LMFDB - Embedded newform 315.4.d.d.64.13

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			64.13
Root			$-3.18874i$ of defining polynomial
Character	$\chi$	$=$	315.64
Dual form			315.4.d.d.64.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.18874i q^{2} +3.20940 q^{4} +(-10.6651 - 3.35486i) q^{5} -7.00000i q^{7} +24.5345i q^{8} +O(q^{10})$	\(q+2.18874i q^{2} +3.20940 q^{4} +(-10.6651 - 3.35486i) q^{5} -7.00000i q^{7} +24.5345i q^{8} +(7.34293 - 23.3432i) q^{10} +38.2891 q^{11} -30.6983i q^{13} +15.3212 q^{14} -28.0246 q^{16} +36.2007i q^{17} +63.7569 q^{19} +(-34.2286 - 10.7671i) q^{20} +83.8051i q^{22} +139.745i q^{23} +(102.490 + 71.5600i) q^{25} +67.1908 q^{26} -22.4658i q^{28} +17.9911 q^{29} +318.751 q^{31} +134.937i q^{32} -79.2340 q^{34} +(-23.4840 + 74.6559i) q^{35} -144.735i q^{37} +139.548i q^{38} +(82.3098 - 261.664i) q^{40} +96.6420 q^{41} +306.949i q^{43} +122.885 q^{44} -305.867 q^{46} +105.633i q^{47} -49.0000 q^{49} +(-156.627 + 224.324i) q^{50} -98.5231i q^{52} +188.729i q^{53} +(-408.358 - 128.455i) q^{55} +171.742 q^{56} +39.3779i q^{58} +405.962 q^{59} -217.786 q^{61} +697.663i q^{62} -519.540 q^{64} +(-102.989 + 327.401i) q^{65} +729.073i q^{67} +116.182i q^{68} +(-163.403 - 51.4005i) q^{70} -328.630 q^{71} -546.064i q^{73} +316.788 q^{74} +204.621 q^{76} -268.024i q^{77} +648.965 q^{79} +(298.886 + 94.0186i) q^{80} +211.525i q^{82} -429.083i q^{83} +(121.448 - 386.085i) q^{85} -671.833 q^{86} +939.405i q^{88} +1108.25 q^{89} -214.888 q^{91} +448.498i q^{92} -231.203 q^{94} +(-679.975 - 213.895i) q^{95} -916.936i q^{97} -107.248i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$			2.18874i			0.773838i	0.922114	+	0.386919i	$0.126461\pi$
$2$			2.18874i			0.773838i	−0.922114	+	0.386919i	$0.873539\pi$
$3$		0			0
$3$		0			0
$4$	3.20940			0.401175
$5$	−10.6651	−	3.35486i	−0.953918	−	0.300068i
$5$	−10.6651	−	3.35486i	−0.953918	−	0.300068i
$6$		0			0
$7$		−	7.00000i		−	0.377964i
$7$		−	7.00000i		−	0.377964i
$8$			24.5345i			1.08428i
$9$		0			0
$10$	7.34293	−	23.3432i	0.232204	−	0.738178i
$11$	38.2891			1.04951			0.524755	−	0.851253i	$-0.324157\pi$
$11$	38.2891			1.04951			0.524755	+	0.851253i	$0.324157\pi$
$12$		0			0
$13$		−	30.6983i		−	0.654937i	−0.944862	−	0.327468i	$-0.893804\pi$
$13$		−	30.6983i		−	0.654937i	0.944862	−	0.327468i	$-0.106196\pi$
$14$	15.3212			0.292483
$15$		0			0
$16$	−28.0246			−0.437884
$17$			36.2007i			0.516468i	0.966082	+	0.258234i	$0.0831405\pi$
$17$			36.2007i			0.516468i	−0.966082	+	0.258234i	$0.916860\pi$
$18$		0			0
$19$	63.7569			0.769833			0.384917	−	0.922951i	$-0.374230\pi$
$19$	63.7569			0.769833			0.384917	+	0.922951i	$0.374230\pi$
$20$	−34.2286	−	10.7671i	−0.382688	−	0.120380i
$21$		0			0
$22$			83.8051i			0.812150i
$23$			139.745i			1.26691i	0.773780	+	0.633454i	$0.218363\pi$
$23$			139.745i			1.26691i	−0.773780	+	0.633454i	$0.781637\pi$
$24$		0			0
$25$	102.490	+	71.5600i	0.819919	+	0.572480i
$26$	67.1908			0.506815
$27$		0			0
$28$		−	22.4658i		−	0.151630i
$29$	17.9911			0.115202			0.0576011	−	0.998340i	$-0.481655\pi$
$29$	17.9911			0.115202			0.0576011	+	0.998340i	$0.481655\pi$
$30$		0			0
$31$	318.751			1.84675			0.923376	−	0.383898i	$-0.125419\pi$
$31$	318.751			1.84675			0.923376	+	0.383898i	$0.125419\pi$
$32$			134.937i			0.745431i
$33$		0			0
$34$	−79.2340			−0.399662
$35$	−23.4840	+	74.6559i	−0.113415	+	0.360547i
$36$		0			0
$37$		−	144.735i		−	0.643090i	−0.946894	−	0.321545i	$-0.895798\pi$
$37$		−	144.735i		−	0.643090i	0.946894	−	0.321545i	$-0.104202\pi$
$38$			139.548i			0.595726i
$39$		0			0
$40$	82.3098	−	261.664i	0.325358	−	1.03432i
$41$	96.6420			0.368121			0.184060	−	0.982915i	$-0.441076\pi$
$41$	96.6420			0.368121			0.184060	+	0.982915i	$0.441076\pi$
$42$		0			0
$43$			306.949i			1.08859i	0.838895	+	0.544294i	$0.183202\pi$
$43$			306.949i			1.08859i	−0.838895	+	0.544294i	$0.816798\pi$
$44$	122.885			0.421037
$45$		0			0
$46$	−305.867			−0.980382
$47$			105.633i			0.327833i	0.986474	+	0.163916i	$0.0524127\pi$
$47$			105.633i			0.327833i	−0.986474	+	0.163916i	$0.947587\pi$
$48$		0			0
$49$	−49.0000			−0.142857
$50$	−156.627	+	224.324i	−0.443007	+	0.634484i
$51$		0			0
$52$		−	98.5231i		−	0.262744i
$53$			188.729i			0.489131i	0.969633	+	0.244566i	$0.0786454\pi$
$53$			188.729i			0.489131i	−0.969633	+	0.244566i	$0.921355\pi$
$54$		0			0
$55$	−408.358	−	128.455i	−1.00115	−	0.314924i
$56$	171.742			0.409820
$57$		0			0
$58$			39.3779i			0.0891478i
$59$	405.962			0.895793			0.447897	−	0.894085i	$-0.352173\pi$
$59$	405.962			0.895793			0.447897	+	0.894085i	$0.352173\pi$
$60$		0			0
$61$	−217.786			−0.457126			−0.228563	−	0.973529i	$-0.573403\pi$
$61$	−217.786			−0.457126			−0.228563	+	0.973529i	$0.573403\pi$
$62$			697.663i			1.42909i
$63$		0			0
$64$	−519.540			−1.01473
$65$	−102.989	+	327.401i	−0.196525	+	0.624756i
$66$		0			0
$67$			729.073i			1.32941i	0.747106	+	0.664705i	$0.231443\pi$
$67$			729.073i			1.32941i	−0.747106	+	0.664705i	$0.768557\pi$
$68$			116.182i			0.207194i
$69$		0			0
$70$	−163.403	−	51.4005i	−0.279005	−	0.0877648i
$71$	−328.630			−0.549312			−0.274656	−	0.961543i	$-0.588564\pi$
$71$	−328.630			−0.549312			−0.274656	+	0.961543i	$0.588564\pi$
$72$		0			0
$73$		−	546.064i		−	0.875506i	−0.899095	−	0.437753i	$-0.855774\pi$
$73$		−	546.064i		−	0.875506i	0.899095	−	0.437753i	$-0.144226\pi$
$74$	316.788			0.497647
$75$		0			0
$76$	204.621			0.308838
$77$		−	268.024i		−	0.396677i
$78$		0			0
$79$	648.965			0.924231			0.462115	−	0.886820i	$-0.347091\pi$
$79$	648.965			0.924231			0.462115	+	0.886820i	$0.347091\pi$
$80$	298.886	+	94.0186i	0.417706	+	0.131395i
$81$		0			0
$82$			211.525i			0.284866i
$83$		−	429.083i		−	0.567445i	−0.958906	−	0.283723i	$-0.908430\pi$
$83$		−	429.083i		−	0.567445i	0.958906	−	0.283723i	$-0.0915695\pi$
$84$		0			0
$85$	121.448	−	386.085i	0.154975	−	0.492668i
