LMFDB - Embedded newform 2592.2.s.h.1727.3

Newspace parameters

Level:	$ N $	$=$	$ 2592 = 2^{5} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2592.s (of order $6$, degree $2$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$20.6972242039$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{24})$
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 864)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1727.3
Root			$-0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	2592.1727
Dual form			2592.2.s.h.863.3

$q$-expansion

$f(q)$	$=$	$q+(2.72474 - 1.57313i) q^{5} +(-2.98735 - 1.72474i) q^{7} +O(q^{10})$	$q+(2.72474 - 1.57313i) q^{5} +(-2.98735 - 1.72474i) q^{7} +(-2.28024 + 3.94949i) q^{11} +(3.44949 + 5.97469i) q^{13} -3.46410i q^{17} +4.89898i q^{19} +(1.41421 + 2.44949i) q^{23} +(2.44949 - 4.24264i) q^{25} +(1.89898 + 1.09638i) q^{29} +(-2.20881 + 1.27526i) q^{31} -10.8530 q^{35} -4.89898 q^{37} +(-3.55051 + 2.04989i) q^{41} +(-2.51059 - 1.44949i) q^{43} +(-1.09638 + 1.89898i) q^{47} +(2.44949 + 4.24264i) q^{49} +12.9029i q^{53} +14.3485i q^{55} +(1.09638 + 1.89898i) q^{59} +(-2.00000 + 3.46410i) q^{61} +(18.7980 + 10.8530i) q^{65} +(12.9029 - 7.44949i) q^{67} +13.2207 q^{71} -7.89898 q^{73} +(13.6237 - 7.86566i) q^{77} +(-1.73205 - 1.00000i) q^{79} +(-6.20504 + 10.7474i) q^{83} +(-5.44949 - 9.43879i) q^{85} +5.02118i q^{89} -23.7980i q^{91} +(7.70674 + 13.3485i) q^{95} +(2.50000 - 4.33013i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 12 q^{5}+O(q^{10}) $ 8 * q + 12 * q^5	$ 8 q + 12 q^{5} + 8 q^{13} - 24 q^{29} - 48 q^{41} - 16 q^{61} + 72 q^{65} - 24 q^{73} + 60 q^{77} - 24 q^{85} + 20 q^{97}+O(q^{100}) $ 8 * q + 12 * q^5 + 8 * q^13 - 24 * q^29 - 48 * q^41 - 16 * q^61 + 72 * q^65 - 24 * q^73 + 60 * q^77 - 24 * q^85 + 20 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2592\mathbb{Z}\right)^\times$.

$n$	$325$	$1217$	$2431$
$\chi(n)$	$1$	$e\left(\frac{1}{6}\right)$	$-1$

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			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	2.72474	−	1.57313i	1.21854	−	0.703526i	0.253937	−	0.967221i	$-0.418274\pi$
$5$	2.72474	−	1.57313i	1.21854	−	0.703526i	0.964606	+	0.263695i	$0.0849412\pi$
$6$		0			0
$7$	−2.98735	−	1.72474i	−1.12911	−	0.651892i	−0.185399	−	0.982663i	$-0.559358\pi$
$7$	−2.98735	−	1.72474i	−1.12911	−	0.651892i	−0.943711	+	0.330771i	$0.892691\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.28024	+	3.94949i	−0.687518	+	1.19082i	0.285120	+	0.958492i	$0.407966\pi$
$11$	−2.28024	+	3.94949i	−0.687518	+	1.19082i	−0.972638	+	0.232324i	$0.925367\pi$
$12$		0			0
$13$	3.44949	+	5.97469i	0.956716	+	1.65708i	0.730391	+	0.683030i	$0.239338\pi$
$13$	3.44949	+	5.97469i	0.956716	+	1.65708i	0.226326	+	0.974052i	$0.427329\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	3.46410i		−	0.840168i	−0.907485	−	0.420084i	$-0.862001\pi$
$17$		−	3.46410i		−	0.840168i	0.907485	−	0.420084i	$-0.137999\pi$
$18$		0			0
$19$			4.89898i			1.12390i	0.827170	+	0.561951i	$0.189949\pi$
$19$			4.89898i			1.12390i	−0.827170	+	0.561951i	$0.810051\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.41421	+	2.44949i	0.294884	+	0.510754i	0.974958	−	0.222390i	$-0.0713857\pi$
$23$	1.41421	+	2.44949i	0.294884	+	0.510754i	−0.680074	+	0.733144i	$0.738052\pi$
$24$		0			0
$25$	2.44949	−	4.24264i	0.489898	−	0.848528i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.89898	+	1.09638i	0.352632	+	0.203592i	0.665844	−	0.746091i	$-0.268072\pi$
$29$	1.89898	+	1.09638i	0.352632	+	0.203592i	−0.313212	+	0.949683i	$0.601405\pi$
$30$		0			0
$31$	−2.20881	+	1.27526i	−0.396713	+	0.229043i	−0.685065	−	0.728482i	$-0.740226\pi$
$31$	−2.20881	+	1.27526i	−0.396713	+	0.229043i	0.288352	+	0.957525i	$0.406893\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−10.8530			−1.83449
$36$		0			0
$37$	−4.89898			−0.805387			−0.402694	−	0.915335i	$-0.631926\pi$
$37$	−4.89898			−0.805387			−0.402694	+	0.915335i	$0.631926\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−3.55051	+	2.04989i	−0.554497	+	0.320139i	−0.750934	−	0.660378i	$-0.770396\pi$
$41$	−3.55051	+	2.04989i	−0.554497	+	0.320139i	0.196437	+	0.980516i	$0.437063\pi$
$42$		0			0
$43$	−2.51059	−	1.44949i	−0.382861	−	0.221045i	0.296201	−	0.955126i	$-0.404280\pi$
$43$	−2.51059	−	1.44949i	−0.382861	−	0.221045i	−0.679062	+	0.734080i	$0.737613\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.09638	+	1.89898i	−0.159923	+	0.276995i	−0.934841	−	0.355067i	$-0.884458\pi$
$47$	−1.09638	+	1.89898i	−0.159923	+	0.276995i	0.774918	+	0.632062i	$0.217791\pi$
$48$		0			0
$49$	2.44949	+	4.24264i	0.349927	+	0.606092i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			12.9029i			1.77235i	0.463352	+	0.886174i	$0.346647\pi$
