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Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [525,2,Mod(82,525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(525, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 3, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.82");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 525.bc (of order \(12\), degree \(4\), 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: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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 \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{24}^{5} + 2 \zeta_{24}) q^{2} + \zeta_{24}^{7} q^{3} + ( - \zeta_{24}^{6} + \zeta_{24}^{2}) q^{4} + (2 \zeta_{24}^{4} - 1) q^{6} + (2 \zeta_{24}^{5} - 3 \zeta_{24}) q^{7} + (2 \zeta_{24}^{7} - \zeta_{24}^{3}) q^{8} - \zeta_{24}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{24}^{5} + 2 \zeta_{24}) q^{2} + \zeta_{24}^{7} q^{3} + ( - \zeta_{24}^{6} + \zeta_{24}^{2}) q^{4} + (2 \zeta_{24}^{4} - 1) q^{6} + (2 \zeta_{24}^{5} - 3 \zeta_{24}) q^{7} + (2 \zeta_{24}^{7} - \zeta_{24}^{3}) q^{8} - \zeta_{24}^{2} q^{9} + 6 \zeta_{24}^{4} q^{11} + \zeta_{24}^{5} q^{12} + 5 \zeta_{24}^{3} q^{13} + (5 \zeta_{24}^{6} - 4 \zeta_{24}^{2}) q^{14} + (5 \zeta_{24}^{4} - 5) q^{16} + (\zeta_{24}^{7} - 2 \zeta_{24}^{3}) q^{18} + ( - 6 \zeta_{24}^{6} + 3 \zeta_{24}^{2}) q^{19} + ( - 3 \zeta_{24}^{4} + 1) q^{21} + (6 \zeta_{24}^{5} + 6 \zeta_{24}) q^{22} + ( - 2 \zeta_{24}^{7} - 2 \zeta_{24}^{3}) q^{23} + ( - \zeta_{24}^{6} - \zeta_{24}^{2}) q^{24} + (5 \zeta_{24}^{4} + 5) q^{26} + ( - \zeta_{24}^{5} + \zeta_{24}) q^{27} + (3 \zeta_{24}^{7} - \zeta_{24}^{3}) q^{28} + ( - 2 \zeta_{24}^{4} + 4) q^{31} + (6 \zeta_{24}^{5} - 3 \zeta_{24}) q^{32} + (6 \zeta_{24}^{7} - 6 \zeta_{24}^{3}) q^{33} - q^{36} + (5 \zeta_{24}^{5} - 10 \zeta_{24}) q^{37} - 9 \zeta_{24}^{7} q^{38} + (5 \zeta_{24}^{6} - 5 \zeta_{24}^{2}) q^{39} + ( - 12 \zeta_{24}^{4} + 6) q^{41} + ( - 4 \zeta_{24}^{5} - \zeta_{24}) q^{42} + (4 \zeta_{24}^{7} - 2 \zeta_{24}^{3}) q^{43} + 6 \zeta_{24}^{2} q^{44} - 6 \zeta_{24}^{4} q^{46} - 5 \zeta_{24}^{3} q^{48} + ( - 8 \zeta_{24}^{6} + 5 \zeta_{24}^{2}) q^{49} + 5 \zeta_{24} q^{52} + ( - 2 \zeta_{24}^{6} + \zeta_{24}^{2}) q^{54} + ( - 5 \zeta_{24}^{4} + 4) q^{56} + (3 \zeta_{24}^{5} + 3 \zeta_{24}) q^{57} + (4 \zeta_{24}^{6} + 4 \zeta_{24}^{2}) q^{59} + ( - \zeta_{24}^{4} - 1) q^{61} + ( - 6 \zeta_{24}^{5} + 6 \zeta_{24}) q^{62} + ( - 2 \zeta_{24}^{7} + 3 \zeta_{24}^{3}) q^{63} - \zeta_{24}^{6} q^{64} + (6 \zeta_{24}^{4} - 12) q^{66} + (10 \zeta_{24}^{5} - 5 \zeta_{24}) q^{67} + ( - 2 \zeta_{24}^{6} + 4 \zeta_{24}^{2}) q^{69} + 12 q^{71} + ( - \zeta_{24}^{5} + 2 \zeta_{24}) q^{72} - 7 \zeta_{24}^{7} q^{73} + (15 \zeta_{24}^{6} - 15 \zeta_{24}^{2}) q^{74} + ( - 6 \zeta_{24}^{4} + 3) q^{76} + ( - 6 \zeta_{24}^{5} - 12 \zeta_{24}) q^{77} + (10 \zeta_{24}^{7} - 5 \zeta_{24}^{3}) q^{78} + 11 \zeta_{24}^{2} q^{79} + \zeta_{24}^{4} q^{81} - 18 \zeta_{24}^{5} q^{82} + 12 \zeta_{24}^{3} q^{83} + ( - \zeta_{24}^{6} - 2 \zeta_{24}^{2}) q^{84} + (6 \zeta_{24}^{4} - 6) q^{86} + (6 \zeta_{24}^{7} - 12 \zeta_{24}^{3}) q^{88} + ( - 16 \zeta_{24}^{6} + 8 \zeta_{24}^{2}) q^{89} + ( - 5 \zeta_{24}^{4} - 10) q^{91} + ( - 2 \zeta_{24}^{5} - 2 \zeta_{24}) q^{92} + (2 \zeta_{24}^{7} + 2 \zeta_{24}^{3}) q^{93} + ( - 3 \zeta_{24}^{4} - 3) q^{96} + ( - 5 \zeta_{24}^{5} + 5 \zeta_{24}) q^{97} + ( - 13 \zeta_{24}^{7} + 2 \zeta_{24}^{3}) q^{98} - 6 \zeta_{24}^{6} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 24 q^{11} - 20 q^{16} - 4 q^{21} + 60 q^{26} + 24 q^{31} - 8 q^{36} - 24 q^{46} + 12 q^{56} - 12 q^{61} - 72 q^{66} + 96 q^{71} + 4 q^{81} - 24 q^{86} - 100 q^{91} - 36 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(\zeta_{24}^{6}\) \(1\) \(1 - \zeta_{24}^{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
