LMFDB - Embedded newform 3120.2.a.bc.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3120,2,Mod(1,3120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3120, 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("3120.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3120.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:	$24.9133254306$
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:	no (minimal twist has level 390)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	3120.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} +2.82843 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} +2.82843 q^{7} +1.00000 q^{9} -5.65685 q^{11} -1.00000 q^{13} -1.00000 q^{15} +0.828427 q^{17} -2.82843 q^{19} -2.82843 q^{21} +8.48528 q^{23} +1.00000 q^{25} -1.00000 q^{27} -8.82843 q^{29} -4.00000 q^{31} +5.65685 q^{33} +2.82843 q^{35} -11.6569 q^{37} +1.00000 q^{39} -7.65685 q^{41} -9.65685 q^{43} +1.00000 q^{45} +8.00000 q^{47} +1.00000 q^{49} -0.828427 q^{51} +13.3137 q^{53} -5.65685 q^{55} +2.82843 q^{57} +2.34315 q^{59} +6.00000 q^{61} +2.82843 q^{63} -1.00000 q^{65} -5.65685 q^{67} -8.48528 q^{69} +5.65685 q^{71} -14.4853 q^{73} -1.00000 q^{75} -16.0000 q^{77} -2.34315 q^{79} +1.00000 q^{81} -6.34315 q^{83} +0.828427 q^{85} +8.82843 q^{87} -15.6569 q^{89} -2.82843 q^{91} +4.00000 q^{93} -2.82843 q^{95} -3.17157 q^{97} -5.65685 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 2 q^{25} - 2 q^{27} - 12 q^{29} - 8 q^{31} - 12 q^{37} + 2 q^{39} - 4 q^{41} - 8 q^{43} + 2 q^{45} + 16 q^{47} + 2 q^{49} + 4 q^{51} + 4 q^{53} + 16 q^{59} + 12 q^{61} - 2 q^{65} - 12 q^{73} - 2 q^{75} - 32 q^{77} - 16 q^{79} + 2 q^{81} - 24 q^{83} - 4 q^{85} + 12 q^{87} - 20 q^{89} + 8 q^{93} - 12 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 + 2 * q^25 - 2 * q^27 - 12 * q^29 - 8 * q^31 - 12 * q^37 + 2 * q^39 - 4 * q^41 - 8 * q^43 + 2 * q^45 + 16 * q^47 + 2 * q^49 + 4 * q^51 + 4 * q^53 + 16 * q^59 + 12 * q^61 - 2 * q^65 - 12 * q^73 - 2 * q^75 - 32 * q^77 - 16 * q^79 + 2 * q^81 - 24 * q^83 - 4 * q^85 + 12 * q^87 - 20 * q^89 + 8 * q^93 - 12 * 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
$11$	−5.65685	−1.70561	−0.852803	−	0.522233i	$-0.825099\pi$
$11$	−5.65685	−1.70561	−0.852803	+	0.522233i	$0.825099\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	0.828427	0.200923	0.100462	−	0.994941i	$-0.467968\pi$
$17$	0.828427	0.200923	0.100462	+	0.994941i	$0.467968\pi$
$18$	0	0
$19$	−2.82843	−0.648886	−0.324443	−	0.945905i	$-0.605177\pi$
$19$	−2.82843	−0.648886	−0.324443	+	0.945905i	$0.605177\pi$
$20$	0	0
$21$	−2.82843	−0.617213
$22$	0	0
$23$	8.48528	1.76930	0.884652	−	0.466252i	$-0.154396\pi$
$23$	8.48528	1.76930	0.884652	+	0.466252i	$0.154396\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−8.82843	−1.63940	−0.819699	−	0.572795i	$-0.805859\pi$
$29$	−8.82843	−1.63940	−0.819699	+	0.572795i	$0.805859\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	5.65685	0.984732
$34$	0	0
$35$	2.82843	0.478091
$36$	0	0
$37$	−11.6569	−1.91638	−0.958188	−	0.286141i	$-0.907627\pi$
$37$	−11.6569	−1.91638	−0.958188	+	0.286141i	$0.907627\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	−7.65685	−1.19580	−0.597900	−	0.801571i	$-0.703998\pi$
$41$	−7.65685	−1.19580	−0.597900	+	0.801571i	$0.703998\pi$
$42$	0	0
$43$	−9.65685	−1.47266	−0.736328	−	0.676625i	$-0.763442\pi$
$43$	−9.65685	−1.47266	−0.736328	+	0.676625i	$0.763442\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.828427	−0.116003
$52$	0	0
$53$	13.3137	1.82878	0.914389	−	0.404836i	$-0.132671\pi$
$53$	13.3137	1.82878	0.914389	+	0.404836i	$0.132671\pi$
$54$	0	0
$55$	−5.65685	−0.762770
$56$	0	0
$57$	2.82843	0.374634
$58$	0	0
$59$	2.34315	0.305052	0.152526	−	0.988299i	$-0.451259\pi$
$59$	2.34315	0.305052	0.152526	+	0.988299i	$0.451259\pi$
$60$	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
$67$	−5.65685	−0.691095	−0.345547	−	0.938401i	$-0.612307\pi$
$67$	−5.65685	−0.691095	−0.345547	+	0.938401i	$0.612307\pi$
$68$	0	0
$69$	−8.48528	−1.02151
$70$	0	0
$71$	5.65685	0.671345	0.335673	−	0.941979i	$-0.391036\pi$
$71$	5.65685	0.671345	0.335673	+	0.941979i	$0.391036\pi$
$72$	0	0
$73$	−14.4853	−1.69537	−0.847687	−	0.530497i	$-0.822005\pi$
$73$	−14.4853	−1.69537	−0.847687	+	0.530497i	$0.822005\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−16.0000	−1.82337
$78$	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
$83$	−6.34315	−0.696251	−0.348125	−	0.937448i	$-0.613182\pi$
$83$	−6.34315	−0.696251	−0.348125	+	0.937448i	$0.613182\pi$
