LMFDB - Embedded newform 336.4.bc.d.257.2

Newspace parameters

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	336.bc (of order $6$, degree $2$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$19.8246417619$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
Defining polynomial:	$ x^{12} - x^{11} - 29x^{9} + 6x^{8} - 49x^{7} + 1564x^{6} - 441x^{5} + 486x^{4} - 21141x^{3} - 59049x + 531441 $ x^12 - x^11 - 29x^9 + 6x^8 - 49x^7 + 1564x^6 - 441x^5 + 486x^4 - 21141x^3 - 59049x + 531441
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{3} $
Twist minimal:	no (minimal twist has level 21)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			257.2
Root			$0.00299931 - 3.00000i$ of defining polynomial
Character	$\chi$	$=$	336.257
Dual form			336.4.bc.d.17.2

$q$-expansion

$f(q)$	$=$	$q+(-2.59358 + 4.50260i) q^{3} +(-8.05907 + 13.9587i) q^{5} +(5.67909 - 17.6280i) q^{7} +(-13.5467 - 23.3556i) q^{9} +O(q^{10})$	\(q+(-2.59358 + 4.50260i) q^{3} +(-8.05907 + 13.9587i) q^{5} +(5.67909 - 17.6280i) q^{7} +(-13.5467 - 23.3556i) q^{9} +(-30.8296 + 17.7995i) q^{11} +7.40831i q^{13} +(-41.9486 - 72.4897i) q^{15} +(-14.4601 - 25.0457i) q^{17} +(-30.4580 - 17.5849i) q^{19} +(64.6428 + 71.2903i) q^{21} +(-48.0017 - 27.7138i) q^{23} +(-67.3971 - 116.735i) q^{25} +(140.295 - 0.420792i) q^{27} +68.1510i q^{29} +(154.734 - 89.3356i) q^{31} +(-0.184935 - 184.977i) q^{33} +(200.297 + 221.338i) q^{35} +(116.838 - 202.370i) q^{37} +(-33.3566 - 19.2140i) q^{39} +370.068 q^{41} +187.068 q^{43} +(435.189 - 0.870180i) q^{45} +(87.3726 - 151.334i) q^{47} +(-278.496 - 200.222i) q^{49} +(150.274 - 0.150240i) q^{51} +(-235.715 + 136.090i) q^{53} -573.789i q^{55} +(158.173 - 91.5321i) q^{57} +(48.4354 + 83.8926i) q^{59} +(-333.882 - 192.767i) q^{61} +(-488.647 + 106.164i) q^{63} +(-103.411 - 59.7041i) q^{65} +(-509.009 - 881.630i) q^{67} +(249.280 - 144.254i) q^{69} -125.333i q^{71} +(195.346 - 112.783i) q^{73} +(700.411 - 0.700251i) q^{75} +(138.686 + 644.550i) q^{77} +(-532.154 + 921.718i) q^{79} +(-361.972 + 632.785i) q^{81} +601.040 q^{83} +466.140 q^{85} +(-306.856 - 176.755i) q^{87} +(752.606 - 1303.55i) q^{89} +(130.594 + 42.0725i) q^{91} +(0.928190 + 928.402i) q^{93} +(490.926 - 283.436i) q^{95} -327.463i q^{97} +(833.358 + 478.920i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 3 q^{3} + 56 q^{7} - 3 q^{9}+O(q^{10}) $ 12 * q + 3 * q^3 + 56 * q^7 - 3 * q^9	$ 12 q + 3 q^{3} + 56 q^{7} - 3 q^{9} - 6 q^{15} - 300 q^{19} + 357 q^{21} - 42 q^{25} + 930 q^{31} - 855 q^{33} + 764 q^{37} + 426 q^{39} + 1012 q^{43} + 2367 q^{45} - 336 q^{49} + 1341 q^{51} + 270 q^{57} + 2358 q^{61} - 1071 q^{63} - 792 q^{67} - 2904 q^{73} + 2418 q^{75} - 1674 q^{79} + 837 q^{81} + 348 q^{85} - 1638 q^{87} + 1218 q^{91} - 1479 q^{93} + 3354 q^{99}+O(q^{100}) $ 12 * q + 3 * q^3 + 56 * q^7 - 3 * q^9 - 6 * q^15 - 300 * q^19 + 357 * q^21 - 42 * q^25 + 930 * q^31 - 855 * q^33 + 764 * q^37 + 426 * q^39 + 1012 * q^43 + 2367 * q^45 - 336 * q^49 + 1341 * q^51 + 270 * q^57 + 2358 * q^61 - 1071 * q^63 - 792 * q^67 - 2904 * q^73 + 2418 * q^75 - 1674 * q^79 + 837 * q^81 + 348 * q^85 - 1638 * q^87 + 1218 * q^91 - 1479 * q^93 + 3354 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$.

$n$	$85$	$113$	$127$	$241$
$\chi(n)$	$1$	$-1$	$1$	$e\left(\frac{5}{6}\right)$

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			0
$2$		0			0
$3$	−2.59358	+	4.50260i	−0.499134	+	0.866525i
$3$	−2.59358	+	4.50260i	−0.499134	+	0.866525i
$4$		0			0
$5$	−8.05907	+	13.9587i	−0.720825	+	1.24851i	0.239845	+	0.970811i	$0.422903\pi$
$5$	−8.05907	+	13.9587i	−0.720825	+	1.24851i	−0.960670	+	0.277694i	$0.910430\pi$
$6$		0			0
$7$	5.67909	−	17.6280i	0.306642	−	0.951825i
$7$	5.67909	−	17.6280i	0.306642	−	0.951825i
$8$		0			0
$9$	−13.5467	−	23.3556i	−0.501731	−	0.865024i
$10$		0			0
$11$	−30.8296	+	17.7995i	−0.845043	+	0.487886i	−0.858975	−	0.512017i	$-0.828898\pi$
$11$	−30.8296	+	17.7995i	−0.845043	+	0.487886i	0.0139322	+	0.999903i	$0.495565\pi$
$12$		0			0
$13$			7.40831i			0.158054i	0.996872	+	0.0790268i	$0.0251813\pi$
$13$			7.40831i			0.158054i	−0.996872	+	0.0790268i	$0.974819\pi$
$14$		0			0
$15$	−41.9486	−	72.4897i	−0.722073	−	1.24778i
$16$		0			0
$17$	−14.4601	−	25.0457i	−0.206300	−	0.357322i	0.744246	−	0.667905i	$-0.232809\pi$
$17$	−14.4601	−	25.0457i	−0.206300	−	0.357322i	−0.950546	+	0.310584i	$0.899476\pi$
$18$		0			0
$19$	−30.4580	−	17.5849i	−0.367765	−	0.212329i	0.304716	−	0.952443i	$-0.401438\pi$
$19$	−30.4580	−	17.5849i	−0.367765	−	0.212329i	−0.672482	+	0.740114i	$0.734772\pi$
$20$		0			0
$21$	64.6428	+	71.2903i	0.671725	+	0.740801i
$22$		0			0
$23$	−48.0017	−	27.7138i	−0.435175	−	0.251249i	0.266374	−	0.963870i	$-0.414175\pi$
$23$	−48.0017	−	27.7138i	−0.435175	−	0.251249i	−0.701549	+	0.712621i	$0.747508\pi$
$24$		0			0
$25$	−67.3971	−	116.735i	−0.539177	−	0.933881i
$26$		0			0
$27$	140.295	−	0.420792i	0.999996	−	0.00299931i
$28$		0			0
$29$			68.1510i			0.436390i	0.975905	+	0.218195i	$0.0700169\pi$
$29$			68.1510i			0.436390i	−0.975905	+	0.218195i	$0.929983\pi$
$30$		0			0
$31$	154.734	−	89.3356i	0.896484	−	0.517585i	0.0204262	−	0.999791i	$-0.493498\pi$
$31$	154.734	−	89.3356i	0.896484	−	0.517585i	0.876058	+	0.482206i	$0.160164\pi$
$32$		0			0
$33$	−0.184935	−	184.977i	−0.000975549	−	0.975771i
$34$		0			0
$35$	200.297	+	221.338i	0.967323	+	1.06894i
$36$		0			0
$37$	116.838	−	202.370i	0.519137	−	0.899172i	−0.480615	−	0.876931i	$-0.659587\pi$
$37$	116.838	−	202.370i	0.519137	−	0.899172i	0.999753	−	0.0222405i	$-0.00707996\pi$
