Properties

Label 4025.2.a.p
Level $4025$
Weight $2$
Character orbit 4025.a
Self dual yes
Analytic conductor $32.140$
Analytic rank $0$
Dimension $5$
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: \(5\)
Coefficient field: 5.5.2147108.1
Defining polynomial: \( x^{5} - 2x^{4} - 9x^{3} + 17x^{2} + 16x - 27 \) Copy content Toggle raw display
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_{3} q^{3} + (\beta_{2} + 2) q^{4} + (\beta_{4} - \beta_{2}) q^{6} - q^{7} + ( - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1) q^{8} + ( - \beta_{2} - \beta_1 + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - \beta_{3} q^{3} + (\beta_{2} + 2) q^{4} + (\beta_{4} - \beta_{2}) q^{6} - q^{7} + ( - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1) q^{8} + ( - \beta_{2} - \beta_1 + 3) q^{9} + ( - \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 - 1) q^{11} + ( - \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{12} + ( - \beta_{4} - \beta_{3} + 1) q^{13} + \beta_1 q^{14} + (\beta_{4} + \beta_{3} + 3 \beta_{2} + 1) q^{16} + ( - \beta_{4} - 2 \beta_{2} + 3) q^{17} + (\beta_{4} - \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 4) q^{18} + ( - 2 \beta_1 + 2) q^{19} + \beta_{3} q^{21} + (3 \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 3) q^{22} + q^{23} + ( - \beta_{3} + 2 \beta_1 - 8) q^{24} + (2 \beta_{4} + 2 \beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{26} + (\beta_{4} - \beta_{3} - 2 \beta_1 + 1) q^{27} + ( - \beta_{2} - 2) q^{28} + ( - 3 \beta_{2} + \beta_1) q^{29} + ( - \beta_{3} + 6) q^{31} + ( - 3 \beta_{4} - \beta_{3} - 2 \beta_{2} - \beta_1 - 1) q^{32} + ( - 2 \beta_{4} + 4) q^{33} + (3 \beta_{4} + 4 \beta_{2} - 2 \beta_1 + 1) q^{34} + ( - 2 \beta_{4} - \beta_{2} - 3 \beta_1 + 1) q^{36} + ( - 2 \beta_{3} - 2 \beta_1) q^{37} + (2 \beta_{2} - 2 \beta_1 + 8) q^{38} + ( - \beta_{4} - \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 3) q^{39} + ( - \beta_{4} + \beta_{3} + 1) q^{41} + ( - \beta_{4} + \beta_{2}) q^{42} + ( - 2 \beta_{4} + 2) q^{43} + ( - 3 \beta_{4} - 3 \beta_{3} - 5 \beta_{2} + 3 \beta_1 - 5) q^{44} - \beta_1 q^{46} + (3 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{47} + (\beta_{4} + 2 \beta_{3} - \beta_{2} + 4 \beta_1 - 6) q^{48} + q^{49} + ( - \beta_{4} - 5 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{51} + ( - 3 \beta_{4} - \beta_{3} - \beta_{2} + 4) q^{52} + (2 \beta_{2} - 4) q^{53} + ( - 2 \beta_{3} - \beta_{2} + 7) q^{54} + (\beta_{4} - \beta_{3} + \beta_{2} + \beta_1) q^{56} + (2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2}) q^{57} + (3 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} + 3 \beta_1 - 4) q^{58} + ( - \beta_{4} - 2 \beta_1 + 5) q^{59} + ( - \beta_{4} - 2 \beta_{3} - 2 \beta_1 - 3) q^{61} + (\beta_{4} - \beta_{2} - 6 \beta_1) q^{62} + (\beta_{2} + \beta_1 - 3) q^{63} + (4 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 5) q^{64} + (2 \beta_{4} + 4 \beta_{3} + 4 \beta_{2} - 6 \beta_1 + 2) q^{66} + (\beta_{4} + \beta_{3} - \beta_{2} - 5 \beta_1 + 3) q^{67} + ( - 5 