LMFDB - Embedded newform 810.4.e.a.541.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [810,4,Mod(271,810)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(810, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("810.271");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 810 = 2 \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	810.e (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:	$47.7915471046$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 90)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			541.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	810.541
Dual form			810.4.e.a.271.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 + 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(-2.50000 - 4.33013i) q^{5} +(-7.00000 + 12.1244i) q^{7} +8.00000 q^{8} +O(q^{10})$	\(q+(-1.00000 + 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(-2.50000 - 4.33013i) q^{5} +(-7.00000 + 12.1244i) q^{7} +8.00000 q^{8} +10.0000 q^{10} +(3.00000 - 5.19615i) q^{11} +(-34.0000 - 58.8897i) q^{13} +(-14.0000 - 24.2487i) q^{14} +(-8.00000 + 13.8564i) q^{16} -78.0000 q^{17} +44.0000 q^{19} +(-10.0000 + 17.3205i) q^{20} +(6.00000 + 10.3923i) q^{22} +(60.0000 + 103.923i) q^{23} +(-12.5000 + 21.6506i) q^{25} +136.000 q^{26} +56.0000 q^{28} +(63.0000 - 109.119i) q^{29} +(122.000 + 211.310i) q^{31} +(-16.0000 - 27.7128i) q^{32} +(78.0000 - 135.100i) q^{34} +70.0000 q^{35} -304.000 q^{37} +(-44.0000 + 76.2102i) q^{38} +(-20.0000 - 34.6410i) q^{40} +(-240.000 - 415.692i) q^{41} +(-52.0000 + 90.0666i) q^{43} -24.0000 q^{44} -240.000 q^{46} +(300.000 - 519.615i) q^{47} +(73.5000 + 127.306i) q^{49} +(-25.0000 - 43.3013i) q^{50} +(-136.000 + 235.559i) q^{52} +258.000 q^{53} -30.0000 q^{55} +(-56.0000 + 96.9948i) q^{56} +(126.000 + 218.238i) q^{58} +(267.000 + 462.458i) q^{59} +(-181.000 + 313.501i) q^{61} -488.000 q^{62} +64.0000 q^{64} +(-170.000 + 294.449i) q^{65} +(134.000 + 232.095i) q^{67} +(156.000 + 270.200i) q^{68} +(-70.0000 + 121.244i) q^{70} +972.000 q^{71} +470.000 q^{73} +(304.000 - 526.543i) q^{74} +(-88.0000 - 152.420i) q^{76} +(42.0000 + 72.7461i) q^{77} +(-622.000 + 1077.34i) q^{79} +80.0000 q^{80} +960.000 q^{82} +(198.000 - 342.946i) q^{83} +(195.000 + 337.750i) q^{85} +(-104.000 - 180.133i) q^{86} +(24.0000 - 41.5692i) q^{88} +972.000 q^{89} +952.000 q^{91} +(240.000 - 415.692i) q^{92} +(600.000 + 1039.23i) q^{94} +(-110.000 - 190.526i) q^{95} +(23.0000 - 39.8372i) q^{97} -294.000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 4 q^{4} - 5 q^{5} - 14 q^{7} + 16 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 - 4 * q^4 - 5 * q^5 - 14 * q^7 + 16 * q^8	$ 2 q - 2 q^{2} - 4 q^{4} - 5 q^{5} - 14 q^{7} + 16 q^{8} + 20 q^{10} + 6 q^{11} - 68 q^{13} - 28 q^{14} - 16 q^{16} - 156 q^{17} + 88 q^{19} - 20 q^{20} + 12 q^{22} + 120 q^{23} - 25 q^{25} + 272 q^{26} + 112 q^{28} + 126 q^{29} + 244 q^{31} - 32 q^{32} + 156 q^{34} + 140 q^{35} - 608 q^{37} - 88 q^{38} - 40 q^{40} - 480 q^{41} - 104 q^{43} - 48 q^{44} - 480 q^{46} + 600 q^{47} + 147 q^{49} - 50 q^{50} - 272 q^{52} + 516 q^{53} - 60 q^{55} - 112 q^{56} + 252 q^{58} + 534 q^{59} - 362 q^{61} - 976 q^{62} + 128 q^{64} - 340 q^{65} + 268 q^{67} + 312 q^{68} - 140 q^{70} + 1944 q^{71} + 940 q^{73} + 608 q^{74} - 176 q^{76} + 84 q^{77} - 1244 q^{79} + 160 q^{80} + 1920 q^{82} + 396 q^{83} + 390 q^{85} - 208 q^{86} + 48 q^{88} + 1944 q^{89} + 1904 q^{91} + 480 q^{92} + 1200 q^{94} - 220 q^{95} + 46 q^{97} - 588 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 - 4 * q^4 - 5 * q^5 - 14 * q^7 + 16 * q^8 + 20 * q^10 + 6 * q^11 - 68 * q^13 - 28 * q^14 - 16 * q^16 - 156 * q^17 + 88 * q^19 - 20 * q^20 + 12 * q^22 + 120 * q^23 - 25 * q^25 + 272 * q^26 + 112 * q^28 + 126 * q^29 + 244 * q^31 - 32 * q^32 + 156 * q^34 + 140 * q^35 - 608 * q^37 - 88 * q^38 - 40 * q^40 - 480 * q^41 - 104 * q^43 - 48 * q^44 - 480 * q^46 + 600 * q^47 + 147 * q^49 - 50 * q^50 - 272 * q^52 + 516 * q^53 - 60 * q^55 - 112 * q^56 + 252 * q^58 + 534 * q^59 - 362 * q^61 - 976 * q^62 + 128 * q^64 - 340 * q^65 + 268 * q^67 + 312 * q^68 - 140 * q^70 + 1944 * q^71 + 940 * q^73 + 608 * q^74 - 176 * q^76 + 84 * q^77 - 1244 * q^79 + 160 * q^80 + 1920 * q^82 + 396 * q^83 + 390 * q^85 - 208 * q^86 + 48 * q^88 + 1944 * q^89 + 1904 * q^91 + 480 * q^92 + 1200 * q^94 - 220 * q^95 + 46 * q^97 - 588 * q^98

Display 100 coefficients 10 coefficients

Character values

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

$n$	$487$	$731$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$

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 $

$2$	−1.00000	+	1.73205i	−0.353553	+	0.612372i
$2$	−1.00000	+	1.73205i	−0.353553	+	0.612372i
$3$		0			0
$3$		0			0
$4$	−2.00000	−	3.46410i	−0.250000	−	0.433013i
$5$	−2.50000	−	4.33013i	−0.223607	−	0.387298i
$5$	−2.50000	−	4.33013i	−0.223607	−	0.387298i
$6$		0			0
$7$	−7.00000	+	12.1244i	−0.377964	+	0.654654i	−0.990766	−	0.135583i	$-0.956709\pi$
$7$	−7.00000	+	12.1244i	−0.377964	+	0.654654i	0.612801	+	0.790237i	$0.290043\pi$
$8$	8.00000			0.353553
$9$		0			0
$10$	10.0000			0.316228
$11$	3.00000	−	5.19615i	0.0822304	−	0.142427i	−0.821977	−	0.569520i	$-0.807129\pi$
$11$	3.00000	−	5.19615i	0.0822304	−	0.142427i	0.904208	+	0.427093i	$0.140462\pi$
$12$		0			0
$13$	−34.0000	−	58.8897i	−0.725377	−	1.25639i	−0.958819	−	0.284019i	$-0.908332\pi$
$13$	−34.0000	−	58.8897i	−0.725377	−	1.25639i	0.233441	−	0.972371i	$-0.425001\pi$
$14$	−14.0000	−	24.2487i	−0.267261	−	0.462910i
$15$		0			0
$16$	−8.00000	+	13.8564i	−0.125000	+	0.216506i
$17$	−78.0000			−1.11281			−0.556405	−	0.830911i	$-0.687820\pi$
$17$	−78.0000			−1.11281			−0.556405	+	0.830911i	$0.687820\pi$
$18$		0			0
$19$	44.0000			0.531279			0.265639	−	0.964072i	$-0.414417\pi$
$19$	44.0000			0.531279			0.265639	+	0.964072i	$0.414417\pi$
$20$	−10.0000	+	17.3205i	−0.111803	+	0.193649i
$21$		0			0
$22$	6.00000	+	10.3923i	0.0581456	+	0.100711i
$23$	60.0000	+	103.923i	0.543951	+	0.942150i	0.998672	+	0.0515165i	$0.0164055\pi$
