LMFDB - Embedded newform 648.4.i.u.217.1

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:	$8$
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			217.1
Root			$1.30421 + 0.752986i$ of defining polynomial
Character	$\chi$	$=$	648.217
Dual form			648.4.i.u.433.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-8.99236 - 15.5752i) q^{5} +(-3.64155 + 6.30734i) q^{7} +O(q^{10})$	$q+(-8.99236 - 15.5752i) q^{5} +(-3.64155 + 6.30734i) q^{7} +(2.30734 - 3.99643i) q^{11} +(-14.9301 - 25.8597i) q^{13} -67.9721 q^{17} +111.462 q^{19} +(-109.253 - 189.232i) q^{23} +(-99.2250 + 171.863i) q^{25} +(17.1357 - 29.6798i) q^{29} +(38.8523 + 67.2942i) q^{31} +130.984 q^{35} -347.228 q^{37} +(117.273 + 203.123i) q^{41} +(-26.6899 + 46.2283i) q^{43} +(-192.701 + 333.768i) q^{47} +(144.978 + 251.110i) q^{49} +461.661 q^{53} -82.9938 q^{55} +(3.58164 + 6.20358i) q^{59} +(-208.054 + 360.361i) q^{61} +(-268.514 + 465.079i) q^{65} +(434.637 + 752.814i) q^{67} +585.943 q^{71} -733.324 q^{73} +(16.8046 + 29.1064i) q^{77} +(-585.448 + 1014.03i) q^{79} +(-33.7182 + 58.4016i) q^{83} +(611.229 + 1058.68i) q^{85} +965.886 q^{89} +217.475 q^{91} +(-1002.31 - 1736.05i) q^{95} +(-0.716493 + 1.24100i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/648\mathbb{Z}\right)^\times$.

$n$	$325$	$487$	$569$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$5$	−8.99236	−	15.5752i	−0.804301	−	1.39309i	−0.916762	−	0.399434i	$-0.869207\pi$
$5$	−8.99236	−	15.5752i	−0.804301	−	1.39309i	0.112461	−	0.993656i	$-0.464127\pi$
$6$		0			0
$7$	−3.64155	+	6.30734i	−0.196625	+	0.340564i	−0.947432	−	0.319957i	$-0.896331\pi$
$7$	−3.64155	+	6.30734i	−0.196625	+	0.340564i	0.750807	+	0.660522i	$0.229665\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.30734	−	3.99643i	0.0632445	−	0.109543i	−0.832669	−	0.553770i	$-0.813189\pi$
$11$	2.30734	−	3.99643i	0.0632445	−	0.109543i	0.895914	+	0.444228i	$0.146522\pi$
$12$		0			0
$13$	−14.9301	−	25.8597i	−0.318528	−	0.551707i	0.661653	−	0.749810i	$-0.269855\pi$
$13$	−14.9301	−	25.8597i	−0.318528	−	0.551707i	−0.980181	+	0.198103i	$0.936522\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−67.9721			−0.969744			−0.484872	−	0.874585i	$-0.661134\pi$
$17$	−67.9721			−0.969744			−0.484872	+	0.874585i	$0.661134\pi$
$18$		0			0
$19$	111.462			1.34585			0.672926	−	0.739710i	$-0.265037\pi$
$19$	111.462			1.34585			0.672926	+	0.739710i	$0.265037\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−109.253	−	189.232i	−0.990472	−	1.71555i	−0.614498	−	0.788919i	$-0.710641\pi$
$23$	−109.253	−	189.232i	−0.990472	−	1.71555i	−0.375975	−	0.926630i	$-0.622692\pi$
$24$		0			0
$25$	−99.2250	+	171.863i	−0.793800	+	1.37490i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	17.1357	−	29.6798i	0.109725	−	0.190049i	−0.805934	−	0.592005i	$-0.798336\pi$
$29$	17.1357	−	29.6798i	0.109725	−	0.190049i	0.915659	+	0.401957i	$0.131670\pi$
$30$		0			0
$31$	38.8523	+	67.2942i	0.225099	+	0.389884i	0.956349	−	0.292226i	$-0.0943960\pi$
$31$	38.8523	+	67.2942i	0.225099	+	0.389884i	−0.731250	+	0.682110i	$0.761063\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	130.984			0.632583
$36$		0			0
$37$	−347.228			−1.54281			−0.771404	−	0.636346i	$-0.780445\pi$
$37$	−347.228			−1.54281			−0.771404	+	0.636346i	$0.780445\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	117.273	+	203.123i	0.446707	+	0.773719i	0.998169	−	0.0604809i	$-0.0192634\pi$
$41$	117.273	+	203.123i	0.446707	+	0.773719i	−0.551463	+	0.834200i	$0.685930\pi$
$42$		0			0
$43$	−26.6899	+	46.2283i	−0.0946552	+	0.163948i	−0.909465	−	0.415781i	$-0.863508\pi$
$43$	−26.6899	+	46.2283i	−0.0946552	+	0.163948i	0.814809	+	0.579729i	$0.196842\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−192.701	+	333.768i	−0.598049	+	1.03585i	0.395060	+	0.918655i	$0.370724\pi$
$47$	−192.701	+	333.768i	−0.598049	+	1.03585i	−0.993109	+	0.117196i	$0.962609\pi$
$48$		0			0
$49$	144.978	+	251.110i	0.422677	+	0.732098i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	461.661			1.19649			0.598245	−	0.801313i	$-0.295865\pi$
$53$	461.661			1.19649			0.598245	+	0.801313i	$0.295865\pi$
$54$		0			0
$55$	−82.9938			−0.203471
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.58164	+	6.20358i	0.00790322	+	0.0136888i	0.869950	−	0.493140i	$-0.164151\pi$
$59$	3.58164	+	6.20358i	0.00790322	+	0.0136888i	−0.862047	+	0.506829i	$0.830818\pi$
$60$		0			0
$61$	−208.054	+	360.361i	−0.436699	+	0.756385i	−0.997433	−	0.0716114i	$-0.977186\pi$
$61$	−208.054	+	360.361i	−0.436699	+	0.756385i	0.560734	+	0.827996i	$0.310519\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−268.514	+	465.079i	−0.512385	+	0.887477i
$66$		0			0
$67$	434.637	+	752.814i	0.792528	+	1.37270i	0.924397	+	0.381432i	$0.124569\pi$
$67$	434.637	+	752.814i	0.792528	+	1.37270i	−0.131869	+	0.991267i	$0.542098\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	585.943			0.979417			0.489709	−	0.871886i	$-0.337103\pi$
$71$	585.943			0.979417			0.489709	+	0.871886i	$0.337103\pi$
$72$		0			0
$73$	−733.324			−1.17574			−0.587870	−	0.808955i	$-0.700034\pi$
$73$	−733.324			−1.17574			−0.587870	+	0.808955i	$0.700034\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	16.8046	+	29.1064i	0.0248709	+	0.0430777i
$78$		0			0
$79$	−585.448	+	1014.03i	−0.833772	+	1.44414i	0.0612537	+	0.998122i	$0.480490\pi$
$79$	−585.448	+	1014.03i	−0.833772	+	1.44414i	−0.895026	+	0.446014i	$0.852843\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−33.7182	+	58.4016i	−0.0445910	+	0.0772339i	−0.887459	−	0.460886i	$-0.847532\pi$
$83$	−33.7182	+	58.4016i	−0.0445910	+	0.0772339i	0.842868	+	0.538120i	$0.180865\pi$
$84$		0			0