$53$			12.9029i			1.77235i	−0.463352	+	0.886174i	$0.653353\pi$
$54$		0			0
$55$			14.3485i			1.93475i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.09638	+	1.89898i	0.142736	+	0.247226i	0.928526	−	0.371267i	$-0.121077\pi$
$59$	1.09638	+	1.89898i	0.142736	+	0.247226i	−0.785790	+	0.618493i	$0.787743\pi$
$60$		0			0
$61$	−2.00000	+	3.46410i	−0.256074	+	0.443533i	−0.965187	−	0.261562i	$-0.915762\pi$
$61$	−2.00000	+	3.46410i	−0.256074	+	0.443533i	0.709113	+	0.705095i	$0.249096\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	18.7980	+	10.8530i	2.33160	+	1.34615i
$66$		0			0
$67$	12.9029	−	7.44949i	1.57634	−	0.910100i	0.580975	−	0.813921i	$-0.302671\pi$
$67$	12.9029	−	7.44949i	1.57634	−	0.910100i	0.995364	−	0.0961789i	$-0.0306621\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	13.2207			1.56901			0.784506	−	0.620121i	$-0.212917\pi$
$71$	13.2207			1.56901			0.784506	+	0.620121i	$0.212917\pi$
$72$		0			0
$73$	−7.89898			−0.924506			−0.462253	−	0.886748i	$-0.652959\pi$
$73$	−7.89898			−0.924506			−0.462253	+	0.886748i	$0.652959\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	13.6237	−	7.86566i	1.55257	−	0.896375i
$78$		0			0
$79$	−1.73205	−	1.00000i	−0.194871	−	0.112509i	0.399390	−	0.916781i	$-0.369222\pi$
$79$	−1.73205	−	1.00000i	−0.194871	−	0.112509i	−0.594261	+	0.804272i	$0.702555\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−6.20504	+	10.7474i	−0.681092	+	1.17969i	0.293556	+	0.955942i	$0.405161\pi$
$83$	−6.20504	+	10.7474i	−0.681092	+	1.17969i	−0.974648	+	0.223744i	$0.928172\pi$
$84$		0			0
$85$	−5.44949	−	9.43879i	−0.591080	−	1.02378i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			5.02118i			0.532244i	0.963939	+	0.266122i	$0.0857424\pi$
$89$			5.02118i			0.532244i	−0.963939	+	0.266122i	$0.914258\pi$
$90$		0			0
$91$		−	23.7980i		−	2.49470i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	7.70674	+	13.3485i	0.790695	+	1.36952i
$96$		0			0
$97$	2.50000	−	4.33013i	0.253837	−	0.439658i	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	2.50000	−	4.33013i	0.253837	−	0.439658i	0.964579	+	0.263795i	$0.0849741\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1.37628	−	0.794593i	−0.136945	−	0.0790650i	0.429962	−	0.902847i	$-0.358527\pi$
$101$	−1.37628	−	0.794593i	−0.136945	−	0.0790650i	−0.566907	+	0.823782i	$0.691860\pi$
$102$		0			0
$103$	−8.66025	+	5.00000i	−0.853320	+	0.492665i	−0.861770	−	0.507300i	$-0.830644\pi$
$103$	−8.66025	+	5.00000i	−0.853320	+	0.492665i	0.00844953	+	0.999964i	$0.497310\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	8.66025			0.837218			0.418609	−	0.908166i	$-0.362518\pi$
$107$	8.66025			0.837218			0.418609	+	0.908166i	$0.362518\pi$
$108$		0			0
$109$	4.89898			0.469237			0.234619	−	0.972088i	$-0.424616\pi$
$109$	4.89898			0.469237			0.234619	+	0.972088i	$0.424616\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	12.2474	−	7.07107i	1.15214	−	0.665190i	0.202735	−	0.979234i	$-0.435017\pi$
$113$	12.2474	−	7.07107i	1.15214	−	0.665190i	0.949409	+	0.314044i	$0.101684\pi$
$114$		0			0
$115$	7.70674	+	4.44949i	0.718657	+	0.414917i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−5.97469	+	10.3485i	−0.547699	+	0.948643i
$120$		0			0
$121$	−4.89898	−	8.48528i	−0.445362	−	0.771389i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			0.317837i			0.0284282i
$126$		0			0
$127$		−	9.24745i		−	0.820578i	−0.911955	−	0.410289i	$-0.865428\pi$
$127$		−	9.24745i		−	0.820578i	0.911955	−	0.410289i	$-0.134572\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	7.15855	+	12.3990i	0.625446	+	1.08330i	0.988454	+	0.151518i	$0.0484162\pi$
$131$	7.15855	+	12.3990i	0.625446	+	1.08330i	−0.363009	+	0.931786i	$0.618250\pi$
$132$		0			0
$133$	8.44949	−	14.6349i	0.732664	−	1.26901i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	15.2474	+	8.80312i	1.30268	+	0.752101i	0.980863	−	0.194701i	$-0.0623738\pi$
$137$	15.2474	+	8.80312i	1.30268	+	0.752101i	0.321815	+	0.946803i	$0.395707\pi$
$138$		0			0
$139$	17.3205	−	10.0000i	1.46911	−	0.848189i	0.469706	−	0.882823i	$-0.344360\pi$
$139$	17.3205	−	10.0000i	1.46911	−	0.848189i	0.999400	+	0.0346338i	$0.0110265\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−31.4626			−2.63104
$144$		0			0
$145$	6.89898			0.572929
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−5.72474	+	3.30518i	−0.468989	+	0.270771i	−0.715817	−	0.698288i	$-0.753945\pi$
$149$	−5.72474	+	3.30518i	−0.468989	+	0.270771i	0.246827	+	0.969060i	$0.420612\pi$
$150$		0			0
$151$	6.27647	+	3.62372i	0.510772	+	0.294895i	0.733151	−	0.680066i	$-0.238049\pi$
$151$	6.27647	+	3.62372i	0.510772	+	0.294895i	−0.222379	+	0.974960i	$0.571382\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−4.01229	+	6.94949i	−0.322275	+	0.558196i
$156$		0			0
