Properties

Label 912.3.o.c
Level $912$
Weight $3$
Character orbit 912.o
Analytic conductor $24.850$
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] = [912,3,Mod(721,912)]
 
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, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("912.721");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 912.o (of order \(2\), degree \(1\), 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: \(24.8502001097\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.219615408.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 22x^{4} - 39x^{3} + 112x^{2} - 93x + 39 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5}\cdot 3 \)
Twist minimal: no (minimal twist has level 228)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} - \beta_{3} q^{5} - \beta_1 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} - \beta_{3} q^{5} - \beta_1 q^{7} - 3 q^{9} + ( - \beta_1 + 4) q^{11} + ( - \beta_{5} - \beta_{4} + 4 \beta_{2}) q^{13} + \beta_{4} q^{15} + ( - 2 \beta_{3} - \beta_1 - 8) q^{17} + ( - \beta_{5} + \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 1) q^{19} + \beta_{5} q^{21} + (\beta_{3} - 3 \beta_1 - 6) q^{23} + (2 \beta_{3} + 3 \beta_1 + 5) q^{25} - 3 \beta_{2} q^{27} + ( - 2 \beta_{5} - 2 \beta_{4} - 8 \beta_{2}) q^{29} + ( - \beta_{5} + 3 \beta_{4} + 8 \beta_{2}) q^{31} + (\beta_{5} + 4 \beta_{2}) q^{33} + (4 \beta_{3} - \beta_1 - 2) q^{35} + (3 \beta_{5} + 3 \beta_{4} + 4 \beta_{2}) q^{37} + ( - 3 \beta_{3} - 3 \beta_1 - 12) q^{39} + ( - 4 \beta_{5} + 4 \beta_{4} - 4 \beta_{2}) q^{41} + ( - 2 \beta_{3} + \beta_1 + 36) q^{43} + 3 \beta_{3} q^{45} + (5 \beta_{3} + 4 \beta_1 - 4) q^{47} + ( - 2 \beta_{3} - 5 \beta_1 - 7) q^{49} + (\beta_{5} + 2 \beta_{4} - 8 \beta_{2}) q^{51} + 4 \beta_{4} q^{53} + ( - \beta_1 - 2) q^{55} + ( - 2 \beta_{4} + 3 \beta_{3} + \beta_{2} - 3 \beta_1 + 6) q^{57} + (6 \beta_{5} + 2 \beta_{4} - 24 \beta_{2}) q^{59} + (4 \beta_{3} - 7 \beta_1 + 32) q^{61} + 3 \beta_1 q^{63} + (2 \beta_{5} + 10 \beta_{4} - 28 \beta_{2}) q^{65} + (4 \beta_{4} + 4 \beta_{2}) q^{67} + (3 \beta_{5} - \beta_{4} - 6 \beta_{2}) q^{69} + ( - 2 \beta_{5} - 6 \beta_{4} - 8 \beta_{2}) q^{71} + (14 \beta_{3} - \beta_1 + 12) q^{73} + ( - 3 \beta_{5} - 2 \beta_{4} + 5 \beta_{2}) q^{75} + ( - 2 \beta_{3} - 9 \beta_1 + 42) q^{77} + (\beta_{5} - 7 \beta_{4} + 24 \beta_{2}) q^{79} + 9 q^{81} + (4 \beta_{3} + 66) q^{83} + (16 \beta_{3} + 5 \beta_1 + 58) q^{85} + ( - 6 \beta_{3} - 6 \beta_1 + 24) q^{87} + ( - 2 \beta_{5} + 2 \beta_{4} - 8 \beta_{2}) q^{89} + ( - 2 \beta_{5} + 2 \beta_{4} - 40 \beta_{2}) q^{91} + (9 \beta_{3} - 3 \beta_1 - 24) q^{93} + ( - 4 \beta_{5} - 5 \beta_{3} + 32 \beta_{2} - 6 \beta_1 - 60) q^{95} + (2 \beta_{5} - 2 \beta_{4} + 24 \beta_{2}) q^{97} + (3 \beta_1 - 12) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{5} + 2 q^{7} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{5} + 2 q^{7} - 18 q^{9} + 26 q^{11} - 50 q^{17} + 10 q^{19} - 28 q^{23} + 28 q^{25} - 2 q^{35} - 72 q^{39} + 210 q^{43} + 6 q^{45} - 22 q^{47} - 36 q^{49} - 10 q^{55} + 48 q^{57} + 214 q^{61} - 6 q^{63} + 102 q^{73} + 266 q^{77} + 54 q^{81} + 404 q^{83} + 370 q^{85} + 144 q^{87} - 120 q^{93} - 358 q^{95} - 78 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} + 22x^{4} - 39x^{3} + 112x^{2} - 93x + 39 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{4} + 2\nu^{3} - 9\nu^{2} + 8\nu + 7 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{5} - 5\nu^{4} + 32\nu^{3} - 43\nu^{2} + 98\nu - 42 ) / 19 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{4} + 2\nu^{3} - 13\nu^{2} + 12\nu - 17 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{5} - 5\nu^{4} + 108\nu^{3} - 157\nu^{2} + 934\nu - 441 ) / 38 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -6\nu^{5} + 15\nu^{4} - 172\nu^{3} + 