$85$	611.229	+	1058.68i	0.779966	+	1.35094i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	965.886			1.15038			0.575189	−	0.818020i	$-0.304928\pi$
$89$	965.886			1.15038			0.575189	+	0.818020i	$0.304928\pi$
$90$		0			0
$91$	217.475			0.250522
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1002.31	−	1736.05i	−1.08247	−	1.87489i
$96$		0			0
$97$	−0.716493	+	1.24100i	−0.000749988	+	0.00129902i	−0.866400	−	0.499350i	$-0.833572\pi$
$97$	−0.716493	+	1.24100i	−0.000749988	+	0.00129902i	0.865650	+	0.500649i	$0.166905\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−448.089	+	776.112i	−0.441450	+	0.764614i	−0.997797	−	0.0663357i	$-0.978869\pi$
$101$	−448.089	+	776.112i	−0.441450	+	0.764614i	0.556347	+	0.830950i	$0.312203\pi$
$102$		0			0
$103$	−970.404	−	1680.79i	−0.928318	−	1.60789i	−0.786136	−	0.618053i	$-0.787922\pi$
$103$	−970.404	−	1680.79i	−0.928318	−	1.60789i	−0.142181	−	0.989841i	$-0.545412\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1888.04			−1.70583			−0.852915	−	0.522050i	$-0.825168\pi$
$107$	−1888.04			−1.70583			−0.852915	+	0.522050i	$0.825168\pi$
$108$		0			0
$109$	1201.72			1.05600			0.528001	−	0.849244i	$-0.322942\pi$
$109$	1201.72			1.05600			0.528001	+	0.849244i	$0.322942\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−761.211	−	1318.46i	−0.633706	−	1.09761i	−0.986788	−	0.162018i	$-0.948200\pi$
$113$	−761.211	−	1318.46i	−0.633706	−	1.09761i	0.353082	−	0.935592i	$-0.385134\pi$
$114$		0			0
$115$	−1964.89	+	3403.29i	−1.59328	+	2.75963i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	247.523	−	428.723i	0.190676	−	0.330260i
$120$		0			0
$121$	654.852	+	1134.24i	0.492000	+	0.852169i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1320.98			0.945214
$126$		0			0
$127$	−2300.50			−1.60738			−0.803688	−	0.595051i	$-0.797132\pi$
$127$	−2300.50			−1.60738			−0.803688	+	0.595051i	$0.797132\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−519.167	−	899.224i	−0.346258	−	0.599737i	0.639323	−	0.768938i	$-0.279215\pi$
$131$	−519.167	−	899.224i	−0.346258	−	0.599737i	−0.985582	+	0.169201i	$0.945881\pi$
$132$		0			0
$133$	−405.895	+	703.031i	−0.264628	+	0.458350i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−544.763	+	943.557i	−0.339724	+	0.588420i	−0.984381	−	0.176053i	$-0.943667\pi$
$137$	−544.763	+	943.557i	−0.339724	+	0.588420i	0.644657	+	0.764472i	$0.277000\pi$
$138$		0			0
$139$	959.663	+	1662.18i	0.585594	+	1.01428i	0.994801	+	0.101837i	$0.0324719\pi$
$139$	959.663	+	1662.18i	0.585594	+	1.01428i	−0.409207	+	0.912441i	$0.634195\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−137.795			−0.0805807
$144$		0			0
$145$	−616.360			−0.353006
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1065.88	+	1846.16i	0.586044	+	1.01506i	0.994744	+	0.102390i	$0.0326489\pi$
$149$	1065.88	+	1846.16i	0.586044	+	1.01506i	−0.408700	+	0.912669i	$0.634018\pi$
$150$		0			0
$151$	582.932	−	1009.67i	0.314161	−	0.544143i	−0.665098	−	0.746756i	$-0.731610\pi$
$151$	582.932	−	1009.67i	0.314161	−	0.544143i	0.979259	+	0.202613i	$0.0649434\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	698.748	−	1210.27i	0.362095	−	0.627167i
$156$		0			0
$157$	−18.3881	−	31.8491i	−0.00934733	−	0.0161901i	0.861314	−	0.508073i	$-0.169642\pi$
$157$	−18.3881	−	31.8491i	−0.00934733	−	0.0161901i	−0.870661	+	0.491883i	$0.836309\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	1591.40			0.779007
$162$		0			0
$163$	3208.90			1.54197			0.770983	−	0.636856i	$-0.219765\pi$
$163$	3208.90			1.54197			0.770983	+	0.636856i	$0.219765\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−898.044	−	1555.46i	−0.416124	−	0.720748i	0.579422	−	0.815028i	$-0.303278\pi$
$167$	−898.044	−	1555.46i	−0.416124	−	0.720748i	−0.995546	+	0.0942799i	$0.969945\pi$
$168$		0			0
$169$	652.684	−	1130.48i	0.297080	−	0.514557i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−847.357	+	1467.66i	−0.372389	+	0.644997i	−0.989933	−	0.141540i	$-0.954795\pi$
$173$	−847.357	+	1467.66i	−0.372389	+	0.644997i	0.617543	+	0.786537i	$0.288128\pi$
$174$		0			0
$175$	−722.665	−	1251.69i	−0.312162	−	0.540680i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−327.128			−0.136596			−0.0682981	−	0.997665i	$-0.521757\pi$
$179$	−327.128			−0.136596			−0.0682981	+	0.997665i	$0.521757\pi$
$180$		0			0
$181$	−4449.32			−1.82716			−0.913578	−	0.406663i	$-0.866692\pi$
$181$	−4449.32			−1.82716			−0.913578	+	0.406663i	$0.866692\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	3122.40	+	5408.15i	1.24088	+	2.14927i
$186$		0			0
$187$	−156.835	+	271.646i	−0.0613310	+	0.106228i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1039.48	+	1800.43i	−0.393792	+	0.682067i	−0.992946	−	0.118566i	$-0.962170\pi$
$191$	−1039.48	+	1800.43i	−0.393792	+	0.682067i	0.599154	+	0.800633i	$0.295504\pi$
$192$		0			0
$193$	1443.58	+	2500.36i	0.538401	+	0.932539i	0.998990	+	0.0449251i	$0.0143049\pi$
$193$	1443.58	+	2500.36i	0.538401	+	0.932539i	−0.460589	+	0.887614i	$0.652362\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	2679.74			0.969154			0.484577	−	0.874749i	$-0.338973\pi$
$197$	2679.74			0.969154			0.484577	+	0.874749i	$0.338973\pi$
$198$		0			0
$199$	341.646			0.121702			0.0608508	−	0.998147i	$-0.480619\pi$
$199$	341.646			0.121702			0.0608508	+	0.998147i	$0.480619\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	124.801	+	216.161i	0.0431492	+	0.0747366i
$204$		0			0
$205$	2109.12	−	3653.11i	0.718573	−	1.24461i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	257.182	−	445.452i	0.0851178	−	0.147428i
$210$		0			0
$211$	−787.890	−	1364.66i	−0.257064	−	0.445248i	0.708390	−	0.705821i	$-0.249422\pi$
