Properties

Label 4025.2.a.m
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: 4.4.7537.1
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + 4x + 3 \) 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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + ( - \beta_{3} - 1) q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{3} + \beta_{2} + \beta_1) q^{6} - q^{7} - \beta_{3} q^{8} + (\beta_{3} - \beta_{2} + \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + ( - \beta_{3} - 1) q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{3} + \beta_{2} + \beta_1) q^{6} - q^{7} - \beta_{3} q^{8} + (\beta_{3} - \beta_{2} + \beta_1 + 1) q^{9} + ( - \beta_1 + 3) q^{11} + ( - 2 \beta_{2} - \beta_1 - 1) q^{12} + (\beta_{3} + \beta_{2} + 1) q^{13} + \beta_1 q^{14} + (\beta_{3} - \beta_{2} - 2) q^{16} + ( - \beta_1 - 3) q^{17} + ( - 2 \beta_{2} - 3) q^{18} + ( - \beta_{3} + \beta_{2} + \beta_1 + 2) q^{19} + (\beta_{3} + 1) q^{21} + (\beta_{2} - 3 \beta_1 + 3) q^{22} + q^{23} + ( - \beta_{2} + \beta_1 + 3) q^{24} + ( - 2 \beta_{3} - \beta_{2} - 2 \beta_1) q^{26} + (2 \beta_{2} - \beta_1 - 1) q^{27} + ( - \beta_{2} - 1) q^{28} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{29} + (\beta_{3} - \beta_{2} - \beta_1 + 2) q^{31} + (2 \beta_{3} - \beta_{2} + 3 \beta_1) q^{32} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 - 3) q^{33} + (\beta_{2} + 3 \beta_1 + 3) q^{34} + (2 \beta_{2} + 3 \beta_1 - 2) q^{36} + (2 \beta_{2} + 3 \beta_1 + 1) q^{37} + ( - 3 \beta_1 - 3) q^{38} + ( - \beta_{2} - 2 \beta_1 - 4) q^{39} + ( - \beta_{3} - 2 \beta_{2} - 3 \beta_1 + 6) q^{41} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{42} + ( - \beta_{3} + 4 \beta_{2} - 2 \beta_1 + 1) q^{43} + ( - \beta_{3} + 3 \beta_{2} - 2 \beta_1 + 3) q^{44} - \beta_1 q^{46} + (\beta_{3} + 3 \beta_{2} + \beta_1) q^{47} + (\beta_{3} + 3 \beta_{2} - 1) q^{48} + q^{49} + (4 \beta_{3} + \beta_{2} + \beta_1 + 3) q^{51} + (\beta_{3} + 2 \beta_{2} + \beta_1 + 4) q^{52} + ( - 2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{53} + ( - 2 \beta_{3} + \beta_{2} - \beta_1 + 3) q^{54} + \beta_{3} q^{56} + ( - 2 \beta_{3} - 4 \beta_{2} - \beta_1 + 1) q^{57} + (2 \beta_{2} - 2 \beta_1) q^{58} + ( - \beta_{3} - \beta_{2} + 2 \beta_1 + 3) q^{59} + (5 \beta_{2} - 2 \beta_1 + 2) q^{61} + ( - \beta_1 + 3) q^{62} + ( - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{63} + ( - 3 \beta_{3} - 3 \beta_{2} + \beta_1 - 5) q^{64} + (\beta_{3} + \beta_{2} + 2 \beta_1 - 3) q^{66} + ( - \beta_{3} - \beta_{2} + 1) q^{67} + ( - \beta_{3} - 3 \beta_{2} - 2 \beta_1 - 3) q^{68} + ( - \beta_{3} - 1) q^{69} + ( - 5 \beta_{3} + 3 \beta_{2} - 5 \beta_1) q^{71} + ( - 2 \beta_{3} + \beta_{2} - 3) q^{72} + ( - 2 \beta_{3} - 5 \beta_{2} - 2) q^{73} + ( - 2 \beta_{3} - 3 \beta_{2} - 3 \beta_1 - 9) q^{74} + (2 \beta_{3} + \beta_{2} + \beta_1 + 5) q^{76} + (\beta_1 - 3) q^{77} + (\beta_{3} + 2 \beta_{2} + 5 \beta_1 + 6) q^{78} + (\beta_{2} + 5 \beta_1 - 1) q^{79} + (\beta_{3} - 4 \beta_1 - 2) q^{81} + (3 \beta_{3} + 4 \beta_{2} - 4 \beta_1 + 9) q^{82} + (3 \beta_{3} - 3 \beta_{2} + 4 \beta_1 - 3) q^{83} + (2 \beta_{2} + \beta_1 + 1) q^{84} + ( - 3 \beta_{3} + 3 \beta_{2} - 5 \beta_1 + 6) q^{86} + (2 \beta_{3} - 6 \beta_{2} + 6) q^{87} + ( - 2 \beta_{3} + \beta_{2}) q^{88} + ( - \beta_1 + 9) q^{89} + ( - \beta_{3} - \beta_{2} - 1) q^{91} + (\beta_{2} + 1) q^{92} + ( - 2 \beta_{3} + 4 \beta_{2} + \beta_1 - 5) q^{93} + ( - 4 \beta_{3} - 2 \beta_{2} - 3 \beta_1 - 3) q^{94} + ( - 4 \beta_{3} + \beta_{2} - 4 \beta_1 - 6) q^{96} + (5 \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 4) q^{97} - \beta_1 q^{98} + (3 \beta_{3} - 5 \beta_{2} + 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} - 4 q^{3} + 3 q^{4} - 4 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{2} - 4 q^{3} + 3 q^{4} - 4 q^{7} + 6 q^{9} + 11 q^{11} - 3 q^{12} + 3 q^{13} + q^{14} - 7 q^{16} - 