Properties

Label 64.4.e.a
Level $64$
Weight $4$
Character orbit 64.e
Analytic conductor $3.776$
Analytic rank $0$
Dimension $10$
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] = [64,4,Mod(17,64)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(64, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("64.17");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 64 = 2^{6} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 64.e (of order \(4\), degree \(2\), 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: \(3.77612224037\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} - x^{8} + 6x^{7} + 14x^{6} - 80x^{5} + 56x^{4} + 96x^{3} - 64x^{2} - 512x + 1024 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{20} \)
Twist minimal: no (minimal twist has level 16)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{3} + \beta_{3} q^{5} + ( - \beta_{4} + 3 \beta_1) q^{7} + (\beta_{9} + \beta_{5} - \beta_{4} - \beta_{2} + 5 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{3} + \beta_{3} q^{5} + ( - \beta_{4} + 3 \beta_1) q^{7} + (\beta_{9} + \beta_{5} - \beta_{4} - \beta_{2} + 5 \beta_1) q^{9} + (\beta_{9} - \beta_{7} + 2 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{11} + ( - \beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{4}) q^{13} + (\beta_{8} + 2 \beta_{7} - 2 \beta_{6} + 2 \beta_{3} + 13) q^{15} + ( - 3 \beta_{8} + \beta_{7} + 2 \beta_{6} + 5 \beta_{5} - 2 \beta_{3} + 5 \beta_{2} + \cdots - 4) q^{17}+ \cdots + (4 \beta_{9} - 22 \beta_{8} + 4 \beta_{7} + 64 \beta_{6} + 21 \beta_{5} + \cdots - 518) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 2 q^{3} - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 2 q^{3} - 2 q^{5} - 18 q^{11} - 2 q^{13} + 124 q^{15} - 4 q^{17} + 26 q^{19} + 52 q^{21} - 184 q^{27} - 202 q^{29} - 368 q^{31} - 4 q^{33} - 476 q^{35} - 10 q^{37} + 838 q^{43} + 194 q^{45} + 944 q^{47} + 94 q^{49} + 1500 q^{51} - 378 q^{53} - 1706 q^{59} + 910 q^{61} - 2628 q^{63} - 492 q^{65} - 1942 q^{67} + 580 q^{69} + 2954 q^{75} - 268 q^{77} + 4416 q^{79} + 482 q^{81} + 2562 q^{83} - 12 q^{85} - 3332 q^{91} - 2192 q^{93} - 6900 q^{95} - 4 q^{97} - 4958 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} - x^{8} + 6x^{7} + 14x^{6} - 80x^{5} + 56x^{4} + 96x^{3} - 64x^{2} - 512x + 1024 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - \nu^{9} + 14 \nu^{8} - 7 \nu^{7} - 82 \nu^{6} + 170 \nu^{5} + 120 \nu^{4} - 536 \nu^{3} - 384 \nu^{2} + 2752 \nu - 3072 ) / 1280 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 3 \nu^{9} + 2 \nu^{8} - 101 \nu^{7} + 114 \nu^{6} - 210 \nu^{5} + 120 \nu^{4} - 8 \nu^{3} + 3008 \nu^{2} - 3264 \nu + 7424 ) / 1280 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3 \nu^{9} - 122 \nu^{8} + 341 \nu^{7} - 634 \nu^{6} + 