↑

Properties

Label 960.5.e.f
Level $960$
Weight $5$
Character orbit 960.e
Analytic conductor $99.235$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [960,5,Mod(511,960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(960, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("960.511");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 960.e (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: \(99.2351645605\)
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: \( x^{16} - 4 x^{15} - 9 x^{14} + 18 x^{13} + 263 x^{12} - 444 x^{11} - 1732 x^{10} - 832 x^{9} + \cdots + 16777216 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{64}\cdot 3^{4}\cdot 5^{4} \)
Twist minimal: no (minimal twist has level 60)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{2} q^{5} + (\beta_{4} + \beta_1) q^{7} - 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{2} q^{5} + (\beta_{4} + \beta_1) q^{7} - 27 q^{9} + ( - \beta_{6} - \beta_{3}) q^{11} + ( - \beta_{9} - \beta_{2} + 22) q^{13} + \beta_{5} q^{15} + ( - \beta_{15} - \beta_{12} + \cdots - 5 \beta_{2}) q^{17}+ \cdots + (27 \beta_{6} + 27 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 432 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 432 q^{9} + 352 q^{13} - 288 q^{21} + 2000 q^{25} + 3456 q^{29} - 9376 q^{37} + 1248 q^{41} - 3952 q^{49} + 5184 q^{53} - 11232 q^{57} + 3808 q^{61} + 2400 q^{65} - 9792 q^{69} + 11040 q^{73} + 27456 q^{77} + 11664 q^{81} + 11200 q^{85} + 7584 q^{89} - 19872 q^{93} - 14496 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} - 9 x^{14} + 18 x^{13} + 263 x^{12} - 444 x^{11} - 1732 x^{10} - 832 x^{9} + \cdots + 16777216 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 6617 \nu^{15} + 26278 \nu^{14} + 88297 \nu^{13} + 60460 \nu^{12} - 2080027 \nu^{11} + \cdots + 64036536320 ) / 2575826944 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 9755 \nu^{15} - 13120 \nu^{14} - 127395 \nu^{13} - 144550 \nu^{12} + 2087125 \nu^{11} + \cdots - 48431104000 ) / 454557696 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 71221 \nu^{15} + 10922 \nu^{14} - 239477 \nu^{13} - 2610740 \nu^{12} + 3886575 \nu^{11} + \cdots + 430284210176 ) / 1931870208 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 299085 \nu^{15} - 1193838 \nu^{14} - 4264925 \nu^{13} + 2297780 \nu^{12} + 89306007 \nu^{11} + \cdots - 2740169211904 ) / 7727480832 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 227675 \nu^{15} - 271360 \nu^{14} - 3004675 \nu^{13} - 5861350 \nu^{12} + 52470325 \nu^{11} + \cdots - 1513525084160 ) / 5151653888 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 59581 \nu^{15} - 42768 \nu^{14} + 423301 \nu^{13} + 2095514 \nu^{12} - 6724035 \nu^{11} + \cdots + 1598029824 ) / 965935104 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1048109 \nu^{15} + 1621860 \nu^{14} + 14763733 \nu^{13} + 11390118 \nu^{12} + \cdots + 6256972529664 ) / 15454961664 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1128561 \nu^{15} - 1324556 \nu^{14} - 24820761 \nu^{13} - 10009734 \nu^{12} + \cdots - 9641461284864 ) / 15454961664 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 336 \nu^{15} - 181 \nu^{14} + 3600 \nu^{13} + 12301 \nu^{12} - 42422 \nu^{11} + \cdots + 573898752 ) / 3342336 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 46787 \nu^{15} + 117178 \nu^{14} + 861011 \nu^{13} + 4460 \nu^{12} - 13982409 \nu^{11} + \cdots + 403322175488 ) / 454557696 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 57531 \nu^{15} - 