LMFDB - Embedded newform 7728.2.a.ch.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:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 11x^{4} + 23x^{3} + 9x^{2} - 23x + 2 $ x^6 - 3x^5 - 11x^4 + 23x^3 + 9x^2 - 23*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 3864)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.07044$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -2.48633 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.48633 q^{5} +1.00000 q^{7} +1.00000 q^{9} +5.58607 q^{11} -5.15448 q^{13} -2.48633 q^{15} +3.97727 q^{17} +5.58607 q^{19} +1.00000 q^{21} +1.00000 q^{23} +1.18182 q^{25} +1.00000 q^{27} -1.84306 q^{29} +8.60010 q^{31} +5.58607 q^{33} -2.48633 q^{35} +2.80236 q^{37} -5.15448 q^{39} -4.16621 q^{41} -4.11378 q^{43} -2.48633 q^{45} +4.18895 q^{47} +1.00000 q^{49} +3.97727 q^{51} -8.87475 q^{53} -13.8888 q^{55} +5.58607 q^{57} +2.74332 q^{59} -6.89748 q^{61} +1.00000 q^{63} +12.8157 q^{65} +7.59319 q^{67} +1.00000 q^{69} -5.88345 q^{71} +9.31357 q^{73} +1.18182 q^{75} +5.58607 q^{77} -10.9499 q^{79} +1.00000 q^{81} -10.6227 q^{83} -9.88879 q^{85} -1.84306 q^{87} +5.08921 q^{89} -5.15448 q^{91} +8.60010 q^{93} -13.8888 q^{95} -17.3931 q^{97} +5.58607 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{3} + 2 q^{5} + 6 q^{7} + 6 q^{9}+O(q^{10}) $ 6 * q + 6 * q^3 + 2 * q^5 + 6 * q^7 + 6 * q^9	$ 6 q + 6 q^{3} + 2 q^{5} + 6 q^{7} + 6 q^{9} + 3 q^{11} + 2 q^{15} + 6 q^{17} + 3 q^{19} + 6 q^{21} + 6 q^{23} + 10 q^{25} + 6 q^{27} + 5 q^{29} + 4 q^{31} + 3 q^{33} + 2 q^{35} + q^{37} + 12 q^{41} + 6 q^{43} + 2 q^{45} + 6 q^{47} + 6 q^{49} + 6 q^{51} + 10 q^{53} - 3 q^{55} + 3 q^{57} + 14 q^{59} + 4 q^{61} + 6 q^{63} + 27 q^{65} - 7 q^{67} + 6 q^{69} - 7 q^{71} + 10 q^{73} + 10 q^{75} + 3 q^{77} - 14 q^{79} + 6 q^{81} - 14 q^{83} + 21 q^{85} + 5 q^{87} + 25 q^{89} + 4 q^{93} - 3 q^{95} + 11 q^{97} + 3 q^{99}+O(q^{100}) $ 6 * q + 6 * q^3 + 2 * q^5 + 6 * q^7 + 6 * q^9 + 3 * q^11 + 2 * q^15 + 6 * q^17 + 3 * q^19 + 6 * q^21 + 6 * q^23 + 10 * q^25 + 6 * q^27 + 5 * q^29 + 4 * q^31 + 3 * q^33 + 2 * q^35 + q^37 + 12 * q^41 + 6 * q^43 + 2 * q^45 + 6 * q^47 + 6 * q^49 + 6 * q^51 + 10 * q^53 - 3 * q^55 + 3 * q^57 + 14 * q^59 + 4 * q^61 + 6 * q^63 + 27 * q^65 - 7 * q^67 + 6 * q^69 - 7 * q^71 + 10 * q^73 + 10 * q^75 + 3 * q^77 - 14 * q^79 + 6 * q^81 - 14 * q^83 + 21 * q^85 + 5 * q^87 + 25 * q^89 + 4 * q^93 - 3 * q^95 + 11 * q^97 + 3 * 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
$5$	−2.48633	−1.11192	−0.555960	−	0.831209i	$-0.687649\pi$
$5$	−2.48633	−1.11192	−0.555960	+	0.831209i	$0.687649\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.58607	1.68426	0.842131	−	0.539273i	$-0.181301\pi$
$11$	5.58607	1.68426	0.842131	+	0.539273i	$0.181301\pi$
$12$	0	0
$13$	−5.15448	−1.42959	−0.714797	−	0.699332i	$-0.753481\pi$
$13$	−5.15448	−1.42959	−0.714797	+	0.699332i	$0.753481\pi$
$14$	0	0
$15$	−2.48633	−0.641967
$16$	0	0
$17$	3.97727	0.964629	0.482315	−	0.875998i	$-0.339796\pi$
$17$	3.97727	0.964629	0.482315	+	0.875998i	$0.339796\pi$
$18$	0	0
$19$	5.58607	1.28153	0.640766	−	0.767736i	$-0.278617\pi$
$19$	5.58607	1.28153	0.640766	+	0.767736i	$0.278617\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.18182	0.236364
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−1.84306	−0.342247	−0.171124	−	0.985250i	$-0.554740\pi$
$29$	−1.84306	−0.342247	−0.171124	+	0.985250i	$0.554740\pi$
$30$	0	0
$31$	8.60010	1.54462	0.772312	−	0.635243i	$-0.219100\pi$
$31$	8.60010	1.54462	0.772312	+	0.635243i	$0.219100\pi$
$32$	0	0
$33$	5.58607	0.972409
$34$	0	0
$35$	−2.48633	−0.420266
$36$	0	0
$37$	2.80236	0.460705	0.230352	−	0.973107i	$-0.426012\pi$
$37$	2.80236	0.460705	0.230352	+	0.973107i	$0.426012\pi$
$38$	0	0
$39$	−5.15448	−0.825377
$40$	0	0
$41$	−4.16621	−0.650653	−0.325327	−	0.945602i	$-0.605474\pi$
$41$	−4.16621	−0.650653	−0.325327	+	0.945602i	$0.605474\pi$
$42$	0	0
$43$	−4.11378	−0.627345	−0.313672	−	0.949531i	$-0.601559\pi$
$43$	−4.11378	−0.627345	−0.313672	+	0.949531i	$0.601559\pi$
$44$	0	0
$45$	−2.48633	−0.370640
$46$	0	0
$47$	4.18895	0.611021	0.305510	−	0.952189i	$-0.401173\pi$
$47$	4.18895	0.611021	0.305510	+	0.952189i	$0.401173\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	3.97727	0.556929
$52$	0	0
$53$	−8.87475	−1.21904	−0.609520	−	0.792770i	$-0.708638\pi$
$53$	−8.87475	−1.21904	−0.609520	+	0.792770i	$0.708638\pi$
$54$	0	0
$55$	−13.8888	−1.87276
$56$	0	0
$57$	5.58607	0.739893
$58$	0	0
$59$	2.74332	0.357150	0.178575	−	0.983926i	$-0.442851\pi$
$59$	2.74332	0.357150	0.178575	+	0.983926i	$0.442851\pi$
$60$	0	0
$61$	−6.89748	−0.883132	−0.441566	−	0.897229i	$-0.645577\pi$
$61$	−6.89748	−0.883132	−0.441566	+	0.897229i	$0.645577\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	12.8157	1.58959
