Properties

Label 525.2.b.j
Level $525$
Weight $2$
Character orbit 525.b
Analytic conductor $4.192$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [525,2,Mod(251,525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(525, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.251");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 525.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: \(4.19214610612\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 105)
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 a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_{3} q^{3} - q^{4} + (\beta_{4} + \beta_{2}) q^{6} + \beta_{7} q^{7} - \beta_1 q^{8} + (\beta_{6} + \beta_{4} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - \beta_{3} q^{3} - q^{4} + (\beta_{4} + \beta_{2}) q^{6} + \beta_{7} q^{7} - \beta_1 q^{8} + (\beta_{6} + \beta_{4} + 1) q^{9} + ( - \beta_{6} - \beta_{4}) q^{11} + \beta_{3} q^{12} + ( - 2 \beta_{5} - 2 \beta_{3}) q^{13} + ( - 2 \beta_{6} - \beta_{4}) q^{14} - 5 q^{16} + (\beta_{5} - \beta_{3}) q^{17} + (2 \beta_{7} + \beta_{5} + \beta_{3} - \beta_1) q^{18} + ( - \beta_{6} + \beta_{4} - \beta_{2} - 1) q^{21} + ( - 2 \beta_{7} - \beta_{5} - \beta_{3}) q^{22} + 2 \beta_1 q^{23} + (\beta_{4} + \beta_{2}) q^{24} + ( - 2 \beta_{6} + 2 \beta_{4}) q^{26} + ( - 3 \beta_{5} - 2 \beta_{3}) q^{27} - \beta_{7} q^{28} + ( - 2 \beta_{6} - 2 \beta_{4}) q^{29} + (2 \beta_{6} + 2 \beta_{4} + 4 \beta_{2}) q^{31} + 3 \beta_1 q^{32} + (3 \beta_{5} + \beta_{3}) q^{33} + (\beta_{6} + \beta_{4} + 2 \beta_{2}) q^{34} + ( - \beta_{6} - \beta_{4} - 1) q^{36} + (2 \beta_{6} + 2 \beta_{4} - 4) q^{39} + (\beta_{6} - \beta_{4}) q^{41} + (\beta_{7} + 5 \beta_{5} + 2 \beta_{3} + \beta_1) q^{42} + ( - 2 \beta_{7} - \beta_{5} - \beta_{3}) q^{43} + (\beta_{6} + \beta_{4}) q^{44} + 6 q^{46} + (\beta_{5} - \beta_{3}) q^{47} + 5 \beta_{3} q^{48} + ( - \beta_{6} - \beta_{4} - 2 \beta_{2} + 5) q^{49} + (\beta_{6} + \beta_{4} + 4) q^{51} + (2 \beta_{5} + 2 \beta_{3}) q^{52} + ( - 3 \beta_{6} + 2 \beta_{4} - \beta_{2}) q^{54} + ( - 2 \beta_{6} - \beta_{4}) q^{56} + ( - 4 \beta_{7} - 2 \beta_{5} - 2 \beta_{3}) q^{58} + (2 \beta_{6} - 2 \beta_{4}) q^{59} + ( - 2 \beta_{6} - 2 \beta_{4} - 4 \beta_{2}) q^{61} + ( - 6 \beta_{5} + 6 \beta_{3}) q^{62} + (\beta_{7} - \beta_{5} + \beta_{3} + 4 \beta_1) q^{63} - q^{64} + (3 \beta_{6} - \beta_{4} + 2 \beta_{2}) q^{66} + ( - 2 \beta_{7} - \beta_{5} - \beta_{3}) q^{67} + ( - \beta_{5} + \beta_{3}) q^{68} + ( - 2 \beta_{4} - 2 \beta_{2}) q^{69} + ( - \beta_{6} - \beta_{4}) q^{71} + (2 \beta_{7} + \beta_{5} + \beta_{3} - \beta_1) q^{72} + (4 \beta_{5} + 4 \beta_{3}) q^{73} + (\beta_{5} - \beta_{3} - 4 \beta_1) q^{77} + (4 \beta_{7} + 2 \beta_{5} + 2 \beta_{3} + 4 \beta_1) q^{78} - 8 q^{79} + (2 \beta_{6} + 2 \beta_{4} - 7) q^{81} + ( - 3 \beta_{5} - 3 \beta_{3}) q^{82} + ( - \beta_{5} + \beta_{3}) q^{83} + (\beta_{6} - \beta_{4} + \beta_{2} + 1) q^{84} + (3 \beta_{6} + 3 \beta_{4}) q^{86} + (6 \beta_{5} + 2 \beta_{3}) q^{87} + ( - 2 \beta_{7} - \beta_{5} - \beta_{3}) q^{88} + (3 \beta_{6} - 3 \beta_{4}) q^{89} + ( - 