Properties

Label 637.2.c.a
Level $637$
Weight $2$
Character orbit 637.c
Analytic conductor $5.086$
Analytic rank $1$
Dimension $2$
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.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.08647060876\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(197\) \(248\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
246.1
1.00000i
1.00000i
2.00000i −2.00000 −2.00000 1.00000i 4.00000i 0 0 1.00000 −2.00000
246.2 2.00000i −2.00000 −2.00000 1.00000i 4.00000i 0 0 1.00000 −2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.c.a 2
7.b odd 2 1 637.2.c.c yes 2
7.c even 3 2 637.2.r.c 4
7.d odd 6 2 637.2.r.a 4
13.b even 2 1 inner 637.2.c.a 2
13.d odd 4 1 8281.2.a.a 1
13.d odd 4 1 8281.2.a.k 1
91.b odd 2 1 637.2.c.c yes 2
91.i even 4 1 8281.2.a.b 1
91.i even 4 1 8281.2.a.m 1
91.r even 6 2 637.2.r.c 4
91.s odd 6 2 637.2.r.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.c.a 2 1.a even 1 1 trivial
637.2.c.a 2 13.b even 2 1 inner
637.2.c.c yes 2 7.b odd 2 1
637.2.c.c yes 2 91.b odd 2 1
637.2.r.a 4 7.d odd 6 2
637.2.r.a 4 91.s odd 6 2
637.2.r.c 4 7.c even 3 2
637.2.r.c 4 91.r even 6 2
8281.2.a.a 1 13.d odd 4 1
8281.2.a.b 1 91.i even 4 1
8281.2.a.k 1 13.d odd 4 1
8281.2.a.m 1 91.i even 4 1

Hecke kernels

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

\( T_{2}^{2} + 4 \)
\( T_{3} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + T^{2} \)
$3$ \( ( 2 + T )^{2} \)
$5$ \( 1 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 4 + T^{2} \)
$13$ \( 13 + 4 T + T^{2} \)
$17$ \( ( 6 + T )^{2} \)
$19$ \( 9 + T^{2} \)
$23$ \( ( 3 + T )^{2} \)
$29$ \( ( -3 + T )^{2} \)
$31$ \( 9 + T^{2} \)
$37$ \( 36 + T^{2} \)
$41$ \( 100 + T^{2} \)
$43$ \( ( -1 + T )^{2} \)
$47$ \( 121 + T^{2} \)
$53$ \( ( 9 + T )^{2} \)
$59$ \( 64 + T^{2} \)
$61$ \( ( 8 + T )^{2} \)
$67$ \( 144 + T^{2} \)
$71$ \( 196 + T^{2} \)
$73$ \( 81 + T^{2} \)
$79$ \( ( 9 + T )^{2} \)
$83$ \( 121 + T^{2} \)
$89$ \( 25 + T^{2} \)
$97$ \( 81 + T^{2} \)
show more
show less