Properties

Label 637.2.f.g
Level $637$
Weight $2$
Character orbit 637.f
Analytic conductor $5.086$
Analytic rank $0$
Dimension $8$
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.f (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.08647060876\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.1485512441856.7
Defining polynomial: \(x^{8} + 24 x^{6} + 455 x^{4} + 2904 x^{2} + 14641\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{2} ) q^{2} + \beta_{6} q^{3} + \beta_{2} q^{4} + ( \beta_{1} - \beta_{7} ) q^{5} -\beta_{5} q^{6} -3 q^{8} + \beta_{2} q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{2} ) q^{2} + \beta_{6} q^{3} + \beta_{2} q^{4} + ( \beta_{1} - \beta_{7} ) q^{5} -\beta_{5} q^{6} -3 q^{8} + \beta_{2} q^{9} + \beta_{7} q^{10} + ( -1 + \beta_{2} - \beta_{3} - \beta_{4} ) q^{11} + ( -\beta_{5} + \beta_{6} ) q^{12} + ( \beta_{5} - \beta_{6} + \beta_{7} ) q^{13} + ( -1 + \beta_{2} + \beta_{3} + \beta_{4} ) q^{15} + ( 1 - \beta_{2} ) q^{16} + ( \beta_{1} - 2 \beta_{5} ) q^{17} - q^{18} + 2 \beta_{5} q^{19} + \beta_{1} q^{20} + ( -\beta_{2} + \beta_{4} ) q^{22} + ( 3 - 3 \beta_{2} - \beta_{3} - \beta_{4} ) q^{23} -3 \beta_{6} q^{24} + ( 7 - \beta_{3} ) q^{25} + ( -\beta_{1} + \beta_{6} ) q^{26} + ( -4 \beta_{5} + 4 \beta_{6} ) q^{27} + ( 4 - 4 \beta_{2} + \beta_{3} + \beta_{4} ) q^{29} + ( -\beta_{2} - \beta_{4} ) q^{30} + ( -\beta_{5} + \beta_{6} ) q^{31} -5 \beta_{2} q^{32} + ( 2 \beta_{1} - 2 \beta_{5} ) q^{33} + ( -\beta_{1} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{34} + ( -1 + \beta_{2} ) q^{36} + ( 2 - 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{37} + ( -2 \beta_{5} + 2 \beta_{6} ) q^{38} + ( -2 + \beta_{2} - \beta_{4} ) q^{39} + ( -3 \beta_{1} + 3 \beta_{7} ) q^{40} + ( 5 \beta_{6} - \beta_{7} ) q^{41} + ( -3 \beta_{2} - \beta_{4} ) q^{43} + ( -1 - \beta_{3} ) q^{44} + \beta_{1} q^{45} + ( 3 \beta_{2} + \beta_{4} ) q^{46} + ( 2 \beta_{5} - 2 \beta_{6} ) q^{47} + \beta_{5} q^{48} + ( -7 + 7 \beta_{2} + \beta_{3} + \beta_{4} ) q^{50} + ( 3 + \beta_{3} ) q^{51} + ( -\beta_{1} + \beta_{5} + \beta_{7} ) q^{52} + ( -3 + 2 \beta_{3} ) q^{53} -4 \beta_{6} q^{54} -11 \beta_{6} q^{55} -4 q^{57} + ( 4 \beta_{2} - \beta_{4} ) q^{58} + ( 2 \beta_{1} + 3 \beta_{5} ) q^{59} + ( -1 + \beta_{3} ) q^{60} + ( -\beta_{1} - 3 \beta_{5} ) q^{61} -\beta_{6} q^{62} + 7 q^{64} + ( -11 + 12 \beta_{2} + \beta_{4} ) q^{65} + ( -2 \beta_{1} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{66} + ( -1 + \beta_{2} - \beta_{3} - \beta_{4} ) q^{67} + ( -2 \beta_{6} + \beta_{7} ) q^{68} + ( 2 \beta_{1} + 2 \beta_{5} ) q^{69} -6 \beta_{2} q^{71} -3 \beta_{2} q^{72} + ( -\beta_{1} - 6 \beta_{5} + 6 \beta_{6} + \beta_{7} ) q^{73} + ( 2 \beta_{2} - \beta_{4} ) q^{74} + ( 6 \beta_{6} + 2 \beta_{7} ) q^{75} + 2 \beta_{6} q^{76} + ( 1 - 2 \beta_{2} - \beta_{3} ) q^{78} + ( -7 - \beta_{3} ) q^{79} -\beta_{7} q^{80} + ( 5 - 5 \beta_{2} ) q^{81} + ( \beta_{1} - 5 \beta_{5} ) q^{82} + ( -7 \beta_{5} + 7 \beta_{6} ) q^{83} + ( 