LMFDB - Embedded newform 60.7.g.a.41.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [60,7,Mod(41,60)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(60, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("60.41");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 60 = 2^{2} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	60.g (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:	$13.8032450172$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 202x^{6} + 620x^{5} + 12167x^{4} - 25372x^{3} - 177926x^{2} + 190716x + 977814 $ x^8 - 4x^7 - 202x^6 + 620x^5 + 12167x^4 - 25372x^3 - 177926x^2 + 190716*x + 977814
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{7}\cdot 3^{7}\cdot 5^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			41.7
Root			$10.2694 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	60.41
Dual form			60.7.g.a.41.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(25.0836 - 9.99060i) q^{3} -55.9017i q^{5} +134.460 q^{7} +(529.376 - 501.201i) q^{9} +O(q^{10})$	\(q+(25.0836 - 9.99060i) q^{3} -55.9017i q^{5} +134.460 q^{7} +(529.376 - 501.201i) q^{9} -1155.16i q^{11} -78.2272 q^{13} +(-558.492 - 1402.22i) q^{15} -54.6849i q^{17} +1984.42 q^{19} +(3372.73 - 1343.33i) q^{21} -16917.8i q^{23} -3125.00 q^{25} +(8271.35 - 17860.7i) q^{27} -18486.2i q^{29} +38367.7 q^{31} +(-11540.8 - 28975.6i) q^{33} -7516.52i q^{35} +21242.8 q^{37} +(-1962.22 + 781.537i) q^{39} +88980.3i q^{41} -65565.7 q^{43} +(-28018.0 - 29593.0i) q^{45} +163822. i q^{47} -99569.6 q^{49} +(-546.335 - 1371.69i) q^{51} +97179.1i q^{53} -64575.5 q^{55} +(49776.4 - 19825.5i) q^{57} +286230. i q^{59} +119504. q^{61} +(71179.6 - 67391.2i) q^{63} +4373.03i q^{65} -142653. q^{67} +(-169019. - 424360. i) q^{69} +363327. i q^{71} +576517. q^{73} +(-78386.3 + 31220.6i) q^{75} -155322. i q^{77} -627119. q^{79} +(29036.1 - 530647. i) q^{81} +991316. i q^{83} -3056.98 q^{85} +(-184688. - 463701. i) q^{87} -827592. i q^{89} -10518.4 q^{91} +(962400. - 383316. i) q^{93} -110932. i q^{95} +848738. q^{97} +(-578968. - 611514. i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 20 q^{3} - 560 q^{7} + 1492 q^{9}+O(q^{10}) $ 8 * q - 20 * q^3 - 560 * q^7 + 1492 * q^9	$ 8 q - 20 q^{3} - 560 q^{7} + 1492 q^{9} - 6440 q^{13} + 2000 q^{15} - 15272 q^{19} - 868 q^{21} - 25000 q^{25} - 18620 q^{27} + 35032 q^{31} - 111120 q^{33} + 99880 q^{37} + 39608 q^{39} - 161000 q^{43} - 5500 q^{45} + 202560 q^{49} + 429120 q^{51} - 33000 q^{55} - 27160 q^{57} - 135608 q^{61} + 377240 q^{63} + 404920 q^{67} - 254940 q^{69} - 356960 q^{73} + 62500 q^{75} + 707704 q^{79} - 1198112 q^{81} + 828000 q^{85} - 1528440 q^{87} - 2004112 q^{91} - 467920 q^{93} - 1326320 q^{97} + 2650080 q^{99}+O(q^{100}) $ 8 * q - 20 * q^3 - 560 * q^7 + 1492 * q^9 - 6440 * q^13 + 2000 * q^15 - 15272 * q^19 - 868 * q^21 - 25000 * q^25 - 18620 * q^27 + 35032 * q^31 - 111120 * q^33 + 99880 * q^37 + 39608 * q^39 - 161000 * q^43 - 5500 * q^45 + 202560 * q^49 + 429120 * q^51 - 33000 * q^55 - 27160 * q^57 - 135608 * q^61 + 377240 * q^63 + 404920 * q^67 - 254940 * q^69 - 356960 * q^73 + 62500 * q^75 + 707704 * q^79 - 1198112 * q^81 + 828000 * q^85 - 1528440 * q^87 - 2004112 * q^91 - 467920 * q^93 - 1326320 * q^97 + 2650080 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/60\mathbb{Z}\right)^\times$.

$n$	$31$	$37$	$41$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$	25.0836	−	9.99060i	0.929023	−	0.370022i
$3$	25.0836	−	9.99060i	0.929023	−	0.370022i
$4$		0			0
$5$		−	55.9017i		−	0.447214i
$5$		−	55.9017i		−	0.447214i
$6$		0			0
$7$	134.460			0.392010			0.196005	−	0.980603i	$-0.437203\pi$
$7$	134.460			0.392010			0.196005	+	0.980603i	$0.437203\pi$
$8$		0			0
$9$	529.376	−	501.201i	0.726167	−	0.687519i
$10$		0			0
$11$		−	1155.16i		−	0.867890i	−0.900939	−	0.433945i	$-0.857121\pi$
$11$		−	1155.16i		−	0.867890i	0.900939	−	0.433945i	$-0.142879\pi$
$12$		0			0
$13$	−78.2272			−0.0356064			−0.0178032	−	0.999842i	$-0.505667\pi$
$13$	−78.2272			−0.0356064			−0.0178032	+	0.999842i	$0.505667\pi$
$14$		0			0
$15$	−558.492	−	1402.22i	−0.165479	−	0.415472i
$16$		0			0
$17$		−	54.6849i		−	0.0111306i	−0.999985	−	0.00556532i	$-0.998228\pi$
$17$		−	54.6849i		−	0.0111306i	0.999985	−	0.00556532i	$-0.00177151\pi$
$18$		0			0
$19$	1984.42			0.289316			0.144658	−	0.989482i	$-0.453792\pi$
$19$	1984.42			0.289316			0.144658	+	0.989482i	$0.453792\pi$
$20$		0			0
$21$	3372.73	−	1343.33i	0.364186	−	0.145053i
$22$		0			0
$23$		−	16917.8i		−	1.39047i	−0.718784	−	0.695234i	$-0.755301\pi$
$23$		−	16917.8i		−	1.39047i	0.718784	−	0.695234i	$-0.244699\pi$
$24$		0			0
$25$	−3125.00			−0.200000
$26$		0			0
$27$	8271.35	−	17860.7i	0.420228	−	0.907418i
$28$		0			0
$29$		−	18486.2i		−	0.757973i	−0.925402	−	0.378986i	$-0.876273\pi$
$29$		−	18486.2i		−	0.757973i	0.925402	−	0.378986i	$-0.123727\pi$
$30$		0			0
$31$	38367.7			1.28789			0.643947	−	0.765070i	$-0.277296\pi$
$31$	38367.7			1.28789			0.643947	+	0.765070i	$0.277296\pi$
$32$		0			0
$33$	−11540.8	−	28975.6i	−0.321139	−	0.806290i
$34$		0			0
$35$		−	7516.52i		−	0.175312i
$36$		0			0
$37$	21242.8			0.419378			0.209689	−	0.977768i	$-0.432755\pi$
$37$	21242.8			0.419378			0.209689	+	0.977768i	$0.432755\pi$
$38$		0			0
$39$	−1962.22	+	781.537i	−0.0330791	+	0.0131751i
$40$		0			0
$41$			88980.3i			1.29105i	0.763740	+	0.645524i	$0.223361\pi$
$41$			88980.3i			1.29105i	−0.763740	+	0.645524i	$0.776639\pi$
$42$		0			0
$43$	−65565.7			−0.824654			−0.412327	−	0.911036i	$-0.635284\pi$
$43$	−65565.7			−0.824654			−0.412327	+	0.911036i	$0.635284\pi$
$44$		0			0
$45$	−28018.0	−	29593.0i	−0.307468	−	0.324752i
$46$		0			0
$47$			163822.i			1.57790i	0.614457	+	0.788950i	$0.289375\pi$
$47$			163822.i			1.57790i	−0.614457	+	0.788950i	$0.710625\pi$
$48$		0			0
$49$	−99569.6			−0.846328
$50$		0			0
$51$	−546.335	−	1371.69i	−0.00411859	−	0.0103406i
$52$		0			0
$53$			97179.1i			0.652747i	0.945241	+	0.326374i	$0.105827\pi$
$53$			97179.1i			0.652747i	−0.945241	+	0.326374i	$0.894173\pi$
$54$		0			0
$55$	−64575.5			−0.388132
$56$		0			0
$57$	49776.4	−	19825.5i	0.268781	−	0.107053i
$58$		0			0
