LMFDB - Embedded newform 6240.2.a.bh.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6240,2,Mod(1,6240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6240, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6240.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6240.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:	$49.8266508613$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	6240.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} -2.82843 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.00000 q^{5} -2.82843 q^{7} +1.00000 q^{9} -2.82843 q^{11} +1.00000 q^{13} +1.00000 q^{15} +7.65685 q^{17} -6.82843 q^{19} +2.82843 q^{21} +4.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +3.65685 q^{29} -1.17157 q^{31} +2.82843 q^{33} +2.82843 q^{35} -2.00000 q^{37} -1.00000 q^{39} -2.00000 q^{41} +1.65685 q^{43} -1.00000 q^{45} -1.17157 q^{47} +1.00000 q^{49} -7.65685 q^{51} -2.00000 q^{53} +2.82843 q^{55} +6.82843 q^{57} +8.48528 q^{59} +6.00000 q^{61} -2.82843 q^{63} -1.00000 q^{65} -2.82843 q^{67} -4.00000 q^{69} +10.8284 q^{71} +4.34315 q^{73} -1.00000 q^{75} +8.00000 q^{77} +13.6569 q^{79} +1.00000 q^{81} -6.82843 q^{83} -7.65685 q^{85} -3.65685 q^{87} -5.31371 q^{89} -2.82843 q^{91} +1.17157 q^{93} +6.82843 q^{95} +7.65685 q^{97} -2.82843 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^9	$ 2 q - 2 q^{3} - 2 q^{5} + 2 q^{9} + 2 q^{13} + 2 q^{15} + 4 q^{17} - 8 q^{19} + 8 q^{23} + 2 q^{25} - 2 q^{27} - 4 q^{29} - 8 q^{31} - 4 q^{37} - 2 q^{39} - 4 q^{41} - 8 q^{43} - 2 q^{45} - 8 q^{47} + 2 q^{49} - 4 q^{51} - 4 q^{53} + 8 q^{57} + 12 q^{61} - 2 q^{65} - 8 q^{69} + 16 q^{71} + 20 q^{73} - 2 q^{75} + 16 q^{77} + 16 q^{79} + 2 q^{81} - 8 q^{83} - 4 q^{85} + 4 q^{87} + 12 q^{89} + 8 q^{93} + 8 q^{95} + 4 q^{97}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^9 + 2 * q^13 + 2 * q^15 + 4 * q^17 - 8 * q^19 + 8 * q^23 + 2 * q^25 - 2 * q^27 - 4 * q^29 - 8 * q^31 - 4 * q^37 - 2 * q^39 - 4 * q^41 - 8 * q^43 - 2 * q^45 - 8 * q^47 + 2 * q^49 - 4 * q^51 - 4 * q^53 + 8 * q^57 + 12 * q^61 - 2 * q^65 - 8 * q^69 + 16 * q^71 + 20 * q^73 - 2 * q^75 + 16 * q^77 + 16 * q^79 + 2 * q^81 - 8 * q^83 - 4 * q^85 + 4 * q^87 + 12 * q^89 + 8 * q^93 + 8 * q^95 + 4 * q^97

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
$7$	−2.82843	−1.06904	−0.534522	−	0.845154i	$-0.679509\pi$
$7$	−2.82843	−1.06904	−0.534522	+	0.845154i	$0.679509\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.82843	−0.852803	−0.426401	−	0.904534i	$-0.640219\pi$
$11$	−2.82843	−0.852803	−0.426401	+	0.904534i	$0.640219\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	7.65685	1.85706	0.928530	−	0.371257i	$-0.121073\pi$
$17$	7.65685	1.85706	0.928530	+	0.371257i	$0.121073\pi$
$18$	0	0
$19$	−6.82843	−1.56655	−0.783274	−	0.621676i	$-0.786452\pi$
$19$	−6.82843	−1.56655	−0.783274	+	0.621676i	$0.786452\pi$
$20$	0	0
$21$	2.82843	0.617213
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	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	$-0.389732\pi$
$29$	3.65685	0.679061	0.339530	+	0.940595i	$0.389732\pi$
$30$	0	0
$31$	−1.17157	−0.210421	−0.105210	−	0.994450i	$-0.533552\pi$
$31$	−1.17157	−0.210421	−0.105210	+	0.994450i	$0.533552\pi$
$32$	0	0
$33$	2.82843	0.492366
$34$	0	0
$35$	2.82843	0.478091
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	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
$47$	−1.17157	−0.170891	−0.0854457	−	0.996343i	$-0.527231\pi$
$47$	−1.17157	−0.170891	−0.0854457	+	0.996343i	$0.527231\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−7.65685	−1.07217
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	2.82843	0.381385
$56$	0	0
$57$	6.82843	0.904447
$58$	0	0
$59$	8.48528	1.10469	0.552345	−	0.833616i	$-0.313733\pi$
$59$	8.48528	1.10469	0.552345	+	0.833616i	$0.313733\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	−2.82843	−0.356348
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−2.82843	−0.345547	−0.172774	−	0.984962i	$-0.555273\pi$
$67$	−2.82843	−0.345547	−0.172774	+	0.984962i	$0.555273\pi$
$68$	0	0
$69$	−4.00000	−0.481543
$70$	0	0
$71$	10.8284	1.28510	0.642549	−	0.766245i	$-0.277877\pi$
$71$	10.8284	1.28510	0.642549	+	0.766245i	$0.277877\pi$
$72$	0	0
$73$	4.34315	0.508327	0.254163	−	0.967161i	$-0.418200\pi$
$73$	4.34315	0.508327	0.254163	+	0.967161i	$0.418200\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	8.00000	0.911685
$78$	0	0
$79$	13.6569	1.53652	0.768258	−	0.640140i	$-0.221124\pi$
