LMFDB - Embedded newform 384.5.b.a.319.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,5,Mod(319,384)]

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, 1, 0]))

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

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

chi := DirichletCharacter("384.319");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	384.b (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:	$\Q(\sqrt{3}, \sqrt{-19})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} + 5x^{2} - 4x + 61 $ x^4 - 2x^3 + 5x^2 - 4*x + 61
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			319.1
Root			$-1.23205 + 2.17945i$ of defining polynomial
Character	$\chi$	$=$	384.319
Dual form			384.5.b.a.319.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.19615 q^{3} -30.1993i q^{5} +52.3068i q^{7} +27.0000 q^{9} +O(q^{10})$	$q-5.19615 q^{3} -30.1993i q^{5} +52.3068i q^{7} +27.0000 q^{9} +90.0666 q^{11} -60.3987i q^{13} +156.920i q^{15} -338.000 q^{17} +6.92820 q^{19} -271.794i q^{21} +732.295i q^{23} -287.000 q^{25} -140.296 q^{27} +1298.57i q^{29} -1307.67i q^{31} -468.000 q^{33} +1579.63 q^{35} -241.595i q^{37} +313.841i q^{39} +578.000 q^{41} +2029.96 q^{43} -815.382i q^{45} -2196.89i q^{47} -335.000 q^{49} +1756.30 q^{51} +2446.15i q^{53} -2719.95i q^{55} -36.0000 q^{57} +1198.58 q^{59} -6402.26i q^{61} +1412.28i q^{63} -1824.00 q^{65} +8265.35 q^{67} -3805.12i q^{69} +4289.16i q^{71} +8734.00 q^{73} +1491.30 q^{75} +4711.10i q^{77} +11246.0i q^{79} +729.000 q^{81} +13198.2 q^{83} +10207.4i q^{85} -6747.58i q^{87} +910.000 q^{89} +3159.26 q^{91} +6794.85i q^{93} -209.227i q^{95} +5422.00 q^{97} +2431.80 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 108 q^{9}+O(q^{10}) $ 4 * q + 108 * q^9	$ 4 q + 108 q^{9} - 1352 q^{17} - 1148 q^{25} - 1872 q^{33} + 2312 q^{41} - 1340 q^{49} - 144 q^{57} - 7296 q^{65} + 34936 q^{73} + 2916 q^{81} + 3640 q^{89} + 21688 q^{97}+O(q^{100}) $ 4 * q + 108 * q^9 - 1352 * q^17 - 1148 * q^25 - 1872 * q^33 + 2312 * q^41 - 1340 * q^49 - 144 * q^57 - 7296 * q^65 + 34936 * q^73 + 2916 * q^81 + 3640 * q^89 + 21688 * 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
$3$	−5.19615	−0.577350
$3$	−5.19615	−0.577350
$4$	0	0
$5$	− 30.1993i	− 1.20797i	−0.796994	−	0.603987i	$-0.793578\pi$
$5$	− 30.1993i	− 1.20797i	0.796994	−	0.603987i	$-0.206422\pi$
$6$	0	0
$7$	52.3068i	1.06749i	0.845647	+	0.533743i	$0.179215\pi$
$7$	52.3068i	1.06749i	−0.845647	+	0.533743i	$0.820785\pi$
$8$	0	0
$9$	27.0000	0.333333
$10$	0	0
$11$	90.0666	0.744352	0.372176	−	0.928162i	$-0.378612\pi$
$11$	90.0666	0.744352	0.372176	+	0.928162i	$0.378612\pi$
$12$	0	0
$13$	− 60.3987i	− 0.357389i	−0.983905	−	0.178694i	$-0.942813\pi$
$13$	− 60.3987i	− 0.357389i	0.983905	−	0.178694i	$-0.0571873\pi$
$14$	0	0
$15$	156.920i	0.697424i
$16$	0	0
$17$	−338.000	−1.16955	−0.584775	−	0.811195i	$-0.698817\pi$
$17$	−338.000	−1.16955	−0.584775	+	0.811195i	$0.698817\pi$
$18$	0	0
$19$	6.92820	0.0191917	0.00959585	−	0.999954i	$-0.496945\pi$
$19$	6.92820	0.0191917	0.00959585	+	0.999954i	$0.496945\pi$
$20$	0	0
$21$	− 271.794i	− 0.616313i
$22$	0	0
$23$	732.295i	1.38430i	0.721753	+	0.692150i	$0.243336\pi$
$23$	732.295i	1.38430i	−0.721753	+	0.692150i	$0.756664\pi$
$24$	0	0
$25$	−287.000	−0.459200
$26$	0	0
$27$	−140.296	−0.192450
$28$	0	0
$29$	1298.57i	1.54408i	0.635574	+	0.772040i	$0.280764\pi$
$29$	1298.57i	1.54408i	−0.635574	+	0.772040i	$0.719236\pi$
$30$	0	0
$31$	− 1307.67i	− 1.36074i	−0.732869	−	0.680369i	$-0.761819\pi$
$31$	− 1307.67i	− 1.36074i	0.732869	−	0.680369i	$-0.238181\pi$
$32$	0	0
$33$	−468.000	−0.429752
$34$	0	0
$35$	1579.63	1.28949
$36$	0	0
$37$	− 241.595i	− 0.176475i	−0.996099	−	0.0882377i	$-0.971877\pi$
$37$	− 241.595i	− 0.176475i	0.996099	−	0.0882377i	$-0.0281235\pi$
$38$	0	0
$39$	313.841i	0.206338i
$40$	0	0
$41$	578.000	0.343843	0.171921	−	0.985111i	$-0.445002\pi$
$41$	578.000	0.343843	0.171921	+	0.985111i	$0.445002\pi$
$42$	0	0
$43$	2029.96	1.09787	0.548936	−	0.835865i	$-0.315033\pi$
$43$	2029.96	1.09787	0.548936	+	0.835865i	$0.315033\pi$
$44$	0	0
$45$	− 815.382i	− 0.402658i
$46$	0	0
$47$	− 2196.89i	− 0.994516i	−0.867603	−	0.497258i	$-0.834340\pi$
$47$	− 2196.89i	− 0.994516i	0.867603	−	0.497258i	$-0.165660\pi$
$48$	0	0
$49$	−335.000	−0.139525
$50$	0	0
$51$	1756.30	0.675240
$52$	0	0
$53$	2446.15i	0.870825i	0.900231	+	0.435412i	$0.143397\pi$
$53$	2446.15i	0.870825i	−0.900231	+	0.435412i	$0.856603\pi$
$54$	0	0
$55$	− 2719.95i	− 0.899158i
$56$	0	0
$57$	−36.0000	−0.0110803
$58$	0	0
$59$	1198.58	0.344320	0.172160	−	0.985069i	$-0.444925\pi$
$59$	1198.58	0.344320	0.172160	+	0.985069i	$0.444925\pi$
$60$	0	0
$61$	− 6402.26i	− 1.72058i	−0.509809	−	0.860288i	$-0.670284\pi$
$61$	− 6402.26i	− 1.72058i	0.509809	−	0.860288i	$-0.329716\pi$
$62$	0	0
$63$	1412.28i	0.355828i
$64$	0	0
$65$	−1824.00	−0.431716
$66$	0	0
