Properties

Label 775.2.a.i
Level $775$
Weight $2$
Character orbit 775.a
Self dual yes
Analytic conductor $6.188$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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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: \(5\)
Coefficient field: 5.5.205225.1
Defining polynomial: \( x^{5} - x^{4} - 6x^{3} + 3x^{2} + 7x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + \beta_{3} q^{3} + (\beta_{3} + \beta_{2} - \beta_1 + 2) q^{4} + ( - \beta_{4} - 2 \beta_{3} - 1) q^{6} + ( - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 - 2) q^{7} + ( - 2 \beta_{3} - \beta_{2} + \beta_1 - 4) q^{8} + (\beta_{4} - \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + \beta_{3} q^{3} + (\beta_{3} + \beta_{2} - \beta_1 + 2) q^{4} + ( - \beta_{4} - 2 \beta_{3} - 1) q^{6} + ( - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 - 2) q^{7} + ( - 2 \beta_{3} - \beta_{2} + \beta_1 - 4) q^{8} + (\beta_{4} - \beta_1 + 1) q^{9} + (\beta_{4} - \beta_{3} + \beta_1) q^{11} + (3 \beta_{4} + 3 \beta_{3} + 4) q^{12} + (\beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{13} + (\beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{14} + (\beta_{4} + 2 \beta_{3} - \beta_{2} - 3 \beta_1 + 5) q^{16} + ( - \beta_{4} - 2 \beta_{3} - \beta_{2} - 2 \beta_1 - 3) q^{17} + ( - \beta_{4} - 2 \beta_{3} - \beta_{2} - 5) q^{18} + ( - \beta_{4} - \beta_{3} + \beta_{2} - 1) q^{19} + ( - 2 \beta_{3} - \beta_{2} + \beta_1 - 2) q^{21} + (2 \beta_{3} + \beta_{2} - \beta_1 + 3) q^{22} + (\beta_{4} + \beta_{2} + 2 \beta_1 - 2) q^{23} + ( - 4 \beta_{4} - 5 \beta_{3} + \beta_1 - 8) q^{24} + ( - \beta_{4} - \beta_{3} + \beta_{2} + 1) q^{26} + (\beta_{2} - \beta_1 + 1) q^{27} + ( - \beta_{4} - 4 \beta_{3} + \beta_1 - 5) q^{28} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 - 2) q^{29} + q^{31} + ( - 4 \beta_{4} - 5 \beta_{3} - \beta_{2} + \beta_1 - 9) q^{32} + ( - 3 \beta_{4} + \beta_{2} - 5) q^{33} + (2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 3 \beta_1) q^{34} + (4 \beta_{3} - 3 \beta_1 + 6) q^{36} + ( - 2 \beta_{4} + 4 \beta_{2} + \beta_1 + 1) q^{37} + (3 \beta_{4} + 4 \beta_{3} + \beta_1 + 3) q^{38} + (\beta_{2} - \beta_1 + 2) q^{39} + ( - \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{41} + (\beta_{4} + 4 \beta_{3} + \beta_{2} - 3 \beta_1 + 7) q^{42} + ( - 3 \beta_{4} - \beta_{2} - 5 \beta_1 - 2) q^{43} + ( - 3 \beta_{4} - 2 \beta_{3} - \beta_{2} + 2 \beta_1 - 8) q^{44} + (2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 7) q^{46} + (4 \beta_{4} - \beta_{2} - 2 \beta_1) q^{47} + (3 \beta_{4} + 9 \beta_{3} + \beta_{2} - 4 \beta_1 + 12) q^{48} + (\beta_{4} + 3 \beta_{3} + 2 \beta_{2} + 3 \beta_1 + 1) q^{49} + ( - 2 \beta_{3} - \beta_{2} + 2 \beta_1 - 5) q^{51} + (\beta_{4} + 2 \beta_{3} - 2 \beta_{2} + \beta_1 + 1) q^{52} + (2 \beta_{4} + 2 \beta_{3} + \beta_1 - 4) q^{53} + (\beta_{4} - \beta_{2} + 2 \beta_1 - 4) q^{54} + (3 \beta_{4} + 8 \beta_{3} + 3 \beta_{2} + 11) q^{56} + (\beta_{4} - 2 \beta_{3} - \beta_{2} + 3 \beta_1 - 5) q^{57} + ( - \beta_{4} - \beta_{3} - \beta_{2} - 4 \beta_1) q^{58} + (4 \beta_{4} + \beta_{3} + \beta_{2} + 5 \beta_1 + 2) q^{59} + ( - 3 \beta_{4} - \beta_{3} - 3 \beta_{2} - 