LMFDB - Embedded newform 136.4.s.a.107.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [136,4,Mod(3,136)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(136, base_ring=CyclotomicField(16))

chi = DirichletCharacter(H, H._module([8, 8, 1]))

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

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

chi := DirichletCharacter("136.3");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 136 = 2^{3} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	136.s (of order $16$, degree $8$, 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:	$8.02425976078$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{16})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{16}]$

Embedding invariants

Embedding label			107.1
Root			$-0.382683 + 0.923880i$ of defining polynomial
Character	$\chi$	$=$	136.107
Dual form			136.4.s.a.75.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.08239 + 2.61313i) q^{2} +(-5.14239 - 7.69613i) q^{3} +(-5.65685 + 5.65685i) q^{4} +(14.5449 - 21.7679i) q^{6} +(-20.9050 - 8.65914i) q^{8} +(-22.4538 + 54.2082i) q^{9} +O(q^{10})$	\(q+(1.08239 + 2.61313i) q^{2} +(-5.14239 - 7.69613i) q^{3} +(-5.65685 + 5.65685i) q^{4} +(14.5449 - 21.7679i) q^{6} +(-20.9050 - 8.65914i) q^{8} +(-22.4538 + 54.2082i) q^{9} +(57.2002 + 38.2199i) q^{11} +(72.6256 + 14.4461i) q^{12} -64.0000i q^{16} +(-32.4286 + 62.1400i) q^{17} -165.957 q^{18} +(63.1249 + 152.397i) q^{19} +(-37.9605 + 190.840i) q^{22} +(40.8598 + 205.416i) q^{24} +(-115.485 - 47.8354i) q^{25} +(287.548 - 57.1968i) q^{27} +(167.240 - 69.2731i) q^{32} -636.761i q^{33} +(-197.480 - 17.4802i) q^{34} +(-179.630 - 433.666i) q^{36} +(-329.907 + 329.907i) q^{38} +(78.5432 + 394.864i) q^{41} +(128.530 - 310.300i) q^{43} +(-539.777 + 107.368i) q^{44} +(-492.552 + 329.113i) q^{48} +(-316.891 + 131.260i) q^{49} -353.553i q^{50} +(644.998 - 69.9732i) q^{51} +(460.702 + 689.489i) q^{54} +(848.254 - 1269.50i) q^{57} +(21.4716 + 8.89383i) q^{59} +(362.039 + 362.039i) q^{64} +(1663.94 - 689.226i) q^{66} +354.214i q^{67} +(-168.073 - 534.961i) q^{68} +(938.793 - 938.793i) q^{72} +(-236.551 + 1189.22i) q^{73} +(225.721 + 1134.78i) q^{75} +(-1219.18 - 504.999i) q^{76} +(-798.669 - 798.669i) q^{81} +(-946.813 + 632.641i) q^{82} +(-911.297 + 377.472i) q^{83} +949.973 q^{86} +(-864.818 - 1294.29i) q^{88} +(1146.08 - 1146.08i) q^{89} +(-1393.15 - 930.872i) q^{96} +(-414.437 - 82.4366i) q^{97} +(-686.000 - 686.000i) q^{98} +(-3356.19 + 2242.54i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{3} + 80 q^{6} - 80 q^{9}+O(q^{10}) $ 8 * q - 8 * q^3 + 80 * q^6 - 80 * q^9	$ 8 q - 8 q^{3} + 80 q^{6} - 80 q^{9} + 320 q^{12} + 144 q^{22} - 256 q^{24} + 496 q^{27} - 720 q^{34} - 640 q^{36} - 1440 q^{38} + 160 q^{41} + 208 q^{43} - 1600 q^{44} + 512 q^{48} + 1880 q^{51} - 176 q^{54} + 2840 q^{57} - 920 q^{59} + 5232 q^{66} + 3456 q^{72} - 3312 q^{73} - 2632 q^{81} - 3472 q^{83} - 5120 q^{96} - 5488 q^{98} - 1536 q^{99}+O(q^{100}) $ 8 * q - 8 * q^3 + 80 * q^6 - 80 * q^9 + 320 * q^12 + 144 * q^22 - 256 * q^24 + 496 * q^27 - 720 * q^34 - 640 * q^36 - 1440 * q^38 + 160 * q^41 + 208 * q^43 - 1600 * q^44 + 512 * q^48 + 1880 * q^51 - 176 * q^54 + 2840 * q^57 - 920 * q^59 + 5232 * q^66 + 3456 * q^72 - 3312 * q^73 - 2632 * q^81 - 3472 * q^83 - 5120 * q^96 - 5488 * q^98 - 1536 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/136\mathbb{Z}\right)^\times$.

$n$	$69$	$103$	$105$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{5}{16}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	1.08239	+	2.61313i	0.382683	+	0.923880i
$2$	1.08239	+	2.61313i	0.382683	+	0.923880i
$3$	−5.14239	−	7.69613i	−0.989653	−	1.48112i	−0.872864	−	0.487964i	$-0.837740\pi$
$3$	−5.14239	−	7.69613i	−0.989653	−	1.48112i	−0.116789	−	0.993157i	$-0.537260\pi$
$4$	−5.65685	+	5.65685i	−0.707107	+	0.707107i
$5$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$5$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$6$	14.5449	−	21.7679i	0.989653	−	1.48112i
$7$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$7$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$8$	−20.9050	−	8.65914i	−0.923880	−	0.382683i
$9$	−22.4538	+	54.2082i	−0.831621	+	2.00771i
$10$		0			0
$11$	57.2002	+	38.2199i	1.56786	+	1.04761i	0.969094	+	0.246691i	$0.0793433\pi$
$11$	57.2002	+	38.2199i	1.56786	+	1.04761i	0.598769	+	0.800922i	$0.295657\pi$
$12$	72.6256	+	14.4461i	1.74710	+	0.347520i
$13$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$13$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$14$		0			0
$15$		0			0
$16$		−	64.0000i		−	1.00000i
$17$	−32.4286	+	62.1400i	−0.462653	+	0.886539i
$17$	−32.4286	+	62.1400i	−0.462653	+	0.886539i
$18$	−165.957			−2.17313
$19$	63.1249	+	152.397i	0.762202	+	1.84012i	0.465027	+	0.885296i	$0.346045\pi$
$19$	63.1249	+	152.397i	0.762202	+	1.84012i	0.297175	+	0.954823i	$0.403955\pi$
$20$		0			0
$21$		0			0
$22$	−37.9605	+	190.840i	−0.367873	+	1.84942i
$23$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$23$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$24$	40.8598	+	205.416i	0.347520	+	1.74710i
$25$	−115.485	−	47.8354i	−0.923880	−	0.382683i
$26$		0			0
$27$	287.548	−	57.1968i	2.04958	−	0.407686i
$28$		0			0
$29$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$29$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$30$		0			0
$31$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$31$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$32$	167.240	−	69.2731i	0.923880	−	0.382683i
$33$		−	636.761i		−	3.35897i
$34$	−197.480	−	17.4802i	−0.996105	−	0.0881716i
$35$		0			0
$36$	−179.630	−	433.666i	−0.831621	−	2.00771i
$37$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$37$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$38$	−329.907	+	329.907i	−1.40837	+	1.40837i
$39$		0			0
$40$		0			0