$38$		0			0
$39$	−33.3566	−	19.2140i	−0.136957	−	0.0788899i
$40$		0			0
$41$	370.068			1.40963			0.704816	−	0.709390i	$-0.251030\pi$
$41$	370.068			1.40963			0.704816	+	0.709390i	$0.251030\pi$
$42$		0			0
$43$	187.068			0.663432			0.331716	−	0.943379i	$-0.392372\pi$
$43$	187.068			0.663432			0.331716	+	0.943379i	$0.392372\pi$
$44$		0			0
$45$	435.189	−	0.870180i	1.44165	−	0.00288264i
$46$		0			0
$47$	87.3726	−	151.334i	0.271162	−	0.469666i	−0.697998	−	0.716100i	$-0.745925\pi$
$47$	87.3726	−	151.334i	0.271162	−	0.469666i	0.969160	+	0.246434i	$0.0792588\pi$
$48$		0			0
$49$	−278.496	−	200.222i	−0.811942	−	0.583739i
$50$		0			0
$51$	150.274	−	0.150240i	0.412599	−	0.000412505i
$52$		0			0
$53$	−235.715	+	136.090i	−0.610905	+	0.352706i	−0.773319	−	0.634017i	$-0.781405\pi$
$53$	−235.715	+	136.090i	−0.610905	+	0.352706i	0.162415	+	0.986723i	$0.448072\pi$
$54$		0			0
$55$		−	573.789i		−	1.40672i
$56$		0			0
$57$	158.173	−	91.5321i	0.367553	−	0.212697i
$58$		0			0
$59$	48.4354	+	83.8926i	0.106877	+	0.185117i	0.914504	−	0.404578i	$-0.132581\pi$
$59$	48.4354	+	83.8926i	0.106877	+	0.185117i	−0.807626	+	0.589695i	$0.799248\pi$
$60$		0			0
$61$	−333.882	−	192.767i	−0.700807	−	0.404611i	0.106841	−	0.994276i	$-0.465927\pi$
$61$	−333.882	−	192.767i	−0.700807	−	0.404611i	−0.807648	+	0.589665i	$0.799260\pi$
$62$		0			0
$63$	−488.647	+	106.164i	−0.977203	+	0.212307i
$64$		0			0
$65$	−103.411	−	59.7041i	−0.197331	−	0.113929i
$66$		0			0
$67$	−509.009	−	881.630i	−0.928140	−	1.60759i	−0.786432	−	0.617677i	$-0.788074\pi$
$67$	−509.009	−	881.630i	−0.928140	−	1.60759i	−0.141708	−	0.989908i	$-0.545259\pi$
$68$		0			0
$69$	249.280	−	144.254i	0.434924	−	0.251684i
$70$		0			0
$71$		−	125.333i		−	0.209497i	−0.994499	−	0.104749i	$-0.966596\pi$
$71$		−	125.333i		−	0.209497i	0.994499	−	0.104749i	$-0.0334038\pi$
$72$		0			0
$73$	195.346	−	112.783i	0.313199	−	0.180825i	−0.335158	−	0.942162i	$-0.608790\pi$
$73$	195.346	−	112.783i	0.313199	−	0.180825i	0.648357	+	0.761337i	$0.275456\pi$
$74$		0			0
$75$	700.411	−	0.700251i	1.07835	−	0.00107811i
$76$		0			0
$77$	138.686	+	644.550i	0.205256	+	0.953939i
$78$		0			0
$79$	−532.154	+	921.718i	−0.757874	+	1.31268i	0.186059	+	0.982539i	$0.440428\pi$
$79$	−532.154	+	921.718i	−0.757874	+	1.31268i	−0.943933	+	0.330138i	$0.892905\pi$
$80$		0			0
$81$	−361.972	+	632.785i	−0.496533	+	0.868018i
$82$		0			0
$83$	601.040			0.794852			0.397426	−	0.917634i	$-0.369904\pi$
$83$	601.040			0.794852			0.397426	+	0.917634i	$0.369904\pi$
$84$		0			0
$85$	466.140			0.594824
$86$		0			0
$87$	−306.856	−	176.755i	−0.378143	−	0.217817i
$88$		0			0
$89$	752.606	−	1303.55i	0.896360	−	1.55254i	0.0642474	−	0.997934i	$-0.479535\pi$
$89$	752.606	−	1303.55i	0.896360	−	1.55254i	0.832112	−	0.554607i	$-0.187131\pi$
$90$		0			0
$91$	130.594	+	42.0725i	0.150439	+	0.0484658i
$92$		0			0
$93$	0.928190	+	928.402i	0.00103493	+	1.03517i
$94$		0			0
$95$	490.926	−	283.436i	0.530189	−	0.306105i
$96$		0			0
$97$		−	327.463i		−	0.342771i	−0.985204	−	0.171386i	$-0.945176\pi$
$97$		−	327.463i		−	0.342771i	0.985204	−	0.171386i	$-0.0548244\pi$
$98$		0			0
$99$	833.358	+	478.920i	0.846017	+	0.486195i
$100$		0			0
$101$	−547.845	−	948.895i	−0.539729	−	0.934837i	−0.998918	−	0.0464990i	$-0.985194\pi$
$101$	−547.845	−	948.895i	−0.539729	−	0.934837i	0.459190	−	0.888338i	$-0.348140\pi$
$102$		0			0
$103$	−179.848	−	103.835i	−0.172048	−	0.0993318i	0.411503	−	0.911408i	$-0.365004\pi$
$103$	−179.848	−	103.835i	−0.172048	−	0.0993318i	−0.583551	+	0.812076i	$0.698337\pi$
$104$		0			0
$105$	−1516.08	+	327.797i	−1.40909	+	0.304664i
$106$		0			0
$107$	−1561.25	−	901.391i	−1.41058	−	0.814399i	−0.415138	−	0.909759i	$-0.636267\pi$
$107$	−1561.25	−	901.391i	−1.41058	−	0.814399i	−0.995443	+	0.0953593i	$0.969600\pi$
$108$		0			0
$109$	−141.825	−	245.647i	−0.124627	−	0.215860i	0.796960	−	0.604032i	$-0.206440\pi$
$109$	−141.825	−	245.647i	−0.124627	−	0.215860i	−0.921587	+	0.388172i	$0.873107\pi$
$110$		0			0
$111$	608.160	+	1050.94i	0.520036	+	0.898652i
$112$		0			0
$113$			1037.39i			0.863627i	0.901963	+	0.431814i	$0.142126\pi$
$113$			1037.39i			0.863627i	−0.901963	+	0.431814i	$0.857874\pi$
$114$		0			0
$115$	773.697	−	446.694i	0.627371	−	0.362213i
$116$		0			0
$117$	173.026	−	100.358i	0.136720	−	0.0793003i
$118$		0			0
$119$	−523.626	+	112.667i	−0.403368	+	0.0867915i
$120$		0			0
$121$	−31.8573	+	55.1785i	−0.0239349	+	0.0414564i
$122$		0			0
$123$	−959.799	+	1666.27i	−0.703595	+	1.22148i
$124$		0			0
$125$	157.864			0.112958
$126$		0			0
$127$	−1645.81			−1.14994			−0.574968	−	0.818176i	$-0.694985\pi$
$127$	−1645.81			−1.14994			−0.574968	+	0.818176i	$0.694985\pi$
$128$		0			0
$129$	−485.175	+	842.291i	−0.331142	+	0.574881i
$130$		0			0
$131$	−314.185	+	544.184i	−0.209545	+	0.362943i	−0.951571	−	0.307428i	$-0.900532\pi$
$131$	−314.185	+	544.184i	−0.209545	+	0.362943i	0.742026	+	0.670371i	$0.233865\pi$
$132$		0			0
$133$	−482.961	+	437.048i	−0.314873	+	0.284939i
$134$		0			0
$135$	−1124.78	+	1961.74i	−0.717077	+	1.25066i
$136$		0			0
$137$	432.079	−	249.461i	0.269453	−	0.155569i	−0.359186	−	0.933266i	$-0.616946\pi$
$137$	432.079	−	249.461i	0.269453	−	0.155569i	0.628639	+	0.777697i	$0.283612\pi$
$138$		0			0
$139$		−	1216.65i		−	0.742410i	−0.928551	−	0.371205i	$-0.878945\pi$
$139$		−	1216.65i		−	0.742410i	0.928551	−	0.371205i	$-0.121055\pi$
$140$		0			0
$141$	454.788	+	785.900i	0.271631	+	0.469395i
$142$		0			0
$143$	−131.864	−	228.395i	−0.0771121	−	0.133562i
$144$		0			0
