Properties

Label 525.2.g.e
Level $525$
Weight $2$
Character orbit 525.g
Analytic conductor $4.192$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [525,2,Mod(524,525)]
 
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, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.524");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 525.g (of order \(2\), degree \(1\), 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: \(4.19214610612\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.303595776.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
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.

\(f(q)\) \(=\) \( q - \beta_{6} q^{2} + ( - \beta_{6} - \beta_{4} - \beta_{2}) q^{3} + ( - \beta_{3} - \beta_1 + 1) q^{4} + ( - \beta_{3} + \beta_1 + 1) q^{6} + ( - \beta_{7} + \beta_{2}) q^{7} + ( - \beta_{6} + 2 \beta_{2}) q^{8} + ( - \beta_{5} + \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{2} + ( - \beta_{6} - \beta_{4} - \beta_{2}) q^{3} + ( - \beta_{3} - \beta_1 + 1) q^{4} + ( - \beta_{3} + \beta_1 + 1) q^{6} + ( - \beta_{7} + \beta_{2}) q^{7} + ( - \beta_{6} + 2 \beta_{2}) q^{8} + ( - \beta_{5} + \beta_1 - 1) q^{9} - \beta_{5} q^{11} + (2 \beta_{7} - \beta_{6} - 2 \beta_{4}) q^{12} + (3 \beta_{6} + \beta_{2}) q^{13} + ( - \beta_{5} - 2 \beta_1 + 1) q^{14} + ( - \beta_{3} - \beta_1 + 3) q^{16} + ( - \beta_{7} - \beta_{6} - 2 \beta_{4} - \beta_{2}) q^{17} + (\beta_{6} - 2 \beta_{4} - 2 \beta_{2}) q^{18} + (\beta_{5} + \beta_{3} - \beta_1) q^{19} + ( - 2 \beta_{3} - \beta_1 - 1) q^{21} - \beta_{7} q^{22} + ( - 2 \beta_{6} - 4 \beta_{2}) q^{23} + (2 \beta_{5} - 3 \beta_{3} + \beta_1 - 1) q^{24} + (2 \beta_{3} + 2 \beta_1 - 8) q^{26} + ( - 2 \beta_{7} + 2 \beta_{6} + \beta_{2}) q^{27} + ( - \beta_{7} - \beta_{6} + 4 \beta_{4} + 2 \beta_{2}) q^{28} + ( - \beta_{3} + \beta_1) q^{29} + (\beta_{5} + \beta_{3} - \beta_1) q^{31} + ( - 3 \beta_{6} - 2 \beta_{2}) q^{32} + ( - \beta_{7} - \beta_{4} + \beta_{2}) q^{33} + ( - \beta_{5} - \beta_{3} + \beta_1) q^{34} + (2 \beta_{5} + \beta_{3} + 3 \beta_1 - 5) q^{36} + ( - \beta_{7} - 2 \beta_{6} - 4 \beta_{4} - 2 \beta_{2}) q^{37} + ( - \beta_{7} + 2 \beta_{6} + 4 \beta_{4} + 2 \beta_{2}) q^{38} + (\beta_{5} + 2 \beta_{3} - 3 \beta_1 - 4) q^{39} + 6 q^{41} + (\beta_{7} - 3 \beta_{6} - 2 \beta_{4} + 2 \beta_{2}) q^{42} + ( - 2 \beta_{6} - 4 \beta_{4} - 2 \beta_{2}) q^{43} + (\beta_{5} + \beta_{3} - \beta_1) q^{44} + (2 \beta_{3} + 2 \beta_1 + 2) q^{46} + (2 \beta_{7} - \beta_{6} - 2 \beta_{4} - \beta_{2}) q^{47} + (2 \beta_{7} - 3 \beta_{6} - 4 \beta_{4} - 2 \beta_{2}) q^{48} + (2 \beta_{5} + 2 \beta_{3} - 2 \beta_1 - 1) q^{49} + ( - 2 \beta_{5} - \beta_{3} - 4) q^{51} + (6 \beta_{6} - 6 \beta_{2}) q^{52} + ( - 2 \beta_{6} + 2 \beta_{2}) q^{53} + ( - 2 \beta_{5} + 3 \beta_{3} - \beta_1 - 5) q^{54} + (\beta_{5} + 2 \beta_{3} - 4 \beta_1 + 7) q^{56} + (\beta_{7} - 2 \beta_{6} + 2 \beta_{4}) q^{57} + (2 \beta_{7} - 2 \beta_{6} - 4 \beta_{4} - 2 \beta_{2}) q^{58} + ( - 2 \beta_{3} - 2 \beta_1 - 4) q^{59} + (2 \beta_{5} + 2 \beta_{3} - 2 \beta_1) q^{61} + ( - \beta_{7} + 2 \beta_{6} + 4 \beta_{4} + 2 \beta_{2}) q^{62} + (3 \beta_{7} + 2 \beta_{6} - \beta_{4} + 2 \beta_{2}) q^{63} + (\beta_{3} + \beta_1 + 1) q^{64} + ( - \beta_{5} - \beta_{3} - \beta_1) q^{66} + (2 \beta_{6} + 4 \beta_{4} + 2 \beta_{2}) q^{67} + 3 \beta_{7} q^{68} + ( - 4 \beta_{5} + 2 \beta_{3} + 2 \beta_1 + 6) q^{69} + ( - \beta_{5} + 3 \beta_{3} - 3 \beta_1) q^{71} + (4 \beta_{7} + 5 \beta_{6} - 2 \beta_{2}) q^{72} - 4 \beta_{2} q^{73} + ( - \beta_{5} - 3 \beta_{3} + 3 \beta_1) q^{74} + ( - 3 \beta_{5} + 