LMFDB - Embedded newform 405.4.b.c.244.3

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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 44x^{6} + 567x^{4} + 2024x^{2} + 1900 $ x^8 + 44x^6 + 567x^4 + 2024*x^2 + 1900
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			244.3
Root			$-1.85698i$ of defining polynomial
Character	$\chi$	$=$	405.244
Dual form			405.4.b.c.244.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.85698i q^{2} +4.55164 q^{4} +(-7.77825 + 8.03112i) q^{5} -1.02402i q^{7} -23.3081i q^{8} +O(q^{10})$	\(q-1.85698i q^{2} +4.55164 q^{4} +(-7.77825 + 8.03112i) q^{5} -1.02402i q^{7} -23.3081i q^{8} +(14.9136 + 14.4440i) q^{10} -16.1402 q^{11} +3.05437i q^{13} -1.90159 q^{14} -6.86951 q^{16} -69.2824i q^{17} +12.6207 q^{19} +(-35.4038 + 36.5547i) q^{20} +29.9720i q^{22} -56.6563i q^{23} +(-3.99771 - 124.936i) q^{25} +5.67190 q^{26} -4.66098i q^{28} -196.187 q^{29} +213.360 q^{31} -173.708i q^{32} -128.656 q^{34} +(8.22405 + 7.96510i) q^{35} -299.951i q^{37} -23.4363i q^{38} +(187.190 + 181.296i) q^{40} -139.067 q^{41} -475.161i q^{43} -73.4644 q^{44} -105.209 q^{46} -193.371i q^{47} +341.951 q^{49} +(-232.003 + 7.42365i) q^{50} +13.9024i q^{52} -214.125i q^{53} +(125.543 - 129.624i) q^{55} -23.8680 q^{56} +364.315i q^{58} +149.848 q^{59} -495.145 q^{61} -396.204i q^{62} -377.528 q^{64} +(-24.5300 - 23.7577i) q^{65} +761.598i q^{67} -315.348i q^{68} +(14.7910 - 15.2719i) q^{70} -736.184 q^{71} +701.058i q^{73} -557.001 q^{74} +57.4446 q^{76} +16.5279i q^{77} -780.649 q^{79} +(53.4327 - 55.1698i) q^{80} +258.245i q^{82} -961.505i q^{83} +(556.415 + 538.896i) q^{85} -882.362 q^{86} +376.198i q^{88} -520.561 q^{89} +3.12775 q^{91} -257.879i q^{92} -359.086 q^{94} +(-98.1666 + 101.358i) q^{95} -1155.19i q^{97} -634.996i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 24 q^{4} + 15 q^{5}+O(q^{10}) $ 8 * q - 24 * q^4 + 15 * q^5	$ 8 q - 24 q^{4} + 15 q^{5} + 7 q^{10} - 42 q^{14} + 4 q^{16} + 118 q^{19} - 129 q^{20} + 17 q^{25} - 36 q^{26} + 318 q^{29} - 416 q^{31} + 638 q^{34} + 192 q^{35} - 265 q^{40} + 486 q^{41} + 852 q^{44} - 598 q^{46} + 350 q^{49} + 1143 q^{50} - 162 q^{55} + 1530 q^{56} - 1146 q^{59} - 398 q^{61} + 1640 q^{64} - 1833 q^{65} + 630 q^{70} - 1728 q^{71} + 1218 q^{74} - 3498 q^{76} + 2596 q^{79} + 1923 q^{80} - 233 q^{85} + 480 q^{86} + 1086 q^{89} - 2574 q^{91} + 1238 q^{94} + 1674 q^{95}+O(q^{100}) $ 8 * q - 24 * q^4 + 15 * q^5 + 7 * q^10 - 42 * q^14 + 4 * q^16 + 118 * q^19 - 129 * q^20 + 17 * q^25 - 36 * q^26 + 318 * q^29 - 416 * q^31 + 638 * q^34 + 192 * q^35 - 265 * q^40 + 486 * q^41 + 852 * q^44 - 598 * q^46 + 350 * q^49 + 1143 * q^50 - 162 * q^55 + 1530 * q^56 - 1146 * q^59 - 398 * q^61 + 1640 * q^64 - 1833 * q^65 + 630 * q^70 - 1728 * q^71 + 1218 * q^74 - 3498 * q^76 + 2596 * q^79 + 1923 * q^80 - 233 * q^85 + 480 * q^86 + 1086 * q^89 - 2574 * q^91 + 1238 * q^94 + 1674 * 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 $

$2$		−	1.85698i		−	0.656540i	−0.944584	−	0.328270i	$-0.893534\pi$
$2$		−	1.85698i		−	0.656540i	0.944584	−	0.328270i	$-0.106466\pi$
$3$		0			0
$3$		0			0
$4$	4.55164			0.568955
$5$	−7.77825	+	8.03112i	−0.695708	+	0.718325i
$5$	−7.77825	+	8.03112i	−0.695708	+	0.718325i
$6$		0			0
$7$		−	1.02402i		−	0.0552920i	−0.999618	−	0.0276460i	$-0.991199\pi$
$7$		−	1.02402i		−	0.0552920i	0.999618	−	0.0276460i	$-0.00880112\pi$
$8$		−	23.3081i		−	1.03008i
$9$		0			0
$10$	14.9136	+	14.4440i	0.471609	+	0.456760i
$11$	−16.1402			−0.442405			−0.221203	−	0.975228i	$-0.570998\pi$
$11$	−16.1402			−0.442405			−0.221203	+	0.975228i	$0.570998\pi$
$12$		0			0
$13$			3.05437i			0.0651639i	0.999469	+	0.0325819i	$0.0103730\pi$
$13$			3.05437i			0.0651639i	−0.999469	+	0.0325819i	$0.989627\pi$
$14$	−1.90159			−0.0363015
$15$		0			0
$16$	−6.86951			−0.107336
$17$		−	69.2824i		−	0.988438i	−0.869337	−	0.494219i	$-0.835454\pi$
$17$		−	69.2824i		−	0.988438i	0.869337	−	0.494219i	$-0.164546\pi$
$18$		0			0
$19$	12.6207			0.152388			0.0761941	−	0.997093i	$-0.475723\pi$
$19$	12.6207			0.152388			0.0761941	+	0.997093i	$0.475723\pi$
$20$	−35.4038	+	36.5547i	−0.395826	+	0.408694i
$21$		0			0
$22$			29.9720i			0.290457i
$23$		−	56.6563i		−	0.513637i	−0.966460	−	0.256819i	$-0.917326\pi$
$23$		−	56.6563i		−	0.513637i	0.966460	−	0.256819i	$-0.0826743\pi$
$24$		0			0
$25$	−3.99771	−	124.936i	−0.0319816	−	0.999488i
$26$	5.67190			0.0427827
$27$		0			0
$28$		−	4.66098i		−	0.0314587i
$29$	−196.187			−1.25624			−0.628121	−	0.778116i	$-0.716176\pi$
$29$	−196.187			−1.25624			−0.628121	+	0.778116i	$0.716176\pi$
$30$		0			0
$31$	213.360			1.23615			0.618073	−	0.786121i	$-0.287914\pi$
$31$	213.360			1.23615			0.618073	+	0.786121i	$0.287914\pi$
$32$		−	173.708i		−	0.959612i
$33$		0			0
$34$	−128.656			−0.648950
$35$	8.22405	+	7.96510i	0.0397176	+	0.0384671i
$36$		0			0
$37$		−	299.951i		−	1.33275i	−0.745619	−	0.666373i	$-0.767846\pi$
$37$		−	299.951i		−	1.33275i	0.745619	−	0.666373i	$-0.232154\pi$
$38$		−	23.4363i		−	0.100049i
$39$		0			0
$40$	187.190	+	181.296i	0.739934	+	0.716636i
$41$	−139.067			−0.529723			−0.264862	−	0.964286i	$-0.585326\pi$
$41$	−139.067			−0.529723			−0.264862	+	0.964286i	$0.585326\pi$
$42$		0			0
$43$		−	475.161i		−	1.68515i	−0.538582	−	0.842573i	$-0.681040\pi$
$43$		−	475.161i		−	1.68515i	0.538582	−	0.842573i	$-0.318960\pi$
$44$	−73.4644			−0.251708
$45$		0			0
$46$	−105.209			−0.337224
$47$		−	193.371i		−	0.600130i	−0.953919	−	0.300065i	$-0.902992\pi$
$47$		−	193.371i		−	0.600130i	0.953919	−	0.300065i	$-0.0970084\pi$
$48$		0			0
$49$	341.951			0.996943
$50$	−232.003	+	7.42365i	−0.656205	+	0.0209972i
$51$		0			0
$52$			13.9024i			0.0370753i
$53$		−	214.125i		−	0.554950i	−0.960733	−	0.277475i	$-0.910502\pi$
$53$		−	214.125i		−	0.554950i	0.960733	−	0.277475i	$-0.0894976\pi$
$54$		0			0
$55$	125.543	−	129.624i	0.307785	−	0.317791i
$56$	−23.8680			−0.0569553
$57$		0			0
$58$			364.315i			0.824774i
$59$	149.848			0.330653			0.165326	−	0.986239i	$-0.447132\pi$
$59$	149.848			0.330653			0.165326	+	0.986239i	$0.447132\pi$
$60$		0			0
$61$	−495.145			−1.03929			−0.519646	−	0.854382i	$-0.673936\pi$
