LMFDB - Embedded newform 810.3.b.b.809.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [810,3,Mod(809,810)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(810, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("810.809");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 810 = 2 \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	810.b (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:	$22.0709014132$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 2x^{14} - 230x^{12} - 96x^{10} + 25551x^{8} - 7776x^{6} - 1509030x^{4} + 1062882x^{2} + 43046721 $ x^16 + 2x^14 - 230x^12 - 96x^10 + 25551x^8 - 7776x^6 - 1509030x^4 + 1062882*x^2 + 43046721
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{8} $
Twist minimal:	no (minimal twist has level 90)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			809.12
Root			$-0.173499 - 2.99498i$ of defining polynomial
Character	$\chi$	$=$	810.809
Dual form			810.3.b.b.809.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} +2.00000 q^{4} +(-3.56108 + 3.50980i) q^{5} -3.17976i q^{7} +2.82843 q^{8} +O(q^{10})$	\(q+1.41421 q^{2} +2.00000 q^{4} +(-3.56108 + 3.50980i) q^{5} -3.17976i q^{7} +2.82843 q^{8} +(-5.03613 + 4.96360i) q^{10} -15.6543i q^{11} +1.33596i q^{13} -4.49686i q^{14} +4.00000 q^{16} +23.4761 q^{17} -23.3447 q^{19} +(-7.12217 + 7.01960i) q^{20} -22.1386i q^{22} +28.6849 q^{23} +(0.362627 - 24.9974i) q^{25} +1.88933i q^{26} -6.35952i q^{28} -33.1593i q^{29} +18.5472 q^{31} +5.65685 q^{32} +33.2002 q^{34} +(11.1603 + 11.3234i) q^{35} +3.95544i q^{37} -33.0144 q^{38} +(-10.0723 + 9.92721i) q^{40} -60.3301i q^{41} +2.83257i q^{43} -31.3087i q^{44} +40.5665 q^{46} +71.6735 q^{47} +38.8891 q^{49} +(0.512832 - 35.3516i) q^{50} +2.67192i q^{52} -14.3465 q^{53} +(54.9436 + 55.7464i) q^{55} -8.99372i q^{56} -46.8943i q^{58} +72.3760i q^{59} +11.2385 q^{61} +26.2296 q^{62} +8.00000 q^{64} +(-4.68894 - 4.75746i) q^{65} -3.35443i q^{67} +46.9521 q^{68} +(15.7831 + 16.0137i) q^{70} -40.0554i q^{71} -86.6579i q^{73} +5.59384i q^{74} -46.6894 q^{76} -49.7770 q^{77} -90.5804 q^{79} +(-14.2443 + 14.0392i) q^{80} -85.3196i q^{82} +102.892 q^{83} +(-83.6002 + 82.3962i) q^{85} +4.00586i q^{86} -44.2772i q^{88} -144.459i q^{89} +4.24802 q^{91} +57.3697 q^{92} +101.362 q^{94} +(83.1323 - 81.9351i) q^{95} +74.8437i q^{97} +54.9975 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 32 q^{4}+O(q^{10}) $ 16 * q + 32 * q^4	$ 16 q + 32 q^{4} - 16 q^{10} + 64 q^{16} + 144 q^{19} - 12 q^{25} - 56 q^{31} + 272 q^{34} - 32 q^{40} - 56 q^{46} - 24 q^{49} + 20 q^{55} - 136 q^{61} + 128 q^{64} + 224 q^{70} + 288 q^{76} - 840 q^{79} + 272 q^{85} - 168 q^{91} + 328 q^{94}+O(q^{100}) $ 16 * q + 32 * q^4 - 16 * q^10 + 64 * q^16 + 144 * q^19 - 12 * q^25 - 56 * q^31 + 272 * q^34 - 32 * q^40 - 56 * q^46 - 24 * q^49 + 20 * q^55 - 136 * q^61 + 128 * q^64 + 224 * q^70 + 288 * q^76 - 840 * q^79 + 272 * q^85 - 168 * q^91 + 328 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/810\mathbb{Z}\right)^\times$.

$n$	$487$	$731$
$\chi(n)$	$-1$	$-1$

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 $

$2$	1.41421			0.707107
$2$	1.41421			0.707107
$3$		0			0
$3$		0			0
$4$	2.00000			0.500000
$5$	−3.56108	+	3.50980i	−0.712217	+	0.701960i
$5$	−3.56108	+	3.50980i	−0.712217	+	0.701960i
$6$		0			0
$7$		−	3.17976i		−	0.454251i	−0.973865	−	0.227126i	$-0.927067\pi$
$7$		−	3.17976i		−	0.454251i	0.973865	−	0.227126i	$-0.0729328\pi$
$8$	2.82843			0.353553
$9$		0			0
$10$	−5.03613	+	4.96360i	−0.503613	+	0.496360i
$11$		−	15.6543i		−	1.42312i	−0.702624	−	0.711561i	$-0.747989\pi$
$11$		−	15.6543i		−	1.42312i	0.702624	−	0.711561i	$-0.252011\pi$
$12$		0			0
$13$			1.33596i			0.102766i	0.998679	+	0.0513830i	$0.0163629\pi$
$13$			1.33596i			0.102766i	−0.998679	+	0.0513830i	$0.983637\pi$
$14$		−	4.49686i		−	0.321204i
$15$		0			0
$16$	4.00000			0.250000
$17$	23.4761			1.38094			0.690472	−	0.723359i	$-0.257403\pi$
$17$	23.4761			1.38094			0.690472	+	0.723359i	$0.257403\pi$
$18$		0			0
$19$	−23.3447			−1.22867			−0.614334	−	0.789046i	$-0.710575\pi$
$19$	−23.3447			−1.22867			−0.614334	+	0.789046i	$0.710575\pi$
$20$	−7.12217	+	7.01960i	−0.356108	+	0.350980i
$21$		0			0
$22$		−	22.1386i		−	1.00630i
$23$	28.6849			1.24717			0.623584	−	0.781756i	$-0.285676\pi$
$23$	28.6849			1.24717			0.623584	+	0.781756i	$0.285676\pi$
$24$		0			0
$25$	0.362627	−	24.9974i	0.0145051	−	0.999895i
$26$			1.88933i			0.0726665i
$27$		0			0
$28$		−	6.35952i		−	0.227126i
$29$		−	33.1593i		−	1.14342i	−0.820454	−	0.571712i	$-0.806279\pi$
$29$		−	33.1593i		−	1.14342i	0.820454	−	0.571712i	$-0.193721\pi$
$30$		0			0
$31$	18.5472			0.598296			0.299148	−	0.954207i	$-0.403298\pi$
$31$	18.5472			0.598296			0.299148	+	0.954207i	$0.403298\pi$
$32$	5.65685			0.176777
$33$		0			0
$34$	33.2002			0.976475
$35$	11.1603	+	11.3234i	0.318866	+	0.323525i
$36$		0			0
$37$			3.95544i			0.106904i	0.998570	+	0.0534519i	$0.0170224\pi$
$37$			3.95544i			0.106904i	−0.998570	+	0.0534519i	$0.982978\pi$
$38$	−33.0144			−0.868799
$39$		0			0
$40$	−10.0723	+	9.92721i	−0.251807	+	0.248180i
$41$		−	60.3301i		−	1.47147i	−0.677272	−	0.735733i	$-0.736838\pi$
$41$		−	60.3301i		−	1.47147i	0.677272	−	0.735733i	$-0.263162\pi$
$42$		0			0
$43$			2.83257i			0.0658738i	0.999457	+	0.0329369i	$0.0104860\pi$
$43$			2.83257i			0.0658738i	−0.999457	+	0.0329369i	$0.989514\pi$
$44$		−	31.3087i		−	0.711561i
$45$		0			0
$46$	40.5665			0.881881
$47$	71.6735			1.52497			0.762485	−	0.647006i	$-0.223979\pi$
$47$	71.6735			1.52497			0.762485	+	0.647006i	$0.223979\pi$
$48$		0			0
$49$	38.8891			0.793656
$50$	0.512832	−	35.3516i	0.0102566	−	0.707032i
$51$		0			0
$52$			2.67192i			0.0513830i
$53$	−14.3465			−0.270689			−0.135345	−	0.990799i	$-0.543214\pi$
$53$	−14.3465			−0.270689			−0.135345	+	0.990799i	$0.543214\pi$
$54$		0			0
$55$	54.9436	+	55.7464i	0.998974	+	1.01357i
$56$		−	8.99372i		−	0.160602i
$57$		0			0
$58$		−	46.8943i		−	0.808523i
$59$			72.3760i			1.22671i	0.789807	+	0.613356i	$0.210181\pi$
$59$			72.3760i			1.22671i	−0.789807	+	0.613356i	$0.789819\pi$
$60$		0			0
$61$	11.2385			0.184237			0.0921186	−	0.995748i	$-0.470636\pi$
$61$	11.2385			0.184237			0.0921186	+	0.995748i	$0.470636\pi$
