Properties

Label 4004.2.e.b
Level 4004
Weight 2
Character orbit 4004.e
Analytic conductor 31.972
Analytic rank 0
Dimension 48
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4004.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(31.9721009693\)
Analytic rank: \(0\)
Dimension: \(48\)
Coefficient ring index: multiple of None
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48q + 4q^{7} - 48q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 48q + 4q^{7} - 48q^{9} + 2q^{11} - 48q^{13} + 8q^{15} - 4q^{17} - 10q^{21} + 4q^{23} - 44q^{25} - 10q^{33} + 14q^{35} - 16q^{37} + 10q^{49} - 8q^{53} + 12q^{55} - 16q^{61} - 16q^{63} + 4q^{67} + 16q^{73} + 22q^{77} + 64q^{81} + 4q^{83} + 12q^{87} - 4q^{91} + 36q^{93} - 40q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3849.1 0 3.23317i 0 0.717019i 0 0.284426 2.63042i 0 −7.45341 0
3849.2 0 3.15005i 0 2.09918i 0 −1.59876 + 2.10807i 0 −6.92283 0
3849.3 0 3.04894i 0 4.30537i 0 2.63608 + 0.225969i 0 −6.29604 0
3849.4 0 2.99006i 0 1.25381i 0 2.36377 + 1.18853i 0 −5.94047 0
3849.5 0 2.95250i 0 1.47372i 0 −0.700125 2.55144i 0 −5.71727 0
3849.6 0 2.73693i 0 3.14756i 0 1.11363 + 2.39996i 0 −4.49077 0
3849.7 0 2.42199i 0 0.343428i 0 −2.23070 + 1.42266i 0 −2.86603 0
3849.8 0 2.41288i 0 2.25637i 0 −2.16066 + 1.52694i 0 −2.82198 0
3849.9 0 2.39724i 0 2.83291i 0 −2.55736 0.678178i 0 −2.74675 0
3849.10 0 2.30818i 0 0.586386i 0 2.63094 + 0.279585i 0 −2.32771 0
3849.11 0 1.96820i 0 3.88595i 0 0.0756292 2.64467i 0 −0.873803 0
3849.12 0 1.77770i 0 3.16364i 0 −0.435138 2.60972i 0 −0.160220 0
3849.13 0 1.76248i 0 1.09765i 0 1.82169 1.91871i 0 −0.106338 0
3849.14 0 1.33626i 0 2.57665i 0 −2.48110 0.918779i 0 1.21441 0
3849.15 0 1.12498i 0 0.542613i 0 −1.59076 + 2.11411i 0 1.73443 0
3849.16 0 1.04079i 0 4.22530i 0 2.02914 1.69782i 0 1.91676 0
3849.17 0 1.00653i 0 1.31275i 0 1.40205 + 2.24371i 0 1.98690 0
3849.18 0 0.968020i 0 0.815183i 0 2.53164 0.768635i 0 2.06294 0
3849.19 0 0.950260i 0 3.52969i 0 2.60374 0.469588i 0 2.09701 0
3849.20 0 0.867297i 0 0.687824i 0 −2.28648 1.33116i 0 2.24780 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3849.48
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4004.2.e.b yes 48
7.b odd 2 1 4004.2.e.a 48
11.b odd 2 1 4004.2.e.a 48
77.b even 2 1 inner 4004.2.e.b yes 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4004.2.e.a 48 7.b odd 2 1
4004.2.e.a 48 11.b odd 2 1
4004.2.e.b yes 48 1.a even 1 1 trivial
4004.2.e.b yes 48 77.b even 2 1 inner

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database