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Properties

Label 144.9.q.b
Level $144$
Weight $9$
Character orbit 144.q
Analytic conductor $58.663$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [144,9,Mod(65,144)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("144.65"); S:= CuspForms(chi, 9); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(144, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 9, names="a")
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 144.q (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,-126] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(58.6625198488\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} - 150208 x^{14} - 1927740 x^{13} + 8702363206 x^{12} + 239206241152 x^{11} + \cdots + 81\!\cdots\!61 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{32}\cdot 3^{25} \)
Twist minimal: no (minimal twist has level 18)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + (\beta_{5} + \beta_{2} - 2 \beta_1 - 7) q^{3} + ( - \beta_{5} - \beta_{4} - 2 \beta_{2} + \cdots - 73) q^{5} + ( - \beta_{14} + \beta_{13} + \cdots + 232) q^{7} + (\beta_{15} + \beta_{14} - 3 \beta_{13} + \cdots - 2383) q^{9}+ \cdots + (5916 \beta_{15} - 8880 \beta_{14} + \cdots - 1082316) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 126 q^{3} - 882 q^{5} + 1846 q^{7} - 28662 q^{9} - 45756 q^{11} - 3370 q^{13} - 128754 q^{15} - 362180 q^{19} - 299166 q^{21} - 1311138 q^{23} + 963394 q^{25} + 208656 q^{27} - 2851290 q^{29} - 542438 q^{31}+ \cdots - 366888330 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} - 150208 x^{14} - 1927740 x^{13} + 8702363206 x^{12} + 239206241152 x^{11} + \cdots + 81\!\cdots\!61 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 15\!\cdots\!75 \nu^{15} + \cdots - 22\!\cdots\!25 ) / 38\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 85\!\cdots\!95 \nu^{15} + \cdots + 75\!\cdots\!78 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 43\!\cdots\!64 \nu^{15} + \cdots + 11\!\cdots\!35 ) / 94\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 76\!\cdots\!48 \nu^{15} + \cdots - 57\!\cdots\!59 ) / 94\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 40\!\cdots\!54 \nu^{15} + \cdots + 58\!\cdots\!21 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 60\!\cdots\!63 \nu^{15} + \cdots + 74\!\cdots\!80 ) / 39\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 60\!\cdots\!56 \nu^{15} + \cdots - 12\!\cdots\!03 ) / 39\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 31\!\cdots\!53 \nu^{15} + \cdots - 13\!\cdots\!81 ) / 94\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 84\!\cdots\!83 \nu^{15} + \cdots - 10\!\cdots\!84 ) / 94\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 40\!\cdots\!31 \nu^{15} + \cdots - 77\!\cdots\!06 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 24\!\cdots\!21 \nu^{15} + \cdots - 18\!\cdots\!44 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 52\!\cdots\!27 \nu^{15} + \cdots + 18\!\cdots\!99 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 11\!\cdots\!76 \nu^{15} + \cdots + 17\!\cdots\!57 ) / 60\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 10\!\cdots\!22 \nu^{15} + \cdots + 14\!\cdots\!09 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 51\!\cdots\!15 \nu^{15} + \cdots + 16\!\cdots\!87 ) / 14\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{8} - 3\beta_{5} - 2\beta_{4} - 3\beta_{2} - 8\beta _1 + 7 ) / 9 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 90 \beta_{15} - 30 \beta_{14} + 44 \beta_{13} - 122 \beta_{12} + 90 \beta_{11} - 128 \beta_{10} + \cdots + 507299 ) / 27 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 840 \beta_{15} - 182 \beta_{14} + 7930 \beta_{13} - 7430 \beta_{12} + 1110 \beta_{11} + 5164 \beta_{10} + \cdots + 4536757 ) / 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3625830 \beta_{15} - 1275606 \beta_{14} + 4536320 \beta_{13} - 4487846 \beta_{12} + 3660930 \beta_{11} + \cdots + 17529569576 ) / 27 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 168943320 \beta_{15} - 41248670 \beta_{14} + 1310640334 \beta_{13} - 1357971862 \beta_{12} + \cdots + 939739460064 ) / 27 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 146765172960 \beta_{15} - 54658380624 \beta_{14} + 271482539324 \beta_{13} - 194950088336 \beta_{12} + \cdots + 693774893149451 ) / 27 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 9690336507960 \beta_{15} - 2691335041732 \beta_{14} + 65582561599424 \beta_{13} - 68283525477800 \beta_{12} + \cdots + 53\!\cdots\!99 ) / 27 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 61\!\cdots\!10 \beta_{15} + \cdots + 29\!