Properties

Label 4025.2.a.o
Level $4025$
Weight $2$
Character orbit 4025.a
Self dual yes
Analytic conductor $32.140$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.2777.1
Defining polynomial: \( x^{4} - x^{3} - 4x^{2} + x + 2 \) 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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + 1) q^{2} + (\beta_1 + 1) q^{3} + ( - \beta_{3} - \beta_{2} + 1) q^{4} + ( - \beta_{3} - \beta_{2} + 1) q^{6} - q^{7} + ( - 2 \beta_{2} + \beta_1 + 2) q^{8} + (\beta_{2} + 3 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + 1) q^{2} + (\beta_1 + 1) q^{3} + ( - \beta_{3} - \beta_{2} + 1) q^{4} + ( - \beta_{3} - \beta_{2} + 1) q^{6} - q^{7} + ( - 2 \beta_{2} + \beta_1 + 2) q^{8} + (\beta_{2} + 3 \beta_1) q^{9} + ( - \beta_{3} + 2 \beta_1 - 2) q^{11} + ( - 2 \beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{12} + (2 \beta_{3} - \beta_{2} + 1) q^{13} + (\beta_{3} - 1) q^{14} + ( - 2 \beta_{3} - \beta_{2} + 2 \beta_1 + 2) q^{16} + ( - \beta_{3} - 2 \beta_{2} + 2) q^{17} + (\beta_{3} - 2 \beta_{2} - \beta_1 - 1) q^{18} + (3 \beta_{3} - 3 \beta_{2} - 1) q^{19} + ( - \beta_1 - 1) q^{21} + (2 \beta_{3} - 3 \beta_{2}) q^{22} + q^{23} + ( - 2 \beta_{3} - \beta_{2} + 2 \beta_1 + 6) q^{24} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 - 2) q^{26} + (\beta_{3} + 4 \beta_{2} + 4 \beta_1 + 2) q^{27} + (\beta_{3} + \beta_{2} - 1) q^{28} + ( - \beta_{3} + \beta_1) q^{29} + ( - \beta_{3} + \beta_{2} - 4 \beta_1 + 1) q^{31} + ( - 3 \beta_{3} - \beta_{2} - \beta_1 + 3) q^{32} + ( - \beta_{3} + \beta_{2} + \beta_1 + 2) q^{33} + ( - 4 \beta_{3} - 3 \beta_{2} + 2 \beta_1 + 6) q^{34} + ( - \beta_{3} - 2 \beta_{2} - 4 \beta_1 - 1) q^{36} + (2 \beta_{2} - \beta_1 + 2) q^{37} + ( - 2 \beta_{3} + 3 \beta_1 - 4) q^{38} + (\beta_{3} + \beta_{2} + 2 \beta_1 + 2) q^{39} + ( - 3 \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{41} + (\beta_{3} + \beta_{2} - 1) q^{42} + ( - \beta_{3} + 4 \beta_{2} + 2 \beta_1 + 1) q^{43} + ( - \beta_{3} - \beta_{2} - \beta_1 + 3) q^{44} + ( - \beta_{3} + 1) q^{46} + (\beta_{2} + \beta_1 + 5) q^{47} + ( - 3 \beta_{3} - \beta_{2} + 3 \beta_1 + 7) q^{48} + q^{49} + ( - 3 \beta_{3} - 3 \beta_{2} - \beta_1 + 4) q^{51} + ( - \beta_{3} - \beta_1 - 1) q^{52} + ( - 4 \beta_{3} + 5 \beta_{2} + 3 \beta_1 + 4) q^{53} + (2 \beta_{3} + \beta_{2} - 4 \beta_1 - 4) q^{54} + (2 \beta_{2} - \beta_1 - 2) q^{56} + ( - \beta_1 + 2) q^{57} + ( - 2 \beta_{2} + 2) q^{58} + ( - \beta_{3} + 3 \beta_{2} + \beta_1 - 9) q^{59} + ( - \beta_{2} + \beta_1) q^{61} + (4 \beta_{2} - \beta_1 + 2) q^{62} + ( - \beta_{2} - 3 \beta_1) q^{63} + ( - \beta_{2} - 3 \beta_1 + 6) q^{64} + ( - \beta_{3} - \beta_{2} - \beta_1 + 3) q^{66} + ( - \beta_{3} - \beta_{2} - 