LMFDB - Embedded newform 1170.2.e.g.469.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1170,2,Mod(469,1170)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1170, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1170.469");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1170.e (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:	$9.34249703649$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			469.6
Root			$0.403032 - 0.403032i$ of defining polynomial
Character	$\chi$	$=$	1170.469
Dual form			1170.2.e.g.469.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +(1.67513 + 1.48119i) q^{5} -1.19394i q^{7} -1.00000i q^{8} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000 q^{4} +(1.67513 + 1.48119i) q^{5} -1.19394i q^{7} -1.00000i q^{8} +(-1.48119 + 1.67513i) q^{10} -2.00000 q^{11} +1.00000i q^{13} +1.19394 q^{14} +1.00000 q^{16} +4.54420i q^{17} -4.15633 q^{19} +(-1.67513 - 1.48119i) q^{20} -2.00000i q^{22} +7.11871i q^{23} +(0.612127 + 4.96239i) q^{25} -1.00000 q^{26} +1.19394i q^{28} +10.7308 q^{29} -5.35026 q^{31} +1.00000i q^{32} -4.54420 q^{34} +(1.76845 - 2.00000i) q^{35} +3.92478i q^{37} -4.15633i q^{38} +(1.48119 - 1.67513i) q^{40} -1.03761 q^{41} +10.8872i q^{43} +2.00000 q^{44} -7.11871 q^{46} -1.61213i q^{47} +5.57452 q^{49} +(-4.96239 + 0.612127i) q^{50} -1.00000i q^{52} -4.18664i q^{53} +(-3.35026 - 2.96239i) q^{55} -1.19394 q^{56} +10.7308i q^{58} -2.31265 q^{59} -7.08840 q^{61} -5.35026i q^{62} -1.00000 q^{64} +(-1.48119 + 1.67513i) q^{65} +4.70052i q^{67} -4.54420i q^{68} +(2.00000 + 1.76845i) q^{70} +9.27504 q^{71} -3.58181i q^{73} -3.92478 q^{74} +4.15633 q^{76} +2.38787i q^{77} -15.1998 q^{79} +(1.67513 + 1.48119i) q^{80} -1.03761i q^{82} -1.73813i q^{83} +(-6.73084 + 7.61213i) q^{85} -10.8872 q^{86} +2.00000i q^{88} +14.3127 q^{89} +1.19394 q^{91} -7.11871i q^{92} +1.61213 q^{94} +(-6.96239 - 6.15633i) q^{95} -9.19394i q^{97} +5.57452i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{4}+O(q^{10}) $ 6 * q - 6 * q^4	$ 6 q - 6 q^{4} + 2 q^{10} - 12 q^{11} + 8 q^{14} + 6 q^{16} - 4 q^{19} + 2 q^{25} - 6 q^{26} + 20 q^{29} - 12 q^{31} - 8 q^{34} - 12 q^{35} - 2 q^{40} - 28 q^{41} + 12 q^{44} + 10 q^{49} - 8 q^{50} - 8 q^{56} + 28 q^{59} - 4 q^{61} - 6 q^{64} + 2 q^{65} + 12 q^{70} - 8 q^{71} + 20 q^{74} + 4 q^{76} + 16 q^{79} + 4 q^{85} + 44 q^{89} + 8 q^{91} + 8 q^{94} - 20 q^{95}+O(q^{100}) $ 6 * q - 6 * q^4 + 2 * q^10 - 12 * q^11 + 8 * q^14 + 6 * q^16 - 4 * q^19 + 2 * q^25 - 6 * q^26 + 20 * q^29 - 12 * q^31 - 8 * q^34 - 12 * q^35 - 2 * q^40 - 28 * q^41 + 12 * q^44 + 10 * q^49 - 8 * q^50 - 8 * q^56 + 28 * q^59 - 4 * q^61 - 6 * q^64 + 2 * q^65 + 12 * q^70 - 8 * q^71 + 20 * q^74 + 4 * q^76 + 16 * q^79 + 4 * q^85 + 44 * q^89 + 8 * q^91 + 8 * q^94 - 20 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1170\mathbb{Z}\right)^\times$.

$n$	$911$	$937$	$1081$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$	1.67513	+	1.48119i	0.749141	+	0.662410i
$5$	1.67513	+	1.48119i	0.749141	+	0.662410i
$6$		0			0
$7$		−	1.19394i		−	0.451266i	−0.974212	−	0.225633i	$-0.927555\pi$
$7$		−	1.19394i		−	0.451266i	0.974212	−	0.225633i	$-0.0724450\pi$
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$	−1.48119	+	1.67513i	−0.468395	+	0.529723i
$11$	−2.00000			−0.603023			−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000			−0.603023			−0.301511	+	0.953463i	$0.597491\pi$
$12$		0			0
$13$			1.00000i			0.277350i
$13$			1.00000i			0.277350i
$14$	1.19394			0.319093
$15$		0			0
$16$	1.00000			0.250000
$17$			4.54420i			1.10213i	0.834462	+	0.551065i	$0.185778\pi$
$17$			4.54420i			1.10213i	−0.834462	+	0.551065i	$0.814222\pi$
$18$		0			0
$19$	−4.15633			−0.953526			−0.476763	−	0.879032i	$-0.658190\pi$
$19$	−4.15633			−0.953526			−0.476763	+	0.879032i	$0.658190\pi$
$20$	−1.67513	−	1.48119i	−0.374571	−	0.331205i
$21$		0			0
$22$		−	2.00000i		−	0.426401i
$23$			7.11871i			1.48435i	0.670204	+	0.742177i	$0.266207\pi$
$23$			7.11871i			1.48435i	−0.670204	+	0.742177i	$0.733793\pi$
$24$		0			0
$25$	0.612127	+	4.96239i	0.122425	+	0.992478i
$26$	−1.00000			−0.196116
$27$		0			0
$28$			1.19394i			0.225633i
$29$	10.7308			1.99267			0.996334	−	0.0855539i	$-0.0272660\pi$
$29$	10.7308			1.99267			0.996334	+	0.0855539i	$0.0272660\pi$
$30$		0			0
$31$	−5.35026			−0.960935			−0.480468	−	0.877012i	$-0.659533\pi$
$31$	−5.35026			−0.960935			−0.480468	+	0.877012i	$0.659533\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$	−4.54420			−0.779324
$35$	1.76845	−	2.00000i	0.298923	−	0.338062i
$36$		0			0
$37$			3.92478i			0.645229i	0.946530	+	0.322615i	$0.104562\pi$
$37$			3.92478i			0.645229i	−0.946530	+	0.322615i	$0.895438\pi$
$38$		−	4.15633i		−	0.674245i
$39$		0			0
$40$	1.48119	−	1.67513i	0.234197	−	0.264861i
$41$	−1.03761			−0.162048			−0.0810238	−	0.996712i	$-0.525819\pi$
$41$	−1.03761			−0.162048			−0.0810238	+	0.996712i	$0.525819\pi$
$42$		0			0
$43$			10.8872i			1.66028i	0.557557	+	0.830139i	$0.311739\pi$
$43$			10.8872i			1.66028i	−0.557557	+	0.830139i	$0.688261\pi$
$44$	2.00000			0.301511
$45$		0			0
$46$	−7.11871			−1.04960
$47$		−	1.61213i		−	0.235153i	−0.993064	−	0.117576i	$-0.962487\pi$
$47$		−	1.61213i		−	0.235153i	0.993064	−	0.117576i	$-0.0375125\pi$
$48$		0			0
$49$	5.57452			0.796359
$50$	−4.96239	+	0.612127i	−0.701788	+	0.0865678i
$51$		0			0
$52$		−	1.00000i		−	0.138675i
$53$		−	4.18664i		−	0.575080i	−0.957769	−	0.287540i	$-0.907163\pi$
$53$		−	4.18664i		−	0.575080i	0.957769	−	0.287540i	$-0.0928374\pi$
$54$		0			0
$55$	−3.35026	−	2.96239i	−0.451749	−	0.399448i
$56$	−1.19394			−0.159546
$57$		0			0
$58$			10.7308i			1.40903i
$59$	−2.31265			−0.301081			−0.150541	−	0.988604i	$-0.548101\pi$
$59$	−2.31265			−0.301081			−0.150541	+	0.988604i	$0.548101\pi$
$60$		0			0
$61$	−7.08840			−0.907576			−0.453788	−	0.891110i	$-0.649928\pi$
$61$	−7.08840			−0.907576			−0.453788	+	0.891110i	$0.649928\pi$
$62$		−	5.35026i		−	0.679484i
$63$		0			0
$64$	−1.00000			−0.125000
$65$	−1.48119	+	1.67513i	−0.183720	+	0.207774i
$66$		0			0
$67$			4.70052i			0.574260i	0.957892	+	0.287130i	$0.0927012\pi$
