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Properties

Label 144.3.m.b
Level $144$
Weight $3$
Character orbit 144.m
Analytic conductor $3.924$
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] = [144,3,Mod(19,144)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(144, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("144.19");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 144.m (of order \(4\), degree \(2\), 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: \(3.92371580679\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + ( - \beta_{2} - 1) q^{4} + \beta_{8} q^{5} + (\beta_{12} + \beta_{11}) q^{7} + ( - \beta_{13} + \beta_{7} + \beta_{4}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} + ( - \beta_{2} - 1) q^{4} + \beta_{8} q^{5} + (\beta_{12} + \beta_{11}) q^{7} + ( - \beta_{13} + \beta_{7} + \beta_{4}) q^{8} + ( - \beta_{15} + \beta_{11} + \cdots + \beta_{3}) q^{10}+ \cdots + (6 \beta_{13} + 14 \beta_{10} + \cdots + 6 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 12 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 12 q^{4} + 16 q^{10} + 32 q^{16} - 32 q^{19} + 104 q^{22} - 24 q^{34} + 96 q^{37} - 312 q^{40} - 32 q^{43} - 224 q^{46} + 112 q^{49} - 264 q^{52} - 256 q^{55} + 312 q^{58} - 32 q^{61} + 456 q^{64} - 256 q^{67} + 744 q^{70} + 264 q^{76} - 280 q^{82} + 160 q^{85} - 912 q^{88} + 288 q^{91} - 1104 q^{94}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{14} + 14\nu^{12} + 110\nu^{10} - 472\nu^{8} + 944\nu^{6} + 5504\nu^{4} - 24064\nu^{2} - 20480 ) / 12288 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5\nu^{15} - 14\nu^{13} - 110\nu^{11} + 472\nu^{9} - 944\nu^{7} - 5504\nu^{5} + 24064\nu^{3} + 20480\nu ) / 12288 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{15} + 46\nu^{13} - 122\nu^{11} - 968\nu^{9} + 4528\nu^{7} - 8960\nu^{5} - 46592\nu^{3} + 229376\nu ) / 12288 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{14} - 6\nu^{12} + 10\nu^{10} + 88\nu^{8} - 752\nu^{6} + 1408\nu^{4} + 2560\nu^{2} - 22528 ) / 2048 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -17\nu^{15} - 34\nu^{13} + 326\nu^{11} - 424\nu^{9} - 1360\nu^{7} + 26624\nu^{5} + 9728\nu^{3} - 167936\nu ) / 24576 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -11\nu^{15} + 14\nu^{13} + 170\nu^{11} - 784\nu^{9} + 1328\nu^{7} + 9536\nu^{5} - 30208\nu^{3} - 26624\nu ) / 12288 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -11\nu^{15} + 14\nu^{13} + 170\nu^{11} - 784\nu^{9} + 1328\nu^{7} + 9536\nu^{5} - 17920\nu^{3} - 38912\nu ) / 12288 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{15} - 16\nu^{13} + 38\nu^{11} + 308\nu^{9} - 1696\nu^{7} + 3296\nu^{5} + 13568\nu^{3} - 72704\nu ) / 3072 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -5\nu^{14} + 38\nu^{12} - 34\nu^{10} - 1000\nu^{8} + 4592\nu^{6} - 1792\nu^{4} - 45568\nu^{2} + 182272 ) / 6144 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 11 \nu^{14} + 146 \nu^{12} - 142 \nu^{10} - 2344 \nu^{8} + 11600 \nu^{6} - 