LMFDB - Embedded newform 378.2.h.c.361.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,2,Mod(289,378)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 2]))

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

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

chi := DirichletCharacter("378.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	378.h (of order $3$, degree $2$, not 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:	$3.01834519640$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.309123.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 $ x^6 - 3x^5 + 10x^4 - 15x^3 + 19x^2 - 12*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.1
Root			$0.500000 - 0.224437i$ of defining polynomial
Character	$\chi$	$=$	378.361
Dual form			378.2.h.c.289.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} -1.58836 q^{5} +(-2.64400 - 0.0963576i) q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} -1.58836 q^{5} +(-2.64400 - 0.0963576i) q^{7} +1.00000 q^{8} +(0.794182 - 1.37556i) q^{10} +1.58836 q^{11} +(2.40545 - 4.16635i) q^{13} +(1.40545 - 2.24159i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(2.69963 - 4.67589i) q^{17} +(-3.54944 - 6.14781i) q^{19} +(0.794182 + 1.37556i) q^{20} +(-0.794182 + 1.37556i) q^{22} -0.300372 q^{23} -2.47710 q^{25} +(2.40545 + 4.16635i) q^{26} +(1.23855 + 2.33795i) q^{28} +(-4.13781 - 7.16689i) q^{29} +(1.35600 + 2.34867i) q^{31} +(-0.500000 - 0.866025i) q^{32} +(2.69963 + 4.67589i) q^{34} +(4.19963 + 0.153051i) q^{35} +(0.500000 + 0.866025i) q^{37} +7.09888 q^{38} -1.58836 q^{40} +(-2.93818 + 5.08907i) q^{41} +(-0.833104 - 1.44298i) q^{43} +(-0.794182 - 1.37556i) q^{44} +(0.150186 - 0.260130i) q^{46} +(1.33310 - 2.30900i) q^{47} +(6.98143 + 0.509538i) q^{49} +(1.23855 - 2.14523i) q^{50} -4.81089 q^{52} +(-2.44437 + 4.23377i) q^{53} -2.52290 q^{55} +(-2.64400 - 0.0963576i) q^{56} +8.27561 q^{58} +(3.23855 + 5.60933i) q^{59} +(2.23855 - 3.87728i) q^{61} -2.71201 q^{62} +1.00000 q^{64} +(-3.82072 + 6.61769i) q^{65} +(5.02654 + 8.70623i) q^{67} -5.39926 q^{68} +(-2.23236 + 3.56046i) q^{70} -12.7207 q^{71} +(8.02654 - 13.9024i) q^{73} -1.00000 q^{74} +(-3.54944 + 6.14781i) q^{76} +(-4.19963 - 0.153051i) q^{77} +(-4.19344 + 7.26325i) q^{79} +(0.794182 - 1.37556i) q^{80} +(-2.93818 - 5.08907i) q^{82} +(-1.18292 - 2.04887i) q^{83} +(-4.28799 + 7.42702i) q^{85} +1.66621 q^{86} +1.58836 q^{88} +(-1.60507 - 2.78007i) q^{89} +(-6.76145 + 10.7840i) q^{91} +(0.150186 + 0.260130i) q^{92} +(1.33310 + 2.30900i) q^{94} +(5.63781 + 9.76497i) q^{95} +(0.712008 + 1.23323i) q^{97} +(-3.93199 + 5.79133i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{2} - 3 q^{4} + 2 q^{5} - 4 q^{7} + 6 q^{8}+O(q^{10}) $ 6 * q - 3 * q^2 - 3 * q^4 + 2 * q^5 - 4 * q^7 + 6 * q^8	$ 6 q - 3 q^{2} - 3 q^{4} + 2 q^{5} - 4 q^{7} + 6 q^{8} - q^{10} - 2 q^{11} + 8 q^{13} + 2 q^{14} - 3 q^{16} + 4 q^{17} - 3 q^{19} - q^{20} + q^{22} - 14 q^{23} - 4 q^{25} + 8 q^{26} + 2 q^{28} + 5 q^{29} + 20 q^{31} - 3 q^{32} + 4 q^{34} + 13 q^{35} + 3 q^{37} + 6 q^{38} + 2 q^{40} - 6 q^{43} + q^{44} + 7 q^{46} + 9 q^{47} - 12 q^{49} + 2 q^{50} - 16 q^{52} - 15 q^{53} - 26 q^{55} - 4 q^{56} - 10 q^{58} + 14 q^{59} + 8 q^{61} - 40 q^{62} + 6 q^{64} + 12 q^{65} + q^{67} - 8 q^{68} + 10 q^{70} - 14 q^{71} + 19 q^{73} - 6 q^{74} - 3 q^{76} - 13 q^{77} + 5 q^{79} - q^{80} - 2 q^{83} - 2 q^{85} + 12 q^{86} - 2 q^{88} + 9 q^{89} - 46 q^{91} + 7 q^{92} + 9 q^{94} + 4 q^{95} + 28 q^{97} + 12 q^{98}+O(q^{100}) $ 6 * q - 3 * q^2 - 3 * q^4 + 2 * q^5 - 4 * q^7 + 6 * q^8 - q^10 - 2 * q^11 + 8 * q^13 + 2 * q^14 - 3 * q^16 + 4 * q^17 - 3 * q^19 - q^20 + q^22 - 14 * q^23 - 4 * q^25 + 8 * q^26 + 2 * q^28 + 5 * q^29 + 20 * q^31 - 3 * q^32 + 4 * q^34 + 13 * q^35 + 3 * q^37 + 6 * q^38 + 2 * q^40 - 6 * q^43 + q^44 + 7 * q^46 + 9 * q^47 - 12 * q^49 + 2 * q^50 - 16 * q^52 - 15 * q^53 - 26 * q^55 - 4 * q^56 - 10 * q^58 + 14 * q^59 + 8 * q^61 - 40 * q^62 + 6 * q^64 + 12 * q^65 + q^67 - 8 * q^68 + 10 * q^70 - 14 * q^71 + 19 * q^73 - 6 * q^74 - 3 * q^76 - 13 * q^77 + 5 * q^79 - q^80 - 2 * q^83 - 2 * q^85 + 12 * q^86 - 2 * q^88 + 9 * q^89 - 46 * q^91 + 7 * q^92 + 9 * q^94 + 4 * q^95 + 28 * q^97 + 12 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

$n$	$29$	$325$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$

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 $

$2$	−0.500000	+	0.866025i	−0.353553	+	0.612372i
$2$	−0.500000	+	0.866025i	−0.353553	+	0.612372i
$3$		0			0
$3$		0			0
$4$	−0.500000	−	0.866025i	−0.250000	−	0.433013i
$5$	−1.58836			−0.710338			−0.355169	−	0.934802i	$-0.615577\pi$
$5$	−1.58836			−0.710338			−0.355169	+	0.934802i	$0.615577\pi$
$6$		0			0
$7$	−2.64400	−	0.0963576i	−0.999337	−	0.0364197i
$7$	−2.64400	−	0.0963576i	−0.999337	−	0.0364197i
$8$	1.00000			0.353553
$9$		0			0
$10$	0.794182	−	1.37556i	0.251142	−	0.434991i
$11$	1.58836			0.478910			0.239455	−	0.970907i	$-0.423031\pi$
$11$	1.58836			0.478910			0.239455	+	0.970907i	$0.423031\pi$
$12$		0			0
$13$	2.40545	−	4.16635i	0.667151	−	1.15554i	−0.311547	−	0.950231i	$-0.600847\pi$
$13$	2.40545	−	4.16635i	0.667151	−	1.15554i	0.978697	−	0.205308i	$-0.0658196\pi$
$14$	1.40545	−	2.24159i	0.375621	−	0.599090i
$15$		0			0
$16$	−0.500000	+	0.866025i	−0.125000	+	0.216506i
$17$	2.69963	−	4.67589i	0.654756	−	1.13407i	−0.327199	−	0.944955i	$-0.606105\pi$
$17$	2.69963	−	4.67589i	0.654756	−	1.13407i	0.981955	−	0.189115i	$-0.0605620\pi$
$18$		0			0
$19$	−3.54944	−	6.14781i	−0.814298	−	1.41041i	−0.909831	−	0.414979i	$-0.863789\pi$
$19$	−3.54944	−	6.14781i	−0.814298	−	1.41041i	0.0955331	−	0.995426i	$-0.469544\pi$
$20$	0.794182	+	1.37556i	0.177584	+	0.307585i
$21$		0			0
$22$	−0.794182	+	1.37556i	−0.169320	+	0.293271i
$23$	−0.300372			−0.0626319			−0.0313159	−	0.999510i	$-0.509970\pi$
$23$	−0.300372			−0.0626319			−0.0313159	+	0.999510i	$0.509970\pi$
$24$		0			0
$25$	−2.47710			−0.495420
$26$	2.40545	+	4.16635i	0.471747	+	0.817089i
$27$		0			0
$28$	1.23855	+	2.33795i	0.234064	+	0.441830i
$29$	−4.13781	−	7.16689i	−0.768371	−	1.33086i	−0.938446	−	0.345427i	$-0.887734\pi$
