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Properties

Label 2254.4.a.z
Level $2254$
Weight $4$
Character orbit 2254.a
Self dual yes
Analytic conductor $132.990$
Analytic rank $1$
Dimension $14$
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: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 200x^{12} + 15521x^{10} - 598294x^{8} + 12167812x^{6} - 125559722x^{4} + 539505876x^{2} - 324615200 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 7^{2} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{2} + \beta_1 q^{3} + 4 q^{4} + ( - \beta_{11} - \beta_1) q^{5} - 2 \beta_1 q^{6} - 8 q^{8} + (\beta_{2} + 2) q^{9} + (2 \beta_{11} + 2 \beta_1) q^{10} + (\beta_{6} - 7) q^{11} + 4 \beta_1 q^{12}+ \cdots + (\beta_{7} - 9 \beta_{6} - \beta_{5} + \cdots + 147) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 28 q^{2} + 56 q^{4} - 112 q^{8} + 22 q^{9} - 92 q^{11} - 268 q^{15} + 224 q^{16} - 44 q^{18} + 184 q^{22} + 322 q^{23} + 130 q^{25} + 196 q^{29} + 536 q^{30} - 448 q^{32} + 88 q^{36} + 628 q^{37}+ \cdots + 1800 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 200x^{12} + 15521x^{10} - 598294x^{8} + 12167812x^{6} - 125559722x^{4} + 539505876x^{2} - 324615200 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 29 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 9375001 \nu^{12} - 968337226 \nu^{10} - 3764862845 \nu^{8} + 2941442295390 \nu^{6} + \cdots - 22\!\cdots\!90 ) / 46909523345958 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 27713645 \nu^{12} - 4302043610 \nu^{10} + 221911961945 \nu^{8} - 4282056580806 \nu^{6} + \cdots + 16\!\cdots\!30 ) / 46909523345958 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 240866125 \nu^{12} - 43961224102 \nu^{10} + 2959957112419 \nu^{8} - 90876839031060 \nu^{6} + \cdots + 31\!\cdots\!40 ) / 46909523345958 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 297294097 \nu^{12} + 54860839720 \nu^{10} - 3769878865687 \nu^{8} + \cdots - 62\!\cdots\!14 ) / 46909523345958 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1088341559 \nu^{12} - 195319865540 \nu^{10} + 12889178720825 \nu^{8} + \cdots + 18\!\cdots\!68 ) / 46909523345958 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 14504688349 \nu^{13} - 2819156131780 \nu^{11} + 209505236543569 \nu^{9} + \cdots + 35\!\cdots\!64 \nu ) / 42\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2664829 \nu^{13} - 476336500 \nu^{11} + 31109264149 \nu^{9} - 911634310506 \nu^{7} + \cdots - 31196102624596 \nu ) / 5214397635720 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 29474458061 \nu^{13} + 5013905712770 \nu^{11} - 304467373298261 \nu^{9} + \cdots + 11\!\cdots\!04 \nu ) / 42\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 11044494271 \nu^{13} + 1995760789060 \nu^{11} - 133186880144911 \nu^{9} + \cdots - 41\!\cdots\!56 \nu ) / 12\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 6798822493 \nu^{13} + 1290384706300 \nu^{11} - 92184808093513 \nu^{9} + \cdots - 59\!\cdots\!88 \nu ) / 60\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 1446155205 \nu^{13} + 264460982596 \nu^{11} - 17919884657349 \nu^{9} + \cdots - 10\!\cdots\!24 \nu ) / 813098404663272 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 29 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} - 3\beta_{12} + \beta_{11} - 2\beta_{10} - 2\beta_{9} - 3\beta_{8} + 45\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} + 3\beta_{6} - \beta_{5} + 6\beta_{4} - 13\beta_{3} + 71\beta_{2} + 1317 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 114\beta_{13} - 261\beta_{12} + 17\beta_{11} - 150\beta_{10} - 163\beta_{9} - 274\beta_{8} + 2396\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 146\beta_{7} + 368\beta_{6} - 191\beta_{5} + 298\beta_{4} - 1253\beta_{3} + 4670\beta_{2} + 71393 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 8772 \beta_{13} - 18407 \beta_{12} - 2528 \beta_{11} - 9436 \beta_{10} - 13000 \beta_{9} + \cdots + 138777 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 14396\beta_{7} + 31180\beta_{6} - 23558\beta_{5} + 6347\beta_{4} - 95969\beta_{3} + 302964\beta_{2} + 4215830 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 607869 \beta_{13} - 1230622 \beta_{12} - 380123 \beta_{11} - 577646 \beta_{10} - 1029924 \beta_{9} + \cdots + 8396823 