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Properties

Label 384.3.h.g
Level $384$
Weight $3$
Character orbit 384.h
Analytic conductor $10.463$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [384,3,Mod(65,384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(384, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.65");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 384.h (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: \(10.4632421514\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 28x^{14} + 274x^{12} + 1236x^{10} + 2703x^{8} + 2676x^{6} + 946x^{4} + 64x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{46}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{8} q^{3} + \beta_{2} q^{5} + \beta_1 q^{7} + ( - \beta_{5} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{8} q^{3} + \beta_{2} q^{5} + \beta_1 q^{7} + ( - \beta_{5} - 1) q^{9} - \beta_{3} q^{11} + \beta_{15} q^{13} - \beta_{13} q^{15} + ( - \beta_{11} - \beta_{5} + \beta_{4}) q^{17} - \beta_{9} q^{19} + ( - \beta_{15} - \beta_{7}) q^{21} + ( - \beta_{13} + \beta_{12}) q^{23} + (\beta_{5} + \beta_{4} + 3) q^{25} + (\beta_{10} - \beta_{9} - \beta_{3}) q^{27} + (\beta_{7} - \beta_{6} - \beta_{2}) q^{29} + ( - \beta_{14} - \beta_{13}) q^{31} + (3 \beta_{11} + 2 \beta_{5} - 3 \beta_{4} - 4) q^{33} + (\beta_{10} + 3 \beta_{8} - 3 \beta_{3}) q^{35} + ( - \beta_{15} + \beta_{7} + \beta_{6}) q^{37} + (\beta_{12} + 5 \beta_1) q^{39} + ( - \beta_{11} + 4 \beta_{5} - 4 \beta_{4}) q^{41} + ( - 2 \beta_{10} - \beta_{9} + 6 \beta_{8}) q^{43} + (3 \beta_{6} - 6 \beta_{2}) q^{45} + (\beta_{14} - 4 \beta_{13} + \cdots - \beta_1) q^{47}+ \cdots + (4 \beta_{10} - \beta_{9} + \cdots + 8 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{9} + 48 q^{25} - 64 q^{33} - 80 q^{49} + 32 q^{57} - 288 q^{73} + 400 q^{81} + 224 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 28x^{14} + 274x^{12} + 1236x^{10} + 2703x^{8} + 2676x^{6} + 946x^{4} + 64x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 30 \nu^{14} - 691 \nu^{12} - 3961 \nu^{10} + 5346 \nu^{8} + 107233 \nu^{6} + 290346 \nu^{4} + \cdots + 31667 ) / 3462 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 546 \nu^{14} - 15115 \nu^{12} - 145023 \nu^{10} - 634108 \nu^{8} - 1316949 \nu^{6} - 1189412 \nu^{4} + \cdots - 30203 ) / 3462 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2984 \nu^{14} + 83695 \nu^{12} + 820929 \nu^{10} + 3712317 \nu^{8} + 8150361 \nu^{6} + 8172537 \nu^{4} + \cdots + 157630 ) / 5193 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1015 \nu^{15} + 2208 \nu^{14} - 28091 \nu^{13} + 61936 \nu^{12} - 269128 \nu^{11} + 607264 \nu^{10} + \cdots + 63628 ) / 3462 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1015 \nu^{15} + 2208 \nu^{14} + 28091 \nu^{13} + 61936 \nu^{12} + 269128 \nu^{11} + 607264 \nu^{10} + \cdots + 63628 ) / 3462 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 5936 \nu^{15} + 9902 \nu^{14} - 166576 \nu^{13} + 275467 \nu^{12} - 1637556 \nu^{11} + \cdots + 209431 ) / 10386 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 5936 \nu^{15} - 9902 \nu^{14} - 166576 \nu^{13} - 275467 \nu^{12} - 1637556 \nu^{11} + \cdots - 209431 ) / 10386 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 21559 \nu^{15} + 3128 \nu^{14} + 602609 \nu^{13} + 87358 \nu^{12} + 5877930 \nu^{11} + 850674 \nu^{10} + \cdots + 67252 ) / 20772 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 18625 \nu^{15} - 520835 \nu^{13} - 5081682 \nu^{11} - 22763733 \nu^{9} - 48910902 \nu^{7} + \cdots + 618256 \nu ) / 10386 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 21559 \nu^{15} + 3128 \nu^{14} - 602609 \nu^{13} + 87358 \nu^{12} - 5877930 \nu^{11} + \cdots + 67252 ) / 6924 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 2892 \nu^{15} - 80922 \nu^{13} - 790818 \nu^{11} - 3557804 \nu^{9} - 7735302 \nu^{7} + \cdots - 96798 \nu ) / 577 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 17434 \nu^{15} + 2016 \nu^{14} + 486536 \nu^{13} + 56475 \nu^{12} + 4732924 \nu^{11} + 553179 \nu^{10} + \cdots + 51999 ) / 3462 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 7002 \nu^{15} + 672 \nu^{14} + 195784 \nu^{13} + 18825 \nu^{12} + 1911052 \nu^{11} + 184393 \nu^{10} + \cdots + 17333 ) / 1154 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 10503 \nu^{15} + 5025 \nu^{14} - 293676 \nu^{13} + 140842 \nu^{12} - 2866578 \nu^{11} + \cdots + 145831 ) / 1731 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 58 \nu^{15} - 1622 \nu^{13} - 15837 \nu^{11} - 71163 \nu^{9} - 154461 \nu^{7} - 150243 \nu^{5} + \cdots - 1925 \nu ) / 9 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( -3\beta_{11} + \beta_{10} + 3\beta_{9} - 3\beta_{8} + 3\beta_{7} + 3\beta_{6} ) / 48 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 2 \beta_{14} - 2 \beta_{13} - 8 \beta_{10} - 24 \beta_{8} - 3 \beta_{7} + 3 \beta_{6} + 6 \beta_{5} + \cdots - 168 ) / 48 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 12 \beta_{15} - 2 \beta_{14} + 4 \beta_{13} + 6 \beta_{12} + 16 \beta_{11} - 8 \beta_{10} + \cdots + 2 \beta_1 ) / 32 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 14 \beta_{14} + 14 \beta_{13} + 58 \beta_{10} + 174 \beta_{8} + 27 \beta_{7} - 27 \beta_{6} + \cdots + 708 ) / 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 540 \beta_{15} + 90 \beta_{14} - 120 \beta_{13} - 330 \beta_{12} - 534 \beta_{11} + 326 \beta_{10} + \cdots - 90 \beta_1 ) / 96 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 62 \beta_{14} - 62 \beta_{13} - 273 \beta_{10} - 819 \beta_{8} - 130 \beta_{7} + 130 \beta_{6} + \cdots - 2644 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 7308 \beta_{15} - 1225 \beta_{14} + 1316 \beta_{13} + 4809 \beta_{12} + 6810 \beta_{11} + \cdots + 1225 \beta_1 ) / 96 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 2468 \beta_{14} + 2468 \beta_{13} + 11236 \beta_{10} + 33708 \beta_{8} + 5346 \beta_{7} - 5346 \beta_{6} + \cdots + 99492 ) / 24 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 32556 \beta_{15} + 5484 \beta_{14} - 5376 \beta_{13} - 22044 \beta_{12} - 30008 \beta_{11} + \cdots - 5484 \beta_1 ) / 32 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 65722 \beta_{14} - 65722 \beta_{13} - 303766 \beta_{10} - 911298 \beta_{8} - 144237 \beta_{7} + \cdots - 2601168 ) / 48 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 1304424 \beta_{15} - 220363 \beta_{14} + 208736 \beta_{13} + 893079 \beta_{12} + 1200666 \beta_{11} + \cdots + 220363 \beta_1 ) / 96 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 73098 \beta_{14} + 73098 \beta_{13} + 340002 \beta_{10} + 1020006 \beta_{8} + 161271 \beta_{7} + \cdots + 2875588 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 8716032 \beta_{15} + 1474590 \beta_{14} - 1379820 \beta_{13} - 5993130 \beta_{12} + \cdots - 1474590 \beta_1 ) / 48 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 11721514 \beta_{14} - 11721514 \beta_{13} - 54659524 \beta_{10} - 163978572 \beta_{8} + \cdots - 460174344 ) / 48 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 77686908 \beta_{15} - 13152080 \beta_{14} + 12254364 \beta_{13} + 53506036 \beta_{12} + \cdots + 13152080 \beta_1 ) / 32 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
0.242964i
2.24296i
− 0.242964i
− 2.24296i
− 0.736993i
1.26301i
0.736993i
− 1.26301i
− 0.151206i
− 2.15121i
0.151206i
2.15121i
3.65718i
1.65718i
− 3.65718i
− 1.65718i
0 −2.75782 − 1.18087i 0 −3.68481 0 5.43855 0 6.21110 + 6.51323i 0
65.2 0 −2.75782 − 1.18087i 0 3.68481 0 −5.43855 0 6.21110 + 6.51323i 0
65.3 0 −2.75782 + 1.18087i 0 −3.68481 0 5.43855 0 6.21110 − 6.51323i 0
65.4 0 −2.75782 + 1.18087i 0 3.68481 0 −5.43855 0 6.21110 − 6.51323i 0
65.5 0 −0.628052 − 2.93352i 0 −6.51323 0 −7.64344 0 −8.21110 + 3.68481i 0
