LMFDB - Embedded newform 3840.2.a.bm.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3840,2,Mod(1,3840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3840, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3840 = 2^{8} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3840.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:	$30.6625543762$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1920)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	3840.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} +2.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} +2.00000 q^{7} +1.00000 q^{9} -0.828427 q^{11} -0.828427 q^{13} +1.00000 q^{15} +2.82843 q^{17} +2.00000 q^{21} +5.65685 q^{23} +1.00000 q^{25} +1.00000 q^{27} -3.65685 q^{29} +1.17157 q^{31} -0.828427 q^{33} +2.00000 q^{35} +6.48528 q^{37} -0.828427 q^{39} -3.65685 q^{41} +1.65685 q^{43} +1.00000 q^{45} -1.65685 q^{47} -3.00000 q^{49} +2.82843 q^{51} +11.6569 q^{53} -0.828427 q^{55} +4.82843 q^{59} -9.65685 q^{61} +2.00000 q^{63} -0.828427 q^{65} -9.65685 q^{67} +5.65685 q^{69} +13.6569 q^{71} +9.31371 q^{73} +1.00000 q^{75} -1.65685 q^{77} +12.4853 q^{79} +1.00000 q^{81} +2.82843 q^{85} -3.65685 q^{87} -4.34315 q^{89} -1.65685 q^{91} +1.17157 q^{93} +7.65685 q^{97} -0.828427 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^5 + 4 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} + 2 q^{5} + 4 q^{7} + 2 q^{9} + 4 q^{11} + 4 q^{13} + 2 q^{15} + 4 q^{21} + 2 q^{25} + 2 q^{27} + 4 q^{29} + 8 q^{31} + 4 q^{33} + 4 q^{35} - 4 q^{37} + 4 q^{39} + 4 q^{41} - 8 q^{43} + 2 q^{45} + 8 q^{47} - 6 q^{49} + 12 q^{53} + 4 q^{55} + 4 q^{59} - 8 q^{61} + 4 q^{63} + 4 q^{65} - 8 q^{67} + 16 q^{71} - 4 q^{73} + 2 q^{75} + 8 q^{77} + 8 q^{79} + 2 q^{81} + 4 q^{87} - 20 q^{89} + 8 q^{91} + 8 q^{93} + 4 q^{97} + 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^5 + 4 * q^7 + 2 * q^9 + 4 * q^11 + 4 * q^13 + 2 * q^15 + 4 * q^21 + 2 * q^25 + 2 * q^27 + 4 * q^29 + 8 * q^31 + 4 * q^33 + 4 * q^35 - 4 * q^37 + 4 * q^39 + 4 * q^41 - 8 * q^43 + 2 * q^45 + 8 * q^47 - 6 * q^49 + 12 * q^53 + 4 * q^55 + 4 * q^59 - 8 * q^61 + 4 * q^63 + 4 * q^65 - 8 * q^67 + 16 * q^71 - 4 * q^73 + 2 * q^75 + 8 * q^77 + 8 * q^79 + 2 * q^81 + 4 * q^87 - 20 * q^89 + 8 * q^91 + 8 * q^93 + 4 * q^97 + 4 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.828427	−0.249780	−0.124890	−	0.992171i	$-0.539858\pi$
$11$	−0.828427	−0.249780	−0.124890	+	0.992171i	$0.539858\pi$
$12$	0	0
$13$	−0.828427	−0.229764	−0.114882	−	0.993379i	$-0.536649\pi$
$13$	−0.828427	−0.229764	−0.114882	+	0.993379i	$0.536649\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	2.82843	0.685994	0.342997	−	0.939336i	$-0.388558\pi$
$17$	2.82843	0.685994	0.342997	+	0.939336i	$0.388558\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	5.65685	1.17954	0.589768	−	0.807573i	$-0.299219\pi$
$23$	5.65685	1.17954	0.589768	+	0.807573i	$0.299219\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−3.65685	−0.679061	−0.339530	−	0.940595i	$-0.610268\pi$
$29$	−3.65685	−0.679061	−0.339530	+	0.940595i	$0.610268\pi$
$30$	0	0
$31$	1.17157	0.210421	0.105210	−	0.994450i	$-0.466448\pi$
$31$	1.17157	0.210421	0.105210	+	0.994450i	$0.466448\pi$
$32$	0	0
$33$	−0.828427	−0.144211
$34$	0	0
$35$	2.00000	0.338062
$36$	0	0
$37$	6.48528	1.06617	0.533087	−	0.846061i	$-0.321032\pi$
$37$	6.48528	1.06617	0.533087	+	0.846061i	$0.321032\pi$
$38$	0	0
$39$	−0.828427	−0.132655
$40$	0	0
$41$	−3.65685	−0.571105	−0.285552	−	0.958363i	$-0.592177\pi$
$41$	−3.65685	−0.571105	−0.285552	+	0.958363i	$0.592177\pi$
$42$	0	0
$43$	1.65685	0.252668	0.126334	−	0.991988i	$-0.459679\pi$
$43$	1.65685	0.252668	0.126334	+	0.991988i	$0.459679\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−1.65685	−0.241677	−0.120839	−	0.992672i	$-0.538558\pi$
$47$	−1.65685	−0.241677	−0.120839	+	0.992672i	$0.538558\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	2.82843	0.396059
$52$	0	0
$53$	11.6569	1.60119	0.800596	−	0.599204i	$-0.204516\pi$
$53$	11.6569	1.60119	0.800596	+	0.599204i	$0.204516\pi$
$54$	0	0
$55$	−0.828427	−0.111705
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.82843	0.628608	0.314304	−	0.949322i	$-0.398229\pi$
$59$	4.82843	0.628608	0.314304	+	0.949322i	$0.398229\pi$
$60$	0	0
$61$	−9.65685	−1.23643	−0.618217	−	0.786008i	$-0.712145\pi$
$61$	−9.65685	−1.23643	−0.618217	+	0.786008i	$0.712145\pi$
$62$	0	0
$63$	2.00000	0.251976
$64$	0	0
$65$	−0.828427	−0.102754
$66$	0	0
$67$	−9.65685	−1.17977	−0.589886	−	0.807486i	$-0.700827\pi$
$67$	−9.65685	−1.17977	−0.589886	+	0.807486i	$0.700827\pi$
$68$	0	0
$69$	5.65685	0.681005
$70$	0	0
$71$	13.6569	1.62077	0.810385	−	0.585897i	$-0.199258\pi$
