Properties

Label 260.2.p.c
Level $260$
Weight $2$
Character orbit 260.p
Analytic conductor $2.076$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 260 = 2^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 260.p (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.07611045255\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.3317760000.5
Defining polynomial: \( x^{8} - 7x^{4} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{2} q^{4} + ( - \beta_{5} - \beta_1) q^{5} + ( - 2 \beta_{5} + 2 \beta_1) q^{7} + (\beta_{6} + 2 \beta_{4}) q^{8} + 3 \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{2} q^{4} + ( - \beta_{5} - \beta_1) q^{5} + ( - 2 \beta_{5} + 2 \beta_1) q^{7} + (\beta_{6} + 2 \beta_{4}) q^{8} + 3 \beta_{3} q^{9} + ( - 2 \beta_{3} - \beta_{2}) q^{10} + (\beta_{6} - \beta_{5} + \beta_{4} + \beta_1) q^{11} + (\beta_{6} - \beta_{4} - 2 \beta_{3} - 2) q^{13} + ( - 4 \beta_{3} + 2 \beta_{2}) q^{14} + (\beta_{7} + 4) q^{16} + (\beta_{3} - 1) q^{17} + 3 \beta_{6} q^{18} + ( - \beta_{6} - \beta_{5} - \beta_{4} + \beta_1) q^{19} + ( - 3 \beta_{6} - 2 \beta_{4}) q^{20} + (\beta_{7} - 2 \beta_{3} + \beta_{2} + 2) q^{22} + (2 \beta_{7} + \beta_{3} - 2 \beta_{2} + 1) q^{23} + 5 \beta_{3} q^{25} + (\beta_{7} - 2 \beta_{6} - 2 \beta_1 - 2) q^{26} + ( - 2 \beta_{6} + 4 \beta_{4}) q^{28} + 2 \beta_{3} q^{29} + ( - 4 \beta_{6} + 4 \beta_{5} - 4 \beta_{4} - 4 \beta_1) q^{31} + (2 \beta_{5} + 3 \beta_1) q^{32} + (\beta_{6} - \beta_1) q^{34} + (2 \beta_{3} - 4 \beta_{2}) q^{35} + 3 \beta_{7} q^{36} + ( - 4 \beta_{5} - 4 \beta_1) q^{37} + ( - \beta_{7} - 2 \beta_{3} + \beta_{2} - 2) q^{38} + ( - 3 \beta_{7} - 4) q^{40} + (2 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + 2 \beta_1) q^{41} + ( - 4 \beta_{7} - 2 \beta_{3} + 4 \beta_{2} - 2) q^{43} + ( - \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + \beta_1) q^{44} + ( - 3 \beta_{6} + 3 \beta_{4}) q^{45} + ( - \beta_{6} + 4 \beta_{5} - 4 \beta_{4} - \beta_1) q^{46} + ( - 2 \beta_{5} + 2 \beta_1) q^{47} - 5 \beta_{3} q^{49} + 5 \beta_{6} q^{50} + ( - 2 \beta_{7} + 2 \beta_{5} - 2 \beta_{2} - 3 \beta_1) q^{52} + (2 \beta_{3} + 2) q^{53} + ( - 2 \beta_{7} + \beta_{3} - 2 \beta_{2} - 1) q^{55} + ( - 2 \beta_{7} + 8) q^{56} + 2 \beta_{6} q^{58} + (3 \beta_{6} + 3 \beta_{5} + 3 \beta_{4} - 3 \beta_1) q^{59} - 6 q^{61} + ( - 4 \beta_{7} + 8 \beta_{3} - 4 \beta_{2} - 8) q^{62} + (6 \beta_{6} + 6 \beta_{4}) q^{63} + (4 \beta_{3} + 3 \beta_{2}) q^{64} + (2 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + 2 \beta_1 + 5) q^{65} + ( - 2 \beta_{5} + 2 \beta_1) q^{67} + (\beta_{7} - \beta_{2}) q^{68} + ( - 2 \beta_{6} - 8 \beta_{4}) q^{70} + ( - 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} - 2 \beta_1) q^{71} + (6 \beta_{5} - 3 \beta_1) q^{72} + (2 \beta_{6} - 2 \beta_{4}) q^{73} + ( - 8 \beta_{3} - 4 \beta_{2}) q^{74} + ( - \beta_{6} - 2 \beta_{5} + 2 \beta_{4} - \beta_1) q^{76} + ( - 6 \beta_{3} + 6) q^{77} + ( - 2 \beta_{3} + 4 \beta_{2}) q^{79} + ( - 6 \beta_{5} - \beta_1) q^{80} - 9 q^{81} + (2 \beta_{7} + 4 \beta_{3} + 2 \beta_{2} - 4) q^{82} + ( - 4 \beta_{6} - 4 \beta_{4}) q^{83} + ( - \beta_{6} + \beta_{5} + \beta_{4} + \beta_1) q^{85} + (2 \beta_{6} - 8 \beta_{5} + 8 \beta_{4} + 2 \beta_1) q^{86} + ( - \beta_{7} + 4 \beta_{3} + \beta_{2} + 4) q^{88} + ( - 4 \beta_{6} + 4 \beta_{5} + 4 \beta_{4} + 4 \beta_1) q^{89} + ( - 3 \beta_{7} + 6) q^{90} + (4 \beta_{7} - 4 \beta_{6} + 4 \beta_{5} - 4 \beta_{4} - 4 \beta_1 + 2) q^{91} + ( - \beta_{7} + 8 \beta_{3} - \beta_{2} - 8) q^{92} + ( - 4 \beta_{3} + 2 \beta_{2}) q^{94} + (2 \beta_{7} + \beta_{3} - 2 \beta_{2} + 1) q^{95} + (2 \beta_{5} + 2 \beta_1) q^{97} - 5 \beta_{6} q^{98} + (3 \beta_{6} + 3 \beta_{5} + 3 \beta_{4} - 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{13} + 28 q^{16} - 8 q^{17} + 12 q^{22} - 20 q^{26} - 12 q^{36} - 12 q^{38} - 20 q^{40} + 8 q^{52} + 16 q^{53} + 72 q^{56} - 48 q^{61} - 48 q^{62} + 40 q^{65} - 4 q^{68} + 48 q^{77} - 72 q^{81} - 40 q^{82} + 36 q^{88} + 60 q^{90} - 60 q^{92}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 7x^{4} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} - 3\nu^{2} ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + 7\nu^{3} ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} - 3\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - 3\nu^{3} ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{4} - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + 2\beta_{4} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} + 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{5} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4\beta_{3} + 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 7\beta_{6} + 6\beta_{4} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/260\mathbb{Z}\right)^\times\).

