Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1078,2,Mod(67,1078)] 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("1078.67"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1078, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1078 = 2 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1078.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-2,-2,-2,-4,4,0,4,0,-4,-2,-2,4,0,0,-2,-4,0,-4,8,0,4,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.60787333789\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 154)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 177.2
Root \(0.707107 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 1078.177
Dual form 1078.2.e.m.67.2

$q$-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+(-0.500000 + 0.866025i) q^{2} +(0.207107 + 0.358719i) q^{3} +(-0.500000 - 0.866025i) q^{4} +(-1.70711 + 2.95680i) q^{5} -0.414214 q^{6} +1.00000 q^{8} +(1.41421 - 2.44949i) q^{9} +(-1.70711 - 2.95680i) q^{10} +(-0.500000 - 0.866025i) q^{11} +(0.207107 - 0.358719i) q^{12} -1.82843 q^{13} -1.41421 q^{15} +(-0.500000 + 0.866025i) q^{16} +(-3.82843 - 6.63103i) q^{17} +(1.41421 + 2.44949i) q^{18} +(-1.70711 + 2.95680i) q^{19} +3.41421 q^{20} +1.00000 q^{22} +(-1.12132 + 1.94218i) q^{23} +(0.207107 + 0.358719i) q^{24} +(-3.32843 - 5.76500i) q^{25} +(0.914214 - 1.58346i) q^{26} +2.41421 q^{27} -8.65685 q^{29} +(0.707107 - 1.22474i) q^{30} +(-2.00000 - 3.46410i) q^{31} +(-0.500000 - 0.866025i) q^{32} +(0.207107 - 0.358719i) q^{33} +7.65685 q^{34} -2.82843 q^{36} +(3.29289 - 5.70346i) q^{37} +(-1.70711 - 2.95680i) q^{38} +(-0.378680 - 0.655892i) q^{39} +(-1.70711 + 2.95680i) q^{40} +2.58579 q^{41} +5.65685 q^{43} +(-0.500000 + 0.866025i) q^{44} +(4.82843 + 8.36308i) q^{45} +(-1.12132 - 1.94218i) q^{46} +(3.24264 - 5.61642i) q^{47} -0.414214 q^{48} +6.65685 q^{50} +(1.58579 - 2.74666i) q^{51} +(0.914214 + 1.58346i) q^{52} +(5.94975 + 10.3053i) q^{53} +(-1.20711 + 2.09077i) q^{54} +3.41421 q^{55} -1.41421 q^{57} +(4.32843 - 7.49706i) q^{58} +(-4.20711 - 7.28692i) q^{59} +(0.707107 + 1.22474i) q^{60} +(3.08579 - 5.34474i) q^{61} +4.00000 q^{62} +1.00000 q^{64} +(3.12132 - 5.40629i) q^{65} +(0.207107 + 0.358719i) q^{66} +(-5.62132 - 9.73641i) q^{67} +(-3.82843 + 6.63103i) q^{68} -0.928932 q^{69} +3.07107 q^{71} +(1.41421 - 2.44949i) q^{72} +(-3.29289 - 5.70346i) q^{73} +(3.29289 + 5.70346i) q^{74} +(1.37868 - 2.38794i) q^{75} +3.41421 q^{76} +0.757359 q^{78} +(2.37868 - 4.11999i) q^{79} +(-1.70711 - 2.95680i) q^{80} +(-3.74264 - 6.48244i) q^{81} +(-1.29289 + 2.23936i) q^{82} -16.1421 q^{83} +26.1421 q^{85} +(-2.82843 + 4.89898i) q^{86} +(-1.79289 - 3.10538i) q^{87} +(-0.500000 - 0.866025i) q^{88} +(-2.24264 + 3.88437i) q^{89} -9.65685 q^{90} +2.24264 q^{92} +(0.828427 - 1.43488i) q^{93} +(3.24264 + 5.61642i) q^{94} +(-5.82843 - 10.0951i) q^{95} +(0.207107 - 0.358719i) q^{96} -1.82843 q^{97} -2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 2 q^{3} - 2 q^{4} - 4 q^{5} + 4 q^{6} + 4 q^{8} - 4 q^{10} - 2 q^{11} - 2 q^{12} + 4 q^{13} - 2 q^{16} - 4 q^{17} - 4 q^{19} + 8 q^{20} + 4 q^{22} + 4 q^{23} - 2 q^{24} - 2 q^{25} - 2 q^{26}+ \cdots + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1078\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(981\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.500000 + 0.866025i −0.353553 + 0.612372i
