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Properties

Label 425.2.c.a
Level $425$
Weight $2$
Character orbit 425.c
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(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, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("425.424");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
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: \(3.39364208590\)
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: \( 2 \)
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.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + \beta_1 q^{3} + (\beta_{2} - \beta_1) q^{4} + ( - \beta_{4} + \beta_{3}) q^{6} + (2 \beta_{2} - \beta_1) q^{7} + ( - \beta_{5} + \beta_{3}) q^{8} + (\beta_{2} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} + \beta_1 q^{3} + (\beta_{2} - \beta_1) q^{4} + ( - \beta_{4} + \beta_{3}) q^{6} + (2 \beta_{2} - \beta_1) q^{7} + ( - \beta_{5} + \beta_{3}) q^{8} + (\beta_{2} - \beta_1) q^{9} + (\beta_{4} + \beta_{3}) q^{11} + (\beta_1 - 2) q^{12} + ( - \beta_{5} - \beta_{4} - \beta_{3}) q^{13} + ( - 2 \beta_{5} + 3 \beta_{4} + 3 \beta_{3}) q^{14} + ( - 2 \beta_1 - 1) q^{16} + (2 \beta_{5} - \beta_{4} + \beta_1 + 2) q^{17} + ( - \beta_{5} + 2 \beta_{4} + \beta_{3}) q^{18} - 2 \beta_1 q^{19} + (\beta_{2} + \beta_1 - 1) q^{21} + ( - 2 \beta_{2} + \beta_1 + 2) q^{22} + (4 \beta_{2} - \beta_1 + 4) q^{23} + ( - \beta_{4} + 3 \beta_{3}) q^{24} + (\beta_{2} - \beta_1 - 3) q^{26} + ( - 2 \beta_1 - 2) q^{27} + ( - 4 \beta_{2} + \beta_1 + 4) q^{28} + ( - 2 \beta_{5} - 4 \beta_{3}) q^{29} + ( - 2 \beta_{5} - \beta_{4} - 3 \beta_{3}) q^{31} + ( - 2 \beta_{5} + 3 \beta_{4}) q^{32} + ( - \beta_{5} + \beta_{4} - \beta_{3}) q^{33} + ( - 3 \beta_{4} + \beta_{3} + \cdots - \beta_1) q^{34}+ \cdots + (2 \beta_{5} - 3 \beta_{4} - \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{4} - 4 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{4} - 4 q^{7} - 2 q^{9} - 12 q^{12} - 6 q^{16} + 12 q^{17} - 8 q^{21} + 16 q^{22} + 16 q^{23} - 20 q^{26} - 12 q^{27} + 32 q^{28} - 6 q^{34} + 22 q^{36} + 16 q^{37} - 32 q^{48} + 14 q^{49} + 16 q^{51} - 32 q^{57} - 16 q^{58} - 8 q^{59} - 28 q^{62} + 32 q^{63} + 22 q^{64} + 8 q^{66} - 16 q^{68} - 60 q^{73} + 24 q^{76} - 12 q^{78} - 26 q^{81} + 60 q^{82} + 8 q^{86} - 4 q^{88} - 4 q^{89} + 44 q^{92} - 8 q^{94} - 40 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) : 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
\(\beta_{2}\)\(=\) \( ( -5\nu^{5} + 17\nu^{4} - 20\nu^{3} - 5\nu^{2} + 10\nu + 29 ) / 23 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7\nu^{5} - 10\nu^{4} + 5\nu^{3} + 30\nu^{2} + 32\nu - 13 ) / 23 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -11\nu^{5} + 19\nu^{4} - 21\nu^{3} - 11\nu^{2} - 70\nu + 27 ) / 23 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -14\nu^{5} + 20\nu^{4} - 10\nu^{3} - 37\nu^{2} - 64\nu + 26 ) / 23 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{5} - \beta_{4} + 2\beta_{3} - \beta_{2} + 2\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{2} + 5\beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{5} + 3\beta_{4} - 9\beta_{3} - 3\beta_{2} + 8\beta _1 - 9 \) 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
−0.854638 − 0.854638i
0.403032 − 0.403032i
1.45161 + 1.45161i
1.45161 − 1.45161i
0.403032 + 0.403032i
−0.854638 + 0.854638i
− 2.17009i 0.539189 −2.70928 0 − 1.17009i −4.87936 1.53919i −2.70928 0
424.2 − 1.48119i 1.67513 −0.193937 0 − 2.48119i 1.28726 − 2.67513i −0.193937 0
424.3 − 0.311108i −2.21432 1.90321 0 0.688892i 1.59210 − 1.21432i 1.90321 0
424.4 0.311108i −2.21432 1.90321 0 − 0.688892i 1.59210 1.21432i 1.90321 0
424.5 1.48119i 1.67513 −0.193937 0 2.48119i 1.28726 2.67513i −0.193937 0
424.6 2.17009i 0.539189 −2.70928 0 1.17009i −4.87936 − 1.53919i −2.70928 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 424.6
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} - 4T_{3} + 2 \) acting on \(S_{2}^{\mathrm{new}}(425, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 7 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T^{3} - 4 T + 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( (T^{3} + 2 T^{2} - 12 T + 10)^{2} \) Copy content Toggle raw display
$11$ \( T^{6} + 12 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$13$ \( T^{6} + 24 T^{4} + \cdots + 400 \) Copy content Toggle raw display
$17$ \( T^{6} - 12 T^{5} + \cdots + 4913 \) Copy content Toggle raw display
$19$ \( (T^{3} - 16 T - 16)^{2} \) Copy content Toggle raw display
$23$ \( (T^{3} - 8 T^{2} + \cdots + 214)^{2} \) 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
$37$ \( (T^{3} - 8 T^{2} + \cdots + 160)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + 172 T^{4} + \cdots + 53824 \) Copy content Toggle raw display
$43$ \( T^{6} + 84 T^{4} + \cdots + 4624 \) Copy content Toggle raw display
$47$ \( T^{6} + 100 T^{4} + \cdots + 21904 \) Copy content Toggle raw display
$53$ \( T^{6} + 172 T^{4} + \cdots + 87616 \) 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
$67$ \( T^{6} + 212 T^{4} + \cdots + 211600 \) Copy content Toggle raw display
$71$ \( T^{6} + 340 T^{4} + \cdots + 391876 \) Copy content Toggle raw display
$73$ \( (T^{3} + 30 T^{2} + \cdots + 824)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + 192 T^{4} + \cdots + 244036 \) Copy content Toggle raw display
$83$ \( T^{6} + 300 T^{4} + \cdots + 633616 \) Copy content Toggle raw display
$89$ \( (T^{3} + 2 T^{2} + \cdots - 1396)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + 20 T^{2} + \cdots - 464)^{2} \) Copy content Toggle raw display
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