$66$	0	0
$67$	7.59319	0.927656	0.463828	−	0.885925i	$-0.346476\pi$
$67$	7.59319	0.927656	0.463828	+	0.885925i	$0.346476\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−5.88345	−0.698237	−0.349118	−	0.937079i	$-0.613519\pi$
$71$	−5.88345	−0.698237	−0.349118	+	0.937079i	$0.613519\pi$
$72$	0	0
$73$	9.31357	1.09007	0.545035	−	0.838413i	$-0.316516\pi$
$73$	9.31357	1.09007	0.545035	+	0.838413i	$0.316516\pi$
$74$	0	0
$75$	1.18182	0.136465
$76$	0	0
$77$	5.58607	0.636591
$78$	0	0
$79$	−10.9499	−1.23196	−0.615981	−	0.787761i	$-0.711240\pi$
$79$	−10.9499	−1.23196	−0.615981	+	0.787761i	$0.711240\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−10.6227	−1.16599	−0.582996	−	0.812475i	$-0.698119\pi$
$83$	−10.6227	−1.16599	−0.582996	+	0.812475i	$0.698119\pi$
$84$	0	0
$85$	−9.88879	−1.07259
$86$	0	0
$87$	−1.84306	−0.197597
$88$	0	0
$89$	5.08921	0.539455	0.269727	−	0.962937i	$-0.413066\pi$
$89$	5.08921	0.539455	0.269727	+	0.962937i	$0.413066\pi$
$90$	0	0
$91$	−5.15448	−0.540336
$92$	0	0
$93$	8.60010	0.891789
$94$	0	0
$95$	−13.8888	−1.42496
$96$	0	0
$97$	−17.3931	−1.76600	−0.883000	−	0.469373i	$-0.844480\pi$
$97$	−17.3931	−1.76600	−0.883000	+	0.469373i	$0.844480\pi$
$98$	0	0
$99$	5.58607	0.561421
$100$	0	0
$101$	8.22933	0.818849	0.409425	−	0.912344i	$-0.365729\pi$
$101$	8.22933	0.818849	0.409425	+	0.912344i	$0.365729\pi$
$102$	0	0
$103$	18.4979	1.82265	0.911326	−	0.411685i	$-0.135060\pi$
$103$	18.4979	1.82265	0.911326	+	0.411685i	$0.135060\pi$
$104$	0	0
$105$	−2.48633	−0.242641
$106$	0	0
$107$	3.20639	0.309974	0.154987	−	0.987917i	$-0.450467\pi$
$107$	3.20639	0.309974	0.154987	+	0.987917i	$0.450467\pi$
$108$	0	0
$109$	−4.83615	−0.463219	−0.231609	−	0.972809i	$-0.574399\pi$
$109$	−4.83615	−0.463219	−0.231609	+	0.972809i	$0.574399\pi$
$110$	0	0
$111$	2.80236	0.265988
$112$	0	0
$113$	5.24615	0.493516	0.246758	−	0.969077i	$-0.420635\pi$
$113$	5.24615	0.493516	0.246758	+	0.969077i	$0.420635\pi$
$114$	0	0
$115$	−2.48633	−0.231851
$116$	0	0
$117$	−5.15448	−0.476531
$118$	0	0
$119$	3.97727	0.364596
$120$	0	0
$121$	20.2041	1.83674
$122$	0	0
$123$	−4.16621	−0.375655
$124$	0	0
$125$	9.49324	0.849101
$126$	0	0
$127$	0.504977	0.0448094	0.0224047	−	0.999749i	$-0.492868\pi$
$127$	0.504977	0.0448094	0.0224047	+	0.999749i	$0.492868\pi$
$128$	0	0
$129$	−4.11378	−0.362198
$130$	0	0
$131$	−10.2315	−0.893929	−0.446964	−	0.894552i	$-0.647495\pi$
$131$	−10.2315	−0.893929	−0.446964	+	0.894552i	$0.647495\pi$
$132$	0	0
$133$	5.58607	0.484373
$134$	0	0
$135$	−2.48633	−0.213989
$136$	0	0
$137$	17.1389	1.46427	0.732136	−	0.681158i	$-0.238523\pi$
$137$	17.1389	1.46427	0.732136	+	0.681158i	$0.238523\pi$
$138$	0	0
$139$	18.9267	1.60534	0.802671	−	0.596422i	$-0.203411\pi$
$139$	18.9267	1.60534	0.802671	+	0.596422i	$0.203411\pi$
$140$	0	0
$141$	4.18895	0.352773
$142$	0	0
$143$	−28.7932	−2.40781
$144$	0	0
$145$	4.58245	0.380551
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−4.96857	−0.407041	−0.203521	−	0.979071i	$-0.565238\pi$
$149$	−4.96857	−0.407041	−0.203521	+	0.979071i	$0.565238\pi$
$150$	0	0
$151$	12.6974	1.03330	0.516649	−	0.856197i	$-0.327179\pi$
$151$	12.6974	1.03330	0.516649	+	0.856197i	$0.327179\pi$
$152$	0	0
$153$	3.97727	0.321543
$154$	0	0
$155$	−21.3827	−1.71750
$156$	0	0
$157$	4.23163	0.337721	0.168861	−	0.985640i	$-0.445991\pi$
$157$	4.23163	0.337721	0.168861	+	0.985640i	$0.445991\pi$
$158$	0	0
$159$	−8.87475	−0.703814
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	−0.988263	−0.0774067	−0.0387034	−	0.999251i	$-0.512323\pi$
$163$	−0.988263	−0.0774067	−0.0387034	+	0.999251i	$0.512323\pi$
$164$	0	0
$165$	−13.8888	−1.08124
$166$	0	0
$167$	14.6094	1.13051	0.565254	−	0.824917i	$-0.308778\pi$
$167$	14.6094	1.13051	0.565254	+	0.824917i	$0.308778\pi$
$168$	0	0
$169$	13.5686	1.04374
$170$	0	0
$171$	5.58607	0.427177
$172$	0	0
$173$	20.1221	1.52985	0.764926	−	0.644119i	$-0.222776\pi$
$173$	20.1221	1.52985	0.764926	+	0.644119i	$0.222776\pi$
$174$	0	0
$175$	1.18182	0.0893373
$176$	0	0
$177$	2.74332	0.206200
$178$	0	0
$179$	24.2584	1.81316	0.906579	−	0.422035i	$-0.138684\pi$
$179$	24.2584	1.81316	0.906579	+	0.422035i	$0.138684\pi$
$180$	0	0
$181$	1.45055	0.107819	0.0539093	−	0.998546i	$-0.482832\pi$
$181$	1.45055	0.107819	0.0539093	+	0.998546i	$0.482832\pi$
$182$	0	0
$183$	−6.89748	−0.509877
$184$	0	0
$185$	−6.96758	−0.512267
$186$	0	0
$187$	22.2173	1.62469
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−1.08208	−0.0782967	−0.0391484	−	0.999233i	$-0.512464\pi$
$191$	−1.08208	−0.0782967	−0.0391484	+	0.999233i	$0.512464\pi$
$192$	0	0
$193$	1.29707	0.0933652	0.0466826	−	0.998910i	$-0.485135\pi$
$193$	1.29707	0.0933652	0.0466826	+	0.998910i	$0.485135\pi$