$67$	8265.35	1.84124	0.920622	−	0.390454i	$-0.127682\pi$
$67$	8265.35	1.84124	0.920622	+	0.390454i	$0.127682\pi$
$68$	0	0
$69$	− 3805.12i	− 0.799226i
$70$	0	0
$71$	4289.16i	0.850854i	0.904993	+	0.425427i	$0.139876\pi$
$71$	4289.16i	0.850854i	−0.904993	+	0.425427i	$0.860124\pi$
$72$	0	0
$73$	8734.00	1.63896	0.819478	−	0.573110i	$-0.194263\pi$
$73$	8734.00	1.63896	0.819478	+	0.573110i	$0.194263\pi$
$74$	0	0
$75$	1491.30	0.265119
$76$	0	0
$77$	4711.10i	0.794585i
$78$	0	0
$79$	11246.0i	1.80195i	0.433873	+	0.900974i	$0.357147\pi$
$79$	11246.0i	1.80195i	−0.433873	+	0.900974i	$0.642853\pi$
$80$	0	0
$81$	729.000	0.111111
$82$	0	0
$83$	13198.2	1.91584	0.957920	−	0.287034i	$-0.0926693\pi$
$83$	13198.2	1.91584	0.957920	+	0.287034i	$0.0926693\pi$
$84$	0	0
$85$	10207.4i	1.41279i
$86$	0	0
$87$	− 6747.58i	− 0.891475i
$88$	0	0
$89$	910.000	0.114884	0.0574422	−	0.998349i	$-0.481705\pi$
$89$	910.000	0.114884	0.0574422	+	0.998349i	$0.481705\pi$
$90$	0	0
$91$	3159.26	0.381507
$92$	0	0
$93$	6794.85i	0.785623i
$94$	0	0
$95$	− 209.227i	− 0.0231831i
$96$	0	0
$97$	5422.00	0.576257	0.288128	−	0.957592i	$-0.406967\pi$
$97$	5422.00	0.576257	0.288128	+	0.957592i	$0.406967\pi$
$98$	0	0
$99$	2431.80	0.248117
$100$	0	0
$101$	− 10962.4i	− 1.07464i	−0.843380	−	0.537318i	$-0.819438\pi$
$101$	− 10962.4i	− 1.07464i	0.843380	−	0.537318i	$-0.180562\pi$
$102$	0	0
$103$	5387.60i	0.507833i	0.967226	+	0.253916i	$0.0817188\pi$
$103$	5387.60i	0.507833i	−0.967226	+	0.253916i	$0.918281\pi$
$104$	0	0
$105$	−8208.00	−0.744490
$106$	0	0
$107$	6436.30	0.562171	0.281086	−	0.959683i	$-0.409305\pi$
$107$	6436.30	0.562171	0.281086	+	0.959683i	$0.409305\pi$
$108$	0	0
$109$	60.3987i	0.00508364i	0.999997	+	0.00254182i	$0.000809087\pi$
$109$	60.3987i	0.00508364i	−0.999997	+	0.00254182i	$0.999191\pi$
$110$	0	0
$111$	1255.36i	0.101888i
$112$	0	0
$113$	−3166.00	−0.247944	−0.123972	−	0.992286i	$-0.539563\pi$
$113$	−3166.00	−0.247944	−0.123972	+	0.992286i	$0.539563\pi$
$114$	0	0
$115$	22114.8	1.67220
$116$	0	0
$117$	− 1630.76i	− 0.119130i
$118$	0	0
$119$	− 17679.7i	− 1.24848i
$120$	0	0
$121$	−6529.00	−0.445939
$122$	0	0
$123$	−3003.38	−0.198518
$124$	0	0
$125$	− 10207.4i	− 0.653272i
$126$	0	0
$127$	− 12919.8i	− 0.801028i	−0.916291	−	0.400514i	$-0.868832\pi$
$127$	− 12919.8i	− 0.801028i	0.916291	−	0.400514i	$-0.131168\pi$
$128$	0	0
$129$	−10548.0	−0.633856
$130$	0	0
$131$	22676.0	1.32137	0.660684	−	0.750664i	$-0.270266\pi$
$131$	22676.0	1.32137	0.660684	+	0.750664i	$0.270266\pi$
$132$	0	0
$133$	362.392i	0.0204869i
$134$	0	0
$135$	4236.85i	0.232475i
$136$	0	0
$137$	−15550.0	−0.828494	−0.414247	−	0.910165i	$-0.635955\pi$
$137$	−15550.0	−0.828494	−0.414247	+	0.910165i	$0.635955\pi$
$138$	0	0
$139$	−31502.5	−1.63048	−0.815241	−	0.579122i	$-0.803396\pi$
$139$	−31502.5	−1.63048	−0.815241	+	0.579122i	$0.803396\pi$
$140$	0	0
$141$	11415.3i	0.574184i
$142$	0	0
$143$	− 5439.91i	− 0.266023i
$144$	0	0
$145$	39216.0	1.86521
$146$	0	0
$147$	1740.71	0.0805549
$148$	0	0
$149$	− 22619.3i	− 1.01884i	−0.860518	−	0.509421i	$-0.829860\pi$
$149$	− 22619.3i	− 1.01884i	0.860518	−	0.509421i	$-0.170140\pi$
$150$	0	0
$151$	− 7689.10i	− 0.337226i	−0.985682	−	0.168613i	$-0.946071\pi$
$151$	− 7689.10i	− 0.337226i	0.985682	−	0.168613i	$-0.0539289\pi$
$152$	0	0
$153$	−9126.00	−0.389850
$154$	0	0
$155$	−39490.8	−1.64374
$156$	0	0
$157$	− 1207.97i	− 0.0490070i	−0.999700	−	0.0245035i	$-0.992200\pi$
$157$	− 1207.97i	− 0.0490070i	0.999700	−	0.0245035i	$-0.00780049\pi$
$158$	0	0
$159$	− 12710.5i	− 0.502771i
$160$	0	0
$161$	−38304.0	−1.47772
$162$	0	0
$163$	26749.8	1.00680	0.503402	−	0.864052i	$-0.332081\pi$
$163$	26749.8	1.00680	0.503402	+	0.864052i	$0.332081\pi$
$164$	0	0
$165$	14133.3i	0.519129i
$166$	0	0
$167$	44983.8i	1.61296i	0.591261	+	0.806480i	$0.298630\pi$
$167$	44983.8i	1.61296i	−0.591261	+	0.806480i	$0.701370\pi$
$168$	0	0
$169$	24913.0	0.872273
$170$	0	0
$171$	187.061	0.00639723
$172$	0	0
$173$	− 4197.71i	− 0.140256i	−0.997538	−	0.0701278i	$-0.977659\pi$
$173$	− 4197.71i	− 0.140256i	0.997538	−	0.0701278i	$-0.0223407\pi$
$174$	0	0
$175$	− 15012.0i	− 0.490189i
$176$	0	0
$177$	−6228.00	−0.198793
$178$	0	0
$179$	−12491.6	−0.389861	−0.194931	−	0.980817i	$-0.562448\pi$
$179$	−12491.6	−0.389861	−0.194931	+	0.980817i	$0.562448\pi$
$180$	0	0
$181$	37870.0i	1.15595i	0.816056	+	0.577973i	$0.196156\pi$
$181$	37870.0i	1.15595i	−0.816056	+	0.577973i	$0.803844\pi$
$182$	0	0
$183$	33267.1i	0.993374i
$184$	0	0
$185$	−7296.00	−0.213178
$186$	0	0
$187$	−30442.5	−0.870557
$188$	0	0
$189$	− 7338.44i	− 0.205438i
$190$	0	0
$191$	32848.7i	0.900432i	0.892920	+	0.450216i	$0.148653\pi$
$191$	32848.7i	0.900432i	−0.892920	+	0.450216i	$0.851347\pi$
$192$	0	0
$193$	44830.0	1.20352	0.601761	−	0.798676i	$-0.294466\pi$
$193$	44830.0	1.20352	0.601761	+	0.798676i	$0.294466\pi$
$194$	0	0
$195$	9477.78	0.249251
$196$	0	0
