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Properties

Label 320.2.c.a.129.1
Level $320$
Weight $2$
Character 320.129
Analytic conductor $2.555$
Analytic rank $0$
Dimension $2$
CM discriminant -4
Inner twists $4$

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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(129,320)] 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("320.129"); 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, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 320.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-2,0,0,0,6,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.55521286468\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 160)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 129.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 320.129
Dual form 320.2.c.a.129.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.00000 - 2.00000i) q^{5} +3.00000 q^{9} -4.00000i q^{13} -8.00000i q^{17} +(-3.00000 + 4.00000i) q^{25} +10.0000 q^{29} +12.0000i q^{37} -10.0000 q^{41} +(-3.00000 - 6.00000i) q^{45} +7.00000 q^{49} +4.00000i q^{53} -10.0000 q^{61} +(-8.00000 + 4.00000i) q^{65} +16.0000i q^{73} +9.00000 q^{81} +(-16.0000 + 8.00000i) q^{85} +10.0000 q^{89} +8.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + 6 q^{9} - 6 q^{25} + 20 q^{29} - 20 q^{41} - 6 q^{45} + 14 q^{49} - 20 q^{61} - 16 q^{65} + 18 q^{81} - 32 q^{85} + 20 q^{89}+O(q^{100}) \) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 0 0
\(5\) −1.00000 − 2.00000i −0.447214 − 0.894427i
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) − 4.00000i − 1.10940i −0.832050 − 0.554700i \(-0.812833\pi\)
0.832050 − 0.554700i \(-0.187167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 8.00000i − 1.94029i −0.242536 − 0.970143i \(-0.577979\pi\)
0.242536 − 0.970143i \(-0.422021\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) −3.00000 + 4.00000i −0.600000 + 0.800000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 10.0000 1.85695 0.928477 − 0.371391i \(-0.121119\pi\)
0.928477 + 0.371391i \(0.121119\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 12.0000i 1.97279i 0.164399 + 0.986394i \(0.447432\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −10.0000 −1.56174 −0.780869 − 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) −3.00000 − 6.00000i −0.447214 − 0.894427i
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.00000i 0.549442i 0.961524 + 0.274721i \(0.0885855\pi\)
−0.961524 + 0.274721i \(0.911414\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −8.00000 + 4.00000i −0.992278 + 0.496139i
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 16.0000i 1.87266i 0.351123 + 0.936329i \(0.385800\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) −16.0000 + 8.00000i −1.73544 + 0.867722i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 − 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 8.00000i 0.812277i 0.913812 + 0.406138i \(0.133125\pi\)
−0.913812 + 0.406138i \(0.866875\pi\)
\(98\) 0 0
\(99\) 0 0
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.c.a.129.1 2
3.2 odd 2 2880.2.f.n.1729.2 2
4.3 odd 2 CM 320.2.c.a.129.1 2
5.2 odd 4 1600.2.a.l.1.1 1
5.3 odd 4 1600.2.a.m.1.1 1
5.4 even 2 inner 320.2.c.a.129.2 2
8.3 odd 2 160.2.c.a.129.2 yes 2
8.5 even 2 160.2.c.a.129.2 yes 2
12.11 even 2 2880.2.f.n.1729.2 2
15.14 odd 2 2880.2.f.n.1729.1 2
16.3 odd 4 1280.2.f.c.129.2 2
16.5 even 4 1280.2.f.d.129.1 2
16.11 odd 4 1280.2.f.d.129.1 2
16.13 even 4 1280.2.f.c.129.2 2
20.3 even 4 1600.2.a.m.1.1 1
20.7 even 4 1600.2.a.l.1.1 1
20.19 odd 2 inner 320.2.c.a.129.2 2
24.5 odd 2 1440.2.f.c.289.1 2
24.11 even 2 1440.2.f.c.289.1 2
40.3 even 4 800.2.a.e.1.1 1
40.13 odd 4 800.2.a.e.1.1 1
40.19 odd 2 160.2.c.a.129.1 ✓ 2
40.27 even 4 800.2.a.f.1.1 1
40.29 even 2 160.2.c.a.129.1 ✓ 2
40.37 odd 4 800.2.a.f.1.1 1
60.59 even 2 2880.2.f.n.1729.1 2
80.19 odd 4 1280.2.f.d.129.2 2
80.29 even 4 1280.2.f.d.129.2 2
80.59 odd 4 1280.2.f.c.129.1 2
80.69 even 4 1280.2.f.c.129.1 2
120.29 odd 2 1440.2.f.c.289.2 2
120.53 even 4 7200.2.a.y.1.1 1
120.59 even 2 1440.2.f.c.289.2 2
120.77 even 4 7200.2.a.bb.1.1 1
120.83 odd 4 7200.2.a.y.1.1 1
120.107 odd 4 7200.2.a.bb.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
160.2.c.a.129.1 ✓ 2 40.19 odd 2
160.2.c.a.129.1 ✓ 2 40.29 even 2
160.2.c.a.129.2 yes 2 8.3 odd 2
160.2.c.a.129.2 yes 2 8.5 even 2
320.2.c.a.129.1 2 1.1 even 1 trivial
320.2.c.a.129.1 2 4.3 odd 2 CM
320.2.c.a.129.2 2 5.4 even 2 inner
320.2.c.a.129.2 2 20.19 odd 2 inner
800.2.a.e.1.1 1 40.3 even 4
800.2.a.e.1.1 1 40.13 odd 4
800.2.a.f.1.1 1 40.27 even 4
800.2.a.f.1.1 1 40.37 odd 4
1280.2.f.c.129.1 2 80.59 odd 4
1280.2.f.c.129.1 2 80.69 even 4
1280.2.f.c.129.2 2 16.3 odd 4
1280.2.f.c.129.2 2 16.13 even 4
1280.2.f.d.129.1 2 16.5 even 4
1280.2.f.d.129.1 2 16.11 odd 4
1280.2.f.d.129.2 2 80.19 odd 4
1280.2.f.d.129.2 2 80.29 even 4
1440.2.f.c.289.1 2 24.5 odd 2
1440.2.f.c.289.1 2 24.11 even 2
1440.2.f.c.289.2 2 120.29 odd 2
1440.2.f.c.289.2 2 120.59 even 2
1600.2.a.l.1.1 1 5.2 odd 4
1600.2.a.l.1.1 1 20.7 even 4
1600.2.a.m.1.1 1 5.3 odd 4
1600.2.a.m.1.1 1 20.3 even 4
2880.2.f.n.1729.1 2 15.14 odd 2
2880.2.f.n.1729.1 2 60.59 even 2
2880.2.f.n.1729.2 2 3.2 odd 2
2880.2.f.n.1729.2 2 12.11 even 2
7200.2.a.y.1.1 1 120.53 even 4
7200.2.a.y.1.1 1 120.83 odd 4
7200.2.a.bb.1.1 1 120.77 even 4
7200.2.a.bb.1.1 1 120.107 odd 4