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Properties

Label 308.2.i.b
Level $308$
Weight $2$
Character orbit 308.i
Analytic conductor $2.459$
Analytic rank $0$
Dimension $6$
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] = [308,2,Mod(177,308)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(308, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("308.177");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 308 = 2^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 308.i (of order \(3\), 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: \(2.45939238226\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.1783323.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 5x^{4} - 2x^{3} + 19x^{2} - 12x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{5} - \beta_{4} + 1) q^{3} + (\beta_{4} + \beta_{2} + \beta_1) q^{5} + (\beta_{4} - \beta_{3} + \beta_{2} + \beta_1) q^{7} + (\beta_{5} - \beta_{4} - \beta_{3} + \cdots + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} - \beta_{4} + 1) q^{3} + (\beta_{4} + \beta_{2} + \beta_1) q^{5} + (\beta_{4} - \beta_{3} + \beta_{2} + \beta_1) q^{7} + (\beta_{5} - \beta_{4} - \beta_{3} + \cdots + \beta_1) q^{9}+ \cdots + (\beta_{3} - \beta_{2} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{3} + 2 q^{5} + 2 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{3} + 2 q^{5} + 2 q^{7} - 4 q^{9} - 3 q^{11} - 6 q^{13} + 2 q^{15} + q^{17} + 9 q^{19} - 8 q^{21} + 5 q^{25} - 12 q^{27} + 10 q^{29} + 9 q^{31} + 3 q^{33} - 9 q^{35} - 20 q^{37} + 7 q^{39} - 10 q^{41} + 16 q^{43} - 7 q^{45} + 3 q^{47} + 12 q^{49} - 12 q^{51} + 15 q^{53} - 4 q^{55} + 4 q^{57} + 10 q^{59} + 8 q^{61} + 4 q^{63} - 3 q^{65} + 8 q^{67} - 54 q^{69} + 6 q^{71} - 12 q^{73} + 3 q^{75} - 4 q^{77} - 7 q^{79} + 13 q^{81} - 50 q^{83} - 18 q^{85} + 24 q^{87} + 27 q^{89} - 23 q^{91} - 6 q^{93} + 19 q^{95} - 22 q^{97} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 5x^{4} - 2x^{3} + 19x^{2} - 12x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 5\nu^{4} + 25\nu^{3} - 19\nu^{2} + 12\nu - 60 ) / 83 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4\nu^{5} - 20\nu^{4} + 17\nu^{3} - 76\nu^{2} + 48\nu - 240 ) / 83 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -20\nu^{5} + 17\nu^{4} - 85\nu^{3} - 35\nu^{2} - 323\nu + 204 ) / 249 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -16\nu^{5} - 3\nu^{4} - 68\nu^{3} - 28\nu^{2} - 275\nu - 36 ) / 83 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - 3\beta_{4} - \beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{3} + 4\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -5\beta_{5} + 12\beta_{4} - \beta _1 - 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -6\beta_{5} + 3\beta_{4} + 6\beta_{3} - 17\beta_{2} - 17\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/308\mathbb{Z}\right)^\times\).

