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Properties

Label 78.2.b.a.25.2
Level $78$
Weight $2$
Character 78.25
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(25,78)] 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("78.25"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(78, base_ring=CyclotomicField(2)) 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.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == 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: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 25.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 78.25
Dual form 78.2.b.a.25.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +2.00000i q^{5} +1.00000i q^{6} -2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} -2.00000 q^{10} -1.00000 q^{12} +(-3.00000 - 2.00000i) q^{13} +2.00000 q^{14} +2.00000i q^{15} +1.00000 q^{16} -2.00000 q^{17} +1.00000i q^{18} -6.00000i q^{19} -2.00000i q^{20} -2.00000i q^{21} +4.00000 q^{23} -1.00000i q^{24} +1.00000 q^{25} +(2.00000 - 3.00000i) q^{26} +1.00000 q^{27} +2.00000i q^{28} -10.0000 q^{29} -2.00000 q^{30} +10.0000i q^{31} +1.00000i q^{32} -2.00000i q^{34} +4.00000 q^{35} -1.00000 q^{36} +8.00000i q^{37} +6.00000 q^{38} +(-3.00000 - 2.00000i) q^{39} +2.00000 q^{40} +10.0000i q^{41} +2.00000 q^{42} +4.00000 q^{43} +2.00000i q^{45} +4.00000i q^{46} -12.0000i q^{47} +1.00000 q^{48} +3.00000 q^{49} +1.00000i q^{50} -2.00000 q^{51} +(3.00000 + 2.00000i) q^{52} -6.00000 q^{53} +1.00000i q^{54} -2.00000 q^{56} -6.00000i q^{57} -10.0000i q^{58} +4.00000i q^{59} -2.00000i q^{60} +2.00000 q^{61} -10.0000 q^{62} -2.00000i q^{63} -1.00000 q^{64} +(4.00000 - 6.00000i) q^{65} -2.00000i q^{67} +2.00000 q^{68} +4.00000 q^{69} +4.00000i q^{70} -1.00000i q^{72} -4.00000i q^{73} -8.00000 q^{74} +1.00000 q^{75} +6.00000i q^{76} +(2.00000 - 3.00000i) q^{78} +2.00000i q^{80} +1.00000 q^{81} -10.0000 q^{82} -4.00000i q^{83} +2.00000i q^{84} -4.00000i q^{85} +4.00000i q^{86} -10.0000 q^{87} -6.00000i q^{89} -2.00000 q^{90} +(-4.00000 + 6.00000i) q^{91} -4.00000 q^{92} +10.0000i q^{93} +12.0000 q^{94} +12.0000 q^{95} +1.00000i q^{96} -12.0000i q^{97} +3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{4} + 2 q^{9} - 4 q^{10} - 2 q^{12} - 6 q^{13} + 4 q^{14} + 2 q^{16} - 4 q^{17} + 8 q^{23} + 2 q^{25} + 4 q^{26} + 2 q^{27} - 20 q^{29} - 4 q^{30} + 8 q^{35} - 2 q^{36} + 12 q^{38} - 6 q^{39}+ \cdots + 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(53\) \(67\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 1.00000 0.577350
\(4\) −1.00000 −0.500000
\(5\) 2.00000i 0.894427i 0.894427 + 0.447214i \(0.147584\pi\)
−0.894427 + 0.447214i \(0.852416\pi\)
\(6\) 1.00000i 0.408248i
\(7\) − 2.00000i − 0.755929i −0.925820 − 0.377964i \(-0.876624\pi\)
0.925820 − 0.377964i \(-0.123376\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 1.00000 0.333333
\(10\) −2.00000 −0.632456
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) −1.00000 −0.288675
\(13\) −3.00000 − 2.00000i −0.832050 − 0.554700i
\(14\) 2.00000 0.534522
\(15\) 2.00000i 0.516398i
