LMFDB - Embedded newform 7728.2.a.ce.1.3

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:	$4$
Coefficient field:	4.4.15317.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 5x + 2 $ x^4 - 2x^3 - 4x^2 + 5*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 483)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.329727$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +3.17434 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.17434 q^{5} +1.00000 q^{7} +1.00000 q^{9} -5.06562 q^{11} +4.07644 q^{13} +3.17434 q^{15} +4.22101 q^{17} +5.06562 q^{19} +1.00000 q^{21} -1.00000 q^{23} +5.07644 q^{25} +1.00000 q^{27} +6.68466 q^{29} +2.22101 q^{31} -5.06562 q^{33} +3.17434 q^{35} -1.91023 q^{37} +4.07644 q^{39} -1.37639 q^{41} +3.39535 q^{43} +3.17434 q^{45} +5.81484 q^{47} +1.00000 q^{49} +4.22101 q^{51} -6.57594 q^{53} -16.0800 q^{55} +5.06562 q^{57} +5.67027 q^{59} -14.4862 q^{61} +1.00000 q^{63} +12.9400 q^{65} -13.8842 q^{67} -1.00000 q^{69} +2.95333 q^{71} -2.02520 q^{73} +5.07644 q^{75} -5.06562 q^{77} -7.14831 q^{79} +1.00000 q^{81} +12.7226 q^{83} +13.3989 q^{85} +6.68466 q^{87} -17.0629 q^{89} +4.07644 q^{91} +2.22101 q^{93} +16.0800 q^{95} +2.08977 q^{97} -5.06562 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 5 q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 5 * q^5 + 4 * q^7 + 4 * q^9	$ 4 q + 4 q^{3} + 5 q^{5} + 4 q^{7} + 4 q^{9} - q^{11} + 7 q^{13} + 5 q^{15} + 2 q^{17} + q^{19} + 4 q^{21} - 4 q^{23} + 11 q^{25} + 4 q^{27} + 2 q^{29} - 6 q^{31} - q^{33} + 5 q^{35} + 16 q^{37} + 7 q^{39} + 5 q^{41} - 9 q^{43} + 5 q^{45} + 21 q^{47} + 4 q^{49} + 2 q^{51} + 10 q^{53} - 17 q^{55} + q^{57} + 26 q^{59} + 2 q^{61} + 4 q^{63} + 26 q^{65} - 5 q^{67} - 4 q^{69} + 19 q^{71} + 10 q^{73} + 11 q^{75} - q^{77} + 6 q^{79} + 4 q^{81} + 2 q^{83} - 7 q^{85} + 2 q^{87} - 17 q^{89} + 7 q^{91} - 6 q^{93} + 17 q^{95} + 32 q^{97} - q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 5 * q^5 + 4 * q^7 + 4 * q^9 - q^11 + 7 * q^13 + 5 * q^15 + 2 * q^17 + q^19 + 4 * q^21 - 4 * q^23 + 11 * q^25 + 4 * q^27 + 2 * q^29 - 6 * q^31 - q^33 + 5 * q^35 + 16 * q^37 + 7 * q^39 + 5 * q^41 - 9 * q^43 + 5 * q^45 + 21 * q^47 + 4 * q^49 + 2 * q^51 + 10 * q^53 - 17 * q^55 + q^57 + 26 * q^59 + 2 * q^61 + 4 * q^63 + 26 * q^65 - 5 * q^67 - 4 * q^69 + 19 * q^71 + 10 * q^73 + 11 * q^75 - q^77 + 6 * q^79 + 4 * q^81 + 2 * q^83 - 7 * q^85 + 2 * q^87 - 17 * q^89 + 7 * q^91 - 6 * q^93 + 17 * q^95 + 32 * q^97 - 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
$5$	3.17434	1.41961	0.709804	−	0.704399i	$-0.248783\pi$
$5$	3.17434	1.41961	0.709804	+	0.704399i	$0.248783\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.06562	−1.52734	−0.763671	−	0.645606i	$-0.776605\pi$
$11$	−5.06562	−1.52734	−0.763671	+	0.645606i	$0.776605\pi$
$12$	0	0
$13$	4.07644	1.13060	0.565300	−	0.824885i	$-0.308760\pi$
$13$	4.07644	1.13060	0.565300	+	0.824885i	$0.308760\pi$
$14$	0	0
$15$	3.17434	0.819611
$16$	0	0
$17$	4.22101	1.02374	0.511872	−	0.859062i	$-0.328952\pi$
$17$	4.22101	1.02374	0.511872	+	0.859062i	$0.328952\pi$
$18$	0	0
$19$	5.06562	1.16213	0.581067	−	0.813856i	$-0.302636\pi$
$19$	5.06562	1.16213	0.581067	+	0.813856i	$0.302636\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	5.07644	1.01529
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	6.68466	1.24131	0.620655	−	0.784084i	$-0.286867\pi$
$29$	6.68466	1.24131	0.620655	+	0.784084i	$0.286867\pi$
$30$	0	0
$31$	2.22101	0.398905	0.199452	−	0.979908i	$-0.436084\pi$
$31$	2.22101	0.398905	0.199452	+	0.979908i	$0.436084\pi$
$32$	0	0
$33$	−5.06562	−0.881811
$34$	0	0
$35$	3.17434	0.536562
$36$	0	0
$37$	−1.91023	−0.314041	−0.157020	−	0.987595i	$-0.550189\pi$
$37$	−1.91023	−0.314041	−0.157020	+	0.987595i	$0.550189\pi$
$38$	0	0
$39$	4.07644	0.652753
$40$	0	0
$41$	−1.37639	−0.214957	−0.107478	−	0.994207i	$-0.534278\pi$
$41$	−1.37639	−0.214957	−0.107478	+	0.994207i	$0.534278\pi$
$42$	0	0
$43$	3.39535	0.517786	0.258893	−	0.965906i	$-0.416642\pi$
$43$	3.39535	0.517786	0.258893	+	0.965906i	$0.416642\pi$
$44$	0	0
$45$	3.17434	0.473203
$46$	0	0
$47$	5.81484	0.848182	0.424091	−	0.905620i	$-0.360594\pi$
$47$	5.81484	0.848182	0.424091	+	0.905620i	$0.360594\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	4.22101	0.591059
$52$	0	0
$53$	−6.57594	−0.903275	−0.451637	−	0.892202i	$-0.649160\pi$
$53$	−6.57594	−0.903275	−0.451637	+	0.892202i	$0.649160\pi$
$54$	0	0
$55$	−16.0800	−2.16823
$56$	0	0
$57$	5.06562	0.670958
$58$	0	0
$59$	5.67027	0.738207	0.369103	−	0.929388i	$-0.379665\pi$
$59$	5.67027	0.738207	0.369103	+	0.929388i	$0.379665\pi$
$60$	0	0
$61$	−14.4862	−1.85476	−0.927382	−	0.374115i	$-0.877946\pi$
