LMFDB - Embedded newform 384.4.c.d.383.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,4,Mod(383,384)]

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([1, 0, 1]))

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

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

chi := DirichletCharacter("384.383");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	384.c (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:	$22.6567334422$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 39x^{10} + 549x^{8} + 3500x^{6} + 10236x^{4} + 11952x^{2} + 4624 $ x^12 + 39x^10 + 549x^8 + 3500x^6 + 10236x^4 + 11952*x^2 + 4624
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{30}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			383.12
Root			$-0.910871i$ of defining polynomial
Character	$\chi$	$=$	384.383
Dual form			384.4.c.d.383.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.80343 + 1.98169i) q^{3} -11.9846i q^{5} +22.6995i q^{7} +(19.1458 + 19.0378i) q^{9} +O(q^{10})$	\(q+(4.80343 + 1.98169i) q^{3} -11.9846i q^{5} +22.6995i q^{7} +(19.1458 + 19.0378i) q^{9} -61.9410 q^{11} +71.3317 q^{13} +(23.7497 - 57.5672i) q^{15} +74.3305i q^{17} +108.076i q^{19} +(-44.9834 + 109.036i) q^{21} +24.7059 q^{23} -18.6309 q^{25} +(54.2388 + 129.388i) q^{27} -84.1331i q^{29} +130.377i q^{31} +(-297.529 - 122.748i) q^{33} +272.045 q^{35} +397.468 q^{37} +(342.637 + 141.357i) q^{39} +104.132i q^{41} -161.650i q^{43} +(228.160 - 229.455i) q^{45} -72.8162 q^{47} -172.269 q^{49} +(-147.300 + 357.041i) q^{51} -136.267i q^{53} +742.339i q^{55} +(-214.172 + 519.134i) q^{57} +243.465 q^{59} -358.454 q^{61} +(-432.149 + 434.602i) q^{63} -854.883i q^{65} +449.508i q^{67} +(118.673 + 48.9593i) q^{69} -329.053 q^{71} -925.687 q^{73} +(-89.4923 - 36.9207i) q^{75} -1406.03i q^{77} +55.9579i q^{79} +(4.12647 + 728.988i) q^{81} +928.194 q^{83} +890.823 q^{85} +(166.725 - 404.127i) q^{87} +853.930i q^{89} +1619.20i q^{91} +(-258.366 + 626.256i) q^{93} +1295.25 q^{95} -714.982 q^{97} +(-1185.91 - 1179.22i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 10 q^{3}+O(q^{10}) $ 12 * q + 10 * q^3	$ 12 q + 10 q^{3} + 36 q^{11} + 84 q^{15} + 136 q^{21} - 120 q^{23} - 300 q^{25} - 266 q^{27} - 116 q^{33} + 432 q^{35} + 528 q^{37} + 620 q^{39} + 440 q^{45} - 1248 q^{47} - 948 q^{49} - 1072 q^{51} - 172 q^{57} + 2508 q^{59} + 624 q^{61} + 2744 q^{63} - 24 q^{69} - 2040 q^{71} - 216 q^{73} - 3894 q^{75} - 1076 q^{81} + 4572 q^{83} + 480 q^{85} + 4156 q^{87} - 112 q^{93} - 5448 q^{95} - 48 q^{97} - 6044 q^{99}+O(q^{100}) $ 12 * q + 10 * q^3 + 36 * q^11 + 84 * q^15 + 136 * q^21 - 120 * q^23 - 300 * q^25 - 266 * q^27 - 116 * q^33 + 432 * q^35 + 528 * q^37 + 620 * q^39 + 440 * q^45 - 1248 * q^47 - 948 * q^49 - 1072 * q^51 - 172 * q^57 + 2508 * q^59 + 624 * q^61 + 2744 * q^63 - 24 * q^69 - 2040 * q^71 - 216 * q^73 - 3894 * q^75 - 1076 * q^81 + 4572 * q^83 + 480 * q^85 + 4156 * q^87 - 112 * q^93 - 5448 * q^95 - 48 * q^97 - 6044 * 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
$3$	4.80343	+	1.98169i	0.924420	+	0.381376i
$3$	4.80343	+	1.98169i	0.924420	+	0.381376i
$4$		0			0
$5$		−	11.9846i		−	1.07194i	−0.844238	−	0.535968i	$-0.819947\pi$
$5$		−	11.9846i		−	1.07194i	0.844238	−	0.535968i	$-0.180053\pi$
$6$		0			0
$7$			22.6995i			1.22566i	0.790215	+	0.612830i	$0.209969\pi$
$7$			22.6995i			1.22566i	−0.790215	+	0.612830i	$0.790031\pi$
$8$		0			0
$9$	19.1458	+	19.0378i	0.709105	+	0.705103i
$10$		0			0
$11$	−61.9410			−1.69781			−0.848905	−	0.528545i	$-0.822738\pi$
$11$	−61.9410			−1.69781			−0.848905	+	0.528545i	$0.822738\pi$
$12$		0			0
$13$	71.3317			1.52184			0.760918	−	0.648848i	$-0.224749\pi$
$13$	71.3317			1.52184			0.760918	+	0.648848i	$0.224749\pi$
$14$		0			0
$15$	23.7497	−	57.5672i	0.408810	−	0.990920i
$16$		0			0
$17$			74.3305i			1.06046i	0.847854	+	0.530229i	$0.177894\pi$
$17$			74.3305i			1.06046i	−0.847854	+	0.530229i	$0.822106\pi$
$18$		0			0
$19$			108.076i			1.30496i	0.757805	+	0.652481i	$0.226272\pi$
$19$			108.076i			1.30496i	−0.757805	+	0.652481i	$0.773728\pi$
$20$		0			0
$21$	−44.9834	+	109.036i	−0.467437	+	1.13302i
$22$		0			0
$23$	24.7059			0.223980			0.111990	−	0.993709i	$-0.464278\pi$
$23$	24.7059			0.223980			0.111990	+	0.993709i	$0.464278\pi$
$24$		0			0
$25$	−18.6309			−0.149047
$26$		0			0
$27$	54.2388	+	129.388i	0.386602	+	0.922247i
$28$		0			0
$29$		−	84.1331i		−	0.538728i	−0.963038	−	0.269364i	$-0.913186\pi$
$29$		−	84.1331i		−	0.538728i	0.963038	−	0.269364i	$-0.0868135\pi$
$30$		0			0
$31$			130.377i			0.755367i	0.925935	+	0.377684i	$0.123279\pi$
$31$			130.377i			0.755367i	−0.925935	+	0.377684i	$0.876721\pi$
$32$		0			0
$33$	−297.529	−	122.748i	−1.56949	−	0.647504i
$34$		0			0
$35$	272.045			1.31383
$36$		0			0
$37$	397.468			1.76604			0.883019	−	0.469338i	$-0.155507\pi$
$37$	397.468			1.76604			0.883019	+	0.469338i	$0.155507\pi$
$38$		0			0
$39$	342.637	+	141.357i	1.40682	+	0.580391i
$40$		0			0
$41$			104.132i			0.396649i	0.980136	+	0.198325i	$0.0635500\pi$
$41$			104.132i			0.396649i	−0.980136	+	0.198325i	$0.936450\pi$
$42$		0			0
$43$		−	161.650i		−	0.573289i	−0.958037	−	0.286644i	$-0.907460\pi$
$43$		−	161.650i		−	0.573289i	0.958037	−	0.286644i	$-0.0925398\pi$
$44$		0			0
$45$	228.160	−	229.455i	0.755825	−	0.760116i
$46$		0			0
$47$	−72.8162			−0.225986			−0.112993	−	0.993596i	$-0.536044\pi$
$47$	−72.8162			−0.225986			−0.112993	+	0.993596i	$0.536044\pi$
$48$		0			0
$49$	−172.269			−0.502242
$50$		0			0
$51$	−147.300	+	357.041i	−0.404433	+	0.980310i
$52$		0			0
$53$		−	136.267i		−	0.353164i	−0.984286	−	0.176582i	$-0.943496\pi$
$53$		−	136.267i		−	0.353164i	0.984286	−	0.176582i	$-0.0565041\pi$
$54$		0			0
$55$			742.339i			1.81995i
$56$		0			0
$57$	−214.172	+	519.134i	−0.497681	+	1.20633i
$58$		0			0
$59$	243.465			0.537228			0.268614	−	0.963248i	$-0.413434\pi$
$59$	243.465			0.537228			0.268614	+	0.963248i	$0.413434\pi$
