LMFDB - Embedded newform 198.7.d.a.109.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [198,7,Mod(109,198)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(198, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("198.109");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 198 = 2 \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	198.d (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:	$45.5507085567$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 1781x^{4} + 14500x^{3} + 786532x^{2} - 11444432x + 42080676 $ x^6 - 2x^5 - 1781x^4 + 14500x^3 + 786532x^2 - 11444432*x + 42080676
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 22)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			109.4
Root			$25.9852 + 1.41421i$ of defining polynomial
Character	$\chi$	$=$	198.109
Dual form			198.7.d.a.109.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.65685i q^{2} -32.0000 q^{4} -175.767 q^{5} -412.125i q^{7} -181.019i q^{8} +O(q^{10})$	\(q+5.65685i q^{2} -32.0000 q^{4} -175.767 q^{5} -412.125i q^{7} -181.019i q^{8} -994.286i q^{10} +(-1259.88 + 429.258i) q^{11} -1853.71i q^{13} +2331.33 q^{14} +1024.00 q^{16} -8103.56i q^{17} +9527.12i q^{19} +5624.53 q^{20} +(-2428.25 - 7126.96i) q^{22} -219.542 q^{23} +15268.9 q^{25} +10486.1 q^{26} +13188.0i q^{28} -6798.33i q^{29} +6235.97 q^{31} +5792.62i q^{32} +45840.6 q^{34} +72437.7i q^{35} -59358.7 q^{37} -53893.5 q^{38} +31817.1i q^{40} +33300.1i q^{41} +131981. i q^{43} +(40316.2 - 13736.3i) q^{44} -1241.92i q^{46} -75062.8 q^{47} -52197.7 q^{49} +86373.8i q^{50} +59318.6i q^{52} +231249. q^{53} +(221445. - 75449.3i) q^{55} -74602.5 q^{56} +38457.2 q^{58} -128524. q^{59} -268328. i q^{61} +35276.0i q^{62} -32768.0 q^{64} +325819. i q^{65} +181664. q^{67} +259314. i q^{68} -409770. q^{70} -162021. q^{71} +331034. i q^{73} -335784. i q^{74} -304868. i q^{76} +(176908. + 519228. i) q^{77} -94241.5i q^{79} -179985. q^{80} -188374. q^{82} +676011. i q^{83} +1.42433e6i q^{85} -746596. q^{86} +(77704.1 + 228063. i) q^{88} -288260. q^{89} -763958. q^{91} +7025.36 q^{92} -424619. i q^{94} -1.67455e6i q^{95} -1.47936e6 q^{97} -295275. i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 192 q^{4} - 368 q^{5}+O(q^{10}) $ 6 * q - 192 * q^4 - 368 * q^5	$ 6 q - 192 q^{4} - 368 q^{5} + 1166 q^{11} - 2208 q^{14} + 6144 q^{16} + 11776 q^{20} + 1056 q^{22} - 2156 q^{23} + 59862 q^{25} + 1824 q^{26} - 78468 q^{31} + 28704 q^{34} - 205920 q^{37} - 101472 q^{38} - 37312 q^{44} - 493460 q^{47} - 270762 q^{49} + 531700 q^{53} + 274956 q^{55} + 70656 q^{56} + 509184 q^{58} + 833380 q^{59} - 196608 q^{64} + 537420 q^{67} + 96096 q^{70} + 460372 q^{71} - 249744 q^{77} - 376832 q^{80} + 428640 q^{82} - 1055808 q^{86} - 33792 q^{88} - 2377952 q^{89} - 5068656 q^{91} + 68992 q^{92} + 1351632 q^{97}+O(q^{100}) $ 6 * q - 192 * q^4 - 368 * q^5 + 1166 * q^11 - 2208 * q^14 + 6144 * q^16 + 11776 * q^20 + 1056 * q^22 - 2156 * q^23 + 59862 * q^25 + 1824 * q^26 - 78468 * q^31 + 28704 * q^34 - 205920 * q^37 - 101472 * q^38 - 37312 * q^44 - 493460 * q^47 - 270762 * q^49 + 531700 * q^53 + 274956 * q^55 + 70656 * q^56 + 509184 * q^58 + 833380 * q^59 - 196608 * q^64 + 537420 * q^67 + 96096 * q^70 + 460372 * q^71 - 249744 * q^77 - 376832 * q^80 + 428640 * q^82 - 1055808 * q^86 - 33792 * q^88 - 2377952 * q^89 - 5068656 * q^91 + 68992 * q^92 + 1351632 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/198\mathbb{Z}\right)^\times$.

$n$	$145$	$155$
$\chi(n)$	$-1$	$1$

Coefficient data

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

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

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

$2$			5.65685i			0.707107i
$2$			5.65685i			0.707107i
$3$		0			0
$3$		0			0
$4$	−32.0000			−0.500000
$5$	−175.767			−1.40613			−0.703066	−	0.711124i	$-0.748186\pi$
$5$	−175.767			−1.40613			−0.703066	+	0.711124i	$0.748186\pi$
$6$		0			0
$7$		−	412.125i		−	1.20153i	−0.799426	−	0.600765i	$-0.794863\pi$
$7$		−	412.125i		−	1.20153i	0.799426	−	0.600765i	$-0.205137\pi$
$8$		−	181.019i		−	0.353553i
$9$		0			0
$10$		−	994.286i		−	0.994286i
$11$	−1259.88	+	429.258i	−0.946567	+	0.322508i
$11$	−1259.88	+	429.258i	−0.946567	+	0.322508i
$12$		0			0
$13$		−	1853.71i		−	0.843744i	−0.906655	−	0.421872i	$-0.861373\pi$
$13$		−	1853.71i		−	0.843744i	0.906655	−	0.421872i	$-0.138627\pi$
$14$	2331.33			0.849610
$15$		0			0
$16$	1024.00			0.250000
$17$		−	8103.56i		−	1.64941i	−0.565563	−	0.824705i	$-0.691341\pi$
$17$		−	8103.56i		−	1.64941i	0.565563	−	0.824705i	$-0.308659\pi$
$18$		0			0
$19$			9527.12i			1.38900i	0.719495	+	0.694498i	$0.244373\pi$
$19$			9527.12i			1.38900i	−0.719495	+	0.694498i	$0.755627\pi$
$20$	5624.53			0.703066
$21$		0			0
$22$	−2428.25	−	7126.96i	−0.228048	−	0.669324i
$23$	−219.542			−0.0180441			−0.00902205	−	0.999959i	$-0.502872\pi$
$23$	−219.542			−0.0180441			−0.00902205	+	0.999959i	$0.502872\pi$
$24$		0			0
$25$	15268.9			0.977208
$26$	10486.1			0.596617
$27$		0			0
$28$			13188.0i			0.600765i
$29$		−	6798.33i		−	0.278746i	−0.990240	−	0.139373i	$-0.955491\pi$
$29$		−	6798.33i		−	0.278746i	0.990240	−	0.139373i	$-0.0445087\pi$
$30$		0			0
$31$	6235.97			0.209324			0.104662	−	0.994508i	$-0.466624\pi$
$31$	6235.97			0.209324			0.104662	+	0.994508i	$0.466624\pi$
$32$			5792.62i			0.176777i
$33$		0			0
$34$	45840.6			1.16631
$35$			72437.7i			1.68951i
$36$		0			0
$37$	−59358.7			−1.17187			−0.585935	−	0.810358i	$-0.699273\pi$
$37$	−59358.7			−1.17187			−0.585935	+	0.810358i	$0.699273\pi$
$38$	−53893.5			−0.982168
$39$		0			0
$40$			31817.1i			0.497143i
$41$			33300.1i			0.483163i	0.970380	+	0.241582i	$0.0776662\pi$
$41$			33300.1i			0.483163i	−0.970380	+	0.241582i	$0.922334\pi$
$42$		0			0
$43$			131981.i			1.65999i	0.557771	+	0.829995i	$0.311657\pi$
$43$			131981.i			1.65999i	−0.557771	+	0.829995i	$0.688343\pi$
$44$	40316.2	−	13736.3i	0.473283	−	0.161254i
$45$		0			0
$46$		−	1241.92i		−	0.0127591i
$47$	−75062.8			−0.722988			−0.361494	−	0.932374i	$-0.617733\pi$
$47$	−75062.8			−0.722988			−0.361494	+	0.932374i	$0.617733\pi$
$48$		0			0
$49$	−52197.7			−0.443673
$50$			86373.8i			0.690990i
$51$		0			0
$52$			59318.6i			0.421872i
