Properties

Label 4025.2.a.j
Level 4025
Weight 2
Character orbit 4025.a
Self dual yes
Analytic conductor 32.140
Analytic rank 0
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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 3 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 2\)

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
0.311108
−1.48119
2.17009
−1.90321 −0.688892 1.62222 0 1.31111 1.00000 0.719004 −2.52543 0
1.2 0.193937 −2.48119 −1.96239 0 −0.481194 1.00000 −0.768452 3.15633 0
1.3 2.70928 1.17009 5.34017 0 3.17009 1.00000 9.04945 −1.63090 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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.j 3
5.b even 2 1 161.2.a.c 3
15.d odd 2 1 1449.2.a.m 3
20.d odd 2 1 2576.2.a.v 3
35.c odd 2 1 1127.2.a.f 3
115.c odd 2 1 3703.2.a.c 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
161.2.a.c 3 5.b even 2 1
1127.2.a.f 3 35.c odd 2 1
1449.2.a.m 3 15.d odd 2 1
2576.2.a.v 3 20.d odd 2 1
3703.2.a.c 3 115.c odd 2 1
4025.2.a.j 3 1.a even 1 1 trivial

Atkin-Lehner signs

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

Hecke kernels

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

\( T_{2}^{3} - T_{2}^{2} - 5 T_{2} + 1 \)
\( T_{3}^{3} + 2 T_{3}^{2} - 2 T_{3} - 2 \)
\( T_{11}^{3} - 4 T_{11}^{2} + 4 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} - 3 T^{3} + 2 T^{4} - 4 T^{5} + 8 T^{6} \)
$3$ \( 1 + 2 T + 7 T^{2} + 10 T^{3} + 21 T^{4} + 18 T^{5} + 27 T^{6} \)
$5$ \( \)
$7$ \( ( 1 - T )^{3} \)
$11$ \( 1 - 4 T + 33 T^{2} - 84 T^{3} + 363 T^{4} - 484 T^{5} + 1331 T^{6} \)
$13$ \( 1 + 2 T + 27 T^{2} + 44 T^{3} + 351 T^{4} + 338 T^{5} + 2197 T^{6} \)
$17$ \( 1 + 4 T + 53 T^{2} + 134 T^{3} + 901 T^{4} + 1156 T^{5} + 4913 T^{6} \)
$19$ \( 1 - 8 T + 41 T^{2} - 144 T^{3} + 779 T^{4} - 2888 T^{5} + 6859 T^{6} \)
$23$ \( ( 1 + T )^{3} \)
$29$ \( 1 + 2 T + 79 T^{2} + 120 T^{3} + 2291 T^{4} + 1682 T^{5} + 24389 T^{6} \)
$31$ \( 1 - 16 T + 119 T^{2} - 654 T^{3} + 3689 T^{4} - 15376 T^{5} + 29791 T^{6} \)
$37$ \( 1 - 6 T + 107 T^{2} - 404 T^{3} + 3959 T^{4} - 8214 T^{5} + 50653 T^{6} \)
$41$ \( 1 + 14 T + 135 T^{2} + 996 T^{3} + 5535 T^{4} + 23534 T^{5} + 68921 T^{6} \)
$43$ \( 1 - 8 T + 89 T^{2} - 384 T^{3} + 3827 T^{4} - 14792 T^{5} + 79507 T^{6} \)
$47$ \( 1 + 16 T + 203 T^{2} + 1514 T^{3} + 9541 T^{4} + 35344 T^{5} + 103823 T^{6} \)
$53$ \( 1 + 6 T + 107 T^{2} + 388 T^{3} + 5671 T^{4} + 16854 T^{5} + 148877 T^{6} \)
$59$ \( 1 + 10 T + 75 T^{2} + 210 T^{3} + 4425 T^{4} + 34810 T^{5} + 205379 T^{6} \)
$61$ \( 1 - 10 T + 137 T^{2} - 726 T^{3} + 8357 T^{4} - 37210 T^{5} + 226981 T^{6} \)
$67$ \( 1 + 16 T + 193 T^{2} + 1468 T^{3} + 12931 T^{4} + 71824 T^{5} + 300763 T^{6} \)
$71$ \( 1 - 4 T + 133 T^{2} - 504 T^{3} + 9443 T^{4} - 20164 T^{5} + 357911 T^{6} \)
$73$ \( 1 - 6 T + 111 T^{2} - 660 T^{3} + 8103 T^{4} - 31974 T^{5} + 389017 T^{6} \)
$79$ \( 1 - 16 T + 245 T^{2} - 2100 T^{3} + 19355 T^{4} - 99856 T^{5} + 493039 T^{6} \)
$83$ \( 1 + 20 T + 145 T^{2} + 648 T^{3} + 12035 T^{4} + 137780 T^{5} + 571787 T^{6} \)
$89$ \( 1 - 4 T + 153 T^{2} - 990 T^{3} + 13617 T^{4} - 31684 T^{5} + 704969 T^{6} \)
$97$ \( 1 + 6 T + 269 T^{2} + 1166 T^{3} + 26093 T^{4} + 56454 T^{5} + 912673 T^{6} \)
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