LMFDB - Embedded newform 2280.2.a.l.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2280,2,Mod(1,2280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2280, 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("2280.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2280 = 2^{3} \cdot 3 \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2280.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:	$18.2058916609$
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			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	2280.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} +2.82843 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.00000 q^{5} +2.82843 q^{7} +1.00000 q^{9} -2.82843 q^{11} -4.82843 q^{13} +1.00000 q^{15} +3.65685 q^{17} -1.00000 q^{19} -2.82843 q^{21} +4.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +4.82843 q^{29} +2.82843 q^{33} -2.82843 q^{35} +0.828427 q^{37} +4.82843 q^{39} +4.82843 q^{41} -2.82843 q^{43} -1.00000 q^{45} +4.00000 q^{47} +1.00000 q^{49} -3.65685 q^{51} -2.00000 q^{53} +2.82843 q^{55} +1.00000 q^{57} +8.00000 q^{59} -13.3137 q^{61} +2.82843 q^{63} +4.82843 q^{65} +9.65685 q^{67} -4.00000 q^{69} +8.00000 q^{71} -11.6569 q^{73} -1.00000 q^{75} -8.00000 q^{77} +2.34315 q^{79} +1.00000 q^{81} +8.00000 q^{83} -3.65685 q^{85} -4.82843 q^{87} +12.8284 q^{89} -13.6569 q^{91} +1.00000 q^{95} +0.828427 q^{97} -2.82843 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 4 q^{13} + 2 q^{15} - 4 q^{17} - 2 q^{19} + 8 q^{23} + 2 q^{25} - 2 q^{27} + 4 q^{29} - 4 q^{37} + 4 q^{39} + 4 q^{41} - 2 q^{45} + 8 q^{47} + 2 q^{49} + 4 q^{51} - 4 q^{53} + 2 q^{57} + 16 q^{59} - 4 q^{61} + 4 q^{65} + 8 q^{67} - 8 q^{69} + 16 q^{71} - 12 q^{73} - 2 q^{75} - 16 q^{77} + 16 q^{79} + 2 q^{81} + 16 q^{83} + 4 q^{85} - 4 q^{87} + 20 q^{89} - 16 q^{91} + 2 q^{95} - 4 q^{97}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^9 - 4 * q^13 + 2 * q^15 - 4 * q^17 - 2 * q^19 + 8 * q^23 + 2 * q^25 - 2 * q^27 + 4 * q^29 - 4 * q^37 + 4 * q^39 + 4 * q^41 - 2 * q^45 + 8 * q^47 + 2 * q^49 + 4 * q^51 - 4 * q^53 + 2 * q^57 + 16 * q^59 - 4 * q^61 + 4 * q^65 + 8 * q^67 - 8 * q^69 + 16 * q^71 - 12 * q^73 - 2 * q^75 - 16 * q^77 + 16 * q^79 + 2 * q^81 + 16 * q^83 + 4 * q^85 - 4 * q^87 + 20 * q^89 - 16 * q^91 + 2 * 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	$-0.320491\pi$
$7$	2.82843	1.06904	0.534522	+	0.845154i	$0.320491\pi$
$8$	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
$13$	−4.82843	−1.33916	−0.669582	−	0.742738i	$-0.733527\pi$
$13$	−4.82843	−1.33916	−0.669582	+	0.742738i	$0.733527\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	3.65685	0.886917	0.443459	−	0.896295i	$-0.353751\pi$
$17$	3.65685	0.886917	0.443459	+	0.896295i	$0.353751\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	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
$29$	4.82843	0.896616	0.448308	−	0.893879i	$-0.352027\pi$
$29$	4.82843	0.896616	0.448308	+	0.893879i	$0.352027\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	2.82843	0.492366
$34$	0	0
$35$	−2.82843	−0.478091
$36$	0	0
$37$	0.828427	0.136193	0.0680963	−	0.997679i	$-0.478307\pi$
$37$	0.828427	0.136193	0.0680963	+	0.997679i	$0.478307\pi$
$38$	0	0
$39$	4.82843	0.773167
$40$	0	0
$41$	4.82843	0.754074	0.377037	−	0.926198i	$-0.376943\pi$
$41$	4.82843	0.754074	0.377037	+	0.926198i	$0.376943\pi$
$42$	0	0
$43$	−2.82843	−0.431331	−0.215666	−	0.976467i	$-0.569192\pi$
$43$	−2.82843	−0.431331	−0.215666	+	0.976467i	$0.569192\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	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
$57$	1.00000	0.132453
$58$	0	0
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	−13.3137	−1.70465	−0.852323	−	0.523016i	$-0.824807\pi$
$61$	−13.3137	−1.70465	−0.852323	+	0.523016i	$0.824807\pi$
$62$	0	0
$63$	2.82843	0.356348
$64$	0	0
$65$	4.82843	0.598893
$66$	0	0
$67$	9.65685	1.17977	0.589886	−	0.807486i	$-0.299173\pi$
$67$	9.65685	1.17977	0.589886	+	0.807486i	$0.299173\pi$
$68$	0	0
$69$	−4.00000	−0.481543
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	−11.6569	−1.36433	−0.682166	−	0.731198i	$-0.738962\pi$
$73$	−11.6569	−1.36433	−0.682166	+	0.731198i	$0.738962\pi$
$74$	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	$-0.457920\pi$
$79$	2.34315	0.263624	0.131812	+	0.991275i	$0.457920\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	8.00000	0.878114	0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000	0.878114	0.439057	+	0.898459i	$0.355313\pi$
