Properties

Label 25.20.a.f
Level $25$
Weight $20$
Character orbit 25.a
Self dual yes
Analytic conductor $57.204$
Analytic rank $1$
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] = [25,20,Mod(1,25)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(25, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 20, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("25.1");
 
S:= CuspForms(chi, 20);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 20 \)
Character orbit: \([\chi]\) \(=\) 25.a (trivial)

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: yes
Analytic conductor: \(57.2041741391\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 726881x^{6} + 160513523376x^{4} - 10607307647230976x^{2} + 32429098232548950016 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{21}\cdot 3^{8}\cdot 5^{14} \)
Twist minimal: no (minimal twist has level 5)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{2} + \beta_1) q^{3} + (\beta_{5} + 202593) q^{4} + (\beta_{6} + 3 \beta_{5} + 420717) q^{6} + ( - \beta_{3} + 322 \beta_{2} - 30617 \beta_1) q^{7} + ( - \beta_{4} + 3 \beta_{3} - 656 \beta_{2} + 92624 \beta_1) q^{8} + ( - 9 \beta_{7} - 12 \beta_{6} - 369 \beta_{5} - 43169823) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{2} + \beta_1) q^{3} + (\beta_{5} + 202593) q^{4} + (\beta_{6} + 3 \beta_{5} + 420717) q^{6} + ( - \beta_{3} + 322 \beta_{2} - 30617 \beta_1) q^{7} + ( - \beta_{4} + 3 \beta_{3} - 656 \beta_{2} + 92624 \beta_1) q^{8} + ( - 9 \beta_{7} - 12 \beta_{6} - 369 \beta_{5} - 43169823) q^{9} + (154 \beta_{7} - 372 \beta_{6} + 50 \beta_{5} - 422446908) q^{11} + (459 \beta_{4} + 63 \beta_{3} - 102192 \beta_{2} + 1408704 \beta_1) q^{12} + ( - 1119 \beta_{4} + 68 \beta_{3} + 337345 \beta_{2} + 2593732 \beta_1) q^{13} + ( - 2432 \beta_{7} - 855 \beta_{6} - 149281 \beta_{5} + \cdots - 22156323879) q^{14}+ \cdots + (2123060598 \beta_{7} - 96741022068 \beta_{6} + \cdots + 20\!\cdots\!84) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 1620744 q^{4} + 3365736 q^{6} - 345358584 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 1620744 q^{4} + 3365736 q^{6} - 345358584 q^{9} - 3379575264 q^{11} - 177250591032 q^{14} - 312730276832 q^{16} - 4547188380640 q^{19} - 2983154334624 q^{21} + 6176642779680 q^{24} + 15909228128496 q^{26} + 188222300345040 q^{29} + 72115006686976 q^{31} - 378440221985792 q^{34} - 964020253238712 q^{36} - 30\!\cdots\!28 q^{39}+ \cdots + 16\!\cdots\!72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 726881x^{6} + 160513523376x^{4} - 10607307647230976x^{2} + 32429098232548950016 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -173\nu^{7} + 111026445\nu^{5} - 20083149107568\nu^{3} + 947168012636830720\nu ) / 1505285197258752 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3359\nu^{7} + 2389007679\nu^{5} - 315402560004816\nu^{3} - 24029488671950081024\nu ) / 71680247488512 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 98129 \nu^{7} + 77674135857 \nu^{5} + \cdots + 13\!\cdots\!68 \nu ) / 15\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 4\nu^{2} - 726881 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} - 521985\nu^{4} + 59893845840\nu^{2} - 291065473396544 ) / 51116832 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{6} + 982497\nu^{4} - 232144214832\nu^{2} + 7723079222679680 ) / 204467328 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + 726881 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{4} + 3\beta_{3} - 656\beta_{2} + 1141200\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 1776\beta_{7} + 444\beta_{6} + 374041\beta_{5} + 207328941409 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 319487\beta_{4} + 1499523\beta_{3} - 792634640\beta_{2} + 362710457744\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 927045360\beta_{7} + 436228668\beta_{6} + 135349945545\beta_{5} + 65851160816938001 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 321125173071\beta_{4} + 614088211347\beta_{3} - 502145329633296\beta_{2} + 122197890038322320\beta_1 ) / 8 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−607.943
−490.322
−337.134
−56.6657
56.6657
337.134
490.322
607.943
−1215.89 −18754.1 954092. 0 2.28029e7 1.79878e8 −5.22593e8 −8.10544e8 0
1.2 −980.644 42067.9 437374. 0 −4.12537e7 −4.16374e7 8.52313e7 6.07450e8 0
1.3 −674.268 −35433.1 −69650.8 0 2.38914e7 −1.27830e8 4.00474e8 9.32466e7 0
1.4 −113.331 33157.6 −511444. 0 −3.75780e6 −2.70208e7 1.17381e8 −6.28322e7 0
1.5 113.331 −33157.6 −511444. 0 −3.75780e6 2.70208e7 −1.17381e8 −6.28322e7 0
1.6 674.268 35433.1 −69650.8 0 2.38914e7 1.27830e8 −4.00474e8 9.32466e7 0
1.7 980.644 −42067.9 437374. 0 −4.12537e7 4.16374e7 −8.52313e7 6.07450e8 0
1.8 1215.89 18754.1 954092. 0 2.28029e7 −1.79878e8 5.22593e8 −8.10544e8 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)

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 25.20.a.f 8
5.b even 2 1 inner 25.20.a.f 8
5.c odd 4 2 5.20.b.a 8
15.e even 4 2 45.20.b.b 8
20.e even 4 2 80.20.c.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.20.b.a 8 5.c odd 4 2
25.20.a.f 8 1.a even 1 1 trivial
25.20.a.f 8 5.b even 2 1 inner
45.20.b.b 8 15.e even 4 2
80.20.c.a 8 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 2907524T_{2}^{6} + 2568216374016T_{2}^{4} - 678867689422782464T_{2}^{2} + 8301849147532531204096 \) acting on \(S_{20}^{\mathrm{new}}(\Gamma_0(25))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 2907524 T^{6} + \cdots + 83\!\cdots\!96 \) Copy content Toggle raw display
$3$ \( T^{8} - 4476366576 T^{6} + \cdots + 85\!\cdots\!96 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 66\!\cdots\!16 \) Copy content Toggle raw display
$11$ \( (T^{4} + 1689787632 T^{3} + \cdots + 34\!\cdots\!96)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 77\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( (T^{4} + 2273594190320 T^{3} + \cdots + 35\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 38\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( (T^{4} - 94111150172520 T^{3} + \cdots - 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 36057503343488 T^{3} + \cdots + 16\!\cdots\!56)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 63\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 12\!\cdots\!36)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 88\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 65\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 67\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 41\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots - 62\!\cdots\!04)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 21\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 26\!\cdots\!24)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots - 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 18\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
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