$41$	78.5432	+	394.864i	0.299180	+	1.50408i	0.779173	+	0.626809i	$0.215639\pi$
$41$	78.5432	+	394.864i	0.299180	+	1.50408i	−0.479993	+	0.877272i	$0.659361\pi$
$42$		0			0
$43$	128.530	−	310.300i	0.455831	−	1.10047i	−0.514239	−	0.857647i	$-0.671926\pi$
$43$	128.530	−	310.300i	0.455831	−	1.10047i	0.970070	−	0.242826i	$-0.0780743\pi$
$44$	−539.777	+	107.368i	−1.84942	+	0.367873i
$45$		0			0
$46$		0			0
$47$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$47$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$48$	−492.552	+	329.113i	−1.48112	+	0.989653i
$49$	−316.891	+	131.260i	−0.923880	+	0.382683i
$50$		−	353.553i		−	1.00000i
$51$	644.998	−	69.9732i	1.77094	−	0.192122i
$52$		0			0
$53$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$53$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$54$	460.702	+	689.489i	1.16099	+	1.73755i
$55$		0			0
$56$		0			0
$57$	848.254	−	1269.50i	1.97112	−	2.94999i
$58$		0			0
$59$	21.4716	+	8.89383i	0.0473791	+	0.0196251i	0.406247	−	0.913763i	$-0.366837\pi$
$59$	21.4716	+	8.89383i	0.0473791	+	0.0196251i	−0.358868	+	0.933388i	$0.616837\pi$
$60$		0			0
$61$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$61$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$62$		0			0
$63$		0			0
$64$	362.039	+	362.039i	0.707107	+	0.707107i
$65$		0			0
$66$	1663.94	−	689.226i	3.10328	−	1.28542i
$67$			354.214i			0.645883i	0.946419	+	0.322941i	$0.104672\pi$
$67$			354.214i			0.645883i	−0.946419	+	0.322941i	$0.895328\pi$
$68$	−168.073	−	534.961i	−0.299733	−	0.954023i
$69$		0			0
$70$		0			0
$71$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$71$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$72$	938.793	−	938.793i	1.53664	−	1.53664i
$73$	−236.551	+	1189.22i	−0.379263	+	1.90668i	0.0407577	+	0.999169i	$0.487023\pi$
$73$	−236.551	+	1189.22i	−0.379263	+	1.90668i	−0.420021	+	0.907515i	$0.637977\pi$
$74$		0			0
$75$	225.721	+	1134.78i	0.347520	+	1.74710i
$76$	−1219.18	−	504.999i	−1.84012	−	0.762202i
$77$		0			0
$78$		0			0
$79$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$79$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$80$		0			0
$81$	−798.669	−	798.669i	−1.09557	−	1.09557i
$82$	−946.813	+	632.641i	−1.27510	+	0.851993i
$83$	−911.297	+	377.472i	−1.20515	+	0.499191i	−0.892661	−	0.450728i	$-0.851164\pi$
$83$	−911.297	+	377.472i	−1.20515	+	0.499191i	−0.312494	+	0.949920i	$0.601164\pi$
$84$		0			0
$85$		0			0
$86$	949.973			1.19114
$87$		0			0
$88$	−864.818	−	1294.29i	−1.04761	−	1.56786i
$89$	1146.08	−	1146.08i	1.36499	−	1.36499i	0.497565	−	0.867426i	$-0.334227\pi$
$89$	1146.08	−	1146.08i	1.36499	−	1.36499i	0.867426	−	0.497565i	$-0.165773\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	−1393.15	−	930.872i	−1.48112	−	0.989653i
$97$	−414.437	−	82.4366i	−0.433811	−	0.0862904i	−0.0266459	−	0.999645i	$-0.508483\pi$
$97$	−414.437	−	82.4366i	−0.433811	−	0.0862904i	−0.407165	+	0.913355i	$0.633483\pi$
$98$	−686.000	−	686.000i	−0.707107	−	0.707107i
$99$	−3356.19	+	2242.54i	−3.40717	+	2.27660i
$100$	923.880	−	382.683i	0.923880	−	0.382683i
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$	880.990	+	1609.72i	0.855206	+	1.56261i
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	203.891	−	1025.03i	0.184214	−	0.926108i	−0.772486	−	0.635032i	$-0.780987\pi$
$107$	203.891	−	1025.03i	0.184214	−	0.926108i	0.956700	−	0.291076i	$-0.0940132\pi$
$108$	−1303.06	+	1950.17i	−1.16099	+	1.73755i
$109$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$109$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1990.54	−	1330.03i	−1.65711	−	1.10725i	−0.875018	−	0.484091i	$-0.839150\pi$
$113$	−1990.54	−	1330.03i	−1.65711	−	1.10725i	−0.782097	−	0.623157i	$-0.785850\pi$
$114$	4235.51	+	842.496i	3.47975	+	0.692166i
$115$		0			0
$116$		0			0
$117$		0			0
$118$			65.7346i			0.0512827i
$119$		0			0
$120$		0			0
$121$	1301.74	+	3142.69i	0.978019	+	2.36115i
$122$		0			0
$123$	2635.02	−	2635.02i	1.93164	−	1.93164i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$127$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$128$	−554.185	+	1337.92i	−0.382683	+	0.923880i
$129$	−3049.06	+	606.496i	−2.08105	+	0.413946i
$130$		0			0
$131$	474.412	+	94.3664i	0.316409	+	0.0629376i	0.350740	−	0.936473i	$-0.385930\pi$
$131$	474.412	+	94.3664i	0.316409	+	0.0629376i	−0.0343312	+	0.999411i	$0.510930\pi$
$132$	3602.07	+	3602.07i	2.37515	+	2.37515i
$133$		0			0
$134$	−925.606	+	383.399i	−0.596718	+	0.247169i
$135$		0			0
$136$	1216.00	−	1018.23i	0.766700	−	0.642006i
$137$	25.0097			0.0155965			0.00779827	−	0.999970i	$-0.497518\pi$
$137$	25.0097			0.0155965			0.00779827	+	0.999970i	$0.497518\pi$
$138$		0			0
$139$	1435.38	+	2148.19i	0.875878	+	1.31084i	0.949567	+	0.313563i	$0.101523\pi$
$139$	1435.38	+	2148.19i	0.875878	+	1.31084i	−0.0736895	+	0.997281i	$0.523477\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	3469.33	+	1437.04i	2.00771	+	0.831621i
$145$		0			0
$146$	−3363.63	+	669.067i	−1.90668	+	0.379263i
$147$	2639.77	+	1763.84i	1.48112	+	0.989653i
$148$		0			0
$149$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$149$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$150$	−2720.99	+	1818.11i	−1.48112	+	0.989653i
$151$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$151$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$152$		−	3732.47i		−	1.99173i
$153$	−2640.35	−	3153.18i	−1.39516	−	1.66614i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$157$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	1222.55	−	2951.50i	0.592917	−	1.43143i
$163$	4079.03	−	811.370i	1.96009	−	0.389886i	0.972463	−	0.233056i	$-0.0748726\pi$
$163$	4079.03	−	811.370i	1.96009	−	0.389886i	0.987626	−	0.156830i	$-0.0501274\pi$
$164$	−2677.99	−	1789.38i	−1.27510	−	0.851993i
$165$		0			0
$166$	−1972.76	−	1972.76i	−0.922386	−	0.922386i
$167$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$167$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$168$		0			0