$86$	−671.833			−0.842390
$87$		0			0
$88$			939.405i			1.13796i
$89$	1108.25			1.31993			0.659967	−	0.751295i	$-0.270570\pi$
$89$	1108.25			1.31993			0.659967	+	0.751295i	$0.270570\pi$
$90$		0			0
$91$	−214.888			−0.247543
$92$			448.498i			0.508251i
$93$		0			0
$94$	−231.203			−0.253689
$95$	−679.975	−	213.895i	−0.734358	−	0.231002i
$96$		0			0
$97$		−	916.936i		−	0.959801i	−0.877323	−	0.479901i	$-0.840673\pi$
$97$		−	916.936i		−	0.959801i	0.877323	−	0.479901i	$-0.159327\pi$
$98$		−	107.248i		−	0.110548i
$99$		0			0
$100$	328.931	+	229.665i	0.328931	+	0.229665i
$101$	−1812.80			−1.78594			−0.892970	−	0.450115i	$-0.851383\pi$
$101$	−1812.80			−1.78594			−0.892970	+	0.450115i	$0.851383\pi$
$102$		0			0
$103$			1681.12i			1.60821i	0.594490	+	0.804103i	$0.297354\pi$
$103$			1681.12i			1.60821i	−0.594490	+	0.804103i	$0.702646\pi$
$104$	753.168			0.710136
$105$		0			0
$106$	−413.080			−0.378508
$107$		−	1371.58i		−	1.23921i	−0.784915	−	0.619604i	$-0.787293\pi$
$107$		−	1371.58i		−	1.23921i	0.784915	−	0.619604i	$-0.212707\pi$
$108$		0			0
$109$	−56.5255			−0.0496712			−0.0248356	−	0.999692i	$-0.507906\pi$
$109$	−56.5255			−0.0496712			−0.0248356	+	0.999692i	$0.507906\pi$
$110$	281.154	−	893.792i	0.243700	−	0.774725i
$111$		0			0
$112$			196.172i			0.165505i
$113$		−	250.725i		−	0.208728i	−0.994539	−	0.104364i	$-0.966719\pi$
$113$		−	250.725i		−	0.208728i	0.994539	−	0.104364i	$-0.0332806\pi$
$114$		0			0
$115$	468.826	−	1490.40i	0.380158	−	1.20853i
$116$	57.7406			0.0462162
$117$		0			0
$118$			888.548i			0.693199i
$119$	253.405			0.195206
$120$		0			0
$121$	135.057			0.101470
$122$		−	476.679i		−	0.353742i
$123$		0			0
$124$	1023.00			0.740870
$125$	−852.993	−	1107.04i	−0.610352	−	0.792130i
$126$		0			0
$127$		−	2133.33i		−	1.49057i	−0.666744	−	0.745287i	$-0.732313\pi$
$127$		−	2133.33i		−	1.49057i	0.666744	−	0.745287i	$-0.267687\pi$
$128$		−	57.6415i		−	0.0398034i
$129$		0			0
$130$	−716.598	−	225.416i	−0.483460	−	0.152079i
$131$	236.901			0.158001			0.0790005	−	0.996875i	$-0.474827\pi$
$131$	236.901			0.158001			0.0790005	+	0.996875i	$0.474827\pi$
$132$		0			0
$133$		−	446.298i		−	0.290970i
$134$	−1595.76			−1.02875
$135$		0			0
$136$	−888.165			−0.559997
$137$		−	2607.45i		−	1.62606i	−0.582225	−	0.813028i	$-0.697818\pi$
$137$		−	2607.45i		−	1.62606i	0.582225	−	0.813028i	$-0.302182\pi$
$138$		0			0
$139$	368.116			0.224627			0.112314	−	0.993673i	$-0.464174\pi$
$139$	368.116			0.224627			0.112314	+	0.993673i	$0.464174\pi$
$140$	−75.3696	+	239.600i	−0.0454992	+	0.144642i
$141$		0			0
$142$		−	719.286i		−	0.425078i
$143$		−	1175.41i		−	0.687363i
$144$		0			0
$145$	−191.877	−	60.3576i	−0.109893	−	0.0345685i
$146$	1195.19			0.677500
$147$		0			0
$148$		−	464.513i		−	0.257991i
$149$	1458.55			0.801937			0.400969	−	0.916092i	$-0.368674\pi$
$149$	1458.55			0.801937			0.400969	+	0.916092i	$0.368674\pi$
$150$		0			0
$151$	3516.33			1.89506			0.947532	−	0.319662i	$-0.103569\pi$
$151$	3516.33			1.89506			0.947532	+	0.319662i	$0.103569\pi$
$152$			1564.24i			0.834717i
$153$		0			0
$154$	586.636			0.306964
$155$	−3399.51	−	1069.36i	−1.76165	−	0.554151i
$156$		0			0
$157$		−	1339.74i		−	0.681039i	−0.940237	−	0.340519i	$-0.889397\pi$
$157$		−	1339.74i		−	0.681039i	0.940237	−	0.340519i	$-0.110603\pi$
$158$			1420.42i			0.715205i
$159$		0			0
$160$	452.696	−	1439.12i	0.223680	−	0.711080i
$161$	978.216			0.478846
$162$		0			0
$163$			3850.14i			1.85010i	0.379843	+	0.925051i	$0.375978\pi$
$163$			3850.14i			1.85010i	−0.379843	+	0.925051i	$0.624022\pi$
$164$	310.163			0.147681
$165$		0			0
$166$	939.153			0.439111
$167$		−	2326.18i		−	1.07788i	−0.842345	−	0.538939i	$-0.818825\pi$
$167$		−	2326.18i		−	1.07788i	0.842345	−	0.538939i	$-0.181175\pi$
$168$		0			0
$169$	1254.61			0.571058
$170$	845.040	+	265.819i	0.381245	+	0.119926i
$171$		0			0
$172$			985.121i			0.436714i
$173$		−	3954.46i		−	1.73787i	−0.494924	−	0.868936i	$-0.664804\pi$
$173$		−	3954.46i		−	1.73787i	0.494924	−	0.868936i	$-0.335196\pi$
$174$		0			0
$175$	500.920	−	717.429i	0.216377	−	0.309900i
$176$	−1073.04			−0.459564
$177$		0			0
$178$			2425.67i			1.02141i
$179$	−1784.17			−0.744999			−0.372500	−	0.928032i	$-0.621499\pi$
$179$	−1784.17			−0.744999			−0.372500	+	0.928032i	$0.621499\pi$
$180$		0			0
$181$	−1949.69			−0.800658			−0.400329	−	0.916372i	$-0.631104\pi$
$181$	−1949.69			−0.800658			−0.400329	+	0.916372i	$0.631104\pi$
$182$		−	470.335i		−	0.191558i
$183$		0			0
$184$	−3428.58			−1.37369
$185$	−485.566	+	1543.62i	−0.192971	+	0.613455i
$186$		0			0
$187$			1386.09i			0.542038i
$188$			339.018i			0.131518i
$189$		0			0
$190$	468.163	−	1488.29i	0.178758	−	0.568274i
$191$	1007.85			0.381810			0.190905	−	0.981609i	$-0.438858\pi$
$191$	1007.85			0.381810			0.190905	+	0.981609i	$0.438858\pi$
$192$		0			0
$193$			5188.46i			1.93510i	0.252686	+	0.967548i	$0.418686\pi$
$193$			5188.46i			1.93510i	−0.252686	+	0.967548i	$0.581314\pi$
$194$	2006.94			0.742731
$195$		0			0
$196$	−157.260			−0.0573107
$197$			3394.75i			1.22774i	0.789405	+	0.613872i	$0.210389\pi$
$197$			3394.75i			1.22774i	−0.789405	+	0.613872i	$0.789611\pi$
$198$		0			0
$199$	−2916.06			−1.03876			−0.519381	−	0.854543i	$-0.673837\pi$
$199$	−2916.06			−1.03876			−0.519381	+	0.854543i	$0.673837\pi$
$200$	−1755.69	+	2514.54i	−0.620730	+	0.889023i
$201$		0			0
$202$		−	3967.75i		−	1.38203i
$203$		−	125.938i		−	0.0435423i
$204$		0			0
$205$	−1030.70	−	324.220i	−0.351157	−	0.110461i
$206$	−3679.53			−1.24449
$207$		0			0
$208$			860.307i			0.286786i
$209$	2441.20			0.807948
$210$		0			0
$211$	−1129.68			−0.368581			−0.184290	−	0.982872i	$-0.558999\pi$
$211$	−1129.68			−0.368581			−0.184290	+	0.982872i	$0.558999\pi$
$212$			605.707i			0.196227i
$213$		0			0
$214$	3002.03			0.958946
$215$	1029.77	−	3273.65i	0.326650	−	1.03842i
$216$		0			0
