Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [72,8,Mod(25,72)] 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("72.25"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(72, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 72.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [22] 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: \(22.4917218349\)
Analytic rank: \(0\)
Dimension: \(22\)
Relative dimension: \(11\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 49.1
Character \(\chi\) \(=\) 72.49
Dual form 72.8.i.b.25.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-43.7152 + 16.6126i) q^{3} +(42.6088 - 73.8006i) q^{5} +(328.506 + 568.988i) q^{7} +(1635.04 - 1452.45i) q^{9} +(-4204.28 - 7282.02i) q^{11} +(-2210.46 + 3828.63i) q^{13} +(-636.636 + 3934.05i) q^{15} +28792.2 q^{17} +16009.7 q^{19} +(-23813.1 - 19416.1i) q^{21} +(-54163.8 + 93814.5i) q^{23} +(35431.5 + 61369.1i) q^{25} +(-47347.5 + 90656.3i) q^{27} +(18036.9 + 31240.9i) q^{29} +(-84648.5 + 146616. i) q^{31} +(304764. + 248491. i) q^{33} +55988.9 q^{35} -152754. q^{37} +(33027.4 - 204091. i) q^{39} +(-65342.7 + 113177. i) q^{41} +(251342. + 435337. i) q^{43} +(-37524.1 - 182554. i) q^{45} +(405724. + 702734. i) q^{47} +(195940. - 339378. i) q^{49} +(-1.25866e6 + 478313. i) q^{51} +799128. q^{53} -716557. q^{55} +(-699866. + 265962. i) q^{57} +(687208. - 1.19028e6i) q^{59} +(730674. + 1.26556e6i) q^{61} +(1.36355e6 + 453185. i) q^{63} +(188370. + 326266. i) q^{65} +(-1.67297e6 + 2.89766e6i) q^{67} +(809284. - 5.00092e6i) q^{69} -1.97042e6 q^{71} -936218. q^{73} +(-2.56840e6 - 2.09416e6i) q^{75} +(2.76226e6 - 4.78437e6i) q^{77} +(-2.56270e6 - 4.43873e6i) q^{79} +(563771. - 4.74963e6i) q^{81} +(1.20578e6 + 2.08847e6i) q^{83} +(1.22680e6 - 2.12488e6i) q^{85} +(-1.30748e6 - 1.06606e6i) q^{87} +9.09033e6 q^{89} -2.90459e6 q^{91} +(1.26477e6 - 7.81556e6i) q^{93} +(682152. - 1.18152e6i) q^{95} +(2.71305e6 + 4.69913e6i) q^{97} +(-1.74509e7 - 5.79994e6i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 22 q - 44 q^{3} - 125 q^{5} - 1245 q^{7} - 3766 q^{9} + 1699 q^{11} - 4937 q^{13} - 19349 q^{15} + 26540 q^{17} + 28976 q^{19} - 75315 q^{21} - 18239 q^{23} - 109168 q^{25} - 87680 q^{27} - 3525 q^{29}+ \cdots + 24043955 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/72\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(55\) \(65\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(3\) −43.7152 + 16.6126i −0.934778 + 0.355233i
\(4\) 0 0
