LMFDB - Embedded newform 1323.2.g.f.667.3

Newspace parameters

Level:	$ N $	$=$	$ 1323 = 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1323.g (of order $3$, degree $2$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$10.5642081874$
Analytic rank:	$0$
Dimension:	$10$
Relative dimension:	$5$ over $\Q(\zeta_{3})$
Coefficient field:	10.0.991381711347.1
Defining polynomial:	$x^{10} - 2 x^{9} + 9 x^{8} - 8 x^{7} + 40 x^{6} - 36 x^{5} + 90 x^{4} - 3 x^{3} + 36 x^{2} - 9 x + 9$
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 63)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			667.3
Root			$0.247934 + 0.429435i$ of defining polynomial
Character	$\chi$	$=$	1323.667
Dual form			1323.2.g.f.361.3

$q$-expansion

$f(q)$	$=$	$q+(-0.247934 - 0.429435i) q^{2} +(0.877057 - 1.51911i) q^{4} -3.69258 q^{5} -1.86155 q^{8} +O(q^{10})$	\(q+(-0.247934 - 0.429435i) q^{2} +(0.877057 - 1.51911i) q^{4} -3.69258 q^{5} -1.86155 q^{8} +(0.915516 + 1.58572i) q^{10} +0.892568 q^{11} +(-0.598355 - 1.03638i) q^{13} +(-1.29257 - 2.23880i) q^{16} +(-0.124991 - 0.216492i) q^{17} +(-1.40414 + 2.43204i) q^{19} +(-3.23860 + 5.60943i) q^{20} +(-0.221298 - 0.383300i) q^{22} -2.47772 q^{23} +8.63514 q^{25} +(-0.296705 + 0.513909i) q^{26} +(-2.07128 + 3.58755i) q^{29} +(1.79257 - 3.10483i) q^{31} +(-2.50249 + 4.33444i) q^{32} +(-0.0619793 + 0.107351i) q^{34} +(-2.36568 + 4.09747i) q^{37} +1.39253 q^{38} +6.87391 q^{40} +(-2.39093 - 4.14121i) q^{41} +(-4.98928 + 8.64169i) q^{43} +(0.782834 - 1.35591i) q^{44} +(0.614310 + 1.06402i) q^{46} +(5.08653 + 8.81013i) q^{47} +(-2.14095 - 3.70823i) q^{50} -2.09917 q^{52} +(4.94465 + 8.56438i) q^{53} -3.29588 q^{55} +2.05416 q^{58} +(-0.906186 + 1.56956i) q^{59} +(5.40205 + 9.35663i) q^{61} -1.77776 q^{62} -2.68848 q^{64} +(2.20948 + 3.82692i) q^{65} +(-0.514685 + 0.891460i) q^{67} -0.438499 q^{68} +4.94533 q^{71} +(0.915262 + 1.58528i) q^{73} +2.34613 q^{74} +(2.46302 + 4.26607i) q^{76} +(0.899562 + 1.55809i) q^{79} +(4.77293 + 8.26696i) q^{80} +(-1.18559 + 2.05350i) q^{82} +(6.16156 - 10.6721i) q^{83} +(0.461541 + 0.799412i) q^{85} +4.94806 q^{86} -1.66156 q^{88} +(-1.20370 + 2.08488i) q^{89} +(-2.17310 + 3.76392i) q^{92} +(2.52225 - 4.36867i) q^{94} +(5.18489 - 8.98049i) q^{95} +(-5.52210 + 9.56456i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10q - 2q^{2} - 4q^{4} - 8q^{5} + 6q^{8} + O(q^{10}) $	$ 10q - 2q^{2} - 4q^{4} - 8q^{5} + 6q^{8} + 7q^{10} + 8q^{11} + 8q^{13} + 2q^{16} + 12q^{17} - q^{19} + 5q^{20} - q^{22} + 6q^{23} + 2q^{25} + 11q^{26} - 7q^{29} + 3q^{31} + 2q^{32} - 3q^{34} - 40q^{38} - 6q^{40} + 5q^{41} - 7q^{43} + 10q^{44} + 3q^{46} + 27q^{47} - 19q^{50} - 20q^{52} + 21q^{53} - 4q^{55} + 20q^{58} + 30q^{59} + 14q^{61} - 12q^{62} - 50q^{64} + 11q^{65} - 2q^{67} - 54q^{68} + 6q^{71} - 15q^{73} - 72q^{74} - 5q^{76} - 4q^{79} + 20q^{80} + 5q^{82} + 9q^{83} - 6q^{85} - 16q^{86} + 36q^{88} + 28q^{89} - 27q^{92} + 3q^{94} + 14q^{95} + 12q^{97} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1323\mathbb{Z}\right)^\times$.

$n$	$785$	$1081$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$

Coefficient data

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

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

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

$2$	−0.247934	−	0.429435i	−0.175316	−	0.303656i	0.764955	−	0.644084i	$-0.222761\pi$
$2$	−0.247934	−	0.429435i	−0.175316	−	0.303656i	−0.940271	+	0.340428i	$0.889428\pi$
$3$		0			0
$3$		0			0
$4$	0.877057	−	1.51911i	0.438529	−	0.759554i
$5$	−3.69258			−1.65137			−0.825686	−	0.564130i	$-0.809212\pi$
$5$	−3.69258			−1.65137			−0.825686	+	0.564130i	$0.809212\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$	−1.86155			−0.658156
$9$		0			0
$10$	0.915516	+	1.58572i	0.289512	+	0.501449i
$11$	0.892568			0.269119			0.134560	−	0.990905i	$-0.457038\pi$
$11$	0.892568			0.269119			0.134560	+	0.990905i	$0.457038\pi$
$12$		0			0
$13$	−0.598355	−	1.03638i	−0.165954	−	0.287441i	0.771040	−	0.636787i	$-0.219737\pi$
$13$	−0.598355	−	1.03638i	−0.165954	−	0.287441i	−0.936994	+	0.349346i	$0.886404\pi$
$14$		0			0
$15$		0			0
$16$	−1.29257	−	2.23880i	−0.323143	−	0.559701i
$17$	−0.124991	−	0.216492i	−0.0303149	−	0.0525069i	0.850470	−	0.526024i	$-0.176318\pi$
$17$	−0.124991	−	0.216492i	−0.0303149	−	0.0525069i	−0.880785	+	0.473517i	$0.842984\pi$
$18$		0			0
$19$	−1.40414	+	2.43204i	−0.322131	+	0.557948i	−0.980928	−	0.194374i	$-0.937733\pi$
$19$	−1.40414	+	2.43204i	−0.322131	+	0.557948i	0.658796	+	0.752321i	$0.271066\pi$
$20$	−3.23860	+	5.60943i	−0.724174	+	1.25431i
$21$		0			0
$22$	−0.221298	−	0.383300i	−0.0471809	−	0.0817198i
$23$	−2.47772			−0.516639			−0.258320	−	0.966059i	$-0.583169\pi$
$23$	−2.47772			−0.516639			−0.258320	+	0.966059i	$0.583169\pi$
$24$		0			0
$25$	8.63514			1.72703
$26$	−0.296705	+	0.513909i	−0.0581887	+	0.100786i
$27$		0			0
$28$		0			0
$29$	−2.07128	+	3.58755i	−0.384626	+	0.666192i	−0.991717	−	0.128440i	$-0.959003\pi$
$29$	−2.07128	+	3.58755i	−0.384626	+	0.666192i	0.607091	+	0.794632i	$0.292336\pi$
$30$		0			0
$31$	1.79257	−	3.10483i	0.321956	−	0.557644i	−0.658936	−	0.752199i	$-0.728993\pi$
$31$	1.79257	−	3.10483i	0.321956	−	0.557644i	0.980892	+	0.194555i	$0.0623264\pi$
$32$	−2.50249	+	4.33444i	−0.442382	+	0.766229i
$33$		0			0
$34$	−0.0619793	+	0.107351i	−0.0106294	+	0.0184106i
$35$		0			0
$36$		0			0
$37$	−2.36568	+	4.09747i	−0.388915	+	0.673621i	−0.992304	−	0.123826i	$-0.960483\pi$
$37$	−2.36568	+	4.09747i	−0.388915	+	0.673621i	0.603389	+	0.797447i	$0.293817\pi$
$38$	1.39253			0.225899
