LMFDB - Embedded newform 700.2.c.e.699.2

Newspace parameters

Level:	$ N $	$=$	$ 700 = 2^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	700.c (of order $2$, degree $1$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$5.58952814149$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 140)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			699.2
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	700.699
Dual form			700.2.c.e.699.1

$q$-expansion

$f(q)$	$=$	$q+(-0.366025 + 1.36603i) q^{2} +1.73205i q^{3} +(-1.73205 - 1.00000i) q^{4} +(-2.36603 - 0.633975i) q^{6} +(2.00000 - 1.73205i) q^{7} +(2.00000 - 2.00000i) q^{8} +O(q^{10})$	\(q+(-0.366025 + 1.36603i) q^{2} +1.73205i q^{3} +(-1.73205 - 1.00000i) q^{4} +(-2.36603 - 0.633975i) q^{6} +(2.00000 - 1.73205i) q^{7} +(2.00000 - 2.00000i) q^{8} +3.73205i q^{11} +(1.73205 - 3.00000i) q^{12} +6.46410 q^{13} +(1.63397 + 3.36603i) q^{14} +(2.00000 + 3.46410i) q^{16} -0.464102 q^{17} -6.00000 q^{19} +(3.00000 + 3.46410i) q^{21} +(-5.09808 - 1.36603i) q^{22} +5.46410 q^{23} +(3.46410 + 3.46410i) q^{24} +(-2.36603 + 8.83013i) q^{26} +5.19615i q^{27} +(-5.19615 + 1.00000i) q^{28} +5.92820 q^{29} -6.00000 q^{31} +(-5.46410 + 1.46410i) q^{32} -6.46410 q^{33} +(0.169873 - 0.633975i) q^{34} +2.53590i q^{37} +(2.19615 - 8.19615i) q^{38} +11.1962i q^{39} -3.46410i q^{41} +(-5.83013 + 2.83013i) q^{42} -2.00000 q^{43} +(3.73205 - 6.46410i) q^{44} +(-2.00000 + 7.46410i) q^{46} +1.73205i q^{47} +(-6.00000 + 3.46410i) q^{48} +(1.00000 - 6.92820i) q^{49} -0.803848i q^{51} +(-11.1962 - 6.46410i) q^{52} +2.00000i q^{53} +(-7.09808 - 1.90192i) q^{54} +(0.535898 - 7.46410i) q^{56} -10.3923i q^{57} +(-2.16987 + 8.09808i) q^{58} -3.46410 q^{59} -2.53590i q^{61} +(2.19615 - 8.19615i) q^{62} -8.00000i q^{64} +(2.36603 - 8.83013i) q^{66} +3.46410 q^{67} +(0.803848 + 0.464102i) q^{68} +9.46410i q^{69} +0.535898i q^{71} -0.928203 q^{73} +(-3.46410 - 0.928203i) q^{74} +(10.3923 + 6.00000i) q^{76} +(6.46410 + 7.46410i) q^{77} +(-15.2942 - 4.09808i) q^{78} +2.66025i q^{79} -9.00000 q^{81} +(4.73205 + 1.26795i) q^{82} +8.53590i q^{83} +(-1.73205 - 9.00000i) q^{84} +(0.732051 - 2.73205i) q^{86} +10.2679i q^{87} +(7.46410 + 7.46410i) q^{88} +9.46410i q^{89} +(12.9282 - 11.1962i) q^{91} +(-9.46410 - 5.46410i) q^{92} -10.3923i q^{93} +(-2.36603 - 0.633975i) q^{94} +(-2.53590 - 9.46410i) q^{96} -7.39230 q^{97} +(9.09808 + 3.90192i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} - 6 q^{6} + 8 q^{7} + 8 q^{8}+O(q^{10}) $ 4 * q + 2 * q^2 - 6 * q^6 + 8 * q^7 + 8 * q^8	$ 4 q + 2 q^{2} - 6 q^{6} + 8 q^{7} + 8 q^{8} + 12 q^{13} + 10 q^{14} + 8 q^{16} + 12 q^{17} - 24 q^{19} + 12 q^{21} - 10 q^{22} + 8 q^{23} - 6 q^{26} - 4 q^{29} - 24 q^{31} - 8 q^{32} - 12 q^{33} + 18 q^{34} - 12 q^{38} - 6 q^{42} - 8 q^{43} + 8 q^{44} - 8 q^{46} - 24 q^{48} + 4 q^{49} - 24 q^{52} - 18 q^{54} + 16 q^{56} - 26 q^{58} - 12 q^{62} + 6 q^{66} + 24 q^{68} + 24 q^{73} + 12 q^{77} - 30 q^{78} - 36 q^{81} + 12 q^{82} - 4 q^{86} + 16 q^{88} + 24 q^{91} - 24 q^{92} - 6 q^{94} - 24 q^{96} + 12 q^{97} + 26 q^{98}+O(q^{100}) $ 4 * q + 2 * q^2 - 6 * q^6 + 8 * q^7 + 8 * q^8 + 12 * q^13 + 10 * q^14 + 8 * q^16 + 12 * q^17 - 24 * q^19 + 12 * q^21 - 10 * q^22 + 8 * q^23 - 6 * q^26 - 4 * q^29 - 24 * q^31 - 8 * q^32 - 12 * q^33 + 18 * q^34 - 12 * q^38 - 6 * q^42 - 8 * q^43 + 8 * q^44 - 8 * q^46 - 24 * q^48 + 4 * q^49 - 24 * q^52 - 18 * q^54 + 16 * q^56 - 26 * q^58 - 12 * q^62 + 6 * q^66 + 24 * q^68 + 24 * q^73 + 12 * q^77 - 30 * q^78 - 36 * q^81 + 12 * q^82 - 4 * q^86 + 16 * q^88 + 24 * q^91 - 24 * q^92 - 6 * q^94 - 24 * q^96 + 12 * q^97 + 26 * 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 $

$2$	−0.366025	+	1.36603i	−0.258819	+	0.965926i
$2$	−0.366025	+	1.36603i	−0.258819	+	0.965926i
$3$			1.73205i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$3$			1.73205i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$4$	−1.73205	−	1.00000i	−0.866025	−	0.500000i
$5$		0			0
$5$		0			0
$6$	−2.36603	−	0.633975i	−0.965926	−	0.258819i
$7$	2.00000	−	1.73205i	0.755929	−	0.654654i
$7$	2.00000	−	1.73205i	0.755929	−	0.654654i
$8$	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	$0.809790\pi$
$12$	1.73205	−	3.00000i	0.500000	−	0.866025i
$13$	6.46410			1.79282			0.896410	−	0.443227i	$-0.146166\pi$
$13$	6.46410			1.79282			0.896410	+	0.443227i	$0.146166\pi$
$14$	1.63397	+	3.36603i	0.436698	+	0.899608i
$15$		0			0
$16$	2.00000	+	3.46410i	0.500000	+	0.866025i
$17$	−0.464102			−0.112561			−0.0562806	−	0.998415i	$-0.517924\pi$
$17$	−0.464102			−0.112561			−0.0562806	+	0.998415i	$0.517924\pi$
$18$		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
$22$	−5.09808	−	1.36603i	−1.08691	−	0.291238i
$23$	5.46410			1.13934			0.569672	−	0.821872i	$-0.307070\pi$
$23$	5.46410			1.13934			0.569672	+	0.821872i	$0.307070\pi$
$24$	3.46410	+	3.46410i	0.707107	+	0.707107i
$25$		0			0
$26$	−2.36603	+	8.83013i	−0.464016	+	1.73173i
$27$			5.19615i			1.00000i
$28$	−5.19615	+	1.00000i	−0.981981	+	0.188982i
$29$	5.92820			1.10084			0.550420	−	0.834888i	$-0.314468\pi$
$29$	5.92820			1.10084			0.550420	+	0.834888i	$0.314468\pi$
$30$		0			0
$31$	−6.00000			−1.07763			−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000			−1.07763			−0.538816	+	0.842424i	$0.681128\pi$