$169$			2197.00i			1.00000i
$170$		0			0
$171$	−9678.56			−4.32829
$172$	1028.24	+	2482.40i	0.455831	+	1.10047i
$173$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$173$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$174$		0			0
$175$		0			0
$176$	2446.08	−	3660.81i	1.04761	−	1.56786i
$177$	−41.9673	−	210.984i	−0.0178218	−	0.0895961i
$178$	4235.36	+	1754.34i	1.78345	+	0.738728i
$179$	1814.32	−	4380.16i	0.757590	−	1.82898i	0.247429	−	0.968906i	$-0.420414\pi$
$179$	1814.32	−	4380.16i	0.757590	−	1.82898i	0.510161	−	0.860079i	$-0.329586\pi$
$180$		0			0
$181$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$181$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−4229.91	+	2315.00i	−1.65413	+	0.905292i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$191$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$192$	924.552	−	4648.04i	0.347520	−	1.74710i
$193$	1674.09	−	2505.45i	0.624371	−	0.934437i	−0.375600	−	0.926782i	$-0.622563\pi$
$193$	1674.09	−	2505.45i	0.624371	−	0.934437i	0.999971	−	0.00765530i	$-0.00243678\pi$
$194$	−233.166	−	1172.20i	−0.0862904	−	0.433811i
$195$		0			0
$196$	1050.08	−	2535.13i	0.382683	−	0.923880i
$197$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$197$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$198$	−9492.75	−	6342.85i	−3.40717	−	2.27660i
$199$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$199$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$200$	2000.00	+	2000.00i	0.707107	+	0.707107i
$201$	2726.08	−	1821.51i	0.956630	−	0.639200i
$202$		0			0
$203$		0			0
$204$	−3252.83	+	4044.49i	−1.11639	+	1.38809i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−2213.85	+	11129.8i	−0.732703	+	3.68355i
$210$		0			0
$211$	−601.169	−	3022.28i	−0.196143	−	0.986078i	−0.945923	−	0.324391i	$-0.894841\pi$
$211$	−601.169	−	3022.28i	−0.196143	−	0.986078i	0.749780	−	0.661687i	$-0.230159\pi$
$212$		0			0
$213$		0			0
$214$	2899.23	−	576.692i	0.926108	−	0.184214i
$215$		0			0
$216$	−6506.46	−	1294.22i	−2.04958	−	0.407686i
$217$		0			0
$218$		0			0
$219$	10368.8	−	4294.91i	3.19937	−	1.32522i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$223$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$224$		0			0
$225$	5186.15	−	5186.15i	1.53664	−	1.53664i
$226$	1321.00	−	6641.14i	0.388814	−	1.95470i
$227$	325.126	−	486.586i	0.0950634	−	0.142273i	−0.780889	−	0.624669i	$-0.785234\pi$
$227$	325.126	−	486.586i	0.0950634	−	0.142273i	0.875953	+	0.482397i	$0.160234\pi$
$228$	2382.94	+	11979.8i	0.692166	+	3.47975i
$229$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$229$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	5078.54	+	1010.18i	1.42792	+	0.284032i	0.847722	−	0.530441i	$-0.177974\pi$
$233$	5078.54	+	1010.18i	1.42792	+	0.284032i	0.580202	+	0.814473i	$0.302974\pi$
$234$		0			0
$235$		0			0
$236$	−171.773	+	71.1507i	−0.0473791	+	0.0196251i
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	2843.07	+	4254.95i	0.759909	+	1.13728i	0.986575	+	0.163311i	$0.0522174\pi$
$241$	2843.07	+	4254.95i	0.759909	+	1.13728i	−0.226666	+	0.973973i	$0.572783\pi$
$242$	−6803.24	+	6803.24i	−1.80714	+	1.80714i
$243$	−495.279	+	2489.94i	−0.130750	+	0.657323i
$244$		0			0
$245$		0			0
$246$	9737.76	+	4033.51i	2.52381	+	1.04540i
$247$		0			0
$248$		0			0
$249$	7591.31	+	5072.35i	1.93205	+	1.29095i
$250$		0			0
$251$	−4620.51	−	4620.51i	−1.16193	−	1.16193i	−0.984053	−	0.177875i	$-0.943078\pi$
$251$	−4620.51	−	4620.51i	−1.16193	−	1.16193i	−0.177875	−	0.984053i	$-0.556922\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−4096.00			−1.00000
$257$	−2433.61	−	5875.25i	−0.590678	−	1.42602i	−0.882849	−	0.469658i	$-0.844377\pi$
$257$	−2433.61	−	5875.25i	−0.590678	−	1.42602i	0.292170	−	0.956366i	$-0.405623\pi$
$258$	−4885.13	−	7311.12i	−1.17882	−	1.76423i
$259$		0			0
$260$		0			0
$261$		0			0
$262$	266.908	+	1341.84i	0.0629376	+	0.316409i
$263$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$263$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$264$	−5513.80	+	13311.5i	−1.28542	+	3.10328i
$265$		0			0
$266$		0			0
$267$	−14714.0	−	2926.79i	−3.37259	−	0.670849i
$268$	−2003.74	−	2003.74i	−0.456708	−	0.456708i
$269$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$269$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$	3976.96	+	2075.43i	0.886539	+	0.462653i
$273$		0			0
$274$	27.0704	+	65.3536i	0.00596854	+	0.0144093i
$275$	−4777.49	−	7150.02i	−1.04761	−	1.56786i
$276$		0			0
$277$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$277$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$278$	−4059.86	+	6076.01i	−0.875878	+	1.31084i
$279$		0			0
$280$		0			0
$281$	2198.82	−	5308.42i	0.466799	−	1.12695i	−0.498753	−	0.866744i	$-0.666209\pi$
$281$	2198.82	−	5308.42i	0.466799	−	1.12695i	0.965552	−	0.260209i	$-0.0837914\pi$
$282$		0			0
$283$	7915.05	+	5288.66i	1.66255	+	1.11088i	0.843346	+	0.537371i	$0.180582\pi$
$283$	7915.05	+	5288.66i	1.66255	+	1.11088i	0.819200	+	0.573507i	$0.194418\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$			10621.2i			2.17313i
$289$	−2809.77	−	4030.23i	−0.571905	−	0.820320i
$290$		0			0
$291$	1496.75	+	3613.48i	0.301516	+	0.727924i
$292$	−5389.12	−	8065.39i	−1.08005	−	1.61641i
$293$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$293$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$294$	−1751.87	+	8807.22i	−0.347520	+	1.74710i
$295$		0			0
$296$		0			0
$297$	18633.8	+	7718.39i	3.64056	+	1.50797i
$298$		0			0
$299$		0			0
$300$	−7696.13	−	5142.39i	−1.48112	−	0.989653i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	9753.41	−	4039.99i	1.84012	−	0.762202i
$305$		0			0
$306$	5381.75	−	10312.6i	1.00541	−	1.92657i
$307$	7204.00			1.33926			0.669632	−	0.742693i	$-0.266452\pi$
$307$	7204.00			1.33926			0.669632	+	0.742693i	$0.266452\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$311$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$312$		0			0
