Properties

Label 3420.2.t.b
Level $3420$
Weight $2$
Character orbit 3420.t
Analytic conductor $27.309$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3420.t (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(27.3088374913\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 380)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 1) q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} - 1) q^{5} - 4 q^{7} + 3 q^{11} - 6 \zeta_{6} q^{13} + ( - 2 \zeta_{6} + 2) q^{17} + (3 \zeta_{6} + 2) q^{19} + 4 \zeta_{6} q^{23} - \zeta_{6} q^{25} + \zeta_{6} q^{29} - 5 q^{31} + ( - 4 \zeta_{6} + 4) q^{35} - 4 q^{37} + ( - 2 \zeta_{6} + 2) q^{41} + 6 \zeta_{6} q^{47} + 9 q^{49} - 6 \zeta_{6} q^{53} + (3 \zeta_{6} - 3) q^{55} + (\zeta_{6} - 1) q^{59} + 7 \zeta_{6} q^{61} + 6 q^{65} + 14 \zeta_{6} q^{67} + ( - 15 \zeta_{6} + 15) q^{71} + (12 \zeta_{6} - 12) q^{73} - 12 q^{77} + ( - \zeta_{6} + 1) q^{79} - 16 q^{83} + 2 \zeta_{6} q^{85} + 17 \zeta_{6} q^{89} + 24 \zeta_{6} q^{91} + (2 \zeta_{6} - 5) q^{95} + (12 \zeta_{6} - 12) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{5} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{5} - 8 q^{7} + 6 q^{11} - 6 q^{13} + 2 q^{17} + 7 q^{19} + 4 q^{23} - q^{25} + q^{29} - 10 q^{31} + 4 q^{35} - 8 q^{37} + 2 q^{41} + 6 q^{47} + 18 q^{49} - 6 q^{53} - 3 q^{55} - q^{59} + 7 q^{61} + 12 q^{65} + 14 q^{67} + 15 q^{71} - 12 q^{73} - 24 q^{77} + q^{79} - 32 q^{83} + 2 q^{85} + 17 q^{89} + 24 q^{91} - 8 q^{95} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3420\mathbb{Z}\right)^\times\).

\(n\) \(1711\) \(1901\) \(2737\) \(3061\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1261.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 0 −0.500000 0.866025i 0 −4.00000 0 0 0
3241.1 0 0 0 −0.500000 + 0.866025i 0 −4.00000 0 0 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
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3420.2.t.b 2
3.b odd 2 1 380.2.i.a 2
12.b even 2 1 1520.2.q.g 2
15.d odd 2 1 1900.2.i.b 2
15.e even 4 2 1900.2.s.b 4
19.c even 3 1 inner 3420.2.t.b 2
57.f even 6 1 7220.2.a.a 1
57.h odd 6 1 380.2.i.a 2
57.h odd 6 1 7220.2.a.e 1
228.m even 6 1 1520.2.q.g 2
285.n odd 6 1 1900.2.i.b 2
285.v even 12 2 1900.2.s.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.i.a 2 3.b odd 2 1
380.2.i.a 2 57.h odd 6 1
1520.2.q.g 2 12.b even 2 1
1520.2.q.g 2 228.m even 6 1
1900.2.i.b 2 15.d odd 2 1
1900.2.i.b 2 285.n odd 6 1
1900.2.s.b 4 15.e even 4 2
1900.2.s.b 4 285.v even 12 2
3420.2.t.b 2 1.a even 1 1 trivial
3420.2.t.b 2 19.c even 3 1 inner
7220.2.a.a 1 57.f even 6 1
7220.2.a.e 1 57.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3420, [\chi])\):

\( T_{7} + 4 \) Copy content Toggle raw display
\( T_{11} - 3 \) Copy content Toggle raw display
\( T_{13}^{2} + 6T_{13} + 36 \) Copy content Toggle raw display
\( T_{17}^{2} - 2T_{17} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$7$ \( (T + 4)^{2} \) Copy content Toggle raw display
$11$ \( (T - 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$17$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$19$ \( T^{2} - 7T + 19 \) Copy content Toggle raw display
$23$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$29$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$31$ \( (T + 5)^{2} \) Copy content Toggle raw display
$37$ \( (T + 4)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$53$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$59$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$61$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$67$ \( T^{2} - 14T + 196 \) Copy content Toggle raw display
$71$ \( T^{2} - 15T + 225 \) Copy content Toggle raw display
$73$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$79$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$83$ \( (T + 16)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 17T + 289 \) Copy content Toggle raw display
$97$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
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