Properties

Label 637.2.r.a
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: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
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.

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

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

Hecke kernels

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

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

Hecke characteristic polynomials

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