Properties

Label 4025.2.a.v
Level $4025$
Weight $2$
Character orbit 4025.a
Self dual yes
Analytic conductor $32.140$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - x^{7} - 10x^{6} + 9x^{5} + 28x^{4} - 22x^{3} - 16x^{2} + 7x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{4} + 1) q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{7} + \beta_{6} + \beta_{3} + \beta_1) q^{6} - q^{7} + \beta_{3} q^{8} + ( - \beta_{6} + \beta_{4}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{4} + 1) q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{7} + \beta_{6} + \beta_{3} + \beta_1) q^{6} - q^{7} + \beta_{3} q^{8} + ( - \beta_{6} + \beta_{4}) q^{9} + ( - \beta_{7} + \beta_{3} + \beta_1) q^{11} + (\beta_{5} + \beta_{4} + \beta_{3} - \beta_1 + 2) q^{12} + ( - \beta_{7} + \beta_{5} - \beta_{4} + \beta_1) q^{13} - \beta_1 q^{14} + (\beta_{4} + \beta_{3} - \beta_{2}) q^{16} + ( - \beta_{7} - \beta_{5} - \beta_{4} + \beta_{2}) q^{17} + ( - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{18} + \beta_{5} q^{19} + ( - \beta_{4} - 1) q^{21} + (2 \beta_{4} + \beta_{3} + \beta_{2} + 4) q^{22} + q^{23} + (\beta_{6} + \beta_{3} + \beta_{2} + \beta_1) q^{24} + (\beta_{6} + \beta_{2} + \beta_1 + 3) q^{26} + ( - \beta_{6} - \beta_{4} - \beta_1) q^{27} + ( - \beta_{2} - 1) q^{28} + (\beta_{7} - \beta_{6} - \beta_{4} - 2 \beta_{3} + \beta_1 - 2) q^{29} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \beta_1) q^{31} + ( - \beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 2) q^{32} + ( - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 3 \beta_1 + 1) q^{33} + (2 \beta_{7} - 3 \beta_{6} + 2 \beta_{4} - \beta_{3} - 2 \beta_{2} - 2) q^{34} + (2 \beta_{7} - \beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_{2} - \beta_1 + 3) q^{36} + ( - 3 \beta_{7} + \beta_{6} + \beta_{3} + \beta_{2} - \beta_1 + 1) q^{37} + ( - \beta_{7} + 2 \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 + 1) q^{38} + (\beta_{6} + \beta_{4} + \beta_{2} + \beta_1 + 1) q^{39} + (\beta_{7} - \beta_{6} - \beta_{5} + 3 \beta_{4} - 3 \beta_{2} - 1) q^{41} + (\beta_{7} - \beta_{6} - \beta_{3} - \beta_1) q^{42} + (\beta_{7} - 2 \beta_{6} + 2 \beta_{5} - \beta_{3} - 2 \beta_{2}) q^{43} + (2 \beta_{6} + \beta_{4} + 2 \beta_{3} + \beta_{2} + 3 \beta_1 + 2) q^{44} + \beta_1 q^{46} + (2 \beta_{7} - 2 \beta_{6} - \beta_{5} + 2 \beta_{4} + 3) q^{47} + ( - \beta_{5} + \beta_{2} + 2 \beta_1 + 1) q^{48} + q^{49} + ( - \beta_{7} - \beta_{5} + \beta_{4} - \beta_{2} + \beta_1 - 2) q^{51} + (2 \beta_{7} - \beta_{5} + 3 \beta_{4} + \beta_{3} + \beta_1 + 3) q^{52} + ( - \beta_{6} + \beta_{4} - \beta_{2} + 2 \beta_1 + 2) q^{53} + (\beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_1 - 3) q^{54} - \beta_{3} q^{56} + (\beta_{7} + \beta_{5} + \beta_{2} - \beta_1 + 2) q^{57} + (\beta_{7} - \beta_{6} - \beta_{5} - 4 \beta_{4} - 3 \beta_{3} + \beta_{2} - \beta_1) q^{58} + ( - \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - \beta_{3} + 3 \beta_{2} + \beta_1 + 1) q^{59} + ( - \beta_{6} + \beta_{4} - 2 \beta_{3} + 1) q^{61} + ( - \beta_{7} + 2 \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 3) q^{62} + (\beta_{6} - \beta_{4}) q^{63} + ( - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 6) q^{64} + ( - \beta_{6} + \beta_{5} + 4 \beta_{4} + 2 \beta_{3} + \beta_{2} + 9) q^{66} + (3 \beta_{7} - \beta_{6} - \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2} + 4) q^{67} + (2 \beta_{6} - \beta_{5} - 4 \beta_{4} - \beta_{3} + 2 \beta_{2} - \beta_1) q^{68} + (\beta_{4} + 1) q^{69} + (3 \beta_{6} - \beta_{3} - \beta_1 + 3) q^{71} + ( - \beta_{7} + 2 \beta_{6} + \beta_{5} - 2 \beta_{4} + \beta_{2} + 2 \beta_1) q^{72} + (\beta_{6} + 2 \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 4) q^{73} + (\beta_{5} + 5 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} + \beta_1 - 4) q^{74} + (\beta_{7} - \beta_{6} + 4 \beta_{4} + \beta_{3} - \beta_{2} + 4) q^{76} + (\beta_{7} - \beta_{3} - \beta_1) q^{77} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_1 + 3) q^{78} + ( - 3 \beta_{7} + 2 \beta_{6} + \beta_{5} - 3 \beta_{4} - 2 \beta_{3} + \beta_{2} + 3 \beta_1 - 3) q^{79} + (\beta_{7} + 3 \beta_{6} - \beta_{4} - \beta_{3} - 2 \beta_1 - 1) q^{81} + ( - 2 \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - 4 \beta_1) q^{82} + (2 \beta_{7} - 2 \beta_{6} - \beta_{4} - 3 \beta_{3} - \beta_{2} + \beta_1 + 1) q^{83} + ( - \beta_{5} - \beta_{4} - \beta_{3} + \beta_1 - 2) q^{84} + ( - 2 \beta_{7} + 4 \beta_{6} - 2 \beta_{5} - 6 \beta_{4} - \beta_{3} + 4 \beta_{2} + \cdots + 1) q^{86}+ \cdots + ( - \beta_{7} + 2 \beta_{6} - \beta_{5} + \beta_{3} + 3 \beta_1 + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} + 4 q^{3} + 5 q^{4} - q^{6} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{2} + 4 q^{3} + 5 q^{4} - q^{6} - 8 q^{7} + 3 q^{11} + 9 q^{12} + 5 q^{13} - q^{14} - q^{16} + 5 q^{17} + 2 q^{18} - 2 q^{19} - 4 q^{21} + 21 q^{22} + 8 q^{23} - 6 q^{24} + 18 q^{26} + 7 q^{27} - 5 q^{28} - 9 q^{29} - 3 q^{31} + 6 q^{32} + 4 q^{33} - 10 q^{34} + 16 q^{36} + 6 q^{37} + 4 q^{38} - 2 q^{39} - 7 q^{41} + q^{42} + 8 q^{43} + 4 q^{44} + q^{46} + 22 q^{47} + 9 q^{48} + 8 q^{49} - 12 q^{51} + 11 q^{52} + 21 q^{53} - 15 q^{54} + 8 q^{57} + 16 q^{58} + 14 q^{59} + 8 q^{61} + 12 q^{62} - 40 q^{64} + 55 q^{66} + 21 q^{67} + 3 q^{68} + 4 q^{69} + 11 q^{71} - q^{72} + 26 q^{73} - 41 q^{74} + 21 q^{76} - 3 q^{77} + 17 q^{78} - 16 q^{79} - 20 q^{81} - q^{82} + 20 q^{83} - 9 q^{84} + 14 q^{86} - 29 q^{87} + 32 q^{88} + 15 q^{89} - 5 q^{91} + 5 q^{92} + 19 q^{93} + 21 q^{94} + 52 q^{96} + q^{97} + q^{98} + 15 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} - 10x^{6} + 9x^{5} + 28x^{4} - 22x^{3} - 16x^{2} + 7x + 3 \) : 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
\(\beta_{4}\)\(=\) \( \nu^{4} - \nu^{3} - 5\nu^{2} + 4\nu + 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{6} - \nu^{5} - 8\nu^{4} + 6\nu^{3} + 17\nu^{2} - 7\nu - 7 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{7} - \nu^{6} - 9\nu^{5} + 8\nu^{4} + 21\nu^{3} - 17\nu^{2} - 5\nu + 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{7} - \nu^{6} - 10\nu^{5} + 9\nu^{4} + 27\nu^{3} - 21\nu^{2} - 10\nu + 3 \) 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
\(\nu^{4}\)\(=\) \( \beta_{4} + \beta_{3} + 5\beta_{2} + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{7} + \beta_{6} + \beta_{4} + 7\beta_{3} + \beta_{2} + 19\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -\beta_{7} + \beta_{6} + \beta_{5} + 9\beta_{4} + 9\beta_{3} + 24\beta_{2} + 2\beta _1 + 70 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -10\beta_{7} + 11\beta_{6} + \beta_{5} + 10\beta_{4} + 43\beta_{3} + 10\beta_{2} + 94\beta _1 + 24 \) 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
−2.18675
−2.07739
−0.571357
−0.333224
0.651939
1.09801
2.02014
2.39863