$84$	0	0
$85$	0.828427	0.0898555
$86$	0	0
$87$	8.82843	0.946507
$88$	0	0
$89$	−15.6569	−1.65962	−0.829812	−	0.558044i	$-0.811552\pi$
$89$	−15.6569	−1.65962	−0.829812	+	0.558044i	$0.811552\pi$
$90$	0	0
$91$	−2.82843	−0.296500
$92$	0	0
$93$	4.00000	0.414781
$94$	0	0
$95$	−2.82843	−0.290191
$96$	0	0
$97$	−3.17157	−0.322024	−0.161012	−	0.986952i	$-0.551476\pi$
$97$	−3.17157	−0.322024	−0.161012	+	0.986952i	$0.551476\pi$
$98$	0	0
$99$	−5.65685	−0.568535
$100$	0	0
$101$	16.1421	1.60620	0.803101	−	0.595843i	$-0.203182\pi$
$101$	16.1421	1.60620	0.803101	+	0.595843i	$0.203182\pi$
$102$	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
$107$	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
$109$	8.82843	0.845610	0.422805	−	0.906221i	$-0.361046\pi$
$109$	8.82843	0.845610	0.422805	+	0.906221i	$0.361046\pi$
$110$	0	0
$111$	11.6569	1.10642
$112$	0	0
$113$	6.48528	0.610084	0.305042	−	0.952339i	$-0.401330\pi$
$113$	6.48528	0.610084	0.305042	+	0.952339i	$0.401330\pi$
$114$	0	0
$115$	8.48528	0.791257
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	2.34315	0.214796
$120$	0	0
$121$	21.0000	1.90909
$122$	0	0
$123$	7.65685	0.690395
$124$	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
$129$	9.65685	0.850239
$130$	0	0
$131$	6.14214	0.536641	0.268320	−	0.963330i	$-0.413531\pi$
$131$	6.14214	0.536641	0.268320	+	0.963330i	$0.413531\pi$
$132$	0	0
$133$	−8.00000	−0.693688
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−17.3137	−1.47921	−0.739605	−	0.673041i	$-0.764988\pi$
$137$	−17.3137	−1.47921	−0.739605	+	0.673041i	$0.764988\pi$
$138$	0	0
$139$	−6.34315	−0.538019	−0.269009	−	0.963138i	$-0.586696\pi$
$139$	−6.34315	−0.538019	−0.269009	+	0.963138i	$0.586696\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	0	0
$143$	5.65685	0.473050
$144$	0	0
$145$	−8.82843	−0.733161
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	3.65685	0.299581	0.149791	−	0.988718i	$-0.452140\pi$
$149$	3.65685	0.299581	0.149791	+	0.988718i	$0.452140\pi$
$150$	0	0
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	0	0
$153$	0.828427	0.0669744
$154$	0	0
$155$	−4.00000	−0.321288
$156$	0	0
$157$	5.31371	0.424080	0.212040	−	0.977261i	$-0.431989\pi$
$157$	5.31371	0.424080	0.212040	+	0.977261i	$0.431989\pi$
$158$	0	0
$159$	−13.3137	−1.05585
$160$	0	0
$161$	24.0000	1.89146
$162$	0	0
$163$	−11.3137	−0.886158	−0.443079	−	0.896483i	$-0.646114\pi$
$163$	−11.3137	−0.886158	−0.443079	+	0.896483i	$0.646114\pi$
$164$	0	0
$165$	5.65685	0.440386
$166$	0	0
$167$	−8.97056	−0.694163	−0.347081	−	0.937835i	$-0.612827\pi$
$167$	−8.97056	−0.694163	−0.347081	+	0.937835i	$0.612827\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−2.82843	−0.216295
$172$	0	0
$173$	−9.31371	−0.708108	−0.354054	−	0.935225i	$-0.615197\pi$
$173$	−9.31371	−0.708108	−0.354054	+	0.935225i	$0.615197\pi$
$174$	0	0
$175$	2.82843	0.213809
$176$	0	0
$177$	−2.34315	−0.176122
$178$	0	0
$179$	−7.51472	−0.561676	−0.280838	−	0.959755i	$-0.590612\pi$
$179$	−7.51472	−0.561676	−0.280838	+	0.959755i	$0.590612\pi$
$180$	0	0
$181$	−7.65685	−0.569129	−0.284565	−	0.958657i	$-0.591849\pi$
$181$	−7.65685	−0.569129	−0.284565	+	0.958657i	$0.591849\pi$
$182$	0	0
$183$	−6.00000	−0.443533
$184$	0	0
$185$	−11.6569	−0.857029
$186$	0	0
$187$	−4.68629	−0.342696
$188$	0	0
$189$	−2.82843	−0.205738
$190$	0	0
$191$	−11.3137	−0.818631	−0.409316	−	0.912393i	$-0.634232\pi$
$191$	−11.3137	−0.818631	−0.409316	+	0.912393i	$0.634232\pi$
$192$	0	0
$193$	2.48528	0.178894	0.0894472	−	0.995992i	$-0.471490\pi$
$193$	2.48528	0.178894	0.0894472	+	0.995992i	$0.471490\pi$
$194$	0	0
$195$	1.00000	0.0716115
$196$	0	0
$197$	−13.3137	−0.948562	−0.474281	−	0.880373i	$-0.657292\pi$
$197$	−13.3137	−0.948562	−0.474281	+	0.880373i	$0.657292\pi$
$198$	0	0
$199$	−10.3431	−0.733206	−0.366603	−	0.930377i	$-0.619479\pi$
$199$	−10.3431	−0.733206	−0.366603	+	0.930377i	$0.619479\pi$
$200$	0	0
$201$	5.65685	0.399004
$202$	0	0
$203$	−24.9706	−1.75259
$204$	0	0
$205$	−7.65685	−0.534778
$206$	0	0
$207$	8.48528	0.589768
$208$	0	0
$209$	16.0000	1.10674
$210$	0	0
$211$	−0.686292	−0.0472463	−0.0236231	−	0.999721i	$-0.507520\pi$
$211$	−0.686292	−0.0472463	−0.0236231	+	0.999721i	$0.507520\pi$
$212$	0	0
$213$	−5.65685	−0.387601
$214$	0	0
$215$	−9.65685	−0.658592
$216$	0	0
$217$	−11.3137	−0.768025
$218$	0	0
$219$	14.4853	0.978825
$220$	0	0
$221$	−0.828427	−0.0557260
$222$	0	0
$223$	−10.8284	−0.725125	−0.362563	−	0.931959i	$-0.618098\pi$
$223$	−10.8284	−0.725125	−0.362563	+	0.931959i	$0.618098\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	0	0
