↑

Properties

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

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,-2,0,-2,0,0,0,4,2,0,2,0,0,0,24,-2,0,2,0,0,0,-4,-12] 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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 177.1
Root \(-0.707107 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 1078.177
Dual form 1078.2.e.p.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)
 
\(f(q)\) \(=\) \(q+(-0.500000 + 0.866025i) q^{2} +(-0.707107 - 1.22474i) q^{3} +(-0.500000 - 0.866025i) q^{4} +(-2.12132 + 3.67423i) q^{5} +1.41421 q^{6} +1.00000 q^{8} +(0.500000 - 0.866025i) q^{9} +(-2.12132 - 3.67423i) q^{10} +(0.500000 + 0.866025i) q^{11} +(-0.707107 + 1.22474i) q^{12} +6.00000 q^{15} +(-0.500000 + 0.866025i) q^{16} +(2.82843 + 4.89898i) q^{17} +(0.500000 + 0.866025i) q^{18} +4.24264 q^{20} -1.00000 q^{22} +(-3.00000 + 5.19615i) q^{23} +(-0.707107 - 1.22474i) q^{24} +(-6.50000 - 11.2583i) q^{25} -5.65685 q^{27} +2.00000 q^{29} +(-3.00000 + 5.19615i) q^{30} +(0.707107 + 1.22474i) q^{31} +(-0.500000 - 0.866025i) q^{32} +(0.707107 - 1.22474i) q^{33} -5.65685 q^{34} -1.00000 q^{36} +(5.00000 - 8.66025i) q^{37} +(-2.12132 + 3.67423i) q^{40} -11.3137 q^{41} -8.00000 q^{43} +(0.500000 - 0.866025i) q^{44} +(2.12132 + 3.67423i) q^{45} +(-3.00000 - 5.19615i) q^{46} +(-2.12132 + 3.67423i) q^{47} +1.41421 q^{48} +13.0000 q^{50} +(4.00000 - 6.92820i) q^{51} +(-4.00000 - 6.92820i) q^{53} +(2.82843 - 4.89898i) q^{54} -4.24264 q^{55} +(-1.00000 + 1.73205i) q^{58} +(-0.707107 - 1.22474i) q^{59} +(-3.00000 - 5.19615i) q^{60} +(1.41421 - 2.44949i) q^{61} -1.41421 q^{62} +1.00000 q^{64} +(0.707107 + 1.22474i) q^{66} +(-1.00000 - 1.73205i) q^{67} +(2.82843 - 4.89898i) q^{68} +8.48528 q^{69} -2.00000 q^{71} +(0.500000 - 0.866025i) q^{72} +(-4.24264 - 7.34847i) q^{73} +(5.00000 + 8.66025i) q^{74} +(-9.19239 + 15.9217i) q^{75} +(-8.00000 + 13.8564i) q^{79} +(-2.12132 - 3.67423i) q^{80} +(2.50000 + 4.33013i) q^{81} +(5.65685 - 9.79796i) q^{82} -16.9706 q^{83} -24.0000 q^{85} +(4.00000 - 6.92820i) q^{86} +(-1.41421 - 2.44949i) q^{87} +(0.500000 + 0.866025i) q^{88} +(3.53553 - 6.12372i) q^{89} -4.24264 q^{90} +6.00000 q^{92} +(1.00000 - 1.73205i) q^{93} +(-2.12132 - 3.67423i) q^{94} +(-0.707107 + 1.22474i) q^{96} -9.89949 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 2 q^{4} + 4 q^{8} + 2 q^{9} + 2 q^{11} + 24 q^{15} - 2 q^{16} + 2 q^{18} - 4 q^{22} - 12 q^{23} - 26 q^{25} + 8 q^{29} - 12 q^{30} - 2 q^{32} - 4 q^{36} + 20 q^{37} - 32 q^{43} + 2 q^{44}+ \cdots + 4 q^{99}+O(q^{100}) \) 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
\(3\) −0.707107 − 1.22474i −0.408248 − 0.707107i 0.586445 − 0.809989i \(-0.300527\pi\)