130 \nu^{5} + 120 \nu^{4} + 3848 \nu^{3} - 9728 \nu^{2} + 16064 \nu - 2304 ) / 1280 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7 \nu^{9} - 18 \nu^{8} + 49 \nu^{7} + 174 \nu^{6} - 870 \nu^{5} + 1240 \nu^{4} + 2152 \nu^{3} - 5632 \nu^{2} - 15424 \nu + 31744 ) / 1280 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 17 \nu^{9} + 38 \nu^{8} - 39 \nu^{7} + 86 \nu^{6} + 90 \nu^{5} + 520 \nu^{4} - 792 \nu^{3} + 2752 \nu^{2} - 1856 \nu - 4864 ) / 1280 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 37 \nu^{9} + 198 \nu^{8} - 419 \nu^{7} + 166 \nu^{6} + 50 \nu^{5} + 1560 \nu^{4} - 7992 \nu^{3} + 11392 \nu^{2} - 6976 \nu + 256 ) / 1280 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{9} + 2\nu^{8} + \nu^{7} - 6\nu^{6} - 14\nu^{5} + 80\nu^{4} - 56\nu^{3} - 96\nu^{2} + 320\nu + 416 ) / 32 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -3\nu^{9} + 10\nu^{8} + 3\nu^{7} - 6\nu^{6} - 90\nu^{5} + 184\nu^{4} - 56\nu^{3} - 128\nu^{2} - 896\nu + 1728 ) / 64 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 39 \nu^{9} - 66 \nu^{8} - 47 \nu^{7} + 414 \nu^{6} - 294 \nu^{5} - 1288 \nu^{4} + 1704 \nu^{3} + 2944 \nu^{2} - 11072 \nu + 11264 ) / 256 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + \beta _1 + 3 ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{9} + \beta_{8} + \beta_{6} + 5\beta_{5} - 2\beta_{4} + 3\beta_{3} + \beta_{2} + 5\beta _1 + 9 ) / 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{8} - \beta_{7} - 7\beta_{6} + 5\beta_{5} - \beta_{4} - 3\beta_{3} + 15\beta_{2} + 33\beta _1 - 9 ) / 16 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3 \beta_{9} - \beta_{8} + 4 \beta_{7} + 3 \beta_{6} + 23 \beta_{5} + 6 \beta_{4} + 9 \beta_{3} + 3 \beta_{2} + 43 \beta _1 - 117 ) / 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -6\beta_{8} - 7\beta_{7} - \beta_{6} + 43\beta_{5} + 13\beta_{4} + 3\beta_{3} - 7\beta_{2} + 67\beta _1 + 301 ) / 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 3 \beta_{9} - 15 \beta_{8} + 28 \beta_{7} - 27 \beta_{6} + 17 \beta_{5} + 2 \beta_{4} - 49 \beta_{3} - 43 \beta_{2} - 99 \beta _1 + 117 ) / 16 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 24 \beta_{9} + 22 \beta_{8} + 15 \beta_{7} + 33 \beta_{6} + 133 \beta_{5} - 45 \beta_{4} + 61 \beta_{3} - 233 \beta_{2} - 155 \beta _1 + 859 ) / 16 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 11 \beta_{9} + 119 \beta_{8} - 4 \beta_{7} - 197 \beta_{6} - 17 \beta_{5} - 122 \beta_{4} - 191 \beta_{3} + 91 \beta_{2} + 1043 \beta _1 + 987 ) / 16 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 208 \beta_{9} + 146 \beta_{8} + 121 \beta_{7} + 31 \beta_{6} + 155 \beta_{5} - 75 \beta_{4} + 211 \beta_{3} - 71 \beta_{2} + 3019 \beta _1 - 4187 ) / 16 \) Copy content Toggle raw display

