Properties

Label 1170.4.a.bh
Level $1170$
Weight $4$
Character orbit 1170.a
Self dual yes
Analytic conductor $69.032$
Analytic rank $0$
Dimension $4$
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] = [1170,4,Mod(1,1170)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1170, 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("1170.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1170.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: \(69.0322347067\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 965x^{2} - 12062x - 26832 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + 4 q^{4} + 5 q^{5} + ( - \beta_{2} + 3) q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + 4 q^{4} + 5 q^{5} + ( - \beta_{2} + 3) q^{7} + 8 q^{8} + 10 q^{10} + ( - \beta_{3} + 10) q^{11} - 13 q^{13} + ( - 2 \beta_{2} + 6) q^{14} + 16 q^{16} + (\beta_{2} - \beta_1 + 51) q^{17} + (\beta_{3} - \beta_{2} + 17) q^{19} + 20 q^{20} + ( - 2 \beta_{3} + 20) q^{22} + (\beta_{2} + \beta_1 + 23) q^{23} + 25 q^{25} - 26 q^{26} + ( - 4 \beta_{2} + 12) q^{28} + ( - \beta_{3} + \beta_{2} + \beta_1 + 33) q^{29} + (2 \beta_{3} - 6 \beta_{2} + \beta_1 + 40) q^{31} + 32 q^{32} + (2 \beta_{2} - 2 \beta_1 + 102) q^{34} + ( - 5 \beta_{2} + 15) q^{35} + (3 \beta_{3} - 8 \beta_{2} + 60) q^{37} + (2 \beta_{3} - 2 \beta_{2} + 34) q^{38} + 40 q^{40} + (3 \beta_{3} + 8 \beta_{2} + 20) q^{41} + ( - 6 \beta_{3} + 4 \beta_{2} + 2 \beta_1 + 8) q^{43} + ( - 4 \beta_{3} + 40) q^{44} + (2 \beta_{2} + 2 \beta_1 + 46) q^{46} + (6 \beta_{3} + 8 \beta_{2} + 2 \beta_1 + 156) q^{47} + ( - \beta_{3} + 10 \beta_{2} - 4 \beta_1 + 201) q^{49} + 50 q^{50} - 52 q^{52} + (3 \beta_{3} + 12 \beta_{2} - 2 \beta_1 + 78) q^{53} + ( - 5 \beta_{3} + 50) q^{55} + ( - 8 \beta_{2} + 24) q^{56} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 66) q^{58} + ( - 4 \beta_{3} - 4 \beta_{2} - 4 \beta_1 + 208) q^{59} + (7 \beta_{3} + 16 \beta_{2} + 176) q^{61} + (4 \beta_{3} - 12 \beta_{2} + 2 \beta_1 + 80) q^{62} + 64 q^{64} - 65 q^{65} + (12 \beta_{3} - 2 \beta_{2} - \beta_1 + 136) q^{67} + (4 \beta_{2} - 4 \beta_1 + 204) q^{68} + ( - 10 \beta_{2} + 30) q^{70} + (9 \beta_{3} - 2 \beta_{2} - 4 \beta_1 + 28) q^{71} + (\beta_{3} + 9 \beta_{2} - \beta_1 + 171) q^{73} + (6 \beta_{3} - 16 \beta_{2} + 120) q^{74} + (4 \beta_{3} - 4 \beta_{2} + 68) q^{76} + ( - 21 \beta_{3} - 14 \beta_{2} - 2 \beta_1 + 84) q^{77} + ( - 11 \beta_{3} + 10 \beta_{2} + 2 \beta_1 + 44) q^{79} + 80 q^{80} + (6 \beta_{3} + 16 \beta_{2} + 40) q^{82} + ( - 12 \beta_{3} + 4 \beta_{2} + 8 \beta_1 - 104) q^{83} + (5 \beta_{2} - 5 \beta_1 + 255) q^{85} + ( - 12 \beta_{3} + 8 \beta_{2} + 4 \beta_1 + 16) q^{86} + ( - 8 \beta_{3} + 80) q^{88} + (3 \beta_{3} - 6 \beta_{2} + 4 \beta_1 - 114) q^{89} + (13 \beta_{2} - 39) q^{91} + (4 \beta_{2} + 4 \beta_1 + 92) q^{92} + (12 \beta_{3} + 16 \beta_{2} + 4 \beta_1 + 312) q^{94} + (5 \beta_{3} - 5 \beta_{2} + 85) q^{95} + ( - 20 \beta_{3} - 33 \beta_{2} + \beta_1 + 43) q^{97} + ( - 2 \beta_{3} + 20 \beta_{2} - 8 \beta_1 + 402) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{2} + 16 q^{4} + 20 q^{5} + 13 q^{7} + 32 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{2} + 16 q^{4} + 20 q^{5} + 13 q^{7} + 32 q^{8} + 40 q^{10} + 39 q^{11} - 52 q^{13} + 26 q^{14} + 64 q^{16} + 203 q^{17} + 70 q^{19} + 80 q^{20} + 78 q^{22} + 91 q^{23} + 100 q^{25} - 104 q^{26} + 52 q^{28} + 130 q^{29} + 168 q^{31} + 128 q^{32} + 406 q^{34} + 65 q^{35} + 251 q^{37} + 140 q^{38} + 160 q^{40} + 75 q^{41} + 22 q^{43} + 156 q^{44} + 182 q^{46} + 622 q^{47} + 793 q^{49} + 200 q^{50} - 208 q^{52} + 303 q^{53} + 195 q^{55} + 104 q^{56} + 260 q^{58} + 832 q^{59} + 695 q^{61} + 336 q^{62} + 256 q^{64} - 260 q^{65} + 558 q^{67} + 812 q^{68} + 130 q^{70} + 123 q^{71} + 676 q^{73} + 502 q^{74} + 280 q^{76} + 329 q^{77} + 155 q^{79} + 320 q^{80} + 150 q^{82} - 432 q^{83} + 1015 q^{85} + 44 q^{86} + 312 q^{88} - 447 q^{89} - 169 q^{91} + 364 q^{92} + 1244 q^{94} + 350 q^{95} + 185 q^{97} + 1586 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 965x^{2} - 12062x - 26832 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 4\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 13\nu^{3} - 154\nu^{2} - 10273\nu - 43542 ) / 970 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} - 18\nu^{2} + 1417\nu + 17780 ) / 194 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -26\beta_{3} - 10\beta_{2} + 21\beta _1 + 1934 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -308\beta_{3} + 180\beta_{2} + 1039\beta _1 + 36308 ) / 4 \) 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
−11.9253
36.2995
−2.88436
−21.4898
2.00000 0 4.00000 5.00000 0 −33.1018 8.00000 0 10.0000
1.2 2.00000 0 4.00000 5.00000 0 0.497846 8.00000 0 10.0000
1.3 2.00000 0 4.00000 5.00000 0 18.9837 8.00000 0 10.0000
1.4 2.00000 0 4.00000 5.00000 0 26.6203 8.00000 0 10.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(-1\)
\(13\) \(1\)

