			0
$310$		0			0
$311$		−	235.032i		−	0.755730i	−0.925861	−	0.377865i	$-0.876658\pi$
$311$		−	235.032i		−	0.755730i	0.925861	−	0.377865i	$-0.123342\pi$
$312$		0			0
$313$	−497.651			−1.58994			−0.794969	−	0.606650i	$-0.792513\pi$
$313$	−497.651			−1.58994			−0.794969	+	0.606650i	$0.792513\pi$
$314$	112.111	−	207.518i	0.357042	−	0.660886i
$315$		0			0
$316$	−161.837	+	106.064i	−0.512144	+	0.335647i
$317$	−169.959			−0.536149			−0.268074	−	0.963398i	$-0.586387\pi$
$317$	−169.959			−0.536149			−0.268074	+	0.963398i	$0.586387\pi$
$318$		0			0
$319$			433.382i			1.35857i
$320$		0			0
$321$		0			0
$322$	−11.2725	+	20.8655i	−0.0350078	+	0.0647996i
$323$		−	210.894i		−	0.652924i
$324$		0			0
$325$		0			0
$326$	279.212	+	150.844i	0.856478	+	0.462711i
$327$		0			0
$328$	−339.258	+	28.8451i	−1.03432	+	0.0879424i
$329$	157.678			0.479265
$330$		0			0
$331$		−	10.3086i		−	0.0311440i	−0.999879	−	0.0155720i	$-0.995043\pi$
$331$		−	10.3086i		−	0.0311440i	0.999879	−	0.0155720i	$-0.00495691\pi$
$332$	469.205	−	307.506i	1.41327	−	0.926222i
$333$		0			0
$334$	−458.799	−	247.865i	−1.37365	−	0.742112i
$335$		0			0
$336$		0			0
$337$	109.029			0.323527			0.161763	−	0.986830i	$-0.448282\pi$
$337$	109.029			0.323527			0.161763	+	0.986830i	$0.448282\pi$
$338$	−69.9104	+	129.404i	−0.206835	+	0.382853i
$339$		0			0
$340$		0			0
$341$	−258.150			−0.757038
$342$		0			0
$343$			327.490i			0.954781i
$344$	42.6444	+	501.557i	0.123966	+	1.45801i
$345$		0			0
$346$	280.163	−	518.581i	0.809719	−	1.49879i
$347$		−	173.824i		−	0.500934i	−0.968125	−	0.250467i	$-0.919416\pi$
$347$		−	173.824i		−	0.500934i	0.968125	−	0.250467i	$-0.0805842\pi$
$348$		0			0
$349$	259.244			0.742819			0.371410	−	0.928469i	$-0.378875\pi$
$349$	259.244			0.742819			0.371410	+	0.928469i	$0.378875\pi$
$350$		0			0
$351$		0			0
$352$	−211.378	+	165.740i	−0.600506	+	0.470852i
$353$	−115.400			−0.326912			−0.163456	−	0.986551i	$-0.552264\pi$
$353$	−115.400			−0.326912			−0.163456	+	0.986551i	$0.552264\pi$
$354$		0			0
$355$		0			0
$356$	39.7721	+	60.6859i	0.111719	+	0.170466i
$357$		0			0
$358$	259.081	+	139.968i	0.723691	+	0.390973i
$359$			333.832i			0.929895i	0.885338	+	0.464948i	$0.153927\pi$
$359$			333.832i			0.929895i	−0.885338	+	0.464948i	$0.846073\pi$
$360$		0			0
$361$	59.2154			0.164032
$362$	−68.2275	+	126.289i	−0.188474	+	0.348865i
$363$		0			0
$364$	130.415	−	85.4710i	0.358283	−	0.234810i
$365$		0			0
$366$		0			0
$367$			525.148i			1.43092i	0.698653	+	0.715461i	$0.253783\pi$
$367$			525.148i			1.43092i	−0.698653	+	0.715461i	$0.746217\pi$
$368$	−43.6018	−	18.9769i	−0.118483	−	0.0515677i
$369$		0			0
$370$		0			0
$371$		−	84.6227i		−	0.228093i
$372$		0			0
$373$	417.014			1.11800			0.559000	−	0.829167i	$-0.311185\pi$
$373$	417.014			1.11800			0.559000	+	0.829167i	$0.311185\pi$
$374$	179.311	+	96.8724i	0.479441	+	0.259017i
$375$		0			0
$376$	26.7846	+	315.024i	0.0712357	+	0.837830i
$377$	504.442			1.33804
$378$		0			0
$379$		−	126.758i		−	0.334454i	−0.985918	−	0.167227i	$-0.946519\pi$
$379$		−	126.758i		−	0.334454i	0.985918	−	0.167227i	$-0.0534812\pi$
$380$		0			0
$381$		0			0
$382$	587.421	+	317.353i	1.53775	+	0.830767i
$383$			129.404i			0.337870i	0.985627	+	0.168935i	$0.0540328\pi$
$383$			129.404i			0.337870i	−0.985627	+	0.168935i	$0.945967\pi$
$384$		0			0
$385$		0			0
$386$	−183.191	+	339.086i	−0.474587	+	0.878462i
$387$		0			0
$388$	32.8887	+	50.1830i	0.0847648	+	0.129338i
$389$	87.9088			0.225987			0.112993	−	0.993596i	$-0.463956\pi$
$389$	87.9088			0.225987			0.112993	+	0.993596i	$0.463956\pi$
$390$		0			0
$391$			36.0803i			0.0922769i
$392$	−263.699	+	22.4208i	−0.672702	+	0.0571958i
$393$		0			0
$394$	−34.7560	+	64.3335i	−0.0882133	+	0.163283i
$395$		0			0
$396$		0			0
$397$	239.770			0.603955			0.301978	−	0.953315i	$-0.402353\pi$
$397$	239.770			0.603955			0.301978	+	0.953315i	$0.402353\pi$
$398$	406.019	+	219.351i	1.02015	+	0.551134i
$399$		0			0
$400$		0			0
$401$	−383.043			−0.955218			−0.477609	−	0.878572i	$-0.658497\pi$
$401$	−383.043			−0.955218			−0.477609	+	0.878572i	$0.658497\pi$
