Properties

Label 16.34.a.a
Level 16
Weight 34
Character orbit 16.a
Self dual yes
Analytic conductor 110.373
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 34 \)
Character orbit: \([\chi]\) \(=\) 16.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(110.372526210\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 133005564q^{3} + 538799132550q^{5} + 33347311051768q^{7} + 12131419488402573q^{9} + O(q^{10}) \) \( q + 133005564q^{3} + 538799132550q^{5} + 33347311051768q^{7} + 12131419488402573q^{9} + 85882263625386228q^{11} + 1144054008875905166q^{13} + 71663282507523508200q^{15} - 139113675669385621998q^{17} - 80695000174130231060q^{19} + 4435377914323836037152q^{21} + 14120372378143910765544q^{23} + 173889183409697655049375q^{25} + 874160305210698806986200q^{27} - 1632686905195131326709090q^{29} + 1894078958241443951861728q^{31} + 11422818911091179972972592q^{33} + 17967502267567626543848400q^{35} - 96444218751358368990635098q^{37} + 152165548697000772614343624q^{39} + 641768233498553833164038442q^{41} + 817975597351211427387164884q^{43} + 6536398296951471117588051150q^{45} + 6229246687280441243201826768q^{47} - 6618950565324076329761168583q^{49} - 18502892892519712187334796872q^{51} - 21322120079333214208388446794q^{53} + 46273289142788517805116521400q^{55} - 10732884010140289591585617840q^{57} - 298987905886407341741567881020q^{59} - 455881915835062287556960014658q^{61} + 404550219179240819106827399064q^{63} + 616415307572687704036863753300q^{65} - 1172332419477563429554964377412q^{67} + 1878088092045052124536851486816q^{69} - 2591524145775150288511661030472q^{71} - 2825174388069163226217247688374q^{73} + 23128228912906279679319569722500q^{75} + 2863942558945695063591902251104q^{77} - 920688453939087595198198640720q^{79} + 48828888726639213211634490656121q^{81} + 16199219945453134166417678661804q^{83} - 74954327776507013723964597834900q^{85} - 217156442660892972163010779376760q^{87} - 203491630107372946965013220025510q^{89} + 38151124894006957908596484633488q^{91} + 251923040101435700991757982654592q^{93} - 43478396094943467445859067003000q^{95} + 226806680667600950875216250271842q^{97} + 1041873766653137898400473873964644q^{99} + O(q^{100}) \)

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
0 1.33006e8 0 5.38799e11 0 3.33473e13 0 1.21314e16 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)

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 16.34.a.a 1
4.b odd 2 1 2.34.a.a 1
12.b even 2 1 18.34.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.34.a.a 1 4.b odd 2 1
16.34.a.a 1 1.a even 1 1 trivial
18.34.a.c 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 133005564 \) acting on \(S_{34}^{\mathrm{new}}(\Gamma_0(16))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 133005564 T + 5559060566555523 T^{2} \)
$5$ \( 1 - 538799132550 T + \)\(11\!\cdots\!25\)\( T^{2} \)
$7$ \( 1 - 33347311051768 T + \)\(77\!\cdots\!07\)\( T^{2} \)
$11$ \( 1 - 85882263625386228 T + \)\(23\!\cdots\!31\)\( T^{2} \)
$13$ \( 1 - 1144054008875905166 T + \)\(57\!\cdots\!53\)\( T^{2} \)
$17$ \( 1 + \)\(13\!\cdots\!98\)\( T + \)\(40\!\cdots\!37\)\( T^{2} \)
$19$ \( 1 + 80695000174130231060 T + \)\(15\!\cdots\!59\)\( T^{2} \)
$23$ \( 1 - \)\(14\!\cdots\!44\)\( T + \)\(86\!\cdots\!83\)\( T^{2} \)
$29$ \( 1 + \)\(16\!\cdots\!90\)\( T + \)\(18\!\cdots\!89\)\( T^{2} \)
$31$ \( 1 - \)\(18\!\cdots\!28\)\( T + \)\(16\!\cdots\!91\)\( T^{2} \)
$37$ \( 1 + \)\(96\!\cdots\!98\)\( T + \)\(56\!\cdots\!97\)\( T^{2} \)
$41$ \( 1 - \)\(64\!\cdots\!42\)\( T + \)\(16\!\cdots\!21\)\( T^{2} \)
$43$ \( 1 - \)\(81\!\cdots\!84\)\( T + \)\(80\!\cdots\!43\)\( T^{2} \)
$47$ \( 1 - \)\(62\!\cdots\!68\)\( T + \)\(15\!\cdots\!27\)\( T^{2} \)
$53$ \( 1 + \)\(21\!\cdots\!94\)\( T + \)\(79\!\cdots\!73\)\( T^{2} \)
$59$ \( 1 + \)\(29\!\cdots\!20\)\( T + \)\(27\!\cdots\!79\)\( T^{2} \)
$61$ \( 1 + \)\(45\!\cdots\!58\)\( T + \)\(82\!\cdots\!81\)\( T^{2} \)
$67$ \( 1 + \)\(11\!\cdots\!12\)\( T + \)\(18\!\cdots\!87\)\( T^{2} \)
$71$ \( 1 + \)\(25\!\cdots\!72\)\( T + \)\(12\!\cdots\!11\)\( T^{2} \)
$73$ \( 1 + \)\(28\!\cdots\!74\)\( T + \)\(30\!\cdots\!33\)\( T^{2} \)
$79$ \( 1 + \)\(92\!\cdots\!20\)\( T + \)\(41\!\cdots\!39\)\( T^{2} \)
$83$ \( 1 - \)\(16\!\cdots\!04\)\( T + \)\(21\!\cdots\!63\)\( T^{2} \)
$89$ \( 1 + \)\(20\!\cdots\!10\)\( T + \)\(21\!\cdots\!69\)\( T^{2} \)
$97$ \( 1 - \)\(22\!\cdots\!42\)\( T + \)\(36\!\cdots\!77\)\( T^{2} \)
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