Properties

Label 5445.2.a.cd
Level $5445$
Weight $2$
Character orbit 5445.a
Self dual yes
Analytic conductor $43.479$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5445.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(43.4785439006\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - 4x^{7} - 3x^{6} + 22x^{5} - 3x^{4} - 32x^{3} + 9x^{2} + 8x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 495)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + (\beta_{7} + \beta_{6} - \beta_{3} - \beta_1) q^{4} + q^{5} + (\beta_{7} - \beta_{2} - \beta_1 + 1) q^{7} + (\beta_{7} + \beta_{6} - 2 \beta_{3} - \beta_{2} - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} + (\beta_{7} + \beta_{6} - \beta_{3} - \beta_1) q^{4} + q^{5} + (\beta_{7} - \beta_{2} - \beta_1 + 1) q^{7} + (\beta_{7} + \beta_{6} - 2 \beta_{3} - \beta_{2} - \beta_1) q^{8} + ( - \beta_1 + 1) q^{10} + ( - \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + \beta_{3} + 2) q^{13} + (\beta_{5} + \beta_{4} - 2 \beta_1 + 3) q^{14} + (\beta_{6} + 2 \beta_{4} - 2 \beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{16} + ( - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + 2 \beta_{2} + \beta_1 + 1) q^{17} + ( - \beta_{7} + \beta_{6} + \beta_{3} + 2 \beta_{2} + \beta_1) q^{19} + (\beta_{7} + \beta_{6} - \beta_{3} - \beta_1) q^{20} + ( - \beta_{7} - \beta_{6} + 2 \beta_{5} + 2 \beta_{3} - \beta_{2} - \beta_1 + 2) q^{23} + q^{25} + ( - 3 \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + 1) q^{26} + (\beta_{6} + \beta_{4} - 2 \beta_{3} - 2 \beta_1 + 3) q^{28} + (2 \beta_{7} + 2 \beta_{6} - \beta_{5} + \beta_{4} - 2 \beta_{3} + 1) q^{29} + ( - \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{4} - \beta_{3} - \beta_1 + 1) q^{31} + (2 \beta_{7} + 4 \beta_{6} - \beta_{5} + 5 \beta_{4} - 2 \beta_{3} - \beta_{2} - 3 \beta_1 + 4) q^{32} + ( - 3 \beta_{6} - 5 \beta_{4} + \beta_{2} - 1) q^{34} + (\beta_{7} - \beta_{2} - \beta_1 + 1) q^{35} + (\beta_{6} - \beta_{5} - 2 \beta_{4} + 2 \beta_{2} - \beta_1 - 2) q^{37} + ( - \beta_{7} - \beta_{6} - 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - \beta_{2} + \beta_1 - 2) q^{38} + (\beta_{7} + \beta_{6} - 2 \beta_{3} - \beta_{2} - \beta_1) q^{40} + (\beta_{7} + 3 \beta_{6} + \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{41} + (\beta_{7} + 3 \beta_{6} - 2 \beta_{5} + 3 \beta_{4} - 3 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 1) q^{43} + ( - \beta_{6} + 2 \beta_{4} + 2 \beta_{3} - \beta_{2} + 2) q^{46} + (\beta_{7} - \beta_{6} + \beta_{4} + 2 \beta_{3} + 3) q^{47} + (\beta_{7} - 2 \beta_{6} + 2 \beta_{4} + \beta_{3} - 3 \beta_{2} - \beta_1 + 2) q^{49} + ( - \beta_1 + 1) q^{50} + (\beta_{7} - \beta_{6} - \beta_{5} - 4 \beta_{4} + 3 \beta_{2} + 2 \beta_1 - 3) q^{52} + (4 \beta_{7} + 2 \beta_{6} - \beta_{5} - 2 \beta_{3} - 2 \beta_{2} + \beta_1 - 6) q^{53} + (4 \beta_{7} + 5 \beta_{6} - 3 \beta_{5} - 6 