$402$		0			0
$403$			300.477i			0.745602i
$404$	−260.065	−	396.817i	−0.643725	−	0.982221i
$405$		0			0
$406$	362.474	+	195.826i	0.892792	+	0.482329i
$407$			164.024i			0.403008i
$408$		0			0
$409$	−731.976			−1.78967			−0.894836	−	0.446395i	$-0.852708\pi$
$409$	−731.976			−1.78967			−0.894836	+	0.446395i	$0.852708\pi$
$410$		0			0
$411$		0			0
$412$	−554.488	+	363.398i	−1.34584	+	0.882035i
$413$	−200.944			−0.486546
$414$		0			0
$415$		0			0
$416$	192.915	+	246.037i	0.463739	+	0.591434i
$417$		0			0
$418$	138.622	−	256.589i	0.331632	−	0.613850i
$419$		−	692.847i		−	1.65357i	−0.562516	−	0.826786i	$-0.690167\pi$
$419$		−	692.847i		−	1.65357i	0.562516	−	0.826786i	$-0.309833\pi$
$420$		0			0
$421$	−326.622			−0.775824			−0.387912	−	0.921696i	$-0.626804\pi$
$421$	−326.622			−0.775824			−0.387912	+	0.921696i	$0.626804\pi$
$422$	−313.139	−	169.173i	−0.742035	−	0.400883i
$423$		0			0
$424$	169.067	−	14.3748i	0.398743	−	0.0339027i
$425$		0			0
$426$		0			0
$427$		−	280.853i		−	0.657735i
$428$	−348.679	+	228.516i	−0.814670	+	0.533915i
$429$		0			0
$430$		0			0
$431$		−	367.408i		−	0.852456i	−0.904616	−	0.426228i	$-0.859842\pi$
$431$		−	367.408i		−	0.852456i	0.904616	−	0.426228i	$-0.140158\pi$
$432$		0			0
$433$	−493.622			−1.14000			−0.570002	−	0.821643i	$-0.693058\pi$
$433$	−493.622			−1.14000			−0.570002	+	0.821643i	$0.693058\pi$
$434$	−116.646	+	215.912i	−0.268770	+	0.497493i
$435$		0			0
$436$	−133.422	−	203.581i	−0.306014	−	0.466929i
$437$	51.6300			0.118146
$438$		0			0
$439$		−	658.970i		−	1.50107i	−0.660831	−	0.750535i	$-0.729796\pi$
$439$		−	658.970i		−	1.50107i	0.660831	−	0.750535i	$-0.270204\pi$
$440$		0			0
$441$		0			0
$442$	112.756	−	208.712i	0.255104	−	0.472198i
$443$		−	623.245i		−	1.40687i	−0.710758	−	0.703437i	$-0.751648\pi$
$443$		−	623.245i		−	1.40687i	0.710758	−	0.703437i	$-0.248352\pi$
$444$		0			0
$445$		0			0
$446$	309.477	+	167.195i	0.693896	+	0.374876i
$447$		0			0
$448$	43.1099	+	251.683i	0.0962274	+	0.561793i
$449$	−364.198			−0.811132			−0.405566	−	0.914066i	$-0.632925\pi$
$449$	−364.198			−0.811132			−0.405566	+	0.914066i	$0.632925\pi$
$450$		0			0
$451$			357.251i			0.792132i
$452$	−306.569	−	467.775i	−0.678249	−	1.03490i
$453$		0			0
$454$	−134.770	−	72.8094i	−0.296851	−	0.160373i
$455$		0			0
$456$		0			0
$457$	−215.866			−0.472354			−0.236177	−	0.971710i	$-0.575895\pi$
$457$	−215.866			−0.472354			−0.236177	+	0.971710i	$0.575895\pi$
$458$	−73.8038	+	136.611i	−0.161144	+	0.298277i
$459$		0			0
$460$		0			0
$461$	−27.9142			−0.0605515			−0.0302757	−	0.999542i	$-0.509639\pi$
$461$	−27.9142			−0.0605515			−0.0302757	+	0.999542i	$0.509639\pi$
$462$		0			0
$463$		−	369.545i		−	0.798154i	−0.916917	−	0.399077i	$-0.869331\pi$
$463$		−	369.545i		−	0.798154i	0.916917	−	0.399077i	$-0.130669\pi$
$464$	−329.666	+	757.448i	−0.710487	+	1.63243i
$465$		0			0
$466$	−126.947	+	234.979i	−0.272419	+	0.504248i
$467$			741.443i			1.58767i	0.608131	+	0.793837i	$0.291920\pi$
$467$			741.443i			1.58767i	−0.608131	+	0.793837i	$0.708080\pi$
$468$		0			0
$469$	378.392			0.806807
$470$		0			0
$471$		0			0
$472$	−34.1341	−	401.464i	−0.0723179	−	0.850559i
$473$	528.158			1.11661
$474$		0			0
$475$		0			0
$476$	162.045	−	106.200i	0.340430	−	0.223110i
$477$		0			0
$478$	−26.1484	−	14.1266i	−0.0547037	−	0.0295536i
$479$		−	419.700i		−	0.876201i	−0.898926	−	0.438101i	$-0.855651\pi$
$479$		−	419.700i		−	0.876201i	0.898926	−	0.438101i	$-0.144349\pi$
$480$		0			0
$481$	190.919			0.396920
$482$	119.421	−	221.048i	0.247761	−	0.458606i
$483$		0			0
$484$	−110.815	−	169.085i	−0.228956	−	0.349350i
$485$		0			0
$486$		0			0
$487$		−	195.501i		−	0.401440i	−0.979649	−	0.200720i	$-0.935672\pi$
$487$		−	195.501i		−	0.401440i	0.979649	−	0.200720i	$-0.0643281\pi$
$488$	561.114	−	47.7082i	1.14982	−	0.0977626i
$489$		0			0
$490$		0			0
$491$		−	676.741i		−	1.37829i	−0.724623	−	0.689146i	$-0.757986\pi$
$491$		−	676.741i		−	1.37829i	0.724623	−	0.689146i	$-0.242014\pi$
$492$		0			0
$493$	626.785			1.27137
$494$	−298.661	−	161.351i	−0.604577	−	0.326622i
