Properties

Label 4900.2.e.b
Level $4900$
Weight $2$
Character orbit 4900.e
Analytic conductor $39.127$
Analytic rank $1$
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.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(39.1266969904\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 140)
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.

\(f(q)\) \(=\) \( q + 3 i q^{3} -6 q^{9} +O(q^{10})\) \( q + 3 i q^{3} -6 q^{9} -2 q^{11} + 6 i q^{13} + 2 i q^{17} -9 i q^{23} -9 i q^{27} -3 q^{29} -2 q^{31} -6 i q^{33} -8 i q^{37} -18 q^{39} -5 q^{41} + i q^{43} + 8 i q^{47} -6 q^{51} + 4 i q^{53} -8 q^{59} -7 q^{61} + 3 i q^{67} + 27 q^{69} + 8 q^{71} -14 i q^{73} -4 q^{79} + 9 q^{81} + i q^{83} -9 i q^{87} + 13 q^{89} -6 i q^{93} -10 i q^{97} + 12 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 12q^{9} + O(q^{10}) \) \( 2q - 12q^{9} - 4q^{11} - 6q^{29} - 4q^{31} - 36q^{39} - 10q^{41} - 12q^{51} - 16q^{59} - 14q^{61} + 54q^{69} + 16q^{71} - 8q^{79} + 18q^{81} + 26q^{89} + 24q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(1177\) \(2451\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2549.1
1.00000i
1.00000i
0 3.00000i 0 0 0 0 0 −6.00000 0
2549.2 0 3.00000i 0 0 0 0 0 −6.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4900.2.e.b 2
5.b even 2 1 inner 4900.2.e.b 2
5.c odd 4 1 980.2.a.i 1
5.c odd 4 1 4900.2.a.a 1
7.b odd 2 1 4900.2.e.c 2
7.d odd 6 2 700.2.r.c 4
15.e even 4 1 8820.2.a.k 1
20.e even 4 1 3920.2.a.d 1
35.c odd 2 1 4900.2.e.c 2
35.f even 4 1 980.2.a.a 1
35.f even 4 1 4900.2.a.v 1
35.i odd 6 2 700.2.r.c 4
35.k even 12 2 140.2.i.b 2
35.k even 12 2 700.2.i.a 2
35.l odd 12 2 980.2.i.a 2
105.k odd 4 1 8820.2.a.w 1
105.w odd 12 2 1260.2.s.b 2
140.j odd 4 1 3920.2.a.bi 1
140.x odd 12 2 560.2.q.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.i.b 2 35.k even 12 2
560.2.q.a 2 140.x odd 12 2
700.2.i.a 2 35.k even 12 2
700.2.r.c 4 7.d odd 6 2
700.2.r.c 4 35.i odd 6 2
980.2.a.a 1 35.f even 4 1
980.2.a.i 1 5.c odd 4 1
980.2.i.a 2 35.l odd 12 2
1260.2.s.b 2 105.w odd 12 2
3920.2.a.d 1 20.e even 4 1
3920.2.a.bi 1 140.j odd 4 1
4900.2.a.a 1 5.c odd 4 1
4900.2.a.v 1 35.f even 4 1
4900.2.e.b 2 1.a even 1 1 trivial
4900.2.e.b 2 5.b even 2 1 inner
4900.2.e.c 2 7.b odd 2 1
4900.2.e.c 2 35.c odd 2 1
8820.2.a.k 1 15.e even 4 1
8820.2.a.w 1 105.k odd 4 1

Hecke kernels

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

\( T_{3}^{2} + 9 \)
\( T_{11} + 2 \)
\( T_{19} \)
\( T_{31} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 9 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( 2 + T )^{2} \)
$13$ \( 36 + T^{2} \)
$17$ \( 4 + T^{2} \)
$19$ \( T^{2} \)
$23$ \( 81 + T^{2} \)
$29$ \( ( 3 + T )^{2} \)
$31$ \( ( 2 + T )^{2} \)
$37$ \( 64 + T^{2} \)
$41$ \( ( 5 + T )^{2} \)
$43$ \( 1 + T^{2} \)
$47$ \( 64 + T^{2} \)
$53$ \( 16 + T^{2} \)
$59$ \( ( 8 + T )^{2} \)
$61$ \( ( 7 + T )^{2} \)
$67$ \( 9 + T^{2} \)
$71$ \( ( -8 + T )^{2} \)
$73$ \( 196 + T^{2} \)
$79$ \( ( 4 + T )^{2} \)
$83$ \( 1 + T^{2} \)
$89$ \( ( -13 + T )^{2} \)
$97$ \( 100 + T^{2} \)
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