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Properties

Label 2376.2.a.p.1.1
Level $2376$
Weight $2$
Character 2376.1
Self dual yes
Analytic conductor $18.972$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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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] = [2376,2,Mod(1,2376)] 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("2376.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2376, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2376 = 2^{3} \cdot 3^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2376.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,0,0,2,0,-3,0,0,0,-3,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(18.9724555203\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.564.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.571993\) of defining polynomial
Character \(\chi\) \(=\) 2376.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.67282 q^{5} -4.10083 q^{7} -1.00000 q^{11} -3.81681 q^{13} -5.91764 q^{17} -5.81681 q^{19} +3.67282 q^{23} +2.14399 q^{25} +8.38880 q^{29} +6.96080 q^{31} +10.9608 q^{35} +6.52884 q^{37} +7.24482 q^{41} -10.7737 q^{43} -0.143987 q^{47} +9.81681 q^{49} -4.48963 q^{53} +2.67282 q^{55} +11.8745 q^{59} -1.61515 q^{61} +10.2017 q^{65} -6.96080 q^{67} -0.489634 q^{71} -8.48963 q^{73} +4.10083 q^{77} -15.9176 q^{79} -5.16246 q^{83} +15.8168 q^{85} +6.00000 q^{89} +15.6521 q^{91} +15.5473 q^{95} -7.38485 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{5} - 3 q^{7} - 3 q^{11} + 3 q^{17} - 6 q^{19} + q^{23} + 5 q^{25} + 13 q^{29} + 8 q^{31} + 20 q^{35} + 11 q^{37} + 11 q^{41} - 13 q^{43} + q^{47} + 18 q^{49} + 8 q^{53} - 2 q^{55} + 7 q^{59}+ \cdots - 15 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) −2.67282 −1.19532 −0.597662 − 0.801749i \(-0.703903\pi\)
−0.597662 + 0.801749i \(0.703903\pi\)
\(6\) 0 0
\(7\) −4.10083 −1.54997 −0.774984 − 0.631981i \(-0.782242\pi\)
−0.774984 + 0.631981i \(0.782242\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −3.81681 −1.05859 −0.529296 − 0.848437i \(-0.677544\pi\)
−0.529296 + 0.848437i \(0.677544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.91764 −1.43524 −0.717619 − 0.696436i \(-0.754768\pi\)
−0.717619 + 0.696436i \(0.754768\pi\)
\(18\) 0 0
\(19\) −5.81681 −1.33447 −0.667234 − 0.744848i \(-0.732522\pi\)
−0.667234 + 0.744848i \(0.732522\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.67282 0.765837 0.382918 − 0.923782i \(-0.374919\pi\)
0.382918 + 0.923782i \(0.374919\pi\)
\(24\) 0 0
\(25\) 2.14399 0.428797
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 8.38880 1.55776 0.778881 − 0.627172i \(-0.215788\pi\)
0.778881 + 0.627172i \(0.215788\pi\)
\(30\) 0 0
\(31\) 6.96080 1.25020 0.625098 − 0.780546i \(-0.285059\pi\)
0.625098 + 0.780546i \(0.285059\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 10.9608 1.85271
\(36\) 0 0
\(37\) 6.52884 1.07333 0.536667 − 0.843794i \(-0.319683\pi\)
0.536667 + 0.843794i \(0.319683\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 7.24482 1.13145 0.565725 − 0.824594i \(-0.308596\pi\)
0.565725 + 0.824594i \(0.308596\pi\)
\(42\) 0 0
\(43\) −10.7737 −1.64297 −0.821483 − 0.570232i \(-0.806853\pi\)
−0.821483 + 0.570232i \(0.806853\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −0.143987 −0.0210026 −0.0105013 − 0.999945i \(-0.503343\pi\)
−0.0105013 + 0.999945i \(0.503343\pi\)
\(48\) 0 0
\(49\) 9.81681 1.40240
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −4.48963 −0.616699 −0.308349 − 0.951273i \(-0.599777\pi\)
−0.308349 + 0.951273i \(0.599777\pi\)
\(54\) 0 0
\(55\) 2.67282 0.360403
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 11.8745 1.54593 0.772963 − 0.634451i \(-0.218774\pi\)
0.772963 + 0.634451i \(0.218774\pi\)
\(60\) 0 0
\(61\) −1.61515 −0.206799 −0.103399 − 0.994640i \(-0.532972\pi\)
−0.103399 + 0.994640i \(0.532972\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 10.2017 1.26536
\(66\) 0 0
\(67\) −6.96080 −0.850397 −0.425198 − 0.905100i \(-0.639796\pi\)
−0.425198 + 0.905100i \(0.639796\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −0.489634 −0.0581089 −0.0290544 − 0.999578i \(-0.509250\pi\)
−0.0290544 + 0.999578i \(0.509250\pi\)
\(72\) 0 0
\(73\) −8.48963 −0.993636 −0.496818 − 0.867855i \(-0.665498\pi\)
−0.496818 + 0.867855i \(0.665498\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.10083 0.467333
\(78\) 0 0
\(79\) −15.9176 −1.79087 −0.895437 − 0.445188i \(-0.853137\pi\)
−0.895437 + 0.445188i \(0.853137\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −5.16246 −0.566653 −0.283327 − 0.959023i \(-0.591438\pi\)
−0.283327 + 0.959023i \(0.591438\pi\)
\(84\) 0 0
\(85\) 15.8168 1.71557
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 15.6521 1.64079
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 15.5473 1.59512
\(96\) 0 0
\(97\) −7.38485 −0.749818 −0.374909 − 0.927062i \(-0.622326\pi\)
−0.374909 + 0.927062i \(0.622326\pi\)
\(98\) 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 2376.2.a.p.1.1 yes 3
3.2 odd 2 2376.2.a.k.1.3 ✓ 3
4.3 odd 2 4752.2.a.br.1.1 3
12.11 even 2 4752.2.a.bj.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2376.2.a.k.1.3 ✓ 3 3.2 odd 2
2376.2.a.p.1.1 yes 3 1.1 even 1 trivial
4752.2.a.bj.1.3 3 12.11 even 2
4752.2.a.br.1.1 3 4.3 odd 2