$211$	−787.890	−	1364.66i	−0.257064	−	0.445248i	−0.965454	+	0.260573i	$0.916089\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	960.021			0.304525
$216$		0			0
$217$	−565.930			−0.177041
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1014.83	+	1757.74i	0.308891	+	0.535015i
$222$		0			0
$223$	1491.40	−	2583.18i	0.447854	−	0.775706i	−0.550392	−	0.834906i	$-0.685522\pi$
$223$	1491.40	−	2583.18i	0.447854	−	0.775706i	0.998246	+	0.0592003i	$0.0188551\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1069.90	−	1853.13i	0.312828	−	0.541834i	−0.666145	−	0.745822i	$-0.732057\pi$
$227$	1069.90	−	1853.13i	0.312828	−	0.541834i	0.978973	+	0.203988i	$0.0653902\pi$
$228$		0			0
$229$	−309.701	−	536.417i	−0.0893694	−	0.154792i	0.817875	−	0.575395i	$-0.195152\pi$
$229$	−309.701	−	536.417i	−0.0893694	−	0.154792i	−0.907245	+	0.420603i	$0.861818\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−4272.01			−1.20115			−0.600577	−	0.799567i	$-0.705062\pi$
$233$	−4272.01			−1.20115			−0.600577	+	0.799567i	$0.705062\pi$
$234$		0			0
$235$	6931.34			1.92405
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−3126.31	−	5414.93i	−0.846126	−	1.46553i	−0.884639	−	0.466276i	$-0.845595\pi$
$239$	−3126.31	−	5414.93i	−0.846126	−	1.46553i	0.0385130	−	0.999258i	$-0.487738\pi$
$240$		0			0
$241$	−125.390	+	217.182i	−0.0335148	+	0.0580494i	−0.882296	−	0.470694i	$-0.844003\pi$
$241$	−125.390	+	217.182i	−0.0335148	+	0.0580494i	0.848781	+	0.528744i	$0.177337\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	2607.39	−	4516.14i	0.679919	−	1.17765i
$246$		0			0
$247$	−1664.14	−	2882.38i	−0.428692	−	0.742516i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−7146.55			−1.79716			−0.898579	−	0.438812i	$-0.855399\pi$
$251$	−7146.55			−1.79716			−0.898579	+	0.438812i	$0.855399\pi$
$252$		0			0
$253$	−1008.34			−0.250568
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1513.82	+	2622.01i	0.367429	+	0.636406i	0.989163	−	0.146823i	$-0.0469046\pi$
$257$	1513.82	+	2622.01i	0.367429	+	0.636406i	−0.621733	+	0.783229i	$0.713571\pi$
$258$		0			0
$259$	1264.45	−	2190.08i	0.303355	−	0.525425i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	2609.48	−	4519.75i	0.611816	−	1.05970i	−0.379119	−	0.925348i	$-0.623773\pi$
$263$	2609.48	−	4519.75i	0.611816	−	1.05970i	0.990934	−	0.134348i	$-0.0428939\pi$
$264$		0			0
$265$	−4151.42	−	7190.47i	−0.962339	−	1.66682i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−6038.18			−1.36860			−0.684302	−	0.729199i	$-0.739893\pi$
$269$	−6038.18			−1.36860			−0.684302	+	0.729199i	$0.739893\pi$
$270$		0			0
$271$	3726.70			0.835354			0.417677	−	0.908596i	$-0.362844\pi$
$271$	3726.70			0.835354			0.417677	+	0.908596i	$0.362844\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	457.892	+	793.092i	0.100407	+	0.173910i
$276$		0			0
$277$	286.047	−	495.448i	0.0620466	−	0.107468i	−0.833333	−	0.552771i	$-0.813571\pi$
$277$	286.047	−	495.448i	0.0620466	−	0.107468i	0.895380	+	0.445303i	$0.146904\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1023.99	−	1773.61i	0.217389	−	0.376528i	−0.736620	−	0.676307i	$-0.763579\pi$
$281$	1023.99	−	1773.61i	0.217389	−	0.376528i	0.954009	+	0.299778i	$0.0969127\pi$
$282$		0			0
$283$	−2396.98	−	4151.70i	−0.503484	−	0.872059i	−0.999992	−	0.00402723i	$-0.998718\pi$
$283$	−2396.98	−	4151.70i	−0.503484	−	0.872059i	0.496508	−	0.868032i	$-0.334615\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−1708.22			−0.351335
$288$		0			0
$289$	−292.797			−0.0595963
$290$		0			0
$291$		0			0
$292$		0			0
$293$	34.7484	+	60.1859i	0.00692840	+	0.0120003i	0.869469	−	0.493988i	$-0.164461\pi$
$293$	34.7484	+	60.1859i	0.00692840	+	0.0120003i	−0.862540	+	0.505988i	$0.831128\pi$
$294$		0			0
$295$	64.4148	−	111.570i	0.0127131	−	0.0220198i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−3262.32	+	5650.51i	−0.630987	+	1.09290i
$300$		0			0
$301$	−194.385	−	336.685i	−0.0372232	−	0.0644724i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	7483.60			1.40495
$306$		0			0
$307$	665.210			0.123666			0.0618331	−	0.998087i	$-0.480305\pi$
$307$	665.210			0.123666			0.0618331	+	0.998087i	$0.480305\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−126.282	−	218.727i	−0.0230251	−	0.0398806i	0.854283	−	0.519808i	$-0.173996\pi$
$311$	−126.282	−	218.727i	−0.0230251	−	0.0398806i	−0.877308	+	0.479927i	$0.840663\pi$
$312$		0			0
$313$	4413.24	−	7643.96i	0.796969	−	1.38039i	−0.124612	−	0.992206i	$-0.539769\pi$
$313$	4413.24	−	7643.96i	0.796969	−	1.38039i	0.921581	−	0.388185i	$-0.126898\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	458.545	−	794.224i	0.0812444	−	0.140719i	−0.822540	−	0.568707i	$-0.807444\pi$
$317$	458.545	−	794.224i	0.0812444	−	0.140719i	0.903785	+	0.427987i	$0.140777\pi$
$318$		0			0
$319$	−79.0757	−	136.963i	−0.0138790	−	0.0240391i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−7576.32			−1.30513
$324$		0			0
$325$	5925.76			1.01139
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1403.46	−	2430.86i	−0.235183	−	0.407348i
$330$		0			0
$331$	831.405	−	1440.04i	0.138061	−	0.239128i	−0.788702	−	0.614776i	$-0.789246\pi$
$331$	831.405	−	1440.04i	0.138061	−	0.239128i	0.926763	+	0.375648i	$0.122580\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	7816.83	−	13539.1i	1.27486	−	2.20813i
$336$		0			0
$337$	1201.98	+	2081.89i	0.194291	+	0.336522i	0.946668	−	0.322211i	$-0.104426\pi$
$337$	1201.98	+	2081.89i	0.194291	+	0.336522i	−0.752377	+	0.658733i	$0.771093\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	358.582			0.0569452
$342$		0			0
$343$	−4609.88			−0.725686
$344$		0			0
$345$		0			0
$346$		0			0
$347$	3935.81	+	6817.03i	0.608892	+	1.05463i	0.991423	+	0.130689i	$0.0417191\pi$
