LMFDB - Embedded newform 405.4.b.e.244.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,4,Mod(244,405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("405.244");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	405.b (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:	$23.8957735523$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 91 x^{14} + 3268 x^{12} + 59128 x^{10} + 571975 x^{8} + 2881141 x^{6} + 6555196 x^{4} + \cdots + 614656 $ x^16 + 91x^14 + 3268x^12 + 59128x^10 + 571975x^8 + 2881141x^6 + 6555196x^4 + 4069504*x^2 + 614656
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{12}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			244.15
Root			$5.02371i$ of defining polynomial
Character	$\chi$	$=$	405.244
Dual form			405.4.b.e.244.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.02371i q^{2} -17.2377 q^{4} +(5.75603 + 9.58479i) q^{5} +5.38197i q^{7} -46.4074i q^{8} +O(q^{10})$	\(q+5.02371i q^{2} -17.2377 q^{4} +(5.75603 + 9.58479i) q^{5} +5.38197i q^{7} -46.4074i q^{8} +(-48.1512 + 28.9166i) q^{10} -39.1018 q^{11} +86.6271i q^{13} -27.0375 q^{14} +95.2360 q^{16} -15.4194i q^{17} +26.8412 q^{19} +(-99.2205 - 165.219i) q^{20} -196.436i q^{22} -111.138i q^{23} +(-58.7363 + 110.341i) q^{25} -435.189 q^{26} -92.7727i q^{28} -49.2347 q^{29} -179.877 q^{31} +107.179i q^{32} +77.4624 q^{34} +(-51.5851 + 30.9788i) q^{35} -293.496i q^{37} +134.842i q^{38} +(444.805 - 267.122i) q^{40} +27.6032 q^{41} +60.5030i q^{43} +674.023 q^{44} +558.327 q^{46} -96.2351i q^{47} +314.034 q^{49} +(-554.319 - 295.074i) q^{50} -1493.25i q^{52} +251.203i q^{53} +(-225.071 - 374.782i) q^{55} +249.763 q^{56} -247.341i q^{58} +76.8462 q^{59} +490.091 q^{61} -903.648i q^{62} +223.452 q^{64} +(-830.302 + 498.628i) q^{65} +238.721i q^{67} +265.794i q^{68} +(-155.628 - 259.148i) q^{70} -640.447 q^{71} +769.257i q^{73} +1474.44 q^{74} -462.679 q^{76} -210.445i q^{77} -662.352 q^{79} +(548.181 + 912.817i) q^{80} +138.670i q^{82} -1302.14i q^{83} +(147.791 - 88.7543i) q^{85} -303.950 q^{86} +1814.61i q^{88} +995.544 q^{89} -466.224 q^{91} +1915.77i q^{92} +483.457 q^{94} +(154.499 + 257.267i) q^{95} +814.846i q^{97} +1577.62i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 54 q^{4} - 3 q^{5}+O(q^{10}) $ 16 * q - 54 * q^4 - 3 * q^5	$ 16 q - 54 q^{4} - 3 q^{5} - 10 q^{10} - 90 q^{11} + 102 q^{14} + 146 q^{16} - 4 q^{19} + 6 q^{20} - 71 q^{25} - 468 q^{26} + 516 q^{29} + 38 q^{31} - 212 q^{34} - 267 q^{35} - 44 q^{40} - 576 q^{41} + 1644 q^{44} - 290 q^{46} + 4 q^{49} - 558 q^{50} + 15 q^{55} - 2430 q^{56} + 2202 q^{59} + 20 q^{61} + 322 q^{64} - 339 q^{65} - 636 q^{70} - 2952 q^{71} + 4080 q^{74} - 396 q^{76} + 218 q^{79} + 1266 q^{80} + 704 q^{85} - 6108 q^{86} + 4074 q^{89} - 942 q^{91} + 1078 q^{94} + 1692 q^{95}+O(q^{100}) $ 16 * q - 54 * q^4 - 3 * q^5 - 10 * q^10 - 90 * q^11 + 102 * q^14 + 146 * q^16 - 4 * q^19 + 6 * q^20 - 71 * q^25 - 468 * q^26 + 516 * q^29 + 38 * q^31 - 212 * q^34 - 267 * q^35 - 44 * q^40 - 576 * q^41 + 1644 * q^44 - 290 * q^46 + 4 * q^49 - 558 * q^50 + 15 * q^55 - 2430 * q^56 + 2202 * q^59 + 20 * q^61 + 322 * q^64 - 339 * q^65 - 636 * q^70 - 2952 * q^71 + 4080 * q^74 - 396 * q^76 + 218 * q^79 + 1266 * q^80 + 704 * q^85 - 6108 * q^86 + 4074 * q^89 - 942 * q^91 + 1078 * q^94 + 1692 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/405\mathbb{Z}\right)^\times$.

$n$	$82$	$326$
$\chi(n)$	$-1$	$1$

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 $

$2$			5.02371i			1.77615i	0.459699	+	0.888075i	$0.347957\pi$
$2$			5.02371i			1.77615i	−0.459699	+	0.888075i	$0.652043\pi$
$3$		0			0
$3$		0			0
$4$	−17.2377			−2.15471
$5$	5.75603	+	9.58479i	0.514835	+	0.857289i
$5$	5.75603	+	9.58479i	0.514835	+	0.857289i
$6$		0			0
$7$			5.38197i			0.290599i	0.989388	+	0.145300i	$0.0464146\pi$
$7$			5.38197i			0.290599i	−0.989388	+	0.145300i	$0.953585\pi$
$8$		−	46.4074i		−	2.05094i
$9$		0			0
$10$	−48.1512	+	28.9166i	−1.52267	+	0.914424i
$11$	−39.1018			−1.07178			−0.535892	−	0.844286i	$-0.680025\pi$
$11$	−39.1018			−1.07178			−0.535892	+	0.844286i	$0.680025\pi$
$12$		0			0
$13$			86.6271i			1.84816i	0.382204	+	0.924078i	$0.375165\pi$
$13$			86.6271i			1.84816i	−0.382204	+	0.924078i	$0.624835\pi$
$14$	−27.0375			−0.516148
$15$		0			0
$16$	95.2360			1.48806
$17$		−	15.4194i		−	0.219985i	−0.993932	−	0.109993i	$-0.964917\pi$
$17$		−	15.4194i		−	0.219985i	0.993932	−	0.109993i	$-0.0350827\pi$
$18$		0			0
$19$	26.8412			0.324094			0.162047	−	0.986783i	$-0.448190\pi$
$19$	26.8412			0.324094			0.162047	+	0.986783i	$0.448190\pi$
$20$	−99.2205	−	165.219i	−1.10932	−	1.84721i
$21$		0			0
$22$		−	196.436i		−	1.90365i
$23$		−	111.138i		−	1.00756i	−0.863831	−	0.503781i	$-0.831942\pi$
$23$		−	111.138i		−	1.00756i	0.863831	−	0.503781i	$-0.168058\pi$
$24$		0			0
$25$	−58.7363	+	110.341i	−0.469890	+	0.882725i
$26$	−435.189			−3.28260
$27$		0			0
$28$		−	92.7727i		−	0.626157i
$29$	−49.2347			−0.315264			−0.157632	−	0.987498i	$-0.550386\pi$
$29$	−49.2347			−0.315264			−0.157632	+	0.987498i	$0.550386\pi$
$30$		0			0
$31$	−179.877			−1.04215			−0.521077	−	0.853510i	$-0.674470\pi$
$31$	−179.877			−1.04215			−0.521077	+	0.853510i	$0.674470\pi$
$32$			107.179i			0.592085i
$33$		0			0
$34$	77.4624			0.390726
$35$	−51.5851	+	30.9788i	−0.249128	+	0.149611i
$36$		0			0
$37$		−	293.496i		−	1.30407i	−0.758191	−	0.652033i	$-0.773916\pi$
$37$		−	293.496i		−	1.30407i	0.758191	−	0.652033i	$-0.226084\pi$
$38$			134.842i			0.575640i
$39$		0			0
$40$	444.805	−	267.122i	1.75825	−	1.05589i
$41$	27.6032			0.105144			0.0525719	−	0.998617i	$-0.483258\pi$
$41$	27.6032			0.105144			0.0525719	+	0.998617i	$0.483258\pi$
$42$		0			0
$43$			60.5030i			0.214573i	0.994228	+	0.107286i	$0.0342162\pi$
$43$			60.5030i			0.214573i	−0.994228	+	0.107286i	$0.965784\pi$
$44$	674.023			2.30938
$45$		0			0
$46$	558.327			1.78958
$47$		−	96.2351i		−	0.298667i	−0.988787	−	0.149333i	$-0.952287\pi$
$47$		−	96.2351i		−	0.298667i	0.988787	−	0.149333i	$-0.0477127\pi$
$48$		0			0
$49$	314.034			0.915552
$50$	−554.319	−	295.074i	−1.56785	−	0.834595i
$51$		0			0
$52$		−	1493.25i		−	3.98224i
$53$			251.203i			0.651046i	0.945534	+	0.325523i	$0.105540\pi$
$53$			251.203i			0.651046i	−0.945534	+	0.325523i	$0.894460\pi$
$54$		0			0
