LMFDB - Embedded newform 384.8.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,8,Mod(1,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("384.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	384.a (trivial)

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:	yes
Analytic conductor:	$119.955849786$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 430x^{2} - 2448x + 12138 $ x^4 - 430x^2 - 2448x + 12138
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{15}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-13.4480$ of defining polynomial
Character	$\chi$	$=$	384.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+27.0000 q^{3} -530.466 q^{5} -930.332 q^{7} +729.000 q^{9} +O(q^{10})$	$q+27.0000 q^{3} -530.466 q^{5} -930.332 q^{7} +729.000 q^{9} +4803.24 q^{11} +10210.0 q^{13} -14322.6 q^{15} -25673.6 q^{17} +24444.3 q^{19} -25119.0 q^{21} -19503.5 q^{23} +203269. q^{25} +19683.0 q^{27} -242831. q^{29} +137921. q^{31} +129687. q^{33} +493509. q^{35} +146746. q^{37} +275669. q^{39} +279862. q^{41} +793457. q^{43} -386709. q^{45} +1.15302e6 q^{47} +41974.2 q^{49} -693187. q^{51} -681439. q^{53} -2.54795e6 q^{55} +659995. q^{57} -40009.7 q^{59} -2.32040e6 q^{61} -678212. q^{63} -5.41604e6 q^{65} -1.63837e6 q^{67} -526594. q^{69} +3.79092e6 q^{71} -969505. q^{73} +5.48826e6 q^{75} -4.46860e6 q^{77} -6.24969e6 q^{79} +531441. q^{81} -7.63631e6 q^{83} +1.36190e7 q^{85} -6.55643e6 q^{87} +4.65733e6 q^{89} -9.49866e6 q^{91} +3.72388e6 q^{93} -1.29668e7 q^{95} -5.23448e6 q^{97} +3.50156e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 108 q^{3} - 192 q^{5} - 680 q^{7} + 2916 q^{9}+O(q^{10}) $ 4 * q + 108 * q^3 - 192 * q^5 - 680 * q^7 + 2916 * q^9	$ 4 q + 108 q^{3} - 192 q^{5} - 680 q^{7} + 2916 q^{9} - 4496 q^{11} + 12840 q^{13} - 5184 q^{15} - 14952 q^{17} - 21504 q^{19} - 18360 q^{21} - 54992 q^{23} + 190732 q^{25} + 78732 q^{27} - 242384 q^{29} - 151432 q^{31} - 121392 q^{33} + 273984 q^{35} - 113288 q^{37} + 346680 q^{39} - 239176 q^{41} + 1495328 q^{43} - 139968 q^{45} - 772368 q^{47} - 1577100 q^{49} - 403704 q^{51} - 2389776 q^{53} - 2590080 q^{55} - 580608 q^{57} + 141232 q^{59} - 1231304 q^{61} - 495720 q^{63} - 1041024 q^{65} - 441392 q^{67} - 1484784 q^{69} - 1507504 q^{71} - 1516840 q^{73} + 5149764 q^{75} - 12340448 q^{77} - 9540936 q^{79} + 2125764 q^{81} - 4587600 q^{83} - 6382848 q^{85} - 6544368 q^{87} + 162376 q^{89} - 4681104 q^{91} - 4088664 q^{93} - 29221248 q^{95} + 2726760 q^{97} - 3277584 q^{99}+O(q^{100}) $ 4 * q + 108 * q^3 - 192 * q^5 - 680 * q^7 + 2916 * q^9 - 4496 * q^11 + 12840 * q^13 - 5184 * q^15 - 14952 * q^17 - 21504 * q^19 - 18360 * q^21 - 54992 * q^23 + 190732 * q^25 + 78732 * q^27 - 242384 * q^29 - 151432 * q^31 - 121392 * q^33 + 273984 * q^35 - 113288 * q^37 + 346680 * q^39 - 239176 * q^41 + 1495328 * q^43 - 139968 * q^45 - 772368 * q^47 - 1577100 * q^49 - 403704 * q^51 - 2389776 * q^53 - 2590080 * q^55 - 580608 * q^57 + 141232 * q^59 - 1231304 * q^61 - 495720 * q^63 - 1041024 * q^65 - 441392 * q^67 - 1484784 * q^69 - 1507504 * q^71 - 1516840 * q^73 + 5149764 * q^75 - 12340448 * q^77 - 9540936 * q^79 + 2125764 * q^81 - 4587600 * q^83 - 6382848 * q^85 - 6544368 * q^87 + 162376 * q^89 - 4681104 * q^91 - 4088664 * q^93 - 29221248 * q^95 + 2726760 * q^97 - 3277584 * q^99

Display 100 coefficients 10 coefficients

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
$3$	27.0000	0.577350
$3$	27.0000	0.577350
$4$	0	0
$5$	−530.466	−1.89785	−0.948926	−	0.315499i	$-0.897828\pi$
$5$	−530.466	−1.89785	−0.948926	+	0.315499i	$0.897828\pi$
$6$	0	0
$7$	−930.332	−1.02517	−0.512584	−	0.858637i	$-0.671312\pi$
$7$	−930.332	−1.02517	−0.512584	+	0.858637i	$0.671312\pi$
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	4803.24	1.08808	0.544038	−	0.839060i	$-0.316895\pi$
$11$	4803.24	1.08808	0.544038	+	0.839060i	$0.316895\pi$
$12$	0	0
$13$	10210.0	1.28891	0.644455	−	0.764642i	$-0.277084\pi$
$13$	10210.0	1.28891	0.644455	+	0.764642i	$0.277084\pi$
$14$	0	0
$15$	−14322.6	−1.09573
$16$	0	0
$17$	−25673.6	−1.26741	−0.633703	−	0.773577i	$-0.718466\pi$
$17$	−25673.6	−1.26741	−0.633703	+	0.773577i	$0.718466\pi$
$18$	0	0
$19$	24444.3	0.817597	0.408799	−	0.912625i	$-0.365948\pi$
$19$	24444.3	0.817597	0.408799	+	0.912625i	$0.365948\pi$
$20$	0	0
$21$	−25119.0	−0.591881
$22$	0	0
$23$	−19503.5	−0.334245	−0.167122	−	0.985936i	$-0.553448\pi$
$23$	−19503.5	−0.334245	−0.167122	+	0.985936i	$0.553448\pi$
$24$	0	0
$25$	203269.	2.60184
$26$	0	0
$27$	19683.0	0.192450
$28$	0	0
$29$	−242831.	−1.84889	−0.924445	−	0.381316i	$-0.875471\pi$
$29$	−242831.	−1.84889	−0.924445	+	0.381316i	$0.875471\pi$
$30$	0	0
$31$	137921.	0.831507	0.415753	−	0.909477i	$-0.363518\pi$
$31$	137921.	0.831507	0.415753	+	0.909477i	$0.363518\pi$
$32$	0	0
$33$	129687.	0.628201
$34$	0	0
$35$	493509.	1.94562
$36$	0	0
$37$	146746.	0.476277	0.238139	−	0.971231i	$-0.423463\pi$
$37$	146746.	0.476277	0.238139	+	0.971231i	$0.423463\pi$
$38$	0	0
$39$	275669.	0.744153
$40$	0	0
$41$	279862.	0.634162	0.317081	−	0.948398i	$-0.397297\pi$
$41$	279862.	0.634162	0.317081	+	0.948398i	$0.397297\pi$
$42$	0	0
