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Properties

Label 64.12.b.c
Level $64$
Weight $12$
Character orbit 64.b
Analytic conductor $49.174$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [64,12,Mod(33,64)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(64, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("64.33");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 64 = 2^{6} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 64.b (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: \(49.1739635558\)
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: \( x^{16} - 59842 x^{14} + 2677039267 x^{12} - 53545917289282 x^{10} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{178}\cdot 3^{4} \)
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.

\(f(q)\) \(=\) \( q + \beta_{8} q^{3} + \beta_1 q^{5} + \beta_{10} q^{7} + (\beta_{2} - 77593) q^{9} + (\beta_{13} + 10 \beta_{9} - 179 \beta_{8}) q^{11} + (\beta_{4} - 125 \beta_1) q^{13} + (2 \beta_{12} + \beta_{11} - 33 \beta_{10}) q^{15}+ \cdots + (87 \beta_{15} + \cdots + 2394213 \beta_{8}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 1241488 q^{9} - 1515360 q^{17} - 150953008 q^{25} + 729264576 q^{33} + 4494603936 q^{41} + 16213270160 q^{49} + 33337065152 q^{57} + 116662657536 q^{65} + 48621829664 q^{73} + 452547743056 q^{81}+ \cdots + 171362390944 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 59842 x^{14} + 2677039267 x^{12} - 53545917289282 x^{10} + \cdots + 27\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 82\!\cdots\!01 \nu^{14} + \cdots - 17\!\cdots\!00 ) / 32\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 45\!\cdots\!27 \nu^{14} + \cdots + 38\!\cdots\!80 ) / 15\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 42\!\cdots\!47 \nu^{14} + \cdots - 10\!\cdots\!80 ) / 25\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 58\!\cdots\!21 \nu^{14} + \cdots - 14\!\cdots\!00 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 25\!\cdots\!87 \nu^{14} + \cdots + 70\!\cdots\!00 ) / 24\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 67\!\cdots\!39 \nu^{14} + \cdots - 15\!\cdots\!00 ) / 53\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 23\!\cdots\!01 \nu^{14} + \cdots - 11\!\cdots\!20 ) / 15\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 19\!\cdots\!57 \nu^{15} + \cdots - 17\!\cdots\!00 \nu ) / 32\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 40\!\cdots\!19 \nu^{15} + \cdots + 36\!\cdots\!00 \nu ) / 90\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 28\!\cdots\!49 \nu^{15} + \cdots - 32\!\cdots\!00 \nu ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 38\!\cdots\!83 \nu^{15} + \cdots + 50\!\cdots\!00 \nu ) / 51\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 30\!\cdots\!27 \nu^{15} + \cdots - 29\!\cdots\!00 \nu ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 81\!\cdots\!91 \nu^{15} + \cdots - 73\!\cdots\!00 \nu ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 38\!\cdots\!59 \nu^{15} + \cdots - 89\!\cdots\!00 \nu ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 79\!\cdots\!97 \nu^{15} + \cdots + 72\!\cdots\!00 \nu ) / 16\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 5\beta_{14} + 26\beta_{12} + 65\beta_{11} - 493\beta_{10} - 1952\beta_{9} - 123264\beta_{8} ) / 1048576 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 2976 \beta_{6} - 3295 \beta_{5} - 12672 \beta_{4} + 3904 \beta_{3} - 24960 \beta_{2} + \cdots + 7843610624 ) / 1048576 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7\beta_{15} - 418\beta_{13} - 126065\beta_{9} - 7081582\beta_{8} ) / 1024 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 55296 \beta_{7} - 85035936 \beta_{6} - 97998043 \beta_{5} - 394310016 \beta_{4} + \cdots - 232392434647040 ) / 1048576 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 459564032 \beta_{15} - 7489060393 \beta_{14} - 12317849600 \beta_{13} + \cdots - 215164147532288 \beta_{8} ) / 2097152 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -25842240\beta_{7} - 61492592001\beta_{3} + 333147741246\beta_{2} - 108198205444001792 ) / 8192 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 21067138034688 \beta_{15} - 220869417506905 \beta_{14} + 351886010935296 \beta_{13} + \cdots + 63\!\cdots\!88 \beta_{8} ) / 2097152 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 148961889810432 \beta_{7} + \cdots - 20\!\cdots\!20 ) / 1048576 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 66\!\cdots\!40 \beta_{15} + \cdots + 14\!