$29$	−4.13781	−	7.16689i	−0.768371	−	1.33086i	0.170074	−	0.985431i	$-0.445599\pi$
$30$		0			0
$31$	1.35600	+	2.34867i	0.243545	+	0.421833i	0.961722	−	0.274028i	$-0.0883561\pi$
$31$	1.35600	+	2.34867i	0.243545	+	0.421833i	−0.718176	+	0.695861i	$0.755023\pi$
$32$	−0.500000	−	0.866025i	−0.0883883	−	0.153093i
$33$		0			0
$34$	2.69963	+	4.67589i	0.462982	+	0.801909i
$35$	4.19963	+	0.153051i	0.709867	+	0.0258703i
$36$		0			0
$37$	0.500000	+	0.866025i	0.0821995	+	0.142374i	0.904194	−	0.427121i	$-0.140472\pi$
$37$	0.500000	+	0.866025i	0.0821995	+	0.142374i	−0.821995	+	0.569495i	$0.807139\pi$
$38$	7.09888			1.15159
$39$		0			0
$40$	−1.58836			−0.251142
$41$	−2.93818	+	5.08907i	−0.458866	+	0.794780i	−0.998901	−	0.0468628i	$-0.985078\pi$
$41$	−2.93818	+	5.08907i	−0.458866	+	0.794780i	0.540035	+	0.841643i	$0.318411\pi$
$42$		0			0
$43$	−0.833104	−	1.44298i	−0.127047	−	0.220052i	0.795484	−	0.605974i	$-0.207217\pi$
$43$	−0.833104	−	1.44298i	−0.127047	−	0.220052i	−0.922531	+	0.385922i	$0.873883\pi$
$44$	−0.794182	−	1.37556i	−0.119727	−	0.207374i
$45$		0			0
$46$	0.150186	−	0.260130i	0.0221437	−	0.0383540i
$47$	1.33310	−	2.30900i	0.194453	−	0.336803i	−0.752268	−	0.658857i	$-0.771040\pi$
$47$	1.33310	−	2.30900i	0.194453	−	0.336803i	0.946721	+	0.322055i	$0.104373\pi$
$48$		0			0
$49$	6.98143	+	0.509538i	0.997347	+	0.0727912i
$50$	1.23855	−	2.14523i	0.175157	−	0.303382i
$51$		0			0
$52$	−4.81089			−0.667151
$53$	−2.44437	+	4.23377i	−0.335760	+	0.581553i	−0.983630	−	0.180197i	$-0.942326\pi$
$53$	−2.44437	+	4.23377i	−0.335760	+	0.581553i	0.647871	+	0.761750i	$0.275660\pi$
$54$		0			0
$55$	−2.52290			−0.340188
$56$	−2.64400	−	0.0963576i	−0.353319	−	0.0128763i
$57$		0			0
$58$	8.27561			1.08664
$59$	3.23855	+	5.60933i	0.421623	+	0.730273i	0.996098	−	0.0882491i	$-0.0281271\pi$
$59$	3.23855	+	5.60933i	0.421623	+	0.730273i	−0.574475	+	0.818522i	$0.694794\pi$
$60$		0			0
$61$	2.23855	−	3.87728i	0.286617	−	0.496435i	−0.686383	−	0.727240i	$-0.740803\pi$
$61$	2.23855	−	3.87728i	0.286617	−	0.496435i	0.973000	+	0.230805i	$0.0741360\pi$
$62$	−2.71201			−0.344425
$63$		0			0
$64$	1.00000			0.125000
$65$	−3.82072	+	6.61769i	−0.473902	+	0.820823i
$66$		0			0
$67$	5.02654	+	8.70623i	0.614090	+	1.06363i	0.990543	+	0.137199i	$0.0438101\pi$
$67$	5.02654	+	8.70623i	0.614090	+	1.06363i	−0.376454	+	0.926435i	$0.622857\pi$
$68$	−5.39926			−0.654756
$69$		0			0
$70$	−2.23236	+	3.56046i	−0.266818	+	0.425556i
$71$	−12.7207			−1.50967			−0.754833	−	0.655917i	$-0.772282\pi$
$71$	−12.7207			−1.50967			−0.754833	+	0.655917i	$0.772282\pi$
$72$		0			0
$73$	8.02654	−	13.9024i	0.939436	−	1.62715i	0.172909	−	0.984938i	$-0.444683\pi$
$73$	8.02654	−	13.9024i	0.939436	−	1.62715i	0.766527	−	0.642213i	$-0.221983\pi$
$74$	−1.00000			−0.116248
$75$		0			0
$76$	−3.54944	+	6.14781i	−0.407149	+	0.705203i
$77$	−4.19963	−	0.153051i	−0.478592	−	0.0174418i
$78$		0			0
$79$	−4.19344	+	7.26325i	−0.471799	+	0.817179i	−0.999479	−	0.0322635i	$-0.989728\pi$
$79$	−4.19344	+	7.26325i	−0.471799	+	0.817179i	0.527681	+	0.849443i	$0.323062\pi$
$80$	0.794182	−	1.37556i	0.0887922	−	0.153793i
$81$		0			0
$82$	−2.93818	−	5.08907i	−0.324467	−	0.561994i
$83$	−1.18292	−	2.04887i	−0.129842	−	0.224893i	0.793773	−	0.608214i	$-0.208114\pi$
$83$	−1.18292	−	2.04887i	−0.129842	−	0.224893i	−0.923615	+	0.383321i	$0.874780\pi$
$84$		0			0
$85$	−4.28799	+	7.42702i	−0.465098	+	0.805573i
$86$	1.66621			0.179672
$87$		0			0
$88$	1.58836			0.169320
$89$	−1.60507	−	2.78007i	−0.170138	−	0.294687i	0.768330	−	0.640054i	$-0.221088\pi$
$89$	−1.60507	−	2.78007i	−0.170138	−	0.294687i	−0.938468	+	0.345367i	$0.887755\pi$
$90$		0			0
$91$	−6.76145	+	10.7840i	−0.708793	+	1.13047i
$92$	0.150186	+	0.260130i	0.0156580	+	0.0271204i
$93$		0			0
$94$	1.33310	+	2.30900i	0.137499	+	0.238156i
$95$	5.63781	+	9.76497i	0.578427	+	1.00186i
$96$		0			0
$97$	0.712008	+	1.23323i	0.0722934	+	0.125216i	0.899906	−	0.436084i	$-0.143635\pi$
$97$	0.712008	+	1.23323i	0.0722934	+	0.125216i	−0.827613	+	0.561300i	$0.810302\pi$
$98$	−3.93199	+	5.79133i	−0.397191	+	0.585012i
$99$		0			0
$100$	1.23855	+	2.14523i	0.123855	+	0.214523i
$101$	−12.0334			−1.19737			−0.598685	−	0.800985i	$-0.704310\pi$
$101$	−12.0334			−1.19737			−0.598685	+	0.800985i	$0.704310\pi$
$102$		0			0
$103$	−6.09888			−0.600941			−0.300470	−	0.953791i	$-0.597144\pi$
$103$	−6.09888			−0.600941			−0.300470	+	0.953791i	$0.597144\pi$
$104$	2.40545	−	4.16635i	0.235873	−	0.408545i
$105$		0			0
$106$	−2.44437	−	4.23377i	−0.237418	−	0.411220i
$107$	1.54325	+	2.67299i	0.149192	+	0.258408i	0.930929	−	0.365200i	$-0.118999\pi$
$107$	1.54325	+	2.67299i	0.149192	+	0.258408i	−0.781737	+	0.623608i	$0.785666\pi$
$108$		0			0
$109$	1.14400	−	1.98146i	0.109575	−	0.189789i	−0.806023	−	0.591884i	$-0.798384\pi$
$109$	1.14400	−	1.98146i	0.109575	−	0.189789i	0.915598	+	0.402095i	$0.131718\pi$
$110$	1.26145	−	2.18490i	0.120275	−	0.208322i
$111$		0			0
$112$	1.40545	−	2.24159i	0.132802	−	0.211810i
$113$	9.73236	−	16.8569i	0.915543	−	1.58577i	0.109440	−	0.993993i	$-0.465094\pi$
$113$	9.73236	−	16.8569i	0.915543	−	1.58577i	0.806104	−	0.591774i	$-0.201572\pi$
$114$		0			0
$115$	0.477100			0.0444898
$116$	−4.13781	+	7.16689i	−0.384186	+	0.665429i
$117$		0			0
$118$	−6.47710			−0.596265
$119$	−7.58836	+	12.1029i	−0.695624	+	1.10947i
$120$		0			0
$121$	−8.47710			−0.770645
$122$	2.23855	+	3.87728i	0.202669	+	0.351033i
$123$		0			0
$124$	1.35600	−	2.34867i	0.121773	−	0.210917i
$125$	11.8764			1.06225
$126$		0			0
$127$	−13.4400			−1.19260			−0.596302	−	0.802760i	$-0.703364\pi$
$127$	−13.4400			−1.19260			−0.596302	+	0.802760i	$0.703364\pi$
$128$	−0.500000	+	0.866025i	−0.0441942	+	0.0765466i
$129$		0			0
$130$	−3.82072	−	6.61769i	−0.335100	−	0.580410i
$131$	3.17673			0.277552			0.138776	−	0.990324i	$-0.455683\pi$
$131$	3.17673			0.277552			0.138776	+	0.990324i	$0.455683\pi$
$132$		0			0
$133$	8.79232	+	16.5968i	0.762391	+	1.43913i
$134$	−10.0531			−0.868454
$135$		0			0
$136$	2.69963	−	4.67589i	0.231491	−	0.400955i
$137$	21.2632			1.81664			0.908320	−	0.418275i	$-0.137365\pi$
