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Properties

Label 392.3.k.k
Level $392$
Weight $3$
Character orbit 392.k
Analytic conductor $10.681$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [392,3,Mod(67,392)]
 
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([3, 3, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("392.67");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 392.k (of order \(6\), 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: \(10.6812263629\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 2x^{10} - 12x^{9} + 12x^{8} - 12x^{7} + 148x^{6} - 48x^{5} + 192x^{4} - 768x^{3} + 512x^{2} + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{8} + \beta_{7} + \cdots + \beta_{3}) q^{3}+ \cdots + ( - \beta_{11} - \beta_{8} - 2 \beta_{4} + \cdots - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{8} + \beta_{7} + \cdots + \beta_{3}) q^{3}+ \cdots + ( - 5 \beta_{9} - 21 \beta_{7} + \cdots + 76) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{3} - 4 q^{4} - 56 q^{6} + 36 q^{8} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 6 q^{3} - 4 q^{4} - 56 q^{6} + 36 q^{8} - 8 q^{9} + 30 q^{10} - 14 q^{11} + 14 q^{12} - 40 q^{16} + 82 q^{17} + 38 q^{18} + 94 q^{19} + 56 q^{20} - 132 q^{22} + 38 q^{24} + 116 q^{25} - 22 q^{26} + 60 q^{27} + 58 q^{30} - 60 q^{32} - 146 q^{33} - 188 q^{34} + 200 q^{36} - 46 q^{38} + 142 q^{40} - 120 q^{41} + 40 q^{43} - 86 q^{44} + 84 q^{46} - 184 q^{48} - 256 q^{50} - 106 q^{51} - 20 q^{52} - 82 q^{54} - 372 q^{57} - 130 q^{58} - 62 q^{59} - 290 q^{60} + 68 q^{62} - 328 q^{64} - 64 q^{65} + 208 q^{66} - 178 q^{67} - 4 q^{68} + 52 q^{72} - 54 q^{73} + 120 q^{74} - 140 q^{75} + 172 q^{76} + 4 q^{78} + 12 q^{80} + 206 q^{81} - 54 q^{82} + 392 q^{83} + 208 q^{86} + 90 q^{88} + 26 q^{89} - 156 q^{90} + 388 q^{92} - 262 q^{94} - 64 q^{96} + 184 q^{97} + 872 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 2x^{10} - 12x^{9} + 12x^{8} - 12x^{7} + 148x^{6} - 48x^{5} + 192x^{4} - 768x^{3} + 512x^{2} + 4096 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 6 \nu^{11} - 131 \nu^{10} - 36 \nu^{9} - 350 \nu^{8} + 1356 \nu^{7} - 2172 \nu^{6} + 1740 \nu^{5} + \cdots - 98816 ) / 46592 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 11 \nu^{11} + 48 \nu^{10} - 430 \nu^{9} + 420 \nu^{8} - 244 \nu^{7} + 6756 \nu^{6} + \cdots + 230400 ) / 93184 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 6 \nu^{11} + 51 \nu^{10} - 36 \nu^{9} + 14 \nu^{8} - 828 \nu^{7} + 12 \nu^{6} - 444 \nu^{5} + \cdots - 52224 ) / 46592 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 16 \nu^{11} - 319 \nu^{10} - 96 \nu^{9} + 98 \nu^{8} + 340 \nu^{7} + 396 \nu^{6} - 1548 \nu^{5} + \cdots + 512 ) / 46592 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 59 \nu^{11} - 272 \nu^{10} - 718 \nu^{9} - 924 \nu^{8} + 1868 \nu^{7} - 1884 \nu^{6} + \cdots - 94208 ) / 93184 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 51 \nu^{11} - 24 \nu^{10} - 58 \nu^{9} - 756 \nu^{8} - 60 \nu^{7} + 444 \nu^{6} + 3228 \nu^{5} + \cdots + 24576 ) / 46592 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 51 \nu^{11} - 24 \nu^{10} + 306 \nu^{9} - 756 \nu^{8} + 668 \nu^{7} - 3924 \nu^{6} + 7596 \nu^{5} + \cdots - 115200 ) / 46592 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 123 \nu^{11} - 456 \nu^{10} - 374 \nu^{9} - 1260 \nu^{8} + 3228 \nu^{7} - 1756 \nu^{6} + \cdots + 1024 ) / 93184 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 155 \nu^{11} - 184 \nu^{10} + 890 \nu^{9} + 756 \nu^{8} + 1724 \nu^{7} - 7516 \nu^{6} + \cdots - 463872 ) / 93184 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 81 \nu^{11} + 85 \nu^{10} + 486 \nu^{9} - 98 \nu^{8} + 