$197$	42188.5i	1.08708i	0.839383	+	0.543540i	$0.182916\pi$
$197$	42188.5i	1.08708i	−0.839383	+	0.543540i	$0.817084\pi$
$198$	0	0
$199$	18778.1i	0.474183i	0.971487	+	0.237092i	$0.0761942\pi$
$199$	18778.1i	0.474183i	−0.971487	+	0.237092i	$0.923806\pi$
$200$	0	0
$201$	−42948.0	−1.06304
$202$	0	0
$203$	−67924.1	−1.64828
$204$	0	0
$205$	− 17455.2i	− 0.415353i
$206$	0	0
$207$	19772.0i	0.461434i
$208$	0	0
$209$	624.000	0.0142854
$210$	0	0
$211$	−63704.8	−1.43089	−0.715447	−	0.698667i	$-0.753777\pi$
$211$	−63704.8	−1.43089	−0.715447	+	0.698667i	$0.753777\pi$
$212$	0	0
$213$	− 22287.1i	− 0.491241i
$214$	0	0
$215$	− 61303.6i	− 1.32620i
$216$	0	0
$217$	68400.0	1.45257
$218$	0	0
$219$	−45383.2	−0.946252
$220$	0	0
$221$	20414.8i	0.417984i
$222$	0	0
$223$	− 19405.8i	− 0.390231i	−0.980780	−	0.195116i	$-0.937492\pi$
$223$	− 19405.8i	− 0.390231i	0.980780	−	0.195116i	$-0.0625082\pi$
$224$	0	0
$225$	−7749.00	−0.153067
$226$	0	0
$227$	12089.7	0.234620	0.117310	−	0.993095i	$-0.462573\pi$
$227$	12089.7	0.234620	0.117310	+	0.993095i	$0.462573\pi$
$228$	0	0
$229$	− 91020.8i	− 1.73568i	−0.496844	−	0.867840i	$-0.665508\pi$
$229$	− 91020.8i	− 1.73568i	0.496844	−	0.867840i	$-0.334492\pi$
$230$	0	0
$231$	− 24479.6i	− 0.458754i
$232$	0	0
$233$	45166.0	0.831955	0.415977	−	0.909375i	$-0.363440\pi$
$233$	45166.0	0.831955	0.415977	+	0.909375i	$0.363440\pi$
$234$	0	0
$235$	−66344.5	−1.20135
$236$	0	0
$237$	− 58435.7i	− 1.04036i
$238$	0	0
$239$	12135.2i	0.212447i	0.994342	+	0.106223i	$0.0338759\pi$
$239$	12135.2i	0.212447i	−0.994342	+	0.106223i	$0.966124\pi$
$240$	0	0
$241$	85822.0	1.47763	0.738813	−	0.673910i	$-0.235387\pi$
$241$	85822.0	1.47763	0.738813	+	0.673910i	$0.235387\pi$
$242$	0	0
$243$	−3788.00	−0.0641500
$244$	0	0
$245$	10116.8i	0.168543i
$246$	0	0
$247$	− 418.454i	− 0.00685889i
$248$	0	0
$249$	−68580.0	−1.10611
$250$	0	0
$251$	−58190.0	−0.923636	−0.461818	−	0.886975i	$-0.652803\pi$
$251$	−58190.0	−0.923636	−0.461818	+	0.886975i	$0.652803\pi$
$252$	0	0
$253$	65955.4i	1.03041i
$254$	0	0
$255$	− 53039.1i	− 0.815672i
$256$	0	0
$257$	−121726.	−1.84297	−0.921483	−	0.388420i	$-0.873021\pi$
$257$	−121726.	−1.84297	−0.921483	+	0.388420i	$0.873021\pi$
$258$	0	0
$259$	12637.0	0.188385
$260$	0	0
$261$	35061.4i	0.514693i
$262$	0	0
$263$	− 97918.3i	− 1.41564i	−0.706394	−	0.707819i	$-0.749679\pi$
$263$	− 97918.3i	− 1.41564i	0.706394	−	0.707819i	$-0.250321\pi$
$264$	0	0
$265$	73872.0	1.05193
$266$	0	0
$267$	−4728.50	−0.0663286
$268$	0	0
$269$	14465.5i	0.199907i	0.994992	+	0.0999536i	$0.0318694\pi$
$269$	14465.5i	0.199907i	−0.994992	+	0.0999536i	$0.968131\pi$
$270$	0	0
$271$	55497.5i	0.755675i	0.925872	+	0.377837i	$0.123332\pi$
$271$	55497.5i	0.755675i	−0.925872	+	0.377837i	$0.876668\pi$
$272$	0	0
$273$	−16416.0	−0.220263
$274$	0	0
$275$	−25849.1	−0.341807
$276$	0	0
$277$	− 102013.i	− 1.32953i	−0.747053	−	0.664764i	$-0.768532\pi$
$277$	− 102013.i	− 1.32953i	0.747053	−	0.664764i	$-0.231468\pi$
$278$	0	0
$279$	− 35307.1i	− 0.453579i
$280$	0	0
$281$	−40082.0	−0.507618	−0.253809	−	0.967254i	$-0.581683\pi$
$281$	−40082.0	−0.507618	−0.253809	+	0.967254i	$0.581683\pi$
$282$	0	0
$283$	−48324.2	−0.603381	−0.301691	−	0.953406i	$-0.597551\pi$
$283$	−48324.2	−0.603381	−0.301691	+	0.953406i	$0.597551\pi$
$284$	0	0
$285$	1087.18i	0.0133847i
$286$	0	0
$287$	30233.3i	0.367047i
$288$	0	0
$289$	30723.0	0.367848
$290$	0	0
$291$	−28173.5	−0.332702
$292$	0	0
$293$	127592.i	1.48624i	0.669158	+	0.743120i	$0.266655\pi$
$293$	127592.i	1.48624i	−0.669158	+	0.743120i	$0.733345\pi$
$294$	0	0
$295$	− 36196.3i	− 0.415930i
$296$	0	0
$297$	−12636.0	−0.143251
$298$	0	0
$299$	44229.6	0.494733
$300$	0	0
$301$	106181.i	1.17196i
$302$	0	0
$303$	56962.1i	0.620441i
$304$	0	0
$305$	−193344.	−2.07841
$306$	0	0
$307$	−39788.7	−0.422165	−0.211083	−	0.977468i	$-0.567699\pi$
$307$	−39788.7	−0.422165	−0.211083	+	0.977468i	$0.567699\pi$
$308$	0	0
$309$	− 27994.8i	− 0.293197i
$310$	0	0
$311$	174705.i	1.80627i	0.429352	+	0.903137i	$0.358742\pi$
$311$	174705.i	1.80627i	−0.429352	+	0.903137i	$0.641258\pi$
$312$	0	0
$313$	26930.0	0.274883	0.137441	−	0.990510i	$-0.456112\pi$
$313$	26930.0	0.274883	0.137441	+	0.990510i	$0.456112\pi$
$314$	0	0
$315$	42650.0	0.429831
$316$	0	0
$317$	10841.6i	0.107888i	0.998544	+	0.0539440i	$0.0171793\pi$
$317$	10841.6i	0.107888i	−0.998544	+	0.0539440i	$0.982821\pi$
$318$	0	0
$319$	116958.i	1.14934i
$320$	0	0
$321$	−33444.0	−0.324570
$322$	0	0
$323$	−2341.73	−0.0224457
$324$	0	0
$325$	17334.4i	0.164113i
$326$	0	0
$327$	− 313.841i	− 0.00293504i
$328$	0	0
$329$	114912.	1.06163
$330$	0	0
$331$	127597.	1.16462	0.582309	−	0.812968i	$-0.302149\pi$
$331$	127597.	1.16462	0.582309	+	0.812968i	$0.302149\pi$
$332$	0	0
$333$	− 6523.06i	− 0.0588251i
$334$	0	0
$335$	− 249608.i	− 2.22417i
$336$	0	0
$337$	186482.	1.64201	0.821007	−	0.570917i	$-0.193412\pi$
$337$	186482.	1.64201	0.821007	+	0.570917i	$0.193412\pi$
