LMFDB - Embedded newform 405.4.b.c.244.1

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			244.1
Root			$-4.80346i$ of defining polynomial
Character	$\chi$	$=$	405.244
Dual form			405.4.b.c.244.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.80346i q^{2} -15.0732 q^{4} +(8.38048 + 7.40051i) q^{5} -5.92463i q^{7} +33.9759i q^{8} +O(q^{10})$	\(q-4.80346i q^{2} -15.0732 q^{4} +(8.38048 + 7.40051i) q^{5} -5.92463i q^{7} +33.9759i q^{8} +(35.5481 - 40.2553i) q^{10} -40.2408 q^{11} +42.1218i q^{13} -28.4587 q^{14} +42.6163 q^{16} +93.7609i q^{17} +132.389 q^{19} +(-126.321 - 111.550i) q^{20} +193.295i q^{22} -45.1936i q^{23} +(15.4648 + 124.040i) q^{25} +202.330 q^{26} +89.3033i q^{28} +68.8749 q^{29} -3.17649 q^{31} +67.1020i q^{32} +450.376 q^{34} +(43.8453 - 49.6512i) q^{35} +386.267i q^{37} -635.925i q^{38} +(-251.439 + 284.735i) q^{40} +483.691 q^{41} -22.0728i q^{43} +606.559 q^{44} -217.086 q^{46} -96.2512i q^{47} +307.899 q^{49} +(595.820 - 74.2846i) q^{50} -634.912i q^{52} -487.695i q^{53} +(-337.237 - 297.803i) q^{55} +201.295 q^{56} -330.838i q^{58} -783.294 q^{59} +240.900 q^{61} +15.2581i q^{62} +663.252 q^{64} +(-311.723 + 353.001i) q^{65} +572.876i q^{67} -1413.28i q^{68} +(-238.498 - 210.609i) q^{70} -771.300 q^{71} +651.280i q^{73} +1855.42 q^{74} -1995.53 q^{76} +238.412i q^{77} +763.161 q^{79} +(357.145 + 315.382i) q^{80} -2323.39i q^{82} +741.867i q^{83} +(-693.878 + 785.761i) q^{85} -106.026 q^{86} -1367.22i q^{88} -1001.56 q^{89} +249.556 q^{91} +681.213i q^{92} -462.339 q^{94} +(1109.48 + 979.747i) q^{95} +537.661i q^{97} -1478.98i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	4.80346i		−	1.69828i	−0.528168	−	0.849140i	$-0.677121\pi$
$2$		−	4.80346i		−	1.69828i	0.528168	−	0.849140i	$-0.322879\pi$
$3$		0			0
$3$		0			0
$4$	−15.0732			−1.88415
$5$	8.38048	+	7.40051i	0.749573	+	0.661922i
$5$	8.38048	+	7.40051i	0.749573	+	0.661922i
$6$		0			0
$7$		−	5.92463i		−	0.319900i	−0.987125	−	0.159950i	$-0.948867\pi$
$7$		−	5.92463i		−	0.319900i	0.987125	−	0.159950i	$-0.0511333\pi$
$8$			33.9759i			1.50154i
$9$		0			0
$10$	35.5481	−	40.2553i	1.12413	−	1.27298i
$11$	−40.2408			−1.10301			−0.551503	−	0.834173i	$-0.685945\pi$
$11$	−40.2408			−1.10301			−0.551503	+	0.834173i	$0.685945\pi$
$12$		0			0
$13$			42.1218i			0.898653i	0.893368	+	0.449327i	$0.148336\pi$
$13$			42.1218i			0.898653i	−0.893368	+	0.449327i	$0.851664\pi$
$14$	−28.4587			−0.543279
$15$		0			0
$16$	42.6163			0.665879
$17$			93.7609i			1.33767i	0.743412	+	0.668834i	$0.233206\pi$
$17$			93.7609i			1.33767i	−0.743412	+	0.668834i	$0.766794\pi$
$18$		0			0
$19$	132.389			1.59853			0.799266	−	0.600977i	$-0.205222\pi$
$19$	132.389			1.59853			0.799266	+	0.600977i	$0.205222\pi$
$20$	−126.321	−	111.550i	−1.41231	−	1.24716i
$21$		0			0
$22$			193.295i			1.87321i
$23$		−	45.1936i		−	0.409718i	−0.978791	−	0.204859i	$-0.934326\pi$
$23$		−	45.1936i		−	0.409718i	0.978791	−	0.204859i	$-0.0656736\pi$
$24$		0			0
$25$	15.4648	+	124.040i	0.123719	+	0.992317i
$26$	202.330			1.52616
$27$		0			0
$28$			89.3033i			0.602740i
$29$	68.8749			0.441026			0.220513	−	0.975384i	$-0.429227\pi$
$29$	68.8749			0.441026			0.220513	+	0.975384i	$0.429227\pi$
$30$		0			0
$31$	−3.17649			−0.0184037			−0.00920184	−	0.999958i	$-0.502929\pi$
$31$	−3.17649			−0.0184037			−0.00920184	+	0.999958i	$0.502929\pi$
$32$			67.1020i			0.370690i
$33$		0			0
$34$	450.376			2.27173
$35$	43.8453	−	49.6512i	0.211749	−	0.239788i
$36$		0			0
$37$			386.267i			1.71627i	0.513426	+	0.858134i	$0.328376\pi$
$37$			386.267i			1.71627i	−0.513426	+	0.858134i	$0.671624\pi$
$38$		−	635.925i		−	2.71476i
$39$		0			0
$40$	−251.439	+	284.735i	−0.993901	+	1.12551i
$41$	483.691			1.84243			0.921217	−	0.389050i	$-0.127197\pi$
$41$	483.691			1.84243			0.921217	+	0.389050i	$0.127197\pi$
$42$		0			0
$43$		−	22.0728i		−	0.0782809i	−0.999234	−	0.0391404i	$-0.987538\pi$
$43$		−	22.0728i		−	0.0782809i	0.999234	−	0.0391404i	$-0.0124620\pi$
$44$	606.559			2.07823
$45$		0			0
$46$	−217.086			−0.695816
$47$		−	96.2512i		−	0.298717i	−0.988783	−	0.149358i	$-0.952279\pi$
$47$		−	96.2512i		−	0.298717i	0.988783	−	0.149358i	$-0.0477208\pi$
$48$		0			0
$49$	307.899			0.897664
$50$	595.820	−	74.2846i	1.68523	−	0.210109i
$51$		0			0
$52$		−	634.912i		−	1.69320i
$53$		−	487.695i		−	1.26396i	−0.774984	−	0.631981i	$-0.782242\pi$
$53$		−	487.695i		−	1.26396i	0.774984	−	0.631981i	$-0.217758\pi$
$54$		0			0
$55$	−337.237	−	297.803i	−0.826783	−	0.730104i
$56$	201.295			0.480342
$57$		0			0
$58$		−	330.838i		−	0.748985i
$59$	−783.294			−1.72841			−0.864205	−	0.503141i	$-0.832178\pi$
$59$	−783.294			−1.72841			−0.864205	+	0.503141i	$0.832178\pi$
$60$		0			0
$61$	240.900			0.505640			0.252820	−	0.967513i	$-0.418642\pi$
$61$	240.900			0.505640			0.252820	+	0.967513i	$0.418642\pi$
$62$			15.2581i			0.0312546i
$63$		0			0
$64$	663.252			1.29541
$65$	−311.723	+	353.001i	−0.594838	+	0.673606i
$66$		0			0
$67$			572.876i			1.04460i	0.852763	+	0.522298i	$0.174925\pi$
$67$			572.876i			1.04460i	−0.852763	+	0.522298i	$0.825075\pi$
$68$		−	1413.28i		−	2.52037i
$69$		0			0
$70$	−238.498	−	210.609i	−0.407227	−	0.359609i
$71$	−771.300			−1.28925			−0.644623	−	0.764501i	$-0.722986\pi$
$71$	−771.300			−1.28925			−0.644623	+	0.764501i	$0.722986\pi$
$72$		0			0
$73$			651.280i			1.04420i	0.852884	+	0.522100i	$0.174851\pi$
$73$			651.280i			1.04420i	−0.852884	+	0.522100i	$0.825149\pi$
$74$	1855.42			2.91470
$75$		0			0
$76$	−1995.53			−3.01188
$77$			238.412i			0.352851i
$78$		0			0
$79$	763.161			1.08686			0.543432	−	0.839453i	$-0.317124\pi$
$79$	763.161			1.08686			0.543432	+	0.839453i	$0.317124\pi$
$80$	357.145	+	315.382i	0.499125	+	0.440760i
$81$		0			0
$82$		−	2323.39i		−	3.12897i
$83$			741.867i			0.981091i	0.871416	+	0.490545i	$0.163202\pi$