$55$	−225.071	−	374.782i	−0.551792	−	0.918829i
$56$	249.763			0.596001
$57$		0			0
$58$		−	247.341i		−	0.559957i
$59$	76.8462			0.169568			0.0847841	−	0.996399i	$-0.472980\pi$
$59$	76.8462			0.169568			0.0847841	+	0.996399i	$0.472980\pi$
$60$		0			0
$61$	490.091			1.02868			0.514342	−	0.857585i	$-0.328036\pi$
$61$	490.091			1.02868			0.514342	+	0.857585i	$0.328036\pi$
$62$		−	903.648i		−	1.85102i
$63$		0			0
$64$	223.452			0.436430
$65$	−830.302	+	498.628i	−1.58440	+	0.951495i
$66$		0			0
$67$			238.721i			0.435290i	0.976028	+	0.217645i	$0.0698374\pi$
$67$			238.721i			0.435290i	−0.976028	+	0.217645i	$0.930163\pi$
$68$			265.794i			0.474004i
$69$		0			0
$70$	−155.628	−	259.148i	−0.265731	−	0.442488i
$71$	−640.447			−1.07052			−0.535261	−	0.844687i	$-0.679787\pi$
$71$	−640.447			−1.07052			−0.535261	+	0.844687i	$0.679787\pi$
$72$		0			0
$73$			769.257i			1.23335i	0.787217	+	0.616676i	$0.211521\pi$
$73$			769.257i			1.23335i	−0.787217	+	0.616676i	$0.788479\pi$
$74$	1474.44			2.31622
$75$		0			0
$76$	−462.679			−0.698328
$77$		−	210.445i		−	0.311460i
$78$		0			0
$79$	−662.352			−0.943296			−0.471648	−	0.881787i	$-0.656341\pi$
$79$	−662.352			−0.943296			−0.471648	+	0.881787i	$0.656341\pi$
$80$	548.181	+	912.817i	0.766106	+	1.27570i
$81$		0			0
$82$			138.670i			0.186751i
$83$		−	1302.14i		−	1.72203i	−0.508578	−	0.861016i	$-0.669829\pi$
$83$		−	1302.14i		−	1.72203i	0.508578	−	0.861016i	$-0.330171\pi$
$84$		0			0
$85$	147.791	−	88.7543i	0.188591	−	0.113256i
$86$	−303.950			−0.381113
$87$		0			0
$88$			1814.61i			2.19816i
$89$	995.544			1.18570			0.592851	−	0.805312i	$-0.298002\pi$
$89$	995.544			1.18570			0.592851	+	0.805312i	$0.298002\pi$
$90$		0			0
$91$	−466.224			−0.537073
$92$			1915.77i			2.17100i
$93$		0			0
$94$	483.457			0.530477
$95$	154.499	+	257.267i	0.166855	+	0.277842i
$96$		0			0
$97$			814.846i			0.852939i	0.904502	+	0.426470i	$0.140243\pi$
$97$			814.846i			0.852939i	−0.904502	+	0.426470i	$0.859757\pi$
$98$			1577.62i			1.62616i
$99$		0			0
$100$	1012.48	−	1902.02i	1.01248	−	1.90202i
$101$	−652.604			−0.642936			−0.321468	−	0.946920i	$-0.604176\pi$
$101$	−652.604			−0.642936			−0.321468	+	0.946920i	$0.604176\pi$
$102$		0			0
$103$			153.955i			0.147278i	0.997285	+	0.0736391i	$0.0234613\pi$
$103$			153.955i			0.147278i	−0.997285	+	0.0736391i	$0.976539\pi$
$104$	4020.14			3.79045
$105$		0			0
$106$	−1261.97			−1.15636
$107$		−	348.878i		−	0.315209i	−0.987502	−	0.157604i	$-0.949623\pi$
$107$		−	348.878i		−	0.315209i	0.987502	−	0.157604i	$-0.0503770\pi$
$108$		0			0
$109$	−1125.26			−0.988815			−0.494407	−	0.869230i	$-0.664615\pi$
$109$	−1125.26			−0.988815			−0.494407	+	0.869230i	$0.664615\pi$
$110$	1882.80	−	1130.69i	1.63198	−	0.980065i
$111$		0			0
$112$			512.557i			0.432430i
$113$			352.076i			0.293102i	0.989203	+	0.146551i	$0.0468173\pi$
$113$			352.076i			0.293102i	−0.989203	+	0.146551i	$0.953183\pi$
$114$		0			0
$115$	1065.24	−	639.715i	0.863773	−	0.518728i
$116$	848.692			0.679303
$117$		0			0
$118$			386.053i			0.301179i
$119$	82.9866			0.0639275
$120$		0			0
$121$	197.948			0.148721
$122$			2462.08i			1.82710i
$123$		0			0
$124$	3100.65			2.24554
$125$	−1395.68	+	72.1493i	−0.998666	+	0.0516259i
$126$		0			0
$127$			1502.08i			1.04951i	0.851252	+	0.524757i	$0.175844\pi$
$127$			1502.08i			1.04951i	−0.851252	+	0.524757i	$0.824156\pi$
$128$			1979.99i			1.36725i
$129$		0			0
$130$	−2504.96	−	4171.20i	−1.69000	−	2.81414i
$131$	−1574.71			−1.05025			−0.525127	−	0.851024i	$-0.675982\pi$
$131$	−1574.71			−1.05025			−0.525127	+	0.851024i	$0.675982\pi$
$132$		0			0
$133$			144.458i			0.0941814i
$134$	−1199.26			−0.773140
$135$		0			0
$136$	−715.573			−0.451175
$137$		−	892.298i		−	0.556454i	−0.960515	−	0.278227i	$-0.910253\pi$
$137$		−	892.298i		−	0.556454i	0.960515	−	0.278227i	$-0.0897467\pi$
$138$		0			0
$139$	−652.706			−0.398287			−0.199143	−	0.979970i	$-0.563816\pi$
$139$	−652.706			−0.398287			−0.199143	+	0.979970i	$0.563816\pi$
$140$	889.206	−	534.002i	0.536797	−	0.322367i
$141$		0			0
$142$		−	3217.42i		−	1.90141i
$143$		−	3387.27i		−	1.98082i
$144$		0			0
$145$	−283.397	−	471.905i	−0.162309	−	0.270273i
$146$	−3864.52			−2.19062
$147$		0			0
$148$			5059.19i			2.80988i
$149$	−3437.85			−1.89020			−0.945101	−	0.326779i	$-0.894037\pi$
$149$	−3437.85			−1.89020			−0.945101	+	0.326779i	$0.894037\pi$
$150$		0			0
$151$	1670.92			0.900513			0.450256	−	0.892899i	$-0.351333\pi$
$151$	1670.92			0.900513			0.450256	+	0.892899i	$0.351333\pi$
$152$		−	1245.63i		−	0.664696i
$153$		0			0
$154$	1057.21			0.553199
$155$	−1035.37	−	1724.08i	−0.536537	−	0.893428i
$156$		0			0
$157$			666.105i			0.338605i	0.985564	+	0.169302i	$0.0541515\pi$
$157$			666.105i			0.338605i	−0.985564	+	0.169302i	$0.945849\pi$
$158$		−	3327.46i		−	1.67544i
$159$		0			0
$160$	−1027.29	+	616.925i	−0.507588	+	0.304826i
$161$	598.143			0.292797
$162$		0			0
$163$			889.255i			0.427312i	0.976909	+	0.213656i	$0.0685371\pi$
$163$			889.255i			0.427312i	−0.976909	+	0.213656i	$0.931463\pi$
$164$	−475.815			−0.226554
$165$		0			0
$166$	6541.59			3.05859
$167$		−	403.862i		−	0.187137i	−0.995613	−	0.0935683i	$-0.970173\pi$
$167$		−	403.862i		−	0.187137i	0.995613	−	0.0935683i	$-0.0298273\pi$
$168$		0			0
$169$	−5307.25			−2.41568
$170$	445.876	+	742.461i	0.201160	+	0.334966i
$171$		0			0
$172$		−	1042.93i		−	0.462342i
$173$			236.061i			0.103742i	0.998654	+	0.0518712i	$0.0165185\pi$
$173$			236.061i			0.103742i	−0.998654	+	0.0518712i	$0.983481\pi$
$174$		0			0
$175$	−593.850	−	316.117i	−0.256519	−	0.136550i
$176$	−3723.90			−1.59488
$177$		0			0
$178$			5001.33i			2.10599i
$179$	−2404.31			−1.00395			−0.501973	−	0.864883i	$-0.667392\pi$
$179$	−2404.31			−1.00395			−0.501973	+	0.864883i	$0.667392\pi$
$180$		0			0
$181$	−1218.41			−0.500354			−0.250177	−	0.968200i	$-0.580489\pi$
$181$	−1218.41			−0.500354			−0.250177	+	0.968200i	$0.580489\pi$
$182$		−	2342.18i		−	0.953921i
$183$		0			0
$184$	−5157.64			−2.06645
$185$	2813.10	−	1689.37i	1.11796	−	0.671378i
$186$		0			0
$187$			602.925i			0.235777i
$188$			1658.87i			0.643539i
$189$		0			0
$190$	−1292.43	+	776.156i	−0.493490	+	0.296359i
$191$	175.943			0.0666533			0.0333267	−	0.999445i	$-0.489390\pi$
$191$	175.943			0.0666533			0.0333267	+	0.999445i	$0.489390\pi$
$192$		0			0
