LMFDB - Embedded newform 384.3.g.b.127.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,3,Mod(127,384)]

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, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("384.127");

S:= CuspForms(chi, 3);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$10.4632421514$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{18} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.8
Root			$0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	384.127
Dual form			384.3.g.b.127.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} +8.29253 q^{5} +8.55583i q^{7} -3.00000 q^{9} +O(q^{10})$	$q+1.73205i q^{3} +8.29253 q^{5} +8.55583i q^{7} -3.00000 q^{9} +13.7980i q^{11} -17.0693 q^{13} +14.3631i q^{15} +20.3246 q^{17} -20.4440i q^{19} -14.8191 q^{21} -5.51575i q^{23} +43.7660 q^{25} -5.19615i q^{27} -41.0586 q^{29} +22.2953i q^{31} -23.8988 q^{33} +70.9495i q^{35} +11.6326 q^{37} -29.5649i q^{39} +35.9527 q^{41} +66.8648i q^{43} -24.8776 q^{45} +19.9366i q^{47} -24.2023 q^{49} +35.2032i q^{51} -17.6329 q^{53} +114.420i q^{55} +35.4100 q^{57} -62.1572i q^{59} +47.4836 q^{61} -25.6675i q^{63} -141.548 q^{65} -74.8105i q^{67} +9.55356 q^{69} +16.9150i q^{71} +101.712 q^{73} +75.8050i q^{75} -118.053 q^{77} +0.879320i q^{79} +9.00000 q^{81} -23.2131i q^{83} +168.542 q^{85} -71.1155i q^{87} -16.3089 q^{89} -146.042i q^{91} -38.6167 q^{93} -169.532i q^{95} +188.307 q^{97} -41.3939i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{5} - 24 q^{9}+O(q^{10}) $ 8 * q + 16 * q^5 - 24 * q^9	$ 8 q + 16 q^{5} - 24 q^{9} - 48 q^{13} + 16 q^{17} - 8 q^{25} + 80 q^{29} + 16 q^{37} + 80 q^{41} - 48 q^{45} - 88 q^{49} - 176 q^{53} + 96 q^{57} + 272 q^{61} - 160 q^{65} - 16 q^{73} - 320 q^{77} + 72 q^{81} - 32 q^{85} - 240 q^{89} + 192 q^{93} + 400 q^{97}+O(q^{100}) $ 8 * q + 16 * q^5 - 24 * q^9 - 48 * q^13 + 16 * q^17 - 8 * q^25 + 80 * q^29 + 16 * q^37 + 80 * q^41 - 48 * q^45 - 88 * q^49 - 176 * q^53 + 96 * q^57 + 272 * q^61 - 160 * q^65 - 16 * q^73 - 320 * q^77 + 72 * q^81 - 32 * q^85 - 240 * q^89 + 192 * q^93 + 400 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.73205i	0.577350i
$3$	1.73205i	0.577350i
$4$	0	0
$5$	8.29253	1.65851	0.829253	−	0.558874i	$-0.188766\pi$
$5$	8.29253	1.65851	0.829253	+	0.558874i	$0.188766\pi$
$6$	0	0
$7$	8.55583i	1.22226i	0.791529	+	0.611131i	$0.209285\pi$
$7$	8.55583i	1.22226i	−0.791529	+	0.611131i	$0.790715\pi$
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	13.7980i	1.25436i	0.778874	+	0.627180i	$0.215791\pi$
$11$	13.7980i	1.25436i	−0.778874	+	0.627180i	$0.784209\pi$
$12$	0	0
$13$	−17.0693	−1.31302	−0.656512	−	0.754316i	$-0.727969\pi$
$13$	−17.0693	−1.31302	−0.656512	+	0.754316i	$0.727969\pi$
$14$	0	0
$15$	14.3631i	0.957539i
$16$	0	0
$17$	20.3246	1.19556	0.597781	−	0.801659i	$-0.296049\pi$
$17$	20.3246	1.19556	0.597781	+	0.801659i	$0.296049\pi$
$18$	0	0
$19$	− 20.4440i	− 1.07600i	−0.842946	−	0.537999i	$-0.819181\pi$
$19$	− 20.4440i	− 1.07600i	0.842946	−	0.537999i	$-0.180819\pi$
$20$	0	0
$21$	−14.8191	−0.705673
$22$	0	0
$23$	− 5.51575i	− 0.239815i	−0.992785	−	0.119908i	$-0.961740\pi$
$23$	− 5.51575i	− 0.239815i	0.992785	−	0.119908i	$-0.0382598\pi$
$24$	0	0
$25$	43.7660	1.75064
$26$	0	0
$27$	− 5.19615i	− 0.192450i
$28$	0	0
$29$	−41.0586	−1.41581	−0.707906	−	0.706306i	$-0.750360\pi$
$29$	−41.0586	−1.41581	−0.707906	+	0.706306i	$0.750360\pi$
$30$	0	0
$31$	22.2953i	0.719205i	0.933106	+	0.359602i	$0.117088\pi$
$31$	22.2953i	0.719205i	−0.933106	+	0.359602i	$0.882912\pi$
$32$	0	0
$33$	−23.8988	−0.724205
$34$	0	0
$35$	70.9495i	2.02713i
$36$	0	0
$37$	11.6326	0.314396	0.157198	−	0.987567i	$-0.449754\pi$
$37$	11.6326	0.314396	0.157198	+	0.987567i	$0.449754\pi$
$38$	0	0
$39$	− 29.5649i	− 0.758075i
$40$	0	0
$41$	35.9527	0.876894	0.438447	−	0.898757i	$-0.355529\pi$
$41$	35.9527	0.876894	0.438447	+	0.898757i	$0.355529\pi$
$42$	0	0
$43$	66.8648i	1.55499i	0.628886	+	0.777497i	$0.283511\pi$
$43$	66.8648i	1.55499i	−0.628886	+	0.777497i	$0.716489\pi$
$44$	0	0
$45$	−24.8776	−0.552835
$46$	0	0
$47$	19.9366i	0.424182i	0.977250	+	0.212091i	$0.0680274\pi$
$47$	19.9366i	0.424182i	−0.977250	+	0.212091i	$0.931973\pi$
$48$	0	0
$49$	−24.2023	−0.493924
$50$	0	0
$51$	35.2032i	0.690259i
$52$	0	0
$53$	−17.6329	−0.332697	−0.166348	−	0.986067i	$-0.553198\pi$
$53$	−17.6329	−0.332697	−0.166348	+	0.986067i	$0.553198\pi$
$54$	0	0
$55$	114.420i	2.08036i
$56$	0	0
$57$	35.4100	0.621227
$58$	0	0
$59$	− 62.1572i	− 1.05351i	−0.850017	−	0.526756i	$-0.823408\pi$
$59$	− 62.1572i	− 1.05351i	0.850017	−	0.526756i	$-0.176592\pi$
$60$	0	0
$61$	47.4836	0.778419	0.389210	−	0.921149i	$-0.372748\pi$
$61$	47.4836	0.778419	0.389210	+	0.921149i	$0.372748\pi$
$62$	0	0
$63$	− 25.6675i	− 0.407421i
$64$	0	0
$65$	−141.548	−2.17766
$66$	0	0
$67$	− 74.8105i	− 1.11657i	−0.829648	−	0.558287i	$-0.811459\pi$
$67$	− 74.8105i	− 1.11657i	0.829648	−	0.558287i	$-0.188541\pi$
