LMFDB - Embedded newform 648.4.i.v.433.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [648,4,Mod(217,648)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(648, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("648.217");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 648 = 2^{3} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	648.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$38.2332376837$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	8.0.897122304.10
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 8x^{6} + 51x^{4} - 104x^{2} + 169 $ x^8 - 8x^6 + 51x^4 - 104*x^2 + 169
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			433.2
Root			$-2.07341 + 1.19709i$ of defining polynomial
Character	$\chi$	$=$	648.433
Dual form			648.4.i.v.217.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.90171 + 5.02591i) q^{5} +(13.0614 + 22.6230i) q^{7} +O(q^{10})$	$q+(-2.90171 + 5.02591i) q^{5} +(13.0614 + 22.6230i) q^{7} +(-18.6230 - 32.2560i) q^{11} +(15.1687 - 26.2729i) q^{13} +48.4744 q^{17} +88.1865 q^{19} +(42.2630 - 73.2016i) q^{23} +(45.6601 + 79.0857i) q^{25} +(88.3618 + 153.047i) q^{29} +(-77.6920 + 134.566i) q^{31} -151.601 q^{35} -258.310 q^{37} +(110.143 - 190.774i) q^{41} +(23.8807 + 41.3626i) q^{43} +(64.7495 + 112.149i) q^{47} +(-169.700 + 293.928i) q^{49} -577.029 q^{53} +216.154 q^{55} +(-121.758 + 210.891i) q^{59} +(343.816 + 595.507i) q^{61} +(88.0301 + 152.473i) q^{65} +(-189.195 + 327.695i) q^{67} -332.826 q^{71} +1072.19 q^{73} +(486.484 - 842.615i) q^{77} +(630.104 + 1091.37i) q^{79} +(613.047 + 1061.83i) q^{83} +(-140.659 + 243.628i) q^{85} +1095.81 q^{89} +792.495 q^{91} +(-255.892 + 443.217i) q^{95} +(564.300 + 977.397i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{5}+O(q^{10}) $ 8 * q + 8 * q^5	$ 8 q + 8 q^{5} + 8 q^{11} - 4 q^{13} - 32 q^{17} + 160 q^{19} + 200 q^{23} - 8 q^{25} + 216 q^{29} - 80 q^{31} - 816 q^{35} - 552 q^{37} + 384 q^{41} - 160 q^{43} + 768 q^{47} + 268 q^{49} - 1888 q^{53} + 608 q^{55} + 992 q^{59} + 548 q^{61} + 1328 q^{65} - 464 q^{67} - 3440 q^{71} - 1528 q^{73} + 1728 q^{77} - 688 q^{79} + 2128 q^{83} + 1324 q^{85} - 4224 q^{89} + 3552 q^{91} + 2056 q^{95} + 1816 q^{97}+O(q^{100}) $ 8 * q + 8 * q^5 + 8 * q^11 - 4 * q^13 - 32 * q^17 + 160 * q^19 + 200 * q^23 - 8 * q^25 + 216 * q^29 - 80 * q^31 - 816 * q^35 - 552 * q^37 + 384 * q^41 - 160 * q^43 + 768 * q^47 + 268 * q^49 - 1888 * q^53 + 608 * q^55 + 992 * q^59 + 548 * q^61 + 1328 * q^65 - 464 * q^67 - 3440 * q^71 - 1528 * q^73 + 1728 * q^77 - 688 * q^79 + 2128 * q^83 + 1324 * q^85 - 4224 * q^89 + 3552 * q^91 + 2056 * q^95 + 1816 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/648\mathbb{Z}\right)^\times$.

$n$	$325$	$487$	$569$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−2.90171	+	5.02591i	−0.259537	+	0.449531i	−0.966118	−	0.258101i	$-0.916903\pi$
$5$	−2.90171	+	5.02591i	−0.259537	+	0.449531i	0.706581	+	0.707632i	$0.250237\pi$
$6$		0			0
$7$	13.0614	+	22.6230i	0.705249	+	1.22153i	0.966602	+	0.256283i	$0.0824979\pi$
$7$	13.0614	+	22.6230i	0.705249	+	1.22153i	−0.261353	+	0.965243i	$0.584169\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−18.6230	−	32.2560i	−0.510458	−	0.884140i	−0.999927	−	0.0121186i	$-0.996142\pi$
$11$	−18.6230	−	32.2560i	−0.510458	−	0.884140i	0.489468	−	0.872021i	$-0.337191\pi$
$12$		0			0
$13$	15.1687	−	26.2729i	0.323618	−	0.560522i	−0.657614	−	0.753355i	$-0.728434\pi$
$13$	15.1687	−	26.2729i	0.323618	−	0.560522i	0.981232	+	0.192833i	$0.0617675\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	48.4744			0.691575			0.345787	−	0.938313i	$-0.387612\pi$
$17$	48.4744			0.691575			0.345787	+	0.938313i	$0.387612\pi$
$18$		0			0
$19$	88.1865			1.06481			0.532404	−	0.846490i	$-0.321289\pi$
$19$	88.1865			1.06481			0.532404	+	0.846490i	$0.321289\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	42.2630	−	73.2016i	0.383149	−	0.663634i	−0.608361	−	0.793660i	$-0.708173\pi$
$23$	42.2630	−	73.2016i	0.383149	−	0.663634i	0.991511	+	0.130026i	$0.0415061\pi$
$24$		0			0
$25$	45.6601	+	79.0857i	0.365281	+	0.632685i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	88.3618	+	153.047i	0.565806	+	0.980005i	0.996974	+	0.0777332i	$0.0247682\pi$
$29$	88.3618	+	153.047i	0.565806	+	0.980005i	−0.431168	+	0.902272i	$0.641898\pi$
$30$		0			0
$31$	−77.6920	+	134.566i	−0.450126	+	0.779640i	−0.998393	−	0.0566624i	$-0.981954\pi$
$31$	−77.6920	+	134.566i	−0.450126	+	0.779640i	0.548268	+	0.836303i	$0.315287\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−151.601			−0.732152
$36$		0			0
$37$	−258.310			−1.14773			−0.573864	−	0.818951i	$-0.694556\pi$
$37$	−258.310			−1.14773			−0.573864	+	0.818951i	$0.694556\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	110.143	−	190.774i	0.419549	−	0.726680i	−0.576345	−	0.817206i	$-0.695522\pi$
$41$	110.143	−	190.774i	0.419549	−	0.726680i	0.995894	+	0.0905264i	$0.0288549\pi$
$42$		0			0
$43$	23.8807	+	41.3626i	0.0846925	+	0.146692i	0.905260	−	0.424858i	$-0.139676\pi$
$43$	23.8807	+	41.3626i	0.0846925	+	0.146692i	−0.820568	+	0.571549i	$0.806343\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	64.7495	+	112.149i	0.200951	+	0.348057i	0.948835	−	0.315772i	$-0.102264\pi$
$47$	64.7495	+	112.149i	0.200951	+	0.348057i	−0.747884	+	0.663829i	$0.768930\pi$
$48$		0			0
$49$	−169.700	+	293.928i	−0.494751	+	0.856934i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−577.029			−1.49549			−0.747746	−	0.663985i	$-0.768864\pi$
$53$	−577.029			−1.49549			−0.747746	+	0.663985i	$0.768864\pi$
$54$		0			0
$55$	216.154			0.529931
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−121.758	+	210.891i	−0.268670	+	0.465350i	−0.968519	−	0.248941i	$-0.919917\pi$
$59$	−121.758	+	210.891i	−0.268670	+	0.465350i	0.699849	+	0.714291i	$0.253251\pi$
$60$		0			0
$61$	343.816	+	595.507i	0.721658	+	1.24995i	0.960335	+	0.278849i	$0.0899528\pi$