$39$		0			0
$40$	6.87391			1.08686
$41$	−2.39093	−	4.14121i	−0.373400	−	0.646748i	0.616686	−	0.787209i	$-0.288475\pi$
$41$	−2.39093	−	4.14121i	−0.373400	−	0.646748i	−0.990086	+	0.140461i	$0.955142\pi$
$42$		0			0
$43$	−4.98928	+	8.64169i	−0.760859	+	1.31785i	0.181550	+	0.983382i	$0.441889\pi$
$43$	−4.98928	+	8.64169i	−0.760859	+	1.31785i	−0.942408	+	0.334464i	$0.891445\pi$
$44$	0.782834	−	1.35591i	0.118017	−	0.204411i
$45$		0			0
$46$	0.614310	+	1.06402i	0.0905751	+	0.156881i
$47$	5.08653	+	8.81013i	0.741947	+	1.28509i	0.951608	+	0.307316i	$0.0994308\pi$
$47$	5.08653	+	8.81013i	0.741947	+	1.28509i	−0.209661	+	0.977774i	$0.567236\pi$
$48$		0			0
$49$		0			0
$50$	−2.14095	−	3.70823i	−0.302776	−	0.524423i
$51$		0			0
$52$	−2.09917			−0.291102
$53$	4.94465	+	8.56438i	0.679199	+	1.17641i	0.975222	+	0.221227i	$0.0710061\pi$
$53$	4.94465	+	8.56438i	0.679199	+	1.17641i	−0.296023	+	0.955181i	$0.595661\pi$
$54$		0			0
$55$	−3.29588			−0.444416
$56$		0			0
$57$		0			0
$58$	2.05416			0.269724
$59$	−0.906186	+	1.56956i	−0.117975	+	0.204339i	−0.918965	−	0.394339i	$-0.870974\pi$
$59$	−0.906186	+	1.56956i	−0.117975	+	0.204339i	0.800990	+	0.598678i	$0.204307\pi$
$60$		0			0
$61$	5.40205	+	9.35663i	0.691662	+	1.19799i	0.971293	+	0.237886i	$0.0764546\pi$
$61$	5.40205	+	9.35663i	0.691662	+	1.19799i	−0.279631	+	0.960108i	$0.590212\pi$
$62$	−1.77776			−0.225776
$63$		0			0
$64$	−2.68848			−0.336060
$65$	2.20948	+	3.82692i	0.274052	+	0.474671i
$66$		0			0
$67$	−0.514685	+	0.891460i	−0.0628787	+	0.108909i	−0.895751	−	0.444556i	$-0.853361\pi$
$67$	−0.514685	+	0.891460i	−0.0628787	+	0.108909i	0.832872	+	0.553465i	$0.186695\pi$
$68$	−0.438499			−0.0531758
$69$		0			0
$70$		0			0
$71$	4.94533			0.586903			0.293451	−	0.955974i	$-0.405196\pi$
$71$	4.94533			0.586903			0.293451	+	0.955974i	$0.405196\pi$
$72$		0			0
$73$	0.915262	+	1.58528i	0.107123	+	0.185543i	0.914604	−	0.404351i	$-0.132503\pi$
$73$	0.915262	+	1.58528i	0.107123	+	0.185543i	−0.807480	+	0.589894i	$0.799169\pi$
$74$	2.34613			0.272732
$75$		0			0
$76$	2.46302	+	4.26607i	0.282527	+	0.489352i
$77$		0			0
$78$		0			0
$79$	0.899562	+	1.55809i	0.101209	+	0.175298i	0.912183	−	0.409783i	$-0.134396\pi$
$79$	0.899562	+	1.55809i	0.101209	+	0.175298i	−0.810974	+	0.585082i	$0.801062\pi$
$80$	4.77293	+	8.26696i	0.533630	+	0.924274i
$81$		0			0
$82$	−1.18559	+	2.05350i	−0.130926	+	0.226771i
$83$	6.16156	−	10.6721i	0.676319	−	1.17142i	−0.299763	−	0.954014i	$-0.596908\pi$
$83$	6.16156	−	10.6721i	0.676319	−	1.17142i	0.976082	−	0.217405i	$-0.0697591\pi$
$84$		0			0
$85$	0.461541	+	0.799412i	0.0500611	+	0.0867084i
$86$	4.94806			0.533563
$87$		0			0
$88$	−1.66156			−0.177123
$89$	−1.20370	+	2.08488i	−0.127592	+	0.220997i	−0.922743	−	0.385415i	$-0.874058\pi$
$89$	−1.20370	+	2.08488i	−0.127592	+	0.220997i	0.795151	+	0.606412i	$0.207392\pi$
$90$		0			0
$91$		0			0
$92$	−2.17310	+	3.76392i	−0.226561	+	0.392416i
$93$		0			0
$94$	2.52225	−	4.36867i	0.260150	−	0.450593i
$95$	5.18489	−	8.98049i	0.531958	−	0.921379i
$96$		0			0
$97$	−5.52210	+	9.56456i	−0.560684	+	0.971134i	0.436752	+	0.899582i	$0.356129\pi$
$97$	−5.52210	+	9.56456i	−0.560684	+	0.971134i	−0.997437	+	0.0715522i	$0.977205\pi$
$98$		0			0
$99$		0			0
$100$	7.57351	−	13.1177i	0.757351	−	1.31177i
$101$	−2.59964			−0.258674			−0.129337	−	0.991601i	$-0.541285\pi$
$101$	−2.59964			−0.258674			−0.129337	+	0.991601i	$0.541285\pi$
$102$		0			0
$103$	−9.71155			−0.956908			−0.478454	−	0.878113i	$-0.658803\pi$
$103$	−9.71155			−0.956908			−0.478454	+	0.878113i	$0.658803\pi$
$104$	1.11387	+	1.92927i	0.109224	+	0.189181i
$105$		0			0
$106$	2.45189	−	4.24680i	0.238149	−	0.412486i
$107$	5.45025	−	9.44012i	0.526896	−	0.912610i	−0.472613	−	0.881270i	$-0.656689\pi$
$107$	5.45025	−	9.44012i	0.526896	−	0.912610i	0.999509	−	0.0313403i	$-0.00997757\pi$
$108$		0			0
$109$	−1.06096	−	1.83764i	−0.101622	−	0.176014i	0.810731	−	0.585419i	$-0.199070\pi$
$109$	−1.06096	−	1.83764i	−0.101622	−	0.176014i	−0.912353	+	0.409404i	$0.865737\pi$
$110$	0.817161	+	1.41536i	0.0779132	+	0.134950i
$111$		0			0
$112$		0			0
$113$	−7.91318	−	13.7060i	−0.744409	−	1.28935i	−0.950470	−	0.310816i	$-0.899398\pi$
$113$	−7.91318	−	13.7060i	−0.744409	−	1.28935i	0.206061	−	0.978539i	$-0.433935\pi$
$114$		0			0
$115$	9.14916			0.853164
$116$	3.63325	+	6.29298i	0.337339	+	0.584289i
$117$		0			0
$118$	0.898698			0.0827318
$119$		0			0
$120$		0			0
$121$	−10.2033			−0.927575
$122$	2.67871	−	4.63966i	0.242519	−	0.420055i
$123$		0			0
$124$	−3.14438	−	5.44623i	−0.282374	−	0.489086i
$125$	−13.4230			−1.20059
$126$		0			0
$127$	−1.26946			−0.112647			−0.0563233	−	0.998413i	$-0.517938\pi$
$127$	−1.26946			−0.112647			−0.0563233	+	0.998413i	$0.517938\pi$
$128$	5.67155	+	9.82342i	0.501299	+	0.868275i
$129$		0			0
$130$	1.09561	−	1.89765i	0.0960912	−	0.166435i
$131$	−15.0289			−1.31308			−0.656540	−	0.754291i	$-0.727981\pi$
$131$	−15.0289			−1.31308			−0.656540	+	0.754291i	$0.727981\pi$
$132$		0			0
$133$		0			0
$134$	0.510432			0.0440946
$135$		0			0
$136$	0.232677	+	0.403009i	0.0199519	+	0.0345577i
$137$	0.488493			0.0417347			0.0208674	−	0.999782i	$-0.493357\pi$
$137$	0.488493			0.0417347			0.0208674	+	0.999782i	$0.493357\pi$
$138$		0			0
$139$	4.93487	+	8.54745i	0.418570	+	0.724985i	0.995796	−	0.0915997i	$-0.0291980\pi$
$139$	4.93487	+	8.54745i	0.418570	+	0.724985i	−0.577226	+	0.816585i	$0.695865\pi$
$140$		0			0
$141$		0			0
$142$	−1.22612	−	2.12370i	−0.102893	−	0.178217i
$143$	−0.534073	−	0.925042i	−0.0446614	−	0.0773559i
$144$		0			0
$145$	7.64835	−	13.2473i	0.635161	−	1.10013i
