LMFDB - Embedded newform 315.4.d.d.64.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [315,4,Mod(64,315)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(315, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("315.64");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	315.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:	$18.5856016518$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 144 x^{18} + 8518 x^{16} + 269932 x^{14} + 5002289 x^{12} + 55478700 x^{10} + 361614704 x^{8} + \cdots + 603979776 $ x^20 + 144x^18 + 8518x^16 + 269932x^14 + 5002289x^12 + 55478700x^10 + 361614704x^8 + 1303665216x^6 + 2335301632x^4 + 1949503488*x^2 + 603979776
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{26}\cdot 5^{2}\cdot 7^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			64.19
Root			$-6.44617i$ of defining polynomial
Character	$\chi$	$=$	315.64
Dual form			315.4.d.d.64.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.44617i q^{2} -21.6607 q^{4} +(-10.6013 - 3.55149i) q^{5} -7.00000i q^{7} -74.3987i q^{8} +O(q^{10})$	\(q+5.44617i q^{2} -21.6607 q^{4} +(-10.6013 - 3.55149i) q^{5} -7.00000i q^{7} -74.3987i q^{8} +(19.3420 - 57.7363i) q^{10} -60.4870 q^{11} +64.0698i q^{13} +38.1232 q^{14} +231.902 q^{16} -37.7444i q^{17} +134.779 q^{19} +(229.631 + 76.9279i) q^{20} -329.422i q^{22} +52.6223i q^{23} +(99.7739 + 75.3006i) q^{25} -348.935 q^{26} +151.625i q^{28} +165.252 q^{29} -71.9304 q^{31} +667.787i q^{32} +205.563 q^{34} +(-24.8604 + 74.2089i) q^{35} -48.7993i q^{37} +734.031i q^{38} +(-264.226 + 788.721i) q^{40} +10.5566 q^{41} -425.906i q^{43} +1310.19 q^{44} -286.590 q^{46} -249.258i q^{47} -49.0000 q^{49} +(-410.100 + 543.385i) q^{50} -1387.80i q^{52} -544.007i q^{53} +(641.239 + 214.819i) q^{55} -520.791 q^{56} +899.988i q^{58} -567.390 q^{59} +614.549 q^{61} -391.745i q^{62} -1781.66 q^{64} +(227.543 - 679.221i) q^{65} -201.820i q^{67} +817.573i q^{68} +(-404.154 - 135.394i) q^{70} +525.114 q^{71} +137.016i q^{73} +265.769 q^{74} -2919.42 q^{76} +423.409i q^{77} +234.355 q^{79} +(-2458.45 - 823.597i) q^{80} +57.4929i q^{82} -10.3089i q^{83} +(-134.049 + 400.139i) q^{85} +2319.56 q^{86} +4500.15i q^{88} -20.9202 q^{89} +448.488 q^{91} -1139.84i q^{92} +1357.50 q^{94} +(-1428.83 - 478.667i) q^{95} -1388.80i q^{97} -266.862i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 108 q^{4}+O(q^{10}) $ 20 * q - 108 * q^4	$ 20 q - 108 q^{4} + 112 q^{10} + 620 q^{16} - 72 q^{19} + 428 q^{25} - 48 q^{31} - 232 q^{34} - 16 q^{40} - 368 q^{46} - 980 q^{49} + 1904 q^{55} + 2048 q^{61} - 3180 q^{64} - 756 q^{70} - 8368 q^{76} + 1552 q^{79} - 2616 q^{85} + 1456 q^{91} + 8056 q^{94}+O(q^{100}) $ 20 * q - 108 * q^4 + 112 * q^10 + 620 * q^16 - 72 * q^19 + 428 * q^25 - 48 * q^31 - 232 * q^34 - 16 * q^40 - 368 * q^46 - 980 * q^49 + 1904 * q^55 + 2048 * q^61 - 3180 * q^64 - 756 * q^70 - 8368 * q^76 + 1552 * q^79 - 2616 * q^85 + 1456 * q^91 + 8056 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/315\mathbb{Z}\right)^\times$.

$n$	$127$	$136$	$281$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$			5.44617i			1.92551i	0.270373	+	0.962756i	$0.412853\pi$
$2$			5.44617i			1.92551i	−0.270373	+	0.962756i	$0.587147\pi$
$3$		0			0
$3$		0			0
$4$	−21.6607			−2.70759
$5$	−10.6013	−	3.55149i	−0.948206	−	0.317655i
$5$	−10.6013	−	3.55149i	−0.948206	−	0.317655i
$6$		0			0
$7$		−	7.00000i		−	0.377964i
$7$		−	7.00000i		−	0.377964i
$8$		−	74.3987i		−	3.28799i
$9$		0			0
$10$	19.3420	−	57.7363i	0.611648	−	1.82578i
$11$	−60.4870			−1.65796			−0.828978	−	0.559281i	$-0.811077\pi$
$11$	−60.4870			−1.65796			−0.828978	+	0.559281i	$0.811077\pi$
$12$		0			0
$13$			64.0698i			1.36690i	0.729995	+	0.683452i	$0.239522\pi$
$13$			64.0698i			1.36690i	−0.729995	+	0.683452i	$0.760478\pi$
$14$	38.1232			0.727775
$15$		0			0
$16$	231.902			3.62347
$17$		−	37.7444i		−	0.538492i	−0.963071	−	0.269246i	$-0.913225\pi$
$17$		−	37.7444i		−	0.538492i	0.963071	−	0.269246i	$-0.0867745\pi$
$18$		0			0
$19$	134.779			1.62739			0.813697	−	0.581289i	$-0.197451\pi$
$19$	134.779			1.62739			0.813697	+	0.581289i	$0.197451\pi$
$20$	229.631	+	76.9279i	2.56736	+	0.860080i
$21$		0			0
$22$		−	329.422i		−	3.19241i
$23$			52.6223i			0.477065i	0.971134	+	0.238533i	$0.0766664\pi$
$23$			52.6223i			0.477065i	−0.971134	+	0.238533i	$0.923334\pi$
$24$		0			0
$25$	99.7739	+	75.3006i	0.798191	+	0.602405i
$26$	−348.935			−2.63199
$27$		0			0
$28$			151.625i			1.02337i
$29$	165.252			1.05815			0.529077	−	0.848574i	$-0.322538\pi$
$29$	165.252			1.05815			0.529077	+	0.848574i	$0.322538\pi$
$30$		0			0
$31$	−71.9304			−0.416745			−0.208372	−	0.978050i	$-0.566817\pi$
$31$	−71.9304			−0.416745			−0.208372	+	0.978050i	$0.566817\pi$
$32$			667.787i			3.68904i
$33$		0			0
$34$	205.563			1.03687
$35$	−24.8604	+	74.2089i	−0.120062	+	0.358388i
$36$		0			0
$37$		−	48.7993i		−	0.216826i	−0.994106	−	0.108413i	$-0.965423\pi$
$37$		−	48.7993i		−	0.216826i	0.994106	−	0.108413i	$-0.0345769\pi$
$38$			734.031i			3.13357i
$39$		0			0
$40$	−264.226	+	788.721i	−1.04445	+	3.11769i
$41$	10.5566			0.0402113			0.0201056	−	0.999798i	$-0.493600\pi$
$41$	10.5566			0.0402113			0.0201056	+	0.999798i	$0.493600\pi$
$42$		0			0
$43$		−	425.906i		−	1.51047i	−0.655456	−	0.755233i	$-0.727524\pi$
$43$		−	425.906i		−	1.51047i	0.655456	−	0.755233i	$-0.272476\pi$
$44$	1310.19			4.48907
$45$		0			0
$46$	−286.590			−0.918594
$47$		−	249.258i		−	0.773575i	−0.922169	−	0.386787i	$-0.873585\pi$
$47$		−	249.258i		−	0.773575i	0.922169	−	0.386787i	$-0.126415\pi$
$48$		0			0
$49$	−49.0000			−0.142857
$50$	−410.100	+	543.385i	−1.15994	+	1.53693i
$51$		0			0
$52$		−	1387.80i		−	3.70102i
$53$		−	544.007i		−	1.40991i	−0.709254	−	0.704953i	$-0.750968\pi$
$53$		−	544.007i		−	1.40991i	0.709254	−	0.704953i	$-0.249032\pi$
$54$		0			0
$55$	641.239	+	214.819i	1.57209	+	0.526658i
$56$	−520.791			−1.24274
$57$		0			0
$58$			899.988i			2.03749i
$59$	−567.390			−1.25200			−0.625999	−	0.779824i	$-0.715309\pi$
$59$	−567.390			−1.25200			−0.625999	+	0.779824i	$0.715309\pi$
$60$		0			0
