LMFDB - Embedded newform 648.4.i.q.433.1

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:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-43})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 10x^{2} - 11x + 121 $ x^4 - x^3 - 10x^2 - 11x + 121
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			433.1
Root			$-2.58945 + 2.07237i$ of defining polynomial
Character	$\chi$	$=$	648.433
Dual form			648.4.i.q.217.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.67891 + 8.10411i) q^{5} +(-4.17891 - 7.23808i) q^{7} +O(q^{10})$	$q+(-4.67891 + 8.10411i) q^{5} +(-4.17891 - 7.23808i) q^{7} +(-14.5367 - 25.1783i) q^{11} +(-18.3945 + 31.8603i) q^{13} -28.4313 q^{17} +73.7891 q^{19} +(15.2524 - 26.4179i) q^{23} +(18.7156 + 32.4164i) q^{25} +(-92.4680 - 160.159i) q^{29} +(95.4313 - 165.292i) q^{31} +78.2109 q^{35} +160.495 q^{37} +(-104.789 + 181.500i) q^{41} +(-58.4727 - 101.278i) q^{43} +(140.799 + 243.870i) q^{47} +(136.573 - 236.552i) q^{49} +397.450 q^{53} +272.064 q^{55} +(-189.073 + 327.485i) q^{59} +(148.762 + 257.663i) q^{61} +(-172.133 - 298.143i) q^{65} +(413.968 - 717.014i) q^{67} +729.073 q^{71} +1105.44 q^{73} +(-121.495 + 210.436i) q^{77} +(532.134 + 921.683i) q^{79} +(-295.239 - 511.369i) q^{83} +(133.027 - 230.410i) q^{85} -227.588 q^{89} +307.476 q^{91} +(-345.252 + 597.995i) q^{95} +(-642.827 - 1113.41i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} + 6 q^{7}+O(q^{10}) $ 4 * q + 4 * q^5 + 6 * q^7	$ 4 q + 4 q^{5} + 6 q^{7} + 10 q^{11} + 40 q^{13} + 68 q^{17} + 68 q^{19} - 98 q^{23} - 16 q^{25} - 120 q^{29} + 200 q^{31} + 540 q^{35} + 960 q^{37} - 192 q^{41} + 334 q^{43} - 300 q^{47} + 410 q^{49} + 136 q^{53} + 1588 q^{55} - 620 q^{59} - 200 q^{61} - 1370 q^{65} + 1406 q^{67} + 2780 q^{71} + 3604 q^{73} - 804 q^{77} + 334 q^{79} + 500 q^{83} + 1100 q^{85} + 180 q^{89} + 2820 q^{91} - 1222 q^{95} + 200 q^{97}+O(q^{100}) $ 4 * q + 4 * q^5 + 6 * q^7 + 10 * q^11 + 40 * q^13 + 68 * q^17 + 68 * q^19 - 98 * q^23 - 16 * q^25 - 120 * q^29 + 200 * q^31 + 540 * q^35 + 960 * q^37 - 192 * q^41 + 334 * q^43 - 300 * q^47 + 410 * q^49 + 136 * q^53 + 1588 * q^55 - 620 * q^59 - 200 * q^61 - 1370 * q^65 + 1406 * q^67 + 2780 * q^71 + 3604 * q^73 - 804 * q^77 + 334 * q^79 + 500 * q^83 + 1100 * q^85 + 180 * q^89 + 2820 * q^91 - 1222 * q^95 + 200 * 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
$5$	−4.67891	+	8.10411i	−0.418494	+	0.724853i	−0.995788	−	0.0916830i	$-0.970775\pi$
$5$	−4.67891	+	8.10411i	−0.418494	+	0.724853i	0.577294	+	0.816536i	$0.304109\pi$
$6$		0			0
$7$	−4.17891	−	7.23808i	−0.225640	−	0.390820i	0.730871	−	0.682515i	$-0.239114\pi$
$7$	−4.17891	−	7.23808i	−0.225640	−	0.390820i	−0.956511	+	0.291696i	$0.905781\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−14.5367	−	25.1783i	−0.398453	−	0.690142i	0.595082	−	0.803665i	$-0.297120\pi$
$11$	−14.5367	−	25.1783i	−0.398453	−	0.690142i	−0.993535	+	0.113524i	$0.963786\pi$
$12$		0			0
$13$	−18.3945	+	31.8603i	−0.392441	+	0.679727i	−0.992771	−	0.120025i	$-0.961702\pi$
$13$	−18.3945	+	31.8603i	−0.392441	+	0.679727i	0.600330	+	0.799752i	$0.295036\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−28.4313			−0.405623			−0.202812	−	0.979218i	$-0.565008\pi$
$17$	−28.4313			−0.405623			−0.202812	+	0.979218i	$0.565008\pi$
$18$		0			0
$19$	73.7891			0.890967			0.445484	−	0.895290i	$-0.353032\pi$
$19$	73.7891			0.890967			0.445484	+	0.895290i	$0.353032\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	15.2524	−	26.4179i	0.138276	−	0.239500i	−0.788568	−	0.614947i	$-0.789177\pi$
$23$	15.2524	−	26.4179i	0.138276	−	0.239500i	0.926844	+	0.375447i	$0.122511\pi$
$24$		0			0
$25$	18.7156	+	32.4164i	0.149725	+	0.259331i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−92.4680	−	160.159i	−0.592099	−	1.02555i	−0.993949	−	0.109840i	$-0.964966\pi$
$29$	−92.4680	−	160.159i	−0.592099	−	1.02555i	0.401850	−	0.915705i	$-0.368367\pi$
$30$		0			0
$31$	95.4313	−	165.292i	0.552902	−	0.957654i	−0.445161	−	0.895450i	$-0.646854\pi$
$31$	95.4313	−	165.292i	0.552902	−	0.957654i	0.998063	−	0.0622040i	$-0.0198129\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	78.2109			0.377716
$36$		0			0
$37$	160.495			0.713115			0.356558	−	0.934273i	$-0.383950\pi$
$37$	160.495			0.713115			0.356558	+	0.934273i	$0.383950\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−104.789	+	181.500i	−0.399154	+	0.691355i	−0.993622	−	0.112765i	$-0.964029\pi$
$41$	−104.789	+	181.500i	−0.399154	+	0.691355i	0.594468	+	0.804119i	$0.297363\pi$
$42$		0			0
$43$	−58.4727	−	101.278i	−0.207372	−	0.359179i	0.743514	−	0.668721i	$-0.233158\pi$
$43$	−58.4727	−	101.278i	−0.207372	−	0.359179i	−0.950886	+	0.309541i	$0.899824\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	140.799	+	243.870i	0.436970	+	0.756854i	0.997454	−	0.0713113i	$-0.0227184\pi$
$47$	140.799	+	243.870i	0.436970	+	0.756854i	−0.560484	+	0.828165i	$0.689385\pi$
$48$		0			0
$49$	136.573	−	236.552i	0.398173	−	0.689656i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	397.450			1.03007			0.515037	−	0.857168i	$-0.327778\pi$
$53$	397.450			1.03007			0.515037	+	0.857168i	$0.327778\pi$
$54$		0			0
$55$	272.064			0.667002
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−189.073	+	327.485i	−0.417208	+	0.722625i	−0.995657	−	0.0930929i	$-0.970325\pi$
$59$	−189.073	+	327.485i	−0.417208	+	0.722625i	0.578450	+	0.815718i	$0.303658\pi$
$60$		0			0
$61$	148.762	+	257.663i	0.312246	+	0.540826i	0.978848	−	0.204588i	$-0.0655854\pi$
$61$	148.762	+	257.663i	0.312246	+	0.540826i	−0.666602	+	0.745413i	$0.732252\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−172.133	−	298.143i	−0.328468	−	0.568924i
$66$		0			0
$67$	413.968	−	717.014i	0.754839	−	1.30742i	−0.190615	−	0.981665i	$-0.561048\pi$
$67$	413.968	−	717.014i	0.754839	−	1.30742i	0.945454	−	0.325755i	$-0.105619\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	729.073			1.21866			0.609332	−	0.792915i	$-0.291438\pi$
