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Properties

Label 144.12.c.c
Level $144$
Weight $12$
Character orbit 144.c
Analytic conductor $110.641$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [144,12,Mod(143,144)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(144, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("144.143");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 144.c (of order \(2\), degree \(1\), 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: \(110.641418001\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} - 582998 x^{14} + 1964760 x^{13} + 138496214853 x^{12} - 382372991352 x^{11} + \cdots + 57\!\cdots\!16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{118}\cdot 3^{58} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{6} q^{5} + \beta_{5} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{5} + \beta_{5} q^{7} + \beta_{9} q^{11} + ( - \beta_1 + 218272) q^{13} + (\beta_{12} - 31 \beta_{8} - 54 \beta_{6}) q^{17} + ( - \beta_{7} - 92 \beta_{5} + 29 \beta_{2}) q^{19} + (\beta_{15} + 2 \beta_{14} - \beta_{11}) q^{23} + ( - 5 \beta_{4} + 19 \beta_{3} + \cdots - 16359037) q^{25}+ \cdots + ( - 2425 \beta_{4} + \cdots + 33364742848) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 3492352 q^{13} - 261744592 q^{25} + 542785952 q^{37} + 182984560 q^{49} + 10312913120 q^{61} + 59516841472 q^{73} - 42540225696 q^{85} + 533835885568 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} - 582998 x^{14} + 1964760 x^{13} + 138496214853 x^{12} - 382372991352 x^{11} + \cdots + 57\!\cdots\!16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 25\!\cdots\!39 \nu^{15} + \cdots + 66\!\cdots\!84 ) / 44\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 68\!\cdots\!48 \nu^{15} + \cdots + 19\!\cdots\!68 ) / 74\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 79\!\cdots\!89 \nu^{15} + \cdots + 11\!\cdots\!84 ) / 33\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 51\!\cdots\!17 \nu^{15} + \cdots - 12\!\cdots\!64 ) / 19\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 49\!\cdots\!03 \nu^{15} + \cdots - 11\!\cdots\!88 ) / 80\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 12\!\cdots\!49 \nu^{15} + \cdots - 15\!\cdots\!60 ) / 16\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 65\!\cdots\!87 \nu^{15} + \cdots + 16\!\cdots\!08 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 97\!\cdots\!41 \nu^{15} + \cdots + 38\!\cdots\!04 ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 74\!\cdots\!43 \nu^{15} + \cdots - 81\!\cdots\!12 ) / 66\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 57\!\cdots\!13 \nu^{15} + \cdots - 36\!\cdots\!92 ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 10\!\cdots\!61 \nu^{15} + \cdots - 76\!\cdots\!24 ) / 55\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 70\!\cdots\!27 \nu^{15} + \cdots + 12\!\cdots\!88 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 77\!\cdots\!99 \nu^{15} + \cdots - 35\!\cdots\!44 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 13\!\cdots\!63 \nu^{15} + \cdots + 36\!\cdots\!68 ) / 13\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 80\!\cdots\!29 \nu^{15} + \cdots + 89\!\cdots\!36 ) / 66\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 24 \beta_{15} + 3 \beta_{14} + 57 \beta_{11} - 255 \beta_{9} - 16384 \beta_{8} + \cdots + 161243136 ) / 644972544 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 6816 \beta_{15} + 1014 \beta_{14} + 2592 \beta_{13} - 7776 \beta_{12} - 24798 \beta_{11} + \cdots + 47002857873408 ) / 644972544 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 4760928 \beta_{15} - 3982761 \beta_{14} + 97200 \beta_{13} + 10345968 \beta_{12} + \cdots + 44412809379840 ) / 644972544 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 601651248 \beta_{15} - 279591981 \beta_{14} + 80768016 \beta_{13} - 427435056 \beta_{12} + \cdots + 12\!\cdots\!44 ) / 161243136 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 827417891904 \beta_{15} - 1010403591387 \beta_{14} + 23666936400 \beta_{13} + \cdots + 85\!