$68$	0	0
$69$	9.55356	0.138457
$70$	0	0
$71$	16.9150i	0.238240i	0.992880	+	0.119120i	$0.0380073\pi$
$71$	16.9150i	0.238240i	−0.992880	+	0.119120i	$0.961993\pi$
$72$	0	0
$73$	101.712	1.39331	0.696657	−	0.717404i	$-0.254670\pi$
$73$	101.712	1.39331	0.696657	+	0.717404i	$0.254670\pi$
$74$	0	0
$75$	75.8050i	1.01073i
$76$	0	0
$77$	−118.053	−1.53316
$78$	0	0
$79$	0.879320i	0.0111306i	0.999985	+	0.00556532i	$0.00177151\pi$
$79$	0.879320i	0.0111306i	−0.999985	+	0.00556532i	$0.998228\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	− 23.2131i	− 0.279676i	−0.990174	−	0.139838i	$-0.955342\pi$
$83$	− 23.2131i	− 0.279676i	0.990174	−	0.139838i	$-0.0446582\pi$
$84$	0	0
$85$	168.542	1.98285
$86$	0	0
$87$	− 71.1155i	− 0.817420i
$88$	0	0
$89$	−16.3089	−0.183246	−0.0916231	−	0.995794i	$-0.529205\pi$
$89$	−16.3089	−0.183246	−0.0916231	+	0.995794i	$0.529205\pi$
$90$	0	0
$91$	− 146.042i	− 1.60486i
$92$	0	0
$93$	−38.6167	−0.415233
$94$	0	0
$95$	− 169.532i	− 1.78455i
$96$	0	0
$97$	188.307	1.94131	0.970656	−	0.240474i	$-0.0773028\pi$
$97$	188.307	1.94131	0.970656	+	0.240474i	$0.0773028\pi$
$98$	0	0
$99$	− 41.3939i	− 0.418120i
$100$	0	0
$101$	66.8468	0.661849	0.330925	−	0.943657i	$-0.392639\pi$
$101$	66.8468	0.661849	0.330925	+	0.943657i	$0.392639\pi$
$102$	0	0
$103$	− 166.313i	− 1.61469i	−0.590083	−	0.807343i	$-0.700905\pi$
$103$	− 166.313i	− 1.61469i	0.590083	−	0.807343i	$-0.299095\pi$
$104$	0	0
$105$	−122.888	−1.17036
$106$	0	0
$107$	− 80.1467i	− 0.749035i	−0.927220	−	0.374517i	$-0.877808\pi$
$107$	− 80.1467i	− 0.749035i	0.927220	−	0.374517i	$-0.122192\pi$
$108$	0	0
$109$	−26.8781	−0.246589	−0.123294	−	0.992370i	$-0.539346\pi$
$109$	−26.8781	−0.246589	−0.123294	+	0.992370i	$0.539346\pi$
$110$	0	0
$111$	20.1483i	0.181516i
$112$	0	0
$113$	−79.0958	−0.699963	−0.349981	−	0.936757i	$-0.613812\pi$
$113$	−79.0958	−0.699963	−0.349981	+	0.936757i	$0.613812\pi$
$114$	0	0
$115$	− 45.7395i	− 0.397735i
$116$	0	0
$117$	51.2079	0.437675
$118$	0	0
$119$	173.894i	1.46129i
$120$	0	0
$121$	−69.3837	−0.573419
$122$	0	0
$123$	62.2718i	0.506275i
$124$	0	0
$125$	155.618	1.24494
$126$	0	0
$127$	170.725i	1.34429i	0.740419	+	0.672145i	$0.234627\pi$
$127$	170.725i	1.34429i	−0.740419	+	0.672145i	$0.765373\pi$
$128$	0	0
$129$	−115.813	−0.897777
$130$	0	0
$131$	178.980i	1.36626i	0.730297	+	0.683129i	$0.239381\pi$
$131$	178.980i	1.36626i	−0.730297	+	0.683129i	$0.760619\pi$
$132$	0	0
$133$	174.915	1.31515
$134$	0	0
$135$	− 43.0892i	− 0.319180i
$136$	0	0
$137$	−93.2075	−0.680347	−0.340173	−	0.940363i	$-0.610486\pi$
$137$	−93.2075	−0.680347	−0.340173	+	0.940363i	$0.610486\pi$
$138$	0	0
$139$	− 222.476i	− 1.60055i	−0.599635	−	0.800274i	$-0.704687\pi$
$139$	− 222.476i	− 1.60055i	0.599635	−	0.800274i	$-0.295313\pi$
$140$	0	0
$141$	−34.5312	−0.244902
$142$	0	0
$143$	− 235.522i	− 1.64700i
$144$	0	0
$145$	−340.479	−2.34813
$146$	0	0
$147$	− 41.9196i	− 0.285167i
$148$	0	0
$149$	−94.0043	−0.630901	−0.315451	−	0.948942i	$-0.602156\pi$
$149$	−94.0043	−0.630901	−0.315451	+	0.948942i	$0.602156\pi$
$150$	0	0
$151$	− 82.0081i	− 0.543100i	−0.962424	−	0.271550i	$-0.912464\pi$
$151$	− 82.0081i	− 0.543100i	0.962424	−	0.271550i	$-0.0875362\pi$
$152$	0	0
$153$	−60.9737	−0.398521
$154$	0	0
$155$	184.885i	1.19281i
$156$	0	0
$157$	56.7180	0.361261	0.180631	−	0.983551i	$-0.442186\pi$
$157$	56.7180	0.361261	0.180631	+	0.983551i	$0.442186\pi$
$158$	0	0
$159$	− 30.5411i	− 0.192083i
$160$	0	0
$161$	47.1918	0.293117
$162$	0	0
$163$	− 189.588i	− 1.16312i	−0.813504	−	0.581559i	$-0.802443\pi$
$163$	− 189.588i	− 1.16312i	0.813504	−	0.581559i	$-0.197557\pi$
$164$	0	0
$165$	−198.181	−1.20110
$166$	0	0
$167$	− 312.777i	− 1.87291i	−0.350783	−	0.936457i	$-0.614084\pi$
$167$	− 312.777i	− 1.87291i	0.350783	−	0.936457i	$-0.385916\pi$
$168$	0	0
$169$	122.361	0.724031
$170$	0	0
$171$	61.3319i	0.358666i
$172$	0	0
$173$	182.538	1.05513	0.527567	−	0.849514i	$-0.323104\pi$
$173$	182.538	1.05513	0.527567	+	0.849514i	$0.323104\pi$
$174$	0	0
$175$	374.455i	2.13974i
$176$	0	0
$177$	107.659	0.608245
$178$	0	0
$179$	− 19.7781i	− 0.110492i	−0.998473	−	0.0552461i	$-0.982406\pi$
$179$	− 19.7781i	− 0.110492i	0.998473	−	0.0552461i	$-0.0175943\pi$
$180$	0	0
$181$	−265.750	−1.46823	−0.734117	−	0.679023i	$-0.762403\pi$
$181$	−265.750	−1.46823	−0.734117	+	0.679023i	$0.762403\pi$
$182$	0	0
$183$	82.2440i	0.449421i
$184$	0	0
$185$	96.4640	0.521427
$186$	0	0
$187$	280.438i	1.49967i
$188$	0	0
$189$	44.4574	0.235224
$190$	0	0
$191$	− 288.025i	− 1.50799i	−0.656882	−	0.753993i	$-0.728125\pi$
$191$	− 288.025i	− 1.50799i	0.656882	−	0.753993i	$-0.271875\pi$
$192$	0	0
$193$	281.010	1.45601	0.728005	−	0.685572i	$-0.240448\pi$
$193$	281.010	1.45601	0.728005	+	0.685572i	$0.240448\pi$
$194$	0	0
$195$	− 245.168i	− 1.25727i
$196$	0	0
$197$	33.8330	0.171741	0.0858705	−	0.996306i	$-0.472633\pi$
$197$	33.8330	0.171741	0.0858705	+	0.996306i	$0.472633\pi$
$198$	0	0
$199$	− 84.9700i	− 0.426985i	−0.976945	−	0.213493i	$-0.931516\pi$