\(3\) 0.207107 + 0.358719i 0.119573 + 0.207107i 0.919599 − 0.392859i \(-0.128514\pi\)
−0.800025 + 0.599966i \(0.795181\pi\)
\(4\) −0.500000 − 0.866025i −0.250000 − 0.433013i
\(5\) −1.70711 + 2.95680i −0.763441 + 1.32232i 0.177625 + 0.984098i \(0.443158\pi\)
−0.941067 + 0.338221i \(0.890175\pi\)
\(6\) −0.414214 −0.169102
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.41421 − 2.44949i 0.471405 − 0.816497i
\(10\) −1.70711 − 2.95680i −0.539835 − 0.935021i
\(11\) −0.500000 − 0.866025i −0.150756 − 0.261116i
\(12\) 0.207107 − 0.358719i 0.0597866 − 0.103553i
\(13\) −1.82843 −0.507114 −0.253557 − 0.967320i \(-0.581601\pi\)
−0.253557 + 0.967320i \(0.581601\pi\)
\(14\) 0 0
\(15\) −1.41421 −0.365148
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) −3.82843 − 6.63103i −0.928530 − 1.60826i −0.785783 − 0.618502i \(-0.787740\pi\)
−0.142747 − 0.989759i \(-0.545593\pi\)
\(18\) 1.41421 + 2.44949i 0.333333 + 0.577350i
\(19\) −1.70711 + 2.95680i −0.391637 + 0.678335i −0.992666 − 0.120892i \(-0.961424\pi\)
0.601028 + 0.799228i \(0.294758\pi\)
\(20\) 3.41421 0.763441
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) −1.12132 + 1.94218i −0.233811 + 0.404973i −0.958927 − 0.283654i \(-0.908453\pi\)
0.725115 + 0.688628i \(0.241787\pi\)
\(24\) 0.207107 + 0.358719i 0.0422755 + 0.0732233i
\(25\) −3.32843 − 5.76500i −0.665685 − 1.15300i
\(26\) 0.914214 − 1.58346i 0.179292 − 0.310543i
\(27\) 2.41421 0.464616
\(28\) 0 0
\(29\) −8.65685 −1.60754 −0.803769 − 0.594942i \(-0.797175\pi\)
−0.803769 + 0.594942i \(0.797175\pi\)
\(30\) 0.707107 − 1.22474i 0.129099 − 0.223607i
\(31\) −2.00000 − 3.46410i −0.359211 − 0.622171i 0.628619 − 0.777714i \(-0.283621\pi\)
−0.987829 + 0.155543i \(0.950287\pi\)
\(32\) −0.500000 − 0.866025i −0.0883883 − 0.153093i
\(33\) 0.207107 − 0.358719i 0.0360527 − 0.0624450i
\(34\) 7.65685 1.31314
\(35\) 0 0
\(36\) −2.82843 −0.471405
\(37\) 3.29289 − 5.70346i 0.541348 − 0.937643i −0.457479 − 0.889221i \(-0.651247\pi\)
0.998827 − 0.0484222i \(-0.0154193\pi\)
\(38\) −1.70711 − 2.95680i −0.276929 − 0.479656i
\(39\) −0.378680 − 0.655892i −0.0606373 − 0.105027i
\(40\) −1.70711 + 2.95680i −0.269917 + 0.467510i
\(41\) 2.58579 0.403832 0.201916 − 0.979403i \(-0.435283\pi\)
0.201916 + 0.979403i \(0.435283\pi\)
\(42\) 0 0
\(43\) 5.65685 0.862662 0.431331 − 0.902194i \(-0.358044\pi\)
0.431331 + 0.902194i \(0.358044\pi\)
\(44\) −0.500000 + 0.866025i −0.0753778 + 0.130558i
\(45\) 4.82843 + 8.36308i 0.719779 + 1.24669i
\(46\) −1.12132 − 1.94218i −0.165330 − 0.286359i
\(47\) 3.24264 − 5.61642i 0.472988 − 0.819239i −0.526534 − 0.850154i \(-0.676509\pi\)
0.999522 + 0.0309151i \(0.00984215\pi\)
\(48\) −0.414214 −0.0597866
\(49\) 0 0
\(50\) 6.65685 0.941421
\(51\) 1.58579 − 2.74666i 0.222055 − 0.384610i
\(52\) 0.914214 + 1.58346i 0.126779 + 0.219587i
\(53\) 5.94975 + 10.3053i 0.817261 + 1.41554i 0.907693 + 0.419634i \(0.137842\pi\)
−0.0904325 + 0.995903i \(0.528825\pi\)