$61$	614.549			1.28992			0.644958	−	0.764218i	$-0.276875\pi$
$61$	614.549			1.28992			0.644958	+	0.764218i	$0.276875\pi$
$62$		−	391.745i		−	0.802447i
$63$		0			0
$64$	−1781.66			−3.47981
$65$	227.543	−	679.221i	0.434204	−	1.29611i
$66$		0			0
$67$		−	201.820i		−	0.368004i	−0.982926	−	0.184002i	$-0.941095\pi$
$67$		−	201.820i		−	0.368004i	0.982926	−	0.184002i	$-0.0589053\pi$
$68$			817.573i			1.45802i
$69$		0			0
$70$	−404.154	−	135.394i	−0.690081	−	0.231181i
$71$	525.114			0.877741			0.438871	−	0.898550i	$-0.355379\pi$
$71$	525.114			0.877741			0.438871	+	0.898550i	$0.355379\pi$
$72$		0			0
$73$			137.016i			0.219679i	0.993949	+	0.109839i	$0.0350336\pi$
$73$			137.016i			0.219679i	−0.993949	+	0.109839i	$0.964966\pi$
$74$	265.769			0.417500
$75$		0			0
$76$	−2919.42			−4.40632
$77$			423.409i			0.626649i
$78$		0			0
$79$	234.355			0.333759			0.166880	−	0.985977i	$-0.446631\pi$
$79$	234.355			0.333759			0.166880	+	0.985977i	$0.446631\pi$
$80$	−2458.45	−	823.597i	−3.43579	−	1.15101i
$81$		0			0
$82$			57.4929i			0.0774272i
$83$		−	10.3089i		−	0.0136332i	−0.999977	−	0.00681659i	$-0.997830\pi$
$83$		−	10.3089i		−	0.0136332i	0.999977	−	0.00681659i	$-0.00216981\pi$
$84$		0			0
$85$	−134.049	+	400.139i	−0.171055	+	0.510602i
$86$	2319.56			2.90842
$87$		0			0
$88$			4500.15i			5.45134i
$89$	−20.9202			−0.0249162			−0.0124581	−	0.999922i	$-0.503966\pi$
$89$	−20.9202			−0.0249162			−0.0124581	+	0.999922i	$0.503966\pi$
$90$		0			0
$91$	448.488			0.516641
$92$		−	1139.84i		−	1.29170i
$93$		0			0
$94$	1357.50			1.48953
$95$	−1428.83	−	478.667i	−1.54311	−	0.516950i
$96$		0			0
$97$		−	1388.80i		−	1.45373i	−0.686781	−	0.726864i	$-0.740977\pi$
$97$		−	1388.80i		−	1.45373i	0.686781	−	0.726864i	$-0.259023\pi$
$98$		−	266.862i		−	0.275073i
$99$		0			0
$100$	−2161.18	−	1631.07i	−2.16118	−	1.63107i
$101$	−95.4423			−0.0940283			−0.0470142	−	0.998894i	$-0.514971\pi$
$101$	−95.4423			−0.0940283			−0.0470142	+	0.998894i	$0.514971\pi$
$102$		0			0
$103$		−	368.235i		−	0.352265i	−0.984366	−	0.176132i	$-0.943641\pi$
$103$		−	368.235i		−	0.352265i	0.984366	−	0.176132i	$-0.0563587\pi$
$104$	4766.71			4.49437
$105$		0			0
$106$	2962.75			2.71479
$107$			623.980i			0.563761i	0.959449	+	0.281881i	$0.0909582\pi$
$107$			623.980i			0.563761i	−0.959449	+	0.281881i	$0.909042\pi$
$108$		0			0
$109$	381.664			0.335384			0.167692	−	0.985839i	$-0.446369\pi$
$109$	381.664			0.335384			0.167692	+	0.985839i	$0.446369\pi$
$110$	−1169.94	+	3492.30i	−1.01409	+	3.02707i
$111$		0			0
$112$		−	1623.31i		−	1.36954i
$113$			1189.96i			0.990637i	0.868711	+	0.495318i	$0.164949\pi$
$113$			1189.96i			0.990637i	−0.868711	+	0.495318i	$0.835051\pi$
$114$		0			0
$115$	186.887	−	557.863i	0.151542	−	0.452356i
$116$	−3579.47			−2.86505
$117$		0			0
$118$		−	3090.10i		−	2.41074i
$119$	−264.211			−0.203531
$120$		0			0
$121$	2327.68			1.74882
$122$			3346.94i			2.48375i
$123$		0			0
$124$	1558.07			1.12838
$125$	−790.301	−	1152.63i	−0.565493	−	0.824753i
$126$		0			0
$127$			1681.29i			1.17472i	0.809324	+	0.587362i	$0.199834\pi$
$127$			1681.29i			1.17472i	−0.809324	+	0.587362i	$0.800166\pi$
$128$		−	4360.95i		−	3.01138i
$129$		0			0
$130$	3699.15	+	1239.24i	2.49567	+	0.836064i
$131$	−41.0906			−0.0274053			−0.0137027	−	0.999906i	$-0.504362\pi$
$131$	−41.0906			−0.0274053			−0.0137027	+	0.999906i	$0.504362\pi$
$132$		0			0
$133$		−	943.455i		−	0.615097i
$134$	1099.15			0.708596
$135$		0			0
$136$	−2808.14			−1.77056
$137$			1265.83i			0.789394i	0.918811	+	0.394697i	$0.129150\pi$
$137$			1265.83i			0.789394i	−0.918811	+	0.394697i	$0.870850\pi$
$138$		0			0
$139$	576.135			0.351562			0.175781	−	0.984429i	$-0.443755\pi$
$139$	576.135			0.351562			0.175781	+	0.984429i	$0.443755\pi$
$140$	538.495	−	1607.42i	0.325080	−	0.970370i
$141$		0			0
$142$			2859.86i			1.69010i
$143$		−	3875.39i		−	2.26627i
$144$		0			0
$145$	−1751.88	−	586.889i	−1.00335	−	0.336128i
$146$	−746.214			−0.422994
$147$		0			0
$148$			1057.03i			0.587076i
$149$	1588.69			0.873496			0.436748	−	0.899584i	$-0.356130\pi$
$149$	1588.69			0.873496			0.436748	+	0.899584i	$0.356130\pi$
$150$		0			0
$151$	1317.30			0.709935			0.354968	−	0.934879i	$-0.384492\pi$
$151$	1317.30			0.709935			0.354968	+	0.934879i	$0.384492\pi$
$152$		−	10027.4i		−	5.35086i
$153$		0			0
$154$	−2305.96			−1.20662
$155$	762.554	+	255.460i	0.395160	+	0.132381i
$156$		0			0
$157$		−	1803.58i		−	0.916826i	−0.888739	−	0.458413i	$-0.848418\pi$
$157$		−	1803.58i		−	0.916826i	0.888739	−	0.458413i	$-0.151582\pi$
$158$			1276.34i			0.642657i
$159$		0			0
$160$	2371.64	−	7079.39i	1.17184	−	3.49797i
$161$	368.356			0.180314
$162$		0			0
$163$		−	324.426i		−	0.155896i	−0.996957	−	0.0779479i	$-0.975163\pi$
$163$		−	324.426i		−	0.155896i	0.996957	−	0.0779479i	$-0.0248368\pi$
$164$	−228.663			−0.108876
$165$		0			0
$166$	56.1443			0.0262508
$167$		−	3644.85i		−	1.68891i	−0.535630	−	0.844453i	$-0.679926\pi$
$167$		−	3644.85i		−	1.68891i	0.535630	−	0.844453i	$-0.320074\pi$
$168$		0			0
$169$	−1907.94			−0.868428
$170$	−2179.22	−	730.053i	−0.983170	−	0.329368i
$171$		0			0
$172$			9225.44i			4.08973i
$173$		−	193.361i		−	0.0849765i	−0.999097	−	0.0424882i	$-0.986471\pi$
$173$		−	193.361i		−	0.0849765i	0.999097	−	0.0424882i	$-0.0135285\pi$
$174$		0			0
$175$	527.104	−	698.417i	0.227688	−	0.301688i
$176$	−14027.0			−6.00755
$177$		0			0
$178$		−	113.935i		−	0.0479764i
$179$	3498.06			1.46066			0.730328	−	0.683096i	$-0.239367\pi$
$179$	3498.06			1.46066			0.730328	+	0.683096i	$0.239367\pi$
$180$		0			0
$181$	−590.623			−0.242545			−0.121273	−	0.992619i	$-0.538698\pi$
$181$	−590.623			−0.242545			−0.121273	+	0.992619i	$0.538698\pi$
$182$			2442.54i			0.994799i
$183$		0			0
$184$	3915.03			1.56858
$185$	−173.310	+	517.334i	−0.0688757	+	0.205596i
$186$		0			0
$187$			2283.05i			0.892797i
$188$			5399.11i			2.09452i
$189$		0			0
$190$	2606.90	−	7781.66i	0.995392	−	2.97127i
$191$	−1398.87			−0.529940			−0.264970	−	0.964257i	$-0.585362\pi$
$191$	−1398.87			−0.529940			−0.264970	+	0.964257i	$0.585362\pi$
$192$		0			0
