Properties

Label 775.2.j.a
Level $775$
Weight $2$
Character orbit 775.j
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.j (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.18840615665\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{2} + (2 \zeta_{10} - 2) q^{3} - 2 \zeta_{10}^{3} q^{4} + ( - 2 \zeta_{10}^{2} + \zeta_{10} - 2) q^{5} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10}) q^{6} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 3) q^{7} + (4 \zeta_{10}^{2} - 5 \zeta_{10} + 4) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{2} + (2 \zeta_{10} - 2) q^{3} - 2 \zeta_{10}^{3} q^{4} + ( - 2 \zeta_{10}^{2} + \zeta_{10} - 2) q^{5} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10}) q^{6} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 3) q^{7} + (4 \zeta_{10}^{2} - 5 \zeta_{10} + 4) q^{9} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} + 2) q^{10} + (\zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} - 1) q^{11} + (4 \zeta_{10}^{2} - 4 \zeta_{10} + 4) q^{12} + ( - \zeta_{10}^{2} - 3 \zeta_{10} - 1) q^{13} + (6 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 6) q^{14} + ( - 4 \zeta_{10}^{3} + 6 \zeta_{10}^{2} - 6 \zeta_{10} + 4) q^{15} + 4 \zeta_{10} q^{16} + ( - 2 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 2 \zeta_{10}) q^{17} + (8 \zeta_{10}^{3} - 8 \zeta_{10}^{2} + 2) q^{18} + ( - 4 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 4 \zeta_{10}) q^{19} + (2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} - 2) q^{20} + (4 \zeta_{10}^{3} + 2 \zeta_{10} - 2) q^{21} + ( - 2 \zeta_{10}^{3} - 6 \zeta_{10} + 6) q^{22} + (7 \zeta_{10}^{3} - 5 \zeta_{10}^{2} + 5 \zeta_{10} - 7) q^{23} + 5 \zeta_{10}^{2} q^{25} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 8) q^{26} + (8 \zeta_{10}^{3} - 12 \zeta_{10}^{2} + 12 \zeta_{10} - 8) q^{27} + ( - 6 \zeta_{10}^{3} - 4 \zeta_{10} + 4) q^{28} + (\zeta_{10}^{3} + 2 \zeta_{10} - 2) q^{29} + (8 \zeta_{10}^{3} - 4 \zeta_{10} + 4) q^{30} + \zeta_{10}^{2} q^{31} - 8 q^{32} + (4 \zeta_{10}^{3} - 10 \zeta_{10}^{2} + 4 \zeta_{10}) q^{33} + (4 \zeta_{10}^{2} - 6 \zeta_{10} + 4) q^{34} + ( - 4 \zeta_{10}^{2} - 3 \zeta_{10} - 4) q^{35} + (2 \zeta_{10}^{3} - 10 \zeta_{10}^{2} + 10 \zeta_{10} - 2) q^{36} + ( - 5 \zeta_{10}^{2} - 2 \zeta_{10} - 5) q^{37} + (8 \zeta_{10}^{2} - 4 \zeta_{10} + 8) q^{38} + ( - 2 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10} + 2) q^{39} + ( - 5 \zeta_{10}^{2} - 2 \zeta_{10} - 5) q^{41} + ( - 4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} - 4 \zeta_{10}) q^{42} + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} - 1) q^{43} + (6 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 6 \zeta_{10}) q^{44} + (6 \zeta_{10}^{3} - 13 \zeta_{10}^{2} + 6 \zeta_{10}) q^{45} + ( - 14 \zeta_{10}^{3} - 4 \zeta_{10} + 4) q^{46} + (2 \zeta_{10}^{3} + 8 \zeta_{10} - 8) q^{47} + (8 \zeta_{10}^{2} - 8 \zeta_{10}) q^{48} + ( - 8 \zeta_{10}^{3} + 8 \zeta_{10}^{2} + 6) q^{49} - 10 \zeta_{10} q^{50} + (6 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 4) q^{51} + (8 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 6 \zeta_{10} - 8) q^{52} + ( - 3 \zeta_{10}^{3} - 9 \zeta_{10} + 9) q^{53} + ( - 16 \zeta_{10}^{3} + 8 \zeta_{10} - 8) q^{54} + (\zeta_{10}^{3} - \zeta_{10}^{2} + 7) q^{55} + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 