$229$	4.14214	0.273720	0.136860	−	0.990590i	$-0.456299\pi$
$229$	4.14214	0.273720	0.136860	+	0.990590i	$0.456299\pi$
$230$	0	0
$231$	16.0000	1.05272
$232$	0	0
$233$	5.51472	0.361281	0.180641	−	0.983549i	$-0.442183\pi$
$233$	5.51472	0.361281	0.180641	+	0.983549i	$0.442183\pi$
$234$	0	0
$235$	8.00000	0.521862
$236$	0	0
$237$	2.34315	0.152204
$238$	0	0
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	0	0
$241$	5.31371	0.342286	0.171143	−	0.985246i	$-0.445254\pi$
$241$	5.31371	0.342286	0.171143	+	0.985246i	$0.445254\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	2.82843	0.179969
$248$	0	0
$249$	6.34315	0.401981
$250$	0	0
$251$	−10.8284	−0.683484	−0.341742	−	0.939794i	$-0.611017\pi$
$251$	−10.8284	−0.683484	−0.341742	+	0.939794i	$0.611017\pi$
$252$	0	0
$253$	−48.0000	−3.01773
$254$	0	0
$255$	−0.828427	−0.0518781
$256$	0	0
$257$	−4.82843	−0.301189	−0.150595	−	0.988596i	$-0.548119\pi$
$257$	−4.82843	−0.301189	−0.150595	+	0.988596i	$0.548119\pi$
$258$	0	0
$259$	−32.9706	−2.04869
$260$	0	0
$261$	−8.82843	−0.546466
$262$	0	0
$263$	−16.4853	−1.01653	−0.508263	−	0.861202i	$-0.669712\pi$
$263$	−16.4853	−1.01653	−0.508263	+	0.861202i	$0.669712\pi$
$264$	0	0
$265$	13.3137	0.817855
$266$	0	0
$267$	15.6569	0.958184
$268$	0	0
$269$	−14.4853	−0.883183	−0.441592	−	0.897216i	$-0.645586\pi$
$269$	−14.4853	−0.883183	−0.441592	+	0.897216i	$0.645586\pi$
$270$	0	0
$271$	−7.31371	−0.444276	−0.222138	−	0.975015i	$-0.571304\pi$
$271$	−7.31371	−0.444276	−0.222138	+	0.975015i	$0.571304\pi$
$272$	0	0
$273$	2.82843	0.171184
$274$	0	0
$275$	−5.65685	−0.341121
$276$	0	0
$277$	26.0000	1.56219	0.781094	−	0.624413i	$-0.214662\pi$
$277$	26.0000	1.56219	0.781094	+	0.624413i	$0.214662\pi$
$278$	0	0
$279$	−4.00000	−0.239474
$280$	0	0
$281$	8.34315	0.497710	0.248855	−	0.968541i	$-0.419946\pi$
$281$	8.34315	0.497710	0.248855	+	0.968541i	$0.419946\pi$
$282$	0	0
$283$	17.6569	1.04959	0.524796	−	0.851228i	$-0.324142\pi$
$283$	17.6569	1.04959	0.524796	+	0.851228i	$0.324142\pi$
$284$	0	0
$285$	2.82843	0.167542
$286$	0	0
$287$	−21.6569	−1.27836
$288$	0	0
$289$	−16.3137	−0.959630
$290$	0	0
$291$	3.17157	0.185921
$292$	0	0
$293$	−16.6274	−0.971384	−0.485692	−	0.874130i	$-0.661432\pi$
$293$	−16.6274	−0.971384	−0.485692	+	0.874130i	$0.661432\pi$
$294$	0	0
$295$	2.34315	0.136423
$296$	0	0
$297$	5.65685	0.328244
$298$	0	0
$299$	−8.48528	−0.490716
$300$	0	0
$301$	−27.3137	−1.57434
$302$	0	0
$303$	−16.1421	−0.927341
$304$	0	0
$305$	6.00000	0.343559
$306$	0	0
$307$	21.6569	1.23602	0.618011	−	0.786169i	$-0.287939\pi$
$307$	21.6569	1.23602	0.618011	+	0.786169i	$0.287939\pi$
$308$	0	0
$309$	−1.65685	−0.0942551
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	−30.9706	−1.75056	−0.875280	−	0.483617i	$-0.839323\pi$
$313$	−30.9706	−1.75056	−0.875280	+	0.483617i	$0.839323\pi$
$314$	0	0
$315$	2.82843	0.159364
$316$	0	0
$317$	25.3137	1.42176	0.710880	−	0.703314i	$-0.248297\pi$
$317$	25.3137	1.42176	0.710880	+	0.703314i	$0.248297\pi$
$318$	0	0
$319$	49.9411	2.79617
$320$	0	0
$321$	−4.00000	−0.223258
$322$	0	0
$323$	−2.34315	−0.130376
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	−8.82843	−0.488213
$328$	0	0
$329$	22.6274	1.24749
$330$	0	0
$331$	8.48528	0.466393	0.233197	−	0.972430i	$-0.425081\pi$
$331$	8.48528	0.466393	0.233197	+	0.972430i	$0.425081\pi$
$332$	0	0
$333$	−11.6569	−0.638792
$334$	0	0
$335$	−5.65685	−0.309067
$336$	0	0
$337$	10.9706	0.597605	0.298802	−	0.954315i	$-0.403413\pi$
$337$	10.9706	0.597605	0.298802	+	0.954315i	$0.403413\pi$
$338$	0	0
$339$	−6.48528	−0.352232
$340$	0	0
$341$	22.6274	1.22534
$342$	0	0
$343$	−16.9706	−0.916324
$344$	0	0
$345$	−8.48528	−0.456832
$346$	0	0
$347$	9.65685	0.518407	0.259204	−	0.965823i	$-0.416540\pi$
$347$	9.65685	0.518407	0.259204	+	0.965823i	$0.416540\pi$
$348$	0	0
$349$	12.1421	0.649954	0.324977	−	0.945722i	$-0.394644\pi$
$349$	12.1421	0.649954	0.324977	+	0.945722i	$0.394644\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	5.31371	0.282820	0.141410	−	0.989951i	$-0.454836\pi$
$353$	5.31371	0.282820	0.141410	+	0.989951i	$0.454836\pi$
$354$	0	0
$355$	5.65685	0.300235
$356$	0	0
$357$	−2.34315	−0.124012
$358$	0	0
$359$	28.2843	1.49279	0.746393	−	0.665505i	$-0.231784\pi$
$359$	28.2843	1.49279	0.746393	+	0.665505i	$0.231784\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	0	0
$363$	−21.0000	−1.10221
$364$	0	0
$365$	−14.4853	−0.758194
$366$	0	0
$367$	−25.6569	−1.33928	−0.669638	−	0.742687i	$-0.733551\pi$
$367$	−25.6569	−1.33928	−0.669638	+	0.742687i	$0.733551\pi$
