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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [504,3,Mod(197,504)] 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("504.197"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(504, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1, 0])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 504.n (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.7330053238\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 197.1
Character \(\chi\) \(=\) 504.197
Dual form 504.3.n.a.197.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.99917 - 0.0576227i) q^{2} +(3.99336 + 0.230395i) q^{4} +7.03286 q^{5} -2.64575 q^{7} +(-7.97013 - 0.690708i) q^{8} +(-14.0599 - 0.405253i) q^{10} +9.57215 q^{11} +24.0850i q^{13} +(5.28931 + 0.152455i) q^{14} +(15.8938 + 1.84010i) q^{16} -5.54031i q^{17} +26.1010i q^{19} +(28.0848 + 1.62034i) q^{20} +(-19.1363 - 0.551573i) q^{22} -8.90911i q^{23} +24.4612 q^{25} +(1.38785 - 48.1501i) q^{26} +(-10.5654 - 0.609569i) q^{28} +13.0879 q^{29} -32.7051 q^{31} +(-31.6684 - 4.59452i) q^{32} +(-0.319248 + 11.0760i) q^{34} -18.6072 q^{35} +12.2654i q^{37} +(1.50401 - 52.1804i) q^{38} +(-56.0528 - 4.85765i) q^{40} +62.2015i q^{41} -58.9283i q^{43} +(38.2250 + 2.20538i) q^{44} +(-0.513367 + 17.8108i) q^{46} -45.9737i q^{47} +7.00000 q^{49} +(-48.9020 - 1.40952i) q^{50} +(-5.54908 + 96.1802i) q^{52} +62.4348 q^{53} +67.3196 q^{55} +(21.0870 + 1.82744i) q^{56} +(-26.1649 - 0.754161i) q^{58} +9.53218 q^{59} +60.4030i q^{61} +(65.3831 + 1.88456i) q^{62} +(63.0458 + 11.0101i) q^{64} +169.387i q^{65} +17.5241i q^{67} +(1.27646 - 22.1244i) q^{68} +(37.1990 + 1.07220i) q^{70} +107.042i q^{71} +105.146 q^{73} +(0.706763 - 24.5205i) q^{74} +(-6.01356 + 104.231i) q^{76} -25.3255 q^{77} +135.446 q^{79} +(111.779 + 12.9412i) q^{80} +(3.58422 - 124.351i) q^{82} -35.6795 q^{83} -38.9642i q^{85} +(-3.39561 + 117.808i) q^{86} +(-76.2912 - 6.61155i) q^{88} -28.5340i q^{89} -63.7230i q^{91} +(2.05262 - 35.5773i) q^{92} +(-2.64913 + 91.9093i) q^{94} +183.565i q^{95} +0.191171 q^{97} +(-13.9942 - 0.403359i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q - 8 q^{4} - 56 q^{10} + 140 q^{16} + 132 q^{22} + 240 q^{25} - 28 q^{28} + 256 q^{31} - 96 q^{34} + 216 q^{40} + 28 q^{46} + 336 q^{49} + 48 q^{52} - 512 q^{55} - 196 q^{58} + 208 q^{64} + 168 q^{70}+ \cdots - 24 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/504\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(127\) \(253\) \(281\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.99917 − 0.0576227i −0.999585 − 0.0288114i
