Properties

Label 4025.2.a.k
Level $4025$
Weight $2$
Character orbit 4025.a
Self dual yes
Analytic conductor $32.140$
Analytic rank $1$
Dimension $3$
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: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) 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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of \(\nu = \zeta_{14} + \zeta_{14}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) 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
1.80194
0.445042
−1.24698
−0.801938 −1.69202 −1.35690 0 1.35690 1.00000 2.69202 −0.137063 0
1.2 0.554958 3.04892 −1.69202 0 1.69202 1.00000 −2.04892 6.29590 0
1.3 2.24698 −1.35690 3.04892 0 −3.04892 1.00000 2.35690 −1.15883 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.k 3
5.b even 2 1 805.2.a.f 3
15.d odd 2 1 7245.2.a.ba 3
35.c odd 2 1 5635.2.a.r 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
805.2.a.f 3 5.b even 2 1
4025.2.a.k 3 1.a even 1 1 trivial
5635.2.a.r 3 35.c odd 2 1
7245.2.a.ba 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4025))\):

\( T_{2}^{3} - 2T_{2}^{2} - T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{3} - 7T_{3} - 7 \) Copy content Toggle raw display
\( T_{11}^{3} + 7T_{11}^{2} + 14T_{11} + 7 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 2T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{3} - 7T - 7 \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( (T - 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} + 7 T^{2} + 14 T + 7 \) Copy content Toggle raw display
$13$ \( T^{3} + 5 T^{2} - 22 T - 97 \) Copy content Toggle raw display
$17$ \( T^{3} + 5 T^{2} - 8 T - 41 \) Copy content Toggle raw display
$19$ \( T^{3} + 6 T^{2} + 5 T - 13 \) Copy content Toggle raw display
$23$ \( (T - 1)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} + 12 T^{2} + 20 T - 104 \) Copy content Toggle raw display
$31$ \( T^{3} + 4 T^{2} + 3 T - 1 \) Copy content Toggle raw display
$37$ \( T^{3} + 3 T^{2} - 18 T - 27 \) Copy content Toggle raw display
$41$ \( T^{3} - T^{2} - 65 T + 169 \) Copy content Toggle raw display
$43$ \( T^{3} + 2T^{2} - T - 1 \) Copy content Toggle raw display
$47$ \( T^{3} + 12 T^{2} - T - 41 \) Copy content Toggle raw display
$53$ \( T^{3} - 9 T^{2} - 148 T + 1373 \) Copy content Toggle raw display
$59$ \( T^{3} + 21 T^{2} + 126 T + 203 \) Copy content Toggle raw display
$61$ \( T^{3} + 9 T^{2} - 22 T - 211 \) Copy content Toggle raw display
$67$ \( T^{3} + 13 T^{2} - 16 T - 377 \) Copy content Toggle raw display
$71$ \( T^{3} + 16 T^{2} - 15 T - 463 \) Copy content Toggle raw display
$73$ \( T^{3} - 9 T^{2} - 22 T + 29 \) Copy content Toggle raw display
$79$ \( T^{3} + 18 T^{2} + 45 T - 351 \) Copy content Toggle raw display
$83$ \( T^{3} - 29 T^{2} + 194 T + 211 \) Copy content Toggle raw display
$89$ \( T^{3} - 7 T^{2} - 14 T + 91 \) Copy content Toggle raw display
$97$ \( T^{3} + T^{2} - 149 T + 83 \) Copy content Toggle raw display
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