$32$	−5.46410	+	1.46410i	−0.965926	+	0.258819i
$33$	−6.46410			−1.12526
$34$	0.169873	−	0.633975i	0.0291330	−	0.108726i
$35$		0			0
$36$		0			0
$37$			2.53590i			0.416899i	0.978033	+	0.208450i	$0.0668417\pi$
$37$			2.53590i			0.416899i	−0.978033	+	0.208450i	$0.933158\pi$
$38$	2.19615	−	8.19615i	0.356263	−	1.32959i
$39$			11.1962i			1.79282i
$40$		0			0
$41$		−	3.46410i		−	0.541002i	−0.962720	−	0.270501i	$-0.912811\pi$
$41$		−	3.46410i		−	0.541002i	0.962720	−	0.270501i	$-0.0871893\pi$
$42$	−5.83013	+	2.83013i	−0.899608	+	0.436698i
$43$	−2.00000			−0.304997			−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000			−0.304997			−0.152499	+	0.988304i	$0.548732\pi$
$44$	3.73205	−	6.46410i	0.562628	−	0.974500i
$45$		0			0
$46$	−2.00000	+	7.46410i	−0.294884	+	1.10052i
$47$			1.73205i			0.252646i	0.991989	+	0.126323i	$0.0403175\pi$
$47$			1.73205i			0.252646i	−0.991989	+	0.126323i	$0.959682\pi$
$48$	−6.00000	+	3.46410i	−0.866025	+	0.500000i
$49$	1.00000	−	6.92820i	0.142857	−	0.989743i
$50$		0			0
$51$		−	0.803848i		−	0.112561i
$52$	−11.1962	−	6.46410i	−1.55263	−	0.896410i
$53$			2.00000i			0.274721i	0.990521	+	0.137361i	$0.0438619\pi$
$53$			2.00000i			0.274721i	−0.990521	+	0.137361i	$0.956138\pi$
$54$	−7.09808	−	1.90192i	−0.965926	−	0.258819i
$55$		0			0
$56$	0.535898	−	7.46410i	0.0716124	−	0.997433i
$57$		−	10.3923i		−	1.37649i
$58$	−2.16987	+	8.09808i	−0.284918	+	1.06333i
$59$	−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	$-0.948094\pi$
$61$		−	2.53590i		−	0.324689i	0.986734	−	0.162344i	$-0.0519055\pi$
$62$	2.19615	−	8.19615i	0.278912	−	1.04091i
$63$		0			0
$64$		−	8.00000i		−	1.00000i
$65$		0			0
$66$	2.36603	−	8.83013i	0.291238	−	1.08691i
$67$	3.46410			0.423207			0.211604	−	0.977356i	$-0.432131\pi$
$67$	3.46410			0.423207			0.211604	+	0.977356i	$0.432131\pi$
$68$	0.803848	+	0.464102i	0.0974808	+	0.0562806i
$69$			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
$73$	−0.928203			−0.108638			−0.0543190	−	0.998524i	$-0.517299\pi$
$73$	−0.928203			−0.108638			−0.0543190	+	0.998524i	$0.517299\pi$
$74$	−3.46410	−	0.928203i	−0.402694	−	0.107901i
$75$		0			0
$76$	10.3923	+	6.00000i	1.19208	+	0.688247i
$77$	6.46410	+	7.46410i	0.736653	+	0.850613i
$78$	−15.2942	−	4.09808i	−1.73173	−	0.464016i
$79$			2.66025i			0.299302i	0.988739	+	0.149651i	$0.0478150\pi$
$79$			2.66025i			0.299302i	−0.988739	+	0.149651i	$0.952185\pi$
$80$		0			0
$81$	−9.00000			−1.00000
$82$	4.73205	+	1.26795i	0.522568	+	0.140022i
$83$			8.53590i			0.936937i	0.883480	+	0.468468i	$0.155194\pi$
$83$			8.53590i			0.936937i	−0.883480	+	0.468468i	$0.844806\pi$
$84$	−1.73205	−	9.00000i	−0.188982	−	0.981981i
$85$		0			0
$86$	0.732051	−	2.73205i	0.0789391	−	0.294605i
$87$			10.2679i			1.10084i
$88$	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
$92$	−9.46410	−	5.46410i	−0.986701	−	0.569672i
$93$		−	10.3923i		−	1.07763i
$94$	−2.36603	−	0.633975i	−0.244037	−	0.0653895i
$95$		0			0
$96$	−2.53590	−	9.46410i	−0.258819	−	0.965926i
$97$	−7.39230			−0.750575			−0.375287	−	0.926908i	$-0.622456\pi$
$97$	−7.39230			−0.750575			−0.375287	+	0.926908i	$0.622456\pi$
$98$	9.09808	+	3.90192i	0.919044	+	0.394154i
$99$		0			0
$100$		0			0
$101$			8.53590i			0.849354i	0.905345	+	0.424677i	$0.139612\pi$
$101$			8.53590i			0.849354i	−0.905345	+	0.424677i	$0.860388\pi$
$102$	1.09808	+	0.294229i	0.108726	+	0.0291330i
$103$		−	17.1962i		−	1.69439i	−0.531284	−	0.847194i	$-0.678290\pi$
$103$		−	17.1962i		−	1.69439i	0.531284	−	0.847194i	$-0.321710\pi$
$104$	12.9282	−	12.9282i	1.26771	−	1.26771i
$105$		0			0
$106$	−2.73205	−	0.732051i	−0.265360	−	0.0711031i
$107$	18.3923			1.77805			0.889026	−	0.457857i	$-0.151383\pi$
$107$	18.3923			1.77805			0.889026	+	0.457857i	$0.151383\pi$
$108$	5.19615	−	9.00000i	0.500000	−	0.866025i
$109$	−15.9282			−1.52565			−0.762823	−	0.646608i	$-0.776187\pi$
$109$	−15.9282			−1.52565			−0.762823	+	0.646608i	$0.776187\pi$
$110$		0			0
$111$	−4.39230			−0.416899
$112$	10.0000	+	3.46410i	0.944911	+	0.327327i
$113$		−	1.46410i		−	0.137731i	−0.997626	−	0.0688655i	$-0.978062\pi$
$113$		−	1.46410i		−	0.137731i	0.997626	−	0.0688655i	$-0.0219379\pi$
$114$	14.1962	+	3.80385i	1.32959	+	0.356263i
$115$		0			0
$116$	−10.2679	−	5.92820i	−0.953355	−	0.550420i
$117$		0			0
$118$	1.26795	−	4.73205i	0.116724	−	0.435621i
$119$	−0.928203	+	0.803848i	−0.0850883	+	0.0736886i
$120$		0			0
$121$	−2.92820			−0.266200
$122$	3.46410	+	0.928203i	0.313625	+	0.0840356i
$123$	6.00000			0.541002
$124$	10.3923	+	6.00000i	0.933257	+	0.538816i
$125$		0			0
$126$		0			0
$127$	−8.53590			−0.757438			−0.378719	−	0.925512i	$-0.623635\pi$
$127$	−8.53590			−0.757438			−0.378719	+	0.925512i	$0.623635\pi$
$128$	10.9282	+	2.92820i	0.965926	+	0.258819i
$129$		−	3.46410i		−	0.304997i
$130$		0			0
$131$	9.46410			0.826882			0.413441	−	0.910531i	$-0.364327\pi$
$131$	9.46410			0.826882			0.413441	+	0.910531i	$0.364327\pi$
$132$	11.1962	+	6.46410i	0.974500	+	0.562628i
$133$	−12.0000	+	10.3923i	−1.04053	+	0.901127i
$134$	−1.26795	+	4.73205i	−0.109534	+	0.408787i
$135$		0			0
$136$	−0.928203	+	0.928203i	−0.0795928	+	0.0795928i
$137$			0.392305i			0.0335169i	0.999860	+	0.0167584i	$0.00533462\pi$
