LMFDB - Embedded newform 7728.2.a.by.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7728,2,Mod(1,7728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7728, 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("7728.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7728.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:	$61.7083906820$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x + 6 $ x^3 - x^2 - 7*x + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3864)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.841083$ of defining polynomial
Character	$\chi$	$=$	7728.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +0.841083 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +0.841083 q^{5} -1.00000 q^{7} +1.00000 q^{9} +4.45150 q^{11} -5.29258 q^{13} +0.841083 q^{15} -4.13366 q^{17} -2.13366 q^{19} -1.00000 q^{21} -1.00000 q^{23} -4.29258 q^{25} +1.00000 q^{27} +4.13366 q^{29} +0.451497 q^{31} +4.45150 q^{33} -0.841083 q^{35} +9.81583 q^{37} -5.29258 q^{39} +1.54850 q^{41} +7.29258 q^{43} +0.841083 q^{45} -11.2208 q^{47} +1.00000 q^{49} -4.13366 q^{51} +12.0619 q^{53} +3.74408 q^{55} -2.13366 q^{57} -5.42624 q^{59} +12.6569 q^{61} -1.00000 q^{63} -4.45150 q^{65} +13.4262 q^{67} -1.00000 q^{69} -0.974745 q^{71} +9.03666 q^{73} -4.29258 q^{75} -4.45150 q^{77} +1.86634 q^{79} +1.00000 q^{81} -4.45150 q^{83} -3.47675 q^{85} +4.13366 q^{87} +10.9747 q^{89} +5.29258 q^{91} +0.451497 q^{93} -1.79459 q^{95} +10.4515 q^{97} +4.45150 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + q^5 - 3 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} + q^{5} - 3 q^{7} + 3 q^{9} + 2 q^{11} - 3 q^{13} + q^{15} + 2 q^{17} + 8 q^{19} - 3 q^{21} - 3 q^{23} + 3 q^{27} - 2 q^{29} - 10 q^{31} + 2 q^{33} - q^{35} + 12 q^{37} - 3 q^{39} + 16 q^{41} + 9 q^{43} + q^{45} - 14 q^{47} + 3 q^{49} + 2 q^{51} + 15 q^{53} - 13 q^{55} + 8 q^{57} + 11 q^{59} + 19 q^{61} - 3 q^{63} - 2 q^{65} + 13 q^{67} - 3 q^{69} + 13 q^{71} - 10 q^{73} - 2 q^{77} + 20 q^{79} + 3 q^{81} - 2 q^{83} - 15 q^{85} - 2 q^{87} + 17 q^{89} + 3 q^{91} - 10 q^{93} - 13 q^{95} + 20 q^{97} + 2 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + q^5 - 3 * q^7 + 3 * q^9 + 2 * q^11 - 3 * q^13 + q^15 + 2 * q^17 + 8 * q^19 - 3 * q^21 - 3 * q^23 + 3 * q^27 - 2 * q^29 - 10 * q^31 + 2 * q^33 - q^35 + 12 * q^37 - 3 * q^39 + 16 * q^41 + 9 * q^43 + q^45 - 14 * q^47 + 3 * q^49 + 2 * q^51 + 15 * q^53 - 13 * q^55 + 8 * q^57 + 11 * q^59 + 19 * q^61 - 3 * q^63 - 2 * q^65 + 13 * q^67 - 3 * q^69 + 13 * q^71 - 10 * q^73 - 2 * q^77 + 20 * q^79 + 3 * q^81 - 2 * q^83 - 15 * q^85 - 2 * q^87 + 17 * q^89 + 3 * q^91 - 10 * q^93 - 13 * q^95 + 20 * q^97 + 2 * q^99

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
$5$	0.841083	0.376144	0.188072	−	0.982155i	$-0.439776\pi$
$5$	0.841083	0.376144	0.188072	+	0.982155i	$0.439776\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.45150	1.34218	0.671088	−	0.741377i	$-0.265827\pi$
$11$	4.45150	1.34218	0.671088	+	0.741377i	$0.265827\pi$
$12$	0	0
$13$	−5.29258	−1.46790	−0.733949	−	0.679205i	$-0.762325\pi$
$13$	−5.29258	−1.46790	−0.733949	+	0.679205i	$0.762325\pi$
$14$	0	0
$15$	0.841083	0.217167
$16$	0	0
$17$	−4.13366	−1.00256	−0.501280	−	0.865285i	$-0.667137\pi$
$17$	−4.13366	−1.00256	−0.501280	+	0.865285i	$0.667137\pi$
$18$	0	0
$19$	−2.13366	−0.489496	−0.244748	−	0.969587i	$-0.578705\pi$
$19$	−2.13366	−0.489496	−0.244748	+	0.969587i	$0.578705\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−4.29258	−0.858516
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	4.13366	0.767602	0.383801	−	0.923416i	$-0.374615\pi$
$29$	4.13366	0.767602	0.383801	+	0.923416i	$0.374615\pi$
$30$	0	0
$31$	0.451497	0.0810913	0.0405457	−	0.999178i	$-0.487090\pi$
$31$	0.451497	0.0810913	0.0405457	+	0.999178i	$0.487090\pi$
$32$	0	0
$33$	4.45150	0.774906
$34$	0	0
$35$	−0.841083	−0.142169
$36$	0	0
$37$	9.81583	1.61371	0.806856	−	0.590748i	$-0.201167\pi$
$37$	9.81583	1.61371	0.806856	+	0.590748i	$0.201167\pi$
$38$	0	0
$39$	−5.29258	−0.847491
$40$	0	0
$41$	1.54850	0.241835	0.120918	−	0.992663i	$-0.461416\pi$
$41$	1.54850	0.241835	0.120918	+	0.992663i	$0.461416\pi$
$42$	0	0
$43$	7.29258	1.11211	0.556054	−	0.831146i	$-0.312315\pi$
$43$	7.29258	1.11211	0.556054	+	0.831146i	$0.312315\pi$
$44$	0	0
$45$	0.841083	0.125381
$46$	0	0
$47$	−11.2208	−1.63673	−0.818363	−	0.574702i	$-0.805118\pi$
$47$	−11.2208	−1.63673	−0.818363	+	0.574702i	$0.805118\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−4.13366	−0.578829
$52$	0	0
$53$	12.0619	1.65683	0.828416	−	0.560114i	$-0.189243\pi$
$53$	12.0619	1.65683	0.828416	+	0.560114i	$0.189243\pi$
$54$	0	0
$55$	3.74408	0.504851
$56$	0	0
$57$	−2.13366	−0.282611
$58$	0	0
$59$	−5.42624	−0.706437	−0.353218	−	0.935541i	$-0.614913\pi$
