LMFDB - Embedded newform 384.5.h.f.65.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,5,Mod(65,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("384.65");

S:= CuspForms(chi, 5);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$39.6940658242$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 9 $ x^4 - 4*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			65.4
Root			$-1.58114 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	384.65
Dual form			384.5.h.f.65.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(3.00000 + 8.48528i) q^{3} +37.9473 q^{5} -37.9473 q^{7} +(-63.0000 + 50.9117i) q^{9} +O(q^{10})$	\(q+(3.00000 + 8.48528i) q^{3} +37.9473 q^{5} -37.9473 q^{7} +(-63.0000 + 50.9117i) q^{9} -134.000 q^{11} -321.994i q^{13} +(113.842 + 321.994i) q^{15} -237.588i q^{17} -492.146i q^{19} +(-113.842 - 321.994i) q^{21} -643.988i q^{23} +815.000 q^{25} +(-621.000 - 381.838i) q^{27} +796.894 q^{29} -1100.47 q^{31} +(-402.000 - 1137.03i) q^{33} -1440.00 q^{35} +1609.97i q^{37} +(2732.21 - 965.981i) q^{39} -1289.76i q^{41} +424.264i q^{43} +(-2390.68 + 1931.96i) q^{45} +2575.95i q^{47} -961.000 q^{49} +(2016.00 - 712.764i) q^{51} -1631.74 q^{53} -5084.94 q^{55} +(4176.00 - 1476.44i) q^{57} +1382.00 q^{59} -321.994i q^{61} +(2390.68 - 1931.96i) q^{63} -12218.8i q^{65} -186.676i q^{67} +(5464.42 - 1931.96i) q^{69} +1931.96i q^{71} +4894.00 q^{73} +(2445.00 + 6915.50i) q^{75} +5084.94 q^{77} +1479.95 q^{79} +(1377.00 - 6414.87i) q^{81} +5914.00 q^{83} -9015.83i q^{85} +(2390.68 + 6761.87i) q^{87} -9062.28i q^{89} +12218.8i q^{91} +(-3301.42 - 9337.82i) q^{93} -18675.6i q^{95} +5858.00 q^{97} +(8442.00 - 6822.17i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{3} - 252 q^{9}+O(q^{10}) $ 4 * q + 12 * q^3 - 252 * q^9	$ 4 q + 12 q^{3} - 252 q^{9} - 536 q^{11} + 3260 q^{25} - 2484 q^{27} - 1608 q^{33} - 5760 q^{35} - 3844 q^{49} + 8064 q^{51} + 16704 q^{57} + 5528 q^{59} + 19576 q^{73} + 9780 q^{75} + 5508 q^{81} + 23656 q^{83} + 23432 q^{97} + 33768 q^{99}+O(q^{100}) $ 4 * q + 12 * q^3 - 252 * q^9 - 536 * q^11 + 3260 * q^25 - 2484 * q^27 - 1608 * q^33 - 5760 * q^35 - 3844 * q^49 + 8064 * q^51 + 16704 * q^57 + 5528 * q^59 + 19576 * q^73 + 9780 * q^75 + 5508 * q^81 + 23656 * q^83 + 23432 * q^97 + 33768 * 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
$3$	3.00000	+	8.48528i	0.333333	+	0.942809i
$3$	3.00000	+	8.48528i	0.333333	+	0.942809i
$4$		0			0
$5$	37.9473			1.51789			0.758947	−	0.651153i	$-0.225714\pi$
$5$	37.9473			1.51789			0.758947	+	0.651153i	$0.225714\pi$
$6$		0			0
$7$	−37.9473			−0.774435			−0.387218	−	0.921988i	$-0.626564\pi$
$7$	−37.9473			−0.774435			−0.387218	+	0.921988i	$0.626564\pi$
$8$		0			0
$9$	−63.0000	+	50.9117i	−0.777778	+	0.628539i
$10$		0			0
$11$	−134.000			−1.10744			−0.553719	−	0.832704i	$-0.686792\pi$
$11$	−134.000			−1.10744			−0.553719	+	0.832704i	$0.686792\pi$
$12$		0			0
$13$		−	321.994i		−	1.90529i	−0.304087	−	0.952644i	$-0.598351\pi$
$13$		−	321.994i		−	1.90529i	0.304087	−	0.952644i	$-0.401649\pi$
$14$		0			0
$15$	113.842	+	321.994i	0.505964	+	1.43108i
$16$		0			0
$17$		−	237.588i		−	0.822103i	−0.911612	−	0.411052i	$-0.865162\pi$
$17$		−	237.588i		−	0.822103i	0.911612	−	0.411052i	$-0.134838\pi$
$18$		0			0
$19$		−	492.146i		−	1.36329i	−0.731685	−	0.681643i	$-0.761266\pi$
$19$		−	492.146i		−	1.36329i	0.731685	−	0.681643i	$-0.238734\pi$
$20$		0			0
$21$	−113.842	−	321.994i	−0.258145	−	0.730145i
$22$		0			0
$23$		−	643.988i		−	1.21737i	−0.793413	−	0.608684i	$-0.791698\pi$
$23$		−	643.988i		−	1.21737i	0.793413	−	0.608684i	$-0.208302\pi$
$24$		0			0
$25$	815.000			1.30400
$26$		0			0
$27$	−621.000	−	381.838i	−0.851852	−	0.523783i
$28$		0			0
$29$	796.894			0.947555			0.473778	−	0.880645i	$-0.342890\pi$
$29$	796.894			0.947555			0.473778	+	0.880645i	$0.342890\pi$
$30$		0			0
$31$	−1100.47			−1.14513			−0.572566	−	0.819858i	$-0.694052\pi$
$31$	−1100.47			−1.14513			−0.572566	+	0.819858i	$0.694052\pi$
$32$		0			0
$33$	−402.000	−	1137.03i	−0.369146	−	1.04410i
$34$		0			0
$35$	−1440.00			−1.17551
$36$		0			0
$37$			1609.97i			1.17602i	0.808854	+	0.588009i	$0.200088\pi$
$37$			1609.97i			1.17602i	−0.808854	+	0.588009i	$0.799912\pi$
$38$		0			0
$39$	2732.21	−	965.981i	1.79632	−	0.635096i
$40$		0			0
$41$		−	1289.76i		−	0.767259i	−0.923487	−	0.383630i	$-0.874674\pi$
$41$		−	1289.76i		−	0.767259i	0.923487	−	0.383630i	$-0.125326\pi$
$42$		0			0
$43$			424.264i			0.229456i	0.993397	+	0.114728i	$0.0365996\pi$
$43$			424.264i			0.229456i	−0.993397	+	0.114728i	$0.963400\pi$
$44$		0			0
$45$	−2390.68	+	1931.96i	−1.18058	+	0.954056i
$46$		0			0
$47$			2575.95i			1.16612i	0.812431	+	0.583058i	$0.198144\pi$
$47$			2575.95i			1.16612i	−0.812431	+	0.583058i	$0.801856\pi$
$48$		0			0
$49$	−961.000			−0.400250
$50$		0			0
$51$	2016.00	−	712.764i	0.775087	−	0.274034i
$52$		0			0
$53$	−1631.74			−0.580895			−0.290448	−	0.956891i	$-0.593804\pi$
$53$	−1631.74			−0.580895			−0.290448	+	0.956891i	$0.593804\pi$
$54$		0			0
$55$	−5084.94			−1.68097
$56$		0			0
$57$	4176.00	−	1476.44i	1.28532	−	0.454429i
$58$		0			0
$59$	1382.00			0.397012			0.198506	−	0.980100i	$-0.436391\pi$
$59$	1382.00			0.397012			0.198506	+	0.980100i	$0.436391\pi$
$60$		0			0
$61$		−	321.994i		−	0.0865342i	−0.999064	−	0.0432671i	$-0.986223\pi$
$61$		−	321.994i		−	0.0865342i	0.999064	−	0.0432671i	$-0.0137766\pi$
$62$		0			0
$63$	2390.68	−	1931.96i	0.602339	−	0.486763i
$64$		0			0
$65$		−	12218.8i		−	2.89202i
$66$		0			0
$67$		−	186.676i		−	0.0415853i	−0.999784	−	0.0207926i	$-0.993381\pi$
$67$		−	186.676i		−	0.0415853i	0.999784	−	0.0207926i	$-0.00661898\pi$
$68$		0			0
$69$	5464.42	−	1931.96i	1.14775	−	0.405789i
$70$		0			0
$71$			1931.96i			0.383250i	0.981468	+	0.191625i	$0.0613757\pi$
