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,-6] 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 57)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 769.1
Root \(-0.766044 + 0.642788i\) of defining polynomial
Character \(\chi\) \(=\) 912.769
Dual form 912.2.bo.b.625.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.766044 - 0.642788i) q^{3} +(-0.826352 - 0.300767i) q^{5} +(-1.43969 + 2.49362i) q^{7} +(0.173648 - 0.984808i) q^{9} +(-0.918748 - 1.59132i) q^{11} +(-2.11334 - 1.77330i) q^{13} +(-0.826352 + 0.300767i) q^{15} +(-1.23396 - 6.99811i) q^{17} +(-3.93969 - 1.86516i) q^{19} +(0.500000 + 2.83564i) q^{21} +(6.19846 - 2.25606i) q^{23} +(-3.23783 - 2.71686i) q^{25} +(-0.500000 - 0.866025i) q^{27} +(-0.543233 + 3.08083i) q^{29} +(3.82635 - 6.62744i) q^{31} +(-1.72668 - 0.628461i) q^{33} +(1.93969 - 1.62760i) q^{35} +2.83750 q^{37} -2.75877 q^{39} +(3.05303 - 2.56180i) q^{41} +(-10.7626 - 3.91728i) q^{43} +(-0.439693 + 0.761570i) q^{45} +(-0.383256 + 2.17355i) q^{47} +(-0.645430 - 1.11792i) q^{49} +(-5.44356 - 4.56769i) q^{51} +(-2.53936 + 0.924252i) q^{53} +(0.280592 + 1.59132i) q^{55} +(-4.21688 + 1.10359i) q^{57} +(1.46064 + 8.28368i) q^{59} +(-0.578726 + 0.210639i) q^{61} +(2.20574 + 1.85083i) q^{63} +(1.21301 + 2.10100i) q^{65} +(0.638156 - 3.61916i) q^{67} +(3.29813 - 5.71253i) q^{69} +(-7.00387 - 2.54920i) q^{71} +(-7.66637 + 6.43285i) q^{73} -4.22668 q^{75} +5.29086 q^{77} +(-1.23396 + 1.03541i) q^{79} +(-0.939693 - 0.342020i) q^{81} +(0.492726 - 0.853427i) q^{83} +(-1.08512 + 6.15403i) q^{85} +(1.56418 + 2.70924i) q^{87} +(13.0667 + 10.9643i) q^{89} +(7.46451 - 2.71686i) q^{91} +(-1.32888 - 7.53644i) q^{93} +(2.69459 + 2.72621i) q^{95} +(-1.02481 - 5.81201i) q^{97} +(-1.72668 + 0.628461i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{5} - 3 q^{7} - 3 q^{11} - 6 q^{13} - 6 q^{15} - 12 q^{17} - 18 q^{19} + 3 q^{21} + 9 q^{23} - 3 q^{27} + 12 q^{29} + 24 q^{31} + 3 q^{33} + 6 q^{35} + 12 q^{37} + 6 q^{39} + 6 q^{41} - 18 q^{43}+ \cdots + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(e\left(\frac{4}{9}\right)\) \(1\) \(1\) \(1\)

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
\(3\) 0.766044 − 0.642788i 0.442276 − 0.371114i
\(4\) 0 0
\(5\) −0.826352 − 0.300767i −0.369556 − 0.134507i 0.150565 − 0.988600i \(-0.451891\pi\)
−0.520121 + 0.854093i \(0.674113\pi\)
\(6\) 0 0
\(7\) −1.43969 + 2.49362i −0.544153 + 0.942500i 0.454507 + 0.890743i \(0.349815\pi\)
−0.998660 + 0.0517569i \(0.983518\pi\)