$59$	−5.42624	−0.706437	−0.353218	+	0.935541i	$0.614913\pi$
$60$	0	0
$61$	12.6569	1.62055	0.810276	−	0.586049i	$-0.199317\pi$
$61$	12.6569	1.62055	0.810276	+	0.586049i	$0.199317\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−4.45150	−0.552140
$66$	0	0
$67$	13.4262	1.64028	0.820138	−	0.572165i	$-0.193896\pi$
$67$	13.4262	1.64028	0.820138	+	0.572165i	$0.193896\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−0.974745	−0.115681	−0.0578405	−	0.998326i	$-0.518421\pi$
$71$	−0.974745	−0.115681	−0.0578405	+	0.998326i	$0.518421\pi$
$72$	0	0
$73$	9.03666	1.05766	0.528830	−	0.848728i	$-0.322631\pi$
$73$	9.03666	1.05766	0.528830	+	0.848728i	$0.322631\pi$
$74$	0	0
$75$	−4.29258	−0.495664
$76$	0	0
$77$	−4.45150	−0.507295
$78$	0	0
$79$	1.86634	0.209979	0.104990	−	0.994473i	$-0.466519\pi$
$79$	1.86634	0.209979	0.104990	+	0.994473i	$0.466519\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−4.45150	−0.488615	−0.244308	−	0.969698i	$-0.578561\pi$
$83$	−4.45150	−0.488615	−0.244308	+	0.969698i	$0.578561\pi$
$84$	0	0
$85$	−3.47675	−0.377107
$86$	0	0
$87$	4.13366	0.443175
$88$	0	0
$89$	10.9747	1.16332	0.581660	−	0.813432i	$-0.302403\pi$
$89$	10.9747	1.16332	0.581660	+	0.813432i	$0.302403\pi$
$90$	0	0
$91$	5.29258	0.554813
$92$	0	0
$93$	0.451497	0.0468181
$94$	0	0
$95$	−1.79459	−0.184121
$96$	0	0
$97$	10.4515	1.06119	0.530594	−	0.847626i	$-0.321969\pi$
$97$	10.4515	1.06119	0.530594	+	0.847626i	$0.321969\pi$
$98$	0	0
$99$	4.45150	0.447392
$100$	0	0
$101$	−3.42624	−0.340924	−0.170462	−	0.985364i	$-0.554526\pi$
$101$	−3.42624	−0.340924	−0.170462	+	0.985364i	$0.554526\pi$
$102$	0	0
$103$	3.36433	0.331497	0.165749	−	0.986168i	$-0.446996\pi$
$103$	3.36433	0.331497	0.165749	+	0.986168i	$0.446996\pi$
$104$	0	0
$105$	−0.841083	−0.0820813
$106$	0	0
$107$	17.2421	1.66685	0.833427	−	0.552630i	$-0.186376\pi$
$107$	17.2421	1.66685	0.833427	+	0.552630i	$0.186376\pi$
$108$	0	0
$109$	9.74408	0.933313	0.466657	−	0.884439i	$-0.345458\pi$
$109$	9.74408	0.933313	0.466657	+	0.884439i	$0.345458\pi$
$110$	0	0
$111$	9.81583	0.931677
$112$	0	0
$113$	9.47675	0.891498	0.445749	−	0.895158i	$-0.352937\pi$
$113$	9.47675	0.891498	0.445749	+	0.895158i	$0.352937\pi$
$114$	0	0
$115$	−0.841083	−0.0784314
$116$	0	0
$117$	−5.29258	−0.489299
$118$	0	0
$119$	4.13366	0.378932
$120$	0	0
$121$	8.81583	0.801439
$122$	0	0
$123$	1.54850	0.139624
$124$	0	0
$125$	−7.81583	−0.699069
$126$	0	0
$127$	5.61041	0.497844	0.248922	−	0.968524i	$-0.419924\pi$
$127$	5.61041	0.497844	0.248922	+	0.968524i	$0.419924\pi$
$128$	0	0
$129$	7.29258	0.642076
$130$	0	0
$131$	−3.54850	−0.310034	−0.155017	−	0.987912i	$-0.549543\pi$
$131$	−3.54850	−0.310034	−0.155017	+	0.987912i	$0.549543\pi$
$132$	0	0
$133$	2.13366	0.185012
$134$	0	0
$135$	0.841083	0.0723889
$136$	0	0
$137$	10.4515	0.892932	0.446466	−	0.894801i	$-0.352682\pi$
$137$	10.4515	0.892932	0.446466	+	0.894801i	$0.352682\pi$
$138$	0	0
$139$	2.20541	0.187061	0.0935304	−	0.995616i	$-0.470185\pi$
$139$	2.20541	0.187061	0.0935304	+	0.995616i	$0.470185\pi$
$140$	0	0
$141$	−11.2208	−0.944964
$142$	0	0
$143$	−23.5599	−1.97018
$144$	0	0
$145$	3.47675	0.288729
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−4.31783	−0.353731	−0.176865	−	0.984235i	$-0.556596\pi$
$149$	−4.31783	−0.353731	−0.176865	+	0.984235i	$0.556596\pi$
$150$	0	0
$151$	−11.2208	−0.913138	−0.456569	−	0.889688i	$-0.650922\pi$
$151$	−11.2208	−0.913138	−0.456569	+	0.889688i	$0.650922\pi$
$152$	0	0
$153$	−4.13366	−0.334187
$154$	0	0
$155$	0.379747	0.0305020
$156$	0	0
$157$	−21.7555	−1.73628	−0.868138	−	0.496323i	$-0.834683\pi$
$157$	−21.7555	−1.73628	−0.868138	+	0.496323i	$0.834683\pi$
$158$	0	0
$159$	12.0619	0.956572
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	3.47675	0.272320	0.136160	−	0.990687i	$-0.456524\pi$
$163$	3.47675	0.272320	0.136160	+	0.990687i	$0.456524\pi$
$164$	0	0
$165$	3.74408	0.291476
$166$	0	0
$167$	4.18417	0.323781	0.161890	−	0.986809i	$-0.448241\pi$
$167$	4.18417	0.323781	0.161890	+	0.986809i	$0.448241\pi$
$168$	0	0
$169$	15.0114	1.15472
$170$	0	0
$171$	−2.13366	−0.163165
$172$	0	0
$173$	1.28118	0.0974061	0.0487031	−	0.998813i	$-0.484491\pi$
$173$	1.28118	0.0974061	0.0487031	+	0.998813i	$0.484491\pi$
$174$	0	0
$175$	4.29258	0.324489
$176$	0	0
$177$	−5.42624	−0.407861
$178$	0	0
$179$	17.6936	1.32248	0.661240	−	0.750175i	$-0.270031\pi$
$179$	17.6936	1.32248	0.661240	+	0.750175i	$0.270031\pi$
$180$	0	0
$181$	−10.0832	−0.749475	−0.374737	−	0.927131i	$-0.622267\pi$
$181$	−10.0832	−0.749475	−0.374737	+	0.927131i	$0.622267\pi$
$182$	0	0
$183$	12.6569	0.935626
$184$	0	0
$185$	8.25592	0.606988
$186$	0	0
$187$	−18.4010	−1.34561
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−22.5852	−1.63420	−0.817102	−	0.576493i	$-0.804421\pi$
