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Properties

Label 432.5.e.e.161.1
Level $432$
Weight $5$
Character 432.161
Analytic conductor $44.656$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [432,5,Mod(161,432)] 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("432.161"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(432, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 432.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,38] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(44.6558240522\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 27)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 161.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 432.161
Dual form 432.5.e.e.161.2

$q$-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)
 
\(f(q)\) \(=\) \(q-33.0000i q^{5} +19.0000 q^{7} -123.000i q^{11} +302.000 q^{13} +414.000i q^{17} +304.000 q^{19} -300.000i q^{23} -464.000 q^{25} -678.000i q^{29} -239.000 q^{31} -627.000i q^{35} +740.000 q^{37} +228.000i q^{41} +982.000 q^{43} -2166.00i q^{47} -2040.00 q^{49} -1593.00i q^{53} -4059.00 q^{55} +2922.00i q^{59} -316.000 q^{61} -9966.00i q^{65} -4622.00 q^{67} -1818.00i q^{71} -3031.00 q^{73} -2337.00i q^{77} +10450.0 q^{79} -12633.0i q^{83} +13662.0 q^{85} -7002.00i q^{89} +5738.00 q^{91} -10032.0i q^{95} -6517.00 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 38 q^{7} + 604 q^{13} + 608 q^{19} - 928 q^{25} - 478 q^{31} + 1480 q^{37} + 1964 q^{43} - 4080 q^{49} - 8118 q^{55} - 632 q^{61} - 9244 q^{67} - 6062 q^{73} + 20900 q^{79} + 27324 q^{85} + 11476 q^{91}+ \cdots - 13034 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/432\mathbb{Z}\right)^\times\).

