Properties

Label 775.2.f.c
Level $775$
Weight $2$
Character orbit 775.f
Analytic conductor $6.188$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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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: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.3317760000.2
Defining polynomial: \( x^{8} - 25x^{4} + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{6} \)
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.

\(f(q)\) \(=\) \( q - \beta_{6} q^{2} - \beta_{3} q^{3} - 4 \beta_{2} q^{4} - \beta_{7} q^{6} + \beta_{6} q^{7} + 2 \beta_{4} q^{8} - 2 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{2} - \beta_{3} q^{3} - 4 \beta_{2} q^{4} - \beta_{7} q^{6} + \beta_{6} q^{7} + 2 \beta_{4} q^{8} - 2 \beta_{2} q^{9} - \beta_{7} q^{11} + 4 \beta_1 q^{12} + 6 \beta_{2} q^{14} - 4 q^{16} - \beta_1 q^{17} + 2 \beta_{4} q^{18} + 7 \beta_{2} q^{19} + \beta_{7} q^{21} + 6 \beta_1 q^{22} - 2 \beta_{3} q^{23} - 2 \beta_{5} q^{24} - \beta_1 q^{27} - 4 \beta_{4} q^{28} - \beta_{5} q^{29} + (\beta_{7} - 1) q^{31} + 5 \beta_{4} q^{33} + \beta_{5} q^{34} - 8 q^{36} + 3 \beta_1 q^{37} - 7 \beta_{4} q^{38} + 9 q^{41} - 6 \beta_1 q^{42} - 3 \beta_{3} q^{43} - 4 \beta_{5} q^{44} - 2 \beta_{7} q^{46} + 4 \beta_{3} q^{48} + \beta_{2} q^{49} + 5 q^{51} - 5 \beta_{3} q^{53} + \beta_{5} q^{54} + 12 q^{56} - 7 \beta_1 q^{57} + 6 \beta_{3} q^{58} + 3 \beta_{2} q^{59} + 2 \beta_{7} q^{61} + (\beta_{6} - 6 \beta_1) q^{62} - 2 \beta_{4} q^{63} - 8 \beta_{2} q^{64} - 30 q^{66} + 5 \beta_{6} q^{67} - 4 \beta_{3} q^{68} - 10 \beta_{2} q^{69} - 9 q^{71} + 4 \beta_{6} q^{72} + 3 \beta_{3} q^{73} - 3 \beta_{5} q^{74} + 28 q^{76} - 6 \beta_1 q^{77} + \beta_{5} q^{79} + 11 q^{81} - 9 \beta_{6} q^{82} + 5 \beta_{3} q^{83} + 4 \beta_{5} q^{84} - 3 \beta_{7} q^{86} + 5 \beta_{6} q^{87} + 12 \beta_{3} q^{88} + \beta_{5} q^{89} + 8 \beta_1 q^{92} + ( - 5 \beta_{4} + \beta_{3}) q^{93} - 6 \beta_{6} q^{97} - \beta_{4} q^{98} - 2 \beta_{5} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 32 q^{16} - 8 q^{31} - 64 q^{36} + 72 q^{41} + 40 q^{51} + 96 q^{56} - 240 q^{66} - 72 q^{71} + 224 q^{76} + 88 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 25x^{4} + 625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} ) / 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} ) / 125 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} + 25\nu ) / 25 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} + 10\nu^{4} + 50\nu^{2} - 125 ) / 125 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{7} + 5\nu^{5} + 25\nu^{3} + 125\nu ) / 125 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} - 10\nu^{4} + 50\nu^{2} + 125 ) / 125 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2\nu^{7} + 5\nu^{5} - 25\nu^{3} + 125\nu ) / 125 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + \beta_{5} + 2\beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 5\beta_{6} + 5\beta_{4} + 10\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -25\beta_{6} + 25\beta_{4} + 50 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 25\beta_{7} + 25\beta_{5} - 50\beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 125\beta_{2} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 125\beta_{7} - 125\beta_{5} + 250\beta_1 ) / 4 \) 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
2.15988 0.578737i
−2.15988 + 0.578737i
−0.578737 + 2.15988i
0.578737 2.15988i
2.15988 + 0.578737i
−2.15988 0.578737i
−0.578737 2.15988i
0.578737 + 2.15988i
−1.73205 1.73205i −1.58114 1.58114i 4.00000i 0 5.47723i 1.73205 + 1.73205i 3.46410 3.46410i 2.00000i 0
557.2 −1.73205 1.73205i 1.58114 + 1.58114i 4.00000i 0 5.47723i 1.73205 + 1.73205i 3.46410 3.46410i 2.00000i 0
557.3 1.73205 + 1.73205i −1.58114 1.58114i 4.00000i 0 5.47723i −1.73205 1.73205i −3.46410 + 3.46410i 2.00000i 0
557.4 1.73205 + 1.73205i 1.58114 + 1.58114i 4.00000i 0 5.47723i −1.73205 1.73205i −3.46410 + 3.46410i 2.00000i 0
743.1 −1.73205 + 1.73205i −1.58114 + 1.58114i 4.00000i 0 5.47723i 1.73205 1.73205i 3.46410 + 3.46410i 2.00000i 0
743.2 −1.73205 + 1.73205i 1.58114 1.58114i 4.00000i 0 5.47723i 1.73205 1.73205i 3.46410 + 3.46410i 2.00000i 0
743.3 1.73205 1.73205i −1.58114 + 1.58114i 4.00000i 0 5.47723i −1.73205 + 1.73205i −3.46410 3.46410i 2.00000i 0
743.4 1.73205 1.73205i 1.58114 1.58114i 4.00000i 0 5.47723i −1.73205 + 1.73205i −3.46410 3.46410i 2.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 743.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner
31.b odd 2 1 inner
155.c odd 2 1 inner
155.f even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 775.2.f.c 8
5.b even 2 1 inner 775.2.f.c 8
5.c odd 4 2 inner 775.2.f.c 8
31.b odd 2 1 inner 775.2.f.c 8
155.c odd 2 1 inner 775.2.f.c 8
155.f even 4 2 inner 775.2.f.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
775.2.f.c 8 1.a even 1 1 trivial
775.2.f.c 8 5.b even 2 1 inner
775.2.f.c 8 5.c odd 4 2 inner
775.2.f.c 8 31.b odd 2 1 inner
775.2.f.c 8 155.c odd 2 1 inner
775.2.f.c 8 155.f even 4 2 inner

Hecke kernels

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

\( T_{2}^{4} + 36 \) Copy content Toggle raw display
\( T_{3}^{4} + 25 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 36)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} + 25)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 36)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 30)^{4} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( (T^{4} + 25)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 49)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 400)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 30)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 2 T + 31)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 2025)^{2} \) Copy content Toggle raw display
$41$ \( (T - 9)^{8} \) Copy content Toggle raw display
$43$ \( (T^{4} + 2025)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( (T^{4} + 15625)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 9)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 120)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 22500)^{2} \) Copy content Toggle raw display
$71$ \( (T + 9)^{8} \) Copy content Toggle raw display
$73$ \( (T^{4} + 2025)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 30)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 15625)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 30)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 46656)^{2} \) Copy content Toggle raw display
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