Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [912,2,Mod(289,912)] 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("912.289"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(912, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.bo (of order \(9\), degree \(6\), not 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: \(7.28235666434\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 114)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 529.1
Root \(0.939693 - 0.342020i\) of defining polynomial
Character \(\chi\) \(=\) 912.529
Dual form 912.2.bo.f.481.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.939693 + 0.342020i) q^{3} +(0.386659 - 2.19285i) q^{5} +(-0.326352 - 0.565258i) q^{7} +(0.766044 - 0.642788i) q^{9} +(-0.766044 + 1.32683i) q^{11} +(0.439693 + 0.160035i) q^{13} +(0.386659 + 2.19285i) q^{15} +(-1.61334 - 1.35375i) q^{17} +(2.23396 - 3.74292i) q^{19} +(0.500000 + 0.419550i) q^{21} +(-1.02481 - 5.81201i) q^{23} +(0.0393628 + 0.0143269i) q^{25} +(-0.500000 + 0.866025i) q^{27} +(-6.38326 + 5.35619i) q^{29} +(-4.31908 - 7.48086i) q^{31} +(0.266044 - 1.50881i) q^{33} +(-1.36571 + 0.497079i) q^{35} -4.67499 q^{37} -0.467911 q^{39} +(-3.26604 + 1.18874i) q^{41} +(-1.78699 + 10.1345i) q^{43} +(-1.11334 - 1.92836i) q^{45} +(3.55303 - 2.98135i) q^{47} +(3.28699 - 5.69323i) q^{49} +(1.97906 + 0.720317i) q^{51} +(-2.07532 - 11.7697i) q^{53} +(2.61334 + 2.19285i) q^{55} +(-0.819078 + 4.28125i) q^{57} +(-10.9042 - 9.14971i) q^{59} +(-1.58378 - 8.98205i) q^{61} +(-0.613341 - 0.223238i) q^{63} +(0.520945 - 0.902302i) q^{65} +(-0.190722 + 0.160035i) q^{67} +(2.95084 + 5.11100i) q^{69} +(0.772441 - 4.38073i) q^{71} +(8.54323 - 3.10948i) q^{73} -0.0418891 q^{75} +1.00000 q^{77} +(11.3302 - 4.12386i) q^{79} +(0.173648 - 0.984808i) q^{81} +(1.85457 + 3.21221i) q^{83} +(-3.59240 + 3.01438i) q^{85} +(4.16637 - 7.21637i) q^{87} +(-15.7554 - 5.73448i) q^{89} +(-0.0530334 - 0.300767i) q^{91} +(6.61721 + 5.55250i) q^{93} +(-7.34389 - 6.34597i) q^{95} +(1.89053 + 1.58634i) q^{97} +(0.266044 + 1.50881i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 9 q^{5} - 3 q^{7} - 3 q^{13} + 9 q^{15} - 3 q^{17} + 18 q^{19} + 3 q^{21} + 21 q^{23} + 9 q^{25} - 3 q^{27} - 3 q^{29} - 9 q^{31} - 3 q^{33} - 18 q^{35} - 18 q^{37} - 12 q^{39} - 15 q^{41} - 3 q^{43}+ \cdots - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(e\left(\frac{2}{9}\right)\) \(1\) \(1\) \(1\)

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 0
\(3\) −0.939693 + 0.342020i −0.542532 + 0.197465i
\(4\) 0 0