\(n\) \(271\) \(325\) \(353\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) − 33.0000i − 1.32000i −0.751266 − 0.660000i \(-0.770556\pi\)
0.751266 − 0.660000i \(-0.229444\pi\)
\(6\) 0 0
\(7\) 19.0000 0.387755 0.193878 − 0.981026i \(-0.437894\pi\)
0.193878 + 0.981026i \(0.437894\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 123.000i − 1.01653i −0.861201 − 0.508264i \(-0.830287\pi\)
0.861201 − 0.508264i \(-0.169713\pi\)
\(12\) 0 0
\(13\) 302.000 1.78698 0.893491 − 0.449081i \(-0.148248\pi\)
0.893491 + 0.449081i \(0.148248\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 414.000i 1.43253i 0.697830 + 0.716263i \(0.254149\pi\)
−0.697830 + 0.716263i \(0.745851\pi\)
\(18\) 0 0
\(19\) 304.000 0.842105 0.421053 − 0.907036i \(-0.361661\pi\)
0.421053 + 0.907036i \(0.361661\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 300.000i − 0.567108i −0.958956 − 0.283554i \(-0.908487\pi\)
0.958956 − 0.283554i \(-0.0915135\pi\)
\(24\) 0 0
\(25\) −464.000 −0.742400
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 678.000i − 0.806183i −0.915160 − 0.403092i \(-0.867936\pi\)
0.915160 − 0.403092i \(-0.132064\pi\)
\(30\) 0 0
\(31\) −239.000 −0.248699 −0.124350 − 0.992238i \(-0.539684\pi\)
−0.124350 + 0.992238i \(0.539684\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 627.000i − 0.511837i
\(36\) 0 0
\(37\) 740.000 0.540541 0.270270 − 0.962784i \(-0.412887\pi\)
0.270270 + 0.962784i \(0.412887\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 228.000i 0.135634i 0.997698 + 0.0678168i \(0.0216033\pi\)
−0.997698 + 0.0678168i \(0.978397\pi\)
\(42\) 0 0
\(43\) 982.000 0.531098 0.265549 − 0.964097i \(-0.414447\pi\)
0.265549 + 0.964097i \(0.414447\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 2166.00i − 0.980534i −0.871572 − 0.490267i \(-0.836899\pi\)
0.871572 − 0.490267i \(-0.163101\pi\)
\(48\) 0 0
\(49\) −2040.00 −0.849646
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 1593.00i − 0.567106i −0.958957 − 0.283553i \(-0.908487\pi\)
0.958957 − 0.283553i \(-0.0915131\pi\)
\(54\) 0 0
\(55\) −4059.00 −1.34182
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2922.00i 0.839414i 0.907660 + 0.419707i \(0.137867\pi\)
−0.907660 + 0.419707i \(0.862133\pi\)
\(60\) 0 0
\(61\) −316.000 −0.0849234 −0.0424617 − 0.999098i \(-0.513520\pi\)
−0.0424617 + 0.999098i \(0.513520\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 9966.00i − 2.35882i
\(66\) 0 0
\(67\) −4622.00 −1.02963 −0.514814 − 0.857302i \(-0.672139\pi\)
−0.514814 + 0.857302i \(0.672139\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 1818.00i − 0.360643i −0.983608 − 0.180321i \(-0.942286\pi\)
0.983608 − 0.180321i \(-0.0577138\pi\)
\(72\) 0 0
\(73\) −3031.00 −0.568775 −0.284387 − 0.958709i \(-0.591790\pi\)
−0.284387 + 0.958709i \(0.591790\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 2337.00i − 0.394164i
\(78\) 0 0
\(79\) 10450.0 1.67441 0.837206 − 0.546888i \(-0.184188\pi\)
0.837206 + 0.546888i \(0.184188\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 12633.0i − 1.83379i −0.399125 − 0.916897i \(-0.630686\pi\)
0.399125 − 0.916897i \(-0.369314\pi\)
\(84\) 0 0
\(85\) 13662.0 1.89093
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 7002.00i − 0.883979i −0.897020 − 0.441990i \(-0.854273\pi\)
0.897020 − 0.441990i \(-0.145727\pi\)
\(90\) 0 0
\(91\) 5738.00 0.692911
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 10032.0i − 1.11158i
\(96\) 0 0
\(97\) −6517.00 −0.692635 −0.346317 − 0.938117i \(-0.612568\pi\)
−0.346317 + 0.938117i \(0.612568\pi\)
\(98\) 0 0
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 432.5.e.e.161.1 2
3.2 odd 2 inner 432.5.e.e.161.2 2
4.3 odd 2 27.5.b.c.26.1 ✓ 2
12.11 even 2 27.5.b.c.26.2 yes 2
20.3 even 4 675.5.d.a.674.2 2
20.7 even 4 675.5.d.d.674.1 2
20.19 odd 2 675.5.c.h.26.2 2
36.7 odd 6 81.5.d.b.53.1 4
36.11 even 6 81.5.d.b.53.2 4
36.23 even 6 81.5.d.b.26.1 4
36.31 odd 6 81.5.d.b.26.2 4
60.23 odd 4 675.5.d.d.674.2 2
60.47 odd 4 675.5.d.a.674.1 2
60.59 even 2 675.5.c.h.26.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.5.b.c.26.1 ✓ 2 4.3 odd 2
27.5.b.c.26.2 yes 2 12.11 even 2
81.5.d.b.26.1 4 36.23 even 6
81.5.d.b.26.2 4 36.31 odd 6
81.5.d.b.53.1 4 36.7 odd 6
81.5.d.b.53.2 4 36.11 even 6
432.5.e.e.161.1 2 1.1 even 1 trivial
432.5.e.e.161.2 2 3.2 odd 2 inner
675.5.c.h.26.1 2 60.59 even 2
675.5.c.h.26.2 2 20.19 odd 2
675.5.d.a.674.1 2 60.47 odd 4
675.5.d.a.674.2 2 20.3 even 4
675.5.d.d.674.1 2 20.7 even 4
675.5.d.d.674.2 2 60.23 odd 4