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Properties

Label 900.2.bf.c
Level $900$
Weight $2$
Character orbit 900.bf
Analytic conductor $7.187$
Analytic rank $0$
Dimension $16$
Inner twists $16$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [900,2,Mod(7,900)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("900.7"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(900, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 8, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 900.bf (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.18653618192\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: 16.0.162447943996702457856.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{12} - 15x^{8} - 16x^{4} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{13} - \beta_{3} + \beta_1) q^{2} + ( - 2 \beta_{15} - \beta_{9}) q^{3} + (2 \beta_{11} - \beta_{10}) q^{4} + (\beta_{12} - 2 \beta_{2}) q^{6} + (4 \beta_{13} - 4 \beta_{3}) q^{7} + ( - 3 \beta_{15} - \beta_{14} + \cdots - \beta_{6}) q^{8}+ \cdots + (3 \beta_{7} - 6 \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 12 q^{6} + 4 q^{16} - 96 q^{21} + 112 q^{26} - 36 q^{36} + 56 q^{41} + 32 q^{46} - 80 q^{56} + 48 q^{61} - 28 q^{76} + 72 q^{81} + 4 q^{86}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - x^{12} - 15x^{8} - 16x^{4} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{12} - 15\nu^{8} + 255\nu^{4} + 256 ) / 720 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{13} - 15\nu^{9} + 255\nu^{5} + 256\nu ) / 1440 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{14} - 15\nu^{10} + 255\nu^{6} + 256\nu^{2} ) / 1440 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{12} + 15\nu^{8} - 15\nu^{4} - 256 ) / 240 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{15} + 15\nu^{11} + 225\nu^{7} - 256\nu^{3} ) / 1920 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{14} + 15\nu^{10} + 33\nu^{6} - 256\nu^{2} ) / 576 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{13} + 15\nu^{9} - 15\nu^{5} - 256\nu ) / 240 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -7\nu^{15} - 105\nu^{11} + 345\nu^{7} + 1792\nu^{3} ) / 5760 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{14} + 89\nu^{2} ) / 180 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -\nu^{14} + 91\nu^{2} ) / 180 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -17\nu^{12} - 15\nu^{8} + 255\nu^{4} + 272 ) / 720 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -17\nu^{13} - 15\nu^{9} + 255\nu^{5} + 272\nu ) / 1440 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 11\nu^{15} - 75\nu^{11} - 165\nu^{7} - 176\nu^{3} ) / 2880 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 23\nu^{15} + 105\nu^{11} - 345\nu^{7} - 368\nu^{3} ) / 5760 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + \beta_{10} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{15} - \beta_{14} + 3\beta_{9} - \beta_{6} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} + 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{8} + 6\beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{7} + 5\beta_{4} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 3\beta_{9} + 7\beta_{6} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 3\beta_{12} + 17\beta_{5} + 17 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 6\beta_{13} + 17\beta_{8} + 17\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 23\beta_{11} + 11\beta_{10} + 34\beta_{7} - 11\beta_{4} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 22\beta_{15} - 23\beta_{14} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -45\beta_{12} + 45\beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( -90\beta_{13} + 90\beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( -89\beta_{11} + 91\beta_{10} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 182\beta_{15} + 89\beta_{14} + 93\beta_{9} + 89\beta_{6} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(-1 - \beta_{5}\) \(-1\) \(\beta_{7}\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
−1.14839 + 0.825348i
−1.40721 − 0.140577i
1.40721 + 0.140577i
1.14839 − 0.825348i
0.140577 − 1.40721i
0.825348 + 1.14839i
−0.825348 − 1.14839i
−0.140577 + 1.40721i
0.140577 + 1.40721i
0.825348 − 1.14839i
−0.825348 + 1.14839i
−0.140577 − 1.40721i
−1.14839 − 0.825348i
−1.40721 + 0.140577i
1.40721 − 0.140577i
1.14839 + 0.825348i
−1.40721 − 0.140577i −0.448288 − 1.67303i 1.96048 + 0.395644i 0 0.395644 + 2.41733i −1.03528 − 3.86370i −2.70318 − 0.832353i −2.59808 + 1.50000i 0
7.2 −1.14839 + 0.825348i 0.448288 + 1.67303i 0.637600 − 1.89564i 0 −1.89564 − 1.55130i 1.03528 + 3.86370i 0.832353 + 2.70318i −2.59808 + 1.50000i 0
7.3 1.14839 − 0.825348i −0.448288 − 1.67303i 0.637600 − 1.89564i 0 −1.89564 − 1.55130i −1.03528 − 3.86370i −0.832353 − 2.70318i −2.59808 + 1.50000i 0