$61$	343.816	+	595.507i	0.721658	+	1.24995i	−0.238677	+	0.971099i	$0.576714\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	88.0301	+	152.473i	0.167981	+	0.290952i
$66$		0			0
$67$	−189.195	+	327.695i	−0.344982	+	0.597527i	−0.985351	−	0.170541i	$-0.945448\pi$
$67$	−189.195	+	327.695i	−0.344982	+	0.597527i	0.640368	+	0.768068i	$0.278782\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−332.826			−0.556327			−0.278163	−	0.960534i	$-0.589726\pi$
$71$	−332.826			−0.556327			−0.278163	+	0.960534i	$0.589726\pi$
$72$		0			0
$73$	1072.19			1.71904			0.859520	−	0.511103i	$-0.170763\pi$
$73$	1072.19			1.71904			0.859520	+	0.511103i	$0.170763\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	486.484	−	842.615i	0.720000	−	1.24708i
$78$		0			0
$79$	630.104	+	1091.37i	0.897370	+	1.55429i	0.830843	+	0.556506i	$0.187858\pi$
$79$	630.104	+	1091.37i	0.897370	+	1.55429i	0.0665267	+	0.997785i	$0.478808\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	613.047	+	1061.83i	0.810731	+	1.40423i	0.912353	+	0.409404i	$0.134263\pi$
$83$	613.047	+	1061.83i	0.810731	+	1.40423i	−0.101623	+	0.994823i	$0.532403\pi$
$84$		0			0
$85$	−140.659	+	243.628i	−0.179489	+	0.310884i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1095.81			1.30512			0.652559	−	0.757738i	$-0.273695\pi$
$89$	1095.81			1.30512			0.652559	+	0.757738i	$0.273695\pi$
$90$		0			0
$91$	792.495			0.912923
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−255.892	+	443.217i	−0.276357	+	0.478665i
$96$		0			0
$97$	564.300	+	977.397i	0.590681	+	1.02309i	0.994141	+	0.108092i	$0.0344740\pi$
$97$	564.300	+	977.397i	0.590681	+	1.02309i	−0.403460	+	0.914997i	$0.632193\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−617.823	−	1070.10i	−0.608670	−	1.05425i	−0.991460	−	0.130412i	$-0.958370\pi$
$101$	−617.823	−	1070.10i	−0.608670	−	1.05425i	0.382790	−	0.923835i	$-0.374963\pi$
$102$		0			0
$103$	−607.962	+	1053.02i	−0.581595	+	1.00735i	0.413696	+	0.910415i	$0.364238\pi$
$103$	−607.962	+	1053.02i	−0.581595	+	1.00735i	−0.995291	+	0.0969365i	$0.969096\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1779.08			−1.60739			−0.803694	−	0.595043i	$-0.797135\pi$
$107$	−1779.08			−1.60739			−0.803694	+	0.595043i	$0.797135\pi$
$108$		0			0
$109$	−974.934			−0.856714			−0.428357	−	0.903610i	$-0.640907\pi$
$109$	−974.934			−0.856714			−0.428357	+	0.903610i	$0.640907\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	985.735	−	1707.34i	0.820621	−	1.42136i	−0.0846002	−	0.996415i	$-0.526961\pi$
$113$	985.735	−	1707.34i	0.820621	−	1.42136i	0.905221	−	0.424942i	$-0.139705\pi$
$114$		0			0
$115$	245.270	+	424.820i	0.198883	+	0.344475i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	633.143	+	1096.64i	0.487732	+	0.844777i
$120$		0			0
$121$	−28.1310	+	48.7244i	−0.0211353	+	0.0366074i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1255.40			−0.898290
$126$		0			0
$127$	−604.990			−0.422710			−0.211355	−	0.977409i	$-0.567788\pi$
$127$	−604.990			−0.422710			−0.211355	+	0.977409i	$0.567788\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	704.848	−	1220.83i	0.470098	−	0.814234i	−0.529317	−	0.848424i	$-0.677552\pi$
$131$	704.848	−	1220.83i	0.470098	−	0.814234i	0.999415	+	0.0341899i	$0.0108851\pi$
$132$		0			0
$133$	1151.84	+	1995.04i	0.750955	+	1.30069i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1131.77	+	1960.29i	0.705795	+	1.22247i	0.966404	+	0.257028i	$0.0827433\pi$
$137$	1131.77	+	1960.29i	0.705795	+	1.22247i	−0.260609	+	0.965444i	$0.583923\pi$
$138$		0			0
$139$	−770.251	+	1334.11i	−0.470013	+	0.814087i	−0.999412	−	0.0342862i	$-0.989084\pi$
$139$	−770.251	+	1334.11i	−0.470013	+	0.814087i	0.529399	+	0.848373i	$0.322418\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1129.94			−0.660773
$144$		0			0
$145$	−1025.60			−0.587390
$146$		0			0
$147$		0			0
$148$		0			0
$149$	306.517	−	530.903i	0.168529	−	0.291901i	−0.769374	−	0.638799i	$-0.779432\pi$
$149$	306.517	−	530.903i	0.168529	−	0.291901i	0.937903	+	0.346898i	$0.112765\pi$
$150$		0			0
$151$	−1186.49	−	2055.06i	−0.639438	−	1.10754i	−0.985556	−	0.169348i	$-0.945834\pi$
$151$	−1186.49	−	2055.06i	−0.639438	−	1.10754i	0.346118	−	0.938191i	$-0.387500\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−450.879	−	780.946i	−0.233648	−	0.404691i
$156$		0			0
$157$	884.031	−	1531.19i	0.449385	−	0.778357i	−0.548962	−	0.835848i	$-0.684977\pi$
$157$	884.031	−	1531.19i	0.449385	−	0.778357i	0.998346	+	0.0574908i	$0.0183100\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	2208.05			1.08086
$162$		0			0
$163$	2657.82			1.27715			0.638577	−	0.769558i	$-0.279523\pi$
$163$	2657.82			1.27715			0.638577	+	0.769558i	$0.279523\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1676.26	+	2903.37i	−0.776724	+	1.34532i	0.157097	+	0.987583i	$0.449786\pi$
$167$	−1676.26	+	2903.37i	−0.776724	+	1.34532i	−0.933821	+	0.357742i	$0.883547\pi$
$168$		0			0
$169$	638.324	+	1105.61i	0.290543	+	0.503236i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1.21794	+	2.10954i	0.000535251	+	0.000927083i	0.866293	−	0.499536i	$-0.166496\pi$
$173$	1.21794	+	2.10954i	0.000535251	+	0.000927083i	−0.865758	+	0.500463i	$0.833163\pi$
$174$		0			0
$175$	−1192.77	+	2065.94i	−0.515228	+	0.892401i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−1084.01			−0.452642			−0.226321	−	0.974053i	$-0.572670\pi$
$179$	−1084.01			−0.452642			−0.226321	+	0.974053i	$0.572670\pi$
$180$		0			0
$181$	869.840			0.357208			0.178604	−	0.983921i	$-0.442842\pi$
$181$	869.840			0.357208			0.178604	+	0.983921i	$0.442842\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	749.542	−	1298.24i	0.297878	−	0.515939i
$186$		0			0
$187$	−902.738	−	1563.59i	−0.353020	−	0.611448i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−212.574	−	368.189i	−0.0805304	−	0.139483i	0.822948	−	0.568117i	$-0.192328\pi$
$191$	−212.574	−	368.189i	−0.0805304	−	0.139483i	−0.903478	+	0.428635i	$0.858995\pi$
$192$		0			0