2 \beta_{6} - 2 \beta_{4} - 4 \beta_{2} - 4) q^{91} - 2 \beta_1 q^{92} + (4 \beta_{7} + 2 \beta_{5} + 2 \beta_{3} - 8 \beta_1) q^{93} + (\beta_{6} + \beta_{4} + 2 \beta_{2}) q^{94} + ( - 3 \beta_{4} - 3 \beta_{2}) q^{96} + ( - 4 \beta_{5} - 4 \beta_{3}) q^{97} + (3 \beta_{5} - 3 \beta_{3} - 5 \beta_1) q^{98} + ( - \beta_{6} - \beta_{4} + 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} + 8 q^{9} - 40 q^{16} - 8 q^{21} - 8 q^{36} - 32 q^{39} + 48 q^{46} + 40 q^{49} + 32 q^{51} - 8 q^{64} - 64 q^{79} - 56 q^{81} + 8 q^{84} - 32 q^{91} + 64 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{24}^{4} - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{24}^{7} - 2\zeta_{24}^{3} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{24}^{6} - \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{24}^{6} + \zeta_{24}^{5} + \zeta_{24}^{3} - 2\zeta_{24}^{2} - \zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \zeta_{24}^{6} + \zeta_{24}^{5} - \zeta_{24}^{3} - \zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{24}^{6} + \zeta_{24}^{5} + \zeta_{24}^{3} + 2\zeta_{24}^{2} - \zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -2\zeta_{24}^{7} - \zeta_{24}^{6} + \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( \beta_{7} + \beta_{3} + \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( \beta_{5} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -\beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{2} ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
0.965926 + 0.258819i
−0.258819 + 0.965926i
−0.965926 0.258819i
0.258819 0.965926i
−0.258819 0.965926i
0.965926 0.258819i
0.258819 + 0.965926i
−0.965926 + 0.258819i
1.73205i −1.41421 1.00000i −1.00000 0 −1.73205 + 2.44949i 2.44949 1.00000i 1.73205i 1.00000 + 2.82843i 0
251.2 1.73205i −1.41421 + 1.00000i −1.00000 0 1.73205 + 2.44949i −2.44949 + 1.00000i 1.73205i 1.00000 2.82843i 0
251.3 1.73205i 1.41421 1.00000i −1.00000 0 −1.73205 2.44949i −2.44949 1.00000i 1.73205i 1.00000 2.82843i 0
251.4 1.73205i 1.41421 + 1.00000i −1.00000 0 1.73205 2.44949i 2.44949 + 1.00000i 1.73205i 1.00000 + 2.82843i 0
251.5 1.73205i −1.41421 1.00000i −1.00000 0 1.73205 2.44949i −2.44949 1.00000i 1.73205i 1.00000 + 2.82843i 0
251.6 1.73205i −1.41421 + 1.00000i −1.00000 0 −1.73205 2.44949i 2.44949 + 1.00000i 1.73205i 1.00000 2.82843i 0
251.7 1.73205i 1.41421 1.00000i −1.00000 0 1.73205 + 2.44949i 2.44949 1.00000i 1.73205i 1.00000 2.82843i 0
251.8 1.73205i 1.41421 + 1.00000i −1.00000 0 −1.73205 + 2.44949i −2.44949 + 1.00000i 1.73205i 1.00000 + 2.82843i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 251.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
7.b odd 2 1 inner
15.d odd 2 1 inner
21.c even 2 1 inner
35.c odd 2 1 inner
105.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.b.j 8
3.b odd 2 1 inner 525.2.b.j 8
5.b even 2 1 inner 525.2.b.j 8
5.c odd 4 1 105.2.g.a 4
5.c odd 4 1 105.2.g.c yes 4
7.b odd 2 1 inner 525.2.b.j 8
15.d odd 2 1 inner 525.2.b.j 8
15.e even 4 1 105.2.g.a 4
15.e even 4 1 105.2.g.c yes 4
20.e even 4 1 1680.2.k.a 4
20.e even 4 1 1680.2.k.c 4
21.c even 2 1 inner 525.2.b.j 8
