Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [18,5,Mod(5,18)] 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("18.5"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(18, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 18.d (of order \(6\), degree \(2\), 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: \(1.86065933551\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.221456830464.4
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 38x^{6} - 100x^{5} + 449x^{4} - 736x^{3} + 1900x^{2} - 1548x + 2307 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 5.3
Root \(0.500000 + 2.20403i\) of defining polynomial
Character \(\chi\) \(=\) 18.5
Dual form 18.5.d.a.11.3

$q$-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+(2.44949 - 1.41421i) q^{2} +(-1.67960 - 8.84189i) q^{3} +(4.00000 - 6.92820i) q^{4} +(6.41371 + 3.70296i) q^{5} +(-16.6185 - 19.2828i) q^{6} +(30.1882 + 52.2875i) q^{7} -22.6274i q^{8} +(-75.3579 + 29.7016i) q^{9} +20.9471 q^{10} +(55.2054 - 31.8729i) q^{11} +(-67.9768 - 23.7310i) q^{12} +(-144.994 + 251.137i) q^{13} +(147.892 + 85.3852i) q^{14} +(21.9687 - 62.9288i) q^{15} +(-32.0000 - 55.4256i) q^{16} -460.439i q^{17} +(-142.584 + 179.326i) q^{18} +187.741 q^{19} +(51.3097 - 29.6237i) q^{20} +(411.616 - 354.743i) q^{21} +(90.1501 - 156.144i) q^{22} +(-35.8618 - 20.7048i) q^{23} +(-200.069 + 38.0049i) q^{24} +(-285.076 - 493.766i) q^{25} +820.210i q^{26} +(389.189 + 616.419i) q^{27} +483.012 q^{28} +(-1211.49 + 699.452i) q^{29} +(-35.1827 - 185.212i) q^{30} +(-318.482 + 551.628i) q^{31} +(-156.767 - 90.5097i) q^{32} +(-374.539 - 434.586i) q^{33} +(-651.158 - 1127.84i) q^{34} +447.143i q^{35} +(-95.6529 + 640.901i) q^{36} +847.670 q^{37} +(459.869 - 265.506i) q^{38} +(2464.06 + 860.212i) q^{39} +(83.7884 - 145.126i) q^{40} +(-438.352 - 253.083i) q^{41} +(506.568 - 1451.05i) q^{42} +(-702.429 - 1216.64i) q^{43} -509.966i q^{44} +(-593.308 - 88.5497i) q^{45} -117.124 q^{46} +(2146.72 - 1239.41i) q^{47} +(-436.320 + 376.033i) q^{48} +(-622.158 + 1077.61i) q^{49} +(-1396.58 - 806.317i) q^{50} +(-4071.15 + 773.351i) q^{51} +(1159.95 + 2009.10i) q^{52} -773.208i q^{53} +(1825.06 + 959.517i) q^{54} +472.095 q^{55} +(1183.13 - 683.082i) q^{56} +(-315.329 - 1659.98i) q^{57} +(-1978.35 + 3426.60i) q^{58} +(3894.93 + 2248.74i) q^{59} +(-348.109 - 403.919i) q^{60} +(1301.48 + 2254.22i) q^{61} +1801.61i q^{62} +(-3827.95 - 3043.64i) q^{63} -512.000 q^{64} +(-1859.90 + 1073.81i) q^{65} +(-1532.03 - 534.837i) q^{66} +(1691.55 - 2929.85i) q^{67} +(-3190.01 - 1841.75i) q^{68} +(-122.836 + 351.862i) q^{69} +(632.356 + 1095.27i) q^{70} -2771.07i q^{71} +(672.070 + 1705.15i) q^{72} +3525.16 q^{73} +(2076.36 - 1198.79i) q^{74} +(-3887.01 + 3349.94i) q^{75} +(750.964 - 1300.71i) q^{76} +(3333.11 + 1924.37i) q^{77} +(7252.21 - 1377.62i) q^{78} +(-2914.90 - 5048.75i) q^{79} -473.979i q^{80} +(4796.63 - 4476.50i) q^{81} -1431.65 q^{82} +(-4438.63 + 2562.65i) q^{83} +(-811.265 - 4270.73i) q^{84} +(1704.98 - 2953.12i) q^{85} +(-3441.19 - 1986.77i) q^{86} +(8219.29 + 9537.03i) q^{87} +(-721.200 - 1249.16i) q^{88} +1221.14i q^{89} +(-1578.53 + 622.162i) q^{90} -17508.5 q^{91} +(-286.894 + 165.638i) q^{92} +(5412.35 + 1889.47i) q^{93} +(3505.58 - 6071.84i) q^{94} +(1204.12 + 695.197i) q^{95} +(-536.970 + 1538.14i) q^{96} +(7367.60 + 12761.1i) q^{97} +3519.46i q^{98} +(-3213.49 + 4041.56i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 6 q^{3} + 32 q^{4} + 18 q^{5} + 48 q^{6} - 26 q^{7} - 78 q^{9} - 720 q^{11} - 144 q^{12} + 10 q^{13} + 288 q^{14} + 1134 q^{15} - 256 q^{16} - 384 q^{18} + 100 q^{19} + 144 q^{20} + 438 q^{21} + 336 q^{22}+ \cdots - 22338 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/18\mathbb{Z}\right)^\times\).