$71$	13.6569	1.62077	0.810385	+	0.585897i	$0.199258\pi$
$72$	0	0
$73$	9.31371	1.09009	0.545044	−	0.838408i	$-0.316513\pi$
$73$	9.31371	1.09009	0.545044	+	0.838408i	$0.316513\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−1.65685	−0.188816
$78$	0	0
$79$	12.4853	1.40470	0.702352	−	0.711830i	$-0.252133\pi$
$79$	12.4853	1.40470	0.702352	+	0.711830i	$0.252133\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	2.82843	0.306786
$86$	0	0
$87$	−3.65685	−0.392056
$88$	0	0
$89$	−4.34315	−0.460373	−0.230186	−	0.973147i	$-0.573934\pi$
$89$	−4.34315	−0.460373	−0.230186	+	0.973147i	$0.573934\pi$
$90$	0	0
$91$	−1.65685	−0.173686
$92$	0	0
$93$	1.17157	0.121486
$94$	0	0
$95$	0	0
$96$	0	0
$97$	7.65685	0.777436	0.388718	−	0.921357i	$-0.372918\pi$
$97$	7.65685	0.777436	0.388718	+	0.921357i	$0.372918\pi$
$98$	0	0
$99$	−0.828427	−0.0832601
$100$	0	0
$101$	5.31371	0.528734	0.264367	−	0.964422i	$-0.414837\pi$
$101$	5.31371	0.528734	0.264367	+	0.964422i	$0.414837\pi$
$102$	0	0
$103$	4.34315	0.427943	0.213971	−	0.976840i	$-0.431360\pi$
$103$	4.34315	0.427943	0.213971	+	0.976840i	$0.431360\pi$
$104$	0	0
$105$	2.00000	0.195180
$106$	0	0
$107$	−2.34315	−0.226520	−0.113260	−	0.993565i	$-0.536129\pi$
$107$	−2.34315	−0.226520	−0.113260	+	0.993565i	$0.536129\pi$
$108$	0	0
$109$	−4.00000	−0.383131	−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000	−0.383131	−0.191565	+	0.981480i	$0.561356\pi$
$110$	0	0
$111$	6.48528	0.615556
$112$	0	0
$113$	−20.4853	−1.92709	−0.963547	−	0.267541i	$-0.913789\pi$
$113$	−20.4853	−1.92709	−0.963547	+	0.267541i	$0.913789\pi$
$114$	0	0
$115$	5.65685	0.527504
$116$	0	0
$117$	−0.828427	−0.0765881
$118$	0	0
$119$	5.65685	0.518563
$120$	0	0
$121$	−10.3137	−0.937610
$122$	0	0
$123$	−3.65685	−0.329727
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	15.6569	1.38932	0.694661	−	0.719338i	$-0.255555\pi$
$127$	15.6569	1.38932	0.694661	+	0.719338i	$0.255555\pi$
$128$	0	0
$129$	1.65685	0.145878
$130$	0	0
$131$	14.4853	1.26558	0.632792	−	0.774321i	$-0.281909\pi$
$131$	14.4853	1.26558	0.632792	+	0.774321i	$0.281909\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	1.17157	0.100094	0.0500471	−	0.998747i	$-0.484063\pi$
$137$	1.17157	0.100094	0.0500471	+	0.998747i	$0.484063\pi$
$138$	0	0
$139$	−9.65685	−0.819084	−0.409542	−	0.912291i	$-0.634311\pi$
$139$	−9.65685	−0.819084	−0.409542	+	0.912291i	$0.634311\pi$
$140$	0	0
$141$	−1.65685	−0.139532
$142$	0	0
$143$	0.686292	0.0573906
$144$	0	0
$145$	−3.65685	−0.303685
$146$	0	0
$147$	−3.00000	−0.247436
$148$	0	0
$149$	11.6569	0.954967	0.477483	−	0.878641i	$-0.341549\pi$
$149$	11.6569	0.954967	0.477483	+	0.878641i	$0.341549\pi$
$150$	0	0
$151$	−8.48528	−0.690522	−0.345261	−	0.938507i	$-0.612210\pi$
$151$	−8.48528	−0.690522	−0.345261	+	0.938507i	$0.612210\pi$
$152$	0	0
$153$	2.82843	0.228665
$154$	0	0
$155$	1.17157	0.0941030
$156$	0	0
$157$	−22.4853	−1.79452	−0.897260	−	0.441502i	$-0.854446\pi$
$157$	−22.4853	−1.79452	−0.897260	+	0.441502i	$0.854446\pi$
$158$	0	0
$159$	11.6569	0.924449
$160$	0	0
$161$	11.3137	0.891645
$162$	0	0
$163$	−12.0000	−0.939913	−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000	−0.939913	−0.469956	+	0.882690i	$0.655730\pi$
$164$	0	0
$165$	−0.828427	−0.0644930
$166$	0	0
$167$	21.6569	1.67586	0.837929	−	0.545779i	$-0.183766\pi$
$167$	21.6569	1.67586	0.837929	+	0.545779i	$0.183766\pi$
$168$	0	0
$169$	−12.3137	−0.947208
$170$	0	0
$171$	0	0
$172$	0	0
$173$	4.34315	0.330203	0.165102	−	0.986277i	$-0.447205\pi$
$173$	4.34315	0.330203	0.165102	+	0.986277i	$0.447205\pi$
$174$	0	0
$175$	2.00000	0.151186
$176$	0	0
$177$	4.82843	0.362927
$178$	0	0
$179$	−14.4853	−1.08268	−0.541340	−	0.840804i	$-0.682083\pi$
$179$	−14.4853	−1.08268	−0.541340	+	0.840804i	$0.682083\pi$
$180$	0	0
$181$	−12.0000	−0.891953	−0.445976	−	0.895045i	$-0.647144\pi$
$181$	−12.0000	−0.891953	−0.445976	+	0.895045i	$0.647144\pi$
$182$	0	0
$183$	−9.65685	−0.713855
$184$	0	0
$185$	6.48528	0.476807
$186$	0	0
$187$	−2.34315	−0.171348
$188$	0	0
$189$	2.00000	0.145479
$190$	0	0
$191$	4.68629	0.339088	0.169544	−	0.985523i	$-0.445770\pi$
$191$	4.68629	0.339088	0.169544	+	0.985523i	$0.445770\pi$
$192$	0	0
$193$	−10.9706	−0.789678	−0.394839	−	0.918750i	$-0.629200\pi$
$193$	−10.9706	−0.789678	−0.394839	+	0.918750i	$0.629200\pi$
$194$	0	0
$195$	−0.828427	−0.0593249
$196$	0	0
$197$	−1.31371	−0.0935979	−0.0467989	−	0.998904i	$-0.514902\pi$
$197$	−1.31371	−0.0935979	−0.0467989	+	0.998904i	$0.514902\pi$
$198$	0	0
$199$	13.1716	0.933708	0.466854	−	0.884334i	$-0.345387\pi$
$199$	13.1716	0.933708	0.466854	+	0.884334i	$0.345387\pi$
$200$	0	0
$201$	−9.65685	−0.681142
$202$	0	0
$203$	−7.31371	−0.513322
$204$	0	0
$205$	−3.65685	−0.255406
$206$	0	0