$368$	0	0
$369$	−7.65685	−0.398600
$370$	0	0
$371$	37.6569	1.95505
$372$	0	0
$373$	−2.68629	−0.139091	−0.0695455	−	0.997579i	$-0.522155\pi$
$373$	−2.68629	−0.139091	−0.0695455	+	0.997579i	$0.522155\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	8.82843	0.454687
$378$	0	0
$379$	7.51472	0.386005	0.193003	−	0.981198i	$-0.438177\pi$
$379$	7.51472	0.386005	0.193003	+	0.981198i	$0.438177\pi$
$380$	0	0
$381$	9.65685	0.494736
$382$	0	0
$383$	29.6569	1.51539	0.757697	−	0.652606i	$-0.226324\pi$
$383$	29.6569	1.51539	0.757697	+	0.652606i	$0.226324\pi$
$384$	0	0
$385$	−16.0000	−0.815436
$386$	0	0
$387$	−9.65685	−0.490885
$388$	0	0
$389$	−6.48528	−0.328817	−0.164408	−	0.986392i	$-0.552571\pi$
$389$	−6.48528	−0.328817	−0.164408	+	0.986392i	$0.552571\pi$
$390$	0	0
$391$	7.02944	0.355494
$392$	0	0
$393$	−6.14214	−0.309830
$394$	0	0
$395$	−2.34315	−0.117896
$396$	0	0
$397$	30.2843	1.51992	0.759962	−	0.649968i	$-0.225218\pi$
$397$	30.2843	1.51992	0.759962	+	0.649968i	$0.225218\pi$
$398$	0	0
$399$	8.00000	0.400501
$400$	0	0
$401$	−26.9706	−1.34685	−0.673423	−	0.739258i	$-0.735177\pi$
$401$	−26.9706	−1.34685	−0.673423	+	0.739258i	$0.735177\pi$
$402$	0	0
$403$	4.00000	0.199254
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	65.9411	3.26858
$408$	0	0
$409$	−3.65685	−0.180820	−0.0904099	−	0.995905i	$-0.528818\pi$
$409$	−3.65685	−0.180820	−0.0904099	+	0.995905i	$0.528818\pi$
$410$	0	0
$411$	17.3137	0.854022
$412$	0	0
$413$	6.62742	0.326114
$414$	0	0
$415$	−6.34315	−0.311373
$416$	0	0
$417$	6.34315	0.310625
$418$	0	0
$419$	10.8284	0.529003	0.264502	−	0.964385i	$-0.414793\pi$
$419$	10.8284	0.529003	0.264502	+	0.964385i	$0.414793\pi$
$420$	0	0
$421$	−24.1421	−1.17662	−0.588308	−	0.808637i	$-0.700206\pi$
$421$	−24.1421	−1.17662	−0.588308	+	0.808637i	$0.700206\pi$
$422$	0	0
$423$	8.00000	0.388973
$424$	0	0
$425$	0.828427	0.0401846
$426$	0	0
$427$	16.9706	0.821263
$428$	0	0
$429$	−5.65685	−0.273115
$430$	0	0
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	0	0
$433$	−22.9706	−1.10389	−0.551947	−	0.833879i	$-0.686115\pi$
$433$	−22.9706	−1.10389	−0.551947	+	0.833879i	$0.686115\pi$
$434$	0	0
$435$	8.82843	0.423291
$436$	0	0
$437$	−24.0000	−1.14808
$438$	0	0
$439$	−22.6274	−1.07995	−0.539974	−	0.841682i	$-0.681566\pi$
$439$	−22.6274	−1.07995	−0.539974	+	0.841682i	$0.681566\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−30.3431	−1.44165	−0.720823	−	0.693119i	$-0.756236\pi$
$443$	−30.3431	−1.44165	−0.720823	+	0.693119i	$0.756236\pi$
$444$	0	0
$445$	−15.6569	−0.742206
$446$	0	0
$447$	−3.65685	−0.172963
$448$	0	0
$449$	26.2843	1.24043	0.620216	−	0.784431i	$-0.287045\pi$
$449$	26.2843	1.24043	0.620216	+	0.784431i	$0.287045\pi$
$450$	0	0
$451$	43.3137	2.03956
$452$	0	0
$453$	12.0000	0.563809
$454$	0	0
$455$	−2.82843	−0.132599
$456$	0	0
$457$	20.8284	0.974313	0.487156	−	0.873315i	$-0.338034\pi$
$457$	20.8284	0.974313	0.487156	+	0.873315i	$0.338034\pi$
$458$	0	0
$459$	−0.828427	−0.0386677
$460$	0	0
$461$	14.0000	0.652045	0.326023	−	0.945362i	$-0.394291\pi$
$461$	14.0000	0.652045	0.326023	+	0.945362i	$0.394291\pi$
$462$	0	0
$463$	3.79899	0.176554	0.0882770	−	0.996096i	$-0.471864\pi$
$463$	3.79899	0.176554	0.0882770	+	0.996096i	$0.471864\pi$
$464$	0	0
$465$	4.00000	0.185496
$466$	0	0
$467$	−7.31371	−0.338438	−0.169219	−	0.985578i	$-0.554125\pi$
$467$	−7.31371	−0.338438	−0.169219	+	0.985578i	$0.554125\pi$
$468$	0	0
$469$	−16.0000	−0.738811
$470$	0	0
$471$	−5.31371	−0.244843
$472$	0	0
$473$	54.6274	2.51177
$474$	0	0
$475$	−2.82843	−0.129777
$476$	0	0
$477$	13.3137	0.609593
$478$	0	0
$479$	11.3137	0.516937	0.258468	−	0.966020i	$-0.416782\pi$
$479$	11.3137	0.516937	0.258468	+	0.966020i	$0.416782\pi$
$480$	0	0
$481$	11.6569	0.531507
$482$	0	0
$483$	−24.0000	−1.09204
$484$	0	0
$485$	−3.17157	−0.144014
$486$	0	0
$487$	−16.4853	−0.747019	−0.373510	−	0.927626i	$-0.621846\pi$
$487$	−16.4853	−0.747019	−0.373510	+	0.927626i	$0.621846\pi$
$488$	0	0
$489$	11.3137	0.511624
$490$	0	0
$491$	38.1421	1.72133	0.860665	−	0.509171i	$-0.170048\pi$
$491$	38.1421	1.72133	0.860665	+	0.509171i	$0.170048\pi$
$492$	0	0
$493$	−7.31371	−0.329393
$494$	0	0
$495$	−5.65685	−0.254257
$496$	0	0
$497$	16.0000	0.717698
$498$	0	0
$499$	−0.485281	−0.0217242	−0.0108621	−	0.999941i	$-0.503458\pi$
$499$	−0.485281	−0.0217242	−0.0108621	+	0.999941i	$0.503458\pi$
$500$	0	0
$501$	8.97056	0.400775
$502$	0	0
$503$	23.5147	1.04847	0.524235	−	0.851574i	$-0.324351\pi$
$503$	23.5147	1.04847	0.524235	+	0.851574i	$0.324351\pi$
$504$	0	0
