Properties

Label 245.3.h.a
Level $245$
Weight $3$
Character orbit 245.h
Analytic conductor $6.676$
Analytic rank $0$
Dimension $4$
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] = [245,3,Mod(31,245)]
 
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, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("245.31");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 245.h (of order \(6\), 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: \(6.67576647683\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - 2 \beta_{2} q^{2} + \beta_1 q^{3} + ( - \beta_{3} + \beta_1) q^{5} - 2 \beta_{3} q^{6} - 8 q^{8} - 4 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta_{2} q^{2} + \beta_1 q^{3} + ( - \beta_{3} + \beta_1) q^{5} - 2 \beta_{3} q^{6} - 8 q^{8} - 4 \beta_{2} q^{9} - 2 \beta_1 q^{10} + ( - \beta_{2} + 1) q^{11} - 9 \beta_{3} q^{13} + 5 q^{15} + 16 \beta_{2} q^{16} + 3 \beta_1 q^{17} + (8 \beta_{2} - 8) q^{18} + ( - 6 \beta_{3} + 6 \beta_1) q^{19} - 2 q^{22} - 8 \beta_{2} q^{23} - 8 \beta_1 q^{24} + ( - 5 \beta_{2} + 5) q^{25} + (18 \beta_{3} - 18 \beta_1) q^{26} - 13 \beta_{3} q^{27} + 41 q^{29} - 10 \beta_{2} q^{30} - 18 \beta_1 q^{31} + ( - \beta_{3} + \beta_1) q^{33} - 6 \beta_{3} q^{34} + 28 \beta_{2} q^{37} - 12 \beta_1 q^{38} + ( - 45 \beta_{2} + 45) q^{39} + (8 \beta_{3} - 8 \beta_1) q^{40} + 6 \beta_{3} q^{41} - 82 q^{43} - 4 \beta_1 q^{45} + (16 \beta_{2} - 16) q^{46} + ( - 9 \beta_{3} + 9 \beta_1) q^{47} + 16 \beta_{3} q^{48} - 10 q^{50} + 15 \beta_{2} q^{51} + (74 \beta_{2} - 74) q^{53} + (26 \beta_{3} - 26 \beta_1) q^{54} - \beta_{3} q^{55} + 30 q^{57} - 82 \beta_{2} q^{58} + 42 \beta_1 q^{59} + ( - 36 \beta_{3} + 36 \beta_1) q^{61} + 36 \beta_{3} q^{62} + 64 q^{64} - 45 \beta_{2} q^{65} - 2 \beta_1 q^{66} + (2 \beta_{2} - 2) q^{67} - 8 \beta_{3} q^{69} + 14 q^{71} + 32 \beta_{2} q^{72} + 30 \beta_1 q^{73} + ( - 56 \beta_{2} + 56) q^{74} + ( - 5 \beta_{3} + 5 \beta_1) q^{75} - 90 q^{78} + 19 \beta_{2} q^{79} + 16 \beta_1 q^{80} + ( - 29 \beta_{2} + 29) q^{81} + ( - 12 \beta_{3} + 12 \beta_1) q^{82} + 42 \beta_{3} q^{83} + 15 q^{85} + 164 \beta_{2} q^{86} + 41 \beta_1 q^{87} + (8 \beta_{2} - 8) q^{88} + ( - 48 \beta_{3} + 48 \beta_1) q^{89} + 8 \beta_{3} q^{90} - 90 \beta_{2} q^{93} - 18 \beta_1 q^{94} + ( - 30 \beta_{2} + 30) q^{95} - 27 \beta_{3} q^{97} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 32 q^{8} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} - 32 q^{8} - 8 q^{9} + 2 q^{11} + 20 q^{15} + 32 q^{16} - 16 q^{18} - 8 q^{22} - 16 q^{23} + 10 q^{25} + 164 q^{29} - 20 q^{30} + 56 q^{37} + 90 q^{39} - 328 q^{43} - 32 q^{46} - 40 q^{50} + 30 q^{51} - 148 q^{53} + 120 q^{57} - 164 q^{58} + 256 q^{64} - 90 q^{65} - 4 q^{67} + 56 q^{71} + 64 q^{72} + 112 q^{74} - 360 q^{78} + 38 q^{79} + 58 q^{81} + 60 q^{85} + 328 q^{86} - 16 q^{88} - 180 q^{93} + 60 q^{95} - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 5x^{2} + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{3} \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
