Properties

Label 4004.2.a.h
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: \(x^{9} - 3 x^{8} - 19 x^{7} + 51 x^{6} + 116 x^{5} - 247 x^{4} - 249 x^{3} + 288 x^{2} + 189 x - 14\)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} -\beta_{4} q^{5} + q^{7} + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} -\beta_{4} q^{5} + q^{7} + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{9} + q^{11} - q^{13} + ( -2 + \beta_{1} - \beta_{4} - \beta_{6} + \beta_{7} ) q^{15} + ( -\beta_{2} - \beta_{3} + \beta_{8} ) q^{17} + ( 1 - \beta_{2} + \beta_{4} ) q^{19} + \beta_{1} q^{21} + ( 1 - \beta_{1} + \beta_{5} + \beta_{8} ) q^{23} + ( 1 + \beta_{3} + \beta_{6} ) q^{25} + ( 3 + 2 \beta_{1} + 2 \beta_{3} + \beta_{6} + \beta_{7} ) q^{27} + ( 1 + \beta_{1} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{29} + ( 2 + \beta_{2} + \beta_{4} - \beta_{5} - \beta_{7} - 2 \beta_{8} ) q^{31} + \beta_{1} q^{33} -\beta_{4} q^{35} + ( -\beta_{3} - \beta_{4} - \beta_{8} ) q^{37} -\beta_{1} q^{39} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} ) q^{41} + ( \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} + \beta_{7} + \beta_{8} ) q^{43} + ( -1 - \beta_{2} - 4 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} + \beta_{8} ) q^{45} + ( 1 + \beta_{1} - 2 \beta_{4} + \beta_{5} - \beta_{8} ) q^{47} + q^{49} + ( -2 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{8} ) q^{51} + ( 3 + \beta_{1} - \beta_{2} + \beta_{3} - 3 \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} ) q^{53} -\beta_{4} q^{55} + ( \beta_{1} + \beta_{2} - \beta_{3} + 3 \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{8} ) q^{57} + ( 3 - \beta_{1} + \beta_{2} + \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{59} + ( 5 - 2 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} ) q^{61} + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{63} + \beta_{4} q^{65} + ( 1 - \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{6} + \beta_{7} ) q^{67} + ( -2 + 2 \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{69} + ( \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{8} ) q^{71} + ( 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} + 3 \beta_{7} - \beta_{8} ) q^{73} + ( 4 + \beta_{1} - \beta_{2} + 3 \beta_{3} + 2 \beta_{4} + \beta_{6} ) q^{75} + q^{77} + ( -4 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} - 2 \beta_{7} - \beta_{8} ) q^{79} + ( 8 + 2 \beta_{1} - \beta_{2} + 4 \beta_{3} + 2 \beta_{5} + \beta_{6} + \beta_{8} ) q^{81} + ( 2 - \beta_{1} + \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{83} + ( 1 - \beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{8} ) q^{85} + ( 3 + 3 \beta_{1} + \beta_{2} - \beta_{4} + \beta_{7} + \beta_{8} ) q^{87} + ( 2 \beta_{2} + \beta_{4} - 2 \beta_{5} + \beta_{8} ) q^{89} - q^{91} + ( 1 - 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{93} + ( -8 - \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{95} + ( 2 - \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{97} + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q + 3q^{3} + 9q^{7} + 20q^{9} + O(q^{10}) \) \( 