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Properties

Label 2448.1.de.a.2095.2
Level $2448$
Weight $1$
Character 2448.2095
Analytic conductor $1.222$
Analytic rank $0$
Dimension $8$
Projective image $S_{4}$
CM/RM no
Inner twists $8$

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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] = [2448,1,Mod(319,2448)] 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("2448.319"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2448, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 0, 4, 3])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 2448 = 2^{4} \cdot 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2448.de (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.22171115093\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.2.1591812.1

Embedding invariants

Embedding label 2095.2
Root \(-0.258819 + 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 2448.2095
Dual form 2448.1.de.a.319.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.258819 - 0.965926i) q^{3} +(-0.866025 - 0.500000i) q^{9} +(0.258819 + 0.965926i) q^{11} +(-0.500000 - 0.866025i) q^{13} -1.00000i q^{17} +1.41421 q^{19} +(0.965926 + 0.258819i) q^{23} +(-0.866025 - 0.500000i) q^{25} +(-0.707107 + 0.707107i) q^{27} +(-0.366025 - 1.36603i) q^{29} +(0.258819 - 0.965926i) q^{31} +1.00000 q^{33} +(-0.965926 + 0.258819i) q^{39} +(0.707107 - 1.22474i) q^{43} +(-1.22474 - 0.707107i) q^{47} +(0.866025 - 0.500000i) q^{49} +(-0.965926 - 0.258819i) q^{51} +1.00000i q^{53} +(0.366025 - 1.36603i) q^{57} +(0.707107 + 1.22474i) q^{59} +(-1.36603 + 0.366025i) q^{61} +(0.500000 - 0.866025i) q^{69} +(0.707107 + 0.707107i) q^{71} +(-0.707107 + 0.707107i) q^{75} +(-0.258819 - 0.965926i) q^{79} +(0.500000 + 0.866025i) q^{81} +(-0.707107 + 1.22474i) q^{83} -1.41421 q^{87} +1.00000 q^{89} +(-0.866025 - 0.500000i) q^{93} +(0.258819 - 0.965926i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{13} + 4 q^{29} + 8 q^{33} - 4 q^{57} - 4 q^{61} + 4 q^{69} + 4 q^{81} + 8 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2448\mathbb{Z}\right)^\times\).

\(n\) \(613\) \(1361\) \(1873\) \(2143\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{3}{4}\right)\) \(-1\)

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
\(3\) 0.258819 − 0.965926i 0.258819 − 0.965926i
\(4\) 0 0
\(5\) 0 0 0.258819 − 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(6\) 0 0
\(7\) 0 0 0.965926 − 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(8\) 0 0
\(9\) −0.866025 − 0.500000i −0.866025 − 0.500000i
\(10\) 0 0
\(11\) 0.258819 + 0.965926i 0.258819 + 0.965926i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(12\) 0 0
\(13\) −0.500000 − 0.866025i −0.500000 − 0.866025i 0.500000 − 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 1.00000i − 1.00000i
\(18\) 0 0
\(19\) 1.41421 1.41421 0.707107 − 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.965926 + 0.258819i 0.965926 + 0.258819i 0.707107 − 0.707107i \(-0.250000\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(24\) 0 0
\(25\) −0.866025 − 0.500000i −0.866025 − 0.500000i
\(26\) 0 0
\(27\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(28\) 0 0
\(29\) −0.366025 − 1.36603i −0.366025 − 1.36603i −0.866025 − 0.500000i \(-0.833333\pi\)
0.500000 − 0.866025i \(-0.333333\pi\)
\(30\) 0 0
\(31\) 0.258819 − 0.965926i 0.258819 − 0.965926i −0.707107 − 0.707107i \(-0.750000\pi\)
0.965926 − 0.258819i \(-0.0833333\pi\)
\(32\) 0 0
\(33\) 1.00000 1.00000
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(38\) 0 0
\(39\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(40\) 0 0
\(41\) 0 0 −0.965926 − 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(42\) 0 0
\(43\) 0.707107 − 1.22474i 0.707107 − 1.22474i −0.258819 − 0.965926i \(-0.583333\pi\)
0.965926 − 0.258819i \(-0.0833333\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.22474 − 0.707107i −1.22474 − 0.707107i −0.258819 − 0.965926i \(-0.583333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(48\) 0 0
\(49\) 0.866025 − 0.500000i 0.866025 − 0.500000i
\(50\) 0 0
\(51\) −0.965926 − 0.258819i −0.965926 − 0.258819i
\(52\) 0 0
\(53\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0.366025 − 1.36603i 0.366025 − 1.36603i
\(58\) 0 0
\(59\) 0.707107 + 1.22474i 0.707107 + 1.22474i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(60\) 0 0
\(61\) −1.36603 + 0.366025i −1.36603 + 0.366025i −0.866025 − 0.500000i \(-0.833333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(68\) 0 0
\(69\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(70\) 0 0
\(71\) 0.707107 + 0.707107i 0.707107 + 0.707107i 0.965926 − 0.258819i \(-0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(72\) 0 0
\(73\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(74\) 0 0
\(75\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −0.258819 − 0.965926i −0.258819 − 0.965926i −0.965926 − 0.258819i \(-0.916667\pi\)
0.707107 − 0.707107i \(-0.250000\pi\)
\(80\) 0 0
\(81\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(82\) 0 0
\(83\) −0.707107 + 1.22474i −0.707107 + 1.22474i 0.258819 + 0.965926i \(0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −1.41421 −1.41421
\(88\) 0 0
\(89\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −0.866025 − 0.500000i −0.866025 − 0.500000i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 0.965926 − 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(98\) 0 0
\(99\) 0.258819 − 0.965926i 0.258819 − 0.965926i
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 2448.1.de.a.2095.2 yes 8
4.3 odd 2 inner 2448.1.de.a.2095.1 yes 8
9.4 even 3 inner 2448.1.de.a.463.1 yes 8
17.13 even 4 inner 2448.1.de.a.1951.2 yes 8
36.31 odd 6 inner 2448.1.de.a.463.2 yes 8
68.47 odd 4 inner 2448.1.de.a.1951.1 yes 8
153.13 even 12 inner 2448.1.de.a.319.1 ✓ 8
612.319 odd 12 inner 2448.1.de.a.319.2 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2448.1.de.a.319.1 ✓ 8 153.13 even 12 inner
2448.1.de.a.319.2 yes 8 612.319 odd 12 inner
2448.1.de.a.463.1 yes 8 9.4 even 3 inner
2448.1.de.a.463.2 yes 8 36.31 odd 6 inner
2448.1.de.a.1951.1 yes 8 68.47 odd 4 inner
2448.1.de.a.1951.2 yes 8 17.13 even 4 inner
2448.1.de.a.2095.1 yes 8 4.3 odd 2 inner
2448.1.de.a.2095.2 yes 8 1.1 even 1 trivial