$207$	5.65685	0.393179
$208$	0	0
$209$	0	0
$210$	0	0
$211$	14.3431	0.987423	0.493711	−	0.869626i	$-0.335640\pi$
$211$	14.3431	0.987423	0.493711	+	0.869626i	$0.335640\pi$
$212$	0	0
$213$	13.6569	0.935752
$214$	0	0
$215$	1.65685	0.112997
$216$	0	0
$217$	2.34315	0.159063
$218$	0	0
$219$	9.31371	0.629362
$220$	0	0
$221$	−2.34315	−0.157617
$222$	0	0
$223$	18.9706	1.27036	0.635181	−	0.772363i	$-0.280925\pi$
$223$	18.9706	1.27036	0.635181	+	0.772363i	$0.280925\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−17.6569	−1.17193	−0.585963	−	0.810338i	$-0.699284\pi$
$227$	−17.6569	−1.17193	−0.585963	+	0.810338i	$0.699284\pi$
$228$	0	0
$229$	−21.6569	−1.43113	−0.715563	−	0.698549i	$-0.753830\pi$
$229$	−21.6569	−1.43113	−0.715563	+	0.698549i	$0.753830\pi$
$230$	0	0
$231$	−1.65685	−0.109013
$232$	0	0
$233$	11.7990	0.772978	0.386489	−	0.922294i	$-0.373688\pi$
$233$	11.7990	0.772978	0.386489	+	0.922294i	$0.373688\pi$
$234$	0	0
$235$	−1.65685	−0.108081
$236$	0	0
$237$	12.4853	0.811006
$238$	0	0
$239$	−22.6274	−1.46365	−0.731823	−	0.681495i	$-0.761330\pi$
$239$	−22.6274	−1.46365	−0.731823	+	0.681495i	$0.761330\pi$
$240$	0	0
$241$	20.6274	1.32873	0.664364	−	0.747409i	$-0.268702\pi$
$241$	20.6274	1.32873	0.664364	+	0.747409i	$0.268702\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−3.00000	−0.191663
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	8.82843	0.557245	0.278623	−	0.960401i	$-0.410122\pi$
$251$	8.82843	0.557245	0.278623	+	0.960401i	$0.410122\pi$
$252$	0	0
$253$	−4.68629	−0.294625
$254$	0	0
$255$	2.82843	0.177123
$256$	0	0
$257$	14.8284	0.924972	0.462486	−	0.886627i	$-0.346958\pi$
$257$	14.8284	0.924972	0.462486	+	0.886627i	$0.346958\pi$
$258$	0	0
$259$	12.9706	0.805952
$260$	0	0
$261$	−3.65685	−0.226354
$262$	0	0
$263$	29.6569	1.82872	0.914360	−	0.404902i	$-0.132694\pi$
$263$	29.6569	1.82872	0.914360	+	0.404902i	$0.132694\pi$
$264$	0	0
$265$	11.6569	0.716075
$266$	0	0
$267$	−4.34315	−0.265796
$268$	0	0
$269$	7.65685	0.466847	0.233423	−	0.972375i	$-0.425007\pi$
$269$	7.65685	0.466847	0.233423	+	0.972375i	$0.425007\pi$
$270$	0	0
$271$	8.48528	0.515444	0.257722	−	0.966219i	$-0.417028\pi$
$271$	8.48528	0.515444	0.257722	+	0.966219i	$0.417028\pi$
$272$	0	0
$273$	−1.65685	−0.100277
$274$	0	0
$275$	−0.828427	−0.0499560
$276$	0	0
$277$	−23.1716	−1.39224	−0.696122	−	0.717923i	$-0.745093\pi$
$277$	−23.1716	−1.39224	−0.696122	+	0.717923i	$0.745093\pi$
$278$	0	0
$279$	1.17157	0.0701402
$280$	0	0
$281$	−10.9706	−0.654449	−0.327224	−	0.944947i	$-0.606113\pi$
$281$	−10.9706	−0.654449	−0.327224	+	0.944947i	$0.606113\pi$
$282$	0	0
$283$	12.9706	0.771020	0.385510	−	0.922704i	$-0.374026\pi$
$283$	12.9706	0.771020	0.385510	+	0.922704i	$0.374026\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−7.31371	−0.431715
$288$	0	0
$289$	−9.00000	−0.529412
$290$	0	0
$291$	7.65685	0.448853
$292$	0	0
$293$	−17.3137	−1.01148	−0.505739	−	0.862687i	$-0.668780\pi$
$293$	−17.3137	−1.01148	−0.505739	+	0.862687i	$0.668780\pi$
$294$	0	0
$295$	4.82843	0.281122
$296$	0	0
$297$	−0.828427	−0.0480702
$298$	0	0
$299$	−4.68629	−0.271015
$300$	0	0
$301$	3.31371	0.190999
$302$	0	0
$303$	5.31371	0.305265
$304$	0	0
$305$	−9.65685	−0.552950
$306$	0	0
$307$	−4.00000	−0.228292	−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000	−0.228292	−0.114146	+	0.993464i	$0.536413\pi$
$308$	0	0
$309$	4.34315	0.247073
$310$	0	0
$311$	11.3137	0.641542	0.320771	−	0.947157i	$-0.396058\pi$
$311$	11.3137	0.641542	0.320771	+	0.947157i	$0.396058\pi$
$312$	0	0
$313$	22.9706	1.29837	0.649186	−	0.760629i	$-0.275109\pi$
$313$	22.9706	1.29837	0.649186	+	0.760629i	$0.275109\pi$
$314$	0	0
$315$	2.00000	0.112687
$316$	0	0
$317$	13.3137	0.747772	0.373886	−	0.927475i	$-0.378025\pi$
$317$	13.3137	0.747772	0.373886	+	0.927475i	$0.378025\pi$
$318$	0	0
$319$	3.02944	0.169616
$320$	0	0
$321$	−2.34315	−0.130782
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−0.828427	−0.0459529
$326$	0	0
$327$	−4.00000	−0.221201
$328$	0	0
$329$	−3.31371	−0.182691
$330$	0	0
$331$	32.0000	1.75888	0.879440	−	0.476011i	$-0.157918\pi$
$331$	32.0000	1.75888	0.879440	+	0.476011i	$0.157918\pi$
$332$	0	0
$333$	6.48528	0.355391
$334$	0	0
$335$	−9.65685	−0.527610
$336$	0	0
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	0	0
$339$	−20.4853	−1.11261
$340$	0	0
$341$	−0.970563	−0.0525589
$342$	0	0
$343$	−20.0000	−1.07990
$344$	0	0
$345$	5.65685	0.304555
$346$	0	0
$347$	−9.65685	−0.518407	−0.259204	−	0.965823i	$-0.583460\pi$
$347$	−9.65685	−0.518407	−0.259204	+	0.965823i	$0.583460\pi$
$348$	0	0
$349$	−3.02944	−0.162162	−0.0810810	−	0.996708i	$-0.525837\pi$
$349$	−3.02944	−0.162162	−0.0810810	+	0.996708i	$0.525837\pi$
$350$	0	0
$351$	−0.828427	−0.0442182