$43$	793457.	1.52189	0.760946	−	0.648815i	$-0.224735\pi$
$43$	793457.	1.52189	0.760946	+	0.648815i	$0.224735\pi$
$44$	0	0
$45$	−386709.	−0.632617
$46$	0	0
$47$	1.15302e6	1.61992	0.809962	−	0.586482i	$-0.199488\pi$
$47$	1.15302e6	1.61992	0.809962	+	0.586482i	$0.199488\pi$
$48$	0	0
$49$	41974.2	0.0509678
$50$	0	0
$51$	−693187.	−0.731737
$52$	0	0
$53$	−681439.	−0.628726	−0.314363	−	0.949303i	$-0.601791\pi$
$53$	−681439.	−0.628726	−0.314363	+	0.949303i	$0.601791\pi$
$54$	0	0
$55$	−2.54795e6	−2.06501
$56$	0	0
$57$	659995.	0.472040
$58$	0	0
$59$	−40009.7	−0.0253620	−0.0126810	−	0.999920i	$-0.504037\pi$
$59$	−40009.7	−0.0253620	−0.0126810	+	0.999920i	$0.504037\pi$
$60$	0	0
$61$	−2.32040e6	−1.30890	−0.654452	−	0.756103i	$-0.727101\pi$
$61$	−2.32040e6	−1.30890	−0.654452	+	0.756103i	$0.727101\pi$
$62$	0	0
$63$	−678212.	−0.341722
$64$	0	0
$65$	−5.41604e6	−2.44616
$66$	0	0
$67$	−1.63837e6	−0.665505	−0.332753	−	0.943014i	$-0.607977\pi$
$67$	−1.63837e6	−0.665505	−0.332753	+	0.943014i	$0.607977\pi$
$68$	0	0
$69$	−526594.	−0.192976
$70$	0	0
$71$	3.79092e6	1.25701	0.628507	−	0.777804i	$-0.283666\pi$
$71$	3.79092e6	1.25701	0.628507	+	0.777804i	$0.283666\pi$
$72$	0	0
$73$	−969505.	−0.291689	−0.145845	−	0.989308i	$-0.546590\pi$
$73$	−969505.	−0.291689	−0.145845	+	0.989308i	$0.546590\pi$
$74$	0	0
$75$	5.48826e6	1.50217
$76$	0	0
$77$	−4.46860e6	−1.11546
$78$	0	0
$79$	−6.24969e6	−1.42614	−0.713072	−	0.701091i	$-0.752697\pi$
$79$	−6.24969e6	−1.42614	−0.713072	+	0.701091i	$0.752697\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	−7.63631e6	−1.46592	−0.732960	−	0.680272i	$-0.761862\pi$
$83$	−7.63631e6	−1.46592	−0.732960	+	0.680272i	$0.761862\pi$
$84$	0	0
$85$	1.36190e7	2.40535
$86$	0	0
$87$	−6.55643e6	−1.06746
$88$	0	0
$89$	4.65733e6	0.700280	0.350140	−	0.936697i	$-0.386134\pi$
$89$	4.65733e6	0.700280	0.350140	+	0.936697i	$0.386134\pi$
$90$	0	0
$91$	−9.49866e6	−1.32135
$92$	0	0
$93$	3.72388e6	0.480071
$94$	0	0
$95$	−1.29668e7	−1.55168
$96$	0	0
$97$	−5.23448e6	−0.582334	−0.291167	−	0.956672i	$-0.594044\pi$
$97$	−5.23448e6	−0.582334	−0.291167	+	0.956672i	$0.594044\pi$
$98$	0	0
$99$	3.50156e6	0.362692
$100$	0	0
$101$	−514976.	−0.0497351	−0.0248675	−	0.999691i	$-0.507916\pi$
$101$	−514976.	−0.0497351	−0.0248675	+	0.999691i	$0.507916\pi$
$102$	0	0
$103$	−6.04823e6	−0.545379	−0.272689	−	0.962102i	$-0.587913\pi$
$103$	−6.04823e6	−0.545379	−0.272689	+	0.962102i	$0.587913\pi$
$104$	0	0
$105$	1.33247e7	1.12330
$106$	0	0
$107$	−1.00697e7	−0.794647	−0.397324	−	0.917679i	$-0.630061\pi$
$107$	−1.00697e7	−0.794647	−0.397324	+	0.917679i	$0.630061\pi$
$108$	0	0
$109$	−1.47212e6	−0.108881	−0.0544403	−	0.998517i	$-0.517337\pi$
$109$	−1.47212e6	−0.108881	−0.0544403	+	0.998517i	$0.517337\pi$
$110$	0	0
$111$	3.96214e6	0.274979
$112$	0	0
$113$	−8.75618e6	−0.570874	−0.285437	−	0.958397i	$-0.592139\pi$
$113$	−8.75618e6	−0.570874	−0.285437	+	0.958397i	$0.592139\pi$
$114$	0	0
$115$	1.03459e7	0.634347
$116$	0	0
$117$	7.44307e6	0.429637
$118$	0	0
$119$	2.38850e7	1.29930
$120$	0	0
$121$	3.58390e6	0.183911
$122$	0	0
$123$	7.55627e6	0.366134
$124$	0	0
$125$	−6.63845e7	−3.04006
$126$	0	0
$127$	−2.05516e7	−0.890293	−0.445146	−	0.895458i	$-0.646848\pi$
$127$	−2.05516e7	−0.890293	−0.445146	+	0.895458i	$0.646848\pi$
$128$	0	0
$129$	2.14234e7	0.878665
$130$	0	0
$131$	−3.06403e7	−1.19081	−0.595406	−	0.803425i	$-0.703009\pi$
$131$	−3.06403e7	−1.19081	−0.595406	+	0.803425i	$0.703009\pi$
$132$	0	0
$133$	−2.27413e7	−0.838174
$134$	0	0
$135$	−1.04412e7	−0.365242
$136$	0	0
$137$	1.19457e7	0.396906	0.198453	−	0.980110i	$-0.436408\pi$
$137$	1.19457e7	0.396906	0.198453	+	0.980110i	$0.436408\pi$
$138$	0	0
$139$	2.07490e7	0.655308	0.327654	−	0.944798i	$-0.393742\pi$
$139$	2.07490e7	0.655308	0.327654	+	0.944798i	$0.393742\pi$
$140$	0	0
$141$	3.11316e7	0.935264
$142$	0	0
$143$	4.90409e7	1.40243
$144$	0	0
$145$	1.28813e8	3.50892
$146$	0	0
$147$	1.13330e6	0.0294263
$148$	0	0
$149$	−7.75243e7	−1.91993	−0.959966	−	0.280118i	$-0.909627\pi$
$149$	−7.75243e7	−1.91993	−0.959966	+	0.280118i	$0.909627\pi$
$150$	0	0
$151$	6.43978e7	1.52213	0.761065	−	0.648676i	$-0.224677\pi$
$151$	6.43978e7	1.52213	0.761065	+	0.648676i	$0.224677\pi$
$152$	0	0
$153$	−1.87161e7	−0.422469
$154$	0	0
$155$	−7.31626e7	−1.57808
$156$	0	0
$157$	2.72664e7	0.562315	0.281157	−	0.959662i	$-0.409282\pi$
$157$	2.72664e7	0.562315	0.281157	+	0.959662i	$0.409282\pi$
$158$	0	0
$159$	−1.83989e7	−0.362995
$160$	0	0
$161$	1.81447e7	0.342657
$162$	0	0
$163$	−5.03321e7	−0.910309	−0.455154	−	0.890413i	$-0.650416\pi$
$163$	−5.03321e7	−0.910309	−0.455154	+	0.890413i	$0.650416\pi$
$164$	0	0
$165$	−6.87947e7	−1.19223
$166$	0	0
$167$	−6.57706e7	−1.09276	−0.546379	−	0.837538i	$-0.683994\pi$
$167$	−6.57706e7	−1.09276	−0.546379	+	0.837538i	$0.683994\pi$
$168$	0	0
$169$	4.14949e7	0.661289
$170$	0	0
$171$	1.78199e7	0.272532
$172$	0	0
$173$	6.83635e7	1.00384	0.501918	−	0.864915i	$-0.332628\pi$
$173$	6.83635e7	1.00384	0.501918	+	0.864915i	$0.332628\pi$
$174$	0	0
$175$	−1.89107e8	−2.66732
$176$	0	0
$177$	−1.08026e6	−0.0146427
$178$	0	0
$179$	−1.03905e8	−1.35410	−0.677049	−	0.735938i	$-0.736742\pi$
$179$	−1.03905e8	−1.35410	−0.677049	+	0.735938i	$0.736742\pi$
