LMFDB - Embedded newform 384.5.e.b.257.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,5,Mod(257,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("384.257");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	384.e (of order $2$, degree $1$, 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:	$39.6940658242$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 32 x^{14} + 356 x^{13} + 1348 x^{12} - 8992 x^{11} + 22064 x^{10} + \cdots + 21479188203 $ x^16 - 4x^15 - 32x^14 + 356x^13 + 1348x^12 - 8992x^11 + 22064x^10 + 391324x^9 + 724325x^8 - 2262056x^7 + 45109352x^6 + 210288484x^5 + 385003542x^4 + 866573964x^3 + 12462543468x^2 + 25672770504*x + 21479188203
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{54}\cdot 3^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			257.4
Root			$-3.05642 + 3.40211i$ of defining polynomial
Character	$\chi$	$=$	384.257
Dual form			384.5.e.b.257.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-7.95805 + 4.20350i) q^{3} +19.3338i q^{5} +67.6413 q^{7} +(45.6611 - 66.9034i) q^{9} +O(q^{10})$	\(q+(-7.95805 + 4.20350i) q^{3} +19.3338i q^{5} +67.6413 q^{7} +(45.6611 - 66.9034i) q^{9} -77.9559i q^{11} -79.2239 q^{13} +(-81.2697 - 153.859i) q^{15} -309.300i q^{17} -468.110 q^{19} +(-538.293 + 284.331i) q^{21} +617.359i q^{23} +251.204 q^{25} +(-82.1450 + 724.357i) q^{27} -939.786i q^{29} -1103.99 q^{31} +(327.688 + 620.377i) q^{33} +1307.77i q^{35} +720.359 q^{37} +(630.468 - 333.018i) q^{39} -2300.12i q^{41} -2032.68 q^{43} +(1293.50 + 882.804i) q^{45} -2151.47i q^{47} +2174.35 q^{49} +(1300.14 + 2461.42i) q^{51} -2378.30i q^{53} +1507.19 q^{55} +(3725.25 - 1967.70i) q^{57} -2586.95i q^{59} +3135.05 q^{61} +(3088.58 - 4525.43i) q^{63} -1531.70i q^{65} +2472.03 q^{67} +(-2595.07 - 4912.97i) q^{69} -7746.80i q^{71} -10238.3 q^{73} +(-1999.09 + 1055.94i) q^{75} -5273.04i q^{77} +5314.05 q^{79} +(-2391.12 - 6109.77i) q^{81} +10678.6i q^{83} +5979.95 q^{85} +(3950.39 + 7478.86i) q^{87} +2024.39i q^{89} -5358.81 q^{91} +(8785.63 - 4640.64i) q^{93} -9050.36i q^{95} +16858.0 q^{97} +(-5215.51 - 3559.55i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 8 q^{3} + 80 q^{7}+O(q^{10}) $ 16 * q - 8 * q^3 + 80 * q^7	$ 16 q - 8 q^{3} + 80 q^{7} + 416 q^{15} + 816 q^{19} + 608 q^{21} - 2000 q^{25} + 280 q^{27} + 592 q^{31} - 496 q^{33} + 2240 q^{37} - 16 q^{39} + 368 q^{43} + 800 q^{45} + 3984 q^{49} - 352 q^{51} + 1920 q^{55} + 560 q^{57} - 3520 q^{61} - 816 q^{63} - 3536 q^{67} - 10784 q^{69} + 3680 q^{73} + 5112 q^{75} - 14448 q^{79} - 624 q^{81} - 11136 q^{85} - 14944 q^{87} - 22944 q^{91} + 13760 q^{93} + 3264 q^{97} - 26976 q^{99}+O(q^{100}) $ 16 * q - 8 * q^3 + 80 * q^7 + 416 * q^15 + 816 * q^19 + 608 * q^21 - 2000 * q^25 + 280 * q^27 + 592 * q^31 - 496 * q^33 + 2240 * q^37 - 16 * q^39 + 368 * q^43 + 800 * q^45 + 3984 * q^49 - 352 * q^51 + 1920 * q^55 + 560 * q^57 - 3520 * q^61 - 816 * q^63 - 3536 * q^67 - 10784 * q^69 + 3680 * q^73 + 5112 * q^75 - 14448 * q^79 - 624 * q^81 - 11136 * q^85 - 14944 * q^87 - 22944 * q^91 + 13760 * q^93 + 3264 * q^97 - 26976 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$.

$n$	$127$	$133$	$257$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−7.95805	+	4.20350i	−0.884228	+	0.467056i
$3$	−7.95805	+	4.20350i	−0.884228	+	0.467056i
$4$		0			0
$5$			19.3338i			0.773353i	0.922216	+	0.386676i	$0.126377\pi$
$5$			19.3338i			0.773353i	−0.922216	+	0.386676i	$0.873623\pi$
$6$		0			0
$7$	67.6413			1.38044			0.690218	−	0.723602i	$-0.257515\pi$
$7$	67.6413			1.38044			0.690218	+	0.723602i	$0.257515\pi$
$8$		0			0
$9$	45.6611	−	66.9034i	0.563718	−	0.825968i
$10$		0			0
$11$		−	77.9559i		−	0.644264i	−0.946695	−	0.322132i	$-0.895601\pi$
$11$		−	77.9559i		−	0.644264i	0.946695	−	0.322132i	$-0.104399\pi$
$12$		0			0
$13$	−79.2239			−0.468781			−0.234390	−	0.972143i	$-0.575309\pi$
$13$	−79.2239			−0.468781			−0.234390	+	0.972143i	$0.575309\pi$
$14$		0			0
$15$	−81.2697	−	153.859i	−0.361199	−	0.683820i
$16$		0			0
$17$		−	309.300i		−	1.07024i	−0.844775	−	0.535121i	$-0.820266\pi$
$17$		−	309.300i		−	1.07024i	0.844775	−	0.535121i	$-0.179734\pi$
$18$		0			0
$19$	−468.110			−1.29670			−0.648352	−	0.761340i	$-0.724542\pi$
$19$	−468.110			−1.29670			−0.648352	+	0.761340i	$0.724542\pi$
$20$		0			0
$21$	−538.293	+	284.331i	−1.22062	+	0.644741i
$22$		0			0
$23$			617.359i			1.16703i	0.812102	+	0.583515i	$0.198323\pi$
$23$			617.359i			1.16703i	−0.812102	+	0.583515i	$0.801677\pi$
$24$		0			0
$25$	251.204			0.401926
$26$		0			0
$27$	−82.1450	+	724.357i	−0.112682	+	0.993631i
$28$		0			0
$29$		−	939.786i		−	1.11746i	−0.829349	−	0.558731i	$-0.811288\pi$
$29$		−	939.786i		−	1.11746i	0.829349	−	0.558731i	$-0.188712\pi$
$30$		0			0
$31$	−1103.99			−1.14880			−0.574398	−	0.818576i	$-0.694764\pi$
$31$	−1103.99			−1.14880			−0.574398	+	0.818576i	$0.694764\pi$
$32$		0			0
$33$	327.688	+	620.377i	0.300907	+	0.569676i
$34$		0			0
$35$			1307.77i			1.06756i
$36$		0			0
$37$	720.359			0.526193			0.263097	−	0.964769i	$-0.415256\pi$
$37$	720.359			0.526193			0.263097	+	0.964769i	$0.415256\pi$
$38$		0			0
$39$	630.468	−	333.018i	0.414509	−	0.218947i
$40$		0			0
$41$		−	2300.12i		−	1.36830i	−0.729340	−	0.684151i	$-0.760173\pi$
$41$		−	2300.12i		−	1.36830i	0.729340	−	0.684151i	$-0.239827\pi$
$42$		0			0
$43$	−2032.68			−1.09934			−0.549669	−	0.835382i	$-0.685246\pi$
$43$	−2032.68			−1.09934			−0.549669	+	0.835382i	$0.685246\pi$
$44$		0			0
$45$	1293.50	+	882.804i	0.638764	+	0.435952i
$46$		0			0
$47$		−	2151.47i		−	0.973958i	−0.873414	−	0.486979i	$-0.838099\pi$
$47$		−	2151.47i		−	0.973958i	0.873414	−	0.486979i	$-0.161901\pi$
$48$		0			0
$49$	2174.35			0.905602
$50$		0			0
$51$	1300.14	+	2461.42i	0.499863	+	0.946338i
$52$		0			0
$53$		−	2378.30i		−	0.846671i	−0.905973	−	0.423335i	$-0.860859\pi$
$53$		−	2378.30i		−	0.846671i	0.905973	−	0.423335i	$-0.139141\pi$
$54$		0			0