16768 \nu^{4} + \cdots + 348160 ) / 12288 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -13\nu^{15} + 86\nu^{13} + 14\nu^{11} - 1704\nu^{9} + 7280\nu^{7} - 2304\nu^{5} - 76288\nu^{3} + 176128\nu ) / 8192 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( -5\nu^{14} + 62\nu^{12} - 82\nu^{10} - 1336\nu^{8} + 6128\nu^{6} - 8320\nu^{4} - 62464\nu^{2} + 246784 ) / 3072 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 47\nu^{14} - 122\nu^{12} - 554\nu^{10} + 4072\nu^{8} - 10640\nu^{6} - 33152\nu^{4} + 162304\nu^{2} - 77824 ) / 12288 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - \beta_{8} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{14} + \beta_{12} + 2\beta_{11} + \beta_{6} - \beta_{3} + \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{13} + 2\beta_{10} - \beta_{9} - 3\beta_{8} + 3\beta_{7} + 2\beta_{5} - \beta_{4} + \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{15} - 2\beta_{14} + 6\beta_{11} - 4\beta_{6} + 10\beta_{3} + 2\beta_{2} - 30 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -2\beta_{13} - 12\beta_{10} - 4\beta_{9} - 12\beta_{8} + 10\beta_{7} - 8\beta_{5} - 18\beta_{4} - 32\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 4\beta_{15} - 14\beta_{14} + 14\beta_{12} + 16\beta_{11} - 14\beta_{6} + 14\beta_{3} - 38\beta_{2} + 110 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 10\beta_{13} - 20\beta_{10} - 54\beta_{9} + 14\beta_{8} + 14\beta_{7} - 44\beta_{5} - 18\beta_{4} + 118\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 36\beta_{15} + 24\beta_{14} - 36\beta_{12} - 44\beta_{11} - 68\beta_{6} + 328\beta_{3} + 64\beta_{2} + 488 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -72\beta_{13} - 176\beta_{10} + 108\beta_{9} - 92\beta_{8} - 88\beta_{7} - 168\beta_{5} - 288\beta_{4} + 540\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -164\beta_{14} + 396\beta_{12} + 40\beta_{11} + 196\beta_{6} - 60\beta_{3} + 444\beta_{2} + 3268 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 396 \beta_{13} + 680 \beta_{10} + 404 \beta_{9} - 484 \beta_{8} - 316 \beta_{7} + 104 \beta_{5} + \cdots + 3196 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 792\beta_{15} - 88\beta_{14} + 96\beta_{12} + 968\beta_{11} + 720\beta_{6} + 4056\beta_{3} + 2904\beta_{2} - 1768 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 696 \beta_{13} - 144 \beta_{10} + 1936 \beta_{9} - 5456 \beta_{8} + 1048 \beta_{7} + \cdots - 4160 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(\beta_{3}\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
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
−1.99750 − 0.0999235i
1.66730 − 1.10459i
−1.64663 − 1.13516i
−0.136762 − 1.99532i
−1.99532 − 0.136762i 0 3.96259 + 0.545766i −0.227650 + 0.227650i 0 3.90219 −7.83199 − 1.63091i 0 0.485368 − 0.423100i
19.2 −1.13516 − 1.64663i 0 −1.42280 + 3.73840i 2.41234 − 2.41234i 0 −11.8718 7.77089 − 1.90087i 0 −6.71063 − 1.23383i
19.3 −1.10459 + 1.66730i 0 −1.55976 − 3.68336i −4.23991 + 4.23991i 0 −0.262225 7.86415 + 1.46802i 0 −2.38583 − 11.7525i
