Properties

Label 1380.2.r.b
Level $1380$
Weight $2$
Character orbit 1380.r
Analytic conductor $11.019$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1380.r (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.0193554789\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -1 - \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{3} + ( -\zeta_{8} + 2 \zeta_{8}^{3} ) q^{5} + ( 3 - 3 \zeta_{8}^{2} ) q^{7} + ( 2 \zeta_{8} + \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{9} +O(q^{10})\) \( q + ( -1 - \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{3} + ( -\zeta_{8} + 2 \zeta_{8}^{3} ) q^{5} + ( 3 - 3 \zeta_{8}^{2} ) q^{7} + ( 2 \zeta_{8} + \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{9} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{11} + ( -3 - 3 \zeta_{8}^{2} ) q^{13} + ( 1 + 3 \zeta_{8} - 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{15} + 2 \zeta_{8} q^{17} + 4 \zeta_{8}^{2} q^{19} + ( -6 + 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{21} -\zeta_{8}^{3} q^{23} + ( 4 - 3 \zeta_{8}^{2} ) q^{25} + ( -1 - 5 \zeta_{8} + \zeta_{8}^{2} ) q^{27} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{29} -4 q^{31} + ( -2 - 2 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{33} + ( 3 \zeta_{8} + 9 \zeta_{8}^{3} ) q^{35} + ( 4 - 4 \zeta_{8}^{2} ) q^{37} + ( 3 \zeta_{8} + 6 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{39} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{41} + ( 5 + 5 \zeta_{8}^{2} ) q^{43} + ( -6 - 2 \zeta_{8} + 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{45} -8 \zeta_{8} q^{47} -11 \zeta_{8}^{2} q^{49} + ( -2 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{51} + ( -2 - 6 \zeta_{8}^{2} ) q^{55} + ( 4 - 4 \zeta_{8} - 4 \zeta_{8}^{2} ) q^{57} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{59} + 2 q^{61} + ( 3 + 3 \zeta_{8}^{2} - 12 \zeta_{8}^{3} ) q^{63} + ( 9 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{65} + ( 7 - 7 \zeta_{8}^{2} ) q^{67} + ( -\zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{69} + ( -11 \zeta_{8} - 11 \zeta_{8}^{3} ) q^{71} + ( 1 + \zeta_{8}^{2} ) q^{73} + ( -7 + 3 \zeta_{8} - \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{75} + 12 \zeta_{8} q^{77} -10 \zeta_{8}^{2} q^{79} + ( 7 + 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{81} -12 \zeta_{8}^{3} q^{83} + ( -4 - 2 \zeta_{8}^{2} ) q^{85} + ( -2 - 4 \zeta_{8} + 2 \zeta_{8}^{2} ) q^{87} + ( 7 \zeta_{8} - 7 \zeta_{8}^{3} ) q^{89} -18 q^{91} + ( 4 + 4 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{93} + ( -8 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{95} + ( 12 - 12 \zeta_{8}^{2} ) q^{97} + ( -2 \zeta_{8} + 8 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{3} + 12q^{7} + O(q^{10}) \) \( 4q - 4q^{3} + 12q^{7} - 12q^{13} + 4q^{15} - 24q^{21} + 16q^{25} - 4q^{27} - 16q^{31} - 8q^{33} + 16q^{37} + 20q^{43} - 24q^{45} - 8q^{51} - 8q^{55} + 16q^{57} + 8q^{61} + 12q^{63} + 28q^{67} + 4q^{73} - 28q^{75} + 28q^{81} - 16q^{85} - 8q^{87} - 72q^{91} + 16q^{93} + 48q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1380\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(461\) \(691\) \(1201\)
\(\chi(n)\) \(-\zeta_{8}^{2}\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
737.1
0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
−0.707107 0.707107i
0 −1.70711 + 0.292893i 0 −2.12132 0.707107i 0 3.00000 + 3.00000i 0 2.82843 1.00000i 0
737.2 0 −0.292893 + 1.70711i 0 2.12132 + 0.707107i 0 3.00000 + 3.00000i 0 −2.82843 1.00000i 0
1013.1 0 −1.70711 0.292893i 0 −2.12132 + 0.707107i 0 3.00000 3.00000i 0 2.82843 + 1.00000i 0
1013.2 0 −0.292893 1.70711i 0 2.12132 0.707107i 0 3.00000 3.00000i 0 −2.82843 + 1.00000i 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
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1380.2.r.b 4
3.b odd 2 1 inner 1380.2.r.b 4
5.c odd 4 1 inner 1380.2.r.b 4
15.e even 4 1 inner 1380.2.r.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.2.r.b 4 1.a even 1 1 trivial
1380.2.r.b 4 3.b odd 2 1 inner
1380.2.r.b 4 5.c odd 4 1 inner
1380.2.r.b 4 15.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} - 6 T_{7} + 18 \) acting on \(S_{2}^{\mathrm{new}}(1380, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 9 + 12 T + 8 T^{2} + 4 T^{3} + T^{4} \)
$5$ \( 25 - 8 T^{2} + T^{4} \)
$7$ \( ( 18 - 6 T + T^{2} )^{2} \)
$11$ \( ( 8 + T^{2} )^{2} \)
$13$ \( ( 18 + 6 T + T^{2} )^{2} \)
$17$ \( 16 + T^{4} \)
$19$ \( ( 16 + T^{2} )^{2} \)
$23$ \( 1 + T^{4} \)
$29$ \( ( -8 + T^{2} )^{2} \)
$31$ \( ( 4 + T )^{4} \)
$37$ \( ( 32 - 8 T + T^{2} )^{2} \)
$41$ \( ( 8 + T^{2} )^{2} \)
$43$ \( ( 50 - 10 T + T^{2} )^{2} \)
$47$ \( 4096 + T^{4} \)
$53$ \( T^{4} \)
$59$ \( ( -2 + T^{2} )^{2} \)
$61$ \( ( -2 + T )^{4} \)
$67$ \( ( 98 - 14 T + T^{2} )^{2} \)
$71$ \( ( 242 + T^{2} )^{2} \)
$73$ \( ( 2 - 2 T + T^{2} )^{2} \)
$79$ \( ( 100 + T^{2} )^{2} \)
$83$ \( 20736 + T^{4} \)
$89$ \( ( -98 + T^{2} )^{2} \)
$97$ \( ( 288 - 24 T + T^{2} )^{2} \)
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