Properties

Label 16.32.a.b
Level $16$
Weight $32$
Character orbit 16.a
Self dual yes
Analytic conductor $97.403$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 32 \)
Character orbit: \([\chi]\) \(=\) 16.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(97.4034125104\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
Defining polynomial: \(x^{2} - x - 4573872\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{7}\cdot 3 \)
Twist minimal: no (minimal twist has level 1)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 192\sqrt{18295489}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -8681580 - 27 \beta ) q^{3} + ( -9695609010 - 88660 \beta ) q^{5} + ( -15128763788600 + 4495554 \beta ) q^{7} + ( -50633228151963 + 468805320 \beta ) q^{9} +O(q^{10})\) \( q +(-8681580 - 27 \beta) q^{3} +(-9695609010 - 88660 \beta) q^{5} +(-15128763788600 + 4495554 \beta) q^{7} +(-50633228151963 + 468805320 \beta) q^{9} +(3891176872559388 - 1812556825 \beta) q^{11} +(37354476525130310 - 291313526868 \beta) q^{13} +(1698672911337290520 + 1031490326070 \beta) q^{15} +(8612303914493544690 + 2817834255816 \beta) q^{17} +(6185281664011082020 + 9862862047785 \beta) q^{19} +(49477478708035580832 + 369448110596880 \beta) q^{21} +(-\)\(94\!\cdots\!40\)\( + 1159851136203238 \beta) q^{23} +(\)\(73\!\cdots\!75\)\( + 1719225389653200 \beta) q^{25} +(-\)\(27\!\cdots\!40\)\( + 13974307969763970 \beta) q^{27} +(\)\(64\!\cdots\!70\)\( + 14637022336528060 \beta) q^{29} +(-\)\(62\!\cdots\!32\)\( - 150788823188772600 \beta) q^{31} +(-\)\(77\!\cdots\!40\)\( - 89325918478319976 \beta) q^{33} +(-\)\(12\!\cdots\!40\)\( + 1297729063629934460 \beta) q^{35} +(-\)\(41\!\cdots\!30\)\( + 2014015257098656956 \beta) q^{37} +(\)\(49\!\cdots\!56\)\( + 1520490822408173070 \beta) q^{39} +(\)\(43\!\cdots\!42\)\( + 11515965821552939600 \beta) q^{41} +(\)\(91\!\cdots\!00\)\( + 21416940670024423503 \beta) q^{43} +(-\)\(27\!\cdots\!70\)\( - 56211076574893620 \beta) q^{45} +(-\)\(47\!\cdots\!60\)\( - 32105786139859115476 \beta) q^{47} +(\)\(84\!\cdots\!93\)\( - \)\(13\!\cdots\!00\)\( \beta) q^{49} +(-\)\(12\!\cdots\!72\)\( - \)\(25\!\cdots\!10\)\( \beta) q^{51} +(\)\(97\!\cdots\!30\)\( - 75254035496850878148 \beta) q^{53} +(\)\(70\!\cdots\!20\)\( - \)\(32\!\cdots\!30\)\( \beta) q^{55} +(-\)\(23\!\cdots\!20\)\( - \)\(25\!\cdots\!40\)\( \beta) q^{57} +(\)\(99\!\cdots\!60\)\( + \)\(27\!\cdots\!55\)\( \beta) q^{59} +(-\)\(60\!\cdots\!38\)\( + \)\(70\!\cdots\!00\)\( \beta) q^{61} +(\)\(21\!\cdots\!80\)\( - \)\(73\!\cdots\!02\)\( \beta) q^{63} +(\)\(17\!