$193$			2215.26i			0.826207i	0.910684	+	0.413103i	$0.135555\pi$
$193$			2215.26i			0.826207i	−0.910684	+	0.413103i	$0.864445\pi$
$194$	−4093.55			−1.51495
$195$		0			0
$196$	−5413.22			−1.97275
$197$			1054.08i			0.381218i	0.981666	+	0.190609i	$0.0610462\pi$
$197$			1054.08i			0.381218i	−0.981666	+	0.190609i	$0.938954\pi$
$198$		0			0
$199$	3484.04			1.24109			0.620546	−	0.784170i	$-0.286911\pi$
$199$	3484.04			1.24109			0.620546	+	0.784170i	$0.286911\pi$
$200$	5120.62	+	2725.80i	1.81041	+	0.963715i
$201$		0			0
$202$		−	3278.49i		−	1.14195i
$203$		−	264.980i		−	0.0916155i
$204$		0			0
$205$	158.885	+	264.571i	0.0541316	+	0.0901386i
$206$	−773.426			−0.261588
$207$		0			0
$208$			8250.01i			2.75017i
$209$	−1049.54			−0.347359
$210$		0			0
$211$	−4196.90			−1.36932			−0.684660	−	0.728863i	$-0.740049\pi$
$211$	−4196.90			−1.36932			−0.684660	+	0.728863i	$0.740049\pi$
$212$		−	4330.16i		−	1.40282i
$213$		0			0
$214$	1752.66			0.559858
$215$	−579.908	+	348.257i	−0.183951	+	0.110469i
$216$		0			0
$217$		−	968.091i		−	0.302849i
$218$		−	5653.01i		−	1.75628i
$219$		0			0
$220$	3879.70	+	6460.37i	1.18895	+	1.97981i
$221$	1335.73			0.406567
$222$		0			0
$223$			3034.56i			0.911251i	0.890172	+	0.455625i	$0.150584\pi$
$223$			3034.56i			0.911251i	−0.890172	+	0.455625i	$0.849416\pi$
$224$	−576.834			−0.172059
$225$		0			0
$226$	−1768.73			−0.520593
$227$			3236.16i			0.946219i	0.881004	+	0.473109i	$0.156868\pi$
$227$			3236.16i			0.946219i	−0.881004	+	0.473109i	$0.843132\pi$
$228$		0			0
$229$	2974.56			0.858360			0.429180	−	0.903219i	$-0.358803\pi$
$229$	2974.56			0.858360			0.429180	+	0.903219i	$0.358803\pi$
$230$	3213.75	+	5351.44i	0.921340	+	1.53419i
$231$		0			0
$232$			2284.86i			0.646587i
$233$			2927.42i			0.823097i	0.911388	+	0.411549i	$0.135012\pi$
$233$			2927.42i			0.823097i	−0.911388	+	0.411549i	$0.864988\pi$
$234$		0			0
$235$	922.392	−	553.932i	0.256044	−	0.153764i
$236$	−1324.65			−0.365370
$237$		0			0
$238$			416.901i			0.113545i
$239$	−2878.69			−0.779110			−0.389555	−	0.921003i	$-0.627371\pi$
$239$	−2878.69			−0.779110			−0.389555	+	0.921003i	$0.627371\pi$
$240$		0			0
$241$	−1117.58			−0.298713			−0.149357	−	0.988783i	$-0.547720\pi$
$241$	−1117.58			−0.298713			−0.149357	+	0.988783i	$0.547720\pi$
$242$			994.435i			0.264152i
$243$		0			0
$244$	−8448.03			−2.21651
$245$	1807.59	+	3009.95i	0.471358	+	0.784893i
$246$		0			0
$247$			2325.17i			0.598976i
$248$			8347.60i			2.13739i
$249$		0			0
$250$	−362.457	−	7011.49i	−0.0916953	−	1.77378i
$251$	−4612.00			−1.15979			−0.579894	−	0.814692i	$-0.696906\pi$
$251$	−4612.00			−1.15979			−0.579894	+	0.814692i	$0.696906\pi$
$252$		0			0
$253$			4345.71i			1.07989i
$254$	−7546.04			−1.86410
$255$		0			0
$256$	−8159.28			−1.99201
$257$		−	3787.72i		−	0.919345i	−0.888089	−	0.459672i	$-0.847967\pi$
$257$		−	3787.72i		−	0.919345i	0.888089	−	0.459672i	$-0.152033\pi$
$258$		0			0
$259$	1579.59			0.378960
$260$	14312.5	−	8595.18i	3.41393	−	2.05020i
$261$		0			0
$262$		−	7910.89i		−	1.86541i
$263$			4989.56i			1.16985i	0.811089	+	0.584923i	$0.198875\pi$
$263$			4989.56i			1.16985i	−0.811089	+	0.584923i	$0.801125\pi$
$264$		0			0
$265$	−2407.73	+	1445.93i	−0.558135	+	0.335181i
$266$	−725.717			−0.167280
$267$		0			0
$268$		−	4114.99i		−	0.937922i
$269$	5845.57			1.32495			0.662473	−	0.749086i	$-0.269507\pi$
$269$	5845.57			1.32495			0.662473	+	0.749086i	$0.269507\pi$
$270$		0			0
$271$	2766.09			0.620030			0.310015	−	0.950732i	$-0.399666\pi$
$271$	2766.09			0.620030			0.310015	+	0.950732i	$0.399666\pi$
$272$		−	1468.48i		−	0.327351i
$273$		0			0
$274$	4482.65			0.988346
$275$	2296.69	−	4314.51i	0.503621	−	0.946091i
$276$		0			0
$277$			1893.80i			0.410785i	0.978680	+	0.205393i	$0.0658471\pi$
$277$			1893.80i			0.410785i	−0.978680	+	0.205393i	$0.934153\pi$
$278$		−	3279.01i		−	0.707417i
$279$		0			0
$280$	1437.65	+	2393.93i	0.306842	+	0.510945i
$281$	4484.38			0.952012			0.476006	−	0.879442i	$-0.342084\pi$
$281$	4484.38			0.952012			0.476006	+	0.879442i	$0.342084\pi$
$282$		0			0
$283$		−	5797.64i		−	1.21779i	−0.793251	−	0.608894i	$-0.791613\pi$
$283$		−	5797.64i		−	1.21779i	0.793251	−	0.608894i	$-0.208387\pi$
$284$	11039.8			2.30666
$285$		0			0
$286$	17016.7			3.51824
$287$			148.560i			0.0305547i
$288$		0			0
$289$	4675.24			0.951607
$290$	2370.71	−	1423.70i	0.480045	−	0.288285i
$291$		0			0
$292$		−	13260.2i		−	2.65752i
$293$			7910.26i			1.57721i	0.614900	+	0.788605i	$0.289196\pi$
$293$			7910.26i			1.57721i	−0.614900	+	0.788605i	$0.710804\pi$
$294$		0			0
$295$	442.329	+	736.554i	0.0872996	+	0.145369i
$296$	−13620.4			−2.67456
$297$		0			0
$298$		−	17270.8i		−	3.35728i
$299$	9627.59			1.86213
$300$		0			0
$301$	−325.626			−0.0623546
$302$			8394.21i			1.59945i
$303$		0			0
$304$	2556.24			0.482272
$305$	2820.98	+	4697.42i	0.529602	+	0.881880i
$306$		0			0
$307$			4426.37i			0.822887i	0.911435	+	0.411443i	$0.134975\pi$
$307$			4426.37i			0.822887i	−0.911435	+	0.411443i	$0.865025\pi$
$308$			3627.58i			0.671105i
$309$		0			0
$310$	8661.27	−	5201.42i	1.58686	−	0.952971i
$311$	9697.24			1.76810			0.884052	−	0.467389i	$-0.154805\pi$
$311$	9697.24			1.76810			0.884052	+	0.467389i	$0.154805\pi$
$312$		0			0
$313$		−	4523.09i		−	0.816806i	−0.912802	−	0.408403i	$-0.866086\pi$
$313$		−	4523.09i		−	0.816806i	0.912802	−	0.408403i	$-0.133914\pi$
$314$	−3346.32			−0.601413
$315$		0			0
$316$	11417.4			2.03253
$317$		−	5322.21i		−	0.942981i	−0.881871	−	0.471491i	$-0.843716\pi$
$317$		−	5322.21i		−	0.942981i	0.881871	−	0.471491i	$-0.156284\pi$
$318$		0			0
$319$	1925.17			0.337895
$320$	1286.20	+	2141.74i	0.224689	+	0.374147i
$321$		0			0
$322$			3004.90i			0.520051i
$323$		−	413.874i		−	0.0712958i
$324$		0			0
$325$	−9558.48	−	5088.15i	−1.63141	−	0.868430i
$326$	−4467.36			−0.758970
$327$		0			0
$328$		−	1280.99i		−	0.215643i
$329$	517.934			0.0867922
$330$		0			0
$331$	−7833.11			−1.30075			−0.650373	−	0.759615i	$-0.725387\pi$
$331$	−7833.11			−1.30075			−0.650373	+	0.759615i	$0.725387\pi$
$332$			22445.9i			3.71048i
$333$		0			0
$334$	2028.89			0.332383
$335$	−2288.09	+	1374.08i	−0.373169	+	0.224102i
$336$		0			0
$337$			9928.08i			1.60480i	0.596787	+	0.802399i	$0.296444\pi$
$337$			9928.08i			1.60480i	−0.596787	+	0.802399i	$0.703556\pi$
$338$		−	26662.1i		−	4.29061i
