Properties

Label 637.2.h.b
Level $637$
Weight $2$
Character orbit 637.h
Analytic conductor $5.086$
Analytic rank $1$
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: \(1\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} - 1) q^{2} + ( - \beta_{3} - \beta_1) q^{3} + ( - 2 \beta_{3} + 1) q^{4} + (2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{5} + (\beta_{3} + 2 \beta_{2} + \beta_1) q^{6} + (\beta_{3} - 3) q^{8} + (\beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - 1) q^{2} + ( - \beta_{3} - \beta_1) q^{3} + ( - 2 \beta_{3} + 1) q^{4} + (2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{5} + (\beta_{3} + 2 \beta_{2} + \beta_1) q^{6} + (\beta_{3} - 3) q^{8} + (\beta_{2} + 1) q^{9} + ( - 3 \beta_{3} - 5 \beta_{2} - 3 \beta_1) q^{10} + (\beta_{3} + 2 \beta_{2} + \beta_1) q^{11} + ( - \beta_{3} - 4 \beta_{2} - \beta_1) q^{12} + (\beta_{2} - 3) q^{13} + (4 \beta_{2} + \beta_1 + 4) q^{15} + 3 q^{16} + ( - 2 \beta_{3} - 3) q^{17} + ( - \beta_{2} - \beta_1 - 1) q^{18} + ( - 6 \beta_{2} - 6) q^{19} + (4 \beta_{3} + 9 \beta_{2} + 4 \beta_1) q^{20} + ( - 3 \beta_{3} - 4 \beta_{2} - 3 \beta_1) q^{22} - \beta_{3} q^{23} + (3 \beta_{3} + 2 \beta_{2} + 3 \beta_1) q^{24} + ( - 4 \beta_{2} - 4 \beta_1 - 4) q^{25} + ( - 4 \beta_{3} - \beta_{2} - \beta_1 + 3) q^{26} - 4 \beta_{3} q^{27} + ( - 7 \beta_{2} - 2 \beta_1 - 7) q^{29} + ( - 6 \beta_{2} - 5 \beta_1 - 6) q^{30} + ( - 4 \beta_{2} - \beta_1 - 4) q^{31} + (\beta_{3} + 3) q^{32} + (2 \beta_{2} + 2 \beta_1 + 2) q^{33} + ( - \beta_{3} - 1) q^{34} + (\beta_{2} + 2 \beta_1 + 1) q^{36} + ( - 6 \beta_{3} - 1) q^{37} + (6 \beta_{2} + 6 \beta_1 + 6) q^{38} + (3 \beta_{3} + 4 \beta_1) q^{39} + ( - 7 \beta_{3} - 7 \beta_{2} - 7 \beta_1) q^{40} + ( - 3 \beta_{2} - 2 \beta_1 - 3) q^{41} + ( - \beta_{3} + 2 \beta_{2} - \beta_1) q^{43} + (5 \beta_{3} + 6 \beta_{2} + 5 \beta_1) q^{44} + (2 \beta_{3} - 1) q^{45} + (\beta_{3} - 2) q^{46} + (4 \beta_{3} + 2 \beta_{2} + 4 \beta_1) q^{47} + ( - 3 \beta_{3} - 3 \beta_1) q^{48} + (12 \beta_{2} + 8 \beta_1 + 12) q^{50} + (3 \beta_{3} - 4 \beta_{2} + 3 \beta_1) q^{51} + (8 \beta_{3} + \beta_{2} + 2 \beta_1 - 3) q^{52} + (3 \beta_{2} + 3) q^{53} + (4 \beta_{3} - 8) q^{54} + ( - 6 \beta_{2} - 5 \beta_1 - 6) q^{55} + 6 \beta_{3} q^{57} + (11 \beta_{2} + 9 \beta_1 + 11) q^{58} + ( - 3 \beta_{3} - 6) q^{59} + (8 \beta_{2} + 9 \beta_1 + 8) q^{60} + ( - 7 \beta_{2} - 2 \beta_1 - 7) q^{61} + (6 \beta_{2} + 5 \beta_1 + 6) q^{62} + (2 \beta_{3} - 7) q^{64} + ( - 6 \beta_{3} - 4 \beta_{2} - 8 \beta_1 - 1) q^{65} + ( - 6 \beta_{2} - 4 \beta_1 - 6) q^{66} + ( - 3 \beta_{3} - 3 \beta_1) q^{67} + (4 \beta_{3} + 5) q^{68} - 2 \beta_{2} q^{69} + (4 \beta_{3} - 6 \beta_{2} + 