Properties

Label 128.4.a.h
Level $128$
Weight $4$
Character orbit 128.a
Self dual yes
Analytic conductor $7.552$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [128,4,Mod(1,128)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(128, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("128.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 128.a (trivial)

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: yes
Analytic conductor: \(7.55224448073\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = 4\sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 2) q^{3} + ( - 2 \beta + 2) q^{5} + (2 \beta - 4) q^{7} + (4 \beta + 25) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 2) q^{3} + ( - 2 \beta + 2) q^{5} + (2 \beta - 4) q^{7} + (4 \beta + 25) q^{9} + ( - \beta + 46) q^{11} + (6 \beta + 50) q^{13} + ( - 2 \beta - 92) q^{15} + ( - 12 \beta + 46) q^{17} + ( - 7 \beta + 2) q^{19} + 88 q^{21} + ( - 18 \beta + 4) q^{23} + ( - 8 \beta + 71) q^{25} + (6 \beta + 188) q^{27} + ( - 10 \beta + 42) q^{29} + (16 \beta - 192) q^{31} + (44 \beta + 44) q^{33} + (12 \beta - 200) q^{35} + (14 \beta - 86) q^{37} + (62 \beta + 388) q^{39} + ( - 8 \beta - 150) q^{41} + ( - 5 \beta + 150) q^{43} + ( - 42 \beta - 334) q^{45} + ( - 44 \beta - 8) q^{47} + ( - 16 \beta - 135) q^{49} + (22 \beta - 484) q^{51} + (14 \beta - 6) q^{53} + ( - 94 \beta + 188) q^{55} + ( - 12 \beta - 332) q^{57} + ( - 33 \beta - 322) q^{59} + (70 \beta + 146) q^{61} + (34 \beta + 284) q^{63} + ( - 88 \beta - 476) q^{65} + ( - 83 \beta - 86) q^{67} + ( - 32 \beta - 856) q^{69} + (42 \beta + 204) q^{71} + (52 \beta + 206) q^{73} + (55 \beta - 242) q^{75} + (96 \beta - 280) q^{77} + (36 \beta - 200) q^{79} + (92 \beta - 11) q^{81} + (13 \beta + 474) q^{83} + ( - 116 \beta + 1244) q^{85} + (22 \beta - 396) q^{87} + (116 \beta + 286) q^{89} + (76 \beta + 376) q^{91} + ( - 160 \beta + 384) q^{93} + ( - 18 \beta + 676) q^{95} + ( - 92 \beta + 1102) q^{97} + (159 \beta + 958) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{3} + 4 q^{5} - 8 q^{7} + 50 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{3} + 4 q^{5} - 8 q^{7} + 50 q^{9} + 92 q^{11} + 100 q^{13} - 184 q^{15} + 92 q^{17} + 4 q^{19} + 176 q^{21} + 8 q^{23} + 142 q^{25} + 376 q^{27} + 84 q^{29} - 384 q^{31} + 88 q^{33} - 400 q^{35} - 172 q^{37} + 776 q^{39} - 300 q^{41} + 300 q^{43} - 668 q^{45} - 16 q^{47} - 270 q^{49} - 968 q^{51} - 12 q^{53} + 376 q^{55} - 664 q^{57} - 644 q^{59} + 292 q^{61} + 568 q^{63} - 952 q^{65} - 172 q^{67} - 1712 q^{69} + 408 q^{71} + 412 q^{73} - 484 q^{75} - 560 q^{77} - 400 q^{79} - 22 q^{81} + 948 q^{83} + 2488 q^{85} - 792 q^{87} + 572 q^{89} + 752 q^{91} + 768 q^{93} + 1352 q^{95} + 2204 q^{97} + 1916 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.73205
1.73205
0 −4.92820 0 15.8564 0 −17.8564 0 −2.71281 0
1.2 0 8.92820 0 −11.8564 0 9.85641 0 52.7128 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 128.4.a.h yes 2
3.b odd 2 1 1152.4.a.q 2
4.b odd 2 1 128.4.a.f yes 2
8.b even 2 1 128.4.a.e 2
8.d odd 2 1 128.4.a.g yes 2
12.b even 2 1 1152.4.a.r 2
16.e even 4 2 256.4.b.i 4
16.f odd 4 2 256.4.b.h 4
24.f even 2 1 1152.4.a.t 2
24.h odd 2 1 1152.4.a.s 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.4.a.e 2 8.b even 2 1
128.4.a.f yes 2 4.b odd 2 1
128.4.a.g yes 2 8.d odd 2 1
128.4.a.h yes 2 1.a even 1 1 trivial
256.4.b.h 4 16.f odd 4 2
256.4.b.i 4 16.e even 4 2
1152.4.a.q 2 3.b odd 2 1
1152.4.a.r 2 12.b even 2 1
1152.4.a.s 2 24.h odd 2 1
1152.4.a.t 2 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(128))\):

\( T_{3}^{2} - 4T_{3} - 44 \) Copy content Toggle raw display
\( T_{5}^{2} - 4T_{5} - 188 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 4T - 44 \) Copy content Toggle raw display
$5$ \( T^{2} - 4T - 188 \) Copy content Toggle raw display
$7$ \( T^{2} + 8T - 176 \) Copy content Toggle raw display
$11$ \( T^{2} - 92T + 2068 \) Copy content Toggle raw display
$13$ \( T^{2} - 100T + 772 \) Copy content Toggle raw display
$17$ \( T^{2} - 92T - 4796 \) Copy content Toggle raw display
$19$ \( T^{2} - 4T - 2348 \) Copy content Toggle raw display
$23$ \( T^{2} - 8T - 15536 \) Copy content Toggle raw display
$29$ \( T^{2} - 84T - 3036 \) Copy content Toggle raw display
$31$ \( T^{2} + 384T + 24576 \) Copy content Toggle raw display
$37$ \( T^{2} + 172T - 2012 \) Copy content Toggle raw display
$41$ \( T^{2} + 300T + 19428 \) Copy content Toggle raw display
$43$ \( T^{2} - 300T + 21300 \) Copy content Toggle raw display
$47$ \( T^{2} + 16T - 92864 \) Copy content Toggle raw display
$53$ \( T^{2} + 12T - 9372 \) Copy content Toggle raw display
$59$ \( T^{2} + 644T + 51412 \) Copy content Toggle raw display
$61$ \( T^{2} - 292T - 213884 \) Copy content Toggle raw display
$67$ \( T^{2} + 172T - 323276 \) Copy content Toggle raw display
$71$ \( T^{2} - 408T - 43056 \) Copy content Toggle raw display
$73$ \( T^{2} - 412T - 87356 \) Copy content Toggle raw display
$79$ \( T^{2} + 400T - 22208 \) Copy content Toggle raw display
$83$ \( T^{2} - 948T + 216564 \) Copy content Toggle raw display
$89$ \( T^{2} - 572T - 564092 \) Copy content Toggle raw display
$97$ \( T^{2} - 2204 T + 808132 \) Copy content Toggle raw display
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