$338$	0	0
$339$	16451.0	0.143151
$340$	0	0
$341$	− 117777.i	− 1.01287i
$342$	0	0
$343$	108066.i	0.918544i
$344$	0	0
$345$	−114912.	−0.965444
$346$	0	0
$347$	49224.9	0.408814	0.204407	−	0.978886i	$-0.434473\pi$
$347$	49224.9	0.408814	0.204407	+	0.978886i	$0.434473\pi$
$348$	0	0
$349$	− 209825.i	− 1.72269i	−0.508023	−	0.861343i	$-0.669624\pi$
$349$	− 209825.i	− 1.72269i	0.508023	−	0.861343i	$-0.330376\pi$
$350$	0	0
$351$	8473.70i	0.0687795i
$352$	0	0
$353$	−67486.0	−0.541582	−0.270791	−	0.962638i	$-0.587285\pi$
$353$	−67486.0	−0.541582	−0.270791	+	0.962638i	$0.587285\pi$
$354$	0	0
$355$	129530.	1.02781
$356$	0	0
$357$	91866.4i	0.720809i
$358$	0	0
$359$	116016.i	0.900183i	0.892983	+	0.450091i	$0.148609\pi$
$359$	116016.i	0.900183i	−0.892983	+	0.450091i	$0.851391\pi$
$360$	0	0
$361$	−130273.	−0.999632
$362$	0	0
$363$	33925.7	0.257463
$364$	0	0
$365$	− 263761.i	− 1.97982i
$366$	0	0
$367$	− 77884.8i	− 0.578257i	−0.957290	−	0.289128i	$-0.906635\pi$
$367$	− 77884.8i	− 0.578257i	0.957290	−	0.289128i	$-0.0933654\pi$
$368$	0	0
$369$	15606.0	0.114614
$370$	0	0
$371$	−127950.	−0.929593
$372$	0	0
$373$	259835.i	1.86758i	0.357816	+	0.933792i	$0.383521\pi$
$373$	259835.i	1.86758i	−0.357816	+	0.933792i	$0.616479\pi$
$374$	0	0
$375$	53039.1i	0.377167i
$376$	0	0
$377$	78432.0	0.551837
$378$	0	0
$379$	116761.	0.812867	0.406433	−	0.913680i	$-0.366772\pi$
$379$	116761.	0.812867	0.406433	+	0.913680i	$0.366772\pi$
$380$	0	0
$381$	67133.1i	0.462474i
$382$	0	0
$383$	158803.i	1.08259i	0.840834	+	0.541293i	$0.182065\pi$
$383$	158803.i	1.08259i	−0.840834	+	0.541293i	$0.817935\pi$
$384$	0	0
$385$	142272.	0.959838
$386$	0	0
$387$	54809.0	0.365957
$388$	0	0
$389$	− 121673.i	− 0.804073i	−0.915624	−	0.402037i	$-0.868302\pi$
$389$	− 121673.i	− 0.804073i	0.915624	−	0.402037i	$-0.131698\pi$
$390$	0	0
$391$	− 247516.i	− 1.61901i
$392$	0	0
$393$	−117828.	−0.762893
$394$	0	0
$395$	339621.	2.17671
$396$	0	0
$397$	237850.i	1.50911i	0.656234	+	0.754557i	$0.272148\pi$
$397$	237850.i	1.50911i	−0.656234	+	0.754557i	$0.727852\pi$
$398$	0	0
$399$	− 1883.04i	− 0.0118281i
$400$	0	0
$401$	−58130.0	−0.361503	−0.180751	−	0.983529i	$-0.557853\pi$
$401$	−58130.0	−0.361503	−0.180751	+	0.983529i	$0.557853\pi$
$402$	0	0
$403$	−78981.5	−0.486312
$404$	0	0
$405$	− 22015.3i	− 0.134219i
$406$	0	0
$407$	− 21759.6i	− 0.131360i
$408$	0	0
$409$	65186.0	0.389680	0.194840	−	0.980835i	$-0.437581\pi$
$409$	65186.0	0.389680	0.194840	+	0.980835i	$0.437581\pi$
$410$	0	0
$411$	80800.2	0.478331
$412$	0	0
$413$	62693.8i	0.367557i
$414$	0	0
$415$	− 398578.i	− 2.31428i
$416$	0	0
$417$	163692.	0.941359
$418$	0	0
$419$	267519.	1.52379	0.761897	−	0.647698i	$-0.224268\pi$
$419$	267519.	1.52379	0.761897	+	0.647698i	$0.224268\pi$
$420$	0	0
$421$	− 263399.i	− 1.48610i	−0.669233	−	0.743052i	$-0.733377\pi$
$421$	− 263399.i	− 1.48610i	0.669233	−	0.743052i	$-0.266623\pi$
$422$	0	0
$423$	− 59315.9i	− 0.331505i
$424$	0	0
$425$	97006.0	0.537057
$426$	0	0
$427$	334882.	1.83669
$428$	0	0
$429$	28266.6i	0.153588i
$430$	0	0
$431$	− 105346.i	− 0.567104i	−0.958957	−	0.283552i	$-0.908487\pi$
$431$	− 105346.i	− 0.567104i	0.958957	−	0.283552i	$-0.0915129\pi$
$432$	0	0
$433$	83758.0	0.446736	0.223368	−	0.974734i	$-0.428295\pi$
$433$	83758.0	0.446736	0.223368	+	0.974734i	$0.428295\pi$
$434$	0	0
$435$	−203772.	−1.07688
$436$	0	0
$437$	5073.49i	0.0265671i
$438$	0	0
$439$	95983.0i	0.498041i	0.968498	+	0.249020i	$0.0801087\pi$
$439$	95983.0i	0.498041i	−0.968498	+	0.249020i	$0.919891\pi$
$440$	0	0
$441$	−9045.00	−0.0465084
$442$	0	0
$443$	222679.	1.13468	0.567339	−	0.823484i	$-0.307973\pi$
$443$	222679.	1.13468	0.567339	+	0.823484i	$0.307973\pi$
$444$	0	0
$445$	− 27481.4i	− 0.138777i
$446$	0	0
$447$	117533.i	0.588229i
$448$	0	0
$449$	−88658.0	−0.439770	−0.219885	−	0.975526i	$-0.570568\pi$
$449$	−88658.0	−0.439770	−0.219885	+	0.975526i	$0.570568\pi$
$450$	0	0
$451$	52058.5	0.255940
$452$	0	0
$453$	39953.7i	0.194698i
$454$	0	0
$455$	− 95407.6i	− 0.460851i
$456$	0	0
$457$	−29086.0	−0.139268	−0.0696340	−	0.997573i	$-0.522183\pi$
$457$	−29086.0	−0.139268	−0.0696340	+	0.997573i	$0.522183\pi$
$458$	0	0
$459$	47420.1	0.225080
$460$	0	0
$461$	277139.i	1.30406i	0.758195	+	0.652028i	$0.226082\pi$
$461$	277139.i	1.30406i	−0.758195	+	0.652028i	$0.773918\pi$
$462$	0	0
$463$	279371.i	1.30322i	0.758553	+	0.651611i	$0.225907\pi$
$463$	279371.i	1.30322i	−0.758553	+	0.651611i	$0.774093\pi$
$464$	0	0
$465$	205200.	0.949011
$466$	0	0
$467$	−254702.	−1.16788	−0.583939	−	0.811797i	$-0.698489\pi$
$467$	−254702.	−1.16788	−0.583939	+	0.811797i	$0.698489\pi$
$468$	0	0
$469$	432334.i	1.96550i
$470$	0	0
$471$	6276.81i	0.0282942i
$472$	0	0
$473$	182832.	0.817203
$474$	0	0
$475$	−1988.39	−0.00881283
$476$	0	0
$477$	66046.0i	0.290275i
$478$	0	0
$479$	52620.6i	0.229343i	0.993403	+	0.114671i	$0.0365815\pi$
$479$	52620.6i	0.229343i	−0.993403	+	0.114671i	$0.963418\pi$