$137$			0.392305i			0.0335169i	−0.999860	+	0.0167584i	$0.994665\pi$
$138$	−12.9282	−	3.46410i	−1.10052	−	0.294884i
$139$	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
$142$	−0.732051	−	0.196152i	−0.0614323	−	0.0164607i
$143$			24.1244i			2.01738i
$144$		0			0
$145$		0			0
$146$	0.339746	−	1.26795i	0.0281176	−	0.104936i
$147$	12.0000	+	1.73205i	0.989743	+	0.142857i
$148$	2.53590	−	4.39230i	0.208450	−	0.361045i
$149$	−16.9282			−1.38681			−0.693406	−	0.720547i	$-0.743891\pi$
$149$	−16.9282			−1.38681			−0.693406	+	0.720547i	$0.743891\pi$
$150$		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
$154$	−12.5622	+	6.09808i	−1.01229	+	0.491397i
$155$		0			0
$156$	11.1962	−	19.3923i	0.896410	−	1.55263i
$157$	19.8564			1.58471			0.792357	−	0.610058i	$-0.208854\pi$
$157$	19.8564			1.58471			0.792357	+	0.610058i	$0.208854\pi$
$158$	−3.63397	−	0.973721i	−0.289103	−	0.0774650i
$159$	−3.46410			−0.274721
$160$		0			0
$161$	10.9282	−	9.46410i	0.861263	−	0.745876i
$162$	3.29423	−	12.2942i	0.258819	−	0.965926i
$163$	−20.7846			−1.62798			−0.813988	−	0.580881i	$-0.802708\pi$
$163$	−20.7846			−1.62798			−0.813988	+	0.580881i	$0.802708\pi$
$164$	−3.46410	+	6.00000i	−0.270501	+	0.468521i
$165$		0			0
$166$	−11.6603	−	3.12436i	−0.905011	−	0.242497i
$167$		−	5.19615i		−	0.402090i	−0.979582	−	0.201045i	$-0.935566\pi$
$167$		−	5.19615i		−	0.402090i	0.979582	−	0.201045i	$-0.0644338\pi$
$168$	12.9282	+	0.928203i	0.997433	+	0.0716124i
$169$	28.7846			2.21420
$170$		0			0
$171$		0			0
$172$	3.46410	+	2.00000i	0.264135	+	0.152499i
$173$	20.3205			1.54494			0.772470	−	0.635051i	$-0.219021\pi$
$173$	20.3205			1.54494			0.772470	+	0.635051i	$0.219021\pi$
$174$	−14.0263	−	3.75833i	−1.06333	−	0.284918i
$175$		0			0
$176$	−12.9282	+	7.46410i	−0.974500	+	0.562628i
$177$		−	6.00000i		−	0.450988i
$178$	−12.9282	−	3.46410i	−0.969010	−	0.259645i
$179$		−	14.3923i		−	1.07573i	−0.843031	−	0.537866i	$-0.819231\pi$
$179$		−	14.3923i		−	1.07573i	0.843031	−	0.537866i	$-0.180769\pi$
$180$		0			0
$181$		−	12.9282i		−	0.960946i	−0.877010	−	0.480473i	$-0.840465\pi$
$181$		−	12.9282i		−	0.960946i	0.877010	−	0.480473i	$-0.159535\pi$
$182$	10.5622	+	21.7583i	0.782921	+	1.61283i
$183$	4.39230			0.324689
$184$	10.9282	−	10.9282i	0.805638	−	0.805638i
$185$		0			0
$186$	14.1962	+	3.80385i	1.04091	+	0.278912i
$187$		−	1.73205i		−	0.126660i
$188$	1.73205	−	3.00000i	0.126323	−	0.218797i
$189$	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	$-0.0368896\pi$
$192$	13.8564			1.00000
$193$		−	2.53590i		−	0.182538i	−0.995826	−	0.0912690i	$-0.970908\pi$
$193$		−	2.53590i		−	0.182538i	0.995826	−	0.0912690i	$-0.0290923\pi$
$194$	2.70577	−	10.0981i	0.194263	−	0.725000i
$195$		0			0
$196$	−8.66025	+	11.0000i	−0.618590	+	0.785714i
$197$		−	21.3205i		−	1.51902i	−0.650494	−	0.759512i	$-0.725438\pi$
$197$		−	21.3205i		−	1.51902i	0.650494	−	0.759512i	$-0.274562\pi$
$198$		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
$202$	−11.6603	−	3.12436i	−0.820413	−	0.219829i
$203$	11.8564	−	10.2679i	0.832157	−	0.720669i
$204$	−0.803848	+	1.39230i	−0.0562806	+	0.0974808i
$205$		0			0
$206$	23.4904	+	6.29423i	1.63665	+	0.438540i
$207$		0			0
$208$	12.9282	+	22.3923i	0.896410	+	1.55263i
$209$		−	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	$-0.0796753\pi$
$212$	2.00000	−	3.46410i	0.137361	−	0.237915i
$213$	−0.928203			−0.0635994
$214$	−6.73205	+	25.1244i	−0.460194	+	1.71747i
$215$		0			0
$216$	10.3923	+	10.3923i	0.707107	+	0.707107i
$217$	−12.0000	+	10.3923i	−0.814613	+	0.705476i
$218$	5.83013	−	21.7583i	0.394866	−	1.47366i
$219$		−	1.60770i		−	0.108638i
$220$		0			0
$221$	−3.00000			−0.201802
$222$	1.60770	−	6.00000i	0.107901	−	0.402694i
$223$		−	10.2679i		−	0.687593i	−0.939044	−	0.343796i	$-0.888287\pi$
$223$		−	10.2679i		−	0.687593i	0.939044	−	0.343796i	$-0.111713\pi$
$224$	−8.39230	+	12.3923i	−0.560734	+	0.827996i
$225$		0			0
$226$	2.00000	+	0.535898i	0.133038	+	0.0356474i
$227$		−	3.33975i		−	0.221667i	−0.993839	−	0.110833i	$-0.964648\pi$
$227$		−	3.33975i		−	0.221667i	0.993839	−	0.110833i	$-0.0353520\pi$
$228$	−10.3923	+	18.0000i	−0.688247	+	1.19208i
$229$		−	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
$233$		−	22.9282i		−	1.50208i	−0.660259	−	0.751038i	$-0.729553\pi$
$233$		−	22.9282i		−	1.50208i	0.660259	−	0.751038i	$-0.270447\pi$
$234$		0			0
$235$		0			0
$236$	6.00000	+	3.46410i	0.390567	+	0.225494i
$237$	−4.60770			−0.299302
$238$	−0.758330	−	1.56218i	−0.0491552	−	0.101261i
$239$		−	27.9808i		−	1.80993i	−0.425491	−	0.904963i	$-0.639899\pi$
$239$		−	27.9808i		−	1.80993i	0.425491	−	0.904963i	$-0.360101\pi$
$240$		0			0
$241$			4.39230i			0.282933i	0.989943	+	0.141467i	$0.0451818\pi$
$241$			4.39230i			0.282933i	−0.989943	+	0.141467i	$0.954818\pi$
$242$	1.07180	−	4.00000i	0.0688977	−	0.257130i
$243$		0			0
$244$	−2.53590	+	4.39230i	−0.162344	+	0.281189i
$245$		0			0
$246$	−2.19615	+	8.19615i	−0.140022	+	0.522568i
$247$	−38.7846			−2.46781
$248$	−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	$-0.481340\pi$
$251$	1.85641			0.117175			0.0585877	+	0.998282i	$0.481340\pi$
$252$		0			0
$253$			20.3923i			1.28205i
$254$	3.12436	−	11.6603i	0.196040	−	0.731629i
$255$		0			0
$256$	−8.00000	+	13.8564i	−0.500000	+	0.866025i