$137$	21.2632			1.81664			0.908320	+	0.418275i	$0.137365\pi$
$138$		0			0
$139$	6.52654	−	11.3043i	0.553574	−	0.958818i	−0.444439	−	0.895809i	$-0.646597\pi$
$139$	6.52654	−	11.3043i	0.553574	−	0.958818i	0.998013	−	0.0630092i	$-0.0200698\pi$
$140$	−1.96727	−	3.71351i	−0.166264	−	0.313849i
$141$		0			0
$142$	6.36033	−	11.0164i	0.533747	−	0.924478i
$143$	3.82072	−	6.61769i	0.319505	−	0.553399i
$144$		0			0
$145$	6.57234	+	11.3836i	0.545803	+	0.945359i
$146$	8.02654	+	13.9024i	0.664281	+	1.15057i
$147$		0			0
$148$	0.500000	−	0.866025i	0.0410997	−	0.0711868i
$149$	−5.20877			−0.426719			−0.213360	−	0.976974i	$-0.568441\pi$
$149$	−5.20877			−0.426719			−0.213360	+	0.976974i	$0.568441\pi$
$150$		0			0
$151$	−0.522900			−0.0425530			−0.0212765	−	0.999774i	$-0.506773\pi$
$151$	−0.522900			−0.0425530			−0.0212765	+	0.999774i	$0.506773\pi$
$152$	−3.54944	−	6.14781i	−0.287898	−	0.498654i
$153$		0			0
$154$	2.23236	−	3.56046i	0.179889	−	0.286910i
$155$	−2.15383	−	3.73054i	−0.173000	−	0.299644i
$156$		0			0
$157$	−4.43199	−	7.67643i	−0.353711	−	0.612646i	0.633185	−	0.774000i	$-0.281747\pi$
$157$	−4.43199	−	7.67643i	−0.353711	−	0.612646i	−0.986897	+	0.161354i	$0.948414\pi$
$158$	−4.19344	−	7.26325i	−0.333612	−	0.577833i
$159$		0			0
$160$	0.794182	+	1.37556i	0.0627856	+	0.108748i
$161$	0.794182	+	0.0289431i	0.0625903	+	0.00228104i
$162$		0			0
$163$	10.9814	+	19.0204i	0.860132	+	1.48979i	0.871801	+	0.489860i	$0.162952\pi$
$163$	10.9814	+	19.0204i	0.860132	+	1.48979i	−0.0116689	+	0.999932i	$0.503714\pi$
$164$	5.87636			0.458866
$165$		0			0
$166$	2.36584			0.183624
$167$	−1.65019	+	2.85821i	−0.127695	+	0.221175i	−0.922783	−	0.385319i	$-0.874091\pi$
$167$	−1.65019	+	2.85821i	−0.127695	+	0.221175i	0.795088	+	0.606494i	$0.207425\pi$
$168$		0			0
$169$	−5.07234	−	8.78555i	−0.390180	−	0.675812i
$170$	−4.28799	−	7.42702i	−0.328874	−	0.569626i
$171$		0			0
$172$	−0.833104	+	1.44298i	−0.0635236	+	0.110026i
$173$	9.55377	−	16.5476i	0.726360	−	1.25809i	−0.232052	−	0.972703i	$-0.574544\pi$
$173$	9.55377	−	16.5476i	0.726360	−	1.25809i	0.958412	−	0.285389i	$-0.0921227\pi$
$174$		0			0
$175$	6.54944	+	0.238687i	0.495091	+	0.0180431i
$176$	−0.794182	+	1.37556i	−0.0598637	+	0.103687i
$177$		0			0
$178$	3.21015			0.240611
$179$	8.03706	−	13.9206i	0.600718	−	1.04047i	−0.391994	−	0.919968i	$-0.628215\pi$
$179$	8.03706	−	13.9206i	0.600718	−	1.04047i	0.992712	−	0.120507i	$-0.0384520\pi$
$180$		0			0
$181$	8.05308			0.598581			0.299291	−	0.954162i	$-0.403250\pi$
$181$	8.05308			0.598581			0.299291	+	0.954162i	$0.403250\pi$
$182$	−5.95853	−	11.2476i	−0.441676	−	0.833728i
$183$		0			0
$184$	−0.300372			−0.0221437
$185$	−0.794182	−	1.37556i	−0.0583894	−	0.101133i
$186$		0			0
$187$	4.28799	−	7.42702i	0.313569	−	0.543118i
$188$	−2.66621			−0.194453
$189$		0			0
$190$	−11.2756			−0.818019
$191$	−11.9814	+	20.7524i	−0.866946	+	1.50159i	−0.00184390	+	0.999998i	$0.500587\pi$
$191$	−11.9814	+	20.7524i	−0.866946	+	1.50159i	−0.865102	+	0.501596i	$0.832746\pi$
$192$		0			0
$193$	−4.88255	−	8.45682i	−0.351453	−	0.608735i	0.635051	−	0.772470i	$-0.280979\pi$
$193$	−4.88255	−	8.45682i	−0.351453	−	0.608735i	−0.986504	+	0.163735i	$0.947646\pi$
$194$	−1.42402			−0.102238
$195$		0			0
$196$	−3.04944	−	6.30087i	−0.217817	−	0.450062i
$197$	18.2436			1.29980			0.649900	−	0.760020i	$-0.274811\pi$
$197$	18.2436			1.29980			0.649900	+	0.760020i	$0.274811\pi$
$198$		0			0
$199$	9.04944	−	15.6741i	0.641498	−	1.11111i	−0.343601	−	0.939116i	$-0.611647\pi$
$199$	9.04944	−	15.6741i	0.641498	−	1.11111i	0.985098	−	0.171991i	$-0.0550200\pi$
$200$	−2.47710			−0.175157
$201$		0			0
$202$	6.01671	−	10.4212i	0.423334	−	0.733236i
$203$	10.2498	+	19.3479i	0.719392	+	1.35796i
$204$		0			0
$205$	4.66690	−	8.08330i	0.325950	−	0.564562i
$206$	3.04944	−	5.28179i	0.212465	−	0.368000i
$207$		0			0
$208$	2.40545	+	4.16635i	0.166788	+	0.288885i
$209$	−5.63781	−	9.76497i	−0.389975	−	0.675457i
$210$		0			0
$211$	0.166208	−	0.287880i	0.0114422	−	0.0198185i	−0.860248	−	0.509877i	$-0.829691\pi$
$211$	0.166208	−	0.287880i	0.0114422	−	0.0198185i	0.871690	+	0.490058i	$0.163024\pi$
$212$	4.88874			0.335760
$213$		0			0
$214$	−3.08650			−0.210989
$215$	1.32327	+	2.29197i	0.0902464	+	0.156311i
$216$		0			0
$217$	−3.35896	−	6.34053i	−0.228021	−	0.430423i
$218$	1.14400	+	1.98146i	0.0774812	+	0.134201i
$219$		0			0
$220$	1.26145	+	2.18490i	0.0850469	+	0.147306i
$221$	−12.9876	−	22.4952i	−0.873642	−	1.51319i
$222$		0			0
$223$	3.16621	+	5.48403i	0.212025	+	0.367238i	0.952348	−	0.305013i	$-0.0986609\pi$
$223$	3.16621	+	5.48403i	0.212025	+	0.367238i	−0.740323	+	0.672251i	$0.765328\pi$
$224$	1.23855	+	2.33795i	0.0827541	+	0.156211i
$225$		0			0
$226$	9.73236	+	16.8569i	0.647387	+	1.12131i
$227$	23.3090			1.54707			0.773537	−	0.633751i	$-0.218485\pi$
$227$	23.3090			1.54707			0.773537	+	0.633751i	$0.218485\pi$
$228$		0			0
$229$	−4.95420			−0.327383			−0.163691	−	0.986512i	$-0.552340\pi$
$229$	−4.95420			−0.327383			−0.163691	+	0.986512i	$0.552340\pi$
$230$	−0.238550	+	0.413181i	−0.0157295	+	0.0272443i
$231$		0			0
$232$	−4.13781	−	7.16689i	−0.271660	−	0.470529i
$233$	7.13781	+	12.3630i	0.467613	+	0.809930i	0.999315	−	0.0370017i	$-0.0117807\pi$
$233$	7.13781	+	12.3630i	0.467613	+	0.809930i	−0.531702	+	0.846932i	$0.678447\pi$
$234$		0			0
$235$	−2.11745	+	3.66754i	−0.138127	+	0.239244i
$236$	3.23855	−	5.60933i	0.210812	−	0.365136i
$237$		0			0
$238$	−6.68725	−	12.6232i	−0.433470	−	0.818239i
$239$	−2.48762	+	4.30868i	−0.160911	+	0.278706i	−0.935196	−	0.354132i	$-0.884776\pi$
$239$	−2.48762	+	4.30868i	−0.160911	+	0.278706i	0.774285	+	0.632837i	$0.218110\pi$
$240$		0			0
$241$	−13.0000			−0.837404			−0.418702	−	0.908124i	$-0.637515\pi$
$241$	−13.0000			−0.837404			−0.418702	+	0.908124i	$0.637515\pi$
$242$	4.23855	−	7.34138i	0.272464	−	0.471922i
$243$		0			0
$244$	−4.47710			−0.286617
$245$	−11.0891	−	0.809332i	−0.708454	−	0.0517063i
$246$		0			0
$247$	−34.1520			−2.17304
$248$	1.35600	+	2.34867i	0.0861063	+	0.149141i
$249$		0			0
$250$	−5.93818	+	10.2852i	−0.375563	+	0.650495i
$251$	−2.43268			−0.153549			−0.0767746	−	0.997048i	$-0.524462\pi$