$193$	0.854672	−	1.48034i	0.000318760	−	0.000552108i	−0.865866	−	0.500276i	$-0.833232\pi$
$193$	0.854672	−	1.48034i	0.000318760	−	0.000552108i	0.866185	+	0.499724i	$0.166565\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	3079.44			1.11371			0.556855	−	0.830610i	$-0.312008\pi$
$197$	3079.44			1.11371			0.556855	+	0.830610i	$0.312008\pi$
$198$		0			0
$199$	3133.50			1.11622			0.558110	−	0.829767i	$-0.311527\pi$
$199$	3133.50			1.11622			0.558110	+	0.829767i	$0.311527\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−2308.25	+	3998.01i	−0.798068	+	1.38229i
$204$		0			0
$205$	639.209	+	1107.14i	0.217777	+	0.377201i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1642.30	−	2844.54i	−0.543540	−	0.941440i
$210$		0			0
$211$	1393.87	−	2414.26i	0.454778	−	0.787699i	−0.543897	−	0.839152i	$-0.683052\pi$
$211$	1393.87	−	2414.26i	0.454778	−	0.787699i	0.998675	+	0.0514532i	$0.0163853\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−277.180			−0.0879233
$216$		0			0
$217$	−4059.06			−1.26980
$218$		0			0
$219$		0			0
$220$		0			0
$221$	735.291	−	1273.56i	0.223806	−	0.387643i
$222$		0			0
$223$	573.301	+	992.987i	0.172157	+	0.298185i	0.939174	−	0.343442i	$-0.111593\pi$
$223$	573.301	+	992.987i	0.172157	+	0.298185i	−0.767016	+	0.641627i	$0.778260\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−464.070	−	803.794i	−0.135689	−	0.235021i	0.790171	−	0.612886i	$-0.209992\pi$
$227$	−464.070	−	803.794i	−0.135689	−	0.235021i	−0.925861	+	0.377865i	$0.876658\pi$
$228$		0			0
$229$	1758.88	−	3046.47i	0.507554	−	0.879110i	−0.492408	−	0.870365i	$-0.663883\pi$
$229$	1758.88	−	3046.47i	0.507554	−	0.879110i	0.999962	−	0.00874485i	$-0.00278361\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−976.524			−0.274567			−0.137284	−	0.990532i	$-0.543837\pi$
$233$	−976.524			−0.274567			−0.137284	+	0.990532i	$0.543837\pi$
$234$		0			0
$235$	−751.538			−0.208617
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−2270.49	+	3932.60i	−0.614500	+	1.06435i	0.375972	+	0.926631i	$0.377309\pi$
$239$	−2270.49	+	3932.60i	−0.614500	+	1.06435i	−0.990472	+	0.137714i	$0.956024\pi$
$240$		0			0
$241$	−3610.19	−	6253.04i	−0.964951	−	1.67134i	−0.709748	−	0.704456i	$-0.751191\pi$
$241$	−3610.19	−	6253.04i	−0.964951	−	1.67134i	−0.255203	−	0.966888i	$-0.582142\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−984.839	−	1705.79i	−0.256812	−	0.444812i
$246$		0			0
$247$	1337.67	−	2316.91i	0.344591	−	0.596849i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−1285.44			−0.323251			−0.161626	−	0.986852i	$-0.551674\pi$
$251$	−1285.44			−0.323251			−0.161626	+	0.986852i	$0.551674\pi$
$252$		0			0
$253$	−3148.25			−0.782327
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1302.25	−	2255.56i	0.316078	−	0.547462i	−0.663588	−	0.748098i	$-0.730967\pi$
$257$	1302.25	−	2255.56i	0.316078	−	0.547462i	0.979666	+	0.200635i	$0.0643007\pi$
$258$		0			0
$259$	−3373.89	−	5843.75i	−0.809433	−	1.40198i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	2676.84	+	4636.43i	0.627609	+	1.08705i	0.988030	+	0.154261i	$0.0492996\pi$
$263$	2676.84	+	4636.43i	0.627609	+	1.08705i	−0.360421	+	0.932790i	$0.617367\pi$
$264$		0			0
$265$	1674.37	−	2900.10i	0.388136	−	0.672270i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	7141.58			1.61870			0.809349	−	0.587328i	$-0.199820\pi$
$269$	7141.58			1.61870			0.809349	+	0.587328i	$0.199820\pi$
$270$		0			0
$271$	−6334.52			−1.41991			−0.709953	−	0.704249i	$-0.751284\pi$
$271$	−6334.52			−1.41991			−0.709953	+	0.704249i	$0.751284\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1700.66	−	2945.62i	0.372922	−	0.645919i
$276$		0			0
$277$	−3545.09	−	6140.28i	−0.768967	−	1.33189i	−0.938124	−	0.346301i	$-0.887438\pi$
$277$	−3545.09	−	6140.28i	−0.768967	−	1.33189i	0.169157	−	0.985589i	$-0.445896\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−3370.51	−	5837.89i	−0.715543	−	1.23936i	−0.962750	−	0.270394i	$-0.912846\pi$
$281$	−3370.51	−	5837.89i	−0.715543	−	1.23936i	0.247207	−	0.968963i	$-0.420487\pi$
$282$		0			0
$283$	−1779.83	+	3082.76i	−0.373852	+	0.647530i	−0.990154	−	0.139979i	$-0.955296\pi$
$283$	−1779.83	+	3082.76i	−0.373852	+	0.647530i	0.616303	+	0.787509i	$0.288630\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	5754.50			1.18354
$288$		0			0
$289$	−2563.23			−0.521725
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1077.71	−	1866.64i	0.214881	−	0.372185i	−0.738355	−	0.674413i	$-0.764397\pi$
$293$	1077.71	−	1866.64i	0.214881	−	0.372185i	0.953236	+	0.302227i	$0.0977302\pi$
$294$		0			0
$295$	−706.612	−	1223.89i	−0.139459	−	0.241551i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1282.14	−	2220.74i	−0.247988	−	0.429527i
$300$		0			0
$301$	−623.831	+	1080.51i	−0.119458	+	0.206908i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−3990.62			−0.749187
$306$		0			0
$307$	1552.85			0.288684			0.144342	−	0.989528i	$-0.453893\pi$
$307$	1552.85			0.288684			0.144342	+	0.989528i	$0.453893\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−4207.44	+	7287.50i	−0.767145	+	1.32873i	0.171960	+	0.985104i	$0.444990\pi$
$311$	−4207.44	+	7287.50i	−0.767145	+	1.32873i	−0.939105	+	0.343630i	$0.888343\pi$
$312$		0			0
$313$	2852.13	+	4940.03i	0.515053	+	0.892098i	0.999847	+	0.0174701i	$0.00556120\pi$
$313$	2852.13	+	4940.03i	0.515053	+	0.892098i	−0.484794	+	0.874628i	$0.661105\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	0.181688	+	0.314692i	3.21911e−5	+	5.57567e-5i	0.866041	−	0.499972i	$-0.166656\pi$
$317$	0.181688	+	0.314692i	3.21911e−5	+	5.57567e-5i	−0.866009	+	0.500028i	$0.833323\pi$
$318$		0			0
$319$	3291.12	−	5700.39i	0.577641	−	1.00050i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	4274.79			0.736395
$324$		0			0
$325$	2770.41			0.472846
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1691.44	+	2929.66i	−0.283441	+	0.490933i
$330$		0			0
$331$	−4467.17	−	7737.36i	−0.741806	−	1.28485i	−0.951672	−	0.307115i	$-0.900636\pi$