$62$	26.2296			0.423059
$63$		0			0
$64$	8.00000			0.125000
$65$	−4.68894	−	4.75746i	−0.0721376	−	0.0731917i
$66$		0			0
$67$		−	3.35443i		−	0.0500661i	−0.999687	−	0.0250330i	$-0.992031\pi$
$67$		−	3.35443i		−	0.0500661i	0.999687	−	0.0250330i	$-0.00796909\pi$
$68$	46.9521			0.690472
$69$		0			0
$70$	15.7831	+	16.0137i	0.225472	+	0.228767i
$71$		−	40.0554i		−	0.564160i	−0.959391	−	0.282080i	$-0.908976\pi$
$71$		−	40.0554i		−	0.564160i	0.959391	−	0.282080i	$-0.0910244\pi$
$72$		0			0
$73$		−	86.6579i		−	1.18709i	−0.804799	−	0.593547i	$-0.797727\pi$
$73$		−	86.6579i		−	1.18709i	0.804799	−	0.593547i	$-0.202273\pi$
$74$			5.59384i			0.0755925i
$75$		0			0
$76$	−46.6894			−0.614334
$77$	−49.7770			−0.646455
$78$		0			0
$79$	−90.5804			−1.14659			−0.573294	−	0.819350i	$-0.694335\pi$
$79$	−90.5804			−1.14659			−0.573294	+	0.819350i	$0.694335\pi$
$80$	−14.2443	+	14.0392i	−0.178054	+	0.175490i
$81$		0			0
$82$		−	85.3196i		−	1.04048i
$83$	102.892			1.23967			0.619833	−	0.784734i	$-0.287200\pi$
$83$	102.892			1.23967			0.619833	+	0.784734i	$0.287200\pi$
$84$		0			0
$85$	−83.6002	+	82.3962i	−0.983531	+	0.969367i
$86$			4.00586i			0.0465798i
$87$		0			0
$88$		−	44.2772i		−	0.503150i
$89$		−	144.459i		−	1.62313i	−0.584259	−	0.811567i	$-0.698615\pi$
$89$		−	144.459i		−	1.62313i	0.584259	−	0.811567i	$-0.301385\pi$
$90$		0			0
$91$	4.24802			0.0466816
$92$	57.3697			0.623584
$93$		0			0
$94$	101.362			1.07832
$95$	83.1323	−	81.9351i	0.875077	−	0.862475i
$96$		0			0
$97$			74.8437i			0.771584i	0.922586	+	0.385792i	$0.126072\pi$
$97$			74.8437i			0.771584i	−0.922586	+	0.385792i	$0.873928\pi$
$98$	54.9975			0.561199
$99$		0			0
$100$	0.725253	−	49.9947i	0.00725253	−	0.499947i
$101$			16.9570i			0.167891i	0.996470	+	0.0839455i	$0.0267522\pi$
$101$			16.9570i			0.167891i	−0.996470	+	0.0839455i	$0.973248\pi$
$102$		0			0
$103$			105.653i			1.02576i	0.858461	+	0.512879i	$0.171421\pi$
$103$			105.653i			1.02576i	−0.858461	+	0.512879i	$0.828579\pi$
$104$			3.77866i			0.0363333i
$105$		0			0
$106$	−20.2891			−0.191406
$107$	−58.8159			−0.549681			−0.274840	−	0.961490i	$-0.588625\pi$
$107$	−58.8159			−0.549681			−0.274840	+	0.961490i	$0.588625\pi$
$108$		0			0
$109$	−184.365			−1.69142			−0.845710	−	0.533643i	$-0.820823\pi$
$109$	−184.365			−1.69142			−0.845710	+	0.533643i	$0.820823\pi$
$110$	77.7020	+	78.8373i	0.706382	+	0.716703i
$111$		0			0
$112$		−	12.7190i		−	0.113563i
$113$	−17.1551			−0.151815			−0.0759073	−	0.997115i	$-0.524185\pi$
$113$	−17.1551			−0.151815			−0.0759073	+	0.997115i	$0.524185\pi$
$114$		0			0
$115$	−102.149	+	100.678i	−0.888254	+	0.875461i
$116$		−	66.3186i		−	0.571712i
$117$		0			0
$118$			102.355i			0.867416i
$119$		−	74.6482i		−	0.627296i
$120$		0			0
$121$	−124.058			−1.02528
$122$	15.8936			0.130275
$123$		0			0
$124$	37.0943			0.299148
$125$	86.4444	+	90.2905i	0.691555	+	0.722324i
$126$		0			0
$127$		−	17.8557i		−	0.140596i	−0.997526	−	0.0702979i	$-0.977605\pi$
$127$		−	17.8557i		−	0.140596i	0.997526	−	0.0702979i	$-0.0223950\pi$
$128$	11.3137			0.0883883
$129$		0			0
$130$	−6.63117	−	6.72806i	−0.0510090	−	0.0517543i
$131$		−	212.823i		−	1.62460i	−0.583239	−	0.812301i	$-0.698215\pi$
$131$		−	212.823i		−	1.62460i	0.583239	−	0.812301i	$-0.301785\pi$
$132$		0			0
$133$			74.2305i			0.558124i
$134$		−	4.74387i		−	0.0354020i
$135$		0			0
$136$	66.4003			0.488238
$137$	−197.548			−1.44195			−0.720977	−	0.692959i	$-0.756307\pi$
$137$	−197.548			−1.44195			−0.720977	+	0.692959i	$0.756307\pi$
$138$		0			0
$139$	−51.5742			−0.371037			−0.185519	−	0.982641i	$-0.559396\pi$
$139$	−51.5742			−0.371037			−0.185519	+	0.982641i	$0.559396\pi$
$140$	22.3206	+	22.6468i	0.159433	+	0.161763i
$141$		0			0
$142$		−	56.6469i		−	0.398922i
$143$	20.9135			0.146249
$144$		0			0
$145$	116.382	+	118.083i	0.802638	+	0.814366i
$146$		−	122.553i		−	0.839403i
$147$		0			0
$148$			7.91089i			0.0534519i
$149$			225.983i			1.51666i	0.651868	+	0.758332i	$0.273985\pi$
$149$			225.983i			1.51666i	−0.651868	+	0.758332i	$0.726015\pi$
$150$		0			0
$151$	−53.6516			−0.355308			−0.177654	−	0.984093i	$-0.556851\pi$
$151$	−53.6516			−0.355308			−0.177654	+	0.984093i	$0.556851\pi$
$152$	−66.0287			−0.434400
$153$		0			0
$154$	−70.3954			−0.457113
$155$	−66.0480	+	65.0968i	−0.426116	+	0.419979i
$156$		0			0
$157$		−	153.382i		−	0.976953i	−0.872577	−	0.488477i	$-0.837553\pi$
$157$		−	153.382i		−	0.976953i	0.872577	−	0.488477i	$-0.162447\pi$
$158$	−128.100			−0.810760
$159$		0			0
$160$	−20.1445	+	19.8544i	−0.125903	+	0.124090i
$161$		−	91.2109i		−	0.566528i
$162$		0			0
$163$			272.064i			1.66910i	0.550929	+	0.834552i	$0.314273\pi$
$163$			272.064i			1.66910i	−0.550929	+	0.834552i	$0.685727\pi$
$164$		−	120.660i		−	0.735733i
$165$		0			0
$166$	145.512			0.876576
$167$	160.173			0.959122			0.479561	−	0.877508i	$-0.340796\pi$
$167$	160.173			0.959122			0.479561	+	0.877508i	$0.340796\pi$
$168$		0			0
$169$	167.215			0.989439
$170$	−118.228	+	116.526i	−0.695462	+	0.685446i
$171$		0			0
$172$			5.66514i			0.0329369i
$173$	30.4179			0.175826			0.0879130	−	0.996128i	$-0.471980\pi$
$173$	30.4179			0.175826			0.0879130	+	0.996128i	$0.471980\pi$
$174$		0			0
$175$	−79.4856	−	1.15307i	−0.454204	−	0.00658895i
$176$		−	62.6174i		−	0.355781i
$177$		0			0
$178$		−	204.296i		−	1.14773i
$179$			261.709i			1.46206i	0.682344	+	0.731031i	$0.260961\pi$
$179$			261.709i			1.46206i	−0.682344	+	0.731031i	$0.739039\pi$
$180$		0			0
$181$	−158.159			−0.873806			−0.436903	−	0.899509i	$-0.643925\pi$
$181$	−158.159			−0.873806			−0.436903	+	0.899509i	$0.643925\pi$
$182$	6.00761			0.0330089
$183$		0			0
$184$	81.1330			0.440940
$185$	−13.8828	−	14.0857i	−0.0750422	−	0.0761387i
$186$		0			0
$187$		−	367.502i		−	1.96525i
$188$	143.347			0.762485
$189$		0			0
$190$	117.567	−	115.874i	0.618773	−	0.609862i
$191$		−	123.681i		−	0.647542i	−0.946135	−	0.323771i	$-0.895049\pi$
$191$		−	123.681i		−	0.647542i	0.946135	−	0.323771i	$-0.104951\pi$
$192$		0			0
$193$		−	44.7888i		−	0.232066i	−0.993245	−	0.116033i	$-0.962982\pi$
$193$		−	44.7888i		−	0.232066i	0.993245	−	0.116033i	$-0.0370179\pi$
$194$			105.845i			0.545592i
$195$		0			0
$196$	77.7783			0.396828
$197$	203.271			1.03183			0.515917	−	0.856638i	$-0.327451\pi$