$145$	−951.300	−	549.233i	−0.544835	−	0.314561i
$146$		0			0
$147$	1623.82	−	734.663i	0.911092	−	0.412204i
$148$		0			0
$149$	−2010.18	−	1160.58i	−1.10524	−	0.638111i	−0.167648	−	0.985847i	$-0.553617\pi$
$149$	−2010.18	−	1160.58i	−1.10524	−	0.638111i	−0.937592	+	0.347736i	$0.886950\pi$
$150$		0			0
$151$	488.726	+	846.497i	0.263390	+	0.456205i	0.967141	−	0.254242i	$-0.0818260\pi$
$151$	488.726	+	846.497i	0.263390	+	0.456205i	−0.703750	+	0.710447i	$0.748493\pi$
$152$		0			0
$153$	−389.070	+	677.012i	−0.205585	+	0.357733i
$154$		0			0
$155$			2879.85i			1.49235i
$156$		0			0
$157$	−143.752	+	82.9950i	−0.0730740	+	0.0421893i	−0.536092	−	0.844160i	$-0.680100\pi$
$157$	−143.752	+	82.9950i	−0.0730740	+	0.0421893i	0.463018	+	0.886349i	$0.346767\pi$
$158$		0			0
$159$	−1.41397	−	1414.29i	−0.000705251	−	0.705412i
$160$		0			0
$161$	−761.145	+	688.786i	−0.372588	+	0.337168i
$162$		0			0
$163$	488.511	−	846.127i	0.234743	−	0.406587i	−0.724455	−	0.689322i	$-0.757908\pi$
$163$	488.511	−	846.127i	0.234743	−	0.406587i	0.959198	+	0.282735i	$0.0912417\pi$
$164$		0			0
$165$	2583.54	+	1488.16i	1.21896	+	0.702142i
$166$		0			0
$167$	−1.00709			−0.000466651			−0.000233326	−	1.00000i	$-0.500074\pi$
$167$	−1.00709			−0.000466651			−0.000233326		1.00000i	$0.500074\pi$
$168$		0			0
$169$	2142.12			0.975019
$170$		0			0
$171$	1.89874	+	949.584i	0.000849123	+	0.424658i
$172$		0			0
$173$	−1978.27	+	3426.47i	−0.869395	+	1.50584i	−0.00677983	+	0.999977i	$0.502158\pi$
$173$	−1978.27	+	3426.47i	−0.869395	+	1.50584i	−0.862616	+	0.505860i	$0.831175\pi$
$174$		0			0
$175$	−2440.57	+	525.130i	−1.05423	+	0.226835i
$176$		0			0
$177$	−503.356	+	0.503241i	−0.213754	+	0.000213706i
$178$		0			0
$179$	2423.54	−	1399.23i	1.01198	−	0.584266i	0.100208	−	0.994967i	$-0.468049\pi$
$179$	2423.54	−	1399.23i	1.01198	−	0.584266i	0.911770	+	0.410701i	$0.134716\pi$
$180$		0			0
$181$		−	1506.74i		−	0.618758i	−0.950939	−	0.309379i	$-0.899879\pi$
$181$		−	1506.74i		−	0.618758i	0.950939	−	0.309379i	$-0.100121\pi$
$182$		0			0
$183$	1733.90	−	1003.38i	0.700403	−	0.405312i
$184$		0			0
$185$	1883.21	+	3261.82i	0.748414	+	1.29629i
$186$		0			0
$187$	891.599	+	514.765i	0.348664	+	0.201301i
$188$		0			0
$189$	789.332	−	2475.52i	0.303786	−	0.952740i
$190$		0			0
$191$	−3184.51	−	1838.58i	−1.20640	−	0.696518i	−0.244433	−	0.969666i	$-0.578602\pi$
$191$	−3184.51	−	1838.58i	−1.20640	−	0.696518i	−0.961972	+	0.273148i	$0.911935\pi$
$192$		0			0
$193$	64.7335	+	112.122i	0.0241431	+	0.0418171i	0.877845	−	0.478946i	$-0.158981\pi$
$193$	64.7335	+	112.122i	0.0241431	+	0.0418171i	−0.853701	+	0.520763i	$0.825648\pi$
$194$		0			0
$195$	537.026	−	310.769i	0.197217	−	0.114126i
$196$		0			0
$197$		−	3044.81i		−	1.10119i	−0.834774	−	0.550593i	$-0.814402\pi$
$197$		−	3044.81i		−	1.10119i	0.834774	−	0.550593i	$-0.185598\pi$
$198$		0			0
$199$	−3458.29	+	1996.64i	−1.23192	+	0.711248i	−0.967429	−	0.253141i	$-0.918536\pi$
$199$	−3458.29	+	1996.64i	−1.23192	+	0.711248i	−0.264488	+	0.964389i	$0.585203\pi$
$200$		0			0
$201$	5289.78	−	5.28857i	1.85628	−	0.00185586i
$202$		0			0
$203$	1201.37	+	387.035i	0.415367	+	0.133815i
$204$		0			0
$205$	−2982.40	+	5165.67i	−1.01610	+	1.75993i
$206$		0			0
$207$	2.99240	+	1496.54i	0.00100476	+	0.502496i
$208$		0			0
$209$	1252.01			0.414370
$210$		0			0
$211$	4383.67			1.43026			0.715129	−	0.698992i	$-0.246368\pi$
$211$	4383.67			1.43026			0.715129	+	0.698992i	$0.246368\pi$
$212$		0			0
$213$	564.324	+	325.061i	0.181534	+	0.104567i
$214$		0			0
$215$	−1507.59	+	2611.23i	−0.478219	+	0.828299i
$216$		0			0
$217$	−696.065	−	3235.00i	−0.217751	−	1.01201i
$218$		0			0
$219$	1.17181	+	1172.07i	0.000361568	+	0.361650i
$220$		0			0
$221$	185.546	−	107.125i	0.0564759	−	0.0326064i
$222$		0			0
$223$		−	4851.53i		−	1.45687i	−0.685114	−	0.728436i	$-0.740247\pi$
$223$		−	4851.53i		−	1.45687i	0.685114	−	0.728436i	$-0.259753\pi$
$224$		0			0
$225$	−1813.42	+	3155.48i	−0.537308	+	0.934958i
$226$		0			0
$227$	1184.05	+	2050.83i	0.346203	+	0.599642i	0.985572	−	0.169259i	$-0.0541374\pi$
$227$	1184.05	+	2050.83i	0.346203	+	0.599642i	−0.639368	+	0.768901i	$0.720804\pi$
$228$		0			0
$229$	−3737.27	−	2157.72i	−1.07845	−	0.622646i	−0.147975	−	0.988991i	$-0.547276\pi$
$229$	−3737.27	−	2157.72i	−1.07845	−	0.622646i	−0.930479	+	0.366345i	$0.880609\pi$
$230$		0			0
$231$	−3261.84	−	1047.24i	−0.929062	−	0.298284i
$232$		0			0
$233$	−4826.98	−	2786.86i	−1.35719	−	0.783576i	−0.367949	−	0.929846i	$-0.619940\pi$
$233$	−4826.98	−	2786.86i	−1.35719	−	0.783576i	−0.989245	+	0.146270i	$0.953273\pi$
$234$		0			0
$235$	1408.28	+	2439.22i	0.390920	+	0.677094i
$236$		0			0
$237$	−2769.94	−	4786.62i	−0.759186	−	1.31192i
$238$		0			0
$239$		−	4683.70i		−	1.26763i	−0.773485	−	0.633814i	$-0.781488\pi$
$239$		−	4683.70i		−	1.26763i	0.773485	−	0.633814i	$-0.218512\pi$
$240$		0			0
$241$	−1896.77	+	1095.10i	−0.506977	+	0.292703i	−0.731590	−	0.681745i	$-0.761222\pi$
$241$	−1896.77	+	1095.10i	−0.506977	+	0.292703i	0.224613	+	0.974448i	$0.427888\pi$
$242$		0			0
$243$	−1910.37	−	3270.99i	−0.504323	−	0.863515i
$244$		0			0
$245$	5039.26	−	2273.84i	1.31407	−	0.592940i
$246$		0			0
$247$	130.275	−	225.642i	0.0335594	−	0.0581266i
$248$		0			0
$249$	−1558.84	+	2706.24i	−0.396738	+	0.688759i
$250$		0			0
$251$	−2240.70			−0.563473			−0.281736	−	0.959492i	$-0.590910\pi$
$251$	−2240.70			−0.563473			−0.281736	+	0.959492i	$0.590910\pi$
$252$		0			0
$253$	1973.16			0.490323
$254$		0			0
$255$	−1208.97	+	2098.84i	−0.296897	+	0.515429i
$256$		0			0