$61$	−495.145			−1.03929			−0.519646	+	0.854382i	$0.673936\pi$
$62$		−	396.204i		−	0.811580i
$63$		0			0
$64$	−377.528			−0.737360
$65$	−24.5300	−	23.7577i	−0.0468089	−	0.0453350i
$66$		0			0
$67$			761.598i			1.38872i	0.719630	+	0.694358i	$0.244311\pi$
$67$			761.598i			1.38872i	−0.719630	+	0.694358i	$0.755689\pi$
$68$		−	315.348i		−	0.562377i
$69$		0			0
$70$	14.7910	−	15.2719i	0.0252552	−	0.0260762i
$71$	−736.184			−1.23055			−0.615274	−	0.788313i	$-0.710955\pi$
$71$	−736.184			−1.23055			−0.615274	+	0.788313i	$0.710955\pi$
$72$		0			0
$73$			701.058i			1.12401i	0.827135	+	0.562004i	$0.189969\pi$
$73$			701.058i			1.12401i	−0.827135	+	0.562004i	$0.810031\pi$
$74$	−557.001			−0.875001
$75$		0			0
$76$	57.4446			0.0867020
$77$			16.5279i			0.0244615i
$78$		0			0
$79$	−780.649			−1.11177			−0.555886	−	0.831259i	$-0.687621\pi$
$79$	−780.649			−1.11177			−0.555886	+	0.831259i	$0.687621\pi$
$80$	53.4327	−	55.1698i	0.0746745	−	0.0771022i
$81$		0			0
$82$			258.245i			0.347785i
$83$		−	961.505i		−	1.27155i	−0.771873	−	0.635776i	$-0.780680\pi$
$83$		−	961.505i		−	1.27155i	0.771873	−	0.635776i	$-0.219320\pi$
$84$		0			0
$85$	556.415	+	538.896i	0.710020	+	0.687664i
$86$	−882.362			−1.10637
$87$		0			0
$88$			376.198i			0.455714i
$89$	−520.561			−0.619993			−0.309996	−	0.950738i	$-0.600328\pi$
$89$	−520.561			−0.619993			−0.309996	+	0.950738i	$0.600328\pi$
$90$		0			0
$91$	3.12775			0.00360304
$92$		−	257.879i		−	0.292236i
$93$		0			0
$94$	−359.086			−0.394010
$95$	−98.1666	+	101.358i	−0.106018	+	0.109464i
$96$		0			0
$97$		−	1155.19i		−	1.20919i	−0.796532	−	0.604596i	$-0.793334\pi$
$97$		−	1155.19i		−	1.20919i	0.796532	−	0.604596i	$-0.206666\pi$
$98$		−	634.996i		−	0.654533i
$99$		0			0
$100$	−18.1961	−	568.664i	−0.0181961	−	0.568664i
$101$	1388.75			1.36817			0.684086	−	0.729401i	$-0.260201\pi$
$101$	1388.75			1.36817			0.684086	+	0.729401i	$0.260201\pi$
$102$		0			0
$103$		−	1003.13i		−	0.959622i	−0.877372	−	0.479811i	$-0.840705\pi$
$103$		−	1003.13i		−	0.959622i	0.877372	−	0.479811i	$-0.159295\pi$
$104$	71.1916			0.0671242
$105$		0			0
$106$	−397.625			−0.364347
$107$			1626.48i			1.46952i	0.678329	+	0.734758i	$0.262704\pi$
$107$			1626.48i			1.46952i	−0.678329	+	0.734758i	$0.737296\pi$
$108$		0			0
$109$	268.408			0.235861			0.117930	−	0.993022i	$-0.462374\pi$
$109$	268.408			0.235861			0.117930	+	0.993022i	$0.462374\pi$
$110$	−240.709	−	233.130i	−0.208642	−	0.202073i
$111$		0			0
$112$			7.03453i			0.00593483i
$113$		−	1190.42i		−	0.991022i	−0.868602	−	0.495511i	$-0.834981\pi$
$113$		−	1190.42i		−	0.991022i	0.868602	−	0.495511i	$-0.165019\pi$
$114$		0			0
$115$	455.014	+	440.687i	0.368959	+	0.357342i
$116$	−892.972			−0.714745
$117$		0			0
$118$		−	278.264i		−	0.217087i
$119$	−70.9468			−0.0546528
$120$		0			0
$121$	−1070.49			−0.804278
$122$			919.472i			0.682337i
$123$		0			0
$124$	971.135			0.703311
$125$	1034.47	+	939.678i	0.740207	+	0.672379i
$126$		0			0
$127$			396.224i			0.276844i	0.990373	+	0.138422i	$0.0442031\pi$
$127$			396.224i			0.276844i	−0.990373	+	0.138422i	$0.955797\pi$
$128$		−	688.605i		−	0.475505i
$129$		0			0
$130$	−44.1174	+	45.5517i	−0.0297643	+	0.0307319i
$131$	995.717			0.664093			0.332046	−	0.943263i	$-0.392261\pi$
$131$	995.717			0.664093			0.332046	+	0.943263i	$0.392261\pi$
$132$		0			0
$133$		−	12.9238i		−	0.00842585i
$134$	1414.27			0.911748
$135$		0			0
$136$	−1614.84			−1.01817
$137$			2198.88i			1.37126i	0.727949	+	0.685631i	$0.240474\pi$
$137$			2198.88i			1.37126i	−0.727949	+	0.685631i	$0.759526\pi$
$138$		0			0
$139$	475.975			0.290444			0.145222	−	0.989399i	$-0.453610\pi$
$139$	475.975			0.290444			0.145222	+	0.989399i	$0.453610\pi$
$140$	37.4329	+	36.2543i	0.0225975	+	0.0218860i
$141$		0			0
$142$			1367.08i			0.807905i
$143$		−	49.2982i		−	0.0288288i
$144$		0			0
$145$	1525.99	−	1575.60i	0.873977	−	0.902390i
$146$	1301.85			0.737957
$147$		0			0
$148$		−	1365.27i		−	0.758271i
$149$	1440.38			0.791948			0.395974	−	0.918262i	$-0.370407\pi$
$149$	1440.38			0.791948			0.395974	+	0.918262i	$0.370407\pi$
$150$		0			0
$151$	−347.351			−0.187199			−0.0935994	−	0.995610i	$-0.529837\pi$
$151$	−347.351			−0.187199			−0.0935994	+	0.995610i	$0.529837\pi$
$152$		−	294.163i		−	0.156972i
$153$		0			0
$154$	30.6920			0.0160599
$155$	−1659.56	+	1713.52i	−0.859996	+	0.887954i
$156$		0			0
$157$		−	520.092i		−	0.264381i	−0.991224	−	0.132191i	$-0.957799\pi$
$157$		−	520.092i		−	0.264381i	0.991224	−	0.132191i	$-0.0422011\pi$
$158$			1449.65i			0.729923i
$159$		0			0
$160$	1395.07	+	1351.15i	0.689313	+	0.667609i
$161$	−58.0174			−0.0284001
$162$		0			0
$163$			3756.90i			1.80530i	0.430380	+	0.902648i	$0.358380\pi$
$163$			3756.90i			1.80530i	−0.430380	+	0.902648i	$0.641620\pi$
$164$	−632.984			−0.301389
$165$		0			0
$166$	−1785.49			−0.834826
$167$			936.744i			0.434056i	0.976165	+	0.217028i	$0.0696364\pi$
$167$			936.744i			0.434056i	−0.976165	+	0.217028i	$0.930364\pi$
$168$		0			0
$169$	2187.67			0.995754
$170$	1000.72	−	1033.25i	0.451479	−	0.466157i
$171$		0			0
$172$		−	2162.76i		−	0.958772i
$173$			2493.98i			1.09603i	0.836467	+	0.548017i	$0.184617\pi$
$173$			2493.98i			1.09603i	−0.836467	+	0.548017i	$0.815383\pi$
$174$		0			0
$175$	−127.937	+	4.09374i	−0.0552637	+	0.00176833i
$176$	110.875			0.0474860
$177$		0			0
$178$			966.670i			0.407050i
$179$	2769.95			1.15662			0.578312	−	0.815816i	$-0.303712\pi$
$179$	2769.95			1.15662			0.578312	+	0.815816i	$0.303712\pi$
$180$		0			0
$181$	−477.139			−0.195942			−0.0979708	−	0.995189i	$-0.531235\pi$
$181$	−477.139			−0.195942			−0.0979708	+	0.995189i	$0.531235\pi$
$182$		−	5.80815i		−	0.00236554i
$183$		0			0
$184$	−1320.55			−0.529089
$185$	2408.94	+	2333.09i	0.957344	+	0.927201i
$186$		0			0
$187$			1118.23i			0.437290i
$188$		−	880.156i		−	0.341447i
$189$		0			0
$190$	188.219	+	182.293i	0.0718677	+	0.0696049i
$191$	5099.95			1.93204			0.966019	−	0.258469i	$-0.0832181\pi$
$191$	5099.95			1.93204			0.966019	+	0.258469i	$0.0832181\pi$
$192$		0			0
$193$		−	4538.00i		−	1.69250i	−0.532786	−	0.846250i	$-0.678855\pi$
$193$		−	4538.00i		−	1.69250i	0.532786	−	0.846250i	$-0.321145\pi$
$194$	−2145.16			−0.793884
$195$		0			0
