LMFDB - Embedded newform 384.3.e.a.257.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,3,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, 3, names="a")

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

chi := DirichletCharacter("384.257");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 3 $
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:	$10.4632421514$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 18x^{6} + 99x^{4} + 170x^{2} + 81 $ x^8 + 18x^6 + 99x^4 + 170*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{10}\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			257.3
Root			$2.55118i$ of defining polynomial
Character	$\chi$	$=$	384.257
Dual form			384.3.e.a.257.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.57844 - 2.55118i) q^{3} +1.31534i q^{5} +10.2329 q^{7} +(-4.01705 + 8.05378i) q^{9} +O(q^{10})$	\(q+(-1.57844 - 2.55118i) q^{3} +1.31534i q^{5} +10.2329 q^{7} +(-4.01705 + 8.05378i) q^{9} +16.6620i q^{11} -18.7454 q^{13} +(3.35566 - 2.07618i) q^{15} +4.38114i q^{17} +11.5544 q^{19} +(-16.1520 - 26.1059i) q^{21} +16.7490i q^{23} +23.2699 q^{25} +(26.8873 - 2.46419i) q^{27} +12.5498i q^{29} +20.3167 q^{31} +(42.5079 - 26.3001i) q^{33} +13.4597i q^{35} +18.5778 q^{37} +(29.5885 + 47.8230i) q^{39} +78.6737i q^{41} +36.4860 q^{43} +(-10.5934 - 5.28377i) q^{45} -19.9175i q^{47} +55.7118 q^{49} +(11.1771 - 6.91537i) q^{51} -81.3064i q^{53} -21.9162 q^{55} +(-18.2380 - 29.4775i) q^{57} +29.9477i q^{59} -72.0687 q^{61} +(-41.1060 + 82.4133i) q^{63} -24.6565i q^{65} +56.3520 q^{67} +(42.7298 - 26.4374i) q^{69} -136.465i q^{71} -80.8141 q^{73} +(-36.7301 - 59.3657i) q^{75} +170.501i q^{77} +86.0317 q^{79} +(-48.7266 - 64.7048i) q^{81} +80.4263i q^{83} -5.76268 q^{85} +(32.0168 - 19.8091i) q^{87} +131.830i q^{89} -191.820 q^{91} +(-32.0687 - 51.8315i) q^{93} +15.1980i q^{95} -20.4375 q^{97} +(-134.192 - 66.9323i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{3} - 8 q^{7}+O(q^{10}) $ 8 * q - 4 * q^3 - 8 * q^7	$ 8 q - 4 q^{3} - 8 q^{7} - 16 q^{15} + 24 q^{19} + 16 q^{21} - 40 q^{25} + 44 q^{27} + 56 q^{31} + 8 q^{33} + 32 q^{37} + 104 q^{39} - 136 q^{43} - 80 q^{45} + 72 q^{49} - 176 q^{51} - 192 q^{55} - 40 q^{57} - 160 q^{61} - 264 q^{63} + 280 q^{67} + 80 q^{69} - 80 q^{73} + 348 q^{75} + 408 q^{79} + 72 q^{81} + 192 q^{85} + 368 q^{87} - 336 q^{91} + 160 q^{93} + 96 q^{97} - 432 q^{99}+O(q^{100}) $ 8 * q - 4 * q^3 - 8 * q^7 - 16 * q^15 + 24 * q^19 + 16 * q^21 - 40 * q^25 + 44 * q^27 + 56 * q^31 + 8 * q^33 + 32 * q^37 + 104 * q^39 - 136 * q^43 - 80 * q^45 + 72 * q^49 - 176 * q^51 - 192 * q^55 - 40 * q^57 - 160 * q^61 - 264 * q^63 + 280 * q^67 + 80 * q^69 - 80 * q^73 + 348 * q^75 + 408 * q^79 + 72 * q^81 + 192 * q^85 + 368 * q^87 - 336 * q^91 + 160 * q^93 + 96 * q^97 - 432 * 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
$3$	−1.57844	−	2.55118i	−0.526147	−	0.850394i
$3$	−1.57844	−	2.55118i	−0.526147	−	0.850394i
$4$		0			0
$5$			1.31534i			0.263067i	0.991312	+	0.131534i	$0.0419902\pi$
$5$			1.31534i			0.263067i	−0.991312	+	0.131534i	$0.958010\pi$
$6$		0			0
$7$	10.2329			1.46184			0.730920	−	0.682463i	$-0.239091\pi$
$7$	10.2329			1.46184			0.730920	+	0.682463i	$0.239091\pi$
$8$		0			0
$9$	−4.01705	+	8.05378i	−0.446339	+	0.894864i
$10$		0			0
$11$			16.6620i			1.51473i	0.652991	+	0.757366i	$0.273514\pi$
$11$			16.6620i			1.51473i	−0.652991	+	0.757366i	$0.726486\pi$
$12$		0			0
$13$	−18.7454			−1.44196			−0.720978	−	0.692958i	$-0.756307\pi$
$13$	−18.7454			−1.44196			−0.720978	+	0.692958i	$0.756307\pi$
$14$		0			0
$15$	3.35566	−	2.07618i	0.223711	−	0.138412i
$16$		0			0
$17$			4.38114i			0.257714i	0.991663	+	0.128857i	$0.0411308\pi$
$17$			4.38114i			0.257714i	−0.991663	+	0.128857i	$0.958869\pi$
$18$		0			0
$19$	11.5544			0.608129			0.304064	−	0.952652i	$-0.401656\pi$
$19$	11.5544			0.608129			0.304064	+	0.952652i	$0.401656\pi$
$20$		0			0
$21$	−16.1520	−	26.1059i	−0.769143	−	1.24314i
$22$		0			0
$23$			16.7490i			0.728219i	0.931356	+	0.364109i	$0.118627\pi$
$23$			16.7490i			0.728219i	−0.931356	+	0.364109i	$0.881373\pi$
$24$		0			0
$25$	23.2699			0.930796
$26$		0			0
$27$	26.8873	−	2.46419i	0.995827	−	0.0912663i
$28$		0			0
$29$			12.5498i			0.432752i	0.976310	+	0.216376i	$0.0694237\pi$
$29$			12.5498i			0.432752i	−0.976310	+	0.216376i	$0.930576\pi$
$30$		0			0
$31$	20.3167			0.655377			0.327688	−	0.944786i	$-0.393730\pi$
$31$	20.3167			0.655377			0.327688	+	0.944786i	$0.393730\pi$
$32$		0			0
$33$	42.5079	−	26.3001i	1.28812	−	0.796971i
$34$		0			0
$35$			13.4597i			0.384562i
$36$		0			0
$37$	18.5778			0.502103			0.251052	−	0.967974i	$-0.419224\pi$
$37$	18.5778			0.502103			0.251052	+	0.967974i	$0.419224\pi$
$38$		0			0
$39$	29.5885	+	47.8230i	0.758681	+	1.22623i
$40$		0			0
$41$			78.6737i			1.91887i	0.281930	+	0.959435i	$0.409025\pi$
$41$			78.6737i			1.91887i	−0.281930	+	0.959435i	$0.590975\pi$
$42$		0			0
$43$	36.4860			0.848511			0.424255	−	0.905543i	$-0.360536\pi$
$43$	36.4860			0.848511			0.424255	+	0.905543i	$0.360536\pi$
$44$		0			0
$45$	−10.5934	−	5.28377i	−0.235410	−	0.117417i
$46$		0			0
$47$		−	19.9175i		−	0.423777i	−0.977294	−	0.211888i	$-0.932039\pi$
$47$		−	19.9175i		−	0.423777i	0.977294	−	0.211888i	$-0.0679613\pi$
$48$		0			0
$49$	55.7118			1.13698
$50$		0			0
$51$	11.1771	−	6.91537i	0.219158	−	0.135596i
$52$		0			0
$53$		−	81.3064i		−	1.53408i	−0.641598	−	0.767041i	$-0.721728\pi$
$53$		−	81.3064i		−	1.53408i	0.641598	−	0.767041i	$-0.278272\pi$
$54$		0			0
$55$	−21.9162			−0.398476
$56$		0			0
$57$	−18.2380	−	29.4775i	−0.319965	−	0.517149i
$58$		0			0
$59$			29.9477i			0.507589i	0.967258	+	0.253794i	$0.0816787\pi$
$59$			29.9477i			0.507589i	−0.967258	+	0.253794i	$0.918321\pi$
$60$		0			0
$61$	−72.0687			−1.18145			−0.590727	−	0.806872i	$-0.701159\pi$
$61$	−72.0687			−1.18145			−0.590727	+	0.806872i	$0.701159\pi$
$62$		0			0
