Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [114,2,Mod(25,114)] 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("114.25"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(114, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 14])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 114 = 2 \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 114.i (of order \(9\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.910294583043\)
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 25.1
Root \(-0.766044 + 0.642788i\) of defining polynomial
Character \(\chi\) \(=\) 114.25
Dual form 114.2.i.d.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} +(0.939693 + 0.342020i) q^{3} +(0.173648 - 0.984808i) q^{4} +(0.386659 + 2.19285i) q^{5} +(-0.939693 + 0.342020i) q^{6} +(0.326352 - 0.565258i) q^{7} +(0.500000 + 0.866025i) q^{8} +(0.766044 + 0.642788i) q^{9} +(-1.70574 - 1.43128i) q^{10} +(0.766044 + 1.32683i) q^{11} +(0.500000 - 0.866025i) q^{12} +(0.439693 - 0.160035i) q^{13} +(0.113341 + 0.642788i) q^{14} +(-0.386659 + 2.19285i) q^{15} +(-0.939693 - 0.342020i) q^{16} +(-1.61334 + 1.35375i) q^{17} -1.00000 q^{18} +(-2.23396 - 3.74292i) q^{19} +2.22668 q^{20} +(0.500000 - 0.419550i) q^{21} +(-1.43969 - 0.524005i) q^{22} +(1.02481 - 5.81201i) q^{23} +(0.173648 + 0.984808i) q^{24} +(0.0393628 - 0.0143269i) q^{25} +(-0.233956 + 0.405223i) q^{26} +(0.500000 + 0.866025i) q^{27} +(-0.500000 - 0.419550i) q^{28} +(-6.38326 - 5.35619i) q^{29} +(-1.11334 - 1.92836i) q^{30} +(4.31908 - 7.48086i) q^{31} +(0.939693 - 0.342020i) q^{32} +(0.266044 + 1.50881i) q^{33} +(0.365715 - 2.07407i) q^{34} +(1.36571 + 0.497079i) q^{35} +(0.766044 - 0.642788i) q^{36} -4.67499 q^{37} +(4.11721 + 1.43128i) q^{38} +0.467911 q^{39} +(-1.70574 + 1.43128i) q^{40} +(-3.26604 - 1.18874i) q^{41} +(-0.113341 + 0.642788i) q^{42} +(1.78699 + 10.1345i) q^{43} +(1.43969 - 0.524005i) q^{44} +(-1.11334 + 1.92836i) q^{45} +(2.95084 + 5.11100i) q^{46} +(-3.55303 - 2.98135i) q^{47} +(-0.766044 - 0.642788i) q^{48} +(3.28699 + 5.69323i) q^{49} +(-0.0209445 + 0.0362770i) q^{50} +(-1.97906 + 0.720317i) q^{51} +(-0.0812519 - 0.460802i) q^{52} +(-2.07532 + 11.7697i) q^{53} +(-0.939693 - 0.342020i) q^{54} +(-2.61334 + 2.19285i) q^{55} +0.652704 q^{56} +(-0.819078 - 4.28125i) q^{57} +8.33275 q^{58} +(10.9042 - 9.14971i) q^{59} +(2.09240 + 0.761570i) q^{60} +(-1.58378 + 8.98205i) q^{61} +(1.50000 + 8.50692i) q^{62} +(0.613341 - 0.223238i) q^{63} +(-0.500000 + 0.866025i) q^{64} +(0.520945 + 0.902302i) q^{65} +(-1.17365 - 0.984808i) q^{66} +(0.190722 + 0.160035i) q^{67} +(1.05303 + 1.82391i) q^{68} +(2.95084 - 5.11100i) q^{69} +(-1.36571 + 0.497079i) q^{70} +(-0.772441 - 4.38073i) q^{71} +(-0.173648 + 0.984808i) q^{72} +(8.54323 + 3.10948i) q^{73} +(3.58125 - 3.00503i) q^{74} +0.0418891 q^{75} +(-4.07398 + 1.55007i) q^{76} +1.00000 q^{77} +(-0.358441 + 0.300767i) q^{78} +(-11.3302 - 4.12386i) q^{79} +(0.386659 - 2.19285i) q^{80} +(0.173648 + 0.984808i) q^{81} +(3.26604 - 1.18874i) q^{82} +(-1.85457 + 3.21221i) q^{83} +(-0.326352 - 0.565258i) q^{84} +(-3.59240 - 3.01438i) q^{85} +(-7.88326 - 6.61484i) q^{86} +(-4.16637 - 7.21637i) q^{87} +(-0.766044 + 1.32683i) q^{88} +(-15.7554 + 5.73448i) q^{89} +(-0.386659 - 2.19285i) q^{90} +(0.0530334 - 