Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [592,2,Mod(33,592)] 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("592.33"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(592, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 0, 10])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 592 = 2^{4} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 592.bc (of order \(9\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,3] 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: \(4.72714379966\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{9})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 24x^{10} + 264x^{8} - 1687x^{6} + 6600x^{4} - 15000x^{2} + 15625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 74)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 49.2
Root \(2.20976 - 0.342020i\) of defining polynomial
Character \(\chi\) \(=\) 592.49
Dual form 592.2.bc.d.145.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(2.41262 + 2.02443i) q^{3} +(-1.43969 + 0.524005i) q^{5} +(-4.53424 + 1.65033i) q^{7} +(1.20148 + 6.81391i) q^{9} +(-0.546896 - 0.947252i) q^{11} +(0.307284 - 1.74269i) q^{13} +(-4.53424 - 1.65033i) q^{15} +(0.511542 + 2.90110i) q^{17} +(0.632167 + 0.530451i) q^{19} +(-14.2804 - 5.19762i) q^{21} +(-0.121620 + 0.210653i) q^{23} +(-2.03209 + 1.70513i) q^{25} +(-6.17139 + 10.6892i) q^{27} +(2.78587 + 4.82526i) q^{29} +5.73495 q^{31} +(0.598191 - 3.39251i) q^{33} +(5.66313 - 4.75193i) q^{35} +(5.84038 + 1.69999i) q^{37} +(4.26932 - 3.58238i) q^{39} +(0.505653 - 2.86770i) q^{41} -4.37987 q^{43} +(-5.30028 - 9.18036i) q^{45} +(1.13249 - 1.96153i) q^{47} +(12.4734 - 10.4664i) q^{49} +(-4.63890 + 8.03481i) q^{51} +(-5.77859 - 2.10324i) q^{53} +(1.28373 + 1.07717i) q^{55} +(0.451318 + 2.55955i) q^{57} +(8.29301 + 3.01841i) q^{59} +(-2.04075 + 11.5737i) q^{61} +(-16.6930 - 28.9131i) q^{63} +(0.470787 + 2.66996i) q^{65} +(8.44220 - 3.07271i) q^{67} +(-0.719875 + 0.262013i) q^{69} +(0.933535 + 0.783329i) q^{71} +8.79418 q^{73} -8.35455 q^{75} +(4.04303 + 3.39251i) q^{77} +(2.02379 - 0.736599i) q^{79} +(-17.0233 + 6.19599i) q^{81} +(-0.884528 - 5.01641i) q^{83} +(-2.25665 - 3.90864i) q^{85} +(-3.04716 + 17.2813i) q^{87} +(-8.43486 - 3.07004i) q^{89} +(1.48272 + 8.40891i) q^{91} +(13.8362 + 11.6100i) q^{93} +(-1.18809 - 0.432428i) q^{95} +(-2.17954 + 3.77507i) q^{97} +(5.79741 - 4.86460i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{3} - 6 q^{5} - 6 q^{7} - 3 q^{9} - 3 q^{11} - 6 q^{13} - 6 q^{15} - 3 q^{17} + 3 q^{19} - 33 q^{21} + 21 q^{23} - 6 q^{25} - 3 q^{27} + 6 q^{29} - 42 q^{31} + 57 q^{33} + 9 q^{35} - 3 q^{37}+ \cdots + 33 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/592\mathbb{Z}\right)^\times\).

