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Properties

Label 2254.4.a.v
Level $2254$
Weight $4$
Character orbit 2254.a
Self dual yes
Analytic conductor $132.990$
Analytic rank $1$
Dimension $11$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2254,4,Mod(1,2254)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2254, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2254.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2254 = 2 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2254.a (trivial)

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: yes
Analytic conductor: \(132.990305153\)
Analytic rank: \(1\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - 4 x^{10} - 212 x^{9} + 487 x^{8} + 16315 x^{7} - 9025 x^{6} - 516068 x^{5} - 504693 x^{4} + \cdots - 11394027 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 322)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + (\beta_1 - 2) q^{3} + 4 q^{4} + (\beta_{7} - \beta_1 - 3) q^{5} + (2 \beta_1 - 4) q^{6} + 8 q^{8} + (\beta_{2} - 2 \beta_1 + 16) q^{9} + (2 \beta_{7} - 2 \beta_1 - 6) q^{10} + ( - \beta_{7} + \beta_{3} + 1) q^{11}+ \cdots + ( - \beta_{10} + 15 \beta_{9} + \cdots - 135) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q + 22 q^{2} - 18 q^{3} + 44 q^{4} - 33 q^{5} - 36 q^{6} + 88 q^{8} + 171 q^{9} - 66 q^{10} + 8 q^{11} - 72 q^{12} - 185 q^{13} - 186 q^{15} + 176 q^{16} - 107 q^{17} + 342 q^{18} - 114 q^{19} - 132 q^{20}+ \cdots - 1729 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{11} - 4 x^{10} - 212 x^{9} + 487 x^{8} + 16315 x^{7} - 9025 x^{6} - 516068 x^{5} - 504693 x^{4} + \cdots - 11394027 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 39 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3909755756509 \nu^{10} - 478551021030 \nu^{9} + 984400070792690 \nu^{8} + \cdots + 67\!\cdots\!50 ) / 35\!\cdots\!99 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 45471636080483 \nu^{10} - 439975861834335 \nu^{9} + \cdots - 20\!\cdots\!47 ) / 35\!\cdots\!99 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 6082314394691 \nu^{10} - 36646775167673 \nu^{9} + \cdots - 21\!\cdots\!25 ) / 39\!\cdots\!11 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 21403547944868 \nu^{10} + 104211878420485 \nu^{9} + \cdots + 11\!\cdots\!47 ) / 11\!\cdots\!33 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 75976042778014 \nu^{10} + 421064220460935 \nu^{9} + \cdots + 56\!\cdots\!33 ) / 35\!\cdots\!99 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 89813051301055 \nu^{10} - 549260334843342 \nu^{9} + \cdots - 60\!\cdots\!75 ) / 35\!\cdots\!99 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 101458777559057 \nu^{10} + 592039573313451 \nu^{9} + \cdots + 60\!\cdots\!06 ) / 35\!\cdots\!99 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 50288762161984 \nu^{10} - 297871757887759 \nu^{9} + \cdots - 31\!\cdots\!38 ) / 11\!\cdots\!33 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 39 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{9} - 2\beta_{8} - 5\beta_{7} + \beta_{5} + \beta_{4} - \beta_{3} + 70\beta _1 + 81 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 13 \beta_{10} + \beta_{9} - 3 \beta_{8} - 9 \beta_{7} - 9 \beta_{6} + 14 \beta_{5} + 6 \beta_{4} + \cdots + 2674 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 79 \beta_{10} + 269 \beta_{9} - 270 \beta_{8} - 634 \beta_{7} - 16 \beta_{6} + 146 \beta_{5} + \cdots + 10101 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 1773 \beta_{10} + 332 \beta_{9} - 1066 \beta_{8} - 2463 \beta_{7} - 1157 \beta_{6} + 1520 \beta_{5} + \cdots + 208480 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 13494 \beta_{10} + 21431 \beta_{9} - 29919 \beta_{8} - 67268 \beta_{7} - 3104 \beta_{6} + \cdots + 1066780 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 191560 \beta_{10} + 51060 \beta_{9} - 176131 \beta_{8} - 365652 \beta_{7} - 118652 \beta_{6} + \cdots + 17226713 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1638145 \beta_{10} + 1711582 \beta_{9} - 3166639 \beta_{8} - 6767035 \beta_{7} - 433455 \beta_{6} + \cdots + 107022154 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 19113357 \beta_{10} + 6207430 \beta_{9} - 23034329 \beta_{8} - 44500692 \beta_{7} - 11326078 \beta_{6} + \cdots + 1479682169 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−8.28452
