Properties

Label 4900.2.a.e
Level $4900$
Weight $2$
Character orbit 4900.a
Self dual yes
Analytic conductor $39.127$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 20)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2 q^{3} + q^{9} + O(q^{10}) \) \( q - 2 q^{3} + q^{9} + 2 q^{13} - 6 q^{17} + 4 q^{19} - 6 q^{23} + 4 q^{27} + 6 q^{29} + 4 q^{31} - 2 q^{37} - 4 q^{39} - 6 q^{41} + 10 q^{43} - 6 q^{47} + 12 q^{51} + 6 q^{53} - 8 q^{57} - 12 q^{59} - 2 q^{61} - 2 q^{67} + 12 q^{69} - 12 q^{71} + 2 q^{73} + 8 q^{79} - 11 q^{81} + 6 q^{83} - 12 q^{87} + 6 q^{89} - 8 q^{93} + 2 q^{97} + O(q^{100}) \)

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
0
0 −2.00000 0 0 0 0 0 1.00000 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

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4900.2.a.e 1
5.b even 2 1 980.2.a.h 1
5.c odd 4 2 4900.2.e.f 2
7.b odd 2 1 100.2.a.a 1
15.d odd 2 1 8820.2.a.g 1
20.d odd 2 1 3920.2.a.h 1
21.c even 2 1 900.2.a.b 1
28.d even 2 1 400.2.a.c 1
35.c odd 2 1 20.2.a.a 1
35.f even 4 2 100.2.c.a 2
35.i odd 6 2 980.2.i.i 2
35.j even 6 2 980.2.i.c 2
56.e even 2 1 1600.2.a.w 1
56.h odd 2 1 1600.2.a.c 1
84.h odd 2 1 3600.2.a.be 1
105.g even 2 1 180.2.a.a 1
105.k odd 4 2 900.2.d.c 2
140.c even 2 1 80.2.a.b 1
140.j odd 4 2 400.2.c.b 2
280.c odd 2 1 320.2.a.f 1
280.n even 2 1 320.2.a.a 1
280.s even 4 2 1600.2.c.d 2
280.y odd 4 2 1600.2.c.e 2
315.z even 6 2 1620.2.i.b 2
315.bg odd 6 2 1620.2.i.h 2
385.h even 2 1 2420.2.a.a 1
420.o odd 2 1 720.2.a.h 1
420.w even 4 2 3600.2.f.j 2
455.h odd 2 1 3380.2.a.c 1
455.u even 4 2 3380.2.f.b 2
560.be even 4 2 1280.2.d.g 2
560.bf odd 4 2 1280.2.d.c 2
595.b odd 2 1 5780.2.a.f 1
595.u odd 4 2 5780.2.c.a 2
665.g even 2 1 7220.2.a.f 1
840.b odd 2 1 2880.2.a.f 1
840.u even 2 1 2880.2.a.m 1
1540.b odd 2 1 9680.2.a.ba 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.2.a.a 1 35.c odd 2 1
80.2.a.b 1 140.c even 2 1
100.2.a.a 1 7.b odd 2 1
100.2.c.a 2 35.f even 4 2
180.2.a.a 1 105.g even 2 1
320.2.a.a 1 280.n even 2 1
320.2.a.f 1 280.c odd 2 1
400.2.a.c 1 28.d even 2 1
400.2.c.b 2 140.j odd 4 2
720.2.a.h 1 420.o odd 2 1
900.2.a.b 1 21.c even 2 1
900.2.d.c 2 105.k odd 4 2
980.2.a.h 1 5.b even 2 1
980.2.i.c 2 35.j even 6 2
980.2.i.i 2 35.i odd 6 2
1280.2.d.c 2 560.bf odd 4 2
1280.2.d.g 2 560.be even 4 2
1600.2.a.c 1 56.h odd 2 1
1600.2.a.w 1 56.e even 2 1
1600.2.c.d 2 280.s even 4 2
1600.2.c.e 2 280.y odd 4 2
1620.2.i.b 2 315.z even 6 2
1620.2.i.h 2 315.bg odd 6 2
2420.2.a.a 1 385.h even 2 1
2880.2.a.f 1 840.b odd 2 1
2880.2.a.m 1 840.u even 2 1
3380.2.a.c 1 455.h odd 2 1
3380.2.f.b 2 455.u even 4 2
3600.2.a.be 1 84.h odd 2 1
3600.2.f.j 2 420.w even 4 2
3920.2.a.h 1 20.d odd 2 1
4900.2.a.e 1 1.a even 1 1 trivial
4900.2.e.f 2 5.c odd 4 2
5780.2.a.f 1 595.b odd 2 1
5780.2.c.a 2 595.u odd 4 2
7220.2.a.f 1 665.g even 2 1
8820.2.a.g 1 15.d odd 2 1
9680.2.a.ba 1 1540.b odd 2 1

Hecke kernels

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

\( T_{3} + 2 \)
\( T_{11} \)
\( T_{13} - 2 \)
\( T_{19} - 4 \)
\( T_{23} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 2 + T \)
$5$ \( T \)
$7$ \( T \)
$11$ \( T \)
$13$ \( -2 + T \)
$17$ \( 6 + T \)
$19$ \( -4 + T \)
$23$ \( 6 + T \)
$29$ \( -6 + T \)
$31$ \( -4 + T \)
$37$ \( 2 + T \)
$41$ \( 6 + T \)
$43$ \( -10 + T \)
$47$ \( 6 + T \)
$53$ \( -6 + T \)
$59$ \( 12 + T \)
$61$ \( 2 + T \)
$67$ \( 2 + T \)
$71$ \( 12 + T \)
$73$ \( -2 + T \)
$79$ \( -8 + T \)
$83$ \( -6 + T \)
$89$ \( -6 + T \)
$97$ \( -2 + T \)
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