$313$	1330.20	+	6687.38i	0.240216	+	1.20765i	0.892984	+	0.450088i	$0.148607\pi$
$313$	1330.20	+	6687.38i	0.240216	+	1.20765i	−0.652769	+	0.757557i	$0.726393\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$317$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−8937.26	+	3701.94i	−1.55399	+	0.643682i
$322$		0			0
$323$	−11517.0	−	1019.44i	−1.98397	−	0.175614i
$324$	9035.91			1.54937
$325$		0			0
$326$	6535.33	+	9780.81i	1.11030	+	1.66168i
$327$		0			0
$328$	1777.23	−	8934.74i	0.299180	−	1.50408i
$329$		0			0
$330$		0			0
$331$	−10140.0	−	4200.12i	−1.68382	−	0.697461i	−0.684322	−	0.729180i	$-0.739902\pi$
$331$	−10140.0	−	4200.12i	−1.68382	−	0.697461i	−0.999497	+	0.0317191i	$0.989902\pi$
$332$	3019.77	−	7290.38i	0.499191	−	1.20515i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−5753.54	+	3844.39i	−0.930016	+	0.621417i	−0.925571	−	0.378574i	$-0.876415\pi$
$337$	−5753.54	+	3844.39i	−0.930016	+	0.621417i	−0.00444445	+	0.999990i	$0.501415\pi$
$338$	−5741.04	+	2378.02i	−0.923880	+	0.382683i
$339$			22159.0i			3.55018i
$340$		0			0
$341$		0			0
$342$	−10476.0	−	25291.3i	−1.65637	−	3.99882i
$343$		0			0
$344$	−5373.86	+	5373.86i	−0.842265	+	0.842265i
$345$		0			0
$346$		0			0
$347$	2217.57	+	11148.5i	0.343070	+	1.72473i	0.638724	+	0.769436i	$0.279463\pi$
$347$	2217.57	+	11148.5i	0.343070	+	1.72473i	−0.295654	+	0.955295i	$0.595537\pi$
$348$		0			0
$349$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$349$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$350$		0			0
$351$		0			0
$352$	12213.8	+	2429.47i	1.84942	+	0.367873i
$353$	−9376.55	−	9376.55i	−1.41378	−	1.41378i	−0.724461	−	0.689316i	$-0.757911\pi$
$353$	−9376.55	−	9376.55i	−1.41378	−	1.41378i	−0.689316	−	0.724461i	$-0.742089\pi$
$354$	505.902	−	338.033i	0.0759559	−	0.0507521i
$355$		0			0
$356$			12966.4i			1.93039i
$357$		0			0
$358$	13409.7			1.97968
$359$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$359$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$360$		0			0
$361$	−14390.0	+	14390.0i	−2.09798	+	2.09798i
$362$		0			0
$363$	17492.5	−	26179.3i	2.52924	−	3.78528i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$367$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$368$		0			0
$369$	−23168.4	−	4608.49i	−3.26856	−	0.650158i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$	−10627.8	−	8547.55i	−1.46939	−	1.18177i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	2871.83	−	14437.7i	0.389224	−	1.95676i	0.125984	−	0.992032i	$-0.459791\pi$
$379$	2871.83	−	14437.7i	0.389224	−	1.95676i	0.263240	−	0.964730i	$-0.415209\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$383$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$384$	13146.6	−	2615.03i	1.74710	−	0.347520i
$385$		0			0
$386$	8359.08	+	1662.73i	1.10224	+	0.219250i
$387$	13934.8	+	13934.8i	1.83035	+	1.83035i
$388$	2810.74	−	1878.08i	0.367767	−	0.245734i
$389$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$389$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$390$		0			0
$391$		0			0
$392$	7761.20			1.00000
$393$	−1713.35	−	4136.40i	−0.219917	−	0.530926i
$394$		0			0
$395$		0			0
$396$	6299.79	−	31671.2i	0.799436	−	4.01903i
$397$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$397$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$398$		0			0
$399$		0			0
$400$	−3061.47	+	7391.04i	−0.382683	+	0.923880i
$401$	3969.99	−	789.680i	0.494394	−	0.0983410i	0.0584046	−	0.998293i	$-0.481399\pi$
$401$	3969.99	−	789.680i	0.494394	−	0.0983410i	0.435989	+	0.899952i	$0.356399\pi$
$402$	7710.51	+	5152.00i	0.956630	+	0.639200i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	−14089.6	−	4122.34i	−1.70965	−	0.500211i
$409$	−14127.6			−1.70798			−0.853990	−	0.520289i	$-0.825824\pi$
$409$	−14127.6			−1.70798			−0.853990	+	0.520289i	$0.825824\pi$
$410$		0			0
$411$	−128.610	−	192.478i	−0.0154352	−	0.0231004i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	9151.50	−	22093.7i	1.07470	−	2.59456i
$418$	−31479.7	+	6261.71i	−3.68355	+	0.732703i
$419$	−5572.79	−	3723.62i	−0.649758	−	0.434154i	0.186527	−	0.982450i	$-0.440277\pi$
$419$	−5572.79	−	3723.62i	−0.649758	−	0.434154i	−0.836285	+	0.548296i	$0.815277\pi$
$420$		0			0
$421$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$421$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$422$	7246.90	−	4842.23i	0.835957	−	0.558568i
$423$		0			0
$424$		0			0
$425$	6717.51	−	5625.00i	0.766700	−	0.642006i
$426$		0			0
$427$		0			0
$428$	4645.07	+	6951.84i	0.524598	+	0.785116i
$429$		0			0
$430$		0			0
$431$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$431$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$432$	−3660.60	−	18403.1i	−0.407686	−	2.04958i
$433$	14808.1	+	6133.71i	1.64349	+	0.680756i	0.996644	−	0.0818641i	$-0.0260873\pi$
$433$	14808.1	+	6133.71i	1.64349	+	0.680756i	0.646847	+	0.762620i	$0.276087\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	22446.3	+	22446.3i	2.44869	+	2.44869i
$439$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$439$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$440$		0			0
$441$		−	20125.4i		−	2.17313i
$442$		0			0
$443$	3539.69			0.379629			0.189814	−	0.981820i	$-0.439211\pi$
$443$	3539.69			0.379629			0.189814	+	0.981820i	$0.439211\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	2089.93	+	10506.8i	0.219666	+	1.10434i	0.920420	+	0.390932i	$0.127847\pi$
$449$	2089.93	+	10506.8i	0.219666	+	1.10434i	−0.700754	+	0.713403i	$0.747153\pi$
$450$	19165.5	+	7938.61i	2.00771	+	0.831621i
$451$	−10599.0	+	25588.2i	−1.10662	+	2.67162i
$452$	18784.0	−	3736.37i	1.95470	−	0.388814i
$453$		0			0
$454$	1623.43	+	322.919i	0.167822	+	0.0333818i
$455$		0			0
$456$	−28725.5	+	19193.8i	−2.94999	+	1.97112i
$457$	−16660.7	+	6901.09i	−1.70537	+	0.706388i	−0.999997	−	0.00244611i	$-0.999221\pi$
$457$	−16660.7	+	6901.09i	−1.70537	+	0.706388i	−0.705375	+	0.708834i	$0.749221\pi$
