Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [342,2,Mod(293,342)] 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("342.293"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(342, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 342.n (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [18,-18,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.73088374913\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 5 x^{17} + 13 x^{16} - 30 x^{15} + 54 x^{14} - 69 x^{13} + 66 x^{12} + 36 x^{11} - 243 x^{10} + \cdots + 19683 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 335.4
Root \(1.73155 - 0.0417686i\) of defining polynomial
Character \(\chi\) \(=\) 342.335
Dual form 342.2.n.f.293.4

$q$-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 q^{2} +(-0.829601 - 1.52045i) q^{3} +1.00000 q^{4} +(0.820646 + 0.473800i) q^{5} +(0.829601 + 1.52045i) q^{6} +(1.09312 - 1.89335i) q^{7} -1.00000 q^{8} +(-1.62352 + 2.52273i) q^{9} +(-0.820646 - 0.473800i) q^{10} +(5.44935 + 3.14618i) q^{11} +(-0.829601 - 1.52045i) q^{12} -5.87479i q^{13} +(-1.09312 + 1.89335i) q^{14} +(0.0395800 - 1.64081i) q^{15} +1.00000 q^{16} +(0.936586 - 0.540738i) q^{17} +(1.62352 - 2.52273i) q^{18} +(-3.68177 + 2.33336i) q^{19} +(0.820646 + 0.473800i) q^{20} +(-3.78559 - 0.0913166i) q^{21} +(-5.44935 - 3.14618i) q^{22} -4.35455i q^{23} +(0.829601 + 1.52045i) q^{24} +(-2.05103 - 3.55248i) q^{25} +5.87479i q^{26} +(5.18256 + 0.375626i) q^{27} +(1.09312 - 1.89335i) q^{28} +(-1.13349 - 1.96327i) q^{29} +(-0.0395800 + 1.64081i) q^{30} +(3.60409 - 2.08083i) q^{31} -1.00000 q^{32} +(0.262823 - 10.8955i) q^{33} +(-0.936586 + 0.540738i) q^{34} +(1.79413 - 1.03584i) q^{35} +(-1.62352 + 2.52273i) q^{36} -7.47293i q^{37} +(3.68177 - 2.33336i) q^{38} +(-8.93232 + 4.87373i) q^{39} +(-0.820646 - 0.473800i) q^{40} +(2.61686 - 4.53253i) q^{41} +(3.78559 + 0.0913166i) q^{42} +8.12860 q^{43} +(5.44935 + 3.14618i) q^{44} +(-2.52761 + 1.30104i) q^{45} +4.35455i q^{46} +(-9.12185 + 5.26650i) q^{47} +(-0.829601 - 1.52045i) q^{48} +(1.11016 + 1.92286i) q^{49} +(2.05103 + 3.55248i) q^{50} +(-1.59916 - 0.975434i) q^{51} -5.87479i q^{52} +(-5.21491 + 9.03248i) q^{53} +(-5.18256 - 0.375626i) q^{54} +(2.98132 + 5.16380i) q^{55} +(-1.09312 + 1.89335i) q^{56} +(6.60216 + 3.66218i) q^{57} +(1.13349 + 1.96327i) q^{58} +(1.90260 - 3.29540i) q^{59} +(0.0395800 - 1.64081i) q^{60} +(-5.19504 - 8.99808i) q^{61} +(-3.60409 + 2.08083i) q^{62} +(3.00169 + 5.83155i) q^{63} +1.00000 q^{64} +(2.78348 - 4.82112i) q^{65} +(-0.262823 + 10.8955i) q^{66} +11.5779i q^{67} +(0.936586 - 0.540738i) q^{68} +(-6.62087 + 3.61254i) q^{69} +(-1.79413 + 1.03584i) q^{70} +(6.13727 + 10.6301i) q^{71} +(1.62352 - 2.52273i) q^{72} +(6.44130 + 11.1567i) q^{73} +7.47293i q^{74} +(-3.69983 + 6.06562i) q^{75} +(-3.68177 + 2.33336i) q^{76} +(11.9136 - 6.87833i) q^{77} +(8.93232 - 4.87373i) q^{78} +1.21941i q^{79} +(0.820646 + 0.473800i) q^{80} +(-3.72833 - 8.19143i) q^{81} +(-2.61686 + 4.53253i) q^{82} +(3.86795 + 2.23316i) q^{83} +(-3.78559 - 0.0913166i) q^{84} +1.02481 q^{85} -8.12860 q^{86} +(-2.04470 + 3.35215i) q^{87} +(-5.44935 - 3.14618i) q^{88} +(-2.71650 + 4.70511i) q^{89} +(2.52761 - 1.30104i) q^{90} +(-11.1230 - 6.42187i) q^{91} -4.35455i q^{92} +(-6.15375 - 3.75358i) q^{93} +(9.12185 - 5.26650i) q^{94} +(-4.12697 + 0.170443i) q^{95} +(0.829601 + 1.52045i) q^{96} -2.16176i q^{97} +(-1.11016 - 1.92286i) q^{98} +(-16.7841 + 8.63932i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 18 q^{2} - q^{3} + 18 q^{4} + 3 q^{5} + q^{6} - 18 q^{8} + 5 q^{9} - 3 q^{10} + 15 q^{11} - q^{12} - 17 q^{15} + 18 q^{16} - 15 q^{17} - 5 q^{18} + 6 q^{19} + 3 q^{20} + q^{21} - 15 q^{22} + q^{24}+ \cdots - 56 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/342\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(325\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −0.829601 − 1.52045i −0.478970 − 0.877831i
\(4\) 1.00000 0.500000
\(5\) 0.820646 + 0.473800i 0.367004 + 0.211890i 0.672149 − 0.740416i \(-0.265372\pi\)
−0.305145 + 0.952306i \(0.598705\pi\)
\(6\) 0.829601 + 1.52045i 0.338683 + 0.620720i
\(7\) 1.09312 − 1.89335i 0.413162 − 0.715617i −0.582072 − 0.813137i \(-0.697758\pi\)
0.995234 + 0.0975202i \(0.0310911\pi\)
\(8\) −1.00000 −0.353553
\(9\) −1.62352 + 2.52273i −0.541175 + 0.840910i
\(10\) −0.820646 − 0.473800i −0.259511 − 0.149829i
\(11\) 5.44935 + 3.14618i 1.64304 + 0.948609i 0.979745 + 0.200251i \(0.0641759\pi\)
0.663295 + 0.748358i \(0.269157\pi\)
\(12\) −0.829601 − 1.52045i −0.239485 − 0.438916i
\(13\) − 5.87479i − 1.62937i −0.579901 − 0.814687i \(-0.696909\pi\)
0.579901 − 0.814687i \(-0.303091\pi\)
\(14\) −1.09312 + 1.89335i −0.292150 + 0.506018i
\(15\) 0.0395800 − 1.64081i 0.0102195 − 0.423656i
\(16\) 1.00000 0.250000
\(17\) 0.936586 − 0.540738i 0.227155 − 0.131148i −0.382104 − 0.924119i \(-0.624800\pi\)
0.609259 + 0.792971i \(0.291467\pi\)
\(18\) 1.62352 − 2.52273i 0.382668 − 0.594613i
\(19\) −3.68177 + 2.33336i −0.844655 + 0.535310i
\(20\) 0.820646 + 0.473800i 0.183502 + 0.105945i
\(21\) −3.78559 − 0.0913166i −0.826083 − 0.0199269i
\(22\) −5.44935 − 3.14618i −1.16180 − 0.670768i
\(23\) − 4.35455i − 0.907987i −0.891005 − 0.453993i \(-0.849999\pi\)
0.891005 − 0.453993i \(-0.150001\pi\)
