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Properties

Label 48.3.l.a
Level $48$
Weight $3$
Character orbit 48.l
Analytic conductor $1.308$
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] = [48,3,Mod(19,48)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(48, 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("48.19");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 48.l (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: \(1.30790526893\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{9} \)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} - \beta_{5} q^{3} + ( - \beta_{2} + 1) q^{4} + (\beta_{12} + \beta_{10} + \cdots + \beta_1) q^{5}+ \cdots - 3 \beta_{8} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} - \beta_{5} q^{3} + ( - \beta_{2} + 1) q^{4} + (\beta_{12} + \beta_{10} + \cdots + \beta_1) q^{5}+ \cdots + (3 \beta_{15} + 3 \beta_{14} + \cdots + 6) q^{99}+O(q^{100}) \) 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 \( 16 q + 12 q^{4} - 12 q^{8} - 56 q^{10} + 32 q^{11} - 24 q^{12} - 44 q^{14} + 32 q^{16} + 12 q^{18} - 32 q^{19} + 80 q^{20} + 32 q^{22} - 128 q^{23} + 36 q^{24} - 100 q^{26} - 120 q^{28} + 32 q^{29} + 72 q^{30} + 160 q^{32} + 96 q^{34} + 96 q^{35} + 12 q^{36} - 96 q^{37} + 168 q^{38} + 48 q^{40} - 60 q^{42} + 160 q^{43} + 88 q^{44} + 136 q^{46} - 144 q^{48} + 112 q^{49} - 236 q^{50} - 96 q^{51} - 48 q^{52} - 160 q^{53} - 36 q^{54} - 256 q^{55} - 224 q^{56} + 144 q^{58} - 128 q^{59} - 72 q^{60} - 32 q^{61} - 276 q^{62} - 408 q^{64} - 32 q^{65} + 72 q^{66} + 320 q^{67} - 448 q^{68} + 96 q^{69} - 384 q^{70} + 512 q^{71} + 60 q^{72} + 348 q^{74} + 192 q^{75} + 72 q^{76} + 224 q^{77} + 396 q^{78} + 552 q^{80} - 144 q^{81} - 40 q^{82} - 160 q^{83} + 72 q^{84} + 160 q^{85} + 528 q^{86} + 480 q^{88} - 24 q^{90} - 480 q^{91} + 496 q^{92} + 312 q^{94} - 480 q^{96} - 440 q^{98} + 96 q^{99}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{14} - 6 \nu^{12} - 4 \nu^{11} + 10 \nu^{10} + 56 \nu^{9} + 88 \nu^{8} - 128 \nu^{7} + \cdots - 20480 ) / 4096 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - \nu^{15} + 6 \nu^{13} + 4 \nu^{12} - 10 \nu^{11} - 56 \nu^{10} - 88 \nu^{9} + 128 \nu^{8} + \cdots + 24576 \nu ) / 16384 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 13 \nu^{15} - 512 \nu^{14} - 1362 \nu^{13} - 972 \nu^{12} + 2750 \nu^{11} + 9896 \nu^{10} + \cdots - 5505024 ) / 245760 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 81 \nu^{15} + 268 \nu^{14} + 218 \nu^{13} - 588 \nu^{12} - 2310 \nu^{11} - 1616 \nu^{10} + \cdots - 180224 ) / 245760 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 131 \nu^{15} - 88 \nu^{14} + 1122 \nu^{13} + 2268 \nu^{12} + 610 \nu^{11} - 9944 \nu^{10} + \cdots + 7716864 ) / 245760 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 661 \nu^{15} - 280 \nu^{14} + 5486 \nu^{13} + 11556 \nu^{12} + 4334 \nu^{11} - 49608 \nu^{10} + \cdots + 32063488 ) / 737280 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 347 \nu^{15} - 626 \nu^{14} + 1234 \nu^{13} + 4536 \nu^{12} + 5530 \nu^{11} - 11868 \nu^{10} + \cdots + 10231808 ) / 368640 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 293 \nu^{15} + 980 \nu^{14} + 674 \nu^{13} - 2124 \nu^{12} - 7774 \nu^{11} - 4128 \nu^{10} + \cdots - 32768 ) / 245760 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 443 \nu^{15} + 1670 \nu^{14} + 1982 \nu^{13} - 2448 \nu^{12} - 13642 \nu^{11} - 12636 \nu^{10} + \cdots + 4612096 ) / 368640 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 959 \nu^{15} + 2168 \nu^{14} - 778 \nu^{13} - 9612 \nu^{12} - 21514 \nu^{11} + 10296 \nu^{10} + \cdots - 9961472 ) / 737280 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1249 \nu^{15} + 2776 \nu^{14} - 2486 \nu^{13} - 14868 \nu^{12} - 23798 \nu^{11} + 25704 \nu^{10} + \cdots - 29753344 ) / 737280 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 471 \nu^{15} + 524 \nu^{14} - 2474 \nu^{13} - 6372 \nu^{12} - 4362 \nu^{11} + 23936 \nu^{10} + \cdots - 17022976 ) / 245760 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 929 \nu^{15} - 1328 \nu^{14} + 4822 \nu^{13} + 13860 \nu^{12} + 12310 \nu^{11} - 42840 \nu^{10} + \cdots + 36306944 ) / 368640 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 2261 \nu^{15} - 2588 \nu^{14} + 13246 \nu^{13} + 34236 \nu^{12} + 23326 \nu^{11} + \cdots + 91160576 ) / 737280 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{14} - \beta_{12} + \beta_{10} - \beta_{9} + \beta_{7} - \beta_{3} + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{15} - 2\beta_{12} + 2\beta_{11} + 2\beta_{8} + 2\beta_{7} - 2\beta_{6} + \beta_{5} + 2\beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} - \beta_{14} + \beta_{12} + 5\beta_{8} - 2\beta_{7} - \beta_{6} + 3\beta_{5} - 4\beta_{3} + \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{15} - 2 \beta_{14} + \beta_{13} + \beta_{12} + \beta_{11} - \beta_{9} - 5 \beta_{8} + 5 \beta_{7} + \cdots - 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 6 \beta_{15} + \beta_{14} - 8 \beta_{13} - 11 \beta_{12} + 2 \beta_{11} + 9 \beta_{10} - 5 \beta_{9} + \cdots - 24 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 3 \beta_{15} + 6 \beta_{13} + 16 \beta_{12} + 8 \beta_{11} - 16 \beta_{10} + 2 \beta_{9} + 24 \beta_{8} + \cdots - 45 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2 \beta_{15} - 4 \beta_{14} - 4 \beta_{13} - 20 \beta_{12} - 16 \beta_{11} + 2 \beta_{10} + 42 \beta_{9} + \cdots - 88 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 4 \beta_{15} + 48 \beta_{14} - 22 \beta_{13} + 138 \beta_{12} + 6 \beta_{11} + 4 \beta_{10} - 46 \beta_{9} + \cdots - 38 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 36 \beta_{15} + 110 \beta_{14} + 24 \beta_{13} - 82 \beta_{12} - 44 \beta_{11} - 22 \beta_{10} + \cdots - 484 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 190 \beta_{15} + 8 \beta_{14} + 88 \beta_{13} + 44 \beta_{12} - 132 \beta_{11} - 56 \beta_{10} + \cdots + 678 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 428 \beta_{15} + 244 \beta_{14} + 80 \beta_{13} + 124 \beta_{12} - 64 \beta_{11} - 224 \beta_{10} + \cdots - 264 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 556 \beta_{15} + 648 \beta_{14} + 1036 \beta_{13} + 2092 \beta_{12} - 468 \beta_{11} + 192 \beta_{10} + \cdots + 3176 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 648 \beta_{15} + 220 \beta_{14} - 512 \beta_{13} - 3668 \beta_{12} - 456 \beta_{11} - 612 \beta_{10} + \cdots + 6368 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 460 \beta_{15} - 1248 \beta_{14} + 3208 \beta_{13} + 320 \beta_{12} - 832 \beta_{11} + 672 \beta_{10} + \cdots + 5524 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/48\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(31\) \(37\)
\(\chi(n)\) \(1\) \(-1\) \(-\beta_{8}\)

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
1.84258 + 0.777752i
1.80398 − 0.863518i
1.78012 + 0.911682i
0.125358 − 1.99607i
−0.455024 + 1.94755i
−1.25564 + 1.55672i
−1.87459 − 0.697079i
−1.96679 + 0.362960i
1.84258 − 0.777752i
1.80398 + 0.863518i
1.78012 − 0.911682i
0.125358 + 1.99607i
−0.455024 − 1.94755i
−1.25564 − 1.55672i
−1.87459 + 0.697079i
−1.96679 − 0.362960i
−1.84258 + 0.777752i −1.22474 + 1.22474i 2.79020 − 2.86614i −4.78830 + 4.78830i 1.30414 − 3.20924i −10.3302 −2.91202 + 7.45118i − 3.00000i 5.09872 − 12.5469i