$505$	16.1421	0.718316
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	−37.3137	−1.65390	−0.826951	−	0.562275i	$-0.809926\pi$
$509$	−37.3137	−1.65390	−0.826951	+	0.562275i	$0.809926\pi$
$510$	0	0
$511$	−40.9706	−1.81243
$512$	0	0
$513$	2.82843	0.124878
$514$	0	0
$515$	1.65685	0.0730097
$516$	0	0
$517$	−45.2548	−1.99031
$518$	0	0
$519$	9.31371	0.408826
$520$	0	0
$521$	26.9706	1.18160	0.590801	−	0.806817i	$-0.298812\pi$
$521$	26.9706	1.18160	0.590801	+	0.806817i	$0.298812\pi$
$522$	0	0
$523$	10.6274	0.464704	0.232352	−	0.972632i	$-0.425358\pi$
$523$	10.6274	0.464704	0.232352	+	0.972632i	$0.425358\pi$
$524$	0	0
$525$	−2.82843	−0.123443
$526$	0	0
$527$	−3.31371	−0.144347
$528$	0	0
$529$	49.0000	2.13043
$530$	0	0
$531$	2.34315	0.101684
$532$	0	0
$533$	7.65685	0.331655
$534$	0	0
$535$	4.00000	0.172935
$536$	0	0
$537$	7.51472	0.324284
$538$	0	0
$539$	−5.65685	−0.243658
$540$	0	0
$541$	14.4853	0.622771	0.311385	−	0.950284i	$-0.399207\pi$
$541$	14.4853	0.622771	0.311385	+	0.950284i	$0.399207\pi$
$542$	0	0
$543$	7.65685	0.328587
$544$	0	0
$545$	8.82843	0.378168
$546$	0	0
$547$	−0.686292	−0.0293437	−0.0146719	−	0.999892i	$-0.504670\pi$
$547$	−0.686292	−0.0293437	−0.0146719	+	0.999892i	$0.504670\pi$
$548$	0	0
$549$	6.00000	0.256074
$550$	0	0
$551$	24.9706	1.06378
$552$	0	0
$553$	−6.62742	−0.281826
$554$	0	0
$555$	11.6569	0.494806
$556$	0	0
$557$	10.6863	0.452793	0.226396	−	0.974035i	$-0.427306\pi$
$557$	10.6863	0.452793	0.226396	+	0.974035i	$0.427306\pi$
$558$	0	0
$559$	9.65685	0.408441
$560$	0	0
$561$	4.68629	0.197855
$562$	0	0
$563$	30.3431	1.27881	0.639406	−	0.768870i	$-0.279181\pi$
$563$	30.3431	1.27881	0.639406	+	0.768870i	$0.279181\pi$
$564$	0	0
$565$	6.48528	0.272838
$566$	0	0
$567$	2.82843	0.118783
$568$	0	0
$569$	31.6569	1.32712	0.663562	−	0.748121i	$-0.269044\pi$
$569$	31.6569	1.32712	0.663562	+	0.748121i	$0.269044\pi$
$570$	0	0
$571$	−20.9706	−0.877591	−0.438795	−	0.898587i	$-0.644595\pi$
$571$	−20.9706	−0.877591	−0.438795	+	0.898587i	$0.644595\pi$
$572$	0	0
$573$	11.3137	0.472637
$574$	0	0
$575$	8.48528	0.353861
$576$	0	0
$577$	−23.4558	−0.976480	−0.488240	−	0.872710i	$-0.662361\pi$
$577$	−23.4558	−0.976480	−0.488240	+	0.872710i	$0.662361\pi$
$578$	0	0
$579$	−2.48528	−0.103285
$580$	0	0
$581$	−17.9411	−0.744323
$582$	0	0
$583$	−75.3137	−3.11918
$584$	0	0
$585$	−1.00000	−0.0413449
$586$	0	0
$587$	−2.62742	−0.108445	−0.0542226	−	0.998529i	$-0.517268\pi$
$587$	−2.62742	−0.108445	−0.0542226	+	0.998529i	$0.517268\pi$
$588$	0	0
$589$	11.3137	0.466173
$590$	0	0
$591$	13.3137	0.547653
$592$	0	0
$593$	−0.343146	−0.0140913	−0.00704565	−	0.999975i	$-0.502243\pi$
$593$	−0.343146	−0.0140913	−0.00704565	+	0.999975i	$0.502243\pi$
$594$	0	0
$595$	2.34315	0.0960596
$596$	0	0
$597$	10.3431	0.423317
$598$	0	0
$599$	−40.0000	−1.63436	−0.817178	−	0.576386i	$-0.804463\pi$
$599$	−40.0000	−1.63436	−0.817178	+	0.576386i	$0.804463\pi$
$600$	0	0
$601$	29.3137	1.19573	0.597866	−	0.801596i	$-0.296016\pi$
$601$	29.3137	1.19573	0.597866	+	0.801596i	$0.296016\pi$
$602$	0	0
$603$	−5.65685	−0.230365
$604$	0	0
$605$	21.0000	0.853771
$606$	0	0
$607$	−28.9706	−1.17588	−0.587939	−	0.808905i	$-0.700061\pi$
$607$	−28.9706	−1.17588	−0.587939	+	0.808905i	$0.700061\pi$
$608$	0	0
$609$	24.9706	1.01186
$610$	0	0
$611$	−8.00000	−0.323645
$612$	0	0
$613$	22.2843	0.900053	0.450027	−	0.893015i	$-0.351414\pi$
$613$	22.2843	0.900053	0.450027	+	0.893015i	$0.351414\pi$
$614$	0	0
$615$	7.65685	0.308754
$616$	0	0
$617$	2.00000	0.0805170	0.0402585	−	0.999189i	$-0.487182\pi$
$617$	2.00000	0.0805170	0.0402585	+	0.999189i	$0.487182\pi$
$618$	0	0
$619$	34.8284	1.39987	0.699936	−	0.714205i	$-0.253212\pi$
$619$	34.8284	1.39987	0.699936	+	0.714205i	$0.253212\pi$
$620$	0	0
$621$	−8.48528	−0.340503
$622$	0	0
$623$	−44.2843	−1.77421
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−16.0000	−0.638978
$628$	0	0
$629$	−9.65685	−0.385044
$630$	0	0
$631$	33.6569	1.33986	0.669929	−	0.742425i	$-0.266324\pi$
$631$	33.6569	1.33986	0.669929	+	0.742425i	$0.266324\pi$
$632$	0	0
$633$	0.686292	0.0272776
$634$	0	0
$635$	−9.65685	−0.383221
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	5.65685	0.223782
$640$	0	0
$641$	−4.62742	−0.182772	−0.0913860	−	0.995816i	$-0.529130\pi$
$641$	−4.62742	−0.182772	−0.0913860	+	0.995816i	$0.529130\pi$
$642$	0	0
$643$	−39.5980	−1.56159	−0.780796	−	0.624786i	$-0.785186\pi$
$643$	−39.5980	−1.56159	−0.780796	+	0.624786i	$0.785186\pi$
$644$	0	0
$645$	9.65685	0.380238
$646$	0	0
$647$	−8.48528	−0.333591	−0.166795	−	0.985992i	$-0.553342\pi$
