Properties

Label 136.4.a.c
Level $136$
Weight $4$
Character orbit 136.a
Self dual yes
Analytic conductor $8.024$
Analytic rank $0$
Dimension $3$
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] = [136,4,Mod(1,136)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(136, 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("136.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 136 = 2^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 136.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: \(8.02425976078\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1556.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 9x + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 3) q^{3} + ( - \beta_1 + 1) q^{5} + (\beta_{2} - \beta_1 + 4) q^{7} + ( - 6 \beta_{2} + 3 \beta_1 + 22) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 3) q^{3} + ( - \beta_1 + 1) q^{5} + (\beta_{2} - \beta_1 + 4) q^{7} + ( - 6 \beta_{2} + 3 \beta_1 + 22) q^{9} + (\beta_{2} + 2 \beta_1 + 23) q^{11} + ( - 2 \beta_{2} + \beta_1 + 17) q^{13} + (6 \beta_{2} - 2 \beta_1 + 20) q^{15} + 17 q^{17} + (10 \beta_{2} + 38) q^{19} + (6 \beta_{2} - 5 \beta_1 - 11) q^{21} + (17 \beta_{2} - 11 \beta_1 + 18) q^{23} + (8 \beta_{2} - 8 \beta_1 - 25) q^{25} + ( - 34 \beta_{2} + 24 \beta_1 + 174) q^{27} + (16 \beta_{2} + 5 \beta_1 - 25) q^{29} + ( - 17 \beta_{2} + 5 \beta_1 + 104) q^{31} + ( - 34 \beta_{2} + \beta_1 - 5) q^{33} + (2 \beta_{2} - 12 \beta_1 + 86) q^{35} + (28 \beta_{2} - 15 \beta_1 - 185) q^{37} + ( - 30 \beta_{2} + 8 \beta_1 + 114) q^{39} + ( - 24 \beta_{2} + 30 \beta_1 + 8) q^{41} + (22 \beta_{2} - 12 \beta_1 + 14) q^{43} + (12 \beta_{2} + 5 \beta_1 - 173) q^{45} + ( - 12 \beta_{2} - 20 \beta_1 + 16) q^{47} + (2 \beta_{2} - 13 \beta_1 - 222) q^{49} + ( - 17 \beta_{2} + 51) q^{51} + (40 \beta_{2} + 30 \beta_1 - 196) q^{53} + ( - 22 \beta_{2} - 10 \beta_1 - 192) q^{55} + ( - 8 \beta_{2} - 30 \beta_1 - 286) q^{57} + (54 \beta_{2} + 26) q^{59} + ( - 68 \beta_{2} + 45 \beta_1 - 197) q^{61} + (37 \beta_{2} - \beta_1 - 296) q^{63} + (4 \beta_{2} - 8 \beta_1 - 48) q^{65} + ( - 8 \beta_{2} + 12 \beta_1 - 112) q^{67} + (110 \beta_{2} - 73 \beta_1 - 439) q^{69} + ( - 25 \beta_{2} + 63 \beta_1 - 102) q^{71} + (92 \beta_{2} - 52 \beta_1 - 206) q^{73} + (105 \beta_{2} - 40 \beta_1 - 259) q^{75} + (18 \beta_{2} + \beta_1 - 49) q^{77} + (85 \beta_{2} + 81 \beta_1 - 278) q^{79} + ( - 282 \beta_{2} + 69 \beta_1 + 880) q^{81} + ( - 70 \beta_{2} - 20 \beta_1 + 322) q^{83} + ( - 17 \beta_1 + 17) q^{85} + (38 \beta_{2} - 38 \beta_1 - 800) q^{87} + ( - 50 \beta_{2} + 37 \beta_1 + 601) q^{89} + (22 \beta_{2} - 10 \beta_1 - 60) q^{91} + ( - 190 \beta_{2} + 61 \beta_1 + 907) q^{93} + ( - 60 \beta_{2} - 48 \beta_1 - 132) q^{95} + (24 \beta_{2} + 40 \beta_1 + 258) q^{97} + ( - 131 \beta_{2} + 50 \beta_1 + 707) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 8 q^{3} + 2 q^{5} + 12 q^{7} + 63 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 8 q^{3} + 2 q^{5} + 12 q^{7} + 63 q^{9} + 72 q^{11} + 50 q^{13} + 64 q^{15} + 51 q^{17} + 124 q^{19} - 32 q^{21} + 60 q^{23} - 75 q^{25} + 512 q^{27} - 54 q^{29} + 300 q^{31} - 48 q^{33} + 248 q^{35} - 542 q^{37} + 320 q^{39} + 30 q^{41} + 52 q^{43} - 502 q^{45} + 16 q^{47} - 677 q^{49} + 136 q^{51} - 518 q^{53} - 608 q^{55} - 896 q^{57} + 132 q^{59} - 614 q^{61} - 852 q^{63} - 