Properties

Label 637.2.e.f
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: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} + \beta_1) q^{3} + ( - 3 \beta_{2} + \beta_1 - 3) q^{5} - 2 q^{6} - 2 \beta_{3} q^{8} + (\beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{3} + \beta_1) q^{3} + ( - 3 \beta_{2} + \beta_1 - 3) q^{5} - 2 q^{6} - 2 \beta_{3} q^{8} + (\beta_{2} + 1) q^{9} + ( - 3 \beta_{3} + 2 \beta_{2} - 3 \beta_1) q^{10} + (3 \beta_{3} + 3 \beta_1) q^{11} - q^{13} + ( - 3 \beta_{3} - 2) q^{15} + (4 \beta_{2} + 4) q^{16} + (\beta_{3} + \beta_1) q^{17} + (\beta_{3} + \beta_1) q^{18} + (3 \beta_{2} + 3 \beta_1 + 3) q^{19} - 6 q^{22} + (3 \beta_{2} + 2 \beta_1 + 3) q^{23} + 4 \beta_{2} q^{24} + ( - 6 \beta_{3} + 6 \beta_{2} - 6 \beta_1) q^{25} - \beta_1 q^{26} + 4 \beta_{3} q^{27} + (2 \beta_{3} + 3) q^{29} + (6 \beta_{2} - 2 \beta_1 + 6) q^{30} + (3 \beta_{3} - \beta_{2} + 3 \beta_1) q^{31} + ( - 6 \beta_{2} - 6) q^{33} - 2 q^{34} + (2 \beta_{2} - 3 \beta_1 + 2) q^{37} + (3 \beta_{3} + 6 \beta_{2} + 3 \beta_1) q^{38} + ( - \beta_{3} - \beta_1) q^{39} + (4 \beta_{2} - 6 \beta_1 + 4) q^{40} + ( - 2 \beta_{3} + 6) q^{41} - 5 q^{43} + (\beta_{3} - 3 \beta_{2} + \beta_1) q^{45} + (3 \beta_{3} + 4 \beta_{2} + 3 \beta_1) q^{46} + ( - 3 \beta_{2} + \beta_1 - 3) q^{47} + 4 \beta_{3} q^{48} + (6 \beta_{3} + 12) q^{50} + ( - 2 \beta_{2} - 2) q^{51} + (2 \beta_{3} - 3 \beta_{2} + 2 \beta_1) q^{53} + ( - 8 \beta_{2} - 8) q^{54} + ( - 9 \beta_{3} - 6) q^{55} + (3 \beta_{3} - 6) q^{57} + ( - 4 \beta_{2} + 3 \beta_1 - 4) q^{58} + ( - 4 \beta_{3} + 6 \beta_{2} - 4 \beta_1) q^{59} + ( - 6 \beta_{2} - 6) q^{61} + ( - \beta_{3} - 6) q^{62} + 8 q^{64} + (3 \beta_{2} - \beta_1 + 3) q^{65} + ( - 6 \beta_{3} - 6 \beta_1) q^{66} + ( - 6 \beta_{3} - 6 \beta_{2} - 6 \beta_1) q^{67} + (3 \beta_{3} - 4) q^{69} + (5 \beta_{3} - 6) q^{71} + 2 \beta_1 q^{72} + ( - 3 \beta_{3} - 5 \beta_{2} - 3 \beta_1) q^{73} + (2 \beta_{3} - 6 \beta_{2} + 2 \beta_1) q^{74} + (12 \beta_{2} - 6 \beta_1 + 12) q^{75} + 2 q^{78} + ( - 7 \beta_{2} - 6 \beta_1 - 7) q^{79} + (4 \beta_{3} - 12 \beta_{2} + 4 \beta_1) q^{80} - 5 \beta_{2} q^{81} + (4 \beta_{2} + 6 \beta_1 + 4) q^{82} + ( - 3 \beta_{3} + 9) q^{83} + ( - 3 \beta_{3} - 2) q^{85} - 5 \beta_1 q^{86} + (3 \beta_{3} - 4 \beta_{2} + 3 \beta_1) q^{87} + 12 \beta_{2} q^{88} + ( - 3 \beta_{2} + \beta_1 - 3) q^{89} + ( - 3 \beta_{3} - 2) q^{90} + ( - 6 \beta_{2} + \beta_1 - 6) q^{93} + ( - 3 \beta_{3} + 2 \beta_{2} - 3 \beta_1) q^{94} + ( - 6 \beta_{3} - 3 \beta_{2} - 6 \beta_1) q^{95} + ( - 9 \beta_{3} - 1) q^{97} + 3 \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{5} - 8 q^{6} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{5} - 8 q^{6} + 2 q^{9} - 4 q^{10} - 4 q^{13} - 8 q^{15} + 8 q^{16} + 6 q^{19} - 24 q^{22} + 6 q^{23} - 8 q^{24} - 12 q^{25} + 12 q^{29} + 12 q^{30} + 2 q^{31} - 12 q^{33} - 8 q^{34} + 4 q^{37} - 12 q^{38} + 8 q^{40} + 24 q^{41} - 20 q^{43} + 6 q^{45} - 8 q^{46} - 6 q^{47} + 48 q^{50} - 4 q^{51} + 6 q^{53} - 16 q^{54} - 24 q^{55} - 24 q^{57} - 8 q^{58} - 12 q^{59} - 12 q^{61} - 24 q^{62} + 32 q^{64} + 6 q^{65} + 12 q^{67} - 16 q^{69} - 24 q^{71} + 10 q^{73} + 12 q^{74} + 24 q^{75} + 8 q^{78} - 14 q^{79} + 24 q^{80} + 10 q^{81} + 8 q^{82} + 36 q^{83} - 8 q^{85} + 8 q^{87} - 24 q^{88} - 6 q^{89} - 8 q^{90} - 12 q^{93} - 4 q^{94} + 6 q^{95} - 4 q^{97}+O(q^{100}) \) 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 \(\left(\mathbb{Z}/637\mathbb{Z}\right)^\times\).