$146$	0.453849	−	0.786090i	0.0375609	−	0.0650573i
$147$		0			0
$148$	4.14967	+	7.18744i	0.341101	+	0.590804i
$149$	−21.0240			−1.72235			−0.861175	−	0.508309i	$-0.830271\pi$
$149$	−21.0240			−1.72235			−0.861175	+	0.508309i	$0.830271\pi$
$150$		0			0
$151$	1.49838			0.121937			0.0609683	−	0.998140i	$-0.480581\pi$
$151$	1.49838			0.121937			0.0609683	+	0.998140i	$0.480581\pi$
$152$	2.61387	−	4.52735i	0.212013	−	0.367217i
$153$		0			0
$154$		0			0
$155$	−6.61922	+	11.4648i	−0.531669	+	0.920877i
$156$		0			0
$157$	−8.33982	+	14.4450i	−0.665590	+	1.15284i	0.313535	+	0.949577i	$0.398487\pi$
$157$	−8.33982	+	14.4450i	−0.665590	+	1.15284i	−0.979125	+	0.203259i	$0.934847\pi$
$158$	0.446064	−	0.772606i	0.0354870	−	0.0614652i
$159$		0			0
$160$	9.24065	−	16.0053i	0.730538	−	1.26533i
$161$		0			0
$162$		0			0
$163$	−3.34135	+	5.78738i	−0.261714	+	0.453303i	−0.966698	−	0.255921i	$-0.917621\pi$
$163$	−3.34135	+	5.78738i	−0.261714	+	0.453303i	0.704983	+	0.709224i	$0.250954\pi$
$164$	−8.38793			−0.654987
$165$		0			0
$166$	−6.11064			−0.474278
$167$	8.81549	+	15.2689i	0.682163	+	1.18154i	0.974319	+	0.225170i	$0.0722939\pi$
$167$	8.81549	+	15.2689i	0.682163	+	1.18154i	−0.292156	+	0.956371i	$0.594373\pi$
$168$		0			0
$169$	5.78394	−	10.0181i	0.444919	−	0.770622i
$170$	0.228863	−	0.396403i	0.0175530	−	0.0304027i
$171$		0			0
$172$	8.75178	+	15.1585i	0.667317	+	1.15583i
$173$	1.94342	+	3.36611i	0.147756	+	0.255920i	0.930398	−	0.366552i	$-0.119462\pi$
$173$	1.94342	+	3.36611i	0.147756	+	0.255920i	−0.782642	+	0.622472i	$0.786128\pi$
$174$		0			0
$175$		0			0
$176$	−1.15371	−	1.99829i	−0.0869642	−	0.150626i
$177$		0			0
$178$	1.19376			0.0894759
$179$	−3.66758	−	6.35244i	−0.274128	−	0.474804i	0.695787	−	0.718248i	$-0.255056\pi$
$179$	−3.66758	−	6.35244i	−0.274128	−	0.474804i	−0.969915	+	0.243445i	$0.921723\pi$
$180$		0			0
$181$	−11.2566			−0.836693			−0.418346	−	0.908288i	$-0.637390\pi$
$181$	−11.2566			−0.836693			−0.418346	+	0.908288i	$0.637390\pi$
$182$		0			0
$183$		0			0
$184$	4.61238			0.340029
$185$	8.73545	−	15.1302i	0.642243	−	1.11240i
$186$		0			0
$187$	−0.111563	−	0.193234i	−0.00815833	−	0.0141306i
$188$	17.8447			1.30146
$189$		0			0
$190$	−5.14204			−0.373043
$191$	−11.9230	−	20.6512i	−0.862715	−	1.49427i	−0.869298	−	0.494288i	$-0.835429\pi$
$191$	−11.9230	−	20.6512i	−0.862715	−	1.49427i	0.00658302	−	0.999978i	$-0.497905\pi$
$192$		0			0
$193$	−2.96728	+	5.13948i	−0.213589	+	0.369948i	−0.952835	−	0.303488i	$-0.901849\pi$
$193$	−2.96728	+	5.13948i	−0.213589	+	0.369948i	0.739246	+	0.673436i	$0.235182\pi$
$194$	5.47647			0.393188
$195$		0			0
$196$		0			0
$197$	15.4682			1.10206			0.551032	−	0.834484i	$-0.314234\pi$
$197$	15.4682			1.10206			0.551032	+	0.834484i	$0.314234\pi$
$198$		0			0
$199$	−7.74818	−	13.4202i	−0.549254	−	0.951336i	−0.998326	−	0.0578402i	$-0.981579\pi$
$199$	−7.74818	−	13.4202i	−0.549254	−	0.951336i	0.449072	−	0.893496i	$-0.351755\pi$
$200$	−16.0747			−1.13665
$201$		0			0
$202$	0.644540	+	1.11638i	0.0453497	+	0.0785480i
$203$		0			0
$204$		0			0
$205$	8.82870	+	15.2917i	0.616623	+	1.06802i
$206$	2.40783	+	4.17048i	0.167761	+	0.290571i
$207$		0			0
$208$	−1.54684	+	2.67920i	−0.107254	+	0.185769i
$209$	−1.25329	+	2.17076i	−0.0866918	+	0.150155i
$210$		0			0
$211$	0.771898	+	1.33697i	0.0531397	+	0.0920406i	0.891372	−	0.453273i	$-0.149744\pi$
$211$	0.771898	+	1.33697i	0.0531397	+	0.0920406i	−0.838232	+	0.545314i	$0.816410\pi$
$212$	17.3469			1.19139
$213$		0			0
$214$	−5.40522			−0.369493
$215$	18.4233	−	31.9101i	1.25646	−	2.17625i
$216$		0			0
$217$		0			0
$218$	−0.526098	+	0.911229i	−0.0356319	+	0.0617162i
$219$		0			0
$220$	−2.89068	+	5.00680i	−0.194889	+	0.337558i
$221$	−0.149579	+	0.259078i	−0.0100617	+	0.0174275i
$222$		0			0
$223$	2.72171	−	4.71414i	0.182259	−	0.315682i	−0.760390	−	0.649466i	$-0.774992\pi$
$223$	2.72171	−	4.71414i	0.182259	−	0.315682i	0.942649	+	0.333784i	$0.108326\pi$
$224$		0			0
$225$		0			0
$226$	−3.92389	+	6.79638i	−0.261014	+	0.452089i
$227$	−16.0764			−1.06703			−0.533513	−	0.845792i	$-0.679128\pi$
$227$	−16.0764			−1.06703			−0.533513	+	0.845792i	$0.679128\pi$
$228$		0			0
$229$	9.96840			0.658730			0.329365	−	0.944203i	$-0.393165\pi$
$229$	9.96840			0.658730			0.329365	+	0.944203i	$0.393165\pi$
$230$	−2.26839	−	3.92897i	−0.149573	−	0.259068i
$231$		0			0
$232$	3.85578	−	6.67840i	0.253144	−	0.438458i
$233$	−8.27045	+	14.3248i	−0.541815	+	0.938451i	0.456985	+	0.889474i	$0.348929\pi$
$233$	−8.27045	+	14.3248i	−0.541815	+	0.938451i	−0.998800	+	0.0489765i	$0.984404\pi$
$234$		0			0
$235$	−18.7824	−	32.5321i	−1.22523	−	2.12216i
$236$	1.58955	+	2.75319i	0.103471	+	0.179217i
$237$		0			0
$238$		0			0
$239$	11.0119	+	19.0732i	0.712303	+	1.23375i	0.963990	+	0.265937i	$0.0856813\pi$
$239$	11.0119	+	19.0732i	0.712303	+	1.23375i	−0.251687	+	0.967809i	$0.580985\pi$
$240$		0			0
$241$	−16.7201			−1.07703			−0.538517	−	0.842615i	$-0.681015\pi$
$241$	−16.7201			−1.07703			−0.538517	+	0.842615i	$0.681015\pi$
$242$	2.52975	+	4.38166i	0.162619	+	0.281664i
$243$		0			0
$244$	18.9516			1.21325
$245$		0			0
$246$		0			0
$247$	3.36069			0.213836
$248$	−3.33696	+	5.77978i	−0.211897	+	0.367017i
$249$		0			0
$250$	3.32803	+	5.76432i	0.210483	+	0.364568i
$251$	−8.53099			−0.538471			−0.269236	−	0.963074i	$-0.586771\pi$
$251$	−8.53099			−0.538471			−0.269236	+	0.963074i	$0.586771\pi$
$252$		0			0
$253$	−2.21153			−0.139038
$254$	0.314743	+	0.545151i	0.0197488	+	0.0342058i
$255$		0			0
$256$	0.123861	−	0.214533i	0.00774131	−	0.0134083i
