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Properties

Label 88.5.h.a
Level $88$
Weight $5$
Character orbit 88.h
Analytic conductor $9.097$
Analytic rank $0$
Dimension $12$
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] = [88,5,Mod(65,88)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(88, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("88.65");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 88 = 2^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 88.h (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: \(9.09655675138\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 378x^{10} + 49709x^{8} + 2770310x^{6} + 62444900x^{4} + 470757120x^{2} + 33918976 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{32}\cdot 5^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 1) q^{3} + \beta_{2} q^{5} + \beta_{5} q^{7} + (\beta_{3} - \beta_{2} + 2 \beta_1 + 42) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 1) q^{3} + \beta_{2} q^{5} + \beta_{5} q^{7} + (\beta_{3} - \beta_{2} + 2 \beta_1 + 42) q^{9} + ( - \beta_{8} - \beta_{2} + \beta_1 + 1) q^{11} + ( - \beta_{9} - \beta_{5}) q^{13} + ( - \beta_{4} - 6 \beta_1 + 38) q^{15} + (\beta_{10} - \beta_{9} + \cdots - 3 \beta_{5}) q^{17}+ \cdots + (19 \beta_{11} - 25 \beta_{10} + \cdots + 1372) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 8 q^{3} + 492 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 8 q^{3} + 492 q^{9} + 12 q^{11} + 480 q^{15} + 1608 q^{23} + 1484 q^{25} + 2048 q^{27} - 2136 q^{31} + 1568 q^{33} - 288 q^{37} - 8264 q^{45} - 6024 q^{47} - 9940 q^{49} + 3576 q^{53} - 8344 q^{55} - 7704 q^{59} + 10312 q^{67} - 616 q^{69} + 8904 q^{71} - 3544 q^{75} + 9120 q^{77} + 15340 q^{81} + 4224 q^{89} + 23904 q^{91} + 23480 q^{93} - 21280 q^{97} + 14420 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 378x^{10} + 49709x^{8} + 2770310x^{6} + 62444900x^{4} + 470757120x^{2} + 33918976 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 15750551 \nu^{10} + 5569914118 \nu^{8} + 651625074603 \nu^{6} + 28597444221338 \nu^{4} + \cdots - 17\!\cdots\!88 ) / 121905028169408 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 154890597 \nu^{10} - 54350671458 \nu^{8} - 6312454914113 \nu^{6} + \cdots - 88\!\cdots\!92 ) / 243810056338816 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 378659757 \nu^{10} - 136795780626 \nu^{8} - 16317655108521 \nu^{6} + \cdots + 13\!\cdots\!96 ) / 243810056338816 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 580404387 \nu^{10} - 231663190190 \nu^{8} - 31996106215335 \nu^{6} + \cdots - 88\!\cdots\!64 ) / 243810056338816 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 145344619097 \nu^{11} + 56395222028810 \nu^{9} + \cdots + 10\!\cdots\!64 \nu ) / 44\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 4040679 \nu^{11} - 1543185910 \nu^{9} - 205465973851 \nu^{7} - 11569646905162 \nu^{5} + \cdots - 18\!\cdots\!88 \nu ) / 5324641241920 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 399648362261 \nu^{11} + 53323137140 \nu^{10} - 151417808769570 \nu^{9} + \cdots - 11\!\cdots\!40 ) / 44\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 399648362261 \nu^{11} + 53323137140 \nu^{10} + 151417808769570 \nu^{9} + \cdots - 11\!\cdots\!40 ) / 44\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 22020280111 \nu^{11} + 8304633680070 \nu^{9} + \cdots + 11\!\cdots\!32 \nu ) / 17\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1599763168 \nu^{11} + 603579010440 \nu^{9} + 79219262607372 \nu^{7} + \cdots + 74\!\cdots\!56 \nu ) / 866668559641885 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 691710960609 \nu^{11} + 255270266974170 \nu^{9} + \cdots + 17\!