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Properties

Label 78.2.e.a.61.1
Level $78$
Weight $2$
Character 78.61
Analytic conductor $0.623$
Analytic rank $0$
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] = [78,2,Mod(55,78)] 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("78.55"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(78, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 78 = 2 \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 78.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,1,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.622833135766\)
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 61.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 78.61
Dual form 78.2.e.a.55.1

$q$-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} +(-0.500000 + 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{4} +3.00000 q^{5} +(0.500000 + 0.866025i) q^{6} +(-1.00000 - 1.73205i) q^{7} -1.00000 q^{8} +(-0.500000 - 0.866025i) q^{9} +(1.50000 - 2.59808i) q^{10} +(-3.00000 + 5.19615i) q^{11} +1.00000 q^{12} +(-3.50000 - 0.866025i) q^{13} -2.00000 q^{14} +(-1.50000 + 2.59808i) q^{15} +(-0.500000 + 0.866025i) q^{16} +(1.50000 + 2.59808i) q^{17} -1.00000 q^{18} +(-1.00000 - 1.73205i) q^{19} +(-1.50000 - 2.59808i) q^{20} +2.00000 q^{21} +(3.00000 + 5.19615i) q^{22} +(3.00000 - 5.19615i) q^{23} +(0.500000 - 0.866025i) q^{24} +4.00000 q^{25} +(-2.50000 + 2.59808i) q^{26} +1.00000 q^{27} +(-1.00000 + 1.73205i) q^{28} +(-1.50000 + 2.59808i) q^{29} +(1.50000 + 2.59808i) q^{30} -4.00000 q^{31} +(0.500000 + 0.866025i) q^{32} +(-3.00000 - 5.19615i) q^{33} +3.00000 q^{34} +(-3.00000 - 5.19615i) q^{35} +(-0.500000 + 0.866025i) q^{36} +(3.50000 - 6.06218i) q^{37} -2.00000 q^{38} +(2.50000 - 2.59808i) q^{39} -3.00000 q^{40} +(1.50000 - 2.59808i) q^{41} +(1.00000 - 1.73205i) q^{42} +(5.00000 + 8.66025i) q^{43} +6.00000 q^{44} +(-1.50000 - 2.59808i) q^{45} +(-3.00000 - 5.19615i) q^{46} +6.00000 q^{47} +(-0.500000 - 0.866025i) q^{48} +(1.50000 - 2.59808i) q^{49} +(2.00000 - 3.46410i) q^{50} -3.00000 q^{51} +(1.00000 + 3.46410i) q^{52} +3.00000 q^{53} +(0.500000 - 0.866025i) q^{54} +(-9.00000 + 15.5885i) q^{55} +(1.00000 + 1.73205i) q^{56} +2.00000 q^{57} +(1.50000 + 2.59808i) q^{58} +3.00000 q^{60} +(3.50000 + 6.06218i) q^{61} +(-2.00000 + 3.46410i) q^{62} +(-1.00000 + 1.73205i) q^{63} +1.00000 q^{64} +(-10.5000 - 2.59808i) q^{65} -6.00000 q^{66} +(5.00000 - 8.66025i) q^{67} +(1.50000 - 2.59808i) q^{68} +(3.00000 + 5.19615i) q^{69} -6.00000 q^{70} +(-3.00000 - 5.19615i) q^{71} +(0.500000 + 0.866025i) q^{72} -13.0000 q^{73} +(-3.50000 - 6.06218i) q^{74} +(-2.00000 + 3.46410i) q^{75} +(-1.00000 + 1.73205i) q^{76} +12.0000 q^{77} +(-1.00000 - 3.46410i) q^{78} -4.00000 q^{79} +(-1.50000 + 2.59808i) q^{80} +(-0.500000 + 0.866025i) q^{81} +(-1.50000 - 2.59808i) q^{82} -6.00000 q^{83} +(-1.00000 - 1.73205i) q^{84} +(4.50000 + 7.79423i) q^{85} +10.0000 q^{86} +(-1.50000 - 2.59808i) q^{87} +(3.00000 - 5.19615i) q^{88} +(-9.00000 + 15.5885i) q^{89} -3.00000 q^{90} +(2.00000 + 6.92820i) q^{91} -6.00000 q^{92} +(2.00000 - 3.46410i) q^{93} +(3.00000 - 5.19615i) q^{94} +(-3.00000 - 5.19615i) q^{95} -1.00000 q^{96} +(-7.00000 - 12.1244i) q^{97} +(-1.50000 - 2.59808i) q^{98} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{3} - q^{4} + 6 q^{5} + q^{6} - 2 q^{7} - 2 q^{8} - q^{9} + 3 q^{10} - 6 q^{11} + 2 q^{12} - 7 q^{13} - 4 q^{14} - 3 q^{15} - q^{16} + 3 q^{17} - 2 q^{18} - 2 q^{19} - 3 q^{20} + 4 q^{21}+ \cdots + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/78\mathbb{Z}\right)^\times\).

