Properties

Label 96.9.b.a
Level $96$
Weight $9$
Character orbit 96.b
Analytic conductor $39.108$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [96,9,Mod(79,96)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(96, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("96.79");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 96 = 2^{5} \cdot 3 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 96.b (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: \(39.1083465659\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} - 53 x^{14} - 3698 x^{13} + 106330 x^{12} - 1713296 x^{11} + 27269236 x^{10} - 2297763416 x^{9} + 65871759160 x^{8} + \cdots + 67\!\cdots\!08 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{112}\cdot 3^{23} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{2} q^{5} + \beta_{3} q^{7} + 2187 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{2} q^{5} + \beta_{3} q^{7} + 2187 q^{9} + ( - \beta_{6} + 2472) q^{11} + ( - \beta_{9} + 4 \beta_{3} - 8 \beta_{2}) q^{13} + (\beta_{5} + \beta_{3} - 2 \beta_{2}) q^{15} + ( - \beta_{11} + 2 \beta_{6} - 396 \beta_1 + 4830) q^{17} + (\beta_{14} + \beta_{12} + \beta_{6} - 45 \beta_1 - 10472) q^{19} + ( - \beta_{13} - \beta_{8} + \beta_{7} - 12 \beta_{3} - 6 \beta_{2}) q^{21} + ( - 2 \beta_{13} + \beta_{10} + 3 \beta_{9} + \beta_{8} - 2 \beta_{7} - 2 \beta_{5} + \cdots + 165 \beta_{2}) q^{23}+ \cdots + ( - 2187 \beta_{6} + 5406264) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 34992 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 34992 q^{9} + 39552 q^{11} + 77280 q^{17} - 167552 q^{19} - 1604144 q^{25} + 2415744 q^{35} - 2187360 q^{41} - 3525248 q^{43} - 7109552 q^{49} - 13862016 q^{51} - 1550016 q^{57} - 44938752 q^{59} - 52558464 q^{65} - 6892544 q^{67} + 12400160 q^{73} - 12918528 q^{75} + 76527504 q^{81} + 209328000 q^{83} - 152224800 q^{89} - 395802240 q^{91} - 38799136 q^{97} + 86500224 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} - 53 x^{14} - 3698 x^{13} + 106330 x^{12} - 1713296 x^{11} + 27269236 x^{10} - 2297763416 x^{9} + 65871759160 x^{8} + \cdots + 67\!\cdots\!08 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 33\!\cdots\!55 \nu^{15} + \cdots + 11\!\cdots\!04 ) / 25\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 14\!\cdots\!13 \nu^{15} + \cdots - 74\!\cdots\!72 ) / 64\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11\!\cdots\!99 \nu^{15} + \cdots - 13\!\cdots\!76 ) / 64\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 94\!\cdots\!05 \nu^{15} + \cdots + 14\!\cdots\!28 ) / 35\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 39\!\cdots\!11 \nu^{15} + \cdots + 23\!\cdots\!76 ) / 32\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 42\!\cdots\!81 \nu^{15} + \cdots + 33\!\cdots\!88 ) / 27\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 11\!\cdots\!05 \nu^{15} + \cdots - 13\!\cdots\!44 ) / 64\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 11\!\cdots\!83 \nu^{15} + \cdots + 28\!\cdots\!36 ) / 64\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 30\!\cdots\!49 \nu^{15} + \cdots - 20\!\cdots\!20 ) / 16\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 86\!\cdots\!65 \nu^{15} + \cdots + 62\!\cdots\!92 ) / 21\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 17\!\cdots\!35 \nu^{15} + \cdots + 13\!\cdots\!56 ) / 35\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 41\!\cdots\!12 \nu^{15} + \cdots - 20\!\cdots\!56 ) / 58\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 15\!\cdots\!73 \nu^{15} + \cdots + 12\!\cdots\!92 ) / 21\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 13\!\cdots\!87 \nu^{15} + \cdots + 11\!\cdots\!20 ) / 17\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 48\!