$257$	−17.1197			−1.06790			−0.533950	−	0.845516i	$-0.679293\pi$
$257$	−17.1197			−1.06790			−0.533950	+	0.845516i	$0.679293\pi$
$258$		0			0
$259$		0			0
$260$	7.75135			0.480718
$261$		0			0
$262$	3.72617	+	6.45392i	0.230204	+	0.398725i
$263$	−20.5527			−1.26733			−0.633666	−	0.773607i	$-0.718451\pi$
$263$	−20.5527			−1.26733			−0.633666	+	0.773607i	$0.718451\pi$
$264$		0			0
$265$	−18.2585	−	31.6246i	−1.12161	−	1.94269i
$266$		0			0
$267$		0			0
$268$	0.902816	+	1.56372i	0.0551483	+	0.0955196i
$269$	9.92267	+	17.1866i	0.604996	+	1.04788i	0.992052	+	0.125827i	$0.0401585\pi$
$269$	9.92267	+	17.1866i	0.604996	+	1.04788i	−0.387057	+	0.922056i	$0.626508\pi$
$270$		0			0
$271$	−5.32056	+	9.21548i	−0.323201	+	0.559801i	−0.981147	−	0.193265i	$-0.938092\pi$
$271$	−5.32056	+	9.21548i	−0.323201	+	0.559801i	0.657946	+	0.753065i	$0.271426\pi$
$272$	−0.323121	+	0.559663i	−0.0195921	+	0.0339345i
$273$		0			0
$274$	−0.121114	−	0.209776i	−0.00731676	−	0.0126730i
$275$	7.70745			0.464777
$276$		0			0
$277$	−24.8813			−1.49497			−0.747487	−	0.664276i	$-0.768740\pi$
$277$	−24.8813			−1.49497			−0.747487	+	0.664276i	$0.768740\pi$
$278$	2.44705	−	4.23841i	0.146764	−	0.254203i
$279$		0			0
$280$		0			0
$281$	6.83733	−	11.8426i	0.407881	−	0.706470i	−0.586771	−	0.809753i	$-0.699601\pi$
$281$	6.83733	−	11.8426i	0.407881	−	0.706470i	0.994652	+	0.103282i	$0.0329346\pi$
$282$		0			0
$283$	3.16089	−	5.47483i	0.187896	−	0.325445i	−0.756653	−	0.653817i	$-0.773167\pi$
$283$	3.16089	−	5.47483i	0.187896	−	0.325445i	0.944548	+	0.328372i	$0.106500\pi$
$284$	4.33734	−	7.51249i	0.257374	−	0.445784i
$285$		0			0
$286$	−0.264830	+	0.458699i	−0.0156597	+	0.0271234i
$287$		0			0
$288$		0			0
$289$	8.46875	−	14.6683i	0.498162	−	0.862842i
$290$	−7.58515			−0.445415
$291$		0			0
$292$	3.21095			0.187907
$293$	−1.31508	−	2.27778i	−0.0768277	−	0.133069i	0.825052	−	0.565057i	$-0.191146\pi$
$293$	−1.31508	−	2.27778i	−0.0768277	−	0.133069i	−0.901880	+	0.431987i	$0.857812\pi$
$294$		0			0
$295$	3.34616	−	5.79573i	0.194821	−	0.337440i
$296$	4.40382	−	7.62764i	0.255967	−	0.443348i
$297$		0			0
$298$	5.21256	+	9.02841i	0.301955	+	0.523002i
$299$	1.48255	+	2.56786i	0.0857384	+	0.148503i
$300$		0			0
$301$		0			0
$302$	−0.371500	−	0.643457i	−0.0213774	−	0.0370268i
$303$		0			0
$304$	7.25980			0.416378
$305$	−19.9475	−	34.5501i	−1.14219	−	1.97833i
$306$		0			0
$307$	2.79496			0.159517			0.0797583	−	0.996814i	$-0.474585\pi$
$307$	2.79496			0.159517			0.0797583	+	0.996814i	$0.474585\pi$
$308$		0			0
$309$		0			0
$310$	6.56452			0.372840
$311$	7.55013	−	13.0772i	0.428129	−	0.741541i	−0.568578	−	0.822629i	$-0.692506\pi$
$311$	7.55013	−	13.0772i	0.428129	−	0.741541i	0.996707	+	0.0810885i	$0.0258396\pi$
$312$		0			0
$313$	−12.7392	−	22.0650i	−0.720064	−	1.24719i	−0.960974	−	0.276640i	$-0.910779\pi$
$313$	−12.7392	−	22.0650i	−0.720064	−	1.24719i	0.240910	−	0.970548i	$-0.422554\pi$
$314$	8.27090			0.466754
$315$		0			0
$316$	3.15587			0.177531
$317$	16.2605	+	28.1639i	0.913278	+	1.58184i	0.809403	+	0.587253i	$0.199791\pi$
$317$	16.2605	+	28.1639i	0.913278	+	1.58184i	0.103875	+	0.994590i	$0.466876\pi$
$318$		0			0
$319$	−1.84875	+	3.20214i	−0.103510	+	0.179285i
$320$	9.92743			0.554960
$321$		0			0
$322$		0			0
$323$	0.702021			0.0390615
$324$		0			0
$325$	−5.16688	−	8.94931i	−0.286607	−	0.496418i
$326$	3.31373			0.183531
$327$		0			0
$328$	4.45083	+	7.70906i	0.245756	+	0.425661i
$329$		0			0
$330$		0			0
$331$	−9.04741	−	15.6706i	−0.497291	−	0.861333i	0.502704	−	0.864458i	$-0.332338\pi$
$331$	−9.04741	−	15.6706i	−0.497291	−	0.861333i	−0.999995	+	0.00312545i	$0.999005\pi$
$332$	−10.8081	−	18.7201i	−0.593170	−	1.02740i
$333$		0			0
$334$	4.37132	−	7.57135i	0.239188	−	0.414286i
$335$	1.90051	−	3.29179i	0.103836	−	0.179850i
$336$		0			0
$337$	−12.5086	−	21.6656i	−0.681389	−	1.18020i	−0.974557	−	0.224139i	$-0.928043\pi$
$337$	−12.5086	−	21.6656i	−0.681389	−	1.18020i	0.293168	−	0.956061i	$-0.405290\pi$
$338$	−5.73615			−0.312005
$339$		0			0
$340$	1.61919			0.0878130
$341$	1.59999	−	2.77127i	0.0866446	−	0.150073i
$342$		0			0
$343$		0			0
$344$	9.28778	−	16.0869i	0.500764	−	0.867348i
$345$		0			0
$346$	0.963682	−	1.66915i	0.0518078	−	0.0897338i
$347$	5.37444	−	9.30881i	0.288515	−	0.499723i	−0.684940	−	0.728599i	$-0.740172\pi$
$347$	5.37444	−	9.30881i	0.288515	−	0.499723i	0.973456	+	0.228876i	$0.0735051\pi$
$348$		0			0
$349$	1.64301	−	2.84577i	0.0879482	−	0.152331i	−0.818695	−	0.574228i	$-0.805302\pi$
$349$	1.64301	−	2.84577i	0.0879482	−	0.152331i	0.906644	+	0.421897i	$0.138636\pi$
$350$		0			0
$351$		0			0
$352$	−2.23365	+	3.86879i	−0.119054	+	0.206207i
$353$	16.8192			0.895195			0.447598	−	0.894235i	$-0.352280\pi$
$353$	16.8192			0.895195			0.447598	+	0.894235i	$0.352280\pi$
$354$		0			0
$355$	−18.2610			−0.969195
$356$	2.11144	+	3.65711i	0.111906	+	0.193827i
$357$		0			0
$358$	−1.81864	+	3.14997i	−0.0961180	+	0.166481i
$359$	−11.8921	+	20.5978i	−0.627642	+	1.08711i	0.360382	+	0.932805i	$0.382646\pi$
$359$	−11.8921	+	20.5978i	−0.627642	+	1.08711i	−0.988024	+	0.154303i	$0.950687\pi$
$360$		0			0
$361$	5.55680	+	9.62466i	0.292463	+	0.506561i
$362$	2.79088	+	4.83395i	0.146686	+	0.254067i
$363$		0			0
$364$		0			0
$365$	−3.37968	−	5.85377i	−0.176900	−	0.306401i
$366$		0			0
$367$	0.689984			0.0360169			0.0180084	−	0.999838i	$-0.494267\pi$
$367$	0.689984			0.0360169			0.0180084	+	0.999838i	$0.494267\pi$
$368$	3.20263	+	5.54712i	0.166949	+	0.289164i
$369$		0			0
$370$	−8.66327			−0.450382
$371$		0			0
$372$		0			0
$373$	−3.76012			−0.194691			−0.0973457	−	0.995251i	$-0.531035\pi$
