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 19.1
Root \(0.500000 - 5.23712i\) of defining polynomial
Character \(\chi\) \(=\) 54.19
Dual form 54.6.c.b.37.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(2.00000 - 3.46410i) q^{2} +(-8.00000 - 13.8564i) q^{4} +(-33.0434 - 57.2329i) q^{5} +(-57.0952 + 98.8918i) q^{7} -64.0000 q^{8} -264.347 q^{10} +(-192.812 + 333.961i) q^{11} +(-516.287 - 894.236i) q^{13} +(228.381 + 395.567i) q^{14} +(-128.000 + 221.703i) q^{16} -959.020 q^{17} -464.576 q^{19} +(-528.695 + 915.726i) q^{20} +(771.249 + 1335.84i) q^{22} +(1151.70 + 1994.81i) q^{23} +(-621.235 + 1076.01i) q^{25} -4130.30 q^{26} +1827.05 q^{28} +(3549.13 - 6147.28i) q^{29} +(-3881.53 - 6723.00i) q^{31} +(512.000 + 886.810i) q^{32} +(-1918.04 + 3322.14i) q^{34} +7546.48 q^{35} +9317.57 q^{37} +(-929.153 + 1609.34i) q^{38} +(2114.78 + 3662.90i) q^{40} +(-6661.64 - 11538.3i) q^{41} +(1056.29 - 1829.55i) q^{43} +6169.99 q^{44} +9213.62 q^{46} +(1247.40 - 2160.56i) q^{47} +(1883.78 + 3262.80i) q^{49} +(2484.94 + 4304.04i) q^{50} +(-8260.60 + 14307.8i) q^{52} +10044.2 q^{53} +25484.7 q^{55} +(3654.09 - 6329.07i) q^{56} +(-14196.5 - 24589.1i) q^{58} +(2720.33 + 4711.75i) q^{59} +(-17094.4 + 29608.4i) q^{61} -31052.2 q^{62} +4096.00 q^{64} +(-34119.8 + 59097.2i) q^{65} +(-26792.6 - 46406.2i) q^{67} +(7672.16 + 13288.6i) q^{68} +(15093.0 - 26141.8i) q^{70} +970.010 q^{71} -72400.3 q^{73} +(18635.1 - 32277.0i) q^{74} +(3716.61 + 6437.36i) q^{76} +(-22017.3 - 38135.1i) q^{77} +(-16098.9 + 27884.0i) q^{79} +16918.2 q^{80} -53293.1 q^{82} +(-18046.6 + 31257.6i) q^{83} +(31689.3 + 54887.5i) q^{85} +(-4225.17 - 7318.21i) q^{86} +(12340.0 - 21373.5i) q^{88} -42622.2 q^{89} +117910. q^{91} +(18427.2 - 31916.9i) q^{92} +(-4989.60 - 8642.25i) q^{94} +(15351.2 + 26589.0i) q^{95} +(21926.9 - 37978.6i) q^{97} +15070.2 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 12 q^{2} - 48 q^{4} + 54 q^{5} - 132 q^{7} - 384 q^{8} + 432 q^{10} + 315 q^{11} - 744 q^{13} + 528 q^{14} - 768 q^{16} - 2898 q^{17} + 2262 q^{19} + 864 q^{20} - 1260 q^{22} + 3168 q^{23} - 2883 q^{25}+ \cdots - 422856 q^{98}+O(q^{100}) \) 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
\(5\) −33.0434 − 57.2329i −0.591099 − 1.02381i −0.994085 − 0.108607i \(-0.965361\pi\)
0.402986 − 0.915206i \(-0.367972\pi\)
\(6\) 0 0
\(7\) −57.0952 + 98.8918i −0.440407 + 0.762808i −0.997720 − 0.0674952i \(-0.978499\pi\)
0.557312 + 0.830303i \(0.311833\pi\)
\(8\) −64.0000 −0.353553
\(9\) 0 0
\(10\) −264.347 −0.835940
