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Properties

Label 2940.2.k.f
Level $2940$
Weight $2$
Character orbit 2940.k
Analytic conductor $23.476$
Analytic rank $0$
Dimension $8$
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(589,2940)]
 
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, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2940.589");
 
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.k (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: \(8\)
Coefficient field: 8.0.31678304256.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 2x^{6} + 8x^{5} + 32x^{4} - 20x^{3} + 8x^{2} + 8x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} + (\beta_{7} - \beta_1) q^{5} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{3} + (\beta_{7} - \beta_1) q^{5} - q^{9} + (\beta_{6} - \beta_{3} + \beta_1 + 1) q^{11} + ( - \beta_{7} + \beta_{5} - \beta_{4}) q^{13} + ( - \beta_{6} + \beta_{3}) q^{15} + ( - \beta_{7} - \beta_{5}) q^{17} + ( - 2 \beta_{6} - \beta_{2} - 2) q^{19} + (\beta_{7} - \beta_{5} + 2 \beta_{4}) q^{23} + ( - \beta_{5} - \beta_{4} - \beta_{3} + \cdots - 2) q^{25}+ \cdots + ( - \beta_{6} + \beta_{3} - \beta_1 - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{5} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{5} - 8 q^{9} + 8 q^{11} + 2 q^{15} - 8 q^{19} - 12 q^{25} + 12 q^{29} + 4 q^{39} + 24 q^{41} + 2 q^{45} - 4 q^{51} - 20 q^{55} + 28 q^{59} + 32 q^{61} + 26 q^{65} - 12 q^{69} - 28 q^{71} + 8 q^{75} + 16 q^{79} + 8 q^{81} + 16 q^{85} + 16 q^{89} - 22 q^{95} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 2x^{6} + 8x^{5} + 32x^{4} - 20x^{3} + 8x^{2} + 8x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 97\nu^{7} - 173\nu^{6} + 220\nu^{5} + 316\nu^{4} + 4010\nu^{3} - 1148\nu^{2} - 1300\nu - 6130 ) / 2462 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 140\nu^{7} - 237\nu^{6} + 216\nu^{5} + 1116\nu^{4} + 5280\nu^{3} - 1530\nu^{2} + 738\nu + 696 ) / 2462 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 174\nu^{7} - 488\nu^{6} + 585\nu^{5} + 1176\nu^{4} + 4452\nu^{3} - 8760\nu^{2} + 2922\nu + 654 ) / 2462 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -382\nu^{7} + 1227\nu^{6} - 1539\nu^{5} - 2412\nu^{4} - 8076\nu^{3} + 22288\nu^{2} - 5566\nu - 1266 ) / 2462 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 452\nu^{7} - 730\nu^{6} + 416\nu^{5} + 4201\nu^{4} + 15640\nu^{3} - 4588\nu^{2} - 5144\nu + 4076 ) / 2462 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 605\nu^{7} - 1244\nu^{6} + 1461\nu^{5} + 4471\nu^{4} + 19300\nu^{3} - 11272\nu^{2} + 12070\nu + 2656 ) / 2462 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} - 3\beta_{4} + \beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} - \beta_{6} - \beta_{5} - 2\beta_{4} + 8\beta_{3} - \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{6} + 12\beta_{3} - 8\beta_{2} - 12\beta _1 - 20 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8\beta_{7} - 8\beta_{6} + 12\beta_{5} + 26\beta_{4} - 12\beta_{2} - 66\beta _1 - 26 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 24\beta_{7} + 66\beta_{5} + 158\beta_{4} - 120\beta_{3} - 120\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 66\beta_{7} + 66\beta_{6} + 120\beta_{5} + 270\beta_{4} - 572\beta_{3} + 120\beta_{2} + 270 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
589.1
2.13456 + 2.13456i
0.575868 + 0.575868i
−0.285451 − 0.285451i
−1.42497 − 1.42497i
2.13456 − 2.13456i
0.575868 − 0.575868i
−0.285451 + 0.285451i
−1.42497 + 1.42497i
0 − 1.00000i 0 −2.13456 − 0.666078i 0 0 0 −1.00000 0
589.2 0 − 1.00000i 0 −0.575868 + 2.16064i 0 0 0 −1.00000 0
589.3 0 − 1.00000i 0 0.285451 − 2.21777i 0 0 0 −1.00000 0
589.4 0 − 1.00000i 0 1.42497 + 1.72321i 0 0 0 −1.00000 0
589.5 0 1.00000i 0 −2.13456 + 0.666078i 0 0 0 −1.00000 0
589.6 0 1.00000i 0 −0.575868 − 2.16064i 0 0 0 −1.00000 0
589.7 0 1.00000i 0 0.285451 + 2.21777i 0 0 0 −1.00000 0
589.8 0 1.00000i 0 1.42497 − 1.72321i 0 0 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 589.8
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 2940.2.k.f 8
5.b even 2 1 inner 2940.2.k.f 8
7.b odd 2 1 2940.2.k.g 8
7.c even 3 2 420.2.bb.a ✓ 16
7.d odd 6 2 2940.2.bb.i 16
21.h odd 6 2 1260.2.bm.c 16
28.g odd 6 2 1680.2.di.e 16
35.c odd 2 1 2940.2.k.g 8
35.i odd 6 2 2940.2.bb.i 16
35.j even 6 2 420.2.bb.a ✓ 16
35.l odd 12 2 2100.2.q.l 8
35.l odd 12 2 2100.2.q.m 8
105.o odd 6 2 1260.2.bm.c 16
140.p odd 6 2 1680.2.di.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.bb.a ✓ 16 7.c even 3 2
420.2.bb.a ✓ 16 35.j even 6 2
1260.2.bm.c 16 21.h odd 6 2
1260.2.bm.c 16 105.o odd 6 2
1680.2.di.e 16 28.g odd 6 2
1680.2.di.e 16 140.p odd 6 2
2100.2.q.l 8 35.l odd 12 2
2100.2.q.m 8 35.l odd 12 2
2940.2.k.f 8 1.a even 1 1 trivial
2940.2.k.f 8 5.b even 2 1 inner
2940.2.k.g 8 7.b odd 2 1
2940.2.k.g 8 35.c odd 2 1
2940.2.bb.i 16 7.d odd 6 2
2940.2.bb.i 16 35.i odd 6 2