\(5\) 42.6088 − 73.8006i 0.152442 − 0.264037i −0.779683 − 0.626175i \(-0.784620\pi\)
0.932125 + 0.362138i \(0.117953\pi\)
\(6\) 0 0
\(7\) 328.506 + 568.988i 0.361992 + 0.626989i 0.988289 − 0.152596i \(-0.0487633\pi\)
−0.626296 + 0.779585i \(0.715430\pi\)
\(8\) 0 0
\(9\) 1635.04 − 1452.45i 0.747620 − 0.664127i
\(10\) 0 0
\(11\) −4204.28 − 7282.02i −0.952394 − 1.64960i −0.740221 − 0.672363i \(-0.765279\pi\)
−0.212173 − 0.977232i \(-0.568054\pi\)
\(12\) 0 0
\(13\) −2210.46 + 3828.63i −0.279049 + 0.483327i −0.971149 − 0.238475i \(-0.923353\pi\)
0.692100 + 0.721802i \(0.256686\pi\)
\(14\) 0 0
\(15\) −636.636 + 3934.05i −0.0487048 + 0.300968i
\(16\) 0 0
\(17\) 28792.2 1.42136 0.710680 − 0.703516i \(-0.248388\pi\)
0.710680 + 0.703516i \(0.248388\pi\)
\(18\) 0 0
\(19\) 16009.7 0.535481 0.267741 − 0.963491i \(-0.413723\pi\)
0.267741 + 0.963491i \(0.413723\pi\)
\(20\) 0 0
\(21\) −23813.1 − 19416.1i −0.561110 − 0.457504i
\(22\) 0 0
\(23\) −54163.8 + 93814.5i −0.928243 + 1.60776i −0.141983 + 0.989869i \(0.545348\pi\)
−0.786260 + 0.617896i \(0.787985\pi\)
\(24\) 0 0
\(25\) 35431.5 + 61369.1i 0.453523 + 0.785525i
\(26\) 0 0
\(27\) −47347.5 + 90656.3i −0.462939 + 0.886390i
\(28\) 0 0
\(29\) 18036.9 + 31240.9i 0.137331 + 0.237865i 0.926486 − 0.376330i \(-0.122814\pi\)
−0.789154 + 0.614195i \(0.789481\pi\)
\(30\) 0 0
\(31\) −84648.5 + 146616.i −0.510333 + 0.883922i 0.489596 + 0.871950i \(0.337144\pi\)
−0.999928 + 0.0119726i \(0.996189\pi\)
\(32\) 0 0
\(33\) 304764. + 248491.i 1.47627 + 1.20368i
\(34\) 0 0
\(35\) 55988.9 0.220731
\(36\) 0 0
\(37\) −152754. −0.495776 −0.247888 − 0.968789i \(-0.579736\pi\)
−0.247888 + 0.968789i \(0.579736\pi\)
\(38\) 0 0
\(39\) 33027.4 − 204091.i 0.0891554 − 0.550931i
\(40\) 0 0
\(41\) −65342.7 + 113177.i −0.148065 + 0.256457i −0.930512 − 0.366260i \(-0.880638\pi\)
0.782447 + 0.622717i \(0.213971\pi\)
\(42\) 0 0
\(43\) 251342. + 435337.i 0.482087 + 0.834999i 0.999789 − 0.0205624i \(-0.00654569\pi\)
−0.517702 + 0.855561i \(0.673212\pi\)
\(44\) 0 0
\(45\) −37524.1 − 182554.i −0.0613856 − 0.298640i
\(46\) 0 0
\(47\) 405724. + 702734.i 0.570017 + 0.987299i 0.996563 + 0.0828333i \(0.0263969\pi\)
−0.426546 + 0.904466i \(0.640270\pi\)
\(48\) 0 0
\(49\) 195940. − 339378.i 0.237923 − 0.412095i
\(50\) 0 0
\(51\) −1.25866e6 + 478313.i −1.32866 + 0.504913i
\(52\) 0 0
\(53\) 799128. 0.737311 0.368656 − 0.929566i \(-0.379818\pi\)
0.368656 + 0.929566i \(0.379818\pi\)
\(54\) 0 0
\(55\) −716557. −0.580739
\(56\) 0 0
\(57\) −699866. + 265962.i −0.500556 + 0.190220i