$495$		0			0
$496$	−451.184	−	196.370i	−0.909645	−	0.395907i
$497$	−528.158			−1.06269
$498$		0			0
$499$			628.139i			1.25880i	0.777083	+	0.629398i	$0.216698\pi$
$499$			628.139i			1.25880i	−0.777083	+	0.629398i	$0.783302\pi$
$500$		0			0
$501$		0			0
$502$	288.622	+	155.928i	0.574944	+	0.310613i
$503$		−	514.645i		−	1.02315i	−0.859238	−	0.511575i	$-0.829062\pi$
$503$		−	514.645i		−	1.02315i	0.859238	−	0.511575i	$-0.170938\pi$
$504$		0			0
$505$		0			0
$506$	−23.7158	+	43.8979i	−0.0468691	+	0.0867547i
$507$		0			0
$508$	213.569	−	139.968i	0.420412	−	0.275528i
$509$	−802.364			−1.57635			−0.788177	−	0.615449i	$-0.788975\pi$
$509$	−802.364			−1.57635			−0.788177	+	0.615449i	$0.788975\pi$
$510$		0			0
$511$			310.022i			0.606697i
$512$	−495.513	+	128.882i	−0.967799	+	0.251723i
$513$		0			0
$514$	178.813	−	330.983i	0.347886	−	0.643937i
$515$		0			0
$516$		0			0
$517$	331.732			0.641648
$518$	137.187	+	74.1152i	0.264840	+	0.143079i
$519$		0			0
$520$		0			0
$521$	−917.482			−1.76100			−0.880501	−	0.474045i	$-0.842793\pi$
$521$	−917.482			−1.76100			−0.880501	+	0.474045i	$0.842793\pi$
$522$		0			0
$523$		−	957.290i		−	1.83038i	−0.403020	−	0.915191i	$-0.632040\pi$
$523$		−	957.290i		−	1.83038i	0.403020	−	0.915191i	$-0.367960\pi$
$524$	689.161	−	451.660i	1.31519	−	0.861946i
$525$		0			0
$526$	201.555	+	108.890i	0.383184	+	0.207015i
$527$			373.352i			0.708449i
$528$		0			0
$529$	520.167			0.983303
$530$		0			0
$531$		0			0
$532$	−151.970	−	231.882i	−0.285658	−	0.435869i
$533$	415.828			0.780165
$534$		0			0
$535$		0			0
$536$	64.2771	+	755.987i	0.119920	+	1.41042i
$537$		0			0
$538$	210.919	−	390.411i	0.392042	−	0.725670i
$539$			277.685i			0.515185i
$540$		0			0
$541$	−329.014			−0.608159			−0.304080	−	0.952647i	$-0.598349\pi$
$541$	−329.014			−0.608159			−0.304080	+	0.952647i	$0.598349\pi$
$542$	−686.541	−	370.903i	−1.26668	−	0.684322i
$543$		0			0
$544$	239.703	+	305.708i	0.440631	+	0.561963i
$545$		0			0
$546$		0			0
$547$		−	25.7716i		−	0.0471145i	−0.999722	−	0.0235572i	$-0.992501\pi$
$547$		−	25.7716i		−	0.0471145i	0.999722	−	0.0235572i	$-0.00749920\pi$
$548$	532.975	+	813.234i	0.972581	+	1.48400i
$549$		0			0
$550$		0			0
$551$		−	896.913i		−	1.62779i
$552$		0			0
$553$	193.004			0.349013
$554$	−258.973	+	479.359i	−0.467460	+	0.865270i
$555$		0			0
$556$	−409.895	+	268.635i	−0.737220	+	0.483157i
$557$	632.405			1.13538			0.567688	−	0.823244i	$-0.307838\pi$
$557$	632.405			1.13538			0.567688	+	0.823244i	$0.307838\pi$
$558$		0			0
$559$		−	614.757i		−	1.09974i
$560$		0			0
$561$		0			0
$562$	450.354	−	833.605i	0.801341	−	1.48328i
$563$		−	156.675i		−	0.278285i	−0.990272	−	0.139143i	$-0.955565\pi$
$563$		−	156.675i		−	0.278285i	0.990272	−	0.139143i	$-0.0444346\pi$
$564$		0			0
$565$		0			0
$566$	290.028	+	156.687i	0.512417	+	0.276832i
$567$		0			0
$568$	−89.7176	−	1055.20i	−0.157953	−	1.85775i
$569$	−522.581			−0.918421			−0.459210	−	0.888328i	$-0.651868\pi$
$569$	−522.581			−0.918421			−0.459210	+	0.888328i	$0.651868\pi$
$570$		0			0
$571$		−	855.387i		−	1.49805i	−0.662541	−	0.749026i	$-0.730522\pi$
$571$		−	855.387i		−	1.49805i	0.662541	−	0.749026i	$-0.269478\pi$
$572$	274.374	−	179.819i	0.479676	−	0.314368i
$573$		0			0
$574$	298.799	+	161.426i	0.520556	+	0.281229i
$575$		0			0
$576$		0			0
$577$	−484.081			−0.838962			−0.419481	−	0.907764i	$-0.637788\pi$
$577$	−484.081			−0.838962			−0.419481	+	0.907764i	$0.637788\pi$
$578$	−134.631	+	249.202i	−0.232926	+	0.431145i
$579$		0			0
$580$		0			0
$581$	−559.565			−0.963107
$582$		0			0
$583$		−	178.034i		−	0.305375i
$584$	−619.392	+	52.6632i	−1.06060	+	0.0901766i
$585$		0			0
$586$	−259.732	+	480.764i	−0.443228	+	0.820416i
$587$		−	134.666i		−	0.229413i	−0.993399	−	0.114707i	$-0.963407\pi$
$587$		−	134.666i		−	0.229413i	0.993399	−	0.114707i	$-0.0365928\pi$
$588$		0			0
$589$	534.258			0.907060
$590$		0			0
$591$		0			0
$592$	−124.770	+	286.675i	−0.210761	+	0.484249i
$593$	−960.465			−1.61967			−0.809836	−	0.586657i	$-0.800444\pi$