\(n\) \(41\) \(131\) \(157\)
\(\chi(n)\) \(-1\) \(-1\) \(-\beta_{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
103.1
−1.40294 0.178197i
−0.178197 1.40294i
0.178197 + 1.40294i
1.40294 + 0.178197i
−1.40294 + 0.178197i
−0.178197 + 1.40294i
0.178197 1.40294i
1.40294 0.178197i
−1.40294 0.178197i 0 1.93649 + 0.500000i 1.58114 + 1.58114i 0 −2.44949 + 2.44949i −2.62769 1.04655i 3.00000i −1.93649 2.50000i
103.2 −0.178197 1.40294i 0 −1.93649 + 0.500000i 1.58114 + 1.58114i 0 2.44949 2.44949i 1.04655 + 2.62769i 3.00000i 1.93649 2.50000i
103.3 0.178197 + 1.40294i 0 −1.93649 + 0.500000i −1.58114 1.58114i 0 −2.44949 + 2.44949i −1.04655 2.62769i 3.00000i 1.93649 2.50000i
103.4 1.40294 + 0.178197i 0 1.93649 + 0.500000i −1.58114 1.58114i 0 2.44949 2.44949i 2.62769 + 1.04655i 3.00000i −1.93649 2.50000i
207.1 −1.40294 + 0.178197i 0 1.93649 0.500000i 1.58114 1.58114i 0 −2.44949 2.44949i −2.62769 + 1.04655i 3.00000i −1.93649 + 2.50000i
207.2 −0.178197 + 1.40294i 0 −1.93649 0.500000i 1.58114 1.58114i 0 2.44949 + 2.44949i 1.04655 2.62769i 3.00000i 1.93649 + 2.50000i
207.3 0.178197 1.40294i 0 −1.93649 0.500000i −1.58114 + 1.58114i 0 −2.44949 2.44949i −1.04655 + 2.62769i 3.00000i 1.93649 + 2.50000i
207.4 1.40294 0.178197i 0 1.93649 0.500000i −1.58114 + 1.58114i 0 2.44949 + 2.44949i 2.62769 1.04655i 3.00000i −1.93649 + 2.50000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 207.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.c odd 4 1 inner
13.b even 2 1 inner
20.e even 4 1 inner
52.b odd 2 1 inner
65.h odd 4 1 inner
260.p even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 260.2.p.c 8
4.b odd 2 1 inner 260.2.p.c 8
5.c odd 4 1 inner 260.2.p.c 8
13.b even 2 1 inner 260.2.p.c 8
20.e even 4 1 inner 260.2.p.c 8
52.b odd 2 1 inner 260.2.p.c 8
65.h odd 4 1 inner 260.2.p.c 8
260.p even 4 1 inner 260.2.p.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.p.c 8 1.a even 1 1 trivial
260.2.p.c 8 4.b odd 2 1 inner
260.2.p.c 8 5.c odd 4 1 inner
260.2.p.c 8 13.b even 2 1 inner
260.2.p.c 8 20.e even 4 1 inner
260.2.p.c 8 52.b odd 2 1 inner
260.2.p.c 8 65.h odd 4 1 inner
260.2.p.c 8 260.p even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(260, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{7}^{4} + 144 \) Copy content Toggle raw display
\( T_{37}^{4} + 6400 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 7T^{4} + 16 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} + 25)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 144)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 6)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 8 T^{3} + 32 T^{2} + 104 T + 169)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 2 T + 2)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 6)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 900)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 96)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 6400)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 40)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 14400)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 144)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 4 T + 8)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} + 54)^{4} \) Copy content Toggle raw display
$61$ \( (T + 6)^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} + 144)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 24)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 400)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 60)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 2304)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 160)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 400)^{2} \) Copy content Toggle raw display
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