$199$	− 84.9700i	− 0.426985i	0.976945	−	0.213493i	$-0.0684839\pi$
$200$	0	0
$201$	129.576	0.644654
$202$	0	0
$203$	− 351.290i	− 1.73049i
$204$	0	0
$205$	298.138	1.45433
$206$	0	0
$207$	16.5472i	0.0799384i
$208$	0	0
$209$	282.085	1.34969
$210$	0	0
$211$	140.700i	0.666826i	0.942781	+	0.333413i	$0.108200\pi$
$211$	140.700i	0.666826i	−0.942781	+	0.333413i	$0.891800\pi$
$212$	0	0
$213$	−29.2977	−0.137548
$214$	0	0
$215$	554.478i	2.57897i
$216$	0	0
$217$	−190.755	−0.879057
$218$	0	0
$219$	176.170i	0.804431i
$220$	0	0
$221$	−346.926	−1.56980
$222$	0	0
$223$	− 247.529i	− 1.11000i	−0.831851	−	0.554999i	$-0.812719\pi$
$223$	− 247.529i	− 1.11000i	0.831851	−	0.554999i	$-0.187281\pi$
$224$	0	0
$225$	−131.298	−0.583547
$226$	0	0
$227$	129.752i	0.571593i	0.958290	+	0.285797i	$0.0922582\pi$
$227$	129.752i	0.571593i	−0.958290	+	0.285797i	$0.907742\pi$
$228$	0	0
$229$	−294.304	−1.28517	−0.642586	−	0.766214i	$-0.722138\pi$
$229$	−294.304	−1.28517	−0.642586	+	0.766214i	$0.722138\pi$
$230$	0	0
$231$	− 204.474i	− 0.885168i
$232$	0	0
$233$	−239.948	−1.02982	−0.514911	−	0.857244i	$-0.672175\pi$
$233$	−239.948	−1.02982	−0.514911	+	0.857244i	$0.672175\pi$
$234$	0	0
$235$	165.325i	0.703509i
$236$	0	0
$237$	−1.52303	−0.00642628
$238$	0	0
$239$	425.346i	1.77969i	0.456261	+	0.889846i	$0.349188\pi$
$239$	425.346i	1.77969i	−0.456261	+	0.889846i	$0.650812\pi$
$240$	0	0
$241$	264.493	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$241$	264.493	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$242$	0	0
$243$	15.5885i	0.0641500i
$244$	0	0
$245$	−200.698	−0.819176
$246$	0	0
$247$	348.964i	1.41281i
$248$	0	0
$249$	40.2063	0.161471
$250$	0	0
$251$	− 221.125i	− 0.880976i	−0.897759	−	0.440488i	$-0.854805\pi$
$251$	− 221.125i	− 0.880976i	0.897759	−	0.440488i	$-0.145195\pi$
$252$	0	0
$253$	76.1061	0.300815
$254$	0	0
$255$	291.923i	1.14480i
$256$	0	0
$257$	87.0655	0.338776	0.169388	−	0.985549i	$-0.445821\pi$
$257$	87.0655	0.338776	0.169388	+	0.985549i	$0.445821\pi$
$258$	0	0
$259$	99.5270i	0.384274i
$260$	0	0
$261$	123.176	0.471937
$262$	0	0
$263$	277.324i	1.05447i	0.849721	+	0.527233i	$0.176770\pi$
$263$	277.324i	1.05447i	−0.849721	+	0.527233i	$0.823230\pi$
$264$	0	0
$265$	−146.222	−0.551780
$266$	0	0
$267$	− 28.2479i	− 0.105797i
$268$	0	0
$269$	49.8005	0.185132	0.0925659	−	0.995707i	$-0.470493\pi$
$269$	49.8005	0.185132	0.0925659	+	0.995707i	$0.470493\pi$
$270$	0	0
$271$	31.3145i	0.115552i	0.998330	+	0.0577758i	$0.0184009\pi$
$271$	31.3145i	0.115552i	−0.998330	+	0.0577758i	$0.981599\pi$
$272$	0	0
$273$	252.952	0.926566
$274$	0	0
$275$	603.882i	2.19593i
$276$	0	0
$277$	79.1431	0.285715	0.142858	−	0.989743i	$-0.454371\pi$
$277$	79.1431	0.285715	0.142858	+	0.989743i	$0.454371\pi$
$278$	0	0
$279$	− 66.8860i	− 0.239735i
$280$	0	0
$281$	−106.190	−0.377902	−0.188951	−	0.981987i	$-0.560509\pi$
$281$	−106.190	−0.377902	−0.188951	+	0.981987i	$0.560509\pi$
$282$	0	0
$283$	270.251i	0.954949i	0.878646	+	0.477475i	$0.158448\pi$
$283$	270.251i	0.954949i	−0.878646	+	0.477475i	$0.841552\pi$
$284$	0	0
$285$	293.638	1.03031
$286$	0	0
$287$	307.605i	1.07179i
$288$	0	0
$289$	124.088	0.429371
$290$	0	0
$291$	326.158i	1.12082i
$292$	0	0
$293$	−24.2487	−0.0827600	−0.0413800	−	0.999143i	$-0.513175\pi$
$293$	−24.2487	−0.0827600	−0.0413800	+	0.999143i	$0.513175\pi$
$294$	0	0
$295$	− 515.440i	− 1.74726i
$296$	0	0
$297$	71.6963	0.241402
$298$	0	0
$299$	94.1500i	0.314883i
$300$	0	0
$301$	−572.084	−1.90061
$302$	0	0
$303$	115.782i	0.382119i
$304$	0	0
$305$	393.759	1.29101
$306$	0	0
$307$	− 122.865i	− 0.400211i	−0.979774	−	0.200106i	$-0.935871\pi$
$307$	− 122.865i	− 0.400211i	0.979774	−	0.200106i	$-0.0641285\pi$
$308$	0	0
$309$	288.062	0.932239
$310$	0	0
$311$	108.808i	0.349865i	0.984580	+	0.174933i	$0.0559707\pi$
$311$	108.808i	0.349865i	−0.984580	+	0.174933i	$0.944029\pi$
$312$	0	0
$313$	−52.2216	−0.166842	−0.0834211	−	0.996514i	$-0.526585\pi$
$313$	−52.2216	−0.166842	−0.0834211	+	0.996514i	$0.526585\pi$
$314$	0	0
$315$	− 212.848i	− 0.675709i
$316$	0	0
$317$	−94.3251	−0.297555	−0.148778	−	0.988871i	$-0.547534\pi$
$317$	−94.3251	−0.297555	−0.148778	+	0.988871i	$0.547534\pi$
$318$	0	0
$319$	− 566.524i	− 1.77594i
$320$	0	0
$321$	138.818	0.432456
$322$	0	0
$323$	− 415.515i	− 1.28642i
$324$	0	0
$325$	−747.056	−2.29863
$326$	0	0
$327$	− 46.5543i	− 0.142368i
$328$	0	0
$329$	−170.574	−0.518462
$330$	0	0
$331$	406.107i	1.22691i	0.789730	+	0.613454i	$0.210220\pi$
$331$	406.107i	1.22691i	−0.789730	+	0.613454i	$0.789780\pi$
$332$	0	0
$333$	−34.8979	−0.104799
$334$	0	0
$335$	− 620.368i	− 1.85184i
$336$	0	0
$337$	−22.4140	−0.0665105	−0.0332552	−	0.999447i	$-0.510587\pi$
$337$	−22.4140	−0.0665105	−0.0332552	+	0.999447i	$0.510587\pi$
$338$	0	0
$339$	− 136.998i	− 0.404124i
$340$	0	0
$341$	−307.630	−0.902142
$342$	0	0
$343$	212.165i	0.618557i
$344$	0	0
$345$	79.2232	0.229632
$346$	0	0
$347$	− 458.377i	− 1.32097i	−0.750839	−	0.660486i	$-0.770350\pi$
