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Properties

Label 25.18.a.f
Level $25$
Weight $18$
Character orbit 25.a
Self dual yes
Analytic conductor $45.806$
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] = [25,18,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, 18, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("25.1");
 
S:= CuspForms(chi, 18);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 18 \)
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: \(45.8055218361\)
Analytic rank: \(0\)
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: \( x^{8} - 203459x^{6} + 12362849196x^{4} - 237701205446144x^{2} + 1320400799499206656 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{21}\cdot 3^{8}\cdot 5^{14}\cdot 11 \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{2} q^{3} + (\beta_{4} + 72387) q^{4} + (\beta_{6} - 9 \beta_{4} + 44727) q^{6} + (\beta_{3} + 64 \beta_{2} + 18407 \beta_1) q^{7} + (\beta_{5} + 3 \beta_{3} + \cdots + 65895 \beta_1) q^{8}+ \cdots + (3 \beta_{7} + 12 \beta_{6} + \cdots + 29364543) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{2} q^{3} + (\beta_{4} + 72387) q^{4} + (\beta_{6} - 9 \beta_{4} + 44727) q^{6} + (\beta_{3} + 64 \beta_{2} + 18407 \beta_1) q^{7} + (\beta_{5} + 3 \beta_{3} + \cdots + 65895 \beta_1) q^{8}+ \cdots + (1702446174 \beta_{7} + \cdots + 70\!\cdots\!96) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 579096 q^{4} + 357816 q^{6} + 234916344 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 579096 q^{4} + 357816 q^{6} + 234916344 q^{9} + 463296576 q^{11} + 29937907992 q^{14} + 30848001568 q^{16} + 20615713280 q^{19} - 75039699024 q^{21} - 1768741136160 q^{24} - 838901194224 q^{26} + 4079017824720 q^{29} + 11329328658496 q^{31} + 36406243632832 q^{34} + 59729752432728 q^{36} - 40318460422272 q^{39} + 97217252847456 q^{41} + 116357853210912 q^{44} + 10\!\cdots\!36 q^{46}+ \cdots + 56\!\cdots\!68 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 203459x^{6} + 12362849196x^{4} - 237701205446144x^{2} + 1320400799499206656 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -19\nu^{7} + 3990585\nu^{5} - 233779723524\nu^{3} + 2952850120132352\nu ) / 7999594960896 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -30353\nu^{7} + 5463741363\nu^{5} - 242943173908524\nu^{3} + 1147754861965102336\nu ) / 7999594960896 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 4\nu^{2} - 203459 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 117887\nu^{7} - 22025930109\nu^{5} + 1122923251028628\nu^{3} - 12861047218276913920\nu ) / 7999594960896 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} - 8925\nu^{4} + 13674601908\nu^{2} - 261009316476832 ) / 32033232 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -37\nu^{6} + 6788271\nu^{4} - 330797578716\nu^{2} + 3238039981454432 ) / 21355488 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} + 203459 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{5} + 3\beta_{3} + 1412\beta_{2} + 328039\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 12\beta_{7} - 666\beta_{6} + 117547\beta_{4} + 16669866289 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 143227\beta_{5} + 359457\beta_{3} + 314421212\beta_{2} + 31316709117\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -107100\beta_{7} - 122188878\beta_{6} + 12625494933\beta_{4} + 1589405007063119 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 17777883909\beta_{5} + 38584449567\beta_{3} + 45296360157156\beta_{2} + 3162869653005443\beta_1 ) / 8 \) Copy content Toggle raw display

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

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
−330.281
−256.320
−137.667
−98.5951
98.5951
137.667
256.320
330.281
−660.562 11309.3 305271. 0 −7.47047e6 −2.37065e7 −1.15069e8 −1.24087e6 0
1.2 −512.640 −18245.5 131728. 0 9.35337e6 −1.34806e7 −336370. 2.03757e8 0
1.3 −275.335 −2990.60 −55262.8 0 823415. 2.43909e7 5.13044e7 −1.20196e8 0
1.4 −197.190 12817.1 −92188.0 0 −2.52741e6 4.49140e6 4.40247e7 3.51381e7 0
1.5 197.190 −12817.1 −92188.0 0 −2.52741e6 −4.49140e6 −4.40247e7 3.51381e7 0
1.6 275.335 2990.60 −55262.8 0 823415. −2.43909e7 −5.13044e7 −1.20196e8 0
1.7 512.640 18245.5 131728. 0 9.35337e6 1.34806e7 336370. 2.03757e8 0
1.8 660.562 −11309.3 305271. 0 −7.47047e6 2.37065e7 1.15069e8 −1.24087e6 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 25.18.a.f 8
5.b even 2 1 inner 25.18.a.f 8
5.c odd 4 2 5.18.b.a ✓ 8
15.e even 4 2 45.18.b.b 8
20.e even 4 2 80.18.c.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.18.b.a ✓ 8 5.c odd 4 2
25.18.a.f 8 1.a even 1 1 trivial
25.18.a.f 8 5.b even 2 1 inner
45.18.b.b 8 15.e even 4 2
80.18.c.b 8 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 813836T_{2}^{6} + 197805587136T_{2}^{4} - 15212877148553216T_{2}^{2} + 338022604671796903936 \) acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(25))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + \cdots + 33\!\cdots\!36 \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 62\!\cdots\!96 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 12\!\cdots\!16 \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots + 25\!\cdots\!56)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 25\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 24\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 51\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots - 78\!\cdots\!64)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 37\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots - 51\!\cdots\!24)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 28\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 44\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 88\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots - 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots - 56\!\cdots\!44)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 23\!\cdots\!04)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 36\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 53\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots - 35\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 76\!\cdots\!56 \) Copy content Toggle raw display
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