−0.994694 + 0.102882i \(0.967194\pi\)
\(4\) −0.500000 − 0.866025i −0.250000 − 0.433013i
\(5\) −2.12132 + 3.67423i −0.948683 + 1.64317i −0.200480 + 0.979698i \(0.564250\pi\)
−0.748203 + 0.663470i \(0.769083\pi\)
\(6\) 1.41421 0.577350
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 0.500000 − 0.866025i 0.166667 − 0.288675i
\(10\) −2.12132 − 3.67423i −0.670820 − 1.16190i
\(11\) 0.500000 + 0.866025i 0.150756 + 0.261116i
\(12\) −0.707107 + 1.22474i −0.204124 + 0.353553i
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 6.00000 1.54919
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) 2.82843 + 4.89898i 0.685994 + 1.18818i 0.973123 + 0.230285i \(0.0739659\pi\)
−0.287129 + 0.957892i \(0.592701\pi\)
\(18\) 0.500000 + 0.866025i 0.117851 + 0.204124i
\(19\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(20\) 4.24264 0.948683
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) −3.00000 + 5.19615i −0.625543 + 1.08347i 0.362892 + 0.931831i \(0.381789\pi\)
−0.988436 + 0.151642i \(0.951544\pi\)
\(24\) −0.707107 − 1.22474i −0.144338 − 0.250000i
\(25\) −6.50000 − 11.2583i −1.30000 − 2.25167i
\(26\) 0 0
\(27\) −5.65685 −1.08866
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 − 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) −3.00000 + 5.19615i −0.547723 + 0.948683i
\(31\) 0.707107 + 1.22474i 0.127000 + 0.219971i 0.922513 − 0.385966i \(-0.126132\pi\)
−0.795513 + 0.605937i \(0.792798\pi\)
\(32\) −0.500000 − 0.866025i −0.0883883 − 0.153093i
\(33\) 0.707107 − 1.22474i 0.123091 − 0.213201i
\(34\) −5.65685 −0.970143
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) 5.00000 − 8.66025i 0.821995 − 1.42374i −0.0821995 − 0.996616i \(-0.526194\pi\)
0.904194 − 0.427121i \(-0.140472\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −2.12132 + 3.67423i −0.335410 + 0.580948i
\(41\) −11.3137 −1.76690 −0.883452 − 0.468521i \(-0.844787\pi\)
−0.883452 + 0.468521i \(0.844787\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 − 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0.500000 − 0.866025i 0.0753778 − 0.130558i
\(45\) 2.12132 + 3.67423i 0.316228 + 0.547723i
\(46\) −3.00000 − 5.19615i −0.442326 − 0.766131i
\(47\) −2.12132 + 3.67423i −0.309426 + 0.535942i −0.978237 − 0.207491i \(-0.933470\pi\)
0.668811 + 0.743433i \(0.266804\pi\)
\(48\) 1.41421 0.204124
\(49\) 0 0
\(50\) 13.0000 1.83848
\(51\) 4.00000 − 6.92820i 0.560112 − 0.970143i
\(52\) 0 0
\(53\) −4.00000 − 6.92820i −0.549442 − 0.951662i −0.998313 − 0.0580651i \(-0.981507\pi\)
0.448871 − 0.893597i \(-0.351826\pi\)
\(54\) 2.82843 − 4.89898i 0.384900 − 0.666667i
\(55\) −4.24264 −0.572078
\(56\) 0 0
\(57\) 0 0
\(58\) −1.00000 + 1.73205i −0.131306 + 0.227429i
\(59\) −0.707107 − 1.22474i −0.0920575 − 0.159448i 0.816319 − 0.577601i \(-0.196011\pi\)
