Properties

Label 196.6.e.d
Level $196$
Weight $6$
Character orbit 196.e
Analytic conductor $31.435$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Self dual: no
Analytic conductor: \(31.4352286833\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 4)
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.

\(f(q)\) \(=\) \( q + ( -12 + 12 \zeta_{6} ) q^{3} + 54 \zeta_{6} q^{5} + 99 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( -12 + 12 \zeta_{6} ) q^{3} + 54 \zeta_{6} q^{5} + 99 \zeta_{6} q^{9} + ( -540 + 540 \zeta_{6} ) q^{11} + 418 q^{13} -648 q^{15} + ( 594 - 594 \zeta_{6} ) q^{17} + 836 \zeta_{6} q^{19} + 4104 \zeta_{6} q^{23} + ( 209 - 209 \zeta_{6} ) q^{25} -4104 q^{27} -594 q^{29} + ( 4256 - 4256 \zeta_{6} ) q^{31} -6480 \zeta_{6} q^{33} + 298 \zeta_{6} q^{37} + ( -5016 + 5016 \zeta_{6} ) q^{39} -17226 q^{41} -12100 q^{43} + ( -5346 + 5346 \zeta_{6} ) q^{45} -1296 \zeta_{6} q^{47} + 7128 \zeta_{6} q^{51} + ( -19494 + 19494 \zeta_{6} ) q^{53} -29160 q^{55} -10032 q^{57} + ( -7668 + 7668 \zeta_{6} ) q^{59} -34738 \zeta_{6} q^{61} + 22572 \zeta_{6} q^{65} + ( -21812 + 21812 \zeta_{6} ) q^{67} -49248 q^{69} -46872 q^{71} + ( 67562 - 67562 \zeta_{6} ) q^{73} + 2508 \zeta_{6} q^{75} + 76912 \zeta_{6} q^{79} + ( 25191 - 25191 \zeta_{6} ) q^{81} -67716 q^{83} + 32076 q^{85} + ( 7128 - 7128 \zeta_{6} ) q^{87} + 29754 \zeta_{6} q^{89} + 51072 \zeta_{6} q^{93} + ( -45144 + 45144 \zeta_{6} ) q^{95} + 122398 q^{97} -53460 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 12 q^{3} + 54 q^{5} + 99 q^{9} + O(q^{10}) \) \( 2 q - 12 q^{3} + 54 q^{5} + 99 q^{9} - 540 q^{11} + 836 q^{13} - 1296 q^{15} + 594 q^{17} + 836 q^{19} + 4104 q^{23} + 209 q^{25} - 8208 q^{27} - 1188 q^{29} + 4256 q^{31} - 6480 q^{33} + 298 q^{37} - 5016 q^{39} - 34452 q^{41} - 24200 q^{43} - 5346 q^{45} - 1296 q^{47} + 7128 q^{51} - 19494 q^{53} - 58320 q^{55} - 20064 q^{57} - 7668 q^{59} - 34738 q^{61} + 22572 q^{65} - 21812 q^{67} - 98496 q^{69} - 93744 q^{71} + 67562 q^{73} + 2508 q^{75} + 76912 q^{79} + 25191 q^{81} - 135432 q^{83} + 64152 q^{85} + 7128 q^{87} + 29754 q^{89} + 51072 q^{93} - 45144 q^{95} + 244796 q^{97} - 106920 q^{99} + O(q^{100}) \)