$251$	−2.43268			−0.153549			−0.0767746	+	0.997048i	$0.524462\pi$
$252$		0			0
$253$	−0.477100			−0.0299950
$254$	6.71998	−	11.6393i	0.421649	−	0.730318i
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.0312500	−	0.0541266i
$257$	0.987620			0.0616061			0.0308030	−	0.999525i	$-0.490194\pi$
$257$	0.987620			0.0616061			0.0308030	+	0.999525i	$0.490194\pi$
$258$		0			0
$259$	−1.23855	−	2.33795i	−0.0769597	−	0.145273i
$260$	7.64145			0.473902
$261$		0			0
$262$	−1.58836	+	2.75113i	−0.0981295	+	0.169965i
$263$	−17.1854			−1.05970			−0.529848	−	0.848092i	$-0.677751\pi$
$263$	−17.1854			−1.05970			−0.529848	+	0.848092i	$0.677751\pi$
$264$		0			0
$265$	3.88255	−	6.72477i	0.238503	−	0.413099i
$266$	−18.7694	−	0.684031i	−1.15083	−	0.0419407i
$267$		0			0
$268$	5.02654	−	8.70623i	0.307045	−	0.531817i
$269$	−11.4523	+	19.8360i	−0.698262	+	1.20942i	0.270807	+	0.962634i	$0.412709\pi$
$269$	−11.4523	+	19.8360i	−0.698262	+	1.20942i	−0.969069	+	0.246791i	$0.920624\pi$
$270$		0			0
$271$	7.00364	+	12.1307i	0.425441	+	0.736885i	0.996462	−	0.0840504i	$-0.0267857\pi$
$271$	7.00364	+	12.1307i	0.425441	+	0.736885i	−0.571021	+	0.820936i	$0.693452\pi$
$272$	2.69963	+	4.67589i	0.163689	+	0.283518i
$273$		0			0
$274$	−10.6316	+	18.4145i	−0.642279	+	1.11246i
$275$	−3.93454			−0.237261
$276$		0			0
$277$	28.2953			1.70010			0.850049	−	0.526703i	$-0.176572\pi$
$277$	28.2953			1.70010			0.850049	+	0.526703i	$0.176572\pi$
$278$	6.52654	+	11.3043i	0.391436	+	0.677987i
$279$		0			0
$280$	4.19963	+	0.153051i	0.250976	+	0.00914654i
$281$	8.79782	+	15.2383i	0.524834	+	0.909039i	0.999582	+	0.0289175i	$0.00920600\pi$
$281$	8.79782	+	15.2383i	0.524834	+	0.909039i	−0.474748	+	0.880122i	$0.657461\pi$
$282$		0			0
$283$	9.26145	+	16.0413i	0.550536	+	0.953556i	0.998236	+	0.0593725i	$0.0189100\pi$
$283$	9.26145	+	16.0413i	0.550536	+	0.953556i	−0.447700	+	0.894184i	$0.647757\pi$
$284$	6.36033	+	11.0164i	0.377416	+	0.653704i
$285$		0			0
$286$	3.82072	+	6.61769i	0.225924	+	0.391312i
$287$	8.25890	−	13.1724i	0.487508	−	0.777541i
$288$		0			0
$289$	−6.07598	−	10.5239i	−0.357411	−	0.619054i
$290$	−13.1447			−0.771882
$291$		0			0
$292$	−16.0531			−0.939436
$293$	7.04256	−	12.1981i	0.411431	−	0.712619i	−0.583616	−	0.812030i	$-0.698362\pi$
$293$	7.04256	−	12.1981i	0.411431	−	0.712619i	0.995046	+	0.0994108i	$0.0316958\pi$
$294$		0			0
$295$	−5.14400	−	8.90966i	−0.299495	−	0.518741i
$296$	0.500000	+	0.866025i	0.0290619	+	0.0503367i
$297$		0			0
$298$	2.60439	−	4.51093i	0.150868	−	0.261311i
$299$	−0.722528	+	1.25146i	−0.0417849	+	0.0723736i
$300$		0			0
$301$	2.06368	+	3.89550i	0.118949	+	0.224533i
$302$	0.261450	−	0.452845i	0.0150448	−	0.0260583i
$303$		0			0
$304$	7.09888			0.407149
$305$	−3.55563	+	6.15854i	−0.203595	+	0.352637i
$306$		0			0
$307$	−5.85532			−0.334180			−0.167090	−	0.985942i	$-0.553437\pi$
$307$	−5.85532			−0.334180			−0.167090	+	0.985942i	$0.553437\pi$
$308$	1.96727	+	3.71351i	0.112096	+	0.211597i
$309$		0			0
$310$	4.30766			0.244658
$311$	0.405446	+	0.702253i	0.0229907	+	0.0398211i	0.877292	−	0.479957i	$-0.159348\pi$
$311$	0.405446	+	0.702253i	0.0229907	+	0.0398211i	−0.854301	+	0.519778i	$0.826015\pi$
$312$		0			0
$313$	−5.28799	+	9.15907i	−0.298895	+	0.517701i	−0.975883	−	0.218292i	$-0.929951\pi$
$313$	−5.28799	+	9.15907i	−0.298895	+	0.517701i	0.676988	+	0.735994i	$0.263285\pi$
$314$	8.86398			0.500223
$315$		0			0
$316$	8.38688			0.471799
$317$	6.09820	−	10.5624i	0.342509	−	0.593243i	−0.642389	−	0.766379i	$-0.722057\pi$
$317$	6.09820	−	10.5624i	0.342509	−	0.593243i	0.984898	+	0.173136i	$0.0553900\pi$
$318$		0			0
$319$	−6.57234	−	11.3836i	−0.367981	−	0.637361i
$320$	−1.58836			−0.0887922
$321$		0			0
$322$	−0.422156	+	0.673310i	−0.0235259	+	0.0375221i
$323$	−38.3287			−2.13267
$324$		0			0
$325$	−5.95853	+	10.3205i	−0.330520	+	0.572477i
$326$	−21.9629			−1.21641
$327$		0			0
$328$	−2.93818	+	5.08907i	−0.162234	+	0.280997i
$329$	−3.74721	+	5.97654i	−0.206590	+	0.329497i
$330$		0			0
$331$	7.83310	−	13.5673i	0.430546	−	0.745728i	−0.566374	−	0.824148i	$-0.691654\pi$
$331$	7.83310	−	13.5673i	0.430546	−	0.745728i	0.996920	+	0.0784202i	$0.0249876\pi$
$332$	−1.18292	+	2.04887i	−0.0649211	+	0.112447i
$333$		0			0
$334$	−1.65019	−	2.85821i	−0.0902942	−	0.156394i
$335$	−7.98398	−	13.8287i	−0.436211	−	0.755540i
$336$		0			0
$337$	−4.21201	+	7.29541i	−0.229443	+	0.397406i	−0.957643	−	0.287958i	$-0.907024\pi$
$337$	−4.21201	+	7.29541i	−0.229443	+	0.397406i	0.728200	+	0.685364i	$0.240357\pi$
$338$	10.1447			0.551798
$339$		0			0
$340$	8.57598			0.465098
$341$	2.15383	+	3.73054i	0.116636	+	0.202020i
$342$		0			0
$343$	−18.4098	−	2.01993i	−0.994035	−	0.109066i
$344$	−0.833104	−	1.44298i	−0.0449179	−	0.0778002i
$345$		0			0
$346$	9.55377	+	16.5476i	0.513614	+	0.889606i
$347$	0.283662	+	0.491316i	0.0152277	+	0.0263752i	0.873539	−	0.486754i	$-0.161819\pi$
$347$	0.283662	+	0.491316i	0.0152277	+	0.0263752i	−0.858311	+	0.513130i	$0.828486\pi$
$348$		0			0
$349$	−0.00364189	−	0.00630794i	−0.000194946	−	0.000337656i	0.865928	−	0.500169i	$-0.166729\pi$
$349$	−0.00364189	−	0.00630794i	−0.000194946	−	0.000337656i	−0.866123	+	0.499831i	$0.833395\pi$
$350$	−3.48143	+	5.55264i	−0.186090	+	0.296801i
$351$		0			0
$352$	−0.794182	−	1.37556i	−0.0423300	−	0.0733178i
$353$	−6.65383			−0.354148			−0.177074	−	0.984198i	$-0.556663\pi$
$353$	−6.65383			−0.354148			−0.177074	+	0.984198i	$0.556663\pi$
$354$		0			0
$355$	20.2051			1.07237
$356$	−1.60507	+	2.78007i	−0.0850688	+	0.147343i
$357$		0			0
$358$	8.03706	+	13.9206i	0.424772	+	0.735727i
$359$	0.398568	+	0.690339i	0.0210356	+	0.0364347i	0.876352	−	0.481672i	$-0.159970\pi$
$359$	0.398568	+	0.690339i	0.0210356	+	0.0364347i	−0.855316	+	0.518107i	$0.826637\pi$
$360$		0			0
$361$	−15.6971	+	27.1881i	−0.826162	+	1.43095i
$362$	−4.02654	+	6.97418i	−0.211630	+	0.366555i
$363$		0			0
$364$	12.7200	+	0.463566i	0.666708	+	0.0242975i
$365$	−12.7491	+	22.0820i	−0.667317	+	1.15583i
$366$		0			0
$367$	−15.4327			−0.805579			−0.402790	−	0.915293i	$-0.631959\pi$
$367$	−15.4327			−0.805579			−0.402790	+	0.915293i	$0.631959\pi$