440 \nu^{7} + 384 \nu^{6} + 5448 \nu^{5} + \cdots + 99328 ) / 46592 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} - \beta_{8} + \beta_{6} + \beta_{4} - \beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} - \beta_{9} + \beta_{7} + \beta_{6} + \beta_{3} - \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{11} + 2\beta_{8} - 6\beta_{4} - 4\beta_{2} + 2\beta _1 - 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4 \beta_{11} - 2 \beta_{10} + 2 \beta_{9} + 8 \beta_{8} - 10 \beta_{7} + 4 \beta_{6} + \cdots - 2 \beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 6\beta_{10} + 2\beta_{9} + 10\beta_{7} - 6\beta_{6} + 18\beta_{3} - 10\beta _1 - 26 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -12\beta_{11} + 4\beta_{8} - 12\beta_{5} - 84\beta_{4} - 12\beta_{2} - 24\beta _1 - 84 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 12 \beta_{11} - 24 \beta_{10} - 24 \beta_{9} + 60 \beta_{8} - 96 \beta_{7} - 12 \beta_{6} + \cdots + 24 \beta_{2} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 12\beta_{10} + 84\beta_{9} + 28\beta_{7} - 156\beta_{6} - 12\beta_{3} - 28\beta _1 - 84 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 32\beta_{11} - 80\beta_{8} - 168\beta_{5} + 80\beta_{4} + 120\beta_{2} + 80 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 80 \beta_{11} - 136 \beta_{10} - 536 \beta_{9} - 160 \beta_{8} + 200 \beta_{7} + 80 \beta_{6} + \cdots + 536 \beta_{2} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(197\) \(295\) \(297\)
\(\chi(n)\) \(-1\) \(-1\) \(-1 - \beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
−1.61320 + 1.18220i
−1.28408 − 1.53335i
−0.685878 − 1.87872i
0.0421483 + 1.99956i
1.71059 + 1.03628i
1.83041 − 0.805972i
−1.61320 − 1.18220i
−1.28408 + 1.53335i
−0.685878 + 1.87872i
0.0421483 − 1.99956i
1.71059 − 1.03628i
1.83041 + 0.805972i
−1.61320 + 1.18220i −0.717214 + 1.24225i 1.20482 − 3.81424i −2.27256 + 1.31206i −0.311578 − 2.85189i 0 2.56557 + 7.57746i 3.47121 + 6.01231i 2.11497 − 4.80324i
67.2 −1.28408 − 1.53335i 1.46995 − 2.54604i −0.702296 + 3.93786i −7.59793 + 4.38667i −5.79148 + 1.01536i 0 6.93991 − 3.97966i 0.178469 + 0.309118i 16.4826 + 6.01743i
67.3 −0.685878 − 1.87872i 1.46995 − 2.54604i −3.05914 + 2.57714i 7.59793 − 4.38667i −5.79148 − 1.01536i 0 6.93991 + 3.97966i 0.178469 + 0.309118i −13.4525 − 11.2656i
67.4 0.0421483 + 1.99956i −2.25274 + 3.90186i −3.99645 + 0.168556i −6.07099 + 3.50509i −7.89694 − 4.34002i 0 −0.505481 − 7.98401i −5.64968 − 9.78553i −7.26450 − 11.9916i
67.5 1.71059 + 1.03628i −2.25274 + 3.90186i 1.85225 + 3.54530i 6.07099 − 3.50509i −7.89694 + 4.34002i 0 −0.505481 + 7.98401i −5.64968 − 9.78553i 14.0172 + 0.295467i
67.6 1.83041 − 0.805972i −0.717214 + 1.24225i 2.70082 − 2.95052i 2.27256 − 1.31206i −0.311578 + 2.85189i 0 2.56557 − 7.57746i 3.47121 + 6.01231i 3.10224 − 4.23324i
275.1 −1.61320 − 1.18220i −0.717214 − 1.24225i 1.20482 + 3.81424i −2.27256 − 1.31206i −0.311578 + 2.85189i 0 2.56557 − 7.57746i 3.47121 − 6.01231i 2.11497 + 4.80324i
275.2 −1.28408 + 1.53335i 1.46995 + 2.54604i −0.702296 − 3.93786i −7.59793 − 4.38667i −5.79148 − 1.01536i 0 6.93991 + 3.97966i 0.178469 − 0.309118i 16.4826 − 6.01743i
275.3 −0.685878 + 1.87872i 1.46995 + 2.54604i −3.05914 − 2.57714i 7.59793 + 4.38667i −5.79148 + 1.01536i 0 6.93991 − 3.97966i 0.178469 − 0.309118i −13.4525 + 11.2656i
275.4 0.0421483 − 1.99956i −2.25274 − 3.90186i −3.99645 − 0.168556i −6.07099 − 3.50509i −7.89694 + 4.34002i 0 −0.505481 + 7.98401i −5.64968 + 9.78553i −7.26450 + 11.9916i
275.5 1.71059 − 1.03628i −2.25274 − 3.90186i 1.85225 − 3.54530i 6.07099 + 3.50509i −7.89694 − 4.34002i 0 −0.505481 − 7.98401i −5.64968 + 9.78553i 14.0172 − 0.295467i