$83$			741.867i			0.981091i	−0.871416	+	0.490545i	$0.836798\pi$
$84$		0			0
$85$	−693.878	+	785.761i	−0.885432	+	1.00268i
$86$	−106.026			−0.132943
$87$		0			0
$88$		−	1367.22i		−	1.65621i
$89$	−1001.56			−1.19286			−0.596432	−	0.802663i	$-0.703416\pi$
$89$	−1001.56			−1.19286			−0.596432	+	0.802663i	$0.703416\pi$
$90$		0			0
$91$	249.556			0.287479
$92$			681.213i			0.771972i
$93$		0			0
$94$	−462.339			−0.507304
$95$	1109.48	+	979.747i	1.19822	+	1.05810i
$96$		0			0
$97$			537.661i			0.562796i	0.959591	+	0.281398i	$0.0907981\pi$
$97$			537.661i			0.562796i	−0.959591	+	0.281398i	$0.909202\pi$
$98$		−	1478.98i		−	1.52448i
$99$		0			0
$100$	−233.105	−	1869.68i	−0.233105	−	1.86968i
$101$	403.595			0.397616			0.198808	−	0.980038i	$-0.436293\pi$
$101$	403.595			0.397616			0.198808	+	0.980038i	$0.436293\pi$
$102$		0			0
$103$		−	1188.97i		−	1.13740i	−0.822545	−	0.568700i	$-0.807447\pi$
$103$		−	1188.97i		−	1.13740i	0.822545	−	0.568700i	$-0.192553\pi$
$104$	−1431.13			−1.34936
$105$		0			0
$106$	−2342.62			−2.14656
$107$		−	1620.92i		−	1.46449i	−0.681044	−	0.732243i	$-0.738474\pi$
$107$		−	1620.92i		−	1.46449i	0.681044	−	0.732243i	$-0.261526\pi$
$108$		0			0
$109$	−58.8903			−0.0517493			−0.0258746	−	0.999665i	$-0.508237\pi$
$109$	−58.8903			−0.0517493			−0.0258746	+	0.999665i	$0.508237\pi$
$110$	−1430.48	+	1619.91i	−1.23992	+	1.40411i
$111$		0			0
$112$		−	252.486i		−	0.213015i
$113$			1067.87i			0.889001i	0.895779	+	0.444501i	$0.146619\pi$
$113$			1067.87i			0.889001i	−0.895779	+	0.444501i	$0.853381\pi$
$114$		0			0
$115$	334.456	−	378.744i	0.271202	−	0.307114i
$116$	−1038.17			−0.830960
$117$		0			0
$118$			3762.52i			2.93532i
$119$	555.498			0.427920
$120$		0			0
$121$	288.323			0.216621
$122$		−	1157.15i		−	0.858718i
$123$		0			0
$124$	47.8799			0.0346753
$125$	−788.355	+	1153.96i	−0.564101	+	0.825706i
$126$		0			0
$127$		−	282.903i		−	0.197666i	−0.995104	−	0.0988329i	$-0.968489\pi$
$127$		−	282.903i		−	0.197666i	0.995104	−	0.0988329i	$-0.0315109\pi$
$128$		−	2649.09i		−	1.82928i
$129$		0			0
$130$	1695.63	+	1497.35i	1.14397	+	1.01020i
$131$	181.686			0.121176			0.0605878	−	0.998163i	$-0.480702\pi$
$131$	181.686			0.121176			0.0605878	+	0.998163i	$0.480702\pi$
$132$		0			0
$133$		−	784.356i		−	0.511370i
$134$	2751.79			1.77402
$135$		0			0
$136$	−3185.61			−2.00856
$137$			549.560i			0.342716i	0.985209	+	0.171358i	$0.0548155\pi$
$137$			549.560i			0.342716i	−0.985209	+	0.171358i	$0.945185\pi$
$138$		0			0
$139$	2426.25			1.48052			0.740259	−	0.672321i	$-0.234703\pi$
$139$	2426.25			1.48052			0.740259	+	0.672321i	$0.234703\pi$
$140$	−660.890	+	748.404i	−0.398967	+	0.451798i
$141$		0			0
$142$			3704.91i			2.18950i
$143$		−	1695.02i		−	0.991219i
$144$		0			0
$145$	577.205	+	509.710i	0.330581	+	0.291925i
$146$	3128.40			1.77334
$147$		0			0
$148$		−	5822.29i		−	3.23371i
$149$	1847.13			1.01559			0.507793	−	0.861479i	$-0.330461\pi$
$149$	1847.13			1.01559			0.507793	+	0.861479i	$0.330461\pi$
$150$		0			0
$151$	1550.07			0.835384			0.417692	−	0.908589i	$-0.362839\pi$
$151$	1550.07			0.835384			0.417692	+	0.908589i	$0.362839\pi$
$152$			4498.04i			2.40026i
$153$		0			0
$154$	1145.20			0.599240
$155$	−26.6205	−	23.5076i	−0.0137949	−	0.0121818i
$156$		0			0
$157$			1823.32i			0.926858i	0.886134	+	0.463429i	$0.153381\pi$
$157$			1823.32i			0.926858i	−0.886134	+	0.463429i	$0.846619\pi$
$158$		−	3665.81i		−	1.84580i
$159$		0			0
$160$	−496.589	+	562.347i	−0.245368	+	0.277859i
$161$	−267.755			−0.131069
$162$		0			0
$163$			3952.31i			1.89919i	0.313472	+	0.949597i	$0.398508\pi$
$163$			3952.31i			1.89919i	−0.313472	+	0.949597i	$0.601492\pi$
$164$	−7290.78			−3.47143
$165$		0			0
$166$	3563.53			1.66617
$167$			1911.86i			0.885894i	0.896548	+	0.442947i	$0.146067\pi$
$167$			1911.86i			0.885894i	−0.896548	+	0.442947i	$0.853933\pi$
$168$		0			0
$169$	422.753			0.192423
$170$	3774.37	+	3333.02i	1.70283	+	1.50371i
$171$		0			0
$172$			332.709i			0.147493i
$173$		−	3254.32i		−	1.43018i	−0.699031	−	0.715091i	$-0.746385\pi$
$173$		−	3254.32i		−	1.43018i	0.699031	−	0.715091i	$-0.253615\pi$
$174$		0			0
$175$	734.889	−	91.6233i	0.317442	−	0.0395775i
$176$	−1714.91			−0.734469
$177$		0			0
$178$			4810.94i			2.02582i
$179$	−1742.08			−0.727427			−0.363714	−	0.931511i	$-0.618491\pi$
$179$	−1742.08			−0.727427			−0.363714	+	0.931511i	$0.618491\pi$
$180$		0			0
$181$	−573.888			−0.235673			−0.117836	−	0.993033i	$-0.537596\pi$
$181$	−573.888			−0.235673			−0.117836	+	0.993033i	$0.537596\pi$
$182$		−	1198.73i		−	0.488220i
$183$		0			0
$184$	1535.50			0.615208
$185$	−2858.57	+	3237.10i	−1.13604	+	1.28647i
$186$		0			0
$187$		−	3773.01i		−	1.47545i
$188$			1450.82i			0.562828i
$189$		0			0
$190$	4706.17	−	5329.36i	1.79696	−	2.03491i
$191$	−2229.85			−0.844745			−0.422373	−	0.906422i	$-0.638803\pi$
$191$	−2229.85			−0.844745			−0.422373	+	0.906422i	$0.638803\pi$
$192$		0			0
$193$			541.725i			0.202043i	0.994884	+	0.101021i	$0.0322110\pi$
$193$			541.725i			0.202043i	−0.994884	+	0.101021i	$0.967789\pi$
$194$	2582.63			0.955785
$195$		0			0
$196$	−4641.03			−1.69134
$197$			2783.47i			1.00667i	0.864091	+	0.503336i	$0.167894\pi$
$197$			2783.47i			1.00667i	−0.864091	+	0.503336i	$0.832106\pi$
$198$		0			0
$199$	−1247.66			−0.444443			−0.222222	−	0.974996i	$-0.571331\pi$
$199$	−1247.66			−0.444443			−0.222222	+	0.974996i	$0.571331\pi$
$200$	−4214.36	+	525.432i	−1.49000	+	0.185768i
$201$		0			0
$202$		−	1938.65i		−	0.675263i
$203$		−	408.058i		−	0.141084i
$204$		0			0
$205$	4053.56	+	3579.56i	1.38104	+	1.21955i
$206$	−5711.15			−1.93162
$207$		0			0
$208$			1795.07i			0.598394i
$209$	−5327.44			−1.76319
$210$		0			0
$211$	−1598.90			−0.521674			−0.260837	−	0.965383i	$-0.583998\pi$
$211$	−1598.90			−0.521674			−0.260837	+	0.965383i	$0.583998\pi$
$212$			7351.13i			2.38150i
$213$		0			0