$23$	60.0000	+	103.923i	0.543951	+	0.942150i	−0.454721	+	0.890634i	$0.650261\pi$
$24$		0			0
$25$	−12.5000	+	21.6506i	−0.100000	+	0.173205i
$26$	136.000			1.02584
$27$		0			0
$28$	56.0000			0.377964
$29$	63.0000	−	109.119i	0.403407	−	0.698722i	−0.590728	−	0.806871i	$-0.701159\pi$
$29$	63.0000	−	109.119i	0.403407	−	0.698722i	0.994135	+	0.108149i	$0.0344925\pi$
$30$		0			0
$31$	122.000	+	211.310i	0.706834	+	1.22427i	0.966026	+	0.258446i	$0.0832105\pi$
$31$	122.000	+	211.310i	0.706834	+	1.22427i	−0.259192	+	0.965826i	$0.583456\pi$
$32$	−16.0000	−	27.7128i	−0.0883883	−	0.153093i
$33$		0			0
$34$	78.0000	−	135.100i	0.393438	−	0.681454i
$35$	70.0000			0.338062
$36$		0			0
$37$	−304.000			−1.35074			−0.675369	−	0.737480i	$-0.736016\pi$
$37$	−304.000			−1.35074			−0.675369	+	0.737480i	$0.736016\pi$
$38$	−44.0000	+	76.2102i	−0.187835	+	0.325340i
$39$		0			0
$40$	−20.0000	−	34.6410i	−0.0790569	−	0.136931i
$41$	−240.000	−	415.692i	−0.914188	−	1.58342i	−0.808086	−	0.589065i	$-0.799496\pi$
$41$	−240.000	−	415.692i	−0.914188	−	1.58342i	−0.106102	−	0.994355i	$-0.533837\pi$
$42$		0			0
$43$	−52.0000	+	90.0666i	−0.184417	+	0.319419i	−0.943380	−	0.331714i	$-0.892373\pi$
$43$	−52.0000	+	90.0666i	−0.184417	+	0.319419i	0.758963	+	0.651134i	$0.225706\pi$
$44$	−24.0000			−0.0822304
$45$		0			0
$46$	−240.000			−0.769262
$47$	300.000	−	519.615i	0.931053	−	1.61263i	0.149528	−	0.988757i	$-0.452225\pi$
$47$	300.000	−	519.615i	0.931053	−	1.61263i	0.781525	−	0.623874i	$-0.214442\pi$
$48$		0			0
$49$	73.5000	+	127.306i	0.214286	+	0.371154i
$50$	−25.0000	−	43.3013i	−0.0707107	−	0.122474i
$51$		0			0
$52$	−136.000	+	235.559i	−0.362689	+	0.628195i
$53$	258.000			0.668661			0.334330	−	0.942456i	$-0.391490\pi$
$53$	258.000			0.668661			0.334330	+	0.942456i	$0.391490\pi$
$54$		0			0
$55$	−30.0000			−0.0735491
$56$	−56.0000	+	96.9948i	−0.133631	+	0.231455i
$57$		0			0
$58$	126.000	+	218.238i	0.285252	+	0.494071i
$59$	267.000	+	462.458i	0.589160	+	1.02046i	0.994343	+	0.106219i	$0.0338746\pi$
$59$	267.000	+	462.458i	0.589160	+	1.02046i	−0.405183	+	0.914236i	$0.632792\pi$
$60$		0			0
$61$	−181.000	+	313.501i	−0.379913	+	0.658028i	−0.991049	−	0.133497i	$-0.957379\pi$
$61$	−181.000	+	313.501i	−0.379913	+	0.658028i	0.611136	+	0.791525i	$0.290713\pi$
$62$	−488.000			−0.999614
$63$		0			0
$64$	64.0000			0.125000
$65$	−170.000	+	294.449i	−0.324399	+	0.561875i
$66$		0			0
$67$	134.000	+	232.095i	0.244339	+	0.423207i	0.961946	−	0.273241i	$-0.0880957\pi$
$67$	134.000	+	232.095i	0.244339	+	0.423207i	−0.717607	+	0.696449i	$0.754762\pi$
$68$	156.000	+	270.200i	0.278203	+	0.481861i
$69$		0			0
$70$	−70.0000	+	121.244i	−0.119523	+	0.207020i
$71$	972.000			1.62472			0.812360	−	0.583156i	$-0.198182\pi$
$71$	972.000			1.62472			0.812360	+	0.583156i	$0.198182\pi$
$72$		0			0
$73$	470.000			0.753553			0.376776	−	0.926304i	$-0.377033\pi$
$73$	470.000			0.753553			0.376776	+	0.926304i	$0.377033\pi$
$74$	304.000	−	526.543i	0.477558	−	0.827154i
$75$		0			0
$76$	−88.0000	−	152.420i	−0.132820	−	0.230050i
$77$	42.0000	+	72.7461i	0.0621603	+	0.107665i
$78$		0			0
$79$	−622.000	+	1077.34i	−0.885829	+	1.53430i	−0.0410678	+	0.999156i	$0.513076\pi$
$79$	−622.000	+	1077.34i	−0.885829	+	1.53430i	−0.844761	+	0.535144i	$0.820257\pi$
$80$	80.0000			0.111803
$81$		0			0
$82$	960.000			1.29286
$83$	198.000	−	342.946i	0.261847	−	0.453533i	−0.704886	−	0.709321i	$-0.749002\pi$
$83$	198.000	−	342.946i	0.261847	−	0.453533i	0.966733	+	0.255788i	$0.0823350\pi$
$84$		0			0
$85$	195.000	+	337.750i	0.248832	+	0.430990i
$86$	−104.000	−	180.133i	−0.130402	−	0.225864i
$87$		0			0
$88$	24.0000	−	41.5692i	0.0290728	−	0.0503556i
$89$	972.000			1.15766			0.578830	−	0.815448i	$-0.303509\pi$
$89$	972.000			1.15766			0.578830	+	0.815448i	$0.303509\pi$
$90$		0			0
$91$	952.000			1.09667
$92$	240.000	−	415.692i	0.271975	−	0.471075i
$93$		0			0
$94$	600.000	+	1039.23i	0.658354	+	1.14030i
$95$	−110.000	−	190.526i	−0.118797	−	0.205763i
$96$		0			0
$97$	23.0000	−	39.8372i	0.0240752	−	0.0416995i	−0.853737	−	0.520705i	$-0.825669\pi$
$97$	23.0000	−	39.8372i	0.0240752	−	0.0416995i	0.877812	+	0.479005i	$0.159003\pi$
$98$	−294.000			−0.303046
$99$		0			0
$100$	100.000			0.100000
$101$	−753.000	+	1304.23i	−0.741845	+	1.28491i	0.209810	+	0.977742i	$0.432715\pi$
$101$	−753.000	+	1304.23i	−0.741845	+	1.28491i	−0.951655	+	0.307170i	$0.900618\pi$
$102$		0			0
$103$	737.000	+	1276.52i	0.705037	+	1.22116i	0.966678	+	0.255994i	$0.0824028\pi$
$103$	737.000	+	1276.52i	0.705037	+	1.22116i	−0.261642	+	0.965165i	$0.584264\pi$
$104$	−272.000	−	471.118i	−0.256460	−	0.444201i
$105$		0			0
$106$	−258.000	+	446.869i	−0.236407	+	0.409469i
$107$	−924.000			−0.834827			−0.417413	−	0.908717i	$-0.637063\pi$
$107$	−924.000			−0.834827			−0.417413	+	0.908717i	$0.637063\pi$
$108$		0			0
$109$	698.000			0.613360			0.306680	−	0.951813i	$-0.400782\pi$
$109$	698.000			0.613360			0.306680	+	0.951813i	$0.400782\pi$
$110$	30.0000	−	51.9615i	0.0260035	−	0.0450394i
$111$		0			0
$112$	−112.000	−	193.990i	−0.0944911	−	0.163663i
$113$	−111.000	−	192.258i	−0.0924071	−	0.160054i	0.816116	−	0.577888i	$-0.196123\pi$
$113$	−111.000	−	192.258i	−0.0924071	−	0.160054i	−0.908523	+	0.417834i	$0.862789\pi$
$114$		0			0
$115$	300.000	−	519.615i	0.243262	−	0.421342i
$116$	−504.000			−0.403407
$117$		0			0
$118$	−1068.00			−0.833198
$119$	546.000	−	945.700i	0.420603	−	0.728505i
$120$		0			0
$121$	647.500	+	1121.50i	0.486476	+	0.842602i
$122$	−362.000	−	627.002i	−0.268639	−	0.465296i
$123$		0			0
$124$	488.000	−	845.241i	0.353417	−	0.612136i
$125$	125.000			0.0894427
$126$		0			0
$127$	−1906.00			−1.33173			−0.665867	−	0.746071i	$-0.731938\pi$
$127$	−1906.00			−1.33173			−0.665867	+	0.746071i	$0.731938\pi$
$128$	−64.0000	+	110.851i	−0.0441942	+	0.0765466i
$129$		0			0
$130$	−340.000	−	588.897i	−0.229384	−	0.397305i
$131$	1437.00	+	2488.96i	0.958407	+	1.66001i	0.726372	+	0.687301i	$0.241205\pi$
$131$	1437.00	+	2488.96i	0.958407	+	1.66001i	0.232034	+	0.972708i	$0.425462\pi$
$132$		0			0
$133$	−308.000	+	533.472i	−0.200804	+	0.347803i
$134$	−536.000			−0.345547
$135$		0			0
