LMFDB - Embedded newform 384.4.c.a.383.9

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			383.9
Root			$4.03251i$ of defining polynomial
Character	$\chi$	$=$	384.383
Dual form			384.4.c.a.383.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.52186 - 4.54315i) q^{3} -8.01133i q^{5} -12.6015i q^{7} +(-14.2805 - 22.9144i) q^{9} +O(q^{10})$	\(q+(2.52186 - 4.54315i) q^{3} -8.01133i q^{5} -12.6015i q^{7} +(-14.2805 - 22.9144i) q^{9} -37.7546 q^{11} -60.0661 q^{13} +(-36.3967 - 20.2034i) q^{15} +37.0747i q^{17} +127.761i q^{19} +(-57.2506 - 31.7792i) q^{21} +56.9249 q^{23} +60.8186 q^{25} +(-140.117 + 7.09157i) q^{27} -220.677i q^{29} +2.26106i q^{31} +(-95.2117 + 171.525i) q^{33} -100.955 q^{35} +166.549 q^{37} +(-151.478 + 272.890i) q^{39} +154.816i q^{41} +53.4029i q^{43} +(-183.575 + 114.406i) q^{45} -591.855 q^{47} +184.202 q^{49} +(168.436 + 93.4972i) q^{51} -538.619i q^{53} +302.465i q^{55} +(580.437 + 322.195i) q^{57} -586.798 q^{59} -431.489 q^{61} +(-288.756 + 179.956i) q^{63} +481.209i q^{65} -175.719i q^{67} +(143.557 - 258.619i) q^{69} -29.5180 q^{71} +937.448 q^{73} +(153.376 - 276.308i) q^{75} +475.765i q^{77} +409.780i q^{79} +(-321.136 + 654.456i) q^{81} -1299.81 q^{83} +297.018 q^{85} +(-1002.57 - 556.517i) q^{87} -921.475i q^{89} +756.924i q^{91} +(10.2723 + 5.70207i) q^{93} +1023.53 q^{95} -1644.22 q^{97} +(539.153 + 865.123i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.52186	−	4.54315i	0.485332	−	0.874330i
$3$	2.52186	−	4.54315i	0.485332	−	0.874330i
$4$		0			0
$5$		−	8.01133i		−	0.716555i	−0.933615	−	0.358278i	$-0.883364\pi$
$5$		−	8.01133i		−	0.716555i	0.933615	−	0.358278i	$-0.116636\pi$
$6$		0			0
$7$		−	12.6015i		−	0.680418i	−0.940350	−	0.340209i	$-0.889502\pi$
$7$		−	12.6015i		−	0.680418i	0.940350	−	0.340209i	$-0.110498\pi$
$8$		0			0
$9$	−14.2805	−	22.9144i	−0.528906	−	0.848680i
$10$		0			0
$11$	−37.7546			−1.03486			−0.517429	−	0.855726i	$-0.673111\pi$
$11$	−37.7546			−1.03486			−0.517429	+	0.855726i	$0.673111\pi$
$12$		0			0
$13$	−60.0661			−1.28149			−0.640744	−	0.767755i	$-0.721374\pi$
$13$	−60.0661			−1.28149			−0.640744	+	0.767755i	$0.721374\pi$
$14$		0			0
$15$	−36.3967	−	20.2034i	−0.626506	−	0.347767i
$16$		0			0
$17$			37.0747i			0.528938i	0.964394	+	0.264469i	$0.0851967\pi$
$17$			37.0747i			0.528938i	−0.964394	+	0.264469i	$0.914803\pi$
$18$		0			0
$19$			127.761i			1.54265i	0.636441	+	0.771325i	$0.280406\pi$
$19$			127.761i			1.54265i	−0.636441	+	0.771325i	$0.719594\pi$
$20$		0			0
$21$	−57.2506	−	31.7792i	−0.594910	−	0.330228i
$22$		0			0
$23$	56.9249			0.516072			0.258036	−	0.966135i	$-0.416925\pi$
$23$	56.9249			0.516072			0.258036	+	0.966135i	$0.416925\pi$
$24$		0			0
$25$	60.8186			0.486549
$26$		0			0
$27$	−140.117	+	7.09157i	−0.998722	+	0.0505471i
$28$		0			0
$29$		−	220.677i		−	1.41306i	−0.707683	−	0.706530i	$-0.750259\pi$
$29$		−	220.677i		−	1.41306i	0.707683	−	0.706530i	$-0.249741\pi$
$30$		0			0
$31$			2.26106i			0.0130999i	0.999979	+	0.00654997i	$0.00208493\pi$
$31$			2.26106i			0.0130999i	−0.999979	+	0.00654997i	$0.997915\pi$
$32$		0			0
$33$	−95.2117	+	171.525i	−0.502250	+	0.904808i
$34$		0			0
$35$	−100.955			−0.487557
$36$		0			0
$37$	166.549			0.740015			0.370007	−	0.929029i	$-0.379355\pi$
$37$	166.549			0.740015			0.370007	+	0.929029i	$0.379355\pi$
$38$		0			0
$39$	−151.478	+	272.890i	−0.621947	+	1.12044i
$40$		0			0
$41$			154.816i			0.589713i	0.955542	+	0.294857i	$0.0952719\pi$
$41$			154.816i			0.589713i	−0.955542	+	0.294857i	$0.904728\pi$
$42$		0			0
$43$			53.4029i			0.189392i	0.995506	+	0.0946961i	$0.0301879\pi$
$43$			53.4029i			0.189392i	−0.995506	+	0.0946961i	$0.969812\pi$
$44$		0			0
$45$	−183.575	+	114.406i	−0.608126	+	0.378990i
$46$		0			0
$47$	−591.855			−1.83683			−0.918413	−	0.395622i	$-0.870529\pi$
$47$	−591.855			−1.83683			−0.918413	+	0.395622i	$0.870529\pi$
$48$		0			0
$49$	184.202			0.537032
$50$		0			0
$51$	168.436	+	93.4972i	0.462466	+	0.256710i
$52$		0			0
$53$		−	538.619i		−	1.39594i	−0.716126	−	0.697971i	$-0.754086\pi$
$53$		−	538.619i		−	1.39594i	0.716126	−	0.697971i	$-0.245914\pi$
$54$		0			0
$55$			302.465i			0.741533i
$56$		0			0
$57$	580.437	+	322.195i	1.34879	+	0.748697i
$58$		0			0
$59$	−586.798			−1.29482			−0.647412	−	0.762141i	$-0.724148\pi$
$59$	−586.798			−1.29482			−0.647412	+	0.762141i	$0.724148\pi$
$60$		0			0
$61$	−431.489			−0.905681			−0.452840	−	0.891592i	$-0.649589\pi$
$61$	−431.489			−0.905681			−0.452840	+	0.891592i	$0.649589\pi$
$62$		0			0
$63$	−288.756	+	179.956i	−0.577457	+	0.359877i
$64$		0			0
$65$			481.209i			0.918257i
$66$		0			0
$67$		−	175.719i		−	0.320411i	−0.987084	−	0.160205i	$-0.948784\pi$
$67$		−	175.719i		−	0.320411i	0.987084	−	0.160205i	$-0.0512157\pi$
$68$		0			0
$69$	143.557	−	258.619i	0.250466	−	0.451218i
$70$		0			0
$71$	−29.5180			−0.0493400			−0.0246700	−	0.999696i	$-0.507853\pi$
$71$	−29.5180			−0.0493400			−0.0246700	+	0.999696i	$0.507853\pi$
$72$		0			0
$73$	937.448			1.50301			0.751506	−	0.659726i	$-0.229328\pi$
$73$	937.448			1.50301			0.751506	+	0.659726i	$0.229328\pi$
$74$		0			0
$75$	153.376	−	276.308i	0.236138	−	0.425404i
$76$		0			0
$77$			475.765i			0.704136i
$78$		0			0
$79$			409.780i			0.583593i	0.956480	+	0.291796i	$0.0942530\pi$
$79$			409.780i			0.583593i	−0.956480	+	0.291796i	$0.905747\pi$
$80$		0			0
$81$	−321.136	+	654.456i	−0.440516	+	0.897745i
$82$		0			0
$83$	−1299.81			−1.71895			−0.859477	−	0.511174i	$-0.829211\pi$
$83$	−1299.81			−1.71895			−0.859477	+	0.511174i	$0.829211\pi$
$84$		0			0
$85$	297.018			0.379013
$86$		0			0
$87$	−1002.57	−	556.517i	−1.23548	−	0.685803i
$88$		0			0
