Properties

Label 196.4.e.c
Level $196$
Weight $4$
Character orbit 196.e
Analytic conductor $11.564$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 196 = 2^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 196.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.5643743611\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 28)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (4 \zeta_{6} - 4) q^{3} - 6 \zeta_{6} q^{5} + 11 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (4 \zeta_{6} - 4) q^{3} - 6 \zeta_{6} q^{5} + 11 \zeta_{6} q^{9} + ( - 12 \zeta_{6} + 12) q^{11} - 82 q^{13} + 24 q^{15} + ( - 30 \zeta_{6} + 30) q^{17} - 68 \zeta_{6} q^{19} - 216 \zeta_{6} q^{23} + ( - 89 \zeta_{6} + 89) q^{25} - 152 q^{27} + 246 q^{29} + ( - 112 \zeta_{6} + 112) q^{31} + 48 \zeta_{6} q^{33} - 110 \zeta_{6} q^{37} + ( - 328 \zeta_{6} + 328) q^{39} - 246 q^{41} - 172 q^{43} + ( - 66 \zeta_{6} + 66) q^{45} - 192 \zeta_{6} q^{47} + 120 \zeta_{6} q^{51} + (558 \zeta_{6} - 558) q^{53} - 72 q^{55} + 272 q^{57} + (540 \zeta_{6} - 540) q^{59} - 110 \zeta_{6} q^{61} + 492 \zeta_{6} q^{65} + (140 \zeta_{6} - 140) q^{67} + 864 q^{69} - 840 q^{71} + ( - 550 \zeta_{6} + 550) q^{73} + 356 \zeta_{6} q^{75} + 208 \zeta_{6} q^{79} + ( - 311 \zeta_{6} + 311) q^{81} + 516 q^{83} - 180 q^{85} + (984 \zeta_{6} - 984) q^{87} + 1398 \zeta_{6} q^{89} + 448 \zeta_{6} q^{93} + (408 \zeta_{6} - 408) q^{95} + 1586 q^{97} + 132 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{3} - 6 q^{5} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{3} - 6 q^{5} + 11 q^{9} + 12 q^{11} - 164 q^{13} + 48 q^{15} + 30 q^{17} - 68 q^{19} - 216 q^{23} + 89 q^{25} - 304 q^{27} + 492 q^{29} + 112 q^{31} + 48 q^{33} - 110 q^{37} + 328 q^{39} - 492 q^{41} - 344 q^{43} + 66 q^{45} - 192 q^{47} + 120 q^{51} - 558 q^{53} - 144 q^{55} + 544 q^{57} - 540 q^{59} - 110 q^{61} + 492 q^{65} - 140 q^{67} + 1728 q^{69} - 1680 q^{71} + 550 q^{73} + 356 q^{75} + 208 q^{79} + 311 q^{81} + 1032 q^{83} - 360 q^{85} - 984 q^{87} + 1398 q^{89} + 448 q^{93} - 408 q^{95} + 3172 q^{97} + 264 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(99\) \(101\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
165.1
0.500000 + 0.866025i
0.500000 0.866025i
0 −2.00000 + 3.46410i 0 −3.00000 5.19615i 0 0 0 5.50000 + 9.52628i 0
177.1 0 −2.00000 3.46410i 0 −3.00000 + 5.19615i 0 0 0 5.50000 9.52628i 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 196.4.e.c 2
3.b odd 2 1 1764.4.k.k 2
7.b odd 2 1 196.4.e.d 2
7.c even 3 1 28.4.a.b 1
7.c even 3 1 inner 196.4.e.c 2
7.d odd 6 1 196.4.a.b 1
7.d odd 6 1 196.4.e.d 2
21.c even 2 1 1764.4.k.e 2
21.g even 6 1 1764.4.a.k 1
21.g even 6 1 1764.4.k.e 2
21.h odd 6 1 252.4.a.c 1
21.h odd 6 1 1764.4.k.k 2
28.f even 6 1 784.4.a.n 1
28.g odd 6 1 112.4.a.c 1
35.j even 6 1 700.4.a.e 1
35.l odd 12 2 700.4.e.f 2
56.k odd 6 1 448.4.a.m 1
56.p even 6 1 448.4.a.d 1
84.n even 6 1 1008.4.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.4.a.b 1 7.c even 3 1
112.4.a.c 1 28.g odd 6 1
196.4.a.b 1 7.d odd 6 1
196.4.e.c 2 1.a even 1 1 trivial
196.4.e.c 2 7.c even 3 1 inner
196.4.e.d 2 7.b odd 2 1
196.4.e.d 2 7.d odd 6 1
252.4.a.c 1 21.h odd 6 1
448.4.a.d 1 56.p even 6 1
448.4.a.m 1 56.k odd 6 1
700.4.a.e 1 35.j even 6 1
700.4.e.f 2 35.l odd 12 2
784.4.a.n 1 28.f even 6 1
1008.4.a.f 1 84.n even 6 1
1764.4.a.k 1 21.g even 6 1
1764.4.k.e 2 21.c even 2 1
1764.4.k.e 2 21.g even 6 1
1764.4.k.k 2 3.b odd 2 1
1764.4.k.k 2 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(196, [\chi])\):

\( T_{3}^{2} + 4T_{3} + 16 \) Copy content Toggle raw display
\( T_{5}^{2} + 6T_{5} + 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$5$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$13$ \( (T + 82)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 30T + 900 \) Copy content Toggle raw display
$19$ \( T^{2} + 68T + 4624 \) Copy content Toggle raw display
$23$ \( T^{2} + 216T + 46656 \) Copy content Toggle raw display
$29$ \( (T - 246)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 112T + 12544 \) Copy content Toggle raw display
$37$ \( T^{2} + 110T + 12100 \) Copy content Toggle raw display
$41$ \( (T + 246)^{2} \) Copy content Toggle raw display
$43$ \( (T + 172)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 192T + 36864 \) Copy content Toggle raw display
$53$ \( T^{2} + 558T + 311364 \) Copy content Toggle raw display
$59$ \( T^{2} + 540T + 291600 \) Copy content Toggle raw display
$61$ \( T^{2} + 110T + 12100 \) Copy content Toggle raw display
$67$ \( T^{2} + 140T + 19600 \) Copy content Toggle raw display
$71$ \( (T + 840)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 550T + 302500 \) Copy content Toggle raw display
$79$ \( T^{2} - 208T + 43264 \) Copy content Toggle raw display
$83$ \( (T - 516)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 1398 T + 1954404 \) Copy content Toggle raw display
$97$ \( (T - 1586)^{2} \) Copy content Toggle raw display
show more
show less