Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [18,4,Mod(7,18)] 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("18.7"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(18, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 18.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4] 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: \(1.06203438010\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-35})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 8x^{2} - 9x + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 7.2
Root \(-2.31174 - 1.91203i\) of defining polynomial
Character \(\chi\) \(=\) 18.7
Dual form 18.4.c.b.13.2

$q$-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+(1.00000 - 1.73205i) q^{2} +(3.31174 + 4.00405i) q^{3} +(-2.00000 - 3.46410i) q^{4} +(-5.43521 - 9.41407i) q^{5} +(10.2470 - 1.73205i) q^{6} +(-12.4352 + 21.5384i) q^{7} -8.00000 q^{8} +(-5.06479 + 26.5207i) q^{9} -21.7409 q^{10} +(21.3704 - 37.0147i) q^{11} +(7.24695 - 19.4803i) q^{12} +(-7.56479 - 13.1026i) q^{13} +(24.8704 + 43.0768i) q^{14} +(19.6944 - 52.9398i) q^{15} +(-8.00000 + 13.8564i) q^{16} -13.8704 q^{17} +(40.8704 + 35.2932i) q^{18} +143.352 q^{19} +(-21.7409 + 37.6563i) q^{20} +(-127.423 + 21.5384i) q^{21} +(-42.7409 - 74.0293i) q^{22} +(-9.56479 - 16.5667i) q^{23} +(-26.4939 - 32.0324i) q^{24} +(3.41692 - 5.91828i) q^{25} -30.2591 q^{26} +(-122.963 + 67.5500i) q^{27} +99.4817 q^{28} +(-113.046 + 195.802i) q^{29} +(-72.0000 - 87.0514i) q^{30} +(-29.6944 - 51.4321i) q^{31} +(16.0000 + 27.7128i) q^{32} +(218.982 - 37.0147i) q^{33} +(-13.8704 + 24.0243i) q^{34} +270.352 q^{35} +(102.000 - 35.4965i) q^{36} -84.1860 q^{37} +(143.352 - 248.293i) q^{38} +(27.4108 - 73.6821i) q^{39} +(43.4817 + 75.3125i) q^{40} +(-101.630 - 176.028i) q^{41} +(-90.1174 + 242.242i) q^{42} +(-162.945 + 282.229i) q^{43} -170.963 q^{44} +(277.196 - 96.4655i) q^{45} -38.2591 q^{46} +(-5.47180 + 9.47744i) q^{47} +(-81.9756 + 13.8564i) q^{48} +(-137.769 - 238.623i) q^{49} +(-6.83384 - 11.8366i) q^{50} +(-45.9352 - 55.5378i) q^{51} +(-30.2591 + 52.4104i) q^{52} -140.186 q^{53} +(-5.96341 + 280.529i) q^{54} -464.611 q^{55} +(99.4817 - 172.307i) q^{56} +(474.745 + 573.989i) q^{57} +(226.093 + 391.605i) q^{58} +(-57.3704 - 99.3685i) q^{59} +(-222.777 + 37.6563i) q^{60} +(377.528 - 653.898i) q^{61} -118.777 q^{62} +(-508.232 - 438.878i) q^{63} +64.0000 q^{64} +(-82.2325 + 142.431i) q^{65} +(154.870 - 416.302i) q^{66} +(383.723 + 664.627i) q^{67} +(27.7409 + 48.0486i) q^{68} +(34.6578 - 93.1624i) q^{69} +(270.352 - 468.264i) q^{70} +335.854 q^{71} +(40.5183 - 212.166i) q^{72} +167.279 