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(55,342)] 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("342.55"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(342, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 10])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 342.u (of order \(9\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,-9,0,3,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(8)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.73088374913\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 114)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 253.1
Root \(-0.766044 + 0.642788i\) of defining polynomial
Character \(\chi\) \(=\) 342.253
Dual form 342.2.u.a.73.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.766044 - 0.642788i) q^{2} +(0.173648 - 0.984808i) q^{4} +(-0.386659 - 2.19285i) q^{5} +(0.326352 - 0.565258i) q^{7} +(-0.500000 - 0.866025i) q^{8} +(-1.70574 - 1.43128i) q^{10} +(-0.766044 - 1.32683i) q^{11} +(0.439693 - 0.160035i) q^{13} +(-0.113341 - 0.642788i) q^{14} +(-0.939693 - 0.342020i) q^{16} +(1.61334 - 1.35375i) q^{17} +(-2.23396 - 3.74292i) q^{19} -2.22668 q^{20} +(-1.43969 - 0.524005i) q^{22} +(-1.02481 + 5.81201i) q^{23} +(0.0393628 - 0.0143269i) q^{25} +(0.233956 - 0.405223i) q^{26} +(-0.500000 - 0.419550i) q^{28} +(6.38326 + 5.35619i) q^{29} +(4.31908 - 7.48086i) q^{31} +(-0.939693 + 0.342020i) q^{32} +(0.365715 - 2.07407i) q^{34} +(-1.36571 - 0.497079i) q^{35} -4.67499 q^{37} +(-4.11721 - 1.43128i) q^{38} +(-1.70574 + 1.43128i) q^{40} +(3.26604 + 1.18874i) q^{41} +(1.78699 + 10.1345i) q^{43} +(-1.43969 + 0.524005i) q^{44} +(2.95084 + 5.11100i) q^{46} +(3.55303 + 2.98135i) q^{47} +(3.28699 + 5.69323i) q^{49} +(0.0209445 - 0.0362770i) q^{50} +(-0.0812519 - 0.460802i) q^{52} +(2.07532 - 11.7697i) q^{53} +(-2.61334 + 2.19285i) q^{55} -0.652704 q^{56} +8.33275 q^{58} +(-10.9042 + 9.14971i) q^{59} +(-1.58378 + 8.98205i) q^{61} +(-1.50000 - 8.50692i) q^{62} +(-0.500000 + 0.866025i) q^{64} +(-0.520945 - 0.902302i) q^{65} +(0.190722 + 0.160035i) q^{67} +(-1.05303 - 1.82391i) q^{68} +(-1.36571 + 0.497079i) q^{70} +(0.772441 + 4.38073i) q^{71} +(8.54323 + 3.10948i) q^{73} +(-3.58125 + 3.00503i) q^{74} +(-4.07398 + 1.55007i) q^{76} -1.00000 q^{77} +(-11.3302 - 4.12386i) q^{79} +(-0.386659 + 2.19285i) q^{80} +(3.26604 - 1.18874i) q^{82} +(1.85457 - 3.21221i) q^{83} +(-3.59240 - 3.01438i) q^{85} +(7.88326 + 6.61484i) q^{86} +(-0.766044 + 1.32683i) q^{88} +(15.7554 - 5.73448i) q^{89} +(0.0530334 - 0.300767i) q^{91} +(5.54576 + 2.01849i) q^{92} +4.63816 q^{94} +(-7.34389 + 6.34597i) q^{95} +(1.89053 - 1.58634i) q^{97} +(6.17752 + 2.24843i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 9 q^{5} + 3 q^{7} - 3 q^{8} - 3 q^{13} + 6 q^{14} + 3 q^{17} - 18 q^{19} - 3 q^{22} + 21 q^{23} + 9 q^{25} + 6 q^{26} - 3 q^{28} + 3 q^{29} + 9 q^{31} + 12 q^{34} - 18 q^{35} - 18 q^{37} + 6 q^{38}+ \cdots + 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(191\) \(325\)
