Properties

Label 4004.2.a.i
Level $4004$
Weight $2$
Character orbit 4004.a
Self dual yes
Analytic conductor $31.972$
Analytic rank $0$
Dimension $9$
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: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \(x^{9} - 4 x^{8} - 12 x^{7} + 60 x^{6} + 15 x^{5} - 233 x^{4} + 74 x^{3} + 271 x^{2} - 67 x - 87\)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} -\beta_{7} q^{5} + q^{7} + ( 1 + \beta_{3} + \beta_{4} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} -\beta_{7} q^{5} + q^{7} + ( 1 + \beta_{3} + \beta_{4} ) q^{9} - q^{11} + q^{13} + ( -1 + 2 \beta_{1} + \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{8} ) q^{15} + ( -1 + \beta_{1} - \beta_{3} - \beta_{7} ) q^{17} + ( 1 + \beta_{3} + \beta_{4} - \beta_{5} ) q^{19} + \beta_{1} q^{21} + ( -1 - \beta_{4} - \beta_{6} ) q^{23} + ( 5 - \beta_{1} - \beta_{2} + \beta_{5} - \beta_{8} ) q^{25} + ( -1 + 2 \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{8} ) q^{27} + ( -\beta_{4} + \beta_{5} - \beta_{6} ) q^{29} + ( 1 - \beta_{1} + \beta_{2} - \beta_{6} ) q^{31} -\beta_{1} q^{33} -\beta_{7} q^{35} + ( 1 + \beta_{1} - 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} - 2 \beta_{8} ) q^{37} + \beta_{1} q^{39} + ( 2 + \beta_{1} - \beta_{2} - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{41} + ( 1 + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{8} ) q^{43} + ( 4 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - \beta_{7} ) q^{45} + ( 2 - \beta_{1} - \beta_{3} - \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{8} ) q^{47} + q^{49} + ( 3 - \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} ) q^{51} + ( -1 - \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{53} + \beta_{7} q^{55} + ( -2 + 3 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{57} + ( -1 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} ) q^{59} + ( 5 - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{8} ) q^{61} + ( 1 + \beta_{3} + \beta_{4} ) q^{63} -\beta_{7} q^{65} + ( 2 - \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} ) q^{67} + ( 3 - 4 \beta_{1} + \beta_{3} - \beta_{4} - 3 \beta_{6} + 2 \beta_{7} + \beta_{8} ) q^{69} + ( 2 \beta_{1} - 3 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{71} + ( 2 - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} ) q^{73} + ( -4 + 8 \beta_{1} - \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} ) q^{75} - q^{77} + ( 1 - 3 \beta_{1} + \beta_{2} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{79} + ( -1 + \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - 3 \beta_{7} - \beta_{8} ) q^{81} + ( 1 + \beta_{1} - \beta_{2} + \beta_{4} - \beta_{6} + 2 \beta_{7} + \beta_{8} ) q^{83} + ( 6 + 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{85} + ( 4 - \beta_{1} - \beta_{2} - \beta_{4} - 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} ) q^{87} + ( -\beta_{4} - 2 \beta_{6} + \beta_{7} - 2 \beta_{8} ) q^{89} + q^{91} + ( -1 + \beta_{1} - 2 \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{8} ) q^{93} + ( -1 + \beta_{2} + 2 \beta_{3} + \beta_{6} - 2 \beta_{7} - 2 \beta_{8} ) q^{95} + ( 3 - 2 \beta_{3} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{97} + ( -1 - \beta_{3} - \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q + 4q^{3} + 4q^{5} + 9q^{7} + 13q^{9} + O(q^{10}) \) \( 9q + 4q^{3} + 4q^{5} + 9q^{7} + 13q^{9} - 9q^{11} + 9q^{13} - 5q^{17} + 17q^{19} + 4q^{21} - 8q^{23} + 35q^{25} + 4q^{27} - 3q^{29} + 9q^{31} - 4q^{33} + 4q^{35} + 7q^{37} + 4q^{39} + 14q^{41} + 21q^{43} + 43q^{45} + q^{47} + 9q^{49} + 25q^{51} - 8q^{53} - 4q^{55} + 8q^{57} - 4q^{59} + 30q^{61} + 13q^{63} + 4q^{65} + 15q^{67} + 9q^{69} - q^{71} + 18q^{73} - 32q^{75} - 9q^{77} + 13q^{79} + 13q^{81} + 2q^{83} + 45q^{85} + 21q^{87} + 9q^{91} - 2q^{93} + 11q^{95} + 29q^{97} - 13q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - 4 x^{8} - 12 x^{7} + 60 x^{6} + 15 x^{5} - 233 x^{4} + 74 x^{3} + 271 x^{2} - 67 x - 87\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -10 \nu^{8} - 189 \nu^{7} + 627 \nu^{6} + 2658 \nu^{5} - 6837 \nu^{4} - 8299 \nu^{3} + 17667 \nu^{2} + 5318 \nu - 5712 \)\()/681\)
\(\beta_{3}\)\(=\)\((\)\( -46 \nu^{8} + 84 \nu^{7} + 705 \nu^{6} - 1257 \nu^{5} - 2712 \nu^{4} + 5000 \nu^{3} + 2817 \nu^{2} - 5365 \nu - 1623 \)\()/681\)
\(\beta_{4}\)\(=\)\((\)\( 46 \nu^{8} - 84 \nu^{7} - 705 \nu^{6} + 1257 \nu^{5} + 2712 \nu^{4} - 5000 \nu^{3} - 2136 \nu^{2} + 5365 \nu - 1101 \)\()/681\)
\(\beta_{5}\)\(=\)\((\)\( -77 \nu^{8} + 111 \nu^{7} + 1491 \nu^{6} - 1734 \nu^{5} - 9129 \nu^{4} + 7126 \nu^{3} + 20334 \nu^{2} - 6449 \nu - 11703 \)\()/681\)
\(\beta_{6}\)\(=\)\((\)\( 176 \nu^{8} - 351 \nu^{7} - 2727 \nu^{6} + 4839 \nu^{5} + 11235 \nu^{4} - 14926 \nu^{3} - 15249 \nu^{2} + 6374 \nu + 6417 \)\()/681\)
\(\beta_{7}\)\(=\)\((\)\( 298 \nu^{8} - 633 \nu^{7} - 4656 \nu^{6} + 8913 \nu^{5} + 19464 \nu^{4} - 29312 \nu^{3} - 26214 \nu^{2} + 18175 \nu + 9774 \)\()/681\)
\(\beta_{8}\)\(=\)\((\)\( -143 \nu^{8} + 271 \nu^{7} + 2315 \nu^{6} - 3804 \nu^{5} - 10533 \nu^{4} + 12553 \nu^{3} + 16717 \nu^{2} - 8215 \nu - 7044 \)\()/227\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} + \beta_{3} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{8} + 2 \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + 8 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(-\beta_{8} - 3 \beta_{7} + 2 \beta_{6} - \beta_{5} + 11 \beta_{4} + 10 \beta_{3} + \beta_{2} + 26\)
\(\nu^{5}\)\(=\)\(13 \beta_{8} - 2 \beta_{7} + 27 \beta_{6} - 15 \beta_{5} + 17 \beta_{4} + 11 \beta_{3} + \beta_{2} + 74 \beta_{1} - 18\)
\(\nu^{6}\)\(=\)\(-13 \beta_{8} - 46 \beta_{7} + 35 \beta_{6} - 15 \beta_{5} + 116 \beta_{4} + 95 \beta_{3} + 15 \beta_{2} + 5 \beta_{1} + 209\)
\(\nu^{7}\)\(=\)\(139 \beta_{8} - 43 \beta_{7} + 311 \beta_{6} - 171 \beta_{5} + 211 \beta_{4} + 109 \beta_{3} + 15 \beta_{2} + 727 \beta_{1} - 212\)
\(\nu^{8}\)\(=\)\(-133 \beta_{8} - 552 \beta_{7} + 466 \beta_{6} - 182 \beta_{5} + 1220 \beta_{4} + 920 \beta_{3} + 171 \beta_{2} + 135 \beta_{1} + 1876\)

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
−3.11788
−1.90247
−0.980714
−0.554873
0.848270
1.89826
2.10249
2.42810
3.27882
0 −3.11788 0 4.40203 0 1.00000 0 6.72118 0
1.2 0 −1.90247 0 −3.63968 0 1.00000 0 0.619386 0
1.3 0 −0.980714 0 −1.42373 0 1.00000 0 −2.03820 0
1.4 0 −0.554873 0 3.11431 0 1.00000 0 −2.69212 0
1.5 0 0.848270 0 −0.844151 0 1.00000 0 −2.28044 0
1.6 0 1.89826 0 3.07273 0 1.00000 0 0.603374 0
1.7 0 2.10249 0 −4.18458 0 1.00000 0 1.42045 0
