Properties

Label 1003.2.a.f
Level 1003
Weight 2
Character orbit 1003.a
Self dual yes
Analytic conductor 8.009
Analytic rank 1
Dimension 4
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: \(4\)
Coefficient field: 4.4.2225.1
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 3 \nu + 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 4 \beta_{1} + 2\)

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
−1.75660
−0.820249
1.13856
2.43828
−2.75660 1.84224 5.59883 −0.0856374 −5.07830 −0.236068 −9.92054 0.393832 0.236068
1.2 −1.82025 −1.50694 1.31331 2.32719 2.74301 4.23607 1.24995 −0.729126 −4.23607
1.3 0.138564 −2.84224 −1.98080 1.70367 −0.393832 −0.236068 −0.551597 5.07830 0.236068
1.4 1.43828 0.506942 0.0686587 −2.94523 0.729126 4.23607 −2.77782 −2.74301 −4.23607
\(n\): 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.f 4
3.b odd 2 1 9027.2.a.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1003.2.a.f 4 1.a even 1 1 trivial
9027.2.a.i 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1003))\):

\( T_{2}^{4} + 3 T_{2}^{3} - 2 T_{2}^{2} - 7 T_{2} + 1 \)
\( T_{3}^{4} + 2 T_{3}^{3} - 5 T_{3}^{2} - 6 T_{3} + 4 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T + 6 T^{2} + 11 T^{3} + 17 T^{4} + 22 T^{5} + 24 T^{6} + 24 T^{7} + 16 T^{8} \)
$3$ \( 1 + 2 T + 7 T^{2} + 12 T^{3} + 28 T^{4} + 36 T^{5} + 63 T^{6} + 54 T^{7} + 81 T^{8} \)
$5$ \( 1 - T + 12 T^{2} - 4 T^{3} + 71 T^{4} - 20 T^{5} + 300 T^{6} - 125 T^{7} + 625 T^{8} \)
$7$ \( ( 1 - 4 T + 13 T^{2} - 28 T^{3} + 49 T^{4} )^{2} \)
$11$ \( 1 + 11 T + 79 T^{2} + 391 T^{3} + 1480 T^{4} + 4301 T^{5} + 9559 T^{6} + 14641 T^{7} + 14641 T^{8} \)
$13$ \( 1 + 9 T + 77 T^{2} + 373 T^{3} + 1668 T^{4} + 4849 T^{5} + 13013 T^{6} + 19773 T^{7} + 28561 T^{8} \)
$17$ \( ( 1 + T )^{4} \)
$19$ \( 1 - 10 T + 92 T^{2} - 520 T^{3} + 2733 T^{4} - 9880 T^{5} + 33212 T^{6} - 68590 T^{7} + 130321 T^{8} \)
$23$ \( 1 + 3 T + 57 T^{2} + 193 T^{3} + 1568 T^{4} + 4439 T^{5} + 30153 T^{6} + 36501 T^{7} + 279841 T^{8} \)
$29$ \( 1 + 9 T + 126 T^{2} + 740 T^{3} + 5585 T^{4} + 21460 T^{5} + 105966 T^{6} + 219501 T^{7} + 707281 T^{8} \)
$31$ \( 1 + 4 T + 109 T^{2} + 374 T^{3} + 4840 T^{4} + 11594 T^{5} + 104749 T^{6} + 119164 T^{7} + 923521 T^{8} \)
$37$ \( 1 + 32 T + 511 T^{2} + 5224 T^{3} + 37492 T^{4} + 193288 T^{5} + 699559 T^{6} + 1620896 T^{7} + 1874161 T^{8} \)
$41$ \( 1 - 4 T + 86 T^{2} - 648 T^{3} + 3671 T^{4} - 26568 T^{5} + 144566 T^{6} - 275684 T^{7} + 2825761 T^{8} \)
$43$ \( 1 - 13 T + 215 T^{2} - 1635 T^{3} + 14508 T^{4} - 70305 T^{5} + 397535 T^{6} - 1033591 T^{7} + 3418801 T^{8} \)
$47$ \( 1 + 33 T + 583 T^{2} + 6699 T^{3} + 54268 T^{4} + 314853 T^{5} + 1287847 T^{6} + 3426159 T^{7} + 4879681 T^{8} \)
$53$ \( 1 - 6 T + 128 T^{2} - 700 T^{3} + 9881 T^{4} - 37100 T^{5} + 359552 T^{6} - 893262 T^{7} + 7890481 T^{8} \)
$59$ \( ( 1 + T )^{4} \)
$61$ \( 1 + 18 T + 307 T^{2} + 3132 T^{3} + 29688 T^{4} + 191052 T^{5} + 1142347 T^{6} + 4085658 T^{7} + 13845841 T^{8} \)
$67$ \( 1 + 8 T + 251 T^{2} + 1596 T^{3} + 24672 T^{4} + 106932 T^{5} + 1126739 T^{6} + 2406104 T^{7} + 20151121 T^{8} \)
$71$ \( 1 + 5 T + 239 T^{2} + 765 T^{3} + 23456 T^{4} + 54315 T^{5} + 1204799 T^{6} + 1789555 T^{7} + 25411681 T^{8} \)
$73$ \( 1 - 18 T + 345 T^{2} - 3510 T^{3} + 38108 T^{4} - 256230 T^{5} + 1838505 T^{6} - 7002306 T^{7} + 28398241 T^{8} \)
$79$ \( 1 - 27 T + 324 T^{2} - 1956 T^{3} + 10761 T^{4} - 154524 T^{5} + 2022084 T^{6} - 13312053 T^{7} + 38950081 T^{8} \)
$83$ \( 1 + 22 T + 421 T^{2} + 5126 T^{3} + 54364 T^{4} + 425458 T^{5} + 2900269 T^{6} + 12579314 T^{7} + 47458321 T^{8} \)
$89$ \( 1 + 9 T + 243 T^{2} + 1849 T^{3} + 31448 T^{4} + 164561 T^{5} + 1924803 T^{6} + 6344721 T^{7} + 62742241 T^{8} \)
$97$ \( 1 + 31 T + 679 T^{2} + 9567 T^{3} + 110264 T^{4} + 927999 T^{5} + 6388711 T^{6} + 28292863 T^{7} + 88529281 T^{8} \)
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