$647$	−8.48528	−0.333591	−0.166795	+	0.985992i	$0.553342\pi$
$648$	0	0
$649$	−13.2548	−0.520298
$650$	0	0
$651$	11.3137	0.443419
$652$	0	0
$653$	−42.2843	−1.65471	−0.827356	−	0.561678i	$-0.810156\pi$
$653$	−42.2843	−1.65471	−0.827356	+	0.561678i	$0.810156\pi$
$654$	0	0
$655$	6.14214	0.239993
$656$	0	0
$657$	−14.4853	−0.565125
$658$	0	0
$659$	7.51472	0.292732	0.146366	−	0.989231i	$-0.453242\pi$
$659$	7.51472	0.292732	0.146366	+	0.989231i	$0.453242\pi$
$660$	0	0
$661$	−8.14214	−0.316692	−0.158346	−	0.987384i	$-0.550616\pi$
$661$	−8.14214	−0.316692	−0.158346	+	0.987384i	$0.550616\pi$
$662$	0	0
$663$	0.828427	0.0321734
$664$	0	0
$665$	−8.00000	−0.310227
$666$	0	0
$667$	−74.9117	−2.90059
$668$	0	0
$669$	10.8284	0.418651
$670$	0	0
$671$	−33.9411	−1.31028
$672$	0	0
$673$	32.6274	1.25769	0.628847	−	0.777529i	$-0.283527\pi$
$673$	32.6274	1.25769	0.628847	+	0.777529i	$0.283527\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	12.3431	0.474386	0.237193	−	0.971463i	$-0.423773\pi$
$677$	12.3431	0.474386	0.237193	+	0.971463i	$0.423773\pi$
$678$	0	0
$679$	−8.97056	−0.344259
$680$	0	0
$681$	4.00000	0.153280
$682$	0	0
$683$	−33.6569	−1.28784	−0.643922	−	0.765091i	$-0.722694\pi$
$683$	−33.6569	−1.28784	−0.643922	+	0.765091i	$0.722694\pi$
$684$	0	0
$685$	−17.3137	−0.661523
$686$	0	0
$687$	−4.14214	−0.158032
$688$	0	0
$689$	−13.3137	−0.507212
$690$	0	0
$691$	27.7990	1.05752	0.528762	−	0.848770i	$-0.322657\pi$
$691$	27.7990	1.05752	0.528762	+	0.848770i	$0.322657\pi$
$692$	0	0
$693$	−16.0000	−0.607790
$694$	0	0
$695$	−6.34315	−0.240609
$696$	0	0
$697$	−6.34315	−0.240264
$698$	0	0
$699$	−5.51472	−0.208586
$700$	0	0
$701$	0.142136	0.00536839	0.00268419	−	0.999996i	$-0.499146\pi$
$701$	0.142136	0.00536839	0.00268419	+	0.999996i	$0.499146\pi$
$702$	0	0
$703$	32.9706	1.24351
$704$	0	0
$705$	−8.00000	−0.301297
$706$	0	0
$707$	45.6569	1.71710
$708$	0	0
$709$	−7.17157	−0.269334	−0.134667	−	0.990891i	$-0.542996\pi$
$709$	−7.17157	−0.269334	−0.134667	+	0.990891i	$0.542996\pi$
$710$	0	0
$711$	−2.34315	−0.0878748
$712$	0	0
$713$	−33.9411	−1.27111
$714$	0	0
$715$	5.65685	0.211554
$716$	0	0
$717$	16.0000	0.597531
$718$	0	0
$719$	29.6569	1.10601	0.553007	−	0.833177i	$-0.313480\pi$
$719$	29.6569	1.10601	0.553007	+	0.833177i	$0.313480\pi$
$720$	0	0
$721$	4.68629	0.174527
$722$	0	0
$723$	−5.31371	−0.197619
$724$	0	0
$725$	−8.82843	−0.327880
$726$	0	0
$727$	45.9411	1.70386	0.851931	−	0.523654i	$-0.175432\pi$
$727$	45.9411	1.70386	0.851931	+	0.523654i	$0.175432\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−8.00000	−0.295891
$732$	0	0
$733$	−0.343146	−0.0126744	−0.00633719	−	0.999980i	$-0.502017\pi$
$733$	−0.343146	−0.0126744	−0.00633719	+	0.999980i	$0.502017\pi$
$734$	0	0
$735$	−1.00000	−0.0368856
$736$	0	0
$737$	32.0000	1.17874
$738$	0	0
$739$	14.1421	0.520227	0.260113	−	0.965578i	$-0.416240\pi$
$739$	14.1421	0.520227	0.260113	+	0.965578i	$0.416240\pi$
$740$	0	0
$741$	−2.82843	−0.103905
$742$	0	0
$743$	36.2843	1.33114	0.665570	−	0.746335i	$-0.268188\pi$
$743$	36.2843	1.33114	0.665570	+	0.746335i	$0.268188\pi$
$744$	0	0
$745$	3.65685	0.133977
$746$	0	0
$747$	−6.34315	−0.232084
$748$	0	0
$749$	11.3137	0.413394
$750$	0	0
$751$	−11.3137	−0.412843	−0.206422	−	0.978463i	$-0.566182\pi$
$751$	−11.3137	−0.412843	−0.206422	+	0.978463i	$0.566182\pi$
$752$	0	0
$753$	10.8284	0.394610
$754$	0	0
$755$	−12.0000	−0.436725
$756$	0	0
$757$	19.9411	0.724773	0.362386	−	0.932028i	$-0.381962\pi$
$757$	19.9411	0.724773	0.362386	+	0.932028i	$0.381962\pi$
$758$	0	0
$759$	48.0000	1.74229
$760$	0	0
$761$	27.6569	1.00256	0.501280	−	0.865285i	$-0.332863\pi$
$761$	27.6569	1.00256	0.501280	+	0.865285i	$0.332863\pi$
$762$	0	0
$763$	24.9706	0.903995
$764$	0	0
$765$	0.828427	0.0299518
$766$	0	0
$767$	−2.34315	−0.0846061
$768$	0	0
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	0	0
$771$	4.82843	0.173892
$772$	0	0
$773$	−53.3137	−1.91756	−0.958780	−	0.284148i	$-0.908289\pi$
$773$	−53.3137	−1.91756	−0.958780	+	0.284148i	$0.908289\pi$
$774$	0	0
$775$	−4.00000	−0.143684
$776$	0	0
$777$	32.9706	1.18281
$778$	0	0
$779$	21.6569	0.775937
$780$	0	0
$781$	−32.0000	−1.14505
$782$	0	0
$783$	8.82843	0.315502
$784$	0	0
$785$	5.31371	0.189654
$786$	0	0
$787$	24.0000	0.855508	0.427754	−	0.903895i	$-0.359305\pi$
$787$	24.0000	0.855508	0.427754	+	0.903895i	$0.359305\pi$
$788$	0	0
$789$	16.4853	0.586892
$790$	0	0
$791$	18.3431	0.652207
$792$	0	0
$793$	−6.00000	−0.213066
$794$	0	0
$795$	−13.3137	−0.472189
$796$	0	0