$59$			286230.i			1.39367i	0.717233	+	0.696834i	$0.245408\pi$
$59$			286230.i			1.39367i	−0.717233	+	0.696834i	$0.754592\pi$
$60$		0			0
$61$	119504.			0.526496			0.263248	−	0.964728i	$-0.415206\pi$
$61$	119504.			0.526496			0.263248	+	0.964728i	$0.415206\pi$
$62$		0			0
$63$	71179.6	−	67391.2i	0.284665	−	0.269514i
$64$		0			0
$65$			4373.03i			0.0159236i
$66$		0			0
$67$	−142653.			−0.474304			−0.237152	−	0.971473i	$-0.576214\pi$
$67$	−142653.			−0.474304			−0.237152	+	0.971473i	$0.576214\pi$
$68$		0			0
$69$	−169019.	−	424360.i	−0.514504	−	1.29178i
$70$		0			0
$71$			363327.i			1.01513i	0.861613	+	0.507566i	$0.169455\pi$
$71$			363327.i			1.01513i	−0.861613	+	0.507566i	$0.830545\pi$
$72$		0			0
$73$	576517.			1.48199			0.740993	−	0.671513i	$-0.234355\pi$
$73$	576517.			1.48199			0.740993	+	0.671513i	$0.234355\pi$
$74$		0			0
$75$	−78386.3	+	31220.6i	−0.185805	+	0.0740045i
$76$		0			0
$77$		−	155322.i		−	0.340222i
$78$		0			0
$79$	−627119.			−1.27195			−0.635973	−	0.771711i	$-0.719401\pi$
$79$	−627119.			−1.27195			−0.635973	+	0.771711i	$0.719401\pi$
$80$		0			0
$81$	29036.1	−	530647.i	0.0546366	−	0.998506i
$82$		0			0
$83$			991316.i			1.73372i	0.498556	+	0.866858i	$0.333864\pi$
$83$			991316.i			1.73372i	−0.498556	+	0.866858i	$0.666136\pi$
$84$		0			0
$85$	−3056.98			−0.00497778
$86$		0			0
$87$	−184688.	−	463701.i	−0.280467	−	0.704174i
$88$		0			0
$89$		−	827592.i		−	1.17394i	−0.809608	−	0.586970i	$-0.800321\pi$
$89$		−	827592.i		−	1.17394i	0.809608	−	0.586970i	$-0.199679\pi$
$90$		0			0
$91$	−10518.4			−0.0139581
$92$		0			0
$93$	962400.	−	383316.i	1.19648	−	0.476550i
$94$		0			0
$95$		−	110932.i		−	0.129386i
$96$		0			0
$97$	848738.			0.929948			0.464974	−	0.885324i	$-0.346064\pi$
$97$	848738.			0.929948			0.464974	+	0.885324i	$0.346064\pi$
$98$		0			0
$99$	−578968.	−	611514.i	−0.596690	−	0.630233i
$100$		0			0
$101$		−	1.11358e6i		−	1.08083i	−0.841398	−	0.540417i	$-0.818267\pi$
$101$		−	1.11358e6i		−	1.08083i	0.841398	−	0.540417i	$-0.181733\pi$
$102$		0			0
$103$	2.09668e6			1.91876			0.959380	−	0.282118i	$-0.0910370\pi$
$103$	2.09668e6			1.91876			0.959380	+	0.282118i	$0.0910370\pi$
$104$		0			0
$105$	−75094.5	−	188541.i	−0.0648695	−	0.162869i
$106$		0			0
$107$			242367.i			0.197844i	0.995095	+	0.0989218i	$0.0315394\pi$
$107$			242367.i			0.197844i	−0.995095	+	0.0989218i	$0.968461\pi$
$108$		0			0
$109$	−2.23474e6			−1.72563			−0.862816	−	0.505519i	$-0.831301\pi$
$109$	−2.23474e6			−1.72563			−0.862816	+	0.505519i	$0.831301\pi$
$110$		0			0
$111$	532846.	−	212228.i	0.389612	−	0.155179i
$112$		0			0
$113$			245486.i			0.170134i	0.996375	+	0.0850672i	$0.0271105\pi$
$113$			245486.i			0.170134i	−0.996375	+	0.0850672i	$0.972890\pi$
$114$		0			0
$115$	−945735.			−0.621836
$116$		0			0
$117$	−41411.6	+	39207.5i	−0.0258562	+	0.0244800i
$118$		0			0
$119$		−	7352.90i		−	0.00436333i
$120$		0			0
$121$	437163.			0.246767
$122$		0			0
$123$	888967.	+	2.23195e6i	0.477717	+	1.19941i
$124$		0			0
$125$			174693.i			0.0894427i
$126$		0			0
$127$	−484140.			−0.236352			−0.118176	−	0.992993i	$-0.537705\pi$
$127$	−484140.			−0.236352			−0.118176	+	0.992993i	$0.537705\pi$
$128$		0			0
$129$	−1.64463e6	+	655041.i	−0.766122	+	0.305140i
$130$		0			0
$131$			3.46823e6i			1.54274i	0.636385	+	0.771372i	$0.280429\pi$
$131$			3.46823e6i			1.54274i	−0.636385	+	0.771372i	$0.719571\pi$
$132$		0			0
$133$	266824.			0.113415
$134$		0			0
$135$	−998444.	−	462383.i	−0.405810	−	0.187932i
$136$		0			0
$137$			1.74492e6i			0.678600i	0.940678	+	0.339300i	$0.110190\pi$
$137$			1.74492e6i			0.678600i	−0.940678	+	0.339300i	$0.889810\pi$
$138$		0			0
$139$	−3.54561e6			−1.32022			−0.660111	−	0.751168i	$-0.729491\pi$
$139$	−3.54561e6			−1.32022			−0.660111	+	0.751168i	$0.729491\pi$
$140$		0			0
$141$	1.63668e6	+	4.10926e6i	0.583859	+	1.46591i
$142$		0			0
$143$			90365.0i			0.0309024i
$144$		0			0
$145$	−1.03341e6			−0.338976
$146$		0			0
$147$	−2.49757e6	+	994761.i	−0.786258	+	0.313160i
$148$		0			0
$149$		−	4.13539e6i		−	1.25014i	−0.780570	−	0.625069i	$-0.785071\pi$
$149$		−	4.13539e6i		−	1.25014i	0.780570	−	0.625069i	$-0.214929\pi$
$150$		0			0
$151$	3.42874e6			0.995873			0.497937	−	0.867213i	$-0.334091\pi$
$151$	3.42874e6			0.995873			0.497937	+	0.867213i	$0.334091\pi$
$152$		0			0
$153$	−27408.1	−	28948.8i	−0.00765252	−	0.00808270i
$154$		0			0
$155$		−	2.14482e6i		−	0.575964i
$156$		0			0
$157$	5.35519e6			1.38381			0.691904	−	0.721990i	$-0.256772\pi$
$157$	5.35519e6			1.38381			0.691904	+	0.721990i	$0.256772\pi$
$158$		0			0
$159$	970878.	+	2.43760e6i	0.241531	+	0.606417i
$160$		0			0
$161$		−	2.27476e6i		−	0.545078i
$162$		0			0
$163$	590537.			0.136359			0.0681796	−	0.997673i	$-0.478281\pi$
$163$	590537.			0.136359			0.0681796	+	0.997673i	$0.478281\pi$
$164$		0			0
$165$	−1.61979e6	+	645148.i	−0.360584	+	0.143618i
$166$		0			0
$167$		−	4.06892e6i		−	0.873634i	−0.899550	−	0.436817i	$-0.856106\pi$
$167$		−	4.06892e6i		−	0.873634i	0.899550	−	0.436817i	$-0.143894\pi$
$168$		0			0
$169$	−4.82069e6			−0.998732
$170$		0			0
$171$	1.05050e6	−	994593.i	0.210092	−	0.198910i
$172$		0			0
$173$		−	4.80964e6i		−	0.928912i	−0.885596	−	0.464456i	$-0.846250\pi$
$173$		−	4.80964e6i		−	0.928912i	0.885596	−	0.464456i	$-0.153750\pi$
$174$		0			0
$175$	−420186.			−0.0784020
$176$		0			0
$177$	2.85961e6	+	7.17968e6i	0.515688	+	1.29475i
$178$		0			0
$179$		−	616529.i		−	0.107496i	−0.998555	−	0.0537482i	$-0.982883\pi$
$179$		−	616529.i		−	0.107496i	0.998555	−	0.0537482i	$-0.0171168\pi$
$180$		0			0
$181$	169854.			0.0286444			0.0143222	−	0.999897i	$-0.495441\pi$
$181$	169854.			0.0286444			0.0143222	+	0.999897i	$0.495441\pi$
$182$		0			0
$183$	2.99760e6	−	1.19392e6i	0.489126	−	0.194815i
$184$		0			0
$185$		−	1.18751e6i		−	0.187552i
$186$		0			0
$187$	−63169.8			−0.00966017
$188$		0			0
$189$	1.11216e6	−	2.40154e6i	0.164734	−	0.355717i
$190$		0			0
$191$		−	5.63748e6i		−	0.809068i	−0.914523	−	0.404534i	$-0.867434\pi$
$191$		−	5.63748e6i		−	0.809068i	0.914523	−	0.404534i	$-0.132566\pi$
$192$		0			0
$193$	−1.93534e6			−0.269207			−0.134603	−	0.990900i	$-0.542976\pi$
$193$	−1.93534e6			−0.269207			−0.134603	+	0.990900i	$0.542976\pi$
