Properties

Label 3380.2.f.b
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 20)
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.

\(f(q)\) \(=\) \( q -2 q^{3} + i q^{5} + 2 i q^{7} + q^{9} +O(q^{10})\) \( q -2 q^{3} + i q^{5} + 2 i q^{7} + q^{9} -2 i q^{15} + 6 q^{17} + 4 i q^{19} -4 i q^{21} -6 q^{23} - q^{25} + 4 q^{27} + 6 q^{29} + 4 i q^{31} -2 q^{35} + 2 i q^{37} -6 i q^{41} + 10 q^{43} + i q^{45} -6 i q^{47} + 3 q^{49} -12 q^{51} -6 q^{53} -8 i q^{57} + 12 i q^{59} + 2 q^{61} + 2 i q^{63} -2 i q^{67} + 12 q^{69} + 12 i q^{71} + 2 i q^{73} + 2 q^{75} + 8 q^{79} -11 q^{81} -6 i q^{83} + 6 i q^{85} -12 q^{87} -6 i q^{89} -8 i q^{93} -4 q^{95} -2 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{3} + 2q^{9} + O(q^{10}) \) \( 2q - 4q^{3} + 2q^{9} + 12q^{17} - 12q^{23} - 2q^{25} + 8q^{27} + 12q^{29} - 4q^{35} + 20q^{43} + 6q^{49} - 24q^{51} - 12q^{53} + 4q^{61} + 24q^{69} + 4q^{75} + 16q^{79} - 22q^{81} - 24q^{87} - 8q^{95} + O(q^{100}) \)