$180$	0	0
$181$	−6.89023e7	−0.863691	−0.431846	−	0.901948i	$-0.642137\pi$
$181$	−6.89023e7	−0.863691	−0.431846	+	0.901948i	$0.642137\pi$
$182$	0	0
$183$	−6.26507e7	−0.755696
$184$	0	0
$185$	−7.78437e7	−0.903904
$186$	0	0
$187$	−1.23316e8	−1.37903
$188$	0	0
$189$	−1.83117e7	−0.197294
$190$	0	0
$191$	−1.14992e7	−0.119412	−0.0597062	−	0.998216i	$-0.519016\pi$
$191$	−1.14992e7	−0.119412	−0.0597062	+	0.998216i	$0.519016\pi$
$192$	0	0
$193$	6.41140e7	0.641952	0.320976	−	0.947087i	$-0.395989\pi$
$193$	6.41140e7	0.641952	0.320976	+	0.947087i	$0.395989\pi$
$194$	0	0
$195$	−1.46233e8	−1.41229
$196$	0	0
$197$	1.77703e8	1.65601	0.828003	−	0.560723i	$-0.189477\pi$
$197$	1.77703e8	1.65601	0.828003	+	0.560723i	$0.189477\pi$
$198$	0	0
$199$	−1.66111e6	−0.0149422	−0.00747108	−	0.999972i	$-0.502378\pi$
$199$	−1.66111e6	−0.0149422	−0.00747108	+	0.999972i	$0.502378\pi$
$200$	0	0
$201$	−4.42361e7	−0.384230
$202$	0	0
$203$	2.25913e8	1.89542
$204$	0	0
$205$	−1.48457e8	−1.20355
$206$	0	0
$207$	−1.42180e7	−0.111415
$208$	0	0
$209$	1.17412e8	0.889609
$210$	0	0
$211$	2.25200e8	1.65036	0.825181	−	0.564868i	$-0.191073\pi$
$211$	2.25200e8	1.65036	0.825181	+	0.564868i	$0.191073\pi$
$212$	0	0
$213$	1.02355e8	0.725738
$214$	0	0
$215$	−4.20902e8	−2.88833
$216$	0	0
$217$	−1.28313e8	−0.852433
$218$	0	0
$219$	−2.61766e7	−0.168407
$220$	0	0
$221$	−2.62127e8	−1.63357
$222$	0	0
$223$	−4.93794e7	−0.298180	−0.149090	−	0.988824i	$-0.547634\pi$
$223$	−4.93794e7	−0.298180	−0.149090	+	0.988824i	$0.547634\pi$
$224$	0	0
$225$	1.48183e8	0.867280
$226$	0	0
$227$	−3.34336e8	−1.89711	−0.948557	−	0.316608i	$-0.897456\pi$
$227$	−3.34336e8	−1.89711	−0.948557	+	0.316608i	$0.897456\pi$
$228$	0	0
$229$	−1.66610e8	−0.916804	−0.458402	−	0.888745i	$-0.651578\pi$
$229$	−1.66610e8	−0.916804	−0.458402	+	0.888745i	$0.651578\pi$
$230$	0	0
$231$	−1.20652e8	−0.644011
$232$	0	0
$233$	2.77181e8	1.43555	0.717773	−	0.696277i	$-0.245161\pi$
$233$	2.77181e8	1.43555	0.717773	+	0.696277i	$0.245161\pi$
$234$	0	0
$235$	−6.11638e8	−3.07438
$236$	0	0
$237$	−1.68742e8	−0.823385
$238$	0	0
$239$	−2.33182e8	−1.10485	−0.552424	−	0.833564i	$-0.686297\pi$
$239$	−2.33182e8	−1.10485	−0.552424	+	0.833564i	$0.686297\pi$
$240$	0	0
$241$	−8.53512e7	−0.392781	−0.196390	−	0.980526i	$-0.562922\pi$
$241$	−8.53512e7	−0.392781	−0.196390	+	0.980526i	$0.562922\pi$
$242$	0	0
$243$	1.43489e7	0.0641500
$244$	0	0
$245$	−2.22659e7	−0.0967294
$246$	0	0
$247$	2.49575e8	1.05381
$248$	0	0
$249$	−2.06180e8	−0.846349
$250$	0	0
$251$	2.43007e8	0.969974	0.484987	−	0.874521i	$-0.338824\pi$
$251$	2.43007e8	0.969974	0.484987	+	0.874521i	$0.338824\pi$
$252$	0	0
$253$	−9.36798e7	−0.363684
$254$	0	0
$255$	3.67712e8	1.38873
$256$	0	0
$257$	8.64384e7	0.317644	0.158822	−	0.987307i	$-0.449230\pi$
$257$	8.64384e7	0.317644	0.158822	+	0.987307i	$0.449230\pi$
$258$	0	0
$259$	−1.36522e8	−0.488264
$260$	0	0
$261$	−1.77024e8	−0.616296
$262$	0	0
$263$	−4.11540e7	−0.139497	−0.0697487	−	0.997565i	$-0.522220\pi$
$263$	−4.11540e7	−0.139497	−0.0697487	+	0.997565i	$0.522220\pi$
$264$	0	0
$265$	3.61480e8	1.19323
$266$	0	0
$267$	1.25748e8	0.404307
$268$	0	0
$269$	−4.01524e8	−1.25770	−0.628852	−	0.777525i	$-0.716475\pi$
$269$	−4.01524e8	−1.25770	−0.628852	+	0.777525i	$0.716475\pi$
$270$	0	0
$271$	3.33296e8	1.01727	0.508637	−	0.860981i	$-0.330150\pi$
$271$	3.33296e8	1.01727	0.508637	+	0.860981i	$0.330150\pi$
$272$	0	0
$273$	−2.56464e8	−0.762881
$274$	0	0
$275$	9.76348e8	2.83100
$276$	0	0
$277$	1.54553e8	0.436916	0.218458	−	0.975846i	$-0.429897\pi$
$277$	1.54553e8	0.436916	0.218458	+	0.975846i	$0.429897\pi$
$278$	0	0
$279$	1.00545e8	0.277169
$280$	0	0
$281$	3.22342e8	0.866652	0.433326	−	0.901237i	$-0.357340\pi$
$281$	3.22342e8	0.866652	0.433326	+	0.901237i	$0.357340\pi$
$282$	0	0
$283$	−4.98182e8	−1.30658	−0.653289	−	0.757108i	$-0.726611\pi$
$283$	−4.98182e8	−1.30658	−0.653289	+	0.757108i	$0.726611\pi$
$284$	0	0
$285$	−3.50105e8	−0.895862
$286$	0	0
$287$	−2.60365e8	−0.650122
$288$	0	0
$289$	2.48795e8	0.606317
$290$	0	0
$291$	−1.41331e8	−0.336211
$292$	0	0
$293$	4.39648e8	1.02110	0.510550	−	0.859848i	$-0.329442\pi$
$293$	4.39648e8	1.02110	0.510550	+	0.859848i	$0.329442\pi$
$294$	0	0
$295$	2.12238e7	0.0481332
$296$	0	0
$297$	9.45421e7	0.209400
$298$	0	0
$299$	−1.99130e8	−0.430811
$300$	0	0
$301$	−7.38179e8	−1.56019
$302$	0	0
$303$	−1.39044e7	−0.0287145
$304$	0	0
$305$	1.23089e9	2.48411
$306$	0	0
$307$	7.03335e8	1.38732	0.693662	−	0.720301i	$-0.255996\pi$
$307$	7.03335e8	1.38732	0.693662	+	0.720301i	$0.255996\pi$
$308$	0	0
$309$	−1.63302e8	−0.314874
$310$	0	0
$311$	−5.05993e7	−0.0953857	−0.0476929	−	0.998862i	$-0.515187\pi$
$311$	−5.05993e7	−0.0953857	−0.0476929	+	0.998862i	$0.515187\pi$
$312$	0	0
$313$	−4.44273e8	−0.818926	−0.409463	−	0.912327i	$-0.634284\pi$
$313$	−4.44273e8	−0.818926	−0.409463	+	0.912327i	$0.634284\pi$
$314$	0	0
$315$	3.59768e8	0.648538
$316$	0	0
$317$	−7.80797e7	−0.137667	−0.0688337	−	0.997628i	$-0.521928\pi$
$317$	−7.80797e7	−0.137667	−0.0688337	+	0.997628i	$0.521928\pi$
$318$	0	0
$319$	−1.16637e9	−2.01173
$320$	0	0
$321$	−2.71882e8	−0.458790
$322$	0	0
$323$	−6.27573e8	−1.03623
$324$	0	0
$325$	2.07537e9	3.35354
