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Properties

Label 5292.2.f.e
Level $5292$
Weight $2$
Character orbit 5292.f
Analytic conductor $42.257$
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] = [5292,2,Mod(2645,5292)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5292, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5292.2645");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5292.f (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: \(42.2568327497\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: 12.0.17213603549184.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 5x^{10} + 19x^{8} - 28x^{6} + 31x^{4} - 6x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{9} \)
Twist minimal: no (minimal twist has level 756)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{5} - \beta_{7} q^{11} + ( - \beta_{11} - \beta_{10}) q^{13} + ( - \beta_{6} - \beta_{3}) q^{17} + ( - \beta_{10} + \beta_{9}) q^{19} + (\beta_{7} - \beta_{2}) q^{23} + \beta_{4} q^{25} + ( - \beta_{7} + 2 \beta_{2}) q^{29} + (\beta_{11} - \beta_{10}) q^{31} + (2 \beta_{8} + \beta_{4} + 4) q^{37} + (\beta_{6} - 2 \beta_1) q^{41} + ( - \beta_{4} - 1) q^{43} + \beta_1 q^{47} + ( - \beta_{7} + \beta_{5}) q^{53} + (\beta_{11} + 2 \beta_{10} - 2 \beta_{9}) q^{55} + (\beta_{6} - \beta_{3} - \beta_1) q^{59} + (2 \beta_{11} - \beta_{9}) q^{61} + (\beta_{7} + \beta_{2}) q^{65} + (\beta_{8} - 2 \beta_{4}) q^{67} + \beta_{5} q^{71} + ( - \beta_{11} + 3 \beta_{9}) q^{73} + (\beta_{8} + 2 \beta_{4} - 4) q^{79} + 5 \beta_1 q^{83} + (3 \beta_{8} + 4 \beta_{4} - 1) q^{85} + ( - \beta_{3} + \beta_1) q^{89} + (\beta_{7} + \beta_{5} + 2 \beta_{2}) q^{95} + (2 \beta_{11} + 2 \beta_{10} + 2 \beta_{9}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 48 q^{37} - 12 q^{43} - 48 q^{79} - 12 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 5x^{10} + 19x^{8} - 28x^{6} + 31x^{4} - 6x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -114\nu^{11} + 545\nu^{9} - 2071\nu^{7} + 2831\nu^{5} - 3379\nu^{3} + 1213\nu ) / 559 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 342\nu^{11} - 1635\nu^{9} + 6213\nu^{7} - 8493\nu^{5} + 10137\nu^{3} - 285\nu ) / 559 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 402\nu^{11} - 2422\nu^{9} + 9539\nu^{7} - 17809\nu^{5} + 19712\nu^{3} - 7043\nu ) / 559 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -75\nu^{10} + 285\nu^{8} - 1083\nu^{6} + 465\nu^{4} - 90\nu^{2} - 2453 ) / 559 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 846\nu^{11} - 4221\nu^{9} + 15369\nu^{7} - 21009\nu^{5} + 18456\nu^{3} - 705\nu ) / 559 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -1164\nu^{11} + 6212\nu^{9} - 23941\nu^{7} + 39527\nu^{5} - 44887\nu^{3} + 16063\nu ) / 559 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -1620\nu^{11} + 7833\nu^{9} - 29430\nu^{7} + 40230\nu^{5} - 42192\nu^{3} + 1350\nu ) / 559 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 210\nu^{10} - 798\nu^{8} + 2697\nu^{6} - 1302\nu^{4} + 252\nu^{2} + 2620 ) / 559 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 228\nu^{10} - 1090\nu^{8} + 4142\nu^{6} - 5662\nu^{4} + 6758\nu^{2} - 749 ) / 559 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 308\nu^{10} - 1394\nu^{8} + 5409\nu^{6} - 7276\nu^{4} + 9090\nu^{2} - 1002 ) / 559 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 449\nu^{10} - 2377\nu^{8} + 8809\nu^{6} - 13293\nu^{4} + 12166\nu^{2} - 1399 ) / 559 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} + 3\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{11} - 2\beta_{10} + 5\beta_{9} + \beta_{4} + 5 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} + \beta_{5} + 7\beta_{2} ) / 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -2\beta_{11} - 7\beta_{10} + 13\beta_{9} - \beta_{8} - 4\beta_{4} - 13 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 7\beta_{7} + 12\beta_{6} + 5\beta_{5} + 15\beta_{3} + 20\beta_{2} - 72\beta_1 ) / 18 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -5\beta_{8} - 14\beta_{4} - 38 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -23\beta_{7} + 42\beta_{6} - 19\beta_{5} + 57\beta_{3} - 61\beta_{2} - 225\beta_1 ) / 18 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 9\beta_{11} + 75\beta_{10} - 117\beta_{9} - 19\beta_{8} - 47\beta_{4} - 117 ) / 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -25\beta_{7} - 22\beta_{5} - 64\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 23\beta_{11} + 244\beta_{10} - 370\beta_{9} + 66\beta_{8} + 155\beta_{4} + 370 ) / 6 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -244\beta_{7} - 465\beta_{6} - 221\beta_{5} - 663\beta_{3} - 614\beta_{2} + 2307\beta_1 ) / 18 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5292\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\) \(2647\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2645.1