\(n\) \(11\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.44949 − 1.41421i 0.612372 − 0.353553i
\(3\) −1.67960 − 8.84189i −0.186622 − 0.982432i
\(4\) 4.00000 − 6.92820i 0.250000 − 0.433013i
\(5\) 6.41371 + 3.70296i 0.256548 + 0.148118i 0.622759 − 0.782414i \(-0.286012\pi\)
−0.366211 + 0.930532i \(0.619345\pi\)
\(6\) −16.6185 − 19.2828i −0.461624 − 0.535633i
\(7\) 30.1882 + 52.2875i 0.616086 + 1.06709i 0.990193 + 0.139707i \(0.0446161\pi\)
−0.374107 + 0.927386i \(0.622051\pi\)
\(8\) − 22.6274i − 0.353553i
\(9\) −75.3579 + 29.7016i −0.930345 + 0.366686i
\(10\) 20.9471 0.209471
\(11\) 55.2054 − 31.8729i 0.456243 − 0.263412i −0.254220 − 0.967146i \(-0.581819\pi\)
0.710463 + 0.703734i \(0.248485\pi\)
\(12\) −67.9768 − 23.7310i −0.472061 − 0.164798i
\(13\) −144.994 + 251.137i −0.857953 + 1.48602i 0.0159253 + 0.999873i \(0.494931\pi\)
−0.873878 + 0.486145i \(0.838403\pi\)
\(14\) 147.892 + 85.3852i 0.754549 + 0.435639i
\(15\) 21.9687 − 62.9288i 0.0976386 − 0.279683i
\(16\) −32.0000 − 55.4256i −0.125000 − 0.216506i
\(17\) − 460.439i − 1.59321i −0.604498 − 0.796606i \(-0.706626\pi\)
0.604498 − 0.796606i \(-0.293374\pi\)
\(18\) −142.584 + 179.326i −0.440074 + 0.553475i
\(19\) 187.741 0.520058 0.260029 − 0.965601i \(-0.416268\pi\)
0.260029 + 0.965601i \(0.416268\pi\)
\(20\) 51.3097 − 29.6237i 0.128274 − 0.0740592i
\(21\) 411.616 − 354.743i 0.933371 − 0.804406i
\(22\) 90.1501 − 156.144i 0.186260 − 0.322613i
\(23\) −35.8618 − 20.7048i −0.0677916 − 0.0391395i 0.465721 − 0.884932i \(-0.345795\pi\)
−0.533513 + 0.845792i \(0.679128\pi\)
\(24\) −200.069 + 38.0049i −0.347342 + 0.0659808i
\(25\) −285.076 − 493.766i −0.456122 − 0.790026i
\(26\) 820.210i 1.21333i
\(27\) 389.189 + 616.419i 0.533867 + 0.845568i
\(28\) 483.012 0.616086
\(29\) −1211.49 + 699.452i −1.44053 + 0.831691i −0.997885 − 0.0650051i \(-0.979294\pi\)
−0.442646 + 0.896696i \(0.645960\pi\)
\(30\) −35.1827 − 185.212i −0.0390918 − 0.205791i
\(31\) −318.482 + 551.628i −0.331407 + 0.574014i −0.982788 − 0.184737i \(-0.940857\pi\)
0.651381 + 0.758751i \(0.274190\pi\)
\(32\) −156.767 − 90.5097i −0.153093 − 0.0883883i
\(33\) −374.539 − 434.586i −0.343929 − 0.399069i
\(34\) −651.158 − 1127.84i −0.563286 − 0.975640i
\(35\) 447.143i 0.365015i
\(36\) −95.6529 + 640.901i −0.0738063 + 0.494523i