9q + 3q^{3} + 9q^{7} + 20q^{9} + 9q^{11} - 9q^{13} - 9q^{15} + 5q^{17} + 10q^{19} + 3q^{21} + 8q^{23} + 3q^{25} + 27q^{27} + 14q^{29} + 11q^{31} + 3q^{33} - 3q^{39} + 14q^{41} + 8q^{43} + 4q^{45} + 10q^{47} + 9q^{49} - 15q^{51} + 21q^{53} - 8q^{57} + 23q^{59} + 34q^{61} + 20q^{63} + 10q^{67} - 16q^{69} + 4q^{71} + 9q^{73} + 30q^{75} + 9q^{77} - 34q^{79} + 69q^{81} + 15q^{83} + 5q^{85} + 39q^{87} - 9q^{91} + 3q^{93} - 64q^{95} + 15q^{97} + 20q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - 3 x^{8} - 19 x^{7} + 51 x^{6} + 116 x^{5} - 247 x^{4} - 249 x^{3} + 288 x^{2} + 189 x - 14\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -11 \nu^{8} - 1526 \nu^{7} + 1071 \nu^{6} + 27349 \nu^{5} - 11387 \nu^{4} - 132924 \nu^{3} + 26935 \nu^{2} + 141008 \nu + 36543 \)\()/15311\)
\(\beta_{3}\)\(=\)\((\)\( 38 \nu^{8} - 296 \nu^{7} - 916 \nu^{6} + 7131 \nu^{5} + 7323 \nu^{4} - 46071 \nu^{3} - 19277 \nu^{2} + 55726 \nu - 10711 \)\()/15311\)
\(\beta_{4}\)\(=\)\((\)\( 49 \nu^{8} + 1230 \nu^{7} - 1987 \nu^{6} - 20218 \nu^{5} + 18710 \nu^{4} + 86853 \nu^{3} - 61523 \nu^{2} - 69971 \nu + 29301 \)\()/15311\)
\(\beta_{5}\)\(=\)\((\)\( 95 \nu^{8} - 740 \nu^{7} - 2290 \nu^{6} + 10172 \nu^{5} + 25963 \nu^{4} - 30967 \nu^{3} - 117092 \nu^{2} + 16827 \nu + 103366 \)\()/15311\)
\(\beta_{6}\)\(=\)\((\)\( 626 \nu^{8} - 847 \nu^{7} - 9449 \nu^{6} + 9491 \nu^{5} + 32800 \nu^{4} - 14361 \nu^{3} + 7997 \nu^{2} - 64309 \nu - 41874 \)\()/15311\)
\(\beta_{7}\)\(=\)\((\)\( -702 \nu^{8} + 1439 \nu^{7} + 11281 \nu^{6} - 23753 \nu^{5} - 47446 \nu^{4} + 121814 \nu^{3} + 30557 \nu^{2} - 169631 \nu + 17363 \)\()/15311\)
\(\beta_{8}\)\(=\)\((\)\( -979 \nu^{8} + 1985 \nu^{7} + 18764 \nu^{6} - 31010 \nu^{5} - 110094 \nu^{4} + 127655 \nu^{3} + 192431 \nu^{2} - 81863 \nu - 70160 \)\()/15311\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{4} + \beta_{3} - \beta_{2} + \beta_{1} + 5\)
\(\nu^{3}\)\(=\)\(\beta_{7} + \beta_{6} + 2 \beta_{3} + 8 \beta_{1} + 3\)
\(\nu^{4}\)\(=\)\(\beta_{8} + \beta_{6} + 2 \beta_{5} - 9 \beta_{4} + 13 \beta_{3} - 10 \beta_{2} + 11 \beta_{1} + 44\)
\(\nu^{5}\)\(=\)\(\beta_{8} + 11 \beta_{7} + 12 \beta_{6} + 31 \beta_{3} - \beta_{2} + 74 \beta_{1} + 49\)
\(\nu^{6}\)\(=\)\(19 \beta_{8} + 21 \beta_{6} + 29 \beta_{5} - 81 \beta_{4} + 148 \beta_{3} - 95 \beta_{2} + 124 \beta_{1} + 434\)
\(\nu^{7}\)\(=\)\(22 \beta_{8} + 110 \beta_{7} + 133 \beta_{6} + 3 \beta_{5} - 2 \beta_{4} + 394 \beta_{3} - 31 \beta_{2} + 734 \beta_{1} + 673\)
\(\nu^{8}\)\(=\)\(249 \beta_{8} + 5 \beta_{7} + 310 \beta_{6} + 337 \beta_{5} - 741 \beta_{4} + 1649 \beta_{3} - 924 \beta_{2} + 1440 \beta_{1} + 4485\)

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
−2.98175
−2.56319
−1.21819
−0.648104
0.0675528
1.32285
2.29815
3.32951
3.39317
0 −2.98175 0 2.28847 0 1.00000 0 5.89080 0
1.2 0 −2.56319 0 −1.31026 0 1.00000 0 3.56996 0
1.3 0 −1.21819 0 3.23173 0 1.00000 0 −1.51602 0
1.4 0 −0.648104 0 −1.99671 0 1.00000 0 −2.57996 0
1.5 0 0.0675528 0 −1.58844 0 1.00000 0 −2.99544 0
1.6 0 1.32285 0 −0.265081 0 1.00000 0 −1.25008 0
1.7 0 2.29815 0 0.952431 0 1.00000 0 2.28151 0