$373$	−3.76012			−0.194691			−0.0973457	+	0.995251i	$0.531035\pi$
$374$	−0.0553208	+	0.0958184i	−0.00286057	+	0.00495465i
$375$		0			0
$376$	−9.46882	−	16.4005i	−0.488317	−	0.845790i
$377$	4.95744			0.255321
$378$		0			0
$379$	32.8735			1.68860			0.844300	−	0.535872i	$-0.180017\pi$
$379$	32.8735			1.68860			0.844300	+	0.535872i	$0.180017\pi$
$380$	−9.09489	−	15.7528i	−0.466558	−	0.808102i
$381$		0			0
$382$	−5.91222	+	10.2403i	−0.302495	+	0.523937i
$383$	−1.07267			−0.0548109			−0.0274055	−	0.999624i	$-0.508725\pi$
$383$	−1.07267			−0.0548109			−0.0274055	+	0.999624i	$0.508725\pi$
$384$		0			0
$385$		0			0
$386$	2.94276			0.149782
$387$		0			0
$388$	9.68640	+	16.7773i	0.491752	+	0.851740i
$389$	23.7436			1.20385			0.601925	−	0.798553i	$-0.294401\pi$
$389$	23.7436			1.20385			0.601925	+	0.798553i	$0.294401\pi$
$390$		0			0
$391$	0.309693	+	0.536405i	0.0156619	+	0.0271271i
$392$		0			0
$393$		0			0
$394$	−3.83510	−	6.64258i	−0.193209	−	0.334648i
$395$	−3.32170	−	5.75336i	−0.167133	−	0.289483i
$396$		0			0
$397$	0.0160489	−	0.0277975i	0.000805471	−	0.00139512i	−0.865622	−	0.500697i	$-0.833077\pi$
$397$	0.0160489	−	0.0277975i	0.000805471	−	0.00139512i	0.866428	+	0.499302i	$0.166410\pi$
$398$	−3.84208	+	6.65467i	−0.192586	+	0.333569i
$399$		0			0
$400$	−11.1616	−	19.3324i	−0.558078	−	0.966619i
$401$	−24.5256			−1.22475			−0.612374	−	0.790568i	$-0.709785\pi$
$401$	−24.5256			−1.22475			−0.612374	+	0.790568i	$0.709785\pi$
$402$		0			0
$403$	−4.29039			−0.213719
$404$	−2.28004	+	3.94914i	−0.113436	+	0.196477i
$405$		0			0
$406$		0			0
$407$	−2.11153	+	3.65728i	−0.104665	+	0.181284i
$408$		0			0
$409$	13.3948	−	23.2006i	0.662333	−	1.14719i	−0.317669	−	0.948202i	$-0.602900\pi$
$409$	13.3948	−	23.2006i	0.662333	−	1.14719i	0.980001	−	0.198992i	$-0.0637667\pi$
$410$	4.37787	−	7.58269i	0.216208	−	0.374483i
$411$		0			0
$412$	−8.51759	+	14.7529i	−0.419631	+	0.726823i
$413$		0			0
$414$		0			0
$415$	−22.7520	+	39.4077i	−1.11685	+	1.93445i
$416$	5.98952			0.293660
$417$		0			0
$418$	1.24293			0.0607938
$419$	−10.5262	−	18.2320i	−0.514240	−	0.890689i	−0.999864	−	0.0165215i	$-0.994741\pi$
$419$	−10.5262	−	18.2320i	−0.514240	−	0.890689i	0.485624	−	0.874168i	$-0.338593\pi$
$420$		0			0
$421$	−7.44533	+	12.8957i	−0.362863	+	0.628498i	−0.988431	−	0.151672i	$-0.951534\pi$
$421$	−7.44533	+	12.8957i	−0.362863	+	0.628498i	0.625568	+	0.780170i	$0.284867\pi$
$422$	0.382760	−	0.662959i	0.0186325	−	0.0322724i
$423$		0			0
$424$	−9.20469	−	15.9430i	−0.447019	−	0.774260i
$425$	−1.07932	−	1.86944i	−0.0523547	−	0.0906809i
$426$		0			0
$427$		0			0
$428$	−9.56037	−	16.5590i	−0.462118	−	0.800412i
$429$		0			0
$430$	−18.2711			−0.881110
$431$	7.95192	+	13.7731i	0.383031	+	0.663428i	0.991494	−	0.130154i	$-0.0415471\pi$
$431$	7.95192	+	13.7731i	0.383031	+	0.663428i	−0.608463	+	0.793582i	$0.708214\pi$
$432$		0			0
$433$	16.3658			0.786490			0.393245	−	0.919434i	$-0.371352\pi$
$433$	16.3658			0.786490			0.393245	+	0.919434i	$0.371352\pi$
$434$		0			0
$435$		0			0
$436$	−3.72210			−0.178256
$437$	3.47905	−	6.02590i	0.166426	−	0.288258i
$438$		0			0
$439$	−7.77236	−	13.4621i	−0.370954	−	0.642512i	0.618758	−	0.785582i	$-0.287636\pi$
$439$	−7.77236	−	13.4621i	−0.370954	−	0.642512i	−0.989713	+	0.143070i	$0.954303\pi$
$440$	6.13543			0.292495
$441$		0			0
$442$	0.148343			0.00705594
$443$	0.895027	+	1.55023i	0.0425240	+	0.0736537i	0.886504	−	0.462721i	$-0.153127\pi$
$443$	0.895027	+	1.55023i	0.0425240	+	0.0736537i	−0.843980	+	0.536375i	$0.819793\pi$
$444$		0			0
$445$	4.44477	−	7.69857i	0.210702	−	0.364947i
$446$	−2.69922			−0.127812
$447$		0			0
$448$		0			0
$449$	−13.5666			−0.640250			−0.320125	−	0.947375i	$-0.603725\pi$
$449$	−13.5666			−0.640250			−0.320125	+	0.947375i	$0.603725\pi$
$450$		0			0
$451$	−2.13407	−	3.69631i	−0.100489	−	0.174053i
$452$	−27.7612			−1.30578
$453$		0			0
$454$	3.98588	+	6.90375i	0.187067	+	0.324009i
$455$		0			0
$456$		0			0
$457$	−1.28459	−	2.22497i	−0.0600905	−	0.104080i	0.834415	−	0.551136i	$-0.185806\pi$
$457$	−1.28459	−	2.22497i	−0.0600905	−	0.104080i	−0.894506	+	0.447057i	$0.852472\pi$
$458$	−2.47151	−	4.28078i	−0.115486	−	0.200028i
$459$		0			0
$460$	8.02434	−	13.8986i	0.374137	−	0.648024i
$461$	18.0934	−	31.3388i	0.842695	−	1.45959i	−0.0449122	−	0.998991i	$-0.514301\pi$
$461$	18.0934	−	31.3388i	0.842695	−	1.45959i	0.887608	−	0.460600i	$-0.152366\pi$
$462$		0			0
$463$	8.19224	+	14.1894i	0.380726	+	0.659436i	0.991166	−	0.132626i	$-0.0423409\pi$
$463$	8.19224	+	14.1894i	0.380726	+	0.659436i	−0.610440	+	0.792062i	$0.709008\pi$
$464$	10.7091			0.497158
$465$		0			0
$466$	8.20210			0.379955
$467$	−4.35022	+	7.53480i	−0.201304	+	0.348669i	−0.948949	−	0.315430i	$-0.897851\pi$
$467$	−4.35022	+	7.53480i	−0.201304	+	0.348669i	0.747645	+	0.664099i	$0.231185\pi$
$468$		0			0
$469$		0			0
$470$	−9.31361	+	16.1316i	−0.429605	+	0.744097i
$471$		0			0
$472$	1.68691	−	2.92181i	0.0776462	−	0.134487i
$473$	−4.45328	+	7.71330i	−0.204762	+	0.354658i
$474$		0			0
$475$	−12.1249	+	21.0010i	−0.556330	+	0.963591i
$476$		0			0
$477$		0			0
$478$	5.46047	−	9.45782i	0.249756	−	0.432591i
$479$	−17.7674			−0.811813			−0.405907	−	0.913915i	$-0.633044\pi$
$479$	−17.7674			−0.811813			−0.405907	+	0.913915i	$0.633044\pi$
$480$		0			0
$481$	5.66207			0.258168
$482$	4.14548	+	7.18018i	0.188821	+	0.327048i
$483$		0			0
$484$	−8.94890	+	15.4999i	−0.406768	+	0.704543i
$485$	20.3908	−	35.3179i	0.925898	−	1.60370i
$486$		0			0
$487$	8.32763	+	14.4239i	0.377361	+	0.653608i	0.990677	−	0.136229i	$-0.0434983\pi$