$60$		0			0
$61$	−358.454			−0.752383			−0.376192	−	0.926542i	$-0.622767\pi$
$61$	−358.454			−0.752383			−0.376192	+	0.926542i	$0.622767\pi$
$62$		0			0
$63$	−432.149	+	434.602i	−0.864216	+	0.869122i
$64$		0			0
$65$		−	854.883i		−	1.63131i
$66$		0			0
$67$			449.508i			0.819645i	0.912165	+	0.409822i	$0.134409\pi$
$67$			449.508i			0.819645i	−0.912165	+	0.409822i	$0.865591\pi$
$68$		0			0
$69$	118.673	+	48.9593i	0.207052	+	0.0854205i
$70$		0			0
$71$	−329.053			−0.550020			−0.275010	−	0.961441i	$-0.588681\pi$
$71$	−329.053			−0.550020			−0.275010	+	0.961441i	$0.588681\pi$
$72$		0			0
$73$	−925.687			−1.48416			−0.742079	−	0.670313i	$-0.766160\pi$
$73$	−925.687			−1.48416			−0.742079	+	0.670313i	$0.766160\pi$
$74$		0			0
$75$	−89.4923	−	36.9207i	−0.137782	−	0.0568431i
$76$		0			0
$77$		−	1406.03i		−	2.08094i
$78$		0			0
$79$			55.9579i			0.0796931i	0.999206	+	0.0398465i	$0.0126869\pi$
$79$			55.9579i			0.0796931i	−0.999206	+	0.0398465i	$0.987313\pi$
$80$		0			0
$81$	4.12647	+	728.988i	0.00566045	+	0.999984i
$82$		0			0
$83$	928.194			1.22750			0.613750	−	0.789500i	$-0.289660\pi$
$83$	928.194			1.22750			0.613750	+	0.789500i	$0.289660\pi$
$84$		0			0
$85$	890.823			1.13674
$86$		0			0
$87$	166.725	−	404.127i	0.205458	−	0.498011i
$88$		0			0
$89$			853.930i			1.01704i	0.861051	+	0.508519i	$0.169807\pi$
$89$			853.930i			1.01704i	−0.861051	+	0.508519i	$0.830193\pi$
$90$		0			0
$91$			1619.20i			1.86525i
$92$		0			0
$93$	−258.366	+	626.256i	−0.288079	+	0.698277i
$94$		0			0
$95$	1295.25			1.39884
$96$		0			0
$97$	−714.982			−0.748406			−0.374203	−	0.927347i	$-0.622084\pi$
$97$	−714.982			−0.748406			−0.374203	+	0.927347i	$0.622084\pi$
$98$		0			0
$99$	−1185.91	−	1179.22i	−1.20393	−	1.19713i
$100$		0			0
$101$			1702.69i			1.67746i	0.544547	+	0.838730i	$0.316701\pi$
$101$			1702.69i			1.67746i	−0.544547	+	0.838730i	$0.683299\pi$
$102$		0			0
$103$		−	655.930i		−	0.627482i	−0.949509	−	0.313741i	$-0.898418\pi$
$103$		−	655.930i		−	0.627482i	0.949509	−	0.313741i	$-0.101582\pi$
$104$		0			0
$105$	1306.75	+	539.108i	1.21453	+	0.501062i
$106$		0			0
$107$	202.008			0.182513			0.0912563	−	0.995827i	$-0.470912\pi$
$107$	202.008			0.182513			0.0912563	+	0.995827i	$0.470912\pi$
$108$		0			0
$109$	588.724			0.517335			0.258667	−	0.965966i	$-0.416717\pi$
$109$	588.724			0.517335			0.258667	+	0.965966i	$0.416717\pi$
$110$		0			0
$111$	1909.21	+	787.657i	1.63256	+	0.673524i
$112$		0			0
$113$		−	2228.76i		−	1.85544i	−0.373280	−	0.927719i	$-0.621767\pi$
$113$		−	2228.76i		−	1.85544i	0.373280	−	0.927719i	$-0.378233\pi$
$114$		0			0
$115$		−	296.091i		−	0.240092i
$116$		0			0
$117$	1365.71	+	1358.00i	1.07914	+	1.07305i
$118$		0			0
$119$	−1687.27			−1.29976
$120$		0			0
$121$	2505.69			1.88256
$122$		0			0
$123$	−206.356	+	500.188i	−0.151272	+	0.366670i
$124$		0			0
$125$		−	1274.79i		−	0.912167i
$126$		0			0
$127$		−	1345.93i		−	0.940409i	−0.882558	−	0.470204i	$-0.844180\pi$
$127$		−	1345.93i		−	0.940409i	0.882558	−	0.470204i	$-0.155820\pi$
$128$		0			0
$129$	320.340	−	776.475i	0.218638	−	0.529960i
$130$		0			0
$131$	−679.531			−0.453213			−0.226607	−	0.973986i	$-0.572763\pi$
$131$	−679.531			−0.453213			−0.226607	+	0.973986i	$0.572763\pi$
$132$		0			0
$133$	−2453.27			−1.59944
$134$		0			0
$135$	1550.66	−	650.031i	0.988590	−	0.414413i
$136$		0			0
$137$			614.487i			0.383205i	0.981473	+	0.191603i	$0.0613685\pi$
$137$			614.487i			0.383205i	−0.981473	+	0.191603i	$0.938631\pi$
$138$		0			0
$139$		−	3163.89i		−	1.93063i	−0.261088	−	0.965315i	$-0.584081\pi$
$139$		−	3163.89i		−	1.93063i	0.261088	−	0.965315i	$-0.415919\pi$
$140$		0			0
$141$	−349.768	−	144.299i	−0.208906	−	0.0861855i
$142$		0			0
$143$	−4418.36			−2.58379
$144$		0			0
$145$	−1008.30			−0.577483
$146$		0			0
$147$	−827.482	−	341.383i	−0.464283	−	0.191543i
$148$		0			0
$149$		−	1068.79i		−	0.587640i	−0.955861	−	0.293820i	$-0.905073\pi$
$149$		−	1068.79i		−	0.587640i	0.955861	−	0.293820i	$-0.0949266\pi$
$150$		0			0
$151$		−	1343.30i		−	0.723949i	−0.932188	−	0.361974i	$-0.882103\pi$
$151$		−	1343.30i		−	0.723949i	0.932188	−	0.361974i	$-0.117897\pi$
$152$		0			0
$153$	−1415.09	+	1423.12i	−0.747732	+	0.751977i
$154$		0			0
$155$	1562.52			0.809706
$156$		0			0
$157$	−2626.27			−1.33503			−0.667513	−	0.744598i	$-0.732641\pi$
$157$	−2626.27			−1.33503			−0.667513	+	0.744598i	$0.732641\pi$
$158$		0			0
$159$	270.038	−	654.548i	0.134688	−	0.326472i
$160$		0			0
$161$			560.813i			0.274523i
$162$		0			0
$163$			967.832i			0.465070i	0.972588	+	0.232535i	$0.0747021\pi$
$163$			967.832i			0.465070i	−0.972588	+	0.232535i	$0.925298\pi$
$164$		0			0
$165$	−1471.08	+	3565.77i	−0.694083	+	1.68239i
$166$		0			0
$167$	−66.2186			−0.0306835			−0.0153418	−	0.999882i	$-0.504884\pi$
$167$	−66.2186			−0.0306835			−0.0153418	+	0.999882i	$0.504884\pi$
$168$		0			0
$169$	2891.22			1.31598
$170$		0			0
$171$	−2057.52	+	2069.20i	−0.920132	+	0.925355i
$172$		0			0
$173$		−	3061.46i		−	1.34543i	−0.739904	−	0.672713i	$-0.765129\pi$
$173$		−	3061.46i		−	1.34543i	0.739904	−	0.672713i	$-0.234871\pi$
$174$		0			0
$175$		−	422.913i		−	0.182681i
$176$		0			0
$177$	1169.47	+	482.471i	0.496625	+	0.204886i
$178$		0			0
$179$	−949.274			−0.396380			−0.198190	−	0.980164i	$-0.563506\pi$
$179$	−949.274			−0.396380			−0.198190	+	0.980164i	$0.563506\pi$
$180$		0			0
$181$	−874.923			−0.359296			−0.179648	−	0.983731i	$-0.557496\pi$
$181$	−874.923			−0.359296			−0.179648	+	0.983731i	$0.557496\pi$
$182$		0			0
$183$	−1721.81	−	710.344i	−0.695518	−	0.286941i
$184$		0			0
$185$		−	4763.50i		−	1.89308i
$186$		0			0
$187$		−	4604.11i		−	1.80046i
$188$		0			0
$189$	−2937.04	+	1231.20i	−1.13036	+	0.473843i
$190$		0			0
$191$	−683.182			−0.258813			−0.129407	−	0.991592i	$-0.541307\pi$
$191$	−683.182			−0.258813			−0.129407	+	0.991592i	$0.541307\pi$
$192$		0			0
$193$	3029.86			1.13002			0.565011	−	0.825083i	$-0.308872\pi$
$193$	3029.86			1.13002			0.565011	+	0.825083i	$0.308872\pi$