$71$			1931.96i			0.383250i	−0.981468	+	0.191625i	$0.938624\pi$
$72$		0			0
$73$	4894.00			0.918371			0.459186	−	0.888340i	$-0.348141\pi$
$73$	4894.00			0.918371			0.459186	+	0.888340i	$0.348141\pi$
$74$		0			0
$75$	2445.00	+	6915.50i	0.434667	+	1.22942i
$76$		0			0
$77$	5084.94			0.857639
$78$		0			0
$79$	1479.95			0.237133			0.118566	−	0.992946i	$-0.462170\pi$
$79$	1479.95			0.237133			0.118566	+	0.992946i	$0.462170\pi$
$80$		0			0
$81$	1377.00	−	6414.87i	0.209877	−	0.977728i
$82$		0			0
$83$	5914.00			0.858470			0.429235	−	0.903193i	$-0.358783\pi$
$83$	5914.00			0.858470			0.429235	+	0.903193i	$0.358783\pi$
$84$		0			0
$85$		−	9015.83i		−	1.24787i
$86$		0			0
$87$	2390.68	+	6761.87i	0.315852	+	0.893364i
$88$		0			0
$89$		−	9062.28i		−	1.14408i	−0.820225	−	0.572041i	$-0.806152\pi$
$89$		−	9062.28i		−	1.14408i	0.820225	−	0.572041i	$-0.193848\pi$
$90$		0			0
$91$			12218.8i			1.47552i
$92$		0			0
$93$	−3301.42	−	9337.82i	−0.381711	−	1.07964i
$94$		0			0
$95$		−	18675.6i		−	2.06932i
$96$		0			0
$97$	5858.00			0.622595			0.311298	−	0.950312i	$-0.399236\pi$
$97$	5858.00			0.622595			0.311298	+	0.950312i	$0.399236\pi$
$98$		0			0
$99$	8442.00	−	6822.17i	0.861341	−	0.696068i
$100$		0			0
$101$	−14837.4			−1.45451			−0.727253	−	0.686370i	$-0.759203\pi$
$101$	−14837.4			−1.45451			−0.727253	+	0.686370i	$0.759203\pi$
$102$		0			0
$103$	2846.05			0.268267			0.134134	−	0.990963i	$-0.457175\pi$
$103$	2846.05			0.268267			0.134134	+	0.990963i	$0.457175\pi$
$104$		0			0
$105$	−4320.00	−	12218.8i	−0.391837	−	1.10828i
$106$		0			0
$107$	−21274.0			−1.85815			−0.929077	−	0.369887i	$-0.879396\pi$
$107$	−21274.0			−1.85815			−0.929077	+	0.369887i	$0.879396\pi$
$108$		0			0
$109$			7405.86i			0.623336i	0.950191	+	0.311668i	$0.100888\pi$
$109$			7405.86i			0.623336i	−0.950191	+	0.311668i	$0.899112\pi$
$110$		0			0
$111$	−13661.0	+	4829.91i	−1.10876	+	0.392006i
$112$		0			0
$113$		−	3461.99i		−	0.271125i	−0.990769	−	0.135562i	$-0.956716\pi$
$113$		−	3461.99i		−	0.271125i	0.990769	−	0.135562i	$-0.0432842\pi$
$114$		0			0
$115$		−	24437.6i		−	1.84783i
$116$		0			0
$117$	16393.2	+	20285.6i	1.19755	+	1.48189i
$118$		0			0
$119$			9015.83i			0.636666i
$120$		0			0
$121$	3315.00			0.226419
$122$		0			0
$123$	10944.0	−	3869.29i	0.723379	−	0.255753i
$124$		0			0
$125$	7209.99			0.461440
$126$		0			0
$127$	21364.3			1.32459			0.662296	−	0.749242i	$-0.269582\pi$
$127$	21364.3			1.32459			0.662296	+	0.749242i	$0.269582\pi$
$128$		0			0
$129$	−3600.00	+	1272.79i	−0.216333	+	0.0764853i
$130$		0			0
$131$	−17338.0			−1.01031			−0.505157	−	0.863027i	$-0.668565\pi$
$131$	−17338.0			−1.01031			−0.505157	+	0.863027i	$0.668565\pi$
$132$		0			0
$133$			18675.6i			1.05578i
$134$		0			0
$135$	−23565.3	−	14489.7i	−1.29302	−	0.795046i
$136$		0			0
$137$		−	27560.2i		−	1.46839i	−0.678939	−	0.734195i	$-0.737560\pi$
$137$		−	27560.2i		−	1.46839i	0.678939	−	0.734195i	$-0.262440\pi$
$138$		0			0
$139$			19668.9i			1.01801i	0.860765	+	0.509003i	$0.169986\pi$
$139$			19668.9i			1.01801i	−0.860765	+	0.509003i	$0.830014\pi$
$140$		0			0
$141$	−21857.7	+	7727.85i	−1.09942	+	0.388705i
$142$		0			0
$143$			43147.2i			2.10999i
$144$		0			0
$145$	30240.0			1.43829
$146$		0			0
$147$	−2883.00	−	8154.36i	−0.133417	−	0.377359i
$148$		0			0
$149$	−9372.99			−0.422188			−0.211094	−	0.977466i	$-0.567703\pi$
$149$	−9372.99			−0.422188			−0.211094	+	0.977466i	$0.567703\pi$
$150$		0			0
$151$	23489.4			1.03019			0.515096	−	0.857133i	$-0.327756\pi$
$151$	23489.4			1.03019			0.515096	+	0.857133i	$0.327756\pi$
$152$		0			0
$153$	12096.0	+	14968.0i	0.516724	+	0.639414i
$154$		0			0
$155$	−41760.0			−1.73819
$156$		0			0
$157$		−	4185.92i		−	0.169821i	−0.996389	−	0.0849105i	$-0.972940\pi$
$157$		−	4185.92i		−	0.169821i	0.996389	−	0.0849105i	$-0.0270604\pi$
$158$		0			0
$159$	−4895.21	−	13845.7i	−0.193632	−	0.547673i
$160$		0			0
$161$			24437.6i			0.942773i
$162$		0			0
$163$		−	44955.0i		−	1.69201i	−0.533175	−	0.846005i	$-0.679001\pi$
$163$		−	44955.0i		−	1.69201i	0.533175	−	0.846005i	$-0.320999\pi$
$164$		0			0
$165$	−15254.8	−	43147.2i	−0.560324	−	1.58484i
$166$		0			0
$167$		−	40571.2i		−	1.45474i	−0.686246	−	0.727370i	$-0.740743\pi$
$167$		−	40571.2i		−	1.45474i	0.686246	−	0.727370i	$-0.259257\pi$
$168$		0			0
$169$	−75119.0			−2.63012
$170$		0			0
$171$	25056.0	+	31005.2i	0.856879	+	1.06033i
$172$		0			0
$173$	23261.7			0.777230			0.388615	−	0.921400i	$-0.372954\pi$
$173$	23261.7			0.777230			0.388615	+	0.921400i	$0.372954\pi$
$174$		0			0
$175$	−30927.1			−1.00986
$176$		0			0
$177$	4146.00	+	11726.7i	0.132337	+	0.374307i
$178$		0			0
$179$	−6586.00			−0.205549			−0.102775	−	0.994705i	$-0.532772\pi$
$179$	−6586.00			−0.205549			−0.102775	+	0.994705i	$0.532772\pi$
$180$		0			0
$181$		−	30589.4i		−	0.933714i	−0.884333	−	0.466857i	$-0.845386\pi$
$181$		−	30589.4i		−	0.933714i	0.884333	−	0.466857i	$-0.154614\pi$
$182$		0			0
$183$	2732.21	−	965.981i	0.0815852	−	0.0288447i
$184$		0			0
$185$			61094.0i			1.78507i
$186$		0			0
$187$			31836.8i			0.910429i
$188$		0			0
$189$	23565.3	+	14489.7i	0.659704	+	0.405636i
$190$		0			0
$191$		−	11591.8i		−	0.317748i	−0.987299	−	0.158874i	$-0.949214\pi$
$191$		−	11591.8i		−	0.317748i	0.987299	−	0.158874i	$-0.0507864\pi$
$192$		0			0
$193$	−24386.0			−0.654675			−0.327338	−	0.944907i	$-0.606151\pi$
$193$	−24386.0			−0.654675			−0.327338	+	0.944907i	$0.606151\pi$
$194$		0			0
$195$	103680.	−	36656.4i	2.72663	−	0.964008i
$196$		0			0
$197$	64244.8			1.65541			0.827705	−	0.561163i	$-0.189646\pi$
$197$	64244.8			1.65541			0.827705	+	0.561163i	$0.189646\pi$
$198$		0			0
$199$	34114.7			0.861459			0.430730	−	0.902481i	$-0.358256\pi$
