↑

Properties

Label 144.5.o.c
Level $144$
Weight $5$
Character orbit 144.o
Analytic conductor $14.885$
Analytic rank $0$
Dimension $16$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [144,5,Mod(31,144)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(144, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 2]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("144.31");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 144.o (of order \(6\), 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: \(14.8852746841\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 6 x^{15} + 12 x^{14} - 10 x^{13} + 276 x^{12} + 5772 x^{11} - 27606 x^{10} + \cdots + 192872105271 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{18} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{5} + \beta_{2}) q^{3} + \beta_{8} q^{5} + (\beta_{15} + \beta_{7} + \beta_1 - 2) q^{7} + (\beta_{12} - \beta_{9} + \beta_{5} + \cdots + 8) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} + \beta_{2}) q^{3} + \beta_{8} q^{5} + (\beta_{15} + \beta_{7} + \beta_1 - 2) q^{7} + (\beta_{12} - \beta_{9} + \beta_{5} + \cdots + 8) q^{9}+ \cdots + ( - 9 \beta_{15} + 27 \beta_{14} + \cdots - 3762) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 15 q^{3} - 3 q^{5} - 39 q^{7} + 159 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 15 q^{3} - 3 q^{5} - 39 q^{7} + 159 q^{9} - 234 q^{11} - 5 q^{13} - 111 q^{15} + 498 q^{17} - 15 q^{21} - 891 q^{23} - 853 q^{25} - 1620 q^{27} - 141 q^{29} - 1665 q^{31} - 1620 q^{33} + 2084 q^{37} - 87 q^{39} - 378 q^{41} - 459 q^{45} - 5931 q^{47} + 3667 q^{49} - 663 q^{51} + 10188 q^{53} - 1803 q^{57} + 8082 q^{59} - 847 q^{61} - 9837 q^{63} + 411 q^{65} + 15540 q^{67} - 11637 q^{69} - 6098 q^{73} + 16119 q^{75} - 12519 q^{77} + 9507 q^{79} - 14409 q^{81} - 29817 q^{83} + 4350 q^{85} + 24915 q^{87} + 29688 q^{89} + 13347 q^{93} + 53820 q^{95} - 13658 q^{97} - 54765 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 6 x^{15} + 12 x^{14} - 10 x^{13} + 276 x^{12} + 5772 x^{11} - 27606 x^{10} + \cdots + 192872105271 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 491105 \nu^{15} - 7592491 \nu^{14} + 58332434 \nu^{13} - 256190127 \nu^{12} + \cdots + 12\!\cdots\!50 ) / 20\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1655318 \nu^{15} - 153721 \nu^{14} - 125255497 \nu^{13} + 1123555530 \nu^{12} + \cdots + 17\!\cdots\!09 ) / 20\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2834435 \nu^{15} + 5000093 \nu^{14} + 1745399624 \nu^{13} - 28595026545 \nu^{12} + \cdots + 43\!\cdots\!36 ) / 20\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3151285 \nu^{15} - 37139129 \nu^{14} + 350639140 \nu^{13} - 2521213635 \nu^{12} + \cdots + 42\!\cdots\!28 ) / 20\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3151285 \nu^{15} - 37139129 \nu^{14} + 350639140 \nu^{13} - 2521213635 \nu^{12} + \cdots + 51\!\cdots\!92 ) / 20\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 74185 \nu^{15} + 1927637 \nu^{14} - 23470132 \nu^{13} + 138815823 \nu^{12} + \cdots - 11\!\cdots\!84 ) / 293224620986496 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 12064129 \nu^{15} - 265407839 \nu^{14} + 1300261954 \nu^{13} + 3839576775 \nu^{12} + \cdots - 17\!\cdots\!76 ) / 10\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 36817454 \nu^{15} - 391283515 \nu^{14} + 1719908285 \nu^{13} - 4020762030 \nu^{12} + \cdots + 15\!\cdots\!91 ) / 20\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 48117929 \nu^{15} + 253144861 \nu^{14} - 1047620240 \nu^{13} + 11380797879 \nu^{12} + \cdots - 38\!\cdots\!32 ) / 20\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 16030889 \nu^{15} + 177340963 \nu^{14} - 887297546 \nu^{13} + 2166600579 \nu^{12} + \cdots - 46\!\cdots\!10 ) / 69\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 49946713 \nu^{15} + 413484904 \nu^{14} - 6743985377 \nu^{13} + 15846743265 \nu^{12} + \cdots + 73\!