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Properties

Label 196.6.e.k
Level $196$
Weight $6$
Character orbit 196.e
Analytic conductor $31.435$
Analytic rank $0$
Dimension $4$
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] = [196,6,Mod(165,196)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(196, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("196.165");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 196 = 2^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 196.e (of order \(3\), degree \(2\), 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: \(31.4352286833\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{109})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 28x^{2} + 27x + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 28)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} + \beta_{2} - 14 \beta_1 + 14) q^{3} + ( - 6 \beta_{2} + 21 \beta_1) q^{5} + (28 \beta_{2} - 62 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + \beta_{2} - 14 \beta_1 + 14) q^{3} + ( - 6 \beta_{2} + 21 \beta_1) q^{5} + (28 \beta_{2} - 62 \beta_1) q^{9} + ( - 21 \beta_{3} - 21 \beta_{2} + \cdots - 330) q^{11}+ \cdots + (10542 \beta_{3} + 84552) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 28 q^{3} + 42 q^{5} - 124 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 28 q^{3} + 42 q^{5} - 124 q^{9} - 660 q^{11} + 1288 q^{13} + 3792 q^{15} - 210 q^{17} + 3724 q^{19} - 24 q^{23} - 2480 q^{25} - 2072 q^{27} + 11064 q^{29} + 2800 q^{31} + 13818 q^{33} + 13238 q^{37} + 19480 q^{39} - 8232 q^{41} + 26864 q^{43} + 39228 q^{45} + 8064 q^{47} - 2292 q^{51} + 53958 q^{53} - 82656 q^{55} + 100348 q^{57} + 36036 q^{59} + 83986 q^{61} + 76308 q^{65} + 2660 q^{67} + 63420 q^{69} + 143136 q^{71} + 31318 q^{73} + 89656 q^{75} - 51136 q^{79} - 30370 q^{81} - 12432 q^{83} + 53964 q^{85} + 4200 q^{87} + 166278 q^{89} + 56938 q^{93} - 66432 q^{95} - 16520 q^{97} + 338208 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 28x^{2} + 27x + 729 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} + 28\nu^{2} - 28\nu + 729 ) / 756 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 28\nu^{2} + 1540\nu - 729 ) / 756 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 41 ) / 14 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + 55\beta _1 - 55 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 14\beta_{3} - 41 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/196\mathbb{Z}\right)^\times\).

\(n\) \(99\) \(101\)
\(\chi(n)\) \(1\) \(-\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
165.1
2.86008 + 4.95380i
−2.36008 − 4.08777i
2.86008 − 4.95380i
−2.36008 + 4.08777i
0 1.77985 − 3.08278i 0 −20.8209 − 36.0629i 0 0 0 115.164 + 199.470i 0
165.2 0 12.2202 − 21.1659i 0 41.8209 + 72.4360i 0 0 0 −177.164 − 306.858i 0
177.1 0 1.77985 + 3.08278i 0 −20.8209 + 36.0629i 0 0 0 115.164 − 199.470i 0
177.2 0 12.2202 + 21.1659i 0 41.8209 − 72.4360i 0 0 0 −177.164 + 306.858i 0
\(n\): e.g. 2-40 or 990-1000
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 196.6.e.k 4
7.b odd 2 1 28.6.e.b ✓ 4
7.c even 3 1 196.6.a.h 2
7.c even 3 1 inner 196.6.e.k 4
7.d odd 6 1 28.6.e.b ✓ 4
7.d odd 6 1 196.6.a.j 2
21.c even 2 1 252.6.k.d 4
21.g even 6 1 252.6.k.d 4
28.d even 2 1 112.6.i.e 4
28.f even 6 1 112.6.i.e 4
28.f even 6 1 784.6.a.o 2
28.g odd 6 1 784.6.a.bd 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.6.e.b ✓ 4 7.b odd 2 1
28.6.e.b ✓ 4 7.d odd 6 1
112.6.i.e 4 28.d even 2 1
112.6.i.e 4 28.f even 6 1
196.6.a.h 2 7.c even 3 1
196.6.a.j 2 7.d odd 6 1
196.6.e.k 4 1.a even 1 1 trivial
196.6.e.k 4 7.c even 3 1 inner
252.6.k.d 4 21.c even 2 1
252.6.k.d 4 21.g even 6 1
784.6.a.o 2 28.f even 6 1
784.6.a.bd 2 28.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 28T_{3}^{3} + 697T_{3}^{2} - 2436T_{3} + 7569 \) acting on \(S_{6}^{\mathrm{new}}(196, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 28 T^{3} + \cdots + 7569 \) Copy content Toggle raw display
$5$ \( T^{4} - 42 T^{3} + \cdots + 12131289 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 3700410561 \) Copy content Toggle raw display
$13$ \( (T^{2} - 644 T - 147452)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 2678994081 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 11959250986225 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 5547141326169 \) Copy content Toggle raw display
$29$ \( (T^{2} - 5532 T - 4654908)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 370117150388041 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 11\!\cdots\!69 \) Copy content Toggle raw display
$41$ \( (T^{2} + 4116 T - 68342940)^{2} \) Copy content Toggle raw display
$43$ \( (T - 6716)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 27\!\cdots\!69 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 48\!\cdots\!09 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 323868145417569 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 31\!\cdots\!09 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 14\!\cdots\!41 \) Copy content Toggle raw display
$71$ \( (T^{2} - 71568 T - 491520960)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 99\!\cdots\!61 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 17\!\cdots\!89 \) Copy content Toggle raw display
$83$ \( (T^{2} + 6216 T - 1440901872)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 43\!\cdots\!25 \) Copy content Toggle raw display
$97$ \( (T^{2} + 8260 T - 5263077500)^{2} \) Copy content Toggle raw display
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