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Properties

Label 78.2.i.a.49.1
Level $78$
Weight $2$
Character 78.49
Analytic conductor $0.623$
Analytic rank $0$
Dimension $4$
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(43,78)] 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("78.43"); 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, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 78 = 2 \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 78.i (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-2] 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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 49.1
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 78.49
Dual form 78.2.i.a.43.1

$q$-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+(-0.866025 + 0.500000i) q^{2} +(-0.500000 - 0.866025i) q^{3} +(0.500000 - 0.866025i) q^{4} -3.73205i q^{5} +(0.866025 + 0.500000i) q^{6} +(2.36603 + 1.36603i) q^{7} +1.00000i q^{8} +(-0.500000 + 0.866025i) q^{9} +(1.86603 + 3.23205i) q^{10} +(1.09808 - 0.633975i) q^{11} -1.00000 q^{12} +(-2.59808 - 2.50000i) q^{13} -2.73205 q^{14} +(-3.23205 + 1.86603i) q^{15} +(-0.500000 - 0.866025i) q^{16} +(-2.86603 + 4.96410i) q^{17} -1.00000i q^{18} +(4.09808 + 2.36603i) q^{19} +(-3.23205 - 1.86603i) q^{20} -2.73205i q^{21} +(-0.633975 + 1.09808i) q^{22} +(2.09808 + 3.63397i) q^{23} +(0.866025 - 0.500000i) q^{24} -8.92820 q^{25} +(3.50000 + 0.866025i) q^{26} +1.00000 q^{27} +(2.36603 - 1.36603i) q^{28} +(2.23205 + 3.86603i) q^{29} +(1.86603 - 3.23205i) q^{30} -1.46410i q^{31} +(0.866025 + 0.500000i) q^{32} +(-1.09808 - 0.633975i) q^{33} -5.73205i q^{34} +(5.09808 - 8.83013i) q^{35} +(0.500000 + 0.866025i) q^{36} +(3.06218 - 1.76795i) q^{37} -4.73205 q^{38} +(-0.866025 + 3.50000i) q^{39} +3.73205 q^{40} +(8.13397 - 4.69615i) q^{41} +(1.36603 + 2.36603i) q^{42} +(-4.83013 + 8.36603i) q^{43} -1.26795i q^{44} +(3.23205 + 1.86603i) q^{45} +(-3.63397 - 2.09808i) q^{46} +2.19615i q^{47} +(-0.500000 + 0.866025i) q^{48} +(0.232051 + 0.401924i) q^{49} +(7.73205 - 4.46410i) q^{50} +5.73205 q^{51} +(-3.46410 + 1.00000i) q^{52} -6.46410 q^{53} +(-0.866025 + 0.500000i) q^{54} +(-2.36603 - 4.09808i) q^{55} +(-1.36603 + 2.36603i) q^{56} -4.73205i q^{57} +(-3.86603 - 2.23205i) q^{58} +(-6.92820 - 4.00000i) q^{59} +3.73205i q^{60} +(4.59808 - 7.96410i) q^{61} +(0.732051 + 1.26795i) q^{62} +(-2.36603 + 1.36603i) q^{63} -1.00000 q^{64} +(-9.33013 + 9.69615i) q^{65} +1.26795 q^{66} +(-11.3660 + 6.56218i) q^{67} +(2.86603 + 4.96410i) q^{68} +(2.09808 - 3.63397i) q^{69} +10.1962i q^{70} +(4.09808 + 2.36603i) q^{71} +(-0.866025 - 0.500000i) q^{72} -6.26795i q^{73} +(-1.76795 + 3.06218i) q^{74} +(4.46410 + 7.73205i) q^{75} +(4.09808 - 2.36603i) q^{76} +3.46410 q^{77} +(-1.00000 - 3.46410i) q^{78} -2.53590 q^{79} +(-3.23205 + 