$339$		0			0
$340$	−2547.58	+	1529.92i	−0.406358	+	0.244034i
$341$	7033.49			1.11696
$342$		0			0
$343$			3536.14i			0.556658i
$344$	2807.79			0.440075
$345$		0			0
$346$	−1185.90			−0.184262
$347$			4040.33i			0.625062i	0.949908	+	0.312531i	$0.101177\pi$
$347$			4040.33i			0.625062i	−0.949908	+	0.312531i	$0.898823\pi$
$348$		0			0
$349$	−4799.72			−0.736169			−0.368085	−	0.929792i	$-0.619986\pi$
$349$	−4799.72			−0.736169			−0.368085	+	0.929792i	$0.619986\pi$
$350$	1588.08	−	2983.33i	0.242533	−	0.455617i
$351$		0			0
$352$		−	4190.88i		−	0.634588i
$353$		−	3189.48i		−	0.480903i	−0.970661	−	0.240451i	$-0.922705\pi$
$353$		−	3189.48i		−	0.480903i	0.970661	−	0.240451i	$-0.0772955\pi$
$354$		0			0
$355$	−3686.43	−	6138.55i	−0.551142	−	0.917747i
$356$	−17160.9			−2.55484
$357$		0			0
$358$		−	12078.5i		−	1.78316i
$359$	1159.83			0.170511			0.0852554	−	0.996359i	$-0.472829\pi$
$359$	1159.83			0.170511			0.0852554	+	0.996359i	$0.472829\pi$
$360$		0			0
$361$	−6138.55			−0.894963
$362$		−	6120.96i		−	0.888703i
$363$		0			0
$364$	8036.62			1.15724
$365$	−7373.16	+	4427.87i	−1.05734	+	0.634973i
$366$		0			0
$367$			11875.2i			1.68904i	0.535524	+	0.844520i	$0.320114\pi$
$367$			11875.2i			1.68904i	−0.535524	+	0.844520i	$0.679886\pi$
$368$		−	10584.4i		−	1.49932i
$369$		0			0
$370$	8486.91	+	14132.2i	1.19247	+	1.98567i
$371$	−1351.97			−0.189194
$372$		0			0
$373$			2643.26i			0.366924i	0.983027	+	0.183462i	$0.0587305\pi$
$373$			2643.26i			0.366924i	−0.983027	+	0.183462i	$0.941270\pi$
$374$	−3028.92			−0.418774
$375$		0			0
$376$	−4466.02			−0.612546
$377$		−	4265.06i		−	0.582657i
$378$		0			0
$379$	−10452.7			−1.41667			−0.708335	−	0.705876i	$-0.750553\pi$
$379$	−10452.7			−1.41667			−0.708335	+	0.705876i	$0.750553\pi$
$380$	−2663.19	−	4434.68i	−0.359524	−	0.598669i
$381$		0			0
$382$			883.887i			0.118386i
$383$			10879.8i			1.45153i	0.687945	+	0.725763i	$0.258513\pi$
$383$			10879.8i			1.45153i	−0.687945	+	0.725763i	$0.741487\pi$
$384$		0			0
$385$	2017.07	−	1211.33i	0.267011	−	0.160350i
$386$	−11128.8			−1.46747
$387$		0			0
$388$		−	14046.1i		−	1.83784i
$389$	−1677.66			−0.218665			−0.109333	−	0.994005i	$-0.534871\pi$
$389$	−1677.66			−0.218665			−0.109333	+	0.994005i	$0.534871\pi$
$390$		0			0
$391$	−1713.68			−0.221649
$392$		−	14573.5i		−	1.87774i
$393$		0			0
$394$	−5295.38			−0.677100
$395$	−3812.52	−	6348.50i	−0.485642	−	0.808678i
$396$		0			0
$397$			13147.5i			1.66210i	0.556199	+	0.831049i	$0.312259\pi$
$397$			13147.5i			1.66210i	−0.556199	+	0.831049i	$0.687741\pi$
$398$			17502.8i			2.20437i
$399$		0			0
$400$	−5593.81	+	10508.4i	−0.699226	+	1.31355i
$401$	−6521.91			−0.812191			−0.406095	−	0.913831i	$-0.633110\pi$
$401$	−6521.91			−0.812191			−0.406095	+	0.913831i	$0.633110\pi$
$402$		0			0
$403$		−	15582.2i		−	1.92606i
$404$	11249.4			1.38534
$405$		0			0
$406$	1331.18			0.162723
$407$			11476.2i			1.39768i
$408$		0			0
$409$	−5642.67			−0.682181			−0.341091	−	0.940030i	$-0.610796\pi$
$409$	−5642.67			−0.682181			−0.341091	+	0.940030i	$0.610796\pi$
$410$	−1329.13	+	798.191i	−0.160100	+	0.0961459i
$411$		0			0
$412$		−	2653.83i		−	0.317342i
$413$			413.584i			0.0492764i
$414$		0			0
$415$	12480.8	−	7495.17i	1.47628	−	0.886562i
$416$	−9284.59			−1.09427
$417$		0			0
$418$		−	5272.57i		−	0.616961i
$419$	9621.73			1.12184			0.560922	−	0.827869i	$-0.310447\pi$
$419$	9621.73			1.12184			0.560922	+	0.827869i	$0.310447\pi$
$420$		0			0
$421$	10030.3			1.16115			0.580577	−	0.814205i	$-0.302827\pi$
$421$	10030.3			1.16115			0.580577	+	0.814205i	$0.302827\pi$
$422$		−	21084.0i		−	2.43212i
$423$		0			0
$424$	11657.7			1.33525
$425$	1701.38	+	905.676i	0.194186	+	0.103369i
$426$		0			0
$427$			2637.66i			0.298935i
$428$			6013.85i			0.679183i
$429$		0			0
$430$	−1749.54	−	2913.29i	−0.196210	−	0.326724i
$431$	3867.23			0.432200			0.216100	−	0.976371i	$-0.430666\pi$
$431$	3867.23			0.432200			0.216100	+	0.976371i	$0.430666\pi$
$432$		0			0
$433$			2345.27i			0.260292i	0.991495	+	0.130146i	$0.0415446\pi$
$433$			2345.27i			0.260292i	−0.991495	+	0.130146i	$0.958455\pi$
$434$	4863.41			0.537906
$435$		0			0
$436$	19396.9			2.13061
$437$		−	2983.08i		−	0.326545i
$438$		0			0
$439$	14117.9			1.53487			0.767437	−	0.641125i	$-0.221532\pi$
$439$	14117.9			1.53487			0.767437	+	0.641125i	$0.221532\pi$
$440$	−17392.7	+	10445.0i	−1.88446	+	1.13169i
$441$		0			0
$442$			6710.34i			0.722123i
$443$		−	17796.6i		−	1.90867i	−0.298731	−	0.954337i	$-0.596563\pi$
$443$		−	17796.6i		−	1.90867i	0.298731	−	0.954337i	$-0.403437\pi$
$444$		0			0
$445$	5730.38	+	9542.08i	0.610441	+	1.01649i
$446$	−15244.7			−1.61852
$447$		0			0
$448$			1202.61i			0.126826i
$449$	11518.0			1.21062			0.605310	−	0.795990i	$-0.293049\pi$
$449$	11518.0			1.21062			0.605310	+	0.795990i	$0.293049\pi$
$450$		0			0
$451$	−1079.33			−0.112691
$452$		−	6068.97i		−	0.631550i
$453$		0			0
$454$	−16257.5			−1.68063
$455$	−2683.60	−	4468.66i	−0.276504	−	0.460427i
$456$		0			0
$457$			9857.71i			1.00902i	0.863405	+	0.504512i	$0.168328\pi$
$457$			9857.71i			1.00902i	−0.863405	+	0.504512i	$0.831672\pi$
$458$			14943.3i			1.52458i
$459$		0			0
$460$	−18362.2	+	11027.2i	−1.86118	+	1.11771i
$461$	−16081.6			−1.62472			−0.812358	−	0.583159i	$-0.801816\pi$
$461$	−16081.6			−1.62472			−0.812358	+	0.583159i	$0.801816\pi$
$462$		0			0
$463$		−	18895.8i		−	1.89668i	−0.317263	−	0.948338i	$-0.602764\pi$
$463$		−	18895.8i		−	1.89668i	0.317263	−	0.948338i	$-0.397236\pi$
$464$	−4688.92			−0.469133
$465$		0			0
$466$	−14706.5			−1.46194
$467$			10368.7i			1.02742i	0.857963	+	0.513712i	$0.171730\pi$
$467$			10368.7i			1.02742i	−0.857963	+	0.513712i	$0.828270\pi$
$468$		0			0
$469$	−1284.79			−0.126495
$470$	2782.79	+	4633.83i	0.273108	+	0.454772i
$471$		0			0
$472$		−	3566.23i		−	0.347774i
$473$		−	2365.77i		−	0.229976i
$474$		0			0
$475$	−1576.55	+	2961.67i	−0.152289	+	0.286086i
$476$	−1430.50			−0.137745
$477$		0			0
$478$		−	14461.7i		−	1.38382i
$479$	9554.02			0.911345			0.455672	−	0.890148i	$-0.349399\pi$
$479$	9554.02			0.911345			0.455672	+	0.890148i	$0.349399\pi$
$480$		0			0
$481$	25424.7			2.41012
$482$		−	5614.42i		−	0.530559i
$483$		0			0
$484$	−3412.17			−0.320451
$485$	−7810.13	+	4690.28i	−0.731216	+	0.439123i
$486$		0			0