$84$	0	0
$85$	−3.65685	−0.396642
$86$	0	0
$87$	−4.82843	−0.517662
$88$	0	0
$89$	12.8284	1.35981	0.679905	−	0.733300i	$-0.262021\pi$
$89$	12.8284	1.35981	0.679905	+	0.733300i	$0.262021\pi$
$90$	0	0
$91$	−13.6569	−1.43163
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	0.828427	0.0841140	0.0420570	−	0.999115i	$-0.486609\pi$
$97$	0.828427	0.0841140	0.0420570	+	0.999115i	$0.486609\pi$
$98$	0	0
$99$	−2.82843	−0.284268
$100$	0	0
$101$	7.65685	0.761885	0.380943	−	0.924599i	$-0.375599\pi$
$101$	7.65685	0.761885	0.380943	+	0.924599i	$0.375599\pi$
$102$	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
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	0	0
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	0	0
$111$	−0.828427	−0.0786308
$112$	0	0
$113$	−3.65685	−0.344008	−0.172004	−	0.985096i	$-0.555024\pi$
$113$	−3.65685	−0.344008	−0.172004	+	0.985096i	$0.555024\pi$
$114$	0	0
$115$	−4.00000	−0.373002
$116$	0	0
$117$	−4.82843	−0.446388
$118$	0	0
$119$	10.3431	0.948155
$120$	0	0
$121$	−3.00000	−0.272727
$122$	0	0
$123$	−4.82843	−0.435365
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	0	0
$129$	2.82843	0.249029
$130$	0	0
$131$	10.8284	0.946084	0.473042	−	0.881040i	$-0.343156\pi$
$131$	10.8284	0.946084	0.473042	+	0.881040i	$0.343156\pi$
$132$	0	0
$133$	−2.82843	−0.245256
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−10.9706	−0.937278	−0.468639	−	0.883390i	$-0.655256\pi$
$137$	−10.9706	−0.937278	−0.468639	+	0.883390i	$0.655256\pi$
$138$	0	0
$139$	9.65685	0.819084	0.409542	−	0.912291i	$-0.365689\pi$
$139$	9.65685	0.819084	0.409542	+	0.912291i	$0.365689\pi$
$140$	0	0
$141$	−4.00000	−0.336861
$142$	0	0
$143$	13.6569	1.14204
$144$	0	0
$145$	−4.82843	−0.400979
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	4.34315	0.355804	0.177902	−	0.984048i	$-0.443069\pi$
$149$	4.34315	0.355804	0.177902	+	0.984048i	$0.443069\pi$
$150$	0	0
$151$	19.3137	1.57173	0.785864	−	0.618400i	$-0.212219\pi$
$151$	19.3137	1.57173	0.785864	+	0.618400i	$0.212219\pi$
$152$	0	0
$153$	3.65685	0.295639
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	0	0
$159$	2.00000	0.158610
$160$	0	0
$161$	11.3137	0.891645
$162$	0	0
$163$	2.82843	0.221540	0.110770	−	0.993846i	$-0.464668\pi$
$163$	2.82843	0.221540	0.110770	+	0.993846i	$0.464668\pi$
$164$	0	0
$165$	−2.82843	−0.220193
$166$	0	0
$167$	5.65685	0.437741	0.218870	−	0.975754i	$-0.429763\pi$
$167$	5.65685	0.437741	0.218870	+	0.975754i	$0.429763\pi$
$168$	0	0
$169$	10.3137	0.793362
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	0.343146	0.0260889	0.0130444	−	0.999915i	$-0.495848\pi$
$173$	0.343146	0.0260889	0.0130444	+	0.999915i	$0.495848\pi$
$174$	0	0
$175$	2.82843	0.213809
$176$	0	0
$177$	−8.00000	−0.601317
$178$	0	0
$179$	11.3137	0.845626	0.422813	−	0.906217i	$-0.361043\pi$
$179$	11.3137	0.845626	0.422813	+	0.906217i	$0.361043\pi$
$180$	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
$183$	13.3137	0.984178
$184$	0	0
$185$	−0.828427	−0.0609072
$186$	0	0
$187$	−10.3431	−0.756366
$188$	0	0
$189$	−2.82843	−0.205738
$190$	0	0
$191$	13.1716	0.953062	0.476531	−	0.879158i	$-0.341894\pi$
$191$	13.1716	0.953062	0.476531	+	0.879158i	$0.341894\pi$
$192$	0	0
$193$	25.7990	1.85705	0.928526	−	0.371267i	$-0.121077\pi$
$193$	25.7990	1.85705	0.928526	+	0.371267i	$0.121077\pi$
$194$	0	0
$195$	−4.82843	−0.345771
$196$	0	0
$197$	−25.3137	−1.80353	−0.901764	−	0.432230i	$-0.857727\pi$
$197$	−25.3137	−1.80353	−0.901764	+	0.432230i	$0.857727\pi$
$198$	0	0
$199$	−24.9706	−1.77012	−0.885058	−	0.465481i	$-0.845882\pi$
$199$	−24.9706	−1.77012	−0.885058	+	0.465481i	$0.845882\pi$
$200$	0	0
$201$	−9.65685	−0.681142
$202$	0	0
$203$	13.6569	0.958523
$204$	0	0
$205$	−4.82843	−0.337232
$206$	0	0
$207$	4.00000	0.278019
$208$	0	0
$209$	2.82843	0.195646
$210$	0	0
$211$	26.6274	1.83311	0.916553	−	0.399912i	$-0.130959\pi$
$211$	26.6274	1.83311	0.916553	+	0.399912i	$0.130959\pi$
$212$	0	0
$213$	−8.00000	−0.548151
$214$	0	0
$215$	2.82843	0.192897
$216$	0	0
$217$	0	0
$218$	0	0
$219$	11.6569	0.787697
$220$	0	0
$221$	−17.6569	−1.18773
$222$	0	0
$223$	−20.9706	−1.40429	−0.702146	−	0.712033i	$-0.747775\pi$
$223$	−20.9706	−1.40429	−0.702146	+	0.712033i	$0.747775\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−9.65685	−0.640948	−0.320474	−	0.947257i	$-0.603842\pi$