1.86603i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(-4.69615 + 8.13397i) q^{82} +0.196152i q^{83} +(-2.36603 - 1.36603i) q^{84} +(18.5263 + 10.6962i) q^{85} -9.66025i q^{86} +(2.23205 - 3.86603i) q^{87} +(0.633975 + 1.09808i) q^{88} +(-8.19615 + 4.73205i) q^{89} -3.73205 q^{90} +(-2.73205 - 9.46410i) q^{91} +4.19615 q^{92} +(-1.26795 + 0.732051i) q^{93} +(-1.09808 - 1.90192i) q^{94} +(8.83013 - 15.2942i) q^{95} -1.00000i q^{96} +(-5.19615 - 3.00000i) q^{97} +(-0.401924 - 0.232051i) q^{98} +1.26795i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} + 2 q^{4} + 6 q^{7} - 2 q^{9} + 4 q^{10} - 6 q^{11} - 4 q^{12} - 4 q^{14} - 6 q^{15} - 2 q^{16} - 8 q^{17} + 6 q^{19} - 6 q^{20} - 6 q^{22} - 2 q^{23} - 8 q^{25} + 14 q^{26} + 4 q^{27} + 6 q^{28}+ \cdots - 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(53\) \(67\)
\(\chi(n)\) \(1\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.866025 + 0.500000i −0.612372 + 0.353553i
\(3\) −0.500000 − 0.866025i −0.288675 − 0.500000i
\(4\) 0.500000 − 0.866025i 0.250000 − 0.433013i
\(5\) − 3.73205i − 1.66902i −0.550990 − 0.834512i \(-0.685750\pi\)
0.550990 − 0.834512i \(-0.314250\pi\)
\(6\) 0.866025 + 0.500000i 0.353553 + 0.204124i
\(7\) 2.36603 + 1.36603i 0.894274 + 0.516309i 0.875338 − 0.483512i \(-0.160639\pi\)
0.0189356 + 0.999821i \(0.493972\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 1.86603 + 3.23205i 0.590089 + 1.02206i
\(11\) 1.09808 − 0.633975i 0.331082 − 0.191151i −0.325239 − 0.945632i \(-0.605445\pi\)
0.656322 + 0.754481i \(0.272111\pi\)
\(12\) −1.00000 −0.288675
\(13\) −2.59808 − 2.50000i −0.720577 − 0.693375i
\(14\) −2.73205 −0.730171
\(15\) −3.23205 + 1.86603i −0.834512 + 0.481806i
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) −2.86603 + 4.96410i −0.695113 + 1.20397i 0.275029 + 0.961436i \(0.411312\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) 4.09808 + 2.36603i 0.940163 + 0.542803i 0.890011 − 0.455938i \(-0.150696\pi\)
0.0501517 + 0.998742i \(0.484030\pi\)
\(20\) −3.23205 − 1.86603i −0.722709 − 0.417256i
\(21\) − 2.73205i − 0.596182i
\(22\) −0.633975 + 1.09808i −0.135164 + 0.234111i
\(23\) 2.09808 + 3.63397i 0.437479 + 0.757736i 0.997494 − 0.0707462i \(-0.0225381\pi\)
−0.560015 + 0.828482i \(0.689205\pi\)
\(24\) 0.866025 − 0.500000i 0.176777 − 0.102062i
\(25\) −8.92820 −1.78564
\(26\) 3.50000 + 0.866025i 0.686406 + 0.169842i
\(27\) 1.00000 0.192450
\(28\) 2.36603 − 1.36603i 0.447137 − 0.258155i
\(29\) 2.23205 + 3.86603i 0.414481 + 0.717903i 0.995374 − 0.0960774i \(-0.0306296\pi\)
−0.580892 + 0.813980i \(0.697296\pi\)
\(30\) 1.86603 − 3.23205i 0.340688 − 0.590089i
\(31\) − 1.46410i − 0.262960i −0.991319 − 0.131480i \(-0.958027\pi\)
0.991319 − 0.131480i \(-0.0419730\pi\)
\(32\) 0.866025 + 0.500000i 0.153093 + 0.0883883i
\(33\) −1.09808 − 0.633975i −0.191151 − 0.110361i
\(34\) − 5.73205i − 0.983039i
\(35\) 5.09808 − 8.83013i 0.861732 − 1.49256i
