Properties

Label 4004.2.a.j
Level $4004$
Weight $2$
Character orbit 4004.a
Self dual yes
Analytic conductor $31.972$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4004.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.9721009693\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 2 x^{9} - 18 x^{8} + 32 x^{7} + 93 x^{6} - 119 x^{5} - 174 x^{4} + 107 x^{3} + 51 x^{2} - 17 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} + \beta_{3} q^{5} - q^{7} + ( 1 + \beta_{2} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} + \beta_{3} q^{5} - q^{7} + ( 1 + \beta_{2} ) q^{9} - q^{11} - q^{13} + ( \beta_{1} - \beta_{6} + \beta_{8} ) q^{15} + ( 1 - \beta_{1} + \beta_{2} - \beta_{5} - \beta_{6} + \beta_{8} - \beta_{9} ) q^{17} + ( 1 + \beta_{5} + \beta_{8} ) q^{19} + \beta_{1} q^{21} + ( -\beta_{1} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} ) q^{23} + ( 3 - \beta_{1} + \beta_{3} + \beta_{7} ) q^{25} + ( -1 - 2 \beta_{1} - \beta_{7} - \beta_{8} ) q^{27} + ( 1 - \beta_{1} + \beta_{2} - \beta_{4} + \beta_{6} - \beta_{9} ) q^{29} + ( -2 - \beta_{2} + \beta_{4} - \beta_{8} ) q^{31} + \beta_{1} q^{33} -\beta_{3} q^{35} + ( 1 + \beta_{1} + \beta_{4} - \beta_{6} - \beta_{8} ) q^{37} + \beta_{1} q^{39} + ( 1 - 2 \beta_{1} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{41} + ( 3 + \beta_{3} + \beta_{4} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{43} + ( -2 - \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{9} ) q^{45} + ( -\beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} - \beta_{7} + 2 \beta_{8} ) q^{47} + q^{49} + ( 2 - 3 \beta_{1} + 2 \beta_{3} - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{51} + ( 2 - \beta_{4} - \beta_{5} + \beta_{8} ) q^{53} -\beta_{3} q^{55} + ( 1 - \beta_{2} + 3 \beta_{3} + \beta_{6} - 2 \beta_{8} + 2 \beta_{9} ) q^{57} + ( -3 - \beta_{1} + \beta_{6} - \beta_{8} - \beta_{9} ) q^{59} + ( 1 - \beta_{3} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{61} + ( -1 - \beta_{2} ) q^{63} -\beta_{3} q^{65} + ( 2 - \beta_{1} - \beta_{3} + \beta_{4} + \beta_{7} - \beta_{9} ) q^{67} + ( 6 - \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{69} + ( -3 + \beta_{1} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} - 2 \beta_{8} - \beta_{9} ) q^{71} + ( 3 - 3 \beta_{1} - \beta_{2} + 2 \beta_{5} + \beta_{6} - 2 \beta_{8} ) q^{73} + ( 4 - \beta_{2} - \beta_{4} + \beta_{5} + \beta_{7} + \beta_{9} ) q^{75} + q^{77} + ( 4 + \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{8} ) q^{79} + ( 5 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{8} - 2 \beta_{9} ) q^{81} + ( 2 - 2 \beta_{1} + \beta_{3} - \beta_{6} + \beta_{7} - \beta_{8} ) q^{83} + ( 3 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{8} + 2 \beta_{9} ) q^{85} + ( -\beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{5} - \beta_{6} - 2 \beta_{8} + \beta_{9} ) q^{87} + ( 4 + \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{89} + q^{91} + ( 4 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{8} - \beta_{9} ) q^{93} + ( 2 - 3 \beta_{1} - \beta_{2} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{8} ) q^{95} + ( -1 - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{8} ) q^{97} + ( -1 - \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 2q^{3} - 2q^{5} - 10q^{7} + 10q^{9} + O(q^{10}) \) \( 10q - 2q^{3} - 2q^{5} - 10q^{7} + 10q^{9} - 10q^{11} - 10q^{13} - 4q^{15} + 9q^{17} + 3q^{19} + 2q^{21} - 4q^{23} + 22q^{25} - 8q^{27} + 13q^{29} - 17q^{31} + 2q^{33} + 2q^{35} + 11q^{37} + 2q^{39} + 8q^{41} + 21q^{43} - 15q^{45} + q^{47} + 10q^{49} + 19q^{51} + 22q^{53} + 2q^{55} + 8q^{57} - 24q^{59} + 16q^{61} - 10q^{63} + 2q^{65} + 19q^{67} + 51q^{69} - 25q^{71} + 22q^{73} + 28q^{75} + 10q^{77} + 41q^{79} + 54q^{81} + 8q^{83} + 9q^{85} - 5q^{87} + 36q^{89} + 10q^{91} + 12q^{93} + 13q^{95} - 5q^{97} - 10q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 2 x^{9} - 18 x^{8} + 32 x^{7} + 93 x^{6} - 119 x^{5} - 174 x^{4} + 107 x^{3} + 51 x^{2} - 17 x - 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\((\)\( 8 \nu^{9} + 6 \nu^{8} - 215 \nu^{7} - 54 \nu^{6} + 1783 \nu^{5} - 130 \nu^{4} - 4362 \nu^{3} + 148 \nu^{2} + 1715 \nu - 26 \)\()/125\)
\(\beta_{4}\)\(=\)\((\)\( -13 \nu^{9} + 34 \nu^{8} + 240 \nu^{7} - 631 \nu^{6} - 1263 \nu^{5} + 3330 \nu^{4} + 2132 \nu^{3} - 5628 \nu^{2} - 515 \nu + 1061 \)\()/125\)
\(\beta_{5}\)\(=\)\((\)\( -31 \nu^{9} + 83 \nu^{8} + 505 \nu^{7} - 1322 \nu^{6} - 2056 \nu^{5} + 4935 \nu^{4} + 2509 \nu^{3} - 4836 \nu^{2} + 845 \nu + 432 \)\()/125\)
\(\beta_{6}\)\(=\)\((\)\( -31 \nu^{9} + 33 \nu^{8} + 630 \nu^{7} - 572 \nu^{6} - 3906 \nu^{5} + 2460 \nu^{4} + 7959 \nu^{3} - 2286 \nu^{2} - 2055 \nu + 357 \)\()/125\)
\(\beta_{7}\)\(=\)\((\)\( 53 \nu^{9} - 104 \nu^{8} - 940 \nu^{7} + 1611 \nu^{6} + 4728 \nu^{5} - 5430 \nu^{4} - 8542 \nu^{3} + 3593 \nu^{2} + 1290 \nu - 466 \)\()/125\)
\(\beta_{8}\)\(=\)\((\)\( -53 \nu^{9} + 104 \nu^{8} + 940 \nu^{7} - 1611 \nu^{6} - 4728 \nu^{5} + 5430 \nu^{4} + 8667 \nu^{3} - 3593 \nu^{2} - 2290 \nu + 341 \)\()/125\)
\(\beta_{9}\)\(=\)\((\)\( -59 \nu^{9} + 112 \nu^{8} + 1070 \nu^{7} - 1758 \nu^{6} - 5634 \nu^{5} + 6115 \nu^{4} + 10951 \nu^{3} - 4029 \nu^{2} - 3570 \nu - 27 \)\()/125\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{8} + \beta_{7} + 8 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(-2 \beta_{9} + 2 \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 11 \beta_{2} - 2 \beta_{1} + 32\)
