Properties

Label 637.2.q.f
Level $637$
Weight $2$
Character orbit 637.q
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.q (of order \(6\), degree \(2\), 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: \(x^{4} - x^{3} - x^{2} - 2 x + 4\)
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 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.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{3} ) q^{2} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{3} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{4} + ( \beta_{1} - \beta_{3} ) q^{5} + ( -2 + \beta_{1} - \beta_{2} ) q^{6} + ( 1 - 2 \beta_{2} ) q^{8} + ( -3 + \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{3} ) q^{2} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{3} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{4} + ( \beta_{1} - \beta_{3} ) q^{5} + ( -2 + \beta_{1} - \beta_{2} ) q^{6} + ( 1 - 2 \beta_{2} ) q^{8} + ( -3 + \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{9} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{10} + ( -3 + 2 \beta_{2} - \beta_{3} ) q^{11} -5 q^{12} + ( -3 - \beta_{2} ) q^{13} + ( -3 + \beta_{2} + \beta_{3} ) q^{15} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{16} + ( 3 - 3 \beta_{2} ) q^{17} + ( 1 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{18} + ( 3 - 3 \beta_{1} ) q^{19} + ( -3 + 2 \beta_{1} - \beta_{2} ) q^{20} + ( -\beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{22} + ( -2 + 4 \beta_{1} - 2 \beta_{3} ) q^{23} + ( 1 - 2 \beta_{2} + 3 \beta_{3} ) q^{24} + ( 2 + \beta_{1} + \beta_{3} ) q^{25} + ( -4 + \beta_{1} + 3 \beta_{3} ) q^{26} -5 q^{27} + ( -1 + 2 \beta_{1} + 6 \beta_{2} - \beta_{3} ) q^{29} + ( -3 - 2 \beta_{1} + \beta_{2} + 4 \beta_{3} ) q^{30} + ( -5 + 10 \beta_{2} ) q^{31} + ( 4 - \beta_{1} + 3 \beta_{2} ) q^{32} + ( -5 \beta_{1} - 5 \beta_{2} ) q^{33} + ( 3 \beta_{1} - 3 \beta_{3} ) q^{34} + ( 2 - 4 \beta_{1} - 7 \beta_{2} + 2 \beta_{3} ) q^{36} + ( 8 - 4 \beta_{2} ) q^{37} + ( 9 - 3 \beta_{1} - 3 \beta_{3} ) q^{38} + ( 4 - 7 \beta_{1} - 8 \beta_{2} + 5 \beta_{3} ) q^{39} + ( -2 + \beta_{1} + \beta_{3} ) q^{40} + ( -6 + 4 \beta_{2} - 2 \beta_{3} ) q^{41} + ( -4 + 3 \beta_{1} + 7 \beta_{2} - 6 \beta_{3} ) q^{43} + ( 3 - 4 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} ) q^{44} + ( -\beta_{1} - \beta_{2} ) q^{45} + ( -8 + 6 \beta_{1} - 2 \beta_{2} ) q^{46} + ( 5 - 2 \beta_{1} - 10 \beta_{2} + 2 \beta_{3} ) q^{47} + ( 6 - \beta_{1} - 7 \beta_{2} + 2 \beta_{3} ) q^{48} + ( -1 + \beta_{2} - \beta_{3} ) q^{50} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{51} + ( -1 - 2 \beta_{1} - 3 \beta_{2} + 7 \beta_{3} ) q^{52} + ( -5 + 4 \beta_{1} + 4 \beta_{3} ) q^{53} + ( -5 + 5 \beta_{3} ) q^{54} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{55} + ( 3 + 3 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} ) q^{57} + ( 