Properties

Label 16.48.a.a
Level 16
Weight 48
Character orbit 16.a
Self dual yes
Analytic conductor 223.852
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 48 \)
Character orbit: \([\chi]\) \(=\) 16.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(223.852260248\)
Analytic rank: \(1\)
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

\(f(q)\) \(=\) \( q + 196634580372q^{3} + 20669962168980750q^{5} + 51172881836896522696q^{7} + 12076343839115024370597q^{9} + O(q^{10}) \) \( q + 196634580372q^{3} + 20669962168980750q^{5} + 51172881836896522696q^{7} + 12076343839115024370597q^{9} - 5297430319653102012090852q^{11} - 125094137018739417081409402q^{13} + 4064429337402644731195839000q^{15} - 44830025442540925573659584526q^{17} + 1111319860561156308410383992100q^{19} + 10062358146424088287113934122912q^{21} + 178238089260084834809121532218072q^{23} - 283295399693004794853630397015625q^{25} - 2853653550830246541421912587394680q^{27} - 30534112362977188191390732250983690q^{29} + 110801717324244256756593954244061728q^{31} - 1041657987954897538755593553359956944q^{33} + 1057741531646373274518609629562102000q^{35} + 6203767833063528570721355158143068654q^{37} - 24597833139677296378218826724605457544q^{39} - 35669771283594680742424472586368472678q^{41} - 338085414102589873498086573597898665668q^{43} + 249617570294111306560850061725759007750q^{45} + 2298386467820046045287393257826261887536q^{47} - 2624674481263329603593254666205508347127q^{49} - 8815133240960118497613035779645874523672q^{51} - 29595651085041661560230876234553121751202q^{53} - 109497684300041220259762369217245239099000q^{55} + 218523914440512523147376470653969663061200q^{57} - 402897770467707194917911296294114858644020q^{59} + 575153391792220796561057524320411384118742q^{61} + 617981316300766453265286945678925325569512q^{63} - 2585691079738638133004233998762095807011500q^{65} + 1868033100301939556103984556344225386618836q^{67} + 35047771887963861461812551205650266114882784q^{69} - 55201984112681452831941304628692429011115512q^{71} - 79182696974319051005003915162106408285767302q^{73} - 55705672039952015459827558277951183002312500q^{75} - 271084775786801164287626553493206987231976992q^{77} - 88275129763458787700378919632160122138091440q^{79} - 882222632967742841806650442067058936377219799q^{81} + 737730557224622917773365649736040155174606692q^{83} - 926634929931765436852019322219384849891874500q^{85} - 6004062371525516748622590259261984867366132680q^{87} - 8277033948537162260782072894676071243017411990q^{89} - 6401427492148495240956316985608715680560787792q^{91} + 21787449190549732089363877936773246358145202816q^{93} + 22970939475436063098422531384533294973052075000q^{95} - 61954178934472753824888537150641827226577342046q^{97} - 63973590003883872689251426636780212535281478644q^{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 1.96635e11 0 2.06700e16 0 5.11729e19 0 1.20763e22 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 16.48.a.a 1
4.b odd 2 1 2.48.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.48.a.a 1 4.b odd 2 1
16.48.a.a 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 196634580372 \) acting on \(S_{48}^{\mathrm{new}}(\Gamma_0(16))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 196634580372 T + \)\(26\!\cdots\!87\)\( T^{2} \)
$5$ \( 1 - 20669962168980750 T + \)\(71\!\cdots\!25\)\( T^{2} \)
$7$ \( 1 - 51172881836896522696 T + \)\(52\!\cdots\!43\)\( T^{2} \)
$11$ \( 1 + \)\(52\!\cdots\!52\)\( T + \)\(88\!\cdots\!71\)\( T^{2} \)
$13$ \( 1 + \)\(12\!\cdots\!02\)\( T + \)\(22\!\cdots\!17\)\( T^{2} \)
$17$ \( 1 + \)\(44\!\cdots\!26\)\( T + \)\(67\!\cdots\!73\)\( T^{2} \)
$19$ \( 1 - \)\(11\!\cdots\!00\)\( T + \)\(12\!\cdots\!39\)\( T^{2} \)
$23$ \( 1 - \)\(17\!\cdots\!72\)\( T + \)\(10\!\cdots\!47\)\( T^{2} \)
$29$ \( 1 + \)\(30\!\cdots\!90\)\( T + \)\(54\!\cdots\!09\)\( T^{2} \)
$31$ \( 1 - \)\(11\!\cdots\!28\)\( T + \)\(12\!\cdots\!11\)\( T^{2} \)
$37$ \( 1 - \)\(62\!\cdots\!54\)\( T + \)\(50\!\cdots\!33\)\( T^{2} \)
$41$ \( 1 + \)\(35\!\cdots\!78\)\( T + \)\(63\!\cdots\!81\)\( T^{2} \)
$43$ \( 1 + \)\(33\!\cdots\!68\)\( T + \)\(59\!\cdots\!07\)\( T^{2} \)
$47$ \( 1 - \)\(22\!\cdots\!36\)\( T + \)\(38\!\cdots\!63\)\( T^{2} \)
$53$ \( 1 + \)\(29\!\cdots\!02\)\( T + \)\(10\!\cdots\!37\)\( T^{2} \)
$59$ \( 1 + \)\(40\!\cdots\!20\)\( T + \)\(16\!\cdots\!19\)\( T^{2} \)
$61$ \( 1 - \)\(57\!\cdots\!42\)\( T + \)\(81\!\cdots\!21\)\( T^{2} \)
$67$ \( 1 - \)\(18\!\cdots\!36\)\( T + \)\(66\!\cdots\!23\)\( T^{2} \)
$71$ \( 1 + \)\(55\!\cdots\!12\)\( T + \)\(10\!\cdots\!91\)\( T^{2} \)
$73$ \( 1 + \)\(79\!\cdots\!02\)\( T + \)\(37\!\cdots\!97\)\( T^{2} \)
$79$ \( 1 + \)\(88\!\cdots\!40\)\( T + \)\(15\!\cdots\!59\)\( T^{2} \)
$83$ \( 1 - \)\(73\!\cdots\!92\)\( T + \)\(15\!\cdots\!27\)\( T^{2} \)
$89$ \( 1 + \)\(82\!\cdots\!90\)\( T + \)\(41\!\cdots\!29\)\( T^{2} \)
$97$ \( 1 + \)\(61\!\cdots\!46\)\( T + \)\(23\!\cdots\!13\)\( T^{2} \)
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