\(36\) 0.500000 + 0.866025i 0.0833333 + 0.144338i
\(37\) 3.06218 − 1.76795i 0.503419 − 0.290649i −0.226705 − 0.973963i \(-0.572795\pi\)
0.730124 + 0.683314i \(0.239462\pi\)
\(38\) −4.73205 −0.767640
\(39\) −0.866025 + 3.50000i −0.138675 + 0.560449i
\(40\) 3.73205 0.590089
\(41\) 8.13397 − 4.69615i 1.27031 − 0.733416i 0.295267 − 0.955415i \(-0.404592\pi\)
0.975047 + 0.221999i \(0.0712582\pi\)
\(42\) 1.36603 + 2.36603i 0.210782 + 0.365086i
\(43\) −4.83013 + 8.36603i −0.736587 + 1.27581i 0.217436 + 0.976075i \(0.430231\pi\)
−0.954023 + 0.299732i \(0.903103\pi\)
\(44\) − 1.26795i − 0.191151i
\(45\) 3.23205 + 1.86603i 0.481806 + 0.278171i
\(46\) −3.63397 − 2.09808i −0.535800 − 0.309344i
\(47\) 2.19615i 0.320342i 0.987089 + 0.160171i \(0.0512045\pi\)
−0.987089 + 0.160171i \(0.948795\pi\)
\(48\) −0.500000 + 0.866025i −0.0721688 + 0.125000i
\(49\) 0.232051 + 0.401924i 0.0331501 + 0.0574177i
\(50\) 7.73205 − 4.46410i 1.09348 − 0.631319i
\(51\) 5.73205 0.802648
\(52\) −3.46410 + 1.00000i −0.480384 + 0.138675i
\(53\) −6.46410 −0.887913 −0.443956 − 0.896048i \(-0.646425\pi\)
−0.443956 + 0.896048i \(0.646425\pi\)
\(54\) −0.866025 + 0.500000i −0.117851 + 0.0680414i
\(55\) −2.36603 − 4.09808i −0.319035 − 0.552584i
\(56\) −1.36603 + 2.36603i −0.182543 + 0.316173i
\(57\) − 4.73205i − 0.626775i
\(58\) −3.86603 − 2.23205i −0.507634 − 0.293083i
\(59\) −6.92820 − 4.00000i −0.901975 − 0.520756i −0.0241347 − 0.999709i \(-0.507683\pi\)
−0.877841 + 0.478953i \(0.841016\pi\)
\(60\) 3.73205i 0.481806i
\(61\) 4.59808 − 7.96410i 0.588723 − 1.01970i −0.405677 − 0.914017i \(-0.632964\pi\)
0.994400 − 0.105682i \(-0.0337026\pi\)
\(62\) 0.732051 + 1.26795i 0.0929705 + 0.161030i
\(63\) −2.36603 + 1.36603i −0.298091 + 0.172103i
\(64\) −1.00000 −0.125000
\(65\) −9.33013 + 9.69615i −1.15726 + 1.20266i
\(66\) 1.26795 0.156074
\(67\) −11.3660 + 6.56218i −1.38858 + 0.801698i −0.993155 − 0.116800i \(-0.962736\pi\)
−0.395426 + 0.918498i \(0.629403\pi\)
\(68\) 2.86603 + 4.96410i 0.347557 + 0.601986i
\(69\) 2.09808 − 3.63397i 0.252579 − 0.437479i
\(70\) 10.1962i 1.21867i
\(71\) 4.09808 + 2.36603i 0.486352 + 0.280796i 0.723060 − 0.690785i \(-0.242735\pi\)
−0.236708 + 0.971581i \(0.576068\pi\)
\(72\) −0.866025 − 0.500000i −0.102062 − 0.0589256i
\(73\) − 6.26795i − 0.733608i −0.930298 − 0.366804i \(-0.880452\pi\)
0.930298 − 0.366804i \(-0.119548\pi\)
\(74\) −1.76795 + 3.06218i −0.205520 + 0.355971i
\(75\) 4.46410 + 7.73205i 0.515470 + 0.892820i
\(76\) 4.09808 − 2.36603i 0.470082 − 0.271402i
\(77\) 3.46410 0.394771
\(78\) −1.00000 − 3.46410i −0.113228 − 0.392232i
\(79\) −2.53590 −0.285311 −0.142655 − 0.989772i \(-0.545564\pi\)
−0.142655 + 0.989772i \(0.545564\pi\)
\(80\) −3.23205 + 1.86603i −0.361354 + 0.208628i
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) −4.69615 + 8.13397i −0.518603 + 0.898247i
\(83\) 0.196152i 0.0215305i 0.999942 + 0.0107653i \(0.00342676\pi\)
