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: \( x^{4} + 1 \) Copy content Toggle raw display
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.

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

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 \( T_{3}^{4} + 6T_{3}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(760, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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