19.4 −0.0999235 − 1.99750i 0 −3.98003 + 0.399195i −6.01265 + 6.01265i 0 8.23187 1.19509 + 7.91023i 0 12.6111 + 11.4095i
19.5 0.0999235 + 1.99750i 0 −3.98003 + 0.399195i 6.01265 − 6.01265i 0 8.23187 −1.19509 − 7.91023i 0 12.6111 + 11.4095i
19.6 1.10459 − 1.66730i 0 −1.55976 − 3.68336i 4.23991 − 4.23991i 0 −0.262225 −7.86415 − 1.46802i 0 −2.38583 − 11.7525i
19.7 1.13516 + 1.64663i 0 −1.42280 + 3.73840i −2.41234 + 2.41234i 0 −11.8718 −7.77089 + 1.90087i 0 −6.71063 − 1.23383i
19.8 1.99532 + 0.136762i 0 3.96259 + 0.545766i 0.227650 − 0.227650i 0 3.90219 7.83199 + 1.63091i 0 0.485368 − 0.423100i
91.1 −1.99532 + 0.136762i 0 3.96259 − 0.545766i −0.227650 − 0.227650i 0 3.90219 −7.83199 + 1.63091i 0 0.485368 + 0.423100i
91.2 −1.13516 + 1.64663i 0 −1.42280 − 3.73840i 2.41234 + 2.41234i 0 −11.8718 7.77089 + 1.90087i 0 −6.71063 + 1.23383i
91.3 −1.10459 − 1.66730i 0 −1.55976 + 3.68336i −4.23991 − 4.23991i 0 −0.262225 7.86415 − 1.46802i 0 −2.38583 + 11.7525i
91.4 −0.0999235 + 1.99750i 0 −3.98003 − 0.399195i −6.01265 − 6.01265i 0 8.23187 1.19509 − 7.91023i 0 12.6111 − 11.4095i
91.5 0.0999235 − 1.99750i 0 −3.98003 − 0.399195i 6.01265 + 6.01265i 0 8.23187 −1.19509 + 7.91023i 0 12.6111 − 11.4095i
91.6 1.10459 + 1.66730i 0 −1.55976 + 3.68336i 4.23991 + 4.23991i 0 −0.262225 −7.86415 + 1.46802i 0 −2.38583 + 11.7525i
91.7 1.13516 − 1.64663i 0 −1.42280 − 3.73840i −2.41234 − 2.41234i 0 −11.8718 −7.77089 − 1.90087i 0 −6.71063 + 1.23383i
91.8 1.99532 − 0.136762i 0 3.96259 − 0.545766i 0.227650 + 0.227650i 0 3.90219 7.83199 − 1.63091i 0 0.485368 + 0.423100i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.8
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 144.3.m.b ✓ 16
3.b odd 2 1 inner 144.3.m.b ✓ 16
4.b odd 2 1 576.3.m.b 16
8.b even 2 1 1152.3.m.e 16
8.d odd 2 1 1152.3.m.d 16
12.b even 2 1 576.3.m.b 16
16.e even 4 1 576.3.m.b 16
16.e even 4 1 1152.3.m.d 16
16.f odd 4 1 inner 144.3.m.b ✓ 16
16.f odd 4 1 1152.3.m.e 16
24.f even 2 1 1152.3.m.d 16
24.h odd 2 1 1152.3.m.e 16
48.i odd 4 1 576.3.m.b 16
48.i odd 4 1 1152.3.m.d 16
48.k even 4 1 inner 144.3.m.b ✓ 16
48.k even 4 1 1152.3.m.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.3.m.b ✓ 16 1.a even 1 1 trivial
144.3.m.b ✓ 16 3.b odd 2 1 inner
144.3.m.b ✓ 16 16.f odd 4 1 inner
144.3.m.b ✓ 16 48.k even 4 1 inner
576.3.m.b 16 4.b odd 2 1
576.3.m.b 16 12.b even 2 1
576.3.m.b 16 16.e even 4 1
576.3.m.b 16 48.i odd 4 1
1152.3.m.d 16 8.d odd 2 1
1152.3.m.d 16 16.e even 4 1
1152.3.m.d 16 24.f even 2 1
1152.3.m.d 16 48.i odd 4 1
1152.3.m.e 16 8.b even 2 1
1152.3.m.e 16 16.f odd 4 1
1152.3.m.e 16 24.h odd 2 1
1152.3.m.e 16 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} + 6656T_{5}^{12} + 7641216T_{5}^{8} + 915505152T_{5}^{4} + 9834496 \) acting on \(S_{3}^{\mathrm{new}}(144, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 6 T^{14} + \cdots + 65536 \) 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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