Properties

Label 775.2.a.g
Level $775$
Weight $2$
Character orbit 775.a
Self dual yes
Analytic conductor $6.188$
Analytic rank $0$
Dimension $4$
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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.20308.1
Defining polynomial: \( x^{4} - x^{3} - 8x^{2} + 4x + 12 \) 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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{3} q^{3} + (\beta_{2} + \beta_1 + 2) q^{4} + ( - \beta_{2} + 2) q^{6} - \beta_{2} q^{7} + (2 \beta_{3} + \beta_{2} + 3 \beta_1 + 2) q^{8} + ( - 2 \beta_{2} - \beta_1 + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{3} q^{3} + (\beta_{2} + \beta_1 + 2) q^{4} + ( - \beta_{2} + 2) q^{6} - \beta_{2} q^{7} + (2 \beta_{3} + \beta_{2} + 3 \beta_1 + 2) q^{8} + ( - 2 \beta_{2} - \beta_1 + 2) q^{9} + (\beta_{2} + 2 \beta_1 - 2) q^{11} + 2 q^{12} + (\beta_{2} - 4) q^{13} + ( - 2 \beta_{3} - 2 \beta_1 + 2) q^{14} + (2 \beta_{3} + 3 \beta_{2} + 5 \beta_1 + 2) q^{16} + ( - \beta_{3} - \beta_{2} - 2 \beta_1) q^{17} + ( - 4 \beta_{3} - \beta_{2} - 3 \beta_1) q^{18} + ( - 2 \beta_{3} - \beta_1 + 1) q^{19} + ( - 2 \beta_{3} - \beta_{2}) q^{21} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 6) q^{22} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{23} + (2 \beta_{2} + 2 \beta_1 - 4) q^{24} + (2 \beta_{3} - 2 \beta_1 - 2) q^{26} + ( - 3 \beta_{3} - \beta_{2} - 2) q^{27} + ( - 2 \beta_{2} - 4) q^{28} + (2 \beta_{3} + \beta_{2} + 2) q^{29} + q^{31} + (2 \beta_{3} + 5 \beta_{2} + 7 \beta_1 + 6) q^{32} + (4 \beta_{3} - \beta_{2} + 4) q^{33} + ( - 2 \beta_{3} - 3 \beta_{2} - 4 \beta_1 - 4) q^{34} + ( - 2 \beta_{3} - 3 \beta_{2} - 3 \beta_1 - 6) q^{36} + (\beta_{3} + \beta_{2} - 2) q^{37} - 3 \beta_{2} q^{38} + (6 \beta_{3} + \beta_{2}) q^{39} + ( - 2 \beta_{3} + \beta_{2} - \beta_1 + 3) q^{41} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 6) q^{42} + (\beta_{3} + \beta_{2} + 2 \beta_1 - 2) q^{43} + (4 \beta_{3} + 2 \beta_{2} + 8 \beta_1 + 4) q^{44} + (2 \beta_{3} - 4 \beta_{2} - 6) q^{46} + (2 \beta_{3} + 4) q^{47} + (4 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{48} + ( - 2 \beta_{3} - 2 \beta_1 - 1) q^{49} + ( - 2 \beta_{3} - \beta_{2} - \beta_1 + 1) q^{51} + ( - 2 \beta_{2} - 4 \beta_1 - 4) q^{52} + (3 \beta_{3} - 2 \beta_{2} - 2) q^{53} + ( - 2 \beta_{3} - 3 \beta_{2} - 4 \beta_1 + 8) q^{54} - 4 \beta_1 q^{56} + ( - \beta_{3} - 3 \beta_{2} - 2 \beta_1 + 8) q^{57} + (2 \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 6) q^{58} + ( - 4 \beta_{3} - \beta_{2} - 3 \beta_1 + 3) q^{59} + (2 \beta_{3} + 6) q^{61} + \beta_1 q^{62} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 10) q^{63} + (6 \beta_{3} + 3 \beta_{2} + 13 \beta_1 + 10) q^{64} + ( - 2 \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 