−7.59976
−5.30453
−3.68799
−1.83354
−1.55871
1.12842
3.85313
8.10595
9.21174
9.96981
2.00000 −10.2845 4.00000 16.3087 −20.5690 0 8.00000 78.7713 32.6174
1.2 2.00000 −9.59976 4.00000 −12.3184 −19.1995 0 8.00000 65.1554 −24.6368
1.3 2.00000 −7.30453 4.00000 1.24799 −14.6091 0 8.00000 26.3562 2.49597
1.4 2.00000 −5.68799 4.00000 −15.4130 −11.3760 0 8.00000 5.35322 −30.8260
1.5 2.00000 −3.83354 4.00000 −17.7022 −7.66708 0 8.00000 −12.3040 −35.4044
1.6 2.00000 −3.55871 4.00000 17.9984 −7.11742 0 8.00000 −14.3356 35.9968
1.7 2.00000 −0.871578 4.00000 6.15720 −1.74316 0 8.00000 −26.2404 12.3144
1.8 2.00000 1.85313 4.00000 −2.81000 3.70626 0 8.00000 −23.5659 −5.62000
1.9 2.00000 6.10595 4.00000 5.20425 12.2119 0 8.00000 10.2826 10.4085
1.10 2.00000 7.21174 4.00000 −16.2698 14.4235 0 8.00000 25.0092 −32.5396
1.11 2.00000 7.96981 4.00000 −15.4032 15.9396 0 8.00000 36.5179 −30.8063
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(7\) \( -1 \)
\(23\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2254.4.a.v 11
7.b odd 2 1 2254.4.a.y 11
7.d odd 6 2 322.4.e.a ✓ 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.4.e.a ✓ 22 7.d odd 6 2
2254.4.a.v 11 1.a even 1 1 trivial
2254.4.a.y 11 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{11} + 18 T_{3}^{10} - 72 T_{3}^{9} - 2729 T_{3}^{8} - 4981 T_{3}^{7} + 132809 T_{3}^{6} + \cdots + 31720591 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2254))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{11} \) Copy content Toggle raw display
$3$ \( T^{11} + 18 T^{10} + \cdots + 31720591 \) Copy content Toggle raw display
$5$ \( T^{11} + \cdots - 27782491131 \) Copy content Toggle raw display
$7$ \( T^{11} \) Copy content Toggle raw display
$11$ \( T^{11} + \cdots + 764423984891043 \) Copy content Toggle raw display
$13$ \( T^{11} + \cdots + 72\!\cdots\!13 \) Copy content Toggle raw display
$17$ \( T^{11} + \cdots - 49\!\cdots\!23 \) Copy content Toggle raw display
$19$ \( T^{11} + \cdots - 11\!\cdots\!97 \) Copy content Toggle raw display
$23$ \( (T - 23)^{11} \) Copy content Toggle raw display
$29$ \( T^{11} + \cdots - 59\!\cdots\!13 \) Copy content Toggle raw display
$31$ \( T^{11} + \cdots + 14\!\cdots\!03 \) Copy content Toggle raw display
$37$ \( T^{11} + \cdots - 29\!\cdots\!21 \) Copy content Toggle raw display
$41$ \( T^{11} + \cdots + 22\!\cdots\!13 \) Copy content Toggle raw display
$43$ \( T^{11} + \cdots - 47\!\cdots\!91 \) Copy content Toggle raw display
$47$ \( T^{11} + \cdots - 11\!\cdots\!99 \) Copy content Toggle raw display
$53$ \( T^{11} + \cdots + 83\!\cdots\!33 \) Copy content Toggle raw display
$59$ \( T^{11} + \cdots + 10\!\cdots\!17 \) Copy content Toggle raw display
$61$ \( T^{11} + \cdots - 32\!\cdots\!31 \) Copy content Toggle raw display
$67$ \( T^{11} + \cdots + 83\!\cdots\!63 \) Copy content Toggle raw display
$71$ \( T^{11} + \cdots + 26\!\cdots\!21 \) Copy content Toggle raw display
$73$ \( T^{11} + \cdots + 10\!\cdots\!79 \) Copy content Toggle raw display
$79$ \( T^{11} + \cdots + 81\!\cdots\!47 \) Copy content Toggle raw display
$83$ \( T^{11} + \cdots - 72\!\cdots\!99 \) Copy content Toggle raw display
$89$ \( T^{11} + \cdots + 53\!\cdots\!01 \) Copy content Toggle raw display
$97$ \( T^{11} + \cdots - 17\!\cdots\!41 \) Copy content Toggle raw display
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