$214$	−7786.01			−2.48711
$215$	163.350	−	184.981i	0.0518158	−	0.0586772i
$216$		0			0
$217$			18.8195i			0.00588733i
$218$			282.877i			0.0878847i
$219$		0			0
$220$	5083.25	+	4488.85i	1.55779	+	1.37563i
$221$	−3949.38			−1.20210
$222$		0			0
$223$			667.796i			0.200533i	0.994961	+	0.100267i	$0.0319696\pi$
$223$			667.796i			0.200533i	−0.994961	+	0.100267i	$0.968030\pi$
$224$	397.554			0.118584
$225$		0			0
$226$	5129.49			1.50977
$227$			4252.45i			1.24337i	0.783267	+	0.621686i	$0.213552\pi$
$227$			4252.45i			1.24337i	−0.783267	+	0.621686i	$0.786448\pi$
$228$		0			0
$229$	−5294.00			−1.52768			−0.763838	−	0.645409i	$-0.776687\pi$
$229$	−5294.00			−1.52768			−0.763838	+	0.645409i	$0.776687\pi$
$230$	−1819.28	−	1606.55i	−0.521565	−	0.460576i
$231$		0			0
$232$			2340.09i			0.662218i
$233$			972.936i			0.273559i	0.990602	+	0.136779i	$0.0436751\pi$
$233$			972.936i			0.273559i	−0.990602	+	0.136779i	$0.956325\pi$
$234$		0			0
$235$	712.308	−	806.631i	0.197727	−	0.223910i
$236$	11806.8			3.25659
$237$		0			0
$238$		−	2668.31i		−	0.726727i
$239$	1506.34			0.407686			0.203843	−	0.979004i	$-0.434657\pi$
$239$	1506.34			0.407686			0.203843	+	0.979004i	$0.434657\pi$
$240$		0			0
$241$	812.835			0.217259			0.108629	−	0.994082i	$-0.465354\pi$
$241$	812.835			0.217259			0.108629	+	0.994082i	$0.465354\pi$
$242$		−	1384.95i		−	0.367883i
$243$		0			0
$244$	−3631.13			−0.952703
$245$	2580.34	+	2278.61i	0.672864	+	0.594184i
$246$		0			0
$247$			5576.47i			1.43653i
$248$		−	107.924i		−	0.0276338i
$249$		0			0
$250$	5543.00	+	3786.83i	1.40228	+	0.958001i
$251$	−37.2158			−0.00935873			−0.00467937	−	0.999989i	$-0.501489\pi$
$251$	−37.2158			−0.00935873			−0.00467937	+	0.999989i	$0.501489\pi$
$252$		0			0
$253$			1818.63i			0.451922i
$254$	−1358.91			−0.335692
$255$		0			0
$256$	−7418.77			−1.81122
$257$		−	2483.80i		−	0.602862i	−0.953488	−	0.301431i	$-0.902536\pi$
$257$		−	2483.80i		−	0.602862i	0.953488	−	0.301431i	$-0.0974642\pi$
$258$		0			0
$259$	2288.49			0.549034
$260$	4698.67	−	5320.86i	1.12077	−	1.26918i
$261$		0			0
$262$		−	872.723i		−	0.205790i
$263$			3677.92i			0.862320i	0.902275	+	0.431160i	$0.141896\pi$
$263$			3677.92i			0.862320i	−0.902275	+	0.431160i	$0.858104\pi$
$264$		0			0
$265$	3609.19	−	4087.11i	0.836645	−	0.947432i
$266$	−3767.62			−0.868450
$267$		0			0
$268$		−	8635.09i		−	1.96818i
$269$	−3140.57			−0.711836			−0.355918	−	0.934517i	$-0.615832\pi$
$269$	−3140.57			−0.711836			−0.355918	+	0.934517i	$0.615832\pi$
$270$		0			0
$271$	−8287.34			−1.85764			−0.928819	−	0.370534i	$-0.879175\pi$
$271$	−8287.34			−1.85764			−0.928819	+	0.370534i	$0.879175\pi$
$272$			3995.74i			0.890725i
$273$		0			0
$274$	2639.79			0.582027
$275$	−622.317	−	4991.46i	−0.136462	−	1.09453i
$276$		0			0
$277$		−	5606.80i		−	1.21617i	−0.793871	−	0.608086i	$-0.791937\pi$
$277$		−	5606.80i		−	1.21617i	0.793871	−	0.608086i	$-0.208063\pi$
$278$		−	11654.4i		−	2.51433i
$279$		0			0
$280$	1686.95	+	1489.69i	0.360051	+	0.317949i
$281$	2782.64			0.590742			0.295371	−	0.955383i	$-0.404557\pi$
$281$	2782.64			0.590742			0.295371	+	0.955383i	$0.404557\pi$
$282$		0			0
$283$		−	7672.81i		−	1.61166i	−0.592144	−	0.805832i	$-0.701718\pi$
$283$		−	7672.81i		−	1.61166i	0.592144	−	0.805832i	$-0.298282\pi$
$284$	11626.0			2.42914
$285$		0			0
$286$	−8141.94			−1.68337
$287$		−	2865.69i		−	0.589394i
$288$		0			0
$289$	−3878.10			−0.789354
$290$	2448.37	−	2772.58i	0.495770	−	0.561419i
$291$		0			0
$292$		−	9816.89i		−	1.96743i
$293$		−	569.568i		−	0.113565i	−0.998387	−	0.0567825i	$-0.981916\pi$
$293$		−	569.568i		−	0.113565i	0.998387	−	0.0567825i	$-0.0180841\pi$
$294$		0			0
$295$	−6564.37	−	5796.77i	−1.29557	−	1.14407i
$296$	−13123.8			−2.57704
$297$		0			0
$298$		−	8872.59i		−	1.72475i
$299$	1903.64			0.368195
$300$		0			0
$301$	−130.773			−0.0250420
$302$		−	7445.70i		−	1.41871i
$303$		0			0
$304$	5641.93			1.06443
$305$	2018.85	+	1782.78i	0.379014	+	0.334694i
$306$		0			0
$307$			1722.19i			0.320165i	0.987104	+	0.160082i	$0.0511760\pi$
$307$			1722.19i			0.320165i	−0.987104	+	0.160082i	$0.948824\pi$
$308$		−	3593.64i		−	0.664826i
$309$		0			0
$310$	−112.918	+	127.870i	−0.0206881	+	0.0234276i
$311$	−2836.40			−0.517162			−0.258581	−	0.965990i	$-0.583255\pi$
$311$	−2836.40			−0.517162			−0.258581	+	0.965990i	$0.583255\pi$
$312$		0			0
$313$		−	2592.65i		−	0.468195i	−0.972213	−	0.234098i	$-0.924786\pi$
$313$		−	2592.65i		−	0.468195i	0.972213	−	0.234098i	$-0.0752135\pi$
$314$	8758.24			1.57406
$315$		0			0
$316$	−11503.3			−2.04782
$317$		−	8039.90i		−	1.42450i	−0.701927	−	0.712249i	$-0.747677\pi$
$317$		−	8039.90i		−	1.42450i	0.701927	−	0.712249i	$-0.252323\pi$
$318$		0			0
$319$	−2771.58			−0.486454
$320$	5558.37	+	4908.40i	0.971007	+	0.857463i
$321$		0			0
$322$			1286.15i			0.222591i
$323$			12412.9i			2.13831i
$324$		0			0
$325$	−5224.78	+	651.406i	−0.891749	+	0.111180i
$326$	18984.8			3.22536
$327$		0			0
$328$			16433.8i			2.76648i
$329$	−570.253			−0.0955594
$330$		0			0
$331$	3354.77			0.557083			0.278542	−	0.960424i	$-0.410149\pi$
$331$	3354.77			0.557083			0.278542	+	0.960424i	$0.410149\pi$
$332$		−	11182.3i		−	1.84853i
$333$		0			0
$334$	9183.55			1.50450
$335$	−4239.58	+	4800.98i	−0.691441	+	0.783001i
$336$		0			0
$337$			3335.40i			0.539142i	0.962981	+	0.269571i	$0.0868819\pi$
$337$			3335.40i			0.539142i	−0.962981	+	0.269571i	$0.913118\pi$
$338$		−	2030.67i		−	0.326787i
$339$		0			0
$340$	10459.0	−	11843.9i	1.66829	−	1.88920i
$341$	127.824			0.0202994
$342$		0			0
$343$		−	3856.33i		−	0.607063i
$344$	749.946			0.117542
$345$		0			0
$346$	−15632.0			−2.42885
$347$		−	10296.7i		−	1.59296i	−0.604664	−	0.796481i	$-0.706693\pi$
$347$		−	10296.7i		−	1.59296i	0.604664	−	0.796481i	$-0.293307\pi$
$348$		0			0
$349$	741.982			0.113803			0.0569017	−	0.998380i	$-0.481878\pi$