$53$	231249.			1.55329			0.776645	−	0.629938i	$-0.216920\pi$
$53$	231249.			1.55329			0.776645	+	0.629938i	$0.216920\pi$
$54$		0			0
$55$	221445.	−	75449.3i	1.33100	−	0.453489i
$56$	−74602.5			−0.424805
$57$		0			0
$58$	38457.2			0.197103
$59$	−128524.			−0.625788			−0.312894	−	0.949788i	$-0.601299\pi$
$59$	−128524.			−0.625788			−0.312894	+	0.949788i	$0.601299\pi$
$60$		0			0
$61$		−	268328.i		−	1.18216i	−0.806613	−	0.591079i	$-0.798702\pi$
$61$		−	268328.i		−	1.18216i	0.806613	−	0.591079i	$-0.201298\pi$
$62$			35276.0i			0.148014i
$63$		0			0
$64$	−32768.0			−0.125000
$65$			325819.i			1.18642i
$66$		0			0
$67$	181664.			0.604010			0.302005	−	0.953306i	$-0.402344\pi$
$67$	181664.			0.604010			0.302005	+	0.953306i	$0.402344\pi$
$68$			259314.i			0.824705i
$69$		0			0
$70$	−409770.			−1.19466
$71$	−162021.			−0.452684			−0.226342	−	0.974048i	$-0.572677\pi$
$71$	−162021.			−0.452684			−0.226342	+	0.974048i	$0.572677\pi$
$72$		0			0
$73$			331034.i			0.850950i	0.904970	+	0.425475i	$0.139893\pi$
$73$			331034.i			0.850950i	−0.904970	+	0.425475i	$0.860107\pi$
$74$		−	335784.i		−	0.828637i
$75$		0			0
$76$		−	304868.i		−	0.694498i
$77$	176908.	+	519228.i	0.387503	+	1.13733i
$78$		0			0
$79$		−	94241.5i		−	0.191144i	−0.995423	−	0.0955720i	$-0.969532\pi$
$79$		−	94241.5i		−	0.191144i	0.995423	−	0.0955720i	$-0.0304680\pi$
$80$	−179985.			−0.351533
$81$		0			0
$82$	−188374.			−0.341648
$83$			676011.i			1.18228i	0.806570	+	0.591138i	$0.201321\pi$
$83$			676011.i			1.18228i	−0.806570	+	0.591138i	$0.798679\pi$
$84$		0			0
$85$			1.42433e6i			2.31929i
$86$	−746596.			−1.17379
$87$		0			0
$88$	77704.1	+	228063.i	0.114024	+	0.334662i
$89$	−288260.			−0.408898			−0.204449	−	0.978877i	$-0.565540\pi$
$89$	−288260.			−0.408898			−0.204449	+	0.978877i	$0.565540\pi$
$90$		0			0
$91$	−763958.			−1.01378
$92$	7025.36			0.00902205
$93$		0			0
$94$		−	424619.i		−	0.511230i
$95$		−	1.67455e6i		−	1.95311i
$96$		0			0
$97$	−1.47936e6			−1.62091			−0.810453	−	0.585804i	$-0.800779\pi$
$97$	−1.47936e6			−1.62091			−0.810453	+	0.585804i	$0.800779\pi$
$98$		−	295275.i		−	0.313724i
$99$		0			0
$100$	−488604.			−0.488604
$101$			1.80675e6i			1.75362i	0.480838	+	0.876809i	$0.340332\pi$
$101$			1.80675e6i			1.75362i	−0.480838	+	0.876809i	$0.659668\pi$
$102$		0			0
$103$	1.25034e6			1.14424			0.572119	−	0.820171i	$-0.306122\pi$
$103$	1.25034e6			1.14424			0.572119	+	0.820171i	$0.306122\pi$
$104$	−335557.			−0.298309
$105$		0			0
$106$			1.30814e6i			1.09834i
$107$			1.19558e6i			0.975953i	0.872857	+	0.487976i	$0.162265\pi$
$107$			1.19558e6i			0.975953i	−0.872857	+	0.487976i	$0.837735\pi$
$108$		0			0
$109$			643224.i			0.496687i	0.968672	+	0.248343i	$0.0798862\pi$
$109$			643224.i			0.496687i	−0.968672	+	0.248343i	$0.920114\pi$
$110$	426806.	+	1.25268e6i	0.320665	+	0.941158i
$111$		0			0
$112$		−	422016.i		−	0.300382i
$113$	1.33637e6			0.926169			0.463084	−	0.886314i	$-0.346743\pi$
$113$	1.33637e6			0.926169			0.463084	+	0.886314i	$0.346743\pi$
$114$		0			0
$115$	38588.2			0.0253724
$116$			217547.i			0.139373i
$117$		0			0
$118$		−	727040.i		−	0.442499i
$119$	−3.33967e6			−1.98182
$120$		0			0
$121$	1.40304e6	−	1.08163e6i	0.791977	−	0.610551i
$122$	1.51789e6			0.835912
$123$		0			0
$124$	−199551.			−0.104662
$125$	62595.1			0.0320487
$126$		0			0
$127$		−	2.16447e6i		−	1.05667i	−0.849035	−	0.528337i	$-0.822816\pi$
$127$		−	2.16447e6i		−	1.05667i	0.849035	−	0.528337i	$-0.177184\pi$
$128$		−	185364.i		−	0.0883883i
$129$		0			0
$130$	−1.84311e6			−0.838923
$131$		−	2.78752e6i		−	1.23995i	−0.784622	−	0.619974i	$-0.787143\pi$
$131$		−	2.78752e6i		−	1.23995i	0.784622	−	0.619974i	$-0.212857\pi$
$132$		0			0
$133$	3.92636e6			1.66892
$134$			1.02765e6i			0.427099i
$135$		0			0
$136$	−1.46690e6			−0.583155
$137$	2.96546e6			1.15327			0.576634	−	0.817003i	$-0.304366\pi$
$137$	2.96546e6			1.15327			0.576634	+	0.817003i	$0.304366\pi$
$138$		0			0
$139$			789181.i			0.293855i	0.989147	+	0.146927i	$0.0469383\pi$
$139$			789181.i			0.293855i	−0.989147	+	0.146927i	$0.953062\pi$
$140$		−	2.31801e6i		−	0.844755i
$141$		0			0
$142$		−	916526.i		−	0.320096i
$143$	795719.	+	2.33545e6i	0.272114	+	0.798660i
$144$		0			0
$145$			1.19492e6i			0.391954i
$146$	−1.87261e6			−0.601713
$147$		0			0
$148$	1.89948e6			0.585935
$149$		−	5.69493e6i		−	1.72159i	−0.508953	−	0.860794i	$-0.669967\pi$
$149$		−	5.69493e6i		−	1.72159i	0.508953	−	0.860794i	$-0.330033\pi$
$150$		0			0
$151$			2.77930e6i			0.807244i	0.914926	+	0.403622i	$0.132249\pi$
$151$			2.77930e6i			0.807244i	−0.914926	+	0.403622i	$0.867751\pi$
$152$	1.72459e6			0.491084
$153$		0			0
$154$	−2.93720e6	+	1.00074e6i	−0.804212	+	0.274006i
$155$	−1.09607e6			−0.294337
$156$		0			0
$157$	7.04317e6			1.81999			0.909996	−	0.414617i	$-0.136085\pi$
$157$	7.04317e6			1.81999			0.909996	+	0.414617i	$0.136085\pi$
$158$	533110.			0.135159
$159$		0			0
$160$		−	1.01815e6i		−	0.248571i
$161$			90478.9i			0.0216805i
$162$		0			0
$163$	−697459.			−0.161048			−0.0805241	−	0.996753i	$-0.525659\pi$
$163$	−697459.			−0.161048			−0.0805241	+	0.996753i	$0.525659\pi$
$164$		−	1.06560e6i		−	0.241582i
$165$		0			0
$166$	−3.82409e6			−0.835996
$167$		−	708308.i		−	0.152080i	−0.997105	−	0.0760401i	$-0.975772\pi$
$167$		−	708308.i		−	0.152080i	0.997105	−	0.0760401i	$-0.0242277\pi$
$168$		0			0
$169$	1.39058e6			0.288096
$170$	−8.05725e6			−1.63999
$171$		0			0
$172$		−	4.22339e6i		−	0.829995i
$173$			6.46711e6i			1.24903i	0.781014	+	0.624514i	$0.214703\pi$
$173$			6.46711e6i			1.24903i	−0.781014	+	0.624514i	$0.785297\pi$
$174$		0			0
$175$		−	6.29268e6i		−	1.17414i
$176$	−1.29012e6	+	439561.i	−0.236642	+	0.0806271i
$177$		0			0
$178$		−	1.63065e6i		−	0.289135i
$179$	1.08249e7			1.88741			0.943705	−	0.330790i	$-0.107315\pi$
$179$	1.08249e7			1.88741			0.943705	+	0.330790i	$0.107315\pi$
$180$		0			0
$181$	−3.56684e6			−0.601517			−0.300759	−	0.953700i	$-0.597240\pi$
$181$	−3.56684e6			−0.601517			−0.300759	+	0.953700i	$0.597240\pi$
$182$		−	4.32160e6i		−	0.716853i
$183$		0			0
$184$			39741.4i			0.00637955i
$185$	1.04333e7			1.64780
$186$		0			0
$187$	3.47852e6	+	1.02095e7i	0.531949	+	1.56128i
$188$	2.40201e6			0.361494
$189$		0			0
$190$	9.47268e6			1.38106
$191$	3.27361e6			0.469815			0.234907	−	0.972018i	$-0.424521\pi$
