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Properties

Label 1152.3.m.d
Level $1152$
Weight $3$
Character orbit 1152.m
Analytic conductor $31.390$
Analytic rank $0$
Dimension $16$
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] = [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: \( x^{16} - 6x^{14} + 10x^{12} + 88x^{10} - 752x^{8} + 1408x^{6} + 2560x^{4} - 24576x^{2} + 65536 \) 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 144)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{5} - \beta_{8} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{5} - \beta_{8} q^{7} + (\beta_{14} + \beta_{4}) q^{11} - \beta_{15} q^{13} + (\beta_{14} + \beta_{11} - \beta_{10}) q^{17} + (\beta_{12} - \beta_{2} + 2 \beta_1 - 2) q^{19} + ( - \beta_{14} - \beta_{11} + \cdots + \beta_{3}) q^{23}+ \cdots + (2 \beta_{15} - 2 \beta_{12} + 8 \beta_{8}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 32 q^{19} - 96 q^{37} - 32 q^{43} + 112 q^{49} + 256 q^{55} + 32 q^{61} - 256 q^{67} - 160 q^{85} + 288 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 6x^{14} + 10x^{12} + 88x^{10} - 752x^{8} + 1408x^{6} + 2560x^{4} - 24576x^{2} + 65536 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -5\nu^{14} + 14\nu^{12} + 110\nu^{10} - 472\nu^{8} + 944\nu^{6} + 5504\nu^{4} - 24064\nu^{2} - 20480 ) / 12288 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{14} - 22\nu^{12} - 214\nu^{10} - 712\nu^{8} + 4880\nu^{6} - 13696\nu^{4} + 22016\nu^{2} + 409600 ) / 12288 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -17\nu^{15} - 34\nu^{13} + 326\nu^{11} - 424\nu^{9} - 1360\nu^{7} + 26624\nu^{5} + 9728\nu^{3} - 167936\nu ) / 24576 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11\nu^{15} - 14\nu^{13} - 170\nu^{11} + 784\nu^{9} - 1328\nu^{7} - 9536\nu^{5} + 30208\nu^{3} + 26624\nu ) / 12288 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{15} + 6\nu^{13} - 62\nu^{11} - 48\nu^{9} + 816\nu^{7} - 4032\nu^{5} - 1024\nu^{3} + 51200\nu ) / 1024 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 13\nu^{14} - 94\nu^{12} - 94\nu^{10} + 344\nu^{8} - 4144\nu^{6} + 3200\nu^{4} + 3584\nu^{2} + 188416 ) / 12288 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -3\nu^{15} - 6\nu^{13} + 114\nu^{11} + 8\nu^{9} - 880\nu^{7} + 6656\nu^{5} - 512\nu^{3} - 94208\nu ) / 2048 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -7\nu^{14} + 74\nu^{12} - 70\nu^{10} - 1448\nu^{8} + 6928\nu^{6} - 6784\nu^{4} - 70144\nu^{2} + 237568 ) / 4096 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 11\nu^{15} - 74\nu^{13} - 98\nu^{11} + 1912\nu^{9} - 6416\nu^{7} - 2048\nu^{5} + 94720\nu^{3} - 151552\nu ) / 6144 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -7\nu^{15} + 46\nu^{13} + 34\nu^{11} - 1088\nu^{9} + 4336\nu^{7} - 1088\nu^{5} - 51200\nu^{3} + 137216\nu ) / 3072 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -7\nu^{15} + 55\nu^{13} + 4\nu^{11} - 1142\nu^{9} + 4600\nu^{7} - 2672\nu^{5} - 51968\nu^{3} + 135680\nu ) / 3072 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 37\nu^{14} + 146\nu^{12} - 1006\nu^{10} - 616\nu^{8} + 10832\nu^{6} - 76672\nu^{4} - 64000\nu^{2} + 778240 ) / 12288 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -47\nu^{14} + 122\nu^{12} + 554\nu^{10} - 4072\nu^{8} + 10640\nu^{6} + 33152\nu^{4} - 150016\nu^{2} + 65536 ) / 12288 