↑

Properties

Label 1078.2.e.n.177.1
Level $1078$
Weight $2$
Character 1078.177
Analytic conductor $8.608$
Analytic rank $0$
Dimension $4$
Inner twists $2$

Related objects

Downloads

Learn more

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,-2,2,4,0,4,-6,2,-2,-2,4,0,-24,-2,-4,-6,-10,-4,0,4,-8] 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: \(\Q(\sqrt{-3}, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 2x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 154)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 177.1
Root \(0.809017 + 1.40126i\) of defining polynomial
Character \(\chi\) \(=\) 1078.177
Dual form 1078.2.e.n.67.1

$q$-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} +(-1.61803 - 2.80252i) q^{3} +(-0.500000 - 0.866025i) q^{4} +(1.61803 - 2.80252i) q^{5} +3.23607 q^{6} +1.00000 q^{8} +(-3.73607 + 6.47106i) q^{9} +(1.61803 + 2.80252i) q^{10} +(-0.500000 - 0.866025i) q^{11} +(-1.61803 + 2.80252i) q^{12} -1.23607 q^{13} -10.4721 q^{15} +(-0.500000 + 0.866025i) q^{16} +(-3.23607 - 5.60503i) q^{17} +(-3.73607 - 6.47106i) q^{18} +(-1.38197 + 2.39364i) q^{19} -3.23607 q^{20} +1.00000 q^{22} +(-2.00000 + 3.46410i) q^{23} +(-1.61803 - 2.80252i) q^{24} +(-2.73607 - 4.73901i) q^{25} +(0.618034 - 1.07047i) q^{26} +14.4721 q^{27} -4.47214 q^{29} +(5.23607 - 9.06914i) q^{30} +(1.00000 + 1.73205i) q^{31} +(-0.500000 - 0.866025i) q^{32} +(-1.61803 + 2.80252i) q^{33} +6.47214 q^{34} +7.47214 q^{36} +(5.47214 - 9.47802i) q^{37} +(-1.38197 - 2.39364i) q^{38} +(2.00000 + 3.46410i) q^{39} +(1.61803 - 2.80252i) q^{40} -6.47214 q^{41} -1.52786 q^{43} +(-0.500000 + 0.866025i) q^{44} +(12.0902 + 20.9408i) q^{45} +(-2.00000 - 3.46410i) q^{46} +(-1.00000 + 1.73205i) q^{47} +3.23607 q^{48} +5.47214 q^{50} +(-10.4721 + 18.1383i) q^{51} +(0.618034 + 1.07047i) q^{52} +(0.236068 + 0.408882i) q^{53} +(-7.23607 + 12.5332i) q^{54} -3.23607 q^{55} +8.94427 q^{57} +(2.23607 - 3.87298i) q^{58} +(3.61803 + 6.26662i) q^{59} +(5.23607 + 9.06914i) q^{60} +(-2.61803 + 4.53457i) q^{61} -2.00000 q^{62} +1.00000 q^{64} +(-2.00000 + 3.46410i) q^{65} +(-1.61803 - 2.80252i) q^{66} +(7.70820 + 13.3510i) q^{67} +(-3.23607 + 5.60503i) q^{68} +12.9443 q^{69} -2.47214 q^{71} +(-3.73607 + 6.47106i) q^{72} +(-2.47214 - 4.28187i) q^{73} +(5.47214 + 9.47802i) q^{74} +(-8.85410 + 15.3358i) q^{75} +2.76393 q^{76} -4.00000 q^{78} +(1.61803 + 2.80252i) q^{80} +(-12.2082 - 21.1452i) q^{81} +(3.23607 - 5.60503i) q^{82} -10.1803 q^{83} -20.9443 q^{85} +(0.763932 - 1.32317i) q^{86} +(7.23607 + 12.5332i) q^{87} +(-0.500000 - 0.866025i) q^{88} +(5.00000 - 8.66025i) q^{89} -24.1803 q^{90} +4.00000 q^{92} +(3.23607 - 5.60503i) q^{93} +(-1.00000 - 1.73205i) q^{94} +(4.47214 + 7.74597i) q^{95} +(-1.61803 + 2.80252i) q^{96} -3.52786 q^{97} +7.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 2 q^{3} - 2 q^{4} + 2 q^{5} + 4 q^{6} + 4 q^{8} - 6 q^{9} + 2 q^{10} - 2 q^{11} - 2 q^{12} + 4 q^{13} - 24 q^{15} - 2 q^{16} - 4 q^{17} - 6 q^{18} - 10 q^{19} - 4 q^{20} + 4 q^{22} - 8 q^{23}+ \cdots + 12 q^{99}+O(q^{100}) \) 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