$191$	−22.5852	−1.63420	−0.817102	+	0.576493i	$0.804421\pi$
$192$	0	0
$193$	−4.58516	−0.330047	−0.165024	−	0.986290i	$-0.552770\pi$
$193$	−4.58516	−0.330047	−0.165024	+	0.986290i	$0.552770\pi$
$194$	0	0
$195$	−4.45150	−0.318778
$196$	0	0
$197$	18.1956	1.29638	0.648191	−	0.761478i	$-0.275526\pi$
$197$	18.1956	1.29638	0.648191	+	0.761478i	$0.275526\pi$
$198$	0	0
$199$	−1.93809	−0.137387	−0.0686937	−	0.997638i	$-0.521883\pi$
$199$	−1.93809	−0.137387	−0.0686937	+	0.997638i	$0.521883\pi$
$200$	0	0
$201$	13.4262	0.947014
$202$	0	0
$203$	−4.13366	−0.290126
$204$	0	0
$205$	1.30242	0.0909649
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−9.49799	−0.656990
$210$	0	0
$211$	−10.1337	−0.697630	−0.348815	−	0.937192i	$-0.613416\pi$
$211$	−10.1337	−0.697630	−0.348815	+	0.937192i	$0.613416\pi$
$212$	0	0
$213$	−0.974745	−0.0667884
$214$	0	0
$215$	6.13366	0.418312
$216$	0	0
$217$	−0.451497	−0.0306496
$218$	0	0
$219$	9.03666	0.610641
$220$	0	0
$221$	21.8777	1.47166
$222$	0	0
$223$	−5.69357	−0.381269	−0.190635	−	0.981661i	$-0.561055\pi$
$223$	−5.69357	−0.381269	−0.190635	+	0.981661i	$0.561055\pi$
$224$	0	0
$225$	−4.29258	−0.286172
$226$	0	0
$227$	−0.112421	−0.00746166	−0.00373083	−	0.999993i	$-0.501188\pi$
$227$	−0.112421	−0.00746166	−0.00373083	+	0.999993i	$0.501188\pi$
$228$	0	0
$229$	5.47675	0.361914	0.180957	−	0.983491i	$-0.442081\pi$
$229$	5.47675	0.361914	0.180957	+	0.983491i	$0.442081\pi$
$230$	0	0
$231$	−4.45150	−0.292887
$232$	0	0
$233$	9.47675	0.620843	0.310421	−	0.950599i	$-0.399530\pi$
$233$	9.47675	0.620843	0.310421	+	0.950599i	$0.399530\pi$
$234$	0	0
$235$	−9.43764	−0.615644
$236$	0	0
$237$	1.86634	0.121232
$238$	0	0
$239$	−12.7906	−0.827353	−0.413677	−	0.910424i	$-0.635756\pi$
$239$	−12.7906	−0.827353	−0.413677	+	0.910424i	$0.635756\pi$
$240$	0	0
$241$	−1.81583	−0.116968	−0.0584839	−	0.998288i	$-0.518627\pi$
$241$	−1.81583	−0.116968	−0.0584839	+	0.998288i	$0.518627\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0.841083	0.0537348
$246$	0	0
$247$	11.2926	0.718530
$248$	0	0
$249$	−4.45150	−0.282102
$250$	0	0
$251$	13.0465	0.823488	0.411744	−	0.911300i	$-0.364920\pi$
$251$	13.0465	0.823488	0.411744	+	0.911300i	$0.364920\pi$
$252$	0	0
$253$	−4.45150	−0.279863
$254$	0	0
$255$	−3.47675	−0.217723
$256$	0	0
$257$	−10.1238	−0.631507	−0.315753	−	0.948841i	$-0.602257\pi$
$257$	−10.1238	−0.631507	−0.315753	+	0.948841i	$0.602257\pi$
$258$	0	0
$259$	−9.81583	−0.609926
$260$	0	0
$261$	4.13366	0.255867
$262$	0	0
$263$	−12.8623	−0.793125	−0.396562	−	0.918008i	$-0.629797\pi$
$263$	−12.8623	−0.793125	−0.396562	+	0.918008i	$0.629797\pi$
$264$	0	0
$265$	10.1451	0.623206
$266$	0	0
$267$	10.9747	0.671644
$268$	0	0
$269$	2.70742	0.165074	0.0825372	−	0.996588i	$-0.473698\pi$
$269$	2.70742	0.165074	0.0825372	+	0.996588i	$0.473698\pi$
$270$	0	0
$271$	−28.5753	−1.73583	−0.867914	−	0.496715i	$-0.834539\pi$
$271$	−28.5753	−1.73583	−0.867914	+	0.496715i	$0.834539\pi$
$272$	0	0
$273$	5.29258	0.320322
$274$	0	0
$275$	−19.1084	−1.15228
$276$	0	0
$277$	−21.8272	−1.31147	−0.655736	−	0.754991i	$-0.727641\pi$
$277$	−21.8272	−1.31147	−0.655736	+	0.754991i	$0.727641\pi$
$278$	0	0
$279$	0.451497	0.0270304
$280$	0	0
$281$	13.2208	0.788689	0.394344	−	0.918963i	$-0.370972\pi$
$281$	13.2208	0.788689	0.394344	+	0.918963i	$0.370972\pi$
$282$	0	0
$283$	1.69357	0.100672	0.0503361	−	0.998732i	$-0.483971\pi$
$283$	1.69357	0.100672	0.0503361	+	0.998732i	$0.483971\pi$
$284$	0	0
$285$	−1.79459	−0.106302
$286$	0	0
$287$	−1.54850	−0.0914052
$288$	0	0
$289$	0.0871666	0.00512745
$290$	0	0
$291$	10.4515	0.612678
$292$	0	0
$293$	31.4376	1.83661	0.918303	−	0.395877i	$-0.129559\pi$
$293$	31.4376	1.83661	0.918303	+	0.395877i	$0.129559\pi$
$294$	0	0
$295$	−4.56392	−0.265722
$296$	0	0
$297$	4.45150	0.258302
$298$	0	0
$299$	5.29258	0.306078
$300$	0	0
$301$	−7.29258	−0.420337
$302$	0	0
$303$	−3.42624	−0.196832
$304$	0	0
$305$	10.6455	0.609560
$306$	0	0
$307$	−24.3505	−1.38976	−0.694878	−	0.719128i	$-0.744541\pi$
$307$	−24.3505	−1.38976	−0.694878	+	0.719128i	$0.744541\pi$
$308$	0	0
$309$	3.36433	0.191390
$310$	0	0
$311$	−18.0212	−1.02189	−0.510945	−	0.859613i	$-0.670705\pi$
$311$	−18.0212	−1.02189	−0.510945	+	0.859613i	$0.670705\pi$
$312$	0	0
$313$	−2.26733	−0.128157	−0.0640784	−	0.997945i	$-0.520411\pi$
$313$	−2.26733	−0.128157	−0.0640784	+	0.997945i	$0.520411\pi$
$314$	0	0
$315$	−0.841083	−0.0473896
$316$	0	0
$317$	−0.205413	−0.0115372	−0.00576858	−	0.999983i	$-0.501836\pi$
$317$	−0.205413	−0.0115372	−0.00576858	+	0.999983i	$0.501836\pi$
$318$	0	0
$319$	18.4010	1.03026
$320$	0	0
$321$	17.2421	0.962359
$322$	0	0
$323$	8.81984	0.490749
$324$	0	0
$325$	22.7188	1.26021
$326$	0	0
$327$	9.74408	0.538849
$328$	0	0
$329$	11.2208	0.618624
$330$	0	0
$331$	12.9030	0.709213	0.354606	−	0.935016i	$-0.384615\pi$