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 77 \nu^{15} + 470 \nu^{13} + 398 \nu^{11} - 10408 \nu^{9} + 41072 \nu^{7} - 256 \nu^{5} + \cdots + 987136 \nu ) / 24576 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 79 \nu^{14} - 490 \nu^{12} - 202 \nu^{10} + 10760 \nu^{8} - 43024 \nu^{6} + 16256 \nu^{4} + \cdots - 1163264 ) / 12288 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{10} + \beta_{9} - \beta_{7} - \beta_{5} ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{15} + 2\beta_{13} + \beta_{12} - \beta_{6} + \beta_{2} + 2\beta _1 + 6 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{14} + 4\beta_{11} + \beta_{10} + 2\beta_{7} + \beta_{5} + 4\beta_{4} + 4\beta_{3} ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{15} + 3\beta_{13} + 5\beta_{8} - \beta_{2} - 2\beta _1 + 8 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 8\beta_{14} - 5\beta_{10} - \beta_{9} - 7\beta_{7} - 7\beta_{5} + 32\beta_{4} + 24\beta_{3} ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 3\beta_{15} - 12\beta_{13} - 7\beta_{12} + 14\beta_{8} - 5\beta_{6} + 7\beta_{2} + 38\beta _1 - 114 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 4\beta_{14} - 12\beta_{11} + 8\beta_{10} - 7\beta_{9} + 17\beta_{7} + 14\beta_{5} + 52\beta_{4} + 28\beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -8\beta_{15} - 39\beta_{13} - 23\beta_{12} + 17\beta_{8} - 5\beta_{6} - 8\beta_{2} - 16\beta _1 + 238 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 128\beta_{14} - 88\beta_{11} + 17\beta_{10} + 79\beta_{9} - 35\beta_{7} - 41\beta_{5} + 72\beta_{4} - 32\beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -23\beta_{15} - 90\beta_{13} + 5\beta_{12} - 60\beta_{8} - 49\beta_{6} + 69\beta_{2} + 658\beta _1 + 1014 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 180 \beta_{14} + 36 \beta_{11} + 229 \beta_{10} + 54 \beta_{9} + 132 \beta_{7} + 37 \beta_{5} + \cdots - 140 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 101\beta_{15} + 209\beta_{13} + 108\beta_{12} + 47\beta_{8} - 92\beta_{6} - 79\beta_{2} - 22\beta _1 + 2888 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 8 \beta_{14} + 800 \beta_{11} - 163 \beta_{10} + 497 \beta_{9} - 201 \beta_{7} - 513 \beta_{5} + \cdots + 168 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 773\beta_{15} + 492\beta_{13} + 511\beta_{12} + 706\beta_{8} - 195\beta_{6} + 369\beta_{2} + 5514\beta _1 - 2878 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 1316 \beta_{14} + 620 \beta_{11} + 568 \beta_{10} - 1145 \beta_{9} + 735 \beta_{7} + \cdots + 1668 \beta_{3} \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
415.1
−1.99750 + 0.0999235i
1.66730 + 1.10459i
1.64663 − 1.13516i
−0.136762 + 1.99532i
0.136762 − 1.99532i
−1.64663 + 1.13516i
−1.66730 − 1.10459i
1.99750 − 0.0999235i
−1.99750 − 0.0999235i
1.66730 − 1.10459i
1.64663 + 1.13516i
−0.136762 − 1.99532i
0.136762 + 1.99532i
−1.64663 − 1.13516i
−1.66730 + 1.10459i
1.99750 + 0.0999235i
0 0 0 −6.01265 − 6.01265i 0 −8.23187 0 0 0
415.2 0 0 0 −4.23991 − 4.23991i 0 0.262225 0 0 0
415.3 0 0 0 −2.41234 − 2.41234i 0 11.8718 0 0 0
415.4 0 0 0 −0.227650 − 0.227650i 0 −3.90219 0 0 0
415.5 0 0 0 0.227650 + 0.227650i 0 −3.90219 0 0 0
415.6 0 0 0 2.41234 + 2.41234i 0 11.8718 0 0 0
415.7 0 0 0 4.23991 + 4.23991i 0 0.262225 0 0 0
415.8 0 0 0 6.01265 + 6.01265i 0 −8.23187 0 0 0