$191$	3.27361e6			0.469815			0.234907	+	0.972018i	$0.424521\pi$
$192$		0			0
$193$		−	2.87534e6i		−	0.399961i	−0.979800	−	0.199981i	$-0.935912\pi$
$193$		−	2.87534e6i		−	0.399961i	0.979800	−	0.199981i	$-0.0640879\pi$
$194$		−	8.36851e6i		−	1.14615i
$195$		0			0
$196$	1.67033e6			0.221837
$197$		−	6.01219e6i		−	0.786383i	−0.919457	−	0.393192i	$-0.871371\pi$
$197$		−	6.01219e6i		−	0.786383i	0.919457	−	0.393192i	$-0.128629\pi$
$198$		0			0
$199$	4.86533e6			0.617381			0.308690	−	0.951163i	$-0.400109\pi$
$199$	4.86533e6			0.617381			0.308690	+	0.951163i	$0.400109\pi$
$200$		−	2.76396e6i		−	0.345495i
$201$		0			0
$202$	−1.02205e7			−1.24000
$203$	−2.80176e6			−0.334922
$204$		0			0
$205$		−	5.85304e6i		−	0.679392i
$206$			7.07299e6i			0.809098i
$207$		0			0
$208$		−	1.89819e6i		−	0.210936i
$209$	−4.08960e6	−	1.20030e7i	−0.447962	−	1.31478i
$210$		0			0
$211$		−	4.94713e6i		−	0.526630i	−0.964710	−	0.263315i	$-0.915184\pi$
$211$		−	4.94713e6i		−	0.526630i	0.964710	−	0.263315i	$-0.0848158\pi$
$212$	−7.39998e6			−0.776645
$213$		0			0
$214$	−6.76324e6			−0.690103
$215$		−	2.31978e7i		−	2.33417i
$216$		0			0
$217$		−	2.57000e6i		−	0.251509i
$218$	−3.63862e6			−0.351211
$219$		0			0
$220$	−7.08623e6	+	2.41438e6i	−0.665499	+	0.226745i
$221$	−1.50216e7			−1.39168
$222$		0			0
$223$	−1.37738e7			−1.24205			−0.621027	−	0.783789i	$-0.713284\pi$
$223$	−1.37738e7			−1.24205			−0.621027	+	0.783789i	$0.713284\pi$
$224$	2.38728e6			0.212402
$225$		0			0
$226$			7.55963e6i			0.654900i
$227$			1.60283e7i			1.37028i	0.728411	+	0.685140i	$0.240259\pi$
$227$			1.60283e7i			1.37028i	−0.728411	+	0.685140i	$0.759741\pi$
$228$		0			0
$229$	−4.33803e6			−0.361232			−0.180616	−	0.983554i	$-0.557809\pi$
$229$	−4.33803e6			−0.361232			−0.180616	+	0.983554i	$0.557809\pi$
$230$			218288.i			0.0179410i
$231$		0			0
$232$	−1.23063e6			−0.0985516
$233$			6.96714e6i			0.550791i	0.961331	+	0.275395i	$0.0888088\pi$
$233$			6.96714e6i			0.550791i	−0.961331	+	0.275395i	$0.911191\pi$
$234$		0			0
$235$	1.31935e7			1.01662
$236$	4.11276e6			0.312894
$237$		0			0
$238$		−	1.88921e7i		−	1.40136i
$239$			4.71730e6i			0.345541i	0.984962	+	0.172771i	$0.0552719\pi$
$239$			4.71730e6i			0.345541i	−0.984962	+	0.172771i	$0.944728\pi$
$240$		0			0
$241$		−	6.19630e6i		−	0.442671i	−0.975198	−	0.221336i	$-0.928958\pi$
$241$		−	6.19630e6i		−	0.442671i	0.975198	−	0.221336i	$-0.0710416\pi$
$242$	6.11861e6	+	7.93677e6i	0.431725	+	0.560012i
$243$		0			0
$244$			8.58648e6i			0.591079i
$245$	9.17461e6			0.623864
$246$		0			0
$247$	1.76605e7			1.17196
$248$		−	1.12883e6i		−	0.0740072i
$249$		0			0
$250$			354091.i			0.0226619i
$251$	−8.78682e6			−0.555662			−0.277831	−	0.960630i	$-0.589616\pi$
$251$	−8.78682e6			−0.555662			−0.277831	+	0.960630i	$0.589616\pi$
$252$		0			0
$253$	276597.	−	94240.5i	0.0170799	−	0.00581937i
$254$	1.22441e7			0.747182
$255$		0			0
$256$	1.04858e6			0.0625000
$257$	−5.89939e6			−0.347542			−0.173771	−	0.984786i	$-0.555595\pi$
$257$	−5.89939e6			−0.347542			−0.173771	+	0.984786i	$0.555595\pi$
$258$		0			0
$259$			2.44632e7i			1.40804i
$260$		−	1.04262e7i		−	0.593208i
$261$		0			0
$262$	1.57686e7			0.876776
$263$			1.62207e7i			0.891666i	0.895116	+	0.445833i	$0.147092\pi$
$263$			1.62207e7i			0.891666i	−0.895116	+	0.445833i	$0.852908\pi$
$264$		0			0
$265$	−4.06459e7			−2.18413
$266$			2.22108e7i			1.18010i
$267$		0			0
$268$	−5.81324e6			−0.302005
$269$	9.03612e6			0.464221			0.232111	−	0.972689i	$-0.425437\pi$
$269$	9.03612e6			0.464221			0.232111	+	0.972689i	$0.425437\pi$
$270$		0			0
$271$		−	2.17153e6i		−	0.109108i	−0.998511	−	0.0545542i	$-0.982626\pi$
$271$		−	2.17153e6i		−	0.109108i	0.998511	−	0.0545542i	$-0.0173738\pi$
$272$		−	8.29804e6i		−	0.412353i
$273$		0			0
$274$			1.67752e7i			0.815484i
$275$	−1.92370e7	+	6.55429e6i	−0.924992	+	0.315158i
$276$		0			0
$277$		−	2.28964e6i		−	0.107728i	−0.998548	−	0.0538639i	$-0.982846\pi$
$277$		−	2.28964e6i		−	0.107728i	0.998548	−	0.0538639i	$-0.0171537\pi$
$278$	−4.46428e6			−0.207787
$279$		0			0
$280$	1.31126e7			0.597332
$281$		−	8.71219e6i		−	0.392653i	−0.980539	−	0.196326i	$-0.937099\pi$
$281$		−	8.71219e6i		−	0.392653i	0.980539	−	0.196326i	$-0.0629012\pi$
$282$		0			0
$283$			1.05751e7i			0.466581i	0.972407	+	0.233290i	$0.0749492\pi$
$283$			1.05751e7i			0.466581i	−0.972407	+	0.233290i	$0.925051\pi$
$284$	5.18466e6			0.226342
$285$		0			0
$286$	−1.32113e7	+	4.50127e6i	−0.564738	+	0.192414i
$287$	1.37238e7			0.580535
$288$		0			0
$289$	−4.15300e7			−1.72056
$290$	−6.75949e6			−0.277153
$291$		0			0
$292$		−	1.05931e7i		−	0.425475i
$293$		−	1.52726e6i		−	0.0607170i	−0.999539	−	0.0303585i	$-0.990335\pi$
$293$		−	1.52726e6i		−	0.0607170i	0.999539	−	0.0303585i	$-0.00966489\pi$
$294$		0			0
$295$	2.25902e7			0.879941
$296$			1.07451e7i			0.414319i
$297$		0			0
$298$	3.22154e7			1.21735
$299$			406967.i			0.0152246i
$300$		0			0
$301$	5.43926e7			1.99453
$302$	−1.57221e7			−0.570808
$303$		0			0
$304$			9.75577e6i			0.347249i
$305$			4.71630e7i			1.66227i
$306$		0			0
$307$		−	7.67760e6i		−	0.265345i	−0.991160	−	0.132672i	$-0.957644\pi$
$307$		−	7.67760e6i		−	0.265345i	0.991160	−	0.132672i	$-0.0423558\pi$
$308$	−5.66106e6	−	1.66153e7i	−0.193752	−	0.568664i
$309$		0			0
$310$		−	6.20034e6i		−	0.208128i
$311$	−2.19337e7			−0.729172			−0.364586	−	0.931170i	$-0.618790\pi$
$311$	−2.19337e7			−0.729172			−0.364586	+	0.931170i	$0.618790\pi$
$312$		0			0
$313$	−3.09060e7			−1.00788			−0.503941	−	0.863738i	$-0.668117\pi$
$313$	−3.09060e7			−1.00788			−0.503941	+	0.863738i	$0.668117\pi$
$314$			3.98422e7i			1.28693i
$315$		0			0
$316$			3.01573e6i			0.0955720i
$317$	−4.89664e7			−1.53717			−0.768583	−	0.639750i	$-0.779038\pi$
$317$	−4.89664e7			−1.53717			−0.768583	+	0.639750i	$0.779038\pi$
$318$		0			0
$319$	2.91824e6	+	8.56509e6i	0.0898979	+	0.263852i
$320$	5.75952e6			0.175767
$321$		0			0
$322$	−511826.			−0.0153304
$323$	7.72035e7			2.29102
$324$		0			0
$325$		−	2.83040e7i		−	0.824513i
$326$		−	3.94542e6i		−	0.113878i
$327$		0			0
$328$	6.02796e6			0.170824
$329$			3.09352e7i			0.868692i
$330$		0			0
$331$	2.46737e7			0.680377			0.340188	−	0.940357i	$-0.389509\pi$
$331$	2.46737e7			0.680377			0.340188	+	0.940357i	$0.389509\pi$
