↑

Properties

Label 320.2.f.a.289.3
Level $320$
Weight $2$
Character 320.289
Analytic conductor $2.555$
Analytic rank $0$
Dimension $4$
CM discriminant -40
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [320,2,Mod(289,320)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("320.289"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(320, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 320.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.55521286468\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 289.3
Root \(-0.618034i\) of defining polynomial
Character \(\chi\) \(=\) 320.289
Dual form 320.2.f.a.289.4

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.23607 q^{5} -4.47214i q^{7} -3.00000 q^{9} -2.00000i q^{11} +4.47214 q^{13} +6.00000i q^{19} -4.47214i q^{23} +5.00000 q^{25} -10.0000i q^{35} +4.47214 q^{37} +2.00000 q^{41} -6.70820 q^{45} +13.4164i q^{47} -13.0000 q^{49} -13.4164 q^{53} -4.47214i q^{55} +14.0000i q^{59} +13.4164i q^{63} +10.0000 q^{65} -8.94427 q^{77} +9.00000 q^{81} -14.0000 q^{89} -20.0000i q^{91} +13.4164i q^{95} +6.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{9} + 20 q^{25} + 8 q^{41} - 52 q^{49} + 40 q^{65} + 36 q^{81} - 56 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 0 0
\(5\) 2.23607 1.00000
\(6\) 0 0
\(7\) − 4.47214i − 1.69031i −0.534522 − 0.845154i \(-0.679509\pi\)
0.534522 − 0.845154i \(-0.320491\pi\)
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) − 2.00000i − 0.603023i −0.953463 − 0.301511i \(-0.902509\pi\)
0.953463 − 0.301511i \(-0.0974911\pi\)
\(12\) 0 0
\(13\) 4.47214 1.24035 0.620174 − 0.784465i \(-0.287062\pi\)
0.620174 + 0.784465i \(0.287062\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 6.00000i 1.37649i 0.725476 + 0.688247i \(0.241620\pi\)
−0.725476 + 0.688247i \(0.758380\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 4.47214i − 0.932505i −0.884652 − 0.466252i \(-0.845604\pi\)
0.884652 − 0.466252i \(-0.154396\pi\)
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 10.0000i − 1.69031i
\(36\) 0 0
\(37\) 4.47214 0.735215 0.367607 − 0.929981i \(-0.380177\pi\)
0.367607 + 0.929981i \(0.380177\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) −6.70820 −1.00000
\(46\) 0 0
\(47\) 13.4164i 1.95698i 0.206284 + 0.978492i \(0.433863\pi\)
−0.206284 + 0.978492i \(0.566137\pi\)
\(48\) 0 0
\(49\) −13.0000 −1.85714
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −13.4164 −1.84289 −0.921443 − 0.388514i \(-0.872988\pi\)
−0.921443 + 0.388514i \(0.872988\pi\)
\(54\) 0 0
\(55\) − 4.47214i − 0.603023i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 14.0000i 1.82264i 0.411693 + 0.911322i \(0.364937\pi\)
−0.411693 + 0.911322i \(0.635063\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 13.4164i 1.69031i
\(64\) 0 0
\(65\) 10.0000 1.24035
\(66\) 0 0
\(67\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −8.94427 −1.01929
