Properties

Label 153.4.d.b
Level $153$
Weight $4$
Character orbit 153.d
Analytic conductor $9.027$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [153,4,Mod(118,153)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(153, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("153.118");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 153 = 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 153.d (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(9.02729223088\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.4669632.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 74x^{2} + 1072 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 17)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 1) q^{2} + ( - \beta_{2} + 1) q^{4} - \beta_{3} q^{5} + ( - \beta_{3} + \beta_1) q^{7} + (7 \beta_{2} + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 1) q^{2} + ( - \beta_{2} + 1) q^{4} - \beta_{3} q^{5} + ( - \beta_{3} + \beta_1) q^{7} + (7 \beta_{2} + 1) q^{8} + (\beta_{3} + 6 \beta_1) q^{10} + 7 \beta_1 q^{11} + ( - 14 \beta_{2} + 42) q^{13} + 8 \beta_1 q^{14} + (7 \beta_{2} - 63) q^{16} + (\beta_{3} - 14 \beta_{2} - 8 \beta_1 - 21) q^{17} - 28 q^{19} + (\beta_{3} + 6 \beta_1) q^{20} + ( - 7 \beta_{3} + 14 \beta_1) q^{22} + ( - 7 \beta_{3} - 7 \beta_1) q^{23} + ( - 48 \beta_{2} - 91) q^{25} + ( - 42 \beta_{2} + 154) q^{26} + 8 \beta_1 q^{28} + 7 \beta_{3} q^{29} + (\beta_{3} + 7 \beta_1) q^{31} + (7 \beta_{2} - 127) q^{32} + (7 \beta_{3} + 21 \beta_{2} - 22 \beta_1 + 91) q^{34} + ( - 20 \beta_{2} - 224) q^{35} + (7 \beta_{3} + 28 \beta_1) q^{37} + (28 \beta_{2} - 28) q^{38} + ( - 15 \beta_{3} - 42 \beta_1) q^{40} + (14 \beta_{3} - 20 \beta_1) q^{41} + ( - 76 \beta_{2} - 92) q^{43} + ( - 7 \beta_{3} + 14 \beta_1) q^{44} + (14 \beta_{3} + 28 \beta_1) q^{46} + ( - 28 \beta_{2} - 224) q^{47} + (14 \beta_{2} + 71) q^{49} + (91 \beta_{2} + 293) q^{50} + ( - 42 \beta_{2} + 154) q^{52} + (92 \beta_{2} + 98) q^{53} + (196 \beta_{2} - 56) q^{55} + ( - 8 \beta_{3} - 48 \beta_1) q^{56} + ( - 7 \beta_{3} - 42 \beta_1) q^{58} + ( - 140 \beta_{2} + 364) q^{59} + (21 \beta_{3} - 64 \beta_1) q^{61} + ( - 8 \beta_{3} + 8 \beta_1) q^{62} + (71 \beta_{2} + 321) q^{64} + ( - 14 \beta_{3} + 84 \beta_1) q^{65} + ( - 64 \beta_{2} - 28) q^{67} + (7 \beta_{3} + 21 \beta_{2} - 22 \beta_1 + 91) q^{68} + (224 \beta_{2} - 64) q^{70} + (7 \beta_{3} - 21 \beta_1) q^{71} + ( - 56 \beta_{3} + 48 \beta_1) q^{73} + ( - 35 \beta_{3} + 14 \beta_1) q^{74} + (28 \beta_{2} - 28) q^{76} + (238 \beta_{2} - 336) q^{77} + ( - 21 \beta_{3} - 77 \beta_1) q^{79} + (49 \beta_{3} - 42 \beta_1) q^{80} + (6 \beta_{3} - 124 \beta_1) q^{82} + ( - 84 \beta_{2} + 756) q^{83} + (49 \beta_{3} - 176 \beta_{2} + 84 \beta_1 + 280) q^{85} + (92 \beta_{2} + 516) q^{86} + (49 \beta_{3} - 42 \beta_1) q^{88} + (42 \beta_{2} - 518) q^{89} + ( - 28 \beta_{3} + 140 \beta_1) q^{91} + (14 \beta_{3} + 28 \beta_1) q^{92} + 224 \beta_{2} q^{94} + 28 \beta_{3} q^{95} + (22 \beta_{3} - 28 \beta_1) q^{97} + ( - 71 \beta_{2} - 41) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 2 q^{4} + 18 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} + 2 q^{4} + 18 q^{8} + 140 q^{13} - 238 q^{16} - 112 q^{17} - 112 q^{19} - 460 q^{25} + 532 q^{26} - 494 q^{32} + 406 q^{34} - 936 q^{35} - 56 q^{38} - 520 q^{43} - 952 q^{47} + 312 q^{49} + 1354 q^{50} + 532 q^{52} + 576 q^{53} + 168 q^{55} + 1176 q^{59} + 1426 q^{64} - 240 q^{67} + 406 q^{68} + 192 q^{70} - 56 q^{76} - 868 q^{77} + 2856 q^{83} + 768 q^{85} + 