$197$	203.271			1.03183			0.515917	+	0.856638i	$0.327451\pi$
$198$		0			0
$199$	−77.5958			−0.389929			−0.194964	−	0.980810i	$-0.562459\pi$
$199$	−77.5958			−0.389929			−0.194964	+	0.980810i	$0.562459\pi$
$200$	1.02566	−	70.7032i	0.00512832	−	0.353516i
$201$		0			0
$202$			23.9808i			0.118717i
$203$	−105.439			−0.519402
$204$		0			0
$205$	211.746	+	214.840i	1.03291	+	1.04800i
$206$			149.416i			0.725320i
$207$		0			0
$208$			5.34383i			0.0256915i
$209$			365.446i			1.74854i
$210$		0			0
$211$	−132.330			−0.627157			−0.313578	−	0.949562i	$-0.601528\pi$
$211$	−132.330			−0.627157			−0.313578	+	0.949562i	$0.601528\pi$
$212$	−28.6931			−0.135345
$213$		0			0
$214$	−83.1782			−0.388683
$215$	−9.94176	−	10.0870i	−0.0462407	−	0.0469164i
$216$		0			0
$217$		−	58.9755i		−	0.271777i
$218$	−260.731			−1.19601
$219$		0			0
$220$	109.887	+	111.493i	0.499487	+	0.506786i
$221$			31.3630i			0.141914i
$222$		0			0
$223$			349.183i			1.56584i	0.622122	+	0.782921i	$0.286271\pi$
$223$			349.183i			1.56584i	−0.622122	+	0.782921i	$0.713729\pi$
$224$		−	17.9874i		−	0.0803010i
$225$		0			0
$226$	−24.2609			−0.107349
$227$	−218.334			−0.961825			−0.480912	−	0.876769i	$-0.659694\pi$
$227$	−218.334			−0.961825			−0.480912	+	0.876769i	$0.659694\pi$
$228$		0			0
$229$	380.456			1.66138			0.830690	−	0.556735i	$-0.187946\pi$
$229$	380.456			1.66138			0.830690	+	0.556735i	$0.187946\pi$
$230$	−144.461	+	142.380i	−0.628090	+	0.619045i
$231$		0			0
$232$		−	93.7887i		−	0.404261i
$233$	−232.991			−0.999963			−0.499981	−	0.866036i	$-0.666660\pi$
$233$	−232.991			−0.999963			−0.499981	+	0.866036i	$0.666660\pi$
$234$		0			0
$235$	−255.235	+	251.560i	−1.08611	+	1.07047i
$236$			144.752i			0.613356i
$237$		0			0
$238$		−	105.568i		−	0.443565i
$239$			157.493i			0.658965i	0.944162	+	0.329482i	$0.106874\pi$
$239$			157.493i			0.658965i	−0.944162	+	0.329482i	$0.893126\pi$
$240$		0			0
$241$	−115.151			−0.477806			−0.238903	−	0.971043i	$-0.576788\pi$
$241$	−115.151			−0.477806			−0.238903	+	0.971043i	$0.576788\pi$
$242$	−175.445			−0.724980
$243$		0			0
$244$	22.4769			0.0921186
$245$	−138.487	+	136.493i	−0.565255	+	0.557114i
$246$		0			0
$247$		−	31.1875i		−	0.126265i
$248$	52.4593			0.211529
$249$		0			0
$250$	122.251	+	127.690i	0.489003	+	0.510760i
$251$			157.563i			0.627740i	0.949466	+	0.313870i	$0.101626\pi$
$251$			157.563i			0.627740i	−0.949466	+	0.313870i	$0.898374\pi$
$252$		0			0
$253$		−	449.043i		−	1.77487i
$254$		−	25.2517i		−	0.0994162i
$255$		0			0
$256$	16.0000			0.0625000
$257$	−20.1945			−0.0785777			−0.0392889	−	0.999228i	$-0.512509\pi$
$257$	−20.1945			−0.0785777			−0.0392889	+	0.999228i	$0.512509\pi$
$258$		0			0
$259$	12.5774			0.0485612
$260$	−9.37789	−	9.51491i	−0.0360688	−	0.0365958i
$261$		0			0
$262$		−	300.977i		−	1.14877i
$263$	101.679			0.386614			0.193307	−	0.981138i	$-0.438079\pi$
$263$	101.679			0.386614			0.193307	+	0.981138i	$0.438079\pi$
$264$		0			0
$265$	51.0892	−	50.3534i	0.192789	−	0.190013i
$266$			104.978i			0.394653i
$267$		0			0
$268$		−	6.70885i		−	0.0250330i
$269$		−	83.9984i		−	0.312262i	−0.987736	−	0.156131i	$-0.950098\pi$
$269$		−	83.9984i		−	0.312262i	0.987736	−	0.156131i	$-0.0499022\pi$
$270$		0			0
$271$	349.591			1.29000			0.645001	−	0.764182i	$-0.276857\pi$
$271$	349.591			1.29000			0.645001	+	0.764182i	$0.276857\pi$
$272$	93.9042			0.345236
$273$		0			0
$274$	−279.375			−1.01962
$275$	−391.317	−	5.67668i	−1.42297	−	0.0206425i
$276$		0			0
$277$		−	326.401i		−	1.17834i	−0.808008	−	0.589172i	$-0.799454\pi$
$277$		−	326.401i		−	1.17834i	0.808008	−	0.589172i	$-0.200546\pi$
$278$	−72.9369			−0.262363
$279$		0			0
$280$	31.5661	+	32.0274i	0.112736	+	0.114383i
$281$		−	108.743i		−	0.386986i	−0.981102	−	0.193493i	$-0.938018\pi$
$281$		−	108.743i		−	0.386986i	0.981102	−	0.193493i	$-0.0619816\pi$
$282$		0			0
$283$			380.702i			1.34524i	0.739990	+	0.672618i	$0.234830\pi$
$283$			380.702i			1.34524i	−0.739990	+	0.672618i	$0.765170\pi$
$284$		−	80.1108i		−	0.282080i
$285$		0			0
$286$	29.5762			0.103413
$287$	−191.835			−0.668415
$288$		0			0
$289$	262.125			0.907007
$290$	164.590	+	166.995i	0.567550	+	0.575843i
$291$		0			0
$292$		−	173.316i		−	0.593547i
$293$	−101.317			−0.345790			−0.172895	−	0.984940i	$-0.555312\pi$
$293$	−101.317			−0.345790			−0.172895	+	0.984940i	$0.555312\pi$
$294$		0			0
$295$	−254.025	−	257.737i	−0.861102	−	0.873685i
$296$			11.1877i			0.0377962i
$297$		0			0
$298$			319.588i			1.07244i
$299$			38.3218i			0.128166i
$300$		0			0
$301$	9.00690			0.0299232
$302$	−75.8748			−0.251241
$303$		0			0
$304$	−93.3787			−0.307167
$305$	−40.0211	+	39.4448i	−0.131217	+	0.129327i
$306$		0			0
$307$		−	115.566i		−	0.376436i	−0.982127	−	0.188218i	$-0.939729\pi$
$307$		−	115.566i		−	0.376436i	0.982127	−	0.188218i	$-0.0602711\pi$
$308$	−99.5541			−0.323228
$309$		0			0
$310$	−93.4060	+	92.0608i	−0.301310	+	0.296970i
$311$			338.675i			1.08899i	0.838765	+	0.544494i	$0.183278\pi$
$311$			338.675i			1.08899i	−0.838765	+	0.544494i	$0.816722\pi$
$312$		0			0
$313$			267.167i			0.853569i	0.904353	+	0.426785i	$0.140354\pi$
$313$			267.167i			0.853569i	−0.904353	+	0.426785i	$0.859646\pi$
$314$		−	216.914i		−	0.690810i
$315$		0			0
$316$	−181.161			−0.573294
$317$	293.696			0.926485			0.463243	−	0.886231i	$-0.346686\pi$
$317$	293.696			0.926485			0.463243	+	0.886231i	$0.346686\pi$
$318$		0			0
$319$	−519.087			−1.62723
$320$	−28.4887	+	28.0784i	−0.0890271	+	0.0877450i
$321$		0			0
$322$		−	128.992i		−	0.400595i
$323$	−548.041			−1.69672
$324$		0			0
$325$	33.3954	+	0.484454i	0.102755	+	0.00149063i
$326$			384.757i			1.18023i
$327$		0			0
$328$		−	170.639i		−	0.520242i
$329$		−	227.905i		−	0.692719i
$330$		0			0
$331$	−47.2025			−0.142606			−0.0713028	−	0.997455i	$-0.522716\pi$
$331$	−47.2025			−0.142606			−0.0713028	+	0.997455i	$0.522716\pi$
$332$	205.785			0.619833
$333$		0			0
$334$	226.519			0.678202
$335$	11.7734	+	11.9454i	0.0351444	+	0.0356579i
$336$		0			0
$337$		−	468.253i		−	1.38947i	−0.719264	−	0.694737i	$-0.755521\pi$
$337$		−	468.253i		−	1.38947i	0.719264	−	0.694737i	$-0.244479\pi$
$338$	236.478			0.699639
$339$		0			0
$340$	−167.200	+	164.792i	−0.491766	+	0.484684i