4 \beta_1) q^{71} + ( - 3 \beta_{2} - \beta_1 - 3) q^{72} + (5 \beta_{2} - 4 \beta_1 + 5) q^{73} + (5 \beta_{3} - 11) q^{74} + (4 \beta_{3} - 8) q^{75} + ( - 6 \beta_{2} - 12 \beta_1 - 6) q^{76} + ( - 3 \beta_{3} - 8 \beta_{2} - 4 \beta_1 - 2) q^{78} + (3 \beta_{3} + 6 \beta_{2} + 3 \beta_1) q^{79} + (6 \beta_{3} + 3 \beta_{2} + 6 \beta_1) q^{80} - 5 \beta_{2} q^{81} + (7 \beta_{2} + 5 \beta_1 + 7) q^{82} + (5 \beta_{3} - 6) q^{83} + ( - 4 \beta_{3} + 5 \beta_{2} - 4 \beta_1) q^{85} + ( - \beta_{3} - \beta_1) q^{86} + (7 \beta_{3} - 4) q^{87} + ( - 5 \beta_{3} - 8 \beta_{2} - 5 \beta_1) q^{88} + (8 \beta_{3} + 4) q^{89} + ( - 3 \beta_{3} + 5) q^{90} + ( - \beta_{3} + 4) q^{92} + (4 \beta_{3} - 2) q^{93} + ( - 6 \beta_{3} - 10 \beta_{2} - 6 \beta_1) q^{94} + ( - 12 \beta_{3} + 6) q^{95} + ( - 3 \beta_{3} + 2 \beta_{2} - 3 \beta_1) q^{96} + (2 \beta_{3} - 8 \beta_{2} + 2 \beta_1) q^{97} + (\beta_{3} - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 4 q^{4} - 2 q^{5} - 4 q^{6} - 12 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 4 q^{4} - 2 q^{5} - 4 q^{6} - 12 q^{8} + 2 q^{9} + 10 q^{10} - 4 q^{11} + 8 q^{12} - 14 q^{13} + 8 q^{15} + 12 q^{16} - 12 q^{17} - 2 q^{18} - 12 q^{19} - 18 q^{20} + 8 q^{22} - 4 q^{24} - 8 q^{25} + 14 q^{26} - 14 q^{29} - 12 q^{30} - 8 q^{31} + 12 q^{32} + 4 q^{33} - 4 q^{34} + 2 q^{36} - 4 q^{37} + 12 q^{38} + 14 q^{40} - 6 q^{41} - 4 q^{43} - 12 q^{44} - 4 q^{45} - 8 q^{46} - 4 q^{47} + 24 q^{50} + 8 q^{51} - 14 q^{52} + 6 q^{53} - 32 q^{54} - 12 q^{55} + 22 q^{58} - 24 q^{59} + 16 q^{60} - 14 q^{61} + 12 q^{62} - 28 q^{64} + 4 q^{65} - 12 q^{66} + 20 q^{68} + 4 q^{69} + 12 q^{71} - 6 q^{72} + 10 q^{73} - 44 q^{74} - 32 q^{75} - 12 q^{76} + 8 q^{78} - 12 q^{79} - 6 q^{80} + 10 q^{81} + 14 q^{82} - 24 q^{83} - 10 q^{85} - 16 q^{87} + 16 q^{88} + 16 q^{89} + 20 q^{90} + 16 q^{92} - 8 q^{93} + 20 q^{94} + 24 q^{95} - 4 q^{96} + 16 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
165.1
0.707107 1.22474i
−0.707107 + 1.22474i
0.707107 + 1.22474i
−0.707107 1.22474i
−2.41421 0.707107 + 1.22474i 3.82843 −1.91421 3.31552i −1.70711 2.95680i 0 −4.41421 0.500000 0.866025i 4.62132 + 8.00436i
165.2 0.414214 −0.707107 1.22474i −1.82843 0.914214 + 1.58346i −0.292893 0.507306i 0 −1.58579 0.500000 0.866025i 0.378680 + 0.655892i
471.1 −2.41421 0.707107 1.22474i 3.82843 −1.91421 + 3.31552i −1.70711 + 2.95680i 0 −4.41421 0.500000 + 0.866025i 4.62132 8.00436i
471.2 0.414214 −0.707107 + 1.22474i −1.82843 0.914214 1.58346i −0.292893 + 0.507306i 0 −1.58579 0.500000 + 0.866025i 0.378680 0.655892i
\(n\): 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.b 4
7.b odd 2 1 637.2.h.c 4
7.c even 3 1 637.2.f.f yes 4
7.c even 3 1 637.2.g.f 4
