Properties

Label 1040.2.dh.c
Level $1040$
Weight $2$
Character orbit 1040.dh
Analytic conductor $8.304$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1040,2,Mod(289,1040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1040, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1040.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1040.dh (of order \(6\), degree \(2\), 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: \(8.30444181021\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.513226913958144.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6x^{10} + 28x^{8} - 46x^{6} + 58x^{4} - 8x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 260)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{10} + \beta_{2}) q^{3} - \beta_{9} q^{5} + \beta_1 q^{7} + ( - \beta_{5} + \beta_{4} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{10} + \beta_{2}) q^{3} - \beta_{9} q^{5} + \beta_1 q^{7} + ( - \beta_{5} + \beta_{4} + 1) q^{9} + \beta_{4} q^{11} + (\beta_{11} - \beta_{9} + \beta_{7} + \beta_{6} + \beta_{2}) q^{13} + ( - \beta_{11} - \beta_{10} - \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2}) q^{15} + (2 \beta_{2} - \beta_1) q^{17} + (\beta_{8} - \beta_{7} + \beta_{5} - 2 \beta_{4} - 2) q^{19} + q^{21} + ( - \beta_{11} + \beta_{9} + \beta_{8} + \beta_{6}) q^{23} + (\beta_{11} + \beta_{10} - \beta_{9} + \beta_{7} + \beta_{6} + \beta_{3} + \beta_1) q^{25} + (2 \beta_{11} - 2 \beta_{9} + 2 \beta_{7} + \beta_{6} + \beta_1) q^{27} + ( - \beta_{11} - \beta_{9} - \beta_{8} - \beta_{5} + 2 \beta_{4} + \beta_{3}) q^{29} + (\beta_{11} + \beta_{9} + \beta_{7} + 2) q^{31} - \beta_{2} q^{33} + (\beta_{7} + \beta_{4} + \beta_{2} - 2 \beta_1 + 1) q^{35} + ( - 2 \beta_{11} - 2 \beta_{10} + 2 \beta_{9} + 2 \beta_{8} + \beta_{6} + 2 \beta_{2}) q^{37} + ( - \beta_{11} - \beta_{9} - \beta_{8} - 2 \beta_{5} + 3 \beta_{4} + \beta_{3} + 3) q^{39} + (\beta_{11} + \beta_{9} + \beta_{8} + 2 \beta_{5} - \beta_{4} - 2 \beta_{3}) q^{41} + ( - \beta_{8} - \beta_{7} + \beta_{2} - 2 \beta_1) q^{43} + (2 \beta_{7} + 2 \beta_{4} - 3 \beta_{2} + \beta_1 + 2) q^{45} + (2 \beta_{11} + 4 \beta_{10} - 2 \beta_{9} + 2 \beta_{7} + 2 \beta_{6} + 2 \beta_1) q^{47} + (2 \beta_{11} + 2 \beta_{9} + 2 \beta_{8} - \beta_{5} + \beta_{4} + \beta_{3}) q^{49} + ( - 2 \beta_{3} + 7) q^{51} + ( - \beta_{11} - 3 \beta_{10} + \beta_{9} - \beta_{7} + \beta_{6} + \beta_1) q^{53} + (\beta_{9} + \beta_{8}) q^{55} + ( - 3 \beta_{11} + 2 \beta_{10} + 3 \beta_{9} - 3 \beta_{7} - \beta_{6} - \beta_1) q^{57} + (\beta_{8} - \beta_{7} - 2 \beta_{5} + \beta_{4} + 1) q^{59} + (\beta_{8} - \beta_{7} - \beta_{5} + 4 \beta_{4} + 4) q^{61} + ( - \beta_{10} + 3 \beta_{6} + \beta_{2}) q^{63} + (2 \beta_{11} + 2 \beta_{10} - \beta_{9} + 2 \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \cdots - 3) q^{65}+ \cdots + (\beta_{3} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 4 q^{9} - 6 q^{11} - 10 q^{15} - 10 q^{19} + 12 q^{21} + 4 q^{25} - 10 q^{29} + 24 q^{31} + 6 q^{35} + 18 q^{39} + 2 q^{41} + 12 q^{45} - 4 q^{49} + 76 q^{51} + 2 q^{59} + 22 q^{61} - 40 q^{65} - 26 q^{69} + 14 q^{71} + 8 q^{75} - 8 q^{79} - 22 q^{81} + 14 q^{85} + 2 q^{89} - 58 q^{91} + 20 q^{95} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6x^{10} + 28x^{8} - 46x^{6} + 58x^{4} - 8x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -9\nu^{11} + 42\nu^{9} - 196\nu^{7} + 87\nu^{5} - 12\nu^{3} - 732\nu ) / 394 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -9\nu^{11} + 42\nu^{9} - 196\nu^{7} + 87\nu^{5} - 12\nu^{3} - 1520\nu ) / 394 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 24\nu^{10} - 112\nu^{8} + 457\nu^{6} - 232\nu^{4} + 32\nu^{2} + 376 ) / 197 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -56\nu^{10} + 327\nu^{8} - 1526\nu^{6} + 2380\nu^{4} - 3161\nu^{2} + 42 ) / 394 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -96\nu^{10} + 