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Properties

Label 1458.4.a.h
Level $1458$
Weight $4$
Character orbit 1458.a
Self dual yes
Analytic conductor $86.025$
Analytic rank $1$
Dimension $12$
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: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} - 84 x^{10} + 218 x^{9} + 2661 x^{8} - 5916 x^{7} - 38663 x^{6} + 75069 x^{5} + \cdots - 160299 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{14} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + 4 q^{4} + (\beta_{4} - 1) q^{5} + (\beta_{8} - 3) q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + 4 q^{4} + (\beta_{4} - 1) q^{5} + (\beta_{8} - 3) q^{7} + 8 q^{8} + (2 \beta_{4} - 2) q^{10} + ( - \beta_{11} + \beta_{10} - 2 \beta_{8} + \cdots - 3) q^{11}+ \cdots + ( - 2 \beta_{11} + 24 \beta_{10} + \cdots - 126) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 24 q^{2} + 48 q^{4} - 15 q^{5} - 42 q^{7} + 96 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 24 q^{2} + 48 q^{4} - 15 q^{5} - 42 q^{7} + 96 q^{8} - 30 q^{10} - 33 q^{11} - 117 q^{13} - 84 q^{14} + 192 q^{16} - 102 q^{17} - 171 q^{19} - 60 q^{20} - 66 q^{22} - 174 q^{23} + 75 q^{25} - 234 q^{26} - 168 q^{28} - 573 q^{29} - 372 q^{31} + 384 q^{32} - 204 q^{34} - 624 q^{35} - 555 q^{37} - 342 q^{38} - 120 q^{40} - 852 q^{41} - 1002 q^{43} - 132 q^{44} - 348 q^{46} - 306 q^{47} - 678 q^{49} + 150 q^{50} - 468 q^{52} - 897 q^{53} - 1953 q^{55} - 336 q^{56} - 1146 q^{58} - 795 q^{59} - 2667 q^{61} - 744 q^{62} + 768 q^{64} - 1020 q^{65} - 2640 q^{67} - 408 q^{68} - 1248 q^{70} - 120 q^{71} - 3036 q^{73} - 1110 q^{74} - 684 q^{76} - 984 q^{77} - 1944 q^{79} - 240 q^{80} - 1704 q^{82} - 54 q^{83} - 2916 q^{85} - 2004 q^{86} - 264 q^{88} - 1065 q^{89} - 2859 q^{91} - 696 q^{92} - 612 q^{94} - 1038 q^{95} - 3690 q^{97} - 1356 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} - 84 x^{10} + 218 x^{9} + 2661 x^{8} - 5916 x^{7} - 38663 x^{6} + 75069 x^{5} + \cdots - 160299 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 483092425 \nu^{11} + 6547935408 \nu^{10} - 50441235508 \nu^{9} - 550054209938 \nu^{8} + \cdots - 930562872756255 ) / 61300409700528 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 883275495 \nu^{11} + 47064223588 \nu^{10} - 126143895176 \nu^{9} + \cdots + 230027476480245 ) / 61300409700528 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 4268201 \nu^{11} + 10992920 \nu^{10} + 386648860 \nu^{9} - 772914038 \nu^{8} + \cdots - 1320893754057 ) / 112892098896 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2530632623 \nu^{11} + 4131545664 \nu^{10} + 207766593020 \nu^{9} + \cdots + 101470506466617 ) / 61300409700528 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 5248735 \nu^{11} + 18365086 \nu^{10} + 350080574 \nu^{9} - 1037433364 \nu^{8} + \cdots + 665690370111 ) / 56446049448 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5767934801 \nu^{11} - 37505154088 \nu^{10} - 336844649052 \nu^{9} + 2315297295494 \nu^{8} + \cdots + 679469869520865 ) / 61300409700528 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2986833435 \nu^{11} + 2191284124 \nu^{10} + 272242440328 \nu^{9} + \cdots - 736618210253559 ) / 30650204850264 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1707163627 \nu^{11} - 1750054217 \nu^{10} - 143293099587 \nu^{9} + 128915069041 \nu^{8} + \cdots + 558175599185670 ) / 15325102425132 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 8016642805 \nu^{11} - 31075759104 \nu^{10} - 565628205076 \nu^{9} + 1917761429590 \nu^{8} + \cdots + 15\!\cdots\!09 ) / 61300409700528 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2165017008 \nu^{11} - 6685860347 \nu^{10} - 166616138873 \nu^{9} + 427593341553 \nu^{8} + \cdots + 414932309597097 ) / 15325102425132 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 7184722691 \nu^{11} - 4938199530 \nu^{10} + 676955114126 \nu^{9} + \cdots - 580316676454053 ) / 30650204850264 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{10} + \beta_{7} + \beta_{5} - \beta_{4} + \beta _1 + 2 ) / 9 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{9} - 3\beta_{8} - 4\beta_{6} - 2\beta_{5} - 4\beta_{4} - 2\beta_{3} - \beta_{2} + \beta _1 + 130 ) / 9 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 3 \beta_{11} + 23 \beta_{10} + 5 \beta_{9} - 3 \beta_{8} + 23 \beta_{7} - 7 \beta_{6} + 21 \beta_{5} + \cdots + 95 ) / 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 9 \beta_{11} + 