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Properties

Label 1840.2.e.d
Level $1840$
Weight $2$
Character orbit 1840.e
Analytic conductor $14.692$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1840,2,Mod(369,1840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1840, 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("1840.369");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1840.e (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: \(14.6924739719\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.527896576.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 2x^{6} + 2x^{5} + 7x^{4} - 10x^{3} + 8x^{2} + 4x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 115)
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.

\(f(q)\) \(=\) \( q + (\beta_{4} + \beta_{2} - \beta_1) q^{3} + ( - \beta_{6} - 1) q^{5} + (\beta_{7} - \beta_{6} - \beta_{4}) q^{7} + ( - \beta_{7} + \beta_{5} - \beta_{2} + \cdots - 2) q^{9} + ( - \beta_{7} + \beta_{5} + \beta_{3} + \cdots - 1) q^{11}+ \cdots + (2 \beta_{7} - 2 \beta_{5} + 4 \beta_{3} + \cdots + 12) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 6 q^{5} - 8 q^{9} - 4 q^{11} - 6 q^{15} - 8 q^{19} - 4 q^{21} - 16 q^{25} - 8 q^{29} - 28 q^{35} - 16 q^{39} - 16 q^{41} + 24 q^{45} - 20 q^{51} + 16 q^{55} - 16 q^{61} - 14 q^{65} + 4 q^{69} + 48 q^{71}+ \cdots + 72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 2x^{6} + 2x^{5} + 7x^{4} - 10x^{3} + 8x^{2} + 4x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 25\nu^{7} - 46\nu^{6} + 80\nu^{5} - 124\nu^{4} + 398\nu^{3} - 205\nu^{2} + 354\nu - 740 ) / 467 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -27\nu^{7} + 31\nu^{6} + 7\nu^{5} - 221\nu^{4} - 187\nu^{3} + 128\nu^{2} - 401\nu - 882 ) / 467 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -108\nu^{7} + 124\nu^{6} + 28\nu^{5} - 417\nu^{4} - 748\nu^{3} + 512\nu^{2} + 731\nu - 726 ) / 467 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -142\nu^{7} + 336\nu^{6} - 361\nu^{5} - 211\nu^{4} - 897\nu^{3} + 2005\nu^{2} - 1469\nu - 280 ) / 467 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 232\nu^{7} - 595\nu^{6} + 649\nu^{5} + 325\nu^{4} + 1209\nu^{3} - 3210\nu^{2} + 2650\nu + 418 ) / 467 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -269\nu^{7} + 551\nu^{6} - 674\nu^{5} - 403\nu^{4} - 1742\nu^{3} + 2486\nu^{2} - 3286\nu - 537 ) / 467 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 284\nu^{7} - 672\nu^{6} + 722\nu^{5} + 422\nu^{4} + 1794\nu^{3} - 3543\nu^{2} + 2471\nu + 560 ) / 467 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} + \beta_{5} - \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{5} + 4\beta_{4} - \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{7} + \beta_{6} - 2\beta_{5} + \beta_{4} + \beta_{3} - 3\beta_{2} + 2\beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 5\beta_{7} - 5\beta_{5} + 2\beta_{3} - 3\beta_{2} - 5\beta _1 - 12 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 10\beta_{7} - 5\beta_{6} - 17\beta_{5} - 7\beta_{4} + 5\beta_{3} + 10\beta_{2} - 17\beta _1 - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -5\beta_{7} - 6\beta_{6} - 11\beta_{5} - 21\beta_{4} + 12\beta_{2} - 12\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -51\beta_{7} - 22\beta_{6} + 15\beta_{5} - 38\beta_{4} - 22\beta_{3} + 51\beta_{2} - 15\beta _1 + 38 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times\).