\(n\) \(53\) \(67\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(3\) −0.500000 + 0.866025i −0.288675 + 0.500000i
\(4\) −0.500000 − 0.866025i −0.250000 − 0.433013i
\(5\) 3.00000 1.34164 0.670820 − 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) 0.500000 + 0.866025i 0.204124 + 0.353553i
\(7\) −1.00000 − 1.73205i −0.377964 − 0.654654i 0.612801 − 0.790237i \(-0.290043\pi\)
−0.990766 + 0.135583i \(0.956709\pi\)
\(8\) −1.00000 −0.353553
\(9\) −0.500000 − 0.866025i −0.166667 − 0.288675i
\(10\) 1.50000 − 2.59808i 0.474342 − 0.821584i
\(11\) −3.00000 + 5.19615i −0.904534 + 1.56670i −0.0829925 + 0.996550i \(0.526448\pi\)
−0.821541 + 0.570149i \(0.806886\pi\)
\(12\) 1.00000 0.288675
\(13\) −3.50000 − 0.866025i −0.970725 − 0.240192i
\(14\) −2.00000 −0.534522
\(15\) −1.50000 + 2.59808i −0.387298 + 0.670820i
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) 1.50000 + 2.59808i 0.363803 + 0.630126i 0.988583 − 0.150675i \(-0.0481447\pi\)
−0.624780 + 0.780801i \(0.714811\pi\)
\(18\) −1.00000 −0.235702
\(19\) −1.00000 − 1.73205i −0.229416 − 0.397360i 0.728219 − 0.685344i \(-0.240348\pi\)
−0.957635 + 0.287984i \(0.907015\pi\)
\(20\) −1.50000 − 2.59808i −0.335410 − 0.580948i
\(21\) 2.00000 0.436436
\(22\) 3.00000 + 5.19615i 0.639602 + 1.10782i
\(23\) 3.00000 − 5.19615i 0.625543 − 1.08347i −0.362892 − 0.931831i \(-0.618211\pi\)
0.988436 − 0.151642i \(-0.0484560\pi\)
\(24\) 0.500000 − 0.866025i 0.102062 − 0.176777i
\(25\) 4.00000 0.800000
\(26\) −2.50000 + 2.59808i −0.490290 + 0.509525i
\(27\) 1.00000 0.192450
\(28\) −1.00000 + 1.73205i −0.188982 + 0.327327i
\(29\) −1.50000 + 2.59808i −0.278543 + 0.482451i −0.971023 − 0.238987i \(-0.923185\pi\)
0.692480 + 0.721437i \(0.256518\pi\)
\(30\) 1.50000 + 2.59808i 0.273861 + 0.474342i
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0.500000 + 0.866025i 0.0883883 + 0.153093i
\(33\) −3.00000 − 5.19615i −0.522233 − 0.904534i
\(34\) 3.00000 0.514496
\(35\) −3.00000 − 5.19615i −0.507093 − 0.878310i
\(36\) −0.500000 + 0.866025i −0.0833333 + 0.144338i
\(37\) 3.50000 − 6.06218i 0.575396 − 0.996616i −0.420602 − 0.907245i \(-0.638181\pi\)
0.995998 − 0.0893706i \(-0.0284856\pi\)
\(38\) −2.00000 −0.324443
\(39\) 2.50000 − 2.59808i 0.400320 − 0.416025i
\(40\) −3.00000 −0.474342
\(41\) 1.50000 − 2.59808i 0.234261 − 0.405751i −0.724797 − 0.688963i \(-0.758066\pi\)
0.959058 + 0.283211i \(0.0913998\pi\)
\(42\) 1.00000 − 1.73205i 0.154303 − 0.267261i
\(43\) 5.00000 + 8.66025i 0.762493 + 1.32068i 0.941562 + 0.336840i \(0.109358\pi\)
