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Properties

Label 392.4.i.l
Level $392$
Weight $4$
Character orbit 392.i
Analytic conductor $23.129$
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] = [392,4,Mod(177,392)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(392, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("392.177");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 392.i (of order \(3\), degree \(2\), not 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: \(23.1287487223\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-19})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 4x^{2} - 5x + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{3} + \beta_1 + 1) q^{3} + (\beta_{3} - \beta_{2} + 11 \beta_1) q^{5} + (2 \beta_{3} - 2 \beta_{2} + 31 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + \beta_1 + 1) q^{3} + (\beta_{3} - \beta_{2} + 11 \beta_1) q^{5} + (2 \beta_{3} - 2 \beta_{2} + 31 \beta_1) q^{9} + ( - 6 \beta_{3} - 18 \beta_1 - 18) q^{11} + ( - 5 \beta_{2} + 21) q^{13} + ( - 12 \beta_{2} - 68) q^{15} + (6 \beta_{3} - 4 \beta_1 - 4) q^{17} + (13 \beta_{3} - 13 \beta_{2} - 59 \beta_1) q^{19} + (4 \beta_{3} - 4 \beta_{2} - 52 \beta_1) q^{23} + ( - 22 \beta_{3} - 53 \beta_1 - 53) q^{25} + ( - 6 \beta_{2} - 118) q^{27} + (22 \beta_{2} - 28) q^{29} + (14 \beta_{3} - 10 \beta_1 - 10) q^{31} + ( - 24 \beta_{3} + 24 \beta_{2} - 360 \beta_1) q^{33} + ( - 10 \beta_{3} + 10 \beta_{2} + 252 \beta_1) q^{37} + (16 \beta_{3} - 264 \beta_1 - 264) q^{39} + ( - 18 \beta_{2} - 272) q^{41} + ( - 14 \beta_{2} + 206) q^{43} + ( - 53 \beta_{3} - 455 \beta_1 - 455) q^{45} + (14 \beta_{3} - 14 \beta_{2} - 250 \beta_1) q^{47} + (2 \beta_{3} - 2 \beta_{2} + 338 \beta_1) q^{51} + (72 \beta_{3} - 134 \beta_1 - 134) q^{53} + (84 \beta_{2} + 540) q^{55} + (46 \beta_{2} - 682) q^{57} + ( - 13 \beta_{3} + 99 \beta_1 + 99) q^{59} + ( - 31 \beta_{3} + 31 \beta_{2} - 173 \beta_1) q^{61} + ( - 34 \beta_{3} + 34 \beta_{2} - 54 \beta_1) q^{65} + (68 \beta_{3} - 504 \beta_1 - 504) q^{67} + (48 \beta_{2} - 176) q^{69} + ( - 44 \beta_{2} - 612) q^{71} + (36 \beta_{3} - 358 \beta_1 - 358) q^{73} + ( - 75 \beta_{3} + 75 \beta_{2} - 1307 \beta_1) q^{75} + (36 \beta_{3} - 36 \beta_{2} + 292 \beta_1) q^{79} + ( - 70 \beta_{3} + 377 \beta_1 + 377) q^{81} + ( - 47 \beta_{2} - 615) q^{83} + ( - 62 \beta_{2} - 298) q^{85} + ( - 6 \beta_{3} + 1226 \beta_1 + 1226) q^{87} + (160 \beta_{3} - 160 \beta_{2} + 298 \beta_1) q^{89} + (4 \beta_{3} - 4 \beta_{2} + 788 \beta_1) q^{93} + ( - 84 \beta_{3} - 92 \beta_1 - 92) q^{95} + ( - 70 \beta_{2} - 428) q^{97} + (222 \beta_{2} + 1242) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 22 q^{5} - 62 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} - 22 q^{5} - 62 q^{9} - 36 q^{11} + 84 q^{13} - 272 q^{15} - 8 q^{17} + 118 q^{19} + 104 q^{23} - 106 q^{25} - 472 q^{27} - 112 q^{29} - 20 q^{31} + 720 q^{33} - 504 q^{37} - 528 q^{39} - 1088 q^{41} + 824 q^{43} - 910 q^{45} + 500 q^{47} - 676 q^{51} - 268 q^{53} + 2160 q^{55} - 2728 q^{57} + 198 q^{59} + 346 q^{61} + 108 q^{65} - 1008 q^{67} - 704 q^{69} - 2448 q^{71} - 716 q^{73} + 2614 q^{75} - 584 q^{79} + 754 q^{81} - 2460 q^{83} - 1192 q^{85} + 2452 q^{87} - 596 q^{89} - 1576 q^{93} - 184 q^{95} - 1712 q^{97} + 4968 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 4x^{2} - 5x + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 4\nu^{2} - 4\nu - 25 ) / 20 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{3} + 2\nu^{2} + 18\nu + 5 ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11\nu^{3} + 4\nu^{2} + 36\nu - 95 ) / 20 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - 3\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} + 27\beta _1 + 27 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4\beta_{3} - 2\beta_{2} + 21 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/392\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(295\) \(297\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