$458$		0			0
$459$	−5770.57	+	19723.0i	−0.586813	+	2.00565i
$460$		0			0
$461$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$461$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$462$		0			0
$463$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$463$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$464$		0			0
$465$		0			0
$466$	2857.23	+	14364.3i	0.284032	+	1.42792i
$467$	1017.94	+	421.646i	0.100867	+	0.0417804i	0.432546	−	0.901612i	$-0.357615\pi$
$467$	1017.94	+	421.646i	0.100867	+	0.0417804i	−0.331679	+	0.943392i	$0.607615\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	−371.851	−	371.851i	−0.0362624	−	0.0362624i
$473$	19211.6	−	12836.8i	1.86755	−	1.24786i
$474$		0			0
$475$		−	20619.2i		−	1.99173i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$479$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$480$		0			0
$481$		0			0
$482$	−8041.40	+	12034.8i	−0.759909	+	1.13728i
$483$		0			0
$484$	−25141.5	−	10414.0i	−2.36115	−	0.978019i
$485$		0			0
$486$	−7042.60	+	1400.86i	−0.657323	+	0.130750i
$487$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$487$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$488$		0			0
$489$	−27220.4	−	27220.4i	−2.51728	−	2.51728i
$490$		0			0
$491$	3776.58	−	1564.31i	0.347117	−	0.143781i	−0.202311	−	0.979321i	$-0.564845\pi$
$491$	3776.58	−	1564.31i	0.347117	−	0.143781i	0.549429	+	0.835540i	$0.314845\pi$
$492$			29811.8i			2.73175i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	−5037.92	+	25327.3i	−0.453322	+	2.27901i
$499$	−4445.91	+	6653.77i	−0.398850	+	0.596921i	−0.975481	−	0.220084i	$-0.929367\pi$
$499$	−4445.91	+	6653.77i	−0.398850	+	0.596921i	0.576631	+	0.817005i	$0.304367\pi$
$500$		0			0
$501$		0			0
$502$	7072.77	−	17075.2i	0.628831	−	1.51813i
$503$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$503$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	16908.4	−	11297.8i	1.48112	−	0.989653i
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$	−4433.48	−	10703.4i	−0.382683	−	0.923880i
$513$	26868.1	+	40210.9i	2.31238	+	3.46073i
$514$	12718.7	−	12718.7i	1.09143	−	1.09143i
$515$		0			0
$516$	13817.2	−	20679.0i	1.17882	−	1.76423i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	10941.7	+	7311.00i	0.920084	+	0.614781i	0.922827	−	0.385215i	$-0.125873\pi$
$521$	10941.7	+	7311.00i	0.920084	+	0.614781i	−0.00274220	+	0.999996i	$0.500873\pi$
$522$		0			0
$523$	16601.4	+	16601.4i	1.38801	+	1.38801i	0.829493	+	0.558517i	$0.188629\pi$
$523$	16601.4	+	16601.4i	1.38801	+	1.38801i	0.558517	+	0.829493i	$0.311371\pi$
$524$	−3217.49	+	2149.86i	−0.268238	+	0.179231i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	−40752.7			−3.35897
$529$	−4656.11	−	11240.8i	−0.382683	−	0.923880i
$530$		0			0
$531$	−964.237	+	964.237i	−0.0788029	+	0.0788029i
$532$		0			0
$533$		0			0
$534$	−8278.21	−	41617.4i	−0.670849	−	3.37259i
$535$		0			0
$536$	3067.19	−	7404.85i	0.247169	−	0.596718i
$537$	−43040.2	+	8561.22i	−3.45870	+	0.687978i
$538$		0			0
$539$	−23143.0	−	4603.42i	−1.84942	−	0.367873i
$540$		0			0
$541$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$541$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$542$		0			0
$543$		0			0
$544$	−1118.73	+	12638.7i	−0.0881716	+	0.996105i
$545$		0			0
$546$		0			0
$547$	9622.65	+	14401.3i	0.752166	+	1.12570i	0.988083	+	0.153923i	$0.0491908\pi$
$547$	9622.65	+	14401.3i	0.752166	+	1.12570i	−0.235917	+	0.971773i	$0.575809\pi$
$548$	−141.476	+	141.476i	−0.0110284	+	0.0110284i
$549$		0			0
$550$	13512.8	−	20223.3i	1.04761	−	1.56786i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−20271.7	−	4032.30i	−1.54625	−	0.307567i
$557$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$557$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	39568.4	+	20649.3i	2.97786	+	1.55404i
$562$	16251.6			1.21980
$563$	10183.3	+	24584.6i	0.762298	+	1.84035i	0.463795	+	0.885943i	$0.346488\pi$
$563$	10183.3	+	24584.6i	0.762298	+	1.84035i	0.298504	+	0.954408i	$0.403512\pi$
$564$		0			0
$565$		0			0
$566$	−5252.76	+	26407.4i	−0.390089	+	1.96111i
$567$		0			0
$568$		0			0
$569$	7516.59	+	3113.47i	0.553800	+	0.229391i	0.641991	−	0.766712i	$-0.278109\pi$
$569$	7516.59	+	3113.47i	0.553800	+	0.229391i	−0.0881913	+	0.996104i	$0.528109\pi$
$570$		0			0
$571$	−17742.9	+	3529.29i	−1.30038	+	0.258662i	−0.796252	−	0.604965i	$-0.793187\pi$
$571$	−17742.9	+	3529.29i	−1.30038	+	0.258662i	−0.504131	+	0.863627i	$0.668187\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−27754.6	+	11496.3i	−2.00771	+	0.831621i
$577$		−	19651.9i		−	1.41789i	−0.705266	−	0.708943i	$-0.749173\pi$
$577$		−	19651.9i		−	1.41789i	0.705266	−	0.708943i	$-0.250827\pi$
$578$	7490.24	−	11704.6i	0.539019	−	0.842294i
$579$	−27891.1			−2.00192
$580$		0			0
$581$		0			0
$582$	−7822.40	+	7822.40i	−0.557129	+	0.557129i
$583$		0			0
$584$	15242.7	−	22812.4i	1.08005	−	1.61641i
$585$		0			0
$586$		0			0
$587$	3960.77	−	9562.15i	0.278498	−	0.672355i	−0.721296	−	0.692627i	$-0.756453\pi$
$587$	3960.77	−	9562.15i	0.278498	−	0.672355i	0.999795	+	0.0202721i	$0.00645325\pi$
$588$	−24910.6	+	4955.02i	−1.74710	+	0.347520i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	26658.6	−	11042.3i	1.84610	−	0.764679i	0.906825	−	0.421507i	$-0.138499\pi$
$593$	26658.6	−	11042.3i	1.84610	−	0.764679i	0.939273	−	0.343171i	$-0.111501\pi$
$594$			57046.9i			3.94051i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$599$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$600$	5107.48	−	25677.0i	0.347520	−	1.74710i
$601$	10632.1	−	15912.0i	0.721615	−	1.07997i	−0.271455	−	0.962451i	$-0.587505\pi$
$601$	10632.1	−	15912.0i	0.721615	−	1.07997i	0.993070	−	0.117522i	$-0.0374951\pi$
$602$		0			0
$603$	−19201.3	−	7953.45i	−1.29675	−	0.537130i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$607$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$608$	21114.0	+	21114.0i	1.40837	+	1.40837i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	32773.2	+	2900.96i	2.16467	+	0.191608i