0.300767i) q^{91} +(-5.54576 - 2.01849i) q^{92} +(6.61721 - 5.55250i) q^{93} +4.63816 q^{94} +(7.34389 - 6.34597i) q^{95} +1.00000 q^{96} +(1.89053 - 1.58634i) q^{97} +(-6.17752 - 2.24843i) q^{98} +(-0.266044 + 1.50881i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 9 q^{5} + 3 q^{7} + 3 q^{8} + 3 q^{12} - 3 q^{13} - 6 q^{14} - 9 q^{15} - 3 q^{17} - 6 q^{18} - 18 q^{19} + 3 q^{21} - 3 q^{22} - 21 q^{23} + 9 q^{25} - 6 q^{26} + 3 q^{27} - 3 q^{28} - 3 q^{29} + 9 q^{31}+ \cdots + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/114\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(97\)
\(\chi(n)\) \(1\) \(e\left(\frac{7}{9}\right)\)

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
\(3\) 0.939693 + 0.342020i 0.542532 + 0.197465i
\(4\) 0.173648 − 0.984808i 0.0868241 − 0.492404i
\(5\) 0.386659 + 2.19285i 0.172919 + 0.980674i 0.940518 + 0.339743i \(0.110340\pi\)
−0.767599 + 0.640930i \(0.778549\pi\)
\(6\) −0.939693 + 0.342020i −0.383628 + 0.139629i
\(7\) 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
\(9\) 0.766044 + 0.642788i 0.255348 + 0.214263i
\(10\) −1.70574 − 1.43128i −0.539401 − 0.452612i
\(11\) 0.766044 + 1.32683i 0.230971 + 0.400054i 0.958094 − 0.286453i \(-0.0924764\pi\)
−0.727123 + 0.686507i \(0.759143\pi\)
\(12\) 0.500000 − 0.866025i 0.144338 − 0.250000i
\(13\) 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
\(15\) −0.386659 + 2.19285i −0.0998350 + 0.566192i
\(16\) −0.939693 − 0.342020i −0.234923 − 0.0855050i
\(17\) −1.61334 + 1.35375i −0.391293 + 0.328333i −0.817116 − 0.576473i \(-0.804429\pi\)
0.425824 + 0.904806i \(0.359984\pi\)
\(18\) −1.00000 −0.235702
\(19\) −2.23396 − 3.74292i −0.512505 − 0.858685i
\(20\) 2.22668 0.497901
\(21\) 0.500000 − 0.419550i 0.109109 − 0.0915533i
\(22\) −1.43969 − 0.524005i −0.306943 − 0.111718i
\(23\) 1.02481 − 5.81201i 0.213689 − 1.21189i −0.669479 − 0.742831i \(-0.733483\pi\)
0.883168 − 0.469058i \(-0.155406\pi\)
\(24\) 0.173648 + 0.984808i 0.0354458 + 0.201023i
\(25\) 0.0393628 − 0.0143269i 0.00787257 − 0.00286538i
\(26\) −0.233956 + 0.405223i −0.0458825 + 0.0794708i
\(27\) 0.500000 + 0.866025i 0.0962250 + 0.166667i
\(28\) −0.500000 − 0.419550i −0.0944911 − 0.0792875i
\(29\) −6.38326 − 5.35619i −1.18534 − 0.994619i −0.999928 − 0.0119582i \(-0.996193\pi\)
−0.185412 − 0.982661i \(-0.559362\pi\)
\(30\) −1.11334 − 1.92836i −0.203267 − 0.352069i
\(31\) 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
\(33\) 0.266044 + 1.50881i 0.0463124 + 0.262651i
\(34\) 0.365715 − 2.07407i 0.0627195 − 0.355700i
\(35\) 1.36571 + 0.497079i 0.230848 + 0.0840218i
\(36\) 0.766044 − 0.642788i 0.127674 − 0.107131i
\(37\) −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
\(39\) 0.467911 0.0749257
\(40\) −1.70574 + 1.43128i −0.269701 + 0.226306i
\(41\) −3.26604 − 1.18874i −0.510070 − 0.185650i 0.0741475 − 0.997247i \(-0.476376\pi\)
−0.584218 + 0.811597i \(0.698599\pi\)
\(42\) −0.113341 + 0.642788i −0.0174889 + 0.0991843i
\(43\) 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
\(45\) −1.11334 + 1.92836i −0.165967 + 0.287463i
\(46\) 2.95084 + 5.11100i 0.435077 + 0.753576i
\(47\) −3.55303 − 2.98135i −0.518263 − 0.434874i 0.345763 − 0.938322i \(-0.387620\pi\)
−0.864026 + 0.503448i \(0.832065\pi\)
\(48\) −0.766044 − 0.642788i −0.110569 − 0.0927784i