$227$	−9.65685	−0.640948	−0.320474	+	0.947257i	$0.603842\pi$
$228$	0	0
$229$	20.6274	1.36310	0.681549	−	0.731772i	$-0.261307\pi$
$229$	20.6274	1.36310	0.681549	+	0.731772i	$0.261307\pi$
$230$	0	0
$231$	8.00000	0.526361
$232$	0	0
$233$	11.6569	0.763666	0.381833	−	0.924231i	$-0.375293\pi$
$233$	11.6569	0.763666	0.381833	+	0.924231i	$0.375293\pi$
$234$	0	0
$235$	−4.00000	−0.260931
$236$	0	0
$237$	−2.34315	−0.152204
$238$	0	0
$239$	−11.7990	−0.763213	−0.381607	−	0.924325i	$-0.624629\pi$
$239$	−11.7990	−0.763213	−0.381607	+	0.924325i	$0.624629\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	4.82843	0.307225
$248$	0	0
$249$	−8.00000	−0.506979
$250$	0	0
$251$	8.48528	0.535586	0.267793	−	0.963476i	$-0.413706\pi$
$251$	8.48528	0.535586	0.267793	+	0.963476i	$0.413706\pi$
$252$	0	0
$253$	−11.3137	−0.711287
$254$	0	0
$255$	3.65685	0.229001
$256$	0	0
$257$	−22.9706	−1.43286	−0.716432	−	0.697657i	$-0.754226\pi$
$257$	−22.9706	−1.43286	−0.716432	+	0.697657i	$0.754226\pi$
$258$	0	0
$259$	2.34315	0.145596
$260$	0	0
$261$	4.82843	0.298872
$262$	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
$267$	−12.8284	−0.785087
$268$	0	0
$269$	1.51472	0.0923540	0.0461770	−	0.998933i	$-0.485296\pi$
$269$	1.51472	0.0923540	0.0461770	+	0.998933i	$0.485296\pi$
$270$	0	0
$271$	−16.9706	−1.03089	−0.515444	−	0.856923i	$-0.672373\pi$
$271$	−16.9706	−1.03089	−0.515444	+	0.856923i	$0.672373\pi$
$272$	0	0
$273$	13.6569	0.826550
$274$	0	0
$275$	−2.82843	−0.170561
$276$	0	0
$277$	−14.0000	−0.841178	−0.420589	−	0.907251i	$-0.638177\pi$
$277$	−14.0000	−0.841178	−0.420589	+	0.907251i	$0.638177\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−20.1421	−1.20158	−0.600790	−	0.799407i	$-0.705147\pi$
$281$	−20.1421	−1.20158	−0.600790	+	0.799407i	$0.705147\pi$
$282$	0	0
$283$	−8.48528	−0.504398	−0.252199	−	0.967675i	$-0.581154\pi$
$283$	−8.48528	−0.504398	−0.252199	+	0.967675i	$0.581154\pi$
$284$	0	0
$285$	−1.00000	−0.0592349
$286$	0	0
$287$	13.6569	0.806139
$288$	0	0
$289$	−3.62742	−0.213377
$290$	0	0
$291$	−0.828427	−0.0485633
$292$	0	0
$293$	3.65685	0.213636	0.106818	−	0.994279i	$-0.465934\pi$
$293$	3.65685	0.213636	0.106818	+	0.994279i	$0.465934\pi$
$294$	0	0
$295$	−8.00000	−0.465778
$296$	0	0
$297$	2.82843	0.164122
$298$	0	0
$299$	−19.3137	−1.11694
$300$	0	0
$301$	−8.00000	−0.461112
$302$	0	0
$303$	−7.65685	−0.439875
$304$	0	0
$305$	13.3137	0.762341
$306$	0	0
$307$	−17.6569	−1.00773	−0.503865	−	0.863782i	$-0.668089\pi$
$307$	−17.6569	−1.00773	−0.503865	+	0.863782i	$0.668089\pi$
$308$	0	0
$309$	−9.65685	−0.549359
$310$	0	0
$311$	1.85786	0.105350	0.0526749	−	0.998612i	$-0.483225\pi$
$311$	1.85786	0.105350	0.0526749	+	0.998612i	$0.483225\pi$
$312$	0	0
$313$	2.00000	0.113047	0.0565233	−	0.998401i	$-0.481998\pi$
$313$	2.00000	0.113047	0.0565233	+	0.998401i	$0.481998\pi$
$314$	0	0
$315$	−2.82843	−0.159364
$316$	0	0
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	0	0
$319$	−13.6569	−0.764637
$320$	0	0
$321$	−4.00000	−0.223258
$322$	0	0
$323$	−3.65685	−0.203473
$324$	0	0
$325$	−4.82843	−0.267833
$326$	0	0
$327$	−14.0000	−0.774202
$328$	0	0
$329$	11.3137	0.623745
$330$	0	0
$331$	−14.3431	−0.788371	−0.394185	−	0.919031i	$-0.628973\pi$
$331$	−14.3431	−0.788371	−0.394185	+	0.919031i	$0.628973\pi$
$332$	0	0
$333$	0.828427	0.0453975
$334$	0	0
$335$	−9.65685	−0.527610
$336$	0	0
$337$	−2.48528	−0.135382	−0.0676910	−	0.997706i	$-0.521563\pi$
$337$	−2.48528	−0.135382	−0.0676910	+	0.997706i	$0.521563\pi$
$338$	0	0
$339$	3.65685	0.198613
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−16.9706	−0.916324
$344$	0	0
$345$	4.00000	0.215353
$346$	0	0
$347$	27.3137	1.46628	0.733138	−	0.680080i	$-0.238055\pi$
$347$	27.3137	1.46628	0.733138	+	0.680080i	$0.238055\pi$
$348$	0	0
$349$	17.3137	0.926782	0.463391	−	0.886154i	$-0.346633\pi$
$349$	17.3137	0.926782	0.463391	+	0.886154i	$0.346633\pi$
$350$	0	0
$351$	4.82843	0.257722
$352$	0	0
$353$	−31.6569	−1.68492	−0.842462	−	0.538756i	$-0.818895\pi$
$353$	−31.6569	−1.68492	−0.842462	+	0.538756i	$0.818895\pi$
$354$	0	0
$355$	−8.00000	−0.424596
$356$	0	0
$357$	−10.3431	−0.547417
$358$	0	0
$359$	8.48528	0.447836	0.223918	−	0.974608i	$-0.428115\pi$
$359$	8.48528	0.447836	0.223918	+	0.974608i	$0.428115\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	3.00000	0.157459
$364$	0	0
$365$	11.6569	0.610148
$366$	0	0
$367$	−22.1421	−1.15581	−0.577905	−	0.816104i	$-0.696130\pi$
