LMFDB - Embedded newform 6240.2.a.br.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:	$0$
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:	$\Z[a_1, \ldots, a_{7}]$
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)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} -2.82843 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{5} -2.82843 q^{7} +1.00000 q^{9} -2.82843 q^{11} +1.00000 q^{13} -1.00000 q^{15} -3.65685 q^{17} +1.17157 q^{19} -2.82843 q^{21} -4.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -7.65685 q^{29} +6.82843 q^{31} -2.82843 q^{33} +2.82843 q^{35} -2.00000 q^{37} +1.00000 q^{39} -2.00000 q^{41} +9.65685 q^{43} -1.00000 q^{45} +6.82843 q^{47} +1.00000 q^{49} -3.65685 q^{51} -2.00000 q^{53} +2.82843 q^{55} +1.17157 q^{57} +8.48528 q^{59} +6.00000 q^{61} -2.82843 q^{63} -1.00000 q^{65} -2.82843 q^{67} -4.00000 q^{69} -5.17157 q^{71} +15.6569 q^{73} +1.00000 q^{75} +8.00000 q^{77} -2.34315 q^{79} +1.00000 q^{81} +1.17157 q^{83} +3.65685 q^{85} -7.65685 q^{87} +17.3137 q^{89} -2.82843 q^{91} +6.82843 q^{93} -1.17157 q^{95} -3.65685 q^{97} -2.82843 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$17$	−3.65685	−0.886917	−0.443459	−	0.896295i	$-0.646249\pi$
$17$	−3.65685	−0.886917	−0.443459	+	0.896295i	$0.646249\pi$
$18$	0	0
$19$	1.17157	0.268777	0.134389	−	0.990929i	$-0.457093\pi$
$19$	1.17157	0.268777	0.134389	+	0.990929i	$0.457093\pi$
$20$	0	0
$21$	−2.82843	−0.617213
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	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	$-0.751722\pi$
$29$	−7.65685	−1.42184	−0.710921	+	0.703272i	$0.751722\pi$
$30$	0	0
$31$	6.82843	1.22642	0.613211	−	0.789919i	$-0.289878\pi$
$31$	6.82843	1.22642	0.613211	+	0.789919i	$0.289878\pi$
$32$	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
$43$	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
$47$	6.82843	0.996028	0.498014	−	0.867169i	$-0.334063\pi$
$47$	6.82843	0.996028	0.498014	+	0.867169i	$0.334063\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.65685	−0.512062
$52$	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
$57$	1.17157	0.155179
$58$	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
$71$	−5.17157	−0.613753	−0.306876	−	0.951749i	$-0.599284\pi$
$71$	−5.17157	−0.613753	−0.306876	+	0.951749i	$0.599284\pi$
$72$	0	0
$73$	15.6569	1.83250	0.916248	−	0.400611i	$-0.131202\pi$
$73$	15.6569	1.83250	0.916248	+	0.400611i	$0.131202\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	8.00000	0.911685
$78$	0	0
$79$	−2.34315	−0.263624	−0.131812	−	0.991275i	$-0.542080\pi$
$79$	−2.34315	−0.263624	−0.131812	+	0.991275i	$0.542080\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.17157	0.128597	0.0642984	−	0.997931i	$-0.479519\pi$
$83$	1.17157	0.128597	0.0642984	+	0.997931i	$0.479519\pi$
$84$	0	0
$85$	3.65685	0.396642
$86$	0	0
$87$	−7.65685	−0.820901
$88$	0	0
$89$	17.3137	1.83525	0.917625	−	0.397448i	$-0.130104\pi$
$89$	17.3137	1.83525	0.917625	+	0.397448i	$0.130104\pi$
$90$	0	0
$91$	−2.82843	−0.296500
$92$	0	0
$93$	6.82843	0.708075
$94$	0	0
$95$	−1.17157	−0.120201
$96$	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
$99$	−2.82843	−0.284268
$100$	0	0
$101$	−7.65685	−0.761885	−0.380943	−	0.924599i	$-0.624401\pi$
$101$	−7.65685	−0.761885	−0.380943	+	0.924599i	$0.624401\pi$
$102$	0	0
$103$	9.65685	0.951518	0.475759	−	0.879576i	$-0.342173\pi$
$103$	9.65685	0.951518	0.475759	+	0.879576i	$0.342173\pi$
$104$	0	0
$105$	2.82843	0.276026
$106$	0	0
$107$	15.3137	1.48043	0.740216	−	0.672369i	$-0.234723\pi$
$107$	15.3137	1.48043	0.740216	+	0.672369i	$0.234723\pi$
$108$	0	0
$109$	−4.34315	−0.415998	−0.207999	−	0.978129i	$-0.566695\pi$
$109$	−4.34315	−0.415998	−0.207999	+	0.978129i	$0.566695\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	0	0
$113$	−0.343146	−0.0322804	−0.0161402	−	0.999870i	$-0.505138\pi$
