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Properties

Label 8752.2.a.s
Level $8752$
Weight $2$
Character orbit 8752.a
Self dual yes
Analytic conductor $69.885$
Analytic rank $1$
Dimension $18$
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] = [8752,2,Mod(1,8752)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8752, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8752.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8752 = 2^{4} \cdot 547 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8752.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: \(69.8850718490\)
Analytic rank: \(1\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 4 x^{17} - 18 x^{16} + 84 x^{15} + 116 x^{14} - 708 x^{13} - 282 x^{12} + 3104 x^{11} + \cdots + 328 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 547)
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 a basis \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{9} + 1) q^{3} + ( - \beta_{3} - 2) q^{5} + (\beta_{17} - \beta_{16} + \cdots - \beta_1) q^{7}+ \cdots + (\beta_{17} - \beta_{15} + \beta_{14} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{9} + 1) q^{3} + ( - \beta_{3} - 2) q^{5} + (\beta_{17} - \beta_{16} + \cdots - \beta_1) q^{7}+ \cdots + ( - 3 \beta_{17} + \beta_{15} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 10 q^{3} - 27 q^{5} + 11 q^{7} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 10 q^{3} - 27 q^{5} + 11 q^{7} + 14 q^{9} - 2 q^{11} - 25 q^{13} - 9 q^{15} - 30 q^{17} - 4 q^{19} - 16 q^{21} + 26 q^{23} + 31 q^{25} + 37 q^{27} - 18 q^{29} + 5 q^{31} - 10 q^{33} + 9 q^{35} - 18 q^{37} - 7 q^{39} - 17 q^{41} - 8 q^{43} - 44 q^{45} + 52 q^{47} + 29 q^{49} - 19 q^{51} - 60 q^{53} - 11 q^{55} + 4 q^{57} + 8 q^{59} - 26 q^{61} + q^{63} - 6 q^{65} - 12 q^{67} - 38 q^{69} + q^{71} - 2 q^{73} + 17 q^{75} - 73 q^{77} - 18 q^{79} + 18 q^{81} + 43 q^{83} + 51 q^{85} - 3 q^{87} - 28 q^{89} + q^{91} - 60 q^{93} + 18 q^{95} - 34 q^{97} - 15 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 4 x^{17} - 18 x^{16} + 84 x^{15} + 116 x^{14} - 708 x^{13} - 282 x^{12} + 3104 x^{11} + \cdots + 328 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 73030826 \nu^{17} - 256120405 \nu^{16} - 1356797574 \nu^{15} + 5113465236 \nu^{14} + \cdots - 59655036146 ) / 2338032863 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 197952229 \nu^{17} - 3918595304 \nu^{16} + 22600044356 \nu^{15} + 63480744292 \nu^{14} + \cdots + 1867879847532 ) / 4676065726 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 377442591 \nu^{17} - 3691417402 \nu^{16} + 2195758068 \nu^{15} + 67864507870 \nu^{14} + \cdots + 875081238676 ) / 4676065726 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 280914143 \nu^{17} - 701368576 \nu^{16} - 6767277373 \nu^{15} + 16268509010 \nu^{14} + \cdots - 245184253071 ) / 2338032863 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 918896525 \nu^{17} - 5543735306 \nu^{16} - 8092976468 \nu^{15} + 105501268198 \nu^{14} + \cdots + 742560783748 ) / 4676065726 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1002969289 \nu^{17} - 5357577374 \nu^{16} - 11353710084 \nu^{15} + 102147035436 \nu^{14} + \cdots + 677849390104 ) / 4676065726 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 685736770 \nu^{17} + 2668474422 \nu^{16} + 11910611780 \nu^{15} - 53542188699 \nu^{14} + \cdots - 42018281445 ) / 2338032863 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1386154783 \nu^{17} + 6430527688 \nu^{16} + 19874972614 \nu^{15} - 125741499426 \nu^{14} + \cdots - 374413808574 ) / 4676065726 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1665088329 \nu^{17} - 6609375872 \nu^{16} - 28840683926 \nu^{15} + 133345072732 \nu^{14} + \cdots - 107954465010 ) / 4676065726 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1832524509 \nu^{17} - 6639698786 \nu^{16} - 33972458906 \nu^{15} + 135523924258 \nu^{14} + \cdots - 180269480248 ) / 4676065726 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1216895826 \nu^{17} - 5844979198 \nu^{16} - 17002403395 \nu^{15} + 114758539026 \nu^{14} + \cdots + 306757426577 ) / 2338032863 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 2694174873 \nu^{17} + 8898905406 \nu^{16} + 54157234818 \nu^{15} - 186122321980 \nu^{14} + \cdots + 952441749148 ) / 4676065726 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 1410676829 \nu^{17} - 5054336188 \nu^{16} - 26608639913 \nu^{15} + 103627914575 \nu^{14} + \cdots - 361693721127 ) / 2338032863 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 1422023073 \nu^{17} - 4606595223 \nu^{16} - 28914727965 \nu^{15} + 96834645667 \nu^{14} + \cdots - 485286282032 ) / 2338032863 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 3038182645 \nu^{17} - 11663900722 \nu^{16} - 54084465652 \nu^{15} + 236680213784 \nu^{14} + \cdots - 317577900352 ) / 4676065726 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{14} - \beta_{11} + \beta_{10} + \beta_{7} - \beta_{6} + \beta_{2} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{15} + 2\beta_{11} - \beta_{10} + \beta_{9} - \beta_{5} - \beta_{3} + 8\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{17} - \beta_{16} - \beta_{15} - 9 \beta_{14} - \beta_{13} + \beta_{12} - 5 \beta_{11} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2 \beta_{17} + \beta_{16} - 11 \beta_{15} - 2 \beta_{13} - 3 \beta_{12} + 24 \beta_{11} - 13 \beta_{10} + \cdots + 87 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 13 \beta_{17} - 14 \beta_{16} - 11 \beta_{15} - 69 \beta_{14} - 16 \beta_{13} + 18 \beta_{12} + \cdots + 16 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 19 \beta_{17} + 10 \beta_{16} - 88 \beta_{15} - 5 \beta_{14} - 35 \beta_{13} - 27 \beta_{12} + \cdots + 532 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 129 \beta_{17} - 142 \beta_{16} - 90 \beta_{15} - 504 \beta_{14} - 177 \beta_{13} + 215 \beta_{12} + \cdots + 177 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 110 \beta_{17} + 52 \beta_{16} - 631 \beta_{15} - 100 \beta_{14} - 405 \beta_{13} - 123 \beta_{12} + \cdots + 3349 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1143 \beta_{17} - 1271 \beta_{16} - 668 \beta_{15} - 3612 \beta_{14} - 1672 \beta_{13} + 2131 \beta_{12} + \cdots + 1668 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 363 \beta_{17} + 9 \beta_{16} - 4324 \beta_{15} - 1303 \beta_{14} - 3942 \beta_{13} + 112 \beta_{12} + \cdots + 21527 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 9516 \beta_{17} - 10684 \beta_{16} - 4773 \beta_{15} - 25720 \beta_{14} - 14522 \beta_{13} + \cdots + 14428 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 1271 \beta_{17} - 3636 \beta_{16} - 29054 \beta_{15} - 14050 \beta_{14} - 35039 \beta_{13} + \cdots + 140724 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 76316 \beta_{17} - 86672 \beta_{16} - 33603 \beta_{15} - 183102 \beta_{14} - 120050 \beta_{13} + \cdots + 118635 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 39322 \beta_{17} - 57635 \beta_{16} - 193666 \beta_{15} - 136266 \beta_{14} - 295219 \beta_{13} + \cdots + 933162 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 597910 \beta_{17} - 688126 \beta_{16} - 235431 \beta_{15} - 1307213 \beta_{14} - 962160 \beta_{13} + \cdots + 944762 \) Copy content Toggle raw display