$71$	729.073			1.21866			0.609332	+	0.792915i	$0.291438\pi$
$72$		0			0
$73$	1105.44			1.77236			0.886178	−	0.463344i	$-0.153351\pi$
$73$	1105.44			1.77236			0.886178	+	0.463344i	$0.153351\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−121.495	+	210.436i	−0.179814	+	0.311447i
$78$		0			0
$79$	532.134	+	921.683i	0.757845	+	1.31263i	0.943948	+	0.330095i	$0.107081\pi$
$79$	532.134	+	921.683i	0.757845	+	1.31263i	−0.186103	+	0.982530i	$0.559586\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−295.239	−	511.369i	−0.390442	−	0.676266i	0.602066	−	0.798447i	$-0.294345\pi$
$83$	−295.239	−	511.369i	−0.390442	−	0.676266i	−0.992508	+	0.122181i	$0.961011\pi$
$84$		0			0
$85$	133.027	−	230.410i	0.169751	−	0.294017i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−227.588			−0.271059			−0.135529	−	0.990773i	$-0.543274\pi$
$89$	−227.588			−0.271059			−0.135529	+	0.990773i	$0.543274\pi$
$90$		0			0
$91$	307.476			0.354201
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−345.252	+	597.995i	−0.372865	+	0.645821i
$96$		0			0
$97$	−642.827	−	1113.41i	−0.672878	−	1.16546i	−0.977084	−	0.212853i	$-0.931725\pi$
$97$	−642.827	−	1113.41i	−0.672878	−	1.16546i	0.304206	−	0.952606i	$-0.401609\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−842.367	−	1459.02i	−0.829888	−	1.43741i	−0.898126	−	0.439739i	$-0.855071\pi$
$101$	−842.367	−	1459.02i	−0.829888	−	1.43741i	0.0682378	−	0.997669i	$-0.478262\pi$
$102$		0			0
$103$	427.201	−	739.935i	0.408674	−	0.707844i	−0.586068	−	0.810262i	$-0.699325\pi$
$103$	427.201	−	739.935i	0.408674	−	0.707844i	0.994741	+	0.102418i	$0.0326580\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1369.73			1.23754			0.618768	−	0.785574i	$-0.287632\pi$
$107$	1369.73			1.23754			0.618768	+	0.785574i	$0.287632\pi$
$108$		0			0
$109$	1475.06			1.29620			0.648099	−	0.761556i	$-0.275564\pi$
$109$	1475.06			1.29620			0.648099	+	0.761556i	$0.275564\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	4.50943	−	7.81057i	0.00375409	−	0.00650227i	−0.864142	−	0.503248i	$-0.832138\pi$
$113$	4.50943	−	7.81057i	0.00375409	−	0.00650227i	0.867896	+	0.496745i	$0.165472\pi$
$114$		0			0
$115$	142.729	+	247.213i	0.115735	+	0.200459i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	118.812	+	205.788i	0.0915248	+	0.158526i
$120$		0			0
$121$	242.867	−	420.658i	0.182470	−	0.316047i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1520.00			−1.08762
$126$		0			0
$127$	2076.05			1.45055			0.725274	−	0.688460i	$-0.241713\pi$
$127$	2076.05			1.45055			0.725274	+	0.688460i	$0.241713\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−887.207	+	1536.69i	−0.591723	+	1.02489i	0.402278	+	0.915518i	$0.368219\pi$
$131$	−887.207	+	1536.69i	−0.591723	+	1.02489i	−0.994000	+	0.109376i	$0.965115\pi$
$132$		0			0
$133$	−308.358	−	534.091i	−0.201038	−	0.348208i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−605.344	−	1048.49i	−0.377504	−	0.653855i	0.613195	−	0.789932i	$-0.289884\pi$
$137$	−605.344	−	1048.49i	−0.377504	−	0.653855i	−0.990698	+	0.136076i	$0.956551\pi$
$138$		0			0
$139$	366.614	−	634.994i	0.223711	−	0.387478i	−0.732221	−	0.681067i	$-0.761516\pi$
$139$	366.614	−	634.994i	0.223711	−	0.387478i	0.955932	+	0.293589i	$0.0948496\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1069.59			0.625477
$144$		0			0
$145$	1730.60			0.991160
$146$		0			0
$147$		0			0
$148$		0			0
$149$	257.781	−	446.489i	0.141733	−	0.245489i	−0.786416	−	0.617697i	$-0.788066\pi$
$149$	257.781	−	446.489i	0.141733	−	0.245489i	0.928149	+	0.372208i	$0.121399\pi$
$150$		0			0
$151$	576.945	+	999.299i	0.310935	+	0.538555i	0.978565	−	0.205938i	$-0.0660246\pi$
$151$	576.945	+	999.299i	0.310935	+	0.538555i	−0.667630	+	0.744493i	$0.732691\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	893.028	+	1546.77i	0.462773	+	0.801546i
$156$		0			0
$157$	684.754	−	1186.03i	0.348085	−	0.602901i	−0.637824	−	0.770182i	$-0.720165\pi$
$157$	684.754	−	1186.03i	0.348085	−	0.602901i	0.985909	+	0.167281i	$0.0534987\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−254.953			−0.124802
$162$		0			0
$163$	−3796.90			−1.82452			−0.912259	−	0.409613i	$-0.865664\pi$
$163$	−3796.90			−1.82452			−0.912259	+	0.409613i	$0.865664\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1024.45	−	1774.39i	0.474695	−	0.822196i	−0.524885	−	0.851173i	$-0.675892\pi$
$167$	1024.45	−	1774.39i	0.474695	−	0.822196i	0.999580	+	0.0289774i	$0.00922508\pi$
$168$		0			0
$169$	421.782	+	730.547i	0.191981	+	0.332520i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1012.61	−	1753.90i	−0.445016	−	0.770789i	0.553038	−	0.833156i	$-0.313469\pi$
$173$	−1012.61	−	1753.90i	−0.445016	−	0.770789i	−0.998053	+	0.0623667i	$0.980135\pi$
$174$		0			0
$175$	156.422	−	270.931i	0.0675679	−	0.117031i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	3142.50			1.31219			0.656094	−	0.754679i	$-0.272208\pi$
$179$	3142.50			1.31219			0.656094	+	0.754679i	$0.272208\pi$
$180$		0			0
$181$	37.6685			0.0154689			0.00773446	−	0.999970i	$-0.497538\pi$
$181$	37.6685			0.0154689			0.00773446	+	0.999970i	$0.497538\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−750.943	+	1300.67i	−0.298435	+	0.516904i
$186$		0			0
$187$	413.298	+	715.852i	0.161622	+	0.279937i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2117.06	+	3666.86i	0.802016	+	1.38913i	0.918287	+	0.395915i	$0.129573\pi$
$191$	2117.06	+	3666.86i	0.802016	+	1.38913i	−0.116271	+	0.993218i	$0.537094\pi$
$192$		0			0
$193$	−1914.65	+	3316.27i	−0.714090	+	1.23684i	0.249219	+	0.968447i	$0.419826\pi$
$193$	−1914.65	+	3316.27i	−0.714090	+	1.23684i	−0.963309	+	0.268394i	$0.913507\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1072.38			−0.387838			−0.193919	−	0.981018i	$-0.562120\pi$
$197$	−1072.38			−0.387838			−0.193919	+	0.981018i	$0.562120\pi$
$198$		0			0
