Properties

Label 23.18.a.a
Level $23$
Weight $18$
Character orbit 23.a
Self dual yes
Analytic conductor $42.141$
Analytic rank $1$
Dimension $14$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 23 \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 23.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(42.1410800892\)
Analytic rank: \(1\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial: \( x^{14} - 327680 x^{12} - 2885829 x^{11} + 40317445636 x^{10} + 536194434472 x^{9} + \cdots + 12\!\cdots\!92 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{33}\cdot 3^{12} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + \beta_1 - 760) q^{3} + (\beta_{3} - 3 \beta_{2} + 26 \beta_1 + 56174) q^{4} + ( - \beta_{5} + \beta_{3} - 6 \beta_{2} + 119 \beta_1 - 25932) q^{5} + (2 \beta_{5} + \beta_{4} + 4 \beta_{3} + 6 \beta_{2} + 3820 \beta_1 - 166643) q^{6} + (\beta_{8} - 2 \beta_{5} - \beta_{4} - 22 \beta_{3} + 54 \beta_{2} + 4696 \beta_1 - 2832083) q^{7} + (\beta_{10} - 2 \beta_{8} - \beta_{7} + 5 \beta_{5} - 5 \beta_{4} - 75 \beta_{3} + \cdots - 4947158) q^{8}+ \cdots + ( - 2 \beta_{12} + \beta_{11} - 2 \beta_{10} + \beta_{9} - 4 \beta_{8} + \cdots + 56866396) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + \beta_1 - 760) q^{3} + (\beta_{3} - 3 \beta_{2} + 26 \beta_1 + 56174) q^{4} + ( - \beta_{5} + \beta_{3} - 6 \beta_{2} + 119 \beta_1 - 25932) q^{5} + (2 \beta_{5} + \beta_{4} + 4 \beta_{3} + 6 \beta_{2} + 3820 \beta_1 - 166643) q^{6} + (\beta_{8} - 2 \beta_{5} - \beta_{4} - 22 \beta_{3} + 54 \beta_{2} + 4696 \beta_1 - 2832083) q^{7} + (\beta_{10} - 2 \beta_{8} - \beta_{7} + 5 \beta_{5} - 5 \beta_{4} - 75 \beta_{3} + \cdots - 4947158) q^{8}+ \cdots + (558719835 \beta_{13} + 1397873302 \beta_{12} + \cdots + 25\!\cdots\!58) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 10640 q^{3} + 786432 q^{4} - 363048 q^{5} - 2333030 q^{6} - 39649066 q^{7} - 69259896 q^{8} + 796129528 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 10640 q^{3} + 786432 q^{4} - 363048 q^{5} - 2333030 q^{6} - 39649066 q^{7} - 69259896 q^{8} + 796129528 q^{9} - 312719540 q^{10} - 45399620 q^{11} - 8621310628 q^{12} - 10510197306 q^{13} - 12286634640 q^{14} - 16443659490 q^{15} + 65383333632 q^{16} - 35705720330 q^{17} + 27658188862 q^{18} - 84895273414 q^{19} + 331348024336 q^{20} + 185190266362 q^{21} + 270540900120 q^{22} - 1096353793934 q^{23} + 1697198124384 q^{24} + 525715171346 q^{25} + 4272672484934 q^{26} - 3706093330604 q^{27} - 9883598189096 q^{28} - 4114009788386 q^{29} - 14194804268004 q^{30} + 3718266369468 q^{31} - 29197309605632 q^{32} - 16110579243626 q^{33} - 31423174598564 q^{34} + 13804822380504 q^{35} + 51950006703548 q^{36} - 58067881808868 q^{37} - 76590705469880 q^{38} + 69866971570764 q^{39} - 129282722434320 q^{40} - 74370388815170 q^{41} - 430581394397552 q^{42} - 127444248270174 q^{43} - 563872902913048 q^{44} - 602432292081270 q^{45} - 749727107945564 q^{47} - 17\!\cdots\!72 q^{48}+ \cdots + 35\!\cdots\!38 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 327680 x^{12} - 2885829 x^{11} + 40317445636 x^{10} + 536194434472 x^{9} + \cdots + 12\!\cdots\!92 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 34\!\cdots\!13 \nu^{13} + \cdots + 69\!\cdots\!68 ) / 32\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 34\!\cdots\!13 \nu^{13} + \cdots - 13\!\cdots\!92 ) / 10\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 20\!\cdots\!59 \nu^{13} + \cdots + 27\!\cdots\!84 ) / 16\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 42\!