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1 - 1) q^{68} - \beta_{3} q^{69} + (\beta_{4} + 3 \beta_{3} + 1) q^{71} + (\beta_{4} + 5 \beta_{3} + 4 \beta_{2} + 2 \beta_1 + 6) q^{72} + (\beta_{4} + \beta_{3} + 4 \beta_{2} - 1) q^{73} + (2 \beta_{4} + 8) q^{74} + ( - 2 \beta_{4} + 2 \beta_{3} - 6 \beta_1 + 4) q^{76} + (\beta_{4} + \beta_{3} + \beta_{2} - \beta_1 + 1) q^{77} + (4 \beta_{4} + \beta_{2} - 2 \beta_1 - 7) q^{78} + (\beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 7) q^{79} + (3 \beta_{4} - \beta_{3} + \beta_{2} - \beta_1) q^{81} + (2 \beta_{3} + 3 \beta_{2} - 2 \beta_1 + 1) q^{82} + ( - 2 \beta_{4} - 2) q^{83} + (\beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{84} + (2 \beta_{4} + 4 \beta_{3} + 4 \beta_{2} - 4 \beta_1 + 2) q^{86} + ( - \beta_{4} - 3 \beta_{3} + 4 \beta_{2} - 6 \beta_1 + 3) q^{87} + (5 \beta_{4} - \beta_{3} + 3 \beta_{2} + 5 \beta_1 - 3) q^{88} + ( - \beta_{4} - 2 \beta_{3} + 2 \beta_1 - 5) q^{89} + (\beta_{4} + \beta_{3} - 1) q^{91} + (\beta_{2} + 2) q^{92} + ( - 6 \beta_{3} - \beta_{2} - \beta_1 + 6) q^{93} + ( - \beta_{4} - 2 \beta_{3} + 3 \beta_{2} + 4 \beta_1 - 8) q^{94} + ( - 2 \beta_{4} - \beta_{3} - 3 \beta_{2} + 4 \beta_1 - 1) q^{96} + (3 \beta_{4} + 2 \beta_{2} - 2 \beta_1 + 3) q^{97} - \beta_1 q^{98} + (\beta_{4} - \beta_{3} + \beta_{2} + 3 \beta_1 - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 2 q^{2} + 12 q^{4} - 3 q^{6} - 5 q^{7} - 3 q^{8} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 2 q^{2} + 12 q^{4} - 3 q^{6} - 5 q^{7} - 3 q^{8} + 11 q^{9} - 4 q^{11} - 3 q^{12} + 6 q^{13} + 2 q^{14} + 10 q^{16} + 12 q^{17} + 19 q^{18} + 6 q^{19} - 14 q^{22} + 5 q^{23} - 36 q^{24} + q^{26} - 12 q^{28} - 4 q^{29} + 30 q^{31} - 8 q^{32} + 22 q^{33} + 6 q^{34} - q^{36} - 4 q^{37} + 40 q^{38} + 16 q^{39} + 6 q^{41} + 3 q^{42} + 12 q^{43} - 26 q^{44} - 2 q^{46} - 10 q^{47} - 25 q^{48} + 5 q^{49} - 4 q^{51} + 21 q^{52} - 16 q^{53} + 33 q^{54} + 3 q^{56} - 6 q^{57} - 13 q^{58} + 22 q^{59} - 18 q^{61} - 15 q^{62} - 11 q^{63} + 25 q^{64} + 4 q^{66} + 2 q^{67} - 12 q^{68} + 4 q^{71} + 41 q^{72} + 2 q^{73} + 38 q^{74} + 10 q^{76} + 4 q^{77} - 41 q^{78} + 30 q^{79} - 3 q^{81} + 7 q^{82} - 8 q^{83} + 3 q^{84} + 8 q^{86} + 12 q^{87} - 4 q^{88} - 20 q^{89} - 6 q^{91} + 12 q^{92} + 26 q^{93} - 25 q^{94} - q^{96} + 12 q^{97} - 2 q^{98} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 2x^{4} - 9x^{3} + 17x^{2} + 16x - 27 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} - \nu^{3} - 8\nu^{2} + 5\nu + 11 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} + \nu^{3} - 10\nu^{2} - 5\nu + 19 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} - \beta_{3} + \beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + \beta_{3} + 9\beta_{2} + 21 \) 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
2.69017
2.11948
1.23828
−1.50216
−2.54577
−2.69017 0.269842 5.23702 0 −0.725921 −1.00000 −8.70812 −2.92719 0
1.2 −2.11948 1.84074 2.49221 0 −3.90141 −1.00000 −1.04322 0.388311 0
1.3 −1.23828 −2.68857 −0.466664 0 3.32920 −1.00000 3.05442 4.22838 0