$593$	−960.465			−1.61967			−0.809836	+	0.586657i	$0.800444\pi$
$594$		0			0
$595$		0			0
$596$	266.178	+	406.145i	0.446608	+	0.681452i
$597$		0			0
$598$	51.0956	+	27.6043i	0.0854442	+	0.0461611i
$599$		−	492.073i		−	0.821491i	−0.911750	−	0.410746i	$-0.865268\pi$
$599$		−	492.073i		−	0.821491i	0.911750	−	0.410746i	$-0.134732\pi$
$600$		0			0
$601$	152.187			0.253223			0.126611	−	0.991952i	$-0.459590\pi$
$601$	152.187			0.253223			0.126611	+	0.991952i	$0.459590\pi$
$602$	238.650	−	441.742i	0.396429	−	0.733791i
$603$		0			0
$604$	379.302	−	248.586i	0.627984	−	0.411566i
$605$		0			0
$606$		0			0
$607$		−	1032.50i		−	1.70100i	−0.525978	−	0.850498i	$-0.676301\pi$
$607$		−	1032.50i		−	1.70100i	0.525978	−	0.850498i	$-0.323699\pi$
$608$	437.461	−	343.010i	0.719508	−	0.564160i
$609$		0			0
$610$		0			0
$611$		−	386.124i		−	0.631955i
$612$		0			0
$613$	774.029			1.26269			0.631345	−	0.775502i	$-0.282503\pi$
$613$	774.029			1.26269			0.631345	+	0.775502i	$0.282503\pi$
$614$	233.863	+	126.344i	0.380885	+	0.205772i
$615$		0			0
$616$	266.962	−	22.6981i	0.433379	−	0.0368476i
$617$	−594.857			−0.964112			−0.482056	−	0.876141i	$-0.660110\pi$
$617$	−594.857			−0.964112			−0.482056	+	0.876141i	$0.660110\pi$
$618$		0			0
$619$		−	158.333i		−	0.255788i	−0.991788	−	0.127894i	$-0.959178\pi$
$619$		−	158.333i		−	0.255788i	0.991788	−	0.127894i	$-0.0408217\pi$
$620$		0			0
$621$		0			0
$622$	−413.569	−	223.430i	−0.664902	−	0.359212i
$623$		−	72.3729i		−	0.116168i
$624$		0			0
$625$		0			0
$626$	−473.085	+	875.680i	−0.755726	+	1.39885i
$627$		0			0
$628$	−258.578	−	394.548i	−0.411748	−	0.628262i
$629$	237.222			0.377142
$630$		0			0
$631$			832.689i			1.31963i	0.751426	+	0.659817i	$0.229366\pi$
$631$			832.689i			1.31963i	−0.751426	+	0.659817i	$0.770634\pi$
$632$	32.7854	+	385.602i	0.0518757	+	0.610130i
$633$		0			0
$634$	−161.569	+	299.065i	−0.254841	+	0.471711i
$635$		0			0
$636$		0			0
$637$	323.215			0.507402
$638$	762.592	+	411.989i	1.19529	+	0.645751i
$639$		0			0
$640$		0			0
$641$	23.7158			0.0369981			0.0184990	−	0.999829i	$-0.494111\pi$
$641$	23.7158			0.0369981			0.0184990	+	0.999829i	$0.494111\pi$
$642$		0			0
$643$		−	637.188i		−	0.990961i	−0.868619	−	0.495480i	$-0.834992\pi$
$643$		−	637.188i		−	0.990961i	0.868619	−	0.495480i	$-0.165008\pi$
$644$	25.9994	+	39.6709i	0.0403717	+	0.0616008i
$645$		0			0
$646$	−371.096	−	200.484i	−0.574451	−	0.310346i
$647$			462.353i			0.714610i	0.933988	+	0.357305i	$0.116304\pi$
$647$			462.353i			0.714610i	−0.933988	+	0.357305i	$0.883696\pi$
$648$		0			0
$649$	−422.756			−0.651396
$650$		0			0
$651$		0			0
$652$	530.857	−	347.911i	0.814198	−	0.533606i
$653$	−661.697			−1.01332			−0.506659	−	0.862146i	$-0.669120\pi$
$653$	−661.697			−1.01332			−0.506659	+	0.862146i	$0.669120\pi$
$654$		0			0
$655$		0			0
$656$	−271.754	+	624.390i	−0.414260	+	0.951813i
$657$		0			0
$658$	149.895	−	277.455i	0.227803	−	0.421664i
$659$		−	248.647i		−	0.377310i	−0.982043	−	0.188655i	$-0.939587\pi$
$659$		−	248.647i		−	0.377310i	0.982043	−	0.188655i	$-0.0604127\pi$
$660$		0			0
$661$	−1022.35			−1.54668			−0.773339	−	0.633993i	$-0.781415\pi$
$661$	−1022.35			−1.54668			−0.773339	+	0.633993i	$0.781415\pi$
$662$	−18.1394	−	9.79977i	−0.0274009	−	0.0148033i
$663$		0			0
$664$	−95.0527	−	1117.95i	−0.143152	−	1.68366i
$665$		0			0
$666$		0			0
$667$			153.446i			0.230054i
$668$	−872.301	+	571.686i	−1.30584	+	0.855817i
$669$		0			0
$670$		0			0
$671$		−	590.873i		−	0.880586i
$672$		0			0
$673$	−683.215			−1.01518			−0.507589	−	0.861599i	$-0.669463\pi$
$673$	−683.215			−1.01518			−0.507589	+	0.861599i	$0.669463\pi$
$674$	103.646	−	191.850i	0.153778	−	0.284644i
$675$		0			0
$676$	161.244	+	246.033i	0.238527	+	0.363954i
$677$	707.751			1.04542			0.522711	−	0.852510i	$-0.324921\pi$
$677$	707.751			1.04542			0.522711	+	0.852510i	$0.324921\pi$
$678$		0			0
$679$		−	59.8473i		−	0.0881404i
$680$		0			0
$681$		0			0
$682$	−245.407	+	454.248i	−0.359834	+	0.666053i
$683$		−	597.701i		−	0.875112i	−0.899191	−	0.437556i	$-0.855844\pi$
$683$		−	597.701i		−	0.875112i	0.899191	−	0.437556i	$-0.144156\pi$