$332$		−	2.16323e7i		−	0.591138i
$333$		0			0
$334$	4.00679e6			0.107537
$335$	−3.19304e7			−0.849317
$336$		0			0
$337$			5.73399e7i			1.49819i	0.662462	+	0.749095i	$0.269512\pi$
$337$			5.73399e7i			1.49819i	−0.662462	+	0.749095i	$0.730488\pi$
$338$			7.86633e6i			0.203715i
$339$		0			0
$340$		−	4.55787e7i		−	1.15964i
$341$	−7.85658e6	+	2.67684e6i	−0.198139	+	0.0675087i
$342$		0			0
$343$		−	2.69741e7i		−	0.668443i
$344$	2.38911e7			0.586895
$345$		0			0
$346$	−3.65835e7			−0.883196
$347$			7.09887e7i			1.69903i	0.527566	+	0.849514i	$0.323105\pi$
$347$			7.09887e7i			1.69903i	−0.527566	+	0.849514i	$0.676895\pi$
$348$		0			0
$349$			7.62194e7i			1.79304i	0.443005	+	0.896519i	$0.353912\pi$
$349$			7.62194e7i			1.79304i	−0.443005	+	0.896519i	$0.646088\pi$
$350$	3.55968e7			0.830245
$351$		0			0
$352$	−2.48653e6	−	7.29801e6i	−0.0570119	−	0.167331i
$353$	6.84796e6			0.155682			0.0778408	−	0.996966i	$-0.475197\pi$
$353$	6.84796e6			0.155682			0.0778408	+	0.996966i	$0.475197\pi$
$354$		0			0
$355$	2.84778e7			0.636533
$356$	9.22433e6			0.204449
$357$		0			0
$358$			6.12351e7i			1.33460i
$359$		−	1.85553e7i		−	0.401036i	−0.979690	−	0.200518i	$-0.935737\pi$
$359$		−	1.85553e7i		−	0.401036i	0.979690	−	0.200518i	$-0.0642626\pi$
$360$		0			0
$361$	−4.37201e7			−0.929307
$362$		−	2.01771e7i		−	0.425337i
$363$		0			0
$364$	2.44466e7			0.506892
$365$		−	5.81847e7i		−	1.19655i
$366$		0			0
$367$	−3.44501e7			−0.696934			−0.348467	−	0.937321i	$-0.613298\pi$
$367$	−3.44501e7			−0.696934			−0.348467	+	0.937321i	$0.613298\pi$
$368$	−224811.			−0.00451102
$369$		0			0
$370$			5.90196e7i			1.16517i
$371$		−	9.53035e7i		−	1.86632i
$372$		0			0
$373$			2.21873e6i			0.0427540i	0.999771	+	0.0213770i	$0.00680503\pi$
$373$			2.21873e6i			0.0427540i	−0.999771	+	0.0213770i	$0.993195\pi$
$374$	−5.77537e7	+	1.96775e7i	−1.10399	+	0.376144i
$375$		0			0
$376$			1.35878e7i			0.255615i
$377$	−1.26021e7			−0.235190
$378$		0			0
$379$	−4.32864e7			−0.795122			−0.397561	−	0.917576i	$-0.630143\pi$
$379$	−4.32864e7			−0.795122			−0.397561	+	0.917576i	$0.630143\pi$
$380$			5.35855e7i			0.976555i
$381$		0			0
$382$			1.85183e7i			0.332209i
$383$	1.83762e7			0.327085			0.163542	−	0.986536i	$-0.447708\pi$
$383$	1.83762e7			0.327085			0.163542	+	0.986536i	$0.447708\pi$
$384$		0			0
$385$	−3.10945e7	−	9.12629e7i	−0.544881	−	1.59923i
$386$	1.62654e7			0.282815
$387$		0			0
$388$	4.73394e7			0.810453
$389$	−9.86287e6			−0.167554			−0.0837769	−	0.996485i	$-0.526698\pi$
$389$	−9.86287e6			−0.167554			−0.0837769	+	0.996485i	$0.526698\pi$
$390$		0			0
$391$			1.77907e6i			0.0297621i
$392$			9.44880e6i			0.156862i
$393$		0			0
$394$	3.40101e7			0.556057
$395$			1.65645e7i			0.268774i
$396$		0			0
$397$	−8.82757e7			−1.41081			−0.705406	−	0.708803i	$-0.749235\pi$
$397$	−8.82757e7			−1.41081			−0.705406	+	0.708803i	$0.749235\pi$
$398$			2.75225e7i			0.436554i
$399$		0			0
$400$	1.56353e7			0.244302
$401$	−8.24767e7			−1.27908			−0.639541	−	0.768757i	$-0.720875\pi$
$401$	−8.24767e7			−1.27908			−0.639541	+	0.768757i	$0.720875\pi$
$402$		0			0
$403$		−	1.15597e7i		−	0.176616i
$404$		−	5.78162e7i		−	0.876809i
$405$		0			0
$406$		−	1.58492e7i		−	0.236825i
$407$	7.47849e7	−	2.54802e7i	1.10925	−	0.377938i
$408$		0			0
$409$		−	8.96469e7i		−	1.31028i	−0.755506	−	0.655142i	$-0.772609\pi$
$409$		−	8.96469e7i		−	1.31028i	0.755506	−	0.655142i	$-0.227391\pi$
$410$	3.31098e7			0.480402
$411$		0			0
$412$	−4.00109e7			−0.572119
$413$			5.29678e7i			0.751903i
$414$		0			0
$415$		−	1.18820e8i		−	1.66244i
$416$	1.07378e7			0.149154
$417$		0			0
$418$	6.78994e7	−	2.31342e7i	0.929687	−	0.316757i
$419$	−1.01853e8			−1.38463			−0.692315	−	0.721595i	$-0.743409\pi$
$419$	−1.01853e8			−1.38463			−0.692315	+	0.721595i	$0.743409\pi$
$420$		0			0
$421$	5.27316e7			0.706683			0.353342	−	0.935494i	$-0.385045\pi$
$421$	5.27316e7			0.706683			0.353342	+	0.935494i	$0.385045\pi$
$422$	2.79852e7			0.372384
$423$		0			0
$424$		−	4.18606e7i		−	0.549171i
$425$		−	1.23732e8i		−	1.61182i
$426$		0			0
$427$	−1.10584e8			−1.42040
$428$		−	3.82587e7i		−	0.487976i
$429$		0			0
$430$	1.31227e8			1.65050
$431$		−	1.47790e6i		−	0.0184592i	−0.999957	−	0.00922959i	$-0.997062\pi$
$431$		−	1.47790e6i		−	0.0184592i	0.999957	−	0.00922959i	$-0.00293791\pi$
$432$		0			0
$433$	2.90742e7			0.358133			0.179066	−	0.983837i	$-0.442692\pi$
$433$	2.90742e7			0.358133			0.179066	+	0.983837i	$0.442692\pi$
$434$	1.45381e7			0.177844
$435$		0			0
$436$		−	2.05832e7i		−	0.248343i
$437$		−	2.09161e6i		−	0.0250632i
$438$		0			0
$439$			5.95615e7i			0.703999i	0.936000	+	0.352000i	$0.114498\pi$
$439$			5.95615e7i			0.703999i	−0.936000	+	0.352000i	$0.885502\pi$
$440$	−1.36578e7	−	4.00858e7i	−0.160333	−	0.470579i
$441$		0			0
$442$		−	8.49750e7i		−	0.984067i
$443$	8.59153e7			0.988233			0.494116	−	0.869396i	$-0.335492\pi$
$443$	8.59153e7			0.988233			0.494116	+	0.869396i	$0.335492\pi$
$444$		0			0
$445$	5.06665e7			0.574965
$446$		−	7.79166e7i		−	0.878264i
$447$		0			0
$448$			1.35045e7i			0.150191i
$449$	−1.20600e8			−1.33232			−0.666162	−	0.745807i	$-0.732064\pi$
$449$	−1.20600e8			−1.33232			−0.666162	+	0.745807i	$0.732064\pi$
$450$		0			0
$451$	−1.42944e7	−	4.19541e7i	−0.155824	−	0.457346i
$452$	−4.27637e7			−0.463084
$453$		0			0
$454$	−9.06697e7			−0.968935
$455$	1.34278e8			1.42551
$456$		0			0
$457$			5.87263e7i			0.615296i	0.951500	+	0.307648i	$0.0995420\pi$
$457$			5.87263e7i			0.615296i	−0.951500	+	0.307648i	$0.900458\pi$
$458$		−	2.45396e7i		−	0.255430i
$459$		0			0
$460$	−1.23482e6			−0.0126862
$461$		−	4.94494e7i		−	0.504729i	−0.967632	−	0.252364i	$-0.918792\pi$
$461$		−	4.94494e7i		−	0.504729i	0.967632	−	0.252364i	$-0.0812081\pi$
$462$		0			0
$463$	−1.06727e8			−1.07531			−0.537653	−	0.843166i	$-0.680689\pi$
$463$	−1.06727e8			−1.07531			−0.537653	+	0.843166i	$0.680689\pi$
$464$		−	6.96150e6i		−	0.0696865i
$465$		0			0
$466$	−3.94121e7			−0.389468
$467$	−3.59851e7			−0.353324			−0.176662	−	0.984272i	$-0.556530\pi$
$467$	−3.59851e7			−0.353324			−0.176662	+	0.984272i	$0.556530\pi$
$468$		0			0
$469$		−	7.48681e7i		−	0.725735i
$470$			7.46339e7i			0.718857i
$471$		0			0
$472$			2.32653e7i			0.221250i
