Properties

Label 775.2.f.a
Level $775$
Weight $2$
Character orbit 775.f
Analytic conductor $6.188$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 775 = 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 775.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.18840615665\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{2} - 1) q^{3} + \beta_{2} q^{4} + ( - \beta_{3} - \beta_1) q^{6} + 2 \beta_1 q^{7} - \beta_{3} q^{8} - \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{2} - 1) q^{3} + \beta_{2} q^{4} + ( - \beta_{3} - \beta_1) q^{6} + 2 \beta_1 q^{7} - \beta_{3} q^{8} - \beta_{2} q^{9} + (2 \beta_{3} + 2 \beta_1) q^{11} + ( - \beta_{2} + 1) q^{12} + ( - 3 \beta_{2} - 3) q^{13} + 6 \beta_{2} q^{14} + 5 q^{16} + ( - 5 \beta_{2} + 5) q^{17} - \beta_{3} q^{18} + 2 \beta_{2} q^{19} + ( - 2 \beta_{3} - 2 \beta_1) q^{21} + (6 \beta_{2} - 6) q^{22} + (\beta_{2} + 1) q^{23} + (\beta_{3} - \beta_1) q^{24} + ( - 3 \beta_{3} - 3 \beta_1) q^{26} + (4 \beta_{2} - 4) q^{27} + 2 \beta_{3} q^{28} + ( - 4 \beta_{3} + 4 \beta_1) q^{29} + (\beta_{3} + \beta_1 + 5) q^{31} + 3 \beta_1 q^{32} - 4 \beta_{3} q^{33} + ( - 5 \beta_{3} + 5 \beta_1) q^{34} + q^{36} + ( - 3 \beta_{2} + 3) q^{37} + 2 \beta_{3} q^{38} + 6 \beta_{2} q^{39} - 6 q^{41} + ( - 6 \beta_{2} + 6) q^{42} + (3 \beta_{2} + 3) q^{43} + (2 \beta_{3} - 2 \beta_1) q^{44} + (\beta_{3} + \beta_1) q^{46} - 6 \beta_1 q^{47} + ( - 5 \beta_{2} - 5) q^{48} + 5 \beta_{2} q^{49} - 10 q^{51} + ( - 3 \beta_{2} + 3) q^{52} + (\beta_{2} + 1) q^{53} + (4 \beta_{3} - 4 \beta_1) q^{54} + 6 q^{56} + ( - 2 \beta_{2} + 2) q^{57} + (12 \beta_{2} + 12) q^{58} + 12 \beta_{2} q^{59} + (2 \beta_{3} + 2 \beta_1) q^{61} + (3 \beta_{2} + 5 \beta_1 - 3) q^{62} - 2 \beta_{3} q^{63} - \beta_{2} q^{64} + 12 q^{66} - 2 \beta_1 q^{67} + (5 \beta_{2} + 5) q^{68} - 2 \beta_{2} q^{69} + 6 q^{71} - \beta_1 q^{72} + ( - 3 \beta_{2} - 3) q^{73} + ( - 3 \beta_{3} + 3 \beta_1) q^{74} - 2 q^{76} + (12 \beta_{2} - 12) q^{77} + 6 \beta_{3} q^{78} + (4 \beta_{3} - 4 \beta_1) q^{79} + 5 q^{81} - 6 \beta_1 q^{82} + ( - 7 \beta_{2} - 7) q^{83} + ( - 2 \beta_{3} + 2 \beta_1) q^{84} + (3 \beta_{3} + 3 \beta_1) q^{86} - 8 \beta_1 q^{87} + (6 \beta_{2} + 6) q^{88} + (4 \beta_{3} - 4 \beta_1) q^{89} + ( - 6 \beta_{3} - 6 \beta_1) q^{91} + (\beta_{2} - 1) q^{92} + ( - 2 \beta_{3} - 5 \beta_{2} - 5) q^{93} - 18 \beta_{2} q^{94} + ( - 3 \beta_{3} - 3 \beta_1) q^{96} + 5 \beta_{3} q^{98} + ( - 2 \beta_{3} + 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} + 4 q^{12} - 12 q^{13} + 20 q^{16} + 20 q^{17} - 24 q^{22} + 4 q^{23} - 16 q^{27} + 20 q^{31} + 4 q^{36} + 12 q^{37} - 24 q^{41} + 24 q^{42} + 12 q^{43} - 20 q^{48} - 40 q^{51} + 12 q^{52} + 4 q^{53} + 24 q^{56} + 8 q^{57} + 48 q^{58} - 12 q^{62} + 48 q^{66} + 20 q^{68} + 24 q^{71} - 12 q^{73} - 8 q^{76} - 48 q^{77} + 20 q^{81} - 28 q^{83} + 24 q^{88} - 4 q^{92} - 20 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9 \) : Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/775\mathbb{Z}\right)^\times\).

\(n\) \(251\) \(652\)
\(\chi(n)\) \(-1\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
557.1
−1.22474 1.22474i
1.22474 + 1.22474i
−1.22474 + 1.22474i
1.22474 1.22474i
−1.22474 1.22474i −1.00000 1.00000i 1.00000i 0 2.44949i −2.44949 2.44949i −1.22474 + 1.22474i 1.00000i 0
557.2 1.22474 + 1.22474i −1.00000 1.00000i 1.00000i 0 2.44949i 2.44949 + 2.44949i 1.22474 1.22474i 1.00000i 0
743.1 −1.22474 + 1.22474i −1.00000 + 1.00000i 1.00000i 0 2.44949i −2.44949 + 2.44949i −1.22474 1.22474i 1.00000i 0
743.2 1.22474 1.22474i −1.00000 + 1.00000i 1.00000i 0 2.44949i 2.44949 2.44949i 1.22474 + 1.22474i 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
5.c odd 4 1 inner
155.c odd 2 1 inner
155.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 775.2.f.a 4
5.b even 2 1 775.2.f.b yes 4
5.c odd 4 1 inner 775.2.f.a 4
5.c odd 4 1 775.2.f.b yes 4
31.b odd 2 1 775.2.f.b yes 4
155.c odd 2 1 inner 775.2.f.a 4
155.f even 4 1 inner 775.2.f.a 4
155.f even 4 1 775.2.f.b yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
775.2.f.a 4 1.a even 1 1 trivial
775.2.f.a 4 5.c odd 4 1 inner
775.2.f.a 4 155.c odd 2 1 inner
775.2.f.a 4 155.f even 4 1 inner
775.2.f.b yes 4 5.b even 2 1
775.2.f.b yes 4 5.c odd 4 1
775.2.f.b yes 4 31.b odd 2 1
775.2.f.b yes 4 155.f even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(775, [\chi])\):

\( T_{2}^{4} + 9 \) Copy content Toggle raw display
\( T_{3}^{2} + 2T_{3} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 9 \) Copy content Toggle raw display
$3$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 144 \) Copy content Toggle raw display
$11$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 6 T + 18)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 10 T + 50)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 96)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 10 T + 31)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 6 T + 18)^{2} \) Copy content Toggle raw display
$41$ \( (T + 6)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 6 T + 18)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 11664 \) Copy content Toggle raw display
$53$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 144 \) Copy content Toggle raw display
$71$ \( (T - 6)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 6 T + 18)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 96)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 14 T + 98)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 96)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
show more
show less