$487$		−	17514.2i		−	1.62966i	−0.579699	−	0.814831i	$-0.696830\pi$
$487$		−	17514.2i		−	1.62966i	0.579699	−	0.814831i	$-0.303170\pi$
$488$		−	22743.8i		−	2.10977i
$489$		0			0
$490$	−15121.1	+	9080.81i	−1.39409	+	0.837203i
$491$	−3866.54			−0.355386			−0.177693	−	0.984086i	$-0.556863\pi$
$491$	−3866.54			−0.355386			−0.177693	+	0.984086i	$0.556863\pi$
$492$		0			0
$493$			759.169i			0.0693534i
$494$	−11681.0			−1.06387
$495$		0			0
$496$	−17130.7			−1.55079
$497$		−	3446.87i		−	0.311093i
$498$		0			0
$499$	−4855.96			−0.435637			−0.217818	−	0.975989i	$-0.569894\pi$
$499$	−4855.96			−0.435637			−0.217818	+	0.975989i	$0.569894\pi$
$500$	24058.3	−	1243.69i	2.15184	−	0.111239i
$501$		0			0
$502$		−	23169.3i		−	2.05996i
$503$		−	5481.79i		−	0.485927i	−0.970035	−	0.242963i	$-0.921881\pi$
$503$		−	5481.79i		−	0.485927i	0.970035	−	0.242963i	$-0.0781195\pi$
$504$		0			0
$505$	−3756.41	−	6255.07i	−0.331006	−	0.551182i
$506$	−21831.6			−1.91805
$507$		0			0
$508$		−	25892.4i		−	2.26140i
$509$	20427.1			1.77881			0.889405	−	0.457120i	$-0.151119\pi$
$509$	20427.1			1.77881			0.889405	+	0.457120i	$0.151119\pi$
$510$		0			0
$511$	−4140.12			−0.358411
$512$		−	25150.0i		−	2.17086i
$513$		0			0
$514$	19028.4			1.63289
$515$	−1475.63	+	886.171i	−0.126260	+	0.0758240i
$516$		0			0
$517$			3762.96i			0.320106i
$518$			7935.39i			0.673091i
$519$		0			0
$520$	23140.0	+	38532.2i	1.95146	+	3.24951i
$521$	−5768.55			−0.485076			−0.242538	−	0.970142i	$-0.577980\pi$
$521$	−5768.55			−0.485076			−0.242538	+	0.970142i	$0.577980\pi$
$522$		0			0
$523$		−	1753.72i		−	0.146625i	−0.997309	−	0.0733126i	$-0.976643\pi$
$523$		−	1753.72i		−	0.146625i	0.997309	−	0.0733126i	$-0.0233571\pi$
$524$	27144.4			2.26299
$525$		0			0
$526$	−25066.1			−2.07782
$527$			2773.58i			0.229258i
$528$		0			0
$529$	−184.728			−0.0151827
$530$	−7263.96	−	12095.7i	−0.595332	−	0.991332i
$531$		0			0
$532$		−	2490.13i		−	0.202934i
$533$			2391.18i			0.194322i
$534$		0			0
$535$	3343.92	−	2008.15i	0.270225	−	0.162280i
$536$	11078.4			0.892751
$537$		0			0
$538$			29366.4i			2.35330i
$539$	−12279.3			−0.981274
$540$		0			0
$541$	7146.18			0.567908			0.283954	−	0.958838i	$-0.408354\pi$
$541$	7146.18			0.567908			0.283954	+	0.958838i	$0.408354\pi$
$542$			13896.1i			1.10127i
$543$		0			0
$544$	1652.63			0.130250
$545$	−6477.06	−	10785.4i	−0.509076	−	0.847701i
$546$		0			0
$547$		−	7407.89i		−	0.579047i	−0.957171	−	0.289523i	$-0.906503\pi$
$547$		−	7407.89i		−	0.579047i	0.957171	−	0.289523i	$-0.0934968\pi$
$548$			15381.1i			1.19900i
$549$		0			0
$550$	21674.9	+	11537.9i	1.68040	+	0.894506i
$551$	−1321.52			−0.102175
$552$		0			0
$553$		−	3564.76i		−	0.274121i
$554$	−9513.91			−0.729616
$555$		0			0
$556$	11251.1			0.858192
$557$			11116.3i			0.845626i	0.906217	+	0.422813i	$0.138957\pi$
$557$			11116.3i			0.845626i	−0.906217	+	0.422813i	$0.861043\pi$
$558$		0			0
$559$	−5241.20			−0.396564
$560$	−4912.75	+	2950.30i	−0.370717	+	0.222630i
$561$		0			0
$562$			22528.2i			1.69092i
$563$		−	11206.6i		−	0.838901i	−0.907778	−	0.419450i	$-0.862223\pi$
$563$		−	11206.6i		−	0.838901i	0.907778	−	0.419450i	$-0.137777\pi$
$564$		0			0
$565$	−3374.57	+	2026.56i	−0.251273	+	0.150899i
$566$	29125.7			2.16298
$567$		0			0
$568$			29721.5i			2.19557i
$569$	−15873.8			−1.16953			−0.584765	−	0.811203i	$-0.698813\pi$
$569$	−15873.8			−1.16953			−0.584765	+	0.811203i	$0.698813\pi$
$570$		0			0
$571$	−2388.45			−0.175050			−0.0875250	−	0.996162i	$-0.527896\pi$
$571$	−2388.45			−0.175050			−0.0875250	+	0.996162i	$0.527896\pi$
$572$			58388.7i			4.26810i
$573$		0			0
$574$	−746.320			−0.0542697
$575$	12263.1	+	6527.85i	0.889401	+	0.473444i
$576$		0			0
$577$			10429.0i			0.752455i	0.926527	+	0.376227i	$0.122779\pi$
$577$			10429.0i			0.752455i	−0.926527	+	0.376227i	$0.877221\pi$
$578$			23487.1i			1.69020i
$579$		0			0
$580$	4885.10	+	8134.54i	0.349729	+	0.582359i
$581$	7008.09			0.500421
$582$		0			0
$583$		−	9822.50i		−	0.697781i
$584$	35699.2			2.52953
$585$		0			0
$586$	−39738.9			−2.80136
$587$			3148.05i			0.221352i	0.993857	+	0.110676i	$0.0353016\pi$
$587$			3148.05i			0.221352i	−0.993857	+	0.110676i	$0.964698\pi$
$588$		0			0
$589$	−4828.10			−0.337756
$590$	−3700.24	+	2222.13i	−0.258197	+	0.155057i
$591$		0			0
$592$		−	27951.4i		−	1.94053i
$593$		−	12123.4i		−	0.839542i	−0.907630	−	0.419771i	$-0.862110\pi$
$593$		−	12123.4i		−	0.839542i	0.907630	−	0.419771i	$-0.137890\pi$
$594$		0			0
$595$	477.673	+	795.409i	0.0329121	+	0.0548043i
$596$	59260.6			4.07283
$597$		0			0
$598$			48366.2i			3.30743i
$599$	15842.8			1.08067			0.540333	−	0.841451i	$-0.318298\pi$
$599$	15842.8			1.08067			0.540333	+	0.841451i	$0.318298\pi$
$600$		0			0
$601$	−25690.1			−1.74363			−0.871815	−	0.489836i	$-0.837057\pi$
$601$	−25690.1			−1.74363			−0.871815	+	0.489836i	$0.837057\pi$
$602$		−	1635.85i		−	0.110751i
$603$		0			0
$604$	−28802.8			−1.94034
$605$	1139.40	+	1897.29i	0.0765670	+	0.127497i
$606$		0			0
$607$			14523.3i			0.971139i	0.874198	+	0.485570i	$0.161388\pi$
$607$			14523.3i			0.971139i	−0.874198	+	0.485570i	$0.838612\pi$
$608$			2876.81i			0.191891i
$609$		0			0
$610$	−23598.5	+	14171.8i	−1.56635	+	0.940653i
$611$	8336.56			0.551982
$612$		0			0
$613$		−	26426.7i		−	1.74121i	−0.491981	−	0.870606i	$-0.663727\pi$
$613$		−	26426.7i		−	1.74121i	0.491981	−	0.870606i	$-0.336273\pi$
$614$	−22236.8			−1.46157
$615$		0			0
$616$	−9766.19			−0.638784
$617$		−	3757.77i		−	0.245190i	−0.992457	−	0.122595i	$-0.960878\pi$
$617$		−	3757.77i		−	0.245190i	0.992457	−	0.122595i	$-0.0391216\pi$
$618$		0			0
$619$	23480.5			1.52465			0.762327	−	0.647192i	$-0.224057\pi$
$619$	23480.5			1.52465			0.762327	+	0.647192i	$0.224057\pi$
$620$	17847.4	+	29719.1i	1.15608	+	1.92508i
$621$		0			0
$622$			48716.1i			3.14042i
$623$			5357.99i			0.344564i
$624$		0			0
$625$	−8725.10	−	12962.0i	−0.558407	−	0.829567i
$626$	22722.7			1.45077
$627$		0			0
$628$		−	11482.1i		−	0.729595i
$629$	−4525.52			−0.286875
$630$		0			0
$631$	8654.12			0.545983			0.272991	−	0.962016i	$-0.411987\pi$
$631$	8654.12			0.545983			0.272991	+	0.962016i	$0.411987\pi$
$632$			30738.0i			1.93464i
$633$		0			0
$634$	26737.2			1.67488
$635$	−14397.2	+	8646.04i	−0.899738	+	0.540327i
$636$		0			0
$637$			27203.9i			1.69208i
$638$			9671.48i			0.600153i
$639$		0			0