\(5\) 0.386659 − 2.19285i 0.172919 − 0.980674i −0.767599 − 0.640930i \(-0.778549\pi\)
0.940518 − 0.339743i \(-0.110340\pi\)
\(6\) 0 0
\(7\) −0.326352 − 0.565258i −0.123349 − 0.213647i 0.797737 − 0.603005i \(-0.206030\pi\)
−0.921087 + 0.389358i \(0.872697\pi\)
\(8\) 0 0
\(9\) 0.766044 − 0.642788i 0.255348 − 0.214263i
\(10\) 0 0
\(11\) −0.766044 + 1.32683i −0.230971 + 0.400054i −0.958094 − 0.286453i \(-0.907524\pi\)
0.727123 + 0.686507i \(0.240857\pi\)
\(12\) 0 0
\(13\) 0.439693 + 0.160035i 0.121949 + 0.0443857i 0.402274 − 0.915519i \(-0.368220\pi\)
−0.280325 + 0.959905i \(0.590442\pi\)
\(14\) 0 0
\(15\) 0.386659 + 2.19285i 0.0998350 + 0.566192i
\(16\) 0 0
\(17\) −1.61334 − 1.35375i −0.391293 − 0.328333i 0.425824 − 0.904806i \(-0.359984\pi\)
−0.817116 + 0.576473i \(0.804429\pi\)
\(18\) 0 0
\(19\) 2.23396 − 3.74292i 0.512505 − 0.858685i
\(20\) 0 0
\(21\) 0.500000 + 0.419550i 0.109109 + 0.0915533i
\(22\) 0 0
\(23\) −1.02481 − 5.81201i −0.213689 − 1.21189i −0.883168 − 0.469058i \(-0.844594\pi\)
0.669479 − 0.742831i \(-0.266517\pi\)
\(24\) 0 0
\(25\) 0.0393628 + 0.0143269i 0.00787257 + 0.00286538i
\(26\) 0 0
\(27\) −0.500000 + 0.866025i −0.0962250 + 0.166667i
\(28\) 0 0
\(29\) −6.38326 + 5.35619i −1.18534 + 0.994619i −0.185412 + 0.982661i \(0.559362\pi\)
−0.999928 + 0.0119582i \(0.996193\pi\)
\(30\) 0 0
\(31\) −4.31908 − 7.48086i −0.775729 − 1.34360i −0.934384 − 0.356268i \(-0.884049\pi\)
0.158654 − 0.987334i \(-0.449284\pi\)
\(32\) 0 0
\(33\) 0.266044 − 1.50881i 0.0463124 − 0.262651i
\(34\) 0 0
\(35\) −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 \(0.625551\pi\)
\(38\) 0 0
\(39\) −0.467911 −0.0749257
\(40\) 0 0
\(41\) −3.26604 + 1.18874i −0.510070 + 0.185650i −0.584218 − 0.811597i \(-0.698599\pi\)
0.0741475 + 0.997247i \(0.476376\pi\)
\(42\) 0 0
\(43\) −1.78699 + 10.1345i −0.272513 + 1.54550i 0.474238 + 0.880396i \(0.342723\pi\)
−0.746752 + 0.665103i \(0.768388\pi\)
\(44\) 0 0
\(45\) −1.11334 − 1.92836i −0.165967 − 0.287463i
\(46\) 0 0
\(47\) 3.55303 − 2.98135i 0.518263 − 0.434874i −0.345763 − 0.938322i \(-0.612380\pi\)
0.864026 + 0.503448i \(0.167935\pi\)
\(48\) 0 0
\(49\) 3.28699 − 5.69323i 0.469570 − 0.813319i
\(50\) 0 0
\(51\) 1.97906 + 0.720317i 0.277123 + 0.100865i
\(52\) 0 0
\(53\) −2.07532 − 11.7697i −0.285067 − 1.61670i −0.705045 − 0.709163i \(-0.749073\pi\)
0.419977 − 0.907535i \(-0.362038\pi\)
\(54\) 0 0
\(55\) 2.61334 + 2.19285i 0.352383 + 0.295684i
\(56\) 0 0
\(57\) −0.819078 + 4.28125i −0.108490 + 0.567066i
\(58\) 0 0
\(59\) −10.9042 − 9.14971i −1.41961 − 1.19119i −0.951551 − 0.307492i \(-0.900510\pi\)