$67$			4.70052i			0.574260i	−0.957892	+	0.287130i	$0.907299\pi$
$68$		−	4.54420i		−	0.551065i
$69$		0			0
$70$	2.00000	+	1.76845i	0.239046	+	0.211370i
$71$	9.27504			1.10074			0.550372	−	0.834919i	$-0.314486\pi$
$71$	9.27504			1.10074			0.550372	+	0.834919i	$0.314486\pi$
$72$		0			0
$73$		−	3.58181i		−	0.419219i	−0.977785	−	0.209610i	$-0.932781\pi$
$73$		−	3.58181i		−	0.419219i	0.977785	−	0.209610i	$-0.0672193\pi$
$74$	−3.92478			−0.456246
$75$		0			0
$76$	4.15633			0.476763
$77$			2.38787i			0.272123i
$78$		0			0
$79$	−15.1998			−1.71011			−0.855056	−	0.518535i	$-0.826478\pi$
$79$	−15.1998			−1.71011			−0.855056	+	0.518535i	$0.826478\pi$
$80$	1.67513	+	1.48119i	0.187285	+	0.165603i
$81$		0			0
$82$		−	1.03761i		−	0.114585i
$83$		−	1.73813i		−	0.190785i	−0.995440	−	0.0953925i	$-0.969589\pi$
$83$		−	1.73813i		−	0.190785i	0.995440	−	0.0953925i	$-0.0304106\pi$
$84$		0			0
$85$	−6.73084	+	7.61213i	−0.730062	+	0.825651i
$86$	−10.8872			−1.17399
$87$		0			0
$88$			2.00000i			0.213201i
$89$	14.3127			1.51714			0.758569	−	0.651593i	$-0.225899\pi$
$89$	14.3127			1.51714			0.758569	+	0.651593i	$0.225899\pi$
$90$		0			0
$91$	1.19394			0.125159
$92$		−	7.11871i		−	0.742177i
$93$		0			0
$94$	1.61213			0.166278
$95$	−6.96239	−	6.15633i	−0.714326	−	0.631626i
$96$		0			0
$97$		−	9.19394i		−	0.933503i	−0.884389	−	0.466751i	$-0.845424\pi$
$97$		−	9.19394i		−	0.933503i	0.884389	−	0.466751i	$-0.154576\pi$
$98$			5.57452i			0.563111i
$99$		0			0
$100$	−0.612127	−	4.96239i	−0.0612127	−	0.496239i
$101$	−13.8945			−1.38255			−0.691275	−	0.722592i	$-0.742951\pi$
$101$	−13.8945			−1.38255			−0.691275	+	0.722592i	$0.742951\pi$
$102$		0			0
$103$		−	7.79877i		−	0.768436i	−0.923242	−	0.384218i	$-0.874471\pi$
$103$		−	7.79877i		−	0.768436i	0.923242	−	0.384218i	$-0.125529\pi$
$104$	1.00000			0.0980581
$105$		0			0
$106$	4.18664			0.406643
$107$			11.5369i			1.11531i	0.830071	+	0.557657i	$0.188300\pi$
$107$			11.5369i			1.11531i	−0.830071	+	0.557657i	$0.811700\pi$
$108$		0			0
$109$	1.58181			0.151510			0.0757549	−	0.997126i	$-0.475863\pi$
$109$	1.58181			0.151510			0.0757549	+	0.997126i	$0.475863\pi$
$110$	2.96239	−	3.35026i	0.282453	−	0.319435i
$111$		0			0
$112$		−	1.19394i		−	0.112816i
$113$			14.4690i			1.36113i	0.732689	+	0.680563i	$0.238265\pi$
$113$			14.4690i			1.36113i	−0.732689	+	0.680563i	$0.761735\pi$
$114$		0			0
$115$	−10.5442	+	11.9248i	−0.983252	+	1.11199i
$116$	−10.7308			−0.996334
$117$		0			0
$118$		−	2.31265i		−	0.212897i
$119$	5.42548			0.497353
$120$		0			0
$121$	−7.00000			−0.636364
$122$		−	7.08840i		−	0.641753i
$123$		0			0
$124$	5.35026			0.480468
$125$	−6.32487	+	9.21933i	−0.565713	+	0.824602i
$126$		0			0
$127$		−	2.18664i		−	0.194033i	−0.995283	−	0.0970166i	$-0.969070\pi$
$127$		−	2.18664i		−	0.194033i	0.995283	−	0.0970166i	$-0.0309300\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$	−1.67513	−	1.48119i	−0.146919	−	0.129909i
$131$	−5.64244			−0.492983			−0.246491	−	0.969145i	$-0.579278\pi$
$131$	−5.64244			−0.492983			−0.246491	+	0.969145i	$0.579278\pi$
$132$		0			0
$133$			4.96239i			0.430294i
$134$	−4.70052			−0.406063
$135$		0			0
$136$	4.54420			0.389662
$137$		−	2.46310i		−	0.210436i	−0.994449	−	0.105218i	$-0.966446\pi$
$137$		−	2.46310i		−	0.210436i	0.994449	−	0.105218i	$-0.0335541\pi$
$138$		0			0
$139$	16.6253			1.41014			0.705070	−	0.709138i	$-0.250916\pi$
$139$	16.6253			1.41014			0.705070	+	0.709138i	$0.250916\pi$
$140$	−1.76845	+	2.00000i	−0.149461	+	0.169031i
$141$		0			0
$142$			9.27504i			0.778344i
$143$		−	2.00000i		−	0.167248i
$144$		0			0
$145$	17.9756	+	15.8945i	1.49279	+	1.31996i
$146$	3.58181			0.296433
$147$		0			0
$148$		−	3.92478i		−	0.322615i
$149$	22.0508			1.80647			0.903235	−	0.429146i	$-0.141185\pi$
$149$	22.0508			1.80647			0.903235	+	0.429146i	$0.141185\pi$
$150$		0			0
$151$	−2.96239			−0.241076			−0.120538	−	0.992709i	$-0.538462\pi$
$151$	−2.96239			−0.241076			−0.120538	+	0.992709i	$0.538462\pi$
$152$			4.15633i			0.337122i
$153$		0			0
$154$	−2.38787			−0.192420
$155$	−8.96239	−	7.92478i	−0.719876	−	0.636533i
$156$		0			0
$157$		−	20.3634i		−	1.62518i	−0.582836	−	0.812590i	$-0.698057\pi$
$157$		−	20.3634i		−	1.62518i	0.582836	−	0.812590i	$-0.301943\pi$
$158$		−	15.1998i		−	1.20923i
$159$		0			0
$160$	−1.48119	+	1.67513i	−0.117099	+	0.132431i
$161$	8.49929			0.669838
$162$		0			0
$163$		−	25.3258i		−	1.98367i	−0.127521	−	0.991836i	$-0.540702\pi$
$163$		−	25.3258i		−	1.98367i	0.127521	−	0.991836i	$-0.459298\pi$
$164$	1.03761			0.0810238
$165$		0			0
$166$	1.73813			0.134905
$167$		−	6.70052i		−	0.518502i	−0.965810	−	0.259251i	$-0.916524\pi$
$167$		−	6.70052i		−	0.518502i	0.965810	−	0.259251i	$-0.0834757\pi$
$168$		0			0
$169$	−1.00000			−0.0769231
$170$	−7.61213	−	6.73084i	−0.583823	−	0.516232i
$171$		0			0
$172$		−	10.8872i		−	0.830139i
$173$		−	0.649738i		−	0.0493987i	−0.999695	−	0.0246993i	$-0.992137\pi$
$173$		−	0.649738i		−	0.0493987i	0.999695	−	0.0246993i	$-0.00786284\pi$
$174$		0			0
$175$	5.92478	−	0.730841i	0.447871	−	0.0552464i
$176$	−2.00000			−0.150756
$177$		0			0
$178$			14.3127i			1.07278i
$179$	6.10554			0.456349			0.228175	−	0.973620i	$-0.426724\pi$
$179$	6.10554			0.456349			0.228175	+	0.973620i	$0.426724\pi$
$180$		0			0
$181$	22.9380			1.70496			0.852482	−	0.522756i	$-0.175096\pi$
$181$	22.9380			1.70496			0.852482	+	0.522756i	$0.175096\pi$
$182$			1.19394i			0.0885005i
$183$		0			0
$184$	7.11871			0.524799
$185$	−5.81336	+	6.57452i	−0.427407	+	0.483368i
$186$		0			0
$187$		−	9.08840i		−	0.664609i
$188$			1.61213i			0.117576i
$189$		0			0
$190$	6.15633	−	6.96239i	0.446627	−	0.505105i
$191$	−19.0132			−1.37574			−0.687872	−	0.725832i	$-0.741455\pi$
$191$	−19.0132			−1.37574			−0.687872	+	0.725832i	$0.741455\pi$
$192$		0			0
$193$		−	4.41819i		−	0.318028i	−0.987276	−	0.159014i	$-0.949168\pi$
$193$		−	4.41819i		−	0.318028i	0.987276	−	0.159014i	$-0.0508315\pi$
$194$	9.19394			0.660086
$195$		0			0
$196$	−5.57452			−0.398180
$197$		−	5.22425i		−	0.372213i	−0.982530	−	0.186106i	$-0.940413\pi$