\(n\) \(737\) \(1151\) \(1201\) \(1381\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
369.1
−1.07037 + 1.07037i
1.47984 + 1.47984i
−0.199724 + 0.199724i
0.790245 + 0.790245i
0.790245 − 0.790245i
−0.199724 − 0.199724i
1.47984 − 1.47984i
−1.07037 − 1.07037i
0 − 3.14073i 0 −2.07037 − 0.844739i 0 − 1.20647i 0 −6.86420 0
369.2 0 − 1.95969i 0 0.479844 − 2.18398i 0 − 2.28394i 0 −0.840379 0
369.3 0 − 1.39945i 0 −1.19972 + 1.88697i 0 4.60747i 0 1.04155 0
369.4 0 − 0.580491i 0 −0.209755 + 2.22621i 0 − 0.315061i 0 2.66303 0
369.5 0 0.580491i 0 −0.209755 − 2.22621i 0 0.315061i 0 2.66303 0
369.6 0 1.39945i 0 −1.19972 − 1.88697i 0 − 4.60747i 0 1.04155 0
369.7 0 1.95969i 0 0.479844 + 2.18398i 0 2.28394i 0 −0.840379 0
369.8 0 3.14073i 0 −2.07037 + 0.844739i 0 1.20647i 0 −6.86420 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 369.8
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 1840.2.e.d 8
4.b odd 2 1 115.2.b.b ✓ 8
5.b even 2 1 inner 1840.2.e.d 8
5.c odd 4 1 9200.2.a.ck 4
5.c odd 4 1 9200.2.a.cq 4
12.b even 2 1 1035.2.b.e 8
20.d odd 2 1 115.2.b.b ✓ 8
20.e even 4 1 575.2.a.i 4
20.e even 4 1 575.2.a.j 4
60.h even 2 1 1035.2.b.e 8
60.l odd 4 1 5175.2.a.bv 4
60.l odd 4 1 5175.2.a.bw 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.2.b.b ✓ 8 4.b odd 2 1
115.2.b.b ✓ 8 20.d odd 2 1
575.2.a.i 4 20.e even 4 1
575.2.a.j 4 20.e even 4 1
1035.2.b.e 8 12.b even 2 1
1035.2.b.e 8 60.h even 2 1
1840.2.e.d 8 1.a even 1 1 trivial
1840.2.e.d 8 5.b even 2 1 inner
5175.2.a.bv 4 60.l odd 4 1
5175.2.a.bw 4 60.l odd 4 1
9200.2.a.ck 4 5.c odd 4 1
9200.2.a.cq 4 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1840, [\chi])\):

\( T_{3}^{8} + 16T_{3}^{6} + 70T_{3}^{4} + 96T_{3}^{2} + 25 \) Copy content Toggle raw display
\( T_{7}^{8} + 28T_{7}^{6} + 152T_{7}^{4} + 176T_{7}^{2} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 16 T^{6} + \cdots + 25 \) Copy content Toggle raw display
$5$ \( T^{8} + 6 T^{7} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{8} + 28 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( (T^{4} + 2 T^{3} - 16 T^{2} + \cdots - 28)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 64 T^{6} + \cdots + 3721 \) Copy content Toggle raw display
$17$ \( T^{8} + 80 T^{6} + \cdots + 400 \) Copy content Toggle raw display
$19$ \( (T^{4} + 4 T^{3} - 6 T^{2} + \cdots - 20)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} + 4 T^{3} - 22 T^{2} + \cdots + 5)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 74 T^{2} + \cdots - 167)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 148 T^{6} + \cdots + 226576 \) Copy content Toggle raw display
$41$ \( (T^{4} + 8 T^{3} + \cdots + 2485)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 252 T^{6} + \cdots + 3857296 \) Copy content Toggle raw display
$47$ \( T^{8} + 280 T^{6} + \cdots + 20367169 \) Copy content Toggle raw display
$53$ \( T^{8} + 224 T^{6} + \cdots + 4804864 \) Copy content Toggle raw display
$59$ \( (T^{4} - 20 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 8 T^{3} + \cdots - 2756)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 20 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$71$ \( (T^{4} - 24 T^{3} + \cdots - 7435)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 368 T^{6} + \cdots + 69538921 \) Copy content Toggle raw display
$79$ \( (T^{4} - 24 T^{3} + \cdots + 28)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 576 T^{6} + \cdots + 223442704 \) Copy content Toggle raw display
$89$ \( (T^{4} - 8 T^{3} + \cdots + 2380)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 460 T^{6} + \cdots + 21864976 \) Copy content Toggle raw display
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