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Properties

Label 225.2.e.d
Level $225$
Weight $2$
Character orbit 225.e
Analytic conductor $1.797$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [225,2,Mod(76,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.76");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 225.e (of order \(3\), degree \(2\), 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: \(1.79663404548\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
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: \( 3^{2} \)
Twist minimal: no (minimal twist has level 45)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{7} q^{2} + ( - \beta_{7} + \beta_{6}) q^{3} - \beta_{2} q^{4} + ( - \beta_{2} - \beta_1 - 1) q^{6} + (\beta_{7} - 2 \beta_{5} + 2 \beta_{3}) q^{7} - \beta_{6} q^{8} + ( - \beta_{4} - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{2} + ( - \beta_{7} + \beta_{6}) q^{3} - \beta_{2} q^{4} + ( - \beta_{2} - \beta_1 - 1) q^{6} + (\beta_{7} - 2 \beta_{5} + 2 \beta_{3}) q^{7} - \beta_{6} q^{8} + ( - \beta_{4} - \beta_{2}) q^{9} + ( - \beta_{4} + \beta_{2} + 3 \beta_1 - 3) q^{11} + ( - \beta_{7} + 2 \beta_{6} + 3 \beta_{5} - 2 \beta_{3}) q^{12} + (2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + \beta_{3}) q^{13} - \beta_{2} q^{14} + (2 \beta_{4} - 2 \beta_{2} - \beta_1 + 1) q^{16} + ( - 2 \beta_{6} + \beta_{5}) q^{17} + (\beta_{7} + \beta_{6} + 3 \beta_{5} - \beta_{3}) q^{18} + (\beta_{4} + 1) q^{19} + ( - 2 \beta_{4} + \beta_{2} - 3 \beta_1 + 3) q^{21} + (4 \beta_{7} - 4 \beta_{6} - 4 \beta_{5} - \beta_{3}) q^{22} + ( - \beta_{7} + \beta_{6} + \beta_{5}) q^{23} + (\beta_{4} - \beta_1 - 1) q^{24} + (\beta_{4} + 3) q^{26} + (3 \beta_{6} + 3 \beta_{5} - 3 \beta_{3}) q^{27} + 3 \beta_{6} q^{28} + ( - 2 \beta_{4} + 2 \beta_{2} + 3 \beta_1 - 3) q^{29} + (\beta_{2} + \beta_1) q^{31} + ( - \beta_{7} + \beta_{6} + \beta_{5} + 4 \beta_{3}) q^{32} + (2 \beta_{7} - 4 \beta_{6} - 3 \beta_{5} - 2 \beta_{3}) q^{33} + ( - \beta_{4} + \beta_{2} - \beta_1 + 1) q^{34} + (6 \beta_1 - 3) q^{36} - 3 \beta_{5} q^{37} + ( - 2 \beta_{7} - \beta_{5} + \beta_{3}) q^{38} + (\beta_{4} + \beta_{2} - 3 \beta_1 + 3) q^{39} + (\beta_{2} - 6 \beta_1) q^{41} + ( - \beta_{7} + 2 \beta_{6} + 3 \beta_{5} - 2 \beta_{3}) q^{42} + ( - 4 \beta_{7} - \beta_{5} + \beta_{3}) q^{43} + (3 \beta_{4} + 3) q^{44} + ( - \beta_{4} - 2) q^{46} + ( - 3 \beta_{7} + \beta_{5} - \beta_{3}) q^{47} + ( - 4 \beta_{7} + 3 \beta_{6} + 6 \beta_{5} - \beta_{3}) q^{48} + (3 \beta_{2} + \beta_1) q^{49} + (2 \beta_{4} + \beta_{2} - \beta_1 - 4) q^{51} + ( - 3 \beta_{5} + 3 \beta_{3}) q^{52} + (2 \beta_{6} + 2 \beta_{5}) q^{53} + (3 \beta_{2} + 6 \beta_1 - 3) q^{54} + (2 \beta_{4} - 2 \beta_{2} + 3 \beta_1 - 3) q^{56} + ( - 2 \beta_{7} + \beta_{3}) q^{57} + (5 \beta_{7} - 5 \beta_{6} - 5 \beta_{5} - 2 \beta_{3}) q^{58} + (2 \beta_{2} - 6 \beta_1) q^{59} + ( - 5 \beta_{4} + 5 \beta_{2} + 2 \beta_1 - 2) q^{61} + ( - 2 \beta_{6} - 3 \beta_{5}) q^{62} + (3 \beta_{7} + 3 \beta_{6} - 3 \beta_{5} + 3 \beta_{3}) q^{63} + ( - \beta_{4} - 4) q^{64} + (5 \beta_{4} - \beta_{2} - 3 \beta_1 + 9) q^{66} + (\beta_{7} - \beta_{6} - \beta_{5} + 5 \beta_{3}) q^{67} + (4 \beta_{7} - 4 \beta_{6} - 4 \beta_{5} + 5 \beta_{3}) q^{68} + ( - \beta_{4} + \beta_1 - 2) q^{69} + ( - 3 \beta_{4} + 9) q^{71} + (\beta_{7} - 2 \beta_{6} + 2 \beta_{3}) q^{72} + 6 \beta_{5} q^{73} + (3 \beta_{4} - 3 \beta_{2} - 3 \beta_1 + 3) q^{74} + ( - \beta_{2} - 