\(\nu^{5}\)\(=\)\(-2 \beta_{9} + 14 \beta_{8} + 13 \beta_{7} + 2 \beta_{5} - \beta_{4} - 2 \beta_{3} - \beta_{2} + 75 \beta_{1} + 7\)
\(\nu^{6}\)\(=\)\(-31 \beta_{9} + 31 \beta_{8} - 15 \beta_{7} - 17 \beta_{6} - 13 \beta_{5} + 13 \beta_{4} - 19 \beta_{3} + 117 \beta_{2} - 40 \beta_{1} + 300\)
\(\nu^{7}\)\(=\)\(-34 \beta_{9} + 166 \beta_{8} + 145 \beta_{7} - 8 \beta_{6} + 32 \beta_{5} - 15 \beta_{4} - 43 \beta_{3} - 25 \beta_{2} + 754 \beta_{1} + 11\)
\(\nu^{8}\)\(=\)\(-377 \beta_{9} + 372 \beta_{8} - 185 \beta_{7} - 228 \beta_{6} - 137 \beta_{5} + 145 \beta_{4} - 269 \beta_{3} + 1236 \beta_{2} - 577 \beta_{1} + 2996\)
\(\nu^{9}\)\(=\)\(-427 \beta_{9} + 1849 \beta_{8} + 1566 \beta_{7} - 175 \beta_{6} + 413 \beta_{5} - 185 \beta_{4} - 637 \beta_{3} - 426 \beta_{2} + 7826 \beta_{1} - 492\)

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
3.21473
2.78432
2.16679
0.672934
0.357256
−0.0962745
−0.497051
−1.59443
−1.69862
−3.30965
0 −3.21473 0 −2.32236 0 −1.00000 0 7.33449 0
1.2 0 −2.78432 0 2.77311 0 −1.00000 0 4.75245 0
1.3 0 −2.16679 0 −1.26627 0 −1.00000 0 1.69497 0
1.4 0 −0.672934 0 0.538228 0 −1.00000 0 −2.54716 0
1.5 0 −0.357256 0 3.31742 0 −1.00000 0 −2.87237 0
1.6 0 0.0962745 0 −1.48698 0 −1.00000 0 −2.99073 0
1.7 0 0.497051 0 −2.93955 0 −1.00000 0 −2.75294 0
1.8 0 1.59443 0 4.37026 0 −1.00000 0 −0.457782 0
1.9 0 1.69862 0 −3.84237 0 −1.00000 0 −0.114675 0
1.10 0 3.30965 0 −1.14150 0 −1.00000 0 7.95375 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(1\)
\(11\) \(1\)
\(13\) \(1\)

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 4004.2.a.j 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4004.2.a.j 10 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{10} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4004))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \)
$3$ \( -2 + 17 T + 51 T^{2} - 107 T^{3} - 174 T^{4} + 119 T^{5} + 93 T^{6} - 32 T^{7} - 18 T^{8} + 2 T^{9} + T^{10} \)
$5$ \( 1220 + 757 T - 3387 T^{2} - 4189 T^{3} - 568 T^{4} + 1059 T^{5} + 331 T^{6} - 84 T^{7} - 34 T^{8} + 2 T^{9} + T^{10} \)
$7$ \( ( 1 + T )^{10} \)
$11$ \( ( 1 + T )^{10} \)
$13$ \( ( 1 + T )^{10} \)
$17$ \( -2937904 - 1757066 T + 696341 T^{2} + 410532 T^{3} - 63323 T^{4} - 31103 T^{5} + 3426 T^{6} + 941 T^{7} - 101 T^{8} - 9 T^{9} + T^{10} \)
$19$ \( 30352 + 255648 T + 588623 T^{2} + 323704 T^{3} - 153065 T^{4} - 21753 T^{5} + 8164 T^{6} + 457 T^{7} - 157 T^{8} - 3 T^{9} + T^{10} \)
