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Properties

Label 196.8.e.b
Level $196$
Weight $8$
Character orbit 196.e
Analytic conductor $61.227$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [196,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("196.165");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 196 = 2^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 8 \)
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: \(61.2274649949\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{1009})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 253x^{2} + 252x + 63504 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - 7 \beta_1) q^{3} + ( - 11 \beta_{3} - 11 \beta_{2} + \cdots + 147) q^{5} + (14 \beta_{3} + 14 \beta_{2} + \cdots + 1129) q^{9} + ( - 182 \beta_{2} + 1746 \beta_1) q^{11} + (201 \beta_{3} - 8085) q^{13}+ \cdots + (229922 \beta_{3} + 4542166) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 14 q^{3} + 294 q^{5} + 2258 q^{9} + 3492 q^{11} - 32340 q^{13} - 48512 q^{15} + 29232 q^{17} + 3206 q^{19} + 9360 q^{23} - 131146 q^{25} - 36344 q^{27} + 369408 q^{29} - 165060 q^{31} - 342832 q^{33}+ \cdots + 18168664 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 253x^{2} + 252x + 63504 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} + 253\nu^{2} - 253\nu + 63504 ) / 63756 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 253\nu^{2} + 127765\nu - 63504 ) / 63756 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{3} + 757 ) / 253 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + 505\beta _1 - 505 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 253\beta_{3} - 757 ) / 2 \) Copy content Toggle raw display

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

Character values

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

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

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
8.19119 − 14.1876i
−7.69119 + 13.3215i
8.19119 + 14.1876i
−7.69119 − 13.3215i
0 −19.3824 + 33.5713i 0 248.206 + 429.906i 0 0 0 342.147 + 592.615i 0
165.2 0 12.3824 − 21.4469i 0 −101.206 − 175.294i 0 0 0 786.853 + 1362.87i 0
177.1 0 −19.3824 − 33.5713i 0 248.206 − 429.906i 0 0 0 342.147 − 592.615i 0
177.2 0 12.3824 + 21.4469i 0 −101.206 + 175.294i 0 0 0 786.853 − 1362.87i 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 196.8.e.b 4
7.b odd 2 1 196.8.e.c 4
7.c even 3 1 28.8.a.b ✓ 2
7.c even 3 1 inner 196.8.e.b 4
7.d odd 6 1 196.8.a.a 2
7.d odd 6 1 196.8.e.c 4
21.h odd 6 1 252.8.a.f 2
28.g odd 6 1 112.8.a.h 2
56.k odd 6 1 448.8.a.q 2
56.p even 6 1 448.8.a.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.8.a.b ✓ 2 7.c even 3 1
112.8.a.h 2 28.g odd 6 1
196.8.a.a 2 7.d odd 6 1
196.8.e.b 4 1.a even 1 1 trivial
196.8.e.b 4 7.c even 3 1 inner
196.8.e.c 4 7.b odd 2 1
196.8.e.c 4 7.d odd 6 1
252.8.a.f 2 21.h odd 6 1
448.8.a.o 2 56.p even 6 1
448.8.a.q 2 56.k odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 14T_{3}^{3} + 1156T_{3}^{2} - 13440T_{3} + 921600 \) acting on \(S_{8}^{\mathrm{new}}(196, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 14 T^{3} + \cdots + 921600 \) Copy content Toggle raw display
$5$ \( T^{4} + \cdots + 10096230400 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 922555576960000 \) Copy content Toggle raw display
$13$ \( (T^{2} + 16170 T + 24602616)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 44\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 21\!\cdots\!84 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 79\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( (T^{2} - 184704 T + 5339156348)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 44\!\cdots\!36 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{2} + 116760 T - 89232832116)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 294428 T - 9812264768)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 46\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 77\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{2} + \cdots - 5444851901440)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 14\!\cdots\!24 \) Copy content Toggle raw display
$83$ \( (T^{2} + \cdots + 84534867832880)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{2} + \cdots + 174783333299356)^{2} \) Copy content Toggle raw display
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