$341$		−	290.344i		−	0.851448i
$342$		0			0
$343$		−	279.466i		−	0.814770i
$344$			8.01172i			0.0232899i
$345$		0			0
$346$	43.0174			0.124328
$347$	−57.0054			−0.164281			−0.0821404	−	0.996621i	$-0.526176\pi$
$347$	−57.0054			−0.164281			−0.0821404	+	0.996621i	$0.526176\pi$
$348$		0			0
$349$	−179.131			−0.513268			−0.256634	−	0.966509i	$-0.582613\pi$
$349$	−179.131			−0.513268			−0.256634	+	0.966509i	$0.582613\pi$
$350$	−112.410	−	1.63068i	−0.321170	−	0.00465909i
$351$		0			0
$352$		−	88.5543i		−	0.251575i
$353$	−22.8170			−0.0646375			−0.0323188	−	0.999478i	$-0.510289\pi$
$353$	−22.8170			−0.0646375			−0.0323188	+	0.999478i	$0.510289\pi$
$354$		0			0
$355$	140.586	+	142.641i	0.396018	+	0.401804i
$356$		−	288.918i		−	0.811567i
$357$		0			0
$358$			370.113i			1.03383i
$359$			598.539i			1.66724i	0.552339	+	0.833619i	$0.313735\pi$
$359$			598.539i			1.66724i	−0.552339	+	0.833619i	$0.686265\pi$
$360$		0			0
$361$	183.974			0.509623
$362$	−223.670			−0.617874
$363$		0			0
$364$	8.49605			0.0233408
$365$	304.152	+	308.596i	0.833293	+	0.845469i
$366$		0			0
$367$		−	18.7000i		−	0.0509536i	−0.999675	−	0.0254768i	$-0.991890\pi$
$367$		−	18.7000i		−	0.0509536i	0.999675	−	0.0254768i	$-0.00811039\pi$
$368$	114.739			0.311792
$369$		0			0
$370$	−19.6333	−	19.9201i	−0.0530629	−	0.0538382i
$371$			45.6185i			0.122961i
$372$		0			0
$373$			288.883i			0.774485i	0.921978	+	0.387243i	$0.126572\pi$
$373$			288.883i			0.774485i	−0.921978	+	0.387243i	$0.873428\pi$
$374$		−	519.727i		−	1.38964i
$375$		0			0
$376$	202.723			0.539158
$377$	44.2994			0.117505
$378$		0			0
$379$	427.427			1.12778			0.563888	−	0.825852i	$-0.309305\pi$
$379$	427.427			1.12778			0.563888	+	0.825852i	$0.309305\pi$
$380$	166.265	−	163.870i	0.437539	−	0.431237i
$381$		0			0
$382$		−	174.911i		−	0.457881i
$383$	178.280			0.465484			0.232742	−	0.972539i	$-0.425230\pi$
$383$	178.280			0.465484			0.232742	+	0.972539i	$0.425230\pi$
$384$		0			0
$385$	177.260	−	174.707i	0.460416	−	0.453785i
$386$		−	63.3409i		−	0.164096i
$387$		0			0
$388$			149.687i			0.385792i
$389$			368.910i			0.948356i	0.880429	+	0.474178i	$0.157255\pi$
$389$			368.910i			0.948356i	−0.880429	+	0.474178i	$0.842745\pi$
$390$		0			0
$391$	673.407			1.72227
$392$	109.995			0.280600
$393$		0			0
$394$	287.469			0.729617
$395$	322.564	−	317.919i	0.816619	−	0.804859i
$396$		0			0
$397$		−	520.661i		−	1.31149i	−0.754983	−	0.655745i	$-0.772355\pi$
$397$		−	520.661i		−	1.31149i	0.754983	−	0.655745i	$-0.227645\pi$
$398$	−109.737			−0.275721
$399$		0			0
$400$	1.45051	−	99.9895i	0.00362627	−	0.249974i
$401$			0.323245i			0.000806096i	1.00000		0.000403048i	$0.000128294\pi$
$401$			0.323245i			0.000806096i	−1.00000		0.000403048i	$0.999872\pi$
$402$		0			0
$403$			24.7782i			0.0614844i
$404$			33.9140i			0.0839455i
$405$		0			0
$406$	−149.113			−0.367273
$407$	61.9199			0.152137
$408$		0			0
$409$	9.52823			0.0232964			0.0116482	−	0.999932i	$-0.496292\pi$
$409$	9.52823			0.0232964			0.0116482	+	0.999932i	$0.496292\pi$
$410$	299.455	+	303.830i	0.730377	+	0.741050i
$411$		0			0
$412$			211.306i			0.512879i
$413$	230.138			0.557236
$414$		0			0
$415$	−366.408	+	361.131i	−0.882911	+	0.870196i
$416$			7.55732i			0.0181666i
$417$		0			0
$418$			516.818i			1.23641i
$419$		−	390.835i		−	0.932780i	−0.884579	−	0.466390i	$-0.845554\pi$
$419$		−	390.835i		−	0.932780i	0.884579	−	0.466390i	$-0.154446\pi$
$420$		0			0
$421$	427.682			1.01587			0.507935	−	0.861395i	$-0.330409\pi$
$421$	427.682			1.01587			0.507935	+	0.861395i	$0.330409\pi$
$422$	−187.143			−0.443467
$423$		0			0
$424$	−40.5781			−0.0957031
$425$	8.51304	−	586.840i	0.0200307	−	1.38080i
$426$		0			0
$427$		−	35.7356i		−	0.0836900i
$428$	−117.632			−0.274840
$429$		0			0
$430$	−14.0598	−	14.2652i	−0.0326971	−	0.0331749i
$431$		−	396.306i		−	0.919503i	−0.888048	−	0.459752i	$-0.847938\pi$
$431$		−	396.306i		−	0.919503i	0.888048	−	0.459752i	$-0.152062\pi$
$432$		0			0
$433$			700.722i			1.61830i	0.587606	+	0.809148i	$0.300071\pi$
$433$			700.722i			1.61830i	−0.587606	+	0.809148i	$0.699929\pi$
$434$		−	83.4040i		−	0.192175i
$435$		0			0
$436$	−368.730			−0.845710
$437$	−669.639			−1.53235
$438$		0			0
$439$	−95.1798			−0.216811			−0.108405	−	0.994107i	$-0.534574\pi$
$439$	−95.1798			−0.216811			−0.108405	+	0.994107i	$0.534574\pi$
$440$	155.404	+	157.675i	0.353191	+	0.358352i
$441$		0			0
$442$			44.3540i			0.100348i
$443$	237.685			0.536536			0.268268	−	0.963344i	$-0.413549\pi$
$443$	237.685			0.536536			0.268268	+	0.963344i	$0.413549\pi$
$444$		0			0
$445$	507.022	+	514.430i	1.13937	+	1.15602i
$446$			493.819i			1.10722i
$447$		0			0
$448$		−	25.4381i		−	0.0567814i
$449$			574.978i			1.28057i	0.768136	+	0.640287i	$0.221185\pi$
$449$			574.978i			1.28057i	−0.768136	+	0.640287i	$0.778815\pi$
$450$		0			0
$451$	−944.428			−2.09408
$452$	−34.3101			−0.0759073
$453$		0			0
$454$	−308.771			−0.680113
$455$	−15.1276	+	14.9097i	−0.0332474	+	0.0327686i
$456$		0			0
$457$		−	552.639i		−	1.20927i	−0.796501	−	0.604637i	$-0.793318\pi$
$457$		−	552.639i		−	1.20927i	0.796501	−	0.604637i	$-0.206682\pi$
$458$	538.046			1.17477
$459$		0			0
$460$	−204.298	+	201.356i	−0.444127	+	0.437731i
$461$		−	753.521i		−	1.63454i	−0.576258	−	0.817268i	$-0.695488\pi$
$461$		−	753.521i		−	1.63454i	0.576258	−	0.817268i	$-0.304512\pi$
$462$		0			0
$463$		−	393.957i		−	0.850878i	−0.904987	−	0.425439i	$-0.860120\pi$
$463$		−	393.957i		−	0.850878i	0.904987	−	0.425439i	$-0.139880\pi$
$464$		−	132.637i		−	0.285856i
$465$		0			0
$466$	−329.499			−0.707080
$467$	−144.091			−0.308546			−0.154273	−	0.988028i	$-0.549304\pi$
$467$	−144.091			−0.308546			−0.154273	+	0.988028i	$0.549304\pi$
$468$		0			0
$469$	−10.6663			−0.0227426
$470$	−360.957	+	355.759i	−0.767995	+	0.756934i
$471$		0			0
$472$			204.710i			0.433708i
$473$	44.3421			0.0937464
$474$		0			0
$475$	−8.46540	+	583.556i	−0.0178219	+	1.22854i
$476$		−	149.296i		−	0.313648i
$477$		0			0
$478$			222.728i			0.465959i
$479$		−	160.218i		−	0.334484i	−0.985916	−	0.167242i	$-0.946514\pi$
$479$		−	160.218i		−	0.334484i	0.985916	−	0.167242i	$-0.0534861\pi$
$480$		0			0
$481$	−5.28431			−0.0109861
$482$	−162.848			−0.337860
$483$		0			0
$484$	−248.117			−0.512638