$157$	−2.89898	−	5.02118i	−0.231364	−	0.400734i	0.726846	−	0.686801i	$-0.240985\pi$
$157$	−2.89898	−	5.02118i	−0.231364	−	0.400734i	−0.958210	+	0.286067i	$0.907652\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	9.75663i		−	0.768930i
$162$		0			0
$163$		−	6.00000i		−	0.469956i	−0.972001	−	0.234978i	$-0.924498\pi$
$163$		−	6.00000i		−	0.469956i	0.972001	−	0.234978i	$-0.0755019\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−5.83183	−	10.1010i	−0.451280	−	0.781640i	0.547186	−	0.837011i	$-0.315699\pi$
$167$	−5.83183	−	10.1010i	−0.451280	−	0.781640i	−0.998466	+	0.0553709i	$0.982366\pi$
$168$		0			0
$169$	−17.2980	+	29.9609i	−1.33061	+	2.30469i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	7.62372	+	4.40156i	0.579621	+	0.334644i	0.760983	−	0.648772i	$-0.224717\pi$
$173$	7.62372	+	4.40156i	0.579621	+	0.334644i	−0.181362	+	0.983416i	$0.558050\pi$
$174$		0			0
$175$	−14.6349	+	8.44949i	−1.10630	+	0.638721i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−8.66025			−0.647298			−0.323649	−	0.946177i	$-0.604910\pi$
$179$	−8.66025			−0.647298			−0.323649	+	0.946177i	$0.604910\pi$
$180$		0			0
$181$	9.79796			0.728277			0.364138	−	0.931345i	$-0.381364\pi$
$181$	9.79796			0.728277			0.364138	+	0.931345i	$0.381364\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−13.3485	+	7.70674i	−0.981399	+	0.566611i
$186$		0			0
$187$	13.6814	+	7.89898i	1.00049	+	0.577631i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−5.33902	+	9.24745i	−0.386318	+	0.669122i	−0.991951	−	0.126622i	$-0.959586\pi$
$191$	−5.33902	+	9.24745i	−0.386318	+	0.669122i	0.605633	+	0.795744i	$0.292920\pi$
$192$		0			0
$193$	−5.94949	−	10.3048i	−0.428254	−	0.741757i	0.568464	−	0.822708i	$-0.307538\pi$
$193$	−5.94949	−	10.3048i	−0.428254	−	0.741757i	−0.996718	+	0.0809508i	$0.974204\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			1.87492i			0.133582i	0.997767	+	0.0667911i	$0.0212761\pi$
$197$			1.87492i			0.133582i	−0.997767	+	0.0667911i	$0.978724\pi$
$198$		0			0
$199$			5.65153i			0.400626i	0.979732	+	0.200313i	$0.0641960\pi$
$199$			5.65153i			0.400626i	−0.979732	+	0.200313i	$0.935804\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−3.78194	−	6.55051i	−0.265440	−	0.459756i
$204$		0			0
$205$	−6.44949	+	11.1708i	−0.450452	+	0.780206i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−19.3485	−	11.1708i	−1.33836	−	0.772703i
$210$		0			0
$211$	−17.7491	+	10.2474i	−1.22190	+	0.705463i	−0.965322	−	0.261061i	$-0.915928\pi$
$211$	−17.7491	+	10.2474i	−1.22190	+	0.705463i	−0.256576	+	0.966524i	$0.582594\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−9.12096			−0.622044
$216$		0			0
$217$	8.79796			0.597244
$218$		0			0
$219$		0			0
$220$		0			0
$221$	20.6969	−	11.9494i	1.39223	−	0.803802i
$222$		0			0
$223$	−17.1455	−	9.89898i	−1.14815	−	0.662885i	−0.199714	−	0.979854i	$-0.564001\pi$
$223$	−17.1455	−	9.89898i	−1.14815	−	0.662885i	−0.948436	+	0.316969i	$0.897335\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	11.4887	−	19.8990i	0.762531	−	1.32074i	−0.179012	−	0.983847i	$-0.557290\pi$
$227$	11.4887	−	19.8990i	0.762531	−	1.32074i	0.941542	−	0.336895i	$-0.109377\pi$
$228$		0			0
$229$	−1.00000	−	1.73205i	−0.0660819	−	0.114457i	0.831092	−	0.556136i	$-0.187717\pi$
$229$	−1.00000	−	1.73205i	−0.0660819	−	0.114457i	−0.897173	+	0.441679i	$0.854383\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			7.56388i			0.495526i	0.968821	+	0.247763i	$0.0796954\pi$
$233$			7.56388i			0.495526i	−0.968821	+	0.247763i	$0.920305\pi$
$234$		0			0
$235$			6.89898i			0.450040i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−3.60697	−	6.24745i	−0.233315	−	0.404114i	0.725466	−	0.688258i	$-0.241624\pi$
$239$	−3.60697	−	6.24745i	−0.233315	−	0.404114i	−0.958782	+	0.284144i	$0.908291\pi$
$240$		0			0
$241$	0.101021	−	0.174973i	0.00650730	−	0.0112710i	−0.862753	−	0.505625i	$-0.831262\pi$
$241$	0.101021	−	0.174973i	0.00650730	−	0.0112710i	0.869261	+	0.494354i	$0.164595\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	13.3485	+	7.70674i	0.852802	+	0.492366i
$246$		0			0
$247$	−29.2699	+	16.8990i	−1.86240	+	1.07526i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	17.3205			1.09326			0.546630	−	0.837374i	$-0.315910\pi$
$251$	17.3205			1.09326			0.546630	+	0.837374i	$0.315910\pi$
$252$		0			0
$253$	−12.8990			−0.810952
$254$		0			0
$255$		0			0
$256$		0			0
$257$	19.8990	−	11.4887i	1.24126	−	0.716644i	0.271913	−	0.962322i	$-0.412344\pi$
$257$	19.8990	−	11.4887i	1.24126	−	0.716644i	0.969352	+	0.245678i	$0.0790105\pi$
$258$		0			0
$259$	14.6349	+	8.44949i	0.909371	+	0.525026i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−15.2706	+	26.4495i	−0.941627	+	1.63095i	−0.179258	+	0.983802i	$0.557370\pi$
