Properties

Label 1045.2.a.h
Level $1045$
Weight $2$
Character orbit 1045.a
Self dual yes
Analytic conductor $8.344$
Analytic rank $0$
Dimension $7$
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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \(x^{7} - x^{6} - 10 x^{5} + 8 x^{4} + 27 x^{3} - 16 x^{2} - 18 x + 11\)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - x^{6} - 10 x^{5} + 8 x^{4} + 27 x^{3} - 16 x^{2} - 18 x + 11\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( -\nu^{6} + 11 \nu^{4} + 2 \nu^{3} - 33 \nu^{2} - 10 \nu + 21 \)
\(\beta_{4}\)\(=\)\( -2 \nu^{6} + \nu^{5} + 20 \nu^{4} - 5 \nu^{3} - 52 \nu^{2} - \nu + 27 \)
\(\beta_{5}\)\(=\)\( -3 \nu^{6} + \nu^{5} + 30 \nu^{4} - 3 \nu^{3} - 78 \nu^{2} - 10 \nu + 41 \)
\(\beta_{6}\)\(=\)\( 4 \nu^{6} - \nu^{5} - 41 \nu^{4} + 2 \nu^{3} + 111 \nu^{2} + 15 \nu - 62 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{6} + \beta_{5} + \beta_{3} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-\beta_{5} + \beta_{4} + \beta_{3} + 7 \beta_{2} + \beta_{1} + 14\)
\(\nu^{5}\)\(=\)\(9 \beta_{6} + 7 \beta_{5} + 3 \beta_{4} + 9 \beta_{3} + 28 \beta_{1} + 1\)
\(\nu^{6}\)\(=\)\(2 \beta_{6} - 9 \beta_{5} + 11 \beta_{4} + 12 \beta_{3} + 44 \beta_{2} + 11 \beta_{1} + 76\)

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
−2.40300
−1.54354
−1.08185
0.719047
0.745312
1.97792
2.58611
−2.40300 −1.17935 3.77440 −1.00000 2.83397 −2.15598 −4.26389 −1.60914 2.40300
1.2 −1.54354 2.58921 0.382516 −1.00000 −3.99655 −3.41636 2.49665 3.70400 1.54354
1.3 −1.08185 −0.870874 −0.829591 −1.00000 0.942160 4.02863 3.06121 −2.24158 1.08185
1.4 0.719047 0.163114 −1.48297 −1.00000 0.117287 1.05678 −2.50442 −2.97339 −0.719047
1.5 0.745312 −2.05058 −1.44451 −1.00000 −1.52832 −3.86782 −2.56724 1.20489 −0.745312
1.6 1.97792 2.65412 1.91218 −1.00000 5.24965 1.06653 −0.173707 4.04436 −1.97792
1.7 2.58611 1.69436 4.68797 −1.00000 4.38180 2.28823 6.95140 −0.129144 −2.58611
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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.h 7
3.b odd 2 1 9405.2.a.bd 7
5.b even 2 1 5225.2.a.m 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.2.a.h 7 1.a even 1 1 trivial
5225.2.a.m 7 5.b even 2 1
9405.2.a.bd 7 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{7} - T_{2}^{6} - 10 T_{2}^{5} + 8 T_{2}^{4} + 27 T_{2}^{3} - 16 T_{2}^{2} - 18 T_{2} + 11 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1045))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 11 - 18 T - 16 T^{2} + 27 T^{3} + 8 T^{4} - 10 T^{5} - T^{6} + T^{7} \)
$3$ \( 4 - 20 T - 31 T^{2} + 17 T^{3} + 20 T^{4} - 7 T^{5} - 3 T^{6} + T^{7} \)
$5$ \( ( 1 + T )^{7} \)
$7$ \( 296 - 460 T + 3 T^{2} + 205 T^{3} - 18 T^{4} - 27 T^{5} + T^{6} + T^{7} \)
$11$ \( ( -1 + T )^{7} \)
$13$ \( -1184 - 184 T + 1133 T^{2} + 526 T^{3} - 55 T^{4} - 49 T^{5} - T^{6} + T^{7} \)
$17$ \( 1096 + 428 T - 1829 T^{2} + 616 T^{3} + 141 T^{4} - 61 T^{5} - T^{6} + T^{7} \)
$19$ \( ( 1 + T )^{7} \)
$23$ \( -7724 - 1508 T + 3449 T^{2} + 326 T^{3} - 341 T^{4} - 38 T^{5} + 8 T^{6} + T^{7} \)
$29$ \( -368 + 960 T - 435 T^{2} - 420 T^{3} + 253 T^{4} + T^{5} - 11 T^{6} + T^{7} \)
$31$ \( 8300 - 13500 T - 7261 T^{2} + 1474 T^{3} + 547 T^{4} - 83 T^{5} - 7 T^{6} + T^{7} \)
$37$ \( -20896 + 8216 T + 18113 T^{2} - 1999 T^{3} - 1170 T^{4} - 10 T^{5} + 17 T^{6} + T^{7} \)
$41$ \( -1328 + 520 T + 897 T^{2} - 543 T^{3} - 35 T^{4} + 80 T^{5} - 17 T^{6} + T^{7} \)
$43$ \( 2872 - 16172 T + 6573 T^{2} + 3343 T^{3} - 510 T^{4} - 159 T^{5} + 3 T^{6} + T^{7} \)
$47$ \( 35972 - 95404 T - 31135 T^{2} + 3760 T^{3} + 1239 T^{4} - 82 T^{5} - 14 T^{6} + T^{7} \)
$53$ \( 1211488 - 493272 T - 86071 T^{2} + 20236 T^{3} + 1551 T^{4} - 263 T^{5} - 7 T^{6} + T^{7} \)
$59$ \( -106388 + 139916 T + 50927 T^{2} - 18274 T^{3} + 119 T^{4} + 357 T^{5} - 35 T^{6} + T^{7} \)
$61$ \( 1052 - 12172 T + 28087 T^{2} - 16456 T^{3} + 3035 T^{4} - 103 T^{5} - 17 T^{6} + T^{7} \)
$67$ \( -1748 - 11912 T - 17537 T^{2} + 4510 T^{3} + 542 T^{4} - 133 T^{5} - 4 T^{6} + T^{7} \)
$71$ \( 14492 - 35628 T - 19889 T^{2} + 1982 T^{3} + 1135 T^{4} - 110 T^{5} - 10 T^{6} + T^{7} \)
$73$ \( 85096 - 145236 T + 82651 T^{2} - 20270 T^{3} + 1850 T^{4} + 69 T^{5} - 22 T^{6} + T^{7} \)
$79$ \( 19664 - 26336 T - 9323 T^{2} + 3017 T^{3} + 646 T^{4} - 106 T^{5} - 11 T^{6} + T^{7} \)
$83$ \( 2216 - 15716 T + 3111 T^{2} + 5798 T^{3} - 2997 T^{4} + 533 T^{5} - 39 T^{6} + T^{7} \)
$89$ \( -20716 - 39380 T - 19859 T^{2} + 664 T^{3} + 1628 T^{4} - 81 T^{5} - 18 T^{6} + T^{7} \)
$97$ \( -6460112 - 1303672 T + 202011 T^{2} + 46188 T^{3} - 1394 T^{4} - 415 T^{5} + 4 T^{6} + T^{7} \)
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