$63$	−41.1060	+	82.4133i	−0.652476	+	1.30815i
$64$		0			0
$65$		−	24.6565i		−	0.379331i
$66$		0			0
$67$	56.3520			0.841074			0.420537	−	0.907275i	$-0.361842\pi$
$67$	56.3520			0.841074			0.420537	+	0.907275i	$0.361842\pi$
$68$		0			0
$69$	42.7298	−	26.4374i	0.619273	−	0.383150i
$70$		0			0
$71$		−	136.465i		−	1.92204i	−0.276479	−	0.961020i	$-0.589168\pi$
$71$		−	136.465i		−	1.92204i	0.276479	−	0.961020i	$-0.410832\pi$
$72$		0			0
$73$	−80.8141			−1.10704			−0.553521	−	0.832835i	$-0.686716\pi$
$73$	−80.8141			−1.10704			−0.553521	+	0.832835i	$0.686716\pi$
$74$		0			0
$75$	−36.7301	−	59.3657i	−0.489735	−	0.791543i
$76$		0			0
$77$			170.501i			2.21429i
$78$		0			0
$79$	86.0317			1.08901			0.544504	−	0.838758i	$-0.316718\pi$
$79$	86.0317			1.08901			0.544504	+	0.838758i	$0.316718\pi$
$80$		0			0
$81$	−48.7266	−	64.7048i	−0.601563	−	0.798825i
$82$		0			0
$83$			80.4263i			0.968992i	0.874793	+	0.484496i	$0.160997\pi$
$83$			80.4263i			0.968992i	−0.874793	+	0.484496i	$0.839003\pi$
$84$		0			0
$85$	−5.76268			−0.0677962
$86$		0			0
$87$	32.0168	−	19.8091i	0.368009	−	0.227691i
$88$		0			0
$89$			131.830i			1.48124i	0.671923	+	0.740621i	$0.265468\pi$
$89$			131.830i			1.48124i	−0.671923	+	0.740621i	$0.734532\pi$
$90$		0			0
$91$	−191.820			−2.10791
$92$		0			0
$93$	−32.0687	−	51.8315i	−0.344824	−	0.557328i
$94$		0			0
$95$			15.1980i			0.159979i
$96$		0			0
$97$	−20.4375			−0.210696			−0.105348	−	0.994435i	$-0.533596\pi$
$97$	−20.4375			−0.210696			−0.105348	+	0.994435i	$0.533596\pi$
$98$		0			0
$99$	−134.192	−	66.9323i	−1.35548	−	0.676083i
$100$		0			0
$101$			143.374i			1.41954i	0.704431	+	0.709772i	$0.251202\pi$
$101$			143.374i			1.41954i	−0.704431	+	0.709772i	$0.748798\pi$
$102$		0			0
$103$	−113.748			−1.10435			−0.552173	−	0.833730i	$-0.686201\pi$
$103$	−113.748			−1.10435			−0.552173	+	0.833730i	$0.686201\pi$
$104$		0			0
$105$	34.3381	−	21.2453i	0.327029	−	0.202336i
$106$		0			0
$107$		−	6.26024i		−	0.0585069i	−0.999572	−	0.0292535i	$-0.990687\pi$
$107$		−	6.26024i		−	0.0585069i	0.999572	−	0.0292535i	$-0.00931300\pi$
$108$		0			0
$109$	147.657			1.35465			0.677327	−	0.735682i	$-0.263138\pi$
$109$	147.657			1.35465			0.677327	+	0.735682i	$0.263138\pi$
$110$		0			0
$111$	−29.3240	−	47.3954i	−0.264180	−	0.426986i
$112$		0			0
$113$		−	51.3905i		−	0.454783i	−0.973803	−	0.227392i	$-0.926980\pi$
$113$		−	51.3905i		−	0.454783i	0.973803	−	0.227392i	$-0.0730197\pi$
$114$		0			0
$115$	−22.0306			−0.191571
$116$		0			0
$117$	75.3013	−	150.971i	0.643601	−	1.29035i
$118$		0			0
$119$			44.8317i			0.376737i
$120$		0			0
$121$	−156.624			−1.29441
$122$		0			0
$123$	200.711	−	124.182i	1.63179	−	1.00961i
$124$		0			0
$125$			63.4912i			0.507929i
$126$		0			0
$127$	−145.579			−1.14629			−0.573147	−	0.819452i	$-0.694278\pi$
$127$	−145.579			−1.14629			−0.573147	+	0.819452i	$0.694278\pi$
$128$		0			0
$129$	−57.5909	−	93.0823i	−0.446441	−	0.721568i
$130$		0			0
$131$			45.3993i			0.346560i	0.984873	+	0.173280i	$0.0554365\pi$
$131$			45.3993i			0.346560i	−0.984873	+	0.173280i	$0.944563\pi$
$132$		0			0
$133$	118.235			0.888987
$134$		0			0
$135$	3.24124	+	35.3659i	0.0240092	+	0.261969i
$136$		0			0
$137$		−	179.157i		−	1.30771i	−0.756618	−	0.653857i	$-0.773150\pi$
$137$		−	179.157i		−	1.30771i	0.756618	−	0.653857i	$-0.226850\pi$
$138$		0			0
$139$	−50.8841			−0.366073			−0.183036	−	0.983106i	$-0.558593\pi$
$139$	−50.8841			−0.366073			−0.183036	+	0.983106i	$0.558593\pi$
$140$		0			0
$141$	−50.8131	+	31.4386i	−0.360377	+	0.222969i
$142$		0			0
$143$		−	312.337i		−	2.18418i
$144$		0			0
$145$	−16.5072			−0.113843
$146$		0			0
$147$	−87.9378	−	142.131i	−0.598216	−	0.966877i
$148$		0			0
$149$			76.0410i			0.510342i	0.966896	+	0.255171i	$0.0821318\pi$
$149$			76.0410i			0.510342i	−0.966896	+	0.255171i	$0.917868\pi$
$150$		0			0
$151$	−179.918			−1.19151			−0.595754	−	0.803167i	$-0.703147\pi$
$151$	−179.918			−1.19151			−0.595754	+	0.803167i	$0.703147\pi$
$152$		0			0
$153$	−35.2847	−	17.5993i	−0.230619	−	0.115028i
$154$		0			0
$155$			26.7233i			0.172408i
$156$		0			0
$157$	67.2180			0.428140			0.214070	−	0.976818i	$-0.431328\pi$
$157$	67.2180			0.428140			0.214070	+	0.976818i	$0.431328\pi$
$158$		0			0
$159$	−207.427	+	128.337i	−1.30457	+	0.807153i
$160$		0			0
$161$			171.391i			1.06454i
$162$		0			0
$163$	102.723			0.630204			0.315102	−	0.949058i	$-0.397961\pi$
$163$	102.723			0.630204			0.315102	+	0.949058i	$0.397961\pi$
$164$		0			0
$165$	34.5934	+	55.9122i	0.209657	+	0.338862i
$166$		0			0
$167$		−	80.7935i		−	0.483793i	−0.970302	−	0.241897i	$-0.922231\pi$
$167$		−	80.7935i		−	0.483793i	0.970302	−	0.241897i	$-0.0777695\pi$
$168$		0			0
$169$	182.391			1.07924
$170$		0			0
$171$	−46.4148	+	93.0569i	−0.271431	+	0.544192i
$172$		0			0
$173$		−	169.185i		−	0.977950i	−0.872298	−	0.488975i	$-0.837371\pi$
$173$		−	169.185i		−	0.977950i	0.872298	−	0.488975i	$-0.162629\pi$
$174$		0			0
$175$	238.118			1.36067
$176$		0			0
$177$	76.4021	−	47.2707i	0.431650	−	0.267066i
$178$		0			0
$179$		−	170.982i		−	0.955205i	−0.878576	−	0.477602i	$-0.841506\pi$
$179$		−	170.982i		−	0.955205i	0.878576	−	0.477602i	$-0.158494\pi$
$180$		0			0
$181$	−39.6292			−0.218946			−0.109473	−	0.993990i	$-0.534916\pi$
$181$	−39.6292			−0.218946			−0.109473	+	0.993990i	$0.534916\pi$
$182$		0			0
$183$	113.756	+	183.860i	0.621618	+	1.00470i
$184$		0			0
$185$			24.4361i			0.132087i
$186$		0			0
$187$	−72.9988			−0.390368
$188$		0			0
$189$	275.135	−	25.2158i	1.45574	−	0.133417i
$190$		0			0
$191$			239.917i			1.25611i	0.778170	+	0.628054i	$0.216148\pi$
$191$			239.917i			1.25611i	−0.778170	+	0.628054i	$0.783852\pi$
$192$		0			0
$193$	−13.4972			−0.0699337			−0.0349669	−	0.999388i	$-0.511133\pi$
$193$	−13.4972			−0.0699337			−0.0349669	+	0.999388i	$0.511133\pi$
$194$		0			0
$195$	−62.9033	+	38.9189i	−0.322581	+	0.199584i
$196$		0			0
$197$			16.6040i			0.0842843i	0.999112	+	0.0421422i	$0.0134182\pi$
