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Properties

Label 1450.2.j.h
Level $1450$
Weight $2$
Character orbit 1450.j
Analytic conductor $11.578$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1450,2,Mod(157,1450)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1450, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1450.157"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1450 = 2 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1450.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,-12,0,0,4,0,28,0,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.5783082931\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 18x^{10} + 119x^{8} + 346x^{6} + 397x^{4} + 80x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 290)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{7} q^{2} - \beta_{2} q^{3} - q^{4} - \beta_{11} q^{6} - \beta_{10} q^{7} + \beta_{7} q^{8} + ( - \beta_{8} + \beta_{6} + \beta_{5} + \cdots + 3) q^{9} + ( - \beta_{10} - \beta_{6} + \cdots - \beta_1) q^{11}+ \cdots + (4 \beta_{11} - 8 \beta_{10} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{4} + 4 q^{7} + 28 q^{9} + 10 q^{11} - 2 q^{13} + 4 q^{14} + 12 q^{16} - 16 q^{19} - 16 q^{21} + 10 q^{22} - 4 q^{23} - 2 q^{26} - 12 q^{27} - 4 q^{28} + 20 q^{29} + 18 q^{31} + 6 q^{33} - 28 q^{36}+ \cdots + 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 18x^{10} + 119x^{8} + 346x^{6} + 397x^{4} + 80x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + 9\nu^{4} + 19\nu^{2} + 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{9} + \nu^{8} + 14\nu^{7} + 13\nu^{6} + 64\nu^{5} + 53\nu^{4} + 99\nu^{3} + 68\nu^{2} + 18\nu + 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{9} - \nu^{8} + 14\nu^{7} - 13\nu^{6} + 64\nu^{5} - 53\nu^{4} + 99\nu^{3} - 68\nu^{2} + 18\nu - 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{11} + 18 \nu^{9} + \nu^{8} + 118 \nu^{7} + 14 \nu^{6} + 335 \nu^{5} + 64 \nu^{4} + 362 \nu^{3} + \cdots + 14 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{11} + 18\nu^{9} + 119\nu^{7} + 344\nu^{5} + 379\nu^{3} + 42\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{11} + 18 \nu^{9} - \nu^{8} + 118 \nu^{7} - 14 \nu^{6} + 335 \nu^{5} - 64 \nu^{4} + 362 \nu^{3} + \cdots - 18 ) / 4 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - \nu^{11} - \nu^{10} - 18 \nu^{9} - 17 \nu^{8} - 118 \nu^{7} - 105 \nu^{6} - 333 \nu^{5} - 280 \nu^{4} + \cdots - 24 ) / 4 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - \nu^{11} + \nu^{10} - 18 \nu^{9} + 17 \nu^{8} - 118 \nu^{7} + 105 \nu^{6} - 333 \nu^{5} + 280 \nu^{4} + \cdots + 24 ) / 4 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -3\nu^{11} - 54\nu^{9} - 355\nu^{7} - 1014\nu^{5} - 1099\nu^{3} - 122\nu ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + \beta_{8} + \beta_{7} + \beta_{6} - 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{8} + \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - 6\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -9\beta_{11} + \beta_{10} + \beta_{9} - 8\beta_{8} - 9\beta_{7} - 8\beta_{6} + 26\beta _1 - 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 9\beta_{8} - 9\beta_{6} - 9\beta_{5} + 9\beta_{4} + 11\beta_{3} + 35\beta_{2} - 80 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 64\beta_{11} - 9\beta_{10} - 9\beta_{9} + 53\beta_{8} + 68\beta_{7} + 53\beta_{6} - 141\beta _1 + 53 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -64\beta_{8} + 64\beta_{6} + 62\beta_{5} - 62\beta_{4} - 90\beta_{3} - 205\beta_{2} + 441 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 419 \beta_{11} + 62 \beta_{10} + 62 \beta_{9} - 329 \beta_{8} - 475 \beta_{7} - 329 \beta_{6} + \cdots - 329 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2 \beta_{10} - 2 \beta_{9} + 423 \beta_{8} - 423 \beta_{6} - 389 \beta_{5} + 389 \beta_{4} + 655 \beta_{3} + \cdots - 2481 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 2643 \beta_{11} - 389 \beta_{10} - 389 \beta_{9} + 1988 \beta_{8} + 3179 \beta_{7} + 1988 \beta_{6} + \cdots + 1988 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1450\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1277\)