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1220970 \beta_{7} + 2350009 \beta_{6} - 2298513 \beta_{5} - 451828 \beta_{4} - 6830172 \beta_{3} + \cdots + 259787732 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 40758268 \beta_{13} - 81067693 \beta_{12} - 36909404 \beta_{11} - 35579638 \beta_{10} + \cdots + 521329989 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 95764806 \beta_{7} + 168826332 \beta_{6} - 196623864 \beta_{5} - 79547697 \beta_{4} - 471706827 \beta_{3} + \cdots + 16393179810 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 2706436733 \beta_{13} - 5327445870 \beta_{12} - 3080944471 \beta_{11} - 2220842062 \beta_{10} + \cdots + 32934233237 \beta_1 \) 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
−8.18737
−7.18301
−5.47743
−4.57720
−4.25950
−3.40048
−0.843641
0.843641
3.40048
4.25950
4.57720
5.47743
7.18301
8.18737
−2.00000 −8.18737 4.00000 0.929896 16.3747 0 −8.00000 40.0330 −1.85979
1.2 −2.00000 −7.18301 4.00000 20.3352 14.3660 0 −8.00000 24.5957 −40.6703
1.3 −2.00000 −5.47743 4.00000 −6.20172 10.9549 0 −8.00000 3.00226 12.4034
1.4 −2.00000 −4.57720 4.00000 14.7602 9.15439 0 −8.00000 −6.04928 −29.5203
1.5 −2.00000 −4.25950 4.00000 −9.71073 8.51900 0 −8.00000 −8.85666 19.4215
1.6 −2.00000 −3.40048 4.00000 −0.220651 6.80097 0 −8.00000 −15.4367 0.441301
1.7 −2.00000 −0.843641 4.00000 −13.2267 1.68728 0 −8.00000 −26.2883 26.4534
1.8 −2.00000 0.843641 4.00000 13.2267 −1.68728 0 −8.00000 −26.2883 −26.4534
1.9 −2.00000 3.40048 4.00000 0.220651 −6.80097 0 −8.00000 −15.4367 −0.441301
1.10 −2.00000 4.25950 4.00000 9.71073 −8.51900 0 −8.00000 −8.85666 −19.4215
1.11 −2.00000 4.57720 4.00000 −14.7602 −9.15439 0 −8.00000 −6.04928 29.5203
1.12 −2.00000 5.47743 4.00000 6.20172 −10.9549 0 −8.00000 3.00226 −12.4034
1.13 −2.00000 7.18301 4.00000 −20.3352 −14.3660 0 −8.00000 24.5957 40.6703
1.14 −2.00000 8.18737 4.00000 −0.929896 −16.3747 0 −8.00000 40.0330 1.85979
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
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.z ✓ 14
7.b odd 2 1 inner 2254.4.a.z ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2254.4.a.z ✓ 14 1.a even 1 1 trivial
2254.4.a.z ✓ 14 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{14} - 200 T_{3}^{12} + 15521 T_{3}^{10} - 598294 T_{3}^{8} + 12167812 T_{3}^{6} + \cdots - 324615200 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2254))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 2)^{14} \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots - 324615200 \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots - 2406514688 \) Copy content Toggle raw display
$7$ \( T^{14} \) Copy content Toggle raw display
$11$ \( (T^{7} + 46 T^{6} + \cdots - 225917440)^{2} \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots - 32\!\cdots\!32 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots - 22\!\cdots\!12 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots - 23\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T - 23)^{14} \) Copy content Toggle raw display
$29$ \( (T^{7} + \cdots + 22432350296392)^{2} \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots - 49\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{7} + \cdots + 148197986333696)^{2} \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots - 55\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{7} + \cdots - 230437133548544)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots - 97\!\cdots\!92 \) Copy content Toggle raw display
$53$ \( (T^{7} + \cdots - 32\!\cdots\!44)^{2} \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots - 76\!\cdots\!68 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots - 68\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{7} + \cdots - 14\!\cdots\!24)^{2} \) Copy content Toggle raw display
$71$ \( (T^{7} + \cdots - 26\!\cdots\!40)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots - 14\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{7} + \cdots - 47\!\cdots\!72)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots - 23\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots - 26\!\cdots\!32 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots - 89\!\cdots\!00 \) Copy content Toggle raw display
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