$199$	34114.7			0.861459			0.430730	+	0.902481i	$0.358256\pi$
$200$		0			0
$201$	1584.00	−	560.029i	0.0392070	−	0.0138618i
$202$		0			0
$203$	−30240.0			−0.733820
$204$		0			0
$205$		−	48943.1i		−	1.16462i
$206$		0			0
$207$	32786.5	+	40571.2i	0.765164	+	0.946842i
$208$		0			0
$209$			65947.6i			1.50975i
$210$		0			0
$211$			17666.4i			0.396810i	0.980120	+	0.198405i	$0.0635760\pi$
$211$			17666.4i			0.396810i	−0.980120	+	0.198405i	$0.936424\pi$
$212$		0			0
$213$	−16393.2	+	5795.89i	−0.361331	+	0.127750i
$214$		0			0
$215$			16099.7i			0.348290i
$216$		0			0
$217$	41760.0			0.886831
$218$		0			0
$219$	14682.0	+	41527.0i	0.306124	+	0.865849i
$220$		0			0
$221$	−76501.8			−1.56634
$222$		0			0
$223$	−43146.1			−0.867625			−0.433812	−	0.901003i	$-0.642832\pi$
$223$	−43146.1			−0.867625			−0.433812	+	0.901003i	$0.642832\pi$
$224$		0			0
$225$	−51345.0	+	41493.0i	−1.01422	+	0.819615i
$226$		0			0
$227$	−32678.0			−0.634167			−0.317084	−	0.948398i	$-0.602704\pi$
$227$	−32678.0			−0.634167			−0.317084	+	0.948398i	$0.602704\pi$
$228$		0			0
$229$			55704.9i			1.06224i	0.847297	+	0.531120i	$0.178229\pi$
$229$			55704.9i			1.06224i	−0.847297	+	0.531120i	$0.821771\pi$
$230$		0			0
$231$	15254.8	+	43147.2i	0.285880	+	0.808590i
$232$		0			0
$233$			47687.3i			0.878397i	0.898390	+	0.439198i	$0.144737\pi$
$233$			47687.3i			0.878397i	−0.898390	+	0.439198i	$0.855263\pi$
$234$		0			0
$235$			97750.4i			1.77004i
$236$		0			0
$237$	4439.84	+	12557.8i	0.0790443	+	0.223571i
$238$		0			0
$239$		−	74702.6i		−	1.30780i	−0.756583	−	0.653898i	$-0.773133\pi$
$239$		−	74702.6i		−	1.30780i	0.756583	−	0.653898i	$-0.226867\pi$
$240$		0			0
$241$	766.000			0.0131885			0.00659424	−	0.999978i	$-0.497901\pi$
$241$	766.000			0.0131885			0.00659424	+	0.999978i	$0.497901\pi$
$242$		0			0
$243$	58563.0	−	7560.39i	0.991770	−	0.128036i
$244$		0			0
$245$	−36467.4			−0.607537
$246$		0			0
$247$	−158468.			−2.59745
$248$		0			0
$249$	17742.0	+	50182.0i	0.286157	+	0.809373i
$250$		0			0
$251$	−2054.00			−0.0326027			−0.0163013	−	0.999867i	$-0.505189\pi$
$251$	−2054.00			−0.0326027			−0.0163013	+	0.999867i	$0.505189\pi$
$252$		0			0
$253$			86294.3i			1.34816i
$254$		0			0
$255$	76501.8	−	27047.5i	1.17650	−	0.415955i
$256$		0			0
$257$		−	18667.6i		−	0.282633i	−0.989964	−	0.141316i	$-0.954866\pi$
$257$		−	18667.6i		−	0.282633i	0.989964	−	0.141316i	$-0.0451335\pi$
$258$		0			0
$259$		−	61094.0i		−	0.910750i
$260$		0			0
$261$	−50204.3	+	40571.2i	−0.736987	+	0.595576i
$262$		0			0
$263$			92090.2i			1.33138i	0.746228	+	0.665690i	$0.231863\pi$
$263$			92090.2i			1.33138i	−0.746228	+	0.665690i	$0.768137\pi$
$264$		0			0
$265$	−61920.0			−0.881737
$266$		0			0
$267$	76896.0	−	27186.8i	1.07865	−	0.381361i
$268$		0			0
$269$	7627.41			0.105408			0.0527039	−	0.998610i	$-0.483216\pi$
$269$	7627.41			0.105408			0.0527039	+	0.998610i	$0.483216\pi$
$270$		0			0
$271$	53999.1			0.735271			0.367636	−	0.929970i	$-0.380167\pi$
$271$	53999.1			0.735271			0.367636	+	0.929970i	$0.380167\pi$
$272$		0			0
$273$	−103680.	+	36656.4i	−1.39114	+	0.491841i
$274$		0			0
$275$	−109210.			−1.44410
$276$		0			0
$277$			23505.5i			0.306345i	0.988199	+	0.153173i	$0.0489490\pi$
$277$			23505.5i			0.306345i	−0.988199	+	0.153173i	$0.951051\pi$
$278$		0			0
$279$	69329.8	−	56026.9i	0.890659	−	0.719761i
$280$		0			0
$281$			118624.i			1.50231i	0.660123	+	0.751157i	$0.270504\pi$
$281$			118624.i			1.50231i	−0.660123	+	0.751157i	$0.729496\pi$
$282$		0			0
$283$		−	134763.i		−	1.68267i	−0.540515	−	0.841334i	$-0.681771\pi$
$283$		−	134763.i		−	1.68267i	0.540515	−	0.841334i	$-0.318229\pi$
$284$		0			0
$285$	158468.	−	56026.9i	1.95098	−	0.689774i
$286$		0			0
$287$			48943.1i			0.594193i
$288$		0			0
$289$	27073.0			0.324146
$290$		0			0
$291$	17574.0	+	49706.8i	0.207532	+	0.586989i
$292$		0			0
$293$	34494.1			0.401800			0.200900	−	0.979612i	$-0.435613\pi$
$293$	34494.1			0.401800			0.200900	+	0.979612i	$0.435613\pi$
$294$		0			0
$295$	52443.2			0.602622
$296$		0			0
$297$	83214.0	+	51166.2i	0.943373	+	0.580057i
$298$		0			0
$299$	−207360.			−2.31944
$300$		0			0
$301$		−	16099.7i		−	0.177699i
$302$		0			0
$303$	−44512.2	−	125900.i	−0.484835	−	1.37132i
$304$		0			0
$305$		−	12218.8i		−	0.131350i
$306$		0			0
$307$		−	23402.4i		−	0.248304i	−0.992263	−	0.124152i	$-0.960379\pi$
$307$		−	23402.4i		−	0.248304i	0.992263	−	0.124152i	$-0.0396210\pi$
$308$		0			0
$309$	8538.15	+	24149.5i	0.0894225	+	0.252925i
$310$		0			0
$311$		−	19963.6i		−	0.206404i	−0.994660	−	0.103202i	$-0.967091\pi$
$311$		−	19963.6i		−	0.206404i	0.994660	−	0.103202i	$-0.0329088\pi$
$312$		0			0
$313$	−27458.0			−0.280272			−0.140136	−	0.990132i	$-0.544754\pi$
$313$	−27458.0			−0.280272			−0.140136	+	0.990132i	$0.544754\pi$
$314$		0			0
$315$	90720.0	−	73312.8i	0.914286	−	0.738854i
$316$		0			0
$317$	−168752.			−1.67931			−0.839653	−	0.543123i	$-0.817242\pi$
$317$	−168752.			−1.67931			−0.839653	+	0.543123i	$0.817242\pi$
$318$		0			0
$319$	−106784.			−1.04936
$320$		0			0
$321$	−63822.0	−	180516.i	−0.619385	−	1.75188i
$322$		0			0
$323$	−116928.			−1.12076
$324$		0			0
$325$		−	262425.i		−	2.48450i
$326$		0			0
$327$	−62840.8	+	22217.6i	−0.587687	+	0.207779i
$328$		0			0
$329$		−	97750.4i		−	0.903081i
$330$		0			0
$331$			79065.9i			0.721661i	0.932631	+	0.360830i	$0.117507\pi$
$331$			79065.9i			0.721661i	−0.932631	+	0.360830i	$0.882493\pi$
$332$		0			0
$333$	−81966.2	−	101428.i	−0.739174	−	0.914681i
$334$		0			0
$335$		−	7083.86i		−	0.0631220i
$336$		0			0
$337$	63454.0			0.558726			0.279363	−	0.960186i	$-0.409877\pi$
$337$	63454.0			0.558726			0.279363	+	0.960186i	$0.409877\pi$
$338$		0			0
$339$	29376.0	−	10386.0i	0.255619	−	0.0903750i
$340$		0			0
$341$	147463.			1.26816
$342$		0			0
