Properties

Label 4900.2.e.q
Level $4900$
Weight $2$
Character orbit 4900.e
Analytic conductor $39.127$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 980)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{3} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{9} +O(q^{10})\) \( q + ( \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{3} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{9} + ( -1 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{11} + ( \zeta_{8} + 5 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{13} + ( \zeta_{8} - 5 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{17} + ( -2 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{19} + ( -\zeta_{8} - 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{23} + ( \zeta_{8} - \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{27} + ( -1 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{29} + ( -6 - \zeta_{8} + \zeta_{8}^{3} ) q^{31} + ( \zeta_{8} + 3 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{33} + ( \zeta_{8} - 2 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{37} + ( -7 - 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{39} + ( -2 - \zeta_{8} + \zeta_{8}^{3} ) q^{41} + ( 4 \zeta_{8} - 6 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{43} + ( -7 \zeta_{8} - \zeta_{8}^{2} - 7 \zeta_{8}^{3} ) q^{47} + ( 3 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{51} + ( 3 \zeta_{8} + 8 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{53} + ( -6 \zeta_{8} - 10 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{57} + ( 2 - \zeta_{8} + \zeta_{8}^{3} ) q^{59} + ( -8 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{61} + ( 5 \zeta_{8} - 4 \zeta_{8}^{2} + 5 \zeta_{8}^{3} ) q^{67} + ( 4 + 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{69} + ( -2 - 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{71} + ( 2 \zeta_{8} + 8 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{73} + ( 1 + 10 \zeta_{8} - 10 \zeta_{8}^{3} ) q^{79} + ( -1 - 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{81} -8 \zeta_{8}^{2} q^{83} + ( 3 \zeta_{8} + 7 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{87} + ( -12 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{89} + ( -7 \zeta_{8} - 8 \zeta_{8}^{2} - 7 \zeta_{8}^{3} ) q^{93} + ( 9 \zeta_{8} - 3 \zeta_{8}^{2} + 9 \zeta_{8}^{3} ) q^{97} + ( -8 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + O(q^{10}) \) \( 4 q - 4 q^{11} - 8 q^{19} - 4 q^{29} - 24 q^{31} - 28 q^{39} - 8 q^{41} + 12 q^{51} + 8 q^{59} - 32 q^{61} + 16 q^{69} - 8 q^{71} + 4 q^{79} - 4 q^{81} - 32 q^{99} + O(q^{100}) \)

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
0.707107 0.707107i
−0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
0 2.41421i 0 0 0 0 0 −2.82843 0
2549.2 0 0.414214i 0 0 0 0 0 2.82843 0
2549.3 0 0.414214i 0 0 0 0 0 2.82843 0
2549.4 0 2.41421i 0 0 0 0 0 −2.82843 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.q 4
5.b even 2 1 inner 4900.2.e.q 4
5.c odd 4 1 980.2.a.j 2
5.c odd 4 1 4900.2.a.z 2
7.b odd 2 1 4900.2.e.r 4
15.e even 4 1 8820.2.a.bg 2
20.e even 4 1 3920.2.a.bx 2
35.c odd 2 1 4900.2.e.r 4
35.f even 4 1 980.2.a.k yes 2
35.f even 4 1 4900.2.a.x 2
35.k even 12 2 980.2.i.k 4
35.l odd 12 2 980.2.i.l 4
105.k odd 4 1 8820.2.a.bl 2
140.j odd 4 1 3920.2.a.bo 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
980.2.a.j 2 5.c odd 4 1
980.2.a.k yes 2 35.f even 4 1
980.2.i.k 4 35.k even 12 2
980.2.i.l 4 35.l odd 12 2
3920.2.a.bo 2 140.j odd 4 1
3920.2.a.bx 2 20.e even 4 1
4900.2.a.x 2 35.f even 4 1
4900.2.a.z 2 5.c odd 4 1
4900.2.e.q 4 1.a even 1 1 trivial
4900.2.e.q 4 5.b even 2 1 inner
4900.2.e.r 4 7.b odd 2 1
4900.2.e.r 4 35.c odd 2 1
8820.2.a.bg 2 15.e even 4 1
8820.2.a.bl 2 105.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 \(S_{2}^{\mathrm{new}}(4900, [\chi])\):

\( T_{3}^{4} + 6 T_{3}^{2} + 1 \)
\( T_{11}^{2} + 2 T_{11} - 7 \)
\( T_{19}^{2} + 4 T_{19} - 28 \)
\( T_{31}^{2} + 12 T_{31} + 34 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 1 + 6 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( -7 + 2 T + T^{2} )^{2} \)
$13$ \( 529 + 54 T^{2} + T^{4} \)
$17$ \( 529 + 54 T^{2} + T^{4} \)
$19$ \( ( -28 + 4 T + T^{2} )^{2} \)
$23$ \( 4 + 12 T^{2} + T^{4} \)
$29$ \( ( -31 + 2 T + T^{2} )^{2} \)
$31$ \( ( 34 + 12 T + T^{2} )^{2} \)
$37$ \( 4 + 12 T^{2} + T^{4} \)
$41$ \( ( 2 + 4 T + T^{2} )^{2} \)
$43$ \( 16 + 136 T^{2} + T^{4} \)
$47$ \( 9409 + 198 T^{2} + T^{4} \)
$53$ \( 2116 + 164 T^{2} + T^{4} \)
$59$ \( ( 2 - 4 T + T^{2} )^{2} \)
$61$ \( ( 56 + 16 T + T^{2} )^{2} \)
$67$ \( 1156 + 132 T^{2} + T^{4} \)
$71$ \( ( -68 + 4 T + T^{2} )^{2} \)
$73$ \( 3136 + 144 T^{2} + T^{4} \)
$79$ \( ( -199 - 2 T + T^{2} )^{2} \)
$83$ \( ( 64 + T^{2} )^{2} \)
$89$ \( ( -288 + T^{2} )^{2} \)
$97$ \( 23409 + 342 T^{2} + T^{4} \)
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