$326$	0	0
$327$	−3.97472e7	−0.0628622
$328$	0	0
$329$	−1.07269e9	−1.66069
$330$	0	0
$331$	5.34081e8	0.809485	0.404743	−	0.914431i	$-0.367361\pi$
$331$	5.34081e8	0.809485	0.404743	+	0.914431i	$0.367361\pi$
$332$	0	0
$333$	1.06978e8	0.158759
$334$	0	0
$335$	8.69102e8	1.26303
$336$	0	0
$337$	−2.55509e7	−0.0363665	−0.0181832	−	0.999835i	$-0.505788\pi$
$337$	−2.55509e7	−0.0363665	−0.0181832	+	0.999835i	$0.505788\pi$
$338$	0	0
$339$	−2.36417e8	−0.329594
$340$	0	0
$341$	6.62469e8	0.904743
$342$	0	0
$343$	7.27118e8	0.972917
$344$	0	0
$345$	2.79340e8	0.366240
$346$	0	0
$347$	−1.01189e9	−1.30011	−0.650055	−	0.759887i	$-0.725254\pi$
$347$	−1.01189e9	−1.30011	−0.650055	+	0.759887i	$0.725254\pi$
$348$	0	0
$349$	1.15695e9	1.45689	0.728443	−	0.685107i	$-0.240244\pi$
$349$	1.15695e9	1.45689	0.728443	+	0.685107i	$0.240244\pi$
$350$	0	0
$351$	2.00963e8	0.248051
$352$	0	0
$353$	−1.31242e9	−1.58804	−0.794022	−	0.607890i	$-0.792016\pi$
$353$	−1.31242e9	−1.58804	−0.794022	+	0.607890i	$0.792016\pi$
$354$	0	0
$355$	−2.01095e9	−2.38563
$356$	0	0
$357$	6.44894e8	0.750153
$358$	0	0
$359$	−4.62022e7	−0.0527026	−0.0263513	−	0.999653i	$-0.508389\pi$
$359$	−4.62022e7	−0.0527026	−0.0263513	+	0.999653i	$0.508389\pi$
$360$	0	0
$361$	−2.96349e8	−0.331535
$362$	0	0
$363$	9.67653e7	0.106181
$364$	0	0
$365$	5.14289e8	0.553583
$366$	0	0
$367$	8.17215e8	0.862989	0.431494	−	0.902116i	$-0.357986\pi$
$367$	8.17215e8	0.862989	0.431494	+	0.902116i	$0.357986\pi$
$368$	0	0
$369$	2.04019e8	0.211387
$370$	0	0
$371$	6.33965e8	0.644550
$372$	0	0
$373$	2.08683e8	0.208212	0.104106	−	0.994566i	$-0.466802\pi$
$373$	2.08683e8	0.208212	0.104106	+	0.994566i	$0.466802\pi$
$374$	0	0
$375$	−1.79238e9	−1.75518
$376$	0	0
$377$	−2.47930e9	−2.38305
$378$	0	0
$379$	−1.11653e9	−1.05350	−0.526748	−	0.850022i	$-0.676589\pi$
$379$	−1.11653e9	−1.05350	−0.526748	+	0.850022i	$0.676589\pi$
$380$	0	0
$381$	−5.54894e8	−0.514011
$382$	0	0
$383$	6.90286e8	0.627818	0.313909	−	0.949453i	$-0.398361\pi$
$383$	6.90286e8	0.627818	0.313909	+	0.949453i	$0.398361\pi$
$384$	0	0
$385$	2.37044e9	2.11698
$386$	0	0
$387$	5.78431e8	0.507298
$388$	0	0
$389$	−1.85656e9	−1.59914	−0.799569	−	0.600574i	$-0.794939\pi$
$389$	−1.85656e9	−1.59914	−0.799569	+	0.600574i	$0.794939\pi$
$390$	0	0
$391$	5.00725e8	0.423624
$392$	0	0
$393$	−8.27288e8	−0.687516
$394$	0	0
$395$	3.31524e9	2.70661
$396$	0	0
$397$	−1.59743e7	−0.0128131	−0.00640655	−	0.999979i	$-0.502039\pi$
$397$	−1.59743e7	−0.0128131	−0.00640655	+	0.999979i	$0.502039\pi$
$398$	0	0
$399$	−6.14015e8	−0.483920
$400$	0	0
$401$	7.07966e8	0.548286	0.274143	−	0.961689i	$-0.411606\pi$
$401$	7.07966e8	0.548286	0.274143	+	0.961689i	$0.411606\pi$
$402$	0	0
$403$	1.40817e9	1.07174
$404$	0	0
$405$	−2.81911e8	−0.210872
$406$	0	0
$407$	7.04855e8	0.518226
$408$	0	0
$409$	−1.54308e9	−1.11521	−0.557606	−	0.830105i	$-0.688280\pi$
$409$	−1.54308e9	−1.11521	−0.557606	+	0.830105i	$0.688280\pi$
$410$	0	0
$411$	3.22533e8	0.229154
$412$	0	0
$413$	3.72223e7	0.0260003
$414$	0	0
$415$	4.05080e9	2.78210
$416$	0	0
$417$	5.60223e8	0.378342
$418$	0	0
$419$	−3.64486e8	−0.242065	−0.121032	−	0.992649i	$-0.538620\pi$
$419$	−3.64486e8	−0.242065	−0.121032	+	0.992649i	$0.538620\pi$
$420$	0	0
$421$	1.97063e9	1.28712	0.643560	−	0.765396i	$-0.277457\pi$
$421$	1.97063e9	1.28712	0.643560	+	0.765396i	$0.277457\pi$
$422$	0	0
$423$	8.40552e8	0.539975
$424$	0	0
$425$	−5.21864e9	−3.29759
$426$	0	0
$427$	2.15874e9	1.34185
$428$	0	0
$429$	1.32410e9	0.809695
$430$	0	0
$431$	−1.29837e9	−0.781136	−0.390568	−	0.920574i	$-0.627721\pi$
$431$	−1.29837e9	−0.781136	−0.390568	+	0.920574i	$0.627721\pi$
$432$	0	0
$433$	2.48225e9	1.46939	0.734696	−	0.678397i	$-0.237325\pi$
$433$	2.48225e9	1.46939	0.734696	+	0.678397i	$0.237325\pi$
$434$	0	0
$435$	3.47796e9	2.02587
$436$	0	0
$437$	−4.76748e8	−0.273278
$438$	0	0
$439$	8.40045e8	0.473889	0.236945	−	0.971523i	$-0.423854\pi$
$439$	8.40045e8	0.473889	0.236945	+	0.971523i	$0.423854\pi$
$440$	0	0
$441$	3.05992e7	0.0169893
$442$	0	0
$443$	−9.91489e8	−0.541845	−0.270922	−	0.962601i	$-0.587329\pi$
$443$	−9.91489e8	−0.541845	−0.270922	+	0.962601i	$0.587329\pi$
$444$	0	0
$445$	−2.47055e9	−1.32903
$446$	0	0
$447$	−2.09316e9	−1.10847
$448$	0	0
$449$	2.22228e8	0.115861	0.0579305	−	0.998321i	$-0.481550\pi$
$449$	2.22228e8	0.115861	0.0579305	+	0.998321i	$0.481550\pi$
$450$	0	0
$451$	1.34424e9	0.690017
$452$	0	0
$453$	1.73874e9	0.878802
$454$	0	0
$455$	5.03871e9	2.50772
$456$	0	0
$457$	2.72029e8	0.133324	0.0666620	−	0.997776i	$-0.478765\pi$
$457$	2.72029e8	0.133324	0.0666620	+	0.997776i	$0.478765\pi$
$458$	0	0
$459$	−5.05334e8	−0.243912
$460$	0	0
$461$	−2.29484e9	−1.09093	−0.545467	−	0.838132i	$-0.683648\pi$
$461$	−2.29484e9	−1.09093	−0.545467	+	0.838132i	$0.683648\pi$
$462$	0	0
$463$	1.05198e9	0.492579	0.246289	−	0.969196i	$-0.420789\pi$
$463$	1.05198e9	0.492579	0.246289	+	0.969196i	$0.420789\pi$
$464$	0	0
$465$	−1.97539e9	−0.911103
$466$	0	0
$467$	−1.36844e9	−0.621750	−0.310875	−	0.950451i	$-0.600622\pi$
$467$	−1.36844e9	−0.621750	−0.310875	+	0.950451i	$0.600622\pi$
$468$	0	0
$469$	1.52423e9	0.682254
$470$	0	0
$471$	7.36193e8	0.324653
$472$	0	0
$473$	3.81116e9	1.65594
