Properties

Label 1078.2.a.q.1.1
Level $1078$
Weight $2$
Character 1078.1
Self dual yes
Analytic conductor $8.608$
Analytic rank $1$
Dimension $2$
Inner twists $2$

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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(1,1078)] 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("1078.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1078, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1078 = 2 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1078.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2,0,2,0,0,0,-2,-2,0,-2,0,0,0,-4,2,0,2,0,0,0,2,-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: yes
Analytic conductor: \(8.60787333789\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 1078.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.41421 q^{3} +1.00000 q^{4} +1.41421 q^{5} +1.41421 q^{6} -1.00000 q^{8} -1.00000 q^{9} -1.41421 q^{10} -1.00000 q^{11} -1.41421 q^{12} +2.82843 q^{13} -2.00000 q^{15} +1.00000 q^{16} -5.65685 q^{17} +1.00000 q^{18} +2.82843 q^{19} +1.41421 q^{20} +1.00000 q^{22} -2.00000 q^{23} +1.41421 q^{24} -3.00000 q^{25} -2.82843 q^{26} +5.65685 q^{27} -6.00000 q^{29} +2.00000 q^{30} -1.41421 q^{31} -1.00000 q^{32} +1.41421 q^{33} +5.65685 q^{34} -1.00000 q^{36} -2.00000 q^{37} -2.82843 q^{38} -4.00000 q^{39} -1.41421 q^{40} +5.65685 q^{41} +4.00000 q^{43} -1.00000 q^{44} -1.41421 q^{45} +2.00000 q^{46} +4.24264 q^{47} -1.41421 q^{48} +3.00000 q^{50} +8.00000 q^{51} +2.82843 q^{52} -12.0000 q^{53} -5.65685 q^{54} -1.41421 q^{55} -4.00000 q^{57} +6.00000 q^{58} +4.24264 q^{59} -2.00000 q^{60} -5.65685 q^{61} +1.41421 q^{62} +1.00000 q^{64} +4.00000 q^{65} -1.41421 q^{66} -2.00000 q^{67} -5.65685 q^{68} +2.82843 q^{69} -10.0000 q^{71} +1.00000 q^{72} -14.1421 q^{73} +2.00000 q^{74} +4.24264 q^{75} +2.82843 q^{76} +4.00000 q^{78} -8.00000 q^{79} +1.41421 q^{80} -5.00000 q^{81} -5.65685 q^{82} -2.82843 q^{83} -8.00000 q^{85} -4.00000 q^{86} +8.48528 q^{87} +1.00000 q^{88} -7.07107 q^{89} +1.41421 q^{90} -2.00000 q^{92} +2.00000 q^{93} -4.24264 q^{94} +4.00000 q^{95} +1.41421 q^{96} -15.5563 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{9} - 2 q^{11} - 4 q^{15} + 2 q^{16} + 2 q^{18} + 2 q^{22} - 4 q^{23} - 6 q^{25} - 12 q^{29} + 4 q^{30} - 2 q^{32} - 2 q^{36} - 4 q^{37} - 8 q^{39} + 8 q^{43} - 2 q^{44}+ \cdots + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.41421 −0.816497 −0.408248 − 0.912871i \(-0.633860\pi\)
−0.408248 + 0.912871i \(0.633860\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.41421 0.632456 0.316228 − 0.948683i \(-0.397584\pi\)
0.316228 + 0.948683i \(0.397584\pi\)
\(6\) 1.41421 0.577350
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) −1.00000 −0.333333
\(10\) −1.41421 −0.447214
\(11\) −1.00000 −0.301511
\(12\) −1.41421 −0.408248
\(13\) 2.82843 0.784465 0.392232 − 0.919866i \(-0.371703\pi\)
0.392232 + 0.919866i \(0.371703\pi\)
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) 1.00000 0.250000
\(17\) −5.65685 −1.37199 −0.685994 − 0.727607i \(-0.740633\pi\)
−0.685994 + 0.727607i \(0.740633\pi\)
\(18\) 1.00000 0.235702
\(19\) 2.82843 0.648886 0.324443 − 0.945905i \(-0.394823\pi\)
0.324443 + 0.945905i \(0.394823\pi\)
\(20\) 1.41421 0.316228
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) −2.00000 −0.417029 −0.208514 − 0.978019i \(-0.566863\pi\)
−0.208514 + 0.978019i \(0.566863\pi\)