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$	7797.56	+	18825.0i	0.512514	+	1.23732i
$615$		0			0
$616$		0			0
$617$	3413.30	−	17159.8i	0.222713	−	1.11966i	−0.693958	−	0.720015i	$-0.744135\pi$
$617$	3413.30	−	17159.8i	0.222713	−	1.11966i	0.916672	−	0.399641i	$-0.130865\pi$
$618$		0			0
$619$	705.955	+	3549.07i	0.0458396	+	0.230451i	0.996913	−	0.0785136i	$-0.0250174\pi$
$619$	705.955	+	3549.07i	0.0458396	+	0.230451i	−0.951073	+	0.308965i	$0.900017\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	11048.5	+	11048.5i	0.707107	+	0.707107i
$626$	−16035.2	+	10714.4i	−1.02379	+	0.684076i
$627$	97040.5	−	40195.5i	6.18090	−	2.56021i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$631$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$632$		0			0
$633$	−20168.4	+	20168.4i	−1.26639	+	1.26639i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	29541.5	+	5876.17i	1.82031	+	0.362082i	0.982849	−	0.184413i	$-0.0590382\pi$
$641$	29541.5	+	5876.17i	1.82031	+	0.362082i	0.837462	+	0.546495i	$0.184038\pi$
$642$	−19347.2	−	19347.2i	−1.18937	−	1.18937i
$643$	2294.79	−	1533.33i	0.140743	−	0.0940415i	−0.483205	−	0.875507i	$-0.660528\pi$
$643$	2294.79	−	1533.33i	0.140743	−	0.0940415i	0.623948	+	0.781466i	$0.285528\pi$
$644$		0			0
$645$		0			0
$646$	−9801.99	−	31198.8i	−0.596988	−	1.90016i
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$	9780.40	+	23612.0i	0.592917	+	1.43143i
$649$	888.258	+	1329.37i	0.0537245	+	0.0804043i
$650$		0			0
$651$		0			0
$652$	−18484.7	+	27664.3i	−1.11030	+	1.66168i
$653$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$653$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$654$		0			0
$655$		0			0
$656$	25271.3	−	5026.77i	1.50408	−	0.299180i
$657$	−59154.1	−	39525.5i	−3.51267	−	2.34709i
$658$		0			0
$659$	21039.3	+	21039.3i	1.24366	+	1.24366i	0.958471	+	0.285191i	$0.0920570\pi$
$659$	21039.3	+	21039.3i	1.24366	+	1.24366i	0.285191	+	0.958471i	$0.407943\pi$
$660$		0			0
$661$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$661$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$662$		−	31043.2i		−	1.82255i
$663$		0			0
$664$	22319.3			1.30445
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	246.583	−	49.0484i	0.0141234	−	0.00280933i	−0.188024	−	0.982165i	$-0.560208\pi$
$673$	246.583	−	49.0484i	0.0141234	−	0.00280933i	0.202147	+	0.979355i	$0.435208\pi$
$674$	−16273.5	−	10873.6i	−0.930016	−	0.621417i
$675$	−35943.5	−	7149.60i	−2.04958	−	0.407686i
$676$	−12428.1	−	12428.1i	−0.707107	−	0.707107i
$677$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$677$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$678$	−57904.2	+	23984.7i	−3.27994	+	1.35859i
$679$		0			0
$680$		0			0
$681$	−5416.76			−0.304803
$682$		0			0
$683$	−15910.5	−	23811.7i	−0.891358	−	1.33401i	−0.942112	−	0.335300i	$-0.891162\pi$
$683$	−15910.5	−	23811.7i	−0.891358	−	1.33401i	0.0507540	−	0.998711i	$-0.483838\pi$
$684$	54750.2	−	54750.2i	3.06056	−	3.06056i
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−19859.2	−	8225.95i	−1.10047	−	0.455831i
$689$		0			0
$690$		0			0
$691$	18124.1	+	12110.2i	0.997793	+	0.666704i	0.943345	−	0.331813i	$-0.107660\pi$
$691$	18124.1	+	12110.2i	0.997793	+	0.666704i	0.0544477	+	0.998517i	$0.482660\pi$
$692$		0			0
$693$		0			0
$694$	−26732.1	+	17861.8i	−1.46216	+	0.976982i
$695$		0			0
$696$		0			0
$697$	−27083.9	−	7924.21i	−1.47184	−	0.430632i
$698$		0			0
$699$	−18341.3	−	44279.9i	−0.992464	−	2.39602i
$700$		0			0
$701$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$701$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$702$		0			0
$703$		0			0
$704$	6871.58	+	34545.8i	0.367873	+	1.84942i
$705$		0			0
$706$	14353.0	−	34651.2i	0.765130	−	1.84719i
$707$		0			0
$708$	1430.91	+	956.102i	0.0759559	+	0.0507521i
$709$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$709$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$710$		0			0
$711$		0			0
$712$	−33882.9	+	14034.8i	−1.78345	+	0.738728i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	14514.6	+	35041.3i	0.757590	+	1.82898i
$717$		0			0
$718$		0			0
$719$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$719$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$720$		0			0
$721$		0			0
$722$	−53178.7	−	22027.3i	−2.74114	−	1.13542i
$723$	18126.5	−	43761.2i	0.932408	−	2.25103i
$724$		0			0
$725$		0			0
$726$	87343.5	+	17373.7i	4.46504	+	0.888152i
$727$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$727$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$728$		0			0
$729$	−6465.04	+	2677.91i	−0.328458	+	0.136052i
$730$		0			0
$731$	15114.0	+	18049.5i	0.764721	+	0.913249i
$732$		0			0
$733$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$733$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−13538.0	+	20261.1i	−0.676635	+	1.01266i
$738$	−13034.8	−	65530.2i	−0.650158	−	3.26856i
$739$	−16345.1	−	6770.36i	−0.813619	−	0.337012i	−0.0632220	−	0.997999i	$-0.520138\pi$
$739$	−16345.1	−	6770.36i	−0.813619	−	0.337012i	−0.750397	+	0.660988i	$0.770138\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$743$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		−	57875.4i		−	2.83474i
$748$	10832.4	−	37023.6i	0.529506	−	1.80978i
$749$		0			0
$750$		0			0
$751$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$751$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$752$		0			0
$753$	−11799.6	+	59320.5i	−0.571050	+	2.87086i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$757$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$758$	40835.9	−	8122.76i	1.95676	−	0.389224i
$759$		0			0
$760$		0			0
$761$	−4900.48	−	4900.48i	−0.233433	−	0.233433i	0.580691	−	0.814124i	$-0.302782\pi$
$761$	−4900.48	−	4900.48i	−0.233433	−	0.233433i	−0.814124	+	0.580691i	$0.802782\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	21063.2	+	31523.3i	0.989653	+	1.48112i