82.1
−0.965926 − 0.258819i
0.965926 + 0.258819i
0.258819 + 0.965926i
−0.258819 − 0.965926i
0.258819 − 0.965926i
−0.258819 + 0.965926i
−0.965926 + 0.258819i
0.965926 − 0.258819i
−1.67303 + 0.448288i 0.258819 − 0.965926i 0.866025 − 0.500000i 0 1.73205i 2.38014 − 1.15539i 1.22474 − 1.22474i −0.866025 − 0.500000i 0
82.2 1.67303 − 0.448288i −0.258819 + 0.965926i 0.866025 − 0.500000i 0 1.73205i −2.38014 + 1.15539i −1.22474 + 1.22474i −0.866025 − 0.500000i 0
157.1 −0.448288 + 1.67303i −0.965926 + 0.258819i −0.866025 − 0.500000i 0 − 1.73205i 1.15539 − 2.38014i −1.22474 + 1.22474i 0.866025 − 0.500000i 0
157.2 0.448288 − 1.67303i 0.965926 − 0.258819i −0.866025 − 0.500000i 0 − 1.73205i −1.15539 + 2.38014i 1.22474 − 1.22474i 0.866025 − 0.500000i 0
418.1 −0.448288 − 1.67303i −0.965926 − 0.258819i −0.866025 + 0.500000i 0 1.73205i 1.15539 + 2.38014i −1.22474 − 1.22474i 0.866025 + 0.500000i 0
418.2 0.448288 + 1.67303i 0.965926 + 0.258819i −0.866025 + 0.500000i 0 1.73205i −1.15539 − 2.38014i 1.22474 + 1.22474i 0.866025 + 0.500000i 0
493.1 −1.67303 − 0.448288i 0.258819 + 0.965926i 0.866025 + 0.500000i 0 − 1.73205i 2.38014 + 1.15539i 1.22474 + 1.22474i −0.866025 + 0.500000i 0
493.2 1.67303 + 0.448288i −0.258819 − 0.965926i 0.866025 + 0.500000i 0 − 1.73205i −2.38014 − 1.15539i −1.22474 − 1.22474i −0.866025 + 0.500000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 82.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner
7.d odd 6 1 inner
35.i odd 6 1 inner
35.k even 12 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.bc.b ✓ 8
5.b even 2 1 inner 525.2.bc.b ✓ 8
5.c odd 4 2 inner 525.2.bc.b ✓ 8
7.d odd 6 1 inner 525.2.bc.b ✓ 8
35.i odd 6 1 inner 525.2.bc.b ✓ 8
35.k even 12 2 inner 525.2.bc.b ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.bc.b ✓ 8 1.a even 1 1 trivial
525.2.bc.b ✓ 8 5.b even 2 1 inner
525.2.bc.b ✓ 8 5.c odd 4 2 inner
525.2.bc.b ✓ 8 7.d odd 6 1 inner
525.2.bc.b ✓ 8 35.i odd 6 1 inner
525.2.bc.b ✓ 8 35.k even 12 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 9T_{2}^{4} + 81 \) acting on \(S_{2}^{\mathrm{new}}(525, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 9T^{4} + 81 \) Copy content Toggle raw display
$3$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 23T^{4} + 2401 \) Copy content Toggle raw display
$11$ \( (T^{2} - 6 T + 36)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 625)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( (T^{4} + 27 T^{2} + 729)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 144 T^{4} + 20736 \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( (T^{2} - 6 T + 12)^{4} \) Copy content Toggle raw display
$37$ \( T^{8} - 5625 T^{4} + 31640625 \) Copy content Toggle raw display
$41$ \( (T^{2} + 108)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 144)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( (T^{4} + 48 T^{2} + 2304)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 3 T + 3)^{4} \) Copy content Toggle raw display
$67$ \( T^{8} - 5625 T^{4} + 31640625 \) Copy content Toggle raw display
$71$ \( (T - 12)^{8} \) Copy content Toggle raw display
$73$ \( T^{8} - 2401 T^{4} + 5764801 \) Copy content Toggle raw display
$79$ \( (T^{4} - 121 T^{2} + 14641)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 20736)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 192 T^{2} + 36864)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 625)^{2} \) Copy content Toggle raw display
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