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Properties

Label 2940.2.d.a
Level $2940$
Weight $2$
Character orbit 2940.d
Analytic conductor $23.476$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2940,2,Mod(881,2940)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2940, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 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("2940.881");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2940.d (of order \(2\), degree \(1\), 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: \(23.4760181943\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: 10.0.29471584693248.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} + 2x^{8} - 4x^{7} + 13x^{6} - 36x^{5} + 39x^{4} - 36x^{3} + 54x^{2} - 162x + 243 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 420)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{3} - q^{5} + (\beta_{8} - \beta_{6} - \beta_{4}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{3} - q^{5} + (\beta_{8} - \beta_{6} - \beta_{4}) q^{9} + ( - \beta_{8} - \beta_{7} + \beta_{4}) q^{11} + (\beta_{9} + 2 \beta_{8} + \cdots - \beta_{4}) q^{13}+ \cdots + ( - \beta_{8} + \beta_{7} - \beta_{4} + \cdots + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 2 q^{3} - 10 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 2 q^{3} - 10 q^{5} + 2 q^{15} + 12 q^{17} + 10 q^{25} - 8 q^{27} - 16 q^{33} + 2 q^{37} + 6 q^{39} + 8 q^{41} - 26 q^{43} + 28 q^{47} + 4 q^{51} + 18 q^{57} - 14 q^{67} + 14 q^{69} - 2 q^{75} - 2 q^{79} - 28 q^{81} + 8 q^{83} - 12 q^{85} - 34 q^{87} + 56 q^{89} + 22 q^{93} + 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} + 2x^{8} - 4x^{7} + 13x^{6} - 36x^{5} + 39x^{4} - 36x^{3} + 54x^{2} - 162x + 243 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - 2\nu^{6} + 2\nu^{5} - 4\nu^{4} + 4\nu^{3} - 18\nu^{2} + 21\nu - 18 ) / 18 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{8} + 2\nu^{7} - 11\nu^{6} + 22\nu^{5} - 4\nu^{4} + 18\nu^{3} - 75\nu^{2} + 198\nu - 81 ) / 108 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5\nu^{9} - \nu^{8} + \nu^{7} + 7\nu^{6} + 74\nu^{5} - 126\nu^{4} - 39\nu^{3} + 117\nu^{2} + 135\nu - 1053 ) / 324 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{9} + 2\nu^{8} - 2\nu^{7} + 4\nu^{6} - 13\nu^{5} + 36\nu^{4} - 39\nu^{3} + 36\nu^{2} - 54\nu + 162 ) / 81 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -5\nu^{9} + 4\nu^{8} - 25\nu^{7} + 8\nu^{6} - 68\nu^{5} + 102\nu^{4} - 195\nu^{3} + 216\nu^{2} - 135\nu + 810 ) / 324 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5\nu^{9} + 11\nu^{8} - 5\nu^{7} - 5\nu^{6} + 62\nu^{5} - 42\nu^{4} - 75\nu^{3} - 63\nu^{2} + 81\nu - 1053 ) / 324 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -8\nu^{9} + 7\nu^{8} - 16\nu^{7} + 23\nu^{6} - 50\nu^{5} + 108\nu^{4} - 114\nu^{3} + 153\nu^{2} + 405 ) / 324 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 13\nu^{9} - 8\nu^{8} + 17\nu^{7} - 16\nu^{6} + 124\nu^{5} - 234\nu^{4} + 75\nu^{3} - 360\nu^{2} + 135\nu - 1458 ) / 324 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{9} - \beta_{8} + \beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} + \beta_{7} - \beta_{6} - 2\beta_{5} - \beta_{4} - \beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{8} - \beta_{6} + 2\beta_{5} - \beta_{4} + 2\beta_{3} - 2\beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{8} - 2\beta_{6} + \beta_{5} + 2\beta_{4} + 4\beta_{3} - 2\beta_{2} - 6\beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 6\beta_{9} + 10\beta_{8} - 2\beta_{7} - 6\beta_{6} + 2\beta_{5} - 2\beta_{4} - 2\beta_{3} - 4\beta_{2} + 6\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 2 \beta_{9} - 6 \beta_{8} - 8 \beta_{7} - 8 \beta_{6} + 18 \beta_{5} + 10 \beta_{4} - 4 \beta_{3} + \cdots - 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 13 \beta_{9} + \beta_{8} + 24 \beta_{7} - 8 \beta_{6} - 8 \beta_{5} - 3 \beta_{4} - 14 \beta_{3} + \cdots + 44 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 5 \beta_{9} - 44 \beta_{8} + 17 \beta_{7} + 5 \beta_{6} + 12 \beta_{5} - 21 \beta_{4} - 8 \beta_{3} + \cdots - 1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2940\mathbb{Z}\right)^\times\).