$194$	0	0
$195$	12.8157	0.917752
$196$	0	0
$197$	5.19534	0.370153	0.185076	−	0.982724i	$-0.440747\pi$
$197$	5.19534	0.370153	0.185076	+	0.982724i	$0.440747\pi$
$198$	0	0
$199$	−3.58114	−0.253860	−0.126930	−	0.991912i	$-0.540512\pi$
$199$	−3.58114	−0.253860	−0.126930	+	0.991912i	$0.540512\pi$
$200$	0	0
$201$	7.59319	0.535582
$202$	0	0
$203$	−1.84306	−0.129357
$204$	0	0
$205$	10.3586	0.723474
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	31.2041	2.15844
$210$	0	0
$211$	−19.9911	−1.37625	−0.688123	−	0.725594i	$-0.741565\pi$
$211$	−19.9911	−1.37625	−0.688123	+	0.725594i	$0.741565\pi$
$212$	0	0
$213$	−5.88345	−0.403127
$214$	0	0
$215$	10.2282	0.697557
$216$	0	0
$217$	8.60010	0.583813
$218$	0	0
$219$	9.31357	0.629352
$220$	0	0
$221$	−20.5007	−1.37903
$222$	0	0
$223$	−27.4107	−1.83556	−0.917779	−	0.397091i	$-0.870020\pi$
$223$	−27.4107	−1.83556	−0.917779	+	0.397091i	$0.870020\pi$
$224$	0	0
$225$	1.18182	0.0787881
$226$	0	0
$227$	18.7001	1.24117	0.620583	−	0.784141i	$-0.286896\pi$
$227$	18.7001	1.24117	0.620583	+	0.784141i	$0.286896\pi$
$228$	0	0
$229$	−16.2800	−1.07581	−0.537906	−	0.843005i	$-0.680784\pi$
$229$	−16.2800	−1.07581	−0.537906	+	0.843005i	$0.680784\pi$
$230$	0	0
$231$	5.58607	0.367536
$232$	0	0
$233$	5.27412	0.345519	0.172759	−	0.984964i	$-0.444732\pi$
$233$	5.27412	0.345519	0.172759	+	0.984964i	$0.444732\pi$
$234$	0	0
$235$	−10.4151	−0.679405
$236$	0	0
$237$	−10.9499	−0.711274
$238$	0	0
$239$	−6.07035	−0.392658	−0.196329	−	0.980538i	$-0.562902\pi$
$239$	−6.07035	−0.392658	−0.196329	+	0.980538i	$0.562902\pi$
$240$	0	0
$241$	−15.3383	−0.988030	−0.494015	−	0.869453i	$-0.664471\pi$
$241$	−15.3383	−0.988030	−0.494015	+	0.869453i	$0.664471\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−2.48633	−0.158846
$246$	0	0
$247$	−28.7932	−1.83207
$248$	0	0
$249$	−10.6227	−0.673185
$250$	0	0
$251$	−15.1433	−0.955835	−0.477917	−	0.878405i	$-0.658608\pi$
$251$	−15.1433	−0.955835	−0.477917	+	0.878405i	$0.658608\pi$
$252$	0	0
$253$	5.58607	0.351193
$254$	0	0
$255$	−9.88879	−0.619260
$256$	0	0
$257$	24.1995	1.50952	0.754762	−	0.655998i	$-0.227752\pi$
$257$	24.1995	1.50952	0.754762	+	0.655998i	$0.227752\pi$
$258$	0	0
$259$	2.80236	0.174130
$260$	0	0
$261$	−1.84306	−0.114082
$262$	0	0
$263$	−9.51990	−0.587022	−0.293511	−	0.955956i	$-0.594824\pi$
$263$	−9.51990	−0.587022	−0.293511	+	0.955956i	$0.594824\pi$
$264$	0	0
$265$	22.0655	1.35547
$266$	0	0
$267$	5.08921	0.311454
$268$	0	0
$269$	18.8245	1.14775	0.573874	−	0.818944i	$-0.305440\pi$
$269$	18.8245	1.14775	0.573874	+	0.818944i	$0.305440\pi$
$270$	0	0
$271$	6.20839	0.377133	0.188566	−	0.982060i	$-0.439616\pi$
$271$	6.20839	0.377133	0.188566	+	0.982060i	$0.439616\pi$
$272$	0	0
$273$	−5.15448	−0.311963
$274$	0	0
$275$	6.60174	0.398100
$276$	0	0
$277$	−18.3161	−1.10051	−0.550253	−	0.834998i	$-0.685469\pi$
$277$	−18.3161	−1.10051	−0.550253	+	0.834998i	$0.685469\pi$
$278$	0	0
$279$	8.60010	0.514875
$280$	0	0
$281$	0.962077	0.0573927	0.0286963	−	0.999588i	$-0.490864\pi$
$281$	0.962077	0.0573927	0.0286963	+	0.999588i	$0.490864\pi$
$282$	0	0
$283$	2.75663	0.163865	0.0819323	−	0.996638i	$-0.473891\pi$
$283$	2.75663	0.163865	0.0819323	+	0.996638i	$0.473891\pi$
$284$	0	0
$285$	−13.8888	−0.822701
$286$	0	0
$287$	−4.16621	−0.245924
$288$	0	0
$289$	−1.18134	−0.0694908
$290$	0	0
$291$	−17.3931	−1.01960
$292$	0	0
$293$	8.89036	0.519380	0.259690	−	0.965692i	$-0.416380\pi$
$293$	8.89036	0.519380	0.259690	+	0.965692i	$0.416380\pi$
$294$	0	0
$295$	−6.82079	−0.397122
$296$	0	0
$297$	5.58607	0.324136
$298$	0	0
$299$	−5.15448	−0.298091
$300$	0	0
$301$	−4.11378	−0.237114
$302$	0	0
$303$	8.22933	0.472763
$304$	0	0
$305$	17.1494	0.981972
$306$	0	0
$307$	2.37067	0.135301	0.0676506	−	0.997709i	$-0.478450\pi$
$307$	2.37067	0.135301	0.0676506	+	0.997709i	$0.478450\pi$
$308$	0	0
$309$	18.4979	1.05231
$310$	0	0
$311$	34.4292	1.95230	0.976151	−	0.217093i	$-0.0696574\pi$
$311$	34.4292	1.95230	0.976151	+	0.217093i	$0.0696574\pi$
$312$	0	0
$313$	17.4861	0.988371	0.494185	−	0.869357i	$-0.335466\pi$
$313$	17.4861	0.988371	0.494185	+	0.869357i	$0.335466\pi$
$314$	0	0
$315$	−2.48633	−0.140089
$316$	0	0
$317$	30.7191	1.72535	0.862677	−	0.505755i	$-0.168786\pi$
$317$	30.7191	1.72535	0.862677	+	0.505755i	$0.168786\pi$
$318$	0	0
$319$	−10.2954	−0.576434
$320$	0	0
$321$	3.20639	0.178963
$322$	0	0
$323$	22.2173	1.23620
$324$	0	0
$325$	−6.09167	−0.337905
$326$	0	0
$327$	−4.83615	−0.267439
$328$	0	0
$329$	4.18895	0.230944
$330$	0	0
$331$	−15.6222	−0.858671	−0.429336	−	0.903145i	$-0.641252\pi$
$331$	−15.6222	−0.858671	−0.429336	+	0.903145i	$0.641252\pi$
$332$	0	0
$333$	2.80236	0.153568
$334$	0	0
$335$	−18.8792	−1.03148
$336$	0	0
