Properties

Label 5625.2.a.v
Level $5625$
Weight $2$
Character orbit 5625.a
Self dual yes
Analytic conductor $44.916$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5625 = 3^{2} \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5625.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.9158511370\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: 8.8.33620000000.1
Defining polynomial: \( x^{8} - 10x^{6} + 30x^{4} - 25x^{2} + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 225)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + (\beta_{4} + \beta_{2} + 1) q^{7} + \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + (\beta_{4} + \beta_{2} + 1) q^{7} + \beta_{3} q^{8} + ( - \beta_{7} - \beta_{5} - \beta_1) q^{11} + (\beta_{6} - \beta_{2} - 2) q^{13} + (\beta_{5} + \beta_{3} + 2 \beta_1) q^{14} + (\beta_{4} - \beta_{2} - 1) q^{16} + (\beta_{7} - \beta_{5} - \beta_{3} - \beta_1) q^{17} + (\beta_{6} - \beta_{2} - 3) q^{19} + ( - 2 \beta_{6} - 4 \beta_{4} - \beta_{2} - 2) q^{22} + ( - 2 \beta_{7} + \beta_{5} - 2 \beta_1) q^{23} + (\beta_{7} + \beta_{5} - 2 \beta_{3} - 2 \beta_1) q^{26} + (\beta_{6} + 2 \beta_{4} + \beta_{2} + 5) q^{28} + (3 \beta_{7} - 2 \beta_{3} + \beta_1) q^{29} + ( - 3 \beta_{6} - 2 \beta_{4} - \beta_{2} - 4) q^{31} + (\beta_{5} - 3 \beta_{3} - 2 \beta_1) q^{32} + ( - 3 \beta_{4} - 2 \beta_{2} - 5) q^{34} + ( - 2 \beta_{6} - 2 \beta_{4} - 3 \beta_{2} - 2) q^{37} + (\beta_{7} + \beta_{5} - 2 \beta_{3} - 3 \beta_1) q^{38} + (\beta_{7} + \beta_{5} + 3 \beta_{3} - 2 \beta_1) q^{41} + ( - 2 \beta_{6} - \beta_{4} + 5) q^{43} + ( - 4 \beta_{5} + \beta_{3} - 3 \beta_1) q^{44} + ( - \beta_{6} + \beta_{4} - 2 \beta_{2} - 4) q^{46} + (\beta_{7} - 2 \beta_{5} + 2 \beta_{3} - \beta_1) q^{47} + (2 \beta_{6} + 2 \beta_{4} + \beta_{2} - 1) q^{49} + (2 \beta_{4} - 2 \beta_{2} - 5) q^{52} + (\beta_{7} + 5 \beta_{5} - 3 \beta_{3} + 3 \beta_1) q^{53} + (\beta_{7} + \beta_{5} - 2 \beta_{3} + 3 \beta_1) q^{56} + (3 \beta_{6} + \beta_{4} - \beta_{2} - 2) q^{58} + ( - 2 \beta_{7} + 6 \beta_{5} - \beta_{3} + 3 \beta_1) q^{59} + (2 \beta_{4} - 2 \beta_{2} - 3) q^{61} + ( - 3 \beta_{7} - 5 \beta_{5} + 2 \beta_{3} - 8 \beta_1) q^{62} + (\beta_{6} - 2 \beta_{4} - 3 \beta_{2} - 7) q^{64} + (2 \beta_{6} - 5 \beta_{4} + 2 \beta_{2} + 3) q^{67} + ( - 2 \beta_{7} - \beta_{5} - 5 \beta_1) q^{68} + ( - 2 \beta_{3} + 7 \beta_1) q^{71} + ( - 2 \beta_{6} - 9 \beta_{4} + 2 \beta_{2} - 8) q^{73} + ( - 2 \beta_{7} - 4 \beta_{5} - \beta_{3} - 7 \beta_1) q^{74} + (2 \beta_{4} - 3 \beta_{2} - 6) q^{76} + ( - 3 \beta_{5} - 4 \beta_1) q^{77} + (3 \beta_{6} - 2 \beta_{2} - 7) q^{79} + (2 \beta_{6} + 7 \beta_{4} + \beta_{2} - 4) q^{82} + ( - \beta_{7} + \beta_{5} + 3 \beta_{3}) q^{83} + ( - 2 \beta_{7} - 3 \beta_{5} + 2 \beta_{3} + 3 \beta_1) q^{86} + ( - 3 \beta_{4} - 4) q^{88} + ( - \beta_{7} + 2 \beta_{5} + 3 \beta_{3} + 5 \beta_1) q^{89} + ( - 2 \beta_{6} - \beta_{2} - 5) q^{91} + (3 \beta_{7} - 2 \beta_{5} - \beta_{3} - 3 \beta_1) q^{92} + ( - \beta_{6} - 3 \beta_{4} + \beta_{2} - 2) q^{94} + (4 \beta_{6} + 4 \beta_{4} - 2 \beta_{2} + 5) q^{97} + (2 \beta_{7} + 4 \beta_{5} - \beta_{3} + 2 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{4} - 10 q^{13} - 8 q^{16} - 18 q^{19} + 30 q^{28} - 26 q^{31} - 20 q^{34} + 40 q^{43} - 30 q^{46} - 16 q^{49} - 40 q^{52} - 10 q^{58} - 24 q^{61} - 34 q^{64} + 40 q^{67} - 40 q^{73} - 44 q^{76} - 42 q^{79} - 60 q^{82} - 20 q^{88} - 40 q^{91} - 10 q^{94} + 40 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 10x^{6} + 30x^{4} - 25x^{2} + 5 \) : 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
\(\beta_{4}\)\(=\) \( \nu^{4} - 5\nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} - 5\nu^{3} + 2\nu \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{6} - 8\nu^{4} + 17\nu^{2} - 6 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{7} - 9\nu^{5} + 23\nu^{3} - 13\nu \) 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