$194$		0			0
$195$	43689.2	+	109691.i	0.00589211	+	0.0147934i
$196$		0			0
$197$		−	5.63667e6i		−	0.737265i	−0.929575	−	0.368632i	$-0.879826\pi$
$197$		−	5.63667e6i		−	0.737265i	0.929575	−	0.368632i	$-0.120174\pi$
$198$		0			0
$199$	661790.			0.0839772			0.0419886	−	0.999118i	$-0.486631\pi$
$199$	661790.			0.0839772			0.0419886	+	0.999118i	$0.486631\pi$
$200$		0			0
$201$	−3.57825e6	+	1.42519e6i	−0.440639	+	0.175503i
$202$		0			0
$203$		−	2.48565e6i		−	0.297133i
$204$		0			0
$205$	4.97415e6			0.577374
$206$		0			0
$207$	−8.47923e6	−	8.95588e6i	−0.955972	−	1.00971i
$208$		0			0
$209$		−	2.29232e6i		−	0.251095i
$210$		0			0
$211$	726505.			0.0773376			0.0386688	−	0.999252i	$-0.487688\pi$
$211$	726505.			0.0773376			0.0386688	+	0.999252i	$0.487688\pi$
$212$		0			0
$213$	3.62985e6	+	9.11355e6i	0.375621	+	0.943080i
$214$		0			0
$215$			3.66524e6i			0.368796i
$216$		0			0
$217$	5.15890e6			0.504868
$218$		0			0
$219$	1.44611e7	−	5.75976e6i	1.37680	−	0.548368i
$220$		0			0
$221$			4277.84i			0.000396322i
$222$		0			0
$223$	19464.1			0.00175517			0.000877585	−	1.00000i	$-0.499721\pi$
$223$	19464.1			0.00175517			0.000877585		1.00000i	$0.499721\pi$
$224$		0			0
$225$	−1.65430e6	+	1.56625e6i	−0.145233	+	0.137504i
$226$		0			0
$227$		−	1.74686e6i		−	0.149342i	−0.997208	−	0.0746708i	$-0.976209\pi$
$227$		−	1.74686e6i		−	0.149342i	0.997208	−	0.0746708i	$-0.0237906\pi$
$228$		0			0
$229$	803494.			0.0669077			0.0334539	−	0.999440i	$-0.489349\pi$
$229$	803494.			0.0669077			0.0334539	+	0.999440i	$0.489349\pi$
$230$		0			0
$231$	−1.55177e6	−	3.89605e6i	−0.125890	−	0.316074i
$232$		0			0
$233$			1.41647e7i			1.11980i	0.828561	+	0.559900i	$0.189160\pi$
$233$			1.41647e7i			1.11980i	−0.828561	+	0.559900i	$0.810840\pi$
$234$		0			0
$235$	9.15795e6			0.705659
$236$		0			0
$237$	−1.57304e7	+	6.26530e6i	−1.18167	+	0.470648i
$238$		0			0
$239$			1.51609e7i			1.11054i	0.831672	+	0.555268i	$0.187384\pi$
$239$			1.51609e7i			1.11054i	−0.831672	+	0.555268i	$0.812616\pi$
$240$		0			0
$241$	−2.44786e7			−1.74878			−0.874391	−	0.485222i	$-0.838739\pi$
$241$	−2.44786e7			−1.74878			−0.874391	+	0.485222i	$0.838739\pi$
$242$		0			0
$243$	−4.57316e6	−	1.36006e7i	−0.318711	−	0.947852i
$244$		0			0
$245$			5.56611e6i			0.378489i
$246$		0			0
$247$	−155235.			−0.0103015
$248$		0			0
$249$	9.90384e6	+	2.48658e7i	0.641513	+	1.61066i
$250$		0			0
$251$			5.38840e6i			0.340752i	0.985379	+	0.170376i	$0.0544982\pi$
$251$			5.38840e6i			0.340752i	−0.985379	+	0.170376i	$0.945502\pi$
$252$		0			0
$253$	−1.95428e7			−1.20677
$254$		0			0
$255$	−76680.0	+	30541.0i	−0.00462447	+	0.00184189i
$256$		0			0
$257$			2.81725e7i			1.65969i	0.557998	+	0.829843i	$0.311570\pi$
$257$			2.81725e7i			1.65969i	−0.557998	+	0.829843i	$0.688430\pi$
$258$		0			0
$259$	2.85629e6			0.164401
$260$		0			0
$261$	−9.26530e6	−	9.78614e6i	−0.521120	−	0.550415i
$262$		0			0
$263$			1.14892e7i			0.631570i	0.948831	+	0.315785i	$0.102268\pi$
$263$			1.14892e7i			0.631570i	−0.948831	+	0.315785i	$0.897732\pi$
$264$		0			0
$265$	5.43248e6			0.291917
$266$		0			0
$267$	−8.26814e6	−	2.07590e7i	−0.434384	−	1.09062i
$268$		0			0
$269$			1.46873e7i			0.754546i	0.926102	+	0.377273i	$0.123138\pi$
$269$			1.46873e7i			0.754546i	−0.926102	+	0.377273i	$0.876862\pi$
$270$		0			0
$271$	−1.77493e6			−0.0891812			−0.0445906	−	0.999005i	$-0.514198\pi$
$271$	−1.77493e6			−0.0891812			−0.0445906	+	0.999005i	$0.514198\pi$
$272$		0			0
$273$	−263839.	+	105085.i	−0.0129674	+	0.00516479i
$274$		0			0
$275$			3.60988e6i			0.173578i
$276$		0			0
$277$	2.74584e7			1.29192			0.645960	−	0.763371i	$-0.276457\pi$
$277$	2.74584e7			1.29192			0.645960	+	0.763371i	$0.276457\pi$
$278$		0			0
$279$	2.03109e7	−	1.92299e7i	0.935226	−	0.885452i
$280$		0			0
$281$		−	2.73409e6i		−	0.123224i	−0.998100	−	0.0616118i	$-0.980376\pi$
$281$		−	2.73409e6i		−	0.123224i	0.998100	−	0.0616118i	$-0.0196241\pi$
$282$		0			0
$283$	−3.51523e7			−1.55094			−0.775469	−	0.631386i	$-0.782487\pi$
$283$	−3.51523e7			−1.55094			−0.775469	+	0.631386i	$0.782487\pi$
$284$		0			0
$285$	−1.10828e6	−	2.78259e6i	−0.0478758	−	0.120203i
$286$		0			0
$287$			1.19642e7i			0.506104i
$288$		0			0
$289$	2.41346e7			0.999876
$290$		0			0
$291$	2.12894e7	−	8.47941e6i	0.863943	−	0.344102i
$292$		0			0
$293$		−	4.66113e7i		−	1.85306i	−0.376227	−	0.926528i	$-0.622779\pi$
$293$		−	4.66113e7i		−	1.85306i	0.376227	−	0.926528i	$-0.377221\pi$
$294$		0			0
$295$	1.60007e7			0.623267
$296$		0			0
$297$	−2.06320e7	−	9.55475e6i	−0.787539	−	0.364712i
$298$		0			0
$299$			1.32343e6i			0.0495095i
$300$		0			0
$301$	−8.81594e6			−0.323273
$302$		0			0
$303$	−1.11254e7	−	2.79327e7i	−0.399932	−	1.00412i
$304$		0			0
$305$		−	6.68050e6i		−	0.235456i
$306$		0			0
$307$	3.11960e7			1.07816			0.539081	−	0.842254i	$-0.318772\pi$
$307$	3.11960e7			1.07816			0.539081	+	0.842254i	$0.318772\pi$
$308$		0			0
$309$	5.25923e7	−	2.09471e7i	1.78257	−	0.709984i
$310$		0			0
$311$			3.51987e7i			1.17016i	0.810975	+	0.585081i	$0.198937\pi$
$311$			3.51987e7i			1.17016i	−0.810975	+	0.585081i	$0.801063\pi$
$312$		0			0
$313$	−3.60591e7			−1.17593			−0.587965	−	0.808886i	$-0.700071\pi$
$313$	−3.60591e7			−1.17593			−0.587965	+	0.808886i	$0.700071\pi$
$314$		0			0
$315$	−3.76728e6	−	3.97906e6i	−0.120530	−	0.127306i
$316$		0			0
$317$		−	5.74502e7i		−	1.80349i	−0.432269	−	0.901745i	$-0.642287\pi$
$317$		−	5.74502e7i		−	1.80349i	0.432269	−	0.901745i	$-0.357713\pi$
$318$		0			0
$319$	−2.13545e7			−0.657837
$320$		0			0
$321$	2.42139e6	+	6.07944e6i	0.0732065	+	0.183801i
$322$		0			0
$323$		−	108518.i		−	0.00322027i
$324$		0			0
$325$	244460.			0.00712127
$326$		0			0
$327$	−5.60554e7	+	2.23264e7i	−1.60315	+	0.638522i
$328$		0			0
$329$			2.20275e7i			0.618553i
$330$		0			0
$331$	3.26327e7			0.899849			0.449924	−	0.893067i	$-0.351451\pi$
$331$	3.26327e7			0.899849			0.449924	+	0.893067i	$0.351451\pi$
$332$		0			0
$333$	1.12454e7	−	1.06469e7i	0.304539	−	0.288330i
$334$		0			0
$335$			7.97454e6i			0.212115i
$336$		0			0
$337$	4.34020e7			1.13402			0.567009	−	0.823712i	$-0.308100\pi$
$337$	4.34020e7			1.13402			0.567009	+	0.823712i	$0.308100\pi$