−0.179069 + 0.983836i \(0.557309\pi\)
\(44\) 6.00000 0.904534
\(45\) −1.50000 − 2.59808i −0.223607 − 0.387298i
\(46\) −3.00000 − 5.19615i −0.442326 − 0.766131i
\(47\) 6.00000 0.875190 0.437595 − 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) −0.500000 − 0.866025i −0.0721688 − 0.125000i
\(49\) 1.50000 − 2.59808i 0.214286 − 0.371154i
\(50\) 2.00000 − 3.46410i 0.282843 − 0.489898i
\(51\) −3.00000 −0.420084
\(52\) 1.00000 + 3.46410i 0.138675 + 0.480384i
\(53\) 3.00000 0.412082 0.206041 − 0.978543i \(-0.433942\pi\)
0.206041 + 0.978543i \(0.433942\pi\)
\(54\) 0.500000 − 0.866025i 0.0680414 − 0.117851i
\(55\) −9.00000 + 15.5885i −1.21356 + 2.10195i
\(56\) 1.00000 + 1.73205i 0.133631 + 0.231455i
\(57\) 2.00000 0.264906
\(58\) 1.50000 + 2.59808i 0.196960 + 0.341144i
\(59\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(60\) 3.00000 0.387298
\(61\) 3.50000 + 6.06218i 0.448129 + 0.776182i 0.998264 − 0.0588933i \(-0.0187572\pi\)
−0.550135 + 0.835076i \(0.685424\pi\)
\(62\) −2.00000 + 3.46410i −0.254000 + 0.439941i
\(63\) −1.00000 + 1.73205i −0.125988 + 0.218218i
\(64\) 1.00000 0.125000
\(65\) −10.5000 − 2.59808i −1.30236 − 0.322252i
\(66\) −6.00000 −0.738549
\(67\) 5.00000 − 8.66025i 0.610847 − 1.05802i −0.380251 − 0.924883i \(-0.624162\pi\)
0.991098 − 0.133135i \(-0.0425044\pi\)
\(68\) 1.50000 − 2.59808i 0.181902 − 0.315063i
\(69\) 3.00000 + 5.19615i 0.361158 + 0.625543i
\(70\) −6.00000 −0.717137
\(71\) −3.00000 − 5.19615i −0.356034 − 0.616670i 0.631260 − 0.775571i \(-0.282538\pi\)
−0.987294 + 0.158901i \(0.949205\pi\)
\(72\) 0.500000 + 0.866025i 0.0589256 + 0.102062i
\(73\) −13.0000 −1.52153 −0.760767 − 0.649025i \(-0.775177\pi\)
−0.760767 + 0.649025i \(0.775177\pi\)
\(74\) −3.50000 − 6.06218i −0.406867 − 0.704714i
\(75\) −2.00000 + 3.46410i −0.230940 + 0.400000i
\(76\) −1.00000 + 1.73205i −0.114708 + 0.198680i
\(77\) 12.0000 1.36753
\(78\) −1.00000 − 3.46410i −0.113228 − 0.392232i
\(79\) −4.00000 −0.450035 −0.225018 − 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) −1.50000 + 2.59808i −0.167705 + 0.290474i
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) −1.50000 − 2.59808i −0.165647 − 0.286910i
\(83\) −6.00000 −0.658586 −0.329293 − 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) −1.00000 − 1.73205i −0.109109 − 0.188982i
\(85\) 4.50000 + 7.79423i 0.488094 + 0.845403i
\(86\) 10.0000 1.07833
\(87\) −1.50000 − 2.59808i −0.160817 − 0.278543i
\(88\) 3.00000 − 5.19615i 0.319801 − 0.553912i