$347$	3935.81	+	6817.03i	0.608892	+	1.05463i	−0.382531	+	0.923943i	$0.624948\pi$
$348$		0			0
$349$	5737.04	−	9936.85i	0.879934	−	1.52409i	0.0285217	−	0.999593i	$-0.490920\pi$
$349$	5737.04	−	9936.85i	0.879934	−	1.52409i	0.851412	−	0.524497i	$-0.175747\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−2010.59	+	3482.45i	−0.303153	+	0.525077i	−0.976848	−	0.213933i	$-0.931373\pi$
$353$	−2010.59	+	3482.45i	−0.303153	+	0.525077i	0.673695	+	0.739009i	$0.264706\pi$
$354$		0			0
$355$	−5269.01	−	9126.19i	−0.787746	−	1.36442i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1662.93			−0.244473			−0.122237	−	0.992501i	$-0.539007\pi$
$359$	−1662.93			−0.244473			−0.122237	+	0.992501i	$0.539007\pi$
$360$		0			0
$361$	5564.84			0.811319
$362$		0			0
$363$		0			0
$364$		0			0
$365$	6594.31	+	11421.7i	0.945649	+	1.63791i
$366$		0			0
$367$	1469.32	−	2544.93i	0.208986	−	0.361974i	−0.742409	−	0.669946i	$-0.766317\pi$
$367$	1469.32	−	2544.93i	0.208986	−	0.361974i	0.951395	+	0.307972i	$0.0996504\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−1681.16	+	2911.85i	−0.235260	+	0.407482i
$372$		0			0
$373$	3809.77	+	6598.71i	0.528853	+	0.916001i	0.999434	+	0.0336437i	$0.0107112\pi$
$373$	3809.77	+	6598.71i	0.528853	+	0.916001i	−0.470581	+	0.882357i	$0.655956\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1023.35			−0.139802
$378$		0			0
$379$	−13118.3			−1.77795			−0.888975	−	0.457955i	$-0.848582\pi$
$379$	−13118.3			−1.77795			−0.888975	+	0.457955i	$0.848582\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	4301.35	+	7450.15i	0.573861	+	0.993956i	0.996164	+	0.0875014i	$0.0278882\pi$
$383$	4301.35	+	7450.15i	0.573861	+	0.993956i	−0.422304	+	0.906454i	$0.638778\pi$
$384$		0			0
$385$	302.226	−	523.470i	0.0400074	−	0.0692948i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−6460.79	+	11190.4i	−0.842095	+	1.45855i	0.0460245	+	0.998940i	$0.485345\pi$
$389$	−6460.79	+	11190.4i	−0.842095	+	1.45855i	−0.888120	+	0.459612i	$0.847989\pi$
$390$		0			0
$391$	7426.17	+	12862.5i	0.960505	+	1.66364i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	21058.2			2.68242
$396$		0			0
$397$	5511.90			0.696812			0.348406	−	0.937344i	$-0.386723\pi$
$397$	5511.90			0.696812			0.348406	+	0.937344i	$0.386723\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1116.65	−	1934.10i	−0.139060	−	0.240859i	0.788081	−	0.615572i	$-0.211075\pi$
$401$	−1116.65	−	1934.10i	−0.139060	−	0.240859i	−0.927141	+	0.374712i	$0.877741\pi$
$402$		0			0
$403$	1160.14	−	2009.42i	0.143401	−	0.248378i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−801.173	+	1387.67i	−0.0975741	+	0.169003i
$408$		0			0
$409$	−1112.66	−	1927.18i	−0.134517	−	0.232990i	0.790896	−	0.611950i	$-0.209615\pi$
$409$	−1112.66	−	1927.18i	−0.134517	−	0.232990i	−0.925413	+	0.378961i	$0.876281\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−52.1708			−0.00621588
$414$		0			0
$415$	1212.82			0.143458
$416$		0			0
$417$		0			0
$418$		0			0
$419$	2710.56	+	4694.83i	0.316037	+	0.547392i	0.979657	−	0.200677i	$-0.0643143\pi$
$419$	2710.56	+	4694.83i	0.316037	+	0.547392i	−0.663620	+	0.748070i	$0.730981\pi$
$420$		0			0
$421$	−6410.19	+	11102.8i	−0.742075	+	1.28531i	0.209474	+	0.977814i	$0.432825\pi$
$421$	−6410.19	+	11102.8i	−0.742075	+	1.28531i	−0.951549	+	0.307497i	$0.900509\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	6744.53	−	11681.9i	0.769783	−	1.33330i
$426$		0			0
$427$	−1515.28	−	2624.54i	−0.171732	−	0.297448i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−10070.7			−1.12549			−0.562747	−	0.826629i	$-0.690255\pi$
$431$	−10070.7			−1.12549			−0.562747	+	0.826629i	$0.690255\pi$
$432$		0			0
$433$	−5708.09			−0.633518			−0.316759	−	0.948506i	$-0.602595\pi$
$433$	−5708.09			−0.633518			−0.316759	+	0.948506i	$0.602595\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−12177.6	−	21092.2i	−1.33303	−	2.30888i
$438$		0			0
$439$	1489.79	−	2580.40i	0.161968	−	0.280537i	−0.773606	−	0.633667i	$-0.781549\pi$
$439$	1489.79	−	2580.40i	0.161968	−	0.280537i	0.935574	+	0.353129i	$0.114882\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−5246.18	+	9086.65i	−0.562649	+	0.974537i	0.434615	+	0.900616i	$0.356884\pi$
$443$	−5246.18	+	9086.65i	−0.562649	+	0.974537i	−0.997264	+	0.0739204i	$0.976449\pi$
$444$		0			0
$445$	−8685.59	−	15043.9i	−0.925251	−	1.60258i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−4846.95			−0.509447			−0.254724	−	0.967014i	$-0.581985\pi$
$449$	−4846.95			−0.509447			−0.254724	+	0.967014i	$0.581985\pi$
$450$		0			0
$451$	1082.36			0.113007
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1955.61	−	3387.22i	−0.201495	−	0.349000i
$456$		0			0
$457$	−2570.87	+	4452.87i	−0.263151	+	0.455791i	−0.967078	−	0.254482i	$-0.918095\pi$
$457$	−2570.87	+	4452.87i	−0.263151	+	0.455791i	0.703926	+	0.710273i	$0.251428\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	1996.33	−	3457.75i	0.201689	−	0.349335i	−0.747384	−	0.664392i	$-0.768690\pi$
$461$	1996.33	−	3457.75i	0.201689	−	0.349335i	0.949073	+	0.315057i	$0.102024\pi$
$462$		0			0
$463$	4753.53	+	8233.35i	0.477139	+	0.826428i	0.999657	−	0.0261998i	$-0.00834060\pi$
$463$	4753.53	+	8233.35i	0.477139	+	0.826428i	−0.522518	+	0.852628i	$0.675007\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−1373.54			−0.136102			−0.0680511	−	0.997682i	$-0.521678\pi$
$467$	−1373.54			−0.136102			−0.0680511	+	0.997682i	$0.521678\pi$
$468$		0			0
$469$	−6331.00			−0.623323
$470$		0			0
$471$		0			0
$472$		0			0
$473$	123.166	+	213.329i	0.0119728	+	0.0207376i
$474$		0			0