275.6 1.83041 + 0.805972i −0.717214 − 1.24225i 2.70082 + 2.95052i 2.27256 + 1.31206i −0.311578 − 2.85189i 0 2.56557 + 7.57746i 3.47121 − 6.01231i 3.10224 + 4.23324i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
8.d odd 2 1 inner
56.k odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 392.3.k.k 12
7.b odd 2 1 56.3.k.c ✓ 12
7.c even 3 1 392.3.g.l 6
7.c even 3 1 inner 392.3.k.k 12
7.d odd 6 1 56.3.k.c ✓ 12
7.d odd 6 1 392.3.g.k 6
8.d odd 2 1 inner 392.3.k.k 12
28.d even 2 1 224.3.o.c 12
28.f even 6 1 224.3.o.c 12
28.f even 6 1 1568.3.g.k 6
28.g odd 6 1 1568.3.g.i 6
56.e even 2 1 56.3.k.c ✓ 12
56.h odd 2 1 224.3.o.c 12
56.j odd 6 1 224.3.o.c 12
56.j odd 6 1 1568.3.g.k 6
56.k odd 6 1 392.3.g.l 6
56.k odd 6 1 inner 392.3.k.k 12
56.m even 6 1 56.3.k.c ✓ 12
56.m even 6 1 392.3.g.k 6
56.p even 6 1 1568.3.g.i 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.k.c ✓ 12 7.b odd 2 1
56.3.k.c ✓ 12 7.d odd 6 1
56.3.k.c ✓ 12 56.e even 2 1
56.3.k.c ✓ 12 56.m even 6 1
224.3.o.c 12 28.d even 2 1
224.3.o.c 12 28.f even 6 1
224.3.o.c 12 56.h odd 2 1
224.3.o.c 12 56.j odd 6 1
392.3.g.k 6 7.d odd 6 1
392.3.g.k 6 56.m even 6 1
392.3.g.l 6 7.c even 3 1
392.3.g.l 6 56.k odd 6 1
392.3.k.k 12 1.a even 1 1 trivial
392.3.k.k 12 7.c even 3 1 inner
392.3.k.k 12 8.d odd 2 1 inner
392.3.k.k 12 56.k odd 6 1 inner
1568.3.g.i 6 28.g odd 6 1
1568.3.g.i 6 56.p even 6 1
1568.3.g.k 6 28.f even 6 1
1568.3.g.k 6 56.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(392, [\chi])\):

\( T_{3}^{6} + 3T_{3}^{5} + 20T_{3}^{4} + 5T_{3}^{3} + 178T_{3}^{2} + 209T_{3} + 361 \) Copy content Toggle raw display
\( T_{5}^{12} - 133T_{5}^{10} + 13038T_{5}^{8} - 566489T_{5}^{6} + 18167550T_{5}^{4} - 121144597T_{5}^{2} + 678446209 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 2 T^{10} + \cdots + 4096 \) Copy content Toggle raw display
$3$ \( (T^{6} + 3 T^{5} + \cdots + 361)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 678446209 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( (T^{6} + 7 T^{5} + \cdots + 82369)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 64 T^{4} + \cdots + 5488)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 41 T^{5} + \cdots + 2627641)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} - 47 T^{5} + \cdots + 5139289)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 282099132781249 \) Copy content Toggle raw display
$29$ \( (T^{6} + 3208 T^{4} + \cdots + 747251568)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 27\!\cdots\!09 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 50\!\cdots\!49 \) Copy content Toggle raw display
$41$ \( (T^{3} + 30 T^{2} + \cdots - 364)^{4} \) Copy content Toggle raw display
$43$ \( (T^{3} - 10 T^{2} + \cdots - 16408)^{4} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 7867032061329 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 18\!\cdots\!49 \) Copy content Toggle raw display
$59$ \( (T^{6} + 31 T^{5} + \cdots + 19158129)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 31\!\cdots\!69 \) Copy content Toggle raw display
$67$ \( (T^{6} + 89 T^{5} + \cdots + 102323854161)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 22416 T^{4} + \cdots + 203138627328)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 27 T^{5} + \cdots + 16072081)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 73\!\cdots\!09 \) Copy content Toggle raw display
$83$ \( (T^{3} - 98 T^{2} + \cdots + 823528)^{4} \) Copy content Toggle raw display
$89$ \( (T^{6} - 13 T^{5} + \cdots + 86434209)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} - 46 T^{2} + \cdots + 407276)^{4} \) Copy content Toggle raw display
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