Properties

Label 8025.2.a.s
Level 8025
Weight 2
Character orbit 8025.a
Self dual yes
Analytic conductor 64.080
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8025 = 3 \cdot 5^{2} \cdot 107 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8025.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.0799476221\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 321)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

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

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
−0.618034
1.61803
−0.618034 1.00000 −1.61803 0 −0.618034 2.00000 2.23607 1.00000 0
1.2 1.61803 1.00000 0.618034 0 1.61803 2.00000 −2.23607 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 8025.2.a.s 2
5.b even 2 1 321.2.a.a 2
15.d odd 2 1 963.2.a.a 2
20.d odd 2 1 5136.2.a.y 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
321.2.a.a 2 5.b even 2 1
963.2.a.a 2 15.d odd 2 1
5136.2.a.y 2 20.d odd 2 1
8025.2.a.s 2 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(107\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8025))\):

\( T_{2}^{2} - T_{2} - 1 \)
\( T_{7} - 2 \)
\( T_{11}^{2} + 6 T_{11} + 4 \)
\( T_{13} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + 3 T^{2} - 2 T^{3} + 4 T^{4} \)
$3$ \( ( 1 - T )^{2} \)
$5$ 1
$7$ \( ( 1 - 2 T + 7 T^{2} )^{2} \)
$11$ \( 1 + 6 T + 26 T^{2} + 66 T^{3} + 121 T^{4} \)
$13$ \( ( 1 - T + 13 T^{2} )^{2} \)
$17$ \( 1 + 2 T + 15 T^{2} + 34 T^{3} + 289 T^{4} \)
$19$ \( 1 + 8 T + 49 T^{2} + 152 T^{3} + 361 T^{4} \)
$23$ \( 1 - 4 T + 30 T^{2} - 92 T^{3} + 529 T^{4} \)
$29$ \( 1 + 2 T + 54 T^{2} + 58 T^{3} + 841 T^{4} \)
$31$ \( ( 1 + 6 T + 31 T^{2} )^{2} \)
$37$ \( ( 1 + T + 37 T^{2} )^{2} \)
$41$ \( 1 + 2 T^{2} + 1681 T^{4} \)
$43$ \( 1 - 6 T + 90 T^{2} - 258 T^{3} + 1849 T^{4} \)
$47$ \( 1 + 6 T + 98 T^{2} + 282 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 14 T + 150 T^{2} + 742 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 8 T + 54 T^{2} - 472 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 2 T - 57 T^{2} - 122 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 10 T + 154 T^{2} + 670 T^{3} + 4489 T^{4} \)
$71$ \( 1 + 8 T + 153 T^{2} + 568 T^{3} + 5041 T^{4} \)
$73$ \( 1 + 2 T + 22 T^{2} + 146 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 4 T - 18 T^{2} + 316 T^{3} + 6241 T^{4} \)
$83$ \( 1 - 12 T + 182 T^{2} - 996 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 10 T + 198 T^{2} + 890 T^{3} + 7921 T^{4} \)
$97$ \( 1 - 2 T + 150 T^{2} - 194 T^{3} + 9409 T^{4} \)
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