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

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.f 4
7.b odd 2 1 637.2.e.g 4
7.c even 3 1 91.2.a.c 2
7.c even 3 1 inner 637.2.e.f 4
7.d odd 6 1 637.2.a.g 2
7.d odd 6 1 637.2.e.g 4
21.g even 6 1 5733.2.a.s 2
21.h odd 6 1 819.2.a.h 2
28.g odd 6 1 1456.2.a.q 2
35.j even 6 1 2275.2.a.j 2
56.k odd 6 1 5824.2.a.bk 2
56.p even 6 1 5824.2.a.bl 2
91.r even 6 1 1183.2.a.d 2
91.s odd 6 1 8281.2.a.v 2
91.z odd 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.c even 3 1
637.2.a.g 2 7.d odd 6 1
637.2.e.f 4 1.a even 1 1 trivial
637.2.e.f 4 7.c even 3 1 inner
637.2.e.g 4 7.b odd 2 1
637.2.e.g 4 7.d odd 6 1
819.2.a.h 2 21.h odd 6 1
1183.2.a.d 2 91.r even 6 1
1183.2.c.d 4 91.z odd 12 2
1456.2.a.q 2 28.g odd 6 1
2275.2.a.j 2 35.j even 6 1
5733.2.a.s 2 21.g even 6 1
5824.2.a.bk 2 56.k odd 6 1
5824.2.a.bl 2 56.p even 6 1
8281.2.a.v 2 91.s odd 6 1

Hecke kernels

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

\( T_{2}^{4} + 2T_{2}^{2} + 4 \) Copy content Toggle raw display
\( T_{3}^{4} + 2T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{5}^{4} + 6T_{5}^{3} + 29T_{5}^{2} + 42T_{5} + 49 \) Copy content Toggle raw display

Hecke characteristic polynomials

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