Properties

Label 1045.2.a.d
Level $1045$
Weight $2$
Character orbit 1045.a
Self dual yes
Analytic conductor $8.344$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1045.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.34436701122\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\Q(\zeta_{22})^+\)
Defining polynomial: \( x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 \) 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{4} + \beta_1 - 1) q^{2} + ( - \beta_{4} + \beta_{3} - \beta_{2} - 2) q^{3} + ( - \beta_{3} - \beta_{2} + 1) q^{4} + q^{5} + (3 \beta_{4} - 2 \beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{6} + (\beta_{4} - \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 1) q^{7} + ( - 2 \beta_{4} + \beta_{3}) q^{8} + (2 \beta_{4} + \beta_{2} + \beta_1 + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} + \beta_1 - 1) q^{2} + ( - \beta_{4} + \beta_{3} - \beta_{2} - 2) q^{3} + ( - \beta_{3} - \beta_{2} + 1) q^{4} + q^{5} + (3 \beta_{4} - 2 \beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{6} + (\beta_{4} - \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 1) q^{7} + ( - 2 \beta_{4} + \beta_{3}) q^{8} + (2 \beta_{4} + \beta_{2} + \beta_1 + 2) q^{9} + ( - \beta_{4} + \beta_1 - 1) q^{10} - q^{11} + ( - \beta_{4} + 3 \beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{12} + (2 \beta_{4} + \beta_{3} - \beta_{2} + \beta_1) q^{13} + (3 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - \beta_1) q^{14} + ( - \beta_{4} + \beta_{3} - \beta_{2} - 2) q^{15} + (2 \beta_{4} - \beta_{3} + \beta_{2} + \beta_1) q^{16} + ( - 2 \beta_{4} + 3 \beta_{3} - \beta_{2} + \beta_1 - 2) q^{17} + ( - 2 \beta_{4} + 2 \beta_{3} + \beta_{2} - 2) q^{18} + q^{19} + ( - \beta_{3} - \beta_{2} + 1) q^{20} + (2 \beta_{4} - 3 \beta_{3} + \beta_{2} + 2 \beta_1 + 3) q^{21} + (\beta_{4} - \beta_1 + 1) q^{22} + ( - \beta_{3} + 2 \beta_{2} - 1) q^{23} + (3 \beta_{4} - 2 \beta_{2} + 2) q^{24} + q^{25} + (\beta_{3} + 4 \beta_{2} - 3 \beta_1 - 2) q^{26} + ( - 2 \beta_{4} - 4 \beta_{3} - 3 \beta_1 - 1) q^{27} + (\beta_{4} + 3 \beta_{3} + 3 \beta_{2} - \beta_1 - 4) q^{28} + (\beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 3 \beta_1 + 3) q^{29} + (3 \beta_{4} - 2 \beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{30} + (2 \beta_{4} - \beta_{3} + 6 \beta_{2} - \beta_1 + 1) q^{31} + (2 \beta_{4} + \beta_{3} - \beta_1) q^{32} + (\beta_{4} - \beta_{3} + \beta_{2} + 2) q^{33} + (6 \beta_{4} - 5 \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 4) q^{34} + (\beta_{4} - \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 1) q^{35} + (3 \beta_{4} - 4 \beta_{3} - 3 \beta_{2} - 4 \beta_1 + 1) q^{36} + (4 \beta_{4} - 4 \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 1) q^{37} + ( - \beta_{4} + \beta_1 - 1) q^{38} + ( - 6 \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 - 3) q^{39} + ( - 2 \beta_{4} + \beta_{3}) q^{40} + (2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + \beta_1 + 4) q^{41} + ( - 10 \beta_{4} + 5 \beta_{3} - 2 \beta_{2} + 4 \beta_1 - 2) q^{42} + (3 \beta_{4} - 6 \beta_{3} - \beta_{2} - 1) q^{43} + (\beta_{3} + \beta_{2} - 1) q^{44} + (2 \beta_{4} + \beta_{2} + \beta_1 + 2) q^{45} + (\beta_{4} + \beta_{3} - 3 \beta_{2} + 3) q^{46} + ( - \beta_{4} + 3 \beta_{3} - 5 \beta_{2} + \beta_1 - 6) q^{47} + ( - 2 \beta_{4} - 3 \beta_{3} + \beta_{2} - 3 \beta_1 - 3) q^{48} + ( - 5 \beta_{4} + 2 \beta_{3} - 5 \beta_{2} + 3 \beta_1 + 1) q^{49} + ( - \beta_{4} + \beta_1 - 1) q^{50} + (2 \beta_{4} - \beta_{3} - 3 \beta_{2} - \beta_1 + 5) q^{51} + (7 \beta_{4} - 3 \beta_{3} - \beta_{2} - 5 \beta_1 + 3) q^{52} + ( - 4 \beta_{4} + \beta_{3} - 3 \beta_{2} + 7 \beta_1 - 4) q^{53} + ( - 4 \beta_{4} + 2 \beta_{3} - 6 \beta_{2} + 5 \beta_1) q^{54} - q^{55} + (8 \beta_{4} - 6 \beta_{3} + 5 \beta_{2} - 6 \beta_1 + 5) q^{56} + ( - \beta_{4} + \beta_{3} - \beta_{2} - 2) q^{57} + ( - 6 \beta_{4} + 3 \beta_{3} + \beta_{2} + 4 \beta_1 - 9) q^{58} + ( - 3 \beta_{3} + 3 \beta_{2} - \beta_1 - 2) q^{59} + ( - \beta_{4} + 3 \beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{60} + ( - 3 \beta_{4} + 3 \beta_{3} - 4 \beta_{2} + 3 \beta_1 - 2) q^{61} + (4 \beta_{4} + 3 \beta_{3} - 5 \beta_{2} + 2) q^{62} + ( - 6 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} + \beta_1 - 7) q^{63} + ( - \beta_{4} + 3 \beta_{3} + \beta_{2} - 5 \beta_1 - 3) q^{64} + (2 \beta_{4} + \beta_{3} - \beta_{2} + \beta_1) q^{65} + ( - 3 \beta_{4} + 2 \beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{66} + (6 \beta_{4} - 5 \beta_{3} + 2 \beta_{2} - 5 \beta_1 - 2) q^{67} + ( - 5 \beta_{4} + 5 \beta_{3} + \beta_{2} + \beta_1 - 7) q^{68} + (4 \beta_{4} - 5 \beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{69} + (3 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - \beta_1) q^{70} + ( - \beta_{4} - 2 \beta_{3} - 1) q^{71} + ( - 4 \beta_{4} + 3 \beta_{3} + 2 \beta_1 - 7) q^{72} + (\beta_{4} - \beta_{3} + \beta_1 - 3) q^{73} + ( - 3 \beta_{4} + 8 \beta_{3} - 2 \beta_{2} + \beta_1 - 7) q^{74} + ( - \beta_{4} + \beta_{3} - \beta_{2} - 2) q^{75} + ( - \beta_{3} - \beta_{2} + 1) q^{76} + ( - \beta_{4} + \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 1) q^{77} + (5 \beta_{4} - 7 \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 7) q^{78} + (2 \beta_{4} + 4 \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 1) q^{79} + (2 \beta_{4} - \beta_{3} + \beta_{2} + \beta_1) q^{80} + (3 \beta_{4} + 2 \beta_{3} + 6 \beta_{2} + 3 \beta_1 + 1) q^{81} + ( - 7 \beta_{4} + 4 \beta_{3} - 2 \beta_{2} + 4 \beta_1 - 3) q^{82} + ( - 4 \beta_{4} + 5 \beta_{3} - 3 \beta_{2} + 5 \beta_1 - 10) q^{83} + (2 \beta_{4} - 9 \beta_{3} - 