\(49\) 3.28699 + 5.69323i 0.469570 + 0.813319i
\(50\) −0.0209445 + 0.0362770i −0.00296200 + 0.00513034i
\(51\) −1.97906 + 0.720317i −0.277123 + 0.100865i
\(52\) −0.0812519 − 0.460802i −0.0112676 − 0.0639018i
\(53\) −2.07532 + 11.7697i −0.285067 + 1.61670i 0.419977 + 0.907535i \(0.362038\pi\)
−0.705045 + 0.709163i \(0.749073\pi\)
\(54\) −0.939693 − 0.342020i −0.127876 − 0.0465430i
\(55\) −2.61334 + 2.19285i −0.352383 + 0.295684i
\(56\) 0.652704 0.0872212
\(57\) −0.819078 − 4.28125i −0.108490 − 0.567066i
\(58\) 8.33275 1.09414
\(59\) 10.9042 − 9.14971i 1.41961 − 1.19119i 0.468055 − 0.883699i \(-0.344955\pi\)
0.951551 − 0.307492i \(-0.0994896\pi\)
\(60\) 2.09240 + 0.761570i 0.270127 + 0.0983183i
\(61\) −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
\(63\) 0.613341 − 0.223238i 0.0772737 − 0.0281253i
\(64\) −0.500000 + 0.866025i −0.0625000 + 0.108253i
\(65\) 0.520945 + 0.902302i 0.0646152 + 0.111917i
\(66\) −1.17365 − 0.984808i −0.144466 − 0.121221i
\(67\) 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
\(69\) 2.95084 − 5.11100i 0.355239 − 0.615292i
\(70\) −1.36571 + 0.497079i −0.163234 + 0.0594124i
\(71\) −0.772441 − 4.38073i −0.0916719 − 0.519897i −0.995716 − 0.0924590i \(-0.970527\pi\)
0.904045 − 0.427438i \(-0.140584\pi\)
\(72\) −0.173648 + 0.984808i −0.0204646 + 0.116061i
\(73\) 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
\(75\) 0.0418891 0.00483693
\(76\) −4.07398 + 1.55007i −0.467317 + 0.177805i
\(77\) 1.00000 0.113961
\(78\) −0.358441 + 0.300767i −0.0405854 + 0.0340552i
\(79\) −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
\(81\) 0.173648 + 0.984808i 0.0192942 + 0.109423i
\(82\) 3.26604 − 1.18874i 0.360674 − 0.131275i
\(83\) −1.85457 + 3.21221i −0.203566 + 0.352586i −0.949675 − 0.313238i \(-0.898586\pi\)
0.746109 + 0.665824i \(0.231920\pi\)
\(84\) −0.326352 − 0.565258i −0.0356079 − 0.0616747i
\(85\) −3.59240 − 3.01438i −0.389650 − 0.326955i
\(86\) −7.88326 − 6.61484i −0.850073 − 0.713296i
\(87\) −4.16637 − 7.21637i −0.446682 − 0.773676i
\(88\) −0.766044 + 1.32683i −0.0816606 + 0.141440i
\(89\) −15.7554 + 5.73448i −1.67007 + 0.607854i −0.991896 − 0.127051i \(-0.959449\pi\)
−0.678169 + 0.734906i \(0.737226\pi\)
\(90\) −0.386659 − 2.19285i −0.0407575 − 0.231147i
\(91\) 0.0530334 − 0.300767i 0.00555941 − 0.0315290i
\(92\) −5.54576 − 2.01849i −0.578185 − 0.210442i
\(93\) 6.61721 − 5.55250i 0.686173 − 0.575767i
\(94\) 4.63816 0.478389
\(95\) 7.34389 − 6.34597i 0.753468 − 0.651083i
\(96\) 1.00000 0.102062
\(97\) 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
\(99\) −0.266044 + 1.50881i −0.0267385 + 0.151641i
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 114.2.i.d.25.1 ✓ 6
3.2 odd 2 342.2.u.a.253.1 6
4.3 odd 2 912.2.bo.f.481.1 6
19.4 even 9 2166.2.a.o.1.3 3
19.15 odd 18 2166.2.a.u.1.3 3
19.16 even 9 inner 114.2.i.d.73.1 yes 6
57.23 odd 18 6498.2.a.bs.1.1 3
57.35 odd 18 342.2.u.a.73.1 6
57.53 even 18 6498.2.a.bn.1.1 3
76.35 odd 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 1.1 even 1 trivial
114.2.i.d.73.1 yes 6 19.16 even 9 inner
342.2.u.a.73.1 6 57.35 odd 18
342.2.u.a.253.1 6 3.2 odd 2
912.2.bo.f.481.1 6 4.3 odd 2
912.2.bo.f.529.1 6 76.35 odd 18
2166.2.a.o.1.3 3 19.4 even 9
2166.2.a.u.1.3 3 19.15 odd 18
6498.2.a.bn.1.1 3 57.53 even 18
6498.2.a.bs.1.1 3 57.23 odd 18