$349$	741.982			0.113803			0.0569017	+	0.998380i	$0.481878\pi$
$350$	−440.109	−	3530.01i	−0.0672137	−	0.539106i
$351$		0			0
$352$		−	2700.24i		−	0.408873i
$353$			8617.66i			1.29935i	0.760211	+	0.649677i	$0.225096\pi$
$353$			8617.66i			1.29935i	−0.760211	+	0.649677i	$0.774904\pi$
$354$		0			0
$355$	−6463.86	−	5708.01i	−0.966383	−	0.853380i
$356$	15096.7			2.24754
$357$		0			0
$358$			8368.03i			1.23537i
$359$	8233.30			1.21041			0.605205	−	0.796070i	$-0.293091\pi$
$359$	8233.30			1.21041			0.605205	+	0.796070i	$0.293091\pi$
$360$		0			0
$361$	10667.9			1.55531
$362$			2756.65i			0.400238i
$363$		0			0
$364$	−3761.62			−0.541654
$365$	−4819.81	+	5458.04i	−0.691179	+	0.782704i
$366$		0			0
$367$		−	1578.47i		−	0.224510i	−0.993679	−	0.112255i	$-0.964193\pi$
$367$		−	1578.47i		−	0.224510i	0.993679	−	0.112255i	$-0.0358074\pi$
$368$		−	1925.98i		−	0.272823i
$369$		0			0
$370$	15549.3	+	13731.0i	2.18478	+	1.92931i
$371$	−2889.41			−0.404341
$372$		0			0
$373$			10137.6i			1.40725i	0.710573	+	0.703624i	$0.248436\pi$
$373$			10137.6i			1.40725i	−0.710573	+	0.703624i	$0.751564\pi$
$374$	−18123.5			−2.50573
$375$		0			0
$376$	3270.22			0.448534
$377$			2901.14i			0.396329i
$378$		0			0
$379$	4602.87			0.623835			0.311917	−	0.950109i	$-0.399029\pi$
$379$	4602.87			0.623835			0.311917	+	0.950109i	$0.399029\pi$
$380$	−16723.5	−	14767.9i	−2.25762	−	1.99363i
$381$		0			0
$382$			10711.0i			1.43461i
$383$			5802.14i			0.774088i	0.922061	+	0.387044i	$0.126504\pi$
$383$			5802.14i			0.774088i	−0.922061	+	0.387044i	$0.873496\pi$
$384$		0			0
$385$	−1764.37	+	1998.01i	−0.233560	+	0.264488i
$386$	2602.15			0.343125
$387$		0			0
$388$		−	8104.29i		−	1.06039i
$389$	2060.06			0.268507			0.134254	−	0.990947i	$-0.457136\pi$
$389$	2060.06			0.268507			0.134254	+	0.990947i	$0.457136\pi$
$390$		0			0
$391$	4237.39			0.548067
$392$			10461.2i			1.34788i
$393$		0			0
$394$	13370.3			1.70961
$395$	6395.65	+	5647.78i	0.814684	+	0.719420i
$396$		0			0
$397$		−	6429.56i		−	0.812822i	−0.913690	−	0.406411i	$-0.866780\pi$
$397$		−	6429.56i		−	0.812822i	0.913690	−	0.406411i	$-0.133220\pi$
$398$			5993.08i			0.754789i
$399$		0			0
$400$	659.053	+	5286.11i	0.0823816	+	0.660764i
$401$	−1519.51			−0.189229			−0.0946143	−	0.995514i	$-0.530162\pi$
$401$	−1519.51			−0.189229			−0.0946143	+	0.995514i	$0.530162\pi$
$402$		0			0
$403$		−	133.799i		−	0.0165385i
$404$	−6083.48			−0.749169
$405$		0			0
$406$	−1960.09			−0.239600
$407$		−	15543.7i		−	1.89305i
$408$		0			0
$409$	2637.37			0.318850			0.159425	−	0.987210i	$-0.449036\pi$
$409$	2637.37			0.318850			0.159425	+	0.987210i	$0.449036\pi$
$410$	17194.3	−	19471.1i	2.07113	−	2.34539i
$411$		0			0
$412$			17921.5i			2.14304i
$413$			4640.72i			0.552918i
$414$		0			0
$415$	−5490.20	+	6217.20i	−0.649406	+	0.735399i
$416$	−2826.46			−0.333121
$417$		0			0
$418$			25590.1i			2.99439i
$419$	−7959.45			−0.928030			−0.464015	−	0.885827i	$-0.653592\pi$
$419$	−7959.45			−0.928030			−0.464015	+	0.885827i	$0.653592\pi$
$420$		0			0
$421$	−11153.8			−1.29122			−0.645612	−	0.763666i	$-0.723398\pi$
$421$	−11153.8			−1.29122			−0.645612	+	0.763666i	$0.723398\pi$
$422$			7680.27i			0.885947i
$423$		0			0
$424$	16569.9			1.89789
$425$	−11630.1	+	1449.99i	−1.32739	+	0.165494i
$426$		0			0
$427$		−	1427.24i		−	0.161754i
$428$			24432.4i			2.75931i
$429$		0			0
$430$	−888.549	−	784.647i	−0.0996503	−	0.0879978i
$431$	1935.32			0.216290			0.108145	−	0.994135i	$-0.465509\pi$
$431$	1935.32			0.216290			0.108145	+	0.994135i	$0.465509\pi$
$432$		0			0
$433$		−	10586.2i		−	1.17492i	−0.809254	−	0.587459i	$-0.800128\pi$
$433$		−	10586.2i		−	1.17492i	0.809254	−	0.587459i	$-0.199872\pi$
$434$	90.3987			0.00999833
$435$		0			0
$436$	887.667			0.0975035
$437$		−	5983.14i		−	0.654948i
$438$		0			0
$439$	8290.92			0.901376			0.450688	−	0.892681i	$-0.351179\pi$
$439$	8290.92			0.901376			0.450688	+	0.892681i	$0.351179\pi$
$440$	10118.1	−	11458.0i	1.09628	−	1.24145i
$441$		0			0
$442$			18970.7i			2.04150i
$443$		−	11437.7i		−	1.22668i	−0.789818	−	0.613341i	$-0.789825\pi$
$443$		−	11437.7i		−	1.22668i	0.789818	−	0.613341i	$-0.210175\pi$
$444$		0			0
$445$	−8393.54	−	7412.05i	−0.894139	−	0.789583i
$446$	3207.73			0.340561
$447$		0			0
$448$		−	3929.52i		−	0.414403i
$449$	2039.73			0.214389			0.107194	−	0.994238i	$-0.465813\pi$
$449$	2039.73			0.214389			0.107194	+	0.994238i	$0.465813\pi$
$450$		0			0
$451$	−19464.1			−2.03221
$452$		−	16096.3i		−	1.67501i
$453$		0			0
$454$	20426.5			2.11159
$455$	2091.40	+	1846.84i	0.215486	+	0.190289i
$456$		0			0
$457$		−	17873.6i		−	1.82952i	−0.403992	−	0.914762i	$-0.632378\pi$
$457$		−	17873.6i		−	1.82952i	0.403992	−	0.914762i	$-0.367622\pi$
$458$			25429.5i			2.59442i
$459$		0			0
$460$	−5041.33	+	5708.89i	−0.510985	+	0.578649i
$461$	−10408.0			−1.05152			−0.525758	−	0.850634i	$-0.676218\pi$
$461$	−10408.0			−1.05152			−0.525758	+	0.850634i	$0.676218\pi$
$462$		0			0
$463$		−	5058.09i		−	0.507709i	−0.967242	−	0.253854i	$-0.918302\pi$
$463$		−	5058.09i		−	0.507709i	0.967242	−	0.253854i	$-0.0816984\pi$
$464$	2935.19			0.293670
$465$		0			0
$466$	4673.46			0.464579
$467$		−	12238.9i		−	1.21274i	−0.795184	−	0.606368i	$-0.792626\pi$
$467$		−	12238.9i		−	1.21274i	0.795184	−	0.606368i	$-0.207374\pi$
$468$		0			0
$469$	3394.08			0.334166
$470$	−3874.62	−	3421.54i	−0.380261	−	0.335796i
$471$		0			0
$472$		−	26613.1i		−	2.59527i
$473$			888.229i			0.0863442i
$474$		0			0
$475$	2047.37	+	16421.5i	0.197768	+	1.58625i
$476$	−8373.15			−0.806266
$477$		0			0
$478$		−	7235.64i		−	0.692365i
$479$	13956.6			1.33130			0.665650	−	0.746264i	$-0.268154\pi$
$479$	13956.6			1.33130			0.665650	+	0.746264i	$0.268154\pi$
$480$		0			0
$481$	−16270.3			−1.54233
$482$		−	3904.42i		−	0.368966i
$483$		0			0
$484$	−4345.95			−0.408147
$485$	−3978.97	+	4505.86i	−0.372527	+	0.421857i