$194$		0			0
$195$	1694.11	−	4106.37i	0.622142	−	1.50802i
$196$		0			0
$197$		−	2661.33i		−	0.962496i	−0.876584	−	0.481248i	$-0.840184\pi$
$197$		−	2661.33i		−	0.962496i	0.876584	−	0.481248i	$-0.159816\pi$
$198$		0			0
$199$			2975.37i			1.05989i	0.848031	+	0.529946i	$0.177788\pi$
$199$			2975.37i			1.05989i	−0.848031	+	0.529946i	$0.822212\pi$
$200$		0			0
$201$	−890.784	+	2159.18i	−0.312592	+	0.757696i
$202$		0			0
$203$	1909.78			0.660298
$204$		0			0
$205$	1247.98			0.425183
$206$		0			0
$207$	473.015	+	470.345i	0.158825	+	0.157929i
$208$		0			0
$209$		−	6694.32i		−	2.21558i
$210$		0			0
$211$		−	2601.38i		−	0.848751i	−0.905486	−	0.424376i	$-0.860494\pi$
$211$		−	2601.38i		−	0.848751i	0.905486	−	0.424376i	$-0.139506\pi$
$212$		0			0
$213$	−1580.58	−	652.080i	−0.508450	−	0.209764i
$214$		0			0
$215$	−1937.31			−0.614529
$216$		0			0
$217$	−2959.50			−0.925824
$218$		0			0
$219$	−4446.47	−	1834.42i	−1.37199	−	0.566022i
$220$		0			0
$221$			5302.13i			1.61384i
$222$		0			0
$223$		−	4031.63i		−	1.21066i	−0.795973	−	0.605332i	$-0.793040\pi$
$223$		−	4031.63i		−	1.21066i	0.795973	−	0.605332i	$-0.206960\pi$
$224$		0			0
$225$	−356.705	−	354.691i	−0.105690	−	0.105094i
$226$		0			0
$227$	4990.76			1.45924			0.729622	−	0.683851i	$-0.239696\pi$
$227$	4990.76			1.45924			0.729622	+	0.683851i	$0.239696\pi$
$228$		0			0
$229$	5091.07			1.46912			0.734558	−	0.678546i	$-0.237390\pi$
$229$	5091.07			1.46912			0.734558	+	0.678546i	$0.237390\pi$
$230$		0			0
$231$	2786.31	−	6753.78i	0.793619	−	1.92366i
$232$		0			0
$233$		−	229.872i		−	0.0646326i	−0.999478	−	0.0323163i	$-0.989712\pi$
$233$		−	229.872i		−	0.0646326i	0.999478	−	0.0323163i	$-0.0102884\pi$
$234$		0			0
$235$			872.674i			0.242243i
$236$		0			0
$237$	−110.891	+	268.790i	−0.0303930	+	0.0736699i
$238$		0			0
$239$	−1523.39			−0.412300			−0.206150	−	0.978520i	$-0.566093\pi$
$239$	−1523.39			−0.412300			−0.206150	+	0.978520i	$0.566093\pi$
$240$		0			0
$241$	1736.76			0.464210			0.232105	−	0.972691i	$-0.425439\pi$
$241$	1736.76			0.464210			0.232105	+	0.972691i	$0.425439\pi$
$242$		0			0
$243$	−1424.80	+	3509.82i	−0.376137	+	0.926564i
$244$		0			0
$245$			2064.58i			0.538371i
$246$		0			0
$247$			7709.23i			1.98594i
$248$		0			0
$249$	4458.51	+	1839.39i	1.13473	+	0.468139i
$250$		0			0
$251$	−1246.85			−0.313547			−0.156774	−	0.987635i	$-0.550109\pi$
$251$	−1246.85			−0.313547			−0.156774	+	0.987635i	$0.550109\pi$
$252$		0			0
$253$	−1530.31			−0.380275
$254$		0			0
$255$	4279.00	+	1765.33i	1.05083	+	0.433527i
$256$		0			0
$257$			6505.35i			1.57896i	0.613777	+	0.789479i	$0.289649\pi$
$257$			6505.35i			1.57896i	−0.613777	+	0.789479i	$0.710351\pi$
$258$		0			0
$259$			9022.35i			2.16456i
$260$		0			0
$261$	1601.71	−	1610.80i	0.379859	−	0.382015i
$262$		0			0
$263$	7974.40			1.86967			0.934834	−	0.355086i	$-0.115549\pi$
$263$	7974.40			1.86967			0.934834	+	0.355086i	$0.115549\pi$
$264$		0			0
$265$	−1633.11			−0.378569
$266$		0			0
$267$	−1692.22	+	4101.79i	−0.387874	+	0.940171i
$268$		0			0
$269$		−	569.640i		−	0.129114i	−0.997914	−	0.0645568i	$-0.979437\pi$
$269$		−	569.640i		−	0.129114i	0.997914	−	0.0645568i	$-0.0205634\pi$
$270$		0			0
$271$		−	6729.09i		−	1.50835i	−0.656673	−	0.754175i	$-0.728037\pi$
$271$		−	6729.09i		−	1.50835i	0.656673	−	0.754175i	$-0.271963\pi$
$272$		0			0
$273$	−3208.74	+	7777.70i	−0.711362	+	1.72428i
$274$		0			0
$275$	1154.02			0.253054
$276$		0			0
$277$	−3054.27			−0.662502			−0.331251	−	0.943543i	$-0.607471\pi$
$277$	−3054.27			−0.662502			−0.331251	+	0.943543i	$0.607471\pi$
$278$		0			0
$279$	−2482.09	+	2496.18i	−0.532612	+	0.535635i
$280$		0			0
$281$			673.673i			0.143018i	0.997440	+	0.0715088i	$0.0227814\pi$
$281$			673.673i			0.143018i	−0.997440	+	0.0715088i	$0.977219\pi$
$282$		0			0
$283$		−	1972.26i		−	0.414270i	−0.978312	−	0.207135i	$-0.933586\pi$
$283$		−	1972.26i		−	0.414270i	0.978312	−	0.207135i	$-0.0664139\pi$
$284$		0			0
$285$	6221.62	+	2566.77i	1.29311	+	0.533482i
$286$		0			0
$287$	−2363.74			−0.486157
$288$		0			0
$289$	−612.028			−0.124573
$290$		0			0
$291$	−3434.36	−	1416.87i	−0.691842	−	0.285424i
$292$		0			0
$293$		−	3641.01i		−	0.725974i	−0.931794	−	0.362987i	$-0.881757\pi$
$293$		−	3641.01i		−	0.725974i	0.931794	−	0.362987i	$-0.118243\pi$
$294$		0			0
$295$		−	2917.84i		−	0.575874i
$296$		0			0
$297$	−3359.61	−	8014.40i	−0.656377	−	1.56580i
$298$		0			0
$299$	1762.32			0.340861
$300$		0			0
$301$	3669.38			0.702657
$302$		0			0
$303$	−3374.19	+	8178.72i	−0.639743	+	1.55068i
$304$		0			0
$305$			4295.94i			0.806507i
$306$		0			0
$307$			2847.42i			0.529351i	0.964337	+	0.264676i	$0.0852649\pi$
$307$			2847.42i			0.529351i	−0.964337	+	0.264676i	$0.914735\pi$
$308$		0			0
$309$	1299.85	−	3150.71i	0.239307	−	0.580057i
$310$		0			0
$311$	−8784.69			−1.60172			−0.800859	−	0.598853i	$-0.795623\pi$
$311$	−8784.69			−1.60172			−0.800859	+	0.598853i	$0.795623\pi$
$312$		0			0
$313$	−4790.21			−0.865043			−0.432522	−	0.901624i	$-0.642376\pi$
$313$	−4790.21			−0.865043			−0.432522	+	0.901624i	$0.642376\pi$
$314$		0			0
$315$	5208.53	+	5179.13i	0.931643	+	0.926385i
$316$		0			0
$317$		−	6390.34i		−	1.13223i	−0.824326	−	0.566116i	$-0.808446\pi$
$317$		−	6390.34i		−	1.13223i	0.824326	−	0.566116i	$-0.191554\pi$
$318$		0			0
$319$			5211.29i			0.914659i
$320$		0			0
$321$	970.331	+	400.316i	0.168718	+	0.0696059i
$322$		0			0
$323$	−8033.33			−1.38386
$324$		0			0
$325$	−1328.98			−0.226826
$326$		0			0
$327$	2827.89	+	1166.67i	0.478235	+	0.197299i
$328$		0			0
$329$		−	1652.89i		−	0.276982i
$330$		0			0
$331$			3472.47i			0.576629i	0.957536	+	0.288315i	$0.0930949\pi$
$331$			3472.47i			0.576629i	−0.957536	+	0.288315i	$0.906905\pi$
$332$		0			0
$333$	7609.87	+	7566.91i	1.25231	+	1.24524i
$334$		0			0
$335$	5387.18			0.878607
$336$		0			0
$337$	672.232			0.108661			0.0543305	−	0.998523i	$-0.482698\pi$
$337$	672.232			0.108661			0.0543305	+	0.998523i	$0.482698\pi$