$217$		−	2231.25i		−	0.698006i
$218$		−	123.720i		−	0.0384374i
$219$		0			0
$220$	−1310.58	−	412.262i	−0.401634	−	0.126340i
$221$	1111.30			0.338254
$222$		0			0
$223$		−	787.987i		−	0.236626i	−0.992976	−	0.118313i	$-0.962251\pi$
$223$		−	787.987i		−	0.236626i	0.992976	−	0.118313i	$-0.0377486\pi$
$224$	944.562			0.281746
$225$		0			0
$226$	548.773			0.161521
$227$		−	879.537i		−	0.257167i	−0.991699	−	0.128584i	$-0.958957\pi$
$227$		−	879.537i		−	0.257167i	0.991699	−	0.128584i	$-0.0410430\pi$
$228$		0			0
$229$	−4014.49			−1.15845			−0.579224	−	0.815168i	$-0.696644\pi$
$229$	−4014.49			−1.15845			−0.579224	+	0.815168i	$0.696644\pi$
$230$	3262.10	+	1026.14i	0.935204	+	0.294181i
$231$		0			0
$232$			441.403i			0.124912i
$233$		−	3092.44i		−	0.869497i	−0.900552	−	0.434748i	$-0.856837\pi$
$233$		−	3092.44i		−	0.869497i	0.900552	−	0.434748i	$-0.143163\pi$
$234$		0			0
$235$	354.384	−	1126.59i	0.0983721	−	0.312726i
$236$	1302.89			0.359370
$237$		0			0
$238$			554.638i			0.151058i
$239$	−6416.47			−1.73660			−0.868299	−	0.496042i	$-0.834786\pi$
$239$	−6416.47			−1.73660			−0.868299	+	0.496042i	$0.834786\pi$
$240$		0			0
$241$	−620.723			−0.165910			−0.0829550	−	0.996553i	$-0.526436\pi$
$241$	−620.723			−0.165910			−0.0829550	+	0.996553i	$0.526436\pi$
$242$			295.606i			0.0785217i
$243$		0			0
$244$	−698.963			−0.183387
$245$	522.591	+	164.388i	0.136274	+	0.0428668i
$246$		0			0
$247$		−	1957.23i		−	0.504192i
$248$			7820.39i			2.00240i
$249$		0			0
$250$	2423.02	−	1866.98i	0.612981	−	0.472314i
$251$	4672.65			1.17504			0.587520	−	0.809210i	$-0.300104\pi$
$251$	4672.65			1.17504			0.587520	+	0.809210i	$0.300104\pi$
$252$		0			0
$253$			5350.72i			1.32963i
$254$	4669.32			1.15346
$255$		0			0
$256$	−4030.16			−0.983925
$257$			6778.08i			1.64516i	0.568653	+	0.822578i	$0.307465\pi$
$257$			6778.08i			1.64516i	−0.568653	+	0.822578i	$0.692535\pi$
$258$		0			0
$259$	−1013.15			−0.243065
$260$	−330.531	+	1050.76i	−0.0788411	+	0.250636i
$261$		0			0
$262$			518.515i			0.122267i
$263$			2624.42i			0.615318i	0.951497	+	0.307659i	$0.0995456\pi$
$263$			2624.42i			0.615318i	−0.951497	+	0.307659i	$0.900454\pi$
$264$		0			0
$265$	633.160	−	2012.82i	0.146773	−	0.466591i
$266$	976.833			0.225163
$267$		0			0
$268$			2339.89i			0.533326i
$269$	−6906.55			−1.56543			−0.782713	−	0.622383i	$-0.786165\pi$
$269$	−6906.55			−1.56543			−0.782713	+	0.622383i	$0.786165\pi$
$270$		0			0
$271$	−3131.68			−0.701978			−0.350989	−	0.936380i	$-0.614155\pi$
$271$	−3131.68			−0.701978			−0.350989	+	0.936380i	$0.614155\pi$
$272$		−	1014.51i		−	0.226153i
$273$		0			0
$274$	5707.04			1.25830
$275$	3924.25	+	2739.97i	0.860512	+	0.600823i
$276$		0			0
$277$		−	2579.62i		−	0.559547i	−0.960066	−	0.279774i	$-0.909741\pi$
$277$		−	2579.62i		−	0.559547i	0.960066	−	0.279774i	$-0.0902594\pi$
$278$			805.712i			0.173825i
$279$		0			0
$280$	−1831.65	−	576.169i	−0.390935	−	0.122974i
$281$	−7511.13			−1.59458			−0.797289	−	0.603598i	$-0.793733\pi$
$281$	−7511.13			−1.59458			−0.797289	+	0.603598i	$0.793733\pi$
$282$		0			0
$283$			3545.52i			0.744732i	0.928086	+	0.372366i	$0.121453\pi$
$283$			3545.52i			0.744732i	−0.928086	+	0.372366i	$0.878547\pi$
$284$	−1054.70			−0.220370
$285$		0			0
$286$	2572.68			0.531907
$287$		−	676.494i		−	0.139137i
$288$		0			0
$289$	3602.51			0.733261
$290$	132.107	−	419.970i	0.0267504	−	0.0850397i
$291$		0			0
$292$		−	1752.54i		−	0.351231i
$293$			7116.99i			1.41904i	0.704685	+	0.709520i	$0.251088\pi$
$293$			7116.99i			1.41904i	−0.704685	+	0.709520i	$0.748912\pi$
$294$		0			0
$295$	−4329.64	−	1361.95i	−0.854513	−	0.268799i
$296$	3551.01			0.697291
$297$		0			0
$298$			3192.38i			0.620570i
$299$	4289.94			0.829745
$300$		0			0
$301$	2148.64			0.411447
$302$			7696.34i			1.46647i
$303$		0			0
$304$	−1786.76			−0.337098
$305$	2322.72	+	730.643i	0.436061	+	0.137169i
$306$		0			0
$307$			3726.77i			0.692827i	0.938082	+	0.346414i	$0.112601\pi$
$307$			3726.77i			0.692827i	−0.938082	+	0.346414i	$0.887399\pi$
$308$		−	860.195i		−	0.159137i
$309$		0			0
$310$	2340.56	−	7440.67i	0.428823	−	1.36323i
$311$	−1064.23			−0.194041			−0.0970204	−	0.995282i	$-0.530931\pi$
$311$	−1064.23			−0.194041			−0.0970204	+	0.995282i	$0.530931\pi$
$312$		0			0
$313$			3090.63i			0.558124i	0.960273	+	0.279062i	$0.0900235\pi$
$313$			3090.63i			0.558124i	−0.960273	+	0.279062i	$0.909976\pi$
$314$	2932.35			0.527014
$315$		0			0
$316$	2082.79			0.370778
$317$		−	2811.07i		−	0.498061i	−0.968496	−	0.249031i	$-0.919888\pi$
$317$		−	2811.07i		−	0.498061i	0.968496	−	0.249031i	$-0.0801120\pi$
$318$		0			0
$319$	688.863			0.120906
$320$	5540.96	+	1742.98i	0.967966	+	0.304487i
$321$		0			0
$322$			2141.07i			0.370549i
$323$			2308.04i			0.397594i
$324$		0			0
$325$	2196.77	−	3146.26i	0.374938	−	0.536995i
$326$	−8426.98			−1.43168
$327$		0			0
$328$			2371.06i			0.399147i
$329$	739.430			0.123909
$330$		0			0
$331$	4666.48			0.774904			0.387452	−	0.921890i	$-0.373355\pi$
$331$	4666.48			0.774904			0.387452	+	0.921890i	$0.373355\pi$
$332$		−	1377.10i		−	0.227645i
$333$		0			0
$334$	5091.42			0.834102
$335$	2445.94	−	7775.66i	0.398913	−	1.26815i
$336$		0			0
$337$		−	424.171i		−	0.0685641i	−0.999412	−	0.0342820i	$-0.989086\pi$
$337$		−	424.171i		−	0.0685641i	0.999412	−	0.0342820i	$-0.0109144\pi$
$338$			2746.03i			0.441906i
$339$		0			0
$340$	389.775	−	1239.10i	0.0621722	−	0.197646i
$341$	12204.7			1.93818
$342$		0			0
$343$			343.000i			0.0539949i
$344$	−7530.84			−1.18034
$345$		0			0
$346$	8655.30			1.34483
$347$			148.130i			0.0229166i	0.999934	+	0.0114583i	$0.00364737\pi$
$347$			148.130i			0.0229166i	−0.999934	+	0.0114583i	$0.996353\pi$
$348$		0			0
$349$	2551.08			0.391278			0.195639	−	0.980676i	$-0.437322\pi$
$349$	2551.08			0.391278			0.195639	+	0.980676i	$0.437322\pi$
$350$	1570.27	+	1096.39i	0.239812	+	0.167441i