$352$	0	0
$353$	−25.4558	−1.35488	−0.677439	−	0.735579i	$-0.736910\pi$
$353$	−25.4558	−1.35488	−0.677439	+	0.735579i	$0.736910\pi$
$354$	0	0
$355$	13.6569	0.724831
$356$	0	0
$357$	5.65685	0.299392
$358$	0	0
$359$	16.9706	0.895672	0.447836	−	0.894116i	$-0.352195\pi$
$359$	16.9706	0.895672	0.447836	+	0.894116i	$0.352195\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	−10.3137	−0.541329
$364$	0	0
$365$	9.31371	0.487502
$366$	0	0
$367$	−2.68629	−0.140223	−0.0701116	−	0.997539i	$-0.522336\pi$
$367$	−2.68629	−0.140223	−0.0701116	+	0.997539i	$0.522336\pi$
$368$	0	0
$369$	−3.65685	−0.190368
$370$	0	0
$371$	23.3137	1.21039
$372$	0	0
$373$	−30.4853	−1.57847	−0.789234	−	0.614093i	$-0.789522\pi$
$373$	−30.4853	−1.57847	−0.789234	+	0.614093i	$0.789522\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	3.02944	0.156024
$378$	0	0
$379$	−24.2843	−1.24740	−0.623700	−	0.781664i	$-0.714371\pi$
$379$	−24.2843	−1.24740	−0.623700	+	0.781664i	$0.714371\pi$
$380$	0	0
$381$	15.6569	0.802125
$382$	0	0
$383$	6.34315	0.324120	0.162060	−	0.986781i	$-0.448186\pi$
$383$	6.34315	0.324120	0.162060	+	0.986781i	$0.448186\pi$
$384$	0	0
$385$	−1.65685	−0.0844411
$386$	0	0
$387$	1.65685	0.0842226
$388$	0	0
$389$	−13.3137	−0.675032	−0.337516	−	0.941320i	$-0.609587\pi$
$389$	−13.3137	−0.675032	−0.337516	+	0.941320i	$0.609587\pi$
$390$	0	0
$391$	16.0000	0.809155
$392$	0	0
$393$	14.4853	0.730686
$394$	0	0
$395$	12.4853	0.628203
$396$	0	0
$397$	6.48528	0.325487	0.162743	−	0.986668i	$-0.447966\pi$
$397$	6.48528	0.325487	0.162743	+	0.986668i	$0.447966\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	36.6274	1.82909	0.914543	−	0.404489i	$-0.132551\pi$
$401$	36.6274	1.82909	0.914543	+	0.404489i	$0.132551\pi$
$402$	0	0
$403$	−0.970563	−0.0483472
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−5.37258	−0.266309
$408$	0	0
$409$	−2.68629	−0.132829	−0.0664143	−	0.997792i	$-0.521156\pi$
$409$	−2.68629	−0.132829	−0.0664143	+	0.997792i	$0.521156\pi$
$410$	0	0
$411$	1.17157	0.0577894
$412$	0	0
$413$	9.65685	0.475183
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−9.65685	−0.472898
$418$	0	0
$419$	−2.48528	−0.121414	−0.0607070	−	0.998156i	$-0.519336\pi$
$419$	−2.48528	−0.121414	−0.0607070	+	0.998156i	$0.519336\pi$
$420$	0	0
$421$	−16.9706	−0.827095	−0.413547	−	0.910483i	$-0.635710\pi$
$421$	−16.9706	−0.827095	−0.413547	+	0.910483i	$0.635710\pi$
$422$	0	0
$423$	−1.65685	−0.0805590
$424$	0	0
$425$	2.82843	0.137199
$426$	0	0
$427$	−19.3137	−0.934656
$428$	0	0
$429$	0.686292	0.0331345
$430$	0	0
$431$	−21.6569	−1.04317	−0.521587	−	0.853198i	$-0.674660\pi$
$431$	−21.6569	−1.04317	−0.521587	+	0.853198i	$0.674660\pi$
$432$	0	0
$433$	−32.6274	−1.56797	−0.783987	−	0.620777i	$-0.786817\pi$
$433$	−32.6274	−1.56797	−0.783987	+	0.620777i	$0.786817\pi$
$434$	0	0
$435$	−3.65685	−0.175333
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−22.1421	−1.05679	−0.528393	−	0.849000i	$-0.677205\pi$
$439$	−22.1421	−1.05679	−0.528393	+	0.849000i	$0.677205\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	−17.6569	−0.838902	−0.419451	−	0.907778i	$-0.637777\pi$
$443$	−17.6569	−0.838902	−0.419451	+	0.907778i	$0.637777\pi$
$444$	0	0
$445$	−4.34315	−0.205885
$446$	0	0
$447$	11.6569	0.551350
$448$	0	0
$449$	1.31371	0.0619977	0.0309989	−	0.999519i	$-0.490131\pi$
$449$	1.31371	0.0619977	0.0309989	+	0.999519i	$0.490131\pi$
$450$	0	0
$451$	3.02944	0.142651
$452$	0	0
$453$	−8.48528	−0.398673
$454$	0	0
$455$	−1.65685	−0.0776745
$456$	0	0
$457$	−14.9706	−0.700293	−0.350147	−	0.936695i	$-0.613868\pi$
$457$	−14.9706	−0.700293	−0.350147	+	0.936695i	$0.613868\pi$
$458$	0	0
$459$	2.82843	0.132020
$460$	0	0
$461$	19.6569	0.915511	0.457755	−	0.889078i	$-0.348654\pi$
$461$	19.6569	0.915511	0.457755	+	0.889078i	$0.348654\pi$
$462$	0	0
$463$	−24.6274	−1.14453	−0.572267	−	0.820068i	$-0.693936\pi$
$463$	−24.6274	−1.14453	−0.572267	+	0.820068i	$0.693936\pi$
$464$	0	0
$465$	1.17157	0.0543304
$466$	0	0
$467$	−12.6863	−0.587052	−0.293526	−	0.955951i	$-0.594829\pi$
$467$	−12.6863	−0.587052	−0.293526	+	0.955951i	$0.594829\pi$
$468$	0	0
$469$	−19.3137	−0.891824
$470$	0	0
$471$	−22.4853	−1.03607
$472$	0	0
$473$	−1.37258	−0.0631114
$474$	0	0
$475$	0	0
$476$	0	0
$477$	11.6569	0.533731
$478$	0	0
$479$	−10.3431	−0.472590	−0.236295	−	0.971681i	$-0.575933\pi$
$479$	−10.3431	−0.472590	−0.236295	+	0.971681i	$0.575933\pi$
$480$	0	0
$481$	−5.37258	−0.244969
$482$	0	0
$483$	11.3137	0.514792
$484$	0	0
$485$	7.65685	0.347680
$486$	0	0
$487$	−2.97056	−0.134609	−0.0673045	−	0.997732i	$-0.521440\pi$
$487$	−2.97056	−0.134609	−0.0673045	+	0.997732i	$0.521440\pi$
$488$	0	0
$489$	−12.0000	−0.542659
$490$	0	0
$491$	27.1716	1.22624	0.613118	−	0.789991i	$-0.289915\pi$