$197$			16.6040i			0.0842843i	−0.999112	+	0.0421422i	$0.986582\pi$
$198$		0			0
$199$	143.674			0.721979			0.360990	−	0.932570i	$-0.382439\pi$
$199$	143.674			0.721979			0.360990	+	0.932570i	$0.382439\pi$
$200$		0			0
$201$	−88.9483	−	143.764i	−0.442529	−	0.715244i
$202$		0			0
$203$			128.421i			0.632614i
$204$		0			0
$205$	−103.482			−0.504792
$206$		0			0
$207$	−134.893	−	67.2817i	−0.651657	−	0.325032i
$208$		0			0
$209$			192.521i			0.921151i
$210$		0			0
$211$	83.6553			0.396471			0.198235	−	0.980154i	$-0.436479\pi$
$211$	83.6553			0.396471			0.198235	+	0.980154i	$0.436479\pi$
$212$		0			0
$213$	−348.146	+	215.402i	−1.63449	+	1.01128i
$214$		0			0
$215$			47.9913i			0.223215i
$216$		0			0
$217$	207.898			0.958056
$218$		0			0
$219$	127.560	+	206.171i	0.582467	+	0.941422i
$220$		0			0
$221$		−	82.1263i		−	0.371612i
$222$		0			0
$223$	146.898			0.658734			0.329367	−	0.944202i	$-0.393165\pi$
$223$	146.898			0.658734			0.329367	+	0.944202i	$0.393165\pi$
$224$		0			0
$225$	−93.4763	+	187.410i	−0.415450	+	0.832936i
$226$		0			0
$227$		−	28.5250i		−	0.125661i	−0.998024	−	0.0628304i	$-0.979987\pi$
$227$		−	28.5250i		−	0.125661i	0.998024	−	0.0628304i	$-0.0200127\pi$
$228$		0			0
$229$	−345.118			−1.50707			−0.753533	−	0.657410i	$-0.771652\pi$
$229$	−345.118			−1.50707			−0.753533	+	0.657410i	$0.771652\pi$
$230$		0			0
$231$	434.978	−	269.125i	1.88302	−	1.16504i
$232$		0			0
$233$			105.272i			0.451809i	0.974149	+	0.225905i	$0.0725338\pi$
$233$			105.272i			0.451809i	−0.974149	+	0.225905i	$0.927466\pi$
$234$		0			0
$235$	26.1982			0.111482
$236$		0			0
$237$	−135.796	−	219.482i	−0.572978	−	0.926086i
$238$		0			0
$239$		−	280.456i		−	1.17346i	−0.809783	−	0.586729i	$-0.800415\pi$
$239$		−	280.456i		−	1.17346i	0.809783	−	0.586729i	$-0.199585\pi$
$240$		0			0
$241$	−369.833			−1.53458			−0.767289	−	0.641302i	$-0.778395\pi$
$241$	−369.833			−1.53458			−0.767289	+	0.641302i	$0.778395\pi$
$242$		0			0
$243$	−88.1616	+	226.443i	−0.362805	+	0.931865i
$244$		0			0
$245$			73.2798i			0.299101i
$246$		0			0
$247$	−216.593			−0.876894
$248$		0			0
$249$	205.182	−	126.948i	0.824025	−	0.509832i
$250$		0			0
$251$		−	27.4959i		−	0.109545i	−0.998499	−	0.0547727i	$-0.982557\pi$
$251$		−	27.4959i		−	0.109545i	0.998499	−	0.0547727i	$-0.0174434\pi$
$252$		0			0
$253$	−279.073			−1.10306
$254$		0			0
$255$	9.09604	+	14.7016i	0.0356708	+	0.0576534i
$256$		0			0
$257$			4.92447i			0.0191614i	0.999954	+	0.00958068i	$0.00304967\pi$
$257$			4.92447i			0.0191614i	−0.999954	+	0.00958068i	$0.996950\pi$
$258$		0			0
$259$	190.105			0.733995
$260$		0			0
$261$	−101.073	−	50.4132i	−0.387254	−	0.193154i
$262$		0			0
$263$			263.575i			1.00219i	0.865394	+	0.501093i	$0.167068\pi$
$263$			263.575i			1.00219i	−0.865394	+	0.501093i	$0.832932\pi$
$264$		0			0
$265$	106.945			0.403567
$266$		0			0
$267$	336.323	−	208.087i	1.25964	−	0.779351i
$268$		0			0
$269$		−	236.023i		−	0.877409i	−0.898631	−	0.438704i	$-0.855438\pi$
$269$		−	236.023i		−	0.877409i	0.898631	−	0.438704i	$-0.144562\pi$
$270$		0			0
$271$	433.288			1.59885			0.799424	−	0.600767i	$-0.205138\pi$
$271$	433.288			1.59885			0.799424	+	0.600767i	$0.205138\pi$
$272$		0			0
$273$	302.776	+	489.367i	1.10907	+	1.79255i
$274$		0			0
$275$			387.724i			1.40991i
$276$		0			0
$277$	244.067			0.881107			0.440553	−	0.897726i	$-0.354782\pi$
$277$	244.067			0.881107			0.440553	+	0.897726i	$0.354782\pi$
$278$		0			0
$279$	−81.6131	+	163.626i	−0.292520	+	0.586473i
$280$		0			0
$281$		−	431.627i		−	1.53604i	−0.640426	−	0.768020i	$-0.721242\pi$
$281$		−	431.627i		−	1.53604i	0.640426	−	0.768020i	$-0.278758\pi$
$282$		0			0
$283$	−35.1843			−0.124326			−0.0621631	−	0.998066i	$-0.519800\pi$
$283$	−35.1843			−0.124326			−0.0621631	+	0.998066i	$0.519800\pi$
$284$		0			0
$285$	38.7728	−	23.9891i	0.136045	−	0.0841723i
$286$		0			0
$287$			805.058i			2.80508i
$288$		0			0
$289$	269.806			0.933583
$290$		0			0
$291$	32.2594	+	52.1397i	0.110857	+	0.179174i
$292$		0			0
$293$		−	214.900i		−	0.733446i	−0.930330	−	0.366723i	$-0.880480\pi$
$293$		−	214.900i		−	0.733446i	0.930330	−	0.366723i	$-0.119520\pi$
$294$		0			0
$295$	−39.3914			−0.133530
$296$		0			0
$297$	41.0585	+	447.998i	0.138244	+	1.50841i
$298$		0			0
$299$		−	313.968i		−	1.05006i
$300$		0			0
$301$	373.356			1.24039
$302$		0			0
$303$	365.773	−	226.307i	1.20717	−	0.746889i
$304$		0			0
$305$		−	94.7946i		−	0.310802i
$306$		0			0
$307$	425.639			1.38645			0.693223	−	0.720723i	$-0.256190\pi$
$307$	425.639			1.38645			0.693223	+	0.720723i	$0.256190\pi$
$308$		0			0
$309$	179.544	+	290.191i	0.581048	+	0.939128i
$310$		0			0
$311$		−	7.67121i		−	0.0246663i	−0.999924	−	0.0123331i	$-0.996074\pi$
$311$		−	7.67121i		−	0.0246663i	0.999924	−	0.0123331i	$-0.00392586\pi$
$312$		0			0
$313$	313.959			1.00306			0.501532	−	0.865139i	$-0.332770\pi$
$313$	313.959			1.00306			0.501532	+	0.865139i	$0.332770\pi$
$314$		0			0
$315$	−108.401	−	54.0682i	−0.344131	−	0.171645i
$316$		0			0
$317$		−	361.930i		−	1.14174i	−0.821042	−	0.570868i	$-0.806607\pi$
$317$		−	361.930i		−	1.14174i	0.821042	−	0.570868i	$-0.193393\pi$
$318$		0			0
$319$	−209.105			−0.655503
$320$		0			0
$321$	−15.9710	+	9.88142i	−0.0497539	+	0.0307832i
$322$		0			0
$323$			50.6216i			0.156723i
$324$		0			0
$325$	−436.204			−1.34217
$326$		0			0
$327$	−233.068	−	376.701i	−0.712748	−	1.15199i
$328$		0			0
$329$		−	203.813i		−	0.619493i
$330$		0			0
$331$	−283.000			−0.854984			−0.427492	−	0.904019i	$-0.640603\pi$
$331$	−283.000			−0.854984			−0.427492	+	0.904019i	$0.640603\pi$
$332$		0			0
$333$	−74.6280	+	149.622i	−0.224108	+	0.449314i
$334$		0			0
$335$			74.1218i			0.221259i
$336$		0			0
$337$	242.847			0.720614			0.360307	−	0.932834i	$-0.382672\pi$
$337$	242.847			0.720614			0.360307	+	0.932834i	$0.382672\pi$
$338$		0			0
$339$	−131.106	+	81.1168i	−0.386745	+	0.239283i
$340$		0			0
$341$			338.517i			0.992720i
