Properties

Label 1045.2.a.k
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: \( x^{9} - 3x^{8} - 9x^{7} + 29x^{6} + 23x^{5} - 84x^{4} - 23x^{3} + 89x^{2} + 8x - 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 3x^{8} - 9x^{7} + 29x^{6} + 23x^{5} - 84x^{4} - 23x^{3} + 89x^{2} + 8x - 27 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu - 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{8} - 2\nu^{7} - 9\nu^{6} + 16\nu^{5} + 21\nu^{4} - 31\nu^{3} - 12\nu^{2} + 15\nu - 1 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{8} - 4\nu^{7} - 5\nu^{6} + 32\nu^{5} - 7\nu^{4} - 59\nu^{3} + 24\nu^{2} + 25\nu - 7 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{7} - 3\nu^{6} - 7\nu^{5} + 24\nu^{4} + 8\nu^{3} - 45\nu^{2} + 19 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\nu^{7} + 2\nu^{6} + 9\nu^{5} - 16\nu^{4} - 21\nu^{3} + 31\nu^{2} + 13\nu - 15 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{8} - 4\nu^{7} - 3\nu^{6} + 30\nu^{5} - 25\nu^{4} - 47\nu^{3} + 64\nu^{2} + 15\nu - 27 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} + \beta_{6} - \beta_{4} + 7\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{8} + \beta_{7} + 2\beta_{6} - \beta_{5} - \beta_{4} + 7\beta_{3} + \beta_{2} + 27\beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 12\beta_{8} + \beta_{7} + 11\beta_{6} - 2\beta_{5} - 10\beta_{4} + \beta_{3} + 44\beta_{2} + 2\beta _1 + 89 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 26\beta_{8} + 10\beta_{7} + 24\beta_{6} - 13\beta_{5} - 13\beta_{4} + 44\beta_{3} + 16\beta_{2} + 155\beta _1 + 85 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 107\beta_{8} + 13\beta_{7} + 94\beta_{6} - 28\beta_{5} - 77\beta_{4} + 16\beta_{3} + 277\beta_{2} + 36\beta _1 + 564 \) 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
−2.35228
−1.35249
−1.24377
−0.730132
0.682920
1.22961
1.92391
2.14345
2.69878
−2.35228 0.154676 3.53320 1.00000 −0.363841 1.56958 −3.60652 −2.97608 −2.35228
1.2 −1.35249 −2.73889 −0.170779 1.00000 3.70431 2.99075 2.93595 4.50151 −1.35249
1.3 −1.24377 3.09454 −0.453038 1.00000 −3.84889 3.83759 3.05101 6.57616 −1.24377
1.4 −0.730132 0.820975 −1.46691 1.00000 −0.599420 −1.34825 2.53130 −2.32600 −0.730132
1.5 0.682920 2.48419 −1.53362 1.00000 1.69650 0.856948 −2.41318 3.17118 0.682920
1.6 1.22961 −2.38474 −0.488068 1.00000 −2.93229 1.13367 −3.05934 2.68698 1.22961
1.7 1.92391 0.621031 1.70143 1.00000 1.19481 5.15278 −0.574421 −2.61432 1.92391
1.8 2.14345 2.53942 2.59437 1.00000 5.44311 −1.53582 1.27399 3.44866 2.14345
1.9 2.69878 −1.59120 5.28341 1.00000 −4.29429 0.342738 8.86121 −0.468091 2.69878
\(n\): 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.k 9
3.b odd 2 1 9405.2.a.bh 9
5.b even 2 1 5225.2.a.p 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.2.a.k 9 1.a even 1 1 trivial
5225.2.a.p 9 5.b even 2 1
9405.2.a.bh 9 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{9} - 3T_{2}^{8} - 9T_{2}^{7} + 29T_{2}^{6} + 23T_{2}^{5} - 84T_{2}^{4} - 23T_{2}^{3} + 89T_{2}^{2} + 8T_{2} - 27 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1045))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} - 3 T^{8} - 9 T^{7} + 29 T^{6} + \cdots - 27 \) Copy content Toggle raw display
$3$ \( T^{9} - 3 T^{8} - 15 T^{7} + 46 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( (T - 1)^{9} \) Copy content Toggle raw display
$7$ \( T^{9} - 13 T^{8} + 55 T^{7} - 56 T^{6} + \cdots - 64 \) Copy content Toggle raw display
$11$ \( (T - 1)^{9} \) Copy content Toggle raw display
$13$ \( T^{9} - 5 T^{8} - 29 T^{7} + 125 T^{6} + \cdots - 64 \) Copy content Toggle raw display
$17$ \( T^{9} - 13 T^{8} + 35 T^{7} + \cdots + 5904 \) Copy content Toggle raw display
$19$ \( (T - 1)^{9} \) Copy content Toggle raw display
$23$ \( T^{9} - 8 T^{8} - 72 T^{7} + 559 T^{6} + \cdots - 192 \) Copy content Toggle raw display
$29$ \( T^{9} + 3 T^{8} - 81 T^{7} + \cdots - 1680 \) Copy content Toggle raw display
$31$ \( T^{9} + 9 T^{8} - 85 T^{7} - 1163 T^{6} + \cdots + 64 \) Copy content Toggle raw display
$37$ \( T^{9} + 7 T^{8} - 162 T^{7} + \cdots - 32192 \) Copy content Toggle raw display
$41$ \( T^{9} - 9 T^{8} - 164 T^{7} + \cdots + 1696176 \) Copy content Toggle raw display
$43$ \( T^{9} - 23 T^{8} + 85 T^{7} + \cdots + 29632 \) Copy content Toggle raw display
$47$ \( T^{9} - 20 T^{8} + 80 T^{7} + \cdots - 576 \) Copy content Toggle raw display
$53$ \( T^{9} + 5 T^{8} - 195 T^{7} + \cdots + 768192 \) Copy content Toggle raw display
$59$ \( T^{9} - 19 T^{8} - 91 T^{7} + \cdots - 3341760 \) Copy content Toggle raw display
$61$ \( T^{9} - T^{8} - 211 T^{7} + \cdots - 23824 \) Copy content Toggle raw display
$67$ \( T^{9} + 10 T^{8} - 241 T^{7} + \cdots + 46441456 \) Copy content Toggle raw display
$71$ \( T^{9} - 298 T^{7} + \cdots - 34636224 \) Copy content Toggle raw display
$73$ \( T^{9} - 12 T^{8} + \cdots - 160279408 \) Copy content Toggle raw display
$79$ \( T^{9} + 21 T^{8} - 146 T^{7} + \cdots - 71054080 \) Copy content Toggle raw display
$83$ \( T^{9} - 47 T^{8} + 777 T^{7} + \cdots + 909504 \) Copy content Toggle raw display
$89$ \( T^{9} + 2 T^{8} - 369 T^{7} + \cdots + 37547280 \) Copy content Toggle raw display
$97$ \( T^{9} + 32 T^{8} + \cdots - 165140288 \) Copy content Toggle raw display
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