$197$		−	5.22425i		−	0.372213i	0.982530	−	0.186106i	$-0.0595869\pi$
$198$		0			0
$199$	19.1998			1.36104			0.680519	−	0.732730i	$-0.261754\pi$
$199$	19.1998			1.36104			0.680519	+	0.732730i	$0.261754\pi$
$200$	4.96239	−	0.612127i	0.350894	−	0.0432839i
$201$		0			0
$202$		−	13.8945i		−	0.977611i
$203$		−	12.8119i		−	0.899222i
$204$		0			0
$205$	−1.73813	−	1.53690i	−0.121397	−	0.107342i
$206$	7.79877			0.543366
$207$		0			0
$208$			1.00000i			0.0693375i
$209$	8.31265			0.574998
$210$		0			0
$211$	9.86414			0.679076			0.339538	−	0.940592i	$-0.389729\pi$
$211$	9.86414			0.679076			0.339538	+	0.940592i	$0.389729\pi$
$212$			4.18664i			0.287540i
$213$		0			0
$214$	−11.5369			−0.788647
$215$	−16.1260	+	18.2374i	−1.09978	+	1.24378i
$216$		0			0
$217$			6.38787i			0.433637i
$218$			1.58181i			0.107134i
$219$		0			0
$220$	3.35026	+	2.96239i	0.225875	+	0.199724i
$221$	−4.54420			−0.305676
$222$		0			0
$223$			18.9076i			1.26615i	0.774091	+	0.633074i	$0.218207\pi$
$223$			18.9076i			1.26615i	−0.774091	+	0.633074i	$0.781793\pi$
$224$	1.19394			0.0797732
$225$		0			0
$226$	−14.4690			−0.962462
$227$		−	19.3258i		−	1.28270i	−0.767248	−	0.641350i	$-0.778375\pi$
$227$		−	19.3258i		−	1.28270i	0.767248	−	0.641350i	$-0.221625\pi$
$228$		0			0
$229$	−11.5672			−0.764383			−0.382192	−	0.924083i	$-0.624831\pi$
$229$	−11.5672			−0.764383			−0.382192	+	0.924083i	$0.624831\pi$
$230$	−11.9248	−	10.5442i	−0.786297	−	0.695264i
$231$		0			0
$232$		−	10.7308i		−	0.704514i
$233$			10.7816i			0.706328i	0.935561	+	0.353164i	$0.114894\pi$
$233$			10.7816i			0.706328i	−0.935561	+	0.353164i	$0.885106\pi$
$234$		0			0
$235$	2.38787	−	2.70052i	0.155768	−	0.176163i
$236$	2.31265			0.150541
$237$		0			0
$238$			5.42548i			0.351682i
$239$	21.9248			1.41820			0.709098	−	0.705110i	$-0.249102\pi$
$239$	21.9248			1.41820			0.709098	+	0.705110i	$0.249102\pi$
$240$		0			0
$241$	−4.23743			−0.272957			−0.136478	−	0.990643i	$-0.543578\pi$
$241$	−4.23743			−0.272957			−0.136478	+	0.990643i	$0.543578\pi$
$242$		−	7.00000i		−	0.449977i
$243$		0			0
$244$	7.08840			0.453788
$245$	9.33804	+	8.25694i	0.596586	+	0.527517i
$246$		0			0
$247$		−	4.15633i		−	0.264461i
$248$			5.35026i			0.339742i
$249$		0			0
$250$	−9.21933	−	6.32487i	−0.583082	−	0.400020i
$251$	−1.95509			−0.123404			−0.0617022	−	0.998095i	$-0.519653\pi$
$251$	−1.95509			−0.123404			−0.0617022	+	0.998095i	$0.519653\pi$
$252$		0			0
$253$		−	14.2374i		−	0.895099i
$254$	2.18664			0.137202
$255$		0			0
$256$	1.00000			0.0625000
$257$			6.78163i			0.423026i	0.977375	+	0.211513i	$0.0678391\pi$
$257$			6.78163i			0.423026i	−0.977375	+	0.211513i	$0.932161\pi$
$258$		0			0
$259$	4.68594			0.291170
$260$	1.48119	−	1.67513i	0.0918598	−	0.103887i
$261$		0			0
$262$		−	5.64244i		−	0.348591i
$263$		−	9.96968i		−	0.614757i	−0.951587	−	0.307378i	$-0.900548\pi$
$263$		−	9.96968i		−	0.614757i	0.951587	−	0.307378i	$-0.0994518\pi$
$264$		0			0
$265$	6.20123	−	7.01317i	0.380939	−	0.430816i
$266$	−4.96239			−0.304264
$267$		0			0
$268$		−	4.70052i		−	0.287130i
$269$	9.64244			0.587910			0.293955	−	0.955819i	$-0.405028\pi$
$269$	9.64244			0.587910			0.293955	+	0.955819i	$0.405028\pi$
$270$		0			0
$271$	21.1392			1.28411			0.642057	−	0.766657i	$-0.278081\pi$
$271$	21.1392			1.28411			0.642057	+	0.766657i	$0.278081\pi$
$272$			4.54420i			0.275532i
$273$		0			0
$274$	2.46310			0.148801
$275$	−1.22425	−	9.92478i	−0.0738253	−	0.598487i
$276$		0			0
$277$		−	7.48612i		−	0.449797i	−0.974382	−	0.224899i	$-0.927795\pi$
$277$		−	7.48612i		−	0.449797i	0.974382	−	0.224899i	$-0.0722051\pi$
$278$			16.6253i			0.997119i
$279$		0			0
$280$	−2.00000	−	1.76845i	−0.119523	−	0.105685i
$281$	18.1622			1.08347			0.541733	−	0.840551i	$-0.317768\pi$
$281$	18.1622			1.08347			0.541733	+	0.840551i	$0.317768\pi$
$282$		0			0
$283$		−	11.8134i		−	0.702231i	−0.936332	−	0.351116i	$-0.885802\pi$
$283$		−	11.8134i		−	0.702231i	0.936332	−	0.351116i	$-0.114198\pi$
$284$	−9.27504			−0.550372
$285$		0			0
$286$	2.00000			0.118262
$287$			1.23884i			0.0731265i
$288$		0			0
$289$	−3.64974			−0.214690
$290$	−15.8945	+	17.9756i	−0.933355	+	1.05556i
$291$		0			0
$292$			3.58181i			0.209610i
$293$			12.7005i			0.741973i	0.928638	+	0.370986i	$0.120980\pi$
$293$			12.7005i			0.741973i	−0.928638	+	0.370986i	$0.879020\pi$
$294$		0			0
$295$	−3.87399	−	3.42548i	−0.225553	−	0.199439i
$296$	3.92478			0.228123
$297$		0			0
$298$			22.0508i			1.27737i
$299$	−7.11871			−0.411686
$300$		0			0
$301$	12.9986			0.749226
$302$		−	2.96239i		−	0.170466i
$303$		0			0
$304$	−4.15633			−0.238382
$305$	−11.8740	−	10.4993i	−0.679903	−	0.601188i
$306$		0			0
$307$			6.98683i			0.398759i	0.979922	+	0.199380i	$0.0638927\pi$
$307$			6.98683i			0.398759i	−0.979922	+	0.199380i	$0.936107\pi$
$308$		−	2.38787i		−	0.136062i
$309$		0			0
$310$	7.92478	−	8.96239i	0.450097	−	0.509029i
$311$	7.53690			0.427379			0.213689	−	0.976902i	$-0.431452\pi$
$311$	7.53690			0.427379			0.213689	+	0.976902i	$0.431452\pi$
$312$		0			0
$313$		−	21.7137i		−	1.22733i	−0.789566	−	0.613665i	$-0.789694\pi$
$313$		−	21.7137i		−	1.22733i	0.789566	−	0.613665i	$-0.210306\pi$
$314$	20.3634			1.14918
$315$		0			0
$316$	15.1998			0.855056
$317$			28.3634i			1.59305i	0.604606	+	0.796525i	$0.293331\pi$
$317$			28.3634i			1.59305i	−0.604606	+	0.796525i	$0.706669\pi$
$318$		0			0
$319$	−21.4617			−1.20162
$320$	−1.67513	−	1.48119i	−0.0936427	−	0.0828013i
$321$		0			0
$322$			8.49929i			0.473647i
$323$		−	18.8872i		−	1.05091i
$324$		0			0
$325$	−4.96239	+	0.612127i	−0.275264	+	0.0339547i
$326$	25.3258			1.40267
$327$		0			0
$328$			1.03761i			0.0572925i
$329$	−1.92478			−0.106116
$330$		0			0
$331$	12.6801			0.696959			0.348479	−	0.937316i	$-0.386698\pi$
$331$	12.6801			0.696959			0.348479	+	0.937316i	$0.386698\pi$
$332$			1.73813i			0.0953925i
$333$		0			0
$334$	6.70052			0.366636
$335$	−6.96239	+	7.87399i	−0.380396	+	0.430202i
$336$		0			0
$337$			18.1768i			0.990153i	0.868849	+	0.495077i	$0.164860\pi$
$337$			18.1768i			0.990153i	−0.868849	+	0.495077i	$0.835140\pi$
$338$		−	1.00000i		−	0.0543928i
$339$		0			0
$340$	6.73084	−	7.61213i	0.365031	−	0.412826i