35.c odd 2 1 inner 525.2.b.j 8
35.f even 4 1 105.2.g.a 4
35.f even 4 1 105.2.g.c yes 4
35.k even 12 2 735.2.p.a 8
35.k even 12 2 735.2.p.c 8
35.l odd 12 2 735.2.p.a 8
35.l odd 12 2 735.2.p.c 8
60.l odd 4 1 1680.2.k.a 4
60.l odd 4 1 1680.2.k.c 4
105.g even 2 1 inner 525.2.b.j 8
105.k odd 4 1 105.2.g.a 4
105.k odd 4 1 105.2.g.c yes 4
105.w odd 12 2 735.2.p.a 8
105.w odd 12 2 735.2.p.c 8
105.x even 12 2 735.2.p.a 8
105.x even 12 2 735.2.p.c 8
140.j odd 4 1 1680.2.k.a 4
140.j odd 4 1 1680.2.k.c 4
420.w even 4 1 1680.2.k.a 4
420.w even 4 1 1680.2.k.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.g.a 4 5.c odd 4 1
105.2.g.a 4 15.e even 4 1
105.2.g.a 4 35.f even 4 1
105.2.g.a 4 105.k odd 4 1
105.2.g.c yes 4 5.c odd 4 1
105.2.g.c yes 4 15.e even 4 1
105.2.g.c yes 4 35.f even 4 1
105.2.g.c yes 4 105.k odd 4 1
525.2.b.j 8 1.a even 1 1 trivial
525.2.b.j 8 3.b odd 2 1 inner
525.2.b.j 8 5.b even 2 1 inner
525.2.b.j 8 7.b odd 2 1 inner
525.2.b.j 8 15.d odd 2 1 inner
525.2.b.j 8 21.c even 2 1 inner
525.2.b.j 8 35.c odd 2 1 inner
525.2.b.j 8 105.g even 2 1 inner
735.2.p.a 8 35.k even 12 2
735.2.p.a 8 35.l odd 12 2
735.2.p.a 8 105.w odd 12 2
735.2.p.a 8 105.x even 12 2
735.2.p.c 8 35.k even 12 2
735.2.p.c 8 35.l odd 12 2
735.2.p.c 8 105.w odd 12 2
735.2.p.c 8 105.x even 12 2
1680.2.k.a 4 20.e even 4 1
1680.2.k.a 4 60.l odd 4 1
1680.2.k.a 4 140.j odd 4 1
1680.2.k.a 4 420.w even 4 1
1680.2.k.c 4 20.e even 4 1
1680.2.k.c 4 60.l odd 4 1
1680.2.k.c 4 140.j odd 4 1
1680.2.k.c 4 420.w even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(525, [\chi])\):

\( T_{2}^{2} + 3 \) Copy content Toggle raw display
\( T_{11}^{2} + 8 \) Copy content Toggle raw display
\( T_{17}^{2} - 8 \) Copy content Toggle raw display
\( T_{37} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 3)^{4} \) Copy content Toggle raw display
$3$ \( (T^{4} - 2 T^{2} + 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} - 10 T^{2} + 49)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 8)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 16)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 8)^{4} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( (T^{2} + 12)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 32)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 96)^{4} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( (T^{2} - 12)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 24)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} - 8)^{4} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( (T^{2} - 48)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 96)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 24)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 8)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 64)^{4} \) Copy content Toggle raw display
$79$ \( (T + 8)^{8} \) Copy content Toggle raw display
$83$ \( (T^{2} - 8)^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} - 108)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 64)^{4} \) Copy content Toggle raw display
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