$113$	−0.343146	−0.0322804	−0.0161402	+	0.999870i	$0.505138\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	10.3431	0.948155
$120$	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
$127$	−1.65685	−0.147022	−0.0735110	−	0.997294i	$-0.523420\pi$
$127$	−1.65685	−0.147022	−0.0735110	+	0.997294i	$0.523420\pi$
$128$	0	0
$129$	9.65685	0.850239
$130$	0	0
$131$	16.0000	1.39793	0.698963	−	0.715158i	$-0.253645\pi$
$131$	16.0000	1.39793	0.698963	+	0.715158i	$0.253645\pi$
$132$	0	0
$133$	−3.31371	−0.287335
$134$	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
$141$	6.82843	0.575057
$142$	0	0
$143$	−2.82843	−0.236525
$144$	0	0
$145$	7.65685	0.635867
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−9.31371	−0.763009	−0.381504	−	0.924367i	$-0.624594\pi$
$149$	−9.31371	−0.763009	−0.381504	+	0.924367i	$0.624594\pi$
$150$	0	0
$151$	−6.82843	−0.555690	−0.277845	−	0.960626i	$-0.589620\pi$
$151$	−6.82843	−0.555690	−0.277845	+	0.960626i	$0.589620\pi$
$152$	0	0
$153$	−3.65685	−0.295639
$154$	0	0
$155$	−6.82843	−0.548472
$156$	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
$163$	22.1421	1.73431	0.867153	−	0.498042i	$-0.165947\pi$
$163$	22.1421	1.73431	0.867153	+	0.498042i	$0.165947\pi$
$164$	0	0
$165$	2.82843	0.220193
$166$	0	0
$167$	−4.48528	−0.347081	−0.173541	−	0.984827i	$-0.555521\pi$
$167$	−4.48528	−0.347081	−0.173541	+	0.984827i	$0.555521\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	1.17157	0.0895924
$172$	0	0
$173$	20.6274	1.56827	0.784137	−	0.620588i	$-0.213106\pi$
$173$	20.6274	1.56827	0.784137	+	0.620588i	$0.213106\pi$
$174$	0	0
$175$	−2.82843	−0.213809
$176$	0	0
$177$	8.48528	0.637793
$178$	0	0
$179$	13.6569	1.02076	0.510381	−	0.859949i	$-0.329505\pi$
$179$	13.6569	1.02076	0.510381	+	0.859949i	$0.329505\pi$
$180$	0	0
$181$	−5.31371	−0.394965	−0.197482	−	0.980306i	$-0.563277\pi$
$181$	−5.31371	−0.394965	−0.197482	+	0.980306i	$0.563277\pi$
$182$	0	0
$183$	6.00000	0.443533
$184$	0	0
$185$	2.00000	0.147043
$186$	0	0
$187$	10.3431	0.756366
$188$	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
$193$	15.6569	1.12701	0.563503	−	0.826114i	$-0.309454\pi$
$193$	15.6569	1.12701	0.563503	+	0.826114i	$0.309454\pi$
$194$	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
$203$	21.6569	1.52001
$204$	0	0
$205$	2.00000	0.139686
$206$	0	0
$207$	−4.00000	−0.278019
$208$	0	0
$209$	−3.31371	−0.229214
$210$	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
$213$	−5.17157	−0.354350
$214$	0	0
$215$	−9.65685	−0.658592
$216$	0	0
$217$	−19.3137	−1.31110
$218$	0	0
$219$	15.6569	1.05799
$220$	0	0
$221$	−3.65685	−0.245987
$222$	0	0
$223$	−5.17157	−0.346314	−0.173157	−	0.984894i	$-0.555397\pi$
$223$	−5.17157	−0.346314	−0.173157	+	0.984894i	$0.555397\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	17.1716	1.13972	0.569859	−	0.821743i	$-0.306998\pi$
$227$	17.1716	1.13972	0.569859	+	0.821743i	$0.306998\pi$
$228$	0	0
$229$	8.34315	0.551331	0.275665	−	0.961254i	$-0.411102\pi$
$229$	8.34315	0.551331	0.275665	+	0.961254i	$0.411102\pi$
$230$	0	0
$231$	8.00000	0.526361
$232$	0	0
$233$	10.9706	0.718705	0.359353	−	0.933202i	$-0.382997\pi$
$233$	10.9706	0.718705	0.359353	+	0.933202i	$0.382997\pi$
$234$	0	0
$235$	−6.82843	−0.445437
$236$	0	0
$237$	−2.34315	−0.152204
$238$	0	0
$239$	7.51472	0.486087	0.243043	−	0.970015i	$-0.421854\pi$
$239$	7.51472	0.486087	0.243043	+	0.970015i	$0.421854\pi$
$240$	0	0
$241$	21.3137	1.37294	0.686468	−	0.727160i	$-0.259160\pi$
$241$	21.3137	1.37294	0.686468	+	0.727160i	$0.259160\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	1.17157	0.0745454
$248$	0	0
$249$	1.17157	0.0742454
$250$	0	0
$251$	8.97056	0.566217	0.283108	−	0.959088i	$-0.408634\pi$
$251$	8.97056	0.566217	0.283108	+	0.959088i	$0.408634\pi$
$252$	0	0
$253$	11.3137	0.711287
$254$	0	0
$255$	3.65685	0.229001
$256$	0	0
$257$	−14.9706	−0.933838	−0.466919	−	0.884300i	$-0.654636\pi$
$257$	−14.9706	−0.933838	−0.466919	+	0.884300i	$0.654636\pi$
$258$	0	0
