Properties

Label 23.8.a.a
Level $23$
Weight $8$
Character orbit 23.a
Self dual yes
Analytic conductor $7.185$
Analytic rank $1$
Dimension $5$
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: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \( x^{5} - 2x^{4} - 104x^{3} + 200x^{2} + 2037x - 3704 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 3) q^{2} + (\beta_{2} - 14) q^{3} + (\beta_{4} - 3 \beta_{2} - 7 \beta_1 + 51) q^{4} + ( - 4 \beta_{4} + \beta_{3} - 2 \beta_{2} - 14 \beta_1 - 13) q^{5} + ( - 3 \beta_{4} - 6 \beta_{3} - 5 \beta_{2} - 42 \beta_1 + 100) q^{6} + (13 \beta_{4} + 5 \beta_{3} + \beta_{2} - 7 \beta_1 - 232) q^{7} + (10 \beta_{4} + 20 \beta_{3} + 34 \beta_{2} + 50 \beta_1 - 1190) q^{8} + (6 \beta_{4} - 21 \beta_{3} - 11 \beta_{2} - 36 \beta_1 - 446) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 3) q^{2} + (\beta_{2} - 14) q^{3} + (\beta_{4} - 3 \beta_{2} - 7 \beta_1 + 51) q^{4} + ( - 4 \beta_{4} + \beta_{3} - 2 \beta_{2} - 14 \beta_1 - 13) q^{5} + ( - 3 \beta_{4} - 6 \beta_{3} - 5 \beta_{2} - 42 \beta_1 + 100) q^{6} + (13 \beta_{4} + 5 \beta_{3} + \beta_{2} - 7 \beta_1 - 232) q^{7} + (10 \beta_{4} + 20 \beta_{3} + 34 \beta_{2} + 50 \beta_1 - 1190) q^{8} + (6 \beta_{4} - 21 \beta_{3} - 11 \beta_{2} - 36 \beta_1 - 446) q^{9} + ( - 45 \beta_{4} - 8 \beta_{3} + 51 \beta_{2} + 35 \beta_1 - 2267) q^{10} + ( - 17 \beta_{4} - 69 \beta_{3} - 81 \beta_{2} - 85 \beta_1 - 262) q^{11} + ( - 21 \beta_{4} + 96 \beta_{3} + 83 \beta_{2} + 387 \beta_1 - 5539) q^{12} + (25 \beta_{4} + 55 \beta_{3} - 76 \beta_{2} + 385 \beta_1 - 3814) q^{13} + (69 \beta_{4} - 40 \beta_{3} - 55 \beta_{2} - 57 \beta_1 - 1367) q^{14} + (129 \beta_{4} + 171 \beta_{3} - 141 \beta_{2} + 657 \beta_1 - 4698) q^{15} + ( - 200 \beta_{4} - 424 \beta_{3} - 136 \beta_{2} - 1376 \beta_1 + 6184) q^{16} + (186 \beta_{4} - \beta_{3} - 68 \beta_{2} + 112 \beta_1 - 951) q^{17} + (150 \beta_{4} + 330 \beta_{3} + 340 \beta_{2} + 180 \beta_1 - 4964) q^{18} + ( - 558 \beta_{4} - 28 \beta_{3} + 414 \beta_{2} + 30 \beta_1 - 7828) q^{19} + (74 \beta_{4} - 428 \beta_{3} + 58 \beta_{2} - 2686 \beta_1 + 20242) q^{20} + ( - 270 \beta_{4} - 51 \beta_{3} - 74 \beta_{2} - 252 \beta_1 + 6625) q^{21} + (367 \beta_{4} + 1280 \beta_{3} + 1315 \beta_{2} + 2367 \beta_1 - 14771) q^{22} + 12167 q^{23} + ( - 126 \beta_{4} - 924 \beta_{3} - 1758 \beta_{2} - 4734 \beta_1 + 71970) q^{24} + (325 \beta_{4} - 442 \beta_{3} + 181 \beta_{2} + 2015 \beta_1 + 11234) q^{25} + (538 \beta_{4} - 154 \beta_{3} - 1320 \beta_{2} - 3071 \beta_1 + 68969) q^{26} + ( - 165 \beta_{4} + 204 \beta_{3} - 1550 \beta_{2} + 3141 \beta_1 + 21469) q^{27} + ( - 804 \beta_{4} + 308 \beta_{3} + 540 \beta_{2} + 2492 \beta_1 + 18504) q^{28} + (1618 \beta_{4} + 805 \beta_{3} + 1495 \beta_{2} + 6364 \beta_1 - 29291) q^{29} + (1257 \beta_{4} - 948 \beta_{3} - 3063 \beta_{2} - 2127 \beta_1 + 103755) q^{30} + ( - 3537 \beta_{4} - 2320 \beta_{3} + 936 \beta_{2} + \cdots - 