$368$	0.150186	−	0.260130i	0.00782898	−	0.0135602i
$369$		0			0
$370$	1.58836			0.0825751
$371$	6.87085	−	10.9585i	0.356717	−	0.568939i
$372$		0			0
$373$	10.2422			0.530321			0.265160	−	0.964204i	$-0.414575\pi$
$373$	10.2422			0.530321			0.265160	+	0.964204i	$0.414575\pi$
$374$	4.28799	+	7.42702i	0.221727	+	0.384042i
$375$		0			0
$376$	1.33310	−	2.30900i	0.0687496	−	0.119078i
$377$	−39.8131			−2.05048
$378$		0			0
$379$	25.0087			1.28461			0.642304	−	0.766450i	$-0.277979\pi$
$379$	25.0087			1.28461			0.642304	+	0.766450i	$0.277979\pi$
$380$	5.63781	−	9.76497i	0.289213	−	0.500932i
$381$		0			0
$382$	−11.9814	−	20.7524i	−0.613023	−	1.06179i
$383$	6.26695			0.320226			0.160113	−	0.987099i	$-0.448814\pi$
$383$	6.26695			0.320226			0.160113	+	0.987099i	$0.448814\pi$
$384$		0			0
$385$	6.67054	+	0.243101i	0.339962	+	0.0123896i
$386$	9.76509			0.497030
$387$		0			0
$388$	0.712008	−	1.23323i	0.0361467	−	0.0626080i
$389$	21.6342			1.09690			0.548448	−	0.836185i	$-0.315219\pi$
$389$	21.6342			1.09690			0.548448	+	0.836185i	$0.315219\pi$
$390$		0			0
$391$	−0.810892	+	1.40451i	−0.0410086	+	0.0710290i
$392$	6.98143	+	0.509538i	0.352615	+	0.0257356i
$393$		0			0
$394$	−9.12178	+	15.7994i	−0.459549	+	0.795962i
$395$	6.66071	−	11.5367i	0.335137	−	0.580473i
$396$		0			0
$397$	2.05308	+	3.55605i	0.103041	+	0.178473i	0.912936	−	0.408102i	$-0.133809\pi$
$397$	2.05308	+	3.55605i	0.103041	+	0.178473i	−0.809895	+	0.586575i	$0.800476\pi$
$398$	9.04944	+	15.6741i	0.453608	+	0.785671i
$399$		0			0
$400$	1.23855	−	2.14523i	0.0619275	−	0.107262i
$401$	−16.7417			−0.836041			−0.418021	−	0.908438i	$-0.637276\pi$
$401$	−16.7417			−0.836041			−0.418021	+	0.908438i	$0.637276\pi$
$402$		0			0
$403$	13.0472			0.649926
$404$	6.01671	+	10.4212i	0.299343	+	0.518476i
$405$		0			0
$406$	−21.8807	−	0.797418i	−1.08592	−	0.0395752i
$407$	0.794182	+	1.37556i	0.0393661	+	0.0681842i
$408$		0			0
$409$	4.38255	+	7.59079i	0.216703	+	0.375341i	0.953798	−	0.300449i	$-0.0971364\pi$
$409$	4.38255	+	7.59079i	0.216703	+	0.375341i	−0.737095	+	0.675789i	$0.763803\pi$
$410$	4.66690	+	8.08330i	0.230482	+	0.399206i
$411$		0			0
$412$	3.04944	+	5.28179i	0.150235	+	0.260215i
$413$	−8.02221	−	15.1431i	−0.394747	−	0.745144i
$414$		0			0
$415$	1.87890	+	3.25436i	0.0922318	+	0.159750i
$416$	−4.81089			−0.235873
$417$		0			0
$418$	11.2756			0.551508
$419$	0.210149	−	0.363988i	0.0102664	−	0.0177820i	−0.860847	−	0.508865i	$-0.830065\pi$
$419$	0.210149	−	0.363988i	0.0102664	−	0.0177820i	0.871113	+	0.491083i	$0.163399\pi$
$420$		0			0
$421$	3.28799	+	5.69497i	0.160247	+	0.277556i	0.934957	−	0.354761i	$-0.115438\pi$
$421$	3.28799	+	5.69497i	0.160247	+	0.277556i	−0.774710	+	0.632316i	$0.782104\pi$
$422$	0.166208	+	0.287880i	0.00809086	+	0.0140138i
$423$		0			0
$424$	−2.44437	+	4.23377i	−0.118709	+	0.205610i
$425$	−6.68725	+	11.5827i	−0.324379	+	0.561841i
$426$		0			0
$427$	−6.29232	+	10.0358i	−0.304507	+	0.485667i
$428$	1.54325	−	2.67299i	0.0745959	−	0.129204i
$429$		0			0
$430$	−2.64654			−0.127628
$431$	−11.0439	+	19.1287i	−0.531968	+	0.921395i	0.467336	+	0.884080i	$0.345214\pi$
$431$	−11.0439	+	19.1287i	−0.531968	+	0.921395i	−0.999304	+	0.0373155i	$0.988119\pi$
$432$		0			0
$433$	−9.43268			−0.453306			−0.226653	−	0.973976i	$-0.572778\pi$
$433$	−9.43268			−0.453306			−0.226653	+	0.973976i	$0.572778\pi$
$434$	7.17054	+	0.261323i	0.344197	+	0.0125439i
$435$		0			0
$436$	−2.28799			−0.109575
$437$	1.06615	+	1.84663i	0.0510010	+	0.0883363i
$438$		0			0
$439$	15.6032	−	27.0256i	0.744701	−	1.28986i	−0.205634	−	0.978629i	$-0.565926\pi$
$439$	15.6032	−	27.0256i	0.744701	−	1.28986i	0.950334	−	0.311231i	$-0.100741\pi$
$440$	−2.52290			−0.120275
$441$		0			0
$442$	25.9752			1.23552
$443$	6.52723	−	11.3055i	0.310118	−	0.537140i	−0.668270	−	0.743919i	$-0.732965\pi$
$443$	6.52723	−	11.3055i	0.310118	−	0.537140i	0.978388	+	0.206779i	$0.0662981\pi$
$444$		0			0
$445$	2.54944	+	4.41576i	0.120855	+	0.209327i
$446$	−6.33242			−0.299849
$447$		0			0
$448$	−2.64400	−	0.0963576i	−0.124917	−	0.00455247i
$449$	9.91706			0.468015			0.234008	−	0.972235i	$-0.424816\pi$
$449$	9.91706			0.468015			0.234008	+	0.972235i	$0.424816\pi$
$450$		0			0
$451$	−4.66690	+	8.08330i	−0.219756	+	0.380628i
$452$	−19.4647			−0.915543
$453$		0			0
$454$	−11.6545	+	20.1862i	−0.546974	+	0.947386i
$455$	10.7396	−	17.1290i	0.503482	−	0.803019i
$456$		0			0
$457$	12.2615	−	21.2375i	0.573566	−	0.993446i	−0.422629	−	0.906303i	$-0.638893\pi$
$457$	12.2615	−	21.2375i	0.573566	−	0.993446i	0.996196	−	0.0871436i	$-0.0277739\pi$
$458$	2.47710	−	4.29046i	0.115747	−	0.200480i
$459$		0			0
$460$	−0.238550	−	0.413181i	−0.0111224	−	0.0192646i
$461$	−1.75526	−	3.04020i	−0.0817506	−	0.141596i	0.822251	−	0.569125i	$-0.192718\pi$
$461$	−1.75526	−	3.04020i	−0.0817506	−	0.141596i	−0.904002	+	0.427528i	$0.859384\pi$
$462$		0			0
$463$	8.69413	−	15.0587i	0.404050	−	0.699836i	−0.590160	−	0.807286i	$-0.700935\pi$
$463$	8.69413	−	15.0587i	0.404050	−	0.699836i	0.994210	+	0.107451i	$0.0342687\pi$
$464$	8.27561			0.384186
$465$		0			0
$466$	−14.2756			−0.661305
$467$	−6.69894	−	11.6029i	−0.309990	−	0.536918i	0.668370	−	0.743829i	$-0.266992\pi$
$467$	−6.69894	−	11.6029i	−0.309990	−	0.536918i	−0.978360	+	0.206911i	$0.933659\pi$
$468$		0			0
$469$	−12.4512	−	23.5036i	−0.574945	−	1.08529i
$470$	−2.11745	−	3.66754i	−0.0976709	−	0.169171i
$471$		0			0
$472$	3.23855	+	5.60933i	0.149066	+	0.258190i
$473$	−1.32327	−	2.29197i	−0.0608441	−	0.105385i
$474$		0			0
$475$	8.79232	+	15.2287i	0.403419	+	0.698743i
$476$	14.2756	+	0.520259i	0.654322	+	0.0238460i
$477$		0			0
$478$	−2.48762	−	4.30868i	−0.113781	−	0.197075i
$479$	−20.8058			−0.950641			−0.475321	−	0.879813i	$-0.657668\pi$
$479$	−20.8058			−0.950641			−0.475321	+	0.879813i	$0.657668\pi$
$480$		0			0
$481$	4.81089			0.219358
$482$	6.50000	−	11.2583i	0.296067	−	0.512803i
$483$		0			0
$484$	4.23855	+	7.34138i	0.192661	+	0.333699i
$485$	−1.13093	−	1.95882i	−0.0513528	−	0.0889456i
$486$		0			0
$487$	16.2472	−	28.1410i	0.736231	−	1.27519i	−0.217950	−	0.975960i	$-0.569937\pi$