−1.56052 + 0.900969i
−1.56052 − 0.900969i
−1.07992 + 0.623490i
−1.07992 − 0.623490i
−0.385418 − 0.222521i
−0.385418 + 0.222521i
0.385418 − 0.222521i
0.385418 + 0.222521i
1.07992 + 0.623490i
1.07992 − 0.623490i
1.56052 + 0.900969i
1.56052 − 0.900969i
0 0 0 −3.12105 0 0 0 0 0
2645.2 0 0 0 −3.12105 0 0 0 0 0
2645.3 0 0 0 −2.15983 0 0 0 0 0
2645.4 0 0 0 −2.15983 0 0 0 0 0
2645.5 0 0 0 −0.770835 0 0 0 0 0
2645.6 0 0 0 −0.770835 0 0 0 0 0
2645.7 0 0 0 0.770835 0 0 0 0 0
2645.8 0 0 0 0.770835 0 0 0 0 0
2645.9 0 0 0 2.15983 0 0 0 0 0
2645.10 0 0 0 2.15983 0 0 0 0 0
2645.11 0 0 0 3.12105 0 0 0 0 0
2645.12 0 0 0 3.12105 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2645.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5292.2.f.e 12
3.b odd 2 1 inner 5292.2.f.e 12
7.b odd 2 1 inner 5292.2.f.e 12
7.c even 3 1 756.2.t.e ✓ 12
7.d odd 6 1 756.2.t.e ✓ 12
21.c even 2 1 inner 5292.2.f.e 12
21.g even 6 1 756.2.t.e ✓ 12
21.h odd 6 1 756.2.t.e ✓ 12
63.g even 3 1 2268.2.bm.i 12
63.h even 3 1 2268.2.w.i 12
63.i even 6 1 2268.2.w.i 12
63.j odd 6 1 2268.2.w.i 12
63.k odd 6 1 2268.2.bm.i 12
63.n odd 6 1 2268.2.bm.i 12
63.s even 6 1 2268.2.bm.i 12
63.t odd 6 1 2268.2.w.i 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.t.e ✓ 12 7.c even 3 1
756.2.t.e ✓ 12 7.d odd 6 1
756.2.t.e ✓ 12 21.g even 6 1
756.2.t.e ✓ 12 21.h odd 6 1
2268.2.w.i 12 63.h even 3 1
2268.2.w.i 12 63.i even 6 1
2268.2.w.i 12 63.j odd 6 1
2268.2.w.i 12 63.t odd 6 1
2268.2.bm.i 12 63.g even 3 1
2268.2.bm.i 12 63.k odd 6 1
2268.2.bm.i 12 63.n odd 6 1
2268.2.bm.i 12 63.s even 6 1
5292.2.f.e 12 1.a even 1 1 trivial
5292.2.f.e 12 3.b odd 2 1 inner
5292.2.f.e 12 7.b odd 2 1 inner
5292.2.f.e 12 21.c 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 \(S_{2}^{\mathrm{new}}(5292, [\chi])\):

\( T_{5}^{6} - 15T_{5}^{4} + 54T_{5}^{2} - 27 \) Copy content Toggle raw display
\( T_{13}^{6} + 42T_{13}^{4} + 441T_{13}^{2} + 1323 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{6} - 15 T^{4} + \cdots - 27)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( (T^{6} + 45 T^{4} + \cdots + 729)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 42 T^{4} + \cdots + 1323)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 114 T^{4} + \cdots - 49923)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 51 T^{4} + \cdots + 27)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + 54 T^{4} + \cdots + 729)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 153 T^{4} + \cdots + 123201)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 126 T^{4} + \cdots + 1323)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} - 12 T^{2} + \cdots + 377)^{4} \) Copy content Toggle raw display
$41$ \( (T^{6} - 186 T^{4} + \cdots - 136107)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} + 3 T^{2} - 18 T - 13)^{4} \) Copy content Toggle raw display
$47$ \( (T^{6} - 15 T^{4} + \cdots - 27)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 234 T^{4} + \cdots + 123201)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 303 T^{4} + \cdots - 344763)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 177 T^{4} + \cdots + 22707)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} - 147 T - 497)^{4} \) Copy content Toggle raw display
$71$ \( (T^{6} + 189 T^{4} + \cdots + 35721)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 123 T^{4} + \cdots + 4563)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 12 T^{2} + \cdots - 377)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} - 375 T^{4} + \cdots - 421875)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 27)^{6} \) Copy content Toggle raw display
$97$ \( (T^{6} + 204 T^{4} + \cdots + 1728)^{2} \) Copy content Toggle raw display
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