65.6 0 −0.628052 − 2.93352i 0 6.51323 0 7.64344 0 −8.21110 + 3.68481i 0
65.7 0 −0.628052 + 2.93352i 0 −6.51323 0 −7.64344 0 −8.21110 − 3.68481i 0
65.8 0 −0.628052 + 2.93352i 0 6.51323 0 7.64344 0 −8.21110 − 3.68481i 0
65.9 0 0.628052 − 2.93352i 0 −6.51323 0 7.64344 0 −8.21110 − 3.68481i 0
65.10 0 0.628052 − 2.93352i 0 6.51323 0 −7.64344 0 −8.21110 − 3.68481i 0
65.11 0 0.628052 + 2.93352i 0 −6.51323 0 7.64344 0 −8.21110 + 3.68481i 0
65.12 0 0.628052 + 2.93352i 0 6.51323 0 −7.64344 0 −8.21110 + 3.68481i 0
65.13 0 2.75782 − 1.18087i 0 −3.68481 0 −5.43855 0 6.21110 − 6.51323i 0
65.14 0 2.75782 − 1.18087i 0 3.68481 0 5.43855 0 6.21110 − 6.51323i 0
65.15 0 2.75782 + 1.18087i 0 −3.68481 0 −5.43855 0 6.21110 + 6.51323i 0
65.16 0 2.75782 + 1.18087i 0 3.68481 0 5.43855 0 6.21110 + 6.51323i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.3.h.g ✓ 16
3.b odd 2 1 inner 384.3.h.g ✓ 16
4.b odd 2 1 inner 384.3.h.g ✓ 16
8.b even 2 1 inner 384.3.h.g ✓ 16
8.d odd 2 1 inner 384.3.h.g ✓ 16
12.b even 2 1 inner 384.3.h.g ✓ 16
16.e even 4 1 768.3.e.o 8
16.e even 4 1 768.3.e.p 8
16.f odd 4 1 768.3.e.o 8
16.f odd 4 1 768.3.e.p 8
24.f even 2 1 inner 384.3.h.g ✓ 16
24.h odd 2 1 inner 384.3.h.g ✓ 16
48.i odd 4 1 768.3.e.o 8
48.i odd 4 1 768.3.e.p 8
48.k even 4 1 768.3.e.o 8
48.k even 4 1 768.3.e.p 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.3.h.g ✓ 16 1.a even 1 1 trivial
384.3.h.g ✓ 16 3.b odd 2 1 inner
384.3.h.g ✓ 16 4.b odd 2 1 inner
384.3.h.g ✓ 16 8.b even 2 1 inner
384.3.h.g ✓ 16 8.d odd 2 1 inner
384.3.h.g ✓ 16 12.b even 2 1 inner
384.3.h.g ✓ 16 24.f even 2 1 inner
384.3.h.g ✓ 16 24.h odd 2 1 inner
768.3.e.o 8 16.e even 4 1
768.3.e.o 8 16.f odd 4 1
768.3.e.o 8 48.i odd 4 1
768.3.e.o 8 48.k even 4 1
768.3.e.p 8 16.e even 4 1
768.3.e.p 8 16.f odd 4 1
768.3.e.p 8 48.i odd 4 1
768.3.e.p 8 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(384, [\chi])\):

\( T_{5}^{4} - 56T_{5}^{2} + 576 \) Copy content Toggle raw display
\( T_{11}^{4} - 320T_{11}^{2} + 432 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} + 4 T^{6} + \cdots + 6561)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} - 56 T^{2} + 576)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} - 88 T^{2} + 1728)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} - 320 T^{2} + 432)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 208)^{8} \) Copy content Toggle raw display
$17$ \( (T^{4} + 352 T^{2} + 1024)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} + 1000 T^{2} + 139968)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 2048 T^{2} + 995328)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} - 1144 T^{2} + 10816)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 4120 T^{2} + 3195072)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 928 T^{2} + 2304)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 4352 T^{2} + 1327104)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 2344 T^{2} + 914112)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} + 11264 T^{2} + 28311552)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} - 3832 T^{2} + 2096704)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} - 2592 T^{2} + 314928)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} + 8352 T^{2} + 186624)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 360 T^{2} + 15552)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 12800 T^{2} + 31961088)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 36 T - 7164)^{8} \) Copy content Toggle raw display
$79$ \( (T^{4} - 15448 T^{2} + 1453248)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 8192 T^{2} + 10112688)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} + 20192 T^{2} + 9216)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 28 T - 5004)^{8} \) Copy content Toggle raw display
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