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: \(x^{5} - x^{4} - 4 x^{3} + 3 x^{2} + 3 x - 1\)
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.

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

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 \( T_{2}^{5} + 3 T_{2}^{4} - 3 T_{2}^{3} - 15 T_{2}^{2} - 10 T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1045))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 10 T - 15 T^{2} - 3 T^{3} + 3 T^{4} + T^{5} \)
$3$ \( -23 - 35 T - 6 T^{2} + 13 T^{3} + 7 T^{4} + T^{5} \)
$5$ \( ( -1 + T )^{5} \)
$7$ \( -253 - 231 T - 22 T^{2} + 33 T^{3} + 11 T^{4} + T^{5} \)
$11$ \( ( 1 + T )^{5} \)
$13$ \( -529 + 322 T + 47 T^{2} - 37 T^{3} - T^{4} + T^{5} \)
$17$ \( 23 - 10 T - 59 T^{2} - 25 T^{3} + 3 T^{4} + T^{5} \)
$19$ \( ( -1 + T )^{5} \)
$23$ \( -1 + 12 T - 29 T^{2} + 8 T^{3} + 8 T^{4} + T^{5} \)
$29$ \( 1441 - 1980 T + 649 T^{2} - 33 T^{3} - 11 T^{4} + T^{5} \)
$31$ \( 10649 + 2062 T - 595 T^{2} - 111 T^{3} + 5 T^{4} + T^{5} \)
$37$ \( 3917 + 37 T - 438 T^{2} - 38 T^{3} + 9 T^{4} + T^{5} \)
$41$ \( 593 - 915 T + 181 T^{2} + 46 T^{3} - 15 T^{4} + T^{5} \)
$43$ \( 28753 - 755 T - 1784 T^{2} - 115 T^{3} + 13 T^{4} + T^{5} \)
$47$ \( -989 - 3538 T - 625 T^{2} + 72 T^{3} + 20 T^{4} + T^{5} \)
$53$ \( 3917 + 742 T - 683 T^{2} - 155 T^{3} + 5 T^{4} + T^{5} \)
$59$ \( 197 - 248 T - 43 T^{2} + 65 T^{3} + 17 T^{4} + T^{5} \)
$61$ \( 241 + 342 T + 59 T^{2} - 47 T^{3} - 3 T^{4} + T^{5} \)
$67$ \( -11881 - 11336 T - 1066 T^{2} + 175 T^{3} + 28 T^{4} + T^{5} \)
$71$ \( -23 - 116 T - 109 T^{2} - 12 T^{3} + 6 T^{4} + T^{5} \)
$73$ \( 23 + 148 T + 186 T^{2} + 87 T^{3} + 16 T^{4} + T^{5} \)
$79$ \( -15203 + 5105 T + 422 T^{2} - 146 T^{3} - 3 T^{4} + T^{5} \)
$83$ \( -737 + 792 T + 1111 T^{2} + 341 T^{3} + 33 T^{4} + T^{5} \)
$89$ \( 9791 + 6308 T - 1442 T^{2} - 111 T^{3} + 16 T^{4} + T^{5} \)
$97$ \( 3323 + 34 T - 466 T^{2} - 25 T^{3} + 14 T^{4} + T^{5} \)
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