645\nu^{8} - 3010\nu^{6} + 5656\nu^{4} - 6235\nu^{2} + 860 ) / 394 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -103\nu^{11} + 612\nu^{9} - 2856\nu^{7} + 4673\nu^{5} - 5916\nu^{3} + 816\nu ) / 394 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 33 \nu^{11} + 103 \nu^{10} + 154 \nu^{9} - 612 \nu^{8} - 653 \nu^{7} + 2856 \nu^{6} + 319 \nu^{5} - 4673 \nu^{4} - 44 \nu^{3} + 5916 \nu^{2} - 1305 \nu - 816 ) / 394 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 33 \nu^{11} - 103 \nu^{10} + 154 \nu^{9} + 612 \nu^{8} - 653 \nu^{7} - 2856 \nu^{6} + 319 \nu^{5} + 4673 \nu^{4} - 44 \nu^{3} - 5916 \nu^{2} - 1305 \nu + 816 ) / 394 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 212 \nu^{11} - 9 \nu^{10} - 1252 \nu^{9} + 42 \nu^{8} + 5777 \nu^{7} - 196 \nu^{6} - 9010 \nu^{5} + 87 \nu^{4} + 10658 \nu^{3} - 12 \nu^{2} - 159 \nu - 732 ) / 394 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -112\nu^{11} + 654\nu^{9} - 3052\nu^{7} + 4760\nu^{5} - 6125\nu^{3} + 84\nu ) / 197 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 179 \nu^{11} - 112 \nu^{10} + 1098 \nu^{9} + 654 \nu^{8} - 5124 \nu^{7} - 3052 \nu^{6} + 8691 \nu^{5} + 4760 \nu^{4} - 10614 \nu^{3} - 5928 \nu^{2} + 1464 \nu + 84 ) / 394 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} + \beta_{9} + \beta_{8} - 4\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{10} + 2\beta_{6} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4\beta_{8} - 4\beta_{7} - \beta_{5} - 13\beta_{4} - 13 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -\beta_{11} - 6\beta_{10} + \beta_{9} + \beta_{8} + 16\beta_{6} + 6\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -8\beta_{11} - 8\beta_{9} - 8\beta_{7} - 3\beta_{3} - 24 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 6\beta_{8} + 6\beta_{7} + 21\beta_{2} - 65\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -65\beta_{11} - 65\beta_{9} - 65\beta_{8} + 28\beta_{5} + 188\beta_{4} - 28\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 14\beta_{11} + 40\beta_{10} - 14\beta_{9} + 14\beta_{7} - 133\beta_{6} - 133\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -266\beta_{8} + 266\beta_{7} + 121\beta_{5} + 757\beta_{4} + 757 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 121\beta_{11} + 318\beta_{10} - 121\beta_{9} - 121\beta_{8} - 1092\beta_{6} - 318\beta_{2} ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1040\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(417\) \(561\) \(911\)
\(\chi(n)\) \(1\) \(-1\) \(-1 - \beta_{4}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
1.14430 + 0.660662i
0.323103 + 0.186544i
−1.75675 1.01426i
1.75675 + 1.01426i
−0.323103 0.186544i
−1.14430 0.660662i
1.14430 0.660662i
0.323103 0.186544i
−1.75675 + 1.01426i
1.75675 1.01426i
−0.323103 + 0.186544i
−1.14430 + 0.660662i
0 −2.57937 + 1.48920i 0 0.254102 + 2.22158i 0 −0.290769 0.167875i 0 2.93543 5.08432i 0
289.2 0 −1.24744 + 0.720209i 0 1.86081 1.23992i 0 −0.601232 0.347122i 0 −0.462598 + 0.801244i 0
289.3 0 −0.201864 + 0.116546i 0 −2.11491 + 0.726062i 0 −3.71537 2.14507i 0 −1.47283 + 2.55102i 0
289.4 0 0.201864 0.116546i 0 −2.11491 0.726062i 0 3.71537 + 2.14507i 0 −1.47283 + 2.55102i 0
289.5 0 1.24744 0.720209i 0 1.86081 + 1.23992i 0 0.601232 + 0.347122i 0 −0.462598 + 0.801244i 0
289.6 0 2.57937 1.48920i 0 0.254102 2.22158i 0 0.290769 + 0.167875i 0 2.93543 5.08432i 0
529.1 0 −2.57937 1.48920i 0 0.254102 2.22158i 0 −0.290769 + 0.167875i 0 2.93543 + 5.08432i 0
529.2 0 −1.24744 0.720209i 0 1.86081 + 1.23992i 0 −0.601232 + 0.347122i 0 −0.462598 0.801244i 0
529.3 0 −0.201864 0.116546i 0 −2.11491 0.726062i 0 −3.71537 + 2.14507i 0 −1.47283 2.55102i 0
529.4 0 0.201864 + 0.116546i 0 −2.11491 + 0.726062i 0 3.71537 2.14507i 0 −1.47283 2.55102i 0
529.5 0 1.24744 + 0.720209i 0 1.86081 1.23992i 0 0.601232 0.347122i 0 −0.462598 0.801244i 0
