Properties

Label 4025.2.a.h
Level $4025$
Weight $2$
Character orbit 4025.a
Self dual yes
Analytic conductor $32.140$
Analytic rank $0$
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: \(0\)
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 805)
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.

\(f(q)\) \(=\) \( q + ( - \beta - 1) q^{2} + (2 \beta - 1) q^{3} + 3 \beta q^{4} + ( - 3 \beta - 1) q^{6} + q^{7} + ( - 4 \beta - 1) q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 1) q^{2} + (2 \beta - 1) q^{3} + 3 \beta q^{4} + ( - 3 \beta - 1) q^{6} + q^{7} + ( - 4 \beta - 1) q^{8} + 2 q^{9} + 6 q^{11} + (3 \beta + 6) q^{12} - 3 q^{13} + ( - \beta - 1) q^{14} + (3 \beta + 5) q^{16} - 2 \beta q^{17} + ( - 2 \beta - 2) q^{18} + ( - 4 \beta + 6) q^{19} + (2 \beta - 1) q^{21} + ( - 6 \beta - 6) q^{22} - q^{23} + ( - 6 \beta - 7) q^{24} + (3 \beta + 3) q^{26} + ( - 2 \beta + 1) q^{27} + 3 \beta q^{28} + ( - 4 \beta + 5) q^{29} + (4 \beta - 3) q^{31} + ( - 3 \beta - 6) q^{32} + (12 \beta - 6) q^{33} + (4 \beta + 2) q^{34} + 6 \beta q^{36} + ( - 2 \beta + 6) q^{37} + (2 \beta - 2) q^{38} + ( - 6 \beta + 3) q^{39} + ( - 6 \beta + 3) q^{41} + ( - 3 \beta - 1) q^{42} + 6 \beta q^{43} + 18 \beta q^{44} + (\beta + 1) q^{46} + ( - 2 \beta + 3) q^{47} + (13 \beta + 1) q^{48} + q^{49} + ( - 2 \beta - 4) q^{51} - 9 \beta q^{52} + (2 \beta + 10) q^{53} + (3 \beta + 1) q^{54} + ( - 4 \beta - 1) q^{56} + (8 \beta - 14) q^{57} + (3 \beta - 1) q^{58} + (4 \beta - 4) q^{59} + (6 \beta + 2) q^{61} + ( - 5 \beta - 1) q^{62} + 2 q^{63} + (6 \beta - 1) q^{64} + ( - 18 \beta - 6) q^{66} + (2 \beta - 2) q^{67} + ( - 6 \beta - 6) q^{68} + ( - 2 \beta + 1) q^{69} + (2 \beta - 3) q^{71} + ( - 8 \beta - 2) q^{72} - 9 q^{73} + ( - 2 \beta - 4) q^{74} + (6 \beta - 12) q^{76} + 6 q^{77} + (9 \beta + 3) q^{78} + (12 \beta - 6) q^{79} - 11 q^{81} + (9 \beta + 3) q^{82} + 6 \beta q^{83} + (3 \beta + 6) q^{84} + ( - 12 \beta - 6) q^{86} + (6 \beta - 13) q^{87} + ( - 24 \beta - 6) q^{88} + (2 \beta - 8) q^{89} - 3 q^{91} - 3 \beta q^{92} + ( - 2 \beta + 11) q^{93} + (\beta - 1) q^{94} - 15 \beta q^{96} + (6 \beta + 4) q^{97} + ( - \beta - 1) q^{98} + 12 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} + 3 q^{4} - 5 q^{6} + 2 q^{7} - 6 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{2} + 3 q^{4} - 5 q^{6} + 2 q^{7} - 6 q^{8} + 4 q^{9} + 12 q^{11} + 15 q^{12} - 6 q^{13} - 3 q^{14} + 13 q^{16} - 2 q^{17} - 6 q^{18} + 8 q^{19} - 18 q^{22} - 2 q^{23} - 20 q^{24} + 9 q^{26} + 3 q^{28} + 6 q^{29} - 2 q^{31} - 15 q^{32} + 8 q^{34} + 6 q^{36} + 10 q^{37} - 2 q^{38} - 5 q^{42} + 6 q^{43} + 18 q^{44} + 3 q^{46} + 4 q^{47} + 15 q^{48} + 2 q^{49} - 10 q^{51} - 9 q^{52} + 22 q^{53} + 5 q^{54} - 6 q^{56} - 20 q^{57} + q^{58} - 4 q^{59} + 10 q^{61} - 7 q^{62} + 4 q^{63} + 4 q^{64} - 30 q^{66} - 2 q^{67} - 18 q^{68} - 4 q^{71} - 12 q^{72} - 18 q^{73} - 10 q^{74} - 18 q^{76} + 12 q^{77} + 15 q^{78} - 22 q^{81} + 15 q^{82} + 6 q^{83} + 15 q^{84} - 24 q^{86} - 20 q^{87} - 36 q^{88} - 14 q^{89} - 6 q^{91} - 3 q^{92} + 20 q^{93} - q^{94} - 15 q^{96} + 14 q^{97} - 3 q^{98} + 24 q^{99}+O(q^{100}) \) 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 2.23607 4.85410 0 −5.85410 1.00000 −7.47214 2.00000 0
1.2 −0.381966 −2.23607 −1.85410 0 0.854102 1.00000 1.47214 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.h 2
5.b even 2 1 805.2.a.e 2
15.d odd 2 1 7245.2.a.v 2
35.c odd 2 1 5635.2.a.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
805.2.a.e 2 5.b even 2 1
4025.2.a.h 2 1.a even 1 1 trivial
5635.2.a.q 2 35.c odd 2 1
7245.2.a.v 2 15.d odd 2 1

Hecke kernels

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

\( T_{2}^{2} + 3T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{2} - 5 \) Copy content Toggle raw display
\( T_{11} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 3T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} - 5 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( (T - 6)^{2} \) Copy content Toggle raw display
$13$ \( (T + 3)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$19$ \( T^{2} - 8T - 4 \) Copy content Toggle raw display
$23$ \( (T + 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 6T - 11 \) Copy content Toggle raw display
$31$ \( T^{2} + 2T - 19 \) Copy content Toggle raw display
$37$ \( T^{2} - 10T + 20 \) Copy content Toggle raw display
$41$ \( T^{2} - 45 \) Copy content Toggle raw display
$43$ \( T^{2} - 6T - 36 \) Copy content Toggle raw display
$47$ \( T^{2} - 4T - 1 \) Copy content Toggle raw display
$53$ \( T^{2} - 22T + 116 \) Copy content Toggle raw display
$59$ \( T^{2} + 4T - 16 \) Copy content Toggle raw display
$61$ \( T^{2} - 10T - 20 \) Copy content Toggle raw display
$67$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$71$ \( T^{2} + 4T - 1 \) Copy content Toggle raw display
$73$ \( (T + 9)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 180 \) Copy content Toggle raw display
$83$ \( T^{2} - 6T - 36 \) Copy content Toggle raw display
$89$ \( T^{2} + 14T + 44 \) Copy content Toggle raw display
$97$ \( T^{2} - 14T + 4 \) Copy content Toggle raw display
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