$193$		−	529.184i		−	0.197365i	−0.995119	−	0.0986826i	$-0.968537\pi$
$193$		−	529.184i		−	0.197365i	0.995119	−	0.0986826i	$-0.0314629\pi$
$194$	7563.66			2.79917
$195$		0			0
$196$	1061.38			0.386799
$197$		−	3841.24i		−	1.38922i	−0.719385	−	0.694612i	$-0.755576\pi$
$197$		−	3841.24i		−	1.38922i	0.719385	−	0.694612i	$-0.244424\pi$
$198$		0			0
$199$	3052.58			1.08740			0.543698	−	0.839281i	$-0.317024\pi$
$199$	3052.58			1.08740			0.543698	+	0.839281i	$0.317024\pi$
$200$	5602.26	−	7423.05i	1.98070	−	2.62444i
$201$		0			0
$202$		−	519.795i		−	0.181053i
$203$		−	1156.76i		−	0.399945i
$204$		0			0
$205$	−111.913	−	37.4916i	−0.0381286	−	0.0127733i
$206$	2005.47			0.678289
$207$		0			0
$208$			14857.9i			4.95293i
$209$	−8152.40			−2.69815
$210$		0			0
$211$	−3340.97			−1.09006			−0.545028	−	0.838418i	$-0.683481\pi$
$211$	−3340.97			−1.09006			−0.545028	+	0.838418i	$0.683481\pi$
$212$			11783.6i			3.81745i
$213$		0			0
$214$	−3398.30			−1.08553
$215$	−1512.60	+	4515.14i	−0.479807	+	1.43223i
$216$		0			0
$217$			503.513i			0.157515i
$218$			2078.61i			0.645785i
$219$		0			0
$220$	−13889.7	−	4653.14i	−4.25657	−	1.42597i
$221$	2418.28			0.736068
$222$		0			0
$223$			5475.60i			1.64428i	0.569289	+	0.822138i	$0.307219\pi$
$223$			5475.60i			1.64428i	−0.569289	+	0.822138i	$0.692781\pi$
$224$	4674.51			1.39432
$225$		0			0
$226$	−6480.72			−1.90748
$227$			527.894i			0.154350i	0.997018	+	0.0771752i	$0.0245901\pi$
$227$			527.894i			0.154350i	−0.997018	+	0.0771752i	$0.975410\pi$
$228$		0			0
$229$	−1038.38			−0.299642			−0.149821	−	0.988713i	$-0.547870\pi$
$229$	−1038.38			−0.299642			−0.149821	+	0.988713i	$0.547870\pi$
$230$	3038.21	+	1017.82i	0.871017	+	0.291796i
$231$		0			0
$232$		−	12294.5i		−	3.47920i
$233$			1854.95i			0.521554i	0.965399	+	0.260777i	$0.0839787\pi$
$233$			1854.95i			0.521554i	−0.965399	+	0.260777i	$0.916021\pi$
$234$		0			0
$235$	−885.236	+	2642.45i	−0.245730	+	0.733508i
$236$	12290.1			3.38990
$237$		0			0
$238$		−	1438.94i		−	0.391901i
$239$	−4071.39			−1.10191			−0.550955	−	0.834535i	$-0.685736\pi$
$239$	−4071.39			−1.10191			−0.550955	+	0.834535i	$0.685736\pi$
$240$		0			0
$241$	−5734.52			−1.53275			−0.766375	−	0.642394i	$-0.777941\pi$
$241$	−5734.52			−1.53275			−0.766375	+	0.642394i	$0.777941\pi$
$242$			12676.9i			3.36737i
$243$		0			0
$244$	−13311.6			−3.49257
$245$	519.462	+	174.023i	0.135458	+	0.0453792i
$246$		0			0
$247$			8635.28i			2.22449i
$248$			5351.53i			1.37025i
$249$		0			0
$250$	6277.40	−	4304.11i	1.58807	−	1.08886i
$251$	4371.18			1.09923			0.549614	−	0.835419i	$-0.314775\pi$
$251$	4371.18			1.09923			0.549614	+	0.835419i	$0.314775\pi$
$252$		0			0
$253$		−	3182.96i		−	0.790953i
$254$	−9156.57			−2.26195
$255$		0			0
$256$	9497.13			2.31864
$257$		−	6667.99i		−	1.61844i	−0.587509	−	0.809218i	$-0.699891\pi$
$257$		−	6667.99i		−	1.61844i	0.587509	−	0.809218i	$-0.300109\pi$
$258$		0			0
$259$	−341.595			−0.0819524
$260$	−4928.75	+	14712.4i	−1.17565	+	3.50933i
$261$		0			0
$262$		−	223.786i		−	0.0527693i
$263$		−	4668.18i		−	1.09450i	−0.836970	−	0.547248i	$-0.815675\pi$
$263$		−	4668.18i		−	1.09450i	0.836970	−	0.547248i	$-0.184325\pi$
$264$		0			0
$265$	−1932.03	+	5767.16i	−0.447864	+	1.33688i
$266$	5138.22			1.18438
$267$		0			0
$268$			4371.58i			0.996406i
$269$	153.845			0.0348701			0.0174351	−	0.999848i	$-0.494450\pi$
$269$	153.845			0.0348701			0.0174351	+	0.999848i	$0.494450\pi$
$270$		0			0
$271$	−3774.11			−0.845982			−0.422991	−	0.906134i	$-0.639020\pi$
$271$	−3774.11			−0.845982			−0.422991	+	0.906134i	$0.639020\pi$
$272$		−	8753.01i		−	1.95121i
$273$		0			0
$274$	−6893.91			−1.51999
$275$	−6035.02	−	4554.71i	−1.32337	−	0.998760i
$276$		0			0
$277$		−	7464.31i		−	1.61909i	−0.587060	−	0.809544i	$-0.699715\pi$
$277$		−	7464.31i		−	1.61909i	0.587060	−	0.809544i	$-0.300285\pi$
$278$			3137.73i			0.676937i
$279$		0			0
$280$	5521.04	+	1849.58i	1.17838	+	0.394763i
$281$	5596.80			1.18818			0.594088	−	0.804400i	$-0.297513\pi$
$281$	5596.80			1.18818			0.594088	+	0.804400i	$0.297513\pi$
$282$		0			0
$283$		−	8132.88i		−	1.70830i	−0.520025	−	0.854151i	$-0.674077\pi$
$283$		−	8132.88i		−	1.70830i	0.520025	−	0.854151i	$-0.325923\pi$
$284$	−11374.4			−2.37657
$285$		0			0
$286$	21106.0			4.36372
$287$		−	73.8961i		−	0.0151984i
$288$		0			0
$289$	3488.36			0.710026
$290$	3196.30	−	9541.02i	0.647217	−	1.93196i
$291$		0			0
$292$		−	2967.88i		−	0.594801i
$293$			4134.83i			0.824434i	0.911086	+	0.412217i	$0.135245\pi$
$293$			4134.83i			0.824434i	−0.911086	+	0.412217i	$0.864755\pi$
$294$		0			0
$295$	6015.05	+	2015.08i	1.18715	+	0.397703i
$296$	−3630.60			−0.712921
$297$		0			0
$298$			8652.30i			1.68193i
$299$	−3371.50			−0.652102
$300$		0			0
$301$	−2981.34			−0.570903
$302$			7174.23i			1.36699i
$303$		0			0
$304$	31255.6			5.89681
$305$	−6515.00	−	2182.56i	−1.22311	−	0.409748i
$306$		0			0
$307$		−	9663.06i		−	1.79642i	−0.439569	−	0.898209i	$-0.644869\pi$
$307$		−	9663.06i		−	1.79642i	0.439569	−	0.898209i	$-0.355131\pi$
$308$		−	9171.36i		−	1.69671i
$309$		0			0
$310$	−1391.28	+	4153.00i	−0.254901	+	0.760885i
$311$	7930.53			1.44598			0.722989	−	0.690860i	$-0.242768\pi$
$311$	7930.53			1.44598			0.722989	+	0.690860i	$0.242768\pi$
$312$		0			0
$313$			1247.21i			0.225228i	0.993639	+	0.112614i	$0.0359223\pi$
$313$			1247.21i			0.225228i	−0.993639	+	0.112614i	$0.964078\pi$
$314$	9822.63			1.76536
$315$		0			0
$316$	−5076.30			−0.903684
$317$			2884.20i			0.511019i	0.966807	+	0.255510i	$0.0822432\pi$
$317$			2884.20i			0.511019i	−0.966807	+	0.255510i	$0.917757\pi$
$318$		0			0
$319$	−9995.58			−1.75437
$320$	18887.9	+	6327.56i	3.29958	+	1.10538i
$321$		0			0
$322$			2006.13i			0.347196i
$323$		−	5087.17i		−	0.876340i
$324$		0			0
$325$	−4824.49	+	6392.49i	−0.823429	+	1.09105i
$326$	1766.88			0.300179
$327$		0			0
$328$		−	785.396i		−	0.132214i
$329$	−1744.81			−0.292384
$330$		0			0
$331$	−5668.46			−0.941289			−0.470645	−	0.882323i	$-0.655979\pi$
$331$	−5668.46			−0.941289			−0.470645	+	0.882323i	$0.655979\pi$
$332$			223.299i			0.0369131i
$333$		0			0
$334$	19850.5			3.25201
