Properties

Label 10000.2.a.bo
Level $10000$
Weight $2$
Character orbit 10000.a
Self dual yes
Analytic conductor $79.850$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 10000 = 2^{4} \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 10000.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(79.8504020213\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \( x^{12} - x^{11} - 20 x^{10} + 11 x^{9} + 144 x^{8} - 29 x^{7} - 440 x^{6} + 4 x^{5} + 556 x^{4} + 15 x^{3} - 285 x^{2} + 45 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 5 \)
Twist minimal: no (minimal twist has level 5000)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{6} q^{3} + (\beta_{5} - \beta_{2} + \beta_1 + 1) q^{7} + ( - \beta_{10} - \beta_{9} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{3} + (\beta_{5} - \beta_{2} + \beta_1 + 1) q^{7} + ( - \beta_{10} - \beta_{9} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + 2) q^{9} + (\beta_{10} + \beta_{8} + \beta_{6} - \beta_{3} - \beta_{2} + \beta_1) q^{11} + (\beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} + 2 \beta_{2} - 2) q^{13} + ( - \beta_{10} - \beta_{8} + 2 \beta_{7} + \beta_{2}) q^{17} + (\beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} + \beta_{2} - 2) q^{19} + (2 \beta_{11} - \beta_{9} + \beta_{8} - 3 \beta_{7} + \beta_{6} - 2 \beta_{5} + \beta_{2} - \beta_1 - 1) q^{21} + ( - \beta_{11} - \beta_{10} - 2 \beta_{8} - \beta_{7} - \beta_{6} - \beta_{2} - \beta_1 + 1) q^{23} + (\beta_{10} + \beta_{9} - 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{2} + \beta_1 - 1) q^{27} + ( - \beta_{11} - \beta_{8} + \beta_{7} - \beta_{6} + 2 \beta_{5} + \beta_{4} + \beta_{3} - \beta_1 + 1) q^{29} + (\beta_{11} + \beta_{10} - \beta_{9} - \beta_{7} - 2 \beta_{5} + \beta_{2} - 4) q^{31} + ( - \beta_{11} - 2 \beta_{8} - \beta_{7} - 2 \beta_{6} + 2 \beta_{3} - 2 \beta_{2} - \beta_1 - 1) q^{33} + (\beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} + \beta_{4} - 2 \beta_{3} - \beta_1 - 2) q^{37} + ( - 2 \beta_{11} - \beta_{10} - 3 \beta_{8} + \beta_{7} - 3 \beta_{6} + 5 \beta_{5} + 2 \beta_{4} + \cdots + 1) q^{39}+ \cdots + (\beta_{8} + \beta_{7} + 4 \beta_{6} - 7 \beta_{5} - \beta_{4} - 3 \beta_{3} + 11 \beta_{2} + \cdots - 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{3} + 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 4 q^{3} + 26 q^{9} + q^{11} - 4 q^{13} - 8 q^{17} - 9 q^{19} + 12 q^{21} - 37 q^{27} + 8 q^{29} - 33 q^{31} - 26 q^{33} - 6 q^{37} - 14 q^{39} + 27 q^{41} - 50 q^{43} + 18 q^{47} + 12 q^{49} + 5 q^{51} - 22 q^{53} - 36 q^{57} - 33 q^{59} - 8 q^{61} - 26 q^{63} - 41 q^{67} + 3 q^{69} - 19 q^{71} + 5 q^{73} + 13 q^{77} - 58 q^{79} + 68 q^{81} - 18 q^{83} - 48 q^{87} + 44 q^{89} - 46 q^{91} - 10 q^{93} - 22 q^{97} - 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} - 20 x^{10} + 11 x^{9} + 144 x^{8} - 29 x^{7} - 440 x^{6} + 4 x^{5} + 556 x^{4} + 15 x^{3} - 285 x^{2} + 45 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 307 \nu^{11} - 8733 \nu^{10} - 678 \nu^{9} + 180827 \nu^{8} - 27635 \nu^{7} - 1274457 \nu^{6} + 159642 \nu^{5} + 3473001 \nu^{4} - 410492 \nu^{3} - 3087181 \nu^{2} + \cdots + 648531 ) / 63003 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 802 \nu^{11} - 6738 \nu^{10} + 24756 \nu^{9} + 135204 \nu^{8} - 214775 \nu^{7} - 939928 \nu^{6} + 642787 \nu^{5} + 2541583 \nu^{4} - 590618 \nu^{3} + \cdots + 568359 ) / 63003 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1351 \nu^{11} - 13969 \nu^{10} + 48406 \nu^{9} + 261955 \nu^{8} - 433991 \nu^{7} - 1723439 \nu^{6} + 1296397 \nu^{5} + 4411093 \nu^{4} - 1154914 \nu^{3} + \cdots + 577755 ) / 63003 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2231 \nu^{11} + 529 \nu^{10} + 42209 \nu^{9} + 21083 \nu^{8} - 281581 \nu^{7} - 323536 \nu^{6} + 746510 \nu^{5} + 1275695 \nu^{4} - 692225 \nu^{3} - 1604243 \nu^{2} + \cdots + 536010 ) / 63003 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3607 \nu^{11} - 10598 \nu^{10} - 56193 \nu^{9} + 160053 \nu^{8} + 293370 \nu^{7} - 848573 \nu^{6} - 513662 \nu^{5} + 1905609 \nu^{4} - 8724 \nu^{3} - 1635878 \nu^{2} + \cdots + 348303 ) / 63003 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 5601 \nu^{11} - 1703 \nu^{10} + 118948 \nu^{9} + 78063 \nu^{8} - 861089 \nu^{7} - 798674 \nu^{6} + 2409179 \nu^{5} + 2723982 \nu^{4} - 2248279 \nu^{3} + \cdots + 888045 ) / 63003 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 7021 \nu^{11} + 11191 \nu^{10} + 129472 \nu^{9} - 144518 \nu^{8} - 854757 \nu^{7} + 587287 \nu^{6} + 2324195 \nu^{5} - 910206 \nu^{4} - 2310683 \nu^{3} + \cdots - 140112 ) / 63003 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 9252 \nu^{11} - 11720 \nu^{10} - 171681 \nu^{9} + 123435 \nu^{8} + 1136338 \nu^{7} - 263751 \nu^{6} - 3070705 \nu^{5} - 365489 \nu^{4} + 3065911 \nu^{3} + \cdots - 269892 ) / 63003 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 11122 \nu^{11} + 4724 \nu^{10} - 234641 \nu^{9} - 183751 \nu^{8} + 1678758 \nu^{7} + 1757068 \nu^{6} - 4587698 \nu^{5} - 5542082 \nu^{4} + 4101418 \nu^{3} + \cdots - 1338864 ) / 63003 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 20291 \nu^{11} + 14554 \nu^{10} + 393757 \nu^{9} - 88219 \nu^{8} - 2649292 \nu^{7} - 451469 \nu^{6} + 6888818 \nu^{5} + 2888145 \nu^{4} - 5681070 \nu^{3} + \cdots + 608583 ) / 63003 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 28451 \nu^{11} - 22984 \nu^{10} - 557759 \nu^{9} + 196840 \nu^{8} + 3801976 \nu^{7} + 15784 \nu^{6} - 10129172 \nu^{5} - 2213534 \nu^{4} + 8855582 \nu^{3} + \cdots - 914526 ) / 63003 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 2\beta_{10} + \beta_{9} + 3\beta_{8} + 3\beta_{6} + 2\beta_{5} - 2\beta_{4} - 2\beta_{3} - 4\beta_{2} + 3\beta _1 + 2 ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3 \beta_{10} + 4 \beta_{9} + 2 \beta_{8} + 2 \beta_{6} + 3 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 6 \beta_{2} + 2 \beta _1 + 18 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3 \beta_{10} + 2 \beta_{9} + 5 \beta_{8} + \beta_{7} + 4 \beta_{6} + 3 \beta_{5} - \beta_{4} - 2 \beta_{3} - 6 \beta_{2} + 4 \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 5 \beta_{11} + 32 \beta_{10} + 41 \beta_{9} + 33 \beta_{8} - 5 \beta_{7} + 28 \beta_{6} + 37 \beta_{5} + 18 \beta_{4} + 8 \beta_{3} - 74 \beta_{2} + 33 \beta _1 + 122 ) / 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 10 \beta_{11} + 133 \beta_{10} + 99 \beta_{9} + 207 \beta_{8} + 20 \beta_{7} + 147 \beta_{6} + 153 \beta_{5} + 2 \beta_{4} - 63 \beta_{3} - 306 \beta_{2} + 172 \beta _1 + 238 ) / 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 15 \beta_{11} + 69 \beta_{10} + 80 \beta_{9} + 81 \beta_{8} - 16 \beta_{7} + 56 \beta_{6} + 88 \beta_{5} + 37 \beta_{4} + \beta_{3} - 164 \beta_{2} + 78 \beta _1 + 204 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 175 \beta_{11} + 1302 \beta_{10} + 1031 \beta_{9} + 1853 \beta_{8} - 65 \beta_{7} + 1168 \beta_{6} + 1647 \beta_{5} + 318 \beta_{4} - 477 \beta_{3} - 3129 \beta_{2} + 1588 \beta _1 + 2567 ) / 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 875 \beta_{11} + 3753 \beta_{10} + 3944 \beta_{9} + 4517 \beta_{8} - 1075 \beta_{7} + 2657 \beta_{6} + 5078 \beta_{5} + 2047 \beta_{4} - 423 \beta_{3} - 8866 \beta_{2} + 4212 \beta _1 + 9383 ) / 5 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 463 \beta_{11} + 2662 \beta_{10} + 2182 \beta_{9} + 3516 \beta_{8} - 511 \beta_{7} + 1966 \beta_{6} + 3589 \beta_{5} + 1043 \beta_{4} - 833 \beta_{3} - 6398 \beta_{2} + 3055 \beta _1 + 5343 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 9580 \beta_{11} + 40577 \beta_{10} + 39526 \beta_{9} + 48468 \beta_{8} - 13430 \beta_{7} + 25073 \beta_{6} + 57032 \beta_{5} + 22963 \beta_{4} - 7372 \beta_{3} - 94789 \beta_{2} + 44143 \beta _1 + 90952 ) / 5 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 27580 \beta_{11} + 138588 \beta_{10} + 115539 \beta_{9} + 173197 \beta_{8} - 39595 \beta_{7} + 86647 \beta_{6} + 194933 \beta_{5} + 67367 \beta_{4} - 39963 \beta_{3} - 328846 \beta_{2} + \cdots + 275263 ) / 5 \) Copy content Toggle raw display

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
−0.504162
−2.17259
0.571452
2.75578
3.24312
1.91609
−2.46481
−1.99544
−0.969395
−1.14928
1.01267
0.756559
0 −3.42065 0 0 0 −3.76868 0 8.70082 0
1.2 0 −3.13445 0 0 0 −2.92885 0 6.82478 0
1.3 0 −2.94508 0 0 0 3.90152 0 5.67351 0
1.4 0 −2.81354 0 0 0 0.0468346 0 4.91602 0
1.5 0 −0.602175 0 0 0 3.10675 0 −2.63738 0
1.6 0 −0.444623 0 0 0 −2.92309 0 −2.80231 0
1.7 0 −0.208054 0 0 0 0.439180 0 −2.95671 0
1.8 0 0.0330070 0 0 0 2.74120 0 −2.99891 0
1.9 0 1.78053 0 0 0 −4.82418 0 0.170304 0
1.10 0 2.38971 0 0 0 2.31613 0 2.71069 0
1.11 0 2.62870 0 0 0 0.0349473 0 3.91007 0
1.12 0 2.73663 0 0 0 1.85824 0 4.48912 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(1\)

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 10000.2.a.bo 12
4.b odd 2 1 5000.2.a.p yes 12
5.b even 2 1 10000.2.a.bp 12
20.d odd 2 1 5000.2.a.o 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5000.2.a.o 12 20.d odd 2 1