$341$	10.7005			0.579466
$342$		0			0
$343$		−	15.0132i		−	0.810635i
$344$	10.8872			0.586997
$345$		0			0
$346$	0.649738			0.0349301
$347$			10.7612i			0.577689i	0.957376	+	0.288845i	$0.0932711\pi$
$347$			10.7612i			0.577689i	−0.957376	+	0.288845i	$0.906729\pi$
$348$		0			0
$349$	12.8061			0.685493			0.342746	−	0.939428i	$-0.388643\pi$
$349$	12.8061			0.685493			0.342746	+	0.939428i	$0.388643\pi$
$350$	0.730841	+	5.92478i	0.0390651	+	0.316693i
$351$		0			0
$352$		−	2.00000i		−	0.106600i
$353$			21.5369i			1.14629i	0.819453	+	0.573147i	$0.194278\pi$
$353$			21.5369i			1.14629i	−0.819453	+	0.573147i	$0.805722\pi$
$354$		0			0
$355$	15.5369	+	13.7381i	0.824613	+	0.729144i
$356$	−14.3127			−0.758569
$357$		0			0
$358$			6.10554i			0.322688i
$359$	14.8265			0.782514			0.391257	−	0.920281i	$-0.372040\pi$
$359$	14.8265			0.782514			0.391257	+	0.920281i	$0.372040\pi$
$360$		0			0
$361$	−1.72496			−0.0907874
$362$			22.9380i			1.20559i
$363$		0			0
$364$	−1.19394			−0.0625793
$365$	5.30536	−	6.00000i	0.277695	−	0.314054i
$366$		0			0
$367$			37.5125i			1.95813i	0.203536	+	0.979067i	$0.434757\pi$
$367$			37.5125i			1.95813i	−0.203536	+	0.979067i	$0.565243\pi$
$368$			7.11871i			0.371089i
$369$		0			0
$370$	−6.57452	−	5.81336i	−0.341793	−	0.302222i
$371$	−4.99859			−0.259514
$372$		0			0
$373$		−	18.1866i		−	0.941669i	−0.882222	−	0.470834i	$-0.843953\pi$
$373$		−	18.1866i		−	0.941669i	0.882222	−	0.470834i	$-0.156047\pi$
$374$	9.08840			0.469950
$375$		0			0
$376$	−1.61213			−0.0831391
$377$			10.7308i			0.552666i
$378$		0			0
$379$	24.6312			1.26522			0.632609	−	0.774471i	$-0.281984\pi$
$379$	24.6312			1.26522			0.632609	+	0.774471i	$0.281984\pi$
$380$	6.96239	+	6.15633i	0.357163	+	0.315813i
$381$		0			0
$382$		−	19.0132i		−	0.972799i
$383$		−	1.14903i		−	0.0587127i	−0.999569	−	0.0293564i	$-0.990654\pi$
$383$		−	1.14903i		−	0.0587127i	0.999569	−	0.0293564i	$-0.00934577\pi$
$384$		0			0
$385$	−3.53690	+	4.00000i	−0.180257	+	0.203859i
$386$	4.41819			0.224880
$387$		0			0
$388$			9.19394i			0.466751i
$389$	−3.50659			−0.177791			−0.0888955	−	0.996041i	$-0.528334\pi$
$389$	−3.50659			−0.177791			−0.0888955	+	0.996041i	$0.528334\pi$
$390$		0			0
$391$	−32.3488			−1.63595
$392$		−	5.57452i		−	0.281556i
$393$		0			0
$394$	5.22425			0.263194
$395$	−25.4617	−	22.5139i	−1.28112	−	1.13280i
$396$		0			0
$397$		−	28.8627i		−	1.44858i	−0.689496	−	0.724289i	$-0.742168\pi$
$397$		−	28.8627i		−	1.44858i	0.689496	−	0.724289i	$-0.257832\pi$
$398$			19.1998i			0.962400i
$399$		0			0
$400$	0.612127	+	4.96239i	0.0306063	+	0.248119i
$401$	−17.0376			−0.850818			−0.425409	−	0.905001i	$-0.639870\pi$
$401$	−17.0376			−0.850818			−0.425409	+	0.905001i	$0.639870\pi$
$402$		0			0
$403$		−	5.35026i		−	0.266516i
$404$	13.8945			0.691275
$405$		0			0
$406$	12.8119			0.635846
$407$		−	7.84955i		−	0.389088i
$408$		0			0
$409$	−6.77575			−0.335039			−0.167520	−	0.985869i	$-0.553576\pi$
$409$	−6.77575			−0.335039			−0.167520	+	0.985869i	$0.553576\pi$
$410$	1.53690	−	1.73813i	0.0759023	−	0.0858404i
$411$		0			0
$412$			7.79877i			0.384218i
$413$			2.76116i			0.135868i
$414$		0			0
$415$	2.57452	−	2.91160i	0.126378	−	0.142925i
$416$	−1.00000			−0.0490290
$417$		0			0
$418$			8.31265i			0.406585i
$419$	−20.7572			−1.01406			−0.507028	−	0.861930i	$-0.669256\pi$
$419$	−20.7572			−1.01406			−0.507028	+	0.861930i	$0.669256\pi$
$420$		0			0
$421$	−36.9683			−1.80172			−0.900862	−	0.434106i	$-0.857064\pi$
$421$	−36.9683			−1.80172			−0.900862	+	0.434106i	$0.857064\pi$
$422$			9.86414i			0.480179i
$423$		0			0
$424$	−4.18664			−0.203321
$425$	−22.5501	+	2.78163i	−1.09384	+	0.134929i
$426$		0			0
$427$			8.46310i			0.409558i
$428$		−	11.5369i		−	0.557657i
$429$		0			0
$430$	−18.2374	−	16.1260i	−0.879487	−	0.777665i
$431$	−16.6253			−0.800813			−0.400406	−	0.916338i	$-0.631131\pi$
$431$	−16.6253			−0.800813			−0.400406	+	0.916338i	$0.631131\pi$
$432$		0			0
$433$		−	4.89701i		−	0.235336i	−0.993053	−	0.117668i	$-0.962458\pi$
$433$		−	4.89701i		−	0.235336i	0.993053	−	0.117668i	$-0.0375418\pi$
$434$	−6.38787			−0.306628
$435$		0			0
$436$	−1.58181			−0.0757549
$437$		−	29.5877i		−	1.41537i
$438$		0			0
$439$	−4.62530			−0.220754			−0.110377	−	0.993890i	$-0.535206\pi$
$439$	−4.62530			−0.220754			−0.110377	+	0.993890i	$0.535206\pi$
$440$	−2.96239	+	3.35026i	−0.141226	+	0.159717i
$441$		0			0
$442$		−	4.54420i		−	0.216145i
$443$		−	33.1900i		−	1.57690i	−0.615097	−	0.788451i	$-0.710883\pi$
$443$		−	33.1900i		−	1.57690i	0.615097	−	0.788451i	$-0.289117\pi$
$444$		0			0
$445$	23.9756	+	21.1998i	1.13655	+	1.00497i
$446$	−18.9076			−0.895302
$447$		0			0
$448$			1.19394i			0.0564082i
$449$	6.06063			0.286019			0.143010	−	0.989721i	$-0.454322\pi$
$449$	6.06063			0.286019			0.143010	+	0.989721i	$0.454322\pi$
$450$		0			0
$451$	2.07522			0.0977184
$452$		−	14.4690i		−	0.680563i
$453$		0			0
$454$	19.3258			0.907006
$455$	2.00000	+	1.76845i	0.0937614	+	0.0829063i
$456$		0			0
$457$			36.6820i			1.71591i	0.513725	+	0.857955i	$0.328265\pi$
$457$			36.6820i			1.71591i	−0.513725	+	0.857955i	$0.671735\pi$
$458$		−	11.5672i		−	0.540501i
$459$		0			0
$460$	10.5442	−	11.9248i	0.491626	−	0.555996i
$461$	7.28963			0.339512			0.169756	−	0.985486i	$-0.445702\pi$
$461$	7.28963			0.339512			0.169756	+	0.985486i	$0.445702\pi$
$462$		0			0
$463$		−	30.5950i		−	1.42187i	−0.703258	−	0.710935i	$-0.748272\pi$
$463$		−	30.5950i		−	1.42187i	0.703258	−	0.710935i	$-0.251728\pi$
$464$	10.7308			0.498167
$465$		0			0
$466$	−10.7816			−0.499449
$467$			10.1359i			0.469032i	0.972112	+	0.234516i	$0.0753504\pi$
$467$			10.1359i			0.469032i	−0.972112	+	0.234516i	$0.924650\pi$
$468$		0			0
$469$	5.61213			0.259144
$470$	2.70052	+	2.38787i	0.124566	+	0.110144i
$471$		0			0
$472$			2.31265i			0.106448i
$473$		−	21.7743i		−	1.00118i
$474$		0			0
$475$	−2.54420	−	20.6253i	−0.116736	−	0.946354i
$476$	−5.42548			−0.248677
$477$		0			0
$478$			21.9248i			1.00282i
$479$	10.3272			0.471864			0.235932	−	0.971770i	$-0.424186\pi$
$479$	10.3272			0.471864			0.235932	+	0.971770i	$0.424186\pi$