10 \beta_{2} - \beta_{4} ) q^{85} + ( 3 - \beta_{3} ) q^{86} + ( -2 \beta_{1} + 5 \beta_{5} ) q^{87} + ( 3 - 3 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} ) q^{88} + ( \beta_{6} - 4 \beta_{7} ) q^{89} + ( -\beta_{1} + \beta_{7} ) q^{90} + ( 3 - \beta_{3} ) q^{92} + ( 2 - 2 \beta_{2} ) q^{93} + 2 \beta_{6} q^{94} + ( 2 \beta_{2} + 2 \beta_{4} ) q^{95} + ( 5 \beta_{5} - 5 \beta_{6} ) q^{96} + 3 \beta_{5} q^{97} + ( -1 - \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} + 4 q^{4} - 24 q^{8} + 4 q^{9} + O(q^{10}) \) \( 8 q - 4 q^{2} + 4 q^{4} - 24 q^{8} + 4 q^{9} - 4 q^{11} - 4 q^{15} + 4 q^{16} - 8 q^{18} - 4 q^{22} + 12 q^{23} + 56 q^{25} + 16 q^{29} - 4 q^{30} - 20 q^{32} - 4 q^{36} + 8 q^{37} - 12 q^{39} - 12 q^{43} - 8 q^{44} + 12 q^{46} - 28 q^{50} + 24 q^{51} - 24 q^{53} - 32 q^{57} + 16 q^{58} - 8 q^{60} + 56 q^{64} - 40 q^{65} - 4 q^{67} - 24 q^{71} - 12 q^{72} + 8 q^{74} - 56 q^{79} + 20 q^{81} + 40 q^{85} + 24 q^{86} + 12 q^{88} + 24 q^{92} + 8 q^{93} + 8 q^{95} - 8 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 24 x^{6} + 455 x^{4} + 2904 x^{2} + 14641\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 24 \nu^{6} + 455 \nu^{4} + 10920 \nu^{2} + 69696 \)\()/55055\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{6} + 2556 \)\()/455\)
\(\beta_{4}\)\(=\)\((\)\( -167 \nu^{6} - 5460 \nu^{4} - 75985 \nu^{2} - 484968 \)\()/55055\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} - 3011 \nu \)\()/5005\)
\(\beta_{6}\)\(=\)\((\)\( -191 \nu^{7} - 5915 \nu^{5} - 86905 \nu^{3} - 554664 \nu \)\()/605605\)
\(\beta_{7}\)\(=\)\((\)\( 24 \nu^{7} + 455 \nu^{5} + 10920 \nu^{3} + 69696 \nu \)\()/55055\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} + \beta_{3} + 12 \beta_{2} - 12\)
\(\nu^{3}\)\(=\)\(13 \beta_{7} + 11 \beta_{6} - 11 \beta_{5} - 13 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-24 \beta_{4} - 167 \beta_{2}\)
\(\nu^{5}\)\(=\)\(-191 \beta_{7} - 264 \beta_{6}\)
\(\nu^{6}\)\(=\)\(-455 \beta_{3} + 2556\)
\(\nu^{7}\)\(=\)\(5005 \beta_{5} + 3011 \beta_{1}\)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
295.1
−1.34203 2.32446i
2.04914 + 3.54921i
−2.04914 3.54921i
1.34203 + 2.32446i
−1.34203 + 2.32446i
2.04914 3.54921i
−2.04914 + 3.54921i
1.34203 2.32446i
−0.500000 + 0.866025i −0.707107 + 1.22474i 0.500000 + 0.866025i −2.68406 −0.707107 1.22474i 0 −3.00000 0.500000 + 0.866025i 1.34203 2.32446i
295.2 −0.500000 + 0.866025i −0.707107 + 1.22474i 0.500000 + 0.866025i 4.09827 −0.707107 1.22474i 0 −3.00000 0.500000 + 0.866025i −2.04914 + 3.54921i
295.3 −0.500000 + 0.866025i 0.707107 1.22474i 0.500000 + 0.866025i −4.09827 0.707107 + 1.22474i 0 −3.00000 0.500000 + 0.866025i 2.04914 3.54921i
295.4 −0.500000 + 0.866025i 0.707107 1.22474i 0.500000 + 0.866025i 2.68406 0.707107 + 1.22474i 0 −3.00000 0.500000 + 0.866025i −1.34203 + 2.32446i
393.1 −0.500000 0.866025i −0.707107 1.22474i 0.500000 0.866025i −2.68406 −0.707107 + 1.22474i 0 −3.00000 0.500000 0.866025i 1.34203 + 2.32446i