\(8\) 0 0
\(9\) 0.173648 − 0.984808i 0.0578827 − 0.328269i
\(10\) 0 0
\(11\) −0.918748 − 1.59132i −0.277013 − 0.479801i 0.693628 − 0.720333i \(-0.256011\pi\)
−0.970641 + 0.240533i \(0.922678\pi\)
\(12\) 0 0
\(13\) −2.11334 − 1.77330i −0.586135 − 0.491826i 0.300820 − 0.953681i \(-0.402740\pi\)
−0.886955 + 0.461855i \(0.847184\pi\)
\(14\) 0 0
\(15\) −0.826352 + 0.300767i −0.213363 + 0.0776578i
\(16\) 0 0
\(17\) −1.23396 − 6.99811i −0.299278 − 1.69729i −0.649286 − 0.760544i \(-0.724932\pi\)
0.350008 − 0.936747i \(-0.386179\pi\)
\(18\) 0 0
\(19\) −3.93969 − 1.86516i −0.903827 − 0.427897i
\(20\) 0 0
\(21\) 0.500000 + 2.83564i 0.109109 + 0.618788i
\(22\) 0 0
\(23\) 6.19846 − 2.25606i 1.29247 − 0.470420i 0.397932 − 0.917415i \(-0.369728\pi\)
0.894537 + 0.446995i \(0.147506\pi\)
\(24\) 0 0
\(25\) −3.23783 − 2.71686i −0.647565 − 0.543372i
\(26\) 0 0
\(27\) −0.500000 − 0.866025i −0.0962250 − 0.166667i
\(28\) 0 0
\(29\) −0.543233 + 3.08083i −0.100876 + 0.572096i 0.891912 + 0.452209i \(0.149364\pi\)
−0.992788 + 0.119886i \(0.961747\pi\)
\(30\) 0 0
\(31\) 3.82635 − 6.62744i 0.687233 − 1.19032i −0.285496 − 0.958380i \(-0.592158\pi\)
0.972729 − 0.231943i \(-0.0745082\pi\)
\(32\) 0 0
\(33\) −1.72668 − 0.628461i −0.300577 − 0.109401i
\(34\) 0 0
\(35\) 1.93969 − 1.62760i 0.327868 − 0.275114i
\(36\) 0 0
\(37\) 2.83750 0.466481 0.233241 − 0.972419i \(-0.425067\pi\)
0.233241 + 0.972419i \(0.425067\pi\)
\(38\) 0 0
\(39\) −2.75877 −0.441757
\(40\) 0 0
\(41\) 3.05303 − 2.56180i 0.476804 − 0.400086i −0.372465 − 0.928046i \(-0.621487\pi\)
0.849269 + 0.527960i \(0.177043\pi\)
\(42\) 0 0
\(43\) −10.7626 − 3.91728i −1.64129 − 0.597380i −0.654024 − 0.756474i \(-0.726920\pi\)
−0.987263 + 0.159094i \(0.949143\pi\)
\(44\) 0 0
\(45\) −0.439693 + 0.761570i −0.0655455 + 0.113528i
\(46\) 0 0
\(47\) −0.383256 + 2.17355i −0.0559036 + 0.317045i −0.999917 − 0.0128613i \(-0.995906\pi\)
0.944014 + 0.329906i \(0.107017\pi\)
\(48\) 0 0
\(49\) −0.645430 − 1.11792i −0.0922042 − 0.159702i
\(50\) 0 0
\(51\) −5.44356 − 4.56769i −0.762251 − 0.639605i
\(52\) 0 0
\(53\) −2.53936 + 0.924252i −0.348808 + 0.126956i −0.510482 − 0.859888i \(-0.670533\pi\)
0.161674 + 0.986844i \(0.448311\pi\)
\(54\) 0 0
\(55\) 0.280592 + 1.59132i 0.0378351 + 0.214573i
\(56\) 0 0
\(57\) −4.21688 + 1.10359i −0.558540 + 0.146174i
\(58\) 0 0
\(59\) 1.46064 + 8.28368i 0.190159 + 1.07844i 0.919146 + 0.393917i \(0.128880\pi\)
−0.728987 + 0.684527i \(0.760009\pi\)
\(60\) 0 0