\(3\) 0 0
\(4\) 3.99336 + 0.230395i 0.998340 + 0.0575988i
\(5\) 7.03286 1.40657 0.703286 − 0.710907i \(-0.251715\pi\)
0.703286 + 0.710907i \(0.251715\pi\)
\(6\) 0 0
\(7\) −2.64575 −0.377964
\(8\) −7.97013 − 0.690708i −0.996266 − 0.0863384i
\(9\) 0 0
\(10\) −14.0599 − 0.405253i −1.40599 − 0.0405253i
\(11\) 9.57215 0.870195 0.435098 − 0.900383i \(-0.356714\pi\)
0.435098 + 0.900383i \(0.356714\pi\)
\(12\) 0 0
\(13\) 24.0850i 1.85270i 0.376669 + 0.926348i \(0.377069\pi\)
−0.376669 + 0.926348i \(0.622931\pi\)
\(14\) 5.28931 + 0.152455i 0.377808 + 0.0108897i
\(15\) 0 0
\(16\) 15.8938 + 1.84010i 0.993365 + 0.115006i
\(17\) − 5.54031i − 0.325900i −0.986634 − 0.162950i \(-0.947899\pi\)
0.986634 − 0.162950i \(-0.0521010\pi\)
\(18\) 0 0
\(19\) 26.1010i 1.37374i 0.726781 + 0.686870i \(0.241016\pi\)
−0.726781 + 0.686870i \(0.758984\pi\)
\(20\) 28.0848 + 1.62034i 1.40424 + 0.0810169i
\(21\) 0 0
\(22\) −19.1363 − 0.551573i −0.869834 − 0.0250715i
\(23\) − 8.90911i − 0.387353i −0.981065 − 0.193676i \(-0.937959\pi\)
0.981065 − 0.193676i \(-0.0620412\pi\)
\(24\) 0 0
\(25\) 24.4612 0.978447
\(26\) 1.38785 − 48.1501i 0.0533787 − 1.85193i
\(27\) 0 0
\(28\) −10.5654 − 0.609569i −0.377337 − 0.0217703i
\(29\) 13.0879 0.451307 0.225654 − 0.974208i \(-0.427548\pi\)
0.225654 + 0.974208i \(0.427548\pi\)
\(30\) 0 0
\(31\) −32.7051 −1.05500 −0.527502 − 0.849554i \(-0.676871\pi\)
−0.527502 + 0.849554i \(0.676871\pi\)
\(32\) −31.6684 − 4.59452i −0.989639 − 0.143579i
\(33\) 0 0
\(34\) −0.319248 + 11.0760i −0.00938964 + 0.325765i
\(35\) −18.6072 −0.531635
\(36\) 0 0
\(37\) 12.2654i 0.331496i 0.986168 + 0.165748i \(0.0530039\pi\)
−0.986168 + 0.165748i \(0.946996\pi\)
\(38\) 1.50401 − 52.1804i 0.0395793 − 1.37317i
\(39\) 0 0
\(40\) −56.0528 − 4.85765i −1.40132 − 0.121441i
\(41\) 62.2015i 1.51711i 0.651609 + 0.758555i \(0.274094\pi\)
−0.651609 + 0.758555i \(0.725906\pi\)
\(42\) 0 0
\(43\) − 58.9283i − 1.37042i −0.728343 − 0.685212i \(-0.759709\pi\)
0.728343 − 0.685212i \(-0.240291\pi\)
\(44\) 38.2250 + 2.20538i 0.868751 + 0.0501222i
\(45\) 0 0
\(46\) −0.513367 + 17.8108i −0.0111602 + 0.387192i
\(47\) − 45.9737i − 0.978165i −0.872238 − 0.489082i \(-0.837332\pi\)
0.872238 − 0.489082i \(-0.162668\pi\)
\(48\) 0 0
\(49\) 7.00000 0.142857
\(50\) −48.9020 − 1.40952i −0.978041 − 0.0281904i
\(51\) 0 0
\(52\) −5.54908 + 96.1802i −0.106713 + 1.84962i
\(53\) 62.4348 1.17802 0.589008 − 0.808127i \(-0.299519\pi\)
0.589008 + 0.808127i \(0.299519\pi\)
\(54\) 0 0
\(55\) 67.3196 1.22399
\(56\) 21.0870 + 1.82744i 0.376553 + 0.0326329i