\cdots\!75 \nu^{15} + \cdots - 35\!\cdots\!92 ) / 35\!\cdots\!04 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( - 20 \beta_{15} - 21 \beta_{14} - 14 \beta_{12} - 11 \beta_{11} + 18 \beta_{10} + 90 \beta_{9} - 18 \beta_{8} - 36 \beta_{7} - 17 \beta_{6} - 76 \beta_{5} + 32 \beta_{4} + 122 \beta_{3} - 2458 \beta_{2} + 6158 \beta _1 + 221184 ) / 442368 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 100 \beta_{15} - 681 \beta_{14} + 226 \beta_{13} + 227 \beta_{12} + 224 \beta_{11} + 450 \beta_{10} + 684 \beta_{9} + 478 \beta_{8} + 1628 \beta_{7} + 3371 \beta_{6} + 1618 \beta_{5} - 119 \beta_{4} + \cdots + 4700160 ) / 442368 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 676 \beta_{15} + 2343 \beta_{14} + 690 \beta_{13} + 371 \beta_{12} - 7840 \beta_{11} - 1278 \beta_{10} + 4284 \beta_{9} - 3234 \beta_{8} - 10788 \beta_{7} + 10427 \beta_{6} + 5058 \beta_{5} + \cdots + 118683648 ) / 147456 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 82556 \beta_{15} + 5073 \beta_{14} - 280910 \beta_{13} - 1505 \beta_{12} - 296090 \beta_{11} + 91710 \beta_{10} - 120600 \beta_{9} - 79454 \beta_{8} + 873956 \beta_{7} + \cdots - 7843792896 ) / 442368 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1478900 \beta_{15} - 8324115 \beta_{14} - 2604110 \beta_{13} + 1770791 \beta_{12} + 12634922 \beta_{11} - 3255066 \beta_{10} - 3424464 \beta_{9} - 5214470 \beta_{8} + \cdots + 186829092864 ) / 442368 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 19766180 \beta_{15} + 47058855 \beta_{14} - 3587442 \beta_{13} - 22138511 \beta_{12} - 66611486 \beta_{11} + 3998610 \beta_{10} - 17639064 \beta_{9} + \cdots - 749621753856 ) / 147456 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1753705492 \beta_{15} + 493260981 \beta_{14} + 188361082 \beta_{13} + 551122555 \beta_{12} + 3374690110 \beta_{11} + 2347334838 \beta_{10} + \cdots + 371766963935232 ) / 442368 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 27903888460 \beta_{15} - 44480115555 \beta_{14} + 38098673002 \beta_{13} - 5761001725 \beta_{12} - 3698114626 \beta_{11} - 3798491994 \beta_{10} + \cdots - 91\!\cdots\!28 ) / 442368 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 32573375924 \beta_{15} + 39722882205 \beta_{14} - 47798864838 \beta_{13} + 20072827211 \beta_{12} - 25105280650 \beta_{11} + 101146012326 \beta_{10} + \cdots + 477924820850688 ) / 49152 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 11666730974260 \beta_{15} + 29043577810365 \beta_{14} - 87659865014 \beta_{13} + 1732287546739 \beta_{12} - 13399210312466 \beta_{11} + \cdots + 21\!\cdots\!20 ) / 442368 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 403035787648876 \beta_{15} - 260186595326763 \beta_{14} - 536438883825110 \beta_{13} - 213667699060909 \beta_{12} + \cdots - 10\!\cdots\!44 ) / 442368 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 46\!\cdots\!36 \beta_{15} + \cdots + 68\!\cdots\!36 ) / 147456 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 26\!\cdots\!72 \beta_{15} + \cdots - 28\!\cdots\!16 ) / 442368 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 58\!\cdots\!92 \beta_{15} + \cdots + 61\!\cdots\!68 ) / 442368 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 22\!\cdots\!24 \beta_{15} + \cdots - 69\!\cdots\!12 ) / 147456 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/96\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(37\) \(65\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
−8.27881 21.1409i
18.5990 + 4.76321i
−21.2244 + 11.6505i
10.3061 17.5416i
10.3061 + 17.5416i
−21.2244 11.6505i
18.5990 4.76321i
−8.27881 + 21.1409i
6.36882 + 1.58881i
4.35613 4.60039i
−4.80096 1.91918i
−1.32591 + 5.48291i
