Properties

Label 23.8.a.b
Level $23$
Weight $8$
Character orbit 23.a
Self dual yes
Analytic conductor $7.185$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 23 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 23.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.18485558613\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - 832x^{6} - 1059x^{5} + 203052x^{4} + 678328x^{3} - 13424272x^{2} - 73308944x - 37372224 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4}\cdot 5 \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} - \beta_1 + 5) q^{3} + (\beta_{3} + \beta_{2} + 2 \beta_1 + 80) q^{4} + ( - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} + 12 \beta_1 + 56) q^{5} + (3 \beta_{7} + 4 \beta_{6} + 3 \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + 16 \beta_1 - 220) q^{6} + ( - 6 \beta_{6} - \beta_{5} + 5 \beta_{4} - 4 \beta_{3} - \beta_{2} + 9 \beta_1 + 183) q^{7} + ( - 5 \beta_{7} + 8 \beta_{6} + \beta_{5} - 6 \beta_{4} + \beta_{3} - 16 \beta_{2} + \cdots + 395) q^{8}+ \cdots + (\beta_{7} - 9 \beta_{6} - 7 \beta_{5} - 6 \beta_{4} + 5 \beta_{3} - \beta_{2} - 31 \beta_1 + 1736) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} - \beta_1 + 5) q^{3} + (\beta_{3} + \beta_{2} + 2 \beta_1 + 80) q^{4} + ( - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} + 12 \beta_1 + 56) q^{5} + (3 \beta_{7} + 4 \beta_{6} + 3 \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + 16 \beta_1 - 220) q^{6} + ( - 6 \beta_{6} - \beta_{5} + 5 \beta_{4} - 4 \beta_{3} - \beta_{2} + 9 \beta_1 + 183) q^{7} + ( - 5 \beta_{7} + 8 \beta_{6} + \beta_{5} - 6 \beta_{4} + \beta_{3} - 16 \beta_{2} + \cdots + 395) q^{8}+ \cdots + ( - 51954 \beta_{7} - 3384 \beta_{6} - 10116 \beta_{5} + \cdots - 8083478) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 40 q^{3} + 640 q^{4} + 444 q^{5} - 1745 q^{6} + 1446 q^{7} + 3177 q^{8} + 13878 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 40 q^{3} + 640 q^{4} + 444 q^{5} - 1745 q^{6} + 1446 q^{7} + 3177 q^{8} + 13878 q^{9} + 19502 q^{10} + 7588 q^{11} + 22733 q^{12} + 19862 q^{13} + 17544 q^{14} - 12770 q^{15} + 64336 q^{16} + 42070 q^{17} - 59129 q^{18} + 1050 q^{19} + 3364 q^{20} - 7698 q^{21} - 128220 q^{22} - 97336 q^{23} - 621188 q^{24} + 49496 q^{25} - 371761 q^{26} - 69500 q^{27} + 143050 q^{28} - 102578 q^{29} - 671470 q^{30} + 304172 q^{31} - 612824 q^{32} + 747242 q^{33} - 524530 q^{34} + 531048 q^{35} + 1868983 q^{36} + 286472 q^{37} - 762932 q^{38} + 1032828 q^{39} + 2105286 q^{40} + 1324414 q^{41} - 1886168 q^{42} + 2052578 q^{43} - 867298 q^{44} + 2087442 q^{45} + 675556 q^{47} + 1411151 q^{48} - 55404 q^{49} + 1458528 q^{50} + 2775482 q^{51} - 1695409 q^{52} + 203654 q^{53} - 9897559 q^{54} - 1024444 q^{55} - 5766846 q^{56} + 3908648 q^{57} - 5039991 q^{58} - 748892 q^{59} - 18153300 q^{60} + 61822 q^{61} - 4939277 q^{62} + 1411632 q^{63} + 2702267 q^{64} - 1571618 q^{65} + 3791866 q^{66} + 3235604 q^{67} + 4914980 q^{68} - 486680 q^{69} + 10871764 q^{70} - 4951664 q^{71} - 7940241 q^{72} + 11019370 q^{73} + 356954 q^{74} - 13607220 q^{75} + 21973240 q^{76} - 5284888 q^{77} - 1506779 q^{78} + 4202464 q^{79} + 8785886 q^{80} + 10294096 q^{81} + 32636759 q^{82} + 518568 q^{83} + 7629190 q^{84} + 9854220 q^{85} - 14681386 q^{86} + 4862532 q^{87} + 20589740 q^{88} + 4203864 q^{89} + 49021076 q^{90} + 2488406 q^{91} - 7786880 q^{92} - 23367842 q^{93} + 12314327 q^{94} - 44485300 q^{95} - 45317009 q^{96} + 18621134 q^{97} + 35756 q^{98} - 64729930 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 832x^{6} - 1059x^{5} + 203052x^{4} + 678328x^{3} - 13424272x^{2} - 73308944x - 37372224 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 70454057 \nu^{7} + 418096898 \nu^{6} + 53704469932 \nu^{5} - 276596270045 \nu^{4} - 11462075003394 \nu^{3} + \cdots + 209986539079392 ) / 3315013604640 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 70454057 \nu^{7} - 418096898 \nu^{6} - 53704469932 \nu^{5} + 276596270045 \nu^{4} + 11462075003394 \nu^{3} + \cdots - 899509368844512 ) / 3315013604640 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 99425533 \nu^{7} + 632516202 \nu^{6} + 81664904028 \nu^{5} - 385294374305 \nu^{4} - 19245461052586 \nu^{3} + \cdots + 10\!\cdots\!08 ) / 3315013604640 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11355547 \nu^{7} - 155754618 \nu^{6} - 9395550780 \nu^{5} + 95325842855 \nu^{4} + 2172142308418 \nu^{3} - 11800181584396 \nu^{2} + \cdots + 20717221503936 ) / 331501360464 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 66289297 \nu^{7} + 447046798 \nu^{6} + 51731639972 \nu^{5} - 294409331845 \nu^{4} - 11272965745974 \nu^{3} + \cdots + 178479916668432 ) / 1657506802320 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 169439777 \nu^{7} - 1061508378 \nu^{6} - 133842936252 \nu^{5} + 651322391125 \nu^{4} + 29673399783434 \nu^{3} + \cdots - 12\!