$263$	−15.2706	+	26.4495i	−0.941627	+	1.63095i	−0.762368	+	0.647143i	$0.775963\pi$
$264$		0			0
$265$	20.2980	+	35.1571i	1.24689	+	2.15968i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	10.3923i		−	0.633630i	−0.948487	−	0.316815i	$-0.897387\pi$
$269$		−	10.3923i		−	0.633630i	0.948487	−	0.316815i	$-0.102613\pi$
$270$		0			0
$271$			17.4495i			1.05998i	0.848004	+	0.529991i	$0.177805\pi$
$271$			17.4495i			1.05998i	−0.848004	+	0.529991i	$0.822195\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	11.1708	+	19.3485i	0.673627	+	1.16676i
$276$		0			0
$277$	−2.89898	+	5.02118i	−0.174183	+	0.301693i	−0.939878	−	0.341510i	$-0.889062\pi$
$277$	−2.89898	+	5.02118i	−0.174183	+	0.301693i	0.765695	+	0.643203i	$0.222395\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−18.7980	−	10.8530i	−1.12139	−	0.647436i	−0.179636	−	0.983733i	$-0.557492\pi$
$281$	−18.7980	−	10.8530i	−1.12139	−	0.647436i	−0.941756	+	0.336297i	$0.890825\pi$
$282$		0			0
$283$	11.1708	−	6.44949i	0.664038	−	0.383382i	−0.129776	−	0.991543i	$-0.541426\pi$
$283$	11.1708	−	6.44949i	0.664038	−	0.383382i	0.793814	+	0.608161i	$0.208092\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	14.1421			0.834784
$288$		0			0
$289$	5.00000			0.294118
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−24.7980	+	14.3171i	−1.44871	+	0.836414i	−0.998405	−	0.0564592i	$-0.982019\pi$
$293$	−24.7980	+	14.3171i	−1.44871	+	0.836414i	−0.450307	+	0.892874i	$0.648686\pi$
$294$		0			0
$295$	5.97469	+	3.44949i	0.347860	+	0.200837i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−9.75663	+	16.8990i	−0.564241	+	0.977293i
$300$		0			0
$301$	5.00000	+	8.66025i	0.288195	+	0.499169i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			12.5851i			0.720618i
$306$		0			0
$307$		−	18.0000i		−	1.02731i	−0.857996	−	0.513657i	$-0.828290\pi$
$307$		−	18.0000i		−	1.02731i	0.857996	−	0.513657i	$-0.171710\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	4.87832	+	8.44949i	0.276624	+	0.479127i	0.970544	−	0.240926i	$-0.0774511\pi$
$311$	4.87832	+	8.44949i	0.276624	+	0.479127i	−0.693920	+	0.720052i	$0.744118\pi$
$312$		0			0
$313$	9.74745	−	16.8831i	0.550958	−	0.954288i	−0.447247	−	0.894410i	$-0.647596\pi$
$313$	9.74745	−	16.8831i	0.550958	−	0.954288i	0.998206	−	0.0598776i	$-0.0190710\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	13.3763	+	7.72280i	0.751286	+	0.433755i	0.826159	−	0.563438i	$-0.190522\pi$
$317$	13.3763	+	7.72280i	0.751286	+	0.433755i	−0.0748721	+	0.997193i	$0.523855\pi$
$318$		0			0
$319$	−8.66025	+	5.00000i	−0.484881	+	0.279946i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	16.9706			0.944267
$324$		0			0
$325$	33.7980			1.87477
$326$		0			0
$327$		0			0
$328$		0			0
$329$	6.55051	−	3.78194i	0.361141	−	0.208505i
$330$		0			0
$331$	−24.8523	−	14.3485i	−1.36600	−	0.788663i	−0.375590	−	0.926786i	$-0.622560\pi$
$331$	−24.8523	−	14.3485i	−1.36600	−	0.788663i	−0.990415	+	0.138123i	$0.955893\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	23.4381	−	40.5959i	1.28056	−	2.21799i
$336$		0			0
$337$	−11.8990	−	20.6096i	−0.648179	−	1.12268i	−0.983558	−	0.180595i	$-0.942198\pi$
$337$	−11.8990	−	20.6096i	−0.648179	−	1.12268i	0.335379	−	0.942083i	$-0.391136\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	11.6315i		−	0.629884i
$342$		0			0
$343$			7.24745i			0.391325i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−11.4012	−	19.7474i	−0.612048	−	1.06010i	−0.990895	−	0.134639i	$-0.957013\pi$
$347$	−11.4012	−	19.7474i	−0.612048	−	1.06010i	0.378847	−	0.925460i	$-0.376321\pi$
$348$		0			0
$349$	−7.55051	+	13.0779i	−0.404170	+	0.700042i	−0.994224	−	0.107321i	$-0.965773\pi$
$349$	−7.55051	+	13.0779i	−0.404170	+	0.700042i	0.590055	+	0.807363i	$0.299106\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	12.2474	+	7.07107i	0.651866	+	0.376355i	0.789171	−	0.614174i	$-0.210511\pi$
$353$	12.2474	+	7.07107i	0.651866	+	0.376355i	−0.137305	+	0.990529i	$0.543844\pi$
$354$		0			0
$355$	36.0231	−	20.7980i	1.91191	−	1.10384i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	0.349945			0.0184694			0.00923470	−	0.999957i	$-0.497060\pi$
$359$	0.349945			0.0184694			0.00923470	+	0.999957i	$0.497060\pi$
$360$		0			0
$361$	−5.00000			−0.263158
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−21.5227	+	12.4261i	−1.12655	+	0.650414i
$366$		0			0
$367$	−11.6476	−	6.72474i	−0.608000	−	0.351029i	0.164182	−	0.986430i	$-0.447501\pi$
$367$	−11.6476	−	6.72474i	−0.608000	−	0.351029i	−0.772182	+	0.635401i	$0.780835\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	22.2542	−	38.5454i	1.15538	−	2.00118i
$372$		0			0
$373$	7.24745	+	12.5529i	0.375259	+	0.649967i	0.990366	−	0.138477i	$-0.0442205\pi$
$373$	7.24745	+	12.5529i	0.375259	+	0.649967i	−0.615107	+	0.788444i	$0.710887\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			15.1278i			0.779119i
