LMFDB - Embedded newform 3840.2.a.bg.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:	$1$
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)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} -2.00000 q^{7} +1.00000 q^{9} -4.82843 q^{11} +4.82843 q^{13} -1.00000 q^{15} -2.82843 q^{17} +2.00000 q^{21} +5.65685 q^{23} +1.00000 q^{25} -1.00000 q^{27} +7.65685 q^{29} -6.82843 q^{31} +4.82843 q^{33} -2.00000 q^{35} -10.4853 q^{37} -4.82843 q^{39} +7.65685 q^{41} +9.65685 q^{43} +1.00000 q^{45} -9.65685 q^{47} -3.00000 q^{49} +2.82843 q^{51} +0.343146 q^{53} -4.82843 q^{55} +0.828427 q^{59} +1.65685 q^{61} -2.00000 q^{63} +4.82843 q^{65} -1.65685 q^{67} -5.65685 q^{69} -2.34315 q^{71} -13.3137 q^{73} -1.00000 q^{75} +9.65685 q^{77} +4.48528 q^{79} +1.00000 q^{81} -2.82843 q^{85} -7.65685 q^{87} -15.6569 q^{89} -9.65685 q^{91} +6.82843 q^{93} -3.65685 q^{97} -4.82843 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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}+ \cdots - 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

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	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.82843	−1.45583	−0.727913	−	0.685670i	$-0.759509\pi$
$11$	−4.82843	−1.45583	−0.727913	+	0.685670i	$0.759509\pi$
$12$	0	0
$13$	4.82843	1.33916	0.669582	−	0.742738i	$-0.266473\pi$
$13$	4.82843	1.33916	0.669582	+	0.742738i	$0.266473\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−2.82843	−0.685994	−0.342997	−	0.939336i	$-0.611442\pi$
$17$	−2.82843	−0.685994	−0.342997	+	0.939336i	$0.611442\pi$
$18$	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
$29$	7.65685	1.42184	0.710921	−	0.703272i	$-0.248278\pi$
$29$	7.65685	1.42184	0.710921	+	0.703272i	$0.248278\pi$
$30$	0	0
$31$	−6.82843	−1.22642	−0.613211	−	0.789919i	$-0.710122\pi$
$31$	−6.82843	−1.22642	−0.613211	+	0.789919i	$0.710122\pi$
$32$	0	0
$33$	4.82843	0.840521
$34$	0	0
$35$	−2.00000	−0.338062
$36$	0	0
$37$	−10.4853	−1.72377	−0.861885	−	0.507104i	$-0.830716\pi$
$37$	−10.4853	−1.72377	−0.861885	+	0.507104i	$0.830716\pi$
$38$	0	0
$39$	−4.82843	−0.773167
$40$	0	0
$41$	7.65685	1.19580	0.597900	−	0.801571i	$-0.296002\pi$
$41$	7.65685	1.19580	0.597900	+	0.801571i	$0.296002\pi$
$42$	0	0
$43$	9.65685	1.47266	0.736328	−	0.676625i	$-0.236558\pi$
$43$	9.65685	1.47266	0.736328	+	0.676625i	$0.236558\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−9.65685	−1.40860	−0.704298	−	0.709904i	$-0.748738\pi$
$47$	−9.65685	−1.40860	−0.704298	+	0.709904i	$0.748738\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	2.82843	0.396059
$52$	0	0
$53$	0.343146	0.0471347	0.0235673	−	0.999722i	$-0.492498\pi$
$53$	0.343146	0.0471347	0.0235673	+	0.999722i	$0.492498\pi$
$54$	0	0
$55$	−4.82843	−0.651065
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0.828427	0.107852	0.0539260	−	0.998545i	$-0.482826\pi$
$59$	0.828427	0.107852	0.0539260	+	0.998545i	$0.482826\pi$
$60$	0	0
$61$	1.65685	0.212138	0.106069	−	0.994359i	$-0.466173\pi$
$61$	1.65685	0.212138	0.106069	+	0.994359i	$0.466173\pi$
$62$	0	0
$63$	−2.00000	−0.251976
$64$	0	0
$65$	4.82843	0.598893
$66$	0	0
$67$	−1.65685	−0.202417	−0.101208	−	0.994865i	$-0.532271\pi$
$67$	−1.65685	−0.202417	−0.101208	+	0.994865i	$0.532271\pi$
$68$	0	0
$69$	−5.65685	−0.681005
$70$	0	0
$71$	−2.34315	−0.278080	−0.139040	−	0.990287i	$-0.544402\pi$
$71$	−2.34315	−0.278080	−0.139040	+	0.990287i	$0.544402\pi$
$72$	0	0
$73$	−13.3137	−1.55825	−0.779126	−	0.626868i	$-0.784337\pi$
$73$	−13.3137	−1.55825	−0.779126	+	0.626868i	$0.784337\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	9.65685	1.10050
$78$	0	0
$79$	4.48528	0.504634	0.252317	−	0.967645i	$-0.418807\pi$
$79$	4.48528	0.504634	0.252317	+	0.967645i	$0.418807\pi$
$80$	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
$87$	−7.65685	−0.820901
$88$	0	0
$89$	−15.6569	−1.65962	−0.829812	−	0.558044i	$-0.811552\pi$
$89$	−15.6569	−1.65962	−0.829812	+	0.558044i	$0.811552\pi$
$90$	0	0
$91$	−9.65685	−1.01231
$92$	0	0
$93$	6.82843	0.708075
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−3.65685	−0.371297	−0.185649	−	0.982616i	$-0.559439\pi$
$97$	−3.65685	−0.371297	−0.185649	+	0.982616i	$0.559439\pi$
$98$	0	0
$99$	−4.82843	−0.485275
$100$	0	0
