Properties

Label 23.12.a.a
Level $23$
Weight $12$
Character orbit 23.a
Self dual yes
Analytic conductor $17.672$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 23 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 23.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.6718931529\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - 2672x^{6} - 1234x^{5} + 2202967x^{4} + 2386582x^{3} - 543567396x^{2} - 1204011928x + 23305583840 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{11}\cdot 3^{2} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 4) q^{2} + (\beta_{2} - \beta_1 - 124) q^{3} + ( - \beta_{4} + \beta_{3} + 9 \beta_1 + 640) q^{4} + ( - \beta_{7} + 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 22 \beta_1 - 1433) q^{5} + (4 \beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_{4} - 5 \beta_{3} - 24 \beta_{2} + \cdots + 4327) q^{6}+ \cdots + ( - 4 \beta_{7} - 28 \beta_{6} + 13 \beta_{5} - 26 \beta_{4} + \cdots + 81120) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 4) q^{2} + (\beta_{2} - \beta_1 - 124) q^{3} + ( - \beta_{4} + \beta_{3} + 9 \beta_1 + 640) q^{4} + ( - \beta_{7} + 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 22 \beta_1 - 1433) q^{5} + (4 \beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_{4} - 5 \beta_{3} - 24 \beta_{2} + \cdots + 4327) q^{6}+ \cdots + (4909916 \beta_{7} - 1188214 \beta_{6} + \cdots + 23278335882) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 32 q^{2} - 992 q^{3} + 5120 q^{4} - 11466 q^{5} + 34618 q^{6} - 54118 q^{7} - 155568 q^{8} + 648814 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 32 q^{2} - 992 q^{3} + 5120 q^{4} - 11466 q^{5} + 34618 q^{6} - 54118 q^{7} - 155568 q^{8} + 648814 q^{9} + 517892 q^{10} - 291462 q^{11} - 884188 q^{12} - 2211306 q^{13} - 939584 q^{14} - 2205330 q^{15} - 8561344 q^{16} - 5775330 q^{17} - 51349034 q^{18} - 21015588 q^{19} - 65503576 q^{20} - 36171230 q^{21} - 83047784 q^{22} + 51490744 q^{23} - 129286728 q^{24} - 36491644 q^{25} - 119299562 q^{26} - 394617320 q^{27} - 392796032 q^{28} - 322285430 q^{29} - 646885140 q^{30} - 415184840 q^{31} + 31831744 q^{32} - 549306602 q^{33} - 28224252 q^{34} + 603721008 q^{35} + 690703676 q^{36} + 176642018 q^{37} + 554685496 q^{38} + 2251149264 q^{39} + 1337904816 q^{40} + 357962218 q^{41} + 340644280 q^{42} + 2500461376 q^{43} + 5064743472 q^{44} + 385017072 q^{45} - 205962976 q^{46} + 261795200 q^{47} + 4421752784 q^{48} + 2656605924 q^{49} + 1642758328 q^{50} + 6771514570 q^{51} + 3841657212 q^{52} + 3542935060 q^{53} + 18173306686 q^{54} - 10100187604 q^{55} + 7995463104 q^{56} - 14761628752 q^{57} - 9113565454 q^{58} + 930905396 q^{59} + 19344914040 q^{60} - 25338655048 q^{61} + 4385691666 q^{62} - 25499316044 q^{63} - 34067008768 q^{64} - 25954746658 q^{65} + 13172584012 q^{66} - 3123467482 q^{67} - 37358480280 q^{68} - 6384852256 q^{69} - 35719175696 q^{70} - 52612263236 q^{71} - 9100886376 q^{72} - 67014176274 q^{73} + 10171443276 q^{74} - 87540153860 q^{75} + 17955918576 q^{76} - 44516617816 q^{77} - 25596104778 q^{78} - 27683357604 q^{79} + 74357773216 q^{80} + 55141240264 q^{81} + 73615849126 q^{82} - 12253964262 q^{83} + 168565479344 q^{84} + 58779027600 q^{85} + 90522557252 q^{86} - 129275944888 q^{87} + 33736356800 q^{88} + 10662817760 q^{89} + 450294422856 q^{90} - 28336741418 q^{91} + 32954076160 q^{92} + 164368292014 q^{93} + 285145948346 q^{94} - 64104297380 q^{95} + 208023008864 q^{96} - 124519454530 q^{97} + 215615498272 q^{98} + 186256571332 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2672x^{6} - 1234x^{5} + 2202967x^{4} + 2386582x^{3} - 543567396x^{2} - 1204011928x + 23305583840 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 10909 \nu^{7} + 5556513046 \nu^{6} - 87447461660 \nu^{5} - 8309753879862 \nu^{4} + 127008950486609 \nu^{3} + \cdots - 11\!\cdots\!32 ) / 351247704839136 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 77166089 \nu^{7} + 2418025262 \nu^{6} + 155618608148 \nu^{5} - 3874884628830 \nu^{4} - 91423325240435 \nu^{3} + \cdots - 16\!\cdots\!92 ) / 50178243548448 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 77166089 \nu^{7} + 2418025262 \nu^{6} + 155618608148 \nu^{5} - 3874884628830 \nu^{4} - 91423325240435 \nu^{3} + \cdots - 35\!\cdots\!36 ) / 50178243548448 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 537325315 \nu^{7} - 524855422 \nu^{6} - 1278748655404 \nu^{5} + 7982075029386 \nu^{4} + 794767707999049 \nu^{3} + \cdots + 11\!\cdots\!00 ) / 175623852419568 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 197139097 \nu^{7} - 216622390 \nu^{6} - 372484370980 \nu^{5} - 1154094613170 \nu^{4} + 168914805034723 \nu^{3} + \cdots - 19\!