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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1728,2,Mod(145,1728)] 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("1728.145"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1728, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([0, 9, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1728.bc (of order \(12\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [72,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.7981494693\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(18\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 144)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 145.6
Character \(\chi\) \(=\) 1728.145
Dual form 1728.2.bc.e.1585.6

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.430214 - 1.60558i) q^{5} +(3.62762 - 2.09441i) q^{7} +(-4.63241 - 1.24125i) q^{11} +(3.28323 - 0.879738i) q^{13} +2.14142 q^{17} +(-1.03156 - 1.03156i) q^{19} +(0.405884 + 0.234337i) q^{23} +(1.93733 - 1.11852i) q^{25} +(1.75557 - 6.55186i) q^{29} +(-3.18054 + 5.50886i) q^{31} +(-4.92339 - 4.92339i) q^{35} +(-0.728237 + 0.728237i) q^{37} +(2.52351 + 1.45695i) q^{41} +(-10.6289 - 2.84802i) q^{43} +(-4.61716 - 7.99715i) q^{47} +(5.27308 - 9.13324i) q^{49} +(-1.17892 + 1.17892i) q^{53} +7.97171i q^{55} +(0.397061 + 1.48185i) q^{59} +(-2.01826 + 7.53224i) q^{61} +(-2.82498 - 4.89300i) q^{65} +(9.82470 - 2.63252i) q^{67} +8.27863i q^{71} -8.16641i q^{73} +(-19.4043 + 5.19937i) q^{77} +(-3.63065 - 6.28847i) q^{79} +(2.03413 - 7.59149i) q^{83} +(-0.921267 - 3.43821i) q^{85} +11.1968i q^{89} +(10.0678 - 10.0678i) q^{91} +(-1.21246 + 2.10004i) q^{95} +(-5.67759 - 9.83387i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q - 4 q^{5} - 2 q^{11} - 16 q^{13} + 16 q^{17} - 28 q^{19} - 4 q^{29} - 28 q^{31} - 16 q^{35} + 16 q^{37} + 10 q^{43} - 56 q^{47} + 4 q^{49} + 8 q^{53} - 14 q^{59} - 32 q^{61} + 64 q^{65} + 18 q^{67}+ \cdots + 40 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −0.430214 − 1.60558i −0.192397 − 0.718037i −0.992925 − 0.118741i \(-0.962114\pi\)
0.800528 − 0.599296i \(-0.204553\pi\)
\(6\) 0 0
\(7\) 3.62762 − 2.09441i 1.37111 − 0.791611i 0.380043 − 0.924969i \(-0.375909\pi\)
0.991068 + 0.133358i \(0.0425759\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −4.63241 − 1.24125i −1.39673 − 0.374251i −0.519557 − 0.854436i \(-0.673903\pi\)
−0.877168 + 0.480184i \(0.840570\pi\)
\(12\) 0 0
\(13\) 3.28323 − 0.879738i 0.910603 − 0.243995i 0.227039 − 0.973886i \(-0.427096\pi\)
0.683564 + 0.729890i \(0.260429\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.14142 0.519370 0.259685 − 0.965693i \(-0.416381\pi\)
0.259685 + 0.965693i \(0.416381\pi\)
\(18\) 0 0
\(19\) −1.03156 − 1.03156i −0.236656 − 0.236656i 0.578808 − 0.815464i \(-0.303518\pi\)
−0.815464 + 0.578808i \(0.803518\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.405884 + 0.234337i 0.0846327 + 0.0488627i 0.541719 − 0.840560i \(-0.317774\pi\)
−0.457086 + 0.889422i \(0.651107\pi\)
\(24\) 0 0
\(25\) 1.93733 − 1.11852i 0.387465 − 0.223703i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.75557 − 6.55186i 0.326000 − 1.21665i −0.587302 − 0.809368i \(-0.699810\pi\)
0.913302 − 0.407282i \(-0.133523\pi\)
\(30\) 0 0
\(31\) −3.18054 + 5.50886i −0.571242 + 0.989421i 0.425196 + 0.905101i \(0.360205\pi\)
−0.996439 + 0.0843198i \(0.973128\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4.92339 − 4.92339i −0.832204 − 0.832204i
\(36\) 0 0
\(37\) −0.728237 + 0.728237i −0.119721 + 0.119721i −0.764429 − 0.644708i \(-0.776979\pi\)
