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Properties

Label 144.4.s.c
Level $144$
Weight $4$
Character orbit 144.s
Analytic conductor $8.496$
Analytic rank $0$
Dimension $10$
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] = [144,4,Mod(47,144)]
 
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, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("144.47");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 144.s (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: \(8.49627504083\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2 x^{9} + 15 x^{8} + 26 x^{7} + 101 x^{6} + 396 x^{5} + 1292 x^{4} + 2864 x^{3} + 7860 x^{2} + \cdots + 26368 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{6} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{6} + 2 \beta_{2} - 2) q^{3} + ( - \beta_{8} - \beta_{4} + 1) q^{5} + ( - \beta_{9} - \beta_{8} - 2 \beta_{6} + \cdots - 2) q^{7}+ \cdots + (\beta_{9} + \beta_{8} + \beta_{7} + \cdots + 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} + 2 \beta_{2} - 2) q^{3} + ( - \beta_{8} - \beta_{4} + 1) q^{5} + ( - \beta_{9} - \beta_{8} - 2 \beta_{6} + \cdots - 2) q^{7}+ \cdots + ( - 18 \beta_{9} + 3 \beta_{8} + \cdots + 114) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 6 q^{3} + 9 q^{5} - 33 q^{7} + 60 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 6 q^{3} + 9 q^{5} - 33 q^{7} + 60 q^{9} + 27 q^{11} + 55 q^{13} + 3 q^{15} - 69 q^{21} - 153 q^{23} + 56 q^{25} - 216 q^{27} - 99 q^{29} - 351 q^{31} - 585 q^{33} + 990 q^{35} + 1100 q^{37} - 471 q^{39} + 783 q^{41} + 333 q^{43} + 9 q^{45} - 603 q^{47} + 442 q^{49} + 762 q^{51} - 2046 q^{57} - 423 q^{59} - 325 q^{61} - 81 q^{63} - 3735 q^{65} - 753 q^{67} - 999 q^{69} + 5040 q^{71} + 736 q^{73} - 3456 q^{75} + 2565 q^{77} - 237 q^{79} + 900 q^{81} - 1323 q^{83} + 1452 q^{85} + 5577 q^{87} - 2061 q^{93} - 2322 q^{95} + 553 q^{97} - 2799 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2 x^{9} + 15 x^{8} + 26 x^{7} + 101 x^{6} + 396 x^{5} + 1292 x^{4} + 2864 x^{3} + 7860 x^{2} + \cdots + 26368 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 29 \nu^{9} + 272 \nu^{8} - 487 \nu^{7} + 4348 \nu^{6} - 3117 \nu^{5} + 37158 \nu^{4} + \cdots - 69952 ) / 1049760 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 29 \nu^{9} - 272 \nu^{8} + 487 \nu^{7} - 4348 \nu^{6} + 3117 \nu^{5} - 37158 \nu^{4} + \cdots + 69952 ) / 1049760 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3 \nu^{9} - 4 \nu^{8} + 159 \nu^{7} + 304 \nu^{6} + 2529 \nu^{5} + 5034 \nu^{4} + 30702 \nu^{3} + \cdots + 434384 ) / 58320 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 49 \nu^{9} - 832 \nu^{8} + 1091 \nu^{7} - 7868 \nu^{6} - 21279 \nu^{5} - 59790 \nu^{4} + \cdots - 377920 ) / 209952 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3 \nu^{9} + 2 \nu^{8} + 159 \nu^{7} - 98 \nu^{6} + 1881 \nu^{5} + 2316 \nu^{4} + 10794 \nu^{3} + \cdots + 24944 ) / 11664 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 31 \nu^{9} + 28 \nu^{8} - 83 \nu^{7} - 688 \nu^{6} + 1947 \nu^{5} - 6378 \nu^{4} + 18946 \nu^{3} + \cdots + 170872 ) / 87480 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 563 \nu^{9} - 3676 \nu^{8} + 10991 \nu^{7} - 62264 \nu^{6} - 43419 \nu^{5} + 133206 \nu^{4} + \cdots + 5797376 ) / 1049760 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 89 \nu^{9} + 422 \nu^{8} - 1327 \nu^{7} + 778 \nu^{6} + 3003 \nu^{5} + 3048 \nu^{4} + \cdots + 26408 ) / 87480 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1523 \nu^{9} + 10244 \nu^{8} - 41329 \nu^{7} + 41896 \nu^{6} - 62139 \nu^{5} - 537834 \nu^{4} + \cdots - 22297504 ) / 1049760 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{9} - \beta_{8} + \beta_{7} - 2\beta_{6} + 4\beta_{3} + 2\beta_{2} - 10 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{9} + 2\beta_{8} - 5\beta_{6} + \beta_{5} + 5\beta_{4} + 11\beta_{3} - 14\beta_{2} - 4\beta _1 - 44 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 6 \beta_{9} + 15 \beta_{8} - 2 \beta_{7} - 14 \beta_{6} - 2 \beta_{5} + 4 \beta_{4} + 2 \beta_{3} + \cdots - 97 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 33 \beta_{9} + 21 \beta_{8} - 25 \beta_{7} + 75 \beta_{6} - 20 \beta_{5} - 43 \beta_{4} - 98 \beta_{3} + \cdots - 37 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 70 \beta_{9} - 110 \beta_{8} - 68 \beta_{7} + 444 \beta_{6} - 66 \beta_{5} - 187 \beta_{4} + \cdots + 1408 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 124 \beta_{9} - 529 \beta_{8} + 71 \beta_{7} + 684 \beta_{6} + 321 \beta_{5} - 116 \beta_{4} + \cdots + 7457 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 1368 \beta_{9} - 693 \beta_{8} + 860 \beta_{7} - 3128 \beta_{6} + 1662 \beta_{5} + 1229 \beta_{4} + \cdots + 14907 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 3785 \beta_{9} + 1132 \beta_{8} + 2492 \beta_{7} - 21551 \beta_{6} + 1873 \beta_{5} + 5103 \beta_{4} + \cdots - 46020 ) / 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(1\) \(\beta_{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