$338$		0			0
$339$	4416.71	−	10705.7i	0.707619	−	1.71520i
$340$		0			0
$341$		−	8075.68i		−	1.28247i
$342$		0			0
$343$			3875.51i			0.610082i
$344$		0			0
$345$	586.759	−	1422.25i	0.0915653	−	0.221946i
$346$		0			0
$347$	606.106			0.0937679			0.0468839	−	0.998900i	$-0.485071\pi$
$347$	606.106			0.0937679			0.0468839	+	0.998900i	$0.485071\pi$
$348$		0			0
$349$	7881.87			1.20890			0.604451	−	0.796643i	$-0.293393\pi$
$349$	7881.87			1.20890			0.604451	+	0.796643i	$0.293393\pi$
$350$		0			0
$351$	3868.95	+	9229.44i	0.588345	+	1.40351i
$352$		0			0
$353$			528.013i			0.0796128i	0.999207	+	0.0398064i	$0.0126741\pi$
$353$			528.013i			0.0796128i	−0.999207	+	0.0398064i	$0.987326\pi$
$354$		0			0
$355$			3943.58i			0.589587i
$356$		0			0
$357$	−8104.67	−	3343.64i	−1.20153	−	0.495698i
$358$		0			0
$359$	9239.79			1.35838			0.679188	−	0.733964i	$-0.262332\pi$
$359$	9239.79			1.35838			0.679188	+	0.733964i	$0.262332\pi$
$360$		0			0
$361$	−4821.37			−0.702926
$362$		0			0
$363$	12035.9	+	4965.49i	1.74028	+	0.717963i
$364$		0			0
$365$			11094.0i			1.59092i
$366$		0			0
$367$			8258.02i			1.17456i	0.809382	+	0.587282i	$0.199802\pi$
$367$			8258.02i			1.17456i	−0.809382	+	0.587282i	$0.800198\pi$
$368$		0			0
$369$	−1982.43	+	1993.69i	−0.279678	+	0.281266i
$370$		0			0
$371$	3093.19			0.432859
$372$		0			0
$373$	2343.62			0.325330			0.162665	−	0.986681i	$-0.447991\pi$
$373$	2343.62			0.325330			0.162665	+	0.986681i	$0.447991\pi$
$374$		0			0
$375$	2526.24	−	6123.37i	0.347878	−	0.843226i
$376$		0			0
$377$		−	6001.36i		−	0.819856i
$378$		0			0
$379$		−	778.226i		−	0.105474i	−0.998608	−	0.0527372i	$-0.983205\pi$
$379$		−	778.226i		−	0.105474i	0.998608	−	0.0527372i	$-0.0167946\pi$
$380$		0			0
$381$	2667.21	−	6465.07i	0.358649	−	0.869333i
$382$		0			0
$383$	9755.66			1.30154			0.650772	−	0.759274i	$-0.274446\pi$
$383$	9755.66			1.30154			0.650772	+	0.759274i	$0.274446\pi$
$384$		0			0
$385$	−16850.8			−2.23063
$386$		0			0
$387$	3077.46	−	3094.93i	0.404227	−	0.406522i
$388$		0			0
$389$		−	3566.20i		−	0.464816i	−0.972618	−	0.232408i	$-0.925340\pi$
$389$		−	3566.20i		−	0.464816i	0.972618	−	0.232408i	$-0.0746605\pi$
$390$		0			0
$391$			1836.40i			0.237521i
$392$		0			0
$393$	−3264.08	−	1346.62i	−0.418960	−	0.172845i
$394$		0			0
$395$	670.634			0.0854259
$396$		0			0
$397$	−8958.31			−1.13251			−0.566253	−	0.824232i	$-0.691607\pi$
$397$	−8958.31			−1.13251			−0.566253	+	0.824232i	$0.691607\pi$
$398$		0			0
$399$	−11784.1	−	4861.61i	−1.47855	−	0.609987i
$400$		0			0
$401$			12946.7i			1.61229i	0.591716	+	0.806146i	$0.298451\pi$
$401$			12946.7i			1.61229i	−0.591716	+	0.806146i	$0.701549\pi$
$402$		0			0
$403$			9300.01i			1.14955i
$404$		0			0
$405$	8736.64	−	49.4541i	1.07192	−	0.00606764i
$406$		0			0
$407$	−24619.6			−2.99840
$408$		0			0
$409$	−7979.27			−0.964669			−0.482335	−	0.875987i	$-0.660211\pi$
$409$	−7979.27			−0.964669			−0.482335	+	0.875987i	$0.660211\pi$
$410$		0			0
$411$	−1217.72	+	2951.64i	−0.146145	+	0.354243i
$412$		0			0
$413$			5526.55i			0.658459i
$414$		0			0
$415$		−	11124.0i		−	1.31580i
$416$		0			0
$417$	6269.83	−	15197.5i	0.736295	−	1.78471i
$418$		0			0
$419$	8671.29			1.01103			0.505514	−	0.862819i	$-0.331303\pi$
$419$	8671.29			1.01103			0.505514	+	0.862819i	$0.331303\pi$
$420$		0			0
$421$	2908.89			0.336747			0.168373	−	0.985723i	$-0.446149\pi$
$421$	2908.89			0.336747			0.168373	+	0.985723i	$0.446149\pi$
$422$		0			0
$423$	−1394.13	−	1386.26i	−0.160248	−	0.159343i
$424$		0			0
$425$		−	1384.85i		−	0.158059i
$426$		0			0
$427$		−	8136.75i		−	0.922166i
$428$		0			0
$429$	−21223.3	−	8755.80i	−2.38851	−	0.985394i
$430$		0			0
$431$	4968.52			0.555279			0.277639	−	0.960685i	$-0.410448\pi$
$431$	4968.52			0.555279			0.277639	+	0.960685i	$0.410448\pi$
$432$		0			0
$433$	9208.60			1.02203			0.511013	−	0.859573i	$-0.329271\pi$
$433$	9208.60			1.02203			0.511013	+	0.859573i	$0.329271\pi$
$434$		0			0
$435$	−4843.31	−	1998.14i	−0.533837	−	0.220238i
$436$		0			0
$437$			2670.11i			0.292285i
$438$		0			0
$439$			16534.4i			1.79760i	0.438361	+	0.898799i	$0.355559\pi$
$439$			16534.4i			1.79760i	−0.438361	+	0.898799i	$0.644441\pi$
$440$		0			0
$441$	−3298.23	−	3279.62i	−0.356142	−	0.354132i
$442$		0			0
$443$	−9024.44			−0.967865			−0.483933	−	0.875105i	$-0.660792\pi$
$443$	−9024.44			−0.967865			−0.483933	+	0.875105i	$0.660792\pi$
$444$		0			0
$445$	10234.0			1.09020
$446$		0			0
$447$	2118.00	−	5133.83i	0.224111	−	0.543226i
$448$		0			0
$449$			8971.07i			0.942920i	0.881888	+	0.471460i	$0.156273\pi$
$449$			8971.07i			0.942920i	−0.881888	+	0.471460i	$0.843727\pi$
$450$		0			0
$451$		−	6450.01i		−	0.673435i
$452$		0			0
$453$	2662.00	−	6452.45i	0.276096	−	0.669233i
$454$		0			0
$455$	19405.5			1.99943
$456$		0			0
$457$	11467.8			1.17383			0.586915	−	0.809648i	$-0.300342\pi$
$457$	11467.8			1.17383			0.586915	+	0.809648i	$0.300342\pi$
$458$		0			0
$459$	−9617.45	+	4031.60i	−0.978005	+	0.409976i
$460$		0			0
$461$			14270.2i			1.44171i	0.693086	+	0.720855i	$0.256251\pi$
$461$			14270.2i			1.44171i	−0.693086	+	0.720855i	$0.743749\pi$
$462$		0			0
$463$			557.295i			0.0559389i	0.999609	+	0.0279694i	$0.00890411\pi$
$463$			557.295i			0.0559389i	−0.999609	+	0.0279694i	$0.991096\pi$
$464$		0			0
$465$	7505.44	+	3096.42i	0.748508	+	0.308802i
$466$		0			0
$467$	−9659.10			−0.957109			−0.478554	−	0.878058i	$-0.658839\pi$
$467$	−9659.10			−0.957109			−0.478554	+	0.878058i	$0.658839\pi$
$468$		0			0
$469$	−10203.6			−1.00461
$470$		0			0
$471$	−12615.1	−	5204.44i	−1.23412	−	0.509146i
$472$		0			0
$473$			10012.8i			0.973336i
$474$		0			0
$475$		−	2013.55i		−	0.194501i
$476$		0			0
$477$	2594.22	−	2608.94i	0.249017	−	0.250430i
$478$		0			0
$479$	4353.23			0.415249			0.207625	−	0.978209i	$-0.433427\pi$
$479$	4353.23			0.415249			0.207625	+	0.978209i	$0.433427\pi$
$480$		0			0
$481$	28352.1			2.68762
$482$		0			0