$199$	−1078.53			−0.384196			−0.192098	−	0.981376i	$-0.561529\pi$
$199$	−1078.53			−0.384196			−0.192098	+	0.981376i	$0.561529\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−772.831	+	1338.58i	−0.267202	+	0.462808i
$204$		0			0
$205$	−980.597	−	1698.44i	−0.334087	−	0.578656i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1072.65	−	1857.89i	−0.355009	−	0.614893i
$210$		0			0
$211$	1808.08	−	3131.69i	0.589922	−	1.02177i	−0.404321	−	0.914617i	$-0.632492\pi$
$211$	1808.08	−	3131.69i	0.589922	−	1.02177i	0.994242	−	0.107157i	$-0.0341747\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1094.35			0.347136
$216$		0			0
$217$	−1595.19			−0.499027
$218$		0			0
$219$		0			0
$220$		0			0
$221$	522.980	−	905.828i	0.159183	−	0.275713i
$222$		0			0
$223$	430.574	+	745.777i	0.129298	+	0.223950i	0.923405	−	0.383828i	$-0.125394\pi$
$223$	430.574	+	745.777i	0.129298	+	0.223950i	−0.794107	+	0.607778i	$0.792061\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1542.98	+	2672.52i	0.451150	+	0.781414i	0.998458	−	0.0555169i	$-0.0176807\pi$
$227$	1542.98	+	2672.52i	0.451150	+	0.781414i	−0.547308	+	0.836931i	$0.684347\pi$
$228$		0			0
$229$	−3095.65	+	5361.81i	−0.893301	+	1.54724i	−0.0574083	+	0.998351i	$0.518284\pi$
$229$	−3095.65	+	5361.81i	−0.893301	+	1.54724i	−0.835893	+	0.548892i	$0.815050\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−4325.48			−1.21619			−0.608093	−	0.793866i	$-0.708065\pi$
$233$	−4325.48			−1.21619			−0.608093	+	0.793866i	$0.708065\pi$
$234$		0			0
$235$	−2635.13			−0.731477
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1559.24	−	2700.68i	0.422003	−	0.730931i	−0.574132	−	0.818763i	$-0.694660\pi$
$239$	1559.24	−	2700.68i	0.422003	−	0.730931i	0.996135	+	0.0878315i	$0.0279937\pi$
$240$		0			0
$241$	−361.462	−	626.071i	−0.0966134	−	0.167339i	0.813667	−	0.581331i	$-0.197468\pi$
$241$	−361.462	−	626.071i	−0.0966134	−	0.167339i	−0.910281	+	0.413991i	$0.864134\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1278.03	+	2213.61i	0.333267	+	0.577235i
$246$		0			0
$247$	−1357.32	+	2350.94i	−0.349652	+	0.605615i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−6373.89			−1.60285			−0.801427	−	0.598093i	$-0.795925\pi$
$251$	−6373.89			−1.60285			−0.801427	+	0.598093i	$0.795925\pi$
$252$		0			0
$253$	−886.877			−0.220385
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1902.41	−	3295.07i	0.461748	−	0.799771i	−0.537300	−	0.843391i	$-0.680556\pi$
$257$	1902.41	−	3295.07i	0.461748	−	0.799771i	0.999048	+	0.0436203i	$0.0138892\pi$
$258$		0			0
$259$	−670.695	−	1161.68i	−0.160907	−	0.278699i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	729.311	+	1263.20i	0.170993	+	0.296169i	0.938767	−	0.344552i	$-0.111969\pi$
$263$	729.311	+	1263.20i	0.170993	+	0.296169i	−0.767774	+	0.640721i	$0.778636\pi$
$264$		0			0
$265$	−1859.63	+	3220.98i	−0.431080	+	0.746653i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−2240.99			−0.507940			−0.253970	−	0.967212i	$-0.581736\pi$
$269$	−2240.99			−0.507940			−0.253970	+	0.967212i	$0.581736\pi$
$270$		0			0
$271$	−3077.92			−0.689927			−0.344963	−	0.938616i	$-0.612109\pi$
$271$	−3077.92			−0.689927			−0.344963	+	0.938616i	$0.612109\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	544.128	−	942.457i	0.119317	−	0.206663i
$276$		0			0
$277$	−1545.35	−	2676.63i	−0.335203	−	0.580588i	0.648321	−	0.761367i	$-0.275471\pi$
$277$	−1545.35	−	2676.63i	−0.335203	−	0.580588i	−0.983524	+	0.180779i	$0.942138\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−2420.90	−	4193.12i	−0.513946	−	0.890180i	−0.999869	−	0.0161788i	$-0.994850\pi$
$281$	−2420.90	−	4193.12i	−0.513946	−	0.890180i	0.485923	−	0.874001i	$-0.338483\pi$
$282$		0			0
$283$	−359.725	+	623.062i	−0.0755598	+	0.130873i	−0.901330	−	0.433134i	$-0.857408\pi$
$283$	−359.725	+	623.062i	−0.0755598	+	0.130873i	0.825770	+	0.564007i	$0.190741\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	1751.62			0.360260
$288$		0			0
$289$	−4104.66			−0.835470
$290$		0			0
$291$		0			0
$292$		0			0
$293$	2625.01	−	4546.65i	0.523395	−	0.906548i	−0.476234	−	0.879319i	$-0.657998\pi$
$293$	2625.01	−	4546.65i	0.523395	−	0.906548i	0.999629	−	0.0272288i	$-0.00866827\pi$
$294$		0			0
$295$	−1769.31	−	3064.54i	−0.349198	−	0.604829i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	561.120	+	971.889i	0.108530	+	0.187979i
$300$		0			0
$301$	−488.704	+	846.460i	−0.0935829	+	0.162090i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−2784.17			−0.522692
$306$		0			0
$307$	−5823.45			−1.08261			−0.541306	−	0.840826i	$-0.682070\pi$
$307$	−5823.45			−1.08261			−0.541306	+	0.840826i	$0.682070\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−196.407	+	340.187i	−0.0358110	+	0.0620265i	−0.883375	−	0.468666i	$-0.844735\pi$
$311$	−196.407	+	340.187i	−0.0358110	+	0.0620265i	0.847564	+	0.530693i	$0.178068\pi$
$312$		0			0
$313$	−558.235	−	966.891i	−0.100809	−	0.174607i	0.811209	−	0.584756i	$-0.198810\pi$
$313$	−558.235	−	966.891i	−0.100809	−	0.174607i	−0.912018	+	0.410150i	$0.865476\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−654.340	−	1133.35i	−0.115935	−	0.200805i	0.802218	−	0.597031i	$-0.203653\pi$
$317$	−654.340	−	1133.35i	−0.115935	−	0.200805i	−0.918153	+	0.396226i	$0.870320\pi$
$318$		0			0
$319$	−2688.36	+	4656.38i	−0.471848	+	0.817264i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−2097.92			−0.361397
$324$		0			0
$325$	−1377.06			−0.235033
$326$		0			0
$327$		0			0
$328$		0			0
$329$	1176.77	−	2038.22i	0.197196	−	0.341553i
$330$		0			0
$331$	3099.55	+	5368.58i	0.514703	+	0.891492i	0.999854	+	0.0170616i	$0.00543113\pi$
$331$	3099.55	+	5368.58i	0.514703	+	0.891492i	−0.485151	+	0.874430i	$0.661236\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	3873.84	+	6709.68i	0.631792	+	1.09430i
$336$		0			0
$337$	−403.851	+	699.491i	−0.0652794	+	0.113067i	−0.896818	−	0.442400i	$-0.854127\pi$