$89$		−	921.475i		−	1.09748i	−0.835992	−	0.548742i	$-0.815107\pi$
$89$		−	921.475i		−	1.09748i	0.835992	−	0.548742i	$-0.184893\pi$
$90$		0			0
$91$			756.924i			0.871947i
$92$		0			0
$93$	10.2723	+	5.70207i	0.0114537	+	0.00635781i
$94$		0			0
$95$	1023.53			1.10539
$96$		0			0
$97$	−1644.22			−1.72108			−0.860542	−	0.509379i	$-0.829875\pi$
$97$	−1644.22			−1.72108			−0.860542	+	0.509379i	$0.829875\pi$
$98$		0			0
$99$	539.153	+	865.123i	0.547343	+	0.878264i
$100$		0			0
$101$		−	1708.51i		−	1.68320i	−0.540105	−	0.841598i	$-0.681615\pi$
$101$		−	1708.51i		−	1.68320i	0.540105	−	0.841598i	$-0.318385\pi$
$102$		0			0
$103$		−	354.419i		−	0.339048i	−0.985526	−	0.169524i	$-0.945777\pi$
$103$		−	354.419i		−	0.339048i	0.985526	−	0.169524i	$-0.0542230\pi$
$104$		0			0
$105$	−254.594	+	458.653i	−0.236627	+	0.426286i
$106$		0			0
$107$	532.454			0.481068			0.240534	−	0.970641i	$-0.422677\pi$
$107$	532.454			0.481068			0.240534	+	0.970641i	$0.422677\pi$
$108$		0			0
$109$	−2099.43			−1.84485			−0.922427	−	0.386171i	$-0.873797\pi$
$109$	−2099.43			−1.84485			−0.922427	+	0.386171i	$0.873797\pi$
$110$		0			0
$111$	420.014	−	756.659i	0.359153	−	0.647017i
$112$		0			0
$113$		−	510.544i		−	0.425026i	−0.977158	−	0.212513i	$-0.931835\pi$
$113$		−	510.544i		−	0.425026i	0.977158	−	0.212513i	$-0.0681647\pi$
$114$		0			0
$115$		−	456.044i		−	0.369794i
$116$		0			0
$117$	857.772	+	1376.38i	0.677787	+	1.08757i
$118$		0			0
$119$	467.198			0.359899
$120$		0			0
$121$	94.4098			0.0709315
$122$		0			0
$123$	703.354	+	390.425i	0.515604	+	0.286207i
$124$		0			0
$125$		−	1488.65i		−	1.06519i
$126$		0			0
$127$			989.228i			0.691180i	0.938386	+	0.345590i	$0.112321\pi$
$127$			989.228i			0.691180i	−0.938386	+	0.345590i	$0.887679\pi$
$128$		0			0
$129$	242.617	+	134.674i	0.165591	+	0.0919180i
$130$		0			0
$131$	−519.522			−0.346495			−0.173248	−	0.984878i	$-0.555426\pi$
$131$	−519.522			−0.346495			−0.173248	+	0.984878i	$0.555426\pi$
$132$		0			0
$133$	1609.98			1.04965
$134$		0			0
$135$	56.8129	+	1122.52i	0.0362198	+	0.715639i
$136$		0			0
$137$		−	2731.83i		−	1.70362i	−0.523853	−	0.851809i	$-0.675506\pi$
$137$		−	2731.83i		−	1.70362i	0.523853	−	0.851809i	$-0.324494\pi$
$138$		0			0
$139$			182.811i			0.111553i	0.998443	+	0.0557765i	$0.0177634\pi$
$139$			182.811i			0.111553i	−0.998443	+	0.0557765i	$0.982237\pi$
$140$		0			0
$141$	−1492.57	+	2688.89i	−0.891470	+	1.60599i
$142$		0			0
$143$	2267.77			1.32616
$144$		0			0
$145$	−1767.92			−1.01254
$146$		0			0
$147$	464.531	−	836.857i	0.260638	−	0.469543i
$148$		0			0
$149$			935.337i			0.514267i	0.966376	+	0.257134i	$0.0827780\pi$
$149$			935.337i			0.514267i	−0.966376	+	0.257134i	$0.917222\pi$
$150$		0			0
$151$		−	3637.89i		−	1.96058i	−0.197572	−	0.980288i	$-0.563306\pi$
$151$		−	3637.89i		−	1.96058i	0.197572	−	0.980288i	$-0.436694\pi$
$152$		0			0
$153$	849.544	−	529.445i	0.448899	−	0.279759i
$154$		0			0
$155$	18.1141			0.00938682
$156$		0			0
$157$	2194.94			1.11577			0.557883	−	0.829919i	$-0.311614\pi$
$157$	2194.94			1.11577			0.557883	+	0.829919i	$0.311614\pi$
$158$		0			0
$159$	−2447.03	−	1358.32i	−1.22051	−	0.677495i
$160$		0			0
$161$		−	717.340i		−	0.351145i
$162$		0			0
$163$		−	2938.89i		−	1.41222i	−0.708103	−	0.706109i	$-0.750449\pi$
$163$		−	2938.89i		−	1.41222i	0.708103	−	0.706109i	$-0.249551\pi$
$164$		0			0
$165$	1374.14	+	762.773i	0.648345	+	0.359889i
$166$		0			0
$167$	3106.90			1.43964			0.719818	−	0.694163i	$-0.244225\pi$
$167$	3106.90			1.43964			0.719818	+	0.694163i	$0.244225\pi$
$168$		0			0
$169$	1410.94			0.642211
$170$		0			0
$171$	2927.56	−	1824.49i	1.30922	−	0.815918i
$172$		0			0
$173$			2882.13i			1.26662i	0.773900	+	0.633308i	$0.218303\pi$
$173$			2882.13i			1.26662i	−0.773900	+	0.633308i	$0.781697\pi$
$174$		0			0
$175$		−	766.406i		−	0.331056i
$176$		0			0
$177$	−1479.82	+	2665.91i	−0.628419	+	1.13210i
$178$		0			0
$179$	2571.39			1.07371			0.536857	−	0.843673i	$-0.319611\pi$
$179$	2571.39			1.07371			0.536857	+	0.843673i	$0.319611\pi$
$180$		0			0
$181$	563.098			0.231242			0.115621	−	0.993293i	$-0.463114\pi$
$181$	563.098			0.231242			0.115621	+	0.993293i	$0.463114\pi$
$182$		0			0
$183$	−1088.15	+	1960.32i	−0.439556	+	0.791864i
$184$		0			0
$185$		−	1334.28i		−	0.530261i
$186$		0			0
$187$		−	1399.74i		−	0.547376i
$188$		0			0
$189$	89.3645	+	1765.68i	0.0343932	+	0.679548i
$190$		0			0
$191$	2637.29			0.999098			0.499549	−	0.866286i	$-0.333499\pi$
$191$	2637.29			0.999098			0.499549	+	0.866286i	$0.333499\pi$
$192$		0			0
$193$	1498.22			0.558777			0.279388	−	0.960178i	$-0.409868\pi$
$193$	1498.22			0.558777			0.279388	+	0.960178i	$0.409868\pi$
$194$		0			0
$195$	2186.21	+	1213.54i	0.802860	+	0.445659i
$196$		0			0
$197$		−	394.173i		−	0.142557i	−0.997456	−	0.0712783i	$-0.977292\pi$
$197$		−	394.173i		−	0.142557i	0.997456	−	0.0712783i	$-0.0227079\pi$
$198$		0			0
$199$			3133.05i			1.11606i	0.829821	+	0.558030i	$0.188443\pi$
$199$			3133.05i			1.11606i	−0.829821	+	0.558030i	$0.811557\pi$
$200$		0			0
$201$	−798.320	−	443.139i	−0.280145	−	0.155506i
$202$		0			0
$203$	−2780.87			−0.961471
$204$		0			0
$205$	1240.28			0.422562
$206$		0			0
$207$	−812.915	−	1304.40i	−0.272954	−	0.437981i
$208$		0			0
$209$		−	4823.56i		−	1.59642i
$210$		0			0
$211$			1665.67i			0.543458i	0.962374	+	0.271729i	$0.0875955\pi$
$211$			1665.67i			0.543458i	−0.962374	+	0.271729i	$0.912405\pi$
$212$		0			0
$213$	−74.4401	+	134.105i	−0.0239462	+	0.0431394i
$214$		0			0
$215$	427.828			0.135710
$216$		0			0
$217$	28.4927			0.00891343
$218$		0			0
$219$	2364.11	−	4258.97i	0.729460	−	1.31413i
$220$		0			0
$221$		−	2226.94i		−	0.677828i