$491$	27.1716	1.22624	0.613118	+	0.789991i	$0.289915\pi$
$492$	0	0
$493$	−10.3431	−0.465832
$494$	0	0
$495$	−0.828427	−0.0372350
$496$	0	0
$497$	27.3137	1.22519
$498$	0	0
$499$	6.62742	0.296684	0.148342	−	0.988936i	$-0.452606\pi$
$499$	6.62742	0.296684	0.148342	+	0.988936i	$0.452606\pi$
$500$	0	0
$501$	21.6569	0.967557
$502$	0	0
$503$	−9.65685	−0.430578	−0.215289	−	0.976550i	$-0.569069\pi$
$503$	−9.65685	−0.430578	−0.215289	+	0.976550i	$0.569069\pi$
$504$	0	0
$505$	5.31371	0.236457
$506$	0	0
$507$	−12.3137	−0.546871
$508$	0	0
$509$	18.0000	0.797836	0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000	0.797836	0.398918	+	0.916987i	$0.369386\pi$
$510$	0	0
$511$	18.6274	0.824028
$512$	0	0
$513$	0	0
$514$	0	0
$515$	4.34315	0.191382
$516$	0	0
$517$	1.37258	0.0603661
$518$	0	0
$519$	4.34315	0.190643
$520$	0	0
$521$	−5.31371	−0.232798	−0.116399	−	0.993203i	$-0.537135\pi$
$521$	−5.31371	−0.232798	−0.116399	+	0.993203i	$0.537135\pi$
$522$	0	0
$523$	37.9411	1.65905	0.829525	−	0.558470i	$-0.188611\pi$
$523$	37.9411	1.65905	0.829525	+	0.558470i	$0.188611\pi$
$524$	0	0
$525$	2.00000	0.0872872
$526$	0	0
$527$	3.31371	0.144347
$528$	0	0
$529$	9.00000	0.391304
$530$	0	0
$531$	4.82843	0.209536
$532$	0	0
$533$	3.02944	0.131219
$534$	0	0
$535$	−2.34315	−0.101303
$536$	0	0
$537$	−14.4853	−0.625086
$538$	0	0
$539$	2.48528	0.107049
$540$	0	0
$541$	−12.0000	−0.515920	−0.257960	−	0.966156i	$-0.583050\pi$
$541$	−12.0000	−0.515920	−0.257960	+	0.966156i	$0.583050\pi$
$542$	0	0
$543$	−12.0000	−0.514969
$544$	0	0
$545$	−4.00000	−0.171341
$546$	0	0
$547$	−25.6569	−1.09701	−0.548504	−	0.836148i	$-0.684802\pi$
$547$	−25.6569	−1.09701	−0.548504	+	0.836148i	$0.684802\pi$
$548$	0	0
$549$	−9.65685	−0.412144
$550$	0	0
$551$	0	0
$552$	0	0
$553$	24.9706	1.06186
$554$	0	0
$555$	6.48528	0.275285
$556$	0	0
$557$	23.6569	1.00237	0.501187	−	0.865339i	$-0.332897\pi$
$557$	23.6569	1.00237	0.501187	+	0.865339i	$0.332897\pi$
$558$	0	0
$559$	−1.37258	−0.0580541
$560$	0	0
$561$	−2.34315	−0.0989277
$562$	0	0
$563$	28.9706	1.22096	0.610482	−	0.792030i	$-0.290976\pi$
$563$	28.9706	1.22096	0.610482	+	0.792030i	$0.290976\pi$
$564$	0	0
$565$	−20.4853	−0.861822
$566$	0	0
$567$	2.00000	0.0839921
$568$	0	0
$569$	−5.31371	−0.222762	−0.111381	−	0.993778i	$-0.535527\pi$
$569$	−5.31371	−0.222762	−0.111381	+	0.993778i	$0.535527\pi$
$570$	0	0
$571$	38.6274	1.61651	0.808254	−	0.588835i	$-0.200413\pi$
$571$	38.6274	1.61651	0.808254	+	0.588835i	$0.200413\pi$
$572$	0	0
$573$	4.68629	0.195773
$574$	0	0
$575$	5.65685	0.235907
$576$	0	0
$577$	−5.02944	−0.209378	−0.104689	−	0.994505i	$-0.533385\pi$
$577$	−5.02944	−0.209378	−0.104689	+	0.994505i	$0.533385\pi$
$578$	0	0
$579$	−10.9706	−0.455921
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−9.65685	−0.399946
$584$	0	0
$585$	−0.828427	−0.0342512
$586$	0	0
$587$	−29.6569	−1.22407	−0.612035	−	0.790831i	$-0.709649\pi$
$587$	−29.6569	−1.22407	−0.612035	+	0.790831i	$0.709649\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−1.31371	−0.0540387
$592$	0	0
$593$	−39.7990	−1.63435	−0.817174	−	0.576391i	$-0.804461\pi$
$593$	−39.7990	−1.63435	−0.817174	+	0.576391i	$0.804461\pi$
$594$	0	0
$595$	5.65685	0.231908
$596$	0	0
$597$	13.1716	0.539077
$598$	0	0
$599$	−45.6569	−1.86549	−0.932744	−	0.360539i	$-0.882593\pi$
$599$	−45.6569	−1.86549	−0.932744	+	0.360539i	$0.882593\pi$
$600$	0	0
$601$	−27.9411	−1.13974	−0.569871	−	0.821734i	$-0.693007\pi$
$601$	−27.9411	−1.13974	−0.569871	+	0.821734i	$0.693007\pi$
$602$	0	0
$603$	−9.65685	−0.393258
$604$	0	0
$605$	−10.3137	−0.419312
$606$	0	0
$607$	−24.6274	−0.999596	−0.499798	−	0.866142i	$-0.666592\pi$
$607$	−24.6274	−0.999596	−0.499798	+	0.866142i	$0.666592\pi$
$608$	0	0
$609$	−7.31371	−0.296366
$610$	0	0
$611$	1.37258	0.0555288
$612$	0	0
$613$	−9.51472	−0.384296	−0.192148	−	0.981366i	$-0.561545\pi$
$613$	−9.51472	−0.384296	−0.192148	+	0.981366i	$0.561545\pi$
$614$	0	0
$615$	−3.65685	−0.147459
$616$	0	0
$617$	−2.82843	−0.113868	−0.0569341	−	0.998378i	$-0.518132\pi$
$617$	−2.82843	−0.113868	−0.0569341	+	0.998378i	$0.518132\pi$
$618$	0	0
$619$	−44.9706	−1.80752	−0.903760	−	0.428040i	$-0.859204\pi$
$619$	−44.9706	−1.80752	−0.903760	+	0.428040i	$0.859204\pi$
$620$	0	0
$621$	5.65685	0.227002
$622$	0	0
$623$	−8.68629	−0.348009
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	18.3431	0.731389
$630$	0	0
$631$	−19.5147	−0.776869	−0.388434	−	0.921476i	$-0.626984\pi$
$631$	−19.5147	−0.776869	−0.388434	+	0.921476i	$0.626984\pi$
$632$	0	0
$633$	14.3431	0.570089
$634$	0	0
$635$	15.6569	0.621323
$636$	0	0
$637$	2.48528	0.0984704
$638$	0	0
$639$	13.6569	0.540257
$640$	0	0
$641$	23.6569	0.934390	0.467195	−	0.884154i	$-0.345265\pi$