$337$	−403.851	+	699.491i	−0.0652794	+	0.113067i	0.831538	+	0.555467i	$0.187461\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−5549.03			−0.881223
$342$		0			0
$343$	−5149.64			−0.810655
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−746.608	+	1293.16i	−0.115504	+	0.200059i	−0.917981	−	0.396624i	$-0.870182\pi$
$347$	−746.608	+	1293.16i	−0.115504	+	0.200059i	0.802477	+	0.596683i	$0.203515\pi$
$348$		0			0
$349$	249.305	+	431.809i	0.0382378	+	0.0662299i	0.884511	−	0.466519i	$-0.154492\pi$
$349$	249.305	+	431.809i	0.0382378	+	0.0662299i	−0.846273	+	0.532749i	$0.821159\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	6002.66	+	10396.9i	0.905069	+	1.56762i	0.820825	+	0.571180i	$0.193514\pi$
$353$	6002.66	+	10396.9i	0.905069	+	1.56762i	0.0842436	+	0.996445i	$0.473153\pi$
$354$		0			0
$355$	−3411.27	+	5908.49i	−0.510004	+	0.883352i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	6842.79			1.00599			0.502993	−	0.864291i	$-0.332232\pi$
$359$	6842.79			1.00599			0.502993	+	0.864291i	$0.332232\pi$
$360$		0			0
$361$	−1414.17			−0.206177
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−5172.26	+	8958.61i	−0.741721	+	1.28470i
$366$		0			0
$367$	56.0606	+	97.0999i	0.00797368	+	0.0138108i	0.869985	−	0.493079i	$-0.164129\pi$
$367$	56.0606	+	97.0999i	0.00797368	+	0.0138108i	−0.862011	+	0.506890i	$0.830795\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−1660.91	−	2876.78i	−0.232426	−	0.402574i
$372$		0			0
$373$	−988.085	+	1711.41i	−0.137161	+	0.237570i	−0.926421	−	0.376489i	$-0.877131\pi$
$373$	−988.085	+	1711.41i	−0.137161	+	0.237570i	0.789260	+	0.614059i	$0.210464\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	6803.63			0.929455
$378$		0			0
$379$	−669.717			−0.0907679			−0.0453840	−	0.998970i	$-0.514451\pi$
$379$	−669.717			−0.0907679			−0.0453840	+	0.998970i	$0.514451\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	5620.25	−	9734.57i	0.749821	−	1.29873i	−0.198087	−	0.980185i	$-0.563473\pi$
$383$	5620.25	−	9734.57i	0.749821	−	1.29873i	0.947908	−	0.318544i	$-0.103194\pi$
$384$		0			0
$385$	−1136.93	−	1969.22i	−0.150502	−	0.260677i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	1882.99	+	3261.43i	0.245427	+	0.425092i	0.962252	−	0.272161i	$-0.0877385\pi$
$389$	1882.99	+	3261.43i	0.245427	+	0.425092i	−0.716825	+	0.697254i	$0.754405\pi$
$390$		0			0
$391$	−433.644	+	751.093i	−0.0560878	+	0.0971468i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−9959.22			−1.26861
$396$		0			0
$397$	−2005.69			−0.253558			−0.126779	−	0.991931i	$-0.540464\pi$
$397$	−2005.69			−0.253558			−0.126779	+	0.991931i	$0.540464\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	2777.83	−	4811.34i	0.345930	−	0.599169i	−0.639592	−	0.768715i	$-0.720897\pi$
$401$	2777.83	−	4811.34i	0.345930	−	0.599169i	0.985522	+	0.169546i	$0.0542300\pi$
$402$		0			0
$403$	3510.83	+	6080.93i	0.433962	+	0.751645i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−2333.08	−	4041.01i	−0.284143	−	0.492150i
$408$		0			0
$409$	7438.24	−	12883.4i	0.899260	−	1.55756i	0.0708190	−	0.997489i	$-0.477439\pi$
$409$	7438.24	−	12883.4i	0.899260	−	1.55756i	0.828441	−	0.560076i	$-0.189228\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	3160.48			0.376555
$414$		0			0
$415$	5525.59			0.653592
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−4749.82	+	8226.92i	−0.553803	+	0.959216i	0.444192	+	0.895932i	$0.353491\pi$
$419$	−4749.82	+	8226.92i	−0.553803	+	0.959216i	−0.997996	+	0.0632842i	$0.979843\pi$
$420$		0			0
$421$	2998.70	+	5193.90i	0.347144	+	0.601271i	0.985741	−	0.168270i	$-0.0538182\pi$
$421$	2998.70	+	5193.90i	0.347144	+	0.601271i	−0.638597	+	0.769541i	$0.720485\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−532.109	−	921.640i	−0.0607320	−	0.105191i
$426$		0			0
$427$	1243.32	−	2153.50i	0.140910	−	0.244064i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	15607.1			1.74424			0.872121	−	0.489291i	$-0.162744\pi$
$431$	15607.1			1.74424			0.872121	+	0.489291i	$0.162744\pi$
$432$		0			0
$433$	−1204.46			−0.133678			−0.0668390	−	0.997764i	$-0.521291\pi$
$433$	−1204.46			−0.133678			−0.0668390	+	0.997764i	$0.521291\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1125.46	−	1949.35i	0.123199	−	0.213387i
$438$		0			0
$439$	−8492.29	−	14709.1i	−0.923269	−	1.59915i	−0.794322	−	0.607497i	$-0.792174\pi$
$439$	−8492.29	−	14709.1i	−0.923269	−	1.59915i	−0.128947	−	0.991651i	$-0.541160\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−3584.85	−	6209.14i	−0.384473	−	0.665926i	0.607223	−	0.794531i	$-0.292283\pi$
$443$	−3584.85	−	6209.14i	−0.384473	−	0.665926i	−0.991696	+	0.128605i	$0.958950\pi$
$444$		0			0
$445$	1064.86	−	1844.39i	0.113437	−	0.196478i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−4648.71			−0.488611			−0.244305	−	0.969698i	$-0.578560\pi$
$449$	−4648.71			−0.488611			−0.244305	+	0.969698i	$0.578560\pi$
$450$		0			0
$451$	6093.16			0.636177
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1438.65	+	2491.82i	−0.148231	+	0.256744i
$456$		0			0
$457$	−472.024	−	817.569i	−0.0483158	−	0.0836855i	0.840856	−	0.541259i	$-0.182052\pi$
$457$	−472.024	−	817.569i	−0.0483158	−	0.0836855i	−0.889172	+	0.457573i	$0.848719\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	2549.51	+	4415.88i	0.257576	+	0.446134i	0.965592	−	0.260062i	$-0.0837430\pi$
$461$	2549.51	+	4415.88i	0.257576	+	0.446134i	−0.708016	+	0.706196i	$0.750410\pi$
$462$		0			0
$463$	458.859	−	794.767i	0.0460583	−	0.0797753i	−0.842077	−	0.539357i	$-0.818667\pi$
$463$	458.859	−	794.767i	0.0460583	−	0.0797753i	0.888135	+	0.459582i	$0.152001\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−19073.2			−1.88994			−0.944969	−	0.327160i	$-0.893908\pi$
$467$	−19073.2			−1.88994			−0.944969	+	0.327160i	$0.893908\pi$
$468$		0			0
$469$	−6919.74			−0.681287
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−1700.00	+	2944.49i	−0.165256	+	0.286232i
