Properties

Label 2944.1.p.a.2897.1
Level $2944$
Weight $1$
Character 2944.2897
Analytic conductor $1.469$
Analytic rank $0$
Dimension $4$
Projective image $D_{8}$
CM discriminant -23
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2944,1,Mod(689,2944)] 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("2944.689"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2944, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([0, 5, 4])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 2944 = 2^{7} \cdot 23 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2944.p (of order \(8\), degree \(4\), not 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: \(1.46924739719\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 736)
Projective image: \(D_{8}\)
Projective field: Galois closure of 8.0.600953971539968.26

Embedding invariants

Embedding label 2897.1
Root \(-0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 2944.2897
Dual form 2944.1.p.a.689.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.70711 + 0.707107i) q^{3} +(1.70711 + 1.70711i) q^{9} +(0.707107 - 1.70711i) q^{13} +(0.707107 + 0.707107i) q^{23} +(-0.707107 + 0.707107i) q^{25} +(1.00000 + 2.41421i) q^{27} +(-0.707107 - 0.292893i) q^{29} -1.41421 q^{31} +(2.41421 - 2.41421i) q^{39} +(-1.41421 - 1.41421i) q^{41} -1.00000i q^{49} +(0.707107 + 1.70711i) q^{59} +(0.707107 + 1.70711i) q^{69} +(-1.00000 + 1.00000i) q^{71} +(-1.70711 + 0.707107i) q^{75} +2.41421i q^{81} +(-1.00000 - 1.00000i) q^{87} +(-2.41421 - 1.00000i) q^{93} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 4 q^{9} + 4 q^{27} + 4 q^{39} - 4 q^{71} - 4 q^{75} - 4 q^{87} - 4 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2944\mathbb{Z}\right)^\times\).

\(n\) \(645\) \(1151\) \(2305\)
\(\chi(n)\) \(e\left(\frac{3}{8}\right)\) \(1\) \(-1\)

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
\(3\) 1.70711 + 0.707107i 1.70711 + 0.707107i 1.00000 \(0\)
0.707107 + 0.707107i \(0.250000\pi\)
\(4\) 0 0
\(5\) 0 0 −0.382683 − 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(6\) 0 0
\(7\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(8\) 0 0
\(9\) 1.70711 + 1.70711i 1.70711 + 1.70711i
\(10\) 0 0
\(11\) 0 0 0.923880 − 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(12\) 0 0
\(13\) 0.707107 − 1.70711i 0.707107 − 1.70711i − 1.00000i \(-0.5\pi\)
0.707107 − 0.707107i \(-0.250000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 0 0 0.382683 − 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(24\) 0 0
\(25\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(26\) 0 0
\(27\) 1.00000 + 2.41421i 1.00000 + 2.41421i
\(28\) 0 0
\(29\) −0.707107 − 0.292893i −0.707107 − 0.292893i − 1.00000i \(-0.5\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(30\) 0 0
\(31\) −1.41421 −1.41421 −0.707107 − 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 −0.382683 − 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(38\) 0 0
\(39\) 2.41421 − 2.41421i 2.41421 − 2.41421i
\(40\) 0 0
\(41\) −1.41421 − 1.41421i −1.41421 − 1.41421i −0.707107 − 0.707107i \(-0.750000\pi\)
−0.707107 − 0.707107i \(-0.750000\pi\)
\(42\) 0 0
\(43\) 0 0 0.923880 − 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) − 1.00000i − 1.00000i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 0.923880 − 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0.707107 + 1.70711i 0.707107 + 1.70711i 0.707107 + 0.707107i \(0.250000\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 0 0 −0.923880 − 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 −0.923880 − 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(68\) 0 0
\(69\) 0.707107 + 1.70711i 0.707107 + 1.70711i
\(70\) 0 0
\(71\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(74\) 0 0
\(75\) −1.70711 + 0.707107i −1.70711 + 0.707107i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 2.41421i 2.41421i
\(82\) 0 0
\(83\) 0 0 0.382683 − 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −1.00000 − 1.00000i −1.00000 − 1.00000i
\(88\) 0 0
\(89\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −2.41421 − 1.00000i −2.41421 − 1.00000i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 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 2944.1.p.a.2897.1 4
4.3 odd 2 736.1.p.a.413.1 ✓ 4
23.22 odd 2 CM 2944.1.p.a.2897.1 4
32.11 odd 8 736.1.p.a.597.1 yes 4
32.21 even 8 inner 2944.1.p.a.689.1 4
92.91 even 2 736.1.p.a.413.1 ✓ 4
736.459 even 8 736.1.p.a.597.1 yes 4
736.597 odd 8 inner 2944.1.p.a.689.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
736.1.p.a.413.1 ✓ 4 4.3 odd 2
736.1.p.a.413.1 ✓ 4 92.91 even 2
736.1.p.a.597.1 yes 4 32.11 odd 8
736.1.p.a.597.1 yes 4 736.459 even 8
2944.1.p.a.689.1 4 32.21 even 8 inner
2944.1.p.a.689.1 4 736.597 odd 8 inner
2944.1.p.a.2897.1 4 1.1 even 1 trivial
2944.1.p.a.2897.1 4 23.22 odd 2 CM