$257$	−6.00000			−0.374270			−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000			−0.374270			−0.187135	+	0.982334i	$0.559920\pi$
$258$	4.73205	+	1.26795i	0.294605	+	0.0789391i
$259$	4.39230	+	5.07180i	0.272925	+	0.315146i
$260$		0			0
$261$		0			0
$262$	−3.46410	+	12.9282i	−0.214013	+	0.798707i
$263$	4.53590			0.279695			0.139848	−	0.990173i	$-0.455339\pi$
$263$	4.53590			0.279695			0.139848	+	0.990173i	$0.455339\pi$
$264$	−12.9282	+	12.9282i	−0.795676	+	0.795676i
$265$		0			0
$266$	−9.80385	−	20.1962i	−0.601112	−	1.23831i
$267$	−16.3923			−1.00319
$268$	−6.00000	−	3.46410i	−0.366508	−	0.211604i
$269$		−	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	$-0.475459\pi$
$271$	2.53590			0.154045			0.0770224	+	0.997029i	$0.475459\pi$
$272$	−0.928203	−	1.60770i	−0.0562806	−	0.0974808i
$273$	19.3923	+	22.3923i	1.17368	+	1.35524i
$274$	−0.535898	−	0.143594i	−0.0323748	−	0.00867480i
$275$		0			0
$276$	9.46410	−	16.3923i	0.569672	−	0.986701i
$277$			24.7846i			1.48916i	0.667532	+	0.744581i	$0.267351\pi$
$277$			24.7846i			1.48916i	−0.667532	+	0.744581i	$0.732649\pi$
$278$	−2.53590	+	9.46410i	−0.152093	+	0.567619i
$279$		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	$0.443418\pi$
$282$	1.09808	−	4.09808i	0.0653895	−	0.244037i
$283$		−	12.1244i		−	0.720718i	−0.932814	−	0.360359i	$-0.882654\pi$
$283$		−	12.1244i		−	0.720718i	0.932814	−	0.360359i	$-0.117346\pi$
$284$	0.535898	−	0.928203i	0.0317997	−	0.0550787i
$285$		0			0
$286$	−32.9545	−	8.83013i	−1.94864	−	0.522136i
$287$	−6.00000	−	6.92820i	−0.354169	−	0.408959i
$288$		0			0
$289$	−16.7846			−0.987330
$290$		0			0
$291$		−	12.8038i		−	0.750575i
$292$	1.60770	+	0.928203i	0.0940832	+	0.0543190i
$293$	−14.3205			−0.836613			−0.418307	−	0.908306i	$-0.637376\pi$
$293$	−14.3205			−0.836613			−0.418307	+	0.908306i	$0.637376\pi$
$294$	−6.75833	+	15.7583i	−0.394154	+	0.919044i
$295$		0			0
$296$	5.07180	+	5.07180i	0.294792	+	0.294792i
$297$	−19.3923			−1.12526
$298$	6.19615	−	23.1244i	0.358933	−	1.33956i
$299$	35.3205			2.04264
$300$		0			0
$301$	−4.00000	+	3.46410i	−0.230556	+	0.199667i
$302$	−6.56218	−	1.75833i	−0.377611	−	0.101181i
$303$	−14.7846			−0.849354
$304$	−12.0000	−	20.7846i	−0.688247	−	1.19208i
$305$		0			0
$306$		0			0
$307$			1.73205i			0.0988534i	0.998778	+	0.0494267i	$0.0157394\pi$
$307$			1.73205i			0.0988534i	−0.998778	+	0.0494267i	$0.984261\pi$
$308$	−3.73205	−	19.3923i	−0.212653	−	1.10498i
$309$	29.7846			1.69439
$310$		0			0
$311$	−19.8564			−1.12595			−0.562977	−	0.826473i	$-0.690344\pi$
$311$	−19.8564			−1.12595			−0.562977	+	0.826473i	$0.690344\pi$
$312$	22.3923	+	22.3923i	1.26771	+	1.26771i
$313$	−24.4641			−1.38279			−0.691396	−	0.722476i	$-0.743004\pi$
$313$	−24.4641			−1.38279			−0.691396	+	0.722476i	$0.743004\pi$
$314$	−7.26795	+	27.1244i	−0.410154	+	1.53072i
$315$		0			0
$316$	2.66025	−	4.60770i	0.149651	−	0.259203i
$317$			16.9282i			0.950783i	0.879774	+	0.475391i	$0.157694\pi$
$317$			16.9282i			0.950783i	−0.879774	+	0.475391i	$0.842306\pi$
$318$	1.26795	−	4.73205i	0.0711031	−	0.265360i
$319$			22.1244i			1.23873i
$320$		0			0
$321$			31.8564i			1.77805i
$322$	8.92820	+	18.3923i	0.497549	+	1.02496i
$323$	2.78461			0.154940
$324$	15.5885	+	9.00000i	0.866025	+	0.500000i
$325$		0			0
$326$	7.60770	−	28.3923i	0.421351	−	1.57250i
$327$		−	27.5885i		−	1.52565i
$328$	−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	$-0.0492521\pi$
$332$	8.53590	−	14.7846i	0.468468	−	0.811411i
$333$		0			0
$334$	7.09808	+	1.90192i	0.388389	+	0.104069i
$335$		0			0
$336$	−6.00000	+	17.3205i	−0.327327	+	0.944911i
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	−10.5359	+	39.3205i	−0.573077	+	2.13875i
$339$	2.53590			0.137731
$340$		0			0
$341$		−	22.3923i		−	1.21261i
$342$		0			0
$343$	−10.0000	−	15.5885i	−0.539949	−	0.841698i
$344$	−4.00000	+	4.00000i	−0.215666	+	0.215666i
$345$		0			0
$346$	−7.43782	+	27.7583i	−0.399860	+	1.49230i
$347$	−14.2487			−0.764911			−0.382455	−	0.923974i	$-0.624921\pi$
$347$	−14.2487			−0.764911			−0.382455	+	0.923974i	$0.624921\pi$
$348$	10.2679	−	17.7846i	0.550420	−	0.953355i
$349$			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
$352$	−5.46410	−	20.3923i	−0.291238	−	1.08691i
$353$	13.3923			0.712800			0.356400	−	0.934333i	$-0.384004\pi$
$353$	13.3923			0.712800			0.356400	+	0.934333i	$0.384004\pi$
$354$	8.19615	+	2.19615i	0.435621	+	0.116724i
$355$		0			0
$356$	9.46410	−	16.3923i	0.501596	−	0.868790i
$357$	−1.39230	−	1.60770i	−0.0736886	−	0.0850883i
$358$	19.6603	+	5.26795i	1.03908	+	0.278420i
$359$			9.32051i			0.491918i	0.969280	+	0.245959i	$0.0791028\pi$
$359$			9.32051i			0.491918i	−0.969280	+	0.245959i	$0.920897\pi$
$360$		0			0
$361$	17.0000			0.894737
$362$	17.6603	+	4.73205i	0.928202	+	0.248711i
$363$		−	5.07180i		−	0.266200i
$364$	−33.5885	+	6.46410i	−1.76051	+	0.338811i
$365$		0			0
$366$	−1.60770	+	6.00000i	−0.0840356	+	0.313625i
$367$		−	36.1244i		−	1.88568i	−0.333251	−	0.942838i	$-0.608146\pi$
$367$		−	36.1244i		−	1.88568i	0.333251	−	0.942838i	$-0.391854\pi$
$368$	10.9282	+	18.9282i	0.569672	+	0.986701i
$369$		0			0
$370$		0			0
$371$	3.46410	+	4.00000i	0.179847	+	0.207670i
$372$	−10.3923	+	18.0000i	−0.538816	+	0.933257i
$373$		−	24.3923i		−	1.26299i	−0.775382	−	0.631493i	$-0.782443\pi$