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 −6.00000 + 10.3923i 0 27.0000 + 46.7654i 0 0 0 49.5000 + 85.7365i 0
177.1 0 −6.00000 10.3923i 0 27.0000 46.7654i 0 0 0 49.5000 85.7365i 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.d 2
7.b odd 2 1 196.6.e.g 2
7.c even 3 1 196.6.a.e 1
7.c even 3 1 inner 196.6.e.d 2
7.d odd 6 1 4.6.a.a 1
7.d odd 6 1 196.6.e.g 2
21.g even 6 1 36.6.a.a 1
28.f even 6 1 16.6.a.b 1
28.g odd 6 1 784.6.a.d 1
35.i odd 6 1 100.6.a.b 1
35.k even 12 2 100.6.c.b 2
56.j odd 6 1 64.6.a.f 1
56.m even 6 1 64.6.a.b 1
63.i even 6 1 324.6.e.d 2
63.k odd 6 1 324.6.e.a 2
63.s even 6 1 324.6.e.d 2
63.t odd 6 1 324.6.e.a 2
77.i even 6 1 484.6.a.a 1
84.j odd 6 1 144.6.a.c 1
91.s odd 6 1 676.6.a.a 1
91.bb even 12 2 676.6.d.a 2
105.p even 6 1 900.6.a.h 1
105.w odd 12 2 900.6.d.a 2
112.v even 12 2 256.6.b.c 2
112.x odd 12 2 256.6.b.g 2
140.s even 6 1 400.6.a.d 1
140.x odd 12 2 400.6.c.f 2
168.ba even 6 1 576.6.a.bc 1
168.be odd 6 1 576.6.a.bd 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.6.a.a 1 7.d odd 6 1
16.6.a.b 1 28.f even 6 1
36.6.a.a 1 21.g even 6 1
64.6.a.b 1 56.m even 6 1
64.6.a.f 1 56.j odd 6 1
100.6.a.b 1 35.i odd 6 1
100.6.c.b 2 35.k even 12 2
144.6.a.c 1 84.j odd 6 1
196.6.a.e 1 7.c even 3 1
196.6.e.d 2 1.a even 1 1 trivial
196.6.e.d 2 7.c even 3 1 inner
196.6.e.g 2 7.b odd 2 1
196.6.e.g 2 7.d odd 6 1
256.6.b.c 2 112.v even 12 2
256.6.b.g 2 112.x odd 12 2
324.6.e.a 2 63.k odd 6 1
324.6.e.a 2 63.t odd 6 1
324.6.e.d 2 63.i even 6 1
324.6.e.d 2 63.s even 6 1
400.6.a.d 1 140.s even 6 1
400.6.c.f 2 140.x odd 12 2
484.6.a.a 1 77.i even 6 1
576.6.a.bc 1 168.ba even 6 1
576.6.a.bd 1 168.be odd 6 1
676.6.a.a 1 91.s odd 6 1
676.6.d.a 2 91.bb even 12 2
784.6.a.d 1 28.g odd 6 1
900.6.a.h 1 105.p even 6 1
900.6.d.a 2 105.w odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 12 T_{3} + 144 \) acting on \(S_{6}^{\mathrm{new}}(196, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 144 + 12 T + T^{2} \)
$5$ \( 2916 - 54 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 291600 + 540 T + T^{2} \)
$13$ \( ( -418 + T )^{2} \)
$17$ \( 352836 - 594 T + T^{2} \)
$19$ \( 698896 - 836 T + T^{2} \)
$23$ \( 16842816 - 4104 T + T^{2} \)
$29$ \( ( 594 + T )^{2} \)
$31$ \( 18113536 - 4256 T + T^{2} \)
$37$ \( 88804 - 298 T + T^{2} \)
$41$ \( ( 17226 + T )^{2} \)
$43$ \( ( 12100 + T )^{2} \)
$47$ \( 1679616 + 1296 T + T^{2} \)
$53$ \( 380016036 + 19494 T + T^{2} \)
$59$ \( 58798224 + 7668 T + T^{2} \)
$61$ \( 1206728644 + 34738 T + T^{2} \)
$67$ \( 475763344 + 21812 T + T^{2} \)
$71$ \( ( 46872 + T )^{2} \)
$73$ \( 4564623844 - 67562 T + T^{2} \)
$79$ \( 5915455744 - 76912 T + T^{2} \)
$83$ \( ( 67716 + T )^{2} \)
$89$ \( 885300516 - 29754 T + T^{2} \)
$97$ \( ( -122398 + T )^{2} \)
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