$367$	−22.1421	−1.15581	−0.577905	+	0.816104i	$0.696130\pi$
$368$	0	0
$369$	4.82843	0.251358
$370$	0	0
$371$	−5.65685	−0.293689
$372$	0	0
$373$	11.1716	0.578442	0.289221	−	0.957262i	$-0.406604\pi$
$373$	11.1716	0.578442	0.289221	+	0.957262i	$0.406604\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−23.3137	−1.20072
$378$	0	0
$379$	−1.65685	−0.0851069	−0.0425534	−	0.999094i	$-0.513549\pi$
$379$	−1.65685	−0.0851069	−0.0425534	+	0.999094i	$0.513549\pi$
$380$	0	0
$381$	−12.0000	−0.614779
$382$	0	0
$383$	30.6274	1.56499	0.782494	−	0.622658i	$-0.213947\pi$
$383$	30.6274	1.56499	0.782494	+	0.622658i	$0.213947\pi$
$384$	0	0
$385$	8.00000	0.407718
$386$	0	0
$387$	−2.82843	−0.143777
$388$	0	0
$389$	32.6274	1.65428	0.827138	−	0.561999i	$-0.189968\pi$
$389$	32.6274	1.65428	0.827138	+	0.561999i	$0.189968\pi$
$390$	0	0
$391$	14.6274	0.739740
$392$	0	0
$393$	−10.8284	−0.546222
$394$	0	0
$395$	−2.34315	−0.117896
$396$	0	0
$397$	16.6274	0.834506	0.417253	−	0.908790i	$-0.362993\pi$
$397$	16.6274	0.834506	0.417253	+	0.908790i	$0.362993\pi$
$398$	0	0
$399$	2.82843	0.141598
$400$	0	0
$401$	7.17157	0.358131	0.179066	−	0.983837i	$-0.442693\pi$
$401$	7.17157	0.358131	0.179066	+	0.983837i	$0.442693\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−2.34315	−0.116145
$408$	0	0
$409$	−6.97056	−0.344672	−0.172336	−	0.985038i	$-0.555132\pi$
$409$	−6.97056	−0.344672	−0.172336	+	0.985038i	$0.555132\pi$
$410$	0	0
$411$	10.9706	0.541138
$412$	0	0
$413$	22.6274	1.11342
$414$	0	0
$415$	−8.00000	−0.392705
$416$	0	0
$417$	−9.65685	−0.472898
$418$	0	0
$419$	3.79899	0.185593	0.0927964	−	0.995685i	$-0.470419\pi$
$419$	3.79899	0.185593	0.0927964	+	0.995685i	$0.470419\pi$
$420$	0	0
$421$	−4.34315	−0.211672	−0.105836	−	0.994384i	$-0.533752\pi$
$421$	−4.34315	−0.211672	−0.105836	+	0.994384i	$0.533752\pi$
$422$	0	0
$423$	4.00000	0.194487
$424$	0	0
$425$	3.65685	0.177383
$426$	0	0
$427$	−37.6569	−1.82234
$428$	0	0
$429$	−13.6569	−0.659359
$430$	0	0
$431$	−0.970563	−0.0467504	−0.0233752	−	0.999727i	$-0.507441\pi$
$431$	−0.970563	−0.0467504	−0.0233752	+	0.999727i	$0.507441\pi$
$432$	0	0
$433$	−1.51472	−0.0727927	−0.0363964	−	0.999337i	$-0.511588\pi$
$433$	−1.51472	−0.0727927	−0.0363964	+	0.999337i	$0.511588\pi$
$434$	0	0
$435$	4.82843	0.231505
$436$	0	0
$437$	−4.00000	−0.191346
$438$	0	0
$439$	36.2843	1.73175	0.865877	−	0.500257i	$-0.166761\pi$
$439$	36.2843	1.73175	0.865877	+	0.500257i	$0.166761\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−13.6569	−0.648857	−0.324428	−	0.945910i	$-0.605172\pi$
$443$	−13.6569	−0.648857	−0.324428	+	0.945910i	$0.605172\pi$
$444$	0	0
$445$	−12.8284	−0.608126
$446$	0	0
$447$	−4.34315	−0.205424
$448$	0	0
$449$	17.5147	0.826571	0.413285	−	0.910602i	$-0.364381\pi$
$449$	17.5147	0.826571	0.413285	+	0.910602i	$0.364381\pi$
$450$	0	0
$451$	−13.6569	−0.643076
$452$	0	0
$453$	−19.3137	−0.907437
$454$	0	0
$455$	13.6569	0.640243
$456$	0	0
$457$	−17.3137	−0.809901	−0.404951	−	0.914339i	$-0.632711\pi$
$457$	−17.3137	−0.809901	−0.404951	+	0.914339i	$0.632711\pi$
$458$	0	0
$459$	−3.65685	−0.170687
$460$	0	0
$461$	30.2843	1.41048	0.705240	−	0.708969i	$-0.250839\pi$
$461$	30.2843	1.41048	0.705240	+	0.708969i	$0.250839\pi$
$462$	0	0
$463$	−14.1421	−0.657241	−0.328620	−	0.944462i	$-0.606584\pi$
$463$	−14.1421	−0.657241	−0.328620	+	0.944462i	$0.606584\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−10.3431	−0.478624	−0.239312	−	0.970943i	$-0.576922\pi$
$467$	−10.3431	−0.478624	−0.239312	+	0.970943i	$0.576922\pi$
$468$	0	0
$469$	27.3137	1.26123
$470$	0	0
$471$	−2.00000	−0.0921551
$472$	0	0
$473$	8.00000	0.367840
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	0	0
$479$	19.7990	0.904639	0.452319	−	0.891856i	$-0.350597\pi$
$479$	19.7990	0.904639	0.452319	+	0.891856i	$0.350597\pi$
$480$	0	0
$481$	−4.00000	−0.182384
$482$	0	0
$483$	−11.3137	−0.514792
$484$	0	0
$485$	−0.828427	−0.0376169
$486$	0	0
$487$	−6.34315	−0.287435	−0.143718	−	0.989619i	$-0.545906\pi$
$487$	−6.34315	−0.287435	−0.143718	+	0.989619i	$0.545906\pi$
$488$	0	0
$489$	−2.82843	−0.127906
$490$	0	0
$491$	7.51472	0.339135	0.169567	−	0.985519i	$-0.445763\pi$
$491$	7.51472	0.339135	0.169567	+	0.985519i	$0.445763\pi$
$492$	0	0
$493$	17.6569	0.795225
$494$	0	0
$495$	2.82843	0.127128
$496$	0	0
$497$	22.6274	1.01498
$498$	0	0
$499$	9.65685	0.432300	0.216150	−	0.976360i	$-0.430650\pi$
$499$	9.65685	0.432300	0.216150	+	0.976360i	$0.430650\pi$
$500$	0	0