$335$	−716.763	+	2139.55i	−0.116898	+	0.348944i
$336$		0			0
$337$			8556.34i			1.38307i	0.722344	+	0.691534i	$0.243065\pi$
$337$			8556.34i			1.38307i	−0.722344	+	0.691534i	$0.756935\pi$
$338$		−	10390.9i		−	1.67217i
$339$		0			0
$340$	2903.60	−	8667.31i	0.463146	−	1.38250i
$341$	4350.86			0.690945
$342$		0			0
$343$			343.000i			0.0539949i
$344$	−31686.9			−4.96640
$345$		0			0
$346$	1053.07			0.163623
$347$		−	6794.69i		−	1.05118i	−0.850739	−	0.525588i	$-0.823845\pi$
$347$		−	6794.69i		−	1.05118i	0.850739	−	0.525588i	$-0.176155\pi$
$348$		0			0
$349$	−8922.38			−1.36849			−0.684246	−	0.729251i	$-0.739869\pi$
$349$	−8922.38			−1.36849			−0.684246	+	0.729251i	$0.739869\pi$
$350$	3803.70	+	2870.70i	0.580903	+	0.438415i
$351$		0			0
$352$		−	40392.4i		−	6.11626i
$353$			8668.14i			1.30696i	0.756942	+	0.653482i	$0.226693\pi$
$353$			8668.14i			1.30696i	−0.756942	+	0.653482i	$0.773307\pi$
$354$		0			0
$355$	−5566.88	−	1864.94i	−0.832280	−	0.278819i
$356$	453.148			0.0674629
$357$		0			0
$358$			19051.0i			2.81251i
$359$	−3084.48			−0.453462			−0.226731	−	0.973957i	$-0.572804\pi$
$359$	−3084.48			−0.453462			−0.226731	+	0.973957i	$0.572804\pi$
$360$		0			0
$361$	11306.5			1.64841
$362$		−	3216.63i		−	0.467023i
$363$		0			0
$364$	−9714.59			−1.39885
$365$	486.612	−	1452.55i	0.0697820	−	0.208301i
$366$		0			0
$367$		−	7926.72i		−	1.12744i	−0.825965	−	0.563721i	$-0.809369\pi$
$367$		−	7926.72i		−	1.12744i	0.825965	−	0.563721i	$-0.190631\pi$
$368$			12203.2i			1.72863i
$369$		0			0
$370$	−2817.49	−	943.876i	−0.395877	−	0.132621i
$371$	−3808.05			−0.532895
$372$		0			0
$373$			9724.91i			1.34996i	0.737834	+	0.674982i	$0.235849\pi$
$373$			9724.91i			1.34996i	−0.737834	+	0.674982i	$0.764151\pi$
$374$	−12433.9			−1.71909
$375$		0			0
$376$	−18544.5			−2.54350
$377$			10587.6i			1.44640i
$378$		0			0
$379$	793.694			0.107571			0.0537854	−	0.998553i	$-0.482871\pi$
$379$	793.694			0.107571			0.0537854	+	0.998553i	$0.482871\pi$
$380$	30949.6	+	10368.3i	4.17810	+	1.39969i
$381$		0			0
$382$		−	7618.46i		−	1.02040i
$383$		−	2732.63i		−	0.364571i	−0.983246	−	0.182286i	$-0.941650\pi$
$383$		−	2732.63i		−	0.364571i	0.983246	−	0.182286i	$-0.0583495\pi$
$384$		0			0
$385$	1503.73	−	4488.67i	0.199058	−	0.594192i
$386$	2882.02			0.380029
$387$		0			0
$388$			30082.5i			3.93610i
$389$	11011.1			1.43518			0.717588	−	0.696468i	$-0.245246\pi$
$389$	11011.1			1.43518			0.717588	+	0.696468i	$0.245246\pi$
$390$		0			0
$391$	1986.20			0.256896
$392$			3645.54i			0.469713i
$393$		0			0
$394$	20920.0			2.67497
$395$	−2484.46	−	832.308i	−0.316473	−	0.106020i
$396$		0			0
$397$		−	9688.40i		−	1.22480i	−0.790547	−	0.612402i	$-0.790204\pi$
$397$		−	9688.40i		−	1.22480i	0.790547	−	0.612402i	$-0.209796\pi$
$398$			16624.9i			2.09379i
$399$		0			0
$400$	23137.7	+	17462.3i	2.89222	+	2.18279i
$401$	7442.12			0.926788			0.463394	−	0.886152i	$-0.346632\pi$
$401$	7442.12			0.926788			0.463394	+	0.886152i	$0.346632\pi$
$402$		0			0
$403$		−	4608.57i		−	0.569650i
$404$	2067.35			0.254590
$405$		0			0
$406$	6299.92			0.770098
$407$			2951.72i			0.359488i
$408$		0			0
$409$	10839.1			1.31042			0.655208	−	0.755449i	$-0.272581\pi$
$409$	10839.1			1.31042			0.655208	+	0.755449i	$0.272581\pi$
$410$	204.185	−	609.498i	0.0245951	−	0.0734170i
$411$		0			0
$412$			7976.24i			0.953789i
$413$			3971.73i			0.473211i
$414$		0			0
$415$	−36.6121	+	109.288i	−0.00433064	+	0.0129271i
$416$	−42784.9			−5.04256
$417$		0			0
$418$		−	44399.3i		−	5.19532i
$419$	−475.629			−0.0554558			−0.0277279	−	0.999616i	$-0.508827\pi$
$419$	−475.629			−0.0554558			−0.0277279	+	0.999616i	$0.508827\pi$
$420$		0			0
$421$	6879.42			0.796396			0.398198	−	0.917300i	$-0.369636\pi$
$421$	6879.42			0.796396			0.398198	+	0.917300i	$0.369636\pi$
$422$		−	18195.5i		−	2.09891i
$423$		0			0
$424$	−40473.4			−4.63576
$425$	2842.18	−	3765.91i	0.324390	−	0.429820i
$426$		0			0
$427$		−	4301.84i		−	0.487543i
$428$		−	13515.9i		−	1.52644i
$429$		0			0
$430$	−24590.2	−	8237.87i	−2.75778	−	0.923873i
$431$	3993.61			0.446324			0.223162	−	0.974781i	$-0.428362\pi$
$431$	3993.61			0.446324			0.223162	+	0.974781i	$0.428362\pi$
$432$		0			0
$433$			13362.5i			1.48306i	0.670922	+	0.741528i	$0.265898\pi$
$433$			13362.5i			1.48306i	−0.670922	+	0.741528i	$0.734102\pi$
$434$	−2742.22			−0.303296
$435$		0			0
$436$	−8267.13			−0.908082
$437$			7092.39i			0.776373i
$438$		0			0
$439$	−16047.5			−1.74466			−0.872330	−	0.488918i	$-0.837392\pi$
$439$	−16047.5			−1.74466			−0.872330	+	0.488918i	$0.837392\pi$
$440$	15982.2	−	47707.4i	1.73164	−	5.16900i
$441$		0			0
$442$			13170.3i			1.41731i
$443$		−	16538.8i		−	1.77378i	−0.461985	−	0.886888i	$-0.652863\pi$
$443$		−	16538.8i		−	1.77378i	0.461985	−	0.886888i	$-0.347137\pi$
$444$		0			0
$445$	221.781	+	74.2979i	0.0236257	+	0.00791474i
$446$	−29821.0			−3.16607
$447$		0			0
$448$			12471.6i			1.31525i
$449$	−2790.61			−0.293312			−0.146656	−	0.989188i	$-0.546851\pi$
$449$	−2790.61			−0.293312			−0.146656	+	0.989188i	$0.546851\pi$
$450$		0			0
$451$	−638.536			−0.0666685
$452$		−	25775.4i		−	2.68224i
$453$		0			0
$454$	−2875.00			−0.297204
$455$	−4754.55	−	1592.80i	−0.489883	−	0.164114i
$456$		0			0
$457$			8003.62i			0.819242i	0.912256	+	0.409621i	$0.134339\pi$
$457$			8003.62i			0.819242i	−0.912256	+	0.409621i	$0.865661\pi$
$458$		−	5655.18i		−	0.576964i
$459$		0			0
$460$	−4048.12	+	12083.7i	−0.410314	+	1.22480i
$461$	11025.5			1.11390			0.556948	−	0.830547i	$-0.311972\pi$
$461$	11025.5			1.11390			0.556948	+	0.830547i	$0.311972\pi$
$462$		0			0
$463$		−	539.353i		−	0.0541379i	−0.999634	−	0.0270689i	$-0.991383\pi$
$463$		−	539.353i		−	0.0541379i	0.999634	−	0.0270689i	$-0.00861737\pi$
$464$	38322.2			3.83418
$465$		0			0
$466$	−10102.4			−1.00426
$467$			10642.7i			1.05457i	0.849688	+	0.527286i	$0.176790\pi$
$467$			10642.7i			1.05457i	−0.849688	+	0.527286i	$0.823210\pi$
$468$		0			0
$469$	−1412.74			−0.139093
$470$	−14391.2	−	4821.15i	−1.41238	−	0.473155i
$471$		0			0
$472$			42213.1i			4.11656i
$473$			25761.8i			2.50429i