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 1) q^{56} + (\beta_{6} + \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 6 \beta_1 + 3) q^{58} + ( - \beta_{7} - 2 \beta_{6} + \beta_{4} + \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 4) q^{59} + (\beta_{7} + \beta_{6} - 2 \beta_{5} - 5 \beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1 - 4) q^{61} + (3 \beta_{7} + \beta_{6} - \beta_{4} - 4 \beta_{3} - 2 \beta_1 + 1) q^{62} + (3 \beta_{7} + 5 \beta_{6} - 2 \beta_{5} + 5 \beta_{4} - \beta_{3} - 6 \beta_{2} - 7 \beta_1 + 5) q^{64} + ( - \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + \beta_{3} + 2) q^{65} + (\beta_{7} + 4 \beta_{6} - 2 \beta_{4} - \beta_{3} + 2 \beta_{2} - 2) q^{67} + (2 \beta_{7} - 2 \beta_{6} + \beta_{5} - 8 \beta_{4} + 4 \beta_{2} + 2 \beta_1 - 3) q^{68} + (\beta_{5} + \beta_{4} - 2 \beta_1 + 3) q^{70} + ( - 2 \beta_{6} + 3 \beta_{5} - 6 \beta_{4} - \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 5) q^{71} + ( - 3 \beta_{7} - \beta_{6} + \beta_{4} + 2 \beta_{3} - \beta_{2} - 3 \beta_1 + 6) q^{73} + (\beta_{7} + \beta_{6} - \beta_{5} - 5 \beta_{4} - \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 1) q^{74} + ( - \beta_{6} + 2 \beta_{3} + \beta_{2} + 6 \beta_1 - 4) q^{76} + ( - 4 \beta_{5} + \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 4 \beta_1 - 4) q^{79} + (\beta_{6} + 2 \beta_{4} - 2 \beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{80} + (4 \beta_{6} - 2 \beta_{5} + 6 \beta_{4} + \beta_{3} - 3 \beta_{2} - 5 \beta_1 + 5) q^{82} + ( - 2 \beta_{6} - 4 \beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_{2} + \beta_1 + 9) q^{83} + ( - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + 2 \beta_{2} + \beta_1 + 1) q^{85} + (6 \beta_{6} - 2 \beta_{5} + 9 \beta_{4} - 3 \beta_{3} - 4 \beta_{2} - 4 \beta_1 - 2) q^{86} + (\beta_{6} - \beta_{5} + 4 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 7) q^{89} + ( - \beta_{7} - 4 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + 5 \beta_{3} + \beta_{2} - 3 \beta_1 + 8) q^{91} + ( - 3 \beta_{5} + 3 \beta_{4} + \beta_{2} + 3 \beta_1 - 2) q^{92} + ( - 3 \beta_{7} - 5 \beta_{6} + 2 \beta_{5} - \beta_{4} + 5 \beta_{3} - \beta_1 + 4) q^{94} + ( - \beta_{7} + \beta_{6} + \beta_{3} + 2 \beta_{2} + \beta_1) q^{95} + ( - 3 \beta_{6} - 3 \beta_{5} - 2 \beta_{4} + \beta_{3} + 3 \beta_{2} + 2 \beta_1 + 1) q^{97} + ( - \beta_{7} - \beta_{6} + 3 \beta_{5} + 5 \beta_{4} + 2 \beta_{3} + 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} + 6 q^{4} + 8 q^{5} + 8 q^{7} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{2} + 6 q^{4} + 8 q^{5} + 8 q^{7} + 12 q^{8} + 4 q^{10} + 6 q^{13} + 14 q^{14} + 14 q^{16} + 8 q^{17} - 2 q^{19} + 6 q^{20} + 4 q^{23} + 8 q^{25} - 2 q^{26} + 24 q^{28} + 22 q^{29} + 10 q^{31} + 28 q^{32} - 2 q^{34} + 8 q^{35} - 14 q^{37} - 20 q^{38} + 12 q^{40} + 22 q^{41} + 14 q^{43} - 2 q^{46} + 10 q^{47} + 4 q^{50} - 10 q^{52} - 18 q^{53} + 