$338$		0			0
$339$	2.45256e6	+	6.15769e6i	0.0629535	+	0.158059i
$340$		0			0
$341$		−	4.43209e7i		−	1.11775i
$342$		0			0
$343$	−2.92071e7			−0.723779
$344$		0			0
$345$	−2.37225e7	+	9.44846e6i	−0.577700	+	0.230093i
$346$		0			0
$347$			3.50726e7i			0.839421i	0.907658	+	0.419710i	$0.137868\pi$
$347$			3.50726e7i			0.839421i	−0.907658	+	0.419710i	$0.862132\pi$
$348$		0			0
$349$	1.41882e7			0.333772			0.166886	−	0.985976i	$-0.446629\pi$
$349$	1.41882e7			0.333772			0.166886	+	0.985976i	$0.446629\pi$
$350$		0			0
$351$	−647045.	+	1.39719e6i	−0.0149628	+	0.0323099i
$352$		0			0
$353$		−	1.86131e7i		−	0.423149i	−0.977362	−	0.211575i	$-0.932141\pi$
$353$		−	1.86131e7i		−	0.423149i	0.977362	−	0.211575i	$-0.0678591\pi$
$354$		0			0
$355$	2.03106e7			0.453980
$356$		0			0
$357$	−73459.9	−	184437.i	−0.00161453	−	0.00405363i
$358$		0			0
$359$			5.28748e6i			0.114279i	0.998366	+	0.0571394i	$0.0181979\pi$
$359$			5.28748e6i			0.114279i	−0.998366	+	0.0571394i	$0.981802\pi$
$360$		0			0
$361$	−4.31080e7			−0.916296
$362$		0			0
$363$	1.09656e7	−	4.36752e6i	0.229252	−	0.0913094i
$364$		0			0
$365$		−	3.22283e7i		−	0.662764i
$366$		0			0
$367$	1.00133e6			0.0202571			0.0101286	−	0.999949i	$-0.496776\pi$
$367$	1.00133e6			0.0202571			0.0101286	+	0.999949i	$0.496776\pi$
$368$		0			0
$369$	4.45970e7	+	4.71040e7i	0.887619	+	0.937516i
$370$		0			0
$371$			1.30666e7i			0.255884i
$372$		0			0
$373$	4.27333e7			0.823455			0.411728	−	0.911307i	$-0.364926\pi$
$373$	4.27333e7			0.823455			0.411728	+	0.911307i	$0.364926\pi$
$374$		0			0
$375$	1.74529e6	+	4.38193e6i	0.0330958	+	0.0830943i
$376$		0			0
$377$			1.44612e6i			0.0269887i
$378$		0			0
$379$	−1.05217e8			−1.93272			−0.966358	−	0.257202i	$-0.917199\pi$
$379$	−1.05217e8			−1.93272			−0.966358	+	0.257202i	$0.917199\pi$
$380$		0			0
$381$	−1.21440e7	+	4.83685e6i	−0.219577	+	0.0874556i
$382$		0			0
$383$			6.60163e7i			1.17505i	0.809208	+	0.587523i	$0.199897\pi$
$383$			6.60163e7i			1.17505i	−0.809208	+	0.587523i	$0.800103\pi$
$384$		0			0
$385$	−8.68279e6			−0.152152
$386$		0			0
$387$	−3.47089e7	+	3.28616e7i	−0.598836	+	0.566965i
$388$		0			0
$389$		−	6.54648e7i		−	1.11214i	−0.831136	−	0.556070i	$-0.812309\pi$
$389$		−	6.54648e7i		−	1.11214i	0.831136	−	0.556070i	$-0.187691\pi$
$390$		0			0
$391$	−925149.			−0.0154768
$392$		0			0
$393$	3.46497e7	+	8.69957e7i	0.570850	+	1.43324i
$394$		0			0
$395$			3.50570e7i			0.568831i
$396$		0			0
$397$	5.36739e7			0.857811			0.428905	−	0.903349i	$-0.358899\pi$
$397$	5.36739e7			0.857811			0.428905	+	0.903349i	$0.358899\pi$
$398$		0			0
$399$	6.69291e6	−	2.66573e6i	0.105365	−	0.0419660i
$400$		0			0
$401$		−	8.82767e6i		−	0.136903i	−0.997654	−	0.0684515i	$-0.978194\pi$
$401$		−	8.82767e6i		−	0.136903i	0.997654	−	0.0684515i	$-0.0218058\pi$
$402$		0			0
$403$	−3.00139e6			−0.0458572
$404$		0			0
$405$	−2.96641e7	−	1.62317e6i	−0.446546	−	0.0244342i
$406$		0			0
$407$		−	2.45388e7i		−	0.363974i
$408$		0			0
$409$	−1.11796e8			−1.63402			−0.817011	−	0.576622i	$-0.804371\pi$
$409$	−1.11796e8			−1.63402			−0.817011	+	0.576622i	$0.804371\pi$
$410$		0			0
$411$	1.74328e7	+	4.37689e7i	0.251097	+	0.630435i
$412$		0			0
$413$			3.84863e7i			0.546332i
$414$		0			0
$415$	5.54162e7			0.775341
$416$		0			0
$417$	−8.89368e7	+	3.54228e7i	−1.22652	+	0.488512i
$418$		0			0
$419$		−	2.09574e7i		−	0.284902i	−0.989802	−	0.142451i	$-0.954502\pi$
$419$		−	2.09574e7i		−	0.284902i	0.989802	−	0.142451i	$-0.0454983\pi$
$420$		0			0
$421$	−1.87381e7			−0.251118			−0.125559	−	0.992086i	$-0.540073\pi$
$421$	−1.87381e7			−0.251118			−0.125559	+	0.992086i	$0.540073\pi$
$422$		0			0
$423$	8.21080e7	+	8.67236e7i	1.08484	+	1.14582i
$424$		0			0
$425$			170890.i			0.00222613i
$426$		0			0
$427$	1.60685e7			0.206392
$428$		0			0
$429$	902801.	+	2.26668e6i	0.0114346	+	0.0287090i
$430$		0			0
$431$		−	7.51443e6i		−	0.0938565i	−0.998898	−	0.0469283i	$-0.985057\pi$
$431$		−	7.51443e6i		−	0.0938565i	0.998898	−	0.0469283i	$-0.0149432\pi$
$432$		0			0
$433$	6.88652e7			0.848274			0.424137	−	0.905598i	$-0.360578\pi$
$433$	6.88652e7			0.848274			0.424137	+	0.905598i	$0.360578\pi$
$434$		0			0
$435$	−2.59217e7	+	1.03244e7i	−0.314916	+	0.125429i
$436$		0			0
$437$		−	3.35721e7i		−	0.402285i
$438$		0			0
$439$	−1.30096e8			−1.53769			−0.768846	−	0.639434i	$-0.779169\pi$
$439$	−1.30096e8			−1.53769			−0.768846	+	0.639434i	$0.779169\pi$
$440$		0			0
$441$	−5.27097e7	+	4.99044e7i	−0.614575	+	0.581866i
$442$		0			0
$443$		−	1.19372e8i		−	1.37306i	−0.727100	−	0.686531i	$-0.759132\pi$
$443$		−	1.19372e8i		−	1.37306i	0.727100	−	0.686531i	$-0.240868\pi$
$444$		0			0
$445$	−4.62638e7			−0.525002
$446$		0			0
$447$	−4.13151e7	−	1.03731e8i	−0.462579	−	1.16141i
$448$		0			0
$449$		−	5.58450e7i		−	0.616943i	−0.951234	−	0.308471i	$-0.900183\pi$
$449$		−	5.58450e7i		−	0.616943i	0.951234	−	0.308471i	$-0.0998174\pi$
$450$		0			0
$451$	1.02787e8			1.12049
$452$		0			0
$453$	8.60053e7	−	3.42552e7i	0.925189	−	0.368495i
$454$		0			0
$455$			587996.i			0.00624223i
$456$		0			0
$457$	1.17246e8			1.22843			0.614213	−	0.789140i	$-0.289473\pi$
$457$	1.17246e8			1.22843			0.614213	+	0.789140i	$0.289473\pi$
$458$		0			0
$459$	−976711.	−	452318.i	−0.0101002	−	0.00467741i
$460$		0			0
$461$		−	1.77582e8i		−	1.81258i	−0.422657	−	0.906290i	$-0.638903\pi$
$461$		−	1.77582e8i		−	1.81258i	0.422657	−	0.906290i	$-0.361097\pi$
$462$		0			0
$463$	1.25613e8			1.26558			0.632791	−	0.774323i	$-0.281909\pi$
$463$	1.25613e8			1.26558			0.632791	+	0.774323i	$0.281909\pi$
$464$		0			0
$465$	−2.14280e7	−	5.37998e7i	−0.213120	−	0.535084i
$466$		0			0
$467$		−	2.66195e7i		−	0.261366i	−0.991424	−	0.130683i	$-0.958283\pi$
$467$		−	2.66195e7i		−	0.261366i	0.991424	−	0.130683i	$-0.0417170\pi$
$468$		0			0
$469$	−1.91810e7			−0.185932
$470$		0			0
$471$	1.34327e8	−	5.35016e7i	1.28559	−	0.512040i
$472$		0			0
$473$			7.57390e7i			0.715709i
$474$		0			0
$475$	−6.20131e6			−0.0578632
$476$		0			0
$477$	4.87062e7	+	5.14442e7i	0.448776	+	0.474003i
$478$		0			0
$479$		−	1.63522e8i		−	1.48789i	−0.668242	−	0.743944i	$-0.732953\pi$
$479$		−	1.63522e8i		−	1.48789i	0.668242	−	0.743944i	$-0.267047\pi$
