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Properties

Label 2025.2.b.o
Level $2025$
Weight $2$
Character orbit 2025.b
Analytic conductor $16.170$
Analytic rank $0$
Dimension $8$
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] = [2025,2,Mod(649,2025)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2025, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2025.649");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2025 = 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2025.b (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: \(16.1697064093\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.34810603776.6
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 12x^{6} + 42x^{4} + 49x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 225)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{6} + \beta_{4} - 1) q^{4} + ( - \beta_{7} + \beta_{2}) q^{7} + (\beta_{7} + \beta_{3} - 2 \beta_{2} - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{6} + \beta_{4} - 1) q^{4} + ( - \beta_{7} + \beta_{2}) q^{7} + (\beta_{7} + \beta_{3} - 2 \beta_{2} - \beta_1) q^{8} + (\beta_{5} + \beta_{4}) q^{11} + (2 \beta_{2} - \beta_1) q^{13} + ( - \beta_{6} - \beta_{5} + 2 \beta_{4}) q^{14} + (3 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + 1) q^{16} + (\beta_{7} - 2 \beta_{3} + \beta_{2}) q^{17} + ( - 2 \beta_{6} - \beta_{4} + 1) q^{19} + (\beta_{7} - 2 \beta_1) q^{22} + ( - \beta_{7} + 5 \beta_{3} + \cdots + \beta_1) q^{23}+ \cdots + ( - 3 \beta_{7} + 6 \beta_{3} + \cdots + 4 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} + 2 q^{11} + 6 q^{14} + 8 q^{16} - 4 q^{19} + 20 q^{26} + 2 q^{29} - 8 q^{31} - 18 q^{34} + 10 q^{41} + 44 q^{44} + 6 q^{49} + 60 q^{56} + 34 q^{59} - 26 q^{61} - 38 q^{64} + 16 q^{71} + 80 q^{74} + 22 q^{76} + 14 q^{79} + 68 q^{86} - 18 q^{89} - 34 q^{91} - 6 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 12x^{6} + 42x^{4} + 49x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 8\nu^{3} + 10\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 10\nu^{5} + 23\nu^{3} + 11\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{6} - 19\nu^{4} - 35\nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} + 11\nu^{4} + 31\nu^{2} + 18 ) / 3 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -2\nu^{6} - 19\nu^{4} - 38\nu^{2} - 9 ) / 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} - 8\nu^{5} - 4\nu^{3} + 24\nu ) / 3 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{6} + \beta_{4} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{3} - 2\beta_{2} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9\beta_{6} + 2\beta_{5} - 8\beta_{4} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{7} - 8\beta_{3} + 19\beta_{2} + 30\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -68\beta_{6} - 19\beta_{5} + 57\beta_{4} - 90 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 57\beta_{7} + 60\beta_{3} - 144\beta_{2} - 196\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2025\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(1702\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
− 2.63372i
− 1.63372i
− 1.47325i
− 0.473255i
0.473255i
1.47325i
1.63372i
2.63372i
− 2.63372i 0 −4.93650 0 0 − 1.79743i 7.73393i 0 0
649.2 − 1.63372i 0 −0.669052 0 0 0.505348i − 2.17440i 0 0
649.3 − 1.47325i 0 −0.170479 0 0 3.86583i − 2.69535i 0 0
649.4 − 0.473255i 0 1.77603 0 0 2.56305i − 1.78702i 0 0
649.5 0.473255i 0 1.77603 0 0 − 2.56305i 1.78702i 0 0
649.6 1.47325i 0 −0.170479 0 0 − 3.86583i 2.69535i 0 0
649.7 1.63372i 0 −0.669052 0 0 − 0.505348i 2.17440i 0 0
649.8 2.63372i 0 −4.93650 0 0 1.79743i − 7.73393i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 649.8
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 2025.2.b.o 8
3.b odd 2 1 2025.2.b.n 8
5.b even 2 1 inner 2025.2.b.o 8
5.c odd 4 1 2025.2.a.p 4
5.c odd 4 1 2025.2.a.z 4
9.c even 3 2 675.2.k.c 16
9.d odd 6 2 225.2.k.c 16
15.d odd 2 1 2025.2.b.n 8
15.e even 4 1 2025.2.a.q 4
15.e even 4 1 2025.2.a.y 4
45.h odd 6 2 225.2.k.c 16
45.j even 6 2 675.2.k.c 16
45.k odd 12 2 675.2.e.c 8
45.k odd 12 2 675.2.e.e 8
45.l even 12 2 225.2.e.c ✓ 8
45.l even 12 2 225.2.e.e yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.2.e.c ✓ 8 45.l even 12 2
225.2.e.e yes 8 45.l even 12 2
225.2.k.c 16 9.d odd 6 2
225.2.k.c 16 45.h odd 6 2
675.2.e.c 8 45.k odd 12 2
675.2.e.e 8 45.k odd 12 2
675.2.k.c 16 9.c even 3 2
675.2.k.c 16 45.j even 6 2
2025.2.a.p 4 5.c odd 4 1
2025.2.a.q 4 15.e even 4 1
2025.2.a.y 4 15.e even 4 1
2025.2.a.z 4 5.c odd 4 1
2025.2.b.n 8 3.b odd 2 1
2025.2.b.n 8 15.d odd 2 1
2025.2.b.o 8 1.a even 1 1 trivial
2025.2.b.o 8 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2025, [\chi])\):

\( T_{2}^{8} + 12T_{2}^{6} + 42T_{2}^{4} + 49T_{2}^{2} + 9 \) Copy content Toggle raw display
\( T_{11}^{4} - T_{11}^{3} - 25T_{11}^{2} - 41T_{11} - 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 12 T^{6} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 25 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$11$ \( (T^{4} - T^{3} - 25 T^{2} + \cdots - 9)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 64 T^{6} + \cdots + 11449 \) Copy content Toggle raw display
$17$ \( T^{8} + 81 T^{6} + \cdots + 91809 \) Copy content Toggle raw display
$19$ \( (T^{4} + 2 T^{3} - 27 T^{2} + \cdots - 25)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 111 T^{6} + \cdots + 59049 \) Copy content Toggle raw display
$29$ \( (T^{4} - T^{3} - 40 T^{2} + \cdots - 129)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 4 T^{3} + \cdots + 243)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 199 T^{6} + \cdots + 418609 \) Copy content Toggle raw display
$41$ \( (T^{4} - 5 T^{3} + \cdots - 207)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 196 T^{6} + \cdots + 452929 \) Copy content Toggle raw display
$47$ \( T^{8} + 186 T^{6} + \cdots + 145161 \) Copy content Toggle raw display
$53$ \( T^{8} + 228 T^{6} + \cdots + 221841 \) Copy content Toggle raw display
$59$ \( (T^{4} - 17 T^{3} + \cdots - 2313)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 13 T^{3} - 3 T^{2} + \cdots - 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 217 T^{6} + \cdots + 59049 \) Copy content Toggle raw display
$71$ \( (T^{4} - 8 T^{3} + \cdots + 381)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 196 T^{6} + \cdots + 12769 \) Copy content Toggle raw display
$79$ \( (T^{4} - 7 T^{3} + \cdots + 207)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 324 T^{6} + \cdots + 531441 \) Copy content Toggle raw display
$89$ \( (T^{4} + 9 T^{3} + \cdots + 2025)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 199 T^{6} + \cdots + 908209 \) Copy content Toggle raw display
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