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Properties

Label 128.4.b.c.65.2
Level $128$
Weight $4$
Character 128.65
Analytic conductor $7.552$
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] = [128,4,Mod(65,128)] 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("128.65"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(128, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 128.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,54] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.55224448073\)
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: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 65.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 128.65
Dual form 128.4.b.c.65.1

$q$-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)
 
\(f(q)\) \(=\) \(q+4.00000i q^{5} +27.0000 q^{9} +92.0000i q^{13} +94.0000 q^{17} +109.000 q^{25} +284.000i q^{29} -396.000i q^{37} -230.000 q^{41} +108.000i q^{45} -343.000 q^{49} -572.000i q^{53} -468.000i q^{61} -368.000 q^{65} -1098.00 q^{73} +729.000 q^{81} +376.000i q^{85} +1670.00 q^{89} -594.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 54 q^{9} + 188 q^{17} + 218 q^{25} - 460 q^{41} - 686 q^{49} - 736 q^{65} - 2196 q^{73} + 1458 q^{81} + 3340 q^{89} - 1188 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(127\)
\(\chi(n)\) \(-1\) \(1\)

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
\(5\) 4.00000i 0.357771i 0.983870 + 0.178885i \(0.0572491\pi\)
−0.983870 + 0.178885i \(0.942751\pi\)
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 27.0000 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 92.0000i 1.96279i 0.192012 + 0.981393i \(0.438499\pi\)
−0.192012 + 0.981393i \(0.561501\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 94.0000 1.34108 0.670540 − 0.741874i \(-0.266063\pi\)
0.670540 + 0.741874i \(0.266063\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 109.000 0.872000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 284.000i 1.81853i 0.416214 + 0.909267i \(0.363357\pi\)
−0.416214 + 0.909267i \(0.636643\pi\)
\(30\) 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
\(37\) − 396.000i − 1.75951i −0.475424 − 0.879757i \(-0.657705\pi\)
0.475424 − 0.879757i \(-0.342295\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −230.000 −0.876097 −0.438048 − 0.898951i \(-0.644330\pi\)
−0.438048 + 0.898951i \(0.644330\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 108.000i 0.357771i
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −343.000 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 572.000i − 1.48246i −0.671253 − 0.741229i \(-0.734243\pi\)
0.671253 − 0.741229i \(-0.265757\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) − 468.000i − 0.982316i −0.871071 − 0.491158i \(-0.836574\pi\)
0.871071 − 0.491158i \(-0.163426\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −368.000 −0.702227
\(66\) 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
\(73\) −1098.00 −1.76043 −0.880214 − 0.474578i \(-0.842601\pi\)
−0.880214 + 0.474578i \(0.842601\pi\)
\(74\) 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
\(81\) 729.000 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 376.000i 0.479799i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1670.00 1.98898 0.994492 − 0.104809i \(-0.0334231\pi\)
0.994492 + 0.104809i \(0.0334231\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −594.000 −0.621769 −0.310884 − 0.950448i \(-0.600625\pi\)
−0.310884 + 0.950448i \(0.600625\pi\)
\(98\) 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 128.4.b.c.65.2 yes 2
3.2 odd 2 1152.4.d.d.577.1 2
4.3 odd 2 CM 128.4.b.c.65.2 yes 2
8.3 odd 2 inner 128.4.b.c.65.1 ✓ 2
8.5 even 2 inner 128.4.b.c.65.1 ✓ 2
12.11 even 2 1152.4.d.d.577.1 2
16.3 odd 4 256.4.a.e.1.1 1
16.5 even 4 256.4.a.d.1.1 1
16.11 odd 4 256.4.a.d.1.1 1
16.13 even 4 256.4.a.e.1.1 1
24.5 odd 2 1152.4.d.d.577.2 2
24.11 even 2 1152.4.d.d.577.2 2
48.5 odd 4 2304.4.a.j.1.1 1
48.11 even 4 2304.4.a.j.1.1 1
48.29 odd 4 2304.4.a.g.1.1 1
48.35 even 4 2304.4.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.4.b.c.65.1 ✓ 2 8.3 odd 2 inner
128.4.b.c.65.1 ✓ 2 8.5 even 2 inner
128.4.b.c.65.2 yes 2 1.1 even 1 trivial
128.4.b.c.65.2 yes 2 4.3 odd 2 CM
256.4.a.d.1.1 1 16.5 even 4
256.4.a.d.1.1 1 16.11 odd 4
256.4.a.e.1.1 1 16.3 odd 4
256.4.a.e.1.1 1 16.13 even 4
1152.4.d.d.577.1 2 3.2 odd 2
1152.4.d.d.577.1 2 12.11 even 2
1152.4.d.d.577.2 2 24.5 odd 2
1152.4.d.d.577.2 2 24.11 even 2
2304.4.a.g.1.1 1 48.29 odd 4
2304.4.a.g.1.1 1 48.35 even 4
2304.4.a.j.1.1 1 48.5 odd 4
2304.4.a.j.1.1 1 48.11 even 4