100100 \nu^{14} - 799955 \nu^{13} - 602498 \nu^{12} + 14462477 \nu^{11} + \cdots - 435343065088 ) / 454557696 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 108453 \nu^{15} - 194448 \nu^{14} - 1278461 \nu^{13} - 1601290 \nu^{12} + 24352491 \nu^{11} + \cdots - 671431524352 ) / 454557696 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 9273 \nu^{15} - 9372 \nu^{14} - 116577 \nu^{13} - 182662 \nu^{12} + 1863135 \nu^{11} + \cdots - 47028633600 ) / 26738688 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 165367 \nu^{15} + 93284 \nu^{14} + 2303151 \nu^{13} + 4580698 \nu^{12} - 36515153 \nu^{11} + \cdots + 1047581425664 ) / 454557696 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 225257 \nu^{15} + 235856 \nu^{14} + 2712833 \nu^{13} + 4934802 \nu^{12} + \cdots + 1192816541696 ) / 454557696 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 21 \beta_{15} - 9 \beta_{14} + 6 \beta_{13} + 20 \beta_{12} - 11 \beta_{11} + 12 \beta_{10} + \cdots + 480 ) / 1920 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 14 \beta_{15} - 6 \beta_{14} - 16 \beta_{13} + 30 \beta_{12} - 34 \beta_{11} - 18 \beta_{10} + \cdots + 4080 ) / 1920 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3 \beta_{15} - 7 \beta_{14} + 8 \beta_{13} - 20 \beta_{12} - 43 \beta_{11} + 40 \beta_{10} + \cdots + 4720 ) / 640 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 154 \beta_{15} - 126 \beta_{14} + 64 \beta_{13} + 170 \beta_{12} - 514 \beta_{11} - 432 \beta_{10} + \cdots - 41520 ) / 1920 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 102 \beta_{15} + 102 \beta_{14} - 126 \beta_{13} - 84 \beta_{12} - 36 \beta_{11} - 144 \beta_{10} + \cdots + 5616 ) / 384 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 195 \beta_{15} - 65 \beta_{14} + 280 \beta_{13} - 300 \beta_{12} + 75 \beta_{11} - 175 \beta_{10} + \cdots - 21520 ) / 320 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 4549 \beta_{15} + 4161 \beta_{14} - 454 \beta_{13} + 260 \beta_{12} + 2739 \beta_{11} + \cdots + 101280 ) / 1920 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 6728 \beta_{15} + 9888 \beta_{14} - 4252 \beta_{13} - 10350 \beta_{12} + 28112 \beta_{11} + \cdots - 873840 ) / 1920 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 47 \beta_{15} - 4163 \beta_{14} - 708 \beta_{13} - 3360 \beta_{12} + 8813 \beta_{11} + 8700 \beta_{10} + \cdots - 1037840 ) / 640 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 19464 \beta_{15} + 20340 \beta_{14} - 888 \beta_{13} + 1122 \beta_{12} - 12432 \beta_{11} + \cdots - 568464 ) / 384 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 306072 \beta_{15} - 227688 \beta_{14} - 277698 \beta_{13} - 84040 \beta_{12} + 280498 \beta_{11} + \cdots - 71292720 ) / 1920 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 144445 \beta_{15} + 29335 \beta_{14} - 96690 \beta_{13} + 65090 \beta_{12} - 6255 \beta_{11} + \cdots - 4252400 ) / 320 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 301697 \beta_{15} + 892827 \beta_{14} + 2350202 \beta_{13} - 110500 \beta_{12} + 1619733 \beta_{11} + \cdots - 120437760 ) / 1920 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 1568678 \beta_{15} - 4023618 \beta_{14} - 8297048 \beta_{13} - 2140210 \beta_{12} + \cdots - 137204880 ) / 1920 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 1129145 \beta_{15} + 1616261 \beta_{14} + 986016 \beta_{13} + 510228 \beta_{12} + 1542985 \beta_{11} + \cdots - 65397456 ) / 128 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/960\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(577\) \(641\) \(901\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