$378$		0			0
$379$		−	8.00000i		−	0.410932i	−0.978664	−	0.205466i	$-0.934129\pi$
$379$		−	8.00000i		−	0.410932i	0.978664	−	0.205466i	$-0.0658711\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	12.2672	+	21.2474i	0.626826	+	1.08569i	0.988185	+	0.153267i	$0.0489795\pi$
$383$	12.2672	+	21.2474i	0.626826	+	1.08569i	−0.361359	+	0.932427i	$0.617687\pi$
$384$		0			0
$385$	24.7474	−	42.8638i	1.26125	−	2.18454i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−18.7702	−	10.8370i	−0.951685	−	0.549455i	−0.0580807	−	0.998312i	$-0.518498\pi$
$389$	−18.7702	−	10.8370i	−0.951685	−	0.549455i	−0.893604	+	0.448857i	$0.851831\pi$
$390$		0			0
$391$	8.48528	−	4.89898i	0.429119	−	0.247752i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−6.29253			−0.316611
$396$		0			0
$397$	−20.6969			−1.03875			−0.519375	−	0.854547i	$-0.673835\pi$
$397$	−20.6969			−1.03875			−0.519375	+	0.854547i	$0.673835\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−19.5959	+	11.3137i	−0.978573	+	0.564980i	−0.901839	−	0.432072i	$-0.857783\pi$
$401$	−19.5959	+	11.3137i	−0.978573	+	0.564980i	−0.0767343	+	0.997052i	$0.524449\pi$
$402$		0			0
$403$	−15.2385	−	8.79796i	−0.759084	−	0.438258i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	11.1708	−	19.3485i	0.553718	−	0.959068i
$408$		0			0
$409$	3.50000	+	6.06218i	0.173064	+	0.299755i	0.939490	−	0.342578i	$-0.111300\pi$
$409$	3.50000	+	6.06218i	0.173064	+	0.299755i	−0.766426	+	0.642333i	$0.777967\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	7.56388i		−	0.372194i
$414$		0			0
$415$			39.0454i			1.91666i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−9.58166	−	16.5959i	−0.468095	−	0.810764i	0.531241	−	0.847221i	$-0.321726\pi$
$419$	−9.58166	−	16.5959i	−0.468095	−	0.810764i	−0.999335	+	0.0364574i	$0.988393\pi$
$420$		0			0
$421$	5.10102	−	8.83523i	0.248609	−	0.430603i	−0.714531	−	0.699603i	$-0.753360\pi$
$421$	5.10102	−	8.83523i	0.248609	−	0.430603i	0.963140	+	0.269001i	$0.0866934\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−14.6969	−	8.48528i	−0.712906	−	0.411597i
$426$		0			0
$427$	11.9494	−	6.89898i	0.578271	−	0.333865i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	6.57826			0.316864			0.158432	−	0.987370i	$-0.449356\pi$
$431$	6.57826			0.316864			0.158432	+	0.987370i	$0.449356\pi$
$432$		0			0
$433$	−26.3939			−1.26841			−0.634204	−	0.773165i	$-0.718672\pi$
$433$	−26.3939			−1.26841			−0.634204	+	0.773165i	$0.718672\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−12.0000	+	6.92820i	−0.574038	+	0.331421i
$438$		0			0
$439$	−4.54442	−	2.62372i	−0.216894	−	0.125224i	0.387618	−	0.921820i	$-0.373298\pi$
$439$	−4.54442	−	2.62372i	−0.216894	−	0.125224i	−0.604511	+	0.796597i	$0.706631\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−3.00340	+	5.20204i	−0.142696	+	0.247156i	−0.928511	−	0.371305i	$-0.878910\pi$
$443$	−3.00340	+	5.20204i	−0.142696	+	0.247156i	0.785815	+	0.618462i	$0.212244\pi$
$444$		0			0
$445$	7.89898	+	13.6814i	0.374448	+	0.648562i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			5.37113i			0.253479i	0.991936	+	0.126740i	$0.0404512\pi$
$449$			5.37113i			0.253479i	−0.991936	+	0.126740i	$0.959549\pi$
$450$		0			0
$451$		−	18.6969i		−	0.880404i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−37.4373	−	64.8434i	−1.75509	−	3.03990i
$456$		0			0
$457$	−12.5000	+	21.6506i	−0.584725	+	1.01277i	0.410184	+	0.912003i	$0.365464\pi$
$457$	−12.5000	+	21.6506i	−0.584725	+	1.01277i	−0.994910	+	0.100771i	$0.967869\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−10.6237	−	6.13361i	−0.494796	−	0.285671i	0.231766	−	0.972772i	$-0.425550\pi$
$461$	−10.6237	−	6.13361i	−0.494796	−	0.285671i	−0.726562	+	0.687101i	$0.758883\pi$
$462$		0			0
$463$	14.7618	−	8.52270i	0.686037	−	0.396084i	−0.116089	−	0.993239i	$-0.537036\pi$
$463$	14.7618	−	8.52270i	0.686037	−	0.396084i	0.802126	+	0.597155i	$0.203702\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−17.1455			−0.793401			−0.396700	−	0.917948i	$-0.629845\pi$
$467$	−17.1455			−0.793401			−0.396700	+	0.917948i	$0.629845\pi$
$468$		0			0
$469$	−51.3939			−2.37315
$470$		0			0
$471$		0			0
$472$		0			0
$473$	11.4495	−	6.61037i	0.526448	−	0.303945i
$474$		0			0
$475$	20.7846	+	12.0000i	0.953663	+	0.550598i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	9.58166	−	16.5959i	0.437797	−	0.758287i	−0.559722	−	0.828680i	$-0.689092\pi$
$479$	9.58166	−	16.5959i	0.437797	−	0.758287i	0.997519	+	0.0703935i	$0.0224255\pi$
$480$		0			0
$481$	−16.8990	−	29.2699i	−0.770527	−	1.33459i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	15.7313i		−	0.714323i
$486$		0			0
$487$		−	15.7980i		−	0.715874i	−0.933746	−	0.357937i	$-0.883480\pi$
