Properties

Label 392.4.i.m
Level $392$
Weight $4$
Character orbit 392.i
Analytic conductor $23.129$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.11163123.4
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 14x^{4} + 49x^{2} + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3 \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} - 2 \beta_1) q^{3} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 1) q^{5} + ( - 7 \beta_{5} + \beta_{4} + 7 \beta_{3} + \beta_{2} + 4 \beta_1 - 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} - 2 \beta_1) q^{3} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 1) q^{5} + ( - 7 \beta_{5} + \beta_{4} + 7 \beta_{3} + \beta_{2} + 4 \beta_1 - 4) q^{9} - 3 \beta_{4} q^{11} + (\beta_{3} + 3 \beta_{2} + 8) q^{13} + ( - 14 \beta_{3} - \beta_{2} + 38) q^{15} + ( - 13 \beta_{5} - \beta_{4} - 15 \beta_1) q^{17} + ( - 2 \beta_{5} - 3 \beta_{4} + 2 \beta_{3} - 3 \beta_{2} + 28 \beta_1 - 28) q^{19} + (7 \beta_{5} + 2 \beta_{4} - 7 \beta_{3} + 2 \beta_{2} - 64 \beta_1 + 64) q^{23} + (14 \beta_{5} + 10 \beta_{4} - 92 \beta_1) q^{25} + ( - 20 \beta_{3} - 7 \beta_{2} + 152) q^{27} + ( - 7 \beta_{3} + 3 \beta_{2} + 60) q^{29} + ( - 26 \beta_{5} - \beta_{4} - 122 \beta_1) q^{31} + ( - 24 \beta_{5} + 24 \beta_{3} + 27 \beta_1 - 27) q^{33} + ( - 14 \beta_{5} - 12 \beta_{4} + 14 \beta_{3} - 12 \beta_{2} - 187 \beta_1 + 187) q^{37} + ( - 21 \beta_{5} + \beta_{4} + 38 \beta_1) q^{39} + (23 \beta_{3} - 11 \beta_{2} - 8) q^{41} + (56 \beta_{3} + 108) q^{43} + (89 \beta_{5} + 13 \beta_{4} - 436 \beta_1) q^{45} + (50 \beta_{5} + 3 \beta_{4} - 50 \beta_{3} + 3 \beta_{2} + 86 \beta_1 - 86) q^{47} + (42 \beta_{5} - 13 \beta_{4} - 42 \beta_{3} - 13 \beta_{2} - 312 \beta_1 + 312) q^{51} + ( - 28 \beta_{5} - 34 \beta_{4} - 273 \beta_1) q^{53} + ( - 3 \beta_{3} + 30 \beta_{2} + 540) q^{55} + ( - 14 \beta_{3} - 2 \beta_{2} + 83) q^{57} + (29 \beta_{5} - 590 \beta_1) q^{59} + (66 \beta_{5} + 4 \beta_{4} - 66 \beta_{3} + 4 \beta_{2} - 185 \beta_1 + 185) q^{61} + (7 \beta_{5} + 39 \beta_{4} - 7 \beta_{3} + 39 \beta_{2} - 568 \beta_1 + 568) q^{65} + ( - 7 \beta_{5} + 34 \beta_{4} + 174 \beta_1) q^{67} + (83 \beta_{3} + 7 \beta_{2} - 299) q^{69} + (14 \beta_{3} - 6 \beta_{2} + 220) q^{71} + ( - 98 \beta_{5} - 38 \beta_{4} - 193 \beta_1) q^{73} + ( - 82 \beta_{5} + 14 \beta_{4} + 82 \beta_{3} + 