$797$	16.6274	0.588973	0.294487	−	0.955656i	$-0.404851\pi$
$797$	16.6274	0.588973	0.294487	+	0.955656i	$0.404851\pi$
$798$	0	0
$799$	6.62742	0.234461
$800$	0	0
$801$	−15.6569	−0.553208
$802$	0	0
$803$	81.9411	2.89164
$804$	0	0
$805$	24.0000	0.845889
$806$	0	0
$807$	14.4853	0.509906
$808$	0	0
$809$	13.3137	0.468085	0.234043	−	0.972226i	$-0.424804\pi$
$809$	13.3137	0.468085	0.234043	+	0.972226i	$0.424804\pi$
$810$	0	0
$811$	1.85786	0.0652384	0.0326192	−	0.999468i	$-0.489615\pi$
$811$	1.85786	0.0652384	0.0326192	+	0.999468i	$0.489615\pi$
$812$	0	0
$813$	7.31371	0.256503
$814$	0	0
$815$	−11.3137	−0.396302
$816$	0	0
$817$	27.3137	0.955586
$818$	0	0
$819$	−2.82843	−0.0988332
$820$	0	0
$821$	34.2843	1.19653	0.598265	−	0.801299i	$-0.295857\pi$
$821$	34.2843	1.19653	0.598265	+	0.801299i	$0.295857\pi$
$822$	0	0
$823$	52.9706	1.84644	0.923219	−	0.384275i	$-0.125548\pi$
$823$	52.9706	1.84644	0.923219	+	0.384275i	$0.125548\pi$
$824$	0	0
$825$	5.65685	0.196946
$826$	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
$829$	−53.3137	−1.85166	−0.925831	−	0.377938i	$-0.876633\pi$
$829$	−53.3137	−1.85166	−0.925831	+	0.377938i	$0.876633\pi$
$830$	0	0
$831$	−26.0000	−0.901930
$832$	0	0
$833$	0.828427	0.0287033
$834$	0	0
$835$	−8.97056	−0.310439
$836$	0	0
$837$	4.00000	0.138260
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	48.9411	1.68763
$842$	0	0
$843$	−8.34315	−0.287353
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	59.3970	2.04090
$848$	0	0
$849$	−17.6569	−0.605982
$850$	0	0
$851$	−98.9117	−3.39065
$852$	0	0
$853$	38.2843	1.31083	0.655414	−	0.755270i	$-0.272494\pi$
$853$	38.2843	1.31083	0.655414	+	0.755270i	$0.272494\pi$
$854$	0	0
$855$	−2.82843	−0.0967302
$856$	0	0
$857$	−20.8284	−0.711486	−0.355743	−	0.934584i	$-0.615772\pi$
$857$	−20.8284	−0.711486	−0.355743	+	0.934584i	$0.615772\pi$
$858$	0	0
$859$	−37.9411	−1.29453	−0.647267	−	0.762263i	$-0.724088\pi$
$859$	−37.9411	−1.29453	−0.647267	+	0.762263i	$0.724088\pi$
$860$	0	0
$861$	21.6569	0.738064
$862$	0	0
$863$	28.2843	0.962808	0.481404	−	0.876499i	$-0.340127\pi$
$863$	28.2843	0.962808	0.481404	+	0.876499i	$0.340127\pi$
$864$	0	0
$865$	−9.31371	−0.316676
$866$	0	0
$867$	16.3137	0.554043
$868$	0	0
$869$	13.2548	0.449639
$870$	0	0
$871$	5.65685	0.191675
$872$	0	0
$873$	−3.17157	−0.107341
$874$	0	0
$875$	2.82843	0.0956183
$876$	0	0
$877$	−51.2548	−1.73075	−0.865376	−	0.501122i	$-0.832921\pi$
$877$	−51.2548	−1.73075	−0.865376	+	0.501122i	$0.832921\pi$
$878$	0	0
$879$	16.6274	0.560829
$880$	0	0
$881$	−10.2843	−0.346486	−0.173243	−	0.984879i	$-0.555425\pi$
$881$	−10.2843	−0.346486	−0.173243	+	0.984879i	$0.555425\pi$
$882$	0	0
$883$	−31.3137	−1.05379	−0.526895	−	0.849930i	$-0.676644\pi$
$883$	−31.3137	−1.05379	−0.526895	+	0.849930i	$0.676644\pi$
$884$	0	0
$885$	−2.34315	−0.0787640
$886$	0	0
$887$	40.4853	1.35936	0.679681	−	0.733508i	$-0.262118\pi$
$887$	40.4853	1.35936	0.679681	+	0.733508i	$0.262118\pi$
$888$	0	0
$889$	−27.3137	−0.916072
$890$	0	0
$891$	−5.65685	−0.189512
$892$	0	0
$893$	−22.6274	−0.757198
$894$	0	0
$895$	−7.51472	−0.251189
$896$	0	0
$897$	8.48528	0.283315
$898$	0	0
$899$	35.3137	1.17778
$900$	0	0
$901$	11.0294	0.367444
$902$	0	0
$903$	27.3137	0.908943
$904$	0	0
$905$	−7.65685	−0.254522
$906$	0	0
$907$	−8.28427	−0.275075	−0.137537	−	0.990497i	$-0.543919\pi$
$907$	−8.28427	−0.275075	−0.137537	+	0.990497i	$0.543919\pi$
$908$	0	0
$909$	16.1421	0.535401
$910$	0	0
$911$	−24.9706	−0.827312	−0.413656	−	0.910433i	$-0.635748\pi$
$911$	−24.9706	−0.827312	−0.413656	+	0.910433i	$0.635748\pi$
$912$	0	0
$913$	35.8823	1.18753
$914$	0	0
$915$	−6.00000	−0.198354
$916$	0	0
$917$	17.3726	0.573693
$918$	0	0
$919$	−41.9411	−1.38351	−0.691755	−	0.722132i	$-0.743162\pi$
$919$	−41.9411	−1.38351	−0.691755	+	0.722132i	$0.743162\pi$
$920$	0	0
$921$	−21.6569	−0.713618
$922$	0	0
$923$	−5.65685	−0.186198
$924$	0	0
$925$	−11.6569	−0.383275
$926$	0	0
$927$	1.65685	0.0544182
$928$	0	0
$929$	−33.5980	−1.10231	−0.551157	−	0.834402i	$-0.685813\pi$
$929$	−33.5980	−1.10231	−0.551157	+	0.834402i	$0.685813\pi$
$930$	0	0
$931$	−2.82843	−0.0926980
$932$	0	0
$933$	−24.0000	−0.785725
$934$	0	0
$935$	−4.68629	−0.153258
$936$	0	0
$937$	16.6274	0.543194	0.271597	−	0.962411i	$-0.412448\pi$
$937$	16.6274	0.543194	0.271597	+	0.962411i	$0.412448\pi$
$938$	0	0
$939$	30.9706	1.01069
$940$	0	0
$941$	54.9706	1.79199	0.895995	−	0.444065i	$-0.146464\pi$
$941$	54.9706	1.79199	0.895995	+	0.444065i	$0.146464\pi$
$942$	0	0
$943$	−64.9706	−2.11573
$944$	0	0
$945$	−2.82843	−0.0920087
$946$	0	0