$485$	−262.686	−	266.524i	−0.541621	−	0.549535i
$486$		0			0
$487$		−	184.489i		−	0.378827i	−0.981897	−	0.189414i	$-0.939341\pi$
$487$		−	184.489i		−	0.378827i	0.981897	−	0.189414i	$-0.0606587\pi$
$488$	31.7872			0.0651377
$489$		0			0
$490$	−195.851	+	193.030i	−0.399696	+	0.393939i
$491$			83.3423i			0.169740i	0.996392	+	0.0848700i	$0.0270475\pi$
$491$			83.3423i			0.169740i	−0.996392	+	0.0848700i	$0.972953\pi$
$492$		0			0
$493$		−	778.449i		−	1.57900i
$494$		−	44.1058i		−	0.0892830i
$495$		0			0
$496$	74.1886			0.149574
$497$	−127.366			−0.256271
$498$		0			0
$499$	−402.812			−0.807238			−0.403619	−	0.914927i	$-0.632248\pi$
$499$	−402.812			−0.807238			−0.403619	+	0.914927i	$0.632248\pi$
$500$	172.889	+	180.581i	0.345778	+	0.361162i
$501$		0			0
$502$			222.827i			0.443879i
$503$	743.290			1.47771			0.738857	−	0.673862i	$-0.235366\pi$
$503$	743.290			1.47771			0.738857	+	0.673862i	$0.235366\pi$
$504$		0			0
$505$	−59.5156	−	60.3852i	−0.117853	−	0.119575i
$506$		−	635.042i		−	1.25502i
$507$		0			0
$508$		−	35.7113i		−	0.0702979i
$509$			97.6240i			0.191796i	0.995391	+	0.0958978i	$0.0305722\pi$
$509$			97.6240i			0.191796i	−0.995391	+	0.0958978i	$0.969428\pi$
$510$		0			0
$511$	−275.551			−0.539239
$512$	22.6274			0.0441942
$513$		0			0
$514$	−28.5593			−0.0555628
$515$	−370.821	−	376.239i	−0.720040	−	0.730562i
$516$		0			0
$517$		−	1122.00i		−	2.17022i
$518$	17.7871			0.0343380
$519$		0			0
$520$	−13.2623	−	13.4561i	−0.0255045	−	0.0258772i
$521$			630.912i			1.21096i	0.795859	+	0.605482i	$0.207019\pi$
$521$			630.912i			1.21096i	−0.795859	+	0.605482i	$0.792981\pi$
$522$		0			0
$523$			441.750i			0.844647i	0.906445	+	0.422323i	$0.138785\pi$
$523$			441.750i			0.844647i	−0.906445	+	0.422323i	$0.861215\pi$
$524$		−	425.646i		−	0.812301i
$525$		0			0
$526$	143.797			0.273377
$527$	435.414			0.826213
$528$		0			0
$529$	293.821			0.555427
$530$	72.2510	−	71.2105i	0.136323	−	0.134359i
$531$		0			0
$532$			148.461i			0.279062i
$533$	80.5985			0.151217
$534$		0			0
$535$	209.448	−	206.432i	0.391492	−	0.385854i
$536$		−	9.48775i		−	0.0177010i
$537$		0			0
$538$		−	118.792i		−	0.220802i
$539$		−	608.784i		−	1.12947i
$540$		0			0
$541$	365.131			0.674919			0.337459	−	0.941340i	$-0.390432\pi$
$541$	365.131			0.674919			0.337459	+	0.941340i	$0.390432\pi$
$542$	494.396			0.912169
$543$		0			0
$544$	132.801			0.244119
$545$	656.538	−	647.083i	1.20466	−	1.18731i
$546$		0			0
$547$			364.069i			0.665574i	0.943002	+	0.332787i	$0.107989\pi$
$547$			364.069i			0.665574i	−0.943002	+	0.332787i	$0.892011\pi$
$548$	−395.095			−0.720977
$549$		0			0
$550$	−553.406	−	8.02804i	−1.00619	−	0.0145964i
$551$			774.093i			1.40489i
$552$		0			0
$553$			288.024i			0.520839i
$554$		−	461.601i		−	0.833215i
$555$		0			0
$556$	−103.148			−0.185519
$557$	901.292			1.61812			0.809059	−	0.587728i	$-0.199977\pi$
$557$	901.292			1.61812			0.809059	+	0.587728i	$0.199977\pi$
$558$		0			0
$559$	−3.78420			−0.00676958
$560$	44.6413	+	45.2935i	0.0797165	+	0.0808813i
$561$		0			0
$562$		−	153.786i		−	0.273640i
$563$	−81.6432			−0.145015			−0.0725073	−	0.997368i	$-0.523100\pi$
$563$	−81.6432			−0.145015			−0.0725073	+	0.997368i	$0.523100\pi$
$564$		0			0
$565$	61.0906	−	60.2108i	0.108125	−	0.106568i
$566$			538.394i			0.951225i
$567$		0			0
$568$		−	113.294i		−	0.199461i
$569$			99.5184i			0.174900i	0.996169	+	0.0874502i	$0.0278719\pi$
$569$			99.5184i			0.174900i	−0.996169	+	0.0874502i	$0.972128\pi$
$570$		0			0
$571$	−402.277			−0.704514			−0.352257	−	0.935903i	$-0.614586\pi$
$571$	−402.277			−0.704514			−0.352257	+	0.935903i	$0.614586\pi$
$572$	41.8271			0.0731243
$573$		0			0
$574$	−271.296			−0.472641
$575$	10.4019	−	717.046i	0.0180903	−	1.24704i
$576$		0			0
$577$		−	278.698i		−	0.483012i	−0.970399	−	0.241506i	$-0.922359\pi$
$577$		−	278.698i		−	0.483012i	0.970399	−	0.241506i	$-0.0776414\pi$
$578$	370.701			0.641351
$579$		0			0
$580$	232.765	+	236.166i	0.401319	+	0.407183i
$581$		−	327.173i		−	0.563120i
$582$		0			0
$583$			224.585i			0.385224i
$584$		−	245.106i		−	0.419701i
$585$		0			0
$586$	−143.283			−0.244511
$587$	325.927			0.555243			0.277621	−	0.960691i	$-0.410454\pi$
$587$	325.927			0.555243			0.277621	+	0.960691i	$0.410454\pi$
$588$		0			0
$589$	−432.978			−0.735106
$590$	−359.246	−	364.495i	−0.608891	−	0.617788i
$591$		0			0
$592$			15.8218i			0.0267260i
$593$	−113.167			−0.190838			−0.0954190	−	0.995437i	$-0.530419\pi$
$593$	−113.167			−0.190838			−0.0954190	+	0.995437i	$0.530419\pi$
$594$		0			0
$595$	262.000	+	265.828i	0.440336	+	0.446770i
$596$			451.966i			0.758332i
$597$		0			0
$598$			54.1952i			0.0906273i
$599$		−	283.109i		−	0.472636i	−0.971676	−	0.236318i	$-0.924059\pi$
$599$		−	283.109i		−	0.472636i	0.971676	−	0.236318i	$-0.0759407\pi$
$600$		0			0
$601$	523.593			0.871203			0.435601	−	0.900140i	$-0.356536\pi$
$601$	523.593			0.871203			0.435601	+	0.900140i	$0.356536\pi$
$602$	12.7377			0.0211589
$603$		0			0
$604$	−107.303			−0.177654
$605$	441.782	−	435.420i	0.730219	−	0.719703i
$606$		0			0
$607$			1018.05i			1.67718i	0.544761	+	0.838591i	$0.316620\pi$
$607$			1018.05i			1.67718i	−0.544761	+	0.838591i	$0.683380\pi$
$608$	−132.057			−0.217200
$609$		0			0
$610$	−56.5984	+	55.7833i	−0.0927843	+	0.0914481i
$611$			95.7528i			0.156715i
$612$		0			0
$613$		−	537.130i		−	0.876231i	−0.898919	−	0.438116i	$-0.855646\pi$
$613$		−	537.130i		−	0.876231i	0.898919	−	0.438116i	$-0.144354\pi$
$614$		−	163.435i		−	0.266180i
$615$		0			0
$616$	−140.791			−0.228556
$617$	1136.98			1.84275			0.921375	−	0.388676i	$-0.127067\pi$
$617$	1136.98			1.84275			0.921375	+	0.388676i	$0.127067\pi$
$618$		0			0
$619$	−234.742			−0.379228			−0.189614	−	0.981859i	$-0.560724\pi$
$619$	−234.742			−0.379228			−0.189614	+	0.981859i	$0.560724\pi$
$620$	−132.096	+	130.194i	−0.213058	+	0.209990i
$621$		0			0
$622$			478.959i			0.770031i
$623$	−459.345			−0.737311
$624$		0			0
$625$	−624.737	−	18.1294i	−0.999579	−	0.0290071i
$626$			377.832i			0.603565i
$627$		0			0
$628$		−	306.763i		−	0.488477i
$629$			92.8582i			0.147628i
$630$		0			0
$631$	131.866			0.208980			0.104490	−	0.994526i	$-0.466679\pi$
$631$	131.866			0.208980			0.104490	+	0.994526i	$0.466679\pi$
$632$	−256.200			−0.405380