36767) q^{31}+ \cdots + (22488 \beta_{4} + 56148 \beta_{3} - 34946 \beta_{2} + \cdots + 5972578) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 16 q^{2} - 68 q^{3} + 256 q^{4} - 56 q^{5} + 538 q^{6} - 1156 q^{7} - 5952 q^{8} - 2195 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 16 q^{2} - 68 q^{3} + 256 q^{4} - 56 q^{5} + 538 q^{6} - 1156 q^{7} - 5952 q^{8} - 2195 q^{9} - 11260 q^{10} - 1318 q^{11} - 28012 q^{12} - 19662 q^{13} - 6848 q^{14} - 24600 q^{15} + 32448 q^{16} - 5002 q^{17} - 24650 q^{18} - 38314 q^{19} + 104440 q^{20} + 33280 q^{21} - 74872 q^{22} + 60835 q^{23} + 361992 q^{24} + 54959 q^{25} + 345430 q^{26} + 100900 q^{27} + 90800 q^{28} - 150634 q^{29} + 515724 q^{30} - 179940 q^{31} - 404032 q^{32} - 619688 q^{33} + 32116 q^{34} - 374032 q^{35} + 510524 q^{36} - 752672 q^{37} + 456808 q^{38} - 207996 q^{39} - 1082576 q^{40} - 1192910 q^{41} - 250520 q^{42} - 932646 q^{43} + 2467104 q^{44} - 389952 q^{45} - 194672 q^{46} - 1008460 q^{47} - 1916464 q^{48} - 2005219 q^{49} + 1571224 q^{50} - 211520 q^{51} - 1740516 q^{52} + 897104 q^{53} + 1844686 q^{54} + 1203168 q^{55} + 3050144 q^{56} + 3137192 q^{57} + 5685090 q^{58} + 1020972 q^{59} - 1479384 q^{60} - 2758364 q^{61} + 2661794 q^{62} + 1135132 q^{63} + 5173248 q^{64} - 1350472 q^{65} + 11693212 q^{66} - 1523138 q^{67} + 2501304 q^{68} - 827356 q^{69} - 2794240 q^{70} + 3044884 q^{71} - 6740904 q^{72} - 8872022 q^{73} + 1408492 q^{74} + 1960276 q^{75} - 17963952 q^{76} - 3501672 q^{77} - 15280362 q^{78} - 4437540 q^{79} + 12197536 q^{80} - 7995203 q^{81} - 7738154 q^{82} - 4637362 q^{83} + 2663744 q^{84} - 8625728 q^{85} - 3025868 q^{86} + 17151068 q^{87} - 41815040 q^{88} + 6381402 q^{89} - 4970376 q^{90} + 3240808 q^{91} + 3114752 q^{92} + 7185076 q^{93} - 13893974 q^{94} + 15762704 q^{95} + 40696544 q^{96} - 6432034 q^{97} + 22652640 q^{98} + 30201754 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 2x^{4} - 104x^{3} + 200x^{2} + 2037x - 3704 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5\nu^{4} + 13\nu^{3} - 693\nu^{2} - 791\nu + 16130 ) / 194 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -8\nu^{4} + 18\nu^{3} + 682\nu^{2} - 1140\nu - 6893 ) / 97 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 15\nu^{4} + 39\nu^{3} - 1303\nu^{2} - 2761\nu + 15410 ) / 194 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} - 3\beta_{2} + \beta _1 + 171 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 11\beta_{4} + 10\beta_{3} - \beta_{2} + 135\beta _1 + 55 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 55\beta_{4} - 13\beta_{3} - 129\beta_{2} + 52\beta _1 + 5485 ) / 2 \) 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
−8.79636
−5.07229
1.79887
5.20719
8.86257
−21.5927 −62.6937 338.245 256.889 1353.73 −226.020 −4539.77 1743.49 −5546.93
1.2 −14.1446 6.23602 72.0689 178.126 −88.2058 368.361 791.122 −2148.11 −2519.52
1.3 −0.402251 50.9104 −127.838 −384.733 −20.4788 −88.4463 102.911 404.864 154.759
