Properties

Label 245.2.e.f
Level $245$
Weight $2$
Character orbit 245.e
Analytic conductor $1.956$
Analytic rank $0$
Dimension $4$
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] = [245,2,Mod(116,245)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(245, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("245.116");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 245.e (of order \(3\), degree \(2\), 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: \(1.95633484952\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{3} + ( - \beta_{2} - 1) q^{5} + (\beta_{3} + 2) q^{6} - 2 \beta_{3} q^{8} + 2 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{3} + ( - \beta_{2} - 1) q^{5} + (\beta_{3} + 2) q^{6} - 2 \beta_{3} q^{8} + 2 \beta_1 q^{9} + ( - \beta_{3} - \beta_1) q^{10} + (2 \beta_{3} - 3 \beta_{2} + 2 \beta_1) q^{11} + ( - \beta_{3} + 3) q^{13} + (\beta_{3} + 1) q^{15} + (4 \beta_{2} + 4) q^{16} + (3 \beta_{3} - \beta_{2} + 3 \beta_1) q^{17} + 4 \beta_{2} q^{18} + ( - 6 \beta_{2} - 6) q^{19} + ( - 3 \beta_{3} - 4) q^{22} + ( - 6 \beta_{2} - \beta_1 - 6) q^{23} + (2 \beta_{3} - 4 \beta_{2} + 2 \beta_1) q^{24} + \beta_{2} q^{25} + (2 \beta_{2} + 3 \beta_1 + 2) q^{26} + ( - \beta_{3} + 1) q^{27} + (4 \beta_{3} - 3) q^{29} + ( - 2 \beta_{2} + \beta_1 - 2) q^{30} + (3 \beta_{3} + 6 \beta_{2} + 3 \beta_1) q^{31} + (7 \beta_{2} - 5 \beta_1 + 7) q^{33} + ( - \beta_{3} - 6) q^{34} + (2 \beta_{2} - 3 \beta_1 + 2) q^{37} + ( - 6 \beta_{3} - 6 \beta_1) q^{38} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{39} - 2 \beta_1 q^{40} + (3 \beta_{3} - 2) q^{41} + 2 q^{43} + ( - 2 \beta_{3} - 2 \beta_1) q^{45} + ( - 6 \beta_{3} - 2 \beta_{2} - 6 \beta_1) q^{46} + (3 \beta_{2} + 3 \beta_1 + 3) q^{47} + ( - 4 \beta_{3} - 4) q^{48} + \beta_{3} q^{50} + (7 \beta_{2} - 4 \beta_1 + 7) q^{51} + (3 \beta_{3} + 3 \beta_1) q^{53} + (2 \beta_{2} + \beta_1 + 2) q^{54} + ( - 2 \beta_{3} - 3) q^{55} + (6 \beta_{3} + 6) q^{57} + ( - 8 \beta_{2} - 3 \beta_1 - 8) q^{58} + ( - 3 \beta_{3} + 2 \beta_{2} - 3 \beta_1) q^{59} + 2 \beta_1 q^{61} + (6 \beta_{3} - 6) q^{62} + 8 q^{64} + ( - 3 \beta_{2} - \beta_1 - 3) q^{65} + (7 \beta_{3} - 10 \beta_{2} + 7 \beta_1) q^{66} + ( - 3 \beta_{3} - 4 \beta_{2} - 3 \beta_1) q^{67} + (5 \beta_{3} + 4) q^{69} + ( - 2 \beta_{3} - 6) q^{71} + (8 \beta_{2} + 8) q^{72} + ( - 6 \beta_{3} - 6 \beta_1) q^{73} + (2 \beta_{3} - 6 \beta_{2} + 2 \beta_1) q^{74} + ( - \beta_{2} + \beta_1 - 1) q^{75} + (\beta_{3} + 4) q^{78} + (7 \beta_{2} - 6 \beta_1 + 7) q^{79} - 4 \beta_{2} q^{80} + (6 \beta_{3} - \beta_{2} + 6 \beta_1) q^{81} + ( - 6 \beta_{2} - 2 \beta_1 - 6) q^{82} + ( - 3 \beta_{3} - 1) q^{85} + 2 \beta_1 q^{86} + ( - \beta_{3} + 5 \beta_{2} - \beta_1) q^{87} + ( - 6 \beta_{3} + 8 \beta_{2} - 6 \beta_1) q^{88} + (8 \beta_{2} + 8) q^{89} + 4 q^{90} + 3 \beta_1 q^{93} + (3 \beta_{3} + 6 \beta_{2} + 3 \beta_1) q^{94} + 6 \beta_{2} q^{95} + ( - 3 \beta_{3} + 9) q^{97} + ( - 6 \beta_{3} - 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} - 2 q^{5} + 8 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} - 2 q^{5} + 8 q^{6} + 6 q^{11} + 12 q^{13} + 4 q^{15} + 8 q^{16} + 2 q^{17} - 8 q^{18} - 12 q^{19} - 16 q^{22} - 12 q^{23} + 8 q^{24} - 2 q^{25} + 4 q^{26} + 4 q^{27} - 12 q^{29} - 4 q^{30} - 12 q^{31} + 14 q^{33} - 24 q^{34} + 4 q^{37} - 2 q^{39} - 8 q^{41} + 8 q^{43} + 4 q^{46} + 6 q^{47} - 16 q^{48} + 14 q^{51} + 4 q^{54} - 12 q^{55} + 24 q^{57} - 16 q^{58} - 4 q^{59} - 24 q^{62} + 32 q^{64} - 6 q^{65} + 20 q^{66} + 8 q^{67} + 16 q^{69} - 24 q^{71} + 16 q^{72} + 12 q^{74} - 2 q^{75} + 16 q^{78} + 14 q^{79} + 8 q^{80} + 2 q^{81} - 12 q^{82} - 4 q^{85} - 10 q^{87} - 16 q^{88} + 16 q^{89} + 16 q^{90} - 12 q^{94} - 12 q^{95} + 36 q^{97} - 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/245\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\)
