Properties

Label 144.3.q.d
Level $144$
Weight $3$
Character orbit 144.q
Analytic conductor $3.924$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 144.q (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.92371580679\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} - 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 q^{3} + ( 1 - \beta_{1} + \beta_{2} ) q^{5} + ( \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{7} + 9 q^{9} +O(q^{10})\) \( q + 3 q^{3} + ( 1 - \beta_{1} + \beta_{2} ) q^{5} + ( \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{7} + 9 q^{9} + ( 6 + 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{11} + ( -7 + \beta_{1} + 7 \beta_{2} - 2 \beta_{3} ) q^{13} + ( 3 - 3 \beta_{1} + 3 \beta_{2} ) q^{15} + ( 8 - 16 \beta_{2} + 2 \beta_{3} ) q^{17} + ( -2 - 4 \beta_{1} + 2 \beta_{3} ) q^{19} + ( 3 \beta_{1} + 9 \beta_{2} + 3 \beta_{3} ) q^{21} + ( -5 + \beta_{1} - 5 \beta_{2} ) q^{23} + ( -2 \beta_{1} + 10 \beta_{2} - 2 \beta_{3} ) q^{25} + 27 q^{27} + ( 2 - 3 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{29} + ( -37 + \beta_{1} + 37 \beta_{2} - 2 \beta_{3} ) q^{31} + ( 18 + 6 \beta_{1} - 9 \beta_{2} - 6 \beta_{3} ) q^{33} + ( 29 - 58 \beta_{2} ) q^{35} + ( -30 + 4 \beta_{1} - 2 \beta_{3} ) q^{37} + ( -21 + 3 \beta_{1} + 21 \beta_{2} - 6 \beta_{3} ) q^{39} + ( -23 - 23 \beta_{2} ) q^{41} + ( 4 \beta_{1} - 5 \beta_{2} + 4 \beta_{3} ) q^{43} + ( 9 - 9 \beta_{1} + 9 \beta_{2} ) q^{45} + ( -58 - 3 \beta_{1} + 29 \beta_{2} + 3 \beta_{3} ) q^{47} + ( -56 - 6 \beta_{1} + 56 \beta_{2} + 12 \beta_{3} ) q^{49} + ( 24 - 48 \beta_{2} + 6 \beta_{3} ) q^{51} + ( 32 - 64 \beta_{2} - 8 \beta_{3} ) q^{53} + ( -55 - 2 \beta_{1} + \beta_{3} ) q^{55} + ( -6 - 12 \beta_{1} + 6 \beta_{3} ) q^{57} + ( 3 - 2 \beta_{1} + 3 \beta_{2} ) q^{59} + ( -\beta_{1} - 31 \beta_{2} - \beta_{3} ) q^{61} + ( 9 \beta_{1} + 27 \beta_{2} + 9 \beta_{3} ) q^{63} + ( -78 + 10 \beta_{1} + 39 \beta_{2} - 10 \beta_{3} ) q^{65} + ( 11 + 10 \beta_{1} - 11 \beta_{2} - 20 \beta_{3} ) q^{67} + ( -15 + 3 \beta_{1} - 15 \beta_{2} ) q^{69} + ( 24 - 48 \beta_{2} - 2 \beta_{3} ) q^{71} + ( 10 + 12 \beta_{1} - 6 \beta_{3} ) q^{73} + ( -6 \beta_{1} + 30 \beta_{2} - 6 \beta_{3} ) q^{75} + ( 73 + 15 \beta_{1} + 73 \beta_{2} ) q^{77} + ( -7 \beta_{1} + 43 \beta_{2} - 7 \beta_{3} ) q^{79} + 81 q^{81} + ( 22 - 14 \beta_{1} - 11 \beta_{2} + 14 \beta_{3} ) q^{83} + ( 88 - 10 \beta_{1} - 88 \beta_{2} + 20 \beta_{3} ) q^{85} + ( 6 - 9 \beta_{1} - 3 \beta_{2} + 9 \beta_{3} ) q^{87} + ( 24 - 48 \beta_{2} + 6 \beta_{3} ) q^{89} + ( 75 - 8 \beta_{1} + 4 \beta_{3} ) q^{91} + ( -111 + 3 \beta_{1} + 111 \beta_{2} - 6 \beta_{3} ) q^{93} + ( 62 - 4 \beta_{1} + 62 \beta_{2} ) q^{95} + ( -2 \beta_{1} + 121 \beta_{2} - 2 \beta_{3} ) q^{97} + ( 54 + 18 \beta_{1} - 27 \beta_{2} - 18 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 12q^{3} + 6q^{5} + 6q^{7} + 36q^{9} + O(q^{10}) \) \( 4q + 12q^{3} + 6q^{5} + 6q^{7} + 36q^{9} + 18q^{11} - 14q^{13} + 18q^{15} - 8q^{19} + 18q^{21} - 30q^{23} + 20q^{25} + 108q^{27} + 6q^{29} - 74q^{31} + 54q^{33} - 120q^{37} - 42q^{39} - 138q^{41} - 10q^{43} + 54q^{45} - 174q^{47} - 112q^{49} - 220q^{55} - 24q^{57} + 18q^{59} - 62q^{61} + 54q^{63} - 234q^{65} + 22q^{67} - 90q^{69} + 40q^{73} + 60q^{75} + 438q^{77} + 86q^{79} + 324q^{81} + 66q^{83} + 176q^{85} + 18q^{87} + 300q^{91} - 222q^{93} + 372q^{95} + 242q^{97} + 162q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 4 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( 2 \nu^{3} \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/4\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(\beta_{3}\)\(/2\)