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

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
0.826129
−0.924759
2.59964
0.763493
−0.957552
−0.735255
1.35726
1.98431
−1.15793
−2.24960
−1.87675
1.74487
2.72204
−1.52216
1.04467
−2.50138
0.523506
2.35947
0 −2.26041 0 −0.786316 0 5.06179 0 2.10946 0
1.2 0 −2.22306 0 −3.95421 0 3.08028 0 1.94201 0
1.3 0 −2.09484 0 −3.02323 0 0.561390 0 1.38836 0
1.4 0 −1.83524 0 −1.51218 0 −1.20167 0 0.368111 0
1.5 0 −0.564538 0 −4.10274 0 −4.97202 0 −2.68130 0
1.6 0 −0.544167 0 0.962787 0 3.25298 0 −2.70388 0
1.7 0 −0.387927 0 −1.46929 0 −0.194696 0 −2.84951 0
1.8 0 0.150114 0 2.87852 0 2.68467 0 −2.97747 0
1.9 0 0.220625 0 −0.421419 0 0.645304 0 −2.95132 0
1.10 0 0.790850 0 −3.96974 0 4.97706 0 −2.37456 0
1.11 0 1.13919 0 −1.30620 0 1.71403 0 −1.70224 0
1.12 0 1.27304 0 −3.61409 0 −4.28084 0 −1.37936 0
1.13 0 1.76734 0 0.469688 0 −1.03831 0 0.123492 0
1.14 0 2.58636 0 1.24712 0 0.899316 0 3.68924 0
1.15 0 2.71791 0 0.714085 0 2.03236 0 4.38705 0
1.16 0 3.08733 0 −3.57921 0 −1.44216 0 6.53160 0
1.17 0 3.08736 0 −1.35183 0 −3.60927 0 6.53179 0
1.18 0 3.09007 0 −4.18174 0 2.82979 0 6.54852 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(547\) \(-1\)

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 8752.2.a.s 18
4.b odd 2 1 547.2.a.b ✓ 18
12.b even 2 1 4923.2.a.l 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
547.2.a.b ✓ 18 4.b odd 2 1
4923.2.a.l 18 12.b even 2 1
8752.2.a.s 18 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8752))\):

\( T_{3}^{18} - 10 T_{3}^{17} + 16 T_{3}^{16} + 141 T_{3}^{15} - 499 T_{3}^{14} - 485 T_{3}^{13} + \cdots - 32 \) Copy content Toggle raw display
\( T_{5}^{18} + 27 T_{5}^{17} + 304 T_{5}^{16} + 1771 T_{5}^{15} + 4872 T_{5}^{14} - 1378 T_{5}^{13} + \cdots - 15872 \) Copy content Toggle raw display
\( T_{7}^{18} - 11 T_{7}^{17} - 17 T_{7}^{16} + 591 T_{7}^{15} - 1324 T_{7}^{14} - 9436 T_{7}^{13} + \cdots - 58576 \) Copy content Toggle raw display
\( T_{11}^{18} + 2 T_{11}^{17} - 113 T_{11}^{16} - 224 T_{11}^{15} + 4921 T_{11}^{14} + 9574 T_{11}^{13} + \cdots - 82432 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} \) Copy content Toggle raw display
$3$ \( T^{18} - 10 T^{17} + \cdots - 32 \) Copy content Toggle raw display
$5$ \( T^{18} + 27 T^{17} + \cdots - 15872 \) Copy content Toggle raw display
$7$ \( T^{18} - 11 T^{17} + \cdots - 58576 \) Copy content Toggle raw display
$11$ \( T^{18} + 2 T^{17} + \cdots - 82432 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots - 160401143 \) Copy content Toggle raw display
$17$ \( T^{18} + 30 T^{17} + \cdots - 5744344 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots - 584801324 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 8189636608 \) Copy content Toggle raw display
$29$ \( T^{18} + 18 T^{17} + \cdots - 13479926 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 1756132711264 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 5114786922536 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 341990380184 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 23810212304 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 35899903339 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots - 971885299094 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots - 1458822574688 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 75258329816 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 12533716163723 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots - 368406703332424 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots - 12033371727622 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 1079280536 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots - 94\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 4077961316512 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots - 28869767458 \) Copy content Toggle raw display
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