$136$	−624.000			−0.393438
$137$	−399.000	+	691.088i	−0.248824	+	0.430976i	−0.963200	−	0.268787i	$-0.913377\pi$
$137$	−399.000	+	691.088i	−0.248824	+	0.430976i	0.714376	+	0.699762i	$0.246711\pi$
$138$		0			0
$139$	350.000	+	606.218i	0.213573	+	0.369919i	0.952830	−	0.303504i	$-0.0981566\pi$
$139$	350.000	+	606.218i	0.213573	+	0.369919i	−0.739257	+	0.673423i	$0.764823\pi$
$140$	−140.000	−	242.487i	−0.0845154	−	0.146385i
$141$		0			0
$142$	−972.000	+	1683.55i	−0.574426	+	0.994934i
$143$	−408.000			−0.238592
$144$		0			0
$145$	−630.000			−0.360818
$146$	−470.000	+	814.064i	−0.266421	+	0.461455i
$147$		0			0
$148$	608.000	+	1053.09i	0.337684	+	0.584887i
$149$	57.0000	+	98.7269i	0.0313397	+	0.0542820i	0.881270	−	0.472613i	$-0.156689\pi$
$149$	57.0000	+	98.7269i	0.0313397	+	0.0542820i	−0.849930	+	0.526895i	$0.823356\pi$
$150$		0			0
$151$	−532.000	+	921.451i	−0.286712	+	0.496600i	−0.973023	−	0.230708i	$-0.925896\pi$
$151$	−532.000	+	921.451i	−0.286712	+	0.496600i	0.686311	+	0.727308i	$0.259229\pi$
$152$	352.000			0.187835
$153$		0			0
$154$	−168.000			−0.0879080
$155$	610.000	−	1056.55i	0.316106	−	0.547511i
$156$		0			0
$157$	974.000	+	1687.02i	0.495119	+	0.857571i	0.999984	−	0.00562710i	$-0.00179117\pi$
$157$	974.000	+	1687.02i	0.495119	+	0.857571i	−0.504865	+	0.863198i	$0.668458\pi$
$158$	−1244.00	−	2154.67i	−0.626375	−	1.08491i
$159$		0			0
$160$	−80.0000	+	138.564i	−0.0395285	+	0.0684653i
$161$	−1680.00			−0.822376
$162$		0			0
$163$	2060.00			0.989887			0.494944	−	0.868925i	$-0.335189\pi$
$163$	2060.00			0.989887			0.494944	+	0.868925i	$0.335189\pi$
$164$	−960.000	+	1662.77i	−0.457094	+	0.791710i
$165$		0			0
$166$	396.000	+	685.892i	0.185154	+	0.320696i
$167$	−624.000	−	1080.80i	−0.289141	−	0.500807i	0.684464	−	0.729047i	$-0.260036\pi$
$167$	−624.000	−	1080.80i	−0.289141	−	0.500807i	−0.973605	+	0.228240i	$0.926703\pi$
$168$		0			0
$169$	−1213.50	+	2101.84i	−0.552344	+	0.956688i
$170$	−780.000			−0.351902
$171$		0			0
$172$	416.000			0.184417
$173$	−573.000	+	992.465i	−0.251817	+	0.436160i	−0.964026	−	0.265807i	$-0.914362\pi$
$173$	−573.000	+	992.465i	−0.251817	+	0.436160i	0.712209	+	0.701968i	$0.247695\pi$
$174$		0			0
$175$	−175.000	−	303.109i	−0.0755929	−	0.130931i
$176$	48.0000	+	83.1384i	0.0205576	+	0.0356068i
$177$		0			0
$178$	−972.000	+	1683.55i	−0.409295	+	0.708919i
$179$	1146.00			0.478525			0.239263	−	0.970955i	$-0.423094\pi$
$179$	1146.00			0.478525			0.239263	+	0.970955i	$0.423094\pi$
$180$		0			0
$181$	−118.000			−0.0484579			−0.0242289	−	0.999706i	$-0.507713\pi$
$181$	−118.000			−0.0484579			−0.0242289	+	0.999706i	$0.507713\pi$
$182$	−952.000	+	1648.91i	−0.387730	+	0.671569i
$183$		0			0
$184$	480.000	+	831.384i	0.192316	+	0.333100i
$185$	760.000	+	1316.36i	0.302034	+	0.523138i
$186$		0			0
$187$	−234.000	+	405.300i	−0.0915068	+	0.158494i
$188$	−2400.00			−0.931053
$189$		0			0
$190$	440.000			0.168005
$191$	846.000	−	1465.31i	0.320494	−	0.555112i	−0.660096	−	0.751181i	$-0.729484\pi$
$191$	846.000	−	1465.31i	0.320494	−	0.555112i	0.980590	+	0.196069i	$0.0628177\pi$
$192$		0			0
$193$	−1675.00	−	2901.19i	−0.624711	−	1.08203i	−0.988597	−	0.150587i	$-0.951884\pi$
$193$	−1675.00	−	2901.19i	−0.624711	−	1.08203i	0.363886	−	0.931443i	$-0.381450\pi$
$194$	46.0000	+	79.6743i	0.0170238	+	0.0294860i
$195$		0			0
$196$	294.000	−	509.223i	0.107143	−	0.185577i
$197$	3606.00			1.30415			0.652073	−	0.758156i	$-0.273899\pi$
$197$	3606.00			1.30415			0.652073	+	0.758156i	$0.273899\pi$
$198$		0			0
$199$	2696.00			0.960374			0.480187	−	0.877166i	$-0.340569\pi$
$199$	2696.00			0.960374			0.480187	+	0.877166i	$0.340569\pi$
$200$	−100.000	+	173.205i	−0.0353553	+	0.0612372i
$201$		0			0
$202$	−1506.00	−	2608.47i	−0.524563	−	0.908570i
$203$	882.000	+	1527.67i	0.304947	+	0.528184i
$204$		0			0
$205$	−1200.00	+	2078.46i	−0.408837	+	0.708127i
$206$	−2948.00			−0.997072
$207$		0			0
$208$	1088.00			0.362689
$209$	132.000	−	228.631i	0.0436872	−	0.0756685i
$210$		0			0
$211$	2.00000	+	3.46410i	0.000652539	+	0.00113023i	0.866351	−	0.499435i	$-0.166459\pi$
$211$	2.00000	+	3.46410i	0.000652539	+	0.00113023i	−0.865699	+	0.500565i	$0.833126\pi$
$212$	−516.000	−	893.738i	−0.167165	−	0.289539i
$213$		0			0
$214$	924.000	−	1600.41i	0.295156	−	0.511225i
$215$	520.000			0.164947
$216$		0			0
$217$	−3416.00			−1.06863
$218$	−698.000	+	1208.97i	−0.216856	+	0.375605i
$219$		0			0
$220$	60.0000	+	103.923i	0.0183873	+	0.0318477i
$221$	2652.00	+	4593.40i	0.807207	+	1.39812i
$222$		0			0
$223$	581.000	−	1006.32i	0.174469	−	0.302190i	−0.765508	−	0.643426i	$-0.777512\pi$
$223$	581.000	−	1006.32i	0.174469	−	0.302190i	0.939977	+	0.341237i	$0.110846\pi$
$224$	448.000			0.133631
$225$		0			0
$226$	444.000			0.130683
$227$	−1200.00	+	2078.46i	−0.350867	+	0.607719i	−0.986402	−	0.164353i	$-0.947446\pi$
$227$	−1200.00	+	2078.46i	−0.350867	+	0.607719i	0.635535	+	0.772072i	$0.280780\pi$
$228$		0			0
$229$	1157.00	+	2003.98i	0.333872	+	0.578283i	0.983268	−	0.182167i	$-0.0583113\pi$
$229$	1157.00	+	2003.98i	0.333872	+	0.578283i	−0.649395	+	0.760451i	$0.724978\pi$
$230$	600.000	+	1039.23i	0.172012	+	0.297934i
$231$		0			0
$232$	504.000	−	872.954i	0.142626	−	0.247035i
$233$	18.0000			0.00506103			0.00253051	−	0.999997i	$-0.499195\pi$
$233$	18.0000			0.00506103			0.00253051	+	0.999997i	$0.499195\pi$
$234$		0			0
$235$	−3000.00			−0.832759
$236$	1068.00	−	1849.83i	0.294580	−	0.510228i
$237$		0			0
$238$	1092.00	+	1891.40i	0.297411	+	0.515131i
$239$	−2934.00	−	5081.84i	−0.794078	−	1.37538i	−0.923423	−	0.383784i	$-0.874621\pi$
$239$	−2934.00	−	5081.84i	−0.794078	−	1.37538i	0.129345	−	0.991600i	$-0.458713\pi$
$240$		0			0
$241$	2165.00	−	3749.89i	0.578672	−	1.00229i	−0.416960	−	0.908925i	$-0.636905\pi$
$241$	2165.00	−	3749.89i	0.578672	−	1.00229i	0.995632	−	0.0933643i	$-0.0297621\pi$
$242$	−2590.00			−0.687981
$243$		0			0
$244$	1448.00			0.379913
$245$	367.500	−	636.529i	0.0958315	−	0.165985i
$246$		0			0
$247$	−1496.00	−	2591.15i	−0.385377	−	0.667493i
$248$	976.000	+	1690.48i	0.249903	+	0.432846i
$249$		0			0