\cdots\!76 ) / 644972544 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 470188001179296 \beta_{15} - 253679570602326 \beta_{14} + 49191034213728 \beta_{13} + \cdots + 60\!\cdots\!44 ) / 644972544 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 13\!\cdots\!60 \beta_{15} + \cdots + 14\!\cdots\!52 ) / 644972544 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 97\!\cdots\!40 \beta_{15} + \cdots + 93\!\cdots\!88 ) / 80621568 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 21\!\cdots\!28 \beta_{15} + \cdots + 23\!\cdots\!60 ) / 644972544 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 12\!\cdots\!16 \beta_{15} + \cdots + 92\!\cdots\!28 ) / 644972544 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 33\!\cdots\!20 \beta_{15} + \cdots + 34\!\cdots\!72 ) / 644972544 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 44\!\cdots\!44 \beta_{15} + \cdots + 28\!\cdots\!68 ) / 161243136 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 48\!\cdots\!16 \beta_{15} + \cdots + 50\!\cdots\!44 ) / 644972544 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 25\!\cdots\!72 \beta_{15} + \cdots + 14\!\cdots\!00 ) / 644972544 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 69\!\cdots\!68 \beta_{15} + \cdots + 72\!\cdots\!00 ) / 644972544 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
143.1
352.826 − 0.707107i
350.376 − 0.707107i
−352.138 + 0.707107i
−349.689 + 0.707107i
150.940 + 0.707107i
148.490 + 0.707107i
−150.627 − 0.707107i
−148.178 − 0.707107i
−148.178 + 0.707107i
−150.627 + 0.707107i
148.490 − 0.707107i
150.940 − 0.707107i
−349.689 − 0.707107i
−352.138 − 0.707107i
350.376 + 0.707107i
352.826 + 0.707107i
0 0 0 − 13363.4i 0 − 38839.2i 0 0 0
143.2 0 0 0 − 13363.4i 0 38839.2i 0 0 0
143.3 0 0 0 − 7915.04i 0 − 52093.4i 0 0 0
143.4 0 0 0 − 7915.04i 0 52093.4i 0 0 0
143.5 0 0 0 − 3124.52i 0 − 59143.7i 0 0 0
143.6 0 0 0 − 3124.52i 0 59143.7i 0 0 0
143.7 0 0 0 − 3123.70i 0 − 11974.2i 0 0 0
143.8 0 0 0 − 3123.70i 0 11974.2i 0 0 0
143.9 0 0 0 3123.70i 0 − 11974.2i 0 0 0
143.10 0 0 0 3123.70i 0 11974.2i 0 0 0
143.11 0 0 0 3124.52i 0 − 59143.7i 0 0 0
143.12 0 0 0 3124.52i 0 59143.7i 0 0 0
143.13 0 0 0 7915.04i 0 − 52093.4i 0 0 0
143.14 0 0 0 7915.04i 0 52093.4i 0 0 0
143.15 0 0 0 13363.4i 0 − 38839.2i 0 0 0
143.16 0 0 0 13363.4i 0 38839.2i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 143.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.12.c.c ✓ 16
3.b odd 2 1 inner 144.12.c.c ✓ 16
4.b odd 2 1 inner 144.12.c.c ✓ 16
12.b even 2 1 inner 144.12.c.c ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.12.c.c ✓ 16 1.a even 1 1 trivial
144.12.c.c ✓ 16 3.b odd 2 1 inner
144.12.c.c ✓ 16 4.b odd 2 1 inner
144.12.c.c ✓ 16 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 260748648 T_{5}^{6} + \cdots + 10\!\cdots\!00 \) acting on \(S_{12}^{\mathrm{new}}(144, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + \cdots + 20\!\cdots\!44)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 22\!\cdots\!56)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots + 13\!\cdots\!44)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 25\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 68\!\cdots\!04)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 97\!\cdots\!64)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 79\!\cdots\!64)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 31\!\cdots\!76)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots - 36\!\cdots\!36)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 10\!\cdots\!64)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 42\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 51\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 87\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 88\!\cdots\!48)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 14\!\cdots\!04)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 66\!\cdots\!72)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 91\!\cdots\!56)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 72\!\cdots\!36)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots - 66\!\cdots\!72)^{4} \) Copy content Toggle raw display
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