Properties

Label 96.9.e.a
Level $96$
Weight $9$
Character orbit 96.e
Analytic conductor $39.108$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [96,9,Mod(65,96)]
 
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([0, 0, 1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("96.65");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 96 = 2^{5} \cdot 3 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 96.e (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: \(39.1083465659\)
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: \( x^{16} + 76 x^{14} - 248 x^{13} + 4938 x^{12} - 55200 x^{11} + 274396 x^{10} + 6509208 x^{9} - 68126957 x^{8} - 386972864 x^{7} - 10265857452 x^{6} + \cdots + 889067248974921 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{110}\cdot 3^{30} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + \beta_{3} q^{5} + ( - 3 \beta_{2} + \beta_1) q^{7} + (\beta_{6} + \beta_{4} - \beta_{3} - 435) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + \beta_{3} q^{5} + ( - 3 \beta_{2} + \beta_1) q^{7} + (\beta_{6} + \beta_{4} - \beta_{3} - 435) q^{9} + ( - \beta_{11} - 23 \beta_{2}) q^{11} + (\beta_{5} - 6366) q^{13} + ( - \beta_{11} + \beta_{9} - \beta_{7} - \beta_{2} + 4 \beta_1) q^{15} + ( - \beta_{12} - 3 \beta_{6} - 2 \beta_{4} - 18 \beta_{3}) q^{17} + (\beta_{13} - 2 \beta_{9} + 2 \beta_{7} - 143 \beta_{2} - \beta_1) q^{19} + (\beta_{14} - \beta_{12} + \beta_{10} - 3 \beta_{6} + 3 \beta_{5} + 7 \beta_{4} - 44 \beta_{3} + \cdots - 17478) q^{21}+ \cdots + (69 \beta_{15} - 768 \beta_{13} + 4491 \beta_{11} + 1071 \beta_{9} + \cdots + 14355 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 6960 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 6960 q^{9} - 101856 q^{13} - 279648 q^{21} - 1773424 q^{25} + 2428992 q^{33} + 4039456 q^{37} + 10154112 q^{45} + 2562992 q^{49} - 15023520 q^{57} - 28444384 q^{61} - 47615616 q^{69} - 12069600 q^{73} + 13926672 q^{81} + 136312832 q^{85} - 24251616 q^{93} + 141063840 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 76 x^{14} - 248 x^{13} + 4938 x^{12} - 55200 x^{11} + 274396 x^{10} + 6509208 x^{9} - 68126957 x^{8} - 386972864 x^{7} - 10265857452 x^{6} + \cdots + 889067248974921 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 26\!\cdots\!87 \nu^{15} + \cdots + 69\!\cdots\!34 ) / 93\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 19\!\cdots\!27 \nu^{15} + \cdots + 44\!\cdots\!54 ) / 28\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 27\!\cdots\!49 \nu^{15} + \cdots - 33\!\cdots\!83 ) / 17\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 78\!\cdots\!84 \nu^{15} + \cdots + 10\!\cdots\!48 ) / 42\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 10\!\cdots\!80 \nu^{15} + \cdots + 38\!\cdots\!60 ) / 33\!\cdots\!95 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 15\!\cdots\!62 \nu^{15} + \cdots + 16\!\cdots\!98 ) / 35\!\cdots\!15 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 79\!\cdots\!56 \nu^{15} + \cdots + 16\!\cdots\!22 ) / 14\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 18\!\cdots\!67 \nu^{15} + \cdots + 18\!\cdots\!96 ) / 14\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 22\!\cdots\!67 \nu^{15} + \cdots - 50\!\cdots\!04 ) / 14\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 42\!\cdots\!24 \nu^{15} + \cdots + 12\!\cdots\!88 ) / 17\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 67\!\cdots\!83 \nu^{15} + \cdots + 13\!\cdots\!26 ) / 25\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 16\!\cdots\!72 \nu^{15} + \cdots - 25\!\cdots\!44 ) / 58\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 90\!\cdots\!15 \nu^{15} + \cdots - 21\!\cdots\!82 ) / 28\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 73\!\cdots\!67 \nu^{15} + \cdots + 10\!\cdots\!09 ) / 17\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 86\!