$222$		0			0
$223$			2675.69i			0.803488i	0.915752	+	0.401744i	$0.131596\pi$
$223$			2675.69i			0.803488i	−0.915752	+	0.401744i	$0.868404\pi$
$224$		0			0
$225$	−868.518	−	1393.62i	−0.257339	−	0.412924i
$226$		0			0
$227$	47.7540			0.0139627			0.00698137	−	0.999976i	$-0.497778\pi$
$227$	47.7540			0.0139627			0.00698137	+	0.999976i	$0.497778\pi$
$228$		0			0
$229$	1392.28			0.401767			0.200884	−	0.979615i	$-0.435619\pi$
$229$	1392.28			0.401767			0.200884	+	0.979615i	$0.435619\pi$
$230$		0			0
$231$	2161.47	+	1199.81i	0.615647	+	0.341740i
$232$		0			0
$233$			1627.49i			0.457598i	0.973474	+	0.228799i	$0.0734799\pi$
$233$			1627.49i			0.457598i	−0.973474	+	0.228799i	$0.926520\pi$
$234$		0			0
$235$			4741.54i			1.31619i
$236$		0			0
$237$	1861.69	+	1033.41i	0.510253	+	0.283236i
$238$		0			0
$239$	−2858.90			−0.773752			−0.386876	−	0.922132i	$-0.626446\pi$
$239$	−2858.90			−0.773752			−0.386876	+	0.922132i	$0.626446\pi$
$240$		0			0
$241$	−1021.31			−0.272982			−0.136491	−	0.990641i	$-0.543582\pi$
$241$	−1021.31			−0.272982			−0.136491	+	0.990641i	$0.543582\pi$
$242$		0			0
$243$	2163.43	+	3109.42i	0.571128	+	0.820861i
$244$		0			0
$245$		−	1475.70i		−	0.384813i
$246$		0			0
$247$		−	7674.10i		−	1.97689i
$248$		0			0
$249$	−3277.95	+	5905.26i	−0.834263	+	1.50293i
$250$		0			0
$251$	−3017.49			−0.758814			−0.379407	−	0.925230i	$-0.623872\pi$
$251$	−3017.49			−0.758814			−0.379407	+	0.925230i	$0.623872\pi$
$252$		0			0
$253$	−2149.18			−0.534062
$254$		0			0
$255$	749.037	−	1349.40i	0.183947	−	0.331383i
$256$		0			0
$257$		−	1097.93i		−	0.266486i	−0.991083	−	0.133243i	$-0.957461\pi$
$257$		−	1097.93i		−	0.266486i	0.991083	−	0.133243i	$-0.0425390\pi$
$258$		0			0
$259$		−	2098.77i		−	0.503519i
$260$		0			0
$261$	−5056.68	+	3151.37i	−1.19924	+	0.747376i
$262$		0			0
$263$	2785.15			0.653002			0.326501	−	0.945197i	$-0.394130\pi$
$263$	2785.15			0.653002			0.326501	+	0.945197i	$0.394130\pi$
$264$		0			0
$265$	−4315.05			−1.00027
$266$		0			0
$267$	−4186.40	−	2323.83i	−0.959564	−	0.532644i
$268$		0			0
$269$		−	4034.08i		−	0.914358i	−0.889375	−	0.457179i	$-0.848860\pi$
$269$		−	4034.08i		−	0.914358i	0.889375	−	0.457179i	$-0.151140\pi$
$270$		0			0
$271$		−	2226.42i		−	0.499060i	−0.968367	−	0.249530i	$-0.919724\pi$
$271$		−	2226.42i		−	0.499060i	0.968367	−	0.249530i	$-0.0802761\pi$
$272$		0			0
$273$	3438.82	+	1908.85i	0.762370	+	0.423184i
$274$		0			0
$275$	−2296.18			−0.503509
$276$		0			0
$277$	−6418.65			−1.39227			−0.696136	−	0.717910i	$-0.745099\pi$
$277$	−6418.65			−1.39227			−0.696136	+	0.717910i	$0.745099\pi$
$278$		0			0
$279$	51.8107	−	32.2890i	0.0111177	−	0.00692864i
$280$		0			0
$281$		−	3917.54i		−	0.831675i	−0.909439	−	0.415838i	$-0.863488\pi$
$281$		−	3917.54i		−	0.831675i	0.909439	−	0.415838i	$-0.136512\pi$
$282$		0			0
$283$			4219.70i			0.886343i	0.896437	+	0.443171i	$0.146147\pi$
$283$			4219.70i			0.886343i	−0.896437	+	0.443171i	$0.853853\pi$
$284$		0			0
$285$	2581.21	−	4650.08i	0.536483	−	0.966480i
$286$		0			0
$287$	1950.92			0.401251
$288$		0			0
$289$	3538.46			0.720225
$290$		0			0
$291$	−4146.49	+	7469.94i	−0.835297	+	1.50480i
$292$		0			0
$293$		−	766.953i		−	0.152921i	−0.997073	−	0.0764605i	$-0.975638\pi$
$293$		−	766.953i		−	0.152921i	0.997073	−	0.0764605i	$-0.0243619\pi$
$294$		0			0
$295$			4701.03i			0.927812i
$296$		0			0
$297$	5290.05	−	267.739i	1.03354	−	0.0523091i
$298$		0			0
$299$	−3419.26			−0.661341
$300$		0			0
$301$	672.957			0.128866
$302$		0			0
$303$	−7762.01	−	4308.61i	−1.47167	−	0.816908i
$304$		0			0
$305$			3456.80i			0.648970i
$306$		0			0
$307$		−	7296.62i		−	1.35648i	−0.734839	−	0.678242i	$-0.762742\pi$
$307$		−	7296.62i		−	1.35648i	0.734839	−	0.678242i	$-0.237258\pi$
$308$		0			0
$309$	−1610.18	−	893.794i	−0.296440	−	0.164551i
$310$		0			0
$311$	−8641.39			−1.57559			−0.787795	−	0.615938i	$-0.788777\pi$
$311$	−8641.39			−1.57559			−0.787795	+	0.615938i	$0.788777\pi$
$312$		0			0
$313$	−2171.54			−0.392149			−0.196075	−	0.980589i	$-0.562819\pi$
$313$	−2171.54			−0.392149			−0.196075	+	0.980589i	$0.562819\pi$
$314$		0			0
$315$	1441.68	+	2313.32i	0.257872	+	0.413780i
$316$		0			0
$317$		−	1682.16i		−	0.298043i	−0.988834	−	0.149021i	$-0.952388\pi$
$317$		−	1682.16i		−	0.298043i	0.988834	−	0.149021i	$-0.0476123\pi$
$318$		0			0
$319$			8331.58i			1.46232i
$320$		0			0
$321$	1342.77	−	2419.02i	0.233478	−	0.420613i
$322$		0			0
$323$	−4736.70			−0.815967
$324$		0			0
$325$	−3653.14			−0.623506
$326$		0			0
$327$	−5294.47	+	9538.04i	−0.895366	+	1.61301i
$328$		0			0
$329$			7458.26i			1.24981i
$330$		0			0
$331$			2389.41i			0.396779i	0.980123	+	0.198389i	$0.0635711\pi$
$331$			2389.41i			0.396779i	−0.980123	+	0.198389i	$0.936429\pi$
$332$		0			0
$333$	−2378.40	−	3816.37i	−0.391398	−	0.628036i
$334$		0			0
$335$	−1407.75			−0.229592
$336$		0			0
$337$	11043.0			1.78501			0.892505	−	0.451037i	$-0.148946\pi$
$337$	11043.0			1.78501			0.892505	+	0.451037i	$0.148946\pi$
$338$		0			0
$339$	−2319.48	−	1287.52i	−0.371613	−	0.206278i
$340$		0			0
$341$		−	85.3653i		−	0.0135566i
$342$		0			0
$343$		−	6643.54i		−	1.04582i
$344$		0			0
$345$	−2071.88	−	1150.08i	−0.323322	−	0.179473i
$346$		0			0
$347$	8790.06			1.35987			0.679935	−	0.733272i	$-0.262008\pi$
$347$	8790.06			1.35987			0.679935	+	0.733272i	$0.262008\pi$
$348$		0			0
$349$	2636.08			0.404316			0.202158	−	0.979353i	$-0.435205\pi$
$349$	2636.08			0.404316			0.202158	+	0.979353i	$0.435205\pi$
$350$		0			0
$351$	8416.27	−	425.963i	1.27985	−	0.0647756i
$352$		0			0
$353$			10828.6i			1.63272i	0.577544	+	0.816359i	$0.304011\pi$
$353$			10828.6i			1.63272i	−0.577544	+	0.816359i	$0.695989\pi$