$487$		−	15.7980i		−	0.715874i	0.933746	−	0.357937i	$-0.116520\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−1.50170	−	2.60102i	−0.0677708	−	0.117382i	0.830149	−	0.557542i	$-0.188255\pi$
$491$	−1.50170	−	2.60102i	−0.0677708	−	0.117382i	−0.897920	+	0.440159i	$0.854922\pi$
$492$		0			0
$493$	3.79796	−	6.57826i	0.171051	−	0.296270i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−39.4949	−	22.8024i	−1.77159	−	1.02283i
$498$		0			0
$499$	−4.41761	+	2.55051i	−0.197760	+	0.114177i	−0.595610	−	0.803274i	$-0.703090\pi$
$499$	−4.41761	+	2.55051i	−0.197760	+	0.114177i	0.397850	+	0.917450i	$0.369756\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	31.1127			1.38725			0.693623	−	0.720338i	$-0.256013\pi$
$503$	31.1127			1.38725			0.693623	+	0.720338i	$0.256013\pi$
$504$		0			0
$505$	−5.00000			−0.222497
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−38.1186	+	22.0078i	−1.68958	+	0.975478i	−0.734742	+	0.678347i	$0.762697\pi$
$509$	−38.1186	+	22.0078i	−1.68958	+	0.975478i	−0.954837	+	0.297131i	$0.903970\pi$
$510$		0			0
$511$	23.5970	+	13.6237i	1.04387	+	0.602678i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−15.7313	+	27.2474i	−0.693205	+	1.20067i
$516$		0			0
$517$	−5.00000	−	8.66025i	−0.219900	−	0.380878i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			0.921404i			0.0403674i	0.999796	+	0.0201837i	$0.00642511\pi$
$521$			0.921404i			0.0403674i	−0.999796	+	0.0201837i	$0.993575\pi$
$522$		0			0
$523$			31.3939i			1.37276i	0.727244	+	0.686379i	$0.240801\pi$
$523$			31.3939i			1.37276i	−0.727244	+	0.686379i	$0.759199\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	4.41761	+	7.65153i	0.192434	+	0.333306i
$528$		0			0
$529$	7.50000	−	12.9904i	0.326087	−	0.564799i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−24.4949	−	14.1421i	−1.06099	−	0.612564i
$534$		0			0
$535$	23.5970	−	13.6237i	1.02019	−	0.589005i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−22.3417			−0.962325
$540$		0			0
$541$	−20.0000			−0.859867			−0.429934	−	0.902861i	$-0.641463\pi$
$541$	−20.0000			−0.859867			−0.429934	+	0.902861i	$0.641463\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	13.3485	−	7.70674i	0.571786	−	0.330121i
$546$		0			0
$547$	−3.63907	−	2.10102i	−0.155596	−	0.0898332i	0.420181	−	0.907440i	$-0.361967\pi$
$547$	−3.63907	−	2.10102i	−0.155596	−	0.0898332i	−0.575776	+	0.817607i	$0.695300\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−5.37113	+	9.30306i	−0.228818	+	0.396324i
$552$		0			0
$553$	3.44949	+	5.97469i	0.146687	+	0.254070i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	29.8735i		−	1.26578i	−0.774242	−	0.632890i	$-0.781869\pi$
$557$		−	29.8735i		−	1.26578i	0.774242	−	0.632890i	$-0.218131\pi$
$558$		0			0
$559$		−	20.0000i		−	0.845910i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−2.74094	−	4.74745i	−0.115517	−	0.200081i	0.802469	−	0.596693i	$-0.203519\pi$
$563$	−2.74094	−	4.74745i	−0.115517	−	0.200081i	−0.917986	+	0.396612i	$0.870186\pi$
$564$		0			0
$565$	22.2474	−	38.5337i	0.935957	−	1.62113i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−16.8990	−	9.75663i	−0.708442	−	0.409019i	0.102042	−	0.994780i	$-0.467462\pi$
$569$	−16.8990	−	9.75663i	−0.708442	−	0.409019i	−0.810484	+	0.585761i	$0.800796\pi$
$570$		0			0
$571$	0.174973	−	0.101021i	0.00732238	−	0.00422758i	−0.496334	−	0.868131i	$-0.665321\pi$
$571$	0.174973	−	0.101021i	0.00732238	−	0.00422758i	0.503657	+	0.863904i	$0.331988\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	13.8564			0.577852
$576$		0			0
$577$	15.7980			0.657678			0.328839	−	0.944386i	$-0.393343\pi$
$577$	15.7980			0.657678			0.328839	+	0.944386i	$0.393343\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	37.0732	−	21.4042i	1.53806	−	0.887997i
$582$		0			0
$583$	−50.9599	−	29.4217i	−2.11054	−	1.21852i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	16.7402	−	28.9949i	0.690942	−	1.19675i	−0.280587	−	0.959829i	$-0.590529\pi$
$587$	16.7402	−	28.9949i	0.690942	−	1.19675i	0.971529	−	0.236919i	$-0.0761376\pi$
$588$		0			0
$589$	−6.24745	−	10.8209i	−0.257422	−	0.445867i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	42.7764i		−	1.75661i	−0.478097	−	0.878307i	$-0.658673\pi$
$593$		−	42.7764i		−	1.75661i	0.478097	−	0.878307i	$-0.341327\pi$
$594$		0			0
$595$			37.5959i			1.54128i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	18.4169	+	31.8990i	0.752493	+	1.30336i	0.946611	+	0.322379i	$0.104482\pi$
$599$	18.4169	+	31.8990i	0.752493	+	1.30336i	−0.194117	+	0.980978i	$0.562184\pi$
$600$		0			0
$601$	−19.8485	+	34.3786i	−0.809636	+	1.40233i	0.103480	+	0.994631i	$0.467002\pi$
$601$	−19.8485	+	34.3786i	−0.809636	+	1.40233i	−0.913116	+	0.407699i	$0.866331\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−26.6969	−	15.4135i	−1.08538	−	0.626647i