$347$	− 458.377i	− 1.32097i	0.750839	−	0.660486i	$-0.229650\pi$
$348$	0	0
$349$	−282.960	−0.810774	−0.405387	−	0.914145i	$-0.632863\pi$
$349$	−282.960	−0.810774	−0.405387	+	0.914145i	$0.632863\pi$
$350$	0	0
$351$	88.6947i	0.252692i
$352$	0	0
$353$	−175.820	−0.498073	−0.249036	−	0.968494i	$-0.580114\pi$
$353$	−175.820	−0.498073	−0.249036	+	0.968494i	$0.580114\pi$
$354$	0	0
$355$	140.268i	0.395122i
$356$	0	0
$357$	−301.193	−0.843677
$358$	0	0
$359$	− 53.5791i	− 0.149246i	−0.997212	−	0.0746228i	$-0.976225\pi$
$359$	− 53.5791i	− 0.149246i	0.997212	−	0.0746228i	$-0.0237753\pi$
$360$	0	0
$361$	−56.9552	−0.157771
$362$	0	0
$363$	− 120.176i	− 0.331063i
$364$	0	0
$365$	843.449	2.31082
$366$	0	0
$367$	62.4258i	0.170097i	0.996377	+	0.0850487i	$0.0271046\pi$
$367$	62.4258i	0.170097i	−0.996377	+	0.0850487i	$0.972895\pi$
$368$	0	0
$369$	−107.858	−0.292298
$370$	0	0
$371$	− 150.864i	− 0.406643i
$372$	0	0
$373$	370.552	0.993437	0.496719	−	0.867912i	$-0.334538\pi$
$373$	370.552	0.993437	0.496719	+	0.867912i	$0.334538\pi$
$374$	0	0
$375$	269.538i	0.718768i
$376$	0	0
$377$	700.841	1.85900
$378$	0	0
$379$	68.7286i	0.181342i	0.995881	+	0.0906709i	$0.0289012\pi$
$379$	68.7286i	0.181342i	−0.995881	+	0.0906709i	$0.971099\pi$
$380$	0	0
$381$	−295.704	−0.776126
$382$	0	0
$383$	− 155.858i	− 0.406939i	−0.979081	−	0.203470i	$-0.934778\pi$
$383$	− 155.858i	− 0.406939i	0.979081	−	0.203470i	$-0.0652218\pi$
$384$	0	0
$385$	−978.958	−2.54275
$386$	0	0
$387$	− 200.594i	− 0.518332i
$388$	0	0
$389$	481.792	1.23854	0.619269	−	0.785179i	$-0.287429\pi$
$389$	481.792	1.23854	0.619269	+	0.785179i	$0.287429\pi$
$390$	0	0
$391$	− 112.105i	− 0.286714i
$392$	0	0
$393$	−310.002	−0.788810
$394$	0	0
$395$	7.29179i	0.0184602i
$396$	0	0
$397$	422.315	1.06377	0.531883	−	0.846818i	$-0.321485\pi$
$397$	422.315	1.06377	0.531883	+	0.846818i	$0.321485\pi$
$398$	0	0
$399$	302.962i	0.759303i
$400$	0	0
$401$	−641.025	−1.59857	−0.799283	−	0.600955i	$-0.794787\pi$
$401$	−641.025	−1.59857	−0.799283	+	0.600955i	$0.794787\pi$
$402$	0	0
$403$	− 380.566i	− 0.944333i
$404$	0	0
$405$	74.6328	0.184278
$406$	0	0
$407$	160.507i	0.394365i
$408$	0	0
$409$	−278.562	−0.681081	−0.340541	−	0.940230i	$-0.610610\pi$
$409$	−278.562	−0.681081	−0.340541	+	0.940230i	$0.610610\pi$
$410$	0	0
$411$	− 161.440i	− 0.392798i
$412$	0	0
$413$	531.807	1.28767
$414$	0	0
$415$	− 192.496i	− 0.463845i
$416$	0	0
$417$	385.340	0.924077
$418$	0	0
$419$	505.611i	1.20671i	0.797473	+	0.603354i	$0.206169\pi$
$419$	505.611i	1.20671i	−0.797473	+	0.603354i	$0.793831\pi$
$420$	0	0
$421$	179.846	0.427189	0.213594	−	0.976922i	$-0.431483\pi$
$421$	179.846	0.427189	0.213594	+	0.976922i	$0.431483\pi$
$422$	0	0
$423$	− 59.8097i	− 0.141394i
$424$	0	0
$425$	889.526	2.09300
$426$	0	0
$427$	406.262i	0.951432i
$428$	0	0
$429$	407.935	0.950898
$430$	0	0
$431$	580.645i	1.34720i	0.739094	+	0.673602i	$0.235254\pi$
$431$	580.645i	1.34720i	−0.739094	+	0.673602i	$0.764746\pi$
$432$	0	0
$433$	−425.621	−0.982959	−0.491479	−	0.870889i	$-0.663544\pi$
$433$	−425.621	−0.982959	−0.491479	+	0.870889i	$0.663544\pi$
$434$	0	0
$435$	− 589.727i	− 1.35570i
$436$	0	0
$437$	−112.764	−0.258041
$438$	0	0
$439$	− 129.991i	− 0.296107i	−0.988979	−	0.148053i	$-0.952699\pi$
$439$	− 129.991i	− 0.296107i	0.988979	−	0.148053i	$-0.0473008\pi$
$440$	0	0
$441$	72.6069	0.164641
$442$	0	0
$443$	− 205.564i	− 0.464027i	−0.972713	−	0.232014i	$-0.925469\pi$
$443$	− 205.564i	− 0.464027i	0.972713	−	0.232014i	$-0.0745314\pi$
$444$	0	0
$445$	−135.242	−0.303915
$446$	0	0
$447$	− 162.820i	− 0.364251i
$448$	0	0
$449$	415.190	0.924698	0.462349	−	0.886698i	$-0.347007\pi$
$449$	415.190	0.924698	0.462349	+	0.886698i	$0.347007\pi$
$450$	0	0
$451$	496.073i	1.09994i
$452$	0	0
$453$	142.042	0.313559
$454$	0	0
$455$	− 1211.06i	− 2.66167i
$456$	0	0
$457$	−598.927	−1.31056	−0.655281	−	0.755385i	$-0.727450\pi$
$457$	−598.927	−1.31056	−0.655281	+	0.755385i	$0.727450\pi$
$458$	0	0
$459$	− 105.610i	− 0.230086i
$460$	0	0
$461$	−376.804	−0.817361	−0.408681	−	0.912677i	$-0.634011\pi$
$461$	−376.804	−0.817361	−0.408681	+	0.912677i	$0.634011\pi$
$462$	0	0
$463$	− 218.577i	− 0.472088i	−0.971742	−	0.236044i	$-0.924149\pi$
$463$	− 218.577i	− 0.472088i	0.971742	−	0.236044i	$-0.0758510\pi$
$464$	0	0
$465$	−320.230	−0.688666
$466$	0	0
$467$	− 443.608i	− 0.949911i	−0.880010	−	0.474955i	$-0.842464\pi$
$467$	− 443.608i	− 0.949911i	0.880010	−	0.474955i	$-0.157536\pi$
$468$	0	0
$469$	640.066	1.36475
$470$	0	0
$471$	98.2385i	0.208574i
$472$	0	0
$473$	−922.597	−1.95052
$474$	0	0
$475$	− 894.751i	− 1.88369i
$476$	0	0
$477$	52.8988	0.110899
$478$	0	0
$479$	− 222.133i	− 0.463743i	−0.972746	−	0.231871i	$-0.925515\pi$
$479$	− 222.133i	− 0.463743i	0.972746	−	0.231871i	$-0.0744849\pi$
$480$	0	0
$481$	−198.561	−0.412809
$482$	0	0
$483$	81.7387i	0.169231i
$484$	0	0
$485$	1561.54	3.21968
$486$	0	0
$487$	− 403.795i	− 0.829148i	−0.910016	−	0.414574i	$-0.863931\pi$