3 \beta_1 + 9) q^{67} + ( - 7 \beta_{3} - 5 \beta_{2} + 3 \beta_1 + 13) q^{68} + (\beta_1 + 1) q^{69} + (\beta_{3} - \beta_{2} + 2 \beta_1 - 3) q^{71} + ( - 3 \beta_{3} + 5 \beta_{2} + 4 \beta_1 + 5) q^{72} + (4 \beta_{3} - 3 \beta_{2} - \beta_1 - 8) q^{73} + (3 \beta_{2} - 2 \beta_1) q^{74} + ( - 2 \beta_{3} + \beta_{2} + 2) q^{76} + (\beta_{3} - 2 \beta_1 + 2) q^{77} + ( - \beta_{3} - \beta_1 - 1) q^{78} + (5 \beta_{3} + \beta_{2} + 3 \beta_1 - 8) q^{79} + (5 \beta_{3} + 6 \beta_{2} + 6 \beta_1 + 6) q^{81} + (3 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 3) q^{82} + ( - 3 \beta_{3} - \beta_{2} + \beta_1 + 3) q^{83} + (2 \beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{84} + (3 \beta_{3} + \beta_{2} - 4 \beta_1 - 1) q^{86} + ( - \beta_{3} + \beta_1 + 2) q^{87} + ( - 8 \beta_{3} + 5 \beta_{2} + \beta_1 + 6) q^{88} + (4 \beta_{3} + 2 \beta_{2} - \beta_1) q^{89} + ( - 2 \beta_{3} + \beta_{2} - 1) q^{91} + ( - \beta_{3} - \beta_{2} + 1) q^{92} + ( - 4 \beta_{2} - 7 \beta_1 - 8) q^{93} + ( - 4 \beta_{3} - \beta_1 + 4) q^{94} + ( - 4 \beta_{3} - 5 \beta_{2} - 3 \beta_1 + 2) q^{96} + (4 \beta_{2} + 6 \beta_1 - 1) q^{97} + ( - \beta_{3} + 1) q^{98} + (3 \beta_{3} + \beta_{2} - 2 \beta_1 + 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 5 q^{3} + 2 q^{4} + 2 q^{6} - 4 q^{7} + 9 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} + 5 q^{3} + 2 q^{4} + 2 q^{6} - 4 q^{7} + 9 q^{8} + 3 q^{9} - 8 q^{11} + 3 q^{12} + 8 q^{13} - 2 q^{14} + 6 q^{16} + 6 q^{17} - 3 q^{18} + 2 q^{19} - 5 q^{21} + 4 q^{22} + 4 q^{23} + 22 q^{24} - 11 q^{26} + 14 q^{27} - 2 q^{28} - q^{29} - 2 q^{31} + 5 q^{32} + 7 q^{33} + 18 q^{34} - 10 q^{36} + 7 q^{37} - 17 q^{38} + 12 q^{39} - 9 q^{41} - 2 q^{42} + 4 q^{43} + 9 q^{44} + 2 q^{46} + 21 q^{47} + 25 q^{48} + 4 q^{49} + 9 q^{51} - 7 q^{52} + 11 q^{53} - 16 q^{54} - 9 q^{56} + 7 q^{57} + 8 q^{58} - 37 q^{59} + q^{61} + 7 q^{62} - 3 q^{63} + 21 q^{64} + 9 q^{66} + 31 q^{67} + 41 q^{68} + 5 q^{69} - 8 q^{71} + 18 q^{72} - 25 q^{73} - 2 q^{74} + 4 q^{76} + 8 q^{77} - 7 q^{78} - 19 q^{79} + 40 q^{81} + 16 q^{82} + 7 q^{83} - 3 q^{84} - 2 q^{86} + 7 q^{87} + 9 q^{88} + 7 q^{89} - 8 q^{91} + 2 q^{92} - 39 q^{93} + 7 q^{94} - 3 q^{96} + 2 q^{97} + 2 q^{98} + 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 4x^{2} + x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 3\nu + 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 4\beta _1 + 1 \) 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
−0.679643
2.36234
−1.50848
0.825785
−1.26308 0.320357 −0.404635 0 −0.404635 −1.00000 3.03724 −2.89737 0
1.2 −0.515722 3.36234 −1.73403 0 −1.73403 −1.00000 1.92572 8.30533 0
1.3 1.18264 −0.508481 −0.601352 0 −0.601352 −1.00000 −3.07647 −2.74145 0
