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Properties

Label 576.8.c.d
Level $576$
Weight $8$
Character orbit 576.c
Analytic conductor $179.934$
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] = [576,8,Mod(575,576)]
 
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([1, 0, 1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("576.575");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 576.c (of order \(2\), degree \(1\), 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: \(179.933774679\)
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: \( x^{8} - 4x^{7} - 1078x^{6} + 3248x^{5} + 435239x^{4} - 875896x^{3} - 77990030x^{2} + 78428520x + 5236407012 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{28}\cdot 3^{12} \)
Twist minimal: no (minimal twist has level 144)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{4} - \beta_{3}) q^{5} + ( - \beta_{5} + \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} - \beta_{3}) q^{5} + ( - \beta_{5} + \beta_1) q^{7} + ( - 2 \beta_{7} - \beta_{6}) q^{11} + ( - 7 \beta_{2} - 3040) q^{13} + ( - 23 \beta_{4} - 34 \beta_{3}) q^{17} + (40 \beta_{5} - 53 \beta_1) q^{19} + ( - 7 \beta_{7} + 5 \beta_{6}) q^{23} + ( - 46 \beta_{2} - 9229) q^{25} + ( - 347 \beta_{4} - 65 \beta_{3}) q^{29} + (295 \beta_{5} - 21 \beta_1) q^{31} + (96 \beta_{7} - 53 \beta_{6}) q^{35} + ( - 108 \beta_{2} - 135530) q^{37} + (3035 \beta_{4} + 990 \beta_{3}) q^{41} + (382 \beta_{5} - 1992 \beta_1) q^{43} + (45 \beta_{7} + 485 \beta_{6}) q^{47} + ( - 280 \beta_{2} + 297751) q^{49} + ( - 1935 \beta_{4} + 4935 \beta_{3}) q^{53} + ( - 1326 \beta_{5} + 6642 \beta_1) q^{55} + (750 \beta_{7} - 970 \beta_{6}) q^{59} + ( - 1300 \beta_{2} - 1535198) q^{61} + (23648 \beta_{4} + 5938 \beta_{3}) q^{65} + ( - 2490 \beta_{5} - 9395 \beta_1) q^{67} + ( - 1227 \beta_{7} + 2039 \beta_{6}) q^{71} + ( - 2956 \beta_{2} + 952480) q^{73} + ( - 33312 \beta_{4} - 8016 \beta_{3}) q^{77} + (4025 \beta_{5} + 14837 \beta_1) q^{79} + (1136 \beta_{7} - 3275 \beta_{6}) q^{83} + ( - 1267 \beta_{2} - 2847078) q^{85} + (26515 \beta_{4} - 16220 \beta_{3}) q^{89} + (5980 \beta_{5} - 28303 \beta_1) q^{91} + ( - 4178 \beta_{7} + 1964 \beta_{6}) q^{95} + (4202 \beta_{2} + 5247040) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 24320 q^{13} - 73832 q^{25} - 1084240 q^{37} + 2382008 q^{49} - 12281584 q^{61} + 7619840 q^{73} - 22776624 q^{85} + 41976320 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 1078x^{6} + 3248x^{5} + 435239x^{4} - 875896x^{3} - 77990030x^{2} + 78428520x + 5236407012 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 256 \nu^{6} - 768 \nu^{5} - 345728 \nu^{4} + 692736 \nu^{3} + 130777216 \nu^{2} + \cdots - 15030904704 ) / 1164225 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 172832 \nu^{7} - 604912 \nu^{6} - 195895168 \nu^{5} + 491250200 \nu^{4} + \cdots + 11866046900244 ) / 150052691425 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 4717100 \nu^{7} - 1239745241 \nu^{6} + 7615078723 \nu^{5} + 1011678330733 \nu^{4} + \cdots + 25\!\cdots\!44 ) / 450158074275 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 19746 \nu^{7} - 69111 \nu^{6} - 16100847 \nu^{5} + 40424895 \nu^{4} + 4391545473 \nu^{3} + \cdots + 201232021572 ) / 279167798 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 12418400 \nu^{7} - 46557672 \nu^{6} - 13399487584 \nu^{5} + 37808043336 \nu^{4} + \cdots + 425181070091448 ) / 150052691425 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 71193532 \nu^{7} - 582139488 \nu^{6} + 59683836968 \nu^{5} + 528937672200 \nu^{4} + \cdots + 14\!\cdots\!56 ) / 150052691425 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3029512 \nu^{7} - 77108640 \nu^{6} - 2234096144 \nu^{5} + 59913365232 \nu^{4} + \cdots + 12\!