$473$	−5.66539e7	−	1.66280e8i	−0.535360	−	1.57129i
$474$		0			0
$475$			1.45468e8i			1.35734i
$476$	1.06870e8			0.990908
$477$		0			0
$478$	−2.66851e7			−0.244335
$479$			9.11006e7i			0.828924i	0.910067	+	0.414462i	$0.136030\pi$
$479$			9.11006e7i			0.828924i	−0.910067	+	0.414462i	$0.863970\pi$
$480$		0			0
$481$			1.10034e8i			0.988759i
$482$	3.50516e7			0.313016
$483$		0			0
$484$	−4.48971e7	+	3.46121e7i	−0.395988	+	0.305276i
$485$	2.60021e8			2.27921
$486$		0			0
$487$	2.02981e8			1.75739			0.878694	−	0.477386i	$-0.158416\pi$
$487$	2.02981e8			1.75739			0.878694	+	0.477386i	$0.158416\pi$
$488$	−4.85725e7			−0.417956
$489$		0			0
$490$			5.18995e7i			0.441138i
$491$			2.26588e7i			0.191422i	0.995409	+	0.0957111i	$0.0305125\pi$
$491$			2.26588e7i			0.191422i	−0.995409	+	0.0957111i	$0.969488\pi$
$492$		0			0
$493$	−5.50907e7			−0.459767
$494$			9.99027e7i			0.828698i
$495$		0			0
$496$	6.38563e6			0.0523310
$497$			6.67727e7i			0.543913i
$498$		0			0
$499$	1.96136e8			1.57854			0.789269	−	0.614048i	$-0.210460\pi$
$499$	1.96136e8			1.57854			0.789269	+	0.614048i	$0.210460\pi$
$500$	−2.00304e6			−0.0160243
$501$		0			0
$502$		−	4.97058e7i		−	0.392912i
$503$		−	4.99226e7i		−	0.392277i	−0.980576	−	0.196139i	$-0.937160\pi$
$503$		−	4.99226e7i		−	0.392277i	0.980576	−	0.196139i	$-0.0628403\pi$
$504$		0			0
$505$		−	3.17567e8i		−	2.46582i
$506$	533105.	+	1.56467e6i	0.00411491	+	0.0120773i
$507$		0			0
$508$			6.92632e7i			0.528337i
$509$	6.80263e7			0.515850			0.257925	−	0.966165i	$-0.416961\pi$
$509$	6.80263e7			0.515850			0.257925	+	0.966165i	$0.416961\pi$
$510$		0			0
$511$	1.36427e8			1.02244
$512$			5.93164e6i			0.0441942i
$513$		0			0
$514$		−	3.33720e7i		−	0.245749i
$515$	−2.19768e8			−1.60895
$516$		0			0
$517$	9.45702e7	−	3.22213e7i	0.684357	−	0.233170i
$518$	−1.38385e8			−0.995632
$519$		0			0
$520$	5.89796e7			0.419461
$521$	2.00252e7			0.141600			0.0708000	−	0.997491i	$-0.477445\pi$
$521$	2.00252e7			0.141600			0.0708000	+	0.997491i	$0.477445\pi$
$522$		0			0
$523$			2.66789e8i			1.86493i	0.361262	+	0.932464i	$0.382346\pi$
$523$			2.66789e8i			1.86493i	−0.361262	+	0.932464i	$0.617654\pi$
$524$			8.92005e7i			0.619974i
$525$		0			0
$526$	−9.17581e7			−0.630503
$527$		−	5.05335e7i		−	0.345261i
$528$		0			0
$529$	−1.47988e8			−0.999674
$530$		−	2.29928e8i		−	1.54441i
$531$		0			0
$532$	−1.25644e8			−0.834459
$533$	6.17286e7			0.407666
$534$		0			0
$535$		−	2.10144e8i		−	1.37232i
$536$		−	3.28846e7i		−	0.213550i
$537$		0			0
$538$			5.11160e7i			0.328254i
$539$	6.57629e7	−	2.24063e7i	0.419966	−	0.143088i
$540$		0			0
$541$			2.06398e8i			1.30351i	0.758431	+	0.651753i	$0.225966\pi$
$541$			2.06398e8i			1.30351i	−0.758431	+	0.651753i	$0.774034\pi$
$542$	1.22840e7			0.0771513
$543$		0			0
$544$	4.69408e7			0.291577
$545$		−	1.13057e8i		−	0.698408i
$546$		0			0
$547$			1.40391e8i			0.857781i	0.903357	+	0.428890i	$0.141095\pi$
$547$			1.40391e8i			0.857781i	−0.903357	+	0.428890i	$0.858905\pi$
$548$	−9.48947e7			−0.576634
$549$		0			0
$550$	−3.70767e7	−	1.08821e8i	−0.222850	−	0.654068i
$551$	6.47685e7			0.387177
$552$		0			0
$553$	−3.88392e7			−0.229665
$554$	1.29522e7			0.0761751
$555$		0			0
$556$		−	2.52538e7i		−	0.146927i
$557$			2.17888e8i			1.26086i	0.776246	+	0.630430i	$0.217122\pi$
$557$			2.17888e8i			1.26086i	−0.776246	+	0.630430i	$0.782878\pi$
$558$		0			0
$559$	2.44654e8			1.40061
$560$			7.41762e7i			0.422377i
$561$		0			0
$562$	4.92836e7			0.277647
$563$		−	9.17850e7i		−	0.514335i	−0.966367	−	0.257168i	$-0.917211\pi$
$563$		−	9.17850e7i		−	0.514335i	0.966367	−	0.257168i	$-0.0827893\pi$
$564$		0			0
$565$	−2.34888e8			−1.30232
$566$	−5.98220e7			−0.329922
$567$		0			0
$568$			2.93288e7i			0.160048i
$569$			2.89969e8i			1.57403i	0.616931	+	0.787017i	$0.288376\pi$
$569$			2.89969e8i			1.57403i	−0.616931	+	0.787017i	$0.711624\pi$
$570$		0			0
$571$		−	1.72236e8i		−	0.925160i	−0.886578	−	0.462580i	$-0.846924\pi$
$571$		−	1.72236e8i		−	0.925160i	0.886578	−	0.462580i	$-0.153076\pi$
$572$	−2.54630e7	−	7.47343e7i	−0.136057	−	0.399330i
$573$		0			0
$574$			7.76335e7i			0.410500i
$575$	−3.35217e6			−0.0176328
$576$		0			0
$577$	1.38310e8			0.719988			0.359994	−	0.932955i	$-0.382779\pi$
$577$	1.38310e8			0.719988			0.359994	+	0.932955i	$0.382779\pi$
$578$		−	2.34929e8i		−	1.21662i
$579$		0			0
$580$		−	3.82374e7i		−	0.195977i
$581$	2.78601e8			1.42054
$582$		0			0
$583$	−2.91346e8	+	9.92657e7i	−1.47029	+	0.500949i
$584$	5.99236e7			0.300856
$585$		0			0
$586$	8.63949e6			0.0429334
$587$	−8.51407e7			−0.420942			−0.210471	−	0.977600i	$-0.567500\pi$
$587$	−8.51407e7			−0.420942			−0.210471	+	0.977600i	$0.567500\pi$
$588$		0			0
$589$			5.94108e7i			0.290750i
$590$			1.27789e8i			0.622212i
$591$		0			0
$592$	−6.07834e7			−0.292968
$593$		−	3.19267e8i		−	1.53105i	−0.643406	−	0.765525i	$-0.722479\pi$
$593$		−	3.19267e8i		−	1.53105i	0.643406	−	0.765525i	$-0.277521\pi$
$594$		0			0
$595$	5.87003e8			2.78670
$596$			1.82238e8i			0.860794i
$597$		0			0
$598$	−2.30215e6			−0.0107654
$599$	−1.44229e7			−0.0671076			−0.0335538	−	0.999437i	$-0.510683\pi$
$599$	−1.44229e7			−0.0671076			−0.0335538	+	0.999437i	$0.510683\pi$
$600$		0			0
$601$			7.86679e7i			0.362388i	0.983447	+	0.181194i	$0.0579963\pi$
$601$			7.86679e7i			0.362388i	−0.983447	+	0.181194i	$0.942004\pi$
$602$			3.07691e8i			1.41034i
$603$		0			0
$604$		−	8.89377e7i		−	0.403622i
$605$	−2.46607e8	+	1.90114e8i	−1.11362	+	0.858516i
$606$		0			0
$607$			3.71631e7i			0.166167i	0.996543	+	0.0830837i	$0.0264769\pi$
$607$			3.71631e7i			0.166167i	−0.996543	+	0.0830837i	$0.973523\pi$
$608$	−5.51870e7			−0.245542
$609$		0			0
$610$	−2.66794e8			−1.17540
$611$			1.39144e8i			0.610017i
$612$		0			0
$613$		−	2.11808e8i		−	0.919518i	−0.888044	−	0.459759i	$-0.847936\pi$
$613$		−	2.11808e8i		−	0.919518i	0.888044	−	0.459759i	$-0.152064\pi$
$614$	4.34311e7			0.187627
$615$		0			0
$616$	9.39903e7	−	3.20238e7i	0.402106	−	0.137003i
$617$	−1.28156e8			−0.545609			−0.272805	−	0.962069i	$-0.587951\pi$
$617$	−1.28156e8			−0.545609			−0.272805	+	0.962069i	$0.587951\pi$
$618$		0			0
$619$	−7.02259e7			−0.296091			−0.148046	−	0.988981i	$-0.547298\pi$
$619$	−7.02259e7			−0.296091			−0.148046	+	0.988981i	$0.547298\pi$