\(37\) 847.670 0.619189 0.309594 − 0.950869i \(-0.399807\pi\)
0.309594 + 0.950869i \(0.399807\pi\)
\(38\) 459.869 − 265.506i 0.318469 − 0.183868i
\(39\) 2464.06 + 860.212i 1.62002 + 0.565557i
\(40\) 83.7884 − 145.126i 0.0523677 − 0.0907036i
\(41\) −438.352 − 253.083i −0.260769 − 0.150555i 0.363916 − 0.931432i \(-0.381439\pi\)
−0.624685 + 0.780877i \(0.714773\pi\)
\(42\) 506.568 − 1451.05i 0.287170 − 0.822592i
\(43\) −702.429 − 1216.64i −0.379897 − 0.658001i 0.611150 − 0.791515i \(-0.290707\pi\)
−0.991047 + 0.133514i \(0.957374\pi\)
\(44\) − 509.966i − 0.263412i
\(45\) −593.308 − 88.5497i −0.292991 − 0.0437282i
\(46\) −117.124 −0.0553516
\(47\) 2146.72 − 1239.41i 0.971806 − 0.561073i 0.0720198 − 0.997403i \(-0.477056\pi\)
0.899786 + 0.436331i \(0.143722\pi\)
\(48\) −436.320 + 376.033i −0.189375 + 0.163209i
\(49\) −622.158 + 1077.61i −0.259125 + 0.448817i
\(50\) −1396.58 − 806.317i −0.558633 − 0.322527i
\(51\) −4071.15 + 773.351i −1.56522 + 0.297328i
\(52\) 1159.95 + 2009.10i 0.428976 + 0.743009i
\(53\) − 773.208i − 0.275261i −0.990484 − 0.137630i \(-0.956051\pi\)
0.990484 − 0.137630i \(-0.0439486\pi\)
\(54\) 1825.06 + 959.517i 0.625879 + 0.329052i
\(55\) 472.095 0.156065
\(56\) 1183.13 − 683.082i 0.377274 − 0.217819i
\(57\) −315.329 − 1659.98i −0.0970541 − 0.510921i
\(58\) −1978.35 + 3426.60i −0.588094 + 1.01861i
\(59\) 3894.93 + 2248.74i 1.11891 + 0.646004i 0.941123 − 0.338065i \(-0.109772\pi\)
0.177789 + 0.984069i \(0.443106\pi\)
\(60\) −348.109 − 403.919i −0.0966968 − 0.112200i
\(61\) 1301.48 + 2254.22i 0.349765 + 0.605811i 0.986208 − 0.165513i \(-0.0529280\pi\)
−0.636442 + 0.771324i \(0.719595\pi\)
\(62\) 1801.61i 0.468681i
\(63\) −3827.95 − 3043.64i −0.964461 − 0.766853i
\(64\) −512.000 −0.125000
\(65\) −1859.90 + 1073.81i −0.440213 + 0.254157i
\(66\) −1532.03 − 534.837i −0.351705 − 0.122782i
\(67\) 1691.55 − 2929.85i 0.376821 − 0.652673i −0.613777 − 0.789479i \(-0.710351\pi\)
0.990598 + 0.136807i \(0.0436840\pi\)
\(68\) −3190.01 − 1841.75i −0.689881 − 0.398303i
\(69\) −122.836 + 351.862i −0.0258005 + 0.0739050i
\(70\) 632.356 + 1095.27i 0.129052 + 0.223525i
\(71\) − 2771.07i − 0.549706i −0.961486 − 0.274853i \(-0.911371\pi\)
0.961486 − 0.274853i \(-0.0886292\pi\)
\(72\) 672.070 + 1705.15i 0.129643 + 0.328926i
\(73\) 3525.16 0.661506 0.330753 − 0.943717i \(-0.392697\pi\)