$373$		−	24.3923i		−	1.26299i	0.775382	−	0.631493i	$-0.217557\pi$
$374$	2.36603	+	0.633975i	0.122344	+	0.0327820i
$375$		0			0
$376$	3.46410	+	3.46410i	0.178647	+	0.178647i
$377$	38.3205			1.97361
$378$	−17.4904	+	8.49038i	−0.899608	+	0.436698i
$379$		−	26.3923i		−	1.35568i	−0.735209	−	0.677841i	$-0.762916\pi$
$379$		−	26.3923i		−	1.35568i	0.735209	−	0.677841i	$-0.237084\pi$
$380$		0			0
$381$		−	14.7846i		−	0.757438i
$382$	4.36603	+	1.16987i	0.223385	+	0.0598559i
$383$			20.5359i			1.04934i	0.851307	+	0.524668i	$0.175810\pi$
$383$			20.5359i			1.04934i	−0.851307	+	0.524668i	$0.824190\pi$
$384$	−5.07180	+	18.9282i	−0.258819	+	0.965926i
$385$		0			0
$386$	3.46410	+	0.928203i	0.176318	+	0.0472443i
$387$		0			0
$388$	12.8038	+	7.39230i	0.650017	+	0.375287i
$389$	−6.85641			−0.347634			−0.173817	−	0.984778i	$-0.555610\pi$
$389$	−6.85641			−0.347634			−0.173817	+	0.984778i	$0.555610\pi$
$390$		0			0
$391$	−2.53590			−0.128246
$392$	−11.8564	−	15.8564i	−0.598839	−	0.800869i
$393$			16.3923i			0.826882i
$394$	29.1244	+	7.80385i	1.46726	+	0.393152i
$395$		0			0
$396$		0			0
$397$	−5.53590			−0.277839			−0.138919	−	0.990304i	$-0.544363\pi$
$397$	−5.53590			−0.277839			−0.138919	+	0.990304i	$0.544363\pi$
$398$	−1.26795	+	4.73205i	−0.0635566	+	0.237196i
$399$	−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	$0.703823\pi$
$402$	−8.19615	−	2.19615i	−0.408787	−	0.109534i
$403$	−38.7846			−1.93200
$404$	8.53590	−	14.7846i	0.424677	−	0.735562i
$405$		0			0
$406$	9.68653	+	19.9545i	0.480735	+	0.990324i
$407$	−9.46410			−0.469118
$408$	−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
$412$	−17.1962	+	29.7846i	−0.847194	+	1.46738i
$413$	−6.92820	+	6.00000i	−0.340915	+	0.295241i
$414$		0			0
$415$		0			0
$416$	−35.3205	+	9.46410i	−1.73173	+	0.464016i
$417$			12.0000i			0.587643i
$418$	30.5885	+	8.19615i	1.49613	+	0.400887i
$419$	−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	$0.346772\pi$
$422$	9.83013	+	2.63397i	0.478523	+	0.128220i
$423$		0			0
$424$	4.00000	+	4.00000i	0.194257	+	0.194257i
$425$		0			0
$426$	0.339746	−	1.26795i	0.0164607	−	0.0614323i
$427$	−4.39230	−	5.07180i	−0.212559	−	0.245441i
$428$	−31.8564	−	18.3923i	−1.53984	−	0.889026i
$429$	−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	$0.860765\pi$
$432$	−18.0000	+	10.3923i	−0.866025	+	0.500000i
$433$	−4.14359			−0.199128			−0.0995642	−	0.995031i	$-0.531745\pi$
$433$	−4.14359			−0.199128			−0.0995642	+	0.995031i	$0.531745\pi$
$434$	−9.80385	−	20.1962i	−0.470600	−	0.969446i
$435$		0			0
$436$	27.5885	+	15.9282i	1.32125	+	0.762823i
$437$	−32.7846			−1.56830
$438$	2.19615	+	0.588457i	0.104936	+	0.0281176i
$439$	−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
$442$	1.09808	−	4.09808i	0.0522302	−	0.194926i
$443$	26.0000			1.23530			0.617649	−	0.786454i	$-0.288085\pi$
$443$	26.0000			1.23530			0.617649	+	0.786454i	$0.288085\pi$
$444$	7.60770	+	4.39230i	0.361045	+	0.208450i
$445$		0			0
$446$	14.0263	+	3.75833i	0.664164	+	0.177962i
$447$		−	29.3205i		−	1.38681i
$448$	−13.8564	−	16.0000i	−0.654654	−	0.755929i
$449$	1.92820			0.0909975			0.0454988	−	0.998964i	$-0.485512\pi$
$449$	1.92820			0.0909975			0.0454988	+	0.998964i	$0.485512\pi$
$450$		0			0
$451$	12.9282			0.608765
$452$	−1.46410	+	2.53590i	−0.0688655	+	0.119279i
$453$	−8.32051			−0.390932
$454$	4.56218	+	1.22243i	0.214114	+	0.0573716i
$455$		0			0
$456$	−20.7846	−	20.7846i	−0.973329	−	0.973329i
$457$		−	27.4641i		−	1.28472i	−0.766405	−	0.642358i	$-0.777956\pi$
$457$		−	27.4641i		−	1.28472i	0.766405	−	0.642358i	$-0.222044\pi$
$458$	21.1244	+	5.66025i	0.987076	+	0.264486i
$459$		−	2.41154i		−	0.112561i
$460$		0			0
$461$			27.7128i			1.29071i	0.763881	+	0.645357i	$0.223291\pi$
$461$			27.7128i			1.29071i	−0.763881	+	0.645357i	$0.776709\pi$
$462$	−10.5622	−	21.7583i	−0.491397	−	1.01229i
$463$	−4.39230			−0.204128			−0.102064	−	0.994778i	$-0.532545\pi$
$463$	−4.39230			−0.204128			−0.102064	+	0.994778i	$0.532545\pi$
$464$	11.8564	+	20.5359i	0.550420	+	0.953355i
$465$		0			0
$466$	31.3205	+	8.39230i	1.45089	+	0.388766i
$467$			22.5167i			1.04195i	0.853573	+	0.520973i	$0.174431\pi$
$467$			22.5167i			1.04195i	−0.853573	+	0.520973i	$0.825569\pi$
$468$		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
$473$		−	7.46410i		−	0.343200i
$474$	1.68653	−	6.29423i	0.0774650	−	0.289103i
$475$		0			0
$476$	2.41154	−	0.464102i	0.110533	−	0.0212721i
$477$		0			0
$478$	38.2224	+	10.2417i	1.74825	+	0.468443i
$479$	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
$482$	−6.00000	−	1.60770i	−0.273293	−	0.0732285i
$483$	16.3923	+	18.9282i	0.745876	+	0.861263i
$484$	5.07180	+	2.92820i	0.230536	+	0.133100i
$485$		0			0
$486$		0			0
$487$	28.7846			1.30436			0.652178	−	0.758066i	$-0.273856\pi$
$487$	28.7846			1.30436			0.652178	+	0.758066i	$0.273856\pi$
$488$	−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	$-0.279746\pi$
$492$	−10.3923	−	6.00000i	−0.468521	−	0.270501i
$493$	−2.75129			−0.123912
$494$	14.1962	−	52.9808i	0.638715	−	2.38372i
$495$		0			0
$496$	−12.0000	−	20.7846i	−0.538816	−	0.933257i
$497$	0.928203	+	1.07180i	0.0416356	+	0.0480767i
$498$	5.41154	−	20.1962i	0.242497	−	0.905011i
$499$			5.58846i			0.250174i	0.992146	+	0.125087i	$0.0399210\pi$
$499$			5.58846i			0.250174i	−0.992146	+	0.125087i	$0.960079\pi$