$769$	−12798.1	+	12798.1i	−0.600144	+	0.600144i	−0.940351	−	0.340207i	$-0.889503\pi$
$769$	−12798.1	+	12798.1i	−0.600144	+	0.600144i	0.340207	+	0.940351i	$0.389503\pi$
$770$		0			0
$771$	−32702.1	+	48942.2i	−1.52755	+	2.28613i
$772$	4702.90	+	23643.1i	0.219250	+	1.10224i
$773$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$773$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$774$	−21330.5	+	51496.4i	−0.990580	+	2.39147i
$775$		0			0
$776$	7949.97	+	5312.00i	0.367767	+	0.245734i
$777$		0			0
$778$		0			0
$779$	−55218.0	+	36895.5i	−2.53965	+	1.69694i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	8400.67	+	20281.0i	0.382683	+	0.923880i
$785$		0			0
$786$	8954.42	−	8954.42i	0.406353	−	0.406353i
$787$	329.824	−	1658.14i	0.0149390	−	0.0751033i	−0.972597	−	0.232499i	$-0.925310\pi$
$787$	329.824	−	1658.14i	0.0149390	−	0.0751033i	0.987536	+	0.157396i	$0.0503098\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	89579.7	−	17818.5i	4.01903	−	0.799436i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$797$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$798$		0			0
$799$		0			0
$800$	−22627.4			−1.00000
$801$	36393.1	+	87860.8i	1.60535	+	3.87567i
$802$	6360.62	+	9519.34i	0.280051	+	0.419127i
$803$	−58982.7	+	58982.7i	−2.59210	+	2.59210i
$804$	−5117.03	+	25725.0i	−0.224457	+	1.12842i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	22166.9	−	4409.27i	0.963345	−	0.191621i	0.311730	−	0.950171i	$-0.399092\pi$
$809$	22166.9	−	4409.27i	0.963345	−	0.191621i	0.651615	+	0.758550i	$0.274092\pi$
$810$		0			0
$811$	41290.3	+	8213.15i	1.78779	+	0.355614i	0.974176	−	0.225791i	$-0.0724968\pi$
$811$	41290.3	+	8213.15i	1.78779	+	0.355614i	0.813614	+	0.581405i	$0.197497\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	−4478.29	−	41279.9i	−0.192122	−	1.77094i
$817$	55402.3			2.37244
$818$	−15291.6	−	36917.2i	−0.653616	−	1.57797i
$819$		0			0
$820$		0			0
$821$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$821$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$822$	363.763	−	544.410i	0.0154352	−	0.0231004i
$823$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$823$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$824$		0			0
$825$	−30459.8	+	73536.4i	−1.28542	+	3.10328i
$826$		0			0
$827$	12446.8	+	8316.72i	0.523361	+	0.349698i	0.789019	−	0.614369i	$-0.210589\pi$
$827$	12446.8	+	8316.72i	0.523361	+	0.349698i	−0.265658	+	0.964067i	$0.585589\pi$
$828$		0			0
$829$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$829$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	2119.81	−	23948.2i	0.0881716	−	0.996105i
$834$	67639.1			2.80833
$835$		0			0
$836$	−50436.0	−	75482.8i	−2.08656	−	3.12276i
$837$		0			0
$838$	3698.34	−	18592.8i	0.152455	−	0.766442i
$839$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$839$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$840$		0			0
$841$	22532.5	+	9333.27i	0.923880	+	0.382683i
$842$		0			0
$843$	−52161.5	+	10375.6i	−2.13112	+	0.423907i
$844$	20497.3	+	13695.9i	0.835957	+	0.558568i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$		−	88111.6i		−	3.56182i
$850$	21969.8	+	11465.3i	0.886539	+	0.462653i
$851$		0			0
$852$		0			0
$853$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$853$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$854$		0			0
$855$		0			0
$856$	−13138.2	+	19662.8i	−0.524598	+	0.785116i
$857$	5538.18	+	27842.3i	0.220748	+	1.10977i	0.919102	+	0.394020i	$0.128916\pi$
$857$	5538.18	+	27842.3i	0.220748	+	1.10977i	−0.698354	+	0.715752i	$0.746084\pi$
$858$		0			0
$859$	830.298	−	2004.52i	0.0329795	−	0.0796196i	−0.906532	−	0.422138i	$-0.861280\pi$
$859$	830.298	−	2004.52i	0.0329795	−	0.0796196i	0.939511	+	0.342518i	$0.111280\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$863$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$864$	44127.3	−	29484.9i	1.73755	−	1.16099i
$865$		0			0
$866$			45334.5i			1.77890i
$867$	−16568.3	+	42349.3i	−0.649006	+	1.65889i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	13774.4	−	20614.9i	0.534013	−	0.799206i
$874$		0			0
$875$		0			0
$876$	−34359.3	+	82950.7i	−1.32522	+	3.19937i
$877$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$877$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	41065.6	−	27439.1i	1.57041	−	1.04932i	0.602471	−	0.798141i	$-0.294183\pi$
$881$	41065.6	−	27439.1i	1.57041	−	1.04932i	0.967942	−	0.251176i	$-0.0808172\pi$
$882$	52590.1	−	21783.5i	2.00771	−	0.831621i
$883$		−	50260.1i		−	1.91550i	−0.287602	−	0.957750i	$-0.592858\pi$
$883$		−	50260.1i		−	1.91550i	0.287602	−	0.957750i	$-0.407142\pi$
$884$		0			0
$885$		0			0
$886$	3831.33	+	9249.65i	0.145278	+	0.350731i
$887$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$887$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−15158.9	−	76209.1i	−0.569970	−	2.86543i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−25193.5	+	16833.7i	−0.936210	+	0.625556i
$899$		0			0
$900$			58674.5i			2.17313i
$901$		0			0
$902$	−78337.4			−2.89174
$903$		0			0
$904$	30095.2	+	45040.7i	1.10725	+	1.65711i
$905$		0			0
$906$		0			0
$907$	−14052.4	+	21030.9i	−0.514445	+	0.769922i	−0.994208	−	0.107477i	$-0.965723\pi$
$907$	−14052.4	+	21030.9i	−0.514445	+	0.769922i	0.479762	+	0.877399i	$0.340723\pi$
$908$	913.354	+	4591.74i	0.0333818	+	0.167822i
$909$		0			0
$910$		0			0
$911$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$911$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$912$	−81248.1	−	54288.3i	−2.94999	−	1.97112i
$913$	−66553.3	−	13238.3i	−2.41248	−	0.479872i
$914$	−36066.8	−	36066.8i	−1.30524	−	1.30524i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	−57784.8	+	6268.84i	−2.07754	+	0.225384i
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	−37045.8	−	55442.9i	−1.32541	−	1.98361i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	14797.2	+	9887.15i	0.522582	+	0.349178i	0.788716	−	0.614758i	$-0.210746\pi$