$196$	1556.44			0.567215
$197$		−	835.536i		−	0.302180i	−0.988520	−	0.151090i	$-0.951722\pi$
$197$		−	835.536i		−	0.302180i	0.988520	−	0.151090i	$-0.0482783\pi$
$198$		0			0
$199$	5045.21			1.79721			0.898607	−	0.438753i	$-0.144580\pi$
$199$	5045.21			1.79721			0.898607	+	0.438753i	$0.144580\pi$
$200$	−2912.02	+	93.1789i	−1.02956	+	0.0329437i
$201$		0			0
$202$		−	2578.87i		−	0.898261i
$203$			200.900i			0.0694602i
$204$		0			0
$205$	1081.70	−	1116.87i	0.368533	−	0.380514i
$206$	−1862.78			−0.630031
$207$		0			0
$208$		−	20.9820i		−	0.00699444i
$209$	−203.700			−0.0674173
$210$		0			0
$211$	−4048.83			−1.32101			−0.660504	−	0.750822i	$-0.729658\pi$
$211$	−4048.83			−1.32101			−0.660504	+	0.750822i	$0.729658\pi$
$212$		−	974.620i		−	0.315741i
$213$		0			0
$214$	3020.34			0.964797
$215$	3816.07	+	3695.92i	1.21048	+	1.17237i
$216$		0			0
$217$		−	218.485i		−	0.0683490i
$218$		−	498.427i		−	0.154852i
$219$		0			0
$220$	571.424	−	590.001i	0.175116	−	0.180808i
$221$	211.614			0.0644105
$222$		0			0
$223$		−	1976.14i		−	0.593419i	−0.954968	−	0.296709i	$-0.904111\pi$
$223$		−	1976.14i		−	0.593419i	0.954968	−	0.296709i	$-0.0958893\pi$
$224$	−177.881			−0.0530589
$225$		0			0
$226$	−2210.59			−0.650646
$227$		−	742.725i		−	0.217165i	−0.994087	−	0.108582i	$-0.965369\pi$
$227$		−	742.725i		−	0.217165i	0.994087	−	0.108582i	$-0.0346311\pi$
$228$		0			0
$229$	1151.56			0.332301			0.166151	−	0.986100i	$-0.446866\pi$
$229$	1151.56			0.332301			0.166151	+	0.986100i	$0.446866\pi$
$230$	818.346	−	844.950i	0.234609	−	0.242236i
$231$		0			0
$232$			4572.75i			1.29403i
$233$			1954.50i			0.549544i	0.961509	+	0.274772i	$0.0886024\pi$
$233$			1954.50i			0.549544i	−0.961509	+	0.274772i	$0.911398\pi$
$234$		0			0
$235$	1552.99	+	1504.09i	0.431088	+	0.417515i
$236$	682.052			0.188126
$237$		0			0
$238$			131.746i			0.0358817i
$239$	1567.50			0.424239			0.212120	−	0.977244i	$-0.431963\pi$
$239$	1567.50			0.424239			0.212120	+	0.977244i	$0.431963\pi$
$240$		0			0
$241$	−7402.25			−1.97851			−0.989255	−	0.146197i	$-0.953297\pi$
$241$	−7402.25			−1.97851			−0.989255	+	0.146197i	$0.953297\pi$
$242$			1987.88i			0.528041i
$243$		0			0
$244$	−2253.72			−0.591310
$245$	−2659.78	+	2746.25i	−0.693581	+	0.716129i
$246$		0			0
$247$			38.5482i			0.00993021i
$248$		−	4973.01i		−	1.27333i
$249$		0			0
$250$	1744.96	−	1920.99i	0.441444	−	0.485976i
$251$	−6720.19			−1.68994			−0.844970	−	0.534814i	$-0.820382\pi$
$251$	−6720.19			−1.68994			−0.844970	+	0.534814i	$0.820382\pi$
$252$		0			0
$253$			914.445i			0.227236i
$254$	735.779			0.181759
$255$		0			0
$256$	−4298.95			−1.04955
$257$		−	224.241i		−	0.0544270i	−0.999630	−	0.0272135i	$-0.991337\pi$
$257$		−	224.241i		−	0.0544270i	0.999630	−	0.0272135i	$-0.00866340\pi$
$258$		0			0
$259$	−307.156			−0.0736902
$260$	−111.652	−	108.136i	−0.0266321	−	0.0257936i
$261$		0			0
$262$		−	1849.02i		−	0.436004i
$263$		−	3935.69i		−	0.922757i	−0.887203	−	0.461379i	$-0.847355\pi$
$263$		−	3935.69i		−	0.922757i	0.887203	−	0.461379i	$-0.152645\pi$
$264$		0			0
$265$	1719.66	+	1665.52i	0.398634	+	0.386083i
$266$	−23.9993			−0.00553191
$267$		0			0
$268$			3466.52i			0.790116i
$269$	3509.03			0.795352			0.397676	−	0.917526i	$-0.369817\pi$
$269$	3509.03			0.795352			0.397676	+	0.917526i	$0.369817\pi$
$270$		0			0
$271$	−3557.73			−0.797480			−0.398740	−	0.917064i	$-0.630552\pi$
$271$	−3557.73			−0.797480			−0.398740	+	0.917064i	$0.630552\pi$
$272$			475.936i			0.106095i
$273$		0			0
$274$	4083.27			0.900290
$275$	64.5238	+	2016.49i	0.0141488	+	0.442179i
$276$		0			0
$277$		−	4502.64i		−	0.976668i	−0.872657	−	0.488334i	$-0.837605\pi$
$277$		−	4502.64i		−	0.976668i	0.872657	−	0.488334i	$-0.162395\pi$
$278$		−	883.875i		−	0.190688i
$279$		0			0
$280$	185.651	−	191.687i	0.0396243	−	0.0409124i
$281$	8534.68			1.81187			0.905936	−	0.423414i	$-0.139168\pi$
$281$	8534.68			1.81187			0.905936	+	0.423414i	$0.139168\pi$
$282$		0			0
$283$			4611.25i			0.968589i	0.874905	+	0.484294i	$0.160924\pi$
$283$			4611.25i			0.968589i	−0.874905	+	0.484294i	$0.839076\pi$
$284$	−3350.84			−0.700126
$285$		0			0
$286$	−91.5457			−0.0189273
$287$			142.408i			0.0292895i
$288$		0			0
$289$	112.947			0.0229895
$290$	−2925.85	−	2833.73i	−0.592456	−	0.573801i
$291$		0			0
$292$			3190.96i			0.639510i
$293$			7923.38i			1.57983i	0.613219	+	0.789913i	$0.289874\pi$
$293$			7923.38i			1.57983i	−0.613219	+	0.789913i	$0.710126\pi$
$294$		0			0
$295$	−1165.55	+	1203.44i	−0.230038	+	0.237516i
$296$	−6991.28			−1.37284
$297$		0			0
$298$		−	2674.75i		−	0.519946i
$299$	173.050			0.0334706
$300$		0			0
$301$	−486.575			−0.0931752
$302$			645.022i			0.122904i
$303$		0			0
$304$	−86.6977			−0.0163568
$305$	3851.36	−	3976.57i	0.723043	−	0.746549i
$306$		0			0
$307$			3908.89i			0.726684i	0.931656	+	0.363342i	$0.118364\pi$
$307$			3908.89i			0.726684i	−0.931656	+	0.363342i	$0.881636\pi$
$308$			75.2292i			0.0139175i
$309$		0			0
$310$	3181.96	+	3081.77i	0.582978	+	0.564622i
$311$	−5864.44			−1.06927			−0.534633	−	0.845084i	$-0.679550\pi$
$311$	−5864.44			−1.06927			−0.534633	+	0.845084i	$0.679550\pi$
$312$		0			0
$313$		−	2580.04i		−	0.465918i	−0.972487	−	0.232959i	$-0.925159\pi$
$313$		−	2580.04i		−	0.465918i	0.972487	−	0.232959i	$-0.0748407\pi$
$314$	−965.799			−0.173577
$315$		0			0
$316$	−3553.23			−0.632547
$317$		−	5113.14i		−	0.905939i	−0.891526	−	0.452969i	$-0.850365\pi$
$317$		−	5113.14i		−	0.905939i	0.891526	−	0.452969i	$-0.149635\pi$
$318$		0			0
$319$	3166.50			0.555768
$320$	2936.51	−	3031.97i	0.512987	−	0.529664i
$321$		0			0
$322$			107.737i			0.0186458i
$323$		−	874.389i		−	0.150626i
$324$		0			0
$325$	381.601	−	12.2105i	0.0651306	−	0.00208405i
$326$	6976.48			1.18525
$327$		0			0
$328$			3241.39i			0.545659i
$329$	−198.017			−0.0331824
$330$		0			0
$331$	−9659.43			−1.60402			−0.802009	−	0.597311i	$-0.796236\pi$
$331$	−9659.43			−1.60402			−0.802009	+	0.597311i	$0.796236\pi$
$332$		−	4376.42i		−	0.723456i
$333$		0			0
$334$	1739.51			0.284976
$335$	−6116.48	−	5923.89i	−0.997549	−	0.966140i
$336$		0			0
$337$			10014.6i			1.61878i	0.587269	+	0.809392i	$0.300203\pi$