$101$	−17.3137	−1.72278	−0.861389	−	0.507946i	$-0.830405\pi$
$101$	−17.3137	−1.72278	−0.861389	+	0.507946i	$0.830405\pi$
$102$	0	0
$103$	−15.6569	−1.54272	−0.771358	−	0.636402i	$-0.780422\pi$
$103$	−15.6569	−1.54272	−0.771358	+	0.636402i	$0.780422\pi$
$104$	0	0
$105$	2.00000	0.195180
$106$	0	0
$107$	13.6569	1.32026	0.660129	−	0.751152i	$-0.270502\pi$
$107$	13.6569	1.32026	0.660129	+	0.751152i	$0.270502\pi$
$108$	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
$111$	10.4853	0.995219
$112$	0	0
$113$	−3.51472	−0.330637	−0.165318	−	0.986240i	$-0.552865\pi$
$113$	−3.51472	−0.330637	−0.165318	+	0.986240i	$0.552865\pi$
$114$	0	0
$115$	5.65685	0.527504
$116$	0	0
$117$	4.82843	0.446388
$118$	0	0
$119$	5.65685	0.518563
$120$	0	0
$121$	12.3137	1.11943
$122$	0	0
$123$	−7.65685	−0.690395
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−4.34315	−0.385392	−0.192696	−	0.981259i	$-0.561723\pi$
$127$	−4.34315	−0.385392	−0.192696	+	0.981259i	$0.561723\pi$
$128$	0	0
$129$	−9.65685	−0.850239
$130$	0	0
$131$	2.48528	0.217140	0.108570	−	0.994089i	$-0.465373\pi$
$131$	2.48528	0.217140	0.108570	+	0.994089i	$0.465373\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	6.82843	0.583392	0.291696	−	0.956511i	$-0.405780\pi$
$137$	6.82843	0.583392	0.291696	+	0.956511i	$0.405780\pi$
$138$	0	0
$139$	−1.65685	−0.140533	−0.0702663	−	0.997528i	$-0.522385\pi$
$139$	−1.65685	−0.140533	−0.0702663	+	0.997528i	$0.522385\pi$
$140$	0	0
$141$	9.65685	0.813254
$142$	0	0
$143$	−23.3137	−1.94959
$144$	0	0
$145$	7.65685	0.635867
$146$	0	0
$147$	3.00000	0.247436
$148$	0	0
$149$	0.343146	0.0281116	0.0140558	−	0.999901i	$-0.495526\pi$
$149$	0.343146	0.0281116	0.0140558	+	0.999901i	$0.495526\pi$
$150$	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
$155$	−6.82843	−0.548472
$156$	0	0
$157$	−5.51472	−0.440122	−0.220061	−	0.975486i	$-0.570626\pi$
$157$	−5.51472	−0.440122	−0.220061	+	0.975486i	$0.570626\pi$
$158$	0	0
$159$	−0.343146	−0.0272132
$160$	0	0
$161$	−11.3137	−0.891645
$162$	0	0
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	0	0
$165$	4.82843	0.375893
$166$	0	0
$167$	−10.3431	−0.800377	−0.400188	−	0.916433i	$-0.631055\pi$
$167$	−10.3431	−0.800377	−0.400188	+	0.916433i	$0.631055\pi$
$168$	0	0
$169$	10.3137	0.793362
$170$	0	0
$171$	0	0
$172$	0	0
$173$	15.6569	1.19037	0.595184	−	0.803589i	$-0.297079\pi$
$173$	15.6569	1.19037	0.595184	+	0.803589i	$0.297079\pi$
$174$	0	0
$175$	−2.00000	−0.151186
$176$	0	0
$177$	−0.828427	−0.0622684
$178$	0	0
$179$	−2.48528	−0.185759	−0.0928793	−	0.995677i	$-0.529607\pi$
$179$	−2.48528	−0.185759	−0.0928793	+	0.995677i	$0.529607\pi$
$180$	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
$183$	−1.65685	−0.122478
$184$	0	0
$185$	−10.4853	−0.770893
$186$	0	0
$187$	13.6569	0.998688
$188$	0	0
$189$	2.00000	0.145479
$190$	0	0
$191$	−27.3137	−1.97635	−0.988175	−	0.153328i	$-0.951001\pi$
$191$	−27.3137	−1.97635	−0.988175	+	0.153328i	$0.951001\pi$
$192$	0	0
$193$	22.9706	1.65346	0.826729	−	0.562601i	$-0.190199\pi$
$193$	22.9706	1.65346	0.826729	+	0.562601i	$0.190199\pi$
$194$	0	0
$195$	−4.82843	−0.345771
$196$	0	0
$197$	21.3137	1.51854	0.759269	−	0.650776i	$-0.225556\pi$
$197$	21.3137	1.51854	0.759269	+	0.650776i	$0.225556\pi$
$198$	0	0
$199$	−18.8284	−1.33471	−0.667356	−	0.744739i	$-0.732574\pi$
$199$	−18.8284	−1.33471	−0.667356	+	0.744739i	$0.732574\pi$
$200$	0	0
$201$	1.65685	0.116865
$202$	0	0
$203$	−15.3137	−1.07481
$204$	0	0
$205$	7.65685	0.534778
$206$	0	0
$207$	5.65685	0.393179
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−25.6569	−1.76629	−0.883145	−	0.469099i	$-0.844579\pi$
$211$	−25.6569	−1.76629	−0.883145	+	0.469099i	$0.844579\pi$
$212$	0	0
$213$	2.34315	0.160550
$214$	0	0
$215$	9.65685	0.658592
$216$	0	0
$217$	13.6569	0.927088
$218$	0	0
$219$	13.3137	0.899657
$220$	0	0
$221$	−13.6569	−0.918659
$222$	0	0
$223$	14.9706	1.00250	0.501252	−	0.865302i	$-0.332873\pi$
$223$	14.9706	1.00250	0.501252	+	0.865302i	$0.332873\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	6.34315	0.421009	0.210505	−	0.977593i	$-0.432489\pi$
$227$	6.34315	0.421009	0.210505	+	0.977593i	$0.432489\pi$
$228$	0	0
$229$	−10.3431	−0.683494	−0.341747	−	0.939792i	$-0.611019\pi$