0.251202 + 2.90505i
2.22708 + 2.73757i
−1.92524 + 0.665541i
−1.46035 − 1.71659i
1.90731 − 2.85951i
0.251202 − 2.90505i
2.22708 − 2.73757i
−1.92524 − 0.665541i
−1.46035 + 1.71659i
1.90731 + 2.85951i
0 −5.03169 + 1.29696i 0 −16.1214 + 9.30769i 0 −10.2118 − 5.89578i 0 23.6358 − 13.0518i 0
47.2 0 −4.74160 − 2.12537i 0 13.2594 − 7.65534i 0 −4.22536 − 2.43952i 0 17.9656 + 20.1553i 0
47.3 0 −1.15275 + 5.06667i 0 5.73939 − 3.31364i 0 25.3681 + 14.6463i 0 −24.3423 − 11.6812i 0
47.4 0 2.97323 + 4.26145i 0 −3.70432 + 2.13869i 0 −29.1403 − 16.8241i 0 −9.31984 + 25.3405i 0
47.5 0 4.95282 − 1.57150i 0 5.32688 − 3.07547i 0 1.70936 + 0.986900i 0 22.0608 − 15.5667i 0
95.1 0 −5.03169 − 1.29696i 0 −16.1214 − 9.30769i 0 −10.2118 + 5.89578i 0 23.6358 + 13.0518i 0
95.2 0 −4.74160 + 2.12537i 0 13.2594 + 7.65534i 0 −4.22536 + 2.43952i 0 17.9656 − 20.1553i 0
95.3 0 −1.15275 − 5.06667i 0 5.73939 + 3.31364i 0 25.3681 − 14.6463i 0 −24.3423 + 11.6812i 0
95.4 0 2.97323 − 4.26145i 0 −3.70432 − 2.13869i 0 −29.1403 + 16.8241i 0 −9.31984 − 25.3405i 0
95.5 0 4.95282 + 1.57150i 0 5.32688 + 3.07547i 0 1.70936 − 0.986900i 0 22.0608 + 15.5667i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.5
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.4.s.c ✓ 10
3.b odd 2 1 432.4.s.c 10
4.b odd 2 1 144.4.s.d yes 10
9.c even 3 1 432.4.s.d 10
9.c even 3 1 1296.4.c.e 10
9.d odd 6 1 144.4.s.d yes 10
9.d odd 6 1 1296.4.c.f 10
12.b even 2 1 432.4.s.d 10
36.f odd 6 1 432.4.s.c 10
36.f odd 6 1 1296.4.c.f 10
36.h even 6 1 inner 144.4.s.c ✓ 10
36.h even 6 1 1296.4.c.e 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.4.s.c ✓ 10 1.a even 1 1 trivial
144.4.s.c ✓ 10 36.h even 6 1 inner
144.4.s.d yes 10 4.b odd 2 1
144.4.s.d yes 10 9.d odd 6 1
432.4.s.c 10 3.b odd 2 1
432.4.s.c 10 36.f odd 6 1
432.4.s.d 10 9.c even 3 1
432.4.s.d 10 12.b even 2 1
1296.4.c.e 10 9.c even 3 1
1296.4.c.e 10 36.h even 6 1
1296.4.c.f 10 9.d odd 6 1
1296.4.c.f 10 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 \(S_{4}^{\mathrm{new}}(144, [\chi])\):

\( T_{5}^{10} - 9 T_{5}^{9} - 300 T_{5}^{8} + 2943 T_{5}^{7} + 89136 T_{5}^{6} - 1358937 T_{5}^{5} + \cdots + 2469692592 \) Copy content Toggle raw display
\( T_{7}^{10} + 33 T_{7}^{9} - 534 T_{7}^{8} - 29601 T_{7}^{7} + 553266 T_{7}^{6} + 22321737 T_{7}^{5} + \cdots + 12527233200 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + 6 T^{9} + \cdots + 14348907 \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 2469692592 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 12527233200 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 22979185782921 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 18\!\cdots\!84 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 71\!\cdots\!28 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 68\!\cdots\!72 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 83\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 49\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{5} - 550 T^{4} + \cdots - 2882353280)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 13\!\cdots\!03 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 46\!\cdots\!87 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 48\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 16\!\cdots\!68 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 16\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 64\!\cdots\!83 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots + 77392428682176)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots + 54363390126368)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 46\!\cdots\!52 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 17\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 19\!\cdots\!72 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 47\!\cdots\!01 \) Copy content Toggle raw display
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