\cdots\!65 ) / 20\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 20264939 \nu^{15} - 171189958 \nu^{14} + 512470271 \nu^{13} - 779794665 \nu^{12} + \cdots + 87\!\cdots\!41 ) / 69\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 64201678 \nu^{15} - 479160593 \nu^{14} + 797982475 \nu^{13} - 2786371806 \nu^{12} + \cdots + 29\!\cdots\!01 ) / 20\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 110892160 \nu^{15} + 1096225115 \nu^{14} - 2002914979 \nu^{13} - 22948266732 \nu^{12} + \cdots + 17\!\cdots\!31 ) / 20\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 37968260 \nu^{15} - 231458371 \nu^{14} - 70837525 \nu^{13} + 2637321780 \nu^{12} + \cdots + 15\!\cdots\!21 ) / 69\!\cdots\!72 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{5} + \beta_{4} + 4 ) / 9 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{14} + \beta_{13} - \beta_{10} - 6\beta_{5} + \beta_{4} - \beta_{3} + 35\beta _1 + 28 ) / 9 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3 \beta_{15} + 4 \beta_{14} + 4 \beta_{13} - 6 \beta_{12} - 3 \beta_{11} - 4 \beta_{10} - 6 \beta_{9} + \cdots + 163 ) / 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 24 \beta_{15} + 10 \beta_{14} + 28 \beta_{13} + 6 \beta_{12} - 51 \beta_{11} - 13 \beta_{10} + \cdots - 1124 ) / 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 9 \beta_{15} - 53 \beta_{14} + 154 \beta_{13} + 261 \beta_{12} - 234 \beta_{11} - 109 \beta_{10} + \cdots - 14492 ) / 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 102 \beta_{15} - 656 \beta_{14} + 604 \beta_{13} + 1497 \beta_{12} - 993 \beta_{11} + 1451 \beta_{10} + \cdots - 284 ) / 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 4665 \beta_{15} - 9557 \beta_{14} + 1882 \beta_{13} + 4200 \beta_{12} - 1518 \beta_{11} + \cdots + 456922 ) / 9 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 20088 \beta_{15} - 56798 \beta_{14} + 6562 \beta_{13} - 8586 \beta_{12} + 15795 \beta_{11} + \cdots + 2542219 ) / 9 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 126033 \beta_{15} - 277595 \beta_{14} + 63460 \beta_{13} - 280833 \beta_{12} + 209220 \beta_{11} + \cdots + 9766606 ) / 9 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 310674 \beta_{15} - 1237031 \beta_{14} + 543853 \beta_{13} - 2860707 \beta_{12} + 1120647 \beta_{11} + \cdots - 35188838 ) / 9 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 611676 \beta_{15} - 5672339 \beta_{14} + 3599506 \beta_{13} - 18548424 \beta_{12} + 3280221 \beta_{11} + \cdots - 942333509 ) / 9 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 6438756 \beta_{15} - 13165802 \beta_{14} + 3246382 \beta_{13} - 76131870 \beta_{12} + 4453314 \beta_{11} + \cdots - 8467304009 ) / 9 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 37758990 \beta_{15} + 12275416 \beta_{14} - 159444080 \beta_{13} - 153907986 \beta_{12} + \cdots - 52030671344 ) / 9 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 753887664 \beta_{15} + 444979855 \beta_{14} - 2195955143 \beta_{13} + 460072566 \beta_{12} + \cdots - 230727256526 ) / 9 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 4270602561 \beta_{15} + 5382998092 \beta_{14} - 19063629278 \beta_{13} + 6813818724 \beta_{12} + \cdots - 552549643511 ) / 9 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
5.86888 + 1.81494i
4.04285 + 4.21201i
4.64877 − 3.69952i
−0.578380 + 4.95063i
−3.95867 + 1.55294i
−1.24850 − 4.68447i
−4.18384 + 0.357482i
−1.59112 − 4.50401i
5.86888 − 1.81494i
4.04285 − 4.21201i
4.64877 + 3.69952i
−0.578380 − 4.95063i
−3.95867 − 1.55294i
−1.24850 + 4.68447i
−4.18384 − 0.357482i
−1.59112 + 4.50401i
0 −8.87510 − 1.49416i 0 13.9505 − 24.1629i 0 −3.38319 + 1.95328i 0 76.5349 + 26.5217i 0
31.2 0 −8.21199 + 3.68283i 0 −17.2620 + 29.8986i 0 −29.8352 + 17.2254i 0 53.8735 − 60.4867i 0
31.3 0 −2.26928 − 8.70921i 0 −5.92396 + 10.2606i 0 3.53139 − 2.03885i 0 −70.7007 + 39.5273i 0