6) q^{66} + (\beta_{2} + 2 \beta_1 + 2) q^{67} + ( - 4 \beta_{3} - 4 \beta_{2} - 10 \beta_1 - 6) q^{68} + (2 \beta_{3} - \beta_{2} - 2 \beta_1 + 6) q^{69} + ( - 2 \beta_{2} - \beta_1 + 1) q^{71} + (2 \beta_{3} - 3 \beta_{2} - 9 \beta_1 - 2) q^{72} + ( - 3 \beta_{3} - 4 \beta_1 - 2) q^{73} + (2 \beta_{3} + \beta_{2} - 4) q^{74} + ( - 2 \beta_{3} - 4 \beta_1 + 4) q^{76} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 2) q^{77} + (2 \beta_{3} + 6 \beta_{2} + 2 \beta_1 - 14) q^{78} + ( - 3 \beta_{2} - 4) q^{79} + ( - \beta_{2} + 9) q^{81} + (2 \beta_{3} - 3 \beta_{2} + 4 \beta_1 - 2) q^{82} + ( - \beta_{3} + 3 \beta_{2} + 4) q^{83} - 2 \beta_{2} q^{84} + (2 \beta_{3} + 3 \beta_{2} + 2 \beta_1 + 4) q^{86} + (5 \beta_{2} + 2 \beta_1 - 10) q^{87} + (8 \beta_{2} + 12 \beta_1 + 8) q^{88} + (\beta_{2} - 4 \beta_1 - 2) q^{89} + (2 \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 6) q^{91} + ( - 4 \beta_{3} - 10 \beta_1 + 4) q^{92} - \beta_{3} q^{93} + (2 \beta_{2} + 4 \beta_1 - 4) q^{94} + (4 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 4) q^{96} + (2 \beta_{3} + 2 \beta_1 - 4) q^{97} + ( - 4 \beta_{2} - 3 \beta_1 - 4) q^{98} + ( - 6 \beta_{3} + 4 \beta_{2} - 2 \beta_1 - 14) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} + q^{3} + 9 q^{4} + 8 q^{6} + 9 q^{8} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} + q^{3} + 9 q^{4} + 8 q^{6} + 9 q^{8} + 7 q^{9} - 6 q^{11} + 8 q^{12} - 16 q^{13} + 8 q^{14} + 11 q^{16} - q^{17} + q^{18} + 5 q^{19} + 2 q^{21} + 24 q^{22} - 14 q^{24} - 12 q^{26} - 5 q^{27} - 16 q^{28} + 6 q^{29} + 4 q^{31} + 29 q^{32} + 12 q^{33} - 18 q^{34} - 25 q^{36} - 9 q^{37} - 6 q^{39} + 13 q^{41} + 24 q^{42} - 7 q^{43} + 20 q^{44} - 26 q^{46} + 14 q^{47} - 2 q^{48} - 4 q^{49} + 5 q^{51} - 20 q^{52} - 11 q^{53} + 30 q^{54} - 4 q^{56} + 31 q^{57} - 22 q^{58} + 13 q^{59} + 22 q^{61} + q^{62} + 40 q^{63} + 47 q^{64} - 20 q^{66} + 10 q^{67} - 30 q^{68} + 20 q^{69} + 3 q^{71} - 19 q^{72} - 9 q^{73} - 18 q^{74} + 14 q^{76} - 8 q^{77} - 56 q^{78} - 16 q^{79} + 36 q^{81} - 6 q^{82} + 17 q^{83} + 16 q^{86} - 38 q^{87} + 44 q^{88} - 12 q^{89} - 24 q^{91} + 10 q^{92} + q^{93} - 12 q^{94} + 14 q^{96} - 16 q^{97} - 19 q^{98} - 52 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 8x^{2} + 4x + 12 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} - \nu^{2} - 6\nu + 2 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} + \beta_{2} + 7\beta _1 + 2 \) 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
−2.27244
−1.15729
1.62946
2.80027
−2.27244 0.632112 3.16400 0 −1.43644 −3.43644 −2.64511 −2.60043 0
1.2 −1.15729 −3.02722 −0.660672 0 3.50338 1.50338 3.07918 6.16405 0
1.3 1.62946 3.05273 0.655151 0 4.97431 2.97431 −2.19138 6.31916 0
1.4 2.80027 0.342376 5.84153 0 0.958747 −1.04125 10.7573 −2.88278 0
\(n\): e.g. 2-40 or 990-1000