$500$		0			0
$501$	9.00000			0.402090
$502$	−0.679492	+	2.53590i	−0.0303272	+	0.113183i
$503$			15.5885i			0.695055i	0.937670	+	0.347527i	$0.112979\pi$
$503$			15.5885i			0.695055i	−0.937670	+	0.347527i	$0.887021\pi$
$504$		0			0
$505$		0			0
$506$	−27.8564	−	7.46410i	−1.23837	−	0.331820i
$507$			49.8564i			2.21420i
$508$	14.7846	+	8.53590i	0.655961	+	0.378719i
$509$		−	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
$513$		−	31.1769i		−	1.37649i
$514$	2.19615	−	8.19615i	0.0968681	−	0.361517i
$515$		0			0
$516$	−3.46410	+	6.00000i	−0.152499	+	0.264135i
$517$	−6.46410			−0.284291
$518$	−8.53590	+	4.14359i	−0.375046	+	0.182059i
$519$			35.1962i			1.54494i
$520$		0			0
$521$		−	34.3923i		−	1.50675i	−0.657589	−	0.753377i	$-0.728424\pi$
$521$		−	34.3923i		−	1.50675i	0.657589	−	0.753377i	$-0.271576\pi$
$522$		0			0
$523$			24.2487i			1.06032i	0.847897	+	0.530161i	$0.177869\pi$
$523$			24.2487i			1.06032i	−0.847897	+	0.530161i	$0.822131\pi$
$524$	−16.3923	−	9.46410i	−0.716101	−	0.413441i
$525$		0			0
$526$	−1.66025	+	6.19615i	−0.0723905	+	0.270165i
$527$	2.78461			0.121300
$528$	−12.9282	−	22.3923i	−0.562628	−	0.974500i
$529$	6.85641			0.298105
$530$		0			0
$531$		0			0
$532$	31.1769	−	6.00000i	1.35169	−	0.260133i
$533$		−	22.3923i		−	0.969918i
$534$	6.00000	−	22.3923i	0.259645	−	0.969010i
$535$		0			0
$536$	6.92820	−	6.92820i	0.299253	−	0.299253i
$537$	24.9282			1.07573
$538$	16.3923	+	4.39230i	0.706722	+	0.189366i
$539$	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	$0.758742\pi$
$542$	−0.928203	+	3.46410i	−0.0398697	+	0.148796i
$543$	22.3923			0.960946
$544$	2.53590	−	0.679492i	0.108726	−	0.0291330i
$545$		0			0
$546$	−37.6865	+	18.2942i	−1.61283	+	0.782921i
$547$	14.5359			0.621510			0.310755	−	0.950490i	$-0.399418\pi$
$547$	14.5359			0.621510			0.310755	+	0.950490i	$0.399418\pi$
$548$	0.392305	−	0.679492i	0.0167584	−	0.0290265i
$549$		0			0
$550$		0			0
$551$	−35.5692			−1.51530
$552$	18.9282	+	18.9282i	0.805638	+	0.805638i
$553$	4.60770	+	5.32051i	0.195939	+	0.226251i
$554$	−33.8564	−	9.07180i	−1.43842	−	0.385424i
$555$		0			0
$556$	−12.0000	−	6.92820i	−0.508913	−	0.293821i
$557$		−	5.85641i		−	0.248144i	−0.992273	−	0.124072i	$-0.960405\pi$
$557$		−	5.85641i		−	0.248144i	0.992273	−	0.124072i	$-0.0395954\pi$
$558$		0			0
$559$	−12.9282			−0.546805
$560$		0			0
$561$	3.00000			0.126660
$562$	−2.16987	+	8.09808i	−0.0915306	+	0.341597i
$563$		−	43.1769i		−	1.81969i	−0.414948	−	0.909845i	$-0.636200\pi$
$563$		−	43.1769i		−	1.81969i	0.414948	−	0.909845i	$-0.363800\pi$
$564$	5.19615	+	3.00000i	0.218797	+	0.126323i
$565$		0			0
$566$	16.5622	+	4.43782i	0.696160	+	0.186536i
$567$	−18.0000	+	15.5885i	−0.755929	+	0.654654i
$568$	1.07180	+	1.07180i	0.0449716	+	0.0449716i
$569$	−20.9282			−0.877356			−0.438678	−	0.898644i	$-0.644553\pi$
$569$	−20.9282			−0.877356			−0.438678	+	0.898644i	$0.644553\pi$
$570$		0			0
$571$		−	41.3205i		−	1.72921i	−0.502453	−	0.864605i	$-0.667569\pi$
$571$		−	41.3205i		−	1.72921i	0.502453	−	0.864605i	$-0.332431\pi$
$572$	24.1244	−	41.7846i	1.00869	−	1.74710i
$573$	5.53590			0.231265
$574$	11.6603	−	5.66025i	0.486690	−	0.236254i
$575$		0			0
$576$		0			0
$577$	1.39230			0.0579624			0.0289812	−	0.999580i	$-0.490774\pi$
$577$	1.39230			0.0579624			0.0289812	+	0.999580i	$0.490774\pi$
$578$	6.14359	−	22.9282i	0.255540	−	0.953688i
$579$	4.39230			0.182538
$580$		0			0
$581$	14.7846	+	17.0718i	0.613369	+	0.708257i
$582$	17.4904	+	4.68653i	0.725000	+	0.194263i
$583$	−7.46410			−0.309132
$584$	−1.85641	+	1.85641i	−0.0768186	+	0.0768186i
$585$		0			0
$586$	5.24167	−	19.5622i	0.216531	−	0.808106i
$587$			27.4641i			1.13356i	0.823868	+	0.566782i	$0.191812\pi$
$587$			27.4641i			1.13356i	−0.823868	+	0.566782i	$0.808188\pi$
$588$	−19.0526	−	15.0000i	−0.785714	−	0.618590i
$589$	36.0000			1.48335
$590$		0			0
$591$	36.9282			1.51902
$592$	−8.78461	+	5.07180i	−0.361045	+	0.208450i
$593$	23.5359			0.966504			0.483252	−	0.875481i	$-0.339456\pi$
$593$	23.5359			0.966504			0.483252	+	0.875481i	$0.339456\pi$
$594$	7.09808	−	26.4904i	0.291238	−	1.08691i
$595$		0			0
$596$	29.3205	+	16.9282i	1.20101	+	0.693406i
$597$			6.00000i			0.245564i
$598$	−12.9282	+	48.2487i	−0.528674	+	1.97304i
$599$		−	14.1244i		−	0.577106i	−0.957464	−	0.288553i	$-0.906826\pi$
$599$		−	14.1244i		−	0.577106i	0.957464	−	0.288553i	$-0.0931741\pi$
$600$		0			0
$601$		−	26.7846i		−	1.09257i	−0.837600	−	0.546284i	$-0.816042\pi$
$601$		−	26.7846i		−	1.09257i	0.837600	−	0.546284i	$-0.183958\pi$
$602$	−3.26795	−	6.73205i	−0.133192	−	0.274378i
$603$		0			0
$604$	4.80385	−	8.32051i	0.195466	−	0.338557i
$605$		0			0
$606$	5.41154	−	20.1962i	0.219829	−	0.820413i
$607$			18.8038i			0.763225i	0.924322	+	0.381612i	$0.124631\pi$
$607$			18.8038i			0.763225i	−0.924322	+	0.381612i	$0.875369\pi$
$608$	32.7846	−	8.78461i	1.32959	−	0.356263i
$609$	17.7846	+	20.5359i	0.720669	+	0.832157i
$610$		0			0
$611$			11.1962i			0.452948i
$612$		0			0
$613$		−	10.0000i		−	0.403896i	−0.979396	−	0.201948i	$-0.935273\pi$
$613$		−	10.0000i		−	0.403896i	0.979396	−	0.201948i	$-0.0647272\pi$
$614$	−2.36603	−	0.633975i	−0.0954850	−	0.0255851i
$615$		0			0
$616$	27.8564	+	2.00000i	1.12237	+	0.0805823i