$250$	−125.000	+	216.506i	−0.0316228	+	0.0547723i
$251$	−498.000			−0.125233			−0.0626165	−	0.998038i	$-0.519944\pi$
$251$	−498.000			−0.125233			−0.0626165	+	0.998038i	$0.519944\pi$
$252$		0			0
$253$	720.000			0.178917
$254$	1906.00	−	3301.29i	0.470839	−	0.815517i
$255$		0			0
$256$	−128.000	−	221.703i	−0.0312500	−	0.0541266i
$257$	321.000	+	555.988i	0.0779122	+	0.134948i	0.902349	−	0.431006i	$-0.141841\pi$
$257$	321.000	+	555.988i	0.0779122	+	0.134948i	−0.824437	+	0.565954i	$0.808508\pi$
$258$		0			0
$259$	2128.00	−	3685.80i	0.510531	−	0.884265i
$260$	1360.00			0.324399
$261$		0			0
$262$	−5748.00			−1.35539
$263$	−3984.00	+	6900.49i	−0.934084	+	1.61788i	−0.157823	+	0.987467i	$0.550448\pi$
$263$	−3984.00	+	6900.49i	−0.934084	+	1.61788i	−0.776260	+	0.630413i	$0.782886\pi$
$264$		0			0
$265$	−645.000	−	1117.17i	−0.149517	−	0.258971i
$266$	−616.000	−	1066.94i	−0.141990	−	0.245934i
$267$		0			0
$268$	536.000	−	928.379i	0.122169	−	0.211604i
$269$	4218.00			0.956045			0.478022	−	0.878348i	$-0.341354\pi$
$269$	4218.00			0.956045			0.478022	+	0.878348i	$0.341354\pi$
$270$		0			0
$271$	848.000			0.190082			0.0950412	−	0.995473i	$-0.469702\pi$
$271$	848.000			0.190082			0.0950412	+	0.995473i	$0.469702\pi$
$272$	624.000	−	1080.80i	0.139101	−	0.240931i
$273$		0			0
$274$	−798.000	−	1382.18i	−0.175945	−	0.304746i
$275$	75.0000	+	129.904i	0.0164461	+	0.0284854i
$276$		0			0
$277$	752.000	−	1302.50i	0.163117	−	0.282526i	−0.772868	−	0.634567i	$-0.781179\pi$
$277$	752.000	−	1302.50i	0.163117	−	0.282526i	0.935985	+	0.352040i	$0.114512\pi$
$278$	−1400.00			−0.302037
$279$		0			0
$280$	560.000			0.119523
$281$	−654.000	+	1132.76i	−0.138841	+	0.240480i	−0.927058	−	0.374917i	$-0.877671\pi$
$281$	−654.000	+	1132.76i	−0.138841	+	0.240480i	0.788217	+	0.615397i	$0.211004\pi$
$282$		0			0
$283$	2966.00	+	5137.26i	0.623005	+	1.07908i	0.988923	+	0.148429i	$0.0474217\pi$
$283$	2966.00	+	5137.26i	0.623005	+	1.07908i	−0.365918	+	0.930647i	$0.619245\pi$
$284$	−1944.00	−	3367.11i	−0.406180	−	0.703525i
$285$		0			0
$286$	408.000	−	706.677i	0.0843551	−	0.146107i
$287$	6720.00			1.38212
$288$		0			0
$289$	1171.00			0.238347
$290$	630.000	−	1091.19i	0.127569	−	0.220955i
$291$		0			0
$292$	−940.000	−	1628.13i	−0.188388	−	0.326298i
$293$	2613.00	+	4525.85i	0.521000	+	0.902399i	0.999702	+	0.0244213i	$0.00777430\pi$
$293$	2613.00	+	4525.85i	0.521000	+	0.902399i	−0.478701	+	0.877978i	$0.658892\pi$
$294$		0			0
$295$	1335.00	−	2312.29i	0.263480	−	0.456361i
$296$	−2432.00			−0.477558
$297$		0			0
$298$	−228.000			−0.0443211
$299$	4080.00	−	7066.77i	0.789139	−	1.36683i
$300$		0			0
$301$	−728.000	−	1260.93i	−0.139406	−	0.241458i
$302$	−1064.00	−	1842.90i	−0.202736	−	0.351149i
$303$		0			0
$304$	−352.000	+	609.682i	−0.0664098	+	0.115025i
$305$	1810.00			0.339804
$306$		0			0
$307$	4448.00			0.826908			0.413454	−	0.910525i	$-0.364322\pi$
$307$	4448.00			0.826908			0.413454	+	0.910525i	$0.364322\pi$
$308$	168.000	−	290.985i	0.0310802	−	0.0538324i
$309$		0			0
$310$	1220.00	+	2113.10i	0.223520	+	0.387149i
$311$	−4566.00	−	7908.54i	−0.832521	−	1.44197i	−0.896033	−	0.443988i	$-0.853563\pi$
$311$	−4566.00	−	7908.54i	−0.832521	−	1.44197i	0.0635115	−	0.997981i	$-0.479770\pi$
$312$		0			0
$313$	1085.00	−	1879.28i	0.195936	−	0.339370i	−0.751271	−	0.659994i	$-0.770559\pi$
$313$	1085.00	−	1879.28i	0.195936	−	0.339370i	0.947207	+	0.320623i	$0.103892\pi$
$314$	−3896.00			−0.700204
$315$		0			0
$316$	4976.00			0.885829
$317$	3837.00	−	6645.88i	0.679834	−	1.17751i	−0.295197	−	0.955437i	$-0.595385\pi$
$317$	3837.00	−	6645.88i	0.679834	−	1.17751i	0.975031	−	0.222071i	$-0.0712815\pi$
$318$		0			0
$319$	−378.000	−	654.715i	−0.0663446	−	0.114912i
$320$	−160.000	−	277.128i	−0.0279508	−	0.0484123i
$321$		0			0
$322$	1680.00	−	2909.85i	0.290754	−	0.503600i
$323$	−3432.00			−0.591212
$324$		0			0
$325$	1700.00			0.290151
$326$	−2060.00	+	3568.02i	−0.349978	+	0.606180i
$327$		0			0
$328$	−1920.00	−	3325.54i	−0.323214	−	0.559823i
$329$	4200.00	+	7274.61i	0.703810	+	1.21903i
$330$		0			0
$331$	−4798.00	+	8310.38i	−0.796743	+	1.38000i	0.124983	+	0.992159i	$0.460112\pi$
$331$	−4798.00	+	8310.38i	−0.796743	+	1.38000i	−0.921726	+	0.387841i	$0.873221\pi$
$332$	−1584.00			−0.261847
$333$		0			0
$334$	2496.00			0.408907
$335$	670.000	−	1160.47i	0.109272	−	0.189264i
$336$		0			0
$337$	−6079.00	−	10529.1i	−0.982624	−	1.70195i	−0.652053	−	0.758173i	$-0.726092\pi$
$337$	−6079.00	−	10529.1i	−0.982624	−	1.70195i	−0.330570	−	0.943781i	$-0.607241\pi$
$338$	−2427.00	−	4203.69i	−0.390566	−	0.676481i
$339$		0			0
$340$	780.000	−	1351.00i	0.124416	−	0.215495i
$341$	1464.00			0.232493
$342$		0			0
$343$	−6860.00			−1.07990
$344$	−416.000	+	720.533i	−0.0652012	+	0.112932i
$345$		0			0
$346$	−1146.00	−	1984.93i	−0.178062	−	0.308412i
$347$	5160.00	+	8937.38i	0.798280	+	1.38266i	0.920735	+	0.390188i	$0.127590\pi$
$347$	5160.00	+	8937.38i	0.798280	+	1.38266i	−0.122455	+	0.992474i	$0.539077\pi$
$348$		0			0
$349$	1079.00	−	1868.88i	0.165494	−	0.286645i	−0.771336	−	0.636428i	$-0.780411\pi$
$349$	1079.00	−	1868.88i	0.165494	−	0.286645i	0.936831	+	0.349783i	$0.113745\pi$
$350$	700.000			0.106904
$351$		0			0
$352$	−192.000			−0.0290728
$353$	−165.000	+	285.788i	−0.0248784	+	0.0430906i	−0.878197	−	0.478300i	$-0.841253\pi$
$353$	−165.000	+	285.788i	−0.0248784	+	0.0430906i	0.853318	+	0.521391i	$0.174587\pi$
$354$		0			0
$355$	−2430.00	−	4208.88i	−0.363299	−	0.629252i
$356$	−1944.00	−	3367.11i	−0.289415	−	0.501282i
$357$		0			0
$358$	−1146.00	+	1984.93i	−0.169184	+	0.293036i
$359$	8664.00			1.27373			0.636864	−	0.770976i	$-0.280231\pi$
$359$	8664.00			1.27373			0.636864	+	0.770976i	$0.280231\pi$
$360$		0			0
$361$	−4923.00			−0.717743
$362$	118.000	−	204.382i	0.0171324	−	0.0296743i
$363$		0			0
$364$	−1904.00	−	3297.82i	−0.274167	−	0.474871i
$365$	−1175.00	−	2035.16i	−0.168499	−	0.291850i
$366$		0			0
$367$	−1891.00	+	3275.31i	−0.268963	+	0.465857i	−0.968594	−	0.248647i	$-0.920014\pi$
$367$	−1891.00	+	3275.31i	−0.268963	+	0.465857i	0.699631	+	0.714504i	$0.253348\pi$
$368$	−1920.00			−0.271975