$354$		0			0
$355$			236.478i			0.0353548i
$356$		0			0
$357$	1178.21	−	2122.55i	0.174670	−	0.314670i
$358$		0			0
$359$	−278.615			−0.0409603			−0.0204802	−	0.999790i	$-0.506519\pi$
$359$	−278.615			−0.0409603			−0.0204802	+	0.999790i	$0.506519\pi$
$360$		0			0
$361$	−9463.85			−1.37977
$362$		0			0
$363$	238.088	−	428.918i	0.0344253	−	0.0620176i
$364$		0			0
$365$		−	7510.20i		−	1.07699i
$366$		0			0
$367$			12082.9i			1.71860i	0.511476	+	0.859298i	$0.329099\pi$
$367$			12082.9i			1.71860i	−0.511476	+	0.859298i	$0.670901\pi$
$368$		0			0
$369$	3547.52	−	2210.85i	0.500478	−	0.311903i
$370$		0			0
$371$	−6787.41			−0.949824
$372$		0			0
$373$	−2101.82			−0.291764			−0.145882	−	0.989302i	$-0.546602\pi$
$373$	−2101.82			−0.291764			−0.145882	+	0.989302i	$0.546602\pi$
$374$		0			0
$375$	−6763.18	−	3754.17i	−0.931331	−	0.516973i
$376$		0			0
$377$			13255.2i			1.81082i
$378$		0			0
$379$			2146.64i			0.290938i	0.989363	+	0.145469i	$0.0464691\pi$
$379$			2146.64i			0.290938i	−0.989363	+	0.145469i	$0.953531\pi$
$380$		0			0
$381$	4494.21	+	2494.69i	0.604319	+	0.335451i
$382$		0			0
$383$	−8881.17			−1.18487			−0.592437	−	0.805617i	$-0.701834\pi$
$383$	−8881.17			−1.18487			−0.592437	+	0.805617i	$0.701834\pi$
$384$		0			0
$385$	3811.51			0.504552
$386$		0			0
$387$	1223.69	−	762.618i	0.160733	−	0.100171i
$388$		0			0
$389$			3095.15i			0.403420i	0.979445	+	0.201710i	$0.0646498\pi$
$389$			3095.15i			0.403420i	−0.979445	+	0.201710i	$0.935350\pi$
$390$		0			0
$391$			2110.48i			0.272970i
$392$		0			0
$393$	−1310.16	+	2360.27i	−0.168165	+	0.302951i
$394$		0			0
$395$	3282.88			0.418176
$396$		0			0
$397$	10755.4			1.35969			0.679847	−	0.733354i	$-0.262046\pi$
$397$	10755.4			1.35969			0.679847	+	0.733354i	$0.262046\pi$
$398$		0			0
$399$	4060.14	−	7314.39i	0.509427	−	0.917738i
$400$		0			0
$401$		−	8333.04i		−	1.03774i	−0.854854	−	0.518868i	$-0.826354\pi$
$401$		−	8333.04i		−	1.03774i	0.854854	−	0.518868i	$-0.173646\pi$
$402$		0			0
$403$		−	135.813i		−	0.0167874i
$404$		0			0
$405$	5243.06	+	2572.73i	0.643283	+	0.315654i
$406$		0			0
$407$	−6288.00			−0.765810
$408$		0			0
$409$	8921.31			1.07856			0.539279	−	0.842127i	$-0.318697\pi$
$409$	8921.31			1.07856			0.539279	+	0.842127i	$0.318697\pi$
$410$		0			0
$411$	−12411.1	−	6889.27i	−1.48952	−	0.826820i
$412$		0			0
$413$			7394.54i			0.881021i
$414$		0			0
$415$			10413.2i			1.23173i
$416$		0			0
$417$	830.540	+	461.024i	0.0975341	+	0.0541402i
$418$		0			0
$419$	6810.32			0.794048			0.397024	−	0.917808i	$-0.370043\pi$
$419$	6810.32			0.794048			0.397024	+	0.917808i	$0.370043\pi$
$420$		0			0
$421$	−8336.91			−0.965121			−0.482561	−	0.875863i	$-0.660293\pi$
$421$	−8336.91			−0.965121			−0.482561	+	0.875863i	$0.660293\pi$
$422$		0			0
$423$	8451.96	+	13562.0i	0.971509	+	1.55888i
$424$		0			0
$425$			2254.83i			0.257354i
$426$		0			0
$427$			5437.42i			0.616241i
$428$		0			0
$429$	5719.00	−	10302.8i	0.643627	−	1.15950i
$430$		0			0
$431$	−755.175			−0.0843979			−0.0421990	−	0.999109i	$-0.513436\pi$
$431$	−755.175			−0.0843979			−0.0421990	+	0.999109i	$0.513436\pi$
$432$		0			0
$433$	−2736.36			−0.303697			−0.151849	−	0.988404i	$-0.548523\pi$
$433$	−2736.36			−0.303697			−0.151849	+	0.988404i	$0.548523\pi$
$434$		0			0
$435$	−4458.44	+	8031.92i	−0.491416	+	0.885290i
$436$		0			0
$437$			7272.78i			0.796120i
$438$		0			0
$439$		−	15732.1i		−	1.71037i	−0.518322	−	0.855186i	$-0.673443\pi$
$439$		−	15732.1i		−	1.71037i	0.518322	−	0.855186i	$-0.326557\pi$
$440$		0			0
$441$	−2630.49	−	4220.87i	−0.284039	−	0.455768i
$442$		0			0
$443$	−1568.89			−0.168263			−0.0841313	−	0.996455i	$-0.526812\pi$
$443$	−1568.89			−0.168263			−0.0841313	+	0.996455i	$0.526812\pi$
$444$		0			0
$445$	−7382.24			−0.786408
$446$		0			0
$447$	4249.38	+	2358.79i	0.449639	+	0.249590i
$448$		0			0
$449$		−	8882.13i		−	0.933571i	−0.884371	−	0.466786i	$-0.845412\pi$
$449$		−	8882.13i		−	0.933571i	0.884371	−	0.466786i	$-0.154588\pi$
$450$		0			0
$451$		−	5845.03i		−	0.610270i
$452$		0			0
$453$	−16527.5	−	9174.23i	−1.71419	−	0.951530i
$454$		0			0
$455$	6063.97			0.624798
$456$		0			0
$457$	6441.62			0.659357			0.329679	−	0.944093i	$-0.393060\pi$
$457$	6441.62			0.659357			0.329679	+	0.944093i	$0.393060\pi$
$458$		0			0
$459$	−262.918	−	5194.79i	−0.0267363	−	0.528262i
$460$		0			0
$461$		−	4112.54i		−	0.415488i	−0.978183	−	0.207744i	$-0.933388\pi$
$461$		−	4112.54i		−	0.415488i	0.978183	−	0.207744i	$-0.0666122\pi$
$462$		0			0
$463$		−	16061.5i		−	1.61219i	−0.591788	−	0.806094i	$-0.701578\pi$
$463$		−	16061.5i		−	1.61219i	0.591788	−	0.806094i	$-0.298422\pi$
$464$		0			0
$465$	45.6811	−	82.2950i	0.00455572	−	0.00820718i
$466$		0			0
$467$	12155.4			1.20447			0.602233	−	0.798321i	$-0.294278\pi$
$467$	12155.4			1.20447			0.602233	+	0.798321i	$0.294278\pi$
$468$		0			0
$469$	−2214.33			−0.218013
$470$		0			0
$471$	5535.33	−	9971.95i	0.541517	−	0.975548i
$472$		0			0
$473$		−	2016.20i		−	0.195994i
$474$		0			0
$475$			7770.24i			0.750575i
$476$		0			0
$477$	−12342.1	+	7691.73i	−1.18471	+	0.738323i
$478$		0			0
$479$	−3121.62			−0.297767			−0.148883	−	0.988855i	$-0.547568\pi$
$479$	−3121.62			−0.297767			−0.148883	+	0.988855i	$0.547568\pi$
$480$		0			0
$481$	−10004.0			−0.948320
$482$		0			0
$483$	−3258.99	−	1809.03i	−0.307017	−	0.170422i
$484$		0			0
$485$			13172.4i			1.23325i
$486$		0			0
$487$			1582.51i			0.147249i	0.997286	+	0.0736244i	$0.0234566\pi$
$487$			1582.51i			0.147249i	−0.997286	+	0.0736244i	$0.976543\pi$
$488$		0			0
$489$	−13351.8	−	7411.46i	−1.23474	−	0.685394i
$490$		0			0
$491$	12211.6			1.12241			0.561205	−	0.827677i	$-0.310338\pi$
