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: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
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:

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

Atkin-Lehner signs

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

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

$p$ $F_p(T)$
$2$ \( \)
$3$ \( 1 - 3 T + 8 T^{2} - 21 T^{3} + 41 T^{4} - 85 T^{5} + 168 T^{6} - 327 T^{7} + 639 T^{8} - 1094 T^{9} + 1917 T^{10} - 2943 T^{11} + 4536 T^{12} - 6885 T^{13} + 9963 T^{14} - 15309 T^{15} + 17496 T^{16} - 19683 T^{17} + 19683 T^{18} \)
$5$ \( 1 + 21 T^{2} + 4 T^{3} + 231 T^{4} + 125 T^{5} + 1723 T^{6} + 1475 T^{7} + 10031 T^{8} + 9583 T^{9} + 50155 T^{10} + 36875 T^{11} + 215375 T^{12} + 78125 T^{13} + 721875 T^{14} + 62500 T^{15} + 1640625 T^{16} + 1953125 T^{18} \)
$7$ \( ( 1 - T )^{9} \)
$11$ \( ( 1 - T )^{9} \)
$13$ \( ( 1 + T )^{9} \)
$17$ \( 1 - 5 T + 58 T^{2} - 269 T^{3} + 1851 T^{4} - 6863 T^{5} + 35928 T^{6} - 115557 T^{7} + 557691 T^{8} - 1709086 T^{9} + 9480747 T^{10} - 33395973 T^{11} + 176514264 T^{12} - 573204623 T^{13} + 2628155307 T^{14} - 6493006061 T^{15} + 23799643034 T^{16} - 34878787205 T^{17} + 118587876497 T^{18} \)
$19$ \( 1 - 10 T + 129 T^{2} - 858 T^{3} + 6799 T^{4} - 36687 T^{5} + 230863 T^{6} - 1094153 T^{7} + 5856445 T^{8} - 24273225 T^{9} + 111272455 T^{10} - 394989233 T^{11} + 1583489317 T^{12} - 4781086527 T^{13} + 16834997101 T^{14} - 40365365898 T^{15} + 115309454331 T^{16} - 169835630410 T^{17} + 322687697779 T^{18} \)
$23$ \( 1 - 8 T + 146 T^{2} - 896 T^{3} + 9956 T^{4} - 51621 T^{5} + 438850 T^{6} - 1963688 T^{7} + 13741055 T^{8} - 53051974 T^{9} + 316044265 T^{10} - 1038790952 T^{11} + 5339487950 T^{12} - 14445672261 T^{13} + 64080230908 T^{14} - 132640156544 T^{15} + 497104515262 T^{16} - 626487882248 T^{17} + 1801152661463 T^{18} \)
$29$ \( 1 - 14 T + 214 T^{2} - 1810 T^{3} + 15904 T^{4} - 98781 T^{5} + 676330 T^{6} - 3606286 T^{7} + 22533647 T^{8} - 111646538 T^{9} + 653475763 T^{10} - 3032886526 T^{11} + 16495012370 T^{12} - 69865924461 T^{13} + 326209313696 T^{14} - 1076630211010 T^{15} + 3691473530126 T^{16} - 7003449781454 T^{17} + 14507145975869 T^{18} \)
$31$ \( 1 - 11 T + 93 T^{2} - 474 T^{3} + 4027 T^{4} - 25667 T^{5} + 195325 T^{6} - 1011266 T^{7} + 7205150 T^{8} - 36845532 T^{9} + 223359650 T^{10} - 971826626 T^{11} + 5818927075 T^{12} - 23704013507 T^{13} + 115289591077 T^{14} - 420676744794 T^{15} + 2558673112323 T^{16} - 9381801411851 T^{17} + 26439622160671 T^{18} \)
$37$ \( 1 + 145 T^{2} - 175 T^{3} + 9615 T^{4} - 26440 T^{5} + 384903 T^{6} - 1904977 T^{7} + 12041144 T^{8} - 86185728 T^{9} + 445522328 T^{10} - 2607913513 T^{11} + 19496491659 T^{12} - 49552816840 T^{13} + 666742146555 T^{14} - 449002121575 T^{15} + 13765122184285 T^{16} + 129961739795077 T^{18} \)
$41$ \( 1 - 14 T + 283 T^{2} - 3147 T^{3} + 38143 T^{4} - 345712 T^{5} + 3189999 T^{6} - 24091885 T^{7} + 183199126 T^{8} - 1168958292 T^{9} + 7511164166 T^{10} - 40498458685 T^{11} + 219857921079 T^{12} - 976899486832 T^{13} + 4419103074743 T^{14} - 14948578046427 T^{15} + 55115459508323 T^{16} - 111788953207694 T^{17} + 327381934393961 T^{18} \)
$43$ \( 1 - 8 T + 176 T^{2} - 946 T^{3} + 13274 T^{4} - 55927 T^{5} + 667442 T^{6} - 2392933 T^{7} + 27280255 T^{8} - 92929925 T^{9} + 1173050965 T^{10} - 4424533117 T^{11} + 53066311094 T^{12} - 191203283527 T^{13} + 1951390072382 T^{14} - 5980009444354 T^{15} + 47840075554832 T^{16} - 93505602220808 T^{17} + 502592611936843 T^{18} \)
$47$ \( 1 - 10 T + 182 T^{2} - 1176 T^{3} + 16608 T^{4} - 94125 T^{5} + 1162654 T^{6} - 5625600 T^{7} + 62630179 T^{8} - 270886898 T^{9} + 2943618413 T^{10} - 12426950400 T^{11} + 120710226242 T^{12} - 459299974125 T^{13} + 3808961876256 T^{14} - 12676357226904 T^{15} + 92205407924266 T^{16} - 238112866617610 T^{17} + 1119130473102767 T^{18} \)
