LMFDB - Embedded newform 700.2.g.g.251.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [700,2,Mod(251,700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(700, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("700.251");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 700 = 2^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	700.g (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$5.58952814149$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 140)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			251.4
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	700.251
Dual form			700.2.g.g.251.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.36603 + 0.366025i) q^{2} +1.73205 q^{3} +(1.73205 + 1.00000i) q^{4} +(2.36603 + 0.633975i) q^{6} +(1.73205 - 2.00000i) q^{7} +(2.00000 + 2.00000i) q^{8} +O(q^{10})$	\(q+(1.36603 + 0.366025i) q^{2} +1.73205 q^{3} +(1.73205 + 1.00000i) q^{4} +(2.36603 + 0.633975i) q^{6} +(1.73205 - 2.00000i) q^{7} +(2.00000 + 2.00000i) q^{8} +3.73205i q^{11} +(3.00000 + 1.73205i) q^{12} -6.46410i q^{13} +(3.09808 - 2.09808i) q^{14} +(2.00000 + 3.46410i) q^{16} -0.464102i q^{17} -6.00000 q^{19} +(3.00000 - 3.46410i) q^{21} +(-1.36603 + 5.09808i) q^{22} +5.46410i q^{23} +(3.46410 + 3.46410i) q^{24} +(2.36603 - 8.83013i) q^{26} -5.19615 q^{27} +(5.00000 - 1.73205i) q^{28} -5.92820 q^{29} +6.00000 q^{31} +(1.46410 + 5.46410i) q^{32} +6.46410i q^{33} +(0.169873 - 0.633975i) q^{34} +2.53590 q^{37} +(-8.19615 - 2.19615i) q^{38} -11.1962i q^{39} +3.46410i q^{41} +(5.36603 - 3.63397i) q^{42} -2.00000i q^{43} +(-3.73205 + 6.46410i) q^{44} +(-2.00000 + 7.46410i) q^{46} -1.73205 q^{47} +(3.46410 + 6.00000i) q^{48} +(-1.00000 - 6.92820i) q^{49} -0.803848i q^{51} +(6.46410 - 11.1962i) q^{52} -2.00000 q^{53} +(-7.09808 - 1.90192i) q^{54} +(7.46410 - 0.535898i) q^{56} -10.3923 q^{57} +(-8.09808 - 2.16987i) q^{58} -3.46410 q^{59} +2.53590i q^{61} +(8.19615 + 2.19615i) q^{62} +8.00000i q^{64} +(-2.36603 + 8.83013i) q^{66} -3.46410i q^{67} +(0.464102 - 0.803848i) q^{68} +9.46410i q^{69} +0.535898i q^{71} +0.928203i q^{73} +(3.46410 + 0.928203i) q^{74} +(-10.3923 - 6.00000i) q^{76} +(7.46410 + 6.46410i) q^{77} +(4.09808 - 15.2942i) q^{78} -2.66025i q^{79} -9.00000 q^{81} +(-1.26795 + 4.73205i) q^{82} +8.53590 q^{83} +(8.66025 - 3.00000i) q^{84} +(0.732051 - 2.73205i) q^{86} -10.2679 q^{87} +(-7.46410 + 7.46410i) q^{88} +9.46410i q^{89} +(-12.9282 - 11.1962i) q^{91} +(-5.46410 + 9.46410i) q^{92} +10.3923 q^{93} +(-2.36603 - 0.633975i) q^{94} +(2.53590 + 9.46410i) q^{96} -7.39230i q^{97} +(1.16987 - 9.83013i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 6 q^{6} + 8 q^{8}+O(q^{10}) $ 4 * q + 2 * q^2 + 6 * q^6 + 8 * q^8	$ 4 q + 2 q^{2} + 6 q^{6} + 8 q^{8} + 12 q^{12} + 2 q^{14} + 8 q^{16} - 24 q^{19} + 12 q^{21} - 2 q^{22} + 6 q^{26} + 20 q^{28} + 4 q^{29} + 24 q^{31} - 8 q^{32} + 18 q^{34} + 24 q^{37} - 12 q^{38} + 18 q^{42} - 8 q^{44} - 8 q^{46} - 4 q^{49} + 12 q^{52} - 8 q^{53} - 18 q^{54} + 16 q^{56} - 22 q^{58} + 12 q^{62} - 6 q^{66} - 12 q^{68} + 16 q^{77} + 6 q^{78} - 36 q^{81} - 12 q^{82} + 48 q^{83} - 4 q^{86} - 48 q^{87} - 16 q^{88} - 24 q^{91} - 8 q^{92} - 6 q^{94} + 24 q^{96} + 22 q^{98}+O(q^{100}) $ 4 * q + 2 * q^2 + 6 * q^6 + 8 * q^8 + 12 * q^12 + 2 * q^14 + 8 * q^16 - 24 * q^19 + 12 * q^21 - 2 * q^22 + 6 * q^26 + 20 * q^28 + 4 * q^29 + 24 * q^31 - 8 * q^32 + 18 * q^34 + 24 * q^37 - 12 * q^38 + 18 * q^42 - 8 * q^44 - 8 * q^46 - 4 * q^49 + 12 * q^52 - 8 * q^53 - 18 * q^54 + 16 * q^56 - 22 * q^58 + 12 * q^62 - 6 * q^66 - 12 * q^68 + 16 * q^77 + 6 * q^78 - 36 * q^81 - 12 * q^82 + 48 * q^83 - 4 * q^86 - 48 * q^87 - 16 * q^88 - 24 * q^91 - 8 * q^92 - 6 * q^94 + 24 * q^96 + 22 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/700\mathbb{Z}\right)^\times$.

$n$	$101$	$351$	$477$
$\chi(n)$	$-1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	1.36603	+	0.366025i	0.965926	+	0.258819i
$2$	1.36603	+	0.366025i	0.965926	+	0.258819i
$3$	1.73205			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$3$	1.73205			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$4$	1.73205	+	1.00000i	0.866025	+	0.500000i
$5$		0			0
$5$		0			0
$6$	2.36603	+	0.633975i	0.965926	+	0.258819i
$7$	1.73205	−	2.00000i	0.654654	−	0.755929i
$7$	1.73205	−	2.00000i	0.654654	−	0.755929i
$8$	2.00000	+	2.00000i	0.707107	+	0.707107i
$9$		0			0
$10$		0			0
$11$			3.73205i			1.12526i	0.826710	+	0.562628i	$0.190210\pi$
$11$			3.73205i			1.12526i	−0.826710	+	0.562628i	$0.809790\pi$
$12$	3.00000	+	1.73205i	0.866025	+	0.500000i
$13$		−	6.46410i		−	1.79282i	−0.443227	−	0.896410i	$-0.646166\pi$
$13$		−	6.46410i		−	1.79282i	0.443227	−	0.896410i	$-0.353834\pi$
$14$	3.09808	−	2.09808i	0.827996	−	0.560734i
$15$		0			0
$16$	2.00000	+	3.46410i	0.500000	+	0.866025i
$17$		−	0.464102i		−	0.112561i	−0.998415	−	0.0562806i	$-0.982076\pi$
$17$		−	0.464102i		−	0.112561i	0.998415	−	0.0562806i	$-0.0179241\pi$
$18$		0			0
$19$	−6.00000			−1.37649			−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000			−1.37649			−0.688247	+	0.725476i	$0.741620\pi$
$20$		0			0
$21$	3.00000	−	3.46410i	0.654654	−	0.755929i
$22$	−1.36603	+	5.09808i	−0.291238	+	1.08691i
$23$			5.46410i			1.13934i	0.821872	+	0.569672i	$0.192930\pi$
$23$			5.46410i			1.13934i	−0.821872	+	0.569672i	$0.807070\pi$
$24$	3.46410	+	3.46410i	0.707107	+	0.707107i
$25$		0			0
$26$	2.36603	−	8.83013i	0.464016	−	1.73173i
$27$	−5.19615			−1.00000
$28$	5.00000	−	1.73205i	0.944911	−	0.327327i
$29$	−5.92820			−1.10084			−0.550420	−	0.834888i	$-0.685532\pi$
$29$	−5.92820			−1.10084			−0.550420	+	0.834888i	$0.685532\pi$
$30$		0			0
$31$	6.00000			1.07763			0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000			1.07763			0.538816	+	0.842424i	$0.318872\pi$