$337$	33.6472	1.83288	0.916440	−	0.400173i	$-0.131050\pi$
$337$	33.6472	1.83288	0.916440	+	0.400173i	$0.131050\pi$
$338$	0	0
$339$	5.24615	0.284932
$340$	0	0
$341$	48.0407	2.60155
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−2.48633	−0.133859
$346$	0	0
$347$	−14.9200	−0.800946	−0.400473	−	0.916309i	$-0.631154\pi$
$347$	−14.9200	−0.800946	−0.400473	+	0.916309i	$0.631154\pi$
$348$	0	0
$349$	−20.5930	−1.10232	−0.551160	−	0.834400i	$-0.685814\pi$
$349$	−20.5930	−1.10232	−0.551160	+	0.834400i	$0.685814\pi$
$350$	0	0
$351$	−5.15448	−0.275126
$352$	0	0
$353$	−21.9843	−1.17010	−0.585052	−	0.810995i	$-0.698926\pi$
$353$	−21.9843	−1.17010	−0.585052	+	0.810995i	$0.698926\pi$
$354$	0	0
$355$	14.6282	0.776383
$356$	0	0
$357$	3.97727	0.210499
$358$	0	0
$359$	23.9900	1.26614	0.633072	−	0.774093i	$-0.281794\pi$
$359$	23.9900	1.26614	0.633072	+	0.774093i	$0.281794\pi$
$360$	0	0
$361$	12.2041	0.642323
$362$	0	0
$363$	20.2041	1.06044
$364$	0	0
$365$	−23.1566	−1.21207
$366$	0	0
$367$	37.5673	1.96100	0.980499	−	0.196526i	$-0.0629660\pi$
$367$	37.5673	1.96100	0.980499	+	0.196526i	$0.0629660\pi$
$368$	0	0
$369$	−4.16621	−0.216884
$370$	0	0
$371$	−8.87475	−0.460754
$372$	0	0
$373$	−20.5445	−1.06375	−0.531877	−	0.846821i	$-0.678513\pi$
$373$	−20.5445	−1.06375	−0.531877	+	0.846821i	$0.678513\pi$
$374$	0	0
$375$	9.49324	0.490229
$376$	0	0
$377$	9.50000	0.489275
$378$	0	0
$379$	−29.2707	−1.50353	−0.751766	−	0.659429i	$-0.770798\pi$
$379$	−29.2707	−1.50353	−0.751766	+	0.659429i	$0.770798\pi$
$380$	0	0
$381$	0.504977	0.0258707
$382$	0	0
$383$	17.4775	0.893060	0.446530	−	0.894769i	$-0.352660\pi$
$383$	17.4775	0.893060	0.446530	+	0.894769i	$0.352660\pi$
$384$	0	0
$385$	−13.8888	−0.707838
$386$	0	0
$387$	−4.11378	−0.209115
$388$	0	0
$389$	8.42258	0.427042	0.213521	−	0.976938i	$-0.431507\pi$
$389$	8.42258	0.427042	0.213521	+	0.976938i	$0.431507\pi$
$390$	0	0
$391$	3.97727	0.201139
$392$	0	0
$393$	−10.2315	−0.516110
$394$	0	0
$395$	27.2251	1.36984
$396$	0	0
$397$	−5.29576	−0.265787	−0.132893	−	0.991130i	$-0.542427\pi$
$397$	−5.29576	−0.265787	−0.132893	+	0.991130i	$0.542427\pi$
$398$	0	0
$399$	5.58607	0.279653
$400$	0	0
$401$	28.6548	1.43095	0.715477	−	0.698636i	$-0.246209\pi$
$401$	28.6548	1.43095	0.715477	+	0.698636i	$0.246209\pi$
$402$	0	0
$403$	−44.3290	−2.20819
$404$	0	0
$405$	−2.48633	−0.123547
$406$	0	0
$407$	15.6542	0.775948
$408$	0	0
$409$	−25.3278	−1.25238	−0.626190	−	0.779671i	$-0.715387\pi$
$409$	−25.3278	−1.25238	−0.626190	+	0.779671i	$0.715387\pi$
$410$	0	0
$411$	17.1389	0.845398
$412$	0	0
$413$	2.74332	0.134990
$414$	0	0
$415$	26.4115	1.29649
$416$	0	0
$417$	18.9267	0.926845
$418$	0	0
$419$	20.8059	1.01643	0.508216	−	0.861229i	$-0.330305\pi$
$419$	20.8059	1.01643	0.508216	+	0.861229i	$0.330305\pi$
$420$	0	0
$421$	10.3139	0.502668	0.251334	−	0.967900i	$-0.419131\pi$
$421$	10.3139	0.502668	0.251334	+	0.967900i	$0.419131\pi$
$422$	0	0
$423$	4.18895	0.203674
$424$	0	0
$425$	4.70042	0.228004
$426$	0	0
$427$	−6.89748	−0.333793
$428$	0	0
$429$	−28.7932	−1.39015
$430$	0	0
$431$	−1.85804	−0.0894989	−0.0447494	−	0.998998i	$-0.514249\pi$
$431$	−1.85804	−0.0894989	−0.0447494	+	0.998998i	$0.514249\pi$
$432$	0	0
$433$	0.815349	0.0391832	0.0195916	−	0.999808i	$-0.493763\pi$
$433$	0.815349	0.0391832	0.0195916	+	0.999808i	$0.493763\pi$
$434$	0	0
$435$	4.58245	0.219711
$436$	0	0
$437$	5.58607	0.267218
$438$	0	0
$439$	−12.4995	−0.596570	−0.298285	−	0.954477i	$-0.596415\pi$
$439$	−12.4995	−0.596570	−0.298285	+	0.954477i	$0.596415\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−9.83457	−0.467255	−0.233627	−	0.972326i	$-0.575060\pi$
$443$	−9.83457	−0.467255	−0.233627	+	0.972326i	$0.575060\pi$
$444$	0	0
$445$	−12.6534	−0.599830
$446$	0	0
$447$	−4.96857	−0.235005
$448$	0	0
$449$	−10.3194	−0.487002	−0.243501	−	0.969901i	$-0.578296\pi$
$449$	−10.3194	−0.487002	−0.243501	+	0.969901i	$0.578296\pi$
$450$	0	0
$451$	−23.2727	−1.09587
$452$	0	0
$453$	12.6974	0.596575
$454$	0	0
$455$	12.8157	0.600810
$456$	0	0
$457$	−37.4293	−1.75087	−0.875434	−	0.483338i	$-0.839424\pi$
$457$	−37.4293	−1.75087	−0.875434	+	0.483338i	$0.839424\pi$
$458$	0	0
$459$	3.97727	0.185643
$460$	0	0
$461$	14.6431	0.681998	0.340999	−	0.940064i	$-0.389235\pi$
$461$	14.6431	0.681998	0.340999	+	0.940064i	$0.389235\pi$
$462$	0	0
$463$	31.1788	1.44900	0.724500	−	0.689274i	$-0.242071\pi$
$463$	31.1788	1.44900	0.724500	+	0.689274i	$0.242071\pi$
$464$	0	0
$465$	−21.3827	−0.991597
$466$	0	0
$467$	32.6787	1.51219	0.756096	−	0.654461i	$-0.227105\pi$
$467$	32.6787	1.51219	0.756096	+	0.654461i	$0.227105\pi$
$468$	0	0
$469$	7.59319	0.350621
$470$	0	0
$471$	4.23163	0.194983
$472$	0	0
$473$	−22.9798	−1.05661
$474$	0	0
$475$	6.60174	0.302908
$476$	0	0
