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 67.1
Root \(-0.707107 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 1078.67
Dual form 1078.2.e.m.177.1

$q$-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+(-0.500000 - 0.866025i) q^{2} +(-1.20711 + 2.09077i) q^{3} +(-0.500000 + 0.866025i) q^{4} +(-0.292893 - 0.507306i) q^{5} +2.41421 q^{6} +1.00000 q^{8} +(-1.41421 - 2.44949i) q^{9} +(-0.292893 + 0.507306i) q^{10} +(-0.500000 + 0.866025i) q^{11} +(-1.20711 - 2.09077i) q^{12} +3.82843 q^{13} +1.41421 q^{15} +(-0.500000 - 0.866025i) q^{16} +(1.82843 - 3.16693i) q^{17} +(-1.41421 + 2.44949i) q^{18} +(-0.292893 - 0.507306i) q^{19} +0.585786 q^{20} +1.00000 q^{22} +(3.12132 + 5.40629i) q^{23} +(-1.20711 + 2.09077i) q^{24} +(2.32843 - 4.03295i) q^{25} +(-1.91421 - 3.31552i) q^{26} -0.414214 q^{27} +2.65685 q^{29} +(-0.707107 - 1.22474i) q^{30} +(-2.00000 + 3.46410i) q^{31} +(-0.500000 + 0.866025i) q^{32} +(-1.20711 - 2.09077i) q^{33} -3.65685 q^{34} +2.82843 q^{36} +(4.70711 + 8.15295i) q^{37} +(-0.292893 + 0.507306i) q^{38} +(-4.62132 + 8.00436i) q^{39} +(-0.292893 - 0.507306i) q^{40} +5.41421 q^{41} -5.65685 q^{43} +(-0.500000 - 0.866025i) q^{44} +(-0.828427 + 1.43488i) q^{45} +(3.12132 - 5.40629i) q^{46} +(-5.24264 - 9.08052i) q^{47} +2.41421 q^{48} -4.65685 q^{50} +(4.41421 + 7.64564i) q^{51} +(-1.91421 + 3.31552i) q^{52} +(-3.94975 + 6.84116i) q^{53} +(0.207107 + 0.358719i) q^{54} +0.585786 q^{55} +1.41421 q^{57} +(-1.32843 - 2.30090i) q^{58} +(-2.79289 + 4.83743i) q^{59} +(-0.707107 + 1.22474i) q^{60} +(5.91421 + 10.2437i) q^{61} +4.00000 q^{62} +1.00000 q^{64} +(-1.12132 - 1.94218i) q^{65} +(-1.20711 + 2.09077i) q^{66} +(-1.37868 + 2.38794i) q^{67} +(1.82843 + 3.16693i) q^{68} -15.0711 q^{69} -11.0711 q^{71} +(-1.41421 - 2.44949i) q^{72} +(-4.70711 + 8.15295i) q^{73} +(4.70711 - 8.15295i) q^{74} +(5.62132 + 9.73641i) q^{75} +0.585786 q^{76} +9.24264 q^{78} +(6.62132 + 11.4685i) q^{79} +(-0.292893 + 0.507306i) q^{80} +(4.74264 - 8.21449i) q^{81} +(-2.70711 - 4.68885i) q^{82} +12.1421 q^{83} -2.14214 q^{85} +(2.82843 + 4.89898i) q^{86} +(-3.20711 + 5.55487i) q^{87} +(-0.500000 + 0.866025i) q^{88} +(6.24264 + 10.8126i) q^{89} +1.65685 q^{90} -6.24264 q^{92} +(-4.82843 - 8.36308i) q^{93} +(-5.24264 + 9.08052i) q^{94} +(-0.171573 + 0.297173i) q^{95} +(-1.20711 - 2.09077i) q^{96} +3.82843 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 \(\left(\mathbb{Z}/1078\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(981\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\)

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
\(3\) −1.20711 + 2.09077i −0.696923 + 1.20711i 0.272605 + 0.962126i \(0.412115\pi\)
−0.969528 + 0.244981i \(0.921218\pi\)
\(4\) −0.500000 + 0.866025i −0.250000 + 0.433013i
\(5\) −0.292893 − 0.507306i −0.130986 − 0.226874i 0.793071 − 0.609129i \(-0.208481\pi\)
−0.924057 + 0.382255i \(0.875148\pi\)
\(6\) 2.41421 0.985599