$79$	13.6569	1.53652	0.768258	+	0.640140i	$0.221124\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−6.82843	−0.749517	−0.374759	−	0.927122i	$-0.622274\pi$
$83$	−6.82843	−0.749517	−0.374759	+	0.927122i	$0.622274\pi$
$84$	0	0
$85$	−7.65685	−0.830502
$86$	0	0
$87$	−3.65685	−0.392056
$88$	0	0
$89$	−5.31371	−0.563252	−0.281626	−	0.959524i	$-0.590874\pi$
$89$	−5.31371	−0.563252	−0.281626	+	0.959524i	$0.590874\pi$
$90$	0	0
$91$	−2.82843	−0.296500
$92$	0	0
$93$	1.17157	0.121486
$94$	0	0
$95$	6.82843	0.700582
$96$	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
$99$	−2.82843	−0.284268
$100$	0	0
$101$	3.65685	0.363871	0.181935	−	0.983311i	$-0.441764\pi$
$101$	3.65685	0.363871	0.181935	+	0.983311i	$0.441764\pi$
$102$	0	0
$103$	1.65685	0.163255	0.0816274	−	0.996663i	$-0.473988\pi$
$103$	1.65685	0.163255	0.0816274	+	0.996663i	$0.473988\pi$
$104$	0	0
$105$	−2.82843	−0.276026
$106$	0	0
$107$	7.31371	0.707043	0.353521	−	0.935426i	$-0.384984\pi$
$107$	7.31371	0.707043	0.353521	+	0.935426i	$0.384984\pi$
$108$	0	0
$109$	−15.6569	−1.49965	−0.749827	−	0.661634i	$-0.769863\pi$
$109$	−15.6569	−1.49965	−0.749827	+	0.661634i	$0.769863\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	0	0
$113$	−11.6569	−1.09658	−0.548292	−	0.836287i	$-0.684722\pi$
$113$	−11.6569	−1.09658	−0.548292	+	0.836287i	$0.684722\pi$
$114$	0	0
$115$	−4.00000	−0.373002
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	−21.6569	−1.98528
$120$	0	0
$121$	−3.00000	−0.272727
$122$	0	0
$123$	2.00000	0.180334
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−9.65685	−0.856907	−0.428454	−	0.903564i	$-0.640941\pi$
$127$	−9.65685	−0.856907	−0.428454	+	0.903564i	$0.640941\pi$
$128$	0	0
$129$	−1.65685	−0.145878
$130$	0	0
$131$	−16.0000	−1.39793	−0.698963	−	0.715158i	$-0.746355\pi$
$131$	−16.0000	−1.39793	−0.698963	+	0.715158i	$0.746355\pi$
$132$	0	0
$133$	19.3137	1.67471
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	0	0
$139$	−16.9706	−1.43942	−0.719712	−	0.694273i	$-0.755726\pi$
$139$	−16.9706	−1.43942	−0.719712	+	0.694273i	$0.755726\pi$
$140$	0	0
$141$	1.17157	0.0986642
$142$	0	0
$143$	−2.82843	−0.236525
$144$	0	0
$145$	−3.65685	−0.303685
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	13.3137	1.09070	0.545351	−	0.838208i	$-0.316396\pi$
$149$	13.3137	1.09070	0.545351	+	0.838208i	$0.316396\pi$
$150$	0	0
$151$	1.17157	0.0953412	0.0476706	−	0.998863i	$-0.484820\pi$
$151$	1.17157	0.0953412	0.0476706	+	0.998863i	$0.484820\pi$
$152$	0	0
$153$	7.65685	0.619020
$154$	0	0
$155$	1.17157	0.0941030
$156$	0	0
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	0	0
$159$	2.00000	0.158610
$160$	0	0
$161$	−11.3137	−0.891645
$162$	0	0
$163$	6.14214	0.481089	0.240545	−	0.970638i	$-0.422674\pi$
$163$	6.14214	0.481089	0.240545	+	0.970638i	$0.422674\pi$
$164$	0	0
$165$	−2.82843	−0.220193
$166$	0	0
$167$	−12.4853	−0.966140	−0.483070	−	0.875582i	$-0.660478\pi$
$167$	−12.4853	−0.966140	−0.483070	+	0.875582i	$0.660478\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−6.82843	−0.522183
$172$	0	0
$173$	−24.6274	−1.87239	−0.936194	−	0.351484i	$-0.885677\pi$
$173$	−24.6274	−1.87239	−0.936194	+	0.351484i	$0.885677\pi$
$174$	0	0
$175$	−2.82843	−0.213809
$176$	0	0
$177$	−8.48528	−0.637793
$178$	0	0
$179$	−2.34315	−0.175135	−0.0875675	−	0.996159i	$-0.527909\pi$
$179$	−2.34315	−0.175135	−0.0875675	+	0.996159i	$0.527909\pi$
$180$	0	0
$181$	17.3137	1.28692	0.643459	−	0.765481i	$-0.277499\pi$
$181$	17.3137	1.28692	0.643459	+	0.765481i	$0.277499\pi$
$182$	0	0
$183$	−6.00000	−0.443533
$184$	0	0
$185$	2.00000	0.147043
$186$	0	0
$187$	−21.6569	−1.58371
$188$	0	0
$189$	2.82843	0.205738
$190$	0	0
$191$	5.65685	0.409316	0.204658	−	0.978834i	$-0.434392\pi$
$191$	5.65685	0.409316	0.204658	+	0.978834i	$0.434392\pi$
$192$	0	0
$193$	4.34315	0.312626	0.156313	−	0.987708i	$-0.450039\pi$
$193$	4.34315	0.312626	0.156313	+	0.987708i	$0.450039\pi$
$194$	0	0
$195$	1.00000	0.0716115
$196$	0	0
$197$	−22.0000	−1.56744	−0.783718	−	0.621117i	$-0.786679\pi$
$197$	−22.0000	−1.56744	−0.783718	+	0.621117i	$0.786679\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	2.82843	0.199502
$202$	0	0
$203$	−10.3431	−0.725947
$204$	0	0
$205$	2.00000	0.139686
$206$	0	0
$207$	4.00000	0.278019
$208$	0	0
$209$	19.3137	1.33596
$210$	0	0
$211$	−5.65685	−0.389434	−0.194717	−	0.980859i	$-0.562379\pi$
$211$	−5.65685	−0.389434	−0.194717	+	0.980859i	$0.562379\pi$
$212$	0	0
$213$	−10.8284	−0.741952
$214$	0	0
$215$	−1.65685	−0.112997
$216$	0	0
$217$	3.31371	0.224949
$218$	0	0