177.1
−1.63746 − 1.52274i
2.13746 + 0.656712i
−1.63746 + 1.52274i
2.13746 − 0.656712i
0 −3.27492 − 5.67232i 0 −1.72508 + 2.98793i 0 0 0 −7.95017 + 13.7701i 0
177.2 0 4.27492 + 7.40437i 0 −9.27492 + 16.0646i 0 0 0 −23.0498 + 39.9235i 0
361.1 0 −3.27492 + 5.67232i 0 −1.72508 − 2.98793i 0 0 0 −7.95017 − 13.7701i 0
361.2 0 4.27492 − 7.40437i 0 −9.27492 − 16.0646i 0 0 0 −23.0498 − 39.9235i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 392.4.i.l 4
7.b odd 2 1 392.4.i.i 4
7.c even 3 1 56.4.a.c ✓ 2
7.c even 3 1 inner 392.4.i.l 4
7.d odd 6 1 392.4.a.h 2
7.d odd 6 1 392.4.i.i 4
21.h odd 6 1 504.4.a.i 2
28.f even 6 1 784.4.a.t 2
28.g odd 6 1 112.4.a.h 2
35.j even 6 1 1400.4.a.i 2
35.l odd 12 2 1400.4.g.h 4
56.k odd 6 1 448.4.a.r 2
56.p even 6 1 448.4.a.s 2
84.n even 6 1 1008.4.a.x 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.4.a.c ✓ 2 7.c even 3 1
112.4.a.h 2 28.g odd 6 1
392.4.a.h 2 7.d odd 6 1
392.4.i.i 4 7.b odd 2 1
392.4.i.i 4 7.d odd 6 1
392.4.i.l 4 1.a even 1 1 trivial
392.4.i.l 4 7.c even 3 1 inner
448.4.a.r 2 56.k odd 6 1
448.4.a.s 2 56.p even 6 1
504.4.a.i 2 21.h odd 6 1
784.4.a.t 2 28.f even 6 1
1008.4.a.x 2 84.n even 6 1
1400.4.a.i 2 35.j even 6 1
1400.4.g.h 4 35.l odd 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(392, [\chi])\):

\( T_{3}^{4} - 2T_{3}^{3} + 60T_{3}^{2} + 112T_{3} + 3136 \) Copy content Toggle raw display
\( T_{5}^{4} + 22T_{5}^{3} + 420T_{5}^{2} + 1408T_{5} + 4096 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 2 T^{3} + \cdots + 3136 \) Copy content Toggle raw display
$5$ \( T^{4} + 22 T^{3} + \cdots + 4096 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 36 T^{3} + \cdots + 2985984 \) Copy content Toggle raw display
$13$ \( (T^{2} - 42 T - 984)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 8 T^{3} + \cdots + 4145296 \) Copy content Toggle raw display
$19$ \( T^{4} - 118 T^{3} + \cdots + 37847104 \) Copy content Toggle raw display
$23$ \( T^{4} - 104 T^{3} + \cdots + 3211264 \) Copy content Toggle raw display
$29$ \( (T^{2} + 56 T - 26804)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 20 T^{3} + \cdots + 122589184 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 3341302416 \) Copy content Toggle raw display
$41$ \( (T^{2} + 544 T + 55516)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 412 T + 31264)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 2634563584 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 77024011024 \) Copy content Toggle raw display
$59$ \( T^{4} - 198 T^{3} + \cdots + 28224 \) Copy content Toggle raw display
$61$ \( T^{4} - 346 T^{3} + \cdots + 617423104 \) Copy content Toggle raw display
$67$ \( T^{4} + 1008 T^{3} + \cdots + 91240704 \) Copy content Toggle raw display
$71$ \( (T^{2} + 1224 T + 264192)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 2947621264 \) Copy content Toggle raw display
$79$ \( T^{4} + 584 T^{3} + \cdots + 129777664 \) Copy content Toggle raw display
$83$ \( (T^{2} + 1230 T + 252312)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 1877985196816 \) Copy content Toggle raw display
$97$ \( (T^{2} + 856 T - 96116)^{2} \) Copy content Toggle raw display
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