$487$	16.2472	−	28.1410i	0.736231	−	1.27519i	0.954181	−	0.299230i	$-0.0967298\pi$
$488$	2.23855	−	3.87728i	0.101334	−	0.175516i
$489$		0			0
$490$	6.24543	−	9.19874i	0.282140	−	0.415557i
$491$	9.66071	−	16.7328i	0.435982	−	0.755142i	−0.561394	−	0.827549i	$-0.689735\pi$
$491$	9.66071	−	16.7328i	0.435982	−	0.755142i	0.997375	+	0.0724067i	$0.0230679\pi$
$492$		0			0
$493$	−44.6822			−2.01238
$494$	17.0760	−	29.5765i	0.768285	−	1.33071i
$495$		0			0
$496$	−2.71201			−0.121773
$497$	33.6334	+	1.22573i	1.50866	+	0.0549816i
$498$		0			0
$499$	−11.1506			−0.499169			−0.249585	−	0.968353i	$-0.580294\pi$
$499$	−11.1506			−0.499169			−0.249585	+	0.968353i	$0.580294\pi$
$500$	−5.93818	−	10.2852i	−0.265563	−	0.459969i
$501$		0			0
$502$	1.21634	−	2.10676i	0.0542878	−	0.0940293i
$503$	40.7651			1.81763			0.908813	−	0.417204i	$-0.136990\pi$
$503$	40.7651			1.81763			0.908813	+	0.417204i	$0.136990\pi$
$504$		0			0
$505$	19.1135			0.850537
$506$	0.238550	−	0.413181i	0.0106048	−	0.0183681i
$507$		0			0
$508$	6.71998	+	11.6393i	0.298151	+	0.516413i
$509$	−1.44506			−0.0640510			−0.0320255	−	0.999487i	$-0.510196\pi$
$509$	−1.44506			−0.0640510			−0.0320255	+	0.999487i	$0.510196\pi$
$510$		0			0
$511$	−22.5617	+	35.9844i	−0.998073	+	1.59186i
$512$	1.00000			0.0441942
$513$		0			0
$514$	−0.493810	+	0.855304i	−0.0217810	+	0.0377259i
$515$	9.68725			0.426871
$516$		0			0
$517$	2.11745	−	3.66754i	0.0931255	−	0.161298i
$518$	2.64400	+	0.0963576i	0.116171	+	0.00423371i
$519$		0			0
$520$	−3.82072	+	6.61769i	−0.167550	+	0.290205i
$521$	−9.64214	+	16.7007i	−0.422430	+	0.731670i	−0.996177	−	0.0873630i	$-0.972156\pi$
$521$	−9.64214	+	16.7007i	−0.422430	+	0.731670i	0.573747	+	0.819033i	$0.305489\pi$
$522$		0			0
$523$	−18.3454	−	31.7752i	−0.802189	−	1.38943i	−0.918173	−	0.396180i	$-0.870335\pi$
$523$	−18.3454	−	31.7752i	−0.802189	−	1.38943i	0.115984	−	0.993251i	$-0.462998\pi$
$524$	−1.58836	−	2.75113i	−0.0693880	−	0.120184i
$525$		0			0
$526$	8.59269	−	14.8830i	0.374659	−	0.648929i
$527$	14.6428			0.637851
$528$		0			0
$529$	−22.9098			−0.996077
$530$	3.88255	+	6.72477i	0.168647	+	0.292105i
$531$		0			0
$532$	9.97710	−	15.9128i	0.432562	−	0.689907i
$533$	14.1353	+	24.4830i	0.612266	+	1.06048i
$534$		0			0
$535$	−2.45125	−	4.24568i	−0.105977	−	0.183557i
$536$	5.02654	+	8.70623i	0.217114	+	0.376052i
$537$		0			0
$538$	−11.4523	−	19.8360i	−0.493745	−	0.855192i
$539$	11.0891	+	0.809332i	0.477639	+	0.0348604i
$540$		0			0
$541$	−1.62543	−	2.81532i	−0.0698825	−	0.121040i	0.828967	−	0.559298i	$-0.188929\pi$
$541$	−1.62543	−	2.81532i	−0.0698825	−	0.121040i	−0.898849	+	0.438258i	$0.855596\pi$
$542$	−14.0073			−0.601664
$543$		0			0
$544$	−5.39926			−0.231491
$545$	−1.81708	+	3.14728i	−0.0778352	+	0.134815i
$546$		0			0
$547$	−2.95853	−	5.12432i	−0.126498	−	0.219100i	0.795820	−	0.605534i	$-0.207040\pi$
$547$	−2.95853	−	5.12432i	−0.126498	−	0.219100i	−0.922317	+	0.386433i	$0.873707\pi$
$548$	−10.6316	−	18.4145i	−0.454160	−	0.786628i
$549$		0			0
$550$	1.96727	−	3.40741i	0.0838846	−	0.145292i
$551$	−29.3738	+	50.8769i	−1.25137	+	2.16743i
$552$		0			0
$553$	11.7873	−	18.7999i	0.501247	−	0.799454i
$554$	−14.1476	+	24.5044i	−0.601076	+	1.04109i
$555$		0			0
$556$	−13.0531			−0.553574
$557$	−12.8040	+	22.1772i	−0.542523	+	0.939678i	0.456235	+	0.889859i	$0.349198\pi$
$557$	−12.8040	+	22.1772i	−0.542523	+	0.939678i	−0.998758	+	0.0498188i	$0.984136\pi$
$558$		0			0
$559$	−8.01594			−0.339038
$560$	−2.23236	+	3.56046i	−0.0943344	+	0.150457i
$561$		0			0
$562$	−17.5956			−0.742228
$563$	−23.3189	−	40.3895i	−0.982773	−	1.70221i	−0.651443	−	0.758698i	$-0.725836\pi$
$563$	−23.3189	−	40.3895i	−0.982773	−	1.70221i	−0.331330	−	0.943515i	$-0.607497\pi$
$564$		0			0
$565$	−15.4585	+	26.7750i	−0.650345	+	1.12643i
$566$	−18.5229			−0.778576
$567$		0			0
$568$	−12.7207			−0.533747
$569$	15.5989	−	27.0181i	0.653939	−	1.13266i	−0.328219	−	0.944602i	$-0.606449\pi$
$569$	15.5989	−	27.0181i	0.653939	−	1.13266i	0.982159	−	0.188054i	$-0.0602182\pi$
$570$		0			0
$571$	7.83812	+	13.5760i	0.328015	+	0.568139i	0.982118	−	0.188267i	$-0.0602869\pi$
$571$	7.83812	+	13.5760i	0.328015	+	0.568139i	−0.654103	+	0.756406i	$0.726954\pi$
$572$	−7.64145			−0.319505
$573$		0			0
$574$	7.27816	+	13.7386i	0.303785	+	0.573438i
$575$	0.744051			0.0310291
$576$		0			0
$577$	6.99567	−	12.1169i	0.291234	−	0.504431i	−0.682868	−	0.730542i	$-0.739268\pi$
$577$	6.99567	−	12.1169i	0.291234	−	0.504431i	0.974102	+	0.226110i	$0.0726010\pi$
$578$	12.1520			0.505455
$579$		0			0
$580$	6.57234	−	11.3836i	0.272902	−	0.472680i
$581$	2.93021	+	5.53120i	0.121565	+	0.229473i
$582$		0			0
$583$	−3.88255	+	6.72477i	−0.160799	+	0.278511i
$584$	8.02654	−	13.9024i	0.332141	−	0.575285i
$585$		0			0
$586$	7.04256	+	12.1981i	0.290926	+	0.503898i
$587$	−1.44801	−	2.50803i	−0.0597658	−	0.103517i	0.834594	−	0.550865i	$-0.185702\pi$
$587$	−1.44801	−	2.50803i	−0.0597658	−	0.103517i	−0.894360	+	0.447348i	$0.852369\pi$
$588$		0			0
$589$	9.62612	−	16.6729i	0.396637	−	0.686996i
$590$	10.2880			0.423550
$591$		0			0
$592$	−1.00000			−0.0410997
$593$	2.04394	+	3.54021i	0.0839346	+	0.145379i	0.904937	−	0.425546i	$-0.139918\pi$
$593$	2.04394	+	3.54021i	0.0839346	+	0.145379i	−0.821002	+	0.570925i	$0.806585\pi$
$594$		0			0
$595$	12.0531	−	19.2238i	0.494128	−	0.788100i
$596$	2.60439	+	4.51093i	0.106680	+	0.184775i
$597$		0			0
$598$	−0.722528	−	1.25146i	−0.0295464	−	0.0511758i
$599$	−9.88255	−	17.1171i	−0.403790	−	0.699385i	0.590390	−	0.807118i	$-0.298974\pi$
$599$	−9.88255	−	17.1171i	−0.403790	−	0.699385i	−0.994180	+	0.107734i	$0.965641\pi$
$600$		0			0
$601$	−13.4320	−	23.2649i	−0.547902	−	0.948994i	−0.998418	−	0.0562261i	$-0.982093\pi$
$601$	−13.4320	−	23.2649i	−0.547902	−	0.948994i	0.450516	−	0.892768i	$-0.351240\pi$
$602$	−4.40545	−	0.160552i	−0.179553	−	0.00654360i
$603$		0			0
$604$	0.261450	+	0.452845i	0.0106383	+	0.0184260i
$605$	13.4647			0.547419
$606$		0			0
$607$	−15.2422			−0.618661			−0.309331	−	0.950955i	$-0.600105\pi$
$607$	−15.2422			−0.618661			−0.309331	+	0.950955i	$0.600105\pi$
$608$	−3.54944	+	6.14781i	−0.143949	+	0.249327i
$609$		0			0
