Properties

Label 1003.2.a.e
Level 1003
Weight 2
Character orbit 1003.a
Self dual yes
Analytic conductor 8.009
Analytic rank 1
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1003 = 17 \cdot 59 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1003.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.00899532273\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} + \nu - 3 \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + \beta_{1} + 6\)\()/2\)

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
−0.254102
2.11491
−1.86081
−1.00000 −1.68133 −1.00000 −4.18953 1.68133 1.68133 3.00000 −0.173127 4.18953
1.2 −1.00000 0.357926 −1.00000 2.58774 −0.357926 −0.357926 3.00000 −2.87189 −2.58774
1.3 −1.00000 3.32340 −1.00000 −2.39821 −3.32340 −3.32340 3.00000 8.04502 2.39821
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 1003.2.a.e 3
3.b odd 2 1 9027.2.a.h 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1003.2.a.e 3 1.a even 1 1 trivial
9027.2.a.h 3 3.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(17\) \(1\)
\(59\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1003))\):

\( T_{2} + 1 \)
\( T_{3}^{3} - 2 T_{3}^{2} - 5 T_{3} + 2 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + 2 T^{2} )^{3} \)
$3$ \( 1 - 2 T + 4 T^{2} - 10 T^{3} + 12 T^{4} - 18 T^{5} + 27 T^{6} \)
$5$ \( 1 + 4 T + 8 T^{2} + 14 T^{3} + 40 T^{4} + 100 T^{5} + 125 T^{6} \)
$7$ \( 1 + 2 T + 16 T^{2} + 26 T^{3} + 112 T^{4} + 98 T^{5} + 343 T^{6} \)
$11$ \( ( 1 + 11 T^{2} )^{3} \)
$13$ \( 1 + 4 T + 23 T^{2} + 48 T^{3} + 299 T^{4} + 676 T^{5} + 2197 T^{6} \)
$17$ \( ( 1 + T )^{3} \)
$19$ \( 1 + 8 T + 72 T^{2} + 308 T^{3} + 1368 T^{4} + 2888 T^{5} + 6859 T^{6} \)
$23$ \( 1 - 2 T + 49 T^{2} - 108 T^{3} + 1127 T^{4} - 1058 T^{5} + 12167 T^{6} \)
$29$ \( 1 + 44 T^{2} + 106 T^{3} + 1276 T^{4} + 24389 T^{6} \)
$31$ \( 1 + 18 T + 185 T^{2} + 1228 T^{3} + 5735 T^{4} + 17298 T^{5} + 29791 T^{6} \)
$37$ \( 1 - 4 T + 31 T^{2} - 424 T^{3} + 1147 T^{4} - 5476 T^{5} + 50653 T^{6} \)
$41$ \( 1 - 12 T + 68 T^{2} - 242 T^{3} + 2788 T^{4} - 20172 T^{5} + 68921 T^{6} \)
$43$ \( 1 + 6 T + 125 T^{2} + 500 T^{3} + 5375 T^{4} + 11094 T^{5} + 79507 T^{6} \)
$47$ \( 1 + 2 T + 33 T^{2} - 324 T^{3} + 1551 T^{4} + 4418 T^{5} + 103823 T^{6} \)
$53$ \( 1 + 28 T + 408 T^{2} + 3654 T^{3} + 21624 T^{4} + 78652 T^{5} + 148877 T^{6} \)
$59$ \( ( 1 + T )^{3} \)
$61$ \( 1 + 2 T + 123 T^{2} + 20 T^{3} + 7503 T^{4} + 7442 T^{5} + 226981 T^{6} \)
$67$ \( 1 + 4 T + 105 T^{2} + 280 T^{3} + 7035 T^{4} + 17956 T^{5} + 300763 T^{6} \)
$71$ \( 1 + 22 T + 349 T^{2} + 3348 T^{3} + 24779 T^{4} + 110902 T^{5} + 357911 T^{6} \)
$73$ \( 1 + 4 T + 199 T^{2} + 568 T^{3} + 14527 T^{4} + 21316 T^{5} + 389017 T^{6} \)
$79$ \( 1 - 6 T + 56 T^{2} + 434 T^{3} + 4424 T^{4} - 37446 T^{5} + 493039 T^{6} \)
$83$ \( 1 + 185 T^{2} + 64 T^{3} + 15355 T^{4} + 571787 T^{6} \)
$89$ \( 1 + 20 T + 379 T^{2} + 3744 T^{3} + 33731 T^{4} + 158420 T^{5} + 704969 T^{6} \)
$97$ \( 1 - 22 T + 375 T^{2} - 4300 T^{3} + 36375 T^{4} - 206998 T^{5} + 912673 T^{6} \)
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