$477$	−8.87475	−0.406347
$478$	0	0
$479$	9.19487	0.420124	0.210062	−	0.977688i	$-0.432633\pi$
$479$	9.19487	0.420124	0.210062	+	0.977688i	$0.432633\pi$
$480$	0	0
$481$	−14.4447	−0.658621
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	43.2449	1.96365
$486$	0	0
$487$	5.17476	0.234491	0.117245	−	0.993103i	$-0.462594\pi$
$487$	5.17476	0.234491	0.117245	+	0.993103i	$0.462594\pi$
$488$	0	0
$489$	−0.988263	−0.0446908
$490$	0	0
$491$	−16.7889	−0.757675	−0.378837	−	0.925463i	$-0.623676\pi$
$491$	−16.7889	−0.757675	−0.378837	+	0.925463i	$0.623676\pi$
$492$	0	0
$493$	−7.33034	−0.330142
$494$	0	0
$495$	−13.8888	−0.624255
$496$	0	0
$497$	−5.88345	−0.263909
$498$	0	0
$499$	−35.8694	−1.60574	−0.802868	−	0.596157i	$-0.796693\pi$
$499$	−35.8694	−1.60574	−0.802868	+	0.596157i	$0.796693\pi$
$500$	0	0
$501$	14.6094	0.652699
$502$	0	0
$503$	27.1698	1.21144	0.605720	−	0.795678i	$-0.292885\pi$
$503$	27.1698	1.21144	0.605720	+	0.795678i	$0.292885\pi$
$504$	0	0
$505$	−20.4608	−0.910494
$506$	0	0
$507$	13.5686	0.602604
$508$	0	0
$509$	6.54934	0.290294	0.145147	−	0.989410i	$-0.453634\pi$
$509$	6.54934	0.290294	0.145147	+	0.989410i	$0.453634\pi$
$510$	0	0
$511$	9.31357	0.412008
$512$	0	0
$513$	5.58607	0.246631
$514$	0	0
$515$	−45.9918	−2.02664
$516$	0	0
$517$	23.3997	1.02912
$518$	0	0
$519$	20.1221	0.883260
$520$	0	0
$521$	−4.08140	−0.178809	−0.0894047	−	0.995995i	$-0.528496\pi$
$521$	−4.08140	−0.178809	−0.0894047	+	0.995995i	$0.528496\pi$
$522$	0	0
$523$	26.7355	1.16906	0.584530	−	0.811372i	$-0.301279\pi$
$523$	26.7355	1.16906	0.584530	+	0.811372i	$0.301279\pi$
$524$	0	0
$525$	1.18182	0.0515789
$526$	0	0
$527$	34.2049	1.48999
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	2.74332	0.119050
$532$	0	0
$533$	21.4746	0.930170
$534$	0	0
$535$	−7.97214	−0.344666
$536$	0	0
$537$	24.2584	1.04683
$538$	0	0
$539$	5.58607	0.240609
$540$	0	0
$541$	−36.5692	−1.57223	−0.786117	−	0.618077i	$-0.787912\pi$
$541$	−36.5692	−1.57223	−0.786117	+	0.618077i	$0.787912\pi$
$542$	0	0
$543$	1.45055	0.0622491
$544$	0	0
$545$	12.0242	0.515062
$546$	0	0
$547$	26.9567	1.15258	0.576292	−	0.817244i	$-0.304499\pi$
$547$	26.9567	1.15258	0.576292	+	0.817244i	$0.304499\pi$
$548$	0	0
$549$	−6.89748	−0.294377
$550$	0	0
$551$	−10.2954	−0.438601
$552$	0	0
$553$	−10.9499	−0.465638
$554$	0	0
$555$	−6.96758	−0.295757
$556$	0	0
$557$	−39.6641	−1.68062	−0.840310	−	0.542106i	$-0.817627\pi$
$557$	−39.6641	−1.68062	−0.840310	+	0.542106i	$0.817627\pi$
$558$	0	0
$559$	21.2044	0.896849
$560$	0	0
$561$	22.2173	0.938014
$562$	0	0
$563$	39.4716	1.66353	0.831764	−	0.555129i	$-0.187331\pi$
$563$	39.4716	1.66353	0.831764	+	0.555129i	$0.187331\pi$
$564$	0	0
$565$	−13.0436	−0.548750
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−35.2655	−1.47841	−0.739204	−	0.673482i	$-0.764798\pi$
$569$	−35.2655	−1.47841	−0.739204	+	0.673482i	$0.764798\pi$
$570$	0	0
$571$	14.4493	0.604684	0.302342	−	0.953200i	$-0.402232\pi$
$571$	14.4493	0.604684	0.302342	+	0.953200i	$0.402232\pi$
$572$	0	0
$573$	−1.08208	−0.0452046
$574$	0	0
$575$	1.18182	0.0492854
$576$	0	0
$577$	36.0733	1.50175	0.750876	−	0.660443i	$-0.229632\pi$
$577$	36.0733	1.50175	0.750876	+	0.660443i	$0.229632\pi$
$578$	0	0
$579$	1.29707	0.0539044
$580$	0	0
$581$	−10.6227	−0.440703
$582$	0	0
$583$	−49.5749	−2.05318
$584$	0	0
$585$	12.8157	0.529864
$586$	0	0
$587$	−15.1433	−0.625032	−0.312516	−	0.949913i	$-0.601172\pi$
$587$	−15.1433	−0.625032	−0.312516	+	0.949913i	$0.601172\pi$
$588$	0	0
$589$	48.0407	1.97948
$590$	0	0
$591$	5.19534	0.213708
$592$	0	0
$593$	7.73756	0.317744	0.158872	−	0.987299i	$-0.449214\pi$
$593$	7.73756	0.317744	0.158872	+	0.987299i	$0.449214\pi$
$594$	0	0
$595$	−9.88879	−0.405401
$596$	0	0
$597$	−3.58114	−0.146566
$598$	0	0
$599$	−7.73552	−0.316065	−0.158032	−	0.987434i	$-0.550515\pi$
$599$	−7.73552	−0.316065	−0.158032	+	0.987434i	$0.550515\pi$
$600$	0	0
$601$	7.28082	0.296991	0.148495	−	0.988913i	$-0.452557\pi$
$601$	7.28082	0.296991	0.148495	+	0.988913i	$0.452557\pi$
$602$	0	0
$603$	7.59319	0.309219
$604$	0	0
$605$	−50.2341	−2.04231
$606$	0	0
$607$	17.6503	0.716405	0.358203	−	0.933644i	$-0.383390\pi$
$607$	17.6503	0.716405	0.358203	+	0.933644i	$0.383390\pi$
$608$	0	0
$609$	−1.84306	−0.0746845
$610$	0	0
$611$	−21.5918	−0.873511
$612$	0	0
$613$	−25.4707	−1.02875	−0.514375	−	0.857565i	$-0.671976\pi$
$613$	−25.4707	−1.02875	−0.514375	+	0.857565i	$0.671976\pi$
$614$	0	0
$615$	10.3586	0.417698
$616$	0	0
$617$	3.26123	0.131292	0.0656462	−	0.997843i	$-0.479089\pi$
$617$	3.26123	0.131292	0.0656462	+	0.997843i	$0.479089\pi$
$618$	0	0
$619$	−16.6910	−0.670870	−0.335435	−	0.942063i	$-0.608883\pi$
$619$	−16.6910	−0.670870	−0.335435	+	0.942063i	$0.608883\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	5.08921	0.203895