$53$ \( 1 - 21 T + 277 T^{2} - 2968 T^{3} + 30947 T^{4} - 288185 T^{5} + 2342249 T^{6} - 18000300 T^{7} + 137833141 T^{8} - 1040546905 T^{9} + 7305156473 T^{10} - 50562842700 T^{11} + 348707004373 T^{12} - 2273918266985 T^{13} + 12941895921871 T^{14} - 65783823830872 T^{15} + 325394985734849 T^{16} - 1307453498638581 T^{17} + 3299763591802133 T^{18} \)
$59$ \( 1 - 23 T + 518 T^{2} - 8163 T^{3} + 115526 T^{4} - 1388946 T^{5} + 15076132 T^{6} - 146537445 T^{7} + 1293518891 T^{8} - 10417971342 T^{9} + 76317614569 T^{10} - 510096846045 T^{11} + 3096320914028 T^{12} - 16830360091506 T^{13} + 82592344566274 T^{14} - 344319696111483 T^{15} + 1289121469136242 T^{16} - 3377100064899383 T^{17} + 8662995818654939 T^{18} \)
$61$ \( 1 - 34 T + 860 T^{2} - 15381 T^{3} + 233681 T^{4} - 2948067 T^{5} + 33192584 T^{6} - 328091222 T^{7} + 2968578671 T^{8} - 24128224246 T^{9} + 181083298931 T^{10} - 1220827437062 T^{11} + 7534085908904 T^{12} - 40818466939347 T^{13} + 197366108213981 T^{14} - 792434878046541 T^{15} + 2702758838978060 T^{16} - 6518048641907554 T^{17} + 11694146092834141 T^{18} \)
$67$ \( 1 - 10 T + 350 T^{2} - 3424 T^{3} + 56851 T^{4} - 534274 T^{5} + 5808712 T^{6} - 52969882 T^{7} + 447475317 T^{8} - 3957619294 T^{9} + 29980846239 T^{10} - 237781800298 T^{11} + 1747045647256 T^{12} - 10766220021154 T^{13} + 76755962458057 T^{14} - 309729500546656 T^{15} + 2121249061863050 T^{16} - 4060676775566410 T^{17} + 27206534396294947 T^{18} \)
$71$ \( 1 - 4 T + 386 T^{2} - 1012 T^{3} + 74320 T^{4} - 132554 T^{5} + 9430158 T^{6} - 11810548 T^{7} + 876200831 T^{8} - 881477364 T^{9} + 62210259001 T^{10} - 59536972468 T^{11} + 3375157279938 T^{12} - 3368419963274 T^{13} + 134090325366320 T^{14} - 129637487328052 T^{15} + 3510716381138926 T^{16} - 2583014124983044 T^{17} + 45848500718449031 T^{18} \)
$73$ \( 1 - 9 T + 193 T^{2} - 663 T^{3} + 12610 T^{4} + 34115 T^{5} + 857410 T^{6} - 120365 T^{7} + 134510578 T^{8} - 427210620 T^{9} + 9819272194 T^{10} - 641425085 T^{11} + 333547065970 T^{12} + 968805991715 T^{13} + 26141432787730 T^{14} - 100334592029607 T^{15} + 2132147914185721 T^{16} - 7258140827046729 T^{17} + 58871586708267913 T^{18} \)
$79$ \( 1 + 34 T + 954 T^{2} + 18742 T^{3} + 323460 T^{4} + 4647733 T^{5} + 60285514 T^{6} + 685620581 T^{7} + 7129338937 T^{8} + 66150889905 T^{9} + 563217776023 T^{10} + 4278958046021 T^{11} + 29723109537046 T^{12} + 181029576816373 T^{13} + 995304662820540 T^{14} + 4555945091374582 T^{15} + 18320529172795686 T^{16} + 51581699536823074 T^{17} + 119851595982618319 T^{18} \)
$83$ \( 1 - 15 T + 486 T^{2} - 7454 T^{3} + 124002 T^{4} - 1678655 T^{5} + 21032850 T^{6} - 233678476 T^{7} + 2484495939 T^{8} - 22761826039 T^{9} + 206213162937 T^{10} - 1609811021164 T^{11} + 12026310202950 T^{12} - 79666147838255 T^{13} + 488448917813286 T^{14} - 2437013543092526 T^{15} + 13188120780958722 T^{16} - 33784383482085615 T^{17} + 186940255267540403 T^{18} \)
$89$ \( 1 + 263 T^{2} - 80 T^{3} + 42883 T^{4} - 86953 T^{5} + 5161687 T^{6} - 14370001 T^{7} + 526563995 T^{8} - 1611468791 T^{9} + 46864195555 T^{10} - 113824777921 T^{11} + 3638829322703 T^{12} - 5455626081673 T^{13} + 239461221351467 T^{14} - 39758503276880 T^{15} + 11632841077524127 T^{16} + 350356403707485209 T^{18} \)
$97$ \( 1 - 15 T + 783 T^{2} - 9737 T^{3} + 277122 T^{4} - 2896117 T^{5} + 58631534 T^{6} - 517666299 T^{7} + 8231099808 T^{8} - 61045311040 T^{9} + 798416681376 T^{10} - 4870722207291 T^{11} + 53511418030382 T^{12} - 256391155701877 T^{13} + 2379740906700354 T^{14} - 8110648411993673 T^{15} + 63265056746362479 T^{16} - 117561503915654415 T^{17} + 760231058654565217 T^{18} \)
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