Properties

Label 4004.2.m.a
Level 4004
Weight 2
Character orbit 4004.m
Analytic conductor 31.972
Analytic rank 0
Dimension 2
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.m (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(31.9721009693\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + 2 q^{3} + i q^{5} + i q^{7} + q^{9} +O(q^{10})\) \( q + 2 q^{3} + i q^{5} + i q^{7} + q^{9} -i q^{11} + ( 2 + 3 i ) q^{13} + 2 i q^{15} -4 q^{17} + 5 i q^{19} + 2 i q^{21} -9 q^{23} + 4 q^{25} -4 q^{27} + 5 q^{29} + 7 i q^{31} -2 i q^{33} - q^{35} + 2 i q^{37} + ( 4 + 6 i ) q^{39} -6 i q^{41} -11 q^{43} + i q^{45} -i q^{47} - q^{49} -8 q^{51} - q^{53} + q^{55} + 10 i q^{57} + 4 i q^{59} -6 q^{61} + i q^{63} + ( -3 + 2 i ) q^{65} + 10 i q^{67} -18 q^{69} + 4 i q^{71} + 3 i q^{73} + 8 q^{75} + q^{77} + 11 q^{79} -11 q^{81} -9 i q^{83} -4 i q^{85} + 10 q^{87} -15 i q^{89} + ( -3 + 2 i ) q^{91} + 14 i q^{93} -5 q^{95} + 7 i q^{97} -i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{3} + 2q^{9} + O(q^{10}) \) \( 2q + 4q^{3} + 2q^{9} + 4q^{13} - 8q^{17} - 18q^{23} + 8q^{25} - 8q^{27} + 10q^{29} - 2q^{35} + 8q^{39} - 22q^{43} - 2q^{49} - 16q^{51} - 2q^{53} + 2q^{55} - 12q^{61} - 6q^{65} - 36q^{69} + 16q^{75} + 2q^{77} + 22q^{79} - 22q^{81} + 20q^{87} - 6q^{91} - 10q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4004\mathbb{Z}\right)^\times\).

\(n\) \(365\) \(925\) \(2003\) \(3433\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2157.1
1.00000i
1.00000i
0 2.00000 0 1.00000i 0 1.00000i 0 1.00000 0
2157.2 0 2.00000 0 1.00000i 0 1.00000i 0 1.00000 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
13.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.m.a 2
13.b even 2 1 inner 4004.2.m.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4004.2.m.a 2 1.a even 1 1 trivial
4004.2.m.a 2 13.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 2 \) acting on \(S_{2}^{\mathrm{new}}(4004, [\chi])\).