$474$		0			0
$475$	1381.01	+	2391.98i	0.133400	+	0.231056i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−9163.06	−	15870.9i	−0.874052	−	1.51390i	−0.857769	−	0.514036i	$-0.828150\pi$
$479$	−9163.06	−	15870.9i	−0.874052	−	1.51390i	−0.0162836	−	0.999867i	$-0.505183\pi$
$480$		0			0
$481$	−2952.24	+	5113.42i	−0.279855	+	0.484724i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	12030.9			1.12638
$486$		0			0
$487$	18087.7			1.68302			0.841512	−	0.540238i	$-0.181666\pi$
$487$	18087.7			1.68302			0.841512	+	0.540238i	$0.181666\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	1930.14	−	3343.10i	0.177405	−	0.307275i	−0.763586	−	0.645706i	$-0.776563\pi$
$491$	1930.14	−	3343.10i	0.177405	−	0.307275i	0.940991	+	0.338432i	$0.109896\pi$
$492$		0			0
$493$	2628.98	+	4553.53i	0.240169	+	0.415985i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−3046.73	−	5277.09i	−0.274979	−	0.476278i
$498$		0			0
$499$	7222.33	−	12509.4i	0.647928	−	1.12224i	−0.335689	−	0.941973i	$-0.608969\pi$
$499$	7222.33	−	12509.4i	0.647928	−	1.12224i	0.983617	−	0.180271i	$-0.0576974\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−2074.88			−0.183925			−0.0919625	−	0.995762i	$-0.529314\pi$
$503$	−2074.88			−0.183925			−0.0919625	+	0.995762i	$0.529314\pi$
$504$		0			0
$505$	15765.4			1.38921
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−2489.86	+	4312.56i	−0.216819	+	0.375542i	−0.953834	−	0.300335i	$-0.902902\pi$
$509$	−2489.86	+	4312.56i	−0.216819	+	0.375542i	0.737015	+	0.675877i	$0.236235\pi$
$510$		0			0
$511$	−4619.54	−	8001.27i	−0.399914	−	0.692672i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	3997.67	+	6924.17i	0.342055	+	0.592457i
$516$		0			0
$517$	4093.50	−	7090.15i	0.348224	−	0.603142i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	8212.49			0.690587			0.345293	−	0.938495i	$-0.387779\pi$
$521$	8212.49			0.690587			0.345293	+	0.938495i	$0.387779\pi$
$522$		0			0
$523$	12630.6			1.05602			0.528009	−	0.849239i	$-0.322939\pi$
$523$	12630.6			1.05602			0.528009	+	0.849239i	$0.322939\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−2713.23	+	4699.46i	−0.224270	+	0.388447i
$528$		0			0
$529$	5618.23	+	9731.06i	0.461760	+	0.799791i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−3855.09	−	6677.22i	−0.313288	−	0.542631i
$534$		0			0
$535$	−6408.82	+	11100.4i	−0.517902	+	0.897032i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−7941.32			−0.634614
$540$		0			0
$541$	22132.0			1.75883			0.879416	−	0.476054i	$-0.157933\pi$
$541$	22132.0			1.75883			0.879416	+	0.476054i	$0.157933\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−6901.69	+	11954.1i	−0.542451	+	0.939553i
$546$		0			0
$547$	2271.34	+	3934.07i	0.177542	+	0.307511i	0.941038	−	0.338301i	$-0.109852\pi$
$547$	2271.34	+	3934.07i	0.177542	+	0.307511i	−0.763496	+	0.645812i	$0.776519\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−6823.13	−	11818.0i	−0.527541	−	0.913728i
$552$		0			0
$553$	4447.48	−	7703.26i	0.342000	−	0.592361i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	22330.0			1.69866			0.849330	−	0.527862i	$-0.177006\pi$
$557$	22330.0			1.69866			0.849330	+	0.527862i	$0.177006\pi$
$558$		0			0
$559$	4302.31			0.325525
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−7093.39	+	12286.1i	−0.530996	+	0.919712i	0.468350	+	0.883543i	$0.344849\pi$
$563$	−7093.39	+	12286.1i	−0.530996	+	0.919712i	−0.999346	+	0.0361685i	$0.988485\pi$
$564$		0			0
$565$	42.1985	+	73.0899i	0.00314213	+	0.00544233i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	5469.16	+	9472.86i	0.402951	+	0.697932i	0.994081	−	0.108645i	$-0.0346511\pi$
$569$	5469.16	+	9472.86i	0.402951	+	0.697932i	−0.591130	+	0.806577i	$0.701318\pi$
$570$		0			0
$571$	6070.81	−	10514.9i	0.444931	−	0.770642i	−0.553117	−	0.833104i	$-0.686562\pi$
$571$	6070.81	−	10514.9i	0.444931	−	0.770642i	0.998047	+	0.0624614i	$0.0198950\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1141.83			0.0828132
$576$		0			0
$577$	−14080.0			−1.01587			−0.507936	−	0.861395i	$-0.669592\pi$
$577$	−14080.0			−1.01587			−0.507936	+	0.861395i	$0.669592\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−2467.56	+	4273.93i	−0.176199	+	0.305185i
$582$		0			0
$583$	−5777.62	−	10007.1i	−0.410437	−	0.710898i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−12597.5	−	21819.5i	−0.885781	−	1.53422i	−0.844816	−	0.535057i	$-0.820290\pi$
$587$	−12597.5	−	21819.5i	−0.885781	−	1.53422i	−0.0409650	−	0.999161i	$-0.513043\pi$
$588$		0			0
$589$	7041.79	−	12196.7i	0.492618	−	0.853239i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	24086.1			1.66796			0.833978	−	0.551798i	$-0.186058\pi$
$593$	24086.1			1.66796			0.833978	+	0.551798i	$0.186058\pi$
$594$		0			0
$595$	−2223.64			−0.153210
$596$		0			0
$597$		0			0
$598$		0			0
$599$	13917.4	−	24105.7i	0.949332	−	1.64429i	0.202495	−	0.979283i	$-0.435095\pi$
$599$	13917.4	−	24105.7i	0.949332	−	1.64429i	0.746836	−	0.665008i	$-0.231572\pi$
$600$		0			0
$601$	8056.88	+	13954.9i	0.546833	+	0.947143i	0.998489	+	0.0549509i	$0.0175002\pi$
$601$	8056.88	+	13954.9i	0.546833	+	0.947143i	−0.451656	+	0.892192i	$0.649166\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2272.71	+	3936.44i	0.152725	+	0.264528i
$606$		0			0
$607$	1462.99	−	2533.97i	0.0978268	−	0.169441i	−0.812958	−	0.582322i	$-0.802144\pi$
$607$	1462.99	−	2533.97i	0.0978268	−	0.169441i	0.910785	+	0.412881i	$0.135478\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−10359.7			−0.685939
$612$		0			0
$613$	−11350.9			−0.747896			−0.373948	−	0.927450i	$-0.621996\pi$
$613$	−11350.9			−0.747896			−0.373948	+	0.927450i	$0.621996\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−7907.60	+	13696.4i	−0.515961	+	0.893671i	0.483867	+	0.875142i	$0.339232\pi$
$617$	−7907.60	+	13696.4i	−0.515961	+	0.893671i	−0.999828	+	0.0185297i	$0.994101\pi$
$618$		0			0