\cdots\!05 ) / 27 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 17\!\cdots\!00 \beta_{15} + \cdots + 97\!\cdots\!61 ) / 9 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 26\!\cdots\!70 \beta_{15} + \cdots + 12\!\cdots\!78 ) / 27 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 27\!\cdots\!60 \beta_{15} + \cdots + 15\!\cdots\!86 ) / 27 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 12\!\cdots\!00 \beta_{15} + \cdots + 58\!\cdots\!47 ) / 27 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 14\!\cdots\!80 \beta_{15} + \cdots + 79\!\cdots\!25 ) / 27 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 54\!\cdots\!90 \beta_{15} + \cdots + 27\!\cdots\!83 ) / 27 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 24\!\cdots\!60 \beta_{15} + \cdots + 13\!\cdots\!65 ) / 9 \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
−13.0653 − 0.866025i
71.4012 − 0.866025i
−197.435 − 0.866025i
164.354 − 0.866025i
220.333 − 0.866025i
−2.97990 − 0.866025i
−93.2991 − 0.866025i
−147.309 − 0.866025i
−13.0653 + 0.866025i
71.4012 + 0.866025i
−197.435 + 0.866025i
164.354 + 0.866025i
220.333 + 0.866025i
−2.97990 + 0.866025i
−93.2991 + 0.866025i
−147.309 + 0.866025i
0 −77.7361 − 22.7616i 0 −115.044 + 66.4206i 0 1060.32 − 1836.53i 0 5524.82 + 3538.80i 0
65.2 0 −59.0222 + 55.4742i 0 265.055 − 153.030i 0 −770.107 + 1333.86i 0 406.233 − 6548.41i 0
65.3 0 −36.7596 − 72.1785i 0 −944.709 + 545.428i 0 −1250.70 + 2166.27i 0 −3858.46 + 5306.50i 0
65.4 0 −22.8219 + 77.7185i 0 683.343 − 394.528i 0 −89.7132 + 155.388i 0 −5519.32 − 3547.36i 0
65.5 0 −7.38176 − 80.6629i 0 935.250 − 539.967i 0 2056.09 − 3561.25i 0 −6452.02 + 1190.87i 0
65.6 0 24.1444 − 77.3178i 0 −69.6596 + 40.2180i 0 −370.849 + 642.329i 0 −5395.10 − 3733.58i 0
65.7 0 47.1686 + 65.8492i 0 −476.096 + 274.874i 0 1631.36 − 2825.60i 0 −2111.24 + 6212.04i 0
65.8 0 69.4086 + 41.7547i 0 −719.139 + 415.195i 0 −1343.41 + 2326.85i 0 3074.10 + 5796.26i 0
113.1 0 −77.7361 + 22.7616i 0 −115.044 − 66.4206i 0 1060.32 + 1836.53i 0 5524.82 − 3538.80i 0
113.2 0 −59.0222 − 55.4742i 0 265.055 + 153.030i 0 −770.107 − 1333.86i 0 406.233 + 6548.41i 0
113.3 0 −36.7596 + 72.1785i 0 −944.709 − 545.428i 0 −1250.70 − 2166.27i 0 −3858.46 − 5306.50i 0
113.4 0 −22.8219 − 77.7185i 0 683.343 + 394.528i 0 −89.7132 − 155.388i 0 −5519.32 + 3547.36i 0
113.5 0 −7.38176 + 80.6629i 0 935.250 + 539.967i 0 2056.09 + 3561.25i 0 −6452.02 − 1190.87i 0
113.6 0 24.1444 + 77.3178i 0 −69.6596 − 40.2180i 0 −370.849 − 642.329i 0 −5395.10 + 3733.58i 0
113.7 0 47.1686 − 65.8492i 0 −476.096 − 274.874i 0 1631.36 + 2825.60i 0 −2111.24 − 6212.04i 0
113.8 0 69.4086 − 41.7547i 0 −719.139 − 415.195i 0 −1343.41 − 2326.85i 0 3074.10 − 5796.26i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.9.q.b 16
3.b odd 2 1 432.9.q.c 16
4.b odd 2 1 18.9.d.a ✓ 16
9.c even 3 1 432.9.q.c 16
9.d odd 6 1 inner 144.9.q.b 16
12.b even 2 1 54.9.d.a 16
36.f odd 6 1 54.9.d.a 16
36.f odd 6 1 162.9.b.c 16
36.h even 6 1 18.9.d.a ✓ 16
36.h even 6 1 162.9.b.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.9.d.a ✓ 16 4.b odd 2 1
18.9.d.a ✓ 16 36.h even 6 1
54.9.d.a 16 12.b even 2 1
54.9.d.a 16 36.f odd 6 1
144.9.q.b 16 1.a even 1 1 trivial
144.9.q.b 16 9.d odd 6 1 inner
162.9.b.c 16 36.f odd 6 1
162.9.b.c 16 36.h even 6 1
432.9.q.c 16 3.b odd 2 1
432.9.q.c 16 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} + 882 T_{5}^{15} - 1655235 T_{5}^{14} - 1688626926 T_{5}^{13} + 2254411584990 T_{5}^{12} + \cdots + 19\!\cdots\!00 \) acting on \(S_{9}^{\mathrm{new}}(144, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 34\!\cdots\!81 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 52\!\cdots\!49 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 22\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots - 11\!\cdots\!40)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 12\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 73\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots - 17\!\cdots\!56)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 24\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 23\!\cdots\!25 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 18\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 73\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 23\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 63\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 26\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 79\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 27\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 23\!\cdots\!25 \) Copy content Toggle raw display
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