$475$	−11059.8	+	19156.2i	−1.06834	+	1.85041i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	8553.88	−	14815.7i	0.815943	−	1.41325i	−0.0927063	−	0.995694i	$-0.529552\pi$
$479$	8553.88	−	14815.7i	0.815943	−	1.41325i	0.908649	−	0.417561i	$-0.137115\pi$
$480$		0			0
$481$	5184.15	+	8979.21i	0.491428	+	0.851178i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	25.7718			0.00241286
$486$		0			0
$487$	−4627.18			−0.430549			−0.215275	−	0.976554i	$-0.569065\pi$
$487$	−4627.18			−0.430549			−0.215275	+	0.976554i	$0.569065\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−4580.83	−	7934.23i	−0.421039	−	0.729261i	0.575003	−	0.818152i	$-0.305001\pi$
$491$	−4580.83	−	7934.23i	−0.421039	−	0.729261i	−0.996041	+	0.0888910i	$0.971668\pi$
$492$		0			0
$493$	−1164.75	+	2017.40i	−0.106405	+	0.184298i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−2133.74	+	3695.74i	−0.192578	+	0.333555i
$498$		0			0
$499$	−4149.54	−	7187.21i	−0.372262	−	0.644777i	0.617651	−	0.786452i	$-0.288084\pi$
$499$	−4149.54	−	7187.21i	−0.372262	−	0.644777i	−0.989913	+	0.141675i	$0.954751\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	9568.57			0.848194			0.424097	−	0.905617i	$-0.360592\pi$
$503$	9568.57			0.848194			0.424097	+	0.905617i	$0.360592\pi$
$504$		0			0
$505$	16117.5			1.42024
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−4374.51	−	7576.88i	−0.380937	−	0.659802i	0.610260	−	0.792202i	$-0.291065\pi$
$509$	−4374.51	−	7576.88i	−0.380937	−	0.659802i	−0.991196	+	0.132400i	$0.957732\pi$
$510$		0			0
$511$	2670.43	−	4625.33i	0.231180	−	0.400416i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−17452.4	+	30228.5i	−1.49329	+	2.58646i
$516$		0			0
$517$	889.253	+	1540.23i	0.0756467	+	0.131024i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−4974.67			−0.418319			−0.209160	−	0.977882i	$-0.567073\pi$
$521$	−4974.67			−0.418319			−0.209160	+	0.977882i	$0.567073\pi$
$522$		0			0
$523$	10144.9			0.848196			0.424098	−	0.905616i	$-0.360591\pi$
$523$	10144.9			0.848196			0.424098	+	0.905616i	$0.360591\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−2640.87	−	4574.12i	−0.218289	−	0.378087i
$528$		0			0
$529$	−17789.0	+	30811.5i	−1.46207	+	2.53238i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	3501.80	−	6065.29i	0.284577	−	0.492903i
$534$		0			0
$535$	16977.9	+	29406.7i	1.37200	+	2.37638i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	1338.06			0.106928
$540$		0			0
$541$	−9130.53			−0.725604			−0.362802	−	0.931866i	$-0.618180\pi$
$541$	−9130.53			−0.725604			−0.362802	+	0.931866i	$0.618180\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−10806.3	−	18717.1i	−0.849344	−	1.47111i
$546$		0			0
$547$	1877.16	−	3251.33i	0.146730	−	0.254144i	−0.783287	−	0.621660i	$-0.786458\pi$
$547$	1877.16	−	3251.33i	0.146730	−	0.254144i	0.930017	+	0.367516i	$0.119792\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	1909.98	−	3308.18i	0.147673	−	0.255777i
$552$		0			0
$553$	−4263.87	−	7385.24i	−0.327881	−	0.567906i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−9750.59			−0.741734			−0.370867	−	0.928686i	$-0.620939\pi$
$557$	−9750.59			−0.741734			−0.370867	+	0.928686i	$0.620939\pi$
$558$		0			0
$559$	1593.93			0.120601
$560$		0			0
$561$		0			0
$562$		0			0
$563$	1443.17	+	2499.64i	0.108033	+	0.187118i	0.914973	−	0.403515i	$-0.132212\pi$
$563$	1443.17	+	2499.64i	0.108033	+	0.187118i	−0.806941	+	0.590633i	$0.798878\pi$
$564$		0			0
$565$	−13690.2	+	23712.1i	−1.01938	+	1.76562i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−3754.03	+	6502.18i	−0.276586	+	0.479060i	−0.970534	−	0.240964i	$-0.922536\pi$
$569$	−3754.03	+	6502.18i	−0.276586	+	0.479060i	0.693948	+	0.720025i	$0.255870\pi$
$570$		0			0
$571$	−4580.16	−	7933.07i	−0.335681	−	0.581416i	0.647935	−	0.761696i	$-0.275633\pi$
$571$	−4580.16	−	7933.07i	−0.335681	−	0.581416i	−0.983615	+	0.180280i	$0.942300\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	43362.6			3.14495
$576$		0			0
$577$	−21578.4			−1.55688			−0.778441	−	0.627718i	$-0.783989\pi$
$577$	−21578.4			−1.55688			−0.778441	+	0.627718i	$0.783989\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−245.573	−	425.344i	−0.0175354	−	0.0303722i
$582$		0			0
$583$	1065.21	−	1845.00i	0.0756715	−	0.131067i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−11377.1	+	19705.6i	−0.799968	+	1.38559i	0.119668	+	0.992814i	$0.461817\pi$
$587$	−11377.1	+	19705.6i	−0.799968	+	1.38559i	−0.919636	+	0.392772i	$0.871516\pi$
$588$		0			0
$589$	4330.57	+	7500.76i	0.302951	+	0.524726i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−11462.9			−0.793801			−0.396901	−	0.917862i	$-0.629914\pi$
$593$	−11462.9			−0.793801			−0.396901	+	0.917862i	$0.629914\pi$
$594$		0			0
$595$	−8903.28			−0.613443
$596$		0			0
$597$		0			0
$598$		0			0
$599$	3000.09	+	5196.30i	0.204642	+	0.354449i	0.950018	−	0.312194i	$-0.101064\pi$
$599$	3000.09	+	5196.30i	0.204642	+	0.354449i	−0.745377	+	0.666643i	$0.767730\pi$
$600$		0			0
$601$	6437.41	−	11149.9i	0.436917	−	0.756763i	−0.560533	−	0.828132i	$-0.689404\pi$
$601$	6437.41	−	11149.9i	0.436917	−	0.756763i	0.997450	+	0.0713693i	$0.0227369\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	11777.3	−	20398.9i	0.791433	−	1.37080i
$606$		0			0
$607$	11823.0	+	20478.1i	0.790580	+	1.36932i	0.925608	+	0.378483i	$0.123554\pi$
$607$	11823.0	+	20478.1i	0.790580	+	1.36932i	−0.135029	+	0.990842i	$0.543113\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	11508.2			0.761982
$612$		0			0
$613$	1047.98			0.0690500			0.0345250	−	0.999404i	$-0.489008\pi$
$613$	1047.98			0.0690500			0.0345250	+	0.999404i	$0.489008\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	999.343	+	1730.91i	0.0652059	+	0.112940i	0.896785	−	0.442466i	$-0.145896\pi$