Character values

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

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

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
1.22474 0.707107i
−1.22474 + 0.707107i
1.22474 + 0.707107i
−1.22474 0.707107i
0 3.00000 0 −3.39898 + 1.96240i 0 6.39898 11.0834i 0 9.00000 0
65.2 0 3.00000 0 6.39898 3.69445i 0 −3.39898 + 5.88721i 0 9.00000 0
113.1 0 3.00000 0 −3.39898 1.96240i 0 6.39898 + 11.0834i 0 9.00000 0
113.2 0 3.00000 0 6.39898 + 3.69445i 0 −3.39898 5.88721i 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
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 144.3.q.d 4
3.b odd 2 1 432.3.q.c 4
4.b odd 2 1 72.3.m.a 4
8.b even 2 1 576.3.q.c 4
8.d odd 2 1 576.3.q.h 4
9.c even 3 1 432.3.q.c 4
9.c even 3 1 1296.3.e.c 4
9.d odd 6 1 inner 144.3.q.d 4
9.d odd 6 1 1296.3.e.c 4
12.b even 2 1 216.3.m.a 4
24.f even 2 1 1728.3.q.e 4
24.h odd 2 1 1728.3.q.f 4
36.f odd 6 1 216.3.m.a 4
36.f odd 6 1 648.3.e.b 4
36.h even 6 1 72.3.m.a 4
36.h even 6 1 648.3.e.b 4
72.j odd 6 1 576.3.q.c 4
72.l even 6 1 576.3.q.h 4
72.n even 6 1 1728.3.q.f 4
72.p odd 6 1 1728.3.q.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.3.m.a 4 4.b odd 2 1
72.3.m.a 4 36.h even 6 1
144.3.q.d 4 1.a even 1 1 trivial
144.3.q.d 4 9.d odd 6 1 inner
216.3.m.a 4 12.b even 2 1
216.3.m.a 4 36.f odd 6 1
432.3.q.c 4 3.b odd 2 1
432.3.q.c 4 9.c even 3 1
576.3.q.c 4 8.b even 2 1
576.3.q.c 4 72.j odd 6 1
576.3.q.h 4 8.d odd 2 1
576.3.q.h 4 72.l even 6 1
648.3.e.b 4 36.f odd 6 1
648.3.e.b 4 36.h even 6 1
1296.3.e.c 4 9.c even 3 1
1296.3.e.c 4 9.d odd 6 1
1728.3.q.e 4 24.f even 2 1
1728.3.q.e 4 72.p odd 6 1
1728.3.q.f 4 24.h odd 2 1
1728.3.q.f 4 72.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 6 T_{5}^{3} - 17 T_{5}^{2} + 174 T_{5} + 841 \) acting on \(S_{3}^{\mathrm{new}}(144, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -3 + T )^{4} \)
$5$ \( 841 + 174 T - 17 T^{2} - 6 T^{3} + T^{4} \)
$7$ \( 7569 + 522 T + 123 T^{2} - 6 T^{3} + T^{4} \)
$11$ \( 10201 + 1818 T + 7 T^{2} - 18 T^{3} + T^{4} \)
$13$ \( 2209 - 658 T + 243 T^{2} + 14 T^{3} + T^{4} \)
$17$ \( 4096 + 640 T^{2} + T^{4} \)
$19$ \( ( -380 + 4 T + T^{2} )^{2} \)
$23$ \( 1849 + 1290 T + 343 T^{2} + 30 T^{3} + T^{4} \)
$29$ \( 81225 + 1710 T - 273 T^{2} - 6 T^{3} + T^{4} \)
$31$ \( 1620529 + 94202 T + 4203 T^{2} + 74 T^{3} + T^{4} \)
$37$ \( ( 516 + 60 T + T^{2} )^{2} \)
$41$ \( ( 1587 + 69 T + T^{2} )^{2} \)
$43$ \( 2283121 - 15110 T + 1611 T^{2} + 10 T^{3} + T^{4} \)
$47$ \( 4995225 + 388890 T + 12327 T^{2} + 174 T^{3} + T^{4} \)
$53$ \( 1048576 + 10240 T^{2} + T^{4} \)
$59$ \( 10201 + 1818 T + 7 T^{2} - 18 T^{3} + T^{4} \)
$61$ \( 748225 + 53630 T + 2979 T^{2} + 62 T^{3} + T^{4} \)
$67$ \( 89851441 + 208538 T + 9963 T^{2} - 22 T^{3} + T^{4} \)
$71$ \( 2560000 + 3712 T^{2} + T^{4} \)
$73$ \( ( -3356 - 20 T + T^{2} )^{2} \)
$79$ \( 8151025 + 245530 T + 10251 T^{2} - 86 T^{3} + T^{4} \)
$83$ \( 34916281 + 389994 T - 4457 T^{2} - 66 T^{3} + T^{4} \)
$89$ \( 331776 + 5760 T^{2} + T^{4} \)
$97$ \( 203262049 - 3450194 T + 44307 T^{2} - 242 T^{3} + T^{4} \)
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