\cdots\!61 \nu^{13} + \cdots - 64\!\cdots\!04 ) / 10\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 46\!\cdots\!71 \nu^{13} + \cdots - 89\!\cdots\!64 ) / 32\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 92\!\cdots\!19 \nu^{13} + \cdots + 64\!\cdots\!36 ) / 32\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 15\!\cdots\!41 \nu^{13} + \cdots - 38\!\cdots\!48 ) / 16\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 39\!\cdots\!18 \nu^{13} + \cdots + 13\!\cdots\!32 ) / 32\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 96\!\cdots\!91 \nu^{13} + \cdots - 22\!\cdots\!44 ) / 64\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 59\!\cdots\!03 \nu^{13} + \cdots + 14\!\cdots\!12 ) / 32\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 65\!\cdots\!21 \nu^{13} + \cdots - 31\!\cdots\!24 ) / 32\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 16\!\cdots\!63 \nu^{13} + \cdots - 10\!\cdots\!36 ) / 64\!\cdots\!56 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 3\beta_{2} + 26\beta _1 + 187246 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - \beta_{10} + 2 \beta_{8} + \beta_{7} - 5 \beta_{5} + 5 \beta_{4} + 75 \beta_{3} - 682 \beta_{2} + 326327 \beta _1 + 4947158 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 224 \beta_{13} + 52 \beta_{12} + 188 \beta_{11} - 141 \beta_{10} - 156 \beta_{9} - 278 \beta_{8} + 57 \beta_{7} + 164 \beta_{6} - 793 \beta_{5} + 529 \beta_{4} + 204557 \beta_{3} - 925236 \beta_{2} + \cdots + 30559248350 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1550 \beta_{13} + 1512 \beta_{12} - 19056 \beta_{11} - 81111 \beta_{10} + 5626 \beta_{9} + 106338 \beta_{8} + 65113 \beta_{7} - 9526 \beta_{6} + 243407 \beta_{5} + 401561 \beta_{4} + \cdots + 584907929538 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 34269328 \beta_{13} + 10469852 \beta_{12} + 22949020 \beta_{11} - 30852049 \beta_{10} - 28993180 \beta_{9} - 53972054 \beta_{8} + 12031829 \beta_{7} + \cdots + 28\!\cdots\!38 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 208669648 \beta_{13} - 10556340 \beta_{12} - 7166358444 \beta_{11} - 22293003221 \beta_{10} + 695911292 \beta_{9} + 19633624650 \beta_{8} + \cdots + 19\!\cdots\!66 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 2124859092442 \beta_{13} + 808945409280 \beta_{12} + 1147017349400 \beta_{11} - 2563793962981 \beta_{10} - 2086630915686 \beta_{9} + \cdots + 13\!\cdots\!06 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 116691460912192 \beta_{13} - 5954355365780 \beta_{12} - 995205479856724 \beta_{11} + \cdots + 27\!\cdots\!02 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 49\!\cdots\!68 \beta_{13} + \cdots + 28\!\cdots\!46 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 12\!\cdots\!02 \beta_{13} + \cdots + 18\!\cdots\!22 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 55\!\cdots\!92 \beta_{13} + \cdots + 29\!\cdots\!78 ) / 8 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 39\!\cdots\!28 \beta_{13} + \cdots + 46\!\cdots\!78 ) / 8 \) 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
346.114
306.641
259.571
177.245
120.434
87.1216
−2.15800
−33.2496
−56.8892
−107.083
−236.962
−262.027
−279.726
−319.031
−692.227 −17580.4 348107. −370975. 1.21696e7 −1.89338e7 −1.50237e8 1.79930e8 2.56799e8
1.2 −613.281 −4261.96 245042. 1.52257e6 2.61378e6 6.06383e6 −6.98954e7 −1.10976e8 −9.33761e8
1.3 −519.142 13379.6 138436. 329626. −6.94591e6 1.47770e7 −3.82310e6 4.98736e7 −1.71123e8
1.4 −354.490 16551.9 −5409.13 −1.34244e6 −5.86748e6 4.33996e6 4.83811e7 1.44825e8 4.75880e8