−1.32591 5.48291i
−4.80096 + 1.91918i
4.35613 + 4.60039i
6.36882 1.58881i
0 −46.7654 0 1184.17i 0 2680.57i 0 2187.00 0
79.2 0 −46.7654 0 646.509i 0 3735.34i 0 2187.00 0
79.3 0 −46.7654 0 272.483i 0 799.092i 0 2187.00 0
79.4 0 −46.7654 0 1.90021i 0 3072.34i 0 2187.00 0
79.5 0 −46.7654 0 1.90021i 0 3072.34i 0 2187.00 0
79.6 0 −46.7654 0 272.483i 0 799.092i 0 2187.00 0
79.7 0 −46.7654 0 646.509i 0 3735.34i 0 2187.00 0
79.8 0 −46.7654 0 1184.17i 0 2680.57i 0 2187.00 0
79.9 0 46.7654 0 939.156i 0 1910.28i 0 2187.00 0
79.10 0 46.7654 0 788.857i 0 3473.61i 0 2187.00 0
79.11 0 46.7654 0 715.724i 0 1549.07i 0 2187.00 0
79.12 0 46.7654 0 126.597i 0 585.111i 0 2187.00 0
79.13 0 46.7654 0 126.597i 0 585.111i 0 2187.00 0
79.14 0 46.7654 0 715.724i 0 1549.07i 0 2187.00 0
79.15 0 46.7654 0 788.857i 0 3473.61i 0 2187.00 0
79.16 0 46.7654 0 939.156i 0 1910.28i 0 2187.00 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 96.9.b.a 16
3.b odd 2 1 288.9.b.c 16
4.b odd 2 1 24.9.b.a 16
8.b even 2 1 24.9.b.a 16
8.d odd 2 1 inner 96.9.b.a 16
12.b even 2 1 72.9.b.d 16
24.f even 2 1 288.9.b.c 16
24.h odd 2 1 72.9.b.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.9.b.a 16 4.b odd 2 1
24.9.b.a 16 8.b even 2 1
72.9.b.d 16 12.b even 2 1
72.9.b.d 16 24.h odd 2 1
96.9.b.a 16 1.a even 1 1 trivial
96.9.b.a 16 8.d odd 2 1 inner
288.9.b.c 16 3.b odd 2 1
288.9.b.c 16 24.f even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(96, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{2} - 2187)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} + 3927072 T^{14} + \cdots + 70\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{16} + 49673184 T^{14} + \cdots + 21\!\cdots\!28 \) Copy content Toggle raw display
$11$ \( (T^{8} - 19776 T^{7} + \cdots + 76\!\cdots\!28)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + 7059575808 T^{14} + \cdots + 52\!\cdots\!68 \) Copy content Toggle raw display
$17$ \( (T^{8} - 38640 T^{7} + \cdots - 48\!\cdots\!28)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 83776 T^{7} + \cdots + 72\!\cdots\!56)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + 527725518720 T^{14} + \cdots + 15\!\cdots\!28 \) Copy content Toggle raw display
$29$ \( T^{16} + 5214422746656 T^{14} + \cdots + 28\!\cdots\!32 \) Copy content Toggle raw display
$31$ \( T^{16} + 6032112849120 T^{14} + \cdots + 17\!\cdots\!88 \) Copy content Toggle raw display
$37$ \( T^{16} + 31066208159616 T^{14} + \cdots + 11\!\cdots\!72 \) Copy content Toggle raw display
$41$ \( (T^{8} + 1093680 T^{7} + \cdots - 14\!\cdots\!32)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 1762624 T^{7} + \cdots + 21\!\cdots\!16)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + 182235605915520 T^{14} + \cdots + 51\!\cdots\!72 \) Copy content Toggle raw display
$53$ \( T^{16} + 578138645574432 T^{14} + \cdots + 31\!\cdots\!72 \) Copy content Toggle raw display
$59$ \( (T^{8} + 22469376 T^{7} + \cdots + 13\!\cdots\!56)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 49\!\cdots\!08 \) Copy content Toggle raw display
$67$ \( (T^{8} + 3446272 T^{7} + \cdots + 12\!\cdots\!96)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 12\!\cdots\!28 \) Copy content Toggle raw display
$73$ \( (T^{8} - 6200080 T^{7} + \cdots - 58\!\cdots\!28)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 27\!\cdots\!08 \) Copy content Toggle raw display
$83$ \( (T^{8} - 104664000 T^{7} + \cdots - 36\!\cdots\!92)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 76112400 T^{7} + \cdots - 51\!\cdots\!08)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 19399568 T^{7} + \cdots + 21\!\cdots\!52)^{2} \) Copy content Toggle raw display
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