$32$	1.46410	+	5.46410i	0.258819	+	0.965926i
$33$			6.46410i			1.12526i
$34$	0.169873	−	0.633975i	0.0291330	−	0.108726i
$35$		0			0
$36$		0			0
$37$	2.53590			0.416899			0.208450	−	0.978033i	$-0.433158\pi$
$37$	2.53590			0.416899			0.208450	+	0.978033i	$0.433158\pi$
$38$	−8.19615	−	2.19615i	−1.32959	−	0.356263i
$39$		−	11.1962i		−	1.79282i
$40$		0			0
$41$			3.46410i			0.541002i	0.962720	+	0.270501i	$0.0871893\pi$
$41$			3.46410i			0.541002i	−0.962720	+	0.270501i	$0.912811\pi$
$42$	5.36603	−	3.63397i	0.827996	−	0.560734i
$43$		−	2.00000i		−	0.304997i	−0.988304	−	0.152499i	$-0.951268\pi$
$43$		−	2.00000i		−	0.304997i	0.988304	−	0.152499i	$-0.0487319\pi$
$44$	−3.73205	+	6.46410i	−0.562628	+	0.974500i
$45$		0			0
$46$	−2.00000	+	7.46410i	−0.294884	+	1.10052i
$47$	−1.73205			−0.252646			−0.126323	−	0.991989i	$-0.540318\pi$
$47$	−1.73205			−0.252646			−0.126323	+	0.991989i	$0.540318\pi$
$48$	3.46410	+	6.00000i	0.500000	+	0.866025i
$49$	−1.00000	−	6.92820i	−0.142857	−	0.989743i
$50$		0			0
$51$		−	0.803848i		−	0.112561i
$52$	6.46410	−	11.1962i	0.896410	−	1.55263i
$53$	−2.00000			−0.274721			−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000			−0.274721			−0.137361	+	0.990521i	$0.543862\pi$
$54$	−7.09808	−	1.90192i	−0.965926	−	0.258819i
$55$		0			0
$56$	7.46410	−	0.535898i	0.997433	−	0.0716124i
$57$	−10.3923			−1.37649
$58$	−8.09808	−	2.16987i	−1.06333	−	0.284918i
$59$	−3.46410			−0.450988			−0.225494	−	0.974245i	$-0.572400\pi$
$59$	−3.46410			−0.450988			−0.225494	+	0.974245i	$0.572400\pi$
$60$		0			0
$61$			2.53590i			0.324689i	0.986734	+	0.162344i	$0.0519055\pi$
$61$			2.53590i			0.324689i	−0.986734	+	0.162344i	$0.948094\pi$
$62$	8.19615	+	2.19615i	1.04091	+	0.278912i
$63$		0			0
$64$			8.00000i			1.00000i
$65$		0			0
$66$	−2.36603	+	8.83013i	−0.291238	+	1.08691i
$67$		−	3.46410i		−	0.423207i	−0.977356	−	0.211604i	$-0.932131\pi$
$67$		−	3.46410i		−	0.423207i	0.977356	−	0.211604i	$-0.0678686\pi$
$68$	0.464102	−	0.803848i	0.0562806	−	0.0974808i
$69$			9.46410i			1.13934i
$70$		0			0
$71$			0.535898i			0.0635994i	0.999494	+	0.0317997i	$0.0101239\pi$
$71$			0.535898i			0.0635994i	−0.999494	+	0.0317997i	$0.989876\pi$
$72$		0			0
$73$			0.928203i			0.108638i	0.998524	+	0.0543190i	$0.0172988\pi$
$73$			0.928203i			0.108638i	−0.998524	+	0.0543190i	$0.982701\pi$
$74$	3.46410	+	0.928203i	0.402694	+	0.107901i
$75$		0			0
$76$	−10.3923	−	6.00000i	−1.19208	−	0.688247i
$77$	7.46410	+	6.46410i	0.850613	+	0.736653i
$78$	4.09808	−	15.2942i	0.464016	−	1.73173i
$79$		−	2.66025i		−	0.299302i	−0.988739	−	0.149651i	$-0.952185\pi$
$79$		−	2.66025i		−	0.299302i	0.988739	−	0.149651i	$-0.0478150\pi$
$80$		0			0
$81$	−9.00000			−1.00000
$82$	−1.26795	+	4.73205i	−0.140022	+	0.522568i
$83$	8.53590			0.936937			0.468468	−	0.883480i	$-0.344806\pi$
$83$	8.53590			0.936937			0.468468	+	0.883480i	$0.344806\pi$
$84$	8.66025	−	3.00000i	0.944911	−	0.327327i
$85$		0			0
$86$	0.732051	−	2.73205i	0.0789391	−	0.294605i
$87$	−10.2679			−1.10084
$88$	−7.46410	+	7.46410i	−0.795676	+	0.795676i
$89$			9.46410i			1.00319i	0.865102	+	0.501596i	$0.167254\pi$
$89$			9.46410i			1.00319i	−0.865102	+	0.501596i	$0.832746\pi$
$90$		0			0
$91$	−12.9282	−	11.1962i	−1.35524	−	1.17368i
$92$	−5.46410	+	9.46410i	−0.569672	+	0.986701i
$93$	10.3923			1.07763
$94$	−2.36603	−	0.633975i	−0.244037	−	0.0653895i
$95$		0			0
$96$	2.53590	+	9.46410i	0.258819	+	0.965926i
$97$		−	7.39230i		−	0.750575i	−0.926908	−	0.375287i	$-0.877544\pi$
$97$		−	7.39230i		−	0.750575i	0.926908	−	0.375287i	$-0.122456\pi$
$98$	1.16987	−	9.83013i	0.118175	−	0.992993i
$99$		0			0
$100$		0			0
$101$		−	8.53590i		−	0.849354i	−0.905345	−	0.424677i	$-0.860388\pi$
$101$		−	8.53590i		−	0.849354i	0.905345	−	0.424677i	$-0.139612\pi$
$102$	0.294229	−	1.09808i	0.0291330	−	0.108726i
$103$	−17.1962			−1.69439			−0.847194	−	0.531284i	$-0.821710\pi$
$103$	−17.1962			−1.69439			−0.847194	+	0.531284i	$0.821710\pi$
$104$	12.9282	−	12.9282i	1.26771	−	1.26771i
$105$		0			0
$106$	−2.73205	−	0.732051i	−0.265360	−	0.0711031i
$107$		−	18.3923i		−	1.77805i	−0.457857	−	0.889026i	$-0.651383\pi$
$107$		−	18.3923i		−	1.77805i	0.457857	−	0.889026i	$-0.348617\pi$
$108$	−9.00000	−	5.19615i	−0.866025	−	0.500000i
$109$	15.9282			1.52565			0.762823	−	0.646608i	$-0.223813\pi$
$109$	15.9282			1.52565			0.762823	+	0.646608i	$0.223813\pi$
$110$		0			0
$111$	4.39230			0.416899
$112$	10.3923	+	2.00000i	0.981981	+	0.188982i
$113$	1.46410			0.137731			0.0688655	−	0.997626i	$-0.478062\pi$
$113$	1.46410			0.137731			0.0688655	+	0.997626i	$0.478062\pi$
$114$	−14.1962	−	3.80385i	−1.32959	−	0.356263i
$115$		0			0
$116$	−10.2679	−	5.92820i	−0.953355	−	0.550420i
$117$		0			0
$118$	−4.73205	−	1.26795i	−0.435621	−	0.116724i
$119$	−0.928203	−	0.803848i	−0.0850883	−	0.0736886i
$120$		0			0
$121$	−2.92820			−0.266200
$122$	−0.928203	+	3.46410i	−0.0840356	+	0.313625i
$123$			6.00000i			0.541002i
$124$	10.3923	+	6.00000i	0.933257	+	0.538816i
$125$		0			0
$126$		0			0
$127$			8.53590i			0.757438i	0.925512	+	0.378719i	$0.123635\pi$
$127$			8.53590i			0.757438i	−0.925512	+	0.378719i	$0.876365\pi$
$128$	−2.92820	+	10.9282i	−0.258819	+	0.965926i
$129$		−	3.46410i		−	0.304997i
$130$		0			0
$131$	−9.46410			−0.826882			−0.413441	−	0.910531i	$-0.635673\pi$
$131$	−9.46410			−0.826882			−0.413441	+	0.910531i	$0.635673\pi$
$132$	−6.46410	+	11.1962i	−0.562628	+	0.974500i
$133$	−10.3923	+	12.0000i	−0.901127	+	1.04053i
$134$	1.26795	−	4.73205i	0.109534	−	0.408787i
$135$		0			0
$136$	0.928203	−	0.928203i	0.0795928	−	0.0795928i
$137$	0.392305			0.0335169			0.0167584	−	0.999860i	$-0.494665\pi$
$137$	0.392305			0.0335169			0.0167584	+	0.999860i	$0.494665\pi$
$138$	−3.46410	+	12.9282i	−0.294884	+	1.10052i
$139$	6.92820			0.587643			0.293821	−	0.955860i	$-0.405073\pi$
