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

Related objects

Downloads

Learn more about

Newspace parameters

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

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: \( 1 \)
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.

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

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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
13.b Even 1 no

Hecke kernels

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