$55$	1507.19			0.498243
$56$		0			0
$57$	3725.25	−	1967.70i	1.14658	−	0.605634i
$58$		0			0
$59$		−	2586.95i		−	0.743162i	−0.928401	−	0.371581i	$-0.878816\pi$
$59$		−	2586.95i		−	0.743162i	0.928401	−	0.371581i	$-0.121184\pi$
$60$		0			0
$61$	3135.05			0.842529			0.421265	−	0.906938i	$-0.361586\pi$
$61$	3135.05			0.842529			0.421265	+	0.906938i	$0.361586\pi$
$62$		0			0
$63$	3088.58	−	4525.43i	0.778176	−	1.14019i
$64$		0			0
$65$		−	1531.70i		−	0.362533i
$66$		0			0
$67$	2472.03			0.550687			0.275343	−	0.961346i	$-0.411208\pi$
$67$	2472.03			0.550687			0.275343	+	0.961346i	$0.411208\pi$
$68$		0			0
$69$	−2595.07	−	4912.97i	−0.545068	−	1.03192i
$70$		0			0
$71$		−	7746.80i		−	1.53676i	−0.639994	−	0.768380i	$-0.721063\pi$
$71$		−	7746.80i		−	1.53676i	0.639994	−	0.768380i	$-0.278937\pi$
$72$		0			0
$73$	−10238.3			−1.92123			−0.960617	−	0.277877i	$-0.910369\pi$
$73$	−10238.3			−1.92123			−0.960617	+	0.277877i	$0.910369\pi$
$74$		0			0
$75$	−1999.09	+	1055.94i	−0.355394	+	0.187722i
$76$		0			0
$77$		−	5273.04i		−	0.889365i
$78$		0			0
$79$	5314.05			0.851474			0.425737	−	0.904847i	$-0.360015\pi$
$79$	5314.05			0.851474			0.425737	+	0.904847i	$0.360015\pi$
$80$		0			0
$81$	−2391.12	−	6109.77i	−0.364445	−	0.931225i
$82$		0			0
$83$			10678.6i			1.55010i	0.631903	+	0.775048i	$0.282274\pi$
$83$			10678.6i			1.55010i	−0.631903	+	0.775048i	$0.717726\pi$
$84$		0			0
$85$	5979.95			0.827675
$86$		0			0
$87$	3950.39	+	7478.86i	0.521917	+	0.988091i
$88$		0			0
$89$			2024.39i			0.255572i	0.991802	+	0.127786i	$0.0407871\pi$
$89$			2024.39i			0.255572i	−0.991802	+	0.127786i	$0.959213\pi$
$90$		0			0
$91$	−5358.81			−0.647121
$92$		0			0
$93$	8785.63	−	4640.64i	1.01580	−	0.536552i
$94$		0			0
$95$		−	9050.36i		−	1.00281i
$96$		0			0
$97$	16858.0			1.79169			0.895846	−	0.444365i	$-0.146571\pi$
$97$	16858.0			1.79169			0.895846	+	0.444365i	$0.146571\pi$
$98$		0			0
$99$	−5215.51	−	3559.55i	−0.532141	−	0.363183i
$100$		0			0
$101$		−	19505.3i		−	1.91210i	−0.293209	−	0.956048i	$-0.594723\pi$
$101$		−	19505.3i		−	1.91210i	0.293209	−	0.956048i	$-0.405277\pi$
$102$		0			0
$103$	8458.43			0.797288			0.398644	−	0.917106i	$-0.369481\pi$
$103$	8458.43			0.797288			0.398644	+	0.917106i	$0.369481\pi$
$104$		0			0
$105$	−5497.19	−	10407.3i	−0.498612	−	0.943969i
$106$		0			0
$107$			11493.8i			1.00392i	0.864892	+	0.501958i	$0.167387\pi$
$107$			11493.8i			1.00392i	−0.864892	+	0.501958i	$0.832613\pi$
$108$		0			0
$109$	5163.95			0.434639			0.217319	−	0.976101i	$-0.430269\pi$
$109$	5163.95			0.434639			0.217319	+	0.976101i	$0.430269\pi$
$110$		0			0
$111$	−5732.65	+	3028.03i	−0.465275	+	0.245762i
$112$		0			0
$113$			4118.50i			0.322539i	0.986910	+	0.161269i	$0.0515588\pi$
$113$			4118.50i			0.322539i	−0.986910	+	0.161269i	$0.948441\pi$
$114$		0			0
$115$	−11935.9			−0.902526
$116$		0			0
$117$	−3617.45	+	5300.35i	−0.264260	+	0.387198i
$118$		0			0
$119$		−	20921.5i		−	1.47740i
$120$		0			0
$121$	8563.88			0.584924
$122$		0			0
$123$	9668.55	+	18304.4i	0.639074	+	1.20989i
$124$		0			0
$125$			16940.4i			1.08418i
$126$		0			0
$127$	−6955.17			−0.431221			−0.215611	−	0.976479i	$-0.569174\pi$
$127$	−6955.17			−0.431221			−0.215611	+	0.976479i	$0.569174\pi$
$128$		0			0
$129$	16176.1	−	8544.36i	0.972066	−	0.513453i
$130$		0			0
$131$		−	12734.3i		−	0.742048i	−0.928623	−	0.371024i	$-0.879007\pi$
$131$		−	12734.3i		−	0.742048i	0.928623	−	0.371024i	$-0.120993\pi$
$132$		0			0
$133$	−31663.6			−1.79002
$134$		0			0
$135$	−14004.6	−	1588.18i	−0.768427	−	0.0871427i
$136$		0			0
$137$		−	23048.5i		−	1.22801i	−0.789302	−	0.614006i	$-0.789557\pi$
$137$		−	23048.5i		−	1.22801i	0.789302	−	0.614006i	$-0.210443\pi$
$138$		0			0
$139$	13779.7			0.713199			0.356600	−	0.934257i	$-0.383936\pi$
$139$	13779.7			0.713199			0.356600	+	0.934257i	$0.383936\pi$
$140$		0			0
$141$	9043.72	+	17121.5i	0.454893	+	0.861200i
$142$		0			0
$143$			6175.97i			0.302018i
$144$		0			0
$145$	18169.6			0.864192
$146$		0			0
$147$	−17303.6	+	9139.89i	−0.800759	+	0.422967i
$148$		0			0
$149$		−	861.178i		−	0.0387901i	−0.999812	−	0.0193950i	$-0.993826\pi$
$149$		−	861.178i		−	0.0387901i	0.999812	−	0.0193950i	$-0.00617402\pi$
$150$		0			0
$151$	−11857.7			−0.520052			−0.260026	−	0.965602i	$-0.583731\pi$
$151$	−11857.7			−0.520052			−0.260026	+	0.965602i	$0.583731\pi$
$152$		0			0
$153$	−20693.2	−	14123.0i	−0.883985	−	0.603314i
$154$		0			0
$155$		−	21344.4i		−	0.888424i
$156$		0			0
$157$	24128.3			0.978874			0.489437	−	0.872039i	$-0.337202\pi$
$157$	24128.3			0.978874			0.489437	+	0.872039i	$0.337202\pi$
$158$		0			0
$159$	9997.18	+	18926.6i	0.395443	+	0.748650i
$160$		0			0
$161$			41759.0i			1.61101i
$162$		0			0
$163$	18099.6			0.681230			0.340615	−	0.940203i	$-0.389365\pi$
$163$	18099.6			0.681230			0.340615	+	0.940203i	$0.389365\pi$
$164$		0			0
$165$	−11994.3	+	6335.46i	−0.440560	+	0.232707i
$166$		0			0
$167$			13135.2i			0.470981i	0.971877	+	0.235490i	$0.0756696\pi$
$167$			13135.2i			0.470981i	−0.971877	+	0.235490i	$0.924330\pi$
$168$		0			0
$169$	−22284.6			−0.780245
$170$		0			0
$171$	−21374.5	+	31318.2i	−0.730975	+	1.07104i
$172$		0			0
$173$		−	37599.9i		−	1.25630i	−0.778092	−	0.628151i	$-0.783812\pi$
$173$		−	37599.9i		−	1.25630i	0.778092	−	0.628151i	$-0.216188\pi$
$174$		0			0
$175$	16991.8			0.554833
$176$		0			0
$177$	10874.2	+	20587.1i	0.347098	+	0.657125i
$178$		0			0
$179$		−	30293.4i		−	0.945456i	−0.881208	−	0.472728i	$-0.843269\pi$
$179$		−	30293.4i		−	0.945456i	0.881208	−	0.472728i	$-0.156731\pi$
$180$		0			0
$181$	−41419.0			−1.26428			−0.632138	−	0.774855i	$-0.717823\pi$
$181$	−41419.0			−1.26428			−0.632138	+	0.774855i	$0.717823\pi$
$182$		0			0
$183$	−24948.9	+	13178.2i	−0.744988	+	0.393508i
$184$		0			0
$185$			13927.3i			0.406933i
$186$		0			0
$187$	−24111.8			−0.689518
$188$		0			0
$189$	−5556.40	+	48996.5i	−0.155550	+	1.37164i
$190$		0			0
$191$			26217.3i			0.718655i	0.933211	+	0.359328i	$0.116994\pi$