$684$		0			0
$685$		0			0
$686$	576.260	+	311.324i	0.840030	+	0.453825i
$687$		0			0
$688$	923.093	+	401.760i	1.34170	+	0.583953i
$689$	−207.225			−0.300762
$690$		0			0
$691$			1280.71i			1.85342i	0.375777	+	0.926710i	$0.377376\pi$
$691$			1280.71i			1.85342i	−0.375777	+	0.926710i	$0.622624\pi$
$692$	−646.178	−	985.964i	−0.933783	−	1.42480i
$693$		0			0
$694$	−305.866	−	165.244i	−0.440729	−	0.238103i
$695$		0			0
$696$		0			0
$697$	516.679			0.741290
$698$	246.447	−	456.173i	0.353075	−	0.653543i
$699$		0			0
$700$		0			0
$701$	312.568			0.445889			0.222944	−	0.974831i	$-0.428433\pi$
$701$	312.568			0.445889			0.222944	+	0.974831i	$0.428433\pi$
$702$		0			0
$703$		−	339.459i		−	0.482872i
$704$	90.6969	+	529.505i	0.128831	+	0.752138i
$705$		0			0
$706$	−109.703	+	203.061i	−0.155387	+	0.287622i
$707$			473.237i			0.669360i
$708$		0			0
$709$	265.120			0.373935			0.186967	−	0.982366i	$-0.440134\pi$
$709$	265.120			0.373935			0.186967	+	0.982366i	$0.440134\pi$
$710$		0			0
$711$		0			0
$712$	144.593	−	12.2939i	0.203081	−	0.0172667i
$713$	−91.4021			−0.128194
$714$		0			0
$715$		0			0
$716$	492.584	−	322.828i	0.687967	−	0.450877i
$717$		0			0
$718$	587.421	+	317.353i	0.818135	+	0.441996i
$719$			430.705i			0.599033i	0.954091	+	0.299517i	$0.0968254\pi$
$719$			430.705i			0.599033i	−0.954091	+	0.299517i	$0.903175\pi$
$720$		0			0
$721$	661.273			0.917160
$722$	56.2923	−	104.197i	0.0779671	−	0.144317i
$723$		0			0
$724$	157.362	+	240.110i	0.217351	+	0.331644i
$725$		0			0
$726$		0			0
$727$			679.243i			0.934310i	0.884176	+	0.467155i	$0.154721\pi$
$727$			679.243i			0.934310i	−0.884176	+	0.467155i	$0.845279\pi$
$728$	−26.4198	−	310.734i	−0.0362910	−	0.426832i
$729$		0			0
$730$		0			0
$731$		−	763.855i		−	1.04494i
$732$		0			0
$733$	1045.41			1.42620			0.713101	−	0.701061i	$-0.247290\pi$
$733$	1045.41			1.42620			0.713101	+	0.701061i	$0.247290\pi$
$734$	924.066	+	499.225i	1.25895	+	0.680143i
$735$		0			0
$736$	−74.8418	+	58.6828i	−0.101687	+	0.0797321i
$737$	796.082			1.08017
$738$		0			0
$739$		−	846.147i		−	1.14499i	−0.819908	−	0.572495i	$-0.805976\pi$
$739$		−	846.147i		−	1.14499i	0.819908	−	0.572495i	$-0.194024\pi$
$740$		0			0
$741$		0			0
$742$	−148.904	−	80.4453i	−0.200680	−	0.108417i
$743$		−	322.828i		−	0.434493i	−0.976117	−	0.217246i	$-0.930293\pi$
$743$		−	322.828i		−	0.434493i	0.976117	−	0.217246i	$-0.0697075\pi$
$744$		0			0
$745$		0			0
$746$	396.429	−	733.790i	0.531406	−	0.983633i
$747$		0			0
$748$	340.919	−	223.430i	0.455774	−	0.298703i
$749$	415.828			0.555178
$750$		0			0
$751$		−	716.488i		−	0.954045i	−0.878891	−	0.477023i	$-0.841716\pi$
$751$		−	716.488i		−	0.954045i	0.878891	−	0.477023i	$-0.158284\pi$
$752$	579.788	+	252.342i	0.770994	+	0.335562i
$753$		0			0
$754$	479.541	−	887.630i	0.635996	−	1.17723i
$755$		0			0
$756$		0			0
$757$	375.636			0.496217			0.248108	−	0.968732i	$-0.420191\pi$
$757$	375.636			0.496217			0.248108	+	0.968732i	$0.420191\pi$
$758$	−223.047	−	120.501i	−0.294257	−	0.158972i
$759$		0			0
$760$		0			0
$761$	−97.6836			−0.128362			−0.0641811	−	0.997938i	$-0.520444\pi$
$761$	−97.6836			−0.128362			−0.0641811	+	0.997938i	$0.520444\pi$
$762$		0			0
$763$			242.787i			0.318201i
$764$	1116.85	−	731.955i	1.46184	−	0.958056i
$765$		0			0
$766$	227.703	+	123.016i	0.297263	+	0.160596i
$767$			492.073i			0.641556i
$768$		0			0
$769$	−665.651			−0.865605			−0.432803	−	0.901489i	$-0.642475\pi$
$769$	−665.651			−0.865605			−0.432803	+	0.901489i	$0.642475\pi$
$770$		0			0
$771$		0			0
$772$	422.518	+	644.695i	0.547303	+	0.835097i
$773$	−167.735			−0.216992			−0.108496	−	0.994097i	$-0.534604\pi$
$773$	−167.735			−0.216992			−0.108496	+	0.994097i	$0.534604\pi$
$774$		0			0
$775$		0			0
$776$	119.569	−	10.1662i	0.154083	−	0.0131008i
$777$		0			0
$778$	83.5692	−	154.687i	0.107415	−	0.198826i
$779$		−	739.355i		−	0.949108i
$780$		0			0
$781$	−1111.17			−1.42275
$782$	63.4879	+	34.2992i	0.0811865	+	0.0438609i
$783$		0			0
$784$	−211.230	+	485.326i	−0.269426	+	0.619039i
$785$		0			0