\cdots\!48 ) / 6002107657 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -3\beta_{7} + 6\beta_{6} + 64\beta_{4} - 144\beta_{2} + 5184 ) / 10368 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 138\beta_{7} + 156\beta_{6} - 64\beta_{4} - 864\beta_{3} - 144\beta_{2} - 81\beta _1 + 2814912 ) / 10368 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 2223 \beta_{7} + 5094 \beta_{6} - 1296 \beta_{5} + 52000 \beta_{4} - 1296 \beta_{3} + \cdots + 4219776 ) / 10368 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 17889 \beta_{7} + 22758 \beta_{6} - 648 \beta_{5} + 8480 \beta_{4} - 118800 \beta_{3} + \cdots + 194559408 ) / 2592 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 910893 \beta_{7} + 2406066 \beta_{6} - 1175040 \beta_{5} + 24018016 \beta_{4} - 1185840 \beta_{3} + \cdots + 1938561984 ) / 10368 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 27885438 \beta_{7} + 39753396 \beta_{6} - 3518640 \beta_{5} + 42626240 \beta_{4} + \cdots + 218820216960 ) / 10368 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 311724879 \beta_{7} + 955740486 \beta_{6} - 676901232 \beta_{5} + 9521479456 \beta_{4} + \cdots + 759090714624 ) / 10368 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
575.1
−14.7145 + 0.707107i
−17.1640 + 0.707107i
18.1640 − 0.707107i
15.7145 − 0.707107i
15.7145 + 0.707107i
18.1640 + 0.707107i
−17.1640 − 0.707107i
−14.7145 − 0.707107i
0 0 0 − 376.565i 0 − 925.855i 0 0 0
575.2 0 0 0 − 376.565i 0 925.855i 0 0 0
575.3 0 0 0 − 181.403i 0 − 440.881i 0 0 0
575.4 0 0 0 − 181.403i 0 440.881i 0 0 0
575.5 0 0 0 181.403i 0 − 440.881i 0 0 0
575.6 0 0 0 181.403i 0 440.881i 0 0 0
575.7 0 0 0 376.565i 0 − 925.855i 0 0 0
575.8 0 0 0 376.565i 0 925.855i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 575.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.8.c.d 8
3.b odd 2 1 inner 576.8.c.d 8
4.b odd 2 1 inner 576.8.c.d 8
8.b even 2 1 144.8.c.c ✓ 8
8.d odd 2 1 144.8.c.c ✓ 8
12.b even 2 1 inner 576.8.c.d 8
24.f even 2 1 144.8.c.c ✓ 8
24.h odd 2 1 144.8.c.c ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.8.c.c ✓ 8 8.b even 2 1
144.8.c.c ✓ 8 8.d odd 2 1
144.8.c.c ✓ 8 24.f even 2 1
144.8.c.c ✓ 8 24.h odd 2 1
576.8.c.d 8 1.a even 1 1 trivial
576.8.c.d 8 3.b odd 2 1 inner
576.8.c.d 8 4.b odd 2 1 inner
576.8.c.d 8 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 174708T_{5}^{2} + 4666256100 \) acting on \(S_{8}^{\mathrm{new}}(576, [\chi])\). 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
$5$ \( (T^{4} + 174708 T^{2} + 4666256100)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 1051584 T^{2} + 166620708864)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots + 437756195045376)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 6080 T - 59406224)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + \cdots + 73\!\cdots\!64)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 34\!\cdots\!44)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots + 48\!\cdots\!16)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 13\!\cdots\!64)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 16\!\cdots\!04)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 271060 T + 2027396836)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 17\!\cdots\!64)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + \cdots + 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 32\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 32\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 3070396 T - 10816540796)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 44\!\cdots\!16)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + \cdots - 11334420475136)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 73\!\cdots\!44)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 14\!\cdots\!36)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 90\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + \cdots + 2794670120896)^{4} \) Copy content Toggle raw display
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