$191$			26217.3i			0.718655i	−0.933211	+	0.359328i	$0.883006\pi$
$192$		0			0
$193$	−24871.7			−0.667716			−0.333858	−	0.942623i	$-0.608351\pi$
$193$	−24871.7			−0.667716			−0.333858	+	0.942623i	$0.608351\pi$
$194$		0			0
$195$	6438.51	+	12189.4i	0.169323	+	0.320561i
$196$		0			0
$197$		−	44861.5i		−	1.15595i	−0.816053	−	0.577977i	$-0.803842\pi$
$197$		−	44861.5i		−	1.15595i	0.816053	−	0.577977i	$-0.196158\pi$
$198$		0			0
$199$	1782.37			0.0450081			0.0225041	−	0.999747i	$-0.492836\pi$
$199$	1782.37			0.0450081			0.0225041	+	0.999747i	$0.492836\pi$
$200$		0			0
$201$	−19672.6	+	10391.2i	−0.486932	+	0.257201i
$202$		0			0
$203$		−	63568.4i		−	1.54258i
$204$		0			0
$205$	44470.0			1.05818
$206$		0			0
$207$	41303.4	+	28189.3i	0.963929	+	0.657876i
$208$		0			0
$209$			36492.0i			0.835420i
$210$		0			0
$211$	27516.8			0.618064			0.309032	−	0.951052i	$-0.399995\pi$
$211$	27516.8			0.618064			0.309032	+	0.951052i	$0.399995\pi$
$212$		0			0
$213$	32563.7	+	61649.5i	0.717753	+	1.35885i
$214$		0			0
$215$		−	39299.4i		−	0.850176i
$216$		0			0
$217$	−74675.5			−1.58584
$218$		0			0
$219$	81476.5	−	43036.5i	1.69881	−	0.897323i
$220$		0			0
$221$			24504.0i			0.501709i
$222$		0			0
$223$	−58322.1			−1.17280			−0.586399	−	0.810023i	$-0.699455\pi$
$223$	−58322.1			−1.17280			−0.586399	+	0.810023i	$0.699455\pi$
$224$		0			0
$225$	11470.2	−	16806.4i	0.226573	−	0.331978i
$226$		0			0
$227$		−	96363.0i		−	1.87007i	−0.354552	−	0.935036i	$-0.615367\pi$
$227$		−	96363.0i		−	1.87007i	0.354552	−	0.935036i	$-0.384633\pi$
$228$		0			0
$229$	−63619.0			−1.21315			−0.606577	−	0.795025i	$-0.707458\pi$
$229$	−63619.0			−1.21315			−0.606577	+	0.795025i	$0.707458\pi$
$230$		0			0
$231$	22165.2	+	41963.1i	0.415383	+	0.786401i
$232$		0			0
$233$			104862.i			1.93155i	0.259380	+	0.965775i	$0.416482\pi$
$233$			104862.i			1.93155i	−0.259380	+	0.965775i	$0.583518\pi$
$234$		0			0
$235$	41596.2			0.753213
$236$		0			0
$237$	−42289.4	+	22337.6i	−0.752897	+	0.397686i
$238$		0			0
$239$		−	54132.9i		−	0.947688i	−0.880609	−	0.473844i	$-0.842866\pi$
$239$		−	54132.9i		−	0.947688i	0.880609	−	0.473844i	$-0.157134\pi$
$240$		0			0
$241$	18650.0			0.321103			0.160552	−	0.987027i	$-0.448673\pi$
$241$	18650.0			0.321103			0.160552	+	0.987027i	$0.448673\pi$
$242$		0			0
$243$	44711.1	+	38570.7i	0.757186	+	0.653199i
$244$		0			0
$245$			42038.5i			0.700350i
$246$		0			0
$247$	37085.6			0.607870
$248$		0			0
$249$	−44887.6	−	84980.9i	−0.723981	−	1.37064i
$250$		0			0
$251$			1998.43i			0.0317206i	0.999874	+	0.0158603i	$0.00504871\pi$
$251$			1998.43i			0.0317206i	−0.999874	+	0.0158603i	$0.994951\pi$
$252$		0			0
$253$	48126.8			0.751875
$254$		0			0
$255$	−47588.7	+	25136.7i	−0.731853	+	0.386570i
$256$		0			0
$257$			68298.2i			1.03405i	0.855969	+	0.517027i	$0.172961\pi$
$257$			68298.2i			1.03405i	−0.855969	+	0.517027i	$0.827039\pi$
$258$		0			0
$259$	48726.0			0.726376
$260$		0			0
$261$	−62874.8	−	42911.7i	−0.922987	−	0.629933i
$262$		0			0
$263$			84260.2i			1.21818i	0.793102	+	0.609089i	$0.208465\pi$
$263$			84260.2i			1.21818i	−0.793102	+	0.609089i	$0.791535\pi$
$264$		0			0
$265$	45981.6			0.654775
$266$		0			0
$267$	−8509.52	−	16110.2i	−0.119367	−	0.225984i
$268$		0			0
$269$			84312.3i			1.16516i	0.812773	+	0.582581i	$0.197957\pi$
$269$			84312.3i			1.16516i	−0.812773	+	0.582581i	$0.802043\pi$
$270$		0			0
$271$	−97886.0			−1.33285			−0.666426	−	0.745571i	$-0.732177\pi$
$271$	−97886.0			−1.33285			−0.666426	+	0.745571i	$0.732177\pi$
$272$		0			0
$273$	42645.7	−	22525.8i	0.572203	−	0.302242i
$274$		0			0
$275$		−	19582.8i		−	0.258946i
$276$		0			0
$277$	104604.			1.36329			0.681646	−	0.731682i	$-0.261264\pi$
$277$	104604.			1.36329			0.681646	+	0.731682i	$0.261264\pi$
$278$		0			0
$279$	−50409.6	+	73860.8i	−0.647596	+	0.948868i
$280$		0			0
$281$			76354.4i			0.966988i	0.875348	+	0.483494i	$0.160633\pi$
$281$			76354.4i			0.966988i	−0.875348	+	0.483494i	$0.839367\pi$
$282$		0			0
$283$	−129472.			−1.61660			−0.808300	−	0.588771i	$-0.799612\pi$
$283$	−129472.			−1.61660			−0.808300	+	0.588771i	$0.799612\pi$
$284$		0			0
$285$	38043.2	+	72023.2i	0.468368	+	0.886713i
$286$		0			0
$287$		−	155583.i		−	1.88885i
$288$		0			0
$289$	−12145.5			−0.145418
$290$		0			0
$291$	−134157.	+	70862.8i	−1.58426	+	0.836820i
$292$		0			0
$293$		−	100946.i		−	1.17585i	−0.808915	−	0.587926i	$-0.799945\pi$
$293$		−	100946.i		−	1.17585i	0.808915	−	0.587926i	$-0.200055\pi$
$294$		0			0
$295$	50015.6			0.574726
$296$		0			0
$297$	56467.9	+	6403.69i	0.640161	+	0.0725968i
$298$		0			0
$299$		−	48909.6i		−	0.547081i
$300$		0			0
$301$	−137493.			−1.51757
$302$		0			0
$303$	81990.6	+	155224.i	0.893056	+	1.69073i
$304$		0			0
$305$			60612.5i			0.651572i
$306$		0			0
$307$	−11808.0			−0.125285			−0.0626426	−	0.998036i	$-0.519953\pi$
$307$	−11808.0			−0.125285			−0.0626426	+	0.998036i	$0.519953\pi$
$308$		0			0
$309$	−67312.6	+	35555.0i	−0.704985	+	0.372378i
$310$		0			0
$311$			36362.0i			0.375947i	0.982174	+	0.187974i	$0.0601920\pi$
$311$			36362.0i			0.375947i	−0.982174	+	0.187974i	$0.939808\pi$
$312$		0			0
$313$	−66566.3			−0.679463			−0.339731	−	0.940522i	$-0.610336\pi$
$313$	−66566.3			−0.679463			−0.339731	+	0.940522i	$0.610336\pi$
$314$		0			0
$315$	87493.9	+	59714.0i	0.881773	+	0.601804i
$316$		0			0
$317$		−	105833.i		−	1.05318i	−0.850119	−	0.526591i	$-0.823470\pi$
$317$		−	105833.i		−	1.05318i	0.850119	−	0.526591i	$-0.176530\pi$
$318$		0			0
$319$	−73261.8			−0.719940
$320$		0			0
$321$	−48314.4	−	91468.5i	−0.468885	−	0.887691i
$322$		0			0
$323$			144787.i			1.38779i
$324$		0			0
$325$	−19901.3			−0.188415
$326$		0			0
$327$	−41094.9	+	21706.7i	−0.384320	+	0.203001i
$328$		0			0
$329$		−	145528.i		−	1.34449i
$330$		0			0
$331$	−58819.5			−0.536865			−0.268432	−	0.963298i	$-0.586506\pi$
$331$	−58819.5			−0.536865			−0.268432	+	0.963298i	$0.586506\pi$