$487$	8.32763	+	14.4239i	0.377361	+	0.653608i	−0.613316	+	0.789837i	$0.710165\pi$
$488$	−10.0562	−	17.4178i	−0.455222	−	0.788467i
$489$		0			0
$490$		0			0
$491$	3.21021	+	5.56025i	0.144875	+	0.250930i	0.929326	−	0.369260i	$-0.120389\pi$
$491$	3.21021	+	5.56025i	0.144875	+	0.250930i	−0.784451	+	0.620190i	$0.787055\pi$
$492$		0			0
$493$	1.03557			0.0466396
$494$	−0.833230	−	1.44320i	−0.0374888	−	0.0649325i
$495$		0			0
$496$	−9.26814			−0.416152
$497$		0			0
$498$		0			0
$499$	11.1459			0.498960			0.249480	−	0.968380i	$-0.419740\pi$
$499$	11.1459			0.498960			0.249480	+	0.968380i	$0.419740\pi$
$500$	−11.7728	+	20.3911i	−0.526495	+	0.911916i
$501$		0			0
$502$	2.11512	+	3.66350i	0.0944026	+	0.163510i
$503$	−17.7223			−0.790200			−0.395100	−	0.918638i	$-0.629290\pi$
$503$	−17.7223			−0.790200			−0.395100	+	0.918638i	$0.629290\pi$
$504$		0			0
$505$	9.59939			0.427167
$506$	0.548314	+	0.949708i	0.0243755	+	0.0422197i
$507$		0			0
$508$	−1.11339	+	1.92845i	−0.0493988	+	0.0855612i
$509$	31.0823			1.37770			0.688848	−	0.724906i	$-0.258117\pi$
$509$	31.0823			1.37770			0.688848	+	0.724906i	$0.258117\pi$
$510$		0			0
$511$		0			0
$512$	22.5634			0.997169
$513$		0			0
$514$	4.24456	+	7.35180i	0.187220	+	0.324274i
$515$	35.8607			1.58021
$516$		0			0
$517$	4.54008	+	7.86365i	0.199672	+	0.345843i
$518$		0			0
$519$		0			0
$520$	−4.11304	−	7.12399i	−0.180369	−	0.312408i
$521$	−2.37986	−	4.12203i	−0.104263	−	0.180590i	0.809174	−	0.587570i	$-0.199915\pi$
$521$	−2.37986	−	4.12203i	−0.104263	−	0.180590i	−0.913437	+	0.406980i	$0.866582\pi$
$522$		0			0
$523$	−20.1258	+	34.8588i	−0.880038	+	1.52427i	−0.0287402	+	0.999587i	$0.509150\pi$
$523$	−20.1258	+	34.8588i	−0.880038	+	1.52427i	−0.851298	+	0.524683i	$0.824184\pi$
$524$	−13.1812	+	22.8305i	−0.575823	+	0.997355i
$525$		0			0
$526$	5.09571	+	8.82602i	0.222183	+	0.384833i
$527$	−0.896226			−0.0390402
$528$		0			0
$529$	−16.8609			−0.733084
$530$	−9.05381	+	15.6817i	−0.393272	+	0.681168i
$531$		0			0
$532$		0			0
$533$	−2.86125	+	4.95583i	−0.123935	+	0.214661i
$534$		0			0
$535$	−20.1255	+	34.8584i	−0.870101	+	1.50706i
$536$	0.958109	−	1.65949i	0.0413840	−	0.0716792i
$537$		0			0
$538$	4.92033	−	8.52227i	0.212131	−	0.367421i
$539$		0			0
$540$		0			0
$541$	12.0547	−	20.8794i	0.518273	−	0.897675i	−0.481502	−	0.876445i	$-0.659908\pi$
$541$	12.0547	−	20.8794i	0.518273	−	0.897675i	0.999775	−	0.0212301i	$-0.00675826\pi$
$542$	5.27659			0.226649
$543$		0			0
$544$	1.25116			0.0536431
$545$	3.91769	+	6.78564i	0.167815	+	0.290665i
$546$		0			0
$547$	−6.17751	+	10.6998i	−0.264131	+	0.457489i	−0.967336	−	0.253499i	$-0.918419\pi$
$547$	−6.17751	+	10.6998i	−0.264131	+	0.457489i	0.703204	+	0.710988i	$0.251752\pi$
$548$	0.428436	−	0.742073i	0.0183019	−	0.0316998i
$549$		0			0
$550$	−1.91094	−	3.30985i	−0.0814828	−	0.141132i
$551$	−5.81671	−	10.0748i	−0.247800	−	0.429203i
$552$		0			0
$553$		0			0
$554$	6.16893	+	10.6849i	0.262093	+	0.453958i
$555$		0			0
$556$	17.3127			0.734220
$557$	−4.03845	−	6.99479i	−0.171114	−	0.296379i	0.767695	−	0.640815i	$-0.221403\pi$
$557$	−4.03845	−	6.99479i	−0.171114	−	0.296379i	−0.938810	+	0.344436i	$0.888070\pi$
$558$		0			0
$559$	11.9415			0.505070
$560$		0			0
$561$		0			0
$562$	−6.78083			−0.286032
$563$	−22.6064	+	39.1554i	−0.952744	+	1.65020i	−0.213296	+	0.976988i	$0.568420\pi$
$563$	−22.6064	+	39.1554i	−0.952744	+	1.65020i	−0.739448	+	0.673214i	$0.764913\pi$
$564$		0			0
$565$	29.2200	+	50.6106i	1.22930	+	2.12920i
$566$	−3.13477			−0.131764
$567$		0			0
$568$	−9.20596			−0.386274
$569$	11.2149	+	19.4248i	0.470155	+	0.814332i	0.999418	−	0.0341263i	$-0.0108648\pi$
$569$	11.2149	+	19.4248i	0.470155	+	0.814332i	−0.529263	+	0.848458i	$0.677532\pi$
$570$		0			0
$571$	10.9134	−	18.9026i	0.456713	−	0.791050i	−0.542072	−	0.840332i	$-0.682360\pi$
$571$	10.9134	−	18.9026i	0.456713	−	0.791050i	0.998785	+	0.0492820i	$0.0156933\pi$
$572$	−1.87365			−0.0783413
$573$		0			0
$574$		0			0
$575$	−21.3954			−0.892251
$576$		0			0
$577$	16.1022	+	27.8898i	0.670342	+	1.16107i	0.977807	+	0.209508i	$0.0671861\pi$
$577$	16.1022	+	27.8898i	0.670342	+	1.16107i	−0.307465	+	0.951559i	$0.599481\pi$
$578$	−8.39877			−0.349343
$579$		0			0
$580$	−13.4161	−	23.2373i	−0.557072	−	0.964878i
$581$		0			0
$582$		0			0
$583$	4.41343	+	7.64429i	0.182786	+	0.316594i
$584$	−1.70380	−	2.95107i	−0.0705039	−	0.122116i
$585$		0			0
$586$	−0.652105	+	1.12948i	−0.0269382	+	0.0466584i
$587$	−9.72304	+	16.8408i	−0.401313	+	0.695094i	−0.993885	−	0.110424i	$-0.964779\pi$
$587$	−9.72304	+	16.8408i	−0.401313	+	0.695094i	0.592572	+	0.805518i	$0.298113\pi$
$588$		0			0
$589$	5.03404	+	8.71921i	0.207424	+	0.359269i
$590$	−3.31851			−0.136621
$591$		0			0
$592$	12.2313			0.502701
$593$	−14.4202	+	24.9766i	−0.592168	+	1.02566i	0.401772	+	0.915740i	$0.368394\pi$
$593$	−14.4202	+	24.9766i	−0.592168	+	1.02566i	−0.993940	+	0.109925i	$0.964939\pi$
$594$		0			0
$595$		0			0
$596$	−18.4392	+	31.9377i	−0.755300	+	1.30822i
$597$		0			0
$598$	0.735152	−	1.27332i	0.0300626	−	0.0520699i
$599$	−23.4994	+	40.7022i	−0.960161	+	1.66305i	−0.238072	+	0.971247i	$0.576516\pi$
$599$	−23.4994	+	40.7022i	−0.960161	+	1.66305i	−0.722089	+	0.691800i	$0.756818\pi$
$600$		0			0
$601$	7.80843	−	13.5246i	0.318512	−	0.551680i	−0.661665	−	0.749799i	$-0.730150\pi$
$601$	7.80843	−	13.5246i	0.318512	−	0.551680i	0.980178	+	0.198119i	$0.0634834\pi$
$602$		0			0
$603$		0			0
$604$	1.31417	−	2.27620i	0.0534727	−	0.0926174i
$605$	37.6766			1.53177
$606$		0			0
$607$	28.6532			1.16300			0.581500	−	0.813547i	$-0.302466\pi$
$607$	28.6532			1.16300			0.581500	+	0.813547i	$0.302466\pi$
