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Properties

Label 96.8.f.c
Level $96$
Weight $8$
Character orbit 96.f
Analytic conductor $29.989$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [96,8,Mod(47,96)]
 
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, 1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("96.47");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 96 = 2^{5} \cdot 3 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 96.f (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: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 274 x^{18} + 45864 x^{16} - 5031360 x^{14} + 776389632 x^{12} - 102828146688 x^{10} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{100}\cdot 3^{20} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 6) q^{3} - \beta_{2} q^{5} + \beta_{6} q^{7} + ( - \beta_{12} - \beta_{10} + \cdots + 254) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 6) q^{3} - \beta_{2} q^{5} + \beta_{6} q^{7} + ( - \beta_{12} - \beta_{10} + \cdots + 254) q^{9}+ \cdots + (396 \beta_{13} + 2854 \beta_{12} + \cdots - 506927) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 120 q^{3} + 5076 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 120 q^{3} + 5076 q^{9} - 133744 q^{19} + 711596 q^{25} - 85320 q^{27} + 263640 q^{33} + 2292080 q^{43} + 2495228 q^{49} + 7417536 q^{51} - 5067120 q^{57} - 3654640 q^{67} + 19892680 q^{73} + 8612088 q^{75} - 6817932 q^{81} - 67826976 q^{91} + 5561800 q^{97} - 10152864 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 274 x^{18} + 45864 x^{16} - 5031360 x^{14} + 776389632 x^{12} - 102828146688 x^{10} + \cdots + 11\!\cdots\!24 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 95422545 \nu^{19} + 620241440 \nu^{18} + 15140695730 \nu^{17} - 87283107392 \nu^{16} + \cdots - 22\!\cdots\!72 ) / 64\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4315447 \nu^{19} - 1526664414 \nu^{17} + 147120293272 \nu^{15} - 16892295429184 \nu^{13} + \cdots - 74\!\cdots\!00 \nu ) / 55\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3849289 \nu^{19} + 1707402274 \nu^{17} - 237168431720 \nu^{15} + \cdots + 10\!\cdots\!24 \nu ) / 44\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 9145853 \nu^{19} - 304947402 \nu^{17} - 78679972216 \nu^{15} + 6878081814848 \nu^{13} + \cdots + 10\!\cdots\!44 \nu ) / 71\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 477112725 \nu^{19} + 2480965760 \nu^{18} - 75703478650 \nu^{17} - 349132429568 \nu^{16} + \cdots - 90\!\cdots\!88 ) / 32\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 36324251 \nu^{18} + 5638441958 \nu^{16} - 604247544120 \nu^{14} + 69006563638080 \nu^{12} + \cdots + 97\!\cdots\!96 ) / 11\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 45707085 \nu^{19} - 425960080 \nu^{18} - 11604132970 \nu^{17} + 198661832224 \nu^{16} + \cdots + 11\!\cdots\!84 ) / 24\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 364320555 \nu^{19} - 933237760 \nu^{18} + 95522253830 \nu^{17} - 3579600607232 \nu^{16} + \cdots - 22\!\cdots\!52 ) / 21\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 10974793 \nu^{19} + 2662861346 \nu^{17} - 554151274088 \nu^{15} + \cdots + 36\!\cdots\!80 \nu ) / 55\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1856342025 \nu^{19} - 12935686400 \nu^{18} + 407692327330 \nu^{17} + \cdots + 30\!\cdots\!00 ) / 64\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 93533229 \nu^{18} + 29602031658 \nu^{16} - 3225014713864 \nu^{14} + \cdots + 18\!\cdots\!48 ) / 11\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1132589283 \nu^{19} - 620241440 \nu^{18} - 126749399798 \nu^{17} + 87283107392 \nu^{16} + \cdots + 22\!\cdots\!72 ) / 21\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 95422545 \nu^{19} - 71615427520 \nu^{18} + 15140695730 \nu^{17} + 17655879459712 \nu^{16} + \cdots + 82\!\cdots\!12 ) / 32\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 410865295 \nu^{18} + 90664600846 \nu^{16} - 11408765166424 \nu^{14} + \cdots + 43\!\cdots\!64 ) / 55\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 14512057 \nu^{19} - 805585420 \nu^{18} + 5265563646 \nu^{17} + 146984890584 \nu^{16} + \cdots + 59\!