Properties

Label 3330.2.d.d
Level $3330$
Weight $2$
Character orbit 3330.d
Analytic conductor $26.590$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3330.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(26.5901838731\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1110)
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.

\(f(q)\) \(=\) \( q -i q^{2} - q^{4} + ( -1 + 2 i ) q^{5} + i q^{7} + i q^{8} +O(q^{10})\) \( q -i q^{2} - q^{4} + ( -1 + 2 i ) q^{5} + i q^{7} + i q^{8} + ( 2 + i ) q^{10} -5 q^{11} + q^{14} + q^{16} -i q^{17} + ( 1 - 2 i ) q^{20} + 5 i q^{22} + 4 i q^{23} + ( -3 - 4 i ) q^{25} -i q^{28} -3 q^{29} + q^{31} -i q^{32} - q^{34} + ( -2 - i ) q^{35} -i q^{37} + ( -2 - i ) q^{40} + q^{41} -7 i q^{43} + 5 q^{44} + 4 q^{46} + 4 i q^{47} + 6 q^{49} + ( -4 + 3 i ) q^{50} + 3 i q^{53} + ( 5 - 10 i ) q^{55} - q^{56} + 3 i q^{58} -8 q^{59} + 5 q^{61} -i q^{62} - q^{64} -4 i q^{67} + i q^{68} + ( -1 + 2 i ) q^{70} + 6 q^{71} -10 i q^{73} - q^{74} -5 i q^{77} + ( -1 + 2 i ) q^{80} -i q^{82} -8 i q^{83} + ( 2 + i ) q^{85} -7 q^{86} -5 i q^{88} + 8 q^{89} -4 i q^{92} + 4 q^{94} -i q^{97} -6 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} - 2q^{5} + O(q^{10}) \) \( 2q - 2q^{4} - 2q^{5} + 4q^{10} - 10q^{11} + 2q^{14} + 2q^{16} + 2q^{20} - 6q^{25} - 6q^{29} + 2q^{31} - 2q^{34} - 4q^{35} - 4q^{40} + 2q^{41} + 10q^{44} + 8q^{46} + 12q^{49} - 8q^{50} + 10q^{55} - 2q^{56} - 16q^{59} + 10q^{61} - 2q^{64} - 2q^{70} + 12q^{71} - 2q^{74} - 2q^{80} + 4q^{85} - 14q^{86} + 16q^{89} + 8q^{94} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3330\mathbb{Z}\right)^\times\).

\(n\) \(371\) \(631\) \(667\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1999.1
1.00000i
1.00000i
1.00000i 0 −1.00000 −1.00000 + 2.00000i 0 1.00000i 1.00000i 0 2.00000 + 1.00000i
1999.2 1.00000i 0 −1.00000 −1.00000 2.00000i 0 1.00000i 1.00000i 0 2.00000 1.00000i
\(n\): 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 3330.2.d.d 2
3.b odd 2 1 1110.2.d.c 2
5.b even 2 1 inner 3330.2.d.d 2
15.d odd 2 1 1110.2.d.c 2
15.e even 4 1 5550.2.a.d 1
15.e even 4 1 5550.2.a.bo 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.d.c 2 3.b odd 2 1
1110.2.d.c 2 15.d odd 2 1
3330.2.d.d 2 1.a even 1 1 trivial
3330.2.d.d 2 5.b even 2 1 inner
5550.2.a.d 1 15.e even 4 1
5550.2.a.bo 1 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3330, [\chi])\):

\( T_{7}^{2} + 1 \)
\( T_{11} + 5 \)
\( T_{17}^{2} + 1 \)
\( T_{29} + 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 5 + 2 T + T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( ( 5 + T )^{2} \)
$13$ \( T^{2} \)
$17$ \( 1 + T^{2} \)
$19$ \( T^{2} \)
$23$ \( 16 + T^{2} \)
$29$ \( ( 3 + T )^{2} \)
$31$ \( ( -1 + T )^{2} \)
$37$ \( 1 + T^{2} \)
$41$ \( ( -1 + T )^{2} \)
$43$ \( 49 + T^{2} \)
$47$ \( 16 + T^{2} \)
$53$ \( 9 + T^{2} \)
$59$ \( ( 8 + T )^{2} \)
$61$ \( ( -5 + T )^{2} \)
$67$ \( 16 + T^{2} \)
$71$ \( ( -6 + T )^{2} \)
$73$ \( 100 + T^{2} \)
$79$ \( T^{2} \)
$83$ \( 64 + T^{2} \)
$89$ \( ( -8 + T )^{2} \)
$97$ \( 1 + T^{2} \)
show more
show less