$474$	0	0
$475$	4.96876e9	2.12726
$476$	0	0
$477$	−4.96769e8	−0.209575
$478$	0	0
$479$	−7.98936e8	−0.332152	−0.166076	−	0.986113i	$-0.553110\pi$
$479$	−7.98936e8	−0.332152	−0.166076	+	0.986113i	$0.553110\pi$
$480$	0	0
$481$	1.49827e9	0.613879
$482$	0	0
$483$	4.89907e8	0.197833
$484$	0	0
$485$	2.77671e9	1.10518
$486$	0	0
$487$	−2.57453e9	−1.01006	−0.505029	−	0.863102i	$-0.668518\pi$
$487$	−2.57453e9	−1.01006	−0.505029	+	0.863102i	$0.668518\pi$
$488$	0	0
$489$	−1.35897e9	−0.525567
$490$	0	0
$491$	1.77530e9	0.676842	0.338421	−	0.940995i	$-0.390107\pi$
$491$	1.77530e9	0.676842	0.338421	+	0.940995i	$0.390107\pi$
$492$	0	0
$493$	6.23434e9	2.34329
$494$	0	0
$495$	−1.85746e9	−0.688336
$496$	0	0
$497$	−3.52681e9	−1.28865
$498$	0	0
$499$	−3.41248e9	−1.22947	−0.614735	−	0.788734i	$-0.710737\pi$
$499$	−3.41248e9	−1.22947	−0.614735	+	0.788734i	$0.710737\pi$
$500$	0	0
$501$	−1.77581e9	−0.630904
$502$	0	0
$503$	−1.25973e9	−0.441356	−0.220678	−	0.975347i	$-0.570827\pi$
$503$	−1.25973e9	−0.441356	−0.220678	+	0.975347i	$0.570827\pi$
$504$	0	0
$505$	2.73177e8	0.0943898
$506$	0	0
$507$	1.12036e9	0.381795
$508$	0	0
$509$	−4.42432e9	−1.48708	−0.743540	−	0.668692i	$-0.766855\pi$
$509$	−4.42432e9	−1.48708	−0.743540	+	0.668692i	$0.766855\pi$
$510$	0	0
$511$	9.01962e8	0.299030
$512$	0	0
$513$	4.81137e8	0.157347
$514$	0	0
$515$	3.20838e9	1.03505
$516$	0	0
$517$	5.53823e9	1.76260
$518$	0	0
$519$	1.84581e9	0.579565
$520$	0	0
$521$	−3.01665e9	−0.934529	−0.467264	−	0.884118i	$-0.654760\pi$
$521$	−3.01665e9	−0.934529	−0.467264	+	0.884118i	$0.654760\pi$
$522$	0	0
$523$	−5.45237e9	−1.66659	−0.833297	−	0.552826i	$-0.813549\pi$
$523$	−5.45237e9	−1.66659	−0.833297	+	0.552826i	$0.813549\pi$
$524$	0	0
$525$	−5.10590e9	−1.53998
$526$	0	0
$527$	−3.54094e9	−1.05386
$528$	0	0
$529$	−3.02444e9	−0.888280
$530$	0	0
$531$	−2.91670e7	−0.00845399
$532$	0	0
$533$	2.85738e9	0.817378
$534$	0	0
$535$	5.34164e9	1.50812
$536$	0	0
$537$	−2.80543e9	−0.781789
$538$	0	0
$539$	2.01612e8	0.0554569
$540$	0	0
$541$	−1.27780e9	−0.346955	−0.173478	−	0.984838i	$-0.555500\pi$
$541$	−1.27780e9	−0.346955	−0.173478	+	0.984838i	$0.555500\pi$
$542$	0	0
$543$	−1.86036e9	−0.498652
$544$	0	0
$545$	7.80909e8	0.206639
$546$	0	0
$547$	−5.90809e9	−1.54345	−0.771724	−	0.635958i	$-0.780605\pi$
$547$	−5.90809e9	−1.54345	−0.771724	+	0.635958i	$0.780605\pi$
$548$	0	0
$549$	−1.69157e9	−0.436301
$550$	0	0
$551$	−5.93582e9	−1.51165
$552$	0	0
$553$	5.81428e9	1.46204
$554$	0	0
$555$	−2.10178e9	−0.521869
$556$	0	0
$557$	−1.13395e9	−0.278035	−0.139018	−	0.990290i	$-0.544394\pi$
$557$	−1.13395e9	−0.278035	−0.139018	+	0.990290i	$0.544394\pi$
$558$	0	0
$559$	8.10117e9	1.96158
$560$	0	0
$561$	−3.32954e9	−0.796186
$562$	0	0
$563$	2.21605e9	0.523360	0.261680	−	0.965155i	$-0.415724\pi$
$563$	2.21605e9	0.523360	0.261680	+	0.965155i	$0.415724\pi$
$564$	0	0
$565$	4.64485e9	1.08343
$566$	0	0
$567$	−4.94416e8	−0.113907
$568$	0	0
$569$	−1.95125e9	−0.444039	−0.222019	−	0.975042i	$-0.571265\pi$
$569$	−1.95125e9	−0.444039	−0.222019	+	0.975042i	$0.571265\pi$
$570$	0	0
$571$	−6.28413e9	−1.41260	−0.706299	−	0.707914i	$-0.749637\pi$
$571$	−6.28413e9	−1.41260	−0.706299	+	0.707914i	$0.749637\pi$
$572$	0	0
$573$	−3.10477e8	−0.0689427
$574$	0	0
$575$	−3.96445e9	−0.869652
$576$	0	0
$577$	−5.95604e9	−1.29075	−0.645375	−	0.763866i	$-0.723299\pi$
$577$	−5.95604e9	−1.29075	−0.645375	+	0.763866i	$0.723299\pi$
$578$	0	0
$579$	1.73108e9	0.370631
$580$	0	0
$581$	7.10430e9	1.50281
$582$	0	0
$583$	−3.27311e9	−0.684102
$584$	0	0
$585$	−3.94829e9	−0.815387
$586$	0	0
$587$	−2.03096e9	−0.414447	−0.207223	−	0.978294i	$-0.566443\pi$
$587$	−2.03096e9	−0.414447	−0.207223	+	0.978294i	$0.566443\pi$
$588$	0	0
$589$	3.37139e9	0.679838
$590$	0	0
$591$	4.79797e9	0.956096
$592$	0	0
$593$	7.80592e8	0.153721	0.0768604	−	0.997042i	$-0.475510\pi$
$593$	7.80592e8	0.153721	0.0768604	+	0.997042i	$0.475510\pi$
$594$	0	0
$595$	−1.26702e10	−2.46588
$596$	0	0
$597$	−4.48500e7	−0.00862686
$598$	0	0
$599$	1.85326e9	0.352325	0.176162	−	0.984361i	$-0.443632\pi$
$599$	1.85326e9	0.352325	0.176162	+	0.984361i	$0.443632\pi$
$600$	0	0
$601$	1.50306e9	0.282434	0.141217	−	0.989979i	$-0.454898\pi$
$601$	1.50306e9	0.282434	0.141217	+	0.989979i	$0.454898\pi$
$602$	0	0
$603$	−1.19438e9	−0.221835
$604$	0	0
$605$	−1.90114e9	−0.349035
$606$	0	0
$607$	−1.89042e9	−0.343082	−0.171541	−	0.985177i	$-0.554875\pi$
$607$	−1.89042e9	−0.343082	−0.171541	+	0.985177i	$0.554875\pi$
$608$	0	0
$609$	6.09966e9	1.09432
$610$	0	0
$611$	1.17723e10	2.08794
$612$	0	0
$613$	−8.21094e9	−1.43973	−0.719865	−	0.694114i	$-0.755796\pi$
$613$	−8.21094e9	−1.43973	−0.719865	+	0.694114i	$0.755796\pi$
$614$	0	0
$615$	−4.00834e9	−0.694868
$616$	0	0
$617$	9.13340e9	1.56543	0.782717	−	0.622378i	$-0.213833\pi$
$617$	9.13340e9	1.56543	0.782717	+	0.622378i	$0.213833\pi$
$618$	0	0
$619$	−3.93564e9	−0.666957	−0.333479	−	0.942758i	$-0.608222\pi$
$619$	−3.93564e9	−0.666957	−0.333479	+	0.942758i	$0.608222\pi$
$620$	0	0
$621$	−3.83887e8	−0.0643254
$622$	0	0
$623$	−4.33286e9	−0.717904
$624$	0	0
$625$	1.93343e10	3.16773
$626$	0	0
$627$	3.17011e9	0.513616
$628$	0	0
$629$	−3.76750e9	−0.603637