$331$	12.9030	0.709213	0.354606	+	0.935016i	$0.384615\pi$
$332$	0	0
$333$	9.81583	0.537904
$334$	0	0
$335$	11.2926	0.616980
$336$	0	0
$337$	29.8272	1.62479	0.812396	−	0.583106i	$-0.198163\pi$
$337$	29.8272	1.62479	0.812396	+	0.583106i	$0.198163\pi$
$338$	0	0
$339$	9.47675	0.514707
$340$	0	0
$341$	2.00984	0.108839
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−0.841083	−0.0452824
$346$	0	0
$347$	22.2575	1.19484	0.597422	−	0.801927i	$-0.296192\pi$
$347$	22.2575	1.19484	0.597422	+	0.801927i	$0.296192\pi$
$348$	0	0
$349$	15.4262	0.825748	0.412874	−	0.910788i	$-0.364525\pi$
$349$	15.4262	0.825748	0.412874	+	0.910788i	$0.364525\pi$
$350$	0	0
$351$	−5.29258	−0.282497
$352$	0	0
$353$	29.1605	1.55206	0.776028	−	0.630699i	$-0.217232\pi$
$353$	29.1605	1.55206	0.776028	+	0.630699i	$0.217232\pi$
$354$	0	0
$355$	−0.819841	−0.0435127
$356$	0	0
$357$	4.13366	0.218777
$358$	0	0
$359$	−27.3757	−1.44484	−0.722418	−	0.691457i	$-0.756969\pi$
$359$	−27.3757	−1.44484	−0.722418	+	0.691457i	$0.756969\pi$
$360$	0	0
$361$	−14.4475	−0.760394
$362$	0	0
$363$	8.81583	0.462711
$364$	0	0
$365$	7.60058	0.397832
$366$	0	0
$367$	−2.38959	−0.124735	−0.0623677	−	0.998053i	$-0.519865\pi$
$367$	−2.38959	−0.124735	−0.0623677	+	0.998053i	$0.519865\pi$
$368$	0	0
$369$	1.54850	0.0806118
$370$	0	0
$371$	−12.0619	−0.626223
$372$	0	0
$373$	−8.27716	−0.428575	−0.214288	−	0.976771i	$-0.568743\pi$
$373$	−8.27716	−0.428575	−0.214288	+	0.976771i	$0.568743\pi$
$374$	0	0
$375$	−7.81583	−0.403608
$376$	0	0
$377$	−21.8777	−1.12676
$378$	0	0
$379$	23.8990	1.22761	0.613804	−	0.789458i	$-0.289638\pi$
$379$	23.8990	1.22761	0.613804	+	0.789458i	$0.289638\pi$
$380$	0	0
$381$	5.61041	0.287430
$382$	0	0
$383$	−2.91283	−0.148839	−0.0744194	−	0.997227i	$-0.523710\pi$
$383$	−2.91283	−0.148839	−0.0744194	+	0.997227i	$0.523710\pi$
$384$	0	0
$385$	−3.74408	−0.190816
$386$	0	0
$387$	7.29258	0.370703
$388$	0	0
$389$	18.2070	0.923130	0.461565	−	0.887106i	$-0.347288\pi$
$389$	18.2070	0.923130	0.461565	+	0.887106i	$0.347288\pi$
$390$	0	0
$391$	4.13366	0.209048
$392$	0	0
$393$	−3.54850	−0.178998
$394$	0	0
$395$	1.56974	0.0789824
$396$	0	0
$397$	5.48815	0.275443	0.137721	−	0.990471i	$-0.456022\pi$
$397$	5.48815	0.275443	0.137721	+	0.990471i	$0.456022\pi$
$398$	0	0
$399$	2.13366	0.106817
$400$	0	0
$401$	−34.2070	−1.70821	−0.854107	−	0.520097i	$-0.825896\pi$
$401$	−34.2070	−1.70821	−0.854107	+	0.520097i	$0.825896\pi$
$402$	0	0
$403$	−2.38959	−0.119034
$404$	0	0
$405$	0.841083	0.0417937
$406$	0	0
$407$	43.6951	2.16589
$408$	0	0
$409$	14.7188	0.727799	0.363899	−	0.931438i	$-0.381445\pi$
$409$	14.7188	0.727799	0.363899	+	0.931438i	$0.381445\pi$
$410$	0	0
$411$	10.4515	0.515534
$412$	0	0
$413$	5.42624	0.267008
$414$	0	0
$415$	−3.74408	−0.183790
$416$	0	0
$417$	2.20541	0.108000
$418$	0	0
$419$	14.8411	0.725034	0.362517	−	0.931977i	$-0.381917\pi$
$419$	14.8411	0.725034	0.362517	+	0.931977i	$0.381917\pi$
$420$	0	0
$421$	−34.8312	−1.69757	−0.848785	−	0.528737i	$-0.822666\pi$
$421$	−34.8312	−1.69757	−0.848785	+	0.528737i	$0.822666\pi$
$422$	0	0
$423$	−11.2208	−0.545575
$424$	0	0
$425$	17.7441	0.860714
$426$	0	0
$427$	−12.6569	−0.612511
$428$	0	0
$429$	−23.5599	−1.13748
$430$	0	0
$431$	12.5639	0.605183	0.302591	−	0.953120i	$-0.402148\pi$
$431$	12.5639	0.605183	0.302591	+	0.953120i	$0.402148\pi$
$432$	0	0
$433$	−22.3911	−1.07605	−0.538025	−	0.842929i	$-0.680829\pi$
$433$	−22.3911	−1.07605	−0.538025	+	0.842929i	$0.680829\pi$
$434$	0	0
$435$	3.47675	0.166697
$436$	0	0
$437$	2.13366	0.102067
$438$	0	0
$439$	15.1802	0.724509	0.362255	−	0.932079i	$-0.382007\pi$
$439$	15.1802	0.724509	0.362255	+	0.932079i	$0.382007\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	0.635669	0.0302016	0.0151008	−	0.999886i	$-0.495193\pi$
$443$	0.635669	0.0302016	0.0151008	+	0.999886i	$0.495193\pi$
$444$	0	0
$445$	9.23067	0.437576
$446$	0	0
$447$	−4.31783	−0.204227
$448$	0	0
$449$	−6.01140	−0.283696	−0.141848	−	0.989888i	$-0.545304\pi$
$449$	−6.01140	−0.283696	−0.141848	+	0.989888i	$0.545304\pi$
$450$	0	0
$451$	6.89316	0.324586
$452$	0	0
$453$	−11.2208	−0.527201
$454$	0	0
$455$	4.45150	0.208689
$456$	0	0
$457$	−11.3024	−0.528705	−0.264352	−	0.964426i	$-0.585158\pi$
$457$	−11.3024	−0.528705	−0.264352	+	0.964426i	$0.585158\pi$
$458$	0	0
$459$	−4.13366	−0.192943
$460$	0	0
$461$	−8.59656	−0.400382	−0.200191	−	0.979757i	$-0.564156\pi$
$461$	−8.59656	−0.400382	−0.200191	+	0.979757i	$0.564156\pi$
$462$	0	0
$463$	31.4278	1.46057	0.730287	−	0.683140i	$-0.239386\pi$
$463$	31.4278	1.46057	0.730287	+	0.683140i	$0.239386\pi$
$464$	0	0
$465$	0.379747	0.0176103
$466$	0	0
$467$	17.0872	0.790700	0.395350	−	0.918531i	$-0.370623\pi$
$467$	17.0872	0.790700	0.395350	+	0.918531i	$0.370623\pi$
$468$	0	0
