Properties

Label 16.7.c.b
Level 16
Weight 7
Character orbit 16.c
Analytic conductor 3.681
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 16.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.68086533792\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{5} \)
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.

\(f(q)\) \(=\) \( q + ( 16 - 32 \zeta_{6} ) q^{3} -150 q^{5} + ( 352 - 704 \zeta_{6} ) q^{7} -39 q^{9} +O(q^{10})\) \( q + ( 16 - 32 \zeta_{6} ) q^{3} -150 q^{5} + ( 352 - 704 \zeta_{6} ) q^{7} -39 q^{9} + ( -528 + 1056 \zeta_{6} ) q^{11} + 154 q^{13} + ( -2400 + 4800 \zeta_{6} ) q^{15} + 7458 q^{17} + ( 1232 - 2464 \zeta_{6} ) q^{19} -16896 q^{21} + ( 4896 - 9792 \zeta_{6} ) q^{23} + 6875 q^{25} + ( 11040 - 22080 \zeta_{6} ) q^{27} -10758 q^{29} + ( -1152 + 2304 \zeta_{6} ) q^{31} + 25344 q^{33} + ( -52800 + 105600 \zeta_{6} ) q^{35} -11350 q^{37} + ( 2464 - 4928 \zeta_{6} ) q^{39} + 67122 q^{41} + ( 45936 - 91872 \zeta_{6} ) q^{43} + 5850 q^{45} + ( -40128 + 80256 \zeta_{6} ) q^{47} -254063 q^{49} + ( 119328 - 238656 \zeta_{6} ) q^{51} + 109962 q^{53} + ( 79200 - 158400 \zeta_{6} ) q^{55} -59136 q^{57} + ( -176592 + 353184 \zeta_{6} ) q^{59} + 306746 q^{61} + ( -13728 + 27456 \zeta_{6} ) q^{63} -23100 q^{65} + ( -127216 + 254432 \zeta_{6} ) q^{67} -235008 q^{69} + ( -214944 + 429888 \zeta_{6} ) q^{71} + 165682 q^{73} + ( 110000 - 220000 \zeta_{6} ) q^{75} + 557568 q^{77} + ( 440000 - 880000 \zeta_{6} ) q^{79} -558351 q^{81} + ( 285648 - 571296 \zeta_{6} ) q^{83} -1118700 q^{85} + ( -172128 + 344256 \zeta_{6} ) q^{87} + 471954 q^{89} + ( 54208 - 108416 \zeta_{6} ) q^{91} + 55296 q^{93} + ( -184800 + 369600 \zeta_{6} ) q^{95} + 910594 q^{97} + ( 20592 - 41184 \zeta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 300q^{5} - 78q^{9} + O(q^{10}) \) \( 2q - 300q^{5} - 78q^{9} + 308q^{13} + 14916q^{17} - 33792q^{21} + 13750q^{25} - 21516q^{29} + 50688q^{33} - 22700q^{37} + 134244q^{41} + 11700q^{45} - 508126q^{49} + 219924q^{53} - 118272q^{57} + 613492q^{61} - 46200q^{65} - 470016q^{69} + 331364q^{73} + 1115136q^{77} - 1116702q^{81} - 2237400q^{85} + 943908q^{89} + 110592q^{93} + 1821188q^{97} + O(q^{100}) \)

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 27.7128i 0 −150.000 0 609.682i 0 −39.0000 0
15.2 0 27.7128i 0 −150.000 0 609.682i 0 −39.0000 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 16.7.c.b 2
3.b odd 2 1 144.7.g.f 2
4.b odd 2 1 inner 16.7.c.b 2
5.b even 2 1 400.7.b.c 2
5.c odd 4 2 400.7.h.b 4
8.b even 2 1 64.7.c.d 2
8.d odd 2 1 64.7.c.d 2
12.b even 2 1 144.7.g.f 2
16.e even 4 2 256.7.d.e 4
16.f odd 4 2 256.7.d.e 4
20.d odd 2 1 400.7.b.c 2
20.e even 4 2 400.7.h.b 4
24.f even 2 1 576.7.g.d 2
24.h odd 2 1 576.7.g.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.7.c.b 2 1.a even 1 1 trivial
16.7.c.b 2 4.b odd 2 1 inner
64.7.c.d 2 8.b even 2 1
64.7.c.d 2 8.d odd 2 1
144.7.g.f 2 3.b odd 2 1
144.7.g.f 2 12.b even 2 1
256.7.d.e 4 16.e even 4 2
256.7.d.e 4 16.f odd 4 2
400.7.b.c 2 5.b even 2 1
400.7.b.c 2 20.d odd 2 1
400.7.h.b 4 5.c odd 4 2
400.7.h.b 4 20.e even 4 2
576.7.g.d 2 24.f even 2 1
576.7.g.d 2 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 768 \) acting on \(S_{7}^{\mathrm{new}}(16, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 690 T^{2} + 531441 T^{4} \)
$5$ \( ( 1 + 150 T + 15625 T^{2} )^{2} \)
$7$ \( 1 + 136414 T^{2} + 13841287201 T^{4} \)
$11$ \( 1 - 2706770 T^{2} + 3138428376721 T^{4} \)
$13$ \( ( 1 - 154 T + 4826809 T^{2} )^{2} \)
$17$ \( ( 1 - 7458 T + 24137569 T^{2} )^{2} \)
$19$ \( 1 - 89538290 T^{2} + 2213314919066161 T^{4} \)
$23$ \( 1 - 224159330 T^{2} + 21914624432020321 T^{4} \)
$29$ \( ( 1 + 10758 T + 594823321 T^{2} )^{2} \)
$31$ \( 1 - 1771026050 T^{2} + 787662783788549761 T^{4} \)
$37$ \( ( 1 + 11350 T + 2565726409 T^{2} )^{2} \)
$41$ \( ( 1 - 67122 T + 4750104241 T^{2} )^{2} \)
$43$ \( 1 - 6312377810 T^{2} + 39959630797262576401 T^{4} \)
$47$ \( 1 - 16727661506 T^{2} + \)\(11\!\cdots\!41\)\( T^{4} \)
$53$ \( ( 1 - 109962 T + 22164361129 T^{2} )^{2} \)
$59$ \( 1 + 9193136110 T^{2} + \)\(17\!\cdots\!81\)\( T^{4} \)
$61$ \( ( 1 - 306746 T + 51520374361 T^{2} )^{2} \)
$67$ \( 1 - 132365032370 T^{2} + \)\(81\!\cdots\!61\)\( T^{4} \)
$71$ \( 1 - 117597798434 T^{2} + \)\(16\!\cdots\!41\)\( T^{4} \)
$73$ \( ( 1 - 165682 T + 151334226289 T^{2} )^{2} \)
$79$ \( 1 + 94625088958 T^{2} + \)\(59\!\cdots\!41\)\( T^{4} \)
$83$ \( 1 - 409096407026 T^{2} + \)\(10\!\cdots\!61\)\( T^{4} \)
$89$ \( ( 1 - 471954 T + 496981290961 T^{2} )^{2} \)
$97$ \( ( 1 - 910594 T + 832972004929 T^{2} )^{2} \)
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