$608$	−7.02769	−	12.1723i	−0.285010	−	0.493652i
$609$		0			0
$610$	−9.89134	+	17.1323i	−0.400489	+	0.693667i
$611$	6.08711	−	10.5432i	0.246258	−	0.426531i
$612$		0			0
$613$	14.6734	+	25.4151i	0.592653	+	1.02651i	0.993873	+	0.110524i	$0.0352529\pi$
$613$	14.6734	+	25.4151i	0.592653	+	1.02651i	−0.401220	+	0.915982i	$0.631414\pi$
$614$	−0.692965	−	1.20025i	−0.0279658	−	0.0484382i
$615$		0			0
$616$		0			0
$617$	−2.06401	−	3.57497i	−0.0830938	−	0.143923i	0.821484	−	0.570232i	$-0.193147\pi$
$617$	−2.06401	−	3.57497i	−0.0830938	−	0.143923i	−0.904577	+	0.426310i	$0.859813\pi$
$618$		0			0
$619$	−22.7130			−0.912912			−0.456456	−	0.889746i	$-0.650881\pi$
$619$	−22.7130			−0.912912			−0.456456	+	0.889746i	$0.650881\pi$
$620$	11.6109	+	20.1106i	0.466304	+	0.807662i
$621$		0			0
$622$	−7.48774			−0.300231
$623$		0			0
$624$		0			0
$625$	6.38996			0.255598
$626$	−6.31698	+	10.9413i	−0.252477	+	0.437304i
$627$		0			0
$628$	14.6290	+	25.3382i	0.583761	+	1.01110i
$629$	1.18276			0.0471597
$630$		0			0
$631$	−38.6411			−1.53828			−0.769138	−	0.639082i	$-0.779314\pi$
$631$	−38.6411			−1.53828			−0.769138	+	0.639082i	$0.779314\pi$
$632$	−1.67458	−	2.90045i	−0.0666110	−	0.115374i
$633$		0			0
$634$	8.06304	−	13.9656i	0.320224	−	0.554645i
$635$	4.68759			0.186021
$636$		0			0
$637$		0			0
$638$	1.83348			0.0725881
$639$		0			0
$640$	−20.9427	−	36.2737i	−0.827831	−	1.43385i
$641$	28.4726			1.12460			0.562301	−	0.826933i	$-0.309916\pi$
$641$	28.4726			1.12460			0.562301	+	0.826933i	$0.309916\pi$
$642$		0			0
$643$	8.52125	+	14.7592i	0.336045	+	0.582048i	0.983685	−	0.179899i	$-0.0575771\pi$
$643$	8.52125	+	14.7592i	0.336045	+	0.582048i	−0.647640	+	0.761947i	$0.724244\pi$
$644$		0			0
$645$		0			0
$646$	−0.174055	−	0.301472i	−0.00684810	−	0.0118613i
$647$	1.68809	+	2.92386i	0.0663657	+	0.114949i	0.897299	−	0.441423i	$-0.145526\pi$
$647$	1.68809	+	2.92386i	0.0663657	+	0.114949i	−0.830933	+	0.556372i	$0.812193\pi$
$648$		0			0
$649$	−0.808833	+	1.40094i	−0.0317495	+	0.0549917i
$650$	−2.56209	+	4.43768i	−0.100494	+	0.174060i
$651$		0			0
$652$	5.86110	+	10.1517i	0.229538	+	0.397572i
$653$	18.3451			0.717899			0.358950	−	0.933357i	$-0.383135\pi$
$653$	18.3451			0.717899			0.358950	+	0.933357i	$0.383135\pi$
$654$		0			0
$655$	55.4954			2.16838
$656$	−6.18090	+	10.7056i	−0.241324	+	0.417985i
$657$		0			0
$658$		0			0
$659$	13.9248	−	24.1184i	0.542432	−	0.939519i	−0.456332	−	0.889810i	$-0.650837\pi$
$659$	13.9248	−	24.1184i	0.542432	−	0.939519i	0.998764	−	0.0497098i	$-0.0158297\pi$
$660$		0			0
$661$	19.5071	−	33.7872i	0.758737	−	1.31417i	−0.184758	−	0.982784i	$-0.559150\pi$
$661$	19.5071	−	33.7872i	0.758737	−	1.31417i	0.943495	−	0.331387i	$-0.107516\pi$
$662$	−4.48633	+	7.77054i	−0.174366	+	0.302011i
$663$		0			0
$664$	−11.4700	+	19.8667i	−0.445123	+	0.770976i
$665$		0			0
$666$		0			0
$667$	5.13203	−	8.88894i	0.198713	−	0.344181i
$668$	30.9268			1.19659
$669$		0			0
$670$	−1.88481			−0.0728165
$671$	4.82170	+	8.35143i	0.186140	+	0.322404i
$672$		0			0
$673$	24.6154	−	42.6352i	0.948856	−	1.64347i	0.201014	−	0.979588i	$-0.435576\pi$
$673$	24.6154	−	42.6352i	0.948856	−	1.64347i	0.747841	−	0.663878i	$-0.231090\pi$
$674$	−6.20264	+	10.7433i	−0.238917	+	0.413816i
$675$		0			0
$676$	−10.1457	−	17.5729i	−0.390219	−	0.675879i
$677$	11.6958	+	20.2577i	0.449505	+	0.778565i	0.998354	−	0.0573564i	$-0.0182671\pi$
$677$	11.6958	+	20.2577i	0.449505	+	0.778565i	−0.548849	+	0.835922i	$0.684934\pi$
$678$		0			0
$679$		0			0
$680$	−0.859180	−	1.48814i	−0.0329480	−	0.0570677i
$681$		0			0
$682$	−1.58677			−0.0607607
$683$	15.1632	+	26.2634i	0.580204	+	1.00494i	0.995455	+	0.0952356i	$0.0303604\pi$
$683$	15.1632	+	26.2634i	0.580204	+	1.00494i	−0.415251	+	0.909707i	$0.636306\pi$
$684$		0			0
$685$	−1.80380			−0.0689196
$686$		0			0
$687$		0			0
$688$	25.7961			0.983466
$689$	5.91731	−	10.2491i	0.225432	−	0.390459i
$690$		0			0
$691$	−2.05665	−	3.56223i	−0.0782387	−	0.135513i	0.824251	−	0.566224i	$-0.191596\pi$
$691$	−2.05665	−	3.56223i	−0.0782387	−	0.135513i	−0.902490	+	0.430711i	$0.858263\pi$
$692$	6.81797			0.259180
$693$		0			0
$694$	−5.33003			−0.202325
$695$	−18.2224	−	31.5621i	−0.691215	−	1.19722i
$696$		0			0
$697$	−0.597691	+	1.03523i	−0.0226392	+	0.0392122i
$698$	−1.62943			−0.0616749
$699$		0			0
$700$		0			0
$701$	−29.1835			−1.10225			−0.551123	−	0.834424i	$-0.685800\pi$
$701$	−29.1835			−1.10225			−0.551123	+	0.834424i	$0.685800\pi$
$702$		0			0
$703$	−6.64347	−	11.5068i	−0.250563	−	0.433988i
$704$	−2.39965			−0.0904403
$705$		0			0
$706$	−4.17005	−	7.22274i	−0.156942	−	0.271831i
$707$		0			0
$708$		0			0
$709$	21.2309	+	36.7729i	0.797342	+	1.38104i	0.921341	+	0.388755i	$0.127095\pi$
$709$	21.2309	+	36.7729i	0.797342	+	1.38104i	−0.123999	+	0.992282i	$0.539572\pi$
$710$	4.52753	+	7.84192i	0.169915	+	0.294302i
$711$		0			0
$712$	2.24075	−	3.88109i	0.0839757	−	0.145450i
$713$	−4.44149	+	7.69288i	−0.166335	+	0.288101i
$714$		0			0
$715$	1.97211	+	3.41579i	0.0737526	+	0.127743i
$716$	−12.8667			−0.480852
$717$		0			0
$718$	11.7938			0.440142
$719$	−5.57126	+	9.64970i	−0.207773	+	0.359873i	−0.951013	−	0.309152i	$-0.899955\pi$
$719$	−5.57126	+	9.64970i	−0.207773	+	0.359873i	0.743240	+	0.669025i	$0.233288\pi$
$720$		0			0
$721$		0			0
$722$	2.75544	−	4.77256i	0.102547	−	0.177616i
$723$		0			0
$724$	−9.87264	+	17.0999i	−0.366914	+	0.635513i
$725$	−17.8858	+	30.9790i	−0.664260	+	1.15053i
$726$		0			0
$727$	14.3410	−	24.8393i	0.531878	−	0.921239i	−0.467430	−	0.884030i	$-0.654820\pi$