$947$	−30.3431	−0.986020	−0.493010	−	0.870024i	$-0.664103\pi$
$947$	−30.3431	−0.986020	−0.493010	+	0.870024i	$0.664103\pi$
$948$	0	0
$949$	14.4853	0.470212
$950$	0	0
$951$	−25.3137	−0.820853
$952$	0	0
$953$	−27.8579	−0.902405	−0.451202	−	0.892422i	$-0.649005\pi$
$953$	−27.8579	−0.902405	−0.451202	+	0.892422i	$0.649005\pi$
$954$	0	0
$955$	−11.3137	−0.366103
$956$	0	0
$957$	−49.9411	−1.61437
$958$	0	0
$959$	−48.9706	−1.58134
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	4.00000	0.128898
$964$	0	0
$965$	2.48528	0.0800040
$966$	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
$969$	2.34315	0.0752727
$970$	0	0
$971$	−15.5147	−0.497891	−0.248946	−	0.968517i	$-0.580084\pi$
$971$	−15.5147	−0.497891	−0.248946	+	0.968517i	$0.580084\pi$
$972$	0	0
$973$	−17.9411	−0.575166
$974$	0	0
$975$	1.00000	0.0320256
$976$	0	0
$977$	−8.34315	−0.266921	−0.133460	−	0.991054i	$-0.542609\pi$
$977$	−8.34315	−0.266921	−0.133460	+	0.991054i	$0.542609\pi$
$978$	0	0
$979$	88.5685	2.83066
$980$	0	0
$981$	8.82843	0.281870
$982$	0	0
$983$	2.34315	0.0747347	0.0373674	−	0.999302i	$-0.488103\pi$
$983$	2.34315	0.0747347	0.0373674	+	0.999302i	$0.488103\pi$
$984$	0	0
$985$	−13.3137	−0.424210
$986$	0	0
$987$	−22.6274	−0.720239
$988$	0	0
$989$	−81.9411	−2.60558
$990$	0	0
$991$	42.9117	1.36313	0.681567	−	0.731755i	$-0.261299\pi$
$991$	42.9117	1.36313	0.681567	+	0.731755i	$0.261299\pi$
$992$	0	0
$993$	−8.48528	−0.269272
$994$	0	0
$995$	−10.3431	−0.327900
$996$	0	0
$997$	61.3137	1.94182	0.970912	−	0.239435i	$-0.0769623\pi$
$997$	61.3137	1.94182	0.970912	+	0.239435i	$0.0769623\pi$
$998$	0	0
$999$	11.6569	0.368807

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3120.2.a.bc.1.2		2
3.2	odd	2		9360.2.a.ch.1.2		2
4.3	odd	2		390.2.a.h.1.1	✓	2
12.11	even	2		1170.2.a.o.1.1		2
20.3	even	4		1950.2.e.o.1249.2		4
20.7	even	4		1950.2.e.o.1249.3		4
20.19	odd	2		1950.2.a.bd.1.2		2
52.31	even	4		5070.2.b.q.1351.1		4
52.47	even	4		5070.2.b.q.1351.4		4
52.51	odd	2		5070.2.a.bc.1.2		2
60.23	odd	4		5850.2.e.bk.5149.4		4
60.47	odd	4		5850.2.e.bk.5149.1		4
60.59	even	2		5850.2.a.cl.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.a.h.1.1	✓	2	4.3	odd	2
1170.2.a.o.1.1		2	12.11	even	2
1950.2.a.bd.1.2		2	20.19	odd	2
1950.2.e.o.1249.2		4	20.3	even	4
1950.2.e.o.1249.3		4	20.7	even	4
3120.2.a.bc.1.2		2	1.1	even	1	trivial
5070.2.a.bc.1.2		2	52.51	odd	2
5070.2.b.q.1351.1		4	52.31	even	4
5070.2.b.q.1351.4		4	52.47	even	4
5850.2.a.cl.1.2		2	60.59	even	2
5850.2.e.bk.5149.1		4	60.47	odd	4
5850.2.e.bk.5149.4		4	60.23	odd	4
9360.2.a.ch.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:	\( N \)	\(=\)	\( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3120.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} +2.82843 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} +2.82843 q^{7} +1.00000 q^{9} -5.65685 q^{11} -1.00000 q^{13} -1.00000 q^{15} +0.828427 q^{17} -2.82843 q^{19} -2.82843 q^{21} +8.48528 q^{23} +1.00000 q^{25} -1.00000 q^{27} -8.82843 q^{29} -4.00000 q^{31} +5.65685 q^{33} +2.82843 q^{35} -11.6569 q^{37} +1.00000 q^{39} -7.65685 q^{41} -9.65685 q^{43} +1.00000 q^{45} +8.00000 q^{47} +1.00000 q^{49} -0.828427 q^{51} +13.3137 q^{53} -5.65685 q^{55} +2.82843 q^{57} +2.34315 q^{59} +6.00000 q^{61} +2.82843 q^{63} -1.00000 q^{65} -5.65685 q^{67} -8.48528 q^{69} +5.65685 q^{71} -14.4853 q^{73} -1.00000 q^{75} -16.0000 q^{77} -2.34315 q^{79} +1.00000 q^{81} -6.34315 q^{83} +0.828427 q^{85} +8.82843 q^{87} -15.6569 q^{89} -2.82843 q^{91} +4.00000 q^{93} -2.82843 q^{95} -3.17157 q^{97} -5.65685 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{5} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^9	\( 2 q - 2 q^{3} + 2 q^{5} + 2 q^{9} - 2 q^{13} - 2 q^{15} - 4 q^{17} + 2 q^{25} - 2 q^{27} - 12 q^{29} - 8 q^{31} - 12 q^{37} + 2 q^{39} - 4 q^{41} - 8 q^{43} + 2 q^{45} + 16 q^{47} + 2 q^{49} + 4 q^{51} + 4 q^{53} + 16 q^{59} + 12 q^{61} - 2 q^{65} - 12 q^{73} - 2 q^{75} - 32 q^{77} - 16 q^{79} + 2 q^{81} - 24 q^{83} - 4 q^{85} + 12 q^{87} - 20 q^{89} + 8 q^{93} - 12 q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^9 - 2 * q^13 - 2 * q^15 - 4 * q^17 + 2 * q^25 - 2 * q^27 - 12 * q^29 - 8 * q^31 - 12 * q^37 + 2 * q^39 - 4 * q^41 - 8 * q^43 + 2 * q^45 + 16 * q^47 + 2 * q^49 + 4 * q^51 + 4 * q^53 + 16 * q^59 + 12 * q^61 - 2 * q^65 - 12 * q^73 - 2 * q^75 - 32 * q^77 - 16 * q^79 + 2 * q^81 - 24 * q^83 - 4 * q^85 + 12 * q^87 - 20 * q^89 + 8 * q^93 - 12 * q^97

Introduction

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