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Properties

Label 5.26.b.a
Level $5$
Weight $26$
Character orbit 5.b
Analytic conductor $19.800$
Analytic rank $0$
Dimension $12$
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] = [5,26,Mod(4,5)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 26, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5.4");
 
S:= CuspForms(chi, 26);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5 \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 5.b (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: \(19.7998389976\)
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: \( x^{12} + 71168091 x^{10} + \cdots + 10\!\cdots\!36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{44}\cdot 3^{20}\cdot 5^{29} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{3} - 19 \beta_1) q^{3} + (\beta_{2} - 13890962) q^{4} + ( - \beta_{4} + 38 \beta_{3} + \cdots + 45795255) q^{5}+ \cdots + ( - \beta_{9} - \beta_{8} + \cdots - 329038836753) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{3} - 19 \beta_1) q^{3} + (\beta_{2} - 13890962) q^{4} + ( - \beta_{4} + 38 \beta_{3} + \cdots + 45795255) q^{5}+ \cdots + (385515624231 \beta_{11} + \cdots - 83\!\cdots\!56) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 166691544 q^{4} + 549543060 q^{5} + 10591544184 q^{6} - 3948466041036 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 166691544 q^{4} + 549543060 q^{5} + 10591544184 q^{6} - 3948466041036 q^{9} + 4435846671960 q^{10} - 1090673824176 q^{11} - 890646861445848 q^{14} + 443085522435120 q^{15} + 22\!\cdots\!32 q^{16}+ \cdots - 10\!\cdots\!72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 71168091 x^{10} + \cdots + 10\!\cdots\!36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 4\nu^{2} + 47445394 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 17\!\cdots\!09 \nu^{11} + \cdots - 26\!\cdots\!04 \nu ) / 36\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 30\!\cdots\!81 \nu^{11} + \cdots - 15\!\cdots\!52 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 30\!\cdots\!81 \nu^{11} + \cdots - 58\!\cdots\!52 ) / 56\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 46\!\cdots\!27 \nu^{11} + \cdots + 70\!\cdots\!84 ) / 56\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 31\!\cdots\!21 \nu^{11} + \cdots - 53\!\cdots\!68 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 11\!\cdots\!11 \nu^{11} + \cdots - 37\!\cdots\!88 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 12\!\cdots\!93 \nu^{11} + \cdots + 38\!\cdots\!44 ) / 56\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 32\!\cdots\!51 \nu^{11} + \cdots - 41\!\cdots\!92 ) / 56\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 23\!\cdots\!71 \nu^{11} + \cdots - 30\!\cdots\!68 ) / 37\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} - 47445394 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} - 34\beta_{4} - 78567\beta_{3} - 80838931\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 23 \beta_{11} - 23 \beta_{10} - 124 \beta_{9} + 98 \beta_{8} - 222 \beta_{6} + 65469 \beta_{5} + \cdots + 19\!\cdots\!68 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 133434 \beta_{11} + 14826 \beta_{10} + 1528688 \beta_{9} - 32252507 \beta_{7} + \cdots + 19\!\cdots\!57 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 822527133 \beta_{11} + 822527133 \beta_{10} - 1588508268 \beta_{9} - 5604157590 \beta_{8} + \cdots - 45\!\cdots\!48 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 10015603660062 \beta_{11} - 1112844851118 \beta_{10} - 64099974788304 \beta_{9} + \cdots - 48\!\cdots\!59 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 23\!\cdots\!11 \beta_{11} + \cdots + 11\!\cdots\!28 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 45\!\cdots\!90 \beta_{11} + \cdots + 12\!\cdots\!29 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 63\!\cdots\!05 \beta_{11} + \cdots - 30\!\cdots\!08 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 16\!\cdots\!70 \beta_{11} + \cdots - 33\!\cdots\!99 \beta_1 ) / 8 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
