Properties

Label 16.9.c.a
Level 16
Weight 9
Character orbit 16.c
Analytic conductor 6.518
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 16.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.51805776098\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-35}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 24\sqrt{-35}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{3} -510 q^{5} + 18 \beta q^{7} -13599 q^{9} +O(q^{10})\) \( q -\beta q^{3} -510 q^{5} + 18 \beta q^{7} -13599 q^{9} -135 \beta q^{11} -27710 q^{13} + 510 \beta q^{15} + 50370 q^{17} -765 \beta q^{19} + 362880 q^{21} -1242 \beta q^{23} -130525 q^{25} + 7038 \beta q^{27} + 54978 q^{29} -8280 \beta q^{31} -2721600 q^{33} -9180 \beta q^{35} + 793730 q^{37} + 27710 \beta q^{39} -75582 q^{41} -3519 \beta q^{43} + 6935490 q^{45} -20196 \beta q^{47} -767039 q^{49} -50370 \beta q^{51} + 11166210 q^{53} + 68850 \beta q^{55} -15422400 q^{57} + 153765 \beta q^{59} -23826622 q^{61} -244782 \beta q^{63} + 14132100 q^{65} -52785 \beta q^{67} -25038720 q^{69} + 71010 \beta q^{71} + 6516610 q^{73} + 130525 \beta q^{75} + 48988800 q^{77} + 343620 \beta q^{79} + 52663041 q^{81} -517293 \beta q^{83} -25688700 q^{85} -54978 \beta q^{87} + 86795778 q^{89} -498780 \beta q^{91} -166924800 q^{93} + 390150 \beta q^{95} -46670270 q^{97} + 1835865 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 1020q^{5} - 27198q^{9} + O(q^{10}) \) \( 2q - 1020q^{5} - 27198q^{9} - 55420q^{13} + 100740q^{17} + 725760q^{21} - 261050q^{25} + 109956q^{29} - 5443200q^{33} + 1587460q^{37} - 151164q^{41} + 13870980q^{45} - 1534078q^{49} + 22332420q^{53} - 30844800q^{57} - 47653244q^{61} + 28264200q^{65} - 50077440q^{69} + 13033220q^{73} + 97977600q^{77} + 105326082q^{81} - 51377400q^{85} + 173591556q^{89} - 333849600q^{93} - 93340540q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/16\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(15\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
15.1
0.500000 + 2.95804i
0.500000 2.95804i
0 141.986i 0 −510.000 0 2555.75i 0 −13599.0 0
15.2 0 141.986i 0 −510.000 0 2555.75i 0 −13599.0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 16.9.c.a 2
3.b odd 2 1 144.9.g.g 2
4.b odd 2 1 inner 16.9.c.a 2
5.b even 2 1 400.9.b.c 2
5.c odd 4 2 400.9.h.b 4
8.b even 2 1 64.9.c.d 2
8.d odd 2 1 64.9.c.d 2
12.b even 2 1 144.9.g.g 2
16.e even 4 2 256.9.d.f 4
16.f odd 4 2 256.9.d.f 4
20.d odd 2 1 400.9.b.c 2
20.e even 4 2 400.9.h.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.9.c.a 2 1.a even 1 1 trivial
16.9.c.a 2 4.b odd 2 1 inner
64.9.c.d 2 8.b even 2 1
64.9.c.d 2 8.d odd 2 1
144.9.g.g 2 3.b odd 2 1
144.9.g.g 2 12.b even 2 1
256.9.d.f 4 16.e even 4 2
256.9.d.f 4 16.f odd 4 2
400.9.b.c 2 5.b even 2 1
400.9.b.c 2 20.d odd 2 1
400.9.h.b 4 5.c odd 4 2
400.9.h.b 4 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 20160 \) acting on \(S_{9}^{\mathrm{new}}(16, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - 78 T + 6561 T^{2} )( 1 + 78 T + 6561 T^{2} ) \)
$5$ \( ( 1 + 510 T + 390625 T^{2} )^{2} \)
$7$ \( 1 - 4997762 T^{2} + 33232930569601 T^{4} \)
$11$ \( 1 - 61301762 T^{2} + 45949729863572161 T^{4} \)
$13$ \( ( 1 + 27710 T + 815730721 T^{2} )^{2} \)
$17$ \( ( 1 - 50370 T + 6975757441 T^{2} )^{2} \)
$19$ \( 1 - 22168990082 T^{2} + \)\(28\!\cdots\!81\)\( T^{4} \)
$23$ \( 1 - 125523880322 T^{2} + \)\(61\!\cdots\!61\)\( T^{4} \)
$29$ \( ( 1 - 54978 T + 500246412961 T^{2} )^{2} \)
$31$ \( 1 - 323644730882 T^{2} + \)\(72\!\cdots\!81\)\( T^{4} \)
$37$ \( ( 1 - 793730 T + 3512479453921 T^{2} )^{2} \)
$41$ \( ( 1 + 75582 T + 7984925229121 T^{2} )^{2} \)
$43$ \( 1 - 23126751997442 T^{2} + \)\(13\!\cdots\!01\)\( T^{4} \)
$47$ \( 1 - 39399744456962 T^{2} + \)\(56\!\cdots\!21\)\( T^{4} \)
$53$ \( ( 1 - 11166210 T + 62259690411361 T^{2} )^{2} \)
$59$ \( 1 + 182995617327358 T^{2} + \)\(21\!\cdots\!41\)\( T^{4} \)
$61$ \( ( 1 + 23826622 T + 191707312997281 T^{2} )^{2} \)
$67$ \( 1 - 755964429617282 T^{2} + \)\(16\!\cdots\!81\)\( T^{4} \)
$71$ \( 1 - 1189851873275522 T^{2} + \)\(41\!\cdots\!21\)\( T^{4} \)
$73$ \( ( 1 - 6516610 T + 806460091894081 T^{2} )^{2} \)
$79$ \( 1 - 653831579109122 T^{2} + \)\(23\!\cdots\!21\)\( T^{4} \)
$83$ \( 1 + 890071220357758 T^{2} + \)\(50\!\cdots\!81\)\( T^{4} \)
$89$ \( ( 1 - 86795778 T + 3936588805702081 T^{2} )^{2} \)
$97$ \( ( 1 + 46670270 T + 7837433594376961 T^{2} )^{2} \)
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