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Properties

Label 312.2.bf.b
Level $312$
Weight $2$
Character orbit 312.bf
Analytic conductor $2.491$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [312,2,Mod(49,312)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(312, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("312.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 312.bf (of order \(6\), degree \(2\), 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: \(2.49133254306\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.649638144.4
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 14x^{6} + 75x^{4} - 170x^{2} + 169 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + ( - \beta_{6} - \beta_{2}) q^{5} + (\beta_{7} - \beta_{4} + \beta_{3} + \cdots + 1) q^{7}+ \cdots + ( - \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + ( - \beta_{6} - \beta_{2}) q^{5} + (\beta_{7} - \beta_{4} + \beta_{3} + \cdots + 1) q^{7}+ \cdots + ( - \beta_{7} + \beta_{6} + \beta_{3} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{3} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{3} - 4 q^{9} - 6 q^{11} - 6 q^{13} + 6 q^{15} + 12 q^{17} + 6 q^{19} - 2 q^{23} - 4 q^{25} + 8 q^{27} + 6 q^{29} + 6 q^{33} + 10 q^{35} - 24 q^{41} - 8 q^{43} - 6 q^{45} + 18 q^{49} - 24 q^{51} + 4 q^{53} - 10 q^{55} - 36 q^{59} + 2 q^{61} - 28 q^{65} + 36 q^{67} - 2 q^{69} + 6 q^{71} + 2 q^{75} + 56 q^{77} - 12 q^{79} - 4 q^{81} + 24 q^{85} + 6 q^{87} - 12 q^{89} + 38 q^{91} - 6 q^{93} - 34 q^{95} - 66 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 14x^{6} + 75x^{4} - 170x^{2} + 169 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} - 25\nu^{5} + 315\nu^{3} - 182\nu^{2} - 740\nu + 546 ) / 364 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + 14\nu^{5} - 62\nu^{3} + 79\nu - 26 ) / 52 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5\nu^{7} + 13\nu^{6} - 57\nu^{5} - 221\nu^{4} + 245\nu^{3} + 1365\nu^{2} - 122\nu - 2392 ) / 364 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{7} - 39\nu^{6} + 16\nu^{5} + 481\nu^{4} + 126\nu^{3} - 1911\nu^{2} - 673\nu + 2262 ) / 364 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11\nu^{7} - 89\nu^{5} + 91\nu^{4} + 175\nu^{3} - 819\nu^{2} + 314\nu + 1729 ) / 364 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5\nu^{7} - 13\nu^{6} - 57\nu^{5} + 130\nu^{4} + 245\nu^{3} - 546\nu^{2} - 304\nu + 845 ) / 182 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -10\nu^{7} - 52\nu^{6} + 114\nu^{5} + 611\nu^{4} - 490\nu^{3} - 2457\nu^{2} + 790\nu + 2743 ) / 364 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{7} - \beta_{6} + 2\beta_{5} - \beta_{4} + 3\beta_{3} + \beta_{2} + 3\beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{6} + \beta_{4} + \beta_{3} - \beta_{2} - \beta _1 + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{7} - 2\beta_{6} + 9\beta_{5} + 2\beta_{4} + 14\beta_{3} + 14\beta_{2} + 15\beta _1 + 8 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{6} + 2\beta_{5} + 5\beta_{4} + 3\beta_{3} - 5\beta_{2} - 3\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -13\beta_{7} + 11\beta_{6} + 45\beta_{5} + 31\beta_{4} + 63\beta_{3} + 129\beta_{2} + 54\beta _1 + 38 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -7\beta_{7} - 99\beta_{6} + 47\beta_{5} + 65\beta_{4} + 25\beta_{3} - 65\beta_{2} - 18\beta _1 + 44 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -210\beta_{7} + 199\beta_{6} + 230\beta_{5} + 231\beta_{4} + 251\beta_{3} + 809\beta_{2} + 63\beta _1 + 90 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/312\mathbb{Z}\right)^\times\).