148 q^{65} - 332 q^{67} - 1280 q^{69} - 268 q^{71} - 578 q^{73} - 712 q^{75} - 128 q^{77} - 668 q^{79} + 2427 q^{81} + 876 q^{83} + 34 q^{85} - 2400 q^{87} + 1790 q^{89} - 168 q^{91} + 2592 q^{93} - 504 q^{95} + 838 q^{97} + 2040 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 9x + 11 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 4\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{2} + 2\nu - 13 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{2} - \beta _1 + 25 ) / 4 \) 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
2.80979
−3.08060
1.27082
0 −5.40937 0 −9.23914 0 2.17022 0 2.26124 0
1.2 0 3.18096 0 14.3224 0 17.1415 0 −16.8815 0
1.3 0 10.2284 0 −3.08327 0 −7.31168 0 77.6202 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(17\) \(-1\)

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 136.4.a.c 3
3.b odd 2 1 1224.4.a.f 3
4.b odd 2 1 272.4.a.g 3
8.b even 2 1 1088.4.a.s 3
8.d odd 2 1 1088.4.a.z 3
12.b even 2 1 2448.4.a.bf 3
17.b even 2 1 2312.4.a.c 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.4.a.c 3 1.a even 1 1 trivial
272.4.a.g 3 4.b odd 2 1
1088.4.a.s 3 8.b even 2 1
1088.4.a.z 3 8.d odd 2 1
1224.4.a.f 3 3.b odd 2 1
2312.4.a.c 3 17.b even 2 1
2448.4.a.bf 3 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} - 8T_{3}^{2} - 40T_{3} + 176 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(136))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - 8 T^{2} - 40 T + 176 \) Copy content Toggle raw display
$5$ \( T^{3} - 2 T^{2} - 148 T - 408 \) Copy content Toggle raw display
$7$ \( T^{3} - 12 T^{2} - 104 T + 272 \) Copy content Toggle raw display
$11$ \( T^{3} - 72 T^{2} + 952 T + 4752 \) Copy content Toggle raw display
$13$ \( T^{3} - 50 T^{2} + 556 T - 1496 \) Copy content Toggle raw display
$17$ \( (T - 17)^{3} \) Copy content Toggle raw display
$19$ \( T^{3} - 124 T^{2} - 1008 T + 151488 \) Copy content Toggle raw display
$23$ \( T^{3} - 60 T^{2} - 23624 T + 1168976 \) Copy content Toggle raw display
$29$ \( T^{3} + 54 T^{2} - 23156 T - 1826616 \) Copy content Toggle raw display
$31$ \( T^{3} - 300 T^{2} + 13528 T - 122448 \) Copy content Toggle raw display
$37$ \( T^{3} + 542 T^{2} + 40876 T - 451992 \) Copy content Toggle raw display
$41$ \( T^{3} - 30 T^{2} - 127188 T + 13345816 \) Copy content Toggle raw display
$43$ \( T^{3} - 52 T^{2} - 34800 T + 2509632 \) Copy content Toggle raw display
$47$ \( T^{3} - 16 T^{2} - 82560 T + 1737728 \) Copy content Toggle raw display
$53$ \( T^{3} + 518 T^{2} + \cdots - 97851384 \) Copy content Toggle raw display
$59$ \( T^{3} - 132 T^{2} - 173040 T + 2838592 \) Copy content Toggle raw display
$61$ \( T^{3} + 614 T^{2} + \cdots - 115549176 \) Copy content Toggle raw display
$67$ \( T^{3} + 332 T^{2} + 16944 T + 78912 \) Copy content Toggle raw display
$71$ \( T^{3} + 268 T^{2} + \cdots + 104748848 \) Copy content Toggle raw display
$73$ \( T^{3} + 578 T^{2} + \cdots + 17940632 \) Copy content Toggle raw display
$79$ \( T^{3} + 668 T^{2} + \cdots - 976074672 \) Copy content Toggle raw display
$83$ \( T^{3} - 876 T^{2} + \cdots + 211499712 \) Copy content Toggle raw display
$89$ \( T^{3} - 1790 T^{2} + \cdots - 72949736 \) Copy content Toggle raw display
$97$ \( T^{3} - 838 T^{2} + \cdots + 60169976 \) Copy content Toggle raw display
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