$483$	−1111.35	+	2693.82i	−0.104696	+	0.253775i
$484$		0			0
$485$			8568.78i			0.802244i
$486$		0			0
$487$		−	12368.5i		−	1.15087i	−0.817849	−	0.575433i	$-0.804833\pi$
$487$		−	12368.5i		−	1.15087i	0.817849	−	0.575433i	$-0.195167\pi$
$488$		0			0
$489$	−1917.94	+	4648.91i	−0.177367	+	0.429920i
$490$		0			0
$491$	−19510.2			−1.79324			−0.896621	−	0.442798i	$-0.853986\pi$
$491$	−19510.2			−1.79324			−0.896621	+	0.442798i	$0.853986\pi$
$492$		0			0
$493$	6253.66			0.571299
$494$		0			0
$495$	−14132.5	+	14212.7i	−1.28325	+	1.29053i
$496$		0			0
$497$		−	7469.36i		−	0.674138i
$498$		0			0
$499$			1269.62i			0.113899i	0.998377	+	0.0569497i	$0.0181375\pi$
$499$			1269.62i			0.113899i	−0.998377	+	0.0569497i	$0.981863\pi$
$500$		0			0
$501$	−318.076	−	131.224i	−0.0283645	−	0.0117020i
$502$		0			0
$503$	14901.0			1.32088			0.660442	−	0.750877i	$-0.270369\pi$
$503$	14901.0			1.32088			0.660442	+	0.750877i	$0.270369\pi$
$504$		0			0
$505$	20406.0			1.79813
$506$		0			0
$507$	13887.8	+	5729.48i	1.21652	+	0.501884i
$508$		0			0
$509$			1853.05i			0.161365i	0.996740	+	0.0806826i	$0.0257100\pi$
$509$			1853.05i			0.161365i	−0.996740	+	0.0806826i	$0.974290\pi$
$510$		0			0
$511$		−	21012.7i		−	1.81907i
$512$		0			0
$513$	−13983.7	+	5861.90i	−1.20350	+	0.504501i
$514$		0			0
$515$	−7861.07			−0.672621
$516$		0			0
$517$	4510.31			0.383681
$518$		0			0
$519$	6066.85	−	14705.5i	0.513112	−	1.24374i
$520$		0			0
$521$			1482.78i			0.124686i	0.998055	+	0.0623432i	$0.0198573\pi$
$521$			1482.78i			0.124686i	−0.998055	+	0.0623432i	$0.980143\pi$
$522$		0			0
$523$		−	16235.8i		−	1.35744i	−0.734396	−	0.678721i	$-0.762535\pi$
$523$		−	16235.8i		−	1.35744i	0.734396	−	0.678721i	$-0.237465\pi$
$524$		0			0
$525$	838.082	−	2031.43i	0.0696703	−	0.168874i
$526$		0			0
$527$	−9690.99			−0.801036
$528$		0			0
$529$	−11556.6			−0.949833
$530$		0			0
$531$	4661.35	+	4635.03i	0.380951	+	0.378801i
$532$		0			0
$533$			7427.88i			0.603635i
$534$		0			0
$535$		−	2420.99i		−	0.195642i
$536$		0			0
$537$	−4559.77	−	1881.16i	−0.366422	−	0.151170i
$538$		0			0
$539$	10670.5			0.852712
$540$		0			0
$541$	−13039.6			−1.03626			−0.518131	−	0.855301i	$-0.673372\pi$
$541$	−13039.6			−1.03626			−0.518131	+	0.855301i	$0.673372\pi$
$542$		0			0
$543$	−4202.63	−	1733.82i	−0.332140	−	0.137027i
$544$		0			0
$545$		−	7055.63i		−	0.554550i
$546$		0			0
$547$		−	18647.7i		−	1.45762i	−0.684715	−	0.728811i	$-0.740074\pi$
$547$		−	18647.7i		−	1.45762i	0.684715	−	0.728811i	$-0.259926\pi$
$548$		0			0
$549$	−6862.91	−	6824.17i	−0.533519	−	0.530507i
$550$		0			0
$551$	9092.75			0.703020
$552$		0			0
$553$	−1270.22			−0.0976766
$554$		0			0
$555$	9439.77	−	22881.1i	0.721975	−	1.75000i
$556$		0			0
$557$			6780.36i			0.515786i	0.966173	+	0.257893i	$0.0830283\pi$
$557$			6780.36i			0.515786i	−0.966173	+	0.257893i	$0.916972\pi$
$558$		0			0
$559$		−	11530.8i		−	0.872451i
$560$		0			0
$561$	9123.90	−	22115.5i	0.686651	−	1.66438i
$562$		0			0
$563$	−5028.45			−0.376419			−0.188209	−	0.982129i	$-0.560268\pi$
$563$	−5028.45			−0.376419			−0.188209	+	0.982129i	$0.560268\pi$
$564$		0			0
$565$	−26710.9			−1.98891
$566$		0			0
$567$	−16547.7	+	93.6689i	−1.22564	+	0.00693779i
$568$		0			0
$569$		−	3264.38i		−	0.240510i	−0.992743	−	0.120255i	$-0.961629\pi$
$569$		−	3264.38i		−	0.240510i	0.992743	−	0.120255i	$-0.0383712\pi$
$570$		0			0
$571$		−	3622.18i		−	0.265470i	−0.991152	−	0.132735i	$-0.957624\pi$
$571$		−	3622.18i		−	0.265470i	0.991152	−	0.132735i	$-0.0423759\pi$
$572$		0			0
$573$	−3281.62	−	1353.85i	−0.239252	−	0.0987050i
$574$		0			0
$575$	−460.294			−0.0333836
$576$		0			0
$577$	−9604.14			−0.692939			−0.346469	−	0.938061i	$-0.612620\pi$
$577$	−9604.14			−0.692939			−0.346469	+	0.938061i	$0.612620\pi$
$578$		0			0
$579$	14553.7	+	6004.24i	1.04462	+	0.430963i
$580$		0			0
$581$			21069.6i			1.50450i
$582$		0			0
$583$			8440.51i			0.599606i
$584$		0			0
$585$	16275.1	−	16367.5i	1.15024	−	1.15677i
$586$		0			0
$587$	−8942.98			−0.628818			−0.314409	−	0.949288i	$-0.601806\pi$
$587$	−8942.98			−0.628818			−0.314409	+	0.949288i	$0.601806\pi$
$588$		0			0
$589$	−14090.6			−0.985726
$590$		0			0
$591$	5273.92	−	12783.5i	0.367073	−	0.889751i
$592$		0			0
$593$		−	9534.33i		−	0.660250i	−0.943937	−	0.330125i	$-0.892909\pi$
$593$		−	9534.33i		−	0.660250i	0.943937	−	0.330125i	$-0.107091\pi$
$594$		0			0
$595$			20221.3i			1.39326i
$596$		0			0
$597$	−5896.25	+	14292.0i	−0.404217	+	0.979786i
$598$		0			0
$599$	−5691.25			−0.388210			−0.194105	−	0.980981i	$-0.562180\pi$
$599$	−5691.25			−0.388210			−0.194105	+	0.980981i	$0.562180\pi$
$600$		0			0
$601$	13566.3			0.920769			0.460385	−	0.887720i	$-0.347712\pi$
$601$	13566.3			0.920769			0.460385	+	0.887720i	$0.347712\pi$
$602$		0			0
$603$	−8557.64	+	8606.21i	−0.577934	+	0.581214i
$604$		0			0
$605$		−	30029.7i		−	2.01799i
$606$		0			0
$607$			813.549i			0.0544002i	0.999630	+	0.0272001i	$0.00865913\pi$
$607$			813.549i			0.0544002i	−0.999630	+	0.0272001i	$0.991341\pi$
$608$		0			0
$609$	9173.50	+	3784.59i	0.610393	+	0.251821i
$610$		0			0
$611$	−5194.11			−0.343913
$612$		0			0
$613$	25165.4			1.65811			0.829055	−	0.559167i	$-0.188879\pi$
$613$	25165.4			1.65811			0.829055	+	0.559167i	$0.188879\pi$
$614$		0			0
$615$	5994.56	+	2473.10i	0.393047	+	0.162154i
$616$		0			0
$617$			24993.6i			1.63080i	0.578898	+	0.815400i	$0.303483\pi$
$617$			24993.6i			1.63080i	−0.578898	+	0.815400i	$0.696517\pi$
$618$		0			0
$619$		−	7970.28i		−	0.517532i	−0.965940	−	0.258766i	$-0.916684\pi$
$619$		−	7970.28i		−	0.517532i	0.965940	−	0.258766i	$-0.0833159\pi$
$620$		0			0
$621$	1340.02	+	3196.64i	0.0865911	+	0.206565i
$622$		0			0
$623$	−19383.8			−1.24654
$624$		0			0
$625$	−17606.8			−1.12683
$626$		0			0
$627$	13266.0	−	32155.7i	0.844968	−	2.04813i
$628$		0			0
$629$			29544.0i			1.87281i
$630$		0			0
$631$			4608.28i			0.290734i	0.989378	+	0.145367i	$0.0464362\pi$
$631$			4608.28i			0.290734i	−0.989378	+	0.145367i	$0.953564\pi$
