↑

Properties

Label 1152.3.m.f
Level $1152$
Weight $3$
Character orbit 1152.m
Analytic conductor $31.390$
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] = [1152,3,Mod(415,1152)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1152, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1152.415");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1152.m (of order \(4\), degree \(2\), 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: \(31.3897264543\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 6 x^{14} - 4 x^{13} + 10 x^{12} + 56 x^{11} + 88 x^{10} - 128 x^{9} - 496 x^{8} - 512 x^{7} + \cdots + 65536 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{28} \)
Twist minimal: no (minimal twist has level 48)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{11} q^{5} + \beta_{3} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{11} q^{5} + \beta_{3} q^{7} + ( - \beta_{10} - \beta_{5} + 2 \beta_{2} + 2) q^{11} + ( - \beta_{15} - \beta_{14} + \cdots + \beta_{3}) q^{13}+ \cdots + ( - 2 \beta_{14} + 2 \beta_{13} + \cdots + 6 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 32 q^{11} + 32 q^{19} + 128 q^{23} + 32 q^{29} + 96 q^{35} + 96 q^{37} - 160 q^{43} + 112 q^{49} - 160 q^{53} - 256 q^{55} - 128 q^{59} + 32 q^{61} + 32 q^{65} - 320 q^{67} - 512 q^{71} + 224 q^{77} - 160 q^{83} - 160 q^{85} + 480 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 6 x^{14} - 4 x^{13} + 10 x^{12} + 56 x^{11} + 88 x^{10} - 128 x^{9} - 496 x^{8} - 512 x^{7} + \cdots + 65536 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 5 \nu^{15} - 18 \nu^{14} - 134 \nu^{13} - 168 \nu^{12} + 170 \nu^{11} + 1156 \nu^{10} + \cdots - 753664 ) / 12288 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 347 \nu^{15} - 626 \nu^{14} + 1234 \nu^{13} + 4536 \nu^{12} + 5530 \nu^{11} - 11868 \nu^{10} + \cdots + 10231808 ) / 368640 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 47 \nu^{15} + 110 \nu^{14} - 90 \nu^{13} - 528 \nu^{12} - 610 \nu^{11} + 1684 \nu^{10} + \cdots - 1228800 ) / 40960 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 53 \nu^{15} - 89 \nu^{14} + 226 \nu^{13} + 714 \nu^{12} + 730 \nu^{11} - 2082 \nu^{10} + \cdots + 1974272 ) / 30720 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 751 \nu^{15} - 448 \nu^{14} + 6626 \nu^{13} + 14076 \nu^{12} + 5306 \nu^{11} - 54888 \nu^{10} + \cdots + 41107456 ) / 368640 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 119 \nu^{15} + 1230 \nu^{14} + 3430 \nu^{13} + 2064 \nu^{12} - 9010 \nu^{11} - 25772 \nu^{10} + \cdots + 14581760 ) / 122880 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 197 \nu^{15} - 244 \nu^{14} - 2572 \nu^{13} - 3852 \nu^{12} + 998 \nu^{11} + 22056 \nu^{10} + \cdots - 12972032 ) / 92160 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1063 \nu^{15} + 2242 \nu^{14} - 2666 \nu^{13} - 14184 \nu^{12} - 21122 \nu^{11} + 26652 \nu^{10} + \cdots - 26214400 ) / 368640 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 995 \nu^{15} - 3884 \nu^{14} - 4022 \nu^{13} + 5076 \nu^{12} + 30658 \nu^{11} + 28992 \nu^{10} + \cdots - 8568832 ) / 368640 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 134 \nu^{15} + 20 \nu^{14} - 1153 \nu^{13} - 2232 \nu^{12} - 622 \nu^{11} + 9756 \nu^{10} + \cdots - 6232064 ) / 46080 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 545 \nu^{15} + 1574 \nu^{14} + 302 \nu^{13} - 5256 \nu^{12} - 12838 \nu^{11} + 1188 \nu^{10} + \cdots - 6053888 ) / 