$606$		0			0
$607$	3.63907	−	2.10102i	0.147705	−	0.0852778i	−0.424326	−	0.905510i	$-0.639489\pi$
$607$	3.63907	−	2.10102i	0.147705	−	0.0852778i	0.572031	+	0.820232i	$0.306156\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−15.1278			−0.612003
$612$		0			0
$613$	30.0000			1.21169			0.605844	−	0.795583i	$-0.292835\pi$
$613$	30.0000			1.21169			0.605844	+	0.795583i	$0.292835\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−0.247449	+	0.142865i	−0.00996191	+	0.00575151i	−0.504973	−	0.863135i	$-0.668497\pi$
$617$	−0.247449	+	0.142865i	−0.00996191	+	0.00575151i	0.495011	+	0.868887i	$0.335164\pi$
$618$		0			0
$619$	15.2385	+	8.79796i	0.612488	+	0.353620i	0.773938	−	0.633261i	$-0.218284\pi$
$619$	15.2385	+	8.79796i	0.612488	+	0.353620i	−0.161451	+	0.986881i	$0.551617\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	8.66025	−	15.0000i	0.346966	−	0.600962i
$624$		0			0
$625$	12.7474	+	22.0792i	0.509898	+	0.883169i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			16.9706i			0.676661i
$630$		0			0
$631$		−	27.7423i		−	1.10441i	−0.833710	−	0.552203i	$-0.813787\pi$
$631$		−	27.7423i		−	1.10441i	0.833710	−	0.552203i	$-0.186213\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−14.5475	−	25.1969i	−0.577298	−	0.999910i
$636$		0			0
$637$	−16.8990	+	29.2699i	−0.669562	+	1.15972i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	17.7526	+	10.2494i	0.701184	+	0.404829i	0.807788	−	0.589473i	$-0.200665\pi$
$641$	17.7526	+	10.2494i	0.701184	+	0.404829i	−0.106604	+	0.994302i	$0.533998\pi$
$642$		0			0
$643$	18.5276	−	10.6969i	0.730659	−	0.421846i	−0.0880043	−	0.996120i	$-0.528049\pi$
$643$	18.5276	−	10.6969i	0.730659	−	0.421846i	0.818663	+	0.574274i	$0.194716\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	20.7846			0.817127			0.408564	−	0.912730i	$-0.366030\pi$
$647$	20.7846			0.817127			0.408564	+	0.912730i	$0.366030\pi$
$648$		0			0
$649$	−10.0000			−0.392534
$650$		0			0
$651$		0			0
$652$		0			0
$653$	1.07321	−	0.619620i	0.0419981	−	0.0242476i	−0.478854	−	0.877895i	$-0.658948\pi$
$653$	1.07321	−	0.619620i	0.0419981	−	0.0242476i	0.520852	+	0.853647i	$0.325614\pi$
$654$		0			0
$655$	39.0105	+	22.5227i	1.52427	+	0.880035i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	8.71563	−	15.0959i	0.339513	−	0.588053i	−0.644828	−	0.764327i	$-0.723071\pi$
$659$	8.71563	−	15.0959i	0.339513	−	0.588053i	0.984341	+	0.176274i	$0.0564045\pi$
$660$		0			0
$661$	−22.2474	−	38.5337i	−0.865325	−	1.49879i	−0.866724	−	0.498789i	$-0.833778\pi$
$661$	−22.2474	−	38.5337i	−0.865325	−	1.49879i	0.00139818	−	0.999999i	$-0.499555\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	53.1687i		−	2.06179i
$666$		0			0
$667$			6.20204i			0.240144i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−9.12096	−	15.7980i	−0.352111	−	0.609873i
$672$		0			0
$673$	−1.29796	+	2.24813i	−0.0500326	+	0.0866591i	−0.889957	−	0.456044i	$-0.849266\pi$
$673$	−1.29796	+	2.24813i	−0.0500326	+	0.0866591i	0.839924	+	0.542703i	$0.182599\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	39.4949	+	22.8024i	1.51791	+	0.876367i	0.999778	+	0.0210677i	$0.00670654\pi$
$677$	39.4949	+	22.8024i	1.51791	+	0.876367i	0.518134	+	0.855299i	$0.326627\pi$
$678$		0			0
$679$	−14.9367	+	8.62372i	−0.573219	+	0.330948i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	10.3923			0.397650			0.198825	−	0.980035i	$-0.436287\pi$
$683$	10.3923			0.397650			0.198825	+	0.980035i	$0.436287\pi$
$684$		0			0
$685$	55.3939			2.11649
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−77.0908	+	44.5084i	−2.93693	+	1.69564i
$690$		0			0
$691$	−3.46410	−	2.00000i	−0.131781	−	0.0760836i	0.432660	−	0.901557i	$-0.357575\pi$
$691$	−3.46410	−	2.00000i	−0.131781	−	0.0760836i	−0.564441	+	0.825473i	$0.690908\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	31.4626	−	54.4949i	1.19345	−	2.06711i
$696$		0			0
$697$	7.10102	+	12.2993i	0.268970	+	0.465870i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			13.2528i			0.500553i	0.968174	+	0.250276i	$0.0805215\pi$
$701$			13.2528i			0.500553i	−0.968174	+	0.250276i	$0.919478\pi$
$702$		0			0
$703$		−	24.0000i		−	0.905177i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	2.74094	+	4.74745i	0.103084	+	0.178546i
$708$		0			0
$709$	20.0000	−	34.6410i	0.751116	−	1.30097i	−0.196167	−	0.980571i	$-0.562849\pi$
$709$	20.0000	−	34.6410i	0.751116	−	1.30097i	0.947282	−	0.320400i	$-0.103817\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−6.24745	−	3.60697i	−0.233969	−	0.135082i
$714$		0			0
$715$	−85.7277	+	49.4949i	−3.20603	+	1.85100i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−6.92820			−0.258378			−0.129189	−	0.991620i	$-0.541237\pi$
$719$	−6.92820			−0.258378			−0.129189	+	0.991620i	$0.541237\pi$
