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Properties

Label 384.7.h.f
Level $384$
Weight $7$
Character orbit 384.h
Analytic conductor $88.341$
Analytic rank $0$
Dimension $8$
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] = [384,7,Mod(65,384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(384, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.65");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 384.h (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: \(88.3407681100\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 172x^{6} + 13179x^{4} - 522628x^{2} + 8755681 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{26}\cdot 3^{6} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{4} + 9) q^{3} - \beta_{6} q^{5} + ( - \beta_{6} - \beta_{3}) q^{7} + ( - 3 \beta_{7} - 15 \beta_{5} + \cdots - 207) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} + 9) q^{3} - \beta_{6} q^{5} + ( - \beta_{6} - \beta_{3}) q^{7} + ( - 3 \beta_{7} - 15 \beta_{5} + \cdots - 207) q^{9}+ \cdots + (84 \beta_{7} - 9057 \beta_{5} + \cdots - 211122) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 72 q^{3} - 1656 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 72 q^{3} - 1656 q^{9} + 1648 q^{11} - 200 q^{25} + 21384 q^{27} - 22608 q^{33} + 136320 q^{35} + 105208 q^{49} + 275328 q^{51} - 391104 q^{57} + 836624 q^{59} - 1964944 q^{73} - 59400 q^{75} + 166536 q^{81} - 587024 q^{83} - 1477232 q^{97} - 1688976 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 172x^{6} + 13179x^{4} - 522628x^{2} + 8755681 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 4304\nu^{6} - 469248\nu^{4} + 18250656\nu^{2} - 251351264 ) / 97081 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 10144\nu^{6} - 1244544\nu^{4} + 74403840\nu^{2} - 1830375232 ) / 97081 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -31800\nu^{7} + 5753664\nu^{5} - 475691952\nu^{3} + 27076321320\nu ) / 287262679 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 37990 \nu^{7} - 47344 \nu^{6} - 4705618 \nu^{5} + 9433292 \nu^{4} + 279940646 \nu^{3} + \cdots + 16218154722 ) / 287262679 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 37990 \nu^{7} - 47344 \nu^{6} + 4705618 \nu^{5} + 9433292 \nu^{4} - 279940646 \nu^{3} + \cdots + 16218154722 ) / 287262679 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -293636\nu^{7} + 47925144\nu^{5} - 2781189072\nu^{3} + 71694324452\nu ) / 861788037 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -240320\nu^{7} + 26137616\nu^{5} - 1288141264\nu^{3} + 26931123856\nu ) / 287262679 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - 4\beta_{5} + 4\beta_{4} + 2\beta_{3} ) / 96 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 48\beta_{5} + 48\beta_{4} + \beta_{2} - 2\beta _1 + 8256 ) / 192 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 127\beta_{7} + 90\beta_{6} - 544\beta_{5} + 544\beta_{4} + 63\beta_{3} ) / 96 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2718\beta_{5} + 2718\beta_{4} + 23\beta_{2} - 34\beta _1 + 38712 ) / 48 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5056\beta_{7} + 11970\beta_{6} - 31372\beta_{5} + 31372\beta_{4} - 95\beta_{3} ) / 96 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 163632\beta_{5} + 163632\beta_{4} + 965\beta_{2} - 336\beta _1 - 1152256 ) / 32 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -133523\beta_{7} + 819468\beta_{6} - 944428\beta_{5} + 944428\beta_{4} - 123892\beta_{3} ) / 96 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
7.10120 + 0.847848i
−7.10120 + 0.847848i
7.10120 − 0.847848i
−7.10120 − 0.847848i
−6.86097 + 3.28347i
6.86097 + 3.28347i
−6.86097 − 3.28347i
6.86097 − 3.28347i
0 −4.41641 − 26.6364i 0 −127.030 0 −467.888 0 −689.991 + 235.274i 0
65.2 0 −4.41641 − 26.6364i 0 127.030 0 467.888 0 −689.991 + 235.274i 0
65.3 0 −4.41641 + 26.6364i 0 −127.030 0 −467.888 0 −689.991 − 235.274i 0
65.4 0 −4.41641 + 26.6364i 0 127.030 0 467.888 0 −689.991 − 235.274i 0
65.5 0 22.4164 − 15.0501i 0 −122.733 0 206.594 0 275.991 − 674.737i 0
65.6 0 22.4164 − 15.0501i 0 122.733 0 −206.594 0 275.991 − 674.737i 0
65.7 0 22.4164 + 15.0501i 0 −122.733 0 206.594 0 275.991 + 674.737i 0
65.8 0 22.4164 + 15.0501i 0 122.733 0 −206.594 0 275.991 + 674.737i 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
8.d odd 2 1 inner
12.b even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.7.h.f yes 8
3.b odd 2 1 384.7.h.e ✓ 8
4.b odd 2 1 384.7.h.e ✓ 8
8.b even 2 1 384.7.h.e ✓ 8
8.d odd 2 1 inner 384.7.h.f yes 8
12.b even 2 1 inner 384.7.h.f yes 8
24.f even 2 1 384.7.h.e ✓ 8
24.h odd 2 1 inner 384.7.h.f yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.7.h.e ✓ 8 3.b odd 2 1
384.7.h.e ✓ 8 4.b odd 2 1
384.7.h.e ✓ 8 8.b even 2 1
384.7.h.e ✓ 8 24.f even 2 1
384.7.h.f yes 8 1.a even 1 1 trivial
384.7.h.f yes 8 8.d odd 2 1 inner
384.7.h.f yes 8 12.b even 2 1 inner
384.7.h.f yes 8 24.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{7}^{\mathrm{new}}(384, [\chi])\):

\( T_{5}^{4} - 31200T_{5}^{2} + 243072000 \) Copy content Toggle raw display
\( T_{11}^{2} - 412T_{11} - 79244 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - 36 T^{3} + \cdots + 531441)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} - 31200 T^{2} + 243072000)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 261600 T^{2} + 9343687680)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 412 T - 79244)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots + 1562505707520)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + \cdots + 231275569987584)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 6011206401024)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots + 87\!\cdots\!20)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 51\!\cdots\!80)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 33\!\cdots\!80)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 45\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 62\!\cdots\!44)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 34\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + \cdots + 93\!\cdots\!20)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 10\!\cdots\!80)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 209156 T - 29855566796)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 28\!\cdots\!20)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 60\!\cdots\!04)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 16\!\cdots\!20)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 491236 T + 49489033924)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 26\!\cdots\!80)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 146756 T + 1115288884)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 25\!\cdots\!64)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + \cdots - 1772532994364)^{4} \) Copy content Toggle raw display
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