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Properties

Label 768.5.b.i
Level $768$
Weight $5$
Character orbit 768.b
Analytic conductor $79.388$
Analytic rank $0$
Dimension $16$
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] = [768,5,Mod(127,768)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(768, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("768.127");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 768.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: \(79.3881316484\)
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: \( x^{16} - 8 x^{15} + 104 x^{14} - 588 x^{13} + 3846 x^{12} - 15796 x^{11} + 66992 x^{10} - 203224 x^{9} + \cdots + 2196513 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{70}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 384)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + ( - \beta_{10} + \beta_{9}) q^{5} + (\beta_{14} + \beta_{10} + \beta_{9}) q^{7} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + ( - \beta_{10} + \beta_{9}) q^{5} + (\beta_{14} + \beta_{10} + \beta_{9}) q^{7} + 27 q^{9} + ( - \beta_{3} + \beta_{2} + 36) q^{11} + (3 \beta_{15} + 2 \beta_{14} + \cdots + \beta_{8}) q^{13}+ \cdots + ( - 27 \beta_{3} + 27 \beta_{2} + 972) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 432 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 432 q^{9} + 576 q^{11} - 480 q^{17} + 192 q^{19} - 2672 q^{25} + 9024 q^{35} - 1440 q^{41} + 12224 q^{43} - 2480 q^{49} + 6336 q^{51} + 7488 q^{57} + 15360 q^{59} - 1344 q^{65} + 12288 q^{67} + 8480 q^{73} + 21888 q^{75} + 11664 q^{81} + 13248 q^{83} - 18720 q^{89} - 30272 q^{91} + 13088 q^{97} + 15552 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 104 x^{14} - 588 x^{13} + 3846 x^{12} - 15796 x^{11} + 66992 x^{10} - 203224 x^{9} + \cdots + 2196513 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 1889914 \nu^{14} + 13229398 \nu^{13} - 174465892 \nu^{12} + 874813178 \nu^{11} + \cdots - 957752553078 ) / 17163657725 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 40995992134 \nu^{14} + 286971944938 \nu^{13} - 4357532588292 \nu^{12} + \cdots - 16\!\cdots\!78 ) / 34474922906435 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1335934914602 \nu^{14} + 9351544402214 \nu^{13} - 129208024044656 \nu^{12} + \cdots - 16\!\cdots\!54 ) / 517123843596525 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 938407424 \nu^{14} - 6568851968 \nu^{13} + 89672932672 \nu^{12} - 452642520448 \nu^{11} + \cdots + 11\!\cdots\!48 ) / 136408294275 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1124881716374 \nu^{14} + 7874172014618 \nu^{13} - 104988564599912 \nu^{12} + \cdots - 80\!\cdots\!38 ) / 103424768719305 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 8645389659576 \nu^{14} - 60517727617032 \nu^{13} + 816905103453628 \nu^{12} + \cdots + 90\!\cdots\!52 ) / 517123843596525 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 9697062176674 \nu^{14} + 67879435236718 \nu^{13} - 916023978250472 \nu^{12} + \cdots - 10\!\cdots\!98 ) / 517123843596525 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 485392506323600 \nu^{15} + \cdots + 89\!\cdots\!72 ) / 12\!\cdots\!85 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 9336597664 \nu^{15} + 70024482480 \nu^{14} - 914871856016 \nu^{13} + \cdots + 46\!\cdots\!64 ) / 11637724052625 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 37\!\cdots\!12 \nu^{15} + \cdots + 23\!\cdots\!62 ) / 30\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 76\!\cdots\!68 \nu^{15} + \cdots + 29\!\cdots\!18 ) / 61\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 37\!\cdots\!32 \nu^{15} + \cdots - 20\!