$343$	127579.			1.08440
$344$		0			0
$345$	207360.	−	73312.8i	1.74216	−	0.615945i
$346$		0			0
$347$	−28166.0			−0.233919			−0.116960	−	0.993137i	$-0.537315\pi$
$347$	−28166.0			−0.233919			−0.116960	+	0.993137i	$0.537315\pi$
$348$		0			0
$349$		−	67296.7i		−	0.552514i	−0.961084	−	0.276257i	$-0.910906\pi$
$349$		−	67296.7i		−	0.552514i	0.961084	−	0.276257i	$-0.0890940\pi$
$350$		0			0
$351$	−122949.	+	199958.i	−0.997957	+	1.62302i
$352$		0			0
$353$		−	27831.7i		−	0.223352i	−0.993745	−	0.111676i	$-0.964378\pi$
$353$		−	27831.7i		−	0.223352i	0.993745	−	0.111676i	$-0.0356220\pi$
$354$		0			0
$355$			73312.8i			0.581732i
$356$		0			0
$357$	−76501.8	+	27047.5i	−0.600254	+	0.212222i
$358$		0			0
$359$		−	147473.i		−	1.14426i	−0.820164	−	0.572129i	$-0.806118\pi$
$359$		−	147473.i		−	1.14426i	0.820164	−	0.572129i	$-0.193882\pi$
$360$		0			0
$361$	−111887.			−0.858549
$362$		0			0
$363$	9945.00	+	28128.7i	0.0754730	+	0.213470i
$364$		0			0
$365$	185714.			1.39399
$366$		0			0
$367$	216565.			1.60789			0.803946	−	0.594702i	$-0.202730\pi$
$367$	216565.			1.60789			0.803946	+	0.594702i	$0.202730\pi$
$368$		0			0
$369$	65664.0	+	81255.1i	0.482253	+	0.596757i
$370$		0			0
$371$	61920.0			0.449866
$372$		0			0
$373$			163895.i			1.17801i	0.808131	+	0.589003i	$0.200479\pi$
$373$			163895.i			1.17801i	−0.808131	+	0.589003i	$0.799521\pi$
$374$		0			0
$375$	21630.0	+	61178.8i	0.153813	+	0.435049i
$376$		0			0
$377$		−	256595.i		−	1.80537i
$378$		0			0
$379$		−	259836.i		−	1.80893i	−0.426550	−	0.904464i	$-0.640271\pi$
$379$		−	259836.i		−	1.80893i	0.426550	−	0.904464i	$-0.359729\pi$
$380$		0			0
$381$	64093.0	+	181283.i	0.441531	+	1.24884i
$382$		0			0
$383$			222820.i			1.51899i	0.650511	+	0.759497i	$0.274555\pi$
$383$			222820.i			1.51899i	−0.650511	+	0.759497i	$0.725445\pi$
$384$		0			0
$385$	192960.			1.30180
$386$		0			0
$387$	−21600.0	−	26728.6i	−0.144222	−	0.178466i
$388$		0			0
$389$	−59008.1			−0.389953			−0.194977	−	0.980808i	$-0.562463\pi$
$389$	−59008.1			−0.389953			−0.194977	+	0.980808i	$0.562463\pi$
$390$		0			0
$391$	−153004.			−1.00080
$392$		0			0
$393$	−52014.0	−	147118.i	−0.336771	−	0.952533i
$394$		0			0
$395$	56160.0			0.359942
$396$		0			0
$397$			236665.i			1.50160i	0.660530	+	0.750799i	$0.270331\pi$
$397$			236665.i			1.50160i	−0.660530	+	0.750799i	$0.729669\pi$
$398$		0			0
$399$	−158468.	+	56026.9i	−0.995396	+	0.351926i
$400$		0			0
$401$		−	193906.i		−	1.20587i	−0.797789	−	0.602937i	$-0.793997\pi$
$401$		−	193906.i		−	1.20587i	0.797789	−	0.602937i	$-0.206003\pi$
$402$		0			0
$403$			354345.i			2.18181i
$404$		0			0
$405$	52253.5	−	243427.i	0.318570	−	1.48409i
$406$		0			0
$407$		−	215736.i		−	1.30237i
$408$		0			0
$409$	−125086.			−0.747760			−0.373880	−	0.927477i	$-0.621973\pi$
$409$	−125086.			−0.747760			−0.373880	+	0.927477i	$0.621973\pi$
$410$		0			0
$411$	233856.	−	82680.6i	1.38441	−	0.489463i
$412$		0			0
$413$	−52443.2			−0.307460
$414$		0			0
$415$	224421.			1.30307
$416$		0			0
$417$	−166896.	+	59006.6i	−0.959785	+	0.339335i
$418$		0			0
$419$	72346.0			0.412085			0.206042	−	0.978543i	$-0.433942\pi$
$419$	72346.0			0.412085			0.206042	+	0.978543i	$0.433942\pi$
$420$		0			0
$421$			254053.i			1.43338i	0.697394	+	0.716688i	$0.254343\pi$
$421$			254053.i			1.43338i	−0.697394	+	0.716688i	$0.745657\pi$
$422$		0			0
$423$	−131146.	−	162285.i	−0.732950	−	0.906979i
$424$		0			0
$425$		−	193634.i		−	1.07202i
$426$		0			0
$427$			12218.8i			0.0670151i
$428$		0			0
$429$	−366116.	+	129442.i	−1.98932	+	0.703330i
$430$		0			0
$431$		−	280779.i		−	1.51150i	−0.654858	−	0.755752i	$-0.727271\pi$
$431$		−	280779.i		−	1.51150i	0.654858	−	0.755752i	$-0.272729\pi$
$432$		0			0
$433$	303266.			1.61751			0.808757	−	0.588143i	$-0.200141\pi$
$433$	303266.			1.61751			0.808757	+	0.588143i	$0.200141\pi$
$434$		0			0
$435$	90720.0	+	256595.i	0.479429	+	1.35603i
$436$		0			0
$437$	−316936.			−1.65962
$438$		0			0
$439$	235387.			1.22139			0.610694	−	0.791866i	$-0.290890\pi$
$439$	235387.			1.22139			0.610694	+	0.791866i	$0.290890\pi$
$440$		0			0
$441$	60543.0	−	48926.1i	0.311305	−	0.251573i
$442$		0			0
$443$	151354.			0.771235			0.385617	−	0.922659i	$-0.373989\pi$
$443$	151354.			0.771235			0.385617	+	0.922659i	$0.373989\pi$
$444$		0			0
$445$		−	343889.i		−	1.73660i
$446$		0			0
$447$	−28119.0	−	79532.5i	−0.140729	−	0.398042i
$448$		0			0
$449$		−	86312.3i		−	0.428134i	−0.976819	−	0.214067i	$-0.931329\pi$
$449$		−	86312.3i		−	0.428134i	0.976819	−	0.214067i	$-0.0686711\pi$
$450$		0			0
$451$			172828.i			0.849692i
$452$		0			0
$453$	70468.2	+	199314.i	0.343397	+	0.971274i
$454$		0			0
$455$			463671.i			2.23969i
$456$		0			0
$457$	30626.0			0.146642			0.0733209	−	0.997308i	$-0.476640\pi$
$457$	30626.0			0.146642			0.0733209	+	0.997308i	$0.476640\pi$
$458$		0			0
$459$	−90720.0	+	147542.i	−0.430604	+	0.700310i
$460$		0			0
$461$	326233.			1.53506			0.767532	−	0.641011i	$-0.221485\pi$
$461$	326233.			1.53506			0.767532	+	0.641011i	$0.221485\pi$
$462$		0			0
$463$	−305286.			−1.42412			−0.712058	−	0.702121i	$-0.752237\pi$
$463$	−305286.			−1.42412			−0.712058	+	0.702121i	$0.752237\pi$
$464$		0			0
$465$	−125280.	−	354345.i	−0.579396	−	1.63878i
$466$		0			0
$467$	−299558.			−1.37356			−0.686779	−	0.726866i	$-0.740976\pi$
$467$	−299558.			−1.37356			−0.686779	+	0.726866i	$0.740976\pi$
$468$		0			0
$469$			7083.86i			0.0322051i
$470$		0			0
$471$	35518.7	−	12557.8i	0.160109	−	0.0566070i
$472$		0			0
$473$		−	56851.4i		−	0.254108i
$474$		0			0
$475$		−	401099.i		−	1.77773i
$476$		0			0
$477$	102799.	−	83074.4i	0.451808	−	0.365116i
$478$		0			0
$479$		−	43791.2i		−	0.190860i	−0.995436	−	0.0954301i	$-0.969577\pi$
$479$		−	43791.2i		−	0.190860i	0.995436	−	0.0954301i	$-0.0304226\pi$