$219$	−4.34315	−0.293483
$220$	0	0
$221$	7.65685	0.515056
$222$	0	0
$223$	10.8284	0.725125	0.362563	−	0.931959i	$-0.381902\pi$
$223$	10.8284	0.725125	0.362563	+	0.931959i	$0.381902\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−22.8284	−1.51518	−0.757588	−	0.652733i	$-0.773622\pi$
$227$	−22.8284	−1.51518	−0.757588	+	0.652733i	$0.773622\pi$
$228$	0	0
$229$	19.6569	1.29896	0.649481	−	0.760378i	$-0.274986\pi$
$229$	19.6569	1.29896	0.649481	+	0.760378i	$0.274986\pi$
$230$	0	0
$231$	−8.00000	−0.526361
$232$	0	0
$233$	−22.9706	−1.50485	−0.752426	−	0.658677i	$-0.771116\pi$
$233$	−22.9706	−1.50485	−0.752426	+	0.658677i	$0.771116\pi$
$234$	0	0
$235$	1.17157	0.0764250
$236$	0	0
$237$	−13.6569	−0.887108
$238$	0	0
$239$	−24.4853	−1.58382	−0.791911	−	0.610637i	$-0.790913\pi$
$239$	−24.4853	−1.58382	−0.791911	+	0.610637i	$0.790913\pi$
$240$	0	0
$241$	−1.31371	−0.0846234	−0.0423117	−	0.999104i	$-0.513472\pi$
$241$	−1.31371	−0.0846234	−0.0423117	+	0.999104i	$0.513472\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−6.82843	−0.434482
$248$	0	0
$249$	6.82843	0.432734
$250$	0	0
$251$	24.9706	1.57613	0.788064	−	0.615593i	$-0.211084\pi$
$251$	24.9706	1.57613	0.788064	+	0.615593i	$0.211084\pi$
$252$	0	0
$253$	−11.3137	−0.711287
$254$	0	0
$255$	7.65685	0.479491
$256$	0	0
$257$	18.9706	1.18335	0.591676	−	0.806176i	$-0.298467\pi$
$257$	18.9706	1.18335	0.591676	+	0.806176i	$0.298467\pi$
$258$	0	0
$259$	5.65685	0.351500
$260$	0	0
$261$	3.65685	0.226354
$262$	0	0
$263$	−9.65685	−0.595467	−0.297734	−	0.954649i	$-0.596231\pi$
$263$	−9.65685	−0.595467	−0.297734	+	0.954649i	$0.596231\pi$
$264$	0	0
$265$	2.00000	0.122859
$266$	0	0
$267$	5.31371	0.325194
$268$	0	0
$269$	−7.65685	−0.466847	−0.233423	−	0.972375i	$-0.574993\pi$
$269$	−7.65685	−0.466847	−0.233423	+	0.972375i	$0.574993\pi$
$270$	0	0
$271$	4.48528	0.272461	0.136231	−	0.990677i	$-0.456501\pi$
$271$	4.48528	0.272461	0.136231	+	0.990677i	$0.456501\pi$
$272$	0	0
$273$	2.82843	0.171184
$274$	0	0
$275$	−2.82843	−0.170561
$276$	0	0
$277$	−29.3137	−1.76129	−0.880645	−	0.473777i	$-0.842890\pi$
$277$	−29.3137	−1.76129	−0.880645	+	0.473777i	$0.842890\pi$
$278$	0	0
$279$	−1.17157	−0.0701402
$280$	0	0
$281$	−24.6274	−1.46915	−0.734574	−	0.678528i	$-0.762618\pi$
$281$	−24.6274	−1.46915	−0.734574	+	0.678528i	$0.762618\pi$
$282$	0	0
$283$	−17.6569	−1.04959	−0.524796	−	0.851228i	$-0.675858\pi$
$283$	−17.6569	−1.04959	−0.524796	+	0.851228i	$0.675858\pi$
$284$	0	0
$285$	−6.82843	−0.404481
$286$	0	0
$287$	5.65685	0.333914
$288$	0	0
$289$	41.6274	2.44867
$290$	0	0
$291$	−7.65685	−0.448853
$292$	0	0
$293$	−9.31371	−0.544113	−0.272056	−	0.962281i	$-0.587704\pi$
$293$	−9.31371	−0.544113	−0.272056	+	0.962281i	$0.587704\pi$
$294$	0	0
$295$	−8.48528	−0.494032
$296$	0	0
$297$	2.82843	0.164122
$298$	0	0
$299$	4.00000	0.231326
$300$	0	0
$301$	−4.68629	−0.270113
$302$	0	0
$303$	−3.65685	−0.210081
$304$	0	0
$305$	−6.00000	−0.343559
$306$	0	0
$307$	−21.1716	−1.20833	−0.604163	−	0.796861i	$-0.706492\pi$
$307$	−21.1716	−1.20833	−0.604163	+	0.796861i	$0.706492\pi$
$308$	0	0
$309$	−1.65685	−0.0942551
$310$	0	0
$311$	4.68629	0.265735	0.132868	−	0.991134i	$-0.457581\pi$
$311$	4.68629	0.265735	0.132868	+	0.991134i	$0.457581\pi$
$312$	0	0
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	0	0
$315$	2.82843	0.159364
$316$	0	0
$317$	−2.68629	−0.150877	−0.0754386	−	0.997150i	$-0.524036\pi$
$317$	−2.68629	−0.150877	−0.0754386	+	0.997150i	$0.524036\pi$
$318$	0	0
$319$	−10.3431	−0.579105
$320$	0	0
$321$	−7.31371	−0.408211
$322$	0	0
$323$	−52.2843	−2.90917
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	15.6569	0.865826
$328$	0	0
$329$	3.31371	0.182691
$330$	0	0
$331$	−18.1421	−0.997182	−0.498591	−	0.866837i	$-0.666149\pi$
$331$	−18.1421	−0.997182	−0.498591	+	0.866837i	$0.666149\pi$
$332$	0	0
$333$	−2.00000	−0.109599
$334$	0	0
$335$	2.82843	0.154533
$336$	0	0
$337$	−6.00000	−0.326841	−0.163420	−	0.986557i	$-0.552253\pi$
$337$	−6.00000	−0.326841	−0.163420	+	0.986557i	$0.552253\pi$
$338$	0	0
$339$	11.6569	0.633113
$340$	0	0
$341$	3.31371	0.179447
$342$	0	0
$343$	16.9706	0.916324
$344$	0	0
$345$	4.00000	0.215353
$346$	0	0
$347$	10.6274	0.570510	0.285255	−	0.958452i	$-0.407922\pi$
$347$	10.6274	0.570510	0.285255	+	0.958452i	$0.407922\pi$
$348$	0	0
$349$	−2.97056	−0.159011	−0.0795053	−	0.996834i	$-0.525334\pi$
$349$	−2.97056	−0.159011	−0.0795053	+	0.996834i	$0.525334\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	17.3137	0.921516	0.460758	−	0.887526i	$-0.347578\pi$