34 q^{56} + 12 q^{58} + 2 q^{59} - 14 q^{61} + 30 q^{62} + 30 q^{64} + 6 q^{65} + 10 q^{67} + 6 q^{68} + 14 q^{70} - 2 q^{71} + 16 q^{73} + 24 q^{74} - 22 q^{76} - 16 q^{79} + 14 q^{80} + 10 q^{82} + 46 q^{83} + 8 q^{85} - 28 q^{86} + 38 q^{89} + 8 q^{91} - 24 q^{92} - 10 q^{94} - 2 q^{95} - 4 q^{97} + 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 3x^{6} + 22x^{5} - 3x^{4} - 32x^{3} + 9x^{2} + 8x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{7} + 4\nu^{6} + 3\nu^{5} - 22\nu^{4} + 4\nu^{3} + 30\nu^{2} - 13\nu - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{7} - 4\nu^{6} - 3\nu^{5} + 22\nu^{4} - 3\nu^{3} - 32\nu^{2} + 10\nu + 6 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 4\nu^{7} - 17\nu^{6} - 8\nu^{5} + 91\nu^{4} - 34\nu^{3} - 124\nu^{2} + 66\nu + 19 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 7\nu^{7} - 29\nu^{6} - 16\nu^{5} + 154\nu^{4} - 48\nu^{3} - 206\nu^{2} + 101\nu + 30 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -7\nu^{7} + 30\nu^{6} + 13\nu^{5} - 159\nu^{4} + 62\nu^{3} + 214\nu^{2} - 115\nu - 33 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 8\nu^{7} - 34\nu^{6} - 16\nu^{5} + 181\nu^{4} - 65\nu^{3} - 245\nu^{2} + 124\nu + 38 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{6} - \beta_{3} + \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{7} + 2\beta_{6} - \beta_{3} + \beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8\beta_{7} + 9\beta_{6} + 2\beta_{4} - 6\beta_{3} + 3\beta_{2} + 10\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 20\beta_{7} + 23\beta_{6} + \beta_{5} + 5\beta_{4} - 12\beta_{3} + 14\beta_{2} + 36\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 64\beta_{7} + 79\beta_{6} + 4\beta_{5} + 25\beta_{4} - 44\beta_{3} + 43\beta_{2} + 94\beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 178\beta_{7} + 225\beta_{6} + 19\beta_{5} + 71\beta_{4} - 114\beta_{3} + 151\beta_{2} + 301\beta _1 + 30 \) 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
3.04904
2.13569
1.48523
0.909121
−0.205878
−0.226007
−1.39491
−1.75229
−2.04904 0 2.19858 1.00000 0 −0.589936 −0.406903 0 −2.04904
1.2 −1.13569 0 −0.710206 1.00000 0 4.10132 3.07796 0 −1.13569
1.3 −0.485227 0 −1.76455 1.00000 0 −1.61947 1.82666 0 −0.485227
1.4 0.0908791 0 −1.99174 1.00000 0 −2.45732 −0.362766 0 0.0908791
1.5 1.20588 0 −0.545859 1.00000 0 4.31539 −3.06999 0 1.20588
1.6 1.22601 0 −0.496906 1.00000 0 −0.451695 −3.06123 0 1.22601
1.7 2.39491 0 3.73560 1.00000 0 1.96984 4.15660 0 2.39491
1.8 2.75229 0 5.57509 1.00000 0 2.73187 9.83966 0 2.75229
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(-1\)
\(11\) \(1\)

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 5445.2.a.cd 8
3.b odd 2 1 5445.2.a.ca 8
11.b odd 2 1 5445.2.a.cb 8
11.c even 5 2 495.2.n.g 16
33.d even 2 1 5445.2.a.cc 8
33.h odd 10 2 495.2.n.h yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
495.2.n.g 16 11.c even 5 2