$617$	999.343	+	1730.91i	0.0652059	+	0.112940i	−0.831579	+	0.555406i	$0.812563\pi$
$618$		0			0
$619$	7623.25	−	13203.8i	0.494999	−	0.857363i	−0.504985	−	0.863128i	$-0.668502\pi$
$619$	7623.25	−	13203.8i	0.494999	−	0.857363i	0.999983	+	0.00576553i	$0.00183524\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−3517.32	+	6092.17i	−0.226193	+	0.391778i
$624$		0			0
$625$	524.427	+	908.334i	0.0335633	+	0.0581334i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	23601.8			1.49613
$630$		0			0
$631$	−13870.7			−0.875094			−0.437547	−	0.899195i	$-0.644153\pi$
$631$	−13870.7			−0.875094			−0.437547	+	0.899195i	$0.644153\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	20686.9	+	35830.8i	1.29281	+	2.23922i
$636$		0			0
$637$	4329.08	−	7498.19i	0.269269	−	0.466388i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−9879.67	+	17112.1i	−0.608773	+	1.05443i	0.382670	+	0.923885i	$0.375005\pi$
$641$	−9879.67	+	17112.1i	−0.608773	+	1.05443i	−0.991443	+	0.130540i	$0.958329\pi$
$642$		0			0
$643$	5847.34	+	10127.9i	0.358626	+	0.621159i	0.987732	−	0.156161i	$-0.0499120\pi$
$643$	5847.34	+	10127.9i	0.358626	+	0.621159i	−0.629105	+	0.777320i	$0.716579\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	3214.68			0.195335			0.0976676	−	0.995219i	$-0.468862\pi$
$647$	3214.68			0.195335			0.0976676	+	0.995219i	$0.468862\pi$
$648$		0			0
$649$	33.0563			0.00199934
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−4623.74	−	8008.56i	−0.277092	−	0.479937i	0.693569	−	0.720390i	$-0.256037\pi$
$653$	−4623.74	−	8008.56i	−0.277092	−	0.479937i	−0.970661	+	0.240453i	$0.922704\pi$
$654$		0			0
$655$	−9337.07	+	16172.3i	−0.556992	+	0.964738i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−3991.01	+	6912.62i	−0.235914	+	0.408615i	−0.959538	−	0.281579i	$-0.909142\pi$
$659$	−3991.01	+	6912.62i	−0.235914	+	0.408615i	0.723624	+	0.690195i	$0.242475\pi$
$660$		0			0
$661$	8516.73	+	14751.4i	0.501154	+	0.868023i	0.999999	+	0.00133254i	$0.000424159\pi$
$661$	8516.73	+	14751.4i	0.501154	+	0.868023i	−0.498846	+	0.866691i	$0.666243\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	14599.8			0.851363
$666$		0			0
$667$	−7488.50			−0.434717
$668$		0			0
$669$		0			0
$670$		0			0
$671$	960.106	+	1662.95i	0.0552377	+	0.0956744i
$672$		0			0
$673$	−9263.71	+	16045.2i	−0.530594	+	0.919016i	0.468769	+	0.883321i	$0.344698\pi$
$673$	−9263.71	+	16045.2i	−0.530594	+	0.919016i	−0.999363	+	0.0356948i	$0.988636\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−16481.6	+	28547.1i	−0.935659	+	1.62061i	−0.162205	+	0.986757i	$0.551860\pi$
$677$	−16481.6	+	28547.1i	−0.935659	+	1.62061i	−0.773454	+	0.633852i	$0.781473\pi$
$678$		0			0
$679$	−5.21828	−	9.03833i	−0.000294933	−	0.000510838i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−9336.54			−0.523064			−0.261532	−	0.965195i	$-0.584228\pi$
$683$	−9336.54			−0.523064			−0.261532	+	0.965195i	$0.584228\pi$
$684$		0			0
$685$	19594.8			1.09296
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−6892.65	−	11938.4i	−0.381116	−	0.660112i
$690$		0			0
$691$	−14424.1	+	24983.3i	−0.794096	+	1.37541i	0.129316	+	0.991603i	$0.458722\pi$
$691$	−14424.1	+	24983.3i	−0.794096	+	1.37541i	−0.923412	+	0.383811i	$0.874611\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	17259.3	−	29893.9i	0.941987	−	1.63157i
$696$		0			0
$697$	−7971.29	−	13806.7i	−0.433191	−	0.750309i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−28870.8			−1.55554			−0.777772	−	0.628547i	$-0.783650\pi$
$701$	−28870.8			−1.55554			−0.777772	+	0.628547i	$0.783650\pi$
$702$		0			0
$703$	−38702.8			−2.07639
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−3263.47	−	5652.50i	−0.173600	−	0.300685i
$708$		0			0
$709$	−7616.82	+	13192.7i	−0.403463	+	0.698819i	−0.994141	−	0.108088i	$-0.965527\pi$
$709$	−7616.82	+	13192.7i	−0.403463	+	0.698819i	0.590678	+	0.806907i	$0.298860\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	8489.48	−	14704.2i	0.445909	−	0.772338i
$714$		0			0
$715$	1239.11	+	2146.19i	0.0648111	+	0.112256i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−9946.12			−0.515894			−0.257947	−	0.966159i	$-0.583046\pi$
$719$	−9946.12			−0.515894			−0.257947	+	0.966159i	$0.583046\pi$
$720$		0			0
$721$	14135.1			0.730122
$722$		0			0
$723$		0			0
$724$		0			0
$725$	3400.57	+	5889.96i	0.174199	+	0.301721i
$726$		0			0
$727$	15203.7	−	26333.6i	0.775619	−	1.34341i	−0.158827	−	0.987306i	$-0.550771\pi$
$727$	15203.7	−	26333.6i	0.775619	−	1.34341i	0.934446	−	0.356105i	$-0.115895\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1814.17	−	3142.23i	0.0917913	−	0.158987i
$732$		0			0
$733$	−7537.91	−	13056.0i	−0.379835	−	0.657893i	0.611203	−	0.791474i	$-0.290686\pi$
$733$	−7537.91	−	13056.0i	−0.379835	−	0.657893i	−0.991038	+	0.133580i	$0.957353\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	4011.43			0.200492
$738$		0			0
$739$	19253.6			0.958399			0.479200	−	0.877706i	$-0.340927\pi$
$739$	19253.6			0.958399			0.479200	+	0.877706i	$0.340927\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−7866.50	−	13625.2i	−0.388417	−	0.672758i	0.603820	−	0.797121i	$-0.293645\pi$
$743$	−7866.50	−	13625.2i	−0.388417	−	0.672758i	−0.992237	+	0.124363i	$0.960311\pi$
$744$		0			0
$745$	19169.6	−	33202.7i	0.942712	−	1.63282i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	6875.39	−	11908.5i	0.335409	−	0.580945i
$750$		0			0
$751$	−6769.31	−	11724.8i	−0.328916	−	0.569699i	0.653381	−	0.757029i	$-0.273350\pi$
$751$	−6769.31	−	11724.8i	−0.328916	−	0.569699i	−0.982297	+	0.187330i	$0.940017\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−20967.7			−1.01072