$351$		0			0
$352$			5166.63i			0.782337i
$353$			103.057i			0.0155388i	0.999970	+	0.00776938i	$0.00247309\pi$
$353$			103.057i			0.0155388i	−0.999970	+	0.00776938i	$0.997527\pi$
$354$		0			0
$355$	3504.88	+	1102.51i	0.523998	+	0.164831i
$356$	3556.81			0.529524
$357$		0			0
$358$		−	3905.08i		−	0.576509i
$359$	−5888.89			−0.865749			−0.432874	−	0.901454i	$-0.642501\pi$
$359$	−5888.89			−0.865749			−0.432874	+	0.901454i	$0.642501\pi$
$360$		0			0
$361$	−2794.06			−0.407357
$362$		−	4267.37i		−	0.619580i
$363$		0			0
$364$	−689.662			−0.0993079
$365$	−1831.97	+	5823.84i	−0.262711	+	0.835161i
$366$		0			0
$367$		−	7923.29i		−	1.12696i	−0.826131	−	0.563478i	$-0.809463\pi$
$367$		−	7923.29i		−	1.12696i	0.826131	−	0.563478i	$-0.190537\pi$
$368$		−	3916.30i		−	0.554759i
$369$		0			0
$370$	−3378.59	−	1062.78i	−0.474715	−	0.149328i
$371$	1321.10			0.184874
$372$		0			0
$373$		−	2276.35i		−	0.315992i	−0.987440	−	0.157996i	$-0.949497\pi$
$373$		−	2276.35i		−	0.315992i	0.987440	−	0.157996i	$-0.0505034\pi$
$374$	−3033.80			−0.419449
$375$		0			0
$376$	−2591.65			−0.355463
$377$		−	552.296i		−	0.0754501i
$378$		0			0
$379$	−5754.55			−0.779925			−0.389963	−	0.920831i	$-0.627512\pi$
$379$	−5754.55			−0.779925			−0.389963	+	0.920831i	$0.627512\pi$
$380$	−2182.31	−	686.476i	−0.294606	−	0.0926723i
$381$		0			0
$382$			2205.93i			0.295459i
$383$		−	7706.66i		−	1.02818i	−0.857737	−	0.514089i	$-0.828130\pi$
$383$		−	7706.66i		−	1.02818i	0.857737	−	0.514089i	$-0.171870\pi$
$384$		0			0
$385$	−899.183	+	2858.51i	−0.119030	+	0.378398i
$386$	−11356.2			−1.49745
$387$		0			0
$388$		−	2942.81i		−	0.385048i
$389$	−7109.10			−0.926596			−0.463298	−	0.886203i	$-0.653334\pi$
$389$	−7109.10			−0.926596			−0.463298	+	0.886203i	$0.653334\pi$
$390$		0			0
$391$	−5058.87			−0.654317
$392$		−	1202.19i		−	0.154897i
$393$		0			0
$394$	−7430.23			−0.950076
$395$	−6921.29	−	2177.19i	−0.881640	−	0.277332i
$396$		0			0
$397$		−	10541.5i		−	1.33266i	−0.745659	−	0.666328i	$-0.767865\pi$
$397$		−	10541.5i		−	1.33266i	0.745659	−	0.666328i	$-0.232135\pi$
$398$		−	6382.50i		−	0.803834i
$399$		0			0
$400$	−2872.23	−	2005.44i	−0.359029	−	0.250680i
$401$	12793.4			1.59320			0.796599	−	0.604509i	$-0.206631\pi$
$401$	12793.4			1.59320			0.796599	+	0.604509i	$0.206631\pi$
$402$		0			0
$403$		−	9785.10i		−	1.20951i
$404$	−5817.99			−0.716474
$405$		0			0
$406$	275.645			0.0336947
$407$		−	5541.79i		−	0.674929i
$408$		0			0
$409$	4187.96			0.506311			0.253156	−	0.967426i	$-0.418531\pi$
$409$	4187.96			0.506311			0.253156	+	0.967426i	$0.418531\pi$
$410$	709.636	−	2255.94i	0.0854790	−	0.271739i
$411$		0			0
$412$			5395.37i			0.645172i
$413$		−	2841.74i		−	0.338578i
$414$		0			0
$415$	−1439.51	+	4576.22i	−0.170272	+	0.541296i
$416$	4142.35			0.488210
$417$		0			0
$418$			5343.15i			0.625221i
$419$	1165.65			0.135909			0.0679546	−	0.997688i	$-0.478353\pi$
$419$	1165.65			0.135909			0.0679546	+	0.997688i	$0.478353\pi$
$420$		0			0
$421$	−9835.71			−1.13863			−0.569315	−	0.822120i	$-0.692791\pi$
$421$	−9835.71			−1.13863			−0.569315	+	0.822120i	$0.692791\pi$
$422$		−	2472.59i		−	0.285222i
$423$		0			0
$424$	−4630.38			−0.530356
$425$	−2590.52	+	3710.20i	−0.295667	+	0.423461i
$426$		0			0
$427$			1524.50i			0.172777i
$428$		−	4401.93i		−	0.497139i
$429$		0			0
$430$	7165.18	+	2253.90i	0.803571	+	0.252774i
$431$	5897.82			0.659137			0.329568	−	0.944132i	$-0.393097\pi$
$431$	5897.82			0.659137			0.329568	+	0.944132i	$0.393097\pi$
$432$		0			0
$433$			15493.7i			1.71958i	0.510649	+	0.859790i	$0.329405\pi$
$433$			15493.7i			1.71958i	−0.510649	+	0.859790i	$0.670595\pi$
$434$	4883.64			0.540144
$435$		0			0
$436$	−181.413			−0.0199268
$437$			8909.72i			0.975308i
$438$		0			0
$439$	1649.59			0.179341			0.0896705	−	0.995971i	$-0.471419\pi$
$439$	1649.59			0.179341			0.0896705	+	0.995971i	$0.471419\pi$
$440$	3151.57	−	10018.9i	0.341467	−	1.08552i
$441$		0			0
$442$			2432.35i			0.261754i
$443$		−	14931.4i		−	1.60138i	−0.599078	−	0.800691i	$-0.704466\pi$
$443$		−	14931.4i		−	1.60138i	0.599078	−	0.800691i	$-0.295534\pi$
$444$		0			0
$445$	−11819.6	−	3718.02i	−1.25911	−	0.396070i
$446$	1724.70			0.183110
$447$		0			0
$448$			3636.78i			0.383531i
$449$	−11883.4			−1.24903			−0.624514	−	0.781014i	$-0.714703\pi$
$449$	−11883.4			−1.24903			−0.624514	+	0.781014i	$0.714703\pi$
$450$		0			0
$451$	3700.34			0.386346
$452$		−	804.676i		−	0.0837363i
$453$		0			0
$454$	1925.08			0.199006
$455$	2291.81	+	720.920i	0.236136	+	0.0742797i
$456$		0			0
$457$			4953.25i			0.507010i	0.967334	+	0.253505i	$0.0815834\pi$
$457$			4953.25i			0.507010i	−0.967334	+	0.253505i	$0.918417\pi$
$458$		−	8786.68i		−	0.896451i
$459$		0			0
$460$	1504.65	−	4783.29i	0.152510	−	0.484830i
$461$	793.410			0.0801579			0.0400789	−	0.999197i	$-0.487239\pi$
$461$	793.410			0.0801579			0.0400789	+	0.999197i	$0.487239\pi$
$462$		0			0
$463$			10841.0i			1.08817i	0.839029	+	0.544087i	$0.183124\pi$
$463$			10841.0i			1.08817i	−0.839029	+	0.544087i	$0.816876\pi$
$464$	−504.193			−0.0504452
$465$		0			0
$466$	6768.57			0.672849
$467$		−	16996.8i		−	1.68420i	−0.539323	−	0.842099i	$-0.681320\pi$
$467$		−	16996.8i		−	1.68420i	0.539323	−	0.842099i	$-0.318680\pi$
$468$		0			0
$469$	5103.51			0.502470
$470$	2465.81	+	775.655i	0.241999	+	0.0761241i
$471$		0			0
$472$			9960.09i			0.971293i
$473$			11752.8i			1.14248i
$474$		0			0
$475$	6534.43	+	4562.44i	0.631201	+	0.440714i
$476$	813.276			0.0783119
$477$		0			0
$478$		−	14044.0i		−	1.34385i
$479$	−12747.3			−1.21595			−0.607975	−	0.793956i	$-0.708018\pi$
$479$	−12747.3			−1.21595			−0.607975	+	0.793956i	$0.708018\pi$
$480$		0			0
$481$	−4443.13			−0.421183
$482$		−	1358.60i		−	0.128387i
$483$		0			0
$484$	433.452			0.0407074
$485$	−3076.19	+	9779.24i	−0.288006	+	0.915572i
$486$		0			0
$487$		−	18646.9i		−	1.73505i	−0.497389	−	0.867527i	$-0.665708\pi$