$480$		0			0
$481$	518400.			2.24065
$482$		0			0
$483$	−207360.	+	73312.8i	−0.888855	+	0.314258i
$484$		0			0
$485$	222295.			0.945033
$486$		0			0
$487$	152662.			0.643685			0.321842	−	0.946793i	$-0.395698\pi$
$487$	152662.			0.643685			0.321842	+	0.946793i	$0.395698\pi$
$488$		0			0
$489$	381456.	−	134865.i	1.59524	−	0.564003i
$490$		0			0
$491$	367142.			1.52290			0.761449	−	0.648224i	$-0.224488\pi$
$491$	367142.			1.52290			0.761449	+	0.648224i	$0.224488\pi$
$492$		0			0
$493$		−	189332.i		−	0.778988i
$494$		0			0
$495$	320351.	−	258883.i	1.30742	−	1.05656i
$496$		0			0
$497$		−	73312.8i		−	0.296802i
$498$		0			0
$499$			60160.6i			0.241608i	0.992676	+	0.120804i	$0.0385473\pi$
$499$			60160.6i			0.241608i	−0.992676	+	0.120804i	$0.961453\pi$
$500$		0			0
$501$	344258.	−	121714.i	1.37154	−	0.484913i
$502$		0			0
$503$			263391.i			1.04103i	0.853851	+	0.520517i	$0.174261\pi$
$503$			263391.i			1.04103i	−0.853851	+	0.520517i	$0.825739\pi$
$504$		0			0
$505$	−563040.			−2.20778
$506$		0			0
$507$	−225357.	−	637406.i	−0.876708	−	2.47971i
$508$		0			0
$509$	15975.8			0.0616634			0.0308317	−	0.999525i	$-0.490184\pi$
$509$	15975.8			0.0616634			0.0308317	+	0.999525i	$0.490184\pi$
$510$		0			0
$511$	−185714.			−0.711219
$512$		0			0
$513$	−187920.	+	305623.i	−0.714066	+	1.16132i
$514$		0			0
$515$	108000.			0.407201
$516$		0			0
$517$		−	345177.i		−	1.29140i
$518$		0			0
$519$	69785.1	+	197382.i	0.259077	+	0.732779i
$520$		0			0
$521$			210707.i			0.776252i	0.921606	+	0.388126i	$0.126877\pi$
$521$			210707.i			0.776252i	−0.921606	+	0.388126i	$0.873123\pi$
$522$		0			0
$523$			92812.0i			0.339313i	0.985503	+	0.169657i	$0.0542659\pi$
$523$			92812.0i			0.339313i	−0.985503	+	0.169657i	$0.945734\pi$
$524$		0			0
$525$	−92781.2	−	262425.i	−0.336621	−	0.952109i
$526$		0			0
$527$			261459.i			0.941418i
$528$		0			0
$529$	−134879.			−0.481984
$530$		0			0
$531$	−87066.0	+	70360.0i	−0.308787	+	0.249538i
$532$		0			0
$533$	−415296.			−1.46185
$534$		0			0
$535$	−807292.			−2.82048
$536$		0			0
$537$	−19758.0	−	55884.1i	−0.0685164	−	0.193794i
$538$		0			0
$539$	128774.			0.443252
$540$		0			0
$541$			249545.i			0.852618i	0.904578	+	0.426309i	$0.140186\pi$
$541$			249545.i			0.852618i	−0.904578	+	0.426309i	$0.859814\pi$
$542$		0			0
$543$	259560.	−	91768.2i	0.880314	−	0.311238i
$544$		0			0
$545$			281033.i			0.946158i
$546$		0			0
$547$		−	258309.i		−	0.863306i	−0.902040	−	0.431653i	$-0.857930\pi$
$547$		−	258309.i		−	0.863306i	0.902040	−	0.431653i	$-0.142070\pi$
$548$		0			0
$549$	16393.2	+	20285.6i	0.0543902	+	0.0673044i
$550$		0			0
$551$		−	392188.i		−	1.29179i
$552$		0			0
$553$	−56160.0			−0.183644
$554$		0			0
$555$	−518400.	+	183282.i	−1.68298	+	0.595023i
$556$		0			0
$557$	−151448.			−0.488149			−0.244075	−	0.969756i	$-0.578484\pi$
$557$	−151448.			−0.488149			−0.244075	+	0.969756i	$0.578484\pi$
$558$		0			0
$559$	136610.			0.437180
$560$		0			0
$561$	−270144.	+	95510.3i	−0.858360	+	0.303476i
$562$		0			0
$563$	−238118.			−0.751234			−0.375617	−	0.926775i	$-0.622569\pi$
$563$	−238118.			−0.751234			−0.375617	+	0.926775i	$0.622569\pi$
$564$		0			0
$565$		−	131373.i		−	0.411539i
$566$		0			0
$567$	−52253.5	+	243427.i	−0.162536	+	0.757187i
$568$		0			0
$569$		−	163868.i		−	0.506138i	−0.967448	−	0.253069i	$-0.918560\pi$
$569$		−	163868.i		−	0.506138i	0.967448	−	0.253069i	$-0.0814400\pi$
$570$		0			0
$571$		−	207126.i		−	0.635275i	−0.948212	−	0.317638i	$-0.897110\pi$
$571$		−	207126.i		−	0.635275i	0.948212	−	0.317638i	$-0.102890\pi$
$572$		0			0
$573$	98359.5	−	34775.3i	0.299576	−	0.105916i
$574$		0			0
$575$		−	524850.i		−	1.58745i
$576$		0			0
$577$	−306146.			−0.919553			−0.459777	−	0.888035i	$-0.652071\pi$
$577$	−306146.			−0.919553			−0.459777	+	0.888035i	$0.652071\pi$
$578$		0			0
$579$	−73158.0	−	206922.i	−0.218225	−	0.617234i
$580$		0			0
$581$	−224421.			−0.664830
$582$		0			0
$583$	218653.			0.643306
$584$		0			0
$585$	622080.	+	769785.i	1.81775	+	2.24935i
$586$		0			0
$587$	−460954.			−1.33777			−0.668885	−	0.743366i	$-0.733228\pi$
$587$	−460954.			−1.33777			−0.668885	+	0.743366i	$0.733228\pi$
$588$		0			0
$589$			541594.i			1.56114i
$590$		0			0
$591$	192734.	+	545135.i	0.551804	+	1.56074i
$592$		0			0
$593$			254966.i			0.725057i	0.931973	+	0.362529i	$0.118087\pi$
$593$			254966.i			0.725057i	−0.931973	+	0.362529i	$0.881913\pi$
$594$		0			0
$595$			342127.i			0.966391i
$596$		0			0
$597$	102344.	+	289472.i	0.287153	+	0.812192i
$598$		0			0
$599$		−	245359.i		−	0.683831i	−0.939731	−	0.341916i	$-0.888924\pi$
$599$		−	245359.i		−	0.683831i	0.939731	−	0.341916i	$-0.111076\pi$
$600$		0			0
$601$	74206.0			0.205442			0.102721	−	0.994710i	$-0.467245\pi$
$601$	74206.0			0.205442			0.102721	+	0.994710i	$0.467245\pi$
$602$		0			0
$603$	9504.00	+	11760.6i	0.0261380	+	0.0323441i
$604$		0			0
$605$	125795.			0.343680
$606$		0			0
$607$	−299518.			−0.812917			−0.406458	−	0.913669i	$-0.633236\pi$
$607$	−299518.			−0.812917			−0.406458	+	0.913669i	$0.633236\pi$
$608$		0			0
$609$	−90720.0	−	256595.i	−0.244607	−	0.691852i
$610$		0			0
$611$	829440.			2.22179
$612$		0			0
$613$			206398.i			0.549268i	0.961549	+	0.274634i	$0.0885567\pi$
$613$			206398.i			0.549268i	−0.961549	+	0.274634i	$0.911443\pi$
$614$		0			0
$615$	415296.	−	146829.i	1.09801	−	0.388206i
$616$		0			0
$617$			716667.i			1.88255i	0.337638	+	0.941276i	$0.390372\pi$
$617$			716667.i			1.88255i	−0.337638	+	0.941276i	$0.609628\pi$
$618$		0			0
$619$			153329.i			0.400169i	0.979779	+	0.200084i	$0.0641216\pi$
$619$			153329.i			0.400169i	−0.979779	+	0.200084i	$0.935878\pi$
$620$		0			0
$621$	−245899.	+	399916.i	−0.637636	+	1.03702i
$622$		0			0
$623$			343889.i			0.886018i
$624$		0			0