$480$	0	0
$481$	−14592.0	−0.0630703
$482$	0	0
$483$	199033.	0.853162
$484$	0	0
$485$	− 163741.i	− 0.696103i
$486$	0	0
$487$	147557.i	0.622162i	0.950383	+	0.311081i	$0.100691\pi$
$487$	147557.i	0.622162i	−0.950383	+	0.311081i	$0.899309\pi$
$488$	0	0
$489$	−138996.	−0.581279
$490$	0	0
$491$	78240.2	0.324539	0.162270	−	0.986746i	$-0.448119\pi$
$491$	78240.2	0.324539	0.162270	+	0.986746i	$0.448119\pi$
$492$	0	0
$493$	− 438917.i	− 1.80588i
$494$	0	0
$495$	− 73438.7i	− 0.299719i
$496$	0	0
$497$	−224352.	−0.908275
$498$	0	0
$499$	−298044.	−1.19696	−0.598480	−	0.801138i	$-0.704229\pi$
$499$	−298044.	−1.19696	−0.598480	+	0.801138i	$0.704229\pi$
$500$	0	0
$501$	− 233743.i	− 0.931243i
$502$	0	0
$503$	− 132964.i	− 0.525530i	−0.964860	−	0.262765i	$-0.915366\pi$
$503$	− 132964.i	− 0.525530i	0.964860	−	0.262765i	$-0.0846344\pi$
$504$	0	0
$505$	−331056.	−1.29813
$506$	0	0
$507$	−129452.	−0.503607
$508$	0	0
$509$	12411.9i	0.0479075i	0.999713	+	0.0239538i	$0.00762545\pi$
$509$	12411.9i	0.0479075i	−0.999713	+	0.0239538i	$0.992375\pi$
$510$	0	0
$511$	456847.i	1.74956i
$512$	0	0
$513$	−972.000	−0.00369344
$514$	0	0
$515$	162702.	0.613449
$516$	0	0
$517$	− 197866.i	− 0.740270i
$518$	0	0
$519$	21811.9i	0.0809766i
$520$	0	0
$521$	170018.	0.626353	0.313177	−	0.949695i	$-0.398607\pi$
$521$	170018.	0.626353	0.313177	+	0.949695i	$0.398607\pi$
$522$	0	0
$523$	401621.	1.46829	0.734147	−	0.678991i	$-0.237582\pi$
$523$	401621.	1.46829	0.734147	+	0.678991i	$0.237582\pi$
$524$	0	0
$525$	78004.9i	0.283011i
$526$	0	0
$527$	441992.i	1.59145i
$528$	0	0
$529$	−256415.	−0.916288
$530$	0	0
$531$	32361.6	0.114773
$532$	0	0
$533$	− 34910.4i	− 0.122886i
$534$	0	0
$535$	− 194372.i	− 0.679088i
$536$	0	0
$537$	64908.0	0.225087
$538$	0	0
$539$	−30172.3	−0.103856
$540$	0	0
$541$	− 138857.i	− 0.474430i	−0.971457	−	0.237215i	$-0.923765\pi$
$541$	− 138857.i	− 0.474430i	0.971457	−	0.237215i	$-0.0762345\pi$
$542$	0	0
$543$	− 196778.i	− 0.667386i
$544$	0	0
$545$	1824.00	0.00614090
$546$	0	0
$547$	−99094.1	−0.331187	−0.165593	−	0.986194i	$-0.552954\pi$
$547$	−99094.1	−0.331187	−0.165593	+	0.986194i	$0.552954\pi$
$548$	0	0
$549$	− 172861.i	− 0.573525i
$550$	0	0
$551$	8996.77i	0.0296335i
$552$	0	0
$553$	−588240.	−1.92355
$554$	0	0
$555$	37911.1	0.123078
$556$	0	0
$557$	199104.i	0.641756i	0.947120	+	0.320878i	$0.103978\pi$
$557$	199104.i	0.641756i	−0.947120	+	0.320878i	$0.896022\pi$
$558$	0	0
$559$	− 122607.i	− 0.392367i
$560$	0	0
$561$	158184.	0.502617
$562$	0	0
$563$	−572457.	−1.80603	−0.903017	−	0.429605i	$-0.858653\pi$
$563$	−572457.	−1.80603	−0.903017	+	0.429605i	$0.858653\pi$
$564$	0	0
$565$	95611.1i	0.299510i
$566$	0	0
$567$	38131.6i	0.118609i
$568$	0	0
$569$	563330.	1.73996	0.869978	−	0.493090i	$-0.164133\pi$
$569$	563330.	1.73996	0.869978	+	0.493090i	$0.164133\pi$
$570$	0	0
$571$	−113117.	−0.346940	−0.173470	−	0.984839i	$-0.555498\pi$
$571$	−113117.	−0.346940	−0.173470	+	0.984839i	$0.555498\pi$
$572$	0	0
$573$	− 170687.i	− 0.519865i
$574$	0	0
$575$	− 210169.i	− 0.635671i
$576$	0	0
$577$	155858.	0.468142	0.234071	−	0.972220i	$-0.424795\pi$
$577$	155858.	0.468142	0.234071	+	0.972220i	$0.424795\pi$
$578$	0	0
$579$	−232944.	−0.694854
$580$	0	0
$581$	690357.i	2.04513i
$582$	0	0
$583$	220316.i	0.648200i
$584$	0	0
$585$	−49248.0	−0.143905
$586$	0	0
$587$	−270914.	−0.786239	−0.393119	−	0.919487i	$-0.628604\pi$
$587$	−270914.	−0.786239	−0.393119	+	0.919487i	$0.628604\pi$
$588$	0	0
$589$	− 9059.80i	− 0.0261149i
$590$	0	0
$591$	− 219218.i	− 0.627626i
$592$	0	0
$593$	−418078.	−1.18891	−0.594454	−	0.804130i	$-0.702632\pi$
$593$	−418078.	−1.18891	−0.594454	+	0.804130i	$0.702632\pi$
$594$	0	0
$595$	−533915.	−1.50813
$596$	0	0
$597$	− 97574.1i	− 0.273770i
$598$	0	0
$599$	− 539701.i	− 1.50418i	−0.659060	−	0.752090i	$-0.729046\pi$
$599$	− 539701.i	− 1.50418i	0.659060	−	0.752090i	$-0.270954\pi$
$600$	0	0
$601$	−439490.	−1.21675	−0.608373	−	0.793651i	$-0.708178\pi$
$601$	−439490.	−1.21675	−0.608373	+	0.793651i	$0.708178\pi$
$602$	0	0
$603$	223164.	0.613748
$604$	0	0
$605$	197171.i	0.538683i
$606$	0	0
$607$	− 256879.i	− 0.697189i	−0.937274	−	0.348595i	$-0.886659\pi$
$607$	− 256879.i	− 0.697189i	0.937274	−	0.348595i	$-0.113341\pi$
$608$	0	0
$609$	352944.	0.951637
$610$	0	0
$611$	−132689.	−0.355429
$612$	0	0
$613$	222871.i	0.593107i	0.955016	+	0.296553i	$0.0958373\pi$
$613$	222871.i	0.593107i	−0.955016	+	0.296553i	$0.904163\pi$
$614$	0	0
$615$	90700.0i	0.239804i
$616$	0	0
$617$	588718.	1.54645	0.773227	−	0.634129i	$-0.218641\pi$
$617$	588718.	1.54645	0.773227	+	0.634129i	$0.218641\pi$
$618$	0	0
$619$	−417570.	−1.08980	−0.544901	−	0.838500i	$-0.683433\pi$
$619$	−417570.	−1.08980	−0.544901	+	0.838500i	$0.683433\pi$
$620$	0	0
$621$	− 102738.i	− 0.266409i
$622$	0	0
$623$	47599.2i	0.122638i
$624$	0	0
$625$	−487631.	−1.24834
$626$	0	0
$627$	−3242.40	−0.00824767
$628$	0	0
