Properties

Label 4900.2.a.y
Level $4900$
Weight $2$
Character orbit 4900.a
Self dual yes
Analytic conductor $39.127$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4900.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(39.1266969904\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 196)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta q^{3} + 5 q^{9} +O(q^{10})\) \( q + 2 \beta q^{3} + 5 q^{9} + 4 q^{11} -3 \beta q^{13} -\beta q^{17} + 2 \beta q^{19} + 4 q^{23} + 4 \beta q^{27} + 8 q^{29} + 8 \beta q^{33} + 8 q^{37} -12 q^{39} -5 \beta q^{41} + 4 q^{43} -4 \beta q^{47} -4 q^{51} -10 q^{53} + 8 q^{57} + 10 \beta q^{59} -5 \beta q^{61} + 8 \beta q^{69} + 5 \beta q^{73} + 8 q^{79} + q^{81} + 10 \beta q^{83} + 16 \beta q^{87} + 5 \beta q^{89} + \beta q^{97} + 20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 10q^{9} + O(q^{10}) \) \( 2q + 10q^{9} + 8q^{11} + 8q^{23} + 16q^{29} + 16q^{37} - 24q^{39} + 8q^{43} - 8q^{51} - 20q^{53} + 16q^{57} + 16q^{79} + 2q^{81} + 40q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.41421
1.41421
0 −2.82843 0 0 0 0 0 5.00000 0
1.2 0 2.82843 0 0 0 0 0 5.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(7\) \(1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4900.2.a.y 2
5.b even 2 1 196.2.a.c 2
5.c odd 4 2 4900.2.e.p 4
7.b odd 2 1 inner 4900.2.a.y 2
15.d odd 2 1 1764.2.a.l 2
20.d odd 2 1 784.2.a.m 2
35.c odd 2 1 196.2.a.c 2
35.f even 4 2 4900.2.e.p 4
35.i odd 6 2 196.2.e.b 4
35.j even 6 2 196.2.e.b 4
40.e odd 2 1 3136.2.a.bs 2
40.f even 2 1 3136.2.a.br 2
60.h even 2 1 7056.2.a.cr 2
105.g even 2 1 1764.2.a.l 2
105.o odd 6 2 1764.2.k.l 4
105.p even 6 2 1764.2.k.l 4
140.c even 2 1 784.2.a.m 2
140.p odd 6 2 784.2.i.l 4
140.s even 6 2 784.2.i.l 4
280.c odd 2 1 3136.2.a.br 2
280.n even 2 1 3136.2.a.bs 2
420.o odd 2 1 7056.2.a.cr 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
196.2.a.c 2 5.b even 2 1
196.2.a.c 2 35.c odd 2 1
196.2.e.b 4 35.i odd 6 2
196.2.e.b 4 35.j even 6 2
784.2.a.m 2 20.d odd 2 1
784.2.a.m 2 140.c even 2 1
784.2.i.l 4 140.p odd 6 2
784.2.i.l 4 140.s even 6 2
1764.2.a.l 2 15.d odd 2 1
1764.2.a.l 2 105.g even 2 1
1764.2.k.l 4 105.o odd 6 2
1764.2.k.l 4 105.p even 6 2
3136.2.a.br 2 40.f even 2 1
3136.2.a.br 2 280.c odd 2 1
3136.2.a.bs 2 40.e odd 2 1
3136.2.a.bs 2 280.n even 2 1
4900.2.a.y 2 1.a even 1 1 trivial
4900.2.a.y 2 7.b odd 2 1 inner
4900.2.e.p 4 5.c odd 4 2
4900.2.e.p 4 35.f even 4 2
7056.2.a.cr 2 60.h even 2 1
7056.2.a.cr 2 420.o odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4900))\):

\( T_{3}^{2} - 8 \)
\( T_{11} - 4 \)
\( T_{13}^{2} - 18 \)
\( T_{19}^{2} - 8 \)
\( T_{23} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -8 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( -4 + T )^{2} \)
$13$ \( -18 + T^{2} \)
$17$ \( -2 + T^{2} \)
$19$ \( -8 + T^{2} \)
$23$ \( ( -4 + T )^{2} \)
$29$ \( ( -8 + T )^{2} \)
$31$ \( T^{2} \)
$37$ \( ( -8 + T )^{2} \)
$41$ \( -50 + T^{2} \)
$43$ \( ( -4 + T )^{2} \)
$47$ \( -32 + T^{2} \)
$53$ \( ( 10 + T )^{2} \)
$59$ \( -200 + T^{2} \)
$61$ \( -50 + T^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( -50 + T^{2} \)
$79$ \( ( -8 + T )^{2} \)
$83$ \( -200 + T^{2} \)
$89$ \( -50 + T^{2} \)
$97$ \( -2 + T^{2} \)
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