Properties

Label 912.6.a.k
Level $912$
Weight $6$
Character orbit 912.a
Self dual yes
Analytic conductor $146.270$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [912,6,Mod(1,912)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(912, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("912.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 912.a (trivial)

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: yes
Analytic conductor: \(146.270043669\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.286833.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 154x + 760 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 228)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 9 q^{3} + ( - \beta_{2} - 2 \beta_1 - 2) q^{5} + (6 \beta_{2} - 5 \beta_1 + 13) q^{7} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 9 q^{3} + ( - \beta_{2} - 2 \beta_1 - 2) q^{5} + (6 \beta_{2} - 5 \beta_1 + 13) q^{7} + 81 q^{9} + ( - 26 \beta_{2} + \beta_1 + 223) q^{11} + (51 \beta_{2} - 5 \beta_1 - 315) q^{13} + (9 \beta_{2} + 18 \beta_1 + 18) q^{15} + ( - 18 \beta_{2} + 39 \beta_1 - 261) q^{17} + 361 q^{19} + ( - 54 \beta_{2} + 45 \beta_1 - 117) q^{21} + ( - 17 \beta_{2} - 109 \beta_1 + 395) q^{23} + ( - 66 \beta_{2} - 37 \beta_1 - 1700) q^{25} - 729 q^{27} + ( - 452 \beta_{2} - 16 \beta_1 - 1990) q^{29} + (201 \beta_{2} + 347 \beta_1 + 1987) q^{31} + (234 \beta_{2} - 9 \beta_1 - 2007) q^{33} + ( - 336 \beta_{2} - 35 \beta_1 + 3127) q^{35} + (855 \beta_{2} + 157 \beta_1 - 3421) q^{37} + ( - 459 \beta_{2} + 45 \beta_1 + 2835) q^{39} + ( - 410 \beta_{2} + 168 \beta_1 - 2334) q^{41} + (552 \beta_{2} + 741 \beta_1 + 2147) q^{43} + ( - 81 \beta_{2} - 162 \beta_1 - 162) q^{45} + (157 \beta_{2} + 434 \beta_1 + 2156) q^{47} + ( - 528 \beta_{2} + 273 \beta_1 - 1012) q^{49} + (162 \beta_{2} - 351 \beta_1 + 2349) q^{51} + (1388 \beta_{2} + 1356 \beta_1 + 3558) q^{53} + (288 \beta_{2} - 717 \beta_1 + 89) q^{55} - 3249 q^{57} + ( - 72 \beta_{2} + 1340 \beta_1 + 13184) q^{59} + ( - 1020 \beta_{2} - 971 \beta_1 - 14571) q^{61} + (486 \beta_{2} - 405 \beta_1 + 1053) q^{63} + ( - 818 \beta_{2} + 1116 \beta_1 + 1668) q^{65} + ( - 534 \beta_{2} - 1738 \beta_1 + 24842) q^{67} + (153 \beta_{2} + 981 \beta_1 - 3555) q^{69} + ( - 2168 \beta_{2} + 380 \beta_1 + 25220) q^{71} + (450 \beta_{2} - 515 \beta_1 - 22139) q^{73} + (594 \beta_{2} + 333 \beta_1 + 15300) q^{75} + (864 \beta_{2} - 2497 \beta_1 - 22819) q^{77} + (1467 \beta_{2} - 641 \beta_1 + 28139) q^{79} + 6561 q^{81} + ( - 3744 \beta_{2} - 852 \beta_1 + 21876) q^{83} + (2262 \beta_{2} + 909 \beta_1 - 25425) q^{85} + (4068 \beta_{2} + 144 \beta_1 + 17910) q^{87} + ( - 9696 \beta_{2} - 2164 \beta_1 + 43010) q^{89} + ( - 1416 \beta_{2} + 4298 \beta_1 + 52526) q^{91} + ( - 1809 \beta_{2} - 3123 \beta_1 - 17883) q^{93} + ( - 361 \beta_{2} - 722 \beta_1 - 722) q^{95} + (10578 \beta_{2} + 2494 \beta_1 - 26536) q^{97} + ( - 2106 \beta_{2} + 81 \beta_1 + 18063) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 27 q^{3} - 5 q^{5} + 33 q^{7} + 243 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 27 q^{3} - 5 q^{5} + 33 q^{7} + 243 q^{9} + 695 q^{11} - 996 q^{13} + 45 q^{15} - 765 q^{17} + 1083 q^{19} - 297 q^{21} + 1202 q^{23} - 5034 q^{25} - 2187 q^{27} - 5518 q^{29} + 5760 q^{31} - 6255 q^{33} + 9717 q^{35} - 11118 q^{37} + 8964 q^{39} - 6592 q^{41} + 5889 q^{43} - 405 q^{45} + 6311 q^{47} - 