$619$	6164.41	+	10677.1i	0.400272	+	0.693292i	0.993759	−	0.111552i	$-0.0355823\pi$
$619$	6164.41	+	10677.1i	0.400272	+	0.693292i	−0.593486	+	0.804844i	$0.702249\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	951.068	+	1647.30i	0.0611617	+	0.105935i
$624$		0			0
$625$	4772.50	−	8266.21i	0.305440	−	0.529037i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−4563.08			−0.289256
$630$		0			0
$631$	25545.6			1.61165			0.805826	−	0.592152i	$-0.201722\pi$
$631$	25545.6			1.61165			0.805826	+	0.592152i	$0.201722\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−9713.64	+	16824.5i	−0.607046	+	1.05143i
$636$		0			0
$637$	5024.41	+	8702.54i	0.312519	+	0.541298i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	6361.66	+	11018.7i	0.391998	+	0.678960i	0.992713	−	0.120503i	$-0.0384508\pi$
$641$	6361.66	+	11018.7i	0.391998	+	0.678960i	−0.600715	+	0.799463i	$0.705118\pi$
$642$		0			0
$643$	7088.18	−	12277.1i	0.434729	−	0.752972i	−0.562545	−	0.826767i	$-0.690178\pi$
$643$	7088.18	−	12277.1i	0.434729	−	0.752972i	0.997273	+	0.0737946i	$0.0235109\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	25409.1			1.54395			0.771975	−	0.635653i	$-0.219269\pi$
$647$	25409.1			1.54395			0.771975	+	0.635653i	$0.219269\pi$
$648$		0			0
$649$	10994.0			0.664952
$650$		0			0
$651$		0			0
$652$		0			0
$653$	6748.22	−	11688.3i	0.404408	−	0.700455i	−0.589845	−	0.807517i	$-0.700811\pi$
$653$	6748.22	−	11688.3i	0.404408	−	0.700455i	0.994252	+	0.107062i	$0.0341443\pi$
$654$		0			0
$655$	−8302.32	−	14380.0i	−0.495265	−	0.857824i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−6047.54	−	10474.7i	−0.357479	−	0.619172i	0.630060	−	0.776547i	$-0.283030\pi$
$659$	−6047.54	−	10474.7i	−0.357479	−	0.619172i	−0.987539	+	0.157374i	$0.949697\pi$
$660$		0			0
$661$	5002.37	−	8664.35i	0.294356	−	0.509840i	−0.680479	−	0.732768i	$-0.738228\pi$
$661$	5002.37	−	8664.35i	0.294356	−	0.509840i	0.974835	+	0.222928i	$0.0715614\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	5771.11			0.336533
$666$		0			0
$667$	−5641.42			−0.327491
$668$		0			0
$669$		0			0
$670$		0			0
$671$	4325.02	−	7491.15i	0.248831	−	0.430988i
$672$		0			0
$673$	−4955.53	−	8583.23i	−0.283836	−	0.491618i	0.688490	−	0.725246i	$-0.258274\pi$
$673$	−4955.53	−	8583.23i	−0.283836	−	0.491618i	−0.972326	+	0.233627i	$0.924940\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	9424.73	+	16324.1i	0.535040	+	0.926716i	0.999161	+	0.0409446i	$0.0130367\pi$
$677$	9424.73	+	16324.1i	0.535040	+	0.926716i	−0.464122	+	0.885771i	$0.653630\pi$
$678$		0			0
$679$	−5372.63	+	9305.67i	−0.303656	+	0.525948i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−17224.5			−0.964973			−0.482486	−	0.875903i	$-0.660266\pi$
$683$	−17224.5			−0.964973			−0.482486	+	0.875903i	$0.660266\pi$
$684$		0			0
$685$	11329.4			0.631932
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−7310.91	+	12662.9i	−0.404243	+	0.700170i
$690$		0			0
$691$	3977.32	+	6888.91i	0.218964	+	0.379257i	0.954492	−	0.298238i	$-0.0963989\pi$
$691$	3977.32	+	6888.91i	0.218964	+	0.379257i	−0.735527	+	0.677495i	$0.763066\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	3430.71	+	5942.16i	0.187243	+	0.324315i
$696$		0			0
$697$	2979.29	−	5160.28i	0.161906	−	0.280430i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	3771.49			0.203206			0.101603	−	0.994825i	$-0.467603\pi$
$701$	3771.49			0.203206			0.101603	+	0.994825i	$0.467603\pi$
$702$		0			0
$703$	11842.8			0.635362
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−7040.35	+	12194.2i	−0.374512	+	0.648673i
$708$		0			0
$709$	−141.333	−	244.795i	−0.00748640	−	0.0129668i	0.862258	−	0.506469i	$-0.169050\pi$
$709$	−141.333	−	244.795i	−0.00748640	−	0.0129668i	−0.869744	+	0.493503i	$0.835716\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−2911.10	−	5042.18i	−0.152906	−	0.264840i
$714$		0			0
$715$	−5004.49	+	8668.04i	−0.261759	+	0.453379i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	8779.02			0.455358			0.227679	−	0.973736i	$-0.426886\pi$
$719$	8779.02			0.455358			0.227679	+	0.973736i	$0.426886\pi$
$720$		0			0
$721$	−7140.94			−0.368852
$722$		0			0
$723$		0			0
$724$		0			0
$725$	3461.19	−	5994.96i	0.177304	−	0.307100i
$726$		0			0
$727$	1242.74	+	2152.49i	0.0633985	+	0.109809i	0.895982	−	0.444089i	$-0.146473\pi$
$727$	1242.74	+	2152.49i	0.0633985	+	0.109809i	−0.832584	+	0.553899i	$0.813139\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1662.45	+	2879.45i	0.0841150	+	0.145691i
$732$		0			0
$733$	5089.78	−	8815.75i	0.256474	−	0.444225i	−0.708821	−	0.705388i	$-0.750773\pi$
$733$	5089.78	−	8815.75i	0.256474	−	0.444225i	0.965295	+	0.261163i	$0.0841060\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−24071.0			−1.20307
$738$		0			0
$739$	3318.26			0.165175			0.0825874	−	0.996584i	$-0.473682\pi$
$739$	3318.26			0.165175			0.0825874	+	0.996584i	$0.473682\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	1333.34	−	2309.41i	0.0658350	−	0.114030i	−0.831229	−	0.555930i	$-0.812362\pi$
$743$	1333.34	−	2309.41i	0.0658350	−	0.114030i	0.897064	+	0.441900i	$0.145696\pi$
$744$		0			0
$745$	2412.26	+	4178.16i	0.118629	+	0.205471i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−5723.96	−	9914.18i	−0.279237	−	0.483653i
$750$		0			0
$751$	−9768.80	+	16920.1i	−0.474659	+	0.822133i	−0.999579	−	0.0290186i	$-0.990762\pi$
$751$	−9768.80	+	16920.1i	−0.474659	+	0.822133i	0.524920	+	0.851151i	$0.324095\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−10797.9			−0.520498
$756$		0			0
$757$	−30037.2			−1.44217			−0.721084	−	0.692848i	$-0.756356\pi$
$757$	−30037.2			−1.44217			−0.721084	+	0.692848i	$0.756356\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	20577.1	−	35640.6i	0.980184	−	1.69773i	0.318544	−	0.947908i	$-0.396806\pi$
$761$	20577.1	−	35640.6i	0.980184	−	1.69773i	0.661641	−	0.749821i	$-0.269860\pi$
$762$		0			0