$610$	−3.55563	−	6.15854i	−0.143963	−	0.249352i
$611$	−6.41342	−	11.1084i	−0.259459	−	0.449396i
$612$		0			0
$613$	−1.36033	+	2.35617i	−0.0549434	+	0.0951648i	−0.892189	−	0.451662i	$-0.850831\pi$
$613$	−1.36033	+	2.35617i	−0.0549434	+	0.0951648i	0.837246	+	0.546827i	$0.184165\pi$
$614$	2.92766	−	5.07085i	0.118151	−	0.204643i
$615$		0			0
$616$	−4.19963	−	0.153051i	−0.169208	−	0.00616660i
$617$	9.21812	−	15.9663i	0.371108	−	0.642777i	−0.618629	−	0.785684i	$-0.712311\pi$
$617$	9.21812	−	15.9663i	0.371108	−	0.642777i	0.989736	+	0.142906i	$0.0456448\pi$
$618$		0			0
$619$	0.107546			0.00432262			0.00216131	−	0.999998i	$-0.499312\pi$
$619$	0.107546			0.00432262			0.00216131	+	0.999998i	$0.499312\pi$
$620$	−2.15383	+	3.73054i	−0.0864998	+	0.149822i
$621$		0			0
$622$	−0.810892			−0.0325138
$623$	3.97593	+	7.50516i	0.159292	+	0.300688i
$624$		0			0
$625$	−6.47848			−0.259139
$626$	−5.28799	−	9.15907i	−0.211351	−	0.366070i
$627$		0			0
$628$	−4.43199	+	7.67643i	−0.176856	+	0.306323i
$629$	5.39926			0.215282
$630$		0			0
$631$	35.7266			1.42225			0.711126	−	0.703064i	$-0.248185\pi$
$631$	35.7266			1.42225			0.711126	+	0.703064i	$0.248185\pi$
$632$	−4.19344	+	7.26325i	−0.166806	+	0.288916i
$633$		0			0
$634$	6.09820	+	10.5624i	0.242190	+	0.419486i
$635$	21.3475			0.847152
$636$		0			0
$637$	18.9164	−	27.8615i	0.749494	−	1.10391i
$638$	13.1447			0.520403
$639$		0			0
$640$	0.794182	−	1.37556i	0.0313928	−	0.0543739i
$641$	−17.3128			−0.683813			−0.341906	−	0.939734i	$-0.611073\pi$
$641$	−17.3128			−0.683813			−0.341906	+	0.939734i	$0.611073\pi$
$642$		0			0
$643$	14.4821	−	25.0838i	0.571119	−	0.989207i	−0.425332	−	0.905037i	$-0.639843\pi$
$643$	14.4821	−	25.0838i	0.571119	−	0.989207i	0.996451	−	0.0841700i	$-0.0268239\pi$
$644$	−0.372026	−	0.702253i	−0.0146599	−	0.0276727i
$645$		0			0
$646$	19.1643	−	33.1936i	0.754011	−	1.30599i
$647$	−1.27816	+	2.21384i	−0.0502497	+	0.0870350i	−0.890056	−	0.455851i	$-0.849335\pi$
$647$	−1.27816	+	2.21384i	−0.0502497	+	0.0870350i	0.839807	+	0.542886i	$0.182668\pi$
$648$		0			0
$649$	5.14400	+	8.90966i	0.201920	+	0.349735i
$650$	−5.95853	−	10.3205i	−0.233713	−	0.404802i
$651$		0			0
$652$	10.9814	−	19.0204i	0.430066	−	0.744896i
$653$	−29.9766			−1.17308			−0.586538	−	0.809922i	$-0.699509\pi$
$653$	−29.9766			−1.17308			−0.586538	+	0.809922i	$0.699509\pi$
$654$		0			0
$655$	−5.04580			−0.197156
$656$	−2.93818	−	5.08907i	−0.114717	−	0.198695i
$657$		0			0
$658$	−3.30223	−	6.23345i	−0.128734	−	0.243005i
$659$	7.63162	+	13.2183i	0.297286	+	0.514914i	0.975514	−	0.219937i	$-0.0705853\pi$
$659$	7.63162	+	13.2183i	0.297286	+	0.514914i	−0.678228	+	0.734851i	$0.737252\pi$
$660$		0			0
$661$	13.6261	+	23.6011i	0.529994	+	0.917977i	0.999388	+	0.0349881i	$0.0111393\pi$
$661$	13.6261	+	23.6011i	0.529994	+	0.917977i	−0.469393	+	0.882989i	$0.655527\pi$
$662$	7.83310	+	13.5673i	0.304442	+	0.527309i
$663$		0			0
$664$	−1.18292	−	2.04887i	−0.0459061	−	0.0795117i
$665$	−13.9654	−	26.3618i	−0.541555	−	1.02227i
$666$		0			0
$667$	1.24288	+	2.15273i	0.0481245	+	0.0833541i
$668$	3.30037			0.127695
$669$		0			0
$670$	15.9680			0.616896
$671$	3.55563	−	6.15854i	0.137264	−	0.237748i
$672$		0			0
$673$	23.2280	+	40.2320i	0.895372	+	1.55083i	0.833344	+	0.552755i	$0.186423\pi$
$673$	23.2280	+	40.2320i	0.895372	+	1.55083i	0.0620280	+	0.998074i	$0.480243\pi$
$674$	−4.21201	−	7.29541i	−0.162240	−	0.281009i
$675$		0			0
$676$	−5.07234	+	8.78555i	−0.195090	+	0.337906i
$677$	2.54944	−	4.41576i	0.0979830	−	0.169712i	−0.812867	−	0.582450i	$-0.802094\pi$
$677$	2.54944	−	4.41576i	0.0979830	−	0.169712i	0.910850	+	0.412738i	$0.135428\pi$
$678$		0			0
$679$	−1.76371	−	3.32927i	−0.0676852	−	0.127766i
$680$	−4.28799	+	7.42702i	−0.164437	+	0.284813i
$681$		0			0
$682$	−4.30766			−0.164949
$683$	7.77197	−	13.4614i	0.297386	−	0.515088i	−0.678151	−	0.734923i	$-0.737218\pi$
$683$	7.77197	−	13.4614i	0.297386	−	0.515088i	0.975537	+	0.219835i	$0.0705518\pi$
$684$		0			0
$685$	−33.7738			−1.29043
$686$	10.9542	−	14.9334i	0.418233	−	0.570159i
$687$		0			0
$688$	1.66621			0.0635236
$689$	11.7596	+	20.3682i	0.448005	+	0.775967i
$690$		0			0
$691$	−11.6483	+	20.1755i	−0.443123	+	0.767512i	−0.997919	−	0.0644744i	$-0.979463\pi$
$691$	−11.6483	+	20.1755i	−0.443123	+	0.767512i	0.554796	+	0.831986i	$0.312796\pi$
$692$	−19.1075			−0.726360
$693$		0			0
$694$	−0.567323			−0.0215353
$695$	−10.3665	+	17.9553i	−0.393225	+	0.681085i
$696$		0			0
$697$	15.8640	+	27.4772i	0.600891	+	1.04077i
$698$	0.00728378			0.000275695
$699$		0			0
$700$	−3.06801	−	5.79133i	−0.115960	−	0.218892i
$701$	45.6464			1.72404			0.862020	−	0.506874i	$-0.169199\pi$
$701$	45.6464			1.72404			0.862020	+	0.506874i	$0.169199\pi$
$702$		0			0
$703$	3.54944	−	6.14781i	0.133870	−	0.231869i
$704$	1.58836			0.0598637
$705$		0			0
$706$	3.32691	−	5.76238i	0.125210	−	0.216870i
$707$	31.8163	+	1.15951i	1.19658	+	0.0436079i
$708$		0			0
$709$	−9.00069	+	15.5897i	−0.338028	+	0.585482i	−0.984062	−	0.177827i	$-0.943093\pi$
$709$	−9.00069	+	15.5897i	−0.338028	+	0.585482i	0.646034	+	0.763309i	$0.276427\pi$
$710$	−10.1025	+	17.4981i	−0.379141	+	0.656692i
$711$		0			0
$712$	−1.60507	−	2.78007i	−0.0601527	−	0.104188i
$713$	−0.407305	−	0.705474i	−0.0152537	−	0.0264202i
$714$		0			0
$715$	−6.06870	+	10.5113i	−0.226957	+	0.393100i
$716$	−16.0741			−0.600718
$717$		0			0
$718$	−0.797135			−0.0297488
$719$	−18.4389	−	31.9371i	−0.687654	−	1.19105i	−0.972595	−	0.232506i	$-0.925307\pi$
$719$	−18.4389	−	31.9371i	−0.687654	−	1.19105i	0.284941	−	0.958545i	$-0.408026\pi$
$720$		0			0
$721$	16.1254	+	0.587674i	0.600542	+	0.0218861i
$722$	−15.6971	−	27.1881i	−0.584185	−	1.01184i
$723$		0			0
$724$	−4.02654	−	6.97418i	−0.149645	−	0.259193i
$725$	10.2498	+	17.7531i	0.380666	+	0.659334i
$726$		0			0
$727$	15.2429	+	26.4014i	0.565327	+	0.979175i	0.997019	+	0.0771543i	$0.0245834\pi$
$727$	15.2429	+	26.4014i	0.565327	+	0.979175i	−0.431692	+	0.902021i	$0.642083\pi$
$728$	−6.76145	+	10.7840i	−0.250596	+	0.399683i
$729$		0			0
$730$	−12.7491	−	22.0820i	−0.471864	−	0.817293i
$731$	−8.99628			−0.332739
$732$		0			0