\(\chi(n)\) \(1\) \(e\left(\frac{7}{9}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.766044 − 0.642788i 0.541675 − 0.454519i
\(3\) 0 0
\(4\) 0.173648 − 0.984808i 0.0868241 − 0.492404i
\(5\) −0.386659 − 2.19285i −0.172919 − 0.980674i −0.940518 − 0.339743i \(-0.889660\pi\)
0.767599 − 0.640930i \(-0.221451\pi\)
\(6\) 0 0
\(7\) 0.326352 − 0.565258i 0.123349 − 0.213647i −0.797737 − 0.603005i \(-0.793970\pi\)
0.921087 + 0.389358i \(0.127303\pi\)
\(8\) −0.500000 − 0.866025i −0.176777 − 0.306186i
\(9\) 0 0
\(10\) −1.70574 − 1.43128i −0.539401 − 0.452612i
\(11\) −0.766044 − 1.32683i −0.230971 − 0.400054i 0.727123 − 0.686507i \(-0.240857\pi\)
−0.958094 + 0.286453i \(0.907524\pi\)
\(12\) 0 0
\(13\) 0.439693 − 0.160035i 0.121949 − 0.0443857i −0.280325 − 0.959905i \(-0.590442\pi\)
0.402274 + 0.915519i \(0.368220\pi\)
\(14\) −0.113341 − 0.642788i −0.0302916 − 0.171792i
\(15\) 0 0
\(16\) −0.939693 − 0.342020i −0.234923 − 0.0855050i
\(17\) 1.61334 − 1.35375i 0.391293 − 0.328333i −0.425824 − 0.904806i \(-0.640016\pi\)
0.817116 + 0.576473i \(0.195571\pi\)
\(18\) 0 0
\(19\) −2.23396 − 3.74292i −0.512505 − 0.858685i
\(20\) −2.22668 −0.497901
\(21\) 0 0
\(22\) −1.43969 − 0.524005i −0.306943 − 0.111718i
\(23\) −1.02481 + 5.81201i −0.213689 + 1.21189i 0.669479 + 0.742831i \(0.266517\pi\)
−0.883168 + 0.469058i \(0.844594\pi\)
\(24\) 0 0
\(25\) 0.0393628 − 0.0143269i 0.00787257 − 0.00286538i
\(26\) 0.233956 − 0.405223i 0.0458825 − 0.0794708i
\(27\) 0 0
\(28\) −0.500000 − 0.419550i −0.0944911 − 0.0792875i
\(29\) 6.38326 + 5.35619i 1.18534 + 0.994619i 0.999928 + 0.0119582i \(0.00380651\pi\)
0.185412 + 0.982661i \(0.440638\pi\)
\(30\) 0 0
\(31\) 4.31908 − 7.48086i 0.775729 − 1.34360i −0.158654 − 0.987334i \(-0.550716\pi\)
0.934384 − 0.356268i \(-0.115951\pi\)
\(32\) −0.939693 + 0.342020i −0.166116 + 0.0604612i
\(33\) 0 0
\(34\) 0.365715 − 2.07407i 0.0627195 − 0.355700i
\(35\) −1.36571 − 0.497079i −0.230848 − 0.0840218i
\(36\) 0 0
\(37\) −4.67499 −0.768564 −0.384282 − 0.923216i \(-0.625551\pi\)
−0.384282 + 0.923216i \(0.625551\pi\)
\(38\) −4.11721 − 1.43128i −0.667900 − 0.232185i
\(39\) 0 0
\(40\) −1.70574 + 1.43128i −0.269701 + 0.226306i
\(41\) 3.26604 + 1.18874i 0.510070 + 0.185650i 0.584218 − 0.811597i \(-0.301401\pi\)
−0.0741475 + 0.997247i \(0.523624\pi\)
\(42\) 0 0
\(43\) 1.78699 + 10.1345i 0.272513 + 1.54550i 0.746752 + 0.665103i \(0.231612\pi\)
−0.474238 + 0.880396i \(0.657277\pi\)
\(44\) −1.43969 + 0.524005i −0.217042 + 0.0789968i
\(45\) 0 0
\(46\) 2.95084 + 5.11100i 0.435077 + 0.753576i
\(47\) 3.55303 + 2.98135i 0.518263 + 0.434874i 0.864026 − 0.503448i \(-0.167935\pi\)
−0.345763 + 0.938322i \(0.612380\pi\)
\(48\) 0 0
\(49\) 3.28699 + 5.69323i 0.469570 + 0.813319i
\(50\) 0.0209445 − 0.0362770i 0.00296200 − 0.00513034i
\(51\) 0 0
\(52\) −0.0812519 − 0.460802i −0.0112676 − 0.0639018i
\(53\) 2.07532 − 11.7697i 0.285067 − 1.61670i −0.419977 − 0.907535i \(-0.637962\pi\)
0.705045 − 0.709163i \(-0.250927\pi\)