$229$	−10.3431	−0.683494	−0.341747	+	0.939792i	$0.611019\pi$
$230$	0	0
$231$	−9.65685	−0.635374
$232$	0	0
$233$	−27.7990	−1.82117	−0.910586	−	0.413319i	$-0.864369\pi$
$233$	−27.7990	−1.82117	−0.910586	+	0.413319i	$0.864369\pi$
$234$	0	0
$235$	−9.65685	−0.629944
$236$	0	0
$237$	−4.48528	−0.291350
$238$	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
$241$	−24.6274	−1.58639	−0.793196	−	0.608967i	$-0.791584\pi$
$241$	−24.6274	−1.58639	−0.793196	+	0.608967i	$0.791584\pi$
$242$	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
$251$	−3.17157	−0.200188	−0.100094	−	0.994978i	$-0.531914\pi$
$251$	−3.17157	−0.200188	−0.100094	+	0.994978i	$0.531914\pi$
$252$	0	0
$253$	−27.3137	−1.71720
$254$	0	0
$255$	2.82843	0.177123
$256$	0	0
$257$	9.17157	0.572107	0.286053	−	0.958214i	$-0.407656\pi$
$257$	9.17157	0.572107	0.286053	+	0.958214i	$0.407656\pi$
$258$	0	0
$259$	20.9706	1.30305
$260$	0	0
$261$	7.65685	0.473947
$262$	0	0
$263$	−18.3431	−1.13109	−0.565543	−	0.824719i	$-0.691334\pi$
$263$	−18.3431	−1.13109	−0.565543	+	0.824719i	$0.691334\pi$
$264$	0	0
$265$	0.343146	0.0210793
$266$	0	0
$267$	15.6569	0.958184
$268$	0	0
$269$	−3.65685	−0.222962	−0.111481	−	0.993767i	$-0.535559\pi$
$269$	−3.65685	−0.222962	−0.111481	+	0.993767i	$0.535559\pi$
$270$	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
$273$	9.65685	0.584459
$274$	0	0
$275$	−4.82843	−0.291165
$276$	0	0
$277$	−28.8284	−1.73213	−0.866066	−	0.499929i	$-0.833359\pi$
$277$	−28.8284	−1.73213	−0.866066	+	0.499929i	$0.833359\pi$
$278$	0	0
$279$	−6.82843	−0.408807
$280$	0	0
$281$	22.9706	1.37031	0.685154	−	0.728398i	$-0.259735\pi$
$281$	22.9706	1.37031	0.685154	+	0.728398i	$0.259735\pi$
$282$	0	0
$283$	20.9706	1.24657	0.623285	−	0.781995i	$-0.285798\pi$
$283$	20.9706	1.24657	0.623285	+	0.781995i	$0.285798\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−15.3137	−0.903940
$288$	0	0
$289$	−9.00000	−0.529412
$290$	0	0
$291$	3.65685	0.214369
$292$	0	0
$293$	5.31371	0.310430	0.155215	−	0.987881i	$-0.450393\pi$
$293$	5.31371	0.310430	0.155215	+	0.987881i	$0.450393\pi$
$294$	0	0
$295$	0.828427	0.0482329
$296$	0	0
$297$	4.82843	0.280174
$298$	0	0
$299$	27.3137	1.57959
$300$	0	0
$301$	−19.3137	−1.11322
$302$	0	0
$303$	17.3137	0.994647
$304$	0	0
$305$	1.65685	0.0948712
$306$	0	0
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	0	0
$309$	15.6569	0.890687
$310$	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
$313$	−10.9706	−0.620093	−0.310046	−	0.950721i	$-0.600345\pi$
$313$	−10.9706	−0.620093	−0.310046	+	0.950721i	$0.600345\pi$
$314$	0	0
$315$	−2.00000	−0.112687
$316$	0	0
$317$	−9.31371	−0.523110	−0.261555	−	0.965189i	$-0.584235\pi$
$317$	−9.31371	−0.523110	−0.261555	+	0.965189i	$0.584235\pi$
$318$	0	0
$319$	−36.9706	−2.06995
$320$	0	0
$321$	−13.6569	−0.762251
$322$	0	0
$323$	0	0
$324$	0	0
$325$	4.82843	0.267833
$326$	0	0
$327$	4.00000	0.221201
$328$	0	0
$329$	19.3137	1.06480
$330$	0	0
$331$	−32.0000	−1.75888	−0.879440	−	0.476011i	$-0.842082\pi$
$331$	−32.0000	−1.75888	−0.879440	+	0.476011i	$0.842082\pi$
$332$	0	0
$333$	−10.4853	−0.574590
$334$	0	0
$335$	−1.65685	−0.0905236
$336$	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
$339$	3.51472	0.190893
$340$	0	0
$341$	32.9706	1.78546
$342$	0	0
$343$	20.0000	1.07990
$344$	0	0
$345$	−5.65685	−0.304555
$346$	0	0
$347$	−1.65685	−0.0889446	−0.0444723	−	0.999011i	$-0.514161\pi$
$347$	−1.65685	−0.0889446	−0.0444723	+	0.999011i	$0.514161\pi$
$348$	0	0
$349$	−36.9706	−1.97899	−0.989494	−	0.144571i	$-0.953820\pi$
$349$	−36.9706	−1.97899	−0.989494	+	0.144571i	$0.953820\pi$
$350$	0	0
$351$	−4.82843	−0.257722
$352$	0	0
$353$	25.4558	1.35488	0.677439	−	0.735579i	$-0.263090\pi$
$353$	25.4558	1.35488	0.677439	+	0.735579i	$0.263090\pi$
$354$	0	0
$355$	−2.34315	−0.124361
$356$	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
$363$	−12.3137	−0.646302
$364$	0	0
$365$	−13.3137	−0.696871
$366$	0	0
$367$	25.3137	1.32136	0.660682	−	0.750666i	$-0.270267\pi$
$367$	25.3137	1.32136	0.660682	+	0.750666i	$0.270267\pi$
$368$	0	0
$369$	7.65685	0.398600
$370$	0	0
$371$	−0.686292	−0.0356305
$372$	0	0
$373$	−13.5147	−0.699766	−0.349883	−	0.936793i	$-0.613779\pi$
$373$	−13.5147	−0.699766	−0.349883	+	0.936793i	$0.613779\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	36.9706	1.90408