$480$		0			0
$481$	−3.92478			−0.178954
$482$		−	4.23743i		−	0.193010i
$483$		0			0
$484$	7.00000			0.318182
$485$	13.6180	−	15.4010i	0.618362	−	0.699326i
$486$		0			0
$487$		−	12.5198i		−	0.567325i	−0.958924	−	0.283662i	$-0.908451\pi$
$487$		−	12.5198i		−	0.567325i	0.958924	−	0.283662i	$-0.0915495\pi$
$488$			7.08840i			0.320877i
$489$		0			0
$490$	−8.25694	+	9.33804i	−0.373011	+	0.421850i
$491$	−26.1465			−1.17997			−0.589987	−	0.807413i	$-0.700867\pi$
$491$	−26.1465			−1.17997			−0.589987	+	0.807413i	$0.700867\pi$
$492$		0			0
$493$			48.7631i			2.19618i
$494$	4.15633			0.187002
$495$		0			0
$496$	−5.35026			−0.240234
$497$		−	11.0738i		−	0.496728i
$498$		0			0
$499$	−24.9438			−1.11664			−0.558320	−	0.829626i	$-0.688554\pi$
$499$	−24.9438			−1.11664			−0.558320	+	0.829626i	$0.688554\pi$
$500$	6.32487	−	9.21933i	0.282857	−	0.412301i
$501$		0			0
$502$		−	1.95509i		−	0.0872601i
$503$			10.0303i			0.447230i	0.974678	+	0.223615i	$0.0717858\pi$
$503$			10.0303i			0.447230i	−0.974678	+	0.223615i	$0.928214\pi$
$504$		0			0
$505$	−23.2750	−	20.5804i	−1.03573	−	0.915816i
$506$	14.2374			0.632931
$507$		0			0
$508$			2.18664i			0.0970166i
$509$	16.8119			0.745176			0.372588	−	0.927997i	$-0.378470\pi$
$509$	16.8119			0.745176			0.372588	+	0.927997i	$0.378470\pi$
$510$		0			0
$511$	−4.27645			−0.189179
$512$			1.00000i			0.0441942i
$513$		0			0
$514$	−6.78163			−0.299125
$515$	11.5515	−	13.0640i	0.509020	−	0.575667i
$516$		0			0
$517$			3.22425i			0.141803i
$518$			4.68594i			0.205888i
$519$		0			0
$520$	1.67513	+	1.48119i	0.0734593	+	0.0649547i
$521$	14.8510			0.650633			0.325316	−	0.945605i	$-0.394529\pi$
$521$	14.8510			0.650633			0.325316	+	0.945605i	$0.394529\pi$
$522$		0			0
$523$		−	10.5990i		−	0.463460i	−0.972780	−	0.231730i	$-0.925561\pi$
$523$		−	10.5990i		−	0.463460i	0.972780	−	0.231730i	$-0.0744386\pi$
$524$	5.64244			0.246491
$525$		0			0
$526$	9.96968			0.434699
$527$		−	24.3127i		−	1.05908i
$528$		0			0
$529$	−27.6761			−1.20331
$530$	7.01317	+	6.20123i	0.304633	+	0.269364i
$531$		0			0
$532$		−	4.96239i		−	0.215147i
$533$		−	1.03761i		−	0.0449439i
$534$		0			0
$535$	−17.0884	+	19.3258i	−0.738796	+	0.835528i
$536$	4.70052			0.203032
$537$		0			0
$538$			9.64244i			0.415715i
$539$	−11.1490			−0.480223
$540$		0			0
$541$	−12.1925			−0.524197			−0.262099	−	0.965041i	$-0.584415\pi$
$541$	−12.1925			−0.524197			−0.262099	+	0.965041i	$0.584415\pi$
$542$			21.1392i			0.908006i
$543$		0			0
$544$	−4.54420			−0.194831
$545$	2.64974	+	2.34297i	0.113502	+	0.100362i
$546$		0			0
$547$			4.12127i			0.176213i	0.996111	+	0.0881064i	$0.0280815\pi$
$547$			4.12127i			0.176213i	−0.996111	+	0.0881064i	$0.971918\pi$
$548$			2.46310i			0.105218i
$549$		0			0
$550$	9.92478	−	1.22425i	0.423194	−	0.0522024i
$551$	−44.6009			−1.90006
$552$		0			0
$553$			18.1476i			0.771715i
$554$	7.48612			0.318055
$555$		0			0
$556$	−16.6253			−0.705070
$557$		−	0.111420i		−	0.00472100i	−0.999997	−	0.00236050i	$-0.999249\pi$
$557$		−	0.111420i		−	0.00472100i	0.999997	−	0.00236050i	$-0.000751371\pi$
$558$		0			0
$559$	−10.8872			−0.460478
$560$	1.76845	−	2.00000i	0.0747307	−	0.0845154i
$561$		0			0
$562$			18.1622i			0.766126i
$563$		−	11.7889i		−	0.496844i	−0.968652	−	0.248422i	$-0.920088\pi$
$563$		−	11.7889i		−	0.496844i	0.968652	−	0.248422i	$-0.0799119\pi$
$564$		0			0
$565$	−21.4314	+	24.2374i	−0.901624	+	1.01968i
$566$	11.8134			0.496552
$567$		0			0
$568$		−	9.27504i		−	0.389172i
$569$	−27.6991			−1.16121			−0.580604	−	0.814186i	$-0.697183\pi$
$569$	−27.6991			−1.16121			−0.580604	+	0.814186i	$0.697183\pi$
$570$		0			0
$571$	27.4471			1.14863			0.574313	−	0.818636i	$-0.305269\pi$
$571$	27.4471			1.14863			0.574313	+	0.818636i	$0.305269\pi$
$572$			2.00000i			0.0836242i
$573$		0			0
$574$	−1.23884			−0.0517083
$575$	−35.3258	+	4.35756i	−1.47319	+	0.181723i
$576$		0			0
$577$		−	8.98286i		−	0.373961i	−0.982364	−	0.186981i	$-0.940130\pi$
$577$		−	8.98286i		−	0.373961i	0.982364	−	0.186981i	$-0.0598702\pi$
$578$		−	3.64974i		−	0.151809i
$579$		0			0
$580$	−17.9756	−	15.8945i	−0.746395	−	0.659982i
$581$	−2.07522			−0.0860947
$582$		0			0
$583$			8.37328i			0.346786i
$584$	−3.58181			−0.148216
$585$		0			0
$586$	−12.7005			−0.524654
$587$		−	41.6531i		−	1.71921i	−0.510963	−	0.859603i	$-0.670711\pi$
$587$		−	41.6531i		−	1.71921i	0.510963	−	0.859603i	$-0.329289\pi$
$588$		0			0
$589$	22.2374			0.916277
$590$	3.42548	−	3.87399i	0.141025	−	0.159490i
$591$		0			0
$592$			3.92478i			0.161307i
$593$			15.6121i			0.641113i	0.947229	+	0.320557i	$0.103870\pi$
$593$			15.6121i			0.641113i	−0.947229	+	0.320557i	$0.896130\pi$
$594$		0			0
$595$	9.08840	+	8.03620i	0.372588	+	0.329452i
$596$	−22.0508			−0.903235
$597$		0			0
$598$		−	7.11871i		−	0.291106i
$599$	9.92478			0.405515			0.202758	−	0.979229i	$-0.435010\pi$
$599$	9.92478			0.405515			0.202758	+	0.979229i	$0.435010\pi$
$600$		0			0
$601$	−15.6775			−0.639499			−0.319749	−	0.947502i	$-0.603599\pi$
$601$	−15.6775			−0.639499			−0.319749	+	0.947502i	$0.603599\pi$
$602$			12.9986i			0.529783i
$603$		0			0
$604$	2.96239			0.120538
$605$	−11.7259	−	10.3684i	−0.476726	−	0.421534i
$606$		0			0
$607$			32.5139i			1.31970i	0.751398	+	0.659849i	$0.229380\pi$
$607$			32.5139i			1.31970i	−0.751398	+	0.659849i	$0.770620\pi$
$608$		−	4.15633i		−	0.168561i
$609$		0			0
$610$	10.4993	−	11.8740i	0.425104	−	0.480764i
$611$	1.61213			0.0652197
$612$		0			0
$613$			29.9511i			1.20971i	0.796334	+	0.604857i	$0.206770\pi$
$613$			29.9511i			1.20971i	−0.796334	+	0.604857i	$0.793230\pi$
$614$	−6.98683			−0.281965
$615$		0			0
$616$	2.38787			0.0962102
$617$		−	15.1490i		−	0.609877i	−0.952372	−	0.304939i	$-0.901364\pi$
$617$		−	15.1490i		−	0.609877i	0.952372	−	0.304939i	$-0.0986359\pi$
$618$		0			0
$619$	−2.79621			−0.112389			−0.0561947	−	0.998420i	$-0.517897\pi$
$619$	−2.79621			−0.112389			−0.0561947	+	0.998420i	$0.517897\pi$
$620$	8.96239	+	7.92478i	0.359938	+	0.318267i
$621$		0			0
$622$			7.53690i			0.302202i
$623$		−	17.0884i		−	0.684632i
$624$		0			0
$625$	−24.2506	+	6.07522i	−0.970024	+	0.243009i