6) q^{61} + (\beta_1 - 1) q^{62} + ( - \beta_{4} + 3 \beta_{2} + 4 \beta_1 - 2) q^{63} + (6 \beta_{4} + 10 \beta_{3} + 3 \beta_{2} + 11) q^{64} + (4 \beta_{4} + 4 \beta_{3} - \beta_1 + 8) q^{66} + (2 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} + \beta_1 + 5) q^{67} + ( - 4 \beta_{4} - 7 \beta_{3} - \beta_{2} - 7) q^{68} + ( - 2 \beta_{4} - 3 \beta_{3} + \beta_{2} - 3) q^{69} + (\beta_{4} + 3 \beta_{2} + 2 \beta_1) q^{71} + ( - 2 \beta_{4} - 7 \beta_{3} - \beta_{2} + 6 \beta_1 - 9) q^{72} + (3 \beta_{4} + \beta_{3}) q^{73} + (6 \beta_{4} + 7 \beta_{3} + \beta_{2} + 7 \beta_1 + 4) q^{74} + ( - 5 \beta_{4} - 8 \beta_{3} - \beta_{2} - 5) q^{76} + (3 \beta_{3} + \beta_{2} - 2 \beta_1) q^{77} + (\beta_{4} - \beta_{2} + 3 \beta_1 - 5) q^{78} + ( - 4 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 3 \beta_1 + 1) q^{79} + ( - \beta_{4} + 2 \beta_{3} + 4 \beta_1 - 3) q^{81} + (\beta_{4} + \beta_{3} + 2 \beta_{2} + 3 \beta_1 + 5) q^{82} + (\beta_{4} - 4 \beta_{2} - 2 \beta_1) q^{83} + ( - 4 \beta_{4} - 7 \beta_{3} - \beta_{2} + 5 \beta_1 - 17) q^{84} + (2 \beta_{4} - 3 \beta_{3} - 5 \beta_{2} - 10) q^{86} + ( - 2 \beta_{4} - \beta_{3} - \beta_1 - 1) q^{87} + (4 \beta_{4} + 4 \beta_{3} - 4 \beta_1 + 13) q^{88} + ( - 5 \beta_{4} - 2 \beta_{3} - 3) q^{89} + (\beta_{4} - \beta_{3} - \beta_1 - 4) q^{91} + ( - 2 \beta_{4} - 4 \beta_{3} - 4 \beta_{2} + 5 \beta_1 - 11) q^{92} + \beta_{3} q^{93} + ( - 5 \beta_{4} - 7 \beta_{3} - 2 \beta_{2} - 5 \beta_1 - 10) q^{94} + ( - 3 \beta_{4} - 14 \beta_{3} - 4 \beta_{2} + 8 \beta_1 - 20) q^{96} + (3 \beta_{4} - 4 \beta_{3} - 6 \beta_{2} - 7) q^{97} + ( - 2 \beta_{4} - 2 \beta_{3} + 3 \beta_{2} + 2 \beta_1 + 4) q^{98} + (\beta_{4} - 5 \beta_{3} - 3 \beta_{2} + \beta_1 - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 4 q^{2} - q^{3} + 6 q^{4} - q^{6} - 6 q^{7} - 15 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 4 q^{2} - q^{3} + 6 q^{4} - q^{6} - 6 q^{7} - 15 q^{8} + 2 q^{9} + 11 q^{12} - 4 q^{13} + 2 q^{14} + 20 q^{16} - 11 q^{17} - 19 q^{18} - 4 q^{19} - 5 q^{21} + 10 q^{22} - 12 q^{23} - 26 q^{24} + 6 q^{26} + 2 q^{27} - 18 q^{28} - 6 q^{29} + 5 q^{31} - 29 q^{32} - 21 q^{33} - 5 q^{34} + 23 q^{36} + 2 q^{37} + 6 q^{38} + 7 q^{39} - 2 q^{41} + 24 q^{42} - 7 q^{43} - 28 q^{44} + 27 q^{46} - 8 q^{47} + 39 q^{48} - q^{49} - 19 q^{51} + 6 q^{52} - 25 q^{53} - 18 q^{54} + 35 q^{56} - 20 q^{57} + q^{58} + 4 q^{59} - 17 q^{61} - 4 q^{62} - 10 q^{63} + 27 q^{64} + 27 q^{66} + 13 q^{67} - 18 q^{68} - 10 q^{69} - 6 q^{71} - 26 q^{72} - 7 q^{73} + 6 q^{74} - 5 q^{76} - 7 q^{77} - 22 q^{78} + 12 q^{79} - 11 q^{81} + 21 q^{82} + 4 q^{83} - 63 q^{84} - 41 q^{86} - q^{87} + 49 q^{88} - 3 q^{89} - 22 q^{91} - 34 q^{92} - q^{93} - 34 q^{94} - 64 q^{96} - 25 q^{97} + 22 q^{98} + 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - x^{4} - 6x^{3} + 3x^{2} + 7x - 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - \nu^{2} - 4\nu + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{3} + 2\nu^{2} + 3\nu - 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - \nu^{3} - 5\nu^{2} + \nu + 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 2\beta_{2} + 5\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 6\beta_{3} + 7\beta_{2} + 9\beta _1 + 14 \) 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