$630$	0	0
$631$	−5.29696e9	−0.839313	−0.419656	−	0.907683i	$-0.637849\pi$
$631$	−5.29696e9	−0.839313	−0.419656	+	0.907683i	$0.637849\pi$
$632$	0	0
$633$	6.08040e9	0.952837
$634$	0	0
$635$	1.09019e10	1.68964
$636$	0	0
$637$	4.28555e8	0.0656929
$638$	0	0
$639$	2.76358e9	0.419005
$640$	0	0
$641$	9.15727e9	1.37329	0.686646	−	0.726992i	$-0.259082\pi$
$641$	9.15727e9	1.37329	0.686646	+	0.726992i	$0.259082\pi$
$642$	0	0
$643$	−1.18605e9	−0.175940	−0.0879698	−	0.996123i	$-0.528038\pi$
$643$	−1.18605e9	−0.175940	−0.0879698	+	0.996123i	$0.528038\pi$
$644$	0	0
$645$	−1.13644e10	−1.66758
$646$	0	0
$647$	6.58435e9	0.955757	0.477879	−	0.878426i	$-0.341406\pi$
$647$	6.58435e9	0.955757	0.477879	+	0.878426i	$0.341406\pi$
$648$	0	0
$649$	−1.92176e8	−0.0275958
$650$	0	0
$651$	−3.46444e9	−0.492153
$652$	0	0
$653$	−1.87870e9	−0.264035	−0.132018	−	0.991247i	$-0.542146\pi$
$653$	−1.87870e9	−0.264035	−0.132018	+	0.991247i	$0.542146\pi$
$654$	0	0
$655$	1.62536e10	2.25999
$656$	0	0
$657$	−7.06769e8	−0.0972297
$658$	0	0
$659$	−1.00667e10	−1.37022	−0.685108	−	0.728441i	$-0.740245\pi$
$659$	−1.00667e10	−1.37022	−0.685108	+	0.728441i	$0.740245\pi$
$660$	0	0
$661$	−7.98681e9	−1.07564	−0.537821	−	0.843059i	$-0.680753\pi$
$661$	−7.98681e9	−1.07564	−0.537821	+	0.843059i	$0.680753\pi$
$662$	0	0
$663$	−7.07742e9	−0.943143
$664$	0	0
$665$	1.20635e10	1.59073
$666$	0	0
$667$	4.73605e9	0.617982
$668$	0	0
$669$	−1.33324e9	−0.172154
$670$	0	0
$671$	−1.11454e10	−1.42419
$672$	0	0
$673$	5.79369e9	0.732660	0.366330	−	0.930485i	$-0.380614\pi$
$673$	5.79369e9	0.732660	0.366330	+	0.930485i	$0.380614\pi$
$674$	0	0
$675$	4.00094e9	0.500724
$676$	0	0
$677$	1.23248e9	0.152657	0.0763287	−	0.997083i	$-0.475680\pi$
$677$	1.23248e9	0.152657	0.0763287	+	0.997083i	$0.475680\pi$
$678$	0	0
$679$	4.86980e9	0.596990
$680$	0	0
$681$	−9.02708e9	−1.09530
$682$	0	0
$683$	8.18833e9	0.983383	0.491691	−	0.870770i	$-0.336379\pi$
$683$	8.18833e9	0.983383	0.491691	+	0.870770i	$0.336379\pi$
$684$	0	0
$685$	−6.33676e9	−0.753269
$686$	0	0
$687$	−4.49847e9	−0.529317
$688$	0	0
$689$	−6.95747e9	−0.810372
$690$	0	0
$691$	−1.52637e9	−0.175990	−0.0879948	−	0.996121i	$-0.528046\pi$
$691$	−1.52637e9	−0.175990	−0.0879948	+	0.996121i	$0.528046\pi$
$692$	0	0
$693$	−3.25761e9	−0.371820
$694$	0	0
$695$	−1.10066e10	−1.24368
$696$	0	0
$697$	−7.18507e9	−0.803741
$698$	0	0
$699$	7.48388e9	0.828813
$700$	0	0
$701$	3.00901e9	0.329921	0.164961	−	0.986300i	$-0.447250\pi$
$701$	3.00901e9	0.329921	0.164961	+	0.986300i	$0.447250\pi$
$702$	0	0
$703$	3.58710e9	0.389403
$704$	0	0
$705$	−1.65142e10	−1.77499
$706$	0	0
$707$	4.79099e8	0.0509867
$708$	0	0
$709$	6.60947e9	0.696474	0.348237	−	0.937407i	$-0.386780\pi$
$709$	6.60947e9	0.696474	0.348237	+	0.937407i	$0.386780\pi$
$710$	0	0
$711$	−4.55602e9	−0.475381
$712$	0	0
$713$	−2.68995e9	−0.277927
$714$	0	0
$715$	−2.60145e10	−2.66161
$716$	0	0
$717$	−6.29591e9	−0.637884
$718$	0	0
$719$	−9.58311e9	−0.961513	−0.480757	−	0.876854i	$-0.659638\pi$
$719$	−9.58311e9	−0.961513	−0.480757	+	0.876854i	$0.659638\pi$
$720$	0	0
$721$	5.62686e9	0.559104
$722$	0	0
$723$	−2.30448e9	−0.226772
$724$	0	0
$725$	−4.93599e10	−4.81052
$726$	0	0
$727$	−7.40647e9	−0.714893	−0.357447	−	0.933934i	$-0.616353\pi$
$727$	−7.40647e9	−0.714893	−0.357447	+	0.933934i	$0.616353\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	−2.03709e10	−1.92886
$732$	0	0
$733$	1.58074e10	1.48251	0.741255	−	0.671223i	$-0.234231\pi$
$733$	1.58074e10	1.48251	0.741255	+	0.671223i	$0.234231\pi$
$734$	0	0
$735$	−6.01178e8	−0.0558467
$736$	0	0
$737$	−7.86950e9	−0.724121
$738$	0	0
$739$	9.33337e9	0.850713	0.425356	−	0.905026i	$-0.360149\pi$
$739$	9.33337e9	0.850713	0.425356	+	0.905026i	$0.360149\pi$
$740$	0	0
$741$	6.73853e9	0.608417
$742$	0	0
$743$	−1.19221e10	−1.06633	−0.533163	−	0.846012i	$-0.678997\pi$
$743$	−1.19221e10	−1.06633	−0.533163	+	0.846012i	$0.678997\pi$
$744$	0	0
$745$	4.11240e10	3.64374
$746$	0	0
$747$	−5.56687e9	−0.488640
$748$	0	0
$749$	9.36818e9	0.814646
$750$	0	0
$751$	−1.55698e10	−1.34135	−0.670677	−	0.741750i	$-0.733996\pi$
$751$	−1.55698e10	−1.34135	−0.670677	+	0.741750i	$0.733996\pi$
$752$	0	0
$753$	6.56118e9	0.560015
$754$	0	0
$755$	−3.41608e10	−2.88878
$756$	0	0
$757$	2.36270e9	0.197958	0.0989790	−	0.995090i	$-0.468442\pi$
$757$	2.36270e9	0.197958	0.0989790	+	0.995090i	$0.468442\pi$
$758$	0	0
$759$	−2.52935e9	−0.209973
$760$	0	0
$761$	−1.55817e10	−1.28165	−0.640823	−	0.767689i	$-0.721407\pi$
$761$	−1.55817e10	−1.28165	−0.640823	+	0.767689i	$0.721407\pi$
$762$	0	0
$763$	1.36956e9	0.111621
$764$	0	0
$765$	9.92823e9	0.801783
$766$	0	0
$767$	−4.08497e8	−0.0326893
$768$	0	0
$769$	−4.29713e9	−0.340750	−0.170375	−	0.985379i	$-0.554498\pi$
$769$	−4.29713e9	−0.340750	−0.170375	+	0.985379i	$0.554498\pi$
$770$	0	0
$771$	2.33384e9	0.183392
$772$	0	0
$773$	−6.93932e9	−0.540367	−0.270184	−	0.962809i	$-0.587084\pi$
$773$	−6.93932e9	−0.540367	−0.270184	+	0.962809i	$0.587084\pi$
$774$	0	0
$775$	2.80351e10	2.16345
$776$	0	0
$777$	−3.68610e9	−0.281899
$778$	0	0
$779$	6.84102e9	0.518489
$780$	0	0
$781$	1.82087e10	1.36773
$782$	0	0
$783$	−4.77964e9	−0.355819
$784$	0	0
$785$	−1.44639e10	−1.06719