$61$	−14.4862	−1.85476	−0.927382	+	0.374115i	$0.877946\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	12.9400	1.60501
$66$	0	0
$67$	−13.8842	−1.69623	−0.848113	−	0.529816i	$-0.822261\pi$
$67$	−13.8842	−1.69623	−0.848113	+	0.529816i	$0.822261\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	2.95333	0.350496	0.175248	−	0.984524i	$-0.443927\pi$
$71$	2.95333	0.350496	0.175248	+	0.984524i	$0.443927\pi$
$72$	0	0
$73$	−2.02520	−0.237032	−0.118516	−	0.992952i	$-0.537814\pi$
$73$	−2.02520	−0.237032	−0.118516	+	0.992952i	$0.537814\pi$
$74$	0	0
$75$	5.07644	0.586177
$76$	0	0
$77$	−5.06562	−0.577281
$78$	0	0
$79$	−7.14831	−0.804248	−0.402124	−	0.915585i	$-0.631728\pi$
$79$	−7.14831	−0.804248	−0.402124	+	0.915585i	$0.631728\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.7226	1.39648	0.698242	−	0.715862i	$-0.253966\pi$
$83$	12.7226	1.39648	0.698242	+	0.715862i	$0.253966\pi$
$84$	0	0
$85$	13.3989	1.45332
$86$	0	0
$87$	6.68466	0.716671
$88$	0	0
$89$	−17.0629	−1.80867	−0.904334	−	0.426826i	$-0.859632\pi$
$89$	−17.0629	−1.80867	−0.904334	+	0.426826i	$0.859632\pi$
$90$	0	0
$91$	4.07644	0.427327
$92$	0	0
$93$	2.22101	0.230308
$94$	0	0
$95$	16.0800	1.64977
$96$	0	0
$97$	2.08977	0.212184	0.106092	−	0.994356i	$-0.466166\pi$
$97$	2.08977	0.212184	0.106092	+	0.994356i	$0.466166\pi$
$98$	0	0
$99$	−5.06562	−0.509114
$100$	0	0
$101$	13.4312	1.33645	0.668227	−	0.743957i	$-0.267054\pi$
$101$	13.4312	1.33645	0.668227	+	0.743957i	$0.267054\pi$
$102$	0	0
$103$	−2.72964	−0.268960	−0.134480	−	0.990916i	$-0.542936\pi$
$103$	−2.72964	−0.268960	−0.134480	+	0.990916i	$0.542936\pi$
$104$	0	0
$105$	3.17434	0.309784
$106$	0	0
$107$	2.36558	0.228689	0.114344	−	0.993441i	$-0.463523\pi$
$107$	2.36558	0.228689	0.114344	+	0.993441i	$0.463523\pi$
$108$	0	0
$109$	−18.1179	−1.73538	−0.867691	−	0.497104i	$-0.834397\pi$
$109$	−18.1179	−1.73538	−0.867691	+	0.497104i	$0.834397\pi$
$110$	0	0
$111$	−1.91023	−0.181311
$112$	0	0
$113$	−6.08001	−0.571959	−0.285979	−	0.958236i	$-0.592319\pi$
$113$	−6.08001	−0.571959	−0.285979	+	0.958236i	$0.592319\pi$
$114$	0	0
$115$	−3.17434	−0.296009
$116$	0	0
$117$	4.07644	0.376867
$118$	0	0
$119$	4.22101	0.386939
$120$	0	0
$121$	14.6605	1.33277
$122$	0	0
$123$	−1.37639	−0.124105
$124$	0	0
$125$	0.242644	0.0217027
$126$	0	0
$127$	−5.42763	−0.481624	−0.240812	−	0.970572i	$-0.577414\pi$
$127$	−5.42763	−0.481624	−0.240812	+	0.970572i	$0.577414\pi$
$128$	0	0
$129$	3.39535	0.298944
$130$	0	0
$131$	0.585698	0.0511727	0.0255863	−	0.999673i	$-0.491855\pi$
$131$	0.585698	0.0511727	0.0255863	+	0.999673i	$0.491855\pi$
$132$	0	0
$133$	5.06562	0.439245
$134$	0	0
$135$	3.17434	0.273204
$136$	0	0
$137$	19.6605	1.67971	0.839856	−	0.542810i	$-0.182640\pi$
$137$	19.6605	1.67971	0.839856	+	0.542810i	$0.182640\pi$
$138$	0	0
$139$	9.98667	0.847059	0.423529	−	0.905882i	$-0.360791\pi$
$139$	9.98667	0.847059	0.423529	+	0.905882i	$0.360791\pi$
$140$	0	0
$141$	5.81484	0.489698
$142$	0	0
$143$	−20.6497	−1.72681
$144$	0	0
$145$	21.2194	1.76217
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	2.01064	0.164718	0.0823592	−	0.996603i	$-0.473755\pi$
$149$	2.01064	0.164718	0.0823592	+	0.996603i	$0.473755\pi$
$150$	0	0
$151$	7.43951	0.605418	0.302709	−	0.953083i	$-0.402109\pi$
$151$	7.43951	0.605418	0.302709	+	0.953083i	$0.402109\pi$
$152$	0	0
$153$	4.22101	0.341248
$154$	0	0
$155$	7.05023	0.566288
$156$	0	0
$157$	17.5041	1.39698	0.698488	−	0.715621i	$-0.253856\pi$
$157$	17.5041	1.39698	0.698488	+	0.715621i	$0.253856\pi$
$158$	0	0
$159$	−6.57594	−0.521506
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	3.95439	0.309732	0.154866	−	0.987935i	$-0.450505\pi$
$163$	3.95439	0.309732	0.154866	+	0.987935i	$0.450505\pi$
$164$	0	0
$165$	−16.0800	−1.25183
$166$	0	0
$167$	3.08726	0.238899	0.119450	−	0.992840i	$-0.461887\pi$
$167$	3.08726	0.238899	0.119450	+	0.992840i	$0.461887\pi$
$168$	0	0
$169$	3.61736	0.278258
$170$	0	0
$171$	5.06562	0.387378
$172$	0	0
$173$	−23.9265	−1.81910	−0.909549	−	0.415596i	$-0.863573\pi$
$173$	−23.9265	−1.81910	−0.909549	+	0.415596i	$0.863573\pi$
$174$	0	0
$175$	5.07644	0.383743
$176$	0	0
$177$	5.67027	0.426204
$178$	0	0
$179$	17.5014	1.30811	0.654057	−	0.756445i	$-0.273065\pi$
$179$	17.5014	1.30811	0.654057	+	0.756445i	$0.273065\pi$
$180$	0	0
$181$	14.9738	1.11299	0.556497	−	0.830850i	$-0.312145\pi$
$181$	14.9738	1.11299	0.556497	+	0.830850i	$0.312145\pi$
$182$	0	0
$183$	−14.4862	−1.07085
$184$	0	0
$185$	−6.06373	−0.445815
$186$	0	0
$187$	−21.3820	−1.56361
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−15.4395	−1.11716	−0.558582	−	0.829449i	$-0.688654\pi$
$191$	−15.4395	−1.11716	−0.558582	+	0.829449i	$0.688654\pi$
$192$	0	0
$193$	17.4395	1.25532	0.627662	−	0.778486i	$-0.284012\pi$
$193$	17.4395	1.25532	0.627662	+	0.778486i	$0.284012\pi$
