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Properties

Label 2254.4.a.ba
Level $2254$
Weight $4$
Character orbit 2254.a
Self dual yes
Analytic conductor $132.990$
Analytic rank $1$
Dimension $16$
Inner twists $2$

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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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 254 x^{14} + 25397 x^{12} - 1272932 x^{10} + 33773972 x^{8} - 463782026 x^{6} + \cdots + 3521235600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2}\cdot 7^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + \beta_1 q^{3} + 4 q^{4} + ( - \beta_{10} - \beta_1) q^{5} + 2 \beta_1 q^{6} + 8 q^{8} + (\beta_{4} + \beta_{3} + 5) q^{9} + ( - 2 \beta_{10} - 2 \beta_1) q^{10} + (\beta_{7} - \beta_{5} - \beta_{3} - 15) q^{11}+ \cdots + (18 \beta_{8} - 49 \beta_{7} + \cdots - 446) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 32 q^{2} + 64 q^{4} + 128 q^{8} + 76 q^{9} - 232 q^{11} - 388 q^{15} + 256 q^{16} + 152 q^{18} - 464 q^{22} - 368 q^{23} + 320 q^{25} - 632 q^{29} - 776 q^{30} + 512 q^{32} + 304 q^{36} - 1368 q^{37}+ \cdots - 6420 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 254 x^{14} + 25397 x^{12} - 1272932 x^{10} + 33773972 x^{8} - 463782026 x^{6} + \cdots + 3521235600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 884046507739661 \nu^{14} + \cdots - 54\!\cdots\!72 ) / 17\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 52\!\cdots\!96 \nu^{14} + \cdots + 18\!\cdots\!44 ) / 85\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 52\!\cdots\!96 \nu^{14} + \cdots - 46\!\cdots\!16 ) / 85\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 24\!\cdots\!15 \nu^{14} + \cdots - 11\!\cdots\!70 ) / 28\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 30\!\cdots\!80 \nu^{14} + \cdots - 48\!\cdots\!02 ) / 28\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 28\!\cdots\!83 \nu^{14} + \cdots + 14\!\cdots\!04 ) / 57\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 73\!\cdots\!12 \nu^{14} + \cdots - 93\!\cdots\!64 ) / 85\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1048567246 \nu^{15} + 70281423179 \nu^{13} + 21846467051068 \nu^{11} + \cdots - 65\!\cdots\!72 \nu ) / 37\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 10\!\cdots\!94 \nu^{15} + \cdots - 11\!\cdots\!08 \nu ) / 84\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 43\!\cdots\!33 \nu^{15} + \cdots - 52\!\cdots\!24 \nu ) / 16\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 38\!\cdots\!87 \nu^{15} + \cdots + 40\!\cdots\!56 \nu ) / 84\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 73\!\cdots\!71 \nu^{15} + \cdots + 22\!\cdots\!48 \nu ) / 56\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 28\!\cdots\!51 \nu^{15} + \cdots + 70\!\cdots\!88 \nu ) / 16\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 13\!\cdots\!28 \nu^{15} + \cdots - 77\!\cdots\!00 \nu ) / 56\!\cdots\!96 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} + 32 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{15} + \beta_{14} + 3\beta_{13} + 4\beta_{12} + 6\beta_{11} + 8\beta_{10} + 4\beta_{9} + 55\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -7\beta_{8} + 8\beta_{7} - 8\beta_{6} + 5\beta_{5} + 71\beta_{4} + 84\beta_{3} - 10\beta_{2} + 1741 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 244 \beta_{15} + 89 \beta_{14} + 204 \beta_{13} + 427 \beta_{12} + 509 \beta_{11} + 607 \beta_{10} + \cdots + 3387 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 1056 \beta_{8} + 1350 \beta_{7} - 683 \beta_{6} + 1126 \beta_{5} + 4681 \beta_{4} + 5832 \beta_{3} + \cdots + 109203 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 16108 \beta_{15} + 5007 \beta_{14} + 12026 \beta_{13} + 37608 \beta_{12} + 38247 \beta_{11} + \cdots + 220087 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 110272 \beta_{8} + 147821 \beta_{7} - 39810 \beta_{6} + 134404 \beta_{5} + 307763 \beta_{4} + \cdots + 7292232 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1015071 \beta_{15} + 187407 \beta_{14} + 727060 \beta_{13} + 3123704 \beta_{12} + 2841581 \beta_{11} + \cdots + 14834340 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 10025625 \beta_{8} + 13794429 \beta_{7} - 1635271 \beta_{6} + 13145282 \beta_{5} + 20461702 \beta_{4} + \cdots + 505266859 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 63293132 \beta_{15} - 726952 \beta_{14} + 46574035 \beta_{13} + 252488153 \beta_{12} + \cdots + 1027910857 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 853511410 \beta_{8} + 1194500531 \beta_{7} - 12598366 \beta_{6} + 1172568144 \beta_{5} + \cdots + 35894083796 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 3950042783 \beta_{15} - 1100286821 \beta_{14} + 3158043164 \beta_{13} + 20096581290 \beta_{12} + \cdots + 72774204864 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 70120196049 \beta_{8} + 99328068825 \beta_{7} + 7027470531 \beta_{6} + 99427862064 \beta_{5} + \cdots + 2595825417647 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 247950515730 \beta_{15} - 153365977592 \beta_{14} + 224060595765 \beta_{13} + 1584006190849 \beta_{12} + \cdots + 5239530295693 \beta_1 \) 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
−8.75239
−7.85787
−7.01725
−5.94774
−3.83399
−3.37356
−2.08969
−0.764849
0.764849
2.08969
3.37356
3.83399
5.94774
7.01725
7.85787
8.75239
2.00000 −8.75239 4.00000 6.01551 −17.5048 0 8.00000 49.6042 12.0310
1.2 2.00000 −7.85787 4.00000 15.2684 −15.7157 0 8.00000 34.7462 30.5368
1.3 2.00000 −7.01725 4.00000 15.6050 −14.0345 0 8.00000 22.2418 31.2101
1.4 2.00000 −5.94774 4.00000 −7.10437 −11.8955 0 8.00000 8.37564 −14.2087
1.5 2.00000 −3.83399 4.00000 4.35542 −7.66798 0 8.00000 −12.3005 8.71084
1.6 2.00000 −3.37356 4.00000 −2.84894 −6.74711 0 8.00000 −15.6191 −5.69789
1.7 2.00000 −2.08969 4.00000 −21.7453 −4.17939 0 8.00000 −22.6332 −43.4907
1.8 2.00000 −0.764849 4.00000 −9.83634 −1.52970 0 8.00000 −26.4150 −19.6727
1.9 2.00000 0.764849 4.00000 9.83634 1.52970 0 8.00000 −26.4150 19.6727
1.10 2.00000 2.08969 4.00000 21.7453 4.17939 0 8.00000 −22.6332 43.4907
1.11 2.00000 3.37356 4.00000 2.84894 6.74711 0 8.00000 −15.6191 5.69789
1.12 2.00000 3.83399 4.00000 −4.35542 7.66798 0 8.00000 −12.3005 −8.71084
1.13 2.00000 5.94774 4.00000 7.10437 11.8955 0 8.00000 8.37564 14.2087
1.14 2.00000 7.01725 4.00000 −15.6050 14.0345 0 8.00000 22.2418 −31.2101
1.15 2.00000 7.85787 4.00000 −15.2684 15.7157 0 8.00000 34.7462 −30.5368
1.16 2.00000 8.75239 4.00000 −6.01551 17.5048 0 8.00000 49.6042 −12.0310
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2254.4.a.ba ✓ 16
7.b odd 2 1 inner 2254.4.a.ba ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2254.4.a.ba ✓ 16 1.a even 1 1 trivial
2254.4.a.ba ✓ 16 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - 254 T_{3}^{14} + 25397 T_{3}^{12} - 1272932 T_{3}^{10} + 33773972 T_{3}^{8} + \cdots + 3521235600 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2254))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 3521235600 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 730369218318336 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} + 116 T^{7} + \cdots - 22010625792)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 21\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T + 23)^{16} \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots - 20\!\cdots\!92)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 36\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots - 91\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots - 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 18\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 95\!\cdots\!80)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots - 29\!\cdots\!60)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 13\!\cdots\!60)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 17\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 24\!\cdots\!68)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 60\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 18\!\cdots\!64 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 58\!\cdots\!36 \) Copy content Toggle raw display
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