\(n\) \(45\) \(57\) \(155\)
\(\chi(n)\) \(-\beta_{4}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
177.1
0.356769 + 0.617942i
−0.956115 − 1.65604i
1.09935 + 1.90412i
0.356769 − 0.617942i
−0.956115 + 1.65604i
1.09935 − 1.90412i
0 −0.745432 − 1.29113i 0 0.143231 − 0.248083i 0 2.63409 − 0.248083i 0 0.388663 − 0.673184i 0
177.2 0 0.828310 + 1.43468i 0 1.45611 − 2.52206i 0 0.799494 − 2.52206i 0 0.127804 − 0.221364i 0
177.3 0 1.41712 + 2.45453i 0 −0.599346 + 1.03810i 0 −2.43359 + 1.03810i 0 −2.51647 + 4.35865i 0
221.1 0 −0.745432 + 1.29113i 0 0.143231 + 0.248083i 0 2.63409 + 0.248083i 0 0.388663 + 0.673184i 0
221.2 0 0.828310 − 1.43468i 0 1.45611 + 2.52206i 0 0.799494 + 2.52206i 0 0.127804 + 0.221364i 0
221.3 0 1.41712 − 2.45453i 0 −0.599346 − 1.03810i 0 −2.43359 − 1.03810i 0 −2.51647 − 4.35865i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 177.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 308.2.i.b ✓ 6
3.b odd 2 1 2772.2.s.e 6
4.b odd 2 1 1232.2.q.j 6
7.b odd 2 1 2156.2.i.j 6
7.c even 3 1 inner 308.2.i.b ✓ 6
7.c even 3 1 2156.2.a.g 3
7.d odd 6 1 2156.2.a.k 3
7.d odd 6 1 2156.2.i.j 6
21.h odd 6 1 2772.2.s.e 6
28.f even 6 1 8624.2.a.cg 3
28.g odd 6 1 1232.2.q.j 6
28.g odd 6 1 8624.2.a.cp 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
308.2.i.b ✓ 6 1.a even 1 1 trivial
308.2.i.b ✓ 6 7.c even 3 1 inner
1232.2.q.j 6 4.b odd 2 1
1232.2.q.j 6 28.g odd 6 1
2156.2.a.g 3 7.c even 3 1
2156.2.a.k 3 7.d odd 6 1
2156.2.i.j 6 7.b odd 2 1
2156.2.i.j 6 7.d odd 6 1
2772.2.s.e 6 3.b odd 2 1
2772.2.s.e 6 21.h odd 6 1
8624.2.a.cg 3 28.f even 6 1
8624.2.a.cp 3 28.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} - 3T_{3}^{5} + 11T_{3}^{4} - 8T_{3}^{3} + 25T_{3}^{2} - 14T_{3} + 49 \) acting on \(S_{2}^{\mathrm{new}}(308, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 3 T^{5} + \cdots + 49 \) Copy content Toggle raw display
$5$ \( T^{6} - 2 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{6} - 2 T^{5} + \cdots + 343 \) Copy content Toggle raw display
$11$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$13$ \( (T^{3} + 3 T^{2} - 2 T - 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} - T^{5} + \cdots + 81 \) Copy content Toggle raw display
$19$ \( T^{6} - 9 T^{5} + \cdots + 38809 \) Copy content Toggle raw display
$23$ \( T^{6} + 75 T^{4} + \cdots + 49 \) Copy content Toggle raw display
$29$ \( (T^{3} - 5 T^{2} - 14 T + 67)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} - 9 T^{5} + \cdots + 2209 \) Copy content Toggle raw display
$37$ \( T^{6} + 20 T^{5} + \cdots + 28224 \) Copy content Toggle raw display
$41$ \( (T^{3} + 5 T^{2} + \cdots - 201)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} - 8 T^{2} - 35 T + 37)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} - 3 T^{5} + \cdots + 4489 \) Copy content Toggle raw display
$53$ \( T^{6} - 15 T^{5} + \cdots + 10609 \) Copy content Toggle raw display
$59$ \( T^{6} - 10 T^{5} + \cdots + 22201 \) Copy content Toggle raw display
$61$ \( T^{6} - 8 T^{5} + \cdots + 64 \) Copy content Toggle raw display
$67$ \( T^{6} - 8 T^{5} + \cdots + 287296 \) Copy content Toggle raw display
$71$ \( (T^{3} - 3 T^{2} + \cdots + 755)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 12 T^{5} + \cdots + 657721 \) Copy content Toggle raw display
$79$ \( T^{6} + 7 T^{5} + \cdots + 2653641 \) Copy content Toggle raw display
$83$ \( (T^{3} + 25 T^{2} + \cdots + 313)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} - 27 T^{5} + \cdots + 112225 \) Copy content Toggle raw display
$97$ \( (T^{3} + 11 T^{2} + \cdots - 45)^{2} \) Copy content Toggle raw display
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