$469$	−13.4262	−0.619966
$470$	0	0
$471$	−21.7555	−1.00244
$472$	0	0
$473$	32.4629	1.49265
$474$	0	0
$475$	9.15892	0.420240
$476$	0	0
$477$	12.0619	0.552277
$478$	0	0
$479$	−34.0326	−1.55499	−0.777496	−	0.628888i	$-0.783510\pi$
$479$	−34.0326	−1.55499	−0.777496	+	0.628888i	$0.783510\pi$
$480$	0	0
$481$	−51.9511	−2.36876
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	8.79057	0.399159
$486$	0	0
$487$	20.7188	0.938859	0.469430	−	0.882970i	$-0.344460\pi$
$487$	20.7188	0.938859	0.469430	+	0.882970i	$0.344460\pi$
$488$	0	0
$489$	3.47675	0.157224
$490$	0	0
$491$	−23.1491	−1.04470	−0.522352	−	0.852730i	$-0.674945\pi$
$491$	−23.1491	−1.04470	−0.522352	+	0.852730i	$0.674945\pi$
$492$	0	0
$493$	−17.0872	−0.769567
$494$	0	0
$495$	3.74408	0.168284
$496$	0	0
$497$	0.974745	0.0437233
$498$	0	0
$499$	9.01542	0.403585	0.201793	−	0.979428i	$-0.435323\pi$
$499$	9.01542	0.403585	0.201793	+	0.979428i	$0.435323\pi$
$500$	0	0
$501$	4.18417	0.186935
$502$	0	0
$503$	33.0579	1.47398	0.736989	−	0.675904i	$-0.236247\pi$
$503$	33.0579	1.47398	0.736989	+	0.675904i	$0.236247\pi$
$504$	0	0
$505$	−2.88175	−0.128236
$506$	0	0
$507$	15.0114	0.666680
$508$	0	0
$509$	13.7752	0.610573	0.305287	−	0.952261i	$-0.401248\pi$
$509$	13.7752	0.610573	0.305287	+	0.952261i	$0.401248\pi$
$510$	0	0
$511$	−9.03666	−0.399758
$512$	0	0
$513$	−2.13366	−0.0942035
$514$	0	0
$515$	2.82968	0.124691
$516$	0	0
$517$	−49.9495	−2.19678
$518$	0	0
$519$	1.28118	0.0562375
$520$	0	0
$521$	31.1703	1.36560	0.682798	−	0.730607i	$-0.260763\pi$
$521$	31.1703	1.36560	0.682798	+	0.730607i	$0.260763\pi$
$522$	0	0
$523$	−28.9030	−1.26384	−0.631920	−	0.775034i	$-0.717733\pi$
$523$	−28.9030	−1.26384	−0.631920	+	0.775034i	$0.717733\pi$
$524$	0	0
$525$	4.29258	0.187344
$526$	0	0
$527$	−1.86634	−0.0812989
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−5.42624	−0.235479
$532$	0	0
$533$	−8.19557	−0.354990
$534$	0	0
$535$	14.5020	0.626976
$536$	0	0
$537$	17.6936	0.763534
$538$	0	0
$539$	4.45150	0.191740
$540$	0	0
$541$	−19.0465	−0.818873	−0.409436	−	0.912339i	$-0.634275\pi$
$541$	−19.0465	−0.818873	−0.409436	+	0.912339i	$0.634275\pi$
$542$	0	0
$543$	−10.0832	−0.432710
$544$	0	0
$545$	8.19557	0.351060
$546$	0	0
$547$	−15.2322	−0.651283	−0.325642	−	0.945493i	$-0.605580\pi$
$547$	−15.2322	−0.651283	−0.325642	+	0.945493i	$0.605580\pi$
$548$	0	0
$549$	12.6569	0.540184
$550$	0	0
$551$	−8.81984	−0.375738
$552$	0	0
$553$	−1.86634	−0.0793647
$554$	0	0
$555$	8.25592	0.350444
$556$	0	0
$557$	−17.1198	−0.725390	−0.362695	−	0.931908i	$-0.618143\pi$
$557$	−17.1198	−0.725390	−0.362695	+	0.931908i	$0.618143\pi$
$558$	0	0
$559$	−38.5966	−1.63246
$560$	0	0
$561$	−18.4010	−0.776890
$562$	0	0
$563$	−13.3431	−0.562344	−0.281172	−	0.959657i	$-0.590723\pi$
$563$	−13.3431	−0.562344	−0.281172	+	0.959657i	$0.590723\pi$
$564$	0	0
$565$	7.97073	0.335331
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−12.0733	−0.506140	−0.253070	−	0.967448i	$-0.581440\pi$
$569$	−12.0733	−0.506140	−0.253070	+	0.967448i	$0.581440\pi$
$570$	0	0
$571$	−12.8426	−0.537448	−0.268724	−	0.963217i	$-0.586602\pi$
$571$	−12.8426	−0.537448	−0.268724	+	0.963217i	$0.586602\pi$
$572$	0	0
$573$	−22.5852	−0.943509
$574$	0	0
$575$	4.29258	0.179013
$576$	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
$579$	−4.58516	−0.190553
$580$	0	0
$581$	4.45150	0.184679
$582$	0	0
$583$	53.6936	2.22376
$584$	0	0
$585$	−4.45150	−0.184047
$586$	0	0
$587$	−6.10258	−0.251881	−0.125940	−	0.992038i	$-0.540195\pi$
$587$	−6.10258	−0.251881	−0.125940	+	0.992038i	$0.540195\pi$
$588$	0	0
$589$	−0.963343	−0.0396939
$590$	0	0
$591$	18.1956	0.748466
$592$	0	0
$593$	−1.75548	−0.0720889	−0.0360445	−	0.999350i	$-0.511476\pi$
$593$	−1.75548	−0.0720889	−0.0360445	+	0.999350i	$0.511476\pi$
$594$	0	0
$595$	3.47675	0.142533
$596$	0	0
$597$	−1.93809	−0.0793207
$598$	0	0
$599$	−13.2421	−0.541056	−0.270528	−	0.962712i	$-0.587198\pi$
$599$	−13.2421	−0.541056	−0.270528	+	0.962712i	$0.587198\pi$
$600$	0	0
$601$	29.3757	1.19826	0.599131	−	0.800651i	$-0.295513\pi$
$601$	29.3757	1.19826	0.599131	+	0.800651i	$0.295513\pi$
$602$	0	0
$603$	13.4262	0.546759
$604$	0	0
$605$	7.41484	0.301456
$606$	0	0
$607$	27.1491	1.10195	0.550974	−	0.834523i	$-0.314257\pi$
$607$	27.1491	1.10195	0.550974	+	0.834523i	$0.314257\pi$
$608$	0	0
$609$	−4.13366	−0.167504
$610$	0	0
$611$	59.3871	2.40255
$612$	0	0
$613$	10.5753	0.427133	0.213567	−	0.976929i	$-0.431492\pi$
$613$	10.5753	0.427133	0.213567	+	0.976929i	$0.431492\pi$
$614$	0	0
$615$	1.30242	0.0525186
$616$	0	0
$617$	8.14507	0.327908	0.163954	−	0.986468i	$-0.447575\pi$
$617$	8.14507	0.327908	0.163954	+	0.986468i	$0.447575\pi$
$618$	0	0
$619$	−14.1026	−0.566831	−0.283415	−	0.958997i	$-0.591467\pi$