\(3\) −1.61803 − 2.80252i −0.934172 − 1.61803i −0.776103 − 0.630606i \(-0.782806\pi\)
−0.158069 − 0.987428i \(-0.550527\pi\)
\(4\) −0.500000 − 0.866025i −0.250000 − 0.433013i
\(5\) 1.61803 − 2.80252i 0.723607 − 1.25332i −0.235938 − 0.971768i \(-0.575816\pi\)
0.959545 − 0.281556i \(-0.0908504\pi\)
\(6\) 3.23607 1.32112
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) −3.73607 + 6.47106i −1.24536 + 2.15702i
\(10\) 1.61803 + 2.80252i 0.511667 + 0.886234i
\(11\) −0.500000 − 0.866025i −0.150756 − 0.261116i
\(12\) −1.61803 + 2.80252i −0.467086 + 0.809017i
\(13\) −1.23607 −0.342824 −0.171412 − 0.985199i \(-0.554833\pi\)
−0.171412 + 0.985199i \(0.554833\pi\)
\(14\) 0 0
\(15\) −10.4721 −2.70389
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) −3.23607 − 5.60503i −0.784862 − 1.35942i −0.929082 − 0.369875i \(-0.879401\pi\)
0.144220 − 0.989546i \(-0.453933\pi\)
\(18\) −3.73607 − 6.47106i −0.880600 − 1.52524i
\(19\) −1.38197 + 2.39364i −0.317045 + 0.549138i −0.979870 − 0.199636i \(-0.936024\pi\)
0.662825 + 0.748774i \(0.269357\pi\)
\(20\) −3.23607 −0.723607
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) −2.00000 + 3.46410i −0.417029 + 0.722315i −0.995639 − 0.0932891i \(-0.970262\pi\)
0.578610 + 0.815604i \(0.303595\pi\)
\(24\) −1.61803 − 2.80252i −0.330280 − 0.572061i
\(25\) −2.73607 − 4.73901i −0.547214 − 0.947802i
\(26\) 0.618034 − 1.07047i 0.121206 − 0.209936i
\(27\) 14.4721 2.78516
\(28\) 0 0
\(29\) −4.47214 −0.830455 −0.415227 − 0.909718i \(-0.636298\pi\)
−0.415227 + 0.909718i \(0.636298\pi\)
\(30\) 5.23607 − 9.06914i 0.955971 − 1.65579i
\(31\) 1.00000 + 1.73205i 0.179605 + 0.311086i 0.941745 − 0.336327i \(-0.109185\pi\)
−0.762140 + 0.647412i \(0.775851\pi\)
\(32\) −0.500000 − 0.866025i −0.0883883 − 0.153093i
\(33\) −1.61803 + 2.80252i −0.281664 + 0.487856i
\(34\) 6.47214 1.10996
\(35\) 0 0
\(36\) 7.47214 1.24536
\(37\) 5.47214 − 9.47802i 0.899614 − 1.55818i 0.0716249 − 0.997432i \(-0.477182\pi\)
0.827989 − 0.560745i \(-0.189485\pi\)
\(38\) −1.38197 − 2.39364i −0.224184 − 0.388299i
\(39\) 2.00000 + 3.46410i 0.320256 + 0.554700i
\(40\) 1.61803 − 2.80252i 0.255834 − 0.443117i
\(41\) −6.47214 −1.01078 −0.505389 − 0.862892i \(-0.668651\pi\)
−0.505389 + 0.862892i \(0.668651\pi\)
\(42\) 0 0
\(43\) −1.52786 −0.232997 −0.116499 − 0.993191i \(-0.537167\pi\)
−0.116499 + 0.993191i \(0.537167\pi\)
\(44\) −0.500000 + 0.866025i −0.0753778 + 0.130558i
\(45\) 12.0902 + 20.9408i 1.80230 + 3.12167i
\(46\) −2.00000 − 3.46410i −0.294884 − 0.510754i
\(47\) −1.00000 + 1.73205i −0.145865 + 0.252646i −0.929695 − 0.368329i \(-0.879930\pi\)
0.783830 + 0.620975i \(0.213263\pi\)
\(48\) 3.23607 0.467086
\(49\) 0 0
\(50\) 5.47214 0.773877
\(51\) −10.4721 + 18.1383i −1.46639 + 2.53987i
\(52\) 0.618034 + 1.07047i 0.0857059 + 0.148447i
\(53\) 0.236068 + 0.408882i 0.0324264 + 0.0561642i 0.881783 − 0.471655i \(-0.156343\pi\)
−0.849357 + 0.527819i \(0.823010\pi\)
\(54\) −7.23607 + 12.5332i −0.984704 + 1.70556i
\(55\) −3.23607 −0.436351
\(56\) 0 0
\(57\) 8.94427 1.18470
\(58\) 2.23607 − 3.87298i 0.293610 − 0.508548i
\(59\) 3.61803 + 6.26662i 0.471028 + 0.815844i 0.999451 − 0.0331370i \(-0.0105498\pi\)