−2.18675 2.66680 2.78189 0 −5.83163 −1.00000 −1.70979 4.11181 0
1.2 −2.07739 −0.298210 2.31556 0 0.619500 −1.00000 −0.655538 −2.91107 0
1.3 −0.571357 −1.62458 −1.67355 0 0.928216 −1.00000 2.09891 −0.360734 0
1.4 −0.333224 0.161242 −1.88896 0 −0.0537298 −1.00000 1.29590 −2.97400 0
1.5 0.651939 2.38619 −1.57498 0 1.55565 −1.00000 −2.33067 2.69390 0
1.6 1.09801 0.493654 −0.794374 0 0.542037 −1.00000 −3.06825 −2.75631 0
1.7 2.02014 −1.91409 2.08098 0 −3.86674 −1.00000 0.163592 0.663753 0
1.8 2.39863 2.12900 3.75344 0 5.10670 −1.00000 4.20585 1.53265 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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.v yes 8
5.b even 2 1 4025.2.a.u 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4025.2.a.u 8 5.b even 2 1
4025.2.a.v yes 8 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4025))\):

\( T_{2}^{8} - T_{2}^{7} - 10T_{2}^{6} + 9T_{2}^{5} + 28T_{2}^{4} - 22T_{2}^{3} - 16T_{2}^{2} + 7T_{2} + 3 \) Copy content Toggle raw display
\( T_{3}^{8} - 4T_{3}^{7} - 4T_{3}^{6} + 27T_{3}^{5} - 3T_{3}^{4} - 47T_{3}^{3} + 15T_{3}^{2} + 5T_{3} - 1 \) Copy content Toggle raw display
\( T_{11}^{8} - 3T_{11}^{7} - 30T_{11}^{6} + 43T_{11}^{5} + 297T_{11}^{4} - 28T_{11}^{3} - 982T_{11}^{2} - 908T_{11} - 183 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{7} - 10 T^{6} + 9 T^{5} + \cdots + 3 \) Copy content Toggle raw display
$3$ \( T^{8} - 4 T^{7} - 4 T^{6} + 27 T^{5} + \cdots - 1 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T + 1)^{8} \) Copy content Toggle raw display
$11$ \( T^{8} - 3 T^{7} - 30 T^{6} + 43 T^{5} + \cdots - 183 \) Copy content Toggle raw display
$13$ \( T^{8} - 5 T^{7} - 30 T^{6} + 193 T^{5} + \cdots + 19 \) Copy content Toggle raw display
$17$ \( T^{8} - 5 T^{7} - 43 T^{6} + \cdots + 4083 \) Copy content Toggle raw display
$19$ \( T^{8} + 2 T^{7} - 29 T^{6} - 44 T^{5} + \cdots + 675 \) Copy content Toggle raw display
$23$ \( (T - 1)^{8} \) Copy content Toggle raw display
$29$ \( T^{8} + 9 T^{7} - 55 T^{6} + \cdots + 7311 \) Copy content Toggle raw display
$31$ \( T^{8} + 3 T^{7} - 46 T^{6} - 157 T^{5} + \cdots + 97 \) Copy content Toggle raw display
$37$ \( T^{8} - 6 T^{7} - 168 T^{6} + \cdots - 210905 \) Copy content Toggle raw display
$41$ \( T^{8} + 7 T^{7} - 182 T^{6} + \cdots + 209409 \) Copy content Toggle raw display
$43$ \( T^{8} - 8 T^{7} - 158 T^{6} + \cdots + 356869 \) Copy content Toggle raw display
$47$ \( T^{8} - 22 T^{7} + 101 T^{6} + \cdots - 269835 \) Copy content Toggle raw display
$53$ \( T^{8} - 21 T^{7} + 121 T^{6} + \cdots + 957 \) Copy content Toggle raw display
$59$ \( T^{8} - 14 T^{7} - 207 T^{6} + \cdots - 2725305 \) Copy content Toggle raw display
$61$ \( T^{8} - 8 T^{7} - 84 T^{6} + \cdots + 23917 \) Copy content Toggle raw display
$67$ \( T^{8} - 21 T^{7} + 23 T^{6} + \cdots + 46337 \) Copy content Toggle raw display
$71$ \( T^{8} - 11 T^{7} - 175 T^{6} + \cdots + 88041 \) Copy content Toggle raw display
$73$ \( T^{8} - 26 T^{7} + 132 T^{6} + \cdots + 40969 \) Copy content Toggle raw display
$79$ \( T^{8} + 16 T^{7} - 164 T^{6} + \cdots - 1052537 \) Copy content Toggle raw display
$83$ \( T^{8} - 20 T^{7} - 59 T^{6} + \cdots + 195159 \) Copy content Toggle raw display
$89$ \( T^{8} - 15 T^{7} - 354 T^{6} + \cdots + 19143423 \) Copy content Toggle raw display
$97$ \( T^{8} - T^{7} - 230 T^{6} + \cdots + 124907 \) Copy content Toggle raw display
show more
show less