$332$		0			0
$333$	32892.4	−	48194.4i	0.296624	−	0.434619i
$334$		0			0
$335$			47793.8i			0.425875i
$336$		0			0
$337$	19308.9			0.170020			0.0850098	−	0.996380i	$-0.472908\pi$
$337$	19308.9			0.170020			0.0850098	+	0.996380i	$0.472908\pi$
$338$		0			0
$339$	−17312.1	−	32775.2i	−0.150644	−	0.285198i
$340$		0			0
$341$			86062.8i			0.740127i
$342$		0			0
$343$	−15330.9			−0.130310
$344$		0			0
$345$	94986.5	−	50172.6i	0.798039	−	0.421530i
$346$		0			0
$347$		−	49750.9i		−	0.413182i	−0.978427	−	0.206591i	$-0.933763\pi$
$347$		−	49750.9i		−	0.413182i	0.978427	−	0.206591i	$-0.0662370\pi$
$348$		0			0
$349$	−149030.			−1.22355			−0.611777	−	0.791030i	$-0.709545\pi$
$349$	−149030.			−1.22355			−0.611777	+	0.791030i	$0.709545\pi$
$350$		0			0
$351$	6507.85	−	57386.4i	0.0528230	−	0.465795i
$352$		0			0
$353$		−	98810.0i		−	0.792960i	−0.918043	−	0.396480i	$-0.870232\pi$
$353$		−	98810.0i		−	0.792960i	0.918043	−	0.396480i	$-0.129768\pi$
$354$		0			0
$355$	149775.			1.18846
$356$		0			0
$357$	87943.4	+	166494.i	0.690028	+	1.30636i
$358$		0			0
$359$		−	233349.i		−	1.81058i	−0.424798	−	0.905288i	$-0.639655\pi$
$359$		−	233349.i		−	1.81058i	0.424798	−	0.905288i	$-0.360345\pi$
$360$		0			0
$361$	88806.4			0.681444
$362$		0			0
$363$	−68151.7	+	35998.3i	−0.517206	+	0.273192i
$364$		0			0
$365$		−	197944.i		−	1.48579i
$366$		0			0
$367$	187106.			1.38917			0.694587	−	0.719409i	$-0.255587\pi$
$367$	187106.			1.38917			0.694587	+	0.719409i	$0.255587\pi$
$368$		0			0
$369$	−153886.	−	105026.i	−1.13017	−	0.771336i
$370$		0			0
$371$		−	160871.i		−	1.16877i
$372$		0			0
$373$	33982.5			0.244252			0.122126	−	0.992515i	$-0.461029\pi$
$373$	33982.5			0.244252			0.122126	+	0.992515i	$0.461029\pi$
$374$		0			0
$375$	−71208.8	−	134812.i	−0.506374	−	0.958665i
$376$		0			0
$377$			74453.5i			0.523845i
$378$		0			0
$379$	−150006.			−1.04431			−0.522156	−	0.852850i	$-0.674872\pi$
$379$	−150006.			−1.04431			−0.522156	+	0.852850i	$0.674872\pi$
$380$		0			0
$381$	55349.6	−	29236.1i	0.381298	−	0.201405i
$382$		0			0
$383$		−	71272.7i		−	0.485876i	−0.970042	−	0.242938i	$-0.921889\pi$
$383$		−	71272.7i		−	0.485876i	0.970042	−	0.242938i	$-0.0781112\pi$
$384$		0			0
$385$	101948.			0.687792
$386$		0			0
$387$	−92814.3	+	135993.i	−0.619717	+	0.908018i
$388$		0			0
$389$		−	90246.1i		−	0.596388i	−0.954505	−	0.298194i	$-0.903616\pi$
$389$		−	90246.1i		−	0.596388i	0.954505	−	0.298194i	$-0.0963843\pi$
$390$		0			0
$391$	190949.			1.24901
$392$		0			0
$393$	53528.6	+	101340.i	0.346578	+	0.656139i
$394$		0			0
$395$			102741.i			0.658489i
$396$		0			0
$397$	145813.			0.925158			0.462579	−	0.886578i	$-0.346924\pi$
$397$	145813.			0.925158			0.462579	+	0.886578i	$0.346924\pi$
$398$		0			0
$399$	251981.	−	133098.i	1.58278	−	0.836038i
$400$		0			0
$401$		−	108331.i		−	0.673695i	−0.941559	−	0.336848i	$-0.890639\pi$
$401$		−	108331.i		−	0.673695i	0.941559	−	0.336848i	$-0.109361\pi$
$402$		0			0
$403$	87462.6			0.538533
$404$		0			0
$405$	118125.	−	46229.5i	0.720165	−	0.281844i
$406$		0			0
$407$		−	56156.2i		−	0.339007i
$408$		0			0
$409$	294079.			1.75800			0.878998	−	0.476825i	$-0.158213\pi$
$409$	294079.			1.75800			0.878998	+	0.476825i	$0.158213\pi$
$410$		0			0
$411$	96884.6	+	183421.i	0.573550	+	1.08584i
$412$		0			0
$413$		−	174985.i		−	1.02589i
$414$		0			0
$415$	−206458.			−1.19877
$416$		0			0
$417$	−109660.	+	57923.1i	−0.630631	+	0.333104i
$418$		0			0
$419$		−	17811.7i		−	0.101456i	−0.998713	−	0.0507281i	$-0.983846\pi$
$419$		−	17811.7i		−	0.101456i	0.998713	−	0.0507281i	$-0.0161542\pi$
$420$		0			0
$421$	349782.			1.97348			0.986741	−	0.162304i	$-0.0518924\pi$
$421$	349782.			1.97348			0.986741	+	0.162304i	$0.0518924\pi$
$422$		0			0
$423$	−143941.	−	98238.6i	−0.804457	−	0.549037i
$424$		0			0
$425$		−	77697.3i		−	0.430158i
$426$		0			0
$427$	212059.			1.16306
$428$		0			0
$429$	−25960.7	−	49148.7i	−0.141059	−	0.267053i
$430$		0			0
$431$			149894.i			0.806920i	0.914997	+	0.403460i	$0.132193\pi$
$431$			149894.i			0.806920i	−0.914997	+	0.403460i	$0.867807\pi$
$432$		0			0
$433$	−248766.			−1.32683			−0.663415	−	0.748252i	$-0.730893\pi$
$433$	−248766.			−1.32683			−0.663415	+	0.748252i	$0.730893\pi$
$434$		0			0
$435$	−144595.	+	76376.1i	−0.764143	+	0.403626i
$436$		0			0
$437$		−	288992.i		−	1.51329i
$438$		0			0
$439$	120416.			0.624822			0.312411	−	0.949947i	$-0.398863\pi$
$439$	120416.			0.624822			0.312411	+	0.949947i	$0.398863\pi$
$440$		0			0
$441$	99283.3	−	145471.i	0.510504	−	0.747998i
$442$		0			0
$443$			168777.i			0.860014i	0.902826	+	0.430007i	$0.141489\pi$
$443$			168777.i			0.860014i	−0.902826	+	0.430007i	$0.858511\pi$
$444$		0			0
$445$	−39139.1			−0.197647
$446$		0			0
$447$	3619.97	+	6853.30i	0.0181171	+	0.0342993i
$448$		0			0
$449$		−	1278.50i		−	0.00634173i	−0.999995	−	0.00317087i	$-0.998991\pi$
$449$		−	1278.50i		−	0.00634173i	0.999995	−	0.00317087i	$-0.00100932\pi$
$450$		0			0
$451$	−179308.			−0.881548
$452$		0			0
$453$	94364.2	−	49843.9i	0.459844	−	0.242893i
$454$		0			0
$455$		−	103606.i		−	0.500453i
$456$		0			0
$457$	−202856.			−0.971306			−0.485653	−	0.874152i	$-0.661418\pi$
$457$	−202856.			−0.971306			−0.485653	+	0.874152i	$0.661418\pi$
$458$		0			0
$459$	224044.	+	25407.5i	1.06343	+	0.120597i
$460$		0			0
$461$		−	65727.6i		−	0.309276i	−0.987971	−	0.154638i	$-0.950579\pi$
$461$		−	65727.6i		−	0.309276i	0.987971	−	0.154638i	$-0.0494211\pi$
$462$		0			0
$463$	266165.			1.24162			0.620812	−	0.783960i	$-0.286803\pi$
$463$	266165.			1.24162			0.620812	+	0.783960i	$0.286803\pi$
$464$		0			0
$465$	89721.2	+	169860.i	0.414944	+	0.785569i
$466$		0			0
$467$			83490.5i			0.382828i	0.981509	+	0.191414i	$0.0613072\pi$
$467$			83490.5i			0.382828i	−0.981509	+	0.191414i	$0.938693\pi$
$468$		0			0
$469$	167212.			0.760187
$470$		0			0
$471$	−192014.	+	101423.i	−0.865547	+	0.457189i
$472$		0			0
$473$			158459.i			0.708264i
$474$		0			0