$625$	−235775.			−0.603584
$626$		0			0
$627$	−559584.	+	197843.i	−1.42341	+	0.503252i
$628$		0			0
$629$	382509.			0.966809
$630$		0			0
$631$	448955.			1.12757			0.563786	−	0.825921i	$-0.309344\pi$
$631$	448955.			1.12757			0.563786	+	0.825921i	$0.309344\pi$
$632$		0			0
$633$	−149904.	+	52999.1i	−0.374116	+	0.132270i
$634$		0			0
$635$	810720.			2.01059
$636$		0			0
$637$			309436.i			0.762592i
$638$		0			0
$639$	−98359.5	−	121714.i	−0.240888	−	0.298083i
$640$		0			0
$641$		−	73482.5i		−	0.178841i	−0.995994	−	0.0894207i	$-0.971498\pi$
$641$		−	73482.5i		−	0.178841i	0.995994	−	0.0894207i	$-0.0285016\pi$
$642$		0			0
$643$		−	260753.i		−	0.630677i	−0.948979	−	0.315338i	$-0.897882\pi$
$643$		−	260753.i		−	0.630677i	0.948979	−	0.315338i	$-0.102118\pi$
$644$		0			0
$645$	−136610.	+	48299.1i	−0.328371	+	0.116097i
$646$		0			0
$647$		−	226040.i		−	0.539978i	−0.962863	−	0.269989i	$-0.912980\pi$
$647$		−	226040.i		−	0.539978i	0.962863	−	0.269989i	$-0.0870201\pi$
$648$		0			0
$649$	−185188.			−0.439667
$650$		0			0
$651$	125280.	+	354345.i	0.295610	+	0.836113i
$652$		0			0
$653$	144390.			0.338618			0.169309	−	0.985563i	$-0.445846\pi$
$653$	144390.			0.338618			0.169309	+	0.985563i	$0.445846\pi$
$654$		0			0
$655$	−657931.			−1.53355
$656$		0			0
$657$	−308322.	+	249162.i	−0.714289	+	0.577232i
$658$		0			0
$659$	383174.			0.882318			0.441159	−	0.897429i	$-0.354567\pi$
$659$	383174.			0.882318			0.441159	+	0.897429i	$0.354567\pi$
$660$		0			0
$661$		−	57636.9i		−	0.131916i	−0.997822	−	0.0659580i	$-0.978990\pi$
$661$		−	57636.9i		−	0.131916i	0.997822	−	0.0659580i	$-0.0210103\pi$
$662$		0			0
$663$	−229505.	−	649139.i	−0.522115	−	1.47676i
$664$		0			0
$665$			708691.i			1.60256i
$666$		0			0
$667$		−	513190.i		−	1.15352i
$668$		0			0
$669$	−129438.	−	366107.i	−0.289208	−	0.818005i
$670$		0			0
$671$			43147.2i			0.0958313i
$672$		0			0
$673$	679778.			1.50085			0.750424	−	0.660956i	$-0.229849\pi$
$673$	679778.			1.50085			0.750424	+	0.660956i	$0.229849\pi$
$674$		0			0
$675$	−506115.	−	311198.i	−1.11081	−	0.683013i
$676$		0			0
$677$	6716.68			0.0146547			0.00732735	−	0.999973i	$-0.497668\pi$
$677$	6716.68			0.0146547			0.00732735	+	0.999973i	$0.497668\pi$
$678$		0			0
$679$	−222295.			−0.482160
$680$		0			0
$681$	−98034.0	−	277282.i	−0.211389	−	0.597899i
$682$		0			0
$683$	552634.			1.18467			0.592333	−	0.805693i	$-0.298207\pi$
$683$	552634.			1.18467			0.592333	+	0.805693i	$0.298207\pi$
$684$		0			0
$685$		−	1.04584e6i		−	2.22886i
$686$		0			0
$687$	−472672.	+	167115.i	−1.00149	+	0.354080i
$688$		0			0
$689$			525409.i			1.10677i
$690$		0			0
$691$			327719.i			0.686349i	0.939272	+	0.343174i	$0.111502\pi$
$691$			327719.i			0.686349i	−0.939272	+	0.343174i	$0.888498\pi$
$692$		0			0
$693$	−320351.	+	258883.i	−0.667053	+	0.539060i
$694$		0			0
$695$			746382.i			1.54522i
$696$		0			0
$697$	−306432.			−0.630766
$698$		0			0
$699$	−404640.	+	143062.i	−0.828160	+	0.292799i
$700$		0			0
$701$	−838750.			−1.70685			−0.853427	−	0.521212i	$-0.825480\pi$
$701$	−838750.			−1.70685			−0.853427	+	0.521212i	$0.825480\pi$
$702$		0			0
$703$	792340.			1.60325
$704$		0			0
$705$	−829440.	+	293251.i	−1.66881	+	0.590013i
$706$		0			0
$707$	563040.			1.12642
$708$		0			0
$709$		−	3541.93i		−	0.00704608i	−0.999994	−	0.00352304i	$-0.998879\pi$
$709$		−	3541.93i		−	0.00704608i	0.999994	−	0.00352304i	$-0.00112142\pi$
$710$		0			0
$711$	−93236.6	+	75346.5i	−0.184437	+	0.149047i
$712$		0			0
$713$			708691.i			1.39405i
$714$		0			0
$715$			1.63732e6i			3.20274i
$716$		0			0
$717$	633872.	−	224108.i	1.23300	−	0.435932i
$718$		0			0
$719$			1.02909e6i			1.99066i	0.0965451	+	0.995329i	$0.469221\pi$
$719$			1.02909e6i			1.99066i	−0.0965451	+	0.995329i	$0.530779\pi$
$720$		0			0
$721$	−108000.			−0.207756
$722$		0			0
$723$	2298.00	+	6499.73i	0.00439616	+	0.0124342i
$724$		0			0
$725$	649469.			1.23561
$726$		0			0
$727$	821901.			1.55507			0.777537	−	0.628838i	$-0.216469\pi$
$727$	821901.			1.55507			0.777537	+	0.628838i	$0.216469\pi$
$728$		0			0
$729$	239841.	+	474242.i	0.451303	+	0.892371i
$730$		0			0
$731$	100800.			0.188637
$732$		0			0
$733$		−	118816.i		−	0.221139i	−0.993868	−	0.110570i	$-0.964732\pi$
$733$		−	118816.i		−	0.221139i	0.993868	−	0.110570i	$-0.0352675\pi$
$734$		0			0
$735$	−109402.	−	309436.i	−0.202512	−	0.572791i
$736$		0			0
$737$			25014.6i			0.0460531i
$738$		0			0
$739$		−	349169.i		−	0.639363i	−0.947525	−	0.319681i	$-0.896424\pi$
$739$		−	349169.i		−	0.639363i	0.947525	−	0.319681i	$-0.103576\pi$
$740$		0			0
$741$	−475404.	−	1.34465e6i	−0.865818	−	2.44890i
$742$		0			0
$743$		−	532578.i		−	0.964729i	−0.875971	−	0.482365i	$-0.839778\pi$
$743$		−	532578.i		−	0.964729i	0.875971	−	0.482365i	$-0.160222\pi$
$744$		0			0
$745$	−355680.			−0.640836
$746$		0			0
$747$	−372582.	+	301092.i	−0.667699	+	0.539582i
$748$		0			0
$749$	807292.			1.43902
$750$		0			0
$751$	753444.			1.33589			0.667946	−	0.744210i	$-0.267174\pi$
$751$	753444.			1.33589			0.667946	+	0.744210i	$0.267174\pi$
$752$		0			0
$753$	−6162.00	−	17428.8i	−0.0108676	−	0.0307381i
$754$		0			0
$755$	891360.			1.56372
$756$		0			0
$757$			87904.3i			0.153398i	0.997054	+	0.0766988i	$0.0244380\pi$
$757$			87904.3i			0.153398i	−0.997054	+	0.0766988i	$0.975562\pi$
$758$		0			0
$759$	−732232.	+	258883.i	−1.27106	+	0.449386i
$760$		0			0
$761$		−	1.06243e6i		−	1.83455i	−0.398257	−	0.917274i	$-0.630385\pi$
$761$		−	1.06243e6i		−	1.83455i	0.398257	−	0.917274i	$-0.369615\pi$
$762$		0			0
$763$		−	281033.i		−	0.482734i
$764$		0			0
$765$	459011.	+	567997.i	0.784332	+	0.970562i
$766$		0			0
$767$		−	444995.i		−	0.756423i
$768$		0			0
$769$	259006.			0.437983			0.218991	−	0.975727i	$-0.429723\pi$