\(n\) \(1081\) \(1177\) \(1471\) \(1961\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
881.1
1.72689 − 0.133595i
1.72689 + 0.133595i
1.15038 − 1.29484i
1.15038 + 1.29484i
0.527154 − 1.64988i
0.527154 + 1.64988i
−1.08831 − 1.34743i
−1.08831 + 1.34743i
−1.31611 − 1.12599i
−1.31611 + 1.12599i
0 −1.72689 − 0.133595i 0 −1.00000 0 0 0 2.96430 + 0.461407i 0
881.2 0 −1.72689 + 0.133595i 0 −1.00000 0 0 0 2.96430 − 0.461407i 0
881.3 0 −1.15038 − 1.29484i 0 −1.00000 0 0 0 −0.353244 + 2.97913i 0
881.4 0 −1.15038 + 1.29484i 0 −1.00000 0 0 0 −0.353244 − 2.97913i 0
881.5 0 −0.527154 − 1.64988i 0 −1.00000 0 0 0 −2.44422 + 1.73948i 0
881.6 0 −0.527154 + 1.64988i 0 −1.00000 0 0 0 −2.44422 − 1.73948i 0
881.7 0 1.08831 − 1.34743i 0 −1.00000 0 0 0 −0.631142 − 2.93286i 0
881.8 0 1.08831 + 1.34743i 0 −1.00000 0 0 0 −0.631142 + 2.93286i 0
881.9 0 1.31611 − 1.12599i 0 −1.00000 0 0 0 0.464299 − 2.96385i 0
881.10 0 1.31611 + 1.12599i 0 −1.00000 0 0 0 0.464299 + 2.96385i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 881.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2940.2.d.a 10
3.b odd 2 1 2940.2.d.b 10
7.b odd 2 1 2940.2.d.b 10
7.c even 3 1 420.2.bh.b yes 10
7.d odd 6 1 420.2.bh.a ✓ 10
21.c even 2 1 inner 2940.2.d.a 10
21.g even 6 1 420.2.bh.b yes 10
21.h odd 6 1 420.2.bh.a ✓ 10
35.i odd 6 1 2100.2.bi.k 10
35.j even 6 1 2100.2.bi.j 10
35.k even 12 2 2100.2.bo.h 20
35.l odd 12 2 2100.2.bo.g 20
105.o odd 6 1 2100.2.bi.k 10
105.p even 6 1 2100.2.bi.j 10
105.w odd 12 2 2100.2.bo.g 20
105.x even 12 2 2100.2.bo.h 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.bh.a ✓ 10 7.d odd 6 1
420.2.bh.a ✓ 10 21.h odd 6 1
420.2.bh.b yes 10 7.c even 3 1
420.2.bh.b yes 10 21.g even 6 1
2100.2.bi.j 10 35.j even 6 1
2100.2.bi.j 10 105.p even 6 1
2100.2.bi.k 10 35.i odd 6 1
2100.2.bi.k 10 105.o odd 6 1
2100.2.bo.g 20 35.l odd 12 2
2100.2.bo.g 20 105.w odd 12 2
2100.2.bo.h 20 35.k even 12 2
2100.2.bo.h 20 105.x even 12 2
2940.2.d.a 10 1.a even 1 1 trivial
2940.2.d.a 10 21.c even 2 1 inner
2940.2.d.b 10 3.b odd 2 1
2940.2.d.b 10 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2940, [\chi])\):

\( T_{11}^{10} + 44T_{11}^{8} + 600T_{11}^{6} + 2732T_{11}^{4} + 4192T_{11}^{2} + 1728 \) Copy content Toggle raw display
\( T_{17}^{5} - 6T_{17}^{4} - 34T_{17}^{3} + 186T_{17}^{2} - 80T_{17} - 264 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + 2 T^{9} + \cdots + 243 \) Copy content Toggle raw display
$5$ \( (T + 1)^{10} \) Copy content Toggle raw display
$7$ \( T^{10} \) Copy content Toggle raw display
$11$ \( T^{10} + 44 T^{8} + \cdots + 1728 \) Copy content Toggle raw display
$13$ \( T^{10} + 81 T^{8} + \cdots + 3888 \) Copy content Toggle raw display
$17$ \( (T^{5} - 6 T^{4} + \cdots - 264)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} + 33 T^{8} + \cdots + 192 \) Copy content Toggle raw display
$23$ \( T^{10} + 130 T^{8} + \cdots + 2462508 \) Copy content Toggle raw display
$29$ \( T^{10} + 142 T^{8} + \cdots + 8748 \) Copy content Toggle raw display
$31$ \( T^{10} + 201 T^{8} + \cdots + 1978032 \) Copy content Toggle raw display
$37$ \( (T^{5} - T^{4} - 67 T^{3} + \cdots - 112)^{2} \) Copy content Toggle raw display
$41$ \( (T^{5} - 4 T^{4} + \cdots - 1338)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} + 13 T^{4} + \cdots + 1559)^{2} \) Copy content Toggle raw display
$47$ \( (T^{5} - 14 T^{4} + \cdots - 17184)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 113246208 \) Copy content Toggle raw display
$59$ \( (T^{5} - 160 T^{3} + \cdots - 11184)^{2} \) Copy content Toggle raw display
$61$ \( T^{10} + 150 T^{8} + \cdots + 1338672 \) Copy content Toggle raw display
$67$ \( (T^{5} + 7 T^{4} + \cdots + 6759)^{2} \) Copy content Toggle raw display
$71$ \( T^{10} + 288 T^{8} + \cdots + 88259328 \) Copy content Toggle raw display
$73$ \( T^{10} + 509 T^{8} + \cdots + 1572528 \) Copy content Toggle raw display
$79$ \( (T^{5} + T^{4} + \cdots - 11768)^{2} \) Copy content Toggle raw display
$83$ \( (T^{5} - 4 T^{4} + \cdots + 28794)^{2} \) Copy content Toggle raw display
$89$ \( (T^{5} - 28 T^{4} + \cdots + 522)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + 212 T^{8} + \cdots + 1051392 \) Copy content Toggle raw display
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