$139$	6.92820			0.587643			0.293821	+	0.955860i	$0.405073\pi$
$140$		0			0
$141$	−3.00000			−0.252646
$142$	−0.196152	+	0.732051i	−0.0164607	+	0.0614323i
$143$	24.1244			2.01738
$144$		0			0
$145$		0			0
$146$	−0.339746	+	1.26795i	−0.0281176	+	0.104936i
$147$	−1.73205	−	12.0000i	−0.142857	−	0.989743i
$148$	4.39230	+	2.53590i	0.361045	+	0.208450i
$149$	16.9282			1.38681			0.693406	−	0.720547i	$-0.256109\pi$
$149$	16.9282			1.38681			0.693406	+	0.720547i	$0.256109\pi$
$150$		0			0
$151$			4.80385i			0.390932i	0.980711	+	0.195466i	$0.0626219\pi$
$151$			4.80385i			0.390932i	−0.980711	+	0.195466i	$0.937378\pi$
$152$	−12.0000	−	12.0000i	−0.973329	−	0.973329i
$153$		0			0
$154$	7.83013	+	11.5622i	0.630970	+	0.931707i
$155$		0			0
$156$	11.1962	−	19.3923i	0.896410	−	1.55263i
$157$			19.8564i			1.58471i	0.610058	+	0.792357i	$0.291146\pi$
$157$			19.8564i			1.58471i	−0.610058	+	0.792357i	$0.708854\pi$
$158$	0.973721	−	3.63397i	0.0774650	−	0.289103i
$159$	−3.46410			−0.274721
$160$		0			0
$161$	10.9282	+	9.46410i	0.861263	+	0.745876i
$162$	−12.2942	−	3.29423i	−0.965926	−	0.258819i
$163$		−	20.7846i		−	1.62798i	−0.580881	−	0.813988i	$-0.697292\pi$
$163$		−	20.7846i		−	1.62798i	0.580881	−	0.813988i	$-0.302708\pi$
$164$	−3.46410	+	6.00000i	−0.270501	+	0.468521i
$165$		0			0
$166$	11.6603	+	3.12436i	0.905011	+	0.242497i
$167$	5.19615			0.402090			0.201045	−	0.979582i	$-0.435566\pi$
$167$	5.19615			0.402090			0.201045	+	0.979582i	$0.435566\pi$
$168$	12.9282	−	0.928203i	0.997433	−	0.0716124i
$169$	−28.7846			−2.21420
$170$		0			0
$171$		0			0
$172$	2.00000	−	3.46410i	0.152499	−	0.264135i
$173$		−	20.3205i		−	1.54494i	−0.635051	−	0.772470i	$-0.719021\pi$
$173$		−	20.3205i		−	1.54494i	0.635051	−	0.772470i	$-0.280979\pi$
$174$	−14.0263	−	3.75833i	−1.06333	−	0.284918i
$175$		0			0
$176$	−12.9282	+	7.46410i	−0.974500	+	0.562628i
$177$	−6.00000			−0.450988
$178$	−3.46410	+	12.9282i	−0.259645	+	0.969010i
$179$			14.3923i			1.07573i	0.843031	+	0.537866i	$0.180769\pi$
$179$			14.3923i			1.07573i	−0.843031	+	0.537866i	$0.819231\pi$
$180$		0			0
$181$			12.9282i			0.960946i	0.877010	+	0.480473i	$0.159535\pi$
$181$			12.9282i			0.960946i	−0.877010	+	0.480473i	$0.840465\pi$
$182$	−13.5622	−	20.0263i	−1.00530	−	1.48445i
$183$			4.39230i			0.324689i
$184$	−10.9282	+	10.9282i	−0.805638	+	0.805638i
$185$		0			0
$186$	14.1962	+	3.80385i	1.04091	+	0.278912i
$187$	1.73205			0.126660
$188$	−3.00000	−	1.73205i	−0.218797	−	0.126323i
$189$	−9.00000	+	10.3923i	−0.654654	+	0.755929i
$190$		0			0
$191$		−	3.19615i		−	0.231265i	−0.993292	−	0.115633i	$-0.963110\pi$
$191$		−	3.19615i		−	0.231265i	0.993292	−	0.115633i	$-0.0368896\pi$
$192$			13.8564i			1.00000i
$193$	2.53590			0.182538			0.0912690	−	0.995826i	$-0.470908\pi$
$193$	2.53590			0.182538			0.0912690	+	0.995826i	$0.470908\pi$
$194$	2.70577	−	10.0981i	0.194263	−	0.725000i
$195$		0			0
$196$	5.19615	−	13.0000i	0.371154	−	0.928571i
$197$	−21.3205			−1.51902			−0.759512	−	0.650494i	$-0.774562\pi$
$197$	−21.3205			−1.51902			−0.759512	+	0.650494i	$0.774562\pi$
$198$		0			0
$199$	3.46410			0.245564			0.122782	−	0.992434i	$-0.460818\pi$
$199$	3.46410			0.245564			0.122782	+	0.992434i	$0.460818\pi$
$200$		0			0
$201$		−	6.00000i		−	0.423207i
$202$	3.12436	−	11.6603i	0.219829	−	0.820413i
$203$	−10.2679	+	11.8564i	−0.720669	+	0.832157i
$204$	0.803848	−	1.39230i	0.0562806	−	0.0974808i
$205$		0			0
$206$	−23.4904	−	6.29423i	−1.63665	−	0.438540i
$207$		0			0
$208$	22.3923	−	12.9282i	1.55263	−	0.896410i
$209$		−	22.3923i		−	1.54891i
$210$		0			0
$211$		−	7.19615i		−	0.495404i	−0.968836	−	0.247702i	$-0.920325\pi$
$211$		−	7.19615i		−	0.495404i	0.968836	−	0.247702i	$-0.0796753\pi$
$212$	−3.46410	−	2.00000i	−0.237915	−	0.137361i
$213$			0.928203i			0.0635994i
$214$	6.73205	−	25.1244i	0.460194	−	1.71747i
$215$		0			0
$216$	−10.3923	−	10.3923i	−0.707107	−	0.707107i
$217$	10.3923	−	12.0000i	0.705476	−	0.814613i
$218$	21.7583	+	5.83013i	1.47366	+	0.394866i
$219$			1.60770i			0.108638i
$220$		0			0
$221$	−3.00000			−0.201802
$222$	6.00000	+	1.60770i	0.402694	+	0.107901i
$223$	−10.2679			−0.687593			−0.343796	−	0.939044i	$-0.611713\pi$
$223$	−10.2679			−0.687593			−0.343796	+	0.939044i	$0.611713\pi$
$224$	13.4641	+	6.53590i	0.899608	+	0.436698i
$225$		0			0
$226$	2.00000	+	0.535898i	0.133038	+	0.0356474i
$227$	3.33975			0.221667			0.110833	−	0.993839i	$-0.464648\pi$
$227$	3.33975			0.221667			0.110833	+	0.993839i	$0.464648\pi$
$228$	−18.0000	−	10.3923i	−1.19208	−	0.688247i
$229$		−	15.4641i		−	1.02190i	−0.859611	−	0.510948i	$-0.829294\pi$
$229$		−	15.4641i		−	1.02190i	0.859611	−	0.510948i	$-0.170706\pi$
$230$		0			0
$231$	12.9282	+	11.1962i	0.850613	+	0.736653i
$232$	−11.8564	−	11.8564i	−0.778411	−	0.778411i
$233$	22.9282			1.50208			0.751038	−	0.660259i	$-0.229553\pi$
$233$	22.9282			1.50208			0.751038	+	0.660259i	$0.229553\pi$
$234$		0			0
$235$		0			0
$236$	−6.00000	−	3.46410i	−0.390567	−	0.225494i
$237$		−	4.60770i		−	0.299302i
$238$	−0.973721	−	1.43782i	−0.0631169	−	0.0932002i
$239$			27.9808i			1.80993i	0.425491	+	0.904963i	$0.360101\pi$
$239$			27.9808i			1.80993i	−0.425491	+	0.904963i	$0.639899\pi$
$240$		0			0
$241$		−	4.39230i		−	0.282933i	−0.989943	−	0.141467i	$-0.954818\pi$
$241$		−	4.39230i		−	0.282933i	0.989943	−	0.141467i	$-0.0451818\pi$
$242$	−4.00000	−	1.07180i	−0.257130	−	0.0688977i
$243$		0			0
$244$	−2.53590	+	4.39230i	−0.162344	+	0.281189i
$245$		0			0
$246$	−2.19615	+	8.19615i	−0.140022	+	0.522568i
$247$			38.7846i			2.46781i
$248$	12.0000	+	12.0000i	0.762001	+	0.762001i
$249$	14.7846			0.936937
$250$		0			0
$251$	−1.85641			−0.117175			−0.0585877	−	0.998282i	$-0.518660\pi$
$251$	−1.85641			−0.117175			−0.0585877	+	0.998282i	$0.518660\pi$
$252$		0			0
$253$	−20.3923			−1.28205
$254$	−3.12436	+	11.6603i	−0.196040	+	0.731629i