$378$	0	0
$379$	−32.2843	−1.65833	−0.829166	−	0.559003i	$-0.811184\pi$
$379$	−32.2843	−1.65833	−0.829166	+	0.559003i	$0.811184\pi$
$380$	0	0
$381$	4.34315	0.222506
$382$	0	0
$383$	−17.6569	−0.902223	−0.451112	−	0.892468i	$-0.648972\pi$
$383$	−17.6569	−0.902223	−0.451112	+	0.892468i	$0.648972\pi$
$384$	0	0
$385$	9.65685	0.492159
$386$	0	0
$387$	9.65685	0.490885
$388$	0	0
$389$	9.31371	0.472224	0.236112	−	0.971726i	$-0.424127\pi$
$389$	9.31371	0.472224	0.236112	+	0.971726i	$0.424127\pi$
$390$	0	0
$391$	−16.0000	−0.809155
$392$	0	0
$393$	−2.48528	−0.125366
$394$	0	0
$395$	4.48528	0.225679
$396$	0	0
$397$	−10.4853	−0.526241	−0.263121	−	0.964763i	$-0.584752\pi$
$397$	−10.4853	−0.526241	−0.263121	+	0.964763i	$0.584752\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−8.62742	−0.430833	−0.215416	−	0.976522i	$-0.569111\pi$
$401$	−8.62742	−0.430833	−0.215416	+	0.976522i	$0.569111\pi$
$402$	0	0
$403$	−32.9706	−1.64238
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	50.6274	2.50951
$408$	0	0
$409$	−25.3137	−1.25168	−0.625841	−	0.779951i	$-0.715244\pi$
$409$	−25.3137	−1.25168	−0.625841	+	0.779951i	$0.715244\pi$
$410$	0	0
$411$	−6.82843	−0.336821
$412$	0	0
$413$	−1.65685	−0.0815285
$414$	0	0
$415$	0	0
$416$	0	0
$417$	1.65685	0.0811365
$418$	0	0
$419$	−14.4853	−0.707652	−0.353826	−	0.935311i	$-0.615120\pi$
$419$	−14.4853	−0.707652	−0.353826	+	0.935311i	$0.615120\pi$
$420$	0	0
$421$	16.9706	0.827095	0.413547	−	0.910483i	$-0.364290\pi$
$421$	16.9706	0.827095	0.413547	+	0.910483i	$0.364290\pi$
$422$	0	0
$423$	−9.65685	−0.469532
$424$	0	0
$425$	−2.82843	−0.137199
$426$	0	0
$427$	−3.31371	−0.160362
$428$	0	0
$429$	23.3137	1.12560
$430$	0	0
$431$	10.3431	0.498212	0.249106	−	0.968476i	$-0.419863\pi$
$431$	10.3431	0.498212	0.249106	+	0.968476i	$0.419863\pi$
$432$	0	0
$433$	12.6274	0.606835	0.303417	−	0.952858i	$-0.401872\pi$
$433$	12.6274	0.606835	0.303417	+	0.952858i	$0.401872\pi$
$434$	0	0
$435$	−7.65685	−0.367118
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−6.14214	−0.293148	−0.146574	−	0.989200i	$-0.546825\pi$
$439$	−6.14214	−0.293148	−0.146574	+	0.989200i	$0.546825\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	6.34315	0.301372	0.150686	−	0.988582i	$-0.451852\pi$
$443$	6.34315	0.301372	0.150686	+	0.988582i	$0.451852\pi$
$444$	0	0
$445$	−15.6569	−0.742206
$446$	0	0
$447$	−0.343146	−0.0162302
$448$	0	0
$449$	−21.3137	−1.00586	−0.502928	−	0.864328i	$-0.667744\pi$
$449$	−21.3137	−1.00586	−0.502928	+	0.864328i	$0.667744\pi$
$450$	0	0
$451$	−36.9706	−1.74088
$452$	0	0
$453$	8.48528	0.398673
$454$	0	0
$455$	−9.65685	−0.452720
$456$	0	0
$457$	18.9706	0.887405	0.443703	−	0.896174i	$-0.353665\pi$
$457$	18.9706	0.887405	0.443703	+	0.896174i	$0.353665\pi$
$458$	0	0
$459$	2.82843	0.132020
$460$	0	0
$461$	8.34315	0.388579	0.194290	−	0.980944i	$-0.437760\pi$
$461$	8.34315	0.388579	0.194290	+	0.980944i	$0.437760\pi$
$462$	0	0
$463$	−20.6274	−0.958637	−0.479319	−	0.877641i	$-0.659116\pi$
$463$	−20.6274	−0.958637	−0.479319	+	0.877641i	$0.659116\pi$
$464$	0	0
$465$	6.82843	0.316661
$466$	0	0
$467$	35.3137	1.63412	0.817062	−	0.576550i	$-0.195601\pi$
$467$	35.3137	1.63412	0.817062	+	0.576550i	$0.195601\pi$
$468$	0	0
$469$	3.31371	0.153013
$470$	0	0
$471$	5.51472	0.254105
$472$	0	0
$473$	−46.6274	−2.14393
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0.343146	0.0157116
$478$	0	0
$479$	21.6569	0.989527	0.494763	−	0.869028i	$-0.335255\pi$
$479$	21.6569	0.989527	0.494763	+	0.869028i	$0.335255\pi$
$480$	0	0
$481$	−50.6274	−2.30841
$482$	0	0
$483$	11.3137	0.514792
$484$	0	0
$485$	−3.65685	−0.166049
$486$	0	0
$487$	−30.9706	−1.40341	−0.701705	−	0.712468i	$-0.747578\pi$
$487$	−30.9706	−1.40341	−0.701705	+	0.712468i	$0.747578\pi$
$488$	0	0
$489$	−12.0000	−0.542659
$490$	0	0
$491$	−32.8284	−1.48153	−0.740763	−	0.671766i	$-0.765536\pi$
$491$	−32.8284	−1.48153	−0.740763	+	0.671766i	$0.765536\pi$
$492$	0	0
$493$	−21.6569	−0.975376
$494$	0	0
$495$	−4.82843	−0.217022
$496$	0	0
$497$	4.68629	0.210209
$498$	0	0
$499$	38.6274	1.72920	0.864600	−	0.502460i	$-0.167572\pi$
$499$	38.6274	1.72920	0.864600	+	0.502460i	$0.167572\pi$
$500$	0	0
$501$	10.3431	0.462098
$502$	0	0
