Properties

Label 3420.2.f.a
Level $3420$
Weight $2$
Character orbit 3420.f
Analytic conductor $27.309$
Analytic rank $0$
Dimension $4$
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.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(27.3088374913\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{5})\)
Defining polynomial: \(x^{4} + 6 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 380)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{5} + ( \beta_{1} - \beta_{3} ) q^{7} +O(q^{10})\) \( q -\beta_{2} q^{5} + ( \beta_{1} - \beta_{3} ) q^{7} + ( 3 - \beta_{2} ) q^{11} + ( -2 \beta_{1} - \beta_{3} ) q^{13} -2 \beta_{3} q^{17} + q^{19} + ( -3 \beta_{1} + \beta_{3} ) q^{23} + 5 q^{25} -2 \beta_{2} q^{29} -4 q^{31} + ( 3 \beta_{1} + \beta_{3} ) q^{35} + 3 \beta_{1} q^{37} + 6 q^{41} + ( -3 \beta_{1} + 3 \beta_{3} ) q^{43} + ( -3 \beta_{1} + 3 \beta_{3} ) q^{47} - q^{49} -3 \beta_{1} q^{53} + ( 5 - 3 \beta_{2} ) q^{55} + ( -6 + 2 \beta_{2} ) q^{59} + ( -5 - 3 \beta_{2} ) q^{61} + ( -3 \beta_{1} + 4 \beta_{3} ) q^{65} -3 \beta_{3} q^{67} + ( 6 + 2 \beta_{2} ) q^{71} + 6 \beta_{1} q^{73} + ( 6 \beta_{1} - 2 \beta_{3} ) q^{77} + ( -2 - 6 \beta_{2} ) q^{79} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{83} + ( 2 \beta_{1} + 4 \beta_{3} ) q^{85} + ( -6 + 4 \beta_{2} ) q^{89} + ( -2 - 6 \beta_{2} ) q^{91} -\beta_{2} q^{95} + ( \beta_{1} - 4 \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + O(q^{10}) \) \( 4 q + 12 q^{11} + 4 q^{19} + 20 q^{25} - 16 q^{31} + 24 q^{41} - 4 q^{49} + 20 q^{55} - 24 q^{59} - 20 q^{61} + 24 q^{71} - 8 q^{79} - 24 q^{89} - 8 q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 6 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} + 5 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} - 5 \beta_{1}\)

Character values

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

\(n\) \(1711\) \(1901\) \(2737\) \(3061\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1369.1
0.874032i
0.874032i
2.28825i
2.28825i
0 0 0 −2.23607 0 2.82843i 0 0 0
1369.2 0 0 0 −2.23607 0 2.82843i 0 0 0
1369.3 0 0 0 2.23607 0 2.82843i 0 0 0
1369.4 0 0 0 2.23607 0 2.82843i 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
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 3420.2.f.a 4
3.b odd 2 1 380.2.c.a 4
5.b even 2 1 inner 3420.2.f.a 4
12.b even 2 1 1520.2.d.f 4
15.d odd 2 1 380.2.c.a 4
15.e even 4 2 1900.2.a.j 4
60.h even 2 1 1520.2.d.f 4
60.l odd 4 2 7600.2.a.ce 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.c.a 4 3.b odd 2 1
380.2.c.a 4 15.d odd 2 1
1520.2.d.f 4 12.b even 2 1
1520.2.d.f 4 60.h even 2 1
1900.2.a.j 4 15.e even 4 2
3420.2.f.a 4 1.a even 1 1 trivial
3420.2.f.a 4 5.b even 2 1 inner
7600.2.a.ce 4 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 8 \) acting on \(S_{2}^{\mathrm{new}}(3420, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( -5 + T^{2} )^{2} \)
$7$ \( ( 8 + T^{2} )^{2} \)
$11$ \( ( 4 - 6 T + T^{2} )^{2} \)
$13$ \( 484 + 46 T^{2} + T^{4} \)
$17$ \( 64 + 56 T^{2} + T^{4} \)
$19$ \( ( -1 + T )^{4} \)
$23$ \( 64 + 56 T^{2} + T^{4} \)
$29$ \( ( -20 + T^{2} )^{2} \)
$31$ \( ( 4 + T )^{4} \)
$37$ \( 324 + 54 T^{2} + T^{4} \)
$41$ \( ( -6 + T )^{4} \)
$43$ \( ( 72 + T^{2} )^{2} \)
$47$ \( ( 72 + T^{2} )^{2} \)
$53$ \( 324 + 54 T^{2} + T^{4} \)
$59$ \( ( 16 + 12 T + T^{2} )^{2} \)
$61$ \( ( -20 + 10 T + T^{2} )^{2} \)
$67$ \( 324 + 126 T^{2} + T^{4} \)
$71$ \( ( 16 - 12 T + T^{2} )^{2} \)
$73$ \( 5184 + 216 T^{2} + T^{4} \)
$79$ \( ( -176 + 4 T + T^{2} )^{2} \)
$83$ \( 5184 + 216 T^{2} + T^{4} \)
$89$ \( ( -44 + 12 T + T^{2} )^{2} \)
$97$ \( 3844 + 214 T^{2} + T^{4} \)
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