1.8 0 2.42810 0 0.790216 0 1.00000 0 2.89567 0
1.9 0 3.27882 0 2.71285 0 1.00000 0 7.75069 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
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.i 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4004.2.a.i 9 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{9} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4004))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} \)
$3$ \( -87 - 67 T + 271 T^{2} + 74 T^{3} - 233 T^{4} + 15 T^{5} + 60 T^{6} - 12 T^{7} - 4 T^{8} + T^{9} \)
$5$ \( -1653 + 177 T + 3319 T^{2} - 294 T^{3} - 1343 T^{4} + 269 T^{5} + 136 T^{6} - 32 T^{7} - 4 T^{8} + T^{9} \)
$7$ \( ( -1 + T )^{9} \)
$11$ \( ( 1 + T )^{9} \)
$13$ \( ( -1 + T )^{9} \)
$17$ \( -2916 - 17577 T - 13302 T^{2} + 1615 T^{3} + 3309 T^{4} + 324 T^{5} - 245 T^{6} - 41 T^{7} + 5 T^{8} + T^{9} \)
$19$ \( -23112 + 109401 T - 106094 T^{2} + 31827 T^{3} + 3431 T^{4} - 3488 T^{5} + 437 T^{6} + 61 T^{7} - 17 T^{8} + T^{9} \)
$23$ \( 2736 - 8736 T - 8844 T^{2} + 4480 T^{3} + 4371 T^{4} - 231 T^{5} - 572 T^{6} - 65 T^{7} + 8 T^{8} + T^{9} \)
$29$ \( -53184 + 96720 T - 13312 T^{2} - 36948 T^{3} + 6456 T^{4} + 4533 T^{5} - 264 T^{6} - 128 T^{7} + 3 T^{8} + T^{9} \)
$31$ \( -598608 + 353168 T + 257144 T^{2} - 46744 T^{3} - 23737 T^{4} + 2897 T^{5} + 810 T^{6} - 90 T^{7} - 9 T^{8} + T^{9} \)
$37$ \( 891328 - 1171232 T + 363568 T^{2} + 110436 T^{3} - 74271 T^{4} + 7366 T^{5} + 1409 T^{6} - 186 T^{7} - 7 T^{8} + T^{9} \)
$41$ \( -1251072 - 1297728 T + 762336 T^{2} + 236384 T^{3} - 110912 T^{4} + 795 T^{5} + 2651 T^{6} - 154 T^{7} - 14 T^{8} + T^{9} \)
$43$ \( -31422 + 356561 T - 449641 T^{2} + 124198 T^{3} + 31290 T^{4} - 19041 T^{5} + 2530 T^{6} + 8 T^{7} - 21 T^{8} + T^{9} \)
$47$ \( -2123136 + 999312 T + 2643888 T^{2} - 1059700 T^{3} - 78908 T^{4} + 33521 T^{5} + 576 T^{6} - 326 T^{7} - T^{8} + T^{9} \)
$53$ \( 145422 + 258963 T - 56508 T^{2} - 166708 T^{3} + 35508 T^{4} + 10477 T^{5} - 1106 T^{6} - 190 T^{7} + 8 T^{8} + T^{9} \)
$59$ \( 7749264 + 1529616 T - 1836212 T^{2} - 387396 T^{3} + 93249 T^{4} + 18848 T^{5} - 1268 T^{6} - 264 T^{7} + 4 T^{8} + T^{9} \)
$61$ \( 6189536 - 6082715 T - 1076219 T^{2} + 789253 T^{3} - 1329 T^{4} - 29792 T^{5} + 2697 T^{6} + 167 T^{7} - 30 T^{8} + T^{9} \)
$67$ \( -254075797 + 64646675 T + 20653217 T^{2} - 2632365 T^{3} - 487554 T^{4} + 42770 T^{5} + 4565 T^{6} - 329 T^{7} - 15 T^{8} + T^{9} \)
$71$ \( -17857872 + 14641296 T + 1192632 T^{2} - 1412080 T^{3} + 4299 T^{4} + 35183 T^{5} - 305 T^{6} - 331 T^{7} + T^{8} + T^{9} \)
$73$ \( -3281792 - 1616256 T + 1625360 T^{2} + 166776 T^{3} - 148746 T^{4} - 1591 T^{5} + 3669 T^{6} - 150 T^{7} - 18 T^{8} + T^{9} \)
$79$ \( 43551736 - 78242687 T + 22937659 T^{2} + 61958 T^{3} - 616824 T^{4} + 33597 T^{5} + 5138 T^{6} - 364 T^{7} - 13 T^{8} + T^{9} \)
$83$ \( 32568006 + 15751395 T - 3162857 T^{2} - 1741891 T^{3} + 47801 T^{4} + 43960 T^{5} - 39 T^{6} - 383 T^{7} - 2 T^{8} + T^{9} \)
$89$ \( 133893 + 2113533 T - 3391551 T^{2} - 1135944 T^{3} + 89301 T^{4} + 34945 T^{5} - 602 T^{6} - 342 T^{7} + T^{9} \)
$97$ \( 869184 - 515744 T - 8805448 T^{2} + 2871640 T^{3} - 60561 T^{4} - 67002 T^{5} + 7259 T^{6} - 4 T^{7} - 29 T^{8} + T^{9} \)
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