−0.999942 + 0.0107653i \(0.996573\pi\)
\(84\) −2.36603 − 1.36603i −0.258155 − 0.149046i
\(85\) 18.5263 + 10.6962i 2.00946 + 1.16016i
\(86\) − 9.66025i − 1.04169i
\(87\) 2.23205 − 3.86603i 0.239301 − 0.414481i
\(88\) 0.633975 + 1.09808i 0.0675819 + 0.117055i
\(89\) −8.19615 + 4.73205i −0.868790 + 0.501596i −0.866946 − 0.498402i \(-0.833920\pi\)
−0.00184433 + 0.999998i \(0.500587\pi\)
\(90\) −3.73205 −0.393393
\(91\) −2.73205 − 9.46410i −0.286397 − 0.992107i
\(92\) 4.19615 0.437479
\(93\) −1.26795 + 0.732051i −0.131480 + 0.0759101i
\(94\) −1.09808 − 1.90192i −0.113258 − 0.196168i
\(95\) 8.83013 − 15.2942i 0.905952 − 1.56915i
\(96\) − 1.00000i − 0.102062i
\(97\) −5.19615 − 3.00000i −0.527589 − 0.304604i 0.212445 − 0.977173i \(-0.431857\pi\)
−0.740034 + 0.672569i \(0.765191\pi\)
\(98\) −0.401924 − 0.232051i −0.0406004 − 0.0234407i
\(99\) 1.26795i 0.127434i
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.i.a.49.1 yes 4
3.2 odd 2 234.2.l.c.127.2 4
4.3 odd 2 624.2.bv.e.49.1 4
5.2 odd 4 1950.2.y.b.49.1 4
5.3 odd 4 1950.2.y.g.49.2 4
5.4 even 2 1950.2.bc.d.751.2 4
12.11 even 2 1872.2.by.h.1297.2 4
13.2 odd 12 1014.2.a.i.1.1 2
13.3 even 3 1014.2.b.e.337.1 4
13.4 even 6 inner 78.2.i.a.43.1 ✓ 4
13.5 odd 4 1014.2.e.i.991.1 4
13.6 odd 12 1014.2.e.i.529.1 4
13.7 odd 12 1014.2.e.g.529.2 4
13.8 odd 4 1014.2.e.g.991.2 4
13.9 even 3 1014.2.i.a.823.2 4
13.10 even 6 1014.2.b.e.337.4 4
13.11 odd 12 1014.2.a.k.1.2 2
13.12 even 2 1014.2.i.a.361.2 4
39.2 even 12 3042.2.a.y.1.2 2
39.11 even 12 3042.2.a.p.1.1 2
39.17 odd 6 234.2.l.c.199.2 4
39.23 odd 6 3042.2.b.i.1351.1 4
39.29 odd 6 3042.2.b.i.1351.4 4
52.11 even 12 8112.2.a.bp.1.2 2
52.15 even 12 8112.2.a.bj.1.1 2
52.43 odd 6 624.2.bv.e.433.2 4
65.4 even 6 1950.2.bc.d.901.2 4
65.17 odd 12 1950.2.y.g.199.2 4
65.43 odd 12 1950.2.y.b.199.1 4
156.95 even 6 1872.2.by.h.433.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.2.i.a.43.1 ✓ 4 13.4 even 6 inner
78.2.i.a.49.1 yes 4 1.1 even 1 trivial
234.2.l.c.127.2 4 3.2 odd 2
234.2.l.c.199.2 4 39.17 odd 6
624.2.bv.e.49.1 4 4.3 odd 2
624.2.bv.e.433.2 4 52.43 odd 6
1014.2.a.i.1.1 2 13.2 odd 12
1014.2.a.k.1.2 2 13.11 odd 12
1014.2.b.e.337.1 4 13.3 even 3
1014.2.b.e.337.4 4 13.10 even 6
1014.2.e.g.529.2 4 13.7 odd 12
1014.2.e.g.991.2 4 13.8 odd 4
1014.2.e.i.529.1 4 13.6 odd 12
1014.2.e.i.991.1 4 13.5 odd 4
1014.2.i.a.361.2 4 13.12 even 2
1014.2.i.a.823.2 4 13.9 even 3
1872.2.by.h.433.1 4 156.95 even 6
1872.2.by.h.1297.2 4 12.11 even 2
1950.2.y.b.49.1 4 5.2 odd 4
1950.2.y.b.199.1 4 65.43 odd 12
1950.2.y.g.49.2 4 5.3 odd 4
1950.2.y.g.199.2 4 65.17 odd 12
1950.2.bc.d.751.2 4 5.4 even 2
1950.2.bc.d.901.2 4 65.4 even 6
3042.2.a.p.1.1 2 39.11 even 12
3042.2.a.y.1.2 2 39.2 even 12
3042.2.b.i.1351.1 4 39.23 odd 6
3042.2.b.i.1351.4 4 39.29 odd 6
8112.2.a.bj.1.1 2 52.15 even 12
8112.2.a.bp.1.2 2 52.11 even 12