$501$	−5.65685	−0.252730
$502$	0	0
$503$	−3.02944	−0.135076	−0.0675380	−	0.997717i	$-0.521514\pi$
$503$	−3.02944	−0.135076	−0.0675380	+	0.997717i	$0.521514\pi$
$504$	0	0
$505$	−7.65685	−0.340726
$506$	0	0
$507$	−10.3137	−0.458048
$508$	0	0
$509$	−34.7696	−1.54113	−0.770567	−	0.637359i	$-0.780027\pi$
$509$	−34.7696	−1.54113	−0.770567	+	0.637359i	$0.780027\pi$
$510$	0	0
$511$	−32.9706	−1.45853
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	−9.65685	−0.425532
$516$	0	0
$517$	−11.3137	−0.497576
$518$	0	0
$519$	−0.343146	−0.0150624
$520$	0	0
$521$	−16.8284	−0.737267	−0.368633	−	0.929575i	$-0.620174\pi$
$521$	−16.8284	−0.737267	−0.368633	+	0.929575i	$0.620174\pi$
$522$	0	0
$523$	−18.6274	−0.814520	−0.407260	−	0.913312i	$-0.633516\pi$
$523$	−18.6274	−0.814520	−0.407260	+	0.913312i	$0.633516\pi$
$524$	0	0
$525$	−2.82843	−0.123443
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	8.00000	0.347170
$532$	0	0
$533$	−23.3137	−1.00983
$534$	0	0
$535$	−4.00000	−0.172935
$536$	0	0
$537$	−11.3137	−0.488223
$538$	0	0
$539$	−2.82843	−0.121829
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	0	0
$543$	−17.3137	−0.743002
$544$	0	0
$545$	−14.0000	−0.599694
$546$	0	0
$547$	−17.6569	−0.754953	−0.377476	−	0.926019i	$-0.623208\pi$
$547$	−17.6569	−0.754953	−0.377476	+	0.926019i	$0.623208\pi$
$548$	0	0
$549$	−13.3137	−0.568215
$550$	0	0
$551$	−4.82843	−0.205698
$552$	0	0
$553$	6.62742	0.281826
$554$	0	0
$555$	0.828427	0.0351648
$556$	0	0
$557$	16.6274	0.704526	0.352263	−	0.935901i	$-0.385412\pi$
$557$	16.6274	0.704526	0.352263	+	0.935901i	$0.385412\pi$
$558$	0	0
$559$	13.6569	0.577623
$560$	0	0
$561$	10.3431	0.436688
$562$	0	0
$563$	−20.9706	−0.883804	−0.441902	−	0.897063i	$-0.645696\pi$
$563$	−20.9706	−0.883804	−0.441902	+	0.897063i	$0.645696\pi$
$564$	0	0
$565$	3.65685	0.153845
$566$	0	0
$567$	2.82843	0.118783
$568$	0	0
$569$	−1.79899	−0.0754176	−0.0377088	−	0.999289i	$-0.512006\pi$
$569$	−1.79899	−0.0754176	−0.0377088	+	0.999289i	$0.512006\pi$
$570$	0	0
$571$	4.97056	0.208012	0.104006	−	0.994577i	$-0.466834\pi$
$571$	4.97056	0.208012	0.104006	+	0.994577i	$0.466834\pi$
$572$	0	0
$573$	−13.1716	−0.550250
$574$	0	0
$575$	4.00000	0.166812
$576$	0	0
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	0	0
$579$	−25.7990	−1.07217
$580$	0	0
$581$	22.6274	0.938743
$582$	0	0
$583$	5.65685	0.234283
$584$	0	0
$585$	4.82843	0.199631
$586$	0	0
$587$	−24.9706	−1.03065	−0.515323	−	0.856996i	$-0.672328\pi$
$587$	−24.9706	−1.03065	−0.515323	+	0.856996i	$0.672328\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	25.3137	1.04127
$592$	0	0
$593$	38.9706	1.60033	0.800165	−	0.599780i	$-0.204745\pi$
$593$	38.9706	1.60033	0.800165	+	0.599780i	$0.204745\pi$
$594$	0	0
$595$	−10.3431	−0.424028
$596$	0	0
$597$	24.9706	1.02198
$598$	0	0
$599$	−32.9706	−1.34714	−0.673570	−	0.739123i	$-0.735240\pi$
$599$	−32.9706	−1.34714	−0.673570	+	0.739123i	$0.735240\pi$
$600$	0	0
$601$	−8.34315	−0.340324	−0.170162	−	0.985416i	$-0.554429\pi$
$601$	−8.34315	−0.340324	−0.170162	+	0.985416i	$0.554429\pi$
$602$	0	0
$603$	9.65685	0.393258
$604$	0	0
$605$	3.00000	0.121967
$606$	0	0
$607$	6.34315	0.257460	0.128730	−	0.991680i	$-0.458910\pi$
$607$	6.34315	0.257460	0.128730	+	0.991680i	$0.458910\pi$
$608$	0	0
$609$	−13.6569	−0.553404
$610$	0	0
$611$	−19.3137	−0.781349
$612$	0	0
$613$	−4.62742	−0.186900	−0.0934498	−	0.995624i	$-0.529789\pi$
$613$	−4.62742	−0.186900	−0.0934498	+	0.995624i	$0.529789\pi$
$614$	0	0
$615$	4.82843	0.194701
$616$	0	0
$617$	−9.02944	−0.363511	−0.181756	−	0.983344i	$-0.558178\pi$
$617$	−9.02944	−0.363511	−0.181756	+	0.983344i	$0.558178\pi$
$618$	0	0
$619$	−38.3431	−1.54114	−0.770571	−	0.637355i	$-0.780029\pi$
$619$	−38.3431	−1.54114	−0.770571	+	0.637355i	$0.780029\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	0	0
$623$	36.2843	1.45370
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−2.82843	−0.112956
$628$	0	0
$629$	3.02944	0.120792
$630$	0	0
$631$	22.6274	0.900783	0.450392	−	0.892831i	$-0.351284\pi$
$631$	22.6274	0.900783	0.450392	+	0.892831i	$0.351284\pi$
$632$	0	0
$633$	−26.6274	−1.05834
$634$	0	0
$635$	−12.0000	−0.476205
$636$	0	0
$637$	−4.82843	−0.191309
$638$	0	0
$639$	8.00000	0.316475
$640$	0	0
$641$	28.8284	1.13865	0.569327	−	0.822111i	$-0.307204\pi$
$641$	28.8284	1.13865	0.569327	+	0.822111i	$0.307204\pi$
$642$	0	0
$643$	28.7696	1.13456	0.567280	−	0.823525i	$-0.307996\pi$