$786$		0			0
$787$		−	185.097i		−	0.235193i	−0.993061	−	0.117597i	$-0.962481\pi$
$787$		−	185.097i		−	0.235193i	0.993061	−	0.117597i	$-0.0375190\pi$
$788$	80.1626	+	122.315i	0.101729	+	0.155223i
$789$		0			0
$790$		0			0
$791$			557.860i			0.705259i
$792$		0			0
$793$	−687.756			−0.867284
$794$	227.934	−	421.907i	0.287071	−	0.531369i
$795$		0			0
$796$	771.953	−	505.920i	0.969790	−	0.635578i
$797$	−529.145			−0.663921			−0.331960	−	0.943293i	$-0.607710\pi$
$797$	−529.145			−0.663921			−0.331960	+	0.943293i	$0.607710\pi$
$798$		0			0
$799$		−	479.771i		−	0.600465i
$800$		0			0
$801$		0			0
$802$	−364.134	+	674.013i	−0.454033	+	0.840415i
$803$			652.242i			0.812256i
$804$		0			0
$805$		0			0
$806$	528.729	+	285.645i	0.655991	+	0.354398i
$807$		0			0
$808$	−945.478	+	80.3883i	−1.17015	+	0.0994905i
$809$	1306.81			1.61533			0.807667	−	0.589638i	$-0.200730\pi$
$809$	1306.81			1.61533			0.807667	+	0.589638i	$0.200730\pi$
$810$		0			0
$811$			582.743i			0.718549i	0.933232	+	0.359274i	$0.116976\pi$
$811$			582.743i			0.718549i	−0.933232	+	0.359274i	$0.883024\pi$
$812$	689.161	−	451.660i	0.848720	−	0.556231i
$813$		0			0
$814$	288.622	+	155.928i	0.354572	+	0.191557i
$815$		0			0
$816$		0			0
$817$	−1093.06			−1.33789
$818$	−695.843	+	1288.01i	−0.850663	+	1.57458i
$819$		0			0
$820$		0			0
$821$	−224.659			−0.273641			−0.136821	−	0.990596i	$-0.543688\pi$
$821$	−224.659			−0.273641			−0.136821	+	0.990596i	$0.543688\pi$
$822$		0			0
$823$			680.427i			0.826764i	0.910558	+	0.413382i	$0.135653\pi$
$823$			680.427i			0.826764i	−0.910558	+	0.413382i	$0.864347\pi$
$824$	112.330	+	1321.15i	0.136322	+	1.60334i
$825$		0			0
$826$	−191.024	+	353.586i	−0.231264	+	0.428070i
$827$			594.207i			0.718509i	0.933240	+	0.359255i	$0.116969\pi$
$827$			594.207i			0.718509i	−0.933240	+	0.359255i	$0.883031\pi$
$828$		0			0
$829$	546.029			0.658659			0.329330	−	0.944215i	$-0.393177\pi$
$829$	546.029			0.658659			0.329330	+	0.944215i	$0.393177\pi$
$830$		0			0
$831$		0			0
$832$	616.325	−	105.568i	0.740776	−	0.126885i
$833$	401.605			0.482119
$834$		0			0
$835$		0			0
$836$	−319.723	−	487.846i	−0.382444	−	0.583548i
$837$		0			0
$838$	−1219.15	−	658.645i	−1.45484	−	0.785973i
$839$		−	279.613i		−	0.333269i	−0.986019	−	0.166634i	$-0.946710\pi$
$839$		−	279.613i		−	0.333269i	0.986019	−	0.166634i	$-0.0532900\pi$
$840$		0			0
$841$	1824.65			2.16963
$842$	−310.499	+	574.733i	−0.368763	+	0.682581i
$843$		0			0
$844$	−595.362	+	390.186i	−0.705406	+	0.462306i
$845$		0			0
$846$		0			0
$847$			201.648i			0.238073i
$848$	135.427	−	311.160i	0.159702	−	0.366934i
$849$		0			0
$850$		0			0
$851$			58.0755i			0.0682438i
$852$		0			0
$853$	422.173			0.494927			0.247463	−	0.968897i	$-0.420403\pi$
$853$	422.173			0.494927			0.247463	+	0.968897i	$0.420403\pi$
$854$	−494.196	−	266.989i	−0.578684	−	0.312633i
$855$		0			0
$856$	70.6363	+	830.780i	0.0825190	+	0.970537i
$857$	1609.18			1.87768			0.938842	−	0.344347i	$-0.111900\pi$
$857$	1609.18			1.87768			0.938842	+	0.344347i	$0.111900\pi$
$858$		0			0
$859$			1146.70i			1.33493i	0.744643	+	0.667463i	$0.232620\pi$
$859$			1146.70i			1.33493i	−0.744643	+	0.667463i	$0.767380\pi$
$860$		0			0
$861$		0			0
$862$	−646.502	−	349.272i	−0.750003	−	0.405188i
$863$		−	1318.54i		−	1.52786i	−0.645300	−	0.763929i	$-0.723268\pi$
$863$		−	1318.54i		−	1.52786i	0.645300	−	0.763929i	$-0.276732\pi$
$864$		0			0
$865$		0			0
$866$	−469.255	+	868.591i	−0.541865	+	1.00299i
$867$		0			0
$868$	269.037	+	410.508i	0.309951	+	0.472935i
$869$	406.053			0.467265
$870$		0			0
$871$		−	926.612i		−	1.06385i
$872$	−485.063	+	41.2420i	−0.556265	+	0.0472959i
$873$		0			0
$874$	49.0813	−	90.8496i	0.0561571	−	0.103947i
$875$		0			0
$876$		0			0
$877$	1194.61			1.36216			0.681079	−	0.732210i	$-0.261511\pi$
$877$	1194.61			1.36216			0.681079	+	0.732210i	$0.261511\pi$
$878$	−1159.54	−	626.440i	−1.32066	−	0.713485i
$879$		0			0
$880$		0			0
$881$	1008.88			1.14516			0.572579	−	0.819850i	$-0.305943\pi$
$881$	1008.88			1.14516			0.572579	+	0.819850i	$0.305943\pi$
$882$		0			0