− 5297.67i
− 4680.77i
− 3156.95i
− 2840.59i
− 1460.97i
− 1011.64i
1011.64i
1460.97i
2840.59i
3156.95i
4680.77i
5297.67i
− 10595.3i 1.42235e6i −7.87068e7 5.09516e8 + 1.96001e8i 1.50702e10 − 3.45094e10i 4.78405e11i −1.17578e12 2.07670e12 − 5.39850e12i
4.2 − 9361.55i − 531494.i −5.40841e7 −5.44720e8 + 3.61012e7i −4.97561e9 1.37587e10i 1.92189e11i 5.64803e11 3.37963e11 + 5.09942e12i
4.3 − 6313.90i − 71116.0i −6.31093e6 3.25528e8 − 4.38241e8i −4.49019e8 3.34630e10i − 1.72013e11i 8.42231e11 −2.76701e12 − 2.05535e12i
4.4 − 5681.18i − 1.63997e6i 1.27857e6 4.20624e8 + 3.47992e8i −9.31697e9 − 6.01315e10i − 1.97893e11i −1.84221e12 1.97701e12 − 2.38965e12i
4.5 − 2921.94i 1.33312e6i 2.50167e7 −5.18558e8 − 1.70647e8i 3.89530e9 − 4.60230e10i − 1.71141e11i −9.29927e11 −4.98620e11 + 1.51520e12i
4.6 − 2023.27i 529751.i 2.94608e7 8.23814e7 + 5.39663e8i 1.07183e9 2.78390e10i − 1.27497e11i 5.66653e11 1.09188e12 − 1.66680e11i
4.7 2023.27i − 529751.i 2.94608e7 8.23814e7 − 5.39663e8i 1.07183e9 − 2.78390e10i 1.27497e11i 5.66653e11 1.09188e12 + 1.66680e11i
4.8 2921.94i − 1.33312e6i 2.50167e7 −5.18558e8 + 1.70647e8i 3.89530e9 4.60230e10i 1.71141e11i −9.29927e11 −4.98620e11 − 1.51520e12i
4.9 5681.18i 1.63997e6i 1.27857e6 4.20624e8 − 3.47992e8i −9.31697e9 6.01315e10i 1.97893e11i −1.84221e12 1.97701e12 + 2.38965e12i
4.10 6313.90i 71116.0i −6.31093e6 3.25528e8 + 4.38241e8i −4.49019e8 − 3.34630e10i 1.72013e11i 8.42231e11 −2.76701e12 + 2.05535e12i
4.11 9361.55i 531494.i −5.40841e7 −5.44720e8 − 3.61012e7i −4.97561e9 − 1.37587e10i − 1.92189e11i 5.64803e11 3.37963e11 − 5.09942e12i
4.12 10595.3i − 1.42235e6i −7.87068e7 5.09516e8 − 1.96001e8i 1.50702e10 3.45094e10i − 4.78405e11i −1.17578e12 2.07670e12 + 5.39850e12i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.12
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 5.26.b.a ✓ 12
3.b odd 2 1 45.26.b.b 12
5.b even 2 1 inner 5.26.b.a ✓ 12
5.c odd 4 2 25.26.a.f 12
15.d odd 2 1 45.26.b.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.26.b.a ✓ 12 1.a even 1 1 trivial
5.26.b.a ✓ 12 5.b even 2 1 inner
25.26.a.f 12 5.c odd 4 2
45.26.b.b 12 3.b odd 2 1
45.26.b.b 12 15.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{26}^{\mathrm{new}}(5, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + \cdots + 44\!\cdots\!56 \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 38\!\cdots\!96 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 70\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 14\!\cdots\!36 \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots - 68\!\cdots\!36)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 63\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 22\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots - 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 15\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots - 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots - 29\!\cdots\!36)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 23\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 13\!\cdots\!64)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 18\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 43\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 41\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 44\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 66\!\cdots\!64)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 75\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 14\!\cdots\!64)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 98\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 27\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 32\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
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