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Properties

Label 4000.2.c.f
Level $4000$
Weight $2$
Character orbit 4000.c
Analytic conductor $31.940$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4000,2,Mod(1249,4000)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4000, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4000.1249");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4000 = 2^{5} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4000.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(31.9401608085\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 22x^{10} + 179x^{8} + 646x^{6} + 929x^{4} + 252x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{11} q^{7} + (\beta_{6} + \beta_{4} + \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{11} q^{7} + (\beta_{6} + \beta_{4} + \beta_{3}) q^{9} + (\beta_{4} + \beta_{3} + \beta_{2} - 2) q^{11} + ( - \beta_{11} + \beta_{10} + \cdots - \beta_1) q^{13}+ \cdots + ( - 6 \beta_{6} - \beta_{4} - 4 \beta_{3} + \cdots + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 8 q^{9} - 26 q^{11} + 18 q^{19} - 4 q^{29} - 24 q^{31} + 52 q^{39} - 10 q^{41} + 6 q^{49} - 36 q^{51} + 50 q^{59} - 12 q^{61} + 24 q^{69} - 68 q^{71} + 32 q^{79} - 36 q^{81} + 6 q^{89} - 52 q^{91} + 86 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 22x^{10} + 179x^{8} + 646x^{6} + 929x^{4} + 252x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{10} - 33\nu^{8} - 165\nu^{6} - 189\nu^{4} + 274\nu^{2} + 36 ) / 46 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4\nu^{10} + 66\nu^{8} + 353\nu^{6} + 631\nu^{4} + 119\nu^{2} + 20 ) / 46 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{10} - 38\nu^{8} - 75\nu^{6} + 556\nu^{4} + 1883\nu^{2} + 560 ) / 92 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5\nu^{11} + 94\nu^{9} + 631\nu^{7} + 1818\nu^{5} + 2121\nu^{3} + 784\nu ) / 184 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -5\nu^{10} - 94\nu^{8} - 631\nu^{6} - 1818\nu^{4} - 2029\nu^{2} - 324 ) / 92 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -5\nu^{11} - 94\nu^{9} - 631\nu^{7} - 1818\nu^{5} - 2029\nu^{3} - 232\nu ) / 92 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -3\nu^{10} - 84\nu^{8} - 811\nu^{6} - 3216\nu^{4} - 4511\nu^{2} - 544 ) / 92 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -7\nu^{11} - 150\nu^{9} - 1187\nu^{7} - 4146\nu^{5} - 5619\nu^{3} - 978\nu ) / 92 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 13\nu^{11} + 318\nu^{9} + 2855\nu^{7} + 11222\nu^{5} + 17125\nu^{3} + 4044\nu ) / 184 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 15\nu^{11} + 328\nu^{9} + 2629\nu^{7} + 9180\nu^{5} + 12159\nu^{3} + 1846\nu ) / 92 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{4} + \beta_{3} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 2\beta_{5} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} - 9\beta_{6} - 6\beta_{4} - 8\beta_{3} - \beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{10} + 4\beta_{9} - 11\beta_{7} - 16\beta_{5} + 39\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -11\beta_{8} + 70\beta_{6} + 37\beta_{4} + 61\beta_{3} + 15\beta_{2} - 82 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -4\beta_{11} - 24\beta_{10} - 58\beta_{9} + 96\beta_{7} + 116\beta_{5} - 261\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 92\beta_{8} - 521\beta_{6} - 237\beta_{4} - 459\beta_{3} - 158\beta_{2} + 475 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 66\beta_{11} + 222\beta_{10} + 604\beta_{9} - 771\beta_{7} - 824\beta_{5} + 1784\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -705\beta_{8} + 3809\beta_{6} + 1562\beta_{4} + 3434\beta_{3} + 1441\beta_{2} - 2883 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -736\beta_{11} - 1872\beta_{10} - 5490\beta_{9} + 5955\beta_{7} + 5858\beta_{5} - 12393\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4000\mathbb{Z}\right)^\times\).