$883$			210.869i			0.238809i	0.992846	+	0.119405i	$0.0380986\pi$
$883$			210.869i			0.238809i	−0.992846	+	0.119405i	$0.961901\pi$
$884$	−260.065	−	396.817i	−0.294191	−	0.448888i
$885$		0			0
$886$	−1096.68	−	592.479i	−1.23779	−	0.668712i
$887$		−	1682.25i		−	1.89657i	−0.317425	−	0.948283i	$-0.602818\pi$
$887$		−	1682.25i		−	1.89657i	0.317425	−	0.948283i	$-0.397182\pi$
$888$		0			0
$889$	−254.699			−0.286500
$890$		0			0
$891$		0			0
$892$	588.401	−	385.624i	0.659642	−	0.432314i
$893$	−686.541			−0.768803
$894$		0			0
$895$		0			0
$896$	483.851	+	163.402i	0.540012	+	0.182368i
$897$		0			0
$898$	−346.220	+	640.853i	−0.385545	+	0.713645i
$899$			1587.83i			1.76622i
$900$		0			0
$901$	−257.484			−0.285775
$902$	628.629	+	339.616i	0.696928	+	0.376514i
$903$		0			0
$904$	−1114.55	+	94.7631i	−1.23290	+	0.104826i
$905$		0			0
$906$		0			0
$907$			1198.24i			1.32111i	0.750779	+	0.660554i	$0.229678\pi$
$907$			1198.24i			1.32111i	−0.750779	+	0.660554i	$0.770322\pi$
$908$	−256.235	+	167.930i	−0.282197	+	0.184945i
$909$		0			0
$910$		0			0
$911$			647.021i			0.710232i	0.934822	+	0.355116i	$0.115559\pi$
$911$			647.021i			0.710232i	−0.934822	+	0.355116i	$0.884441\pi$
$912$		0			0
$913$	−1177.24			−1.28942
$914$	−205.210	+	379.844i	−0.224519	+	0.415584i
$915$		0			0
$916$	170.224	+	259.735i	0.185834	+	0.283553i
$917$	−821.881			−0.896272
$918$		0			0
$919$			1316.77i			1.43283i	0.697672	+	0.716417i	$0.254219\pi$
$919$			1316.77i			1.43283i	−0.697672	+	0.716417i	$0.745781\pi$
$920$		0			0
$921$		0			0
$922$	−26.5363	+	49.1187i	−0.0287812	+	0.0532740i
$923$			1293.36i			1.40126i
$924$		0			0
$925$		0			0
$926$	−650.262	−	351.303i	−0.702227	−	0.379377i
$927$		0			0
$928$	1019.44	+	1300.15i	1.09853	+	1.40102i
$929$	29.9973			0.0322898			0.0161449	−	0.999870i	$-0.494861\pi$
$929$	29.9973			0.0322898			0.0161449	+	0.999870i	$0.494861\pi$
$930$		0			0
$931$		−	574.687i		−	0.617279i
$932$	292.796	+	446.760i	0.314159	+	0.479356i
$933$		0			0
$934$	1304.66	+	704.843i	1.39686	+	0.754650i
$935$		0			0
$936$		0			0
$937$	−1023.87			−1.09271			−0.546356	−	0.837553i	$-0.683985\pi$
$937$	−1023.87			−1.09271			−0.546356	+	0.837553i	$0.683985\pi$
$938$	359.713	−	665.830i	0.383490	−	0.709840i
$939$		0			0
$940$		0			0
$941$	−746.535			−0.793343			−0.396671	−	0.917961i	$-0.629835\pi$
$941$	−746.535			−0.793343			−0.396671	+	0.917961i	$0.629835\pi$
$942$		0			0
$943$			126.491i			0.134136i
$944$	−738.876	−	321.583i	−0.782708	−	0.340660i
$945$		0			0
$946$	502.086	−	929.361i	0.530746	−	0.982411i
$947$		−	1070.22i		−	1.13011i	−0.825053	−	0.565056i	$-0.808855\pi$
$947$		−	1070.22i		−	1.13011i	0.825053	−	0.565056i	$-0.191145\pi$
$948$		0			0
$949$	759.187			0.799986
$950$		0			0
$951$		0			0
$952$	−32.8275	−	386.096i	−0.0344826	−	0.405563i
$953$	−121.117			−0.127091			−0.0635453	−	0.997979i	$-0.520241\pi$
$953$	−121.117			−0.127091			−0.0635453	+	0.997979i	$0.520241\pi$
$954$		0			0
$955$		0			0
$956$	−49.7151	+	32.5821i	−0.0520033	+	0.0340817i
$957$		0			0
$958$	−738.516	−	398.982i	−0.770894	−	0.416474i
$959$		−	969.849i		−	1.01131i
$960$		0			0
$961$	15.1868			0.0158031
$962$	181.494	−	335.946i	0.188663	−	0.349216i
$963$		0			0
$964$	−275.437	−	420.272i	−0.285723	−	0.435967i
$965$		0			0
$966$		0			0
$967$		−	1543.20i		−	1.59587i	−0.602747	−	0.797933i	$-0.705927\pi$
$967$		−	1543.20i		−	1.59587i	0.602747	−	0.797933i	$-0.294073\pi$
$968$	−402.872	+	34.2538i	−0.416190	+	0.0353861i
$969$		0			0
$970$		0			0
$971$			945.350i			0.973584i	0.873518	+	0.486792i	$0.161833\pi$
$971$			945.350i			0.973584i	−0.873518	+	0.486792i	$0.838167\pi$
$972$		0			0
$973$	488.833			0.502398
$974$	−344.010	−	185.851i	−0.353193	−	0.190812i
$975$		0			0
$976$	449.466	−	1032.70i	0.460519	−	1.05810i
$977$	−419.744			−0.429626			−0.214813	−	0.976655i	$-0.568914\pi$
$977$	−419.744			−0.429626			−0.214813	+	0.976655i	$0.568914\pi$
$978$		0			0
$979$		−	152.262i		−	0.155528i
$980$		0			0
$981$		0			0
$982$	−1190.81	−	643.335i	−1.21264	−	0.655127i
$983$			645.254i			0.656413i	0.944606	+	0.328206i	$0.106444\pi$