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) −1.41421 − 2.44949i −0.471405 − 0.816497i
\(10\) −0.292893 + 0.507306i −0.0926210 + 0.160424i
\(11\) −0.500000 + 0.866025i −0.150756 + 0.261116i
\(12\) −1.20711 − 2.09077i −0.348462 − 0.603553i
\(13\) 3.82843 1.06181 0.530907 − 0.847430i \(-0.321851\pi\)
0.530907 + 0.847430i \(0.321851\pi\)
\(14\) 0 0
\(15\) 1.41421 0.365148
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) 1.82843 − 3.16693i 0.443459 − 0.768093i −0.554485 − 0.832194i \(-0.687085\pi\)
0.997943 + 0.0641009i \(0.0204179\pi\)
\(18\) −1.41421 + 2.44949i −0.333333 + 0.577350i
\(19\) −0.292893 − 0.507306i −0.0671943 − 0.116384i 0.830471 − 0.557062i \(-0.188071\pi\)
−0.897665 + 0.440678i \(0.854738\pi\)
\(20\) 0.585786 0.130986
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 3.12132 + 5.40629i 0.650840 + 1.12729i 0.982919 + 0.184037i \(0.0589166\pi\)
−0.332079 + 0.943252i \(0.607750\pi\)
\(24\) −1.20711 + 2.09077i −0.246400 + 0.426777i
\(25\) 2.32843 − 4.03295i 0.465685 − 0.806591i
\(26\) −1.91421 − 3.31552i −0.375408 − 0.650226i
\(27\) −0.414214 −0.0797154
\(28\) 0 0
\(29\) 2.65685 0.493365 0.246683 − 0.969096i \(-0.420659\pi\)
0.246683 + 0.969096i \(0.420659\pi\)
\(30\) −0.707107 − 1.22474i −0.129099 − 0.223607i
\(31\) −2.00000 + 3.46410i −0.359211 + 0.622171i −0.987829 − 0.155543i \(-0.950287\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) −0.500000 + 0.866025i −0.0883883 + 0.153093i
\(33\) −1.20711 − 2.09077i −0.210130 − 0.363956i
\(34\) −3.65685 −0.627145
\(35\) 0 0
\(36\) 2.82843 0.471405
\(37\) 4.70711 + 8.15295i 0.773844 + 1.34034i 0.935442 + 0.353480i \(0.115002\pi\)
−0.161599 + 0.986857i \(0.551665\pi\)
\(38\) −0.292893 + 0.507306i −0.0475136 + 0.0822959i
\(39\) −4.62132 + 8.00436i −0.740003 + 1.28172i
\(40\) −0.292893 − 0.507306i −0.0463105 − 0.0802121i
\(41\) 5.41421 0.845558 0.422779 − 0.906233i \(-0.361055\pi\)
0.422779 + 0.906233i \(0.361055\pi\)
\(42\) 0 0
\(43\) −5.65685 −0.862662 −0.431331 − 0.902194i \(-0.641956\pi\)
−0.431331 + 0.902194i \(0.641956\pi\)
\(44\) −0.500000 − 0.866025i −0.0753778 − 0.130558i
\(45\) −0.828427 + 1.43488i −0.123495 + 0.213899i
\(46\) 3.12132 − 5.40629i 0.460214 − 0.797113i
\(47\) −5.24264 − 9.08052i −0.764718 − 1.32453i −0.940396 − 0.340082i \(-0.889545\pi\)
0.175678 − 0.984448i \(-0.443788\pi\)
\(48\) 2.41421 0.348462
\(49\) 0 0
\(50\) −4.65685 −0.658579
\(51\) 4.41421 + 7.64564i 0.618114 + 1.07060i
\(52\) −1.91421 + 3.31552i −0.265454 + 0.459779i
\(53\) −3.94975 + 6.84116i −0.542540 + 0.939706i 0.456218 + 0.889868i \(0.349204\pi\)
−0.998757 + 0.0498379i \(0.984130\pi\)
\(54\) 0.207107 + 0.358719i 0.0281837 + 0.0488155i
\(55\) 0.585786 0.0789874
\(56\) 0 0
\(57\) 1.41421 0.187317
\(58\) −1.32843 − 2.30090i −0.174431 − 0.302123i
\(59\) −2.79289 + 4.83743i −0.363604 + 0.629780i −0.988551 − 0.150887i \(-0.951787\pi\)
0.624947 + 0.780667i \(0.285120\pi\)
