Properties

Label 637.2.f.d
Level $637$
Weight $2$
Character orbit 637.f
Analytic conductor $5.086$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.f (of order \(3\), degree \(2\), 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: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 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.

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

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{3} + \zeta_{12} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} ) / 3 \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
295.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 + 1.50000i 1.36603 2.36603i −0.500000 0.866025i −1.73205 2.36603 + 4.09808i 0 −1.73205 −2.23205 3.86603i 1.50000 2.59808i
295.2 0.866025 1.50000i −0.366025 + 0.633975i −0.500000 0.866025i 1.73205 0.633975 + 1.09808i 0 1.73205 1.23205 + 2.13397i 1.50000 2.59808i
393.1 −0.866025 1.50000i 1.36603 + 2.36603i −0.500000 + 0.866025i −1.73205 2.36603 4.09808i 0 −1.73205 −2.23205 + 3.86603i 1.50000 + 2.59808i
393.2 0.866025 + 1.50000i −0.366025 0.633975i −0.500000 + 0.866025i 1.73205 0.633975 1.09808i 0 1.73205 1.23205 2.13397i 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
13.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.f.d 4
7.b odd 2 1 91.2.f.b 4
7.c even 3 1 637.2.g.d 4
7.c even 3 1 637.2.h.e 4
7.d odd 6 1 637.2.g.e 4
7.d odd 6 1 637.2.h.d 4
13.c even 3 1 inner 637.2.f.d 4
13.c even 3 1 8281.2.a.r 2
13.e even 6 1 8281.2.a.t 2
21.c even 2 1 819.2.o.b 4
28.d even 2 1 1456.2.s.o 4
91.g even 3 1 637.2.h.e 4
91.h even 3 1 637.2.g.d 4
91.m odd 6 1 637.2.h.d 4
91.n odd 6 1 91.2.f.b 4
91.n odd 6 1 1183.2.a.f 2
91.t odd 6 1 1183.2.a.e 2
91.v odd 6 1 637.2.g.e 4
91.bc even 12 2 1183.2.c.e 4
273.bn even 6 1 819.2.o.b 4
364.v even 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.b odd 2 1
91.2.f.b 4 91.n odd 6 1
637.2.f.d 4 1.a even 1 1 trivial
637.2.f.d 4 13.c even 3 1 inner
637.2.g.d 4 7.c even 3 1
637.2.g.d 4 91.h even 3 1
637.2.g.e 4 7.d odd 6 1
637.2.g.e 4 91.v odd 6 1
637.2.h.d 4 7.d odd 6 1
637.2.h.d 4 91.m odd 6 1
637.2.h.e 4 7.c even 3 1
637.2.h.e 4 91.g even 3 1
819.2.o.b 4 21.c even 2 1
819.2.o.b 4 273.bn even 6 1
1183.2.a.e 2 91.t odd 6 1
1183.2.a.f 2 91.n odd 6 1
1183.2.c.e 4 91.bc even 12 2
1456.2.s.o 4 28.d even 2 1
1456.2.s.o 4 364.v even 6 1
8281.2.a.r 2 13.c even 3 1
8281.2.a.t 2 13.e even 6 1

Hecke kernels

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

\( T_{2}^{4} + 3T_{2}^{2} + 9 \) Copy content Toggle raw display
\( T_{3}^{4} - 2T_{3}^{3} + 6T_{3}^{2} + 4T_{3} + 4 \) Copy content Toggle raw display
\( T_{5}^{2} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$3$ \( T^{4} - 2 T^{3} + 6 T^{2} + 4 T + 4 \) Copy content Toggle raw display
$5$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 6 T^{3} + 30 T^{2} + 36 T + 36 \) Copy content Toggle raw display
$13$ \( T^{4} + 4 T^{3} + 3 T^{2} + 52 T + 169 \) Copy content Toggle raw display
$17$ \( T^{4} + 12 T^{3} + 111 T^{2} + \cdots + 1089 \) Copy content Toggle raw display
$19$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 6 T^{3} + 30 T^{2} + 36 T + 36 \) Copy content Toggle raw display
$29$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 2 T - 26)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 7 T + 49)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 27T^{2} + 729 \) Copy content Toggle raw display
$43$ \( T^{4} + 10 T^{3} + 102 T^{2} - 20 T + 4 \) Copy content Toggle raw display
$47$ \( (T^{2} + 12 T - 12)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 6 T - 39)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 18 T^{3} + 246 T^{2} + \cdots + 6084 \) Copy content Toggle raw display
$61$ \( T^{4} + 20 T^{3} + 327 T^{2} + \cdots + 5329 \) Copy content Toggle raw display
$67$ \( T^{4} - 2 T^{3} + 30 T^{2} + 52 T + 676 \) Copy content Toggle raw display
$71$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 4 T - 23)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 22 T + 94)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 6 T - 18)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 12 T^{3} + 156 T^{2} + \cdots + 144 \) Copy content Toggle raw display
$97$ \( T^{4} + 8 T^{3} + 156 T^{2} + \cdots + 8464 \) Copy content Toggle raw display
show more
show less