$619$	−14.1026	−0.566831	−0.283415	+	0.958997i	$0.591467\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−10.9747	−0.439694
$624$	0	0
$625$	14.8891	0.595566
$626$	0	0
$627$	−9.49799	−0.379313
$628$	0	0
$629$	−40.5753	−1.61784
$630$	0	0
$631$	−24.9861	−0.994683	−0.497341	−	0.867555i	$-0.665690\pi$
$631$	−24.9861	−0.994683	−0.497341	+	0.867555i	$0.665690\pi$
$632$	0	0
$633$	−10.1337	−0.402777
$634$	0	0
$635$	4.71882	0.187261
$636$	0	0
$637$	−5.29258	−0.209700
$638$	0	0
$639$	−0.974745	−0.0385603
$640$	0	0
$641$	13.1491	0.519357	0.259679	−	0.965695i	$-0.416383\pi$
$641$	13.1491	0.519357	0.259679	+	0.965695i	$0.416383\pi$
$642$	0	0
$643$	23.0677	0.909703	0.454851	−	0.890567i	$-0.349692\pi$
$643$	23.0677	0.909703	0.454851	+	0.890567i	$0.349692\pi$
$644$	0	0
$645$	6.13366	0.241513
$646$	0	0
$647$	−3.88758	−0.152836	−0.0764182	−	0.997076i	$-0.524348\pi$
$647$	−3.88758	−0.152836	−0.0764182	+	0.997076i	$0.524348\pi$
$648$	0	0
$649$	−24.1549	−0.948163
$650$	0	0
$651$	−0.451497	−0.0176956
$652$	0	0
$653$	7.56974	0.296227	0.148113	−	0.988970i	$-0.452680\pi$
$653$	7.56974	0.296227	0.148113	+	0.988970i	$0.452680\pi$
$654$	0	0
$655$	−2.98458	−0.116617
$656$	0	0
$657$	9.03666	0.352554
$658$	0	0
$659$	−25.2535	−0.983736	−0.491868	−	0.870670i	$-0.663686\pi$
$659$	−25.2535	−0.983736	−0.491868	+	0.870670i	$0.663686\pi$
$660$	0	0
$661$	25.8158	1.00412	0.502060	−	0.864833i	$-0.332576\pi$
$661$	25.8158	1.00412	0.502060	+	0.864833i	$0.332576\pi$
$662$	0	0
$663$	21.8777	0.849661
$664$	0	0
$665$	1.79459	0.0695911
$666$	0	0
$667$	−4.13366	−0.160056
$668$	0	0
$669$	−5.69357	−0.220126
$670$	0	0
$671$	56.3422	2.17507
$672$	0	0
$673$	−46.9861	−1.81118	−0.905591	−	0.424151i	$-0.860573\pi$
$673$	−46.9861	−1.81118	−0.905591	+	0.424151i	$0.860573\pi$
$674$	0	0
$675$	−4.29258	−0.165221
$676$	0	0
$677$	28.9046	1.11089	0.555446	−	0.831552i	$-0.312547\pi$
$677$	28.9046	1.11089	0.555446	+	0.831552i	$0.312547\pi$
$678$	0	0
$679$	−10.4515	−0.401092
$680$	0	0
$681$	−0.112421	−0.00430799
$682$	0	0
$683$	39.4475	1.50942	0.754708	−	0.656061i	$-0.227779\pi$
$683$	39.4475	1.50942	0.754708	+	0.656061i	$0.227779\pi$
$684$	0	0
$685$	8.79057	0.335871
$686$	0	0
$687$	5.47675	0.208951
$688$	0	0
$689$	−63.8386	−2.43206
$690$	0	0
$691$	−44.3194	−1.68599	−0.842995	−	0.537922i	$-0.819210\pi$
$691$	−44.3194	−1.68599	−0.842995	+	0.537922i	$0.819210\pi$
$692$	0	0
$693$	−4.45150	−0.169098
$694$	0	0
$695$	1.85493	0.0703617
$696$	0	0
$697$	−6.40099	−0.242455
$698$	0	0
$699$	9.47675	0.358444
$700$	0	0
$701$	−19.4262	−0.733719	−0.366860	−	0.930276i	$-0.619567\pi$
$701$	−19.4262	−0.733719	−0.366860	+	0.930276i	$0.619567\pi$
$702$	0	0
$703$	−20.9437	−0.789905
$704$	0	0
$705$	−9.43764	−0.355442
$706$	0	0
$707$	3.42624	0.128857
$708$	0	0
$709$	−47.2421	−1.77421	−0.887107	−	0.461565i	$-0.847288\pi$
$709$	−47.2421	−1.77421	−0.887107	+	0.461565i	$0.847288\pi$
$710$	0	0
$711$	1.86634	0.0699931
$712$	0	0
$713$	−0.451497	−0.0169087
$714$	0	0
$715$	−19.8158	−0.741070
$716$	0	0
$717$	−12.7906	−0.477673
$718$	0	0
$719$	−11.1377	−0.415365	−0.207683	−	0.978196i	$-0.566592\pi$
$719$	−11.1377	−0.415365	−0.207683	+	0.978196i	$0.566592\pi$
$720$	0	0
$721$	−3.36433	−0.125294
$722$	0	0
$723$	−1.81583	−0.0675314
$724$	0	0
$725$	−17.7441	−0.658998
$726$	0	0
$727$	39.2030	1.45396	0.726979	−	0.686660i	$-0.240924\pi$
$727$	39.2030	1.45396	0.726979	+	0.686660i	$0.240924\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−30.1451	−1.11496
$732$	0	0
$733$	30.9258	1.14227	0.571135	−	0.820856i	$-0.306503\pi$
$733$	30.9258	1.14227	0.571135	+	0.820856i	$0.306503\pi$
$734$	0	0
$735$	0.841083	0.0310238
$736$	0	0
$737$	59.7669	2.20154
$738$	0	0
$739$	−10.1337	−0.372773	−0.186386	−	0.982477i	$-0.559678\pi$
$739$	−10.1337	−0.372773	−0.186386	+	0.982477i	$0.559678\pi$
$740$	0	0
$741$	11.2926	0.414843
$742$	0	0
$743$	−43.0887	−1.58077	−0.790386	−	0.612609i	$-0.790120\pi$
$743$	−43.0887	−1.58077	−0.790386	+	0.612609i	$0.790120\pi$
$744$	0	0
$745$	−3.63166	−0.133054
$746$	0	0
$747$	−4.45150	−0.162872
$748$	0	0
$749$	−17.2421	−0.630012
$750$	0	0
$751$	46.4531	1.69510	0.847548	−	0.530719i	$-0.178078\pi$
$751$	46.4531	1.69510	0.847548	+	0.530719i	$0.178078\pi$
$752$	0	0
$753$	13.0465	0.475441
$754$	0	0
$755$	−9.43764	−0.343471
$756$	0	0
$757$	−21.2812	−0.773478	−0.386739	−	0.922189i	$-0.626398\pi$
$757$	−21.2812	−0.773478	−0.386739	+	0.922189i	$0.626398\pi$
$758$	0	0
$759$	−4.45150	−0.161579
$760$	0	0
$761$	12.9960	0.471104	0.235552	−	0.971862i	$-0.424310\pi$
$761$	12.9960	0.471104	0.235552	+	0.971862i	$0.424310\pi$
$762$	0	0
$763$	−9.74408	−0.352759
$764$	0	0
$765$	−3.47675	−0.125702
$766$	0	0
$767$	28.7188	1.03698
$768$	0	0
$769$	23.5615	0.849648	0.424824	−	0.905276i	$-0.360336\pi$
$769$	23.5615	0.849648	0.424824	+	0.905276i	$0.360336\pi$
$770$	0	0
$771$	−10.1238	−0.364601
