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Properties

Label 196.6.e.l
Level $196$
Weight $6$
Character orbit 196.e
Analytic conductor $31.435$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4 x^{7} - 1166 x^{6} + 3512 x^{5} + 513939 x^{4} - 1033736 x^{3} - 101466410 x^{2} + \cdots + 7574050372 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2}\cdot 7^{4} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{6} - \beta_{5} + \beta_{2}) q^{3} + (7 \beta_{3} + \beta_{2}) q^{5} + ( - \beta_{7} - \beta_{4} - 370 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{6} - \beta_{5} + \beta_{2}) q^{3} + (7 \beta_{3} + \beta_{2}) q^{5} + ( - \beta_{7} - \beta_{4} - 370 \beta_1) q^{9} + (\beta_{7} - 111 \beta_1 + 111) q^{11} + ( - 42 \beta_{6} + 15 \beta_{5}) q^{13} + ( - 8 \beta_{4} - 956) q^{15} + ( - 7 \beta_{6} - 80 \beta_{5} + \cdots + 80 \beta_{2}) q^{17}+ \cdots + (259 \beta_{4} + 16603) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 1480 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 1480 q^{9} + 444 q^{11} - 7648 q^{15} - 3408 q^{23} - 11904 q^{25} + 41448 q^{29} - 13732 q^{37} - 25608 q^{39} - 57992 q^{43} - 181852 q^{51} - 528 q^{53} + 179080 q^{57} - 110220 q^{65} - 195384 q^{67} + 43344 q^{71} + 117304 q^{79} - 182456 q^{81} - 182600 q^{85} - 250152 q^{93} + 255552 q^{95} + 132824 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4 x^{7} - 1166 x^{6} + 3512 x^{5} + 513939 x^{4} - 1033736 x^{3} - 101466410 x^{2} + \cdots + 7574050372 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -2\nu^{6} + 6\nu^{5} + 2929\nu^{4} - 5868\nu^{3} - 1204253\nu^{2} + 1207188\nu + 154804482 ) / 2780050 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 6342 \nu^{7} - 85096 \nu^{6} + 8103275 \nu^{5} + 137621616 \nu^{4} - 4133910386 \nu^{3} + \cdots + 29772446498102 ) / 149139952325 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1812 \nu^{7} + 6342 \nu^{6} + 2223256 \nu^{5} - 5573995 \nu^{4} - 1091175064 \nu^{3} + \cdots - 87630052846 ) / 21305707475 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 3624 \nu^{7} + 12684 \nu^{6} + 4446512 \nu^{5} - 11147990 \nu^{4} - 2182350128 \nu^{3} + \cdots - 324400058017 ) / 21305707475 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 739557 \nu^{7} + 65623087 \nu^{6} + 452390441 \nu^{5} - 56639445545 \nu^{4} + \cdots - 15\!\cdots\!56 ) / 298279904650 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 16254 \nu^{7} - 56889 \nu^{6} - 14098777 \nu^{5} + 35389165 \nu^{4} + 4089892863 \nu^{3} + \cdots + 199229885932 ) / 3277801150 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 851776 \nu^{7} + 2981216 \nu^{6} + 994446838 \nu^{5} - 2493570135 \nu^{4} + \cdots - 21491134416808 ) / 42611414950 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} - 2\beta_{3} + 7 ) / 14 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} + 12\beta_{3} - 28\beta_{2} + 28\beta _1 + 4095 ) / 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6\beta_{7} + 4\beta_{6} + 295\beta_{4} - 1746\beta_{3} - 42\beta_{2} + 42\beta _1 + 6139 ) / 14 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 12 \beta_{7} + 64 \beta_{6} + 112 \beta_{5} + 589 \beta_{4} + 4728 \beta_{3} - 16520 \beta_{2} + \cdots + 1168867 ) / 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5880 \beta_{7} + 11912 \beta_{6} + 280 \beta_{5} + 87299 \beta_{4} - 849622 \beta_{3} - 41230 \beta_{2} + \cdots + 2911937 ) / 14 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 17610 \beta_{7} + 117728 \beta_{6} + 164864 \beta_{5} + 260425 \beta_{4} + 1065348 \beta_{3} + \cdots + 324678221 ) / 14 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 3626098 \beta_{7} + 12421976 \beta_{6} + 576044 \beta_{5} + 25891027 \beta_{4} - 348415762 \beta_{3} + \cdots + 1126189155 ) / 14 \) 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
16.9466 − 1.22474i
−15.9466 + 1.22474i
−17.3608 − 1.22474i
18.3608 + 1.22474i
16.9466 + 1.22474i
−15.9466 − 1.22474i
−17.3608 + 1.22474i
18.3608 − 1.22474i
0 −14.6044 + 25.2955i 0 49.2526 + 85.3080i 0 0 0 −305.076 − 528.407i 0
165.2 0 −9.65464 + 16.7223i 0 −24.9936 − 43.2902i 0 0 0 −64.9240 − 112.452i 0
165.3 0 9.65464 − 16.7223i 0 24.9936 + 43.2902i 0 0 0 −64.9240 − 112.452i 0
165.4 0 14.6044 − 25.2955i 0 −49.2526 − 85.3080i 0 0 0 −305.076 − 528.407i 0
177.1 0 −14.6044 − 25.2955i 0 49.2526 − 85.3080i 0 0 0 −305.076 + 528.407i 0
177.2 0 −9.65464 − 16.7223i 0 −24.9936 + 43.2902i 0 0 0 −64.9240 + 112.452i 0
177.3 0 9.65464 + 16.7223i 0 24.9936 − 43.2902i 0 0 0 −64.9240 + 112.452i 0
177.4 0 14.6044 + 25.2955i 0 −49.2526 + 85.3080i 0 0 0 −305.076 + 528.407i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 165.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 196.6.e.l 8
7.b odd 2 1 inner 196.6.e.l 8
7.c even 3 1 196.6.a.k ✓ 4
7.c even 3 1 inner 196.6.e.l 8
7.d odd 6 1 196.6.a.k ✓ 4
7.d odd 6 1 inner 196.6.e.l 8
28.f even 6 1 784.6.a.bi 4
28.g odd 6 1 784.6.a.bi 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
196.6.a.k ✓ 4 7.c even 3 1
196.6.a.k ✓ 4 7.d odd 6 1
196.6.e.l 8 1.a even 1 1 trivial
196.6.e.l 8 7.b odd 2 1 inner
196.6.e.l 8 7.c even 3 1 inner
196.6.e.l 8 7.d odd 6 1 inner
784.6.a.bi 4 28.f even 6 1
784.6.a.bi 4 28.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 1226T_{3}^{6} + 1184980T_{3}^{4} + 389985696T_{3}^{2} + 101185065216 \) acting on \(S_{6}^{\mathrm{new}}(196, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 101185065216 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 587857653842176 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} - 222 T^{3} + \cdots + 2056803904)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 745074 T^{2} + 11601874944)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 17\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 63\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots + 8792221268224)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 10362 T + 24016784)^{4} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 21\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 915554310064384)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 20\!\cdots\!24)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 14498 T - 176356136)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 16\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 127384721082256)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 12\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 47\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 36\!\cdots\!56)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 10836 T - 321527808)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 91\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 28\!\cdots\!76)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 25\!\cdots\!16)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 24\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 79\!\cdots\!84)^{2} \) Copy content Toggle raw display
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