$194$	0	0
$195$	12.9400	0.926653
$196$	0	0
$197$	22.1698	1.57953	0.789765	−	0.613409i	$-0.210202\pi$
$197$	22.1698	1.57953	0.789765	+	0.613409i	$0.210202\pi$
$198$	0	0
$199$	−14.9246	−1.05798	−0.528989	−	0.848629i	$-0.677429\pi$
$199$	−14.9246	−1.05798	−0.528989	+	0.848629i	$0.677429\pi$
$200$	0	0
$201$	−13.8842	−0.979316
$202$	0	0
$203$	6.68466	0.469171
$204$	0	0
$205$	−4.36914	−0.305154
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−25.6605	−1.77497
$210$	0	0
$211$	3.70880	0.255325	0.127662	−	0.991818i	$-0.459253\pi$
$211$	3.70880	0.255325	0.127662	+	0.991818i	$0.459253\pi$
$212$	0	0
$213$	2.95333	0.202359
$214$	0	0
$215$	10.7780	0.735053
$216$	0	0
$217$	2.22101	0.150772
$218$	0	0
$219$	−2.02520	−0.136851
$220$	0	0
$221$	17.2067	1.15745
$222$	0	0
$223$	−5.71883	−0.382961	−0.191480	−	0.981496i	$-0.561329\pi$
$223$	−5.71883	−0.382961	−0.191480	+	0.981496i	$0.561329\pi$
$224$	0	0
$225$	5.07644	0.338429
$226$	0	0
$227$	−26.5437	−1.76176	−0.880882	−	0.473336i	$-0.843050\pi$
$227$	−26.5437	−1.76176	−0.880882	+	0.473336i	$0.843050\pi$
$228$	0	0
$229$	−3.70818	−0.245044	−0.122522	−	0.992466i	$-0.539098\pi$
$229$	−3.70818	−0.245044	−0.122522	+	0.992466i	$0.539098\pi$
$230$	0	0
$231$	−5.06562	−0.333293
$232$	0	0
$233$	22.1950	1.45404	0.727021	−	0.686616i	$-0.240904\pi$
$233$	22.1950	1.45404	0.727021	+	0.686616i	$0.240904\pi$
$234$	0	0
$235$	18.4583	1.20409
$236$	0	0
$237$	−7.14831	−0.464333
$238$	0	0
$239$	20.2921	1.31259	0.656293	−	0.754506i	$-0.272124\pi$
$239$	20.2921	1.31259	0.656293	+	0.754506i	$0.272124\pi$
$240$	0	0
$241$	14.5158	0.935043	0.467522	−	0.883982i	$-0.345147\pi$
$241$	14.5158	0.935043	0.467522	+	0.883982i	$0.345147\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	3.17434	0.202801
$246$	0	0
$247$	20.6497	1.31391
$248$	0	0
$249$	12.7226	0.806260
$250$	0	0
$251$	−4.03791	−0.254871	−0.127435	−	0.991847i	$-0.540675\pi$
$251$	−4.03791	−0.254871	−0.127435	+	0.991847i	$0.540675\pi$
$252$	0	0
$253$	5.06562	0.318473
$254$	0	0
$255$	13.3989	0.839073
$256$	0	0
$257$	−16.7676	−1.04593	−0.522966	−	0.852354i	$-0.675174\pi$
$257$	−16.7676	−1.04593	−0.522966	+	0.852354i	$0.675174\pi$
$258$	0	0
$259$	−1.91023	−0.118696
$260$	0	0
$261$	6.68466	0.413770
$262$	0	0
$263$	22.0308	1.35848	0.679240	−	0.733917i	$-0.262310\pi$
$263$	22.0308	1.35848	0.679240	+	0.733917i	$0.262310\pi$
$264$	0	0
$265$	−20.8743	−1.28230
$266$	0	0
$267$	−17.0629	−1.04423
$268$	0	0
$269$	−17.9902	−1.09688	−0.548442	−	0.836188i	$-0.684779\pi$
$269$	−17.9902	−1.09688	−0.548442	+	0.836188i	$0.684779\pi$
$270$	0	0
$271$	−0.940007	−0.0571014	−0.0285507	−	0.999592i	$-0.509089\pi$
$271$	−0.940007	−0.0571014	−0.0285507	+	0.999592i	$0.509089\pi$
$272$	0	0
$273$	4.07644	0.246717
$274$	0	0
$275$	−25.7153	−1.55069
$276$	0	0
$277$	3.73339	0.224317	0.112159	−	0.993690i	$-0.464223\pi$
$277$	3.73339	0.224317	0.112159	+	0.993690i	$0.464223\pi$
$278$	0	0
$279$	2.22101	0.132968
$280$	0	0
$281$	21.7180	1.29559	0.647794	−	0.761816i	$-0.275692\pi$
$281$	21.7180	1.29559	0.647794	+	0.761816i	$0.275692\pi$
$282$	0	0
$283$	24.0534	1.42982	0.714912	−	0.699215i	$-0.246467\pi$
$283$	24.0534	1.42982	0.714912	+	0.699215i	$0.246467\pi$
$284$	0	0
$285$	16.0800	0.952497
$286$	0	0
$287$	−1.37639	−0.0812459
$288$	0	0
$289$	0.816901	0.0480530
$290$	0	0
$291$	2.08977	0.122504
$292$	0	0
$293$	24.0021	1.40222	0.701109	−	0.713054i	$-0.252688\pi$
$293$	24.0021	1.40222	0.701109	+	0.713054i	$0.252688\pi$
$294$	0	0
$295$	17.9994	1.04796
$296$	0	0
$297$	−5.06562	−0.293937
$298$	0	0
$299$	−4.07644	−0.235747
$300$	0	0
$301$	3.39535	0.195705
$302$	0	0
$303$	13.4312	0.771602
$304$	0	0
$305$	−45.9840	−2.63304
$306$	0	0
$307$	−31.9678	−1.82450	−0.912249	−	0.409637i	$-0.865655\pi$
$307$	−31.9678	−1.82450	−0.912249	+	0.409637i	$0.865655\pi$
$308$	0	0
$309$	−2.72964	−0.155284
$310$	0	0
$311$	21.1889	1.20151	0.600756	−	0.799432i	$-0.294866\pi$
$311$	21.1889	1.20151	0.600756	+	0.799432i	$0.294866\pi$
$312$	0	0
$313$	−14.3935	−0.813567	−0.406783	−	0.913525i	$-0.633350\pi$
$313$	−14.3935	−0.813567	−0.406783	+	0.913525i	$0.633350\pi$
$314$	0	0
$315$	3.17434	0.178854
$316$	0	0
$317$	−24.3516	−1.36772	−0.683862	−	0.729612i	$-0.739701\pi$
$317$	−24.3516	−1.36772	−0.683862	+	0.729612i	$0.739701\pi$
$318$	0	0
$319$	−33.8619	−1.89590
$320$	0	0
$321$	2.36558	0.132034
$322$	0	0
$323$	21.3820	1.18973
$324$	0	0
$325$	20.6938	1.14789
$326$	0	0
$327$	−18.1179	−1.00192
$328$	0	0
$329$	5.81484	0.320583
$330$	0	0
$331$	9.09082	0.499677	0.249838	−	0.968288i	$-0.419622\pi$
$331$	9.09082	0.499677	0.249838	+	0.968288i	$0.419622\pi$
$332$	0	0
$333$	−1.91023	−0.104680
$334$	0	0
$335$	−44.0732	−2.40798
$336$	0	0
$337$	−35.4620	−1.93174	−0.965870	−	0.259028i	$-0.916598\pi$