$486$		0			0
$487$			12379.8i			1.15191i	0.817480	+	0.575957i	$0.195370\pi$
$487$			12379.8i			1.15191i	−0.817480	+	0.575957i	$0.804630\pi$
$488$			8184.79i			0.759238i
$489$		0			0
$490$	10945.2	−	12394.6i	1.00909	−	1.14271i
$491$	5404.13			0.496711			0.248356	−	0.968669i	$-0.420110\pi$
$491$	5404.13			0.496711			0.248356	+	0.968669i	$0.420110\pi$
$492$		0			0
$493$			6457.77i			0.589946i
$494$	26786.3			2.43962
$495$		0			0
$496$	−135.370			−0.0122546
$497$			4569.66i			0.412429i
$498$		0			0
$499$	17671.3			1.58532			0.792661	−	0.609662i	$-0.208695\pi$
$499$	17671.3			1.58532			0.792661	+	0.609662i	$0.208695\pi$
$500$	11883.0	−	17393.9i	1.06285	−	1.55576i
$501$		0			0
$502$			178.765i			0.0158937i
$503$		−	363.988i		−	0.0322653i	−0.999870	−	0.0161326i	$-0.994865\pi$
$503$		−	363.988i		−	0.0322653i	0.999870	−	0.0161326i	$-0.00513540\pi$
$504$		0			0
$505$	3382.32	+	2986.81i	0.298042	+	0.263191i
$506$	8735.70			0.767489
$507$		0			0
$508$			4264.26i			0.372433i
$509$	−5123.45			−0.446155			−0.223078	−	0.974801i	$-0.571610\pi$
$509$	−5123.45			−0.446155			−0.223078	+	0.974801i	$0.571610\pi$
$510$		0			0
$511$	3858.59			0.334039
$512$			14443.1i			1.24668i
$513$		0			0
$514$	−11930.9			−1.02383
$515$	8798.95	−	9964.10i	0.752870	−	0.852564i
$516$		0			0
$517$			3873.23i			0.329486i
$518$		−	10992.7i		−	0.932413i
$519$		0			0
$520$	−11993.5	−	10591.1i	−1.01145	−	0.893173i
$521$	6075.64			0.510899			0.255450	−	0.966822i	$-0.417776\pi$
$521$	6075.64			0.510899			0.255450	+	0.966822i	$0.417776\pi$
$522$		0			0
$523$		−	4504.94i		−	0.376649i	−0.982107	−	0.188325i	$-0.939694\pi$
$523$		−	4504.94i		−	0.376649i	0.982107	−	0.188325i	$-0.0603057\pi$
$524$	−2738.60			−0.228313
$525$		0			0
$526$	17666.7			1.46446
$527$		−	297.830i		−	0.0246180i
$528$		0			0
$529$	10124.5			0.832131
$530$	−19632.3	−	17336.6i	−1.60900	−	1.42086i
$531$		0			0
$532$			11822.8i			0.963500i
$533$			20373.9i			1.65571i
$534$		0			0
$535$	11995.6	−	13584.1i	0.969375	−	1.09774i
$536$	−19464.0			−1.56850
$537$		0			0
$538$			15085.6i			1.20890i
$539$	−12390.1			−0.990128
$540$		0			0
$541$	11565.9			0.919144			0.459572	−	0.888141i	$-0.348003\pi$
$541$	11565.9			0.919144			0.459572	+	0.888141i	$0.348003\pi$
$542$			39807.9i			3.15479i
$543$		0			0
$544$	−6291.54			−0.495859
$545$	−493.529	−	435.819i	−0.0387898	−	0.0342540i
$546$		0			0
$547$		−	23357.6i		−	1.82578i	−0.408211	−	0.912888i	$-0.633847\pi$
$547$		−	23357.6i		−	1.82578i	0.408211	−	0.912888i	$-0.366153\pi$
$548$		−	8283.64i		−	0.645729i
$549$		0			0
$550$	−23976.3	+	2989.27i	−1.85882	+	0.231751i
$551$	9118.28			0.704994
$552$		0			0
$553$		−	4521.44i		−	0.347688i
$554$	−26932.0			−2.06540
$555$		0			0
$556$	−36571.5			−2.78952
$557$		−	6186.93i		−	0.470644i	−0.971917	−	0.235322i	$-0.924385\pi$
$557$		−	6186.93i		−	0.470644i	0.971917	−	0.235322i	$-0.0756145\pi$
$558$		0			0
$559$	929.748			0.0703473
$560$	1868.52	−	2115.95i	0.140999	−	0.159670i
$561$		0			0
$562$		−	13366.3i		−	1.00324i
$563$			4194.56i			0.313996i	0.987599	+	0.156998i	$0.0501816\pi$
$563$			4194.56i			0.313996i	−0.987599	+	0.156998i	$0.949818\pi$
$564$		0			0
$565$	−7902.82	+	8949.30i	−0.588450	+	0.666371i
$566$	−36856.0			−2.73706
$567$		0			0
$568$		−	26205.6i		−	1.93585i
$569$	−5116.20			−0.376946			−0.188473	−	0.982078i	$-0.560354\pi$
$569$	−5116.20			−0.376946			−0.188473	+	0.982078i	$0.560354\pi$
$570$		0			0
$571$	−157.057			−0.0115108			−0.00575538	−	0.999983i	$-0.501832\pi$
$571$	−157.057			−0.0115108			−0.00575538	+	0.999983i	$0.501832\pi$
$572$			25549.4i			1.86761i
$573$		0			0
$574$	−13765.2			−1.00096
$575$	5605.80	−	698.911i	0.406571	−	0.0506897i
$576$		0			0
$577$		−	16135.6i		−	1.16418i	−0.813124	−	0.582090i	$-0.802235\pi$
$577$		−	16135.6i		−	1.16418i	0.813124	−	0.582090i	$-0.197765\pi$
$578$			18628.3i			1.34054i
$579$		0			0
$580$	−8700.34	−	7682.97i	−0.622865	−	0.550031i
$581$	4395.29			0.313851
$582$		0			0
$583$			19625.2i			1.39416i
$584$	−22127.9			−1.56791
$585$		0			0
$586$	−2735.90			−0.192865
$587$		−	11196.6i		−	0.787277i	−0.919265	−	0.393638i	$-0.871216\pi$
$587$		−	11196.6i		−	0.787277i	0.919265	−	0.393638i	$-0.128784\pi$
$588$		0			0
$589$	−420.532			−0.0294189
$590$	−27844.6	+	31531.7i	−1.94295	+	2.20024i
$591$		0			0
$592$			16461.3i			1.14283i
$593$		−	6631.85i		−	0.459254i	−0.973279	−	0.229627i	$-0.926249\pi$
$593$		−	6631.85i		−	0.459254i	0.973279	−	0.229627i	$-0.0737506\pi$
$594$		0			0
$595$	4655.34	+	4110.97i	0.320757	+	0.283249i
$596$	−27842.1			−1.91352
$597$		0			0
$598$		−	9144.04i		−	0.625297i
$599$	27939.8			1.90583			0.952913	−	0.303242i	$-0.0980691\pi$
$599$	27939.8			1.90583			0.952913	+	0.303242i	$0.0980691\pi$
$600$		0			0
$601$	−20484.4			−1.39031			−0.695154	−	0.718861i	$-0.744664\pi$
$601$	−20484.4			−1.39031			−0.695154	+	0.718861i	$0.744664\pi$
$602$			628.165i			0.0425284i
$603$		0			0
$604$	−23364.6			−1.57399
$605$	2416.28	+	2133.74i	0.162373	+	0.143386i
$606$		0			0
$607$			27376.1i			1.83058i	0.402795	+	0.915290i	$0.368039\pi$
$607$			27376.1i			1.83058i	−0.402795	+	0.915290i	$0.631961\pi$
$608$			8883.57i			0.592560i
$609$		0			0
$610$	8563.52	−	9697.48i	0.568404	−	0.643672i
$611$	4054.28			0.268443
$612$		0			0
$613$		−	19526.4i		−	1.28656i	−0.765630	−	0.643281i	$-0.777573\pi$
$613$		−	19526.4i		−	1.28656i	0.765630	−	0.643281i	$-0.222427\pi$
$614$	8272.47			0.543729
$615$		0			0
$616$	−8100.27			−0.529820
$617$		−	7120.77i		−	0.464622i	−0.972642	−	0.232311i	$-0.925371\pi$
$617$		−	7120.77i		−	0.464622i	0.972642	−	0.232311i	$-0.0746286\pi$
$618$		0			0
$619$	−27419.3			−1.78041			−0.890207	−	0.455556i	$-0.849441\pi$
$619$	−27419.3			−1.78041			−0.890207	+	0.455556i	$0.849441\pi$
$620$	401.256	+	354.336i	0.0259917	+	0.0229524i
$621$		0			0
$622$			13624.5i			0.878286i
$623$			5933.86i			0.381597i
