Properties

Label 3380.2.f.a
Level $3380$
Weight $2$
Character orbit 3380.f
Analytic conductor $26.989$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3380.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(26.9894358832\)
Analytic rank: \(0\)
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 260)
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.

\(f(q)\) \(=\) \( q -3 q^{3} -i q^{5} + 3 i q^{7} + 6 q^{9} +O(q^{10})\) \( q -3 q^{3} -i q^{5} + 3 i q^{7} + 6 q^{9} + 3 i q^{11} + 3 i q^{15} + 7 q^{17} -i q^{19} -9 i q^{21} + 7 q^{23} - q^{25} -9 q^{27} -5 q^{29} + 4 i q^{31} -9 i q^{33} + 3 q^{35} -3 i q^{37} -7 i q^{41} + 9 q^{43} -6 i q^{45} + 8 i q^{47} -2 q^{49} -21 q^{51} -6 q^{53} + 3 q^{55} + 3 i q^{57} + 5 i q^{59} -5 q^{61} + 18 i q^{63} -13 i q^{67} -21 q^{69} + 3 i q^{71} -14 i q^{73} + 3 q^{75} -9 q^{77} -8 q^{79} + 9 q^{81} -12 i q^{83} -7 i q^{85} + 15 q^{87} + 7 i q^{89} -12 i q^{93} - q^{95} + 11 i q^{97} + 18 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} + 12 q^{9} + O(q^{10}) \) \( 2 q - 6 q^{3} + 12 q^{9} + 14 q^{17} + 14 q^{23} - 2 q^{25} - 18 q^{27} - 10 q^{29} + 6 q^{35} + 18 q^{43} - 4 q^{49} - 42 q^{51} - 12 q^{53} + 6 q^{55} - 10 q^{61} - 42 q^{69} + 6 q^{75} - 18 q^{77} - 16 q^{79} + 18 q^{81} + 30 q^{87} - 2 q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3380\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(1691\) \(1861\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3041.1
1.00000i
1.00000i
0 −3.00000 0 1.00000i 0 3.00000i 0 6.00000 0
3041.2 0 −3.00000 0 1.00000i 0 3.00000i 0 6.00000 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 3380.2.f.a 2
13.b even 2 1 inner 3380.2.f.a 2
13.d odd 4 1 3380.2.a.a 1
13.d odd 4 1 3380.2.a.b 1
13.f odd 12 2 260.2.i.d 2
39.k even 12 2 2340.2.q.a 2
52.l even 12 2 1040.2.q.b 2
65.o even 12 2 1300.2.bb.e 4
65.s odd 12 2 1300.2.i.a 2
65.t even 12 2 1300.2.bb.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.i.d 2 13.f odd 12 2
1040.2.q.b 2 52.l even 12 2
1300.2.i.a 2 65.s odd 12 2
1300.2.bb.e 4 65.o even 12 2
1300.2.bb.e 4 65.t even 12 2
2340.2.q.a 2 39.k even 12 2
3380.2.a.a 1 13.d odd 4 1
3380.2.a.b 1 13.d odd 4 1
3380.2.f.a 2 1.a even 1 1 trivial
3380.2.f.a 2 13.b 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 \(S_{2}^{\mathrm{new}}(3380, [\chi])\):

\( T_{3} + 3 \)
\( T_{19}^{2} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( 3 + T )^{2} \)
$5$ \( 1 + T^{2} \)
$7$ \( 9 + T^{2} \)
$11$ \( 9 + T^{2} \)
$13$ \( T^{2} \)
$17$ \( ( -7 + T )^{2} \)
$19$ \( 1 + T^{2} \)
$23$ \( ( -7 + T )^{2} \)
$29$ \( ( 5 + T )^{2} \)
$31$ \( 16 + T^{2} \)
$37$ \( 9 + T^{2} \)
$41$ \( 49 + T^{2} \)
$43$ \( ( -9 + T )^{2} \)
$47$ \( 64 + T^{2} \)
$53$ \( ( 6 + T )^{2} \)
$59$ \( 25 + T^{2} \)
$61$ \( ( 5 + T )^{2} \)
$67$ \( 169 + T^{2} \)
$71$ \( 9 + T^{2} \)
$73$ \( 196 + T^{2} \)
$79$ \( ( 8 + T )^{2} \)
$83$ \( 144 + T^{2} \)
$89$ \( 49 + T^{2} \)
$97$ \( 121 + T^{2} \)
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