\(24\) 0.829601 + 1.52045i 0.169342 + 0.310360i
\(25\) −2.05103 − 3.55248i −0.410205 − 0.710497i
\(26\) 5.87479i 1.15214i
\(27\) 5.18256 + 0.375626i 0.997384 + 0.0722893i
\(28\) 1.09312 − 1.89335i 0.206581 − 0.357809i
\(29\) −1.13349 − 1.96327i −0.210484 − 0.364570i 0.741382 − 0.671083i \(-0.234171\pi\)
−0.951866 + 0.306514i \(0.900837\pi\)
\(30\) −0.0395800 + 1.64081i −0.00722628 + 0.299570i
\(31\) 3.60409 − 2.08083i 0.647315 − 0.373727i −0.140112 − 0.990136i \(-0.544746\pi\)
0.787427 + 0.616408i \(0.211413\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0.262823 − 10.8955i 0.0457517 − 1.89667i
\(34\) −0.936586 + 0.540738i −0.160623 + 0.0927358i
\(35\) 1.79413 − 1.03584i 0.303264 − 0.175090i
\(36\) −1.62352 + 2.52273i −0.270587 + 0.420455i
\(37\) − 7.47293i − 1.22854i −0.789095 − 0.614271i \(-0.789450\pi\)
0.789095 − 0.614271i \(-0.210550\pi\)
\(38\) 3.68177 − 2.33336i 0.597262 − 0.378522i
\(39\) −8.93232 + 4.87373i −1.43032 + 0.780422i
\(40\) −0.820646 − 0.473800i −0.129755 − 0.0749144i
\(41\) 2.61686 − 4.53253i 0.408684 − 0.707862i −0.586058 − 0.810269i \(-0.699321\pi\)
0.994743 + 0.102407i \(0.0326545\pi\)
\(42\) 3.78559 + 0.0913166i 0.584129 + 0.0140905i
\(43\) 8.12860 1.23960 0.619800 − 0.784760i \(-0.287214\pi\)
0.619800 + 0.784760i \(0.287214\pi\)
\(44\) 5.44935 + 3.14618i 0.821520 + 0.474305i
\(45\) −2.52761 + 1.30104i −0.376794 + 0.193948i
\(46\) 4.35455i 0.642044i
\(47\) −9.12185 + 5.26650i −1.33056 + 0.768198i −0.985385 − 0.170341i \(-0.945513\pi\)
−0.345173 + 0.938539i \(0.612180\pi\)
\(48\) −0.829601 − 1.52045i −0.119743 − 0.219458i
\(49\) 1.11016 + 1.92286i 0.158595 + 0.274694i
\(50\) 2.05103 + 3.55248i 0.290059 + 0.502397i
\(51\) −1.59916 − 0.975434i −0.223927 − 0.136588i
\(52\) − 5.87479i − 0.814687i
\(53\) −5.21491 + 9.03248i −0.716322 + 1.24071i 0.246125 + 0.969238i \(0.420843\pi\)
−0.962447 + 0.271469i \(0.912491\pi\)
\(54\) −5.18256 − 0.375626i −0.705257 − 0.0511163i
\(55\) 2.98132 + 5.16380i 0.402001 + 0.696287i
\(56\) −1.09312 + 1.89335i −0.146075 + 0.253009i
\(57\) 6.60216 + 3.66218i 0.874477 + 0.485067i
\(58\) 1.13349 + 1.96327i 0.148835 + 0.257790i
\(59\) 1.90260 − 3.29540i 0.247697 − 0.429024i −0.715189 − 0.698931i \(-0.753660\pi\)
0.962886 + 0.269907i \(0.0869929\pi\)
\(60\) 0.0395800 − 1.64081i 0.00510975 − 0.211828i
\(61\) −5.19504 − 8.99808i −0.665157 − 1.15209i −0.979243 − 0.202691i \(-0.935031\pi\)
0.314086 − 0.949395i \(-0.398302\pi\)
\(62\) −3.60409 + 2.08083i −0.457720 + 0.264265i
\(63\) 3.00169 + 5.83155i 0.378177 + 0.734706i
\(64\) 1.00000 0.125000
\(65\) 2.78348 − 4.82112i 0.345248 − 0.597987i
\(66\) −0.262823 + 10.8955i −0.0323513 + 1.34115i