$624$		0			0
$625$	−15146.7	+	3836.50i	−0.969387	+	0.245536i
$626$	−12453.7			−0.795126
$627$		0			0
$628$		−	27483.3i		−	1.74634i
$629$	−36216.7			−2.29580
$630$		0			0
$631$	16582.0			1.04615			0.523075	−	0.852287i	$-0.324785\pi$
$631$	16582.0			1.04615			0.523075	+	0.852287i	$0.324785\pi$
$632$			25929.1i			1.63197i
$633$		0			0
$634$	−38619.3			−2.41920
$635$	2093.63	−	2370.86i	0.130839	−	0.148165i
$636$		0			0
$637$			12969.3i			0.806689i
$638$			13313.2i			0.826135i
$639$		0			0
$640$	19604.6	−	22200.6i	1.21084	−	1.37118i
$641$	18822.5			1.15982			0.579909	−	0.814681i	$-0.303088\pi$
$641$	18822.5			1.15982			0.579909	+	0.814681i	$0.303088\pi$
$642$		0			0
$643$		−	2537.28i		−	0.155615i	−0.996968	−	0.0778075i	$-0.975208\pi$
$643$		−	2537.28i		−	0.155615i	0.996968	−	0.0778075i	$-0.0247920\pi$
$644$	4035.94			0.246954
$645$		0			0
$646$	59624.9			3.63144
$647$		−	27469.1i		−	1.66912i	−0.550917	−	0.834560i	$-0.685722\pi$
$647$		−	27469.1i		−	1.66912i	0.550917	−	0.834560i	$-0.314278\pi$
$648$		0			0
$649$	31520.4			1.90644
$650$	3129.00	+	25097.0i	0.188815	+	1.51444i
$651$		0			0
$652$		−	59574.0i		−	3.57837i
$653$			11173.5i			0.669606i	0.942288	+	0.334803i	$0.108670\pi$
$653$			11173.5i			0.669606i	−0.942288	+	0.334803i	$0.891330\pi$
$654$		0			0
$655$	1522.62	+	1344.57i	0.0908299	+	0.0802088i
$656$	20613.1			1.22684
$657$		0			0
$658$			2739.19i			0.162287i
$659$	−9701.80			−0.573488			−0.286744	−	0.958007i	$-0.592573\pi$
$659$	−9701.80			−0.573488			−0.286744	+	0.958007i	$0.592573\pi$
$660$		0			0
$661$	21351.4			1.25639			0.628195	−	0.778056i	$-0.283794\pi$
$661$	21351.4			1.25639			0.628195	+	0.778056i	$0.283794\pi$
$662$		−	16114.5i		−	0.946083i
$663$		0			0
$664$	−25205.6			−1.47315
$665$	5804.64	−	6573.28i	0.338487	−	0.383309i
$666$		0			0
$667$		−	3112.71i		−	0.180696i
$668$		−	28817.9i		−	1.66916i
$669$		0			0
$670$	23061.3	+	20364.6i	1.32975	+	1.17426i
$671$	−9694.00			−0.557724
$672$		0			0
$673$			9988.13i			0.572086i	0.958217	+	0.286043i	$0.0923401\pi$
$673$			9988.13i			0.572086i	−0.958217	+	0.286043i	$0.907660\pi$
$674$	16021.5			0.915614
$675$		0			0
$676$	−6372.24			−0.362554
$677$			4153.60i			0.235799i	0.993026	+	0.117899i	$0.0376160\pi$
$677$			4153.60i			0.235799i	−0.993026	+	0.117899i	$0.962384\pi$
$678$		0			0
$679$	3185.44			0.180038
$680$	−26697.0	−	23575.2i	−1.50556	−	1.32951i
$681$		0			0
$682$		−	613.999i		−	0.0344740i
$683$		−	2014.15i		−	0.112840i	−0.998407	−	0.0564198i	$-0.982031\pi$
$683$		−	2014.15i		−	0.112840i	0.998407	−	0.0564198i	$-0.0179685\pi$
$684$		0			0
$685$	−4067.03	+	4605.57i	−0.226851	+	0.256890i
$686$	−18523.7			−1.03096
$687$		0			0
$688$		−	940.662i		−	0.0521256i
$689$	20542.6			1.13586
$690$		0			0
$691$	6459.81			0.355634			0.177817	−	0.984064i	$-0.443097\pi$
$691$	6459.81			0.355634			0.177817	+	0.984064i	$0.443097\pi$
$692$			49053.1i			2.69468i
$693$		0			0
$694$	−49459.9			−2.70529
$695$	20333.2	+	17955.5i	1.10976	+	0.979988i
$696$		0			0
$697$			45351.2i			2.46456i
$698$		−	3564.08i		−	0.193270i
$699$		0			0
$700$	−11077.1	+	1381.06i	−0.598110	+	0.0745701i
$701$	14805.8			0.797727			0.398863	−	0.917010i	$-0.369405\pi$
$701$	14805.8			0.797727			0.398863	+	0.917010i	$0.369405\pi$
$702$		0			0
$703$			51137.5i			2.74351i
$704$	−26689.8			−1.42885
$705$		0			0
$706$	41394.6			2.20666
$707$		−	2391.15i		−	0.127197i
$708$		0			0
$709$	−2824.99			−0.149640			−0.0748201	−	0.997197i	$-0.523838\pi$
$709$	−2824.99			−0.149640			−0.0748201	+	0.997197i	$0.523838\pi$
$710$	−27418.2	+	31048.9i	−1.44928	+	1.64119i
$711$		0			0
$712$		−	34028.9i		−	1.79113i
$713$			143.557i			0.00754032i
$714$		0			0
$715$	12544.0	−	14205.0i	0.656110	−	0.742991i
$716$	26258.8			1.37058
$717$		0			0
$718$		−	39548.3i		−	2.05561i
$719$	−3441.37			−0.178500			−0.0892499	−	0.996009i	$-0.528447\pi$
$719$	−3441.37			−0.178500			−0.0892499	+	0.996009i	$0.528447\pi$
$720$		0			0
$721$	−7044.18			−0.363854
$722$		−	51242.6i		−	2.64135i
$723$		0			0
$724$	8650.35			0.444044
$725$	1065.14	+	8543.22i	0.0545631	+	0.437638i
$726$		0			0
$727$			31950.7i			1.62997i	0.579486	+	0.814983i	$0.303254\pi$
$727$			31950.7i			1.62997i	−0.579486	+	0.814983i	$0.696746\pi$
$728$			8478.90i			0.431661i
$729$		0			0
$730$	26217.5	+	23151.8i	1.32925	+	1.17381i
$731$	2069.57			0.104714
$732$		0			0
$733$		−	27110.6i		−	1.36610i	−0.730371	−	0.683051i	$-0.760653\pi$
$733$		−	27110.6i		−	1.36610i	0.730371	−	0.683051i	$-0.239347\pi$
$734$	−7582.11			−0.381282
$735$		0			0
$736$	3032.58			0.151878
$737$		−	23053.0i		−	1.15220i
$738$		0			0
$739$	−20566.5			−1.02375			−0.511876	−	0.859059i	$-0.671049\pi$
$739$	−20566.5			−1.02375			−0.511876	+	0.859059i	$0.671049\pi$
$740$	43087.9	−	48793.6i	2.14046	−	2.42390i
$741$		0			0
$742$			13879.2i			0.686685i
$743$		−	23354.6i		−	1.15316i	−0.817041	−	0.576580i	$-0.804387\pi$
$743$		−	23354.6i		−	1.15316i	0.817041	−	0.576580i	$-0.195613\pi$
$744$		0			0
$745$	15479.8	+	13669.7i	0.761256	+	0.672239i
$746$	48695.4			2.38990
$747$		0			0
$748$			56871.5i			2.77998i
$749$	−9603.33			−0.468489
$750$		0			0
$751$	−2747.62			−0.133505			−0.0667523	−	0.997770i	$-0.521264\pi$
$751$	−2747.62			−0.133505			−0.0667523	+	0.997770i	$0.521264\pi$
$752$		−	4101.87i		−	0.198909i
$753$		0			0
$754$	13935.5			0.673078
$755$	12990.3	+	11471.3i	0.626181	+	0.552959i
$756$		0			0
$757$		−	17838.0i		−	0.856449i	−0.903672	−	0.428224i	$-0.859139\pi$
$757$		−	17838.0i		−	0.856449i	0.903672	−	0.428224i	$-0.140861\pi$
$758$		−	22109.7i		−	1.05945i
$759$		0			0
$760$	−33287.8	+	37695.7i	−1.58878	+	1.79917i
$761$	−6545.12			−0.311775			−0.155887	−	0.987775i	$-0.549824\pi$
$761$	−6545.12			−0.311775			−0.155887	+	0.987775i	$0.549824\pi$
$762$		0			0
$763$			348.903i			0.0165546i
$764$	33611.0			1.59163
$765$		0			0