$331$	−4467.17	−	7737.36i	−0.741806	−	1.28485i	0.209866	−	0.977730i	$-0.432697\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1097.98	−	1901.75i	−0.179071	−	0.310161i
$336$		0			0
$337$	4762.05	−	8248.11i	0.769749	−	1.33324i	−0.167950	−	0.985795i	$-0.553715\pi$
$337$	4762.05	−	8248.11i	0.769749	−	1.33324i	0.937699	−	0.347448i	$-0.112952\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	5787.43			0.919081
$342$		0			0
$343$	94.0633			0.0148074
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−4404.05	+	7628.04i	−0.681331	+	1.18010i	0.293244	+	0.956038i	$0.405265\pi$
$347$	−4404.05	+	7628.04i	−0.681331	+	1.18010i	−0.974575	+	0.224062i	$0.928068\pi$
$348$		0			0
$349$	−1155.62	−	2001.59i	−0.177246	−	0.306998i	0.763690	−	0.645583i	$-0.223385\pi$
$349$	−1155.62	−	2001.59i	−0.177246	−	0.306998i	−0.940936	+	0.338584i	$0.890052\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	82.8041	+	143.421i	0.0124850	+	0.0216247i	0.872200	−	0.489149i	$-0.162692\pi$
$353$	82.8041	+	143.421i	0.0124850	+	0.0216247i	−0.859715	+	0.510773i	$0.829359\pi$
$354$		0			0
$355$	965.765	−	1672.75i	0.144387	−	0.250086i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−7007.10			−1.03014			−0.515071	−	0.857148i	$-0.672234\pi$
$359$	−7007.10			−1.03014			−0.515071	+	0.857148i	$0.672234\pi$
$360$		0			0
$361$	917.854			0.133817
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−3111.17	+	5388.71i	−0.446154	+	0.772762i
$366$		0			0
$367$	−1457.76	−	2524.91i	−0.207342	−	0.359126i	0.743535	−	0.668698i	$-0.233148\pi$
$367$	−1457.76	−	2524.91i	−0.207342	−	0.359126i	−0.950876	+	0.309571i	$0.899815\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−7536.80	−	13054.1i	−1.05469	−	1.82678i
$372$		0			0
$373$	−2043.56	+	3539.54i	−0.283676	+	0.491342i	−0.972287	−	0.233789i	$-0.924887\pi$
$373$	−2043.56	+	3539.54i	−0.283676	+	0.491342i	0.688611	+	0.725131i	$0.258221\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	5361.32			0.732419
$378$		0			0
$379$	5785.10			0.784065			0.392032	−	0.919951i	$-0.371772\pi$
$379$	5785.10			0.784065			0.392032	+	0.919951i	$0.371772\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	6485.46	−	11233.1i	0.865252	−	1.49866i	−0.00154474	−	0.999999i	$-0.500492\pi$
$383$	6485.46	−	11233.1i	0.865252	−	1.49866i	0.866797	−	0.498662i	$-0.166175\pi$
$384$		0			0
$385$	2823.27	+	4890.05i	0.373733	+	0.647325i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	2278.08	+	3945.75i	0.296923	+	0.514286i	0.975430	−	0.220307i	$-0.0707061\pi$
$389$	2278.08	+	3945.75i	0.296923	+	0.514286i	−0.678507	+	0.734594i	$0.737373\pi$
$390$		0			0
$391$	2048.67	−	3548.40i	0.264976	−	0.458952i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−7313.52			−0.931603
$396$		0			0
$397$	3193.40			0.403709			0.201854	−	0.979416i	$-0.435303\pi$
$397$	3193.40			0.403709			0.201854	+	0.979416i	$0.435303\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−7515.98	+	13018.1i	−0.935986	+	1.62117i	−0.163119	+	0.986606i	$0.552155\pi$
$401$	−7515.98	+	13018.1i	−0.935986	+	1.62117i	−0.772867	+	0.634568i	$0.781178\pi$
$402$		0			0
$403$	2356.97	+	4082.39i	0.291337	+	0.504611i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	4810.51	+	8332.04i	0.585867	+	1.01475i
$408$		0			0
$409$	316.934	−	548.947i	0.0383164	−	0.0663659i	−0.846231	−	0.532816i	$-0.821134\pi$
$409$	316.934	−	548.947i	0.0383164	−	0.0663659i	0.884548	+	0.466450i	$0.154467\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−6361.30			−0.757916
$414$		0			0
$415$	−7115.54			−0.841658
$416$		0			0
$417$		0			0
$418$		0			0
$419$	6994.94	−	12115.6i	0.815574	−	1.41261i	−0.0933417	−	0.995634i	$-0.529755\pi$
$419$	6994.94	−	12115.6i	0.815574	−	1.41261i	0.908915	−	0.416981i	$-0.136912\pi$
$420$		0			0
$421$	−4426.72	−	7667.30i	−0.512458	−	0.887604i	−0.999896	−	0.0144458i	$-0.995402\pi$
$421$	−4426.72	−	7667.30i	−0.512458	−	0.887604i	0.487437	−	0.873158i	$-0.337932\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	2213.35	+	3833.63i	0.252619	+	0.437549i
$426$		0			0
$427$	−8981.42	+	15556.3i	−1.01790	+	1.76305i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	16509.1			1.84504			0.922522	−	0.385945i	$-0.126125\pi$
$431$	16509.1			1.84504			0.922522	+	0.385945i	$0.126125\pi$
$432$		0			0
$433$	2218.19			0.246188			0.123094	−	0.992395i	$-0.460718\pi$
$433$	2218.19			0.246188			0.123094	+	0.992395i	$0.460718\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	3727.02	−	6455.39i	0.407981	−	0.706643i
$438$		0			0
$439$	722.909	+	1252.12i	0.0785936	+	0.136128i	0.902643	−	0.430389i	$-0.141624\pi$
$439$	722.909	+	1252.12i	0.0785936	+	0.136128i	−0.824050	+	0.566517i	$0.808290\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−492.812	−	853.576i	−0.0528538	−	0.0915454i	0.838388	−	0.545074i	$-0.183498\pi$
$443$	−492.812	−	853.576i	−0.0528538	−	0.0915454i	−0.891242	+	0.453528i	$0.850165\pi$
$444$		0			0
$445$	−3179.72	+	5507.44i	−0.338726	+	0.586691i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−6682.65			−0.702391			−0.351196	−	0.936302i	$-0.614225\pi$
$449$	−6682.65			−0.702391			−0.351196	+	0.936302i	$0.614225\pi$
$450$		0			0
$451$	−8204.79			−0.856649
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−2299.59	+	3983.01i	−0.236937	+	0.410388i
$456$		0			0
$457$	−2382.72	−	4126.99i	−0.243893	−	0.422434i	0.717927	−	0.696118i	$-0.245091\pi$
$457$	−2382.72	−	4126.99i	−0.243893	−	0.422434i	−0.961820	+	0.273684i	$0.911758\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	4917.33	+	8517.06i	0.496796	+	0.860476i	0.999993	−	0.00369591i	$-0.00117645\pi$
$461$	4917.33	+	8517.06i	0.496796	+	0.860476i	−0.503197	+	0.864172i	$0.667843\pi$
$462$		0			0
$463$	2790.25	−	4832.86i	0.280074	−	0.485102i	−0.691329	−	0.722540i	$-0.742974\pi$
$463$	2790.25	−	4832.86i	0.280074	−	0.485102i	0.971403	+	0.237438i	$0.0763077\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−4459.02			−0.441839			−0.220920	−	0.975292i	$-0.570906\pi$