184320 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 1417 \nu^{15} - 4300 \nu^{14} - 1186 \nu^{13} + 13356 \nu^{12} + 35366 \nu^{11} + \cdots + 13975552 ) / 368640 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 265 \nu^{15} + 526 \nu^{14} - 578 \nu^{13} - 2856 \nu^{12} - 4406 \nu^{11} + 6228 \nu^{10} + \cdots - 5570560 ) / 61440 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 1181 \nu^{15} + 2693 \nu^{14} - 2326 \nu^{13} - 13986 \nu^{12} - 22786 \nu^{11} + 22218 \nu^{10} + \cdots - 25296896 ) / 184320 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 2519 \nu^{15} - 4382 \nu^{14} + 9178 \nu^{13} + 33552 \nu^{12} + 37810 \nu^{11} + \cdots + 80101376 ) / 368640 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{12} + 2\beta_{11} - 2\beta_{8} - 2\beta_{6} + \beta_{4} + 4\beta_{2} - \beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 2 \beta_{14} + \beta_{12} + 2 \beta_{11} + \beta_{10} - \beta_{9} - \beta_{7} - 2 \beta_{5} + \cdots + 6 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - \beta_{15} - 4 \beta_{13} + \beta_{12} + \beta_{11} + \beta_{10} - 2 \beta_{9} + \beta_{8} + 2 \beta_{7} + \cdots + 6 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{15} - \beta_{14} + \beta_{13} - \beta_{12} + 3 \beta_{11} - \beta_{10} + \beta_{9} - \beta_{8} + \cdots + 8 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 8 \beta_{15} - 8 \beta_{14} - 16 \beta_{13} + 2 \beta_{12} + 18 \beta_{11} + 8 \beta_{10} - 4 \beta_{9} + \cdots - 80 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 12 \beta_{15} + 4 \beta_{14} + 2 \beta_{13} + 9 \beta_{12} + 2 \beta_{11} + 9 \beta_{10} + 5 \beta_{9} + \cdots - 102 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 15 \beta_{15} - 4 \beta_{14} - 8 \beta_{13} - 3 \beta_{12} - 47 \beta_{11} - \beta_{10} - 10 \beta_{9} + \cdots - 182 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 13 \beta_{15} + 21 \beta_{14} + 9 \beta_{13} + 26 \beta_{12} + 23 \beta_{11} - 24 \beta_{10} + \cdots - 146 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 14 \beta_{15} + 24 \beta_{14} + 32 \beta_{13} - 84 \beta_{12} - 120 \beta_{11} + 66 \beta_{10} + \cdots - 276 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 88 \beta_{15} + 38 \beta_{14} + 80 \beta_{13} + 5 \beta_{12} - 166 \beta_{11} - 55 \beta_{10} + \cdots - 786 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 167 \beta_{15} + 48 \beta_{14} + 36 \beta_{13} + 143 \beta_{12} - 105 \beta_{11} + 27 \beta_{10} + \cdots + 1210 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 127 \beta_{15} + 165 \beta_{14} + 227 \beta_{13} - 175 \beta_{12} - 131 \beta_{11} - 207 \beta_{10} + \cdots + 8 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 520 \beta_{15} - 904 \beta_{14} + 32 \beta_{13} - 322 \beta_{12} + 174 \beta_{11} + 296 \beta_{10} + \cdots + 6448 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 204 \beta_{15} + 484 \beta_{14} - 426 \beta_{13} - 25 \beta_{12} - 210 \beta_{11} - 113 \beta_{10} + \cdots + 7174 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 1033 \beta_{15} - 1244 \beta_{14} + 920 \beta_{13} - 565 \beta_{12} + 1631 \beta_{11} + 1233 \beta_{10} + \cdots + 4502 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(-1\) \(1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
415.1
−1.87459 + 0.697079i
1.84258 − 0.777752i
−0.455024 − 1.94755i
−1.25564 − 1.55672i
1.78012 − 0.911682i
−1.96679 − 0.362960i
0.125358 + 1.99607i