$257$	−555.785	+	962.648i	−0.134898	+	0.233651i	−0.925559	−	0.378604i	$-0.876404\pi$
$257$	−555.785	+	962.648i	−0.134898	+	0.233651i	0.790660	+	0.612255i	$0.209737\pi$
$258$		0			0
$259$	−2903.85	−	3208.90i	−0.696665	−	0.769851i
$260$		0			0
$261$	1591.71	−	923.223i	0.377488	−	0.218950i
$262$		0			0
$263$	−1782.86	+	1029.34i	−0.418007	+	0.241337i	−0.694224	−	0.719759i	$-0.744252\pi$
$263$	−1782.86	+	1029.34i	−0.418007	+	0.241337i	0.276217	+	0.961095i	$0.410919\pi$
$264$		0			0
$265$		−	4387.04i		−	1.01696i
$266$		0			0
$267$	3917.42	+	6769.54i	0.897912	+	1.55164i
$268$		0			0
$269$	−2414.62	−	4182.24i	−0.547294	−	0.947940i	−0.998459	−	0.0554999i	$-0.982325\pi$
$269$	−2414.62	−	4182.24i	−0.547294	−	0.947940i	0.451165	−	0.892441i	$-0.351009\pi$
$270$		0			0
$271$	191.772	+	110.720i	0.0429865	+	0.0248183i	0.521339	−	0.853350i	$-0.325433\pi$
$271$	191.772	+	110.720i	0.0429865	+	0.0248183i	−0.478353	+	0.878168i	$0.658766\pi$
$272$		0			0
$273$	−528.141	+	478.894i	−0.117086	+	0.106168i
$274$		0			0
$275$	4155.65	+	2399.27i	0.911255	+	0.526113i
$276$		0			0
$277$	−1233.58	−	2136.62i	−0.267576	−	0.463455i	0.700660	−	0.713496i	$-0.252889\pi$
$277$	−1233.58	−	2136.62i	−0.267576	−	0.463455i	−0.968235	+	0.250041i	$0.919556\pi$
$278$		0			0
$279$	−4182.63	−	2403.70i	−0.897517	−	0.515792i
$280$		0			0
$281$			4174.76i			0.886282i	0.896452	+	0.443141i	$0.146136\pi$
$281$			4174.76i			0.886282i	−0.896452	+	0.443141i	$0.853864\pi$
$282$		0			0
$283$	5628.39	−	3249.55i	1.18224	−	0.682565i	0.225706	−	0.974195i	$-0.427531\pi$
$283$	5628.39	−	3249.55i	1.18224	−	0.682565i	0.956531	+	0.291630i	$0.0941977\pi$
$284$		0			0
$285$	2.94488	+	2945.55i	0.000612069	+	0.612209i
$286$		0			0
$287$	2101.65	−	6523.57i	0.432252	−	1.34172i
$288$		0			0
$289$	2038.31	−	3530.46i	0.414881	−	0.718595i
$290$		0			0
$291$	1474.43	+	849.300i	0.297020	+	0.171089i
$292$		0			0
$293$	5637.32			1.12401			0.562007	−	0.827133i	$-0.310030\pi$
$293$	5637.32			1.12401			0.562007	+	0.827133i	$0.310030\pi$
$294$		0			0
$295$	−1561.38			−0.308159
$296$		0			0
$297$	−4317.76	+	2510.16i	−0.843576	+	0.490418i
$298$		0			0
$299$	205.312	−	355.611i	0.0397108	−	0.0687810i
$300$		0			0
$301$	1062.38	−	3297.64i	0.203436	−	0.631472i
$302$		0			0
$303$	5693.37	−	5.69207i	1.07946	−	0.00107921i
$304$		0			0
$305$	5381.56	−	3107.05i	1.01032	−	0.583308i
$306$		0			0
$307$		−	3442.95i		−	0.640064i	−0.947407	−	0.320032i	$-0.896306\pi$
$307$		−	3442.95i		−	0.640064i	0.947407	−	0.320032i	$-0.103694\pi$
$308$		0			0
$309$	933.976	−	540.477i	0.171948	−	0.0995037i
$310$		0			0
$311$	75.7324	+	131.172i	0.0138083	+	0.0239167i	0.872847	−	0.487994i	$-0.162271\pi$
$311$	75.7324	+	131.172i	0.0138083	+	0.0239167i	−0.859039	+	0.511911i	$0.828938\pi$
$312$		0			0
$313$	8335.31	+	4812.40i	1.50524	+	0.869050i	0.999982	+	0.00608123i	$0.00193573\pi$
$313$	8335.31	+	4812.40i	1.50524	+	0.869050i	0.505257	+	0.862969i	$0.331398\pi$
$314$		0			0
$315$	2456.13	−	7676.47i	0.439325	−	1.37308i
$316$		0			0
$317$	7866.93	+	4541.98i	1.39385	+	0.804741i	0.993739	−	0.111726i	$-0.0356377\pi$
$317$	7866.93	+	4541.98i	1.39385	+	0.804741i	0.400112	+	0.916466i	$0.368971\pi$
$318$		0			0
$319$	−1213.05	−	2101.07i	−0.212909	−	0.368768i
$320$		0			0
$321$	8107.83	−	4691.87i	1.40977	−	0.815809i
$322$		0			0
$323$			1017.12i			0.175214i
$324$		0			0
$325$	864.811	−	499.299i	0.147603	−	0.0852188i
$326$		0			0
$327$	1473.88	−	1.47355i	0.249254	−	0.000249197i
$328$		0			0
$329$	−2171.52	−	2399.65i	−0.363890	−	0.402118i
$330$		0			0
$331$	−702.788	+	1217.26i	−0.116703	+	0.202136i	−0.918459	−	0.395516i	$-0.870566\pi$
$331$	−702.788	+	1217.26i	−0.116703	+	0.202136i	0.801756	+	0.597651i	$0.203899\pi$
$332$		0			0
$333$	−6309.25	+	12.6156i	−1.03827	+	0.00207607i
$334$		0			0
$335$	16408.6			2.67611
$336$		0			0
$337$	7983.35			1.29045			0.645223	−	0.763994i	$-0.276764\pi$
$337$	7983.35			1.29045			0.645223	+	0.763994i	$0.276764\pi$
$338$		0			0
$339$	−4670.97	−	2690.56i	−0.748354	−	0.431066i
$340$		0			0
$341$	−3180.25	+	5508.36i	−0.505045	+	0.874764i
$342$		0			0
$343$	−5111.13	+	3772.26i	−0.804592	+	0.593828i
$344$		0			0
$345$	4.64112	+	4642.18i	0.000724260	+	0.724425i
$346$		0			0
$347$	−2268.41	+	1309.67i	−0.350935	+	0.202612i	−0.665097	−	0.746757i	$-0.731610\pi$
$347$	−2268.41	+	1309.67i	−0.350935	+	0.202612i	0.314162	+	0.949369i	$0.398276\pi$
$348$		0			0
$349$			6032.33i			0.925224i	0.886561	+	0.462612i	$0.153088\pi$
$349$			6032.33i			0.925224i	−0.886561	+	0.462612i	$0.846912\pi$
$350$		0			0
$351$	3.11736	+	1039.35i	0.000474052	+	0.158053i
$352$		0			0
$353$	−2658.15	−	4604.06i	−0.400791	−	0.694190i	0.593031	−	0.805180i	$-0.297931\pi$
$353$	−2658.15	−	4604.06i	−0.400791	−	0.694190i	−0.993822	+	0.110990i	$0.964598\pi$
$354$		0			0
$355$	1749.49	+	1010.07i	0.261558	+	0.151011i
$356$		0			0
$357$	850.770	−	2649.89i	0.126128	−	0.392849i
$358$		0			0
$359$	−1612.51	−	930.982i	−0.237061	−	0.136867i	0.376764	−	0.926309i	$-0.377037\pi$
$359$	−1612.51	−	930.982i	−0.237061	−	0.136867i	−0.613825	+	0.789442i	$0.710370\pi$
$360$		0			0
$361$	−2811.04	−	4868.87i	−0.409832	−	0.709851i
$362$		0			0
$363$	−165.822	−	286.550i	−0.0239763	−	0.0414325i
$364$		0			0
$365$			3635.70i			0.521373i
$366$		0			0
$367$	1675.89	−	967.574i	0.238367	−	0.137621i	−0.376059	−	0.926596i	$-0.622721\pi$
$367$	1675.89	−	967.574i	0.238367	−	0.137621i	0.614426	+	0.788975i	$0.289388\pi$
$368$		0			0
$369$	−5013.21	−	8643.18i	−0.707255	−	1.21937i
$370$		0			0
$371$	1060.36	+	4928.06i	0.148385	+	0.689629i