$772$	0	0
$773$	54.6991	1.96739	0.983696	−	0.179841i	$-0.0575582\pi$
$773$	54.6991	1.96739	0.983696	+	0.179841i	$0.0575582\pi$
$774$	0	0
$775$	−1.93809	−0.0696182
$776$	0	0
$777$	−9.81583	−0.352141
$778$	0	0
$779$	−3.30398	−0.118377
$780$	0	0
$781$	−4.33908	−0.155264
$782$	0	0
$783$	4.13366	0.147725
$784$	0	0
$785$	−18.2982	−0.653089
$786$	0	0
$787$	−6.47274	−0.230728	−0.115364	−	0.993323i	$-0.536803\pi$
$787$	−6.47274	−0.230728	−0.115364	+	0.993323i	$0.536803\pi$
$788$	0	0
$789$	−12.8623	−0.457911
$790$	0	0
$791$	−9.47675	−0.336955
$792$	0	0
$793$	−66.9877	−2.37880
$794$	0	0
$795$	10.1451	0.359808
$796$	0	0
$797$	−10.5118	−0.372349	−0.186174	−	0.982517i	$-0.559609\pi$
$797$	−10.5118	−0.372349	−0.186174	+	0.982517i	$0.559609\pi$
$798$	0	0
$799$	46.3831	1.64092
$800$	0	0
$801$	10.9747	0.387774
$802$	0	0
$803$	40.2267	1.41957
$804$	0	0
$805$	0.841083	0.0296443
$806$	0	0
$807$	2.70742	0.0953057
$808$	0	0
$809$	−30.9976	−1.08982	−0.544908	−	0.838496i	$-0.683435\pi$
$809$	−30.9976	−1.08982	−0.544908	+	0.838496i	$0.683435\pi$
$810$	0	0
$811$	−17.2307	−0.605051	−0.302525	−	0.953141i	$-0.597830\pi$
$811$	−17.2307	−0.605051	−0.302525	+	0.953141i	$0.597830\pi$
$812$	0	0
$813$	−28.5753	−1.00218
$814$	0	0
$815$	2.92424	0.102432
$816$	0	0
$817$	−15.5599	−0.544372
$818$	0	0
$819$	5.29258	0.184938
$820$	0	0
$821$	10.1238	0.353324	0.176662	−	0.984272i	$-0.443470\pi$
$821$	10.1238	0.353324	0.176662	+	0.984272i	$0.443470\pi$
$822$	0	0
$823$	30.5966	1.06653	0.533265	−	0.845949i	$-0.320965\pi$
$823$	30.5966	1.06653	0.533265	+	0.845949i	$0.320965\pi$
$824$	0	0
$825$	−19.1084	−0.665269
$826$	0	0
$827$	−20.3797	−0.708673	−0.354337	−	0.935118i	$-0.615293\pi$
$827$	−20.3797	−0.708673	−0.354337	+	0.935118i	$0.615293\pi$
$828$	0	0
$829$	19.9088	0.691462	0.345731	−	0.938334i	$-0.387631\pi$
$829$	19.9088	0.691462	0.345731	+	0.938334i	$0.387631\pi$
$830$	0	0
$831$	−21.8272	−0.757178
$832$	0	0
$833$	−4.13366	−0.143223
$834$	0	0
$835$	3.51923	0.121788
$836$	0	0
$837$	0.451497	0.0156060
$838$	0	0
$839$	−18.7807	−0.648383	−0.324191	−	0.945991i	$-0.605092\pi$
$839$	−18.7807	−0.648383	−0.324191	+	0.945991i	$0.605092\pi$
$840$	0	0
$841$	−11.9128	−0.410787
$842$	0	0
$843$	13.2208	0.455350
$844$	0	0
$845$	12.6258	0.434342
$846$	0	0
$847$	−8.81583	−0.302915
$848$	0	0
$849$	1.69357	0.0581231
$850$	0	0
$851$	−9.81583	−0.336482
$852$	0	0
$853$	−50.3911	−1.72536	−0.862680	−	0.505750i	$-0.831216\pi$
$853$	−50.3911	−1.72536	−0.862680	+	0.505750i	$0.831216\pi$
$854$	0	0
$855$	−1.79459	−0.0613736
$856$	0	0
$857$	5.50783	0.188144	0.0940720	−	0.995565i	$-0.470012\pi$
$857$	5.50783	0.188144	0.0940720	+	0.995565i	$0.470012\pi$
$858$	0	0
$859$	51.3447	1.75186	0.875928	−	0.482441i	$-0.160250\pi$
$859$	51.3447	1.75186	0.875928	+	0.482441i	$0.160250\pi$
$860$	0	0
$861$	−1.54850	−0.0527728
$862$	0	0
$863$	28.4108	0.967116	0.483558	−	0.875312i	$-0.339344\pi$
$863$	28.4108	0.967116	0.483558	+	0.875312i	$0.339344\pi$
$864$	0	0
$865$	1.07758	0.0366387
$866$	0	0
$867$	0.0871666	0.00296033
$868$	0	0
$869$	8.30800	0.281829
$870$	0	0
$871$	−71.0595	−2.40776
$872$	0	0
$873$	10.4515	0.353730
$874$	0	0
$875$	7.81583	0.264223
$876$	0	0
$877$	55.1506	1.86230	0.931152	−	0.364630i	$-0.118805\pi$
$877$	55.1506	1.86230	0.931152	+	0.364630i	$0.118805\pi$
$878$	0	0
$879$	31.4376	1.06037
$880$	0	0
$881$	54.2901	1.82908	0.914540	−	0.404494i	$-0.132552\pi$
$881$	54.2901	1.82908	0.914540	+	0.404494i	$0.132552\pi$
$882$	0	0
$883$	−48.2184	−1.62268	−0.811339	−	0.584576i	$-0.801261\pi$
$883$	−48.2184	−1.62268	−0.811339	+	0.584576i	$0.801261\pi$
$884$	0	0
$885$	−4.56392	−0.153414
$886$	0	0
$887$	29.2421	0.981853	0.490926	−	0.871201i	$-0.336658\pi$
$887$	29.2421	0.981853	0.490926	+	0.871201i	$0.336658\pi$
$888$	0	0
$889$	−5.61041	−0.188167
$890$	0	0
$891$	4.45150	0.149131
$892$	0	0
$893$	23.9415	0.801171
$894$	0	0
$895$	14.8818	0.497442
$896$	0	0
$897$	5.29258	0.176714
$898$	0	0
$899$	1.86634	0.0622458
$900$	0	0
$901$	−49.8599	−1.66107
$902$	0	0
$903$	−7.29258	−0.242682
$904$	0	0
$905$	−8.48077	−0.281910
$906$	0	0
$907$	36.5461	1.21349	0.606746	−	0.794896i	$-0.292475\pi$
$907$	36.5461	1.21349	0.606746	+	0.794896i	$0.292475\pi$
$908$	0	0
$909$	−3.42624	−0.113641
$910$	0	0
$911$	−33.6625	−1.11529	−0.557644	−	0.830080i	$-0.688295\pi$
$911$	−33.6625	−1.11529	−0.557644	+	0.830080i	$0.688295\pi$
$912$	0	0
$913$	−19.8158	−0.655808
$914$	0	0
$915$	10.6455	0.351930
$916$	0	0
$917$	3.54850	0.117182
$918$	0	0
$919$	−22.8932	−0.755176	−0.377588	−	0.925974i	$-0.623246\pi$
$919$	−22.8932	−0.755176	−0.377588	+	0.925974i	$0.623246\pi$
$920$	0	0
$921$	−24.3505	−0.802376
$922$	0	0
$923$	5.15892	0.169808
$924$	0	0
$925$	−42.1352	−1.38540
$926$	0	0
$927$	3.36433	0.110499
$928$	0	0
$929$	58.3212	1.91346	0.956728	−	0.290982i	$-0.0939821\pi$