23 \beta_{10} + 81 \beta_{9} - 93 \beta_{8} + 5 \beta_{7} - 105 \beta_{6} + \cdots + 2893 ) / 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 141 \beta_{11} + 534 \beta_{10} + 289 \beta_{9} - 144 \beta_{8} + 678 \beta_{7} - 317 \beta_{6} + \cdots + 4337 ) / 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 645 \beta_{11} + 1030 \beta_{10} + 2731 \beta_{9} - 2832 \beta_{8} + 517 \beta_{7} - 2711 \beta_{6} + \cdots + 73282 ) / 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 5499 \beta_{11} + 12694 \beta_{10} + 11943 \beta_{9} - 5685 \beta_{8} + 20299 \beta_{7} + \cdots + 162635 ) / 9 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 28992 \beta_{11} + 35169 \beta_{10} + 89117 \beta_{9} - 84393 \beta_{8} + 26871 \beta_{7} + \cdots + 1979092 ) / 9 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 196320 \beta_{11} + 310127 \beta_{10} + 436775 \beta_{9} - 201288 \beta_{8} + 604967 \beta_{7} + \cdots + 5592065 ) / 9 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 1100583 \beta_{11} + 1081211 \beta_{10} + 2881395 \beta_{9} - 2468622 \beta_{8} + 1119884 \beta_{7} + \cdots + 55588891 ) / 9 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 6691146 \beta_{11} + 7751271 \beta_{10} + 15070834 \beta_{9} - 6666552 \beta_{8} + 18003531 \beta_{7} + \cdots + 184608044 ) / 9 \) 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
−5.09348
2.26518
2.83003
4.63561
1.28872
5.64664
0.202756
−4.50760
5.39897
−3.61988
−4.02029
−2.02668
2.00000 0 4.00000 −19.9081 0 −18.7985 8.00000 0 −39.8162
1.2 2.00000 0 4.00000 −16.7110 0 10.3947 8.00000 0 −33.4219
1.3 2.00000 0 4.00000 −14.4354 0 1.11687 8.00000 0 −28.8709
1.4 2.00000 0 4.00000 −7.39985 0 11.9119 8.00000 0 −14.7997
1.5 2.00000 0 4.00000 −3.67149 0 0.904412 8.00000 0 −7.34298
1.6 2.00000 0 4.00000 −1.14586 0 2.70850 8.00000 0 −2.29173
1.7 2.00000 0 4.00000 1.17173 0 26.0678 8.00000 0 2.34346
1.8 2.00000 0 4.00000 4.52754 0 −24.4739 8.00000 0 9.05509
1.9 2.00000 0 4.00000 6.81687 0 −33.5777 8.00000 0 13.6337
1.10 2.00000 0 4.00000 7.06572 0 5.02029 8.00000 0 14.1314
1.11 2.00000 0 4.00000 7.57519 0 −3.78677 8.00000 0 15.1504
1.12 2.00000 0 4.00000 21.1147 0 −19.4876 8.00000 0 42.2293
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
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.h 12
3.b odd 2 1 1458.4.a.e 12
27.e even 9 2 54.4.e.a ✓ 24
27.f odd 18 2 162.4.e.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.4.e.a ✓ 24 27.e even 9 2
162.4.e.a 24 27.f odd 18 2
1458.4.a.e 12 3.b odd 2 1
1458.4.a.h 12 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 15 T_{5}^{11} - 675 T_{5}^{10} - 9447 T_{5}^{9} + 128439 T_{5}^{8} + 1329318 T_{5}^{7} + \cdots + 6110399349 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1458))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 6110399349 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots - 50539227821 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 13\!\cdots\!19 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 26\!\cdots\!19 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 59\!\cdots\!57 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 12\!\cdots\!19 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots - 19\!\cdots\!47 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 13\!\cdots\!37 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 31\!\cdots\!57 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 54\!\cdots\!43 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots - 30\!\cdots\!47 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots - 52\!\cdots\!32 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 88\!\cdots\!69 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots - 60\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots - 16\!\cdots\!77 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots - 30\!\cdots\!99 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots - 25\!\cdots\!57 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots - 10\!\cdots\!67 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 14\!\cdots\!79 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots - 13\!\cdots\!87 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots - 34\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 34\!\cdots\!27 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots - 31\!\cdots\!21 \) Copy content Toggle raw display
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