\(89\) −9.00000 + 15.5885i −0.953998 + 1.65237i −0.217354 + 0.976093i \(0.569742\pi\)
−0.736644 + 0.676280i \(0.763591\pi\)
\(90\) −3.00000 −0.316228
\(91\) 2.00000 + 6.92820i 0.209657 + 0.726273i
\(92\) −6.00000 −0.625543
\(93\) 2.00000 − 3.46410i 0.207390 − 0.359211i
\(94\) 3.00000 − 5.19615i 0.309426 − 0.535942i
\(95\) −3.00000 − 5.19615i −0.307794 − 0.533114i
\(96\) −1.00000 −0.102062
\(97\) −7.00000 − 12.1244i −0.710742 − 1.23104i −0.964579 − 0.263795i \(-0.915026\pi\)
0.253837 − 0.967247i \(-0.418307\pi\)
\(98\) −1.50000 − 2.59808i −0.151523 − 0.262445i
\(99\) 6.00000 0.603023
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 78.2.e.a.61.1 yes 2
3.2 odd 2 234.2.h.a.217.1 2
4.3 odd 2 624.2.q.g.529.1 2
5.2 odd 4 1950.2.z.g.1699.2 4
5.3 odd 4 1950.2.z.g.1699.1 4
5.4 even 2 1950.2.i.m.451.1 2
12.11 even 2 1872.2.t.c.1153.1 2
13.2 odd 12 1014.2.i.b.361.2 4
13.3 even 3 inner 78.2.e.a.55.1 ✓ 2
13.4 even 6 1014.2.a.f.1.1 1
13.5 odd 4 1014.2.i.b.823.1 4
13.6 odd 12 1014.2.b.c.337.2 2
13.7 odd 12 1014.2.b.c.337.1 2
13.8 odd 4 1014.2.i.b.823.2 4
13.9 even 3 1014.2.a.c.1.1 1
13.10 even 6 1014.2.e.a.991.1 2
13.11 odd 12 1014.2.i.b.361.1 4
13.12 even 2 1014.2.e.a.529.1 2
39.17 odd 6 3042.2.a.h.1.1 1
39.20 even 12 3042.2.b.h.1351.2 2
39.29 odd 6 234.2.h.a.55.1 2
39.32 even 12 3042.2.b.h.1351.1 2
39.35 odd 6 3042.2.a.i.1.1 1
52.3 odd 6 624.2.q.g.289.1 2
52.35 odd 6 8112.2.a.m.1.1 1
52.43 odd 6 8112.2.a.c.1.1 1
65.3 odd 12 1950.2.z.g.1849.2 4
65.29 even 6 1950.2.i.m.601.1 2
65.42 odd 12 1950.2.z.g.1849.1 4
156.107 even 6 1872.2.t.c.289.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.2.e.a.55.1 ✓ 2 13.3 even 3 inner
78.2.e.a.61.1 yes 2 1.1 even 1 trivial
234.2.h.a.55.1 2 39.29 odd 6
234.2.h.a.217.1 2 3.2 odd 2
624.2.q.g.289.1 2 52.3 odd 6
624.2.q.g.529.1 2 4.3 odd 2
1014.2.a.c.1.1 1 13.9 even 3
1014.2.a.f.1.1 1 13.4 even 6
1014.2.b.c.337.1 2 13.7 odd 12
1014.2.b.c.337.2 2 13.6 odd 12
1014.2.e.a.529.1 2 13.12 even 2
1014.2.e.a.991.1 2 13.10 even 6
1014.2.i.b.361.1 4 13.11 odd 12
1014.2.i.b.361.2 4 13.2 odd 12
1014.2.i.b.823.1 4 13.5 odd 4
1014.2.i.b.823.2 4 13.8 odd 4
1872.2.t.c.289.1 2 156.107 even 6
1872.2.t.c.1153.1 2 12.11 even 2
1950.2.i.m.451.1 2 5.4 even 2
1950.2.i.m.601.1 2 65.29 even 6
1950.2.z.g.1699.1 4 5.3 odd 4
1950.2.z.g.1699.2 4 5.2 odd 4
1950.2.z.g.1849.1 4 65.42 odd 12
1950.2.z.g.1849.2 4 65.3 odd 12
3042.2.a.h.1.1 1 39.17 odd 6
3042.2.a.i.1.1 1 39.35 odd 6
3042.2.b.h.1351.1 2 39.32 even 12
3042.2.b.h.1351.2 2 39.20 even 12
8112.2.a.c.1.1 1 52.43 odd 6
8112.2.a.m.1.1 1 52.35 odd 6