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Properties

Label 1053.2.b.k
Level $1053$
Weight $2$
Character orbit 1053.b
Analytic conductor $8.408$
Analytic rank $0$
Dimension $12$
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] = [1053,2,Mod(649,1053)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1053, 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("1053.649");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1053 = 3^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1053.b (of order \(2\), degree \(1\), 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: \(8.40824733284\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 10x^{10} + 15x^{8} + 146x^{6} + 481x^{4} - 15x^{2} + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + (\beta_{2} - 1) q^{4} - \beta_{4} q^{5} - \beta_{6} q^{7} + ( - \beta_{5} - \beta_{4} + \beta_{3}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} + (\beta_{2} - 1) q^{4} - \beta_{4} q^{5} - \beta_{6} q^{7} + ( - \beta_{5} - \beta_{4} + \beta_{3}) q^{8} + ( - \beta_{2} + \beta_1 + 1) q^{10} + \beta_{5} q^{11} + ( - \beta_{10} + \beta_1) q^{13} - \beta_{8} q^{14} + ( - \beta_{2} + \beta_1 + 2) q^{16} + ( - \beta_{8} + \beta_{7}) q^{17} - \beta_{9} q^{19} + (2 \beta_{5} - \beta_{3}) q^{20} + (\beta_{2} - 3) q^{22} + ( - \beta_{11} - \beta_{8} - \beta_{7}) q^{23} + ( - \beta_{2} + \beta_1) q^{25} + ( - \beta_{11} - 2 \beta_{8} + \cdots + \beta_{4}) q^{26}+ \cdots + (6 \beta_{5} - \beta_{3}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 8 q^{4} + 4 q^{10} - 4 q^{13} + 16 q^{16} - 32 q^{22} - 8 q^{25} - 52 q^{40} + 24 q^{43} + 36 q^{49} + 20 q^{52} + 4 q^{55} - 56 q^{61} - 4 q^{64} - 80 q^{79} + 4 q^{82} + 32 q^{88} + 12 q^{91} + 76 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 10x^{10} + 15x^{8} + 146x^{6} + 481x^{4} - 15x^{2} + 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 4874\nu^{10} - 58361\nu^{8} + 193967\nu^{6} + 401378\nu^{4} - 8004\nu^{2} - 797771 ) / 2120449 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 7678\nu^{10} - 100637\nu^{8} + 356892\nu^{6} + 748884\nu^{4} - 15219\nu^{2} - 6556544 ) / 2120449 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -197122\nu^{10} + 1880029\nu^{8} - 1618239\nu^{6} - 36090728\nu^{4} - 87182878\nu^{2} - 318156 ) / 19084041 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 207068\nu^{10} - 2090483\nu^{8} + 3613308\nu^{6} + 27531784\nu^{4} + 105162953\nu^{2} - 765978 ) / 19084041 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 351640\nu^{10} - 3510952\nu^{8} + 5201535\nu^{6} + 50966318\nu^{4} + 172236961\nu^{2} - 832953 ) / 19084041 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 100429\nu^{11} - 1021114\nu^{9} + 1760091\nu^{7} + 13685084\nu^{5} + 46221313\nu^{3} + 11259549\nu ) / 12722694 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -316577\nu^{11} + 3233372\nu^{9} - 4751529\nu^{7} - 52741750\nu^{5} - 120618929\nu^{3} + 36050217\nu ) / 38168082 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2209\nu^{11} - 22654\nu^{9} + 38919\nu^{7} + 309932\nu^{5} + 1012495\nu^{3} - 310161\nu ) / 124326 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 80933\nu^{11} - 787670\nu^{9} + 984223\nu^{7} + 12079572\nu^{5} + 42012431\nu^{3} + 10209735\nu ) / 4240898 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 220151\nu^{11} - 2115317\nu^{9} + 2402016\nu^{7} + 33840259\nu^{5} + 118221776\nu^{3} + 28720539\nu ) / 6361347 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1107589 \nu^{11} - 11305774 \nu^{9} + 18646005 \nu^{7} + 161899190 \nu^{5} + 484032820 \nu^{3} - 147496206 \nu ) / 19084041 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{11} + \beta_{8} - 2\beta_{7} + 3\beta_{6} ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} + 2\beta_{4} + \beta_{2} - 3\beta _1 + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -7\beta_{11} + 6\beta_{10} - 12\beta_{9} + 19\beta_{8} - 8\beta_{7} + 3\beta_{6} ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -19\beta_{5} + 22\beta_{4} - 11\beta_{3} + 7\beta_{2} - 12\beta _1 + 17 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -16\beta_{11} + 57\beta_{10} - 129\beta_{9} + 34\beta_{8} - 41\beta_{7} + 60\beta_{6} ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -173\beta_{5} + 222\beta_{4} - 74\beta_{3} + 11\beta_{2} - 11\beta _1 + 30 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 74\beta_{11} + 444\beta_{10} - 1053\beta_{9} - 173\beta_{8} + 148\beta_{7} + 600\beta_{6} ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -1275\beta_{5} + 1604\beta_{4} - 584\beta_{3} - 329\beta_{2} + 543\beta _1 - 814 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 2369\beta_{11} + 2961\beta_{10} - 6786\beta_{9} - 5750\beta_{8} + 4195\beta_{7} + 3357\beta_{6} ) / 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -6819\beta_{5} + 8563\beta_{4} - 3142\beta_{3} - 4952\beta_{2} + 8793\beta _1 - 12010 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 28304\beta_{11} + 11796\beta_{10} - 27441\beta_{9} - 68921\beta_{8} + 49552\beta_{7} + 14484\beta_{6} ) / 6 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1053\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(730\)
\(\chi(n)\) \(1\) \(-1\)