$337$	−35.4620	−1.93174	−0.965870	+	0.259028i	$0.916598\pi$
$338$	0	0
$339$	−6.08001	−0.330221
$340$	0	0
$341$	−11.2508	−0.609264
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−3.17434	−0.170901
$346$	0	0
$347$	0.799628	0.0429263	0.0214631	−	0.999770i	$-0.493168\pi$
$347$	0.799628	0.0429263	0.0214631	+	0.999770i	$0.493168\pi$
$348$	0	0
$349$	1.20311	0.0644011	0.0322006	−	0.999481i	$-0.489748\pi$
$349$	1.20311	0.0644011	0.0322006	+	0.999481i	$0.489748\pi$
$350$	0	0
$351$	4.07644	0.217584
$352$	0	0
$353$	−1.03334	−0.0549991	−0.0274996	−	0.999622i	$-0.508754\pi$
$353$	−1.03334	−0.0549991	−0.0274996	+	0.999622i	$0.508754\pi$
$354$	0	0
$355$	9.37489	0.497567
$356$	0	0
$357$	4.22101	0.223399
$358$	0	0
$359$	−12.2329	−0.645627	−0.322813	−	0.946463i	$-0.604629\pi$
$359$	−12.2329	−0.645627	−0.322813	+	0.946463i	$0.604629\pi$
$360$	0	0
$361$	6.66051	0.350553
$362$	0	0
$363$	14.6605	0.769477
$364$	0	0
$365$	−6.42869	−0.336493
$366$	0	0
$367$	20.0388	1.04602	0.523008	−	0.852328i	$-0.324810\pi$
$367$	20.0388	1.04602	0.523008	+	0.852328i	$0.324810\pi$
$368$	0	0
$369$	−1.37639	−0.0716522
$370$	0	0
$371$	−6.57594	−0.341406
$372$	0	0
$373$	−25.6803	−1.32968	−0.664838	−	0.746988i	$-0.731499\pi$
$373$	−25.6803	−1.32968	−0.664838	+	0.746988i	$0.731499\pi$
$374$	0	0
$375$	0.242644	0.0125301
$376$	0	0
$377$	27.2496	1.40343
$378$	0	0
$379$	−10.7782	−0.553639	−0.276819	−	0.960922i	$-0.589280\pi$
$379$	−10.7782	−0.553639	−0.276819	+	0.960922i	$0.589280\pi$
$380$	0	0
$381$	−5.42763	−0.278066
$382$	0	0
$383$	−16.2877	−0.832262	−0.416131	−	0.909305i	$-0.636614\pi$
$383$	−16.2877	−0.832262	−0.416131	+	0.909305i	$0.636614\pi$
$384$	0	0
$385$	−16.0800	−0.819513
$386$	0	0
$387$	3.39535	0.172595
$388$	0	0
$389$	22.2160	1.12640	0.563198	−	0.826322i	$-0.309571\pi$
$389$	22.2160	1.12640	0.563198	+	0.826322i	$0.309571\pi$
$390$	0	0
$391$	−4.22101	−0.213466
$392$	0	0
$393$	0.585698	0.0295445
$394$	0	0
$395$	−22.6912	−1.14172
$396$	0	0
$397$	−25.3518	−1.27237	−0.636185	−	0.771536i	$-0.719489\pi$
$397$	−25.3518	−1.27237	−0.636185	+	0.771536i	$0.719489\pi$
$398$	0	0
$399$	5.06562	0.253598
$400$	0	0
$401$	−37.3622	−1.86578	−0.932891	−	0.360160i	$-0.882722\pi$
$401$	−37.3622	−1.86578	−0.932891	+	0.360160i	$0.882722\pi$
$402$	0	0
$403$	9.05380	0.451002
$404$	0	0
$405$	3.17434	0.157734
$406$	0	0
$407$	9.67652	0.479647
$408$	0	0
$409$	−29.7899	−1.47302	−0.736508	−	0.676429i	$-0.763527\pi$
$409$	−29.7899	−1.47302	−0.736508	+	0.676429i	$0.763527\pi$
$410$	0	0
$411$	19.6605	0.969782
$412$	0	0
$413$	5.67027	0.279016
$414$	0	0
$415$	40.3858	1.98246
$416$	0	0
$417$	9.98667	0.489050
$418$	0	0
$419$	2.46784	0.120562	0.0602809	−	0.998181i	$-0.480800\pi$
$419$	2.46784	0.120562	0.0602809	+	0.998181i	$0.480800\pi$
$420$	0	0
$421$	−15.1942	−0.740519	−0.370260	−	0.928928i	$-0.620731\pi$
$421$	−15.1942	−0.740519	−0.370260	+	0.928928i	$0.620731\pi$
$422$	0	0
$423$	5.81484	0.282727
$424$	0	0
$425$	21.4277	1.03940
$426$	0	0
$427$	−14.4862	−0.701035
$428$	0	0
$429$	−20.6497	−0.996977
$430$	0	0
$431$	15.3289	0.738369	0.369184	−	0.929356i	$-0.379637\pi$
$431$	15.3289	0.738369	0.369184	+	0.929356i	$0.379637\pi$
$432$	0	0
$433$	−1.93895	−0.0931799	−0.0465899	−	0.998914i	$-0.514835\pi$
$433$	−1.93895	−0.0931799	−0.0465899	+	0.998914i	$0.514835\pi$
$434$	0	0
$435$	21.2194	1.01739
$436$	0	0
$437$	−5.06562	−0.242322
$438$	0	0
$439$	−4.04649	−0.193129	−0.0965643	−	0.995327i	$-0.530785\pi$
$439$	−4.04649	−0.193129	−0.0965643	+	0.995327i	$0.530785\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−17.8871	−0.849844	−0.424922	−	0.905230i	$-0.639698\pi$
$443$	−17.8871	−0.849844	−0.424922	+	0.905230i	$0.639698\pi$
$444$	0	0
$445$	−54.1636	−2.56760
$446$	0	0
$447$	2.01064	0.0951002
$448$	0	0
$449$	33.1765	1.56569	0.782847	−	0.622214i	$-0.213767\pi$
$449$	33.1765	1.56569	0.782847	+	0.622214i	$0.213767\pi$
$450$	0	0
$451$	6.97229	0.328312
$452$	0	0
$453$	7.43951	0.349538
$454$	0	0
$455$	12.9400	0.606637
$456$	0	0
$457$	29.0828	1.36043	0.680217	−	0.733010i	$-0.261885\pi$
$457$	29.0828	1.36043	0.680217	+	0.733010i	$0.261885\pi$
$458$	0	0
$459$	4.22101	0.197020
$460$	0	0
$461$	−33.3560	−1.55354	−0.776772	−	0.629782i	$-0.783144\pi$
$461$	−33.3560	−1.55354	−0.776772	+	0.629782i	$0.783144\pi$
$462$	0	0
$463$	1.87795	0.0872759	0.0436380	−	0.999047i	$-0.486105\pi$
$463$	1.87795	0.0872759	0.0436380	+	0.999047i	$0.486105\pi$
$464$	0	0
$465$	7.05023	0.326947
$466$	0	0
$467$	−12.9275	−0.598214	−0.299107	−	0.954220i	$-0.596689\pi$
$467$	−12.9275	−0.598214	−0.299107	+	0.954220i	$0.596689\pi$
$468$	0	0
$469$	−13.8842	−0.641113
$470$	0	0
$471$	17.5041	0.806545
$472$	0	0
$473$	−17.1995	−0.790836
$474$	0	0
$475$	25.7153	1.17990
$476$	0	0
$477$	−6.57594	−0.301092
$478$	0	0