−0.908377 + 0.418153i \(0.862678\pi\)
\(60\) −3.00000 − 5.19615i −0.387298 − 0.670820i
\(61\) 1.41421 − 2.44949i 0.181071 − 0.313625i −0.761174 − 0.648547i \(-0.775377\pi\)
0.942246 + 0.334922i \(0.108710\pi\)
\(62\) −1.41421 −0.179605
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0.707107 + 1.22474i 0.0870388 + 0.150756i
\(67\) −1.00000 − 1.73205i −0.122169 − 0.211604i 0.798454 − 0.602056i \(-0.205652\pi\)
−0.920623 + 0.390453i \(0.872318\pi\)
\(68\) 2.82843 − 4.89898i 0.342997 − 0.594089i
\(69\) 8.48528 1.02151
\(70\) 0 0
\(71\) −2.00000 −0.237356 −0.118678 − 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) 0.500000 − 0.866025i 0.0589256 − 0.102062i
\(73\) −4.24264 − 7.34847i −0.496564 − 0.860073i 0.503429 − 0.864037i \(-0.332072\pi\)
−0.999992 + 0.00396356i \(0.998738\pi\)
\(74\) 5.00000 + 8.66025i 0.581238 + 1.00673i
\(75\) −9.19239 + 15.9217i −1.06145 + 1.83848i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −8.00000 + 13.8564i −0.900070 + 1.55897i −0.0726692 + 0.997356i \(0.523152\pi\)
−0.827401 + 0.561611i \(0.810182\pi\)
\(80\) −2.12132 − 3.67423i −0.237171 − 0.410792i
\(81\) 2.50000 + 4.33013i 0.277778 + 0.481125i
\(82\) 5.65685 − 9.79796i 0.624695 − 1.08200i
\(83\) −16.9706 −1.86276 −0.931381 − 0.364047i \(-0.881395\pi\)
−0.931381 + 0.364047i \(0.881395\pi\)
\(84\) 0 0
\(85\) −24.0000 −2.60317
\(86\) 4.00000 − 6.92820i 0.431331 − 0.747087i
\(87\) −1.41421 − 2.44949i −0.151620 − 0.262613i
\(88\) 0.500000 + 0.866025i 0.0533002 + 0.0923186i
\(89\) 3.53553 − 6.12372i 0.374766 − 0.649113i −0.615526 − 0.788116i \(-0.711056\pi\)
0.990292 + 0.139003i \(0.0443898\pi\)
\(90\) −4.24264 −0.447214
\(91\) 0 0
\(92\) 6.00000 0.625543
\(93\) 1.00000 − 1.73205i 0.103695 − 0.179605i
\(94\) −2.12132 − 3.67423i −0.218797 − 0.378968i
\(95\) 0 0
\(96\) −0.707107 + 1.22474i −0.0721688 + 0.125000i
\(97\) −9.89949 −1.00514 −0.502571 − 0.864536i \(-0.667612\pi\)
−0.502571 + 0.864536i \(0.667612\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
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.p.177.1 4
7.2 even 3 1078.2.a.u.1.2 yes 2
7.3 odd 6 inner 1078.2.e.p.67.2 4
7.4 even 3 inner 1078.2.e.p.67.1 4
7.5 odd 6 1078.2.a.u.1.1 ✓ 2
7.6 odd 2 inner 1078.2.e.p.177.2 4
21.2 odd 6 9702.2.a.cp.1.1 2
21.5 even 6 9702.2.a.cp.1.2 2
28.19 even 6 8624.2.a.bs.1.2 2
28.23 odd 6 8624.2.a.bs.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1078.2.a.u.1.1 ✓ 2 7.5 odd 6
1078.2.a.u.1.2 yes 2 7.2 even 3
1078.2.e.p.67.1 4 7.4 even 3 inner
1078.2.e.p.67.2 4 7.3 odd 6 inner
1078.2.e.p.177.1 4 1.1 even 1 trivial
1078.2.e.p.177.2 4 7.6 odd 2 inner
8624.2.a.bs.1.1 2 28.23 odd 6
8624.2.a.bs.1.2 2 28.19 even 6
9702.2.a.cp.1.1 2 21.2 odd 6
9702.2.a.cp.1.2 2 21.5 even 6