$487$	− 403.795i	− 0.829148i	0.910016	−	0.414574i	$-0.136069\pi$
$488$	0	0
$489$	328.376	0.671526
$490$	0	0
$491$	808.347i	1.64633i	0.567803	+	0.823164i	$0.307793\pi$
$491$	808.347i	1.64633i	−0.567803	+	0.823164i	$0.692207\pi$
$492$	0	0
$493$	−834.498	−1.69269
$494$	0	0
$495$	− 343.260i	− 0.693454i
$496$	0	0
$497$	−144.722	−0.291192
$498$	0	0
$499$	− 132.172i	− 0.264874i	−0.991191	−	0.132437i	$-0.957720\pi$
$499$	− 132.172i	− 0.264874i	0.991191	−	0.132437i	$-0.0422802\pi$
$500$	0	0
$501$	541.745	1.08133
$502$	0	0
$503$	− 92.8360i	− 0.184565i	−0.995733	−	0.0922823i	$-0.970584\pi$
$503$	− 92.8360i	− 0.184565i	0.995733	−	0.0922823i	$-0.0294162\pi$
$504$	0	0
$505$	554.329	1.09768
$506$	0	0
$507$	211.936i	0.418020i
$508$	0	0
$509$	−927.417	−1.82204	−0.911018	−	0.412366i	$-0.864703\pi$
$509$	−927.417	−1.82204	−0.911018	+	0.412366i	$0.864703\pi$
$510$	0	0
$511$	870.231i	1.70300i
$512$	0	0
$513$	−106.230	−0.207076
$514$	0	0
$515$	− 1379.15i	− 2.67796i
$516$	0	0
$517$	−275.084	−0.532077
$518$	0	0
$519$	316.165i	0.609182i
$520$	0	0
$521$	527.145	1.01179	0.505897	−	0.862594i	$-0.331162\pi$
$521$	527.145	1.01179	0.505897	+	0.862594i	$0.331162\pi$
$522$	0	0
$523$	− 972.249i	− 1.85898i	−0.368843	−	0.929492i	$-0.620246\pi$
$523$	− 972.249i	− 1.85898i	0.368843	−	0.929492i	$-0.379754\pi$
$524$	0	0
$525$	−648.575	−1.23538
$526$	0	0
$527$	453.143i	0.859854i
$528$	0	0
$529$	498.577	0.942489
$530$	0	0
$531$	186.472i	0.351171i
$532$	0	0
$533$	−613.687	−1.15138
$534$	0	0
$535$	− 664.619i	− 1.24228i
$536$	0	0
$537$	34.2567	0.0637927
$538$	0	0
$539$	− 333.942i	− 0.619559i
$540$	0	0
$541$	−4.66591	−0.00862461	−0.00431231	−	0.999991i	$-0.501373\pi$
$541$	−4.66591	−0.00862461	−0.00431231	+	0.999991i	$0.501373\pi$
$542$	0	0
$543$	− 460.293i	− 0.847685i
$544$	0	0
$545$	−222.888	−0.408968
$546$	0	0
$547$	386.796i	0.707123i	0.935411	+	0.353561i	$0.115029\pi$
$547$	386.796i	0.707123i	−0.935411	+	0.353561i	$0.884971\pi$
$548$	0	0
$549$	−142.451	−0.259473
$550$	0	0
$551$	839.399i	1.52341i
$552$	0	0
$553$	−7.52332	−0.0136046
$554$	0	0
$555$	167.081i	0.301046i
$556$	0	0
$557$	−219.630	−0.394308	−0.197154	−	0.980373i	$-0.563170\pi$
$557$	−219.630	−0.394308	−0.197154	+	0.980373i	$0.563170\pi$
$558$	0	0
$559$	− 1141.34i	− 2.04175i
$560$	0	0
$561$	−485.732	−0.865833
$562$	0	0
$563$	266.807i	0.473902i	0.971522	+	0.236951i	$0.0761480\pi$
$563$	266.807i	0.473902i	−0.971522	+	0.236951i	$0.923852\pi$
$564$	0	0
$565$	−655.904	−1.16089
$566$	0	0
$567$	77.0025i	0.135807i
$568$	0	0
$569$	−972.046	−1.70834	−0.854170	−	0.519993i	$-0.825934\pi$
$569$	−972.046	−1.70834	−0.854170	+	0.519993i	$0.825934\pi$
$570$	0	0
$571$	340.194i	0.595786i	0.954599	+	0.297893i	$0.0962838\pi$
$571$	340.194i	0.595786i	−0.954599	+	0.297893i	$0.903716\pi$
$572$	0	0
$573$	498.875	0.870637
$574$	0	0
$575$	− 241.402i	− 0.419830i
$576$	0	0
$577$	65.0584	0.112753	0.0563764	−	0.998410i	$-0.482045\pi$
$577$	65.0584	0.112753	0.0563764	+	0.998410i	$0.482045\pi$
$578$	0	0
$579$	486.723i	0.840628i
$580$	0	0
$581$	198.608	0.341838
$582$	0	0
$583$	− 243.299i	− 0.417322i
$584$	0	0
$585$	424.643	0.725886
$586$	0	0
$587$	162.144i	0.276226i	0.990417	+	0.138113i	$0.0441036\pi$
$587$	162.144i	0.276226i	−0.990417	+	0.138113i	$0.955896\pi$
$588$	0	0
$589$	455.805	0.773862
$590$	0	0
$591$	58.6004i	0.0991547i
$592$	0	0
$593$	−793.227	−1.33765	−0.668825	−	0.743420i	$-0.733203\pi$
$593$	−793.227	−1.33765	−0.668825	+	0.743420i	$0.733203\pi$
$594$	0	0
$595$	1442.02i	2.42356i
$596$	0	0
$597$	147.172	0.246520
$598$	0	0
$599$	432.194i	0.721525i	0.932658	+	0.360763i	$0.117484\pi$
$599$	432.194i	0.721525i	−0.932658	+	0.360763i	$0.882516\pi$
$600$	0	0
$601$	936.310	1.55792	0.778960	−	0.627074i	$-0.215748\pi$
$601$	936.310	1.55792	0.778960	+	0.627074i	$0.215748\pi$
$602$	0	0
$603$	224.431i	0.372191i
$604$	0	0
$605$	−575.366	−0.951018
$606$	0	0
$607$	223.972i	0.368982i	0.982834	+	0.184491i	$0.0590637\pi$
$607$	223.972i	0.368982i	−0.982834	+	0.184491i	$0.940936\pi$
$608$	0	0
$609$	608.453	0.999101
$610$	0	0
$611$	− 340.304i	− 0.556962i
$612$	0	0
$613$	1010.40	1.64828	0.824141	−	0.566385i	$-0.191658\pi$
$613$	1010.40	1.64828	0.824141	+	0.566385i	$0.191658\pi$
$614$	0	0
$615$	516.391i	0.839660i
$616$	0	0
$617$	−400.206	−0.648633	−0.324316	−	0.945949i	$-0.605134\pi$
$617$	−400.206	−0.648633	−0.324316	+	0.945949i	$0.605134\pi$
$618$	0	0
$619$	417.349i	0.674231i	0.941463	+	0.337116i	$0.109451\pi$
$619$	417.349i	0.674231i	−0.941463	+	0.337116i	$0.890549\pi$
$620$	0	0
$621$	−28.6607	−0.0461525
$622$	0	0
$623$	− 139.536i	− 0.223975i
$624$	0	0
$625$	196.315	0.314104
$626$	0	0
$627$	488.585i	0.779243i
$628$	0	0
$629$	236.428	0.375880
$630$	0	0
$631$	718.033i	1.13793i	0.822362	+	0.568965i	$0.192656\pi$
$631$	718.033i	1.13793i	−0.822362	+	0.568965i	$0.807344\pi$
$632$	0	0
$633$	−243.700	−0.384992
$634$	0	0
$635$	1415.74i	2.22951i
$636$	0	0
$637$	413.116	0.648534
$638$	0	0
$639$	− 50.7451i	− 0.0794133i