$727$	14.3410	−	24.8393i	0.531878	−	0.921239i	0.999308	−	0.0372089i	$-0.0118467\pi$
$728$		0			0
$729$		0			0
$730$	−1.67588	+	2.90270i	−0.0620269	+	0.107434i
$731$	2.49447			0.0922614
$732$		0			0
$733$	25.0528			0.925348			0.462674	−	0.886529i	$-0.346890\pi$
$733$	25.0528			0.925348			0.462674	+	0.886529i	$0.346890\pi$
$734$	−0.171071	−	0.296303i	−0.00631433	−	0.0109367i
$735$		0			0
$736$	6.20047	−	10.7395i	0.228552	−	0.395864i
$737$	−0.459391	+	0.795689i	−0.0169219	+	0.0293096i
$738$		0			0
$739$	13.7608	+	23.8344i	0.506198	+	0.876761i	0.999974	+	0.00717223i	$0.00228301\pi$
$739$	13.7608	+	23.8344i	0.506198	+	0.876761i	−0.493776	+	0.869589i	$0.664384\pi$
$740$	−15.3230	−	26.5402i	−0.563284	−	0.975637i
$741$		0			0
$742$		0			0
$743$	7.00608	+	12.1349i	0.257028	+	0.445186i	0.965444	−	0.260609i	$-0.0839233\pi$
$743$	7.00608	+	12.1349i	0.257028	+	0.445186i	−0.708416	+	0.705795i	$0.750590\pi$
$744$		0			0
$745$	77.6326			2.84424
$746$	0.932261	+	1.61472i	0.0341325	+	0.0591192i
$747$		0			0
$748$	−0.391390			−0.0143106
$749$		0			0
$750$		0			0
$751$	−52.2594			−1.90697			−0.953486	−	0.301436i	$-0.902534\pi$
$751$	−52.2594			−1.90697			−0.953486	+	0.301436i	$0.902534\pi$
$752$	13.1494	−	22.7755i	0.479511	−	0.830537i
$753$		0			0
$754$	−1.22912	−	2.12889i	−0.0447618	−	0.0775298i
$755$	−5.53289			−0.201363
$756$		0			0
$757$	−43.3447			−1.57539			−0.787694	−	0.616066i	$-0.788725\pi$
$757$	−43.3447			−1.57539			−0.787694	+	0.616066i	$0.788725\pi$
$758$	−8.15047	−	14.1170i	−0.296038	−	0.512753i
$759$		0			0
$760$	−9.65191	+	16.7176i	−0.350112	+	0.606411i
$761$	−17.2510			−0.625348			−0.312674	−	0.949860i	$-0.601225\pi$
$761$	−17.2510			−0.625348			−0.312674	+	0.949860i	$0.601225\pi$
$762$		0			0
$763$		0			0
$764$	−41.8285			−1.51330
$765$		0			0
$766$	0.265952	+	0.460642i	0.00960922	+	0.0166437i
$767$	2.16889			0.0783139
$768$		0			0
$769$	10.6727	+	18.4856i	0.384867	+	0.666609i	0.991751	−	0.128182i	$-0.0409141\pi$
$769$	10.6727	+	18.4856i	0.384867	+	0.666609i	−0.606884	+	0.794790i	$0.707581\pi$
$770$		0			0
$771$		0			0
$772$	5.20495	+	9.01523i	0.187330	+	0.324465i
$773$	−6.57357	−	11.3858i	−0.236435	−	0.409517i	0.723254	−	0.690582i	$-0.242646\pi$
$773$	−6.57357	−	11.3858i	−0.236435	−	0.409517i	−0.959689	+	0.281065i	$0.909312\pi$
$774$		0			0
$775$	15.4791	−	26.8106i	0.556027	−	0.963066i
$776$	10.2796	−	17.8049i	0.369018	−	0.639158i
$777$		0			0
$778$	−5.88685	−	10.1963i	−0.211054	−	0.365556i
$779$	13.4288			0.481136
$780$		0			0
$781$	4.41405			0.157947
$782$	0.153567	−	0.265986i	0.00549155	−	0.00951164i
$783$		0			0
$784$		0			0
$785$	30.7954	−	53.3393i	1.09914	−	1.90376i
$786$		0			0
$787$	−14.0650	+	24.3614i	−0.501364	+	0.868389i	0.498634	+	0.866812i	$0.333835\pi$
$787$	−14.0650	+	24.3614i	−0.501364	+	0.868389i	−0.999999	+	0.00157623i	$0.999498\pi$
$788$	13.5665	−	23.4979i	0.483287	−	0.837077i
$789$		0			0
$790$	−1.64713	+	2.85291i	−0.0586021	+	0.101502i
$791$		0			0
$792$		0			0
$793$	6.46470	−	11.1972i	0.229568	−	0.397624i
$794$	−0.0159163			−0.000564847
$795$		0			0
$796$

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:	\( N \)	\(=\)	\( 1323 = 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1323.g (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.247934 - 0.429435i) q^{2} +(0.877057 - 1.51911i) q^{4} -3.69258 q^{5} -1.86155 q^{8} +O(q^{10})\)	\(q+(-0.247934 - 0.429435i) q^{2} +(0.877057 - 1.51911i) q^{4} -3.69258 q^{5} -1.86155 q^{8} +(0.915516 + 1.58572i) q^{10} +0.892568 q^{11} +(-0.598355 - 1.03638i) q^{13} +(-1.29257 - 2.23880i) q^{16} +(-0.124991 - 0.216492i) q^{17} +(-1.40414 + 2.43204i) q^{19} +(-3.23860 + 5.60943i) q^{20} +(-0.221298 - 0.383300i) q^{22} -2.47772 q^{23} +8.63514 q^{25} +(-0.296705 + 0.513909i) q^{26} +(-2.07128 + 3.58755i) q^{29} +(1.79257 - 3.10483i) q^{31} +(-2.50249 + 4.33444i) q^{32} +(-0.0619793 + 0.107351i) q^{34} +(-2.36568 + 4.09747i) q^{37} +1.39253 q^{38} +6.87391 q^{40} +(-2.39093 - 4.14121i) q^{41} +(-4.98928 + 8.64169i) q^{43} +(0.782834 - 1.35591i) q^{44} +(0.614310 + 1.06402i) q^{46} +(5.08653 + 8.81013i) q^{47} +(-2.14095 - 3.70823i) q^{50} -2.09917 q^{52} +(4.94465 + 8.56438i) q^{53} -3.29588 q^{55} +2.05416 q^{58} +(-0.906186 + 1.56956i) q^{59} +(5.40205 + 9.35663i) q^{61} -1.77776 q^{62} -2.68848 q^{64} +(2.20948 + 3.82692i) q^{65} +(-0.514685 + 0.891460i) q^{67} -0.438499 q^{68} +4.94533 q^{71} +(0.915262 + 1.58528i) q^{73} +2.34613 q^{74} +(2.46302 + 4.26607i) q^{76} +(0.899562 + 1.55809i) q^{79} +(4.77293 + 8.26696i) q^{80} +(-1.18559 + 2.05350i) q^{82} +(6.16156 - 10.6721i) q^{83} +(0.461541 + 0.799412i) q^{85} +4.94806 q^{86} -1.66156 q^{88} +(-1.20370 + 2.08488i) q^{89} +(-2.17310 + 3.76392i) q^{92} +(2.52225 - 4.36867i) q^{94} +(5.18489 - 8.98049i) q^{95} +(-5.52210 + 9.56456i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10q - 2q^{2} - 4q^{4} - 8q^{5} + 6q^{8} + O(q^{10}) \)	\( 10q - 2q^{2} - 4q^{4} - 8q^{5} + 6q^{8} + 7q^{10} + 8q^{11} + 8q^{13} + 2q^{16} + 12q^{17} - q^{19} + 5q^{20} - q^{22} + 6q^{23} + 2q^{25} + 11q^{26} - 7q^{29} + 3q^{31} + 2q^{32} - 3q^{34} - 40q^{38} - 6q^{40} + 5q^{41} - 7q^{43} + 10q^{44} + 3q^{46} + 27q^{47} - 19q^{50} - 20q^{52} + 21q^{53} - 4q^{55} + 20q^{58} + 30q^{59} + 14q^{61} - 12q^{62} - 50q^{64} + 11q^{65} - 2q^{67} - 54q^{68} + 6q^{71} - 15q^{73} - 72q^{74} - 5q^{76} - 4q^{79} + 20q^{80} + 5q^{82} + 9q^{83} - 6q^{85} - 16q^{86} + 36q^{88} + 28q^{89} - 27q^{92} + 3q^{94} + 14q^{95} + 12q^{97} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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