\(11\) −192.812 + 333.961i −0.480455 + 0.832173i −0.999749 − 0.0224231i \(-0.992862\pi\)
0.519293 + 0.854596i \(0.326195\pi\)
\(12\) 0 0
\(13\) −516.287 − 894.236i −0.847292 − 1.46755i −0.883616 − 0.468212i \(-0.844898\pi\)
0.0363242 − 0.999340i \(-0.488435\pi\)
\(14\) 228.381 + 395.567i 0.311415 + 0.539386i
\(15\) 0 0
\(16\) −128.000 + 221.703i −0.125000 + 0.216506i
\(17\) −959.020 −0.804832 −0.402416 − 0.915457i \(-0.631829\pi\)
−0.402416 + 0.915457i \(0.631829\pi\)
\(18\) 0 0
\(19\) −464.576 −0.295239 −0.147619 − 0.989044i \(-0.547161\pi\)
−0.147619 + 0.989044i \(0.547161\pi\)
\(20\) −528.695 + 915.726i −0.295549 + 0.511906i
\(21\) 0 0
\(22\) 771.249 + 1335.84i 0.339733 + 0.588435i
\(23\) 1151.70 + 1994.81i 0.453963 + 0.786288i 0.998628 − 0.0523662i \(-0.0166763\pi\)
−0.544664 + 0.838654i \(0.683343\pi\)
\(24\) 0 0
\(25\) −621.235 + 1076.01i −0.198795 + 0.344323i
\(26\) −4130.30 −1.19825
\(27\) 0 0
\(28\) 1827.05 0.440407
\(29\) 3549.13 − 6147.28i 0.783659 − 1.35734i −0.146138 − 0.989264i \(-0.546684\pi\)
0.929797 − 0.368073i \(-0.119982\pi\)
\(30\) 0 0
\(31\) −3881.53 − 6723.00i −0.725435 − 1.25649i −0.958795 − 0.284100i \(-0.908305\pi\)
0.233360 − 0.972391i \(-0.425028\pi\)
\(32\) 512.000 + 886.810i 0.0883883 + 0.153093i
\(33\) 0 0
\(34\) −1918.04 + 3322.14i −0.284551 + 0.492857i
\(35\) 7546.48 1.04130
\(36\) 0 0
\(37\) 9317.57 1.11892 0.559459 − 0.828858i \(-0.311009\pi\)
0.559459 + 0.828858i \(0.311009\pi\)
\(38\) −929.153 + 1609.34i −0.104383 + 0.180796i
\(39\) 0 0
\(40\) 2114.78 + 3662.90i 0.208985 + 0.361972i
\(41\) −6661.64 − 11538.3i −0.618901 − 1.07197i −0.989687 − 0.143250i \(-0.954245\pi\)
0.370785 − 0.928719i \(-0.379089\pi\)
\(42\) 0 0
\(43\) 1056.29 − 1829.55i 0.0871190 − 0.150895i −0.819173 − 0.573546i \(-0.805567\pi\)
0.906292 + 0.422652i \(0.138901\pi\)
\(44\) 6169.99 0.480455
\(45\) 0 0
\(46\) 9213.62 0.642001
\(47\) 1247.40 − 2160.56i 0.0823686 − 0.142667i −0.821898 − 0.569634i \(-0.807085\pi\)
0.904267 + 0.426968i \(0.140418\pi\)
\(48\) 0 0
\(49\) 1883.78 + 3262.80i 0.112083 + 0.194133i
\(50\) 2484.94 + 4304.04i 0.140569 + 0.243473i
\(51\) 0 0
\(52\) −8260.60 + 14307.8i −0.423646 + 0.733776i
\(53\) 10044.2 0.491163 0.245582 − 0.969376i \(-0.421021\pi\)
0.245582 + 0.969376i \(0.421021\pi\)
\(54\) 0 0
\(55\) 25484.7 1.13599
\(56\) 3654.09 − 6329.07i 0.155707 − 0.269693i
\(57\) 0 0
\(58\) −14196.5 − 24589.1i −0.554131 − 0.959783i
\(59\) 2720.33 + 4711.75i 0.101740 + 0.176219i 0.912402 − 0.409296i \(-0.134226\pi\)