\(61\) −0.578726 + 0.210639i −0.0740982 + 0.0269696i −0.378803 − 0.925477i \(-0.623664\pi\)
0.304705 + 0.952447i \(0.401442\pi\)
\(62\) 0 0
\(63\) 2.20574 + 1.85083i 0.277897 + 0.233183i
\(64\) 0 0
\(65\) 1.21301 + 2.10100i 0.150456 + 0.260597i
\(66\) 0 0
\(67\) 0.638156 − 3.61916i 0.0779631 − 0.442151i −0.920691 − 0.390292i \(-0.872374\pi\)
0.998654 − 0.0518592i \(-0.0165147\pi\)
\(68\) 0 0
\(69\) 3.29813 − 5.71253i 0.397049 − 0.687708i
\(70\) 0 0
\(71\) −7.00387 − 2.54920i −0.831206 − 0.302534i −0.108853 − 0.994058i \(-0.534718\pi\)
−0.722354 + 0.691523i \(0.756940\pi\)
\(72\) 0 0
\(73\) −7.66637 + 6.43285i −0.897281 + 0.752908i −0.969657 − 0.244469i \(-0.921386\pi\)
0.0723759 + 0.997377i \(0.476942\pi\)
\(74\) 0 0
\(75\) −4.22668 −0.488055
\(76\) 0 0
\(77\) 5.29086 0.602949
\(78\) 0 0
\(79\) −1.23396 + 1.03541i −0.138831 + 0.116493i −0.709559 − 0.704646i \(-0.751106\pi\)
0.570728 + 0.821139i \(0.306661\pi\)
\(80\) 0 0
\(81\) −0.939693 − 0.342020i −0.104410 − 0.0380022i
\(82\) 0 0
\(83\) 0.492726 − 0.853427i 0.0540837 − 0.0936757i −0.837716 − 0.546106i \(-0.816110\pi\)
0.891800 + 0.452430i \(0.149443\pi\)
\(84\) 0 0
\(85\) −1.08512 + 6.15403i −0.117698 + 0.667499i
\(86\) 0 0
\(87\) 1.56418 + 2.70924i 0.167697 + 0.290461i
\(88\) 0 0
\(89\) 13.0667 + 10.9643i 1.38507 + 1.16221i 0.967294 + 0.253659i \(0.0816341\pi\)
0.417774 + 0.908551i \(0.362810\pi\)
\(90\) 0 0
\(91\) 7.46451 − 2.71686i 0.782493 − 0.284804i
\(92\) 0 0
\(93\) −1.32888 − 7.53644i −0.137798 − 0.781493i
\(94\) 0 0
\(95\) 2.69459 + 2.72621i 0.276459 + 0.279703i
\(96\) 0 0
\(97\) −1.02481 − 5.81201i −0.104054 − 0.590121i −0.991594 − 0.129389i \(-0.958698\pi\)
0.887540 − 0.460731i \(-0.152413\pi\)
\(98\) 0 0
\(99\) −1.72668 + 0.628461i −0.173538 + 0.0631627i
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.b.769.1 6
4.3 odd 2 57.2.i.a.28.1 ✓ 6
12.11 even 2 171.2.u.a.28.1 6
19.17 even 9 inner 912.2.bo.b.625.1 6
76.51 even 18 1083.2.a.n.1.1 3
76.55 odd 18 57.2.i.a.55.1 yes 6
76.63 odd 18 1083.2.a.m.1.3 3
228.131 even 18 171.2.u.a.55.1 6
228.203 odd 18 3249.2.a.x.1.3 3
228.215 even 18 3249.2.a.w.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
57.2.i.a.28.1 ✓ 6 4.3 odd 2
57.2.i.a.55.1 yes 6 76.55 odd 18
171.2.u.a.28.1 6 12.11 even 2
171.2.u.a.55.1 6 228.131 even 18
912.2.bo.b.625.1 6 19.17 even 9 inner
912.2.bo.b.769.1 6 1.1 even 1 trivial
1083.2.a.m.1.3 3 76.63 odd 18
1083.2.a.n.1.1 3 76.51 even 18
3249.2.a.w.1.1 3 228.215 even 18
3249.2.a.x.1.3 3 228.203 odd 18