Properties

Label 23.14.a.b
Level $23$
Weight $14$
Character orbit 23.a
Self dual yes
Analytic conductor $24.663$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 23 \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 23.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(24.6631136589\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial: \( x^{14} - 6 x^{13} - 91997 x^{12} + 766599 x^{11} + 3278769040 x^{10} - 30986318669 x^{9} - 56829440072404 x^{8} + 496745885608086 x^{7} + \cdots - 45\!\cdots\!18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{19}\cdot 3^{4}\cdot 11 \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 5) q^{2} + (\beta_{2} - 2 \beta_1 + 148) q^{3} + (\beta_{3} - \beta_{2} - 14 \beta_1 + 4979) q^{4} + (\beta_{5} + \beta_{3} - 2 \beta_{2} - 18 \beta_1 + 3734) q^{5} + ( - \beta_{7} + \beta_{5} + 8 \beta_{3} + 2 \beta_{2} - 42 \beta_1 + 22241) q^{6} + ( - \beta_{8} + 3 \beta_{5} + 10 \beta_{3} + 21 \beta_{2} + 641 \beta_1 + 13341) q^{7} + ( - \beta_{9} + 2 \beta_{8} + \beta_{7} - \beta_{6} + 32 \beta_{3} + 199 \beta_{2} + \cdots + 170326) q^{8}+ \cdots + ( - \beta_{13} - \beta_{12} - \beta_{11} - \beta_{10} + 4 \beta_{9} + 2 \beta_{8} + \cdots + 860832) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 5) q^{2} + (\beta_{2} - 2 \beta_1 + 148) q^{3} + (\beta_{3} - \beta_{2} - 14 \beta_1 + 4979) q^{4} + (\beta_{5} + \beta_{3} - 2 \beta_{2} - 18 \beta_1 + 3734) q^{5} + ( - \beta_{7} + \beta_{5} + 8 \beta_{3} + 2 \beta_{2} - 42 \beta_1 + 22241) q^{6} + ( - \beta_{8} + 3 \beta_{5} + 10 \beta_{3} + 21 \beta_{2} + 641 \beta_1 + 13341) q^{7} + ( - \beta_{9} + 2 \beta_{8} + \beta_{7} - \beta_{6} + 32 \beta_{3} + 199 \beta_{2} + \cdots + 170326) q^{8}+ \cdots + (8101192 \beta_{13} - 26305856 \beta_{12} + \cdots - 676685387934) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 64 q^{2} + 2056 q^{3} + 69632 q^{4} + 52182 q^{5} + 311167 q^{6} + 190602 q^{7} + 2356809 q^{8} + 12050784 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 64 q^{2} + 2056 q^{3} + 69632 q^{4} + 52182 q^{5} + 311167 q^{6} + 190602 q^{7} + 2356809 q^{8} + 12050784 q^{9} + 3585670 q^{10} + 1070730 q^{11} - 8508331 q^{12} + 23949638 q^{13} - 119280968 q^{14} - 44834930 q^{15} + 256829072 q^{16} + 69487470 q^{17} + 92449927 q^{18} + 111438548 q^{19} + 1129282316 q^{20} + 621345174 q^{21} + 2278933028 q^{22} + 2072502446 q^{23} + 8776950724 q^{24} + 5548551686 q^{25} - 925154105 q^{26} - 2006600744 q^{27} + 10886499970 q^{28} + 6082889362 q^{29} + 33591682946 q^{30} + 15979895560 q^{31} + 39045677992 q^{32} + 48341340746 q^{33} + 26300859414 q^{34} + 71251965504 q^{35} + 134660338135 q^{36} + 52356093690 q^{37} + 96969962716 q^{38} + 35694630240 q^{39} + 30337594230 q^{40} + 116782373266 q^{41} + 47161428352 q^{42} + 551363512 q^{43} - 18191926218 q^{44} + 66956385060 q^{45} + 9474296896 q^{46} - 89763073312 q^{47} + 7373438519 q^{48} + 198965141586 q^{49} - 353559739256 q^{50} - 849385907902 q^{51} + 290946305159 q^{52} - 255252512096 q^{53} - 20138610103 q^{54} - 308239853444 q^{55} - 1741462242990 q^{56} - 373036556464 q^{57} - 2063171638367 q^{58} - 844368470500 q^{59} - 3864457510716 q^{60} - 660411924036 q^{61} - 3066592203813 q^{62} - 2044550744028 q^{63} - 149179140181 q^{64} + 25563523898 q^{65} - 3128765504558 q^{66} + 343438236966 q^{67} - 687566878740 q^{68} + 304361787784 q^{69} + 2831163146300 q^{70} + 525250335580 q^{71} - 1782771811281 q^{72} + 6080256001118 q^{73} + 1193156509458 q^{74} + 3035968085076 q^{75} + 11140697506136 q^{76} - 905513956696 q^{77} + 15392222627509 q^{78} + 2029462022780 q^{79} + 6389606776510 q^{80} + 11017226960590 q^{81} + 5032544493407 q^{82} + 1645588044714 q^{83} - 8835767120594 q^{84} + 8689341605448 q^{85} - 5028664556794 q^{86} + 14107817502696 q^{87} + 35486297142892 q^{88} + 3557834996156 q^{89} - 20611184383708 q^{90} + 20574193795614 q^{91} + 10308035022848 q^{92} + 32845521705562 q^{93} - 4653170522585 q^{94} + 35338742719324 q^{95} + 44121425602615 q^{96} + 20411381883630 q^{97} - 10415391287228 q^{98} - 9767188111540 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 6 x^{13} - 91997 x^{12} + 766599 x^{11} + 3278769040 x^{10} - 30986318669 x^{9} - 56829440072404 x^{8} + 496745885608086 x^{7} + \cdots - 45\!