$369$		0			0
$370$	−3040.00			−0.427141
$371$	−1806.00	+	3128.08i	−0.252730	+	0.437741i
$372$		0			0
$373$	−5638.00	−	9765.30i	−0.782640	−	1.35557i	−0.930399	−	0.366548i	$-0.880539\pi$
$373$	−5638.00	−	9765.30i	−0.782640	−	1.35557i	0.147759	−	0.989023i	$-0.452794\pi$
$374$	−468.000	−	810.600i	−0.0647051	−	0.112073i
$375$		0			0
$376$	2400.00	−	4156.92i	0.329177	−	0.570151i
$377$	−8568.00			−1.17049
$378$		0			0
$379$	980.000			0.132821			0.0664106	−	0.997792i	$-0.478845\pi$
$379$	980.000			0.132821			0.0664106	+	0.997792i	$0.478845\pi$
$380$	−440.000	+	762.102i	−0.0593987	+	0.102882i
$381$		0			0
$382$	1692.00	+	2930.63i	0.226624	+	0.392524i
$383$	−2100.00	−	3637.31i	−0.280170	−	0.485268i	0.691257	−	0.722609i	$-0.257057\pi$
$383$	−2100.00	−	3637.31i	−0.280170	−	0.485268i	−0.971426	+	0.237341i	$0.923724\pi$
$384$		0			0
$385$	210.000	−	363.731i	0.0277989	−	0.0481492i
$386$	6700.00			0.883474
$387$		0			0
$388$	−184.000			−0.0240752
$389$	6669.00	−	11551.0i	0.869233	−	1.50556i	0.00645168	−	0.999979i	$-0.497946\pi$
$389$	6669.00	−	11551.0i	0.869233	−	1.50556i	0.862782	−	0.505577i	$-0.168720\pi$
$390$		0			0
$391$	−4680.00	−	8106.00i	−0.605314	−	1.04843i
$392$	588.000	+	1018.45i	0.0757614	+	0.131223i
$393$		0			0
$394$	−3606.00	+	6245.78i	−0.461085	+	0.798623i
$395$	6220.00			0.792309
$396$		0			0
$397$	−7192.00			−0.909209			−0.454605	−	0.890693i	$-0.650219\pi$
$397$	−7192.00			−0.909209			−0.454605	+	0.890693i	$0.650219\pi$
$398$	−2696.00	+	4669.61i	−0.339543	+	0.588106i
$399$		0			0
$400$	−200.000	−	346.410i	−0.0250000	−	0.0433013i
$401$	−1158.00	−	2005.71i	−0.144209	−	0.249777i	0.784869	−	0.619662i	$-0.212730\pi$
$401$	−1158.00	−	2005.71i	−0.144209	−	0.249777i	−0.929078	+	0.369885i	$0.879397\pi$
$402$		0			0
$403$	8296.00	−	14369.1i	1.02544	−	1.77612i
$404$	6024.00			0.741845
$405$		0			0
$406$	−3528.00			−0.431260
$407$	−912.000	+	1579.63i	−0.111072	+	0.192382i
$408$		0			0
$409$	6179.00	+	10702.3i	0.747022	+	1.29388i	0.949244	+	0.314540i	$0.101850\pi$
$409$	6179.00	+	10702.3i	0.747022	+	1.29388i	−0.202222	+	0.979340i	$0.564816\pi$
$410$	−2400.00	−	4156.92i	−0.289092	−	0.500721i
$411$		0			0
$412$	2948.00	−	5106.09i	0.352518	−	0.610580i
$413$	−7476.00			−0.890726
$414$		0			0
$415$	−1980.00			−0.234203
$416$	−1088.00	+	1884.47i	−0.128230	+	0.222100i
$417$		0			0
$418$	264.000	+	457.261i	0.0308915	+	0.0535057i
$419$	1653.00	+	2863.08i	0.192731	+	0.333820i	0.946154	−	0.323716i	$-0.104932\pi$
$419$	1653.00	+	2863.08i	0.192731	+	0.333820i	−0.753423	+	0.657536i	$0.771599\pi$
$420$		0			0
$421$	7253.00	−	12562.6i	0.839643	−	1.45430i	−0.0505509	−	0.998721i	$-0.516098\pi$
$421$	7253.00	−	12562.6i	0.839643	−	1.45430i	0.890194	−	0.455582i	$-0.150569\pi$
$422$	−8.00000			−0.000922829
$423$		0			0
$424$	2064.00			0.236407
$425$	975.000	−	1688.75i	0.111281	−	0.192744i
$426$		0			0
$427$	−2534.00	−	4389.02i	−0.287187	−	0.497422i
$428$	1848.00	+	3200.83i	0.208707	+	0.361491i
$429$		0			0
$430$	−520.000	+	900.666i	−0.0583177	+	0.101009i
$431$	6480.00			0.724201			0.362100	−	0.932139i	$-0.382060\pi$
$431$	6480.00			0.724201			0.362100	+	0.932139i	$0.382060\pi$
$432$		0			0
$433$	11894.0			1.32007			0.660034	−	0.751236i	$-0.270542\pi$
$433$	11894.0			1.32007			0.660034	+	0.751236i	$0.270542\pi$
$434$	3416.00	−	5916.69i	0.377819	−	0.654401i
$435$		0			0
$436$	−1396.00	−	2417.94i	−0.153340	−	0.265593i
$437$	2640.00	+	4572.61i	0.288989	+	0.500544i
$438$		0			0
$439$	6344.00	−	10988.1i	0.689710	−	1.19461i	−0.282222	−	0.959349i	$-0.591071\pi$
$439$	6344.00	−	10988.1i	0.689710	−	1.19461i	0.971932	−	0.235263i	$-0.0755952\pi$
$440$	−240.000			−0.0260035
$441$		0			0
$442$	−10608.0			−1.14156
$443$	2484.00	−	4302.41i	0.266407	−	0.461431i	−0.701524	−	0.712646i	$-0.747497\pi$
$443$	2484.00	−	4302.41i	0.266407	−	0.461431i	0.967931	+	0.251215i	$0.0808301\pi$
$444$		0			0
$445$	−2430.00	−	4208.88i	−0.258861	−	0.448360i
$446$	1162.00	+	2012.64i	0.123368	+	0.213680i
$447$		0			0
$448$	−448.000	+	775.959i	−0.0472456	+	0.0818317i
$449$	11508.0			1.20957			0.604784	−	0.796389i	$-0.293259\pi$
$449$	11508.0			1.20957			0.604784	+	0.796389i	$0.293259\pi$
$450$		0			0
$451$	−2880.00			−0.300696
$452$	−444.000	+	769.031i	−0.0462035	+	0.0800269i
$453$		0			0
$454$	−2400.00	−	4156.92i	−0.248100	−	0.429722i
$455$	−2380.00	−	4122.28i	−0.245222	−	0.424737i
$456$		0			0
$457$	−541.000	+	937.039i	−0.0553762	+	0.0959144i	−0.892385	−	0.451276i	$-0.850969\pi$
$457$	−541.000	+	937.039i	−0.0553762	+	0.0959144i	0.837008	+	0.547190i	$0.184302\pi$
$458$	−4628.00			−0.472166
$459$		0			0
$460$	−2400.00			−0.243262
$461$	−5619.00	+	9732.39i	−0.567685	+	0.983260i	0.429109	+	0.903253i	$0.358828\pi$
$461$	−5619.00	+	9732.39i	−0.567685	+	0.983260i	−0.996794	+	0.0800071i	$0.974506\pi$
$462$		0			0
$463$	1151.00	+	1993.59i	0.115532	+	0.200108i	0.917992	−	0.396598i	$-0.129809\pi$
$463$	1151.00	+	1993.59i	0.115532	+	0.200108i	−0.802460	+	0.596706i	$0.796476\pi$
$464$	1008.00	+	1745.91i	0.100852	+	0.174680i
$465$		0			0
$466$	−18.0000	+	31.1769i	−0.00178934	+	0.00309923i
$467$	−15876.0			−1.57313			−0.786567	−	0.617505i	$-0.788144\pi$
$467$	−15876.0			−1.57313			−0.786567	+	0.617505i	$0.788144\pi$
$468$		0			0
$469$	−3752.00			−0.369406
$470$	3000.00	−	5196.15i	0.294425	−	0.509959i
$471$		0			0
$472$	2136.00	+	3699.66i	0.208300	+	0.360785i
$473$	312.000	+	540.400i	0.0303293	+	0.0525319i
$474$		0			0
$475$	−550.000	+	952.628i	−0.0531279	+	0.0920201i
$476$	−4368.00			−0.420603
$477$		0			0
$478$	11736.0			1.12300
$479$	2322.00	−	4021.82i	0.221492	−	0.383636i	−0.733769	−	0.679399i	$-0.762241\pi$
$479$	2322.00	−	4021.82i	0.221492	−	0.383636i	0.955261	+	0.295763i	$0.0955738\pi$
$480$		0			0
$481$	10336.0	+	17902.5i	0.979794	+	1.69705i
$482$	4330.00	+	7499.78i	0.409183	+	0.708725i
$483$		0			0
$484$	2590.00	−	4486.01i	0.243238	−	0.421301i
$485$	−230.000			−0.0215335
$486$		0			0
$487$	2426.00			0.225734			0.112867	−	0.993610i	$-0.463997\pi$
$487$	2426.00			0.225734			0.112867	+	0.993610i	$0.463997\pi$
$488$	−1448.00	+	2508.01i	−0.134319	+	0.232648i
