Properties

Label 1.18.a.a
Level 1
Weight 18
Character orbit 1.a
Self dual yes
Analytic conductor 1.832
Analytic rank 1
Dimension 1
CM no
Inner twists 1

Related objects

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Newspace parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 18 \)
Character orbit: \([\chi]\) = 1.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.83222087345\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 528q^{2} - 4284q^{3} + 147712q^{4} - 1025850q^{5} + 2261952q^{6} + 3225992q^{7} - 8785920q^{8} - 110787507q^{9} + O(q^{10}) \) \( q - 528q^{2} - 4284q^{3} + 147712q^{4} - 1025850q^{5} + 2261952q^{6} + 3225992q^{7} - 8785920q^{8} - 110787507q^{9} + 541648800q^{10} - 753618228q^{11} - 632798208q^{12} + 2541064526q^{13} - 1703323776q^{14} + 4394741400q^{15} - 14721941504q^{16} - 5429742318q^{17} + 58495803696q^{18} + 1487499860q^{19} - 151530355200q^{20} - 13820149728q^{21} + 397910424384q^{22} - 317091823464q^{23} + 37638881280q^{24} + 289428769375q^{25} - 1341682069728q^{26} + 1027850138280q^{27} + 476517730304q^{28} + 2433410602590q^{29} - 2320423459200q^{30} - 8849722053088q^{31} + 8924773220352q^{32} + 3228500488752q^{33} + 2866903943904q^{34} - 3309383893200q^{35} - 16364644233984q^{36} + 12691652946662q^{37} - 785399926080q^{38} - 10885920429384q^{39} + 9013036032000q^{40} + 48864151002282q^{41} + 7297039056384q^{42} - 91019974317844q^{43} - 111318455694336q^{44} + 113651364055950q^{45} + 167424482788992q^{46} - 49304994276048q^{47} + 63068797403136q^{48} - 222223489603143q^{49} - 152818390230000q^{50} + 23261016090312q^{51} + 375345723264512q^{52} + 22940453195766q^{53} - 542704873011840q^{54} + 773099259193800q^{55} - 28343307632640q^{56} - 6372449400240q^{57} - 1284840798167520q^{58} + 32695090729980q^{59} + 649156041676800q^{60} - 1308285854869378q^{61} + 4672653244030464q^{62} - 357399611281944q^{63} - 2782645943533568q^{64} - 2606751043997100q^{65} - 1704648258061056q^{66} + 5196143861984132q^{67} - 802038097276416q^{68} + 1358421371719776q^{69} + 1747354695609600q^{70} - 3709489877412408q^{71} + 973370173501440q^{72} + 3402372968272586q^{73} - 6701192755837536q^{74} - 1239912848002500q^{75} + 219721579320320q^{76} - 2431166374582176q^{77} + 5747765986714752q^{78} + 2366533941308240q^{79} + 15102503691878400q^{80} + 9903806719952121q^{81} - 25800271729204896q^{82} - 29766750443172204q^{83} - 2041401956622336q^{84} + 5570101156920300q^{85} + 48058546439821632q^{86} - 10424731021495560q^{87} + 6621229461749760q^{88} + 29167184100574170q^{89} - 60007920221541600q^{90} + 8197453832359792q^{91} - 46838267427514368q^{92} + 37912209275428992q^{93} + 26033036977753344q^{94} - 1525951731381000q^{95} - 38233728475987968q^{96} - 63769879140957598q^{97} + 117334002510459504q^{98} + 83491484709877596q^{99} + O(q^{100}) \)

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
0
−528.000 −4284.00 147712. −1.02585e6 2.26195e6 3.22599e6 −8.78592e6 −1.10788e8 5.41649e8
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 1.18.a.a 1
3.b odd 2 1 9.18.a.b 1
4.b odd 2 1 16.18.a.b 1
5.b even 2 1 25.18.a.a 1
5.c odd 4 2 25.18.b.a 2
7.b odd 2 1 49.18.a.a 1
8.b even 2 1 64.18.a.d 1
8.d odd 2 1 64.18.a.b 1
11.b odd 2 1 121.18.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.18.a.a 1 1.a even 1 1 trivial
9.18.a.b 1 3.b odd 2 1
16.18.a.b 1 4.b odd 2 1
25.18.a.a 1 5.b even 2 1
25.18.b.a 2 5.c odd 4 2
49.18.a.a 1 7.b odd 2 1
64.18.a.b 1 8.d odd 2 1
64.18.a.d 1 8.b even 2 1
121.18.a.b 1 11.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{18}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 528 T + 131072 T^{2} \)
$3$ \( 1 + 4284 T + 129140163 T^{2} \)
$5$ \( 1 + 1025850 T + 762939453125 T^{2} \)
$7$ \( 1 - 3225992 T + 232630513987207 T^{2} \)
$11$ \( 1 + 753618228 T + 505447028499293771 T^{2} \)
$13$ \( 1 - 2541064526 T + 8650415919381337933 T^{2} \)
$17$ \( 1 + 5429742318 T + \)\(82\!\cdots\!77\)\( T^{2} \)
$19$ \( 1 - 1487499860 T + \)\(54\!\cdots\!39\)\( T^{2} \)
$23$ \( 1 + 317091823464 T + \)\(14\!\cdots\!03\)\( T^{2} \)
$29$ \( 1 - 2433410602590 T + \)\(72\!\cdots\!09\)\( T^{2} \)
$31$ \( 1 + 8849722053088 T + \)\(22\!\cdots\!11\)\( T^{2} \)
$37$ \( 1 - 12691652946662 T + \)\(45\!\cdots\!17\)\( T^{2} \)
$41$ \( 1 - 48864151002282 T + \)\(26\!\cdots\!81\)\( T^{2} \)
$43$ \( 1 + 91019974317844 T + \)\(58\!\cdots\!43\)\( T^{2} \)
$47$ \( 1 + 49304994276048 T + \)\(26\!\cdots\!87\)\( T^{2} \)
$53$ \( 1 - 22940453195766 T + \)\(20\!\cdots\!13\)\( T^{2} \)
$59$ \( 1 - 32695090729980 T + \)\(12\!\cdots\!19\)\( T^{2} \)
$61$ \( 1 + 1308285854869378 T + \)\(22\!\cdots\!21\)\( T^{2} \)
$67$ \( 1 - 5196143861984132 T + \)\(11\!\cdots\!27\)\( T^{2} \)
$71$ \( 1 + 3709489877412408 T + \)\(29\!\cdots\!91\)\( T^{2} \)
$73$ \( 1 - 3402372968272586 T + \)\(47\!\cdots\!53\)\( T^{2} \)
$79$ \( 1 - 2366533941308240 T + \)\(18\!\cdots\!59\)\( T^{2} \)
$83$ \( 1 + 29766750443172204 T + \)\(42\!\cdots\!23\)\( T^{2} \)
$89$ \( 1 - 29167184100574170 T + \)\(13\!\cdots\!29\)\( T^{2} \)
$97$ \( 1 + 63769879140957598 T + \)\(59\!\cdots\!37\)\( T^{2} \)
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