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 −2.00000 0 1.00000i 0 2.00000i 0 1.00000 0
3041.2 0 −2.00000 0 1.00000i 0 2.00000i 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 3380.2.f.b 2
13.b even 2 1 inner 3380.2.f.b 2
13.d odd 4 1 20.2.a.a 1
13.d odd 4 1 3380.2.a.c 1
39.f even 4 1 180.2.a.a 1
52.f even 4 1 80.2.a.b 1
65.f even 4 1 100.2.c.a 2
65.g odd 4 1 100.2.a.a 1
65.k even 4 1 100.2.c.a 2
91.i even 4 1 980.2.a.h 1
91.z odd 12 2 980.2.i.i 2
91.bb even 12 2 980.2.i.c 2
104.j odd 4 1 320.2.a.f 1
104.m even 4 1 320.2.a.a 1
117.y odd 12 2 1620.2.i.h 2
117.z even 12 2 1620.2.i.b 2
143.g even 4 1 2420.2.a.a 1
156.l odd 4 1 720.2.a.h 1
195.j odd 4 1 900.2.d.c 2
195.n even 4 1 900.2.a.b 1
195.u odd 4 1 900.2.d.c 2
208.l even 4 1 1280.2.d.g 2
208.m odd 4 1 1280.2.d.c 2
208.r odd 4 1 1280.2.d.c 2
208.s even 4 1 1280.2.d.g 2
221.g odd 4 1 5780.2.a.f 1
221.h odd 4 1 5780.2.c.a 2
221.i odd 4 1 5780.2.c.a 2
247.i even 4 1 7220.2.a.f 1
260.l odd 4 1 400.2.c.b 2
260.s odd 4 1 400.2.c.b 2
260.u even 4 1 400.2.a.c 1
273.o odd 4 1 8820.2.a.g 1
312.w odd 4 1 2880.2.a.f 1
312.y even 4 1 2880.2.a.m 1
364.p odd 4 1 3920.2.a.h 1
455.n odd 4 1 4900.2.e.f 2
455.u even 4 1 4900.2.a.e 1
455.w odd 4 1 4900.2.e.f 2
520.t even 4 1 1600.2.a.w 1
520.x odd 4 1 1600.2.c.e 2
520.y even 4 1 1600.2.c.d 2
520.bj even 4 1 1600.2.c.d 2
520.bk odd 4 1 1600.2.c.e 2
520.bo odd 4 1 1600.2.a.c 1
572.k odd 4 1 9680.2.a.ba 1
780.u even 4 1 3600.2.f.j 2
780.bb odd 4 1 3600.2.a.be 1
780.bn even 4 1 3600.2.f.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.2.a.a 1 13.d odd 4 1
80.2.a.b 1 52.f even 4 1
100.2.a.a 1 65.g odd 4 1
100.2.c.a 2 65.f even 4 1
100.2.c.a 2 65.k even 4 1
180.2.a.a 1 39.f even 4 1
320.2.a.a 1 104.m even 4 1
320.2.a.f 1 104.j odd 4 1
400.2.a.c 1 260.u even 4 1
400.2.c.b 2 260.l odd 4 1
400.2.c.b 2 260.s odd 4 1
720.2.a.h 1 156.l odd 4 1
900.2.a.b 1 195.n even 4 1
900.2.d.c 2 195.j odd 4 1
900.2.d.c 2 195.u odd 4 1
980.2.a.h 1 91.i even 4 1
980.2.i.c 2 91.bb even 12 2
980.2.i.i 2 91.z odd 12 2
1280.2.d.c 2 208.m odd 4 1
1280.2.d.c 2 208.r odd 4 1
1280.2.d.g 2 208.l even 4 1
1280.2.d.g 2 208.s even 4 1
1600.2.a.c 1 520.bo odd 4 1
1600.2.a.w 1 520.t even 4 1
1600.2.c.d 2 520.y even 4 1
1600.2.c.d 2 520.bj even 4 1
1600.2.c.e 2 520.x odd 4 1
1600.2.c.e 2 520.bk odd 4 1
1620.2.i.b 2 117.z even 12 2
1620.2.i.h 2 117.y odd 12 2
2420.2.a.a 1 143.g even 4 1
2880.2.a.f 1 312.w odd 4 1
2880.2.a.m 1 312.y even 4 1
3380.2.a.c 1 13.d odd 4 1
3380.2.f.b 2 1.a even 1 1 trivial
3380.2.f.b 2 13.b even 2 1 inner
3600.2.a.be 1 780.bb odd 4 1
3600.2.f.j 2 780.u even 4 1
3600.2.f.j 2 780.bn even 4 1
3920.2.a.h 1 364.p odd 4 1
4900.2.a.e 1 455.u even 4 1
4900.2.e.f 2 455.n odd 4 1
4900.2.e.f 2 455.w odd 4 1
5780.2.a.f 1 221.g odd 4 1
5780.2.c.a 2 221.h odd 4 1
5780.2.c.a 2 221.i odd 4 1
7220.2.a.f 1 247.i even 4 1
8820.2.a.g 1 273.o odd 4 1
9680.2.a.ba 1 572.k odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3380, [\chi])\):

\( T_{3} + 2 \)
\( T_{19}^{2} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( 2 + T )^{2} \)
$5$ \( 1 + T^{2} \)
$7$ \( 4 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( T^{2} \)
$17$ \( ( -6 + T )^{2} \)
$19$ \( 16 + T^{2} \)
$23$ \( ( 6 + T )^{2} \)
$29$ \( ( -6 + T )^{2} \)
$31$ \( 16 + T^{2} \)
$37$ \( 4 + T^{2} \)
$41$ \( 36 + T^{2} \)
$43$ \( ( -10 + T )^{2} \)
$47$ \( 36 + T^{2} \)
$53$ \( ( 6 + T )^{2} \)
$59$ \( 144 + T^{2} \)
$61$ \( ( -2 + T )^{2} \)
$67$ \( 4 + T^{2} \)
$71$ \( 144 + T^{2} \)
$73$ \( 4 + T^{2} \)
$79$ \( ( -8 + T )^{2} \)
$83$ \( 36 + T^{2} \)
$89$ \( 36 + T^{2} \)
$97$ \( 4 + T^{2} \)
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