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 1170.4.a.bh yes 4
3.b odd 2 1 1170.4.a.bg 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1170.4.a.bg 4 3.b odd 2 1
1170.4.a.bh yes 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1170))\):

\( T_{7}^{4} - 13T_{7}^{3} - 998T_{7}^{2} + 17228T_{7} - 8328 \) Copy content Toggle raw display
\( T_{11}^{4} - 39T_{11}^{3} - 3572T_{11}^{2} + 126672T_{11} - 876096 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T - 5)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 13 T^{3} - 998 T^{2} + \cdots - 8328 \) Copy content Toggle raw display
$11$ \( T^{4} - 39 T^{3} - 3572 T^{2} + \cdots - 876096 \) Copy content Toggle raw display
$13$ \( (T + 13)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 203 T^{3} + \cdots - 65220480 \) Copy content Toggle raw display
$19$ \( T^{4} - 70 T^{3} - 3164 T^{2} + \cdots + 79488 \) Copy content Toggle raw display
$23$ \( T^{4} - 91 T^{3} - 14252 T^{2} + \cdots + 761760 \) Copy content Toggle raw display
$29$ \( T^{4} - 130 T^{3} - 12616 T^{2} + \cdots + 5180416 \) Copy content Toggle raw display
$31$ \( T^{4} - 168 T^{3} + \cdots + 157975248 \) Copy content Toggle raw display
$37$ \( T^{4} - 251 T^{3} + \cdots - 27818952 \) Copy content Toggle raw display
$41$ \( T^{4} - 75 T^{3} - 107954 T^{2} + \cdots - 42437160 \) Copy content Toggle raw display
$43$ \( T^{4} - 22 T^{3} + \cdots - 895862784 \) Copy content Toggle raw display
$47$ \( T^{4} - 622 T^{3} + \cdots - 32049561600 \) Copy content Toggle raw display
$53$ \( T^{4} - 303 T^{3} + \cdots + 3228673344 \) Copy content Toggle raw display
$59$ \( T^{4} - 832 T^{3} + \cdots - 2851179264 \) Copy content Toggle raw display
$61$ \( T^{4} - 695 T^{3} + \cdots - 24502902840 \) Copy content Toggle raw display
$67$ \( T^{4} - 558 T^{3} + \cdots + 7539849824 \) Copy content Toggle raw display
$71$ \( T^{4} - 123 T^{3} + \cdots + 6787558400 \) Copy content Toggle raw display
$73$ \( T^{4} - 676 T^{3} + \cdots - 27079920 \) Copy content Toggle raw display
$79$ \( T^{4} - 155 T^{3} + \cdots - 36231623232 \) Copy content Toggle raw display
$83$ \( T^{4} + 432 T^{3} + \cdots - 92155737856 \) Copy content Toggle raw display
$89$ \( T^{4} + 447 T^{3} + \cdots + 15361415352 \) Copy content Toggle raw display
$97$ \( T^{4} - 185 T^{3} + \cdots + 653175138840 \) Copy content Toggle raw display
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