991.1 0 0 0 −6.01265 + 6.01265i 0 −8.23187 0 0 0
991.2 0 0 0 −4.23991 + 4.23991i 0 0.262225 0 0 0
991.3 0 0 0 −2.41234 + 2.41234i 0 11.8718 0 0 0
991.4 0 0 0 −0.227650 + 0.227650i 0 −3.90219 0 0 0
991.5 0 0 0 0.227650 − 0.227650i 0 −3.90219 0 0 0
991.6 0 0 0 2.41234 − 2.41234i 0 11.8718 0 0 0
991.7 0 0 0 4.23991 − 4.23991i 0 0.262225 0 0 0
991.8 0 0 0 6.01265 − 6.01265i 0 −8.23187 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
3.b odd 2 1 inner
16.f odd 4 1 inner
48.k even 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.d 16
3.b odd 2 1 inner 1152.3.m.d 16
4.b odd 2 1 1152.3.m.e 16
8.b even 2 1 576.3.m.b 16
8.d odd 2 1 144.3.m.b ✓ 16
12.b even 2 1 1152.3.m.e 16
16.e even 4 1 144.3.m.b ✓ 16
16.e even 4 1 1152.3.m.e 16
16.f odd 4 1 576.3.m.b 16
16.f odd 4 1 inner 1152.3.m.d 16
24.f even 2 1 144.3.m.b ✓ 16
24.h odd 2 1 576.3.m.b 16
48.i odd 4 1 144.3.m.b ✓ 16
48.i odd 4 1 1152.3.m.e 16
48.k even 4 1 576.3.m.b 16
48.k even 4 1 inner 1152.3.m.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.3.m.b ✓ 16 8.d odd 2 1
144.3.m.b ✓ 16 16.e even 4 1
144.3.m.b ✓ 16 24.f even 2 1
144.3.m.b ✓ 16 48.i odd 4 1
576.3.m.b 16 8.b even 2 1
576.3.m.b 16 16.f odd 4 1
576.3.m.b 16 24.h odd 2 1
576.3.m.b 16 48.k even 4 1
1152.3.m.d 16 1.a even 1 1 trivial
1152.3.m.d 16 3.b odd 2 1 inner
1152.3.m.d 16 16.f odd 4 1 inner
1152.3.m.d 16 48.k even 4 1 inner
1152.3.m.e 16 4.b odd 2 1
1152.3.m.e 16 12.b even 2 1
1152.3.m.e 16 16.e even 4 1
1152.3.m.e 16 48.i odd 4 1

Hecke kernels

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

\( T_{5}^{16} + 6656T_{5}^{12} + 7641216T_{5}^{8} + 915505152T_{5}^{4} + 9834496 \) Copy content Toggle raw display
\( T_{7}^{4} - 112T_{7}^{2} - 352T_{7} + 100 \) 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
$5$ \( T^{16} + 6656 T^{12} + \cdots + 9834496 \) Copy content Toggle raw display
$7$ \( (T^{4} - 112 T^{2} + \cdots + 100)^{4} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{8} + 4352 T^{5} + \cdots + 35760400)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 1104 T^{6} + \cdots + 472105984)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 16 T^{7} + \cdots + 6629867776)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 1632 T^{6} + \cdots + 467943424)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{8} + 4032 T^{6} + \cdots + 11588953104)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 48 T^{7} + \cdots + 137744899600)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 7248 T^{6} + \cdots + 198844646400)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 16 T^{7} + \cdots + 21036601600)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 15982596734976)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 51506974970896)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 7615833702400)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 51\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 534697177190400)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 14624 T^{6} + \cdots + 56939504400)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 60\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 236165588582400)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 12384 T^{2} + \cdots + 29866240)^{4} \) Copy content Toggle raw display
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