$475$	−117591.			−0.521179
$476$		0			0
$477$	−159116.	−	108596.i	−0.699323	−	0.477283i
$478$		0			0
$479$		−	203102.i		−	0.885205i	−0.896718	−	0.442603i	$-0.854055\pi$
$479$		−	203102.i		−	0.885205i	0.896718	−	0.442603i	$-0.145945\pi$
$480$		0			0
$481$	−57069.6			−0.246669
$482$		0			0
$483$	−175534.	−	332320.i	−0.752432	−	1.42450i
$484$		0			0
$485$			325930.i			1.38561i
$486$		0			0
$487$	360891.			1.52166			0.760831	−	0.648950i	$-0.224792\pi$
$487$	360891.			1.52166			0.760831	+	0.648950i	$0.224792\pi$
$488$		0			0
$489$	−144038.	+	76081.7i	−0.602362	+	0.318172i
$490$		0			0
$491$			102193.i			0.423895i	0.977281	+	0.211947i	$0.0679805\pi$
$491$			102193.i			0.423895i	−0.977281	+	0.211947i	$0.932019\pi$
$492$		0			0
$493$	−290676.			−1.19596
$494$		0			0
$495$	68819.8	−	100836.i	0.280868	−	0.411533i
$496$		0			0
$497$		−	524004.i		−	2.12140i
$498$		0			0
$499$	76271.0			0.306308			0.153154	−	0.988202i	$-0.451057\pi$
$499$	76271.0			0.306308			0.153154	+	0.988202i	$0.451057\pi$
$500$		0			0
$501$	−55213.8	−	104530.i	−0.219974	−	0.416454i
$502$		0			0
$503$		−	285141.i		−	1.12700i	−0.826116	−	0.563500i	$-0.809454\pi$
$503$		−	285141.i		−	1.12700i	0.826116	−	0.563500i	$-0.190546\pi$
$504$		0			0
$505$	377112.			1.47873
$506$		0			0
$507$	177342.	−	93673.3i	0.689914	−	0.364418i
$508$		0			0
$509$			169808.i			0.655425i	0.944777	+	0.327713i	$0.106278\pi$
$509$			169808.i			0.655425i	−0.944777	+	0.327713i	$0.893722\pi$
$510$		0			0
$511$	−692529.			−2.65214
$512$		0			0
$513$	38452.9	−	339079.i	0.146115	−	1.28845i
$514$		0			0
$515$			163534.i			0.616585i
$516$		0			0
$517$	−167720.			−0.627486
$518$		0			0
$519$	158051.	+	299222.i	0.586763	+	1.11086i
$520$		0			0
$521$		−	169037.i		−	0.622738i	−0.950289	−	0.311369i	$-0.899213\pi$
$521$		−	169037.i		−	0.622738i	0.950289	−	0.311369i	$-0.100787\pi$
$522$		0			0
$523$	−467508.			−1.70917			−0.854586	−	0.519309i	$-0.826189\pi$
$523$	−467508.			−1.70917			−0.854586	+	0.519309i	$0.826189\pi$
$524$		0			0
$525$	−135221.	+	71424.9i	−0.490598	+	0.259138i
$526$		0			0
$527$			341465.i			1.22949i
$528$		0			0
$529$	−101291.			−0.361960
$530$		0			0
$531$	−173076.	−	118123.i	−0.613828	−	0.418934i
$532$		0			0
$533$			182224.i			0.641434i
$534$		0			0
$535$	−222220.			−0.776381
$536$		0			0
$537$	127338.	+	241076.i	0.441581	+	0.835999i
$538$		0			0
$539$		−	169504.i		−	0.583447i
$540$		0			0
$541$	−228195.			−0.779673			−0.389837	−	0.920884i	$-0.627468\pi$
$541$	−228195.			−0.779673			−0.389837	+	0.920884i	$0.627468\pi$
$542$		0			0
$543$	329614.	−	174105.i	1.11791	−	0.590488i
$544$		0			0
$545$			99838.8i			0.336129i
$546$		0			0
$547$	190129.			0.635440			0.317720	−	0.948185i	$-0.397083\pi$
$547$	190129.			0.635440			0.317720	+	0.948185i	$0.397083\pi$
$548$		0			0
$549$	143150.	−	209745.i	0.474948	−	0.695902i
$550$		0			0
$551$			439923.i			1.44902i
$552$		0			0
$553$	359449.			1.17540
$554$		0			0
$555$	−58543.4	−	110834.i	−0.190060	−	0.359821i
$556$		0			0
$557$			183226.i			0.590579i	0.955408	+	0.295289i	$0.0954160\pi$
$557$			183226.i			0.590579i	−0.955408	+	0.295289i	$0.904584\pi$
$558$		0			0
$559$	161037.			0.515349
$560$		0			0
$561$	191883.	−	101354.i	0.609691	−	0.322044i
$562$		0			0
$563$			339876.i			1.07227i	0.844133	+	0.536134i	$0.180116\pi$
$563$			339876.i			1.07227i	−0.844133	+	0.536134i	$0.819884\pi$
$564$		0			0
$565$	−79626.3			−0.249436
$566$		0			0
$567$	−161739.	−	413273.i	−0.503093	−	1.28550i
$568$		0			0
$569$			263919.i			0.815165i	0.913168	+	0.407582i	$0.133628\pi$
$569$			263919.i			0.815165i	−0.913168	+	0.407582i	$0.866372\pi$
$570$		0			0
$571$	−168475.			−0.516730			−0.258365	−	0.966047i	$-0.583184\pi$
$571$	−168475.			−0.516730			−0.258365	+	0.966047i	$0.583184\pi$
$572$		0			0
$573$	−110204.	−	208638.i	−0.335652	−	0.635455i
$574$		0			0
$575$			155083.i			0.469060i
$576$		0			0
$577$	−452005.			−1.35766			−0.678831	−	0.734295i	$-0.737513\pi$
$577$	−452005.			−1.35766			−0.678831	+	0.734295i	$0.737513\pi$
$578$		0			0
$579$	197931.	−	104548.i	0.590413	−	0.311860i
$580$		0			0
$581$			722315.i			2.13981i
$582$		0			0
$583$	−185402.			−0.545479
$584$		0			0
$585$	−102476.	−	69939.2i	−0.299440	−	0.204366i
$586$		0			0
$587$		−	4585.69i		−	0.0133085i	−0.999978	−	0.00665424i	$-0.997882\pi$
$587$		−	4585.69i		−	0.0133085i	0.999978	−	0.00665424i	$-0.00211813\pi$
$588$		0			0
$589$	516791.			1.48965
$590$		0			0
$591$	188575.	+	357010.i	0.539895	+	1.02213i
$592$		0			0
$593$			277349.i			0.788710i	0.918958	+	0.394355i	$0.129032\pi$
$593$			277349.i			0.788710i	−0.918958	+	0.394355i	$0.870968\pi$
$594$		0			0
$595$	404492.			1.14255
$596$		0			0
$597$	−14184.2	+	7492.18i	−0.0397974	+	0.0210213i
$598$		0			0
$599$			354316.i			0.987501i	0.869604	+	0.493751i	$0.164374\pi$
$599$			354316.i			0.987501i	−0.869604	+	0.493751i	$0.835626\pi$
$600$		0			0
$601$	−286642.			−0.793582			−0.396791	−	0.917909i	$-0.629876\pi$
$601$	−286642.			−0.793582			−0.396791	+	0.917909i	$0.629876\pi$
$602$		0			0
$603$	112876.	−	165387.i	0.310432	−	0.454849i
$604$		0			0
$605$			165572.i			0.452353i
$606$		0			0
$607$	−690944.			−1.87528			−0.937639	−	0.347611i	$-0.886993\pi$
$607$	−690944.			−1.87528			−0.937639	+	0.347611i	$0.886993\pi$
$608$		0			0
$609$	267210.	+	505880.i	0.720473	+	1.36400i
$610$		0			0
$611$			170448.i			0.456572i
$612$		0			0
$613$	139902.			0.372309			0.186154	−	0.982520i	$-0.440398\pi$
$613$	139902.			0.372309			0.186154	+	0.982520i	$0.440398\pi$
$614$		0			0
$615$	−353895.	+	186930.i	−0.935672	+	0.494229i
$616$		0			0
$617$			280737.i			0.737444i	0.929540	+	0.368722i	$0.120205\pi$
$617$			280737.i			0.737444i	−0.929540	+	0.368722i	$0.879795\pi$
$618$		0			0
$619$	−403456.			−1.05297			−0.526483	−	0.850185i	$-0.676490\pi$
$619$	−403456.			−1.05297			−0.526483	+	0.850185i	$0.676490\pi$
$620$		0			0
$621$	−447188.	−	50713.0i	−1.15960	−	0.131503i
$622$		0			0