$640$	−18977.8	+	11396.9i	−1.17213	+	0.703908i
$641$	19177.6			1.18170			0.590850	−	0.806781i	$-0.298792\pi$
$641$	19177.6			1.18170			0.590850	+	0.806781i	$0.298792\pi$
$642$		0			0
$643$			9835.57i			0.603230i	0.953430	+	0.301615i	$0.0975258\pi$
$643$			9835.57i			0.603230i	−0.953430	+	0.301615i	$0.902474\pi$
$644$	−10310.6			−0.630892
$645$		0			0
$646$	2079.18			0.126632
$647$		−	7621.18i		−	0.463090i	−0.972824	−	0.231545i	$-0.925622\pi$
$647$		−	7621.18i		−	0.463090i	0.972824	−	0.231545i	$-0.0743781\pi$
$648$		0			0
$649$	−3004.82			−0.181741
$650$	25561.4	−	48019.1i	1.54246	−	2.89763i
$651$		0			0
$652$		−	15328.7i		−	0.920733i
$653$		−	4396.32i		−	0.263463i	−0.991285	−	0.131731i	$-0.957946\pi$
$653$		−	4396.32i		−	0.263463i	0.991285	−	0.131731i	$-0.0420537\pi$
$654$		0			0
$655$	−9064.08	−	15093.3i	−0.540707	−	0.900371i
$656$	2628.82			0.156460
$657$		0			0
$658$			2601.95i			0.154156i
$659$	−10820.3			−0.639601			−0.319801	−	0.947485i	$-0.603616\pi$
$659$	−10820.3			−0.639601			−0.319801	+	0.947485i	$0.603616\pi$
$660$		0			0
$661$	−28915.5			−1.70149			−0.850744	−	0.525581i	$-0.823848\pi$
$661$	−28915.5			−1.70149			−0.850744	+	0.525581i	$0.823848\pi$
$662$		−	39351.3i		−	2.31032i
$663$		0			0
$664$	−60429.0			−3.53178
$665$	−1384.60	+	831.507i	−0.0807407	+	0.0484879i
$666$		0			0
$667$			5471.87i			0.317649i
$668$			6961.65i			0.403225i
$669$		0			0
$670$	−6903.00	−	11494.7i	−0.398039	−	0.662804i
$671$	−19163.4			−1.10253
$672$		0			0
$673$			5061.16i			0.289886i	0.989440	+	0.144943i	$0.0462999\pi$
$673$			5061.16i			0.289886i	−0.989440	+	0.144943i	$0.953700\pi$
$674$	−49875.8			−2.85036
$675$		0			0
$676$	91484.6			5.20509
$677$		−	12059.3i		−	0.684601i	−0.939591	−	0.342300i	$-0.888794\pi$
$677$		−	12059.3i		−	0.684601i	0.939591	−	0.342300i	$-0.111206\pi$
$678$		0			0
$679$	−4385.48			−0.247863
$680$	−4118.86	−	6858.61i	−0.232281	−	0.386788i
$681$		0			0
$682$			35334.2i			1.98390i
$683$		−	6110.33i		−	0.342321i	−0.985243	−	0.171160i	$-0.945248\pi$
$683$		−	6110.33i		−	0.342321i	0.985243	−	0.171160i	$-0.0547516\pi$
$684$		0			0
$685$	8552.49	−	5136.10i	0.477042	−	0.286482i
$686$	−17764.5			−0.988708
$687$		0			0
$688$			5762.06i			0.319297i
$689$	−21761.0			−1.20323
$690$		0			0
$691$	27209.7			1.49798			0.748991	−	0.662580i	$-0.230538\pi$
$691$	27209.7			1.49798			0.748991	+	0.662580i	$0.230538\pi$
$692$		−	4069.15i		−	0.223535i
$693$		0			0
$694$	−20297.5			−1.11020
$695$	−3757.00	−	6256.05i	−0.205052	−	0.341447i
$696$		0			0
$697$		−	425.624i		−	0.0231300i
$698$		−	24112.4i		−	1.30755i
$699$		0			0
$700$	10236.6	+	5449.12i	0.552724	+	0.294225i
$701$	−30424.7			−1.63927			−0.819633	−	0.572888i	$-0.805823\pi$
$701$	−30424.7			−1.63927			−0.819633	+	0.572888i	$0.805823\pi$
$702$		0			0
$703$		−	7877.77i		−	0.422640i
$704$	−8737.37			−0.467759
$705$		0			0
$706$	16023.0			0.854156
$707$		−	3512.30i		−	0.186837i
$708$		0			0
$709$	−1846.41			−0.0978045			−0.0489022	−	0.998804i	$-0.515572\pi$
$709$	−1846.41			−0.0978045			−0.0489022	+	0.998804i	$0.515572\pi$
$710$	30838.3	−	18519.6i	1.63006	−	0.978911i
$711$		0			0
$712$		−	46200.6i		−	2.43180i
$713$			19991.2i			1.05004i
$714$		0			0
$715$	32466.3	−	19497.2i	1.69814	−	1.01980i
$716$	41444.6			2.16321
$717$		0			0
$718$			5826.64i			0.302853i
$719$	−13301.3			−0.689923			−0.344961	−	0.938617i	$-0.612108\pi$
$719$	−13301.3			−0.689923			−0.344961	+	0.938617i	$0.612108\pi$
$720$		0			0
$721$	−828.583			−0.0427989
$722$		−	30838.3i		−	1.58959i
$723$		0			0
$724$	21002.6			1.07812
$725$	2891.86	−	5432.59i	0.148140	−	0.278292i
$726$		0			0
$727$		−	25915.1i		−	1.32206i	−0.750358	−	0.661031i	$-0.770119\pi$
$727$		−	25915.1i		−	1.32206i	0.750358	−	0.661031i	$-0.229881\pi$
$728$			21636.3i			1.10150i
$729$		0			0
$730$	−22244.3	−	37040.6i	−1.12781	−	1.87799i
$731$	932.918			0.0472028
$732$		0			0
$733$		−	22247.8i		−	1.12106i	−0.828132	−	0.560532i	$-0.810597\pi$
$733$		−	22247.8i		−	1.12106i	0.828132	−	0.560532i	$-0.189403\pi$
$734$	−59657.3			−2.99999
$735$		0			0
$736$	11911.7			0.596563
$737$		−	9334.41i		−	0.466536i
$738$		0			0
$739$	23675.6			1.17851			0.589256	−	0.807947i	$-0.299421\pi$
$739$	23675.6			1.17851			0.589256	+	0.807947i	$0.299421\pi$
$740$	−48491.2	+	29120.8i	−2.40888	+	1.44663i
$741$		0			0
$742$		−	6791.91i		−	0.336036i
$743$			33555.1i			1.65682i	0.560120	+	0.828411i	$0.310755\pi$
$743$			33555.1i			1.65682i	−0.560120	+	0.828411i	$0.689245\pi$
$744$		0			0
$745$	−19788.4	−	32951.1i	−0.973142	−	1.62045i
$746$	−13279.0			−0.651713
$747$		0			0
$748$		−	10393.0i		−	0.508030i
$749$	1877.65			0.0915993
$750$		0			0
$751$	21889.8			1.06361			0.531804	−	0.846868i	$-0.321514\pi$
$751$	21889.8			1.06361			0.531804	+	0.846868i	$0.321514\pi$
$752$		−	9165.04i		−	0.444434i
$753$		0			0
$754$	21426.4			1.03489
$755$	9617.86	+	16015.4i	0.463615	+	0.772000i
$756$		0			0
$757$			24362.3i			1.16970i	0.811141	+	0.584851i	$0.198847\pi$
$757$			24362.3i			1.16970i	−0.811141	+	0.584851i	$0.801153\pi$
$758$		−	52511.2i		−	2.51622i
$759$		0			0
$760$	11939.1	−	7169.87i	0.569837	−	0.342209i
$761$	−27685.3			−1.31878			−0.659391	−	0.751801i	$-0.729186\pi$
$761$	−27685.3			−1.31878			−0.659391	+	0.751801i	$0.729186\pi$
$762$		0			0
$763$		−	6056.14i		−	0.287349i
$764$	−3032.85			−0.143619
$765$		0			0
$766$	−54657.2			−2.57813
$767$			6656.96i			0.313388i
$768$		0			0
$769$	6996.11			0.328070			0.164035	−	0.986454i	$-0.447549\pi$
$769$	6996.11			0.328070			0.164035	+	0.986454i	$0.447549\pi$
$770$	6085.35	+	10133.2i	0.284806	+	0.474252i
$771$		0			0
$772$		−	38185.9i		−	1.78023i
$773$		−	19650.7i		−	0.914342i	−0.889379	−	0.457171i	$-0.848863\pi$
$773$		−	19650.7i		−	0.914342i	0.889379	−	0.457171i	$-0.151137\pi$
$774$		0			0
$775$	10565.3	−	19847.7i	0.489698	−	0.919936i
$776$	37814.9			1.74932
$777$		0			0
$778$		−	8428.08i		−	0.388382i
$779$	740.901			0.0340764
$780$		0			0
$781$	25042.6			1.14737
$782$		−	8609.05i		−	0.393681i
$783$		0			0
$784$	29907.4			1.36240
$785$	−6384.47	+	3834.12i	−0.290282	+	0.174326i
$786$		0			0
$787$			1312.96i			0.0594689i	0.999558	+	0.0297344i	$0.00946616\pi$
$787$			1312.96i			0.0594689i	−0.999558	+	0.0297344i	$0.990534\pi$
$788$		−	18169.8i		−	0.821413i
$789$		0			0
$790$	31893.0	−	19153.0i	1.43633	−	0.862573i