$629$	81659.0i	0.206397i
$630$	0	0
$631$	204467.i	0.513529i	0.966474	+	0.256765i	$0.0826565\pi$
$631$	204467.i	0.513529i	−0.966474	+	0.256765i	$0.917344\pi$
$632$	0	0
$633$	331020.	0.826127
$634$	0	0
$635$	−390169.	−0.967620
$636$	0	0
$637$	20233.6i	0.0498647i
$638$	0	0
$639$	115807.i	0.283618i
$640$	0	0
$641$	−270578.	−0.658531	−0.329266	−	0.944237i	$-0.606801\pi$
$641$	−270578.	−0.658531	−0.329266	+	0.944237i	$0.606801\pi$
$642$	0	0
$643$	584969.	1.41485	0.707426	−	0.706788i	$-0.249856\pi$
$643$	584969.	1.41485	0.707426	+	0.706788i	$0.249856\pi$
$644$	0	0
$645$	318543.i	0.765681i
$646$	0	0
$647$	96976.8i	0.231664i	0.993269	+	0.115832i	$0.0369535\pi$
$647$	96976.8i	0.231664i	−0.993269	+	0.115832i	$0.963047\pi$
$648$	0	0
$649$	107952.	0.256296
$650$	0	0
$651$	−355417.	−0.838641
$652$	0	0
$653$	− 532082.i	− 1.24782i	−0.781496	−	0.623911i	$-0.785543\pi$
$653$	− 532082.i	− 1.24782i	0.781496	−	0.623911i	$-0.214457\pi$
$654$	0	0
$655$	− 684800.i	− 1.59618i
$656$	0	0
$657$	235818.	0.546319
$658$	0	0
$659$	−253593.	−0.583938	−0.291969	−	0.956428i	$-0.594310\pi$
$659$	−253593.	−0.583938	−0.291969	+	0.956428i	$0.594310\pi$
$660$	0	0
$661$	− 339441.i	− 0.776892i	−0.921471	−	0.388446i	$-0.873012\pi$
$661$	− 339441.i	− 0.776892i	0.921471	−	0.388446i	$-0.126988\pi$
$662$	0	0
$663$	− 106078.i	− 0.241323i
$664$	0	0
$665$	10944.0	0.0247476
$666$	0	0
$667$	−950937.	−2.13747
$668$	0	0
$669$	100836.i	0.225300i
$670$	0	0
$671$	− 576630.i	− 1.28071i
$672$	0	0
$673$	−191570.	−0.422958	−0.211479	−	0.977383i	$-0.567828\pi$
$673$	−191570.	−0.422958	−0.211479	+	0.977383i	$0.567828\pi$
$674$	0	0
$675$	40265.0	0.0883731
$676$	0	0
$677$	− 498923.i	− 1.08857i	−0.838900	−	0.544285i	$-0.816801\pi$
$677$	− 498923.i	− 1.08857i	0.838900	−	0.544285i	$-0.183199\pi$
$678$	0	0
$679$	283607.i	0.615146i
$680$	0	0
$681$	−62820.0	−0.135458
$682$	0	0
$683$	96059.5	0.205920	0.102960	−	0.994685i	$-0.467169\pi$
$683$	96059.5	0.205920	0.102960	+	0.994685i	$0.467169\pi$
$684$	0	0
$685$	469600.i	1.00080i
$686$	0	0
$687$	472958.i	1.00210i
$688$	0	0
$689$	147744.	0.311223
$690$	0	0
$691$	−517474.	−1.08376	−0.541880	−	0.840456i	$-0.682287\pi$
$691$	−517474.	−1.08376	−0.541880	+	0.840456i	$0.682287\pi$
$692$	0	0
$693$	127200.i	0.264862i
$694$	0	0
$695$	951356.i	1.96958i
$696$	0	0
$697$	−195364.	−0.402142
$698$	0	0
$699$	−234689.	−0.480329
$700$	0	0
$701$	− 214204.i	− 0.435904i	−0.975959	−	0.217952i	$-0.930062\pi$
$701$	− 214204.i	− 0.435904i	0.975959	−	0.217952i	$-0.0699377\pi$
$702$	0	0
$703$	− 1673.82i	− 0.00338686i
$704$	0	0
$705$	344736.	0.693599
$706$	0	0
$707$	573406.	1.14716
$708$	0	0
$709$	− 413550.i	− 0.822688i	−0.911480	−	0.411344i	$-0.865059\pi$
$709$	− 413550.i	− 0.822688i	0.911480	−	0.411344i	$-0.134941\pi$
$710$	0	0
$711$	303641.i	0.600649i
$712$	0	0
$713$	957600.	1.88367
$714$	0	0
$715$	−164282.	−0.321349
$716$	0	0
$717$	− 63056.2i	− 0.122656i
$718$	0	0
$719$	734073.i	1.41998i	0.704213	+	0.709989i	$0.251300\pi$
$719$	734073.i	1.41998i	−0.704213	+	0.709989i	$0.748700\pi$
$720$	0	0
$721$	−281808.	−0.542104
$722$	0	0
$723$	−445944.	−0.853108
$724$	0	0
$725$	− 372690.i	− 0.709042i
$726$	0	0
$727$	− 1.03479e6i	− 1.95786i	−0.204199	−	0.978929i	$-0.565459\pi$
$727$	− 1.03479e6i	− 1.95786i	0.204199	−	0.978929i	$-0.434541\pi$
$728$	0	0
$729$	19683.0	0.0370370
$730$	0	0
$731$	−686128.	−1.28402
$732$	0	0
$733$	200222.i	0.372652i	0.982488	+	0.186326i	$0.0596580\pi$
$733$	200222.i	0.372652i	−0.982488	+	0.186326i	$0.940342\pi$
$734$	0	0
$735$	− 52568.3i	− 0.0973082i
$736$	0	0
$737$	744432.	1.37053
$738$	0	0
$739$	526855.	0.964722	0.482361	−	0.875972i	$-0.339779\pi$
$739$	526855.	0.964722	0.482361	+	0.875972i	$0.339779\pi$
$740$	0	0
$741$	2174.35i	0.00395998i
$742$	0	0
$743$	380375.i	0.689024i	0.938782	+	0.344512i	$0.111956\pi$
$743$	380375.i	0.689024i	−0.938782	+	0.344512i	$0.888044\pi$
$744$	0	0
$745$	−683088.	−1.23073
$746$	0	0
$747$	356352.	0.638614
$748$	0	0
$749$	336662.i	0.600110i
$750$	0	0
$751$	591851.i	1.04938i	0.851293	+	0.524690i	$0.175819\pi$
$751$	591851.i	1.04938i	−0.851293	+	0.524690i	$0.824181\pi$
$752$	0	0
$753$	302364.	0.533261
$754$	0	0
$755$	−232206.	−0.407360
$756$	0	0
$757$	− 373445.i	− 0.651681i	−0.945425	−	0.325840i	$-0.894353\pi$
$757$	− 373445.i	− 0.651681i	0.945425	−	0.325840i	$-0.105647\pi$
$758$	0	0
$759$	− 342714.i	− 0.594906i
$760$	0	0
$761$	37154.0	0.0641558	0.0320779	−	0.999485i	$-0.489788\pi$
$761$	37154.0	0.0641558	0.0320779	+	0.999485i	$0.489788\pi$
$762$	0	0
$763$	−3159.26	−0.00542671
$764$	0	0
$765$	275599.i	0.470929i
$766$	0	0
$767$	− 72392.6i	− 0.123056i
$768$	0	0
$769$	−455330.	−0.769970	−0.384985	−	0.922923i	$-0.625793\pi$
$769$	−455330.	−0.769970	−0.384985	+	0.922923i	$0.625793\pi$
$770$	0	0
$771$	632507.	1.06404
$772$	0	0
$773$	57046.5i	0.0954708i	0.998860	+	0.0477354i	$0.0152004\pi$
$773$	57046.5i	0.0954708i	−0.998860	+	0.0477354i	$0.984800\pi$