\cdots\!18 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 30\!\cdots\!73 \nu^{13} + \cdots + 37\!\cdots\!30 ) / 49\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 30\!\cdots\!73 \nu^{13} + \cdots - 61\!\cdots\!50 ) / 49\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 14\!\cdots\!27 \nu^{13} + \cdots + 27\!\cdots\!78 ) / 34\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 43\!\cdots\!45 \nu^{13} + \cdots + 30\!\cdots\!22 ) / 38\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 40\!\cdots\!85 \nu^{13} + \cdots + 19\!\cdots\!78 ) / 34\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 14\!\cdots\!11 \nu^{13} + \cdots - 13\!\cdots\!38 ) / 49\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 79\!\cdots\!71 \nu^{13} + \cdots + 47\!\cdots\!22 ) / 17\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 23\!\cdots\!49 \nu^{13} + \cdots - 96\!\cdots\!06 ) / 34\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 45\!\cdots\!97 \nu^{13} + \cdots - 11\!\cdots\!98 ) / 58\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 23\!\cdots\!89 \nu^{13} + \cdots + 82\!\cdots\!06 ) / 17\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 21\!\cdots\!99 \nu^{13} + \cdots - 87\!\cdots\!94 ) / 63\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 16\!\cdots\!97 \nu^{13} + \cdots + 39\!\cdots\!82 ) / 43\!\cdots\!20 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - \beta_{2} - 4\beta _1 + 13146 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - 2\beta_{8} - \beta_{7} + \beta_{6} - 17\beta_{3} - 214\beta_{2} + 20772\beta _1 - 54931 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 30 \beta_{13} + 90 \beta_{12} + 33 \beta_{11} + 16 \beta_{10} - 47 \beta_{9} + 77 \beta_{8} - 183 \beta_{7} + 87 \beta_{6} + 46 \beta_{5} + 27 \beta_{4} + 27486 \beta_{3} - 33428 \beta_{2} - 238144 \beta _1 + 271985750 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 1752 \beta_{13} + 486 \beta_{12} - 2966 \beta_{11} + 14232 \beta_{10} + 31359 \beta_{9} - 76498 \beta_{8} - 44838 \beta_{7} + 43226 \beta_{6} + 41532 \beta_{5} - 1407 \beta_{4} + \cdots - 3222631492 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 347969 \beta_{13} + 2688720 \beta_{12} + 807427 \beta_{11} + 298662 \beta_{10} - 1519888 \beta_{9} + 3927109 \beta_{8} - 8117809 \beta_{7} + 4343904 \beta_{6} + \cdots + 6304133829202 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 126134555 \beta_{13} + 9979654 \beta_{12} - 81033741 \beta_{11} + 770516254 \beta_{10} + 851180350 \beta_{9} - 2397043973 \beta_{8} - 1402647352 \beta_{7} + \cdots - 119032454667843 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 10749450902 \beta_{13} + 58031381368 \beta_{12} + 9688352671 \beta_{11} - 3094480224 \beta_{10} - 38105521418 \beta_{9} + 150531216947 \beta_{8} + \cdots + 15\!\cdots\!78 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 5560042960635 \beta_{13} + 272630473618 \beta_{12} - 1365580053713 \beta_{11} + 30001486358494 \beta_{10} + 22471236077827 \beta_{9} + \cdots - 37\!\cdots\!34 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 830150932579018 \beta_{13} + \cdots + 40\!\cdots\!68 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 20\!\cdots\!85 \beta_{13} + \cdots - 11\!\cdots\!65 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 35\!\cdots\!89 \beta_{13} + \cdots + 10\!\cdots\!94 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 65\!\cdots\!86 \beta_{13} + \cdots - 33\!\cdots\!46 \) 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
167.770
144.125
129.200
119.960
99.6136
37.0185
16.3008
−15.6227
−43.4497
−72.0951
−110.794
−140.390
−154.604
−171.033
−162.770 −2381.17 18302.2 37078.1 387584. 189022. −1.64565e6 4.07566e6 −6.03522e6
1.2 −139.125 1444.04 11163.8 −50599.5 −200902. 547975. −413454. 490920. 7.03966e6
1.3 −124.200 −983.535 7233.64 −8001.74 122155. −152007. 119029. −626982. 993816.