$632$		0			0
$633$	5155.12	−	12495.5i	0.323693	−	0.784603i
$634$		0			0
$635$	−16130.4			−1.00806
$636$		0			0
$637$	−12288.2			−0.764330
$638$		0			0
$639$	−6300.00	−	6264.44i	−0.390022	−	0.387821i
$640$		0			0
$641$			4200.53i			0.258832i	0.991590	+	0.129416i	$0.0413102\pi$
$641$			4200.53i			0.258832i	−0.991590	+	0.129416i	$0.958690\pi$
$642$		0			0
$643$		−	3546.92i		−	0.217538i	−0.994067	−	0.108769i	$-0.965309\pi$
$643$		−	3546.92i		−	0.217538i	0.994067	−	0.108769i	$-0.0346908\pi$
$644$		0			0
$645$	−9305.75	−	3839.15i	−0.568083	−	0.234366i
$646$		0			0
$647$	2851.58			0.173272			0.0866362	−	0.996240i	$-0.472388\pi$
$647$	2851.58			0.173272			0.0866362	+	0.996240i	$0.472388\pi$
$648$		0			0
$649$	−15080.5			−0.912112
$650$		0			0
$651$	−14215.7	−	5864.79i	−0.855850	−	0.353087i
$652$		0			0
$653$		−	33080.9i		−	1.98247i	−0.132098	−	0.991237i	$-0.542171\pi$
$653$		−	33080.9i		−	1.98247i	0.132098	−	0.991237i	$-0.457829\pi$
$654$		0			0
$655$			8143.92i			0.485816i
$656$		0			0
$657$	−17723.1	−	17623.0i	−1.05242	−	1.04648i
$658$		0			0
$659$	11761.6			0.695244			0.347622	−	0.937635i	$-0.386989\pi$
$659$	11761.6			0.695244			0.347622	+	0.937635i	$0.386989\pi$
$660$		0			0
$661$	−2478.98			−0.145871			−0.0729357	−	0.997337i	$-0.523237\pi$
$661$	−2478.98			−0.145871			−0.0729357	+	0.997337i	$0.523237\pi$
$662$		0			0
$663$	−10507.1	+	25468.4i	−0.615481	+	1.49187i
$664$		0			0
$665$			29401.5i			1.71450i
$666$		0			0
$667$		−	2078.58i		−	0.120664i
$668$		0			0
$669$	7989.42	−	19365.6i	0.461718	−	1.11916i
$670$		0			0
$671$	22203.0			1.27740
$672$		0			0
$673$	5752.56			0.329487			0.164744	−	0.986336i	$-0.447320\pi$
$673$	5752.56			0.329487			0.164744	+	0.986336i	$0.447320\pi$
$674$		0			0
$675$	−1010.52	−	2410.61i	−0.0576221	−	0.137458i
$676$		0			0
$677$		−	25177.1i		−	1.42930i	−0.699483	−	0.714649i	$-0.746586\pi$
$677$		−	25177.1i		−	1.42930i	0.699483	−	0.714649i	$-0.253414\pi$
$678$		0			0
$679$		−	16229.8i		−	0.917292i
$680$		0			0
$681$	23972.8	+	9890.12i	1.34895	+	0.556520i
$682$		0			0
$683$	33891.4			1.89871			0.949356	−	0.314203i	$-0.101737\pi$
$683$	33891.4			1.89871			0.949356	+	0.314203i	$0.101737\pi$
$684$		0			0
$685$	7364.38			0.410772
$686$		0			0
$687$	24454.6	+	10088.9i	1.35808	+	0.560285i
$688$		0			0
$689$		−	9720.15i		−	0.537457i
$690$		0			0
$691$		−	1194.54i		−	0.0657634i	−0.999459	−	0.0328817i	$-0.989532\pi$
$691$		−	1194.54i		−	0.0657634i	0.999459	−	0.0328817i	$-0.0104685\pi$
$692$		0			0
$693$	26767.7	−	26919.7i	1.46728	−	1.47560i
$694$		0			0
$695$	−37918.0			−2.06951
$696$		0			0
$697$	−7740.15			−0.420630
$698$		0			0
$699$	455.534	−	1104.17i	0.0246493	−	0.0597477i
$700$		0			0
$701$			23289.7i			1.25484i	0.778682	+	0.627419i	$0.215889\pi$
$701$			23289.7i			1.25484i	−0.778682	+	0.627419i	$0.784111\pi$
$702$		0			0
$703$			42956.7i			2.30461i
$704$		0			0
$705$	−1729.37	+	4191.83i	−0.0923854	+	0.223934i
$706$		0			0
$707$	−38650.2			−2.05600
$708$		0			0
$709$	11916.8			0.631236			0.315618	−	0.948886i	$-0.397788\pi$
$709$	11916.8			0.631236			0.315618	+	0.948886i	$0.397788\pi$
$710$		0			0
$711$	−1065.31	+	1071.36i	−0.0561918	+	0.0565108i
$712$		0			0
$713$			3221.08i			0.169187i
$714$		0			0
$715$			52952.3i			2.76966i
$716$		0			0
$717$	−7317.48	−	3018.87i	−0.381138	−	0.157241i
$718$		0			0
$719$	−5520.22			−0.286327			−0.143164	−	0.989699i	$-0.545728\pi$
$719$	−5520.22			−0.286327			−0.143164	+	0.989699i	$0.545728\pi$
$720$		0			0
$721$	14889.3			0.769080
$722$		0			0
$723$	8342.41	+	3441.72i	0.429125	+	0.177038i
$724$		0			0
$725$			1567.48i			0.0802961i
$726$		0			0
$727$		−	28589.0i		−	1.45847i	−0.684264	−	0.729234i	$-0.739876\pi$
$727$		−	28589.0i		−	1.45847i	0.684264	−	0.729234i	$-0.260124\pi$
$728$		0			0
$729$	−13799.3	+	14035.7i	−0.701078	+	0.713085i
$730$		0			0
$731$	12015.5			0.607949
$732$		0			0
$733$	−23340.5			−1.17613			−0.588064	−	0.808814i	$-0.700110\pi$
$733$	−23340.5			−1.17613			−0.588064	+	0.808814i	$0.700110\pi$
$734$		0			0
$735$	−4091.34	+	9917.05i	−0.205322	+	0.497681i
$736$		0			0
$737$		−	27843.0i		−	1.39160i
$738$		0			0
$739$			8818.16i			0.438946i	0.975619	+	0.219473i	$0.0704338\pi$
$739$			8818.16i			0.438946i	−0.975619	+	0.219473i	$0.929566\pi$
$740$		0			0
$741$	−15277.3	+	37030.7i	−0.757388	+	1.83584i
$742$		0			0
$743$	25836.1			1.27569			0.637843	−	0.770166i	$-0.279827\pi$
$743$	25836.1			1.27569			0.637843	+	0.770166i	$0.279827\pi$
$744$		0			0
$745$	−12809.0			−0.629912
$746$		0			0
$747$	17771.1	+	17670.8i	0.870427	+	0.865514i
$748$		0			0
$749$			4585.49i			0.223698i
$750$		0			0
$751$		−	19166.2i		−	0.931272i	−0.884976	−	0.465636i	$-0.845826\pi$
$751$		−	19166.2i		−	0.931272i	0.884976	−	0.465636i	$-0.154174\pi$
$752$		0			0
$753$	−5989.14	−	2470.86i	−0.289849	−	0.119579i
$754$		0			0
$755$	−16098.9			−0.776027
$756$		0			0
$757$	18644.0			0.895150			0.447575	−	0.894247i	$-0.352288\pi$
$757$	18644.0			0.895150			0.447575	+	0.894247i	$0.352288\pi$
$758$		0			0
$759$	−7350.73	−	3032.59i	−0.351534	−	0.145028i
$760$		0			0
$761$		−	5444.32i		−	0.259338i	−0.991557	−	0.129669i	$-0.958608\pi$
$761$		−	5444.32i		−	0.259338i	0.991557	−	0.129669i	$-0.0413915\pi$
$762$		0			0
$763$			13363.8i			0.634077i
$764$		0			0
$765$	17055.5	+	16959.3i	0.806071	+	0.801522i
$766$		0			0
$767$	17366.8			0.817573
$768$		0			0
$769$	−5426.95			−0.254487			−0.127244	−	0.991871i	$-0.540613\pi$
$769$	−5426.95			−0.254487			−0.127244	+	0.991871i	$0.540613\pi$
$770$		0			0
$771$	−12891.6	+	31248.0i	−0.602176	+	1.45962i
$772$		0			0
$773$			27580.1i			1.28330i	0.766999	+	0.641648i	$0.221749\pi$
$773$			27580.1i			1.28330i	−0.766999	+	0.641648i	$0.778251\pi$
$774$		0			0
$775$		−	2429.04i		−	0.112586i
$776$		0			0
$777$	−17879.5	+	43338.2i	−0.825511	+	2.00096i
$778$		0			0
$779$	−11254.1			−0.517612
$780$		0			0
$781$	20381.9			0.933831
$782$		0			0
$783$	10885.8	−	4563.28i	0.496840	−	0.208274i