$503$	−1.65685	−0.0738755	−0.0369377	−	0.999318i	$-0.511760\pi$
$503$	−1.65685	−0.0738755	−0.0369377	+	0.999318i	$0.511760\pi$
$504$	0	0
$505$	−17.3137	−0.770450
$506$	0	0
$507$	−10.3137	−0.458048
$508$	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
$511$	26.6274	1.17793
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−15.6569	−0.689923
$516$	0	0
$517$	46.6274	2.05067
$518$	0	0
$519$	−15.6569	−0.687260
$520$	0	0
$521$	17.3137	0.758527	0.379264	−	0.925289i	$-0.376177\pi$
$521$	17.3137	0.758527	0.379264	+	0.925289i	$0.376177\pi$
$522$	0	0
$523$	29.9411	1.30923	0.654617	−	0.755961i	$-0.272830\pi$
$523$	29.9411	1.30923	0.654617	+	0.755961i	$0.272830\pi$
$524$	0	0
$525$	2.00000	0.0872872
$526$	0	0
$527$	19.3137	0.841318
$528$	0	0
$529$	9.00000	0.391304
$530$	0	0
$531$	0.828427	0.0359507
$532$	0	0
$533$	36.9706	1.60137
$534$	0	0
$535$	13.6569	0.590437
$536$	0	0
$537$	2.48528	0.107248
$538$	0	0
$539$	14.4853	0.623925
$540$	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
$547$	14.3431	0.613269	0.306634	−	0.951827i	$-0.400797\pi$
$547$	14.3431	0.613269	0.306634	+	0.951827i	$0.400797\pi$
$548$	0	0
$549$	1.65685	0.0707128
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−8.97056	−0.381467
$554$	0	0
$555$	10.4853	0.445075
$556$	0	0
$557$	12.3431	0.522996	0.261498	−	0.965204i	$-0.415784\pi$
$557$	12.3431	0.522996	0.261498	+	0.965204i	$0.415784\pi$
$558$	0	0
$559$	46.6274	1.97213
$560$	0	0
$561$	−13.6569	−0.576593
$562$	0	0
$563$	4.97056	0.209484	0.104742	−	0.994499i	$-0.466598\pi$
$563$	4.97056	0.209484	0.104742	+	0.994499i	$0.466598\pi$
$564$	0	0
$565$	−3.51472	−0.147865
$566$	0	0
$567$	−2.00000	−0.0839921
$568$	0	0
$569$	17.3137	0.725828	0.362914	−	0.931823i	$-0.381782\pi$
$569$	17.3137	0.725828	0.362914	+	0.931823i	$0.381782\pi$
$570$	0	0
$571$	6.62742	0.277349	0.138674	−	0.990338i	$-0.455716\pi$
$571$	6.62742	0.277349	0.138674	+	0.990338i	$0.455716\pi$
$572$	0	0
$573$	27.3137	1.14105
$574$	0	0
$575$	5.65685	0.235907
$576$	0	0
$577$	−38.9706	−1.62237	−0.811183	−	0.584793i	$-0.801176\pi$
$577$	−38.9706	−1.62237	−0.811183	+	0.584793i	$0.801176\pi$
$578$	0	0
$579$	−22.9706	−0.954624
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−1.65685	−0.0686199
$584$	0	0
$585$	4.82843	0.199631
$586$	0	0
$587$	18.3431	0.757103	0.378551	−	0.925580i	$-0.376422\pi$
$587$	18.3431	0.757103	0.378551	+	0.925580i	$0.376422\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−21.3137	−0.876729
$592$	0	0
$593$	−0.201010	−0.00825450	−0.00412725	−	0.999991i	$-0.501314\pi$
$593$	−0.201010	−0.00825450	−0.00412725	+	0.999991i	$0.501314\pi$
$594$	0	0
$595$	5.65685	0.231908
$596$	0	0
$597$	18.8284	0.770596
$598$	0	0
$599$	34.3431	1.40322	0.701611	−	0.712560i	$-0.252464\pi$
$599$	34.3431	1.40322	0.701611	+	0.712560i	$0.252464\pi$
$600$	0	0
$601$	39.9411	1.62923	0.814616	−	0.580000i	$-0.196948\pi$
$601$	39.9411	1.62923	0.814616	+	0.580000i	$0.196948\pi$
$602$	0	0
$603$	−1.65685	−0.0674723
$604$	0	0
$605$	12.3137	0.500623
$606$	0	0
$607$	−20.6274	−0.837241	−0.418621	−	0.908161i	$-0.637486\pi$
$607$	−20.6274	−0.837241	−0.418621	+	0.908161i	$0.637486\pi$
$608$	0	0
$609$	15.3137	0.620543
$610$	0	0
$611$	−46.6274	−1.88634
$612$	0	0
$613$	−26.4853	−1.06973	−0.534865	−	0.844937i	$-0.679638\pi$
$613$	−26.4853	−1.06973	−0.534865	+	0.844937i	$0.679638\pi$
$614$	0	0
$615$	−7.65685	−0.308754
$616$	0	0
$617$	2.82843	0.113868	0.0569341	−	0.998378i	$-0.481868\pi$
$617$	2.82843	0.113868	0.0569341	+	0.998378i	$0.481868\pi$
$618$	0	0
$619$	11.0294	0.443311	0.221655	−	0.975125i	$-0.428854\pi$
$619$	11.0294	0.443311	0.221655	+	0.975125i	$0.428854\pi$
$620$	0	0
$621$	−5.65685	−0.227002
$622$	0	0
$623$	31.3137	1.25456
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	29.6569	1.18250
$630$	0	0
$631$	36.4853	1.45246	0.726228	−	0.687454i	$-0.241272\pi$
$631$	36.4853	1.45246	0.726228	+	0.687454i	$0.241272\pi$
$632$	0	0
$633$	25.6569	1.01977
$634$	0	0
$635$	−4.34315	−0.172352
$636$	0	0
$637$	−14.4853	−0.573928
$638$	0	0
$639$	−2.34315	−0.0926934
$640$	0	0
$641$	12.3431	0.487525	0.243762	−	0.969835i	$-0.421618\pi$
$641$	12.3431	0.487525	0.243762	+	0.969835i	$0.421618\pi$
$642$	0	0
$643$	−2.62742	−0.103615	−0.0518076	−	0.998657i	$-0.516498\pi$
