Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1232,2,Mod(177,1232)] 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("1232.177"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1232, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1232.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-2,0,4,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.83756952902\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 154)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 529.2
Root \(0.707107 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 1232.529
Dual form 1232.2.q.f.177.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.207107 - 0.358719i) q^{3} +(1.70711 + 2.95680i) q^{5} +(2.62132 - 0.358719i) q^{7} +(1.41421 + 2.44949i) q^{9} +(0.500000 - 0.866025i) q^{11} +1.82843 q^{13} +1.41421 q^{15} +(3.82843 - 6.63103i) q^{17} +(-1.70711 - 2.95680i) q^{19} +(0.414214 - 1.01461i) q^{21} +(1.12132 + 1.94218i) q^{23} +(-3.32843 + 5.76500i) q^{25} +2.41421 q^{27} -8.65685 q^{29} +(-2.00000 + 3.46410i) q^{31} +(-0.207107 - 0.358719i) q^{33} +(5.53553 + 7.13834i) q^{35} +(3.29289 + 5.70346i) q^{37} +(0.378680 - 0.655892i) q^{39} -2.58579 q^{41} -5.65685 q^{43} +(-4.82843 + 8.36308i) q^{45} +(3.24264 + 5.61642i) q^{47} +(6.74264 - 1.88064i) q^{49} +(-1.58579 - 2.74666i) q^{51} +(5.94975 - 10.3053i) q^{53} +3.41421 q^{55} -1.41421 q^{57} +(-4.20711 + 7.28692i) q^{59} +(-3.08579 - 5.34474i) q^{61} +(4.58579 + 5.91359i) q^{63} +(3.12132 + 5.40629i) q^{65} +(5.62132 - 9.73641i) q^{67} +0.928932 q^{69} -3.07107 q^{71} +(3.29289 - 5.70346i) q^{73} +(1.37868 + 2.38794i) q^{75} +(1.00000 - 2.44949i) q^{77} +(-2.37868 - 4.11999i) q^{79} +(-3.74264 + 6.48244i) q^{81} -16.1421 q^{83} +26.1421 q^{85} +(-1.79289 + 3.10538i) q^{87} +(2.24264 + 3.88437i) q^{89} +(4.79289 - 0.655892i) q^{91} +(0.828427 + 1.43488i) q^{93} +(5.82843 - 10.0951i) q^{95} +1.82843 q^{97} +2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} + 4 q^{5} + 2 q^{7} + 2 q^{11} - 4 q^{13} + 4 q^{17} - 4 q^{19} - 4 q^{21} - 4 q^{23} - 2 q^{25} + 4 q^{27} - 12 q^{29} - 8 q^{31} + 2 q^{33} + 8 q^{35} + 16 q^{37} + 10 q^{39} - 16 q^{41}+ \cdots - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1232\mathbb{Z}\right)^\times\).