\cdots\!49 \nu^{15} + \cdots + 12\!\cdots\!66 ) / 18\!\cdots\!70 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( - 9 \beta_{15} - 90 \beta_{13} - 54 \beta_{9} + 9 \beta_{8} + 101 \beta_{7} + 486 \beta_{4} - 66824 \beta_{2} - 351 \beta_1 ) / 995328 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 153 \beta_{15} + 2736 \beta_{14} - 81 \beta_{13} + 144 \beta_{12} - 729 \beta_{11} + 396 \beta_{10} + 153 \beta_{9} + 637 \beta_{7} + 792 \beta_{6} + 1512 \beta_{5} - 28314 \beta_{4} + 31536 \beta_{3} + \cdots - 18911232 ) / 1990656 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2160 \beta_{15} - 270 \beta_{14} + 3816 \beta_{13} - 594 \beta_{12} + 1134 \beta_{11} + 270 \beta_{10} - 3906 \beta_{9} + 2142 \beta_{8} - 40058 \beta_{7} - 594 \beta_{6} + 2268 \beta_{5} + \cdots + 23141376 ) / 497664 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 21330 \beta_{15} - 936 \beta_{14} - 11286 \beta_{13} - 50616 \beta_{12} - 76950 \beta_{11} - 99288 \beta_{10} + 18630 \beta_{9} + 27288 \beta_{8} + 420454 \beta_{7} - 1318968 \beta_{6} + \cdots - 1020211200 ) / 1990656 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 812178 \beta_{15} + 680400 \beta_{14} + 1090962 \beta_{13} + 1499634 \beta_{11} - 254070 \beta_{10} - 796266 \beta_{9} - 1290492 \beta_{8} + 3829138 \beta_{7} + \cdots + 22613852160 ) / 1990656 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 793647 \beta_{15} - 6635448 \beta_{14} - 3514887 \beta_{13} + 9238824 \beta_{12} - 2578635 \beta_{11} + 1819404 \beta_{10} - 4613769 \beta_{9} - 1150074 \beta_{8} + \cdots - 5479778304 ) / 995328 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 73889505 \beta_{15} + 45426528 \beta_{14} - 44071137 \beta_{13} - 5704776 \beta_{12} + 51871023 \beta_{11} + 3545451 \beta_{10} + 115831809 \beta_{9} + \cdots - 4570800482304 ) / 995328 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 246661560 \beta_{15} - 1539749592 \beta_{14} + 2614167576 \beta_{13} - 1774782216 \beta_{12} + 443257272 \beta_{11} + 964290384 \beta_{10} + \cdots + 89586414637056 ) / 1990656 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 22216369248 \beta_{15} - 27929075184 \beta_{14} - 6845330034 \beta_{13} + 2126068128 \beta_{12} - 9320307606 \beta_{11} - 5988577266 \beta_{10} + \cdots + 10\!\cdots\!32 ) / 1990656 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 142934084685 \beta_{15} + 415500861744 \beta_{14} - 268502929281 \beta_{13} + 110815825392 \beta_{12} + 49758863679 \beta_{11} + \cdots + 33\!\cdots\!28 ) / 1990656 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 83358783360 \beta_{15} + 530284527036 \beta_{14} - 987680490894 \beta_{13} + 363716943084 \beta_{12} - 170355317022 \beta_{11} + \cdots + 34\!\cdots\!96 ) / 995328 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 8788731780303 \beta_{15} + 3701283220080 \beta_{14} - 2837249540577 \beta_{13} - 4841665301520 \beta_{12} - 5548766800473 \beta_{11} + \cdots - 18\!\cdots\!96 ) / 995328 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 56140995962808 \beta_{15} + 109002144203784 \beta_{14} + 486889368351021 \beta_{13} - 145622693602152 \beta_{12} + 104270270242059 \beta_{11} + \cdots + 12\!\cdots\!04 ) / 995328 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 20\!\cdots\!87 \beta_{15} + \cdots + 50\!\cdots\!56 ) / 1990656 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 80\!\cdots\!71 \beta_{15} + \cdots - 58\!\cdots\!64 ) / 497664 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
2.48948 + 11.6660i
2.48948 11.6660i
7.66393 + 7.41107i
7.66393 7.41107i
5.52055 + 1.32008i
5.52055 1.32008i
−7.58882 + 2.32651i
−7.58882 2.32651i
7.58882 + 0.912295i
7.58882 0.912295i
−5.52055 0.0941314i
−5.52055 + 0.0941314i
−7.66393 + 8.82528i
−7.66393 8.82528i
−2.48948 + 10.2518i
−2.48948 10.2518i
0 −75.2171 30.0563i 0 517.842i 0 3763.79 0 4754.24 + 4521.50i 0
65.2 0 −75.2171 + 30.0563i 0 517.842i 0 3763.79 0 4754.24 4521.50i 0