$643$	28.7696	1.13456	0.567280	+	0.823525i	$0.307996\pi$
$644$	0	0
$645$	−2.82843	−0.111369
$646$	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
$649$	−22.6274	−0.888204
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−20.6274	−0.807213	−0.403607	−	0.914933i	$-0.632244\pi$
$653$	−20.6274	−0.807213	−0.403607	+	0.914933i	$0.632244\pi$
$654$	0	0
$655$	−10.8284	−0.423102
$656$	0	0
$657$	−11.6569	−0.454777
$658$	0	0
$659$	−7.02944	−0.273828	−0.136914	−	0.990583i	$-0.543718\pi$
$659$	−7.02944	−0.273828	−0.136914	+	0.990583i	$0.543718\pi$
$660$	0	0
$661$	−29.3137	−1.14017	−0.570086	−	0.821585i	$-0.693090\pi$
$661$	−29.3137	−1.14017	−0.570086	+	0.821585i	$0.693090\pi$
$662$	0	0
$663$	17.6569	0.685735
$664$	0	0
$665$	2.82843	0.109682
$666$	0	0
$667$	19.3137	0.747830
$668$	0	0
$669$	20.9706	0.810769
$670$	0	0
$671$	37.6569	1.45373
$672$	0	0
$673$	−49.1127	−1.89316	−0.946578	−	0.322476i	$-0.895485\pi$
$673$	−49.1127	−1.89316	−0.946578	+	0.322476i	$0.895485\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−2.00000	−0.0768662	−0.0384331	−	0.999261i	$-0.512237\pi$
$677$	−2.00000	−0.0768662	−0.0384331	+	0.999261i	$0.512237\pi$
$678$	0	0
$679$	2.34315	0.0899217
$680$	0	0
$681$	9.65685	0.370051
$682$	0	0
$683$	−2.62742	−0.100535	−0.0502677	−	0.998736i	$-0.516007\pi$
$683$	−2.62742	−0.100535	−0.0502677	+	0.998736i	$0.516007\pi$
$684$	0	0
$685$	10.9706	0.419164
$686$	0	0
$687$	−20.6274	−0.786985
$688$	0	0
$689$	9.65685	0.367897
$690$	0	0
$691$	−9.65685	−0.367364	−0.183682	−	0.982986i	$-0.558802\pi$
$691$	−9.65685	−0.367364	−0.183682	+	0.982986i	$0.558802\pi$
$692$	0	0
$693$	−8.00000	−0.303895
$694$	0	0
$695$	−9.65685	−0.366305
$696$	0	0
$697$	17.6569	0.668801
$698$	0	0
$699$	−11.6569	−0.440903
$700$	0	0
$701$	−34.2843	−1.29490	−0.647450	−	0.762108i	$-0.724164\pi$
$701$	−34.2843	−1.29490	−0.647450	+	0.762108i	$0.724164\pi$
$702$	0	0
$703$	−0.828427	−0.0312447
$704$	0	0
$705$	4.00000	0.150649
$706$	0	0
$707$	21.6569	0.814490
$708$	0	0
$709$	14.0000	0.525781	0.262891	−	0.964826i	$-0.415324\pi$
$709$	14.0000	0.525781	0.262891	+	0.964826i	$0.415324\pi$
$710$	0	0
$711$	2.34315	0.0878748
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−13.6569	−0.510737
$716$	0	0
$717$	11.7990	0.440641
$718$	0	0
$719$	33.4558	1.24769	0.623846	−	0.781547i	$-0.285569\pi$
$719$	33.4558	1.24769	0.623846	+	0.781547i	$0.285569\pi$
$720$	0	0
$721$	27.3137	1.01722
$722$	0	0
$723$	22.0000	0.818189
$724$	0	0
$725$	4.82843	0.179323
$726$	0	0
$727$	0.485281	0.0179981	0.00899904	−	0.999960i	$-0.497135\pi$
$727$	0.485281	0.0179981	0.00899904	+	0.999960i	$0.497135\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−10.3431	−0.382555
$732$	0	0
$733$	2.97056	0.109720	0.0548601	−	0.998494i	$-0.482529\pi$
$733$	2.97056	0.109720	0.0548601	+	0.998494i	$0.482529\pi$
$734$	0	0
$735$	1.00000	0.0368856
$736$	0	0
$737$	−27.3137	−1.00611
$738$	0	0
$739$	23.3137	0.857609	0.428804	−	0.903397i	$-0.358935\pi$
$739$	23.3137	0.857609	0.428804	+	0.903397i	$0.358935\pi$
$740$	0	0
$741$	−4.82843	−0.177377
$742$	0	0
$743$	48.0000	1.76095	0.880475	−	0.474093i	$-0.157224\pi$
$743$	48.0000	1.76095	0.880475	+	0.474093i	$0.157224\pi$
$744$	0	0
$745$	−4.34315	−0.159121
$746$	0	0
$747$	8.00000	0.292705
$748$	0	0
$749$	11.3137	0.413394
$750$	0	0
$751$	24.0000	0.875772	0.437886	−	0.899030i	$-0.355727\pi$
$751$	24.0000	0.875772	0.437886	+	0.899030i	$0.355727\pi$
$752$	0	0
$753$	−8.48528	−0.309221
$754$	0	0
$755$	−19.3137	−0.702898
$756$	0	0
$757$	31.6569	1.15059	0.575294	−	0.817947i	$-0.304888\pi$
$757$	31.6569	1.15059	0.575294	+	0.817947i	$0.304888\pi$
$758$	0	0
$759$	11.3137	0.410662
$760$	0	0
$761$	−37.5980	−1.36293	−0.681463	−	0.731853i	$-0.738656\pi$
$761$	−37.5980	−1.36293	−0.681463	+	0.731853i	$0.738656\pi$
$762$	0	0
$763$	39.5980	1.43354
$764$	0	0
$765$	−3.65685	−0.132214
$766$	0	0
$767$	−38.6274	−1.39476
$768$	0	0
$769$	−4.62742	−0.166869	−0.0834345	−	0.996513i	$-0.526589\pi$
$769$	−4.62742	−0.166869	−0.0834345	+	0.996513i	$0.526589\pi$
$770$	0	0
$771$	22.9706	0.827265
$772$	0	0
$773$	−16.6274	−0.598047	−0.299023	−	0.954246i	$-0.596661\pi$
$773$	−16.6274	−0.598047	−0.299023	+	0.954246i	$0.596661\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−2.34315	−0.0840599
$778$	0	0
$779$	−4.82843	−0.172996
$780$	0	0
$781$	−22.6274	−0.809673
$782$	0	0
$783$	−4.82843	−0.172554
$784$	0	0
$785$	−2.00000	−0.0713831
$786$	0	0