−0.810662 + 0.585515i \(0.800892\pi\)
\(60\) 0 0
\(61\) −17094.4 + 29608.4i −0.588206 + 1.01880i 0.406261 + 0.913757i \(0.366832\pi\)
−0.994467 + 0.105046i \(0.966501\pi\)
\(62\) −31052.2 −1.02592
\(63\) 0 0
\(64\) 4096.00 0.125000
\(65\) −34119.8 + 59097.2i −1.00167 + 1.73494i
\(66\) 0 0
\(67\) −26792.6 − 46406.2i −0.729169 − 1.26296i −0.957235 − 0.289311i \(-0.906574\pi\)
0.228066 − 0.973646i \(-0.426760\pi\)
\(68\) 7672.16 + 13288.6i 0.201208 + 0.348503i
\(69\) 0 0
\(70\) 15093.0 − 26141.8i 0.368154 − 0.637661i
\(71\) 970.010 0.0228365 0.0114183 − 0.999935i \(-0.496365\pi\)
0.0114183 + 0.999935i \(0.496365\pi\)
\(72\) 0 0
\(73\) −72400.3 −1.59013 −0.795066 − 0.606523i \(-0.792564\pi\)
−0.795066 + 0.606523i \(0.792564\pi\)
\(74\) 18635.1 − 32277.0i 0.395597 − 0.685195i
\(75\) 0 0
\(76\) 3716.61 + 6437.36i 0.0738097 + 0.127842i
\(77\) −22017.3 − 38135.1i −0.423192 − 0.732990i
\(78\) 0 0
\(79\) −16098.9 + 27884.0i −0.290220 + 0.502676i −0.973862 − 0.227142i \(-0.927062\pi\)
0.683642 + 0.729818i \(0.260395\pi\)
\(80\) 16918.2 0.295549
\(81\) 0 0
\(82\) −53293.1 −0.875258
\(83\) −18046.6 + 31257.6i −0.287541 + 0.498036i −0.973222 − 0.229866i \(-0.926171\pi\)
0.685681 + 0.727902i \(0.259504\pi\)
\(84\) 0 0
\(85\) 31689.3 + 54887.5i 0.475735 + 0.823998i
\(86\) −4225.17 − 7318.21i −0.0616024 − 0.106699i
\(87\) 0 0
\(88\) 12340.0 − 21373.5i 0.169867 − 0.294218i
\(89\) −42622.2 −0.570375 −0.285188 − 0.958472i \(-0.592056\pi\)
−0.285188 + 0.958472i \(0.592056\pi\)
\(90\) 0 0
\(91\) 117910. 1.49261
\(92\) 18427.2 − 31916.9i 0.226982 − 0.393144i
\(93\) 0 0
\(94\) −4989.60 − 8642.25i −0.0582434 − 0.100880i
\(95\) 15351.2 + 26589.0i 0.174515 + 0.302269i
\(96\) 0 0
\(97\) 21926.9 − 37978.6i 0.236618 − 0.409835i −0.723123 − 0.690719i \(-0.757294\pi\)
0.959742 + 0.280884i \(0.0906275\pi\)
\(98\) 15070.2 0.158509
\(99\) 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.19.1 6
3.2 odd 2 18.6.c.b.7.3 ✓ 6
4.3 odd 2 432.6.i.b.289.1 6
9.2 odd 6 162.6.a.j.1.1 3
9.4 even 3 inner 54.6.c.b.37.1 6
9.5 odd 6 18.6.c.b.13.3 yes 6
9.7 even 3 162.6.a.i.1.3 3
12.11 even 2 144.6.i.b.97.1 6
36.23 even 6 144.6.i.b.49.1 6
36.31 odd 6 432.6.i.b.145.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.6.c.b.7.3 ✓ 6 3.2 odd 2
18.6.c.b.13.3 yes 6 9.5 odd 6
54.6.c.b.19.1 6 1.1 even 1 trivial
54.6.c.b.37.1 6 9.4 even 3 inner
144.6.i.b.49.1 6 36.23 even 6
144.6.i.b.97.1 6 12.11 even 2
162.6.a.i.1.3 3 9.7 even 3
162.6.a.j.1.1 3 9.2 odd 6
432.6.i.b.145.1 6 36.31 odd 6
432.6.i.b.289.1 6 4.3 odd 2