$259$	5.65685	0.351500
$260$	0	0
$261$	−7.65685	−0.473947
$262$	0	0
$263$	−1.65685	−0.102166	−0.0510830	−	0.998694i	$-0.516267\pi$
$263$	−1.65685	−0.102166	−0.0510830	+	0.998694i	$0.516267\pi$
$264$	0	0
$265$	2.00000	0.122859
$266$	0	0
$267$	17.3137	1.05958
$268$	0	0
$269$	3.65685	0.222962	0.111481	−	0.993767i	$-0.464441\pi$
$269$	3.65685	0.222962	0.111481	+	0.993767i	$0.464441\pi$
$270$	0	0
$271$	12.4853	0.758427	0.379213	−	0.925309i	$-0.376195\pi$
$271$	12.4853	0.758427	0.379213	+	0.925309i	$0.376195\pi$
$272$	0	0
$273$	−2.82843	−0.171184
$274$	0	0
$275$	−2.82843	−0.170561
$276$	0	0
$277$	−6.68629	−0.401740	−0.200870	−	0.979618i	$-0.564377\pi$
$277$	−6.68629	−0.401740	−0.200870	+	0.979618i	$0.564377\pi$
$278$	0	0
$279$	6.82843	0.408807
$280$	0	0
$281$	20.6274	1.23053	0.615264	−	0.788321i	$-0.289049\pi$
$281$	20.6274	1.23053	0.615264	+	0.788321i	$0.289049\pi$
$282$	0	0
$283$	6.34315	0.377061	0.188530	−	0.982067i	$-0.439628\pi$
$283$	6.34315	0.377061	0.188530	+	0.982067i	$0.439628\pi$
$284$	0	0
$285$	−1.17157	−0.0693980
$286$	0	0
$287$	5.65685	0.333914
$288$	0	0
$289$	−3.62742	−0.213377
$290$	0	0
$291$	−3.65685	−0.214369
$292$	0	0
$293$	13.3137	0.777795	0.388898	−	0.921281i	$-0.372856\pi$
$293$	13.3137	0.777795	0.388898	+	0.921281i	$0.372856\pi$
$294$	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
$301$	−27.3137	−1.57434
$302$	0	0
$303$	−7.65685	−0.439875
$304$	0	0
$305$	−6.00000	−0.343559
$306$	0	0
$307$	26.8284	1.53118	0.765590	−	0.643329i	$-0.222447\pi$
$307$	26.8284	1.53118	0.765590	+	0.643329i	$0.222447\pi$
$308$	0	0
$309$	9.65685	0.549359
$310$	0	0
$311$	−27.3137	−1.54882	−0.774409	−	0.632685i	$-0.781953\pi$
$311$	−27.3137	−1.54882	−0.774409	+	0.632685i	$0.781953\pi$
$312$	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
$317$	−25.3137	−1.42176	−0.710880	−	0.703314i	$-0.751703\pi$
$317$	−25.3137	−1.42176	−0.710880	+	0.703314i	$0.751703\pi$
$318$	0	0
$319$	21.6569	1.21255
$320$	0	0
$321$	15.3137	0.854728
$322$	0	0
$323$	−4.28427	−0.238383
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	−4.34315	−0.240177
$328$	0	0
$329$	−19.3137	−1.06480
$330$	0	0
$331$	−10.1421	−0.557462	−0.278731	−	0.960369i	$-0.589914\pi$
$331$	−10.1421	−0.557462	−0.278731	+	0.960369i	$0.589914\pi$
$332$	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
$339$	−0.343146	−0.0186371
$340$	0	0
$341$	−19.3137	−1.04590
$342$	0	0
$343$	16.9706	0.916324
$344$	0	0
$345$	4.00000	0.215353
$346$	0	0
$347$	34.6274	1.85890	0.929449	−	0.368952i	$-0.120283\pi$
$347$	34.6274	1.85890	0.929449	+	0.368952i	$0.120283\pi$
$348$	0	0
$349$	30.9706	1.65782	0.828908	−	0.559385i	$-0.188963\pi$
$349$	30.9706	1.65782	0.828908	+	0.559385i	$0.188963\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	−5.31371	−0.282820	−0.141410	−	0.989951i	$-0.545164\pi$
$353$	−5.31371	−0.282820	−0.141410	+	0.989951i	$0.545164\pi$
$354$	0	0
$355$	5.17157	0.274479
$356$	0	0
$357$	10.3431	0.547417
$358$	0	0
$359$	0.485281	0.0256122	0.0128061	−	0.999918i	$-0.495924\pi$
$359$	0.485281	0.0256122	0.0128061	+	0.999918i	$0.495924\pi$
$360$	0	0
$361$	−17.6274	−0.927759
$362$	0	0
$363$	−3.00000	−0.157459
$364$	0	0
$365$	−15.6569	−0.819517
$366$	0	0
$367$	31.3137	1.63456	0.817281	−	0.576239i	$-0.195480\pi$
$367$	31.3137	1.63456	0.817281	+	0.576239i	$0.195480\pi$
$368$	0	0
$369$	−2.00000	−0.104116
$370$	0	0
$371$	5.65685	0.293689
$372$	0	0
$373$	−16.6274	−0.860935	−0.430468	−	0.902606i	$-0.641651\pi$
$373$	−16.6274	−0.860935	−0.430468	+	0.902606i	$0.641651\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−7.65685	−0.394348
$378$	0	0
$379$	−10.1421	−0.520967	−0.260483	−	0.965478i	$-0.583882\pi$
$379$	−10.1421	−0.520967	−0.260483	+	0.965478i	$0.583882\pi$
$380$	0	0
$381$	−1.65685	−0.0848832
$382$	0	0
$383$	9.17157	0.468645	0.234323	−	0.972159i	$-0.424713\pi$
$383$	9.17157	0.468645	0.234323	+	0.972159i	$0.424713\pi$
$384$	0	0
$385$	−8.00000	−0.407718