$733$	6.15059			0.227177			0.113589	−	0.993528i	$-0.463765\pi$
$733$	6.15059			0.227177			0.113589	+	0.993528i	$0.463765\pi$
$734$	7.71634	−	13.3651i	0.284815	−	0.493314i
$735$		0			0
$736$	0.150186	+	0.260130i	0.00553593	+	0.00958851i
$737$	7.98398	+	13.8287i	0.294094	+	0.509385i
$738$		0			0
$739$	−20.3912	+	35.3186i	−0.750103	+	1.29922i	0.197670	+	0.980269i	$0.436663\pi$
$739$	−20.3912	+	35.3186i	−0.750103	+	1.29922i	−0.947772	+	0.318947i	$0.896671\pi$
$740$	−0.794182	+	1.37556i	−0.0291947	+	0.0505667i
$741$		0			0
$742$	6.05494	+	11.4296i	0.222284	+	0.419594i
$743$	−7.25271	+	12.5621i	−0.266076	+	0.460858i	−0.967845	−	0.251547i	$-0.919061\pi$
$743$	−7.25271	+	12.5621i	−0.266076	+	0.460858i	0.701769	+	0.712405i	$0.252394\pi$
$744$		0			0
$745$	8.27342			0.303115
$746$	−5.12110	+	8.87000i	−0.187497	+	0.324754i
$747$		0			0
$748$	−8.57598			−0.313569
$749$	−3.82279	−	7.21608i	−0.139682	−	0.263670i
$750$		0			0
$751$	4.18911			0.152863			0.0764314	−	0.997075i	$-0.475647\pi$
$751$	4.18911			0.152863			0.0764314	+	0.997075i	$0.475647\pi$
$752$	1.33310	+	2.30900i	0.0486133	+	0.0842007i
$753$		0			0
$754$	19.9065	−	34.4791i	0.724953	−	1.25566i
$755$	0.830556			0.0302270
$756$		0			0
$757$	2.38688			0.0867525			0.0433763	−	0.999059i	$-0.486189\pi$
$757$	2.38688			0.0867525			0.0433763	+	0.999059i	$0.486189\pi$
$758$	−12.5043	+	21.6581i	−0.454178	+	0.786659i
$759$		0			0
$760$	5.63781	+	9.76497i	0.204505	+	0.354213i
$761$	−3.63416			−0.131738			−0.0658692	−	0.997828i	$-0.520982\pi$
$761$	−3.63416			−0.131738			−0.0658692	+	0.997828i	$0.520982\pi$
$762$		0			0
$763$	−3.21565	+	5.12874i	−0.116414	+	0.185673i
$764$	23.9629			0.866946
$765$		0			0
$766$	−3.13348	+	5.42734i	−0.113217	+	0.196098i
$767$	31.1606			1.12515
$768$		0			0
$769$	−19.9672	+	34.5842i	−0.720035	+	1.24714i	0.240950	+	0.970538i	$0.422541\pi$
$769$	−19.9672	+	34.5842i	−0.720035	+	1.24714i	−0.960985	+	0.276600i	$0.910792\pi$
$770$	−3.54580	+	5.65531i	−0.127782	+	0.203803i
$771$		0			0
$772$	−4.88255	+	8.45682i	−0.175727	+	0.304368i
$773$	−18.0698	+	31.2978i	−0.649925	+	1.12570i	0.333215	+	0.942851i	$0.391867\pi$
$773$	−18.0698	+	31.2978i	−0.649925	+	1.12570i	−0.983140	+	0.182853i	$0.941467\pi$
$774$		0			0
$775$	−3.35896	−	5.81788i	−0.120657	−	0.208985i
$776$	0.712008	+	1.23323i	0.0255596	+	0.0442705i
$777$		0			0
$778$	−10.8171	+	18.7357i	−0.387811	+	0.671709i
$779$	41.7156			1.49462
$780$		0			0
$781$	−20.2051			−0.722994
$782$	−0.810892	−	1.40451i	−0.0289974	−	0.0502251i
$783$		0			0

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:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	378.h (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} -1.58836 q^{5} +(-2.64400 - 0.0963576i) q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} -1.58836 q^{5} +(-2.64400 - 0.0963576i) q^{7} +1.00000 q^{8} +(0.794182 - 1.37556i) q^{10} +1.58836 q^{11} +(2.40545 - 4.16635i) q^{13} +(1.40545 - 2.24159i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(2.69963 - 4.67589i) q^{17} +(-3.54944 - 6.14781i) q^{19} +(0.794182 + 1.37556i) q^{20} +(-0.794182 + 1.37556i) q^{22} -0.300372 q^{23} -2.47710 q^{25} +(2.40545 + 4.16635i) q^{26} +(1.23855 + 2.33795i) q^{28} +(-4.13781 - 7.16689i) q^{29} +(1.35600 + 2.34867i) q^{31} +(-0.500000 - 0.866025i) q^{32} +(2.69963 + 4.67589i) q^{34} +(4.19963 + 0.153051i) q^{35} +(0.500000 + 0.866025i) q^{37} +7.09888 q^{38} -1.58836 q^{40} +(-2.93818 + 5.08907i) q^{41} +(-0.833104 - 1.44298i) q^{43} +(-0.794182 - 1.37556i) q^{44} +(0.150186 - 0.260130i) q^{46} +(1.33310 - 2.30900i) q^{47} +(6.98143 + 0.509538i) q^{49} +(1.23855 - 2.14523i) q^{50} -4.81089 q^{52} +(-2.44437 + 4.23377i) q^{53} -2.52290 q^{55} +(-2.64400 - 0.0963576i) q^{56} +8.27561 q^{58} +(3.23855 + 5.60933i) q^{59} +(2.23855 - 3.87728i) q^{61} -2.71201 q^{62} +1.00000 q^{64} +(-3.82072 + 6.61769i) q^{65} +(5.02654 + 8.70623i) q^{67} -5.39926 q^{68} +(-2.23236 + 3.56046i) q^{70} -12.7207 q^{71} +(8.02654 - 13.9024i) q^{73} -1.00000 q^{74} +(-3.54944 + 6.14781i) q^{76} +(-4.19963 - 0.153051i) q^{77} +(-4.19344 + 7.26325i) q^{79} +(0.794182 - 1.37556i) q^{80} +(-2.93818 - 5.08907i) q^{82} +(-1.18292 - 2.04887i) q^{83} +(-4.28799 + 7.42702i) q^{85} +1.66621 q^{86} +1.58836 q^{88} +(-1.60507 - 2.78007i) q^{89} +(-6.76145 + 10.7840i) q^{91} +(0.150186 + 0.260130i) q^{92} +(1.33310 + 2.30900i) q^{94} +(5.63781 + 9.76497i) q^{95} +(0.712008 + 1.23323i) q^{97} +(-3.93199 + 5.79133i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{2} - 3 q^{4} + 2 q^{5} - 4 q^{7} + 6 q^{8}+O(q^{10}) \) 6 * q - 3 * q^2 - 3 * q^4 + 2 * q^5 - 4 * q^7 + 6 * q^8	\( 6 q - 3 q^{2} - 3 q^{4} + 2 q^{5} - 4 q^{7} + 6 q^{8} - q^{10} - 2 q^{11} + 8 q^{13} + 2 q^{14} - 3 q^{16} + 4 q^{17} - 3 q^{19} - q^{20} + q^{22} - 14 q^{23} - 4 q^{25} + 8 q^{26} + 2 q^{28} + 5 q^{29} + 20 q^{31} - 3 q^{32} + 4 q^{34} + 13 q^{35} + 3 q^{37} + 6 q^{38} + 2 q^{40} - 6 q^{43} + q^{44} + 7 q^{46} + 9 q^{47} - 12 q^{49} + 2 q^{50} - 16 q^{52} - 15 q^{53} - 26 q^{55} - 4 q^{56} - 10 q^{58} + 14 q^{59} + 8 q^{61} - 40 q^{62} + 6 q^{64} + 12 q^{65} + q^{67} - 8 q^{68} + 10 q^{70} - 14 q^{71} + 19 q^{73} - 6 q^{74} - 3 q^{76} - 13 q^{77} + 5 q^{79} - q^{80} - 2 q^{83} - 2 q^{85} + 12 q^{86} - 2 q^{88} + 9 q^{89} - 46 q^{91} + 7 q^{92} + 9 q^{94} + 4 q^{95} + 28 q^{97} + 12 q^{98}+O(q^{100}) \) 6 * q - 3 * q^2 - 3 * q^4 + 2 * q^5 - 4 * q^7 + 6 * q^8 - q^10 - 2 * q^11 + 8 * q^13 + 2 * q^14 - 3 * q^16 + 4 * q^17 - 3 * q^19 - q^20 + q^22 - 14 * q^23 - 4 * q^25 + 8 * q^26 + 2 * q^28 + 5 * q^29 + 20 * q^31 - 3 * q^32 + 4 * q^34 + 13 * q^35 + 3 * q^37 + 6 * q^38 + 2 * q^40 - 6 * q^43 + q^44 + 7 * q^46 + 9 * q^47 - 12 * q^49 + 2 * q^50 - 16 * q^52 - 15 * q^53 - 26 * q^55 - 4 * q^56 - 10 * q^58 + 14 * q^59 + 8 * q^61 - 40 * q^62 + 6 * q^64 + 12 * q^65 + q^67 - 8 * q^68 + 10 * q^70 - 14 * q^71 + 19 * q^73 - 6 * q^74 - 3 * q^76 - 13 * q^77 + 5 * q^79 - q^80 - 2 * q^83 - 2 * q^85 + 12 * q^86 - 2 * q^88 + 9 * q^89 - 46 * q^91 + 7 * q^92 + 9 * q^94 + 4 * q^95 + 28 * q^97 + 12 * q^98

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