\cdots\!96 \beta_{8} ) / 8192 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 74\!\cdots\!08 \beta_{7} + \cdots + 61\!\cdots\!44 ) / 1048576 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 31\!\cdots\!76 \beta_{15} + \cdots + 56\!\cdots\!08 \beta_{8} ) / 2097152 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 48\!\cdots\!28 \beta_{7} + \cdots + 28\!\cdots\!04 ) / 8192 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 11\!\cdots\!00 \beta_{15} + \cdots - 16\!\cdots\!48 \beta_{8} ) / 2097152 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 11\!\cdots\!60 \beta_{7} + \cdots + 54\!\cdots\!04 ) / 1048576 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 15\!\cdots\!08 \beta_{15} + \cdots - 19\!\cdots\!48 \beta_{8} ) / 4096 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/64\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(63\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
33.1
−147.648 + 85.2448i
147.648 + 85.2448i
−151.157 + 87.2703i
151.157 + 87.2703i
13.0944 + 7.56007i
−13.0944 + 7.56007i
7.85417 − 4.53461i
−7.85417 − 4.53461i
−7.85417 + 4.53461i
7.85417 + 4.53461i
−13.0944 − 7.56007i
13.0944 − 7.56007i
151.157 − 87.2703i
−151.157 − 87.2703i
147.648 − 85.2448i
−147.648 − 85.2448i
0 − 803.959i 0 − 9050.16i 0 71874.1 0 −469202. 0
33.2 0 − 803.959i 0 9050.16i 0 −71874.1 0 −469202. 0
33.3 0 − 576.162i 0 − 2270.57i 0 −37732.3 0 −154816. 0
33.4 0 − 576.162i 0 2270.57i 0 37732.3 0 −154816. 0
33.5 0 − 182.481i 0 − 11689.5i 0 −26516.0 0 143848. 0
33.6 0 − 182.481i 0 11689.5i 0 26516.0 0 143848. 0
33.7 0 − 85.7231i 0 − 3056.94i 0 −68336.8 0 169799. 0
33.8 0 − 85.7231i 0 3056.94i 0 68336.8 0 169799. 0
33.9 0 85.7231i 0 − 3056.94i 0 68336.8 0 169799. 0
33.10 0 85.7231i 0 3056.94i 0 −68336.8 0 169799. 0
33.11 0 182.481i 0 − 11689.5i 0 26516.0 0 143848. 0
33.12 0 182.481i 0 11689.5i 0 −26516.0 0 143848. 0
33.13 0 576.162i 0 − 2270.57i 0 37732.3 0 −154816. 0
33.14 0 576.162i 0 2270.57i 0 −37732.3 0 −154816. 0
33.15 0 803.959i 0 − 9050.16i 0 −71874.1 0 −469202. 0
33.16 0 803.959i 0 9050.16i 0 71874.1 0 −469202. 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 33.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 64.12.b.c ✓ 16
4.b odd 2 1 inner 64.12.b.c ✓ 16
8.b even 2 1 inner 64.12.b.c ✓ 16
8.d odd 2 1 inner 64.12.b.c ✓ 16
16.e even 4 1 256.12.a.l 8
16.e even 4 1 256.12.a.n 8
16.f odd 4 1 256.12.a.l 8
16.f odd 4 1 256.12.a.n 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
64.12.b.c ✓ 16 1.a even 1 1 trivial
64.12.b.c ✓ 16 4.b odd 2 1 inner
64.12.b.c ✓ 16 8.b even 2 1 inner
64.12.b.c ✓ 16 8.d odd 2 1 inner
256.12.a.l 8 16.e even 4 1
256.12.a.l 8 16.f odd 4 1
256.12.a.n 8 16.e even 4 1
256.12.a.n 8 16.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 1018960T_{3}^{6} + 254574831456T_{3}^{4} + 8960906844105984T_{3}^{2} + 52503269231657505024 \) acting on \(S_{12}^{\mathrm{new}}(64, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} + \cdots + 52\!\cdots\!24)^{2} \) Copy content Toggle raw display
$5$ \( (T^{8} + \cdots + 53\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + \cdots + 24\!\cdots\!84)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 49\!\cdots\!04)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 16\!\cdots\!56)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + \cdots + 10\!\cdots\!08)^{4} \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 11\!\cdots\!96)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 37\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 32\!\cdots\!04)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 55\!\cdots\!64)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 14\!\cdots\!24)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 14\!\cdots\!04)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 41\!\cdots\!96)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 60\!\cdots\!16)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 30\!\cdots\!16)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 42\!\cdots\!24)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 55\!\cdots\!36)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 96\!\cdots\!88)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 11\!\cdots\!04)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 45\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots - 17\!\cdots\!68)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots - 11\!\cdots\!28)^{4} \) Copy content Toggle raw display
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