65.3 0 −70.4557 39.9624i 0 249.117i 0 −2035.26 0 3367.02 + 5631.16i 0
65.4 0 −70.4557 + 39.9624i 0 249.117i 0 −2035.26 0 3367.02 5631.16i 0
65.5 0 −40.3363 70.2423i 0 1140.75i 0 −1656.42 0 −3306.96 + 5666.63i 0
65.6 0 −40.3363 + 70.2423i 0 1140.75i 0 −1656.42 0 −3306.96 5666.63i 0
65.7 0 −1.83135 80.9793i 0 611.814i 0 −1627.21 0 −6554.29 + 296.603i 0
65.8 0 −1.83135 + 80.9793i 0 611.814i 0 −1627.21 0 −6554.29 296.603i 0
65.9 0 1.83135 80.9793i 0 611.814i 0 1627.21 0 −6554.29 296.603i 0
65.10 0 1.83135 + 80.9793i 0 611.814i 0 1627.21 0 −6554.29 + 296.603i 0
65.11 0 40.3363 70.2423i 0 1140.75i 0 1656.42 0 −3306.96 5666.63i 0
65.12 0 40.3363 + 70.2423i 0 1140.75i 0 1656.42 0 −3306.96 + 5666.63i 0
65.13 0 70.4557 39.9624i 0 249.117i 0 2035.26 0 3367.02 5631.16i 0
65.14 0 70.4557 + 39.9624i 0 249.117i 0 2035.26 0 3367.02 + 5631.16i 0
65.15 0 75.2171 30.0563i 0 517.842i 0 −3763.79 0 4754.24 4521.50i 0
65.16 0 75.2171 + 30.0563i 0 517.842i 0 −3763.79 0 4754.24 + 4521.50i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 96.9.e.a 16
3.b odd 2 1 inner 96.9.e.a 16
4.b odd 2 1 inner 96.9.e.a 16
8.b even 2 1 192.9.e.k 16
8.d odd 2 1 192.9.e.k 16
12.b even 2 1 inner 96.9.e.a 16
24.f even 2 1 192.9.e.k 16
24.h odd 2 1 192.9.e.k 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.9.e.a 16 1.a even 1 1 trivial
96.9.e.a 16 3.b odd 2 1 inner
96.9.e.a 16 4.b odd 2 1 inner
96.9.e.a 16 12.b even 2 1 inner
192.9.e.k 16 8.b even 2 1
192.9.e.k 16 8.d odd 2 1
192.9.e.k 16 24.f even 2 1
192.9.e.k 16 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 2005856T_{5}^{6} + 1057075569024T_{5}^{4} + 188737602250496000T_{5}^{2} + 8106353661468797440000 \) acting on \(S_{9}^{\mathrm{new}}(96, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + 3480 T^{14} + \cdots + 34\!\cdots\!81 \) Copy content Toggle raw display
$5$ \( (T^{8} + 2005856 T^{6} + \cdots + 81\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} - 23699952 T^{6} + \cdots + 42\!\cdots\!04)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 1131638688 T^{6} + \cdots + 49\!\cdots\!84)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 25464 T^{3} + \cdots + 36\!\cdots\!24)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + 45752913920 T^{6} + \cdots + 14\!\cdots\!76)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 39113216496 T^{6} + \cdots + 39\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 459862987392 T^{6} + \cdots + 30\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 954059446880 T^{6} + \cdots + 46\!\cdots\!76)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 1936810920816 T^{6} + \cdots + 38\!\cdots\!84)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 1009864 T^{3} + \cdots + 42\!\cdots\!76)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + 32719860838784 T^{6} + \cdots + 34\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 52596395323632 T^{6} + \cdots + 73\!\cdots\!84)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 106113688975872 T^{6} + \cdots + 95\!\cdots\!64)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 86855382530144 T^{6} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 923054756623776 T^{6} + \cdots + 14\!\cdots\!24)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 7111096 T^{3} + \cdots + 24\!\cdots\!04)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 94\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 37\!\cdots\!44)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 3017400 T^{3} + \cdots + 15\!\cdots\!00)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 32\!\cdots\!84)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 46\!\cdots\!16)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 35265960 T^{3} + \cdots + 26\!\cdots\!00)^{4} \) Copy content Toggle raw display
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