q^{73} +(-84.1860 + 145.814i) q^{74} +(35.0130 - 5.91828i) q^{75} +(-286.704 - 496.586i) q^{76} +(531.492 + 920.570i) q^{77} +(-100.210 - 121.159i) q^{78} +(12.6578 - 21.9239i) q^{79} +173.927 q^{80} +(-677.696 - 268.643i) q^{81} -406.518 q^{82} +(-143.861 + 249.174i) q^{83} +(329.457 + 398.329i) q^{84} +(75.3887 + 130.577i) q^{85} +(325.890 + 564.458i) q^{86} +(-1158.38 + 195.802i) q^{87} +(-170.963 + 296.117i) q^{88} -860.817 q^{89} +(110.113 - 576.583i) q^{90} +376.279 q^{91} +(-38.2591 + 66.2668i) q^{92} +(107.597 - 289.227i) q^{93} +(10.9436 + 18.9549i) q^{94} +(-779.149 - 1349.53i) q^{95} +(-57.9756 + 155.842i) q^{96} +(201.075 - 348.272i) q^{97} -551.076 q^{98} +(873.418 + 754.230i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 3 q^{3} - 8 q^{4} + 9 q^{5} - 19 q^{7} - 32 q^{8} - 51 q^{9} + 36 q^{10} + 24 q^{11} - 12 q^{12} - 61 q^{13} + 38 q^{14} + 171 q^{15} - 32 q^{16} + 6 q^{17} + 102 q^{18} + 266 q^{19} + 36 q^{20}+ \cdots + 1557 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(11\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 − 1.73205i 0.353553 − 0.612372i
\(3\) 3.31174 + 4.00405i 0.637344 + 0.770579i
\(4\) −2.00000 − 3.46410i −0.250000 − 0.433013i
\(5\) −5.43521 − 9.41407i −0.486140 − 0.842020i 0.513733 − 0.857950i \(-0.328262\pi\)
−0.999873 + 0.0159306i \(0.994929\pi\)
\(6\) 10.2470 − 1.73205i 0.697217 − 0.117851i
\(7\) −12.4352 + 21.5384i −0.671438 + 1.16297i 0.306058 + 0.952013i \(0.400990\pi\)
−0.977496 + 0.210953i \(0.932343\pi\)
\(8\) −8.00000 −0.353553
\(9\) −5.06479 + 26.5207i −0.187585 + 0.982248i
\(10\) −21.7409 −0.687506
\(11\) 21.3704 − 37.0147i 0.585766 − 1.01458i −0.409013 − 0.912528i \(-0.634127\pi\)
0.994779 − 0.102048i \(-0.0325396\pi\)
\(12\) 7.24695 − 19.4803i 0.174335 − 0.468623i
\(13\) −7.56479 − 13.1026i −0.161392 − 0.279539i 0.773976 − 0.633215i \(-0.218265\pi\)
−0.935368 + 0.353676i \(0.884932\pi\)
\(14\) 24.8704 + 43.0768i 0.474779 + 0.822341i
\(15\) 19.6944 − 52.9398i 0.339004 − 0.911266i
\(16\) −8.00000 + 13.8564i −0.125000 + 0.216506i
\(17\) −13.8704 −0.197887 −0.0989433 − 0.995093i \(-0.531546\pi\)
−0.0989433 + 0.995093i \(0.531546\pi\)
\(18\) 40.8704 + 35.2932i 0.535181 + 0.462149i
\(19\) 143.352 1.73091 0.865454 − 0.500989i \(-0.167030\pi\)
0.865454 + 0.500989i \(0.167030\pi\)
\(20\) −21.7409 + 37.6563i −0.243070 + 0.421010i
\(21\) −127.423 + 21.5384i −1.32409 + 0.223813i
\(22\) −42.7409 − 74.0293i −0.414199 − 0.717414i
\(23\) −9.56479 − 16.5667i −0.0867129 − 0.150191i 0.819407 − 0.573212i \(-0.194303\pi\)
−0.906120 + 0.423021i \(0.860970\pi\)
\(24\) −26.4939 − 32.0324i −0.225335 − 0.272441i
