Properties

Label 245.2.b.c
Level $245$
Weight $2$
Character orbit 245.b
Analytic conductor $1.956$
Analytic rank $0$
Dimension $2$
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(99,245)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(245, 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("245.99");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 245.b (of order \(2\), degree \(1\), 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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) 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: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{2} + i q^{3} + q^{4} + ( - 2 i + 1) q^{5} - q^{6} + 3 i q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + i q^{2} + i q^{3} + q^{4} + ( - 2 i + 1) q^{5} - q^{6} + 3 i q^{8} + 2 q^{9} + (i + 2) q^{10} + i q^{12} - 2 i q^{13} + (i + 2) q^{15} - q^{16} + 2 i q^{17} + 2 i q^{18} - 6 q^{19} + ( - 2 i + 1) q^{20} + 3 i q^{23} - 3 q^{24} + ( - 4 i - 3) q^{25} + 2 q^{26} + 5 i q^{27} - 7 q^{29} + (2 i - 1) q^{30} + 2 q^{31} + 5 i q^{32} - 2 q^{34} + 2 q^{36} - 8 i q^{37} - 6 i q^{38} + 2 q^{39} + (3 i + 6) q^{40} + 5 q^{41} - 7 i q^{43} + ( - 4 i + 2) q^{45} - 3 q^{46} - i q^{48} + ( - 3 i + 4) q^{50} - 2 q^{51} - 2 i q^{52} - 6 i q^{53} - 5 q^{54} - 6 i q^{57} - 7 i q^{58} - 10 q^{59} + (i + 2) q^{60} + 7 q^{61} + 2 i q^{62} - 7 q^{64} + ( - 2 i - 4) q^{65} - 5 i q^{67} + 2 i q^{68} - 3 q^{69} - 2 q^{71} + 6 i q^{72} + 6 i q^{73} + 8 q^{74} + ( - 3 i + 4) q^{75} - 6 q^{76} + 2 i q^{78} + 2 q^{79} + (2 i - 1) q^{80} + q^{81} + 5 i q^{82} + 11 i q^{83} + (2 i + 4) q^{85} + 7 q^{86} - 7 i q^{87} - 9 q^{89} + (2 i + 4) q^{90} + 3 i q^{92} + 2 i q^{93} + (12 i - 6) q^{95} - 5 q^{96} - 16 i q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} + 2 q^{5} - 2 q^{6} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} + 2 q^{5} - 2 q^{6} + 4 q^{9} + 4 q^{10} + 4 q^{15} - 2 q^{16} - 12 q^{19} + 2 q^{20} - 6 q^{24} - 6 q^{25} + 4 q^{26} - 14 q^{29} - 2 q^{30} + 4 q^{31} - 4 q^{34} + 4 q^{36} + 4 q^{39} + 12 q^{40} + 10 q^{41} + 4 q^{45} - 6 q^{46} + 8 q^{50} - 4 q^{51} - 10 q^{54} - 20 q^{59} + 4 q^{60} + 14 q^{61} - 14 q^{64} - 8 q^{65} - 6 q^{69} - 4 q^{71} + 16 q^{74} + 8 q^{75} - 12 q^{76} + 4 q^{79} - 2 q^{80} + 2 q^{81} + 8 q^{85} + 14 q^{86} - 18 q^{89} + 8 q^{90} - 12 q^{95} - 10 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(197\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
1.00000i
1.00000i
1.00000i 1.00000i 1.00000 1.00000 + 2.00000i −1.00000 0 3.00000i 2.00000 2.00000 1.00000i
99.2 1.00000i 1.00000i 1.00000 1.00000 2.00000i −1.00000 0 3.00000i 2.00000 2.00000 + 1.00000i
\(n\): 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 245.2.b.c 2
3.b odd 2 1 2205.2.d.d 2
5.b even 2 1 inner 245.2.b.c 2
5.c odd 4 1 1225.2.a.d 1
5.c odd 4 1 1225.2.a.f 1
7.b odd 2 1 245.2.b.b 2
7.c even 3 2 35.2.j.a 4
7.d odd 6 2 245.2.j.c 4
15.d odd 2 1 2205.2.d.d 2
21.c even 2 1 2205.2.d.e 2
21.h odd 6 2 315.2.bf.a 4
28.g odd 6 2 560.2.bw.b 4
35.c odd 2 1 245.2.b.b 2
35.f even 4 1 1225.2.a.b 1
35.f even 4 1 1225.2.a.g 1
35.i odd 6 2 245.2.j.c 4
35.j even 6 2 35.2.j.a 4
35.l odd 12 2 175.2.e.a 2
35.l odd 12 2 175.2.e.b 2
105.g even 2 1 2205.2.d.e 2
105.o odd 6 2 315.2.bf.a 4
140.p odd 6 2 560.2.bw.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.j.a 4 7.c even 3 2
35.2.j.a 4 35.j even 6 2
175.2.e.a 2 35.l odd 12 2
175.2.e.b 2 35.l odd 12 2
245.2.b.b 2 7.b odd 2 1
245.2.b.b 2 35.c odd 2 1
245.2.b.c 2 1.a even 1 1 trivial
245.2.b.c 2 5.b even 2 1 inner
245.2.j.c 4 7.d odd 6 2
245.2.j.c 4 35.i odd 6 2
315.2.bf.a 4 21.h odd 6 2
315.2.bf.a 4 105.o odd 6 2
560.2.bw.b 4 28.g odd 6 2
560.2.bw.b 4 140.p odd 6 2
1225.2.a.b 1 35.f even 4 1
1225.2.a.d 1 5.c odd 4 1
1225.2.a.f 1 5.c odd 4 1
1225.2.a.g 1 35.f even 4 1
2205.2.d.d 2 3.b odd 2 1
2205.2.d.d 2 15.d odd 2 1
2205.2.d.e 2 21.c even 2 1
2205.2.d.e 2 105.g even 2 1

Hecke kernels

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

\( T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{19} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

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