$633$		0			0
$634$	415.349			0.655124
$635$	62.6698	+	63.5855i	0.0986925	+	0.100135i
$636$		0			0
$637$			51.9542i			0.0815608i
$638$	−734.100			−1.15063
$639$		0			0
$640$	−40.2891	+	39.7088i	−0.0629517	+	0.0620451i
$641$			536.826i			0.837483i	0.908106	+	0.418741i	$0.137529\pi$
$641$			536.826i			0.837483i	−0.908106	+	0.418741i	$0.862471\pi$
$642$		0			0
$643$		−	585.189i		−	0.910092i	−0.890468	−	0.455046i	$-0.849623\pi$
$643$		−	585.189i		−	0.910092i	0.890468	−	0.455046i	$-0.150377\pi$
$644$		−	182.422i		−	0.283264i
$645$		0			0
$646$	−775.047			−1.19976
$647$	−375.007			−0.579609			−0.289805	−	0.957086i	$-0.593590\pi$
$647$	−375.007			−0.579609			−0.289805	+	0.957086i	$0.593590\pi$
$648$		0			0
$649$	1133.00			1.74576
$650$	47.2283	+	0.685121i	0.0726589	+	0.00105403i
$651$		0			0
$652$			544.128i			0.834552i
$653$	453.112			0.693892			0.346946	−	0.937885i	$-0.387219\pi$
$653$	453.112			0.693892			0.346946	+	0.937885i	$0.387219\pi$
$654$		0			0
$655$	746.965	+	757.880i	1.14040	+	1.15707i
$656$		−	241.320i		−	0.367866i
$657$		0			0
$658$		−	322.306i		−	0.489826i
$659$			347.194i			0.526850i	0.964680	+	0.263425i	$0.0848521\pi$
$659$			347.194i			0.526850i	−0.964680	+	0.263425i	$0.915148\pi$
$660$		0			0
$661$	946.578			1.43204			0.716020	−	0.698080i	$-0.245962\pi$
$661$	946.578			1.43204			0.716020	+	0.698080i	$0.245962\pi$
$662$	−66.7544			−0.100837
$663$		0			0
$664$	291.023			0.438288
$665$	−260.534	−	264.341i	−0.391780	−	0.397505i
$666$		0			0
$667$		−	951.170i		−	1.42604i
$668$	320.347			0.479561
$669$		0			0
$670$	16.6500	+	16.8933i	0.0248508	+	0.0252139i
$671$		−	175.931i		−	0.262192i
$672$		0			0
$673$		−	208.778i		−	0.310220i	−0.987897	−	0.155110i	$-0.950427\pi$
$673$		−	208.778i		−	0.310220i	0.987897	−	0.155110i	$-0.0495732\pi$
$674$		−	662.210i		−	0.982507i
$675$		0			0
$676$	334.430			0.494720
$677$	416.186			0.614751			0.307375	−	0.951588i	$-0.400549\pi$
$677$	416.186			0.614751			0.307375	+	0.951588i	$0.400549\pi$
$678$		0			0
$679$	237.985			0.350493
$680$	−236.457	+	233.052i	−0.347731	+	0.342723i
$681$		0			0
$682$		−	410.608i		−	0.602064i
$683$	883.982			1.29426			0.647132	−	0.762378i	$-0.275968\pi$
$683$	883.982			1.29426			0.647132	+	0.762378i	$0.275968\pi$
$684$		0			0
$685$	703.484	−	693.352i	1.02698	−	1.01219i
$686$		−	395.225i		−	0.576130i
$687$		0			0
$688$			11.3303i			0.0164684i
$689$		−	19.1664i		−	0.0278176i
$690$		0			0
$691$	370.080			0.535571			0.267786	−	0.963479i	$-0.413708\pi$
$691$	370.080			0.535571			0.267786	+	0.963479i	$0.413708\pi$
$692$	60.8358			0.0879130
$693$		0			0
$694$	−80.6178			−0.116164
$695$	183.660	−	181.015i	0.264259	−	0.260453i
$696$		0			0
$697$		−	1416.31i		−	2.03201i
$698$	−253.329			−0.362935
$699$		0			0
$700$	−158.971	−	2.30613i	−0.227102	−	0.00329447i
$701$		−	820.094i		−	1.16989i	−0.811073	−	0.584946i	$-0.801116\pi$
$701$		−	820.094i		−	1.16989i	0.811073	−	0.584946i	$-0.198884\pi$
$702$		0			0
$703$		−	92.3386i		−	0.131349i
$704$		−	125.235i		−	0.177890i
$705$		0			0
$706$	−32.2682			−0.0457056
$707$	53.9191			0.0762647
$708$		0			0
$709$	−280.410			−0.395500			−0.197750	−	0.980252i	$-0.563363\pi$
$709$	−280.410			−0.395500			−0.197750	+	0.980252i	$0.563363\pi$
$710$	198.819	+	201.724i	0.280027	+	0.284119i
$711$		0			0
$712$		−	408.592i		−	0.573865i
$713$	532.023			0.746175
$714$		0			0
$715$	−74.4749	+	73.4023i	−0.104161	+	0.102661i
$716$			523.418i			0.731031i
$717$		0			0
$718$			846.462i			1.17892i
$719$		−	144.017i		−	0.200302i	−0.994972	−	0.100151i	$-0.968067\pi$
$719$		−	144.017i		−	0.200302i	0.994972	−	0.100151i	$-0.0319327\pi$
$720$		0			0
$721$	335.951			0.465952
$722$	260.179			0.360358
$723$		0			0
$724$	−316.318			−0.436903
$725$	−828.895	−	12.0244i	−1.14330	−	0.0165854i
$726$		0			0
$727$		−	567.714i		−	0.780900i	−0.920624	−	0.390450i	$-0.872319\pi$
$727$		−	567.714i		−	0.780900i	0.920624	−	0.390450i	$-0.127681\pi$
$728$	12.0152			0.0165044
$729$		0			0
$730$	430.136	+	436.421i	0.589227	+	0.597837i
$731$			66.4976i			0.0909680i
$732$		0			0
$733$		−	135.891i		−	0.185390i	−0.995695	−	0.0926949i	$-0.970452\pi$
$733$		−	135.891i		−	0.185390i	0.995695	−	0.0926949i	$-0.0295481\pi$
$734$		−	26.4457i		−	0.0360296i
$735$		0			0
$736$	162.266			0.220470
$737$	−52.5113			−0.0712501
$738$		0			0
$739$	−799.789			−1.08226			−0.541129	−	0.840939i	$-0.682003\pi$
$739$	−799.789			−1.08226			−0.541129	+	0.840939i	$0.682003\pi$
$740$	−27.7656	−	28.1713i	−0.0375211	−	0.0380694i
$741$		0			0
$742$			64.5143i			0.0869465i
$743$	−642.938			−0.865327			−0.432663	−	0.901556i	$-0.642426\pi$
$743$	−642.938			−0.865327			−0.432663	+	0.901556i	$0.642426\pi$
$744$		0			0
$745$	−793.155	−	804.744i	−1.06464	−	1.08019i
$746$			408.542i			0.547644i
$747$		0			0
$748$		−	735.004i		−	0.982626i
$749$			187.020i			0.249693i
$750$		0			0
$751$	−804.131			−1.07075			−0.535374	−	0.844615i	$-0.679829\pi$
$751$	−804.131			−1.07075			−0.535374	+	0.844615i	$0.679829\pi$
$752$	286.694			0.381242
$753$		0			0
$754$	62.6488			0.0830887
$755$	191.058	−	188.306i	0.253057	−	0.249412i
$756$		0			0
$757$			998.545i			1.31908i	0.751668	+	0.659541i	$0.229249\pi$
$757$			998.545i			1.31908i	−0.751668	+	0.659541i	$0.770751\pi$
$758$	604.473			0.797458
$759$		0			0
$760$	235.134	−	231.748i	0.309387	−	0.304931i
$761$		−	436.073i		−	0.573027i	−0.958076	−	0.286513i	$-0.907504\pi$
$761$		−	436.073i		−	0.573027i	0.958076	−	0.286513i	$-0.0924963\pi$
$762$		0			0
$763$			586.236i			0.768330i
$764$		−	247.361i		−	0.323771i
$765$		0			0
$766$	252.126			0.329147
$767$	−96.6913			−0.126064
$768$		0			0
$769$	1432.72			1.86310			0.931551	−	0.363612i	$-0.118456\pi$
$769$	1432.72			1.86310			0.931551	+	0.363612i	$0.118456\pi$
$770$	250.684	−	247.074i	0.325563	−	0.320875i
$771$		0			0
$772$		−	89.5776i		−	0.116033i
$773$	−1341.08			−1.73490			−0.867449	−	0.497526i	$-0.834242\pi$
$773$	−1341.08			−1.73490			−0.867449	+	0.497526i	$0.834242\pi$
$774$		0			0
$775$	6.72570	−	463.630i	0.00867832	−	0.598233i
$776$			211.690i			0.272796i
$777$		0			0
$778$			521.718i			0.670589i
$779$			1408.39i			1.80794i
$780$		0			0
$781$	−627.041			−0.802869
$782$	952.342			1.21783
$783$		0			0
$784$	155.557			0.198414
