Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.k (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: \(\Q(\sqrt{-3}, \sqrt{-7})\)
Defining polynomial: \( x^{4} - x^{3} - x^{2} - 2x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + \beta_1) q^{2} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 1) q^{3} + (\beta_{3} + \beta_1 - 1) q^{4} + ( - \beta_{3} + 1) q^{5} + (\beta_{3} + \beta_{2} - 3) q^{6} + (2 \beta_{2} - 1) q^{8} + ( - 2 \beta_{3} + 4 \beta_{2} + \beta_1 - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + \beta_1) q^{2} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 1) q^{3} + (\beta_{3} + \beta_1 - 1) q^{4} + ( - \beta_{3} + 1) q^{5} + (\beta_{3} + \beta_{2} - 3) q^{6} + (2 \beta_{2} - 1) q^{8} + ( - 2 \beta_{3} + 4 \beta_{2} + \beta_1 - 3) q^{9} + ( - \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{10} + (\beta_{3} - 2 \beta_{2} + 3) q^{11} + 5 \beta_{2} q^{12} + (\beta_{2} - 4) q^{13} + ( - \beta_{2} + \beta_1 - 2) q^{15} + (\beta_{3} + \beta_1) q^{16} - 3 q^{17} + ( - \beta_{2} - \beta_1) q^{18} + (3 \beta_1 - 3) q^{19} + (2 \beta_{3} + \beta_{2} - 4) q^{20} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{22} + ( - 2 \beta_{3} - 2 \beta_1 + 4) q^{23} + ( - 3 \beta_{3} + 2 \beta_{2} - 1) q^{24} + ( - 2 \beta_{3} + 4 \beta_{2} + \beta_1 - 3) q^{25} + (3 \beta_{3} - 4 \beta_1 + 1) q^{26} - 5 q^{27} + (2 \beta_{3} - 6 \beta_{2} - \beta_1 + 5) q^{29} + (4 \beta_{3} + \beta_{2} - 2 \beta_1 - 3) q^{30} + (5 \beta_{2} + 5) q^{31} + (\beta_{3} + 6 \beta_{2} - \beta_1 - 3) q^{32} + (5 \beta_{2} + 5 \beta_1) q^{33} + (3 \beta_{3} - 3 \beta_1) q^{34} + ( - 4 \beta_{3} + 7 \beta_{2} + 2 \beta_1 - 5) q^{36} + ( - 8 \beta_{2} + 4) q^{37} + (6 \beta_{3} + 3 \beta_{2} - 3 \beta_1 - 6) q^{38} + (2 \beta_{3} - 6 \beta_{2} - 7 \beta_1 + 3) q^{39} + (\beta_{3} - 2 \beta_1 + 1) q^{40} + ( - 4 \beta_{2} - 2 \beta_1 - 2) q^{41} + (3 \beta_{3} - 7 \beta_{2} - 6 \beta_1 + 3) q^{43} + (4 \beta_{3} - 3 \beta_{2} + 2) q^{44} + (\beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{45} + ( - 6 \beta_{3} - 4 \beta_{2} + 6 \beta_1 + 2) q^{46} + (2 \beta_{3} - 5 \beta_{2} + 8) q^{47} + ( - \beta_{3} + 7 \beta_{2} + 2 \beta_1 - 1) q^{48} + ( - \beta_{2} - \beta_1) q^{50} + (3 \beta_{3} - 6 \beta_{2} - 6 \beta_1 + 3) q^{51} + ( - 5 \beta_{3} + \beta_{2} - 2 \beta_1 + 3) q^{52} + ( - 8 \beta_{3} + 3 \beta_{2} + 4 \beta_1 + 1) q^{53} + (5 \beta_{3} - 5 \beta_1) q^{54} + ( - 2 \beta_{3} + \beta_{2} + \beta_1) q^{55} + (3 \beta_{3} + 6 \beta_{2} - 3 \beta_1 - 3) q^{57} + (\beta_{2} + 3 \beta_1 - 2) q^{58} + (8 \beta_{3} - 10 \beta_{2} - 8 \beta_1 + 5) q^{59} + ( - 5 \beta_1 + 5) q^{60} + ( - 12 \beta_{3} + 10 \beta_{2} + 6 \beta_1 - 4) q^{61} + ( - 10 \beta_{3} + 5 \beta_1 + 5) q^{62} + ( - 2 \beta_{3} - 2 \beta_1 + 9) q^{64} + (4 \beta_{3} - \beta_1 - 3) q^{65} + (5 \beta_{2} - 5) q^{66} + (6 \beta_{3} - \beta_{2} - 4) q^{67} + ( - 3 \beta_{3} - 3 \beta_1 + 3) q^{68} + ( - 2 \beta_{3} - 6 \beta_{2} + 