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 = [2,-1,2,-1,2,-4,0,2,-1,2,-1,2,4,0,8,-1,0,-1,2,-4,0,2,0] 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: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 177.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1078.177
Dual form 1078.2.e.e.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)
 
\(f(q)\) \(=\) \(q+(-0.500000 + 0.866025i) q^{2} +(1.00000 + 1.73205i) q^{3} +(-0.500000 - 0.866025i) q^{4} +(1.00000 - 1.73205i) q^{5} -2.00000 q^{6} +1.00000 q^{8} +(-0.500000 + 0.866025i) q^{9} +(1.00000 + 1.73205i) q^{10} +(-0.500000 - 0.866025i) q^{11} +(1.00000 - 1.73205i) q^{12} +2.00000 q^{13} +4.00000 q^{15} +(-0.500000 + 0.866025i) q^{16} +(-0.500000 - 0.866025i) q^{18} +(1.00000 - 1.73205i) q^{19} -2.00000 q^{20} +1.00000 q^{22} +(1.00000 + 1.73205i) q^{24} +(0.500000 + 0.866025i) q^{25} +(-1.00000 + 1.73205i) q^{26} +4.00000 q^{27} +6.00000 q^{29} +(-2.00000 + 3.46410i) q^{30} +(-2.00000 - 3.46410i) q^{31} +(-0.500000 - 0.866025i) q^{32} +(1.00000 - 1.73205i) q^{33} +1.00000 q^{36} +(-1.00000 + 1.73205i) q^{37} +(1.00000 + 1.73205i) q^{38} +(2.00000 + 3.46410i) q^{39} +(1.00000 - 1.73205i) q^{40} +8.00000 q^{41} +12.0000 q^{43} +(-0.500000 + 0.866025i) q^{44} +(1.00000 + 1.73205i) q^{45} +(-6.00000 + 10.3923i) q^{47} -2.00000 q^{48} -1.00000 q^{50} +(-1.00000 - 1.73205i) q^{52} +(1.00000 + 1.73205i) q^{53} +(-2.00000 + 3.46410i) q^{54} -2.00000 q^{55} +4.00000 q^{57} +(-3.00000 + 5.19615i) q^{58} +(-5.00000 - 8.66025i) q^{59} +(-2.00000 - 3.46410i) q^{60} +(5.00000 - 8.66025i) q^{61} +4.00000 q^{62} +1.00000 q^{64} +(2.00000 - 3.46410i) q^{65} +(1.00000 + 1.73205i) q^{66} +(6.00000 + 10.3923i) q^{67} +4.00000 q^{71} +(-0.500000 + 0.866025i) q^{72} +(-6.00000 - 10.3923i) q^{73} +(-1.00000 - 1.73205i) q^{74} +(-1.00000 + 1.73205i) q^{75} -2.00000 q^{76} -4.00000 q^{78} +(1.00000 + 1.73205i) q^{80} +(5.50000 + 9.52628i) q^{81} +(-4.00000 + 6.92820i) q^{82} -18.0000 q^{83} +(-6.00000 + 10.3923i) q^{86} +(6.00000 + 10.3923i) q^{87} +(-0.500000 - 0.866025i) q^{88} -2.00000 q^{90} +(4.00000 - 6.92820i) q^{93} +(-6.00000 - 10.3923i) q^{94} +(-2.00000 - 3.46410i) q^{95} +(1.00000 - 1.73205i) q^{96} -12.0000 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 2 q^{3} - q^{4} + 2 q^{5} - 4 q^{6} + 2 q^{8} - q^{9} + 2 q^{10} - q^{11} + 2 q^{12} + 4 q^{13} + 8 q^{15} - q^{16} - q^{18} + 2 q^{19} - 4 q^{20} + 2 q^{22} + 2 q^{24} + q^{25} - 2 q^{26}+ \cdots + 2 q^{99}+O(q^{100}) \) 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
\(3\) 1.00000 + 1.73205i 0.577350 + 1.00000i 0.995782 + 0.0917517i \(0.0292466\pi\)
−0.418432 + 0.908248i \(0.637420\pi\)
\(4\) −0.500000 − 0.866025i −0.250000 − 0.433013i
\(5\) 1.00000 − 1.73205i 0.447214 − 0.774597i −0.550990 − 0.834512i \(-0.685750\pi\)
0.998203 + 0.0599153i \(0.0190830\pi\)
\(6\) −2.00000 −0.816497
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 1.00000 + 1.73205i 0.316228 + 0.547723i
\(11\) −0.500000 − 0.866025i −0.150756 − 0.261116i
\(12\) 1.00000 − 1.73205i 0.288675 − 0.500000i
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 4.00000 1.03280
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(18\) −0.500000 − 0.866025i −0.117851 − 0.204124i
\(19\) 1.00000 − 1.73205i 0.229416 − 0.397360i −0.728219 − 0.685344i \(-0.759652\pi\)
0.957635 + 0.287984i \(0.0929851\pi\)
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(24\) 1.00000 + 1.73205i 0.204124 + 0.353553i
\(25\) 0.500000 + 0.866025i 0.100000 + 0.173205i
\(26\) −1.00000 + 1.73205i −0.196116 + 0.339683i
\(27\) 4.00000 0.769800
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) −2.00000 + 3.46410i −0.365148 + 0.632456i
\(31\) −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
\(33\) 1.00000 − 1.73205i 0.174078 − 0.301511i