$785$	538.339	+	546.205i	0.685782	+	0.695803i
$786$		0			0
$787$			1048.34i			1.33207i	0.745921	+	0.666034i	$0.232009\pi$
$787$			1048.34i			1.33207i	−0.745921	+	0.666034i	$0.767991\pi$
$788$	406.543			0.515917
$789$		0			0
$790$	456.175	−	449.606i	0.577437	−	0.569121i
$791$			54.5489i			0.0689620i
$792$		0			0
$793$			15.0141i			0.0189333i
$794$		−	736.326i		−	0.927363i
$795$		0			0
$796$	−155.192			−0.194964
$797$	−1327.16			−1.66519			−0.832594	−	0.553883i	$-0.813145\pi$
$797$	−1327.16			−1.66519			−0.832594	+	0.553883i	$0.813145\pi$
$798$		0			0
$799$	1682.61			2.10590
$800$	2.05133	−	141.406i	0.00256416	−	0.176758i
$801$		0			0
$802$			0.457137i			0.000569996i
$803$	−1356.57			−1.68938
$804$		0			0
$805$	320.132	+	324.810i	0.397680	+	0.403490i
$806$			35.0417i			0.0434761i
$807$		0			0
$808$			47.9616i			0.0593584i
$809$			976.178i			1.20665i	0.797496	+	0.603324i	$0.206157\pi$
$809$			976.178i			1.20665i	−0.797496	+	0.603324i	$0.793843\pi$
$810$		0			0
$811$	−289.065			−0.356431			−0.178215	−	0.983991i	$-0.557032\pi$
$811$	−289.065			−0.356431			−0.178215	+	0.983991i	$0.557032\pi$
$812$	−210.877			−0.259701
$813$		0			0
$814$	87.5679			0.107577
$815$	−954.890	−	968.842i	−1.17164	−	1.18876i
$816$		0			0
$817$		−	66.1255i		−	0.0809369i
$818$	13.4750			0.0164731
$819$		0			0
$820$	423.493	+	429.681i	0.516455	+	0.524001i
$821$		−	238.422i		−	0.290405i	−0.989402	−	0.145202i	$-0.953617\pi$
$821$		−	238.422i		−	0.290405i	0.989402	−	0.145202i	$-0.0463833\pi$
$822$		0			0
$823$		−	421.581i		−	0.512249i	−0.966644	−	0.256125i	$-0.917554\pi$
$823$		−	421.581i		−	0.512249i	0.966644	−	0.256125i	$-0.0824458\pi$
$824$			298.832i			0.362660i
$825$		0			0
$826$	325.465			0.394025
$827$	−944.973			−1.14265			−0.571326	−	0.820723i	$-0.693571\pi$
$827$	−944.973			−1.14265			−0.571326	+	0.820723i	$0.693571\pi$
$828$		0			0
$829$	−609.872			−0.735672			−0.367836	−	0.929891i	$-0.619901\pi$
$829$	−609.872			−0.735672			−0.367836	+	0.929891i	$0.619901\pi$
$830$	−518.179	+	510.717i	−0.624312	+	0.615321i
$831$		0			0
$832$			10.6877i			0.0128457i
$833$	912.963			1.09599
$834$		0			0
$835$	−570.391	+	562.176i	−0.683103	+	0.673265i
$836$			730.891i			0.874272i
$837$		0			0
$838$		−	552.724i		−	0.659575i
$839$			66.6016i			0.0793821i	0.999212	+	0.0396911i	$0.0126374\pi$
$839$			66.6016i			0.0793821i	−0.999212	+	0.0396911i	$0.987363\pi$
$840$		0			0
$841$	−258.539			−0.307418
$842$	604.833			0.718329
$843$		0			0
$844$	−264.660			−0.313578
$845$	−595.467	+	586.892i	−0.704695	+	0.694546i
$846$		0			0
$847$			394.476i			0.465733i
$848$	−57.3861			−0.0676723
$849$		0			0
$850$	12.0393	−	829.916i	0.0141638	−	0.976372i
$851$			113.461i			0.133327i
$852$		0			0
$853$		−	361.920i		−	0.424291i	−0.977238	−	0.212145i	$-0.931955\pi$
$853$		−	361.920i		−	0.424291i	0.977238	−	0.212145i	$-0.0680451\pi$
$854$		−	50.5378i		−	0.0591778i
$855$		0			0
$856$	−166.356			−0.194342
$857$	−1035.43			−1.20821			−0.604103	−	0.796906i	$-0.706468\pi$
$857$	−1035.43			−1.20821			−0.604103	+	0.796906i	$0.706468\pi$
$858$		0			0
$859$	251.841			0.293180			0.146590	−	0.989197i	$-0.453170\pi$
$859$	251.841			0.293180			0.146590	+	0.989197i	$0.453170\pi$
$860$	−19.8835	−	20.1740i	−0.0231204	−	0.0234582i
$861$		0			0
$862$		−	560.461i		−	0.650187i
$863$	872.334			1.01082			0.505408	−	0.862881i	$-0.331342\pi$
$863$	872.334			1.01082			0.505408	+	0.862881i	$0.331342\pi$
$864$		0			0
$865$	−108.321	+	106.761i	−0.125226	+	0.123423i
$866$			990.970i			1.14431i
$867$		0			0
$868$		−	117.951i		−	0.135888i
$869$			1417.98i			1.63173i
$870$		0			0
$871$	4.48137			0.00514509
$872$	−521.462			−0.598007
$873$		0			0
$874$	−947.012			−1.08354
$875$	287.102	−	274.872i	0.328116	−	0.314140i
$876$		0			0
$877$			1468.28i			1.67421i	0.547044	+	0.837104i	$0.315753\pi$
$877$			1468.28i			1.67421i	−0.547044	+	0.837104i	$0.684247\pi$
$878$	−134.605			−0.153308
$879$		0			0
$880$	219.774	+	222.986i	0.249744	+	0.253393i
$881$			727.513i			0.825781i	0.910781	+	0.412891i	$0.135481\pi$
$881$			727.513i			0.825781i	−0.910781	+	0.412891i	$0.864519\pi$
$882$		0			0
$883$			384.077i			0.434968i	0.976064	+	0.217484i	$0.0697850\pi$
$883$			384.077i			0.434968i	−0.976064	+	0.217484i	$0.930215\pi$
$884$			62.7260i			0.0709571i
$885$		0			0
$886$	336.138			0.379388
$887$	464.833			0.524051			0.262026	−	0.965061i	$-0.415609\pi$
$887$	464.833			0.524051			0.262026	+	0.965061i	$0.415609\pi$
$888$		0			0
$889$	−56.7767			−0.0638658
$890$	717.037	+	727.514i	0.805660	+	0.817432i
$891$		0			0
$892$			698.365i			0.782921i
$893$	−1673.20			−1.87368
$894$		0			0
$895$	−918.546	−	931.968i	−1.02631	−	1.04131i
$896$		−	35.9749i		−	0.0401505i
$897$		0			0
$898$			813.141i			0.905503i
$899$		−	615.011i		−	0.684105i
$900$		0			0
$901$	−336.800			−0.373807
$902$	−1335.62			−1.48073
$903$		0			0
$904$	−48.5218			−0.0536746
$905$	563.217	−	555.106i	0.622339	−	0.613376i
$906$		0			0
$907$			246.427i			0.271695i	0.990730	+	0.135847i	$0.0433757\pi$
$907$			246.427i			0.271695i	−0.990730	+	0.135847i	$0.956624\pi$
$908$	−436.668			−0.480912
$909$		0			0
$910$	−21.3936	+	21.0855i	−0.0235095	+	0.0231709i
$911$		−	144.890i		−	0.159045i	−0.996833	−	0.0795224i	$-0.974660\pi$
$911$		−	144.890i		−	0.159045i	0.996833	−	0.0795224i	$-0.0253395\pi$
$912$		0			0
$913$		−	1610.71i		−	1.76420i
$914$		−	781.549i		−	0.855086i
$915$		0			0
$916$	760.912			0.830690
$917$	−676.725			−0.737977
$918$		0			0
$919$	−1421.82			−1.54714			−0.773569	−	0.633712i	$-0.781530\pi$
$919$	−1421.82			−1.54714			−0.773569	+	0.633712i	$0.781530\pi$
$920$	−288.921	+	284.761i	−0.314045	+	0.309522i
$921$		0			0
$922$		−	1065.64i		−	1.15579i
$923$	53.5123			0.0579765
$924$		0			0
$925$	98.8757	+	1.43435i	0.106893	+	0.00155065i
$926$		−	557.139i		−	0.601662i
$927$		0			0
$928$		−	187.577i		−	0.202131i
$929$		−	547.295i		−	0.589123i	−0.955632	−	0.294562i	$-0.904826\pi$
$929$		−	547.295i		−	0.589123i	0.955632	−	0.294562i	$-0.0951736\pi$
$930$		0			0
$931$	−907.854			−0.975139
$932$	−465.983			−0.499981
$933$		0			0
$934$	−203.776			−0.218175
$935$	1289.86	+	1308.71i	1.37953	+	1.39969i
$936$		0			0
$937$			1384.47i			1.47756i	0.673948	+	0.738779i	$0.264597\pi$
