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Properties

Label 96.8.d.a
Level $96$
Weight $8$
Character orbit 96.d
Analytic conductor $29.989$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [96,8,Mod(49,96)]
 
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, 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("96.49");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 96 = 2^{5} \cdot 3 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 96.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: \(29.9889624465\)
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: \( 2^{84}\cdot 3^{16} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{3} - \beta_1) q^{5} + ( - \beta_{4} - 98) q^{7} - 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{3} - \beta_1) q^{5} + ( - \beta_{4} - 98) q^{7} - 729 q^{9} + (\beta_{7} + 4 \beta_{3} + \beta_1) q^{11} + ( - \beta_{10} - \beta_{7} + \cdots + 47 \beta_1) q^{13}+ \cdots + ( - 729 \beta_{7} - 2916 \beta_{3} - 729 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 1372 q^{7} - 10206 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 1372 q^{7} - 10206 q^{9} + 13500 q^{15} - 2908 q^{17} + 143416 q^{23} - 202626 q^{25} + 89468 q^{31} - 474552 q^{39} - 441284 q^{41} + 1056408 q^{47} + 2158134 q^{49} - 4757504 q^{55} + 1551096 q^{57} + 1000188 q^{63} - 2520464 q^{65} - 5172696 q^{71} - 5446196 q^{73} + 14373548 q^{79} + 7440174 q^{81} - 7902036 q^{87} - 11952620 q^{89} + 69327376 q^{95} + 133732 q^{97}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( ( - 45486279 \nu^{13} + 613276227 \nu^{12} - 1623882033 \nu^{11} - 9211813317 \nu^{10} + \cdots - 18\!\cdots\!78 ) / 27\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 115209 \nu^{13} - 1675699 \nu^{12} - 42332287 \nu^{11} + 1713107413 \nu^{10} + \cdots + 23\!\cdots\!30 ) / 121873991991296 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1052435365 \nu^{13} - 12540621657 \nu^{12} + 48847481027 \nu^{11} - 258144870849 \nu^{10} + \cdots + 67\!\cdots\!26 ) / 83\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1256255 \nu^{13} + 7174107 \nu^{12} + 30451335 \nu^{11} - 346825197 \nu^{10} + \cdots - 42\!\cdots\!26 ) / 670306955952128 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 13094433 \nu^{13} - 265555269 \nu^{12} + 1479231783 \nu^{11} - 4067655309 \nu^{10} + \cdots + 52\!\cdots\!26 ) / 13\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 10032627 \nu^{13} - 129048575 \nu^{12} - 932136475 \nu^{11} + 25573970665 \nu^{10} + \cdots + 10\!\cdots\!30 ) / 10\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 10919078081 \nu^{13} - 149147482725 \nu^{12} + 309185801095 \nu^{11} + \cdots + 67\!\cdots\!50 ) / 83\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 6975652531 \nu^{13} + 125588280159 \nu^{12} - 1001303099813 \nu^{11} + \cdots - 64\!\cdots\!18 ) / 41\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 18291801 \nu^{13} + 364218397 \nu^{12} - 2812906927 \nu^{11} + 2169227941 \nu^{10} + \cdots - 51\!\cdots\!34 ) / 10\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 9913107233 \nu^{13} - 92303364677 \nu^{12} - 36015776729 \nu^{11} + 2918196055155 \nu^{10} + \cdots + 53\!\cdots\!34 ) / 41\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 23820530101 \nu^{13} + 345932808105 \nu^{12} - 982915325747 \nu^{11} + \cdots - 17\!\cdots\!02 ) / 83\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 174998109 \nu^{13} - 1084723825 \nu^{12} - 17562372149 \nu^{11} + 152996013671 \nu^{10} + \cdots + 14\!\cdots\!30 ) / 40\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 37834063719 \nu^{13} + 497994825571 \nu^{12} - 1677784565649 \nu^{11} + \cdots - 18\!\cdots\!94 ) / 41\!