511.1
−1.85197 − 2.13780i
2.44021 − 1.43016i
−2.28990 + 1.66022i
2.70166 + 0.837276i
−1.14149 + 2.58786i
1.85226 + 2.13755i
−2.48191 − 1.35651i
2.77114 − 0.566380i
2.70166 − 0.837276i
−2.28990 − 1.66022i
2.44021 + 1.43016i
−1.85197 + 2.13780i
2.77114 + 0.566380i
−2.48191 + 1.35651i
1.85226 − 2.13755i
−1.14149 − 2.58786i
0 − 5.19615i 0 −11.1803 0 − 89.0673i 0 −27.0000 0
511.2 0 − 5.19615i 0 −11.1803 0 − 24.1355i 0 −27.0000 0
511.3 0 − 5.19615i 0 −11.1803 0 12.7755i 0 −27.0000 0
511.4 0 − 5.19615i 0 −11.1803 0 86.5709i 0 −27.0000 0
511.5 0 − 5.19615i 0 11.1803 0 − 61.3317i 0 −27.0000 0
511.6 0 − 5.19615i 0 11.1803 0 − 1.39605i 0 −27.0000 0
511.7 0 − 5.19615i 0 11.1803 0 19.2859i 0 −27.0000 0
511.8 0 − 5.19615i 0 11.1803 0 29.5855i 0 −27.0000 0
511.9 0 5.19615i 0 −11.1803 0 − 86.5709i 0 −27.0000 0
511.10 0 5.19615i 0 −11.1803 0 − 12.7755i 0 −27.0000 0
511.11 0 5.19615i 0 −11.1803 0 24.1355i 0 −27.0000 0
511.12 0 5.19615i 0 −11.1803 0 89.0673i 0 −27.0000 0
511.13 0 5.19615i 0 11.1803 0 − 29.5855i 0 −27.0000 0
511.14 0 5.19615i 0 11.1803 0 − 19.2859i 0 −27.0000 0
511.15 0 5.19615i 0 11.1803 0 1.39605i 0 −27.0000 0
511.16 0 5.19615i 0 11.1803 0 61.3317i 0 −27.0000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 511.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 960.5.e.f 16
4.b odd 2 1 inner 960.5.e.f 16
8.b even 2 1 60.5.c.a ✓ 16
8.d odd 2 1 60.5.c.a ✓ 16
24.f even 2 1 180.5.c.c 16
24.h odd 2 1 180.5.c.c 16
40.e odd 2 1 300.5.c.d 16
40.f even 2 1 300.5.c.d 16
40.i odd 4 2 300.5.f.b 32
40.k even 4 2 300.5.f.b 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.5.c.a ✓ 16 8.b even 2 1
60.5.c.a ✓ 16 8.d odd 2 1
180.5.c.c 16 24.f even 2 1
180.5.c.c 16 24.h odd 2 1
300.5.c.d 16 40.e odd 2 1
300.5.c.d 16 40.f even 2 1
300.5.f.b 32 40.i odd 4 2
300.5.f.b 32 40.k even 4 2
960.5.e.f 16 1.a even 1 1 trivial
960.5.e.f 16 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{16} + 21184 T_{7}^{14} + 157121024 T_{7}^{12} + 484373856256 T_{7}^{10} + 612313337626624 T_{7}^{8} + \cdots + 13\!\cdots\!76 \) acting on \(S_{5}^{\mathrm{new}}(960, [\chi])\). 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
$5$ \( (T^{2} - 125)^{8} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 15\!\cdots\!56 \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots - 236865200916224)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 70\!\cdots\!76 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 55\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 81\!\cdots\!36)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 48\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots - 27\!\cdots\!04)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 12\!\cdots\!64)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 86\!\cdots\!36)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 44\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots - 20\!\cdots\!24)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 57\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots - 68\!\cdots\!96)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 46\!\cdots\!56 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 61\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots - 46\!\cdots\!76)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots - 12\!\cdots\!16)^{2} \) Copy content Toggle raw display
show more
show less