$786$	0	0
$787$	2.00164e10	1.46377	0.731886	−	0.681427i	$-0.238640\pi$
$787$	2.00164e10	1.46377	0.731886	+	0.681427i	$0.238640\pi$
$788$	0	0
$789$	−1.11116e9	−0.0805389
$790$	0	0
$791$	8.14615e9	0.585241
$792$	0	0
$793$	−2.36912e10	−1.68706
$794$	0	0
$795$	9.75996e9	0.688911
$796$	0	0
$797$	−4.77813e9	−0.334313	−0.167157	−	0.985930i	$-0.553459\pi$
$797$	−4.77813e9	−0.334313	−0.167157	+	0.985930i	$0.553459\pi$
$798$	0	0
$799$	−2.96022e10	−2.05310
$800$	0	0
$801$	3.39519e9	0.233427
$802$	0	0
$803$	−4.65676e9	−0.317380
$804$	0	0
$805$	−9.62514e9	−0.650312
$806$	0	0
$807$	−1.08411e10	−0.726136
$808$	0	0
$809$	−2.64254e10	−1.75470	−0.877349	−	0.479853i	$-0.840690\pi$
$809$	−2.64254e10	−1.75470	−0.877349	+	0.479853i	$0.840690\pi$
$810$	0	0
$811$	−2.71149e10	−1.78498	−0.892492	−	0.451063i	$-0.851045\pi$
$811$	−2.71149e10	−1.78498	−0.892492	+	0.451063i	$0.851045\pi$
$812$	0	0
$813$	8.99900e9	0.587323
$814$	0	0
$815$	2.66995e10	1.72763
$816$	0	0
$817$	1.93955e10	1.24430
$818$	0	0
$819$	−6.92452e9	−0.440449
$820$	0	0
$821$	2.07029e10	1.30566	0.652830	−	0.757505i	$-0.273582\pi$
$821$	2.07029e10	1.30566	0.652830	+	0.757505i	$0.273582\pi$
$822$	0	0
$823$	−3.03545e9	−0.189812	−0.0949059	−	0.995486i	$-0.530255\pi$
$823$	−3.03545e9	−0.189812	−0.0949059	+	0.995486i	$0.530255\pi$
$824$	0	0
$825$	2.63614e10	1.63448
$826$	0	0
$827$	−1.59965e9	−0.0983460	−0.0491730	−	0.998790i	$-0.515659\pi$
$827$	−1.59965e9	−0.0983460	−0.0491730	+	0.998790i	$0.515659\pi$
$828$	0	0
$829$	−1.37869e10	−0.840477	−0.420239	−	0.907414i	$-0.638054\pi$
$829$	−1.37869e10	−0.840477	−0.420239	+	0.907414i	$0.638054\pi$
$830$	0	0
$831$	4.17293e9	0.252254
$832$	0	0
$833$	−1.07763e9	−0.0645969
$834$	0	0
$835$	3.48890e10	2.07389
$836$	0	0
$837$	2.71471e9	0.160024
$838$	0	0
$839$	−3.87002e9	−0.226228	−0.113114	−	0.993582i	$-0.536083\pi$
$839$	−3.87002e9	−0.226228	−0.113114	+	0.993582i	$0.536083\pi$
$840$	0	0
$841$	4.17170e10	2.41839
$842$	0	0
$843$	8.70323e9	0.500362
$844$	0	0
$845$	−2.20116e10	−1.25503
$846$	0	0
$847$	−3.33422e9	−0.188539
$848$	0	0
$849$	−1.34509e10	−0.754354
$850$	0	0
$851$	−2.86206e9	−0.159193
$852$	0	0
$853$	9.20942e9	0.508055	0.254027	−	0.967197i	$-0.418245\pi$
$853$	9.20942e9	0.508055	0.254027	+	0.967197i	$0.418245\pi$
$854$	0	0
$855$	−9.45283e9	−0.517226
$856$	0	0
$857$	3.21307e10	1.74376	0.871882	−	0.489715i	$-0.162899\pi$
$857$	3.21307e10	1.74376	0.871882	+	0.489715i	$0.162899\pi$
$858$	0	0
$859$	−3.17662e10	−1.70997	−0.854987	−	0.518650i	$-0.826435\pi$
$859$	−3.17662e10	−1.70997	−0.854987	+	0.518650i	$0.826435\pi$
$860$	0	0
$861$	−7.02984e9	−0.375348
$862$	0	0
$863$	−2.41602e10	−1.27957	−0.639784	−	0.768555i	$-0.720976\pi$
$863$	−2.41602e10	−1.27957	−0.639784	+	0.768555i	$0.720976\pi$
$864$	0	0
$865$	−3.62645e10	−1.90513
$866$	0	0
$867$	6.71747e9	0.350057
$868$	0	0
$869$	−3.00187e10	−1.55175
$870$	0	0
$871$	−1.67278e10	−0.857776
$872$	0	0
$873$	−3.81594e9	−0.194111
$874$	0	0
$875$	6.17596e10	3.11657
$876$	0	0
$877$	4.85014e8	0.0242804	0.0121402	−	0.999926i	$-0.496136\pi$
$877$	4.85014e8	0.0242804	0.0121402	+	0.999926i	$0.496136\pi$
$878$	0	0
$879$	1.18705e10	0.589532
$880$	0	0
$881$	−1.92738e10	−0.949626	−0.474813	−	0.880087i	$-0.657484\pi$
$881$	−1.92738e10	−0.949626	−0.474813	+	0.880087i	$0.657484\pi$
$882$	0	0
$883$	−2.80412e10	−1.37067	−0.685336	−	0.728227i	$-0.740345\pi$
$883$	−2.80412e10	−1.37067	−0.685336	+	0.728227i	$0.740345\pi$
$884$	0	0
$885$	5.73041e8	0.0277897
$886$	0	0
$887$	3.01064e10	1.44853	0.724263	−	0.689524i	$-0.242180\pi$
$887$	3.01064e10	1.44853	0.724263	+	0.689524i	$0.242180\pi$
$888$	0	0
$889$	1.91198e10	0.912699
$890$	0	0
$891$	2.55264e9	0.120897
$892$	0	0
$893$	2.81848e10	1.32445
$894$	0	0
$895$	5.51180e10	2.56988
$896$	0	0
$897$	−5.37651e9	−0.248729
$898$	0	0
$899$	−3.34916e10	−1.53736
$900$	0	0
$901$	1.74950e10	0.796851
$902$	0	0
$903$	−1.99308e10	−0.900779
$904$	0	0
$905$	3.65503e10	1.63916
$906$	0	0
$907$	3.84680e10	1.71188	0.855942	−	0.517072i	$-0.172978\pi$
$907$	3.84680e10	1.71188	0.855942	+	0.517072i	$0.172978\pi$
$908$	0	0
$909$	−3.75418e8	−0.0165784
$910$	0	0
$911$	2.36782e10	1.03761	0.518805	−	0.854892i	$-0.326377\pi$
$911$	2.36782e10	1.03761	0.518805	+	0.854892i	$0.326377\pi$
$912$	0	0
$913$	−3.66790e10	−1.59503
$914$	0	0
$915$	3.32341e10	1.43420
$916$	0	0
$917$	2.85056e10	1.22078
$918$	0	0
$919$	4.49843e10	1.91186	0.955932	−	0.293588i	$-0.0948495\pi$
$919$	4.49843e10	1.91186	0.955932	+	0.293588i	$0.0948495\pi$
$920$	0	0
$921$	1.89900e10	0.800972
$922$	0	0
$923$	3.87052e10	1.62018
$924$	0	0
$925$	2.98289e10	1.23920
$926$	0	0
$927$	−4.40916e9	−0.181793
$928$	0	0
$929$	−7.23071e9	−0.295887	−0.147944	−	0.988996i	$-0.547265\pi$
$929$	−7.23071e9	−0.295887	−0.147944	+	0.988996i	$0.547265\pi$
$930$	0	0
$931$	1.02603e9	0.0416712
$932$	0	0
$933$	−1.36618e9	−0.0550710
$934$	0	0
$935$	6.54151e10	2.61720
$936$	0	0
$937$	4.00467e10	1.59030	0.795148	−	0.606415i	$-0.207393\pi$
$937$	4.00467e10	1.59030	0.795148	+	0.606415i	$0.207393\pi$
$938$	0	0
$939$	−1.19954e10	−0.472807
$940$	0	0
$941$	3.15429e10	1.23406	0.617032	−	0.786938i	$-0.288335\pi$
$941$	3.15429e10	1.23406	0.617032	+	0.786938i	$0.288335\pi$
