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Properties

Label 288.8.d.d
Level $288$
Weight $8$
Character orbit 288.d
Analytic conductor $89.967$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [288,8,Mod(145,288)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(288, 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("288.145");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 288.d (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: \(89.9668873394\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 6 x^{13} - 52 x^{12} + 300 x^{11} - 1005 x^{10} - 23250 x^{9} + 349930 x^{8} + \cdots + 3813237677250 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{88}\cdot 3^{18} \)
Twist minimal: no (minimal twist has level 24)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{5} + (\beta_{3} - 98) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{5} + (\beta_{3} - 98) q^{7} + ( - \beta_{7} - 4 \beta_{2}) q^{11} + (\beta_{10} - \beta_{7} + \cdots + 2 \beta_{2}) q^{13}+ \cdots + (190 \beta_{12} - 464 \beta_{8} + \cdots + 9656) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 1372 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 1372 q^{7} + 2908 q^{17} - 143416 q^{23} - 202626 q^{25} + 89468 q^{31} + 441284 q^{41} - 1056408 q^{47} + 2158134 q^{49} - 4757504 q^{55} + 2520464 q^{65} + 5172696 q^{71} - 5446196 q^{73} + 14373548 q^{79} + 11952620 q^{89} - 69327376 q^{95} + 133732 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 6 x^{13} - 52 x^{12} + 300 x^{11} - 1005 x^{10} - 23250 x^{9} + 349930 x^{8} + \cdots + 3813237677250 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 2324553 \nu^{13} - 167884787 \nu^{12} + 3384637889 \nu^{11} - 28983151979 \nu^{10} + \cdots - 83\!\cdots\!26 ) / 20\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 594447101 \nu^{13} + 7190225169 \nu^{12} - 26859563563 \nu^{11} + 115254715449 \nu^{10} + \cdots - 36\!\cdots\!30 ) / 41\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1256255 \nu^{13} - 7174107 \nu^{12} - 30451335 \nu^{11} + 346825197 \nu^{10} + \cdots + 42\!\cdots\!26 ) / 670306955952128 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 136458837 \nu^{13} + 1839828681 \nu^{12} - 4871646099 \nu^{11} - 27635439951 \nu^{10} + \cdots - 55\!\cdots\!34 ) / 69\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 816519 \nu^{13} + 158560859 \nu^{12} - 2035242449 \nu^{11} + 1703151875 \nu^{10} + \cdots + 25\!\cdots\!58 ) / 251365108482048 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11305833 \nu^{13} + 161988589 \nu^{12} - 2007047455 \nu^{11} + 12731384821 \nu^{10} + \cdots + 14\!\cdots\!94 ) / 10\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1279597987 \nu^{13} + 17493542415 \nu^{12} - 35603446325 \nu^{11} + \cdots - 81\!\cdots\!10 ) / 10\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 8437121 \nu^{13} + 130735397 \nu^{12} - 941992135 \nu^{11} + 11109093229 \nu^{10} + \cdots - 17\!\cdots\!46 ) / 670306955952128 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3419596847 \nu^{13} + 61874225739 \nu^{12} - 498215726857 \nu^{11} + \cdots - 31\!\cdots\!42 ) / 20\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 4751865361 \nu^{13} - 43391939317 \nu^{12} - 25315357513 \nu^{11} + 1417644867651 \nu^{10} + \cdots + 25\!\cdots\!66 ) / 20\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 740127227 \nu^{13} - 10752905607 \nu^{12} + 30563864989 \nu^{11} - 16055626143 \nu^{10} + \cdots + 55\!\cdots\!74 ) / 26\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 20473695 \nu^{13} - 240903595 \nu^{12} - 350099687 \nu^{11} + 14650751165 \nu^{10} + \cdots + 13\!\cdots\!54 ) / 251365108482048 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 19121720115 \nu^{13} - 251757155807 \nu^{12} + 846199751973 \nu^{11} + \cdots + 91\!\cdots\!98 ) / 20\!\cdots\!36 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 22 \beta_{12} + 36 \beta_{11} + 162 \beta_{10} + 54 \beta_{9} - 25 \beta_{8} + 108 \beta_{7} + \cdots + 142168 ) / 331776 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 486 \beta_{13} + 226 \beta_{12} - 729 \beta_{11} - 648 \beta_{10} + 297 \beta_{9} - 104 \beta_{8} + \cdots + 3317912 ) / 331776 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2106 \beta_{13} - 818 \beta_{12} - 9351 \beta_{11} + 7776 \beta_{10} - 1809 \beta_{9} + \cdots + 5973128 ) / 331776 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 1890 \beta_{13} + 4098 \beta_{12} - 38907 \beta_{11} + 756 \beta_{10} - 1017 \beta_{9} + \cdots + 86990616 ) / 110592 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 130086 \beta_{13} + 4398 \beta_{12} - 721521 \beta_{11} + 151308 \beta_{10} + 190917 \beta_{9} + \cdots + 3778788360 ) / 331776 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 