$763$	−6164.16	−	10676.6i	−0.292474	−	0.506579i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−6955.84	−	12047.9i	−0.327459	−	0.567175i
$768$		0			0
$769$	−10939.1	+	18947.0i	−0.512969	+	0.888488i	0.486918	+	0.873448i	$0.338121\pi$
$769$	−10939.1	+	18947.0i	−0.512969	+	0.888488i	−0.999887	+	0.0150403i	$0.995212\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−23321.1			−1.08512			−0.542561	−	0.840016i	$-0.682545\pi$
$773$	−23321.1			−1.08512			−0.542561	+	0.840016i	$0.682545\pi$
$774$		0			0
$775$	7144.23			0.331133
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−7732.29	+	13392.7i	−0.355633	+	0.615974i
$780$		0			0
$781$	−10598.3	−	18356.9i	−0.485581	−	0.841050i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	6407.81	+	11098.6i	0.291343	+	0.504621i
$786$		0			0
$787$	−8155.78	+	14126.2i	−0.369406	+	0.639829i	−0.989473	−	0.144719i	$-0.953772\pi$
$787$	−8155.78	+	14126.2i	−0.369406	+	0.639829i	0.620067	+	0.784549i	$0.287105\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−75.3780			−0.00338829
$792$		0			0
$793$	−10945.6			−0.490152
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−11480.7	+	19885.2i	−0.510249	+	0.883778i	0.489680	+	0.871902i	$0.337114\pi$
$797$	−11480.7	+	19885.2i	−0.510249	+	0.883778i	−0.999929	+	0.0118755i	$0.996220\pi$
$798$		0			0
$799$	−4003.08	−	6933.54i	−0.177245	−	0.306997i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−16069.5	−	27833.2i	−0.706202	−	1.22318i
$804$		0			0
$805$	1192.90	−	2066.17i	0.0522289	−	0.0904630i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	30987.4			1.34667			0.673337	−	0.739336i	$-0.264860\pi$
$809$	30987.4			1.34667			0.673337	+	0.739336i	$0.264860\pi$
$810$		0			0
$811$	−40596.9			−1.75777			−0.878884	−	0.477036i	$-0.841711\pi$
$811$	−40596.9			−1.75777			−0.878884	+	0.477036i	$0.841711\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	17765.4	−	30770.5i	0.763551	−	1.32251i
$816$		0			0
$817$	−4314.65	−	7473.19i	−0.184762	−	0.320017i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	3936.73	+	6818.62i	0.167348	+	0.289855i	0.937487	−	0.348021i	$-0.113146\pi$
$821$	3936.73	+	6818.62i	0.167348	+	0.289855i	−0.770139	+	0.637877i	$0.779813\pi$
$822$		0			0
$823$	−9886.34	+	17123.6i	−0.418732	+	0.725264i	−0.995812	−	0.0914228i	$-0.970859\pi$
$823$	−9886.34	+	17123.6i	−0.418732	+	0.725264i	0.577081	+	0.816687i	$0.304192\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−33412.4			−1.40492			−0.702458	−	0.711726i	$-0.747914\pi$
$827$	−33412.4			−1.40492			−0.702458	+	0.711726i	$0.747914\pi$
$828$		0			0
$829$	681.873			0.0285675			0.0142837	−	0.999898i	$-0.495453\pi$
$829$	681.873			0.0285675			0.0142837	+	0.999898i	$0.495453\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−3882.96	+	6725.48i	−0.161508	+	0.279741i
$834$		0			0
$835$	9586.58	+	16604.4i	0.397314	+	0.688168i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	12617.3	+	21853.8i	0.519187	+	0.899258i	0.999751	+	0.0222988i	$0.00709851\pi$
$839$	12617.3	+	21853.8i	0.519187	+	0.899258i	−0.480564	+	0.876959i	$0.659568\pi$
$840$		0			0
$841$	−4906.16	+	8497.72i	−0.201163	+	0.348424i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−7893.91			−0.321371
$846$		0			0
$847$	−4059.68			−0.164690
$848$		0			0
$849$		0			0
$850$		0			0
$851$	2447.93	−	4239.94i	0.0986064	−	0.170791i
$852$		0			0
$853$	7988.37	+	13836.3i	0.320652	+	0.555386i	0.980623	−	0.195905i	$-0.0627645\pi$
$853$	7988.37	+	13836.3i	0.320652	+	0.555386i	−0.659970	+	0.751292i	$0.729431\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−9807.58	−	16987.2i	−0.390923	−	0.677098i	0.601649	−	0.798761i	$-0.294511\pi$
$857$	−9807.58	−	16987.2i	−0.390923	−	0.677098i	−0.992571	+	0.121663i	$0.961177\pi$
$858$		0			0
$859$	7876.81	−	13643.0i	0.312867	−	0.541902i	−0.666114	−	0.745850i	$-0.732044\pi$
$859$	7876.81	−	13643.0i	0.312867	−	0.541902i	0.978982	+	0.203947i	$0.0653771\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	34943.9			1.37834			0.689169	−	0.724601i	$-0.257976\pi$
$863$	34943.9			1.37834			0.689169	+	0.724601i	$0.257976\pi$
$864$		0			0
$865$	18951.7			0.744946
$866$		0			0
$867$		0			0
$868$		0			0
$869$	15471.0	−	26796.5i	0.603932	−	1.04604i
$870$		0			0
$871$	15229.5	+	26378.3i	0.592459	+	1.02617i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	6351.95	+	11001.9i	0.245412	+	0.425065i
$876$		0			0
$877$	−12989.6	+	22498.7i	−0.500147	+	0.866280i	0.499853	+	0.866110i	$0.333387\pi$
$877$	−12989.6	+	22498.7i	−0.500147	+	0.866280i	−1.00000		0.000169727i	$0.999946\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−36174.8			−1.38338			−0.691692	−	0.722193i	$-0.743134\pi$
$881$	−36174.8			−1.38338			−0.691692	+	0.722193i	$0.743134\pi$
$882$		0			0
$883$	−33756.1			−1.28651			−0.643253	−	0.765654i	$-0.722416\pi$
$883$	−33756.1			−1.28651			−0.643253	+	0.765654i	$0.722416\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	15697.2	−	27188.4i	0.594206	−	1.02920i	−0.399452	−	0.916754i	$-0.630800\pi$
$887$	15697.2	−	27188.4i	0.594206	−	1.02920i	0.993658	−	0.112442i	$-0.0358671\pi$
$888$		0			0
$889$	−8675.62	−	15026.6i	−0.327301	−	0.566903i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	10389.4	+	17995.0i	0.389326	+	0.674332i
$894$		0			0
$895$	−14703.5	+	25467.2i	−0.549143	+	0.951144i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−35297.4			−1.30949
$900$		0			0
$901$	−11300.0			−0.417822
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−176.247	+	305.269i	−0.00647366	+	0.0112127i
$906$		0			0
$907$	−23150.0	−	40097.0i	−0.847502	−	1.46792i	−0.883431	−	0.468562i	$-0.844772\pi$
$907$	−23150.0	−	40097.0i	−0.847502	−	1.46792i	0.0359285	−	0.999354i	$-0.488561\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	26597.6	+	46068.5i	0.967310	+	1.67543i	0.703277	+	0.710916i	$0.251720\pi$
$911$	26597.6	+	46068.5i	0.967310	+	1.67543i	0.264033	+	0.964514i	$0.414947\pi$
$912$		0			0