$766$	27870.3			1.31462
$767$		−	32993.7i		−	1.55324i
$768$		0			0
$769$	11045.8			0.517976			0.258988	−	0.965881i	$-0.416611\pi$
$769$	11045.8			0.517976			0.258988	+	0.965881i	$0.416611\pi$
$770$	9597.34	+	8475.08i	0.449174	+	0.396650i
$771$		0			0
$772$		−	8165.54i		−	0.380679i
$773$		−	16606.1i		−	0.772678i	−0.922357	−	0.386339i	$-0.873739\pi$
$773$		−	16606.1i		−	0.772678i	0.922357	−	0.386339i	$-0.126261\pi$
$774$		0			0
$775$	−49.1238	−	394.010i	−0.00227687	−	0.0182623i
$776$	−18267.5			−0.845060
$777$		0			0
$778$		−	9895.42i		−	0.456000i
$779$	64035.3			2.94519
$780$		0			0
$781$	31037.7			1.42204
$782$		−	20354.1i		−	0.930771i
$783$		0			0
$784$	13121.5			0.597736
$785$	−13493.5	+	15280.3i	−0.613507	+	0.694747i
$786$		0			0
$787$		−	14223.2i		−	0.644223i	−0.946702	−	0.322111i	$-0.895607\pi$
$787$		−	14223.2i		−	0.644223i	0.946702	−	0.322111i	$-0.104393\pi$
$788$		−	41955.9i		−	1.89672i
$789$		0			0
$790$	27128.9	−	30721.3i	1.22178	−	1.38356i
$791$	6326.76			0.284391
$792$		0			0
$793$			10147.1i			0.454395i
$794$	−30884.1			−1.38040
$795$		0			0
$796$	18806.3			0.837399
$797$			2464.92i			0.109551i	0.998499	+	0.0547753i	$0.0174443\pi$
$797$			2464.92i			0.109551i	−0.998499	+	0.0547753i	$0.982556\pi$
$798$		0			0
$799$	9024.59			0.399583
$800$	−8323.31	+	1037.72i	−0.367842	+	0.0458612i
$801$		0			0
$802$			7298.90i			0.321363i
$803$		−	26208.0i		−	1.15176i
$804$		0			0
$805$	−2243.92	−	1981.53i	−0.0982456	−	0.0867573i
$806$	−642.700			−0.0280870
$807$		0			0
$808$			13712.5i			0.597035i
$809$	−11013.8			−0.478647			−0.239323	−	0.970940i	$-0.576926\pi$
$809$	−11013.8			−0.478647			−0.239323	+	0.970940i	$0.576926\pi$
$810$		0			0
$811$	−2186.66			−0.0946783			−0.0473391	−	0.998879i	$-0.515074\pi$
$811$	−2186.66			−0.0946783			−0.0473391	+	0.998879i	$0.515074\pi$
$812$			6150.75i			0.265824i
$813$		0			0
$814$	−74663.5			−3.21493
$815$	−29249.1	+	33122.2i	−1.25712	+	1.42358i
$816$		0			0
$817$		−	2922.20i		−	0.125135i
$818$		−	12668.5i		−	0.541496i
$819$		0			0
$820$	−61100.2	−	53955.5i	−2.60209	−	2.29781i
$821$	−10995.5			−0.467414			−0.233707	−	0.972307i	$-0.575086\pi$
$821$	−10995.5			−0.467414			−0.233707	+	0.972307i	$0.575086\pi$
$822$		0			0
$823$			33494.9i			1.41866i	0.704876	+	0.709330i	$0.251002\pi$
$823$			33494.9i			1.41866i	−0.704876	+	0.709330i	$0.748998\pi$
$824$	40396.2			1.70785
$825$		0			0
$826$	22291.5			0.939009
$827$			33558.7i			1.41107i	0.708677	+	0.705533i	$0.249292\pi$
$827$			33558.7i			1.41107i	−0.708677	+	0.705533i	$0.750708\pi$
$828$		0			0
$829$	40310.6			1.68884			0.844419	−	0.535683i	$-0.179946\pi$
$829$	40310.6			1.68884			0.844419	+	0.535683i	$0.179946\pi$
$830$	29864.1	+	26372.0i	1.24891	+	1.10287i
$831$		0			0
$832$			27937.4i			1.16413i
$833$			28868.9i			1.20078i
$834$		0			0
$835$	−14148.8	+	16022.3i	−0.586393	+	0.664042i
$836$	80301.7			3.32212
$837$		0			0
$838$			38232.9i			1.57605i
$839$	3627.82			0.149280			0.0746402	−	0.997211i	$-0.476219\pi$
$839$	3627.82			0.149280			0.0746402	+	0.997211i	$0.476219\pi$
$840$		0			0
$841$	−19645.2			−0.805496
$842$			53577.1i			2.19286i
$843$		0			0
$844$	24100.6			0.982913
$845$	3542.87	+	3128.59i	0.144235	+	0.127369i
$846$		0			0
$847$		−	1708.21i		−	0.0692971i
$848$		−	20783.7i		−	0.841646i
$849$		0			0
$850$	6964.99	+	55864.6i	0.281055	+	2.25428i
$851$	17456.8			0.703186
$852$		0			0
$853$			18910.0i			0.759045i	0.925183	+	0.379522i	$0.123912\pi$
$853$			18910.0i			0.759045i	−0.925183	+	0.379522i	$0.876088\pi$
$854$	−6855.69			−0.274704
$855$		0			0
$856$	55072.2			2.19898
$857$		−	25284.0i		−	1.00780i	−0.863762	−	0.503900i	$-0.831898\pi$
$857$		−	25284.0i		−	1.00780i	0.863762	−	0.503900i	$-0.168102\pi$
$858$		0			0
$859$	26776.0			1.06354			0.531772	−	0.846887i	$-0.321526\pi$
$859$	26776.0			1.06354			0.531772	+	0.846887i	$0.321526\pi$
$860$	−2462.22	+	2788.26i	−0.0976289	+	0.110557i
$861$		0			0
$862$		−	9296.24i		−	0.367321i
$863$		−	35682.5i		−	1.40747i	−0.710462	−	0.703735i	$-0.751514\pi$
$863$		−	35682.5i		−	1.40747i	0.710462	−	0.703735i	$-0.248486\pi$
$864$		0			0
$865$	24083.7	−	27272.8i	0.946669	−	1.07203i
$866$	−50850.3			−1.99534
$867$		0			0
$868$		−	283.671i		−	0.0110926i
$869$	−30710.2			−1.19882
$870$		0			0
$871$	−24130.6			−0.938730
$872$		−	2000.85i		−	0.0777035i
$873$		0			0
$874$	−28739.8			−1.11228
$875$	6836.78	+	4670.71i	0.264143	+	0.180456i
$876$		0			0
$877$			42535.6i			1.63777i	0.573958	+	0.818885i	$0.305407\pi$
$877$			42535.6i			1.63777i	−0.573958	+	0.818885i	$0.694593\pi$
$878$		−	39825.1i		−	1.53079i
$879$		0			0
$880$	−14371.8	−	12691.2i	−0.550538	−	0.486161i
$881$	−49420.7			−1.88993			−0.944964	−	0.327175i	$-0.893903\pi$
$881$	−49420.7			−1.88993			−0.944964	+	0.327175i	$0.893903\pi$
$882$		0			0
$883$		−	12498.0i		−	0.476321i	−0.971226	−	0.238161i	$-0.923456\pi$
$883$		−	12498.0i		−	0.476321i	0.971226	−	0.238161i	$-0.0765445\pi$
$884$	59529.9			2.26494
$885$		0			0
$886$	−54940.4			−2.08325
$887$		−	36831.2i		−	1.39422i	−0.716966	−	0.697108i	$-0.754470\pi$
$887$		−	36831.2i		−	1.39422i	0.716966	−	0.697108i	$-0.245530\pi$
$888$		0			0
$889$	−1676.09			−0.0632333
$890$	−35603.5	+	40318.0i	−1.34093	+	1.51850i
$891$		0			0
$892$		−	10065.8i		−	0.377835i
$893$		−	12742.6i		−	0.477508i
$894$		0			0
$895$	−14599.5	−	12892.3i	−0.545260	−	0.481500i
$896$	−15694.9			−0.585188
$897$		0			0
$898$		−	9797.74i		−	0.364092i
$899$	−218.780			−0.00811650
$900$		0			0
$901$	45726.7			1.69076
$902$			93495.0i			3.45127i
$903$		0			0
$904$	−36282.0			−1.33487
$905$	−4809.46	−	4247.07i	−0.176654	−	0.155997i
$906$		0			0
$907$		−	41564.4i		−	1.52163i	−0.648966	−	0.760817i	$-0.724798\pi$
$907$		−	41564.4i		−	1.52163i	0.648966	−	0.760817i	$-0.275202\pi$
$908$		−	64098.2i		−	2.34270i
$909$		0			0
$910$	8871.24	−	10046.0i	0.323163	−	0.365956i