\cdots\!40 ) / 50178243548448 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 455908112 \nu^{7} + 12628127309 \nu^{6} + 701437623848 \nu^{5} - 19817912564628 \nu^{4} - 218240615982026 \nu^{3} + \cdots - 42\!\cdots\!40 ) / 43905963104892 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{4} + \beta_{3} + \beta _1 + 2672 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{7} - 6\beta_{6} - 7\beta_{5} - 6\beta_{4} - 3\beta_{3} + 79\beta_{2} + 4124\beta _1 + 3704 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 51 \beta_{7} - 134 \beta_{6} + 97 \beta_{5} - 2372 \beta_{4} + 2567 \beta_{3} + 315 \beta_{2} + 2594 \beta _1 + 5467404 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 2993 \beta_{7} - 4586 \beta_{6} - 5041 \beta_{5} - 1352 \beta_{4} - 185 \beta_{3} + 56895 \beta_{2} + 2306442 \beta _1 + 2278446 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 50961 \beta_{7} - 103810 \beta_{6} + 73187 \beta_{5} - 1309111 \beta_{4} + 1533596 \beta_{3} + 481069 \beta_{2} + 2065085 \beta _1 + 3051054240 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 9150313 \beta_{7} - 11165394 \beta_{6} - 12322917 \beta_{5} + 348606 \beta_{4} + 3189615 \beta_{3} + 150212149 \beta_{2} + 5281351220 \beta _1 + 6436740960 ) / 8 \) 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
35.1157
32.6179
19.5149
5.95627
−9.43512
−16.3063
−33.5241
−33.9392
−74.2314 504.234 3462.31 −3053.92 −37430.0 −27639.7 −104986. 77105.4 226697.
1.2 −69.2357 −806.194 2745.59 −10567.4 55817.4 −39263.4 −48297.9 472801. 731642.
1.3 −43.0298 −312.230 −196.437 −1082.06 13435.2 38616.0 96577.7 −79659.3 46560.9
1.4 −15.9125 −724.922 −1794.79 8501.73 11535.3 38766.9 61148.6 348365. −135284.
1.5 14.8702 536.788 −1826.88 2118.93 7982.17 −68580.9 −57620.3 110995. 31509.0
1.6 28.6126 141.482 −1229.32 −2645.71 4048.18 76456.7 −93772.7 −157130. −75700.7
1.7 63.0482 −462.057 1927.07 5906.93 −29131.8 −36688.5 −7624.20 36349.3 372421.
1.8 63.8784 130.898 2032.45 −10644.5 8361.56 −35785.2 −993.065 −160013. −679953.
\(n\): 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 23.12.a.a 8
3.b odd 2 1 207.12.a.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.12.a.a 8 1.a even 1 1 trivial
207.12.a.a 8 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 32 T_{2}^{7} - 10240 T_{2}^{6} - 243056 T_{2}^{5} + 32897712 T_{2}^{4} + 475545152 T_{2}^{3} - 32355612672 T_{2}^{2} - 118888316928 T_{2} + 6030172585984 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(23))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 32 T^{7} + \cdots + 6030172585984 \) Copy content Toggle raw display
$3$ \( T^{8} + 992 T^{7} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( T^{8} + 11466 T^{7} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{8} + 54118 T^{7} + \cdots - 11\!\cdots\!36 \) Copy content Toggle raw display
$11$ \( T^{8} + 291462 T^{7} + \cdots - 66\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{8} + 2211306 T^{7} + \cdots - 24\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( T^{8} + 5775330 T^{7} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{8} + 21015588 T^{7} + \cdots - 45\!\cdots\!20 \) Copy content Toggle raw display
$23$ \( (T - 6436343)^{8} \) Copy content Toggle raw display
$29$ \( T^{8} + 322285430 T^{7} + \cdots - 12\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{8} + 415184840 T^{7} + \cdots - 20\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{8} - 176642018 T^{7} + \cdots + 26\!\cdots\!32 \) Copy content Toggle raw display
$41$ \( T^{8} - 357962218 T^{7} + \cdots + 99\!\cdots\!80 \) Copy content Toggle raw display
$43$ \( T^{8} - 2500461376 T^{7} + \cdots - 46\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{8} - 261795200 T^{7} + \cdots - 10\!\cdots\!80 \) Copy content Toggle raw display
$53$ \( T^{8} - 3542935060 T^{7} + \cdots + 17\!\cdots\!32 \) Copy content Toggle raw display
$59$ \( T^{8} - 930905396 T^{7} + \cdots - 74\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( T^{8} + 25338655048 T^{7} + \cdots + 16\!\cdots\!08 \) Copy content Toggle raw display
$67$ \( T^{8} + 3123467482 T^{7} + \cdots + 12\!\cdots\!40 \) Copy content Toggle raw display
$71$ \( T^{8} + 52612263236 T^{7} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{8} + 67014176274 T^{7} + \cdots - 39\!\cdots\!08 \) Copy content Toggle raw display
$79$ \( T^{8} + 27683357604 T^{7} + \cdots + 17\!\cdots\!76 \) Copy content Toggle raw display
$83$ \( T^{8} + 12253964262 T^{7} + \cdots + 89\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{8} - 10662817760 T^{7} + \cdots + 22\!\cdots\!20 \) Copy content Toggle raw display
$97$ \( T^{8} + 124519454530 T^{7} + \cdots + 33\!\cdots\!04 \) Copy content Toggle raw display
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