$467$	−4459.02			−0.441839			−0.220920	+	0.975292i	$0.570906\pi$
$468$		0			0
$469$	−9884.58			−0.973193
$470$		0			0
$471$		0			0
$472$		0			0
$473$	889.461	−	1540.59i	0.0864640	−	0.149760i
$474$		0			0
$475$	4026.61	+	6974.29i	0.388954	+	0.673689i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	5134.63	+	8893.44i	0.489785	+	0.848333i	0.999931	−	0.0117551i	$-0.00374184\pi$
$479$	5134.63	+	8893.44i	0.489785	+	0.848333i	−0.510146	+	0.860088i	$0.670409\pi$
$480$		0			0
$481$	−3918.22	+	6786.55i	−0.371425	+	0.643327i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−6549.75			−0.613214
$486$		0			0
$487$	−904.158			−0.0841300			−0.0420650	−	0.999115i	$-0.513394\pi$
$487$	−904.158			−0.0841300			−0.0420650	+	0.999115i	$0.513394\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	10263.5	−	17777.0i	0.943354	−	1.63394i	0.184340	−	0.982863i	$-0.440985\pi$
$491$	10263.5	−	17777.0i	0.943354	−	1.63394i	0.759014	−	0.651074i	$-0.225681\pi$
$492$		0			0
$493$	4283.28	+	7418.87i	0.391297	+	0.677746i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−4347.17	−	7529.52i	−0.392348	−	0.679567i
$498$		0			0
$499$	2087.14	−	3615.04i	0.187241	−	0.324311i	−0.757088	−	0.653313i	$-0.773379\pi$
$499$	2087.14	−	3615.04i	0.187241	−	0.324311i	0.944329	+	0.329001i	$0.106712\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−4835.95			−0.428676			−0.214338	−	0.976760i	$-0.568759\pi$
$503$	−4835.95			−0.428676			−0.214338	+	0.976760i	$0.568759\pi$
$504$		0			0
$505$	7170.97			0.631889
$506$		0			0
$507$		0			0
$508$		0			0
$509$	2602.27	−	4507.26i	0.226608	−	0.392497i	−0.730193	−	0.683241i	$-0.760570\pi$
$509$	2602.27	−	4507.26i	0.226608	−	0.392497i	0.956801	+	0.290745i	$0.0939030\pi$
$510$		0			0
$511$	14004.2	+	24256.0i	1.21235	+	2.09985i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−3528.26	−	6111.12i	−0.301891	−	0.522890i
$516$		0			0
$517$	2411.66	−	4177.12i	0.205154	−	0.355337i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−12174.0			−1.02371			−0.511856	−	0.859071i	$-0.671042\pi$
$521$	−12174.0			−1.02371			−0.511856	+	0.859071i	$0.671042\pi$
$522$		0			0
$523$	11583.5			0.968470			0.484235	−	0.874938i	$-0.339098\pi$
$523$	11583.5			0.968470			0.484235	+	0.874938i	$0.339098\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−3766.07	+	6523.03i	−0.311295	+	0.539179i
$528$		0			0
$529$	2511.19	+	4349.50i	0.206393	+	0.357483i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−3341.45	−	5787.57i	−0.271547	−	0.470333i
$534$		0			0
$535$	5162.39	−	8941.52i	0.417177	−	0.722571i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	12641.3			1.01020
$540$		0			0
$541$	18249.3			1.45027			0.725137	−	0.688605i	$-0.241777\pi$
$541$	18249.3			1.45027			0.725137	+	0.688605i	$0.241777\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	2828.98	−	4899.93i	0.222349	−	0.385119i
$546$		0			0
$547$	4805.00	+	8322.50i	0.375588	+	0.650538i	0.990415	−	0.138124i	$-0.0441073\pi$
$547$	4805.00	+	8322.50i	0.375588	+	0.650538i	−0.614827	+	0.788662i	$0.710774\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	7792.32	+	13496.7i	0.602475	+	1.04352i
$552$		0			0
$553$	−16460.1	+	28509.7i	−1.26574	+	2.19232i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−20247.0			−1.54020			−0.770101	−	0.637923i	$-0.779794\pi$
$557$	−20247.0			−1.54020			−0.770101	+	0.637923i	$0.779794\pi$
$558$		0			0
$559$	1448.95			0.109632
$560$		0			0
$561$		0			0
$562$		0			0
$563$	9172.09	−	15886.5i	0.686603	−	1.18923i	−0.286327	−	0.958132i	$-0.592434\pi$
$563$	9172.09	−	15886.5i	0.686603	−	1.18923i	0.972930	−	0.231099i	$-0.0742322\pi$
$564$		0			0
$565$	5720.64	+	9908.43i	0.425963	+	0.737789i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	11834.3	+	20497.6i	0.871914	+	1.51020i	0.860014	+	0.510271i	$0.170455\pi$
$569$	11834.3	+	20497.6i	0.871914	+	1.51020i	0.0119007	+	0.999929i	$0.496212\pi$
$570$		0			0
$571$	−4425.57	+	7665.30i	−0.324351	+	0.561792i	−0.981381	−	0.192073i	$-0.938479\pi$
$571$	−4425.57	+	7665.30i	−0.324351	+	0.561792i	0.657030	+	0.753864i	$0.271812\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	7718.93			0.559829
$576$		0			0
$577$	16832.2			1.21444			0.607221	−	0.794533i	$-0.292284\pi$
$577$	16832.2			1.21444			0.607221	+	0.794533i	$0.292284\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−16014.5	+	27737.9i	−1.14353	+	1.98066i
$582$		0			0
$583$	10746.0	+	18612.6i	0.763386	+	1.32222i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−11649.3	−	20177.1i	−0.819109	−	1.41874i	−0.906340	−	0.422550i	$-0.861135\pi$
$587$	−11649.3	−	20177.1i	−0.819109	−	1.41874i	0.0872307	−	0.996188i	$-0.472198\pi$
$588$		0			0
$589$	−6851.38	+	11866.9i	−0.479298	+	0.830168i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	8720.33			0.603881			0.301940	−	0.953327i	$-0.402366\pi$
$593$	8720.33			0.603881			0.301940	+	0.953327i	$0.402366\pi$
$594$		0			0
$595$	−7348.79			−0.506338
$596$		0			0
$597$		0			0
$598$		0			0
$599$	183.200	−	317.312i	0.0124964	−	0.0216444i	−0.859710	−	0.510783i	$-0.829355\pi$
$599$	183.200	−	317.312i	0.0124964	−	0.0216444i	0.872206	+	0.489139i	$0.162689\pi$
$600$		0			0
$601$	−2943.48	−	5098.25i	−0.199779	−	0.346027i	0.748678	−	0.662934i	$-0.230689\pi$
$601$	−2943.48	−	5098.25i	−0.199779	−	0.346027i	−0.948457	+	0.316907i	$0.897356\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−163.256	−	282.768i	−0.0109708	−	0.0190019i
$606$		0			0
$607$	9093.56	−	15750.5i	0.608067	−	1.05320i	−0.383492	−	0.923544i	$-0.625279\pi$
$607$	9093.56	−	15750.5i	0.608067	−	1.05320i	0.991559	−	0.129658i	$-0.0413879\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	3928.65			0.260125
$612$		0			0
$613$	1998.47			0.131676			0.0658379	−	0.997830i	$-0.479028\pi$
$613$	1998.47			0.131676			0.0658379	+	0.997830i	$0.479028\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	7900.13	−	13683.4i	0.515474	−	0.892827i	−0.484365	−	0.874866i	$-0.660949\pi$
$617$	7900.13	−	13683.4i	0.515474	−	0.892827i	0.999839	−	0.0179609i	$-0.00571743\pi$