$337$			10014.6i			1.61878i	−0.587269	+	0.809392i	$0.699797\pi$
$338$		−	4062.45i		−	0.653753i
$339$		0			0
$340$	2532.60	+	2452.86i	0.403969	+	0.391250i
$341$	−3443.67			−0.546877
$342$		0			0
$343$		−	701.406i		−	0.110415i
$344$	−11075.1			−1.73584
$345$		0			0
$346$	4631.27			0.719591
$347$			302.277i			0.0467639i	0.999727	+	0.0233819i	$0.00744338\pi$
$347$			302.277i			0.0467639i	−0.999727	+	0.0233819i	$0.992557\pi$
$348$		0			0
$349$	−4268.50			−0.654692			−0.327346	−	0.944904i	$-0.606154\pi$
$349$	−4268.50			−0.654692			−0.327346	+	0.944904i	$0.606154\pi$
$350$	7.60198	+	237.577i	0.00116098	+	0.0362829i
$351$		0			0
$352$			2803.69i			0.424537i
$353$			5331.06i			0.803807i	0.915682	+	0.401904i	$0.131651\pi$
$353$			5331.06i			0.803807i	−0.915682	+	0.401904i	$0.868349\pi$
$354$		0			0
$355$	5726.22	−	5912.38i	0.856102	−	0.883934i
$356$	−2369.40			−0.352748
$357$		0			0
$358$		−	5143.73i		−	0.759370i
$359$	−5000.05			−0.735077			−0.367538	−	0.930008i	$-0.619799\pi$
$359$	−5000.05			−0.735077			−0.367538	+	0.930008i	$0.619799\pi$
$360$		0			0
$361$	−6699.72			−0.976778
$362$			886.036i			0.128644i
$363$		0			0
$364$	14.2364			0.00204997
$365$	−5630.28	−	5453.00i	−0.807403	−	0.781981i
$366$		0			0
$367$		−	11958.3i		−	1.70087i	−0.526084	−	0.850433i	$-0.676340\pi$
$367$		−	11958.3i		−	1.70087i	0.526084	−	0.850433i	$-0.323660\pi$
$368$			389.201i			0.0551318i
$369$		0			0
$370$	4332.49	−	4473.34i	0.608745	−	0.628535i
$371$	−219.269			−0.0306843
$372$		0			0
$373$			8400.04i			1.16605i	0.812453	+	0.583026i	$0.198132\pi$
$373$			8400.04i			1.16605i	−0.812453	+	0.583026i	$0.801868\pi$
$374$	2076.53			0.287099
$375$		0			0
$376$	−4507.12			−0.618183
$377$		−	599.228i		−	0.0818616i
$378$		0			0
$379$	3416.45			0.463038			0.231519	−	0.972830i	$-0.425630\pi$
$379$	3416.45			0.463038			0.231519	+	0.972830i	$0.425630\pi$
$380$	−446.819	+	461.345i	−0.0603192	+	0.0622802i
$381$		0			0
$382$		−	9470.49i		−	1.26846i
$383$		−	6883.21i		−	0.918317i	−0.888354	−	0.459159i	$-0.848151\pi$
$383$		−	6883.21i		−	0.918317i	0.888354	−	0.459159i	$-0.151849\pi$
$384$		0			0
$385$	−132.738	−	128.558i	−0.0175713	−	0.0170180i
$386$	−8426.97			−1.11120
$387$		0			0
$388$		−	5258.00i		−	0.687976i
$389$	−6131.20			−0.799137			−0.399568	−	0.916703i	$-0.630840\pi$
$389$	−6131.20			−0.799137			−0.399568	+	0.916703i	$0.630840\pi$
$390$		0			0
$391$	−3925.29			−0.507699
$392$		−	7970.24i		−	1.02693i
$393$		0			0
$394$	−1551.57			−0.198393
$395$	6072.09	−	6269.49i	0.773468	−	0.798613i
$396$		0			0
$397$			1078.96i			0.136401i	0.997672	+	0.0682006i	$0.0217258\pi$
$397$			1078.96i			0.136401i	−0.997672	+	0.0682006i	$0.978274\pi$
$398$		−	9368.85i		−	1.17994i
$399$		0			0
$400$	27.4623	+	858.249i	0.00343278	+	0.107281i
$401$	1481.85			0.184539			0.0922694	−	0.995734i	$-0.470588\pi$
$401$	1481.85			0.184539			0.0922694	+	0.995734i	$0.470588\pi$
$402$		0			0
$403$			651.680i			0.0805521i
$404$	6321.07			0.778428
$405$		0			0
$406$	373.067			0.0456034
$407$			4841.27i			0.589613i
$408$		0			0
$409$	6855.84			0.828850			0.414425	−	0.910084i	$-0.363983\pi$
$409$	6855.84			0.828850			0.414425	+	0.910084i	$0.363983\pi$
$410$	−2073.99	−	2008.69i	−0.249823	−	0.241957i
$411$		0			0
$412$		−	4565.87i		−	0.545981i
$413$		−	153.447i		−	0.0182824i
$414$		0			0
$415$	7721.96	+	7478.83i	0.913388	+	0.884629i
$416$	530.570			0.0625320
$417$		0			0
$418$			378.266i			0.0442622i
$419$	1750.38			0.204085			0.102043	−	0.994780i	$-0.467462\pi$
$419$	1750.38			0.204085			0.102043	+	0.994780i	$0.467462\pi$
$420$		0			0
$421$	185.281			0.0214490			0.0107245	−	0.999942i	$-0.496586\pi$
$421$	185.281			0.0214490			0.0107245	+	0.999942i	$0.496586\pi$
$422$			7518.58i			0.867296i
$423$		0			0
$424$	−4990.85			−0.571644
$425$	−8655.87	+	276.971i	−0.987933	+	0.0316119i
$426$		0			0
$427$			507.039i			0.0574645i
$428$			7403.17i			0.836088i
$429$		0			0
$430$	6863.23	−	7086.35i	0.769708	−	0.794731i
$431$	2340.26			0.261546			0.130773	−	0.991412i	$-0.458254\pi$
$431$	2340.26			0.261546			0.130773	+	0.991412i	$0.458254\pi$
$432$		0			0
$433$		−	12295.7i		−	1.36465i	−0.731048	−	0.682326i	$-0.760968\pi$
$433$		−	12295.7i		−	1.36465i	0.731048	−	0.682326i	$-0.239032\pi$
$434$	−405.722			−0.0448739
$435$		0			0
$436$	1221.70			0.134194
$437$		−	715.040i		−	0.0782723i
$438$		0			0
$439$	4488.49			0.487981			0.243991	−	0.969778i	$-0.421543\pi$
$439$	4488.49			0.487981			0.243991	+	0.969778i	$0.421543\pi$
$440$	−3021.29	−	2926.16i	−0.327351	−	0.317044i
$441$		0			0
$442$		−	392.963i		−	0.0422881i
$443$			9811.11i			1.05223i	0.850412	+	0.526117i	$0.176353\pi$
$443$			9811.11i			1.05223i	−0.850412	+	0.526117i	$0.823647\pi$
$444$		0			0
$445$	4049.05	−	4180.69i	0.431334	−	0.445356i
$446$	−3669.65			−0.389603
$447$		0			0
$448$			386.598i			0.0407701i
$449$	4230.71			0.444676			0.222338	−	0.974970i	$-0.428631\pi$
$449$	4230.71			0.444676			0.222338	+	0.974970i	$0.428631\pi$
$450$		0			0
$451$	2244.58			0.234352
$452$		−	5418.37i		−	0.563846i
$453$		0			0
$454$	−1379.22			−0.142577
$455$	−24.3284	+	25.1193i	−0.00250666	+	0.00258816i
$456$		0			0
$457$			4686.07i			0.479662i	0.970815	+	0.239831i	$0.0770920\pi$
$457$			4686.07i			0.479662i	−0.970815	+	0.239831i	$0.922908\pi$
$458$		−	2138.42i		−	0.218169i
$459$		0			0
$460$	2071.06	+	2005.85i	0.209921	+	0.203311i
$461$	12995.4			1.31292			0.656460	−	0.754361i	$-0.272053\pi$
$461$	12995.4			1.31292			0.656460	+	0.754361i	$0.272053\pi$
$462$		0			0
$463$			950.920i			0.0954492i	0.998861	+	0.0477246i	$0.0151970\pi$
$463$			950.920i			0.0954492i	−0.998861	+	0.0477246i	$0.984803\pi$
$464$	1347.71			0.134840
$465$		0			0
$466$	3629.47			0.360798
$467$		−	18320.3i		−	1.81534i	−0.419685	−	0.907670i	$-0.637860\pi$
$467$		−	18320.3i		−	1.81534i	0.419685	−	0.907670i	$-0.362140\pi$
$468$		0			0
$469$	779.893			0.0767849
$470$	2793.06	−	2883.86i	0.274116	−	0.283027i
$471$		0			0
$472$		−	3492.66i		−	0.340599i
$473$			7669.19i			0.745518i
$474$		0			0
$475$	−50.4537	−	1576.77i	−0.00487363	−	0.152310i
$476$	−322.924			−0.0310949
$477$		0			0
$478$		−	2910.81i		−	0.278530i
$479$	−10559.3			−1.00724			−0.503619	−	0.863926i	$-0.667998\pi$
