Properties

Label 4900.2.e.h
Level $4900$
Weight $2$
Character orbit 4900.e
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.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(39.1266969904\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 28)
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.

\(f(q)\) \(=\) \( q + i q^{3} + 2 q^{9} +O(q^{10})\) \( q + i q^{3} + 2 q^{9} -3 q^{11} + 2 i q^{13} -3 i q^{17} - q^{19} -3 i q^{23} + 5 i q^{27} + 6 q^{29} + 7 q^{31} -3 i q^{33} -i q^{37} -2 q^{39} -6 q^{41} + 4 i q^{43} + 9 i q^{47} + 3 q^{51} -3 i q^{53} -i q^{57} + 9 q^{59} + q^{61} -7 i q^{67} + 3 q^{69} -i q^{73} + 13 q^{79} + q^{81} + 12 i q^{83} + 6 i q^{87} + 15 q^{89} + 7 i q^{93} + 10 i q^{97} -6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{9} + O(q^{10}) \) \( 2q + 4q^{9} - 6q^{11} - 2q^{19} + 12q^{29} + 14q^{31} - 4q^{39} - 12q^{41} + 6q^{51} + 18q^{59} + 2q^{61} + 6q^{69} + 26q^{79} + 2q^{81} + 30q^{89} - 12q^{99} + O(q^{100}) \)

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 1.00000i 0 0 0 0 0 2.00000 0
2549.2 0 1.00000i 0 0 0 0 0 2.00000 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.h 2
5.b even 2 1 inner 4900.2.e.h 2
5.c odd 4 1 196.2.a.a 1
5.c odd 4 1 4900.2.a.n 1
7.b odd 2 1 4900.2.e.i 2
7.d odd 6 2 700.2.r.b 4
15.e even 4 1 1764.2.a.j 1
20.e even 4 1 784.2.a.g 1
35.c odd 2 1 4900.2.e.i 2
35.f even 4 1 196.2.a.b 1
35.f even 4 1 4900.2.a.g 1
35.i odd 6 2 700.2.r.b 4
35.k even 12 2 28.2.e.a 2
35.k even 12 2 700.2.i.c 2
35.l odd 12 2 196.2.e.a 2
40.i odd 4 1 3136.2.a.v 1
40.k even 4 1 3136.2.a.k 1
60.l odd 4 1 7056.2.a.bw 1
105.k odd 4 1 1764.2.a.a 1
105.w odd 12 2 252.2.k.c 2
105.x even 12 2 1764.2.k.b 2
140.j odd 4 1 784.2.a.d 1
140.w even 12 2 784.2.i.d 2
140.x odd 12 2 112.2.i.b 2
280.s even 4 1 3136.2.a.h 1
280.y odd 4 1 3136.2.a.s 1
280.bp odd 12 2 448.2.i.c 2
280.bv even 12 2 448.2.i.e 2
315.bs even 12 2 2268.2.i.a 2
315.bu odd 12 2 2268.2.i.h 2
315.bw odd 12 2 2268.2.l.a 2
315.cg even 12 2 2268.2.l.h 2
420.w even 4 1 7056.2.a.f 1
420.br even 12 2 1008.2.s.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.2.e.a 2 35.k even 12 2
112.2.i.b 2 140.x odd 12 2
196.2.a.a 1 5.c odd 4 1
196.2.a.b 1 35.f even 4 1
196.2.e.a 2 35.l odd 12 2
252.2.k.c 2 105.w odd 12 2
448.2.i.c 2 280.bp odd 12 2
448.2.i.e 2 280.bv even 12 2
700.2.i.c 2 35.k even 12 2
700.2.r.b 4 7.d odd 6 2
700.2.r.b 4 35.i odd 6 2
784.2.a.d 1 140.j odd 4 1
784.2.a.g 1 20.e even 4 1
784.2.i.d 2 140.w even 12 2
1008.2.s.p 2 420.br even 12 2
1764.2.a.a 1 105.k odd 4 1
1764.2.a.j 1 15.e even 4 1
1764.2.k.b 2 105.x even 12 2
2268.2.i.a 2 315.bs even 12 2
2268.2.i.h 2 315.bu odd 12 2
2268.2.l.a 2 315.bw odd 12 2
2268.2.l.h 2 315.cg even 12 2
3136.2.a.h 1 280.s even 4 1
3136.2.a.k 1 40.k even 4 1
3136.2.a.s 1 280.y odd 4 1
3136.2.a.v 1 40.i odd 4 1
4900.2.a.g 1 35.f even 4 1
4900.2.a.n 1 5.c odd 4 1
4900.2.e.h 2 1.a even 1 1 trivial
4900.2.e.h 2 5.b even 2 1 inner
4900.2.e.i 2 7.b odd 2 1
4900.2.e.i 2 35.c odd 2 1
7056.2.a.f 1 420.w even 4 1
7056.2.a.bw 1 60.l odd 4 1

Hecke kernels

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

\( T_{3}^{2} + 1 \)
\( T_{11} + 3 \)
\( T_{19} + 1 \)
\( T_{31} - 7 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 5 T^{2} + 9 T^{4} \)
$5$ 1
$7$ 1
$11$ \( ( 1 + 3 T + 11 T^{2} )^{2} \)
$13$ \( 1 - 22 T^{2} + 169 T^{4} \)
$17$ \( 1 - 25 T^{2} + 289 T^{4} \)
$19$ \( ( 1 + T + 19 T^{2} )^{2} \)
$23$ \( 1 - 37 T^{2} + 529 T^{4} \)
$29$ \( ( 1 - 6 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 - 7 T + 31 T^{2} )^{2} \)
$37$ \( 1 - 73 T^{2} + 1369 T^{4} \)
$41$ \( ( 1 + 6 T + 41 T^{2} )^{2} \)
$43$ \( 1 - 70 T^{2} + 1849 T^{4} \)
$47$ \( 1 - 13 T^{2} + 2209 T^{4} \)
$53$ \( 1 - 97 T^{2} + 2809 T^{4} \)
$59$ \( ( 1 - 9 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 - T + 61 T^{2} )^{2} \)
$67$ \( 1 - 85 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 + 71 T^{2} )^{2} \)
$73$ \( 1 - 145 T^{2} + 5329 T^{4} \)
$79$ \( ( 1 - 13 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 22 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 - 15 T + 89 T^{2} )^{2} \)
$97$ \( 1 - 94 T^{2} + 9409 T^{4} \)
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