$255$		0			0
$256$	−8.00000	+	13.8564i	−0.500000	+	0.866025i
$257$		−	6.00000i		−	0.374270i	−0.982334	−	0.187135i	$-0.940080\pi$
$257$		−	6.00000i		−	0.374270i	0.982334	−	0.187135i	$-0.0599201\pi$
$258$	1.26795	−	4.73205i	0.0789391	−	0.294605i
$259$	4.39230	−	5.07180i	0.272925	−	0.315146i
$260$		0			0
$261$		0			0
$262$	−12.9282	−	3.46410i	−0.798707	−	0.214013i
$263$			4.53590i			0.279695i	0.990173	+	0.139848i	$0.0446613\pi$
$263$			4.53590i			0.279695i	−0.990173	+	0.139848i	$0.955339\pi$
$264$	−12.9282	+	12.9282i	−0.795676	+	0.795676i
$265$		0			0
$266$	−18.5885	+	12.5885i	−1.13973	+	0.771848i
$267$			16.3923i			1.00319i
$268$	3.46410	−	6.00000i	0.211604	−	0.366508i
$269$		−	12.0000i		−	0.731653i	−0.930683	−	0.365826i	$-0.880786\pi$
$269$		−	12.0000i		−	0.731653i	0.930683	−	0.365826i	$-0.119214\pi$
$270$		0			0
$271$	−2.53590			−0.154045			−0.0770224	−	0.997029i	$-0.524541\pi$
$271$	−2.53590			−0.154045			−0.0770224	+	0.997029i	$0.524541\pi$
$272$	1.60770	−	0.928203i	0.0974808	−	0.0562806i
$273$	−22.3923	−	19.3923i	−1.35524	−	1.17368i
$274$	0.535898	+	0.143594i	0.0323748	+	0.00867480i
$275$		0			0
$276$	−9.46410	+	16.3923i	−0.569672	+	0.986701i
$277$	24.7846			1.48916			0.744581	−	0.667532i	$-0.232649\pi$
$277$	24.7846			1.48916			0.744581	+	0.667532i	$0.232649\pi$
$278$	9.46410	+	2.53590i	0.567619	+	0.152093i
$279$		0			0
$280$		0			0
$281$	5.92820			0.353647			0.176823	−	0.984243i	$-0.443418\pi$
$281$	5.92820			0.353647			0.176823	+	0.984243i	$0.443418\pi$
$282$	−4.09808	−	1.09808i	−0.244037	−	0.0653895i
$283$	−12.1244			−0.720718			−0.360359	−	0.932814i	$-0.617346\pi$
$283$	−12.1244			−0.720718			−0.360359	+	0.932814i	$0.617346\pi$
$284$	−0.535898	+	0.928203i	−0.0317997	+	0.0550787i
$285$		0			0
$286$	32.9545	+	8.83013i	1.94864	+	0.522136i
$287$	6.92820	+	6.00000i	0.408959	+	0.354169i
$288$		0			0
$289$	16.7846			0.987330
$290$		0			0
$291$		−	12.8038i		−	0.750575i
$292$	−0.928203	+	1.60770i	−0.0543190	+	0.0940832i
$293$			14.3205i			0.836613i	0.908306	+	0.418307i	$0.137376\pi$
$293$			14.3205i			0.836613i	−0.908306	+	0.418307i	$0.862624\pi$
$294$	2.02628	−	17.0263i	0.118175	−	0.992993i
$295$		0			0
$296$	5.07180	+	5.07180i	0.294792	+	0.294792i
$297$		−	19.3923i		−	1.12526i
$298$	23.1244	+	6.19615i	1.33956	+	0.358933i
$299$	35.3205			2.04264
$300$		0			0
$301$	−4.00000	−	3.46410i	−0.230556	−	0.199667i
$302$	−1.75833	+	6.56218i	−0.101181	+	0.377611i
$303$		−	14.7846i		−	0.849354i
$304$	−12.0000	−	20.7846i	−0.688247	−	1.19208i
$305$		0			0
$306$		0			0
$307$	−1.73205			−0.0988534			−0.0494267	−	0.998778i	$-0.515739\pi$
$307$	−1.73205			−0.0988534			−0.0494267	+	0.998778i	$0.515739\pi$
$308$	6.46410	+	18.6603i	0.368326	+	1.06327i
$309$	−29.7846			−1.69439
$310$		0			0
$311$	19.8564			1.12595			0.562977	−	0.826473i	$-0.309656\pi$
$311$	19.8564			1.12595			0.562977	+	0.826473i	$0.309656\pi$
$312$	22.3923	−	22.3923i	1.26771	−	1.26771i
$313$			24.4641i			1.38279i	0.722476	+	0.691396i	$0.243004\pi$
$313$			24.4641i			1.38279i	−0.722476	+	0.691396i	$0.756996\pi$
$314$	−7.26795	+	27.1244i	−0.410154	+	1.53072i
$315$		0			0
$316$	2.66025	−	4.60770i	0.149651	−	0.259203i
$317$	16.9282			0.950783			0.475391	−	0.879774i	$-0.342306\pi$
$317$	16.9282			0.950783			0.475391	+	0.879774i	$0.342306\pi$
$318$	−4.73205	−	1.26795i	−0.265360	−	0.0711031i
$319$		−	22.1244i		−	1.23873i
$320$		0			0
$321$		−	31.8564i		−	1.77805i
$322$	11.4641	+	16.9282i	0.638869	+	0.943372i
$323$			2.78461i			0.154940i
$324$	−15.5885	−	9.00000i	−0.866025	−	0.500000i
$325$		0			0
$326$	7.60770	−	28.3923i	0.421351	−	1.57250i
$327$	27.5885			1.52565
$328$	−6.92820	+	6.92820i	−0.382546	+	0.382546i
$329$	−3.00000	+	3.46410i	−0.165395	+	0.190982i
$330$		0			0
$331$		−	5.60770i		−	0.308227i	−0.988053	−	0.154113i	$-0.950748\pi$
$331$		−	5.60770i		−	0.308227i	0.988053	−	0.154113i	$-0.0492521\pi$
$332$	14.7846	+	8.53590i	0.811411	+	0.468468i
$333$		0			0
$334$	7.09808	+	1.90192i	0.388389	+	0.104069i
$335$		0			0
$336$	18.0000	+	3.46410i	0.981981	+	0.188982i
$337$		0			0			−	1.00000i	$-0.5\pi$
$337$		0			0				1.00000i	$0.5\pi$
$338$	−39.3205	−	10.5359i	−2.13875	−	0.573077i
$339$	2.53590			0.137731
$340$		0			0
$341$			22.3923i			1.21261i
$342$		0			0
$343$	−15.5885	−	10.0000i	−0.841698	−	0.539949i
$344$	4.00000	−	4.00000i	0.215666	−	0.215666i
$345$		0			0
$346$	7.43782	−	27.7583i	0.399860	−	1.49230i
$347$			14.2487i			0.764911i	0.923974	+	0.382455i	$0.124921\pi$
$347$			14.2487i			0.764911i	−0.923974	+	0.382455i	$0.875079\pi$
$348$	−17.7846	−	10.2679i	−0.953355	−	0.550420i
$349$			29.3205i			1.56949i	0.619818	+	0.784745i	$0.287206\pi$
$349$			29.3205i			1.56949i	−0.619818	+	0.784745i	$0.712794\pi$
$350$		0			0
$351$			33.5885i			1.79282i
$352$	−20.3923	+	5.46410i	−1.08691	+	0.291238i
$353$		−	13.3923i		−	0.712800i	−0.934333	−	0.356400i	$-0.884004\pi$
$353$		−	13.3923i		−	0.712800i	0.934333	−	0.356400i	$-0.115996\pi$
$354$	−8.19615	−	2.19615i	−0.435621	−	0.116724i
$355$		0			0
$356$	−9.46410	+	16.3923i	−0.501596	+	0.868790i
$357$	−1.60770	−	1.39230i	−0.0850883	−	0.0736886i
$358$	−5.26795	+	19.6603i	−0.278420	+	1.03908i
$359$		−	9.32051i		−	0.491918i	−0.969280	−	0.245959i	$-0.920897\pi$
$359$		−	9.32051i		−	0.491918i	0.969280	−	0.245959i	$-0.0791028\pi$
$360$		0			0
$361$	17.0000			0.894737
$362$	−4.73205	+	17.6603i	−0.248711	+	0.928202i
$363$	−5.07180			−0.266200
$364$	−11.1962	−	32.3205i	−0.586838	−	1.69405i
$365$		0			0
$366$	−1.60770	+	6.00000i	−0.0840356	+	0.313625i
$367$	36.1244			1.88568			0.942838	−	0.333251i	$-0.108146\pi$
$367$	36.1244			1.88568			0.942838	+	0.333251i	$0.108146\pi$
$368$	−18.9282	+	10.9282i	−0.986701	+	0.569672i
$369$		0			0
$370$		0			0
$371$	−3.46410	+	4.00000i	−0.179847	+	0.207670i
$372$	18.0000	+	10.3923i	0.933257	+	0.538816i