$480$		0			0
$481$	−1.66176e6			−0.0149325
$482$		0			0
$483$	−2.27262e7	−	5.70593e7i	−0.201691	−	0.506390i
$484$		0			0
$485$		−	4.74459e7i		−	0.415885i
$486$		0			0
$487$	−6.56192e6			−0.0568125			−0.0284062	−	0.999596i	$-0.509043\pi$
$487$	−6.56192e6			−0.0568125			−0.0284062	+	0.999596i	$0.509043\pi$
$488$		0			0
$489$	1.48128e7	−	5.89982e6i	0.126681	−	0.0504559i
$490$		0			0
$491$			1.29952e8i			1.09784i	0.835874	+	0.548921i	$0.184961\pi$
$491$			1.29952e8i			1.09784i	−0.835874	+	0.548921i	$0.815039\pi$
$492$		0			0
$493$	−1.01092e6			−0.00843673
$494$		0			0
$495$	−3.41847e7	+	3.23653e7i	−0.281849	+	0.266848i
$496$		0			0
$497$			4.88527e7i			0.397942i
$498$		0			0
$499$	−1.09191e8			−0.878788			−0.439394	−	0.898295i	$-0.644807\pi$
$499$	−1.09191e8			−0.878788			−0.439394	+	0.898295i	$0.644807\pi$
$500$		0			0
$501$	−4.06509e7	−	1.02063e8i	−0.323264	−	0.811626i
$502$		0			0
$503$			3.22774e7i			0.253626i	0.991927	+	0.126813i	$0.0404748\pi$
$503$			3.22774e7i			0.253626i	−0.991927	+	0.126813i	$0.959525\pi$
$504$		0			0
$505$	−6.22512e7			−0.483363
$506$		0			0
$507$	−1.20920e8	+	4.81616e7i	−0.927845	+	0.369553i
$508$		0			0
$509$			1.11482e8i			0.845382i	0.906274	+	0.422691i	$0.138914\pi$
$509$			1.11482e8i			0.845382i	−0.906274	+	0.422691i	$0.861086\pi$
$510$		0			0
$511$	7.75183e7			0.580953
$512$		0			0
$513$	1.64138e7	−	3.54432e7i	0.121579	−	0.262531i
$514$		0			0
$515$		−	1.17208e8i		−	0.858095i
$516$		0			0
$517$	1.89241e8			1.36944
$518$		0			0
$519$	−4.80513e7	−	1.20643e8i	−0.343718	−	0.862981i
$520$		0			0
$521$			2.70153e8i			1.91028i	0.296160	+	0.955138i	$0.404294\pi$
$521$			2.70153e8i			1.91028i	−0.296160	+	0.955138i	$0.595706\pi$
$522$		0			0
$523$	−1.29085e8			−0.902341			−0.451171	−	0.892438i	$-0.648993\pi$
$523$	−1.29085e8			−0.902341			−0.451171	+	0.892438i	$0.648993\pi$
$524$		0			0
$525$	−1.05398e7	+	4.19791e6i	−0.0728373	+	0.0290105i
$526$		0			0
$527$		−	2.09813e6i		−	0.0143351i
$528$		0			0
$529$	−1.38177e8			−0.933401
$530$		0			0
$531$	1.43459e8	+	1.51523e8i	0.958172	+	1.01203i
$532$		0			0
$533$		−	6.96068e6i		−	0.0459695i
$534$		0			0
$535$	1.35487e7			0.0884783
$536$		0			0
$537$	−6.15949e6	−	1.54648e7i	−0.0397761	−	0.0998667i
$538$		0			0
$539$			1.15019e8i			0.734519i
$540$		0			0
$541$	−1.19668e8			−0.755765			−0.377883	−	0.925853i	$-0.623348\pi$
$541$	−1.19668e8			−0.755765			−0.377883	+	0.925853i	$0.623348\pi$
$542$		0			0
$543$	4.26055e6	−	1.69694e6i	0.0266113	−	0.0105991i
$544$		0			0
$545$			1.24926e8i			0.771726i
$546$		0			0
$547$	−7.12552e7			−0.435366			−0.217683	−	0.976020i	$-0.569850\pi$
$547$	−7.12552e7			−0.435366			−0.217683	+	0.976020i	$0.569850\pi$
$548$		0			0
$549$	6.32628e7	−	5.98958e7i	0.382324	−	0.361975i
$550$		0			0
$551$		−	3.66844e7i		−	0.219294i
$552$		0			0
$553$	−8.43221e7			−0.498616
$554$		0			0
$555$	−1.18639e7	−	2.97870e7i	−0.0693983	−	0.174240i
$556$		0			0
$557$			2.53675e8i			1.46795i	0.679176	+	0.733976i	$0.262337\pi$
$557$			2.53675e8i			1.46795i	−0.679176	+	0.733976i	$0.737663\pi$
$558$		0			0
$559$	5.12902e6			0.0293629
$560$		0			0
$561$	−1.58453e6	+	631105.i	−0.00897452	+	0.00357448i
$562$		0			0
$563$			4.15634e7i			0.232909i	0.993196	+	0.116454i	$0.0371529\pi$
$563$			4.15634e7i			0.232909i	−0.993196	+	0.116454i	$0.962847\pi$
$564$		0			0
$565$	1.37231e7			0.0760864
$566$		0			0
$567$	3.90418e6	−	7.13506e7i	0.0214181	−	0.391425i
$568$		0			0
$569$			1.69480e8i			0.919986i	0.887923	+	0.459993i	$0.152148\pi$
$569$			1.69480e8i			0.919986i	−0.887923	+	0.459993i	$0.847852\pi$
$570$		0			0
$571$	2.30661e8			1.23898			0.619492	−	0.785003i	$-0.287338\pi$
$571$	2.30661e8			1.23898			0.619492	+	0.785003i	$0.287338\pi$
$572$		0			0
$573$	−5.63218e7	−	1.41408e8i	−0.299373	−	0.751642i
$574$		0			0
$575$			5.28682e7i			0.278094i
$576$		0			0
$577$	9.38558e7			0.488578			0.244289	−	0.969703i	$-0.421446\pi$
$577$	9.38558e7			0.488578			0.244289	+	0.969703i	$0.421446\pi$
$578$		0			0
$579$	−4.85454e7	+	1.93352e7i	−0.250099	+	0.0996125i
$580$		0			0
$581$			1.33292e8i			0.679634i
$582$		0			0
$583$	1.12258e8			0.566513
$584$		0			0
$585$	2.19177e6	+	2.31498e6i	0.0109478	+	0.0115632i
$586$		0			0
$587$		−	2.28514e8i		−	1.12979i	−0.825162	−	0.564897i	$-0.808916\pi$
$587$		−	2.28514e8i		−	1.12979i	0.825162	−	0.564897i	$-0.191084\pi$
$588$		0			0
$589$	7.61376e7			0.372609
$590$		0			0
$591$	−5.63137e7	−	1.41388e8i	−0.272805	−	0.684936i
$592$		0			0
$593$			1.47213e8i			0.705966i	0.935630	+	0.352983i	$0.114833\pi$
$593$			1.47213e8i			0.705966i	−0.935630	+	0.352983i	$0.885167\pi$
$594$		0			0
$595$	−411040.			−0.00195134
$596$		0			0
$597$	1.66001e7	−	6.61169e6i	0.0780167	−	0.0310734i
$598$		0			0
$599$			1.04762e8i			0.487440i	0.969846	+	0.243720i	$0.0783678\pi$
$599$			1.04762e8i			0.487440i	−0.969846	+	0.243720i	$0.921632\pi$
$600$		0			0
$601$	−2.92674e8			−1.34822			−0.674110	−	0.738631i	$-0.735473\pi$
$601$	−2.92674e8			−1.34822			−0.674110	+	0.738631i	$0.735473\pi$
$602$		0			0
$603$	−7.55170e7	+	7.14978e7i	−0.344424	+	0.326092i
$604$		0			0
$605$		−	2.44382e7i		−	0.110358i
$606$		0			0
$607$	−2.94250e8			−1.31568			−0.657840	−	0.753158i	$-0.728530\pi$
$607$	−2.94250e8			−1.31568			−0.657840	+	0.753158i	$0.728530\pi$
$608$		0			0
$609$	−2.48331e7	−	6.23490e7i	−0.109946	−	0.276043i
$610$		0			0
$611$		−	1.28154e7i		−	0.0561833i
$612$		0			0
$613$	−1.10142e8			−0.478158			−0.239079	−	0.971000i	$-0.576845\pi$
$613$	−1.10142e8			−0.478158			−0.239079	+	0.971000i	$0.576845\pi$
$614$		0			0
$615$	1.24770e8	−	4.96948e7i	0.536394	−	0.213641i
$616$		0			0
$617$			3.60731e8i			1.53578i	0.640584	+	0.767888i	$0.278692\pi$
$617$			3.60731e8i			1.53578i	−0.640584	+	0.767888i	$0.721308\pi$
$618$		0			0
$619$	4.02744e8			1.69808			0.849038	−	0.528332i	$-0.177182\pi$
$619$	4.02744e8			1.69808			0.849038	+	0.528332i	$0.177182\pi$
$620$		0			0
$621$	−3.02164e8	−	1.39933e8i	−1.26174	−	0.584314i
$622$		0			0
$623$		−	1.11278e8i		−	0.460197i
$624$		0			0
$625$	9.76562e6			0.0400000
$626$		0			0
$627$	−2.29017e7	−	5.74998e7i	−0.0929106	−	0.233273i
$628$		0			0
$629$		−	1.16166e6i		−	0.00466795i
$630$		0			0