1.4 1.50216 3.04067 0.256481 0 4.56757 −1.00000 −2.61904 6.24568 0
1.5 2.54577 −2.46268 4.48096 0 −6.26943 −1.00000 6.31597 3.06481 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.p 5
5.b even 2 1 161.2.a.d 5
15.d odd 2 1 1449.2.a.r 5
20.d odd 2 1 2576.2.a.bd 5
35.c odd 2 1 1127.2.a.h 5
115.c odd 2 1 3703.2.a.j 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
161.2.a.d 5 5.b even 2 1
1127.2.a.h 5 35.c odd 2 1
1449.2.a.r 5 15.d odd 2 1
2576.2.a.bd 5 20.d odd 2 1
3703.2.a.j 5 115.c odd 2 1
4025.2.a.p 5 1.a even 1 1 trivial

Hecke kernels

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

\( T_{2}^{5} + 2T_{2}^{4} - 9T_{2}^{3} - 17T_{2}^{2} + 16T_{2} + 27 \) Copy content Toggle raw display
\( T_{3}^{5} - 13T_{3}^{3} + 38T_{3} - 10 \) Copy content Toggle raw display
\( T_{11}^{5} + 4T_{11}^{4} - 28T_{11}^{3} - 148T_{11}^{2} - 160T_{11} - 48 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} + 2 T^{4} - 9 T^{3} - 17 T^{2} + \cdots + 27 \) Copy content Toggle raw display
$3$ \( T^{5} - 13 T^{3} + 38 T - 10 \) Copy content Toggle raw display
$5$ \( T^{5} \) Copy content Toggle raw display
$7$ \( (T + 1)^{5} \) Copy content Toggle raw display
$11$ \( T^{5} + 4 T^{4} - 28 T^{3} - 148 T^{2} + \cdots - 48 \) Copy content Toggle raw display
$13$ \( T^{5} - 6 T^{4} - 9 T^{3} + 46 T^{2} + \cdots - 56 \) Copy content Toggle raw display
$17$ \( T^{5} - 12 T^{4} + 6 T^{3} + \cdots + 1536 \) Copy content Toggle raw display
$19$ \( T^{5} - 6 T^{4} - 28 T^{3} + 96 T^{2} + \cdots + 128 \) Copy content Toggle raw display
$23$ \( (T - 1)^{5} \) Copy content Toggle raw display
$29$ \( T^{5} + 4 T^{4} - 111 T^{3} + \cdots - 1452 \) Copy content Toggle raw display
$31$ \( T^{5} - 30 T^{4} + 347 T^{3} + \cdots - 5206 \) Copy content Toggle raw display
$37$ \( T^{5} + 4 T^{4} - 76 T^{3} - 376 T^{2} + \cdots + 32 \) Copy content Toggle raw display
$41$ \( T^{5} - 6 T^{4} - 29 T^{3} + 146 T^{2} + \cdots - 456 \) Copy content Toggle raw display
$43$ \( T^{5} - 12 T^{4} - 24 T^{3} + \cdots - 256 \) Copy content Toggle raw display
$47$ \( T^{5} + 10 T^{4} - 125 T^{3} + \cdots - 11142 \) Copy content Toggle raw display
$53$ \( T^{5} + 16 T^{4} + 52 T^{3} + \cdots + 480 \) Copy content Toggle raw display
$59$ \( T^{5} - 22 T^{4} + 118 T^{3} + \cdots + 1440 \) Copy content Toggle raw display
$61$ \( T^{5} + 18 T^{4} + 34 T^{3} - 438 T^{2} + \cdots + 56 \) Copy content Toggle raw display
$67$ \( T^{5} - 2 T^{4} - 300 T^{3} + \cdots - 61936 \) Copy content Toggle raw display
$71$ \( T^{5} - 4 T^{4} - 101 T^{3} + \cdots - 5184 \) Copy content Toggle raw display
$73$ \( T^{5} - 2 T^{4} - 197 T^{3} + \cdots + 27656 \) Copy content Toggle raw display
$79$ \( T^{5} - 30 T^{4} + 308 T^{3} + \cdots + 1936 \) Copy content Toggle raw display
$83$ \( T^{5} + 8 T^{4} - 56 T^{3} + \cdots + 5376 \) Copy content Toggle raw display
$89$ \( T^{5} + 20 T^{4} + 66 T^{3} + \cdots - 4704 \) Copy content Toggle raw display
$97$ \( T^{5} - 12 T^{4} - 114 T^{3} + \cdots - 4120 \) Copy content Toggle raw display
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