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 $1$

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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

\(f(q)\) \(=\) \( q - 528 q^{2} - 4284 q^{3} + 147712 q^{4} - 1025850 q^{5} + 2261952 q^{6} + 3225992 q^{7} - 8785920 q^{8} - 110787507 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 528 q^{2} - 4284 q^{3} + 147712 q^{4} - 1025850 q^{5} + 2261952 q^{6} + 3225992 q^{7} - 8785920 q^{8} - 110787507 q^{9} + 541648800 q^{10} - 753618228 q^{11} - 632798208 q^{12} + 2541064526 q^{13} - 1703323776 q^{14} + 4394741400 q^{15} - 14721941504 q^{16} - 5429742318 q^{17} + 58495803696 q^{18} + 1487499860 q^{19} - 151530355200 q^{20} - 13820149728 q^{21} + 397910424384 q^{22} - 317091823464 q^{23} + 37638881280 q^{24} + 289428769375 q^{25} - 1341682069728 q^{26} + 1027850138280 q^{27} + 476517730304 q^{28} + 2433410602590 q^{29} - 2320423459200 q^{30} - 8849722053088 q^{31} + 8924773220352 q^{32} + 3228500488752 q^{33} + 2866903943904 q^{34} - 3309383893200 q^{35} - 16364644233984 q^{36} + 12691652946662 q^{37} - 785399926080 q^{38} - 10885920429384 q^{39} + 9013036032000 q^{40} + 48864151002282 q^{41} + 7297039056384 q^{42} - 91019974317844 q^{43} - 111318455694336 q^{44} + 113651364055950 q^{45} + 167424482788992 q^{46} - 49304994276048 q^{47} + 63068797403136 q^{48} - 222223489603143 q^{49} - 152818390230000 q^{50} + 23261016090312 q^{51} + 375345723264512 q^{52} + 22940453195766 q^{53} - 542704873011840 q^{54} + 773099259193800 q^{55} - 28343307632640 q^{56} - 6372449400240 q^{57} - 12\!\cdots\!20 q^{58}+ \cdots + 83\!\cdots\!96 q^{99}+O(q^{100}) \) 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
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

$p$ $F_p(T)$
$2$ \( T + 528 \) Copy content Toggle raw display
$3$ \( T + 4284 \) Copy content Toggle raw display
$5$ \( T + 1025850 \) Copy content Toggle raw display
$7$ \( T - 3225992 \) Copy content Toggle raw display
$11$ \( T + 753618228 \) Copy content Toggle raw display
$13$ \( T - 2541064526 \) Copy content Toggle raw display
$17$ \( T + 5429742318 \) Copy content Toggle raw display
$19$ \( T - 1487499860 \) Copy content Toggle raw display
$23$ \( T + 317091823464 \) Copy content Toggle raw display
$29$ \( T - 2433410602590 \) Copy content Toggle raw display
$31$ \( T + 8849722053088 \) Copy content Toggle raw display
$37$ \( T - 12691652946662 \) Copy content Toggle raw display
$41$ \( T - 48864151002282 \) Copy content Toggle raw display
$43$ \( T + 91019974317844 \) Copy content Toggle raw display
$47$ \( T + 49304994276048 \) Copy content Toggle raw display
$53$ \( T - 22940453195766 \) Copy content Toggle raw display
$59$ \( T - 32695090729980 \) Copy content Toggle raw display
$61$ \( T + 1308285854869378 \) Copy content Toggle raw display
$67$ \( T - 5196143861984132 \) Copy content Toggle raw display
$71$ \( T + 3709489877412408 \) Copy content Toggle raw display
$73$ \( T - 3402372968272586 \) Copy content Toggle raw display
$79$ \( T - 2366533941308240 \) Copy content Toggle raw display
$83$ \( T + 29\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T - 29\!\cdots\!70 \) Copy content Toggle raw display
$97$ \( T + 63\!\cdots\!98 \) Copy content Toggle raw display
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