$791$	−1894.86			−0.0851752
$792$		0			0
$793$			42455.1i			1.90117i
$794$	−66049.1			−2.95214
$795$		0			0
$796$	−60056.8			−2.67419
$797$			2391.63i			0.106294i	0.998587	+	0.0531468i	$0.0169251\pi$
$797$			2391.63i			0.106294i	−0.998587	+	0.0531468i	$0.983075\pi$
$798$		0			0
$799$	−1483.88			−0.0657022
$800$	−11826.2	−	6295.29i	−0.522648	−	0.278215i
$801$		0			0
$802$		−	32764.2i		−	1.44257i
$803$		−	30079.3i		−	1.32189i
$804$		0			0
$805$	3442.93	+	5733.08i	0.150742	+	0.251012i
$806$	78280.4			3.42098
$807$		0			0
$808$			30285.7i			1.31862i
$809$	7059.58			0.306801			0.153400	−	0.988164i	$-0.450978\pi$
$809$	7059.58			0.306801			0.153400	+	0.988164i	$0.450978\pi$
$810$		0			0
$811$	12699.2			0.549849			0.274925	−	0.961466i	$-0.411347\pi$
$811$	12699.2			0.549849			0.274925	+	0.961466i	$0.411347\pi$
$812$			4567.64i			0.197405i
$813$		0			0
$814$	−57653.2			−2.48248
$815$	−8523.32	+	5118.58i	−0.366330	+	0.219995i
$816$		0			0
$817$			1623.97i			0.0695417i
$818$		−	28347.1i		−	1.21166i
$819$		0			0
$820$	−2738.80	−	4560.58i	−0.116638	−	0.194222i
$821$	1624.01			0.0690356			0.0345178	−	0.999404i	$-0.489010\pi$
$821$	1624.01			0.0690356			0.0345178	+	0.999404i	$0.489010\pi$
$822$		0			0
$823$		−	21151.7i		−	0.895870i	−0.894066	−	0.447935i	$-0.852160\pi$
$823$		−	21151.7i		−	0.895870i	0.894066	−	0.447935i	$-0.147840\pi$
$824$	7144.66			0.302058
$825$		0			0
$826$	−2077.73			−0.0875223
$827$		−	18269.2i		−	0.768178i	−0.923296	−	0.384089i	$-0.874516\pi$
$827$		−	18269.2i		−	0.768178i	0.923296	−	0.384089i	$-0.125484\pi$
$828$		0			0
$829$	−19893.4			−0.833446			−0.416723	−	0.909034i	$-0.636821\pi$
$829$	−19893.4			−0.833446			−0.416723	+	0.909034i	$0.636821\pi$
$830$	37653.6	+	62699.7i	1.57467	+	2.62209i
$831$		0			0
$832$			19357.0i			0.806591i
$833$		−	4842.21i		−	0.201408i
$834$		0			0
$835$	3870.93	−	2324.64i	0.160430	−	0.0963445i
$836$	18091.6			0.748457
$837$		0			0
$838$			48336.8i			1.99256i
$839$	11705.8			0.481678			0.240839	−	0.970565i	$-0.422577\pi$
$839$	11705.8			0.481678			0.240839	+	0.970565i	$0.422577\pi$
$840$		0			0
$841$	−21964.9			−0.900608
$842$			50389.2i			2.06238i
$843$		0			0
$844$	72344.8			2.95048
$845$	−30548.7	−	50868.8i	−1.24368	−	2.07094i
$846$		0			0
$847$			1065.35i			0.0432183i
$848$			23923.6i			0.968797i
$849$		0			0
$850$	−4549.85	+	8547.25i	−0.183598	+	0.344904i
$851$	−32618.6			−1.31393
$852$		0			0
$853$		−	1058.28i		−	0.0424792i	−0.999774	−	0.0212396i	$-0.993239\pi$
$853$		−	1058.28i		−	0.0424792i	0.999774	−	0.0212396i	$-0.00676128\pi$
$854$	−13250.8			−0.530953
$855$		0			0
$856$	−16190.5			−0.646473
$857$		−	24035.3i		−	0.958030i	−0.877807	−	0.479015i	$-0.840994\pi$
$857$		−	24035.3i		−	0.958030i	0.877807	−	0.479015i	$-0.159006\pi$
$858$		0			0
$859$	35094.2			1.39395			0.696973	−	0.717098i	$-0.254530\pi$
$859$	35094.2			1.39395			0.696973	+	0.717098i	$0.254530\pi$
$860$	9996.27	−	6003.14i	0.396361	−	0.238030i
$861$		0			0
$862$			19427.9i			0.767651i
$863$			41958.9i			1.65504i	0.561439	+	0.827518i	$0.310248\pi$
$863$			41958.9i			1.65504i	−0.561439	+	0.827518i	$0.689752\pi$
$864$		0			0
$865$	−2262.60	+	1358.78i	−0.0889372	+	0.0534102i
$866$	−11782.0			−0.462318
$867$		0			0
$868$			16687.6i			0.652552i
$869$	25899.1			1.01101
$870$		0			0
$871$	−20679.7			−0.804483
$872$			52220.6i			2.02800i
$873$		0			0
$874$	14986.1			0.579993
$875$	−388.306	−	7511.50i	−0.0150024	−	0.290212i
$876$		0			0
$877$		−	19611.1i		−	0.755097i	−0.925990	−	0.377548i	$-0.876767\pi$
$877$		−	19611.1i		−	0.755097i	0.925990	−	0.377548i	$-0.123233\pi$
$878$			70924.1i			2.72617i
$879$		0			0
$880$	−21434.9	−	35692.7i	−0.821101	−	1.36728i
$881$	10465.2			0.400204			0.200102	−	0.979775i	$-0.435873\pi$
$881$	10465.2			0.400204			0.200102	+	0.979775i	$0.435873\pi$
$882$		0			0
$883$		−	10812.2i		−	0.412074i	−0.978544	−	0.206037i	$-0.933943\pi$
$883$		−	10812.2i		−	0.412074i	0.978544	−	0.206037i	$-0.0660567\pi$
$884$	−23025.0			−0.876033
$885$		0			0
$886$	89405.1			3.39009
$887$		−	36000.6i		−	1.36277i	−0.731923	−	0.681387i	$-0.761377\pi$
$887$		−	36000.6i		−	1.36277i	0.731923	−	0.681387i	$-0.238623\pi$
$888$		0			0
$889$	−8084.17			−0.304988
$890$	−47936.7	+	28787.8i	−1.80544	+	1.08423i
$891$		0			0
$892$		−	52308.7i		−	1.96348i
$893$		−	2583.06i		−	0.0967960i
$894$		0			0
$895$	−13839.3	−	23044.8i	−0.516866	−	0.860672i
$896$	−10656.3			−0.397322
$897$		0			0
$898$			57863.1i			2.15024i
$899$	8856.18			0.328554
$900$		0			0
$901$	3873.40			0.143220
$902$		−	5422.26i		−	0.200157i
$903$		0			0
$904$	16338.9			0.601134
$905$	−7013.23	−	11678.2i	−0.257600	−	0.428948i
$906$		0			0
$907$			5211.64i			0.190794i	0.995439	+	0.0953968i	$0.0304120\pi$
$907$			5211.64i			0.190794i	−0.995439	+	0.0953968i	$0.969588\pi$
$908$		−	55783.9i		−	2.03883i
$909$		0			0
$910$	22449.3	−	13481.6i	0.817787	−	0.491112i
$911$	−16733.1			−0.608555			−0.304277	−	0.952584i	$-0.598415\pi$
$911$	−16733.1			−0.608555			−0.304277	+	0.952584i	$0.598415\pi$
$912$		0			0
$913$			50916.1i			1.84565i
$914$	−49522.3			−1.79218
$915$		0			0
$916$	−51274.5			−1.84952
$917$		−	8475.05i		−	0.305203i
$918$		0			0
$919$	−16722.8			−0.600255			−0.300127	−	0.953899i	$-0.597029\pi$
$919$	−16722.8			−0.600255			−0.300127	+	0.953899i	$0.597029\pi$
$920$	−29687.5	−	49434.9i	−1.06388	−	1.77154i
$921$		0			0
$922$		−	80789.2i		−	2.88574i
$923$		−	55480.0i		−	1.97849i
$924$		0			0
$925$	32384.5	+	17238.9i	1.15113	+	0.612767i
$926$	94926.8			3.36878
$927$		0			0
$928$		−	5276.93i		−	0.186663i
$929$	35344.5			1.24824			0.624120	−	0.781328i	$-0.285457\pi$
$929$	35344.5			1.24824			0.624120	+	0.781328i	$0.285457\pi$
$930$		0			0
$931$	8429.05			0.296725
$932$		−	50461.9i		−	1.77353i
$933$		0			0
$934$	−52089.5			−1.82486
$935$	−5778.90	+	3470.45i	−0.202129	+	0.121386i
$936$		0			0
$937$			2015.96i			0.0702867i	0.999382	+	0.0351433i	$0.0111888\pi$
$937$			2015.96i			0.0702867i	−0.999382	+	0.0351433i	$0.988811\pi$
$938$		−	6454.41i		−	0.224674i
$939$		0			0
$940$	−15899.9	+	9548.50i	−0.551700	+	0.331317i
$941$	39880.3			1.38157			0.690787	−	0.723058i	$-0.257264\pi$
$941$	39880.3			1.38157			0.690787	+	0.723058i	$0.257264\pi$
$942$		0			0
$943$		−	3067.77i		−	0.105939i
$944$	7318.52			0.252328
$945$		0			0
$946$	11885.0			0.408471