2508 q^{49} + 6885 q^{51} + 9286 q^{53} - 21 q^{55} - 9747 q^{57} + 39624 q^{59} - 42693 q^{61} + 2673 q^{63} + 5822 q^{65} + 75060 q^{67} - 10818 q^{69} + 77828 q^{71} - 66867 q^{73} + 45306 q^{75} - 69321 q^{77} + 82950 q^{79} + 19683 q^{81} + 69372 q^{83} - 78537 q^{85} + 49662 q^{87} + 138726 q^{89} + 158994 q^{91} - 51840 q^{93} - 1805 q^{95} - 90186 q^{97} + 56295 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 154x + 760 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{2} - 3\nu + 104 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} + 7\nu - 106 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{2} - 7\beta _1 + 205 ) / 2 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
6.30331
8.64449
−13.9478
0 −9.00000 0 −36.2858 0 −166.831 0 81.0000 0
1.2 0 −9.00000 0 −19.9586 0 92.3678 0 81.0000 0
1.3 0 −9.00000 0 51.2444 0 107.464 0 81.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(19\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.6.a.k 3
4.b odd 2 1 228.6.a.a 3
12.b even 2 1 684.6.a.a 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.6.a.a 3 4.b odd 2 1
684.6.a.a 3 12.b even 2 1
912.6.a.k 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{3} + 5T_{5}^{2} - 2158T_{5} - 37112 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(912))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( (T + 9)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} + 5 T^{2} - 2158 T - 37112 \) Copy content Toggle raw display
$7$ \( T^{3} - 33 T^{2} - 23412 T + 1656000 \) Copy content Toggle raw display
$11$ \( T^{3} - 695 T^{2} + \cdots + 26278976 \) Copy content Toggle raw display
$13$ \( T^{3} + 996 T^{2} + \cdots - 178271216 \) Copy content Toggle raw display
$17$ \( T^{3} + 765 T^{2} + \cdots - 426321684 \) Copy content Toggle raw display
$19$ \( (T - 361)^{3} \) Copy content Toggle raw display
$23$ \( T^{3} - 1202 T^{2} + \cdots - 209746912 \) Copy content Toggle raw display
$29$ \( T^{3} + 5518 T^{2} + \cdots + 10400234888 \) Copy content Toggle raw display
$31$ \( T^{3} - 5760 T^{2} + \cdots + 309849129472 \) Copy content Toggle raw display
$37$ \( T^{3} + 11118 T^{2} + \cdots - 969802262848 \) Copy content Toggle raw display
$41$ \( T^{3} + 6592 T^{2} + \cdots - 220780939552 \) Copy content Toggle raw display
$43$ \( T^{3} - 5889 T^{2} + \cdots + 2706921550192 \) Copy content Toggle raw display
$47$ \( T^{3} - 6311 T^{2} + \cdots + 484607665592 \) Copy content Toggle raw display
$53$ \( T^{3} - 9286 T^{2} + \cdots + 17693656641400 \) Copy content Toggle raw display
$59$ \( T^{3} - 39624 T^{2} + \cdots + 12187656460800 \) Copy content Toggle raw display
$61$ \( T^{3} + 42693 T^{2} + \cdots - 10787695490636 \) Copy content Toggle raw display
$67$ \( T^{3} - 75060 T^{2} + \cdots + 8518946717952 \) Copy content Toggle raw display
$71$ \( T^{3} - 77828 T^{2} + \cdots + 8724141842432 \) Copy content Toggle raw display
$73$ \( T^{3} + 66867 T^{2} + \cdots + 7419579873444 \) Copy content Toggle raw display
$79$ \( T^{3} - 82950 T^{2} + \cdots + 4752096482816 \) Copy content Toggle raw display
$83$ \( T^{3} - 69372 T^{2} + \cdots + 89984667239424 \) Copy content Toggle raw display
$89$ \( T^{3} - 138726 T^{2} + \cdots + 14\!\cdots\!80 \) Copy content Toggle raw display
$97$ \( T^{3} + 90186 T^{2} + \cdots - 15\!\cdots\!68 \) Copy content Toggle raw display
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