$342$		0			0
$343$	68.6810			0.200236
$344$		0			0
$345$	34.7740	+	56.2041i	0.100794	+	0.162910i
$346$		0			0
$347$		−	35.8055i		−	0.103186i	−0.998668	−	0.0515929i	$-0.983570\pi$
$347$		−	35.8055i		−	0.103186i	0.998668	−	0.0515929i	$-0.0164298\pi$
$348$		0			0
$349$	−119.019			−0.341030			−0.170515	−	0.985355i	$-0.554543\pi$
$349$	−119.019			−0.341030			−0.170515	+	0.985355i	$0.554543\pi$
$350$		0			0
$351$	−504.014	+	46.1923i	−1.43594	+	0.131602i
$352$		0			0
$353$		−	158.225i		−	0.448230i	−0.974563	−	0.224115i	$-0.928051\pi$
$353$		−	158.225i		−	0.448230i	0.974563	−	0.224115i	$-0.0719492\pi$
$354$		0			0
$355$	179.497			0.505626
$356$		0			0
$357$	114.374	−	70.7641i	0.320375	−	0.198219i
$358$		0			0
$359$		−	127.881i		−	0.356214i	−0.984011	−	0.178107i	$-0.943003\pi$
$359$		−	127.881i		−	0.356214i	0.984011	−	0.178107i	$-0.0569974\pi$
$360$		0			0
$361$	−227.495			−0.630180
$362$		0			0
$363$	247.221	+	399.576i	0.681051	+	1.10076i
$364$		0			0
$365$		−	106.298i		−	0.291227i
$366$		0			0
$367$	−92.6498			−0.252452			−0.126226	−	0.992002i	$-0.540286\pi$
$367$	−92.6498			−0.252452			−0.126226	+	0.992002i	$0.540286\pi$
$368$		0			0
$369$	−633.620	−	316.036i	−1.71713	−	0.856466i
$370$		0			0
$371$		−	831.998i		−	2.24258i
$372$		0			0
$373$	−9.37918			−0.0251453			−0.0125726	−	0.999921i	$-0.504002\pi$
$373$	−9.37918			−0.0251453			−0.0125726	+	0.999921i	$0.504002\pi$
$374$		0			0
$375$	161.977	−	100.217i	0.431940	−	0.267245i
$376$		0			0
$377$		−	235.251i		−	0.624009i
$378$		0			0
$379$	353.377			0.932392			0.466196	−	0.884681i	$-0.345624\pi$
$379$	353.377			0.932392			0.466196	+	0.884681i	$0.345624\pi$
$380$		0			0
$381$	229.789	+	371.400i	0.603120	+	0.974802i
$382$		0			0
$383$			106.620i			0.278382i	0.990266	+	0.139191i	$0.0444502\pi$
$383$			106.620i			0.278382i	−0.990266	+	0.139191i	$0.955550\pi$
$384$		0			0
$385$	−224.266			−0.582509
$386$		0			0
$387$	−146.566	+	293.850i	−0.378723	+	0.759302i
$388$		0			0
$389$		−	659.144i		−	1.69446i	−0.531227	−	0.847229i	$-0.678269\pi$
$389$		−	659.144i		−	1.69446i	0.531227	−	0.847229i	$-0.321731\pi$
$390$		0			0
$391$	−73.3799			−0.187672
$392$		0			0
$393$	115.822	−	71.6601i	0.294712	−	0.182341i
$394$		0			0
$395$			113.161i			0.286483i
$396$		0			0
$397$	523.170			1.31781			0.658904	−	0.752227i	$-0.271020\pi$
$397$	523.170			1.31781			0.658904	+	0.752227i	$0.271020\pi$
$398$		0			0
$399$	−186.627	−	301.639i	−0.467738	−	0.755989i
$400$		0			0
$401$			255.566i			0.637323i	0.947869	+	0.318661i	$0.103233\pi$
$401$			255.566i			0.637323i	−0.947869	+	0.318661i	$0.896767\pi$
$402$		0			0
$403$	−380.845			−0.945024
$404$		0			0
$405$	85.1086	−	64.0919i	0.210145	−	0.158252i
$406$		0			0
$407$			309.545i			0.760552i
$408$		0			0
$409$	−630.598			−1.54180			−0.770902	−	0.636954i	$-0.780194\pi$
$409$	−630.598			−1.54180			−0.770902	+	0.636954i	$0.780194\pi$
$410$		0			0
$411$	−457.061	+	282.788i	−1.11207	+	0.688050i
$412$		0			0
$413$			306.452i			0.742014i
$414$		0			0
$415$	−105.788			−0.254910
$416$		0			0
$417$	80.3175	+	129.815i	0.192608	+	0.311306i
$418$		0			0
$419$		−	475.721i		−	1.13537i	−0.823245	−	0.567686i	$-0.807839\pi$
$419$		−	475.721i		−	1.13537i	0.823245	−	0.567686i	$-0.192161\pi$
$420$		0			0
$421$	239.614			0.569155			0.284578	−	0.958653i	$-0.408147\pi$
$421$	239.614			0.569155			0.284578	+	0.958653i	$0.408147\pi$
$422$		0			0
$423$	160.411	+	80.0096i	0.379222	+	0.189148i
$424$		0			0
$425$			101.949i			0.239879i
$426$		0			0
$427$	−737.470			−1.72710
$428$		0			0
$429$	−796.828	+	493.006i	−1.85741	+	1.14920i
$430$		0			0
$431$		−	618.539i		−	1.43513i	−0.696494	−	0.717563i	$-0.745258\pi$
$431$		−	618.539i		−	1.43513i	0.696494	−	0.717563i	$-0.254742\pi$
$432$		0			0
$433$	211.763			0.489061			0.244530	−	0.969642i	$-0.421366\pi$
$433$	211.763			0.489061			0.244530	+	0.969642i	$0.421366\pi$
$434$		0			0
$435$	26.0557	+	42.1129i	0.0598981	+	0.0968112i
$436$		0			0
$437$			193.526i			0.442851i
$438$		0			0
$439$	−56.6884			−0.129131			−0.0645654	−	0.997913i	$-0.520566\pi$
$439$	−56.6884			−0.129131			−0.0645654	+	0.997913i	$0.520566\pi$
$440$		0			0
$441$	−223.797	+	448.690i	−0.507476	+	1.01744i
$442$		0			0
$443$		−	687.493i		−	1.55190i	−0.630792	−	0.775952i	$-0.717270\pi$
$443$		−	687.493i		−	1.55190i	0.630792	−	0.775952i	$-0.282730\pi$
$444$		0			0
$445$	−173.401			−0.389666
$446$		0			0
$447$	193.994	−	120.026i	0.433992	−	0.268515i
$448$		0			0
$449$		−	490.455i		−	1.09233i	−0.837679	−	0.546163i	$-0.816088\pi$
$449$		−	490.455i		−	1.09233i	0.837679	−	0.546163i	$-0.183912\pi$
$450$		0			0
$451$	−1310.86			−2.90657
$452$		0			0
$453$	283.989	+	459.002i	0.626908	+	1.01325i
$454$		0			0
$455$		−	252.307i		−	0.554522i
$456$		0			0
$457$	−537.122			−1.17532			−0.587661	−	0.809107i	$-0.699951\pi$
$457$	−537.122			−1.17532			−0.587661	+	0.809107i	$0.699951\pi$
$458$		0			0
$459$	10.7960	+	117.797i	0.0235206	+	0.256639i
$460$		0			0
$461$		−	129.931i		−	0.281847i	−0.990021	−	0.140923i	$-0.954993\pi$
$461$		−	129.931i		−	0.281847i	0.990021	−	0.140923i	$-0.0450071\pi$
$462$		0			0
$463$	262.112			0.566118			0.283059	−	0.959103i	$-0.408651\pi$
$463$	262.112			0.566118			0.283059	+	0.959103i	$0.408651\pi$
$464$		0			0
$465$	68.1759	−	42.1811i	0.146615	−	0.0907121i
$466$		0			0
$467$			418.978i			0.897169i	0.893740	+	0.448585i	$0.148072\pi$
$467$			418.978i			0.897169i	−0.893740	+	0.448585i	$0.851928\pi$
$468$		0			0
$469$	576.643			1.22952
$470$		0			0
$471$	−106.100	−	171.485i	−0.225264	−	0.364087i
$472$		0			0
$473$			607.931i			1.28527i
$474$		0			0
$475$	268.871			0.566043
$476$		0			0
$477$	654.823	+	326.612i	1.37280	+	0.684720i
$478$		0			0
$479$		−	306.699i		−	0.640290i	−0.947369	−	0.320145i	$-0.896268\pi$
$479$		−	306.699i		−	0.640290i	0.947369	−	0.320145i	$-0.103732\pi$
$480$		0			0
$481$	−348.249			−0.724011
$482$		0			0
$483$	437.249	−	270.530i	0.905278	−	0.560104i
$484$		0			0