$631$	−7.78200e7			−0.309744			−0.154872	−	0.987935i	$-0.549497\pi$
$631$	−7.78200e7			−0.309744			−0.154872	+	0.987935i	$0.549497\pi$
$632$		0			0
$633$	1.82234e7	−	7.25822e6i	0.0718484	−	0.0286167i
$634$		0			0
$635$			2.70642e7i			0.105700i
$636$		0			0
$637$	7.78905e6			0.0301347
$638$		0			0
$639$	1.82100e8	+	1.92336e8i	0.697921	+	0.737155i
$640$		0			0
$641$		−	4.82128e8i		−	1.83058i	−0.402797	−	0.915290i	$-0.631962\pi$
$641$		−	4.82128e8i		−	1.83058i	0.402797	−	0.915290i	$-0.368038\pi$
$642$		0			0
$643$	1.57608e7			0.0592850			0.0296425	−	0.999561i	$-0.490563\pi$
$643$	1.57608e7			0.0592850			0.0296425	+	0.999561i	$0.490563\pi$
$644$		0			0
$645$	3.66179e7	+	9.19374e7i	0.136463	+	0.342620i
$646$		0			0
$647$		−	2.10892e8i		−	0.778660i	−0.921098	−	0.389330i	$-0.872707\pi$
$647$		−	2.10892e8i		−	0.778660i	0.921098	−	0.389330i	$-0.127293\pi$
$648$		0			0
$649$	3.30642e8			1.20955
$650$		0			0
$651$	1.29404e8	−	5.15405e7i	0.469034	−	0.186812i
$652$		0			0
$653$			2.03936e8i			0.732410i	0.930534	+	0.366205i	$0.119343\pi$
$653$			2.03936e8i			0.732410i	−0.930534	+	0.366205i	$0.880657\pi$
$654$		0			0
$655$	1.93880e8			0.689936
$656$		0			0
$657$	3.05194e8	−	2.88951e8i	1.07617	−	1.01889i
$658$		0			0
$659$		−	4.80405e8i		−	1.67862i	−0.543656	−	0.839308i	$-0.682960\pi$
$659$		−	4.80405e8i		−	1.67862i	0.543656	−	0.839308i	$-0.317040\pi$
$660$		0			0
$661$	−1.13313e8			−0.392351			−0.196176	−	0.980569i	$-0.562852\pi$
$661$	−1.13313e8			−0.392351			−0.196176	+	0.980569i	$0.562852\pi$
$662$		0			0
$663$	42738.2	+	107304.i	0.000146648	+	0.000368192i
$664$		0			0
$665$		−	1.49159e7i		−	0.0507207i
$666$		0			0
$667$	−3.12746e8			−1.05394
$668$		0			0
$669$	488229.	−	194458.i	0.00163059	−	0.000649452i
$670$		0			0
$671$		−	1.38047e8i		−	0.456940i
$672$		0			0
$673$	−6.78723e7			−0.222663			−0.111331	−	0.993783i	$-0.535511\pi$
$673$	−6.78723e7			−0.222663			−0.111331	+	0.993783i	$0.535511\pi$
$674$		0			0
$675$	−2.58480e7	+	5.58147e7i	−0.0840457	+	0.181484i
$676$		0			0
$677$		−	3.22505e8i		−	1.03937i	−0.854358	−	0.519685i	$-0.826049\pi$
$677$		−	3.22505e8i		−	1.03937i	0.854358	−	0.519685i	$-0.173951\pi$
$678$		0			0
$679$	1.14121e8			0.364549
$680$		0			0
$681$	−1.74522e7	−	4.38176e7i	−0.0552598	−	0.138742i
$682$		0			0
$683$		−	4.17199e8i		−	1.30943i	−0.755877	−	0.654714i	$-0.772789\pi$
$683$		−	4.17199e8i		−	1.30943i	0.755877	−	0.654714i	$-0.227211\pi$
$684$		0			0
$685$	9.75439e7			0.303479
$686$		0			0
$687$	2.01545e7	−	8.02739e6i	0.0621588	−	0.0247574i
$688$		0			0
$689$		−	7.60204e6i		−	0.0232420i
$690$		0			0
$691$	−3.88809e7			−0.117842			−0.0589212	−	0.998263i	$-0.518766\pi$
$691$	−3.88809e7			−0.117842			−0.0589212	+	0.998263i	$0.518766\pi$
$692$		0			0
$693$	−7.78478e7	−	8.22239e7i	−0.233909	−	0.247058i
$694$		0			0
$695$			1.98206e8i			0.590421i
$696$		0			0
$697$	4.86587e6			0.0143702
$698$		0			0
$699$	1.41514e8	+	3.55302e8i	0.414351	+	1.04032i
$700$		0			0
$701$			7.02379e7i			0.203900i	0.994790	+	0.101950i	$0.0325082\pi$
$701$			7.02379e7i			0.203900i	−0.994790	+	0.101950i	$0.967492\pi$
$702$		0			0
$703$	4.21546e7			0.121333
$704$		0			0
$705$	2.29715e8	−	9.14935e7i	0.655573	−	0.261110i
$706$		0			0
$707$		−	1.49732e8i		−	0.423698i
$708$		0			0
$709$	3.27456e8			0.918785			0.459393	−	0.888233i	$-0.348067\pi$
$709$	3.27456e8			0.918785			0.459393	+	0.888233i	$0.348067\pi$
$710$		0			0
$711$	−3.31981e8	+	3.14313e8i	−0.923645	+	0.874486i
$712$		0			0
$713$		−	6.49097e8i		−	1.79078i
$714$		0			0
$715$	5.05156e6			0.0138200
$716$		0			0
$717$	1.51467e8	+	3.80291e8i	0.410923	+	1.03171i
$718$		0			0
$719$			6.46847e8i			1.74026i	0.492819	+	0.870132i	$0.335966\pi$
$719$			6.46847e8i			1.74026i	−0.492819	+	0.870132i	$0.664034\pi$
$720$		0			0
$721$	2.81919e8			0.752173
$722$		0			0
$723$	−6.14012e8	+	2.44556e8i	−1.62466	+	0.647089i
$724$		0			0
$725$			5.77694e7i			0.151595i
$726$		0			0
$727$	−4.36086e8			−1.13493			−0.567465	−	0.823398i	$-0.692076\pi$
$727$	−4.36086e8			−1.13493			−0.567465	+	0.823398i	$0.692076\pi$
$728$		0			0
$729$	−2.50590e8	−	2.95465e8i	−0.646816	−	0.762646i
$730$		0			0
$731$			3.58545e6i			0.00917893i
$732$		0			0
$733$	−4.27585e8			−1.08570			−0.542851	−	0.839829i	$-0.682655\pi$
$733$	−4.27585e8			−1.08570			−0.542851	+	0.839829i	$0.682655\pi$
$734$		0			0
$735$	5.56088e7	+	1.39618e8i	0.140050	+	0.351625i
$736$		0			0
$737$			1.64787e8i			0.411643i
$738$		0			0
$739$	3.65022e8			0.904452			0.452226	−	0.891903i	$-0.350630\pi$
$739$	3.65022e8			0.904452			0.452226	+	0.891903i	$0.350630\pi$
$740$		0			0
$741$	−3.89387e6	+	1.55090e6i	−0.00957032	+	0.00381178i
$742$		0			0
$743$			6.75024e8i			1.64571i	0.568253	+	0.822854i	$0.307620\pi$
$743$			6.75024e8i			1.64571i	−0.568253	+	0.822854i	$0.692380\pi$
$744$		0			0
$745$	−2.31175e8			−0.559079
$746$		0			0
$747$	4.96848e8	+	5.24778e8i	1.19196	+	1.25897i
$748$		0			0
$749$			3.25885e7i			0.0775567i
$750$		0			0
$751$	−5.14564e8			−1.21484			−0.607421	−	0.794380i	$-0.707796\pi$
$751$	−5.14564e8			−1.21484			−0.607421	+	0.794380i	$0.707796\pi$
$752$		0			0
$753$	5.38334e7	+	1.35161e8i	0.126086	+	0.316567i
$754$		0			0
$755$		−	1.91673e8i		−	0.445368i
$756$		0			0
$757$	−2.32695e8			−0.536413			−0.268207	−	0.963361i	$-0.586431\pi$
$757$	−2.32695e8			−0.536413			−0.268207	+	0.963361i	$0.586431\pi$
$758$		0			0
$759$	−4.90204e8	+	1.95245e8i	−1.12112	+	0.446533i
$760$		0			0
$761$		−	3.18044e7i		−	0.0721661i	−0.999349	−	0.0360831i	$-0.988512\pi$
$761$		−	3.18044e7i		−	0.0721661i	0.999349	−	0.0360831i	$-0.0114881\pi$
$762$		0			0
$763$	−3.00482e8			−0.676465
$764$		0			0
$765$	−1.61829e6	+	1.53216e6i	−0.00361470	+	0.00342231i
$766$		0			0
$767$		−	2.23910e7i		−	0.0496234i
$768$		0			0
$769$	6.46145e8			1.42086			0.710430	−	0.703768i	$-0.248501\pi$
$769$	6.46145e8			1.42086			0.710430	+	0.703768i	$0.248501\pi$
$770$		0			0
$771$	2.81460e8	+	7.06668e8i	0.614121	+	1.54189i
$772$		0			0
$773$		−	1.82196e8i		−	0.394457i	−0.980358	−	0.197228i	$-0.936806\pi$
$773$		−	1.82196e8i		−	0.394457i	0.980358	−	0.197228i	$-0.0631941\pi$
$774$		0			0
$775$	−1.19899e8			−0.257579
$776$		0			0