−1.77799
−1.35347
0.418933
1.17073
2.54180
−2.77799 2.60920 5.71723 0 −7.24833 −3.19118 −10.3264 3.80792 0
1.2 −2.35347 −1.91723 3.53883 0 4.51216 0.845802 −3.62160 0.675783 0
1.3 −0.581067 −2.46572 −1.66236 0 1.43275 −1.67419 2.12808 3.07975 0
1.4 0.170728 0.648789 −1.97085 0 0.110767 2.03774 −0.677937 −2.57907 0
1.5 1.54180 0.124960 0.377151 0 0.192664 −4.01817 −2.50211 −2.98438 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 775.2.a.i 5
3.b odd 2 1 6975.2.a.bx 5
5.b even 2 1 775.2.a.j yes 5
5.c odd 4 2 775.2.b.h 10
15.d odd 2 1 6975.2.a.bq 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
775.2.a.i 5 1.a even 1 1 trivial
775.2.a.j yes 5 5.b even 2 1
775.2.b.h 10 5.c odd 4 2
6975.2.a.bq 5 15.d odd 2 1
6975.2.a.bx 5 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{5} + 4T_{2}^{4} - 11T_{2}^{2} - 4T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(775))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} + 4 T^{4} - 11 T^{2} - 4 T + 1 \) Copy content Toggle raw display
$3$ \( T^{5} + T^{4} - 8 T^{3} - 7 T^{2} + 9 T - 1 \) Copy content Toggle raw display
$5$ \( T^{5} \) Copy content Toggle raw display
$7$ \( T^{5} + 6 T^{4} + T^{3} - 35 T^{2} + \cdots + 37 \) Copy content Toggle raw display
$11$ \( T^{5} - 23 T^{3} - 37 T^{2} + 19 T + 37 \) Copy content Toggle raw display
$13$ \( T^{5} + 4 T^{4} - 7 T^{3} - 23 T^{2} + \cdots + 9 \) Copy content Toggle raw display
$17$ \( T^{5} + 11 T^{4} + 13 T^{3} - 117 T^{2} + \cdots - 59 \) Copy content Toggle raw display
$19$ \( T^{5} + 4 T^{4} - 23 T^{3} - 90 T^{2} + \cdots + 135 \) Copy content Toggle raw display
$23$ \( T^{5} + 12 T^{4} + 31 T^{3} + \cdots + 167 \) Copy content Toggle raw display
$29$ \( T^{5} + 6 T^{4} - 13 T^{3} - 70 T^{2} + \cdots + 135 \) Copy content Toggle raw display
$31$ \( (T - 1)^{5} \) Copy content Toggle raw display
$37$ \( T^{5} - 2 T^{4} - 170 T^{3} + \cdots + 27539 \) Copy content Toggle raw display
$41$ \( T^{5} + 2 T^{4} - 67 T^{3} - 151 T^{2} + \cdots - 27 \) Copy content Toggle raw display
$43$ \( T^{5} + 7 T^{4} - 140 T^{3} + \cdots - 11727 \) Copy content Toggle raw display
$47$ \( T^{5} + 8 T^{4} - 175 T^{3} + \cdots + 50279 \) Copy content Toggle raw display
$53$ \( T^{5} + 25 T^{4} + 198 T^{3} + \cdots - 1719 \) Copy content Toggle raw display
$59$ \( T^{5} - 4 T^{4} - 174 T^{3} + \cdots + 4595 \) Copy content Toggle raw display
$61$ \( T^{5} + 17 T^{4} - 23 T^{3} + \cdots - 20609 \) Copy content Toggle raw display
$67$ \( T^{5} - 13 T^{4} - 30 T^{3} + \cdots + 337 \) Copy content Toggle raw display
$71$ \( T^{5} + 6 T^{4} - 67 T^{3} - 142 T^{2} + \cdots - 549 \) Copy content Toggle raw display
$73$ \( T^{5} + 7 T^{4} - 59 T^{3} + \cdots + 5507 \) Copy content Toggle raw display
$79$ \( T^{5} - 12 T^{4} - 186 T^{3} + \cdots - 14985 \) Copy content Toggle raw display
$83$ \( T^{5} - 4 T^{4} - 145 T^{3} + \cdots - 22459 \) Copy content Toggle raw display
$89$ \( T^{5} + 3 T^{4} - 221 T^{3} + \cdots - 39965 \) Copy content Toggle raw display
$97$ \( T^{5} + 25 T^{4} - 143 T^{3} + \cdots + 169317 \) Copy content Toggle raw display
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