Properties

Label 775.2.a.f
Level $775$
Weight $2$
Character orbit 775.a
Self dual yes
Analytic conductor $6.188$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 775 = 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 775.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.18840615665\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{24})^+\)
Defining polynomial: \( x^{4} - 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 155)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} + \beta_1) q^{2} - \beta_1 q^{3} + ( - \beta_{2} - 1) q^{6} + ( - \beta_{3} + \beta_1) q^{7} + ( - 2 \beta_{3} - 2 \beta_1) q^{8} + (\beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + \beta_1) q^{2} - \beta_1 q^{3} + ( - \beta_{2} - 1) q^{6} + ( - \beta_{3} + \beta_1) q^{7} + ( - 2 \beta_{3} - 2 \beta_1) q^{8} + (\beta_{2} - 1) q^{9} + (\beta_{2} - 3) q^{11} + ( - \beta_{3} + \beta_1) q^{13} + 2 \beta_{2} q^{14} - 4 q^{16} + (\beta_{3} - 2 \beta_1) q^{17} - 2 \beta_{3} q^{18} + ( - 3 \beta_{2} - 2) q^{19} + ( - \beta_{2} - 3) q^{21} + ( - 4 \beta_{3} - 2 \beta_1) q^{22} + ( - \beta_{3} - \beta_1) q^{23} + (2 \beta_{2} + 2) q^{24} + 2 \beta_{2} q^{26} + ( - \beta_{3} + 2 \beta_1) q^{27} + ( - 3 \beta_{2} - 3) q^{29} + q^{31} + ( - \beta_{3} + \beta_1) q^{33} + ( - 3 \beta_{2} - 1) q^{34} + ( - \beta_{3} - 2 \beta_1) q^{37} + (\beta_{3} - 5 \beta_1) q^{38} + ( - \beta_{2} - 3) q^{39} + ( - 2 \beta_{2} - 9) q^{41} + ( - 2 \beta_{3} - 4 \beta_1) q^{42} + (7 \beta_{3} + 2 \beta_1) q^{43} - 2 q^{46} + (4 \beta_{3} - 2 \beta_1) q^{47} + 4 \beta_1 q^{48} - q^{49} + (2 \beta_{2} + 5) q^{51} + (2 \beta_{3} + 5 \beta_1) q^{53} + (3 \beta_{2} + 1) q^{54} - 4 \beta_{2} q^{56} + (3 \beta_{3} + 8 \beta_1) q^{57} - 6 \beta_1 q^{58} + (4 \beta_{2} - 3) q^{59} + (6 \beta_{2} - 4) q^{61} + (\beta_{3} + \beta_1) q^{62} + (4 \beta_{3} + 2 \beta_1) q^{63} + 8 q^{64} + 2 \beta_{2} q^{66} + (\beta_{3} - \beta_1) q^{67} + (\beta_{2} + 1) q^{69} + (\beta_{2} - 6) q^{71} + 4 \beta_{3} q^{72} + ( - 4 \beta_{3} + \beta_1) q^{73} + ( - \beta_{2} - 3) q^{74} + 6 \beta_{3} q^{77} + ( - 2 \beta_{3} - 4 \beta_1) q^{78} + (3 \beta_{2} + 1) q^{79} + ( - 5 \beta_{2} - 2) q^{81} + ( - 7 \beta_{3} - 11 \beta_1) q^{82} + ( - 7 \beta_{3} + 2 \beta_1) q^{83} + ( - 5 \beta_{2} + 9) q^{86} + (3 \beta_{3} + 9 \beta_1) q^{87} + (8 \beta_{3} + 4 \beta_1) q^{88} + (3 \beta_{2} - 9) q^{89} + 6 q^{91} - \beta_1 q^{93} + ( - 6 \beta_{2} + 2) q^{94} - 6 \beta_{3} q^{97} + ( - \beta_{3} - \beta_1) q^{98} + ( - 4 \beta_{2} + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{6} - 4 q^{9} - 12 q^{11} - 16 q^{16} - 8 q^{19} - 12 q^{21} + 8 q^{24} - 12 q^{29} + 4 q^{31} - 4 q^{34} - 12 q^{39} - 36 q^{41} - 8 q^{46} - 4 q^{49} + 20 q^{51} + 4 q^{54} - 12 q^{59} - 16 q^{61} + 32 q^{64} + 4 q^{69} - 24 q^{71} - 12 q^{74} + 4 q^{79} - 8 q^{81} + 36 q^{86} - 36 q^{89} + 24 q^{91} + 8 q^{94} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{24} + \zeta_{24}^{-1}\):

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0.517638
−1.93185
1.93185
−0.517638
−1.41421 −0.517638 0 0 0.732051 2.44949 2.82843 −2.73205 0
1.2 −1.41421 1.93185 0 0 −2.73205 −2.44949 2.82843 0.732051 0
1.3 1.41421 −1.93185 0 0 −2.73205 2.44949 −2.82843 0.732051 0
1.4 1.41421 0.517638 0 0 0.732051 −2.44949 −2.82843 −2.73205 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(31\) \(-1\)

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 775.2.a.f 4
3.b odd 2 1 6975.2.a.bk 4
5.b even 2 1 inner 775.2.a.f 4
5.c odd 4 2 155.2.b.a 4
15.d odd 2 1 6975.2.a.bk 4
15.e even 4 2 1395.2.c.c 4
20.e even 4 2 2480.2.d.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
155.2.b.a 4 5.c odd 4 2
775.2.a.f 4 1.a even 1 1 trivial
775.2.a.f 4 5.b even 2 1 inner
1395.2.c.c 4 15.e even 4 2
2480.2.d.b 4 20.e even 4 2
6975.2.a.bk 4 3.b odd 2 1
6975.2.a.bk 4 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 2 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(775))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} - 4T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 6)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 6 T + 6)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 6)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 28T^{2} + 169 \) Copy content Toggle raw display
$19$ \( (T^{2} + 4 T - 23)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 6 T - 18)^{2} \) Copy content Toggle raw display
$31$ \( (T - 1)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} - 12T^{2} + 9 \) Copy content Toggle raw display
$41$ \( (T^{2} + 18 T + 69)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 156T^{2} + 9 \) Copy content Toggle raw display
$47$ \( T^{4} - 112T^{2} + 2704 \) Copy content Toggle raw display
$53$ \( T^{4} - 76T^{2} + 121 \) Copy content Toggle raw display
$59$ \( (T^{2} + 6 T - 39)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 8 T - 92)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 6)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 12 T + 33)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 84T^{2} + 1089 \) Copy content Toggle raw display
$79$ \( (T^{2} - 2 T - 26)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 268 T^{2} + 11881 \) Copy content Toggle raw display
$89$ \( (T^{2} + 18 T + 54)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 144T^{2} + 1296 \) Copy content Toggle raw display
show more
show less