1.4 6.41439 −20.5356 −86.8556 258.464 −131.724 −1375.00 −1378.17 −1765.29 1657.89
1.5 13.7251 −41.9171 60.3796 −364.747 −575.318 165.110 −928.099 −429.959 −5006.20
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.a 5
3.b odd 2 1 207.8.a.b 5
4.b odd 2 1 368.8.a.e 5
5.b even 2 1 575.8.a.a 5
23.b odd 2 1 529.8.a.b 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.8.a.a 5 1.a even 1 1 trivial
207.8.a.b 5 3.b odd 2 1
368.8.a.e 5 4.b odd 2 1
529.8.a.b 5 23.b odd 2 1
575.8.a.a 5 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{5} + 16T_{2}^{4} - 320T_{2}^{3} - 3136T_{2}^{2} + 25680T_{2} + 10816 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(23))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} + 16 T^{4} - 320 T^{3} + \cdots + 10816 \) Copy content Toggle raw display
$3$ \( T^{5} + 68 T^{4} - 2058 T^{3} + \cdots + 17133120 \) Copy content Toggle raw display
$5$ \( T^{5} + 56 T^{4} + \cdots - 1659678082560 \) Copy content Toggle raw display
$7$ \( T^{5} + 1156 T^{4} + \cdots + 1671775278848 \) Copy content Toggle raw display
$11$ \( T^{5} + 1318 T^{4} + \cdots + 18\!\cdots\!80 \) Copy content Toggle raw display
$13$ \( T^{5} + 19662 T^{4} + \cdots - 17\!\cdots\!06 \) Copy content Toggle raw display
$17$ \( T^{5} + 5002 T^{4} + \cdots - 16\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{5} + 38314 T^{4} + \cdots + 51\!\cdots\!60 \) Copy content Toggle raw display
$23$ \( (T - 12167)^{5} \) Copy content Toggle raw display
$29$ \( T^{5} + 150634 T^{4} + \cdots - 20\!\cdots\!50 \) Copy content Toggle raw display
$31$ \( T^{5} + 179940 T^{4} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{5} + 752672 T^{4} + \cdots - 18\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( T^{5} + 1192910 T^{4} + \cdots - 12\!\cdots\!30 \) Copy content Toggle raw display
$43$ \( T^{5} + 932646 T^{4} + \cdots - 97\!\cdots\!48 \) Copy content Toggle raw display
$47$ \( T^{5} + 1008460 T^{4} + \cdots + 17\!\cdots\!12 \) Copy content Toggle raw display
$53$ \( T^{5} - 897104 T^{4} + \cdots + 16\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{5} - 1020972 T^{4} + \cdots + 13\!\cdots\!48 \) Copy content Toggle raw display
$61$ \( T^{5} + 2758364 T^{4} + \cdots + 54\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( T^{5} + 1523138 T^{4} + \cdots + 14\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( T^{5} - 3044884 T^{4} + \cdots - 18\!\cdots\!20 \) Copy content Toggle raw display
$73$ \( T^{5} + 8872022 T^{4} + \cdots - 38\!\cdots\!94 \) Copy content Toggle raw display
$79$ \( T^{5} + 4437540 T^{4} + \cdots + 99\!\cdots\!08 \) Copy content Toggle raw display
$83$ \( T^{5} + 4637362 T^{4} + \cdots + 25\!\cdots\!12 \) Copy content Toggle raw display
$89$ \( T^{5} - 6381402 T^{4} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{5} + 6432034 T^{4} + \cdots + 84\!\cdots\!68 \) Copy content Toggle raw display
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