$624$	0	0
$625$	−29.5124	−1.18050
$626$	0	0
$627$	31.2041	1.24617
$628$	0	0
$629$	11.1457	0.444409
$630$	0	0
$631$	22.0623	0.878288	0.439144	−	0.898417i	$-0.355282\pi$
$631$	22.0623	0.878288	0.439144	+	0.898417i	$0.355282\pi$
$632$	0	0
$633$	−19.9911	−0.794576
$634$	0	0
$635$	−1.25554	−0.0498245
$636$	0	0
$637$	−5.15448	−0.204228
$638$	0	0
$639$	−5.88345	−0.232746
$640$	0	0
$641$	−2.57715	−0.101791	−0.0508957	−	0.998704i	$-0.516208\pi$
$641$	−2.57715	−0.101791	−0.0508957	+	0.998704i	$0.516208\pi$
$642$	0	0
$643$	47.1752	1.86041	0.930204	−	0.367042i	$-0.119629\pi$
$643$	47.1752	1.86041	0.930204	+	0.367042i	$0.119629\pi$
$644$	0	0
$645$	10.2282	0.402735
$646$	0	0
$647$	28.9514	1.13820	0.569099	−	0.822269i	$-0.307292\pi$
$647$	28.9514	1.13820	0.569099	+	0.822269i	$0.307292\pi$
$648$	0	0
$649$	15.3244	0.601534
$650$	0	0
$651$	8.60010	0.337065
$652$	0	0
$653$	27.5076	1.07646	0.538229	−	0.842799i	$-0.319094\pi$
$653$	27.5076	1.07646	0.538229	+	0.842799i	$0.319094\pi$
$654$	0	0
$655$	25.4388	0.993977
$656$	0	0
$657$	9.31357	0.363357
$658$	0	0
$659$	−37.7352	−1.46995	−0.734977	−	0.678092i	$-0.762807\pi$
$659$	−37.7352	−1.46995	−0.734977	+	0.678092i	$0.762807\pi$
$660$	0	0
$661$	−30.3450	−1.18028	−0.590142	−	0.807300i	$-0.700928\pi$
$661$	−30.3450	−1.18028	−0.590142	+	0.807300i	$0.700928\pi$
$662$	0	0
$663$	−20.5007	−0.796182
$664$	0	0
$665$	−13.8888	−0.538584
$666$	0	0
$667$	−1.84306	−0.0713635
$668$	0	0
$669$	−27.4107	−1.05976
$670$	0	0
$671$	−38.5298	−1.48743
$672$	0	0
$673$	−9.49791	−0.366118	−0.183059	−	0.983102i	$-0.558600\pi$
$673$	−9.49791	−0.366118	−0.183059	+	0.983102i	$0.558600\pi$
$674$	0	0
$675$	1.18182	0.0454883
$676$	0	0
$677$	−12.0247	−0.462148	−0.231074	−	0.972936i	$-0.574224\pi$
$677$	−12.0247	−0.462148	−0.231074	+	0.972936i	$0.574224\pi$
$678$	0	0
$679$	−17.3931	−0.667485
$680$	0	0
$681$	18.7001	0.716588
$682$	0	0
$683$	−32.7716	−1.25397	−0.626985	−	0.779031i	$-0.715711\pi$
$683$	−32.7716	−1.25397	−0.626985	+	0.779031i	$0.715711\pi$
$684$	0	0
$685$	−42.6128	−1.62815
$686$	0	0
$687$	−16.2800	−0.621120
$688$	0	0
$689$	45.7447	1.74273
$690$	0	0
$691$	−49.5902	−1.88650	−0.943250	−	0.332083i	$-0.892248\pi$
$691$	−49.5902	−1.88650	−0.943250	+	0.332083i	$0.892248\pi$
$692$	0	0
$693$	5.58607	0.212197
$694$	0	0
$695$	−47.0580	−1.78501
$696$	0	0
$697$	−16.5701	−0.627639
$698$	0	0
$699$	5.27412	0.199485
$700$	0	0
$701$	15.0389	0.568012	0.284006	−	0.958822i	$-0.408336\pi$
$701$	15.0389	0.568012	0.284006	+	0.958822i	$0.408336\pi$
$702$	0	0
$703$	15.6542	0.590408
$704$	0	0
$705$	−10.4151	−0.392255
$706$	0	0
$707$	8.22933	0.309496
$708$	0	0
$709$	2.72603	0.102378	0.0511892	−	0.998689i	$-0.483699\pi$
$709$	2.72603	0.102378	0.0511892	+	0.998689i	$0.483699\pi$
$710$	0	0
$711$	−10.9499	−0.410654
$712$	0	0
$713$	8.60010	0.322076
$714$	0	0
$715$	71.5894	2.67729
$716$	0	0
$717$	−6.07035	−0.226701
$718$	0	0
$719$	−18.3565	−0.684582	−0.342291	−	0.939594i	$-0.611203\pi$
$719$	−18.3565	−0.684582	−0.342291	+	0.939594i	$0.611203\pi$
$720$	0	0
$721$	18.4979	0.688898
$722$	0	0
$723$	−15.3383	−0.570439
$724$	0	0
$725$	−2.17817	−0.0808951
$726$	0	0
$727$	−23.1402	−0.858223	−0.429112	−	0.903251i	$-0.641173\pi$
$727$	−23.1402	−0.858223	−0.429112	+	0.903251i	$0.641173\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−16.3616	−0.605155
$732$	0	0
$733$	−34.3171	−1.26753	−0.633765	−	0.773525i	$-0.718491\pi$
$733$	−34.3171	−1.26753	−0.633765	+	0.773525i	$0.718491\pi$
$734$	0	0
$735$	−2.48633	−0.0917096
$736$	0	0
$737$	42.4161	1.56242
$738$	0	0
$739$	16.9924	0.625077	0.312539	−	0.949905i	$-0.398821\pi$
$739$	16.9924	0.625077	0.312539	+	0.949905i	$0.398821\pi$
$740$	0	0
$741$	−28.7932	−1.05775
$742$	0	0
$743$	−50.5339	−1.85391	−0.926954	−	0.375175i	$-0.877583\pi$
$743$	−50.5339	−1.85391	−0.926954	+	0.375175i	$0.877583\pi$
$744$	0	0
$745$	12.3535	0.452597
$746$	0	0
$747$	−10.6227	−0.388664
$748$	0	0
$749$	3.20639	0.117159
$750$	0	0
$751$	−29.7011	−1.08381	−0.541905	−	0.840440i	$-0.682297\pi$
$751$	−29.7011	−1.08381	−0.541905	+	0.840440i	$0.682297\pi$
$752$	0	0
$753$	−15.1433	−0.551851
$754$	0	0
$755$	−31.5698	−1.14894
$756$	0	0
$757$	53.4804	1.94378	0.971889	−	0.235439i	$-0.0756528\pi$
$757$	53.4804	1.94378	0.971889	+	0.235439i	$0.0756528\pi$
$758$	0	0
$759$	5.58607	0.202761
$760$	0	0
$761$	2.77680	0.100659	0.0503295	−	0.998733i	$-0.483973\pi$
$761$	2.77680	0.100659	0.0503295	+	0.998733i	$0.483973\pi$
$762$	0	0
$763$	−4.83615	−0.175080
$764$	0	0
$765$	−9.88879	−0.357530
$766$	0	0
$767$	−14.1404	−0.510579
$768$	0	0
$769$	18.0874	0.652248	0.326124	−	0.945327i	$-0.394257\pi$
$769$	18.0874	0.652248	0.326124	+	0.945327i	$0.394257\pi$
$770$	0	0