$373$	24.3923			1.26299			0.631493	−	0.775382i	$-0.282443\pi$
$373$	24.3923			1.26299			0.631493	+	0.775382i	$0.282443\pi$
$374$	2.36603	+	0.633975i	0.122344	+	0.0327820i
$375$		0			0
$376$	−3.46410	−	3.46410i	−0.178647	−	0.178647i
$377$			38.3205i			1.97361i
$378$	−16.0981	+	10.9019i	−0.827996	+	0.560734i
$379$			26.3923i			1.35568i	0.735209	+	0.677841i	$0.237084\pi$
$379$			26.3923i			1.35568i	−0.735209	+	0.677841i	$0.762916\pi$
$380$		0			0
$381$			14.7846i			0.757438i
$382$	1.16987	−	4.36603i	0.0598559	−	0.223385i
$383$	20.5359			1.04934			0.524668	−	0.851307i	$-0.324190\pi$
$383$	20.5359			1.04934			0.524668	+	0.851307i	$0.324190\pi$
$384$	−5.07180	+	18.9282i	−0.258819	+	0.965926i
$385$		0			0
$386$	3.46410	+	0.928203i	0.176318	+	0.0472443i
$387$		0			0
$388$	7.39230	−	12.8038i	0.375287	−	0.650017i
$389$	6.85641			0.347634			0.173817	−	0.984778i	$-0.444390\pi$
$389$	6.85641			0.347634			0.173817	+	0.984778i	$0.444390\pi$
$390$		0			0
$391$	2.53590			0.128246
$392$	11.8564	−	15.8564i	0.598839	−	0.800869i
$393$	−16.3923			−0.826882
$394$	−29.1244	−	7.80385i	−1.46726	−	0.393152i
$395$		0			0
$396$		0			0
$397$		−	5.53590i		−	0.277839i	−0.990304	−	0.138919i	$-0.955637\pi$
$397$		−	5.53590i		−	0.277839i	0.990304	−	0.138919i	$-0.0443629\pi$
$398$	4.73205	+	1.26795i	0.237196	+	0.0635566i
$399$	−18.0000	+	20.7846i	−0.901127	+	1.04053i
$400$		0			0
$401$	−23.9282			−1.19492			−0.597459	−	0.801900i	$-0.703823\pi$
$401$	−23.9282			−1.19492			−0.597459	+	0.801900i	$0.703823\pi$
$402$	2.19615	−	8.19615i	0.109534	−	0.408787i
$403$		−	38.7846i		−	1.93200i
$404$	8.53590	−	14.7846i	0.424677	−	0.735562i
$405$		0			0
$406$	−18.3660	+	12.4378i	−0.911491	+	0.617279i
$407$			9.46410i			0.469118i
$408$	1.60770	−	1.60770i	0.0795928	−	0.0795928i
$409$			31.8564i			1.57520i	0.616188	+	0.787599i	$0.288676\pi$
$409$			31.8564i			1.57520i	−0.616188	+	0.787599i	$0.711324\pi$
$410$		0			0
$411$	0.679492			0.0335169
$412$	−29.7846	−	17.1962i	−1.46738	−	0.847194i
$413$	−6.00000	+	6.92820i	−0.295241	+	0.340915i
$414$		0			0
$415$		0			0
$416$	35.3205	−	9.46410i	1.73173	−	0.464016i
$417$	12.0000			0.587643
$418$	8.19615	−	30.5885i	0.400887	−	1.49613i
$419$	−24.2487			−1.18463			−0.592314	−	0.805708i	$-0.701785\pi$
$419$	−24.2487			−1.18463			−0.592314	+	0.805708i	$0.701785\pi$
$420$		0			0
$421$	19.0000			0.926003			0.463002	−	0.886357i	$-0.346772\pi$
$421$	19.0000			0.926003			0.463002	+	0.886357i	$0.346772\pi$
$422$	2.63397	−	9.83013i	0.128220	−	0.478523i
$423$		0			0
$424$	−4.00000	−	4.00000i	−0.194257	−	0.194257i
$425$		0			0
$426$	−0.339746	+	1.26795i	−0.0164607	+	0.0614323i
$427$	5.07180	+	4.39230i	0.245441	+	0.212559i
$428$	18.3923	−	31.8564i	0.889026	−	1.53984i
$429$	41.7846			2.01738
$430$		0			0
$431$			17.5885i			0.847206i	0.905848	+	0.423603i	$0.139235\pi$
$431$			17.5885i			0.847206i	−0.905848	+	0.423603i	$0.860765\pi$
$432$	−10.3923	−	18.0000i	−0.500000	−	0.866025i
$433$			4.14359i			0.199128i	0.995031	+	0.0995642i	$0.0317449\pi$
$433$			4.14359i			0.199128i	−0.995031	+	0.0995642i	$0.968255\pi$
$434$	18.5885	−	12.5885i	0.892275	−	0.604265i
$435$		0			0
$436$	27.5885	+	15.9282i	1.32125	+	0.762823i
$437$		−	32.7846i		−	1.56830i
$438$	−0.588457	+	2.19615i	−0.0281176	+	0.104936i
$439$	−15.7128			−0.749932			−0.374966	−	0.927039i	$-0.622346\pi$
$439$	−15.7128			−0.749932			−0.374966	+	0.927039i	$0.622346\pi$
$440$		0			0
$441$		0			0
$442$	−4.09808	−	1.09808i	−0.194926	−	0.0522302i
$443$			26.0000i			1.23530i	0.786454	+	0.617649i	$0.211915\pi$
$443$			26.0000i			1.23530i	−0.786454	+	0.617649i	$0.788085\pi$
$444$	7.60770	+	4.39230i	0.361045	+	0.208450i
$445$		0			0
$446$	−14.0263	−	3.75833i	−0.664164	−	0.177962i
$447$	29.3205			1.38681
$448$	16.0000	+	13.8564i	0.755929	+	0.654654i
$449$	−1.92820			−0.0909975			−0.0454988	−	0.998964i	$-0.514488\pi$
$449$	−1.92820			−0.0909975			−0.0454988	+	0.998964i	$0.514488\pi$
$450$		0			0
$451$	−12.9282			−0.608765
$452$	2.53590	+	1.46410i	0.119279	+	0.0688655i
$453$			8.32051i			0.390932i
$454$	4.56218	+	1.22243i	0.214114	+	0.0573716i
$455$		0			0
$456$	−20.7846	−	20.7846i	−0.973329	−	0.973329i
$457$	−27.4641			−1.28472			−0.642358	−	0.766405i	$-0.722044\pi$
$457$	−27.4641			−1.28472			−0.642358	+	0.766405i	$0.722044\pi$
$458$	5.66025	−	21.1244i	0.264486	−	0.987076i
$459$			2.41154i			0.112561i
$460$		0			0
$461$		−	27.7128i		−	1.29071i	−0.763881	−	0.645357i	$-0.776709\pi$
$461$		−	27.7128i		−	1.29071i	0.763881	−	0.645357i	$-0.223291\pi$
$462$	13.5622	+	20.0263i	0.630970	+	0.931707i
$463$		−	4.39230i		−	0.204128i	−0.994778	−	0.102064i	$-0.967455\pi$
$463$		−	4.39230i		−	0.204128i	0.994778	−	0.102064i	$-0.0325446\pi$
$464$	−11.8564	−	20.5359i	−0.550420	−	0.953355i
$465$		0			0
$466$	31.3205	+	8.39230i	1.45089	+	0.388766i
$467$	−22.5167			−1.04195			−0.520973	−	0.853573i	$-0.674431\pi$
$467$	−22.5167			−1.04195			−0.520973	+	0.853573i	$0.674431\pi$
$468$		0			0
$469$	−6.92820	−	6.00000i	−0.319915	−	0.277054i
$470$		0			0
$471$			34.3923i			1.58471i
$472$	−6.92820	−	6.92820i	−0.318896	−	0.318896i
$473$	7.46410			0.343200
$474$	1.68653	−	6.29423i	0.0774650	−	0.289103i
$475$		0			0
$476$	−0.803848	−	2.32051i	−0.0368443	−	0.106360i
$477$		0			0
$478$	−10.2417	+	38.2224i	−0.468443	+	1.74825i
$479$	37.1769			1.69866			0.849328	−	0.527865i	$-0.177007\pi$
$479$	37.1769			1.69866			0.849328	+	0.527865i	$0.177007\pi$
$480$		0			0
$481$		−	16.3923i		−	0.747425i
$482$	1.60770	−	6.00000i	0.0732285	−	0.273293i
$483$	18.9282	+	16.3923i	0.861263	+	0.745876i
$484$	−5.07180	−	2.92820i	−0.230536	−	0.133100i
$485$		0			0
$486$		0			0
$487$		−	28.7846i		−	1.30436i	−0.758066	−	0.652178i	$-0.773856\pi$
$487$		−	28.7846i		−	1.30436i	0.758066	−	0.652178i	$-0.226144\pi$
$488$	−5.07180	+	5.07180i	−0.229589	+	0.229589i