$937$			1384.47i			1.47756i	−0.673948	+	0.738779i	$0.735403\pi$
$938$	−15.0844			−0.0160814
$939$		0			0
$940$	−510.471	+	503.119i	−0.543054	+	0.535233i
$941$			701.042i			0.744996i	0.928033	+	0.372498i	$0.121499\pi$
$941$			701.042i			0.744996i	−0.928033	+	0.372498i	$0.878501\pi$
$942$		0			0
$943$		−	1730.56i		−	1.83516i
$944$			289.504i			0.306678i
$945$		0			0
$946$	62.7091			0.0662887
$947$	−655.997			−0.692711			−0.346355	−	0.938103i	$-0.612581\pi$
$947$	−655.997			−0.692711			−0.346355	+	0.938103i	$0.612581\pi$
$948$		0			0
$949$	115.771			0.121993
$950$	−11.9719	+	825.272i	−0.0126020	+	0.868708i
$951$		0			0
$952$		−	211.137i		−	0.221783i
$953$	−290.181			−0.304492			−0.152246	−	0.988343i	$-0.548651\pi$
$953$	−290.181			−0.304492			−0.152246	+	0.988343i	$0.548651\pi$
$954$		0			0
$955$	434.094	+	440.437i	0.454548	+	0.461190i
$956$			314.985i			0.329482i
$957$		0			0
$958$		−	226.582i		−	0.236516i
$959$			628.154i			0.655009i
$960$		0			0
$961$	−617.003			−0.642042
$962$	−7.47314			−0.00776834
$963$		0			0
$964$	−230.302			−0.238903
$965$	157.200	+	159.497i	0.162901	+	0.165281i
$966$		0			0
$967$			1037.15i			1.07254i	0.844047	+	0.536270i	$0.180167\pi$
$967$			1037.15i			1.07254i	−0.844047	+	0.536270i	$0.819833\pi$
$968$	−350.890			−0.362490
$969$		0			0
$970$	−371.494	−	376.923i	−0.382984	−	0.388580i
$971$			1762.50i			1.81514i	0.419901	+	0.907570i	$0.362065\pi$
$971$			1762.50i			1.81514i	−0.419901	+	0.907570i	$0.637935\pi$
$972$		0			0
$973$			163.993i			0.168544i
$974$		−	260.907i		−	0.267871i
$975$		0			0
$976$	44.9539			0.0460593
$977$	−1771.34			−1.81304			−0.906519	−	0.422165i	$-0.861270\pi$
$977$	−1771.34			−1.81304			−0.906519	+	0.422165i	$0.861270\pi$
$978$		0			0
$979$	−2261.41			−2.30992
$980$	−276.975	+	272.986i	−0.282627	+	0.278557i
$981$		0			0
$982$			117.864i			0.120024i
$983$	1525.11			1.55149			0.775745	−	0.631047i	$-0.217374\pi$
$983$	1525.11			1.55149			0.775745	+	0.631047i	$0.217374\pi$
$984$		0			0
$985$	−723.867	+	713.442i	−0.734890	+	0.724306i
$986$		−	1100.89i		−	1.11652i
$987$		0			0
$988$		−	62.3750i		−	0.0631326i
$989$			81.2519i			0.0821556i
$990$		0			0
$991$	−296.858			−0.299554			−0.149777	−	0.988720i	$-0.547856\pi$
$991$	−296.858			−0.299554			−0.149777	+	0.988720i	$0.547856\pi$
$992$	104.919			0.105765
$993$		0			0
$994$	−180.123			−0.181211
$995$	276.325	−	272.346i	0.277714	−	0.273714i
$996$		0			0
$997$			1318.45i			1.32242i	0.750203	+	0.661208i	$0.229956\pi$
$997$			1318.45i			1.32242i	−0.750203	+	0.661208i	$0.770044\pi$
$998$	−569.662			−0.570803
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	810.3.b.b.809.12		16
3.2	odd	2	inner	810.3.b.b.809.5		16
5.4	even	2	inner	810.3.b.b.809.6		16
9.2	odd	6		90.3.j.b.59.8	yes	16
9.4	even	3		90.3.j.b.29.1	✓	16
9.5	odd	6		270.3.j.b.89.5		16
9.7	even	3		270.3.j.b.179.3		16
15.14	odd	2	inner	810.3.b.b.809.11		16
45.2	even	12		450.3.i.e.401.2		16
45.4	even	6		90.3.j.b.29.8	yes	16
45.7	odd	12		1350.3.i.e.1151.6		16
45.13	odd	12		450.3.i.e.101.7		16
45.14	odd	6		270.3.j.b.89.3		16
45.22	odd	12		450.3.i.e.101.2		16
45.23	even	12		1350.3.i.e.251.3		16
45.29	odd	6		90.3.j.b.59.1	yes	16
45.32	even	12		1350.3.i.e.251.6		16
45.34	even	6		270.3.j.b.179.5		16
45.38	even	12		450.3.i.e.401.7		16
45.43	odd	12		1350.3.i.e.1151.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.3.j.b.29.1	✓	16	9.4	even	3
90.3.j.b.29.8	yes	16	45.4	even	6
90.3.j.b.59.1	yes	16	45.29	odd	6
90.3.j.b.59.8	yes	16	9.2	odd	6
270.3.j.b.89.3		16	45.14	odd	6
270.3.j.b.89.5		16	9.5	odd	6
270.3.j.b.179.3		16	9.7	even	3
270.3.j.b.179.5		16	45.34	even	6
450.3.i.e.101.2		16	45.22	odd	12
450.3.i.e.101.7		16	45.13	odd	12
450.3.i.e.401.2		16	45.2	even	12
450.3.i.e.401.7		16	45.38	even	12
810.3.b.b.809.5		16	3.2	odd	2	inner
810.3.b.b.809.6		16	5.4	even	2	inner
810.3.b.b.809.11		16	15.14	odd	2	inner
810.3.b.b.809.12		16	1.1	even	1	trivial
1350.3.i.e.251.3		16	45.23	even	12
1350.3.i.e.251.6		16	45.32	even	12
1350.3.i.e.1151.3		16	45.43	odd	12
1350.3.i.e.1151.6		16	45.7	odd	12

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:	\( N \)	\(=\)	\( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	810.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +2.00000 q^{4} +(-3.56108 + 3.50980i) q^{5} -3.17976i q^{7} +2.82843 q^{8} +O(q^{10})\)	\(q+1.41421 q^{2} +2.00000 q^{4} +(-3.56108 + 3.50980i) q^{5} -3.17976i q^{7} +2.82843 q^{8} +(-5.03613 + 4.96360i) q^{10} -15.6543i q^{11} +1.33596i q^{13} -4.49686i q^{14} +4.00000 q^{16} +23.4761 q^{17} -23.3447 q^{19} +(-7.12217 + 7.01960i) q^{20} -22.1386i q^{22} +28.6849 q^{23} +(0.362627 - 24.9974i) q^{25} +1.88933i q^{26} -6.35952i q^{28} -33.1593i q^{29} +18.5472 q^{31} +5.65685 q^{32} +33.2002 q^{34} +(11.1603 + 11.3234i) q^{35} +3.95544i q^{37} -33.0144 q^{38} +(-10.0723 + 9.92721i) q^{40} -60.3301i q^{41} +2.83257i q^{43} -31.3087i q^{44} +40.5665 q^{46} +71.6735 q^{47} +38.8891 q^{49} +(0.512832 - 35.3516i) q^{50} +2.67192i q^{52} -14.3465 q^{53} +(54.9436 + 55.7464i) q^{55} -8.99372i q^{56} -46.8943i q^{58} +72.3760i q^{59} +11.2385 q^{61} +26.2296 q^{62} +8.00000 q^{64} +(-4.68894 - 4.75746i) q^{65} -3.35443i q^{67} +46.9521 q^{68} +(15.7831 + 16.0137i) q^{70} -40.0554i q^{71} -86.6579i q^{73} +5.59384i q^{74} -46.6894 q^{76} -49.7770 q^{77} -90.5804 q^{79} +(-14.2443 + 14.0392i) q^{80} -85.3196i q^{82} +102.892 q^{83} +(-83.6002 + 82.3962i) q^{85} +4.00586i q^{86} -44.2772i q^{88} -144.459i q^{89} +4.24802 q^{91} +57.3697 q^{92} +101.362 q^{94} +(83.1323 - 81.9351i) q^{95} +74.8437i q^{97} +54.9975 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 32 q^{4}+O(q^{10}) \) 16 * q + 32 * q^4	\( 16 q + 32 q^{4} - 16 q^{10} + 64 q^{16} + 144 q^{19} - 12 q^{25} - 56 q^{31} + 272 q^{34} - 32 q^{40} - 56 q^{46} - 24 q^{49} + 20 q^{55} - 136 q^{61} + 128 q^{64} + 224 q^{70} + 288 q^{76} - 840 q^{79} + 272 q^{85} - 168 q^{91} + 328 q^{94}+O(q^{100}) \) 16 * q + 32 * q^4 - 16 * q^10 + 64 * q^16 + 144 * q^19 - 12 * q^25 - 56 * q^31 + 272 * q^34 - 32 * q^40 - 56 * q^46 - 24 * q^49 + 20 * q^55 - 136 * q^61 + 128 * q^64 + 224 * q^70 + 288 * q^76 - 840 * q^79 + 272 * q^85 - 168 * q^91 + 328 * q^94

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