$386$	0	0
$387$	9.65685	0.490885
$388$	0	0
$389$	−12.3431	−0.625822	−0.312911	−	0.949782i	$-0.601304\pi$
$389$	−12.3431	−0.625822	−0.312911	+	0.949782i	$0.601304\pi$
$390$	0	0
$391$	14.6274	0.739740
$392$	0	0
$393$	16.0000	0.807093
$394$	0	0
$395$	2.34315	0.117896
$396$	0	0
$397$	28.6274	1.43677	0.718384	−	0.695646i	$-0.244882\pi$
$397$	28.6274	1.43677	0.718384	+	0.695646i	$0.244882\pi$
$398$	0	0
$399$	−3.31371	−0.165893
$400$	0	0
$401$	−29.3137	−1.46386	−0.731928	−	0.681382i	$-0.761379\pi$
$401$	−29.3137	−1.46386	−0.731928	+	0.681382i	$0.761379\pi$
$402$	0	0
$403$	6.82843	0.340148
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	5.65685	0.280400
$408$	0	0
$409$	24.6274	1.21775	0.608874	−	0.793267i	$-0.291622\pi$
$409$	24.6274	1.21775	0.608874	+	0.793267i	$0.291622\pi$
$410$	0	0
$411$	−18.0000	−0.887875
$412$	0	0
$413$	−24.0000	−1.18096
$414$	0	0
$415$	−1.17157	−0.0575103
$416$	0	0
$417$	−16.9706	−0.831052
$418$	0	0
$419$	30.6274	1.49625	0.748124	−	0.663559i	$-0.230955\pi$
$419$	30.6274	1.49625	0.748124	+	0.663559i	$0.230955\pi$
$420$	0	0
$421$	14.9706	0.729621	0.364810	−	0.931082i	$-0.381134\pi$
$421$	14.9706	0.729621	0.364810	+	0.931082i	$0.381134\pi$
$422$	0	0
$423$	6.82843	0.332009
$424$	0	0
$425$	−3.65685	−0.177383
$426$	0	0
$427$	−16.9706	−0.821263
$428$	0	0
$429$	−2.82843	−0.136558
$430$	0	0
$431$	−1.85786	−0.0894902	−0.0447451	−	0.998998i	$-0.514248\pi$
$431$	−1.85786	−0.0894902	−0.0447451	+	0.998998i	$0.514248\pi$
$432$	0	0
$433$	13.3137	0.639816	0.319908	−	0.947449i	$-0.396348\pi$
$433$	13.3137	0.639816	0.319908	+	0.947449i	$0.396348\pi$
$434$	0	0
$435$	7.65685	0.367118
$436$	0	0
$437$	−4.68629	−0.224176
$438$	0	0
$439$	−30.6274	−1.46177	−0.730883	−	0.682502i	$-0.760892\pi$
$439$	−30.6274	−1.46177	−0.730883	+	0.682502i	$0.760892\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−26.6274	−1.26511	−0.632553	−	0.774517i	$-0.717993\pi$
$443$	−26.6274	−1.26511	−0.632553	+	0.774517i	$0.717993\pi$
$444$	0	0
$445$	−17.3137	−0.820748
$446$	0	0
$447$	−9.31371	−0.440523
$448$	0	0
$449$	−8.62742	−0.407153	−0.203576	−	0.979059i	$-0.565257\pi$
$449$	−8.62742	−0.407153	−0.203576	+	0.979059i	$0.565257\pi$
$450$	0	0
$451$	5.65685	0.266371
$452$	0	0
$453$	−6.82843	−0.320827
$454$	0	0
$455$	2.82843	0.132599
$456$	0	0
$457$	−34.2843	−1.60375	−0.801875	−	0.597491i	$-0.796164\pi$
$457$	−34.2843	−1.60375	−0.801875	+	0.597491i	$0.796164\pi$
$458$	0	0
$459$	−3.65685	−0.170687
$460$	0	0
$461$	13.3137	0.620081	0.310041	−	0.950723i	$-0.399657\pi$
$461$	13.3137	0.620081	0.310041	+	0.950723i	$0.399657\pi$
$462$	0	0
$463$	−6.14214	−0.285449	−0.142725	−	0.989762i	$-0.545586\pi$
$463$	−6.14214	−0.285449	−0.142725	+	0.989762i	$0.545586\pi$
$464$	0	0
$465$	−6.82843	−0.316661
$466$	0	0
$467$	−4.97056	−0.230010	−0.115005	−	0.993365i	$-0.536688\pi$
$467$	−4.97056	−0.230010	−0.115005	+	0.993365i	$0.536688\pi$
$468$	0	0
$469$	8.00000	0.369406
$470$	0	0
$471$	14.0000	0.645086
$472$	0	0
$473$	−27.3137	−1.25589
$474$	0	0
$475$	1.17157	0.0537555
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	0	0
$479$	−7.51472	−0.343356	−0.171678	−	0.985153i	$-0.554919\pi$
$479$	−7.51472	−0.343356	−0.171678	+	0.985153i	$0.554919\pi$
$480$	0	0
$481$	−2.00000	−0.0911922
$482$	0	0
$483$	11.3137	0.514792
$484$	0	0
$485$	3.65685	0.166049
$486$	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
$489$	22.1421	1.00130
$490$	0	0
$491$	−2.34315	−0.105745	−0.0528723	−	0.998601i	$-0.516838\pi$
$491$	−2.34315	−0.105745	−0.0528723	+	0.998601i	$0.516838\pi$
$492$	0	0
$493$	28.0000	1.26106
$494$	0	0
$495$	2.82843	0.127128
$496$	0	0
$497$	14.6274	0.656129
$498$	0	0
$499$	25.1716	1.12683	0.563417	−	0.826173i	$-0.309486\pi$
$499$	25.1716	1.12683	0.563417	+	0.826173i	$0.309486\pi$
$500$	0	0
$501$	−4.48528	−0.200388
$502$	0	0
$503$	9.65685	0.430578	0.215289	−	0.976550i	$-0.430931\pi$
$503$	9.65685	0.430578	0.215289	+	0.976550i	$0.430931\pi$