\(\chi(n)\) \(\beta_{7}\) \(\beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
157.1
0.280441i
− 0.398749i
− 1.64632i
1.90612i
2.31763i
− 2.45912i
− 0.280441i
0.398749i
1.64632i
− 1.90612i
− 2.31763i
2.45912i
− 1.00000i −2.92135 −1.00000 0 2.92135i −0.915056 − 0.915056i 1.00000i 5.53430 0
157.2 − 1.00000i −2.84100 −1.00000 0 2.84100i 3.46318 + 3.46318i 1.00000i 5.07128 0
157.3 − 1.00000i −0.289618 −1.00000 0 0.289618i −0.632503 − 0.632503i 1.00000i −2.91612 0
157.4 − 1.00000i 0.633275 −1.00000 0 − 0.633275i 0.961303 + 0.961303i 1.00000i −2.59896 0
157.5 − 1.00000i 2.37143 −1.00000 0 − 2.37143i −1.54761 − 1.54761i 1.00000i 2.62368 0
157.6 − 1.00000i 3.04726 −1.00000 0 − 3.04726i 0.670691 + 0.670691i 1.00000i 6.28582 0
1293.1 1.00000i −2.92135 −1.00000 0 − 2.92135i −0.915056 + 0.915056i − 1.00000i 5.53430 0
1293.2 1.00000i −2.84100 −1.00000 0 − 2.84100i 3.46318 − 3.46318i − 1.00000i 5.07128 0
1293.3 1.00000i −0.289618 −1.00000 0 − 0.289618i −0.632503 + 0.632503i − 1.00000i −2.91612 0
1293.4 1.00000i 0.633275 −1.00000 0 0.633275i 0.961303 − 0.961303i − 1.00000i −2.59896 0
1293.5 1.00000i 2.37143 −1.00000 0 2.37143i −1.54761 + 1.54761i − 1.00000i 2.62368 0
1293.6 1.00000i 3.04726 −1.00000 0 3.04726i 0.670691 − 0.670691i − 1.00000i 6.28582 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 157.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
145.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1450.2.j.h 12
5.b even 2 1 290.2.j.f yes 12
5.c odd 4 1 290.2.e.f ✓ 12
5.c odd 4 1 1450.2.e.h 12
29.c odd 4 1 1450.2.e.h 12
145.e even 4 1 inner 1450.2.j.h 12
145.f odd 4 1 290.2.e.f ✓ 12
145.j even 4 1 290.2.j.f yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
290.2.e.f ✓ 12 5.c odd 4 1
290.2.e.f ✓ 12 145.f odd 4 1
290.2.j.f yes 12 5.b even 2 1
290.2.j.f yes 12 145.j even 4 1
1450.2.e.h 12 5.c odd 4 1
1450.2.e.h 12 29.c odd 4 1
1450.2.j.h 12 1.a even 1 1 trivial
1450.2.j.h 12 145.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1450, [\chi])\):

\( T_{3}^{6} - 16T_{3}^{4} + 2T_{3}^{3} + 64T_{3}^{2} - 20T_{3} - 11 \) Copy content Toggle raw display
\( T_{11}^{12} - 10 T_{11}^{11} + 50 T_{11}^{10} - 6 T_{11}^{9} + 105 T_{11}^{8} - 2664 T_{11}^{7} + \cdots + 7840000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$3$ \( (T^{6} - 16 T^{4} + \cdots - 11)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} - 4 T^{11} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( T^{12} - 10 T^{11} + \cdots + 7840000 \) Copy content Toggle raw display
$13$ \( T^{12} + 2 T^{11} + \cdots + 1225 \) Copy content Toggle raw display
$17$ \( T^{12} + 90 T^{10} + \cdots + 484 \) Copy content Toggle raw display
$19$ \( T^{12} + 16 T^{11} + \cdots + 2050624 \) Copy content Toggle raw display
$23$ \( T^{12} + 4 T^{11} + \cdots + 24964 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 594823321 \) Copy content Toggle raw display
$31$ \( T^{12} - 18 T^{11} + \cdots + 1352569 \) Copy content Toggle raw display
$37$ \( (T^{6} - 144 T^{4} + \cdots - 8192)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} - 36 T^{11} + \cdots + 9834496 \) Copy content Toggle raw display
$43$ \( (T^{6} - 6 T^{5} + \cdots - 3467)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 10 T^{5} + \cdots + 452)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} - 18 T^{11} + \cdots + 1352569 \) Copy content Toggle raw display
$59$ \( T^{12} + 234 T^{10} + \cdots + 5134756 \) Copy content Toggle raw display
$61$ \( T^{12} + 4 T^{11} + \cdots + 24964 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 902652006400 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 21844840000 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 429649984 \) Copy content Toggle raw display
$79$ \( T^{12} - 30 T^{11} + \cdots + 52441 \) Copy content Toggle raw display
$83$ \( (T^{2} - 2 T + 2)^{6} \) Copy content Toggle raw display
$89$ \( T^{12} - 20 T^{11} + \cdots + 42198016 \) Copy content Toggle raw display
$97$ \( (T^{6} + 38 T^{5} + \cdots - 494824)^{2} \) Copy content Toggle raw display
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