$491$	12211.6			1.12241			0.561205	+	0.827677i	$0.310338\pi$
$492$		0			0
$493$	8181.55			0.747421
$494$		0			0
$495$	6930.78	−	4319.34i	0.629324	−	0.392201i
$496$		0			0
$497$			371.971i			0.0335718i
$498$		0			0
$499$		−	3871.87i		−	0.347352i	−0.984803	−	0.173676i	$-0.944435\pi$
$499$		−	3871.87i		−	0.347352i	0.984803	−	0.173676i	$-0.0555646\pi$
$500$		0			0
$501$	7835.16	−	14115.1i	0.698701	−	1.25872i
$502$		0			0
$503$	−10985.0			−0.973753			−0.486876	−	0.873471i	$-0.661864\pi$
$503$	−10985.0			−0.973753			−0.486876	+	0.873471i	$0.661864\pi$
$504$		0			0
$505$	−13687.4			−1.20610
$506$		0			0
$507$	3558.19	−	6410.11i	0.311686	−	0.561505i
$508$		0			0
$509$			21191.9i			1.84541i	0.385509	+	0.922704i	$0.374026\pi$
$509$			21191.9i			1.84541i	−0.385509	+	0.922704i	$0.625974\pi$
$510$		0			0
$511$		−	11813.3i		−	1.02268i
$512$		0			0
$513$	−906.025	−	17901.4i	−0.0779766	−	1.54068i
$514$		0			0
$515$	−2839.37			−0.242947
$516$		0			0
$517$	22345.2			1.90086
$518$		0			0
$519$	13094.0	+	7268.33i	1.10744	+	0.614729i
$520$		0			0
$521$		−	8953.89i		−	0.752931i	−0.926431	−	0.376465i	$-0.877139\pi$
$521$		−	8953.89i		−	0.752931i	0.926431	−	0.376465i	$-0.122861\pi$
$522$		0			0
$523$		−	8410.95i		−	0.703223i	−0.936146	−	0.351611i	$-0.885634\pi$
$523$		−	8410.95i		−	0.703223i	0.936146	−	0.351611i	$-0.114366\pi$
$524$		0			0
$525$	−3481.90	−	1932.77i	−0.289453	−	0.160672i
$526$		0			0
$527$	−83.8281			−0.00692905
$528$		0			0
$529$	−8926.55			−0.733669
$530$		0			0
$531$	8379.75	+	13446.1i	0.684840	+	1.09889i
$532$		0			0
$533$		−	9299.21i		−	0.755710i
$534$		0			0
$535$		−	4265.67i		−	0.344712i
$536$		0			0
$537$	6484.69	−	11682.2i	0.521108	−	0.938781i
$538$		0			0
$539$	−6954.47			−0.555752
$540$		0			0
$541$	−2640.19			−0.209816			−0.104908	−	0.994482i	$-0.533455\pi$
$541$	−2640.19			−0.209816			−0.104908	+	0.994482i	$0.533455\pi$
$542$		0			0
$543$	1420.05	−	2558.24i	0.112229	−	0.202182i
$544$		0			0
$545$			16819.2i			1.32194i
$546$		0			0
$547$			133.631i			0.0104455i	0.999986	+	0.00522273i	$0.00166245\pi$
$547$			133.631i			0.0104455i	−0.999986	+	0.00522273i	$0.998338\pi$
$548$		0			0
$549$	6161.87	+	9887.30i	0.479020	+	0.768633i
$550$		0			0
$551$	28193.9			2.17986
$552$		0			0
$553$	5163.85			0.397087
$554$		0			0
$555$	−6061.84	−	3364.87i	−0.463623	−	0.257353i
$556$		0			0
$557$			15450.3i			1.17531i	0.809110	+	0.587657i	$0.199949\pi$
$557$			15450.3i			1.17531i	−0.809110	+	0.587657i	$0.800051\pi$
$558$		0			0
$559$		−	3207.70i		−	0.242704i
$560$		0			0
$561$	−6359.24	−	3529.95i	−0.478587	−	0.265659i
$562$		0			0
$563$	11670.9			0.873662			0.436831	−	0.899544i	$-0.356101\pi$
$563$	11670.9			0.873662			0.436831	+	0.899544i	$0.356101\pi$
$564$		0			0
$565$	−4090.13			−0.304554
$566$		0			0
$567$	8247.13	+	4046.81i	0.610841	+	0.299735i
$568$		0			0
$569$		−	3730.57i		−	0.274857i	−0.990512	−	0.137428i	$-0.956116\pi$
$569$		−	3730.57i		−	0.274857i	0.990512	−	0.137428i	$-0.0438837\pi$
$570$		0			0
$571$			1257.89i			0.0921907i	0.998937	+	0.0460954i	$0.0146778\pi$
$571$			1257.89i			0.0921907i	−0.998937	+	0.0460954i	$0.985322\pi$
$572$		0			0
$573$	6650.87	−	11981.6i	0.484894	−	0.873541i
$574$		0			0
$575$	3462.09			0.251094
$576$		0			0
$577$	3631.50			0.262013			0.131006	−	0.991382i	$-0.458179\pi$
$577$	3631.50			0.262013			0.131006	+	0.991382i	$0.458179\pi$
$578$		0			0
$579$	3778.29	−	6806.62i	0.271192	−	0.488555i
$580$		0			0
$581$			16379.6i			1.16961i
$582$		0			0
$583$			20335.3i			1.44460i
$584$		0			0
$585$	11026.6	−	6871.90i	0.779306	−	0.485672i
$586$		0			0
$587$	15455.9			1.08677			0.543383	−	0.839485i	$-0.317143\pi$
$587$	15455.9			1.08677			0.543383	+	0.839485i	$0.317143\pi$
$588$		0			0
$589$	−288.875			−0.0202086
$590$		0			0
$591$	−1790.79	−	994.048i	−0.124642	−	0.0691873i
$592$		0			0
$593$			15871.0i			1.09906i	0.835474	+	0.549530i	$0.185193\pi$
$593$			15871.0i			1.09906i	−0.835474	+	0.549530i	$0.814807\pi$
$594$		0			0
$595$		−	3742.88i		−	0.257887i
$596$		0			0
$597$	14233.9	+	7901.11i	0.975805	+	0.541660i
$598$		0			0
$599$	−4369.74			−0.298068			−0.149034	−	0.988832i	$-0.547616\pi$
$599$	−4369.74			−0.298068			−0.149034	+	0.988832i	$0.547616\pi$
$600$		0			0
$601$	507.273			0.0344295			0.0172147	−	0.999852i	$-0.494520\pi$
$601$	507.273			0.0344295			0.0172147	+	0.999852i	$0.494520\pi$
$602$		0			0
$603$	−4026.50	+	2509.35i	−0.271926	+	0.169467i
$604$		0			0
$605$		−	756.348i		−	0.0508263i
$606$		0			0
$607$			14525.6i			0.971293i	0.874155	+	0.485647i	$0.161416\pi$
$607$			14525.6i			0.971293i	−0.874155	+	0.485647i	$0.838584\pi$
$608$		0			0
$609$	−7012.95	+	12633.9i	−0.466633	+	0.840643i
$610$		0			0
$611$	35550.4			2.35387
$612$		0			0
$613$	14948.5			0.984932			0.492466	−	0.870332i	$-0.336096\pi$
$613$	14948.5			0.984932			0.492466	+	0.870332i	$0.336096\pi$
$614$		0			0
$615$	3127.82	−	5634.80i	0.205083	−	0.369459i
$616$		0			0
$617$			3250.68i			0.212103i	0.994361	+	0.106052i	$0.0338209\pi$
$617$			3250.68i			0.212103i	−0.994361	+	0.106052i	$0.966179\pi$
$618$		0			0
$619$		−	4111.34i		−	0.266961i	−0.991051	−	0.133480i	$-0.957385\pi$
$619$		−	4111.34i		−	0.266961i	0.991051	−	0.133480i	$-0.0426153\pi$
$620$		0			0
$621$	−7976.14	+	403.687i	−0.515413	+	0.0260860i
$622$		0			0
$623$	−11612.0			−0.746748
$624$		0			0
$625$	−4323.78			−0.276722
$626$		0			0
$627$	−21914.2	−	12164.3i	−1.39580	−	0.774796i
$628$		0			0
$629$			6174.77i			0.391422i
$630$		0			0
$631$		−	25436.9i		−	1.60480i	−0.596790	−	0.802398i	$-0.703557\pi$
$631$		−	25436.9i		−	1.60480i	0.596790	−	0.802398i	$-0.296443\pi$
$632$		0			0