$474$		0			0
$475$	13447.5	+	10149.0i	1.29897	+	0.980350i
$476$	5723.01			0.551079
$477$		0			0
$478$		−	22173.5i		−	2.12174i
$479$	−12976.4			−1.23780			−0.618899	−	0.785471i	$-0.712421\pi$
$479$	−12976.4			−1.23780			−0.618899	+	0.785471i	$0.712421\pi$
$480$		0			0
$481$	3126.56			0.296380
$482$		−	31231.1i		−	2.95133i
$483$		0			0
$484$	−50419.3			−4.73509
$485$	−4932.32	+	14723.1i	−0.461784	+	1.37843i
$486$		0			0
$487$		−	12739.8i		−	1.18541i	−0.805419	−	0.592705i	$-0.798060\pi$
$487$		−	12739.8i		−	1.18541i	0.805419	−	0.592705i	$-0.201940\pi$
$488$		−	45721.6i		−	4.24123i
$489$		0			0
$490$	−947.758	+	2829.08i	−0.0873782	+	0.260826i
$491$	7951.24			0.730824			0.365412	−	0.930846i	$-0.380928\pi$
$491$	7951.24			0.730824			0.365412	+	0.930846i	$0.380928\pi$
$492$		0			0
$493$		−	6237.33i		−	0.569808i
$494$	−47029.2			−4.28329
$495$		0			0
$496$	−16680.8			−1.51006
$497$		−	3675.80i		−	0.331755i
$498$		0			0
$499$	13883.2			1.24549			0.622744	−	0.782426i	$-0.286018\pi$
$499$	13883.2			1.24549			0.622744	+	0.782426i	$0.286018\pi$
$500$	17118.5	+	24966.8i	1.53113	+	2.23310i
$501$		0			0
$502$			23806.2i			2.11658i
$503$			3388.74i			0.300391i	0.988656	+	0.150195i	$0.0479903\pi$
$503$			3388.74i			0.300391i	−0.988656	+	0.150195i	$0.952010\pi$
$504$		0			0
$505$	1011.81	+	338.962i	0.0891583	+	0.0298685i
$506$	17335.0			1.52299
$507$		0			0
$508$		−	36417.9i		−	3.18068i
$509$	−20440.9			−1.78001			−0.890005	−	0.455950i	$-0.849300\pi$
$509$	−20440.9			−1.78001			−0.890005	+	0.455950i	$0.849300\pi$
$510$		0			0
$511$	959.114			0.0830308
$512$			16835.4i			1.45318i
$513$		0			0
$514$	36315.0			3.11632
$515$	−1307.78	+	3903.76i	−0.111899	+	0.334020i
$516$		0			0
$517$			15076.9i			1.28255i
$518$		−	1860.38i		−	0.157800i
$519$		0			0
$520$	−50533.2	−	16928.9i	−4.26159	−	1.42766i
$521$	2391.88			0.201133			0.100566	−	0.994930i	$-0.467935\pi$
$521$	2391.88			0.201133			0.100566	+	0.994930i	$0.467935\pi$
$522$		0			0
$523$		−	4839.38i		−	0.404611i	−0.979322	−	0.202305i	$-0.935157\pi$
$523$		−	4839.38i		−	0.404611i	0.979322	−	0.202305i	$-0.0648433\pi$
$524$	890.052			0.0742025
$525$		0			0
$526$	25423.7			2.10747
$527$			2714.97i			0.224414i
$528$		0			0
$529$	9397.90			0.772409
$530$	−31408.9	−	10522.2i	−2.57418	−	0.862366i
$531$		0			0
$532$			20435.9i			1.66543i
$533$			676.358i			0.0549649i
$534$		0			0
$535$	2216.06	−	6614.99i	0.179081	−	0.534562i
$536$	−15015.2			−1.20999
$537$		0			0
$538$			837.863i			0.0671429i
$539$	2963.86			0.236851
$540$		0			0
$541$	21183.6			1.68347			0.841733	−	0.539894i	$-0.181536\pi$
$541$	21183.6			1.68347			0.841733	+	0.539894i	$0.181536\pi$
$542$		−	20554.4i		−	1.62895i
$543$		0			0
$544$	25205.2			1.98652
$545$	−4046.13	−	1355.48i	−0.318013	−	0.106536i
$546$		0			0
$547$		−	3180.06i		−	0.248574i	−0.992246	−	0.124287i	$-0.960336\pi$
$547$		−	3180.06i		−	0.248574i	0.992246	−	0.124287i	$-0.0396643\pi$
$548$		−	27418.8i		−	2.13736i
$549$		0			0
$550$	24805.7	−	32867.8i	1.92312	−	2.54816i
$551$	22272.5			1.72203
$552$		0			0
$553$		−	1640.48i		−	0.126149i
$554$	40651.9			3.11757
$555$		0			0
$556$	−12479.5			−0.951888
$557$			1196.88i			0.0910476i	0.998963	+	0.0455238i	$0.0144957\pi$
$557$			1196.88i			0.0910476i	−0.998963	+	0.0455238i	$0.985504\pi$
$558$		0			0
$559$	27287.7			2.06466
$560$	−5765.18	+	17209.2i	−0.435041	+	1.29861i
$561$		0			0
$562$			30481.1i			2.28784i
$563$		−	2800.80i		−	0.209662i	−0.994490	−	0.104831i	$-0.966570\pi$
$563$		−	2800.80i		−	0.209662i	0.994490	−	0.104831i	$-0.0334301\pi$
$564$		0			0
$565$	4226.13	−	12615.1i	0.314680	−	0.939328i
$566$	44293.0			3.28935
$567$		0			0
$568$		−	39067.8i		−	2.88600i
$569$	12783.5			0.941849			0.470924	−	0.882174i	$-0.343920\pi$
$569$	12783.5			0.941849			0.470924	+	0.882174i	$0.343920\pi$
$570$		0			0
$571$	−12490.6			−0.915439			−0.457719	−	0.889097i	$-0.651334\pi$
$571$	−12490.6			−0.915439			−0.457719	+	0.889097i	$0.651334\pi$
$572$			83943.8i			6.13613i
$573$		0			0
$574$	402.450			0.0292647
$575$	−3962.49	+	5250.33i	−0.287386	+	0.380789i
$576$		0			0
$577$		−	19884.0i		−	1.43463i	−0.696747	−	0.717317i	$-0.745370\pi$
$577$		−	19884.0i		−	1.43463i	0.696747	−	0.717317i	$-0.254630\pi$
$578$			18998.2i			1.36716i
$579$		0			0
$580$	37947.0	+	12712.5i	2.71666	+	0.910097i
$581$	−72.1626			−0.00515286
$582$		0			0
$583$			32905.3i			2.33756i
$584$	10193.8			0.722301
$585$		0			0
$586$	−22519.0			−1.58746
$587$			586.249i			0.0412216i	0.999788	+	0.0206108i	$0.00656108\pi$
$587$			586.249i			0.0412216i	−0.999788	+	0.0206108i	$0.993439\pi$
$588$		0			0
$589$	−9694.74			−0.678208
$590$	−10974.5	+	32759.0i	−0.765782	+	2.28588i
$591$		0			0
$592$		−	11316.6i		−	0.785661i
$593$			7891.38i			0.546476i	0.961946	+	0.273238i	$0.0880947\pi$
$593$			7891.38i			0.546476i	−0.961946	+	0.273238i	$0.911905\pi$
$594$		0			0
$595$	2800.97	+	938.343i	0.192989	+	0.0646526i
$596$	−34412.3			−2.36507
$597$		0			0
$598$		−	18361.7i		−	1.25563i
$599$	8924.75			0.608774			0.304387	−	0.952549i	$-0.401548\pi$
$599$	8924.75			0.608774			0.304387	+	0.952549i	$0.401548\pi$
$600$		0			0
$601$	−14545.7			−0.987241			−0.493620	−	0.869677i	$-0.664327\pi$
$601$	−14545.7			−0.987241			−0.493620	+	0.869677i	$0.664327\pi$
$602$		−	16236.9i		−	1.09928i
$603$		0			0
$604$	−28533.7			−1.92222
$605$	−24676.4	−	8266.72i	−1.65824	−	0.555521i
$606$		0			0
$607$		−	20253.5i		−	1.35431i	−0.735841	−	0.677154i	$-0.763213\pi$
$607$		−	20253.5i		−	1.35431i	0.735841	−	0.677154i	$-0.236787\pi$
$608$			90003.9i			6.00352i
$609$		0			0
$610$	11886.6	−	35481.8i	0.788975	−	2.35511i
$611$	15969.9			1.05740
$612$		0			0
$613$		−	14610.9i		−	0.962686i	−0.876532	−	0.481343i	$-0.840149\pi$
$613$		−	14610.9i		−	0.962686i	0.876532	−	0.481343i	$-0.159851\pi$
$614$	52626.7			3.45902
$615$		0			0
$616$	31501.1			2.06041
$617$			15791.8i			1.03039i	0.857072	+	0.515197i	$0.172281\pi$
$617$			15791.8i			1.03039i	−0.857072	+	0.515197i	$0.827719\pi$
$618$		0			0
$619$	−561.212			−0.0364410			−0.0182205	−	0.999834i	$-0.505800\pi$
$619$	−561.212			−0.0364410			−0.0182205	+	0.999834i	$0.505800\pi$