$487$		−	18646.9i		−	1.73505i	0.497389	−	0.867527i	$-0.334292\pi$
$488$		−	5343.28i		−	0.495654i
$489$		0			0
$490$	−359.804	+	1143.82i	−0.0331720	+	0.105454i
$491$	−14156.9			−1.30120			−0.650601	−	0.759420i	$-0.725483\pi$
$491$	−14156.9			−1.30120			−0.650601	+	0.759420i	$0.725483\pi$
$492$		0			0
$493$			651.289i			0.0594982i
$494$	4283.87			0.390163
$495$		0			0
$496$	−8932.85			−0.808663
$497$			2300.41i			0.207620i
$498$		0			0
$499$	−4907.11			−0.440225			−0.220112	−	0.975475i	$-0.570642\pi$
$499$	−4907.11			−0.440225			−0.220112	+	0.975475i	$0.570642\pi$
$500$	−2737.59	−	3552.92i	−0.244858	−	0.317783i
$501$		0			0
$502$			10227.2i			0.909290i
$503$			18597.4i			1.64854i	0.566193	+	0.824272i	$0.308416\pi$
$503$			18597.4i			1.64854i	−0.566193	+	0.824272i	$0.691584\pi$
$504$		0			0
$505$	19333.7	+	6081.68i	1.70364	+	0.535903i
$506$	−11711.4			−1.02892
$507$		0			0
$508$		−	6846.72i		−	0.597980i
$509$	17426.9			1.51755			0.758776	−	0.651351i	$-0.225797\pi$
$509$	17426.9			1.51755			0.758776	+	0.651351i	$0.225797\pi$
$510$		0			0
$511$	−3822.45			−0.330910
$512$		−	9282.12i		−	0.801202i
$513$		0			0
$514$	−14835.5			−1.27308
$515$	5639.91	−	17929.3i	0.482571	−	1.53410i
$516$		0			0
$517$			4044.59i			0.344064i
$518$		−	2217.52i		−	0.188093i
$519$		0			0
$520$	−8032.63	−	2526.77i	−0.677412	−	0.213089i
$521$	10927.3			0.918871			0.459435	−	0.888211i	$-0.348052\pi$
$521$	10927.3			0.918871			0.459435	+	0.888211i	$0.348052\pi$
$522$		0			0
$523$		−	18458.9i		−	1.54331i	−0.636041	−	0.771656i	$-0.719429\pi$
$523$		−	18458.9i		−	1.54331i	0.636041	−	0.771656i	$-0.280571\pi$
$524$	760.309			0.0633860
$525$		0			0
$526$	−5744.18			−0.476156
$527$			11539.0i			0.953787i
$528$		0			0
$529$	−7361.72			−0.605056
$530$	4405.55	+	1385.83i	0.361066	+	0.113578i
$531$		0			0
$532$		−	1432.35i		−	0.116730i
$533$		−	2966.75i		−	0.241096i
$534$		0			0
$535$	−4601.44	+	14628.0i	−0.371846	+	1.18210i
$536$	−17887.5			−1.44146
$537$		0			0
$538$		−	15116.7i		−	1.21139i
$539$	−1876.17			−0.149930
$540$		0			0
$541$	13142.8			1.04446			0.522228	−	0.852806i	$-0.325101\pi$
$541$	13142.8			1.04446			0.522228	+	0.852806i	$0.325101\pi$
$542$		−	6854.45i		−	0.543217i
$543$		0			0
$544$	−4884.82			−0.384991
$545$	602.851	+	189.635i	0.0473822	+	0.0149047i
$546$		0			0
$547$			6339.05i			0.495500i	0.968824	+	0.247750i	$0.0796911\pi$
$547$			6339.05i			0.495500i	−0.968824	+	0.247750i	$0.920309\pi$
$548$		−	8368.35i		−	0.652332i
$549$		0			0
$550$	−5997.10	+	8589.17i	−0.464940	+	0.665897i
$551$	1147.06			0.0886864
$552$		0			0
$553$		−	4542.75i		−	0.349326i
$554$	5646.14			0.432999
$555$		0			0
$556$	1181.43			0.0901148
$557$			4748.24i			0.361202i	0.983556	+	0.180601i	$0.0578042\pi$
$557$			4748.24i			0.361202i	−0.983556	+	0.180601i	$0.942196\pi$
$558$		0			0
$559$	9422.81			0.712956
$560$	658.130	−	2092.20i	0.0496626	−	0.157878i
$561$		0			0
$562$		−	16439.9i		−	1.23395i
$563$			13228.5i			0.990257i	0.868820	+	0.495129i	$0.164879\pi$
$563$			13228.5i			0.990257i	−0.868820	+	0.495129i	$0.835121\pi$
$564$		0			0
$565$	−841.147	+	2674.01i	−0.0626325	+	0.199109i
$566$	−7760.23			−0.576302
$567$		0			0
$568$		−	8062.76i		−	0.595609i
$569$	6807.04			0.501522			0.250761	−	0.968049i	$-0.419319\pi$
$569$	6807.04			0.501522			0.250761	+	0.968049i	$0.419319\pi$
$570$		0			0
$571$	18517.5			1.35715			0.678575	−	0.734531i	$-0.262598\pi$
$571$	18517.5			1.35715			0.678575	+	0.734531i	$0.262598\pi$
$572$		−	3772.36i		−	0.275752i
$573$		0			0
$574$	1480.67			0.107669
$575$	−10000.2	+	14322.5i	−0.725280	+	1.03876i
$576$		0			0
$577$			3712.63i			0.267867i	0.990990	+	0.133933i	$0.0427608\pi$
$577$			3712.63i			0.267867i	−0.990990	+	0.133933i	$0.957239\pi$
$578$			7884.98i			0.567425i
$579$		0			0
$580$	−615.810	−	193.712i	−0.0440864	−	0.0138680i
$581$	−3003.58			−0.214474
$582$		0			0
$583$			7226.28i			0.513348i
$584$	13397.4			0.949296
$585$		0			0
$586$	−15577.3			−1.09811
$587$		−	17691.3i		−	1.24395i	−0.783038	−	0.621974i	$-0.786331\pi$
$587$		−	17691.3i		−	1.24395i	0.783038	−	0.621974i	$-0.213669\pi$
$588$		0			0
$589$	20322.5			1.42169
$590$	2980.95	−	9476.48i	0.208007	−	0.661255i
$591$		0			0
$592$			4056.14i			0.281599i
$593$			12475.0i			0.863891i	0.901900	+	0.431945i	$0.142173\pi$
$593$			12475.0i			0.863891i	−0.901900	+	0.431945i	$0.857827\pi$
$594$		0			0
$595$	−2702.59	−	850.137i	−0.186211	−	0.0585752i
$596$	4681.05			0.321717
$597$		0			0
$598$			9389.58i			0.642088i
$599$	13912.8			0.949019			0.474510	−	0.880250i	$-0.342625\pi$
$599$	13912.8			0.949019			0.474510	+	0.880250i	$0.342625\pi$
$600$		0			0
$601$	14295.7			0.970274			0.485137	−	0.874438i	$-0.338770\pi$
$601$	14295.7			0.970274			0.485137	+	0.874438i	$0.338770\pi$
$602$			4702.83i			0.318394i
$603$		0			0
$604$	11285.3			0.760252
$605$	−1440.40	−	453.098i	−0.0967944	−	0.0304480i
$606$		0			0
$607$		−	16170.6i		−	1.08129i	−0.841251	−	0.540645i	$-0.818180\pi$
$607$		−	16170.6i		−	1.08129i	0.841251	−	0.540645i	$-0.181820\pi$
$608$			8603.19i			0.573858i
$609$		0			0
$610$	−1599.19	+	5083.84i	−0.106146	+	0.337440i
$611$	3242.75			0.214710
$612$		0			0
$613$		−	15820.7i		−	1.04240i	−0.853435	−	0.521199i	$-0.825485\pi$
$613$		−	15820.7i		−	1.04240i	0.853435	−	0.521199i	$-0.174515\pi$
$614$	−8156.95			−0.536136
$615$		0			0
$616$	6575.83			0.430110
$617$			23613.4i			1.54074i	0.637594	+	0.770372i	$0.279930\pi$
$617$			23613.4i			1.54074i	−0.637594	+	0.770372i	$0.720070\pi$
$618$		0			0
$619$	−7803.97			−0.506734			−0.253367	−	0.967370i	$-0.581538\pi$
$619$	−7803.97			−0.506734			−0.253367	+	0.967370i	$0.581538\pi$
$620$	−10910.4	−	3432.01i	−0.706729	−	0.222311i
$621$		0			0
$622$		−	2329.32i		−	0.150156i
$623$		−	7757.74i		−	0.498888i
$624$		0			0
$625$	5383.33	+	14668.3i	0.344533	+	0.938774i
$626$	−6764.61			−0.431898
