Properties

Label 3330.2.e.b
Level $3330$
Weight $2$
Character orbit 3330.e
Analytic conductor $26.590$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(26.5901838731\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + ( -2 + i ) q^{5} + 3 i q^{7} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} + ( -2 + i ) q^{5} + 3 i q^{7} + q^{8} + ( -2 + i ) q^{10} + 3 q^{11} -4 q^{13} + 3 i q^{14} + q^{16} -7 q^{17} -4 i q^{19} + ( -2 + i ) q^{20} + 3 q^{22} -6 q^{23} + ( 3 - 4 i ) q^{25} -4 q^{26} + 3 i q^{28} + 9 i q^{29} -5 i q^{31} + q^{32} -7 q^{34} + ( -3 - 6 i ) q^{35} + ( 6 - i ) q^{37} -4 i q^{38} + ( -2 + i ) q^{40} -7 q^{41} + q^{43} + 3 q^{44} -6 q^{46} -8 i q^{47} -2 q^{49} + ( 3 - 4 i ) q^{50} -4 q^{52} -i q^{53} + ( -6 + 3 i ) q^{55} + 3 i q^{56} + 9 i q^{58} -6 i q^{59} -5 i q^{61} -5 i q^{62} + q^{64} + ( 8 - 4 i ) q^{65} -2 i q^{67} -7 q^{68} + ( -3 - 6 i ) q^{70} -12 q^{71} -4 i q^{73} + ( 6 - i ) q^{74} -4 i q^{76} + 9 i q^{77} -4 i q^{79} + ( -2 + i ) q^{80} -7 q^{82} + 14 i q^{83} + ( 14 - 7 i ) q^{85} + q^{86} + 3 q^{88} -6 i q^{89} -12 i q^{91} -6 q^{92} -8 i q^{94} + ( 4 + 8 i ) q^{95} + 7 q^{97} -2 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} - 4q^{5} + 2q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} - 4q^{5} + 2q^{8} - 4q^{10} + 6q^{11} - 8q^{13} + 2q^{16} - 14q^{17} - 4q^{20} + 6q^{22} - 12q^{23} + 6q^{25} - 8q^{26} + 2q^{32} - 14q^{34} - 6q^{35} + 12q^{37} - 4q^{40} - 14q^{41} + 2q^{43} + 6q^{44} - 12q^{46} - 4q^{49} + 6q^{50} - 8q^{52} - 12q^{55} + 2q^{64} + 16q^{65} - 14q^{68} - 6q^{70} - 24q^{71} + 12q^{74} - 4q^{80} - 14q^{82} + 28q^{85} + 2q^{86} + 6q^{88} - 12q^{92} + 8q^{95} + 14q^{97} - 4q^{98} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
739.1
1.00000i
1.00000i
1.00000 0 1.00000 −2.00000 1.00000i 0 3.00000i 1.00000 0 −2.00000 1.00000i
739.2 1.00000 0 1.00000 −2.00000 + 1.00000i 0 3.00000i 1.00000 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
185.d 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.e.b 2
3.b odd 2 1 1110.2.e.a 2
5.b even 2 1 3330.2.e.a 2
15.d odd 2 1 1110.2.e.b yes 2
37.b even 2 1 3330.2.e.a 2
111.d odd 2 1 1110.2.e.b yes 2
185.d even 2 1 inner 3330.2.e.b 2
555.b odd 2 1 1110.2.e.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.e.a 2 3.b odd 2 1
1110.2.e.a 2 555.b odd 2 1
1110.2.e.b yes 2 15.d odd 2 1
1110.2.e.b yes 2 111.d odd 2 1
3330.2.e.a 2 5.b even 2 1
3330.2.e.a 2 37.b even 2 1
3330.2.e.b 2 1.a even 1 1 trivial
3330.2.e.b 2 185.d even 2 1 inner

Hecke kernels

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

\( T_{7}^{2} + 9 \)
\( T_{13} + 4 \)
\( T_{17} + 7 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( 5 + 4 T + T^{2} \)
$7$ \( 9 + T^{2} \)
$11$ \( ( -3 + T )^{2} \)
$13$ \( ( 4 + T )^{2} \)
$17$ \( ( 7 + T )^{2} \)
$19$ \( 16 + T^{2} \)
$23$ \( ( 6 + T )^{2} \)
$29$ \( 81 + T^{2} \)
$31$ \( 25 + T^{2} \)
$37$ \( 37 - 12 T + T^{2} \)
$41$ \( ( 7 + T )^{2} \)
$43$ \( ( -1 + T )^{2} \)
$47$ \( 64 + T^{2} \)
$53$ \( 1 + T^{2} \)
$59$ \( 36 + T^{2} \)
$61$ \( 25 + T^{2} \)
$67$ \( 4 + T^{2} \)
$71$ \( ( 12 + T )^{2} \)
$73$ \( 16 + T^{2} \)
$79$ \( 16 + T^{2} \)
$83$ \( 196 + T^{2} \)
$89$ \( 36 + T^{2} \)
$97$ \( ( -7 + T )^{2} \)
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