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Properties

Label 576.8.d.h
Level $576$
Weight $8$
Character orbit 576.d
Analytic conductor $179.934$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [576,8,Mod(289,576)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(576, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("576.289");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 576.d (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: \(179.933774679\)
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: \( x^{8} - 4405x^{6} + 14555221x^{4} - 21358981620x^{2} + 23510900230416 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{28}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 192)
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.

\(f(q)\) \(=\) \( q + (\beta_{3} - 8 \beta_1) q^{5} + ( - \beta_{4} + 5 \beta_{2}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - 8 \beta_1) q^{5} + ( - \beta_{4} + 5 \beta_{2}) q^{7} + ( - \beta_{7} - 156 \beta_{5}) q^{11} + ( - 4 \beta_{3} + 91 \beta_1) q^{13} + ( - 3 \beta_{6} + 9738) q^{17} + (5 \beta_{7} - 329 \beta_{5}) q^{19} + ( - 130 \beta_{4} - 360 \beta_{2}) q^{23} + (16 \beta_{6} - 39871) q^{25} + (193 \beta_{3} + 7366 \beta_1) q^{29} + ( - 439 \beta_{4} + 1194 \beta_{2}) q^{31} + ( - 13 \beta_{7} - 18363 \beta_{5}) q^{35} + (334 \beta_{3} - 13377 \beta_1) q^{37} + (65 \beta_{6} - 84666) q^{41} + ( - 179 \beta_{7} + 52601 \beta_{5}) q^{43} + ( - 130 \beta_{4} + 4689 \beta_{2}) q^{47} + (42 \beta_{6} - 633163) q^{49} + ( - 2569 \beta_{3} + 65858 \beta_1) q^{53} + ( - 912 \beta_{4} - 25407 \beta_{2}) q^{55} + ( - 336 \beta_{7} + 222903 \beta_{5}) q^{59} + ( - 8682 \beta_{3} - 5195 \beta_1) q^{61} + ( - 123 \beta_{6} + 562608) q^{65} + ( - 244 \beta_{7} - 525643 \beta_{5}) q^{67} + ( - 3950 \beta_{4} + 49797 \beta_{2}) q^{71} + (1010 \beta_{6} - 1519726) q^{73} + (4656 \beta_{3} + 118812 \beta_1) q^{77} + ( - 1451 \beta_{4} - 94534 \beta_{2}) q^{79} + (261 \beta_{7} + 1338444 \beta_{5}) q^{83} + (14346 \beta_{3} - 395028 \beta_1) q^{85} + (1250 \beta_{6} + 5241642) q^{89} + ( - 7 \beta_{7} + 13980 \beta_{5}) q^{91} + (8996 \beta_{4} + 137016 \beta_{2}) q^{95} + ( - 1546 \beta_{6} - 4090274) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 77904 q^{17} - 318968 q^{25} - 677328 q^{41} - 5065304 q^{49} + 4500864 q^{65} - 12157808 q^{73} + 41933136 q^{89} - 32722192 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4405x^{6} + 14555221x^{4} - 21358981620x^{2} + 23510900230416 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 17620\nu^{6} - 58220884\nu^{4} + 256462994020\nu^{2} - 235194428533032 ) / 17643853451421 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -4\nu^{7} + 19412836\nu^{5} - 42785881732\nu^{3} + 94214472774624\nu ) / 2940197162103 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} + 4405\nu^{4} - 9706417\nu^{2} + 10679490810 ) / 1212201 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2153311\nu^{7} - 3566029145\nu^{5} - 7816779551503\nu^{3} + 28701553688751240\nu ) / 8633725622228676 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -4405\nu^{7} + 14555221\nu^{5} - 32079700477\nu^{3} + 23510900230416\nu ) / 17635844439414 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -96\nu^{6} - 1027093692720 ) / 14555221 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 8810\nu^{7} - 29110442\nu^{5} + 106867666586\nu^{3} - 47021800460832\nu ) / 980065720701 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{7} + 24\beta_{5} + 48\beta_{4} + 9\beta_{2} ) / 384 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + 24\beta_{3} + 26430\beta _1 + 211440 ) / 192 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2203\beta_{7} + 79284\beta_{5} ) / 96 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -4405\beta_{6} + 105720\beta_{3} + 58238502\beta _1 - 465908016 ) / 192 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 9710822\beta_{7} + 582120744\beta_{5} - 233059728\beta_{4} + 14500161\beta_{2} ) / 384 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -14555221\beta_{6} - 1027093692720 ) / 96 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -21412340486\beta_{7} - 1795376724648\beta_{5} - 513896171664\beta_{4} + 95948047665\beta_{2} ) / 384 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(325\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
40.2079 + 23.2141i
−40.2079 − 23.2141i
41.0740 − 23.7141i
−41.0740 + 23.7141i
41.0740 + 23.7141i
−41.0740 − 23.7141i
40.2079 − 23.2141i
−40.2079 + 23.2141i
0 0 0 − 435.979i 0 −34.1431 0 0 0
289.2 0 0 0 − 435.979i 0 34.1431 0 0 0
289.3 0 0 0 − 214.276i 0 −616.112 0 0 0
289.4 0 0 0 − 214.276i 0 616.112 0 0 0
289.5 0 0 0 214.276i 0 −616.112 0 0 0
289.6 0 0 0 214.276i 0 616.112 0 0 0
289.7 0 0 0 435.979i 0 −34.1431 0 0 0
289.8 0 0 0 435.979i 0 34.1431 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.8.d.h 8
3.b odd 2 1 192.8.d.c ✓ 8
4.b odd 2 1 inner 576.8.d.h 8
8.b even 2 1 inner 576.8.d.h 8
8.d odd 2 1 inner 576.8.d.h 8
12.b even 2 1 192.8.d.c ✓ 8
24.f even 2 1 192.8.d.c ✓ 8
24.h odd 2 1 192.8.d.c ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
192.8.d.c ✓ 8 3.b odd 2 1
192.8.d.c ✓ 8 12.b even 2 1
192.8.d.c ✓ 8 24.f even 2 1
192.8.d.c ✓ 8 24.h odd 2 1
576.8.d.h 8 1.a even 1 1 trivial
576.8.d.h 8 4.b odd 2 1 inner
576.8.d.h 8 8.b even 2 1 inner
576.8.d.h 8 8.d 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 \(S_{8}^{\mathrm{new}}(576, [\chi])\):

\( T_{5}^{4} + 235992T_{5}^{2} + 8727296400 \) Copy content Toggle raw display
\( T_{7}^{4} - 380760T_{7}^{2} + 442513296 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} + 235992 T^{2} + 8727296400)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 380760 T^{2} + 442513296)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots + 396271131033600)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 6562560 T^{2} + 10277093376)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 19476 T - 87834780)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 25\!\cdots\!36)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 41\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 22\!\cdots\!24)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 50\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 169332 T - 78581998044)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 36\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + \cdots + 44\!\cdots\!24)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 22\!\cdots\!44)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 63\!\cdots\!64)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 10\!\cdots\!44)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 39\!\cdots\!04)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + \cdots - 18394317198524)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 73\!\cdots\!64)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 74\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + \cdots - 4237589143836)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + \cdots - 31779299973500)^{4} \) Copy content Toggle raw display
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