$617$			9.07180i			0.365217i	0.983186	+	0.182608i	$0.0584540\pi$
$617$			9.07180i			0.365217i	−0.983186	+	0.182608i	$0.941546\pi$
$618$	−10.9019	+	40.6865i	−0.438540	+	1.63665i
$619$	−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
$622$	7.26795	−	27.1244i	0.291418	−	1.08759i
$623$	16.3923	+	18.9282i	0.656744	+	0.758342i
$624$	−38.7846	+	22.3923i	−1.55263	+	0.896410i
$625$		0			0
$626$	8.95448	−	33.4186i	0.357893	−	1.33568i
$627$	38.7846			1.54891
$628$	−34.3923	−	19.8564i	−1.37240	−	0.792357i
$629$		−	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
$633$	12.4641			0.495404
$634$	−23.1244	−	6.19615i	−0.918385	−	0.246081i
$635$		0			0
$636$	6.00000	+	3.46410i	0.237915	+	0.137361i
$637$	6.46410	−	44.7846i	0.256117	−	1.77443i
$638$	−30.2224	−	8.09808i	−1.19652	−	0.320606i
$639$		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	$0.531029\pi$
$642$	−43.5167	−	11.6603i	−1.71747	−	0.460194i
$643$		−	31.0526i		−	1.22459i	−0.790628	−	0.612297i	$-0.790246\pi$
$643$		−	31.0526i		−	1.22459i	0.790628	−	0.612297i	$-0.209754\pi$
$644$	−28.3923	+	5.46410i	−1.11881	+	0.215316i
$645$		0			0
$646$	−1.01924	+	3.80385i	−0.0401014	+	0.149660i
$647$		−	10.3923i		−	0.408564i	−0.978912	−	0.204282i	$-0.934514\pi$
$647$		−	10.3923i		−	0.408564i	0.978912	−	0.204282i	$-0.0654859\pi$
$648$	−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
$652$	36.0000	+	20.7846i	1.40987	+	0.813988i
$653$			38.3923i			1.50241i	0.660071	+	0.751203i	$0.270526\pi$
$653$			38.3923i			1.50241i	−0.660071	+	0.751203i	$0.729474\pi$
$654$	37.6865	+	10.0981i	1.47366	+	0.394866i
$655$		0			0
$656$	12.0000	−	6.92820i	0.468521	−	0.270501i
$657$		0			0
$658$	−5.83013	+	2.83013i	−0.227282	+	0.110330i
$659$			20.8038i			0.810403i	0.914227	+	0.405201i	$0.132799\pi$
$659$			20.8038i			0.810403i	−0.914227	+	0.405201i	$0.867201\pi$
$660$		0			0
$661$			15.7128i			0.611158i	0.952167	+	0.305579i	$0.0988499\pi$
$661$			15.7128i			0.611158i	−0.952167	+	0.305579i	$0.901150\pi$
$662$	7.66025	+	2.05256i	0.297724	+	0.0797750i
$663$		−	5.19615i		−	0.201802i
$664$	17.0718	+	17.0718i	0.662514	+	0.662514i
$665$		0			0
$666$		0			0
$667$	32.3923			1.25424
$668$	−5.19615	+	9.00000i	−0.201045	+	0.348220i
$669$	17.7846			0.687593
$670$		0			0
$671$	9.46410			0.365358
$672$	−21.4641	−	14.5359i	−0.827996	−	0.560734i
$673$			49.1769i			1.89563i	0.318820	+	0.947815i	$0.396714\pi$
$673$			49.1769i			1.89563i	−0.318820	+	0.947815i	$0.603286\pi$
$674$		0			0
$675$		0			0
$676$	−49.8564	−	28.7846i	−1.91755	−	1.10710i
$677$	4.60770			0.177088			0.0885441	−	0.996072i	$-0.471779\pi$
$677$	4.60770			0.177088			0.0885441	+	0.996072i	$0.471779\pi$
$678$	−0.928203	+	3.46410i	−0.0356474	+	0.133038i
$679$	−14.7846	+	12.8038i	−0.567381	+	0.491367i
$680$		0			0
$681$	5.78461			0.221667
$682$	30.5885	+	8.19615i	1.17129	+	0.313847i
$683$	27.3205			1.04539			0.522695	−	0.852520i	$-0.324927\pi$
$683$	27.3205			1.04539			0.522695	+	0.852520i	$0.324927\pi$
$684$		0			0
$685$		0			0
$686$	24.9545	−	7.95448i	0.952767	−	0.303704i
$687$	26.7846			1.02190
$688$	−4.00000	−	6.92820i	−0.152499	−	0.264135i
$689$			12.9282i			0.492525i
$690$		0			0
$691$	46.6410			1.77431			0.887154	−	0.461474i	$-0.152679\pi$
$691$	46.6410			1.77431			0.887154	+	0.461474i	$0.152679\pi$
$692$	−35.1962	−	20.3205i	−1.33796	−	0.772470i
$693$		0			0
$694$	5.21539	−	19.4641i	0.197974	−	0.738847i
$695$		0			0
$696$	20.5359	+	20.5359i	0.778411	+	0.778411i
$697$			1.60770i			0.0608958i
$698$	−40.0526	−	10.7321i	−1.51601	−	0.406214i
$699$	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	$0.477231\pi$
$702$	−45.8827	−	12.2942i	−1.73173	−	0.464016i
$703$		−	15.2154i		−	0.573859i
$704$	29.8564			1.12526
$705$		0			0
$706$	−4.90192	+	18.2942i	−0.184486	+	0.688512i
$707$	14.7846	+	17.0718i	0.556032	+	0.642051i
$708$	−6.00000	+	10.3923i	−0.225494	+	0.390567i
$709$	−21.0000			−0.788672			−0.394336	−	0.918966i	$-0.629025\pi$
$709$	−21.0000			−0.788672			−0.394336	+	0.918966i	$0.629025\pi$
$710$		0			0
$711$		0			0
$712$	18.9282	+	18.9282i	0.709364	+	0.709364i
$713$	−32.7846			−1.22779
$714$	2.70577	−	1.31347i	0.101261	−	0.0491552i
$715$		0			0
$716$	−14.3923	+	24.9282i	−0.537866	+	0.931611i
$717$	48.4641			1.80993
$718$	−12.7321	−	3.41154i	−0.475156	−	0.127318i
$719$	−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
$722$	−6.22243	+	23.2224i	−0.231575	+	0.864249i
$723$	−7.60770			−0.282933
$724$	−12.9282	+	22.3923i	−0.480473	+	0.832203i
$725$		0			0
$726$	6.92820	+	1.85641i	0.257130	+	0.0688977i
$727$			34.3923i			1.27554i	0.770227	+	0.637770i	$0.220143\pi$
$727$			34.3923i			1.27554i	−0.770227	+	0.637770i	$0.779857\pi$
$728$	3.46410	−	48.2487i	0.128388	−	1.78822i
$729$	−27.0000			−1.00000
$730$		0			0
$731$	0.928203			0.0343308
$732$	−7.60770	−	4.39230i	−0.281189	−	0.162344i
$733$	18.4641			0.681987			0.340994	−	0.940066i	$-0.389237\pi$
$733$	18.4641			0.681987			0.340994	+	0.940066i	$0.389237\pi$
$734$	49.3468	+	13.2224i	1.82142	+	0.488049i
$735$		0			0
$736$	−29.8564	+	8.00000i	−1.10052	+	0.294884i
$737$			12.9282i			0.476216i
$738$		0			0
$739$			7.73205i			0.284428i	0.989836	+	0.142214i	$0.0454221\pi$
$739$			7.73205i			0.284428i	−0.989836	+	0.142214i	$0.954578\pi$
$740$		0			0
$741$		−	67.1769i		−	2.46781i