\(n\) \(113\) \(149\) \(223\)
\(\chi(n)\) \(e\left(\frac{7}{9}\right)\) \(1\) \(1\)

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
\(3\) 2.41262 + 2.02443i 1.39293 + 1.16880i 0.964139 + 0.265397i \(0.0855031\pi\)
0.428786 + 0.903406i \(0.358941\pi\)
\(4\) 0 0
\(5\) −1.43969 + 0.524005i −0.643850 + 0.234342i −0.643248 − 0.765658i \(-0.722414\pi\)
−0.000601695 1.00000i \(0.500192\pi\)
\(6\) 0 0
\(7\) −4.53424 + 1.65033i −1.71378 + 0.623765i −0.997272 − 0.0738128i \(-0.976483\pi\)
−0.716509 + 0.697578i \(0.754261\pi\)
\(8\) 0 0
\(9\) 1.20148 + 6.81391i 0.400492 + 2.27130i
\(10\) 0 0
\(11\) −0.546896 − 0.947252i −0.164895 − 0.285607i 0.771723 − 0.635959i \(-0.219395\pi\)
−0.936618 + 0.350352i \(0.886062\pi\)
\(12\) 0 0
\(13\) 0.307284 − 1.74269i 0.0852253 − 0.483337i −0.912082 − 0.410007i \(-0.865526\pi\)
0.997308 − 0.0733298i \(-0.0233626\pi\)
\(14\) 0 0
\(15\) −4.53424 − 1.65033i −1.17074 − 0.426113i
\(16\) 0 0
\(17\) 0.511542 + 2.90110i 0.124067 + 0.703619i 0.981858 + 0.189620i \(0.0607255\pi\)
−0.857791 + 0.513999i \(0.828163\pi\)
\(18\) 0 0
\(19\) 0.632167 + 0.530451i 0.145029 + 0.121694i 0.712416 − 0.701758i \(-0.247601\pi\)
−0.567387 + 0.823451i \(0.692046\pi\)
\(20\) 0 0
\(21\) −14.2804 − 5.19762i −3.11623 − 1.13421i
\(22\) 0 0
\(23\) −0.121620 + 0.210653i −0.0253596 + 0.0439241i −0.878427 − 0.477877i \(-0.841406\pi\)
0.853067 + 0.521801i \(0.174740\pi\)
\(24\) 0 0
\(25\) −2.03209 + 1.70513i −0.406418 + 0.341025i
\(26\) 0 0
\(27\) −6.17139 + 10.6892i −1.18768 + 2.05713i
\(28\) 0 0
\(29\) 2.78587 + 4.82526i 0.517322 + 0.896028i 0.999798 + 0.0201189i \(0.00640447\pi\)
−0.482475 + 0.875910i \(0.660262\pi\)
\(30\) 0 0
\(31\) 5.73495 1.03003 0.515013 − 0.857182i \(-0.327787\pi\)
0.515013 + 0.857182i \(0.327787\pi\)
\(32\) 0 0
\(33\) 0.598191 − 3.39251i 0.104132 − 0.590560i
\(34\) 0 0
\(35\) 5.66313 − 4.75193i 0.957243 − 0.803223i
\(36\) 0 0
\(37\) 5.84038 + 1.69999i 0.960152 + 0.279477i
\(38\) 0 0
\(39\) 4.26932 − 3.58238i 0.683638 − 0.573640i
\(40\) 0 0
\(41\) 0.505653 − 2.86770i 0.0789697 − 0.447859i −0.919526 − 0.393029i \(-0.871427\pi\)
0.998496 − 0.0548301i \(-0.0174617\pi\)
\(42\) 0 0
\(43\) −4.37987 −0.667924 −0.333962 − 0.942587i \(-0.608386\pi\)
−0.333962 + 0.942587i \(0.608386\pi\)
\(44\) 0 0
\(45\) −5.30028 − 9.18036i −0.790119 − 1.36853i
\(46\) 0 0
\(47\) 1.13249 − 1.96153i 0.165191 − 0.286119i −0.771532 − 0.636190i \(-0.780509\pi\)
0.936723 + 0.350071i \(0.113843\pi\)
\(48\) 0 0
\(49\) 12.4734 − 10.4664i 1.78192 − 1.49521i
\(50\) 0 0
\(51\) −4.63890 + 8.03481i −0.649576 + 1.12510i
\(52\) 0 0
\(53\) −5.77859 − 2.10324i −0.793751 − 0.288902i −0.0868564 − 0.996221i \(-0.527682\pi\)