\cdots\!80\)\( - \)\(48\!\cdots\!20\)\( \beta) q^{65} +(\)\(48\!\cdots\!60\)\( + \)\(55\!\cdots\!09\)\( \beta) q^{67} +(-\)\(12\!\cdots\!96\)\( + \)\(15\!\cdots\!40\)\( \beta) q^{69} +(-\)\(27\!\cdots\!72\)\( + \)\(59\!\cdots\!50\)\( \beta) q^{71} +(\)\(31\!\cdots\!90\)\( + \)\(99\!\cdots\!12\)\( \beta) q^{73} +(-\)\(37\!\cdots\!00\)\( - \)\(34\!\cdots\!25\)\( \beta) q^{75} +(-\)\(64\!\cdots\!00\)\( + \)\(44\!\cdots\!52\)\( \beta) q^{77} +(\)\(59\!\cdots\!80\)\( - \)\(29\!\cdots\!60\)\( \beta) q^{79} +(-\)\(19\!\cdots\!79\)\( - \)\(33\!\cdots\!60\)\( \beta) q^{81} +(\)\(13\!\cdots\!80\)\( + \)\(39\!\cdots\!33\)\( \beta) q^{83} +(-\)\(25\!\cdots\!60\)\( - \)\(79\!\cdots\!60\)\( \beta) q^{85} +(-\)\(82\!\cdots\!20\)\( - \)\(18\!\cdots\!90\)\( \beta) q^{87} +(-\)\(10\!\cdots\!90\)\( + \)\(16\!\cdots\!80\)\( \beta) q^{89} +(-\)\(14\!\cdots\!12\)\( + \)\(45\!\cdots\!40\)\( \beta) q^{91} +(\)\(32\!\cdots\!60\)\( + \)\(30\!\cdots\!64\)\( \beta) q^{93} +(-\)\(64\!\cdots\!00\)\( - \)\(64\!\cdots\!50\)\( \beta) q^{95} +(-\)\(45\!\cdots\!90\)\( - \)\(10\!\cdots\!24\)\( \beta) q^{97} +(-\)\(77\!\cdots\!44\)\( + \)\(19\!\cdots\!35\)\( \beta) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 17363160q^{3} - 19391218020q^{5} - 30257527577200q^{7} - 101266456303926q^{9} + O(q^{10}) \) \( 2q - 17363160q^{3} - 19391218020q^{5} - 30257527577200q^{7} - 101266456303926q^{9} + 7782353745118776q^{11} + 74708953050260620q^{13} + 3397345822674581040q^{15} + 17224607828987089380q^{17} + 12370563328022164040q^{19} + 98954957416071161664q^{21} - \)\(18\!\cdots\!80\)\(q^{23} + \)\(14\!\cdots\!50\)\(q^{25} - \)\(54\!\cdots\!80\)\(q^{27} + \)\(12\!\cdots\!40\)\(q^{29} - \)\(12\!\cdots\!64\)\(q^{31} - \)\(15\!\cdots\!80\)\(q^{33} - \)\(24\!\cdots\!80\)\(q^{35} - \)\(83\!\cdots\!60\)\(q^{37} + \)\(99\!\cdots\!12\)\(q^{39} + \)\(87\!\cdots\!84\)\(q^{41} + \)\(18\!\cdots\!00\)\(q^{43} - \)\(55\!\cdots\!40\)\(q^{45} - \)\(95\!\cdots\!20\)\(q^{47} + \)\(16\!\cdots\!86\)\(q^{49} - \)\(25\!\cdots\!44\)\(q^{51} + \)\(19\!\cdots\!60\)\(q^{53} + \)\(14\!\cdots\!40\)\(q^{55} - \)\(46\!\cdots\!40\)\(q^{57} + \)\(19\!\cdots\!20\)\(q^{59} - \)\(12\!\cdots\!76\)\(q^{61} + \)\(43\!\cdots\!60\)\(q^{63} + \)\(34\!\cdots\!60\)\(q^{65} + \)\(96\!\cdots\!20\)\(q^{67} - \)\(25\!\cdots\!92\)\(q^{69} - \)\(55\!\cdots\!44\)\(q^{71} + \)\(62\!\cdots\!80\)\(q^{73} - \)\(75\!\cdots\!00\)\(q^{75} - \)\(12\!\cdots\!00\)\(q^{77} + \)\(11\!\cdots\!60\)\(q^{79} - \)\(39\!\cdots\!58\)\(q^{81} + \)\(26\!\cdots\!60\)\(q^{83} - \)\(50\!\cdots\!20\)\(q^{85} - \)\(16\!\cdots\!40\)\(q^{87} - \)\(21\!\cdots\!80\)\(q^{89} - \)\(28\!\cdots\!24\)\(q^{91} + \)\(65\!\cdots\!20\)\(q^{93} - \)\(12\!\cdots\!00\)\(q^{95} - \)\(90\!\cdots\!80\)\(q^{97} - \)\(15\!\cdots\!88\)\(q^{99} + O(q^{100}) \)