$626$	21.7137			0.867854
$627$		0			0
$628$			20.3634i			0.812590i
$629$	−17.8350			−0.711127
$630$		0			0
$631$	13.8134			0.549901			0.274951	−	0.961458i	$-0.411339\pi$
$631$	13.8134			0.549901			0.274951	+	0.961458i	$0.411339\pi$
$632$			15.1998i			0.604616i
$633$		0			0
$634$	−28.3634			−1.12646
$635$	3.23884	−	3.66291i	0.128530	−	0.145358i
$636$		0			0
$637$			5.57452i			0.220870i
$638$		−	21.4617i		−	0.849676i
$639$		0			0
$640$	1.48119	−	1.67513i	0.0585493	−	0.0662154i
$641$	26.8627			1.06101			0.530507	−	0.847681i	$-0.322002\pi$
$641$	26.8627			1.06101			0.530507	+	0.847681i	$0.322002\pi$
$642$		0			0
$643$		−	25.2243i		−	0.994747i	−0.867536	−	0.497374i	$-0.834298\pi$
$643$		−	25.2243i		−	0.994747i	0.867536	−	0.497374i	$-0.165702\pi$
$644$	−8.49929			−0.334919
$645$		0			0
$646$	18.8872			0.743106
$647$		−	11.7323i		−	0.461243i	−0.973044	−	0.230621i	$-0.925924\pi$
$647$		−	11.7323i		−	0.461243i	0.973044	−	0.230621i	$-0.0740758\pi$
$648$		0			0
$649$	4.62530			0.181559
$650$	−0.612127	−	4.96239i	−0.0240096	−	0.194641i
$651$		0			0
$652$			25.3258i			0.991836i
$653$			23.7645i			0.929976i	0.885317	+	0.464988i	$0.153941\pi$
$653$			23.7645i			0.929976i	−0.885317	+	0.464988i	$0.846059\pi$
$654$		0			0
$655$	−9.45183	−	8.35756i	−0.369314	−	0.326557i
$656$	−1.03761			−0.0405119
$657$		0			0
$658$		−	1.92478i		−	0.0750356i
$659$	−33.2809			−1.29644			−0.648220	−	0.761453i	$-0.724486\pi$
$659$	−33.2809			−1.29644			−0.648220	+	0.761453i	$0.724486\pi$
$660$		0			0
$661$	−12.0713			−0.469517			−0.234759	−	0.972054i	$-0.575430\pi$
$661$	−12.0713			−0.469517			−0.234759	+	0.972054i	$0.575430\pi$
$662$			12.6801i			0.492824i
$663$		0			0
$664$	−1.73813			−0.0674527
$665$	−7.35026	+	8.31265i	−0.285031	+	0.322351i
$666$		0			0
$667$			76.3898i			2.95782i
$668$			6.70052i			0.259251i
$669$		0			0
$670$	−7.87399	−	6.96239i	−0.304199	−	0.268981i
$671$	14.1768			0.547289
$672$		0			0
$673$		−	38.8627i		−	1.49805i	−0.662543	−	0.749024i	$-0.730523\pi$
$673$		−	38.8627i		−	1.49805i	0.662543	−	0.749024i	$-0.269477\pi$
$674$	−18.1768			−0.700144
$675$		0			0
$676$	1.00000			0.0384615
$677$			45.1852i			1.73661i	0.496031	+	0.868305i	$0.334790\pi$
$677$			45.1852i			1.73661i	−0.496031	+	0.868305i	$0.665210\pi$
$678$		0			0
$679$	−10.9770			−0.421258
$680$	7.61213	+	6.73084i	0.291912	+	0.258116i
$681$		0			0
$682$			10.7005i			0.409744i
$683$			7.93463i			0.303610i	0.988410	+	0.151805i	$0.0485086\pi$
$683$			7.93463i			0.303610i	−0.988410	+	0.151805i	$0.951491\pi$
$684$		0			0
$685$	3.64832	−	4.12601i	0.139395	−	0.157647i
$686$	15.0132			0.573206
$687$		0			0
$688$			10.8872i			0.415069i
$689$	4.18664			0.159498
$690$		0			0
$691$	37.4069			1.42303			0.711513	−	0.702673i	$-0.248010\pi$
$691$	37.4069			1.42303			0.711513	+	0.702673i	$0.248010\pi$
$692$			0.649738i			0.0246993i
$693$		0			0
$694$	−10.7612			−0.408488
$695$	27.8496	+	24.6253i	1.05639	+	0.934091i
$696$		0			0
$697$		−	4.71511i		−	0.178598i
$698$			12.8061i			0.484717i
$699$		0			0
$700$	−5.92478	+	0.730841i	−0.223936	+	0.0276232i
$701$	−11.0435			−0.417107			−0.208553	−	0.978011i	$-0.566876\pi$
$701$	−11.0435			−0.417107			−0.208553	+	0.978011i	$0.566876\pi$
$702$		0			0
$703$		−	16.3127i		−	0.615243i
$704$	2.00000			0.0753778
$705$		0			0
$706$	−21.5369			−0.810552
$707$			16.5891i			0.623897i
$708$		0			0
$709$	−37.7440			−1.41751			−0.708753	−	0.705457i	$-0.750742\pi$
$709$	−37.7440			−1.41751			−0.708753	+	0.705457i	$0.750742\pi$
$710$	−13.7381	+	15.5369i	−0.515583	+	0.583090i
$711$		0			0
$712$		−	14.3127i		−	0.536389i
$713$		−	38.0870i		−	1.42637i
$714$		0			0
$715$	2.96239	−	3.35026i	0.110787	−	0.125293i
$716$	−6.10554			−0.228175
$717$		0			0
$718$			14.8265i			0.553321i
$719$	47.8007			1.78266			0.891332	−	0.453351i	$-0.149771\pi$
$719$	47.8007			1.78266			0.891332	+	0.453351i	$0.149771\pi$
$720$		0			0
$721$	−9.31124			−0.346769
$722$		−	1.72496i		−	0.0641964i
$723$		0			0
$724$	−22.9380			−0.852482
$725$	6.56864	+	53.2506i	0.243953	+	1.97768i
$726$		0			0
$727$			33.7645i			1.25226i	0.779721	+	0.626128i	$0.215361\pi$
$727$			33.7645i			1.25226i	−0.779721	+	0.626128i	$0.784639\pi$
$728$		−	1.19394i		−	0.0442502i
$729$		0			0
$730$	6.00000	+	5.30536i	0.222070	+	0.196360i
$731$	−49.4734			−1.82984
$732$		0			0
$733$		−	41.8905i		−	1.54726i	−0.633637	−	0.773630i	$-0.718439\pi$
$733$		−	41.8905i		−	1.54726i	0.633637	−	0.773630i	$-0.281561\pi$
$734$	−37.5125			−1.38461
$735$		0			0
$736$	−7.11871			−0.262399
$737$		−	9.40105i		−	0.346292i
$738$		0			0
$739$	−6.14174			−0.225927			−0.112964	−	0.993599i	$-0.536034\pi$
$739$	−6.14174			−0.225927			−0.112964	+	0.993599i	$0.536034\pi$
$740$	5.81336	−	6.57452i	0.213703	−	0.241684i
$741$		0			0
$742$		−	4.99859i		−	0.183504i
$743$			38.4894i			1.41204i	0.708192	+	0.706020i	$0.249511\pi$
$743$			38.4894i			1.41204i	−0.708192	+	0.706020i	$0.750489\pi$
$744$		0			0
$745$	36.9380	+	32.6615i	1.35330	+	1.19662i
$746$	18.1866			0.665860
$747$		0			0
$748$			9.08840i			0.332305i
$749$	13.7743			0.503303
$750$		0			0
$751$	−53.2506			−1.94314			−0.971571	−	0.236748i	$-0.923918\pi$
$751$	−53.2506			−1.94314			−0.971571	+	0.236748i	$0.923918\pi$
$752$		−	1.61213i		−	0.0587882i
$753$		0			0
$754$	−10.7308			−0.390794
$755$	−4.96239	−	4.38787i	−0.180600	−	0.159691i
$756$		0			0
$757$		−	1.59754i		−	0.0580635i	−0.999578	−	0.0290318i	$-0.990758\pi$
$757$		−	1.59754i		−	0.0580635i	0.999578	−	0.0290318i	$-0.00924240\pi$
$758$			24.6312i			0.894645i
$759$		0			0
$760$	−6.15633	+	6.96239i	−0.223313	+	0.252552i
$761$	−1.56134			−0.0565986			−0.0282993	−	0.999599i	$-0.509009\pi$
$761$	−1.56134			−0.0565986			−0.0282993	+	0.999599i	$0.509009\pi$
$762$		0			0
$763$		−	1.88858i		−	0.0683712i
$764$	19.0132			0.687872
$765$		0			0
$766$	1.14903			0.0415162
$767$		−	2.31265i		−	0.0835050i
$768$		0			0
$769$	32.9234			1.18725			0.593623	−	0.804743i	$-0.297697\pi$
$769$	32.9234			1.18725			0.593623	+	0.804743i	$0.297697\pi$
$770$	−4.00000	−	3.53690i	−0.144150	−	0.127461i
$771$		0			0
$772$			4.41819i			0.159014i