$489$		0			0
$490$	735.000	+	1273.06i	0.0677631	+	0.117369i
$491$	117.000	+	202.650i	0.0107538	+	0.0186262i	0.871352	−	0.490658i	$-0.163244\pi$
$491$	117.000	+	202.650i	0.0107538	+	0.0186262i	−0.860598	+	0.509284i	$0.829910\pi$
$492$		0			0
$493$	−4914.00	+	8511.30i	−0.448916	+	0.777545i
$494$	5984.00			0.545006
$495$		0			0
$496$	−3904.00			−0.353417
$497$	−6804.00	+	11784.9i	−0.614087	+	1.06363i
$498$		0			0
$499$	−7102.00	−	12301.0i	−0.637133	−	1.10355i	−0.986059	−	0.166396i	$-0.946787\pi$
$499$	−7102.00	−	12301.0i	−0.637133	−	1.10355i	0.348926	−	0.937150i	$-0.386546\pi$
$500$	−250.000	−	433.013i	−0.0223607	−	0.0387298i
$501$		0			0
$502$	498.000	−	862.561i	0.0442765	−	0.0766892i
$503$	−4920.00			−0.436127			−0.218064	−	0.975935i	$-0.569974\pi$
$503$	−4920.00			−0.436127			−0.218064	+	0.975935i	$0.569974\pi$
$504$		0			0
$505$	7530.00			0.663526
$506$	−720.000	+	1247.08i	−0.0632567	+	0.109564i
$507$		0			0
$508$	3812.00	+	6602.58i	0.332933	+	0.576658i
$509$	−2229.00	−	3860.74i	−0.194104	−	0.336197i	0.752503	−	0.658589i	$-0.228846\pi$
$509$	−2229.00	−	3860.74i	−0.194104	−	0.336197i	−0.946606	+	0.322392i	$0.895513\pi$
$510$		0			0
$511$	−3290.00	+	5698.45i	−0.284816	+	0.493316i
$512$	512.000			0.0441942
$513$		0			0
$514$	−1284.00			−0.110184
$515$	3685.00	−	6382.61i	0.315302	−	0.546119i
$516$		0			0
$517$	−1800.00	−	3117.69i	−0.153122	−	0.265215i
$518$	4256.00	+	7371.61i	0.361000	+	0.625270i
$519$		0			0
$520$	−1360.00	+	2355.59i	−0.114692	+	0.198653i
$521$	−4212.00			−0.354186			−0.177093	−	0.984194i	$-0.556669\pi$
$521$	−4212.00			−0.354186			−0.177093	+	0.984194i	$0.556669\pi$
$522$		0			0
$523$	−11212.0			−0.937412			−0.468706	−	0.883354i	$-0.655280\pi$
$523$	−11212.0			−0.937412			−0.468706	+	0.883354i	$0.655280\pi$
$524$	5748.00	−	9955.83i	0.479203	−	0.830005i
$525$		0			0
$526$	−7968.00	−	13801.0i	−0.660497	−	1.14401i
$527$	−9516.00	−	16482.2i	−0.786572	−	1.36238i
$528$		0			0
$529$	−1116.50	+	1933.83i	−0.0917646	+	0.158941i
$530$	2580.00			0.211449
$531$		0			0
$532$	2464.00			0.200804
$533$	−16320.0	+	28267.1i	−1.32626	+	2.29715i
$534$		0			0
$535$	2310.00	+	4001.04i	0.186673	+	0.323327i
$536$	1072.00	+	1856.76i	0.0863868	+	0.149626i
$537$		0			0
$538$	−4218.00	+	7305.79i	−0.338013	+	0.585455i
$539$	882.000			0.0704832
$540$		0			0
$541$	14018.0			1.11401			0.557006	−	0.830508i	$-0.311950\pi$
$541$	14018.0			1.11401			0.557006	+	0.830508i	$0.311950\pi$
$542$	−848.000	+	1468.78i	−0.0672043	+	0.116401i
$543$		0			0
$544$	1248.00	+	2161.60i	0.0983595	+	0.170364i
$545$	−1745.00	−	3022.43i	−0.137152	−	0.237553i
$546$		0			0
$547$	−9100.00	+	15761.7i	−0.711312	+	1.23203i	0.253052	+	0.967453i	$0.418566\pi$
$547$	−9100.00	+	15761.7i	−0.711312	+	1.23203i	−0.964365	+	0.264577i	$0.914768\pi$
$548$	3192.00			0.248824
$549$		0			0
$550$	−300.000			−0.0232583
$551$	2772.00	−	4801.24i	0.214322	−	0.371216i
$552$		0			0
$553$	−8708.00	−	15082.7i	−0.669624	−	1.15982i
$554$	1504.00	+	2605.00i	0.115341	+	0.199776i
$555$		0			0
$556$	1400.00	−	2424.87i	0.106786	−	0.184959i
$557$	11826.0			0.899612			0.449806	−	0.893126i	$-0.351493\pi$
$557$	11826.0			0.899612			0.449806	+	0.893126i	$0.351493\pi$
$558$		0			0
$559$	7072.00			0.535087
$560$	−560.000	+	969.948i	−0.0422577	+	0.0731925i
$561$		0			0
$562$	−1308.00	−	2265.52i	−0.0981755	−	0.170045i
$563$	−1476.00	−	2556.51i	−0.110490	−	0.191375i	0.805478	−	0.592626i	$-0.201909\pi$
$563$	−1476.00	−	2556.51i	−0.110490	−	0.191375i	−0.915968	+	0.401251i	$0.868575\pi$
$564$		0			0
$565$	−555.000	+	961.288i	−0.0413257	+	0.0715782i
$566$	−11864.0			−0.881062
$567$		0			0
$568$	7776.00			0.574426
$569$	1542.00	−	2670.82i	0.113610	−	0.196778i	−0.803613	−	0.595152i	$-0.797092\pi$
$569$	1542.00	−	2670.82i	0.113610	−	0.196778i	0.917223	+	0.398374i	$0.130425\pi$
$570$		0			0
$571$	2378.00	+	4118.82i	0.174284	+	0.301869i	0.939913	−	0.341413i	$-0.110906\pi$
$571$	2378.00	+	4118.82i	0.174284	+	0.301869i	−0.765629	+	0.643282i	$0.777572\pi$
$572$	816.000	+	1413.35i	0.0596480	+	0.103313i
$573$		0			0
$574$	−6720.00	+	11639.4i	−0.488654	+	0.846374i
$575$	−3000.00			−0.217580
$576$		0			0
$577$	−11014.0			−0.794660			−0.397330	−	0.917676i	$-0.630063\pi$
$577$	−11014.0			−0.794660			−0.397330	+	0.917676i	$0.630063\pi$
$578$	−1171.00	+	2028.23i	−0.0842685	+	0.145957i
$579$		0			0
$580$	1260.00	+	2182.38i	0.0902046	+	0.156239i
$581$	2772.00	+	4801.24i	0.197938	+	0.342839i
$582$		0			0
$583$	774.000	−	1340.61i	0.0549842	−	0.0952355i
$584$	3760.00			0.266421
$585$		0			0
$586$	−10452.0			−0.736806
$587$	−426.000	+	737.854i	−0.0299538	+	0.0518816i	−0.880614	−	0.473835i	$-0.842869\pi$
$587$	−426.000	+	737.854i	−0.0299538	+	0.0518816i	0.850660	+	0.525716i	$0.176203\pi$
$588$		0			0
$589$	5368.00	+	9297.65i	0.375526	+	0.650429i
$590$	2670.00	+	4624.58i	0.186309	+	0.322696i
$591$		0			0
$592$	2432.00	−	4212.35i	0.168842	−	0.292443i
$593$	−15546.0			−1.07656			−0.538278	−	0.842767i	$-0.680925\pi$
$593$	−15546.0			−1.07656			−0.538278	+	0.842767i	$0.680925\pi$
$594$		0			0
$595$	−5460.00			−0.376199
$596$	228.000	−	394.908i	0.0156699	−	0.0271410i
$597$		0			0
$598$	8160.00	+	14133.5i	0.558005	+	0.966494i
$599$	−4308.00	−	7461.67i	−0.293857	−	0.508975i	0.680862	−	0.732412i	$-0.261605\pi$
$599$	−4308.00	−	7461.67i	−0.293857	−	0.508975i	−0.974718	+	0.223437i	$0.928272\pi$
$600$		0			0
$601$	−8755.00	+	15164.1i	−0.594216	+	1.02921i	0.399441	+	0.916759i	$0.369204\pi$
$601$	−8755.00	+	15164.1i	−0.594216	+	1.02921i	−0.993657	+	0.112454i	$0.964129\pi$
$602$	2912.00			0.197150
$603$		0			0
$604$	4256.00			0.286712
$605$	3237.50	−	5607.51i	0.217559	−	0.376823i
$606$		0			0
$607$	6947.00	+	12032.6i	0.464531	+	0.804590i	0.999180	−	0.0404833i	$-0.0128898\pi$
$607$	6947.00	+	12032.6i	0.464531	+	0.804590i	−0.534650	+	0.845074i	$0.679556\pi$
$608$	−704.000	−	1219.36i	−0.0469588	−	0.0813351i
$609$		0			0
$610$	−1810.00	+	3135.01i	−0.120139	+	0.208087i
$611$	−40800.0			−2.70146
$612$		0			0
$613$	−6496.00			−0.428011			−0.214006	−	0.976832i	$-0.568651\pi$
