Properties

Label 637.2.h.g
Level $637$
Weight $2$
Character orbit 637.h
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.h (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: \(\Q(\sqrt{-3}, \sqrt{5})\)
Defining polynomial: \(x^{4} - x^{3} + 2 x^{2} + x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{2} ) q^{2} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{3} -3 \beta_{2} q^{4} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{5} + ( -3 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{6} + ( 1 - 4 \beta_{2} ) q^{8} + ( 1 - 3 \beta_{1} + \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{2} ) q^{2} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{3} -3 \beta_{2} q^{4} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{5} + ( -3 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{6} + ( 1 - 4 \beta_{2} ) q^{8} + ( 1 - 3 \beta_{1} + \beta_{3} ) q^{9} + ( 3 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{10} + ( 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{11} + ( -6 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} ) q^{12} + ( -4 - 3 \beta_{3} ) q^{13} + ( 2 + 3 \beta_{1} + 2 \beta_{3} ) q^{15} + ( 5 - 3 \beta_{2} ) q^{16} + ( 5 + 4 \beta_{2} ) q^{17} + ( -2 - 5 \beta_{1} - 2 \beta_{3} ) q^{18} + ( -3 + 3 \beta_{1} - 3 \beta_{3} ) q^{19} + ( 6 \beta_{1} + 6 \beta_{2} + 3 \beta_{3} ) q^{20} + ( 3 \beta_{1} + 3 \beta_{2} ) q^{22} + ( 2 + 4 \beta_{2} ) q^{23} + ( -9 \beta_{1} - 9 \beta_{2} - 5 \beta_{3} ) q^{24} + ( 3 - 3 \beta_{1} + 3 \beta_{3} ) q^{25} + ( -4 - 3 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{26} + ( 1 + 2 \beta_{2} ) q^{27} + ( 1 - 5 \beta_{1} + \beta_{3} ) q^{29} + ( 5 + 8 \beta_{1} + 5 \beta_{3} ) q^{30} + ( -5 + 6 \beta_{1} - 5 \beta_{3} ) q^{31} + ( 6 - 3 \beta_{2} ) q^{32} + 3 \beta_{1} q^{33} + ( 1 + 3 \beta_{2} ) q^{34} + ( -9 - 6 \beta_{1} - 9 \beta_{3} ) q^{36} + 4 q^{37} + 3 \beta_{1} q^{38} + ( -3 + \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{39} + ( 9 \beta_{1} + 9 \beta_{2} + 5 \beta_{3} ) q^{40} + ( -4 + 2 \beta_{1} - 4 \beta_{3} ) q^{41} + ( 9 \beta_{1} + 9 \beta_{2} - 2 \beta_{3} ) q^{43} + 9 \beta_{3} q^{44} + ( 2 - 5 \beta_{2} ) q^{45} + ( -2 + 6 \beta_{2} ) q^{46} + ( -2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{47} + ( -11 \beta_{1} - 11 \beta_{2} - 8 \beta_{3} ) q^{48} -3 \beta_{1} q^{50} + ( 3 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{51} + ( -9 \beta_{1} + 3 \beta_{2} ) q^{52} + ( -7 + 2 \beta_{1} - 7 \beta_{3} ) q^{53} + ( -1 + 3 \beta_{2} ) q^{54} -3 \beta_{1} q^{55} -3 \beta_{2} q^{57} + ( -4 - 9 \beta_{1} - 4 \beta_{3} ) q^{58} + ( 1 + 2 \beta_{2} ) q^{59} + ( 9 + 15 \beta_{1} + 9 \beta_{3} ) q^{60} + ( 6 + 6 \beta_{3} ) q^{61} + ( 1 + 7 \beta_{1} + \beta_{3} ) q^{62} + ( -1 - 6 \beta_{2} ) q^{64} + ( 3 - \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{65} + ( 3 + 6 \beta_{1} + 3 \beta_{3} ) q^{66} + ( -6 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} ) q^{67} + ( -12 - 3 \beta_{2} ) q^{68} + ( 6 \beta_{1} + 6 \beta_{2} + 2 \beta_{3} ) q^{69} + ( -10 \beta_{1} - 10 \beta_{2} + 2 \beta_{3} ) q^{71} + ( -11 - 11 \beta_{1} - 11 \beta_{3} ) q^{72} + ( 2 + 2 \beta_{3} ) q^{73} + ( 4 - 4 \beta_{2} ) q^{74} + 3 \beta_{2} q^{75} + ( 9 + 9 \beta_{3} ) q^{76} + ( -6 + 3 \beta_{1} + 12 \beta_{2} + 2 \beta_{3} ) q^{78} + 4 \beta_{3} q^{79} + ( 11 \beta_{1} + 11 \beta_{2} + 8 \beta_{3} ) q^{80} + ( -6 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} ) q^{81} + ( -2 - 2 \beta_{3} ) q^{82} + ( -3 - 6 \beta_{2} ) q^{83} + ( -3 \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{85} + ( 16 \beta_{1} + 16 \beta_{2} + 7 \beta_{3} ) q^{86} + ( -4 + 9 \beta_{2} ) q^{87} + ( 3 \beta_{1} + 3 \beta_{2} + 9 \beta_{3} ) q^{88} + ( 13 + 5 \beta_{2} ) q^{89} + ( 7 - 12 \beta_{2} ) q^{90} + ( -12 + 6 \beta_{2} ) q^{92} + ( 1 - 7 \beta_{2} ) q^{93} + ( -3 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{94} + 3 \beta_{2} q^{95} + ( -12 \beta_{1} - 12 \beta_{2} - 9 \beta_{3} ) q^{96} + ( 3 \beta_{1} + 3 \beta_{2} + 14 \beta_{3} ) q^{97} + ( 12 + 3 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{2} + 3 q^{3} + 6 q^{4} - 3 q^{5} + 7 q^{6} + 12 q^{8} - q^{9} + O(q^{10}) \) \( 4 q + 6 q^{2} + 3 q^{3} + 6 q^{4} - 3 q^{5} + 7 q^{6} + 12 q^{8} - q^{9} - 7 q^{10} + 3 q^{11} + 12 q^{12} - 10 q^{13} + 7 q^{15} + 26 q^{16} + 12 q^{17} - 9 q^{18} - 3 q^{19} - 12 q^{20} - 3 q^{22} + 19 q^{24} + 3 q^{25} - 15 q^{26} - 3 q^{29} + 18 q^{30} - 4 q^{31} + 30 q^{32} + 3 q^{33} - 2 q^{34} - 24 q^{36} + 16 q^{37} + 3 q^{38} - 21 q^{39} - 19 q^{40} - 6 q^{41} - 5 q^{43} - 18 q^{44} + 18 q^{45} - 20 q^{46} + 27 q^{48} - 3 q^{50} - q^{51} - 15 q^{52} - 12 q^{53} - 10 q^{54} - 3 q^{55} + 6 q^{57} - 17 q^{58} + 33 q^{60} + 12 q^{61} + 9 q^{62} + 8 q^{64} + 21 q^{65} + 12 q^{66} + 12 q^{67} - 42 q^{68} - 10 q^{69} + 6 q^{71} - 33 q^{72} + 4 q^{73} + 24 q^{74} - 6 q^{75} + 18 q^{76} - 49 q^{78} - 8 q^{79} - 27 q^{80} - 2 q^{81} - 4 q^{82} + q^{85} - 30 q^{86} - 34 q^{87} - 21 q^{88} + 42 q^{89} + 52 q^{90} - 60 q^{92} + 18 q^{93} + 5 q^{94} - 6 q^{95} + 30 q^{96} - 31 q^{97} + 42 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 2 x^{2} + x + 1\):