$929$	58.3212	1.91346	0.956728	+	0.290982i	$0.0939821\pi$
$930$	0	0
$931$	−2.13366	−0.0699280
$932$	0	0
$933$	−18.0212	−0.589989
$934$	0	0
$935$	−15.4768	−0.506144
$936$	0	0
$937$	35.4980	1.15967	0.579834	−	0.814734i	$-0.303117\pi$
$937$	35.4980	1.15967	0.579834	+	0.814734i	$0.303117\pi$
$938$	0	0
$939$	−2.26733	−0.0739914
$940$	0	0
$941$	−54.5150	−1.77714	−0.888569	−	0.458744i	$-0.848300\pi$
$941$	−54.5150	−1.77714	−0.888569	+	0.458744i	$0.848300\pi$
$942$	0	0
$943$	−1.54850	−0.0504262
$944$	0	0
$945$	−0.841083	−0.0273604
$946$	0	0
$947$	−41.3366	−1.34326	−0.671630	−	0.740887i	$-0.734405\pi$
$947$	−41.3366	−1.34326	−0.671630	+	0.740887i	$0.734405\pi$
$948$	0	0
$949$	−47.8272	−1.55254
$950$	0	0
$951$	−0.205413	−0.00666098
$952$	0	0
$953$	−51.2649	−1.66063	−0.830316	−	0.557293i	$-0.811840\pi$
$953$	−51.2649	−1.66063	−0.830316	+	0.557293i	$0.811840\pi$
$954$	0	0
$955$	−18.9960	−0.614696
$956$	0	0
$957$	18.4010	0.594819
$958$	0	0
$959$	−10.4515	−0.337496
$960$	0	0
$961$	−30.7962	−0.993424
$962$	0	0
$963$	17.2421	0.555618
$964$	0	0
$965$	−3.85650	−0.124145
$966$	0	0
$967$	39.6317	1.27447	0.637234	−	0.770670i	$-0.280078\pi$
$967$	39.6317	1.27447	0.637234	+	0.770670i	$0.280078\pi$
$968$	0	0
$969$	8.81984	0.283334
$970$	0	0
$971$	28.3194	0.908813	0.454406	−	0.890795i	$-0.349851\pi$
$971$	28.3194	0.908813	0.454406	+	0.890795i	$0.349851\pi$
$972$	0	0
$973$	−2.20541	−0.0707023
$974$	0	0
$975$	22.7188	0.727585
$976$	0	0
$977$	26.6064	0.851214	0.425607	−	0.904908i	$-0.360061\pi$
$977$	26.6064	0.851214	0.425607	+	0.904908i	$0.360061\pi$
$978$	0	0
$979$	48.8540	1.56138
$980$	0	0
$981$	9.74408	0.311104
$982$	0	0
$983$	31.7148	1.01155	0.505773	−	0.862667i	$-0.331207\pi$
$983$	31.7148	1.01155	0.505773	+	0.862667i	$0.331207\pi$
$984$	0	0
$985$	15.3040	0.487625
$986$	0	0
$987$	11.2208	0.357163
$988$	0	0
$989$	−7.29258	−0.231891
$990$	0	0
$991$	3.94793	0.125410	0.0627050	−	0.998032i	$-0.480027\pi$
$991$	3.94793	0.125410	0.0627050	+	0.998032i	$0.480027\pi$
$992$	0	0
$993$	12.9030	0.409464
$994$	0	0
$995$	−1.63009	−0.0516774
$996$	0	0
$997$	2.35048	0.0744404	0.0372202	−	0.999307i	$-0.488150\pi$
$997$	2.35048	0.0744404	0.0372202	+	0.999307i	$0.488150\pi$
$998$	0	0
$999$	9.81583	0.310559

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.by.1.2		3
4.3	odd	2		3864.2.a.n.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3864.2.a.n.1.2	✓	3	4.3	odd	2
7728.2.a.by.1.2		3	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:	\( N \)	\(=\)	\( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7728.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.841083 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +0.841083 q^{5} -1.00000 q^{7} +1.00000 q^{9} +4.45150 q^{11} -5.29258 q^{13} +0.841083 q^{15} -4.13366 q^{17} -2.13366 q^{19} -1.00000 q^{21} -1.00000 q^{23} -4.29258 q^{25} +1.00000 q^{27} +4.13366 q^{29} +0.451497 q^{31} +4.45150 q^{33} -0.841083 q^{35} +9.81583 q^{37} -5.29258 q^{39} +1.54850 q^{41} +7.29258 q^{43} +0.841083 q^{45} -11.2208 q^{47} +1.00000 q^{49} -4.13366 q^{51} +12.0619 q^{53} +3.74408 q^{55} -2.13366 q^{57} -5.42624 q^{59} +12.6569 q^{61} -1.00000 q^{63} -4.45150 q^{65} +13.4262 q^{67} -1.00000 q^{69} -0.974745 q^{71} +9.03666 q^{73} -4.29258 q^{75} -4.45150 q^{77} +1.86634 q^{79} +1.00000 q^{81} -4.45150 q^{83} -3.47675 q^{85} +4.13366 q^{87} +10.9747 q^{89} +5.29258 q^{91} +0.451497 q^{93} -1.79459 q^{95} +10.4515 q^{97} +4.45150 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + q^5 - 3 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} + q^{5} - 3 q^{7} + 3 q^{9} + 2 q^{11} - 3 q^{13} + q^{15} + 2 q^{17} + 8 q^{19} - 3 q^{21} - 3 q^{23} + 3 q^{27} - 2 q^{29} - 10 q^{31} + 2 q^{33} - q^{35} + 12 q^{37} - 3 q^{39} + 16 q^{41} + 9 q^{43} + q^{45} - 14 q^{47} + 3 q^{49} + 2 q^{51} + 15 q^{53} - 13 q^{55} + 8 q^{57} + 11 q^{59} + 19 q^{61} - 3 q^{63} - 2 q^{65} + 13 q^{67} - 3 q^{69} + 13 q^{71} - 10 q^{73} - 2 q^{77} + 20 q^{79} + 3 q^{81} - 2 q^{83} - 15 q^{85} - 2 q^{87} + 17 q^{89} + 3 q^{91} - 10 q^{93} - 13 q^{95} + 20 q^{97} + 2 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + q^5 - 3 * q^7 + 3 * q^9 + 2 * q^11 - 3 * q^13 + q^15 + 2 * q^17 + 8 * q^19 - 3 * q^21 - 3 * q^23 + 3 * q^27 - 2 * q^29 - 10 * q^31 + 2 * q^33 - q^35 + 12 * q^37 - 3 * q^39 + 16 * q^41 + 9 * q^43 + q^45 - 14 * q^47 + 3 * q^49 + 2 * q^51 + 15 * q^53 - 13 * q^55 + 8 * q^57 + 11 * q^59 + 19 * q^61 - 3 * q^63 - 2 * q^65 + 13 * q^67 - 3 * q^69 + 13 * q^71 - 10 * q^73 - 2 * q^77 + 20 * q^79 + 3 * q^81 - 2 * q^83 - 15 * q^85 - 2 * q^87 + 17 * q^89 + 3 * q^91 - 10 * q^93 - 13 * q^95 + 20 * q^97 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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