$742$	−6.73205	+	3.26795i	−0.247141	+	0.119970i
$743$	−9.60770			−0.352472			−0.176236	−	0.984348i	$-0.556392\pi$
$743$	−9.60770			−0.352472			−0.176236	+	0.984348i	$0.556392\pi$
$744$	−20.7846	−	20.7846i	−0.762001	−	0.762001i
$745$		0			0
$746$	33.3205	+	8.92820i	1.21995	+	0.326885i
$747$		0			0
$748$	−1.73205	+	3.00000i	−0.0633300	+	0.109691i
$749$	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	$0.967488\pi$
$752$	−6.00000	+	3.46410i	−0.218797	+	0.126323i
$753$			3.21539i			0.117175i
$754$	−14.0263	+	52.3468i	−0.510807	+	1.90636i
$755$		0			0
$756$	−5.19615	−	27.0000i	−0.188982	−	0.981981i
$757$		−	10.1436i		−	0.368675i	−0.982863	−	0.184338i	$-0.940986\pi$
$757$		−	10.1436i		−	0.368675i	0.982863	−	0.184338i	$-0.0590140\pi$
$758$	36.0526	+	9.66025i	1.30949	+	0.350876i
$759$	−35.3205			−1.28205
$760$		0			0
$761$		−	6.24871i		−	0.226516i	−0.993566	−	0.113258i	$-0.963871\pi$
$761$		−	6.24871i		−	0.226516i	0.993566	−	0.113258i	$-0.0361286\pi$
$762$	20.1962	+	5.41154i	0.731629	+	0.196040i
$763$	−31.8564	+	27.5885i	−1.15328	+	0.998769i
$764$	−3.19615	+	5.53590i	−0.115633	+	0.200282i
$765$		0			0
$766$	−28.0526	−	7.51666i	−1.01358	−	0.271588i
$767$	−22.3923			−0.808539
$768$	−24.0000	−	13.8564i	−0.866025	−	0.500000i
$769$			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
$772$	−2.53590	+	4.39230i	−0.0912690	+	0.158083i
$773$	−12.4641			−0.448303			−0.224151	−	0.974554i	$-0.571961\pi$
$773$	−12.4641			−0.448303			−0.224151	+	0.974554i	$0.571961\pi$
$774$		0			0
$775$		0			0
$776$	−14.7846	+	14.7846i	−0.530737	+	0.530737i
$777$	−8.78461	+	7.60770i	−0.315146	+	0.272925i
$778$	2.50962	−	9.36603i	0.0899742	−	0.335788i
$779$			20.7846i			0.744686i
$780$		0			0
$781$	−2.00000			−0.0715656
$782$	0.928203	−	3.46410i	0.0331925	−	0.123876i
$783$			30.8038i			1.10084i
$784$	26.0000	−	10.3923i	0.928571	−	0.371154i
$785$		0			0
$786$	−22.3923	−	6.00000i	−0.798707	−	0.214013i

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.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.366025 + 1.36603i) q^{2} +1.73205i q^{3} +(-1.73205 - 1.00000i) q^{4} +(-2.36603 - 0.633975i) q^{6} +(2.00000 - 1.73205i) q^{7} +(2.00000 - 2.00000i) q^{8} +O(q^{10})\)	\(q+(-0.366025 + 1.36603i) q^{2} +1.73205i q^{3} +(-1.73205 - 1.00000i) q^{4} +(-2.36603 - 0.633975i) q^{6} +(2.00000 - 1.73205i) q^{7} +(2.00000 - 2.00000i) q^{8} +3.73205i q^{11} +(1.73205 - 3.00000i) q^{12} +6.46410 q^{13} +(1.63397 + 3.36603i) q^{14} +(2.00000 + 3.46410i) q^{16} -0.464102 q^{17} -6.00000 q^{19} +(3.00000 + 3.46410i) q^{21} +(-5.09808 - 1.36603i) q^{22} +5.46410 q^{23} +(3.46410 + 3.46410i) q^{24} +(-2.36603 + 8.83013i) q^{26} +5.19615i q^{27} +(-5.19615 + 1.00000i) q^{28} +5.92820 q^{29} -6.00000 q^{31} +(-5.46410 + 1.46410i) q^{32} -6.46410 q^{33} +(0.169873 - 0.633975i) q^{34} +2.53590i q^{37} +(2.19615 - 8.19615i) q^{38} +11.1962i q^{39} -3.46410i q^{41} +(-5.83013 + 2.83013i) q^{42} -2.00000 q^{43} +(3.73205 - 6.46410i) q^{44} +(-2.00000 + 7.46410i) q^{46} +1.73205i q^{47} +(-6.00000 + 3.46410i) q^{48} +(1.00000 - 6.92820i) q^{49} -0.803848i q^{51} +(-11.1962 - 6.46410i) q^{52} +2.00000i q^{53} +(-7.09808 - 1.90192i) q^{54} +(0.535898 - 7.46410i) q^{56} -10.3923i q^{57} +(-2.16987 + 8.09808i) q^{58} -3.46410 q^{59} -2.53590i q^{61} +(2.19615 - 8.19615i) q^{62} -8.00000i q^{64} +(2.36603 - 8.83013i) q^{66} +3.46410 q^{67} +(0.803848 + 0.464102i) q^{68} +9.46410i q^{69} +0.535898i q^{71} -0.928203 q^{73} +(-3.46410 - 0.928203i) q^{74} +(10.3923 + 6.00000i) q^{76} +(6.46410 + 7.46410i) q^{77} +(-15.2942 - 4.09808i) q^{78} +2.66025i q^{79} -9.00000 q^{81} +(4.73205 + 1.26795i) q^{82} +8.53590i q^{83} +(-1.73205 - 9.00000i) q^{84} +(0.732051 - 2.73205i) q^{86} +10.2679i q^{87} +(7.46410 + 7.46410i) q^{88} +9.46410i q^{89} +(12.9282 - 11.1962i) q^{91} +(-9.46410 - 5.46410i) q^{92} -10.3923i q^{93} +(-2.36603 - 0.633975i) q^{94} +(-2.53590 - 9.46410i) q^{96} -7.39230 q^{97} +(9.09808 + 3.90192i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} - 6 q^{6} + 8 q^{7} + 8 q^{8}+O(q^{10}) \) 4 * q + 2 * q^2 - 6 * q^6 + 8 * q^7 + 8 * q^8	\( 4 q + 2 q^{2} - 6 q^{6} + 8 q^{7} + 8 q^{8} + 12 q^{13} + 10 q^{14} + 8 q^{16} + 12 q^{17} - 24 q^{19} + 12 q^{21} - 10 q^{22} + 8 q^{23} - 6 q^{26} - 4 q^{29} - 24 q^{31} - 8 q^{32} - 12 q^{33} + 18 q^{34} - 12 q^{38} - 6 q^{42} - 8 q^{43} + 8 q^{44} - 8 q^{46} - 24 q^{48} + 4 q^{49} - 24 q^{52} - 18 q^{54} + 16 q^{56} - 26 q^{58} - 12 q^{62} + 6 q^{66} + 24 q^{68} + 24 q^{73} + 12 q^{77} - 30 q^{78} - 36 q^{81} + 12 q^{82} - 4 q^{86} + 16 q^{88} + 24 q^{91} - 24 q^{92} - 6 q^{94} - 24 q^{96} + 12 q^{97} + 26 q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 - 6 * q^6 + 8 * q^7 + 8 * q^8 + 12 * q^13 + 10 * q^14 + 8 * q^16 + 12 * q^17 - 24 * q^19 + 12 * q^21 - 10 * q^22 + 8 * q^23 - 6 * q^26 - 4 * q^29 - 24 * q^31 - 8 * q^32 - 12 * q^33 + 18 * q^34 - 12 * q^38 - 6 * q^42 - 8 * q^43 + 8 * q^44 - 8 * q^46 - 24 * q^48 + 4 * q^49 - 24 * q^52 - 18 * q^54 + 16 * q^56 - 26 * q^58 - 12 * q^62 + 6 * q^66 + 24 * q^68 + 24 * q^73 + 12 * q^77 - 30 * q^78 - 36 * q^81 + 12 * q^82 - 4 * q^86 + 16 * q^88 + 24 * q^91 - 24 * q^92 - 6 * q^94 - 24 * q^96 + 12 * q^97 + 26 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more