$627$		0			0
$628$		−	4299.77i		−	0.273216i
$629$	5239.51			0.332135
$630$		0			0
$631$	11637.0			0.734170			0.367085	−	0.930187i	$-0.380356\pi$
$631$	11637.0			0.734170			0.367085	+	0.930187i	$0.380356\pi$
$632$			15922.0i			1.00213i
$633$		0			0
$634$	6152.71			0.385419
$635$	−7157.04	+	22752.3i	−0.447273	+	1.42188i
$636$		0			0
$637$			1504.22i			0.0935624i
$638$			1507.75i			0.0935615i
$639$		0			0
$640$	−193.379	+	614.754i	−0.0119437	+	0.0379692i
$641$	12956.9			0.798388			0.399194	−	0.916867i	$-0.369290\pi$
$641$	12956.9			0.798388			0.399194	+	0.916867i	$0.369290\pi$
$642$		0			0
$643$		−	18828.6i		−	1.15478i	−0.816467	−	0.577392i	$-0.804071\pi$
$643$		−	18828.6i		−	1.15478i	0.816467	−	0.577392i	$-0.195929\pi$
$644$	3139.48			0.192101
$645$		0			0
$646$	−5051.71			−0.307673
$647$			9779.98i			0.594267i	0.954836	+	0.297133i	$0.0960306\pi$
$647$			9779.98i			0.594267i	−0.954836	+	0.297133i	$0.903969\pi$
$648$		0			0
$649$	15543.9			0.940144
$650$	6886.37	+	4808.17i	0.415547	+	0.290142i
$651$		0			0
$652$			12356.6i			0.742214i
$653$		−	113.220i		−	0.00678508i	−0.999994	−	0.00339254i	$-0.998920\pi$
$653$		−	113.220i		−	0.00678508i	0.999994	−	0.00339254i	$-0.00107988\pi$
$654$		0			0
$655$	−2526.58	−	794.769i	−0.150720	−	0.0474110i
$656$	−2708.35			−0.161194
$657$		0			0
$658$			1618.42i			0.0958856i
$659$	21374.6			1.26349			0.631743	−	0.775178i	$-0.282340\pi$
$659$	21374.6			1.26349			0.631743	+	0.775178i	$0.282340\pi$
$660$		0			0
$661$	−2246.72			−0.132204			−0.0661022	−	0.997813i	$-0.521056\pi$
$661$	−2246.72			−0.132204			−0.0661022	+	0.997813i	$0.521056\pi$
$662$			10213.7i			0.599650i
$663$		0			0
$664$	10527.3			0.615271
$665$	−1497.27	+	4759.83i	−0.0873106	+	0.277561i
$666$		0			0
$667$			2514.17i			0.145951i
$668$		−	7465.65i		−	0.432417i
$669$		0			0
$670$	17018.9	+	5353.54i	0.981341	+	0.308694i
$671$	−8338.85			−0.479758
$672$		0			0
$673$		−	15724.1i		−	0.900624i	−0.892871	−	0.450312i	$-0.851313\pi$
$673$		−	15724.1i		−	0.900624i	0.892871	−	0.450312i	$-0.148687\pi$
$674$	928.403			0.0530575
$675$		0			0
$676$	4026.55			0.229094
$677$			14211.8i			0.806799i	0.915024	+	0.403400i	$0.132172\pi$
$677$			14211.8i			0.806799i	−0.915024	+	0.403400i	$0.867828\pi$
$678$		0			0
$679$	−6418.55			−0.362771
$680$	9472.39	+	2979.67i	0.534191	+	0.168037i
$681$		0			0
$682$			26712.9i			1.49984i
$683$		−	1053.56i		−	0.0590239i	−0.999564	−	0.0295119i	$-0.990605\pi$
$683$		−	1053.56i		−	0.0590239i	0.999564	−	0.0295119i	$-0.00939531\pi$
$684$		0			0
$685$	−8747.63	+	27808.8i	−0.487927	+	1.55112i
$686$	−750.739			−0.0417833
$687$		0			0
$688$		−	8602.11i		−	0.476675i
$689$	5793.67			0.320350
$690$		0			0
$691$	−36212.8			−1.99363			−0.996815	−	0.0797508i	$-0.974588\pi$
$691$	−36212.8			−1.99363			−0.996815	+	0.0797508i	$0.974588\pi$
$692$		−	12691.4i		−	0.697191i
$693$		0			0
$694$	−324.219			−0.0177337
$695$	−3926.00	−	1234.98i	−0.214276	−	0.0674034i
$696$		0			0
$697$			3498.50i			0.190122i
$698$			5583.66i			0.302786i
$699$		0			0
$700$	1607.65	−	2302.51i	0.0868050	−	0.124324i
$701$	27675.0			1.49111			0.745556	−	0.666443i	$-0.232184\pi$
$701$	27675.0			1.49111			0.745556	+	0.666443i	$0.232184\pi$
$702$		0			0
$703$		−	9227.87i		−	0.495072i
$704$	−19892.7			−1.06497
$705$		0			0
$706$	−225.566			−0.0120245
$707$			12689.6i			0.675022i
$708$		0			0
$709$	−28861.6			−1.52880			−0.764401	−	0.644741i	$-0.776965\pi$
$709$	−28861.6			−1.52880			−0.764401	+	0.644741i	$0.776965\pi$
$710$	−2413.10	+	7671.28i	−0.127552	+	0.405490i
$711$		0			0
$712$			27190.3i			1.43118i
$713$			44543.8i			2.33966i
$714$		0			0
$715$	−3943.34	+	12535.9i	−0.206255	+	0.655687i
$716$	−5726.10			−0.298875
$717$		0			0
$718$		−	12889.3i		−	0.669949i
$719$	13406.8			0.695395			0.347698	−	0.937607i	$-0.386964\pi$
$719$	13406.8			0.695395			0.347698	+	0.937607i	$0.386964\pi$
$720$		0			0
$721$	11767.8			0.607845
$722$		−	6115.48i		−	0.315228i
$723$		0			0
$724$	−6257.32			−0.321204
$725$	1843.90	+	1287.44i	0.0944564	+	0.0659509i
$726$		0			0
$727$		−	2052.02i		−	0.104684i	−0.998629	−	0.0523419i	$-0.983331\pi$
$727$		−	2052.02i		−	0.104684i	0.998629	−	0.0523419i	$-0.0166686\pi$
$728$		−	5272.18i		−	0.268406i
$729$		0			0
$730$	−12746.9	−	4009.71i	−0.646279	−	0.203296i
$731$	−11111.7			−0.562220
$732$		0			0
$733$		−	22230.3i		−	1.12019i	−0.828430	−	0.560093i	$-0.810765\pi$
$733$		−	22230.3i		−	1.12019i	0.828430	−	0.560093i	$-0.189235\pi$
$734$	17342.1			0.872081
$735$		0			0
$736$	−18856.9			−0.944392
$737$			27915.6i			1.39523i
$738$		0			0
$739$	1568.17			0.0780597			0.0390299	−	0.999238i	$-0.487573\pi$
$739$	1568.17			0.0780597			0.0390299	+	0.999238i	$0.487573\pi$
$740$	−1558.38	+	4954.09i	−0.0774149	+	0.246103i
$741$		0			0
$742$			2891.56i			0.143063i
$743$		−	36436.6i		−	1.79910i	−0.436820	−	0.899549i	$-0.643895\pi$
$743$		−	36436.6i		−	1.79910i	0.436820	−	0.899549i	$-0.356105\pi$
$744$		0			0
$745$	−15555.6	−	4893.22i	−0.764982	−	0.240636i
$746$	4982.36			0.244527
$747$		0			0
$748$			4448.52i			0.217452i
$749$	−9601.03			−0.468376
$750$		0			0
$751$	−6257.06			−0.304026			−0.152013	−	0.988379i	$-0.548575\pi$
$751$	−6257.06			−0.304026			−0.152013	+	0.988379i	$0.548575\pi$
$752$		−	2960.32i		−	0.143553i
$753$		0			0
$754$	1208.84			0.0583862
$755$	−37502.1	−	11796.8i	−1.80773	−	0.568648i
$756$		0			0
$757$		−	34671.7i		−	1.66468i	−0.554264	−	0.832341i	$-0.687000\pi$
$757$		−	34671.7i		−	1.66468i	0.554264	−	0.832341i	$-0.313000\pi$
$758$		−	12595.2i		−	0.603536i
$759$		0			0
$760$	5247.82	−	16682.9i	0.250472	−	0.796251i
$761$	−20451.1			−0.974183			−0.487091	−	0.873351i	$-0.661942\pi$
$761$	−20451.1			−0.974183			−0.487091	+	0.873351i	$0.661942\pi$
$762$		0			0
$763$			395.678i			0.0187739i
$764$	3234.60			0.153172
$765$		0			0
$766$	16867.9			0.795643
$767$		−	12462.4i		−	0.586688i