Hecke kernels

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

\( T_{11}^{4} - 4T_{11}^{3} - 6T_{11}^{2} + 22T_{11} + 14 \) Copy content Toggle raw display
\( T_{13}^{8} + 52T_{13}^{6} + 726T_{13}^{4} + 3600T_{13}^{2} + 5625 \) Copy content Toggle raw display
\( T_{19}^{4} + 4T_{19}^{3} - 42T_{19}^{2} - 216T_{19} - 159 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + 2 T^{7} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} - 4 T^{3} - 6 T^{2} + \cdots + 14)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 52 T^{6} + \cdots + 5625 \) Copy content Toggle raw display
$17$ \( T^{8} + 52 T^{6} + \cdots + 4 \) Copy content Toggle raw display
$19$ \( (T^{4} + 4 T^{3} + \cdots - 159)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 60 T^{6} + \cdots + 2916 \) Copy content Toggle raw display
$29$ \( (T^{4} - 6 T^{3} + \cdots + 1098)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 102 T^{2} + \cdots + 1101)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 64 T^{6} + \cdots + 30625 \) Copy content Toggle raw display
$41$ \( (T^{4} - 12 T^{3} + 42 T^{2} + \cdots + 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 280 T^{6} + \cdots + 249001 \) Copy content Toggle raw display
$47$ \( T^{8} + 244 T^{6} + \cdots + 379456 \) Copy content Toggle raw display
$53$ \( T^{8} + 160 T^{6} + \cdots + 9216 \) Copy content Toggle raw display
$59$ \( (T^{4} - 14 T^{3} + \cdots - 11942)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 16 T^{3} + \cdots - 1996)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 252 T^{6} + \cdots + 9 \) Copy content Toggle raw display
$71$ \( (T^{4} + 14 T^{3} + \cdots - 8802)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 232 T^{6} + \cdots + 3530641 \) Copy content Toggle raw display
$79$ \( (T^{4} - 8 T^{3} - 234 T^{2} + \cdots + 33)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 364 T^{6} + \cdots + 27899524 \) Copy content Toggle raw display
$89$ \( (T^{4} - 8 T^{3} + \cdots - 6022)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 196 T^{6} + \cdots + 712336 \) Copy content Toggle raw display
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