$643$	−2.62742	−0.103615	−0.0518076	+	0.998657i	$0.516498\pi$
$644$	0	0
$645$	−9.65685	−0.380238
$646$	0	0
$647$	−17.6569	−0.694163	−0.347081	−	0.937835i	$-0.612827\pi$
$647$	−17.6569	−0.694163	−0.347081	+	0.937835i	$0.612827\pi$
$648$	0	0
$649$	−4.00000	−0.157014
$650$	0	0
$651$	−13.6569	−0.535254
$652$	0	0
$653$	−29.3137	−1.14713	−0.573567	−	0.819159i	$-0.694441\pi$
$653$	−29.3137	−1.14713	−0.573567	+	0.819159i	$0.694441\pi$
$654$	0	0
$655$	2.48528	0.0971080
$656$	0	0
$657$	−13.3137	−0.519417
$658$	0	0
$659$	39.1716	1.52591	0.762954	−	0.646453i	$-0.223748\pi$
$659$	39.1716	1.52591	0.762954	+	0.646453i	$0.223748\pi$
$660$	0	0
$661$	−36.9706	−1.43799	−0.718994	−	0.695016i	$-0.755397\pi$
$661$	−36.9706	−1.43799	−0.718994	+	0.695016i	$0.755397\pi$
$662$	0	0
$663$	13.6569	0.530388
$664$	0	0
$665$	0	0
$666$	0	0
$667$	43.3137	1.67711
$668$	0	0
$669$	−14.9706	−0.578795
$670$	0	0
$671$	−8.00000	−0.308837
$672$	0	0
$673$	33.3137	1.28415	0.642075	−	0.766642i	$-0.278074\pi$
$673$	33.3137	1.28415	0.642075	+	0.766642i	$0.278074\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−5.31371	−0.204222	−0.102111	−	0.994773i	$-0.532560\pi$
$677$	−5.31371	−0.204222	−0.102111	+	0.994773i	$0.532560\pi$
$678$	0	0
$679$	7.31371	0.280674
$680$	0	0
$681$	−6.34315	−0.243070
$682$	0	0
$683$	−19.3137	−0.739019	−0.369509	−	0.929227i	$-0.620474\pi$
$683$	−19.3137	−0.739019	−0.369509	+	0.929227i	$0.620474\pi$
$684$	0	0
$685$	6.82843	0.260901
$686$	0	0
$687$	10.3431	0.394616
$688$	0	0
$689$	1.65685	0.0631211
$690$	0	0
$691$	−38.6274	−1.46946	−0.734728	−	0.678362i	$-0.762690\pi$
$691$	−38.6274	−1.46946	−0.734728	+	0.678362i	$0.762690\pi$
$692$	0	0
$693$	9.65685	0.366834
$694$	0	0
$695$	−1.65685	−0.0628481
$696$	0	0
$697$	−21.6569	−0.820312
$698$	0	0
$699$	27.7990	1.05145
$700$	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
$705$	9.65685	0.363698
$706$	0	0
$707$	34.6274	1.30230
$708$	0	0
$709$	−20.2843	−0.761792	−0.380896	−	0.924618i	$-0.624384\pi$
$709$	−20.2843	−0.761792	−0.380896	+	0.924618i	$0.624384\pi$
$710$	0	0
$711$	4.48528	0.168211
$712$	0	0
$713$	−38.6274	−1.44661
$714$	0	0
$715$	−23.3137	−0.871883
$716$	0	0
$717$	22.6274	0.845036
$718$	0	0
$719$	44.2843	1.65152	0.825762	−	0.564018i	$-0.190745\pi$
$719$	44.2843	1.65152	0.825762	+	0.564018i	$0.190745\pi$
$720$	0	0
$721$	31.3137	1.16618
$722$	0	0
$723$	24.6274	0.915903
$724$	0	0
$725$	7.65685	0.284368
$726$	0	0
$727$	17.0294	0.631587	0.315793	−	0.948828i	$-0.397729\pi$
$727$	17.0294	0.631587	0.315793	+	0.948828i	$0.397729\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−27.3137	−1.01023
$732$	0	0
$733$	17.7990	0.657421	0.328710	−	0.944431i	$-0.393386\pi$
$733$	17.7990	0.657421	0.328710	+	0.944431i	$0.393386\pi$
$734$	0	0
$735$	3.00000	0.110657
$736$	0	0
$737$	8.00000	0.294684
$738$	0	0
$739$	25.9411	0.954260	0.477130	−	0.878833i	$-0.341677\pi$
$739$	25.9411	0.954260	0.477130	+	0.878833i	$0.341677\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	28.9706	1.06283	0.531413	−	0.847113i	$-0.321661\pi$
$743$	28.9706	1.06283	0.531413	+	0.847113i	$0.321661\pi$
$744$	0	0
$745$	0.343146	0.0125719
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−27.3137	−0.998021
$750$	0	0
$751$	−5.45584	−0.199087	−0.0995433	−	0.995033i	$-0.531738\pi$
$751$	−5.45584	−0.199087	−0.0995433	+	0.995033i	$0.531738\pi$
$752$	0	0
$753$	3.17157	0.115579
$754$	0	0
$755$	−8.48528	−0.308811
$756$	0	0
$757$	−17.5147	−0.636583	−0.318292	−	0.947993i	$-0.603109\pi$
$757$	−17.5147	−0.636583	−0.318292	+	0.947993i	$0.603109\pi$
$758$	0	0
$759$	27.3137	0.991425
$760$	0	0
$761$	10.6863	0.387378	0.193689	−	0.981063i	$-0.437955\pi$
$761$	10.6863	0.387378	0.193689	+	0.981063i	$0.437955\pi$
$762$	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
$769$	−9.31371	−0.335861	−0.167930	−	0.985799i	$-0.553708\pi$
$769$	−9.31371	−0.335861	−0.167930	+	0.985799i	$0.553708\pi$
$770$	0	0
$771$	−9.17157	−0.330306
$772$	0	0
$773$	−51.2548	−1.84351	−0.921754	−	0.387775i	$-0.873244\pi$
$773$	−51.2548	−1.84351	−0.921754	+	0.387775i	$0.873244\pi$
$774$	0	0
$775$	−6.82843	−0.245284
$776$	0	0
$777$	−20.9706	−0.752315
$778$	0	0
$779$	0	0
$780$	0	0
$781$	11.3137	0.404836
$782$	0	0
$783$	−7.65685	−0.273634
$784$	0	0
$785$	−5.51472	−0.196829
