Properties

Label 425.2.d.c
Level $425$
Weight $2$
Character orbit 425.d
Analytic conductor $3.394$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [425,2,Mod(101,425)]
 
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([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("425.101");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 425.d (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: \(3.39364208590\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 85)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} - \beta_{4} q^{3} + (\beta_{3} + \beta_1) q^{4} - \beta_{5} q^{6} + ( - \beta_{5} - \beta_{4} + \beta_{2}) q^{7} + (\beta_1 + 1) q^{8} + (\beta_{3} + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} - \beta_{4} q^{3} + (\beta_{3} + \beta_1) q^{4} - \beta_{5} q^{6} + ( - \beta_{5} - \beta_{4} + \beta_{2}) q^{7} + (\beta_1 + 1) q^{8} + (\beta_{3} + \beta_1) q^{9} - \beta_{2} q^{11} + (\beta_{5} + \beta_{4} + \beta_{2}) q^{12} + ( - \beta_{3} + \beta_1 - 1) q^{13} + ( - 2 \beta_{4} + 3 \beta_{2}) q^{14} + ( - 2 \beta_1 - 1) q^{16} + ( - \beta_{5} + \beta_{4} + \cdots + 2 \beta_1) q^{17}+ \cdots + (\beta_{5} + 2 \beta_{4} - 2 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{2} + 2 q^{4} + 6 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{2} + 2 q^{4} + 6 q^{8} + 2 q^{9} - 8 q^{13} - 6 q^{16} + 2 q^{17} + 10 q^{18} - 8 q^{21} - 20 q^{26} - 6 q^{32} - 4 q^{33} + 6 q^{34} + 22 q^{36} - 8 q^{38} - 20 q^{42} - 28 q^{43} + 20 q^{47} - 14 q^{49} + 16 q^{51} + 28 q^{53} + 8 q^{59} - 22 q^{64} + 8 q^{66} - 12 q^{67} + 34 q^{68} + 14 q^{72} + 24 q^{76} + 28 q^{77} - 26 q^{81} - 4 q^{83} + 8 q^{86} - 32 q^{87} + 4 q^{89} + 36 q^{93} + 8 q^{94} - 86 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + 8\nu^{4} - 4\nu^{3} - \nu^{2} + 2\nu + 38 ) / 23 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -4\nu^{5} + 9\nu^{4} - 16\nu^{3} + 19\nu^{2} - 38\nu + 14 ) / 23 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5\nu^{5} - 17\nu^{4} + 20\nu^{3} + 5\nu^{2} - 10\nu - 29 ) / 23 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -14\nu^{5} + 20\nu^{4} - 10\nu^{3} - 37\nu^{2} - 64\nu + 26 ) / 23 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 18\nu^{5} - 29\nu^{4} + 26\nu^{3} + 41\nu^{2} + 102\nu - 40 ) / 23 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} - \beta_{3} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{5} + 4\beta_{4} + 2\beta_{3} + \beta_{2} + 4\beta _1 - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{3} + 5\beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -6\beta_{5} - 8\beta_{4} + 3\beta_{3} - 3\beta_{2} + 8\beta _1 - 9 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
0.403032 0.403032i
0.403032 + 0.403032i
1.45161 + 1.45161i
1.45161 1.45161i
−0.854638 + 0.854638i
−0.854638 0.854638i
−1.48119 1.67513i 0.193937 0 2.48119i 1.28726i 2.67513 0.193937 0
101.2 −1.48119 1.67513i 0.193937 0 2.48119i 1.28726i 2.67513 0.193937 0
101.3 0.311108 2.21432i −1.90321 0 0.688892i 1.59210i −1.21432 −1.90321 0
101.4 0.311108 2.21432i −1.90321 0 0.688892i 1.59210i −1.21432 −1.90321 0
101.5 2.17009 0.539189i 2.70928 0 1.17009i 4.87936i 1.53919 2.70928 0
101.6 2.17009 0.539189i 2.70928 0 1.17009i 4.87936i 1.53919 2.70928 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 425.2.d.c 6
5.b even 2 1 85.2.d.a 6
5.c odd 4 1 425.2.c.a 6
5.c odd 4 1 425.2.c.b 6
15.d odd 2 1 765.2.g.b 6
17.b even 2 1 inner 425.2.d.c 6
17.c even 4 1 7225.2.a.q 3
17.c even 4 1 7225.2.a.r 3
20.d odd 2 1 1360.2.c.f 6
85.c even 2 1 85.2.d.a 6
85.g odd 4 1 425.2.c.a 6
85.g odd 4 1 425.2.c.b 6
85.j even 4 1 1445.2.a.j 3
85.j even 4 1 1445.2.a.k 3
255.h odd 2 1 765.2.g.b 6
340.d odd 2 1 1360.2.c.f 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
85.2.d.a 6 5.b even 2 1
85.2.d.a 6 85.c even 2 1
425.2.c.a 6 5.c odd 4 1
425.2.c.a 6 85.g odd 4 1
425.2.c.b 6 5.c odd 4 1
425.2.c.b 6 85.g odd 4 1
425.2.d.c 6 1.a even 1 1 trivial
425.2.d.c 6 17.b even 2 1 inner
765.2.g.b 6 15.d odd 2 1
765.2.g.b 6 255.h odd 2 1
1360.2.c.f 6 20.d odd 2 1
1360.2.c.f 6 340.d odd 2 1
1445.2.a.j 3 85.j even 4 1
1445.2.a.k 3 85.j even 4 1
7225.2.a.q 3 17.c even 4 1
7225.2.a.r 3 17.c even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(425, [\chi])\):