$489$		−	36.0000i		−	1.62798i
$490$		0			0
$491$		−	34.1244i		−	1.54001i	−0.638037	−	0.770005i	$-0.720254\pi$
$491$		−	34.1244i		−	1.54001i	0.638037	−	0.770005i	$-0.279746\pi$
$492$	−6.00000	+	10.3923i	−0.270501	+	0.468521i
$493$			2.75129i			0.123912i
$494$	−14.1962	+	52.9808i	−0.638715	+	2.38372i
$495$		0			0
$496$	12.0000	+	20.7846i	0.538816	+	0.933257i
$497$	1.07180	+	0.928203i	0.0480767	+	0.0416356i
$498$	20.1962	+	5.41154i	0.905011	+	0.242497i
$499$		−	5.58846i		−	0.250174i	−0.992146	−	0.125087i	$-0.960079\pi$
$499$		−	5.58846i		−	0.250174i	0.992146	−	0.125087i	$-0.0399210\pi$
$500$		0			0
$501$	9.00000			0.402090
$502$	−2.53590	−	0.679492i	−0.113183	−	0.0303272i
$503$	15.5885			0.695055			0.347527	−	0.937670i	$-0.387021\pi$
$503$	15.5885			0.695055			0.347527	+	0.937670i	$0.387021\pi$
$504$		0			0
$505$		0			0
$506$	−27.8564	−	7.46410i	−1.23837	−	0.331820i
$507$	−49.8564			−2.21420
$508$	−8.53590	+	14.7846i	−0.378719	+	0.655961i
$509$		−	1.85641i		−	0.0822838i	−0.999153	−	0.0411419i	$-0.986900\pi$
$509$		−	1.85641i		−	0.0822838i	0.999153	−	0.0411419i	$-0.0130996\pi$
$510$		0			0
$511$	1.85641	+	1.60770i	0.0821226	+	0.0711202i
$512$	−16.0000	+	16.0000i	−0.707107	+	0.707107i
$513$	31.1769			1.37649
$514$	2.19615	−	8.19615i	0.0968681	−	0.361517i
$515$		0			0
$516$	3.46410	−	6.00000i	0.152499	−	0.264135i
$517$		−	6.46410i		−	0.284291i
$518$	7.85641	−	5.32051i	0.345191	−	0.233770i
$519$		−	35.1962i		−	1.54494i
$520$		0			0
$521$			34.3923i			1.50675i	0.657589	+	0.753377i	$0.271576\pi$
$521$			34.3923i			1.50675i	−0.657589	+	0.753377i	$0.728424\pi$
$522$		0			0
$523$	24.2487			1.06032			0.530161	−	0.847897i	$-0.322131\pi$
$523$	24.2487			1.06032			0.530161	+	0.847897i	$0.322131\pi$
$524$	−16.3923	−	9.46410i	−0.716101	−	0.413441i
$525$		0			0
$526$	−1.66025	+	6.19615i	−0.0723905	+	0.270165i
$527$		−	2.78461i		−	0.121300i
$528$	−22.3923	+	12.9282i	−0.974500	+	0.562628i
$529$	−6.85641			−0.298105
$530$		0			0
$531$		0			0
$532$	−30.0000	+	10.3923i	−1.30066	+	0.450564i
$533$	22.3923			0.969918
$534$	−6.00000	+	22.3923i	−0.259645	+	0.969010i
$535$		0			0
$536$	6.92820	−	6.92820i	0.299253	−	0.299253i
$537$			24.9282i			1.07573i
$538$	4.39230	−	16.3923i	0.189366	−	0.706722i
$539$	25.8564	−	3.73205i	1.11371	−	0.160751i
$540$		0			0
$541$	−33.7846			−1.45251			−0.726257	−	0.687423i	$-0.758742\pi$
$541$	−33.7846			−1.45251			−0.726257	+	0.687423i	$0.758742\pi$
$542$	−3.46410	−	0.928203i	−0.148796	−	0.0398697i
$543$			22.3923i			0.960946i
$544$	2.53590	−	0.679492i	0.108726	−	0.0291330i
$545$		0			0
$546$	−23.4904	−	34.6865i	−1.00530	−	1.48445i
$547$		−	14.5359i		−	0.621510i	−0.950490	−	0.310755i	$-0.899418\pi$
$547$		−	14.5359i		−	0.621510i	0.950490	−	0.310755i	$-0.100582\pi$
$548$	0.679492	+	0.392305i	0.0290265	+	0.0167584i
$549$		0			0
$550$		0			0
$551$	35.5692			1.51530
$552$	−18.9282	+	18.9282i	−0.805638	+	0.805638i
$553$	−5.32051	−	4.60770i	−0.226251	−	0.195939i
$554$	33.8564	+	9.07180i	1.43842	+	0.385424i
$555$		0			0
$556$	12.0000	+	6.92820i	0.508913	+	0.293821i
$557$	−5.85641			−0.248144			−0.124072	−	0.992273i	$-0.539595\pi$
$557$	−5.85641			−0.248144			−0.124072	+	0.992273i	$0.539595\pi$
$558$		0			0
$559$	−12.9282			−0.546805
$560$		0			0
$561$	3.00000			0.126660
$562$	8.09808	+	2.16987i	0.341597	+	0.0915306i
$563$	−43.1769			−1.81969			−0.909845	−	0.414948i	$-0.863800\pi$
$563$	−43.1769			−1.81969			−0.909845	+	0.414948i	$0.863800\pi$
$564$	−5.19615	−	3.00000i	−0.218797	−	0.126323i
$565$		0			0
$566$	−16.5622	−	4.43782i	−0.696160	−	0.186536i
$567$	−15.5885	+	18.0000i	−0.654654	+	0.755929i
$568$	−1.07180	+	1.07180i	−0.0449716	+	0.0449716i
$569$	20.9282			0.877356			0.438678	−	0.898644i	$-0.355447\pi$
$569$	20.9282			0.877356			0.438678	+	0.898644i	$0.355447\pi$
$570$		0			0
$571$		−	41.3205i		−	1.72921i	−0.502453	−	0.864605i	$-0.667569\pi$
$571$		−	41.3205i		−	1.72921i	0.502453	−	0.864605i	$-0.332431\pi$
$572$	41.7846	+	24.1244i	1.74710	+	1.00869i
$573$		−	5.53590i		−	0.231265i
$574$	7.26795	+	10.7321i	0.303358	+	0.447947i
$575$		0			0
$576$		0			0
$577$			1.39230i			0.0579624i	0.999580	+	0.0289812i	$0.00922630\pi$
$577$			1.39230i			0.0579624i	−0.999580	+	0.0289812i	$0.990774\pi$
$578$	22.9282	+	6.14359i	0.953688	+	0.255540i
$579$	4.39230			0.182538
$580$		0			0
$581$	14.7846	−	17.0718i	0.613369	−	0.708257i
$582$	4.68653	−	17.4904i	0.194263	−	0.725000i
$583$		−	7.46410i		−	0.309132i
$584$	−1.85641	+	1.85641i	−0.0768186	+	0.0768186i
$585$		0			0
$586$	−5.24167	+	19.5622i	−0.216531	+	0.808106i
$587$	−27.4641			−1.13356			−0.566782	−	0.823868i	$-0.691812\pi$
$587$	−27.4641			−1.13356			−0.566782	+	0.823868i	$0.691812\pi$
$588$	9.00000	−	22.5167i	0.371154	−	0.928571i
$589$	−36.0000			−1.48335
$590$		0			0
$591$	−36.9282			−1.51902
$592$	5.07180	+	8.78461i	0.208450	+	0.361045i
$593$		−	23.5359i		−	0.966504i	−0.875481	−	0.483252i	$-0.839456\pi$
$593$		−	23.5359i		−	0.966504i	0.875481	−	0.483252i	$-0.160544\pi$
$594$	7.09808	−	26.4904i	0.291238	−	1.08691i
$595$		0			0
$596$	29.3205	+	16.9282i	1.20101	+	0.693406i
$597$	6.00000			0.245564
$598$	48.2487	+	12.9282i	1.97304	+	0.528674i
$599$			14.1244i			0.577106i	0.957464	+	0.288553i	$0.0931741\pi$
$599$			14.1244i			0.577106i	−0.957464	+	0.288553i	$0.906826\pi$
$600$		0			0
$601$			26.7846i			1.09257i	0.837600	+	0.546284i	$0.183958\pi$
$601$			26.7846i			1.09257i	−0.837600	+	0.546284i	$0.816042\pi$
$602$	−4.19615	−	6.19615i	−0.171022	−	0.252536i
$603$		0			0
$604$	−4.80385	+	8.32051i	−0.195466	+	0.338557i
$605$		0			0
$606$	5.41154	−	20.1962i	0.219829	−	0.820413i
$607$	−18.8038			−0.763225			−0.381612	−	0.924322i	$-0.624631\pi$
$607$	−18.8038			−0.763225			−0.381612	+	0.924322i	$0.624631\pi$
$608$	−8.78461	−	32.7846i	−0.356263	−	1.32959i