$504$	0	0
$505$	7.65685	0.340726
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	6.68629	0.296365	0.148182	−	0.988960i	$-0.452658\pi$
$509$	6.68629	0.296365	0.148182	+	0.988960i	$0.452658\pi$
$510$	0	0
$511$	−44.2843	−1.95902
$512$	0	0
$513$	1.17157	0.0517262
$514$	0	0
$515$	−9.65685	−0.425532
$516$	0	0
$517$	−19.3137	−0.849416
$518$	0	0
$519$	20.6274	0.905443
$520$	0	0
$521$	−44.6274	−1.95516	−0.977581	−	0.210558i	$-0.932472\pi$
$521$	−44.6274	−1.95516	−0.977581	+	0.210558i	$0.932472\pi$
$522$	0	0
$523$	−15.3137	−0.669622	−0.334811	−	0.942285i	$-0.608672\pi$
$523$	−15.3137	−0.669622	−0.334811	+	0.942285i	$0.608672\pi$
$524$	0	0
$525$	−2.82843	−0.123443
$526$	0	0
$527$	−24.9706	−1.08773
$528$	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
$535$	−15.3137	−0.662069
$536$	0	0
$537$	13.6569	0.589337
$538$	0	0
$539$	−2.82843	−0.121829
$540$	0	0
$541$	−10.9706	−0.471661	−0.235831	−	0.971794i	$-0.575781\pi$
$541$	−10.9706	−0.471661	−0.235831	+	0.971794i	$0.575781\pi$
$542$	0	0
$543$	−5.31371	−0.228033
$544$	0	0
$545$	4.34315	0.186040
$546$	0	0
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	0	0
$549$	6.00000	0.256074
$550$	0	0
$551$	−8.97056	−0.382159
$552$	0	0
$553$	6.62742	0.281826
$554$	0	0
$555$	2.00000	0.0848953
$556$	0	0
$557$	27.9411	1.18390	0.591952	−	0.805973i	$-0.298358\pi$
$557$	27.9411	1.18390	0.591952	+	0.805973i	$0.298358\pi$
$558$	0	0
$559$	9.65685	0.408441
$560$	0	0
$561$	10.3431	0.436688
$562$	0	0
$563$	−36.9706	−1.55812	−0.779062	−	0.626947i	$-0.784304\pi$
$563$	−36.9706	−1.55812	−0.779062	+	0.626947i	$0.784304\pi$
$564$	0	0
$565$	0.343146	0.0144363
$566$	0	0
$567$	−2.82843	−0.118783
$568$	0	0
$569$	−2.68629	−0.112615	−0.0563076	−	0.998413i	$-0.517933\pi$
$569$	−2.68629	−0.112615	−0.0563076	+	0.998413i	$0.517933\pi$
$570$	0	0
$571$	14.6274	0.612138	0.306069	−	0.952009i	$-0.400986\pi$
$571$	14.6274	0.612138	0.306069	+	0.952009i	$0.400986\pi$
$572$	0	0
$573$	5.65685	0.236318
$574$	0	0
$575$	−4.00000	−0.166812
$576$	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
$579$	15.6569	0.650677
$580$	0	0
$581$	−3.31371	−0.137476
$582$	0	0
$583$	5.65685	0.234283
$584$	0	0
$585$	−1.00000	−0.0413449
$586$	0	0
$587$	−2.14214	−0.0884154	−0.0442077	−	0.999022i	$-0.514076\pi$
$587$	−2.14214	−0.0884154	−0.0442077	+	0.999022i	$0.514076\pi$
$588$	0	0
$589$	8.00000	0.329634
$590$	0	0
$591$	−22.0000	−0.904959
$592$	0	0
$593$	−29.3137	−1.20377	−0.601885	−	0.798583i	$-0.705583\pi$
$593$	−29.3137	−1.20377	−0.601885	+	0.798583i	$0.705583\pi$
$594$	0	0
$595$	−10.3431	−0.424028
$596$	0	0
$597$	0	0
$598$	0	0
$599$	26.3431	1.07635	0.538176	−	0.842833i	$-0.319114\pi$
$599$	26.3431	1.07635	0.538176	+	0.842833i	$0.319114\pi$
$600$	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
$607$	10.6274	0.431354	0.215677	−	0.976465i	$-0.430804\pi$
$607$	10.6274	0.431354	0.215677	+	0.976465i	$0.430804\pi$
$608$	0	0
$609$	21.6569	0.877580
$610$	0	0
$611$	6.82843	0.276249
$612$	0	0
$613$	41.3137	1.66864	0.834322	−	0.551277i	$-0.185859\pi$
$613$	41.3137	1.66864	0.834322	+	0.551277i	$0.185859\pi$
$614$	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
$619$	−15.7990	−0.635015	−0.317508	−	0.948256i	$-0.602846\pi$
$619$	−15.7990	−0.635015	−0.317508	+	0.948256i	$0.602846\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	0	0
$623$	−48.9706	−1.96196
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−3.31371	−0.132337
$628$	0	0
$629$	7.31371	0.291617
$630$	0	0
$631$	−34.1421	−1.35918	−0.679588	−	0.733594i	$-0.737842\pi$
$631$	−34.1421	−1.35918	−0.679588	+	0.733594i	$0.737842\pi$
$632$	0	0
$633$	−5.65685	−0.224840
$634$	0	0
$635$	1.65685	0.0657503
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−5.17157	−0.204584
$640$	0	0
$641$	−33.3137	−1.31581	−0.657906	−	0.753100i	$-0.728558\pi$
$641$	−33.3137	−1.31581	−0.657906	+	0.753100i	$0.728558\pi$
$642$	0	0
$643$	16.4853	0.650116	0.325058	−	0.945694i	$-0.394616\pi$