$353$	17.3137	0.921516	0.460758	+	0.887526i	$0.347578\pi$
$354$	0	0
$355$	−10.8284	−0.574713
$356$	0	0
$357$	21.6569	1.14620
$358$	0	0
$359$	16.4853	0.870060	0.435030	−	0.900416i	$-0.356738\pi$
$359$	16.4853	0.870060	0.435030	+	0.900416i	$0.356738\pi$
$360$	0	0
$361$	27.6274	1.45407
$362$	0	0
$363$	3.00000	0.157459
$364$	0	0
$365$	−4.34315	−0.227331
$366$	0	0
$367$	−8.68629	−0.453421	−0.226710	−	0.973962i	$-0.572797\pi$
$367$	−8.68629	−0.453421	−0.226710	+	0.973962i	$0.572797\pi$
$368$	0	0
$369$	−2.00000	−0.104116
$370$	0	0
$371$	5.65685	0.293689
$372$	0	0
$373$	28.6274	1.48227	0.741136	−	0.671355i	$-0.234287\pi$
$373$	28.6274	1.48227	0.741136	+	0.671355i	$0.234287\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	3.65685	0.188338
$378$	0	0
$379$	−18.1421	−0.931899	−0.465949	−	0.884811i	$-0.654287\pi$
$379$	−18.1421	−0.931899	−0.465949	+	0.884811i	$0.654287\pi$
$380$	0	0
$381$	9.65685	0.494736
$382$	0	0
$383$	−14.8284	−0.757697	−0.378849	−	0.925459i	$-0.623680\pi$
$383$	−14.8284	−0.757697	−0.378849	+	0.925459i	$0.623680\pi$
$384$	0	0
$385$	−8.00000	−0.407718
$386$	0	0
$387$	1.65685	0.0842226
$388$	0	0
$389$	−23.6569	−1.19945	−0.599725	−	0.800206i	$-0.704723\pi$
$389$	−23.6569	−1.19945	−0.599725	+	0.800206i	$0.704723\pi$
$390$	0	0
$391$	30.6274	1.54890
$392$	0	0
$393$	16.0000	0.807093
$394$	0	0
$395$	−13.6569	−0.687151
$396$	0	0
$397$	−16.6274	−0.834506	−0.417253	−	0.908790i	$-0.637007\pi$
$397$	−16.6274	−0.834506	−0.417253	+	0.908790i	$0.637007\pi$
$398$	0	0
$399$	−19.3137	−0.966895
$400$	0	0
$401$	−6.68629	−0.333897	−0.166949	−	0.985966i	$-0.553391\pi$
$401$	−6.68629	−0.333897	−0.166949	+	0.985966i	$0.553391\pi$
$402$	0	0
$403$	−1.17157	−0.0583602
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	5.65685	0.280400
$408$	0	0
$409$	−20.6274	−1.01996	−0.509980	−	0.860186i	$-0.670347\pi$
$409$	−20.6274	−1.01996	−0.509980	+	0.860186i	$0.670347\pi$
$410$	0	0
$411$	18.0000	0.887875
$412$	0	0
$413$	−24.0000	−1.18096
$414$	0	0
$415$	6.82843	0.335194
$416$	0	0
$417$	16.9706	0.831052
$418$	0	0
$419$	14.6274	0.714596	0.357298	−	0.933990i	$-0.383698\pi$
$419$	14.6274	0.714596	0.357298	+	0.933990i	$0.383698\pi$
$420$	0	0
$421$	−18.9706	−0.924569	−0.462284	−	0.886732i	$-0.652970\pi$
$421$	−18.9706	−0.924569	−0.462284	+	0.886732i	$0.652970\pi$
$422$	0	0
$423$	−1.17157	−0.0569638
$424$	0	0
$425$	7.65685	0.371412
$426$	0	0
$427$	−16.9706	−0.821263
$428$	0	0
$429$	2.82843	0.136558
$430$	0	0
$431$	30.1421	1.45190	0.725948	−	0.687750i	$-0.241401\pi$
$431$	30.1421	1.45190	0.725948	+	0.687750i	$0.241401\pi$
$432$	0	0
$433$	−9.31371	−0.447588	−0.223794	−	0.974636i	$-0.571844\pi$
$433$	−9.31371	−0.447588	−0.223794	+	0.974636i	$0.571844\pi$
$434$	0	0
$435$	3.65685	0.175333
$436$	0	0
$437$	−27.3137	−1.30659
$438$	0	0
$439$	−14.6274	−0.698129	−0.349064	−	0.937099i	$-0.613501\pi$
$439$	−14.6274	−0.698129	−0.349064	+	0.937099i	$0.613501\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−18.6274	−0.885015	−0.442508	−	0.896765i	$-0.645911\pi$
$443$	−18.6274	−0.885015	−0.442508	+	0.896765i	$0.645911\pi$
$444$	0	0
$445$	5.31371	0.251894
$446$	0	0
$447$	−13.3137	−0.629717
$448$	0	0
$449$	36.6274	1.72855	0.864277	−	0.503016i	$-0.167776\pi$
$449$	36.6274	1.72855	0.864277	+	0.503016i	$0.167776\pi$
$450$	0	0
$451$	5.65685	0.266371
$452$	0	0
$453$	−1.17157	−0.0550453
$454$	0	0
$455$	2.82843	0.132599
$456$	0	0
$457$	22.2843	1.04241	0.521207	−	0.853430i	$-0.325482\pi$
$457$	22.2843	1.04241	0.521207	+	0.853430i	$0.325482\pi$
$458$	0	0
$459$	−7.65685	−0.357391
$460$	0	0
$461$	−9.31371	−0.433783	−0.216891	−	0.976196i	$-0.569592\pi$
$461$	−9.31371	−0.433783	−0.216891	+	0.976196i	$0.569592\pi$
$462$	0	0
$463$	−22.1421	−1.02903	−0.514516	−	0.857481i	$-0.672028\pi$
$463$	−22.1421	−1.02903	−0.514516	+	0.857481i	$0.672028\pi$
$464$	0	0
$465$	−1.17157	−0.0543304
$466$	0	0
$467$	−28.9706	−1.34060	−0.670299	−	0.742091i	$-0.733834\pi$
$467$	−28.9706	−1.34060	−0.670299	+	0.742091i	$0.733834\pi$
$468$	0	0
$469$	8.00000	0.369406
$470$	0	0
$471$	−14.0000	−0.645086
$472$	0	0
$473$	−4.68629	−0.215476
$474$	0	0
$475$	−6.82843	−0.313310
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	0	0
$479$	24.4853	1.11876	0.559381	−	0.828911i	$-0.311039\pi$
$479$	24.4853	1.11876	0.559381	+	0.828911i	$0.311039\pi$
$480$	0	0
$481$	−2.00000	−0.0911922
$482$	0	0
$483$	11.3137	0.514792
$484$	0	0
$485$	−7.65685	−0.347680
$486$	0	0
$487$	25.4558	1.15351	0.576757	−	0.816916i	$-0.304318\pi$