$641$	23.6569	0.934390	0.467195	+	0.884154i	$0.345265\pi$
$642$	0	0
$643$	−42.6274	−1.68106	−0.840531	−	0.541764i	$-0.817757\pi$
$643$	−42.6274	−1.68106	−0.840531	+	0.541764i	$0.817757\pi$
$644$	0	0
$645$	1.65685	0.0652386
$646$	0	0
$647$	6.34315	0.249375	0.124687	−	0.992196i	$-0.460207\pi$
$647$	6.34315	0.249375	0.124687	+	0.992196i	$0.460207\pi$
$648$	0	0
$649$	−4.00000	−0.157014
$650$	0	0
$651$	2.34315	0.0918351
$652$	0	0
$653$	−6.68629	−0.261655	−0.130827	−	0.991405i	$-0.541763\pi$
$653$	−6.68629	−0.261655	−0.130827	+	0.991405i	$0.541763\pi$
$654$	0	0
$655$	14.4853	0.565987
$656$	0	0
$657$	9.31371	0.363362
$658$	0	0
$659$	−44.8284	−1.74627	−0.873134	−	0.487481i	$-0.837916\pi$
$659$	−44.8284	−1.74627	−0.873134	+	0.487481i	$0.837916\pi$
$660$	0	0
$661$	−3.02944	−0.117831	−0.0589157	−	0.998263i	$-0.518764\pi$
$661$	−3.02944	−0.117831	−0.0589157	+	0.998263i	$0.518764\pi$
$662$	0	0
$663$	−2.34315	−0.0910002
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−20.6863	−0.800976
$668$	0	0
$669$	18.9706	0.733444
$670$	0	0
$671$	8.00000	0.308837
$672$	0	0
$673$	10.6863	0.411926	0.205963	−	0.978560i	$-0.433967\pi$
$673$	10.6863	0.411926	0.205963	+	0.978560i	$0.433967\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	17.3137	0.665420	0.332710	−	0.943029i	$-0.392037\pi$
$677$	17.3137	0.665420	0.332710	+	0.943029i	$0.392037\pi$
$678$	0	0
$679$	15.3137	0.587686
$680$	0	0
$681$	−17.6569	−0.676612
$682$	0	0
$683$	−3.31371	−0.126796	−0.0633978	−	0.997988i	$-0.520194\pi$
$683$	−3.31371	−0.126796	−0.0633978	+	0.997988i	$0.520194\pi$
$684$	0	0
$685$	1.17157	0.0447635
$686$	0	0
$687$	−21.6569	−0.826261
$688$	0	0
$689$	−9.65685	−0.367897
$690$	0	0
$691$	−6.62742	−0.252119	−0.126059	−	0.992023i	$-0.540233\pi$
$691$	−6.62742	−0.252119	−0.126059	+	0.992023i	$0.540233\pi$
$692$	0	0
$693$	−1.65685	−0.0629387
$694$	0	0
$695$	−9.65685	−0.366305
$696$	0	0
$697$	−10.3431	−0.391775
$698$	0	0
$699$	11.7990	0.446279
$700$	0	0
$701$	42.0000	1.58632	0.793159	−	0.609015i	$-0.208435\pi$
$701$	42.0000	1.58632	0.793159	+	0.609015i	$0.208435\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−1.65685	−0.0624007
$706$	0	0
$707$	10.6274	0.399685
$708$	0	0
$709$	36.2843	1.36268	0.681342	−	0.731965i	$-0.261397\pi$
$709$	36.2843	1.36268	0.681342	+	0.731965i	$0.261397\pi$
$710$	0	0
$711$	12.4853	0.468235
$712$	0	0
$713$	6.62742	0.248199
$714$	0	0
$715$	0.686292	0.0256658
$716$	0	0
$717$	−22.6274	−0.845036
$718$	0	0
$719$	12.2843	0.458126	0.229063	−	0.973412i	$-0.426434\pi$
$719$	12.2843	0.458126	0.229063	+	0.973412i	$0.426434\pi$
$720$	0	0
$721$	8.68629	0.323494
$722$	0	0
$723$	20.6274	0.767142
$724$	0	0
$725$	−3.65685	−0.135812
$726$	0	0
$727$	−50.9706	−1.89039	−0.945197	−	0.326501i	$-0.894130\pi$
$727$	−50.9706	−1.89039	−0.945197	+	0.326501i	$0.894130\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	4.68629	0.173329
$732$	0	0
$733$	−21.7990	−0.805164	−0.402582	−	0.915384i	$-0.631887\pi$
$733$	−21.7990	−0.805164	−0.402582	+	0.915384i	$0.631887\pi$
$734$	0	0
$735$	−3.00000	−0.110657
$736$	0	0
$737$	8.00000	0.294684
$738$	0	0
$739$	41.9411	1.54283	0.771415	−	0.636333i	$-0.219549\pi$
$739$	41.9411	1.54283	0.771415	+	0.636333i	$0.219549\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	4.97056	0.182352	0.0911761	−	0.995835i	$-0.470937\pi$
$743$	4.97056	0.182352	0.0911761	+	0.995835i	$0.470937\pi$
$744$	0	0
$745$	11.6569	0.427074
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−4.68629	−0.171233
$750$	0	0
$751$	−45.4558	−1.65871	−0.829354	−	0.558724i	$-0.811291\pi$
$751$	−45.4558	−1.65871	−0.829354	+	0.558724i	$0.811291\pi$
$752$	0	0
$753$	8.82843	0.321726
$754$	0	0
$755$	−8.48528	−0.308811
$756$	0	0
$757$	−34.4853	−1.25339	−0.626694	−	0.779265i	$-0.715593\pi$
$757$	−34.4853	−1.25339	−0.626694	+	0.779265i	$0.715593\pi$
$758$	0	0
$759$	−4.68629	−0.170102
$760$	0	0
$761$	33.3137	1.20762	0.603810	−	0.797128i	$-0.293648\pi$
$761$	33.3137	1.20762	0.603810	+	0.797128i	$0.293648\pi$
$762$	0	0
$763$	−8.00000	−0.289619
$764$	0	0
$765$	2.82843	0.102262
$766$	0	0
$767$	−4.00000	−0.144432
$768$	0	0
$769$	13.3137	0.480105	0.240052	−	0.970760i	$-0.422835\pi$
$769$	13.3137	0.480105	0.240052	+	0.970760i	$0.422835\pi$
$770$	0	0
$771$	14.8284	0.534033
$772$	0	0
$773$	39.2548	1.41190	0.705949	−	0.708263i	$-0.250521\pi$
$773$	39.2548	1.41190	0.705949	+	0.708263i	$0.250521\pi$
$774$	0	0
$775$	1.17157	0.0420841
$776$	0	0
$777$	12.9706	0.465316
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−11.3137	−0.404836
$782$	0	0
$783$	−3.65685	−0.130685
$784$	0	0
$785$	−22.4853	−0.802534
$786$	0	0
$787$	−16.6863	−0.594802	−0.297401	−	0.954753i	$-0.596120\pi$
$787$	−16.6863	−0.594802	−0.297401	+	0.954753i	$0.596120\pi$