\(n\) \(309\) \(353\) \(463\) \(673\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)

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
\(3\) 0.207107 − 0.358719i 0.119573 − 0.207107i −0.800025 − 0.599966i \(-0.795181\pi\)
0.919599 + 0.392859i \(0.128514\pi\)
\(4\) 0 0
\(5\) 1.70711 + 2.95680i 0.763441 + 1.32232i 0.941067 + 0.338221i \(0.109825\pi\)
−0.177625 + 0.984098i \(0.556842\pi\)
\(6\) 0 0
\(7\) 2.62132 − 0.358719i 0.990766 − 0.135583i
\(8\) 0 0
\(9\) 1.41421 + 2.44949i 0.471405 + 0.816497i
\(10\) 0 0
\(11\) 0.500000 − 0.866025i 0.150756 − 0.261116i
\(12\) 0 0
\(13\) 1.82843 0.507114 0.253557 − 0.967320i \(-0.418399\pi\)
0.253557 + 0.967320i \(0.418399\pi\)
\(14\) 0 0
\(15\) 1.41421 0.365148
\(16\) 0 0
\(17\) 3.82843 − 6.63103i 0.928530 − 1.60826i 0.142747 − 0.989759i \(-0.454407\pi\)
0.785783 − 0.618502i \(-0.212260\pi\)
\(18\) 0 0
\(19\) −1.70711 − 2.95680i −0.391637 − 0.678335i 0.601028 − 0.799228i \(-0.294758\pi\)
−0.992666 + 0.120892i \(0.961424\pi\)
\(20\) 0 0
\(21\) 0.414214 − 1.01461i 0.0903888 − 0.221406i
\(22\) 0 0
\(23\) 1.12132 + 1.94218i 0.233811 + 0.404973i 0.958927 − 0.283654i \(-0.0915468\pi\)
−0.725115 + 0.688628i \(0.758213\pi\)
\(24\) 0 0
\(25\) −3.32843 + 5.76500i −0.665685 + 1.15300i
\(26\) 0 0
\(27\) 2.41421 0.464616
\(28\) 0 0
\(29\) −8.65685 −1.60754 −0.803769 − 0.594942i \(-0.797175\pi\)
−0.803769 + 0.594942i \(0.797175\pi\)
\(30\) 0 0
\(31\) −2.00000 + 3.46410i −0.359211 + 0.622171i −0.987829 − 0.155543i \(-0.950287\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) 0 0
\(33\) −0.207107 − 0.358719i −0.0360527 − 0.0624450i
\(34\) 0 0
\(35\) 5.53553 + 7.13834i 0.935676 + 1.20660i
\(36\) 0 0
\(37\) 3.29289 + 5.70346i 0.541348 + 0.937643i 0.998827 + 0.0484222i \(0.0154193\pi\)
−0.457479 + 0.889221i \(0.651247\pi\)
\(38\) 0 0
\(39\) 0.378680 − 0.655892i 0.0606373 − 0.105027i
\(40\) 0 0
\(41\) −2.58579 −0.403832 −0.201916 − 0.979403i \(-0.564717\pi\)
−0.201916 + 0.979403i \(0.564717\pi\)
\(42\) 0 0
\(43\) −5.65685 −0.862662 −0.431331 − 0.902194i \(-0.641956\pi\)
−0.431331 + 0.902194i \(0.641956\pi\)
\(44\) 0 0
\(45\) −4.82843 + 8.36308i −0.719779 + 1.24669i
\(46\) 0 0
\(47\) 3.24264 + 5.61642i 0.472988 + 0.819239i 0.999522 − 0.0309151i \(-0.00984215\pi\)
−0.526534 + 0.850154i \(0.676509\pi\)
\(48\) 0 0
\(49\) 6.74264 − 1.88064i 0.963234 − 0.268662i
\(50\) 0 0
\(51\) −1.58579 − 2.74666i −0.222055 − 0.384610i
\(52\) 0 0
\(53\) 5.94975 − 10.3053i 0.817261 − 1.41554i −0.0904325 − 0.995903i \(-0.528825\pi\)
0.907693 − 0.419634i \(-0.137842\pi\)
\(54\) 0 0
\(55\) 3.41421 0.460372
\(56\) 0 0
\(57\) −1.41421 −0.187317
\(58\) 0 0
\(59\) −4.20711 + 7.28692i −0.547719 + 0.948677i 0.450712 + 0.892670i \(0.351170\pi\)
