Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [54,6,Mod(19,54)] 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("54.19"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(54, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 54 = 2 \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 54.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6] 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: \(8.66072626990\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.47347183152.3
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 118x^{4} - 231x^{3} + 3700x^{2} - 3585x + 32331 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{9} \)
Twist minimal: no (minimal twist has level 18)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 19.3
Root \(0.500000 - 4.03013i\) of defining polynomial
Character \(\chi\) \(=\) 54.19
Dual form 54.6.c.b.37.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(2.00000 - 3.46410i) q^{2} +(-8.00000 - 13.8564i) q^{4} +(39.2420 + 67.9691i) q^{5} +(-110.566 + 191.505i) q^{7} -64.0000 q^{8} +313.936 q^{10} +(115.144 - 199.436i) q^{11} +(385.793 + 668.213i) q^{13} +(442.262 + 766.020i) q^{14} +(-128.000 + 221.703i) q^{16} +769.880 q^{17} -383.367 q^{19} +(627.872 - 1087.51i) q^{20} +(-460.577 - 797.743i) q^{22} +(193.178 + 334.594i) q^{23} +(-1517.37 + 2628.15i) q^{25} +3086.34 q^{26} +3538.10 q^{28} +(-394.883 + 683.958i) q^{29} +(-1609.97 - 2788.54i) q^{31} +(512.000 + 886.810i) q^{32} +(1539.76 - 2666.94i) q^{34} -17355.2 q^{35} +2465.33 q^{37} +(-766.733 + 1328.02i) q^{38} +(-2511.49 - 4350.02i) q^{40} +(-4621.73 - 8005.07i) q^{41} +(-5315.76 + 9207.16i) q^{43} -3684.62 q^{44} +1545.42 q^{46} +(976.032 - 1690.54i) q^{47} +(-16046.0 - 27792.4i) q^{49} +(6069.46 + 10512.6i) q^{50} +(6172.68 - 10691.4i) q^{52} +32589.2 q^{53} +18074.0 q^{55} +(7076.19 - 12256.3i) q^{56} +(1579.53 + 2735.83i) q^{58} +(-11916.1 - 20639.2i) q^{59} +(-18804.4 + 32570.2i) q^{61} -12879.7 q^{62} +4096.00 q^{64} +(-30278.5 + 52444.0i) q^{65} +(11525.6 + 19962.9i) q^{67} +(-6159.04 - 10667.8i) q^{68} +(-34710.5 + 60120.3i) q^{70} +66050.4 q^{71} +65130.0 q^{73} +(4930.66 - 8540.16i) q^{74} +(3066.93 + 5312.08i) q^{76} +(25462.0 + 44101.5i) q^{77} +(35433.7 - 61373.0i) q^{79} -20091.9 q^{80} -36973.8 q^{82} +(27643.5 - 47880.0i) q^{83} +(30211.6 + 52328.0i) q^{85} +(21263.0 + 36828.6i) q^{86} +(-7369.24 + 12763.9i) q^{88} -10598.6 q^{89} -170622. q^{91} +(3090.85 - 5353.50i) q^{92} +(-3904.13 - 6762.15i) q^{94} +(-15044.1 - 26057.1i) q^{95} +(41409.9 - 71724.1i) q^{97} -128368. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 12 q^{2} - 48 q^{4} + 54 q^{5} - 132 q^{7} - 384 q^{8} + 432 q^{10} + 315 q^{11} - 744 q^{13} + 528 q^{14} - 768 q^{16} - 2898 q^{17} + 2262 q^{19} + 864 q^{20} - 1260 q^{22} + 3168 q^{23} - 2883 q^{25}+ \cdots - 422856 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/54\mathbb{Z}\right)^\times\).