393.2 −0.500000 0.866025i −0.707107 1.22474i 0.500000 0.866025i 4.09827 −0.707107 + 1.22474i 0 −3.00000 0.500000 0.866025i −2.04914 3.54921i
393.3 −0.500000 0.866025i 0.707107 + 1.22474i 0.500000 0.866025i −4.09827 0.707107 1.22474i 0 −3.00000 0.500000 0.866025i 2.04914 + 3.54921i
393.4 −0.500000 0.866025i 0.707107 + 1.22474i 0.500000 0.866025i 2.68406 0.707107 1.22474i 0 −3.00000 0.500000 0.866025i −1.34203 2.32446i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 393.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
13.c even 3 1 inner
91.n odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.f.g 8
7.b odd 2 1 inner 637.2.f.g 8
7.c even 3 1 637.2.g.h 8
7.c even 3 1 637.2.h.k 8
7.d odd 6 1 637.2.g.h 8
7.d odd 6 1 637.2.h.k 8
13.c even 3 1 inner 637.2.f.g 8
13.c even 3 1 8281.2.a.bv 4
13.e even 6 1 8281.2.a.bn 4
91.g even 3 1 637.2.h.k 8
91.h even 3 1 637.2.g.h 8
91.m odd 6 1 637.2.h.k 8
91.n odd 6 1 inner 637.2.f.g 8
91.n odd 6 1 8281.2.a.bv 4
91.t odd 6 1 8281.2.a.bn 4
91.v odd 6 1 637.2.g.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.f.g 8 1.a even 1 1 trivial
637.2.f.g 8 7.b odd 2 1 inner
637.2.f.g 8 13.c even 3 1 inner
637.2.f.g 8 91.n odd 6 1 inner
637.2.g.h 8 7.c even 3 1
637.2.g.h 8 7.d odd 6 1
637.2.g.h 8 91.h even 3 1
637.2.g.h 8 91.v odd 6 1
637.2.h.k 8 7.c even 3 1
637.2.h.k 8 7.d odd 6 1
637.2.h.k 8 91.g even 3 1
637.2.h.k 8 91.m odd 6 1
8281.2.a.bn 4 13.e even 6 1
8281.2.a.bn 4 91.t odd 6 1
8281.2.a.bv 4 13.c even 3 1
8281.2.a.bv 4 91.n odd 6 1

Hecke kernels

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

\( T_{2}^{2} + T_{2} + 1 \)
\( T_{3}^{4} + 2 T_{3}^{2} + 4 \)
\( T_{5}^{4} - 24 T_{5}^{2} + 121 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{4} \)
$3$ \( ( 4 + 2 T^{2} + T^{4} )^{2} \)
$5$ \( ( 121 - 24 T^{2} + T^{4} )^{2} \)
$7$ \( T^{8} \)
$11$ \( ( 484 - 44 T + 26 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$13$ \( 28561 + 3380 T^{2} + 231 T^{4} + 20 T^{6} + T^{8} \)
$17$ \( 2401 + 1568 T^{2} + 975 T^{4} + 32 T^{6} + T^{8} \)
$19$ \( ( 64 + 8 T^{2} + T^{4} )^{2} \)
$23$ \( ( 196 + 84 T + 50 T^{2} - 6 T^{3} + T^{4} )^{2} \)
$29$ \( ( 49 + 56 T + 71 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$31$ \( ( -2 + T^{2} )^{4} \)
$37$ \( ( 361 + 76 T + 35 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$41$ \( 707281 + 87464 T^{2} + 9975 T^{4} + 104 T^{6} + T^{8} \)
$43$ \( ( 196 - 84 T + 50 T^{2} + 6 T^{3} + T^{4} )^{2} \)
$47$ \( ( -8 + T^{2} )^{4} \)
$53$ \( ( -83 + 6 T + T^{2} )^{4} \)
$59$ \( 38416 + 30576 T^{2} + 24140 T^{4} + 156 T^{6} + T^{8} \)
$61$ \( 28561 + 12168 T^{2} + 5015 T^{4} + 72 T^{6} + T^{8} \)
$67$ \( ( 484 - 44 T + 26 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$71$ \( ( 36 + 6 T + T^{2} )^{4} \)
$73$ \( ( 5329 - 192 T^{2} + T^{4} )^{2} \)
$79$ \( ( 26 + 14 T + T^{2} )^{4} \)
$83$ \( ( -98 + T^{2} )^{4} \)
$89$ \( 1097199376 + 12322128 T^{2} + 105260 T^{4} + 372 T^{6} + T^{8} \)
$97$ \( ( 324 + 18 T^{2} + T^{4} )^{2} \)
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