$784$		0			0
$785$			31474.8i			1.43106i
$786$		0			0
$787$		−	25227.6i		−	1.14265i	−0.820724	−	0.571325i	$-0.806429\pi$
$787$		−	25227.6i		−	1.14265i	0.820724	−	0.571325i	$-0.193571\pi$
$788$		0			0
$789$	38304.4	+	15802.8i	1.72836	+	0.713045i
$790$		0			0
$791$	50591.9			2.27413
$792$		0			0
$793$	−25569.2			−1.14500
$794$		0			0
$795$	−7844.50	−	3236.30i	−0.349957	−	0.144377i
$796$		0			0
$797$			11553.8i			0.513497i	0.966478	+	0.256748i	$0.0826512\pi$
$797$			11553.8i			0.513497i	−0.966478	+	0.256748i	$0.917349\pi$
$798$		0			0
$799$		−	5412.47i		−	0.239649i
$800$		0			0
$801$	−16256.9	+	16349.2i	−0.717116	+	0.721187i
$802$		0			0
$803$	57338.0			2.51982
$804$		0			0
$805$	6721.12			0.294271
$806$		0			0
$807$	1128.85	−	2736.22i	0.0492408	−	0.119355i
$808$		0			0
$809$		−	23754.7i		−	1.03235i	−0.856483	−	0.516175i	$-0.827355\pi$
$809$		−	23754.7i		−	1.03235i	0.856483	−	0.516175i	$-0.172645\pi$
$810$		0			0
$811$		−	8303.89i		−	0.359543i	−0.983708	−	0.179771i	$-0.942464\pi$
$811$		−	8303.89i		−	0.359543i	0.983708	−	0.179771i	$-0.0575358\pi$
$812$		0			0
$813$	13334.9	−	32322.7i	0.575248	−	1.39435i
$814$		0			0
$815$	11599.1			0.498526
$816$		0			0
$817$	17470.5			0.748120
$818$		0			0
$819$	−30825.9	+	31000.9i	−1.31519	+	1.32266i
$820$		0			0
$821$		−	38021.6i		−	1.61628i	−0.588992	−	0.808139i	$-0.700475\pi$
$821$		−	38021.6i		−	1.61628i	0.588992	−	0.808139i	$-0.299525\pi$
$822$		0			0
$823$			16842.2i			0.713345i	0.934230	+	0.356672i	$0.116089\pi$
$823$			16842.2i			0.713345i	−0.934230	+	0.356672i	$0.883911\pi$
$824$		0			0
$825$	5543.25	+	2286.90i	0.233929	+	0.0965088i
$826$		0			0
$827$	17470.7			0.734600			0.367300	−	0.930102i	$-0.380282\pi$
$827$	17470.7			0.734600			0.367300	+	0.930102i	$0.380282\pi$
$828$		0			0
$829$	19906.7			0.834001			0.417001	−	0.908906i	$-0.363081\pi$
$829$	19906.7			0.834001			0.417001	+	0.908906i	$0.363081\pi$
$830$		0			0
$831$	−14671.0	−	6052.60i	−0.612430	−	0.252662i
$832$		0			0
$833$		−	12804.8i		−	0.532607i
$834$		0			0
$835$			793.604i			0.0328908i
$836$		0			0
$837$	−16869.2	+	7071.49i	−0.696635	+	0.292027i
$838$		0			0
$839$	−18413.2			−0.757682			−0.378841	−	0.925462i	$-0.623677\pi$
$839$	−18413.2			−0.757682			−0.378841	+	0.925462i	$0.623677\pi$
$840$		0			0
$841$	17310.6			0.709772
$842$		0			0
$843$	−1335.01	+	3235.94i	−0.0545435	+	0.132208i
$844$		0			0
$845$		−	34650.1i		−	1.41065i
$846$		0			0
$847$			56878.0i			2.30738i
$848$		0			0
$849$	3908.39	−	9473.59i	0.157992	−	0.382960i
$850$		0			0
$851$	9819.81			0.395557
$852$		0			0
$853$	23193.1			0.930969			0.465485	−	0.885056i	$-0.345880\pi$
$853$	23193.1			0.930969			0.465485	+	0.885056i	$0.345880\pi$
$854$		0			0
$855$	24798.6	+	24658.6i	0.991922	+	0.986323i
$856$		0			0
$857$		−	16140.3i		−	0.643341i	−0.946852	−	0.321671i	$-0.895756\pi$
$857$		−	16140.3i		−	0.643341i	0.946852	−	0.321671i	$-0.104244\pi$
$858$		0			0
$859$			2325.65i			0.0923751i	0.998933	+	0.0461875i	$0.0147072\pi$
$859$			2325.65i			0.0923751i	−0.998933	+	0.0461875i	$0.985293\pi$
$860$		0			0
$861$	−11354.0	−	4684.19i	−0.449413	−	0.185408i
$862$		0			0
$863$	−8816.27			−0.347751			−0.173876	−	0.984768i	$-0.555629\pi$
$863$	−8816.27			−0.347751			−0.173876	+	0.984768i	$0.555629\pi$
$864$		0			0
$865$	−36690.4			−1.44221
$866$		0			0
$867$	−2939.83	−	1212.85i	−0.115158	−	0.0475092i
$868$		0			0
$869$		−	3466.09i		−	0.135304i
$870$		0			0
$871$			32064.2i			1.24736i
$872$		0			0
$873$	−13688.9	−	13611.7i	−0.530699	−	0.527703i
$874$		0			0
$875$	28937.2			1.11801
$876$		0			0
$877$	14921.3			0.574524			0.287262	−	0.957852i	$-0.407255\pi$
$877$	14921.3			0.574524			0.287262	+	0.957852i	$0.407255\pi$
$878$		0			0
$879$	7215.34	−	17489.3i	0.276869	−	0.671105i
$880$		0			0
$881$			16861.0i			0.644793i	0.946605	+	0.322397i	$0.104488\pi$
$881$			16861.0i			0.644793i	−0.946605	+	0.322397i	$0.895512\pi$
$882$		0			0
$883$			14458.8i			0.551049i	0.961294	+	0.275524i	$0.0888514\pi$
$883$			14458.8i			0.551049i	−0.961294	+	0.275524i	$0.911149\pi$
$884$		0			0
$885$	5782.23	−	14015.6i	0.219624	−	0.532350i
$886$		0			0
$887$	−19882.3			−0.752629			−0.376314	−	0.926492i	$-0.622809\pi$
$887$	−19882.3			−0.752629			−0.376314	+	0.926492i	$0.622809\pi$
$888$		0			0
$889$	30552.0			1.15262
$890$		0			0
$891$	−255.598	−	45154.3i	−0.00961038	−	1.69778i
$892$		0			0
$893$		−	7869.67i		−	0.294903i
$894$		0			0
$895$			11376.7i			0.424895i
$896$		0			0
$897$	8465.15	+	3492.35i	0.315098	+	0.129996i
$898$		0			0
$899$	10969.0			0.406938
$900$		0			0
$901$	10128.8			0.374516
$902$		0			0
$903$	17625.6	+	7271.57i	0.649550	+	0.267976i
$904$		0			0
$905$			10485.6i			0.385142i
$906$		0			0
$907$			29281.5i			1.07197i	0.844227	+	0.535985i	$0.180060\pi$
$907$			29281.5i			1.07197i	−0.844227	+	0.535985i	$0.819940\pi$
$908$		0			0
$909$	−32415.3	+	32599.3i	−1.18278	+	1.18950i
$910$		0			0
$911$	−37513.5			−1.36430			−0.682150	−	0.731212i	$-0.738955\pi$
$911$	−37513.5			−1.36430			−0.682150	+	0.731212i	$0.738955\pi$
$912$		0			0
$913$	−57493.3			−2.08406
$914$		0			0
$915$	−8513.20	+	20635.2i	−0.307582	+	0.745551i
$916$		0			0
$917$		−	15425.1i		−	0.555485i
$918$		0			0
$919$			22016.4i			0.790266i	0.918624	+	0.395133i	$0.129302\pi$
$919$			22016.4i			0.790266i	−0.918624	+	0.395133i	$0.870698\pi$
$920$		0			0
$921$	−5642.69	+	13677.4i	−0.201882	+	0.489343i
$922$		0			0
$923$	−23471.9			−0.837041
$924$		0			0
$925$	−7405.20			−0.263223
$926$		0			0
$927$	12487.4	−	12558.3i	0.442440	−	0.444951i
$928$		0			0
$929$		−	37944.4i		−	1.34006i	−0.742334	−	0.670030i	$-0.766281\pi$
$929$		−	37944.4i		−	1.34006i	0.742334	−	0.670030i	$-0.233719\pi$
$930$		0			0
$931$		−	18618.1i		−	0.655407i
$932$		0			0
$933$	−42196.6	−	17408.5i	−1.48066	−	0.610856i
$934$		0			0
$935$	−55178.5			−1.92998
$936$		0			0
$937$	30773.5			1.07292			0.536460	−	0.843925i	$-0.319761\pi$
$937$	30773.5			1.07292			0.536460	+	0.843925i	$0.319761\pi$
