Properties

Label 4025.2.a.i
Level $4025$
Weight $2$
Character orbit 4025.a
Self dual yes
Analytic conductor $32.140$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4025 = 5^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4025.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.1397868136\)
Analytic rank: \(1\)
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 161)
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.

\(f(q)\) \(=\) \( q + \beta q^{2} + q^{3} + (\beta - 1) q^{4} + \beta q^{6} + q^{7} + ( - 2 \beta + 1) q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + q^{3} + (\beta - 1) q^{4} + \beta q^{6} + q^{7} + ( - 2 \beta + 1) q^{8} - 2 q^{9} + ( - 4 \beta + 2) q^{11} + (\beta - 1) q^{12} + (2 \beta + 1) q^{13} + \beta q^{14} - 3 \beta q^{16} - 2 \beta q^{18} + (2 \beta - 6) q^{19} + q^{21} + ( - 2 \beta - 4) q^{22} + q^{23} + ( - 2 \beta + 1) q^{24} + (3 \beta + 2) q^{26} - 5 q^{27} + (\beta - 1) q^{28} + (4 \beta + 1) q^{29} - 9 q^{31} + (\beta - 5) q^{32} + ( - 4 \beta + 2) q^{33} + ( - 2 \beta + 2) q^{36} + ( - 6 \beta + 2) q^{37} + ( - 4 \beta + 2) q^{38} + (2 \beta + 1) q^{39} + (2 \beta - 1) q^{41} + \beta q^{42} + 4 \beta q^{43} + (2 \beta - 6) q^{44} + \beta q^{46} + ( - 4 \beta + 1) q^{47} - 3 \beta q^{48} + q^{49} + (\beta + 1) q^{52} + (2 \beta - 10) q^{53} - 5 \beta q^{54} + ( - 2 \beta + 1) q^{56} + (2 \beta - 6) q^{57} + (5 \beta + 4) q^{58} + ( - 4 \beta - 4) q^{59} + ( - 12 \beta + 6) q^{61} - 9 \beta q^{62} - 2 q^{63} + (2 \beta + 1) q^{64} + ( - 2 \beta - 4) q^{66} + ( - 10 \beta + 6) q^{67} + q^{69} + (2 \beta - 9) q^{71} + (4 \beta - 2) q^{72} + ( - 6 \beta + 3) q^{73} + ( - 4 \beta - 6) q^{74} + ( - 6 \beta + 8) q^{76} + ( - 4 \beta + 2) q^{77} + (3 \beta + 2) q^{78} + (2 \beta - 6) q^{79} + q^{81} + (\beta + 2) q^{82} + (4 \beta - 4) q^{83} + (\beta - 1) q^{84} + (4 \beta + 4) q^{86} + (4 \beta + 1) q^{87} + 10 q^{88} + ( - 8 \beta + 4) q^{89} + (2 \beta + 1) q^{91} + (\beta - 1) q^{92} - 9 q^{93} + ( - 3 \beta - 4) q^{94} + (\beta - 5) q^{96} + 6 \beta q^{97} + \beta q^{98} + (8 \beta - 4) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 2 q^{3} - q^{4} + q^{6} + 2 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 2 q^{3} - q^{4} + q^{6} + 2 q^{7} - 4 q^{9} - q^{12} + 4 q^{13} + q^{14} - 3 q^{16} - 2 q^{18} - 10 q^{19} + 2 q^{21} - 10 q^{22} + 2 q^{23} + 7 q^{26} - 10 q^{27} - q^{28} + 6 q^{29} - 18 q^{31} - 9 q^{32} + 2 q^{36} - 2 q^{37} + 4 q^{39} + q^{42} + 4 q^{43} - 10 q^{44} + q^{46} - 2 q^{47} - 3 q^{48} + 2 q^{49} + 3 q^{52} - 18 q^{53} - 5 q^{54} - 10 q^{57} + 13 q^{58} - 12 q^{59} - 9 q^{62} - 4 q^{63} + 4 q^{64} - 10 q^{66} + 2 q^{67} + 2 q^{69} - 16 q^{71} - 16 q^{74} + 10 q^{76} + 7 q^{78} - 10 q^{79} + 2 q^{81} + 5 q^{82} - 4 q^{83} - q^{84} + 12 q^{86} + 6 q^{87} + 20 q^{88} + 4 q^{91} - q^{92} - 18 q^{93} - 11 q^{94} - 9 q^{96} + 6 q^{97} + q^{98}+O(q^{100}) \) 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
−0.618034
1.61803
−0.618034 1.00000 −1.61803 0 −0.618034 1.00000 2.23607 −2.00000 0
1.2 1.61803 1.00000 0.618034 0 1.61803 1.00000 −2.23607 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(7\) \(-1\)
\(23\) \(-1\)

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 4025.2.a.i 2
5.b even 2 1 161.2.a.b 2
15.d odd 2 1 1449.2.a.i 2
20.d odd 2 1 2576.2.a.s 2
35.c odd 2 1 1127.2.a.d 2
115.c odd 2 1 3703.2.a.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
161.2.a.b 2 5.b even 2 1
1127.2.a.d 2 35.c odd 2 1
1449.2.a.i 2 15.d odd 2 1
2576.2.a.s 2 20.d odd 2 1
3703.2.a.b 2 115.c odd 2 1
4025.2.a.i 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4025))\):

\( T_{2}^{2} - T_{2} - 1 \) Copy content Toggle raw display
\( T_{3} - 1 \) Copy content Toggle raw display
\( T_{11}^{2} - 20 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 20 \) Copy content Toggle raw display
$13$ \( T^{2} - 4T - 1 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 10T + 20 \) Copy content Toggle raw display
$23$ \( (T - 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 6T - 11 \) Copy content Toggle raw display
$31$ \( (T + 9)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 2T - 44 \) Copy content Toggle raw display
$41$ \( T^{2} - 5 \) Copy content Toggle raw display
$43$ \( T^{2} - 4T - 16 \) Copy content Toggle raw display
$47$ \( T^{2} + 2T - 19 \) Copy content Toggle raw display
$53$ \( T^{2} + 18T + 76 \) Copy content Toggle raw display
$59$ \( T^{2} + 12T + 16 \) Copy content Toggle raw display
$61$ \( T^{2} - 180 \) Copy content Toggle raw display
$67$ \( T^{2} - 2T - 124 \) Copy content Toggle raw display
$71$ \( T^{2} + 16T + 59 \) Copy content Toggle raw display
$73$ \( T^{2} - 45 \) Copy content Toggle raw display
$79$ \( T^{2} + 10T + 20 \) Copy content Toggle raw display
$83$ \( T^{2} + 4T - 16 \) Copy content Toggle raw display
$89$ \( T^{2} - 80 \) Copy content Toggle raw display
$97$ \( T^{2} - 6T - 36 \) Copy content Toggle raw display
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