1.5 −240.868 −3749.07 −73054.7 725316. 903030. −1.21078e7 4.91676e7 −1.15085e8 −1.74705e8
1.6 −174.243 −15437.5 −100711. −1.53801e6 2.68988e6 −1.10011e7 4.03867e7 1.09177e8 2.67988e8
1.7 4.31600 20224.5 −131053. 240238. 87288.7 −1.90239e7 −1.13133e6 2.79889e8 1.03687e6
1.8 66.4993 758.112 −126650. −509658. 50413.9 2.07983e7 −1.71383e7 −1.28565e8 −3.38919e7
1.9 113.778 −17336.9 −118126. 100283. −1.97257e6 7.81789e6 −2.83534e7 1.71429e8 1.14100e7
1.10 214.166 8587.97 −85204.9 1.16203e6 1.83925e6 −1.80417e6 −4.63192e7 −5.53869e7 2.48868e8
1.11 473.924 2401.23 93531.7 −313091. 1.13800e6 1.02385e7 −1.77912e7 −1.23374e8 −1.48381e8
1.12 524.054 15291.9 143560. −1.30350e6 8.01379e6 −1.81534e7 6.54461e6 1.04702e8 −6.83106e8
1.13 559.452 −22276.2 181915. 331267. −1.24625e7 −227547. 2.84440e7 3.67090e8 1.85328e8
1.14 638.062 −7193.12 276051. 603296. −4.58966e6 −2.24327e7 9.25053e7 −7.73991e7 3.84940e8
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
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 23.18.a.a 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.18.a.a 14 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} - 1310720 T_{2}^{12} + 23086632 T_{2}^{11} + 645079130176 T_{2}^{10} - 17158221903104 T_{2}^{9} + \cdots + 20\!\cdots\!28 \) acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(23))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} - 1310720 T^{12} + \cdots + 20\!\cdots\!28 \) Copy content Toggle raw display
$3$ \( T^{14} + 10640 T^{13} + \cdots - 12\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( T^{14} + 363048 T^{13} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{14} + 39649066 T^{13} + \cdots + 51\!\cdots\!92 \) Copy content Toggle raw display
$11$ \( T^{14} + 45399620 T^{13} + \cdots - 26\!\cdots\!80 \) Copy content Toggle raw display
$13$ \( T^{14} + 10510197306 T^{13} + \cdots + 24\!\cdots\!52 \) Copy content Toggle raw display
$17$ \( T^{14} + 35705720330 T^{13} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{14} + 84895273414 T^{13} + \cdots - 26\!\cdots\!60 \) Copy content Toggle raw display
$23$ \( (T + 78310985281)^{14} \) Copy content Toggle raw display
$29$ \( T^{14} + 4114009788386 T^{13} + \cdots - 42\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{14} - 3718266369468 T^{13} + \cdots - 14\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{14} + 58067881808868 T^{13} + \cdots + 32\!\cdots\!04 \) Copy content Toggle raw display
$41$ \( T^{14} + 74370388815170 T^{13} + \cdots - 27\!\cdots\!80 \) Copy content Toggle raw display
$43$ \( T^{14} + 127444248270174 T^{13} + \cdots - 14\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{14} + 749727107945564 T^{13} + \cdots - 23\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots - 95\!\cdots\!52 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 18\!\cdots\!40 \) Copy content Toggle raw display
$61$ \( T^{14} + 679115861109266 T^{13} + \cdots - 45\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 14\!\cdots\!80 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots - 23\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 63\!\cdots\!80 \) Copy content Toggle raw display
$79$ \( T^{14} - 965348846593408 T^{13} + \cdots - 31\!\cdots\!68 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 18\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots - 82\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots - 17\!\cdots\!00 \) Copy content Toggle raw display
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