$643$	16.4853	0.650116	0.325058	+	0.945694i	$0.394616\pi$
$644$	0	0
$645$	−9.65685	−0.380238
$646$	0	0
$647$	−4.00000	−0.157256	−0.0786281	−	0.996904i	$-0.525054\pi$
$647$	−4.00000	−0.157256	−0.0786281	+	0.996904i	$0.525054\pi$
$648$	0	0
$649$	−24.0000	−0.942082
$650$	0	0
$651$	−19.3137	−0.756964
$652$	0	0
$653$	−21.3137	−0.834070	−0.417035	−	0.908890i	$-0.636931\pi$
$653$	−21.3137	−0.834070	−0.417035	+	0.908890i	$0.636931\pi$
$654$	0	0
$655$	−16.0000	−0.625172
$656$	0	0
$657$	15.6569	0.610832
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	22.9706	0.893451	0.446726	−	0.894671i	$-0.352590\pi$
$661$	22.9706	0.893451	0.446726	+	0.894671i	$0.352590\pi$
$662$	0	0
$663$	−3.65685	−0.142020
$664$	0	0
$665$	3.31371	0.128500
$666$	0	0
$667$	30.6274	1.18590
$668$	0	0
$669$	−5.17157	−0.199945
$670$	0	0
$671$	−16.9706	−0.655141
$672$	0	0
$673$	−17.3137	−0.667394	−0.333697	−	0.942680i	$-0.608296\pi$
$673$	−17.3137	−0.667394	−0.333697	+	0.942680i	$0.608296\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	31.9411	1.22760	0.613799	−	0.789463i	$-0.289641\pi$
$677$	31.9411	1.22760	0.613799	+	0.789463i	$0.289641\pi$
$678$	0	0
$679$	10.3431	0.396934
$680$	0	0
$681$	17.1716	0.658016
$682$	0	0
$683$	14.8284	0.567394	0.283697	−	0.958914i	$-0.408439\pi$
$683$	14.8284	0.567394	0.283697	+	0.958914i	$0.408439\pi$
$684$	0	0
$685$	18.0000	0.687745
$686$	0	0
$687$	8.34315	0.318311
$688$	0	0
$689$	−2.00000	−0.0761939
$690$	0	0
$691$	−8.20101	−0.311981	−0.155991	−	0.987759i	$-0.549857\pi$
$691$	−8.20101	−0.311981	−0.155991	+	0.987759i	$0.549857\pi$
$692$	0	0
$693$	8.00000	0.303895
$694$	0	0
$695$	16.9706	0.643730
$696$	0	0
$697$	7.31371	0.277026
$698$	0	0
$699$	10.9706	0.414945
$700$	0	0
$701$	−4.34315	−0.164038	−0.0820192	−	0.996631i	$-0.526137\pi$
$701$	−4.34315	−0.164038	−0.0820192	+	0.996631i	$0.526137\pi$
$702$	0	0
$703$	−2.34315	−0.0883734
$704$	0	0
$705$	−6.82843	−0.257173
$706$	0	0
$707$	21.6569	0.814490
$708$	0	0
$709$	−23.6569	−0.888452	−0.444226	−	0.895915i	$-0.646521\pi$
$709$	−23.6569	−0.888452	−0.444226	+	0.895915i	$0.646521\pi$
$710$	0	0
$711$	−2.34315	−0.0878748
$712$	0	0
$713$	−27.3137	−1.02291
$714$	0	0
$715$	2.82843	0.105777
$716$	0	0
$717$	7.51472	0.280642
$718$	0	0
$719$	−19.3137	−0.720280	−0.360140	−	0.932898i	$-0.617271\pi$
$719$	−19.3137	−0.720280	−0.360140	+	0.932898i	$0.617271\pi$
$720$	0	0
$721$	−27.3137	−1.01722
$722$	0	0
$723$	21.3137	0.792665
$724$	0	0
$725$	−7.65685	−0.284368
$726$	0	0
$727$	−28.9706	−1.07446	−0.537229	−	0.843436i	$-0.680529\pi$
$727$	−28.9706	−1.07446	−0.537229	+	0.843436i	$0.680529\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−35.3137	−1.30612
$732$	0	0
$733$	18.6863	0.690194	0.345097	−	0.938567i	$-0.387846\pi$
$733$	18.6863	0.690194	0.345097	+	0.938567i	$0.387846\pi$
$734$	0	0
$735$	−1.00000	−0.0368856
$736$	0	0
$737$	8.00000	0.294684
$738$	0	0
$739$	40.7696	1.49973	0.749866	−	0.661590i	$-0.230118\pi$
$739$	40.7696	1.49973	0.749866	+	0.661590i	$0.230118\pi$
$740$	0	0
$741$	1.17157	0.0430388
$742$	0	0
$743$	23.7990	0.873100	0.436550	−	0.899680i	$-0.356200\pi$
$743$	23.7990	0.873100	0.436550	+	0.899680i	$0.356200\pi$
$744$	0	0
$745$	9.31371	0.341228
$746$	0	0
$747$	1.17157	0.0428656
$748$	0	0
$749$	−43.3137	−1.58265
$750$	0	0
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	0	0
$753$	8.97056	0.326905
$754$	0	0
$755$	6.82843	0.248512
$756$	0	0
$757$	9.31371	0.338512	0.169256	−	0.985572i	$-0.445863\pi$
$757$	9.31371	0.338512	0.169256	+	0.985572i	$0.445863\pi$
$758$	0	0
$759$	11.3137	0.410662
$760$	0	0
$761$	41.3137	1.49762	0.748810	−	0.662784i	$-0.230625\pi$
$761$	41.3137	1.49762	0.748810	+	0.662784i	$0.230625\pi$
$762$	0	0
$763$	12.2843	0.444720
$764$	0	0
$765$	3.65685	0.132214
$766$	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
$771$	−14.9706	−0.539152
$772$	0	0
$773$	−9.31371	−0.334991	−0.167495	−	0.985873i	$-0.553568\pi$