−0.528423 + 0.848981i \(0.677216\pi\)
\(60\) 5.23607 + 9.06914i 0.675973 + 1.17082i
\(61\) −2.61803 + 4.53457i −0.335205 + 0.580592i −0.983524 − 0.180777i \(-0.942139\pi\)
0.648319 + 0.761369i \(0.275472\pi\)
\(62\) −2.00000 −0.254000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −2.00000 + 3.46410i −0.248069 + 0.429669i
\(66\) −1.61803 − 2.80252i −0.199166 − 0.344966i
\(67\) 7.70820 + 13.3510i 0.941707 + 1.63108i 0.762214 + 0.647325i \(0.224112\pi\)
0.179493 + 0.983759i \(0.442554\pi\)
\(68\) −3.23607 + 5.60503i −0.392431 + 0.679710i
\(69\) 12.9443 1.55831
\(70\) 0 0
\(71\) −2.47214 −0.293389 −0.146694 − 0.989182i \(-0.546863\pi\)
−0.146694 + 0.989182i \(0.546863\pi\)
\(72\) −3.73607 + 6.47106i −0.440300 + 0.762622i
\(73\) −2.47214 − 4.28187i −0.289342 − 0.501154i 0.684311 − 0.729190i \(-0.260103\pi\)
−0.973653 + 0.228036i \(0.926770\pi\)
\(74\) 5.47214 + 9.47802i 0.636123 + 1.10180i
\(75\) −8.85410 + 15.3358i −1.02238 + 1.77082i
\(76\) 2.76393 0.317045
\(77\) 0 0
\(78\) −4.00000 −0.452911
\(79\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(80\) 1.61803 + 2.80252i 0.180902 + 0.313331i
\(81\) −12.2082 − 21.1452i −1.35647 − 2.34947i
\(82\) 3.23607 − 5.60503i 0.357364 − 0.618972i
\(83\) −10.1803 −1.11744 −0.558719 − 0.829357i \(-0.688707\pi\)
−0.558719 + 0.829357i \(0.688707\pi\)
\(84\) 0 0
\(85\) −20.9443 −2.27173
\(86\) 0.763932 − 1.32317i 0.0823769 − 0.142681i
\(87\) 7.23607 + 12.5332i 0.775788 + 1.34370i
\(88\) −0.500000 − 0.866025i −0.0533002 − 0.0923186i
\(89\) 5.00000 − 8.66025i 0.529999 − 0.917985i −0.469389 − 0.882992i \(-0.655526\pi\)
0.999388 − 0.0349934i \(-0.0111410\pi\)
\(90\) −24.1803 −2.54883
\(91\) 0 0
\(92\) 4.00000 0.417029
\(93\) 3.23607 − 5.60503i 0.335565 − 0.581215i
\(94\) −1.00000 − 1.73205i −0.103142 − 0.178647i
\(95\) 4.47214 + 7.74597i 0.458831 + 0.794719i
\(96\) −1.61803 + 2.80252i −0.165140 + 0.286031i
\(97\) −3.52786 −0.358200 −0.179100 − 0.983831i \(-0.557319\pi\)
−0.179100 + 0.983831i \(0.557319\pi\)
\(98\) 0 0
\(99\) 7.47214 0.750978
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.n.177.1 4
7.2 even 3 1078.2.a.w.1.2 2
7.3 odd 6 1078.2.e.q.67.2 4
7.4 even 3 inner 1078.2.e.n.67.1 4
7.5 odd 6 154.2.a.d.1.1 ✓ 2
7.6 odd 2 1078.2.e.q.177.2 4
21.2 odd 6 9702.2.a.cu.1.2 2
21.5 even 6 1386.2.a.m.1.1 2
28.19 even 6 1232.2.a.p.1.2 2
28.23 odd 6 8624.2.a.bf.1.1 2
35.12 even 12 3850.2.c.q.1849.4 4
35.19 odd 6 3850.2.a.bj.1.2 2
35.33 even 12 3850.2.c.q.1849.1 4
56.5 odd 6 4928.2.a.bt.1.2 2
56.19 even 6 4928.2.a.bk.1.1 2
77.54 even 6 1694.2.a.l.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.a.d.1.1 ✓ 2 7.5 odd 6
1078.2.a.w.1.2 2 7.2 even 3
1078.2.e.n.67.1 4 7.4 even 3 inner
1078.2.e.n.177.1 4 1.1 even 1 trivial
1078.2.e.q.67.2 4 7.3 odd 6
1078.2.e.q.177.2 4 7.6 odd 2
1232.2.a.p.1.2 2 28.19 even 6
1386.2.a.m.1.1 2 21.5 even 6
1694.2.a.l.1.1 2 77.54 even 6
3850.2.a.bj.1.2 2 35.19 odd 6
3850.2.c.q.1849.1 4 35.33 even 12
3850.2.c.q.1849.4 4 35.12 even 12
4928.2.a.bk.1.1 2 56.19 even 6
4928.2.a.bt.1.2 2 56.5 odd 6
8624.2.a.bf.1.1 2 28.23 odd 6
9702.2.a.cu.1.2 2 21.2 odd 6