$487$	25.4558	1.15351	0.576757	+	0.816916i	$0.304318\pi$
$488$	0	0
$489$	−6.14214	−0.277757
$490$	0	0
$491$	13.6569	0.616325	0.308163	−	0.951334i	$-0.400286\pi$
$491$	13.6569	0.616325	0.308163	+	0.951334i	$0.400286\pi$
$492$	0	0
$493$	28.0000	1.26106
$494$	0	0
$495$	2.82843	0.127128
$496$	0	0
$497$	−30.6274	−1.37383
$498$	0	0
$499$	−30.8284	−1.38007	−0.690035	−	0.723776i	$-0.742405\pi$
$499$	−30.8284	−1.38007	−0.690035	+	0.723776i	$0.742405\pi$
$500$	0	0
$501$	12.4853	0.557801
$502$	0	0
$503$	1.65685	0.0738755	0.0369377	−	0.999318i	$-0.488240\pi$
$503$	1.65685	0.0738755	0.0369377	+	0.999318i	$0.488240\pi$
$504$	0	0
$505$	−3.65685	−0.162728
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	29.3137	1.29931	0.649654	−	0.760230i	$-0.274914\pi$
$509$	29.3137	1.29931	0.649654	+	0.760230i	$0.274914\pi$
$510$	0	0
$511$	−12.2843	−0.543424
$512$	0	0
$513$	6.82843	0.301482
$514$	0	0
$515$	−1.65685	−0.0730097
$516$	0	0
$517$	3.31371	0.145737
$518$	0	0
$519$	24.6274	1.08102
$520$	0	0
$521$	0.627417	0.0274876	0.0137438	−	0.999906i	$-0.495625\pi$
$521$	0.627417	0.0274876	0.0137438	+	0.999906i	$0.495625\pi$
$522$	0	0
$523$	−7.31371	−0.319806	−0.159903	−	0.987133i	$-0.551118\pi$
$523$	−7.31371	−0.319806	−0.159903	+	0.987133i	$0.551118\pi$
$524$	0	0
$525$	2.82843	0.123443
$526$	0	0
$527$	−8.97056	−0.390764
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	8.48528	0.368230
$532$	0	0
$533$	−2.00000	−0.0866296
$534$	0	0
$535$	−7.31371	−0.316199
$536$	0	0
$537$	2.34315	0.101114
$538$	0	0
$539$	−2.82843	−0.121829
$540$	0	0
$541$	22.9706	0.987582	0.493791	−	0.869581i	$-0.335611\pi$
$541$	22.9706	0.987582	0.493791	+	0.869581i	$0.335611\pi$
$542$	0	0
$543$	−17.3137	−0.743002
$544$	0	0
$545$	15.6569	0.670666
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	0	0
$549$	6.00000	0.256074
$550$	0	0
$551$	−24.9706	−1.06378
$552$	0	0
$553$	−38.6274	−1.64260
$554$	0	0
$555$	−2.00000	−0.0848953
$556$	0	0
$557$	−39.9411	−1.69236	−0.846180	−	0.532897i	$-0.821103\pi$
$557$	−39.9411	−1.69236	−0.846180	+	0.532897i	$0.821103\pi$
$558$	0	0
$559$	1.65685	0.0700775
$560$	0	0
$561$	21.6569	0.914353
$562$	0	0
$563$	3.02944	0.127676	0.0638378	−	0.997960i	$-0.479666\pi$
$563$	3.02944	0.127676	0.0638378	+	0.997960i	$0.479666\pi$
$564$	0	0
$565$	11.6569	0.490408
$566$	0	0
$567$	−2.82843	−0.118783
$568$	0	0
$569$	−25.3137	−1.06121	−0.530603	−	0.847621i	$-0.678034\pi$
$569$	−25.3137	−1.06121	−0.530603	+	0.847621i	$0.678034\pi$
$570$	0	0
$571$	30.6274	1.28172	0.640859	−	0.767659i	$-0.278578\pi$
$571$	30.6274	1.28172	0.640859	+	0.767659i	$0.278578\pi$
$572$	0	0
$573$	−5.65685	−0.236318
$574$	0	0
$575$	4.00000	0.166812
$576$	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
$579$	−4.34315	−0.180495
$580$	0	0
$581$	19.3137	0.801268
$582$	0	0
$583$	5.65685	0.234283
$584$	0	0
$585$	−1.00000	−0.0413449
$586$	0	0
$587$	−26.1421	−1.07900	−0.539501	−	0.841985i	$-0.681387\pi$
$587$	−26.1421	−1.07900	−0.539501	+	0.841985i	$0.681387\pi$
$588$	0	0
$589$	8.00000	0.329634
$590$	0	0
$591$	22.0000	0.904959
$592$	0	0
$593$	−6.68629	−0.274573	−0.137287	−	0.990531i	$-0.543838\pi$
$593$	−6.68629	−0.274573	−0.137287	+	0.990531i	$0.543838\pi$
$594$	0	0
$595$	21.6569	0.887844
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−37.6569	−1.53862	−0.769309	−	0.638877i	$-0.779399\pi$
$599$	−37.6569	−1.53862	−0.769309	+	0.638877i	$0.779399\pi$
$600$	0	0
$601$	−14.0000	−0.571072	−0.285536	−	0.958368i	$-0.592172\pi$
$601$	−14.0000	−0.571072	−0.285536	+	0.958368i	$0.592172\pi$
$602$	0	0
$603$	−2.82843	−0.115182
$604$	0	0
$605$	3.00000	0.121967
$606$	0	0
$607$	34.6274	1.40548	0.702742	−	0.711445i	$-0.251959\pi$
$607$	34.6274	1.40548	0.702742	+	0.711445i	$0.251959\pi$
$608$	0	0
$609$	10.3431	0.419125
$610$	0	0
$611$	−1.17157	−0.0473968
$612$	0	0
$613$	18.6863	0.754732	0.377366	−	0.926064i	$-0.376830\pi$
$613$	18.6863	0.754732	0.377366	+	0.926064i	$0.376830\pi$
$614$	0	0
$615$	−2.00000	−0.0806478
$616$	0	0
$617$	30.0000	1.20775	0.603877	−	0.797077i	$-0.293622\pi$
$617$	30.0000	1.20775	0.603877	+	0.797077i	$0.293622\pi$
$618$	0	0
$619$	−23.7990	−0.956562	−0.478281	−	0.878207i	$-0.658740\pi$
$619$	−23.7990	−0.956562	−0.478281	+	0.878207i	$0.658740\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	0	0
$623$	15.0294	0.602142
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−19.3137	−0.771315
$628$	0	0
$629$	−15.3137	−0.610598
$630$	0	0
$631$	5.85786	0.233198	0.116599	−	0.993179i	$-0.462801\pi$
$631$	5.85786	0.233198	0.116599	+	0.993179i	$0.462801\pi$