$938$		0			0
$939$	−23009.4	−	9492.69i	−0.799663	−	0.329906i
$940$		0			0
$941$			4127.42i			0.142986i	0.997441	+	0.0714931i	$0.0227764\pi$
$941$			4127.42i			0.142986i	−0.997441	+	0.0714931i	$0.977224\pi$
$942$		0			0
$943$			2572.66i			0.0888414i
$944$		0			0
$945$	14755.4	+	35199.3i	0.507929	+	1.21167i
$946$		0			0
$947$	11313.0			0.388196			0.194098	−	0.980982i	$-0.437822\pi$
$947$	11313.0			0.388196			0.194098	+	0.980982i	$0.437822\pi$
$948$		0			0
$949$	−66030.9			−2.25864
$950$		0			0
$951$	12663.7	−	30695.5i	0.431805	−	1.04666i
$952$		0			0
$953$			25633.6i			0.871304i	0.900115	+	0.435652i	$0.143482\pi$
$953$			25633.6i			0.871304i	−0.900115	+	0.435652i	$0.856518\pi$
$954$		0			0
$955$			8187.67i			0.277431i
$956$		0			0
$957$	−10327.1	+	25032.1i	−0.348829	+	0.845529i
$958$		0			0
$959$	−13948.6			−0.469679
$960$		0			0
$961$	12792.9			0.429420
$962$		0			0
$963$	3867.61	+	3845.78i	0.129421	+	0.128690i
$964$		0			0
$965$		−	36311.7i		−	1.21131i
$966$		0			0
$967$		−	50706.5i		−	1.68626i	−0.537712	−	0.843129i	$-0.680711\pi$
$967$		−	50706.5i		−	1.68626i	0.537712	−	0.843129i	$-0.319289\pi$
$968$		0			0
$969$	−38587.5	−	15919.5i	−1.27927	−	0.527770i
$970$		0			0
$971$	−48466.7			−1.60182			−0.800912	−	0.598782i	$-0.795652\pi$
$971$	−48466.7			−1.60182			−0.800912	+	0.598782i	$0.795652\pi$
$972$		0			0
$973$	71818.8			2.36630
$974$		0			0
$975$	−6383.64	−	2633.61i	−0.209682	−	0.0865058i
$976$		0			0
$977$			33563.4i			1.09907i	0.835472	+	0.549533i	$0.185194\pi$
$977$			33563.4i			1.09907i	−0.835472	+	0.549533i	$0.814806\pi$
$978$		0			0
$979$		−	52893.3i		−	1.72674i
$980$		0			0
$981$	11271.6	+	11208.0i	0.366845	+	0.364774i
$982$		0			0
$983$	−25587.6			−0.830231			−0.415115	−	0.909769i	$-0.636259\pi$
$983$	−25587.6			−0.830231			−0.415115	+	0.909769i	$0.636259\pi$
$984$		0			0
$985$	−31895.0			−1.03173
$986$		0			0
$987$	3275.52	−	7939.56i	0.105634	−	0.256048i
$988$		0			0
$989$		−	3993.71i		−	0.128405i
$990$		0			0
$991$		−	22594.0i		−	0.724239i	−0.932132	−	0.362119i	$-0.882053\pi$
$991$		−	22594.0i		−	0.724239i	0.932132	−	0.362119i	$-0.117947\pi$
$992$		0			0
$993$	−6881.35	+	16679.8i	−0.219912	+	0.533047i
$994$		0			0
$995$	35658.7			1.13614
$996$		0			0
$997$	−5714.27			−0.181517			−0.0907586	−	0.995873i	$-0.528929\pi$
$997$	−5714.27			−0.181517			−0.0907586	+	0.995873i	$0.528929\pi$
$998$		0			0
$999$	21558.2	+	51427.5i	0.682754	+	1.62872i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.4.c.d.383.12	yes	12
3.2	odd	2		384.4.c.a.383.2	yes	12
4.3	odd	2		384.4.c.a.383.1	✓	12
8.3	odd	2		384.4.c.c.383.12	yes	12
8.5	even	2		384.4.c.b.383.1	yes	12
12.11	even	2	inner	384.4.c.d.383.11	yes	12
16.3	odd	4		768.4.f.h.383.7		12
16.5	even	4		768.4.f.g.383.7		12
16.11	odd	4		768.4.f.e.383.6		12
16.13	even	4		768.4.f.f.383.6		12
24.5	odd	2		384.4.c.c.383.11	yes	12
24.11	even	2		384.4.c.b.383.2	yes	12
48.5	odd	4		768.4.f.h.383.8		12
48.11	even	4		768.4.f.f.383.5		12
48.29	odd	4		768.4.f.e.383.5		12
48.35	even	4		768.4.f.g.383.8		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.4.c.a.383.1	✓	12	4.3	odd	2
384.4.c.a.383.2	yes	12	3.2	odd	2
384.4.c.b.383.1	yes	12	8.5	even	2
384.4.c.b.383.2	yes	12	24.11	even	2
384.4.c.c.383.11	yes	12	24.5	odd	2
384.4.c.c.383.12	yes	12	8.3	odd	2
384.4.c.d.383.11	yes	12	12.11	even	2	inner
384.4.c.d.383.12	yes	12	1.1	even	1	trivial
768.4.f.e.383.5		12	48.29	odd	4
768.4.f.e.383.6		12	16.11	odd	4
768.4.f.f.383.5		12	48.11	even	4
768.4.f.f.383.6		12	16.13	even	4
768.4.f.g.383.7		12	16.5	even	4
768.4.f.g.383.8		12	48.35	even	4
768.4.f.h.383.7		12	16.3	odd	4
768.4.f.h.383.8		12	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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	384.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(4.80343 + 1.98169i) q^{3} -11.9846i q^{5} +22.6995i q^{7} +(19.1458 + 19.0378i) q^{9} +O(q^{10})\)	\(q+(4.80343 + 1.98169i) q^{3} -11.9846i q^{5} +22.6995i q^{7} +(19.1458 + 19.0378i) q^{9} -61.9410 q^{11} +71.3317 q^{13} +(23.7497 - 57.5672i) q^{15} +74.3305i q^{17} +108.076i q^{19} +(-44.9834 + 109.036i) q^{21} +24.7059 q^{23} -18.6309 q^{25} +(54.2388 + 129.388i) q^{27} -84.1331i q^{29} +130.377i q^{31} +(-297.529 - 122.748i) q^{33} +272.045 q^{35} +397.468 q^{37} +(342.637 + 141.357i) q^{39} +104.132i q^{41} -161.650i q^{43} +(228.160 - 229.455i) q^{45} -72.8162 q^{47} -172.269 q^{49} +(-147.300 + 357.041i) q^{51} -136.267i q^{53} +742.339i q^{55} +(-214.172 + 519.134i) q^{57} +243.465 q^{59} -358.454 q^{61} +(-432.149 + 434.602i) q^{63} -854.883i q^{65} +449.508i q^{67} +(118.673 + 48.9593i) q^{69} -329.053 q^{71} -925.687 q^{73} +(-89.4923 - 36.9207i) q^{75} -1406.03i q^{77} +55.9579i q^{79} +(4.12647 + 728.988i) q^{81} +928.194 q^{83} +890.823 q^{85} +(166.725 - 404.127i) q^{87} +853.930i q^{89} +1619.20i q^{91} +(-258.366 + 626.256i) q^{93} +1295.25 q^{95} -714.982 q^{97} +(-1185.91 - 1179.22i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 10 q^{3}+O(q^{10}) \) 12 * q + 10 * q^3	\( 12 q + 10 q^{3} + 36 q^{11} + 84 q^{15} + 136 q^{21} - 120 q^{23} - 300 q^{25} - 266 q^{27} - 116 q^{33} + 432 q^{35} + 528 q^{37} + 620 q^{39} + 440 q^{45} - 1248 q^{47} - 948 q^{49} - 1072 q^{51} - 172 q^{57} + 2508 q^{59} + 624 q^{61} + 2744 q^{63} - 24 q^{69} - 2040 q^{71} - 216 q^{73} - 3894 q^{75} - 1076 q^{81} + 4572 q^{83} + 480 q^{85} + 4156 q^{87} - 112 q^{93} - 5448 q^{95} - 48 q^{97} - 6044 q^{99}+O(q^{100}) \) 12 * q + 10 * q^3 + 36 * q^11 + 84 * q^15 + 136 * q^21 - 120 * q^23 - 300 * q^25 - 266 * q^27 - 116 * q^33 + 432 * q^35 + 528 * q^37 + 620 * q^39 + 440 * q^45 - 1248 * q^47 - 948 * q^49 - 1072 * q^51 - 172 * q^57 + 2508 * q^59 + 624 * q^61 + 2744 * q^63 - 24 * q^69 - 2040 * q^71 - 216 * q^73 - 3894 * q^75 - 1076 * q^81 + 4572 * q^83 + 480 * q^85 + 4156 * q^87 - 112 * q^93 - 5448 * q^95 - 48 * q^97 - 6044 * q^99

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