−0.706894 + 0.707319i \(0.749904\pi\)
\(54\) 0 0
\(55\) 1.28373 + 1.07717i 0.173098 + 0.145246i
\(56\) 0 0
\(57\) 0.451318 + 2.55955i 0.0597785 + 0.339021i
\(58\) 0 0
\(59\) 8.29301 + 3.01841i 1.07966 + 0.392963i 0.819780 − 0.572678i \(-0.194096\pi\)
0.259878 + 0.965642i \(0.416318\pi\)
\(60\) 0 0
\(61\) −2.04075 + 11.5737i −0.261292 + 1.48186i 0.518098 + 0.855321i \(0.326640\pi\)
−0.779390 + 0.626539i \(0.784471\pi\)
\(62\) 0 0
\(63\) −16.6930 − 28.9131i −2.10312 − 3.64270i
\(64\) 0 0
\(65\) 0.470787 + 2.66996i 0.0583939 + 0.331168i
\(66\) 0 0
\(67\) 8.44220 − 3.07271i 1.03138 − 0.375391i 0.229773 − 0.973244i \(-0.426202\pi\)
0.801606 + 0.597853i \(0.203979\pi\)
\(68\) 0 0
\(69\) −0.719875 + 0.262013i −0.0866627 + 0.0315427i
\(70\) 0 0
\(71\) 0.933535 + 0.783329i 0.110790 + 0.0929640i 0.696500 − 0.717557i \(-0.254740\pi\)
−0.585710 + 0.810521i \(0.699184\pi\)
\(72\) 0 0
\(73\) 8.79418 1.02928 0.514640 − 0.857406i \(-0.327925\pi\)
0.514640 + 0.857406i \(0.327925\pi\)
\(74\) 0 0
\(75\) −8.35455 −0.964701
\(76\) 0 0
\(77\) 4.04303 + 3.39251i 0.460746 + 0.386612i
\(78\) 0 0
\(79\) 2.02379 − 0.736599i 0.227694 − 0.0828739i −0.225654 − 0.974208i \(-0.572452\pi\)
0.453348 + 0.891334i \(0.350230\pi\)
\(80\) 0 0
\(81\) −17.0233 + 6.19599i −1.89148 + 0.688443i
\(82\) 0 0
\(83\) −0.884528 − 5.01641i −0.0970895 − 0.550622i −0.994087 − 0.108586i \(-0.965368\pi\)
0.896997 − 0.442036i \(-0.145744\pi\)
\(84\) 0 0
\(85\) −2.25665 − 3.90864i −0.244768 − 0.423951i
\(86\) 0 0
\(87\) −3.04716 + 17.2813i −0.326690 + 1.85275i
\(88\) 0 0
\(89\) −8.43486 − 3.07004i −0.894093 − 0.325423i −0.146210 − 0.989254i \(-0.546708\pi\)
−0.747883 + 0.663830i \(0.768930\pi\)
\(90\) 0 0
\(91\) 1.48272 + 8.40891i 0.155431 + 0.881494i
\(92\) 0 0
\(93\) 13.8362 + 11.6100i 1.43475 + 1.20390i
\(94\) 0 0
\(95\) −1.18809 − 0.432428i −0.121895 − 0.0443661i
\(96\) 0 0
\(97\) −2.17954 + 3.77507i −0.221298 + 0.383300i −0.955203 − 0.295953i \(-0.904363\pi\)
0.733904 + 0.679253i \(0.237696\pi\)
\(98\) 0 0
\(99\) 5.79741 − 4.86460i 0.582661 − 0.488911i
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 592.2.bc.d.49.2 12
4.3 odd 2 74.2.f.b.49.1 ✓ 12
12.11 even 2 666.2.x.g.271.2 12
37.34 even 9 inner 592.2.bc.d.145.2 12
148.71 odd 18 74.2.f.b.71.1 yes 12
148.95 odd 18 2738.2.a.q.1.1 6
148.127 odd 18 2738.2.a.t.1.1 6
444.71 even 18 666.2.x.g.145.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.f.b.49.1 ✓ 12 4.3 odd 2
74.2.f.b.71.1 yes 12 148.71 odd 18
592.2.bc.d.49.2 12 1.1 even 1 trivial
592.2.bc.d.145.2 12 37.34 even 9 inner
666.2.x.g.145.2 12 444.71 even 18
666.2.x.g.271.2 12 12.11 even 2
2738.2.a.q.1.1 6 148.95 odd 18
2738.2.a.t.1.1 6 148.127 odd 18