$756$		0			0
$757$	25434.5			1.22118			0.610590	−	0.791947i	$-0.290932\pi$
$757$	25434.5			1.22118			0.610590	+	0.791947i	$0.290932\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−6839.77	−	11846.8i	−0.325810	−	0.564320i	0.655866	−	0.754877i	$-0.272304\pi$
$761$	−6839.77	−	11846.8i	−0.325810	−	0.564320i	−0.981676	+	0.190558i	$0.938970\pi$
$762$		0			0
$763$	−4376.13	+	7579.68i	−0.207636	+	0.359637i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	106.949	−	185.240i	0.00503479	−	0.00872052i
$768$		0			0
$769$	4599.40	+	7966.40i	0.215681	+	0.373571i	0.953483	−	0.301447i	$-0.0974695\pi$
$769$	4599.40	+	7966.40i	0.215681	+	0.373571i	−0.737802	+	0.675017i	$0.764136\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	19726.5			0.917868			0.458934	−	0.888470i	$-0.348231\pi$
$773$	19726.5			0.917868			0.458934	+	0.888470i	$0.348231\pi$
$774$		0			0
$775$	−15420.5			−0.714735
$776$		0			0
$777$		0			0
$778$		0			0
$779$	13071.5	+	22640.5i	0.601201	+	1.04131i
$780$		0			0
$781$	1351.97	−	2341.68i	0.0619428	−	0.107288i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−330.705	+	572.798i	−0.0150361	+	0.0260434i
$786$		0			0
$787$	−7715.21	−	13363.1i	−0.349451	−	0.605266i	0.636701	−	0.771110i	$-0.280298\pi$
$787$	−7715.21	−	13363.1i	−0.349451	−	0.605266i	−0.986152	+	0.165844i	$0.946965\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	11087.9			0.498409
$792$		0			0
$793$	12425.1			0.556404
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−7590.59	−	13147.3i	−0.337356	−	0.584317i	0.646579	−	0.762847i	$-0.276199\pi$
$797$	−7590.59	−	13147.3i	−0.337356	−	0.584317i	−0.983934	+	0.178530i	$0.942866\pi$
$798$		0			0
$799$	13098.3	−	22686.9i	0.579954	−	1.00451i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1692.03	+	2930.68i	−0.0743592	+	0.128794i
$804$		0			0
$805$	−14310.5	−	24786.4i	−0.626556	−	1.08523i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−38480.1			−1.67230			−0.836149	−	0.548503i	$-0.815198\pi$
$809$	−38480.1			−1.67230			−0.836149	+	0.548503i	$0.815198\pi$
$810$		0			0
$811$	32207.1			1.39451			0.697253	−	0.716825i	$-0.254405\pi$
$811$	32207.1			1.39451			0.697253	+	0.716825i	$0.254405\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−28855.6	−	49979.3i	−1.24020	−	2.14810i
$816$		0			0
$817$	−2974.92	+	5152.71i	−0.127392	+	0.220649i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	2930.72	−	5076.16i	0.124583	−	0.215785i	−0.796987	−	0.603997i	$-0.793574\pi$
$821$	2930.72	−	5076.16i	0.124583	−	0.215785i	0.921570	+	0.388212i	$0.126907\pi$
$822$		0			0
$823$	−17025.6	−	29489.2i	−0.721112	−	1.24900i	−0.960555	−	0.278092i	$-0.910298\pi$
$823$	−17025.6	−	29489.2i	−0.721112	−	1.24900i	0.239443	−	0.970910i	$-0.423035\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−17214.4			−0.723824			−0.361912	−	0.932212i	$-0.617876\pi$
$827$	−17214.4			−0.723824			−0.361912	+	0.932212i	$0.617876\pi$
$828$		0			0
$829$	−5144.24			−0.215521			−0.107760	−	0.994177i	$-0.534368\pi$
$829$	−5144.24			−0.215521			−0.107760	+	0.994177i	$0.534368\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−9854.48	−	17068.5i	−0.409889	−	0.709948i
$834$		0			0
$835$	−16151.1	+	27974.5i	−0.669378	+	1.15940i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	7718.56	−	13368.9i	0.317609	−	0.550115i	−0.662379	−	0.749169i	$-0.730453\pi$
$839$	7718.56	−	13368.9i	0.317609	−	0.550115i	0.979989	+	0.199053i	$0.0637867\pi$
$840$		0			0
$841$	11607.2	+	20104.3i	0.475921	+	0.824319i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−23476.7			−0.955765
$846$		0			0
$847$	−9538.70			−0.386958
$848$		0			0
$849$		0			0
$850$		0			0
$851$	37935.7	+	65706.6i	1.52811	+	2.64676i
$852$		0			0
$853$	9905.78	−	17157.3i	0.397617	−	0.688693i	−0.595814	−	0.803122i	$-0.703171\pi$
$853$	9905.78	−	17157.3i	0.397617	−	0.688693i	0.993431	+	0.114429i	$0.0365039\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	18959.8	−	32839.3i	0.755723	−	1.30895i	−0.189291	−	0.981921i	$-0.560619\pi$
$857$	18959.8	−	32839.3i	0.755723	−	1.30895i	0.945014	−	0.327030i	$-0.106048\pi$
$858$		0			0
$859$	−4300.19	−	7448.15i	−0.170804	−	0.295841i	0.767897	−	0.640573i	$-0.221303\pi$
$859$	−4300.19	−	7448.15i	−0.170804	−	0.295841i	−0.938701	+	0.344732i	$0.887970\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−16423.6			−0.647816			−0.323908	−	0.946089i	$-0.604997\pi$
$863$	−16423.6			−0.647816			−0.323908	+	0.946089i	$0.604997\pi$
$864$		0			0
$865$	30478.9			1.19805
$866$		0			0
$867$		0			0
$868$		0			0
$869$	2701.66	+	4679.41i	0.105463	+	0.182667i
$870$		0			0
$871$	12978.4	−	22479.2i	0.504885	−	0.874487i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−4810.40	+	8331.85i	−0.185853	+	0.321906i
$876$		0			0
$877$	−11539.9	−	19987.7i	−0.444328	−	0.769598i	0.553678	−	0.832731i	$-0.313224\pi$
$877$	−11539.9	−	19987.7i	−0.444328	−	0.769598i	−0.998005	+	0.0631332i	$0.979891\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	22691.9			0.867775			0.433887	−	0.900967i	$-0.357142\pi$
$881$	22691.9			0.867775			0.433887	+	0.900967i	$0.357142\pi$
$882$		0			0
$883$	19739.1			0.752291			0.376146	−	0.926561i	$-0.377249\pi$
$883$	19739.1			0.752291			0.376146	+	0.926561i	$0.377249\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	3869.19	+	6701.64i	0.146465	+	0.253685i	0.929919	−	0.367765i	$-0.119877\pi$
$887$	3869.19	+	6701.64i	0.146465	+	0.253685i	−0.783453	+	0.621451i	$0.786544\pi$
$888$		0			0
$889$	8377.39	−	14510.1i	0.316050	−	0.547415i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−21478.9	+	37202.5i	−0.804886	+	1.39410i
$894$		0			0
$895$	2941.66	+	5095.10i	0.109864	+	0.190291i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	2663.04			0.0987958