$479$	−10559.3			−1.00724			−0.503619	+	0.863926i	$0.667998\pi$
$480$		0			0
$481$	916.161			0.0868469
$482$			13745.8i			1.29897i
$483$		0			0
$484$	−4872.50			−0.457597
$485$	9277.46	+	8985.35i	0.868593	+	0.841245i
$486$		0			0
$487$			677.906i			0.0630778i	0.999503	+	0.0315389i	$0.0100408\pi$
$487$			677.906i			0.0630778i	−0.999503	+	0.0315389i	$0.989959\pi$
$488$			11540.9i			1.07056i
$489$		0			0
$490$	5099.73	+	4939.16i	0.470168	+	0.455364i
$491$	−8179.39			−0.751794			−0.375897	−	0.926662i	$-0.622665\pi$
$491$	−8179.39			−0.751794			−0.375897	+	0.926662i	$0.622665\pi$
$492$		0			0
$493$			13592.3i			1.24172i
$494$	71.5831			0.00651958
$495$		0			0
$496$	−1465.68			−0.132683
$497$			753.869i			0.0680395i
$498$		0			0
$499$	−7533.28			−0.675823			−0.337912	−	0.941178i	$-0.609720\pi$
$499$	−7533.28			−0.675823			−0.337912	+	0.941178i	$0.609720\pi$
$500$	4708.54	+	4277.07i	0.421144	+	0.382553i
$501$		0			0
$502$			12479.2i			1.10951i
$503$			5729.79i			0.507910i	0.967216	+	0.253955i	$0.0817315\pi$
$503$			5729.79i			0.507910i	−0.967216	+	0.253955i	$0.918268\pi$
$504$		0			0
$505$	−10802.0	+	11153.2i	−0.951848	+	0.982793i
$506$	1698.10			0.149190
$507$		0			0
$508$			1803.47i			0.157512i
$509$	17391.8			1.51450			0.757250	−	0.653126i	$-0.226543\pi$
$509$	17391.8			1.51450			0.757250	+	0.653126i	$0.226543\pi$
$510$		0			0
$511$	717.899			0.0621487
$512$			2474.21i			0.213566i
$513$		0			0
$514$	−416.409			−0.0357335
$515$	8056.23	+	7802.57i	0.689320	+	0.667616i
$516$		0			0
$517$			3121.05i			0.265501i
$518$			570.382i			0.0483806i
$519$		0			0
$520$	−553.746	+	571.748i	−0.0466988	+	0.0482170i
$521$	−3803.92			−0.319871			−0.159935	−	0.987127i	$-0.551129\pi$
$521$	−3803.92			−0.319871			−0.159935	+	0.987127i	$0.551129\pi$
$522$		0			0
$523$		−	997.925i		−	0.0834345i	−0.999129	−	0.0417172i	$-0.986717\pi$
$523$		−	997.925i		−	0.0834345i	0.999129	−	0.0417172i	$-0.0132829\pi$
$524$	4532.14			0.377839
$525$		0			0
$526$	−7308.49			−0.605828
$527$		−	14782.1i		−	1.22185i
$528$		0			0
$529$	8957.06			0.736177
$530$	3092.83	−	3193.38i	0.253479	−	0.261720i
$531$		0			0
$532$		−	58.8246i		−	0.00479393i
$533$		−	424.763i		−	0.0345188i
$534$		0			0
$535$	−13062.5	−	12651.2i	−1.05559	−	1.02235i
$536$	17751.4			1.43049
$537$		0			0
$538$		−	6516.20i		−	0.522181i
$539$	−5519.17			−0.441053
$540$		0			0
$541$	−4211.74			−0.334707			−0.167354	−	0.985897i	$-0.553522\pi$
$541$	−4211.74			−0.334707			−0.167354	+	0.985897i	$0.553522\pi$
$542$			6606.63i			0.523578i
$543$		0			0
$544$	−12034.9			−0.948517
$545$	−2087.74	+	2155.62i	−0.164090	+	0.169425i
$546$		0			0
$547$			13471.6i			1.05303i	0.850167	+	0.526514i	$0.176501\pi$
$547$			13471.6i			1.05303i	−0.850167	+	0.526514i	$0.823499\pi$
$548$			10008.5i			0.780186i
$549$		0			0
$550$	3744.58	−	119.819i	0.290308	−	0.00928929i
$551$	−2476.01			−0.191436
$552$		0			0
$553$			799.403i			0.0614721i
$554$	−8361.29			−0.641222
$555$		0			0
$556$	2166.47			0.165249
$557$			13291.0i			1.01105i	0.862811	+	0.505526i	$0.168702\pi$
$557$			13291.0i			1.01105i	−0.862811	+	0.505526i	$0.831298\pi$
$558$		0			0
$559$	1451.32			0.109811
$560$	−56.4951	−	54.7163i	−0.00426314	−	0.00412891i
$561$		0			0
$562$		−	15848.7i		−	1.18957i
$563$			13629.1i			1.02024i	0.860103	+	0.510121i	$0.170399\pi$
$563$			13629.1i			1.02024i	−0.860103	+	0.510121i	$0.829601\pi$
$564$		0			0
$565$	9560.42	+	9259.40i	0.711876	+	0.689462i
$566$	8562.99			0.635918
$567$		0			0
$568$			17159.0i			1.26757i
$569$	19977.4			1.47187			0.735935	−	0.677052i	$-0.236743\pi$
$569$	19977.4			1.47187			0.735935	+	0.677052i	$0.236743\pi$
$570$		0			0
$571$	−21864.1			−1.60242			−0.801211	−	0.598383i	$-0.795810\pi$
$571$	−21864.1			−1.60242			−0.801211	+	0.598383i	$0.795810\pi$
$572$		−	224.388i		−	0.0164023i
$573$		0			0
$574$	264.448			0.0192297
$575$	−7078.42	+	226.495i	−0.513375	+	0.0164270i
$576$		0			0
$577$		−	10696.0i		−	0.771714i	−0.922559	−	0.385857i	$-0.873906\pi$
$577$		−	10696.0i		−	0.771714i	0.922559	−	0.385857i	$-0.126094\pi$
$578$		−	209.741i		−	0.0150935i
$579$		0			0
$580$	6945.76	−	7171.56i	0.497253	−	0.513419i
$581$	−984.603			−0.0703067
$582$		0			0
$583$			3456.02i			0.245513i
$584$	16340.3			1.15782
$585$		0			0
$586$	14713.5			1.03722
$587$		−	26421.1i		−	1.85778i	−0.370358	−	0.928889i	$-0.620765\pi$
$587$		−	26421.1i		−	1.85778i	0.370358	−	0.928889i	$-0.379235\pi$
$588$		0			0
$589$	2692.74			0.188374
$590$	2234.77	+	2164.40i	0.155939	+	0.151029i
$591$		0			0
$592$			2060.51i			0.143052i
$593$			25567.0i			1.77051i	0.465110	+	0.885253i	$0.346015\pi$
$593$			25567.0i			1.77051i	−0.465110	+	0.885253i	$0.653985\pi$
$594$		0			0
$595$	551.841	−	569.782i	0.0380223	−	0.0392584i
$596$	6556.07			0.450582
$597$		0			0
$598$		−	321.349i		−	0.0219748i
$599$	−3565.31			−0.243196			−0.121598	−	0.992579i	$-0.538802\pi$
$599$	−3565.31			−0.243196			−0.121598	+	0.992579i	$0.538802\pi$
$600$		0			0
$601$	8979.40			0.609446			0.304723	−	0.952441i	$-0.401436\pi$
$601$	8979.40			0.609446			0.304723	+	0.952441i	$0.401436\pi$
$602$			903.559i			0.0611733i
$603$		0			0
$604$	−1581.01			−0.106508
$605$	8326.56	−	8597.26i	0.559542	−	0.577733i
$606$		0			0
$607$		−	20174.5i		−	1.34903i	−0.738263	−	0.674513i	$-0.764353\pi$
$607$		−	20174.5i		−	1.34903i	0.738263	−	0.674513i	$-0.235647\pi$
$608$		−	2192.31i		−	0.146234i
$609$		0			0
$610$	−7384.39	−	7151.88i	−0.490140	−	0.474707i
$611$	590.628			0.0391068
$612$		0			0
$613$		−	7366.17i		−	0.485346i	−0.970108	−	0.242673i	$-0.921976\pi$
$613$		−	7366.17i		−	0.485346i	0.970108	−	0.242673i	$-0.0780242\pi$
$614$	7258.72			0.477098
$615$		0			0
$616$	385.235			0.0251973
$617$			1688.85i			0.110195i	0.998481	+	0.0550975i	$0.0175470\pi$
$617$			1688.85i			0.110195i	−0.998481	+	0.0550975i	$0.982453\pi$
$618$		0			0
$619$	15073.5			0.978765			0.489383	−	0.872069i	$-0.337222\pi$
$619$	15073.5			0.978765			0.489383	+	0.872069i	$0.337222\pi$
$620$	−7553.73	+	7799.30i	−0.489299	+	0.505206i
$621$		0			0
$622$			10890.1i			0.702017i
$623$			533.066i			0.0342807i
$624$		0			0
$625$	−15593.0	+	998.915i	−0.997954	+	0.0639306i
$626$	−4791.07			−0.305894