13 q^{17} - 10 q^{18} + 8 q^{19} + 4 q^{21} + 8 q^{22} + 4 q^{23} + 14 q^{24} - q^{26} - 7 q^{27} - 3 q^{28} - 2 q^{29} + 8 q^{31} + 4 q^{32} - 12 q^{33} + 14 q^{34} - 7 q^{36} + 5 q^{37} - 15 q^{38} - 17 q^{39} + 23 q^{41} - 2 q^{43} + 7 q^{44} - q^{46} - 2 q^{47} - 7 q^{48} + 4 q^{49} + 12 q^{51} + 15 q^{52} + q^{53} + 10 q^{54} + 7 q^{57} - 4 q^{58} + 15 q^{59} + q^{61} + 11 q^{62} - 6 q^{63} - 16 q^{64} - 11 q^{66} + 5 q^{67} - 11 q^{68} - 4 q^{69} - 8 q^{71} - 13 q^{72} - 3 q^{73} - 36 q^{74} + 20 q^{76} - 11 q^{77} + 27 q^{78} - 12 q^{81} + 28 q^{82} - 5 q^{83} + 3 q^{84} + 16 q^{86} + 30 q^{87} - q^{88} + 35 q^{89} - 3 q^{91} + 3 q^{92} - 23 q^{93} - 13 q^{94} - 29 q^{96} + 11 q^{97} - q^{98} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 5x^{2} + 4x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 4\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 4\beta_1 \) 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
2.15976
1.37933
−0.491918
−2.04717
−2.15976 −2.43525 2.66454 0 5.25956 −1.00000 −1.43525 2.93047 0
1.2 −1.37933 1.89307 −0.0974383 0 −2.61117 −1.00000 2.89307 0.583705 0
1.3 0.491918 −2.84864 −1.75802 0 −1.40130 −1.00000 −1.84864 5.11474 0
1.4 2.04717 −0.609175 2.19091 0 −1.24709 −1.00000 0.390825 −2.62891 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.m 4
5.b even 2 1 805.2.a.i 4
15.d odd 2 1 7245.2.a.bd 4
35.c odd 2 1 5635.2.a.u 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
805.2.a.i 4 5.b even 2 1
4025.2.a.m 4 1.a even 1 1 trivial
5635.2.a.u 4 35.c odd 2 1
7245.2.a.bd 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4025))\):

\( T_{2}^{4} + T_{2}^{3} - 5T_{2}^{2} - 4T_{2} + 3 \) Copy content Toggle raw display
\( T_{3}^{4} + 4T_{3}^{3} - T_{3}^{2} - 15T_{3} - 8 \) Copy content Toggle raw display
\( T_{11}^{4} - 11T_{11}^{3} + 40T_{11}^{2} - 55T_{11} + 24 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + T^{3} - 5 T^{2} - 4 T + 3 \) Copy content Toggle raw display
$3$ \( T^{4} + 4 T^{3} - T^{2} - 15 T - 8 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T + 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 11 T^{3} + 40 T^{2} - 55 T + 24 \) Copy content Toggle raw display
$13$ \( T^{4} - 3 T^{3} - 10 T^{2} + 23 T - 2 \) Copy content Toggle raw display
$17$ \( T^{4} + 13 T^{3} + 58 T^{2} + 101 T + 54 \) Copy content Toggle raw display
$19$ \( T^{4} - 8 T^{3} + 3 T^{2} + 81 T - 108 \) Copy content Toggle raw display
$23$ \( (T - 1)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} + 2 T^{3} - 52 T^{2} + 128 T - 48 \) Copy content Toggle raw display
$31$ \( T^{4} - 8 T^{3} + 3 T^{2} + 23 T + 8 \) Copy content Toggle raw display
$37$ \( T^{4} - 5 T^{3} - 72 T^{2} + 51 T + 526 \) Copy content Toggle raw display
$41$ \( T^{4} - 23 T^{3} + 113 T^{2} + \cdots - 4122 \) Copy content Toggle raw display
$43$ \( T^{4} + 2 T^{3} - 117 T^{2} + \cdots + 692 \) Copy content Toggle raw display
$47$ \( T^{4} + 2 T^{3} - 71 T^{2} - 209 T + 324 \) Copy content Toggle raw display
$53$ \( T^{4} - T^{3} - 62 T^{2} + 55 T + 366 \) Copy content Toggle raw display
$59$ \( T^{4} - 15 T^{3} + 50 T^{2} + \cdots - 228 \) Copy content Toggle raw display
$61$ \( T^{4} - T^{3} - 168 T^{2} + 15 T + 4882 \) Copy content Toggle raw display
$67$ \( T^{4} - 5 T^{3} - 4 T^{2} + 21 T - 4 \) Copy content Toggle raw display
$71$ \( T^{4} + 8 T^{3} - 299 T^{2} + \cdots + 13176 \) Copy content Toggle raw display
$73$ \( T^{4} + 3 T^{3} - 184 T^{2} + \cdots + 7102 \) Copy content Toggle raw display
$79$ \( T^{4} - 147 T^{2} - 121 T + 3436 \) Copy content Toggle raw display
$83$ \( T^{4} + 5 T^{3} - 170 T^{2} + \cdots + 2004 \) Copy content Toggle raw display
$89$ \( T^{4} - 35 T^{3} + 454 T^{2} + \cdots + 5466 \) Copy content Toggle raw display
$97$ \( T^{4} - 11 T^{3} - 211 T^{2} + \cdots - 13114 \) Copy content Toggle raw display
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