\(54\) −1.20711 + 2.09077i −0.164266 + 0.284518i
\(55\) 3.41421 0.460372
\(56\) 0 0
\(57\) −1.41421 −0.187317
\(58\) 4.32843 − 7.49706i 0.568350 − 0.984412i
\(59\) −4.20711 − 7.28692i −0.547719 − 0.948677i −0.998430 − 0.0560070i \(-0.982163\pi\)
0.450712 − 0.892670i \(-0.351170\pi\)
\(60\) 0.707107 + 1.22474i 0.0912871 + 0.158114i
\(61\) 3.08579 − 5.34474i 0.395094 − 0.684324i −0.598019 − 0.801482i \(-0.704045\pi\)
0.993113 + 0.117158i \(0.0373785\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 3.12132 − 5.40629i 0.387152 − 0.670567i
\(66\) 0.207107 + 0.358719i 0.0254931 + 0.0441553i
\(67\) −5.62132 − 9.73641i −0.686754 − 1.18949i −0.972882 − 0.231301i \(-0.925702\pi\)
0.286129 − 0.958191i \(-0.407632\pi\)
\(68\) −3.82843 + 6.63103i −0.464265 + 0.804131i
\(69\) −0.928932 −0.111830
\(70\) 0 0
\(71\) 3.07107 0.364469 0.182234 − 0.983255i \(-0.441667\pi\)
0.182234 + 0.983255i \(0.441667\pi\)
\(72\) 1.41421 − 2.44949i 0.166667 − 0.288675i
\(73\) −3.29289 − 5.70346i −0.385404 − 0.667539i 0.606421 − 0.795144i \(-0.292605\pi\)
−0.991825 + 0.127604i \(0.959271\pi\)
\(74\) 3.29289 + 5.70346i 0.382791 + 0.663014i
\(75\) 1.37868 − 2.38794i 0.159196 − 0.275736i
\(76\) 3.41421 0.391637
\(77\) 0 0
\(78\) 0.757359 0.0857541
\(79\) 2.37868 − 4.11999i 0.267622 − 0.463536i −0.700625 − 0.713530i \(-0.747095\pi\)
0.968247 + 0.249994i \(0.0804287\pi\)
\(80\) −1.70711 − 2.95680i −0.190860 − 0.330580i
\(81\) −3.74264 − 6.48244i −0.415849 − 0.720272i
\(82\) −1.29289 + 2.23936i −0.142776 + 0.247296i
\(83\) −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
\(86\) −2.82843 + 4.89898i −0.304997 + 0.528271i
\(87\) −1.79289 − 3.10538i −0.192218 − 0.332932i
\(88\) −0.500000 − 0.866025i −0.0533002 − 0.0923186i
\(89\) −2.24264 + 3.88437i −0.237719 + 0.411742i −0.960060 − 0.279796i \(-0.909733\pi\)
0.722340 + 0.691538i \(0.243067\pi\)
\(90\) −9.65685 −1.01792
\(91\) 0 0
\(92\) 2.24264 0.233811
\(93\) 0.828427 − 1.43488i 0.0859039 − 0.148790i
\(94\) 3.24264 + 5.61642i 0.334453 + 0.579289i
\(95\) −5.82843 − 10.0951i −0.597984 − 1.03574i
\(96\) 0.207107 − 0.358719i 0.0211377 − 0.0366117i
\(97\) −1.82843 −0.185649 −0.0928243 − 0.995683i \(-0.529589\pi\)
−0.0928243 + 0.995683i \(0.529589\pi\)
\(98\) 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 1078.2.e.m.177.2 4
7.2 even 3 1078.2.a.x.1.1 2
7.3 odd 6 154.2.e.e.67.1 yes 4
7.4 even 3 inner 1078.2.e.m.67.2 4
7.5 odd 6 1078.2.a.t.1.2 2
7.6 odd 2 154.2.e.e.23.1 ✓ 4
21.2 odd 6 9702.2.a.ch.1.1 2
21.5 even 6 9702.2.a.cx.1.2 2
21.17 even 6 1386.2.k.t.991.1 4
21.20 even 2 1386.2.k.t.793.1 4
28.3 even 6 1232.2.q.f.529.2 4
28.19 even 6 8624.2.a.cc.1.1 2
28.23 odd 6 8624.2.a.bh.1.2 2
28.27 even 2 1232.2.q.f.177.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.e.e.23.1 ✓ 4 7.6 odd 2
154.2.e.e.67.1 yes 4 7.3 odd 6
1078.2.a.t.1.2 2 7.5 odd 6
1078.2.a.x.1.1 2 7.2 even 3
1078.2.e.m.67.2 4 7.4 even 3 inner
1078.2.e.m.177.2 4 1.1 even 1 trivial
1232.2.q.f.177.2 4 28.27 even 2
1232.2.q.f.529.2 4 28.3 even 6
1386.2.k.t.793.1 4 21.20 even 2
1386.2.k.t.991.1 4 21.17 even 6
8624.2.a.bh.1.2 2 28.23 odd 6
8624.2.a.cc.1.1 2 28.19 even 6
9702.2.a.ch.1.1 2 21.2 odd 6
9702.2.a.cx.1.2 2 21.5 even 6