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Properties

Label 256.4.e.c
Level $256$
Weight $4$
Character orbit 256.e
Analytic conductor $15.104$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [256,4,Mod(65,256)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(256, 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("256.65");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 256.e (of order \(4\), degree \(2\), 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: \(15.1044889615\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 242x^{12} + 13297x^{8} + 201600x^{4} + 331776 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{40} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{3} + (\beta_{8} - 2 \beta_1 - 2) q^{5} + ( - \beta_{14} + \beta_{11} + \cdots - \beta_{5}) q^{7}+ \cdots + ( - \beta_{7} + \beta_{4} + \cdots - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{3} + (\beta_{8} - 2 \beta_1 - 2) q^{5} + ( - \beta_{14} + \beta_{11} + \cdots - \beta_{5}) q^{7}+ \cdots + ( - 18 \beta_{15} + \cdots + 131 \beta_{5}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 32 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 32 q^{5} - 320 q^{13} - 384 q^{17} - 224 q^{21} - 928 q^{29} - 2432 q^{33} - 640 q^{37} - 896 q^{45} - 2832 q^{49} - 64 q^{53} - 1024 q^{61} + 2208 q^{65} - 32 q^{69} - 1056 q^{77} + 4208 q^{81} - 1824 q^{85} + 3776 q^{93} + 4480 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 242x^{12} + 13297x^{8} + 201600x^{4} + 331776 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -2485\nu^{14} - 608858\nu^{10} - 35048677\nu^{6} - 685063872\nu^{2} ) / 849733632 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1181\nu^{12} + 373066\nu^{8} + 28523501\nu^{4} + 256704192 ) / 17702784 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 635 \nu^{14} + 10164 \nu^{12} + 226822 \nu^{10} + 2041512 \nu^{8} + 23951819 \nu^{6} + \cdots - 661554432 ) / 106216704 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 635 \nu^{14} + 10164 \nu^{12} - 226822 \nu^{10} + 2041512 \nu^{8} - 23951819 \nu^{6} + \cdots - 661554432 ) / 106216704 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -397\nu^{13} - 92150\nu^{9} - 4355545\nu^{5} - 65398896\nu ) / 13277088 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2173\nu^{13} + 525290\nu^{9} + 27378349\nu^{5} + 259144128\nu ) / 35405568 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -113\nu^{14} - 27922\nu^{10} - 1555553\nu^{6} - 17663040\nu^{2} ) / 1755648 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 20449 \nu^{14} - 55230 \nu^{12} + 4739570 \nu^{10} - 12410076 \nu^{8} + 226120081 \nu^{6} + \cdots - 3592812672 ) / 212433408 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 20449 \nu^{14} - 55230 \nu^{12} - 4739570 \nu^{10} - 12410076 \nu^{8} - 226120081 \nu^{6} + \cdots - 3592812672 ) / 212433408 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -103001\nu^{15} - 24581218\nu^{11} - 1293531977\nu^{7} - 18508731840\nu^{3} ) / 5098401792 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -9721\nu^{15} - 2320034\nu^{11} - 120569065\nu^{7} - 1445284032\nu^{3} ) / 283244544 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 58873 \nu^{15} + 148440 \nu^{13} - 13669538 \nu^{11} + 32950320 \nu^{9} + \cdots + 11174367744 \nu ) / 1274600448 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 58873 \nu^{15} + 148440 \nu^{13} + 13669538 \nu^{11} + 32950320 \nu^{9} + \cdots + 11174367744 \nu ) / 1274600448 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 295355 \nu^{15} - 34536 \nu^{13} + 73719430 \nu^{11} + 5728944 \nu^{9} + 