8) q^{57} + ( - 4 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 4 \zeta_{10}) q^{58} + (2 \zeta_{10}^{2} + 9 \zeta_{10} + 2) q^{59} + (4 \zeta_{10}^{3} - 12 \zeta_{10}^{2} + 4 \zeta_{10}) q^{60} + ( - 5 \zeta_{10}^{3} + 11 \zeta_{10}^{2} - 11 \zeta_{10} + 5) q^{61} - 2 \zeta_{10} q^{62} + (2 \zeta_{10}^{2} + 3 \zeta_{10} + 2) q^{63} + ( - 8 \zeta_{10}^{3} + 8 \zeta_{10}^{2} - 8 \zeta_{10} + 8) q^{64} + (7 \zeta_{10}^{3} - \zeta_{10}^{2} + 7 \zeta_{10}) q^{65} + ( - 8 \zeta_{10}^{2} + 20 \zeta_{10} - 8) q^{66} + ( - 9 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 9 \zeta_{10}) q^{67} + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 2) q^{68} + ( - 10 \zeta_{10}^{3} + 6 \zeta_{10}^{2} - 10 \zeta_{10}) q^{69} + ( - 8 \zeta_{10}^{3} + 8 \zeta_{10}^{2} + 14) q^{70} + ( - 5 \zeta_{10}^{3} - \zeta_{10} + 1) q^{71} + (10 \zeta_{10}^{3} - 7 \zeta_{10}^{2} + 7 \zeta_{10} - 10) q^{73} + ( - 10 \zeta_{10}^{3} + 10 \zeta_{10}^{2} + 14) q^{74} + (10 \zeta_{10}^{3} - 10 \zeta_{10}^{2}) q^{75} + (8 \zeta_{10}^{3} - 8 \zeta_{10}^{2} - 4) q^{76} + (9 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10} - 9) q^{77} + (4 \zeta_{10}^{3} + 12 \zeta_{10} - 12) q^{78} + (5 \zeta_{10} - 5) q^{79} + ( - 8 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 8 \zeta_{10}) q^{80} + ( - 12 \zeta_{10}^{3} + 17 \zeta_{10}^{2} - 12 \zeta_{10}) q^{81} + ( - 10 \zeta_{10}^{3} + 10 \zeta_{10}^{2} + 14) q^{82} + ( - \zeta_{10}^{3} + 4 \zeta_{10}^{2} - \zeta_{10}) q^{83} + (4 \zeta_{10}^{2} + 4 \zeta_{10} + 4) q^{84} + (3 \zeta_{10}^{3} - 4 \zeta_{10} + 4) q^{85} + ( - 2 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 6 \zeta_{10} + 2) q^{86} + (2 \zeta_{10}^{2} - 6 \zeta_{10} + 2) q^{87} + ( - 13 \zeta_{10}^{3} + 7 \zeta_{10}^{2} - 7 \zeta_{10} + 13) q^{89} + ( - 12 \zeta_{10}^{2} + 26 \zeta_{10} - 12) q^{90} + ( - 9 \zeta_{10}^{2} - 5 \zeta_{10} - 9) q^{91} + (4 \zeta_{10}^{3} + 10 \zeta_{10}^{2} + 4 \zeta_{10}) q^{92} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2}) q^{93} + ( - 16 \zeta_{10}^{3} + 12 \zeta_{10}^{2} - 16 \zeta_{10}) q^{94} + 10 \zeta_{10}^{3} q^{95} + ( - 16 \zeta_{10} + 16) q^{96} + ( - 9 \zeta_{10}^{3} + \zeta_{10} - 1) q^{97} + (12 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10} - 12) q^{98} + ( - 11 \zeta_{10}^{3} + 11 \zeta_{10}^{2} - 11) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 6 q^{3} - 2 q^{4} - 5 q^{5} - 12 q^{6} + 8 q^{7} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} - 6 q^{3} - 2 q^{4} - 5 q^{5} - 12 q^{6} + 8 q^{7} + 7 q^{9} - 7 q^{11} + 8 q^{12} - 6 q^{13} - 14 q^{14} + 4 q^{16} - 7 q^{17} + 24 q^{18} - 10 q^{19} - 10 q^{20} - 2 q^{21} + 16 q^{22} - 11 q^{23} - 5 q^{25} + 28 q^{26} + 6 q^{28} - 5 q^{29} + 20 q^{30} - q^{31} - 32 q^{32} + 18 q^{33} + 6 q^{34} - 15 q^{35} + 14 q^{36} - 17 q^{37} + 20 q^{38} + 14 q^{39} - 17 q^{41} - 4 q^{42} + 4 q^{43} + 16 q^{44} + 25 q^{45} - 2 q^{46} - 22 q^{47} - 16 q^{48} + 8 q^{49} - 10 q^{50} + 28 q^{51} - 12 q^{52} + 24 q^{53} - 40 q^{54} + 30 q^{55} + 40 q^{57} - 10 q^{58} + 15 q^{59} + 20 q^{60} - 7 q^{61} - 2 q^{62} + 9 q^{63} + 8 q^{64} + 15 q^{65} - 4 q^{66} - 22 q^{67} + 16 q^{68} - 26 q^{69} + 40 q^{70} - 2 q^{71} - 16 q^{73} + 36 q^{74} + 20 q^{75} - 19 q^{77} - 32 q^{78} - 15 q^{79} - 20 q^{80} - 41 q^{81} + 36 q^{82} - 6 q^{83} + 16 q^{84} + 15 q^{85} + 18 q^{86} + 25 q^{89} - 10 q^{90} - 32 q^{91} - 2 q^{92} + 4 q^{93} - 44 q^{94} + 10 q^{95} + 48 q^{96} - 12 q^{97} - 44 q^{98} - 66 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(251\) \(652\)
