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Properties

Label 144.4.i.e
Level $144$
Weight $4$
Character orbit 144.i
Analytic conductor $8.496$
Analytic rank $0$
Dimension $8$
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(49,144)]
 
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([0, 0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("144.49");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 144.i (of order \(3\), degree \(2\), not 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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.5206055409.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + x^{6} + 9x^{5} - 23x^{4} + 27x^{3} + 9x^{2} - 81x + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{3} + (\beta_{4} - \beta_{2} + 2 \beta_1) q^{5} + ( - \beta_{7} - \beta_{6} - \beta_{4} + \cdots - 1) q^{7}+ \cdots + (\beta_{7} + \beta_{5} + 3 \beta_{4} + \cdots - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{3} + (\beta_{4} - \beta_{2} + 2 \beta_1) q^{5} + ( - \beta_{7} - \beta_{6} - \beta_{4} + \cdots - 1) q^{7}+ \cdots + (\beta_{7} + 12 \beta_{6} + 73 \beta_{5} + \cdots + 631) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 3 q^{3} - 5 q^{5} - 3 q^{7} - 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 3 q^{3} - 5 q^{5} - 3 q^{7} - 15 q^{9} - 16 q^{11} + 29 q^{13} - 141 q^{15} - 34 q^{17} + 218 q^{19} + 27 q^{21} - 37 q^{23} + 97 q^{25} + 216 q^{27} - 3 q^{29} - 331 q^{31} + 468 q^{33} + 342 q^{35} - 732 q^{37} + 375 q^{39} - 378 q^{41} - 506 q^{43} - 993 q^{45} - 171 q^{47} + 829 q^{49} - 2193 q^{51} + 820 q^{53} + 2326 q^{55} + 507 q^{57} - 616 q^{59} + 1331 q^{61} + 1899 q^{63} - 815 q^{65} - 1162 q^{67} + 1113 q^{69} + 688 q^{71} - 2614 q^{73} + 1755 q^{75} - 741 q^{77} - 1853 q^{79} - 2547 q^{81} - 1421 q^{83} + 2074 q^{85} - 5877 q^{87} + 1632 q^{89} + 3990 q^{91} + 537 q^{93} - 1292 q^{95} + 2506 q^{97} + 5133 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + x^{6} + 9x^{5} - 23x^{4} + 27x^{3} + 9x^{2} - 81x + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 3\nu^{6} - 8\nu^{5} + 15\nu^{4} - 14\nu^{3} - 30\nu^{2} + 99\nu - 135 ) / 81 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{7} + 15\nu^{6} - 70\nu^{5} + 66\nu^{4} - \nu^{3} - 294\nu^{2} + 522\nu - 432 ) / 81 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 8\nu^{5} + 21\nu^{4} - 5\nu^{3} - 24\nu^{2} + 99\nu - 117 ) / 9 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -16\nu^{7} + 24\nu^{6} - 34\nu^{5} - 60\nu^{4} + 89\nu^{3} - 150\nu^{2} + 360\nu - 27 ) / 81 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -17\nu^{7} + 12\nu^{6} + 82\nu^{5} - 165\nu^{4} + 184\nu^{3} + 6\nu^{2} - 711\nu + 1161 ) / 81 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 22\nu^{7} - 51\nu^{6} - 14\nu^{5} + 240\nu^{4} - 362\nu^{3} + 222\nu^{2} + 396\nu - 1431 ) / 81 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -35\nu^{7} + 84\nu^{6} - 44\nu^{5} - 93\nu^{4} + 490\nu^{3} - 624\nu^{2} - 63\nu + 837 ) / 81 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + 2\beta_{3} - \beta_{2} + 4\beta _1 + 6 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - \beta_{6} - 5\beta_{5} + \beta_{4} - 7\beta_{2} + 2\beta _1 + 10 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{6} - \beta_{5} - \beta_{4} + 3\beta_{3} - 2\beta_{2} - 12\beta _1 - 13 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 5\beta_{7} + 5\beta_{6} - 2\beta_{5} - 2\beta_{4} + \beta_{3} - 7\beta_{2} + 4\beta _1 + 47 ) / 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -5\beta_{7} - \beta_{6} + 22\beta_{5} - 2\beta_{4} + 13\beta_{3} - 7\beta_{2} + 86\beta _1 - 7 ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 3\beta_{7} - \beta_{6} - 5\beta_{5} - 2\beta_{4} - 4\beta_{3} - 10\beta_{2} + 66\beta _1 + 27 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -22\beta_{7} - 58\beta_{6} - 11\beta_{5} - 59\beta_{4} + 23\beta_{3} + 2\beta_{2} + 136\beta _1 - 123 ) / 12 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