\(n\) \(79\) \(145\) \(157\) \(209\)
\(\chi(n)\) \(1\) \(1 + \beta_{2}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−1.42055 + 0.500000i
2.34138 − 0.500000i
1.42055 + 0.500000i
−2.34138 − 0.500000i
−2.34138 + 0.500000i
1.42055 − 0.500000i
2.34138 + 0.500000i
−1.42055 − 0.500000i
0 −0.500000 − 0.866025i 0 − 1.55452i 0 −2.56383 − 1.48023i 0 −0.500000 + 0.866025i 0
49.2 0 −0.500000 − 0.866025i 0 − 0.475353i 0 3.94508 + 2.27769i 0 −0.500000 + 0.866025i 0
49.3 0 −0.500000 − 0.866025i 0 1.28657i 0 1.69781 + 0.980228i 0 −0.500000 + 0.866025i 0
49.4 0 −0.500000 − 0.866025i 0 4.20740i 0 −3.07905 − 1.77769i 0 −0.500000 + 0.866025i 0
121.1 0 −0.500000 + 0.866025i 0 − 4.20740i 0 −3.07905 + 1.77769i 0 −0.500000 − 0.866025i 0
121.2 0 −0.500000 + 0.866025i 0 − 1.28657i 0 1.69781 − 0.980228i 0 −0.500000 − 0.866025i 0
121.3 0 −0.500000 + 0.866025i 0 0.475353i 0 3.94508 − 2.27769i 0 −0.500000 − 0.866025i 0
121.4 0 −0.500000 + 0.866025i 0 1.55452i 0 −2.56383 + 1.48023i 0 −0.500000 − 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 312.2.bf.b ✓ 8
3.b odd 2 1 936.2.bi.c 8
4.b odd 2 1 624.2.bv.g 8
12.b even 2 1 1872.2.by.m 8
13.c even 3 1 4056.2.c.p 8
13.e even 6 1 inner 312.2.bf.b ✓ 8
13.e even 6 1 4056.2.c.p 8
13.f odd 12 1 4056.2.a.bd 4
13.f odd 12 1 4056.2.a.be 4
39.h odd 6 1 936.2.bi.c 8
52.i odd 6 1 624.2.bv.g 8
52.l even 12 1 8112.2.a.cq 4
52.l even 12 1 8112.2.a.cs 4
156.r even 6 1 1872.2.by.m 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.2.bf.b ✓ 8 1.a even 1 1 trivial
312.2.bf.b ✓ 8 13.e even 6 1 inner
624.2.bv.g 8 4.b odd 2 1
624.2.bv.g 8 52.i odd 6 1
936.2.bi.c 8 3.b odd 2 1
936.2.bi.c 8 39.h odd 6 1
1872.2.by.m 8 12.b even 2 1
1872.2.by.m 8 156.r even 6 1
4056.2.a.bd 4 13.f odd 12 1
4056.2.a.be 4 13.f odd 12 1
4056.2.c.p 8 13.c even 3 1
4056.2.c.p 8 13.e even 6 1
8112.2.a.cq 4 52.l even 12 1
8112.2.a.cs 4 52.l even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 22T_{5}^{6} + 81T_{5}^{4} + 88T_{5}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(312, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + 22 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( T^{8} - 23 T^{6} + \cdots + 8836 \) Copy content Toggle raw display
$11$ \( T^{8} + 6 T^{7} + \cdots + 10816 \) Copy content Toggle raw display
$13$ \( T^{8} + 6 T^{7} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( T^{8} - 12 T^{7} + \cdots + 2304 \) Copy content Toggle raw display
$19$ \( T^{8} - 6 T^{7} + \cdots + 141376 \) Copy content Toggle raw display
$23$ \( T^{8} + 2 T^{7} + \cdots + 64 \) Copy content Toggle raw display
$29$ \( T^{8} - 6 T^{7} + \cdots + 1296 \) Copy content Toggle raw display
$31$ \( T^{8} + 118 T^{6} + \cdots + 141376 \) Copy content Toggle raw display
$37$ \( T^{8} - 41 T^{6} + \cdots + 16384 \) Copy content Toggle raw display
$41$ \( T^{8} + 24 T^{7} + \cdots + 3154176 \) Copy content Toggle raw display
$43$ \( T^{8} + 8 T^{7} + \cdots + 481636 \) Copy content Toggle raw display
$47$ \( T^{8} + 160 T^{6} + \cdots + 2096704 \) Copy content Toggle raw display
$53$ \( (T^{4} - 2 T^{3} + \cdots + 208)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 36 T^{7} + \cdots + 262144 \) Copy content Toggle raw display
$61$ \( T^{8} - 2 T^{7} + \cdots + 42003361 \) Copy content Toggle raw display
$67$ \( T^{8} - 36 T^{7} + \cdots + 45796 \) Copy content Toggle raw display
$71$ \( T^{8} - 6 T^{7} + \cdots + 1272384 \) Copy content Toggle raw display
$73$ \( T^{8} + 124 T^{6} + \cdots + 185761 \) Copy content Toggle raw display
$79$ \( (T^{4} + 6 T^{3} + \cdots + 216)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 312 T^{6} + \cdots + 1557504 \) Copy content Toggle raw display
$89$ \( T^{8} + 12 T^{7} + \cdots + 589824 \) Copy content Toggle raw display
$97$ \( T^{8} + 66 T^{7} + \cdots + 33039504 \) Copy content Toggle raw display
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