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Properties

Label 64.17.c.d
Level $64$
Weight $17$
Character orbit 64.c
Analytic conductor $103.888$
Analytic rank $0$
Dimension $6$
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] = [64,17,Mod(63,64)]
 
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([1, 0]))
 
N = Newforms(chi, 17, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("64.63");
 
S:= CuspForms(chi, 17);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 64 = 2^{6} \)
Weight: \( k \) \(=\) \( 17 \)
Character orbit: \([\chi]\) \(=\) 64.c (of order \(2\), degree \(1\), not 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: \(103.887708068\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 5152x^{4} + 242526x^{3} + 17329473x^{2} + 402444531x + 64957563630 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{62}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 4)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - 5 \beta_{2} + 84455) q^{5} + ( - \beta_{4} - 69 \beta_1) q^{7} + ( - 11 \beta_{3} - 204 \beta_{2} - 22929155) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - 5 \beta_{2} + 84455) q^{5} + ( - \beta_{4} - 69 \beta_1) q^{7} + ( - 11 \beta_{3} - 204 \beta_{2} - 22929155) q^{9} + ( - \beta_{5} + 3 \beta_{4} + 1941 \beta_1) q^{11} + ( - 26 \beta_{3} - 6357 \beta_{2} - 424081801) q^{13} + (30 \beta_{5} + 375 \beta_{4} - 31135 \beta_1) q^{15} + ( - 1907 \beta_{3} - 8548 \beta_{2} + 263198006) q^{17} + ( - 159 \beta_{5} + \cdots + 309879 \beta_1) q^{19}+ \cdots + (25237872 \beta_{5} + \cdots - 577363933377 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 506740 q^{5} - 137574522 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 506740 q^{5} - 137574522 q^{9} - 2544478092 q^{13} + 1579205132 q^{17} + 27228321792 q^{21} + 271424476050 q^{25} + 1158411768436 q^{29} - 767957621760 q^{33} - 8581446019212 q^{37} + 1840369253132 q^{41} + 34166370110580 q^{45} - 5527245758202 q^{49} - 130668269409932 q^{53} - 122486852367360 q^{57} + 429486008315508 q^{61} + 12\!\cdots\!00 q^{65}+ \cdots + 41\!\cdots\!52 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 5152x^{4} + 242526x^{3} + 17329473x^{2} + 402444531x + 64957563630 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -97\nu^{5} + 442\nu^{4} + 33782\nu^{3} - 6959844\nu^{2} - 216742365\nu + 62731727586 ) / 9437184 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 9\nu^{5} + 1078\nu^{4} + 60058\nu^{3} + 3727492\nu^{2} + 232753893\nu + 10473576654 ) / 131072 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -157\nu^{5} + 10322\nu^{4} - 669026\nu^{3} - 30391892\nu^{2} + 1919154903\nu + 6264402650 ) / 65536 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 5635 \nu^{5} + 722926 \nu^{4} - 34634974 \nu^{3} + 5223342036 \nu^{2} + 37050577161 \nu + 10768810679910 ) / 3145728 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 162685 \nu^{5} - 757522 \nu^{4} + 1142788642 \nu^{3} + 35513841108 \nu^{2} + 6864738958473 \nu + 83135565024102 ) / 3145728 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} + 3\beta_{4} + 16\beta_{3} - 352\beta_{2} - 1572\beta _1 + 349408 ) / 2097152 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 17\beta_{5} + 1075\beta_{4} - 1008\beta_{3} - 10592\beta_{2} + 62364\beta _1 - 3601163040 ) / 2097152 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -261\beta_{5} - 37647\beta_{4} - 66640\beta_{3} + 2121440\beta_{2} + 34951860\beta _1 - 259708149856 ) / 2097152 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 222107 \beta_{5} - 1404625 \beta_{4} + 3498832 \beta_{3} + 221902624 \beta_{2} + \cdots - 6019589385632 ) / 2097152 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 4557197 \beta_{5} - 103347367 \beta_{4} + 29308208 \beta_{3} + 3296488928 \beta_{2} + \cdots + 14\!\cdots\!08 ) / 2097152 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
63.1
11.8147 + 63.8186i
34.9299 − 57.5840i
−46.2446 − 35.5107i
−46.2446 + 35.5107i
34.9299 + 57.5840i
11.8147 − 63.8186i
0 − 11183.9i 0 −389860. 0 − 1.06777e6i 0 −8.20322e7 0
63.2 0 − 8024.44i 0 664115. 0 6.08943e6i 0 −2.13450e7 0
63.3 0 − 2908.07i 0 −20884.7 0 − 8.01505e6i 0 3.45899e7 0
63.4 0 2908.07i 0 −20884.7 0 8.01505e6i 0 3.45899e7 0
63.5 0 8024.44i 0 664115. 0 − 6.08943e6i 0 −2.13450e7 0
63.6 0 11183.9i 0 −389860. 0 1.06777e6i 0 −8.20322e7 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 63.6
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 64.17.c.d 6
4.b odd 2 1 inner 64.17.c.d 6
8.b even 2 1 4.17.b.b ✓ 6
8.d odd 2 1 4.17.b.b ✓ 6
24.f even 2 1 36.17.d.b 6
24.h odd 2 1 36.17.d.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.17.b.b ✓ 6 8.b even 2 1
4.17.b.b ✓ 6 8.d odd 2 1
36.17.d.b 6 24.f even 2 1
36.17.d.b 6 24.h odd 2 1
64.17.c.d 6 1.a even 1 1 trivial
64.17.c.d 6 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 197927424T_{3}^{4} + 9656364500582400T_{3}^{2} + 68111840207423731138560 \) acting on \(S_{17}^{\mathrm{new}}(64, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + \cdots + 68\!\cdots\!60 \) Copy content Toggle raw display
$5$ \( (T^{3} + \cdots - 54\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( T^{6} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{3} + \cdots - 12\!\cdots\!20)^{2} \) Copy content Toggle raw display
$17$ \( (T^{3} + \cdots - 20\!\cdots\!40)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 10\!\cdots\!60 \) Copy content Toggle raw display
$29$ \( (T^{3} + \cdots + 40\!\cdots\!92)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{3} + \cdots - 48\!\cdots\!80)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} + \cdots + 19\!\cdots\!32)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 13\!\cdots\!60 \) Copy content Toggle raw display
$53$ \( (T^{3} + \cdots - 14\!\cdots\!60)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 63\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{3} + \cdots + 43\!\cdots\!88)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 75\!\cdots\!60 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 44\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{3} + \cdots - 36\!\cdots\!20)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 47\!\cdots\!60 \) Copy content Toggle raw display
$89$ \( (T^{3} + \cdots - 37\!\cdots\!72)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + \cdots + 76\!\cdots\!40)^{2} \) Copy content Toggle raw display
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