7.d odd 6 1 637.2.f.e 4
7.d odd 6 1 637.2.g.g 4
13.c even 3 1 637.2.g.f 4
91.g even 3 1 637.2.f.f yes 4
91.h even 3 1 inner 637.2.h.b 4
91.h even 3 1 8281.2.a.p 2
91.k even 6 1 8281.2.a.x 2
91.l odd 6 1 8281.2.a.y 2
91.m odd 6 1 637.2.f.e 4
91.n odd 6 1 637.2.g.g 4
91.v odd 6 1 637.2.h.c 4
91.v odd 6 1 8281.2.a.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.f.e 4 7.d odd 6 1
637.2.f.e 4 91.m odd 6 1
637.2.f.f yes 4 7.c even 3 1
637.2.f.f yes 4 91.g even 3 1
637.2.g.f 4 7.c even 3 1
637.2.g.f 4 13.c even 3 1
637.2.g.g 4 7.d odd 6 1
637.2.g.g 4 91.n odd 6 1
637.2.h.b 4 1.a even 1 1 trivial
637.2.h.b 4 91.h even 3 1 inner
637.2.h.c 4 7.b odd 2 1
637.2.h.c 4 91.v odd 6 1
8281.2.a.o 2 91.v odd 6 1
8281.2.a.p 2 91.h even 3 1
8281.2.a.x 2 91.k even 6 1
8281.2.a.y 2 91.l odd 6 1

Hecke kernels

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

\( T_{2}^{2} + 2T_{2} - 1 \) Copy content Toggle raw display
\( T_{3}^{4} + 2T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{5}^{4} + 2T_{5}^{3} + 11T_{5}^{2} - 14T_{5} + 49 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 2T^{2} + 4 \) Copy content Toggle raw display
$5$ \( T^{4} + 2 T^{3} + 11 T^{2} - 14 T + 49 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 4 T^{3} + 14 T^{2} + 8 T + 4 \) Copy content Toggle raw display
$13$ \( (T^{2} + 7 T + 13)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 6 T + 1)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 14 T^{3} + 155 T^{2} + \cdots + 1681 \) Copy content Toggle raw display
$31$ \( T^{4} + 8 T^{3} + 50 T^{2} + 112 T + 196 \) Copy content Toggle raw display
$37$ \( (T^{2} + 2 T - 71)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 6 T^{3} + 35 T^{2} + 6 T + 1 \) Copy content Toggle raw display
$43$ \( T^{4} + 4 T^{3} + 14 T^{2} + 8 T + 4 \) Copy content Toggle raw display
$47$ \( T^{4} + 4 T^{3} + 44 T^{2} - 112 T + 784 \) Copy content Toggle raw display
$53$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 12 T + 18)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 14 T^{3} + 155 T^{2} + \cdots + 1681 \) Copy content Toggle raw display
$67$ \( T^{4} + 18T^{2} + 324 \) Copy content Toggle raw display
$71$ \( T^{4} - 12 T^{3} + 140 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$73$ \( T^{4} - 10 T^{3} + 107 T^{2} + \cdots + 49 \) Copy content Toggle raw display
$79$ \( T^{4} + 12 T^{3} + 126 T^{2} + \cdots + 324 \) Copy content Toggle raw display
$83$ \( (T^{2} + 12 T - 14)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 8 T - 112)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 16 T^{3} + 200 T^{2} + \cdots + 3136 \) Copy content Toggle raw display
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