$618$		0			0
$619$	−4883.43	−	8458.35i	−0.317095	−	0.549224i	0.662786	−	0.748809i	$-0.269374\pi$
$619$	−4883.43	−	8458.35i	−0.317095	−	0.549224i	−0.979881	+	0.199585i	$0.936041\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	14312.8	+	24790.5i	0.920433	+	1.59424i
$624$		0			0
$625$	−2064.71	+	3576.19i	−0.132142	+	0.228876i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−12521.4			−0.793739
$630$		0			0
$631$	−9966.80			−0.628799			−0.314399	−	0.949291i	$-0.601803\pi$
$631$	−9966.80			−0.628799			−0.314399	+	0.949291i	$0.601803\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	1755.51	−	3040.63i	0.109709	−	0.190021i
$636$		0			0
$637$	5148.23	+	8917.00i	0.320220	+	0.554638i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	6355.26	+	11007.6i	0.391603	+	0.678277i	0.992661	−	0.120929i	$-0.0385873\pi$
$641$	6355.26	+	11007.6i	0.391603	+	0.678277i	−0.601058	+	0.799205i	$0.705254\pi$
$642$		0			0
$643$	−6405.70	+	11095.0i	−0.392871	+	0.680473i	−0.992827	−	0.119560i	$-0.961851\pi$
$643$	−6405.70	+	11095.0i	−0.392871	+	0.680473i	0.599956	+	0.800033i	$0.295185\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−3729.11			−0.226594			−0.113297	−	0.993561i	$-0.536141\pi$
$647$	−3729.11			−0.226594			−0.113297	+	0.993561i	$0.536141\pi$
$648$		0			0
$649$	9069.97			0.548579
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1653.35	+	2863.68i	−0.0990820	+	0.171615i	−0.911305	−	0.411732i	$-0.864924\pi$
$653$	−1653.35	+	2863.68i	−0.0990820	+	0.171615i	0.812223	+	0.583347i	$0.198257\pi$
$654$		0			0
$655$	4090.53	+	7085.01i	0.244016	+	0.422648i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−8978.50	−	15551.2i	−0.530732	−	0.919256i	−0.999357	−	0.0358581i	$-0.988584\pi$
$659$	−8978.50	−	15551.2i	−0.530732	−	0.919256i	0.468624	−	0.883398i	$-0.344750\pi$
$660$		0			0
$661$	−4847.57	+	8396.23i	−0.285247	+	0.494063i	−0.972669	−	0.232195i	$-0.925409\pi$
$661$	−4847.57	+	8396.23i	−0.285247	+	0.494063i	0.687422	+	0.726258i	$0.258742\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−13369.2			−0.779602
$666$		0			0
$667$	14937.7			0.867153
$668$		0			0
$669$		0			0
$670$		0			0
$671$	12805.8	−	22180.2i	0.736752	−	1.27609i
$672$		0			0
$673$	−10793.6	−	18695.1i	−0.618222	−	1.07079i	−0.989810	−	0.142394i	$-0.954520\pi$
$673$	−10793.6	−	18695.1i	−0.618222	−	1.07079i	0.371588	−	0.928398i	$-0.378813\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	10336.2	+	17902.8i	0.586782	+	1.01634i	0.994651	+	0.103296i	$0.0329389\pi$
$677$	10336.2	+	17902.8i	0.586782	+	1.01634i	−0.407868	+	0.913041i	$0.633728\pi$
$678$		0			0
$679$	−14741.1	+	25532.3i	−0.833153	+	1.44306i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−24403.9			−1.36719			−0.683594	−	0.729862i	$-0.739584\pi$
$683$	−24403.9			−1.36719			−0.683594	+	0.729862i	$0.739584\pi$
$684$		0			0
$685$	−13136.3			−0.732719
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−8752.76	+	15160.2i	−0.483968	+	0.838256i
$690$		0			0
$691$	2929.03	+	5073.24i	0.161253	+	0.279298i	0.935318	−	0.353808i	$-0.115113\pi$
$691$	2929.03	+	5073.24i	0.161253	+	0.279298i	−0.774065	+	0.633106i	$0.781780\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−4470.09	−	7742.43i	−0.243972	−	0.422571i
$696$		0			0
$697$	5339.13	−	9247.65i	0.290149	−	0.502553i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−10530.8			−0.567393			−0.283697	−	0.958914i	$-0.591561\pi$
$701$	−10530.8			−0.567393			−0.283697	+	0.958914i	$0.591561\pi$
$702$		0			0
$703$	−22779.5			−1.22211
$704$		0			0
$705$		0			0
$706$		0			0
$707$	16139.2	−	27954.0i	0.858527	−	1.48701i
$708$		0			0
$709$	1888.39	+	3270.79i	0.100028	+	0.173254i	0.911696	−	0.410865i	$-0.134773\pi$
$709$	1888.39	+	3270.79i	0.100028	+	0.173254i	−0.811668	+	0.584119i	$0.801440\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	6566.99	+	11374.4i	0.344931	+	0.597437i
$714$		0			0
$715$	3278.77	−	5678.99i	0.171495	−	0.297038i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−7109.15			−0.368743			−0.184372	−	0.982857i	$-0.559025\pi$
$719$	−7109.15			−0.368743			−0.184372	+	0.982857i	$0.559025\pi$
$720$		0			0
$721$	−31763.3			−1.64068
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−8069.22	+	13976.3i	−0.413356	+	0.715954i
$726$		0			0
$727$	−1830.18	−	3169.96i	−0.0933667	−	0.161716i	0.815559	−	0.578674i	$-0.196430\pi$
$727$	−1830.18	−	3169.96i	−0.0933667	−	0.161716i	−0.908926	+	0.416958i	$0.863096\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1157.60	+	2005.03i	0.0585712	+	0.101448i
$732$		0			0
$733$	18279.7	−	31661.3i	0.921113	−	1.59541i	0.123416	−	0.992355i	$-0.460615\pi$
$733$	18279.7	−	31661.3i	0.921113	−	1.59541i	0.797697	−	0.603059i	$-0.206052\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	14093.5			0.704396
$738$		0			0
$739$	15735.7			0.783285			0.391642	−	0.920118i	$-0.371907\pi$
$739$	15735.7			0.783285			0.391642	+	0.920118i	$0.371907\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−11847.1	+	20519.9i	−0.584966	+	1.01319i	0.409914	+	0.912124i	$0.365559\pi$
$743$	−11847.1	+	20519.9i	−0.584966	+	1.01319i	−0.994880	+	0.101066i	$0.967775\pi$
$744$		0			0
$745$	1778.85	+	3081.05i	0.0874790	+	0.151518i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−23237.3	−	40248.2i	−1.13361	−	1.96347i
$750$		0			0
$751$	8643.50	−	14971.0i	0.419981	−	0.727429i	−0.575956	−	0.817481i	$-0.695370\pi$
$751$	8643.50	−	14971.0i	0.419981	−	0.727429i	0.995937	+	0.0900521i	$0.0287034\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	13771.4			0.663831
$756$		0			0
$757$	18565.8			0.891394			0.445697	−	0.895184i	$-0.352956\pi$
$757$	18565.8			0.891394			0.445697	+	0.895184i	$0.352956\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−2968.73	+	5142.00i	−0.141415	+	0.244937i	−0.928030	−	0.372507i	$-0.878498\pi$
$761$	−2968.73	+	5142.00i	−0.141415	+	0.244937i	0.786615	+	0.617444i	$0.211832\pi$
$762$		0			0