19.2 −1.80398 − 0.863518i −1.22474 + 1.22474i 2.50867 + 3.11554i 6.49473 − 6.49473i 3.26700 − 1.15182i 3.94273 −1.83527 − 7.78664i − 3.00000i −17.3247 + 6.10803i
19.3 −1.78012 + 0.911682i 1.22474 − 1.22474i 2.33767 − 3.24581i 1.00772 − 1.00772i −1.06362 + 3.29677i 10.0236 −1.20220 + 7.90915i − 3.00000i −0.875146 + 2.71259i
19.4 −0.125358 − 1.99607i 1.22474 − 1.22474i −3.96857 + 0.500444i 3.32679 − 3.32679i −2.59820 − 2.29114i −4.04088 1.49641 + 7.85880i − 3.00000i −7.05755 − 6.22347i
19.5 0.455024 + 1.94755i −1.22474 + 1.22474i −3.58591 + 1.77236i −3.40572 + 3.40572i −2.94254 − 1.82796i 12.1303 −5.08344 − 6.17727i − 3.00000i −8.18251 − 5.08314i
19.6 1.25564 + 1.55672i 1.22474 − 1.22474i −0.846753 + 3.90935i 0.909023 − 0.909023i 3.44442 + 0.368750i −0.654713 −7.14897 + 3.59057i − 3.00000i 2.55650 + 0.273691i
19.7 1.87459 − 0.697079i 1.22474 − 1.22474i 3.02816 − 2.61347i −5.24354 + 5.24354i 1.44215 − 3.14964i −5.32796 3.85476 − 7.01005i − 3.00000i −6.17431 + 13.4846i
19.8 1.96679 + 0.362960i −1.22474 + 1.22474i 3.73652 + 1.42773i 1.69930 − 1.69930i −2.85335 + 1.96428i −5.74280 6.83074 + 4.16426i − 3.00000i 3.95895 − 2.72539i
43.1 −1.84258 − 0.777752i −1.22474 − 1.22474i 2.79020 + 2.86614i −4.78830 − 4.78830i 1.30414 + 3.20924i −10.3302 −2.91202 − 7.45118i 3.00000i 5.09872 + 12.5469i
43.2 −1.80398 + 0.863518i −1.22474 − 1.22474i 2.50867 − 3.11554i 6.49473 + 6.49473i 3.26700 + 1.15182i 3.94273 −1.83527 + 7.78664i 3.00000i −17.3247 − 6.10803i
43.3 −1.78012 − 0.911682i 1.22474 + 1.22474i 2.33767 + 3.24581i 1.00772 + 1.00772i −1.06362 − 3.29677i 10.0236 −1.20220 − 7.90915i 3.00000i −0.875146 − 2.71259i
43.4 −0.125358 + 1.99607i 1.22474 + 1.22474i −3.96857 − 0.500444i 3.32679 + 3.32679i −2.59820 + 2.29114i −4.04088 1.49641 − 7.85880i 3.00000i −7.05755 + 6.22347i
43.5 0.455024 − 1.94755i −1.22474 − 1.22474i −3.58591 − 1.77236i −3.40572 − 3.40572i −2.94254 + 1.82796i 12.1303 −5.08344 + 6.17727i 3.00000i −8.18251 + 5.08314i
43.6 1.25564 − 1.55672i 1.22474 + 1.22474i −0.846753 − 3.90935i 0.909023 + 0.909023i 3.44442 − 0.368750i −0.654713 −7.14897 − 3.59057i 3.00000i 2.55650 − 0.273691i
43.7 1.87459 + 0.697079i 1.22474 + 1.22474i 3.02816 + 2.61347i −5.24354 − 5.24354i 1.44215 + 3.14964i −5.32796 3.85476 + 7.01005i 3.00000i −6.17431 − 13.4846i
43.8 1.96679 − 0.362960i −1.22474 − 1.22474i 3.73652 − 1.42773i 1.69930 + 1.69930i −2.85335 − 1.96428i −5.74280 6.83074 − 4.16426i 3.00000i 3.95895 + 2.72539i
\(n\): 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 48.3.l.a ✓ 16
3.b odd 2 1 144.3.m.c 16
4.b odd 2 1 192.3.l.a 16
8.b even 2 1 384.3.l.a 16
8.d odd 2 1 384.3.l.b 16
12.b even 2 1 576.3.m.c 16
16.e even 4 1 192.3.l.a 16
16.e even 4 1 384.3.l.b 16
16.f odd 4 1 inner 48.3.l.a ✓ 16
16.f odd 4 1 384.3.l.a 16
24.f even 2 1 1152.3.m.c 16
24.h odd 2 1 1152.3.m.f 16
48.i odd 4 1 576.3.m.c 16
48.i odd 4 1 1152.3.m.c 16
48.k even 4 1 144.3.m.c 16
48.k even 4 1 1152.3.m.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.3.l.a ✓ 16 1.a even 1 1 trivial
48.3.l.a ✓ 16 16.f odd 4 1 inner
144.3.m.c 16 3.b odd 2 1
144.3.m.c 16 48.k even 4 1
192.3.l.a 16 4.b odd 2 1
192.3.l.a 16 16.e even 4 1
384.3.l.a 16 8.b even 2 1
384.3.l.a 16 16.f odd 4 1
384.3.l.b 16 8.d odd 2 1
384.3.l.b 16 16.e even 4 1
576.3.m.c 16 12.b even 2 1
576.3.m.c 16 48.i odd 4 1
1152.3.m.c 16 24.f even 2 1
1152.3.m.c 16 48.i odd 4 1
1152.3.m.f 16 24.h odd 2 1
1152.3.m.f 16 48.k even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(48, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 6 T^{14} + \cdots + 65536 \) Copy content Toggle raw display
$3$ \( (T^{4} + 9)^{4} \) 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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