4 \beta_1 - 2) q^{69} + (2 \beta_{3} - 2 \beta_{2} + 2) q^{71} + ( - 4 \beta_{2} - 3 \beta_1 - 1) q^{72} + ( - 2 \beta_{2} - 2) q^{73} + (4 \beta_{3} + 4 \beta_1 - 8) q^{74} + ( - 3 \beta_{3} - 3 \beta_1 - 5) q^{75} + (3 \beta_{2} - 6 \beta_1 + 9) q^{76} + ( - 4 \beta_{3} - 5 \beta_{2} + \beta_1 + 10) q^{78} + 6 \beta_{2} q^{79} + (\beta_{3} + \beta_{2} - 3) q^{80} + (2 \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 2) q^{81} + (4 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{82} + ( - 4 \beta_{3} + 6 \beta_{2} + 4 \beta_1 - 3) q^{83} + (3 \beta_{3} - 3) q^{85} + ( - 2 \beta_{3} - 3 \beta_{2} + 8) q^{86} + (5 \beta_{3} + 5 \beta_1 + 5) q^{87} + ( - \beta_{3} + 6 \beta_{2} + 2 \beta_1 - 1) q^{88} + (5 \beta_{3} - 16 \beta_{2} - 5 \beta_1 + 8) q^{89} + q^{90} + (4 \beta_{3} + 4 \beta_1 - 14) q^{92} + ( - 15 \beta_{3} + 20 \beta_{2} + 15 \beta_1 - 10) q^{93} + ( - 3 \beta_{3} + 2 \beta_{2} + 6 \beta_1 - 3) q^{94} + (3 \beta_{3} + 3 \beta_1 - 9) q^{95} + ( - 10 \beta_{3} + 5 \beta_{2}) q^{96} + ( - 3 \beta_{3} - 5 \beta_{2} + 13) q^{97} + ( - 7 \beta_{3} + 18 \beta_{2} + 7 \beta_1 - 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{3} - 2 q^{4} + 3 q^{5} - 9 q^{6} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{3} - 2 q^{4} + 3 q^{5} - 9 q^{6} - 5 q^{9} - 5 q^{10} + 9 q^{11} + 10 q^{12} - 14 q^{13} - 9 q^{15} + 2 q^{16} - 12 q^{17} - 3 q^{18} - 9 q^{19} - 12 q^{20} - q^{22} + 12 q^{23} - 3 q^{24} - 5 q^{25} + 3 q^{26} - 20 q^{27} + 9 q^{29} - 8 q^{30} + 30 q^{31} + 15 q^{33} - 8 q^{36} - 15 q^{38} - 5 q^{39} + 3 q^{40} - 18 q^{41} - 5 q^{43} + 6 q^{44} + 24 q^{47} + 11 q^{48} - 3 q^{50} - 3 q^{51} + 7 q^{52} + 6 q^{53} + q^{55} - 3 q^{58} + 15 q^{60} - 2 q^{61} + 15 q^{62} + 32 q^{64} - 9 q^{65} - 10 q^{66} - 12 q^{67} + 6 q^{68} - 18 q^{69} + 6 q^{71} - 15 q^{72} - 12 q^{73} - 24 q^{74} - 26 q^{75} + 36 q^{76} + 27 q^{78} + 12 q^{79} - 9 q^{80} + 10 q^{81} - 2 q^{82} - 9 q^{85} + 24 q^{86} + 30 q^{87} + 9 q^{88} + 4 q^{90} - 48 q^{92} - 5 q^{94} - 30 q^{95} + 39 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - x^{2} - 2x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + \nu^{2} - \nu - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + \nu^{2} + \nu + 2 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{3} + \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(197\) \(248\)
\(\chi(n)\) \(\beta_{2}\) \(-\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
459.1
−0.895644 1.09445i
1.39564 + 0.228425i
1.39564 0.228425i
−0.895644 + 1.09445i
2.18890i −0.895644 1.55130i −2.79129 1.89564 1.09445i −3.39564 + 1.96048i 0 1.73205i −0.104356 + 0.180750i −2.39564 4.14938i
459.2 0.456850i 1.39564 + 2.41733i 1.79129 −0.395644 + 0.228425i −1.10436 + 0.637600i 0 1.73205i −2.39564 + 4.14938i −0.104356 0.180750i
569.1 0.456850i 1.39564 2.41733i 1.79129 −0.395644 0.228425i −1.10436 0.637600i 0 1.73205i −2.39564 4.14938i −0.104356 + 0.180750i