$763$	−12734.0	−	22055.9i	−0.604196	−	1.04650i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	3693.80	+	6397.86i	0.173892	+	0.301191i
$768$		0			0
$769$	−1244.77	+	2156.00i	−0.0583713	+	0.101102i	−0.893734	−	0.448596i	$-0.851924\pi$
$769$	−1244.77	+	2156.00i	−0.0583713	+	0.101102i	0.835363	+	0.549698i	$0.185257\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	16109.8			0.749584			0.374792	−	0.927109i	$-0.377714\pi$
$773$	16109.8			0.749584			0.374792	+	0.927109i	$0.377714\pi$
$774$		0			0
$775$	−14189.7			−0.657690
$776$		0			0
$777$		0			0
$778$		0			0
$779$	9713.15	−	16823.7i	0.446739	−	0.773775i
$780$		0			0
$781$	6198.21	+	10735.6i	0.283981	+	0.491870i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	5130.41	+	8886.13i	0.233264	+	0.404025i
$786$		0			0
$787$	−7905.02	+	13691.9i	−0.358047	+	0.620156i	−0.987635	−	0.156773i	$-0.949891\pi$
$787$	−7905.02	+	13691.9i	−0.358047	+	0.620156i	0.629587	+	0.776930i	$0.283224\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	51500.2			2.31497
$792$		0			0
$793$	20860.9			0.934164
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−2092.58	+	3624.45i	−0.0930024	+	0.161085i	−0.908773	−	0.417291i	$-0.862980\pi$
$797$	−2092.58	+	3624.45i	−0.0930024	+	0.161085i	0.815771	+	0.578375i	$0.196313\pi$
$798$		0			0
$799$	3138.69	+	5436.38i	0.138972	+	0.240707i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−19967.3	−	34584.4i	−0.877498	−	1.51987i
$804$		0			0
$805$	−6407.13	+	11097.5i	−0.280524	+	0.485881i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	35154.2			1.52776			0.763878	−	0.645360i	$-0.223293\pi$
$809$	35154.2			1.52776			0.763878	+	0.645360i	$0.223293\pi$
$810$		0			0
$811$	−3028.93			−0.131147			−0.0655734	−	0.997848i	$-0.520888\pi$
$811$	−3028.93			−0.131147			−0.0655734	+	0.997848i	$0.520888\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−7712.22	+	13358.0i	−0.331469	+	0.574121i
$816$		0			0
$817$	2105.96	+	3647.62i	0.0901813	+	0.156199i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−2398.99	−	4155.17i	−0.101980	−	0.176634i	0.810521	−	0.585710i	$-0.199184\pi$
$821$	−2398.99	−	4155.17i	−0.101980	−	0.176634i	−0.912500	+	0.409076i	$0.865851\pi$
$822$		0			0
$823$	20029.4	−	34691.9i	0.848335	−	1.46936i	−0.0343587	−	0.999410i	$-0.510939\pi$
$823$	20029.4	−	34691.9i	0.848335	−	1.46936i	0.882693	−	0.469949i	$-0.155728\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−22168.6			−0.932139			−0.466069	−	0.884748i	$-0.654330\pi$
$827$	−22168.6			−0.932139			−0.466069	+	0.884748i	$0.654330\pi$
$828$		0			0
$829$	13091.9			0.548494			0.274247	−	0.961659i	$-0.411571\pi$
$829$	13091.9			0.548494			0.274247	+	0.961659i	$0.411571\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−8226.09	+	14248.0i	−0.342157	+	0.592634i
$834$		0			0
$835$	−9728.04	−	16849.5i	−0.403177	−	0.698323i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	17395.2	+	30129.3i	0.715790	+	1.23978i	0.962654	+	0.270734i	$0.0872665\pi$
$839$	17395.2	+	30129.3i	0.715790	+	1.23978i	−0.246864	+	0.969050i	$0.579400\pi$
$840$		0			0
$841$	−3421.11	+	5925.54i	−0.140273	+	0.242960i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−7408.93			−0.301627
$846$		0			0
$847$	−1469.72			−0.0596225
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−10916.9	+	18908.7i	−0.439751	+	0.761671i
$852$		0			0
$853$	−23990.4	−	41552.5i	−0.962971	−	1.66792i	−0.714970	−	0.699156i	$-0.753559\pi$
$853$	−23990.4	−	41552.5i	−0.962971	−	1.66792i	−0.248002	−	0.968760i	$-0.579774\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−16007.7	−	27726.2i	−0.638055	−	1.10514i	−0.985859	−	0.167577i	$-0.946406\pi$
$857$	−16007.7	−	27726.2i	−0.638055	−	1.10514i	0.347804	−	0.937567i	$-0.386928\pi$
$858$		0			0
$859$	16256.7	−	28157.5i	0.645719	−	1.11842i	−0.338415	−	0.940997i	$-0.609891\pi$
$859$	16256.7	−	28157.5i	0.645719	−	1.11842i	0.984135	−	0.177422i	$-0.0567758\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	25867.1			1.02031			0.510154	−	0.860083i	$-0.329589\pi$
$863$	25867.1			1.02031			0.510154	+	0.860083i	$0.329589\pi$
$864$		0			0
$865$	−14.1365			−0.000555670
$866$		0			0
$867$		0			0
$868$		0			0
$869$	23468.8	−	40649.2i	0.916140	−	1.58680i
$870$		0			0
$871$	5739.66	+	9941.38i	0.223285	+	0.386740i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−16397.2	−	28400.9i	−0.633518	−	1.09728i
$876$		0			0
$877$	287.352	−	497.708i	0.0110641	−	0.0191635i	−0.860440	−	0.509551i	$-0.829811\pi$
$877$	287.352	−	497.708i	0.0110641	−	0.0191635i	0.871504	+	0.490388i	$0.163145\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−28334.1			−1.08354			−0.541771	−	0.840526i	$-0.682246\pi$
$881$	−28334.1			−1.08354			−0.541771	+	0.840526i	$0.682246\pi$
$882$		0			0
$883$	13424.9			0.511646			0.255823	−	0.966724i	$-0.417654\pi$
$883$	13424.9			0.511646			0.255823	+	0.966724i	$0.417654\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−21881.6	+	37900.0i	−0.828310	+	1.43467i	0.0710540	+	0.997472i	$0.477364\pi$
$887$	−21881.6	+	37900.0i	−0.828310	+	1.43467i	−0.899363	+	0.437202i	$0.855970\pi$
$888$		0			0
$889$	−7902.01	−	13686.7i	−0.298116	−	0.516352i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	5710.03	+	9890.07i	0.213974	+	0.370614i
$894$		0			0
$895$	3145.50	−	5448.16i	0.117477	−	0.203477i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−27460.0			−1.01874
$900$		0			0
$901$	−27971.2			−1.03424
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−2524.02	+	4371.74i	−0.0927087	+	0.160576i
$906$		0			0
$907$	−21398.6	−	37063.4i	−0.783383	−	1.35686i	−0.929961	−	0.367659i	$-0.880159\pi$
$907$	−21398.6	−	37063.4i	−0.783383	−	1.35686i	0.146578	−	0.989199i	$-0.453174\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1003.57	+	1738.24i	0.0364982	+	0.0632167i	0.883697	−	0.468058i	$-0.155046\pi$
$911$	1003.57	+	1738.24i	0.0364982	+	0.0632167i	−0.847199	+	0.531275i	$0.821713\pi$
$912$		0			0