$773$		−	42.8481i		−	1.54114i	−0.637355	−	0.770570i	$-0.719972\pi$
$773$		−	42.8481i		−	1.54114i	0.637355	−	0.770570i	$-0.280028\pi$
$774$		0			0
$775$	−3.27504	−	26.5501i	−0.117643	−	0.953707i
$776$	−9.19394			−0.330043
$777$		0			0
$778$		−	3.50659i		−	0.125717i
$779$	4.31265			0.154517
$780$		0			0
$781$	−18.5501			−0.663774
$782$		−	32.3488i		−	1.15679i
$783$		0			0
$784$	5.57452			0.199090
$785$	30.1622	−	34.1114i	1.07654	−	1.21749i
$786$		0			0
$787$			38.6253i			1.37684i	0.725311	+	0.688422i	$0.241696\pi$
$787$			38.6253i			1.37684i	−0.725311	+	0.688422i	$0.758304\pi$
$788$			5.22425i			0.186106i
$789$		0			0
$790$	22.5139	−	25.4617i	0.801008	−	0.905886i
$791$	17.2750			0.614230
$792$		0			0
$793$		−	7.08840i		−	0.251716i
$794$	28.8627			1.02430
$795$		0			0
$796$	−19.1998			−0.680519
$797$			18.0508i			0.639392i	0.947520	+	0.319696i	$0.103581\pi$
$797$			18.0508i			0.639392i	−0.947520	+	0.319696i	$0.896419\pi$
$798$		0			0
$799$	7.32582			0.259169
$800$	−4.96239	+	0.612127i	−0.175447	+	0.0216420i
$801$		0			0
$802$		−	17.0376i		−	0.601619i
$803$			7.16362i			0.252799i
$804$		0			0
$805$	14.2374	+	12.5891i	0.501803	+	0.443708i
$806$	5.35026			0.188455
$807$		0			0
$808$			13.8945i			0.488805i
$809$	−1.76257			−0.0619687			−0.0309844	−	0.999520i	$-0.509864\pi$
$809$	−1.76257			−0.0619687			−0.0309844	+	0.999520i	$0.509864\pi$
$810$		0			0
$811$	−44.0665			−1.54738			−0.773692	−	0.633562i	$-0.781592\pi$
$811$	−44.0665			−1.54738			−0.773692	+	0.633562i	$0.781592\pi$
$812$			12.8119i			0.449611i
$813$		0			0
$814$	7.84955			0.275127
$815$	37.5125	−	42.4241i	1.31400	−	1.48605i
$816$		0			0
$817$		−	45.2506i		−	1.58312i
$818$		−	6.77575i		−	0.236908i
$819$		0			0
$820$	1.73813	+	1.53690i	0.0606983	+	0.0536710i
$821$	1.54675			0.0539821			0.0269910	−	0.999636i	$-0.491407\pi$
$821$	1.54675			0.0539821			0.0269910	+	0.999636i	$0.491407\pi$
$822$		0			0
$823$		−	20.7250i		−	0.722427i	−0.932483	−	0.361213i	$-0.882363\pi$
$823$		−	20.7250i		−	0.722427i	0.932483	−	0.361213i	$-0.117637\pi$
$824$	−7.79877			−0.271683
$825$		0			0
$826$	−2.76116			−0.0960730
$827$		−	41.7381i		−	1.45138i	−0.688023	−	0.725689i	$-0.741521\pi$
$827$		−	41.7381i		−	1.45138i	0.688023	−	0.725689i	$-0.258479\pi$
$828$		0			0
$829$	26.7757			0.929960			0.464980	−	0.885321i	$-0.346061\pi$
$829$	26.7757			0.929960			0.464980	+	0.885321i	$0.346061\pi$
$830$	2.91160	+	2.57452i	0.101063	+	0.0893627i
$831$		0			0
$832$		−	1.00000i		−	0.0346688i
$833$			25.3317i			0.877692i
$834$		0			0
$835$	9.92478	−	11.2243i	0.343461	−	0.388431i
$836$	−8.31265			−0.287499
$837$		0			0
$838$		−	20.7572i		−	0.717045i
$839$	−48.6253			−1.67873			−0.839366	−	0.543567i	$-0.817073\pi$
$839$	−48.6253			−1.67873			−0.839366	+	0.543567i	$0.817073\pi$
$840$		0			0
$841$	86.1509			2.97072
$842$		−	36.9683i		−	1.27401i
$843$		0			0
$844$	−9.86414			−0.339538
$845$	−1.67513	−	1.48119i	−0.0576263	−	0.0509546i
$846$		0			0
$847$			8.35756i			0.287169i
$848$		−	4.18664i		−	0.143770i
$849$		0			0
$850$	−2.78163	−	22.5501i	−0.0954090	−	0.773461i
$851$	−27.9394			−0.957749
$852$		0			0
$853$			4.53832i			0.155389i	0.996977	+	0.0776945i	$0.0247559\pi$
$853$			4.53832i			0.155389i	−0.996977	+	0.0776945i	$0.975244\pi$
$854$	−8.46310			−0.289601
$855$		0			0
$856$	11.5369			0.394323
$857$			28.2315i			0.964371i	0.876069	+	0.482186i	$0.160157\pi$
$857$			28.2315i			0.964371i	−0.876069	+	0.482186i	$0.839843\pi$
$858$		0			0
$859$	18.1162			0.618115			0.309058	−	0.951043i	$-0.399986\pi$
$859$	18.1162			0.618115			0.309058	+	0.951043i	$0.399986\pi$
$860$	16.1260	−	18.2374i	0.549892	−	0.621891i
$861$		0			0
$862$		−	16.6253i		−	0.566260i
$863$			30.0263i			1.02211i	0.859548	+	0.511054i	$0.170745\pi$
$863$			30.0263i			1.02211i	−0.859548	+	0.511054i	$0.829255\pi$
$864$		0			0
$865$	0.962389	−	1.08840i	0.0327222	−	0.0370066i
$866$	4.89701			0.166407
$867$		0			0
$868$		−	6.38787i		−	0.216819i
$869$	30.3996			1.03124
$870$		0			0
$871$	−4.70052			−0.159271
$872$		−	1.58181i		−	0.0535668i
$873$		0			0
$874$	29.5877			1.00082
$875$	11.0073	+	7.55149i	0.372114	+	0.255287i
$876$		0			0
$877$			22.5355i			0.760969i	0.924787	+	0.380485i	$0.124243\pi$
$877$			22.5355i			0.760969i	−0.924787	+	0.380485i	$0.875757\pi$
$878$		−	4.62530i		−	0.156096i
$879$		0			0
$880$	−3.35026	−	2.96239i	−0.112937	−	0.0998621i
$881$	8.21108			0.276638			0.138319	−	0.990388i	$-0.455830\pi$
$881$	8.21108			0.276638			0.138319	+	0.990388i	$0.455830\pi$
$882$		0			0
$883$		−	6.44851i		−	0.217010i	−0.994096	−	0.108505i	$-0.965394\pi$
$883$		−	6.44851i		−	0.217010i	0.994096	−	0.108505i	$-0.0346063\pi$
$884$	4.54420			0.152838
$885$		0			0
$886$	33.1900			1.11504
$887$			34.6067i			1.16198i	0.813910	+	0.580990i	$0.197335\pi$
$887$			34.6067i			1.16198i	−0.813910	+	0.580990i	$0.802665\pi$
$888$		0			0
$889$	−2.61071			−0.0875605
$890$	−21.1998	+	23.9756i	−0.710619	+	0.803663i
$891$		0			0
$892$		−	18.9076i		−	0.633074i
$893$			6.70052i			0.224224i
$894$		0			0
$895$	10.2276	+	9.04349i	0.341870	+	0.302291i
$896$	−1.19394			−0.0398866
$897$		0			0
$898$			6.06063i			0.202246i
$899$	−57.4128			−1.91482
$900$		0			0
$901$	19.0249			0.633812
$902$			2.07522i			0.0690974i
$903$		0			0
$904$	14.4690			0.481231
$905$	38.4241	+	33.9756i	1.27726	+	1.12939i
$906$		0			0
$907$			20.3371i			0.675282i	0.941275	+	0.337641i	$0.109629\pi$
$907$			20.3371i			0.675282i	−0.941275	+	0.337641i	$0.890371\pi$
$908$			19.3258i			0.641350i
$909$		0			0
$910$	−1.76845	+	2.00000i	−0.0586236	+	0.0662994i
$911$	38.9234			1.28959			0.644794	−	0.764356i	$-0.276943\pi$
$911$	38.9234			1.28959			0.644794	+	0.764356i	$0.276943\pi$
$912$		0			0
$913$			3.47627i			0.115048i
$914$	−36.6820			−1.21333
$915$		0			0
$916$	11.5672			0.382192
$917$			6.73672i			0.222466i
$918$		0			0
$919$	−7.22425			−0.238306			−0.119153	−	0.992876i	$-0.538018\pi$
$919$	−7.22425			−0.238306			−0.119153	+	0.992876i	$0.538018\pi$
$920$	11.9248	+	10.5442i	0.393148	+	0.347632i
$921$		0			0
$922$			7.28963i			0.240071i
$923$			9.27504i			0.305292i
$924$		0			0