$777$	7.16462e7	−	2.85361e7i	0.152732	−	0.0608319i
$778$		0			0
$779$			1.76574e8i			0.373521i
$780$		0			0
$781$	4.19701e8			0.881022
$782$		0			0
$783$	−3.30177e8	−	1.52906e8i	−0.687799	−	0.318522i
$784$		0			0
$785$		−	2.99364e8i		−	0.618858i
$786$		0			0
$787$	−1.57411e8			−0.322932			−0.161466	−	0.986878i	$-0.551622\pi$
$787$	−1.57411e8			−0.322932			−0.161466	+	0.986878i	$0.551622\pi$
$788$		0			0
$789$	1.14784e8	+	2.88190e8i	0.233695	+	0.586743i
$790$		0			0
$791$			3.30080e7i			0.0666944i
$792$		0			0
$793$	−9.34850e6			−0.0187466
$794$		0			0
$795$	1.36266e8	−	5.42737e7i	0.271198	−	0.108016i
$796$		0			0
$797$			5.11553e7i			0.101045i	0.998723	+	0.0505225i	$0.0160887\pi$
$797$			5.11553e7i			0.101045i	−0.998723	+	0.0505225i	$0.983911\pi$
$798$		0			0
$799$	8.95860e6			0.0175631
$800$		0			0
$801$	−4.14790e8	−	4.38107e8i	−0.807106	−	0.852477i
$802$		0			0
$803$		−	6.65971e8i		−	1.28620i
$804$		0			0
$805$	−1.27163e8			−0.243766
$806$		0			0
$807$	1.46735e8	+	3.68411e8i	0.279199	+	0.700990i
$808$		0			0
$809$		−	4.30177e8i		−	0.812459i	−0.913771	−	0.406229i	$-0.866843\pi$
$809$		−	4.30177e8i		−	0.812459i	0.913771	−	0.406229i	$-0.133157\pi$
$810$		0			0
$811$	1.35268e8			0.253590			0.126795	−	0.991929i	$-0.459531\pi$
$811$	1.35268e8			0.253590			0.126795	+	0.991929i	$0.459531\pi$
$812$		0			0
$813$	−4.45216e7	+	1.77326e7i	−0.0828513	+	0.0329990i
$814$		0			0
$815$		−	3.30120e7i		−	0.0609817i
$816$		0			0
$817$	−1.30110e8			−0.238586
$818$		0			0
$819$	−5.56818e6	+	5.27183e6i	−0.0101359	+	0.00959642i
$820$		0			0
$821$			6.01637e8i			1.08719i	0.839348	+	0.543595i	$0.182937\pi$
$821$			6.01637e8i			1.08719i	−0.839348	+	0.543595i	$0.817063\pi$
$822$		0			0
$823$	5.45245e8			0.978119			0.489060	−	0.872250i	$-0.337340\pi$
$823$	5.45245e8			0.978119			0.489060	+	0.872250i	$0.337340\pi$
$824$		0			0
$825$	3.60649e7	+	9.05488e7i	0.0642277	+	0.161258i
$826$		0			0
$827$			2.83751e8i			0.501673i	0.968030	+	0.250836i	$0.0807056\pi$
$827$			2.83751e8i			0.501673i	−0.968030	+	0.250836i	$0.919294\pi$
$828$		0			0
$829$	9.17319e7			0.161011			0.0805057	−	0.996754i	$-0.474346\pi$
$829$	9.17319e7			0.161011			0.0805057	+	0.996754i	$0.474346\pi$
$830$		0			0
$831$	6.88755e8	−	2.74326e8i	1.20022	−	0.478039i
$832$		0			0
$833$			5.44495e6i			0.00942018i
$834$		0			0
$835$	−2.27459e8			−0.390701
$836$		0			0
$837$	3.17353e8	−	6.85274e8i	0.541210	−	1.16866i
$838$		0			0
$839$		−	7.01084e8i		−	1.18709i	−0.804800	−	0.593545i	$-0.797728\pi$
$839$		−	7.01084e8i		−	1.18709i	0.804800	−	0.593545i	$-0.202272\pi$
$840$		0			0
$841$	2.53084e8			0.425477
$842$		0			0
$843$	−2.73152e7	−	6.85808e7i	−0.0455955	−	0.114477i
$844$		0			0
$845$			2.69485e8i			0.446647i
$846$		0			0
$847$	5.87807e7			0.0967352
$848$		0			0
$849$	−8.81747e8	+	3.51193e8i	−1.44086	+	0.573882i
$850$		0			0
$851$		−	3.59382e8i		−	0.583132i
$852$		0			0
$853$	1.36003e8			0.219130			0.109565	−	0.993980i	$-0.465054\pi$
$853$	1.36003e8			0.219130			0.109565	+	0.993980i	$0.465054\pi$
$854$		0			0
$855$	−5.55994e7	−	5.87249e7i	−0.0889553	−	0.0939559i
$856$		0			0
$857$		−	1.07337e9i		−	1.70532i	−0.522465	−	0.852661i	$-0.674987\pi$
$857$		−	1.07337e9i		−	1.70532i	0.522465	−	0.852661i	$-0.325013\pi$
$858$		0			0
$859$	−4.00149e8			−0.631309			−0.315654	−	0.948874i	$-0.602224\pi$
$859$	−4.00149e8			−0.631309			−0.315654	+	0.948874i	$0.602224\pi$
$860$		0			0
$861$	1.19530e8	+	3.00107e8i	0.187270	+	0.470182i
$862$		0			0
$863$			4.98364e8i			0.775379i	0.921790	+	0.387690i	$0.126727\pi$
$863$			4.98364e8i			0.775379i	−0.921790	+	0.387690i	$0.873273\pi$
$864$		0			0
$865$	−2.68867e8			−0.415422
$866$		0			0
$867$	6.05383e8	−	2.41119e8i	0.928908	−	0.369977i
$868$		0			0
$869$			7.24424e8i			1.10391i
$870$		0			0
$871$	1.11593e7			0.0168882
$872$		0			0
$873$	4.49301e8	−	4.25389e8i	0.675297	−	0.639356i
$874$		0			0
$875$			2.34891e7i			0.0350625i
$876$		0			0
$877$	−1.06775e9			−1.58296			−0.791481	−	0.611194i	$-0.790689\pi$
$877$	−1.06775e9			−1.58296			−0.791481	+	0.611194i	$0.790689\pi$
$878$		0			0
$879$	−4.65675e8	−	1.16918e9i	−0.685672	−	1.72153i
$880$		0			0
$881$		−	7.83809e7i		−	0.114626i	−0.998356	−	0.0573129i	$-0.981747\pi$
$881$		−	7.83809e7i		−	0.114626i	0.998356	−	0.0573129i	$-0.0182533\pi$
$882$		0			0
$883$	8.34186e8			1.21166			0.605830	−	0.795594i	$-0.292841\pi$
$883$	8.34186e8			1.21166			0.605830	+	0.795594i	$0.292841\pi$
$884$		0			0
$885$	4.01356e8	−	1.59857e8i	0.579029	−	0.230623i
$886$		0			0
$887$			6.77272e8i			0.970493i	0.874377	+	0.485247i	$0.161270\pi$
$887$			6.77272e8i			0.970493i	−0.874377	+	0.485247i	$0.838730\pi$
$888$		0			0
$889$	−6.50972e7			−0.0926525
$890$		0			0
$891$	−6.12983e8	−	3.35414e7i	−0.866594	−	0.0474185i
$892$		0			0
$893$			3.25092e8i			0.456512i
$894$		0			0
$895$	−3.44650e7			−0.0480739
$896$		0			0
$897$	1.32219e7	+	3.31965e7i	0.0183196	+	0.0459954i
$898$		0			0
$899$		−	7.09273e8i		−	0.976189i
$900$		0			0
$901$	5.31422e6			0.00726550
$902$		0			0
$903$	−2.21136e8	+	8.80765e7i	−0.300328	+	0.119618i
$904$		0			0
$905$		−	9.49511e6i		−	0.0128102i
$906$		0			0
$907$	3.70732e8			0.496865			0.248432	−	0.968649i	$-0.420085\pi$
$907$	3.70732e8			0.496865			0.248432	+	0.968649i	$0.420085\pi$
$908$		0			0
$909$	−5.58129e8	−	5.89504e8i	−0.743093	−	0.784865i
$910$		0			0
$911$		−	5.03944e8i		−	0.666541i	−0.942831	−	0.333271i	$-0.891848\pi$
$911$		−	5.03944e8i		−	0.666541i	0.942831	−	0.333271i	$-0.108152\pi$
$912$		0			0
$913$	1.14513e9			1.50467
$914$		0			0
$915$	−6.67423e7	−	1.67571e8i	−0.0871240	−	0.218744i
$916$		0			0
$917$			4.66336e8i			0.604771i
$918$		0			0
$919$	−3.59392e7			−0.0463043			−0.0231522	−	0.999732i	$-0.507370\pi$
$919$	−3.59392e7			−0.0463043			−0.0231522	+	0.999732i	$0.507370\pi$
$920$		0			0
$921$	7.82509e8	−	3.11667e8i	1.00164	−	0.398944i
$922$		0			0
$923$		−	2.84220e7i		−	0.0361451i
$924$		0			0
$925$	−6.63837e7			−0.0838757
$926$		0			0
$927$	1.10993e9	−	1.05086e9i	1.39334	−	1.31918i
$928$		0			0
$929$		−	4.60827e8i		−	0.574766i	−0.957816	−	0.287383i	$-0.907215\pi$
$929$		−	4.60827e8i		−	0.574766i	0.957816	−	0.287383i	$-0.0927853\pi$
$930$		0			0
$931$	−1.97588e8			−0.244856