$372$		0			0
$373$	−3871.04	+	6704.83i	−0.537359	+	0.930732i	0.461687	+	0.887043i	$0.347244\pi$
$373$	−3871.04	+	6704.83i	−0.537359	+	0.930732i	−0.999045	+	0.0436892i	$0.986089\pi$
$374$		0			0
$375$	−409.432	+	710.797i	−0.0563812	+	0.0978810i
$376$		0			0
$377$	−504.884			−0.0689730
$378$		0			0
$379$	3722.15			0.504470			0.252235	−	0.967666i	$-0.418834\pi$
$379$	3722.15			0.504470			0.252235	+	0.967666i	$0.418834\pi$
$380$		0			0
$381$	4268.53	−	7410.41i	0.573972	−	0.996448i
$382$		0			0
$383$	−3546.73	+	6143.12i	−0.473184	+	0.819578i	−0.999529	−	0.0306926i	$-0.990229\pi$
$383$	−3546.73	+	6143.12i	−0.473184	+	0.819578i	0.526345	+	0.850271i	$0.323562\pi$
$384$		0			0
$385$	−10114.8	−	3258.60i	−1.33895	−	0.431359i
$386$		0			0
$387$	−2534.16	−	4369.09i	−0.332864	−	0.573885i
$388$		0			0
$389$	−6173.12	+	3564.05i	−0.804601	+	0.464537i	−0.845077	−	0.534644i	$-0.820446\pi$
$389$	−6173.12	+	3564.05i	−0.804601	+	0.464537i	0.0404765	+	0.999180i	$0.487112\pi$
$390$		0			0
$391$			1602.98i			0.207330i
$392$		0			0
$393$	−1635.38	−	2826.03i	−0.209908	−	0.362733i
$394$		0			0
$395$	−8577.33	−	14856.4i	−1.09259	−	1.89242i
$396$		0			0
$397$	7738.99	+	4468.11i	0.978360	+	0.564857i	0.901775	−	0.432206i	$-0.142265\pi$
$397$	7738.99	+	4468.11i	0.978360	+	0.564857i	0.0765855	+	0.997063i	$0.475598\pi$
$398$		0			0
$399$	−715.255	−	3308.10i	−0.0897432	−	0.415068i
$400$		0			0
$401$	−7719.60	−	4456.91i	−0.961343	−	0.555032i	−0.0647568	−	0.997901i	$-0.520627\pi$
$401$	−7719.60	−	4456.91i	−0.961343	−	0.555032i	−0.896586	+	0.442869i	$0.853961\pi$
$402$		0			0
$403$	661.826	+	1146.32i	0.0818062	+	0.141692i
$404$		0			0
$405$	−5915.71	−	10152.3i	−0.725812	−	1.24561i
$406$		0			0
$407$			8318.63i			1.01312i
$408$		0			0
$409$	−2680.13	+	1547.37i	−0.324019	+	0.187073i	−0.653183	−	0.757200i	$-0.726567\pi$
$409$	−2680.13	+	1547.37i	−0.324019	+	0.187073i	0.329163	+	0.944273i	$0.393233\pi$
$410$		0			0
$411$	2.59188	+	2592.47i	0.000311066	+	0.311137i
$412$		0			0
$413$	1753.93	−	377.389i	0.208972	−	0.0449639i
$414$		0			0
$415$	−4843.82	+	8389.74i	−0.572949	+	0.992377i
$416$		0			0
$417$	5478.09	+	3155.48i	0.643317	+	0.370562i
$418$		0			0
$419$	−7234.25			−0.843476			−0.421738	−	0.906718i	$-0.638580\pi$
$419$	−7234.25			−0.843476			−0.421738	+	0.906718i	$0.638580\pi$
$420$		0			0
$421$	406.124			0.0470148			0.0235074	−	0.999724i	$-0.492517\pi$
$421$	406.124			0.0470148			0.0235074	+	0.999724i	$0.492517\pi$
$422$		0			0
$423$	−4718.11	+	9.43408i	−0.542323	+	0.00108440i
$424$		0			0
$425$	−1949.14	+	3376.01i	−0.222464	+	0.385319i
$426$		0			0
$427$	−5294.25	+	4790.95i	−0.600016	+	0.542975i
$428$		0			0
$429$	1370.37	−	1.37006i	0.154224	−	0.000154189i
$430$		0			0
$431$	−10590.4	+	6114.37i	−1.18358	+	0.683338i	−0.956839	−	0.290618i	$-0.906139\pi$
$431$	−10590.4	+	6114.37i	−1.18358	+	0.683338i	−0.226737	+	0.973956i	$0.572806\pi$
$432$		0			0
$433$			3252.79i			0.361014i	0.983574	+	0.180507i	$0.0577738\pi$
$433$			3252.79i			0.361014i	−0.983574	+	0.180507i	$0.942226\pi$
$434$		0			0
$435$	4940.24	−	2858.84i	0.544521	−	0.315105i
$436$		0			0
$437$	974.689	+	1688.21i	0.106695	+	0.184801i
$438$		0			0
$439$	13036.8	+	7526.81i	1.41734	+	0.818303i	0.996065	−	0.0886287i	$-0.0282485\pi$
$439$	13036.8	+	7526.81i	1.41734	+	0.818303i	0.421278	+	0.906932i	$0.361582\pi$
$440$		0			0
$441$	−903.614	+	9216.81i	−0.0975720	+	0.995228i
$442$		0			0
$443$	−204.373	−	117.995i	−0.0219189	−	0.0126549i	0.489001	−	0.872283i	$-0.337362\pi$
$443$	−204.373	−	117.995i	−0.0219189	−	0.0126549i	−0.510919	+	0.859629i	$0.670695\pi$
$444$		0			0
$445$	12130.6	+	21010.8i	1.29224	+	2.23822i
$446$		0			0
$447$	10439.2	−	6040.99i	1.10460	−	0.639215i
$448$		0			0
$449$		−	5874.66i		−	0.617466i	−0.951149	−	0.308733i	$-0.900095\pi$
$449$		−	5874.66i		−	0.617466i	0.951149	−	0.308733i	$-0.0999049\pi$
$450$		0			0
$451$	−11409.0	+	6587.02i	−1.19120	+	0.687739i
$452$		0			0
$453$	−5078.98	+	5.07782i	−0.526780	+	0.000526660i
$454$		0			0
$455$	−1639.74	+	1483.86i	−0.168950	+	0.152889i
$456$		0			0
$457$	−153.883	+	266.533i	−0.0157513	+	0.0272821i	−0.873794	−	0.486297i	$-0.838347\pi$
$457$	−153.883	+	266.533i	−0.0157513	+	0.0272821i	0.858042	+	0.513579i	$0.171681\pi$
$458$		0			0
$459$	−2039.23	−	3507.71i	−0.207370	−	0.356701i
$460$		0			0
$461$	−4752.26			−0.480119			−0.240060	−	0.970758i	$-0.577167\pi$
$461$	−4752.26			−0.480119			−0.240060	+	0.970758i	$0.577167\pi$
$462$		0			0
$463$	9529.43			0.956523			0.478261	−	0.878218i	$-0.341267\pi$
$463$	9529.43			0.956523			0.478261	+	0.878218i	$0.341267\pi$
$464$		0			0
$465$	−12966.8	−	7469.10i	−1.29316	−	0.744884i
$466$		0			0
$467$	3269.28	−	5662.56i	0.323949	−	0.561097i	−0.657350	−	0.753586i	$-0.728323\pi$
$467$	3269.28	−	5662.56i	0.323949	−	0.561097i	0.981299	+	0.192489i	$0.0616559\pi$
$468$		0			0
$469$	−18432.1	+	3965.99i	−1.81475	+	0.390474i
$470$		0			0
$471$	−0.862312	−	862.509i	−8.43593e−5	−	0.0843786i
$472$		0			0
$473$	−5767.23	+	3329.71i	−0.560629	+	0.323679i
$474$		0			0
$475$			4740.69i			0.457932i
$476$		0			0
$477$	6371.64	+	3661.70i	0.611609	+	0.351484i
$478$		0			0
$479$	−3671.28	−	6358.85i	−0.350199	−	0.606562i	0.636085	−	0.771619i	$-0.280553\pi$
$479$	−3671.28	−	6358.85i	−0.350199	−	0.606562i	−0.986284	+	0.165057i	$0.947219\pi$
$480$		0			0
$481$	1499.22	+	865.574i	0.142117	+	0.0820515i
$482$		0			0
$483$	−1127.24	−	5213.55i	−0.106193	−	0.491148i
$484$		0			0
$485$	4570.96	+	2639.05i	0.427952	+	0.247078i
$486$		0			0