$633$	7567.41	+	4200.59i	0.475162	+	0.263757i
$634$		0			0
$635$	7925.03			0.495268
$636$		0			0
$637$	−11064.3			−0.688199
$638$		0			0
$639$	421.530	+	676.385i	0.0260962	+	0.0418738i
$640$		0			0
$641$		−	22085.5i		−	1.36088i	−0.732803	−	0.680440i	$-0.761789\pi$
$641$		−	22085.5i		−	1.36088i	0.732803	−	0.680440i	$-0.238211\pi$
$642$		0			0
$643$			930.199i			0.0570505i	0.999593	+	0.0285252i	$0.00908110\pi$
$643$			930.199i			0.0570505i	−0.999593	+	0.0285252i	$0.990919\pi$
$644$		0			0
$645$	1078.92	−	1943.69i	0.0658644	−	0.118655i
$646$		0			0
$647$	−28562.0			−1.73553			−0.867764	−	0.496977i	$-0.834443\pi$
$647$	−28562.0			−1.73553			−0.867764	+	0.496977i	$0.834443\pi$
$648$		0			0
$649$	22154.3			1.33996
$650$		0			0
$651$	71.8547	−	129.447i	0.00432597	−	0.00779328i
$652$		0			0
$653$			9597.11i			0.575136i	0.957760	+	0.287568i	$0.0928467\pi$
$653$			9597.11i			0.575136i	−0.957760	+	0.287568i	$0.907153\pi$
$654$		0			0
$655$			4162.07i			0.248283i
$656$		0			0
$657$	−13387.2	−	21481.0i	−0.794953	−	1.27558i
$658$		0			0
$659$	477.222			0.0282093			0.0141046	−	0.999901i	$-0.495510\pi$
$659$	477.222			0.0282093			0.0141046	+	0.999901i	$0.495510\pi$
$660$		0			0
$661$	−584.379			−0.0343869			−0.0171934	−	0.999852i	$-0.505473\pi$
$661$	−584.379			−0.0343869			−0.0171934	+	0.999852i	$0.505473\pi$
$662$		0			0
$663$	−10117.3	−	5616.02i	−0.592645	−	0.328971i
$664$		0			0
$665$		−	12898.1i		−	0.752130i
$666$		0			0
$667$		−	12562.0i		−	0.729241i
$668$		0			0
$669$	12156.1	+	6747.72i	0.702513	+	0.389958i
$670$		0			0
$671$	16290.7			0.937251
$672$		0			0
$673$	−8219.98			−0.470813			−0.235406	−	0.971897i	$-0.575642\pi$
$673$	−8219.98			−0.470813			−0.235406	+	0.971897i	$0.575642\pi$
$674$		0			0
$675$	−8521.70	+	431.299i	−0.485927	+	0.0245936i
$676$		0			0
$677$		−	23269.5i		−	1.32101i	−0.750824	−	0.660503i	$-0.770343\pi$
$677$		−	23269.5i		−	1.32101i	0.750824	−	0.660503i	$-0.229657\pi$
$678$		0			0
$679$			20719.7i			1.17106i
$680$		0			0
$681$	120.429	−	216.954i	0.00677656	−	0.0122080i
$682$		0			0
$683$	−12757.4			−0.714713			−0.357357	−	0.933968i	$-0.616322\pi$
$683$	−12757.4			−0.714713			−0.357357	+	0.933968i	$0.616322\pi$
$684$		0			0
$685$	−21885.6			−1.22074
$686$		0			0
$687$	3511.14	−	6325.36i	0.194990	−	0.351277i
$688$		0			0
$689$			32352.7i			1.78888i
$690$		0			0
$691$		−	13188.1i		−	0.726049i	−0.931780	−	0.363025i	$-0.881744\pi$
$691$		−	13188.1i		−	0.726049i	0.931780	−	0.363025i	$-0.118256\pi$
$692$		0			0
$693$	10901.9	−	6794.15i	0.597586	−	0.372422i
$694$		0			0
$695$	1464.56			0.0799338
$696$		0			0
$697$	−5739.77			−0.311922
$698$		0			0
$699$	7393.93	+	4104.30i	0.400092	+	0.222087i
$700$		0			0
$701$			686.418i			0.0369838i	0.999829	+	0.0184919i	$0.00588649\pi$
$701$			686.418i			0.0369838i	−0.999829	+	0.0184919i	$0.994114\pi$
$702$		0			0
$703$			21278.5i			1.14158i
$704$		0			0
$705$	21541.6	+	11957.5i	1.15078	+	0.638788i
$706$		0			0
$707$	−21529.8			−1.14528
$708$		0			0
$709$	24361.8			1.29045			0.645224	−	0.763993i	$-0.276764\pi$
$709$	24361.8			1.29045			0.645224	+	0.763993i	$0.276764\pi$
$710$		0			0
$711$	9389.85	−	5851.85i	0.495284	−	0.308666i
$712$		0			0
$713$			128.711i			0.00676051i
$714$		0			0
$715$		−	18167.9i		−	0.950266i
$716$		0			0
$717$	−7209.74	+	12988.4i	−0.375527	+	0.676515i
$718$		0			0
$719$	−10763.6			−0.558293			−0.279147	−	0.960248i	$-0.590052\pi$
$719$	−10763.6			−0.558293			−0.279147	+	0.960248i	$0.590052\pi$
$720$		0			0
$721$	−4466.22			−0.230694
$722$		0			0
$723$	−2575.61	+	4639.98i	−0.132487	+	0.238676i
$724$		0			0
$725$		−	13421.3i		−	0.687523i
$726$		0			0
$727$		−	13784.5i		−	0.703217i	−0.936147	−	0.351609i	$-0.885635\pi$
$727$		−	13784.5i		−	0.703217i	0.936147	−	0.351609i	$-0.114365\pi$
$728$		0			0
$729$	19582.4	−	1987.30i	0.994890	−	0.100965i
$730$		0			0
$731$	−1979.90			−0.100177
$732$		0			0
$733$	−26315.5			−1.32604			−0.663020	−	0.748602i	$-0.730725\pi$
$733$	−26315.5			−1.32604			−0.663020	+	0.748602i	$0.730725\pi$
$734$		0			0
$735$	−6704.34	−	3721.51i	−0.336453	−	0.186762i
$736$		0			0
$737$			6634.21i			0.331580i
$738$		0			0
$739$			37873.7i			1.88526i	0.333843	+	0.942629i	$0.391654\pi$
$739$			37873.7i			1.88526i	−0.333843	+	0.942629i	$0.608346\pi$
$740$		0			0
$741$	−34864.6	−	19353.0i	−1.72845	−	0.959447i
$742$		0			0
$743$	−23306.0			−1.15076			−0.575381	−	0.817886i	$-0.695146\pi$
$743$	−23306.0			−1.15076			−0.575381	+	0.817886i	$0.695146\pi$
$744$		0			0
$745$	7493.29			0.368501
$746$		0			0
$747$	18562.0	+	29784.4i	0.909166	+	1.45884i
$748$		0			0
$749$		−	6709.73i		−	0.327327i
$750$		0			0
$751$		−	18843.1i		−	0.915572i	−0.889062	−	0.457786i	$-0.848642\pi$
$751$		−	18843.1i		−	0.915572i	0.889062	−	0.457786i	$-0.151358\pi$
$752$		0			0
$753$	−7609.68	+	13708.9i	−0.368277	+	0.663454i
$754$		0			0
$755$	−29144.3			−1.40486
$756$		0			0
$757$	−8504.43			−0.408320			−0.204160	−	0.978937i	$-0.565446\pi$
$757$	−8504.43			−0.408320			−0.204160	+	0.978937i	$0.565446\pi$
$758$		0			0
$759$	−5419.92	+	9764.04i	−0.259197	+	0.466946i
$760$		0			0
$761$			26943.9i			1.28346i	0.766929	+	0.641732i	$0.221784\pi$
$761$			26943.9i			1.28346i	−0.766929	+	0.641732i	$0.778216\pi$
$762$		0			0
$763$			26456.0i			1.25527i
$764$		0			0
$765$	−4241.56	−	6805.98i	−0.200462	−	0.321661i
$766$		0			0
$767$	35246.7			1.65930
$768$		0			0
$769$	−20451.5			−0.959040			−0.479520	−	0.877531i	$-0.659189\pi$
$769$	−20451.5			−0.959040			−0.479520	+	0.877531i	$0.659189\pi$
$770$		0			0
$771$	−4988.05	−	2768.81i	−0.232996	−	0.129334i
$772$		0			0
$773$		−	25810.2i		−	1.20094i	−0.799647	−	0.600470i	$-0.794980\pi$