$768$		0			0
$769$	−25022.7			−1.17340			−0.586698	−	0.809806i	$-0.699572\pi$
$769$	−25022.7			−1.17340			−0.586698	+	0.809806i	$0.699572\pi$
$770$	−6256.54	−	1968.08i	−0.292818	−	0.0921100i
$771$		0			0
$772$			16651.8i			0.776312i
$773$			21135.6i			0.983435i	0.870755	+	0.491717i	$0.163631\pi$
$773$			21135.6i			0.983435i	−0.870755	+	0.491717i	$0.836369\pi$
$774$		0			0
$775$	32668.7	+	22809.8i	1.51419	+	1.05723i
$776$	22496.6			1.04070
$777$		0			0
$778$		−	15560.0i		−	0.717035i
$779$	6161.59			0.283392
$780$		0			0
$781$	−12582.9			−0.576508
$782$		−	11072.6i		−	0.506335i
$783$		0			0
$784$	1373.20			0.0625549
$785$	−4494.65	+	14288.5i	−0.204358	+	0.649655i
$786$		0			0
$787$		−	19827.1i		−	0.898044i	−0.893521	−	0.449022i	$-0.851773\pi$
$787$		−	19827.1i		−	0.898044i	0.893521	−	0.449022i	$-0.148227\pi$
$788$			10895.1i			0.492540i
$789$		0			0
$790$	4765.30	−	15148.9i	0.214610	−	0.682247i
$791$	−1755.08			−0.0788916
$792$		0			0
$793$			6685.67i			0.299389i
$794$	23072.7			1.03126
$795$		0			0
$796$	−9358.78			−0.416725
$797$		−	9818.26i		−	0.436362i	−0.975908	−	0.218181i	$-0.929988\pi$
$797$		−	9818.26i		−	0.436362i	0.975908	−	0.218181i	$-0.0700124\pi$
$798$		0			0
$799$	−3823.98			−0.169315
$800$	−9656.12	+	13829.7i	−0.426744	+	0.611193i
$801$		0			0
$802$			28001.5i			1.23288i
$803$		−	20908.3i		−	0.918852i
$804$		0			0
$805$	−10432.8	−	3281.78i	−0.456780	−	0.143686i
$806$	21417.1			0.935961
$807$		0			0
$808$		−	44476.1i		−	1.93646i
$809$	33454.5			1.45389			0.726946	−	0.686695i	$-0.240939\pi$
$809$	33454.5			1.45389			0.726946	+	0.686695i	$0.240939\pi$
$810$		0			0
$811$	−8664.82			−0.375170			−0.187585	−	0.982248i	$-0.560066\pi$
$811$	−8664.82			−0.375170			−0.187585	+	0.982248i	$0.560066\pi$
$812$		−	404.184i		−	0.0174681i
$813$		0			0
$814$	12129.6			0.522286
$815$	12916.7	−	41062.3i	0.555156	−	1.76485i
$816$		0			0
$817$			19570.1i			0.838031i
$818$			9166.38i			0.391803i
$819$		0			0
$820$	−3307.92	−	1040.55i	−0.140875	−	0.0443142i
$821$	−1035.20			−0.0440058			−0.0220029	−	0.999758i	$-0.507004\pi$
$821$	−1035.20			−0.0440058			−0.0220029	+	0.999758i	$0.507004\pi$
$822$		0			0
$823$			24721.7i			1.04708i	0.852002	+	0.523538i	$0.175388\pi$
$823$			24721.7i			1.04708i	−0.852002	+	0.523538i	$0.824612\pi$
$824$	−41245.3			−1.74375
$825$		0			0
$826$	6219.84			0.262005
$827$		−	19994.5i		−	0.840722i	−0.907357	−	0.420361i	$-0.861904\pi$
$827$		−	19994.5i		−	0.840722i	0.907357	−	0.420361i	$-0.138096\pi$
$828$		0			0
$829$	6458.16			0.270568			0.135284	−	0.990807i	$-0.456805\pi$
$829$	6458.16			0.270568			0.135284	+	0.990807i	$0.456805\pi$
$830$	−10016.2	−	3150.73i	−0.418876	−	0.131763i
$831$		0			0
$832$			15949.0i			0.664582i
$833$		−	1773.83i		−	0.0737811i
$834$		0			0
$835$	−7804.02	+	24809.0i	−0.323436	+	1.02821i
$836$	7834.77			0.324128
$837$		0			0
$838$			2551.32i			0.105172i
$839$	−47308.6			−1.94669			−0.973346	−	0.229341i	$-0.926343\pi$
$839$	−47308.6			−1.94669			−0.973346	+	0.229341i	$0.926343\pi$
$840$		0			0
$841$	−24065.3			−0.986728
$842$		−	21527.9i		−	0.881115i
$843$		0			0
$844$	−3625.60			−0.147865
$845$	−13380.6	−	4209.05i	−0.544742	−	0.171356i
$846$		0			0
$847$		−	945.400i		−	0.0383522i
$848$		−	5289.06i		−	0.214183i
$849$		0			0
$850$	−8120.68	−	5669.99i	−0.327691	−	0.228799i
$851$	20226.1			0.814736
$852$		0			0
$853$			634.531i			0.0254700i	0.999919	+	0.0127350i	$0.00405379\pi$
$853$			634.531i			0.0254700i	−0.999919	+	0.0127350i	$0.995946\pi$
$854$	−3336.75			−0.133702
$855$		0			0
$856$	33650.9			1.34365
$857$		−	8778.84i		−	0.349918i	−0.984576	−	0.174959i	$-0.944021\pi$
$857$		−	8778.84i		−	0.349918i	0.984576	−	0.174959i	$-0.0559792\pi$
$858$		0			0
$859$	15693.4			0.623344			0.311672	−	0.950190i	$-0.399111\pi$
$859$	15693.4			0.623344			0.311672	+	0.950190i	$0.399111\pi$
$860$	3304.94	−	10506.4i	0.131044	−	0.416589i
$861$		0			0
$862$			12908.8i			0.510065i
$863$		−	36190.2i		−	1.42749i	−0.700404	−	0.713747i	$-0.746997\pi$
$863$		−	36190.2i		−	1.42749i	0.700404	−	0.713747i	$-0.253003\pi$
$864$		0			0
$865$	−13266.7	+	42174.8i	−0.521480	+	1.65779i
$866$	−33911.7			−1.33068
$867$		0			0
$868$		−	7160.98i		−	0.280022i
$869$	24848.3			0.969989
$870$		0			0
$871$	22381.3			0.870680
$872$		−	1386.82i		−	0.0538576i
$873$		0			0
$874$	−19501.1			−0.754731
$875$	−7749.25	+	5970.95i	−0.299397	+	0.230691i
$876$		0			0
$877$		−	1674.50i		−	0.0644740i	−0.999480	−	0.0322370i	$-0.989737\pi$
$877$		−	1674.50i		−	0.0644740i	0.999480	−	0.0322370i	$-0.0102631\pi$
$878$			3610.53i			0.138781i
$879$		0			0
$880$	11444.1	+	3599.89i	0.438386	+	0.137900i
$881$	−31999.1			−1.22370			−0.611848	−	0.790975i	$-0.709574\pi$
$881$	−31999.1			−1.22370			−0.611848	+	0.790975i	$0.709574\pi$
$882$		0			0
$883$		−	11627.5i		−	0.443145i	−0.975144	−	0.221572i	$-0.928881\pi$
$883$		−	11627.5i		−	0.443145i	0.975144	−	0.221572i	$-0.0711189\pi$
$884$	3566.60			0.135699
$885$		0			0
$886$	32681.0			1.23921
$887$		−	17494.7i		−	0.662250i	−0.943587	−	0.331125i	$-0.892572\pi$
$887$		−	17494.7i		−	0.662250i	0.943587	−	0.331125i	$-0.107428\pi$
$888$		0			0
$889$	−14933.3			−0.563384
$890$	8137.79	−	25870.1i	0.306494	−	0.974346i
$891$		0			0
$892$		−	2528.96i		−	0.0949282i
$893$			6734.83i			0.252377i
$894$		0			0
$895$	19028.4	+	5985.63i	0.710668	+	0.223550i
$896$	−403.491			−0.0150443
$897$		0			0
$898$		−	26009.8i		−	0.966545i
$899$	5734.67			0.212750
$900$		0			0
$901$	−6832.12			−0.252620
$902$			8099.09i			0.298969i
$903$		0			0
$904$	6151.41			0.226320
$905$	20793.7	+	6540.93i	0.763762	+	0.240252i
$906$		0			0
$907$			14261.9i			0.522116i	0.965323	+	0.261058i	$0.0840713\pi$
$907$			14261.9i			0.522116i	−0.965323	+	0.261058i	$0.915929\pi$
$908$		−	2822.78i		−	0.103169i
$909$		0			0
$910$	−1577.91	+	5016.19i	−0.0574804	+	0.182731i
$911$	15247.4			0.554521			0.277260	−	0.960795i	$-0.410574\pi$
