Properties

Label 637.2.r.c
Level $637$
Weight $2$
Character orbit 637.r
Analytic conductor $5.086$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.r (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.08647060876\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \zeta_{12} q^{2} + 2 \zeta_{12}^{2} q^{3} + 2 \zeta_{12}^{2} q^{4} + \zeta_{12} q^{5} + 4 \zeta_{12}^{3} q^{6} + ( -1 + \zeta_{12}^{2} ) q^{9} +O(q^{10})\) \( q + 2 \zeta_{12} q^{2} + 2 \zeta_{12}^{2} q^{3} + 2 \zeta_{12}^{2} q^{4} + \zeta_{12} q^{5} + 4 \zeta_{12}^{3} q^{6} + ( -1 + \zeta_{12}^{2} ) q^{9} + 2 \zeta_{12}^{2} q^{10} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{11} + ( -4 + 4 \zeta_{12}^{2} ) q^{12} + ( -2 + 3 \zeta_{12}^{3} ) q^{13} + 2 \zeta_{12}^{3} q^{15} + ( 4 - 4 \zeta_{12}^{2} ) q^{16} + 6 \zeta_{12}^{2} q^{17} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{18} -3 \zeta_{12} q^{19} + 2 \zeta_{12}^{3} q^{20} -4 q^{22} + ( 3 - 3 \zeta_{12}^{2} ) q^{23} -4 \zeta_{12}^{2} q^{25} + ( -6 - 4 \zeta_{12} + 6 \zeta_{12}^{2} ) q^{26} + 4 q^{27} + 3 q^{29} + ( -4 + 4 \zeta_{12}^{2} ) q^{30} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{31} + ( 8 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{32} -4 \zeta_{12} q^{33} + 12 \zeta_{12}^{3} q^{34} -2 q^{36} -6 \zeta_{12} q^{37} -6 \zeta_{12}^{2} q^{38} + ( -6 \zeta_{12} - 4 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{39} -10 \zeta_{12}^{3} q^{41} + q^{43} -4 \zeta_{12} q^{44} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{45} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{46} -11 \zeta_{12} q^{47} + 8 q^{48} -8 \zeta_{12}^{3} q^{50} + ( -12 + 12 \zeta_{12}^{2} ) q^{51} + ( -6 \zeta_{12} - 4 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{52} + 9 \zeta_{12}^{2} q^{53} + 8 \zeta_{12} q^{54} -2 q^{55} -6 \zeta_{12}^{3} q^{57} + 6 \zeta_{12} q^{58} + ( -8 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{59} + ( -4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{60} + ( 8 - 8 \zeta_{12}^{2} ) q^{61} + 6 q^{62} + 8 q^{64} + ( -3 - 2 \zeta_{12} + 3 \zeta_{12}^{2} ) q^{65} -8 \zeta_{12}^{2} q^{66} + ( 12 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{67} + ( -12 + 12 \zeta_{12}^{2} ) q^{68} + 6 q^{69} + 14 \zeta_{12}^{3} q^{71} + ( -9 \zeta_{12} + 9 \zeta_{12}^{3} ) q^{73} -12 \zeta_{12}^{2} q^{74} + ( 8 - 8 \zeta_{12}^{2} ) q^{75} -6 \zeta_{12}^{3} q^{76} + ( -12 - 8 \zeta_{12}^{3} ) q^{78} + ( 9 - 9 \zeta_{12}^{2} ) q^{79} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{80} + 11 \zeta_{12}^{2} q^{81} + ( 20 - 20 \zeta_{12}^{2} ) q^{82} -11 \zeta_{12}^{3} q^{83} + 6 \zeta_{12}^{3} q^{85} + 2 \zeta_{12} q^{86} + 6 \zeta_{12}^{2} q^{87} + 5 \zeta_{12} q^{89} -2 q^{90} + 6 q^{92} + 6 \zeta_{12} q^{93} -22 \zeta_{12}^{2} q^{94} -3 \zeta_{12}^{2} q^{95} + 16 \zeta_{12} q^{96} + 9 \zeta_{12}^{3} q^{97} -2 \zeta_{12}^{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} + 4q^{4} - 2q^{9} + O(q^{10}) \) \( 4q + 4q^{3} + 4q^{4} - 2q^{9} + 4q^{10} - 8q^{12} - 8q^{13} + 8q^{16} + 12q^{17} - 16q^{22} + 6q^{23} - 8q^{25} - 12q^{26} + 16q^{27} + 12q^{29} - 8q^{30} - 8q^{36} - 12q^{38} - 8q^{39} + 4q^{43} + 32q^{48} - 24q^{51} - 8q^{52} + 18q^{53} - 8q^{55} + 16q^{61} + 24q^{62} + 32q^{64} - 6q^{65} - 16q^{66} - 24q^{68} + 24q^{69} - 24q^{74} + 16q^{75} - 48q^{78} + 18q^{79} + 22q^{81} + 40q^{82} + 12q^{87} - 8q^{90} + 24q^{92} - 44q^{94} - 6q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/637\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(248\)