$487$	3508.78	+	6077.39i	0.326485	+	0.565489i	0.981812	−	0.189857i	$-0.0608024\pi$
$487$	3508.78	+	6077.39i	0.326485	+	0.565489i	−0.655327	+	0.755345i	$0.727469\pi$
$488$		0			0
$489$	2542.77	+	4394.06i	0.235150	+	0.406352i
$490$		0			0
$491$			224.222i			0.0206089i	0.999947	+	0.0103045i	$0.00328007\pi$
$491$			224.222i			0.0206089i	−0.999947	+	0.0103045i	$0.996720\pi$
$492$		0			0
$493$	1706.89	−	985.471i	0.155932	−	0.0900272i
$494$		0			0
$495$	−13401.2	+	7772.96i	−1.21685	+	0.705795i
$496$		0			0
$497$	−2209.38	−	711.777i	−0.199405	−	0.0642406i
$498$		0			0
$499$	−10396.1	+	18006.6i	−0.932651	+	1.61540i	−0.153881	+	0.988089i	$0.549177\pi$
$499$	−10396.1	+	18006.6i	−0.932651	+	1.61540i	−0.778770	+	0.627309i	$0.784156\pi$
$500$		0			0
$501$	2.61196	−	4.53451i	0.000232922	−	0.000404365i
$502$		0			0
$503$	7341.52			0.650780			0.325390	−	0.945580i	$-0.394504\pi$
$503$	7341.52			0.650780			0.325390	+	0.945580i	$0.394504\pi$
$504$		0			0
$505$	17660.5			1.55620
$506$		0			0
$507$	−5555.74	+	9645.09i	−0.486665	+	0.844878i
$508$		0			0
$509$	9956.11	−	17244.5i	0.866988	−	1.50167i	0.00192778	−	0.999998i	$-0.499386\pi$
$509$	9956.11	−	17244.5i	0.866988	−	1.50167i	0.865060	−	0.501669i	$-0.167280\pi$
$510$		0			0
$511$	−878.757	−	4084.07i	−0.0760742	−	0.353559i
$512$		0			0
$513$	−4280.52	−	2454.27i	−0.368400	−	0.211225i
$514$		0			0
$515$	2898.81	−	1673.63i	0.248032	−	0.143202i
$516$		0			0
$517$			6220.75i			0.529184i
$518$		0			0
$519$	−10297.2	−	17794.2i	−0.870900	−	1.50497i
$520$		0			0
$521$	3745.90	+	6488.08i	0.314992	+	0.545582i	0.979436	−	0.201756i	$-0.0646649\pi$
$521$	3745.90	+	6488.08i	0.314992	+	0.545582i	−0.664444	+	0.747338i	$0.731332\pi$
$522$		0			0
$523$	−249.515	−	144.058i	−0.0208614	−	0.0120444i	0.489533	−	0.871985i	$-0.337167\pi$
$523$	−249.515	−	144.058i	−0.0208614	−	0.0120444i	−0.510394	+	0.859940i	$0.670501\pi$
$524$		0			0
$525$	3965.35	−	12350.8i	0.329642	−	1.02673i
$526$		0			0
$527$	−4474.94	−	2583.61i	−0.369889	−	0.213555i
$528$		0			0
$529$	−4547.39	−	7876.32i	−0.373748	−	0.647351i
$530$		0			0
$531$	1303.23	−	2267.71i	0.106507	−	0.185330i
$532$		0			0
$533$			2741.58i			0.222797i
$534$		0			0
$535$	25164.5	−	14528.7i	2.03356	−	1.17408i
$536$		0			0
$537$	14.5379	+	14541.2i	0.00116826	+	1.16853i
$538$		0			0
$539$	12149.8	+	1215.69i	0.970923	+	0.0971496i
$540$		0			0
$541$	−7400.87	+	12818.7i	−0.588149	+	1.01870i	0.406326	+	0.913728i	$0.366810\pi$
$541$	−7400.87	+	12818.7i	−0.588149	+	1.01870i	−0.994475	+	0.104975i	$0.966524\pi$
$542$		0			0
$543$	6784.24	+	3907.85i	0.536169	+	0.308843i
$544$		0			0
$545$	4571.90			0.359337
$546$		0			0
$547$	4036.80			0.315541			0.157771	−	0.987476i	$-0.449569\pi$
$547$	4036.80			0.315541			0.157771	+	0.987476i	$0.449569\pi$
$548$		0			0
$549$	20.8141	+	10409.4i	0.00161807	+	0.809221i
$550$		0			0
$551$	1198.43	−	2075.74i	0.0926585	−	0.160489i
$552$		0			0
$553$	13225.9	+	14615.4i	1.01704	+	1.12388i
$554$		0			0
$555$	−19570.9	+	19.5664i	−1.49683	+	0.00149649i
$556$		0			0
$557$	−14891.1	+	8597.36i	−1.13277	+	0.654007i	−0.944631	−	0.328135i	$-0.893580\pi$
$557$	−14891.1	+	8597.36i	−1.13277	+	0.654007i	−0.188142	+	0.982142i	$0.560246\pi$
$558$		0			0
$559$			1385.86i			0.104858i
$560$		0			0
$561$	−4630.21	+	2679.43i	−0.348463	+	0.201650i
$562$		0			0
$563$	9453.63	+	16374.2i	0.707678	+	1.22573i	0.965716	+	0.259600i	$0.0835907\pi$
$563$	9453.63	+	16374.2i	0.707678	+	1.22573i	−0.258038	+	0.966135i	$0.583076\pi$
$564$		0			0
$565$	−14480.7	−	8360.43i	−1.07824	−	0.622524i
$566$		0			0
$567$	9099.09	+	9974.51i	0.673944	+	0.738783i
$568$		0			0
$569$	6255.57	+	3611.66i	0.460891	+	0.266096i	0.712419	−	0.701754i	$-0.247600\pi$
$569$	6255.57	+	3611.66i	0.460891	+	0.266096i	−0.251528	+	0.967850i	$0.580933\pi$
$570$		0			0
$571$	−4965.17	−	8599.93i	−0.363898	−	0.630290i	0.624700	−	0.780865i	$-0.285221\pi$
$571$	−4965.17	−	8599.93i	−0.363898	−	0.630290i	−0.988599	+	0.150574i	$0.951888\pi$
$572$		0			0
$573$	16537.7	−	9570.08i	1.20571	−	0.697724i
$574$		0			0
$575$			7471.31i			0.541870i
$576$		0			0
$577$	7254.16	−	4188.19i	0.523388	−	0.302178i	−0.214932	−	0.976629i	$-0.568953\pi$
$577$	7254.16	−	4188.19i	0.523388	−	0.302178i	0.738320	+	0.674451i	$0.235620\pi$
$578$		0			0
$579$	−672.730	+	0.672576i	−0.0482862	+	4.82752e-5i
$580$		0			0
$581$	3413.36	−	10595.2i	0.243735	−	0.756560i
$582$		0			0
$583$	4844.67	−	8391.21i	0.344161	−	0.596104i
$584$		0			0
$585$	6.44656	+	3224.01i	0.000455611	+	0.227857i
$586$		0			0
$587$	−21277.2			−1.49609			−0.748043	−	0.663650i	$-0.769006\pi$
$587$	−21277.2			−1.49609			−0.748043	+	0.663650i	$0.769006\pi$
$588$		0			0
$589$	−6283.84			−0.439594
$590$		0			0
$591$	13709.5	+	7896.94i	0.954204	+	0.549639i
$592$		0			0
$593$	1424.49	−	2467.29i	0.0986454	−	0.170859i	−0.812479	−	0.582991i	$-0.801882\pi$
$593$	1424.49	−	2467.29i	0.0986454	−	0.170859i	0.911124	+	0.412132i	$0.135216\pi$
$594$		0			0
$595$	2647.25	−	8217.14i	0.182398	−	0.566168i
$596$		0			0
$597$	−20.7450	−	20749.7i	−0.00142217	−	1.42249i
$598$		0			0
$599$	3844.40	−	2219.57i	0.262234	−	0.151401i	−0.363119	−	0.931743i	$-0.618288\pi$
$599$	3844.40	−	2219.57i	0.262234	−	0.151401i	0.625353	+	0.780342i	$0.284955\pi$
$600$		0			0
$601$		−	7868.29i		−	0.534033i	−0.963692	−	0.267017i	$-0.913962\pi$
$601$		−	7868.29i		−	0.534033i	0.963692	−	0.267017i	$-0.0860379\pi$
$602$		0			0
$603$	−13695.6	+	23831.4i	−0.924924	+	1.60944i
$604$		0			0
$605$	−513.480	−	889.374i	−0.0345057	−	0.0597656i
$606$		0			0