1.35516 − 1.07868i
−1.72895 − 0.103515i
0.172469 + 1.72344i
1.70133 + 0.324778i
1.35516 + 1.07868i
−1.72895 + 0.103515i
0.172469 − 1.72344i
1.70133 − 0.324778i
0 −3.96518 + 3.35817i 0 4.00813 − 6.94228i 0 0.468615 + 0.811666i 0 4.44536 − 26.6315i 0
49.2 0 −2.06310 − 4.76903i 0 −0.845922 + 1.46518i 0 −8.57067 − 14.8448i 0 −18.4873 + 19.6779i 0
49.3 0 2.78092 + 4.38936i 0 −8.65944 + 14.9986i 0 −1.18676 − 2.05553i 0 −11.5330 + 24.4129i 0
49.4 0 4.74736 − 2.11248i 0 2.99723 − 5.19136i 0 7.78882 + 13.4906i 0 18.0749 − 20.0574i 0
97.1 0 −3.96518 − 3.35817i 0 4.00813 + 6.94228i 0 0.468615 − 0.811666i 0 4.44536 + 26.6315i 0
97.2 0 −2.06310 + 4.76903i 0 −0.845922 − 1.46518i 0 −8.57067 + 14.8448i 0 −18.4873 − 19.6779i 0
97.3 0 2.78092 − 4.38936i 0 −8.65944 − 14.9986i 0 −1.18676 + 2.05553i 0 −11.5330 − 24.4129i 0
97.4 0 4.74736 + 2.11248i 0 2.99723 + 5.19136i 0 7.78882 − 13.4906i 0 18.0749 + 20.0574i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.4.i.e 8
3.b odd 2 1 432.4.i.e 8
4.b odd 2 1 72.4.i.a ✓ 8
9.c even 3 1 inner 144.4.i.e 8
9.c even 3 1 1296.4.a.ba 4
9.d odd 6 1 432.4.i.e 8
9.d odd 6 1 1296.4.a.y 4
12.b even 2 1 216.4.i.a 8
36.f odd 6 1 72.4.i.a ✓ 8
36.f odd 6 1 648.4.a.i 4
36.h even 6 1 216.4.i.a 8
36.h even 6 1 648.4.a.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.4.i.a ✓ 8 4.b odd 2 1
72.4.i.a ✓ 8 36.f odd 6 1
144.4.i.e 8 1.a even 1 1 trivial
144.4.i.e 8 9.c even 3 1 inner
216.4.i.a 8 12.b even 2 1
216.4.i.a 8 36.h even 6 1
432.4.i.e 8 3.b odd 2 1
432.4.i.e 8 9.d odd 6 1
648.4.a.h 4 36.h even 6 1
648.4.a.i 4 36.f odd 6 1
1296.4.a.y 4 9.d odd 6 1
1296.4.a.ba 4 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 5T_{5}^{7} + 214T_{5}^{6} - 1951T_{5}^{5} + 31798T_{5}^{4} - 109147T_{5}^{3} + 519121T_{5}^{2} + 708224T_{5} + 1982464 \) acting on \(S_{4}^{\mathrm{new}}(144, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 3 T^{7} + \cdots + 531441 \) Copy content Toggle raw display
$5$ \( T^{8} + 5 T^{7} + \cdots + 1982464 \) Copy content Toggle raw display
$7$ \( T^{8} + 3 T^{7} + \cdots + 352836 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 1515467041 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 19954952644 \) Copy content Toggle raw display
$17$ \( (T^{4} + 17 T^{3} + \cdots + 2567224)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 109 T^{3} + \cdots - 5081408)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 14\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 24\!\cdots\!84 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( (T^{4} + 366 T^{3} + \cdots - 215981856)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 13\!\cdots\!89 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 32\!\cdots\!89 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 96\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( (T^{4} - 410 T^{3} + \cdots - 15963093536)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 20\!\cdots\!89 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 23\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 18\!\cdots\!41 \) Copy content Toggle raw display
$71$ \( (T^{4} - 344 T^{3} + \cdots + 9762389248)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 1307 T^{3} + \cdots - 88005243128)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 97\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 46\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( (T^{4} - 816 T^{3} + \cdots + 22003976592)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 67\!\cdots\!49 \) Copy content Toggle raw display
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