\(60\) −0.707107 + 1.22474i −0.0912871 + 0.158114i
\(61\) 5.91421 + 10.2437i 0.757237 + 1.31157i 0.944254 + 0.329217i \(0.106785\pi\)
−0.187017 + 0.982357i \(0.559882\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −1.12132 − 1.94218i −0.139083 − 0.240898i
\(66\) −1.20711 + 2.09077i −0.148585 + 0.257356i
\(67\) −1.37868 + 2.38794i −0.168433 + 0.291734i −0.937869 − 0.346990i \(-0.887204\pi\)
0.769436 + 0.638723i \(0.220537\pi\)
\(68\) 1.82843 + 3.16693i 0.221729 + 0.384047i
\(69\) −15.0711 −1.81434
\(70\) 0 0
\(71\) −11.0711 −1.31389 −0.656947 − 0.753937i \(-0.728152\pi\)
−0.656947 + 0.753937i \(0.728152\pi\)
\(72\) −1.41421 − 2.44949i −0.166667 − 0.288675i
\(73\) −4.70711 + 8.15295i −0.550925 + 0.954230i 0.447283 + 0.894393i \(0.352392\pi\)
−0.998208 + 0.0598379i \(0.980942\pi\)
\(74\) 4.70711 − 8.15295i 0.547190 − 0.947761i
\(75\) 5.62132 + 9.73641i 0.649094 + 1.12426i
\(76\) 0.585786 0.0671943
\(77\) 0 0
\(78\) 9.24264 1.04652
\(79\) 6.62132 + 11.4685i 0.744957 + 1.29030i 0.950215 + 0.311595i \(0.100863\pi\)
−0.205258 + 0.978708i \(0.565803\pi\)
\(80\) −0.292893 + 0.507306i −0.0327465 + 0.0567185i
\(81\) 4.74264 − 8.21449i 0.526960 − 0.912722i
\(82\) −2.70711 − 4.68885i −0.298950 − 0.517796i
\(83\) 12.1421 1.33277 0.666386 − 0.745607i \(-0.267840\pi\)
0.666386 + 0.745607i \(0.267840\pi\)
\(84\) 0 0
\(85\) −2.14214 −0.232347
\(86\) 2.82843 + 4.89898i 0.304997 + 0.528271i
\(87\) −3.20711 + 5.55487i −0.343838 + 0.595545i
\(88\) −0.500000 + 0.866025i −0.0533002 + 0.0923186i
\(89\) 6.24264 + 10.8126i 0.661719 + 1.14613i 0.980164 + 0.198189i \(0.0635060\pi\)
−0.318445 + 0.947941i \(0.603161\pi\)
\(90\) 1.65685 0.174648
\(91\) 0 0
\(92\) −6.24264 −0.650840
\(93\) −4.82843 − 8.36308i −0.500685 − 0.867211i
\(94\) −5.24264 + 9.08052i −0.540737 + 0.936584i
\(95\) −0.171573 + 0.297173i −0.0176030 + 0.0304893i
\(96\) −1.20711 − 2.09077i −0.123200 − 0.213388i
\(97\) 3.82843 0.388718 0.194359 − 0.980930i \(-0.437737\pi\)
0.194359 + 0.980930i \(0.437737\pi\)
\(98\) 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.67.1 4
7.2 even 3 inner 1078.2.e.m.177.1 4
7.3 odd 6 1078.2.a.t.1.1 2
7.4 even 3 1078.2.a.x.1.2 2
7.5 odd 6 154.2.e.e.23.2 ✓ 4
7.6 odd 2 154.2.e.e.67.2 yes 4
21.5 even 6 1386.2.k.t.793.2 4
21.11 odd 6 9702.2.a.ch.1.2 2
21.17 even 6 9702.2.a.cx.1.1 2
21.20 even 2 1386.2.k.t.991.2 4
28.3 even 6 8624.2.a.cc.1.2 2
28.11 odd 6 8624.2.a.bh.1.1 2
28.19 even 6 1232.2.q.f.177.1 4
28.27 even 2 1232.2.q.f.529.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.e.e.23.2 ✓ 4 7.5 odd 6
154.2.e.e.67.2 yes 4 7.6 odd 2
1078.2.a.t.1.1 2 7.3 odd 6
1078.2.a.x.1.2 2 7.4 even 3
1078.2.e.m.67.1 4 1.1 even 1 trivial
1078.2.e.m.177.1 4 7.2 even 3 inner
1232.2.q.f.177.1 4 28.19 even 6
1232.2.q.f.529.1 4 28.27 even 2
1386.2.k.t.793.2 4 21.5 even 6
1386.2.k.t.991.2 4 21.20 even 2
8624.2.a.bh.1.1 2 28.11 odd 6
8624.2.a.cc.1.2 2 28.3 even 6
9702.2.a.ch.1.2 2 21.11 odd 6
9702.2.a.cx.1.1 2 21.17 even 6