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
2139.16
−2138.16
0 −3.08552e7 0 −8.25073e10 0 −1.14368e13 0 3.34371e14 0
1.2 0 1.34921e7 0 6.31161e10 0 −1.88207e13 0 −4.35638e14 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 16.32.a.b 2
4.b odd 2 1 1.32.a.a 2
12.b even 2 1 9.32.a.a 2
20.d odd 2 1 25.32.a.a 2
20.e even 4 2 25.32.b.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.32.a.a 2 4.b odd 2 1
9.32.a.a 2 12.b even 2 1
16.32.a.b 2 1.a even 1 1 trivial
25.32.a.a 2 20.d odd 2 1
25.32.b.a 4 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 17363160 T_{3} - \)\(41\!\cdots\!84\)\( \) acting on \(S_{32}^{\mathrm{new}}(\Gamma_0(16))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -416300505539184 + 17363160 T + T^{2} \)
$5$ \( -\)\(52\!\cdots\!00\)\( + 19391218020 T + T^{2} \)
$7$ \( \)\(21\!\cdots\!64\)\( + 30257527577200 T + T^{2} \)
$11$ \( \)\(12\!\cdots\!44\)\( - 7782353745118776 T + T^{2} \)
$13$ \( -\)\(55\!\cdots\!04\)\( - 74708953050260620 T + T^{2} \)
$17$ \( \)\(68\!\cdots\!24\)\( - 17224607828987089380 T + T^{2} \)
$19$ \( -\)\(27\!\cdots\!00\)\( - 12370563328022164040 T + T^{2} \)
$23$ \( -\)\(73\!\cdots\!24\)\( + \)\(18\!\cdots\!80\)\( T + T^{2} \)
$29$ \( \)\(39\!\cdots\!00\)\( - \)\(12\!\cdots\!40\)\( T + T^{2} \)
$31$ \( -\)\(11\!\cdots\!76\)\( + \)\(12\!\cdots\!64\)\( T + T^{2} \)
$37$ \( -\)\(25\!\cdots\!56\)\( + \)\(83\!\cdots\!60\)\( T + T^{2} \)
$41$ \( -\)\(70\!\cdots\!36\)\( - \)\(87\!\cdots\!84\)\( T + T^{2} \)
$43$ \( -\)\(22\!\cdots\!64\)\( - \)\(18\!\cdots\!00\)\( T + T^{2} \)
$47$ \( \)\(15\!\cdots\!04\)\( + \)\(95\!\cdots\!20\)\( T + T^{2} \)
$53$ \( \)\(56\!\cdots\!16\)\( - \)\(19\!\cdots\!60\)\( T + T^{2} \)
$59$ \( -\)\(51\!\cdots\!00\)\( - \)\(19\!\cdots\!20\)\( T + T^{2} \)
$61$ \( \)\(36\!\cdots\!44\)\( + \)\(12\!\cdots\!76\)\( T + T^{2} \)
$67$ \( \)\(26\!\cdots\!24\)\( - \)\(96\!\cdots\!20\)\( T + T^{2} \)
$71$ \( -\)\(16\!\cdots\!16\)\( + \)\(55\!\cdots\!44\)\( T + T^{2} \)
$73$ \( \)\(90\!\cdots\!76\)\( - \)\(62\!\cdots\!80\)\( T + T^{2} \)
$79$ \( -\)\(53\!\cdots\!00\)\( - \)\(11\!\cdots\!60\)\( T + T^{2} \)
$83$ \( -\)\(86\!\cdots\!44\)\( - \)\(26\!\cdots\!60\)\( T + T^{2} \)
$89$ \( -\)\(62\!\cdots\!00\)\( + \)\(21\!\cdots\!80\)\( T + T^{2} \)
$97$ \( \)\(19\!\cdots\!04\)\( + \)\(90\!\cdots\!80\)\( T + T^{2} \)
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