$613$	−6496.00			−0.428011			−0.214006	+	0.976832i	$0.568651\pi$
$614$	−4448.00	+	7704.16i	−0.292356	+	0.506376i
$615$		0			0
$616$	336.000	+	581.969i	0.0219770	+	0.0380653i
$617$	−285.000	−	493.634i	−0.0185959	−	0.0322090i	0.856578	−	0.516018i	$-0.172586\pi$
$617$	−285.000	−	493.634i	−0.0185959	−	0.0322090i	−0.875174	+	0.483809i	$0.839253\pi$
$618$		0			0
$619$	1070.00	−	1853.29i	0.0694781	−	0.120340i	−0.829194	−	0.558962i	$-0.811200\pi$
$619$	1070.00	−	1853.29i	0.0694781	−	0.120340i	0.898672	+	0.438622i	$0.144533\pi$
$620$	−4880.00			−0.316106
$621$		0			0
$622$	18264.0			1.17736
$623$	−6804.00	+	11784.9i	−0.437555	+	0.757867i
$624$		0			0
$625$	−312.500	−	541.266i	−0.0200000	−	0.0346410i
$626$	2170.00	+	3758.55i	0.138547	+	0.239971i
$627$		0			0
$628$	3896.00	−	6748.07i	0.247559	−	0.428786i
$629$	23712.0			1.50312
$630$		0			0
$631$	14660.0			0.924890			0.462445	−	0.886648i	$-0.346972\pi$
$631$	14660.0			0.924890			0.462445	+	0.886648i	$0.346972\pi$
$632$	−4976.00	+	8618.68i	−0.313188	+	0.542457i
$633$		0			0
$634$	7674.00	+	13291.8i	0.480715	+	0.832623i
$635$	4765.00	+	8253.22i	0.297785	+	0.515778i
$636$		0			0
$637$	4998.00	−	8656.79i	0.310876	−	0.538453i
$638$	1512.00			0.0938255
$639$		0			0
$640$	640.000			0.0395285
$641$	−228.000	+	394.908i	−0.0140491	+	0.0243337i	−0.872964	−	0.487784i	$-0.837805\pi$
$641$	−228.000	+	394.908i	−0.0140491	+	0.0243337i	0.858915	+	0.512118i	$0.171139\pi$
$642$		0			0
$643$	11726.0	+	20310.0i	0.719173	+	1.24564i	0.961328	+	0.275407i	$0.0888125\pi$
$643$	11726.0	+	20310.0i	0.719173	+	1.24564i	−0.242155	+	0.970238i	$0.577854\pi$
$644$	3360.00	+	5819.69i	0.205594	+	0.356099i
$645$		0			0
$646$	3432.00	−	5944.40i	0.209025	−	0.362042i
$647$	−7224.00			−0.438956			−0.219478	−	0.975617i	$-0.570435\pi$
$647$	−7224.00			−0.438956			−0.219478	+	0.975617i	$0.570435\pi$
$648$		0			0
$649$	3204.00			0.193787
$650$	−1700.00	+	2944.49i	−0.102584	+	0.177680i
$651$		0			0
$652$	−4120.00	−	7136.05i	−0.247472	−	0.428634i
$653$	9573.00	+	16580.9i	0.573691	+	0.993663i	0.996182	+	0.0872955i	$0.0278224\pi$
$653$	9573.00	+	16580.9i	0.573691	+	0.993663i	−0.422491	+	0.906367i	$0.638844\pi$
$654$		0			0
$655$	7185.00	−	12444.8i	0.428612	−	0.742379i
$656$	7680.00			0.457094
$657$		0			0
$658$	−16800.0			−0.995338
$659$	13905.0	−	24084.2i	0.821945	−	1.42365i	−0.0822865	−	0.996609i	$-0.526222\pi$
$659$	13905.0	−	24084.2i	0.821945	−	1.42365i	0.904232	−	0.427042i	$-0.140444\pi$
$660$		0			0
$661$	15299.0	+	26498.6i	0.900245	+	1.55927i	0.827175	+	0.561944i	$0.189946\pi$
$661$	15299.0	+	26498.6i	0.900245	+	1.55927i	0.0730698	+	0.997327i	$0.476720\pi$
$662$	−9596.00	−	16620.8i	−0.563382	−	0.975807i
$663$		0			0
$664$	1584.00	−	2743.57i	0.0925770	−	0.160348i
$665$	3080.00			0.179605
$666$		0			0
$667$	15120.0			0.877734
$668$	−2496.00	+	4323.20i	−0.144571	+	0.250404i
$669$		0			0
$670$	1340.00	+	2320.95i	0.0772667	+	0.133830i
$671$	1086.00	+	1881.01i	0.0624807	+	0.108220i
$672$		0			0
$673$	1889.00	−	3271.84i	0.108196	−	0.187400i	−0.806844	−	0.590765i	$-0.798826\pi$
$673$	1889.00	−	3271.84i	0.108196	−	0.187400i	0.915039	+	0.403365i	$0.132159\pi$
$674$	24316.0			1.38964
$675$		0			0
$676$	9708.00			0.552344
$677$	−13599.0	+	23554.2i	−0.772012	+	1.33716i	0.164447	+	0.986386i	$0.447416\pi$
$677$	−13599.0	+	23554.2i	−0.772012	+	1.33716i	−0.936459	+	0.350778i	$0.885917\pi$
$678$		0			0
$679$	322.000	+	557.720i	0.0181992	+	0.0315219i
$680$	1560.00	+	2702.00i	0.0879754	+	0.152378i
$681$		0			0
$682$	−1464.00	+	2535.72i	−0.0821986	+	0.142372i
$683$	−32316.0			−1.81045			−0.905225	−	0.424933i	$-0.860298\pi$
$683$	−32316.0			−1.81045			−0.905225	+	0.424933i	$0.860298\pi$
$684$		0			0
$685$	3990.00			0.222555
$686$	6860.00	−	11881.9i	0.381802	−	0.661300i
$687$		0			0
$688$	−832.000	−	1441.07i	−0.0461042	−	0.0798548i
$689$	−8772.00	−	15193.5i	−0.485031	−	0.840099i
$690$		0			0
$691$	−14662.0	+	25395.3i	−0.807191	+	1.39810i	0.107611	+	0.994193i	$0.465680\pi$
$691$	−14662.0	+	25395.3i	−0.807191	+	1.39810i	−0.914802	+	0.403902i	$0.867654\pi$
$692$	4584.00			0.251817
$693$		0			0
$694$	−20640.0			−1.12894
$695$	1750.00	−	3031.09i	0.0955126	−	0.165433i
$696$		0			0
$697$	18720.0	+	32424.0i	1.01732	+	1.76205i
$698$	2158.00	+	3737.77i	0.117022	+	0.202688i
$699$		0			0
$700$	−700.000	+	1212.44i	−0.0377964	+	0.0654654i
$701$	−22782.0			−1.22748			−0.613741	−	0.789508i	$-0.710336\pi$
$701$	−22782.0			−1.22748			−0.613741	+	0.789508i	$0.710336\pi$
$702$		0			0
$703$	−13376.0			−0.717618
$704$	192.000	−	332.554i	0.0102788	−	0.0178034i
$705$		0			0
$706$	−330.000	−	571.577i	−0.0175917	−	0.0304697i
$707$	−10542.0	−	18259.3i	−0.560782	−	0.971303i
$708$		0			0
$709$	−13027.0	+	22563.4i	−0.690041	+	1.19519i	0.281783	+	0.959478i	$0.409074\pi$
$709$	−13027.0	+	22563.4i	−0.690041	+	1.19519i	−0.971824	+	0.235708i	$0.924259\pi$
$710$	9720.00			0.513782
$711$		0			0
$712$	7776.00			0.409295
$713$	−14640.0	+	25357.2i	−0.768965	+	1.33189i
$714$		0			0
$715$	1020.00	+	1766.69i	0.0533508	+	0.0924063i
$716$	−2292.00	−	3969.86i	−0.119631	−	0.207208i
$717$		0			0
$718$	−8664.00	+	15006.5i	−0.450331	+	0.779996i
$719$	5976.00			0.309968			0.154984	−	0.987917i	$-0.450467\pi$
$719$	5976.00			0.309968			0.154984	+	0.987917i	$0.450467\pi$
$720$		0			0
$721$	−20636.0			−1.06592
$722$	4923.00	−	8526.89i	0.253761	−	0.439526i
$723$		0			0
$724$	236.000	+	408.764i	0.0121145	+	0.0209829i
$725$	1575.00	+	2727.98i	0.0806814	+	0.139744i
$726$		0			0
$727$	2555.00	−	4425.39i	0.130343	−	0.225762i	−0.793466	−	0.608615i	$-0.791725\pi$
$727$	2555.00	−	4425.39i	0.130343	−	0.225762i	0.923809	+	0.382854i	$0.125059\pi$
$728$	7616.00			0.387730
$729$		0			0
$730$	4700.00			0.238294
$731$	4056.00	−	7025.20i	0.205221	−	0.355453i
$732$		0			0
$733$	−8668.00	−	15013.4i	−0.436780	−	0.756525i	0.560659	−	0.828047i	$-0.310548\pi$
$733$	−8668.00	−	15013.4i	−0.436780	−	0.756525i	−0.997439	+	0.0715214i	$0.977215\pi$
$734$	−3782.00	−	6550.62i	−0.190186	−	0.329411i
$735$		0			0
$736$	1920.00	−	3325.54i	0.0961578	−	0.166550i
$737$	1608.00			0.0803683
$738$		0			0