2 - 3 \beta_{1} - \beta_{2} ) q^{58} + ( 3 - 8 \beta_{1} - 5 \beta_{2} ) q^{59} + ( -5 \beta_{1} + 5 \beta_{3} ) q^{60} + ( -4 + 6 \beta_{1} + 10 \beta_{2} - 12 \beta_{3} ) q^{61} + ( 5 - 10 \beta_{1} + 5 \beta_{3} ) q^{62} + ( 9 - 2 \beta_{1} - 2 \beta_{3} ) q^{64} + ( -1 - 3 \beta_{1} + 4 \beta_{3} ) q^{65} + 5 q^{66} + ( 4 + \beta_{2} - 6 \beta_{3} ) q^{67} + ( -3 + 6 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{68} + ( -8 - 2 \beta_{1} + 6 \beta_{2} + 4 \beta_{3} ) q^{69} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{71} + ( 1 + 3 \beta_{1} + 4 \beta_{2} ) q^{72} + ( 2 - 4 \beta_{2} ) q^{73} + ( 4 + 4 \beta_{1} - 8 \beta_{3} ) q^{74} + ( -3 + 6 \beta_{1} + 11 \beta_{2} - 3 \beta_{3} ) q^{75} + ( 12 - 3 \beta_{2} - 6 \beta_{3} ) q^{76} + ( 5 - 4 \beta_{1} + 5 \beta_{2} + \beta_{3} ) q^{78} -6 q^{79} + ( 3 - \beta_{2} - \beta_{3} ) q^{80} + ( 2 - 4 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{81} + ( -2 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{82} + ( 3 - 4 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} ) q^{83} + ( -3 + 3 \beta_{1} ) q^{85} + ( 3 + 2 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} ) q^{86} + ( -10 + 5 \beta_{1} + 15 \beta_{2} - 10 \beta_{3} ) q^{87} + ( -1 + 2 \beta_{1} + 6 \beta_{2} - \beta_{3} ) q^{88} + ( 11 - 8 \beta_{2} + 5 \beta_{3} ) q^{89} + q^{90} + ( -14 + 4 \beta_{1} + 4 \beta_{3} ) q^{92} + ( -5 + 10 \beta_{2} - 15 \beta_{3} ) q^{93} + ( -3 + 6 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{94} + ( 6 + 3 \beta_{1} - 3 \beta_{2} - 6 \beta_{3} ) q^{95} + ( -5 + 10 \beta_{1} + 10 \beta_{2} - 10 \beta_{3} ) q^{96} + ( 8 - 3 \beta_{1} + 5 \beta_{2} ) q^{97} + ( 9 - 7 \beta_{1} - 18 \beta_{2} + 7 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 3q^{2} + q^{3} + q^{4} - 9q^{6} - 5q^{9} + O(q^{10}) \) \( 4q + 3q^{2} + q^{3} + q^{4} - 9q^{6} - 5q^{9} - 5q^{10} - 9q^{11} - 20q^{12} - 14q^{13} - 9q^{15} - q^{16} + 6q^{17} + 9q^{19} - 12q^{20} - q^{22} - 6q^{23} + 3q^{24} + 10q^{25} - 12q^{26} - 20q^{27} + 9q^{29} - 8q^{30} + 21q^{32} - 15q^{33} - 8q^{36} + 24q^{37} + 30q^{38} - 2q^{39} - 6q^{40} - 18q^{41} - 5q^{43} - 3q^{45} - 30q^{46} + 11q^{48} - 3q^{50} + 6q^{51} - 5q^{52} - 12q^{53} - 15q^{54} + q^{55} + 3q^{58} - 6q^{59} - 2q^{61} + 15q^{62} + 32q^{64} - 3q^{65} + 20q^{66} + 12q^{67} - 3q^{68} - 18q^{69} + 6q^{71} + 15q^{72} + 12q^{74} + 13q^{75} + 36q^{76} + 27q^{78} - 24q^{79} + 9q^{80} + 10q^{81} - 2q^{82} - 9q^{85} - 15q^{87} + 9q^{88} + 33q^{89} + 4q^{90} - 48q^{92} - 15q^{93} - 5q^{94} + 15q^{95} + 39q^{97} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + \nu^{2} - \nu - 2 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + \nu^{2} + \nu + 2 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2}\)
\(\nu^{3}\)\(=\)\(-\beta_{3} + \beta_{2} + \beta_{1} + 2\)

