Properties

Label 425.2.b.e
Level $425$
Weight $2$
Character orbit 425.b
Analytic conductor $3.394$
Analytic rank $0$
Dimension $4$
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(324,425)]
 
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, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("425.324");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 425.b (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: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) 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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + \beta_1) q^{2} + (\beta_{2} - 2 \beta_1) q^{3} + ( - 2 \beta_{3} - 1) q^{4} + \beta_{3} q^{6} + (\beta_{2} + 2 \beta_1) q^{7} + ( - \beta_{2} - 3 \beta_1) q^{8} + (4 \beta_{3} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + \beta_1) q^{2} + (\beta_{2} - 2 \beta_1) q^{3} + ( - 2 \beta_{3} - 1) q^{4} + \beta_{3} q^{6} + (\beta_{2} + 2 \beta_1) q^{7} + ( - \beta_{2} - 3 \beta_1) q^{8} + (4 \beta_{3} - 3) q^{9} + ( - \beta_{3} - 4) q^{11} + (3 \beta_{2} - 2 \beta_1) q^{12} + 2 \beta_{2} q^{13} + ( - 3 \beta_{3} - 4) q^{14} + 3 q^{16} + \beta_1 q^{17} + (\beta_{2} + 5 \beta_1) q^{18} - 2 \beta_{3} q^{19} + 2 q^{21} + ( - 5 \beta_{2} - 6 \beta_1) q^{22} + (\beta_{2} - 2 \beta_1) q^{23} + (\beta_{3} - 4) q^{24} + ( - 2 \beta_{3} - 4) q^{26} + ( - 8 \beta_{2} + 8 \beta_1) q^{27} + ( - 5 \beta_{2} - 6 \beta_1) q^{28} + ( - 2 \beta_{3} + 2) q^{29} - 3 \beta_{3} q^{31} + (\beta_{2} - 3 \beta_1) q^{32} + ( - 2 \beta_{2} + 6 \beta_1) q^{33} + ( - \beta_{3} - 1) q^{34} + (2 \beta_{3} - 13) q^{36} + (6 \beta_{2} + 2 \beta_1) q^{37} + ( - 2 \beta_{2} - 4 \beta_1) q^{38} + (4 \beta_{3} - 4) q^{39} + (6 \beta_{3} + 2) q^{41} + (2 \beta_{2} + 2 \beta_1) q^{42} + ( - 4 \beta_{2} + 2 \beta_1) q^{43} + (9 \beta_{3} + 8) q^{44} + \beta_{3} q^{46} + ( - 2 \beta_{2} + 2 \beta_1) q^{47} + (3 \beta_{2} - 6 \beta_1) q^{48} + ( - 4 \beta_{3} + 1) q^{49} + ( - \beta_{3} + 2) q^{51} + ( - 2 \beta_{2} - 8 \beta_1) q^{52} + (4 \beta_{2} + 6 \beta_1) q^{53} + 8 q^{54} + (5 \beta_{3} + 8) q^{56} + (4 \beta_{2} - 4 \beta_1) q^{57} - 2 \beta_1 q^{58} + (2 \beta_{3} + 12) q^{59} + ( - 4 \beta_{3} + 2) q^{61} + ( - 3 \beta_{2} - 6 \beta_1) q^{62} + (5 \beta_{2} + 2 \beta_1) q^{63} + (2 \beta_{3} + 7) q^{64} + ( - 4 \beta_{3} - 2) q^{66} + (2 \beta_{2} + 6 \beta_1) q^{67} + ( - 2 \beta_{2} - \beta_1) q^{68} + (4 \beta_{3} - 6) q^{69} + 3 \beta_{3} q^{71} + ( - 9 \beta_{2} + \beta_1) q^{72} + (2 \beta_{2} - 2 \beta_1) q^{73} + ( - 8 \beta_{3} - 14) q^{74} + (2 \beta_{3} + 8) q^{76} + ( - 6 \beta_{2} - 10 \beta_1) q^{77} + 4 \beta_1 q^{78} + (\beta_{3} - 4) q^{79} + ( - 12 \beta_{3} + 23) q^{81} + (8 \beta_{2} + 14 \beta_1) q^{82} + ( - 8 \beta_{2} - 2 \beta_1) q^{83} + ( - 4 \beta_{3} - 2) q^{84} + (2 \beta_{3} + 6) q^{86} + (6 \beta_{2} - 8 \beta_1) q^{87} + (7 \beta_{2} + 14 \beta_1) q^{88} + (4 \beta_{3} + 8) q^{89} + ( - 4 \beta_{3} - 4) q^{91} + (3 \beta_{2} - 2 \beta_1) q^{92} + (6 \beta_{2} - 6 \beta_1) q^{93} + 2 q^{94} + (5 \beta_{3} - 8) q^{96} + (4 \beta_{2} + 2 \beta_1) q^{97} + ( - 3 \beta_{2} - 7 \beta_1) q^{98} + ( - 13 \beta_{3} + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 12 q^{9} - 16 q^{11} - 16 q^{14} + 12 q^{16} + 8 q^{21} - 16 q^{24} - 16 q^{26} + 8 q^{29} - 4 q^{34} - 52 q^{36} - 16 q^{39} + 8 q^{41} + 32 q^{44} + 4 q^{49} + 8 q^{51} + 32 q^{54} + 32 q^{56} + 48 q^{59} + 8 q^{61} + 28 q^{64} - 8 q^{66} - 24 q^{69} - 56 q^{74} + 32 q^{76} - 16 q^{79} + 92 q^{81} - 8 q^{84} + 24 q^{86} + 32 q^{89} - 16 q^{91} + 8 q^{94} - 32 q^{96} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 2 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