$774$	0	0
$775$	375301.i	0.624851i
$776$	0	0
$777$	−65664.0	−0.108764
$778$	0	0
$779$	4004.50	0.00659893
$780$	0	0
$781$	386310.i	0.633335i
$782$	0	0
$783$	− 182185.i	− 0.297158i
$784$	0	0
$785$	−36480.0	−0.0591992
$786$	0	0
$787$	−99953.2	−0.161379	−0.0806895	−	0.996739i	$-0.525712\pi$
$787$	−99953.2	−0.161379	−0.0806895	+	0.996739i	$0.525712\pi$
$788$	0	0
$789$	508798.i	0.817319i
$790$	0	0
$791$	− 165603.i	− 0.264677i
$792$	0	0
$793$	−386688.	−0.614914
$794$	0	0
$795$	−383850.	−0.607334
$796$	0	0
$797$	− 517345.i	− 0.814448i	−0.913328	−	0.407224i	$-0.866497\pi$
$797$	− 517345.i	− 0.814448i	0.913328	−	0.407224i	$-0.133503\pi$
$798$	0	0
$799$	742547.i	1.16314i
$800$	0	0
$801$	24570.0	0.0382948
$802$	0	0
$803$	786642.	1.21996
$804$	0	0
$805$	1.15676e6i	1.78505i
$806$	0	0
$807$	− 75164.9i	− 0.115416i
$808$	0	0
$809$	−570046.	−0.870989	−0.435495	−	0.900191i	$-0.643427\pi$
$809$	−570046.	−0.870989	−0.435495	+	0.900191i	$0.643427\pi$
$810$	0	0
$811$	−174418.	−0.265185	−0.132592	−	0.991171i	$-0.542330\pi$
$811$	−174418.	−0.265185	−0.132592	+	0.991171i	$0.542330\pi$
$812$	0	0
$813$	− 288373.i	− 0.436289i
$814$	0	0
$815$	− 807826.i	− 1.21619i
$816$	0	0
$817$	14064.0	0.0210700
$818$	0	0
$819$	85300.0	0.127169
$820$	0	0
$821$	− 438827.i	− 0.651038i	−0.945535	−	0.325519i	$-0.894461\pi$
$821$	− 438827.i	− 0.651038i	0.945535	−	0.325519i	$-0.105539\pi$
$822$	0	0
$823$	− 608380.i	− 0.898205i	−0.893480	−	0.449102i	$-0.851744\pi$
$823$	− 608380.i	− 0.898205i	0.893480	−	0.449102i	$-0.148256\pi$
$824$	0	0
$825$	134316.	0.197342
$826$	0	0
$827$	−727898.	−1.06429	−0.532144	−	0.846654i	$-0.678614\pi$
$827$	−727898.	−1.06429	−0.532144	+	0.846654i	$0.678614\pi$
$828$	0	0
$829$	619509.i	0.901444i	0.892664	+	0.450722i	$0.148833\pi$
$829$	619509.i	0.901444i	−0.892664	+	0.450722i	$0.851167\pi$
$830$	0	0
$831$	530077.i	0.767603i
$832$	0	0
$833$	113230.	0.163182
$834$	0	0
$835$	1.35848e6	1.94841
$836$	0	0
$837$	183461.i	0.261874i
$838$	0	0
$839$	− 1.10064e6i	− 1.56358i	−0.623539	−	0.781792i	$-0.714306\pi$
$839$	− 1.10064e6i	− 1.56358i	0.623539	−	0.781792i	$-0.285694\pi$
$840$	0	0
$841$	−979007.	−1.38418
$842$	0	0
$843$	208272.	0.293073
$844$	0	0
$845$	− 752356.i	− 1.05368i
$846$	0	0
$847$	− 341511.i	− 0.476034i
$848$	0	0
$849$	251100.	0.348362
$850$	0	0
$851$	176919.	0.244295
$852$	0	0
$853$	− 634911.i	− 0.872599i	−0.899802	−	0.436299i	$-0.856289\pi$
$853$	− 634911.i	− 0.872599i	0.899802	−	0.436299i	$-0.143711\pi$
$854$	0	0
$855$	− 5649.13i	− 0.00772769i
$856$	0	0
$857$	−405310.	−0.551856	−0.275928	−	0.961178i	$-0.588985\pi$
$857$	−405310.	−0.551856	−0.275928	+	0.961178i	$0.588985\pi$
$858$	0	0
$859$	−100230.	−0.135835	−0.0679177	−	0.997691i	$-0.521636\pi$
$859$	−100230.	−0.135835	−0.0679177	+	0.997691i	$0.521636\pi$
$860$	0	0
$861$	− 157097.i	− 0.211915i
$862$	0	0
$863$	− 317816.i	− 0.426731i	−0.976972	−	0.213366i	$-0.931557\pi$
$863$	− 317816.i	− 0.426731i	0.976972	−	0.213366i	$-0.0684425\pi$
$864$	0	0
$865$	−126768.	−0.169425
$866$	0	0
$867$	−159641.	−0.212377
$868$	0	0
$869$	1.01289e6i	1.34128i
$870$	0	0
$871$	− 499216.i	− 0.658040i
$872$	0	0
$873$	146394.	0.192086
$874$	0	0
$875$	533915.	0.697358
$876$	0	0
$877$	− 323133.i	− 0.420128i	−0.977688	−	0.210064i	$-0.932633\pi$
$877$	− 323133.i	− 0.420128i	0.977688	−	0.210064i	$-0.0673673\pi$
$878$	0	0
$879$	− 662989.i	− 0.858081i
$880$	0	0
$881$	−1.10419e6	−1.42263	−0.711315	−	0.702873i	$-0.751900\pi$
$881$	−1.10419e6	−1.42263	−0.711315	+	0.702873i	$0.751900\pi$
$882$	0	0
$883$	813877.	1.04385	0.521924	−	0.852992i	$-0.325214\pi$
$883$	813877.	1.04385	0.521924	+	0.852992i	$0.325214\pi$
$884$	0	0
$885$	188081.i	0.240137i
$886$	0	0
$887$	− 221781.i	− 0.281888i	−0.990018	−	0.140944i	$-0.954986\pi$
$887$	− 221781.i	− 0.281888i	0.990018	−	0.140944i	$-0.0450138\pi$
$888$	0	0
$889$	675792.	0.855085
$890$	0	0
$891$	65658.6	0.0827058
$892$	0	0
$893$	− 15220.5i	− 0.0190864i
$894$	0	0
$895$	377237.i	0.470942i
$896$	0	0
$897$	−229824.	−0.285634
$898$	0	0
$899$	1.69810e6	2.10109
$900$	0	0
$901$	− 826797.i	− 1.01847i
$902$	0	0
$903$	− 551732.i	− 0.676632i
$904$	0	0
$905$	1.14365e6	1.39635
$906$	0	0
$907$	577071.	0.701479	0.350739	−	0.936473i	$-0.385930\pi$
$907$	577071.	0.701479	0.350739	+	0.936473i	$0.385930\pi$
$908$	0	0
$909$	− 295984.i	− 0.358212i
$910$	0	0
$911$	922692.i	1.11178i	0.831255	+	0.555891i	$0.187623\pi$
$911$	922692.i	1.11178i	−0.831255	+	0.555891i	$0.812377\pi$
$912$	0	0
$913$	1.18872e6	1.42606
$914$	0	0
$915$	1.00464e6	1.19997
$916$	0	0
$917$	1.18611e6i	1.41054i
$918$	0	0
$919$	1.47845e6i	1.75056i	0.483620	+	0.875278i	$0.339322\pi$
$919$	1.47845e6i	1.75056i	−0.483620	+	0.875278i	$0.660678\pi$
$920$	0	0
$921$	206748.	0.243737
$922$	0	0
$923$	259059.	0.304086
$924$	0	0
$925$	69337.7i	0.0810375i
$926$	0	0
$927$	145465.i	0.169278i
$928$	0	0
$929$	241006.	0.279252	0.139626	−	0.990204i	$-0.455410\pi$