$485$		−	26.8822i		−	0.0554272i
$486$		0			0
$487$	−387.411			−0.795506			−0.397753	−	0.917493i	$-0.630210\pi$
$487$	−387.411			−0.795506			−0.397753	+	0.917493i	$0.630210\pi$
$488$		0			0
$489$	−162.143	−	262.066i	−0.331580	−	0.535922i
$490$		0			0
$491$		−	13.1250i		−	0.0267311i	−0.999911	−	0.0133655i	$-0.995745\pi$
$491$		−	13.1250i		−	0.0267311i	0.999911	−	0.0133655i	$-0.00425451\pi$
$492$		0			0
$493$	−54.9824			−0.111526
$494$		0			0
$495$	88.0385	−	176.508i	0.177855	−	0.356582i
$496$		0			0
$497$		−	1396.43i		−	2.80971i
$498$		0			0
$499$	239.526			0.480012			0.240006	−	0.970771i	$-0.422851\pi$
$499$	239.526			0.480012			0.240006	+	0.970771i	$0.422851\pi$
$500$		0			0
$501$	−206.119	+	127.528i	−0.411415	+	0.254546i
$502$		0			0
$503$			595.462i			1.18382i	0.806004	+	0.591911i	$0.201626\pi$
$503$			595.462i			1.18382i	−0.806004	+	0.591911i	$0.798374\pi$
$504$		0			0
$505$	−188.585			−0.373436
$506$		0			0
$507$	−287.893	−	465.312i	−0.567837	−	0.917775i
$508$		0			0
$509$			600.556i			1.17987i	0.807449	+	0.589937i	$0.200847\pi$
$509$			600.556i			1.17987i	−0.807449	+	0.589937i	$0.799153\pi$
$510$		0			0
$511$	−826.961			−1.61832
$512$		0			0
$513$	310.668	−	28.4724i	0.605591	−	0.0555017i
$514$		0			0
$515$		−	149.616i		−	0.290517i
$516$		0			0
$517$	331.866			0.641908
$518$		0			0
$519$	−431.622	+	267.049i	−0.831642	+	0.514545i
$520$		0			0
$521$		−	394.528i		−	0.757251i	−0.925550	−	0.378626i	$-0.876397\pi$
$521$		−	394.528i		−	0.757251i	0.925550	−	0.378626i	$-0.123603\pi$
$522$		0			0
$523$	639.213			1.22221			0.611103	−	0.791551i	$-0.290726\pi$
$523$	639.213			1.22221			0.611103	+	0.791551i	$0.290726\pi$
$524$		0			0
$525$	−375.855	−	607.482i	−0.715914	−	1.15711i
$526$		0			0
$527$			89.0102i			0.168900i
$528$		0			0
$529$	248.470			0.469697
$530$		0			0
$531$	−241.192	−	120.302i	−0.454223	−	0.226557i
$532$		0			0
$533$		−	1474.77i		−	2.76692i
$534$		0			0
$535$	8.23433			0.0153913
$536$		0			0
$537$	−436.205	+	269.884i	−0.812300	+	0.502578i
$538$		0			0
$539$			928.273i			1.72221i
$540$		0			0
$541$	−606.241			−1.12059			−0.560296	−	0.828292i	$-0.689313\pi$
$541$	−606.241			−1.12059			−0.560296	+	0.828292i	$0.689313\pi$
$542$		0			0
$543$	62.5524	+	101.101i	0.115198	+	0.186190i
$544$		0			0
$545$			194.219i			0.356365i
$546$		0			0
$547$	409.425			0.748492			0.374246	−	0.927330i	$-0.377902\pi$
$547$	409.425			0.748492			0.374246	+	0.927330i	$0.377902\pi$
$548$		0			0
$549$	289.503	−	580.425i	0.527329	−	1.05724i
$550$		0			0
$551$			145.006i			0.263169i
$552$		0			0
$553$	880.352			1.59196
$554$		0			0
$555$	62.3409	−	38.5709i	0.112326	−	0.0694972i
$556$		0			0
$557$			994.040i			1.78463i	0.451411	+	0.892316i	$0.350921\pi$
$557$			994.040i			1.78463i	−0.451411	+	0.892316i	$0.649079\pi$
$558$		0			0
$559$	−683.945			−1.22351
$560$		0			0
$561$	115.224	+	186.233i	0.205391	+	0.331966i
$562$		0			0
$563$		−	29.5954i		−	0.0525673i	−0.999655	−	0.0262837i	$-0.991633\pi$
$563$		−	29.5954i		−	0.0525673i	0.999655	−	0.0262837i	$-0.00836731\pi$
$564$		0			0
$565$	67.5958			0.119639
$566$		0			0
$567$	−498.614	−	662.117i	−0.879389	−	1.16775i
$568$		0			0
$569$			355.324i			0.624471i	0.950005	+	0.312235i	$0.101078\pi$
$569$			355.324i			0.624471i	−0.950005	+	0.312235i	$0.898922\pi$
$570$		0			0
$571$	133.482			0.233769			0.116885	−	0.993145i	$-0.462709\pi$
$571$	133.482			0.233769			0.116885	+	0.993145i	$0.462709\pi$
$572$		0			0
$573$	612.071	−	378.694i	1.06819	−	0.660898i
$574$		0			0
$575$			389.748i			0.677823i
$576$		0			0
$577$	−228.735			−0.396422			−0.198211	−	0.980159i	$-0.563513\pi$
$577$	−228.735			−0.396422			−0.198211	+	0.980159i	$0.563513\pi$
$578$		0			0
$579$	21.3046	+	34.4338i	0.0367954	+	0.0594712i
$580$		0			0
$581$			822.993i			1.41651i
$582$		0			0
$583$	1354.73			2.32372
$584$		0			0
$585$	198.578	+	99.0465i	0.339450	+	0.169310i
$586$		0			0
$587$		−	633.349i		−	1.07896i	−0.841998	−	0.539480i	$-0.818621\pi$
$587$		−	633.349i		−	1.07896i	0.841998	−	0.539480i	$-0.181379\pi$
$588$		0			0
$589$	234.748			0.398553
$590$		0			0
$591$	42.3598	−	26.2084i	0.0716748	−	0.0443459i
$592$		0			0
$593$			189.510i			0.319578i	0.987151	+	0.159789i	$0.0510813\pi$
$593$			189.510i			0.319578i	−0.987151	+	0.159789i	$0.948919\pi$
$594$		0			0
$595$	−58.9688			−0.0991071
$596$		0			0
$597$	−226.781	−	366.538i	−0.379867	−	0.613967i
$598$		0			0
$599$		−	340.082i		−	0.567750i	−0.958861	−	0.283875i	$-0.908380\pi$
$599$		−	340.082i		−	0.567750i	0.958861	−	0.283875i	$-0.0916200\pi$
$600$		0			0
$601$	528.406			0.879211			0.439605	−	0.898191i	$-0.355118\pi$
$601$	528.406			0.879211			0.439605	+	0.898191i	$0.355118\pi$
$602$		0			0
$603$	−226.369	+	453.846i	−0.375404	+	0.752647i
$604$		0			0
$605$		−	206.013i		−	0.340517i
$606$		0			0
$607$	659.649			1.08674			0.543368	−	0.839495i	$-0.317149\pi$
$607$	659.649			1.08674			0.543368	+	0.839495i	$0.317149\pi$
$608$		0			0
$609$	327.624	−	202.704i	0.537971	−	0.332848i
$610$		0			0
$611$			373.362i			0.611067i
$612$		0			0
$613$	−67.7073			−0.110452			−0.0552262	−	0.998474i	$-0.517588\pi$
$613$	−67.7073			−0.110452			−0.0552262	+	0.998474i	$0.517588\pi$
$614$		0			0
$615$	163.341	+	264.002i	0.265595	+	0.429272i
$616$		0			0
$617$			807.698i			1.30907i	0.756031	+	0.654536i	$0.227136\pi$
$617$			807.698i			1.30907i	−0.756031	+	0.654536i	$0.772864\pi$
$618$		0			0
$619$	122.858			0.198479			0.0992393	−	0.995064i	$-0.468359\pi$
$619$	122.858			0.198479			0.0992393	+	0.995064i	$0.468359\pi$
$620$		0			0
$621$	41.2728	+	450.337i	0.0664619	+	0.725180i
$622$		0			0
$623$			1349.01i			2.16534i
$624$		0			0
$625$	498.235			0.797176
$626$		0			0
$627$	491.155	−	303.882i	0.783341	−	0.484661i
$628$		0			0
$629$			81.3921i			0.129399i
$630$		0			0
$631$	141.286			0.223908			0.111954	−	0.993713i	$-0.464289\pi$
$631$	141.286			0.223908			0.111954	+	0.993713i	$0.464289\pi$
$632$		0			0
