Properties

Label 637.2.e.g
Level $637$
Weight $2$
Character orbit 637.e
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.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.08647060876\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

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

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

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
−0.707107 + 1.22474i
0.707107 1.22474i
−0.707107 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i −0.707107 1.22474i 0 2.20711 3.82282i 2.00000 0 −2.82843 0.500000 0.866025i 3.12132 + 5.40629i
79.2 0.707107 1.22474i 0.707107 + 1.22474i 0 0.792893 1.37333i 2.00000 0 2.82843 0.500000 0.866025i −1.12132 1.94218i
508.1 −0.707107 1.22474i −0.707107 + 1.22474i 0 2.20711 + 3.82282i 2.00000 0 −2.82843 0.500000 + 0.866025i 3.12132 5.40629i
508.2 0.707107 + 1.22474i 0.707107 1.22474i 0 0.792893 + 1.37333i 2.00000 0 2.82843 0.500000 + 0.866025i −1.12132 + 1.94218i
\(n\): 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.e.g 4
7.b odd 2 1 637.2.e.f 4
7.c even 3 1 637.2.a.g 2
7.c even 3 1 inner 637.2.e.g 4
7.d odd 6 1 91.2.a.c 2
7.d odd 6 1 637.2.e.f 4
21.g even 6 1 819.2.a.h 2
21.h odd 6 1 5733.2.a.s 2
28.f even 6 1 1456.2.a.q 2
35.i odd 6 1 2275.2.a.j 2
56.j odd 6 1 5824.2.a.bl 2
56.m even 6 1 5824.2.a.bk 2
91.r even 6 1 8281.2.a.v 2
91.s odd 6 1 1183.2.a.d 2
91.bb even 12 2 1183.2.c.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.a.c 2 7.d odd 6 1
637.2.a.g 2 7.c even 3 1
637.2.e.f 4 7.b odd 2 1
637.2.e.f 4 7.d odd 6 1
637.2.e.g 4 1.a even 1 1 trivial
637.2.e.g 4 7.c even 3 1 inner
819.2.a.h 2 21.g even 6 1
1183.2.a.d 2 91.s odd 6 1
1183.2.c.d 4 91.bb even 12 2
1456.2.a.q 2 28.f even 6 1
2275.2.a.j 2 35.i odd 6 1
5733.2.a.s 2 21.h odd 6 1
5824.2.a.bk 2 56.m even 6 1
5824.2.a.bl 2 56.j odd 6 1
8281.2.a.v 2 91.r even 6 1

Hecke kernels

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

\( T_{2}^{4} + 2 T_{2}^{2} + 4 \)
\( T_{3}^{4} + 2 T_{3}^{2} + 4 \)
\( T_{5}^{4} - 6 T_{5}^{3} + 29 T_{5}^{2} - 42 T_{5} + 49 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + 2 T^{2} + T^{4} \)
$3$ \( 4 + 2 T^{2} + T^{4} \)
$5$ \( 49 - 42 T + 29 T^{2} - 6 T^{3} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( 324 + 18 T^{2} + T^{4} \)
$13$ \( ( -1 + T )^{4} \)
$17$ \( 4 + 2 T^{2} + T^{4} \)
$19$ \( 81 - 54 T + 45 T^{2} + 6 T^{3} + T^{4} \)
$23$ \( 1 - 6 T + 35 T^{2} - 6 T^{3} + T^{4} \)
$29$ \( ( 1 - 6 T + T^{2} )^{2} \)
$31$ \( 289 - 34 T + 21 T^{2} + 2 T^{3} + T^{4} \)
$37$ \( 196 + 56 T + 30 T^{2} - 4 T^{3} + T^{4} \)
$41$ \( ( 28 + 12 T + T^{2} )^{2} \)
$43$ \( ( 5 + T )^{4} \)
$47$ \( 49 - 42 T + 29 T^{2} - 6 T^{3} + T^{4} \)
$53$ \( 1 - 6 T + 35 T^{2} - 6 T^{3} + T^{4} \)
$59$ \( 16 - 48 T + 140 T^{2} - 12 T^{3} + T^{4} \)
$61$ \( ( 36 - 6 T + T^{2} )^{2} \)
$67$ \( 1296 + 432 T + 180 T^{2} - 12 T^{3} + T^{4} \)
$71$ \( ( -14 + 12 T + T^{2} )^{2} \)
$73$ \( 49 + 70 T + 93 T^{2} + 10 T^{3} + T^{4} \)
$79$ \( 529 - 322 T + 219 T^{2} + 14 T^{3} + T^{4} \)
$83$ \( ( 63 + 18 T + T^{2} )^{2} \)
$89$ \( 49 - 42 T + 29 T^{2} - 6 T^{3} + T^{4} \)
$97$ \( ( -161 - 2 T + T^{2} )^{2} \)
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