$620$	3.50744e7			0.147169
$621$		0			0
$622$		−	1.24076e8i		−	0.515603i
$623$			1.18799e8i			0.491303i
$624$		0			0
$625$	−2.49578e8			−1.02227
$626$		−	1.74831e8i		−	0.712680i
$627$		0			0
$628$	−2.25382e8			−0.909996
$629$			4.81017e8i			1.93290i
$630$		0			0
$631$	9.23666e7			0.367643			0.183822	−	0.982960i	$-0.441153\pi$
$631$	9.23666e7			0.367643			0.183822	+	0.982960i	$0.441153\pi$
$632$	−1.70595e7			−0.0675796
$633$		0			0
$634$		−	2.76996e8i		−	1.08694i
$635$			3.80442e8i			1.48582i
$636$		0			0
$637$			9.67592e7i			0.374347i
$638$	−4.84515e7	+	1.65081e7i	−0.186571	+	0.0635674i
$639$		0			0
$640$			3.25808e7i			0.124286i
$641$	−1.83403e8			−0.696359			−0.348179	−	0.937428i	$-0.613200\pi$
$641$	−1.83403e8			−0.696359			−0.348179	+	0.937428i	$0.613200\pi$
$642$		0			0
$643$	−3.44975e8			−1.29764			−0.648820	−	0.760942i	$-0.724737\pi$
$643$	−3.44975e8			−1.29764			−0.648820	+	0.760942i	$0.724737\pi$
$644$		−	2.89532e6i		−	0.0108403i
$645$		0			0
$646$			4.36729e8i			1.62000i
$647$	2.10601e7			0.0777585			0.0388793	−	0.999244i	$-0.487621\pi$
$647$	2.10601e7			0.0777585			0.0388793	+	0.999244i	$0.487621\pi$
$648$		0			0
$649$	1.61925e8	−	5.51699e7i	0.592350	−	0.201822i
$650$	1.60112e8			0.583019
$651$		0			0
$652$	2.23187e7			0.0805241
$653$	−1.81407e7			−0.0651502			−0.0325751	−	0.999469i	$-0.510371\pi$
$653$	−1.81407e7			−0.0651502			−0.0325751	+	0.999469i	$0.510371\pi$
$654$		0			0
$655$			4.89952e8i			1.74353i
$656$			3.40993e7i			0.120791i
$657$		0			0
$658$	−1.74996e8			−0.614258
$659$		−	1.83188e8i		−	0.640089i	−0.947403	−	0.320044i	$-0.896302\pi$
$659$		−	1.83188e8i		−	0.640089i	0.947403	−	0.320044i	$-0.103698\pi$
$660$		0			0
$661$	−9.60760e7			−0.332668			−0.166334	−	0.986070i	$-0.553193\pi$
$661$	−9.60760e7			−0.332668			−0.166334	+	0.986070i	$0.553193\pi$
$662$			1.39575e8i			0.481099i
$663$		0			0
$664$	1.22371e8			0.417998
$665$	−6.90123e8			−2.34672
$666$		0			0
$667$			1.49252e6i			0.00502972i
$668$			2.26658e7i			0.0760401i
$669$		0			0
$670$		−	1.80626e8i		−	0.600558i
$671$	1.15182e8	+	3.38061e8i	0.381256	+	1.11899i
$672$		0			0
$673$			2.46880e8i			0.809918i	0.914335	+	0.404959i	$0.132714\pi$
$673$			2.46880e8i			0.809918i	−0.914335	+	0.404959i	$0.867286\pi$
$674$	−3.24363e8			−1.05938
$675$		0			0
$676$	−4.44987e7			−0.144048
$677$		−	9.99216e7i		−	0.322028i	−0.986952	−	0.161014i	$-0.948524\pi$
$677$		−	9.99216e7i		−	0.322028i	0.986952	−	0.161014i	$-0.0514764\pi$
$678$		0			0
$679$			6.09680e8i			1.94757i
$680$	2.57832e8			0.819993
$681$		0			0
$682$	−1.51425e7	−	4.44435e7i	−0.0477359	−	0.140105i
$683$	−2.69471e8			−0.845765			−0.422883	−	0.906184i	$-0.638982\pi$
$683$	−2.69471e8			−0.845765			−0.422883	+	0.906184i	$0.638982\pi$
$684$		0			0
$685$	−5.21228e8			−1.62165
$686$	1.52588e8			0.472660
$687$		0			0
$688$			1.35148e8i			0.414998i
$689$		−	4.28668e8i		−	1.31058i
$690$		0			0
$691$	5.36169e8			1.62505			0.812527	−	0.582923i	$-0.198091\pi$
$691$	5.36169e8			1.62505			0.812527	+	0.582923i	$0.198091\pi$
$692$		−	2.06947e8i		−	0.624514i
$693$		0			0
$694$	−4.01573e8			−1.20139
$695$		−	1.38712e8i		−	0.413198i
$696$		0			0
$697$	2.69849e8			0.796935
$698$	−4.31162e8			−1.26787
$699$		0			0
$700$			2.01366e8i			0.587072i
$701$		−	3.77121e8i		−	1.09478i	−0.836877	−	0.547390i	$-0.815621\pi$
$701$		−	3.77121e8i		−	1.09478i	0.836877	−	0.547390i	$-0.184379\pi$
$702$		0			0
$703$		−	5.65518e8i		−	1.62772i
$704$	4.12838e7	−	1.40659e7i	0.118321	−	0.0403135i
$705$		0			0
$706$			3.87379e7i			0.110084i
$707$	7.44608e8			2.10702
$708$		0			0
$709$	−3.76660e8			−1.05684			−0.528422	−	0.848982i	$-0.677216\pi$
$709$	−3.76660e8			−1.05684			−0.528422	+	0.848982i	$0.677216\pi$
$710$			1.61095e8i			0.450097i
$711$		0			0
$712$			5.21807e7i			0.144567i
$713$	−1.36906e6			−0.00377706
$714$		0			0
$715$	−1.39861e8	−	4.10493e8i	−0.382629	−	1.12302i
$716$	−3.46398e8			−0.943705
$717$		0			0
$718$	1.04964e8			0.283575
$719$	6.49391e8			1.74711			0.873554	−	0.486728i	$-0.161810\pi$
$719$	6.49391e8			1.74711			0.873554	+	0.486728i	$0.161810\pi$
$720$		0			0
$721$		−	5.15296e8i		−	1.37484i
$722$		−	2.47318e8i		−	0.657120i
$723$		0			0
$724$	1.14139e8			0.300759
$725$		−	1.03803e8i		−	0.272393i
$726$		0			0
$727$	5.51243e8			1.43463			0.717314	−	0.696750i	$-0.245371\pi$
$727$	5.51243e8			1.43463			0.717314	+	0.696750i	$0.245371\pi$
$728$			1.38291e8i			0.358427i
$729$		0			0
$730$	3.29142e8			0.846088
$731$	1.06951e9			2.73801
$732$		0			0
$733$		−	7.28139e8i		−	1.84885i	−0.381361	−	0.924426i	$-0.624545\pi$
$733$		−	7.28139e8i		−	1.84885i	0.381361	−	0.924426i	$-0.375455\pi$
$734$		−	1.94879e8i		−	0.492807i
$735$		0			0
$736$		−	1.27173e6i		−	0.00318977i
$737$	−2.28875e8	+	7.79807e7i	−0.571735	+	0.194798i
$738$		0			0
$739$		−	2.49677e8i		−	0.618649i	−0.950957	−	0.309325i	$-0.899897\pi$
$739$		−	2.49677e8i		−	0.618649i	0.950957	−	0.309325i	$-0.100103\pi$
$740$	−3.33865e8			−0.823902
$741$		0			0
$742$	5.39118e8			1.31969
$743$			5.49838e8i			1.34050i	0.742134	+	0.670252i	$0.233814\pi$
$743$			5.49838e8i			1.34050i	−0.742134	+	0.670252i	$0.766186\pi$
$744$		0			0
$745$			1.00098e9i			2.42078i
$746$	−1.25510e7			−0.0302317
$747$		0			0
$748$	−1.11313e8	−	3.26704e8i	−0.265974	−	0.780639i
$749$	4.92730e8			1.17264
$750$		0			0
$751$	1.81933e8			0.429528			0.214764	−	0.976666i	$-0.431102\pi$
$751$	1.81933e8			0.429528			0.214764	+	0.976666i	$0.431102\pi$
$752$	−7.68643e7			−0.180747
$753$		0			0
$754$		−	7.12883e7i		−	0.166305i
$755$		−	4.88508e8i		−	1.13509i
$756$		0			0
$757$	2.89733e8			0.667897			0.333949	−	0.942591i	$-0.391619\pi$
$757$	2.89733e8			0.667897			0.333949	+	0.942591i	$0.391619\pi$
$758$		−	2.44865e8i		−	0.562236i
$759$		0			0
$760$	−3.03126e8			−0.690529
$761$			2.58513e8i			0.586581i	0.956023	+	0.293290i	$0.0947503\pi$
$761$			2.58513e8i			0.586581i	−0.956023	+	0.293290i	$0.905250\pi$
$762$		0			0
$763$	2.65088e8			0.596784
$764$	−1.04756e8			−0.234907
$765$		0			0
$766$			1.03952e8i			0.231284i
$767$			2.38245e8i			0.528005i
$768$		0			0
$769$			1.83751e8i			0.404064i	0.979379	+	0.202032i	$0.0647545\pi$
$769$			1.83751e8i			0.404064i	−0.979379	+	0.202032i	$0.935246\pi$
$770$	5.16261e8	−	1.75897e8i	1.13083	−	0.385289i
$771$		0			0