\cdots\!68 \nu ) / 22\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{11} - 10\beta_{10} + 6\beta_{9} + \beta_{6} + 16\beta_{5} ) / 320 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 5\beta_{8} + 5\beta_{7} - \beta_{4} + 3\beta_{3} - 8\beta_{2} - 31\beta _1 - 1014 ) / 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -226\beta_{11} + 1210\beta_{10} - 1078\beta_{9} + 560\beta_{8} - 560\beta_{7} + 1007\beta_{6} - 1168\beta_{5} ) / 320 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -235\beta_{8} - 235\beta_{7} + 47\beta_{4} - 163\beta_{3} + 226\beta_{2} + 455\beta _1 + 28918 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 22558 \beta_{11} - 170350 \beta_{10} + 195754 \beta_{9} - 100120 \beta_{8} + 100120 \beta_{7} + \cdots + 55904 \beta_{5} ) / 320 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 157319 \beta_{8} + 157319 \beta_{7} - 33391 \beta_{4} + 130005 \beta_{3} - 129016 \beta_{2} + \cdots - 15714490 ) / 16 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 2297214 \beta_{11} + 25886830 \beta_{10} - 33643642 \beta_{9} + 15831400 \beta_{8} + \cdots + 3887008 \beta_{5} ) / 320 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 3245623 \beta_{8} - 3245623 \beta_{7} + 703480 \beta_{4} - 2992431 \beta_{3} + 2606395 \beta_{2} + \cdots + 290871054 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 239466442 \beta_{11} - 4096212490 \beta_{10} + 5646333486 \beta_{9} - 2483614200 \beta_{8} + \cdots - 1892098304 \beta_{5} ) / 320 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 4286103305 \beta_{8} + 4286103305 \beta_{7} - 931264257 \beta_{4} + 4214581723 \beta_{3} + \cdots - 361888014758 ) / 16 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 25625732746 \beta_{11} + 662655667290 \beta_{10} - 939658896478 \beta_{9} + \cdots + 432690746592 \beta_{5} ) / 320 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/88\mathbb{Z}\right)^\times\).

\(n\) \(23\) \(45\) \(57\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
− 0.269725i
0.269725i
− 3.97623i
3.97623i
12.8952i
− 12.8952i
4.97990i
− 4.97990i
− 8.53149i
8.53149i
− 9.91186i
9.91186i
0 −13.6992 0 −35.0360 0 − 60.9590i 0 106.667 0
65.2 0 −13.6992 0 −35.0360 0 60.9590i 0 106.667 0
65.3 0 −11.7450 0 23.4328 0 − 38.1914i 0 56.9448 0
65.4 0 −11.7450 0 23.4328 0 38.1914i 0 56.9448 0
65.5 0 −1.95596 0 6.72383 0 − 35.8576i 0 −77.1742 0
65.6 0 −1.95596 0 6.72383 0 35.8576i 0 −77.1742 0
65.7 0 4.96252 0 −24.3186 0 − 31.8948i 0 −56.3734 0
65.8 0 4.96252 0 −24.3186 0 31.8948i 0 −56.3734 0
65.9 0 9.44639 0 43.3437 0 − 67.1533i 0 8.23432 0
65.10 0 9.44639 0 43.3437 0 67.1533i 0 8.23432 0
65.11 0 16.9912 0 −14.1458 0 − 85.9582i 0 207.701 0
65.12 0 16.9912 0 −14.1458 0 85.9582i 0 207.701 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 88.5.h.a ✓ 12
3.b odd 2 1 792.5.j.a 12
4.b odd 2 1 176.5.h.f 12
8.b even 2 1 704.5.h.k 12
8.d odd 2 1 704.5.h.l 12
11.b odd 2 1 inner 88.5.h.a ✓ 12
33.d even 2 1 792.5.j.a 12
44.c even 2 1 176.5.h.f 12
88.b odd 2 1 704.5.h.k 12
88.g even 2 1 704.5.h.l 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.5.h.a ✓ 12 1.a even 1 1 trivial
88.5.h.a ✓ 12 11.b odd 2 1 inner
176.5.h.f 12 4.b odd 2 1
176.5.h.f 12 44.c even 2 1
704.5.h.k 12 8.b even 2 1
704.5.h.k 12 88.b odd 2 1
704.5.h.l 12 8.d odd 2 1
704.5.h.l 12 88.g even 2 1
792.5.j.a 12 3.b odd 2 1
792.5.j.a 12 33.d even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(88, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} - 4 T^{5} + \cdots - 250668)^{2} \) Copy content Toggle raw display
$5$ \( (T^{6} - 2246 T^{4} + \cdots - 82308988)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 23\!\cdots\!04 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 98\!\cdots\!41 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 14\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 54\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots + 794435299395572)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 21\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 51\!\cdots\!28)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots - 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 57\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 42\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots - 48\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 27\!\cdots\!36)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots + 11\!\cdots\!04)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 27\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 58\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 17\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 89\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 17\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 76\!\cdots\!32)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 17\!\cdots\!84)^{2} \) Copy content Toggle raw display
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