Properties

Label 5265.2.a.ba
Level $5265$
Weight $2$
Character orbit 5265.a
Self dual yes
Analytic conductor $42.041$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5265 = 3^{4} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5265.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(42.0412366642\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - 3x^{7} - 8x^{6} + 31x^{5} - x^{4} - 70x^{3} + 66x^{2} - 19x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 585)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + q^{5} + ( - \beta_{7} + \beta_{4} + \beta_1 - 1) q^{7} + ( - \beta_{3} - \beta_1 + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + q^{5} + ( - \beta_{7} + \beta_{4} + \beta_1 - 1) q^{7} + ( - \beta_{3} - \beta_1 + 1) q^{8} - \beta_1 q^{10} + ( - \beta_{7} + \beta_{6} + \beta_{5} - 1) q^{11} - q^{13} + (\beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{14} + ( - \beta_{6} - \beta_{5} - \beta_{4} - \beta_1 + 2) q^{16} + (\beta_{7} - 2 \beta_{5} + \beta_{4} + \beta_{3} + \beta_1 + 1) q^{17} + (\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 - 2) q^{19} + (\beta_{2} + 1) q^{20} + (2 \beta_{7} + \beta_{6} - \beta_{5} + 2 \beta_{3} - \beta_{2} + \beta_1) q^{22} + ( - \beta_{6} + 2 \beta_{5} - \beta_{4} - 2) q^{23} + q^{25} + \beta_1 q^{26} + (3 \beta_{5} - \beta_{3} - 2 \beta_{2} + \beta_1 - 6) q^{28} + ( - \beta_{6} + \beta_{4} + 2 \beta_1 - 2) q^{29} + (2 \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{4} - \beta_{2} - 2 \beta_1 - 5) q^{31} + ( - \beta_{7} - \beta_{6} + \beta_{4} - \beta_{3} + 2 \beta_{2} + \beta_1) q^{32} + ( - 3 \beta_{7} - \beta_{6} + 4 \beta_{5} - \beta_{3} - 2 \beta_1 - 1) q^{34} + ( - \beta_{7} + \beta_{4} + \beta_1 - 1) q^{35} + (\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{37} + ( - 2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + \beta_{2} + 2 \beta_1 - 2) q^{38} + ( - \beta_{3} - \beta_1 + 1) q^{40} + ( - 2 \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{41} + (2 \beta_{5} - 3 \beta_{4} + \beta_{3} - \beta_{2}) q^{43} + ( - \beta_{7} - \beta_{6} + 4 \beta_{5} + 2 \beta_{4} + \beta_{3} - 2 \beta_{2} - \beta_1 - 2) q^{44} + (2 \beta_{7} + 2 \beta_{6} - 3 \beta_{5} + \beta_{4} - 2 \beta_{2} + 3 \beta_1 - 4) q^{46} + (2 \beta_{7} - 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 3) q^{47} + (3 \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - 2 \beta_1 + 2) q^{49} - \beta_1 q^{50} + ( - \beta_{2} - 1) q^{52} + (2 \beta_{7} + \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + \beta_{2} + 2 \beta_1 - 2) q^{53} + ( - \beta_{7} + \beta_{6} + \beta_{5} - 1) q^{55} + (\beta_{7} + 2 \beta_{6} - 2 \beta_{5} + \beta_{4} + 3 \beta_{3} - \beta_{2} + 6 \beta_1 - 3) q^{56} + ( - \beta_{5} - \beta_{4} - 2 \beta_{2} + 3 \beta_1 - 6) q^{58} + ( - 2 \beta_{7} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 - 2) q^{59} + ( - \beta_{6} - 4 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} + \beta_{2} + 2) q^{61} + (2 \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + 6 \beta_1 + 1) q^{62} + (\beta_{7} + \beta_{6} - \beta_{5} - 2 \beta_{3} - 4) q^{64} - q^{65} + ( - \beta_{7} + \beta_{6} - 2 \beta_{5} - 3 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + \beta_1 - 3) q^{67} + (5 \beta_{7} + 3 \beta_{6} - 5 \beta_{5} - 3 \beta_{4} + \beta_{3} - \beta_{2} + 3 \beta_1) q^{68} + (\beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{70} + ( - 2 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{71} + ( - 3 \beta_{6} + \beta_{5} - 2 \beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{73} + (\beta_{7} + 2 \beta_{6} + 3 \beta_{3} + \beta_1 + 4) q^{74} + (2 \beta_{7} - 4 \beta_{5} + 2 \beta_{4} + \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 5) q^{76} + (4 \beta_{7} - 2 \beta_{6} - \beta_{5} - 3 \beta_{4} - 2 \beta_{3} + \beta_{2} + 2) q^{77} + ( - 2 \beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - 6 \beta_1 + 3) q^{79} + ( - \beta_{6} - \beta_{5} - \beta_{4} - \beta_1 + 2) q^{80} + (\beta_{7} - 4 \beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 - 3) q^{82} + (2 \beta_{6} + \beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_1) q^{83} + (\beta_{7} - 2 \beta_{5} + \beta_{4} + \beta_{3} + \beta_1 + 1) q^{85} + (2 \beta_{7} + 3 \beta_{6} - \beta_{5} + 4 \beta_{4} - 3 \beta_{2} + 2 \beta_1 - 7) q^{86} + (\beta_{7} + 3 \beta_{6} - 3 \beta_{5} - \beta_{4} + 3 \beta_{3} - 2 \beta_{2} + 6 \beta_1 - 3) q^{88} + (3 \beta_{7} + \beta_{6} - \beta_{5} - 2 \beta_{4} - \beta_{3} - 3 \beta_{2} + \beta_1) q^{89} + (\beta_{7} - \beta_{4} - \beta_1 + 1) q^{91} + ( - 5 \beta_{7} - \beta_{6} + 3 \beta_{5} + \beta_{4} + 2 \beta_{3} + 4 \beta_1 - 1) q^{92} + ( - 4 \beta_{7} + 6 \beta_{5} + 4 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - \beta_1 + 2) q^{94} + (\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 - 2) q^{95} + (\beta_{7} + 3 \beta_{6} + \beta_{4} + 2 \beta_{2} + 3 \beta_1 - 6) q^{97} + ( - 4 \beta_{7} - \beta_{6} + 3 \beta_{5} + \beta_{4} - 3 \beta_{3} + 3 \beta_{2} - 4 \beta_1 + 5) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 3 q^{2} + 9 q^{4} + 8 q^{5} - 11 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 3 q^{2} + 9 q^{4} + 8 q^{5} - 11 q^{7} + 6 q^{8} - 3 q^{10} - 6 q^{11} - 8 q^{13} - 10 q^{14} + 11 q^{16} + 2 q^{17} - 10 q^{19} + 9 q^{20} + 3 q^{22} - 6 q^{23} + 8 q^{25} + 3 q^{26} - 34 q^{28} - 14 q^{29} - 31 q^{31} - q^{32} - 7 q^{34} - 11 q^{35} + q^{37} - 9 q^{38} + 6 q^{40} + 12 q^{41} + 15 q^{43} - 16 q^{44} - 32 q^{46} + 18 q^{47} + 17 q^{49} - 3 q^{50} - 9 q^{52} - 2 q^{53} - 6 q^{55} - 16 q^{56} - 42 q^{58} - 24 q^{59} - 9 q^{61} + 20 q^{62} - 30 q^{64} - 8 q^{65} - 18 q^{67} + 14 q^{68} - 10 q^{70} - 10 q^{71} + 6 q^{73} + 37 q^{74} - 53 q^{76} + 34 q^{77} - 3 q^{79} + 11 q^{80} - 34 q^{82} + 10 q^{83} + 2 q^{85} - 60 q^{86} - 14 q^{88} + 13 q^{89} + 11 q^{91} - 5 q^{92} + 17 q^{94} - 10 q^{95} - 34 q^{97} + 30 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} - 8x^{6} + 31x^{5} - x^{4} - 70x^{3} + 66x^{2} - 19x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu + 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{7} - 2\nu^{6} - 10\nu^{5} + 21\nu^{4} + 20\nu^{3} - 51\nu^{2} + 15\nu + 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{7} - 3\nu^{6} - 9\nu^{5} + 31\nu^{4} + 9\nu^{3} - 73\nu^{2} + 43\nu - 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -2\nu^{7} + 5\nu^{6} + 19\nu^{5} - 53\nu^{4} - 29\nu^{3} + 130\nu^{2} - 59\nu + 1 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 3\nu^{7} - 7\nu^{6} - 28\nu^{5} + 74\nu^{4} + 40\nu^{3} - 179\nu^{2} + 92\nu - 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{6} - \beta_{5} - \beta_{4} + 6\beta_{2} - \beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{7} + \beta_{6} - \beta_{4} + 9\beta_{3} - 2\beta_{2} + 27\beta _1 - 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{7} - 9\beta_{6} - 11\beta_{5} - 10\beta_{4} - 2\beta_{3} + 36\beta_{2} - 10\beta _1 + 92 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 12\beta_{7} + 13\beta_{6} - \beta_{5} - 8\beta_{4} + 66\beta_{3} - 23\beta_{2} + 156\beta _1 - 60 \) 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
2.44829
2.25662
1.63404
0.655066
0.494096
0.0672022
−1.96921
−2.58610
−2.44829 0 3.99411 1.00000 0 −3.93452 −4.88214 0 −2.44829