\(\nu^{4}\)\(=\) \( \beta_{4} + 5\beta_{2} + 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} + 5\beta_{3} + 18\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{6} + 8\beta_{4} + 23\beta_{2} + 59 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{7} + 9\beta_{5} + 22\beta_{3} + 83\beta_1 \) 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
−2.16942
−2.03035
−0.936839
−0.541884
0.541884
0.936839
2.03035
2.16942
−2.16942 0 2.70636 0 0 3.32440 −1.53239 0 0
1.2 −2.03035 0 2.12233 0 0 0.504300 −0.248380 0 0
1.3 −0.936839 0 −1.12233 0 0 −2.74037 2.92512 0 0
1.4 −0.541884 0 −1.70636 0 0 −1.08833 2.00842 0 0
1.5 0.541884 0 −1.70636 0 0 −1.08833 −2.00842 0 0
1.6 0.936839 0 −1.12233 0 0 −2.74037 −2.92512 0 0
1.7 2.03035 0 2.12233 0 0 0.504300 0.248380 0 0
1.8 2.16942 0 2.70636 0 0 3.32440 1.53239 0 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
\(3\) \(1\)
\(5\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5625.2.a.v 8
3.b odd 2 1 inner 5625.2.a.v 8
5.b even 2 1 5625.2.a.w 8
15.d odd 2 1 5625.2.a.w 8
25.e even 10 2 225.2.h.e 16
75.h odd 10 2 225.2.h.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.2.h.e 16 25.e even 10 2
225.2.h.e 16 75.h odd 10 2
5625.2.a.v 8 1.a even 1 1 trivial
5625.2.a.v 8 3.b odd 2 1 inner
5625.2.a.w 8 5.b even 2 1
5625.2.a.w 8 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5625))\):

\( T_{2}^{8} - 10T_{2}^{6} + 30T_{2}^{4} - 25T_{2}^{2} + 5 \) Copy content Toggle raw display
\( T_{7}^{4} - 10T_{7}^{2} - 5T_{7} + 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 10 T^{6} + 30 T^{4} - 25 T^{2} + \cdots + 5 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} - 10 T^{2} - 5 T + 5)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 35 T^{6} + 400 T^{4} + \cdots + 125 \) Copy content Toggle raw display
$13$ \( (T^{4} + 5 T^{3} - 10 T^{2} - 60 T - 45)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 50 T^{6} + 790 T^{4} + \cdots + 4205 \) Copy content Toggle raw display
$19$ \( (T^{4} + 9 T^{3} + 11 T^{2} - 61 T - 109)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 95 T^{6} + 2950 T^{4} + \cdots + 142805 \) Copy content Toggle raw display
$29$ \( T^{8} - 160 T^{6} + 7500 T^{4} + \cdots + 125 \) Copy content Toggle raw display
$31$ \( (T^{4} + 13 T^{3} - 16 T^{2} - 378 T + 801)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 90 T^{2} + 195 T - 95)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - 320 T^{6} + 33850 T^{4} + \cdots + 4005125 \) Copy content Toggle raw display
$43$ \( (T^{4} - 20 T^{3} + 115 T^{2} - 150 T - 45)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 100 T^{6} + 1620 T^{4} + \cdots + 12005 \) Copy content Toggle raw display
$53$ \( T^{8} - 310 T^{6} + \cdots + 10210205 \) Copy content Toggle raw display
$59$ \( T^{8} - 445 T^{6} + \cdots + 63190125 \) Copy content Toggle raw display
$61$ \( (T^{4} + 12 T^{3} + 14 T^{2} - 172 T - 319)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 20 T^{3} + 5 T^{2} + 1580 T - 6395)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 550 T^{6} + \cdots + 259560125 \) Copy content Toggle raw display
$73$ \( (T^{4} + 20 T^{3} - 85 T^{2} - 3090 T - 10845)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 21 T^{3} + 36 T^{2} - 1194 T - 5309)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 160 T^{6} + 4530 T^{4} + \cdots + 1805 \) Copy content Toggle raw display
$89$ \( T^{8} - 425 T^{6} + 37000 T^{4} + \cdots + 1128125 \) Copy content Toggle raw display
$97$ \( (T^{4} - 20 T^{3} - 50 T^{2} + 2340 T - 9295)^{2} \) Copy content Toggle raw display
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