31.4 0 −1.91980 + 8.79286i 0 4.09130 − 7.08634i 0 39.2082 − 22.6369i 0 −73.6287 − 33.7610i 0
31.5 0 6.09312 + 6.62374i 0 15.1736 − 26.2814i 0 −71.0685 + 41.0314i 0 −6.74788 + 80.7184i 0
31.6 0 7.42962 − 5.07944i 0 −3.46647 + 6.00410i 0 −57.9454 + 33.4548i 0 29.3985 − 75.4767i 0
31.7 0 7.46617 + 5.02556i 0 −21.9768 + 38.0649i 0 24.5285 − 14.1615i 0 30.4875 + 75.0434i 0
31.8 0 7.78726 − 4.51204i 0 13.9138 − 24.0994i 0 75.4642 − 43.5693i 0 40.2829 − 70.2729i 0
79.1 0 −8.87510 + 1.49416i 0 13.9505 + 24.1629i 0 −3.38319 − 1.95328i 0 76.5349 − 26.5217i 0
79.2 0 −8.21199 − 3.68283i 0 −17.2620 − 29.8986i 0 −29.8352 − 17.2254i 0 53.8735 + 60.4867i 0
79.3 0 −2.26928 + 8.70921i 0 −5.92396 − 10.2606i 0 3.53139 + 2.03885i 0 −70.7007 − 39.5273i 0
79.4 0 −1.91980 − 8.79286i 0 4.09130 + 7.08634i 0 39.2082 + 22.6369i 0 −73.6287 + 33.7610i 0
79.5 0 6.09312 − 6.62374i 0 15.1736 + 26.2814i 0 −71.0685 − 41.0314i 0 −6.74788 − 80.7184i 0
79.6 0 7.42962 + 5.07944i 0 −3.46647 − 6.00410i 0 −57.9454 − 33.4548i 0 29.3985 + 75.4767i 0
79.7 0 7.46617 − 5.02556i 0 −21.9768 − 38.0649i 0 24.5285 + 14.1615i 0 30.4875 − 75.0434i 0
79.8 0 7.78726 + 4.51204i 0 13.9138 + 24.0994i 0 75.4642 + 43.5693i 0 40.2829 + 70.2729i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
36.f odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.5.o.c yes 16
3.b odd 2 1 432.5.o.b 16
4.b odd 2 1 144.5.o.a ✓ 16
9.c even 3 1 144.5.o.a ✓ 16
9.c even 3 1 1296.5.g.l 16
9.d odd 6 1 432.5.o.c 16
9.d odd 6 1 1296.5.g.j 16
12.b even 2 1 432.5.o.c 16
36.f odd 6 1 inner 144.5.o.c yes 16
36.f odd 6 1 1296.5.g.l 16
36.h even 6 1 432.5.o.b 16
36.h even 6 1 1296.5.g.j 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.5.o.a ✓ 16 4.b odd 2 1
144.5.o.a ✓ 16 9.c even 3 1
144.5.o.c yes 16 1.a even 1 1 trivial
144.5.o.c yes 16 36.f odd 6 1 inner
432.5.o.b 16 3.b odd 2 1
432.5.o.b 16 36.h even 6 1
432.5.o.c 16 9.d odd 6 1
432.5.o.c 16 12.b even 2 1
1296.5.g.j 16 9.d odd 6 1
1296.5.g.j 16 36.h even 6 1
1296.5.g.l 16 9.c even 3 1
1296.5.g.l 16 36.f odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(144, [\chi])\):

\( T_{5}^{16} + 3 T_{5}^{15} + 2931 T_{5}^{14} - 28746 T_{5}^{13} + 5991093 T_{5}^{12} + \cdots + 57\!\cdots\!56 \) Copy content Toggle raw display
\( T_{7}^{16} + 39 T_{7}^{15} - 10677 T_{7}^{14} - 436176 T_{7}^{13} + 89132193 T_{7}^{12} + \cdots + 11\!\cdots\!04 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 18\!\cdots\!41 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 57\!\cdots\!56 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 11\!\cdots\!04 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 12\!\cdots\!61 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots - 31\!\cdots\!28)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 33\!\cdots\!44 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 46\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 41\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 72\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots - 45\!\cdots\!32)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 19\!\cdots\!89 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 99\!\cdots\!69 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots - 11\!\cdots\!08)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 53\!\cdots\!29 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 63\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 16\!\cdots\!61 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 14\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots - 76\!\cdots\!60)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 11\!\cdots\!84 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 66\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots - 16\!\cdots\!16)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 12\!\cdots\!81 \) Copy content Toggle raw display
show more
show less