7.4 1.40721 + 0.140577i 0.448288 + 1.67303i 1.96048 + 0.395644i 0 0.395644 + 2.41733i 1.03528 + 3.86370i 2.70318 + 0.832353i −2.59808 + 1.50000i 0
43.1 −0.825348 − 1.14839i 1.67303 − 0.448288i −0.637600 + 1.89564i 0 −1.89564 − 1.55130i −3.86370 + 1.03528i 2.70318 − 0.832353i 2.59808 − 1.50000i 0
43.2 −0.140577 + 1.40721i 1.67303 − 0.448288i −1.96048 − 0.395644i 0 0.395644 + 2.41733i −3.86370 + 1.03528i 0.832353 − 2.70318i 2.59808 − 1.50000i 0
43.3 0.140577 − 1.40721i −1.67303 + 0.448288i −1.96048 − 0.395644i 0 0.395644 + 2.41733i 3.86370 − 1.03528i −0.832353 + 2.70318i 2.59808 − 1.50000i 0
43.4 0.825348 + 1.14839i −1.67303 + 0.448288i −0.637600 + 1.89564i 0 −1.89564 − 1.55130i 3.86370 − 1.03528i −2.70318 + 0.832353i 2.59808 − 1.50000i 0
607.1 −0.825348 + 1.14839i 1.67303 + 0.448288i −0.637600 − 1.89564i 0 −1.89564 + 1.55130i −3.86370 − 1.03528i 2.70318 + 0.832353i 2.59808 + 1.50000i 0
607.2 −0.140577 − 1.40721i 1.67303 + 0.448288i −1.96048 + 0.395644i 0 0.395644 − 2.41733i −3.86370 − 1.03528i 0.832353 + 2.70318i 2.59808 + 1.50000i 0
607.3 0.140577 + 1.40721i −1.67303 − 0.448288i −1.96048 + 0.395644i 0 0.395644 − 2.41733i 3.86370 + 1.03528i −0.832353 − 2.70318i 2.59808 + 1.50000i 0
607.4 0.825348 − 1.14839i −1.67303 − 0.448288i −0.637600 − 1.89564i 0 −1.89564 + 1.55130i 3.86370 + 1.03528i −2.70318 − 0.832353i 2.59808 + 1.50000i 0
643.1 −1.40721 + 0.140577i −0.448288 + 1.67303i 1.96048 − 0.395644i 0 0.395644 − 2.41733i −1.03528 + 3.86370i −2.70318 + 0.832353i −2.59808 − 1.50000i 0
643.2 −1.14839 − 0.825348i 0.448288 − 1.67303i 0.637600 + 1.89564i 0 −1.89564 + 1.55130i 1.03528 − 3.86370i 0.832353 − 2.70318i −2.59808 − 1.50000i 0
643.3 1.14839 + 0.825348i −0.448288 + 1.67303i 0.637600 + 1.89564i 0 −1.89564 + 1.55130i −1.03528 + 3.86370i −0.832353 + 2.70318i −2.59808 − 1.50000i 0
643.4 1.40721 − 0.140577i 0.448288 − 1.67303i 1.96048 − 0.395644i 0 0.395644 − 2.41733i 1.03528 − 3.86370i 2.70318 − 0.832353i −2.59808 − 1.50000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
9.c even 3 1 inner
20.d odd 2 1 inner
20.e even 4 2 inner
36.f odd 6 1 inner
45.j even 6 1 inner
45.k odd 12 2 inner
180.p odd 6 1 inner
180.x even 12 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.2.bf.c ✓ 16
4.b odd 2 1 inner 900.2.bf.c ✓ 16
5.b even 2 1 inner 900.2.bf.c ✓ 16
5.c odd 4 2 inner 900.2.bf.c ✓ 16
9.c even 3 1 inner 900.2.bf.c ✓ 16
20.d odd 2 1 inner 900.2.bf.c ✓ 16
20.e even 4 2 inner 900.2.bf.c ✓ 16
36.f odd 6 1 inner 900.2.bf.c ✓ 16
45.j even 6 1 inner 900.2.bf.c ✓ 16
45.k odd 12 2 inner 900.2.bf.c ✓ 16
180.p odd 6 1 inner 900.2.bf.c ✓ 16
180.x even 12 2 inner 900.2.bf.c ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
900.2.bf.c ✓ 16 1.a even 1 1 trivial
900.2.bf.c ✓ 16 4.b odd 2 1 inner
900.2.bf.c ✓ 16 5.b even 2 1 inner
900.2.bf.c ✓ 16 5.c odd 4 2 inner
900.2.bf.c ✓ 16 9.c even 3 1 inner
900.2.bf.c ✓ 16 20.d odd 2 1 inner
900.2.bf.c ✓ 16 20.e even 4 2 inner
900.2.bf.c ✓ 16 36.f odd 6 1 inner
900.2.bf.c ✓ 16 45.j even 6 1 inner
900.2.bf.c ✓ 16 45.k odd 12 2 inner
900.2.bf.c ✓ 16 180.p odd 6 1 inner
900.2.bf.c ✓ 16 180.x even 12 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} - 256T_{7}^{4} + 65536 \) acting on \(S_{2}^{\mathrm{new}}(900, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - T^{12} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( (T^{8} - 9 T^{4} + 81)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} - 256 T^{4} + 65536)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 7 T^{2} + 49)^{4} \) Copy content Toggle raw display
$13$ \( (T^{8} - 784 T^{4} + 614656)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 3969)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 7)^{8} \) Copy content Toggle raw display
$23$ \( (T^{8} - 256 T^{4} + 65536)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 4 T^{2} + 16)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 28 T^{2} + 784)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 784)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 7 T + 49)^{8} \) Copy content Toggle raw display
$43$ \( (T^{8} - T^{4} + 1)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 1296 T^{4} + 1679616)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 784)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} + 7 T^{2} + 49)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} - 6 T + 36)^{8} \) Copy content Toggle raw display
$67$ \( (T^{8} - 6561 T^{4} + 43046721)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} \) Copy content Toggle raw display
$73$ \( (T^{4} + 30625)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 112 T^{2} + 12544)^{4} \) Copy content Toggle raw display
$83$ \( T^{16} \) Copy content Toggle raw display
$89$ \( (T^{2} + 36)^{8} \) Copy content Toggle raw display
$97$ \( (T^{8} - 49 T^{4} + 2401)^{2} \) Copy content Toggle raw display
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