\cdots\!72 ) / 3315013604640 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + 2\beta _1 + 208 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -5\beta_{7} + 8\beta_{6} + \beta_{5} - 6\beta_{4} + \beta_{3} - 16\beta_{2} + 345\beta _1 + 395 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -72\beta_{7} - 110\beta_{6} - 10\beta_{5} - 25\beta_{4} + 389\beta_{3} + 442\beta_{2} + 672\beta _1 + 71549 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 2255 \beta_{7} + 4809 \beta_{6} + 610 \beta_{5} - 2433 \beta_{4} + 546 \beta_{3} - 9510 \beta_{2} + 133649 \beta _1 + 125645 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 40845 \beta_{7} - 74022 \beta_{6} - 10925 \beta_{5} - 16040 \beta_{4} + 153621 \beta_{3} + 199710 \beta_{2} + 146421 \beta _1 + 27779983 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 865180 \beta_{7} + 2356790 \beta_{6} + 276718 \beta_{5} - 875491 \beta_{4} + 141203 \beta_{3} - 4740018 \beta_{2} + 54123540 \beta _1 + 23127043 \) 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
−21.3077
−14.5712
−7.41631
−6.60982
−0.570902
11.0962
19.4241
19.9556
−21.3077 86.9475 326.017 −188.239 −1852.65 639.155 −4219.28 5372.88 4010.94
1.2 −14.5712 −10.7680 84.3195 −404.860 156.902 −387.911 636.476 −2071.05 5899.30
1.3 −7.41631 65.3727 −72.9984 376.733 −484.824 −902.074 1490.67 2086.59 −2793.97
1.4 −6.60982 −84.4445 −84.3103 −124.345 558.163 −780.885 1403.33 4943.87 821.899
1.5 −0.570902 −37.8447 −127.674 128.909 21.6056 1733.23 145.965 −754.777 −73.5944
1.6 11.0962 60.8046 −4.87416 165.526 674.701 952.148 −1474.40 1510.20 1836.71
1.7 19.4241 36.3647 249.296 −31.9528 706.352 −461.175 2356.08 −864.611 −620.655
1.8 19.9556 −76.4323 270.224 522.229 −1525.25 653.513 2838.16 3654.90 10421.4
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(23\) \(1\)

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 23.8.a.b 8
3.b odd 2 1 207.8.a.f 8
4.b odd 2 1 368.8.a.h 8
5.b even 2 1 575.8.a.b 8
23.b odd 2 1 529.8.a.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.8.a.b 8 1.a even 1 1 trivial
207.8.a.f 8 3.b odd 2 1
368.8.a.h 8 4.b odd 2 1
529.8.a.c 8 23.b odd 2 1
575.8.a.b 8 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 832T_{2}^{6} - 1059T_{2}^{5} + 203052T_{2}^{4} + 678328T_{2}^{3} - 13424272T_{2}^{2} - 73308944T_{2} - 37372224 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(23))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 832 T^{6} + \cdots - 37372224 \) Copy content Toggle raw display
$3$ \( T^{8} - 40 T^{7} + \cdots + 33056528652000 \) Copy content Toggle raw display
$5$ \( T^{8} - 444 T^{7} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{8} - 1446 T^{7} + \cdots + 86\!\cdots\!92 \) Copy content Toggle raw display
$11$ \( T^{8} - 7588 T^{7} + \cdots + 23\!\cdots\!20 \) Copy content Toggle raw display
$13$ \( T^{8} - 19862 T^{7} + \cdots + 39\!\cdots\!68 \) Copy content Toggle raw display
$17$ \( T^{8} - 42070 T^{7} + \cdots - 38\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{8} - 1050 T^{7} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T + 12167)^{8} \) Copy content Toggle raw display
$29$ \( T^{8} + 102578 T^{7} + \cdots - 13\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{8} - 304172 T^{7} + \cdots + 80\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{8} - 286472 T^{7} + \cdots + 16\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( T^{8} - 1324414 T^{7} + \cdots - 38\!\cdots\!40 \) Copy content Toggle raw display
$43$ \( T^{8} - 2052578 T^{7} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{8} - 675556 T^{7} + \cdots + 82\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{8} - 203654 T^{7} + \cdots + 18\!\cdots\!88 \) Copy content Toggle raw display
$59$ \( T^{8} + 748892 T^{7} + \cdots - 15\!\cdots\!60 \) Copy content Toggle raw display
$61$ \( T^{8} - 61822 T^{7} + \cdots + 59\!\cdots\!88 \) Copy content Toggle raw display
$67$ \( T^{8} - 3235604 T^{7} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{8} + 4951664 T^{7} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{8} - 11019370 T^{7} + \cdots + 90\!\cdots\!68 \) Copy content Toggle raw display
$79$ \( T^{8} - 4202464 T^{7} + \cdots + 16\!\cdots\!60 \) Copy content Toggle raw display
$83$ \( T^{8} - 518568 T^{7} + \cdots + 20\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{8} - 4203864 T^{7} + \cdots - 38\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{8} - 18621134 T^{7} + \cdots + 17\!\cdots\!92 \) Copy content Toggle raw display
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