$900$		0			0
$901$	−31380.1			−1.16029
$902$		0			0
$903$		0			0
$904$		0			0
$905$	40009.9	+	69299.1i	1.46958	+	2.54539i
$906$		0			0
$907$	9931.12	−	17201.2i	0.363569	−	0.629720i	−0.624976	−	0.780644i	$-0.714891\pi$
$907$	9931.12	−	17201.2i	0.363569	−	0.629720i	0.988545	+	0.150923i	$0.0482247\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	25094.2	−	43464.5i	0.912633	−	1.58073i	0.102303	−	0.994753i	$-0.467379\pi$
$911$	25094.2	−	43464.5i	0.912633	−	1.58073i	0.810330	−	0.585974i	$-0.199288\pi$
$912$		0			0
$913$	155.599	+	269.505i	0.00564027	+	0.00976924i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	7562.28			0.272332
$918$		0			0
$919$	5586.87			0.200537			0.100269	−	0.994960i	$-0.468030\pi$
$919$	5586.87			0.200537			0.100269	+	0.994960i	$0.468030\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−8748.19	−	15152.3i	−0.311972	−	0.540352i
$924$		0			0
$925$	34453.7	−	59675.5i	1.22468	−	2.12121i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−5121.09	+	8869.98i	−0.180858	+	0.313256i	−0.942173	−	0.335127i	$-0.891221\pi$
$929$	−5121.09	+	8869.98i	−0.180858	+	0.313256i	0.761315	+	0.648383i	$0.224554\pi$
$930$		0			0
$931$	16159.6	+	27989.3i	0.568861	+	0.985296i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	5641.26			0.197314
$936$		0			0
$937$	−52148.1			−1.81815			−0.909074	−	0.416634i	$-0.863210\pi$
$937$	−52148.1			−1.81815			−0.909074	+	0.416634i	$0.863210\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	22218.7	+	38484.0i	0.769724	+	1.33320i	0.937713	+	0.347412i	$0.112939\pi$
$941$	22218.7	+	38484.0i	0.769724	+	1.33320i	−0.167989	+	0.985789i	$0.553727\pi$
$942$		0			0
$943$	25624.9	−	44383.7i	0.884901	−	1.53269i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−5057.66	+	8760.12i	−0.173550	+	0.300597i	−0.939658	−	0.342114i	$-0.888857\pi$
$947$	−5057.66	+	8760.12i	−0.173550	+	0.300597i	0.766109	+	0.642711i	$0.222190\pi$
$948$		0			0
$949$	10948.6	+	18963.5i	0.374507	+	0.648665i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	14693.2			0.499433			0.249716	−	0.968319i	$-0.419663\pi$
$953$	14693.2			0.499433			0.249716	+	0.968319i	$0.419663\pi$
$954$		0			0
$955$	37389.5			1.26691
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−3967.56	−	6872.01i	−0.133597	−	0.231396i
$960$		0			0
$961$	11876.5	−	20570.7i	0.398661	−	0.690500i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	25962.5	−	44968.3i	0.866074	−	1.50008i
$966$		0			0
$967$	−27829.9	−	48202.8i	−0.925491	−	1.60300i	−0.790769	−	0.612114i	$-0.790319\pi$
$967$	−27829.9	−	48202.8i	−0.925491	−	1.60300i	−0.134722	−	0.990883i	$-0.543014\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−52103.0			−1.72200			−0.861001	−	0.508604i	$-0.830162\pi$
$971$	−52103.0			−1.72200			−0.861001	+	0.508604i	$0.830162\pi$
$972$		0			0
$973$	−13978.6			−0.460569
$974$		0			0
$975$		0			0
$976$		0			0
$977$	23067.6	+	39954.3i	0.755372	+	1.30834i	0.945189	+	0.326523i	$0.105877\pi$
$977$	23067.6	+	39954.3i	0.755372	+	1.30834i	−0.189817	+	0.981819i	$0.560789\pi$
$978$		0			0
$979$	2228.63	−	3860.10i	0.0727552	−	0.126016i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	808.890	−	1401.04i	0.0262458	−	0.0454590i	−0.852604	−	0.522557i	$-0.824978\pi$
$983$	808.890	−	1401.04i	0.0262458	−	0.0454590i	0.878850	+	0.477098i	$0.158311\pi$
$984$		0			0
$985$	−24097.2	−	41737.5i	−0.779492	−	1.35012i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	11663.8			0.375013
$990$		0			0
$991$	6422.76			0.205879			0.102939	−	0.994688i	$-0.467175\pi$
$991$	6422.76			0.205879			0.102939	+	0.994688i	$0.467175\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−3072.20	−	5321.21i	−0.0978848	−	0.169541i
$996$		0			0
$997$	−19532.4	+	33831.0i	−0.620457	+	1.07466i	0.368943	+	0.929452i	$0.379720\pi$
$997$	−19532.4	+	33831.0i	−0.620457	+	1.07466i	−0.989401	+	0.145212i	$0.953614\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.4.i.u.217.1		8
3.2	odd	2		648.4.i.v.217.4		8
9.2	odd	6		648.4.a.g.1.1	✓	4
9.4	even	3	inner	648.4.i.u.433.1		8
9.5	odd	6		648.4.i.v.433.4		8
9.7	even	3		648.4.a.j.1.4	yes	4
36.7	odd	6		1296.4.a.bb.1.4		4
36.11	even	6		1296.4.a.x.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.4.a.g.1.1	✓	4	9.2	odd	6
648.4.a.j.1.4	yes	4	9.7	even	3
648.4.i.u.217.1		8	1.1	even	1	trivial
648.4.i.u.433.1		8	9.4	even	3	inner
648.4.i.v.217.4		8	3.2	odd	2
648.4.i.v.433.4		8	9.5	odd	6
1296.4.a.x.1.1		4	36.11	even	6
1296.4.a.bb.1.4		4	36.7	odd	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)

\(f(q)\)	\(=\)	\(q+(-8.99236 - 15.5752i) q^{5} +(-3.64155 + 6.30734i) q^{7} +O(q^{10})\)	\(q+(-8.99236 - 15.5752i) q^{5} +(-3.64155 + 6.30734i) q^{7} +(2.30734 - 3.99643i) q^{11} +(-14.9301 - 25.8597i) q^{13} -67.9721 q^{17} +111.462 q^{19} +(-109.253 - 189.232i) q^{23} +(-99.2250 + 171.863i) q^{25} +(17.1357 - 29.6798i) q^{29} +(38.8523 + 67.2942i) q^{31} +130.984 q^{35} -347.228 q^{37} +(117.273 + 203.123i) q^{41} +(-26.6899 + 46.2283i) q^{43} +(-192.701 + 333.768i) q^{47} +(144.978 + 251.110i) q^{49} +461.661 q^{53} -82.9938 q^{55} +(3.58164 + 6.20358i) q^{59} +(-208.054 + 360.361i) q^{61} +(-268.514 + 465.079i) q^{65} +(434.637 + 752.814i) q^{67} +585.943 q^{71} -733.324 q^{73} +(16.8046 + 29.1064i) q^{77} +(-585.448 + 1014.03i) q^{79} +(-33.7182 + 58.4016i) q^{83} +(611.229 + 1058.68i) q^{85} +965.886 q^{89} +217.475 q^{91} +(-1002.31 - 1736.05i) q^{95} +(-0.716493 + 1.24100i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{5}+O(q^{10}) \) 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

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