\( T_{2}^{3} - T_{2}^{2} - 3T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{6} + 8T_{3}^{4} + 16T_{3}^{2} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{3} - T^{2} - 3 T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{6} + 8 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 28 T^{4} + \cdots + 100 \) Copy content Toggle raw display
$11$ \( T^{6} + 12 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$13$ \( (T^{3} + 4 T^{2} - 4 T - 20)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} - 2 T^{5} + \cdots + 4913 \) Copy content Toggle raw display
$19$ \( (T^{3} - 16 T + 16)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 120 T^{4} + \cdots + 45796 \) Copy content Toggle raw display
$29$ \( T^{6} + 80 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$31$ \( T^{6} + 80 T^{4} + \cdots + 3364 \) Copy content Toggle raw display
$37$ \( T^{6} + 96 T^{4} + \cdots + 25600 \) Copy content Toggle raw display
$41$ \( T^{6} + 172 T^{4} + \cdots + 53824 \) Copy content Toggle raw display
$43$ \( (T^{3} + 14 T^{2} + \cdots + 68)^{2} \) Copy content Toggle raw display
$47$ \( (T^{3} - 10 T^{2} + 148)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} - 14 T^{2} + \cdots + 296)^{2} \) Copy content Toggle raw display
$59$ \( (T^{3} - 4 T^{2} - 8 T + 16)^{2} \) Copy content Toggle raw display
$61$ \( T^{6} + 396 T^{4} + \cdots + 287296 \) Copy content Toggle raw display
$67$ \( (T^{3} + 6 T^{2} + \cdots - 460)^{2} \) Copy content Toggle raw display
$71$ \( T^{6} + 340 T^{4} + \cdots + 391876 \) Copy content Toggle raw display
$73$ \( T^{6} + 332 T^{4} + \cdots + 678976 \) Copy content Toggle raw display
$79$ \( T^{6} + 192 T^{4} + \cdots + 244036 \) Copy content Toggle raw display
$83$ \( (T^{3} + 2 T^{2} + \cdots - 796)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} - 2 T^{2} + \cdots + 1396)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 320 T^{4} + \cdots + 215296 \) Copy content Toggle raw display
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