\(24\) 1.41421 0.288675
\(25\) −3.00000 −0.600000
\(26\) −2.82843 −0.554700
\(27\) 5.65685 1.08866
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 2.00000 0.365148
\(31\) −1.41421 −0.254000 −0.127000 − 0.991903i \(-0.540535\pi\)
−0.127000 + 0.991903i \(0.540535\pi\)
\(32\) −1.00000 −0.176777
\(33\) 1.41421 0.246183
\(34\) 5.65685 0.970143
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) −2.82843 −0.458831
\(39\) −4.00000 −0.640513
\(40\) −1.41421 −0.223607
\(41\) 5.65685 0.883452 0.441726 − 0.897150i \(-0.354366\pi\)
0.441726 + 0.897150i \(0.354366\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) −1.00000 −0.150756
\(45\) −1.41421 −0.210819
\(46\) 2.00000 0.294884
\(47\) 4.24264 0.618853 0.309426 − 0.950923i \(-0.399863\pi\)
0.309426 + 0.950923i \(0.399863\pi\)
\(48\) −1.41421 −0.204124
\(49\) 0 0
\(50\) 3.00000 0.424264
\(51\) 8.00000 1.12022
\(52\) 2.82843 0.392232
\(53\) −12.0000 −1.64833 −0.824163 − 0.566352i \(-0.808354\pi\)
−0.824163 + 0.566352i \(0.808354\pi\)
\(54\) −5.65685 −0.769800
\(55\) −1.41421 −0.190693
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 6.00000 0.787839
\(59\) 4.24264 0.552345 0.276172 − 0.961108i \(-0.410934\pi\)
0.276172 + 0.961108i \(0.410934\pi\)
\(60\) −2.00000 −0.258199
\(61\) −5.65685 −0.724286 −0.362143 − 0.932123i \(-0.617955\pi\)
−0.362143 + 0.932123i \(0.617955\pi\)
\(62\) 1.41421 0.179605
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 4.00000 0.496139
\(66\) −1.41421 −0.174078
\(67\) −2.00000 −0.244339 −0.122169 − 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) −5.65685 −0.685994
\(69\) 2.82843 0.340503
\(70\) 0 0
\(71\) −10.0000 −1.18678 −0.593391 − 0.804914i \(-0.702211\pi\)
−0.593391 + 0.804914i \(0.702211\pi\)
\(72\) 1.00000 0.117851
\(73\) −14.1421 −1.65521 −0.827606 − 0.561310i \(-0.810298\pi\)
−0.827606 + 0.561310i \(0.810298\pi\)
\(74\) 2.00000 0.232495
\(75\) 4.24264 0.489898
\(76\) 2.82843 0.324443
\(77\) 0 0
\(78\) 4.00000 0.452911
\(79\) −8.00000 −0.900070 −0.450035 − 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 1.41421 0.158114
\(81\) −5.00000 −0.555556
\(82\) −5.65685 −0.624695
\(83\) −2.82843 −0.310460 −0.155230 − 0.987878i \(-0.549612\pi\)
−0.155230 + 0.987878i \(0.549612\pi\)
\(84\) 0 0
\(85\) −8.00000 −0.867722
\(86\) −4.00000 −0.431331
\(87\) 8.48528 0.909718
\(88\) 1.00000 0.106600
\(89\) −7.07107 −0.749532 −0.374766 − 0.927119i \(-0.622277\pi\)
−0.374766 + 0.927119i \(0.622277\pi\)
\(90\) 1.41421 0.149071
\(91\) 0 0
\(92\) −2.00000 −0.208514
\(93\) 2.00000 0.207390
\(94\) −4.24264 −0.437595
\(95\) 4.00000 0.410391
\(96\) 1.41421 0.144338
\(97\) −15.5563 −1.57951 −0.789754 − 0.613424i \(-0.789792\pi\)
−0.789754 + 0.613424i \(0.789792\pi\)
\(98\) 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.a.q.1.1 ✓ 2
3.2 odd 2 9702.2.a.dq.1.1 2
4.3 odd 2 8624.2.a.bw.1.2 2
7.2 even 3 1078.2.e.s.67.2 4
7.3 odd 6 1078.2.e.s.177.1 4
7.4 even 3 1078.2.e.s.177.2 4
7.5 odd 6 1078.2.e.s.67.1 4
7.6 odd 2 inner 1078.2.a.q.1.2 yes 2
21.20 even 2 9702.2.a.dq.1.2 2
28.27 even 2 8624.2.a.bw.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1078.2.a.q.1.1 ✓ 2 1.1 even 1 trivial
1078.2.a.q.1.2 yes 2 7.6 odd 2 inner
1078.2.e.s.67.1 4 7.5 odd 6
1078.2.e.s.67.2 4 7.2 even 3
1078.2.e.s.177.1 4 7.3 odd 6
1078.2.e.s.177.2 4 7.4 even 3
8624.2.a.bw.1.1 2 28.27 even 2
8624.2.a.bw.1.2 2 4.3 odd 2
9702.2.a.dq.1.1 2 3.2 odd 2
9702.2.a.dq.1.2 2 21.20 even 2