\(58\) 0 0
\(59\) 687208. − 1.19028e6i 0.435618 − 0.754513i −0.561728 − 0.827322i \(-0.689863\pi\)
0.997346 + 0.0728095i \(0.0231965\pi\)
\(60\) 0 0
\(61\) 730674. + 1.26556e6i 0.412163 + 0.713888i 0.995126 − 0.0986113i \(-0.0314400\pi\)
−0.582963 + 0.812499i \(0.698107\pi\)
\(62\) 0 0
\(63\) 1.36355e6 + 453185.i 0.687033 + 0.228341i
\(64\) 0 0
\(65\) 188370. + 326266.i 0.0850776 + 0.147359i
\(66\) 0 0
\(67\) −1.67297e6 + 2.89766e6i −0.679556 + 1.17703i 0.295559 + 0.955325i \(0.404494\pi\)
−0.975115 + 0.221701i \(0.928839\pi\)
\(68\) 0 0
\(69\) 809284. − 5.00092e6i 0.296571 − 1.83265i
\(70\) 0 0
\(71\) −1.97042e6 −0.653363 −0.326681 − 0.945135i \(-0.605930\pi\)
−0.326681 + 0.945135i \(0.605930\pi\)
\(72\) 0 0
\(73\) −936218. −0.281674 −0.140837 − 0.990033i \(-0.544979\pi\)
−0.140837 + 0.990033i \(0.544979\pi\)
\(74\) 0 0
\(75\) −2.56840e6 − 2.09416e6i −0.702987 − 0.573185i
\(76\) 0 0
\(77\) 2.76226e6 − 4.78437e6i 0.689519 − 1.19428i
\(78\) 0 0
\(79\) −2.56270e6 − 4.43873e6i −0.584795 − 1.01289i −0.994901 − 0.100857i \(-0.967842\pi\)
0.410106 − 0.912038i \(-0.365492\pi\)
\(80\) 0 0
\(81\) 563771. − 4.74963e6i 0.117871 − 0.993029i
\(82\) 0 0
\(83\) 1.20578e6 + 2.08847e6i 0.231470 + 0.400917i 0.958241 − 0.285962i \(-0.0923133\pi\)
−0.726771 + 0.686880i \(0.758980\pi\)
\(84\) 0 0
\(85\) 1.22680e6 − 2.12488e6i 0.216675 − 0.375292i
\(86\) 0 0
\(87\) −1.30748e6 − 1.06606e6i −0.212872 − 0.173566i
\(88\) 0 0
\(89\) 9.09033e6 1.36683 0.683416 − 0.730030i \(-0.260494\pi\)
0.683416 + 0.730030i \(0.260494\pi\)
\(90\) 0 0
\(91\) −2.90459e6 −0.404055
\(92\) 0 0
\(93\) 1.26477e6 − 7.81556e6i 0.163050 − 1.00756i
\(94\) 0 0
\(95\) 682152. − 1.18152e6i 0.0816298 − 0.141387i
\(96\) 0 0
\(97\) 2.71305e6 + 4.69913e6i 0.301826 + 0.522777i 0.976550 − 0.215293i \(-0.0690706\pi\)
−0.674724 + 0.738070i \(0.735737\pi\)
\(98\) 0 0
\(99\) −1.74509e7 − 5.79994e6i −1.80757 − 0.600759i
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 72.8.i.b.49.1 yes 22
3.2 odd 2 216.8.i.b.145.5 22
4.3 odd 2 144.8.i.f.49.11 22
9.2 odd 6 216.8.i.b.73.5 22
9.7 even 3 inner 72.8.i.b.25.1 ✓ 22
12.11 even 2 432.8.i.f.145.5 22
36.7 odd 6 144.8.i.f.97.11 22
36.11 even 6 432.8.i.f.289.5 22
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.8.i.b.25.1 ✓ 22 9.7 even 3 inner
72.8.i.b.49.1 yes 22 1.1 even 1 trivial
144.8.i.f.49.11 22 4.3 odd 2
144.8.i.f.97.11 22 36.7 odd 6
216.8.i.b.73.5 22 9.2 odd 6
216.8.i.b.145.5 22 3.2 odd 2
432.8.i.f.145.5 22 12.11 even 2
432.8.i.f.289.5 22 36.11 even 6