$627$		0			0
$628$		−	2367.27i		−	0.150421i
$629$	−20781.3			−1.31734
$630$		0			0
$631$	−5188.82			−0.327359			−0.163680	−	0.986514i	$-0.552336\pi$
$631$	−5188.82			−0.327359			−0.163680	+	0.986514i	$0.552336\pi$
$632$			18195.5i			1.14522i
$633$		0			0
$634$	−9494.98			−0.594785
$635$	−3182.12	−	3081.93i	−0.198864	−	0.192603i
$636$		0			0
$637$			1044.45i			0.0649647i
$638$		−	5880.12i		−	0.364884i
$639$		0			0
$640$	5530.27	+	5356.14i	0.341567	+	0.330813i
$641$	−13861.3			−0.854117			−0.427058	−	0.904224i	$-0.640450\pi$
$641$	−13861.3			−0.854117			−0.427058	+	0.904224i	$0.640450\pi$
$642$		0			0
$643$		−	27120.4i		−	1.66333i	−0.555274	−	0.831667i	$-0.687387\pi$
$643$		−	27120.4i		−	1.66333i	0.555274	−	0.831667i	$-0.312613\pi$
$644$	−264.074			−0.0161583
$645$		0			0
$646$	−1623.72			−0.0988923
$647$		−	941.957i		−	0.0572367i	−0.999590	−	0.0286184i	$-0.990889\pi$
$647$		−	941.957i		−	0.0572367i	0.999590	−	0.0286184i	$-0.00911075\pi$
$648$		0			0
$649$	−2418.57			−0.146282
$650$	−22.6746	−	708.625i	−0.00136826	−	0.0427608i
$651$		0			0
$652$			17100.0i			1.02713i
$653$		−	6306.43i		−	0.377932i	−0.981984	−	0.188966i	$-0.939486\pi$
$653$		−	6306.43i		−	0.377932i	0.981984	−	0.188966i	$-0.0605136\pi$
$654$		0			0
$655$	−7744.93	+	7996.72i	−0.462015	+	0.477035i
$656$	955.324			0.0568584
$657$		0			0
$658$			367.712i			0.0217856i
$659$	3003.37			0.177533			0.0887667	−	0.996052i	$-0.471707\pi$
$659$	3003.37			0.177533			0.0887667	+	0.996052i	$0.471707\pi$
$660$		0			0
$661$	14926.2			0.878310			0.439155	−	0.898411i	$-0.355278\pi$
$661$	14926.2			0.878310			0.439155	+	0.898411i	$0.355278\pi$
$662$			17937.3i			1.05310i
$663$		0			0
$664$	−22410.9			−1.30980
$665$	103.793	+	100.525i	0.00605250	+	0.00586193i
$666$		0			0
$667$			11115.2i			0.645253i
$668$			4263.72i			0.246958i
$669$		0			0
$670$	−11000.5	+	11358.2i	−0.634310	+	0.654931i
$671$	7991.74			0.459788
$672$		0			0
$673$			12069.9i			0.691321i	0.938360	+	0.345660i	$0.112345\pi$
$673$			12069.9i			0.691321i	−0.938360	+	0.345660i	$0.887655\pi$
$674$	18596.9			1.06280
$675$		0			0
$676$	9957.48			0.566539
$677$			8131.34i			0.461614i	0.973000	+	0.230807i	$0.0741367\pi$
$677$			8131.34i			0.461614i	−0.973000	+	0.230807i	$0.925863\pi$
$678$		0			0
$679$	−1182.94			−0.0668587
$680$	12560.6	−	12969.0i	0.708351	−	0.731379i
$681$		0			0
$682$			6394.81i			0.359047i
$683$		−	28805.3i		−	1.61377i	−0.590710	−	0.806884i	$-0.701152\pi$
$683$		−	28805.3i		−	1.61377i	0.590710	−	0.806884i	$-0.298848\pi$
$684$		0			0
$685$	−17659.5	−	17103.4i	−0.985012	−	0.953998i
$686$	−1302.49			−0.0724919
$687$		0			0
$688$			3264.12i			0.180877i
$689$	654.018			0.0361627
$690$		0			0
$691$	21170.5			1.16551			0.582754	−	0.812649i	$-0.301975\pi$
$691$	21170.5			1.16551			0.582754	+	0.812649i	$0.301975\pi$
$692$			11351.7i			0.623594i
$693$		0			0
$694$	561.321			0.0307024
$695$	−3702.25	+	3822.61i	−0.202064	+	0.208633i
$696$		0			0
$697$			9634.92i			0.523599i
$698$			7926.51i			0.429832i
$699$		0			0
$700$	−582.324	+	18.6332i	−0.0314426	+	0.00100610i
$701$	31508.9			1.69768			0.848841	−	0.528649i	$-0.177301\pi$
$701$	31508.9			1.69768			0.848841	+	0.528649i	$0.177301\pi$
$702$		0			0
$703$		−	3785.57i		−	0.203095i
$704$	6093.39			0.326212
$705$		0			0
$706$	9899.66			0.527732
$707$		−	1422.11i		−	0.0756490i
$708$		0			0
$709$	16548.3			0.876567			0.438283	−	0.898837i	$-0.355587\pi$
$709$	16548.3			0.876567			0.438283	+	0.898837i	$0.355587\pi$
$710$	−10979.1	−	10633.5i	−0.580338	−	0.562066i
$711$		0			0
$712$			12133.3i			0.638644i
$713$		−	12088.2i		−	0.634931i
$714$		0			0
$715$	395.920	+	383.454i	0.0207085	+	0.0200564i
$716$	12607.8			0.658066
$717$		0			0
$718$			9284.97i			0.482607i
$719$	−32635.3			−1.69275			−0.846377	−	0.532584i	$-0.821221\pi$
$719$	−32635.3			−1.69275			−0.846377	+	0.532584i	$0.821221\pi$
$720$		0			0
$721$	−1027.22			−0.0530594
$722$			12441.2i			0.641294i
$723$		0			0
$724$	−2171.76			−0.111482
$725$	784.298	+	24510.8i	0.0401767	+	1.25560i
$726$		0			0
$727$		−	32435.2i		−	1.65468i	−0.561701	−	0.827341i	$-0.689853\pi$
$727$		−	32435.2i		−	1.65468i	0.561701	−	0.827341i	$-0.310147\pi$
$728$		−	72.9018i		−	0.00371143i
$729$		0			0
$730$	−10126.1	+	10455.3i	−0.513402	+	0.530093i
$731$	−32920.3			−1.66566
$732$		0			0
$733$			15278.0i			0.769859i	0.922946	+	0.384930i	$0.125774\pi$
$733$			15278.0i			0.769859i	−0.922946	+	0.384930i	$0.874226\pi$
$734$	−22206.3			−1.11669
$735$		0			0
$736$	−9841.67			−0.492893
$737$		−	12292.3i		−	0.614375i
$738$		0			0
$739$	20611.4			1.02599			0.512993	−	0.858393i	$-0.328537\pi$
$739$	20611.4			1.02599			0.512993	+	0.858393i	$0.328537\pi$
$740$	10964.6	+	10619.4i	0.544685	+	0.527535i
$741$		0			0
$742$			407.177i			0.0201455i
$743$		−	21784.4i		−	1.07563i	−0.843063	−	0.537815i	$-0.819250\pi$
$743$		−	21784.4i		−	1.07563i	0.843063	−	0.537815i	$-0.180750\pi$
$744$		0			0
$745$	−11203.6	+	11567.8i	−0.550964	+	0.568876i
$746$	15598.7			0.765561
$747$		0			0
$748$			5089.79i			0.248798i
$749$	1665.56			0.0812525
$750$		0			0
$751$	18299.2			0.889146			0.444573	−	0.895743i	$-0.353355\pi$
$751$	18299.2			0.889146			0.444573	+	0.895743i	$0.353355\pi$
$752$			1328.37i			0.0644156i
$753$		0			0
$754$	−1112.75			−0.0537455
$755$	2701.78	−	2789.62i	0.130236	−	0.134470i
$756$		0			0
$757$		−	27136.5i		−	1.30290i	−0.758693	−	0.651448i	$-0.774162\pi$
$757$		−	27136.5i		−	1.30290i	0.758693	−	0.651448i	$-0.225838\pi$
$758$		−	6344.27i		−	0.304003i
$759$		0			0
$760$	2362.46	+	2288.08i	0.112757	+	0.109207i
$761$	5534.03			0.263611			0.131806	−	0.991276i	$-0.457922\pi$
$761$	5534.03			0.263611			0.131806	+	0.991276i	$0.457922\pi$
$762$		0			0
$763$		−	274.856i		−	0.0130412i
$764$	23213.1			1.09924
$765$		0			0
$766$	−12782.0			−0.602912
$767$			457.691i			0.0215466i
$768$		0			0
$769$	30272.2			1.41956			0.709782	−	0.704422i	$-0.248794\pi$
$769$	30272.2			1.41956			0.709782	+	0.704422i	$0.248794\pi$
$770$	−238.730	+	246.491i	−0.0111730	+	0.0115363i
$771$		0			0
$772$		−	20655.3i		−	0.962956i
$773$		−	16238.3i		−	0.755562i	−0.925895	−	0.377781i	$-0.876687\pi$
$773$		−	16238.3i		−	0.755562i	0.925895	−	0.377781i	$-0.123313\pi$