$786$	0	0
$787$	39.3137	1.40138	0.700691	−	0.713465i	$-0.252875\pi$
$787$	39.3137	1.40138	0.700691	+	0.713465i	$0.252875\pi$
$788$	0	0
$789$	18.3431	0.653033
$790$	0	0
$791$	7.02944	0.249938
$792$	0	0
$793$	8.00000	0.284088
$794$	0	0
$795$	−0.343146	−0.0121701
$796$	0	0
$797$	10.9706	0.388597	0.194299	−	0.980942i	$-0.437757\pi$
$797$	10.9706	0.388597	0.194299	+	0.980942i	$0.437757\pi$
$798$	0	0
$799$	27.3137	0.966290
$800$	0	0
$801$	−15.6569	−0.553208
$802$	0	0
$803$	64.2843	2.26854
$804$	0	0
$805$	−11.3137	−0.398756
$806$	0	0
$807$	3.65685	0.128727
$808$	0	0
$809$	12.3431	0.433962	0.216981	−	0.976176i	$-0.430379\pi$
$809$	12.3431	0.433962	0.216981	+	0.976176i	$0.430379\pi$
$810$	0	0
$811$	43.5980	1.53093	0.765466	−	0.643476i	$-0.222508\pi$
$811$	43.5980	1.53093	0.765466	+	0.643476i	$0.222508\pi$
$812$	0	0
$813$	−8.48528	−0.297592
$814$	0	0
$815$	12.0000	0.420342
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−9.65685	−0.337438
$820$	0	0
$821$	1.02944	0.0359276	0.0179638	−	0.999839i	$-0.494282\pi$
$821$	1.02944	0.0359276	0.0179638	+	0.999839i	$0.494282\pi$
$822$	0	0
$823$	38.2843	1.33451	0.667253	−	0.744831i	$-0.267470\pi$
$823$	38.2843	1.33451	0.667253	+	0.744831i	$0.267470\pi$
$824$	0	0
$825$	4.82843	0.168104
$826$	0	0
$827$	16.6863	0.580239	0.290120	−	0.956990i	$-0.406305\pi$
$827$	16.6863	0.580239	0.290120	+	0.956990i	$0.406305\pi$
$828$	0	0
$829$	29.9411	1.03990	0.519949	−	0.854197i	$-0.325951\pi$
$829$	29.9411	1.03990	0.519949	+	0.854197i	$0.325951\pi$
$830$	0	0
$831$	28.8284	1.00005
$832$	0	0
$833$	8.48528	0.293998
$834$	0	0
$835$	−10.3431	−0.357939
$836$	0	0
$837$	6.82843	0.236025
$838$	0	0
$839$	−44.2843	−1.52886	−0.764431	−	0.644705i	$-0.776980\pi$
$839$	−44.2843	−1.52886	−0.764431	+	0.644705i	$0.776980\pi$
$840$	0	0
$841$	29.6274	1.02164
$842$	0	0
$843$	−22.9706	−0.791148
$844$	0	0
$845$	10.3137	0.354802
$846$	0	0
$847$	−24.6274	−0.846208
$848$	0	0
$849$	−20.9706	−0.719708
$850$	0	0
$851$	−59.3137	−2.03325
$852$	0	0
$853$	−18.4853	−0.632924	−0.316462	−	0.948605i	$-0.602495\pi$
$853$	−18.4853	−0.632924	−0.316462	+	0.948605i	$0.602495\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−6.14214	−0.209811	−0.104906	−	0.994482i	$-0.533454\pi$
$857$	−6.14214	−0.209811	−0.104906	+	0.994482i	$0.533454\pi$
$858$	0	0
$859$	−54.9117	−1.87356	−0.936781	−	0.349915i	$-0.886210\pi$
$859$	−54.9117	−1.87356	−0.936781	+	0.349915i	$0.886210\pi$
$860$	0	0
$861$	15.3137	0.521890
$862$	0	0
$863$	10.3431	0.352085	0.176042	−	0.984383i	$-0.443670\pi$
$863$	10.3431	0.352085	0.176042	+	0.984383i	$0.443670\pi$
$864$	0	0
$865$	15.6569	0.532349
$866$	0	0
$867$	9.00000	0.305656
$868$	0	0
$869$	−21.6569	−0.734658
$870$	0	0
$871$	−8.00000	−0.271070
$872$	0	0
$873$	−3.65685	−0.123766
$874$	0	0
$875$	−2.00000	−0.0676123
$876$	0	0
$877$	2.48528	0.0839220	0.0419610	−	0.999119i	$-0.486639\pi$
$877$	2.48528	0.0839220	0.0419610	+	0.999119i	$0.486639\pi$
$878$	0	0
$879$	−5.31371	−0.179227
$880$	0	0
$881$	35.6569	1.20131	0.600655	−	0.799508i	$-0.294907\pi$
$881$	35.6569	1.20131	0.600655	+	0.799508i	$0.294907\pi$
$882$	0	0
$883$	−37.9411	−1.27682	−0.638410	−	0.769696i	$-0.720408\pi$
$883$	−37.9411	−1.27682	−0.638410	+	0.769696i	$0.720408\pi$
$884$	0	0
$885$	−0.828427	−0.0278473
$886$	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
$889$	8.68629	0.291329
$890$	0	0
$891$	−4.82843	−0.161758
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−2.48528	−0.0830738
$896$	0	0
$897$	−27.3137	−0.911978
$898$	0	0
$899$	−52.2843	−1.74378
$900$	0	0
$901$	−0.970563	−0.0323341
$902$	0	0
$903$	19.3137	0.642720
$904$	0	0
$905$	−12.0000	−0.398893
$906$	0	0
$907$	−12.9706	−0.430680	−0.215340	−	0.976539i	$-0.569086\pi$
$907$	−12.9706	−0.430680	−0.215340	+	0.976539i	$0.569086\pi$
$908$	0	0
$909$	−17.3137	−0.574259
$910$	0	0
$911$	35.3137	1.17000	0.584998	−	0.811035i	$-0.301095\pi$
$911$	35.3137	1.17000	0.584998	+	0.811035i	$0.301095\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−1.65685	−0.0547739
$916$	0	0
$917$	−4.97056	−0.164142
$918$	0	0
$919$	44.4853	1.46743	0.733717	−	0.679455i	$-0.237784\pi$
$919$	44.4853	1.46743	0.733717	+	0.679455i	$0.237784\pi$
$920$	0	0
$921$	−4.00000	−0.131804
$922$	0	0
$923$	−11.3137	−0.372395