\(n\) \(1377\) \(2501\) \(2751\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1249.1
− 2.71210i
− 2.40554i
− 2.29226i
− 1.81030i
− 0.481965i
− 0.306558i
0.306558i
0.481965i
1.81030i
2.29226i
2.40554i
2.71210i
0 − 2.71210i 0 0 0 0.0156607i 0 −4.35547 0
1249.2 0 − 2.40554i 0 0 0 − 2.20893i 0 −2.78662 0
1249.3 0 − 2.29226i 0 0 0 − 0.938845i 0 −2.25447 0
1249.4 0 − 1.81030i 0 0 0 4.37760i 0 −0.277175 0
1249.5 0 − 0.481965i 0 0 0 − 2.69841i 0 2.76771 0
1249.6 0 − 0.306558i 0 0 0 2.60656i 0 2.90602 0
1249.7 0 0.306558i 0 0 0 − 2.60656i 0 2.90602 0
1249.8 0 0.481965i 0 0 0 2.69841i 0 2.76771 0
1249.9 0 1.81030i 0 0 0 − 4.37760i 0 −0.277175 0
1249.10 0 2.29226i 0 0 0 0.938845i 0 −2.25447 0
1249.11 0 2.40554i 0 0 0 2.20893i 0 −2.78662 0
1249.12 0 2.71210i 0 0 0 − 0.0156607i 0 −4.35547 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1249.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4000.2.c.f 12
4.b odd 2 1 4000.2.c.g 12
5.b even 2 1 inner 4000.2.c.f 12
5.c odd 4 1 4000.2.a.k ✓ 6
5.c odd 4 1 4000.2.a.m yes 6
20.d odd 2 1 4000.2.c.g 12
20.e even 4 1 4000.2.a.l yes 6
20.e even 4 1 4000.2.a.n yes 6
40.i odd 4 1 8000.2.a.bv 6
40.i odd 4 1 8000.2.a.bx 6
40.k even 4 1 8000.2.a.bu 6
40.k even 4 1 8000.2.a.bw 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4000.2.a.k ✓ 6 5.c odd 4 1
4000.2.a.l yes 6 20.e even 4 1
4000.2.a.m yes 6 5.c odd 4 1
4000.2.a.n yes 6 20.e even 4 1
4000.2.c.f 12 1.a even 1 1 trivial
4000.2.c.f 12 5.b even 2 1 inner
4000.2.c.g 12 4.b odd 2 1
4000.2.c.g 12 20.d odd 2 1
8000.2.a.bu 6 40.k even 4 1
8000.2.a.bv 6 40.i odd 4 1
8000.2.a.bw 6 40.k even 4 1
8000.2.a.bx 6 40.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4000, [\chi])\):

\( T_{3}^{12} + 22T_{3}^{10} + 179T_{3}^{8} + 646T_{3}^{6} + 929T_{3}^{4} + 252T_{3}^{2} + 16 \) Copy content Toggle raw display
\( T_{7}^{12} + 39T_{7}^{10} + 515T_{7}^{8} + 2930T_{7}^{6} + 6835T_{7}^{4} + 4079T_{7}^{2} + 1 \) Copy content Toggle raw display
\( T_{11}^{6} + 13T_{11}^{5} + 41T_{11}^{4} - 50T_{11}^{3} - 325T_{11}^{2} - 375T_{11} - 125 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 22 T^{10} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + 39 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( (T^{6} + 13 T^{5} + \cdots - 125)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + 117 T^{10} + \cdots + 390625 \) Copy content Toggle raw display
$17$ \( T^{12} + 148 T^{10} + \cdots + 4000000 \) Copy content Toggle raw display
$19$ \( (T^{6} - 9 T^{5} + \cdots + 125)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 71 T^{10} + \cdots + 4096 \) Copy content Toggle raw display
$29$ \( (T^{6} + 2 T^{5} + \cdots - 2636)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 12 T^{5} + \cdots - 29500)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + 303 T^{10} + \cdots + 64000000 \) Copy content Toggle raw display
$41$ \( (T^{6} + 5 T^{5} + \cdots + 10475)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 588353536 \) Copy content Toggle raw display
$47$ \( T^{12} + 249 T^{10} + \cdots + 6497401 \) Copy content Toggle raw display
$53$ \( T^{12} + 333 T^{10} + \cdots + 97515625 \) Copy content Toggle raw display
$59$ \( (T^{6} - 25 T^{5} + \cdots - 59375)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 6 T^{5} + \cdots - 32220)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + 318 T^{10} + \cdots + 28772496 \) Copy content Toggle raw display
$71$ \( (T^{6} + 34 T^{5} + \cdots - 372500)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + 490 T^{10} + \cdots + 6250000 \) Copy content Toggle raw display
$79$ \( (T^{6} - 16 T^{5} + \cdots + 174500)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 2123366400 \) Copy content Toggle raw display
$89$ \( (T^{6} - 3 T^{5} + \cdots - 142144)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 31506250000 \) Copy content Toggle raw display
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