$620$	−16517.5	−	5533.45i	−1.06993	−	0.358434i
$621$		0			0
$622$			43191.0i			2.78425i
$623$			146.442i			0.00941743i
$624$		0			0
$625$	4284.65	+	15026.1i	0.274218	+	0.961668i
$626$	−6792.50			−0.433679
$627$		0			0
$628$			39067.0i			2.48239i
$629$	−1841.90			−0.116759
$630$		0			0
$631$	16360.8			1.03219			0.516096	−	0.856531i	$-0.327385\pi$
$631$	16360.8			1.03219			0.516096	+	0.856531i	$0.327385\pi$
$632$		−	17435.7i		−	1.09740i
$633$		0			0
$634$	−15707.9			−0.983973
$635$	5971.07	−	17823.8i	0.373157	−	1.11388i
$636$		0			0
$637$		−	3139.42i		−	0.195272i
$638$		−	54437.6i		−	3.37806i
$639$		0			0
$640$	−15487.8	+	46231.6i	−0.956579	+	2.85541i
$641$	30298.1			1.86693			0.933465	−	0.358667i	$-0.116769\pi$
$641$	30298.1			1.86693			0.933465	+	0.358667i	$0.116769\pi$
$642$		0			0
$643$			121.613i			0.00745871i	0.999993	+	0.00372935i	$0.00118709\pi$
$643$			121.613i			0.00745871i	−0.999993	+	0.00372935i	$0.998813\pi$
$644$	−7978.86			−0.488216
$645$		0			0
$646$	27705.6			1.68740
$647$		−	17068.6i		−	1.03715i	−0.855033	−	0.518574i	$-0.826463\pi$
$647$		−	17068.6i		−	1.03715i	0.855033	−	0.518574i	$-0.173537\pi$
$648$		0			0
$649$	34319.7			2.07576
$650$	−34814.6	−	26275.0i	−2.10083	−	1.58552i
$651$		0			0
$652$			7027.31i			0.422102i
$653$		−	2667.60i		−	0.159864i	−0.996800	−	0.0799319i	$-0.974530\pi$
$653$		−	2667.60i		−	0.159864i	0.996800	−	0.0799319i	$-0.0254703\pi$
$654$		0			0
$655$	435.612	+	145.933i	0.0259859	+	0.00870543i
$656$	2448.09			0.145704
$657$		0			0
$658$		−	9502.50i		−	0.562988i
$659$	−11943.1			−0.705976			−0.352988	−	0.935628i	$-0.614834\pi$
$659$	−11943.1			−0.705976			−0.352988	+	0.935628i	$0.614834\pi$
$660$		0			0
$661$	−21867.4			−1.28675			−0.643375	−	0.765551i	$-0.722466\pi$
$661$	−21867.4			−1.28675			−0.643375	+	0.765551i	$0.722466\pi$
$662$		−	30871.4i		−	1.81246i
$663$		0			0
$664$	−766.972			−0.0448258
$665$	−3350.67	+	10001.8i	−0.195389	+	0.583239i
$666$		0			0
$667$			8695.91i			0.504808i
$668$			78950.2i			4.57287i
$669$		0			0
$670$	−11652.4	−	3903.61i	−0.671896	−	0.225089i
$671$	−37172.2			−2.13863
$672$		0			0
$673$			480.080i			0.0274974i	0.999905	+	0.0137487i	$0.00437648\pi$
$673$			480.080i			0.0274974i	−0.999905	+	0.0137487i	$0.995624\pi$
$674$	−46599.3			−2.66311
$675$		0			0
$676$	41327.3			2.35135
$677$			28054.8i			1.59267i	0.604858	+	0.796333i	$0.293230\pi$
$677$			28054.8i			1.59267i	−0.604858	+	0.796333i	$0.706770\pi$
$678$		0			0
$679$	−9721.62			−0.549458
$680$	29769.8	+	9973.07i	1.67885	+	0.562426i
$681$		0			0
$682$			23695.5i			1.33042i
$683$		−	15552.3i		−	0.871291i	−0.900118	−	0.435645i	$-0.856520\pi$
$683$		−	15552.3i		−	0.871291i	0.900118	−	0.435645i	$-0.143480\pi$
$684$		0			0
$685$	4495.57	−	13419.4i	0.250755	−	0.748508i
$686$	−1868.04			−0.103968
$687$		0			0
$688$		−	98768.4i		−	5.47312i
$689$	34854.4			1.92721
$690$		0			0
$691$	20777.6			1.14387			0.571936	−	0.820298i	$-0.306193\pi$
$691$	20777.6			1.14387			0.571936	+	0.820298i	$0.306193\pi$
$692$			4188.33i			0.230082i
$693$		0			0
$694$	37005.0			2.02405
$695$	−6107.77	−	2046.14i	−0.333354	−	0.111675i
$696$		0			0
$697$		−	398.452i		−	0.0216535i
$698$		−	48592.8i		−	2.63505i
$699$		0			0
$700$	−11417.5	+	15128.2i	−0.616485	+	0.816848i
$701$	2132.81			0.114915			0.0574573	−	0.998348i	$-0.481701\pi$
$701$	2132.81			0.114915			0.0574573	+	0.998348i	$0.481701\pi$
$702$		0			0
$703$		−	6577.14i		−	0.352861i
$704$	107768.			5.76938
$705$		0			0
$706$	−47208.1			−2.51657
$707$			668.096i			0.0355394i
$708$		0			0
$709$	2908.28			0.154052			0.0770260	−	0.997029i	$-0.475458\pi$
$709$	2908.28			0.154052			0.0770260	+	0.997029i	$0.475458\pi$
$710$	10156.8	−	30318.2i	0.536868	−	1.60256i
$711$		0			0
$712$			1556.44i			0.0819241i
$713$		−	3785.14i		−	0.198814i
$714$		0			0
$715$	−13763.4	+	41084.0i	−0.719891	+	2.14889i
$716$	−75770.6			−3.95486
$717$		0			0
$718$		−	16798.6i		−	0.873145i
$719$	−12241.1			−0.634931			−0.317466	−	0.948270i	$-0.602832\pi$
$719$	−12241.1			−0.634931			−0.317466	+	0.948270i	$0.602832\pi$
$720$		0			0
$721$	−2577.64			−0.133144
$722$			61576.9i			3.17404i
$723$		0			0
$724$	12793.3			0.656713
$725$	16487.8	+	12443.5i	0.844609	+	0.637437i
$726$		0			0
$727$		−	8776.97i		−	0.447758i	−0.974617	−	0.223879i	$-0.928128\pi$
$727$		−	8776.97i		−	0.447758i	0.974617	−	0.223879i	$-0.0718720\pi$
$728$		−	33367.0i		−	1.69871i
$729$		0			0
$730$	7910.82	+	2650.17i	0.401086	+	0.134366i
$731$	−16075.6			−0.813375
$732$		0			0
$733$		−	3093.56i		−	0.155884i	−0.996958	−	0.0779421i	$-0.975165\pi$
$733$		−	3093.56i		−	0.155884i	0.996958	−	0.0779421i	$-0.0248349\pi$
$734$	43170.3			2.17090
$735$		0			0
$736$	−35140.4			−1.75991
$737$			12207.5i			0.610135i
$738$		0			0
$739$	−18338.3			−0.912837			−0.456419	−	0.889765i	$-0.650868\pi$
$739$	−18338.3			−0.912837			−0.456419	+	0.889765i	$0.650868\pi$
$740$	3754.03	−	11205.8i	0.186487	−	0.556669i
$741$		0			0
$742$		−	20739.3i		−	1.02609i
$743$			3193.45i			0.157680i	0.996887	+	0.0788401i	$0.0251216\pi$
$743$			3193.45i			0.157680i	−0.996887	+	0.0788401i	$0.974878\pi$
$744$		0			0
$745$	−16842.2	−	5642.23i	−0.828255	−	0.277470i
$746$	−52963.5			−2.59937
$747$		0			0
$748$		−	49452.5i		−	2.41733i
$749$	4367.86			0.213082
$750$		0			0
$751$	−13439.8			−0.653029			−0.326515	−	0.945192i	$-0.605874\pi$
$751$	−13439.8			−0.653029			−0.326515	+	0.945192i	$0.605874\pi$
$752$		−	57803.4i		−	2.80302i
$753$		0			0
$754$	−57662.0			−2.78505
$755$	−13965.0	−	4678.37i	−0.673165	−	0.225514i
$756$		0			0
$757$		−	8314.25i		−	0.399190i	−0.979879	−	0.199595i	$-0.936037\pi$
$757$		−	8314.25i		−	0.399190i	0.979879	−	0.199595i	$-0.0639626\pi$
$758$			4322.59i			0.207129i
$759$		0			0
$760$	−35612.2	+	106303.i	−1.69972	+	5.07372i
$761$	22244.7			1.05962			0.529809	−	0.848117i	$-0.322264\pi$
$761$	22244.7			1.05962			0.529809	+	0.848117i	$0.322264\pi$
$762$		0			0
$763$		−	2671.65i		−	0.126763i
$764$	30300.5			1.43486
$765$		0			0
$766$	14882.3			0.701986
$767$		−	36352.5i		−	1.71136i
$768$		0			0