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
−0.386140 + 0.350098i
0.386140 − 0.350098i
2.73354 + 0.691522i
−2.73354 − 0.691522i
−0.615349 + 1.54894i
0.615349 − 1.54894i
0.615349 + 1.54894i
−0.615349 − 1.54894i
−2.73354 + 0.691522i
2.73354 − 0.691522i
0.386140 + 0.350098i
−0.386140 − 0.350098i
− 2.47367i 0 −4.11903 1.50721i 0 − 0.700196i 5.24177i 0 3.72833
649.2 − 2.47367i 0 −4.11903 1.50721i 0 0.700196i 5.24177i 0 3.72833
649.3 − 1.25235i 0 0.431627 − 3.15443i 0 − 1.38304i − 3.04524i 0 −3.95044
649.4 − 1.25235i 0 0.431627 − 3.15443i 0 1.38304i − 3.04524i 0 −3.95044
649.5 − 0.559107i 0 1.68740 2.18584i 0 − 3.09789i − 2.06165i 0 1.22212
649.6 − 0.559107i 0 1.68740 2.18584i 0 3.09789i − 2.06165i 0 1.22212
649.7 0.559107i 0 1.68740 − 2.18584i 0 − 3.09789i 2.06165i 0 1.22212
649.8 0.559107i 0 1.68740 − 2.18584i 0 3.09789i 2.06165i 0 1.22212
649.9 1.25235i 0 0.431627 3.15443i 0 − 1.38304i 3.04524i 0 −3.95044
649.10 1.25235i 0 0.431627 3.15443i 0 1.38304i 3.04524i 0 −3.95044
649.11 2.47367i 0 −4.11903 − 1.50721i 0 − 0.700196i − 5.24177i 0 3.72833
649.12 2.47367i 0 −4.11903 − 1.50721i 0 0.700196i − 5.24177i 0 3.72833
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 649.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.b even 2 1 inner
39.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1053.2.b.k ✓ 12
3.b odd 2 1 inner 1053.2.b.k ✓ 12
13.b even 2 1 inner 1053.2.b.k ✓ 12
39.d odd 2 1 inner 1053.2.b.k ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1053.2.b.k ✓ 12 1.a even 1 1 trivial
1053.2.b.k ✓ 12 3.b odd 2 1 inner
1053.2.b.k ✓ 12 13.b even 2 1 inner
1053.2.b.k ✓ 12 39.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1053, [\chi])\):

\( T_{2}^{6} + 8T_{2}^{4} + 12T_{2}^{2} + 3 \) Copy content Toggle raw display
\( T_{5}^{6} + 17T_{5}^{4} + 81T_{5}^{2} + 108 \) Copy content Toggle raw display
\( T_{17}^{6} - 78T_{17}^{4} + 1089T_{17}^{2} - 3888 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + 8 T^{4} + 12 T^{2} + 3)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{6} + 17 T^{4} + \cdots + 108)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + 12 T^{4} + 24 T^{2} + 9)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + 8 T^{4} + 12 T^{2} + 3)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 2 T^{5} + \cdots + 2197)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 78 T^{4} + \cdots - 3888)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 51 T^{4} + \cdots + 81)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} - 93 T^{4} + \cdots - 8748)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 108 T^{4} + \cdots - 25947)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 159 T^{4} + \cdots + 138384)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 75 T^{4} + \cdots + 1296)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 101 T^{4} + \cdots + 26508)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} - 6 T^{2} + \cdots + 124)^{4} \) Copy content Toggle raw display
$47$ \( (T^{6} + 158 T^{4} + \cdots + 33708)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} - 339 T^{4} + \cdots - 34992)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 233 T^{4} + \cdots + 177147)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 14 T^{2} + \cdots + 67)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + 159 T^{4} + \cdots + 5184)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 281 T^{4} + \cdots + 5547)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 342 T^{4} + \cdots + 186624)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 20 T^{2} + \cdots - 134)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + 305 T^{4} + \cdots + 482403)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 53 T^{4} + \cdots + 12)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 492 T^{4} + \cdots + 3069504)^{2} \) Copy content Toggle raw display
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