$788$	0	0
$789$	29.6569	1.05581
$790$	0	0
$791$	−40.9706	−1.45675
$792$	0	0
$793$	8.00000	0.284088
$794$	0	0
$795$	11.6569	0.413426
$796$	0	0
$797$	−22.9706	−0.813659	−0.406830	−	0.913504i	$-0.633366\pi$
$797$	−22.9706	−0.813659	−0.406830	+	0.913504i	$0.633366\pi$
$798$	0	0
$799$	−4.68629	−0.165789
$800$	0	0
$801$	−4.34315	−0.153458
$802$	0	0
$803$	−7.71573	−0.272282
$804$	0	0
$805$	11.3137	0.398756
$806$	0	0
$807$	7.65685	0.269534
$808$	0	0
$809$	23.6569	0.831731	0.415865	−	0.909426i	$-0.363479\pi$
$809$	23.6569	0.831731	0.415865	+	0.909426i	$0.363479\pi$
$810$	0	0
$811$	35.5980	1.25001	0.625007	−	0.780619i	$-0.285096\pi$
$811$	35.5980	1.25001	0.625007	+	0.780619i	$0.285096\pi$
$812$	0	0
$813$	8.48528	0.297592
$814$	0	0
$815$	−12.0000	−0.420342
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−1.65685	−0.0578952
$820$	0	0
$821$	34.9706	1.22048	0.610241	−	0.792216i	$-0.291073\pi$
$821$	34.9706	1.22048	0.610241	+	0.792216i	$0.291073\pi$
$822$	0	0
$823$	18.2843	0.637350	0.318675	−	0.947864i	$-0.396762\pi$
$823$	18.2843	0.637350	0.318675	+	0.947864i	$0.396762\pi$
$824$	0	0
$825$	−0.828427	−0.0288421
$826$	0	0
$827$	−39.3137	−1.36707	−0.683536	−	0.729917i	$-0.739559\pi$
$827$	−39.3137	−1.36707	−0.683536	+	0.729917i	$0.739559\pi$
$828$	0	0
$829$	−37.9411	−1.31775	−0.658875	−	0.752253i	$-0.728967\pi$
$829$	−37.9411	−1.31775	−0.658875	+	0.752253i	$0.728967\pi$
$830$	0	0
$831$	−23.1716	−0.803813
$832$	0	0
$833$	−8.48528	−0.293998
$834$	0	0
$835$	21.6569	0.749466
$836$	0	0
$837$	1.17157	0.0404955
$838$	0	0
$839$	−12.2843	−0.424100	−0.212050	−	0.977259i	$-0.568014\pi$
$839$	−12.2843	−0.424100	−0.212050	+	0.977259i	$0.568014\pi$
$840$	0	0
$841$	−15.6274	−0.538876
$842$	0	0
$843$	−10.9706	−0.377846
$844$	0	0
$845$	−12.3137	−0.423604
$846$	0	0
$847$	−20.6274	−0.708766
$848$	0	0
$849$	12.9706	0.445149
$850$	0	0
$851$	36.6863	1.25759
$852$	0	0
$853$	−1.51472	−0.0518630	−0.0259315	−	0.999664i	$-0.508255\pi$
$853$	−1.51472	−0.0518630	−0.0259315	+	0.999664i	$0.508255\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	22.1421	0.756361	0.378180	−	0.925732i	$-0.376550\pi$
$857$	22.1421	0.756361	0.378180	+	0.925732i	$0.376550\pi$
$858$	0	0
$859$	−46.9117	−1.60061	−0.800303	−	0.599596i	$-0.795328\pi$
$859$	−46.9117	−1.60061	−0.800303	+	0.599596i	$0.795328\pi$
$860$	0	0
$861$	−7.31371	−0.249251
$862$	0	0
$863$	−21.6569	−0.737208	−0.368604	−	0.929587i	$-0.620164\pi$
$863$	−21.6569	−0.737208	−0.368604	+	0.929587i	$0.620164\pi$
$864$	0	0
$865$	4.34315	0.147671
$866$	0	0
$867$	−9.00000	−0.305656
$868$	0	0
$869$	−10.3431	−0.350867
$870$	0	0
$871$	8.00000	0.271070
$872$	0	0
$873$	7.65685	0.259145
$874$	0	0
$875$	2.00000	0.0676123
$876$	0	0
$877$	−14.4853	−0.489133	−0.244567	−	0.969632i	$-0.578646\pi$
$877$	−14.4853	−0.489133	−0.244567	+	0.969632i	$0.578646\pi$
$878$	0	0
$879$	−17.3137	−0.583977
$880$	0	0
$881$	24.3431	0.820141	0.410071	−	0.912054i	$-0.365504\pi$
$881$	24.3431	0.820141	0.410071	+	0.912054i	$0.365504\pi$
$882$	0	0
$883$	−29.9411	−1.00760	−0.503800	−	0.863821i	$-0.668065\pi$
$883$	−29.9411	−1.00760	−0.503800	+	0.863821i	$0.668065\pi$
$884$	0	0
$885$	4.82843	0.162306
$886$	0	0
$887$	−39.5980	−1.32957	−0.664785	−	0.747035i	$-0.731477\pi$
$887$	−39.5980	−1.32957	−0.664785	+	0.747035i	$0.731477\pi$
$888$	0	0
$889$	31.3137	1.05023
$890$	0	0
$891$	−0.828427	−0.0277534
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−14.4853	−0.484190
$896$	0	0
$897$	−4.68629	−0.156471
$898$	0	0
$899$	−4.28427	−0.142888
$900$	0	0
$901$	32.9706	1.09841
$902$	0	0
$903$	3.31371	0.110273
$904$	0	0
$905$	−12.0000	−0.398893
$906$	0	0
$907$	−20.9706	−0.696316	−0.348158	−	0.937436i	$-0.613193\pi$
$907$	−20.9706	−0.696316	−0.348158	+	0.937436i	$0.613193\pi$
$908$	0	0
$909$	5.31371	0.176245
$910$	0	0
$911$	−12.6863	−0.420316	−0.210158	−	0.977667i	$-0.567398\pi$
$911$	−12.6863	−0.420316	−0.210158	+	0.977667i	$0.567398\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−9.65685	−0.319246
$916$	0	0
$917$	28.9706	0.956692
$918$	0	0
$919$	−27.5147	−0.907627	−0.453813	−	0.891097i	$-0.649937\pi$
$919$	−27.5147	−0.907627	−0.453813	+	0.891097i	$0.649937\pi$
$920$	0	0
$921$	−4.00000	−0.131804
$922$	0	0
$923$	−11.3137	−0.372395
$924$	0	0
$925$	6.48528	0.213235
$926$	0	0
$927$	4.34315	0.142648
$928$	0	0
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	11.3137	0.370394
$934$	0	0
$935$	−2.34315	−0.0766291
$936$	0	0
$937$	−9.31371	−0.304266	−0.152133	−	0.988360i	$-0.548614\pi$
$937$	−9.31371	−0.304266	−0.152133	+	0.988360i	$0.548614\pi$
$938$	0	0
$939$	22.9706	0.749616
$940$	0	0
$941$	−55.2548	−1.80126	−0.900628	−	0.434591i	$-0.856893\pi$
$941$	−55.2548	−1.80126	−0.900628	+	0.434591i	$0.856893\pi$