4397525387 \nu^{7} + \cdots + 93155332608 \nu ) / 5098401792 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 295355 \nu^{15} + 34536 \nu^{13} + 73719430 \nu^{11} - 5728944 \nu^{9} + 4397525387 \nu^{7} + \cdots - 93155332608 \nu ) / 5098401792 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{13} - \beta_{12} - 4\beta_{6} - 16\beta_{5} ) / 32 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{9} - \beta_{8} - \beta_{7} + 2\beta_{4} - 2\beta_{3} - 52\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -8\beta_{15} - 8\beta_{14} - 17\beta_{13} + 17\beta_{12} + 12\beta_{11} - 144\beta_{10} ) / 32 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -13\beta_{9} - 13\beta_{8} - 28\beta_{4} - 28\beta_{3} - 21\beta_{2} - 484 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -120\beta_{15} + 120\beta_{14} + 241\beta_{13} + 241\beta_{12} - 132\beta_{6} + 1552\beta_{5} ) / 32 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -169\beta_{9} + 169\beta_{8} + 329\beta_{7} - 354\beta_{4} + 354\beta_{3} + 5332\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1576\beta_{15} + 1576\beta_{14} + 3345\beta_{13} - 3345\beta_{12} + 3540\beta_{11} + 18320\beta_{10} ) / 32 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 2245\beta_{9} + 2245\beta_{8} + 4472\beta_{4} + 4472\beta_{3} + 4669\beta_{2} + 63940 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 20312\beta_{15} - 20312\beta_{14} - 45521\beta_{13} - 45521\beta_{12} + 57636\beta_{6} - 227088\beta_{5} ) / 32 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 29857\beta_{9} - 29857\beta_{8} - 63537\beta_{7} + 57118\beta_{4} - 57118\beta_{3} - 800692\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 262152 \beta_{15} - 262152 \beta_{14} - 609841 \beta_{13} + 609841 \beta_{12} + \cdots - 2888848 \beta_{10} ) / 32 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( -395197\beta_{9} - 395197\beta_{8} - 736404\beta_{4} - 736404\beta_{3} - 847781\beta_{2} - 10247332 ) / 8 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 3398200 \beta_{15} + 3398200 \beta_{14} + 8086833 \beta_{13} + 8086833 \beta_{12} + \cdots + 37249040 \beta_{5} ) / 32 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 5207449 \beta_{9} + 5207449 \beta_{8} + 11202841 \beta_{7} - 9553178 \beta_{4} + \cdots + 132576916 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 44208104 \beta_{15} + 44208104 \beta_{14} + 106585553 \beta_{13} - 106585553 \beta_{12} + \cdots + 483645840 \beta_{10} ) / 32 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/256\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(255\)
\(\chi(n)\) \(\beta_{1}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
−2.55743 − 2.55743i
−0.826679 − 0.826679i
−1.85032 − 1.85032i
−1.53379 − 1.53379i
1.53379 + 1.53379i
1.85032 + 1.85032i
0.826679 + 0.826679i
2.55743 + 2.55743i
−2.55743 + 2.55743i
−0.826679 + 0.826679i
−1.85032 + 1.85032i
−1.53379 + 1.53379i
1.53379 − 1.53379i
1.85032 − 1.85032i
0.826679 − 0.826679i
2.55743 − 2.55743i
0 −5.63249 − 5.63249i 0 −0.329741 + 0.329741i 0 − 17.2022i 0 36.4499i 0
65.2 0 −3.58521 − 3.58521i 0 −14.4583 + 14.4583i 0 31.2516i 0 − 1.29254i 0
65.3 0 −3.18300 − 3.18300i 0 −3.67026 + 3.67026i 0 − 26.2423i 0 − 6.73704i 0
65.4 0 −1.13572 − 1.13572i 0 10.4583 − 10.4583i 0 10.8978i 0 − 24.4203i 0
65.5 0 1.13572 + 1.13572i 0 10.4583 − 10.4583i 0 − 10.8978i 0 − 24.4203i 0
65.6 0 3.18300 + 3.18300i 0 −3.67026 + 3.67026i 0 26.2423i 0 − 6.73704i 0
65.7 0 3.58521 + 3.58521i 0 −14.4583 + 14.4583i 0 − 31.2516i 0 − 1.29254i 0
65.8 0 5.63249 + 5.63249i 0 −0.329741 + 0.329741i 0 17.2022i 0 36.4499i 0