3 \beta_1) q^{76} - 3 \beta_{3} q^{77} + ( - 4 \beta_{7} + 2 \beta_{6} + \beta_{3}) q^{78} + (2 \beta_{4} - 2 \beta_{2} + 4 \beta_1 - 4) q^{79} + 9 \beta_1 q^{81} + (5 \beta_{6} + 4 \beta_{5}) q^{82} + (\beta_{7} + 6 \beta_{5} - 6 \beta_{3}) q^{83} + ( - 3 \beta_{4} + 3 \beta_1 + 3) q^{84} + ( - 5 \beta_{2} - 9 \beta_1) q^{86} + (4 \beta_{7} - 5 \beta_{6} - 6 \beta_{5} - \beta_{3}) q^{87} + (2 \beta_{7} - \beta_{5} + \beta_{3}) q^{88} + ( - 6 \beta_{4} + 3) q^{89} + (3 \beta_{4} - 3) q^{91} + (\beta_{7} + \beta_{5} - \beta_{3}) q^{92} + ( - 2 \beta_{6} - 3 \beta_{5} + \beta_{3}) q^{93} + ( - 2 \beta_{2} - 5 \beta_1) q^{94} + ( - 5 \beta_{4} + 4 \beta_{2} - 3 \beta_1 - 6) q^{96} + ( - 2 \beta_{7} + 10 \beta_{5} - 10 \beta_{3}) q^{97} + ( - 4 \beta_{6} - 7 \beta_{5}) q^{98} + (6 \beta_{4} - 3 \beta_{2} - 3 \beta_1 + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 12 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 12 q^{6} - 12 q^{11} + 4 q^{16} + 8 q^{19} + 12 q^{21} - 12 q^{24} + 24 q^{26} - 12 q^{29} + 4 q^{31} + 4 q^{34} + 12 q^{39} - 24 q^{41} + 24 q^{44} - 16 q^{46} + 4 q^{49} - 36 q^{51} - 12 q^{56} - 24 q^{59} - 8 q^{61} - 32 q^{64} + 60 q^{66} - 12 q^{69} + 72 q^{71} + 12 q^{74} - 12 q^{76} - 16 q^{79} + 36 q^{81} + 36 q^{84} - 36 q^{86} + 24 q^{89} - 24 q^{91} - 20 q^{94} - 60 q^{96} + 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

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

Display \(\zeta_{24}^j\) in terms of \(\beta_i\)

\(\zeta_{24}\)\(=\) \( ( \beta_{7} + \beta_{6} + 2\beta_{3} ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{4} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( -2\beta_{7} + \beta_{6} + 3\beta_{5} - \beta_{3} ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -\beta_{7} + 2\beta_{6} + \beta_{3} ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( -\beta_{4} + 2\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -\beta_{7} - \beta_{6} + \beta_{3} ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{24}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-\beta_{1}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
76.1
0.965926 − 0.258819i
0.258819 + 0.965926i
−0.258819 − 0.965926i
−0.965926 + 0.258819i
0.965926 + 0.258819i
0.258819 − 0.965926i
−0.258819 + 0.965926i
−0.965926 − 0.258819i
−0.965926 − 1.67303i −0.448288 − 1.67303i −0.866025 + 1.50000i 0 −2.36603 + 2.36603i −0.448288 − 0.776457i −0.517638 −2.59808 + 1.50000i 0
76.2 −0.258819 − 0.448288i 1.67303 − 0.448288i 0.866025 − 1.50000i 0 −0.633975 − 0.633975i 1.67303 + 2.89778i −1.93185 2.59808 − 1.50000i 0
76.3 0.258819 + 0.448288i −1.67303 + 0.448288i 0.866025 − 1.50000i 0 −0.633975 − 0.633975i −1.67303 − 2.89778i 1.93185 2.59808 − 1.50000i 0
76.4 0.965926 + 1.67303i 0.448288 + 1.67303i −0.866025 + 1.50000i 0 −2.36603 + 2.36603i 0.448288 + 0.776457i 0.517638 −2.59808 + 1.50000i 0
151.1 −0.965926 + 1.67303i −0.448288 + 1.67303i −0.866025 − 1.50000i 0 −2.36603 − 2.36603i −0.448288 + 0.776457i −0.517638 −2.59808 − 1.50000i 0
151.2 −0.258819 + 0.448288i 1.67303 + 0.448288i 0.866025 + 1.50000i 0 −0.633975 + 0.633975i 1.67303 − 2.89778i −1.93185 2.59808 + 1.50000i 0
151.3 0.258819 − 0.448288i −1.67303 − 0.448288i 0.866025 + 1.50000i 0 −0.633975 + 0.633975i −1.67303 + 2.89778i 1.93185 2.59808 + 1.50000i 0
151.4 0.965926 − 1.67303i 0.448288 − 1.67303i −0.866025 − 1.50000i 0 −2.36603 − 2.36603i 0.448288 − 0.776457i 0.517638 −2.59808 − 1.50000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 151.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
