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Properties

Label 8752.2.a.r
Level $8752$
Weight $2$
Character orbit 8752.a
Self dual yes
Analytic conductor $69.885$
Analytic rank $1$
Dimension $15$
CM no
Inner twists $1$

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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: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 6 x^{14} - 12 x^{13} + 113 x^{12} + 38 x^{11} - 847 x^{10} - 25 x^{9} + 3173 x^{8} + 324 x^{7} + \cdots - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1094)
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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{3} - \beta_{3} q^{5} + ( - \beta_{4} - 1) q^{7} + (\beta_{2} - \beta_1 + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{3} - \beta_{3} q^{5} + ( - \beta_{4} - 1) q^{7} + (\beta_{2} - \beta_1 + 2) q^{9} + (\beta_{13} - 1) q^{11} + ( - \beta_{12} + \beta_{11} + \cdots + \beta_1) q^{13}+ \cdots + ( - \beta_{14} - \beta_{11} + \beta_{10} + \cdots + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - 9 q^{3} - q^{5} - 11 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q - 9 q^{3} - q^{5} - 11 q^{7} + 18 q^{9} - 8 q^{11} + 8 q^{13} + 14 q^{17} - 23 q^{19} + 17 q^{21} - 17 q^{23} + 12 q^{25} - 30 q^{27} - 5 q^{29} - 26 q^{31} + 4 q^{33} + 6 q^{35} - q^{37} + 13 q^{39} + 7 q^{41} - 10 q^{43} - 29 q^{45} - 9 q^{47} + 18 q^{49} + 4 q^{51} - 14 q^{53} + 3 q^{55} - 5 q^{57} + 2 q^{59} + 21 q^{61} - 5 q^{63} - 20 q^{65} - 5 q^{67} - 26 q^{69} + 6 q^{73} - 10 q^{75} - 34 q^{77} - 17 q^{79} - 5 q^{81} - 14 q^{83} - 21 q^{85} + 27 q^{87} - 15 q^{89} - 2 q^{91} - 41 q^{93} + 32 q^{95} + 12 q^{97} + 15 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 6 x^{14} - 12 x^{13} + 113 x^{12} + 38 x^{11} - 847 x^{10} - 25 x^{9} + 3173 x^{8} + 324 x^{7} + \cdots - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 209276 \nu^{14} + 25009553 \nu^{13} - 147405466 \nu^{12} - 271773813 \nu^{11} + 2367779233 \nu^{10} + \cdots - 858842944 ) / 266854719 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 841919 \nu^{14} - 2142958 \nu^{13} - 33564082 \nu^{12} + 98601438 \nu^{11} + 387858082 \nu^{10} + \cdots - 261668632 ) / 266854719 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1658937 \nu^{14} + 18428052 \nu^{13} - 27676488 \nu^{12} - 274745854 \nu^{11} + \cdots + 250197204 ) / 88951573 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 6223004 \nu^{14} + 48299137 \nu^{13} - 8333087 \nu^{12} - 683531964 \nu^{11} + \cdots + 1848757624 ) / 266854719 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 3880027 \nu^{14} + 19877502 \nu^{13} + 74568551 \nu^{12} - 457195242 \nu^{11} + \cdots + 544589331 ) / 88951573 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 5293270 \nu^{14} + 28411002 \nu^{13} + 91073692 \nu^{12} - 618617090 \nu^{11} + \cdots + 579458067 ) / 88951573 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 16173823 \nu^{14} + 99106682 \nu^{13} + 172089242 \nu^{12} - 1825766142 \nu^{11} + \cdots - 1426116913 ) / 266854719 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 16814783 \nu^{14} + 130119700 \nu^{13} - 749798 \nu^{12} - 1993538949 \nu^{11} + \cdots + 482201353 ) / 266854719 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 8075460 \nu^{14} + 52884921 \nu^{13} + 51962115 \nu^{12} - 856816444 \nu^{11} + \cdots - 384065757 ) / 88951573 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 25027058 \nu^{14} - 147686926 \nu^{13} - 322341283 \nu^{12} + 2901642948 \nu^{11} + \cdots + 537869819 ) / 266854719 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 27701731 \nu^{14} + 190767752 \nu^{13} + 186366770 \nu^{12} - 3356669211 \nu^{11} + \cdots + 1297542104 ) / 266854719 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 9643855 \nu^{14} - 61371491 \nu^{13} - 72815048 \nu^{12} + 1005818484 \nu^{11} - 239134431 \nu^{10} + \cdots + 45526831 ) / 88951573 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{12} + \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} + \beta_{4} + \beta_{3} + \beta_{2} + 7\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{14} - \beta_{13} + 5 \beta_{12} + \beta_{11} - 2 \beta_{10} + 3 \beta_{9} + 3 \beta_{8} + \cdots + 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 2 \beta_{14} - 2 \beta_{13} + 29 \beta_{12} + 12 \beta_{11} - 16 \beta_{10} + 17 \beta_{9} + \cdots + 42 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 15 \beta_{14} - 14 \beta_{13} + 85 \beta_{12} + 20 \beta_{11} - 42 \beta_{10} + 56 \beta_{9} + \cdots + 230 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 41 \beta_{14} - 34 \beta_{13} + 358 \beta_{12} + 126 \beta_{11} - 209 \beta_{10} + 228 \beta_{9} + \cdots + 485 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 197 \beta_{14} - 158 \beta_{13} + 1126 \beta_{12} + 281 \beta_{11} - 625 \beta_{10} + 765 \beta_{9} + \cdots + 2089 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 617 \beta_{14} - 426 \beta_{13} + 4215 \beta_{12} + 1306 \beta_{11} - 2559 \beta_{10} + 2798 \beta_{9} + \cdots + 5326 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2493 \beta_{14} - 1671 \beta_{13} + 13700 \beta_{12} + 3441 \beta_{11} - 8151 \beta_{10} + \cdots + 20276 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 8265 \beta_{14} - 4809 \beta_{13} + 48610 \beta_{12} + 13638 \beta_{11} - 30395 \beta_{10} + \cdots + 57669 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 30843 \beta_{14} - 17307 \beta_{13} + 160524 \beta_{12} + 39481 \beta_{11} - 99717 \beta_{10} + \cdots + 205571 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 104587 \beta_{14} - 51994 \beta_{13} + 554614 \beta_{12} + 143525 \beta_{11} - 354368 \beta_{10} + \cdots + 623148 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 375032 \beta_{14} - 178595 \beta_{13} + 1845698 \beta_{12} + 437903 \beta_{11} - 1177867 \beta_{10} + \cdots + 2144774 \) 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
−2.45308
−1.90953
−1.73236
−1.67718
−1.00484
−0.566530
−0.359098
−0.00607847
0.284129
1.18684
2.31496
2.32849
2.92994
3.31012
3.35422
0 −3.45308 0 −3.43639 0 −4.02871 0 8.92377 0
1.2 0 −2.90953 0 1.58726 0 −0.225008 0 5.46535 0
1.3 0 −2.73236 0 1.12451 0 2.92225 0 4.46580 0
1.4 0 −2.67718 0 −2.27480 0 2.35439 0 4.16730 0
1.5 0 −2.00484 0 2.69588 0 −2.28309 0 1.01937 0
1.6 0 −1.56653 0 −1.10670 0 −4.35042 0 −0.545985 0
1.7 0 −1.35910 0 1.05159 0 −3.35605 0 −1.15285 0
1.8 0 −1.00608 0 3.28895 0 −0.0436829 0 −1.98781 0
1.9 0 −0.715871 0 −3.13634 0 −1.80171 0 −2.48753 0
1.10 0 0.186838 0 1.67339 0 0.961664 0 −2.96509 0
1.11 0 1.31496 0 −4.38013 0 −1.53058 0 −1.27088 0
1.12 0 1.32849 0 3.42344 0 −1.45869 0 −1.23512 0
1.13 0 1.92994 0 −0.885738 0 3.17326 0 0.724668 0
1.14 0 2.31012 0 0.492178 0 −5.00457 0 2.33667 0
1.15 0 2.35422 0 −1.11710 0 3.67095 0 2.54234 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
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.r 15
4.b odd 2 1 1094.2.a.g ✓ 15
12.b even 2 1 9846.2.a.r 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1094.2.a.g ✓ 15 4.b odd 2 1
8752.2.a.r 15 1.a even 1 1 trivial
9846.2.a.r 15 12.b even 2 1