$924$	0	0
$925$	−10.4853	−0.344754
$926$	0	0
$927$	−15.6569	−0.514239
$928$	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
$935$	13.6569	0.446627
$936$	0	0
$937$	13.3137	0.434940	0.217470	−	0.976067i	$-0.430220\pi$
$937$	13.3137	0.434940	0.217470	+	0.976067i	$0.430220\pi$
$938$	0	0
$939$	10.9706	0.358011
$940$	0	0
$941$	35.2548	1.14927	0.574637	−	0.818408i	$-0.305143\pi$
$941$	35.2548	1.14927	0.574637	+	0.818408i	$0.305143\pi$
$942$	0	0
$943$	43.3137	1.41049
$944$	0	0
$945$	2.00000	0.0650600
$946$	0	0
$947$	−45.9411	−1.49289	−0.746443	−	0.665449i	$-0.768240\pi$
$947$	−45.9411	−1.49289	−0.746443	+	0.665449i	$0.768240\pi$
$948$	0	0
$949$	−64.2843	−2.08676
$950$	0	0
$951$	9.31371	0.302018
$952$	0	0
$953$	−33.1716	−1.07453	−0.537266	−	0.843413i	$-0.680543\pi$
$953$	−33.1716	−1.07453	−0.537266	+	0.843413i	$0.680543\pi$
$954$	0	0
$955$	−27.3137	−0.883851
$956$	0	0
$957$	36.9706	1.19509
$958$	0	0
$959$	−13.6569	−0.441003
$960$	0	0
$961$	15.6274	0.504110
$962$	0	0
$963$	13.6569	0.440086
$964$	0	0
$965$	22.9706	0.739449
$966$	0	0
$967$	56.9117	1.83016	0.915078	−	0.403276i	$-0.132129\pi$
$967$	56.9117	1.83016	0.915078	+	0.403276i	$0.132129\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−15.8579	−0.508903	−0.254452	−	0.967086i	$-0.581895\pi$
$971$	−15.8579	−0.508903	−0.254452	+	0.967086i	$0.581895\pi$
$972$	0	0
$973$	3.31371	0.106233
$974$	0	0
$975$	−4.82843	−0.154633
$976$	0	0
$977$	−26.8284	−0.858317	−0.429159	−	0.903229i	$-0.641190\pi$
$977$	−26.8284	−0.858317	−0.429159	+	0.903229i	$0.641190\pi$
$978$	0	0
$979$	75.5980	2.41612
$980$	0	0
$981$	−4.00000	−0.127710
$982$	0	0
$983$	19.0294	0.606945	0.303472	−	0.952840i	$-0.401854\pi$
$983$	19.0294	0.606945	0.303472	+	0.952840i	$0.401854\pi$
$984$	0	0
$985$	21.3137	0.679111
$986$	0	0
$987$	−19.3137	−0.614762
$988$	0	0
$989$	54.6274	1.73705
$990$	0	0
$991$	15.5147	0.492841	0.246421	−	0.969163i	$-0.420746\pi$
$991$	15.5147	0.492841	0.246421	+	0.969163i	$0.420746\pi$
$992$	0	0
$993$	32.0000	1.01549
$994$	0	0
$995$	−18.8284	−0.596901
$996$	0	0
$997$	−12.1421	−0.384545	−0.192273	−	0.981342i	$-0.561586\pi$
$997$	−12.1421	−0.384545	−0.192273	+	0.981342i	$0.561586\pi$
$998$	0	0
$999$	10.4853	0.331740

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3840.2.a.bg.1.1		2
4.3	odd	2		3840.2.a.bm.1.2		2
8.3	odd	2		3840.2.a.bd.1.1		2
8.5	even	2		3840.2.a.bj.1.2		2
16.3	odd	4		1920.2.k.j.961.3	yes	4
16.5	even	4		1920.2.k.k.961.3	yes	4
16.11	odd	4		1920.2.k.j.961.2	✓	4
16.13	even	4		1920.2.k.k.961.2	yes	4
48.5	odd	4		5760.2.k.x.2881.2		4
48.11	even	4		5760.2.k.m.2881.1		4
48.29	odd	4		5760.2.k.x.2881.3		4
48.35	even	4		5760.2.k.m.2881.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1920.2.k.j.961.2	✓	4	16.11	odd	4
1920.2.k.j.961.3	yes	4	16.3	odd	4
1920.2.k.k.961.2	yes	4	16.13	even	4
1920.2.k.k.961.3	yes	4	16.5	even	4
3840.2.a.bd.1.1		2	8.3	odd	2
3840.2.a.bg.1.1		2	1.1	even	1	trivial
3840.2.a.bj.1.2		2	8.5	even	2
3840.2.a.bm.1.2		2	4.3	odd	2
5760.2.k.m.2881.1		4	48.11	even	4
5760.2.k.m.2881.4		4	48.35	even	4
5760.2.k.x.2881.2		4	48.5	odd	4
5760.2.k.x.2881.3		4	48.29	odd	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} -2.00000 q^{7} +1.00000 q^{9} -4.82843 q^{11} +4.82843 q^{13} -1.00000 q^{15} -2.82843 q^{17} +2.00000 q^{21} +5.65685 q^{23} +1.00000 q^{25} -1.00000 q^{27} +7.65685 q^{29} -6.82843 q^{31} +4.82843 q^{33} -2.00000 q^{35} -10.4853 q^{37} -4.82843 q^{39} +7.65685 q^{41} +9.65685 q^{43} +1.00000 q^{45} -9.65685 q^{47} -3.00000 q^{49} +2.82843 q^{51} +0.343146 q^{53} -4.82843 q^{55} +0.828427 q^{59} +1.65685 q^{61} -2.00000 q^{63} +4.82843 q^{65} -1.65685 q^{67} -5.65685 q^{69} -2.34315 q^{71} -13.3137 q^{73} -1.00000 q^{75} +9.65685 q^{77} +4.48528 q^{79} +1.00000 q^{81} -2.82843 q^{85} -7.65685 q^{87} -15.6569 q^{89} -9.65685 q^{91} +6.82843 q^{93} -3.65685 q^{97} -4.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{5} - 4 q^{7} + 2 q^{9} - 4 q^{11} + 4 q^{13} - 2 q^{15} + 4 q^{21} + 2 q^{25} - 2 q^{27} + 4 q^{29} - 8 q^{31} + 4 q^{33} - 4 q^{35} - 4 q^{37} - 4 q^{39} + 4 q^{41} + 8 q^{43} + 2 q^{45}+ \cdots - 4 q^{99}+O(q^{100}) \) 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

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