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Properties

Label 245.4.e.h
Level $245$
Weight $4$
Character orbit 245.e
Analytic conductor $14.455$
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] = [245,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("245.116");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
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: \(14.4554679514\)
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: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + 4 \beta_{2} - \beta_1) q^{2} + ( - \beta_{2} - 4 \beta_1 - 1) q^{3} + ( - 10 \beta_{2} + 8 \beta_1 - 10) q^{4} - 5 \beta_{2} q^{5} + ( - 15 \beta_{3} - 4) q^{6} + (34 \beta_{3} + 24) q^{8}+ \cdots + ( - 136 \beta_{3} + 470) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{2} - 2 q^{3} - 20 q^{4} + 10 q^{5} - 16 q^{6} + 96 q^{8} - 12 q^{9} + 40 q^{10} + 14 q^{11} + 108 q^{12} + 100 q^{13} - 20 q^{15} - 168 q^{16} + 50 q^{17} - 16 q^{18} - 36 q^{19} - 200 q^{20}+ \cdots + 1880 q^{99}+O(q^{100}) \) 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

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

\(\nu\)\(=\) \( \beta_1 \) 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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
116.1
−0.707107 + 1.22474i
0.707107 − 1.22474i
−0.707107 − 1.22474i
0.707107 + 1.22474i
−2.70711 − 4.68885i 2.32843 − 4.03295i −10.6569 + 18.4582i 2.50000 + 4.33013i −25.2132 0 72.0833 2.65685 + 4.60181i 13.5355 − 23.4442i
116.2 −1.29289 − 2.23936i −3.32843 + 5.76500i 0.656854 − 1.13770i 2.50000 + 4.33013i 17.2132 0 −24.0833 −8.65685 − 14.9941i 6.46447 − 11.1968i
226.1 −2.70711 + 4.68885i 2.32843 + 4.03295i −10.6569 − 18.4582i 2.50000 − 4.33013i −25.2132 0 72.0833 2.65685 − 4.60181i 13.5355 + 23.4442i
226.2 −1.29289 + 2.23936i −3.32843 − 5.76500i 0.656854 + 1.13770i 2.50000 − 4.33013i 17.2132 0 −24.0833 −8.65685 + 14.9941i 6.46447 + 11.1968i
\(n\): 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 245.4.e.h 4
7.b odd 2 1 245.4.e.i 4
7.c even 3 1 35.4.a.b ✓ 2
7.c even 3 1 inner 245.4.e.h 4
7.d odd 6 1 245.4.a.k 2
7.d odd 6 1 245.4.e.i 4
21.g even 6 1 2205.4.a.u 2
21.h odd 6 1 315.4.a.f 2
28.g odd 6 1 560.4.a.r 2
35.i odd 6 1 1225.4.a.m 2
35.j even 6 1 175.4.a.c 2
35.l odd 12 2 175.4.b.c 4
56.k odd 6 1 2240.4.a.bo 2
56.p even 6 1 2240.4.a.bn 2
105.o odd 6 1 1575.4.a.z 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.a.b ✓ 2 7.c even 3 1
175.4.a.c 2 35.j even 6 1
175.4.b.c 4 35.l odd 12 2
245.4.a.k 2 7.d odd 6 1
245.4.e.h 4 1.a even 1 1 trivial
245.4.e.h 4 7.c even 3 1 inner
245.4.e.i 4 7.b odd 2 1
245.4.e.i 4 7.d odd 6 1
315.4.a.f 2 21.h odd 6 1
560.4.a.r 2 28.g odd 6 1
1225.4.a.m 2 35.i odd 6 1
1575.4.a.z 2 105.o odd 6 1
2205.4.a.u 2 21.g even 6 1
2240.4.a.bn 2 56.p even 6 1
2240.4.a.bo 2 56.k odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(245, [\chi])\):

\( T_{2}^{4} + 8T_{2}^{3} + 50T_{2}^{2} + 112T_{2} + 196 \) Copy content Toggle raw display
\( T_{3}^{4} + 2T_{3}^{3} + 35T_{3}^{2} - 62T_{3} + 961 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 8 T^{3} + \cdots + 196 \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} + \cdots + 961 \) Copy content Toggle raw display
$5$ \( (T^{2} - 5 T + 25)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 14 T^{3} + \cdots + 3996001 \) Copy content Toggle raw display
$13$ \( (T^{2} - 50 T + 593)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 50 T^{3} + \cdots + 10543009 \) Copy content Toggle raw display
$19$ \( T^{4} + 36 T^{3} + \cdots + 12588304 \) Copy content Toggle raw display
$23$ \( T^{4} + 244 T^{3} + \cdots + 31764496 \) Copy content Toggle raw display
$29$ \( (T^{2} + 26 T - 983)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 3745440000 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 5230760976 \) Copy content Toggle raw display
$41$ \( (T^{2} + 328 T - 3856)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 260 T + 7652)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 350 T^{3} + \cdots + 17833729 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 1022976256 \) Copy content Toggle raw display
$59$ \( (T^{2} - 616 T + 379456)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 336 T^{3} + \cdots + 23970816 \) Copy content Toggle raw display
$67$ \( T^{4} - 152 T^{3} + \cdots + 5837056 \) Copy content Toggle raw display
$71$ \( (T + 952)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 14988615184 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 17966989681 \) Copy content Toggle raw display
$83$ \( (T^{2} + 376 T - 684656)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 216 T^{3} + \cdots + 60715264 \) Copy content Toggle raw display
$97$ \( (T^{2} - 2742 T + 1782841)^{2} \) Copy content Toggle raw display
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