\(54\) 0 0
\(55\) −2.61334 + 2.19285i −0.352383 + 0.295684i
\(56\) −0.652704 −0.0872212
\(57\) 0 0
\(58\) 8.33275 1.09414
\(59\) −10.9042 + 9.14971i −1.41961 + 1.19119i −0.468055 + 0.883699i \(0.655045\pi\)
−0.951551 + 0.307492i \(0.900510\pi\)
\(60\) 0 0
\(61\) −1.58378 + 8.98205i −0.202782 + 1.15003i 0.698110 + 0.715991i \(0.254025\pi\)
−0.900892 + 0.434043i \(0.857086\pi\)
\(62\) −1.50000 − 8.50692i −0.190500 − 1.08038i
\(63\) 0 0
\(64\) −0.500000 + 0.866025i −0.0625000 + 0.108253i
\(65\) −0.520945 − 0.902302i −0.0646152 − 0.111917i
\(66\) 0 0
\(67\) 0.190722 + 0.160035i 0.0233004 + 0.0195514i 0.654363 − 0.756180i \(-0.272937\pi\)
−0.631063 + 0.775732i \(0.717381\pi\)
\(68\) −1.05303 − 1.82391i −0.127699 − 0.221181i
\(69\) 0 0
\(70\) −1.36571 + 0.497079i −0.163234 + 0.0594124i
\(71\) 0.772441 + 4.38073i 0.0916719 + 0.519897i 0.995716 + 0.0924590i \(0.0294727\pi\)
−0.904045 + 0.427438i \(0.859416\pi\)
\(72\) 0 0
\(73\) 8.54323 + 3.10948i 0.999910 + 0.363937i 0.789550 − 0.613687i \(-0.210314\pi\)
0.210360 + 0.977624i \(0.432536\pi\)
\(74\) −3.58125 + 3.00503i −0.416312 + 0.349327i
\(75\) 0 0
\(76\) −4.07398 + 1.55007i −0.467317 + 0.177805i
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) −11.3302 − 4.12386i −1.27475 − 0.463971i −0.386057 − 0.922475i \(-0.626163\pi\)
−0.888692 + 0.458504i \(0.848385\pi\)
\(80\) −0.386659 + 2.19285i −0.0432298 + 0.245168i
\(81\) 0 0
\(82\) 3.26604 − 1.18874i 0.360674 − 0.131275i
\(83\) 1.85457 − 3.21221i 0.203566 − 0.352586i −0.746109 − 0.665824i \(-0.768080\pi\)
0.949675 + 0.313238i \(0.101414\pi\)
\(84\) 0 0
\(85\) −3.59240 − 3.01438i −0.389650 − 0.326955i
\(86\) 7.88326 + 6.61484i 0.850073 + 0.713296i
\(87\) 0 0
\(88\) −0.766044 + 1.32683i −0.0816606 + 0.141440i
\(89\) 15.7554 − 5.73448i 1.67007 − 0.607854i 0.678169 − 0.734906i \(-0.262774\pi\)
0.991896 + 0.127051i \(0.0405513\pi\)
\(90\) 0 0
\(91\) 0.0530334 − 0.300767i 0.00555941 − 0.0315290i
\(92\) 5.54576 + 2.01849i 0.578185 + 0.210442i
\(93\) 0 0
\(94\) 4.63816 0.478389
\(95\) −7.34389 + 6.34597i −0.753468 + 0.651083i
\(96\) 0 0
\(97\) 1.89053 − 1.58634i 0.191954 − 0.161069i −0.541746 − 0.840543i \(-0.682236\pi\)
0.733700 + 0.679474i \(0.237792\pi\)
\(98\) 6.17752 + 2.24843i 0.624024 + 0.227126i
\(99\) 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 342.2.u.a.253.1 6
3.2 odd 2 114.2.i.d.25.1 ✓ 6
12.11 even 2 912.2.bo.f.481.1 6
19.4 even 9 6498.2.a.bs.1.1 3
19.15 odd 18 6498.2.a.bn.1.1 3
19.16 even 9 inner 342.2.u.a.73.1 6
57.23 odd 18 2166.2.a.o.1.3 3
57.35 odd 18 114.2.i.d.73.1 yes 6
57.53 even 18 2166.2.a.u.1.3 3
228.35 even 18 912.2.bo.f.529.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.i.d.25.1 ✓ 6 3.2 odd 2
114.2.i.d.73.1 yes 6 57.35 odd 18
342.2.u.a.73.1 6 19.16 even 9 inner
342.2.u.a.253.1 6 1.1 even 1 trivial
912.2.bo.f.481.1 6 12.11 even 2
912.2.bo.f.529.1 6 228.35 even 18
2166.2.a.o.1.3 3 57.23 odd 18
2166.2.a.u.1.3 3 57.53 even 18
6498.2.a.bn.1.1 3 19.15 odd 18
6498.2.a.bs.1.1 3 19.4 even 9