$739$	−13660.0			−0.679961			−0.339981	−	0.940432i	$-0.610420\pi$
$739$	−13660.0			−0.679961			−0.339981	+	0.940432i	$0.610420\pi$
$740$	3040.00	−	5265.43i	0.151017	−	0.261569i
$741$		0			0
$742$	−3612.00	−	6256.17i	−0.178707	−	0.309530i
$743$	660.000	+	1143.15i	0.0325882	+	0.0564445i	0.881860	−	0.471512i	$-0.156292\pi$
$743$	660.000	+	1143.15i	0.0325882	+	0.0564445i	−0.849271	+	0.527957i	$0.822958\pi$
$744$		0			0
$745$	285.000	−	493.634i	0.0140156	−	0.0242757i
$746$	22552.0			1.10682
$747$		0			0
$748$	1872.00			0.0915068
$749$	6468.00	−	11202.9i	0.315535	−	0.546522i
$750$		0			0
$751$	−7930.00	−	13735.2i	−0.385313	−	0.667381i	0.606500	−	0.795084i	$-0.292573\pi$
$751$	−7930.00	−	13735.2i	−0.385313	−	0.667381i	−0.991813	+	0.127702i	$0.959240\pi$
$752$	4800.00	+	8313.84i	0.232763	+	0.403158i
$753$		0			0
$754$	8568.00	−	14840.2i	0.413830	−	0.716775i
$755$	5320.00			0.256443
$756$		0			0
$757$	22160.0			1.06396			0.531981	−	0.846756i	$-0.321448\pi$
$757$	22160.0			1.06396			0.531981	+	0.846756i	$0.321448\pi$
$758$	−980.000	+	1697.41i	−0.0469594	+	0.0813360i
$759$		0			0
$760$	−880.000	−	1524.20i	−0.0420013	−	0.0727483i
$761$	6558.00	+	11358.8i	0.312388	+	0.541072i	0.978879	−	0.204441i	$-0.0655377\pi$
$761$	6558.00	+	11358.8i	0.312388	+	0.541072i	−0.666491	+	0.745513i	$0.732204\pi$
$762$		0			0
$763$	−4886.00	+	8462.80i	−0.231828	+	0.401539i
$764$	−6768.00			−0.320494
$765$		0			0
$766$	8400.00			0.396220
$767$	18156.0	−	31447.1i	0.854726	−	1.48043i
$768$		0			0
$769$	−16423.0	−	28445.5i	−0.770128	−	1.33390i	−0.937492	−	0.348006i	$-0.886859\pi$
$769$	−16423.0	−	28445.5i	−0.770128	−	1.33390i	0.167364	−	0.985895i	$-0.446474\pi$
$770$	420.000	+	727.461i	0.0196568	+	0.0340466i
$771$		0			0
$772$	−6700.00	+	11604.7i	−0.312355	+	0.541015i
$773$	11982.0			0.557520			0.278760	−	0.960361i	$-0.410077\pi$
$773$	11982.0			0.557520			0.278760	+	0.960361i	$0.410077\pi$
$774$		0			0
$775$	−6100.00			−0.282734
$776$	184.000	−	318.697i	0.00851188	−	0.0147430i
$777$		0			0
$778$	13338.0	+	23102.1i	0.614641	+	1.06459i
$779$	−10560.0	−	18290.5i	−0.485688	−	0.841237i
$780$		0			0
$781$	2916.00	−	5050.66i	0.133601	−	0.231404i
$782$	18720.0

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:	\( N \)	\(=\)	\( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	810.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(-2.50000 - 4.33013i) q^{5} +(-7.00000 + 12.1244i) q^{7} +8.00000 q^{8} +O(q^{10})\)	\(q+(-1.00000 + 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(-2.50000 - 4.33013i) q^{5} +(-7.00000 + 12.1244i) q^{7} +8.00000 q^{8} +10.0000 q^{10} +(3.00000 - 5.19615i) q^{11} +(-34.0000 - 58.8897i) q^{13} +(-14.0000 - 24.2487i) q^{14} +(-8.00000 + 13.8564i) q^{16} -78.0000 q^{17} +44.0000 q^{19} +(-10.0000 + 17.3205i) q^{20} +(6.00000 + 10.3923i) q^{22} +(60.0000 + 103.923i) q^{23} +(-12.5000 + 21.6506i) q^{25} +136.000 q^{26} +56.0000 q^{28} +(63.0000 - 109.119i) q^{29} +(122.000 + 211.310i) q^{31} +(-16.0000 - 27.7128i) q^{32} +(78.0000 - 135.100i) q^{34} +70.0000 q^{35} -304.000 q^{37} +(-44.0000 + 76.2102i) q^{38} +(-20.0000 - 34.6410i) q^{40} +(-240.000 - 415.692i) q^{41} +(-52.0000 + 90.0666i) q^{43} -24.0000 q^{44} -240.000 q^{46} +(300.000 - 519.615i) q^{47} +(73.5000 + 127.306i) q^{49} +(-25.0000 - 43.3013i) q^{50} +(-136.000 + 235.559i) q^{52} +258.000 q^{53} -30.0000 q^{55} +(-56.0000 + 96.9948i) q^{56} +(126.000 + 218.238i) q^{58} +(267.000 + 462.458i) q^{59} +(-181.000 + 313.501i) q^{61} -488.000 q^{62} +64.0000 q^{64} +(-170.000 + 294.449i) q^{65} +(134.000 + 232.095i) q^{67} +(156.000 + 270.200i) q^{68} +(-70.0000 + 121.244i) q^{70} +972.000 q^{71} +470.000 q^{73} +(304.000 - 526.543i) q^{74} +(-88.0000 - 152.420i) q^{76} +(42.0000 + 72.7461i) q^{77} +(-622.000 + 1077.34i) q^{79} +80.0000 q^{80} +960.000 q^{82} +(198.000 - 342.946i) q^{83} +(195.000 + 337.750i) q^{85} +(-104.000 - 180.133i) q^{86} +(24.0000 - 41.5692i) q^{88} +972.000 q^{89} +952.000 q^{91} +(240.000 - 415.692i) q^{92} +(600.000 + 1039.23i) q^{94} +(-110.000 - 190.526i) q^{95} +(23.0000 - 39.8372i) q^{97} -294.000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 4 q^{4} - 5 q^{5} - 14 q^{7} + 16 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 - 4 * q^4 - 5 * q^5 - 14 * q^7 + 16 * q^8	\( 2 q - 2 q^{2} - 4 q^{4} - 5 q^{5} - 14 q^{7} + 16 q^{8} + 20 q^{10} + 6 q^{11} - 68 q^{13} - 28 q^{14} - 16 q^{16} - 156 q^{17} + 88 q^{19} - 20 q^{20} + 12 q^{22} + 120 q^{23} - 25 q^{25} + 272 q^{26} + 112 q^{28} + 126 q^{29} + 244 q^{31} - 32 q^{32} + 156 q^{34} + 140 q^{35} - 608 q^{37} - 88 q^{38} - 40 q^{40} - 480 q^{41} - 104 q^{43} - 48 q^{44} - 480 q^{46} + 600 q^{47} + 147 q^{49} - 50 q^{50} - 272 q^{52} + 516 q^{53} - 60 q^{55} - 112 q^{56} + 252 q^{58} + 534 q^{59} - 362 q^{61} - 976 q^{62} + 128 q^{64} - 340 q^{65} + 268 q^{67} + 312 q^{68} - 140 q^{70} + 1944 q^{71} + 940 q^{73} + 608 q^{74} - 176 q^{76} + 84 q^{77} - 1244 q^{79} + 160 q^{80} + 1920 q^{82} + 396 q^{83} + 390 q^{85} - 208 q^{86} + 48 q^{88} + 1944 q^{89} + 1904 q^{91} + 480 q^{92} + 1200 q^{94} - 220 q^{95} + 46 q^{97} - 588 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 - 4 * q^4 - 5 * q^5 - 14 * q^7 + 16 * q^8 + 20 * q^10 + 6 * q^11 - 68 * q^13 - 28 * q^14 - 16 * q^16 - 156 * q^17 + 88 * q^19 - 20 * q^20 + 12 * q^22 + 120 * q^23 - 25 * q^25 + 272 * q^26 + 112 * q^28 + 126 * q^29 + 244 * q^31 - 32 * q^32 + 156 * q^34 + 140 * q^35 - 608 * q^37 - 88 * q^38 - 40 * q^40 - 480 * q^41 - 104 * q^43 - 48 * q^44 - 480 * q^46 + 600 * q^47 + 147 * q^49 - 50 * q^50 - 272 * q^52 + 516 * q^53 - 60 * q^55 - 112 * q^56 + 252 * q^58 + 534 * q^59 - 362 * q^61 - 976 * q^62 + 128 * q^64 - 340 * q^65 + 268 * q^67 + 312 * q^68 - 140 * q^70 + 1944 * q^71 + 940 * q^73 + 608 * q^74 - 176 * q^76 + 84 * q^77 - 1244 * q^79 + 160 * q^80 + 1920 * q^82 + 396 * q^83 + 390 * q^85 - 208 * q^86 + 48 * q^88 + 1944 * q^89 + 1904 * q^91 + 480 * q^92 + 1200 * q^94 - 220 * q^95 + 46 * q^97 - 588 * q^98

Introduction

L-functions

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