$640$	0	0
$641$	139.926	0.218293	0.109146	−	0.994026i	$-0.465188\pi$
$641$	139.926	0.218293	0.109146	+	0.994026i	$0.465188\pi$
$642$	0	0
$643$	− 777.990i	− 1.20994i	−0.796249	−	0.604969i	$-0.793186\pi$
$643$	− 777.990i	− 1.20994i	0.796249	−	0.604969i	$-0.206814\pi$
$644$	0	0
$645$	−960.384	−1.48897
$646$	0	0
$647$	673.336i	1.04070i	0.853952	+	0.520352i	$0.174199\pi$
$647$	673.336i	1.04070i	−0.853952	+	0.520352i	$0.825801\pi$
$648$	0	0
$649$	857.642	1.32148
$650$	0	0
$651$	− 330.398i	− 0.507524i
$652$	0	0
$653$	−223.410	−0.342128	−0.171064	−	0.985260i	$-0.554721\pi$
$653$	−223.410	−0.342128	−0.171064	+	0.985260i	$0.554721\pi$
$654$	0	0
$655$	1484.20i	2.26595i
$656$	0	0
$657$	−305.136	−0.464438
$658$	0	0
$659$	166.367i	0.252453i	0.992001	+	0.126227i	$0.0402866\pi$
$659$	166.367i	0.252453i	−0.992001	+	0.126227i	$0.959713\pi$
$660$	0	0
$661$	655.773	0.992092	0.496046	−	0.868296i	$-0.334785\pi$
$661$	655.773	0.992092	0.496046	+	0.868296i	$0.334785\pi$
$662$	0	0
$663$	− 600.894i	− 0.906326i
$664$	0	0
$665$	1450.49	2.18119
$666$	0	0
$667$	226.469i	0.339533i
$668$	0	0
$669$	428.734	0.640857
$670$	0	0
$671$	655.176i	0.976418i
$672$	0	0
$673$	−863.477	−1.28303	−0.641513	−	0.767112i	$-0.721693\pi$
$673$	−863.477	−1.28303	−0.641513	+	0.767112i	$0.721693\pi$
$674$	0	0
$675$	− 227.415i	− 0.336911i
$676$	0	0
$677$	−831.520	−1.22824	−0.614121	−	0.789212i	$-0.710489\pi$
$677$	−831.520	−1.22824	−0.614121	+	0.789212i	$0.710489\pi$
$678$	0	0
$679$	1611.13i	2.37279i
$680$	0	0
$681$	−224.737	−0.330010
$682$	0	0
$683$	− 308.273i	− 0.451351i	−0.974202	−	0.225676i	$-0.927541\pi$
$683$	− 308.273i	− 0.451351i	0.974202	−	0.225676i	$-0.0724590\pi$
$684$	0	0
$685$	−772.926	−1.12836
$686$	0	0
$687$	− 509.750i	− 0.741994i
$688$	0	0
$689$	300.982	0.436839
$690$	0	0
$691$	− 666.811i	− 0.964994i	−0.875898	−	0.482497i	$-0.839730\pi$
$691$	− 666.811i	− 0.964994i	0.875898	−	0.482497i	$-0.160270\pi$
$692$	0	0
$693$	354.159	0.511052
$694$	0	0
$695$	− 1844.89i	− 2.65452i
$696$	0	0
$697$	730.722	1.04838
$698$	0	0
$699$	− 415.603i	− 0.594568i
$700$	0	0
$701$	−242.205	−0.345514	−0.172757	−	0.984964i	$-0.555268\pi$
$701$	−242.205	−0.345514	−0.172757	+	0.984964i	$0.555268\pi$
$702$	0	0
$703$	− 237.817i	− 0.338289i
$704$	0	0
$705$	−286.351	−0.406171
$706$	0	0
$707$	571.930i	0.808953i
$708$	0	0
$709$	−1380.24	−1.94674	−0.973371	−	0.229237i	$-0.926377\pi$
$709$	−1380.24	−1.94674	−0.973371	+	0.229237i	$0.926377\pi$
$710$	0	0
$711$	− 2.63796i	− 0.00371021i
$712$	0	0
$713$	122.976	0.172476
$714$	0	0
$715$	− 1953.07i	− 2.73157i
$716$	0	0
$717$	−736.722	−1.02751
$718$	0	0
$719$	968.365i	1.34682i	0.739268	+	0.673411i	$0.235172\pi$
$719$	968.365i	1.34682i	−0.739268	+	0.673411i	$0.764828\pi$
$720$	0	0
$721$	1422.94	1.97357
$722$	0	0
$723$	458.115i	0.633631i
$724$	0	0
$725$	−1796.97	−2.47858
$726$	0	0
$727$	1017.52i	1.39962i	0.714329	+	0.699810i	$0.246732\pi$
$727$	1017.52i	1.39962i	−0.714329	+	0.699810i	$0.753268\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	1359.00i	1.85909i
$732$	0	0
$733$	1026.22	1.40003	0.700016	−	0.714127i	$-0.253176\pi$
$733$	1026.22	1.40003	0.700016	+	0.714127i	$0.253176\pi$
$734$	0	0
$735$	− 347.619i	− 0.472952i
$736$	0	0
$737$	1032.23	1.40059
$738$	0	0
$739$	923.444i	1.24959i	0.780791	+	0.624793i	$0.214817\pi$
$739$	923.444i	1.24959i	−0.780791	+	0.624793i	$0.785183\pi$
$740$	0	0
$741$	−604.424	−0.815686
$742$	0	0
$743$	− 1267.12i	− 1.70542i	−0.522387	−	0.852708i	$-0.674958\pi$
$743$	− 1267.12i	− 1.70542i	0.522387	−	0.852708i	$-0.325042\pi$
$744$	0	0
$745$	−779.533	−1.04635
$746$	0	0
$747$	69.6394i	0.0932254i
$748$	0	0
$749$	685.722	0.915517
$750$	0	0
$751$	− 791.442i	− 1.05385i	−0.849912	−	0.526925i	$-0.823345\pi$
$751$	− 791.442i	− 1.05385i	0.849912	−	0.526925i	$-0.176655\pi$
$752$	0	0
$753$	383.000	0.508632
$754$	0	0
$755$	− 680.054i	− 0.900734i
$756$	0	0
$757$	333.990	0.441202	0.220601	−	0.975364i	$-0.429198\pi$
$757$	333.990	0.441202	0.220601	+	0.975364i	$0.429198\pi$
$758$	0	0
$759$	131.820i	0.173675i
$760$	0	0
$761$	617.122	0.810936	0.405468	−	0.914109i	$-0.367109\pi$
$761$	617.122	0.810936	0.405468	+	0.914109i	$0.367109\pi$
$762$	0	0
$763$	− 229.965i	− 0.301396i
$764$	0	0
$765$	−505.626	−0.660949
$766$	0	0
$767$	1060.98i	1.38329i
$768$	0	0
$769$	−657.604	−0.855142	−0.427571	−	0.903982i	$-0.640631\pi$
$769$	−657.604	−0.855142	−0.427571	+	0.903982i	$0.640631\pi$
$770$	0	0
$771$	150.802i	0.195593i
$772$	0	0
$773$	1314.59	1.70064	0.850319	−	0.526268i	$-0.176409\pi$
$773$	1314.59	1.70064	0.850319	+	0.526268i	$0.176409\pi$
$774$	0	0
$775$	975.779i	1.25907i
$776$	0	0
$777$	−172.386	−0.221861
$778$	0	0
$779$	− 735.015i	− 0.943536i
$780$	0	0
$781$	−233.393	−0.298839
$782$	0	0
$783$	213.347i	0.272473i
$784$	0	0
$785$	470.336	0.599154
$786$	0	0
$787$	301.018i	0.382488i	0.981542	+	0.191244i	$0.0612522\pi$
$787$	301.018i	0.382488i	−0.981542	+	0.191244i	$0.938748\pi$
$788$	0	0