14 \beta_{2} + 472 \beta_1 - 472) q^{75} + (7 \beta_{5} + 46 \beta_{4} - 7 \beta_{3} + 46 \beta_{2} + 44 \beta_1 - 44) q^{79} + (119 \beta_{5} + 7 \beta_{4} - 799 \beta_1) q^{81} + ( - 2 \beta_{3} - 30 \beta_{2} + 800) q^{83} + (140 \beta_{3} - 18 \beta_{2} - 273) q^{85} + (71 \beta_{5} - 7 \beta_{4} - 282 \beta_1) q^{87} + ( - 72 \beta_{5} - 52 \beta_{4} + 72 \beta_{3} - 52 \beta_{2} - 315 \beta_1 + 315) q^{89} + ( - 26 \beta_{4} - 26 \beta_{2} - 449 \beta_1 + 449) q^{93} + (49 \beta_{5} - 56 \beta_{4} + 440 \beta_1) q^{95} + ( - 263 \beta_{3} + 19 \beta_{2} + 20) q^{97} + ( - 147 \beta_{3} + 57 \beta_{2} + 702) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 7 q^{3} - 3 q^{5} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 7 q^{3} - 3 q^{5} - 18 q^{9} + 3 q^{11} + 52 q^{13} + 254 q^{15} - 31 q^{17} - 89 q^{19} + 201 q^{23} - 300 q^{25} + 938 q^{27} + 380 q^{29} - 339 q^{31} - 105 q^{33} + 535 q^{37} + 134 q^{39} - 116 q^{41} + 536 q^{43} - 1410 q^{45} - 205 q^{47} + 965 q^{51} - 757 q^{53} + 3306 q^{55} + 522 q^{57} - 1799 q^{59} + 625 q^{61} + 1750 q^{65} + 495 q^{67} - 1946 q^{69} + 1280 q^{71} - 443 q^{73} - 1484 q^{75} - 79 q^{79} - 2523 q^{81} + 4744 q^{83} - 1954 q^{85} - 910 q^{87} + 821 q^{89} + 1321 q^{93} + 1327 q^{95} + 684 q^{97} + 4620 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 14x^{4} + 49x^{2} + 27 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 7\nu + 3 ) / 6 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{4} + 8\nu^{2} - 27 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{4} + 20\nu^{2} + 27 ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{5} - 2\nu^{4} + 23\nu^{3} - 8\nu^{2} + 81\nu + 27 ) / 6 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{5} + 2\nu^{4} + 23\nu^{3} + 20\nu^{2} + 45\nu + 27 ) / 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -2\beta_{5} + 2\beta_{4} + \beta_{3} + \beta_{2} ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - \beta_{2} - 18 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 14\beta_{5} - 14\beta_{4} - 7\beta_{3} - 7\beta_{2} + 72\beta _1 - 36 ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -2\beta_{3} + 5\beta_{2} + 63 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -40\beta_{5} + 58\beta_{4} + 20\beta_{3} + 29\beta_{2} - 414\beta _1 + 207 ) / 6 \) Copy content Toggle raw display