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −1.00000 + 1.73205i −0.164399 + 0.284747i −0.936442 − 0.350823i \(-0.885902\pi\)
0.772043 + 0.635571i \(0.219235\pi\)
\(38\) 1.00000 + 1.73205i 0.162221 + 0.280976i
\(39\) 2.00000 + 3.46410i 0.320256 + 0.554700i
\(40\) 1.00000 − 1.73205i 0.158114 − 0.273861i
\(41\) 8.00000 1.24939 0.624695 − 0.780869i \(-0.285223\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) 0 0
\(43\) 12.0000 1.82998 0.914991 − 0.403473i \(-0.132197\pi\)
0.914991 + 0.403473i \(0.132197\pi\)
\(44\) −0.500000 + 0.866025i −0.0753778 + 0.130558i
\(45\) 1.00000 + 1.73205i 0.149071 + 0.258199i
\(46\) 0 0
\(47\) −6.00000 + 10.3923i −0.875190 + 1.51587i −0.0186297 + 0.999826i \(0.505930\pi\)
−0.856560 + 0.516047i \(0.827403\pi\)
\(48\) −2.00000 −0.288675
\(49\) 0 0
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −1.00000 − 1.73205i −0.138675 − 0.240192i
\(53\) 1.00000 + 1.73205i 0.137361 + 0.237915i 0.926497 − 0.376303i \(-0.122805\pi\)
−0.789136 + 0.614218i \(0.789471\pi\)
\(54\) −2.00000 + 3.46410i −0.272166 + 0.471405i
\(55\) −2.00000 −0.269680
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) −3.00000 + 5.19615i −0.393919 + 0.682288i
\(59\) −5.00000 − 8.66025i −0.650945 − 1.12747i −0.982894 − 0.184172i \(-0.941040\pi\)
0.331949 − 0.943297i \(-0.392294\pi\)
\(60\) −2.00000 − 3.46410i −0.258199 − 0.447214i
\(61\) 5.00000 − 8.66025i 0.640184 − 1.10883i −0.345207 − 0.938527i \(-0.612191\pi\)
0.985391 − 0.170305i \(-0.0544754\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 2.00000 − 3.46410i 0.248069 − 0.429669i
\(66\) 1.00000 + 1.73205i 0.123091 + 0.213201i
\(67\) 6.00000 + 10.3923i 0.733017 + 1.26962i 0.955588 + 0.294706i \(0.0952216\pi\)
−0.222571 + 0.974916i \(0.571445\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 4.00000 0.474713 0.237356 − 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) −0.500000 + 0.866025i −0.0589256 + 0.102062i
\(73\) −6.00000 − 10.3923i −0.702247 − 1.21633i −0.967676 − 0.252197i \(-0.918847\pi\)
0.265429 − 0.964130i \(-0.414486\pi\)
\(74\) −1.00000 − 1.73205i −0.116248 − 0.201347i
\(75\) −1.00000 + 1.73205i −0.115470 + 0.200000i
\(76\) −2.00000 −0.229416
\(77\) 0 0
\(78\) −4.00000 −0.452911
\(79\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(80\) 1.00000 + 1.73205i 0.111803 + 0.193649i
\(81\) 5.50000 + 9.52628i 0.611111 + 1.05848i
\(82\) −4.00000 + 6.92820i −0.441726 + 0.765092i
\(83\) −18.0000 −1.97576 −0.987878 − 0.155230i \(-0.950388\pi\)
−0.987878 + 0.155230i \(0.950388\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −6.00000 + 10.3923i −0.646997 + 1.12063i
\(87\) 6.00000 + 10.3923i 0.643268 + 1.11417i
\(88\) −0.500000 − 0.866025i −0.0533002 − 0.0923186i
\(89\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(90\) −2.00000 −0.210819
\(91\) 0 0
\(92\) 0 0
\(93\) 4.00000 − 6.92820i 0.414781 − 0.718421i
\(94\) −6.00000 − 10.3923i −0.618853 − 1.07188i
\(95\) −2.00000 − 3.46410i −0.205196 − 0.355409i
\(96\) 1.00000 − 1.73205i 0.102062 − 0.176777i
\(97\) −12.0000 −1.21842 −0.609208 − 0.793011i \(-0.708512\pi\)
−0.609208 + 0.793011i \(0.708512\pi\)
\(98\) 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.e.177.1 2
7.2 even 3 1078.2.a.h.1.1 ✓ 1
7.3 odd 6 1078.2.e.a.67.1 2
7.4 even 3 inner 1078.2.e.e.67.1 2
7.5 odd 6 1078.2.a.l.1.1 yes 1
7.6 odd 2 1078.2.e.a.177.1 2
21.2 odd 6 9702.2.a.t.1.1 1
21.5 even 6 9702.2.a.e.1.1 1
28.19 even 6 8624.2.a.g.1.1 1
28.23 odd 6 8624.2.a.y.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1078.2.a.h.1.1 ✓ 1 7.2 even 3
1078.2.a.l.1.1 yes 1 7.5 odd 6
1078.2.e.a.67.1 2 7.3 odd 6
1078.2.e.a.177.1 2 7.6 odd 2
1078.2.e.e.67.1 2 7.4 even 3 inner
1078.2.e.e.177.1 2 1.1 even 1 trivial
8624.2.a.g.1.1 1 28.19 even 6
8624.2.a.y.1.1 1 28.23 odd 6
9702.2.a.e.1.1 1 21.5 even 6
9702.2.a.t.1.1 1 21.2 odd 6