$789$	−480.340	−0.608796
$790$	0	0
$791$	− 676.731i	− 0.855538i
$792$	0	0
$793$	−810.512	−1.02208
$794$	0	0
$795$	− 253.263i	− 0.318570i
$796$	0	0
$797$	476.193	0.597481	0.298741	−	0.954334i	$-0.403433\pi$
$797$	476.193	0.597481	0.298741	+	0.954334i	$0.403433\pi$
$798$	0	0
$799$	405.202i	0.507137i
$800$	0	0
$801$	48.9267	0.0610821
$802$	0	0
$803$	1403.42i	1.74772i
$804$	0	0
$805$	391.340	0.486136
$806$	0	0
$807$	86.2570i	0.106886i
$808$	0	0
$809$	−692.874	−0.856457	−0.428228	−	0.903671i	$-0.640862\pi$
$809$	−692.874	−0.856457	−0.428228	+	0.903671i	$0.640862\pi$
$810$	0	0
$811$	785.074i	0.968032i	0.875059	+	0.484016i	$0.160822\pi$
$811$	785.074i	0.968032i	−0.875059	+	0.484016i	$0.839178\pi$
$812$	0	0
$813$	−54.2383	−0.0667138
$814$	0	0
$815$	− 1572.17i	− 1.92904i
$816$	0	0
$817$	1366.98	1.67317
$818$	0	0
$819$	438.126i	0.534953i
$820$	0	0
$821$	−527.681	−0.642729	−0.321365	−	0.946956i	$-0.604141\pi$
$821$	−527.681	−0.642729	−0.321365	+	0.946956i	$0.604141\pi$
$822$	0	0
$823$	− 228.552i	− 0.277706i	−0.990313	−	0.138853i	$-0.955658\pi$
$823$	− 228.552i	− 0.277706i	0.990313	−	0.138853i	$-0.0443416\pi$
$824$	0	0
$825$	−1045.95	−1.26782
$826$	0	0
$827$	608.409i	0.735683i	0.929889	+	0.367841i	$0.119903\pi$
$827$	608.409i	0.735683i	−0.929889	+	0.367841i	$0.880097\pi$
$828$	0	0
$829$	151.637	0.182915	0.0914577	−	0.995809i	$-0.470847\pi$
$829$	151.637	0.182915	0.0914577	+	0.995809i	$0.470847\pi$
$830$	0	0
$831$	137.080i	0.164958i
$832$	0	0
$833$	−491.901	−0.590518
$834$	0	0
$835$	− 2593.71i	− 3.10624i
$836$	0	0
$837$	115.850	0.138411
$838$	0	0
$839$	− 653.351i	− 0.778725i	−0.921085	−	0.389363i	$-0.872695\pi$
$839$	− 653.351i	− 0.778725i	0.921085	−	0.389363i	$-0.127305\pi$
$840$	0	0
$841$	844.805	1.00452
$842$	0	0
$843$	− 183.927i	− 0.218182i
$844$	0	0
$845$	1014.68	1.20081
$846$	0	0
$847$	− 593.635i	− 0.700868i
$848$	0	0
$849$	−468.088	−0.551340
$850$	0	0
$851$	− 64.1628i	− 0.0753969i
$852$	0	0
$853$	1372.25	1.60873	0.804367	−	0.594132i	$-0.202505\pi$
$853$	1372.25	1.60873	0.804367	+	0.594132i	$0.202505\pi$
$854$	0	0
$855$	508.596i	0.594849i
$856$	0	0
$857$	−1453.61	−1.69617	−0.848083	−	0.529863i	$-0.822243\pi$
$857$	−1453.61	−1.69617	−0.848083	+	0.529863i	$0.822243\pi$
$858$	0	0
$859$	962.466i	1.12045i	0.828341	+	0.560225i	$0.189285\pi$
$859$	962.466i	1.12045i	−0.828341	+	0.560225i	$0.810715\pi$
$860$	0	0
$861$	−532.787	−0.618801
$862$	0	0
$863$	695.890i	0.806362i	0.915120	+	0.403181i	$0.132095\pi$
$863$	695.890i	0.806362i	−0.915120	+	0.403181i	$0.867905\pi$
$864$	0	0
$865$	1513.70	1.74994
$866$	0	0
$867$	214.927i	0.247897i
$868$	0	0
$869$	−12.1328	−0.0139618
$870$	0	0
$871$	1276.96i	1.46609i
$872$	0	0
$873$	−564.922	−0.647104
$874$	0	0
$875$	1331.44i	1.52165i
$876$	0	0
$877$	338.191	0.385622	0.192811	−	0.981236i	$-0.438240\pi$
$877$	338.191	0.385622	0.192811	+	0.981236i	$0.438240\pi$
$878$	0	0
$879$	− 42.0000i	− 0.0477815i
$880$	0	0
$881$	−459.985	−0.522117	−0.261058	−	0.965323i	$-0.584072\pi$
$881$	−459.985	−0.522117	−0.261058	+	0.965323i	$0.584072\pi$
$882$	0	0
$883$	− 1208.31i	− 1.36842i	−0.729286	−	0.684209i	$-0.760147\pi$
$883$	− 1208.31i	− 1.36842i	0.729286	−	0.684209i	$-0.239853\pi$
$884$	0	0
$885$	892.769	1.00878
$886$	0	0
$887$	− 1500.73i	− 1.69192i	−0.533250	−	0.845958i	$-0.679029\pi$
$887$	− 1500.73i	− 1.69192i	0.533250	−	0.845958i	$-0.320971\pi$
$888$	0	0
$889$	−1460.69	−1.64307
$890$	0	0
$891$	124.182i	0.139373i
$892$	0	0
$893$	407.582	0.456419
$894$	0	0
$895$	− 164.010i	− 0.183252i
$896$	0	0
$897$	−163.073	−0.181798
$898$	0	0
$899$	− 915.415i	− 1.01826i
$900$	0	0
$901$	−358.382	−0.397760
$902$	0	0
$903$	− 990.878i	− 1.09732i
$904$	0	0
$905$	−2203.74	−2.43507
$906$	0	0
$907$	665.128i	0.733328i	0.930353	+	0.366664i	$0.119500\pi$
$907$	665.128i	0.733328i	−0.930353	+	0.366664i	$0.880500\pi$
$908$	0	0
$909$	−200.540	−0.220616
$910$	0	0
$911$	345.687i	0.379459i	0.981836	+	0.189729i	$0.0607611\pi$
$911$	345.687i	0.379459i	−0.981836	+	0.189729i	$0.939239\pi$
$912$	0	0
$913$	320.294	0.350815
$914$	0	0
$915$	682.010i	0.745367i
$916$	0	0
$917$	−1531.32	−1.66993
$918$	0	0
$919$	− 1102.20i	− 1.19934i	−0.800246	−	0.599671i	$-0.795298\pi$
$919$	− 1102.20i	− 1.19934i	0.800246	−	0.599671i	$-0.204702\pi$
$920$	0	0
$921$	212.808	0.231062
$922$	0	0
$923$	− 288.728i	− 0.312815i
$924$	0	0
$925$	509.115	0.550394
$926$	0	0
$927$	498.938i	0.538228i
$928$	0	0
$929$	−1256.03	−1.35202	−0.676010	−	0.736893i	$-0.736292\pi$
$929$	−1256.03	−1.35202	−0.676010	+	0.736893i	$0.736292\pi$
$930$	0	0
$931$	494.790i	0.531461i
$932$	0	0
$933$	−188.461	−0.201995
$934$	0	0
$935$	2325.54i	2.48720i
$936$	0	0
$937$	−822.214	−0.877496	−0.438748	−	0.898610i	$-0.644578\pi$
$937$	−822.214	−0.877496	−0.438748	+	0.898610i	$0.644578\pi$
$938$	0	0
$939$	− 90.4505i	− 0.0963264i
$940$	0	0
$941$	−1185.72	−1.26006	−0.630031	−	0.776570i	$-0.716958\pi$
$941$	−1185.72	−1.26006	−0.630031	+	0.776570i	$0.716958\pi$
$942$	0	0
$943$	− 198.306i	− 0.210293i