−0.998430 + 0.0560070i \(0.982163\pi\)
\(60\) 0 0
\(61\) −3.08579 − 5.34474i −0.395094 − 0.684324i 0.598019 − 0.801482i \(-0.295955\pi\)
−0.993113 + 0.117158i \(0.962621\pi\)
\(62\) 0 0
\(63\) 4.58579 + 5.91359i 0.577755 + 0.745042i
\(64\) 0 0
\(65\) 3.12132 + 5.40629i 0.387152 + 0.670567i
\(66\) 0 0
\(67\) 5.62132 − 9.73641i 0.686754 − 1.18949i −0.286129 − 0.958191i \(-0.592368\pi\)
0.972882 − 0.231301i \(-0.0742982\pi\)
\(68\) 0 0
\(69\) 0.928932 0.111830
\(70\) 0 0
\(71\) −3.07107 −0.364469 −0.182234 − 0.983255i \(-0.558333\pi\)
−0.182234 + 0.983255i \(0.558333\pi\)
\(72\) 0 0
\(73\) 3.29289 − 5.70346i 0.385404 − 0.667539i −0.606421 − 0.795144i \(-0.707395\pi\)
0.991825 + 0.127604i \(0.0407288\pi\)
\(74\) 0 0
\(75\) 1.37868 + 2.38794i 0.159196 + 0.275736i
\(76\) 0 0
\(77\) 1.00000 − 2.44949i 0.113961 − 0.279145i
\(78\) 0 0
\(79\) −2.37868 − 4.11999i −0.267622 − 0.463536i 0.700625 − 0.713530i \(-0.252905\pi\)
−0.968247 + 0.249994i \(0.919571\pi\)
\(80\) 0 0
\(81\) −3.74264 + 6.48244i −0.415849 + 0.720272i
\(82\) 0 0
\(83\) −16.1421 −1.77183 −0.885915 − 0.463848i \(-0.846468\pi\)
−0.885915 + 0.463848i \(0.846468\pi\)
\(84\) 0 0
\(85\) 26.1421 2.83551
\(86\) 0 0
\(87\) −1.79289 + 3.10538i −0.192218 + 0.332932i
\(88\) 0 0
\(89\) 2.24264 + 3.88437i 0.237719 + 0.411742i 0.960060 − 0.279796i \(-0.0902668\pi\)
−0.722340 + 0.691538i \(0.756933\pi\)
\(90\) 0 0
\(91\) 4.79289 − 0.655892i 0.502432 − 0.0687562i
\(92\) 0 0
\(93\) 0.828427 + 1.43488i 0.0859039 + 0.148790i
\(94\) 0 0
\(95\) 5.82843 − 10.0951i 0.597984 − 1.03574i
\(96\) 0 0
\(97\) 1.82843 0.185649 0.0928243 − 0.995683i \(-0.470411\pi\)
0.0928243 + 0.995683i \(0.470411\pi\)
\(98\) 0 0
\(99\) 2.82843 0.284268
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 1232.2.q.f.529.2 4
4.3 odd 2 154.2.e.e.67.1 yes 4
7.2 even 3 inner 1232.2.q.f.177.2 4
7.3 odd 6 8624.2.a.bh.1.2 2
7.4 even 3 8624.2.a.cc.1.1 2
12.11 even 2 1386.2.k.t.991.1 4
28.3 even 6 1078.2.a.x.1.1 2
28.11 odd 6 1078.2.a.t.1.2 2
28.19 even 6 1078.2.e.m.177.2 4
28.23 odd 6 154.2.e.e.23.1 ✓ 4
28.27 even 2 1078.2.e.m.67.2 4
84.11 even 6 9702.2.a.cx.1.2 2
84.23 even 6 1386.2.k.t.793.1 4
84.59 odd 6 9702.2.a.ch.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.e.e.23.1 ✓ 4 28.23 odd 6
154.2.e.e.67.1 yes 4 4.3 odd 2
1078.2.a.t.1.2 2 28.11 odd 6
1078.2.a.x.1.1 2 28.3 even 6
1078.2.e.m.67.2 4 28.27 even 2
1078.2.e.m.177.2 4 28.19 even 6
1232.2.q.f.177.2 4 7.2 even 3 inner
1232.2.q.f.529.2 4 1.1 even 1 trivial
1386.2.k.t.793.1 4 84.23 even 6
1386.2.k.t.991.1 4 12.11 even 2
8624.2.a.bh.1.2 2 7.3 odd 6
8624.2.a.cc.1.1 2 7.4 even 3
9702.2.a.ch.1.1 2 84.59 odd 6
9702.2.a.cx.1.2 2 84.11 even 6