$942$	0	0
$943$	−5.45828e9	−0.211965
$944$	0	0
$945$	9.71374e9	0.374434
$946$	0	0
$947$	−1.45818e10	−0.557940	−0.278970	−	0.960300i	$-0.589993\pi$
$947$	−1.45818e10	−0.557940	−0.278970	+	0.960300i	$0.589993\pi$
$948$	0	0
$949$	−9.89862e9	−0.375961
$950$	0	0
$951$	−2.10815e9	−0.0794823
$952$	0	0
$953$	−1.47800e10	−0.553158	−0.276579	−	0.960991i	$-0.589201\pi$
$953$	−1.47800e10	−0.553158	−0.276579	+	0.960991i	$0.589201\pi$
$954$	0	0
$955$	6.09991e9	0.226627
$956$	0	0
$957$	−3.14921e10	−1.16147
$958$	0	0
$959$	−1.11134e10	−0.406895
$960$	0	0
$961$	−8.49030e9	−0.308597
$962$	0	0
$963$	−7.34083e9	−0.264882
$964$	0	0
$965$	−3.40103e10	−1.21833
$966$	0	0
$967$	3.51653e10	1.25061	0.625304	−	0.780381i	$-0.284975\pi$
$967$	3.51653e10	1.25061	0.625304	+	0.780381i	$0.284975\pi$
$968$	0	0
$969$	−1.69445e10	−0.598266
$970$	0	0
$971$	−4.36232e9	−0.152915	−0.0764576	−	0.997073i	$-0.524361\pi$
$971$	−4.36232e9	−0.152915	−0.0764576	+	0.997073i	$0.524361\pi$
$972$	0	0
$973$	−1.93035e10	−0.671800
$974$	0	0
$975$	5.60349e10	1.93617
$976$	0	0
$977$	−1.19268e10	−0.409159	−0.204579	−	0.978850i	$-0.565583\pi$
$977$	−1.19268e10	−0.409159	−0.204579	+	0.978850i	$0.565583\pi$
$978$	0	0
$979$	2.23702e10	0.761958
$980$	0	0
$981$	−1.07318e9	−0.0362935
$982$	0	0
$983$	−5.38784e9	−0.180916	−0.0904581	−	0.995900i	$-0.528833\pi$
$983$	−5.38784e9	−0.180916	−0.0904581	+	0.995900i	$0.528833\pi$
$984$	0	0
$985$	−9.42652e10	−3.14286
$986$	0	0
$987$	−2.89627e10	−0.958802
$988$	0	0
$989$	−1.54752e10	−0.508685
$990$	0	0
$991$	−3.25517e10	−1.06247	−0.531234	−	0.847225i	$-0.678272\pi$
$991$	−3.25517e10	−1.06247	−0.531234	+	0.847225i	$0.678272\pi$
$992$	0	0
$993$	1.44202e10	0.467357
$994$	0	0
$995$	8.81163e8	0.0283580
$996$	0	0
$997$	3.55618e9	0.113645	0.0568225	−	0.998384i	$-0.481903\pi$
$997$	3.55618e9	0.113645	0.0568225	+	0.998384i	$0.481903\pi$
$998$	0	0
$999$	2.88840e9	0.0916596

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.8.a.r.1.1	yes	4
4.3	odd	2		384.8.a.n.1.1	✓	4
8.3	odd	2		384.8.a.s.1.4	yes	4
8.5	even	2		384.8.a.o.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.8.a.n.1.1	✓	4	4.3	odd	2
384.8.a.o.1.4	yes	4	8.5	even	2
384.8.a.r.1.1	yes	4	1.1	even	1	trivial
384.8.a.s.1.4	yes	4	8.3	odd	2

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:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	384.a (trivial)

\(f(q)\)	\(=\)	\(q+27.0000 q^{3} -530.466 q^{5} -930.332 q^{7} +729.000 q^{9} +O(q^{10})\)	\(q+27.0000 q^{3} -530.466 q^{5} -930.332 q^{7} +729.000 q^{9} +4803.24 q^{11} +10210.0 q^{13} -14322.6 q^{15} -25673.6 q^{17} +24444.3 q^{19} -25119.0 q^{21} -19503.5 q^{23} +203269. q^{25} +19683.0 q^{27} -242831. q^{29} +137921. q^{31} +129687. q^{33} +493509. q^{35} +146746. q^{37} +275669. q^{39} +279862. q^{41} +793457. q^{43} -386709. q^{45} +1.15302e6 q^{47} +41974.2 q^{49} -693187. q^{51} -681439. q^{53} -2.54795e6 q^{55} +659995. q^{57} -40009.7 q^{59} -2.32040e6 q^{61} -678212. q^{63} -5.41604e6 q^{65} -1.63837e6 q^{67} -526594. q^{69} +3.79092e6 q^{71} -969505. q^{73} +5.48826e6 q^{75} -4.46860e6 q^{77} -6.24969e6 q^{79} +531441. q^{81} -7.63631e6 q^{83} +1.36190e7 q^{85} -6.55643e6 q^{87} +4.65733e6 q^{89} -9.49866e6 q^{91} +3.72388e6 q^{93} -1.29668e7 q^{95} -5.23448e6 q^{97} +3.50156e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 108 q^{3} - 192 q^{5} - 680 q^{7} + 2916 q^{9}+O(q^{10}) \) 4 * q + 108 * q^3 - 192 * q^5 - 680 * q^7 + 2916 * q^9	\( 4 q + 108 q^{3} - 192 q^{5} - 680 q^{7} + 2916 q^{9} - 4496 q^{11} + 12840 q^{13} - 5184 q^{15} - 14952 q^{17} - 21504 q^{19} - 18360 q^{21} - 54992 q^{23} + 190732 q^{25} + 78732 q^{27} - 242384 q^{29} - 151432 q^{31} - 121392 q^{33} + 273984 q^{35} - 113288 q^{37} + 346680 q^{39} - 239176 q^{41} + 1495328 q^{43} - 139968 q^{45} - 772368 q^{47} - 1577100 q^{49} - 403704 q^{51} - 2389776 q^{53} - 2590080 q^{55} - 580608 q^{57} + 141232 q^{59} - 1231304 q^{61} - 495720 q^{63} - 1041024 q^{65} - 441392 q^{67} - 1484784 q^{69} - 1507504 q^{71} - 1516840 q^{73} + 5149764 q^{75} - 12340448 q^{77} - 9540936 q^{79} + 2125764 q^{81} - 4587600 q^{83} - 6382848 q^{85} - 6544368 q^{87} + 162376 q^{89} - 4681104 q^{91} - 4088664 q^{93} - 29221248 q^{95} + 2726760 q^{97} - 3277584 q^{99}+O(q^{100}) \) 4 * q + 108 * q^3 - 192 * q^5 - 680 * q^7 + 2916 * q^9 - 4496 * q^11 + 12840 * q^13 - 5184 * q^15 - 14952 * q^17 - 21504 * q^19 - 18360 * q^21 - 54992 * q^23 + 190732 * q^25 + 78732 * q^27 - 242384 * q^29 - 151432 * q^31 - 121392 * q^33 + 273984 * q^35 - 113288 * q^37 + 346680 * q^39 - 239176 * q^41 + 1495328 * q^43 - 139968 * q^45 - 772368 * q^47 - 1577100 * q^49 - 403704 * q^51 - 2389776 * q^53 - 2590080 * q^55 - 580608 * q^57 + 141232 * q^59 - 1231304 * q^61 - 495720 * q^63 - 1041024 * q^65 - 441392 * q^67 - 1484784 * q^69 - 1507504 * q^71 - 1516840 * q^73 + 5149764 * q^75 - 12340448 * q^77 - 9540936 * q^79 + 2125764 * q^81 - 4587600 * q^83 - 6382848 * q^85 - 6544368 * q^87 + 162376 * q^89 - 4681104 * q^91 - 4088664 * q^93 - 29221248 * q^95 + 2726760 * q^97 - 3277584 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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