\(\chi(n)\) \(-1 - \beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
116.1
−0.707107 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
0.707107 1.22474i
−0.707107 1.22474i −1.20711 + 2.09077i 0 −0.500000 0.866025i 3.41421 0 −2.82843 −1.41421 2.44949i −0.707107 + 1.22474i
116.2 0.707107 + 1.22474i 0.207107 0.358719i 0 −0.500000 0.866025i 0.585786 0 2.82843 1.41421 + 2.44949i 0.707107 1.22474i
226.1 −0.707107 + 1.22474i −1.20711 2.09077i 0 −0.500000 + 0.866025i 3.41421 0 −2.82843 −1.41421 + 2.44949i −0.707107 1.22474i
226.2 0.707107 1.22474i 0.207107 + 0.358719i 0 −0.500000 + 0.866025i 0.585786 0 2.82843 1.41421 2.44949i 0.707107 + 1.22474i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.2.e.f 4
7.b odd 2 1 245.2.e.g 4
7.c even 3 1 245.2.a.f yes 2
7.c even 3 1 inner 245.2.e.f 4
7.d odd 6 1 245.2.a.e 2
7.d odd 6 1 245.2.e.g 4
21.g even 6 1 2205.2.a.v 2
21.h odd 6 1 2205.2.a.t 2
28.f even 6 1 3920.2.a.bw 2
28.g odd 6 1 3920.2.a.br 2
35.i odd 6 1 1225.2.a.r 2
35.j even 6 1 1225.2.a.p 2
35.k even 12 2 1225.2.b.i 4
35.l odd 12 2 1225.2.b.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.2.a.e 2 7.d odd 6 1
245.2.a.f yes 2 7.c even 3 1
245.2.e.f 4 1.a even 1 1 trivial
245.2.e.f 4 7.c even 3 1 inner
245.2.e.g 4 7.b odd 2 1
245.2.e.g 4 7.d odd 6 1
1225.2.a.p 2 35.j even 6 1
1225.2.a.r 2 35.i odd 6 1
1225.2.b.i 4 35.k even 12 2
1225.2.b.j 4 35.l odd 12 2
2205.2.a.t 2 21.h odd 6 1
2205.2.a.v 2 21.g even 6 1
3920.2.a.br 2 28.g odd 6 1
3920.2.a.bw 2 28.f even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(245, [\chi])\):

\( T_{2}^{4} + 2T_{2}^{2} + 4 \) Copy content Toggle raw display
\( T_{3}^{4} + 2T_{3}^{3} + 5T_{3}^{2} - 2T_{3} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 2T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} + 5 T^{2} - 2 T + 1 \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 6 T^{3} + 35 T^{2} - 6 T + 1 \) Copy content Toggle raw display
$13$ \( (T^{2} - 6 T + 7)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 2 T^{3} + 21 T^{2} + 34 T + 289 \) Copy content Toggle raw display
$19$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 12 T^{3} + 110 T^{2} + \cdots + 1156 \) Copy content Toggle raw display
$29$ \( (T^{2} + 6 T - 23)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 12 T^{3} + 126 T^{2} + \cdots + 324 \) Copy content Toggle raw display
$37$ \( T^{4} - 4 T^{3} + 30 T^{2} + 56 T + 196 \) Copy content Toggle raw display
$41$ \( (T^{2} + 4 T - 14)^{2} \) Copy content Toggle raw display
$43$ \( (T - 2)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} - 6 T^{3} + 45 T^{2} + 54 T + 81 \) Copy content Toggle raw display
$53$ \( T^{4} + 18T^{2} + 324 \) Copy content Toggle raw display
$59$ \( T^{4} + 4 T^{3} + 30 T^{2} - 56 T + 196 \) Copy content Toggle raw display
$61$ \( T^{4} + 8T^{2} + 64 \) Copy content Toggle raw display
$67$ \( T^{4} - 8 T^{3} + 66 T^{2} + 16 T + 4 \) Copy content Toggle raw display
$71$ \( (T^{2} + 12 T + 28)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 72T^{2} + 5184 \) Copy content Toggle raw display
$79$ \( T^{4} - 14 T^{3} + 219 T^{2} + \cdots + 529 \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} - 8 T + 64)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 18 T + 63)^{2} \) Copy content Toggle raw display
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