$623$			136932.i			0.352801i
$624$		0			0
$625$	−170519.			−0.436530
$626$		0			0
$627$	−153394.	−	290405.i	−0.390188	−	0.738702i
$628$		0			0
$629$		−	222807.i		−	0.563154i
$630$		0			0
$631$	−124703.			−0.313198			−0.156599	−	0.987662i	$-0.550053\pi$
$631$	−124703.			−0.313198			−0.156599	+	0.987662i	$0.550053\pi$
$632$		0			0
$633$	−218980.	+	115667.i	−0.546509	+	0.288670i
$634$		0			0
$635$		−	134470.i		−	0.333486i
$636$		0			0
$637$	−172261.			−0.424529
$638$		0			0
$639$	−518287.	−	353728.i	−1.26931	−	0.866298i
$640$		0			0
$641$		−	25834.8i		−	0.0628766i	−0.999506	−	0.0314383i	$-0.989991\pi$
$641$		−	25834.8i		−	0.0628766i	0.999506	−	0.0314383i	$-0.0100088\pi$
$642$		0			0
$643$	621594.			1.50343			0.751717	−	0.659485i	$-0.229226\pi$
$643$	621594.			1.50343			0.751717	+	0.659485i	$0.229226\pi$
$644$		0			0
$645$	165195.	+	312747.i	0.397080	+	0.751750i
$646$		0			0
$647$			667951.i			1.59564i	0.602893	+	0.797822i	$0.294015\pi$
$647$			667951.i			1.59564i	−0.602893	+	0.797822i	$0.705985\pi$
$648$		0			0
$649$	−201668.			−0.478792
$650$		0			0
$651$	594272.	−	313899.i	1.40224	−	0.740675i
$652$		0			0
$653$			330534.i			0.775157i	0.921837	+	0.387579i	$0.126688\pi$
$653$			330534.i			0.775157i	−0.921837	+	0.387579i	$0.873312\pi$
$654$		0			0
$655$	246202.			0.573865
$656$		0			0
$657$	−467490.	+	684974.i	−1.08303	+	1.58688i
$658$		0			0
$659$			273975.i			0.630869i	0.948947	+	0.315435i	$0.102150\pi$
$659$			273975.i			0.630869i	−0.948947	+	0.315435i	$0.897850\pi$
$660$		0			0
$661$	−841184.			−1.92525			−0.962627	−	0.270831i	$-0.912702\pi$
$661$	−841184.			−1.92525			−0.962627	+	0.270831i	$0.912702\pi$
$662$		0			0
$663$	−103002.	−	195004.i	−0.234326	−	0.443625i
$664$		0			0
$665$		−	612179.i		−	1.38431i
$666$		0			0
$667$	580185.			1.30411
$668$		0			0
$669$	464130.	−	245157.i	1.03702	−	0.547762i
$670$		0			0
$671$		−	244396.i		−	0.542811i
$672$		0			0
$673$	365221.			0.806354			0.403177	−	0.915122i	$-0.367906\pi$
$673$	365221.			0.806354			0.403177	+	0.915122i	$0.367906\pi$
$674$		0			0
$675$	−20635.1	+	181961.i	−0.0452897	+	0.399366i
$676$		0			0
$677$			204902.i			0.447062i	0.974697	+	0.223531i	$0.0717585\pi$
$677$			204902.i			0.447062i	−0.974697	+	0.223531i	$0.928242\pi$
$678$		0			0
$679$	1.14030e6			2.47332
$680$		0			0
$681$	405062.	+	766861.i	0.873429	+	1.65357i
$682$		0			0
$683$		−	651055.i		−	1.39565i	−0.716268	−	0.697825i	$-0.754151\pi$
$683$		−	651055.i		−	1.39565i	0.716268	−	0.697825i	$-0.245849\pi$
$684$		0			0
$685$	445616.			0.949686
$686$		0			0
$687$	506283.	−	267423.i	1.07270	−	0.566610i
$688$		0			0
$689$			188418.i			0.396903i
$690$		0			0
$691$	−81069.8			−0.169786			−0.0848932	−	0.996390i	$-0.527055\pi$
$691$	−81069.8			−0.169786			−0.0848932	+	0.996390i	$0.527055\pi$
$692$		0			0
$693$	−352784.	−	240773.i	−0.734586	−	0.501350i
$694$		0			0
$695$			266415.i			0.551554i
$696$		0			0
$697$	−711426.			−1.46442
$698$		0			0
$699$	−440788.	−	834497.i	−0.902142	−	1.70793i
$700$		0			0
$701$			202875.i			0.412851i	0.978462	+	0.206425i	$0.0661831\pi$
$701$			202875.i			0.412851i	−0.978462	+	0.206425i	$0.933817\pi$
$702$		0			0
$703$	−337207.			−0.682317
$704$		0			0
$705$	−331024.	+	174850.i	−0.666011	+	0.351792i
$706$		0			0
$707$		−	1.31936e6i		−	2.63953i
$708$		0			0
$709$	309617.			0.615932			0.307966	−	0.951397i	$-0.400352\pi$
$709$	309617.			0.615932			0.307966	+	0.951397i	$0.400352\pi$
$710$		0			0
$711$	242645.	−	355528.i	0.479991	−	0.703290i
$712$		0			0
$713$		−	681560.i		−	1.34068i
$714$		0			0
$715$	−119405.			−0.233567
$716$		0			0
$717$	227548.	+	430792.i	0.442623	+	0.837972i
$718$		0			0
$719$			722490.i			1.39757i	0.715331	+	0.698785i	$0.246276\pi$
$719$			722490.i			1.39757i	−0.715331	+	0.698785i	$0.753724\pi$
$720$		0			0
$721$	572140.			1.10061
$722$		0			0
$723$	−148418.	+	78395.3i	−0.283928	+	0.149973i
$724$		0			0
$725$		−	236078.i		−	0.449137i
$726$		0			0
$727$	187688.			0.355113			0.177557	−	0.984111i	$-0.443181\pi$
$727$	187688.			0.355113			0.177557	+	0.984111i	$0.443181\pi$
$728$		0			0
$729$	−517945.	−	119005.i	−0.974606	−	0.223928i
$730$		0			0
$731$			628707.i			1.17656i
$732$		0			0
$733$	−35683.0			−0.0664130			−0.0332065	−	0.999449i	$-0.510572\pi$
$733$	−35683.0			−0.0664130			−0.0332065	+	0.999449i	$0.510572\pi$
$734$		0			0
$735$	−176709.	−	334544.i	−0.327102	−	0.619269i
$736$		0			0
$737$		−	192710.i		−	0.354787i
$738$		0			0
$739$	−136210.			−0.249413			−0.124707	−	0.992194i	$-0.539799\pi$
$739$	−136210.			−0.249413			−0.124707	+	0.992194i	$0.539799\pi$
$740$		0			0
$741$	−295129.	+	155889.i	−0.537496	+	0.283909i
$742$		0			0
$743$		−	73802.1i		−	0.133688i	−0.997763	−	0.0668438i	$-0.978707\pi$
$743$		−	73802.1i		−	0.133688i	0.997763	−	0.0668438i	$-0.0212929\pi$
$744$		0			0
$745$	16649.9			0.0299984
$746$		0			0
$747$	714435.	+	487597.i	1.28033	+	0.873816i
$748$		0			0
$749$			777459.i			1.38584i
$750$		0			0
$751$	231199.			0.409926			0.204963	−	0.978770i	$-0.434293\pi$
$751$	231199.			0.409926			0.204963	+	0.978770i	$0.434293\pi$
$752$		0			0
$753$	−8400.41	−	15903.6i	−0.0148153	−	0.0280483i
$754$		0			0
$755$		−	229255.i		−	0.402183i
$756$		0			0
$757$	26453.5			0.0461627			0.0230813	−	0.999734i	$-0.492652\pi$
$757$	26453.5			0.0461627			0.0230813	+	0.999734i	$0.492652\pi$
$758$		0			0
$759$	−382995.	+	202301.i	−0.664829	+	0.351168i
$760$		0			0
$761$		−	254385.i		−	0.439260i	−0.975583	−	0.219630i	$-0.929515\pi$
$761$		−	254385.i		−	0.439260i	0.975583	−	0.219630i	$-0.0704851\pi$
$762$		0			0
$763$	349296.			0.599991
$764$		0			0
$765$	273051.	−	400079.i	0.466575	−	0.683632i
$766$		0			0
$767$			204948.i			0.348380i
$768$		0			0
$769$	−87292.0			−0.147612			−0.0738060	−	0.997273i	$-0.523515\pi$
$769$	−87292.0			−0.147612			−0.0738060	+	0.997273i	$0.523515\pi$
$770$		0			0
$771$	−287092.	−	543521.i	−0.482961	−	0.914339i
$772$		0			0
