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Properties

Label 825.2.c.c
Level $825$
Weight $2$
Character orbit 825.c
Analytic conductor $6.588$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [825,2,Mod(199,825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(825, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("825.199");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.c (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: \(6.58765816676\)
Analytic rank: \(0\)
Dimension: \(4\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 165)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + \beta_1 q^{3} - q^{4} - \beta_{3} q^{6} - 2 \beta_1 q^{7} + \beta_{2} q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + \beta_1 q^{3} - q^{4} - \beta_{3} q^{6} - 2 \beta_1 q^{7} + \beta_{2} q^{8} - q^{9} - q^{11} - \beta_1 q^{12} + (2 \beta_{2} + 2 \beta_1) q^{13} + 2 \beta_{3} q^{14} - 5 q^{16} - \beta_{2} q^{18} + ( - 2 \beta_{3} - 2) q^{19} + 2 q^{21} - \beta_{2} q^{22} + 4 \beta_{2} q^{23} - \beta_{3} q^{24} + ( - 2 \beta_{3} - 6) q^{26} - \beta_1 q^{27} + 2 \beta_1 q^{28} + 2 \beta_{3} q^{29} + ( - 4 \beta_{3} - 4) q^{31} - 3 \beta_{2} q^{32} - \beta_1 q^{33} + q^{36} + (4 \beta_{2} - 2 \beta_1) q^{37} + ( - 2 \beta_{2} - 6 \beta_1) q^{38} + ( - 2 \beta_{3} - 2) q^{39} + 2 \beta_{3} q^{41} + 2 \beta_{2} q^{42} + ( - 4 \beta_{2} + 2 \beta_1) q^{43} + q^{44} - 12 q^{46} + 4 \beta_{2} q^{47} - 5 \beta_1 q^{48} + 3 q^{49} + ( - 2 \beta_{2} - 2 \beta_1) q^{52} + (4 \beta_{2} - 6 \beta_1) q^{53} + \beta_{3} q^{54} + 2 \beta_{3} q^{56} + ( - 2 \beta_{2} - 2 \beta_1) q^{57} + 6 \beta_1 q^{58} + 4 \beta_{3} q^{59} + 2 q^{61} + ( - 4 \beta_{2} - 12 \beta_1) q^{62} + 2 \beta_1 q^{63} - q^{64} + \beta_{3} q^{66} - 8 \beta_1 q^{67} - 4 \beta_{3} q^{69} + 8 \beta_{3} q^{71} - \beta_{2} q^{72} + ( - 6 \beta_{2} + 2 \beta_1) q^{73} + (2 \beta_{3} - 12) q^{74} + (2 \beta_{3} + 2) q^{76} + 2 \beta_1 q^{77} + ( - 2 \beta_{2} - 6 \beta_1) q^{78} + ( - 2 \beta_{3} + 10) q^{79} + q^{81} + 6 \beta_1 q^{82} + ( - 2 \beta_{2} + 12 \beta_1) q^{83} - 2 q^{84} + ( - 2 \beta_{3} + 12) q^{86} + 2 \beta_{2} q^{87} - \beta_{2} q^{88} + ( - 4 \beta_{3} + 6) q^{89} + (4 \beta_{3} + 4) q^{91} - 4 \beta_{2} q^{92} + ( - 4 \beta_{2} - 4 \beta_1) q^{93} - 12 q^{94} + 3 \beta_{3} q^{96} + 10 \beta_1 q^{97} + 3 \beta_{2} q^{98} + q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 4 q^{9} - 4 q^{11} - 20 q^{16} - 8 q^{19} + 8 q^{21} - 24 q^{26} - 16 q^{31} + 4 q^{36} - 8 q^{39} + 4 q^{44} - 48 q^{46} + 12 q^{49} + 8 q^{61} - 4 q^{64} - 48 q^{74} + 8 q^{76} + 40 q^{79} + 4 q^{81} - 8 q^{84} + 48 q^{86} + 24 q^{89} + 16 q^{91} - 48 q^{94} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{12}^{2} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display

Display \(\zeta_{12}^j\) in terms of \(\beta_i\)

\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{12}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
0.866025 − 0.500000i
−0.866025 + 0.500000i
−0.866025 − 0.500000i
0.866025 + 0.500000i
− 1.73205i − 1.00000i −1.00000 0 −1.73205 2.00000i − 1.73205i −1.00000 0
199.2 − 1.73205i 1.00000i −1.00000 0 1.73205 − 2.00000i − 1.73205i −1.00000 0
199.3 1.73205i − 1.00000i −1.00000 0 1.73205 2.00000i 1.73205i −1.00000 0
199.4 1.73205i 1.00000i −1.00000 0 −1.73205 − 2.00000i 1.73205i −1.00000 0
\(n\): 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 825.2.c.c 4
3.b odd 2 1 2475.2.c.n 4
5.b even 2 1 inner 825.2.c.c 4
5.c odd 4 1 165.2.a.b ✓ 2
5.c odd 4 1 825.2.a.e 2
15.d odd 2 1 2475.2.c.n 4
15.e even 4 1 495.2.a.c 2
15.e even 4 1 2475.2.a.r 2
20.e even 4 1 2640.2.a.x 2
35.f even 4 1 8085.2.a.bd 2
55.e even 4 1 1815.2.a.i 2
55.e even 4 1 9075.2.a.bh 2
60.l odd 4 1 7920.2.a.bz 2
165.l odd 4 1 5445.2.a.s 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.a.b ✓ 2 5.c odd 4 1
495.2.a.c 2 15.e even 4 1
825.2.a.e 2 5.c odd 4 1
825.2.c.c 4 1.a even 1 1 trivial
825.2.c.c 4 5.b even 2 1 inner
1815.2.a.i 2 55.e even 4 1
2475.2.a.r 2 15.e even 4 1
2475.2.c.n 4 3.b odd 2 1
2475.2.c.n 4 15.d odd 2 1
2640.2.a.x 2 20.e even 4 1
5445.2.a.s 2 165.l odd 4 1
7920.2.a.bz 2 60.l odd 4 1
8085.2.a.bd 2 35.f even 4 1
9075.2.a.bh 2 55.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(825, [\chi])\):

\( T_{2}^{2} + 3 \) Copy content Toggle raw display
\( T_{7}^{2} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$11$ \( (T + 1)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 32T^{2} + 64 \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 4 T - 8)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 8 T - 32)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 104T^{2} + 1936 \) Copy content Toggle raw display
$41$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 104T^{2} + 1936 \) Copy content Toggle raw display
$47$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 168T^{2} + 144 \) Copy content Toggle raw display
$59$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$61$ \( (T - 2)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 192)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 224 T^{2} + 10816 \) Copy content Toggle raw display
$79$ \( (T^{2} - 20 T + 88)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 312 T^{2} + 17424 \) Copy content Toggle raw display
$89$ \( (T^{2} - 12 T - 12)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
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