$913$	−8583.62	+	14867.3i	−0.311146	+	0.538921i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	14830.2			0.534065
$918$		0			0
$919$	−32346.8			−1.16107			−0.580534	−	0.814236i	$-0.697156\pi$
$919$	−32346.8			−1.16107			−0.580534	+	0.814236i	$0.697156\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−13411.0	+	23228.5i	−0.478253	+	0.828358i
$924$		0			0
$925$	3003.77	+	5202.68i	0.106771	+	0.184933i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	14478.2	+	25077.0i	0.511319	+	0.885630i	0.999914	+	0.0131195i	$0.00417619\pi$
$929$	14478.2	+	25077.0i	0.511319	+	0.885630i	−0.488595	+	0.872511i	$0.662490\pi$
$930$		0			0
$931$	10077.6	−	17455.0i	0.354759	−	0.614461i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−7735.12			−0.270551
$936$		0			0
$937$	−20401.2			−0.711290			−0.355645	−	0.934621i	$-0.615739\pi$
$937$	−20401.2			−0.711290			−0.355645	+	0.934621i	$0.615739\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−6402.91	+	11090.2i	−0.221816	+	0.384196i	−0.955359	−	0.295446i	$-0.904532\pi$
$941$	−6402.91	+	11090.2i	−0.221816	+	0.384196i	0.733544	+	0.679642i	$0.237865\pi$
$942$		0			0
$943$	3196.56	+	5536.61i	0.110386	+	0.191195i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	23582.3	+	40845.8i	0.809211	+	1.40159i	0.913411	+	0.407038i	$0.133438\pi$
$947$	23582.3	+	40845.8i	0.809211	+	1.40159i	−0.104200	+	0.994556i	$0.533228\pi$
$948$		0			0
$949$	−20334.1	+	35219.7i	−0.695545	+	1.20472i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−22392.5			−0.761138			−0.380569	−	0.924752i	$-0.624272\pi$
$953$	−22392.5			−0.761138			−0.380569	+	0.924752i	$0.624272\pi$
$954$		0			0
$955$	−39622.1			−1.34256
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−5059.35	+	8763.05i	−0.170360	+	0.295072i
$960$		0			0
$961$	−3318.75	−	5748.25i	−0.111401	−	0.192953i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−17916.9	−	31033.0i	−0.597685	−	1.03522i
$966$		0			0
$967$	12684.4	−	21970.1i	0.421824	−	0.730621i	−0.574294	−	0.818649i	$-0.694723\pi$
$967$	12684.4	−	21970.1i	0.421824	−	0.730621i	0.996118	+	0.0880286i	$0.0280567\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	19705.7			0.651271			0.325636	−	0.945495i	$-0.394422\pi$
$971$	19705.7			0.651271			0.325636	+	0.945495i	$0.394422\pi$
$972$		0			0
$973$	−6128.18			−0.201912
$974$		0			0
$975$		0			0
$976$		0			0
$977$	20927.3	−	36247.1i	0.685285	−	1.18695i	−0.288063	−	0.957612i	$-0.593011\pi$
$977$	20927.3	−	36247.1i	0.685285	−	1.18695i	0.973347	−	0.229336i	$-0.0736555\pi$
$978$		0			0
$979$	3308.38	+	5730.28i	0.108004	+	0.187069i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	8754.55	+	15163.3i	0.284056	+	0.491999i	0.972380	−	0.233404i	$-0.0749866\pi$
$983$	8754.55	+	15163.3i	0.284056	+	0.491999i	−0.688324	+	0.725403i	$0.741653\pi$
$984$		0			0
$985$	5017.58	−	8690.70i	0.162308	−	0.281126i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−3567.39			−0.114698
$990$		0			0
$991$	32528.3			1.04268			0.521340	−	0.853349i	$-0.325432\pi$
$991$	32528.3			1.04268			0.521340	+	0.853349i	$0.325432\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	5046.35	−	8740.53i	0.160784	−	0.278486i
$996$		0			0
$997$	−8975.79	−	15546.5i	−0.285121	−	0.493845i	0.687517	−	0.726168i	$-0.258701\pi$
$997$	−8975.79	−	15546.5i	−0.285121	−	0.493845i	−0.972639	+	0.232323i	$0.925367\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.4.i.q.433.1		4
3.2	odd	2		648.4.i.p.433.2		4
9.2	odd	6		648.4.i.p.217.2		4
9.4	even	3		648.4.a.d.1.2	✓	2
9.5	odd	6		648.4.a.e.1.1	yes	2
9.7	even	3	inner	648.4.i.q.217.1		4
36.23	even	6		1296.4.a.p.1.1		2
36.31	odd	6		1296.4.a.n.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.4.a.d.1.2	✓	2	9.4	even	3
648.4.a.e.1.1	yes	2	9.5	odd	6
648.4.i.p.217.2		4	9.2	odd	6
648.4.i.p.433.2		4	3.2	odd	2
648.4.i.q.217.1		4	9.7	even	3	inner
648.4.i.q.433.1		4	1.1	even	1	trivial
1296.4.a.n.1.2		2	36.31	odd	6
1296.4.a.p.1.1		2	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)

\(f(q)\)	\(=\)	\(q+(-4.67891 + 8.10411i) q^{5} +(-4.17891 - 7.23808i) q^{7} +O(q^{10})\)	\(q+(-4.67891 + 8.10411i) q^{5} +(-4.17891 - 7.23808i) q^{7} +(-14.5367 - 25.1783i) q^{11} +(-18.3945 + 31.8603i) q^{13} -28.4313 q^{17} +73.7891 q^{19} +(15.2524 - 26.4179i) q^{23} +(18.7156 + 32.4164i) q^{25} +(-92.4680 - 160.159i) q^{29} +(95.4313 - 165.292i) q^{31} +78.2109 q^{35} +160.495 q^{37} +(-104.789 + 181.500i) q^{41} +(-58.4727 - 101.278i) q^{43} +(140.799 + 243.870i) q^{47} +(136.573 - 236.552i) q^{49} +397.450 q^{53} +272.064 q^{55} +(-189.073 + 327.485i) q^{59} +(148.762 + 257.663i) q^{61} +(-172.133 - 298.143i) q^{65} +(413.968 - 717.014i) q^{67} +729.073 q^{71} +1105.44 q^{73} +(-121.495 + 210.436i) q^{77} +(532.134 + 921.683i) q^{79} +(-295.239 - 511.369i) q^{83} +(133.027 - 230.410i) q^{85} -227.588 q^{89} +307.476 q^{91} +(-345.252 + 597.995i) q^{95} +(-642.827 - 1113.41i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} + 6 q^{7}+O(q^{10}) \) 4 * q + 4 * q^5 + 6 * q^7	\( 4 q + 4 q^{5} + 6 q^{7} + 10 q^{11} + 40 q^{13} + 68 q^{17} + 68 q^{19} - 98 q^{23} - 16 q^{25} - 120 q^{29} + 200 q^{31} + 540 q^{35} + 960 q^{37} - 192 q^{41} + 334 q^{43} - 300 q^{47} + 410 q^{49} + 136 q^{53} + 1588 q^{55} - 620 q^{59} - 200 q^{61} - 1370 q^{65} + 1406 q^{67} + 2780 q^{71} + 3604 q^{73} - 804 q^{77} + 334 q^{79} + 500 q^{83} + 1100 q^{85} + 180 q^{89} + 2820 q^{91} - 1222 q^{95} + 200 q^{97}+O(q^{100}) \) 4 * q + 4 * q^5 + 6 * q^7 + 10 * q^11 + 40 * q^13 + 68 * q^17 + 68 * q^19 - 98 * q^23 - 16 * q^25 - 120 * q^29 + 200 * q^31 + 540 * q^35 + 960 * q^37 - 192 * q^41 + 334 * q^43 - 300 * q^47 + 410 * q^49 + 136 * q^53 + 1588 * q^55 - 620 * q^59 - 200 * q^61 - 1370 * q^65 + 1406 * q^67 + 2780 * q^71 + 3604 * q^73 - 804 * q^77 + 334 * q^79 + 500 * q^83 + 1100 * q^85 + 180 * q^89 + 2820 * q^91 - 1222 * q^95 + 200 * q^97

Introduction

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