Properties

Label 1045.2.a.f
Level $1045$
Weight $2$
Character orbit 1045.a
Self dual yes
Analytic conductor $8.344$
Analytic rank $1$
Dimension $6$
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: \(6\)
Coefficient field: 6.6.7281497.1
Defining polynomial: \(x^{6} - 2 x^{5} - 5 x^{4} + 7 x^{3} + 6 x^{2} - 2 x - 1\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} - 5 x^{4} + 7 x^{3} + 6 x^{2} - 2 x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 2 \)
\(\beta_{3}\)\(=\)\( \nu^{5} - 2 \nu^{4} - 4 \nu^{3} + 6 \nu^{2} + 2 \nu - 1 \)
\(\beta_{4}\)\(=\)\( \nu^{5} - 2 \nu^{4} - 5 \nu^{3} + 7 \nu^{2} + 5 \nu - 1 \)
\(\beta_{5}\)\(=\)\( -\nu^{5} + 3 \nu^{4} + 3 \nu^{3} - 11 \nu^{2} + 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 2\)
\(\nu^{3}\)\(=\)\(-\beta_{4} + \beta_{3} + \beta_{2} + 4 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(\beta_{5} - \beta_{4} + 2 \beta_{3} + 6 \beta_{2} + 7 \beta_{1} + 9\)
\(\nu^{5}\)\(=\)\(2 \beta_{5} - 6 \beta_{4} + 9 \beta_{3} + 10 \beta_{2} + 22 \beta_{1} + 15\)

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
0.497517
−1.79049
−0.748369
1.77015
2.59744
−0.326248
−2.51246 0.895533 4.31247 −1.00000 −2.24999 −0.393051 −5.80998 −2.19802 2.51246
1.2 −2.23198 −1.34246 2.98176 −1.00000 2.99635 4.13295 −2.19127 −1.19780 2.23198
1.3 −0.412130 1.67805 −1.83015 −1.00000 −0.691575 0.0703171 1.57852 −0.184142 0.412130
1.4 0.205229 −3.10246 −1.95788 −1.00000 −0.636714 2.33231 −0.812271 6.62527 −0.205229
1.5 1.21244 1.77266 −0.529980 −1.00000 2.14925 −3.37010 −3.06746 0.142317 −1.21244
1.6 1.73890 −0.901323 1.02379 −1.00000 −1.56731 2.22757 −1.69754 −2.18762 −1.73890
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.f 6
3.b odd 2 1 9405.2.a.z 6
5.b even 2 1 5225.2.a.l 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.2.a.f 6 1.a even 1 1 trivial
5225.2.a.l 6 5.b even 2 1
9405.2.a.z 6 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 2 T_{2}^{5} - 6 T_{2}^{4} - 8 T_{2}^{3} + 11 T_{2}^{2} + 3 T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1045))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + 3 T + 11 T^{2} - 8 T^{3} - 6 T^{4} + 2 T^{5} + T^{6} \)
$3$ \( -10 + T + 19 T^{2} - 2 T^{3} - 9 T^{4} + T^{5} + T^{6} \)
$5$ \( ( 1 + T )^{6} \)
$7$ \( 2 - 25 T - 53 T^{2} + 58 T^{3} - 7 T^{4} - 5 T^{5} + T^{6} \)
$11$ \( ( 1 + T )^{6} \)
$13$ \( 2 + 21 T + 22 T^{2} - 39 T^{3} - 7 T^{4} + 5 T^{5} + T^{6} \)
$17$ \( -1360 + 1329 T + 1226 T^{2} + 25 T^{3} - 73 T^{4} - T^{5} + T^{6} \)
$19$ \( ( 1 + T )^{6} \)
$23$ \( -8912 + 1921 T + 2870 T^{2} + 159 T^{3} - 98 T^{4} - 4 T^{5} + T^{6} \)
$29$ \( 3074 + 2417 T - 44 T^{2} - 347 T^{3} - 41 T^{4} + 9 T^{5} + T^{6} \)
$31$ \( -20 - 203 T - 486 T^{2} + 167 T^{3} + 131 T^{4} + 21 T^{5} + T^{6} \)
$37$ \( -592 - 1717 T - 1631 T^{2} - 622 T^{3} - 82 T^{4} + 3 T^{5} + T^{6} \)
$41$ \( -4210 - 4233 T - 603 T^{2} + 465 T^{3} + 180 T^{4} + 23 T^{5} + T^{6} \)
$43$ \( -326 - 519 T + 207 T^{2} + 222 T^{3} - 41 T^{4} - 7 T^{5} + T^{6} \)
$47$ \( -620 + 3193 T - 1116 T^{2} - 433 T^{3} + 50 T^{4} + 18 T^{5} + T^{6} \)
$53$ \( -4084 - 5189 T - 2224 T^{2} - 259 T^{3} + 61 T^{4} + 17 T^{5} + T^{6} \)
$59$ \( -54968 - 38091 T - 7538 T^{2} + 281 T^{3} + 261 T^{4} + 29 T^{5} + T^{6} \)
$61$ \( 2294 - 4495 T + 1240 T^{2} + 889 T^{3} - 37 T^{4} - 17 T^{5} + T^{6} \)
$67$ \( -73006 - 46067 T + 9604 T^{2} + 1252 T^{3} - 197 T^{4} - 8 T^{5} + T^{6} \)
$71$ \( -5912 - 9959 T - 5772 T^{2} - 1277 T^{3} - 56 T^{4} + 12 T^{5} + T^{6} \)
$73$ \( -4000 + 481 T + 1170 T^{2} + 4 T^{3} - 71 T^{4} - 2 T^{5} + T^{6} \)
$79$ \( 28264 - 28587 T + 7777 T^{2} + 238 T^{3} - 234 T^{4} - 3 T^{5} + T^{6} \)
$83$ \( -82582 + 32921 T + 6286 T^{2} - 1951 T^{3} - 245 T^{4} + 11 T^{5} + T^{6} \)
$89$ \( 3286 - 1705 T - 4526 T^{2} - 876 T^{3} + 65 T^{4} + 22 T^{5} + T^{6} \)
$97$ \( -967268 + 24949 T + 36688 T^{2} - 554 T^{3} - 369 T^{4} + 2 T^{5} + T^{6} \)
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