243\nu^{2} - 902\nu + 411 ) / 38 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} + \beta_{2} + 3 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + \beta_{4} - 3\beta_{3} + \beta_{2} + 3\beta _1 - 33 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -19\beta_{5} - 13\beta_{4} - 9\beta_{3} - 22\beta_{2} + 9\beta _1 - 102 ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -20\beta_{5} - 14\beta_{4} + 18\beta_{3} - 23\beta_{2} - 30\beta _1 + 261 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 149\beta_{5} + 83\beta_{4} + 105\beta_{3} + 296\beta_{2} - 165\beta _1 + 1476 ) / 12 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
721.1
0.500000 + 2.79345i
0.500000 3.19918i
0.500000 0.460304i
0.500000 2.79345i
0.500000 + 3.19918i
0.500000 + 0.460304i
0 1.73205i 0 −7.39180 0 −3.28502 0 −3.00000 0
721.2 0 1.73205i 0 0.556406 0 9.52587 0 −3.00000 0
721.3 0 1.73205i 0 5.83539 0 −5.24085 0 −3.00000 0
721.4 0 1.73205i 0 −7.39180 0 −3.28502 0 −3.00000 0
721.5 0 1.73205i 0 0.556406 0 9.52587 0 −3.00000 0
721.6 0 1.73205i 0 5.83539 0 −5.24085 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 721.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.3.o.c 6
3.b odd 2 1 2736.3.o.m 6
4.b odd 2 1 228.3.h.a 6
12.b even 2 1 684.3.h.e 6
19.b odd 2 1 inner 912.3.o.c 6
57.d even 2 1 2736.3.o.m 6
76.d even 2 1 228.3.h.a 6
228.b odd 2 1 684.3.h.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.3.h.a 6 4.b odd 2 1
228.3.h.a 6 76.d even 2 1
684.3.h.e 6 12.b even 2 1
684.3.h.e 6 228.b odd 2 1
912.3.o.c 6 1.a even 1 1 trivial
912.3.o.c 6 19.b odd 2 1 inner
2736.3.o.m 6 3.b odd 2 1
2736.3.o.m 6 57.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{3} + T_{5}^{2} - 44T_{5} + 24 \) acting on \(S_{3}^{\mathrm{new}}(912, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3)^{3} \) Copy content Toggle raw display
$5$ \( (T^{3} + T^{2} - 44 T + 24)^{2} \) Copy content Toggle raw display
$7$ \( (T^{3} - T^{2} - 64 T - 164)^{2} \) Copy content Toggle raw display
$11$ \( (T^{3} - 13 T^{2} - 8 T + 12)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 792 T^{4} + 98064 T^{2} + \cdots + 2495232 \) Copy content Toggle raw display
$17$ \( (T^{3} + 25 T^{2} - 32 T - 108)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} - 10 T^{5} + 407 T^{4} + \cdots + 47045881 \) Copy content Toggle raw display
$23$ \( (T^{3} + 14 T^{2} - 560 T - 5136)^{2} \) Copy content Toggle raw display
$29$ \( T^{6} + 3168 T^{4} + \cdots + 322486272 \) Copy content Toggle raw display
$31$ \( T^{6} + 3192 T^{4} + 1009296 T^{2} + \cdots + 62208 \) Copy content Toggle raw display
$37$ \( T^{6} + 5976 T^{4} + \cdots + 832267008 \) Copy content Toggle raw display
$41$ \( T^{6} + 10896 T^{4} + \cdots + 29104386048 \) Copy content Toggle raw display
$43$ \( (T^{3} - 105 T^{2} + 3432 T - 35764)^{2} \) Copy content Toggle raw display
$47$ \( (T^{3} + 11 T^{2} - 2084 T - 25272)^{2} \) Copy content Toggle raw display
$53$ \( T^{6} + 4272 T^{4} + \cdots + 63700992 \) Copy content Toggle raw display
$59$ \( T^{6} + 19536 T^{4} + \cdots + 207873257472 \) Copy content Toggle raw display
$61$ \( (T^{3} - 107 T^{2} - 64 T + 104276)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 4320 T^{4} + \cdots + 511377408 \) Copy content Toggle raw display
$71$ \( T^{6} + 11472 T^{4} + \cdots + 48612814848 \) Copy content Toggle raw display
$73$ \( (T^{3} - 51 T^{2} - 7896 T + 115492)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + 19848 T^{4} + \cdots + 60723965952 \) Copy content Toggle raw display
$83$ \( (T^{3} - 202 T^{2} + 12892 T - 259992)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + 3408 T^{4} + \cdots + 956510208 \) Copy content Toggle raw display
$97$ \( T^{6} + 8400 T^{4} + \cdots + 168210432 \) Copy content Toggle raw display
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