$772$			9.20110e7i			0.199981i
$773$	3.75387e6			0.00812720			0.00406360	−	0.999992i	$-0.498707\pi$
$773$	3.75387e6			0.00812720			0.00406360	+	0.999992i	$0.498707\pi$
$774$		0			0
$775$	9.52162e7			0.204553
$776$			2.67792e8i			0.573077i
$777$		0			0
$778$		−	5.57928e7i		−	0.118478i
$779$	−3.17254e8			−0.671112
$780$		0			0
$781$	2.04126e8	−	6.95487e7i	0.428495	−	0.145994i
$782$	−1.00640e7			−0.0210450
$783$		0			0
$784$	−5.34505e7			−0.110918
$785$	−1.23795e9			−2.55915
$786$		0			0
$787$		−	6.46196e8i		−	1.32568i	−0.748759	−	0.662842i	$-0.769350\pi$
$787$		−	6.46196e8i		−	1.32568i	0.748759	−	0.662842i	$-0.230650\pi$
$788$			1.92390e8i			0.393192i
$789$		0			0
$790$	−9.37029e7			−0.190052
$791$		−	5.50749e8i		−	1.11282i
$792$		0			0
$793$	−4.97400e8			−0.997439
$794$		−	4.99363e8i		−	0.997595i
$795$		0			0
$796$	−1.55691e8			−0.308690
$797$	4.26209e8			0.841875			0.420937	−	0.907090i	$-0.361701\pi$
$797$	4.26209e8			0.841875			0.420937	+	0.907090i	$0.361701\pi$
$798$		0			0
$799$			6.08276e8i			1.19250i
$800$			8.84468e7i			0.172748i
$801$		0			0
$802$		−	4.66559e8i		−	0.904447i
$803$	−1.42099e8	−	4.17063e8i	−0.274438	−	0.805481i
$804$		0			0
$805$		−	1.59032e7i		−	0.0304857i
$806$	6.53913e7			0.124886
$807$		0			0
$808$	3.27058e8			0.619998
$809$		−	4.48232e8i		−	0.846559i	−0.905999	−	0.423280i	$-0.860879\pi$
$809$		−	4.48232e8i		−	0.846559i	0.905999	−	0.423280i	$-0.139121\pi$
$810$		0			0
$811$			5.87947e8i			1.10224i	0.834426	+	0.551119i	$0.185799\pi$
$811$			5.87947e8i			1.10224i	−0.834426	+	0.551119i	$0.814201\pi$
$812$	8.96564e7			0.167461
$813$		0			0
$814$	1.44138e8	+	4.23047e8i	0.267242	+	0.784361i
$815$	1.22590e8			0.226455
$816$		0			0
$817$	−1.25740e9			−2.30572
$818$	5.07120e8			0.926511
$819$		0			0
$820$			1.87297e8i			0.339696i
$821$		−	6.93632e8i		−	1.25343i	−0.779249	−	0.626714i	$-0.784399\pi$
$821$		−	6.93632e8i		−	1.25343i	0.779249	−	0.626714i	$-0.215601\pi$
$822$		0			0
$823$	3.30681e8			0.593211			0.296606	−	0.955000i	$-0.404145\pi$
$823$	3.30681e8			0.593211			0.296606	+	0.955000i	$0.404145\pi$
$824$		−	2.26336e8i		−	0.404549i
$825$		0			0
$826$	−2.99631e8			−0.531676
$827$		−	4.07604e7i		−	0.0720646i	−0.999351	−	0.0360323i	$-0.988528\pi$
$827$		−	4.07604e7i		−	0.0720646i	0.999351	−	0.0360323i	$-0.0114719\pi$
$828$		0			0
$829$	−2.93862e7			−0.0515798			−0.0257899	−	0.999667i	$-0.508210\pi$
$829$	−2.93862e7			−0.0515798			−0.0257899	+	0.999667i	$0.508210\pi$
$830$	6.72148e8			1.17552
$831$		0			0
$832$			6.07422e7i			0.105468i
$833$			4.22987e8i			0.731800i
$834$		0			0
$835$			1.24497e8i			0.213845i
$836$	1.30867e8	+	3.84097e8i	0.223981	+	0.657388i
$837$		0			0
$838$		−	5.76170e8i		−	0.979082i
$839$	−1.05633e9			−1.78861			−0.894305	−	0.447458i	$-0.852329\pi$
$839$	−1.05633e9			−1.78861			−0.894305	+	0.447458i	$0.852329\pi$
$840$		0			0
$841$	5.48606e8			0.922301
$842$			2.98295e8i			0.499700i
$843$		0			0
$844$			1.58308e8i			0.263315i
$845$	−2.44418e8			−0.405101
$846$		0			0
$847$	−4.45766e8	−	5.78225e8i	−0.733595	−	0.951584i
$848$	2.36799e8			0.388323
$849$		0			0
$850$	6.99935e8			1.13973
$851$	1.30318e7			0.0211453
$852$		0			0
$853$			5.94560e8i			0.957962i	0.877825	+	0.478981i	$0.158994\pi$
$853$			5.94560e8i			0.957962i	−0.877825	+	0.478981i	$0.841006\pi$
$854$		−	6.25560e8i		−	1.00437i
$855$		0			0
$856$	2.16424e8			0.345051
$857$			7.06965e7i			0.112320i	0.998422	+	0.0561598i	$0.0178856\pi$
$857$			7.06965e7i			0.112320i	−0.998422	+	0.0561598i	$0.982114\pi$
$858$		0			0
$859$	1.00322e9			1.58277			0.791386	−	0.611317i	$-0.209360\pi$
$859$	1.00322e9			1.58277			0.791386	+	0.611317i	$0.209360\pi$
$860$			7.42330e8i			1.16708i
$861$		0			0
$862$	8.36025e6			0.0130526
$863$	−1.96468e7			−0.0305674			−0.0152837	−	0.999883i	$-0.504865\pi$
$863$	−1.96468e7			−0.0305674			−0.0152837	+	0.999883i	$0.504865\pi$
$864$		0			0
$865$		−	1.13670e9i		−	1.75630i
$866$			1.64469e8i			0.253238i
$867$		0			0
$868$			8.22399e7i			0.125754i
$869$	4.04539e7	+	1.18733e8i	0.0616455	+	0.180931i
$870$		0			0
$871$		−	3.36751e8i		−	0.509629i
$872$	1.16436e8			0.175605
$873$		0			0
$874$	1.18319e7			0.0177223
$875$		−	2.57970e7i		−	0.0385075i
$876$		0			0
$877$			5.09819e8i			0.755818i	0.925843	+	0.377909i	$0.123357\pi$
$877$			5.09819e8i			0.755818i	−0.925843	+	0.377909i	$0.876643\pi$
$878$	−3.36931e8			−0.497803
$879$		0			0
$880$	2.26759e8	−	7.72601e7i	0.332749	−	0.113372i
$881$	3.28419e8			0.480287			0.240144	−	0.970737i	$-0.422805\pi$
$881$	3.28419e8			0.480287			0.240144	+	0.970737i	$0.422805\pi$
$882$		0			0
$883$	7.99892e8			1.16185			0.580924	−	0.813958i	$-0.302692\pi$
$883$	7.99892e8			1.16185			0.580924	+	0.813958i	$0.302692\pi$
$884$	4.80691e8			0.695840
$885$		0			0
$886$			4.86010e8i			0.698786i
$887$			3.53129e8i			0.506013i	0.967465	+	0.253007i	$0.0814195\pi$
$887$			3.53129e8i			0.506013i	−0.967465	+	0.253007i	$0.918581\pi$
$888$		0			0
$889$	−8.92033e8			−1.26963
$890$			2.86613e8i			0.406561i
$891$		0			0
$892$	4.40763e8			0.621027
$893$		−	7.15132e8i		−	1.00423i
$894$		0			0
$895$	−1.90266e9			−2.65395
$896$	−7.63930e7			−0.106201
$897$		0			0
$898$		−	6.82219e8i		−	0.942095i
$899$		−	4.23942e7i		−	0.0583482i
$900$		0			0
$901$		−	1.87394e9i		−	2.56201i
$902$	2.37328e8	−	8.08611e7i	0.323393	−	0.110184i
$903$		0			0
$904$		−	2.41908e8i		−	0.327450i
$905$	6.26931e8			0.845813
$906$		0			0
$907$	−1.07465e8			−0.144028			−0.0720140	−	0.997404i	$-0.522943\pi$
$907$	−1.07465e8			−0.144028			−0.0720140	+	0.997404i	$0.522943\pi$
$908$		−	5.12905e8i		−	0.685140i
$909$		0			0
$910$			7.59592e8i			1.00799i
$911$	2.64437e8			0.349758			0.174879	−	0.984590i	$-0.444047\pi$
$911$	2.64437e8			0.349758			0.174879	+	0.984590i	$0.444047\pi$
$912$		0			0
$913$	−2.90183e8	−	8.51692e8i	−0.381294	−	1.11910i
$914$	−3.32206e8			−0.435080
$915$		0			0
$916$	1.38817e8			0.180616
$917$	−1.14880e9			−1.48983
$918$		0			0
$919$			1.39166e9i			1.79303i	0.443015	+	0.896514i	$0.353909\pi$
$919$			1.39166e9i			1.79303i	−0.443015	+	0.896514i	$0.646091\pi$
$920$		−	6.98521e6i		−	0.00897049i
$921$		0			0
$922$	2.79728e8			0.356897
$923$			3.00338e8i			0.381949i
$924$		0			0
$925$	−9.06341e8			−1.14516
$926$		−	6.03740e8i		−	0.760356i
$927$		0			0
