Properties

Label 16.15.c.c
Level $16$
Weight $15$
Character orbit 16.c
Analytic conductor $19.893$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [16,15,Mod(15,16)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(16, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 15, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("16.15");
 
S:= CuspForms(chi, 15);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 15 \)
Character orbit: \([\chi]\) \(=\) 16.c (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(19.8926349043\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 9547x^{2} + 9546x + 91126116 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{25}\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{3} - 21750) q^{5} + ( - \beta_{2} + 245 \beta_1) q^{7} + (226 \beta_{3} - 5000199) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{3} - 21750) q^{5} + ( - \beta_{2} + 245 \beta_1) q^{7} + (226 \beta_{3} - 5000199) q^{9} + ( - 270 \beta_{2} + 3057 \beta_1) q^{11} + ( - 1131 \beta_{3} + 16716154) q^{13} + ( - 9531 \beta_{2} - 52977 \beta_1) q^{15} + ( - 7030 \beta_{3} + 3279138) q^{17} + ( - 112826 \beta_{2} - 80045 \beta_1) q^{19} + (55084 \beta_{3} - 2395550976) q^{21} + ( - 651807 \beta_{2} + 454707 \beta_1) q^{23} + ( - 43500 \beta_{3} - 4222801285) q^{25} + ( - 2154006 \beta_{2} - 7274532 \beta_1) q^{27} + ( - 407767 \beta_{3} - 4375446438) q^{29} + ( - 4065288 \beta_{2} + 4798512 \beta_1) q^{31} + (613662 \beta_{3} - 29549344896) q^{33} + ( - 2344572 \beta_{2} - 12933984 \beta_1) q^{35} + (1942029 \beta_{3} - 34996433590) q^{37} + (10779561 \beta_{2} + 52033891 \beta_1) q^{39} + ( - 2823372 \beta_{3} + 165522102642) q^{41} + (39873372 \beta_{2} - 62935245 \beta_1) q^{43} + ( - 9915699 \beta_{3} + 426883644090) q^{45} + (72228726 \beta_{2} + 107341674 \beta_1) q^{47} + (13424712 \beta_{3} + 91599855697) q^{49} + (67002930 \beta_{2} + 222804948 \beta_1) q^{51} + (33160533 \beta_{3} - 743793406038) q^{53} + ( - 31695057 \beta_{2} - 149697819 \beta_1) q^{55} + ( - 50358406 \beta_{3} + 932608892544) q^{57} + ( - 249381720 \beta_{2} + 339074283 \beta_1) q^{59} + ( - 32995419 \beta_{3} - 1656242712934) q^{61} + ( - 529788573 \beta_{2} - 2943831639 \beta_1) q^{63} + (41315404 \beta_{3} - 1955630580540) q^{65} + ( - 679535978 \beta_{2} + 1229905543 \beta_1) q^{67} + ( - 83653020 \beta_{3} - 3584710764288) q^{69} + ( - 411388821 \beta_{2} + 2286480849 \beta_1) q^{71} + (355525386 \beta_{3} - 3008283147278) q^{73} + (414598500 \beta_{2} - 2864426785 \beta_1) q^{75} + (149257428 \beta_{3} - 7245770902272) q^{77} + (1805219230 \beta_{2} + 3951839050 \beta_1) q^{79} + ( - 1179138954 \beta_{3} + \cdots + 50106626153649) q^{81}+ \cdots + ( - 7140214152 \beta_{2} - 34090631937 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 87000 q^{5} - 20000796 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 87000 q^{5} - 20000796 q^{9} + 66864616 q^{13} + 13116552 q^{17} - 9582203904 q^{21} - 16891205140 q^{25} - 17501785752 q^{29} - 118197379584 q^{33} - 139985734360 q^{37} + 662088410568 q^{41} + 1707534576360 q^{45} + 366399422788 q^{49} - 2975173624152 q^{53} + 3730435570176 q^{57} - 6624970851736 q^{61} - 7822522322160 q^{65} - 14338843057152 q^{69} - 12033132589112 q^{73} - 28983083609088 q^{77} + 200426504614596 q^{81} - 39868454746800 q^{85} + 106307014497096 q^{89} - 166229577732096 q^{93} + 95793165936136 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 9547x^{2} + 9546x + 91126116 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -25760\nu^{3} + 2902288\nu^{2} - 488959152\nu + 13606715664 ) / 15189277 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 73184\nu^{3} + 30397648\nu^{2} + 1427772944\nu + 145786588368 ) / 45567831 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -384\nu^{3} - 5498688 ) / 9547 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 32\beta_{3} + 171\beta_{2} - 597\beta _1 + 6144 ) / 24576 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -32\beta_{3} + 28809\beta_{2} + 28041\beta _1 - 117307392 ) / 24576 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -9547\beta_{3} - 5498688 ) / 384 \) Copy content Toggle raw display

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
15.1
−48.6025 + 84.1819i
49.1025 + 85.0479i
49.1025 85.0479i
−48.6025 84.1819i
0 4273.45i 0 −59268.7 0 1.04417e6i 0 −1.34794e7 0
15.2 0 1141.90i 0 15768.7 0 288003.i 0 3.47902e6 0
15.3 0 1141.90i 0 15768.7 0 288003.i 0 3.47902e6 0
15.4 0 4273.45i 0 −59268.7 0 1.04417e6i 0 −1.34794e7 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.c 4
3.b odd 2 1 144.15.g.g 4
4.b odd 2 1 inner 16.15.c.c 4
8.b even 2 1 64.15.c.c 4
8.d odd 2 1 64.15.c.c 4
12.b even 2 1 144.15.g.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.15.c.c 4 1.a even 1 1 trivial
16.15.c.c 4 4.b odd 2 1 inner
64.15.c.c 4 8.b even 2 1
64.15.c.c 4 8.d odd 2 1
144.15.g.g 4 3.b odd 2 1
144.15.g.g 4 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 19566336T_{3}^{2} + 23813150736384 \) acting on \(S_{15}^{\mathrm{new}}(16, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 19566336 T^{2} + \cdots + 23813150736384 \) Copy content Toggle raw display
$5$ \( (T^{2} + 43500 T - 934589340)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 1173246434304 T^{2} + \cdots + 90\!\cdots\!44 \) Copy content Toggle raw display
$11$ \( T^{4} + 184002652180224 T^{2} + \cdots + 49\!\cdots\!04 \) Copy content Toggle raw display
$13$ \( (T^{2} - 33432308 T - 15\!\cdots\!24)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 6558276 T - 69\!\cdots\!56)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 30\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 37\!\cdots\!04 \) Copy content Toggle raw display
$29$ \( (T^{2} + 8750892876 T - 21\!\cdots\!16)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 12\!\cdots\!44 \) Copy content Toggle raw display
$37$ \( (T^{2} + 69992867180 T - 40\!\cdots\!40)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 331044205284 T + 16\!\cdots\!04)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 39\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 97\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( (T^{2} + 1487586812076 T - 99\!\cdots\!16)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 33\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( (T^{2} + 3312485425868 T + 12\!\cdots\!16)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 54\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 26\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( (T^{2} + 6016566294556 T - 16\!\cdots\!56)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 26\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( (T^{2} - 53153507248548 T - 80\!\cdots\!64)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 47896582968068 T - 16\!\cdots\!44)^{2} \) Copy content Toggle raw display
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