$774$		0			0
$775$	−852.949	−	26656.3i	−0.0395340	−	1.23551i
$776$	−26925.3			−1.24557
$777$		0			0
$778$			11385.5i			0.524666i
$779$	−1755.12			−0.0807236
$780$		0			0
$781$	11882.2			0.544401
$782$			7289.17i			0.333325i
$783$		0			0
$784$	−2349.04			−0.107008
$785$	4176.92	+	4045.40i	0.189912	+	0.183932i
$786$		0			0
$787$			17287.5i			0.783017i	0.920174	+	0.391508i	$0.128047\pi$
$787$			17287.5i			0.783017i	−0.920174	+	0.391508i	$0.871953\pi$
$788$		−	3803.06i		−	0.171927i
$789$		0			0
$790$	−11642.3	−	11275.7i	−0.524322	−	0.507813i
$791$	−1219.02			−0.0547956
$792$		0			0
$793$		−	1512.36i		−	0.0677243i
$794$	2003.60			0.0895529
$795$		0			0
$796$	22964.0			1.02253
$797$			1836.82i			0.0816354i	0.999167	+	0.0408177i	$0.0129963\pi$
$797$			1836.82i			0.0816354i	−0.999167	+	0.0408177i	$0.987004\pi$
$798$		0			0
$799$	−13397.2			−0.593192
$800$	−21702.4	+	694.435i	−0.959121	+	0.0306900i
$801$		0			0
$802$		−	2751.76i		−	0.121157i
$803$		−	11315.2i		−	0.497267i
$804$		0			0
$805$	451.273	−	465.944i	0.0197581	−	0.0204005i
$806$	1210.15			0.0528857
$807$		0			0
$808$		−	32369.0i		−	1.40933i
$809$	−14459.9			−0.628410			−0.314205	−	0.949355i	$-0.601738\pi$
$809$	−14459.9			−0.628410			−0.314205	+	0.949355i	$0.601738\pi$
$810$		0			0
$811$	30701.8			1.32933			0.664664	−	0.747142i	$-0.268575\pi$
$811$	30701.8			1.32933			0.664664	+	0.747142i	$0.268575\pi$
$812$			914.424i			0.0395197i
$813$		0			0
$814$	8990.12			0.387105
$815$	−30172.1	−	29222.1i	−1.29679	−	1.25596i
$816$		0			0
$817$		−	5996.84i		−	0.256797i
$818$		−	12731.1i		−	0.544173i
$819$		0			0
$820$	4923.50	−	5083.57i	0.209678	−	0.216495i
$821$	−29132.0			−1.23838			−0.619192	−	0.785239i	$-0.712540\pi$
$821$	−29132.0			−1.23838			−0.619192	+	0.785239i	$0.712540\pi$
$822$		0			0
$823$		−	35618.9i		−	1.50862i	−0.656516	−	0.754312i	$-0.727971\pi$
$823$		−	35618.9i		−	1.50862i	0.656516	−	0.754312i	$-0.272029\pi$
$824$	−23381.0			−0.988490
$825$		0			0
$826$	−284.948			−0.0120032
$827$		−	9616.94i		−	0.404370i	−0.979347	−	0.202185i	$-0.935196\pi$
$827$		−	9616.94i		−	0.404370i	0.979347	−	0.202185i	$-0.0648042\pi$
$828$		0			0
$829$	−38846.8			−1.62751			−0.813755	−	0.581208i	$-0.802580\pi$
$829$	−38846.8			−1.62751			−0.813755	+	0.581208i	$0.802580\pi$
$830$	13888.0	−	14339.5i	0.580795	−	0.599676i
$831$		0			0
$832$		−	1153.11i		−	0.0480492i
$833$		−	23691.2i		−	0.985417i
$834$		0			0
$835$	−7523.10	−	7286.23i	−0.311794	−	0.301976i
$836$	−927.168			−0.0383574
$837$		0			0
$838$		−	3250.41i		−	0.133990i
$839$	3938.86			0.162079			0.0810396	−	0.996711i	$-0.474176\pi$
$839$	3938.86			0.162079			0.0810396	+	0.996711i	$0.474176\pi$
$840$		0			0
$841$	14100.3			0.578144
$842$		−	344.062i		−	0.0140822i
$843$		0			0
$844$	−18428.8			−0.751594
$845$	−17016.2	+	17569.4i	−0.692753	+	0.715275i
$846$		0			0
$847$			1096.21i			0.0444701i
$848$			1470.93i			0.0595661i
$849$		0			0
$850$	514.328	+	16073.8i	0.0207545	+	0.648618i
$851$	−16994.1			−0.684548
$852$		0			0
$853$		−	27660.3i		−	1.11028i	−0.831756	−	0.555142i	$-0.812664\pi$
$853$		−	27660.3i		−	1.11028i	0.831756	−	0.555142i	$-0.187336\pi$
$854$	941.561			0.0377278
$855$		0			0
$856$	37910.3			1.51372
$857$		−	21674.2i		−	0.863918i	−0.901893	−	0.431959i	$-0.857823\pi$
$857$		−	21674.2i		−	0.863918i	0.901893	−	0.431959i	$-0.142177\pi$
$858$		0			0
$859$	9236.03			0.366856			0.183428	−	0.983033i	$-0.441281\pi$
$859$	9236.03			0.366856			0.183428	+	0.983033i	$0.441281\pi$
$860$	17369.4	+	16822.5i	0.688710	+	0.667025i
$861$		0			0
$862$		−	4345.81i		−	0.171715i
$863$			8055.86i			0.317758i	0.987298	+	0.158879i	$0.0507879\pi$
$863$			8055.86i			0.317758i	−0.987298	+	0.158879i	$0.949212\pi$
$864$		0			0
$865$	−20029.5	−	19398.8i	−0.787309	−	0.762519i
$866$	−22832.9			−0.895950
$867$		0			0
$868$		−	994.464i		−	0.0388875i
$869$	12599.8			0.491853
$870$		0			0
$871$	−2326.20			−0.0904941
$872$		−	6256.08i		−	0.242956i
$873$		0			0
$874$	−1327.81			−0.0513889
$875$	962.251	−	1059.32i	0.0371772	−	0.0409276i
$876$		0			0
$877$			34062.1i			1.31151i	0.754973	+	0.655756i	$0.227650\pi$
$877$			34062.1i			1.31151i	−0.754973	+	0.655756i	$0.772350\pi$
$878$		−	8335.01i		−	0.320379i
$879$		0			0
$880$	−862.416	+	890.453i	−0.0330364	+	0.0341104i
$881$	29946.6			1.14521			0.572603	−	0.819833i	$-0.305934\pi$
$881$	29946.6			1.14521			0.572603	+	0.819833i	$0.305934\pi$
$882$		0			0
$883$			45422.6i			1.73114i	0.500792	+	0.865568i	$0.333042\pi$
$883$			45422.6i			1.73114i	−0.500792	+	0.865568i	$0.666958\pi$
$884$	963.191			0.0366466
$885$		0			0
$886$	18219.0			0.690835
$887$		−	46885.3i		−	1.77481i	−0.460991	−	0.887405i	$-0.652506\pi$
$887$		−	46885.3i		−	1.77481i	0.460991	−	0.887405i	$-0.347494\pi$
$888$		0			0
$889$	405.743			0.0153073
$890$	−7763.44	−	7519.00i	−0.292394	−	0.283188i
$891$		0			0
$892$		−	8994.69i		−	0.337628i
$893$		−	2440.47i		−	0.0914528i
$894$		0			0
$895$	−21545.3	+	22245.8i	−0.804672	+	0.830832i
$896$	−705.147			−0.0262916
$897$		0			0
$898$		−	7856.33i		−	0.291948i
$899$	−41858.4			−1.55290
$900$		0			0
$901$	−14835.1			−0.548534
$902$		−	4168.12i		−	0.153862i
$903$		0			0
$904$	−27746.5			−1.02083
$905$	3711.30	−	3831.96i	0.136318	−	0.140750i
$906$		0			0
$907$			31490.5i			1.15284i	0.817153	+	0.576420i	$0.195551\pi$
$907$			31490.5i			1.15284i	−0.817153	+	0.576420i	$0.804449\pi$
$908$		−	3380.61i		−	0.123557i
$909$		0			0
$910$	46.6460	+	45.1773i	0.00169923	+	0.00164573i
$911$	−40798.1			−1.48376			−0.741878	−	0.670535i	$-0.766065\pi$
$911$	−40798.1			−1.48376			−0.741878	+	0.670535i	$0.766065\pi$
$912$		0			0
$913$			15518.9i			0.562542i
$914$	8701.93			0.314917
$915$		0			0
$916$	5241.47			0.189064
$917$		−	1019.64i		−	0.0367190i
$918$		0			0
$919$	25901.7			0.929725			0.464862	−	0.885383i	$-0.346104\pi$
$919$	25901.7			0.929725			0.464862	+	0.885383i	$0.346104\pi$
$920$	10271.6	−	10605.5i	0.368091	−	0.380058i
$921$		0			0
$922$		−	24132.2i		−	0.861985i
$923$		−	2248.58i		−	0.0801873i
$924$		0			0
$925$	−37474.6	+	1199.11i	−1.33206	+	0.0426234i
$926$	1765.84			0.0626663
$927$		0			0
$928$			34079.3i			1.20550i
