Properties

Label 9025.2.a.k
Level $9025$
Weight $2$
Character orbit 9025.a
Self dual yes
Analytic conductor $72.065$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9025 = 5^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9025.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(72.0649878242\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1805)
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.

\(f(q)\) \(=\) \( q + ( - \beta - 1) q^{2} + ( - \beta - 1) q^{3} + 3 \beta q^{4} + (3 \beta + 2) q^{6} + ( - 2 \beta + 3) q^{7} + ( - 4 \beta - 1) q^{8} + (3 \beta - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 1) q^{2} + ( - \beta - 1) q^{3} + 3 \beta q^{4} + (3 \beta + 2) q^{6} + ( - 2 \beta + 3) q^{7} + ( - 4 \beta - 1) q^{8} + (3 \beta - 1) q^{9} + ( - \beta - 3) q^{11} + ( - 6 \beta - 3) q^{12} - 5 q^{13} + (\beta - 1) q^{14} + (3 \beta + 5) q^{16} + 6 q^{17} + ( - 5 \beta - 2) q^{18} + (\beta - 1) q^{21} + (5 \beta + 4) q^{22} + \beta q^{23} + (9 \beta + 5) q^{24} + (5 \beta + 5) q^{26} + ( - 2 \beta + 1) q^{27} + (3 \beta - 6) q^{28} + ( - 3 \beta + 3) q^{29} + ( - 3 \beta + 9) q^{31} + ( - 3 \beta - 6) q^{32} + (5 \beta + 4) q^{33} + ( - 6 \beta - 6) q^{34} + (6 \beta + 9) q^{36} + (3 \beta - 3) q^{37} + (5 \beta + 5) q^{39} + (10 \beta - 5) q^{41} - \beta q^{42} + (3 \beta - 2) q^{43} + ( - 12 \beta - 3) q^{44} + ( - 2 \beta - 1) q^{46} + (2 \beta + 7) q^{47} + ( - 11 \beta - 8) q^{48} + ( - 8 \beta + 6) q^{49} + ( - 6 \beta - 6) q^{51} - 15 \beta q^{52} + ( - \beta - 4) q^{53} + (3 \beta + 1) q^{54} + ( - 2 \beta + 5) q^{56} + 3 \beta q^{58} + (5 \beta - 8) q^{59} - 3 q^{61} + ( - 3 \beta - 6) q^{62} + (5 \beta - 9) q^{63} + (6 \beta - 1) q^{64} + ( - 14 \beta - 9) q^{66} + (6 \beta - 5) q^{67} + 18 \beta q^{68} + ( - 2 \beta - 1) q^{69} + ( - 2 \beta + 7) q^{71} + ( - 11 \beta - 11) q^{72} + q^{73} - 3 \beta q^{74} + (5 \beta - 7) q^{77} + ( - 15 \beta - 10) q^{78} + 8 q^{79} + ( - 6 \beta + 4) q^{81} + ( - 15 \beta - 5) q^{82} + ( - 4 \beta + 2) q^{83} + 3 q^{84} + ( - 4 \beta - 1) q^{86} + 3 \beta q^{87} + (17 \beta + 7) q^{88} + ( - 6 \beta + 3) q^{89} + (10 \beta - 15) q^{91} + (3 \beta + 3) q^{92} + ( - 3 \beta - 6) q^{93} + ( - 11 \beta - 9) q^{94} + (12 \beta + 9) q^{96} + ( - 5 \beta + 2) q^{97} + (10 \beta + 2) q^{98} - 11 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 7 q^{6} + 4 q^{7} - 6 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 7 q^{6} + 4 q^{7} - 6 q^{8} + q^{9} - 7 q^{11} - 12 q^{12} - 10 q^{13} - q^{14} + 13 q^{16} + 12 q^{17} - 9 q^{18} - q^{21} + 13 q^{22} + q^{23} + 19 q^{24} + 15 q^{26} - 9 q^{28} + 3 q^{29} + 15 q^{31} - 15 q^{32} + 13 q^{33} - 18 q^{34} + 24 q^{36} - 3 q^{37} + 15 q^{39} - q^{42} - q^{43} - 18 q^{44} - 4 q^{46} + 16 q^{47} - 27 q^{48} + 4 q^{49} - 18 q^{51} - 15 q^{52} - 9 q^{53} + 5 q^{54} + 8 q^{56} + 3 q^{58} - 11 q^{59} - 6 q^{61} - 15 q^{62} - 13 q^{63} + 4 q^{64} - 32 q^{66} - 4 q^{67} + 18 q^{68} - 4 q^{69} + 12 q^{71} - 33 q^{72} + 2 q^{73} - 3 q^{74} - 9 q^{77} - 35 q^{78} + 16 q^{79} + 2 q^{81} - 25 q^{82} + 6 q^{84} - 6 q^{86} + 3 q^{87} + 31 q^{88} - 20 q^{91} + 9 q^{92} - 15 q^{93} - 29 q^{94} + 30 q^{96} - q^{97} + 14 q^{98} - 11 q^{99}+O(q^{100}) \) 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
1.61803
−0.618034
−2.61803 −2.61803 4.85410 0 6.85410 −0.236068 −7.47214 3.85410 0
1.2 −0.381966 −0.381966 −1.85410 0 0.145898 4.23607 1.47214 −2.85410 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(19\) \(-1\)

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 9025.2.a.k 2
5.b even 2 1 1805.2.a.e yes 2
19.b odd 2 1 9025.2.a.v 2
95.d odd 2 1 1805.2.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1805.2.a.c 2 95.d odd 2 1
1805.2.a.e yes 2 5.b even 2 1
9025.2.a.k 2 1.a even 1 1 trivial
9025.2.a.v 2 19.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9025))\):

\( T_{2}^{2} + 3T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{2} + 3T_{3} + 1 \) Copy content Toggle raw display
\( T_{7}^{2} - 4T_{7} - 1 \) Copy content Toggle raw display
\( T_{11}^{2} + 7T_{11} + 11 \) Copy content Toggle raw display
\( T_{29}^{2} - 3T_{29} - 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 3T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + 3T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 4T - 1 \) Copy content Toggle raw display
$11$ \( T^{2} + 7T + 11 \) Copy content Toggle raw display
$13$ \( (T + 5)^{2} \) Copy content Toggle raw display
$17$ \( (T - 6)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$29$ \( T^{2} - 3T - 9 \) Copy content Toggle raw display
$31$ \( T^{2} - 15T + 45 \) Copy content Toggle raw display
$37$ \( T^{2} + 3T - 9 \) Copy content Toggle raw display
$41$ \( T^{2} - 125 \) Copy content Toggle raw display
$43$ \( T^{2} + T - 11 \) Copy content Toggle raw display
$47$ \( T^{2} - 16T + 59 \) Copy content Toggle raw display
$53$ \( T^{2} + 9T + 19 \) Copy content Toggle raw display
$59$ \( T^{2} + 11T - 1 \) Copy content Toggle raw display
$61$ \( (T + 3)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 4T - 41 \) Copy content Toggle raw display
$71$ \( T^{2} - 12T + 31 \) Copy content Toggle raw display
$73$ \( (T - 1)^{2} \) Copy content Toggle raw display
$79$ \( (T - 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 20 \) Copy content Toggle raw display
$89$ \( T^{2} - 45 \) Copy content Toggle raw display
$97$ \( T^{2} + T - 31 \) Copy content Toggle raw display
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