\cdots\!82 ) / 27\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 86941390592 \nu^{15} + 652060429440 \nu^{14} - 8385244333248 \nu^{13} + \cdots + 24\!\cdots\!92 ) / 51216760583875 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 26\!\cdots\!56 \nu^{15} + \cdots - 18\!\cdots\!62 ) / 12\!\cdots\!85 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 18996586496 \nu^{15} + 142474398720 \nu^{14} - 1876728815424 \nu^{13} + \cdots + 10\!\cdots\!96 ) / 5691981010375 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 4 \beta_{15} + 21 \beta_{14} - 18 \beta_{13} - \beta_{12} + 13 \beta_{11} + 23 \beta_{10} + 21 \beta_{9} + \cdots + 192 ) / 384 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 8 \beta_{15} + 42 \beta_{14} - 36 \beta_{13} - 2 \beta_{12} + 26 \beta_{11} + 46 \beta_{10} + \cdots - 6912 ) / 768 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 128 \beta_{15} - 314 \beta_{14} + 528 \beta_{13} - 46 \beta_{12} - 266 \beta_{11} - 606 \beta_{10} + \cdots - 7040 ) / 512 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 188 \beta_{15} - 492 \beta_{14} + 810 \beta_{13} - 68 \beta_{12} - 412 \beta_{11} - 932 \beta_{10} + \cdots + 60864 ) / 384 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 20964 \beta_{15} + 14154 \beta_{14} - 40248 \beta_{13} + 4350 \beta_{12} + 20490 \beta_{11} + \cdots + 643968 ) / 1536 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 5397 \beta_{15} + 3952 \beta_{14} - 10740 \beta_{13} + 1144 \beta_{12} + 5468 \beta_{11} + \cdots - 489600 ) / 128 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 685750 \beta_{15} - 279078 \beta_{14} + 1085634 \beta_{13} - 125378 \beta_{12} - 583102 \beta_{11} + \cdots - 22841856 ) / 1536 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 190436 \beta_{15} - 83892 \beta_{14} + 309474 \beta_{13} - 35388 \beta_{12} - 165156 \beta_{11} + \cdots + 9663984 ) / 96 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 6480748 \beta_{15} + 2164118 \beta_{14} - 9678064 \beta_{13} + 1153314 \beta_{12} + \cdots + 278535040 ) / 512 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 60260312 \beta_{15} + 21435372 \beta_{14} - 91613154 \beta_{13} + 10821860 \beta_{12} + \cdots - 2012107392 ) / 768 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 503020796 \beta_{15} - 157006686 \beta_{14} + 737991852 \beta_{13} - 89205306 \beta_{12} + \cdots - 30051355776 ) / 1536 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 366296990 \beta_{15} - 119736834 \beta_{14} + 544002568 \beta_{13} - 65246806 \beta_{12} + \cdots + 8166755264 ) / 128 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 11644562742 \beta_{15} + 3562544130 \beta_{14} - 16998213522 \beta_{13} + 2066971734 \beta_{12} + \cdots + 1048551130368 ) / 1536 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 76211880590 \beta_{15} + 24130618644 \beta_{14} - 112238399838 \beta_{13} + 13553889100 \beta_{12} + \cdots - 1040446914048 ) / 768 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 72803022146 \beta_{15} - 22114869172 \beta_{14} + 106083886282 \beta_{13} - 12937792396 \beta_{12} + \cdots - 11809436661120 ) / 512 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/768\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(511\) \(517\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
0.500000 + 3.21129i
0.500000 − 2.27039i
0.500000 − 2.68460i
0.500000 + 5.62551i
0.500000 − 5.62551i
0.500000 + 2.68460i
0.500000 + 2.27039i
0.500000 − 3.21129i
0.500000 + 1.33456i
0.500000 + 1.74877i
0.500000 + 1.23482i
0.500000 − 3.64904i
0.500000 + 3.64904i
0.500000 − 1.23482i
0.500000 − 1.74877i