\(25\) 3.41692 − 5.91828i 0.0273353 − 0.0473462i
\(26\) −30.2591 −0.228243
\(27\) −122.963 + 67.5500i −0.876456 + 0.481481i
\(28\) 99.4817 0.671438
\(29\) −113.046 + 195.802i −0.723869 + 1.25378i 0.235569 + 0.971858i \(0.424305\pi\)
−0.959438 + 0.281920i \(0.909029\pi\)
\(30\) −72.0000 − 87.0514i −0.438178 − 0.529778i
\(31\) −29.6944 − 51.4321i −0.172041 − 0.297983i 0.767092 − 0.641537i \(-0.221703\pi\)
−0.939133 + 0.343553i \(0.888369\pi\)
\(32\) 16.0000 + 27.7128i 0.0883883 + 0.153093i
\(33\) 218.982 − 37.0147i 1.15515 − 0.195255i
\(34\) −13.8704 + 24.0243i −0.0699635 + 0.121180i
\(35\) 270.352 1.30565
\(36\) 102.000 − 35.4965i 0.472222 − 0.164336i
\(37\) −84.1860 −0.374056 −0.187028 − 0.982355i \(-0.559886\pi\)
−0.187028 + 0.982355i \(0.559886\pi\)
\(38\) 143.352 − 248.293i 0.611968 − 1.05996i
\(39\) 27.4108 − 73.6821i 0.112545 − 0.302528i
\(40\) 43.4817 + 75.3125i 0.171877 + 0.297699i
\(41\) −101.630 − 176.028i −0.387119 − 0.670510i 0.604942 − 0.796270i \(-0.293196\pi\)
−0.992061 + 0.125760i \(0.959863\pi\)
\(42\) −90.1174 + 242.242i −0.331081 + 0.889969i
\(43\) −162.945 + 282.229i −0.577881 + 1.00092i 0.417841 + 0.908520i \(0.362787\pi\)
−0.995722 + 0.0923995i \(0.970546\pi\)
\(44\) −170.963 −0.585766
\(45\) 277.196 − 96.4655i 0.918265 − 0.319560i
\(46\) −38.2591 −0.122631
\(47\) −5.47180 + 9.47744i −0.0169818 + 0.0294133i −0.874391 − 0.485221i \(-0.838739\pi\)
0.857410 + 0.514635i \(0.172072\pi\)
\(48\) −81.9756 + 13.8564i −0.246503 + 0.0416667i
\(49\) −137.769 − 238.623i −0.401659 − 0.695694i
\(50\) −6.83384 − 11.8366i −0.0193290 − 0.0334788i
\(51\) −45.9352 − 55.5378i −0.126122 − 0.152487i
\(52\) −30.2591 + 52.4104i −0.0806959 + 0.139769i
\(53\) −140.186 −0.363321 −0.181661 − 0.983361i \(-0.558147\pi\)
−0.181661 + 0.983361i \(0.558147\pi\)
\(54\) −5.96341 + 280.529i −0.0150281 + 0.706947i
\(55\) −464.611 −1.13906
\(56\) 99.4817 − 172.307i 0.237389 − 0.411170i
\(57\) 474.745 + 573.989i 1.10318 + 1.33380i
\(58\) 226.093 + 391.605i 0.511853 + 0.886555i
\(59\) −57.3704 − 99.3685i −0.126593 − 0.219266i 0.795761 − 0.605610i \(-0.207071\pi\)
−0.922355 + 0.386345i \(0.873738\pi\)
\(60\) −222.777 + 37.6563i −0.479341 + 0.0810234i
\(61\) 377.528 − 653.898i 0.792419 − 1.37251i −0.132047 − 0.991243i \(-0.542155\pi\)
0.924465 − 0.381266i \(-0.124512\pi\)
\(62\) −118.777 −0.243302
\(63\) −508.232 − 438.878i −1.01637 − 0.877674i
\(64\) 64.0000 0.125000
\(65\) −82.2325 + 142.431i −0.156918 + 0.271790i
\(66\) 154.870 − 416.302i 0.288837 − 0.776413i
\(67\) 383.723 + 664.627i 0.699689 + 1.21190i 0.968574 + 0.248725i \(0.0800115\pi\)
−0.268885 + 0.963172i \(0.586655\pi\)