1.2 −2.25662 0 3.09235 1.00000 0 0.706571 −2.46502 0 −2.25662
1.3 −1.63404 0 0.670078 1.00000 0 2.13012 2.17314 0 −1.63404
1.4 −0.655066 0 −1.57089 1.00000 0 −0.777184 2.33917 0 −0.655066
1.5 −0.494096 0 −1.75587 1.00000 0 −4.28539 1.85576 0 −0.494096
1.6 −0.0672022 0 −1.99548 1.00000 0 2.46357 0.268505 0 −0.0672022
1.7 1.96921 0 1.87778 1.00000 0 −3.02828 −0.240686 0 1.96921
1.8 2.58610 0 4.68793 1.00000 0 −4.27489 6.95128 0 2.58610
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(13\) \(1\)

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 5265.2.a.ba 8
3.b odd 2 1 5265.2.a.bf 8
9.c even 3 2 1755.2.i.f 16
9.d odd 6 2 585.2.i.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.i.e 16 9.d odd 6 2
1755.2.i.f 16 9.c even 3 2
5265.2.a.ba 8 1.a even 1 1 trivial
5265.2.a.bf 8 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5265))\):

\( T_{2}^{8} + 3T_{2}^{7} - 8T_{2}^{6} - 31T_{2}^{5} - T_{2}^{4} + 70T_{2}^{3} + 66T_{2}^{2} + 19T_{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{8} + 11T_{7}^{7} + 24T_{7}^{6} - 106T_{7}^{5} - 385T_{7}^{4} + 232T_{7}^{3} + 1360T_{7}^{2} - 30T_{7} - 629 \) Copy content Toggle raw display
\( T_{11}^{8} + 6T_{11}^{7} - 38T_{11}^{6} - 250T_{11}^{5} + 413T_{11}^{4} + 3220T_{11}^{3} - 1002T_{11}^{2} - 11999T_{11} + 634 \) Copy content Toggle raw display
\( T_{17}^{8} - 2T_{17}^{7} - 78T_{17}^{6} + 73T_{17}^{5} + 1532T_{17}^{4} - 748T_{17}^{3} - 5408T_{17}^{2} - 1275T_{17} + 892 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 3 T^{7} - 8 T^{6} - 31 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T - 1)^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 11 T^{7} + 24 T^{6} + \cdots - 629 \) Copy content Toggle raw display
$11$ \( T^{8} + 6 T^{7} - 38 T^{6} - 250 T^{5} + \cdots + 634 \) Copy content Toggle raw display
$13$ \( (T + 1)^{8} \) Copy content Toggle raw display
$17$ \( T^{8} - 2 T^{7} - 78 T^{6} + 73 T^{5} + \cdots + 892 \) Copy content Toggle raw display
$19$ \( T^{8} + 10 T^{7} - 37 T^{6} + \cdots - 1584 \) Copy content Toggle raw display
$23$ \( T^{8} + 6 T^{7} - 79 T^{6} + \cdots + 12015 \) Copy content Toggle raw display
$29$ \( T^{8} + 14 T^{7} + 5 T^{6} + \cdots - 1115 \) Copy content Toggle raw display
$31$ \( T^{8} + 31 T^{7} + 276 T^{6} + \cdots + 50472 \) Copy content Toggle raw display
$37$ \( T^{8} - T^{7} - 120 T^{6} + \cdots - 33660 \) Copy content Toggle raw display
$41$ \( T^{8} - 12 T^{7} - 82 T^{6} + \cdots - 19275 \) Copy content Toggle raw display
$43$ \( T^{8} - 15 T^{7} - 167 T^{6} + \cdots - 7900376 \) Copy content Toggle raw display
$47$ \( T^{8} - 18 T^{7} - 86 T^{6} + \cdots + 651523 \) Copy content Toggle raw display
$53$ \( T^{8} + 2 T^{7} - 282 T^{6} + \cdots + 111438 \) Copy content Toggle raw display
$59$ \( T^{8} + 24 T^{7} + 60 T^{6} + \cdots + 1877418 \) Copy content Toggle raw display
$61$ \( T^{8} + 9 T^{7} - 376 T^{6} + \cdots + 10312831 \) Copy content Toggle raw display
$67$ \( T^{8} + 18 T^{7} - 346 T^{6} + \cdots - 31217363 \) Copy content Toggle raw display
$71$ \( T^{8} + 10 T^{7} - 211 T^{6} + \cdots - 127950 \) Copy content Toggle raw display
$73$ \( T^{8} - 6 T^{7} - 300 T^{6} + \cdots + 1196532 \) Copy content Toggle raw display
$79$ \( T^{8} + 3 T^{7} - 498 T^{6} + \cdots + 131224698 \) Copy content Toggle raw display
$83$ \( T^{8} - 10 T^{7} - 115 T^{6} + \cdots + 685 \) Copy content Toggle raw display
$89$ \( T^{8} - 13 T^{7} - 228 T^{6} + \cdots + 1032219 \) Copy content Toggle raw display
$97$ \( T^{8} + 34 T^{7} + 162 T^{6} + \cdots - 11067218 \) Copy content Toggle raw display
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