\(n\) \(29\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 − 3.46410i 0.353553 − 0.612372i
\(3\) 0 0
\(4\) −8.00000 − 13.8564i −0.250000 − 0.433013i
\(5\) 39.2420 + 67.9691i 0.701982 + 1.21587i 0.967770 + 0.251837i \(0.0810346\pi\)
−0.265788 + 0.964032i \(0.585632\pi\)
\(6\) 0 0
\(7\) −110.566 + 191.505i −0.852854 + 1.47719i 0.0257678 + 0.999668i \(0.491797\pi\)
−0.878622 + 0.477518i \(0.841536\pi\)
\(8\) −64.0000 −0.353553
\(9\) 0 0
\(10\) 313.936 0.992752
\(11\) 115.144 − 199.436i 0.286920 − 0.496960i −0.686153 − 0.727457i \(-0.740702\pi\)
0.973073 + 0.230497i \(0.0740353\pi\)
\(12\) 0 0
\(13\) 385.793 + 668.213i 0.633134 + 1.09662i 0.986907 + 0.161289i \(0.0515652\pi\)
−0.353773 + 0.935331i \(0.615101\pi\)
\(14\) 442.262 + 766.020i 0.603059 + 1.04453i
\(15\) 0 0
\(16\) −128.000 + 221.703i −0.125000 + 0.216506i
\(17\) 769.880 0.646101 0.323051 − 0.946382i \(-0.395292\pi\)
0.323051 + 0.946382i \(0.395292\pi\)
\(18\) 0 0
\(19\) −383.367 −0.243630 −0.121815 − 0.992553i \(-0.538871\pi\)
−0.121815 + 0.992553i \(0.538871\pi\)
\(20\) 627.872 − 1087.51i 0.350991 − 0.607934i
\(21\) 0 0
\(22\) −460.577 − 797.743i −0.202883 − 0.351404i
\(23\) 193.178 + 334.594i 0.0761444 + 0.131886i 0.901583 − 0.432605i \(-0.142406\pi\)
−0.825439 + 0.564491i \(0.809072\pi\)
\(24\) 0 0
\(25\) −1517.37 + 2628.15i −0.485557 + 0.841009i
\(26\) 3086.34 0.895387
\(27\) 0 0
\(28\) 3538.10 0.852854
\(29\) −394.883 + 683.958i −0.0871914 + 0.151020i −0.906323 − 0.422586i \(-0.861123\pi\)
0.819131 + 0.573606i \(0.194456\pi\)
\(30\) 0 0
\(31\) −1609.97 − 2788.54i −0.300893 − 0.521163i 0.675445 − 0.737410i \(-0.263952\pi\)
−0.976339 + 0.216247i \(0.930618\pi\)
\(32\) 512.000 + 886.810i 0.0883883 + 0.153093i
\(33\) 0 0
\(34\) 1539.76 − 2666.94i 0.228431 − 0.395655i
\(35\) −17355.2 −2.39475
\(36\) 0 0
\(37\) 2465.33 0.296054 0.148027 − 0.988983i \(-0.452708\pi\)
0.148027 + 0.988983i \(0.452708\pi\)
\(38\) −766.733 + 1328.02i −0.0861361 + 0.149192i
\(39\) 0 0
\(40\) −2511.49 − 4350.02i −0.248188 − 0.429874i
\(41\) −4621.73 − 8005.07i −0.429383 − 0.743713i 0.567436 − 0.823418i \(-0.307936\pi\)
−0.996819 + 0.0797047i \(0.974602\pi\)
\(42\) 0 0
\(43\) −5315.76 + 9207.16i −0.438424 + 0.759372i −0.997568 − 0.0696983i \(-0.977796\pi\)
0.559145 + 0.829070i \(0.311130\pi\)
\(44\) −3684.62 −0.286920
\(45\) 0 0
\(46\) 1545.42 0.107684
\(47\) 976.032 − 1690.54i 0.0644495 − 0.111630i −0.832000 − 0.554775i \(-0.812804\pi\)
0.896450 + 0.443146i \(0.146138\pi\)
\(48\) 0 0
\(49\) −16046.0 − 27792.4i −0.954720 − 1.65362i
\(50\) 6069.46 + 10512.6i 0.343341 + 0.594683i
\(51\) 0 0
\(52\) 6172.68 − 10691.4i 0.316567 − 0.548310i
\(53\) 32589.2 1.59362 0.796809 − 0.604231i \(-0.206520\pi\)
0.796809 + 0.604231i \(0.206520\pi\)
\(54\) 0 0