$911$	15247.4			0.554521			0.277260	+	0.960795i	$0.410574\pi$
$912$		0			0
$913$		−	16429.2i		−	0.595539i
$914$	−10841.4			−0.392343
$915$		0			0
$916$	−12884.1			−0.464740
$917$		−	1658.31i		−	0.0597187i
$918$		0			0
$919$	−2406.86			−0.0863928			−0.0431964	−	0.999067i	$-0.513754\pi$
$919$	−2406.86			−0.0863928			−0.0431964	+	0.999067i	$0.513754\pi$
$920$	36566.2	+	11502.4i	1.31038	+	0.412199i
$921$		0			0
$922$			1736.57i			0.0620292i
$923$			10088.4i			0.359765i
$924$		0			0
$925$	10357.3	−	14833.9i	0.368156	−	0.527281i
$926$	−23728.2			−0.842071
$927$		0			0
$928$			2427.67i			0.0858752i
$929$	−11356.2			−0.401061			−0.200530	−	0.979687i	$-0.564267\pi$
$929$	−11356.2			−0.401061			−0.200530	+	0.979687i	$0.564267\pi$
$930$		0			0
$931$	−3124.09			−0.109976
$932$		−	9924.88i		−	0.348820i
$933$		0			0
$934$	37201.8			1.30330
$935$	4650.14	−	14782.8i	0.162648	−	0.517059i
$936$		0			0
$937$		−	40140.9i		−	1.39951i	−0.714381	−	0.699757i	$-0.753292\pi$
$937$		−	40140.9i		−	1.39951i	0.714381	−	0.699757i	$-0.246708\pi$
$938$			11170.3i			0.388830i
$939$		0			0
$940$	1137.36	−	3615.67i	0.0394644	−	0.125458i
$941$	−44475.8			−1.54078			−0.770388	−	0.637576i	$-0.779937\pi$
$941$	−44475.8			−1.54078			−0.770388	+	0.637576i	$0.779937\pi$
$942$		0			0
$943$			13505.3i			0.466375i
$944$	−11376.9			−0.392254
$945$		0			0
$946$	−25723.9			−0.884097
$947$		−	7280.66i		−	0.249831i	−0.992167	−	0.124915i	$-0.960134\pi$
$947$		−	7280.66i		−	0.249831i	0.992167	−	0.124915i	$-0.0398660\pi$
$948$		0			0
$949$	−16763.2			−0.573401
$950$	−9986.03	+	14302.2i	−0.341042	+	0.488447i
$951$		0			0
$952$			6217.16i			0.211659i
$953$		−	22114.1i		−	0.751674i	−0.926686	−	0.375837i	$-0.877355\pi$
$953$		−	22114.1i		−	0.751674i	0.926686	−	0.375837i	$-0.122645\pi$
$954$		0			0
$955$	−10748.9	−	3381.21i	−0.364215	−	0.114569i
$956$	−20593.0			−0.696679
$957$		0			0
$958$		−	27900.6i		−	0.940948i
$959$	−18252.2			−0.614591
$960$		0			0
$961$	71810.9			2.41049
$962$		−	9724.87i		−	0.325928i
$963$		0			0
$964$	−1992.15			−0.0665589
$965$	17406.6	−	55335.6i	0.580660	−	1.84592i
$966$		0			0
$967$			34353.8i			1.14245i	0.820795	+	0.571223i	$0.193531\pi$
$967$			34353.8i			1.14245i	−0.820795	+	0.571223i	$0.806469\pi$
$968$			3313.56i			0.110023i
$969$		0			0
$970$	−21404.3	−	6733.00i	−0.708504	−	0.222870i
$971$	7495.37			0.247722			0.123861	−	0.992300i	$-0.460472\pi$
$971$	7495.37			0.247722			0.123861	+	0.992300i	$0.460472\pi$
$972$		0			0
$973$		−	2576.81i		−	0.0849011i
$974$	40813.3			1.34265
$975$		0			0
$976$	6103.37			0.200168
$977$		−	31269.3i		−	1.02394i	−0.859002	−	0.511972i	$-0.828915\pi$
$977$		−	31269.3i		−	1.02394i	0.859002	−	0.511972i	$-0.171085\pi$
$978$		0			0
$979$	42433.9			1.38528
$980$	1677.20	+	527.587i	0.0546697	+	0.0171971i
$981$		0			0
$982$		−	30985.8i		−	1.00692i
$983$		−	25055.2i		−	0.812958i	−0.913660	−	0.406479i	$-0.866756\pi$
$983$		−	25055.2i		−	0.812958i	0.913660	−	0.406479i	$-0.133244\pi$
$984$		0			0
$985$	11388.9	−	36205.4i	0.368407	−	1.17117i
$986$	−1425.51			−0.0460419
$987$		0			0
$988$		−	6281.53i		−	0.202269i
$989$	−42894.6			−1.37914
$990$		0			0
$991$	35098.7			1.12507			0.562536	−	0.826773i	$-0.309826\pi$
$991$	35098.7			1.12507			0.562536	+	0.826773i	$0.309826\pi$
$992$			43011.4i			1.37663i
$993$		0			0
$994$	−5035.00			−0.160665
$995$	31100.1	+	9782.96i	0.990894	+	0.311699i
$996$		0			0
$997$		−	30175.1i		−	0.958530i	−0.877670	−	0.479265i	$-0.840903\pi$
$997$		−	30175.1i		−	0.958530i	0.877670	−	0.479265i	$-0.159097\pi$
$998$		−	10740.4i		−	0.340663i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.4.d.d.64.13	yes	20
3.2	odd	2	inner	315.4.d.d.64.8	yes	20
5.2	odd	4		1575.4.a.bu.1.4		10
5.3	odd	4		1575.4.a.bt.1.7		10
5.4	even	2	inner	315.4.d.d.64.7	✓	20
15.2	even	4		1575.4.a.bu.1.7		10
15.8	even	4		1575.4.a.bt.1.4		10
15.14	odd	2	inner	315.4.d.d.64.14	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.4.d.d.64.7	✓	20	5.4	even	2	inner
315.4.d.d.64.8	yes	20	3.2	odd	2	inner
315.4.d.d.64.13	yes	20	1.1	even	1	trivial
315.4.d.d.64.14	yes	20	15.14	odd	2	inner
1575.4.a.bt.1.4		10	15.8	even	4
1575.4.a.bt.1.7		10	5.3	odd	4
1575.4.a.bu.1.4		10	5.2	odd	4
1575.4.a.bu.1.7		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)

\(f(q)\)	\(=\)	\(q+2.18874i q^{2} +3.20940 q^{4} +(-10.6651 - 3.35486i) q^{5} -7.00000i q^{7} +24.5345i q^{8} +O(q^{10})\)	\(q+2.18874i q^{2} +3.20940 q^{4} +(-10.6651 - 3.35486i) q^{5} -7.00000i q^{7} +24.5345i q^{8} +(7.34293 - 23.3432i) q^{10} +38.2891 q^{11} -30.6983i q^{13} +15.3212 q^{14} -28.0246 q^{16} +36.2007i q^{17} +63.7569 q^{19} +(-34.2286 - 10.7671i) q^{20} +83.8051i q^{22} +139.745i q^{23} +(102.490 + 71.5600i) q^{25} +67.1908 q^{26} -22.4658i q^{28} +17.9911 q^{29} +318.751 q^{31} +134.937i q^{32} -79.2340 q^{34} +(-23.4840 + 74.6559i) q^{35} -144.735i q^{37} +139.548i q^{38} +(82.3098 - 261.664i) q^{40} +96.6420 q^{41} +306.949i q^{43} +122.885 q^{44} -305.867 q^{46} +105.633i q^{47} -49.0000 q^{49} +(-156.627 + 224.324i) q^{50} -98.5231i q^{52} +188.729i q^{53} +(-408.358 - 128.455i) q^{55} +171.742 q^{56} +39.3779i q^{58} +405.962 q^{59} -217.786 q^{61} +697.663i q^{62} -519.540 q^{64} +(-102.989 + 327.401i) q^{65} +729.073i q^{67} +116.182i q^{68} +(-163.403 - 51.4005i) q^{70} -328.630 q^{71} -546.064i q^{73} +316.788 q^{74} +204.621 q^{76} -268.024i q^{77} +648.965 q^{79} +(298.886 + 94.0186i) q^{80} +211.525i q^{82} -429.083i q^{83} +(121.448 - 386.085i) q^{85} -671.833 q^{86} +939.405i q^{88} +1108.25 q^{89} -214.888 q^{91} +448.498i q^{92} -231.203 q^{94} +(-679.975 - 213.895i) q^{95} -916.936i q^{97} -107.248i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 108 q^{4}+O(q^{10}) \) 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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