9.c even 3 1 inner
45.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.2.e.d 8
3.b odd 2 1 675.2.e.d 8
5.b even 2 1 inner 225.2.e.d 8
5.c odd 4 2 45.2.j.a ✓ 8
9.c even 3 1 inner 225.2.e.d 8
9.c even 3 1 2025.2.a.t 4
9.d odd 6 1 675.2.e.d 8
9.d odd 6 1 2025.2.a.r 4
15.d odd 2 1 675.2.e.d 8
15.e even 4 2 135.2.j.a 8
20.e even 4 2 720.2.by.d 8
45.h odd 6 1 675.2.e.d 8
45.h odd 6 1 2025.2.a.r 4
45.j even 6 1 inner 225.2.e.d 8
45.j even 6 1 2025.2.a.t 4
45.k odd 12 2 45.2.j.a ✓ 8
45.k odd 12 2 405.2.b.d 4
45.l even 12 2 135.2.j.a 8
45.l even 12 2 405.2.b.c 4
60.l odd 4 2 2160.2.by.c 8
180.v odd 12 2 2160.2.by.c 8
180.x even 12 2 720.2.by.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.2.j.a ✓ 8 5.c odd 4 2
45.2.j.a ✓ 8 45.k odd 12 2
135.2.j.a 8 15.e even 4 2
135.2.j.a 8 45.l even 12 2
225.2.e.d 8 1.a even 1 1 trivial
225.2.e.d 8 5.b even 2 1 inner
225.2.e.d 8 9.c even 3 1 inner
225.2.e.d 8 45.j even 6 1 inner
405.2.b.c 4 45.l even 12 2
405.2.b.d 4 45.k odd 12 2
675.2.e.d 8 3.b odd 2 1
675.2.e.d 8 9.d odd 6 1
675.2.e.d 8 15.d odd 2 1
675.2.e.d 8 45.h odd 6 1
720.2.by.d 8 20.e even 4 2
720.2.by.d 8 180.x even 12 2
2025.2.a.r 4 9.d odd 6 1
2025.2.a.r 4 45.h odd 6 1
2025.2.a.t 4 9.c even 3 1
2025.2.a.t 4 45.j even 6 1
2160.2.by.c 8 60.l odd 4 2
2160.2.by.c 8 180.v odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 4T_{2}^{6} + 15T_{2}^{4} + 4T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(225, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 4 T^{6} + 15 T^{4} + 4 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} - 9T^{4} + 81 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 12 T^{6} + 135 T^{4} + \cdots + 81 \) Copy content Toggle raw display
$11$ \( (T^{4} + 6 T^{3} + 30 T^{2} + 36 T + 36)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 6 T^{2} + 36)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 28 T^{2} + 4)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 2 T - 2)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} + 4 T^{6} + 15 T^{4} + 4 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$29$ \( (T^{4} + 6 T^{3} + 39 T^{2} - 18 T + 9)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 2 T^{3} + 6 T^{2} + 4 T + 4)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 18)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 12 T^{3} + 111 T^{2} + 396 T + 1089)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 84 T^{6} + 7020 T^{4} + \cdots + 1296 \) Copy content Toggle raw display
$47$ \( T^{8} + 28 T^{6} + 615 T^{4} + \cdots + 28561 \) Copy content Toggle raw display
$53$ \( (T^{4} - 16 T^{2} + 16)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 12 T^{3} + 120 T^{2} + 288 T + 576)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 4 T^{3} + 87 T^{2} - 284 T + 5041)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 84 T^{6} + 5535 T^{4} + \cdots + 2313441 \) Copy content Toggle raw display
$71$ \( (T^{2} - 18 T + 54)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 72)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 8 T^{3} + 60 T^{2} + 32 T + 16)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 172 T^{6} + \cdots + 47458321 \) Copy content Toggle raw display
$89$ \( (T^{2} - 6 T - 99)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} + 336 T^{6} + \cdots + 592240896 \) Copy content Toggle raw display
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