$633$	−132.045	−	213.420i	−0.208602	−	0.337156i
$634$		0			0
$635$		−	191.486i		−	0.301553i
$636$		0			0
$637$	−1044.34			−1.63947
$638$		0			0
$639$	1099.06	+	548.186i	1.71996	+	0.857881i
$640$		0			0
$641$		−	240.601i		−	0.375352i	−0.982231	−	0.187676i	$-0.939904\pi$
$641$		−	240.601i		−	0.375352i	0.982231	−	0.187676i	$-0.0600955\pi$
$642$		0			0
$643$	774.975			1.20525			0.602624	−	0.798025i	$-0.294122\pi$
$643$	774.975			1.20525			0.602624	+	0.798025i	$0.294122\pi$
$644$		0			0
$645$	122.435	−	75.7515i	0.189821	−	0.117444i
$646$		0			0
$647$		−	823.719i		−	1.27314i	−0.771220	−	0.636568i	$-0.780353\pi$
$647$		−	823.719i		−	1.27314i	0.771220	−	0.636568i	$-0.219647\pi$
$648$		0			0
$649$	−498.991			−0.768861
$650$		0			0
$651$	−328.155	−	530.386i	−0.504078	−	0.814724i
$652$		0			0
$653$		−	142.733i		−	0.218581i	−0.994010	−	0.109290i	$-0.965142\pi$
$653$		−	142.733i		−	0.218581i	0.994010	−	0.109290i	$-0.0348579\pi$
$654$		0			0
$655$	−59.7154			−0.0911685
$656$		0			0
$657$	324.634	−	650.859i	0.494116	−	0.990652i
$658$		0			0
$659$			485.418i			0.736598i	0.929707	+	0.368299i	$0.120060\pi$
$659$			485.418i			0.736598i	−0.929707	+	0.368299i	$0.879940\pi$
$660$		0			0
$661$	89.9502			0.136082			0.0680410	−	0.997683i	$-0.478325\pi$
$661$	89.9502			0.136082			0.0680410	+	0.997683i	$0.478325\pi$
$662$		0			0
$663$	−209.519	+	129.632i	−0.316017	+	0.195523i
$664$		0			0
$665$			155.519i			0.233863i
$666$		0			0
$667$	−210.197			−0.315138
$668$		0			0
$669$	−231.869	−	374.763i	−0.346591	−	0.560184i
$670$		0			0
$671$		−	1200.81i		−	1.78958i
$672$		0			0
$673$	779.599			1.15839			0.579197	−	0.815188i	$-0.303366\pi$
$673$	779.599			1.15839			0.579197	+	0.815188i	$0.303366\pi$
$674$		0			0
$675$	625.665	−	57.3415i	0.926911	−	0.0849503i
$676$		0			0
$677$		−	207.151i		−	0.305984i	−0.988227	−	0.152992i	$-0.951109\pi$
$677$		−	207.151i		−	0.305984i	0.988227	−	0.152992i	$-0.0488909\pi$
$678$		0			0
$679$	−209.134			−0.308003
$680$		0			0
$681$	−72.7724	+	45.0250i	−0.106861	+	0.0661160i
$682$		0			0
$683$			680.228i			0.995941i	0.867194	+	0.497970i	$0.165921\pi$
$683$			680.228i			0.995941i	−0.867194	+	0.497970i	$0.834079\pi$
$684$		0			0
$685$	235.652			0.344017
$686$		0			0
$687$	544.748	+	880.458i	0.792938	+	1.28160i
$688$		0			0
$689$			1524.12i			2.21208i
$690$		0			0
$691$	−480.570			−0.695470			−0.347735	−	0.937593i	$-0.613049\pi$
$691$	−480.570			−0.695470			−0.347735	+	0.937593i	$0.613049\pi$
$692$		0			0
$693$	−1373.17	−	684.910i	−1.98149	−	0.988326i
$694$		0			0
$695$		−	66.9297i		−	0.0963018i
$696$		0			0
$697$	−344.680			−0.494520
$698$		0			0
$699$	268.567	−	166.165i	0.384216	−	0.237718i
$700$		0			0
$701$		−	854.903i		−	1.21955i	−0.792575	−	0.609774i	$-0.791260\pi$
$701$		−	854.903i		−	1.21955i	0.792575	−	0.609774i	$-0.208740\pi$
$702$		0			0
$703$	214.656			0.305343
$704$		0			0
$705$	−41.3523	−	66.8364i	−0.0586558	−	0.0948034i
$706$		0			0
$707$			1467.13i			2.07515i
$708$		0			0
$709$	−990.069			−1.39643			−0.698215	−	0.715888i	$-0.746022\pi$
$709$	−990.069			−1.39643			−0.698215	+	0.715888i	$0.746022\pi$
$710$		0			0
$711$	−345.593	+	692.880i	−0.486067	+	0.974514i
$712$		0			0
$713$			340.285i			0.477258i
$714$		0			0
$715$	410.828			0.574585
$716$		0			0
$717$	−715.495	+	442.684i	−0.997901	+	0.617411i
$718$		0			0
$719$		−	502.244i		−	0.698531i	−0.937024	−	0.349266i	$-0.886431\pi$
$719$		−	502.244i		−	0.698531i	0.937024	−	0.349266i	$-0.113569\pi$
$720$		0			0
$721$	−1163.97			−1.61438
$722$		0			0
$723$	583.760	+	943.511i	0.807413	+	1.30499i
$724$		0			0
$725$			292.032i			0.402803i
$726$		0			0
$727$	718.280			0.988005			0.494002	−	0.869461i	$-0.335533\pi$
$727$	718.280			0.988005			0.494002	+	0.869461i	$0.335533\pi$
$728$		0			0
$729$	716.856	−	132.511i	0.983341	−	0.181771i
$730$		0			0
$731$			159.850i			0.218673i
$732$		0			0
$733$	−1362.31			−1.85854			−0.929272	−	0.369397i	$-0.879564\pi$
$733$	−1362.31			−1.85854			−0.929272	+	0.369397i	$0.879564\pi$
$734$		0			0
$735$	186.950	−	115.668i	0.254354	−	0.157371i
$736$		0			0
$737$			938.939i			1.27400i
$738$		0			0
$739$	−1057.01			−1.43033			−0.715164	−	0.698957i	$-0.753648\pi$
$739$	−1057.01			−1.43033			−0.715164	+	0.698957i	$0.753648\pi$
$740$		0			0
$741$	341.879	+	552.568i	0.461375	+	0.745705i
$742$		0			0
$743$			88.0799i			0.118546i	0.998242	+	0.0592731i	$0.0188783\pi$
$743$			88.0799i			0.118546i	−0.998242	+	0.0592731i	$0.981122\pi$
$744$		0			0
$745$	−100.019			−0.134254
$746$		0			0
$747$	−647.736	−	323.076i	−0.867116	−	0.432499i
$748$		0			0
$749$		−	64.0603i		−	0.0855278i
$750$		0			0
$751$	−11.4389			−0.0152315			−0.00761576	−	0.999971i	$-0.502424\pi$
$751$	−11.4389			−0.0152315			−0.00761576	+	0.999971i	$0.502424\pi$
$752$		0			0
$753$	−70.1470	+	43.4006i	−0.0931566	+	0.0576369i
$754$		0			0
$755$		−	236.652i		−	0.313447i
$756$		0			0
$757$	38.6736			0.0510879			0.0255440	−	0.999674i	$-0.491868\pi$
$757$	38.6736			0.0510879			0.0255440	+	0.999674i	$0.491868\pi$
$758$		0			0
$759$	440.501	+	711.966i	0.580370	+	0.938032i
$760$		0			0
$761$		−	814.341i		−	1.07009i	−0.844822	−	0.535047i	$-0.820294\pi$
$761$		−	814.341i		−	1.07009i	0.844822	−	0.535047i	$-0.179706\pi$
$762$		0			0
$763$	1510.96			1.98029
$764$		0			0
$765$	23.1489	−	46.4113i	0.0302601	−	0.0606684i
$766$		0			0
$767$		−	561.383i		−	0.731921i
$768$		0			0
$769$	490.085			0.637302			0.318651	−	0.947872i	$-0.396770\pi$
$769$	490.085			0.637302			0.318651	+	0.947872i	$0.396770\pi$
$770$		0			0
$771$	12.5632	−	7.77298i	0.0162947	−	0.0100817i
$772$		0			0
$773$			747.603i			0.967145i	0.875304	+	0.483572i	$0.160661\pi$
$773$			747.603i			0.967145i	−0.875304	+	0.483572i	$0.839339\pi$
$774$		0			0
$775$	472.767			0.610022
$776$		0			0
$777$	−300.069	−	484.991i	−0.386189	−	0.624184i
$778$		0			0
$779$			909.030i			1.16692i
$780$		0			0
$781$	2273.78			2.91137
$782$		0			0
$783$	30.9251	+	337.430i	0.0394957	+	0.430946i