\(55\) 18074.0 0.805651
\(56\) 7076.19 − 12256.3i 0.301529 − 0.522264i
\(57\) 0 0
\(58\) 1579.53 + 2735.83i 0.0616537 + 0.106787i
\(59\) −11916.1 − 20639.2i −0.445659 − 0.771903i 0.552439 − 0.833553i \(-0.313697\pi\)
−0.998098 + 0.0616498i \(0.980364\pi\)
\(60\) 0 0
\(61\) −18804.4 + 32570.2i −0.647046 + 1.12072i 0.336779 + 0.941584i \(0.390663\pi\)
−0.983825 + 0.179133i \(0.942671\pi\)
\(62\) −12879.7 −0.425528
\(63\) 0 0
\(64\) 4096.00 0.125000
\(65\) −30278.5 + 52444.0i −0.888897 + 1.53962i
\(66\) 0 0
\(67\) 11525.6 + 19962.9i 0.313672 + 0.543295i 0.979154 − 0.203118i \(-0.0651075\pi\)
−0.665483 + 0.746413i \(0.731774\pi\)
\(68\) −6159.04 − 10667.8i −0.161525 − 0.279770i
\(69\) 0 0
\(70\) −34710.5 + 60120.3i −0.846673 + 1.46648i
\(71\) 66050.4 1.55500 0.777498 − 0.628885i \(-0.216489\pi\)
0.777498 + 0.628885i \(0.216489\pi\)
\(72\) 0 0
\(73\) 65130.0 1.43045 0.715227 − 0.698893i \(-0.246324\pi\)
0.715227 + 0.698893i \(0.246324\pi\)
\(74\) 4930.66 − 8540.16i 0.104671 − 0.181295i
\(75\) 0 0
\(76\) 3066.93 + 5312.08i 0.0609074 + 0.105495i
\(77\) 25462.0 + 44101.5i 0.489402 + 0.847669i
\(78\) 0 0
\(79\) 35433.7 − 61373.0i 0.638776 − 1.10639i −0.346925 − 0.937893i \(-0.612774\pi\)
0.985702 − 0.168500i \(-0.0538924\pi\)
\(80\) −20091.9 −0.350991
\(81\) 0 0
\(82\) −36973.8 −0.607239
\(83\) 27643.5 − 47880.0i 0.440451 − 0.762884i −0.557272 − 0.830330i \(-0.688152\pi\)
0.997723 + 0.0674462i \(0.0214851\pi\)
\(84\) 0 0
\(85\) 30211.6 + 52328.0i 0.453551 + 0.785574i
\(86\) 21263.0 + 36828.6i 0.310012 + 0.536957i
\(87\) 0 0
\(88\) −7369.24 + 12763.9i −0.101442 + 0.175702i
\(89\) −10598.6 −0.141831 −0.0709156 − 0.997482i \(-0.522592\pi\)
−0.0709156 + 0.997482i \(0.522592\pi\)
\(90\) 0 0
\(91\) −170622. −2.15988
\(92\) 3090.85 − 5353.50i 0.0380722 − 0.0659430i
\(93\) 0 0
\(94\) −3904.13 − 6762.15i −0.0455727 − 0.0789342i
\(95\) −15044.1 − 26057.1i −0.171024 − 0.296222i
\(96\) 0 0
\(97\) 41409.9 − 71724.1i 0.446864 − 0.773991i −0.551316 − 0.834296i \(-0.685874\pi\)
0.998180 + 0.0603057i \(0.0192076\pi\)
\(98\) −128368. −1.35018
\(99\) 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 54.6.c.b.19.3 6
3.2 odd 2 18.6.c.b.7.1 ✓ 6
4.3 odd 2 432.6.i.b.289.3 6
9.2 odd 6 162.6.a.j.1.3 3
9.4 even 3 inner 54.6.c.b.37.3 6
9.5 odd 6 18.6.c.b.13.1 yes 6
9.7 even 3 162.6.a.i.1.1 3
12.11 even 2 144.6.i.b.97.3 6
36.23 even 6 144.6.i.b.49.3 6
36.31 odd 6 432.6.i.b.145.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.6.c.b.7.1 ✓ 6 3.2 odd 2
18.6.c.b.13.1 yes 6 9.5 odd 6
54.6.c.b.19.3 6 1.1 even 1 trivial
54.6.c.b.37.3 6 9.4 even 3 inner
144.6.i.b.49.3 6 36.23 even 6
144.6.i.b.97.3 6 12.11 even 2
162.6.a.i.1.1 3 9.7 even 3
162.6.a.j.1.3 3 9.2 odd 6
432.6.i.b.145.3 6 36.31 odd 6
432.6.i.b.289.3 6 4.3 odd 2