$771$	24.1995	0.871525
$772$	0	0
$773$	8.32891	0.299570	0.149785	−	0.988719i	$-0.452142\pi$
$773$	8.32891	0.299570	0.149785	+	0.988719i	$0.452142\pi$
$774$	0	0
$775$	10.1638	0.365094
$776$	0	0
$777$	2.80236	0.100534
$778$	0	0
$779$	−23.2727	−0.833833
$780$	0	0
$781$	−32.8653	−1.17601
$782$	0	0
$783$	−1.84306	−0.0658655
$784$	0	0
$785$	−10.5212	−0.375519
$786$	0	0
$787$	36.2009	1.29042	0.645211	−	0.764004i	$-0.276769\pi$
$787$	36.2009	1.29042	0.645211	+	0.764004i	$0.276769\pi$
$788$	0	0
$789$	−9.51990	−0.338918
$790$	0	0
$791$	5.24615	0.186532
$792$	0	0
$793$	35.5529	1.26252
$794$	0	0
$795$	22.0655	0.782584
$796$	0	0
$797$	−34.8246	−1.23355	−0.616775	−	0.787140i	$-0.711561\pi$
$797$	−34.8246	−1.23355	−0.616775	+	0.787140i	$0.711561\pi$
$798$	0	0
$799$	16.6606	0.589408
$800$	0	0
$801$	5.08921	0.179818
$802$	0	0
$803$	52.0262	1.83596
$804$	0	0
$805$	−2.48633	−0.0876315
$806$	0	0
$807$	18.8245	0.662652
$808$	0	0
$809$	13.9755	0.491353	0.245677	−	0.969352i	$-0.420990\pi$
$809$	13.9755	0.491353	0.245677	+	0.969352i	$0.420990\pi$
$810$	0	0
$811$	−52.9478	−1.85925	−0.929625	−	0.368507i	$-0.879869\pi$
$811$	−52.9478	−1.85925	−0.929625	+	0.368507i	$0.879869\pi$
$812$	0	0
$813$	6.20839	0.217738
$814$	0	0
$815$	2.45714	0.0860700
$816$	0	0
$817$	−22.9798	−0.803962
$818$	0	0
$819$	−5.15448	−0.180112
$820$	0	0
$821$	−28.9952	−1.01194	−0.505969	−	0.862552i	$-0.668865\pi$
$821$	−28.9952	−1.01194	−0.505969	+	0.862552i	$0.668865\pi$
$822$	0	0
$823$	27.4595	0.957178	0.478589	−	0.878039i	$-0.341148\pi$
$823$	27.4595	0.957178	0.478589	+	0.878039i	$0.341148\pi$
$824$	0	0
$825$	6.60174	0.229843
$826$	0	0
$827$	36.2253	1.25968	0.629838	−	0.776726i	$-0.283121\pi$
$827$	36.2253	1.25968	0.629838	+	0.776726i	$0.283121\pi$
$828$	0	0
$829$	37.3490	1.29718	0.648592	−	0.761137i	$-0.275358\pi$
$829$	37.3490	1.29718	0.648592	+	0.761137i	$0.275358\pi$
$830$	0	0
$831$	−18.3161	−0.635378
$832$	0	0
$833$	3.97727	0.137804
$834$	0	0
$835$	−36.3237	−1.25703
$836$	0	0
$837$	8.60010	0.297263
$838$	0	0
$839$	36.5700	1.26254	0.631268	−	0.775565i	$-0.282535\pi$
$839$	36.5700	1.26254	0.631268	+	0.775565i	$0.282535\pi$
$840$	0	0
$841$	−25.6031	−0.882867
$842$	0	0
$843$	0.962077	0.0331357
$844$	0	0
$845$	−33.7360	−1.16055
$846$	0	0
$847$	20.2041	0.694222
$848$	0	0
$849$	2.75663	0.0946073
$850$	0	0
$851$	2.80236	0.0960636
$852$	0	0
$853$	17.9871	0.615867	0.307933	−	0.951408i	$-0.400363\pi$
$853$	17.9871	0.615867	0.307933	+	0.951408i	$0.400363\pi$
$854$	0	0
$855$	−13.8888	−0.474987
$856$	0	0
$857$	−48.9378	−1.67168	−0.835842	−	0.548970i	$-0.815020\pi$
$857$	−48.9378	−1.67168	−0.835842	+	0.548970i	$0.815020\pi$
$858$	0	0
$859$	−41.0278	−1.39985	−0.699925	−	0.714217i	$-0.746783\pi$
$859$	−41.0278	−1.39985	−0.699925	+	0.714217i	$0.746783\pi$
$860$	0	0
$861$	−4.16621	−0.141984
$862$	0	0
$863$	−21.3957	−0.728318	−0.364159	−	0.931337i	$-0.618644\pi$
$863$	−21.3957	−0.728318	−0.364159	+	0.931337i	$0.618644\pi$
$864$	0	0
$865$	−50.0300	−1.70107
$866$	0	0
$867$	−1.18134	−0.0401205
$868$	0	0
$869$	−61.1670	−2.07495
$870$	0	0
$871$	−39.1389	−1.32617
$872$	0	0
$873$	−17.3931	−0.588667
$874$	0	0
$875$	9.49324	0.320930
$876$	0	0
$877$	−0.946523	−0.0319618	−0.0159809	−	0.999872i	$-0.505087\pi$
$877$	−0.946523	−0.0319618	−0.0159809	+	0.999872i	$0.505087\pi$
$878$	0	0
$879$	8.89036	0.299864
$880$	0	0
$881$	−15.6489	−0.527226	−0.263613	−	0.964629i	$-0.584914\pi$
$881$	−15.6489	−0.527226	−0.263613	+	0.964629i	$0.584914\pi$
$882$	0	0
$883$	37.2212	1.25259	0.626297	−	0.779585i	$-0.284570\pi$
$883$	37.2212	1.25259	0.626297	+	0.779585i	$0.284570\pi$
$884$	0	0
$885$	−6.82079	−0.229278
$886$	0	0
$887$	−0.511534	−0.0171756	−0.00858781	−	0.999963i	$-0.502734\pi$
$887$	−0.511534	−0.0171756	−0.00858781	+	0.999963i	$0.502734\pi$
$888$	0	0
$889$	0.504977	0.0169364
$890$	0	0
$891$	5.58607	0.187140
$892$	0	0
$893$	23.3997	0.783042
$894$	0	0
$895$	−60.3143	−2.01609
$896$	0	0
$897$	−5.15448	−0.172103
$898$	0	0
$899$	−15.8505	−0.528643
$900$	0	0
$901$	−35.2973	−1.17592
$902$	0	0
$903$	−4.11378	−0.136898
$904$	0	0
$905$	−3.60654	−0.119886
$906$	0	0
$907$	−30.6922	−1.01912	−0.509559	−	0.860436i	$-0.670191\pi$
$907$	−30.6922	−1.01912	−0.509559	+	0.860436i	$0.670191\pi$
$908$	0	0
$909$	8.22933	0.272950
$910$	0	0
$911$	16.4719	0.545737	0.272868	−	0.962051i	$-0.412028\pi$
$911$	16.4719	0.545737	0.272868	+	0.962051i	$0.412028\pi$
$912$	0	0
$913$	−59.3390	−1.96383
$914$	0	0
$915$	17.1494	0.566942
$916$	0	0
$917$	−10.2315	−0.337873
$918$	0	0
$919$	4.23467	0.139689	0.0698445	−	0.997558i	$-0.477750\pi$
$919$	4.23467	0.139689	0.0698445	+	0.997558i	$0.477750\pi$
$920$	0	0
$921$	2.37067	0.0781162
$922$	0	0
$923$	30.3261	0.998195
$924$	0	0
$925$	3.31189	0.108894
$926$	0	0