$784$		0			0
$785$			88.4143i			0.112630i
$786$		0			0
$787$	−975.671			−1.23973			−0.619867	−	0.784707i	$-0.712814\pi$
$787$	−975.671			−1.23973			−0.619867	+	0.784707i	$0.712814\pi$
$788$		0			0
$789$	672.427	−	416.037i	0.852252	−	0.527297i
$790$		0			0
$791$		−	525.873i		−	0.664820i
$792$		0			0
$793$	1350.96			1.70360
$794$		0			0
$795$	−168.807	−	272.837i	−0.212336	−	0.343191i
$796$		0			0
$797$			417.874i			0.524309i	0.965026	+	0.262154i	$0.0844330\pi$
$797$			417.874i			0.524309i	−0.965026	+	0.262154i	$0.915567\pi$
$798$		0			0
$799$	87.2613			0.109213
$800$		0			0
$801$	−1061.73	−	529.569i	−1.32551	−	0.661135i
$802$		0			0
$803$		−	1346.53i		−	1.67687i
$804$		0			0
$805$	−225.437			−0.280046
$806$		0			0
$807$	−602.137	+	372.548i	−0.746143	+	0.461646i
$808$		0			0
$809$			743.849i			0.919468i	0.888057	+	0.459734i	$0.152055\pi$
$809$			743.849i			0.919468i	−0.888057	+	0.459734i	$0.847945\pi$
$810$		0			0
$811$	1248.10			1.53897			0.769483	−	0.638667i	$-0.220514\pi$
$811$	1248.10			1.53897			0.769483	+	0.638667i	$0.220514\pi$
$812$		0			0
$813$	−683.920	−	1105.40i	−0.841229	−	1.35965i
$814$		0			0
$815$			135.116i			0.165786i
$816$		0			0
$817$	421.575			0.516004
$818$		0			0
$819$	770.549	−	1544.87i	0.940841	−	1.88629i
$820$		0			0
$821$			202.487i			0.246635i	0.992367	+	0.123318i	$0.0393534\pi$
$821$			202.487i			0.246635i	−0.992367	+	0.123318i	$0.960647\pi$
$822$		0			0
$823$	525.426			0.638427			0.319214	−	0.947683i	$-0.396581\pi$
$823$	525.426			0.638427			0.319214	+	0.947683i	$0.396581\pi$
$824$		0			0
$825$	989.154	−	611.999i	1.19897	−	0.741817i
$826$		0			0
$827$		−	857.247i		−	1.03657i	−0.855207	−	0.518287i	$-0.826570\pi$
$827$		−	857.247i		−	1.03657i	0.855207	−	0.518287i	$-0.173430\pi$
$828$		0			0
$829$	578.524			0.697857			0.348929	−	0.937149i	$-0.386546\pi$
$829$	578.524			0.697857			0.348929	+	0.937149i	$0.386546\pi$
$830$		0			0
$831$	−385.245	−	622.658i	−0.463592	−	0.749288i
$832$		0			0
$833$			244.081i			0.293015i
$834$		0			0
$835$	106.271			0.127270
$836$		0			0
$837$	546.261	−	50.0642i	0.652641	−	0.0598138i
$838$		0			0
$839$			1450.83i			1.72924i	0.502425	+	0.864621i	$0.332441\pi$
$839$			1450.83i			1.72924i	−0.502425	+	0.864621i	$0.667559\pi$
$840$		0			0
$841$	683.503			0.812726
$842$		0			0
$843$	−1101.16	+	681.298i	−1.30624	+	0.808183i
$844$		0			0
$845$			239.905i			0.283912i
$846$		0			0
$847$	−1602.71			−1.89222
$848$		0			0
$849$	55.5364	+	89.7616i	0.0654139	+	0.105726i
$850$		0			0
$851$			311.161i			0.365641i
$852$		0			0
$853$	−1293.52			−1.51643			−0.758216	−	0.652004i	$-0.773929\pi$
$853$	−1293.52			−1.51643			−0.758216	+	0.652004i	$0.773929\pi$
$854$		0			0
$855$	−122.401	−	61.0510i	−0.143159	−	0.0714047i
$856$		0			0
$857$		−	954.256i		−	1.11348i	−0.830685	−	0.556742i	$-0.812051\pi$
$857$		−	954.256i		−	1.11348i	0.830685	−	0.556742i	$-0.187949\pi$
$858$		0			0
$859$	−1590.52			−1.85159			−0.925796	−	0.378023i	$-0.876604\pi$
$859$	−1590.52			−1.85159			−0.925796	+	0.378023i	$0.876604\pi$
$860$		0			0
$861$	2053.85	−	1270.74i	2.38542	−	1.47588i
$862$		0			0
$863$		−	398.552i		−	0.461821i	−0.972975	−	0.230911i	$-0.925829\pi$
$863$		−	398.552i		−	0.461821i	0.972975	−	0.230911i	$-0.0741705\pi$
$864$		0			0
$865$	222.536			0.257267
$866$		0			0
$867$	−425.872	−	688.323i	−0.491202	−	0.793913i
$868$		0			0
$869$			1433.46i			1.64956i
$870$		0			0
$871$	−1056.34			−1.21279
$872$		0			0
$873$	82.0984	−	164.599i	0.0940417	−	0.188544i
$874$		0			0
$875$			649.697i			0.742511i
$876$		0			0
$877$	515.896			0.588251			0.294125	−	0.955767i	$-0.404972\pi$
$877$	515.896			0.588251			0.294125	+	0.955767i	$0.404972\pi$
$878$		0			0
$879$	−548.248	+	339.207i	−0.623718	+	0.385901i
$880$		0			0
$881$			686.344i			0.779051i	0.921016	+	0.389526i	$0.127361\pi$
$881$			686.344i			0.779051i	−0.921016	+	0.389526i	$0.872639\pi$
$882$		0			0
$883$	733.647			0.830857			0.415429	−	0.909626i	$-0.363632\pi$
$883$	733.647			0.830857			0.415429	+	0.909626i	$0.363632\pi$
$884$		0			0
$885$	62.1770	+	100.495i	0.0702564	+	0.113553i
$886$		0			0
$887$		−	1527.56i		−	1.72217i	−0.508465	−	0.861083i	$-0.669787\pi$
$887$		−	1527.56i		−	1.72217i	0.508465	−	0.861083i	$-0.330213\pi$
$888$		0			0
$889$	−1489.70			−1.67570
$890$		0			0
$891$	1078.11	−	811.885i	1.21001	−	0.911207i
$892$		0			0
$893$		−	230.136i		−	0.257711i
$894$		0			0
$895$	224.899			0.251283
$896$		0			0
$897$	−800.989	+	495.580i	−0.892964	+	0.552486i
$898$		0			0
$899$			254.970i			0.283615i
$900$		0			0
$901$	356.215			0.395355
$902$		0			0
$903$	−589.321	−	952.500i	−0.652626	−	1.05482i
$904$		0			0
$905$		−	52.1258i		−	0.0575976i
$906$		0			0
$907$	174.749			0.192667			0.0963334	−	0.995349i	$-0.469289\pi$
$907$	174.749			0.192667			0.0963334	+	0.995349i	$0.469289\pi$
$908$		0			0
$909$	−1154.70	−	575.940i	−1.27030	−	0.633598i
$910$		0			0
$911$			1459.33i			1.60190i	0.598733	+	0.800949i	$0.295671\pi$
$911$			1459.33i			1.60190i	−0.598733	+	0.800949i	$0.704329\pi$
$912$		0			0
$913$	−1340.07			−1.46776
$914$		0			0
$915$	−241.838	+	149.628i	−0.264304	+	0.163527i
$916$		0			0
$917$			464.566i			0.506615i
$918$		0			0
$919$	1271.02			1.38304			0.691521	−	0.722356i	$-0.256941\pi$
$919$	1271.02			1.38304			0.691521	+	0.722356i	$0.256941\pi$
$920$		0			0
$921$	−671.846	−	1085.88i	−0.729474	−	1.17903i
$922$		0			0
$923$			2558.09i			2.77150i
$924$		0			0
$925$	432.304			0.467356
$926$		0			0
$927$	456.930	−	916.098i	0.492912	−	0.988239i
$928$		0			0
$929$		−	1112.35i		−	1.19736i	−0.800987	−	0.598681i	$-0.795692\pi$
$929$		−	1112.35i		−	1.19736i	0.800987	−	0.598681i	$-0.204308\pi$
$930$		0			0
$931$	643.719			0.691427
$932$		0			0
$933$	−19.5707	+	12.1086i	−0.0209761	+	0.0129781i
$934$		0			0
$935$		−	96.0180i		−	0.102693i
$936$		0			0
$937$	−331.746			−0.354051			−0.177026	−	0.984206i	$-0.556648\pi$
$937$	−331.746			−0.354051			−0.177026	+	0.984206i	$0.556648\pi$