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

Character values

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

\(n\) \(197\) \(248\)
\(\chi(n)\) \(\beta_{3}\) \(\beta_{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
165.1
−0.309017 + 0.535233i
0.809017 1.40126i
−0.309017 0.535233i
0.809017 + 1.40126i
0.381966 0.190983 + 0.330792i −1.85410 −0.190983 0.330792i 0.0729490 + 0.126351i 0 −1.47214 1.42705 2.47172i −0.0729490 0.126351i
165.2 2.61803 1.30902 + 2.26728i 4.85410 −1.30902 2.26728i 3.42705 + 5.93583i 0 7.47214 −1.92705 + 3.33775i −3.42705 5.93583i
471.1 0.381966 0.190983 0.330792i −1.85410 −0.190983 + 0.330792i 0.0729490 0.126351i 0 −1.47214 1.42705 + 2.47172i −0.0729490 + 0.126351i
471.2 2.61803 1.30902 2.26728i 4.85410 −1.30902 + 2.26728i 3.42705 5.93583i 0 7.47214 −1.92705 3.33775i −3.42705 + 5.93583i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.h 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.h.g 4
7.b odd 2 1 637.2.h.f 4
7.c even 3 1 91.2.f.a 4
7.c even 3 1 637.2.g.b 4
7.d odd 6 1 637.2.f.c 4
7.d odd 6 1 637.2.g.c 4
13.c even 3 1 637.2.g.b 4
21.h odd 6 1 819.2.o.c 4
28.g odd 6 1 1456.2.s.h 4
91.g even 3 1 91.2.f.a 4
91.h even 3 1 inner 637.2.h.g 4
91.h even 3 1 1183.2.a.g 2
91.k even 6 1 1183.2.a.c 2
91.l odd 6 1 8281.2.a.n 2
91.m odd 6 1 637.2.f.c 4
91.n odd 6 1 637.2.g.c 4
91.v odd 6 1 637.2.h.f 4
91.v odd 6 1 8281.2.a.bb 2
91.x odd 12 2 1183.2.c.c 4
273.bm odd 6 1 819.2.o.c 4
364.q odd 6 1 1456.2.s.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.f.a 4 7.c even 3 1
91.2.f.a 4 91.g even 3 1
637.2.f.c 4 7.d odd 6 1
637.2.f.c 4 91.m odd 6 1
637.2.g.b 4 7.c even 3 1
637.2.g.b 4 13.c even 3 1
637.2.g.c 4 7.d odd 6 1
637.2.g.c 4 91.n odd 6 1
637.2.h.f 4 7.b odd 2 1
637.2.h.f 4 91.v odd 6 1
637.2.h.g 4 1.a even 1 1 trivial
637.2.h.g 4 91.h even 3 1 inner
819.2.o.c 4 21.h odd 6 1
819.2.o.c 4 273.bm odd 6 1
1183.2.a.c 2 91.k even 6 1
1183.2.a.g 2 91.h even 3 1
1183.2.c.c 4 91.x odd 12 2
1456.2.s.h 4 28.g odd 6 1
1456.2.s.h 4 364.q odd 6 1
8281.2.a.n 2 91.l odd 6 1
8281.2.a.bb 2 91.v odd 6 1

Hecke kernels

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

\( T_{2}^{2} - 3 T_{2} + 1 \)
\( T_{3}^{4} - 3 T_{3}^{3} + 8 T_{3}^{2} - 3 T_{3} + 1 \)
\( T_{5}^{4} + 3 T_{5}^{3} + 8 T_{5}^{2} + 3 T_{5} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 3 T + T^{2} )^{2} \)
$3$ \( 1 - 3 T + 8 T^{2} - 3 T^{3} + T^{4} \)
$5$ \( 1 + 3 T + 8 T^{2} + 3 T^{3} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( 81 + 27 T + 18 T^{2} - 3 T^{3} + T^{4} \)
$13$ \( ( 13 + 5 T + T^{2} )^{2} \)
$17$ \( ( -11 - 6 T + T^{2} )^{2} \)
$19$ \( 81 - 27 T + 18 T^{2} + 3 T^{3} + T^{4} \)
$23$ \( ( -20 + T^{2} )^{2} \)
$29$ \( 841 - 87 T + 38 T^{2} + 3 T^{3} + T^{4} \)
$31$ \( 1681 - 164 T + 57 T^{2} + 4 T^{3} + T^{4} \)
$37$ \( ( -4 + T )^{4} \)
$41$ \( 16 + 24 T + 32 T^{2} + 6 T^{3} + T^{4} \)
$43$ \( 9025 - 475 T + 120 T^{2} + 5 T^{3} + T^{4} \)
$47$ \( 25 + 5 T^{2} + T^{4} \)
$53$ \( 961 + 372 T + 113 T^{2} + 12 T^{3} + T^{4} \)
$59$ \( ( -5 + T^{2} )^{2} \)
$61$ \( ( 36 - 6 T + T^{2} )^{2} \)
$67$ \( 81 + 108 T + 153 T^{2} - 12 T^{3} + T^{4} \)
$71$ \( 13456 + 696 T + 152 T^{2} - 6 T^{3} + T^{4} \)
$73$ \( ( 4 - 2 T + T^{2} )^{2} \)
$79$ \( ( 16 + 4 T + T^{2} )^{2} \)
$83$ \( ( -45 + T^{2} )^{2} \)
$89$ \( ( 79 - 21 T + T^{2} )^{2} \)
$97$ \( 52441 + 7099 T + 732 T^{2} + 31 T^{3} + T^{4} \)
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