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Properties

Label 1152.4.d.d.577.2
Level $1152$
Weight $4$
Character 1152.577
Analytic conductor $67.970$
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] = [1152,4,Mod(577,1152)] 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("1152.577"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1152, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1152.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-188] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(67.9702003266\)
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: no (minimal twist has level 128)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 577.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1152.577
Dual form 1152.4.d.d.577.1

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(4\) 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
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) − 92.0000i − 1.96279i −0.192012 − 0.981393i \(-0.561501\pi\)
0.192012 − 0.981393i \(-0.438499\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −94.0000 −1.34108 −0.670540 − 0.741874i \(-0.733937\pi\)
−0.670540 + 0.741874i \(0.733937\pi\)
\(18\) 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 \(0.342295\pi\)
−0.475424 + 0.879757i \(0.657705\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 230.000 0.876097 0.438048 − 0.898951i \(-0.355670\pi\)
0.438048 + 0.898951i \(0.355670\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 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 \(0.163426\pi\)
−0.871071 + 0.491158i \(0.836574\pi\)
\(62\) 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
\(81\) 0 0
\(82\) 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 \(-0.966577\pi\)
−0.994492 + 0.104809i \(0.966577\pi\)
\(90\) 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 1152.4.d.d.577.2 2
3.2 odd 2 128.4.b.c.65.1 ✓ 2
4.3 odd 2 CM 1152.4.d.d.577.2 2
8.3 odd 2 inner 1152.4.d.d.577.1 2
8.5 even 2 inner 1152.4.d.d.577.1 2
12.11 even 2 128.4.b.c.65.1 ✓ 2
16.3 odd 4 2304.4.a.j.1.1 1
16.5 even 4 2304.4.a.g.1.1 1
16.11 odd 4 2304.4.a.g.1.1 1
16.13 even 4 2304.4.a.j.1.1 1
24.5 odd 2 128.4.b.c.65.2 yes 2
24.11 even 2 128.4.b.c.65.2 yes 2
48.5 odd 4 256.4.a.e.1.1 1
48.11 even 4 256.4.a.e.1.1 1
48.29 odd 4 256.4.a.d.1.1 1
48.35 even 4 256.4.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.4.b.c.65.1 ✓ 2 3.2 odd 2
128.4.b.c.65.1 ✓ 2 12.11 even 2
128.4.b.c.65.2 yes 2 24.5 odd 2
128.4.b.c.65.2 yes 2 24.11 even 2
256.4.a.d.1.1 1 48.29 odd 4
256.4.a.d.1.1 1 48.35 even 4
256.4.a.e.1.1 1 48.5 odd 4
256.4.a.e.1.1 1 48.11 even 4
1152.4.d.d.577.1 2 8.3 odd 2 inner
1152.4.d.d.577.1 2 8.5 even 2 inner
1152.4.d.d.577.2 2 1.1 even 1 trivial
1152.4.d.d.577.2 2 4.3 odd 2 CM
2304.4.a.g.1.1 1 16.5 even 4
2304.4.a.g.1.1 1 16.11 odd 4
2304.4.a.j.1.1 1 16.3 odd 4
2304.4.a.j.1.1 1 16.13 even 4