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −14.0000 −1.48400 −0.741999 − 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 0 0
\(91\) − 20.0000i − 2.09657i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 13.4164i 1.37649i
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 6.00000i 0.603023i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 320.2.f.a.289.3 yes 4
3.2 odd 2 2880.2.d.e.289.1 4
4.3 odd 2 inner 320.2.f.a.289.4 yes 4
5.2 odd 4 1600.2.d.g.801.3 4
5.3 odd 4 1600.2.d.g.801.1 4
5.4 even 2 inner 320.2.f.a.289.2 yes 4
8.3 odd 2 inner 320.2.f.a.289.2 yes 4
8.5 even 2 inner 320.2.f.a.289.1 ✓ 4
12.11 even 2 2880.2.d.e.289.2 4
15.14 odd 2 2880.2.d.e.289.4 4
16.3 odd 4 1280.2.c.c.769.1 2
16.5 even 4 1280.2.c.c.769.2 2
16.11 odd 4 1280.2.c.b.769.2 2
16.13 even 4 1280.2.c.b.769.1 2
20.3 even 4 1600.2.d.g.801.4 4
20.7 even 4 1600.2.d.g.801.2 4
20.19 odd 2 inner 320.2.f.a.289.1 ✓ 4
24.5 odd 2 2880.2.d.e.289.3 4
24.11 even 2 2880.2.d.e.289.4 4
40.3 even 4 1600.2.d.g.801.3 4
40.13 odd 4 1600.2.d.g.801.2 4
40.19 odd 2 CM 320.2.f.a.289.3 yes 4
40.27 even 4 1600.2.d.g.801.1 4
40.29 even 2 inner 320.2.f.a.289.4 yes 4
40.37 odd 4 1600.2.d.g.801.4 4
60.59 even 2 2880.2.d.e.289.3 4
80.3 even 4 6400.2.a.bj.1.1 2
80.13 odd 4 6400.2.a.bi.1.2 2
80.19 odd 4 1280.2.c.c.769.2 2
80.27 even 4 6400.2.a.bi.1.2 2
80.29 even 4 1280.2.c.b.769.2 2
80.37 odd 4 6400.2.a.bj.1.1 2
80.43 even 4 6400.2.a.bi.1.1 2
80.53 odd 4 6400.2.a.bj.1.2 2
80.59 odd 4 1280.2.c.b.769.1 2
80.67 even 4 6400.2.a.bj.1.2 2
80.69 even 4 1280.2.c.c.769.1 2
80.77 odd 4 6400.2.a.bi.1.1 2
120.29 odd 2 2880.2.d.e.289.2 4
120.59 even 2 2880.2.d.e.289.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
320.2.f.a.289.1 ✓ 4 8.5 even 2 inner
320.2.f.a.289.1 ✓ 4 20.19 odd 2 inner
320.2.f.a.289.2 yes 4 5.4 even 2 inner
320.2.f.a.289.2 yes 4 8.3 odd 2 inner
320.2.f.a.289.3 yes 4 1.1 even 1 trivial
320.2.f.a.289.3 yes 4 40.19 odd 2 CM
320.2.f.a.289.4 yes 4 4.3 odd 2 inner
320.2.f.a.289.4 yes 4 40.29 even 2 inner
1280.2.c.b.769.1 2 16.13 even 4
1280.2.c.b.769.1 2 80.59 odd 4
1280.2.c.b.769.2 2 16.11 odd 4
1280.2.c.b.769.2 2 80.29 even 4
1280.2.c.c.769.1 2 16.3 odd 4
1280.2.c.c.769.1 2 80.69 even 4
1280.2.c.c.769.2 2 16.5 even 4
1280.2.c.c.769.2 2 80.19 odd 4
1600.2.d.g.801.1 4 5.3 odd 4
1600.2.d.g.801.1 4 40.27 even 4
1600.2.d.g.801.2 4 20.7 even 4
1600.2.d.g.801.2 4 40.13 odd 4
1600.2.d.g.801.3 4 5.2 odd 4
1600.2.d.g.801.3 4 40.3 even 4
1600.2.d.g.801.4 4 20.3 even 4
1600.2.d.g.801.4 4 40.37 odd 4
2880.2.d.e.289.1 4 3.2 odd 2
2880.2.d.e.289.1 4 120.59 even 2
2880.2.d.e.289.2 4 12.11 even 2
2880.2.d.e.289.2 4 120.29 odd 2
2880.2.d.e.289.3 4 24.5 odd 2
2880.2.d.e.289.3 4 60.59 even 2
2880.2.d.e.289.4 4 15.14 odd 2
2880.2.d.e.289.4 4 24.11 even 2
6400.2.a.bi.1.1 2 80.43 even 4
6400.2.a.bi.1.1 2 80.77 odd 4
6400.2.a.bi.1.2 2 80.13 odd 4
6400.2.a.bi.1.2 2 80.27 even 4
6400.2.a.bj.1.1 2 80.3 even 4
6400.2.a.bj.1.1 2 80.37 odd 4
6400.2.a.bj.1.2 2 80.53 odd 4
6400.2.a.bj.1.2 2 80.67 even 4