Properties

Label 760.2.d.c
Level $760$
Weight $2$
Character orbit 760.d
Analytic conductor $6.069$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 760 = 2^{3} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 760.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.06863055362\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{3} + ( \zeta_{8} - 2 \zeta_{8}^{3} ) q^{5} + ( \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{7} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{9} +O(q^{10})\) \( q + ( \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{3} + ( \zeta_{8} - 2 \zeta_{8}^{3} ) q^{5} + ( \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{7} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{9} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{11} + ( 2 \zeta_{8} - \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{13} + ( 1 + 2 \zeta_{8} + 3 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{15} -\zeta_{8}^{2} q^{17} + q^{19} + ( -3 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{21} + ( 3 \zeta_{8} + \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{23} + ( 4 - 3 \zeta_{8}^{2} ) q^{25} + ( \zeta_{8} - \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{27} + ( -1 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{29} + ( 2 + \zeta_{8} - \zeta_{8}^{3} ) q^{31} + ( \zeta_{8} + 2 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{33} + ( 1 + 2 \zeta_{8} + 3 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{35} + ( 2 \zeta_{8} - 8 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{37} + ( -3 - \zeta_{8} + \zeta_{8}^{3} ) q^{39} + ( -5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{41} + ( 3 \zeta_{8} - 2 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{43} + ( -6 + 2 \zeta_{8}^{2} ) q^{45} -8 \zeta_{8}^{2} q^{47} + ( 4 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{49} + ( 1 + \zeta_{8} - \zeta_{8}^{3} ) q^{51} + ( -2 \zeta_{8} + \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{53} + ( 3 - \zeta_{8}^{2} ) q^{55} + ( \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{57} + ( -13 - \zeta_{8} + \zeta_{8}^{3} ) q^{59} + ( 2 + \zeta_{8} - \zeta_{8}^{3} ) q^{61} + ( -2 \zeta_{8} - 4 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{63} + ( 2 - 2 \zeta_{8} + 6 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{65} + ( 5 \zeta_{8} - \zeta_{8}^{2} + 5 \zeta_{8}^{3} ) q^{67} + ( -7 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{69} + ( -4 + 5 \zeta_{8} - 5 \zeta_{8}^{3} ) q^{71} + ( 2 \zeta_{8} + 11 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{73} + ( 3 + 7 \zeta_{8} + 4 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{75} + ( \zeta_{8} + 2 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{77} + ( -2 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{79} + ( -1 - 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{81} + ( -6 \zeta_{8} + 6 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{83} + ( -2 \zeta_{8} - \zeta_{8}^{3} ) q^{85} + ( -3 \zeta_{8} - 5 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{87} + ( 8 - 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{89} + ( -3 - \zeta_{8} + \zeta_{8}^{3} ) q^{91} + ( 3 \zeta_{8} + 4 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{93} + ( \zeta_{8} - 2 \zeta_{8}^{3} ) q^{95} + ( -4 \zeta_{8} - 2 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{97} -4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 4q^{15} + 4q^{19} - 12q^{21} + 16q^{25} - 4q^{29} + 8q^{31} + 4q^{35} - 12q^{39} - 24q^{45} + 16q^{49} + 4q^{51} + 12q^{55} - 52q^{59} + 8q^{61} + 8q^{65} - 28q^{69} - 16q^{71} + 12q^{75} - 8q^{79} - 4q^{81} + 32q^{89} - 12q^{91} - 16q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/760\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(381\) \(401\) \(457\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
609.1
0.707107 0.707107i
−0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
0 2.41421i 0 2.12132 + 0.707107i 0 2.41421i 0 −2.82843 0
609.2 0 0.414214i 0 −2.12132 + 0.707107i 0 0.414214i 0 2.82843 0
609.3 0 0.414214i 0 −2.12132 0.707107i 0 0.414214i 0 2.82843 0
609.4 0 2.41421i 0 2.12132 0.707107i 0 2.41421i 0 −2.82843 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 760.2.d.c 4
4.b odd 2 1 1520.2.d.d 4
5.b even 2 1 inner 760.2.d.c 4
5.c odd 4 1 3800.2.a.l 2
5.c odd 4 1 3800.2.a.p 2
20.d odd 2 1 1520.2.d.d 4
20.e even 4 1 7600.2.a.x 2
20.e even 4 1 7600.2.a.bc 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.d.c 4 1.a even 1 1 trivial
760.2.d.c 4 5.b even 2 1 inner
1520.2.d.d 4 4.b odd 2 1
1520.2.d.d 4 20.d odd 2 1
3800.2.a.l 2 5.c odd 4 1
3800.2.a.p 2 5.c odd 4 1
7600.2.a.x 2 20.e even 4 1
7600.2.a.bc 2 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 6 T_{3}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(760, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 1 + 6 T^{2} + T^{4} \)
$5$ \( 25 - 8 T^{2} + T^{4} \)
$7$ \( 1 + 6 T^{2} + T^{4} \)
$11$ \( ( -2 + T^{2} )^{2} \)
$13$ \( 49 + 18 T^{2} + T^{4} \)
$17$ \( ( 1 + T^{2} )^{2} \)
$19$ \( ( -1 + T )^{4} \)
$23$ \( 289 + 38 T^{2} + T^{4} \)
$29$ \( ( -7 + 2 T + T^{2} )^{2} \)
$31$ \( ( 2 - 4 T + T^{2} )^{2} \)
$37$ \( 3136 + 144 T^{2} + T^{4} \)
$41$ \( ( -50 + T^{2} )^{2} \)
$43$ \( 196 + 44 T^{2} + T^{4} \)
$47$ \( ( 64 + T^{2} )^{2} \)
$53$ \( 49 + 18 T^{2} + T^{4} \)
$59$ \( ( 167 + 26 T + T^{2} )^{2} \)
$61$ \( ( 2 - 4 T + T^{2} )^{2} \)
$67$ \( 2401 + 102 T^{2} + T^{4} \)
$71$ \( ( -34 + 8 T + T^{2} )^{2} \)
$73$ \( 12769 + 258 T^{2} + T^{4} \)
$79$ \( ( -4 + 4 T + T^{2} )^{2} \)
$83$ \( 1296 + 216 T^{2} + T^{4} \)
$89$ \( ( 46 - 16 T + T^{2} )^{2} \)
$97$ \( 784 + 72 T^{2} + T^{4} \)
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