Character values

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

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

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
2.13755i
2.95906i
0.821510i
2.13755i
2.95906i
0.821510i
0 −4.77144 8.26438i 0 −7.90468 + 13.6913i 0 0 0 −32.0333 + 55.4834i 0
177.2 0 −0.130780 0.226518i 0 9.75047 16.8883i 0 0 0 13.4658 23.3234i 0
177.3 0 1.40222 + 2.42872i 0 −3.34580 + 5.79509i 0 0 0 9.56754 16.5715i 0
361.1 0 −4.77144 + 8.26438i 0 −7.90468 13.6913i 0 0 0 −32.0333 55.4834i 0
361.2 0 −0.130780 + 0.226518i 0 9.75047 + 16.8883i 0 0 0 13.4658 + 23.3234i 0
361.3 0 1.40222 2.42872i 0 −3.34580 5.79509i 0 0 0 9.56754 + 16.5715i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 177.3
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.m 6
7.b odd 2 1 56.4.i.b 6
7.c even 3 1 392.4.a.l 3
7.c even 3 1 inner 392.4.i.m 6
7.d odd 6 1 56.4.i.b 6
7.d odd 6 1 392.4.a.i 3
21.c even 2 1 504.4.s.h 6
21.g even 6 1 504.4.s.h 6
28.d even 2 1 112.4.i.e 6
28.f even 6 1 112.4.i.e 6
28.f even 6 1 784.4.a.be 3
28.g odd 6 1 784.4.a.bb 3
56.e even 2 1 448.4.i.m 6
56.h odd 2 1 448.4.i.j 6
56.j odd 6 1 448.4.i.j 6
56.m even 6 1 448.4.i.m 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.4.i.b 6 7.b odd 2 1
56.4.i.b 6 7.d odd 6 1
112.4.i.e 6 28.d even 2 1
112.4.i.e 6 28.f even 6 1
392.4.a.i 3 7.d odd 6 1
392.4.a.l 3 7.c even 3 1
392.4.i.m 6 1.a even 1 1 trivial
392.4.i.m 6 7.c even 3 1 inner
448.4.i.j 6 56.h odd 2 1
448.4.i.j 6 56.j odd 6 1
448.4.i.m 6 56.e even 2 1
448.4.i.m 6 56.m even 6 1
504.4.s.h 6 21.c even 2 1
504.4.s.h 6 21.g even 6 1
784.4.a.bb 3 28.g odd 6 1
784.4.a.be 3 28.f even 6 1

Hecke kernels

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

\( T_{3}^{6} + 7T_{3}^{5} + 74T_{3}^{4} - 161T_{3}^{3} + 674T_{3}^{2} + 175T_{3} + 49 \) Copy content Toggle raw display
\( T_{5}^{6} + 3T_{5}^{5} + 342T_{5}^{4} + 3127T_{5}^{3} + 117078T_{5}^{2} + 686979T_{5} + 4255969 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 7 T^{5} + 74 T^{4} - 161 T^{3} + \cdots + 49 \) Copy content Toggle raw display
$5$ \( T^{6} + 3 T^{5} + 342 T^{4} + \cdots + 4255969 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} - 3 T^{5} + \cdots + 1492817769 \) Copy content Toggle raw display
$13$ \( (T^{3} - 26 T^{2} - 2244 T - 26328)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + 31 T^{5} + \cdots + 15768076041 \) Copy content Toggle raw display
$19$ \( T^{6} + 89 T^{5} + \cdots + 507555841 \) Copy content Toggle raw display
$23$ \( T^{6} - 201 T^{5} + \cdots + 7342261969 \) Copy content Toggle raw display
$29$ \( (T^{3} - 190 T^{2} + 8476 T + 48504)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 339 T^{5} + \cdots + 6524157303049 \) Copy content Toggle raw display
$37$ \( T^{6} - 535 T^{5} + \cdots + 783910139769 \) Copy content Toggle raw display
$41$ \( (T^{3} + 58 T^{2} - 42436 T - 3860392)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} - 268 T^{2} - 105680 T + 24343488)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 126279990328969 \) Copy content Toggle raw display
$53$ \( T^{6} + 757 T^{5} + \cdots + 55\!\cdots\!81 \) Copy content Toggle raw display
$59$ \( T^{6} + 1799 T^{5} + \cdots + 38\!\cdots\!89 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 273487351727209 \) Copy content Toggle raw display
$67$ \( T^{6} - 495 T^{5} + \cdots + 12\!\cdots\!41 \) Copy content Toggle raw display
$71$ \( (T^{3} - 640 T^{2} + 122304 T - 7291392)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 443 T^{5} + \cdots + 10\!\cdots\!09 \) Copy content Toggle raw display
$79$ \( T^{6} + 79 T^{5} + \cdots + 21\!\cdots\!41 \) Copy content Toggle raw display
$83$ \( (T^{3} - 2372 T^{2} + 1641456 T - 266787264)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} - 821 T^{5} + \cdots + 93\!\cdots\!01 \) Copy content Toggle raw display
$97$ \( (T^{3} - 342 T^{2} - 2726340 T - 217321448)^{2} \) Copy content Toggle raw display
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