1.8 0 3.32951 0 2.67963 0 1.00000 0 8.08561 0
1.9 0 3.39317 0 −3.99177 0 1.00000 0 8.51361 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.h 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4004.2.a.h 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

$p$ $F_p(T)$
$2$ \( T^{9} \)
$3$ \( -14 + 189 T + 288 T^{2} - 249 T^{3} - 247 T^{4} + 116 T^{5} + 51 T^{6} - 19 T^{7} - 3 T^{8} + T^{9} \)
$5$ \( 83 + 311 T - 125 T^{2} - 452 T^{3} + 5 T^{4} + 171 T^{5} + 4 T^{6} - 24 T^{7} + T^{9} \)
$7$ \( ( -1 + T )^{9} \)
$11$ \( ( -1 + T )^{9} \)
$13$ \( ( 1 + T )^{9} \)
$17$ \( 61022 + 157069 T + 44498 T^{2} - 34129 T^{3} - 8325 T^{4} + 2752 T^{5} + 411 T^{6} - 95 T^{7} - 5 T^{8} + T^{9} \)
$19$ \( -19801 - 51339 T + 3857 T^{2} + 31154 T^{3} - 11075 T^{4} - 611 T^{5} + 662 T^{6} - 42 T^{7} - 10 T^{8} + T^{9} \)
$23$ \( -11536 - 122080 T + 76964 T^{2} + 10176 T^{3} - 12613 T^{4} + 733 T^{5} + 576 T^{6} - 61 T^{7} - 8 T^{8} + T^{9} \)
$29$ \( -147280 + 400064 T - 384444 T^{2} + 158216 T^{3} - 19321 T^{4} - 4831 T^{5} + 1438 T^{6} - 47 T^{7} - 14 T^{8} + T^{9} \)
$31$ \( -15554608 - 2692592 T + 3315032 T^{2} - 71368 T^{3} - 148923 T^{4} + 9793 T^{5} + 2254 T^{6} - 186 T^{7} - 11 T^{8} + T^{9} \)
$37$ \( 118400 + 586240 T - 148032 T^{2} - 134392 T^{3} + 12410 T^{4} + 9023 T^{5} - 175 T^{6} - 188 T^{7} + T^{9} \)
$41$ \( -1040000 + 320512 T + 432224 T^{2} - 36824 T^{3} - 42230 T^{4} + 2309 T^{5} + 1445 T^{6} - 86 T^{7} - 14 T^{8} + T^{9} \)
$43$ \( 1770695 - 3285048 T + 1661021 T^{2} - 15742 T^{3} - 107699 T^{4} + 10221 T^{5} + 1806 T^{6} - 211 T^{7} - 8 T^{8} + T^{9} \)
$47$ \( -11326672 - 6222048 T + 5301524 T^{2} - 226384 T^{3} - 204293 T^{4} + 16373 T^{5} + 2584 T^{6} - 241 T^{7} - 10 T^{8} + T^{9} \)
$53$ \( -199204765 - 17592009 T + 18083584 T^{2} + 568286 T^{3} - 524141 T^{4} + 4023 T^{5} + 5936 T^{6} - 200 T^{7} - 21 T^{8} + T^{9} \)
$59$ \( -775936 + 125552 T + 1094656 T^{2} + 78804 T^{3} - 100504 T^{4} - 4421 T^{5} + 2693 T^{6} - 13 T^{7} - 23 T^{8} + T^{9} \)
$61$ \( 262558 + 157753 T - 379519 T^{2} - 88711 T^{3} + 151099 T^{4} - 33072 T^{5} + 1211 T^{6} + 311 T^{7} - 34 T^{8} + T^{9} \)
$67$ \( 235526 + 435639 T + 4606 T^{2} - 263163 T^{3} - 55626 T^{4} + 13904 T^{5} + 1936 T^{6} - 253 T^{7} - 10 T^{8} + T^{9} \)
$71$ \( -19670560 + 2442384 T + 2827664 T^{2} - 449208 T^{3} - 104722 T^{4} + 18585 T^{5} + 1260 T^{6} - 253 T^{7} - 4 T^{8} + T^{9} \)
$73$ \( -41371456 + 12753440 T + 14138360 T^{2} - 1016792 T^{3} - 634711 T^{4} + 57870 T^{5} + 4593 T^{6} - 464 T^{7} - 9 T^{8} + T^{9} \)
$79$ \( 3373045 - 6722294 T + 1443135 T^{2} + 1082440 T^{3} + 7905 T^{4} - 35595 T^{5} - 2746 T^{6} + 243 T^{7} + 34 T^{8} + T^{9} \)
$83$ \( 743353 + 470694 T - 1291590 T^{2} - 708834 T^{3} - 33263 T^{4} + 27639 T^{5} + 2506 T^{6} - 261 T^{7} - 15 T^{8} + T^{9} \)
$89$ \( -318322999 + 113607021 T + 10882147 T^{2} - 5905196 T^{3} - 44233 T^{4} + 92901 T^{5} - 80 T^{6} - 538 T^{7} + T^{9} \)
$97$ \( 336832 + 930720 T + 529720 T^{2} - 11368 T^{3} - 51883 T^{4} - 492 T^{5} + 1903 T^{6} - 90 T^{7} - 15 T^{8} + T^{9} \)
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