Properties

Label 2070.2.d.a
Level $2070$
Weight $2$
Character orbit 2070.d
Analytic conductor $16.529$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2070.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(16.5290332184\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 690)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\zeta_{8}^{2} q^{2} - q^{4} + ( 2 \zeta_{8} + \zeta_{8}^{3} ) q^{5} -2 \zeta_{8}^{2} q^{7} + \zeta_{8}^{2} q^{8} +O(q^{10})\) \( q -\zeta_{8}^{2} q^{2} - q^{4} + ( 2 \zeta_{8} + \zeta_{8}^{3} ) q^{5} -2 \zeta_{8}^{2} q^{7} + \zeta_{8}^{2} q^{8} + ( \zeta_{8} - 2 \zeta_{8}^{3} ) q^{10} + ( 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{11} + ( 2 \zeta_{8} - 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{13} -2 q^{14} + q^{16} + ( -2 \zeta_{8} - 4 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{17} + ( -2 - 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{19} + ( -2 \zeta_{8} - \zeta_{8}^{3} ) q^{20} + ( -3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{22} -\zeta_{8}^{2} q^{23} + ( -4 + 3 \zeta_{8}^{2} ) q^{25} + ( -2 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{26} + 2 \zeta_{8}^{2} q^{28} + ( -2 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{29} + 6 q^{31} -\zeta_{8}^{2} q^{32} + ( -4 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{34} + ( 2 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{35} + ( -\zeta_{8} + 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{37} + ( 3 \zeta_{8} + 2 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{38} + ( -\zeta_{8} + 2 \zeta_{8}^{3} ) q^{40} + ( 4 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{41} + ( -3 \zeta_{8} - 6 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{43} + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{44} - q^{46} + ( 2 \zeta_{8} - 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{47} + 3 q^{49} + ( 3 + 4 \zeta_{8}^{2} ) q^{50} + ( -2 \zeta_{8} + 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{52} + ( -\zeta_{8} + 12 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{53} + ( 3 + 9 \zeta_{8}^{2} ) q^{55} + 2 q^{56} + ( -4 \zeta_{8} + 2 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{58} + ( 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{59} + ( -2 + \zeta_{8} - \zeta_{8}^{3} ) q^{61} -6 \zeta_{8}^{2} q^{62} - q^{64} + ( -6 + 2 \zeta_{8} + 2 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{65} + ( -\zeta_{8} - 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{67} + ( 2 \zeta_{8} + 4 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{68} + ( -4 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{70} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{71} + ( -4 \zeta_{8} - 2 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{73} + ( 2 - \zeta_{8} + \zeta_{8}^{3} ) q^{74} + ( 2 + 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{76} + ( -6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{77} + ( -2 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{79} + ( 2 \zeta_{8} + \zeta_{8}^{3} ) q^{80} + ( -2 \zeta_{8} - 4 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{82} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{83} + ( 6 + 4 \zeta_{8} - 2 \zeta_{8}^{2} - 8 \zeta_{8}^{3} ) q^{85} + ( -6 - 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{86} + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{88} + ( 12 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{89} + ( -4 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{91} + \zeta_{8}^{2} q^{92} + ( -2 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{94} + ( -3 - 4 \zeta_{8} - 9 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{95} -6 \zeta_{8}^{2} q^{97} -3 \zeta_{8}^{2} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} + O(q^{10}) \) \( 4q - 4q^{4} - 8q^{14} + 4q^{16} - 8q^{19} - 16q^{25} - 8q^{26} - 8q^{29} + 24q^{31} - 16q^{34} + 16q^{41} - 4q^{46} + 12q^{49} + 12q^{50} + 12q^{55} + 8q^{56} - 8q^{61} - 4q^{64} - 24q^{65} + 8q^{74} + 8q^{76} - 8q^{79} + 24q^{85} - 24q^{86} + 48q^{89} - 16q^{91} - 8q^{94} - 12q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2070\mathbb{Z}\right)^\times\).

\(n\) \(461\) \(1657\) \(1891\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
829.1
−0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
0.707107 0.707107i
1.00000i 0 −1.00000 −0.707107 2.12132i 0 2.00000i 1.00000i 0 −2.12132 + 0.707107i
829.2 1.00000i 0 −1.00000 0.707107 + 2.12132i 0 2.00000i 1.00000i 0 2.12132 0.707107i
829.3 1.00000i 0 −1.00000 −0.707107 + 2.12132i 0 2.00000i 1.00000i 0 −2.12132 0.707107i
829.4 1.00000i 0 −1.00000 0.707107 2.12132i 0 2.00000i 1.00000i 0 2.12132 + 0.707107i
\(n\): 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 2070.2.d.a 4
3.b odd 2 1 690.2.d.b 4
5.b even 2 1 inner 2070.2.d.a 4
15.d odd 2 1 690.2.d.b 4
15.e even 4 1 3450.2.a.bg 2
15.e even 4 1 3450.2.a.bk 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.d.b 4 3.b odd 2 1
690.2.d.b 4 15.d odd 2 1
2070.2.d.a 4 1.a even 1 1 trivial
2070.2.d.a 4 5.b even 2 1 inner
3450.2.a.bg 2 15.e even 4 1
3450.2.a.bk 2 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2070, [\chi])\):

\( T_{7}^{2} + 4 \)
\( T_{11}^{2} - 18 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( 25 + 8 T^{2} + T^{4} \)
$7$ \( ( 4 + T^{2} )^{2} \)
$11$ \( ( -18 + T^{2} )^{2} \)
$13$ \( 16 + 24 T^{2} + T^{4} \)
$17$ \( 64 + 48 T^{2} + T^{4} \)
$19$ \( ( -14 + 4 T + T^{2} )^{2} \)
$23$ \( ( 1 + T^{2} )^{2} \)
$29$ \( ( -28 + 4 T + T^{2} )^{2} \)
$31$ \( ( -6 + T )^{4} \)
$37$ \( 4 + 12 T^{2} + T^{4} \)
$41$ \( ( 8 - 8 T + T^{2} )^{2} \)
$43$ \( 324 + 108 T^{2} + T^{4} \)
$47$ \( 16 + 24 T^{2} + T^{4} \)
$53$ \( 20164 + 292 T^{2} + T^{4} \)
$59$ \( ( -72 + T^{2} )^{2} \)
$61$ \( ( 2 + 4 T + T^{2} )^{2} \)
$67$ \( 4 + 12 T^{2} + T^{4} \)
$71$ \( ( -32 + T^{2} )^{2} \)
$73$ \( 784 + 72 T^{2} + T^{4} \)
$79$ \( ( -28 + 4 T + T^{2} )^{2} \)
$83$ \( ( 2 + T^{2} )^{2} \)
$89$ \( ( 136 - 24 T + T^{2} )^{2} \)
$97$ \( ( 36 + T^{2} )^{2} \)
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