324.1
0.707107 0.707107i
−0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
2.41421i 0.585786i −3.82843 0 1.41421 3.41421i 4.41421i 2.65685 0
324.2 0.414214i 3.41421i 1.82843 0 −1.41421 0.585786i 1.58579i −8.65685 0
324.3 0.414214i 3.41421i 1.82843 0 −1.41421 0.585786i 1.58579i −8.65685 0
324.4 2.41421i 0.585786i −3.82843 0 1.41421 3.41421i 4.41421i 2.65685 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 425.2.b.e 4
5.b even 2 1 inner 425.2.b.e 4
5.c odd 4 1 85.2.a.b 2
5.c odd 4 1 425.2.a.f 2
15.e even 4 1 765.2.a.i 2
15.e even 4 1 3825.2.a.p 2
20.e even 4 1 1360.2.a.o 2
20.e even 4 1 6800.2.a.ba 2
35.f even 4 1 4165.2.a.q 2
40.i odd 4 1 5440.2.a.bm 2
40.k even 4 1 5440.2.a.ba 2
85.f odd 4 1 1445.2.d.f 4
85.g odd 4 1 1445.2.a.f 2
85.g odd 4 1 7225.2.a.o 2
85.i odd 4 1 1445.2.d.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
85.2.a.b 2 5.c odd 4 1
425.2.a.f 2 5.c odd 4 1
425.2.b.e 4 1.a even 1 1 trivial
425.2.b.e 4 5.b even 2 1 inner
765.2.a.i 2 15.e even 4 1
1360.2.a.o 2 20.e even 4 1
1445.2.a.f 2 85.g odd 4 1
1445.2.d.f 4 85.f odd 4 1
1445.2.d.f 4 85.i odd 4 1
3825.2.a.p 2 15.e even 4 1
4165.2.a.q 2 35.f even 4 1
5440.2.a.ba 2 40.k even 4 1
5440.2.a.bm 2 40.i odd 4 1
6800.2.a.ba 2 20.e even 4 1
7225.2.a.o 2 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 \(S_{2}^{\mathrm{new}}(425, [\chi])\):

\( T_{2}^{4} + 6T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{4} + 12T_{3}^{2} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 6T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} + 12T^{2} + 4 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 12T^{2} + 4 \) Copy content Toggle raw display
$11$ \( (T^{2} + 8 T + 14)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 12T^{2} + 4 \) Copy content Toggle raw display
$29$ \( (T^{2} - 4 T - 4)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 152T^{2} + 4624 \) Copy content Toggle raw display
$41$ \( (T^{2} - 4 T - 68)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 72T^{2} + 784 \) Copy content Toggle raw display
$47$ \( T^{4} + 24T^{2} + 16 \) Copy content Toggle raw display
$53$ \( T^{4} + 136T^{2} + 16 \) Copy content Toggle raw display
$59$ \( (T^{2} - 24 T + 136)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 4 T - 28)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 88T^{2} + 784 \) Copy content Toggle raw display
$71$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 24T^{2} + 16 \) Copy content Toggle raw display
$79$ \( (T^{2} + 8 T + 14)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 264 T^{2} + 15376 \) Copy content Toggle raw display
$89$ \( (T^{2} - 16 T + 32)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 72T^{2} + 784 \) Copy content Toggle raw display
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