$929$	241006.	0.279252	0.139626	+	0.990204i	$0.455410\pi$
$930$	0	0
$931$	−2320.95	−0.00267773
$932$	0	0
$933$	− 907792.i	− 1.04285i
$934$	0	0
$935$	919344.i	1.05161i
$936$	0	0
$937$	−931058.	−1.06047	−0.530234	−	0.847851i	$-0.677896\pi$
$937$	−931058.	−1.06047	−0.530234	+	0.847851i	$0.677896\pi$
$938$	0	0
$939$	−139932.	−0.158704
$940$	0	0
$941$	245490.i	0.277240i	0.990346	+	0.138620i	$0.0442666\pi$
$941$	245490.i	0.277240i	−0.990346	+	0.138620i	$0.955733\pi$
$942$	0	0
$943$	423267.i	0.475982i
$944$	0	0
$945$	−221616.	−0.248163
$946$	0	0
$947$	207853.	0.231770	0.115885	−	0.993263i	$-0.463030\pi$
$947$	207853.	0.231770	0.115885	+	0.993263i	$0.463030\pi$
$948$	0	0
$949$	− 527522.i	− 0.585744i
$950$	0	0
$951$	− 56334.4i	− 0.0622892i
$952$	0	0
$953$	−1.23446e6	−1.35923	−0.679613	−	0.733570i	$-0.737852\pi$
$953$	−1.23446e6	−1.35923	−0.679613	+	0.733570i	$0.737852\pi$
$954$	0	0
$955$	992008.	1.08770
$956$	0	0
$957$	− 607731.i	− 0.663572i
$958$	0	0
$959$	− 813371.i	− 0.884405i
$960$	0	0
$961$	−786479.	−0.851609
$962$	0	0
$963$	173780.	0.187390
$964$	0	0
$965$	− 1.35384e6i	− 1.45382i
$966$	0	0
$967$	− 181034.i	− 0.193601i	−0.995304	−	0.0968003i	$-0.969139\pi$
$967$	− 181034.i	− 0.193601i	0.995304	−	0.0968003i	$-0.0308608\pi$
$968$	0	0
$969$	12168.0	0.0129590
$970$	0	0
$971$	−97708.5	−0.103632	−0.0518160	−	0.998657i	$-0.516501\pi$
$971$	−97708.5	−0.103632	−0.0518160	+	0.998657i	$0.516501\pi$
$972$	0	0
$973$	− 1.64780e6i	− 1.74052i
$974$	0	0
$975$	− 90072.3i	− 0.0947506i
$976$	0	0
$977$	1.67313e6	1.75284	0.876419	−	0.481550i	$-0.159926\pi$
$977$	1.67313e6	1.75284	0.876419	+	0.481550i	$0.159926\pi$
$978$	0	0
$979$	81960.6	0.0855145
$980$	0	0
$981$	1630.76i	0.00169455i
$982$	0	0
$983$	1.03358e6i	1.06964i	0.844966	+	0.534820i	$0.179621\pi$
$983$	1.03358e6i	1.06964i	−0.844966	+	0.534820i	$0.820379\pi$
$984$	0	0
$985$	1.27406e6	1.31316
$986$	0	0
$987$	−597100.	−0.612933
$988$	0	0
$989$	1.48653e6i	1.51978i
$990$	0	0
$991$	− 907889.i	− 0.924454i	−0.886762	−	0.462227i	$-0.847050\pi$
$991$	− 907889.i	− 0.924454i	0.886762	−	0.462227i	$-0.152950\pi$
$992$	0	0
$993$	−663012.	−0.672393
$994$	0	0
$995$	567087.	0.572801
$996$	0	0
$997$	1.47844e6i	1.48735i	0.668542	+	0.743675i	$0.266919\pi$
$997$	1.47844e6i	1.48735i	−0.668542	+	0.743675i	$0.733081\pi$
$998$	0	0
$999$	33894.8i	0.0339627i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.5.b.a.319.1	✓	4
3.2	odd	2		1152.5.b.j.703.4		4
4.3	odd	2	inner	384.5.b.a.319.3	yes	4
8.3	odd	2	inner	384.5.b.a.319.2	yes	4
8.5	even	2	inner	384.5.b.a.319.4	yes	4
12.11	even	2		1152.5.b.j.703.3		4
16.3	odd	4		768.5.g.d.511.3		4
16.5	even	4		768.5.g.d.511.4		4
16.11	odd	4		768.5.g.d.511.2		4
16.13	even	4		768.5.g.d.511.1		4
24.5	odd	2		1152.5.b.j.703.2		4
24.11	even	2		1152.5.b.j.703.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.5.b.a.319.1	✓	4	1.1	even	1	trivial
384.5.b.a.319.2	yes	4	8.3	odd	2	inner
384.5.b.a.319.3	yes	4	4.3	odd	2	inner
384.5.b.a.319.4	yes	4	8.5	even	2	inner
768.5.g.d.511.1		4	16.13	even	4
768.5.g.d.511.2		4	16.11	odd	4
768.5.g.d.511.3		4	16.3	odd	4
768.5.g.d.511.4		4	16.5	even	4
1152.5.b.j.703.1		4	24.11	even	2
1152.5.b.j.703.2		4	24.5	odd	2
1152.5.b.j.703.3		4	12.11	even	2
1152.5.b.j.703.4		4	3.2	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:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	384.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.19615 q^{3} -30.1993i q^{5} +52.3068i q^{7} +27.0000 q^{9} +O(q^{10})\)	\(q-5.19615 q^{3} -30.1993i q^{5} +52.3068i q^{7} +27.0000 q^{9} +90.0666 q^{11} -60.3987i q^{13} +156.920i q^{15} -338.000 q^{17} +6.92820 q^{19} -271.794i q^{21} +732.295i q^{23} -287.000 q^{25} -140.296 q^{27} +1298.57i q^{29} -1307.67i q^{31} -468.000 q^{33} +1579.63 q^{35} -241.595i q^{37} +313.841i q^{39} +578.000 q^{41} +2029.96 q^{43} -815.382i q^{45} -2196.89i q^{47} -335.000 q^{49} +1756.30 q^{51} +2446.15i q^{53} -2719.95i q^{55} -36.0000 q^{57} +1198.58 q^{59} -6402.26i q^{61} +1412.28i q^{63} -1824.00 q^{65} +8265.35 q^{67} -3805.12i q^{69} +4289.16i q^{71} +8734.00 q^{73} +1491.30 q^{75} +4711.10i q^{77} +11246.0i q^{79} +729.000 q^{81} +13198.2 q^{83} +10207.4i q^{85} -6747.58i q^{87} +910.000 q^{89} +3159.26 q^{91} +6794.85i q^{93} -209.227i q^{95} +5422.00 q^{97} +2431.80 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 108 q^{9}+O(q^{10}) \) 4 * q + 108 * q^9	\( 4 q + 108 q^{9} - 1352 q^{17} - 1148 q^{25} - 1872 q^{33} + 2312 q^{41} - 1340 q^{49} - 144 q^{57} - 7296 q^{65} + 34936 q^{73} + 2916 q^{81} + 3640 q^{89} + 21688 q^{97}+O(q^{100}) \) 4 * q + 108 * q^9 - 1352 * q^17 - 1148 * q^25 - 1872 * q^33 + 2312 * q^41 - 1340 * q^49 - 144 * q^57 - 7296 * q^65 + 34936 * q^73 + 2916 * q^81 + 3640 * q^89 + 21688 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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