$769$	259006.			0.437983			0.218991	+	0.975727i	$0.429723\pi$
$770$		0			0
$771$	158400.	−	56002.9i	0.266469	−	0.0942110i
$772$		0			0
$773$	597784.			1.00043			0.500214	−	0.865902i	$-0.333255\pi$
$773$	597784.			1.00043			0.500214	+	0.865902i	$0.333255\pi$
$774$		0			0
$775$	−896885.			−1.49325
$776$		0			0
$777$	518400.	−	183282.i	0.858663	−	0.303583i
$778$		0			0
$779$	−634752.			−1.04599
$780$		0			0
$781$		−	258883.i		−	0.424426i
$782$		0			0
$783$	−494871.	−	304284.i	−0.807177	−	0.496313i
$784$		0			0
$785$		−	158844.i		−	0.257770i
$786$		0			0
$787$		−	916156.i		−	1.47918i	−0.673060	−	0.739588i	$-0.735020\pi$
$787$		−	916156.i		−	1.47918i	0.673060	−	0.739588i	$-0.264980\pi$
$788$		0			0
$789$	−781411.	+	276271.i	−1.25524	+	0.443793i
$790$		0			0
$791$			131373.i			0.209969i
$792$		0			0
$793$	−103680.			−0.164873
$794$		0			0
$795$	−185760.	−	525409.i	−0.293912	−	0.831310i
$796$		0			0
$797$	33735.2			0.0531088			0.0265544	−	0.999647i	$-0.491546\pi$
$797$	33735.2			0.0531088			0.0265544	+	0.999647i	$0.491546\pi$
$798$		0			0
$799$	612015.			0.958668
$800$		0			0
$801$	461376.	+	570924.i	0.719101	+	0.889842i
$802$		0			0
$803$	−655796.			−1.01704
$804$		0			0
$805$			927342.i			1.43103i
$806$		0			0
$807$	22882.2	+	64720.8i	0.0351359	+	0.0993794i
$808$		0			0
$809$		−	914035.i		−	1.39658i	−0.715815	−	0.698290i	$-0.753945\pi$
$809$		−	914035.i		−	1.39658i	0.715815	−	0.698290i	$-0.246055\pi$
$810$		0			0
$811$			1.20567e6i			1.83311i	0.399911	+	0.916554i	$0.369041\pi$
$811$			1.20567e6i			1.83311i	−0.399911	+	0.916554i	$0.630959\pi$
$812$		0			0
$813$	161997.	+	458197.i	0.245090	+	0.693220i
$814$		0			0
$815$		−	1.70592e6i		−	2.56829i
$816$		0			0
$817$	208800.			0.312814
$818$		0			0
$819$	−622080.	−	769785.i	−0.927424	−	1.14763i
$820$		0			0
$821$	−447741.			−0.664263			−0.332132	−	0.943233i	$-0.607768\pi$
$821$	−447741.			−0.664263			−0.332132	+	0.943233i	$0.607768\pi$
$822$		0			0
$823$	327220.			0.483103			0.241552	−	0.970388i	$-0.422344\pi$
$823$	327220.			0.483103			0.241552	+	0.970388i	$0.422344\pi$
$824$		0			0
$825$	−327630.	−	926678.i	−0.481366	−	1.36151i
$826$		0			0
$827$	44966.0			0.0657466			0.0328733	−	0.999460i	$-0.489534\pi$
$827$	44966.0			0.0657466			0.0328733	+	0.999460i	$0.489534\pi$
$828$		0			0
$829$		−	6761.87i		−	0.00983915i	−0.999988	−	0.00491958i	$-0.998434\pi$
$829$		−	6761.87i		−	0.00983915i	0.999988	−	0.00491958i	$-0.00156596\pi$
$830$		0			0
$831$	−199451.	+	70516.6i	−0.288825	+	0.102115i
$832$		0			0
$833$			228322.i			0.329047i
$834$		0			0
$835$		−	1.53957e6i		−	2.20814i
$836$		0			0
$837$	683394.	+	420202.i	0.975484	+	0.599801i
$838$		0			0
$839$			455299.i			0.646804i	0.946262	+	0.323402i	$0.104827\pi$
$839$			455299.i			0.646804i	−0.946262	+	0.323402i	$0.895173\pi$
$840$		0			0
$841$	−72241.0			−0.102139
$842$		0			0
$843$	−1.00656e6	+	355873.i	−1.41640	+	0.500771i
$844$		0			0
$845$	−2.85057e6			−3.99225
$846$		0			0
$847$	−125795.			−0.175347
$848$		0			0
$849$	1.14350e6	−	404290.i	1.58644	−	0.560890i
$850$		0			0
$851$	1.03680e6			1.43165
$852$		0			0
$853$		−	629498.i		−	0.865160i	−0.901596	−	0.432580i	$-0.857603\pi$
$853$		−	629498.i		−	0.865160i	0.901596	−	0.432580i	$-0.142397\pi$
$854$		0			0
$855$	950808.	+	1.17657e6i	1.30065	+	1.60947i
$856$		0			0
$857$			118522.i			0.161376i	0.996739	+	0.0806880i	$0.0257117\pi$
$857$			118522.i			0.161376i	−0.996739	+	0.0806880i	$0.974288\pi$
$858$		0			0
$859$		−	667944.i		−	0.905220i	−0.891709	−	0.452610i	$-0.850493\pi$
$859$		−	667944.i		−	0.905220i	0.891709	−	0.452610i	$-0.149507\pi$
$860$		0			0
$861$	−415296.	+	146829.i	−0.560210	+	0.198064i
$862$		0			0
$863$			233124.i			0.313015i	0.987677	+	0.156507i	$0.0500235\pi$
$863$			233124.i			0.313015i	−0.987677	+	0.156507i	$0.949977\pi$
$864$		0			0
$865$	882720.			1.17975
$866$		0			0
$867$	81219.0	+	229722.i	0.108049	+	0.305608i
$868$		0			0
$869$	−198313.			−0.262610
$870$		0			0
$871$	−60108.6			−0.0792319
$872$		0			0
$873$	−369054.	+	298241.i	−0.484241	+	0.391326i
$874$		0			0
$875$	−273600.			−0.357355
$876$		0			0
$877$			930884.i			1.21031i	0.796108	+	0.605155i	$0.206889\pi$
$877$			930884.i			1.21031i	−0.796108	+	0.605155i	$0.793111\pi$
$878$		0			0
$879$	103482.	+	292692.i	0.133933	+	0.378821i
$880$		0			0
$881$		−	862444.i		−	1.11117i	−0.831461	−	0.555583i	$-0.812495\pi$
$881$		−	862444.i		−	1.11117i	0.831461	−	0.555583i	$-0.187505\pi$
$882$		0			0
$883$		−	402219.i		−	0.515871i	−0.966162	−	0.257936i	$-0.916958\pi$
$883$		−	402219.i		−	0.515871i	0.966162	−	0.257936i	$-0.0830423\pi$
$884$		0			0
$885$	157330.	+	444995.i	0.200874	+	0.568158i
$886$		0			0
$887$		−	465603.i		−	0.591791i	−0.955220	−	0.295896i	$-0.904382\pi$
$887$		−	465603.i		−	0.591791i	0.955220	−	0.295896i	$-0.0956181\pi$
$888$		0			0
$889$	−810720.			−1.02581
$890$		0			0
$891$	−184518.	+	859593.i	−0.232425	+	1.08277i
$892$		0			0
$893$	1.26774e6			1.58975
$894$		0			0
$895$	−249921.			−0.312002
$896$		0			0
$897$	−622080.	−	1.75951e6i	−0.773146	−	2.18679i
$898$		0			0
$899$	−876960.			−1.08508
$900$		0			0
$901$			387681.i			0.477556i
$902$		0			0
$903$	136610.	−	48299.1i	0.167536	−	0.0592329i
$904$		0			0
$905$		−	1.16079e6i		−	1.41728i
$906$		0			0
$907$			1.04785e6i			1.27375i	0.770968	+	0.636874i	$0.219773\pi$
$907$			1.04785e6i			1.27375i	−0.770968	+	0.636874i	$0.780227\pi$
$908$		0			0
$909$	934757.	−	755397.i	1.13128	−	0.914214i
$910$		0			0
$911$			56670.9i			0.0682847i	0.999417	+	0.0341424i	$0.0108700\pi$
$911$			56670.9i			0.0682847i	−0.999417	+	0.0341424i	$0.989130\pi$
$912$		0			0
$913$	−792476.			−0.950702
$914$		0			0
$915$	103680.	−	36656.4i	0.123838	−	0.0437832i
$916$		0			0
$917$	657931.			0.782423
$918$		0			0