3 \beta_{3} - 3 \beta_1) q^{76} + (\beta_{7} + 3 \beta_{6} + 2 \beta_{4} + 3 \beta_{2}) q^{77} + ( - 4 \beta_{7} + 8 \beta_{6} + 10 \beta_{4} + 6 \beta_{2}) q^{78} + (\beta_{3} + \beta_1) q^{79} + ( - \beta_{5} - \beta_{3} - 4 \beta_1 - 3) q^{81} - 6 \beta_{6} q^{82} + ( - 4 \beta_{7} - 4 \beta_{6} - 8 \beta_{4} - 4 \beta_{2}) q^{83} + (\beta_{5} - 4 \beta_{3} + 11) q^{84} + ( - 4 \beta_{3} + 4 \beta_1) q^{86} + (2 \beta_{6} - \beta_{4} - \beta_{2}) q^{87} + (\beta_{7} + 2 \beta_{6} + 4 \beta_{4} + 2 \beta_{2}) q^{88} + (2 \beta_{3} + 2 \beta_1 + 10) q^{89} + (4 \beta_{5} + \beta_{3} + 5 \beta_1) q^{91} + (6 \beta_{6} + 4 \beta_{2}) q^{92} + (\beta_{7} - 2 \beta_{6} + 2 \beta_{4}) q^{93} + (2 \beta_{5} - 4 \beta_{3} + 4 \beta_1) q^{94} + ( - 2 \beta_{5} - \beta_{3} + 3 \beta_1 + 5) q^{96} + ( - 3 \beta_{6} - 5 \beta_{2}) q^{97} + ( - 2 \beta_{7} + 5 \beta_{6} + 8 \beta_{4} + 4 \beta_{2}) q^{98} + ( - 2 \beta_{3} - \beta_1 - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 12 q^{4} + 8 q^{6} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 12 q^{4} + 8 q^{6} - 10 q^{9} + 12 q^{14} + 28 q^{16} - 2 q^{21} - 4 q^{24} - 72 q^{26} - 48 q^{36} - 30 q^{39} + 48 q^{41} + 8 q^{46} - 8 q^{49} - 30 q^{51} - 44 q^{54} + 60 q^{56} - 24 q^{59} + 4 q^{64} + 4 q^{66} + 40 q^{69} - 4 q^{79} - 14 q^{81} + 96 q^{84} + 72 q^{89} - 12 q^{91} + 36 q^{96} - 26 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 5\nu^{6} + 16\nu^{4} - 64\nu^{2} - 63 ) / 144 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - 4\nu^{5} - 2\nu^{3} + 9\nu ) / 54 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11\nu^{6} + 64\nu^{4} + 176\nu^{2} + 351 ) / 144 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + 35\nu ) / 48 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7\nu^{6} + 8\nu^{4} + 40\nu^{2} + 99 ) / 72 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} + 5\nu^{5} + 16\nu^{3} + 18\nu ) / 27 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5\nu^{7} + 16\nu^{5} + 8\nu^{3} + 81\nu ) / 108 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + 4\beta_{4} + 2\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + \beta_{3} - 5\beta _1 - 6 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{7} + 2\beta_{6} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -11\beta_{5} + 9\beta_{3} + 11\beta _1 - 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 15\beta_{7} - 4\beta_{6} - 4\beta_{4} - 34\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 12\beta_{5} - 4\beta_{3} + 4\beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 35\beta_{7} - 52\beta_{4} + 70\beta_{2} ) / 4 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
524.1
−1.26217 + 1.18614i
−1.26217 1.18614i
−0.396143 + 1.68614i
−0.396143 1.68614i
0.396143 + 1.68614i
0.396143 1.68614i
1.26217 + 1.18614i
1.26217 1.18614i
−2.52434 −0.396143 1.68614i 4.37228 0 1.00000 + 4.25639i −1.73205 + 2.00000i −5.98844 −2.68614 + 1.33591i 0
524.2 −2.52434 −0.396143 + 1.68614i 4.37228 0 1.00000 4.25639i −1.73205 2.00000i −5.98844 −2.68614 1.33591i 0
524.3 −0.792287 −1.26217 1.18614i −1.37228 0 1.00000 + 0.939764i 1.73205 2.00000i 2.67181 0.186141 + 2.99422i 0
524.4 −0.792287 −1.26217 + 1.18614i −1.37228 0 1.00000 0.939764i 1.73205 + 2.00000i 2.67181 0.186141 2.99422i 0
524.5 0.792287 1.26217 1.18614i −1.37228 0 1.00000 0.939764i −1.73205 2.00000i −2.67181 0.186141 2.99422i 0
524.6 0.792287 1.26217 + 1.18614i −1.37228 0 1.00000 + 0.939764i −1.73205 + 2.00000i −2.67181 0.186141 + 2.99422i 0
524.7 2.52434 0.396143 1.68614i 4.37228 0 1.00000 4.25639i 1.73205 + 2.00000i 5.98844 −2.68614 1.33591i 0