$720$		0			0
$721$	34.4949			1.28466
$722$		0			0
$723$		0			0
$724$		0			0
$725$	9.30306	−	5.37113i	0.345507	−	0.199479i
$726$		0			0
$727$	−14.9367	−	8.62372i	−0.553973	−	0.319836i	0.196750	−	0.980454i	$-0.436961\pi$
$727$	−14.9367	−	8.62372i	−0.553973	−	0.319836i	−0.750723	+	0.660617i	$0.770295\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−5.02118	+	8.69694i	−0.185715	+	0.321668i
$732$		0			0
$733$	22.2474	+	38.5337i	0.821728	+	1.42328i	0.904394	+	0.426698i	$0.140323\pi$
$733$	22.2474	+	38.5337i	0.821728	+	1.42328i	−0.0826660	+	0.996577i	$0.526343\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			67.9465i			2.50284i
$738$		0			0
$739$		−	39.3939i		−	1.44913i	−0.689208	−	0.724564i	$-0.742041\pi$
$739$		−	39.3939i		−	1.44913i	0.689208	−	0.724564i	$-0.257959\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1.58919	−	2.75255i	−0.0583016	−	0.100981i	0.835402	−	0.549640i	$-0.185235\pi$
$743$	−1.58919	−	2.75255i	−0.0583016	−	0.100981i	−0.893703	+	0.448659i	$0.851902\pi$
$744$		0			0
$745$	−10.3990	+	18.0116i	−0.380989	+	0.659893i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−25.8712	−	14.9367i	−0.945312	−	0.545776i
$750$		0			0
$751$	−22.2935	+	12.8712i	−0.813502	+	0.469676i	−0.848171	−	0.529723i	$-0.822296\pi$
$751$	−22.2935	+	12.8712i	−0.813502	+	0.469676i	0.0346683	+	0.999399i	$0.488963\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	22.8024			0.829864
$756$		0			0
$757$	19.3939			0.704882			0.352441	−	0.935834i	$-0.385352\pi$
$757$	19.3939			0.704882			0.352441	+	0.935834i	$0.385352\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−26.2020	+	15.1278i	−0.949823	+	0.548381i	−0.893026	−	0.450005i	$-0.851422\pi$
$761$	−26.2020	+	15.1278i	−0.949823	+	0.548381i	−0.0567972	+	0.998386i	$0.518089\pi$
$762$		0			0
$763$	−14.6349	−	8.44949i	−0.529821	−	0.305892i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−7.56388	+	13.1010i	−0.273116	+	0.473050i
$768$		0			0
$769$	18.7474	+	32.4715i	0.676050	+	1.17095i	0.976161	+	0.217049i	$0.0696431\pi$
$769$	18.7474	+	32.4715i	0.676050	+	1.17095i	−0.300110	+	0.953904i	$0.597024\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			0.349945i			0.0125867i	0.999980	+	0.00629333i	$0.00200324\pi$
$773$			0.349945i			0.0125867i	−0.999980	+	0.00629333i	$0.997997\pi$
$774$		0			0
$775$			12.4949i			0.448830i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−10.0424	−	17.3939i	−0.359805	−	0.623200i
$780$		0			0
$781$	−30.1464	+	52.2151i	−1.07872	+	1.86840i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−15.7980	−	9.12096i	−0.563853	−	0.325541i
$786$		0			0
$787$	−37.4052	+	21.5959i	−1.33335	+	0.769811i	−0.985812	−	0.167854i	$-0.946316\pi$
$787$	−37.4052	+	21.5959i	−1.33335	+	0.769811i	−0.347540	+	0.937665i	$0.612983\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−48.7832			−1.73453
$792$		0			0
$793$	−27.5959			−0.979960
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−8.72474	+	5.03723i	−0.309046	+	0.178428i	−0.646500	−	0.762914i	$-0.723768\pi$
$797$	−8.72474	+	5.03723i	−0.309046	+	0.178428i	0.337453	+	0.941342i	$0.390434\pi$
$798$		0			0
$799$	6.57826	+	3.79796i	0.232722	+	0.134362i
$800$		0			0
$801$		0			0
$802$

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:	\( N \)	\(=\)	\( 2592 = 2^{5} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2592.s (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.72474 - 1.57313i) q^{5} +(-2.98735 - 1.72474i) q^{7} +O(q^{10})\)	\(q+(2.72474 - 1.57313i) q^{5} +(-2.98735 - 1.72474i) q^{7} +(-2.28024 + 3.94949i) q^{11} +(3.44949 + 5.97469i) q^{13} -3.46410i q^{17} +4.89898i q^{19} +(1.41421 + 2.44949i) q^{23} +(2.44949 - 4.24264i) q^{25} +(1.89898 + 1.09638i) q^{29} +(-2.20881 + 1.27526i) q^{31} -10.8530 q^{35} -4.89898 q^{37} +(-3.55051 + 2.04989i) q^{41} +(-2.51059 - 1.44949i) q^{43} +(-1.09638 + 1.89898i) q^{47} +(2.44949 + 4.24264i) q^{49} +12.9029i q^{53} +14.3485i q^{55} +(1.09638 + 1.89898i) q^{59} +(-2.00000 + 3.46410i) q^{61} +(18.7980 + 10.8530i) q^{65} +(12.9029 - 7.44949i) q^{67} +13.2207 q^{71} -7.89898 q^{73} +(13.6237 - 7.86566i) q^{77} +(-1.73205 - 1.00000i) q^{79} +(-6.20504 + 10.7474i) q^{83} +(-5.44949 - 9.43879i) q^{85} +5.02118i q^{89} -23.7980i q^{91} +(7.70674 + 13.3485i) q^{95} +(2.50000 - 4.33013i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 12 q^{5}+O(q^{10}) \) 8 * q + 12 * q^5	\( 8 q + 12 q^{5} + 8 q^{13} - 24 q^{29} - 48 q^{41} - 16 q^{61} + 72 q^{65} - 24 q^{73} + 60 q^{77} - 24 q^{85} + 20 q^{97}+O(q^{100}) \) 8 * q + 12 * q^5 + 8 * q^13 - 24 * q^29 - 48 * q^41 - 16 * q^61 + 72 * q^65 - 24 * q^73 + 60 * q^77 - 24 * q^85 + 20 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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