Properties

Label 1045.2.a.j
Level $1045$
Weight $2$
Character orbit 1045.a
Self dual yes
Analytic conductor $8.344$
Analytic rank $0$
Dimension $9$
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: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \( x^{9} - 2x^{8} - 11x^{7} + 11x^{6} + 34x^{5} - 20x^{4} - 36x^{3} + 13x^{2} + 10x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 2x^{8} - 11x^{7} + 11x^{6} + 34x^{5} - 20x^{4} - 36x^{3} + 13x^{2} + 10x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -4\nu^{8} + 6\nu^{7} + 47\nu^{6} - 35\nu^{5} - 110\nu^{4} + 112\nu^{3} + 26\nu^{2} - 126\nu + 13 ) / 29 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 8\nu^{8} - 12\nu^{7} - 94\nu^{6} + 41\nu^{5} + 307\nu^{4} - 50\nu^{3} - 400\nu^{2} + 78\nu + 177 ) / 29 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{8} + 22\nu^{7} + 37\nu^{6} - 196\nu^{5} - 94\nu^{4} + 488\nu^{3} + 76\nu^{2} - 288\nu + 9 ) / 29 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{8} - 16\nu^{7} + 39\nu^{6} + 103\nu^{5} - 277\nu^{4} - 202\nu^{3} + 472\nu^{2} + 162\nu - 170 ) / 29 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 14\nu^{8} - 21\nu^{7} - 150\nu^{6} + 50\nu^{5} + 356\nu^{4} + 43\nu^{3} - 178\nu^{2} - 110\nu - 2 ) / 29 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -20\nu^{8} + 30\nu^{7} + 235\nu^{6} - 117\nu^{5} - 695\nu^{4} + 154\nu^{3} + 565\nu^{2} - 108\nu + 7 ) / 29 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 21\nu^{8} - 46\nu^{7} - 196\nu^{6} + 220\nu^{5} + 418\nu^{4} - 327\nu^{3} - 180\nu^{2} + 96\nu - 3 ) / 29 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 16\nu^{8} - 53\nu^{7} - 130\nu^{6} + 372\nu^{5} + 324\nu^{4} - 738\nu^{3} - 249\nu^{2} + 388\nu + 35 ) / 29 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -2\beta_{7} + 2\beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - 2\beta_{3} + 2\beta _1 + 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{8} - 7\beta_{7} - \beta_{6} + 9\beta_{5} + 4\beta_{4} - 9\beta_{3} + 3\beta_{2} + 6\beta _1 + 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -15\beta_{8} - 17\beta_{7} - 9\beta_{6} + 21\beta_{5} + 14\beta_{4} - 33\beta_{3} + 7\beta_{2} + 28\beta _1 + 57 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 51 \beta_{8} - 75 \beta_{7} - 21 \beta_{6} + 99 \beta_{5} + 51 \beta_{4} - 126 \beta_{3} + 34 \beta_{2} + 92 \beta _1 + 155 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 201 \beta_{8} - 243 \beta_{7} - 99 \beta_{6} + 315 \beta_{5} + 188 \beta_{4} - 457 \beta_{3} + 113 \beta_{2} + 362 \beta _1 + 655 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 717 \beta_{8} - 939 \beta_{7} - 315 \beta_{6} + 1239 \beta_{5} + 681 \beta_{4} - 1688 \beta_{3} + 438 \beta_{2} + 1270 \beta _1 + 2203 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 2669 \beta_{8} - 3311 \beta_{7} - 1239 \beta_{6} + 4343 \beta_{5} + 2502 \beta_{4} - 6141 \beta_{3} + 1563 \beta_{2} + 4750 \beta _1 + 8381 ) / 2 \) 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
−0.600178
−1.98926
3.65050
1.56050
0.180277
−1.15934
0.724133
−1.58924
1.22261
−2.21339 0.921304 2.89909 −1.00000 −2.03920 −4.71795 −1.99002 −2.15120 2.21339
1.2 −1.53772 −2.83656 0.364584 −1.00000 4.36184 −2.01727 2.51481 5.04608 1.53772
1.3 −1.31671 2.24751 −0.266275 −1.00000 −2.95932 2.74324 2.98403 2.05132 1.31671