529.6 0 2.57937 + 1.48920i 0 0.254102 + 2.22158i 0 0.290769 0.167875i 0 2.93543 + 5.08432i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
13.c even 3 1 inner
65.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1040.2.dh.c 12
4.b odd 2 1 260.2.ba.a 12
5.b even 2 1 inner 1040.2.dh.c 12
12.b even 2 1 2340.2.de.a 12
13.c even 3 1 inner 1040.2.dh.c 12
20.d odd 2 1 260.2.ba.a 12
20.e even 4 2 1300.2.i.i 12
52.i odd 6 1 3380.2.c.b 6
52.j odd 6 1 260.2.ba.a 12
52.j odd 6 1 3380.2.c.a 6
52.l even 12 2 3380.2.d.c 12
60.h even 2 1 2340.2.de.a 12
65.n even 6 1 inner 1040.2.dh.c 12
156.p even 6 1 2340.2.de.a 12
260.v odd 6 1 260.2.ba.a 12
260.v odd 6 1 3380.2.c.a 6
260.w odd 6 1 3380.2.c.b 6
260.bc even 12 2 3380.2.d.c 12
260.bj even 12 2 1300.2.i.i 12
780.br even 6 1 2340.2.de.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.ba.a 12 4.b odd 2 1
260.2.ba.a 12 20.d odd 2 1
260.2.ba.a 12 52.j odd 6 1
260.2.ba.a 12 260.v odd 6 1
1040.2.dh.c 12 1.a even 1 1 trivial
1040.2.dh.c 12 5.b even 2 1 inner
1040.2.dh.c 12 13.c even 3 1 inner
1040.2.dh.c 12 65.n even 6 1 inner
1300.2.i.i 12 20.e even 4 2
1300.2.i.i 12 260.bj even 12 2
2340.2.de.a 12 12.b even 2 1
2340.2.de.a 12 60.h even 2 1
2340.2.de.a 12 156.p even 6 1
2340.2.de.a 12 780.br even 6 1
3380.2.c.a 6 52.j odd 6 1
3380.2.c.a 6 260.v odd 6 1
3380.2.c.b 6 52.i odd 6 1
3380.2.c.b 6 260.w odd 6 1
3380.2.d.c 12 52.l even 12 2
3380.2.d.c 12 260.bc even 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} - 11T_{3}^{10} + 102T_{3}^{8} - 207T_{3}^{6} + 350T_{3}^{4} - 19T_{3}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(1040, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 11 T^{10} + 102 T^{8} - 207 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T^{6} - T^{4} + 8 T^{3} - 5 T^{2} + 125)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} - 19 T^{10} + 350 T^{8} - 207 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( (T^{2} + T + 1)^{6} \) Copy content Toggle raw display
$13$ \( T^{12} + 6 T^{10} - 57 T^{8} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( T^{12} - 51 T^{10} + 1918 T^{8} + \cdots + 4879681 \) Copy content Toggle raw display
$19$ \( (T^{6} + 5 T^{5} + 42 T^{4} + 21 T^{3} + \cdots + 2809)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} - 59 T^{10} + 2550 T^{8} + \cdots + 5764801 \) Copy content Toggle raw display
$29$ \( (T^{6} + 5 T^{5} + 42 T^{4} + 21 T^{3} + \cdots + 2809)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 6 T^{2} - 4 T + 16)^{4} \) Copy content Toggle raw display
$37$ \( T^{12} - 107 T^{10} + \cdots + 607573201 \) Copy content Toggle raw display
$41$ \( (T^{6} - T^{5} + 78 T^{4} + 435 T^{3} + \cdots + 32041)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} - 99 T^{10} + \cdots + 163047361 \) Copy content Toggle raw display
$47$ \( (T^{6} + 204 T^{4} + 9008 T^{2} + \cdots + 87616)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 80 T^{4} + 2032 T^{2} + \cdots + 16384)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - T^{5} + 126 T^{4} - 93 T^{3} + \cdots + 11881)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - 11 T^{5} + 130 T^{4} + \cdots + 51529)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} - 299 T^{10} + \cdots + 123657019201 \) Copy content Toggle raw display
$71$ \( (T^{6} - 7 T^{5} + 162 T^{4} + 1065 T^{3} + \cdots + 18769)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 324 T^{4} + 29808 T^{2} + \cdots + 746496)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 2 T^{2} - 108 T - 512)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + 128 T^{4} + 2784 T^{2} + \cdots + 3136)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - T^{5} + 126 T^{4} - 93 T^{3} + \cdots + 11881)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} - 299 T^{10} + \cdots + 4499860561 \) Copy content Toggle raw display
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