1.4 −114.960 −157.249 5023.81 57953.0 18077.4 475971. 364215. −1.56960e6 −6.66227e6
1.5 −94.6136 2265.03 759.734 −10060.2 −214303. −210220. 703193. 3.53605e6 951830.
1.6 −32.0185 511.530 −7166.81 −62645.6 −16378.4 −516700. 491767. −1.33266e6 2.00582e6
1.7 −11.3008 −618.076 −8064.29 41118.0 6984.75 −134255. 183709. −1.21231e6 −464666.
1.8 20.6227 2170.76 −7766.70 53936.6 44767.1 −12340.9 −329112. 3.11789e6 1.11232e6
1.9 48.4497 −518.249 −5844.62 −25632.2 −25109.0 136524. −680071. −1.32574e6 −1.24188e6
1.10 77.0951 −2221.44 −2248.35 −42257.5 −171262. −355522. −804900. 3.34048e6 −3.25785e6
1.11 115.794 1474.58 5216.18 −8934.12 170747. 359817. −344581. 580071. −1.03451e6
1.12 145.390 943.197 12946.4 58077.9 137132. 392913. 691241. −704702. 8.44397e6
1.13 159.604 −1789.05 17281.5 24592.6 −285540. −77186.8 1.45072e6 1.60638e6 3.92509e6
1.14 176.033 1915.63 22795.5 −12443.4 337214. −453389. 2.57069e6 2.07532e6 −2.19044e6
\(n\): 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 23.14.a.b 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.14.a.b 14 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} - 64 T_{2}^{13} - 90112 T_{2}^{12} + 4719421 T_{2}^{11} + 3169548060 T_{2}^{10} - 131479864456 T_{2}^{9} - 54547836919384 T_{2}^{8} + \cdots - 40\!\cdots\!28 \) acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(23))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} - 64 T^{13} + \cdots - 40\!\cdots\!28 \) Copy content Toggle raw display
$3$ \( T^{14} - 2056 T^{13} + \cdots - 45\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( T^{14} - 52182 T^{13} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{14} - 190602 T^{13} + \cdots + 32\!\cdots\!16 \) Copy content Toggle raw display
$11$ \( T^{14} - 1070730 T^{13} + \cdots + 16\!\cdots\!20 \) Copy content Toggle raw display
$13$ \( T^{14} - 23949638 T^{13} + \cdots + 82\!\cdots\!96 \) Copy content Toggle raw display
$17$ \( T^{14} - 69487470 T^{13} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{14} - 111438548 T^{13} + \cdots + 88\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T - 148035889)^{14} \) Copy content Toggle raw display
$29$ \( T^{14} - 6082889362 T^{13} + \cdots + 90\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{14} - 15979895560 T^{13} + \cdots - 73\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{14} - 52356093690 T^{13} + \cdots + 97\!\cdots\!32 \) Copy content Toggle raw display
$41$ \( T^{14} - 116782373266 T^{13} + \cdots + 18\!\cdots\!40 \) Copy content Toggle raw display
$43$ \( T^{14} - 551363512 T^{13} + \cdots - 38\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{14} + 89763073312 T^{13} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{14} + 255252512096 T^{13} + \cdots + 10\!\cdots\!72 \) Copy content Toggle raw display
$59$ \( T^{14} + 844368470500 T^{13} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{14} + 660411924036 T^{13} + \cdots + 31\!\cdots\!64 \) Copy content Toggle raw display
$67$ \( T^{14} - 343438236966 T^{13} + \cdots + 27\!\cdots\!60 \) Copy content Toggle raw display
$71$ \( T^{14} - 525250335580 T^{13} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{14} - 6080256001118 T^{13} + \cdots + 63\!\cdots\!80 \) Copy content Toggle raw display
$79$ \( T^{14} - 2029462022780 T^{13} + \cdots + 10\!\cdots\!20 \) Copy content Toggle raw display
$83$ \( T^{14} - 1645588044714 T^{13} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{14} - 3557834996156 T^{13} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{14} - 20411381883630 T^{13} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
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