$787$	−21.9411	−0.782117	−0.391058	−	0.920366i	$-0.627891\pi$
$787$	−21.9411	−0.782117	−0.391058	+	0.920366i	$0.627891\pi$
$788$	0	0
$789$	1.65685	0.0589856
$790$	0	0
$791$	−10.3431	−0.367760
$792$	0	0
$793$	64.2843	2.28280
$794$	0	0
$795$	−2.00000	−0.0709327
$796$	0	0
$797$	−36.3431	−1.28734	−0.643670	−	0.765303i	$-0.722589\pi$
$797$	−36.3431	−1.28734	−0.643670	+	0.765303i	$0.722589\pi$
$798$	0	0
$799$	14.6274	0.517481
$800$	0	0
$801$	12.8284	0.453270
$802$	0	0
$803$	32.9706	1.16351
$804$	0	0
$805$	−11.3137	−0.398756
$806$	0	0
$807$	−1.51472	−0.0533206
$808$	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
$811$	−41.2548	−1.44865	−0.724327	−	0.689457i	$-0.757849\pi$
$811$	−41.2548	−1.44865	−0.724327	+	0.689457i	$0.757849\pi$
$812$	0	0
$813$	16.9706	0.595184
$814$	0	0
$815$	−2.82843	−0.0990755
$816$	0	0
$817$	2.82843	0.0989541
$818$	0	0
$819$	−13.6569	−0.477209
$820$	0	0
$821$	39.6569	1.38403	0.692017	−	0.721881i	$-0.256722\pi$
$821$	39.6569	1.38403	0.692017	+	0.721881i	$0.256722\pi$
$822$	0	0
$823$	13.1716	0.459132	0.229566	−	0.973293i	$-0.426269\pi$
$823$	13.1716	0.459132	0.229566	+	0.973293i	$0.426269\pi$
$824$	0	0
$825$	2.82843	0.0984732
$826$	0	0
$827$	−9.65685	−0.335802	−0.167901	−	0.985804i	$-0.553699\pi$
$827$	−9.65685	−0.335802	−0.167901	+	0.985804i	$0.553699\pi$
$828$	0	0
$829$	46.0000	1.59765	0.798823	−	0.601566i	$-0.205456\pi$
$829$	46.0000	1.59765	0.798823	+	0.601566i	$0.205456\pi$
$830$	0	0
$831$	14.0000	0.485655
$832$	0	0
$833$	3.65685	0.126702
$834$	0	0
$835$	−5.65685	−0.195764
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−38.6274	−1.33357	−0.666783	−	0.745252i	$-0.732329\pi$
$839$	−38.6274	−1.33357	−0.666783	+	0.745252i	$0.732329\pi$
$840$	0	0
$841$	−5.68629	−0.196079
$842$	0	0
$843$	20.1421	0.693732
$844$	0	0
$845$	−10.3137	−0.354802
$846$	0	0
$847$	−8.48528	−0.291558
$848$	0	0
$849$	8.48528	0.291214
$850$	0	0
$851$	3.31371	0.113592
$852$	0	0
$853$	−2.28427	−0.0782120	−0.0391060	−	0.999235i	$-0.512451\pi$
$853$	−2.28427	−0.0782120	−0.0391060	+	0.999235i	$0.512451\pi$
$854$	0	0
$855$	1.00000	0.0341993
$856$	0	0
$857$	16.6274	0.567982	0.283991	−	0.958827i	$-0.408341\pi$
$857$	16.6274	0.567982	0.283991	+	0.958827i	$0.408341\pi$
$858$	0	0
$859$	−47.3137	−1.61432	−0.807161	−	0.590331i	$-0.798997\pi$
$859$	−47.3137	−1.61432	−0.807161	+	0.590331i	$0.798997\pi$
$860$	0	0
$861$	−13.6569	−0.465424
$862$	0	0
$863$	−6.62742	−0.225600	−0.112800	−	0.993618i	$-0.535982\pi$
$863$	−6.62742	−0.225600	−0.112800	+	0.993618i	$0.535982\pi$
$864$	0	0
$865$	−0.343146	−0.0116673
$866$	0	0
$867$	3.62742	0.123194
$868$	0	0
$869$	−6.62742	−0.224820
$870$	0	0
$871$	−46.6274	−1.57991
$872$	0	0
$873$	0.828427	0.0280380
$874$	0	0
$875$	−2.82843	−0.0956183
$876$	0	0
$877$	3.17157	0.107096	0.0535482	−	0.998565i	$-0.482947\pi$
$877$	3.17157	0.107096	0.0535482	+	0.998565i	$0.482947\pi$
$878$	0	0
$879$	−3.65685	−0.123343
$880$	0	0
$881$	−50.2843	−1.69412	−0.847060	−	0.531497i	$-0.821630\pi$
$881$	−50.2843	−1.69412	−0.847060	+	0.531497i	$0.821630\pi$
$882$	0	0
$883$	25.4558	0.856657	0.428329	−	0.903623i	$-0.359103\pi$
$883$	25.4558	0.856657	0.428329	+	0.903623i	$0.359103\pi$
$884$	0	0
$885$	8.00000	0.268917
$886$	0	0
$887$	38.6274	1.29698	0.648491	−	0.761222i	$-0.275400\pi$
$887$	38.6274	1.29698	0.648491	+	0.761222i	$0.275400\pi$
$888$	0	0
$889$	33.9411	1.13835
$890$	0	0
$891$	−2.82843	−0.0947559
$892$	0	0
$893$	−4.00000	−0.133855
$894$	0	0
$895$	−11.3137	−0.378176
$896$	0	0
$897$	19.3137	0.644866
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−7.31371	−0.243655
$902$	0	0
$903$	8.00000	0.266223
$904$	0	0
$905$	−17.3137	−0.575527
$906$	0	0
$907$	23.3137	0.774119	0.387059	−	0.922055i	$-0.373491\pi$
$907$	23.3137	0.774119	0.387059	+	0.922055i	$0.373491\pi$
$908$	0	0
$909$	7.65685	0.253962
$910$	0	0
$911$	54.6274	1.80989	0.904944	−	0.425532i	$-0.139913\pi$
$911$	54.6274	1.80989	0.904944	+	0.425532i	$0.139913\pi$
$912$	0	0
$913$	−22.6274	−0.748858
$914$	0	0
$915$	−13.3137	−0.440138
$916$	0	0
$917$	30.6274	1.01141
$918$	0	0
$919$	−11.3137	−0.373205	−0.186602	−	0.982436i	$-0.559748\pi$
$919$	−11.3137	−0.373205	−0.186602	+	0.982436i	$0.559748\pi$
$920$	0	0
$921$	17.6569	0.581813
$922$	0	0
$923$	−38.6274	−1.27144
$924$	0	0
$925$	0.828427	0.0272385
$926$	0	0
$927$	9.65685	0.317173
$928$	0	0
$929$	39.2548	1.28791	0.643955	−	0.765064i	$-0.277293\pi$
$929$	39.2548	1.28791	0.643955	+	0.765064i	$0.277293\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	−1.85786	−0.0608237