$938$		0			0
$939$	−495.566	−	800.967i	−0.527760	−	0.853000i
$940$		0			0
$941$			36.4001i			0.0386824i	0.999813	+	0.0193412i	$0.00615688\pi$
$941$			36.4001i			0.0386824i	−0.999813	+	0.0193412i	$0.993843\pi$
$942$		0			0
$943$	−1317.71			−1.39736
$944$		0			0
$945$	33.1672	+	361.895i	0.0350976	+	0.382957i
$946$		0			0
$947$		−	675.715i		−	0.713532i	−0.934194	−	0.356766i	$-0.883879\pi$
$947$		−	675.715i		−	0.713532i	0.934194	−	0.356766i	$-0.116121\pi$
$948$		0			0
$949$	1514.89			1.59631
$950$		0			0
$951$	−923.350	+	571.286i	−0.970925	+	0.600721i
$952$		0			0
$953$			611.944i			0.642124i	0.947058	+	0.321062i	$0.104040\pi$
$953$			611.944i			0.642124i	−0.947058	+	0.321062i	$0.895960\pi$
$954$		0			0
$955$	−315.571			−0.330441
$956$		0			0
$957$	330.060	+	533.466i	0.344891	+	0.557435i
$958$		0			0
$959$		−	1833.29i		−	1.91167i
$960$		0			0
$961$	−548.233			−0.570481
$962$		0			0
$963$	50.4186	+	25.1477i	0.0523558	+	0.0261139i
$964$		0			0
$965$		−	17.7534i		−	0.0183973i
$966$		0			0
$967$	−1258.24			−1.30118			−0.650592	−	0.759428i	$-0.725479\pi$
$967$	−1258.24			−1.30118			−0.650592	+	0.759428i	$0.725479\pi$
$968$		0			0
$969$	129.145	−	79.9033i	0.133277	−	0.0824595i
$970$		0			0
$971$		−	1624.87i		−	1.67340i	−0.547663	−	0.836699i	$-0.684482\pi$
$971$		−	1624.87i		−	1.67340i	0.547663	−	0.836699i	$-0.315518\pi$
$972$		0			0
$973$	−520.691			−0.535140
$974$		0			0
$975$	688.522	+	1112.84i	0.706176	+	1.14137i
$976$		0			0
$977$		−	850.205i		−	0.870220i	−0.900377	−	0.435110i	$-0.856710\pi$
$977$		−	850.205i		−	0.870220i	0.900377	−	0.435110i	$-0.143290\pi$
$978$		0			0
$979$	−2196.57			−2.24368
$980$		0			0
$981$	−593.147	+	1189.20i	−0.604635	+	1.21223i
$982$		0			0
$983$			63.7157i			0.0648176i	0.999475	+	0.0324088i	$0.0103178\pi$
$983$			63.7157i			0.0648176i	−0.999475	+	0.0324088i	$0.989682\pi$
$984$		0			0
$985$	−21.8399			−0.0221725
$986$		0			0
$987$	−519.965	+	321.707i	−0.526813	+	0.325945i
$988$		0			0
$989$			611.105i			0.617901i
$990$		0			0
$991$	−1050.59			−1.06013			−0.530066	−	0.847957i	$-0.677833\pi$
$991$	−1050.59			−1.06013			−0.530066	+	0.847957i	$0.677833\pi$
$992$		0			0
$993$	446.698	+	721.984i	0.449847	+	0.727073i
$994$		0			0
$995$			188.980i			0.189929i
$996$		0			0
$997$	54.6960			0.0548606			0.0274303	−	0.999624i	$-0.491268\pi$
$997$	54.6960			0.0548606			0.0274303	+	0.999624i	$0.491268\pi$
$998$		0			0
$999$	499.508	−	45.7793i	0.500008	−	0.0458251i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.3.e.a.257.3	✓	8
3.2	odd	2	inner	384.3.e.a.257.4	yes	8
4.3	odd	2		384.3.e.d.257.6	yes	8
8.3	odd	2		384.3.e.b.257.3	yes	8
8.5	even	2		384.3.e.c.257.6	yes	8
12.11	even	2		384.3.e.d.257.5	yes	8
16.3	odd	4		768.3.h.g.641.4		16
16.5	even	4		768.3.h.h.641.4		16
16.11	odd	4		768.3.h.g.641.13		16
16.13	even	4		768.3.h.h.641.13		16
24.5	odd	2		384.3.e.c.257.5	yes	8
24.11	even	2		384.3.e.b.257.4	yes	8
48.5	odd	4		768.3.h.h.641.14		16
48.11	even	4		768.3.h.g.641.3		16
48.29	odd	4		768.3.h.h.641.3		16
48.35	even	4		768.3.h.g.641.14		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.3.e.a.257.3	✓	8	1.1	even	1	trivial
384.3.e.a.257.4	yes	8	3.2	odd	2	inner
384.3.e.b.257.3	yes	8	8.3	odd	2
384.3.e.b.257.4	yes	8	24.11	even	2
384.3.e.c.257.5	yes	8	24.5	odd	2
384.3.e.c.257.6	yes	8	8.5	even	2
384.3.e.d.257.5	yes	8	12.11	even	2
384.3.e.d.257.6	yes	8	4.3	odd	2
768.3.h.g.641.3		16	48.11	even	4
768.3.h.g.641.4		16	16.3	odd	4
768.3.h.g.641.13		16	16.11	odd	4
768.3.h.g.641.14		16	48.35	even	4
768.3.h.h.641.3		16	48.29	odd	4
768.3.h.h.641.4		16	16.5	even	4
768.3.h.h.641.13		16	16.13	even	4
768.3.h.h.641.14		16	48.5	odd	4

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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	384.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.57844 - 2.55118i) q^{3} +1.31534i q^{5} +10.2329 q^{7} +(-4.01705 + 8.05378i) q^{9} +O(q^{10})\)	\(q+(-1.57844 - 2.55118i) q^{3} +1.31534i q^{5} +10.2329 q^{7} +(-4.01705 + 8.05378i) q^{9} +16.6620i q^{11} -18.7454 q^{13} +(3.35566 - 2.07618i) q^{15} +4.38114i q^{17} +11.5544 q^{19} +(-16.1520 - 26.1059i) q^{21} +16.7490i q^{23} +23.2699 q^{25} +(26.8873 - 2.46419i) q^{27} +12.5498i q^{29} +20.3167 q^{31} +(42.5079 - 26.3001i) q^{33} +13.4597i q^{35} +18.5778 q^{37} +(29.5885 + 47.8230i) q^{39} +78.6737i q^{41} +36.4860 q^{43} +(-10.5934 - 5.28377i) q^{45} -19.9175i q^{47} +55.7118 q^{49} +(11.1771 - 6.91537i) q^{51} -81.3064i q^{53} -21.9162 q^{55} +(-18.2380 - 29.4775i) q^{57} +29.9477i q^{59} -72.0687 q^{61} +(-41.1060 + 82.4133i) q^{63} -24.6565i q^{65} +56.3520 q^{67} +(42.7298 - 26.4374i) q^{69} -136.465i q^{71} -80.8141 q^{73} +(-36.7301 - 59.3657i) q^{75} +170.501i q^{77} +86.0317 q^{79} +(-48.7266 - 64.7048i) q^{81} +80.4263i q^{83} -5.76268 q^{85} +(32.0168 - 19.8091i) q^{87} +131.830i q^{89} -191.820 q^{91} +(-32.0687 - 51.8315i) q^{93} +15.1980i q^{95} -20.4375 q^{97} +(-134.192 - 66.9323i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{3} - 8 q^{7}+O(q^{10}) \) 8 * q - 4 * q^3 - 8 * q^7	\( 8 q - 4 q^{3} - 8 q^{7} - 16 q^{15} + 24 q^{19} + 16 q^{21} - 40 q^{25} + 44 q^{27} + 56 q^{31} + 8 q^{33} + 32 q^{37} + 104 q^{39} - 136 q^{43} - 80 q^{45} + 72 q^{49} - 176 q^{51} - 192 q^{55} - 40 q^{57} - 160 q^{61} - 264 q^{63} + 280 q^{67} + 80 q^{69} - 80 q^{73} + 348 q^{75} + 408 q^{79} + 72 q^{81} + 192 q^{85} + 368 q^{87} - 336 q^{91} + 160 q^{93} + 96 q^{97} - 432 q^{99}+O(q^{100}) \) 8 * q - 4 * q^3 - 8 * q^7 - 16 * q^15 + 24 * q^19 + 16 * q^21 - 40 * q^25 + 44 * q^27 + 56 * q^31 + 8 * q^33 + 32 * q^37 + 104 * q^39 - 136 * q^43 - 80 * q^45 + 72 * q^49 - 176 * q^51 - 192 * q^55 - 40 * q^57 - 160 * q^61 - 264 * q^63 + 280 * q^67 + 80 * q^69 - 80 * q^73 + 348 * q^75 + 408 * q^79 + 72 * q^81 + 192 * q^85 + 368 * q^87 - 336 * q^91 + 160 * q^93 + 96 * q^97 - 432 * q^99

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