0.500000 − 1.33456i
0 −5.19615 0 − 42.6459i 0 78.8689i 0 27.0000 0
127.2 0 −5.19615 0 − 38.1647i 0 − 50.0160i 0 27.0000 0
127.3 0 −5.19615 0 − 24.8934i 0 52.9149i 0 27.0000 0
127.4 0 −5.19615 0 − 18.0608i 0 15.9380i 0 27.0000 0
127.5 0 −5.19615 0 18.0608i 0 − 15.9380i 0 27.0000 0
127.6 0 −5.19615 0 24.8934i 0 − 52.9149i 0 27.0000 0
127.7 0 −5.19615 0 38.1647i 0 50.0160i 0 27.0000 0
127.8 0 −5.19615 0 42.6459i 0 − 78.8689i 0 27.0000 0
127.9 0 5.19615 0 − 32.2453i 0 22.9356i 0 27.0000 0
127.10 0 5.19615 0 − 31.6602i 0 17.9843i 0 27.0000 0
127.11 0 5.19615 0 − 8.12430i 0 42.5107i 0 27.0000 0
127.12 0 5.19615 0 − 2.60435i 0 − 77.5593i 0 27.0000 0
127.13 0 5.19615 0 2.60435i 0 77.5593i 0 27.0000 0
127.14 0 5.19615 0 8.12430i 0 − 42.5107i 0 27.0000 0
127.15 0 5.19615 0 31.6602i 0 − 17.9843i 0 27.0000 0
127.16 0 5.19615 0 32.2453i 0 − 22.9356i 0 27.0000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.16
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 768.5.b.i 16
4.b odd 2 1 768.5.b.h 16
8.b even 2 1 768.5.b.h 16
8.d odd 2 1 inner 768.5.b.i 16
16.e even 4 1 384.5.g.a ✓ 16
16.e even 4 1 384.5.g.b yes 16
16.f odd 4 1 384.5.g.a ✓ 16
16.f odd 4 1 384.5.g.b yes 16
48.i odd 4 1 1152.5.g.c 16
48.i odd 4 1 1152.5.g.f 16
48.k even 4 1 1152.5.g.c 16
48.k even 4 1 1152.5.g.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.5.g.a ✓ 16 16.e even 4 1
384.5.g.a ✓ 16 16.f odd 4 1
384.5.g.b yes 16 16.e even 4 1
384.5.g.b yes 16 16.f odd 4 1
768.5.b.h 16 4.b odd 2 1
768.5.b.h 16 8.b even 2 1
768.5.b.i 16 1.a even 1 1 trivial
768.5.b.i 16 8.d odd 2 1 inner
1152.5.g.c 16 48.i odd 4 1
1152.5.g.c 16 48.k even 4 1
1152.5.g.f 16 48.i odd 4 1
1152.5.g.f 16 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(768, [\chi])\):

\( T_{5}^{16} + 6336 T_{5}^{14} + 16067584 T_{5}^{12} + 20854726656 T_{5}^{10} + 14646386393088 T_{5}^{8} + \cdots + 24\!\cdots\!24 \) Copy content Toggle raw display
\( T_{11}^{8} - 288 T_{11}^{7} - 28096 T_{11}^{6} + 11564544 T_{11}^{5} + 17192448 T_{11}^{4} + \cdots + 38\!\cdots\!84 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{2} - 27)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 24\!\cdots\!24 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 20\!\cdots\!96 \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 38\!\cdots\!84)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 58\!\cdots\!04 \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots - 27\!\cdots\!24)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 53\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 28\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 20\!\cdots\!44 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 40\!\cdots\!44 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 35\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 24\!\cdots\!68)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots - 88\!\cdots\!04)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 52\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 79\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 99\!\cdots\!48)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 38\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 38\!\cdots\!52)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 16\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots - 33\!\cdots\!52)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots - 21\!\cdots\!48)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 37\!\cdots\!68)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 16\!\cdots\!24)^{2} \) Copy content Toggle raw display
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