\(67\) 11.5779i 1.41446i 0.706984 + 0.707230i \(0.250055\pi\)
−0.706984 + 0.707230i \(0.749945\pi\)
\(68\) 0.936586 − 0.540738i 0.113578 − 0.0655741i
\(69\) −6.62087 + 3.61254i −0.797059 + 0.434899i
\(70\) −1.79413 + 1.03584i −0.214440 + 0.123807i
\(71\) 6.13727 + 10.6301i 0.728360 + 1.26156i 0.957576 + 0.288180i \(0.0930503\pi\)
−0.229216 + 0.973375i \(0.573616\pi\)
\(72\) 1.62352 − 2.52273i 0.191334 − 0.297307i
\(73\) 6.44130 + 11.1567i 0.753897 + 1.30579i 0.945921 + 0.324398i \(0.105162\pi\)
−0.192023 + 0.981390i \(0.561505\pi\)
\(74\) 7.47293i 0.868711i
\(75\) −3.69983 + 6.06562i −0.427220 + 0.700398i
\(76\) −3.68177 + 2.33336i −0.422328 + 0.267655i
\(77\) 11.9136 − 6.87833i 1.35768 − 0.783858i
\(78\) 8.93232 − 4.87373i 1.01139 − 0.551842i
\(79\) 1.21941i 0.137194i 0.997644 + 0.0685969i \(0.0218522\pi\)
−0.997644 + 0.0685969i \(0.978148\pi\)
\(80\) 0.820646 + 0.473800i 0.0917510 + 0.0529724i
\(81\) −3.72833 − 8.19143i −0.414259 − 0.910159i
\(82\) −2.61686 + 4.53253i −0.288983 + 0.500534i
\(83\) 3.86795 + 2.23316i 0.424563 + 0.245121i 0.697028 − 0.717044i \(-0.254506\pi\)
−0.272465 + 0.962166i \(0.587839\pi\)
\(84\) −3.78559 − 0.0913166i −0.413042 − 0.00996345i
\(85\) 1.02481 0.111156
\(86\) −8.12860 −0.876530
\(87\) −2.04470 + 3.35215i −0.219215 + 0.359388i
\(88\) −5.44935 − 3.14618i −0.580902 − 0.335384i
\(89\) −2.71650 + 4.70511i −0.287948 + 0.498741i −0.973320 − 0.229453i \(-0.926306\pi\)
0.685372 + 0.728193i \(0.259640\pi\)
\(90\) 2.52761 − 1.30104i 0.266433 − 0.137142i
\(91\) −11.1230 − 6.42187i −1.16601 − 0.673195i
\(92\) − 4.35455i − 0.453993i
\(93\) −6.15375 − 3.75358i −0.638114 − 0.389229i
\(94\) 9.12185 − 5.26650i 0.940847 − 0.543198i
\(95\) −4.12697 + 0.170443i −0.423419 + 0.0174871i
\(96\) 0.829601 + 1.52045i 0.0846708 + 0.155180i
\(97\) − 2.16176i − 0.219493i −0.993960 − 0.109747i \(-0.964996\pi\)
0.993960 − 0.109747i \(-0.0350040\pi\)
\(98\) −1.11016 − 1.92286i −0.112143 − 0.194238i
\(99\) −16.7841 + 8.63932i −1.68687 + 0.868285i
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 342.2.n.f.335.4 yes 18
3.2 odd 2 1026.2.n.f.791.5 18
9.4 even 3 1026.2.j.f.449.5 18
9.5 odd 6 342.2.j.f.221.6 yes 18
19.8 odd 6 342.2.j.f.65.6 ✓ 18
57.8 even 6 1026.2.j.f.521.5 18
171.103 odd 6 1026.2.n.f.179.5 18
171.122 even 6 inner 342.2.n.f.293.4 yes 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
342.2.j.f.65.6 ✓ 18 19.8 odd 6
342.2.j.f.221.6 yes 18 9.5 odd 6
342.2.n.f.293.4 yes 18 171.122 even 6 inner
342.2.n.f.335.4 yes 18 1.1 even 1 trivial
1026.2.j.f.449.5 18 9.4 even 3
1026.2.j.f.521.5 18 57.8 even 6
1026.2.n.f.179.5 18 171.103 odd 6
1026.2.n.f.791.5 18 3.2 odd 2