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Properties

Label 144.3.m.c
Level $144$
Weight $3$
Character orbit 144.m
Analytic conductor $3.924$
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] = [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: \( x^{16} - 6 x^{14} - 4 x^{13} + 10 x^{12} + 56 x^{11} + 88 x^{10} - 128 x^{9} - 496 x^{8} - 512 x^{7} + \cdots + 65536 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12} \)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + ( - \beta_1 + 1) q^{4} + ( - \beta_{13} + \beta_{12} + \cdots - \beta_1) q^{5}+ \cdots + ( - \beta_{15} + \beta_{14} - \beta_{11} + \cdots + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + ( - \beta_1 + 1) q^{4} + ( - \beta_{13} + \beta_{12} + \cdots - \beta_1) q^{5}+ \cdots + (10 \beta_{14} + 2 \beta_{13} + \cdots + 40) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 12 q^{4} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 12 q^{4} + 12 q^{8} - 56 q^{10} - 32 q^{11} + 44 q^{14} + 32 q^{16} - 32 q^{19} - 80 q^{20} + 32 q^{22} + 128 q^{23} + 100 q^{26} - 120 q^{28} - 32 q^{29} - 160 q^{32} + 96 q^{34} - 96 q^{35} - 96 q^{37} - 168 q^{38} + 48 q^{40} + 160 q^{43} - 88 q^{44} + 136 q^{46} + 112 q^{49} + 236 q^{50} - 48 q^{52} + 160 q^{53} - 256 q^{55} + 224 q^{56} + 144 q^{58} + 128 q^{59} - 32 q^{61} + 276 q^{62} - 408 q^{64} + 32 q^{65} + 320 q^{67} + 448 q^{68} - 384 q^{70} - 512 q^{71} - 348 q^{74} + 72 q^{76} - 224 q^{77} - 552 q^{80} - 40 q^{82} + 160 q^{83} + 160 q^{85} - 528 q^{86} + 480 q^{88} - 480 q^{91} - 496 q^{92} + 312 q^{94} + 440 q^{98}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( ( \nu^{14} - 6 \nu^{12} - 4 \nu^{11} + 10 \nu^{10} + 56 \nu^{9} + 88 \nu^{8} - 128 \nu^{7} + \cdots - 20480 ) / 4096 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{15} - 6 \nu^{13} - 4 \nu^{12} + 10 \nu^{11} + 56 \nu^{10} + 88 \nu^{9} - 128 \nu^{8} + \cdots - 24576 \nu ) / 16384 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 19 \nu^{15} - 86 \nu^{14} - 134 \nu^{13} + 48 \nu^{12} + 618 \nu^{11} + 796 \nu^{10} + \cdots - 393216 ) / 40960 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 81 \nu^{15} + 268 \nu^{14} + 218 \nu^{13} - 588 \nu^{12} - 2310 \nu^{11} - 1616 \nu^{10} + \cdots - 180224 ) / 122880 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 172 \nu^{15} - 739 \nu^{14} - 628 \nu^{13} + 1602 \nu^{12} + 6524 \nu^{11} + 5874 \nu^{10} + \cdots - 696320 ) / 184320 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 347 \nu^{15} + 626 \nu^{14} - 1234 \nu^{13} - 4536 \nu^{12} - 5530 \nu^{11} + 11868 \nu^{10} + \cdots - 10231808 ) / 368640 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 697 \nu^{15} - 208 \nu^{14} + 5990 \nu^{13} + 11268 \nu^{12} - 1498 \nu^{11} - 56664 \nu^{10} + \cdots + 41648128 ) / 737280 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 188 \nu^{15} + 323 \nu^{14} - 484 \nu^{13} - 1890 \nu^{12} - 2188 \nu^{11} + 5550 \nu^{10} + \cdots - 4751360 ) / 184320 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 131 \nu^{15} + 88 \nu^{14} - 1122 \nu^{13} - 2268 \nu^{12} - 610 \nu^{11} + 9944 \nu^{10} + \cdots - 7716864 ) / 122880 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1093 \nu^{15} - 2080 \nu^{14} + 4766 \nu^{13} + 15444 \nu^{12} + 14414 \nu^{11} + \cdots + 38699008 ) / 737280 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 187 \nu^{15} - 624 \nu^{14} - 446 \nu^{13} + 1356 \nu^{12} + 5042 \nu^{11} + 2872 \nu^{10} + \cdots + 106496 ) / 122880 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 1249 \nu^{15} - 2776 \nu^{14} + 2486 \nu^{13} + 14868 \nu^{12} + 23798 \nu^{11} + \cdots + 29753344 ) / 737280 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 275 \nu^{15} + 464 \nu^{14} - 1090 \nu^{13} - 3564 \nu^{12} - 3874 \nu^{11} + 10248 \nu^{10} + \cdots - 8830976 ) / 147456 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 1405 \nu^{15} + 76 \nu^{14} - 14126 \nu^{13} - 26172 \nu^{12} - 1358 \nu^{11} + 123216 \nu^{10} + \cdots - 84557824 ) / 737280 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 1037 \nu^{15} + 926 \nu^{14} - 6454 \nu^{13} - 15336 \nu^{12} - 8710 \nu^{11} + 60708 \nu^{10} + \cdots - 41566208 ) / 368640 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{13} - \beta_{9} - 2\beta_{6} - \beta_{4} - 2\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{12} + 2\beta_{11} + \beta_{8} + \beta_{6} - \beta_{5} + 2\beta_{4} - \beta_{3} + \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{14} + \beta_{13} + \beta_{12} + 2\beta_{8} + 2\beta_{7} - 3\beta_{6} + \beta_{4} - \beta_{2} + 2\beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3 \beta_{15} - 2 \beta_{14} + 2 \beta_{13} + 2 \beta_{11} + 2 \beta_{9} - \beta_{8} - 12 \beta_{6} + \cdots + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 2 \beta_{14} + 6 \beta_{11} + 2 \beta_{10} - \beta_{9} + 8 \beta_{8} + 2 \beta_{7} + 10 \beta_{6} + \cdots - 24 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 3 \beta_{15} + 6 \beta_{14} + 2 \beta_{13} + 5 \beta_{12} + 4 \beta_{11} + 2 \beta_{10} - 2 \beta_{9} + \cdots - 23 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 6 \beta_{15} - 3 \beta_{14} - 17 \beta_{13} - 13 \beta_{12} - 2 \beta_{11} - 10 \beta_{10} + 5 \beta_{9} + \cdots - 45 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 11 \beta_{15} - 2 \beta_{14} - 18 \beta_{13} + 22 \beta_{12} - 38 \beta_{11} + 16 \beta_{10} + 16 \beta_{9} + \cdots - 92 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 28 \beta_{15} - 4 \beta_{14} - 26 \beta_{13} - 134 \beta_{12} - 2 \beta_{11} + 2 \beta_{10} + \cdots + 10 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 12 \beta_{15} - 36 \beta_{14} + 4 \beta_{13} + 118 \beta_{12} - 100 \beta_{11} - 52 \beta_{10} + \cdots - 374 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 80 \beta_{15} + 190 \beta_{14} - 306 \beta_{13} - 234 \beta_{12} - 184 \beta_{11} - 88 \beta_{10} + \cdots + 686 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 122 \beta_{15} - 428 \beta_{14} + 348 \beta_{13} + 304 \beta_{12} - 740 \beta_{11} + 48 \beta_{10} + \cdots - 20 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 16 \beta_{15} - 556 \beta_{14} - 272 \beta_{13} - 1536 \beta_{12} + 772 \beta_{11} - 20 \beta_{10} + \cdots + 3824 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 1700 \beta_{15} + 648 \beta_{14} + 1528 \beta_{13} + 3020 \beta_{12} - 1424 \beta_{11} - 840 \beta_{10} + \cdots + 6588 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 1992 \beta_{15} + 460 \beta_{14} + 2596 \beta_{13} - 780 \beta_{12} + 1768 \beta_{11} + 392 \beta_{10} + \cdots + 4276 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(\beta_{6}\) \(1\) \(-1\)

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
−1.96679 + 0.362960i
−1.87459 − 0.697079i
−1.25564 + 1.55672i
