Properties

Label 396.4.j.c
Level $396$
Weight $4$
Character orbit 396.j
Analytic conductor $23.365$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [396,4,Mod(37,396)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(396, base_ring=CyclotomicField(10))
 
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("396.37");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 396 = 2^{2} \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 396.j (of order \(5\), degree \(4\), 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: \(23.3647563623\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 343x^{10} + 44264x^{8} + 2595087x^{6} + 63146941x^{4} + 337629940x^{2} + 294604880 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 132)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{11} + \beta_{5} + \cdots - 3 \beta_1) q^{5}+ \cdots + (\beta_{11} + 2 \beta_{8} + \cdots + 3 \beta_{3}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{11} + \beta_{5} + \cdots - 3 \beta_1) q^{5}+ \cdots + (40 \beta_{11} - 20 \beta_{10} + \cdots - 4) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 16 q^{5} - 14 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 16 q^{5} - 14 q^{7} + 80 q^{11} - 18 q^{13} - 138 q^{17} - 130 q^{19} + 248 q^{23} - 143 q^{25} + 233 q^{29} - 201 q^{31} + 47 q^{35} + 166 q^{37} - 546 q^{41} + 2012 q^{43} + 568 q^{47} - 73 q^{49} + 521 q^{53} - 1122 q^{55} + 194 q^{59} + 72 q^{61} - 2440 q^{65} + 3672 q^{67} + 776 q^{71} - 2503 q^{73} - 2040 q^{77} - 2620 q^{79} + 1310 q^{83} + 3318 q^{85} - 2888 q^{89} - 5254 q^{91} + 5220 q^{95} + 1856 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 343x^{10} + 44264x^{8} + 2595087x^{6} + 63146941x^{4} + 337629940x^{2} + 294604880 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 3608089 \nu^{11} + 76175484 \nu^{10} - 894667961 \nu^{9} + 19697594994 \nu^{8} + \cdots + 71198840950512 ) / 319693216867728 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3608089 \nu^{11} + 76175484 \nu^{10} + 894667961 \nu^{9} + 19697594994 \nu^{8} + \cdots + 71198840950512 ) / 319693216867728 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5815346 \nu^{11} + 38087742 \nu^{10} + 1457635575 \nu^{9} + 9848797497 \nu^{8} + \cdots - 44323883741676 ) / 159846608433864 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 3608089 \nu^{11} + 1524932932 \nu^{10} - 894667961 \nu^{9} + 400982007382 \nu^{8} + \cdots + 10\!\cdots\!16 ) / 319693216867728 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3608089 \nu^{11} + 1524932932 \nu^{10} + 894667961 \nu^{9} + 400982007382 \nu^{8} + \cdots + 10\!\cdots\!16 ) / 319693216867728 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 54733451 \nu^{11} - 519021328 \nu^{10} + 14449261321 \nu^{9} - 134065432600 \nu^{8} + \cdots + 11\!\cdots\!64 ) / 319693216867728 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 54733451 \nu^{11} - 519021328 \nu^{10} - 14449261321 \nu^{9} - 134065432600 \nu^{8} + \cdots + 11\!\cdots\!64 ) / 319693216867728 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 75129075 \nu^{11} - 21211752 \nu^{10} + 21112319943 \nu^{9} - 8014763350 \nu^{8} + \cdots - 841891495854496 ) / 319693216867728 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 75129075 \nu^{11} + 21211752 \nu^{10} + 21112319943 \nu^{9} + 8014763350 \nu^{8} + \cdots + 841891495854496 ) / 319693216867728 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 20594684 \nu^{11} + 369185029 \nu^{10} - 5283997389 \nu^{9} + 96794700651 \nu^{8} + \cdots + 20\!\cdots\!56 ) / 79923304216932 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 20594684 \nu^{11} + 369185029 \nu^{10} + 5283997389 \nu^{9} + 96794700651 \nu^{8} + \cdots + 20\!