$932$		0			0
$933$	3.51656e8	+	8.82911e8i	0.432986	+	1.08711i
$934$		0			0
$935$			3.53130e6i			0.00432016i
$936$		0			0
$937$	−5.28388e8			−0.642295			−0.321147	−	0.947029i	$-0.604068\pi$
$937$	−5.28388e8			−0.642295			−0.321147	+	0.947029i	$0.604068\pi$
$938$		0			0
$939$	−9.04492e8	+	3.60252e8i	−1.09247	+	0.435120i
$940$		0			0
$941$			2.96965e8i			0.356399i	0.983994	+	0.178200i	$0.0570273\pi$
$941$			2.96965e8i			0.356399i	−0.983994	+	0.178200i	$0.942973\pi$
$942$		0			0
$943$	1.50535e9			1.79516
$944$		0			0
$945$	−1.34250e8	−	6.21718e7i	−0.159082	−	0.0736712i
$946$		0			0
$947$			5.43742e8i			0.640240i	0.947377	+	0.320120i	$0.103723\pi$
$947$			5.43742e8i			0.640240i	−0.947377	+	0.320120i	$0.896277\pi$
$948$		0			0
$949$	−4.50993e7			−0.0527681
$950$		0			0
$951$	−5.73962e8	−	1.44106e9i	−0.667332	−	1.67548i
$952$		0			0
$953$			1.43357e9i			1.65630i	0.560506	+	0.828150i	$0.310607\pi$
$953$			1.43357e9i			1.65630i	−0.560506	+	0.828150i	$0.689393\pi$
$954$		0			0
$955$	−3.15145e8			−0.361826
$956$		0			0
$957$	−5.35649e8	+	2.13345e8i	−0.611146	+	0.243414i
$958$		0			0
$959$			2.34621e8i			0.266018i
$960$		0			0
$961$	5.84575e8			0.658673
$962$		0			0
$963$	1.21475e8	+	1.28303e8i	0.136021	+	0.143667i
$964$		0			0
$965$			1.08189e8i			0.120393i
$966$		0			0
$967$	1.86723e8			0.206500			0.103250	−	0.994655i	$-0.467076\pi$
$967$	1.86723e8			0.206500			0.103250	+	0.994655i	$0.467076\pi$
$968$		0			0
$969$	−1.08416e6	−	2.72202e6i	−0.00119157	−	0.00299171i
$970$		0			0
$971$			7.68364e8i			0.839284i	0.907690	+	0.419642i	$0.137844\pi$
$971$			7.68364e8i			0.839284i	−0.907690	+	0.419642i	$0.862156\pi$
$972$		0			0
$973$	−4.76741e8			−0.517540
$974$		0			0
$975$	6.13194e6	−	2.44230e6i	0.00661582	−	0.00263503i
$976$		0			0
$977$			8.60128e7i			0.0922316i	0.998936	+	0.0461158i	$0.0146843\pi$
$977$			8.60128e7i			0.0922316i	−0.998936	+	0.0461158i	$0.985316\pi$
$978$		0			0
$979$	−9.56002e8			−1.01885
$980$		0			0
$981$	−1.18302e9	+	1.12006e9i	−1.25310	+	1.18640i
$982$		0			0
$983$			9.61067e8i			1.01180i	0.862593	+	0.505898i	$0.168839\pi$
$983$			9.61067e8i			1.01180i	−0.862593	+	0.505898i	$0.831161\pi$
$984$		0			0
$985$	−3.15099e8			−0.329715
$986$		0			0
$987$	2.20068e8	+	5.52529e8i	0.228879	+	0.574650i
$988$		0			0
$989$			1.10923e9i			1.14665i
$990$		0			0
$991$	−6.24596e8			−0.641768			−0.320884	−	0.947118i	$-0.603980\pi$
$991$	−6.24596e8			−0.641768			−0.320884	+	0.947118i	$0.603980\pi$
$992$		0			0
$993$	8.18547e8	−	3.26021e8i	0.835980	−	0.332964i
$994$		0			0
$995$		−	3.69952e7i		−	0.0375557i
$996$		0			0
$997$	−1.53088e9			−1.54474			−0.772371	−	0.635171i	$-0.780929\pi$
$997$	−1.53088e9			−1.54474			−0.772371	+	0.635171i	$0.780929\pi$
$998$		0			0
$999$	1.75707e8	−	3.79411e8i	0.176235	−	0.380552i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	60.7.g.a.41.7	✓	8
3.2	odd	2	inner	60.7.g.a.41.8	yes	8
4.3	odd	2		240.7.l.c.161.2		8
5.2	odd	4		300.7.b.e.149.7		16
5.3	odd	4		300.7.b.e.149.10		16
5.4	even	2		300.7.g.h.101.2		8
12.11	even	2		240.7.l.c.161.1		8
15.2	even	4		300.7.b.e.149.9		16
15.8	even	4		300.7.b.e.149.8		16
15.14	odd	2		300.7.g.h.101.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.7.g.a.41.7	✓	8	1.1	even	1	trivial
60.7.g.a.41.8	yes	8	3.2	odd	2	inner
240.7.l.c.161.1		8	12.11	even	2
240.7.l.c.161.2		8	4.3	odd	2
300.7.b.e.149.7		16	5.2	odd	4
300.7.b.e.149.8		16	15.8	even	4
300.7.b.e.149.9		16	15.2	even	4
300.7.b.e.149.10		16	5.3	odd	4
300.7.g.h.101.1		8	15.14	odd	2
300.7.g.h.101.2		8	5.4	even	2

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:	\( N \)	\(=\)	\( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	60.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(25.0836 - 9.99060i) q^{3} -55.9017i q^{5} +134.460 q^{7} +(529.376 - 501.201i) q^{9} +O(q^{10})\)	\(q+(25.0836 - 9.99060i) q^{3} -55.9017i q^{5} +134.460 q^{7} +(529.376 - 501.201i) q^{9} -1155.16i q^{11} -78.2272 q^{13} +(-558.492 - 1402.22i) q^{15} -54.6849i q^{17} +1984.42 q^{19} +(3372.73 - 1343.33i) q^{21} -16917.8i q^{23} -3125.00 q^{25} +(8271.35 - 17860.7i) q^{27} -18486.2i q^{29} +38367.7 q^{31} +(-11540.8 - 28975.6i) q^{33} -7516.52i q^{35} +21242.8 q^{37} +(-1962.22 + 781.537i) q^{39} +88980.3i q^{41} -65565.7 q^{43} +(-28018.0 - 29593.0i) q^{45} +163822. i q^{47} -99569.6 q^{49} +(-546.335 - 1371.69i) q^{51} +97179.1i q^{53} -64575.5 q^{55} +(49776.4 - 19825.5i) q^{57} +286230. i q^{59} +119504. q^{61} +(71179.6 - 67391.2i) q^{63} +4373.03i q^{65} -142653. q^{67} +(-169019. - 424360. i) q^{69} +363327. i q^{71} +576517. q^{73} +(-78386.3 + 31220.6i) q^{75} -155322. i q^{77} -627119. q^{79} +(29036.1 - 530647. i) q^{81} +991316. i q^{83} -3056.98 q^{85} +(-184688. - 463701. i) q^{87} -827592. i q^{89} -10518.4 q^{91} +(962400. - 383316. i) q^{93} -110932. i q^{95} +848738. q^{97} +(-578968. - 611514. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 20 q^{3} - 560 q^{7} + 1492 q^{9}+O(q^{10}) \) 8 * q - 20 * q^3 - 560 * q^7 + 1492 * q^9	\( 8 q - 20 q^{3} - 560 q^{7} + 1492 q^{9} - 6440 q^{13} + 2000 q^{15} - 15272 q^{19} - 868 q^{21} - 25000 q^{25} - 18620 q^{27} + 35032 q^{31} - 111120 q^{33} + 99880 q^{37} + 39608 q^{39} - 161000 q^{43} - 5500 q^{45} + 202560 q^{49} + 429120 q^{51} - 33000 q^{55} - 27160 q^{57} - 135608 q^{61} + 377240 q^{63} + 404920 q^{67} - 254940 q^{69} - 356960 q^{73} + 62500 q^{75} + 707704 q^{79} - 1198112 q^{81} + 828000 q^{85} - 1528440 q^{87} - 2004112 q^{91} - 467920 q^{93} - 1326320 q^{97} + 2650080 q^{99}+O(q^{100}) \) 8 * q - 20 * q^3 - 560 * q^7 + 1492 * q^9 - 6440 * q^13 + 2000 * q^15 - 15272 * q^19 - 868 * q^21 - 25000 * q^25 - 18620 * q^27 + 35032 * q^31 - 111120 * q^33 + 99880 * q^37 + 39608 * q^39 - 161000 * q^43 - 5500 * q^45 + 202560 * q^49 + 429120 * q^51 - 33000 * q^55 - 27160 * q^57 - 135608 * q^61 + 377240 * q^63 + 404920 * q^67 - 254940 * q^69 - 356960 * q^73 + 62500 * q^75 + 707704 * q^79 - 1198112 * q^81 + 828000 * q^85 - 1528440 * q^87 - 2004112 * q^91 - 467920 * q^93 - 1326320 * q^97 + 2650080 * q^99

Introduction

L-functions

Modular forms

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