$23$ \( 1502464 - 6085264 T + 2586688 T^{2} + 451572 T^{3} - 328976 T^{4} + 4491 T^{5} + 12057 T^{6} - 440 T^{7} - 185 T^{8} + 4 T^{9} + T^{10} \)
$29$ \( 1280 + 70304 T + 558384 T^{2} - 863800 T^{3} + 432580 T^{4} - 75794 T^{5} - 3043 T^{6} + 2116 T^{7} - 110 T^{8} - 13 T^{9} + T^{10} \)
$31$ \( -134400 - 432240 T - 139952 T^{2} + 277160 T^{3} + 203064 T^{4} + 33741 T^{5} - 5403 T^{6} - 1702 T^{7} - 30 T^{8} + 17 T^{9} + T^{10} \)
$37$ \( -737920 - 402688 T + 959968 T^{2} + 390984 T^{3} - 130558 T^{4} - 40239 T^{5} + 6328 T^{6} + 1271 T^{7} - 132 T^{8} - 11 T^{9} + T^{10} \)
$41$ \( 88755200 - 45846656 T - 2796160 T^{2} + 4416512 T^{3} - 270968 T^{4} - 143610 T^{5} + 14027 T^{6} + 1873 T^{7} - 214 T^{8} - 8 T^{9} + T^{10} \)
$43$ \( 16264340 - 1406588 T - 9615833 T^{2} + 813185 T^{3} + 864014 T^{4} - 121076 T^{5} - 18877 T^{6} + 4050 T^{7} - 70 T^{8} - 21 T^{9} + T^{10} \)
$47$ \( -1318400 + 7894720 T + 24835312 T^{2} + 1289664 T^{3} - 2192060 T^{4} - 19608 T^{5} + 48051 T^{6} + 218 T^{7} - 380 T^{8} - T^{9} + T^{10} \)
$53$ \( 781420 + 2201212 T - 2834325 T^{2} + 339924 T^{3} + 291380 T^{4} - 51658 T^{5} - 8631 T^{6} + 1938 T^{7} + 34 T^{8} - 22 T^{9} + T^{10} \)
$59$ \( 1570496 + 1016688 T - 1372192 T^{2} - 381932 T^{3} + 255208 T^{4} + 47779 T^{5} - 11392 T^{6} - 2248 T^{7} + 50 T^{8} + 24 T^{9} + T^{10} \)
$61$ \( -873656 - 26966866 T + 20344577 T^{2} + 8308701 T^{3} - 950065 T^{4} - 298683 T^{5} + 20606 T^{6} + 3777 T^{7} - 231 T^{8} - 16 T^{9} + T^{10} \)
$67$ \( -100332940 - 96656477 T - 3036685 T^{2} + 9709933 T^{3} + 205451 T^{4} - 333666 T^{5} + 4838 T^{6} + 4405 T^{7} - 169 T^{8} - 19 T^{9} + T^{10} \)
$71$ \( 37672320 + 334853808 T - 54431920 T^{2} - 25868520 T^{3} + 1502104 T^{4} + 650603 T^{5} - 5625 T^{6} - 6745 T^{7} - 139 T^{8} + 25 T^{9} + T^{10} \)
$73$ \( 1385843456 - 361367168 T - 183315616 T^{2} + 45742368 T^{3} + 2642548 T^{4} - 1069420 T^{5} + 10289 T^{6} + 8687 T^{7} - 298 T^{8} - 22 T^{9} + T^{10} \)
$79$ \( 39555496 + 41156134 T - 13547373 T^{2} - 12263799 T^{3} + 2370718 T^{4} + 349650 T^{5} - 101603 T^{6} + 4890 T^{7} + 382 T^{8} - 41 T^{9} + T^{10} \)
$83$ \( 269203800 - 254545830 T + 48682733 T^{2} + 12358713 T^{3} - 3169613 T^{4} - 241603 T^{5} + 59598 T^{6} + 2261 T^{7} - 427 T^{8} - 8 T^{9} + T^{10} \)
$89$ \( -1328588 + 43335347 T - 110511711 T^{2} - 9288813 T^{3} + 7849450 T^{4} - 380953 T^{5} - 102497 T^{6} + 9922 T^{7} + 88 T^{8} - 36 T^{9} + T^{10} \)
$97$ \( -8187011072 + 1769726144 T + 598088672 T^{2} - 70417384 T^{3} - 15461872 T^{4} + 847181 T^{5} + 162170 T^{6} - 3805 T^{7} - 700 T^{8} + 5 T^{9} + T^{10} \)
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