$947$			35010.2i			1.20135i	0.799493	+	0.600675i	$0.205102\pi$
$947$			35010.2i			1.20135i	−0.799493	+	0.600675i	$0.794898\pi$
$948$		0			0
$949$	−66638.5			−2.27943
$950$	−14878.6	−	7920.13i	−0.508131	−	0.270487i
$951$		0			0
$952$		−	3851.19i		−	0.131111i
$953$			49213.8i			1.67281i	0.548109	+	0.836407i	$0.315348\pi$
$953$			49213.8i			1.67281i	−0.548109	+	0.836407i	$0.684652\pi$
$954$		0			0
$955$	1012.73	+	1686.38i	0.0343155	+	0.0571412i
$956$	49622.0			1.67876
$957$		0			0
$958$			47996.6i			1.61868i
$959$	4802.33			0.161705
$960$		0			0
$961$	2564.57			0.0860855
$962$			127726.i			4.28073i
$963$		0			0
$964$	19264.5			0.643640
$965$	−21232.8	+	12751.1i	−0.708298	+	0.425360i
$966$		0			0
$967$		−	1219.34i		−	0.0405495i	−0.999794	−	0.0202747i	$-0.993546\pi$
$967$		−	1219.34i		−	0.0405495i	0.999794	−	0.0202747i	$-0.00645409\pi$
$968$		−	9186.26i		−	0.305018i
$969$		0			0
$970$	−23562.6	−	39235.8i	−0.779948	−	1.29875i
$971$	−1660.33			−0.0548740			−0.0274370	−	0.999624i	$-0.508735\pi$
$971$	−1660.33			−0.0548740			−0.0274370	+	0.999624i	$0.508735\pi$
$972$		0			0
$973$		−	3512.85i		−	0.115742i
$974$	87986.4			2.89452
$975$		0			0
$976$	46674.3			1.53075
$977$			47307.7i			1.54914i	0.632489	+	0.774569i	$0.282033\pi$
$977$			47307.7i			1.54914i	−0.632489	+	0.774569i	$0.717967\pi$
$978$		0			0
$979$	−38927.5			−1.27082
$980$	−31158.7	−	51884.6i	−1.01564	−	1.69122i
$981$		0			0
$982$		−	19424.4i		−	0.631218i
$983$			41203.0i			1.33690i	0.743757	+	0.668450i	$0.233042\pi$
$983$			41203.0i			1.33690i	−0.743757	+	0.668450i	$0.766958\pi$
$984$		0			0
$985$	−10103.1	+	6067.30i	−0.326814	+	0.196264i
$986$	−3813.84			−0.123182
$987$		0			0
$988$		−	40080.5i		−	1.29062i
$989$	6724.20			0.216195
$990$		0			0
$991$	−29805.3			−0.955395			−0.477698	−	0.878524i	$-0.658529\pi$
$991$	−29805.3			−0.955395			−0.477698	+	0.878524i	$0.658529\pi$
$992$		−	19279.0i		−	0.617044i
$993$		0			0
$994$	17316.1			0.552548
$995$	20054.3	+	33393.8i	0.638957	+	1.06397i
$996$		0			0
$997$			39832.8i			1.26531i	0.774433	+	0.632657i	$0.218036\pi$
$997$			39832.8i			1.26531i	−0.774433	+	0.632657i	$0.781964\pi$
$998$		−	24395.0i		−	0.773756i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.4.b.e.244.15		16
3.2	odd	2		405.4.b.f.244.2		16
5.2	odd	4		2025.4.a.bk.1.2		16
5.3	odd	4		2025.4.a.bk.1.15		16
5.4	even	2	inner	405.4.b.e.244.2		16
9.2	odd	6		135.4.j.a.64.2		32
9.4	even	3		45.4.j.a.34.2	yes	32
9.5	odd	6		135.4.j.a.19.15		32
9.7	even	3		45.4.j.a.4.15	yes	32
15.2	even	4		2025.4.a.bl.1.15		16
15.8	even	4		2025.4.a.bl.1.2		16
15.14	odd	2		405.4.b.f.244.15		16
45.4	even	6		45.4.j.a.34.15	yes	32
45.7	odd	12		225.4.e.g.76.15		32
45.13	odd	12		225.4.e.g.151.2		32
45.14	odd	6		135.4.j.a.19.2		32
45.22	odd	12		225.4.e.g.151.15		32
45.29	odd	6		135.4.j.a.64.15		32
45.34	even	6		45.4.j.a.4.2	✓	32
45.43	odd	12		225.4.e.g.76.2		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.j.a.4.2	✓	32	45.34	even	6
45.4.j.a.4.15	yes	32	9.7	even	3
45.4.j.a.34.2	yes	32	9.4	even	3
45.4.j.a.34.15	yes	32	45.4	even	6
135.4.j.a.19.2		32	45.14	odd	6
135.4.j.a.19.15		32	9.5	odd	6
135.4.j.a.64.2		32	9.2	odd	6
135.4.j.a.64.15		32	45.29	odd	6
225.4.e.g.76.2		32	45.43	odd	12
225.4.e.g.76.15		32	45.7	odd	12
225.4.e.g.151.2		32	45.13	odd	12
225.4.e.g.151.15		32	45.22	odd	12
405.4.b.e.244.2		16	5.4	even	2	inner
405.4.b.e.244.15		16	1.1	even	1	trivial
405.4.b.f.244.2		16	3.2	odd	2
405.4.b.f.244.15		16	15.14	odd	2
2025.4.a.bk.1.2		16	5.2	odd	4
2025.4.a.bk.1.15		16	5.3	odd	4
2025.4.a.bl.1.2		16	15.8	even	4
2025.4.a.bl.1.15		16	15.2	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:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	405.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+5.02371i q^{2} -17.2377 q^{4} +(5.75603 + 9.58479i) q^{5} +5.38197i q^{7} -46.4074i q^{8} +O(q^{10})\)	\(q+5.02371i q^{2} -17.2377 q^{4} +(5.75603 + 9.58479i) q^{5} +5.38197i q^{7} -46.4074i q^{8} +(-48.1512 + 28.9166i) q^{10} -39.1018 q^{11} +86.6271i q^{13} -27.0375 q^{14} +95.2360 q^{16} -15.4194i q^{17} +26.8412 q^{19} +(-99.2205 - 165.219i) q^{20} -196.436i q^{22} -111.138i q^{23} +(-58.7363 + 110.341i) q^{25} -435.189 q^{26} -92.7727i q^{28} -49.2347 q^{29} -179.877 q^{31} +107.179i q^{32} +77.4624 q^{34} +(-51.5851 + 30.9788i) q^{35} -293.496i q^{37} +134.842i q^{38} +(444.805 - 267.122i) q^{40} +27.6032 q^{41} +60.5030i q^{43} +674.023 q^{44} +558.327 q^{46} -96.2351i q^{47} +314.034 q^{49} +(-554.319 - 295.074i) q^{50} -1493.25i q^{52} +251.203i q^{53} +(-225.071 - 374.782i) q^{55} +249.763 q^{56} -247.341i q^{58} +76.8462 q^{59} +490.091 q^{61} -903.648i q^{62} +223.452 q^{64} +(-830.302 + 498.628i) q^{65} +238.721i q^{67} +265.794i q^{68} +(-155.628 - 259.148i) q^{70} -640.447 q^{71} +769.257i q^{73} +1474.44 q^{74} -462.679 q^{76} -210.445i q^{77} -662.352 q^{79} +(548.181 + 912.817i) q^{80} +138.670i q^{82} -1302.14i q^{83} +(147.791 - 88.7543i) q^{85} -303.950 q^{86} +1814.61i q^{88} +995.544 q^{89} -466.224 q^{91} +1915.77i q^{92} +483.457 q^{94} +(154.499 + 257.267i) q^{95} +814.846i q^{97} +1577.62i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 54 q^{4} - 3 q^{5}+O(q^{10}) \) 16 * q - 54 * q^4 - 3 * q^5	\( 16 q - 54 q^{4} - 3 q^{5} - 10 q^{10} - 90 q^{11} + 102 q^{14} + 146 q^{16} - 4 q^{19} + 6 q^{20} - 71 q^{25} - 468 q^{26} + 516 q^{29} + 38 q^{31} - 212 q^{34} - 267 q^{35} - 44 q^{40} - 576 q^{41} + 1644 q^{44} - 290 q^{46} + 4 q^{49} - 558 q^{50} + 15 q^{55} - 2430 q^{56} + 2202 q^{59} + 20 q^{61} + 322 q^{64} - 339 q^{65} - 636 q^{70} - 2952 q^{71} + 4080 q^{74} - 396 q^{76} + 218 q^{79} + 1266 q^{80} + 704 q^{85} - 6108 q^{86} + 4074 q^{89} - 942 q^{91} + 1078 q^{94} + 1692 q^{95}+O(q^{100}) \) 16 * q - 54 * q^4 - 3 * q^5 - 10 * q^10 - 90 * q^11 + 102 * q^14 + 146 * q^16 - 4 * q^19 + 6 * q^20 - 71 * q^25 - 468 * q^26 + 516 * q^29 + 38 * q^31 - 212 * q^34 - 267 * q^35 - 44 * q^40 - 576 * q^41 + 1644 * q^44 - 290 * q^46 + 4 * q^49 - 558 * q^50 + 15 * q^55 - 2430 * q^56 + 2202 * q^59 + 20 * q^61 + 322 * q^64 - 339 * q^65 - 636 * q^70 - 2952 * q^71 + 4080 * q^74 - 396 * q^76 + 218 * q^79 + 1266 * q^80 + 704 * q^85 - 6108 * q^86 + 4074 * q^89 - 942 * q^91 + 1078 * q^94 + 1692 * q^95

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