1.4 −0.0815632 −0.583675 −1.99335 −1.00000 0.0476064 −3.83288 0.325710 −2.65932 0.0815632
1.5 0.287410 −0.930455 −1.91740 −1.00000 −0.267422 0.300889 −1.12590 −2.13425 −0.287410
1.6 0.749568 3.38871 −1.43815 −1.00000 2.54007 1.17594 −2.57713 8.48335 −0.749568
1.7 1.79978 −2.53737 1.23919 −1.00000 −4.56670 0.104026 −1.36928 3.43826 −1.79978
1.8 2.64409 0.185485 4.99120 −1.00000 0.490440 1.87571 7.90900 −2.96560 −2.64409
1.9 2.66854 3.14505 5.12110 −1.00000 8.39270 −4.63172 8.32878 6.89136 −2.66854
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
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.j 9
3.b odd 2 1 9405.2.a.bi 9
5.b even 2 1 5225.2.a.q 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.2.a.j 9 1.a even 1 1 trivial
5225.2.a.q 9 5.b even 2 1
9405.2.a.bi 9 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{9} - 3T_{2}^{8} - 9T_{2}^{7} + 27T_{2}^{6} + 25T_{2}^{5} - 70T_{2}^{4} - 21T_{2}^{3} + 51T_{2}^{2} - 8T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1045))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} - 3 T^{8} - 9 T^{7} + 27 T^{6} + \cdots - 1 \) Copy content Toggle raw display
$3$ \( T^{9} - 3 T^{8} - 17 T^{7} + 46 T^{6} + \cdots - 16 \) Copy content Toggle raw display
$5$ \( (T + 1)^{9} \) Copy content Toggle raw display
$7$ \( T^{9} + 9 T^{8} + 3 T^{7} - 136 T^{6} + \cdots - 32 \) Copy content Toggle raw display
$11$ \( (T + 1)^{9} \) Copy content Toggle raw display
$13$ \( T^{9} + 3 T^{8} - 57 T^{7} + \cdots + 4096 \) Copy content Toggle raw display
$17$ \( T^{9} - 5 T^{8} - 85 T^{7} + \cdots - 69376 \) Copy content Toggle raw display
$19$ \( (T - 1)^{9} \) Copy content Toggle raw display
$23$ \( T^{9} - 4 T^{8} - 158 T^{7} + \cdots - 4067776 \) Copy content Toggle raw display
$29$ \( T^{9} - 3 T^{8} - 149 T^{7} + \cdots + 476032 \) Copy content Toggle raw display
$31$ \( T^{9} + T^{8} - 147 T^{7} + \cdots + 1701376 \) Copy content Toggle raw display
$37$ \( T^{9} - 5 T^{8} - 202 T^{7} + \cdots - 7003648 \) Copy content Toggle raw display
$41$ \( T^{9} + 5 T^{8} - 230 T^{7} + \cdots - 2331392 \) Copy content Toggle raw display
$43$ \( T^{9} + 11 T^{8} - 203 T^{7} + \cdots - 12703712 \) Copy content Toggle raw display
$47$ \( T^{9} - 30 T^{8} + 142 T^{7} + \cdots + 3050368 \) Copy content Toggle raw display
$53$ \( T^{9} + T^{8} - 183 T^{7} + \cdots - 746752 \) Copy content Toggle raw display
$59$ \( T^{9} - 59 T^{8} + 1357 T^{7} + \cdots - 7130816 \) Copy content Toggle raw display
$61$ \( T^{9} + 21 T^{8} - 159 T^{7} + \cdots + 40008976 \) Copy content Toggle raw display
$67$ \( T^{9} + 2 T^{8} - 255 T^{7} + \cdots - 2973104 \) Copy content Toggle raw display
$71$ \( T^{9} - 34 T^{8} + 152 T^{7} + \cdots + 21829696 \) Copy content Toggle raw display
$73$ \( T^{9} + 34 T^{8} + 53 T^{7} + \cdots + 14909696 \) Copy content Toggle raw display
$79$ \( T^{9} + 13 T^{8} - 194 T^{7} + \cdots + 1052672 \) Copy content Toggle raw display
$83$ \( T^{9} - 51 T^{8} + 645 T^{7} + \cdots + 20900704 \) Copy content Toggle raw display
$89$ \( T^{9} - 8 T^{8} - 499 T^{7} + \cdots - 245309648 \) Copy content Toggle raw display
$97$ \( T^{9} + 8 T^{8} - 417 T^{7} + \cdots - 50205184 \) Copy content Toggle raw display
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