$609$	−17.7846	+	20.5359i	−0.720669	+	0.832157i
$610$		0			0
$611$			11.1962i			0.452948i
$612$		0			0
$613$	10.0000			0.403896			0.201948	−	0.979396i	$-0.435273\pi$
$613$	10.0000			0.403896			0.201948	+	0.979396i	$0.435273\pi$
$614$	−2.36603	−	0.633975i	−0.0954850	−	0.0255851i
$615$		0			0
$616$	2.00000	+	27.8564i	0.0805823	+	1.12237i
$617$	9.07180			0.365217			0.182608	−	0.983186i	$-0.441546\pi$
$617$	9.07180			0.365217			0.182608	+	0.983186i	$0.441546\pi$
$618$	−40.6865	−	10.9019i	−1.63665	−	0.438540i
$619$	−35.3205			−1.41965			−0.709826	−	0.704378i	$-0.751226\pi$
$619$	−35.3205			−1.41965			−0.709826	+	0.704378i	$0.751226\pi$
$620$		0			0
$621$		−	28.3923i		−	1.13934i
$622$	27.1244	+	7.26795i	1.08759	+	0.291418i
$623$	18.9282	+	16.3923i	0.758342	+	0.656744i
$624$	38.7846	−	22.3923i	1.55263	−	0.896410i
$625$		0			0
$626$	−8.95448	+	33.4186i	−0.357893	+	1.33568i
$627$		−	38.7846i		−	1.54891i
$628$	−19.8564	+	34.3923i	−0.792357	+	1.37240i
$629$		−	1.17691i		−	0.0469267i
$630$		0			0
$631$		−	26.9090i		−	1.07123i	−0.844463	−	0.535614i	$-0.820080\pi$
$631$		−	26.9090i		−	1.07123i	0.844463	−	0.535614i	$-0.179920\pi$
$632$	5.32051	−	5.32051i	0.211638	−	0.211638i
$633$		−	12.4641i		−	0.495404i
$634$	23.1244	+	6.19615i	0.918385	+	0.246081i
$635$		0			0
$636$	−6.00000	−	3.46410i	−0.237915	−	0.137361i
$637$	−44.7846	+	6.46410i	−1.77443	+	0.256117i
$638$	8.09808	−	30.2224i	0.320606	−	1.19652i
$639$		0			0
$640$		0			0
$641$	−4.92820			−0.194652			−0.0973262	−	0.995253i	$-0.531029\pi$
$641$	−4.92820			−0.194652			−0.0973262	+	0.995253i	$0.531029\pi$
$642$	11.6603	−	43.5167i	0.460194	−	1.71747i
$643$	−31.0526			−1.22459			−0.612297	−	0.790628i	$-0.709754\pi$
$643$	−31.0526			−1.22459			−0.612297	+	0.790628i	$0.709754\pi$
$644$	9.46410	+	27.3205i	0.372938	+	1.07658i
$645$		0			0
$646$	−1.01924	+	3.80385i	−0.0401014	+	0.149660i
$647$	10.3923			0.408564			0.204282	−	0.978912i	$-0.434514\pi$
$647$	10.3923			0.408564			0.204282	+	0.978912i	$0.434514\pi$
$648$	−18.0000	−	18.0000i	−0.707107	−	0.707107i
$649$		−	12.9282i		−	0.507476i
$650$		0			0
$651$	18.0000	−	20.7846i	0.705476	−	0.814613i
$652$	20.7846	−	36.0000i	0.813988	−	1.40987i
$653$	−38.3923			−1.50241			−0.751203	−	0.660071i	$-0.770526\pi$
$653$	−38.3923			−1.50241			−0.751203	+	0.660071i	$0.770526\pi$
$654$	37.6865	+	10.0981i	1.47366	+	0.394866i
$655$		0			0
$656$	−12.0000	+	6.92820i	−0.468521	+	0.270501i
$657$		0			0
$658$	−5.36603	+	3.63397i	−0.209189	+	0.141667i
$659$		−	20.8038i		−	0.810403i	−0.914227	−	0.405201i	$-0.867201\pi$
$659$		−	20.8038i		−	0.810403i	0.914227	−	0.405201i	$-0.132799\pi$
$660$		0			0
$661$		−	15.7128i		−	0.611158i	−0.952167	−	0.305579i	$-0.901150\pi$
$661$		−	15.7128i		−	0.611158i	0.952167	−	0.305579i	$-0.0988499\pi$
$662$	2.05256	−	7.66025i	0.0797750	−	0.297724i
$663$	−5.19615			−0.201802
$664$	17.0718	+	17.0718i	0.662514	+	0.662514i
$665$		0			0
$666$		0			0
$667$		−	32.3923i		−	1.25424i
$668$	9.00000	+	5.19615i	0.348220	+	0.201045i
$669$	−17.7846			−0.687593
$670$		0			0
$671$	−9.46410			−0.365358
$672$	23.3205	+	11.3205i	0.899608	+	0.436698i
$673$	−49.1769			−1.89563			−0.947815	−	0.318820i	$-0.896714\pi$
$673$	−49.1769			−1.89563			−0.947815	+	0.318820i	$0.896714\pi$
$674$		0			0
$675$		0			0
$676$	−49.8564	−	28.7846i	−1.91755	−	1.10710i
$677$			4.60770i			0.177088i	0.996072	+	0.0885441i	$0.0282214\pi$
$677$			4.60770i			0.177088i	−0.996072	+	0.0885441i	$0.971779\pi$
$678$	3.46410	+	0.928203i	0.133038	+	0.0356474i
$679$	−14.7846	−	12.8038i	−0.567381	−	0.491367i
$680$		0			0
$681$	5.78461			0.221667
$682$	−8.19615	+	30.5885i	−0.313847	+	1.17129i
$683$			27.3205i			1.04539i	0.852520	+	0.522695i	$0.175073\pi$
$683$			27.3205i			1.04539i	−0.852520	+	0.522695i	$0.824927\pi$
$684$		0			0
$685$		0			0
$686$	−17.6340	−	19.3660i	−0.673268	−	0.739398i
$687$		−	26.7846i		−	1.02190i
$688$	6.92820	−	4.00000i	0.264135	−	0.152499i
$689$			12.9282i			0.492525i
$690$		0			0
$691$	−46.6410			−1.77431			−0.887154	−	0.461474i	$-0.847321\pi$
$691$	−46.6410			−1.77431			−0.887154	+	0.461474i	$0.847321\pi$
$692$	20.3205	−	35.1962i	0.772470	−	1.33796i
$693$		0			0
$694$	−5.21539	+	19.4641i	−0.197974	+	0.738847i
$695$		0			0
$696$	−20.5359	−	20.5359i	−0.778411	−	0.778411i
$697$	1.60770			0.0608958
$698$	−10.7321	+	40.0526i	−0.406214	+	1.51601i
$699$	39.7128			1.50208
$700$		0			0
$701$	3.78461			0.142943			0.0714714	−	0.997443i	$-0.477231\pi$
$701$	3.78461			0.142943			0.0714714	+	0.997443i	$0.477231\pi$
$702$	−12.2942	+	45.8827i	−0.464016	+	1.73173i
$703$	−15.2154			−0.573859
$704$	−29.8564			−1.12526
$705$		0			0
$706$	4.90192	−	18.2942i	0.184486	−	0.688512i
$707$	−17.0718	−	14.7846i	−0.642051	−	0.556032i
$708$	−10.3923	−	6.00000i	−0.390567	−	0.225494i
$709$	21.0000			0.788672			0.394336	−	0.918966i	$-0.370975\pi$
$709$	21.0000			0.788672			0.394336	+	0.918966i	$0.370975\pi$
$710$		0			0
$711$		0			0
$712$	−18.9282	+	18.9282i	−0.709364	+	0.709364i
$713$			32.7846i			1.22779i
$714$	−1.68653	−	2.49038i	−0.0631169	−	0.0932002i
$715$		0			0
$716$	−14.3923	+	24.9282i	−0.537866	+	0.931611i
$717$			48.4641i			1.80993i
$718$	3.41154	−	12.7321i	0.127318	−	0.475156i
$719$	−45.4641			−1.69552			−0.847762	−	0.530376i	$-0.822051\pi$
$719$	−45.4641			−1.69552			−0.847762	+	0.530376i	$0.822051\pi$
$720$		0			0
$721$	−29.7846	+	34.3923i	−1.10924	+	1.28084i
$722$	23.2224	+	6.22243i	0.864249	+	0.231575i
$723$		−	7.60770i		−	0.282933i
$724$	−12.9282	+	22.3923i	−0.480473	+	0.832203i
$725$		0			0
$726$	−6.92820	−	1.85641i	−0.257130	−	0.0688977i
$727$	−34.3923			−1.27554			−0.637770	−	0.770227i	$-0.720143\pi$
$727$	−34.3923			−1.27554			−0.637770	+	0.770227i	$0.720143\pi$
$728$	−3.46410	−	48.2487i	−0.128388	−	1.78822i
$729$	27.0000			1.00000
$730$		0			0
$731$	−0.928203			−0.0343308
