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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [216,4,Mod(35,216)] 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("216.35"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(216, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 216.l (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.7444125612\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label 35.1
Root \(-1.22474 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 216.35
Dual form 216.4.l.a.179.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-2.44949 - 1.41421i) q^{2} +(4.00000 + 6.92820i) q^{4} -22.6274i q^{8} +(-17.1186 - 9.88344i) q^{11} +(-32.0000 + 55.4256i) q^{16} +24.2022i q^{17} +163.227 q^{19} +(27.9546 + 48.4188i) q^{22} +(62.5000 - 108.253i) q^{25} +(156.767 - 90.5097i) q^{32} +(34.2270 - 59.2830i) q^{34} +(-399.823 - 230.838i) q^{38} +(367.005 - 211.890i) q^{41} +(281.931 - 488.319i) q^{43} -158.135i q^{44} +(-171.500 - 297.047i) q^{49} +(-306.186 + 176.777i) q^{50} +(775.346 - 447.646i) q^{59} -512.000 q^{64} +(456.476 + 790.640i) q^{67} +(-167.678 + 96.8087i) q^{68} -799.089 q^{73} +(652.908 + 1130.87i) q^{76} -1198.63 q^{82} +(590.327 + 340.825i) q^{83} +(-1381.18 + 797.422i) q^{86} +(-223.637 + 387.350i) q^{88} +1329.36i q^{89} +(455.455 - 788.870i) q^{97} +970.151i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 16 q^{4} + 54 q^{11} - 128 q^{16} + 212 q^{19} + 200 q^{22} + 250 q^{25} - 304 q^{34} - 1080 q^{38} + 1566 q^{41} + 290 q^{43} - 686 q^{49} + 2538 q^{59} - 2048 q^{64} - 70 q^{67} - 2160 q^{68} + 860 q^{73}+ \cdots + 1910 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/216\mathbb{Z}\right)^\times\).

\(n\) \(55\) \(109\) \(137\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{1}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.44949 − 1.41421i −0.866025 − 0.500000i
\(3\) 0 0
\(4\) 4.00000 + 6.92820i 0.500000 + 0.866025i
\(5\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(6\) 0 0
\(7\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(8\) − 22.6274i − 1.00000i
\(9\) 0 0
\(10\) 0 0
\(11\) −17.1186 − 9.88344i −0.469224 − 0.270906i 0.246691 − 0.969094i \(-0.420657\pi\)
−0.715915 + 0.698188i \(0.753990\pi\)
\(12\) 0 0
\(13\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −32.0000 + 55.4256i −0.500000 + 0.866025i
\(17\) 24.2022i 0.345288i 0.984984 + 0.172644i \(0.0552309\pi\)
−0.984984 + 0.172644i \(0.944769\pi\)
\(18\) 0 0
\(19\) 163.227 1.97089 0.985443 − 0.170004i \(-0.0543779\pi\)
0.985443 + 0.170004i \(0.0543779\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 27.9546 + 48.4188i 0.270906 + 0.469224i
\(23\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(24\) 0 0
\(25\) 62.5000 − 108.253i 0.500000 − 0.866025i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(30\) 0 0
\(31\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(32\) 156.767 − 90.5097i 0.866025 − 0.500000i
\(33\) 0 0
\(34\) 34.2270 − 59.2830i 0.172644 − 0.299028i
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) −399.823 − 230.838i −1.70684 − 0.985443i
\(39\) 0 0
\(40\) 0 0
\(41\) 367.005 − 211.890i 1.39797 − 0.807116i 0.403786 − 0.914854i \(-0.367694\pi\)
0.994179 + 0.107738i \(0.0343608\pi\)
\(42\) 0 0
\(43\) 281.931 − 488.319i 0.999864 − 1.73181i 0.485624 − 0.874168i \(-0.338592\pi\)
0.514239 − 0.857647i \(-0.328074\pi\)
\(44\) − 158.135i − 0.541813i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(48\) 0 0
\(49\) −171.500 − 297.047i −0.500000 − 0.866025i
\(50\) −306.186 + 176.777i −0.866025 + 0.500000i
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 775.346 − 447.646i 1.71087 − 0.987772i 0.777483 − 0.628904i \(-0.216496\pi\)
0.933388 − 0.358868i \(-0.116837\pi\)
\(60\) 0 0
\(61\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −512.000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 456.476 + 790.640i 0.832350 + 1.44167i 0.896170 + 0.443711i \(0.146338\pi\)
−0.0638199 + 0.997961i \(0.520328\pi\)
\(68\) −167.678 + 96.8087i −0.299028 + 0.172644i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −799.089 −1.28118 −0.640591 − 0.767882i \(-0.721311\pi\)
−0.640591 + 0.767882i \(0.721311\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 652.908 + 1130.87i 0.985443 + 1.70684i
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −1198.63 −1.61423
\(83\) 590.327 + 340.825i 0.780684 + 0.450728i 0.836673 − 0.547703i \(-0.184498\pi\)
−0.0559884 + 0.998431i \(0.517831\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1381.18 + 797.422i −1.73181 + 0.999864i
\(87\) 0 0
\(88\) −223.637 + 387.350i −0.270906 + 0.469224i
\(89\) 1329.36i 1.58328i 0.610988 + 0.791640i \(0.290773\pi\)
−0.610988 + 0.791640i \(0.709227\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 455.455 − 788.870i 0.476746 − 0.825749i −0.522898 − 0.852395i \(-0.675149\pi\)
0.999645 + 0.0266459i \(0.00848265\pi\)
\(98\) 970.151i 1.00000i
\(99\) 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 216.4.l.a.35.1 4
3.2 odd 2 72.4.l.a.11.2 ✓ 4
4.3 odd 2 864.4.p.a.143.2 4
8.3 odd 2 CM 216.4.l.a.35.1 4
8.5 even 2 864.4.p.a.143.2 4
9.4 even 3 72.4.l.a.59.2 yes 4
9.5 odd 6 inner 216.4.l.a.179.1 4
12.11 even 2 288.4.p.a.47.1 4
24.5 odd 2 288.4.p.a.47.1 4
24.11 even 2 72.4.l.a.11.2 ✓ 4
36.23 even 6 864.4.p.a.719.2 4
36.31 odd 6 288.4.p.a.239.1 4
72.5 odd 6 864.4.p.a.719.2 4
72.13 even 6 288.4.p.a.239.1 4
72.59 even 6 inner 216.4.l.a.179.1 4
72.67 odd 6 72.4.l.a.59.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.4.l.a.11.2 ✓ 4 3.2 odd 2
72.4.l.a.11.2 ✓ 4 24.11 even 2
72.4.l.a.59.2 yes 4 9.4 even 3
72.4.l.a.59.2 yes 4 72.67 odd 6
216.4.l.a.35.1 4 1.1 even 1 trivial
216.4.l.a.35.1 4 8.3 odd 2 CM
216.4.l.a.179.1 4 9.5 odd 6 inner
216.4.l.a.179.1 4 72.59 even 6 inner
288.4.p.a.47.1 4 12.11 even 2
288.4.p.a.47.1 4 24.5 odd 2
288.4.p.a.239.1 4 36.31 odd 6
288.4.p.a.239.1 4 72.13 even 6
864.4.p.a.143.2 4 4.3 odd 2
864.4.p.a.143.2 4 8.5 even 2
864.4.p.a.719.2 4 36.23 even 6
864.4.p.a.719.2 4 72.5 odd 6