Properties

Label 637.2.h.d
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: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(197\) \(248\)
\(\chi(n)\) \(-1 + \zeta_{12}\) \(-1 + \zeta_{12}\)

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
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
−1.73205 0.366025 + 0.633975i 1.00000 0.866025 + 1.50000i −0.633975 1.09808i 0 1.73205 1.23205 2.13397i −1.50000 2.59808i
165.2 1.73205 −1.36603 2.36603i 1.00000 −0.866025 1.50000i −2.36603 4.09808i 0 −1.73205 −2.23205 + 3.86603i −1.50000 2.59808i
471.1 −1.73205 0.366025 0.633975i 1.00000 0.866025 1.50000i −0.633975 + 1.09808i 0 1.73205 1.23205 + 2.13397i −1.50000 + 2.59808i
471.2 1.73205 −1.36603 + 2.36603i 1.00000 −0.866025 + 1.50000i −2.36603 + 4.09808i 0 −1.73205 −2.23205 3.86603i −1.50000 + 2.59808i
\(n\): 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.d 4
7.b odd 2 1 637.2.h.e 4
7.c even 3 1 91.2.f.b 4
7.c even 3 1 637.2.g.e 4
7.d odd 6 1 637.2.f.d 4
7.d odd 6 1 637.2.g.d 4
13.c even 3 1 637.2.g.e 4
21.h odd 6 1 819.2.o.b 4
28.g odd 6 1 1456.2.s.o 4
91.g even 3 1 91.2.f.b 4
91.h even 3 1 inner 637.2.h.d 4
91.h even 3 1 1183.2.a.f 2
91.k even 6 1 1183.2.a.e 2
91.l odd 6 1 8281.2.a.t 2
91.m odd 6 1 637.2.f.d 4
91.n odd 6 1 637.2.g.d 4
91.v odd 6 1 637.2.h.e 4
91.v odd 6 1 8281.2.a.r 2
91.x odd 12 2 1183.2.c.e 4
273.bm odd 6 1 819.2.o.b 4
364.q odd 6 1 1456.2.s.o 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.f.b 4 7.c even 3 1
91.2.f.b 4 91.g even 3 1
637.2.f.d 4 7.d odd 6 1
637.2.f.d 4 91.m odd 6 1
637.2.g.d 4 7.d odd 6 1
637.2.g.d 4 91.n odd 6 1
637.2.g.e 4 7.c even 3 1
637.2.g.e 4 13.c even 3 1
637.2.h.d 4 1.a even 1 1 trivial
637.2.h.d 4 91.h even 3 1 inner
637.2.h.e 4 7.b odd 2 1
637.2.h.e 4 91.v odd 6 1
819.2.o.b 4 21.h odd 6 1
819.2.o.b 4 273.bm odd 6 1
1183.2.a.e 2 91.k even 6 1
1183.2.a.f 2 91.h even 3 1
1183.2.c.e 4 91.x odd 12 2
1456.2.s.o 4 28.g odd 6 1
1456.2.s.o 4 364.q odd 6 1
8281.2.a.r 2 91.v odd 6 1
8281.2.a.t 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 \(S_{2}^{\mathrm{new}}(637, [\chi])\):

\( T_{2}^{2} - 3 \)
\( T_{3}^{4} + 2 T_{3}^{3} + 6 T_{3}^{2} - 4 T_{3} + 4 \)
\( T_{5}^{4} + 3 T_{5}^{2} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -3 + T^{2} )^{2} \)
$3$ \( 4 - 4 T + 6 T^{2} + 2 T^{3} + T^{4} \)
$5$ \( 9 + 3 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( 36 + 36 T + 30 T^{2} + 6 T^{3} + T^{4} \)
$13$ \( 169 - 52 T + 3 T^{2} - 4 T^{3} + T^{4} \)
$17$ \( ( 33 + 12 T + T^{2} )^{2} \)
$19$ \( ( 4 + 2 T + T^{2} )^{2} \)
$23$ \( ( 6 - 6 T + T^{2} )^{2} \)
$29$ \( ( 9 - 3 T + T^{2} )^{2} \)
$31$ \( 676 + 52 T + 30 T^{2} - 2 T^{3} + T^{4} \)
$37$ \( ( 7 + T )^{4} \)
$41$ \( 729 + 27 T^{2} + T^{4} \)
$43$ \( 4 - 20 T + 102 T^{2} + 10 T^{3} + T^{4} \)
$47$ \( 144 - 144 T + 156 T^{2} + 12 T^{3} + T^{4} \)
$53$ \( 1521 + 234 T + 75 T^{2} - 6 T^{3} + T^{4} \)
$59$ \( ( 78 - 18 T + T^{2} )^{2} \)
$61$ \( 5329 - 1460 T + 327 T^{2} - 20 T^{3} + T^{4} \)
$67$ \( 676 + 52 T + 30 T^{2} - 2 T^{3} + T^{4} \)
$71$ \( ( 36 + 6 T + T^{2} )^{2} \)
$73$ \( 529 - 92 T + 39 T^{2} + 4 T^{3} + T^{4} \)
$79$ \( 8836 + 2068 T + 390 T^{2} + 22 T^{3} + T^{4} \)
$83$ \( ( -18 - 6 T + T^{2} )^{2} \)
$89$ \( ( -12 - 12 T + T^{2} )^{2} \)
$97$ \( 8464 + 736 T + 156 T^{2} - 8 T^{3} + T^{4} \)
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