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Properties

Label 1458.4.a.j
Level $1458$
Weight $4$
Character orbit 1458.a
Self dual yes
Analytic conductor $86.025$
Analytic rank $0$
Dimension $15$
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] = [1458,4,Mod(1,1458)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1458, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1458.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1458 = 2 \cdot 3^{6} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1458.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: \(86.0247847884\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 339 x^{13} - 1151 x^{12} + 35865 x^{11} + 180141 x^{10} - 1644266 x^{9} - 10786662 x^{8} + \cdots + 33185995624 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{24} \)
Twist minimal: no (minimal twist has level 54)
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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + 4 q^{4} + (\beta_{3} + 1) q^{5} + ( - \beta_{7} + 3) q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + 4 q^{4} + (\beta_{3} + 1) q^{5} + ( - \beta_{7} + 3) q^{7} + 8 q^{8} + (2 \beta_{3} + 2) q^{10} + ( - \beta_{10} + \beta_{3} + 2) q^{11} + (\beta_{9} + 8) q^{13} + ( - 2 \beta_{7} + 6) q^{14} + 16 q^{16} + ( - \beta_{6} + \beta_{3} + 7) q^{17} + ( - \beta_{14} + \beta_{10} + \cdots + 12) q^{19}+ \cdots + (8 \beta_{11} - 4 \beta_{9} + \cdots + 264) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + 30 q^{2} + 60 q^{4} + 15 q^{5} + 42 q^{7} + 120 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 15 q + 30 q^{2} + 60 q^{4} + 15 q^{5} + 42 q^{7} + 120 q^{8} + 30 q^{10} + 33 q^{11} + 117 q^{13} + 84 q^{14} + 240 q^{16} + 102 q^{17} + 171 q^{19} + 60 q^{20} + 66 q^{22} + 174 q^{23} + 600 q^{25} + 234 q^{26} + 168 q^{28} + 573 q^{29} + 372 q^{31} + 480 q^{32} + 204 q^{34} + 624 q^{35} + 555 q^{37} + 342 q^{38} + 120 q^{40} + 852 q^{41} + 1002 q^{43} + 132 q^{44} + 348 q^{46} + 306 q^{47} + 2001 q^{49} + 1200 q^{50} + 468 q^{52} + 897 q^{53} + 1953 q^{55} + 336 q^{56} + 1146 q^{58} + 795 q^{59} + 1587 q^{61} + 744 q^{62} + 960 q^{64} + 1020 q^{65} + 1884 q^{67} + 408 q^{68} + 1248 q^{70} + 120 q^{71} + 2604 q^{73} + 1110 q^{74} + 684 q^{76} + 984 q^{77} + 1944 q^{79} + 240 q^{80} + 1704 q^{82} + 2079 q^{83} + 2916 q^{85} + 2004 q^{86} + 264 q^{88} + 2604 q^{89} + 3399 q^{91} + 696 q^{92} + 612 q^{94} + 1038 q^{95} + 3690 q^{97} + 4002 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 339 x^{13} - 1151 x^{12} + 35865 x^{11} + 180141 x^{10} - 1644266 x^{9} - 10786662 x^{8} + \cdots + 33185995624 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 39\!\cdots\!05 \nu^{14} + \cdots + 29\!\cdots\!28 ) / 45\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 65\!\cdots\!76 \nu^{14} + \cdots + 30\!\cdots\!08 ) / 45\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 18\!\cdots\!81 \nu^{14} + \cdots - 15\!\cdots\!90 ) / 12\!\cdots\!86 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 36\!\cdots\!45 \nu^{14} + \cdots - 34\!\cdots\!80 ) / 12\!\cdots\!86 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 75\!\cdots\!75 \nu^{14} + \cdots - 14\!\cdots\!92 ) / 24\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 14\!\cdots\!81 \nu^{14} + \cdots + 87\!\cdots\!80 ) / 24\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 15\!\cdots\!03 \nu^{14} + \cdots + 12\!\cdots\!64 ) / 24\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 90\!\cdots\!90 \nu^{14} + \cdots + 91\!\cdots\!82 ) / 12\!\cdots\!86 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 65\!\cdots\!49 \nu^{14} + \cdots + 68\!\cdots\!22 ) / 60\!\cdots\!43 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 92\!\cdots\!73 \nu^{14} + \cdots - 49\!\cdots\!04 ) / 80\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 28\!\cdots\!64 \nu^{14} + \cdots - 16\!\cdots\!24 ) / 24\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 96\!\cdots\!99 \nu^{14} + \cdots + 10\!\cdots\!24 ) / 80\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 76\!\cdots\!77 \nu^{14} + \cdots + 62\!\cdots\!98 ) / 40\!\cdots\!62 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 50\!\cdots\!26 \nu^{14} + \cdots - 42\!\cdots\!68 ) / 24\!\cdots\!72 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 3 \beta_{12} - 6 \beta_{11} + 6 \beta_{10} - 3 \beta_{8} + 3 \beta_{7} - 3 \beta_{6} + 3 \beta_{5} + \cdots + 3 ) / 81 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4 \beta_{14} + 5 \beta_{13} + 17 \beta_{12} - 6 \beta_{11} + 5 \beta_{10} - 9 \beta_{9} - 2 \beta_{8} + \cdots + 1221 ) / 27 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 150 \beta_{14} + 222 \beta_{13} + 636 \beta_{12} - 408 \beta_{11} + 294 \beta_{10} - 390 \beta_{9} + \cdots + 18693 ) / 81 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 1220 \beta_{14} + 1354 \beta_{13} + 4300 \beta_{12} - 1233 \beta_{11} + 391 \beta_{10} - 2526 \beta_{9} + \cdots + 155028 ) / 27 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 56103 \beta_{14} + 68334 \beta_{13} + 187458 \beta_{12} - 63471 \beta_{11} + 22425 \beta_{10} + \cdots + 5651856 ) / 81 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 326277 \beta_{14} + 368904 \beta_{13} + 1087296 \beta_{12} - 301464 \beta_{11} + 64737 \beta_{10} + \cdots + 33710457 ) / 27 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 15622965 \beta_{14} + 18142560 \beta_{13} + 50946780 \beta_{12} - 14789742 \beta_{11} + \cdots + 1532327304 ) / 81 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 85536256 \beta_{14} + 97554914 \beta_{13} + 280523966 \beta_{12} - 77827893 \beta_{11} + \cdots + 8472101049 ) / 27 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 4135342704 \beta_{14} + 4748306658 \beta_{13} + 13466261238 \beta_{12} - 3785568951 \beta_{11} + \cdots + 404704171197 ) / 81 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 22365951320 \beta_{14} + 25572138790 \beta_{13} + 73025338021 \beta_{12} - 20309398746 \beta_{11} + \cdots + 2195900321736 ) / 27 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 1083974386344 \beta_{14} + 1241423496132 \beta_{13} + 3531597575847 \beta_{12} - 985905251922 \beta_{11} + \cdots + 106115329126233 ) / 81 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 5847251037777 \beta_{14} + 6689842502325 \beta_{13} + 19068033237213 \beta_{12} + \cdots + 572978641448256 ) / 27 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 283576862805300 \beta_{14} + 324574271828442 \beta_{13} + 924171102777819 \beta_{12} + \cdots + 27\!\cdots\!16 ) / 81 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 15\!\cdots\!55 \beta_{14} + \cdots + 14\!\cdots\!27 ) / 27 \) 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
−8.61540
−5.23606
−6.71810
−2.78882
−1.73555
6.14280
4.59599
16.1700
−4.70587
−2.01851
−4.48162
6.92293
6.13280
−7.37997
3.71541
2.00000 0 4.00000 −18.4078 0 34.7746 8.00000 0 −36.8156
1.2 2.00000 0 4.00000 −17.5886 0 −26.0863 8.00000 0 −35.1771
1.3 2.00000 0 4.00000 −13.7807 0 26.9019 8.00000 0 −27.5613
1.4 2.00000 0 4.00000 −10.1015 0 −8.03419 8.00000 0 −20.2029
1.5 2.00000 0 4.00000 −8.84076 0 −32.7060 8.00000 0 −17.6815
1.6 2.00000 0 4.00000 −7.94244 0 −15.3210 8.00000 0 −15.8849
1.7 2.00000 0 4.00000 −0.857990 0 21.8463 8.00000 0 −1.71598
1.8 2.00000 0 4.00000 −0.733130 0 1.54741 8.00000 0 −1.46626
1.9 2.00000 0 4.00000 3.62116 0 −20.2477 8.00000 0 7.24232
1.10 2.00000 0 4.00000 12.2373 0 18.1798 8.00000 0 24.4746
1.11 2.00000 0 4.00000 12.8582 0 28.5749 8.00000 0 25.7164
1.12 2.00000 0 4.00000 13.6317 0 −13.9856 8.00000 0 27.2634
1.13 2.00000 0 4.00000 14.1026 0 17.2875 8.00000 0 28.2052
1.14 2.00000 0 4.00000 17.3423 0 21.8023 8.00000 0 34.6846
1.15 2.00000 0 4.00000 19.4596 0 −12.5340 8.00000 0 38.9191
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)

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 1458.4.a.j 15
3.b odd 2 1 1458.4.a.i 15
27.e even 9 2 162.4.e.b 30
27.f odd 18 2 54.4.e.b ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.4.e.b ✓ 30 27.f odd 18 2
162.4.e.b 30 27.e even 9 2
1458.4.a.i 15 3.b odd 2 1
1458.4.a.j 15 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{15} - 15 T_{5}^{14} - 1125 T_{5}^{13} + 16197 T_{5}^{12} + 499401 T_{5}^{11} + \cdots - 73586048523336 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1458))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{15} \) Copy content Toggle raw display
$3$ \( T^{15} \) Copy content Toggle raw display
$5$ \( T^{15} + \cdots - 73586048523336 \) Copy content Toggle raw display
$7$ \( T^{15} + \cdots + 23\!\cdots\!36 \) Copy content Toggle raw display
$11$ \( T^{15} + \cdots + 31\!\cdots\!59 \) Copy content Toggle raw display
$13$ \( T^{15} + \cdots - 39\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( T^{15} + \cdots + 40\!\cdots\!09 \) Copy content Toggle raw display
$19$ \( T^{15} + \cdots + 48\!\cdots\!79 \) Copy content Toggle raw display
$23$ \( T^{15} + \cdots - 96\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots - 32\!\cdots\!08 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots + 98\!\cdots\!92 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots - 85\!\cdots\!88 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots - 72\!\cdots\!27 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots + 50\!\cdots\!12 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots - 78\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots + 31\!\cdots\!52 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots - 64\!\cdots\!79 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots - 23\!\cdots\!52 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots - 11\!\cdots\!43 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots - 90\!\cdots\!32 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots + 72\!\cdots\!79 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots - 15\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots + 43\!\cdots\!92 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots + 67\!\cdots\!97 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots + 32\!\cdots\!89 \) Copy content Toggle raw display
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