−0.468055 − 0.883699i \(-0.655045\pi\)
\(60\) 0 0
\(61\) −1.58378 − 8.98205i −0.202782 − 1.15003i −0.900892 − 0.434043i \(-0.857086\pi\)
0.698110 − 0.715991i \(-0.254025\pi\)
\(62\) 0 0
\(63\) −0.613341 − 0.223238i −0.0772737 − 0.0281253i
\(64\) 0 0
\(65\) 0.520945 − 0.902302i 0.0646152 − 0.111917i
\(66\) 0 0
\(67\) −0.190722 + 0.160035i −0.0233004 + 0.0195514i −0.654363 − 0.756180i \(-0.727063\pi\)
0.631063 + 0.775732i \(0.282619\pi\)
\(68\) 0 0
\(69\) 2.95084 + 5.11100i 0.355239 + 0.615292i
\(70\) 0 0
\(71\) 0.772441 − 4.38073i 0.0916719 − 0.519897i −0.904045 − 0.427438i \(-0.859416\pi\)
0.995716 − 0.0924590i \(-0.0294727\pi\)
\(72\) 0 0
\(73\) 8.54323 − 3.10948i 0.999910 − 0.363937i 0.210360 − 0.977624i \(-0.432536\pi\)
0.789550 + 0.613687i \(0.210314\pi\)
\(74\) 0 0
\(75\) −0.0418891 −0.00483693
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) 11.3302 − 4.12386i 1.27475 − 0.463971i 0.386057 − 0.922475i \(-0.373837\pi\)
0.888692 + 0.458504i \(0.151615\pi\)
\(80\) 0 0
\(81\) 0.173648 − 0.984808i 0.0192942 − 0.109423i
\(82\) 0 0
\(83\) 1.85457 + 3.21221i 0.203566 + 0.352586i 0.949675 − 0.313238i \(-0.101414\pi\)
−0.746109 + 0.665824i \(0.768080\pi\)
\(84\) 0 0
\(85\) −3.59240 + 3.01438i −0.389650 + 0.326955i
\(86\) 0 0
\(87\) 4.16637 − 7.21637i 0.446682 − 0.773676i
\(88\) 0 0
\(89\) −15.7554 − 5.73448i −1.67007 − 0.607854i −0.678169 − 0.734906i \(-0.737226\pi\)
−0.991896 + 0.127051i \(0.959449\pi\)
\(90\) 0 0
\(91\) −0.0530334 − 0.300767i −0.00555941 − 0.0315290i
\(92\) 0 0
\(93\) 6.61721 + 5.55250i 0.686173 + 0.575767i
\(94\) 0 0
\(95\) −7.34389 − 6.34597i −0.753468 − 0.651083i
\(96\) 0 0
\(97\) 1.89053 + 1.58634i 0.191954 + 0.161069i 0.733700 − 0.679474i \(-0.237792\pi\)
−0.541746 + 0.840543i \(0.682236\pi\)
\(98\) 0 0
\(99\) 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 912.2.bo.f.529.1 6
4.3 odd 2 114.2.i.d.73.1 yes 6
12.11 even 2 342.2.u.a.73.1 6
19.6 even 9 inner 912.2.bo.f.481.1 6
76.43 odd 18 2166.2.a.o.1.3 3
76.63 odd 18 114.2.i.d.25.1 ✓ 6
76.71 even 18 2166.2.a.u.1.3 3
228.71 odd 18 6498.2.a.bn.1.1 3
228.119 even 18 6498.2.a.bs.1.1 3
228.215 even 18 342.2.u.a.253.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.i.d.25.1 ✓ 6 76.63 odd 18
114.2.i.d.73.1 yes 6 4.3 odd 2
342.2.u.a.73.1 6 12.11 even 2
342.2.u.a.253.1 6 228.215 even 18
912.2.bo.f.481.1 6 19.6 even 9 inner
912.2.bo.f.529.1 6 1.1 even 1 trivial
2166.2.a.o.1.3 3 76.43 odd 18
2166.2.a.u.1.3 3 76.71 even 18
6498.2.a.bn.1.1 3 228.71 odd 18
6498.2.a.bs.1.1 3 228.119 even 18