$911$	−8017.74			−0.291591			−0.145796	−	0.989315i	$-0.546574\pi$
$911$	−8017.74			−0.291591			−0.145796	+	0.989315i	$0.546574\pi$
$912$		0			0
$913$		−	29853.3i		−	1.08215i
$914$	−85855.2			−3.10704
$915$		0			0
$916$	79797.7			2.87837
$917$		−	1076.42i		−	0.0387641i
$918$		0			0
$919$	−10138.1			−0.363900			−0.181950	−	0.983308i	$-0.558241\pi$
$919$	−10138.1			−0.363900			−0.181950	+	0.983308i	$0.558241\pi$
$920$	12868.2	+	11363.5i	0.461143	+	0.407220i
$921$		0			0
$922$			49994.4i			1.78577i
$923$		−	32488.5i		−	1.15858i
$924$		0			0
$925$	−47912.4	+	5973.55i	−1.70308	+	0.212334i
$926$	−24296.3			−0.862232
$927$		0			0
$928$			4621.64i			0.163484i
$929$	48977.5			1.72971			0.864854	−	0.502023i	$-0.167411\pi$
$929$	48977.5			1.72971			0.864854	+	0.502023i	$0.167411\pi$
$930$		0			0
$931$	40762.4			1.43495
$932$		−	14665.3i		−	0.515427i
$933$		0			0
$934$	−58789.0			−2.05957
$935$	27922.2	−	31619.7i	0.976636	−	1.10596i
$936$		0			0
$937$		−	10552.1i		−	0.367901i	−0.982935	−	0.183951i	$-0.941111\pi$
$937$		−	10552.1i		−	0.367901i	0.982935	−	0.183951i	$-0.0588887\pi$
$938$		−	16303.3i		−	0.567508i
$939$		0			0
$940$	−10736.8	+	12158.5i	−0.372548	+	0.421880i
$941$	3733.03			0.129323			0.0646616	−	0.997907i	$-0.479403\pi$
$941$	3733.03			0.129323			0.0646616	+	0.997907i	$0.479403\pi$
$942$		0			0
$943$		−	21859.7i		−	0.754879i
$944$	−33381.1			−1.15091
$945$		0			0
$946$	4266.57			0.146637
$947$			45472.6i			1.56036i	0.625556	+	0.780179i	$0.284872\pi$
$947$			45472.6i			1.56036i	−0.625556	+	0.780179i	$0.715128\pi$
$948$		0			0
$949$	−27433.1			−0.938373
$950$	78880.0	−	9834.47i	2.69390	−	0.335866i
$951$		0			0
$952$			18873.6i			0.642538i
$953$			48198.9i			1.63832i	0.573567	+	0.819159i	$0.305559\pi$
$953$			48198.9i			1.63832i	−0.573567	+	0.819159i	$0.694441\pi$
$954$		0			0
$955$	−18687.2	−	16502.0i	−0.633198	−	0.559155i
$956$	−22705.4			−0.768143
$957$		0			0
$958$		−	67039.9i		−	2.26092i
$959$	3255.94			0.109635
$960$		0			0
$961$	−29780.9			−0.999661
$962$			78153.6i			2.61931i
$963$		0			0
$964$	−12252.0			−0.409348
$965$	−4009.04	+	4539.91i	−0.133736	+	0.151446i
$966$		0			0
$967$		−	35222.2i		−	1.17132i	−0.810555	−	0.585662i	$-0.800834\pi$
$967$		−	35222.2i		−	1.17132i	0.810555	−	0.585662i	$-0.199166\pi$
$968$			9796.04i			0.325265i
$969$		0			0
$970$	21643.7	+	19112.8i	0.716430	+	0.632655i
$971$	18488.0			0.611028			0.305514	−	0.952188i	$-0.401172\pi$
$971$	18488.0			0.611028			0.305514	+	0.952188i	$0.401172\pi$
$972$		0			0
$973$		−	14374.6i		−	0.473618i
$974$	59465.8			1.95627
$975$		0			0
$976$	10266.2			0.336695
$977$			37176.7i			1.21739i	0.793406	+	0.608693i	$0.208306\pi$
$977$			37176.7i			1.21739i	−0.793406	+	0.608693i	$0.791694\pi$
$978$		0			0
$979$	40303.5			1.31574
$980$	−38894.0	−	34346.0i	−1.26778	−	1.11953i
$981$		0			0
$982$		−	25958.5i		−	0.843554i
$983$		−	22458.0i		−	0.728687i	−0.931265	−	0.364343i	$-0.881293\pi$
$983$		−	22458.0i		−	0.728687i	0.931265	−	0.364343i	$-0.118707\pi$
$984$		0			0
$985$	−20599.1	+	23326.8i	−0.666338	+	0.754574i
$986$	31019.6			1.00189
$987$		0			0
$988$		−	84055.3i		−	2.70664i
$989$	−997.552			−0.0320731
$990$		0			0
$991$	−53359.1			−1.71040			−0.855201	−	0.518297i	$-0.826566\pi$
$991$	−53359.1			−1.71040			−0.855201	+	0.518297i	$0.826566\pi$
$992$		−	213.149i		−	0.00682205i
$993$		0			0
$994$	21950.2			0.700420
$995$	−10456.0	−	9233.32i	−0.333143	−	0.294187i
$996$		0			0
$997$		−	38991.0i		−	1.23857i	−0.785166	−	0.619286i	$-0.787422\pi$
$997$		−	38991.0i		−	1.23857i	0.785166	−	0.619286i	$-0.212578\pi$
$998$		−	84883.4i		−	2.69232i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.4.b.c.244.1	yes	8
3.2	odd	2		405.4.b.b.244.8	yes	8
5.2	odd	4		2025.4.a.bd.1.8		8
5.3	odd	4		2025.4.a.bd.1.1		8
5.4	even	2	inner	405.4.b.c.244.8	yes	8
15.2	even	4		2025.4.a.bc.1.1		8
15.8	even	4		2025.4.a.bc.1.8		8
15.14	odd	2		405.4.b.b.244.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.4.b.b.244.1	✓	8	15.14	odd	2
405.4.b.b.244.8	yes	8	3.2	odd	2
405.4.b.c.244.1	yes	8	1.1	even	1	trivial
405.4.b.c.244.8	yes	8	5.4	even	2	inner
2025.4.a.bc.1.1		8	15.2	even	4
2025.4.a.bc.1.8		8	15.8	even	4
2025.4.a.bd.1.1		8	5.3	odd	4
2025.4.a.bd.1.8		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)

\(f(q)\)	\(=\)	\(q-4.80346i q^{2} -15.0732 q^{4} +(8.38048 + 7.40051i) q^{5} -5.92463i q^{7} +33.9759i q^{8} +O(q^{10})\)	\(q-4.80346i q^{2} -15.0732 q^{4} +(8.38048 + 7.40051i) q^{5} -5.92463i q^{7} +33.9759i q^{8} +(35.5481 - 40.2553i) q^{10} -40.2408 q^{11} +42.1218i q^{13} -28.4587 q^{14} +42.6163 q^{16} +93.7609i q^{17} +132.389 q^{19} +(-126.321 - 111.550i) q^{20} +193.295i q^{22} -45.1936i q^{23} +(15.4648 + 124.040i) q^{25} +202.330 q^{26} +89.3033i q^{28} +68.8749 q^{29} -3.17649 q^{31} +67.1020i q^{32} +450.376 q^{34} +(43.8453 - 49.6512i) q^{35} +386.267i q^{37} -635.925i q^{38} +(-251.439 + 284.735i) q^{40} +483.691 q^{41} -22.0728i q^{43} +606.559 q^{44} -217.086 q^{46} -96.2512i q^{47} +307.899 q^{49} +(595.820 - 74.2846i) q^{50} -634.912i q^{52} -487.695i q^{53} +(-337.237 - 297.803i) q^{55} +201.295 q^{56} -330.838i q^{58} -783.294 q^{59} +240.900 q^{61} +15.2581i q^{62} +663.252 q^{64} +(-311.723 + 353.001i) q^{65} +572.876i q^{67} -1413.28i q^{68} +(-238.498 - 210.609i) q^{70} -771.300 q^{71} +651.280i q^{73} +1855.42 q^{74} -1995.53 q^{76} +238.412i q^{77} +763.161 q^{79} +(357.145 + 315.382i) q^{80} -2323.39i q^{82} +741.867i q^{83} +(-693.878 + 785.761i) q^{85} -106.026 q^{86} -1367.22i q^{88} -1001.56 q^{89} +249.556 q^{91} +681.213i q^{92} -462.339 q^{94} +(1109.48 + 979.747i) q^{95} +537.661i q^{97} -1478.98i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 24 q^{4} + 15 q^{5}+O(q^{10}) \) 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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