193.1 0 −5.63249 + 5.63249i 0 −0.329741 − 0.329741i 0 17.2022i 0 − 36.4499i 0
193.2 0 −3.58521 + 3.58521i 0 −14.4583 − 14.4583i 0 − 31.2516i 0 1.29254i 0
193.3 0 −3.18300 + 3.18300i 0 −3.67026 − 3.67026i 0 26.2423i 0 6.73704i 0
193.4 0 −1.13572 + 1.13572i 0 10.4583 + 10.4583i 0 − 10.8978i 0 24.4203i 0
193.5 0 1.13572 − 1.13572i 0 10.4583 + 10.4583i 0 10.8978i 0 24.4203i 0
193.6 0 3.18300 − 3.18300i 0 −3.67026 − 3.67026i 0 − 26.2423i 0 6.73704i 0
193.7 0 3.58521 − 3.58521i 0 −14.4583 − 14.4583i 0 31.2516i 0 1.29254i 0
193.8 0 5.63249 − 5.63249i 0 −0.329741 − 0.329741i 0 − 17.2022i 0 − 36.4499i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
16.e even 4 1 inner
16.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 256.4.e.c ✓ 16
4.b odd 2 1 inner 256.4.e.c ✓ 16
8.b even 2 1 256.4.e.d yes 16
8.d odd 2 1 256.4.e.d yes 16
16.e even 4 1 inner 256.4.e.c ✓ 16
16.e even 4 1 256.4.e.d yes 16
16.f odd 4 1 inner 256.4.e.c ✓ 16
16.f odd 4 1 256.4.e.d yes 16
32.g even 8 1 1024.4.a.k 8
32.g even 8 1 1024.4.a.l 8
32.g even 8 2 1024.4.b.l 16
32.h odd 8 1 1024.4.a.k 8
32.h odd 8 1 1024.4.a.l 8
32.h odd 8 2 1024.4.b.l 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
256.4.e.c ✓ 16 1.a even 1 1 trivial
256.4.e.c ✓ 16 4.b odd 2 1 inner
256.4.e.c ✓ 16 16.e even 4 1 inner
256.4.e.c ✓ 16 16.f odd 4 1 inner
256.4.e.d yes 16 8.b even 2 1
256.4.e.d yes 16 8.d odd 2 1
256.4.e.d yes 16 16.e even 4 1
256.4.e.d yes 16 16.f odd 4 1
1024.4.a.k 8 32.g even 8 1
1024.4.a.k 8 32.h odd 8 1
1024.4.a.l 8 32.g even 8 1
1024.4.a.l 8 32.h odd 8 1
1024.4.b.l 16 32.g even 8 2
1024.4.b.l 16 32.h odd 8 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(256, [\chi])\):

\( T_{3}^{16} + 5104T_{3}^{12} + 4618848T_{3}^{8} + 1122922240T_{3}^{4} + 7269949696 \) Copy content Toggle raw display
\( T_{5}^{8} + 16 T_{5}^{7} + 128 T_{5}^{6} - 1888 T_{5}^{5} + 73288 T_{5}^{4} + 654912 T_{5}^{3} + \cdots + 535824 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 7269949696 \) Copy content Toggle raw display
$5$ \( (T^{8} + 16 T^{7} + \cdots + 535824)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + 2080 T^{6} + \cdots + 23637217536)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 29\!\cdots\!96 \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 20270417143824)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 96 T^{3} + \cdots + 1811088)^{4} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 88\!\cdots\!76 \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 20\!\cdots\!76)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 94\!\cdots\!76)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 16\!\cdots\!44)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 77\!\cdots\!84)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 61\!\cdots\!84)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 12\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 34\!\cdots\!96)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 31\!\cdots\!04)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 83\!\cdots\!96)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 48\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 21\!\cdots\!24)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 14\!\cdots\!64)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 41\!\cdots\!24)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 91\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 14\!\cdots\!84)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 1120 T^{3} + \cdots - 580260467568)^{4} \) Copy content Toggle raw display
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