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Properties

Label 245.2.j.c
Level $245$
Weight $2$
Character orbit 245.j
Analytic conductor $1.956$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [245,2,Mod(79,245)]
 
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([3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("245.79");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 245.j (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: \(1.95633484952\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{3} - \zeta_{12}^{2} q^{4} + ( - \zeta_{12}^{2} + 2 \zeta_{12} + 1) q^{5} + q^{6} - 3 \zeta_{12}^{3} q^{8} + (2 \zeta_{12}^{2} - 2) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12} q^{2} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{3} - \zeta_{12}^{2} q^{4} + ( - \zeta_{12}^{2} + 2 \zeta_{12} + 1) q^{5} + q^{6} - 3 \zeta_{12}^{3} q^{8} + (2 \zeta_{12}^{2} - 2) q^{9} + ( - \zeta_{12}^{3} + 2 \zeta_{12}^{2} + \zeta_{12}) q^{10} - \zeta_{12} q^{12} - 2 \zeta_{12}^{3} q^{13} + ( - \zeta_{12}^{3} + 2) q^{15} + ( - \zeta_{12}^{2} + 1) q^{16} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{17} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{18} + (6 \zeta_{12}^{2} - 6) q^{19} + ( - 2 \zeta_{12}^{3} - 1) q^{20} + 3 \zeta_{12} q^{23} - 3 \zeta_{12}^{2} q^{24} + ( - 4 \zeta_{12}^{3} + \cdots + 4 \zeta_{12}) q^{25} + \cdots - 16 \zeta_{12}^{3} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{4} + 2 q^{5} + 4 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{4} + 2 q^{5} + 4 q^{6} - 4 q^{9} + 4 q^{10} + 8 q^{15} + 2 q^{16} - 12 q^{19} - 4 q^{20} - 6 q^{24} + 6 q^{25} + 4 q^{26} - 28 q^{29} + 2 q^{30} + 4 q^{31} + 8 q^{34} + 8 q^{36} - 4 q^{39} + 12 q^{40} - 20 q^{41} + 4 q^{45} + 6 q^{46} + 16 q^{50} + 4 q^{51} - 10 q^{54} - 20 q^{59} - 4 q^{60} + 14 q^{61} - 28 q^{64} + 8 q^{65} + 12 q^{69} - 8 q^{71} - 16 q^{74} + 8 q^{75} + 24 q^{76} - 4 q^{79} - 2 q^{80} - 2 q^{81} + 16 q^{85} - 14 q^{86} - 18 q^{89} - 16 q^{90} + 12 q^{95} - 10 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(101\) \(197\)
\(\chi(n)\) \(-1 + \zeta_{12}^{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
−0.866025 − 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 − 0.500000i
−0.866025 − 0.500000i −0.866025 + 0.500000i −0.500000 − 0.866025i −1.23205 − 1.86603i 1.00000 0 3.00000i −1.00000 + 1.73205i 0.133975 + 2.23205i
79.2 0.866025 + 0.500000i 0.866025 − 0.500000i −0.500000 − 0.866025i 2.23205 + 0.133975i 1.00000 0 − 3.00000i −1.00000 + 1.73205i 1.86603 + 1.23205i
214.1 −0.866025 + 0.500000i −0.866025 − 0.500000i −0.500000 + 0.866025i −1.23205 + 1.86603i 1.00000 0 − 3.00000i −1.00000 − 1.73205i 0.133975 − 2.23205i
214.2 0.866025 − 0.500000i 0.866025 + 0.500000i −0.500000 + 0.866025i 2.23205 − 0.133975i 1.00000 0 3.00000i −1.00000 − 1.73205i 1.86603 − 1.23205i
\(n\): 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
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.2.j.c 4
5.b even 2 1 inner 245.2.j.c 4
7.b odd 2 1 35.2.j.a ✓ 4
7.c even 3 1 245.2.b.b 2
7.c even 3 1 inner 245.2.j.c 4
7.d odd 6 1 35.2.j.a ✓ 4
7.d odd 6 1 245.2.b.c 2
21.c even 2 1 315.2.bf.a 4
21.g even 6 1 315.2.bf.a 4
21.g even 6 1 2205.2.d.d 2
21.h odd 6 1 2205.2.d.e 2
28.d even 2 1 560.2.bw.b 4
28.f even 6 1 560.2.bw.b 4
35.c odd 2 1 35.2.j.a ✓ 4
35.f even 4 1 175.2.e.a 2
35.f even 4 1 175.2.e.b 2
35.i odd 6 1 35.2.j.a ✓ 4
35.i odd 6 1 245.2.b.c 2
35.j even 6 1 245.2.b.b 2
35.j even 6 1 inner 245.2.j.c 4
35.k even 12 1 175.2.e.a 2
35.k even 12 1 175.2.e.b 2
35.k even 12 1 1225.2.a.d 1
35.k even 12 1 1225.2.a.f 1
35.l odd 12 1 1225.2.a.b 1
35.l odd 12 1 1225.2.a.g 1
105.g even 2 1 315.2.bf.a 4
105.o odd 6 1 2205.2.d.e 2
105.p even 6 1 315.2.bf.a 4
105.p even 6 1 2205.2.d.d 2
140.c even 2 1 560.2.bw.b 4
140.s even 6 1 560.2.bw.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.j.a ✓ 4 7.b odd 2 1
35.2.j.a ✓ 4 7.d odd 6 1
35.2.j.a ✓ 4 35.c odd 2 1
35.2.j.a ✓ 4 35.i odd 6 1
175.2.e.a 2 35.f even 4 1
175.2.e.a 2 35.k even 12 1
175.2.e.b 2 35.f even 4 1
175.2.e.b 2 35.k even 12 1
245.2.b.b 2 7.c even 3 1
245.2.b.b 2 35.j even 6 1
245.2.b.c 2 7.d odd 6 1
245.2.b.c 2 35.i odd 6 1
245.2.j.c 4 1.a even 1 1 trivial
245.2.j.c 4 5.b even 2 1 inner
245.2.j.c 4 7.c even 3 1 inner
245.2.j.c 4 35.j even 6 1 inner
315.2.bf.a 4 21.c even 2 1
315.2.bf.a 4 21.g even 6 1
315.2.bf.a 4 105.g even 2 1
315.2.bf.a 4 105.p even 6 1
560.2.bw.b 4 28.d even 2 1
560.2.bw.b 4 28.f even 6 1
560.2.bw.b 4 140.c even 2 1
560.2.bw.b 4 140.s even 6 1
1225.2.a.b 1 35.l odd 12 1
1225.2.a.d 1 35.k even 12 1
1225.2.a.f 1 35.k even 12 1
1225.2.a.g 1 35.l odd 12 1
2205.2.d.d 2 21.g even 6 1
2205.2.d.d 2 105.p even 6 1
2205.2.d.e 2 21.h odd 6 1
2205.2.d.e 2 105.o odd 6 1

Hecke kernels

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

\( T_{2}^{4} - T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{4} - T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{31}^{2} - 2T_{31} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{4} - 2 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$19$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$29$ \( (T + 7)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 64T^{2} + 4096 \) Copy content Toggle raw display
$41$ \( (T + 5)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 49)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$59$ \( (T^{2} + 10 T + 100)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 7 T + 49)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 25T^{2} + 625 \) Copy content Toggle raw display
$71$ \( (T + 2)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$79$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 121)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 9 T + 81)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 256)^{2} \) Copy content Toggle raw display
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