$983$			645.254i			0.656413i	−0.944606	+	0.328206i	$0.893556\pi$
$984$		0			0
$985$		0			0
$986$	595.844	−	1102.91i	0.604304	−	1.11857i
$987$		0			0
$988$	−567.836	+	372.146i	−0.574733	+	0.376666i
$989$	187.003			0.189083
$990$		0			0
$991$		−	994.611i		−	1.00364i	−0.864971	−	0.501822i	$-0.832663\pi$
$991$		−	994.611i		−	1.00364i	0.864971	−	0.501822i	$-0.167337\pi$
$992$	−774.450	+	607.240i	−0.780695	+	0.612137i
$993$		0			0
$994$	−502.086	+	929.361i	−0.505116	+	0.934971i
$995$		0			0
$996$		0			0
$997$	−1027.03			−1.03012			−0.515062	−	0.857153i	$-0.672231\pi$
$997$	−1027.03			−1.03012			−0.515062	+	0.857153i	$0.672231\pi$
$998$	1105.29	+	597.132i	1.10751	+	0.598328i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.3.c.p.451.5	yes	8
3.2	odd	2	inner	900.3.c.p.451.4	yes	8
4.3	odd	2	inner	900.3.c.p.451.6	yes	8
5.2	odd	4		900.3.f.g.199.15		16
5.3	odd	4		900.3.f.g.199.2		16
5.4	even	2		900.3.c.q.451.4	yes	8
12.11	even	2	inner	900.3.c.p.451.3	✓	8
15.2	even	4		900.3.f.g.199.1		16
15.8	even	4		900.3.f.g.199.16		16
15.14	odd	2		900.3.c.q.451.5	yes	8
20.3	even	4		900.3.f.g.199.13		16
20.7	even	4		900.3.f.g.199.4		16
20.19	odd	2		900.3.c.q.451.3	yes	8
60.23	odd	4		900.3.f.g.199.3		16
60.47	odd	4		900.3.f.g.199.14		16
60.59	even	2		900.3.c.q.451.6	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
900.3.c.p.451.3	✓	8	12.11	even	2	inner
900.3.c.p.451.4	yes	8	3.2	odd	2	inner
900.3.c.p.451.5	yes	8	1.1	even	1	trivial
900.3.c.p.451.6	yes	8	4.3	odd	2	inner
900.3.c.q.451.3	yes	8	20.19	odd	2
900.3.c.q.451.4	yes	8	5.4	even	2
900.3.c.q.451.5	yes	8	15.14	odd	2
900.3.c.q.451.6	yes	8	60.59	even	2
900.3.f.g.199.1		16	15.2	even	4
900.3.f.g.199.2		16	5.3	odd	4
900.3.f.g.199.3		16	60.23	odd	4
900.3.f.g.199.4		16	20.7	even	4
900.3.f.g.199.13		16	20.3	even	4
900.3.f.g.199.14		16	60.47	odd	4
900.3.f.g.199.15		16	5.2	odd	4
900.3.f.g.199.16		16	15.8	even	4

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:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	900.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.950636 - 1.75963i) q^{2} +(-2.19258 - 3.34553i) q^{4} +3.98982i q^{7} +(-7.97124 + 0.677747i) q^{8} +O(q^{10})\)	\(q+(0.950636 - 1.75963i) q^{2} +(-2.19258 - 3.34553i) q^{4} +3.98982i q^{7} +(-7.97124 + 0.677747i) q^{8} +8.39401i q^{11} +9.77033 q^{13} +(7.02060 + 3.79287i) q^{14} +(-6.38516 + 14.6707i) q^{16} +12.1399 q^{17} -17.3719i q^{19} +(14.7703 + 7.97964i) q^{22} +2.97203i q^{23} +(9.28803 - 17.1921i) q^{26} +(13.3481 - 8.74801i) q^{28} +51.6300 q^{29} +30.7541i q^{31} +(19.7450 + 25.1820i) q^{32} +(11.5407 - 21.3618i) q^{34} +19.5407 q^{37} +(-30.5682 - 16.5144i) q^{38} +42.5603 q^{41} -62.9208i q^{43} +(28.0824 - 18.4046i) q^{44} +(5.22967 + 2.82532i) q^{46} -39.5201i q^{47} +33.0813 q^{49} +(-21.4223 - 32.6869i) q^{52} -21.2096 q^{53} +(-2.70409 - 31.8038i) q^{56} +(49.0813 - 90.8496i) q^{58} +50.3640i q^{59} -70.3923 q^{61} +(54.1157 + 29.2359i) q^{62} +(63.0813 - 10.8050i) q^{64} -94.8394i q^{67} +(-26.6178 - 40.6145i) q^{68} +132.376i q^{71} +77.7033 q^{73} +(18.5761 - 34.3843i) q^{74} +(-58.1184 + 38.0894i) q^{76} -33.4906 q^{77} -48.3742i q^{79} +(40.4593 - 74.8903i) q^{82} +140.248i q^{83} +(-110.717 - 59.8148i) q^{86} +(-5.68901 - 66.9106i) q^{88} -18.1394 q^{89} +38.9819i q^{91} +(9.94303 - 6.51642i) q^{92} +(-69.5407 - 37.5692i) q^{94} -15.0000 q^{97} +(31.4483 - 58.2108i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{4}+O(q^{10}) \) 8 * q + 4 * q^4	\( 8 q + 4 q^{4} - 8 q^{13} - 8 q^{16} + 32 q^{22} - 44 q^{28} - 80 q^{34} - 16 q^{37} + 128 q^{46} - 80 q^{49} - 236 q^{52} + 48 q^{58} + 40 q^{61} + 160 q^{64} - 240 q^{73} - 228 q^{76} + 496 q^{82} - 304 q^{88} - 384 q^{94} - 120 q^{97}+O(q^{100}) \) 8 * q + 4 * q^4 - 8 * q^13 - 8 * q^16 + 32 * q^22 - 44 * q^28 - 80 * q^34 - 16 * q^37 + 128 * q^46 - 80 * q^49 - 236 * q^52 + 48 * q^58 + 40 * q^61 + 160 * q^64 - 240 * q^73 - 228 * q^76 + 496 * q^82 - 304 * q^88 - 384 * q^94 - 120 * q^97

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