\(\chi(n)\) \(1\) \(-\zeta_{10}^{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
156.1
−0.309017 + 0.951057i
0.809017 + 0.587785i
0.809017 0.587785i
−0.309017 0.951057i
0.618034 + 1.90211i −2.61803 + 1.90211i −1.61803 + 1.17557i −0.690983 + 2.12663i −5.23607 3.80423i −0.236068 0 2.30902 7.10642i −4.47214
311.1 −1.61803 + 1.17557i −0.381966 + 1.17557i 0.618034 1.90211i −1.80902 1.31433i −0.763932 2.35114i 4.23607 0 1.19098 + 0.865300i 4.47214
466.1 −1.61803 1.17557i −0.381966 1.17557i 0.618034 + 1.90211i −1.80902 + 1.31433i −0.763932 + 2.35114i 4.23607 0 1.19098 0.865300i 4.47214
621.1 0.618034 1.90211i −2.61803 1.90211i −1.61803 1.17557i −0.690983 2.12663i −5.23607 + 3.80423i −0.236068 0 2.30902 + 7.10642i −4.47214
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 775.2.j.a 4
25.d even 5 1 inner 775.2.j.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
775.2.j.a 4 1.a even 1 1 trivial
775.2.j.a 4 25.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 2T_{2}^{3} + 4T_{2}^{2} + 8T_{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(775, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16 \) Copy content Toggle raw display
$3$ \( T^{4} + 6 T^{3} + 16 T^{2} + 16 T + 16 \) Copy content Toggle raw display
$5$ \( T^{4} + 5 T^{3} + 15 T^{2} + 25 T + 25 \) Copy content Toggle raw display
$7$ \( (T^{2} - 4 T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 7 T^{3} + 34 T^{2} + 88 T + 121 \) Copy content Toggle raw display
$13$ \( T^{4} + 6 T^{3} + 16 T^{2} + 11 T + 121 \) Copy content Toggle raw display
$17$ \( T^{4} + 7 T^{3} + 19 T^{2} + 3 T + 1 \) Copy content Toggle raw display
$19$ \( T^{4} + 10 T^{3} + 60 T^{2} + \cdots + 400 \) Copy content Toggle raw display
$23$ \( T^{4} + 11 T^{3} + 51 T^{2} + \cdots + 961 \) Copy content Toggle raw display
$29$ \( T^{4} + 5 T^{3} + 15 T^{2} + 25 T + 25 \) Copy content Toggle raw display
$31$ \( T^{4} + T^{3} + T^{2} + T + 1 \) Copy content Toggle raw display
$37$ \( T^{4} + 17 T^{3} + 114 T^{2} + \cdots + 121 \) Copy content Toggle raw display
$41$ \( T^{4} + 17 T^{3} + 114 T^{2} + \cdots + 121 \) Copy content Toggle raw display
$43$ \( (T^{2} - 2 T - 19)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 22 T^{3} + 244 T^{2} + \cdots + 5776 \) Copy content Toggle raw display
$53$ \( T^{4} - 24 T^{3} + 306 T^{2} + \cdots + 9801 \) Copy content Toggle raw display
$59$ \( T^{4} - 15 T^{3} + 115 T^{2} + \cdots + 9025 \) Copy content Toggle raw display
$61$ \( T^{4} + 7 T^{3} + 199 T^{2} + \cdots + 361 \) Copy content Toggle raw display
$67$ \( T^{4} + 22 T^{3} + 304 T^{2} + \cdots + 10201 \) Copy content Toggle raw display
$71$ \( T^{4} + 2 T^{3} + 24 T^{2} + 133 T + 361 \) Copy content Toggle raw display
$73$ \( T^{4} + 16 T^{3} + 106 T^{2} + \cdots + 3721 \) Copy content Toggle raw display
$79$ \( T^{4} + 15 T^{3} + 100 T^{2} + \cdots + 625 \) Copy content Toggle raw display
$83$ \( T^{4} + 6 T^{3} + 16 T^{2} + 11 T + 121 \) Copy content Toggle raw display
$89$ \( T^{4} - 25 T^{3} + 235 T^{2} + \cdots + 3025 \) Copy content Toggle raw display
$97$ \( T^{4} + 12 T^{3} + 94 T^{2} + \cdots + 7921 \) Copy content Toggle raw display
show more
show less