$632$	0	0
$633$	5.65685	0.224840
$634$	0	0
$635$	9.65685	0.383221
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	10.8284	0.428366
$640$	0	0
$641$	−10.6863	−0.422083	−0.211042	−	0.977477i	$-0.567686\pi$
$641$	−10.6863	−0.422083	−0.211042	+	0.977477i	$0.567686\pi$
$642$	0	0
$643$	0.485281	0.0191376	0.00956881	−	0.999954i	$-0.496954\pi$
$643$	0.485281	0.0191376	0.00956881	+	0.999954i	$0.496954\pi$
$644$	0	0
$645$	1.65685	0.0652386
$646$	0	0
$647$	4.00000	0.157256	0.0786281	−	0.996904i	$-0.474946\pi$
$647$	4.00000	0.157256	0.0786281	+	0.996904i	$0.474946\pi$
$648$	0	0
$649$	−24.0000	−0.942082
$650$	0	0
$651$	−3.31371	−0.129874
$652$	0	0
$653$	1.31371	0.0514094	0.0257047	−	0.999670i	$-0.491817\pi$
$653$	1.31371	0.0514094	0.0257047	+	0.999670i	$0.491817\pi$
$654$	0	0
$655$	16.0000	0.625172
$656$	0	0
$657$	4.34315	0.169442
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	−10.9706	−0.426705	−0.213353	−	0.976975i	$-0.568438\pi$
$661$	−10.9706	−0.426705	−0.213353	+	0.976975i	$0.568438\pi$
$662$	0	0
$663$	−7.65685	−0.297368
$664$	0	0
$665$	−19.3137	−0.748953
$666$	0	0
$667$	14.6274	0.566376
$668$	0	0
$669$	−10.8284	−0.418651
$670$	0	0
$671$	−16.9706	−0.655141
$672$	0	0
$673$	5.31371	0.204828	0.102414	−	0.994742i	$-0.467343\pi$
$673$	5.31371	0.204828	0.102414	+	0.994742i	$0.467343\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−35.9411	−1.38133	−0.690665	−	0.723175i	$-0.742682\pi$
$677$	−35.9411	−1.38133	−0.690665	+	0.723175i	$0.742682\pi$
$678$	0	0
$679$	−21.6569	−0.831114
$680$	0	0
$681$	22.8284	0.874787
$682$	0	0
$683$	−9.17157	−0.350940	−0.175470	−	0.984485i	$-0.556145\pi$
$683$	−9.17157	−0.350940	−0.175470	+	0.984485i	$0.556145\pi$
$684$	0	0
$685$	18.0000	0.687745
$686$	0	0
$687$	−19.6569	−0.749956
$688$	0	0
$689$	−2.00000	−0.0761939
$690$	0	0
$691$	47.7990	1.81836	0.909180	−	0.416404i	$-0.136710\pi$
$691$	47.7990	1.81836	0.909180	+	0.416404i	$0.136710\pi$
$692$	0	0
$693$	8.00000	0.303895
$694$	0	0
$695$	16.9706	0.643730
$696$	0	0
$697$	−15.3137	−0.580048
$698$	0	0
$699$	22.9706	0.868826
$700$	0	0
$701$	−15.6569	−0.591351	−0.295676	−	0.955288i	$-0.595545\pi$
$701$	−15.6569	−0.591351	−0.295676	+	0.955288i	$0.595545\pi$
$702$	0	0
$703$	13.6569	0.515078
$704$	0	0
$705$	−1.17157	−0.0441240
$706$	0	0
$707$	−10.3431	−0.388994
$708$	0	0
$709$	−12.3431	−0.463557	−0.231778	−	0.972769i	$-0.574454\pi$
$709$	−12.3431	−0.463557	−0.231778	+	0.972769i	$0.574454\pi$
$710$	0	0
$711$	13.6569	0.512172
$712$	0	0
$713$	−4.68629	−0.175503
$714$	0	0
$715$	2.82843	0.105777
$716$	0	0
$717$	24.4853	0.914420
$718$	0	0
$719$	−3.31371	−0.123580	−0.0617902	−	0.998089i	$-0.519681\pi$
$719$	−3.31371	−0.123580	−0.0617902	+	0.998089i	$0.519681\pi$
$720$	0	0
$721$	−4.68629	−0.174527
$722$	0	0
$723$	1.31371	0.0488573
$724$	0	0
$725$	3.65685	0.135812
$726$	0	0
$727$	−4.97056	−0.184348	−0.0921740	−	0.995743i	$-0.529382\pi$
$727$	−4.97056	−0.184348	−0.0921740	+	0.995743i	$0.529382\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	12.6863	0.469219
$732$	0	0
$733$	41.3137	1.52596	0.762978	−	0.646424i	$-0.223736\pi$
$733$	41.3137	1.52596	0.762978	+	0.646424i	$0.223736\pi$
$734$	0	0
$735$	1.00000	0.0368856
$736$	0	0
$737$	8.00000	0.294684
$738$	0	0
$739$	32.7696	1.20545	0.602724	−	0.797950i	$-0.294082\pi$
$739$	32.7696	1.20545	0.602724	+	0.797950i	$0.294082\pi$
$740$	0	0
$741$	6.82843	0.250849
$742$	0	0
$743$	15.7990	0.579609	0.289804	−	0.957086i	$-0.406410\pi$
$743$	15.7990	0.579609	0.289804	+	0.957086i	$0.406410\pi$
$744$	0	0
$745$	−13.3137	−0.487777
$746$	0	0
$747$	−6.82843	−0.249839
$748$	0	0
$749$	−20.6863	−0.755861
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	0	0
$753$	−24.9706	−0.909978
$754$	0	0
$755$	−1.17157	−0.0426379
$756$	0	0
$757$	−13.3137	−0.483895	−0.241947	−	0.970289i	$-0.577786\pi$
$757$	−13.3137	−0.483895	−0.241947	+	0.970289i	$0.577786\pi$
$758$	0	0
$759$	11.3137	0.410662
$760$	0	0
$761$	18.6863	0.677378	0.338689	−	0.940898i	$-0.390017\pi$
$761$	18.6863	0.677378	0.338689	+	0.940898i	$0.390017\pi$
$762$	0	0
$763$	44.2843	1.60320
$764$	0	0
$765$	−7.65685	−0.276834
$766$	0	0
$767$	8.48528	0.306386
$768$	0	0
$769$	2.00000	0.0721218	0.0360609	−	0.999350i	$-0.488519\pi$
$769$	2.00000	0.0721218	0.0360609	+	0.999350i	$0.488519\pi$
$770$	0	0
$771$	−18.9706	−0.683208
$772$	0	0
$773$	13.3137	0.478861	0.239430	−	0.970914i	$-0.423039\pi$
$773$	13.3137	0.478861	0.239430	+	0.970914i	$0.423039\pi$
$774$	0	0
$775$	−1.17157	−0.0420841
$776$	0	0
$777$	−5.65685	−0.202939