1.80398 + 0.863518i
−1.87459 − 0.697079i
1.84258 + 0.777752i
−0.455024 + 1.94755i
−1.25564 + 1.55672i
1.78012 + 0.911682i
−1.96679 + 0.362960i
0.125358 − 1.99607i
1.80398 − 0.863518i
0 0 0 −5.24354 − 5.24354i 0 −5.32796 0 0 0
415.2 0 0 0 −4.78830 − 4.78830i 0 −10.3302 0 0 0
415.3 0 0 0 −3.40572 − 3.40572i 0 12.1303 0 0 0
415.4 0 0 0 0.909023 + 0.909023i 0 −0.654713 0 0 0
415.5 0 0 0 1.00772 + 1.00772i 0 10.0236 0 0 0
415.6 0 0 0 1.69930 + 1.69930i 0 −5.74280 0 0 0
415.7 0 0 0 3.32679 + 3.32679i 0 −4.04088 0 0 0
415.8 0 0 0 6.49473 + 6.49473i 0 3.94273 0 0 0
991.1 0 0 0 −5.24354 + 5.24354i 0 −5.32796 0 0 0
991.2 0 0 0 −4.78830 + 4.78830i 0 −10.3302 0 0 0
991.3 0 0 0 −3.40572 + 3.40572i 0 12.1303 0 0 0
991.4 0 0 0 0.909023 − 0.909023i 0 −0.654713 0 0 0
991.5 0 0 0 1.00772 − 1.00772i 0 10.0236 0 0 0
991.6 0 0 0 1.69930 − 1.69930i 0 −5.74280 0 0 0
991.7 0 0 0 3.32679 − 3.32679i 0 −4.04088 0 0 0
991.8 0 0 0 6.49473 − 6.49473i 0 3.94273 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 415.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1152.3.m.f 16
3.b odd 2 1 384.3.l.a 16
4.b odd 2 1 1152.3.m.c 16
8.b even 2 1 144.3.m.c 16
8.d odd 2 1 576.3.m.c 16
12.b even 2 1 384.3.l.b 16
16.e even 4 1 576.3.m.c 16
16.e even 4 1 1152.3.m.c 16
16.f odd 4 1 144.3.m.c 16
16.f odd 4 1 inner 1152.3.m.f 16
24.f even 2 1 192.3.l.a 16
24.h odd 2 1 48.3.l.a ✓ 16
48.i odd 4 1 192.3.l.a 16
48.i odd 4 1 384.3.l.b 16
48.k even 4 1 48.3.l.a ✓ 16
48.k even 4 1 384.3.l.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.3.l.a ✓ 16 24.h odd 2 1
48.3.l.a ✓ 16 48.k even 4 1
144.3.m.c 16 8.b even 2 1
144.3.m.c 16 16.f odd 4 1
192.3.l.a 16 24.f even 2 1
192.3.l.a 16 48.i odd 4 1
384.3.l.a 16 3.b odd 2 1
384.3.l.a 16 48.k even 4 1
384.3.l.b 16 12.b even 2 1
384.3.l.b 16 48.i odd 4 1
576.3.m.c 16 8.d odd 2 1
576.3.m.c 16 16.e even 4 1
1152.3.m.c 16 4.b odd 2 1
1152.3.m.c 16 16.e even 4 1
1152.3.m.f 16 1.a even 1 1 trivial
1152.3.m.f 16 16.f odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1152, [\chi])\):

\( T_{5}^{16} + 32 T_{5}^{13} + 6656 T_{5}^{12} + 8064 T_{5}^{11} + 512 T_{5}^{10} - 518400 T_{5}^{9} + \cdots + 2117472256 \) Copy content Toggle raw display
\( T_{7}^{8} - 224T_{7}^{6} - 448T_{7}^{5} + 13704T_{7}^{4} + 53248T_{7}^{3} - 136576T_{7}^{2} - 720640T_{7} - 400880 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 2117472256 \) Copy content Toggle raw display
$7$ \( (T^{8} - 224 T^{6} + \cdots - 400880)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 25620118503424 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{8} - 1344 T^{6} + \cdots + 816881920)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 10\!\cdots\!96 \) Copy content Toggle raw display
$23$ \( (T^{8} - 64 T^{7} + \cdots - 35037900800)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 38\!\cdots\!04 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 94\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 92\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 96\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 21\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{8} + 256 T^{7} + \cdots + 290924400640)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 98\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 14\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 52\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 409778579046400)^{2} \) Copy content Toggle raw display
show more
show less