\cdots\!76 ) / 10\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 9723524941 \nu^{19} - 133325963456 \nu^{18} + 2914570059882 \nu^{17} + \cdots + 10\!\cdots\!80 ) / 32\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 48454530631 \nu^{19} - 266651926912 \nu^{18} - 8551188260862 \nu^{17} + \cdots + 21\!\cdots\!60 ) / 64\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 390074035 \nu^{19} - 3867170582 \nu^{18} - 33829850730 \nu^{17} + 599532770700 \nu^{16} + \cdots + 15\!\cdots\!28 ) / 67\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 14251286725 \nu^{19} - 900968340608 \nu^{18} + 1271714202330 \nu^{17} + \cdots + 35\!\cdots\!52 ) / 64\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{9} + 2\beta_{5} - 2\beta_{4} - \beta_{2} - 16\beta_1 ) / 512 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{14} + 2 \beta_{13} + 2 \beta_{11} + 2 \beta_{8} + 2 \beta_{7} + 12 \beta_{6} + 18 \beta_{5} + \cdots + 7014 ) / 256 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 2 \beta_{19} + 2 \beta_{18} + 4 \beta_{16} + 2 \beta_{15} - 2 \beta_{14} + 32 \beta_{12} + \cdots - 176 \beta_1 ) / 128 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 8 \beta_{19} - 10 \beta_{17} - 24 \beta_{16} + 14 \beta_{15} - 15 \beta_{14} + 42 \beta_{13} + \cdots - 106546 ) / 64 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 53 \beta_{19} - 13 \beta_{18} - 52 \beta_{17} - 114 \beta_{16} - 73 \beta_{15} + 73 \beta_{14} + \cdots + 26480 \beta_1 ) / 32 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 316 \beta_{19} - 493 \beta_{17} - 828 \beta_{16} + 335 \beta_{15} - 30 \beta_{14} - 1206 \beta_{13} + \cdots - 3257482 ) / 16 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1847 \beta_{19} - 2447 \beta_{18} - 6196 \beta_{17} + 3402 \beta_{16} - 1547 \beta_{15} + \cdots + 8256 \beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 5948 \beta_{19} + 12349 \beta_{17} + 21724 \beta_{16} - 9375 \beta_{15} - 7057 \beta_{14} + \cdots - 1216820844 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 46565 \beta_{19} + 1395 \beta_{18} + 123700 \beta_{17} - 288770 \beta_{16} - 70545 \beta_{15} + \cdots - 49740384 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 133972 \beta_{19} + 590351 \beta_{17} + 1113716 \beta_{16} - 523365 \beta_{15} + 2926271 \beta_{14} + \cdots - 42289780032 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 1129599 \beta_{19} - 205879 \beta_{18} + 31201916 \beta_{17} - 34846694 \beta_{16} + \cdots + 8457060000 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 53135748 \beta_{19} - 35745587 \beta_{17} - 44923300 \beta_{16} + 9177713 \beta_{15} + \cdots - 936141427448 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 577723507 \beta_{19} - 570126773 \beta_{18} + 1239526644 \beta_{17} + 1637695790 \beta_{16} + \cdots + 407587333472 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 8616794676 \beta_{19} + 7177256183 \beta_{17} + 10046115028 \beta_{16} - 2868858845 \beta_{15} + \cdots + 175206440350376 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 12614996247 \beta_{19} - 35598733327 \beta_{18} + 79247106844 \beta_{17} - 70001493718 \beta_{16} + \cdots - 20306214448864 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 99634313912 \beta_{19} + 462369072106 \beta_{17} + 874920987256 \beta_{16} - 412551915150 \beta_{15} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 7898198883286 \beta_{19} + 4674078501126 \beta_{18} + 1381373369128 \beta_{17} + \cdots - 14\!\cdots\!48 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 31072608750256 \beta_{19} - 67354714135748 \beta_{17} - 119173123896368 \beta_{16} + \cdots - 24\!\cdots\!56 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 103489978047164 \beta_{19} - 217975696259484 \beta_{18} + \cdots - 23\!\cdots\!72 \beta_1 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
−3.79334 − 10.6588i
3.79334 − 10.6588i
−3.79334 + 10.6588i
3.79334 + 10.6588i
10.9006 − 3.02933i
−10.9006 − 3.02933i
10.9006 + 3.02933i
−10.9006 + 3.02933i
