Properties

Label 1452.4.a.s
Level $1452$
Weight $4$
Character orbit 1452.a
Self dual yes
Analytic conductor $85.671$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1452,4,Mod(1,1452)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1452, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1452.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1452 = 2^{2} \cdot 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1452.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: \(85.6707733283\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 57x^{4} - 42x^{3} + 603x^{2} + 630x - 882 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 q^{3} + ( - \beta_{4} - 2) q^{5} + (\beta_{5} - \beta_1) q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{3} + ( - \beta_{4} - 2) q^{5} + (\beta_{5} - \beta_1) q^{7} + 9 q^{9} + (\beta_{5} + 2 \beta_{2} + 14 \beta_1) q^{13} + ( - 3 \beta_{4} - 6) q^{15} + (3 \beta_{5} + 2 \beta_{2} + 10 \beta_1) q^{17} + (4 \beta_{5} - \beta_{2} + \beta_1) q^{19} + (3 \beta_{5} - 3 \beta_1) q^{21} + ( - 2 \beta_{4} + 3 \beta_{3} + 26) q^{23} + ( - \beta_{4} - 2 \beta_{3} + 25) q^{25} + 27 q^{27} + ( - 6 \beta_{5} - 3 \beta_{2} - 36 \beta_1) q^{29} + ( - 5 \beta_{4} - 4 \beta_{3} + 25) q^{31} + ( - 9 \beta_{5} + 5 \beta_{2} + 40 \beta_1) q^{35} + ( - 18 \beta_{4} + 3 \beta_{3} + 61) q^{37} + (3 \beta_{5} + 6 \beta_{2} + 42 \beta_1) q^{39} + ( - 3 \beta_{5} - 16 \beta_{2} - 68 \beta_1) q^{41} + ( - 3 \beta_{5} - 7 \beta_{2} - 122 \beta_1) q^{43} + ( - 9 \beta_{4} - 18) q^{45} + ( - 6 \beta_{4} - 6 \beta_{3} + 90) q^{47} + (31 \beta_{4} + 2 \beta_{3} + 218) q^{49} + (9 \beta_{5} + 6 \beta_{2} + 30 \beta_1) q^{51} + ( - 33 \beta_{4} + 6 \beta_{3} + 60) q^{53} + (12 \beta_{5} - 3 \beta_{2} + 3 \beta_1) q^{57} + (2 \beta_{4} - 3 \beta_{3} + 418) q^{59} + ( - 4 \beta_{5} - 7 \beta_{2} + 187 \beta_1) q^{61} + (9 \beta_{5} - 9 \beta_1) q^{63} + (3 \beta_{5} - 282 \beta_1) q^{65} + (23 \beta_{4} + \beta_{3} + 169) q^{67} + ( - 6 \beta_{4} + 9 \beta_{3} + 78) q^{69} + ( - 44 \beta_{4} + 3 \beta_{3} - 196) q^{71} + (14 \beta_{5} - 13 \beta_{2} + 47 \beta_1) q^{73} + ( - 3 \beta_{4} - 6 \beta_{3} + 75) q^{75} + (8 \beta_{5} + 37 \beta_{2} + 99 \beta_1) q^{79} + 81 q^{81} + ( - 12 \beta_{5} + 10 \beta_{2} + 386 \beta_1) q^{83} + ( - 15 \beta_{5} + 12 \beta_{2} - 198 \beta_1) q^{85} + ( - 18 \beta_{5} - 9 \beta_{2} - 108 \beta_1) q^{87} + (5 \beta_{4} - 27 \beta_{3} + 454) q^{89} + ( - 28 \beta_{4} + 31 \beta_{3} + 288) q^{91} + ( - 15 \beta_{4} - 12 \beta_{3} + 75) q^{93} + ( - 42 \beta_{5} + 10 \beta_{2} + 296 \beta_1) q^{95} + (76 \beta_{4} + 8 \beta_{3} + 965) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 18 q^{3} - 12 q^{5} + 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 18 q^{3} - 12 q^{5} + 54 q^{9} - 36 q^{15} + 156 q^{23} + 150 q^{25} + 162 q^{27} + 150 q^{31} + 366 q^{37} - 108 q^{45} + 540 q^{47} + 1308 q^{49} + 360 q^{53} + 2508 q^{59} + 1014 q^{67} + 468 q^{69} - 1176 q^{71} + 450 q^{75} + 486 q^{81} + 2724 q^{89} + 1728 q^{91} + 450 q^{93} + 5790 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 57x^{4} - 42x^{3} + 603x^{2} + 630x - 882 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 8\nu^{5} - 21\nu^{4} - 400\nu^{3} + 714\nu^{2} + 2913\nu - 2625 ) / 21 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -15\nu^{5} + 42\nu^{4} + 743\nu^{3} - 1449\nu^{2} - 5286\nu + 5376 ) / 21 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 36\nu^{5} - 98\nu^{4} - 1800\nu^{3} + 3402\nu^{2} + 13308\nu - 13020 ) / 21 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -66\nu^{5} + 175\nu^{4} + 3300\nu^{3} - 5985\nu^{2} - 24006\nu + 22260 ) / 21 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -69\nu^{5} + 182\nu^{4} + 3457\nu^{3} - 6195\nu^{2} - 25374\nu + 22596 ) / 21 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 2\beta_{4} + \beta_{3} + 12\beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{5} - 2\beta_{4} + \beta_{3} + 2\beta_{2} + 76 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 9\beta_{5} + 22\beta_{4} + 11\beta_{3} - 3\beta_{2} + 204\beta _1 + 84 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 108\beta_{5} - 70\beta_{4} + 49\beta_{3} + 108\beta_{2} + 336\beta _1 + 2724 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 555\beta_{5} + 852\beta_{4} + 468\beta_{3} - 45\beta_{2} + 9636\beta _1 + 5880 ) / 4 \) Copy content Toggle raw display