$778$	0	0
$779$	13.6569	0.489308
$780$	0	0
$781$	−30.6274	−1.09594
$782$	0	0
$783$	−3.65685	−0.130685
$784$	0	0
$785$	−14.0000	−0.499681
$786$	0	0
$787$	−43.7990	−1.56127	−0.780633	−	0.624990i	$-0.785103\pi$
$787$	−43.7990	−1.56127	−0.780633	+	0.624990i	$0.785103\pi$
$788$	0	0
$789$	9.65685	0.343793
$790$	0	0
$791$	32.9706	1.17230
$792$	0	0
$793$	6.00000	0.213066
$794$	0	0
$795$	−2.00000	−0.0709327
$796$	0	0
$797$	−34.0000	−1.20434	−0.602171	−	0.798367i	$-0.705697\pi$
$797$	−34.0000	−1.20434	−0.602171	+	0.798367i	$0.705697\pi$
$798$	0	0
$799$	−8.97056	−0.317356
$800$	0	0
$801$	−5.31371	−0.187751
$802$	0	0
$803$	−12.2843	−0.433503
$804$	0	0
$805$	11.3137	0.398756
$806$	0	0
$807$	7.65685	0.269534
$808$	0	0
$809$	−33.3137	−1.17125	−0.585624	−	0.810583i	$-0.699150\pi$
$809$	−33.3137	−1.17125	−0.585624	+	0.810583i	$0.699150\pi$
$810$	0	0
$811$	46.4264	1.63025	0.815126	−	0.579284i	$-0.196668\pi$
$811$	46.4264	1.63025	0.815126	+	0.579284i	$0.196668\pi$
$812$	0	0
$813$	−4.48528	−0.157306
$814$	0	0
$815$	−6.14214	−0.215150
$816$	0	0
$817$	−11.3137	−0.395817
$818$	0	0
$819$	−2.82843	−0.0988332
$820$	0	0
$821$	13.3137	0.464652	0.232326	−	0.972638i	$-0.425366\pi$
$821$	13.3137	0.464652	0.232326	+	0.972638i	$0.425366\pi$
$822$	0	0
$823$	33.6569	1.17320	0.586602	−	0.809875i	$-0.300465\pi$
$823$	33.6569	1.17320	0.586602	+	0.809875i	$0.300465\pi$
$824$	0	0
$825$	2.82843	0.0984732
$826$	0	0
$827$	−12.4853	−0.434156	−0.217078	−	0.976154i	$-0.569653\pi$
$827$	−12.4853	−0.434156	−0.217078	+	0.976154i	$0.569653\pi$
$828$	0	0
$829$	−32.6274	−1.13320	−0.566599	−	0.823994i	$-0.691741\pi$
$829$	−32.6274	−1.13320	−0.566599	+	0.823994i	$0.691741\pi$
$830$	0	0
$831$	29.3137	1.01688
$832$	0	0
$833$	7.65685	0.265294
$834$	0	0
$835$	12.4853	0.432071
$836$	0	0
$837$	1.17157	0.0404955
$838$	0	0
$839$	−46.1421	−1.59300	−0.796502	−	0.604636i	$-0.793318\pi$
$839$	−46.1421	−1.59300	−0.796502	+	0.604636i	$0.793318\pi$
$840$	0	0
$841$	−15.6274	−0.538876
$842$	0	0
$843$	24.6274	0.848213
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	8.48528	0.291558
$848$	0	0
$849$	17.6569	0.605982
$850$	0	0
$851$	−8.00000	−0.274236
$852$	0	0
$853$	43.2548	1.48102	0.740509	−	0.672047i	$-0.234585\pi$
$853$	43.2548	1.48102	0.740509	+	0.672047i	$0.234585\pi$
$854$	0	0
$855$	6.82843	0.233527
$856$	0	0
$857$	−11.6569	−0.398191	−0.199095	−	0.979980i	$-0.563800\pi$
$857$	−11.6569	−0.398191	−0.199095	+	0.979980i	$0.563800\pi$
$858$	0	0
$859$	−29.6569	−1.01188	−0.505939	−	0.862569i	$-0.668854\pi$
$859$	−29.6569	−1.01188	−0.505939	+	0.862569i	$0.668854\pi$
$860$	0	0
$861$	−5.65685	−0.192785
$862$	0	0
$863$	35.1127	1.19525	0.597625	−	0.801776i	$-0.296111\pi$
$863$	35.1127	1.19525	0.597625	+	0.801776i	$0.296111\pi$
$864$	0	0
$865$	24.6274	0.837357
$866$	0	0
$867$	−41.6274	−1.41374
$868$	0	0
$869$	−38.6274	−1.31035
$870$	0	0
$871$	−2.82843	−0.0958376
$872$	0	0
$873$	7.65685	0.259145
$874$	0	0
$875$	2.82843	0.0956183
$876$	0	0
$877$	38.0000	1.28317	0.641584	−	0.767052i	$-0.278277\pi$
$877$	38.0000	1.28317	0.641584	+	0.767052i	$0.278277\pi$
$878$	0	0
$879$	9.31371	0.314144
$880$	0	0
$881$	−23.9411	−0.806597	−0.403299	−	0.915068i	$-0.632136\pi$
$881$	−23.9411	−0.806597	−0.403299	+	0.915068i	$0.632136\pi$
$882$	0	0
$883$	32.2843	1.08645	0.543226	−	0.839586i	$-0.317203\pi$
$883$	32.2843	1.08645	0.543226	+	0.839586i	$0.317203\pi$
$884$	0	0
$885$	8.48528	0.285230
$886$	0	0
$887$	−4.97056	−0.166895	−0.0834476	−	0.996512i	$-0.526593\pi$
$887$	−4.97056	−0.166895	−0.0834476	+	0.996512i	$0.526593\pi$
$888$	0	0
$889$	27.3137	0.916072
$890$	0	0
$891$	−2.82843	−0.0947559
$892$	0	0
$893$	8.00000	0.267710
$894$	0	0
$895$	2.34315	0.0783227
$896$	0	0
$897$	−4.00000	−0.133556
$898$	0	0
$899$	−4.28427	−0.142888
$900$	0	0
$901$	−15.3137	−0.510174
$902$	0	0
$903$	4.68629	0.155950
$904$	0	0
$905$	−17.3137	−0.575527
$906$	0	0
$907$	−36.9706	−1.22759	−0.613794	−	0.789466i	$-0.710357\pi$
$907$	−36.9706	−1.22759	−0.613794	+	0.789466i	$0.710357\pi$
$908$	0	0
$909$	3.65685	0.121290
$910$	0	0
$911$	−39.5980	−1.31194	−0.655970	−	0.754787i	$-0.727740\pi$
$911$	−39.5980	−1.31194	−0.655970	+	0.754787i	$0.727740\pi$
$912$	0	0
$913$	19.3137	0.639190
$914$	0	0
$915$	6.00000	0.198354
$916$	0	0
$917$	45.2548	1.49445
$918$	0	0
$919$	−36.2843	−1.19691	−0.598454	−	0.801157i	$-0.704218\pi$
$919$	−36.2843	−1.19691	−0.598454	+	0.801157i	$0.704218\pi$
$920$	0	0
$921$	21.1716	0.697627
$922$	0	0
$923$	10.8284	0.356422
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	0	0