5000.2.a.p yes 12 4.b odd 2 1
10000.2.a.bo 12 1.a even 1 1 trivial
10000.2.a.bp 12 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(10000))\):

\( T_{3}^{12} + 4 T_{3}^{11} - 23 T_{3}^{10} - 93 T_{3}^{9} + 195 T_{3}^{8} + 773 T_{3}^{7} - 708 T_{3}^{6} - 2624 T_{3}^{5} + 726 T_{3}^{4} + 2760 T_{3}^{3} + 1165 T_{3}^{2} + 110 T_{3} - 5 \) Copy content Toggle raw display
\( T_{7}^{12} - 48 T_{7}^{10} + 29 T_{7}^{9} + 824 T_{7}^{8} - 919 T_{7}^{7} - 5896 T_{7}^{6} + 9353 T_{7}^{5} + 13407 T_{7}^{4} - 30798 T_{7}^{3} + 12224 T_{7}^{2} - 848 T_{7} + 16 \) Copy content Toggle raw display
\( T_{11}^{12} - T_{11}^{11} - 68 T_{11}^{10} + 48 T_{11}^{9} + 1533 T_{11}^{8} - 1322 T_{11}^{7} - 14022 T_{11}^{6} + 17573 T_{11}^{5} + 44673 T_{11}^{4} - 78417 T_{11}^{3} + 4252 T_{11}^{2} + 29479 T_{11} - 8609 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 4 T^{11} - 23 T^{10} - 93 T^{9} + \cdots - 5 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} - 48 T^{10} + 29 T^{9} + 824 T^{8} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( T^{12} - T^{11} - 68 T^{10} + 48 T^{9} + \cdots - 8609 \) Copy content Toggle raw display
$13$ \( T^{12} + 4 T^{11} - 89 T^{10} + \cdots + 302256 \) Copy content Toggle raw display
$17$ \( T^{12} + 8 T^{11} - 106 T^{10} + \cdots - 4824725 \) Copy content Toggle raw display
$19$ \( T^{12} + 9 T^{11} - 73 T^{10} + \cdots + 4621 \) Copy content Toggle raw display
$23$ \( T^{12} - 143 T^{10} + 41 T^{9} + \cdots - 8514864 \) Copy content Toggle raw display
$29$ \( T^{12} - 8 T^{11} - 117 T^{10} + \cdots - 527024 \) Copy content Toggle raw display
$31$ \( T^{12} + 33 T^{11} + 319 T^{10} + \cdots + 9284400 \) Copy content Toggle raw display
$37$ \( T^{12} + 6 T^{11} - 283 T^{10} + \cdots + 650813616 \) Copy content Toggle raw display
$41$ \( T^{12} - 27 T^{11} + \cdots - 995145329 \) Copy content Toggle raw display
$43$ \( T^{12} + 50 T^{11} + \cdots + 347146281 \) Copy content Toggle raw display
$47$ \( T^{12} - 18 T^{11} - 154 T^{10} + \cdots - 26255664 \) Copy content Toggle raw display
$53$ \( T^{12} + 22 T^{11} + \cdots + 189235456 \) Copy content Toggle raw display
$59$ \( T^{12} + 33 T^{11} + 199 T^{10} + \cdots - 4622139 \) Copy content Toggle raw display
$61$ \( T^{12} + 8 T^{11} - 221 T^{10} + \cdots - 8891600 \) Copy content Toggle raw display
$67$ \( T^{12} + 41 T^{11} + \cdots - 23687194259 \) Copy content Toggle raw display
$71$ \( T^{12} + 19 T^{11} + \cdots - 6051744080 \) Copy content Toggle raw display
$73$ \( T^{12} - 5 T^{11} - 418 T^{10} + \cdots + 927999711 \) Copy content Toggle raw display
$79$ \( T^{12} + 58 T^{11} + \cdots + 39511336720 \) Copy content Toggle raw display
$83$ \( T^{12} + 18 T^{11} + \cdots - 4396075199 \) Copy content Toggle raw display
$89$ \( T^{12} - 44 T^{11} + \cdots - 82788670169 \) Copy content Toggle raw display
$97$ \( T^{12} + 22 T^{11} + \cdots + 171567421 \) Copy content Toggle raw display
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