$928$	3.93802e7			0.0492758
$929$	1.25707e9			1.56788			0.783939	−	0.620838i	$-0.213207\pi$
$929$	1.25707e9			1.56788			0.783939	+	0.620838i	$0.213207\pi$
$930$		0			0
$931$		−	4.97294e8i		−	0.616260i
$932$		−	2.22949e8i		−	0.275395i
$933$		0			0
$934$		−	2.03563e8i		−	0.249837i
$935$	−6.11407e8	−	1.79449e9i	−0.747990	−	2.19536i
$936$		0			0
$937$		−	2.03959e8i		−	0.247927i	−0.992287	−	0.123963i	$-0.960439\pi$
$937$		−	2.03959e8i		−	0.247927i	0.992287	−	0.123963i	$-0.0395605\pi$
$938$	4.23518e8			0.513172
$939$		0			0
$940$	−4.22193e8			−0.508309
$941$		−	9.16620e7i		−	0.110007i	−0.998486	−	0.0550035i	$-0.982483\pi$
$941$		−	9.16620e7i		−	0.110007i	0.998486	−	0.0550035i	$-0.0175170\pi$
$942$		0			0
$943$		−	7.31079e6i		−	0.00871824i
$944$	−1.31608e8			−0.156447
$945$		0			0
$946$	9.40622e8	−	3.20483e8i	1.11107	−	0.378557i
$947$	−1.18415e9			−1.39430			−0.697150	−	0.716925i	$-0.745549\pi$
$947$	−1.18415e9			−1.39430			−0.697150	+	0.716925i	$0.745549\pi$
$948$		0			0
$949$	6.13640e8			0.717984
$950$	−8.22893e8			−0.959782
$951$		0			0
$952$			6.04546e8i			0.700678i
$953$			1.10796e9i			1.28011i	0.768330	+	0.640054i	$0.221088\pi$
$953$			1.10796e9i			1.28011i	−0.768330	+	0.640054i	$0.778912\pi$
$954$		0			0
$955$	−5.75391e8			−0.660622
$956$		−	1.50954e8i		−	0.172771i
$957$		0			0
$958$	−5.15343e8			−0.586138
$959$		−	1.22214e9i		−	1.38569i
$960$		0			0
$961$	−8.48616e8			−0.956183
$962$	−6.22444e8			−0.699158
$963$		0			0
$964$			1.98282e8i			0.221336i
$965$			5.05389e8i			0.562398i
$966$		0			0
$967$			2.09973e8i			0.232212i	0.993237	+	0.116106i	$0.0370412\pi$
$967$			2.09973e8i			0.232212i	−0.993237	+	0.116106i	$0.962959\pi$
$968$	−1.95796e8	−	2.53977e8i	−0.215862	−	0.280006i
$969$		0			0
$970$			1.47090e9i			1.61164i
$971$	−6.83482e8			−0.746568			−0.373284	−	0.927717i	$-0.621768\pi$
$971$	−6.83482e8			−0.746568			−0.373284	+	0.927717i	$0.621768\pi$
$972$		0			0
$973$	3.25241e8			0.353075
$974$			1.14823e9i			1.24266i
$975$		0			0
$976$		−	2.74767e8i		−	0.295540i
$977$	1.51580e9			1.62539			0.812696	−	0.582688i	$-0.197999\pi$
$977$	1.51580e9			1.62539			0.812696	+	0.582688i	$0.197999\pi$
$978$		0			0
$979$	3.63174e8	−	1.23738e8i	0.387049	−	0.131873i
$980$	−2.93588e8			−0.311932
$981$		0			0
$982$	−1.28177e8			−0.135356
$983$	2.07821e7			0.0218790			0.0109395	−	0.999940i	$-0.496518\pi$
$983$	2.07821e7			0.0218790			0.0109395	+	0.999940i	$0.496518\pi$
$984$		0			0
$985$			1.05674e9i			1.10576i
$986$		−	3.11640e8i		−	0.325104i
$987$		0			0
$988$	−5.65135e8			−0.585978
$989$		−	2.89754e7i		−	0.0299530i
$990$		0			0
$991$	8.23277e7			0.0845912			0.0422956	−	0.999105i	$-0.486533\pi$
$991$	8.23277e7			0.0845912			0.0422956	+	0.999105i	$0.486533\pi$
$992$			3.61226e7i			0.0370036i
$993$		0			0
$994$	−3.77723e8			−0.384605
$995$	−8.55162e8			−0.868119
$996$		0			0
$997$			1.31565e8i			0.132756i	0.997795	+	0.0663782i	$0.0211444\pi$
$997$			1.31565e8i			0.132756i	−0.997795	+	0.0663782i	$0.978856\pi$
$998$			1.10951e9i			1.11619i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	198.7.d.a.109.4		6
3.2	odd	2		22.7.b.a.21.3	✓	6
11.10	odd	2	inner	198.7.d.a.109.1		6
12.11	even	2		176.7.h.e.65.2		6
33.32	even	2		22.7.b.a.21.6	yes	6
132.131	odd	2		176.7.h.e.65.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.7.b.a.21.3	✓	6	3.2	odd	2
22.7.b.a.21.6	yes	6	33.32	even	2
176.7.h.e.65.1		6	132.131	odd	2
176.7.h.e.65.2		6	12.11	even	2
198.7.d.a.109.1		6	11.10	odd	2	inner
198.7.d.a.109.4		6	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 198 = 2 \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	198.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+5.65685i q^{2} -32.0000 q^{4} -175.767 q^{5} -412.125i q^{7} -181.019i q^{8} +O(q^{10})\)	\(q+5.65685i q^{2} -32.0000 q^{4} -175.767 q^{5} -412.125i q^{7} -181.019i q^{8} -994.286i q^{10} +(-1259.88 + 429.258i) q^{11} -1853.71i q^{13} +2331.33 q^{14} +1024.00 q^{16} -8103.56i q^{17} +9527.12i q^{19} +5624.53 q^{20} +(-2428.25 - 7126.96i) q^{22} -219.542 q^{23} +15268.9 q^{25} +10486.1 q^{26} +13188.0i q^{28} -6798.33i q^{29} +6235.97 q^{31} +5792.62i q^{32} +45840.6 q^{34} +72437.7i q^{35} -59358.7 q^{37} -53893.5 q^{38} +31817.1i q^{40} +33300.1i q^{41} +131981. i q^{43} +(40316.2 - 13736.3i) q^{44} -1241.92i q^{46} -75062.8 q^{47} -52197.7 q^{49} +86373.8i q^{50} +59318.6i q^{52} +231249. q^{53} +(221445. - 75449.3i) q^{55} -74602.5 q^{56} +38457.2 q^{58} -128524. q^{59} -268328. i q^{61} +35276.0i q^{62} -32768.0 q^{64} +325819. i q^{65} +181664. q^{67} +259314. i q^{68} -409770. q^{70} -162021. q^{71} +331034. i q^{73} -335784. i q^{74} -304868. i q^{76} +(176908. + 519228. i) q^{77} -94241.5i q^{79} -179985. q^{80} -188374. q^{82} +676011. i q^{83} +1.42433e6i q^{85} -746596. q^{86} +(77704.1 + 228063. i) q^{88} -288260. q^{89} -763958. q^{91} +7025.36 q^{92} -424619. i q^{94} -1.67455e6i q^{95} -1.47936e6 q^{97} -295275. i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 192 q^{4} - 368 q^{5}+O(q^{10}) \) 6 * q - 192 * q^4 - 368 * q^5	\( 6 q - 192 q^{4} - 368 q^{5} + 1166 q^{11} - 2208 q^{14} + 6144 q^{16} + 11776 q^{20} + 1056 q^{22} - 2156 q^{23} + 59862 q^{25} + 1824 q^{26} - 78468 q^{31} + 28704 q^{34} - 205920 q^{37} - 101472 q^{38} - 37312 q^{44} - 493460 q^{47} - 270762 q^{49} + 531700 q^{53} + 274956 q^{55} + 70656 q^{56} + 509184 q^{58} + 833380 q^{59} - 196608 q^{64} + 537420 q^{67} + 96096 q^{70} + 460372 q^{71} - 249744 q^{77} - 376832 q^{80} + 428640 q^{82} - 1055808 q^{86} - 33792 q^{88} - 2377952 q^{89} - 5068656 q^{91} + 68992 q^{92} + 1351632 q^{97}+O(q^{100}) \) 6 * q - 192 * q^4 - 368 * q^5 + 1166 * q^11 - 2208 * q^14 + 6144 * q^16 + 11776 * q^20 + 1056 * q^22 - 2156 * q^23 + 59862 * q^25 + 1824 * q^26 - 78468 * q^31 + 28704 * q^34 - 205920 * q^37 - 101472 * q^38 - 37312 * q^44 - 493460 * q^47 - 270762 * q^49 + 531700 * q^53 + 274956 * q^55 + 70656 * q^56 + 509184 * q^58 + 833380 * q^59 - 196608 * q^64 + 537420 * q^67 + 96096 * q^70 + 460372 * q^71 - 249744 * q^77 - 376832 * q^80 + 428640 * q^82 - 1055808 * q^86 - 33792 * q^88 - 2377952 * q^89 - 5068656 * q^91 + 68992 * q^92 + 1351632 * q^97

Introduction

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