0.644708 + 0.764429i \(0.276979\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.52351 + 1.45695i 0.394105 + 0.227537i 0.683937 − 0.729541i \(-0.260266\pi\)
−0.289832 + 0.957078i \(0.593599\pi\)
\(42\) 0 0
\(43\) −10.6289 − 2.84802i −1.62090 − 0.434318i −0.669633 − 0.742692i \(-0.733549\pi\)
−0.951265 + 0.308374i \(0.900215\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.61716 − 7.99715i −0.673482 − 1.16650i −0.976910 − 0.213650i \(-0.931465\pi\)
0.303429 − 0.952854i \(-0.401869\pi\)
\(48\) 0 0
\(49\) 5.27308 − 9.13324i 0.753297 − 1.30475i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.17892 + 1.17892i −0.161937 + 0.161937i −0.783424 − 0.621487i \(-0.786529\pi\)
0.621487 + 0.783424i \(0.286529\pi\)
\(54\) 0 0
\(55\) 7.97171i 1.07491i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0.397061 + 1.48185i 0.0516930 + 0.192921i 0.986944 − 0.161065i \(-0.0514928\pi\)
−0.935251 + 0.353986i \(0.884826\pi\)
\(60\) 0 0
\(61\) −2.01826 + 7.53224i −0.258411 + 0.964404i 0.707749 + 0.706464i \(0.249711\pi\)
−0.966161 + 0.257941i \(0.916956\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.82498 − 4.89300i −0.350395 − 0.606902i
\(66\) 0 0
\(67\) 9.82470 − 2.63252i 1.20028 − 0.321614i 0.397338 − 0.917673i \(-0.369934\pi\)
0.802941 + 0.596059i \(0.203268\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 8.27863i 0.982493i 0.871021 + 0.491246i \(0.163458\pi\)
−0.871021 + 0.491246i \(0.836542\pi\)
\(72\) 0 0
\(73\) − 8.16641i − 0.955806i −0.878413 − 0.477903i \(-0.841397\pi\)
0.878413 − 0.477903i \(-0.158603\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −19.4043 + 5.19937i −2.21133 + 0.592523i
\(78\) 0 0
\(79\) −3.63065 − 6.28847i −0.408480 − 0.707508i 0.586240 − 0.810138i \(-0.300608\pi\)
−0.994720 + 0.102630i \(0.967274\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 2.03413 − 7.59149i 0.223275 − 0.833274i −0.759813 − 0.650142i \(-0.774710\pi\)
0.983088 − 0.183133i \(-0.0586238\pi\)
\(84\) 0 0
\(85\) −0.921267 − 3.43821i −0.0999254 − 0.372927i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 11.1968i 1.18686i 0.804886 + 0.593429i \(0.202226\pi\)
−0.804886 + 0.593429i \(0.797774\pi\)
\(90\) 0 0
\(91\) 10.0678 − 10.0678i 1.05539 − 1.05539i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.21246 + 2.10004i −0.124396 + 0.215460i
\(96\) 0 0
\(97\) −5.67759 − 9.83387i −0.576472 − 0.998479i −0.995880 − 0.0906808i \(-0.971096\pi\)
0.419408 − 0.907798i \(-0.362238\pi\)
\(98\) 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 1728.2.bc.e.145.6 72
3.2 odd 2 576.2.bb.e.337.8 72
4.3 odd 2 432.2.y.e.253.1 72
9.2 odd 6 576.2.bb.e.529.16 72
9.7 even 3 inner 1728.2.bc.e.721.13 72
12.11 even 2 144.2.x.e.13.18 ✓ 72
16.5 even 4 inner 1728.2.bc.e.1009.13 72
16.11 odd 4 432.2.y.e.37.12 72
36.7 odd 6 432.2.y.e.397.12 72
36.11 even 6 144.2.x.e.61.7 yes 72
48.5 odd 4 576.2.bb.e.49.16 72
48.11 even 4 144.2.x.e.85.7 yes 72
144.11 even 12 144.2.x.e.133.18 yes 72
144.43 odd 12 432.2.y.e.181.1 72
144.101 odd 12 576.2.bb.e.241.8 72
144.133 even 12 inner 1728.2.bc.e.1585.6 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
144.2.x.e.13.18 ✓ 72 12.11 even 2
144.2.x.e.61.7 yes 72 36.11 even 6
144.2.x.e.85.7 yes 72 48.11 even 4
144.2.x.e.133.18 yes 72 144.11 even 12
432.2.y.e.37.12 72 16.11 odd 4
432.2.y.e.181.1 72 144.43 odd 12
432.2.y.e.253.1 72 4.3 odd 2
432.2.y.e.397.12 72 36.7 odd 6
576.2.bb.e.49.16 72 48.5 odd 4
576.2.bb.e.241.8 72 144.101 odd 12
576.2.bb.e.337.8 72 3.2 odd 2
576.2.bb.e.529.16 72 9.2 odd 6
1728.2.bc.e.145.6 72 1.1 even 1 trivial
1728.2.bc.e.721.13 72 9.7 even 3 inner
1728.2.bc.e.1009.13 72 16.5 even 4 inner
1728.2.bc.e.1585.6 72 144.133 even 12 inner