\cdots\!56 ) / 79923304216932 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} - \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 8 \beta_{11} + 8 \beta_{10} - 4 \beta_{9} + 4 \beta_{8} + 6 \beta_{7} + 6 \beta_{6} - 5 \beta_{5} + \cdots - 104 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 14 \beta_{11} + 14 \beta_{10} + 4 \beta_{9} + 4 \beta_{8} + 4 \beta_{7} - 4 \beta_{6} - 83 \beta_{5} + \cdots + 112 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 786 \beta_{11} - 786 \beta_{10} + 354 \beta_{9} - 354 \beta_{8} - 448 \beta_{7} - 448 \beta_{6} + \cdots + 8350 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1844 \beta_{11} - 1844 \beta_{10} - 786 \beta_{9} - 786 \beta_{8} - 1336 \beta_{7} + 1336 \beta_{6} + \cdots - 16876 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 76694 \beta_{11} + 76694 \beta_{10} - 33286 \beta_{9} + 33286 \beta_{8} + 32982 \beta_{7} + \cdots - 699000 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 210478 \beta_{11} + 210478 \beta_{10} + 108676 \beta_{9} + 108676 \beta_{8} + 213510 \beta_{7} + \cdots + 2132534 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 7654488 \beta_{11} - 7654488 \beta_{10} + 3347460 \beta_{9} - 3347460 \beta_{8} - 2473986 \beta_{7} + \cdots + 60211744 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 22764318 \beta_{11} - 22764318 \beta_{10} - 13150428 \beta_{9} - 13150428 \beta_{8} - 28160352 \beta_{7} + \cdots - 251712204 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 778094746 \beta_{11} + 778094746 \beta_{10} - 351430574 \beta_{9} + 351430574 \beta_{8} + \cdots - 5333879494 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 2399554960 \beta_{11} + 2399554960 \beta_{10} + 1498581326 \beta_{9} + 1498581326 \beta_{8} + \cdots + 28652335712 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/396\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(199\) \(353\)
\(\chi(n)\) \(\beta_{3}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
2.40560i
9.09156i
10.3216i
8.12963i
1.04017i
8.99158i
2.40560i
9.09156i
10.3216i
8.12963i
1.04017i
8.99158i
0 0 0 −13.7606 + 9.99765i 0 −4.22415 13.0006i 0 0 0
37.2 0 0 0 2.51523 1.82742i 0 9.22490 + 28.3913i 0 0 0
37.3 0 0 0 8.36339 6.07636i 0 −1.79255 5.51689i 0 0 0
181.1 0 0 0 −6.62131 + 20.3783i 0 −22.9893 + 16.7027i 0 0 0
181.2 0 0 0 0.303308 0.933485i 0 14.3799 10.4476i 0 0 0
181.3 0 0 0 1.19997 3.69312i 0 −1.59876 + 1.16157i 0 0 0
289.1 0 0 0 −13.7606 9.99765i 0 −4.22415 + 13.0006i 0 0 0
289.2 0 0 0 2.51523 + 1.82742i 0 9.22490 28.3913i 0 0 0
289.3 0 0 0 8.36339 + 6.07636i 0 −1.79255 + 5.51689i 0 0 0
361.1 0 0 0 −6.62131 20.3783i 0 −22.9893 16.7027i 0 0 0
361.2 0 0 0 0.303308 + 0.933485i 0 14.3799 + 10.4476i 0 0 0
361.3 0 0 0 1.19997 + 3.69312i 0 −1.59876 1.16157i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 396.4.j.c 12
3.b odd 2 1 132.4.i.c 12
11.c even 5 1 inner 396.4.j.c 12
33.f even 10 1 1452.4.a.t 6
33.h odd 10 1 132.4.i.c 12
33.h odd 10 1 1452.4.a.u 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
132.4.i.c 12 3.b odd 2 1
132.4.i.c 12 33.h odd 10 1
396.4.j.c 12 1.a even 1 1 trivial
396.4.j.c 12 11.c even 5 1 inner
1452.4.a.t 6 33.f even 10 1
1452.4.a.u 6 33.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 16 T_{5}^{11} + 387 T_{5}^{10} - 1227 T_{5}^{9} - 21525 T_{5}^{8} - 241843 T_{5}^{7} + \cdots + 1993176025 \) acting on \(S_{4}^{\mathrm{new}}(396, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 1993176025 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 5582545033081 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 55\!\cdots\!81 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 73\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 11\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{6} - 124 T^{5} + \cdots + 18587649476)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 57\!\cdots\!81 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 10\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots - 27670010268564)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 16\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 79\!\cdots\!41 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 45\!\cdots\!21 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 16\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots - 362260302137744)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 79\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 11\!\cdots\!61 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 20\!\cdots\!01 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 769184681261380)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 31\!\cdots\!81 \) Copy content Toggle raw display
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