$479$	43.3983	1.98292	0.991459	−	0.130416i	$-0.0416314\pi$
$479$	43.3983	1.98292	0.991459	+	0.130416i	$0.0416314\pi$
$480$	0	0
$481$	−7.78695	−0.355055
$482$	0	0
$483$	−1.00000	−0.0455016
$484$	0	0
$485$	6.63363	0.301218
$486$	0	0
$487$	−16.9992	−0.770307	−0.385154	−	0.922852i	$-0.625852\pi$
$487$	−16.9992	−0.770307	−0.385154	+	0.922852i	$0.625852\pi$
$488$	0	0
$489$	3.95439	0.178824
$490$	0	0
$491$	−10.9480	−0.494075	−0.247037	−	0.969006i	$-0.579457\pi$
$491$	−10.9480	−0.494075	−0.247037	+	0.969006i	$0.579457\pi$
$492$	0	0
$493$	28.2160	1.27078
$494$	0	0
$495$	−16.0800	−0.722743
$496$	0	0
$497$	2.95333	0.132475
$498$	0	0
$499$	−30.3588	−1.35904	−0.679522	−	0.733655i	$-0.737813\pi$
$499$	−30.3588	−1.35904	−0.679522	+	0.733655i	$0.737813\pi$
$500$	0	0
$501$	3.08726	0.137929
$502$	0	0
$503$	−25.5951	−1.14123	−0.570614	−	0.821219i	$-0.693295\pi$
$503$	−25.5951	−1.14123	−0.570614	+	0.821219i	$0.693295\pi$
$504$	0	0
$505$	42.6352	1.89724
$506$	0	0
$507$	3.61736	0.160652
$508$	0	0
$509$	−8.95422	−0.396889	−0.198444	−	0.980112i	$-0.563589\pi$
$509$	−8.95422	−0.396889	−0.198444	+	0.980112i	$0.563589\pi$
$510$	0	0
$511$	−2.02520	−0.0895898
$512$	0	0
$513$	5.06562	0.223653
$514$	0	0
$515$	−8.66482	−0.381818
$516$	0	0
$517$	−29.4558	−1.29546
$518$	0	0
$519$	−23.9265	−1.05026
$520$	0	0
$521$	15.9405	0.698364	0.349182	−	0.937055i	$-0.386459\pi$
$521$	15.9405	0.698364	0.349182	+	0.937055i	$0.386459\pi$
$522$	0	0
$523$	−28.1102	−1.22917	−0.614587	−	0.788849i	$-0.710677\pi$
$523$	−28.1102	−1.22917	−0.614587	+	0.788849i	$0.710677\pi$
$524$	0	0
$525$	5.07644	0.221554
$526$	0	0
$527$	9.37489	0.408376
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	5.67027	0.246069
$532$	0	0
$533$	−5.61078	−0.243030
$534$	0	0
$535$	7.50914	0.324648
$536$	0	0
$537$	17.5014	0.755241
$538$	0	0
$539$	−5.06562	−0.218192
$540$	0	0
$541$	31.5707	1.35733	0.678666	−	0.734447i	$-0.262558\pi$
$541$	31.5707	1.35733	0.678666	+	0.734447i	$0.262558\pi$
$542$	0	0
$543$	14.9738	0.642587
$544$	0	0
$545$	−57.5124	−2.46356
$546$	0	0
$547$	37.7639	1.61467	0.807333	−	0.590096i	$-0.200910\pi$
$547$	37.7639	1.61467	0.807333	+	0.590096i	$0.200910\pi$
$548$	0	0
$549$	−14.4862	−0.618255
$550$	0	0
$551$	33.8619	1.44257
$552$	0	0
$553$	−7.14831	−0.303977
$554$	0	0
$555$	−6.06373	−0.257391
$556$	0	0
$557$	11.1660	0.473120	0.236560	−	0.971617i	$-0.423980\pi$
$557$	11.1660	0.473120	0.236560	+	0.971617i	$0.423980\pi$
$558$	0	0
$559$	13.8409	0.585409
$560$	0	0
$561$	−21.3820	−0.902750
$562$	0	0
$563$	22.5943	0.952235	0.476117	−	0.879382i	$-0.342044\pi$
$563$	22.5943	0.952235	0.476117	+	0.879382i	$0.342044\pi$
$564$	0	0
$565$	−19.3000	−0.811958
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−40.4210	−1.69454	−0.847268	−	0.531166i	$-0.821754\pi$
$569$	−40.4210	−1.69454	−0.847268	+	0.531166i	$0.821754\pi$
$570$	0	0
$571$	17.1859	0.719206	0.359603	−	0.933105i	$-0.382912\pi$
$571$	17.1859	0.719206	0.359603	+	0.933105i	$0.382912\pi$
$572$	0	0
$573$	−15.4395	−0.644995
$574$	0	0
$575$	−5.07644	−0.211702
$576$	0	0
$577$	31.1465	1.29665	0.648323	−	0.761365i	$-0.275471\pi$
$577$	31.1465	1.29665	0.648323	+	0.761365i	$0.275471\pi$
$578$	0	0
$579$	17.4395	0.724761
$580$	0	0
$581$	12.7226	0.527821
$582$	0	0
$583$	33.3112	1.37961
$584$	0	0
$585$	12.9400	0.535003
$586$	0	0
$587$	11.0144	0.454612	0.227306	−	0.973823i	$-0.427008\pi$
$587$	11.0144	0.454612	0.227306	+	0.973823i	$0.427008\pi$
$588$	0	0
$589$	11.2508	0.463580
$590$	0	0
$591$	22.1698	0.911943
$592$	0	0
$593$	−11.6009	−0.476392	−0.238196	−	0.971217i	$-0.576556\pi$
$593$	−11.6009	−0.476392	−0.238196	+	0.971217i	$0.576556\pi$
$594$	0	0
$595$	13.3989	0.549302
$596$	0	0
$597$	−14.9246	−0.610824
$598$	0	0
$599$	10.1410	0.414350	0.207175	−	0.978304i	$-0.433573\pi$
$599$	10.1410	0.414350	0.207175	+	0.978304i	$0.433573\pi$
$600$	0	0
$601$	39.1210	1.59578	0.797890	−	0.602803i	$-0.205949\pi$
$601$	39.1210	1.59578	0.797890	+	0.602803i	$0.205949\pi$
$602$	0	0
$603$	−13.8842	−0.565408
$604$	0	0
$605$	46.5375	1.89202
$606$	0	0
$607$	−13.8211	−0.560981	−0.280490	−	0.959857i	$-0.590497\pi$
$607$	−13.8211	−0.560981	−0.280490	+	0.959857i	$0.590497\pi$
$608$	0	0
$609$	6.68466	0.270876
$610$	0	0
$611$	23.7038	0.958955
$612$	0	0
$613$	41.6282	1.68135	0.840674	−	0.541541i	$-0.182159\pi$
$613$	41.6282	1.68135	0.840674	+	0.541541i	$0.182159\pi$
$614$	0	0
$615$	−4.36914	−0.176181
$616$	0	0
$617$	2.31552	0.0932192	0.0466096	−	0.998913i	$-0.485158\pi$
$617$	2.31552	0.0932192	0.0466096	+	0.998913i	$0.485158\pi$
$618$	0	0
$619$	−18.3318	−0.736817	−0.368408	−	0.929664i	$-0.620097\pi$
$619$	−18.3318	−0.736817	−0.368408	+	0.929664i	$0.620097\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−17.0629	−0.683612
$624$	0	0