1123794 \beta_{13} + 1021166 \beta_{12} - 2624157 \beta_{11} + 6198444 \beta_{10} + \cdots - 8664130520 ) / 331776 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 7770978 \beta_{13} + 10438734 \beta_{12} - 23270517 \beta_{11} + 5381316 \beta_{10} + \cdots - 376119258744 ) / 331776 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 78894054 \beta_{13} + 17642522 \beta_{12} - 78505263 \beta_{11} + 30115908 \beta_{10} + \cdots - 376262333384 ) / 110592 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 3576992562 \beta_{13} + 50300414 \beta_{12} - 1306825533 \beta_{11} + 623936196 \beta_{10} + \cdots - 5715053251640 ) / 331776 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 30071877426 \beta_{13} + 1476526366 \beta_{12} - 1959497325 \beta_{11} + 2435888700 \beta_{10} + \cdots + 47248584485480 ) / 331776 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 224775131058 \beta_{13} - 26349444818 \beta_{12} + 79149279267 \beta_{11} + \cdots - 265746483735544 ) / 331776 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 182325233538 \beta_{13} - 22389285474 \beta_{12} + 751501611 \beta_{11} + 57625472268 \beta_{10} + \cdots - 270710232693656 ) / 36864 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 10051548981138 \beta_{13} - 4614444617058 \beta_{12} - 10601073350829 \beta_{11} + \cdots + 14\!\cdots\!08 ) / 331776 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/288\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1
−5.80663 + 4.20354i
−6.99438 − 0.299706i
8.85262 − 1.52851i
2.71713 − 7.81354i
−4.99225 − 5.30027i
1.24645 − 7.99620i
7.97707 + 3.91414i
7.97707 − 3.91414i
1.24645 + 7.99620i
−4.99225 + 5.30027i
2.71713 + 7.81354i
8.85262 + 1.52851i
−6.99438 + 0.299706i
−5.80663 − 4.20354i
0 0 0 − 468.400i 0 −81.2421 0 0 0
145.2 0 0 0 − 455.347i 0 743.502 0 0 0
145.3 0 0 0 − 425.308i 0 −1664.03 0 0 0
145.4 0 0 0 − 137.155i 0 −808.153 0 0 0
145.5 0 0 0 − 124.215i 0 −646.373 0 0 0
145.6 0 0 0 − 76.0929i 0 222.735 0 0 0
145.7 0 0 0 − 23.0228i 0 1547.56 0 0 0
145.8 0 0 0 23.0228i 0 1547.56 0 0 0
145.9 0 0 0 76.0929i 0 222.735 0 0 0
145.10 0 0 0 124.215i 0 −646.373 0 0 0
145.11 0 0 0 137.155i 0 −808.153 0 0 0
145.12 0 0 0 425.308i 0 −1664.03 0 0 0
145.13 0 0 0 455.347i 0 743.502 0 0 0
145.14 0 0 0 468.400i 0 −81.2421 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 145.14
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.8.d.d 14
3.b odd 2 1 96.8.d.a 14
4.b odd 2 1 72.8.d.d 14
8.b even 2 1 inner 288.8.d.d 14
8.d odd 2 1 72.8.d.d 14
12.b even 2 1 24.8.d.a ✓ 14
24.f even 2 1 24.8.d.a ✓ 14
24.h odd 2 1 96.8.d.a 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.8.d.a ✓ 14 12.b even 2 1
24.8.d.a ✓ 14 24.f even 2 1
72.8.d.d 14 4.b odd 2 1
72.8.d.d 14 8.d odd 2 1
96.8.d.a 14 3.b odd 2 1
96.8.d.a 14 24.h odd 2 1
288.8.d.d 14 1.a even 1 1 trivial
288.8.d.d 14 8.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{14} + 648188 T_{5}^{12} + 147837846096 T_{5}^{10} + \cdots + 73\!\cdots\!00 \) acting on \(S_{8}^{\mathrm{new}}(288, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 73\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{7} + \cdots - 18\!\cdots\!56)^{2} \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 92\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 82\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( (T^{7} + \cdots - 20\!\cdots\!36)^{2} \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 23\!\cdots\!76 \) Copy content Toggle raw display
$23$ \( (T^{7} + \cdots - 23\!\cdots\!68)^{2} \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 30\!\cdots\!76 \) Copy content Toggle raw display
$31$ \( (T^{7} + \cdots + 33\!\cdots\!12)^{2} \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 88\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( (T^{7} + \cdots - 38\!\cdots\!28)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 14\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( (T^{7} + \cdots + 43\!\cdots\!56)^{2} \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 98\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 33\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 54\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 45\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{7} + \cdots - 12\!\cdots\!68)^{2} \) Copy content Toggle raw display
$73$ \( (T^{7} + \cdots - 49\!\cdots\!48)^{2} \) Copy content Toggle raw display
$79$ \( (T^{7} + \cdots - 64\!\cdots\!80)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 31\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( (T^{7} + \cdots + 78\!\cdots\!80)^{2} \) Copy content Toggle raw display
$97$ \( (T^{7} + \cdots - 18\!\cdots\!04)^{2} \) Copy content Toggle raw display
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