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.g 4
3.b odd 2 1 6975.2.a.bj 4
5.b even 2 1 155.2.a.d 4
5.c odd 4 2 775.2.b.e 8
15.d odd 2 1 1395.2.a.m 4
20.d odd 2 1 2480.2.a.z 4
35.c odd 2 1 7595.2.a.q 4
40.e odd 2 1 9920.2.a.cd 4
40.f even 2 1 9920.2.a.ch 4
155.c odd 2 1 4805.2.a.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
155.2.a.d 4 5.b even 2 1
775.2.a.g 4 1.a even 1 1 trivial
775.2.b.e 8 5.c odd 4 2
1395.2.a.m 4 15.d odd 2 1
2480.2.a.z 4 20.d odd 2 1
4805.2.a.j 4 155.c odd 2 1
6975.2.a.bj 4 3.b odd 2 1
7595.2.a.q 4 35.c odd 2 1
9920.2.a.cd 4 40.e odd 2 1
9920.2.a.ch 4 40.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - T_{2}^{3} - 8T_{2}^{2} + 4T_{2} + 12 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(775))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{3} - 8 T^{2} + 4 T + 12 \) Copy content Toggle raw display
$3$ \( T^{4} - T^{3} - 9 T^{2} + 9 T - 2 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 12 T^{2} + 4 T + 16 \) Copy content Toggle raw display
$11$ \( T^{4} + 6 T^{3} - 16 T^{2} - 124 T - 144 \) Copy content Toggle raw display
$13$ \( T^{4} + 16 T^{3} + 84 T^{2} + 156 T + 64 \) Copy content Toggle raw display
$17$ \( T^{4} + T^{3} - 25 T^{2} + 49 T - 24 \) Copy content Toggle raw display
$19$ \( T^{4} - 5 T^{3} - 21 T^{2} + 81 T + 108 \) Copy content Toggle raw display
$23$ \( T^{4} - 64 T^{2} - 196 T - 24 \) Copy content Toggle raw display
$29$ \( T^{4} - 6 T^{3} - 40 T^{2} + 308 T - 456 \) Copy content Toggle raw display
$31$ \( (T - 1)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} + 9 T^{3} + 7 T^{2} - 7 T - 4 \) Copy content Toggle raw display
$41$ \( T^{4} - 13 T^{3} + 17 T^{2} + \cdots - 294 \) Copy content Toggle raw display
$43$ \( T^{4} + 7 T^{3} - 7 T^{2} - 129 T - 214 \) Copy content Toggle raw display
$47$ \( T^{4} - 14 T^{3} + 36 T^{2} + \cdots - 192 \) Copy content Toggle raw display
$53$ \( T^{4} + 11 T^{3} - 75 T^{2} + \cdots - 2892 \) Copy content Toggle raw display
$59$ \( T^{4} - 13 T^{3} - 65 T^{2} + \cdots + 2484 \) Copy content Toggle raw display
$61$ \( T^{4} - 22 T^{3} + 144 T^{2} + \cdots - 32 \) Copy content Toggle raw display
$67$ \( T^{4} - 10 T^{3} + 8 T^{2} + 36 T - 32 \) Copy content Toggle raw display
$71$ \( T^{4} - 3 T^{3} - 37 T^{2} + 59 T + 384 \) Copy content Toggle raw display
$73$ \( T^{4} + 9 T^{3} - 95 T^{2} - 649 T + 452 \) Copy content Toggle raw display
$79$ \( T^{4} + 16 T^{3} - 12 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$83$ \( T^{4} - 17 T^{3} - 3 T^{2} + 455 T + 738 \) Copy content Toggle raw display
$89$ \( T^{4} + 12 T^{3} - 124 T^{2} + \cdots + 1656 \) Copy content Toggle raw display
$97$ \( T^{4} + 16 T^{3} + 56 T^{2} - 48 T - 16 \) Copy content Toggle raw display
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