524.8 2.52434 0.396143 + 1.68614i 4.37228 0 1.00000 + 4.25639i 1.73205 2.00000i 5.98844 −2.68614 + 1.33591i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 524.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
21.c even 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.g.e 8
3.b odd 2 1 525.2.g.d 8
5.b even 2 1 inner 525.2.g.e 8
5.c odd 4 1 105.2.b.c 4
5.c odd 4 1 525.2.b.g 4
7.b odd 2 1 525.2.g.d 8
15.d odd 2 1 525.2.g.d 8
15.e even 4 1 105.2.b.d yes 4
15.e even 4 1 525.2.b.e 4
20.e even 4 1 1680.2.f.h 4
21.c even 2 1 inner 525.2.g.e 8
35.c odd 2 1 525.2.g.d 8
35.f even 4 1 105.2.b.d yes 4
35.f even 4 1 525.2.b.e 4
35.k even 12 1 735.2.s.g 4
35.k even 12 1 735.2.s.j 4
35.l odd 12 1 735.2.s.h 4
35.l odd 12 1 735.2.s.i 4
60.l odd 4 1 1680.2.f.g 4
105.g even 2 1 inner 525.2.g.e 8
105.k odd 4 1 105.2.b.c 4
105.k odd 4 1 525.2.b.g 4
105.w odd 12 1 735.2.s.h 4
105.w odd 12 1 735.2.s.i 4
105.x even 12 1 735.2.s.g 4
105.x even 12 1 735.2.s.j 4
140.j odd 4 1 1680.2.f.g 4
420.w even 4 1 1680.2.f.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.b.c 4 5.c odd 4 1
105.2.b.c 4 105.k odd 4 1
105.2.b.d yes 4 15.e even 4 1
105.2.b.d yes 4 35.f even 4 1
525.2.b.e 4 15.e even 4 1
525.2.b.e 4 35.f even 4 1
525.2.b.g 4 5.c odd 4 1
525.2.b.g 4 105.k odd 4 1
525.2.g.d 8 3.b odd 2 1
525.2.g.d 8 7.b odd 2 1
525.2.g.d 8 15.d odd 2 1
525.2.g.d 8 35.c odd 2 1
525.2.g.e 8 1.a even 1 1 trivial
525.2.g.e 8 5.b even 2 1 inner
525.2.g.e 8 21.c even 2 1 inner
525.2.g.e 8 105.g even 2 1 inner
735.2.s.g 4 35.k even 12 1
735.2.s.g 4 105.x even 12 1
735.2.s.h 4 35.l odd 12 1
735.2.s.h 4 105.w odd 12 1
735.2.s.i 4 35.l odd 12 1
735.2.s.i 4 105.w odd 12 1
735.2.s.j 4 35.k even 12 1
735.2.s.j 4 105.x even 12 1
1680.2.f.g 4 60.l odd 4 1
1680.2.f.g 4 140.j odd 4 1
1680.2.f.h 4 20.e even 4 1
1680.2.f.h 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 \(S_{2}^{\mathrm{new}}(525, [\chi])\):

\( T_{2}^{4} - 7T_{2}^{2} + 4 \) Copy content Toggle raw display
\( T_{41} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 7 T^{2} + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} + 5 T^{6} + 16 T^{4} + 45 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 2 T^{2} + 49)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 7 T^{2} + 4)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 51 T^{2} + 576)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 21 T^{2} + 36)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 12)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 76 T^{2} + 256)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 19 T^{2} + 16)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 12)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 68 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$41$ \( (T - 6)^{8} \) Copy content Toggle raw display
$43$ \( (T^{4} + 68 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 57 T^{2} + 144)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 76 T^{2} + 256)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 6 T - 24)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 48)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 68 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 184 T^{2} + 16)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 48)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + T - 8)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 336 T^{2} + 9216)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 18 T + 48)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} - 123 T^{2} + 144)^{2} \) Copy content Toggle raw display
show more
show less