Character values

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

\(n\) \(197\) \(248\)
\(\chi(n)\) \(1 - \beta_{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
491.1
1.39564 + 0.228425i
−0.895644 1.09445i
1.39564 0.228425i
−0.895644 + 1.09445i
−0.395644 + 0.228425i 1.39564 + 2.41733i −0.895644 + 1.55130i 0.456850i −1.10436 0.637600i 0 1.73205i −2.39564 + 4.14938i −0.104356 0.180750i
491.2 1.89564 1.09445i −0.895644 1.55130i 1.39564 2.41733i 2.18890i −3.39564 1.96048i 0 1.73205i −0.104356 + 0.180750i −2.39564 4.14938i
589.1 −0.395644 0.228425i 1.39564 2.41733i −0.895644 1.55130i 0.456850i −1.10436 + 0.637600i 0 1.73205i −2.39564 4.14938i −0.104356 + 0.180750i
589.2 1.89564 + 1.09445i −0.895644 + 1.55130i 1.39564 + 2.41733i 2.18890i −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
13.e 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.q.f yes 4
7.b odd 2 1 637.2.q.e 4
7.c even 3 1 637.2.k.f 4
7.c even 3 1 637.2.u.d 4
7.d odd 6 1 637.2.k.d 4
7.d odd 6 1 637.2.u.e 4
13.e even 6 1 inner 637.2.q.f yes 4
13.f odd 12 2 8281.2.a.bq 4
91.k even 6 1 637.2.u.d 4
91.l odd 6 1 637.2.u.e 4
91.p odd 6 1 637.2.k.d 4
91.t odd 6 1 637.2.q.e 4
91.u even 6 1 637.2.k.f 4
91.bc 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.d odd 6 1
637.2.k.d 4 91.p odd 6 1
637.2.k.f 4 7.c even 3 1
637.2.k.f 4 91.u even 6 1
637.2.q.e 4 7.b odd 2 1
637.2.q.e 4 91.t odd 6 1
637.2.q.f yes 4 1.a even 1 1 trivial
637.2.q.f yes 4 13.e even 6 1 inner
637.2.u.d 4 7.c even 3 1
637.2.u.d 4 91.k even 6 1
637.2.u.e 4 7.d odd 6 1
637.2.u.e 4 91.l odd 6 1
8281.2.a.bq 4 13.f odd 12 2
8281.2.a.bs 4 91.bc even 12 2

Hecke kernels

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

\( T_{2}^{4} - 3 T_{2}^{3} + 2 T_{2}^{2} + 3 T_{2} + 1 \)
\( T_{3}^{4} - T_{3}^{3} + 6 T_{3}^{2} + 5 T_{3} + 25 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T + 2 T^{2} - 3 T^{3} + T^{4} \)
$3$ \( 25 + 5 T + 6 T^{2} - T^{3} + T^{4} \)
$5$ \( 1 + 5 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( 25 + 45 T + 32 T^{2} + 9 T^{3} + T^{4} \)
$13$ \( ( 13 + 7 T + T^{2} )^{2} \)
$17$ \( ( 9 - 3 T + T^{2} )^{2} \)
$19$ \( 81 + 81 T + 18 T^{2} - 9 T^{3} + T^{4} \)
$23$ \( 144 - 72 T + 48 T^{2} + 6 T^{3} + T^{4} \)
$29$ \( 225 - 135 T + 66 T^{2} - 9 T^{3} + T^{4} \)
$31$ \( ( 75 + T^{2} )^{2} \)
$37$ \( ( 48 - 12 T + T^{2} )^{2} \)
$41$ \( 400 + 360 T + 128 T^{2} + 18 T^{3} + T^{4} \)
$43$ \( 1681 - 205 T + 66 T^{2} + 5 T^{3} + T^{4} \)
$47$ \( 1681 + 110 T^{2} + T^{4} \)
$53$ \( ( -75 + 6 T + T^{2} )^{2} \)
$59$ \( 11881 - 654 T - 97 T^{2} + 6 T^{3} + T^{4} \)
$61$ \( 35344 - 376 T + 192 T^{2} + 2 T^{3} + T^{4} \)
$67$ \( 2601 + 612 T - 3 T^{2} - 12 T^{3} + T^{4} \)
$71$ \( 16 + 24 T + 8 T^{2} - 6 T^{3} + T^{4} \)
$73$ \( ( 12 + T^{2} )^{2} \)
$79$ \( ( 6 + T )^{4} \)
$83$ \( 625 + 62 T^{2} + T^{4} \)
$89$ \( 2209 - 1551 T + 410 T^{2} - 33 T^{3} + T^{4} \)
$97$ \( 12321 - 4329 T + 618 T^{2} - 39 T^{3} + T^{4} \)
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