$929$	14797.2	+	9887.15i	0.522582	+	0.349178i	−0.266134	+	0.963936i	$0.585746\pi$
$930$		0			0
$931$	−40007.4	−	40007.4i	−1.40837	−	1.40837i
$932$	−34443.0	+	23014.1i	−1.21054	+	0.808854i
$933$		0			0
$934$			3116.40i			0.109178i
$935$		0			0
$936$		0			0
$937$	−4266.79	−	10300.9i	−0.148762	−	0.359143i	0.831879	−	0.554957i	$-0.187265\pi$
$937$	−4266.79	−	10300.9i	−0.148762	−	0.359143i	−0.980641	+	0.195814i	$0.937265\pi$
$938$		0			0
$939$	44626.5	−	44626.5i	1.55094	−	1.55094i
$940$		0			0
$941$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$941$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$942$		0			0
$943$		0			0
$944$	569.205	−	1374.18i	0.0196251	−	0.0473791i
$945$		0			0
$946$	54338.6	+	36307.9i	1.86755	+	1.24786i
$947$	−55530.5	−	11045.7i	−1.90549	−	0.379025i	−0.906428	−	0.422361i	$-0.861202\pi$
$947$	−55530.5	−	11045.7i	−1.90549	−	0.379025i	−0.999061	+	0.0433353i	$0.986202\pi$
$948$		0			0
$949$		0			0
$950$	53880.5	−	22318.0i	1.84012	−	0.762202i
$951$		0			0
$952$		0			0
$953$	16308.3			0.554332			0.277166	−	0.960822i	$-0.410605\pi$
$953$	16308.3			0.554332			0.277166	+	0.960822i	$0.410605\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	11400.5	−	27523.3i	0.382683	−	0.923880i
$962$		0			0
$963$	50987.0	+	34068.4i	1.70616	+	1.14002i
$964$	−40152.4	−	7986.82i	−1.34152	−	0.266844i
$965$		0			0
$966$		0			0
$967$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$967$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$968$		−	76969.9i		−	2.55569i
$969$	51379.2	+	93878.7i	1.70334	+	3.11230i
$970$		0			0
$971$	−21371.3	−	51594.8i	−0.706320	−	1.70521i	−0.708997	−	0.705211i	$-0.750852\pi$
$971$	−21371.3	−	51594.8i	−0.706320	−	1.70521i	0.00267705	−	0.999996i	$-0.499148\pi$
$972$	−11283.5	−	16886.9i	−0.372343	−	0.557251i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−43837.8	−	18158.2i	−1.43551	−	0.594609i	−0.476807	−	0.879008i	$-0.658206\pi$
$977$	−43837.8	−	18158.2i	−1.43551	−	0.594609i	−0.958706	+	0.284399i	$0.908206\pi$
$978$	41667.1	−	100593.i	1.36234	−	3.28898i
$979$	109359.	−	21752.9i	3.57010	−	0.710138i
$980$		0			0
$981$		0			0
$982$	8175.48	+	8175.48i	0.265672	+	0.265672i
$983$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$983$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$984$	−77902.1	+	32268.1i	−2.52381	+	1.04540i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$991$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$992$		0			0
$993$	19819.1	+	99637.3i	0.633373	+	3.18418i
$994$		0			0
$995$		0			0
$996$	−71636.5	+	14249.4i	−2.27901	+	0.453322i
$997$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$997$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$998$	−22199.4	−	4415.73i	−0.704116	−	0.140057i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	136.4.s.a.107.1	yes	8
8.3	odd	2	CM	136.4.s.a.107.1	yes	8
17.7	odd	16	inner	136.4.s.a.75.1	✓	8
136.75	even	16	inner	136.4.s.a.75.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.4.s.a.75.1	✓	8	17.7	odd	16	inner
136.4.s.a.75.1	✓	8	136.75	even	16	inner
136.4.s.a.107.1	yes	8	1.1	even	1	trivial
136.4.s.a.107.1	yes	8	8.3	odd	2	CM

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:	\( N \)	\(=\)	\( 136 = 2^{3} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	136.s (of order \(16\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\(q+(1.08239 + 2.61313i) q^{2} +(-5.14239 - 7.69613i) q^{3} +(-5.65685 + 5.65685i) q^{4} +(14.5449 - 21.7679i) q^{6} +(-20.9050 - 8.65914i) q^{8} +(-22.4538 + 54.2082i) q^{9} +O(q^{10})\)	\(q+(1.08239 + 2.61313i) q^{2} +(-5.14239 - 7.69613i) q^{3} +(-5.65685 + 5.65685i) q^{4} +(14.5449 - 21.7679i) q^{6} +(-20.9050 - 8.65914i) q^{8} +(-22.4538 + 54.2082i) q^{9} +(57.2002 + 38.2199i) q^{11} +(72.6256 + 14.4461i) q^{12} -64.0000i q^{16} +(-32.4286 + 62.1400i) q^{17} -165.957 q^{18} +(63.1249 + 152.397i) q^{19} +(-37.9605 + 190.840i) q^{22} +(40.8598 + 205.416i) q^{24} +(-115.485 - 47.8354i) q^{25} +(287.548 - 57.1968i) q^{27} +(167.240 - 69.2731i) q^{32} -636.761i q^{33} +(-197.480 - 17.4802i) q^{34} +(-179.630 - 433.666i) q^{36} +(-329.907 + 329.907i) q^{38} +(78.5432 + 394.864i) q^{41} +(128.530 - 310.300i) q^{43} +(-539.777 + 107.368i) q^{44} +(-492.552 + 329.113i) q^{48} +(-316.891 + 131.260i) q^{49} -353.553i q^{50} +(644.998 - 69.9732i) q^{51} +(460.702 + 689.489i) q^{54} +(848.254 - 1269.50i) q^{57} +(21.4716 + 8.89383i) q^{59} +(362.039 + 362.039i) q^{64} +(1663.94 - 689.226i) q^{66} +354.214i q^{67} +(-168.073 - 534.961i) q^{68} +(938.793 - 938.793i) q^{72} +(-236.551 + 1189.22i) q^{73} +(225.721 + 1134.78i) q^{75} +(-1219.18 - 504.999i) q^{76} +(-798.669 - 798.669i) q^{81} +(-946.813 + 632.641i) q^{82} +(-911.297 + 377.472i) q^{83} +949.973 q^{86} +(-864.818 - 1294.29i) q^{88} +(1146.08 - 1146.08i) q^{89} +(-1393.15 - 930.872i) q^{96} +(-414.437 - 82.4366i) q^{97} +(-686.000 - 686.000i) q^{98} +(-3356.19 + 2242.54i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{3} + 80 q^{6} - 80 q^{9}+O(q^{10}) \) 8 * q - 8 * q^3 + 80 * q^6 - 80 * q^9	\( 8 q - 8 q^{3} + 80 q^{6} - 80 q^{9} + 320 q^{12} + 144 q^{22} - 256 q^{24} + 496 q^{27} - 720 q^{34} - 640 q^{36} - 1440 q^{38} + 160 q^{41} + 208 q^{43} - 1600 q^{44} + 512 q^{48} + 1880 q^{51} - 176 q^{54} + 2840 q^{57} - 920 q^{59} + 5232 q^{66} + 3456 q^{72} - 3312 q^{73} - 2632 q^{81} - 3472 q^{83} - 5120 q^{96} - 5488 q^{98} - 1536 q^{99}+O(q^{100}) \) 8 * q - 8 * q^3 + 80 * q^6 - 80 * q^9 + 320 * q^12 + 144 * q^22 - 256 * q^24 + 496 * q^27 - 720 * q^34 - 640 * q^36 - 1440 * q^38 + 160 * q^41 + 208 * q^43 - 1600 * q^44 + 512 * q^48 + 1880 * q^51 - 176 * q^54 + 2840 * q^57 - 920 * q^59 + 5232 * q^66 + 3456 * q^72 - 3312 * q^73 - 2632 * q^81 - 3472 * q^83 - 5120 * q^96 - 5488 * q^98 - 1536 * q^99

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