Properties

Label 16.9.c.b
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: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 8 - 16 \zeta_{6} ) q^{3} + 258 q^{5} + ( 1904 - 3808 \zeta_{6} ) q^{7} + 6369 q^{9} +O(q^{10})\) \( q + ( 8 - 16 \zeta_{6} ) q^{3} + 258 q^{5} + ( 1904 - 3808 \zeta_{6} ) q^{7} + 6369 q^{9} + ( 13368 - 26736 \zeta_{6} ) q^{11} + 19138 q^{13} + ( 2064 - 4128 \zeta_{6} ) q^{15} -58686 q^{17} + ( -88088 + 176176 \zeta_{6} ) q^{19} -45696 q^{21} + ( -155952 + 311904 \zeta_{6} ) q^{23} -324061 q^{25} + ( 103440 - 206880 \zeta_{6} ) q^{27} + 842178 q^{29} + ( 606912 - 1213824 \zeta_{6} ) q^{31} -320832 q^{33} + ( 491232 - 982464 \zeta_{6} ) q^{35} + 2548610 q^{37} + ( 153104 - 306208 \zeta_{6} ) q^{39} -4324158 q^{41} + ( -1176072 + 2352144 \zeta_{6} ) q^{43} + 1643202 q^{45} + ( -4176096 + 8352192 \zeta_{6} ) q^{47} -5110847 q^{49} + ( -469488 + 938976 \zeta_{6} ) q^{51} + 1192194 q^{53} + ( 3448944 - 6897888 \zeta_{6} ) q^{55} + 2114112 q^{57} + ( 195288 - 390576 \zeta_{6} ) q^{59} + 8414786 q^{61} + ( 12126576 - 24253152 \zeta_{6} ) q^{63} + 4937604 q^{65} + ( 10060168 - 20120336 \zeta_{6} ) q^{67} + 3742848 q^{69} + ( -17826576 + 35653152 \zeta_{6} ) q^{71} + 12735874 q^{73} + ( -2592488 + 5184976 \zeta_{6} ) q^{75} -76358016 q^{77} + ( -3629600 + 7259200 \zeta_{6} ) q^{79} + 39304449 q^{81} + ( -47962776 + 95925552 \zeta_{6} ) q^{83} -15140988 q^{85} + ( 6737424 - 13474848 \zeta_{6} ) q^{87} -16802814 q^{89} + ( 36438752 - 72877504 \zeta_{6} ) q^{91} -14565888 q^{93} + ( -22726704 + 45453408 \zeta_{6} ) q^{95} + 120994882 q^{97} + ( 85140792 - 170281584 \zeta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 516q^{5} + 12738q^{9} + O(q^{10}) \) \( 2q + 516q^{5} + 12738q^{9} + 38276q^{13} - 117372q^{17} - 91392q^{21} - 648122q^{25} + 1684356q^{29} - 641664q^{33} + 5097220q^{37} - 8648316q^{41} + 3286404q^{45} - 10221694q^{49} + 2384388q^{53} + 4228224q^{57} + 16829572q^{61} + 9875208q^{65} + 7485696q^{69} + 25471748q^{73} - 152716032q^{77} + 78608898q^{81} - 30281976q^{85} - 33605628q^{89} - 29131776q^{93} + 241989764q^{97} + O(q^{100}) \)

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 + 0.866025i
0.500000 0.866025i
0 13.8564i 0 258.000 0 3297.82i 0 6369.00 0
15.2 0 13.8564i 0 258.000 0 3297.82i 0 6369.00 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.b 2
3.b odd 2 1 144.9.g.d 2
4.b odd 2 1 inner 16.9.c.b 2
5.b even 2 1 400.9.b.e 2
5.c odd 4 2 400.9.h.a 4
8.b even 2 1 64.9.c.c 2
8.d odd 2 1 64.9.c.c 2
12.b even 2 1 144.9.g.d 2
16.e even 4 2 256.9.d.d 4
16.f odd 4 2 256.9.d.d 4
20.d odd 2 1 400.9.b.e 2
20.e even 4 2 400.9.h.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.9.c.b 2 1.a even 1 1 trivial
16.9.c.b 2 4.b odd 2 1 inner
64.9.c.c 2 8.b even 2 1
64.9.c.c 2 8.d odd 2 1
144.9.g.d 2 3.b odd 2 1
144.9.g.d 2 12.b even 2 1
256.9.d.d 4 16.e even 4 2
256.9.d.d 4 16.f odd 4 2
400.9.b.e 2 5.b even 2 1
400.9.b.e 2 20.d odd 2 1
400.9.h.a 4 5.c odd 4 2
400.9.h.a 4 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 192 \) acting on \(S_{9}^{\mathrm{new}}(16, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 12930 T^{2} + 43046721 T^{4} \)
$5$ \( ( 1 - 258 T + 390625 T^{2} )^{2} \)
$7$ \( 1 - 653954 T^{2} + 33232930569601 T^{4} \)
$11$ \( 1 + 107392510 T^{2} + 45949729863572161 T^{4} \)
$13$ \( ( 1 - 19138 T + 815730721 T^{2} )^{2} \)
$17$ \( ( 1 + 58686 T + 6975757441 T^{2} )^{2} \)
$19$ \( 1 - 10688638850 T^{2} + \)\(28\!\cdots\!81\)\( T^{4} \)
$23$ \( 1 - 83658891650 T^{2} + \)\(61\!\cdots\!61\)\( T^{4} \)
$29$ \( ( 1 - 842178 T + 500246412961 T^{2} )^{2} \)
$31$ \( 1 - 600755547650 T^{2} + \)\(72\!\cdots\!81\)\( T^{4} \)
$37$ \( ( 1 - 2548610 T + 3512479453921 T^{2} )^{2} \)
$41$ \( ( 1 + 4324158 T + 7984925229121 T^{2} )^{2} \)
$43$ \( 1 - 19226964507650 T^{2} + \)\(13\!\cdots\!01\)\( T^{4} \)
$47$ \( 1 + 4696760080126 T^{2} + \)\(56\!\cdots\!21\)\( T^{4} \)
$53$ \( ( 1 - 1192194 T + 62259690411361 T^{2} )^{2} \)
$59$ \( 1 - 293546462999810 T^{2} + \)\(21\!\cdots\!41\)\( T^{4} \)
$61$ \( ( 1 - 8414786 T + 191707312997281 T^{2} )^{2} \)
$67$ \( 1 - 508514414548610 T^{2} + \)\(16\!\cdots\!81\)\( T^{4} \)
$71$ \( 1 - 338146626840194 T^{2} + \)\(41\!\cdots\!21\)\( T^{4} \)
$73$ \( ( 1 - 12735874 T + 806460091894081 T^{2} )^{2} \)
$79$ \( 1 - 2994695631333122 T^{2} + \)\(23\!\cdots\!21\)\( T^{4} \)
$83$ \( 1 + 2396699180600446 T^{2} + \)\(50\!\cdots\!81\)\( T^{4} \)
$89$ \( ( 1 + 16802814 T + 3936588805702081 T^{2} )^{2} \)
$97$ \( ( 1 - 120994882 T + 7837433594376961 T^{2} )^{2} \)
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