$773$	−9.31371	−0.334991	−0.167495	+	0.985873i	$0.553568\pi$
$774$	0	0
$775$	6.82843	0.245284
$776$	0	0
$777$	5.65685	0.202939
$778$	0	0
$779$	−2.34315	−0.0839519
$780$	0	0
$781$	14.6274	0.523410
$782$	0	0
$783$	−7.65685	−0.273634
$784$	0	0
$785$	−14.0000	−0.499681
$786$	0	0
$787$	4.20101	0.149750	0.0748749	−	0.997193i	$-0.476144\pi$
$787$	4.20101	0.149750	0.0748749	+	0.997193i	$0.476144\pi$
$788$	0	0
$789$	−1.65685	−0.0589856
$790$	0	0
$791$	0.970563	0.0345092
$792$	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
$799$	−24.9706	−0.883395
$800$	0	0
$801$	17.3137	0.611750
$802$	0	0
$803$	−44.2843	−1.56276
$804$	0	0
$805$	−11.3137	−0.398756
$806$	0	0
$807$	3.65685	0.128727
$808$	0	0
$809$	−10.6863	−0.375710	−0.187855	−	0.982197i	$-0.560154\pi$
$809$	−10.6863	−0.375710	−0.187855	+	0.982197i	$0.560154\pi$
$810$	0	0
$811$	38.4264	1.34933	0.674667	−	0.738122i	$-0.264287\pi$
$811$	38.4264	1.34933	0.674667	+	0.738122i	$0.264287\pi$
$812$	0	0
$813$	12.4853	0.437878
$814$	0	0
$815$	−22.1421	−0.775605
$816$	0	0
$817$	11.3137	0.395817
$818$	0	0
$819$	−2.82843	−0.0988332
$820$	0	0
$821$	−9.31371	−0.325051	−0.162525	−	0.986704i	$-0.551964\pi$
$821$	−9.31371	−0.325051	−0.162525	+	0.986704i	$0.551964\pi$
$822$	0	0
$823$	−22.3431	−0.778833	−0.389417	−	0.921062i	$-0.627323\pi$
$823$	−22.3431	−0.778833	−0.389417	+	0.921062i	$0.627323\pi$
$824$	0	0
$825$	−2.82843	−0.0984732
$826$	0	0
$827$	−4.48528	−0.155969	−0.0779843	−	0.996955i	$-0.524848\pi$
$827$	−4.48528	−0.155969	−0.0779843	+	0.996955i	$0.524848\pi$
$828$	0	0
$829$	12.6274	0.438568	0.219284	−	0.975661i	$-0.429628\pi$
$829$	12.6274	0.438568	0.219284	+	0.975661i	$0.429628\pi$
$830$	0	0
$831$	−6.68629	−0.231945
$832$	0	0
$833$	−3.65685	−0.126702
$834$	0	0
$835$	4.48528	0.155220
$836$	0	0
$837$	6.82843	0.236025
$838$	0	0
$839$	17.8579	0.616522	0.308261	−	0.951302i	$-0.400253\pi$
$839$	17.8579	0.616522	0.308261	+	0.951302i	$0.400253\pi$
$840$	0	0
$841$	29.6274	1.02164
$842$	0	0
$843$	20.6274	0.710446
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	8.48528	0.291558
$848$	0	0
$849$	6.34315	0.217696
$850$	0	0
$851$	8.00000	0.274236
$852$	0	0
$853$	−47.2548	−1.61797	−0.808987	−	0.587826i	$-0.799984\pi$
$853$	−47.2548	−1.61797	−0.808987	+	0.587826i	$0.799984\pi$
$854$	0	0
$855$	−1.17157	−0.0400669
$856$	0	0
$857$	−0.343146	−0.0117216	−0.00586082	−	0.999983i	$-0.501866\pi$
$857$	−0.343146	−0.0117216	−0.00586082	+	0.999983i	$0.501866\pi$
$858$	0	0
$859$	18.3431	0.625860	0.312930	−	0.949776i	$-0.398689\pi$
$859$	18.3431	0.625860	0.312930	+	0.949776i	$0.398689\pi$
$860$	0	0
$861$	5.65685	0.192785
$862$	0	0
$863$	27.1127	0.922927	0.461463	−	0.887159i	$-0.347325\pi$
$863$	27.1127	0.922927	0.461463	+	0.887159i	$0.347325\pi$
$864$	0	0
$865$	−20.6274	−0.701353
$866$	0	0
$867$	−3.62742	−0.123194
$868$	0	0
$869$	6.62742	0.224820
$870$	0	0
$871$	−2.82843	−0.0958376
$872$	0	0
$873$	−3.65685	−0.123766
$874$	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
$879$	13.3137	0.449060
$880$	0	0
$881$	43.9411	1.48041	0.740207	−	0.672379i	$-0.234727\pi$
$881$	43.9411	1.48041	0.740207	+	0.672379i	$0.234727\pi$
$882$	0	0
$883$	24.2843	0.817231	0.408615	−	0.912707i	$-0.366012\pi$
$883$	24.2843	0.817231	0.408615	+	0.912707i	$0.366012\pi$
$884$	0	0
$885$	−8.48528	−0.285230
$886$	0	0
$887$	−28.9706	−0.972736	−0.486368	−	0.873754i	$-0.661679\pi$
$887$	−28.9706	−0.972736	−0.486368	+	0.873754i	$0.661679\pi$
$888$	0	0
$889$	4.68629	0.157173
$890$	0	0
$891$	−2.82843	−0.0947559
$892$	0	0
$893$	8.00000	0.267710
$894$	0	0
$895$	−13.6569	−0.456498
$896$	0	0
$897$	−4.00000	−0.133556
$898$	0	0
$899$	−52.2843	−1.74378
$900$	0	0
$901$	7.31371	0.243655
$902$	0	0
$903$	−27.3137	−0.908943
$904$	0	0
$905$	5.31371	0.176634
$906$	0	0
$907$	3.02944	0.100591	0.0502954	−	0.998734i	$-0.483984\pi$
$907$	3.02944	0.100591	0.0502954	+	0.998734i	$0.483984\pi$
$908$	0	0
$909$	−7.65685	−0.253962