\(68\) 27.7409 + 48.0486i 0.0494717 + 0.0856874i
\(69\) 34.6578 − 93.1624i 0.0604682 − 0.162543i
\(70\) 270.352 − 468.264i 0.461618 − 0.799546i
\(71\) 335.854 0.561387 0.280694 − 0.959797i \(-0.409436\pi\)
0.280694 + 0.959797i \(0.409436\pi\)
\(72\) 40.5183 − 212.166i 0.0663212 − 0.347277i
\(73\) 167.279 0.268199 0.134099 − 0.990968i \(-0.457186\pi\)
0.134099 + 0.990968i \(0.457186\pi\)
\(74\) −84.1860 + 145.814i −0.132249 + 0.229062i
\(75\) 35.0130 − 5.91828i 0.0539060 − 0.00911178i
\(76\) −286.704 − 496.586i −0.432727 − 0.749505i
\(77\) 531.492 + 920.570i 0.786612 + 1.36245i
\(78\) −100.210 − 121.159i −0.145469 − 0.175879i
\(79\) 12.6578 − 21.9239i 0.0180267 − 0.0312232i −0.856871 − 0.515530i \(-0.827595\pi\)
0.874898 + 0.484307i \(0.160928\pi\)
\(80\) 173.927 0.243070
\(81\) −677.696 − 268.643i −0.929624 − 0.368510i
\(82\) −406.518 −0.547469
\(83\) −143.861 + 249.174i −0.190250 + 0.329523i −0.945333 − 0.326107i \(-0.894263\pi\)
0.755083 + 0.655629i \(0.227597\pi\)
\(84\) 329.457 + 398.329i 0.427937 + 0.517396i
\(85\) 75.3887 + 130.577i 0.0962006 + 0.166624i
\(86\) 325.890 + 564.458i 0.408624 + 0.707757i
\(87\) −1158.38 + 195.802i −1.42749 + 0.241290i
\(88\) −170.963 + 296.117i −0.207100 + 0.358707i
\(89\) −860.817 −1.02524 −0.512620 − 0.858615i \(-0.671325\pi\)
−0.512620 + 0.858615i \(0.671325\pi\)
\(90\) 110.113 − 576.583i 0.128966 − 0.675302i
\(91\) 376.279 0.433459
\(92\) −38.2591 + 66.2668i −0.0433564 + 0.0750955i
\(93\) 107.597 − 289.227i 0.119971 − 0.322489i
\(94\) 10.9436 + 18.9549i 0.0120079 + 0.0207984i
\(95\) −779.149 − 1349.53i −0.841464 − 1.45746i
\(96\) −57.9756 + 155.842i −0.0616366 + 0.165683i
\(97\) 201.075 − 348.272i 0.210475 − 0.364553i −0.741389 − 0.671076i \(-0.765832\pi\)
0.951863 + 0.306523i \(0.0991657\pi\)
\(98\) −551.076 −0.568032
\(99\) 873.418 + 754.230i 0.886685 + 0.765687i
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 18.4.c.b.7.2 ✓ 4
3.2 odd 2 54.4.c.b.19.2 4
4.3 odd 2 144.4.i.b.97.1 4
9.2 odd 6 162.4.a.g.1.1 2
9.4 even 3 inner 18.4.c.b.13.2 yes 4
9.5 odd 6 54.4.c.b.37.2 4
9.7 even 3 162.4.a.f.1.2 2
12.11 even 2 432.4.i.b.289.2 4
36.7 odd 6 1296.4.a.l.1.2 2
36.11 even 6 1296.4.a.r.1.1 2
36.23 even 6 432.4.i.b.145.2 4
36.31 odd 6 144.4.i.b.49.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.4.c.b.7.2 ✓ 4 1.1 even 1 trivial
18.4.c.b.13.2 yes 4 9.4 even 3 inner
54.4.c.b.19.2 4 3.2 odd 2
54.4.c.b.37.2 4 9.5 odd 6
144.4.i.b.49.1 4 36.31 odd 6
144.4.i.b.97.1 4 4.3 odd 2
162.4.a.f.1.2 2 9.7 even 3
162.4.a.g.1.1 2 9.2 odd 6
432.4.i.b.145.2 4 36.23 even 6
432.4.i.b.289.2 4 12.11 even 2
1296.4.a.l.1.2 2 36.7 odd 6
1296.4.a.r.1.1 2 36.11 even 6