569.2 2.18890i −0.895644 + 1.55130i −2.79129 1.89564 + 1.09445i −3.39564 1.96048i 0 1.73205i −0.104356 0.180750i −2.39564 + 4.14938i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.k 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.k.f 4
7.b odd 2 1 637.2.k.d 4
7.c even 3 1 637.2.q.f yes 4
7.c even 3 1 637.2.u.d 4
7.d odd 6 1 637.2.q.e 4
7.d odd 6 1 637.2.u.e 4
13.e even 6 1 637.2.u.d 4
91.k even 6 1 inner 637.2.k.f 4
91.l odd 6 1 637.2.k.d 4
91.p odd 6 1 637.2.q.e 4
91.t odd 6 1 637.2.u.e 4
91.u even 6 1 637.2.q.f yes 4
91.x odd 12 2 8281.2.a.bq 4
91.ba even 12 2 8281.2.a.bs 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.k.d 4 7.b odd 2 1
637.2.k.d 4 91.l odd 6 1
637.2.k.f 4 1.a even 1 1 trivial
637.2.k.f 4 91.k even 6 1 inner
637.2.q.e 4 7.d odd 6 1
637.2.q.e 4 91.p odd 6 1
637.2.q.f yes 4 7.c even 3 1
637.2.q.f yes 4 91.u even 6 1
637.2.u.d 4 7.c even 3 1
637.2.u.d 4 13.e even 6 1
637.2.u.e 4 7.d odd 6 1
637.2.u.e 4 91.t odd 6 1
8281.2.a.bq 4 91.x odd 12 2
8281.2.a.bs 4 91.ba even 12 2

Hecke kernels

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

\( T_{2}^{4} + 5T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{4} - T_{3}^{3} + 6T_{3}^{2} + 5T_{3} + 25 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 5T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} - T^{3} + 6 T^{2} + 5 T + 25 \) Copy content Toggle raw display
$5$ \( T^{4} - 3 T^{3} + 2 T^{2} + 3 T + 1 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 9 T^{3} + 32 T^{2} - 45 T + 25 \) Copy content Toggle raw display
$13$ \( (T^{2} + 7 T + 13)^{2} \) Copy content Toggle raw display
$17$ \( (T + 3)^{4} \) Copy content Toggle raw display
$19$ \( T^{4} + 9 T^{3} + 18 T^{2} - 81 T + 81 \) Copy content Toggle raw display
$23$ \( (T^{2} - 6 T - 12)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - 9 T^{3} + 66 T^{2} - 135 T + 225 \) Copy content Toggle raw display
$31$ \( (T^{2} - 15 T + 75)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 18 T^{3} + 128 T^{2} + \cdots + 400 \) Copy content Toggle raw display
$43$ \( T^{4} + 5 T^{3} + 66 T^{2} + \cdots + 1681 \) Copy content Toggle raw display
$47$ \( T^{4} - 24 T^{3} + 233 T^{2} + \cdots + 1681 \) Copy content Toggle raw display
$53$ \( T^{4} - 6 T^{3} + 111 T^{2} + \cdots + 5625 \) Copy content Toggle raw display
$59$ \( T^{4} + 230 T^{2} + 11881 \) Copy content Toggle raw display
$61$ \( T^{4} + 2 T^{3} + 192 T^{2} + \cdots + 35344 \) Copy content Toggle raw display
$67$ \( T^{4} + 12 T^{3} - 3 T^{2} + \cdots + 2601 \) Copy content Toggle raw display
$71$ \( T^{4} - 6 T^{3} + 8 T^{2} + 24 T + 16 \) Copy content Toggle raw display
$73$ \( (T^{2} + 6 T + 12)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 6 T + 36)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 62T^{2} + 625 \) Copy content Toggle raw display
$89$ \( T^{4} + 269T^{2} + 2209 \) Copy content Toggle raw display
$97$ \( T^{4} - 39 T^{3} + 618 T^{2} + \cdots + 12321 \) Copy content Toggle raw display
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