$625$	−24.6120	−0.984478
$626$	0	0
$627$	−25.6605	−1.02478
$628$	0	0
$629$	−8.06311	−0.321497
$630$	0	0
$631$	−37.9644	−1.51134	−0.755670	−	0.654953i	$-0.772688\pi$
$631$	−37.9644	−1.51134	−0.755670	+	0.654953i	$0.772688\pi$
$632$	0	0
$633$	3.70880	0.147412
$634$	0	0
$635$	−17.2291	−0.683718
$636$	0	0
$637$	4.07644	0.161514
$638$	0	0
$639$	2.95333	0.116832
$640$	0	0
$641$	5.55174	0.219280	0.109640	−	0.993971i	$-0.465030\pi$
$641$	5.55174	0.219280	0.109640	+	0.993971i	$0.465030\pi$
$642$	0	0
$643$	−39.2894	−1.54942	−0.774711	−	0.632316i	$-0.782105\pi$
$643$	−39.2894	−1.54942	−0.774711	+	0.632316i	$0.782105\pi$
$644$	0	0
$645$	10.7780	0.424383
$646$	0	0
$647$	−18.4287	−0.724506	−0.362253	−	0.932080i	$-0.617992\pi$
$647$	−18.4287	−0.724506	−0.362253	+	0.932080i	$0.617992\pi$
$648$	0	0
$649$	−28.7235	−1.12749
$650$	0	0
$651$	2.22101	0.0870481
$652$	0	0
$653$	42.2708	1.65418	0.827092	−	0.562067i	$-0.189994\pi$
$653$	42.2708	1.65418	0.827092	+	0.562067i	$0.189994\pi$
$654$	0	0
$655$	1.85920	0.0726451
$656$	0	0
$657$	−2.02520	−0.0790107
$658$	0	0
$659$	−47.9174	−1.86660	−0.933298	−	0.359103i	$-0.883083\pi$
$659$	−47.9174	−1.86660	−0.933298	+	0.359103i	$0.883083\pi$
$660$	0	0
$661$	17.9334	0.697528	0.348764	−	0.937211i	$-0.386602\pi$
$661$	17.9334	0.697528	0.348764	+	0.937211i	$0.386602\pi$
$662$	0	0
$663$	17.2067	0.668252
$664$	0	0
$665$	16.0800	0.623556
$666$	0	0
$667$	−6.68466	−0.258831
$668$	0	0
$669$	−5.71883	−0.221103
$670$	0	0
$671$	73.3815	2.83286
$672$	0	0
$673$	−30.4174	−1.17251	−0.586253	−	0.810128i	$-0.699397\pi$
$673$	−30.4174	−1.17251	−0.586253	+	0.810128i	$0.699397\pi$
$674$	0	0
$675$	5.07644	0.195392
$676$	0	0
$677$	31.1083	1.19559	0.597795	−	0.801649i	$-0.296044\pi$
$677$	31.1083	1.19559	0.597795	+	0.801649i	$0.296044\pi$
$678$	0	0
$679$	2.08977	0.0801978
$680$	0	0
$681$	−26.5437	−1.01716
$682$	0	0
$683$	−41.1267	−1.57367	−0.786834	−	0.617164i	$-0.788281\pi$
$683$	−41.1267	−1.57367	−0.786834	+	0.617164i	$0.788281\pi$
$684$	0	0
$685$	62.4092	2.38453
$686$	0	0
$687$	−3.70818	−0.141476
$688$	0	0
$689$	−26.8064	−1.02124
$690$	0	0
$691$	15.1725	0.577191	0.288595	−	0.957451i	$-0.406812\pi$
$691$	15.1725	0.577191	0.288595	+	0.957451i	$0.406812\pi$
$692$	0	0
$693$	−5.06562	−0.192427
$694$	0	0
$695$	31.7011	1.20249
$696$	0	0
$697$	−5.80977	−0.220061
$698$	0	0
$699$	22.1950	0.839491
$700$	0	0
$701$	6.18062	0.233439	0.116719	−	0.993165i	$-0.462762\pi$
$701$	6.18062	0.233439	0.116719	+	0.993165i	$0.462762\pi$
$702$	0	0
$703$	−9.67652	−0.364957
$704$	0	0
$705$	18.4583	0.695179
$706$	0	0
$707$	13.4312	0.505132
$708$	0	0
$709$	16.8330	0.632177	0.316088	−	0.948730i	$-0.397630\pi$
$709$	16.8330	0.632177	0.316088	+	0.948730i	$0.397630\pi$
$710$	0	0
$711$	−7.14831	−0.268083
$712$	0	0
$713$	−2.22101	−0.0831774
$714$	0	0
$715$	−65.5492	−2.45140
$716$	0	0
$717$	20.2921	0.757822
$718$	0	0
$719$	36.7510	1.37058	0.685290	−	0.728270i	$-0.259675\pi$
$719$	36.7510	1.37058	0.685290	+	0.728270i	$0.259675\pi$
$720$	0	0
$721$	−2.72964	−0.101657
$722$	0	0
$723$	14.5158	0.539847
$724$	0	0
$725$	33.9343	1.26029
$726$	0	0
$727$	−42.5612	−1.57851	−0.789253	−	0.614068i	$-0.789532\pi$
$727$	−42.5612	−1.57851	−0.789253	+	0.614068i	$0.789532\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	14.3318	0.530080
$732$	0	0
$733$	−7.67673	−0.283546	−0.141773	−	0.989899i	$-0.545280\pi$
$733$	−7.67673	−0.283546	−0.141773	+	0.989899i	$0.545280\pi$
$734$	0	0
$735$	3.17434	0.117087
$736$	0	0
$737$	70.3321	2.59072
$738$	0	0
$739$	−30.0540	−1.10555	−0.552777	−	0.833329i	$-0.686432\pi$
$739$	−30.0540	−1.10555	−0.552777	+	0.833329i	$0.686432\pi$
$740$	0	0
$741$	20.6497	0.758586
$742$	0	0
$743$	−43.6849	−1.60264	−0.801322	−	0.598234i	$-0.795869\pi$
$743$	−43.6849	−1.60264	−0.801322	+	0.598234i	$0.795869\pi$
$744$	0	0
$745$	6.38247	0.233836
$746$	0	0
$747$	12.7226	0.465494
$748$	0	0
$749$	2.36558	0.0864362
$750$	0	0
$751$	35.0236	1.27803	0.639014	−	0.769195i	$-0.279342\pi$
$751$	35.0236	1.27803	0.639014	+	0.769195i	$0.279342\pi$
$752$	0	0
$753$	−4.03791	−0.147150
$754$	0	0
$755$	23.6155	0.859457
$756$	0	0
$757$	25.1809	0.915214	0.457607	−	0.889155i	$-0.348707\pi$
$757$	25.1809	0.915214	0.457607	+	0.889155i	$0.348707\pi$
$758$	0	0
$759$	5.06562	0.183870
$760$	0	0
$761$	−35.7236	−1.29498	−0.647490	−	0.762074i	$-0.724181\pi$
$761$	−35.7236	−1.29498	−0.647490	+	0.762074i	$0.724181\pi$
$762$	0	0
$763$	−18.1179	−0.655913
$764$	0	0
$765$	13.3989	0.484439
$766$	0	0
$767$	23.1145	0.834617
$768$	0	0
$769$	−42.1421	−1.51968	−0.759842	−	0.650107i	$-0.774724\pi$
$769$	−42.1421	−1.51968	−0.759842	+	0.650107i	$0.774724\pi$
$770$	0	0
$771$	−16.7676	−0.603869
$772$	0	0
$773$	12.5030	0.449702	0.224851	−	0.974393i	$-0.427810\pi$