$925$	−19.4763	+	2.40246i	−0.640376	+	0.0789925i
$926$	30.5950			1.00541
$927$		0			0
$928$			10.7308i			0.352257i
$929$	24.0870			0.790268			0.395134	−	0.918623i	$-0.370698\pi$
$929$	24.0870			0.790268			0.395134	+	0.918623i	$0.370698\pi$
$930$		0			0
$931$	−23.1695			−0.759350
$932$		−	10.7816i		−	0.353164i
$933$		0			0
$934$	−10.1359			−0.331655
$935$	13.4617	−	15.2243i	0.440244	−	0.497886i
$936$		0			0
$937$		−	46.0870i		−	1.50560i	−0.658252	−	0.752798i	$-0.728704\pi$
$937$		−	46.0870i		−	1.50560i	0.658252	−	0.752798i	$-0.271296\pi$
$938$			5.61213i			0.183242i
$939$		0			0
$940$	−2.38787	+	2.70052i	−0.0778838	+	0.0880814i
$941$	10.9887			0.358223			0.179111	−	0.983829i	$-0.442678\pi$
$941$	10.9887			0.358223			0.179111	+	0.983829i	$0.442678\pi$
$942$		0			0
$943$		−	7.38646i		−	0.240536i
$944$	−2.31265			−0.0752704
$945$		0			0
$946$	21.7743			0.707945
$947$		−	32.1378i		−	1.04434i	−0.852842	−	0.522169i	$-0.825123\pi$
$947$		−	32.1378i		−	1.04434i	0.852842	−	0.522169i	$-0.174877\pi$
$948$		0			0
$949$	3.58181			0.116270
$950$	20.6253	−	2.54420i	0.669173	−	0.0825447i
$951$		0			0
$952$		−	5.42548i		−	0.175841i
$953$		−	3.61801i		−	0.117199i	−0.998282	−	0.0585994i	$-0.981337\pi$
$953$		−	3.61801i		−	0.117199i	0.998282	−	0.0585994i	$-0.0186634\pi$
$954$		0			0
$955$	−31.8496	−	28.1622i	−1.03063	−	0.911308i
$956$	−21.9248			−0.709098
$957$		0			0
$958$			10.3272i			0.333658i
$959$	−2.94078			−0.0949627
$960$		0			0
$961$	−2.37470			−0.0766032
$962$		−	3.92478i		−	0.126540i
$963$		0			0
$964$	4.23743			0.136478
$965$	6.54420	−	7.40105i	0.210665	−	0.238248i
$966$		0			0
$967$		−	26.9194i		−	0.865669i	−0.901473	−	0.432835i	$-0.857513\pi$
$967$		−	26.9194i		−	0.865669i	0.901473	−	0.432835i	$-0.142487\pi$
$968$			7.00000i			0.224989i
$969$		0			0
$970$	15.4010	+	13.6180i	0.494498	+	0.437248i
$971$	47.8192			1.53459			0.767296	−	0.641293i	$-0.221602\pi$
$971$	47.8192			1.53459			0.767296	+	0.641293i	$0.221602\pi$
$972$		0			0
$973$		−	19.8496i		−	0.636348i
$974$	12.5198			0.401159
$975$		0			0
$976$	−7.08840			−0.226894
$977$			40.1886i			1.28575i	0.765973	+	0.642873i	$0.222258\pi$
$977$			40.1886i			1.28575i	−0.765973	+	0.642873i	$0.777742\pi$
$978$		0			0
$979$	−28.6253			−0.914869
$980$	−9.33804	−	8.25694i	−0.298293	−	0.263758i
$981$		0			0
$982$		−	26.1465i		−	0.834368i
$983$		−	6.23743i		−	0.198943i	−0.995040	−	0.0994715i	$-0.968285\pi$
$983$		−	6.23743i		−	0.198943i	0.995040	−	0.0994715i	$-0.0317152\pi$
$984$		0			0
$985$	7.73813	−	8.75131i	0.246557	−	0.278840i
$986$	−48.7631			−1.55293
$987$		0			0
$988$			4.15633i			0.132230i
$989$	−77.5026			−2.46444
$990$		0			0
$991$	40.7221			1.29358			0.646791	−	0.762668i	$-0.276111\pi$
$991$	40.7221			1.29358			0.646791	+	0.762668i	$0.276111\pi$
$992$		−	5.35026i		−	0.169871i
$993$		0			0
$994$	11.0738			0.351240
$995$	32.1622	+	28.4387i	1.01961	+	0.901566i
$996$		0			0
$997$		−	30.1866i		−	0.956021i	−0.878354	−	0.478010i	$-0.841358\pi$
$997$		−	30.1866i		−	0.956021i	0.878354	−	0.478010i	$-0.158642\pi$
$998$		−	24.9438i		−	0.789583i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1170.2.e.g.469.6	yes	6
3.2	odd	2		1170.2.e.h.469.1	yes	6
5.2	odd	4		5850.2.a.cq.1.2		3
5.3	odd	4		5850.2.a.cr.1.2		3
5.4	even	2	inner	1170.2.e.g.469.3	✓	6
15.2	even	4		5850.2.a.ct.1.2		3
15.8	even	4		5850.2.a.co.1.2		3
15.14	odd	2		1170.2.e.h.469.4	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1170.2.e.g.469.3	✓	6	5.4	even	2	inner
1170.2.e.g.469.6	yes	6	1.1	even	1	trivial
1170.2.e.h.469.1	yes	6	3.2	odd	2
1170.2.e.h.469.4	yes	6	15.14	odd	2
5850.2.a.co.1.2		3	15.8	even	4
5850.2.a.cq.1.2		3	5.2	odd	4
5850.2.a.cr.1.2		3	5.3	odd	4
5850.2.a.ct.1.2		3	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:	\( N \)	\(=\)	\( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1170.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +(1.67513 + 1.48119i) q^{5} -1.19394i q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +(1.67513 + 1.48119i) q^{5} -1.19394i q^{7} -1.00000i q^{8} +(-1.48119 + 1.67513i) q^{10} -2.00000 q^{11} +1.00000i q^{13} +1.19394 q^{14} +1.00000 q^{16} +4.54420i q^{17} -4.15633 q^{19} +(-1.67513 - 1.48119i) q^{20} -2.00000i q^{22} +7.11871i q^{23} +(0.612127 + 4.96239i) q^{25} -1.00000 q^{26} +1.19394i q^{28} +10.7308 q^{29} -5.35026 q^{31} +1.00000i q^{32} -4.54420 q^{34} +(1.76845 - 2.00000i) q^{35} +3.92478i q^{37} -4.15633i q^{38} +(1.48119 - 1.67513i) q^{40} -1.03761 q^{41} +10.8872i q^{43} +2.00000 q^{44} -7.11871 q^{46} -1.61213i q^{47} +5.57452 q^{49} +(-4.96239 + 0.612127i) q^{50} -1.00000i q^{52} -4.18664i q^{53} +(-3.35026 - 2.96239i) q^{55} -1.19394 q^{56} +10.7308i q^{58} -2.31265 q^{59} -7.08840 q^{61} -5.35026i q^{62} -1.00000 q^{64} +(-1.48119 + 1.67513i) q^{65} +4.70052i q^{67} -4.54420i q^{68} +(2.00000 + 1.76845i) q^{70} +9.27504 q^{71} -3.58181i q^{73} -3.92478 q^{74} +4.15633 q^{76} +2.38787i q^{77} -15.1998 q^{79} +(1.67513 + 1.48119i) q^{80} -1.03761i q^{82} -1.73813i q^{83} +(-6.73084 + 7.61213i) q^{85} -10.8872 q^{86} +2.00000i q^{88} +14.3127 q^{89} +1.19394 q^{91} -7.11871i q^{92} +1.61213 q^{94} +(-6.96239 - 6.15633i) q^{95} -9.19394i q^{97} +5.57452i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{4}+O(q^{10}) \) 6 * q - 6 * q^4	\( 6 q - 6 q^{4} + 2 q^{10} - 12 q^{11} + 8 q^{14} + 6 q^{16} - 4 q^{19} + 2 q^{25} - 6 q^{26} + 20 q^{29} - 12 q^{31} - 8 q^{34} - 12 q^{35} - 2 q^{40} - 28 q^{41} + 12 q^{44} + 10 q^{49} - 8 q^{50} - 8 q^{56} + 28 q^{59} - 4 q^{61} - 6 q^{64} + 2 q^{65} + 12 q^{70} - 8 q^{71} + 20 q^{74} + 4 q^{76} + 16 q^{79} + 4 q^{85} + 44 q^{89} + 8 q^{91} + 8 q^{94} - 20 q^{95}+O(q^{100}) \) 6 * q - 6 * q^4 + 2 * q^10 - 12 * q^11 + 8 * q^14 + 6 * q^16 - 4 * q^19 + 2 * q^25 - 6 * q^26 + 20 * q^29 - 12 * q^31 - 8 * q^34 - 12 * q^35 - 2 * q^40 - 28 * q^41 + 12 * q^44 + 10 * q^49 - 8 * q^50 - 8 * q^56 + 28 * q^59 - 4 * q^61 - 6 * q^64 + 2 * q^65 + 12 * q^70 - 8 * q^71 + 20 * q^74 + 4 * q^76 + 16 * q^79 + 4 * q^85 + 44 * q^89 + 8 * q^91 + 8 * q^94 - 20 * q^95

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