$607$	15144.9	+	8743.92i	1.01271	+	0.584686i	0.911983	−	0.410228i	$-0.134551\pi$
$607$	15144.9	+	8743.92i	1.01271	+	0.584686i	0.100724	+	0.994914i	$0.467884\pi$
$608$		0			0
$609$	−4858.50	+	4405.47i	−0.323278	+	0.293134i
$610$		0			0
$611$	1121.13	+	647.284i	0.0742324	+	0.0428581i
$612$		0			0
$613$	−6422.07	−	11123.3i	−0.423140	−	0.732900i	0.573105	−	0.819482i	$-0.305739\pi$
$613$	−6422.07	−	11123.3i	−0.423140	−	0.732900i	−0.996245	+	0.0865820i	$0.972406\pi$
$614$		0			0
$615$	−15523.8	−	26826.1i	−1.01786	−	1.75892i
$616$		0			0
$617$			23625.5i			1.54153i	0.637117	+	0.770767i	$0.280127\pi$
$617$			23625.5i			1.54153i	−0.637117	+	0.770767i	$0.719873\pi$
$618$		0			0
$619$	−16529.1	+	9543.05i	−1.07328	+	0.619657i	−0.929075	−	0.369891i	$-0.879395\pi$
$619$	−16529.1	+	9543.05i	−1.07328	+	0.619657i	−0.144202	+	0.989548i	$0.546062\pi$
$620$		0			0
$621$	−6746.08	−	3867.92i	−0.435927	−	0.249942i
$622$		0			0
$623$	−18704.9	−	20669.9i	−1.20289	−	1.32925i
$624$		0			0
$625$	7152.40	−	12388.3i	0.457754	−	0.792853i
$626$		0			0
$627$	−3247.18	+	5637.29i	−0.206826	+	0.359062i
$628$		0			0
$629$	−6757.98			−0.428391
$630$		0			0
$631$	−32.3893			−0.00204342			−0.00102171	−	0.999999i	$-0.500325\pi$
$631$	−32.3893			−0.00204342			−0.00102171	+	0.999999i	$0.500325\pi$
$632$		0			0
$633$	−11369.4	+	19737.9i	−0.713891	+	1.23935i
$634$		0			0
$635$	13263.7	−	22973.4i	0.828902	−	1.43570i
$636$		0			0
$637$	1483.31	−	2063.19i	0.0922620	−	0.128330i
$638$		0			0
$639$	−2927.23	+	1697.85i	−0.181220	+	0.105111i
$640$		0			0
$641$	18742.7	−	10821.1i	1.15490	−	0.666784i	0.204827	−	0.978798i	$-0.434337\pi$
$641$	18742.7	−	10821.1i	1.15490	−	0.666784i	0.950078	+	0.312014i	$0.101004\pi$
$642$		0			0
$643$			19867.3i			1.21849i	0.792982	+	0.609246i	$0.208528\pi$
$643$			19867.3i			1.21849i	−0.792982	+	0.609246i	$0.791472\pi$
$644$		0			0
$645$	−7847.24	−	13560.5i	−0.479046	−	0.827820i
$646$		0			0
$647$	11212.2	+	19420.1i	0.681294	+	1.18004i	0.974586	+	0.224012i	$0.0719156\pi$
$647$	11212.2	+	19420.1i	0.681294	+	1.18004i	−0.293293	+	0.956023i	$0.594751\pi$
$648$		0			0
$649$	−2986.49	−	1724.25i	−0.180632	−	0.104288i
$650$		0			0
$651$	16371.2	+	5256.11i	0.985618	+	0.316441i
$652$		0			0
$653$	−17358.5	−	10021.9i	−1.04026	−	0.600594i	−0.120353	−	0.992731i	$-0.538403\pi$
$653$	−17358.5	−	10021.9i	−1.04026	−	0.600594i	−0.919907	+	0.392137i	$0.871736\pi$
$654$		0			0
$655$	−5064.07	−	8771.22i	−0.302091	−	0.523237i
$656$		0			0
$657$	−5280.41	−	3034.59i	−0.313559	−	0.180199i
$658$		0			0
$659$		−	13217.9i		−	0.781327i	−0.920533	−	0.390664i	$-0.872246\pi$
$659$		−	13217.9i		−	0.781327i	0.920533	−	0.390664i	$-0.127754\pi$
$660$		0			0
$661$	8470.90	−	4890.68i	0.498457	−	0.287784i	−0.229619	−	0.973281i	$-0.573748\pi$
$661$	8470.90	−	4890.68i	0.498457	−	0.287784i	0.728076	+	0.685496i	$0.240415\pi$
$662$		0			0
$663$	1.11302	+	1113.28i	6.51979e−5	+	0.0652128i
$664$		0			0
$665$	−2208.41	−	10263.7i	−0.128780	−	0.598511i
$666$		0			0
$667$	1888.72	−	3271.36i	0.109642	−	0.189906i
$668$		0			0
$669$	21844.5	+	12582.8i	1.26242	+	0.727174i
$670$		0			0
$671$	13724.6			0.789617
$672$		0			0
$673$	−4670.73			−0.267524			−0.133762	−	0.991014i	$-0.542706\pi$
$673$	−4670.73			−0.267524			−0.133762	+	0.991014i	$0.542706\pi$
$674$		0			0
$675$	−9504.63	−	16349.1i	−0.541975	−	0.932260i
$676$		0			0
$677$	13521.0	−	23419.1i	0.767584	−	1.32949i	−0.171286	−	0.985221i	$-0.554792\pi$
$677$	13521.0	−	23419.1i	0.767584	−	1.32949i	0.938870	−	0.344273i	$-0.111875\pi$
$678$		0			0
$679$	−5772.53	−	1859.69i	−0.326258	−	0.105108i
$680$		0			0
$681$	−12305.0	+	12.3022i	−0.692406	+	0.000692248i
$682$		0			0
$683$	11596.9	−	6695.45i	0.649694	−	0.375101i	−0.138645	−	0.990342i	$-0.544275\pi$
$683$	11596.9	−	6695.45i	0.649694	−	0.375101i	0.788339	+	0.615241i	$0.210941\pi$
$684$		0			0
$685$			8041.69i			0.448551i
$686$		0			0
$687$	19408.2	−	11231.2i	1.07783	−	0.623724i
$688$		0			0
$689$	−1008.20	−	1746.25i	−0.0557464	−	0.0965557i
$690$		0			0
$691$	−26837.3	−	15494.6i	−1.47748	−	0.853025i	−0.477807	−	0.878465i	$-0.658568\pi$
$691$	−26837.3	−	15494.6i	−1.47748	−	0.853025i	−0.999676	+	0.0254396i	$0.991901\pi$
$692$		0			0
$693$	13175.1	−	11970.6i	0.722197	−	0.656172i
$694$		0			0
$695$	16982.9	+	9805.07i	0.926903	+	0.535147i
$696$		0			0
$697$	−5351.23	−	9268.60i	−0.290807	−	0.503692i
$698$		0			0
$699$	25067.3	−	14506.0i	1.35641	−	0.784933i
$700$		0			0
$701$			2892.67i			0.155855i	0.996959	+	0.0779277i	$0.0248303\pi$
$701$			2892.67i			0.155855i	−0.996959	+	0.0779277i	$0.975170\pi$
$702$		0			0
$703$	−7117.31	+	4109.18i	−0.381841	+	0.220456i
$704$		0			0
$705$	−14635.3	+	14.6320i	−0.781841	+	0.000781662i
$706$		0			0
$707$	−19838.4	+	4268.57i	−1.05530	+	0.227067i
$708$		0			0
$709$	−7965.19	+	13796.1i	−0.421917	+	0.730781i	−0.996127	−	0.0879267i	$-0.971976\pi$
$709$	−7965.19	+	13796.1i	−0.421917	+	0.730781i	0.574210	+	0.818708i	$0.305309\pi$
$710$		0			0
$711$	28736.3	−	57.4595i	1.51574	−	0.00303080i
$712$		0			0
$713$	−9903.30			−0.520171
$714$		0			0
$715$	4250.81			0.222337
$716$		0			0
$717$	21088.8	+	12147.5i	1.09843	+	0.632716i
$718$		0			0
$719$	−5938.87	+	10286.4i	−0.308042	+	0.533545i	−0.977934	−	0.208914i	$-0.933007\pi$
$719$	−5938.87	+	10286.4i	−0.308042	+	0.533545i	0.669892	+	0.742459i	$0.266341\pi$
$720$		0			0
$721$	−2851.78	+	2580.67i	−0.147303	+	0.133300i
$722$		0			0
$723$	−11.3780	−	11380.6i	−0.000585273	−	0.585407i
$724$		0			0
$725$	7955.61	−	4593.18i	0.407537	−	0.235291i
$726$		0			0