$910$	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
$913$	−3.31371	−0.109668
$914$	0	0
$915$	−6.00000	−0.198354
$916$	0	0
$917$	−45.2548	−1.49445
$918$	0	0
$919$	−20.2843	−0.669116	−0.334558	−	0.942375i	$-0.608587\pi$
$919$	−20.2843	−0.669116	−0.334558	+	0.942375i	$0.608587\pi$
$920$	0	0
$921$	26.8284	0.884027
$922$	0	0
$923$	−5.17157	−0.170224
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	0	0
$927$	9.65685	0.317173
$928$	0	0
$929$	28.6274	0.939235	0.469618	−	0.882870i	$-0.344392\pi$
$929$	28.6274	0.939235	0.469618	+	0.882870i	$0.344392\pi$
$930$	0	0
$931$	1.17157	0.0383968
$932$	0	0
$933$	−27.3137	−0.894211
$934$	0	0
$935$	−10.3431	−0.338257
$936$	0	0
$937$	−44.6274	−1.45791	−0.728957	−	0.684559i	$-0.759995\pi$
$937$	−44.6274	−1.45791	−0.728957	+	0.684559i	$0.759995\pi$
$938$	0	0
$939$	−14.0000	−0.456873
$940$	0	0
$941$	−12.6274	−0.411642	−0.205821	−	0.978590i	$-0.565986\pi$
$941$	−12.6274	−0.411642	−0.205821	+	0.978590i	$0.565986\pi$
$942$	0	0
$943$	8.00000	0.260516
$944$	0	0
$945$	2.82843	0.0920087
$946$	0	0
$947$	46.8284	1.52172	0.760860	−	0.648916i	$-0.224778\pi$
$947$	46.8284	1.52172	0.760860	+	0.648916i	$0.224778\pi$
$948$	0	0
$949$	15.6569	0.508243
$950$	0	0
$951$	−25.3137	−0.820853
$952$	0	0
$953$	−11.6569	−0.377603	−0.188801	−	0.982015i	$-0.560460\pi$
$953$	−11.6569	−0.377603	−0.188801	+	0.982015i	$0.560460\pi$
$954$	0	0
$955$	−5.65685	−0.183052
$956$	0	0
$957$	21.6569	0.700067
$958$	0	0
$959$	50.9117	1.64402
$960$	0	0
$961$	15.6274	0.504110
$962$	0	0
$963$	15.3137	0.493477
$964$	0	0
$965$	−15.6569	−0.504012
$966$	0	0
$967$	7.51472	0.241657	0.120829	−	0.992673i	$-0.461445\pi$
$967$	7.51472	0.241657	0.120829	+	0.992673i	$0.461445\pi$
$968$	0	0
$969$	−4.28427	−0.137631
$970$	0	0
$971$	−12.6863	−0.407122	−0.203561	−	0.979062i	$-0.565252\pi$
$971$	−12.6863	−0.407122	−0.203561	+	0.979062i	$0.565252\pi$
$972$	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
$979$	−48.9706	−1.56511
$980$	0	0
$981$	−4.34315	−0.138666
$982$	0	0
$983$	24.7696	0.790026	0.395013	−	0.918676i	$-0.370740\pi$
$983$	24.7696	0.790026	0.395013	+	0.918676i	$0.370740\pi$
$984$	0	0
$985$	22.0000	0.700978
$986$	0	0
$987$	−19.3137	−0.614762
$988$	0	0
$989$	−38.6274	−1.22828
$990$	0	0
$991$	−24.9706	−0.793216	−0.396608	−	0.917988i	$-0.629813\pi$
$991$	−24.9706	−0.793216	−0.396608	+	0.917988i	$0.629813\pi$
$992$	0	0
$993$	−10.1421	−0.321851
$994$	0	0
$995$	0	0
$996$	0	0
$997$	2.68629	0.0850757	0.0425379	−	0.999095i	$-0.486456\pi$
$997$	2.68629	0.0850757	0.0425379	+	0.999095i	$0.486456\pi$
$998$	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.br.1.1	yes	2
4.3	odd	2		6240.2.a.bh.1.2	✓	2

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

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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} -2.82843 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{5} -2.82843 q^{7} +1.00000 q^{9} -2.82843 q^{11} +1.00000 q^{13} -1.00000 q^{15} -3.65685 q^{17} +1.17157 q^{19} -2.82843 q^{21} -4.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -7.65685 q^{29} +6.82843 q^{31} -2.82843 q^{33} +2.82843 q^{35} -2.00000 q^{37} +1.00000 q^{39} -2.00000 q^{41} +9.65685 q^{43} -1.00000 q^{45} +6.82843 q^{47} +1.00000 q^{49} -3.65685 q^{51} -2.00000 q^{53} +2.82843 q^{55} +1.17157 q^{57} +8.48528 q^{59} +6.00000 q^{61} -2.82843 q^{63} -1.00000 q^{65} -2.82843 q^{67} -4.00000 q^{69} -5.17157 q^{71} +15.6569 q^{73} +1.00000 q^{75} +8.00000 q^{77} -2.34315 q^{79} +1.00000 q^{81} +1.17157 q^{83} +3.65685 q^{85} -7.65685 q^{87} +17.3137 q^{89} -2.82843 q^{91} +6.82843 q^{93} -1.17157 q^{95} -3.65685 q^{97} -2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{9}+O(q^{10}) \) 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

L-functions

Modular forms

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Database

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