$919$	−887702.			−1.05108			−0.525540	−	0.850769i	$-0.676137\pi$
$919$	−887702.			−1.05108			−0.525540	+	0.850769i	$0.676137\pi$
$920$		0			0
$921$	198576.	−	70207.2i	0.234103	−	0.0827680i
$922$		0			0
$923$	622080.			0.730202
$924$		0			0
$925$			1.31212e6i			1.53353i
$926$		0			0
$927$	−179301.	+	144897.i	−0.208652	+	0.168617i
$928$		0			0
$929$			255068.i			0.295545i	0.989021	+	0.147773i	$0.0472104\pi$
$929$			255068.i			0.295545i	−0.989021	+	0.147773i	$0.952790\pi$
$930$		0			0
$931$			472953.i			0.545655i
$932$		0			0
$933$	169397.	−	59890.8i	0.194600	−	0.0688014i
$934$		0			0
$935$			1.20812e6i			1.38193i
$936$		0			0
$937$	952706.			1.08512			0.542562	−	0.840015i	$-0.317454\pi$
$937$	952706.			1.08512			0.542562	+	0.840015i	$0.317454\pi$
$938$		0			0
$939$	−82374.0	−	232989.i	−0.0934241	−	0.264243i
$940$		0			0
$941$	691438.			0.780862			0.390431	−	0.920632i	$-0.372326\pi$
$941$	691438.			0.780862			0.390431	+	0.920632i	$0.372326\pi$
$942$		0			0
$943$	−830591.			−0.934037
$944$		0			0
$945$	894240.	+	549846.i	1.00136	+	0.615712i
$946$		0			0
$947$	−1.24172e6			−1.38460			−0.692300	−	0.721610i	$-0.743403\pi$
$947$	−1.24172e6			−1.38460			−0.692300	+	0.721610i	$0.743403\pi$
$948$		0			0
$949$		−	1.57584e6i		−	1.74976i
$950$		0			0
$951$	−506255.	−	1.43191e6i	−0.559769	−	1.58326i
$952$		0			0
$953$		−	101552.i		−	0.111816i	−0.998436	−	0.0559078i	$-0.982195\pi$
$953$		−	101552.i		−	0.111816i	0.998436	−	0.0559078i	$-0.0178053\pi$
$954$		0			0
$955$		−	439877.i		−	0.482308i
$956$		0			0
$957$	−320351.	−	906091.i	−0.349786	−	0.989345i
$958$		0			0
$959$			1.04584e6i			1.13717i
$960$		0			0
$961$	287519.			0.311329
$962$		0			0
$963$	1.34026e6	−	1.08310e6i	1.44523	−	1.16792i
$964$		0			0
$965$	−925384.			−0.993727
$966$		0			0
$967$	675121.			0.721986			0.360993	−	0.932569i	$-0.382438\pi$
$967$	675121.			0.721986			0.360993	+	0.932569i	$0.382438\pi$
$968$		0			0
$969$	−350784.	−	992167.i	−0.373587	−	1.05666i
$970$		0			0
$971$	−125894.			−0.133526			−0.0667631	−	0.997769i	$-0.521267\pi$
$971$	−125894.			−0.133526			−0.0667631	+	0.997769i	$0.521267\pi$
$972$		0			0
$973$		−	746382.i		−	0.788379i
$974$		0			0
$975$	2.22675e6	−	787275.i	2.34241	−	0.828165i
$976$		0			0
$977$		−	1.77319e6i		−	1.85766i	−0.370512	−	0.928828i	$-0.620818\pi$
$977$		−	1.77319e6i		−	1.85766i	0.370512	−	0.928828i	$-0.379182\pi$
$978$		0			0
$979$			1.21435e6i			1.26700i
$980$		0			0
$981$	−377045.	−	466569.i	−0.391791	−	0.484817i
$982$		0			0
$983$		−	839116.i		−	0.868390i	−0.900819	−	0.434195i	$-0.857033\pi$
$983$		−	839116.i		−	0.868390i	0.900819	−	0.434195i	$-0.142967\pi$
$984$		0			0
$985$	2.43792e6			2.51274
$986$		0			0
$987$	829440.	−	293251.i	0.851433	−	0.301027i
$988$		0			0
$989$	273221.			0.279332
$990$		0			0
$991$	−1.48909e6			−1.51626			−0.758131	−	0.652103i	$-0.773887\pi$
$991$	−1.48909e6			−1.51626			−0.758131	+	0.652103i	$0.773887\pi$
$992$		0			0
$993$	−670896.	+	237198.i	−0.680388	+	0.240554i
$994$		0			0
$995$	1.29456e6			1.30760
$996$		0			0
$997$		−	1.59290e6i		−	1.60250i	−0.598327	−	0.801252i	$-0.704168\pi$
$997$		−	1.59290e6i		−	1.60250i	0.598327	−	0.801252i	$-0.295832\pi$
$998$		0			0
$999$	614747.	−	999791.i	0.615978	−	1.00179i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.5.h.f.65.4	yes	4
3.2	odd	2		384.5.h.e.65.3	yes	4
4.3	odd	2		384.5.h.e.65.2	yes	4
8.3	odd	2	inner	384.5.h.f.65.3	yes	4
8.5	even	2		384.5.h.e.65.1	✓	4
12.11	even	2	inner	384.5.h.f.65.1	yes	4
24.5	odd	2	inner	384.5.h.f.65.2	yes	4
24.11	even	2		384.5.h.e.65.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.5.h.e.65.1	✓	4	8.5	even	2
384.5.h.e.65.2	yes	4	4.3	odd	2
384.5.h.e.65.3	yes	4	3.2	odd	2
384.5.h.e.65.4	yes	4	24.11	even	2
384.5.h.f.65.1	yes	4	12.11	even	2	inner
384.5.h.f.65.2	yes	4	24.5	odd	2	inner
384.5.h.f.65.3	yes	4	8.3	odd	2	inner
384.5.h.f.65.4	yes	4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+(3.00000 + 8.48528i) q^{3} +37.9473 q^{5} -37.9473 q^{7} +(-63.0000 + 50.9117i) q^{9} +O(q^{10})\)	\(q+(3.00000 + 8.48528i) q^{3} +37.9473 q^{5} -37.9473 q^{7} +(-63.0000 + 50.9117i) q^{9} -134.000 q^{11} -321.994i q^{13} +(113.842 + 321.994i) q^{15} -237.588i q^{17} -492.146i q^{19} +(-113.842 - 321.994i) q^{21} -643.988i q^{23} +815.000 q^{25} +(-621.000 - 381.838i) q^{27} +796.894 q^{29} -1100.47 q^{31} +(-402.000 - 1137.03i) q^{33} -1440.00 q^{35} +1609.97i q^{37} +(2732.21 - 965.981i) q^{39} -1289.76i q^{41} +424.264i q^{43} +(-2390.68 + 1931.96i) q^{45} +2575.95i q^{47} -961.000 q^{49} +(2016.00 - 712.764i) q^{51} -1631.74 q^{53} -5084.94 q^{55} +(4176.00 - 1476.44i) q^{57} +1382.00 q^{59} -321.994i q^{61} +(2390.68 - 1931.96i) q^{63} -12218.8i q^{65} -186.676i q^{67} +(5464.42 - 1931.96i) q^{69} +1931.96i q^{71} +4894.00 q^{73} +(2445.00 + 6915.50i) q^{75} +5084.94 q^{77} +1479.95 q^{79} +(1377.00 - 6414.87i) q^{81} +5914.00 q^{83} -9015.83i q^{85} +(2390.68 + 6761.87i) q^{87} -9062.28i q^{89} +12218.8i q^{91} +(-3301.42 - 9337.82i) q^{93} -18675.6i q^{95} +5858.00 q^{97} +(8442.00 - 6822.17i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{3} - 252 q^{9}+O(q^{10}) \) 4 * q + 12 * q^3 - 252 * q^9	\( 4 q + 12 q^{3} - 252 q^{9} - 536 q^{11} + 3260 q^{25} - 2484 q^{27} - 1608 q^{33} - 5760 q^{35} - 3844 q^{49} + 8064 q^{51} + 16704 q^{57} + 5528 q^{59} + 19576 q^{73} + 9780 q^{75} + 5508 q^{81} + 23656 q^{83} + 23432 q^{97} + 33768 q^{99}+O(q^{100}) \) 4 * q + 12 * q^3 - 252 * q^9 - 536 * q^11 + 3260 * q^25 - 2484 * q^27 - 1608 * q^33 - 5760 * q^35 - 3844 * q^49 + 8064 * q^51 + 16704 * q^57 + 5528 * q^59 + 19576 * q^73 + 9780 * q^75 + 5508 * q^81 + 23656 * q^83 + 23432 * q^97 + 33768 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

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