0.330753 + 0.943717i \(0.392697\pi\)
\(74\) 2076.36 − 1198.79i 0.379174 − 0.218916i
\(75\) −3887.01 + 3349.94i −0.691025 + 0.595545i
\(76\) 750.964 − 1300.71i 0.130014 − 0.225192i
\(77\) 3333.11 + 1924.37i 0.562170 + 0.324569i
\(78\) 7252.21 − 1377.62i 1.19201 − 0.226434i
\(79\) −2914.90 − 5048.75i −0.467056 − 0.808964i 0.532236 − 0.846596i \(-0.321352\pi\)
−0.999292 + 0.0376317i \(0.988019\pi\)
\(80\) − 473.979i − 0.0740592i
\(81\) 4796.63 − 4476.50i 0.731082 − 0.682289i
\(82\) −1431.65 −0.212917
\(83\) −4438.63 + 2562.65i −0.644307 + 0.371991i −0.786272 − 0.617881i \(-0.787991\pi\)
0.141965 + 0.989872i \(0.454658\pi\)
\(84\) −811.265 − 4270.73i −0.114975 − 0.605263i
\(85\) 1704.98 − 2953.12i 0.235984 − 0.408736i
\(86\) −3441.19 − 1986.77i −0.465277 − 0.268628i
\(87\) 8219.29 + 9537.03i 1.08591 + 1.26001i
\(88\) −721.200 − 1249.16i −0.0931302 − 0.161306i
\(89\) 1221.14i 0.154164i 0.997025 + 0.0770822i \(0.0245604\pi\)
−0.997025 + 0.0770822i \(0.975440\pi\)
\(90\) −1578.53 + 622.162i −0.194880 + 0.0768101i
\(91\) −17508.5 −2.11429
\(92\) −286.894 + 165.638i −0.0338958 + 0.0195698i
\(93\) 5412.35 + 1889.47i 0.625778 + 0.218461i
\(94\) 3505.58 − 6071.84i 0.396738 − 0.687171i
\(95\) 1204.12 + 695.197i 0.133420 + 0.0770301i
\(96\) −536.970 + 1538.14i −0.0582650 + 0.166899i
\(97\) 7367.60 + 12761.1i 0.783038 + 1.35626i 0.930164 + 0.367144i \(0.119664\pi\)
−0.147126 + 0.989118i \(0.547002\pi\)
\(98\) 3519.46i 0.366458i
\(99\) −3213.49 + 4041.56i −0.327874 + 0.412362i
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 18.5.d.a.5.3 ✓ 8
3.2 odd 2 54.5.d.a.17.1 8
4.3 odd 2 144.5.q.b.113.3 8
9.2 odd 6 inner 18.5.d.a.11.3 yes 8
9.4 even 3 162.5.b.c.161.6 8
9.5 odd 6 162.5.b.c.161.3 8
9.7 even 3 54.5.d.a.35.1 8
12.11 even 2 432.5.q.b.17.2 8
36.7 odd 6 432.5.q.b.305.2 8
36.11 even 6 144.5.q.b.65.3 8
36.23 even 6 1296.5.e.e.161.6 8
36.31 odd 6 1296.5.e.e.161.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.5.d.a.5.3 ✓ 8 1.1 even 1 trivial
18.5.d.a.11.3 yes 8 9.2 odd 6 inner
54.5.d.a.17.1 8 3.2 odd 2
54.5.d.a.35.1 8 9.7 even 3
144.5.q.b.65.3 8 36.11 even 6
144.5.q.b.113.3 8 4.3 odd 2
162.5.b.c.161.3 8 9.5 odd 6
162.5.b.c.161.6 8 9.4 even 3
432.5.q.b.17.2 8 12.11 even 2
432.5.q.b.305.2 8 36.7 odd 6
1296.5.e.e.161.3 8 36.31 odd 6
1296.5.e.e.161.6 8 36.23 even 6