2248 q^{86} - 1988 q^{89} + 448 q^{94} - 306 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 74x^{2} + 1072 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} + 40 ) / 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 46\nu ) / 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 6\beta_{2} - 40 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 6\beta_{3} - 46\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/153\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(137\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
118.1
4.44593i
4.44593i
7.36435i
7.36435i
−2.37228 0 −2.37228 19.4389i 0 14.9929i 24.6060 0 46.1145i
118.2 −2.37228 0 −2.37228 19.4389i 0 14.9929i 24.6060 0 46.1145i
118.3 3.37228 0 3.37228 10.1060i 0 17.4703i −15.6060 0 34.0802i
118.4 3.37228 0 3.37228 10.1060i 0 17.4703i −15.6060 0 34.0802i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 153.4.d.b 4
3.b odd 2 1 17.4.b.a 4
12.b even 2 1 272.4.b.d 4
15.d odd 2 1 425.4.d.c 4
15.e even 4 2 425.4.c.c 8
17.b even 2 1 inner 153.4.d.b 4
51.c odd 2 1 17.4.b.a 4
51.f odd 4 2 289.4.a.e 4
204.h even 2 1 272.4.b.d 4
255.h odd 2 1 425.4.d.c 4
255.o even 4 2 425.4.c.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.4.b.a 4 3.b odd 2 1
17.4.b.a 4 51.c odd 2 1
153.4.d.b 4 1.a even 1 1 trivial
153.4.d.b 4 17.b even 2 1 inner
272.4.b.d 4 12.b even 2 1
272.4.b.d 4 204.h even 2 1
289.4.a.e 4 51.f odd 4 2
425.4.c.c 8 15.e even 4 2
425.4.c.c 8 255.o even 4 2
425.4.d.c 4 15.d odd 2 1
425.4.d.c 4 255.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - T_{2} - 8 \) acting on \(S_{4}^{\mathrm{new}}(153, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T - 8)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 480 T^{2} + 38592 \) Copy content Toggle raw display
$7$ \( T^{4} + 530 T^{2} + 68608 \) Copy content Toggle raw display
$11$ \( T^{4} + 3626 T^{2} + \cdots + 2573872 \) Copy content Toggle raw display
$13$ \( (T^{2} - 70 T - 392)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 112 T^{3} + \cdots + 24137569 \) Copy content Toggle raw display
$19$ \( (T + 28)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 28322 T^{2} + \cdots + 10295488 \) Copy content Toggle raw display
$29$ \( T^{4} + 23520 T^{2} + \cdots + 92659392 \) Copy content Toggle raw display
$31$ \( T^{4} + 4274 T^{2} + \cdots + 4390912 \) Copy content Toggle raw display
$37$ \( T^{4} + 86240 T^{2} + \cdots + 1245754048 \) Copy content Toggle raw display
$41$ \( T^{4} + 116960 T^{2} + \cdots + 2799480832 \) Copy content Toggle raw display
$43$ \( (T^{2} + 260 T - 30752)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 476 T + 50176)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 288 T - 49092)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 588 T - 75264)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 482528 T^{2} + \cdots + 7146728128 \) Copy content Toggle raw display
$67$ \( (T^{2} + 120 T - 30192)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 52626 T^{2} + \cdots + 92659392 \) Copy content Toggle raw display
$73$ \( T^{4} + 1611264 T^{2} + \cdots + 647466319872 \) Copy content Toggle raw display
$79$ \( T^{4} + 689234 T^{2} + \cdots + 70925616832 \) Copy content Toggle raw display
$83$ \( (T^{2} - 1428 T + 451584)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 994 T + 232456)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 275552 T^{2} + \cdots + 16878665728 \) Copy content Toggle raw display
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