1.4 2.59615 1.82578 4.74002 0 4.74002 −1.00000 7.11351 0.333489 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.o 4
5.b even 2 1 805.2.a.g 4
15.d odd 2 1 7245.2.a.bf 4
35.c odd 2 1 5635.2.a.s 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
805.2.a.g 4 5.b even 2 1
4025.2.a.o 4 1.a even 1 1 trivial
5635.2.a.s 4 35.c odd 2 1
7245.2.a.bf 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4025))\):

\( T_{2}^{4} - 2T_{2}^{3} - 3T_{2}^{2} + 3T_{2} + 2 \) Copy content Toggle raw display
\( T_{3}^{4} - 5T_{3}^{3} + 5T_{3}^{2} + 2T_{3} - 1 \) Copy content Toggle raw display
\( T_{11}^{4} + 8T_{11}^{3} + 3T_{11}^{2} - 51T_{11} + 41 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2 T^{3} - 3 T^{2} + 3 T + 2 \) Copy content Toggle raw display
$3$ \( T^{4} - 5 T^{3} + 5 T^{2} + 2 T - 1 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T + 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 8 T^{3} + 3 T^{2} - 51 T + 41 \) Copy content Toggle raw display
$13$ \( T^{4} - 8 T^{3} + 7 T^{2} + 21 T + 1 \) Copy content Toggle raw display
$17$ \( T^{4} - 6 T^{3} - 17 T^{2} + 13 T + 31 \) Copy content Toggle raw display
$19$ \( T^{4} - 2 T^{3} - 57 T^{2} + 31 T + 4 \) Copy content Toggle raw display
$23$ \( (T - 1)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} + T^{3} - 8 T^{2} - 4 T + 8 \) Copy content Toggle raw display
$31$ \( T^{4} + 2 T^{3} - 81 T^{2} - 177 T + 134 \) Copy content Toggle raw display
$37$ \( T^{4} - 7 T^{3} - 8 T^{2} + 59 T - 44 \) Copy content Toggle raw display
$41$ \( T^{4} + 9 T^{3} - 13 T^{2} - 9 T + 4 \) Copy content Toggle raw display
$43$ \( T^{4} - 4 T^{3} - 75 T^{2} + \cdots + 1262 \) Copy content Toggle raw display
$47$ \( T^{4} - 21 T^{3} + 157 T^{2} + \cdots + 577 \) Copy content Toggle raw display
$53$ \( T^{4} - 11 T^{3} - 110 T^{2} + \cdots - 2474 \) Copy content Toggle raw display
$59$ \( T^{4} + 37 T^{3} + 472 T^{2} + \cdots + 4226 \) Copy content Toggle raw display
$61$ \( T^{4} - T^{3} - 10 T^{2} + 13 T - 2 \) Copy content Toggle raw display
$67$ \( T^{4} - 31 T^{3} + 310 T^{2} + \cdots - 968 \) Copy content Toggle raw display
$71$ \( T^{4} + 8 T^{3} - 3 T^{2} - 27 T - 16 \) Copy content Toggle raw display
$73$ \( T^{4} + 25 T^{3} + 154 T^{2} + \cdots - 3782 \) Copy content Toggle raw display
$79$ \( T^{4} + 19 T^{3} - 41 T^{2} + \cdots + 601 \) Copy content Toggle raw display
$83$ \( T^{4} - 7 T^{3} - 40 T^{2} - 5 T + 4 \) Copy content Toggle raw display
$89$ \( T^{4} - 7 T^{3} - 102 T^{2} + \cdots - 2458 \) Copy content Toggle raw display
$97$ \( T^{4} - 2 T^{3} - 212 T^{2} + \cdots - 2069 \) Copy content Toggle raw display
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