Character values

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

\(n\) \(5\) \(63\)
\(\chi(n)\) \(-\beta_{1}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
1.28199 + 1.53509i
0.932438 1.76934i
−1.62580 + 1.16481i
−1.56339 1.24732i
1.97476 + 0.316760i
1.28199 1.53509i
0.932438 + 1.76934i
−1.62580 1.16481i
−1.56339 + 1.24732i
1.97476 0.316760i
0 −5.49618 5.49618i 0 −4.66372 + 4.66372i 0 24.8965i 0 33.4160i 0
17.2 0 −1.98356 1.98356i 0 −0.596848 + 0.596848i 0 29.0828i 0 19.1310i 0
17.3 0 −0.756776 0.756776i 0 8.22587 8.22587i 0 2.67171i 0 25.8546i 0
17.4 0 3.27139 + 3.27139i 0 −12.6449 + 12.6449i 0 13.8754i 0 5.59607i 0
17.5 0 5.96513 + 5.96513i 0 8.67959 8.67959i 0 1.63924i 0 44.1656i 0
49.1 0 −5.49618 + 5.49618i 0 −4.66372 4.66372i 0 24.8965i 0 33.4160i 0
49.2 0 −1.98356 + 1.98356i 0 −0.596848 0.596848i 0 29.0828i 0 19.1310i 0
49.3 0 −0.756776 + 0.756776i 0 8.22587 + 8.22587i 0 2.67171i 0 25.8546i 0
49.4 0 3.27139 3.27139i 0 −12.6449 12.6449i 0 13.8754i 0 5.59607i 0
49.5 0 5.96513 5.96513i 0 8.67959 + 8.67959i 0 1.63924i 0 44.1656i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 64.4.e.a 10
3.b odd 2 1 576.4.k.a 10
4.b odd 2 1 16.4.e.a 10
8.b even 2 1 128.4.e.a 10
8.d odd 2 1 128.4.e.b 10
12.b even 2 1 144.4.k.a 10
16.e even 4 1 inner 64.4.e.a 10
16.e even 4 1 128.4.e.a 10
16.f odd 4 1 16.4.e.a 10
16.f odd 4 1 128.4.e.b 10
32.g even 8 2 1024.4.a.m 10
32.g even 8 2 1024.4.b.k 10
32.h odd 8 2 1024.4.a.n 10
32.h odd 8 2 1024.4.b.j 10
48.i odd 4 1 576.4.k.a 10
48.k even 4 1 144.4.k.a 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.4.e.a 10 4.b odd 2 1
16.4.e.a 10 16.f odd 4 1
64.4.e.a 10 1.a even 1 1 trivial
64.4.e.a 10 16.e even 4 1 inner
128.4.e.a 10 8.b even 2 1
128.4.e.a 10 16.e even 4 1
128.4.e.b 10 8.d odd 2 1
128.4.e.b 10 16.f odd 4 1
144.4.k.a 10 12.b even 2 1
144.4.k.a 10 48.k even 4 1
576.4.k.a 10 3.b odd 2 1
576.4.k.a 10 48.i odd 4 1
1024.4.a.m 10 32.g even 8 2
1024.4.a.n 10 32.h odd 8 2
1024.4.b.j 10 32.h odd 8 2
1024.4.b.k 10 32.g even 8 2

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(64, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} - 2 T^{9} + 2 T^{8} + \cdots + 829472 \) Copy content Toggle raw display
$5$ \( T^{10} + 2 T^{9} + 2 T^{8} + \cdots + 202085408 \) Copy content Toggle raw display
$7$ \( T^{10} + 1668 T^{8} + \cdots + 1936000000 \) Copy content Toggle raw display
$11$ \( T^{10} + 18 T^{9} + \cdots + 3810412010528 \) Copy content Toggle raw display
$13$ \( T^{10} + 2 T^{9} + \cdots + 11\!\cdots\!28 \) Copy content Toggle raw display
$17$ \( (T^{5} + 2 T^{4} - 11912 T^{3} + \cdots - 556317664)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} - 26 T^{9} + \cdots + 19\!\cdots\!12 \) Copy content Toggle raw display
$23$ \( T^{10} + 45284 T^{8} + \cdots + 25\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{10} + 202 T^{9} + \cdots + 71\!\cdots\!12 \) Copy content Toggle raw display
$31$ \( (T^{5} + 184 T^{4} - 14912 T^{3} + \cdots - 678952960)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + 10 T^{9} + \cdots + 69\!\cdots\!32 \) Copy content Toggle raw display
$41$ \( T^{10} + 248192 T^{8} + \cdots + 58\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{10} - 838 T^{9} + \cdots + 54\!\cdots\!32 \) Copy content Toggle raw display
$47$ \( (T^{5} - 472 T^{4} + \cdots - 154359955456)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} + 378 T^{9} + \cdots + 63\!\cdots\!52 \) Copy content Toggle raw display
$59$ \( T^{10} + 1706 T^{9} + \cdots + 18\!\cdots\!48 \) Copy content Toggle raw display
$61$ \( T^{10} - 910 T^{9} + \cdots + 96\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{10} + 1942 T^{9} + \cdots + 15\!\cdots\!08 \) Copy content Toggle raw display
$71$ \( T^{10} + 1078692 T^{8} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{10} + 755888 T^{8} + \cdots + 42\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( (T^{5} - 2208 T^{4} + \cdots + 10448447471616)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} - 2562 T^{9} + \cdots + 16\!\cdots\!72 \) Copy content Toggle raw display
$89$ \( T^{10} + 3406512 T^{8} + \cdots + 13\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( (T^{5} + 2 T^{4} + \cdots + 65755091474464)^{2} \) Copy content Toggle raw display
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