Properties

Label 16.15.c.b
Level 16
Weight 15
Character orbit 16.c
Analytic conductor 19.893
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 15 \)
Character orbit: \([\chi]\) \(=\) 16.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(19.8926349043\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3395}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{5}\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 48\sqrt{-3395}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{3} + 107850 q^{5} + 522 \beta q^{7} -3039111 q^{9} +O(q^{10})\) \( q -\beta q^{3} + 107850 q^{5} + 522 \beta q^{7} -3039111 q^{9} + 7425 \beta q^{11} + 59275450 q^{13} -107850 \beta q^{15} -44896350 q^{17} + 306675 \beta q^{19} + 4083125760 q^{21} + 490158 \beta q^{23} + 5528106875 q^{25} -1743858 \beta q^{27} + 30520168602 q^{29} -12396600 \beta q^{31} + 58078944000 q^{33} + 56297700 \beta q^{35} -125971660150 q^{37} -59275450 \beta q^{39} -41444675982 q^{41} -35870679 \beta q^{43} -327768121350 q^{45} -1583604 \beta q^{47} -1453168573871 q^{49} + 44896350 \beta q^{51} + 473374435050 q^{53} + 800786250 \beta q^{55} + 2398836384000 q^{57} -1130436675 \beta q^{59} -58526672422 q^{61} -1586415942 \beta q^{63} + 6392857282500 q^{65} + 1493137935 \beta q^{67} + 3834055088640 q^{69} + 5611095450 \beta q^{71} + 14644904168050 q^{73} -5528106875 \beta q^{75} -30317208768000 q^{77} -2995497900 \beta q^{79} -28176570485199 q^{81} + 8806141827 \beta q^{83} -4842071347500 q^{85} -30520168602 \beta q^{87} + 3585960554322 q^{89} + 30941784900 \beta q^{91} -96967196928000 q^{93} + 33074898750 \beta q^{95} + 65197469070850 q^{97} -22565399175 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 215700q^{5} - 6078222q^{9} + O(q^{10}) \) \( 2q + 215700q^{5} - 6078222q^{9} + 118550900q^{13} - 89792700q^{17} + 8166251520q^{21} + 11056213750q^{25} + 61040337204q^{29} + 116157888000q^{33} - 251943320300q^{37} - 82889351964q^{41} - 655536242700q^{45} - 2906337147742q^{49} + 946748870100q^{53} + 4797672768000q^{57} - 117053344844q^{61} + 12785714565000q^{65} + 7668110177280q^{69} + 29289808336100q^{73} - 60634417536000q^{77} - 56353140970398q^{81} - 9684142695000q^{85} + 7171921108644q^{89} - 193934393856000q^{93} + 130394938141700q^{97} + O(q^{100}) \)

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 + 29.1333i
0.500000 29.1333i
0 2796.80i 0 107850. 0 1.45993e6i 0 −3.03911e6 0
15.2 0 2796.80i 0 107850. 0 1.45993e6i 0 −3.03911e6 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 16.15.c.b 2
3.b odd 2 1 144.15.g.b 2
4.b odd 2 1 inner 16.15.c.b 2
8.b even 2 1 64.15.c.b 2
8.d odd 2 1 64.15.c.b 2
12.b even 2 1 144.15.g.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.15.c.b 2 1.a even 1 1 trivial
16.15.c.b 2 4.b odd 2 1 inner
64.15.c.b 2 8.b even 2 1
64.15.c.b 2 8.d odd 2 1
144.15.g.b 2 3.b odd 2 1
144.15.g.b 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 7822080 \) acting on \(S_{15}^{\mathrm{new}}(16, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 1743858 T^{2} + 22876792454961 T^{4} \)
$5$ \( ( 1 - 107850 T + 6103515625 T^{2} )^{2} \)
$7$ \( 1 + 774945501022 T^{2} + \)\(45\!\cdots\!01\)\( T^{4} \)
$11$ \( 1 - 328263507966482 T^{2} + \)\(14\!\cdots\!81\)\( T^{4} \)
$13$ \( ( 1 - 59275450 T + 3937376385699289 T^{2} )^{2} \)
$17$ \( ( 1 + 44896350 T + 168377826559400929 T^{2} )^{2} \)
$19$ \( 1 - 862350223502568242 T^{2} + \)\(63\!\cdots\!41\)\( T^{4} \)
$23$ \( 1 - 21306379874939894498 T^{2} + \)\(13\!\cdots\!81\)\( T^{4} \)
$29$ \( ( 1 - 30520168602 T + \)\(29\!\cdots\!81\)\( T^{2} )^{2} \)
$31$ \( 1 - \)\(31\!\cdots\!42\)\( T^{2} + \)\(57\!\cdots\!41\)\( T^{4} \)
$37$ \( ( 1 + 125971660150 T + \)\(90\!\cdots\!89\)\( T^{2} )^{2} \)
$41$ \( ( 1 + 41444675982 T + \)\(37\!\cdots\!61\)\( T^{2} )^{2} \)
$43$ \( 1 - \)\(13\!\cdots\!18\)\( T^{2} + \)\(54\!\cdots\!01\)\( T^{4} \)
$47$ \( 1 - \)\(51\!\cdots\!58\)\( T^{2} + \)\(65\!\cdots\!61\)\( T^{4} \)
$53$ \( ( 1 - 473374435050 T + \)\(13\!\cdots\!69\)\( T^{2} )^{2} \)
$59$ \( 1 - \)\(23\!\cdots\!22\)\( T^{2} + \)\(38\!\cdots\!21\)\( T^{4} \)
$61$ \( ( 1 + 58526672422 T + \)\(98\!\cdots\!41\)\( T^{2} )^{2} \)
$67$ \( 1 - \)\(56\!\cdots\!58\)\( T^{2} + \)\(13\!\cdots\!41\)\( T^{4} \)
$71$ \( 1 + \)\(80\!\cdots\!38\)\( T^{2} + \)\(68\!\cdots\!61\)\( T^{4} \)
$73$ \( ( 1 - 14644904168050 T + \)\(12\!\cdots\!09\)\( T^{2} )^{2} \)
$79$ \( 1 - \)\(66\!\cdots\!62\)\( T^{2} + \)\(13\!\cdots\!61\)\( T^{4} \)
$83$ \( 1 - \)\(86\!\cdots\!38\)\( T^{2} + \)\(54\!\cdots\!41\)\( T^{4} \)
$89$ \( ( 1 - 3585960554322 T + \)\(19\!\cdots\!41\)\( T^{2} )^{2} \)
$97$ \( ( 1 - 65197469070850 T + \)\(65\!\cdots\!69\)\( T^{2} )^{2} \)
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