$944$	0	0
$945$	368.664	0.390121
$946$	0	0
$947$	− 1335.34i	− 1.41007i	−0.709170	−	0.705037i	$-0.750930\pi$
$947$	− 1335.34i	− 1.41007i	0.709170	−	0.705037i	$-0.249070\pi$
$948$	0	0
$949$	−1736.15	−1.82945
$950$	0	0
$951$	− 163.376i	− 0.171794i
$952$	0	0
$953$	663.428	0.696147	0.348073	−	0.937467i	$-0.386836\pi$
$953$	663.428	0.696147	0.348073	+	0.937467i	$0.386836\pi$
$954$	0	0
$955$	− 2388.46i	− 2.50100i
$956$	0	0
$957$	981.249	1.02534
$958$	0	0
$959$	− 797.468i	− 0.831562i
$960$	0	0
$961$	463.918	0.482745
$962$	0	0
$963$	240.440i	0.249678i
$964$	0	0
$965$	2330.28	2.41480
$966$	0	0
$967$	− 155.996i	− 0.161320i	−0.996742	−	0.0806600i	$-0.974297\pi$
$967$	− 155.996i	− 0.161320i	0.996742	−	0.0806600i	$-0.0257028\pi$
$968$	0	0
$969$	719.692	0.742716
$970$	0	0
$971$	− 1280.67i	− 1.31892i	−0.751740	−	0.659460i	$-0.770785\pi$
$971$	− 1280.67i	− 1.31892i	0.751740	−	0.659460i	$-0.229215\pi$
$972$	0	0
$973$	1903.47	1.95629
$974$	0	0
$975$	− 1293.94i	− 1.32712i
$976$	0	0
$977$	−217.470	−0.222590	−0.111295	−	0.993787i	$-0.535500\pi$
$977$	−217.470	−0.222590	−0.111295	+	0.993787i	$0.535500\pi$
$978$	0	0
$979$	− 225.030i	− 0.229857i
$980$	0	0
$981$	80.6344	0.0821962
$982$	0	0
$983$	− 1027.63i	− 1.04540i	−0.852515	−	0.522702i	$-0.824924\pi$
$983$	− 1027.63i	− 1.04540i	0.852515	−	0.522702i	$-0.175076\pi$
$984$	0	0
$985$	280.561	0.284833
$986$	0	0
$987$	− 295.443i	− 0.299334i
$988$	0	0
$989$	368.809	0.372911
$990$	0	0
$991$	− 797.792i	− 0.805038i	−0.915412	−	0.402519i	$-0.868135\pi$
$991$	− 797.792i	− 0.805038i	0.915412	−	0.402519i	$-0.131865\pi$
$992$	0	0
$993$	−703.397	−0.708356
$994$	0	0
$995$	− 704.616i	− 0.708157i
$996$	0	0
$997$	−1237.58	−1.24131	−0.620654	−	0.784085i	$-0.713133\pi$
$997$	−1237.58	−1.24131	−0.620654	+	0.784085i	$0.713133\pi$
$998$	0	0
$999$	− 60.4450i	− 0.0605055i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.3.g.b.127.8	yes	8
3.2	odd	2		1152.3.g.c.127.2		8
4.3	odd	2	inner	384.3.g.b.127.4	yes	8
8.3	odd	2		384.3.g.a.127.5	yes	8
8.5	even	2		384.3.g.a.127.1	✓	8
12.11	even	2		1152.3.g.c.127.1		8
16.3	odd	4		768.3.b.e.127.5		8
16.5	even	4		768.3.b.e.127.8		8
16.11	odd	4		768.3.b.f.127.4		8
16.13	even	4		768.3.b.f.127.1		8
24.5	odd	2		1152.3.g.f.127.8		8
24.11	even	2		1152.3.g.f.127.7		8
48.5	odd	4		2304.3.b.t.127.1		8
48.11	even	4		2304.3.b.q.127.1		8
48.29	odd	4		2304.3.b.q.127.8		8
48.35	even	4		2304.3.b.t.127.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.3.g.a.127.1	✓	8	8.5	even	2
384.3.g.a.127.5	yes	8	8.3	odd	2
384.3.g.b.127.4	yes	8	4.3	odd	2	inner
384.3.g.b.127.8	yes	8	1.1	even	1	trivial
768.3.b.e.127.5		8	16.3	odd	4
768.3.b.e.127.8		8	16.5	even	4
768.3.b.f.127.1		8	16.13	even	4
768.3.b.f.127.4		8	16.11	odd	4
1152.3.g.c.127.1		8	12.11	even	2
1152.3.g.c.127.2		8	3.2	odd	2
1152.3.g.f.127.7		8	24.11	even	2
1152.3.g.f.127.8		8	24.5	odd	2
2304.3.b.q.127.1		8	48.11	even	4
2304.3.b.q.127.8		8	48.29	odd	4
2304.3.b.t.127.1		8	48.5	odd	4
2304.3.b.t.127.8		8	48.35	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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	384.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} +8.29253 q^{5} +8.55583i q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q+1.73205i q^{3} +8.29253 q^{5} +8.55583i q^{7} -3.00000 q^{9} +13.7980i q^{11} -17.0693 q^{13} +14.3631i q^{15} +20.3246 q^{17} -20.4440i q^{19} -14.8191 q^{21} -5.51575i q^{23} +43.7660 q^{25} -5.19615i q^{27} -41.0586 q^{29} +22.2953i q^{31} -23.8988 q^{33} +70.9495i q^{35} +11.6326 q^{37} -29.5649i q^{39} +35.9527 q^{41} +66.8648i q^{43} -24.8776 q^{45} +19.9366i q^{47} -24.2023 q^{49} +35.2032i q^{51} -17.6329 q^{53} +114.420i q^{55} +35.4100 q^{57} -62.1572i q^{59} +47.4836 q^{61} -25.6675i q^{63} -141.548 q^{65} -74.8105i q^{67} +9.55356 q^{69} +16.9150i q^{71} +101.712 q^{73} +75.8050i q^{75} -118.053 q^{77} +0.879320i q^{79} +9.00000 q^{81} -23.2131i q^{83} +168.542 q^{85} -71.1155i q^{87} -16.3089 q^{89} -146.042i q^{91} -38.6167 q^{93} -169.532i q^{95} +188.307 q^{97} -41.3939i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{5} - 24 q^{9}+O(q^{10}) \) 8 * q + 16 * q^5 - 24 * q^9	\( 8 q + 16 q^{5} - 24 q^{9} - 48 q^{13} + 16 q^{17} - 8 q^{25} + 80 q^{29} + 16 q^{37} + 80 q^{41} - 48 q^{45} - 88 q^{49} - 176 q^{53} + 96 q^{57} + 272 q^{61} - 160 q^{65} - 16 q^{73} - 320 q^{77} + 72 q^{81} - 32 q^{85} - 240 q^{89} + 192 q^{93} + 400 q^{97}+O(q^{100}) \) 8 * q + 16 * q^5 - 24 * q^9 - 48 * q^13 + 16 * q^17 - 8 * q^25 + 80 * q^29 + 16 * q^37 + 80 * q^41 - 48 * q^45 - 88 * q^49 - 176 * q^53 + 96 * q^57 + 272 * q^61 - 160 * q^65 - 16 * q^73 - 320 * q^77 + 72 * q^81 - 32 * q^85 - 240 * q^89 + 192 * q^93 + 400 * q^97

Introduction

L-functions

Modular forms

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Representations

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Database

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