\(57\) 0 0
\(58\) −26.1649 − 0.754161i −0.451120 − 0.0130028i
\(59\) 9.53218 0.161562 0.0807812 − 0.996732i \(-0.474259\pi\)
0.0807812 + 0.996732i \(0.474259\pi\)
\(60\) 0 0
\(61\) 60.4030i 0.990213i 0.868832 + 0.495107i \(0.164871\pi\)
−0.868832 + 0.495107i \(0.835129\pi\)
\(62\) 65.3831 + 1.88456i 1.05457 + 0.0303961i
\(63\) 0 0
\(64\) 63.0458 + 11.0101i 0.985091 + 0.172032i
\(65\) 169.387i 2.60595i
\(66\) 0 0
\(67\) 17.5241i 0.261553i 0.991412 + 0.130777i \(0.0417470\pi\)
−0.991412 + 0.130777i \(0.958253\pi\)
\(68\) 1.27646 − 22.1244i 0.0187715 − 0.325359i
\(69\) 0 0
\(70\) 37.1990 + 1.07220i 0.531414 + 0.0153171i
\(71\) 107.042i 1.50763i 0.657086 + 0.753815i \(0.271789\pi\)
−0.657086 + 0.753815i \(0.728211\pi\)
\(72\) 0 0
\(73\) 105.146 1.44035 0.720177 − 0.693790i \(-0.244061\pi\)
0.720177 + 0.693790i \(0.244061\pi\)
\(74\) 0.706763 − 24.5205i 0.00955086 − 0.331359i
\(75\) 0 0
\(76\) −6.01356 + 104.231i −0.0791258 + 1.37146i
\(77\) −25.3255 −0.328903
\(78\) 0 0
\(79\) 135.446 1.71451 0.857254 − 0.514895i \(-0.172169\pi\)
0.857254 + 0.514895i \(0.172169\pi\)
\(80\) 111.779 + 12.9412i 1.39724 + 0.161765i
\(81\) 0 0
\(82\) 3.58422 − 124.351i 0.0437100 − 1.51648i
\(83\) −35.6795 −0.429873 −0.214937 − 0.976628i \(-0.568954\pi\)
−0.214937 + 0.976628i \(0.568954\pi\)
\(84\) 0 0
\(85\) − 38.9642i − 0.458403i
\(86\) −3.39561 + 117.808i −0.0394838 + 1.36986i
\(87\) 0 0
\(88\) −76.2912 − 6.61155i −0.866946 − 0.0751313i
\(89\) − 28.5340i − 0.320607i −0.987068 − 0.160304i \(-0.948753\pi\)
0.987068 − 0.160304i \(-0.0512473\pi\)
\(90\) 0 0
\(91\) − 63.7230i − 0.700253i
\(92\) 2.05262 − 35.5773i 0.0223111 − 0.386710i
\(93\) 0 0
\(94\) −2.64913 + 91.9093i −0.0281823 + 0.977759i
\(95\) 183.565i 1.93226i
\(96\) 0 0
\(97\) 0.191171 0.00197083 0.000985417 − 1.00000i \(-0.499686\pi\)
0.000985417 1.00000i \(0.499686\pi\)
\(98\) −13.9942 − 0.403359i −0.142798 − 0.00411591i
\(99\) 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 504.3.n.a.197.1 ✓ 48
3.2 odd 2 inner 504.3.n.a.197.48 yes 48
4.3 odd 2 2016.3.n.a.1457.44 48
8.3 odd 2 2016.3.n.a.1457.5 48
8.5 even 2 inner 504.3.n.a.197.47 yes 48
12.11 even 2 2016.3.n.a.1457.6 48
24.5 odd 2 inner 504.3.n.a.197.2 yes 48
24.11 even 2 2016.3.n.a.1457.43 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.3.n.a.197.1 ✓ 48 1.1 even 1 trivial
504.3.n.a.197.2 yes 48 24.5 odd 2 inner
504.3.n.a.197.47 yes 48 8.5 even 2 inner
504.3.n.a.197.48 yes 48 3.2 odd 2 inner
2016.3.n.a.1457.5 48 8.3 odd 2
2016.3.n.a.1457.6 48 12.11 even 2
2016.3.n.a.1457.43 48 24.11 even 2
2016.3.n.a.1457.44 48 4.3 odd 2