6.74480 + 9.08337i
−6.74480 + 9.08337i
6.74480 − 9.08337i
−6.74480 − 9.08337i
−10.4702 + 4.28654i
10.4702 + 4.28654i
−10.4702 − 4.28654i
10.4702 − 4.28654i
10.0085 − 5.27546i
−10.0085 − 5.27546i
10.0085 + 5.27546i
−10.0085 + 5.27546i
0 −46.6816 − 2.79715i 0 −395.205 0 1395.40i 0 2171.35 + 261.151i 0
47.2 0 −46.6816 − 2.79715i 0 395.205 0 − 1395.40i 0 2171.35 + 261.151i 0
47.3 0 −46.6816 + 2.79715i 0 −395.205 0 − 1395.40i 0 2171.35 − 261.151i 0
47.4 0 −46.6816 + 2.79715i 0 395.205 0 1395.40i 0 2171.35 − 261.151i 0
47.5 0 −35.1794 − 30.8125i 0 −65.0827 0 90.3177i 0 288.180 + 2167.93i 0
47.6 0 −35.1794 − 30.8125i 0 65.0827 0 − 90.3177i 0 288.180 + 2167.93i 0
47.7 0 −35.1794 + 30.8125i 0 −65.0827 0 − 90.3177i 0 288.180 − 2167.93i 0
47.8 0 −35.1794 + 30.8125i 0 65.0827 0 90.3177i 0 288.180 − 2167.93i 0
47.9 0 −15.4671 − 44.1335i 0 −320.900 0 37.9939i 0 −1708.54 + 1365.24i 0
47.10 0 −15.4671 − 44.1335i 0 320.900 0 − 37.9939i 0 −1708.54 + 1365.24i 0
47.11 0 −15.4671 + 44.1335i 0 −320.900 0 − 37.9939i 0 −1708.54 − 1365.24i 0
47.12 0 −15.4671 + 44.1335i 0 320.900 0 37.9939i 0 −1708.54 − 1365.24i 0
47.13 0 24.1914 − 40.0222i 0 −477.520 0 632.635i 0 −1016.56 − 1936.38i 0
47.14 0 24.1914 − 40.0222i 0 477.520 0 − 632.635i 0 −1016.56 − 1936.38i 0
47.15 0 24.1914 + 40.0222i 0 −477.520 0 − 632.635i 0 −1016.56 + 1936.38i 0
47.16 0 24.1914 + 40.0222i 0 477.520 0 632.635i 0 −1016.56 + 1936.38i 0
47.17 0 43.1368 − 18.0615i 0 −277.669 0 − 1066.27i 0 1534.56 − 1558.23i 0
47.18 0 43.1368 − 18.0615i 0 277.669 0 1066.27i 0 1534.56 − 1558.23i 0
47.19 0 43.1368 + 18.0615i 0 −277.669 0 1066.27i 0 1534.56 + 1558.23i 0
47.20 0 43.1368 + 18.0615i 0 277.669 0 − 1066.27i 0 1534.56 + 1558.23i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.d odd 2 1 inner
24.f 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.f.c 20
3.b odd 2 1 inner 96.8.f.c 20
4.b odd 2 1 24.8.f.c ✓ 20
8.b even 2 1 24.8.f.c ✓ 20
8.d odd 2 1 inner 96.8.f.c 20
12.b even 2 1 24.8.f.c ✓ 20
24.f even 2 1 inner 96.8.f.c 20
24.h odd 2 1 24.8.f.c ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.8.f.c ✓ 20 4.b odd 2 1
24.8.f.c ✓ 20 8.b even 2 1
24.8.f.c ✓ 20 12.b even 2 1
24.8.f.c ✓ 20 24.h odd 2 1
96.8.f.c 20 1.a even 1 1 trivial
96.8.f.c 20 3.b odd 2 1 inner
96.8.f.c 20 8.d odd 2 1 inner
96.8.f.c 20 24.f even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} - 568524 T_{5}^{8} + 115131638832 T_{5}^{6} + \cdots - 11\!\cdots\!00 \) acting on \(S_{8}^{\mathrm{new}}(96, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( (T^{10} + \cdots + 50\!\cdots\!07)^{2} \) Copy content Toggle raw display
$5$ \( (T^{10} + \cdots - 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( (T^{10} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} + \cdots + 93\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots + 48\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots + 11\!\cdots\!08)^{2} \) Copy content Toggle raw display
$19$ \( (T^{5} + \cdots + 45\!\cdots\!84)^{4} \) Copy content Toggle raw display
$23$ \( (T^{10} + \cdots - 13\!\cdots\!40)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots - 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 40\!\cdots\!80)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots + 80\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} + \cdots - 15\!\cdots\!20)^{4} \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots - 37\!\cdots\!40)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots - 74\!\cdots\!60)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 45\!\cdots\!20)^{2} \) Copy content Toggle raw display
$67$ \( (T^{5} + \cdots - 31\!\cdots\!60)^{4} \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots - 83\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots + 51\!\cdots\!40)^{4} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 70\!\cdots\!80)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots + 12\!\cdots\!32)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots + 23\!\cdots\!20)^{4} \) Copy content Toggle raw display
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