$773$		−	318085.i		−	0.532334i	−0.963927	−	0.266167i	$-0.914243\pi$
$773$		−	318085.i		−	0.532334i	0.963927	−	0.266167i	$-0.0857573\pi$
$774$		0			0
$775$	−277327.			−0.461731
$776$		0			0
$777$	−387764.	+	204820.i	−0.642282	+	0.339258i
$778$		0			0
$779$			1.07671e6i			1.77428i
$780$		0			0
$781$	−603909.			−0.990078
$782$		0			0
$783$	680740.	+	77198.7i	1.11035	+	0.125918i
$784$		0			0
$785$			466491.i			0.757014i
$786$		0			0
$787$	11698.3			0.0188874			0.00944372	−	0.999955i	$-0.496994\pi$
$787$	11698.3			0.0188874			0.00944372	+	0.999955i	$0.496994\pi$
$788$		0			0
$789$	−354188.	−	670547.i	−0.568958	−	1.07715i
$790$		0			0
$791$			278581.i			0.445244i
$792$		0			0
$793$	−248371.			−0.394961
$794$		0			0
$795$	−365924.	+	193284.i	−0.578970	+	0.305817i
$796$		0			0
$797$		−	210654.i		−	0.331629i	−0.986157	−	0.165814i	$-0.946975\pi$
$797$		−	210654.i		−	0.331629i	0.986157	−	0.165814i	$-0.0530253\pi$
$798$		0			0
$799$	−665450.			−1.04237
$800$		0			0
$801$	135438.	+	92435.8i	0.211094	+	0.144071i
$802$		0			0
$803$			798132.i			1.23778i
$804$		0			0
$805$	−807361.			−1.24588
$806$		0			0
$807$	−354407.	−	670961.i	−0.544196	−	1.03027i
$808$		0			0
$809$			732334.i			1.11895i	0.828846	+	0.559477i	$0.188998\pi$
$809$			732334.i			1.11895i	−0.828846	+	0.559477i	$0.811002\pi$
$810$		0			0
$811$	−440995.			−0.670490			−0.335245	−	0.942131i	$-0.608819\pi$
$811$	−440995.			−0.670490			−0.335245	+	0.942131i	$0.608819\pi$
$812$		0			0
$813$	778982.	−	411464.i	1.17855	−	0.622517i
$814$		0			0
$815$			349934.i			0.526831i
$816$		0			0
$817$	951518.			1.42552
$818$		0			0
$819$	−244689.	+	358523.i	−0.364794	+	0.534501i
$820$		0			0
$821$			813364.i			1.20670i	0.797477	+	0.603349i	$0.206167\pi$
$821$			813364.i			1.20670i	−0.797477	+	0.603349i	$0.793833\pi$
$822$		0			0
$823$	−350711.			−0.517785			−0.258892	−	0.965906i	$-0.583358\pi$
$823$	−350711.			−0.517785			−0.258892	+	0.965906i	$0.583358\pi$
$824$		0			0
$825$	82316.4	+	155841.i	0.120942	+	0.228967i
$826$		0			0
$827$		−	95232.7i		−	0.139244i	−0.997573	−	0.0696218i	$-0.977821\pi$
$827$		−	95232.7i		−	0.139244i	0.997573	−	0.0696218i	$-0.0221793\pi$
$828$		0			0
$829$	−160042.			−0.232876			−0.116438	−	0.993198i	$-0.537148\pi$
$829$	−160042.			−0.232876			−0.116438	+	0.993198i	$0.537148\pi$
$830$		0			0
$831$	−832444.	+	439703.i	−1.20546	+	0.636733i
$832$		0			0
$833$		−	672527.i		−	0.969214i
$834$		0			0
$835$	−253953.			−0.364234
$836$		0			0
$837$	90687.5	−	799685.i	0.129448	−	1.14148i
$838$		0			0
$839$			35266.1i			0.0500995i	0.999686	+	0.0250498i	$0.00797442\pi$
$839$			35266.1i			0.0500995i	−0.999686	+	0.0250498i	$0.992026\pi$
$840$		0			0
$841$	−175916.			−0.248721
$842$		0			0
$843$	−320956.	−	607632.i	−0.451638	−	0.855038i
$844$		0			0
$845$		−	430846.i		−	0.603404i
$846$		0			0
$847$	579272.			0.807450
$848$		0			0
$849$	1.03034e6	−	544235.i	1.42944	−	0.755043i
$850$		0			0
$851$			444720.i			0.614084i
$852$		0			0
$853$	668061.			0.918159			0.459080	−	0.888395i	$-0.348179\pi$
$853$	668061.			0.918159			0.459080	+	0.888395i	$0.348179\pi$
$854$		0			0
$855$	−605500.	−	413250.i	−0.828289	−	0.565302i
$856$		0			0
$857$		−	409636.i		−	0.557746i	−0.960328	−	0.278873i	$-0.910039\pi$
$857$		−	409636.i		−	0.557746i	0.960328	−	0.278873i	$-0.0899608\pi$
$858$		0			0
$859$	96808.2			0.131198			0.0655988	−	0.997846i	$-0.479104\pi$
$859$	96808.2			0.131198			0.0655988	+	0.997846i	$0.479104\pi$
$860$		0			0
$861$	653993.	+	1.23814e6i	0.882200	+	1.67018i
$862$		0			0
$863$		−	627704.i		−	0.842817i	−0.906871	−	0.421408i	$-0.861536\pi$
$863$		−	627704.i		−	0.842817i	0.906871	−	0.421408i	$-0.138464\pi$
$864$		0			0
$865$	726949.			0.971564
$866$		0			0
$867$	96654.4	−	51053.6i	0.128583	−	0.0679185i
$868$		0			0
$869$		−	414261.i		−	0.548574i
$870$		0			0
$871$	−195844.			−0.258151
$872$		0			0
$873$	769757.	−	1.12786e6i	1.01001	−	1.47988i
$874$		0			0
$875$			1.14587e6i			1.49664i
$876$		0			0
$877$	1.29296e6			1.68107			0.840533	−	0.541761i	$-0.182242\pi$
$877$	1.29296e6			1.68107			0.840533	+	0.541761i	$0.182242\pi$
$878$		0			0
$879$	424326.	+	803331.i	0.549189	+	1.03972i
$880$		0			0
$881$		−	1.13499e6i		−	1.46231i	−0.682213	−	0.731153i	$-0.738982\pi$
$881$		−	1.13499e6i		−	1.46231i	0.682213	−	0.731153i	$-0.261018\pi$
$882$		0			0
$883$	216522.			0.277703			0.138851	−	0.990313i	$-0.455659\pi$
$883$	216522.			0.277703			0.138851	+	0.990313i	$0.455659\pi$
$884$		0			0
$885$	−398026.	+	210241.i	−0.508189	+	0.268429i
$886$		0			0
$887$		−	1.24588e6i		−	1.58354i	−0.610821	−	0.791769i	$-0.709160\pi$
$887$		−	1.24588e6i		−	1.58354i	0.610821	−	0.791769i	$-0.290840\pi$
$888$		0			0
$889$	−470457.			−0.595273
$890$		0			0
$891$	−476292.	+	186402.i	−0.599954	+	0.234799i
$892$		0			0
$893$			1.00713e6i			1.26294i
$894$		0			0
$895$	585686.			0.731171
$896$		0			0
$897$	205592.	+	389225.i	0.255518	+	0.483744i
$898$		0			0
$899$			1.03752e6i			1.28374i
$900$		0			0
$901$	−735608.			−0.906143
$902$		0			0
$903$	1.09418e6	−	577952.i	1.34187	−	0.708788i
$904$		0			0
$905$		−	800787.i		−	0.977732i
$906$		0			0
$907$	−71744.4			−0.0872115			−0.0436057	−	0.999049i	$-0.513885\pi$
$907$	−71744.4			−0.0872115			−0.0436057	+	0.999049i	$0.513885\pi$
$908$		0			0
$909$	−1.30497e6	−	890634.i	−1.57933	−	1.07788i
$910$		0			0
$911$		−	463492.i		−	0.558477i	−0.960222	−	0.279239i	$-0.909918\pi$
$911$		−	463492.i		−	0.558477i	0.960222	−	0.279239i	$-0.0900821\pi$
$912$		0			0
$913$	832461.			0.998670
$914$		0			0
$915$	−254785.	−	482357.i	−0.304321	−	0.576138i
$916$		0			0
$917$		−	861364.i		−	1.02435i
$918$		0			0
$919$	−540851.			−0.640393			−0.320196	−	0.947351i	$-0.603749\pi$
$919$	−540851.			−0.640393			−0.320196	+	0.947351i	$0.603749\pi$
$920$		0			0
$921$	93968.7	−	49635.0i	0.110781	−	0.0585152i
$922$		0			0
$923$			613732.i			0.720403i
$924$		0			0
$925$	180957.			0.211491
$926$		0			0
$927$	386222.	−	565898.i	0.449446	−	0.658534i