$773$	12.5030	0.449702	0.224851	+	0.974393i	$0.427810\pi$
$774$	0	0
$775$	11.2748	0.405003
$776$	0	0
$777$	−1.91023	−0.0685293
$778$	0	0
$779$	−6.97229	−0.249808
$780$	0	0
$781$	−14.9605	−0.535328
$782$	0	0
$783$	6.68466	0.238890
$784$	0	0
$785$	55.5639	1.98316
$786$	0	0
$787$	−37.5411	−1.33820	−0.669099	−	0.743174i	$-0.733320\pi$
$787$	−37.5411	−1.33820	−0.669099	+	0.743174i	$0.733320\pi$
$788$	0	0
$789$	22.0308	0.784318
$790$	0	0
$791$	−6.08001	−0.216180
$792$	0	0
$793$	−59.0520	−2.09700
$794$	0	0
$795$	−20.8743	−0.740334
$796$	0	0
$797$	−5.97335	−0.211587	−0.105793	−	0.994388i	$-0.533738\pi$
$797$	−5.97335	−0.211587	−0.105793	+	0.994388i	$0.533738\pi$
$798$	0	0
$799$	24.5445	0.868321
$800$	0	0
$801$	−17.0629	−0.602889
$802$	0	0
$803$	10.2589	0.362029
$804$	0	0
$805$	−3.17434	−0.111881
$806$	0	0
$807$	−17.9902	−0.633286
$808$	0	0
$809$	38.3623	1.34875	0.674374	−	0.738390i	$-0.264414\pi$
$809$	38.3623	1.34875	0.674374	+	0.738390i	$0.264414\pi$
$810$	0	0
$811$	17.7595	0.623619	0.311810	−	0.950145i	$-0.399065\pi$
$811$	17.7595	0.623619	0.311810	+	0.950145i	$0.399065\pi$
$812$	0	0
$813$	−0.940007	−0.0329675
$814$	0	0
$815$	12.5526	0.439698
$816$	0	0
$817$	17.1995	0.601736
$818$	0	0
$819$	4.07644	0.142442
$820$	0	0
$821$	−7.09433	−0.247594	−0.123797	−	0.992308i	$-0.539507\pi$
$821$	−7.09433	−0.247594	−0.123797	+	0.992308i	$0.539507\pi$
$822$	0	0
$823$	−31.6990	−1.10496	−0.552480	−	0.833526i	$-0.686318\pi$
$823$	−31.6990	−1.10496	−0.552480	+	0.833526i	$0.686318\pi$
$824$	0	0
$825$	−25.7153	−0.895292
$826$	0	0
$827$	49.0671	1.70623	0.853115	−	0.521722i	$-0.174710\pi$
$827$	49.0671	1.70623	0.853115	+	0.521722i	$0.174710\pi$
$828$	0	0
$829$	16.4384	0.570931	0.285465	−	0.958389i	$-0.407852\pi$
$829$	16.4384	0.570931	0.285465	+	0.958389i	$0.407852\pi$
$830$	0	0
$831$	3.73339	0.129510
$832$	0	0
$833$	4.22101	0.146249
$834$	0	0
$835$	9.80001	0.339143
$836$	0	0
$837$	2.22101	0.0767692
$838$	0	0
$839$	6.63342	0.229011	0.114506	−	0.993423i	$-0.463472\pi$
$839$	6.63342	0.229011	0.114506	+	0.993423i	$0.463472\pi$
$840$	0	0
$841$	15.6847	0.540850
$842$	0	0
$843$	21.7180	0.748008
$844$	0	0
$845$	11.4827	0.395018
$846$	0	0
$847$	14.6605	0.503741
$848$	0	0
$849$	24.0534	0.825509
$850$	0	0
$851$	1.91023	0.0654820
$852$	0	0
$853$	41.8151	1.43172	0.715861	−	0.698243i	$-0.246034\pi$
$853$	41.8151	1.43172	0.715861	+	0.698243i	$0.246034\pi$
$854$	0	0
$855$	16.0800	0.549925
$856$	0	0
$857$	−36.4689	−1.24575	−0.622877	−	0.782319i	$-0.714036\pi$
$857$	−36.4689	−1.24575	−0.622877	+	0.782319i	$0.714036\pi$
$858$	0	0
$859$	34.5774	1.17976	0.589882	−	0.807489i	$-0.299174\pi$
$859$	34.5774	1.17976	0.589882	+	0.807489i	$0.299174\pi$
$860$	0	0
$861$	−1.37639	−0.0469074
$862$	0	0
$863$	29.2710	0.996395	0.498198	−	0.867063i	$-0.333995\pi$
$863$	29.2710	0.996395	0.498198	+	0.867063i	$0.333995\pi$
$864$	0	0
$865$	−75.9509	−2.58241
$866$	0	0
$867$	0.816901	0.0277434
$868$	0	0
$869$	36.2106	1.22836
$870$	0	0
$871$	−56.5981	−1.91775
$872$	0	0
$873$	2.08977	0.0707279
$874$	0	0
$875$	0.242644	0.00820287
$876$	0	0
$877$	−4.57140	−0.154365	−0.0771826	−	0.997017i	$-0.524592\pi$
$877$	−4.57140	−0.154365	−0.0771826	+	0.997017i	$0.524592\pi$
$878$	0	0
$879$	24.0021	0.809571
$880$	0	0
$881$	11.4285	0.385036	0.192518	−	0.981293i	$-0.438335\pi$
$881$	11.4285	0.385036	0.192518	+	0.981293i	$0.438335\pi$
$882$	0	0
$883$	18.9351	0.637215	0.318608	−	0.947887i	$-0.396785\pi$
$883$	18.9351	0.637215	0.318608	+	0.947887i	$0.396785\pi$
$884$	0	0
$885$	17.9994	0.605042
$886$	0	0
$887$	6.96926	0.234005	0.117002	−	0.993132i	$-0.462671\pi$
$887$	6.96926	0.234005	0.117002	+	0.993132i	$0.462671\pi$
$888$	0	0
$889$	−5.42763	−0.182037
$890$	0	0
$891$	−5.06562	−0.169705
$892$	0	0
$893$	29.4558	0.985700
$894$	0	0
$895$	55.5554	1.85701
$896$	0	0
$897$	−4.07644	−0.136108
$898$	0	0
$899$	14.8467	0.495164
$900$	0	0
$901$	−27.7571	−0.924723
$902$	0	0
$903$	3.39535	0.112990
$904$	0	0
$905$	47.5319	1.58001
$906$	0	0
$907$	−41.9505	−1.39294	−0.696472	−	0.717584i	$-0.745248\pi$
$907$	−41.9505	−1.39294	−0.696472	+	0.717584i	$0.745248\pi$
$908$	0	0
$909$	13.4312	0.445485
$910$	0	0
$911$	10.6040	0.351327	0.175664	−	0.984450i	$-0.443793\pi$
$911$	10.6040	0.351327	0.175664	+	0.984450i	$0.443793\pi$
$912$	0	0
$913$	−64.4477	−2.13291
$914$	0	0
$915$	−45.9840	−1.52019
$916$	0	0
$917$	0.585698	0.0193414
$918$	0	0
$919$	−2.20439	−0.0727160	−0.0363580	−	0.999339i	$-0.511576\pi$
$919$	−2.20439	−0.0727160	−0.0363580	+	0.999339i	$0.511576\pi$
$920$	0	0
$921$	−31.9678	−1.05337
$922$	0	0
$923$	12.0391	0.396271
$924$	0	0
$925$	−9.69719	−0.318842
$926$	0	0
$927$	−2.72964	−0.0896533
$928$	0	0
$929$	−4.27581	−0.140285	−0.0701424	−	0.997537i	$-0.522345\pi$
$929$	−4.27581	−0.140285	−0.0701424	+	0.997537i	$0.522345\pi$