$927$	18.4979	0.607551
$928$	0	0
$929$	−52.9709	−1.73792	−0.868960	−	0.494883i	$-0.835211\pi$
$929$	−52.9709	−1.73792	−0.868960	+	0.494883i	$0.835211\pi$
$930$	0	0
$931$	5.58607	0.183076
$932$	0	0
$933$	34.4292	1.12716
$934$	0	0
$935$	−55.2394	−1.80652
$936$	0	0
$937$	−36.0466	−1.17759	−0.588795	−	0.808282i	$-0.700398\pi$
$937$	−36.0466	−1.17759	−0.588795	+	0.808282i	$0.700398\pi$
$938$	0	0
$939$	17.4861	0.570636
$940$	0	0
$941$	−45.5566	−1.48510	−0.742551	−	0.669790i	$-0.766384\pi$
$941$	−45.5566	−1.48510	−0.742551	+	0.669790i	$0.766384\pi$
$942$	0	0
$943$	−4.16621	−0.135671
$944$	0	0
$945$	−2.48633	−0.0808802
$946$	0	0
$947$	42.4095	1.37812	0.689062	−	0.724702i	$-0.258023\pi$
$947$	42.4095	1.37812	0.689062	+	0.724702i	$0.258023\pi$
$948$	0	0
$949$	−48.0065	−1.55836
$950$	0	0
$951$	30.7191	0.996134
$952$	0	0
$953$	6.71094	0.217389	0.108694	−	0.994075i	$-0.465333\pi$
$953$	6.71094	0.217389	0.108694	+	0.994075i	$0.465333\pi$
$954$	0	0
$955$	2.69041	0.0870597
$956$	0	0
$957$	−10.2954	−0.332805
$958$	0	0
$959$	17.1389	0.553443
$960$	0	0
$961$	42.9618	1.38586
$962$	0	0
$963$	3.20639	0.103325
$964$	0	0
$965$	−3.22494	−0.103815
$966$	0	0
$967$	8.77181	0.282082	0.141041	−	0.990004i	$-0.454955\pi$
$967$	8.77181	0.282082	0.141041	+	0.990004i	$0.454955\pi$
$968$	0	0
$969$	22.2173	0.713722
$970$	0	0
$971$	−29.0192	−0.931270	−0.465635	−	0.884977i	$-0.654174\pi$
$971$	−29.0192	−0.931270	−0.465635	+	0.884977i	$0.654174\pi$
$972$	0	0
$973$	18.9267	0.606762
$974$	0	0
$975$	−6.09167	−0.195090
$976$	0	0
$977$	4.23594	0.135520	0.0677598	−	0.997702i	$-0.478415\pi$
$977$	4.23594	0.135520	0.0677598	+	0.997702i	$0.478415\pi$
$978$	0	0
$979$	28.4286	0.908583
$980$	0	0
$981$	−4.83615	−0.154406
$982$	0	0
$983$	−45.5822	−1.45385	−0.726923	−	0.686719i	$-0.759050\pi$
$983$	−45.5822	−1.45385	−0.726923	+	0.686719i	$0.759050\pi$
$984$	0	0
$985$	−12.9173	−0.411580
$986$	0	0
$987$	4.18895	0.133336
$988$	0	0
$989$	−4.11378	−0.130810
$990$	0	0
$991$	15.0276	0.477366	0.238683	−	0.971098i	$-0.423284\pi$
$991$	15.0276	0.477366	0.238683	+	0.971098i	$0.423284\pi$
$992$	0	0
$993$	−15.6222	−0.495754
$994$	0	0
$995$	8.90389	0.282272
$996$	0	0
$997$	−2.25951	−0.0715594	−0.0357797	−	0.999360i	$-0.511391\pi$
$997$	−2.25951	−0.0715594	−0.0357797	+	0.999360i	$0.511391\pi$
$998$	0	0
$999$	2.80236	0.0886627

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.ch.1.2		6
4.3	odd	2		3864.2.a.w.1.2	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3864.2.a.w.1.2	✓	6	4.3	odd	2
7728.2.a.ch.1.2		6	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.48633 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.48633 q^{5} +1.00000 q^{7} +1.00000 q^{9} +5.58607 q^{11} -5.15448 q^{13} -2.48633 q^{15} +3.97727 q^{17} +5.58607 q^{19} +1.00000 q^{21} +1.00000 q^{23} +1.18182 q^{25} +1.00000 q^{27} -1.84306 q^{29} +8.60010 q^{31} +5.58607 q^{33} -2.48633 q^{35} +2.80236 q^{37} -5.15448 q^{39} -4.16621 q^{41} -4.11378 q^{43} -2.48633 q^{45} +4.18895 q^{47} +1.00000 q^{49} +3.97727 q^{51} -8.87475 q^{53} -13.8888 q^{55} +5.58607 q^{57} +2.74332 q^{59} -6.89748 q^{61} +1.00000 q^{63} +12.8157 q^{65} +7.59319 q^{67} +1.00000 q^{69} -5.88345 q^{71} +9.31357 q^{73} +1.18182 q^{75} +5.58607 q^{77} -10.9499 q^{79} +1.00000 q^{81} -10.6227 q^{83} -9.88879 q^{85} -1.84306 q^{87} +5.08921 q^{89} -5.15448 q^{91} +8.60010 q^{93} -13.8888 q^{95} -17.3931 q^{97} +5.58607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} + 2 q^{5} + 6 q^{7} + 6 q^{9}+O(q^{10}) \) 6 * q + 6 * q^3 + 2 * q^5 + 6 * q^7 + 6 * q^9	\( 6 q + 6 q^{3} + 2 q^{5} + 6 q^{7} + 6 q^{9} + 3 q^{11} + 2 q^{15} + 6 q^{17} + 3 q^{19} + 6 q^{21} + 6 q^{23} + 10 q^{25} + 6 q^{27} + 5 q^{29} + 4 q^{31} + 3 q^{33} + 2 q^{35} + q^{37} + 12 q^{41} + 6 q^{43} + 2 q^{45} + 6 q^{47} + 6 q^{49} + 6 q^{51} + 10 q^{53} - 3 q^{55} + 3 q^{57} + 14 q^{59} + 4 q^{61} + 6 q^{63} + 27 q^{65} - 7 q^{67} + 6 q^{69} - 7 q^{71} + 10 q^{73} + 10 q^{75} + 3 q^{77} - 14 q^{79} + 6 q^{81} - 14 q^{83} + 21 q^{85} + 5 q^{87} + 25 q^{89} + 4 q^{93} - 3 q^{95} + 11 q^{97} + 3 q^{99}+O(q^{100}) \) 6 * q + 6 * q^3 + 2 * q^5 + 6 * q^7 + 6 * q^9 + 3 * q^11 + 2 * q^15 + 6 * q^17 + 3 * q^19 + 6 * q^21 + 6 * q^23 + 10 * q^25 + 6 * q^27 + 5 * q^29 + 4 * q^31 + 3 * q^33 + 2 * q^35 + q^37 + 12 * q^41 + 6 * q^43 + 2 * q^45 + 6 * q^47 + 6 * q^49 + 6 * q^51 + 10 * q^53 - 3 * q^55 + 3 * q^57 + 14 * q^59 + 4 * q^61 + 6 * q^63 + 27 * q^65 - 7 * q^67 + 6 * q^69 - 7 * q^71 + 10 * q^73 + 10 * q^75 + 3 * q^77 - 14 * q^79 + 6 * q^81 - 14 * q^83 + 21 * q^85 + 5 * q^87 + 25 * q^89 + 4 * q^93 - 3 * q^95 + 11 * q^97 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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