495.2.n.h yes 16 33.h odd 10 2
5445.2.a.ca 8 3.b odd 2 1
5445.2.a.cb 8 11.b odd 2 1
5445.2.a.cc 8 33.d even 2 1
5445.2.a.cd 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5445))\):

\( T_{2}^{8} - 4T_{2}^{7} - 3T_{2}^{6} + 24T_{2}^{5} - 8T_{2}^{4} - 32T_{2}^{3} + 14T_{2}^{2} + 10T_{2} - 1 \) Copy content Toggle raw display
\( T_{7}^{8} - 8T_{7}^{7} + 4T_{7}^{6} + 86T_{7}^{5} - 98T_{7}^{4} - 290T_{7}^{3} + 203T_{7}^{2} + 362T_{7} + 101 \) Copy content Toggle raw display
\( T_{23}^{8} - 4T_{23}^{7} - 65T_{23}^{6} + 46T_{23}^{5} + 1309T_{23}^{4} + 3042T_{23}^{3} + 2139T_{23}^{2} + 144T_{23} - 1 \) Copy content Toggle raw display
\( T_{53}^{8} + 18 T_{53}^{7} - 74 T_{53}^{6} - 2934 T_{53}^{5} - 10906 T_{53}^{4} + 75540 T_{53}^{3} + 547489 T_{53}^{2} + 996010 T_{53} + 392695 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 4 T^{7} - 3 T^{6} + 24 T^{5} + \cdots - 1 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T - 1)^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 8 T^{7} + 4 T^{6} + 86 T^{5} + \cdots + 101 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} - 6 T^{7} - 37 T^{6} + \cdots - 1331 \) Copy content Toggle raw display
$17$ \( T^{8} - 8 T^{7} - 34 T^{6} + \cdots + 4231 \) Copy content Toggle raw display
$19$ \( T^{8} + 2 T^{7} - 91 T^{6} + \cdots + 5929 \) Copy content Toggle raw display
$23$ \( T^{8} - 4 T^{7} - 65 T^{6} + 46 T^{5} + \cdots - 1 \) Copy content Toggle raw display
$29$ \( T^{8} - 22 T^{7} + 145 T^{6} + \cdots + 1891 \) Copy content Toggle raw display
$31$ \( T^{8} - 10 T^{7} - 61 T^{6} + \cdots + 23081 \) Copy content Toggle raw display
$37$ \( T^{8} + 14 T^{7} - 3 T^{6} + \cdots + 122881 \) Copy content Toggle raw display
$41$ \( T^{8} - 22 T^{7} + 36 T^{6} + \cdots + 2647555 \) Copy content Toggle raw display
$43$ \( T^{8} - 14 T^{7} - 172 T^{6} + \cdots - 2417279 \) Copy content Toggle raw display
$47$ \( T^{8} - 10 T^{7} - 96 T^{6} + \cdots - 589 \) Copy content Toggle raw display
$53$ \( T^{8} + 18 T^{7} - 74 T^{6} + \cdots + 392695 \) Copy content Toggle raw display
$59$ \( T^{8} - 2 T^{7} - 173 T^{6} + \cdots - 88469 \) Copy content Toggle raw display
$61$ \( T^{8} + 14 T^{7} - 159 T^{6} + \cdots - 462499 \) Copy content Toggle raw display
$67$ \( T^{8} - 10 T^{7} - 173 T^{6} + \cdots - 597971 \) Copy content Toggle raw display
$71$ \( T^{8} + 2 T^{7} - 399 T^{6} + \cdots - 9049805 \) Copy content Toggle raw display
$73$ \( T^{8} - 16 T^{7} - 157 T^{6} + \cdots - 3045251 \) Copy content Toggle raw display
$79$ \( T^{8} + 16 T^{7} - 342 T^{6} + \cdots + 5735705 \) Copy content Toggle raw display
$83$ \( T^{8} - 46 T^{7} + 535 T^{6} + \cdots - 41467789 \) Copy content Toggle raw display
$89$ \( T^{8} - 38 T^{7} + 438 T^{6} + \cdots - 5696725 \) Copy content Toggle raw display
$97$ \( T^{8} + 4 T^{7} - 404 T^{6} + \cdots - 7598959 \) Copy content Toggle raw display
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