$927$	1.65685	0.0544182
$928$	0	0
$929$	−16.6274	−0.545528	−0.272764	−	0.962081i	$-0.587938\pi$
$929$	−16.6274	−0.545528	−0.272764	+	0.962081i	$0.587938\pi$
$930$	0	0
$931$	−6.82843	−0.223793
$932$	0	0
$933$	−4.68629	−0.153422
$934$	0	0
$935$	21.6569	0.708255
$936$	0	0
$937$	0.627417	0.0204968	0.0102484	−	0.999947i	$-0.496738\pi$
$937$	0.627417	0.0204968	0.0102484	+	0.999947i	$0.496738\pi$
$938$	0	0
$939$	14.0000	0.456873
$940$	0	0
$941$	32.6274	1.06362	0.531812	−	0.846863i	$-0.321511\pi$
$941$	32.6274	1.06362	0.531812	+	0.846863i	$0.321511\pi$
$942$	0	0
$943$	−8.00000	−0.260516
$944$	0	0
$945$	−2.82843	−0.0920087
$946$	0	0
$947$	−41.1716	−1.33790	−0.668948	−	0.743309i	$-0.733255\pi$
$947$	−41.1716	−1.33790	−0.668948	+	0.743309i	$0.733255\pi$
$948$	0	0
$949$	4.34315	0.140984
$950$	0	0
$951$	2.68629	0.0871090
$952$	0	0
$953$	−0.343146	−0.0111156	−0.00555779	−	0.999985i	$-0.501769\pi$
$953$	−0.343146	−0.0111156	−0.00555779	+	0.999985i	$0.501769\pi$
$954$	0	0
$955$	−5.65685	−0.183052
$956$	0	0
$957$	10.3431	0.334346
$958$	0	0
$959$	50.9117	1.64402
$960$	0	0
$961$	−29.6274	−0.955723
$962$	0	0
$963$	7.31371	0.235681
$964$	0	0
$965$	−4.34315	−0.139811
$966$	0	0
$967$	−24.4853	−0.787394	−0.393697	−	0.919240i	$-0.628804\pi$
$967$	−24.4853	−0.787394	−0.393697	+	0.919240i	$0.628804\pi$
$968$	0	0
$969$	52.2843	1.67961
$970$	0	0
$971$	35.3137	1.13327	0.566635	−	0.823969i	$-0.308245\pi$
$971$	35.3137	1.13327	0.566635	+	0.823969i	$0.308245\pi$
$972$	0	0
$973$	48.0000	1.53881
$974$	0	0
$975$	−1.00000	−0.0320256
$976$	0	0
$977$	−2.00000	−0.0639857	−0.0319928	−	0.999488i	$-0.510185\pi$
$977$	−2.00000	−0.0639857	−0.0319928	+	0.999488i	$0.510185\pi$
$978$	0	0
$979$	15.0294	0.480343
$980$	0	0
$981$	−15.6569	−0.499885
$982$	0	0
$983$	48.7696	1.55551	0.777754	−	0.628569i	$-0.216359\pi$
$983$	48.7696	1.55551	0.777754	+	0.628569i	$0.216359\pi$
$984$	0	0
$985$	22.0000	0.700978
$986$	0	0
$987$	−3.31371	−0.105477
$988$	0	0
$989$	6.62742	0.210740
$990$	0	0
$991$	−8.97056	−0.284959	−0.142480	−	0.989798i	$-0.545508\pi$
$991$	−8.97056	−0.284959	−0.142480	+	0.989798i	$0.545508\pi$
$992$	0	0
$993$	18.1421	0.575723
$994$	0	0
$995$	0	0
$996$	0	0
$997$	25.3137	0.801693	0.400847	−	0.916145i	$-0.368716\pi$
$997$	25.3137	0.801693	0.400847	+	0.916145i	$0.368716\pi$
$998$	0	0
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6240.2.a.bh.1.1	✓	2
4.3	odd	2		6240.2.a.br.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6240.2.a.bh.1.1	✓	2	1.1	even	1	trivial
6240.2.a.br.1.2	yes	2	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6240.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} -2.82843 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.00000 q^{5} -2.82843 q^{7} +1.00000 q^{9} -2.82843 q^{11} +1.00000 q^{13} +1.00000 q^{15} +7.65685 q^{17} -6.82843 q^{19} +2.82843 q^{21} +4.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +3.65685 q^{29} -1.17157 q^{31} +2.82843 q^{33} +2.82843 q^{35} -2.00000 q^{37} -1.00000 q^{39} -2.00000 q^{41} +1.65685 q^{43} -1.00000 q^{45} -1.17157 q^{47} +1.00000 q^{49} -7.65685 q^{51} -2.00000 q^{53} +2.82843 q^{55} +6.82843 q^{57} +8.48528 q^{59} +6.00000 q^{61} -2.82843 q^{63} -1.00000 q^{65} -2.82843 q^{67} -4.00000 q^{69} +10.8284 q^{71} +4.34315 q^{73} -1.00000 q^{75} +8.00000 q^{77} +13.6569 q^{79} +1.00000 q^{81} -6.82843 q^{83} -7.65685 q^{85} -3.65685 q^{87} -5.31371 q^{89} -2.82843 q^{91} +1.17157 q^{93} +6.82843 q^{95} +7.65685 q^{97} -2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^9	\( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{9} + 2 q^{13} + 2 q^{15} + 4 q^{17} - 8 q^{19} + 8 q^{23} + 2 q^{25} - 2 q^{27} - 4 q^{29} - 8 q^{31} - 4 q^{37} - 2 q^{39} - 4 q^{41} - 8 q^{43} - 2 q^{45} - 8 q^{47} + 2 q^{49} - 4 q^{51} - 4 q^{53} + 8 q^{57} + 12 q^{61} - 2 q^{65} - 8 q^{69} + 16 q^{71} + 20 q^{73} - 2 q^{75} + 16 q^{77} + 16 q^{79} + 2 q^{81} - 8 q^{83} - 4 q^{85} + 4 q^{87} + 12 q^{89} + 8 q^{93} + 8 q^{95} + 4 q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^9 + 2 * q^13 + 2 * q^15 + 4 * q^17 - 8 * q^19 + 8 * q^23 + 2 * q^25 - 2 * q^27 - 4 * q^29 - 8 * q^31 - 4 * q^37 - 2 * q^39 - 4 * q^41 - 8 * q^43 - 2 * q^45 - 8 * q^47 + 2 * q^49 - 4 * q^51 - 4 * q^53 + 8 * q^57 + 12 * q^61 - 2 * q^65 - 8 * q^69 + 16 * q^71 + 20 * q^73 - 2 * q^75 + 16 * q^77 + 16 * q^79 + 2 * q^81 - 8 * q^83 - 4 * q^85 + 4 * q^87 + 12 * q^89 + 8 * q^93 + 8 * q^95 + 4 * q^97

Introduction

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