$928$		0			0
$929$			818980.i			0.948947i	0.880270	+	0.474474i	$0.157362\pi$
$929$			818980.i			0.948947i	−0.880270	+	0.474474i	$0.842638\pi$
$930$		0			0
$931$	−1.01784e6			−1.17430
$932$		0			0
$933$	−152848.	−	289371.i	−0.175588	−	0.332423i
$934$		0			0
$935$		−	466172.i		−	0.533241i
$936$		0			0
$937$	−388827.			−0.442871			−0.221435	−	0.975175i	$-0.571074\pi$
$937$	−388827.			−0.442871			−0.221435	+	0.975175i	$0.571074\pi$
$938$		0			0
$939$	529738.	−	279812.i	0.600800	−	0.317347i
$940$		0			0
$941$			165913.i			0.187371i	0.995602	+	0.0936855i	$0.0298648\pi$
$941$			165913.i			0.187371i	−0.995602	+	0.0936855i	$0.970135\pi$
$942$		0			0
$943$	1.42000e6			1.59685
$944$		0			0
$945$	−947289.	−	107426.i	−1.06076	−	0.120295i
$946$		0			0
$947$		−	764662.i		−	0.852647i	−0.904571	−	0.426324i	$-0.859809\pi$
$947$		−	764662.i		−	0.852647i	0.904571	−	0.426324i	$-0.140191\pi$
$948$		0			0
$949$	811115.			0.900637
$950$		0			0
$951$	444870.	+	842225.i	0.491894	+	0.931252i
$952$		0			0
$953$		−	1.05066e6i		−	1.15685i	−0.815737	−	0.578424i	$-0.803668\pi$
$953$		−	1.05066e6i		−	1.15685i	0.815737	−	0.578424i	$-0.196332\pi$
$954$		0			0
$955$	−506880.			−0.555774
$956$		0			0
$957$	583021.	−	307956.i	0.636591	−	0.336252i
$958$		0			0
$959$		−	1.55903e6i		−	1.69519i
$960$		0			0
$961$	295279.			0.319732
$962$		0			0
$963$	768977.	+	524822.i	0.829202	+	0.565925i
$964$		0			0
$965$		−	480865.i		−	0.516379i
$966$		0			0
$967$	−201023.			−0.214977			−0.107488	−	0.994206i	$-0.534281\pi$
$967$	−201023.			−0.214977			−0.107488	+	0.994206i	$0.534281\pi$
$968$		0			0
$969$	−608611.	−	1.15222e6i	−0.648175	−	1.22712i
$970$		0			0
$971$		−	1.06992e6i		−	1.13478i	−0.823448	−	0.567392i	$-0.807952\pi$
$971$		−	1.06992e6i		−	1.13478i	0.823448	−	0.567392i	$-0.192048\pi$
$972$		0			0
$973$	932079.			0.984525
$974$		0			0
$975$	158376.	−	83655.3i	0.166602	−	0.0880004i
$976$		0			0
$977$			1.04258e6i			1.09224i	0.837706	+	0.546121i	$0.183896\pi$
$977$			1.04258e6i			1.09224i	−0.837706	+	0.546121i	$0.816104\pi$
$978$		0			0
$979$	157813.			0.164656
$980$		0			0
$981$	235792.	−	345485.i	0.245014	−	0.358998i
$982$		0			0
$983$		−	201986.i		−	0.209033i	−0.994523	−	0.104516i	$-0.966671\pi$
$983$		−	201986.i		−	0.209033i	0.994523	−	0.104516i	$-0.0333294\pi$
$984$		0			0
$985$	867343.			0.893961
$986$		0			0
$987$	611729.	+	1.15812e6i	0.627950	+	1.18883i
$988$		0			0
$989$		−	1.25489e6i		−	1.28296i
$990$		0			0
$991$	−905070.			−0.921584			−0.460792	−	0.887508i	$-0.652435\pi$
$991$	−905070.			−0.921584			−0.460792	+	0.887508i	$0.652435\pi$
$992$		0			0
$993$	468088.	−	247248.i	0.474711	−	0.250746i
$994$		0			0
$995$			34459.9i			0.0348071i
$996$		0			0
$997$	−772571.			−0.777227			−0.388614	−	0.921401i	$-0.627046\pi$
$997$	−772571.			−0.777227			−0.388614	+	0.921401i	$0.627046\pi$
$998$		0			0
$999$	−59173.9	+	521797.i	−0.0592924	+	0.522842i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.5.e.b.257.4	yes	16
3.2	odd	2	inner	384.5.e.b.257.3	yes	16
4.3	odd	2		384.5.e.c.257.13	yes	16
8.3	odd	2		384.5.e.a.257.4	yes	16
8.5	even	2		384.5.e.d.257.13	yes	16
12.11	even	2		384.5.e.c.257.14	yes	16
24.5	odd	2		384.5.e.d.257.14	yes	16
24.11	even	2		384.5.e.a.257.3	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.5.e.a.257.3	✓	16	24.11	even	2
384.5.e.a.257.4	yes	16	8.3	odd	2
384.5.e.b.257.3	yes	16	3.2	odd	2	inner
384.5.e.b.257.4	yes	16	1.1	even	1	trivial
384.5.e.c.257.13	yes	16	4.3	odd	2
384.5.e.c.257.14	yes	16	12.11	even	2
384.5.e.d.257.13	yes	16	8.5	even	2
384.5.e.d.257.14	yes	16	24.5	odd	2

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:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	384.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-7.95805 + 4.20350i) q^{3} +19.3338i q^{5} +67.6413 q^{7} +(45.6611 - 66.9034i) q^{9} +O(q^{10})\)	\(q+(-7.95805 + 4.20350i) q^{3} +19.3338i q^{5} +67.6413 q^{7} +(45.6611 - 66.9034i) q^{9} -77.9559i q^{11} -79.2239 q^{13} +(-81.2697 - 153.859i) q^{15} -309.300i q^{17} -468.110 q^{19} +(-538.293 + 284.331i) q^{21} +617.359i q^{23} +251.204 q^{25} +(-82.1450 + 724.357i) q^{27} -939.786i q^{29} -1103.99 q^{31} +(327.688 + 620.377i) q^{33} +1307.77i q^{35} +720.359 q^{37} +(630.468 - 333.018i) q^{39} -2300.12i q^{41} -2032.68 q^{43} +(1293.50 + 882.804i) q^{45} -2151.47i q^{47} +2174.35 q^{49} +(1300.14 + 2461.42i) q^{51} -2378.30i q^{53} +1507.19 q^{55} +(3725.25 - 1967.70i) q^{57} -2586.95i q^{59} +3135.05 q^{61} +(3088.58 - 4525.43i) q^{63} -1531.70i q^{65} +2472.03 q^{67} +(-2595.07 - 4912.97i) q^{69} -7746.80i q^{71} -10238.3 q^{73} +(-1999.09 + 1055.94i) q^{75} -5273.04i q^{77} +5314.05 q^{79} +(-2391.12 - 6109.77i) q^{81} +10678.6i q^{83} +5979.95 q^{85} +(3950.39 + 7478.86i) q^{87} +2024.39i q^{89} -5358.81 q^{91} +(8785.63 - 4640.64i) q^{93} -9050.36i q^{95} +16858.0 q^{97} +(-5215.51 - 3559.55i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{3} + 80 q^{7}+O(q^{10}) \) 16 * q - 8 * q^3 + 80 * q^7	\( 16 q - 8 q^{3} + 80 q^{7} + 416 q^{15} + 816 q^{19} + 608 q^{21} - 2000 q^{25} + 280 q^{27} + 592 q^{31} - 496 q^{33} + 2240 q^{37} - 16 q^{39} + 368 q^{43} + 800 q^{45} + 3984 q^{49} - 352 q^{51} + 1920 q^{55} + 560 q^{57} - 3520 q^{61} - 816 q^{63} - 3536 q^{67} - 10784 q^{69} + 3680 q^{73} + 5112 q^{75} - 14448 q^{79} - 624 q^{81} - 11136 q^{85} - 14944 q^{87} - 22944 q^{91} + 13760 q^{93} + 3264 q^{97} - 26976 q^{99}+O(q^{100}) \) 16 * q - 8 * q^3 + 80 * q^7 + 416 * q^15 + 816 * q^19 + 608 * q^21 - 2000 * q^25 + 280 * q^27 + 592 * q^31 - 496 * q^33 + 2240 * q^37 - 16 * q^39 + 368 * q^43 + 800 * q^45 + 3984 * q^49 - 352 * q^51 + 1920 * q^55 + 560 * q^57 - 3520 * q^61 - 816 * q^63 - 3536 * q^67 - 10784 * q^69 + 3680 * q^73 + 5112 * q^75 - 14448 * q^79 - 624 * q^81 - 11136 * q^85 - 14944 * q^87 - 22944 * q^91 + 13760 * q^93 + 3264 * q^97 - 26976 * q^99

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