5 \beta_{2} - \beta_1 + 8) q^{84} + ( - 2 \beta_{4} + 3 \beta_{3} - \beta_{2} + \beta_1 - 2) q^{85} + ( - 12 \beta_{4} + 9 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 3) q^{86} + ( - 5 \beta_{4} + 10 \beta_{3} + 3 \beta_{2} + 8 \beta_1 - 4) q^{87} + (2 \beta_{4} - \beta_{3}) q^{88} + ( - 2 \beta_{4} + 7 \beta_{3} + \beta_{2} + \beta_1 - 5) q^{89} + ( - 2 \beta_{4} + 2 \beta_{3} + \beta_{2} - 2) q^{90} + ( - 8 \beta_{4} - 3 \beta_{2} + 7 \beta_1 - 8) q^{91} + ( - 4 \beta_{4} + 2 \beta_{3} + \beta_{2} + \beta_1 - 5) q^{92} + (2 \beta_{4} - 10 \beta_{3} - 5 \beta_{2} - 4 \beta_1 - 3) q^{93} + (6 \beta_{4} - 4 \beta_{3} + 7 \beta_{2} - 8 \beta_1 + 3) q^{94} + q^{95} + ( - 5 \beta_{4} + \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 1) q^{96} + (5 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 5 \beta_1) q^{97} + ( - 5 \beta_{4} - 7 \beta_{3} + 2 \beta_{2} + 4 \beta_1 + 2) q^{98} + ( - 2 \beta_{4} - \beta_{2} - \beta_1 - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 3 q^{2} - 7 q^{3} + 5 q^{4} + 5 q^{5} + 2 q^{6} - 11 q^{7} + 3 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 3 q^{2} - 7 q^{3} + 5 q^{4} + 5 q^{5} + 2 q^{6} - 11 q^{7} + 3 q^{8} + 8 q^{9} - 3 q^{10} - 5 q^{11} - 7 q^{12} + q^{13} - 7 q^{15} - 3 q^{16} - 3 q^{17} - 7 q^{18} + 5 q^{19} + 5 q^{20} + 11 q^{21} + 3 q^{22} - 8 q^{23} + 9 q^{24} + 5 q^{25} - 16 q^{26} - 10 q^{27} - 22 q^{28} + 11 q^{29} + 2 q^{30} - 5 q^{31} - 2 q^{32} + 7 q^{33} + 4 q^{34} - 11 q^{35} - 3 q^{36} - 9 q^{37} - 3 q^{38} - 8 q^{39} + 3 q^{40} + 15 q^{41} + 11 q^{42} - 13 q^{43} - 5 q^{44} + 8 q^{45} + 18 q^{46} - 20 q^{47} - 20 q^{48} + 20 q^{49} - 3 q^{50} + 24 q^{51} + q^{52} - 5 q^{53} + 17 q^{54} - 5 q^{55} - 7 q^{57} - 33 q^{58} - 17 q^{59} - 7 q^{60} + 3 q^{61} + 14 q^{62} - 22 q^{63} - 17 q^{64} + q^{65} - 2 q^{66} - 28 q^{67} - 25 q^{68} - 2 q^{69} - 6 q^{71} - 26 q^{72} - 16 q^{73} - 21 q^{74} - 7 q^{75} + 5 q^{76} + 11 q^{77} + 29 q^{78} + 3 q^{79} - 3 q^{80} + q^{81} + 2 q^{82} - 33 q^{83} + 33 q^{84} - 3 q^{85} + 10 q^{86} - 3 q^{88} - 16 q^{89} - 7 q^{90} - 22 q^{91} - 19 q^{92} - 26 q^{93} - 10 q^{94} + 5 q^{95} + 5 q^{96} - 14 q^{97} + 10 q^{98} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{22} + \zeta_{22}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 3\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 4\nu^{2} + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 4\beta_{2} + 6 \) 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
0.284630
−0.830830
−1.68251
1.91899
1.30972
−2.39788 −2.59435 3.74982 1.00000 6.22094 −2.89389 −4.19584 3.73066 −2.39788
1.2 −1.54620 1.51334 0.390736 1.00000 −2.33992 −4.16140 2.48825 −0.709811 −1.54620
1.3 −1.37279 −1.23648 −0.115460 1.00000 1.69742 2.43232 2.90407 −1.47112 −1.37279
1.4 0.0881559 −3.20362 −1.99223 1.00000 −0.282418 −1.95185 −0.351939 7.26315 0.0881559