$934$	0	0
$935$	10.3431	0.338257
$936$	0	0
$937$	−3.65685	−0.119464	−0.0597321	−	0.998214i	$-0.519025\pi$
$937$	−3.65685	−0.119464	−0.0597321	+	0.998214i	$0.519025\pi$
$938$	0	0
$939$	−2.00000	−0.0652675
$940$	0	0
$941$	38.7696	1.26385	0.631926	−	0.775029i	$-0.282265\pi$
$941$	38.7696	1.26385	0.631926	+	0.775029i	$0.282265\pi$
$942$	0	0
$943$	19.3137	0.628941
$944$	0	0
$945$	2.82843	0.0920087
$946$	0	0
$947$	14.6274	0.475327	0.237664	−	0.971348i	$-0.423618\pi$
$947$	14.6274	0.475327	0.237664	+	0.971348i	$0.423618\pi$
$948$	0	0
$949$	56.2843	1.82706
$950$	0	0
$951$	18.0000	0.583690
$952$	0	0
$953$	−8.34315	−0.270261	−0.135131	−	0.990828i	$-0.543145\pi$
$953$	−8.34315	−0.270261	−0.135131	+	0.990828i	$0.543145\pi$
$954$	0	0
$955$	−13.1716	−0.426222
$956$	0	0
$957$	13.6569	0.441463
$958$	0	0
$959$	−31.0294	−1.00199
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	4.00000	0.128898
$964$	0	0
$965$	−25.7990	−0.830499
$966$	0	0
$967$	−49.4558	−1.59039	−0.795196	−	0.606352i	$-0.792632\pi$
$967$	−49.4558	−1.59039	−0.795196	+	0.606352i	$0.792632\pi$
$968$	0	0
$969$	3.65685	0.117475
$970$	0	0
$971$	−50.9117	−1.63383	−0.816917	−	0.576755i	$-0.804319\pi$
$971$	−50.9117	−1.63383	−0.816917	+	0.576755i	$0.804319\pi$
$972$	0	0
$973$	27.3137	0.875637
$974$	0	0
$975$	4.82843	0.154633
$976$	0	0
$977$	51.9411	1.66174	0.830872	−	0.556464i	$-0.187842\pi$
$977$	51.9411	1.66174	0.830872	+	0.556464i	$0.187842\pi$
$978$	0	0
$979$	−36.2843	−1.15965
$980$	0	0
$981$	14.0000	0.446986
$982$	0	0
$983$	−29.6569	−0.945907	−0.472953	−	0.881087i	$-0.656812\pi$
$983$	−29.6569	−0.945907	−0.472953	+	0.881087i	$0.656812\pi$
$984$	0	0
$985$	25.3137	0.806562
$986$	0	0
$987$	−11.3137	−0.360119
$988$	0	0
$989$	−11.3137	−0.359755
$990$	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
$993$	14.3431	0.455166
$994$	0	0
$995$	24.9706	0.791620
$996$	0	0
$997$	30.6863	0.971845	0.485922	−	0.874002i	$-0.338484\pi$
$997$	30.6863	0.971845	0.485922	+	0.874002i	$0.338484\pi$
$998$	0	0
$999$	−0.828427	−0.0262103

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2280.2.a.l.1.2	✓	2
3.2	odd	2		6840.2.a.ba.1.2		2
4.3	odd	2		4560.2.a.bm.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2280.2.a.l.1.2	✓	2	1.1	even	1	trivial
4560.2.a.bm.1.1		2	4.3	odd	2
6840.2.a.ba.1.2		2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2280 = 2^{3} \cdot 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2280.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} +2.82843 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.00000 q^{5} +2.82843 q^{7} +1.00000 q^{9} -2.82843 q^{11} -4.82843 q^{13} +1.00000 q^{15} +3.65685 q^{17} -1.00000 q^{19} -2.82843 q^{21} +4.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +4.82843 q^{29} +2.82843 q^{33} -2.82843 q^{35} +0.828427 q^{37} +4.82843 q^{39} +4.82843 q^{41} -2.82843 q^{43} -1.00000 q^{45} +4.00000 q^{47} +1.00000 q^{49} -3.65685 q^{51} -2.00000 q^{53} +2.82843 q^{55} +1.00000 q^{57} +8.00000 q^{59} -13.3137 q^{61} +2.82843 q^{63} +4.82843 q^{65} +9.65685 q^{67} -4.00000 q^{69} +8.00000 q^{71} -11.6569 q^{73} -1.00000 q^{75} -8.00000 q^{77} +2.34315 q^{79} +1.00000 q^{81} +8.00000 q^{83} -3.65685 q^{85} -4.82843 q^{87} +12.8284 q^{89} -13.6569 q^{91} +1.00000 q^{95} +0.828427 q^{97} -2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^9	\( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{9} - 4 q^{13} + 2 q^{15} - 4 q^{17} - 2 q^{19} + 8 q^{23} + 2 q^{25} - 2 q^{27} + 4 q^{29} - 4 q^{37} + 4 q^{39} + 4 q^{41} - 2 q^{45} + 8 q^{47} + 2 q^{49} + 4 q^{51} - 4 q^{53} + 2 q^{57} + 16 q^{59} - 4 q^{61} + 4 q^{65} + 8 q^{67} - 8 q^{69} + 16 q^{71} - 12 q^{73} - 2 q^{75} - 16 q^{77} + 16 q^{79} + 2 q^{81} + 16 q^{83} + 4 q^{85} - 4 q^{87} + 20 q^{89} - 16 q^{91} + 2 q^{95} - 4 q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^9 - 4 * q^13 + 2 * q^15 - 4 * q^17 - 2 * q^19 + 8 * q^23 + 2 * q^25 - 2 * q^27 + 4 * q^29 - 4 * q^37 + 4 * q^39 + 4 * q^41 - 2 * q^45 + 8 * q^47 + 2 * q^49 + 4 * q^51 - 4 * q^53 + 2 * q^57 + 16 * q^59 - 4 * q^61 + 4 * q^65 + 8 * q^67 - 8 * q^69 + 16 * q^71 - 12 * q^73 - 2 * q^75 - 16 * q^77 + 16 * q^79 + 2 * q^81 + 16 * q^83 + 4 * q^85 - 4 * q^87 + 20 * q^89 - 16 * q^91 + 2 * q^95 - 4 * q^97

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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