Hecke kernels

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

\( T_{3}^{15} + 9 T_{3}^{14} + 9 T_{3}^{13} - 134 T_{3}^{12} - 361 T_{3}^{11} + 594 T_{3}^{10} + 2868 T_{3}^{9} + \cdots + 774 \) Copy content Toggle raw display
\( T_{5}^{15} + T_{5}^{14} - 43 T_{5}^{13} - 22 T_{5}^{12} + 706 T_{5}^{11} + 69 T_{5}^{10} - 5532 T_{5}^{9} + \cdots + 5518 \) Copy content Toggle raw display
\( T_{7}^{15} + 11 T_{7}^{14} - T_{7}^{13} - 383 T_{7}^{12} - 884 T_{7}^{11} + 4451 T_{7}^{10} + \cdots - 2048 \) Copy content Toggle raw display
\( T_{11}^{15} + 8 T_{11}^{14} - 37 T_{11}^{13} - 395 T_{11}^{12} + 417 T_{11}^{11} + 7790 T_{11}^{10} + \cdots - 24512 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} \) Copy content Toggle raw display
$3$ \( T^{15} + 9 T^{14} + \cdots + 774 \) Copy content Toggle raw display
$5$ \( T^{15} + T^{14} + \cdots + 5518 \) Copy content Toggle raw display
$7$ \( T^{15} + 11 T^{14} + \cdots - 2048 \) Copy content Toggle raw display
$11$ \( T^{15} + 8 T^{14} + \cdots - 24512 \) Copy content Toggle raw display
$13$ \( T^{15} - 8 T^{14} + \cdots - 175424 \) Copy content Toggle raw display
$17$ \( T^{15} - 14 T^{14} + \cdots - 3763584 \) Copy content Toggle raw display
$19$ \( T^{15} + \cdots + 110787392 \) Copy content Toggle raw display
$23$ \( T^{15} + 17 T^{14} + \cdots + 695168 \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots + 9190833984 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots + 5001841408 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots - 160926174 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots + 166041984 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots + 1394610544 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots - 9696335327 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots + 496757824 \) Copy content Toggle raw display
$59$ \( T^{15} - 2 T^{14} + \cdots - 31644258 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots + 139215026586 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots + 235334592 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots - 718139776 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots + 6295532891 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots - 36761323136 \) Copy content Toggle raw display
$83$ \( T^{15} + 14 T^{14} + \cdots - 5271742 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots - 1178975872 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots - 4579508723 \) Copy content Toggle raw display
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