$773$		−	25810.2i		−	1.20094i	0.799647	−	0.600470i	$-0.205020\pi$
$774$		0			0
$775$			137.514i			0.00637376i
$776$		0			0
$777$	−9535.05	−	5292.81i	−0.440242	−	0.244374i
$778$		0			0
$779$	−19779.5			−0.909722
$780$		0			0
$781$	1114.44			0.0510599
$782$		0			0
$783$	1564.95	+	30920.6i	0.0714262	+	1.41125i
$784$		0			0
$785$		−	17584.4i		−	0.799508i
$786$		0			0
$787$			9016.03i			0.408370i	0.978932	+	0.204185i	$0.0654544\pi$
$787$			9016.03i			0.408370i	−0.978932	+	0.204185i	$0.934546\pi$
$788$		0			0
$789$	7023.74	−	12653.3i	0.316922	−	0.570939i
$790$		0			0
$791$	−6433.62			−0.289195
$792$		0			0
$793$	25917.9			1.16062
$794$		0			0
$795$	−10881.9	+	19603.9i	−0.485463	+	0.874566i
$796$		0			0
$797$		−	3898.40i		−	0.173260i	−0.996241	−	0.0866302i	$-0.972390\pi$
$797$		−	3898.40i		−	0.173260i	0.996241	−	0.0866302i	$-0.0276099\pi$
$798$		0			0
$799$		−	21942.9i		−	0.971568i
$800$		0			0
$801$	−21115.0	+	13159.1i	−0.931413	+	0.580466i
$802$		0			0
$803$	−35393.0			−1.55541
$804$		0			0
$805$	−5746.85			−0.251615
$806$		0			0
$807$	−18327.5	−	10173.4i	−0.799451	−	0.443767i
$808$		0			0
$809$		−	5567.95i		−	0.241976i	−0.992654	−	0.120988i	$-0.961394\pi$
$809$		−	5567.95i		−	0.241976i	0.992654	−	0.120988i	$-0.0386063\pi$
$810$		0			0
$811$			42428.3i			1.83706i	0.395346	+	0.918532i	$0.370625\pi$
$811$			42428.3i			1.83706i	−0.395346	+	0.918532i	$0.629375\pi$
$812$		0			0
$813$	−10115.0	−	5614.71i	−0.436343	−	0.242210i
$814$		0			0
$815$	−23544.4			−1.01193
$816$		0			0
$817$	−6822.80			−0.292166
$818$		0			0
$819$	17344.4	−	10809.2i	0.740004	−	0.461178i
$820$		0			0
$821$			18734.2i			0.796379i	0.917303	+	0.398190i	$0.130361\pi$
$821$			18734.2i			0.796379i	−0.917303	+	0.398190i	$0.869639\pi$
$822$		0			0
$823$			31710.5i			1.34309i	0.740965	+	0.671543i	$0.234368\pi$
$823$			31710.5i			1.34309i	−0.740965	+	0.671543i	$0.765632\pi$
$824$		0			0
$825$	−5790.64	+	10431.9i	−0.244369	+	0.440233i
$826$		0			0
$827$	5275.05			0.221803			0.110902	−	0.993831i	$-0.464626\pi$
$827$	5275.05			0.221803			0.110902	+	0.993831i	$0.464626\pi$
$828$		0			0
$829$	10279.5			0.430665			0.215332	−	0.976541i	$-0.430917\pi$
$829$	10279.5			0.430665			0.215332	+	0.976541i	$0.430917\pi$
$830$		0			0
$831$	−16186.9	+	29160.9i	−0.675713	+	1.21730i
$832$		0			0
$833$			6829.24i			0.284056i
$834$		0			0
$835$		−	24890.4i		−	1.03158i
$836$		0			0
$837$	−16.0344	−	316.812i	−0.000662164	−	0.0130832i
$838$		0			0
$839$	30259.2			1.24513			0.622564	−	0.782569i	$-0.286091\pi$
$839$	30259.2			1.24513			0.622564	+	0.782569i	$0.286091\pi$
$840$		0			0
$841$	−24309.5			−0.996739
$842$		0			0
$843$	−17798.0	−	9879.48i	−0.727159	−	0.403638i
$844$		0			0
$845$		−	11303.5i		−	0.460180i
$846$		0			0
$847$		−	1189.71i		−	0.0482631i
$848$		0			0
$849$	19170.7	+	10641.5i	0.774956	+	0.430170i
$850$		0			0
$851$	9480.81			0.381901
$852$		0			0
$853$	−18713.7			−0.751167			−0.375584	−	0.926789i	$-0.622558\pi$
$853$	−18713.7			−0.751167			−0.375584	+	0.926789i	$0.622558\pi$
$854$		0			0
$855$	−14616.6	−	23453.7i	−0.584650	−	0.938126i
$856$		0			0
$857$		−	10924.0i		−	0.435423i	−0.976013	−	0.217711i	$-0.930141\pi$
$857$		−	10924.0i		−	0.435423i	0.976013	−	0.217711i	$-0.0698591\pi$
$858$		0			0
$859$		−	33510.2i		−	1.33103i	−0.746385	−	0.665514i	$-0.768212\pi$
$859$		−	33510.2i		−	1.33103i	0.746385	−	0.665514i	$-0.231788\pi$
$860$		0			0
$861$	4919.94	−	8863.33i	0.194740	−	0.350826i
$862$		0			0
$863$	−30253.2			−1.19331			−0.596657	−	0.802497i	$-0.703505\pi$
$863$	−30253.2			−1.19331			−0.596657	+	0.802497i	$0.703505\pi$
$864$		0			0
$865$	23089.7			0.907600
$866$		0			0
$867$	8923.50	−	16075.8i	0.349548	−	0.629714i
$868$		0			0
$869$		−	15471.1i		−	0.603936i
$870$		0			0
$871$			10554.8i			0.410603i
$872$		0			0
$873$	23480.2	+	37676.2i	0.910292	+	1.46065i
$874$		0			0
$875$	−18759.3			−0.724777
$876$		0			0
$877$	11900.1			0.458194			0.229097	−	0.973404i	$-0.426423\pi$
$877$	11900.1			0.458194			0.229097	+	0.973404i	$0.426423\pi$
$878$		0			0
$879$	−3484.38	−	1934.15i	−0.133704	−	0.0742175i
$880$		0			0
$881$		−	8633.31i		−	0.330152i	−0.986281	−	0.165076i	$-0.947213\pi$
$881$		−	8633.31i		−	0.330152i	0.986281	−	0.165076i	$-0.0527869\pi$
$882$		0			0
$883$			27438.6i			1.04573i	0.852414	+	0.522867i	$0.175138\pi$
$883$			27438.6i			1.04573i	−0.852414	+	0.522867i	$0.824862\pi$
$884$		0			0
$885$	21357.5	+	11855.3i	0.811214	+	0.450297i
$886$		0			0
$887$	3124.21			0.118264			0.0591322	−	0.998250i	$-0.481167\pi$
$887$	3124.21			0.118264			0.0591322	+	0.998250i	$0.481167\pi$
$888$		0			0
$889$	12465.8			0.470291
$890$		0			0
$891$	12124.4	−	24708.7i	0.455872	−	0.929038i
$892$		0			0
$893$		−	75615.9i		−	2.83358i
$894$		0			0
$895$		−	20600.3i		−	0.769376i
$896$		0			0
$897$	−8622.88	+	15534.2i	−0.320970	+	0.578230i
$898$		0			0
$899$	498.964			0.0185110
$900$		0			0
$901$	19969.1			0.738367
$902$		0			0
$903$	1697.10	−	3057.35i	0.0625427	−	0.112671i
$904$		0			0
$905$		−	4511.17i		−	0.165698i
$906$		0			0
$907$		−	21850.0i		−	0.799909i	−0.916535	−	0.399955i	$-0.869026\pi$
$907$		−	21850.0i		−	0.799909i	0.916535	−	0.399955i	$-0.130974\pi$
$908$		0			0
$909$	−39149.3	+	24398.3i	−1.42849	+	0.890253i
$910$		0			0
$911$	16739.3			0.608781			0.304390	−	0.952547i	$-0.401547\pi$
$911$	16739.3			0.608781			0.304390	+	0.952547i	$0.401547\pi$
$912$		0			0
$913$	49074.0			1.77887
$914$		0			0
$915$	15704.8	+	8717.56i	0.567414	+	0.314966i
$916$		0			0
$917$			6546.77i			0.235762i
$918$		0			0
$919$			36772.7i			1.31994i	0.751294	+	0.659968i	$0.229430\pi$
$919$			36772.7i			1.31994i	−0.751294	+	0.659968i	$0.770570\pi$
$920$		0			0