1.5 2.22871 −1.47889 2.96714 1.00000 −3.29602 −4.42518 2.15546 −0.812880 2.22871
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(11\) \(1\)
\(19\) \(-1\)

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 1045.2.a.d 5
3.b odd 2 1 9405.2.a.v 5
5.b even 2 1 5225.2.a.j 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.2.a.d 5 1.a even 1 1 trivial
5225.2.a.j 5 5.b even 2 1
9405.2.a.v 5 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{5} + 3T_{2}^{4} - 3T_{2}^{3} - 15T_{2}^{2} - 10T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1045))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} + 3 T^{4} - 3 T^{3} - 15 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{5} + 7 T^{4} + 13 T^{3} - 6 T^{2} + \cdots - 23 \) Copy content Toggle raw display
$5$ \( (T - 1)^{5} \) Copy content Toggle raw display
$7$ \( T^{5} + 11 T^{4} + 33 T^{3} + \cdots - 253 \) Copy content Toggle raw display
$11$ \( (T + 1)^{5} \) Copy content Toggle raw display
$13$ \( T^{5} - T^{4} - 37 T^{3} + 47 T^{2} + \cdots - 529 \) Copy content Toggle raw display
$17$ \( T^{5} + 3 T^{4} - 25 T^{3} - 59 T^{2} + \cdots + 23 \) Copy content Toggle raw display
$19$ \( (T - 1)^{5} \) Copy content Toggle raw display
$23$ \( T^{5} + 8 T^{4} + 8 T^{3} - 29 T^{2} + \cdots - 1 \) Copy content Toggle raw display
$29$ \( T^{5} - 11 T^{4} - 33 T^{3} + \cdots + 1441 \) Copy content Toggle raw display
$31$ \( T^{5} + 5 T^{4} - 111 T^{3} + \cdots + 10649 \) Copy content Toggle raw display
$37$ \( T^{5} + 9 T^{4} - 38 T^{3} + \cdots + 3917 \) Copy content Toggle raw display
$41$ \( T^{5} - 15 T^{4} + 46 T^{3} + \cdots + 593 \) Copy content Toggle raw display
$43$ \( T^{5} + 13 T^{4} - 115 T^{3} + \cdots + 28753 \) Copy content Toggle raw display
$47$ \( T^{5} + 20 T^{4} + 72 T^{3} + \cdots - 989 \) Copy content Toggle raw display
$53$ \( T^{5} + 5 T^{4} - 155 T^{3} + \cdots + 3917 \) Copy content Toggle raw display
$59$ \( T^{5} + 17 T^{4} + 65 T^{3} + \cdots + 197 \) Copy content Toggle raw display
$61$ \( T^{5} - 3 T^{4} - 47 T^{3} + 59 T^{2} + \cdots + 241 \) Copy content Toggle raw display
$67$ \( T^{5} + 28 T^{4} + 175 T^{3} + \cdots - 11881 \) Copy content Toggle raw display
$71$ \( T^{5} + 6 T^{4} - 12 T^{3} - 109 T^{2} + \cdots - 23 \) Copy content Toggle raw display
$73$ \( T^{5} + 16 T^{4} + 87 T^{3} + 186 T^{2} + \cdots + 23 \) Copy content Toggle raw display
$79$ \( T^{5} - 3 T^{4} - 146 T^{3} + \cdots - 15203 \) Copy content Toggle raw display
$83$ \( T^{5} + 33 T^{4} + 341 T^{3} + \cdots - 737 \) Copy content Toggle raw display
$89$ \( T^{5} + 16 T^{4} - 111 T^{3} + \cdots + 9791 \) Copy content Toggle raw display
$97$ \( T^{5} + 14 T^{4} - 25 T^{3} + \cdots + 3323 \) Copy content Toggle raw display
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