$769$	−14250.1			−0.668236			−0.334118	−	0.942531i	$-0.608438\pi$
$769$	−14250.1			−0.668236			−0.334118	+	0.942531i	$0.608438\pi$
$770$	24446.1	+	8189.58i	1.14412	+	0.383288i
$771$		0			0
$772$			11462.5i			0.534385i
$773$			20201.9i			0.939988i	0.882670	+	0.469994i	$0.155744\pi$
$773$			20201.9i			0.939988i	−0.882670	+	0.469994i	$0.844256\pi$
$774$		0			0
$775$	−7176.78	−	5416.40i	−0.332642	−	0.251049i
$776$	−103325.			−4.77984
$777$		0			0
$778$			59968.1i			2.76345i
$779$	1422.81			0.0654396
$780$		0			0
$781$	−31762.6			−1.45526
$782$			10817.2i			0.494656i
$783$		0			0
$784$	−11363.2			−0.517638
$785$	−6405.41	+	19120.3i	−0.291234	+	0.869341i
$786$		0			0
$787$		−	3277.11i		−	0.148433i	−0.997242	−	0.0742163i	$-0.976354\pi$
$787$		−	3277.11i		−	0.148433i	0.997242	−	0.0742163i	$-0.0236455\pi$
$788$			83204.1i			3.76145i
$789$		0			0
$790$	4532.89	−	13530.8i	0.204143	−	0.609371i
$791$	8329.72			0.374426
$792$		0			0
$793$			39374.0i			1.76319i
$794$	52764.6			2.35837
$795$		0			0
$796$	−66121.2			−2.94422
$797$		−	3762.85i		−	0.167236i	−0.996498	−	0.0836180i	$-0.973352\pi$
$797$		−	3762.85i		−	0.167236i	0.996498	−	0.0836180i	$-0.0266476\pi$
$798$		0			0
$799$	−9408.10			−0.416564
$800$	−50284.7	+	66627.7i	−2.22229	+	2.94455i
$801$		0			0
$802$			40531.0i			1.78454i
$803$		−	8287.71i		−	0.364218i
$804$		0			0
$805$	−3905.04	−	1308.21i	−0.170975	−	0.0572775i
$806$	25099.0			1.09687
$807$		0			0
$808$			7100.78i			0.309164i
$809$	−19453.8			−0.845438			−0.422719	−	0.906261i	$-0.638924\pi$
$809$	−19453.8			−0.845438			−0.422719	+	0.906261i	$0.638924\pi$
$810$		0			0
$811$	45456.3			1.96817			0.984085	−	0.177700i	$-0.0568655\pi$
$811$	45456.3			1.96817			0.984085	+	0.177700i	$0.0568655\pi$
$812$			25056.3i			1.08289i
$813$		0			0
$814$	−16075.6			−0.692198
$815$	−1152.20	+	3439.33i	−0.0495210	+	0.147821i
$816$		0			0
$817$		−	57403.3i		−	2.45813i
$818$			59031.7i			2.52322i
$819$		0			0
$820$	2424.12	+	812.095i	0.103237	+	0.0345849i
$821$	−14653.1			−0.622897			−0.311448	−	0.950263i	$-0.600814\pi$
$821$	−14653.1			−0.622897			−0.311448	+	0.950263i	$0.600814\pi$
$822$		0			0
$823$			5569.57i			0.235897i	0.993020	+	0.117948i	$0.0376317\pi$
$823$			5569.57i			0.235897i	−0.993020	+	0.117948i	$0.962368\pi$
$824$	−27396.2			−1.15824
$825$		0			0
$826$	−21630.7			−0.911173
$827$		−	11578.6i		−	0.486852i	−0.969920	−	0.243426i	$-0.921729\pi$
$827$		−	11578.6i		−	0.486852i	0.969920	−	0.243426i	$-0.0782712\pi$
$828$		0			0
$829$	−31832.6			−1.33365			−0.666823	−	0.745216i	$-0.732346\pi$
$829$	−31832.6			−1.33365			−0.666823	+	0.745216i	$0.732346\pi$
$830$	−595.201	−	199.396i	−0.0248912	−	0.00833870i
$831$		0			0
$832$		−	114151.i		−	4.75657i
$833$			1849.48i			0.0769275i
$834$		0			0
$835$	−12944.7	+	38640.1i	−0.536489	+	1.60143i
$836$	176587.			7.30549
$837$		0			0
$838$		−	2590.35i		−	0.106781i
$839$	−16909.4			−0.695802			−0.347901	−	0.937531i	$-0.613105\pi$
$839$	−16909.4			−0.695802			−0.347901	+	0.937531i	$0.613105\pi$
$840$		0			0
$841$	2919.11			0.119690
$842$			37466.5i			1.53347i
$843$		0			0
$844$	72367.8			2.95143
$845$	20226.5	+	6776.01i	0.823449	+	0.275860i
$846$		0			0
$847$		−	16293.8i		−	0.660992i
$848$		−	126156.i		−	5.10875i
$849$		0			0
$850$	20509.8	+	15479.0i	0.827623	+	0.624617i
$851$	2567.93			0.103440
$852$		0			0
$853$			34133.0i			1.37010i	0.728498	+	0.685048i	$0.240219\pi$
$853$			34133.0i			1.37010i	−0.728498	+	0.685048i	$0.759781\pi$
$854$	23428.6			0.938769
$855$		0			0
$856$	46423.3			1.85364
$857$			27751.3i			1.10614i	0.833134	+	0.553072i	$0.186544\pi$
$857$			27751.3i			1.10614i	−0.833134	+	0.553072i	$0.813456\pi$
$858$		0			0
$859$	−23061.8			−0.916016			−0.458008	−	0.888948i	$-0.651437\pi$
$859$	−23061.8			−0.916016			−0.458008	+	0.888948i	$0.651437\pi$
$860$	32764.0	−	97801.4i	1.29912	−	3.87791i
$861$		0			0
$862$			21749.9i			0.859402i
$863$			26058.3i			1.02785i	0.857835	+	0.513924i	$0.171809\pi$
$863$			26058.3i			1.02785i	−0.857835	+	0.513924i	$0.828191\pi$
$864$		0			0
$865$	−686.718	+	2049.87i	−0.0269932	+	0.0805753i
$866$	−72774.6			−2.85564
$867$		0			0
$868$		−	10906.5i		−	0.426486i
$869$	−14175.4			−0.553358
$870$		0			0
$871$	12930.6			0.503027
$872$		−	28395.3i		−	1.10274i
$873$		0			0
$874$	−38626.4			−1.49492
$875$	−8068.39	+	5532.11i	−0.311727	+	0.213736i
$876$		0			0
$877$		−	32589.3i		−	1.25480i	−0.778696	−	0.627402i	$-0.784118\pi$
$877$		−	32589.3i		−	1.25480i	0.778696	−	0.627402i	$-0.215882\pi$
$878$		−	87397.4i		−	3.35936i
$879$		0			0
$880$	148705.	+	49816.9i	5.69640	+	1.90833i
$881$	17236.6			0.659154			0.329577	−	0.944129i	$-0.393094\pi$
$881$	17236.6			0.659154			0.329577	+	0.944129i	$0.393094\pi$
$882$		0			0
$883$			12400.1i			0.472591i	0.971681	+	0.236296i	$0.0759334\pi$
$883$			12400.1i			0.472591i	−0.971681	+	0.236296i	$0.924067\pi$
$884$	−52381.7			−1.99297
$885$		0			0
$886$	90073.1			3.41542
$887$		−	43060.4i		−	1.63002i	−0.579447	−	0.815010i	$-0.696731\pi$
$887$		−	43060.4i		−	1.63002i	0.579447	−	0.815010i	$-0.303269\pi$
$888$		0			0
$889$	11769.0			0.444004
$890$	−404.639	+	1207.86i	−0.0152399	+	0.0454915i
$891$		0			0
$892$		−	118606.i		−	4.45203i
$893$		−	33594.8i		−	1.25891i
$894$		0			0
$895$	−37083.9	−	12423.3i	−1.38500	−	0.463984i
$896$	−30526.6			−1.13820
$897$		0			0
$898$		−	15198.1i		−	0.564775i
$899$	−11886.6			−0.440980
$900$		0			0
$901$	−20533.2			−0.759224
$902$		−	3477.58i		−	0.128371i
$903$		0			0
$904$	88531.4			3.25720
$905$	6261.35	+	2097.59i	0.229983	+	0.0770456i
$906$		0			0
$907$			13578.2i			0.497087i	0.968621	+	0.248544i	$0.0799520\pi$
$907$			13578.2i			0.497087i	−0.968621	+	0.248544i	$0.920048\pi$
$908$		−	11434.6i		−	0.417918i
$909$		0			0
$910$	8674.66	−	25894.1i	0.316002	−	0.943274i
$911$	−977.222			−0.0355399			−0.0177699	−	0.999842i	$-0.505657\pi$
$911$	−977.222			−0.0355399			−0.0177699	+	0.999842i	$0.505657\pi$
$912$		0			0
$913$			623.558i			0.0226032i
$914$	−43589.1			−1.57746
$915$		0			0
$916$	22492.1			0.811308
$917$			287.634i			0.0103582i
$918$		0			0
$919$	3695.14			0.132635			0.0663174	−	0.997799i	$-0.478875\pi$