$942$	0	0
$943$	−20.6863	−0.673638
$944$	0	0
$945$	2.00000	0.0650600
$946$	0	0
$947$	−21.9411	−0.712991	−0.356495	−	0.934297i	$-0.616028\pi$
$947$	−21.9411	−0.712991	−0.356495	+	0.934297i	$0.616028\pi$
$948$	0	0
$949$	−7.71573	−0.250463
$950$	0	0
$951$	13.3137	0.431727
$952$	0	0
$953$	−38.8284	−1.25778	−0.628888	−	0.777496i	$-0.716490\pi$
$953$	−38.8284	−1.25778	−0.628888	+	0.777496i	$0.716490\pi$
$954$	0	0
$955$	4.68629	0.151645
$956$	0	0
$957$	3.02944	0.0979278
$958$	0	0
$959$	2.34315	0.0756641
$960$	0	0
$961$	−29.6274	−0.955723
$962$	0	0
$963$	−2.34315	−0.0755068
$964$	0	0
$965$	−10.9706	−0.353155
$966$	0	0
$967$	44.9117	1.44426	0.722131	−	0.691756i	$-0.243163\pi$
$967$	44.9117	1.44426	0.722131	+	0.691756i	$0.243163\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	44.1421	1.41659	0.708294	−	0.705917i	$-0.249465\pi$
$971$	44.1421	1.41659	0.708294	+	0.705917i	$0.249465\pi$
$972$	0	0
$973$	−19.3137	−0.619169
$974$	0	0
$975$	−0.828427	−0.0265309
$976$	0	0
$977$	−21.1716	−0.677339	−0.338669	−	0.940905i	$-0.609977\pi$
$977$	−21.1716	−0.677339	−0.338669	+	0.940905i	$0.609977\pi$
$978$	0	0
$979$	3.59798	0.114992
$980$	0	0
$981$	−4.00000	−0.127710
$982$	0	0
$983$	−52.9706	−1.68950	−0.844749	−	0.535162i	$-0.820250\pi$
$983$	−52.9706	−1.68950	−0.844749	+	0.535162i	$0.820250\pi$
$984$	0	0
$985$	−1.31371	−0.0418582
$986$	0	0
$987$	−3.31371	−0.105477
$988$	0	0
$989$	9.37258	0.298031
$990$	0	0
$991$	−32.4853	−1.03193	−0.515964	−	0.856610i	$-0.672566\pi$
$991$	−32.4853	−1.03193	−0.515964	+	0.856610i	$0.672566\pi$
$992$	0	0
$993$	32.0000	1.01549
$994$	0	0
$995$	13.1716	0.417567
$996$	0	0
$997$	16.1421	0.511227	0.255613	−	0.966779i	$-0.417723\pi$
$997$	16.1421	0.511227	0.255613	+	0.966779i	$0.417723\pi$
$998$	0	0
$999$	6.48528	0.205185

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3840.2.a.bm.1.1		2
4.3	odd	2		3840.2.a.bg.1.2		2
8.3	odd	2		3840.2.a.bj.1.1		2
8.5	even	2		3840.2.a.bd.1.2		2
16.3	odd	4		1920.2.k.k.961.1	yes	4
16.5	even	4		1920.2.k.j.961.1	✓	4
16.11	odd	4		1920.2.k.k.961.4	yes	4
16.13	even	4		1920.2.k.j.961.4	yes	4
48.5	odd	4		5760.2.k.m.2881.2		4
48.11	even	4		5760.2.k.x.2881.1		4
48.29	odd	4		5760.2.k.m.2881.3		4
48.35	even	4		5760.2.k.x.2881.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1920.2.k.j.961.1	✓	4	16.5	even	4
1920.2.k.j.961.4	yes	4	16.13	even	4
1920.2.k.k.961.1	yes	4	16.3	odd	4
1920.2.k.k.961.4	yes	4	16.11	odd	4
3840.2.a.bd.1.2		2	8.5	even	2
3840.2.a.bg.1.2		2	4.3	odd	2
3840.2.a.bj.1.1		2	8.3	odd	2
3840.2.a.bm.1.1		2	1.1	even	1	trivial
5760.2.k.m.2881.2		4	48.5	odd	4
5760.2.k.m.2881.3		4	48.29	odd	4
5760.2.k.x.2881.1		4	48.11	even	4
5760.2.k.x.2881.4		4	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:	\( N \)	\(=\)	\( 3840 = 2^{8} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3840.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} +2.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} +2.00000 q^{7} +1.00000 q^{9} -0.828427 q^{11} -0.828427 q^{13} +1.00000 q^{15} +2.82843 q^{17} +2.00000 q^{21} +5.65685 q^{23} +1.00000 q^{25} +1.00000 q^{27} -3.65685 q^{29} +1.17157 q^{31} -0.828427 q^{33} +2.00000 q^{35} +6.48528 q^{37} -0.828427 q^{39} -3.65685 q^{41} +1.65685 q^{43} +1.00000 q^{45} -1.65685 q^{47} -3.00000 q^{49} +2.82843 q^{51} +11.6569 q^{53} -0.828427 q^{55} +4.82843 q^{59} -9.65685 q^{61} +2.00000 q^{63} -0.828427 q^{65} -9.65685 q^{67} +5.65685 q^{69} +13.6569 q^{71} +9.31371 q^{73} +1.00000 q^{75} -1.65685 q^{77} +12.4853 q^{79} +1.00000 q^{81} +2.82843 q^{85} -3.65685 q^{87} -4.34315 q^{89} -1.65685 q^{91} +1.17157 q^{93} +7.65685 q^{97} -0.828427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^5 + 4 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{5} + 4 q^{7} + 2 q^{9} + 4 q^{11} + 4 q^{13} + 2 q^{15} + 4 q^{21} + 2 q^{25} + 2 q^{27} + 4 q^{29} + 8 q^{31} + 4 q^{33} + 4 q^{35} - 4 q^{37} + 4 q^{39} + 4 q^{41} - 8 q^{43} + 2 q^{45} + 8 q^{47} - 6 q^{49} + 12 q^{53} + 4 q^{55} + 4 q^{59} - 8 q^{61} + 4 q^{63} + 4 q^{65} - 8 q^{67} + 16 q^{71} - 4 q^{73} + 2 q^{75} + 8 q^{77} + 8 q^{79} + 2 q^{81} + 4 q^{87} - 20 q^{89} + 8 q^{91} + 8 q^{93} + 4 q^{97} + 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^5 + 4 * q^7 + 2 * q^9 + 4 * q^11 + 4 * q^13 + 2 * q^15 + 4 * q^21 + 2 * q^25 + 2 * q^27 + 4 * q^29 + 8 * q^31 + 4 * q^33 + 4 * q^35 - 4 * q^37 + 4 * q^39 + 4 * q^41 - 8 * q^43 + 2 * q^45 + 8 * q^47 - 6 * q^49 + 12 * q^53 + 4 * q^55 + 4 * q^59 - 8 * q^61 + 4 * q^63 + 4 * q^65 - 8 * q^67 + 16 * q^71 - 4 * q^73 + 2 * q^75 + 8 * q^77 + 8 * q^79 + 2 * q^81 + 4 * q^87 - 20 * q^89 + 8 * q^91 + 8 * q^93 + 4 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more