$732$	−4.39230	+	7.60770i	−0.162344	+	0.281189i
$733$		−	18.4641i		−	0.681987i	−0.940066	−	0.340994i	$-0.889237\pi$
$733$		−	18.4641i		−	0.681987i	0.940066	−	0.340994i	$-0.110763\pi$
$734$	49.3468	+	13.2224i	1.82142	+	0.488049i
$735$		0			0
$736$	−29.8564	+	8.00000i	−1.10052	+	0.294884i
$737$	12.9282			0.476216
$738$		0			0
$739$		−	7.73205i		−	0.284428i	−0.989836	−	0.142214i	$-0.954578\pi$
$739$		−	7.73205i		−	0.284428i	0.989836	−	0.142214i	$-0.0454221\pi$
$740$		0			0
$741$			67.1769i			2.46781i
$742$	−6.19615	+	4.19615i	−0.227468	+	0.154046i
$743$		−	9.60770i		−	0.352472i	−0.984348	−	0.176236i	$-0.943608\pi$
$743$		−	9.60770i		−	0.352472i	0.984348	−	0.176236i	$-0.0563922\pi$
$744$	20.7846	+	20.7846i	0.762001	+	0.762001i
$745$		0			0
$746$	33.3205	+	8.92820i	1.21995	+	0.326885i
$747$		0			0
$748$	3.00000	+	1.73205i	0.109691	+	0.0633300i
$749$	−36.7846	−	31.8564i	−1.34408	−	1.16401i
$750$		0			0
$751$			5.58846i			0.203926i	0.994788	+	0.101963i	$0.0325123\pi$
$751$			5.58846i			0.203926i	−0.994788	+	0.101963i	$0.967488\pi$
$752$	−3.46410	−	6.00000i	−0.126323	−	0.218797i
$753$	−3.21539			−0.117175
$754$	−14.0263	+	52.3468i	−0.510807	+	1.90636i
$755$		0			0
$756$	−25.9808	+	9.00000i	−0.944911	+	0.327327i
$757$	−10.1436			−0.368675			−0.184338	−	0.982863i	$-0.559014\pi$
$757$	−10.1436			−0.368675			−0.184338	+	0.982863i	$0.559014\pi$
$758$	−9.66025	+	36.0526i	−0.350876	+	1.30949i
$759$	−35.3205			−1.28205
$760$		0			0
$761$			6.24871i			0.226516i	0.993566	+	0.113258i	$0.0361286\pi$
$761$			6.24871i			0.226516i	−0.993566	+	0.113258i	$0.963871\pi$
$762$	−5.41154	+	20.1962i	−0.196040	+	0.731629i
$763$	27.5885	−	31.8564i	0.998769	−	1.15328i
$764$	3.19615	−	5.53590i	0.115633	−	0.200282i
$765$		0			0
$766$	28.0526	+	7.51666i	1.01358	+	0.271588i
$767$			22.3923i			0.808539i
$768$	−13.8564	+	24.0000i	−0.500000	+	0.866025i
$769$			18.0000i			0.649097i	0.945869	+	0.324548i	$0.105212\pi$
$769$			18.0000i			0.649097i	−0.945869	+	0.324548i	$0.894788\pi$
$770$		0			0
$771$		−	10.3923i		−	0.374270i
$772$	4.39230	+	2.53590i	0.158083	+	0.0912690i
$773$			12.4641i			0.448303i	0.974554	+	0.224151i	$0.0719610\pi$
$773$			12.4641i			0.448303i	−0.974554	+	0.224151i	$0.928039\pi$
$774$		0			0
$775$		0			0
$776$	14.7846	−	14.7846i	0.530737	−	0.530737i
$777$	7.60770	−	8.78461i	0.272925	−	0.315146i
$778$	9.36603	+	2.50962i	0.335788	+	0.0899742i
$779$		−	20.7846i		−	0.744686i
$780$		0			0
$781$	−2.00000			−0.0715656

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:	\( N \)	\(=\)	\( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	700.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.36603 + 0.366025i) q^{2} +1.73205 q^{3} +(1.73205 + 1.00000i) q^{4} +(2.36603 + 0.633975i) q^{6} +(1.73205 - 2.00000i) q^{7} +(2.00000 + 2.00000i) q^{8} +O(q^{10})\)	\(q+(1.36603 + 0.366025i) q^{2} +1.73205 q^{3} +(1.73205 + 1.00000i) q^{4} +(2.36603 + 0.633975i) q^{6} +(1.73205 - 2.00000i) q^{7} +(2.00000 + 2.00000i) q^{8} +3.73205i q^{11} +(3.00000 + 1.73205i) q^{12} -6.46410i q^{13} +(3.09808 - 2.09808i) q^{14} +(2.00000 + 3.46410i) q^{16} -0.464102i q^{17} -6.00000 q^{19} +(3.00000 - 3.46410i) q^{21} +(-1.36603 + 5.09808i) q^{22} +5.46410i q^{23} +(3.46410 + 3.46410i) q^{24} +(2.36603 - 8.83013i) q^{26} -5.19615 q^{27} +(5.00000 - 1.73205i) q^{28} -5.92820 q^{29} +6.00000 q^{31} +(1.46410 + 5.46410i) q^{32} +6.46410i q^{33} +(0.169873 - 0.633975i) q^{34} +2.53590 q^{37} +(-8.19615 - 2.19615i) q^{38} -11.1962i q^{39} +3.46410i q^{41} +(5.36603 - 3.63397i) q^{42} -2.00000i q^{43} +(-3.73205 + 6.46410i) q^{44} +(-2.00000 + 7.46410i) q^{46} -1.73205 q^{47} +(3.46410 + 6.00000i) q^{48} +(-1.00000 - 6.92820i) q^{49} -0.803848i q^{51} +(6.46410 - 11.1962i) q^{52} -2.00000 q^{53} +(-7.09808 - 1.90192i) q^{54} +(7.46410 - 0.535898i) q^{56} -10.3923 q^{57} +(-8.09808 - 2.16987i) q^{58} -3.46410 q^{59} +2.53590i q^{61} +(8.19615 + 2.19615i) q^{62} +8.00000i q^{64} +(-2.36603 + 8.83013i) q^{66} -3.46410i q^{67} +(0.464102 - 0.803848i) q^{68} +9.46410i q^{69} +0.535898i q^{71} +0.928203i q^{73} +(3.46410 + 0.928203i) q^{74} +(-10.3923 - 6.00000i) q^{76} +(7.46410 + 6.46410i) q^{77} +(4.09808 - 15.2942i) q^{78} -2.66025i q^{79} -9.00000 q^{81} +(-1.26795 + 4.73205i) q^{82} +8.53590 q^{83} +(8.66025 - 3.00000i) q^{84} +(0.732051 - 2.73205i) q^{86} -10.2679 q^{87} +(-7.46410 + 7.46410i) q^{88} +9.46410i q^{89} +(-12.9282 - 11.1962i) q^{91} +(-5.46410 + 9.46410i) q^{92} +10.3923 q^{93} +(-2.36603 - 0.633975i) q^{94} +(2.53590 + 9.46410i) q^{96} -7.39230i q^{97} +(1.16987 - 9.83013i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 6 q^{6} + 8 q^{8}+O(q^{10}) \) 4 * q + 2 * q^2 + 6 * q^6 + 8 * q^8	\( 4 q + 2 q^{2} + 6 q^{6} + 8 q^{8} + 12 q^{12} + 2 q^{14} + 8 q^{16} - 24 q^{19} + 12 q^{21} - 2 q^{22} + 6 q^{26} + 20 q^{28} + 4 q^{29} + 24 q^{31} - 8 q^{32} + 18 q^{34} + 24 q^{37} - 12 q^{38} + 18 q^{42} - 8 q^{44} - 8 q^{46} - 4 q^{49} + 12 q^{52} - 8 q^{53} - 18 q^{54} + 16 q^{56} - 22 q^{58} + 12 q^{62} - 6 q^{66} - 12 q^{68} + 16 q^{77} + 6 q^{78} - 36 q^{81} - 12 q^{82} + 48 q^{83} - 4 q^{86} - 48 q^{87} - 16 q^{88} - 24 q^{91} - 8 q^{92} - 6 q^{94} + 24 q^{96} + 22 q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 + 6 * q^6 + 8 * q^8 + 12 * q^12 + 2 * q^14 + 8 * q^16 - 24 * q^19 + 12 * q^21 - 2 * q^22 + 6 * q^26 + 20 * q^28 + 4 * q^29 + 24 * q^31 - 8 * q^32 + 18 * q^34 + 24 * q^37 - 12 * q^38 + 18 * q^42 - 8 * q^44 - 8 * q^46 - 4 * q^49 + 12 * q^52 - 8 * q^53 - 18 * q^54 + 16 * q^56 - 22 * q^58 + 12 * q^62 - 6 * q^66 - 12 * q^68 + 16 * q^77 + 6 * q^78 - 36 * q^81 - 12 * q^82 + 48 * q^83 - 4 * q^86 - 48 * q^87 - 16 * q^88 - 24 * q^91 - 8 * q^92 - 6 * q^94 + 24 * q^96 + 22 * q^98

Introduction

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