−1.93649 1.11803i
1.93649 + 1.11803i
−1.93649 + 1.11803i
1.93649 1.11803i
−1.00000 1.73205i −1.93649 1.11803i 0 −1.93649 + 1.11803i 4.47214i 0 −8.00000 −2.00000 3.46410i 3.87298 + 2.23607i
31.2 −1.00000 1.73205i 1.93649 + 1.11803i 0 1.93649 1.11803i 4.47214i 0 −8.00000 −2.00000 3.46410i −3.87298 2.23607i
166.1 −1.00000 + 1.73205i −1.93649 + 1.11803i 0 −1.93649 1.11803i 4.47214i 0 −8.00000 −2.00000 + 3.46410i 3.87298 2.23607i
166.2 −1.00000 + 1.73205i 1.93649 1.11803i 0 1.93649 + 1.11803i 4.47214i 0 −8.00000 −2.00000 + 3.46410i −3.87298 + 2.23607i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.3.h.a 4
7.b odd 2 1 inner 245.3.h.a 4
7.c even 3 1 35.3.d.b 2
7.c even 3 1 inner 245.3.h.a 4
7.d odd 6 1 35.3.d.b 2
7.d odd 6 1 inner 245.3.h.a 4
21.g even 6 1 315.3.h.a 2
21.h odd 6 1 315.3.h.a 2
28.f even 6 1 560.3.f.b 2
28.g odd 6 1 560.3.f.b 2
35.i odd 6 1 175.3.d.c 2
35.j even 6 1 175.3.d.c 2
35.k even 12 2 175.3.c.c 4
35.l odd 12 2 175.3.c.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.3.d.b 2 7.c even 3 1
35.3.d.b 2 7.d odd 6 1
175.3.c.c 4 35.k even 12 2
175.3.c.c 4 35.l odd 12 2
175.3.d.c 2 35.i odd 6 1
175.3.d.c 2 35.j even 6 1
245.3.h.a 4 1.a even 1 1 trivial
245.3.h.a 4 7.b odd 2 1 inner
245.3.h.a 4 7.c even 3 1 inner
245.3.h.a 4 7.d odd 6 1 inner
315.3.h.a 2 21.g even 6 1
315.3.h.a 2 21.h odd 6 1
560.3.f.b 2 28.f even 6 1
560.3.f.b 2 28.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 2T_{2} + 4 \) acting on \(S_{3}^{\mathrm{new}}(245, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} - 5T^{2} + 25 \) Copy content Toggle raw display
$5$ \( T^{4} - 5T^{2} + 25 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 405)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 45T^{2} + 2025 \) Copy content Toggle raw display
$19$ \( T^{4} - 180 T^{2} + 32400 \) Copy content Toggle raw display
$23$ \( (T^{2} + 8 T + 64)^{2} \) Copy content Toggle raw display
$29$ \( (T - 41)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} - 1620 T^{2} + 2624400 \) Copy content Toggle raw display
$37$ \( (T^{2} - 28 T + 784)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 180)^{2} \) Copy content Toggle raw display
$43$ \( (T + 82)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} - 405 T^{2} + 164025 \) Copy content Toggle raw display
$53$ \( (T^{2} + 74 T + 5476)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 8820 T^{2} + 77792400 \) Copy content Toggle raw display
$61$ \( T^{4} - 6480 T^{2} + 41990400 \) Copy content Toggle raw display
$67$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$71$ \( (T - 14)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 4500 T^{2} + 20250000 \) Copy content Toggle raw display
$79$ \( (T^{2} - 19 T + 361)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 8820)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 11520 T^{2} + 132710400 \) Copy content Toggle raw display
$97$ \( (T^{2} + 3645)^{2} \) Copy content Toggle raw display
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