$921$	−33149.7	−	18401.0i	−1.18601	−	0.658344i
$922$		0			0
$923$	1773.03			0.0632286
$924$		0			0
$925$	10129.3			0.360053
$926$		0			0
$927$	−8121.29	+	5061.27i	−0.287743	+	0.179325i
$928$		0			0
$929$		−	28060.4i		−	0.990991i	−0.868611	−	0.495495i	$-0.834987\pi$
$929$		−	28060.4i		−	0.990991i	0.868611	−	0.495495i	$-0.165013\pi$
$930$		0			0
$931$			23533.8i			0.828452i
$932$		0			0
$933$	−21792.4	+	39259.1i	−0.764683	+	1.37758i
$934$		0			0
$935$	−11213.8			−0.392225
$936$		0			0
$937$	5258.29			0.183331			0.0916653	−	0.995790i	$-0.470781\pi$
$937$	5258.29			0.183331			0.0916653	+	0.995790i	$0.470781\pi$
$938$		0			0
$939$	−5476.32	+	9865.64i	−0.190323	+	0.342868i
$940$		0			0
$941$			37769.5i			1.30845i	0.756300	+	0.654224i	$0.227005\pi$
$941$			37769.5i			1.30845i	−0.756300	+	0.654224i	$0.772995\pi$
$942$		0			0
$943$			8812.91i			0.304335i
$944$		0			0
$945$	14145.5	−	715.928i	0.486934	−	0.0246446i
$946$		0			0
$947$	−9839.83			−0.337647			−0.168823	−	0.985646i	$-0.553997\pi$
$947$	−9839.83			−0.337647			−0.168823	+	0.985646i	$0.553997\pi$
$948$		0			0
$949$	−56308.8			−1.92609
$950$		0			0
$951$	−7642.31	−	4242.17i	−0.260588	−	0.144650i
$952$		0			0
$953$			21903.4i			0.744514i	0.928130	+	0.372257i	$0.121416\pi$
$953$			21903.4i			0.744514i	−0.928130	+	0.372257i	$0.878584\pi$
$954$		0			0
$955$		−	21128.2i		−	0.715909i
$956$		0			0
$957$	37851.7	+	21011.1i	1.27855	+	0.709709i
$958$		0			0
$959$	−34425.1			−1.15917
$960$		0			0
$961$	29785.9			0.999828
$962$		0			0
$963$	−7603.70	−	12200.9i	−0.254440	−	0.408273i
$964$		0			0
$965$		−	12002.7i		−	0.400394i
$966$		0			0
$967$		−	11943.8i		−	0.397193i	−0.980081	−	0.198597i	$-0.936362\pi$
$967$		−	11943.8i		−	0.397193i	0.980081	−	0.198597i	$-0.0636384\pi$
$968$		0			0
$969$	−11945.3	+	21519.6i	−0.396015	+	0.713424i
$970$		0			0
$971$	−15658.6			−0.517517			−0.258758	−	0.965942i	$-0.583313\pi$
$971$	−15658.6			−0.517517			−0.258758	+	0.965942i	$0.583313\pi$
$972$		0			0
$973$	2303.70			0.0759026
$974$		0			0
$975$	−9212.69	+	16596.8i	−0.302607	+	0.545150i
$976$		0			0
$977$			14022.1i			0.459168i	0.973289	+	0.229584i	$0.0737366\pi$
$977$			14022.1i			0.459168i	−0.973289	+	0.229584i	$0.926263\pi$
$978$		0			0
$979$			34789.9i			1.13574i
$980$		0			0
$981$	29980.9	+	48107.2i	0.975755	+	1.56569i
$982$		0			0
$983$	24228.1			0.786120			0.393060	−	0.919513i	$-0.371417\pi$
$983$	24228.1			0.786120			0.393060	+	0.919513i	$0.371417\pi$
$984$		0			0
$985$	−3157.85			−0.102150
$986$		0			0
$987$	33884.0	+	18808.7i	1.09275	+	0.606572i
$988$		0			0
$989$			3039.96i			0.0977401i
$990$		0			0
$991$		−	17211.8i		−	0.551715i	−0.961198	−	0.275858i	$-0.911038\pi$
$991$		−	17211.8i		−	0.551715i	0.961198	−	0.275858i	$-0.0889618\pi$
$992$		0			0
$993$	10855.4	+	6025.75i	0.346916	+	0.192569i
$994$		0			0
$995$	25099.9			0.799719
$996$		0			0
$997$	55019.5			1.74773			0.873864	−	0.486171i	$-0.161607\pi$
$997$	55019.5			1.74773			0.873864	+	0.486171i	$0.161607\pi$
$998$		0			0
$999$	−23336.4	+	1181.10i	−0.739069	+	0.0374056i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.4.c.a.383.9	✓	12
3.2	odd	2		384.4.c.d.383.3	yes	12
4.3	odd	2		384.4.c.d.383.4	yes	12
8.3	odd	2		384.4.c.b.383.9	yes	12
8.5	even	2		384.4.c.c.383.4	yes	12
12.11	even	2	inner	384.4.c.a.383.10	yes	12
16.3	odd	4		768.4.f.f.383.1		12
16.5	even	4		768.4.f.e.383.1		12
16.11	odd	4		768.4.f.g.383.12		12
16.13	even	4		768.4.f.h.383.12		12
24.5	odd	2		384.4.c.b.383.10	yes	12
24.11	even	2		384.4.c.c.383.3	yes	12
48.5	odd	4		768.4.f.f.383.2		12
48.11	even	4		768.4.f.h.383.11		12
48.29	odd	4		768.4.f.g.383.11		12
48.35	even	4		768.4.f.e.383.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.4.c.a.383.9	✓	12	1.1	even	1	trivial
384.4.c.a.383.10	yes	12	12.11	even	2	inner
384.4.c.b.383.9	yes	12	8.3	odd	2
384.4.c.b.383.10	yes	12	24.5	odd	2
384.4.c.c.383.3	yes	12	24.11	even	2
384.4.c.c.383.4	yes	12	8.5	even	2
384.4.c.d.383.3	yes	12	3.2	odd	2
384.4.c.d.383.4	yes	12	4.3	odd	2
768.4.f.e.383.1		12	16.5	even	4
768.4.f.e.383.2		12	48.35	even	4
768.4.f.f.383.1		12	16.3	odd	4
768.4.f.f.383.2		12	48.5	odd	4
768.4.f.g.383.11		12	48.29	odd	4
768.4.f.g.383.12		12	16.11	odd	4
768.4.f.h.383.11		12	48.11	even	4
768.4.f.h.383.12		12	16.13	even	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)

\(f(q)\)	\(=\)	\(q+(2.52186 - 4.54315i) q^{3} -8.01133i q^{5} -12.6015i q^{7} +(-14.2805 - 22.9144i) q^{9} +O(q^{10})\)	\(q+(2.52186 - 4.54315i) q^{3} -8.01133i q^{5} -12.6015i q^{7} +(-14.2805 - 22.9144i) q^{9} -37.7546 q^{11} -60.0661 q^{13} +(-36.3967 - 20.2034i) q^{15} +37.0747i q^{17} +127.761i q^{19} +(-57.2506 - 31.7792i) q^{21} +56.9249 q^{23} +60.8186 q^{25} +(-140.117 + 7.09157i) q^{27} -220.677i q^{29} +2.26106i q^{31} +(-95.2117 + 171.525i) q^{33} -100.955 q^{35} +166.549 q^{37} +(-151.478 + 272.890i) q^{39} +154.816i q^{41} +53.4029i q^{43} +(-183.575 + 114.406i) q^{45} -591.855 q^{47} +184.202 q^{49} +(168.436 + 93.4972i) q^{51} -538.619i q^{53} +302.465i q^{55} +(580.437 + 322.195i) q^{57} -586.798 q^{59} -431.489 q^{61} +(-288.756 + 179.956i) q^{63} +481.209i q^{65} -175.719i q^{67} +(143.557 - 258.619i) q^{69} -29.5180 q^{71} +937.448 q^{73} +(153.376 - 276.308i) q^{75} +475.765i q^{77} +409.780i q^{79} +(-321.136 + 654.456i) q^{81} -1299.81 q^{83} +297.018 q^{85} +(-1002.57 - 556.517i) q^{87} -921.475i q^{89} +756.924i q^{91} +(10.2723 + 5.70207i) q^{93} +1023.53 q^{95} -1644.22 q^{97} +(539.153 + 865.123i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 10 q^{3}+O(q^{10}) \) 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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