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Properties

Label 425.4.c.d
Level $425$
Weight $4$
Character orbit 425.c
Analytic conductor $25.076$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [425,4,Mod(424,425)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(425, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("425.424");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 425.c (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: \(25.0758117524\)
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: \( x^{8} - 4x^{7} - 1104x^{6} + 3326x^{5} + 470009x^{4} - 945566x^{3} - 91296866x^{2} + 91770204x + 6832091620 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + \beta_1 q^{3} + 7 q^{4} + \beta_{6} q^{6} + ( - \beta_{2} - 4 \beta_1) q^{7} - 15 \beta_{5} q^{8} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} + \beta_1 q^{3} + 7 q^{4} + \beta_{6} q^{6} + ( - \beta_{2} - 4 \beta_1) q^{7} - 15 \beta_{5} q^{8} - 8 q^{9} + ( - 4 \beta_{6} - \beta_{4}) q^{11} + 7 \beta_1 q^{12} + (3 \beta_{5} - 2 \beta_{3} + \cdots + \beta_1) q^{13}+ \cdots + (32 \beta_{6} + 8 \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 56 q^{4} - 64 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 56 q^{4} - 64 q^{9} + 328 q^{16} - 28 q^{19} - 532 q^{21} + 28 q^{26} + 70 q^{34} - 448 q^{36} + 1380 q^{49} + 190 q^{51} - 2016 q^{59} + 1336 q^{64} + 532 q^{66} + 2128 q^{69} - 196 q^{76} - 3592 q^{81} - 3724 q^{84} - 1184 q^{86} - 3724 q^{89} + 1792 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 1104x^{6} + 3326x^{5} + 470009x^{4} - 945566x^{3} - 91296866x^{2} + 91770204x + 6832091620 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} - 3\nu^{5} - 1395\nu^{4} + 2795\nu^{3} + 549114\nu^{2} - 550512\nu - 69692910 ) / 325450 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 227 \nu^{7} - 27289 \nu^{6} - 184769 \nu^{5} + 37622445 \nu^{4} + 51451138 \nu^{3} + \cdots + 1864868178920 ) / 17245270050 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 11824 \nu^{7} + 7777778 \nu^{6} - 14002991 \nu^{5} - 6504071055 \nu^{4} + \cdots - 205600627311370 ) / 155207430450 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 49 \nu^{7} - 26323 \nu^{6} + 120418 \nu^{5} + 21622725 \nu^{4} - 55802531 \nu^{3} + \cdots + 590844391235 ) / 274218075 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2234 \nu^{7} - 7819 \nu^{6} - 1807883 \nu^{5} + 4539255 \nu^{4} + 503171221 \nu^{3} + \cdots + 24097933790 ) / 6748149150 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 98 \nu^{7} - 343 \nu^{6} - 81869 \nu^{5} + 205530 \nu^{4} + 24438157 \nu^{3} - 36862937 \nu^{2} + \cdots + 1289847695 ) / 274218075 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1705 \nu^{7} + 20527 \nu^{6} - 1951540 \nu^{5} - 17030430 \nu^{4} + 699324545 \nu^{3} + \cdots - 552688411400 ) / 548436150 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} - \beta_{5} - 2\beta_{2} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{4} + 2\beta_{3} - 2\beta_{2} - 2\beta _1 + 278 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -6\beta_{7} + 845\beta_{6} - 833\beta_{5} + 6\beta_{3} - 540\beta_{2} - 273\beta _1 + 835 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{7} + 1112\beta_{6} - 556\beta_{5} + 535\beta_{4} + 1112\beta_{3} - 1092\beta_{2} - 1668\beta _1 + 71913 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 5570 \beta_{7} + 387671 \beta_{6} - 359561 \beta_{5} - 100 \beta_{4} + 5550 \beta_{3} - 136032 \beta_{2} + \cdots + 359561 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 8340 \beta_{7} + 678001 \beta_{6} - 426100 \beta_{5} + 197061 \beta_{4} + 452952 \beta_{3} + \cdots + 17005538 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 3190166 \beta_{7} + 145771025 \beta_{6} - 119500653 \beta_{5} - 195860 \beta_{4} + 3151246 \beta_{3} + \cdots + 118569095 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/425\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(326\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
424.1
−16.3152 + 1.67945i
17.3152 + 1.67945i
−16.3152 − 2.67945i
17.3152 − 2.67945i
−16.3152 − 1.67945i
17.3152 − 1.67945i
−16.3152 + 2.67945i
17.3152 + 2.67945i
− 1.00000i −4.35890 7.00000 0 4.35890i −1.55903 − 15.0000i −8.00000 0
424.2 − 1.00000i −4.35890 7.00000 0 4.35890i 32.0713 − 15.0000i −8.00000 0
424.3 − 1.00000i 4.35890 7.00000 0 − 4.35890i −32.0713 − 15.0000i −8.00000 0
424.4 − 1.00000i 4.35890 7.00000 0 − 4.35890i 1.55903 − 15.0000i −8.00000 0
424.5 1.00000i −4.35890 7.00000 0 − 4.35890i −1.55903 15.0000i −8.00000 0
424.6 1.00000i −4.35890 7.00000 0 − 4.35890i 32.0713 15.0000i −8.00000 0
424.7 1.00000i 4.35890 7.00000 0 4.35890i −32.0713 15.0000i −8.00000 0
424.8 1.00000i 4.35890 7.00000 0 4.35890i 1.55903 15.0000i −8.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 424.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
17.b even 2 1 inner
85.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 425.4.c.d 8
5.b even 2 1 inner 425.4.c.d 8
5.c odd 4 1 425.4.d.b ✓ 4
5.c odd 4 1 425.4.d.d yes 4
17.b even 2 1 inner 425.4.c.d 8
85.c even 2 1 inner 425.4.c.d 8
85.g odd 4 1 425.4.d.b ✓ 4
85.g odd 4 1 425.4.d.d yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
425.4.c.d 8 1.a even 1 1 trivial
425.4.c.d 8 5.b even 2 1 inner
425.4.c.d 8 17.b even 2 1 inner
425.4.c.d 8 85.c even 2 1 inner
425.4.d.b ✓ 4 5.c odd 4 1
425.4.d.b ✓ 4 85.g odd 4 1
425.4.d.d yes 4 5.c odd 4 1
425.4.d.d yes 4 85.g odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(425, [\chi])\):

\( T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{2} - 19 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 19)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} - 1031 T^{2} + 2500)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 1031 T^{2} + 2500)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 10769 T^{2} + 28729600)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 582622237229761 \) Copy content Toggle raw display
$19$ \( (T^{2} + 7 T - 5360)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 43640 T^{2} + 206554384)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 18096)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 86999 T^{2} + 249324100)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 111596 T^{2} + 1255993600)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 106379 T^{2} + 649230400)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 21904)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} + 50176)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} + 576093 T^{2} + 56850772356)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 504 T - 280320)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} + 999404 T^{2} + 247844665600)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 1068321 T^{2} + 58982400)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 1211331 T^{2} + 366146010000)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 633195 T^{2} + 4517452944)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 151895 T^{2} + 385022884)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 584265 T^{2} + 79179206544)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 931 T - 46550)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} - 1253360 T^{2} + 33609155584)^{2} \) Copy content Toggle raw display
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