$929$	6125.72			0.216338			0.108169	−	0.994132i	$-0.465501\pi$
$929$	6125.72			0.216338			0.108169	+	0.994132i	$0.465501\pi$
$930$		0			0
$931$	4315.65			0.151922
$932$			8896.19i			0.312666i
$933$		0			0
$934$	−34020.4			−1.19184
$935$	−8980.66	−	8697.89i	−0.314117	−	0.304226i
$936$		0			0
$937$		−	32640.5i		−	1.13801i	−0.822333	−	0.569007i	$-0.807328\pi$
$937$		−	32640.5i		−	1.13801i	0.822333	−	0.569007i	$-0.192672\pi$
$938$		−	1448.24i		−	0.0504124i
$939$		0			0
$940$	7068.64	+	6846.07i	0.245270	+	0.237547i
$941$	−23106.0			−0.800460			−0.400230	−	0.916415i	$-0.631070\pi$
$941$	−23106.0			−0.800460			−0.400230	+	0.916415i	$0.631070\pi$
$942$		0			0
$943$			7879.04i			0.272086i
$944$	−1029.38			−0.0354909
$945$		0			0
$946$	14241.5			0.489463
$947$			2309.51i			0.0792491i	0.999215	+	0.0396245i	$0.0126162\pi$
$947$			2309.51i			0.0792491i	−0.999215	+	0.0396245i	$0.987384\pi$
$948$		0			0
$949$	−2141.29			−0.0732447
$950$	−2928.03	+	93.6913i	−0.0999979	+	0.00319973i
$951$		0			0
$952$			1653.63i			0.0562968i
$953$		−	27818.8i		−	0.945580i	−0.881175	−	0.472790i	$-0.843247\pi$
$953$		−	27818.8i		−	0.945580i	0.881175	−	0.472790i	$-0.156753\pi$
$954$		0			0
$955$	−39668.7	+	40958.3i	−1.34413	+	1.38783i
$956$	7134.69			0.241373
$957$		0			0
$958$			19608.4i			0.661293i
$959$	2251.70			0.0758199
$960$		0			0
$961$	15731.3			0.528056
$962$		−	1701.29i		−	0.0570185i
$963$		0			0
$964$	−33692.4			−1.12568
$965$	36445.2	+	35297.7i	1.21577	+	1.17749i
$966$		0			0
$967$			19865.9i			0.660646i	0.943868	+	0.330323i	$0.107158\pi$
$967$			19865.9i			0.660646i	−0.943868	+	0.330323i	$0.892842\pi$
$968$			24951.2i			0.828472i
$969$		0			0
$970$	16685.6	−	17228.0i	0.552311	−	0.570267i
$971$	10529.6			0.348004			0.174002	−	0.984745i	$-0.444330\pi$
$971$	10529.6			0.348004			0.174002	+	0.984745i	$0.444330\pi$
$972$		0			0
$973$		−	487.409i		−	0.0160592i
$974$	1258.86			0.0414131
$975$		0			0
$976$	3401.40			0.111553
$977$			23428.3i			0.767184i	0.923503	+	0.383592i	$0.125313\pi$
$977$			23428.3i			0.767184i	−0.923503	+	0.383592i	$0.874687\pi$
$978$		0			0
$979$	8401.97			0.274288
$980$	−12106.4	+	12499.9i	−0.394616	+	0.407445i
$981$		0			0
$982$			15188.9i			0.493583i
$983$		−	60100.0i		−	1.95004i	−0.222114	−	0.975021i	$-0.571296\pi$
$983$		−	60100.0i		−	1.95004i	0.222114	−	0.975021i	$-0.428704\pi$
$984$		0			0
$985$	6710.29	+	6499.00i	0.217063	+	0.210229i
$986$	25240.6			0.815238
$987$		0			0
$988$			175.457i			0.00564984i
$989$	−26920.9			−0.865555
$990$		0			0
$991$	13491.2			0.432455			0.216228	−	0.976343i	$-0.430625\pi$
$991$	13491.2			0.432455			0.216228	+	0.976343i	$0.430625\pi$
$992$		−	37062.3i		−	1.18622i
$993$		0			0
$994$	1399.92			0.0446707
$995$	−39242.9	+	40518.7i	−1.25034	+	1.29098i
$996$		0			0
$997$			30289.6i			0.962167i	0.876675	+	0.481084i	$0.159757\pi$
$997$			30289.6i			0.962167i	−0.876675	+	0.481084i	$0.840243\pi$
$998$			13989.1i			0.443705i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.4.b.c.244.3	yes	8
3.2	odd	2		405.4.b.b.244.6	yes	8
5.2	odd	4		2025.4.a.bd.1.6		8
5.3	odd	4		2025.4.a.bd.1.3		8
5.4	even	2	inner	405.4.b.c.244.6	yes	8
15.2	even	4		2025.4.a.bc.1.3		8
15.8	even	4		2025.4.a.bc.1.6		8
15.14	odd	2		405.4.b.b.244.3	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.4.b.b.244.3	✓	8	15.14	odd	2
405.4.b.b.244.6	yes	8	3.2	odd	2
405.4.b.c.244.3	yes	8	1.1	even	1	trivial
405.4.b.c.244.6	yes	8	5.4	even	2	inner
2025.4.a.bc.1.3		8	15.2	even	4
2025.4.a.bc.1.6		8	15.8	even	4
2025.4.a.bd.1.3		8	5.3	odd	4
2025.4.a.bd.1.6		8	5.2	odd	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)

\(f(q)\)	\(=\)	\(q-1.85698i q^{2} +4.55164 q^{4} +(-7.77825 + 8.03112i) q^{5} -1.02402i q^{7} -23.3081i q^{8} +O(q^{10})\)	\(q-1.85698i q^{2} +4.55164 q^{4} +(-7.77825 + 8.03112i) q^{5} -1.02402i q^{7} -23.3081i q^{8} +(14.9136 + 14.4440i) q^{10} -16.1402 q^{11} +3.05437i q^{13} -1.90159 q^{14} -6.86951 q^{16} -69.2824i q^{17} +12.6207 q^{19} +(-35.4038 + 36.5547i) q^{20} +29.9720i q^{22} -56.6563i q^{23} +(-3.99771 - 124.936i) q^{25} +5.67190 q^{26} -4.66098i q^{28} -196.187 q^{29} +213.360 q^{31} -173.708i q^{32} -128.656 q^{34} +(8.22405 + 7.96510i) q^{35} -299.951i q^{37} -23.4363i q^{38} +(187.190 + 181.296i) q^{40} -139.067 q^{41} -475.161i q^{43} -73.4644 q^{44} -105.209 q^{46} -193.371i q^{47} +341.951 q^{49} +(-232.003 + 7.42365i) q^{50} +13.9024i q^{52} -214.125i q^{53} +(125.543 - 129.624i) q^{55} -23.8680 q^{56} +364.315i q^{58} +149.848 q^{59} -495.145 q^{61} -396.204i q^{62} -377.528 q^{64} +(-24.5300 - 23.7577i) q^{65} +761.598i q^{67} -315.348i q^{68} +(14.7910 - 15.2719i) q^{70} -736.184 q^{71} +701.058i q^{73} -557.001 q^{74} +57.4446 q^{76} +16.5279i q^{77} -780.649 q^{79} +(53.4327 - 55.1698i) q^{80} +258.245i q^{82} -961.505i q^{83} +(556.415 + 538.896i) q^{85} -882.362 q^{86} +376.198i q^{88} -520.561 q^{89} +3.12775 q^{91} -257.879i q^{92} -359.086 q^{94} +(-98.1666 + 101.358i) q^{95} -1155.19i q^{97} -634.996i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 24 q^{4} + 15 q^{5}+O(q^{10}) \) 8 * q - 24 * q^4 + 15 * q^5	\( 8 q - 24 q^{4} + 15 q^{5} + 7 q^{10} - 42 q^{14} + 4 q^{16} + 118 q^{19} - 129 q^{20} + 17 q^{25} - 36 q^{26} + 318 q^{29} - 416 q^{31} + 638 q^{34} + 192 q^{35} - 265 q^{40} + 486 q^{41} + 852 q^{44} - 598 q^{46} + 350 q^{49} + 1143 q^{50} - 162 q^{55} + 1530 q^{56} - 1146 q^{59} - 398 q^{61} + 1640 q^{64} - 1833 q^{65} + 630 q^{70} - 1728 q^{71} + 1218 q^{74} - 3498 q^{76} + 2596 q^{79} + 1923 q^{80} - 233 q^{85} + 480 q^{86} + 1086 q^{89} - 2574 q^{91} + 1238 q^{94} + 1674 q^{95}+O(q^{100}) \) 8 * q - 24 * q^4 + 15 * q^5 + 7 * q^10 - 42 * q^14 + 4 * q^16 + 118 * q^19 - 129 * q^20 + 17 * q^25 - 36 * q^26 + 318 * q^29 - 416 * q^31 + 638 * q^34 + 192 * q^35 - 265 * q^40 + 486 * q^41 + 852 * q^44 - 598 * q^46 + 350 * q^49 + 1143 * q^50 - 162 * q^55 + 1530 * q^56 - 1146 * q^59 - 398 * q^61 + 1640 * q^64 - 1833 * q^65 + 630 * q^70 - 1728 * q^71 + 1218 * q^74 - 3498 * q^76 + 2596 * q^79 + 1923 * q^80 - 233 * q^85 + 480 * q^86 + 1086 * q^89 - 2574 * q^91 + 1238 * q^94 + 1674 * q^95

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