\(\chi(n)\) \(-1\) \(-1 + \zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
116.1
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
−1.73205 + 1.00000i 1.00000 1.73205i 1.00000 1.73205i −0.866025 + 0.500000i 4.00000i 0 0 −0.500000 0.866025i 1.00000 1.73205i
116.2 1.73205 1.00000i 1.00000 1.73205i 1.00000 1.73205i 0.866025 0.500000i 4.00000i 0 0 −0.500000 0.866025i 1.00000 1.73205i
324.1 −1.73205 1.00000i 1.00000 + 1.73205i 1.00000 + 1.73205i −0.866025 0.500000i 4.00000i 0 0 −0.500000 + 0.866025i 1.00000 + 1.73205i
324.2 1.73205 + 1.00000i 1.00000 + 1.73205i 1.00000 + 1.73205i 0.866025 + 0.500000i 4.00000i 0 0 −0.500000 + 0.866025i 1.00000 + 1.73205i
\(n\): 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
13.b even 2 1 inner
91.r even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.r.c 4
7.b odd 2 1 637.2.r.a 4
7.c even 3 1 637.2.c.a 2
7.c even 3 1 inner 637.2.r.c 4
7.d odd 6 1 637.2.c.c yes 2
7.d odd 6 1 637.2.r.a 4
13.b even 2 1 inner 637.2.r.c 4
91.b odd 2 1 637.2.r.a 4
91.r even 6 1 637.2.c.a 2
91.r even 6 1 inner 637.2.r.c 4
91.s odd 6 1 637.2.c.c yes 2
91.s odd 6 1 637.2.r.a 4
91.z odd 12 1 8281.2.a.a 1
91.z odd 12 1 8281.2.a.k 1
91.bb even 12 1 8281.2.a.b 1
91.bb even 12 1 8281.2.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.c.a 2 7.c even 3 1
637.2.c.a 2 91.r even 6 1
637.2.c.c yes 2 7.d odd 6 1
637.2.c.c yes 2 91.s odd 6 1
637.2.r.a 4 7.b odd 2 1
637.2.r.a 4 7.d odd 6 1
637.2.r.a 4 91.b odd 2 1
637.2.r.a 4 91.s odd 6 1
637.2.r.c 4 1.a even 1 1 trivial
637.2.r.c 4 7.c even 3 1 inner
637.2.r.c 4 13.b even 2 1 inner
637.2.r.c 4 91.r even 6 1 inner
8281.2.a.a 1 91.z odd 12 1
8281.2.a.b 1 91.bb even 12 1
8281.2.a.k 1 91.z odd 12 1
8281.2.a.m 1 91.bb even 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(637, [\chi])\):

\( T_{2}^{4} - 4 T_{2}^{2} + 16 \)
\( T_{3}^{2} - 2 T_{3} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 - 4 T^{2} + T^{4} \)
$3$ \( ( 4 - 2 T + T^{2} )^{2} \)
$5$ \( 1 - T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( 16 - 4 T^{2} + T^{4} \)
$13$ \( ( 13 + 4 T + T^{2} )^{2} \)
$17$ \( ( 36 - 6 T + T^{2} )^{2} \)
$19$ \( 81 - 9 T^{2} + T^{4} \)
$23$ \( ( 9 - 3 T + T^{2} )^{2} \)
$29$ \( ( -3 + T )^{4} \)
$31$ \( 81 - 9 T^{2} + T^{4} \)
$37$ \( 1296 - 36 T^{2} + T^{4} \)
$41$ \( ( 100 + T^{2} )^{2} \)
$43$ \( ( -1 + T )^{4} \)
$47$ \( 14641 - 121 T^{2} + T^{4} \)
$53$ \( ( 81 - 9 T + T^{2} )^{2} \)
$59$ \( 4096 - 64 T^{2} + T^{4} \)
$61$ \( ( 64 - 8 T + T^{2} )^{2} \)
$67$ \( 20736 - 144 T^{2} + T^{4} \)
$71$ \( ( 196 + T^{2} )^{2} \)
$73$ \( 6561 - 81 T^{2} + T^{4} \)
$79$ \( ( 81 - 9 T + T^{2} )^{2} \)
$83$ \( ( 121 + T^{2} )^{2} \)
$89$ \( 625 - 25 T^{2} + T^{4} \)
$97$ \( ( 81 + T^{2} )^{2} \)
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