−0.455024 + 1.94755i
0.125358 − 1.99607i
1.78012 + 0.911682i
1.80398 − 0.863518i
1.84258 + 0.777752i
−1.96679 − 0.362960i
−1.87459 + 0.697079i
−1.25564 − 1.55672i
−0.455024 − 1.94755i
0.125358 + 1.99607i
1.78012 − 0.911682i
1.80398 + 0.863518i
1.84258 − 0.777752i
−1.96679 − 0.362960i 0 3.73652 + 1.42773i −1.69930 + 1.69930i 0 −5.74280 −6.83074 − 4.16426i 0 3.95895 − 2.72539i
19.2 −1.87459 + 0.697079i 0 3.02816 − 2.61347i 5.24354 − 5.24354i 0 −5.32796 −3.85476 + 7.01005i 0 −6.17431 + 13.4846i
19.3 −1.25564 − 1.55672i 0 −0.846753 + 3.90935i −0.909023 + 0.909023i 0 −0.654713 7.14897 − 3.59057i 0 2.55650 + 0.273691i
19.4 −0.455024 − 1.94755i 0 −3.58591 + 1.77236i 3.40572 − 3.40572i 0 12.1303 5.08344 + 6.17727i 0 −8.18251 − 5.08314i
19.5 0.125358 + 1.99607i 0 −3.96857 + 0.500444i −3.32679 + 3.32679i 0 −4.04088 −1.49641 − 7.85880i 0 −7.05755 − 6.22347i
19.6 1.78012 − 0.911682i 0 2.33767 − 3.24581i −1.00772 + 1.00772i 0 10.0236 1.20220 − 7.90915i 0 −0.875146 + 2.71259i
19.7 1.80398 + 0.863518i 0 2.50867 + 3.11554i −6.49473 + 6.49473i 0 3.94273 1.83527 + 7.78664i 0 −17.3247 + 6.10803i
19.8 1.84258 − 0.777752i 0 2.79020 − 2.86614i 4.78830 − 4.78830i 0 −10.3302 2.91202 − 7.45118i 0 5.09872 − 12.5469i
91.1 −1.96679 + 0.362960i 0 3.73652 − 1.42773i −1.69930 − 1.69930i 0 −5.74280 −6.83074 + 4.16426i 0 3.95895 + 2.72539i
91.2 −1.87459 − 0.697079i 0 3.02816 + 2.61347i 5.24354 + 5.24354i 0 −5.32796 −3.85476 − 7.01005i 0 −6.17431 − 13.4846i
91.3 −1.25564 + 1.55672i 0 −0.846753 − 3.90935i −0.909023 − 0.909023i 0 −0.654713 7.14897 + 3.59057i 0 2.55650 − 0.273691i
91.4 −0.455024 + 1.94755i 0 −3.58591 − 1.77236i 3.40572 + 3.40572i 0 12.1303 5.08344 − 6.17727i 0 −8.18251 + 5.08314i
91.5 0.125358 − 1.99607i 0 −3.96857 − 0.500444i −3.32679 − 3.32679i 0 −4.04088 −1.49641 + 7.85880i 0 −7.05755 + 6.22347i
91.6 1.78012 + 0.911682i 0 2.33767 + 3.24581i −1.00772 − 1.00772i 0 10.0236 1.20220 + 7.90915i 0 −0.875146 − 2.71259i
91.7 1.80398 − 0.863518i 0 2.50867 − 3.11554i −6.49473 − 6.49473i 0 3.94273 1.83527 − 7.78664i 0 −17.3247 − 6.10803i
91.8 1.84258 + 0.777752i 0 2.79020 + 2.86614i 4.78830 + 4.78830i 0 −10.3302 2.91202 + 7.45118i 0 5.09872 + 12.5469i
\(n\): 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
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 144.3.m.c 16
3.b odd 2 1 48.3.l.a ✓ 16
4.b odd 2 1 576.3.m.c 16
8.b even 2 1 1152.3.m.f 16
8.d odd 2 1 1152.3.m.c 16
12.b even 2 1 192.3.l.a 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 inner 144.3.m.c 16
16.f odd 4 1 1152.3.m.f 16
24.f even 2 1 384.3.l.b 16
24.h odd 2 1 384.3.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 3.b odd 2 1
48.3.l.a ✓ 16 48.k even 4 1
144.3.m.c 16 1.a even 1 1 trivial
144.3.m.c 16 16.f odd 4 1 inner
192.3.l.a 16 12.b even 2 1
192.3.l.a 16 48.i odd 4 1
384.3.l.a 16 24.h odd 2 1
384.3.l.a 16 48.k even 4 1
384.3.l.b 16 24.f even 2 1
384.3.l.b 16 48.i odd 4 1
576.3.m.c 16 4.b odd 2 1
576.3.m.c 16 16.e even 4 1
1152.3.m.c 16 8.d odd 2 1
1152.3.m.c 16 16.e even 4 1
1152.3.m.f 16 8.b even 2 1
1152.3.m.f 16 16.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} - 32 T_{5}^{13} + 6656 T_{5}^{12} - 8064 T_{5}^{11} + 512 T_{5}^{10} + 518400 T_{5}^{9} + \cdots + 2117472256 \) 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
$5$ \( T^{16} + \cdots + 2117472256 \) 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
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