$727$		−	16795.8i		−	0.856839i	−0.903580	−	0.428419i	$-0.859071\pi$
$727$		−	16795.8i		−	0.856839i	0.903580	−	0.428419i	$-0.140929\pi$
$728$		0			0
$729$	19682.6	−	118.070i	0.999982	−	0.00599859i
$730$		0			0
$731$	−2705.03	−	4685.24i	−0.136866	−	0.237059i
$732$		0			0
$733$	22048.4	+	12729.7i	1.11102	+	0.641447i	0.939093	−	0.343663i	$-0.111668\pi$
$733$	22048.4	+	12729.7i	1.11102	+	0.641447i	0.171926	+	0.985110i	$0.445001\pi$
$734$		0			0
$735$	−2831.54	+	28587.1i	−0.142099	+	1.43463i
$736$		0			0
$737$	31385.1	+	18120.2i	1.56864	+	0.905653i
$738$		0			0
$739$	−9319.48	−	16141.8i	−0.463901	−	0.803499i	0.535251	−	0.844693i	$-0.320217\pi$
$739$	−9319.48	−	16141.8i	−0.463901	−	0.803499i	−0.999151	+	0.0411940i	$0.986884\pi$
$740$		0			0
$741$	678.099	+	1171.79i	0.0336175	+	0.0580930i
$742$		0			0
$743$			14043.3i			0.693401i	0.937976	+	0.346700i	$0.112698\pi$
$743$			14043.3i			0.693401i	−0.937976	+	0.346700i	$0.887302\pi$
$744$		0			0
$745$	32400.4	−	18706.4i	1.59337	−	0.919932i
$746$		0			0
$747$	−8142.12	−	14037.7i	−0.398802	−	0.687566i
$748$		0			0
$749$	−24756.3	+	22402.8i	−1.20771	+	1.09290i
$750$		0			0
$751$	8115.13	−	14055.8i	0.394308	−	0.682961i	−0.598705	−	0.800970i	$-0.704318\pi$
$751$	8115.13	−	14055.8i	0.394308	−	0.682961i	0.993013	+	0.118009i	$0.0376510\pi$
$752$		0			0
$753$	5811.43	−	10089.0i	0.281248	−	0.488263i
$754$		0			0
$755$	−15754.7			−0.759433
$756$		0			0
$757$	−33345.7			−1.60102			−0.800508	−	0.599322i	$-0.795437\pi$
$757$	−33345.7			−1.60102			−0.800508	+	0.599322i	$0.795437\pi$
$758$		0			0
$759$	−5117.55	+	8884.35i	−0.244737	+	0.424877i
$760$		0			0
$761$	−5394.02	+	9342.71i	−0.256942	+	0.445037i	−0.965421	−	0.260695i	$-0.916048\pi$
$761$	−5394.02	+	9342.71i	−0.256942	+	0.445037i	0.708479	+	0.705732i	$0.249382\pi$
$762$		0			0
$763$	−5135.72	+	1105.04i	−0.243677	+	0.0524313i
$764$		0			0
$765$	−6314.68	−	10887.0i	−0.298441	−	0.514537i
$766$		0			0
$767$	−621.503	+	358.825i	−0.0292584	+	0.0168923i
$768$		0			0
$769$		−	35799.6i		−	1.67876i	−0.543543	−	0.839381i	$-0.682918\pi$
$769$		−	35799.6i		−	1.67876i	0.543543	−	0.839381i	$-0.317082\pi$
$770$		0			0
$771$	−2892.94	−	4999.17i	−0.135132	−	0.233516i
$772$		0			0
$773$	18306.3	+	31707.5i	0.851788	+	1.47534i	0.879593	+	0.475727i	$0.157815\pi$
$773$	18306.3	+	31707.5i	0.851788	+	1.47534i	−0.0278053	+	0.999613i	$0.508852\pi$
$774$		0			0
$775$	−20857.2	−	12041.9i	−0.966727	−	0.558140i
$776$		0			0
$777$	21979.7	−	4752.31i	1.01482	−	0.219419i
$778$		0			0
$779$	−11271.5	−	6507.62i	−0.518414	−	0.299306i
$780$		0			0
$781$	2230.86	+	3863.97i	0.102211	+	0.177034i
$782$		0			0
$783$	28.6773	+	9561.27i	0.00130887	+	0.436388i
$784$		0			0
$785$		−	2675.45i		−	0.121644i
$786$		0			0
$787$	34874.9	−	20135.0i	1.57961	−	0.911990i	0.584701	−	0.811249i	$-0.301212\pi$
$787$	34874.9	−	20135.0i	1.57961	−	0.911990i	0.994913	−	0.100742i	$-0.0321216\pi$
$788$		0			0
$789$	−10.6947	−

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:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	336.bc (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.59358 + 4.50260i) q^{3} +(-8.05907 + 13.9587i) q^{5} +(5.67909 - 17.6280i) q^{7} +(-13.5467 - 23.3556i) q^{9} +O(q^{10})\)	\(q+(-2.59358 + 4.50260i) q^{3} +(-8.05907 + 13.9587i) q^{5} +(5.67909 - 17.6280i) q^{7} +(-13.5467 - 23.3556i) q^{9} +(-30.8296 + 17.7995i) q^{11} +7.40831i q^{13} +(-41.9486 - 72.4897i) q^{15} +(-14.4601 - 25.0457i) q^{17} +(-30.4580 - 17.5849i) q^{19} +(64.6428 + 71.2903i) q^{21} +(-48.0017 - 27.7138i) q^{23} +(-67.3971 - 116.735i) q^{25} +(140.295 - 0.420792i) q^{27} +68.1510i q^{29} +(154.734 - 89.3356i) q^{31} +(-0.184935 - 184.977i) q^{33} +(200.297 + 221.338i) q^{35} +(116.838 - 202.370i) q^{37} +(-33.3566 - 19.2140i) q^{39} +370.068 q^{41} +187.068 q^{43} +(435.189 - 0.870180i) q^{45} +(87.3726 - 151.334i) q^{47} +(-278.496 - 200.222i) q^{49} +(150.274 - 0.150240i) q^{51} +(-235.715 + 136.090i) q^{53} -573.789i q^{55} +(158.173 - 91.5321i) q^{57} +(48.4354 + 83.8926i) q^{59} +(-333.882 - 192.767i) q^{61} +(-488.647 + 106.164i) q^{63} +(-103.411 - 59.7041i) q^{65} +(-509.009 - 881.630i) q^{67} +(249.280 - 144.254i) q^{69} -125.333i q^{71} +(195.346 - 112.783i) q^{73} +(700.411 - 0.700251i) q^{75} +(138.686 + 644.550i) q^{77} +(-532.154 + 921.718i) q^{79} +(-361.972 + 632.785i) q^{81} +601.040 q^{83} +466.140 q^{85} +(-306.856 - 176.755i) q^{87} +(752.606 - 1303.55i) q^{89} +(130.594 + 42.0725i) q^{91} +(0.928190 + 928.402i) q^{93} +(490.926 - 283.436i) q^{95} -327.463i q^{97} +(833.358 + 478.920i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 3 q^{3} + 56 q^{7} - 3 q^{9}+O(q^{10}) \) 12 * q + 3 * q^3 + 56 * q^7 - 3 * q^9	\( 12 q + 3 q^{3} + 56 q^{7} - 3 q^{9} - 6 q^{15} - 300 q^{19} + 357 q^{21} - 42 q^{25} + 930 q^{31} - 855 q^{33} + 764 q^{37} + 426 q^{39} + 1012 q^{43} + 2367 q^{45} - 336 q^{49} + 1341 q^{51} + 270 q^{57} + 2358 q^{61} - 1071 q^{63} - 792 q^{67} - 2904 q^{73} + 2418 q^{75} - 1674 q^{79} + 837 q^{81} + 348 q^{85} - 1638 q^{87} + 1218 q^{91} - 1479 q^{93} + 3354 q^{99}+O(q^{100}) \) 12 * q + 3 * q^3 + 56 * q^7 - 3 * q^9 - 6 * q^15 - 300 * q^19 + 357 * q^21 - 42 * q^25 + 930 * q^31 - 855 * q^33 + 764 * q^37 + 426 * q^39 + 1012 * q^43 + 2367 * q^45 - 336 * q^49 + 1341 * q^51 + 270 * q^57 + 2358 * q^61 - 1071 * q^63 - 792 * q^67 - 2904 * q^73 + 2418 * q^75 - 1674 * q^79 + 837 * q^81 + 348 * q^85 - 1638 * q^87 + 1218 * q^91 - 1479 * q^93 + 3354 * q^99

Introduction

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