$919$	3695.14			0.132635			0.0663174	+	0.997799i	$0.478875\pi$
$920$	−41504.3	−	13904.2i	−1.48734	−	0.498268i
$921$		0			0
$922$			60046.5i			2.14482i
$923$			33644.0i			1.19979i
$924$		0			0
$925$	3674.61	−	4868.89i	0.130617	−	0.173068i
$926$	2937.41			0.104243
$927$		0			0
$928$			110353.i			3.90357i
$929$	28459.7			1.00510			0.502548	−	0.864549i	$-0.332396\pi$
$929$	28459.7			1.00510			0.502548	+	0.864549i	$0.332396\pi$
$930$		0			0
$931$	−6604.19			−0.232485
$932$		−	40179.7i		−	1.41215i
$933$		0			0
$934$	−57961.9			−2.03059
$935$	8108.22	−	24203.2i	0.283601	−	0.846556i
$936$		0			0
$937$			48937.4i			1.70621i	0.521743	+	0.853103i	$0.325282\pi$
$937$			48937.4i			1.70621i	−0.521743	+	0.853103i	$0.674718\pi$
$938$		−	7694.03i		−	0.267824i
$939$		0			0
$940$	19174.9	−	57237.4i	0.665336	−	1.98604i
$941$	−32372.5			−1.12148			−0.560741	−	0.827991i	$-0.689484\pi$
$941$	−32372.5			−1.12148			−0.560741	+	0.827991i	$0.689484\pi$
$942$		0			0
$943$			555.511i			0.0191834i
$944$	−131579.			−4.53657
$945$		0			0
$946$	−140303.			−4.82203
$947$		−	20328.9i		−	0.697572i	−0.937202	−	0.348786i	$-0.886594\pi$
$947$		−	20328.9i		−	0.697572i	0.937202	−	0.348786i	$-0.113406\pi$
$948$		0			0
$949$	−8778.61			−0.300280
$950$	−55272.9	+	73237.1i	−1.88767	+	2.50118i
$951$		0			0
$952$			19657.0i			0.669208i
$953$			31183.7i			1.05996i	0.848011	+	0.529979i	$0.177800\pi$
$953$			31183.7i			1.05996i	−0.848011	+	0.529979i	$0.822200\pi$
$954$		0			0
$955$	14829.8	+	4968.06i	0.502492	+	0.168338i
$956$	88189.3			2.98352
$957$		0			0
$958$		−	70671.5i		−	2.38339i
$959$	8860.79			0.298363
$960$		0			0
$961$	−24617.0			−0.826324
$962$			17027.8i			0.570683i
$963$		0			0
$964$	124214.			4.15006
$965$	−1879.39	+	5610.02i	−0.0626940	+	0.187143i
$966$		0			0
$967$		−	36948.1i		−	1.22872i	−0.789027	−	0.614359i	$-0.789415\pi$
$967$		−	36948.1i		−	1.22872i	0.789027	−	0.614359i	$-0.210585\pi$
$968$		−	173176.i		−	5.75010i
$969$		0			0
$970$	−80184.4	−	26862.2i	−2.65419	−	0.889169i
$971$	22035.3			0.728266			0.364133	−	0.931347i	$-0.381365\pi$
$971$	22035.3			0.728266			0.364133	+	0.931347i	$0.381365\pi$
$972$		0			0
$973$		−	4032.95i		−	0.132878i
$974$	69383.0			2.28252
$975$		0			0
$976$	142515.			4.67397
$977$		−	13220.8i		−	0.432929i	−0.976290	−	0.216465i	$-0.930547\pi$
$977$		−	13220.8i		−	0.432929i	0.976290	−	0.216465i	$-0.0694526\pi$
$978$		0			0
$979$	1265.40			0.0413099
$980$	−11251.9	−	3769.47i	−0.366765	−	0.122869i
$981$		0			0
$982$			43303.8i			1.40721i
$983$		−	28323.8i		−	0.919012i	−0.888175	−	0.459506i	$-0.848027\pi$
$983$		−	28323.8i		−	0.919012i	0.888175	−	0.459506i	$-0.151973\pi$
$984$		0			0
$985$	−13642.1	+	40722.0i	−0.441294	+	1.31727i
$986$	33969.6			1.09717
$987$		0			0
$988$		−	187047.i		−	6.02302i
$989$	22412.1			0.720591
$990$		0			0
$991$	7605.32			0.243785			0.121893	−	0.992543i	$-0.461104\pi$
$991$	7605.32			0.243785			0.121893	+	0.992543i	$0.461104\pi$
$992$		−	48034.2i		−	1.53739i
$993$		0			0
$994$	20019.0			0.638798
$995$	−32361.2	−	10841.2i	−1.03108	−	0.345416i
$996$		0			0
$997$		−	14387.3i		−	0.457021i	−0.973541	−	0.228510i	$-0.926615\pi$
$997$		−	14387.3i		−	0.457021i	0.973541	−	0.228510i	$-0.0733855\pi$
$998$			75610.4i			2.39820i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.4.d.d.64.19	yes	20
3.2	odd	2	inner	315.4.d.d.64.2	yes	20
5.2	odd	4		1575.4.a.bu.1.1		10
5.3	odd	4		1575.4.a.bt.1.10		10
5.4	even	2	inner	315.4.d.d.64.1	✓	20
15.2	even	4		1575.4.a.bu.1.10		10
15.8	even	4		1575.4.a.bt.1.1		10
15.14	odd	2	inner	315.4.d.d.64.20	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.4.d.d.64.1	✓	20	5.4	even	2	inner
315.4.d.d.64.2	yes	20	3.2	odd	2	inner
315.4.d.d.64.19	yes	20	1.1	even	1	trivial
315.4.d.d.64.20	yes	20	15.14	odd	2	inner
1575.4.a.bt.1.1		10	15.8	even	4
1575.4.a.bt.1.10		10	5.3	odd	4
1575.4.a.bu.1.1		10	5.2	odd	4
1575.4.a.bu.1.10		10	15.2	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	315.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+5.44617i q^{2} -21.6607 q^{4} +(-10.6013 - 3.55149i) q^{5} -7.00000i q^{7} -74.3987i q^{8} +O(q^{10})\)	\(q+5.44617i q^{2} -21.6607 q^{4} +(-10.6013 - 3.55149i) q^{5} -7.00000i q^{7} -74.3987i q^{8} +(19.3420 - 57.7363i) q^{10} -60.4870 q^{11} +64.0698i q^{13} +38.1232 q^{14} +231.902 q^{16} -37.7444i q^{17} +134.779 q^{19} +(229.631 + 76.9279i) q^{20} -329.422i q^{22} +52.6223i q^{23} +(99.7739 + 75.3006i) q^{25} -348.935 q^{26} +151.625i q^{28} +165.252 q^{29} -71.9304 q^{31} +667.787i q^{32} +205.563 q^{34} +(-24.8604 + 74.2089i) q^{35} -48.7993i q^{37} +734.031i q^{38} +(-264.226 + 788.721i) q^{40} +10.5566 q^{41} -425.906i q^{43} +1310.19 q^{44} -286.590 q^{46} -249.258i q^{47} -49.0000 q^{49} +(-410.100 + 543.385i) q^{50} -1387.80i q^{52} -544.007i q^{53} +(641.239 + 214.819i) q^{55} -520.791 q^{56} +899.988i q^{58} -567.390 q^{59} +614.549 q^{61} -391.745i q^{62} -1781.66 q^{64} +(227.543 - 679.221i) q^{65} -201.820i q^{67} +817.573i q^{68} +(-404.154 - 135.394i) q^{70} +525.114 q^{71} +137.016i q^{73} +265.769 q^{74} -2919.42 q^{76} +423.409i q^{77} +234.355 q^{79} +(-2458.45 - 823.597i) q^{80} +57.4929i q^{82} -10.3089i q^{83} +(-134.049 + 400.139i) q^{85} +2319.56 q^{86} +4500.15i q^{88} -20.9202 q^{89} +448.488 q^{91} -1139.84i q^{92} +1357.50 q^{94} +(-1428.83 - 478.667i) q^{95} -1388.80i q^{97} -266.862i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 108 q^{4}+O(q^{10}) \) 20 * q - 108 * q^4	\( 20 q - 108 q^{4} + 112 q^{10} + 620 q^{16} - 72 q^{19} + 428 q^{25} - 48 q^{31} - 232 q^{34} - 16 q^{40} - 368 q^{46} - 980 q^{49} + 1904 q^{55} + 2048 q^{61} - 3180 q^{64} - 756 q^{70} - 8368 q^{76} + 1552 q^{79} - 2616 q^{85} + 1456 q^{91} + 8056 q^{94}+O(q^{100}) \) 20 * q - 108 * q^4 + 112 * q^10 + 620 * q^16 - 72 * q^19 + 428 * q^25 - 48 * q^31 - 232 * q^34 - 16 * q^40 - 368 * q^46 - 980 * q^49 + 1904 * q^55 + 2048 * q^61 - 3180 * q^64 - 756 * q^70 - 8368 * q^76 + 1552 * q^79 - 2616 * q^85 + 1456 * q^91 + 8056 * q^94

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