$930$	0	0
$931$	5.06562	0.166019
$932$	0	0
$933$	21.1889	0.693693
$934$	0	0
$935$	−67.8738	−2.21971
$936$	0	0
$937$	−30.6645	−1.00177	−0.500883	−	0.865515i	$-0.666991\pi$
$937$	−30.6645	−1.00177	−0.500883	+	0.865515i	$0.666991\pi$
$938$	0	0
$939$	−14.3935	−0.469713
$940$	0	0
$941$	9.24521	0.301385	0.150693	−	0.988581i	$-0.451850\pi$
$941$	9.24521	0.301385	0.150693	+	0.988581i	$0.451850\pi$
$942$	0	0
$943$	1.37639	0.0448215
$944$	0	0
$945$	3.17434	0.103261
$946$	0	0
$947$	−21.8800	−0.711005	−0.355502	−	0.934675i	$-0.615690\pi$
$947$	−21.8800	−0.711005	−0.355502	+	0.934675i	$0.615690\pi$
$948$	0	0
$949$	−8.25562	−0.267989
$950$	0	0
$951$	−24.3516	−0.789656
$952$	0	0
$953$	−38.2800	−1.24001	−0.620005	−	0.784598i	$-0.712870\pi$
$953$	−38.2800	−1.24001	−0.620005	+	0.784598i	$0.712870\pi$
$954$	0	0
$955$	−49.0103	−1.58593
$956$	0	0
$957$	−33.8619	−1.09460
$958$	0	0
$959$	19.6605	0.634871
$960$	0	0
$961$	−26.0671	−0.840875
$962$	0	0
$963$	2.36558	0.0762296
$964$	0	0
$965$	55.3589	1.78207
$966$	0	0
$967$	15.5397	0.499723	0.249862	−	0.968282i	$-0.419615\pi$
$967$	15.5397	0.499723	0.249862	+	0.968282i	$0.419615\pi$
$968$	0	0
$969$	21.3820	0.686889
$970$	0	0
$971$	−23.9345	−0.768094	−0.384047	−	0.923314i	$-0.625470\pi$
$971$	−23.9345	−0.768094	−0.384047	+	0.923314i	$0.625470\pi$
$972$	0	0
$973$	9.98667	0.320158
$974$	0	0
$975$	20.6938	0.662732
$976$	0	0
$977$	−8.13749	−0.260341	−0.130171	−	0.991492i	$-0.541553\pi$
$977$	−8.13749	−0.260341	−0.130171	+	0.991492i	$0.541553\pi$
$978$	0	0
$979$	86.4344	2.76245
$980$	0	0
$981$	−18.1179	−0.578461
$982$	0	0
$983$	−24.2119	−0.772239	−0.386119	−	0.922449i	$-0.626185\pi$
$983$	−24.2119	−0.772239	−0.386119	+	0.922449i	$0.626185\pi$
$984$	0	0
$985$	70.3744	2.24232
$986$	0	0
$987$	5.81484	0.185088
$988$	0	0
$989$	−3.39535	−0.107966
$990$	0	0
$991$	−36.1425	−1.14810	−0.574052	−	0.818819i	$-0.694629\pi$
$991$	−36.1425	−1.14810	−0.574052	+	0.818819i	$0.694629\pi$
$992$	0	0
$993$	9.09082	0.288489
$994$	0	0
$995$	−47.3758	−1.50191
$996$	0	0
$997$	−0.629234	−0.0199280	−0.00996402	−	0.999950i	$-0.503172\pi$
$997$	−0.629234	−0.0199280	−0.00996402	+	0.999950i	$0.503172\pi$
$998$	0	0
$999$	−1.91023	−0.0604371

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.ce.1.3		4
4.3	odd	2		483.2.a.j.1.3	✓	4
12.11	even	2		1449.2.a.o.1.2		4
28.27	even	2		3381.2.a.x.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.a.j.1.3	✓	4	4.3	odd	2
1449.2.a.o.1.2		4	12.11	even	2
3381.2.a.x.1.3		4	28.27	even	2
7728.2.a.ce.1.3		4	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.17434 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.17434 q^{5} +1.00000 q^{7} +1.00000 q^{9} -5.06562 q^{11} +4.07644 q^{13} +3.17434 q^{15} +4.22101 q^{17} +5.06562 q^{19} +1.00000 q^{21} -1.00000 q^{23} +5.07644 q^{25} +1.00000 q^{27} +6.68466 q^{29} +2.22101 q^{31} -5.06562 q^{33} +3.17434 q^{35} -1.91023 q^{37} +4.07644 q^{39} -1.37639 q^{41} +3.39535 q^{43} +3.17434 q^{45} +5.81484 q^{47} +1.00000 q^{49} +4.22101 q^{51} -6.57594 q^{53} -16.0800 q^{55} +5.06562 q^{57} +5.67027 q^{59} -14.4862 q^{61} +1.00000 q^{63} +12.9400 q^{65} -13.8842 q^{67} -1.00000 q^{69} +2.95333 q^{71} -2.02520 q^{73} +5.07644 q^{75} -5.06562 q^{77} -7.14831 q^{79} +1.00000 q^{81} +12.7226 q^{83} +13.3989 q^{85} +6.68466 q^{87} -17.0629 q^{89} +4.07644 q^{91} +2.22101 q^{93} +16.0800 q^{95} +2.08977 q^{97} -5.06562 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 5 q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 5 * q^5 + 4 * q^7 + 4 * q^9	\( 4 q + 4 q^{3} + 5 q^{5} + 4 q^{7} + 4 q^{9} - q^{11} + 7 q^{13} + 5 q^{15} + 2 q^{17} + q^{19} + 4 q^{21} - 4 q^{23} + 11 q^{25} + 4 q^{27} + 2 q^{29} - 6 q^{31} - q^{33} + 5 q^{35} + 16 q^{37} + 7 q^{39} + 5 q^{41} - 9 q^{43} + 5 q^{45} + 21 q^{47} + 4 q^{49} + 2 q^{51} + 10 q^{53} - 17 q^{55} + q^{57} + 26 q^{59} + 2 q^{61} + 4 q^{63} + 26 q^{65} - 5 q^{67} - 4 q^{69} + 19 q^{71} + 10 q^{73} + 11 q^{75} - q^{77} + 6 q^{79} + 4 q^{81} + 2 q^{83} - 7 q^{85} + 2 q^{87} - 17 q^{89} + 7 q^{91} - 6 q^{93} + 17 q^{95} + 32 q^{97} - q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 5 * q^5 + 4 * q^7 + 4 * q^9 - q^11 + 7 * q^13 + 5 * q^15 + 2 * q^17 + q^19 + 4 * q^21 - 4 * q^23 + 11 * q^25 + 4 * q^27 + 2 * q^29 - 6 * q^31 - q^33 + 5 * q^35 + 16 * q^37 + 7 * q^39 + 5 * q^41 - 9 * q^43 + 5 * q^45 + 21 * q^47 + 4 * q^49 + 2 * q^51 + 10 * q^53 - 17 * q^55 + q^57 + 26 * q^59 + 2 * q^61 + 4 * q^63 + 26 * q^65 - 5 * q^67 - 4 * q^69 + 19 * q^71 + 10 * q^73 + 11 * q^75 - q^77 + 6 * q^79 + 4 * q^81 + 2 * q^83 - 7 * q^85 + 2 * q^87 - 17 * q^89 + 7 * q^91 - 6 * q^93 + 17 * q^95 + 32 * q^97 - q^99

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