$913$	22833.5	−	39548.8i	0.827688	−	1.43360i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	36825.2			1.32614
$918$		0			0
$919$	−43558.3			−1.56350			−0.781750	−	0.623592i	$-0.785673\pi$
$919$	−43558.3			−1.56350			−0.781750	+	0.623592i	$0.785673\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−5048.52	+	8744.30i	−0.180037	+	0.311833i
$924$		0			0
$925$	−11794.5	−	20428.6i	−0.419243	−	0.726151i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−15754.0	−	27286.8i	−0.556376	−	0.963671i	−0.997795	−	0.0663701i	$-0.978858\pi$
$929$	−15754.0	−	27286.8i	−0.556376	−	0.963671i	0.441419	−	0.897301i	$-0.354475\pi$
$930$		0			0
$931$	−14965.2	+	25920.5i	−0.526815	+	0.912471i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	10477.9			0.366487
$936$		0			0
$937$	−20675.3			−0.720846			−0.360423	−	0.932789i	$-0.617368\pi$
$937$	−20675.3			−0.720846			−0.360423	+	0.932789i	$0.617368\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−712.769	+	1234.55i	−0.0246925	+	0.0427686i	−0.878108	−	0.478463i	$-0.841194\pi$
$941$	−712.769	+	1234.55i	−0.0246925	+	0.0427686i	0.853415	+	0.521232i	$0.174527\pi$
$942$		0			0
$943$	−9309.97	−	16125.3i	−0.321500	−	0.556854i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	17830.9	+	30884.0i	0.611853	+	1.05976i	0.990928	+	0.134395i	$0.0429091\pi$
$947$	17830.9	+	30884.0i	0.611853	+	1.05976i	−0.379075	+	0.925366i	$0.623758\pi$
$948$		0			0
$949$	16263.6	−	28169.4i	0.556311	−	0.963560i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−10384.9			−0.352991			−0.176496	−	0.984301i	$-0.556476\pi$
$953$	−10384.9			−0.352991			−0.176496	+	0.984301i	$0.556476\pi$
$954$		0			0
$955$	2467.31			0.0836024
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−29565.0	+	51208.2i	−0.995521	+	1.72429i
$960$		0			0
$961$	2823.41	+	4890.29i	0.0947739	+	0.164153i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	4.96002	+	8.59102i	0.000165460	+	0.000286585i
$966$		0			0
$967$	4221.70	−	7312.19i	0.140394	−	0.243169i	−0.787251	−	0.616632i	$-0.788497\pi$
$967$	4221.70	−	7312.19i	0.140394	−	0.243169i	0.927645	+	0.373463i	$0.121830\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	6938.43			0.229315			0.114657	−	0.993405i	$-0.463423\pi$
$971$	6938.43			0.229315			0.114657	+	0.993405i	$0.463423\pi$
$972$		0			0
$973$	−40242.2			−1.32590
$974$		0			0
$975$		0			0
$976$		0			0
$977$	19388.7	−	33582.3i	0.634903	−	1.09969i	−0.351632	−	0.936138i	$-0.614373\pi$
$977$	19388.7	−	33582.3i	0.634903	−	1.09969i	0.986536	−	0.163547i	$-0.0522935\pi$
$978$		0			0
$979$	−20407.2	−	35346.4i	−0.666208	−	1.15391i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−3796.28	−	6575.34i	−0.123176	−	0.213348i	0.797842	−	0.602866i	$-0.205975\pi$
$983$	−3796.28	−	6575.34i	−0.123176	−	0.213348i	−0.921019	+	0.389519i	$0.872641\pi$
$984$		0			0
$985$	−8935.64	+	15477.0i	−0.289049	+	0.500647i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	4037.08			0.129799
$990$		0			0
$991$	5.96741			0.000191282			9.56412e−5	−	1.00000i	$-0.499970\pi$
$991$	5.96741			0.000191282			9.56412e−5		1.00000i	$0.499970\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−9092.51	+	15748.7i	−0.289700	+	0.501776i
$996$		0			0
$997$	−17385.9	−	30113.2i	−0.552272	−	0.956564i	−0.998110	−	0.0614501i	$-0.980427\pi$
$997$	−17385.9	−	30113.2i	−0.552272	−	0.956564i	0.445838	−	0.895114i	$-0.352906\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.4.i.v.433.2		8
3.2	odd	2		648.4.i.u.433.3		8
9.2	odd	6		648.4.i.u.217.3		8
9.4	even	3		648.4.a.g.1.3	✓	4
9.5	odd	6		648.4.a.j.1.2	yes	4
9.7	even	3	inner	648.4.i.v.217.2		8
36.23	even	6		1296.4.a.bb.1.2		4
36.31	odd	6		1296.4.a.x.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.4.a.g.1.3	✓	4	9.4	even	3
648.4.a.j.1.2	yes	4	9.5	odd	6
648.4.i.u.217.3		8	9.2	odd	6
648.4.i.u.433.3		8	3.2	odd	2
648.4.i.v.217.2		8	9.7	even	3	inner
648.4.i.v.433.2		8	1.1	even	1	trivial
1296.4.a.x.1.3		4	36.31	odd	6
1296.4.a.bb.1.2		4	36.23	even	6

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:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	648.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.90171 + 5.02591i) q^{5} +(13.0614 + 22.6230i) q^{7} +O(q^{10})\)	\(q+(-2.90171 + 5.02591i) q^{5} +(13.0614 + 22.6230i) q^{7} +(-18.6230 - 32.2560i) q^{11} +(15.1687 - 26.2729i) q^{13} +48.4744 q^{17} +88.1865 q^{19} +(42.2630 - 73.2016i) q^{23} +(45.6601 + 79.0857i) q^{25} +(88.3618 + 153.047i) q^{29} +(-77.6920 + 134.566i) q^{31} -151.601 q^{35} -258.310 q^{37} +(110.143 - 190.774i) q^{41} +(23.8807 + 41.3626i) q^{43} +(64.7495 + 112.149i) q^{47} +(-169.700 + 293.928i) q^{49} -577.029 q^{53} +216.154 q^{55} +(-121.758 + 210.891i) q^{59} +(343.816 + 595.507i) q^{61} +(88.0301 + 152.473i) q^{65} +(-189.195 + 327.695i) q^{67} -332.826 q^{71} +1072.19 q^{73} +(486.484 - 842.615i) q^{77} +(630.104 + 1091.37i) q^{79} +(613.047 + 1061.83i) q^{83} +(-140.659 + 243.628i) q^{85} +1095.81 q^{89} +792.495 q^{91} +(-255.892 + 443.217i) q^{95} +(564.300 + 977.397i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{5}+O(q^{10}) \) 8 * q + 8 * q^5	\( 8 q + 8 q^{5} + 8 q^{11} - 4 q^{13} - 32 q^{17} + 160 q^{19} + 200 q^{23} - 8 q^{25} + 216 q^{29} - 80 q^{31} - 816 q^{35} - 552 q^{37} + 384 q^{41} - 160 q^{43} + 768 q^{47} + 268 q^{49} - 1888 q^{53} + 608 q^{55} + 992 q^{59} + 548 q^{61} + 1328 q^{65} - 464 q^{67} - 3440 q^{71} - 1528 q^{73} + 1728 q^{77} - 688 q^{79} + 2128 q^{83} + 1324 q^{85} - 4224 q^{89} + 3552 q^{91} + 2056 q^{95} + 1816 q^{97}+O(q^{100}) \) 8 * q + 8 * q^5 + 8 * q^11 - 4 * q^13 - 32 * q^17 + 160 * q^19 + 200 * q^23 - 8 * q^25 + 216 * q^29 - 80 * q^31 - 816 * q^35 - 552 * q^37 + 384 * q^41 - 160 * q^43 + 768 * q^47 + 268 * q^49 - 1888 * q^53 + 608 * q^55 + 992 * q^59 + 548 * q^61 + 1328 * q^65 - 464 * q^67 - 3440 * q^71 - 1528 * q^73 + 1728 * q^77 - 688 * q^79 + 2128 * q^83 + 1324 * q^85 - 4224 * q^89 + 3552 * q^91 + 2056 * q^95 + 1816 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more