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
0.826066
−2.63804
3.55668
7.02078
−2.65070
−6.11480
0 3.00000 0 −14.4612 0 −29.5924 0 9.00000 0
1.2 0 3.00000 0 −14.4612 0 29.5924 0 9.00000 0
1.3 0 3.00000 0 −5.89550 0 −28.1622 0 9.00000 0
1.4 0 3.00000 0 −5.89550 0 28.1622 0 9.00000 0
1.5 0 3.00000 0 14.3567 0 −3.76593 0 9.00000 0
1.6 0 3.00000 0 14.3567 0 3.76593 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( -1 \)
\(11\) \( +1 \)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1452.4.a.s 6
11.b odd 2 1 inner 1452.4.a.s 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1452.4.a.s 6 1.a even 1 1 trivial
1452.4.a.s 6 11.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1452))\):

\( T_{5}^{3} + 6T_{5}^{2} - 207T_{5} - 1224 \) Copy content Toggle raw display
\( T_{7}^{6} - 1683T_{7}^{4} + 718200T_{7}^{2} - 9850032 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T - 3)^{6} \) Copy content Toggle raw display
$5$ \( (T^{3} + 6 T^{2} + \cdots - 1224)^{2} \) Copy content Toggle raw display
$7$ \( T^{6} - 1683 T^{4} + \cdots - 9850032 \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots - 8493273792 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots - 15990504192 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots - 108774759168 \) Copy content Toggle raw display
$23$ \( (T^{3} - 78 T^{2} + \cdots - 18216)^{2} \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots - 133671898188 \) Copy content Toggle raw display
$31$ \( (T^{3} - 75 T^{2} + \cdots + 4283504)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} - 183 T^{2} + \cdots + 12642299)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots - 16\!\cdots\!12 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots - 447970523328 \) Copy content Toggle raw display
$47$ \( (T^{3} - 270 T^{2} + \cdots + 18907776)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} - 180 T^{2} + \cdots + 72201402)^{2} \) Copy content Toggle raw display
$59$ \( (T^{3} - 1254 T^{2} + \cdots - 63358344)^{2} \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots - 560781252145152 \) Copy content Toggle raw display
$67$ \( (T^{3} - 507 T^{2} + \cdots + 29960396)^{2} \) Copy content Toggle raw display
$71$ \( (T^{3} + 588 T^{2} + \cdots - 79195824)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots - 10\!\cdots\!88 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots - 19\!\cdots\!48 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots - 35\!\cdots\!28 \) Copy content Toggle raw display
$89$ \( (T^{3} - 1362 T^{2} + \cdots + 1387816812)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} - 2895 T^{2} + \cdots + 1212723379)^{2} \) Copy content Toggle raw display
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