\cdots\!72 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 18 \beta_{12} - 12 \beta_{11} + 54 \beta_{10} + 15 \beta_{9} + 18 \beta_{8} - 36 \beta_{7} + \cdots + 47381 ) / 110592 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 162 \beta_{13} + 54 \beta_{12} + 243 \beta_{11} - 216 \beta_{10} + 132 \beta_{9} + \cdots + 1105936 ) / 110592 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 702 \beta_{13} - 198 \beta_{12} + 3117 \beta_{11} + 2592 \beta_{10} + 1872 \beta_{9} + \cdots + 1990876 ) / 110592 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 630 \beta_{13} + 918 \beta_{12} + 12969 \beta_{11} + 252 \beta_{10} - 394 \beta_{9} + \cdots + 28991902 ) / 36864 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 43362 \beta_{13} + 13914 \beta_{12} + 240507 \beta_{11} + 50436 \beta_{10} - 20118 \beta_{9} + \cdots + 1259597250 ) / 110592 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 374598 \beta_{13} + 284346 \beta_{12} + 874719 \beta_{11} + 2066148 \beta_{10} - 506166 \beta_{9} + \cdots - 2888309374 ) / 110592 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2590326 \beta_{13} + 5459130 \beta_{12} + 7756839 \beta_{11} + 1793772 \beta_{10} + \cdots - 125374570482 ) / 110592 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 26298018 \beta_{13} + 15037086 \beta_{12} + 26168421 \beta_{11} + 10038636 \beta_{10} + \cdots - 125437119722 ) / 36864 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1192330854 \beta_{13} + 112346442 \beta_{12} + 435608511 \beta_{11} + 207978732 \beta_{10} + \cdots - 1905326206498 ) / 110592 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 10023959142 \beta_{13} + 284804298 \beta_{12} + 653165775 \beta_{11} + 811962900 \beta_{10} + \cdots + 15751669307146 ) / 110592 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 74925043686 \beta_{13} - 18443800998 \beta_{12} - 26383093089 \beta_{11} + 38773564380 \beta_{10} + \cdots - 88550448271466 ) / 110592 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 182325233538 \beta_{13} - 29740798242 \beta_{12} - 751501611 \beta_{11} + 57625472268 \beta_{10} + \cdots - 270665828428634 ) / 36864 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 3350516327046 \beta_{13} - 1505544981462 \beta_{12} + 3533691116943 \beta_{11} + \cdots + 49\!\cdots\!02 ) / 110592 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−5.80663 + 4.20354i
2.71713 − 7.81354i
−4.99225 − 5.30027i
7.97707 − 3.91414i
1.24645 + 7.99620i
8.85262 + 1.52851i
−6.99438 + 0.299706i
−6.99438 − 0.299706i
8.85262 − 1.52851i
1.24645 − 7.99620i
7.97707 + 3.91414i
−4.99225 + 5.30027i
2.71713 + 7.81354i
−5.80663 − 4.20354i
0 − 27.0000i 0 − 468.400i 0 −81.2421 0 −729.000 0
49.2 0 − 27.0000i 0 − 137.155i 0 −808.153 0 −729.000 0
49.3 0 − 27.0000i 0 − 124.215i 0 −646.373 0 −729.000 0
49.4 0 − 27.0000i 0 23.0228i 0 1547.56 0 −729.000 0
49.5 0 − 27.0000i 0 76.0929i 0 222.735 0 −729.000 0
49.6 0 − 27.0000i 0 425.308i 0 −1664.03 0 −729.000 0
49.7 0 − 27.0000i 0 455.347i 0 743.502 0 −729.000 0
49.8 0 27.0000i 0 − 455.347i 0 743.502 0 −729.000 0
49.9 0 27.0000i 0 − 425.308i 0 −1664.03 0 −729.000 0
49.10 0 27.0000i 0 − 76.0929i 0 222.735 0 −729.000 0
49.11 0 27.0000i 0 − 23.0228i 0 1547.56 0 −729.000 0
49.12 0 27.0000i 0 124.215i 0 −646.373 0 −729.000 0
49.13 0 27.0000i 0 137.155i 0 −808.153 0 −729.000 0
49.14 0 27.0000i 0 468.400i 0 −81.2421 0 −729.000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.14
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 96.8.d.a 14
3.b odd 2 1 288.8.d.d 14
4.b odd 2 1 24.8.d.a ✓ 14
8.b even 2 1 inner 96.8.d.a 14
8.d odd 2 1 24.8.d.a ✓ 14
12.b even 2 1 72.8.d.d 14
24.f even 2 1 72.8.d.d 14
24.h odd 2 1 288.8.d.d 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.8.d.a ✓ 14 4.b odd 2 1
24.8.d.a ✓ 14 8.d odd 2 1
72.8.d.d 14 12.b even 2 1
72.8.d.d 14 24.f even 2 1
96.8.d.a 14 1.a even 1 1 trivial
96.8.d.a 14 8.b even 2 1 inner
288.8.d.d 14 3.b odd 2 1
288.8.d.d 14 24.h odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(96, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( (T^{2} + 729)^{7} \) 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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