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 1.00000i 0.235702i
\(19\) − 6.00000i − 1.37649i −0.725476 − 0.688247i \(-0.758380\pi\)
0.725476 − 0.688247i \(-0.241620\pi\)
\(20\) − 2.00000i − 0.447214i
\(21\) − 2.00000i − 0.436436i
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 − 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) − 1.00000i − 0.204124i
\(25\) 1.00000 0.200000
\(26\) 2.00000 − 3.00000i 0.392232 − 0.588348i
\(27\) 1.00000 0.192450
\(28\) 2.00000i 0.377964i
\(29\) −10.0000 −1.85695 −0.928477 − 0.371391i \(-0.878881\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) −2.00000 −0.365148
\(31\) 10.0000i 1.79605i 0.439941 + 0.898027i \(0.354999\pi\)
−0.439941 + 0.898027i \(0.645001\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) − 2.00000i − 0.342997i
\(35\) 4.00000 0.676123
\(36\) −1.00000 −0.166667
\(37\) 8.00000i 1.31519i 0.753371 + 0.657596i \(0.228427\pi\)
−0.753371 + 0.657596i \(0.771573\pi\)
\(38\) 6.00000 0.973329
\(39\) −3.00000 − 2.00000i −0.480384 − 0.320256i
\(40\) 2.00000 0.316228
\(41\) 10.0000i 1.56174i 0.624695 + 0.780869i \(0.285223\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 2.00000 0.308607
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 2.00000i 0.298142i
\(46\) 4.00000i 0.589768i
\(47\) − 12.0000i − 1.75038i −0.483779 − 0.875190i \(-0.660736\pi\)
0.483779 − 0.875190i \(-0.339264\pi\)
\(48\) 1.00000 0.144338
\(49\) 3.00000 0.428571
\(50\) 1.00000i 0.141421i
\(51\) −2.00000 −0.280056
\(52\) 3.00000 + 2.00000i 0.416025 + 0.277350i
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 1.00000i 0.136083i
\(55\) 0 0
\(56\) −2.00000 −0.267261
\(57\) − 6.00000i − 0.794719i
\(58\) − 10.0000i − 1.31306i
\(59\) 4.00000i 0.520756i 0.965507 + 0.260378i \(0.0838471\pi\)
−0.965507 + 0.260378i \(0.916153\pi\)
\(60\) − 2.00000i − 0.258199i
\(61\) 2.00000 0.256074 0.128037 − 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −10.0000 −1.27000
\(63\) − 2.00000i − 0.251976i
\(64\) −1.00000 −0.125000
\(65\) 4.00000 − 6.00000i 0.496139 − 0.744208i
\(66\) 0 0
\(67\) − 2.00000i − 0.244339i −0.992509 − 0.122169i \(-0.961015\pi\)
0.992509 − 0.122169i \(-0.0389851\pi\)
\(68\) 2.00000 0.242536
\(69\) 4.00000 0.481543
\(70\) 4.00000i 0.478091i
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) − 4.00000i − 0.468165i −0.972217 − 0.234082i \(-0.924791\pi\)
0.972217 − 0.234082i \(-0.0752085\pi\)
\(74\) −8.00000 −0.929981
\(75\) 1.00000 0.115470
\(76\) 6.00000i 0.688247i
\(77\) 0 0
\(78\) 2.00000 − 3.00000i 0.226455 − 0.339683i
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 2.00000i 0.223607i
\(81\) 1.00000 0.111111
\(82\) −10.0000 −1.10432
\(83\) − 4.00000i − 0.439057i −0.975606 − 0.219529i \(-0.929548\pi\)
0.975606 − 0.219529i \(-0.0704519\pi\)
\(84\) 2.00000i 0.218218i
\(85\) − 4.00000i − 0.433861i
\(86\) 4.00000i 0.431331i
\(87\) −10.0000 −1.07211
\(88\) 0 0
\(89\) − 6.00000i − 0.635999i −0.948091 − 0.317999i \(-0.896989\pi\)
0.948091 − 0.317999i \(-0.103011\pi\)
\(90\) −2.00000 −0.210819
\(91\) −4.00000 + 6.00000i −0.419314 + 0.628971i
\(92\) −4.00000 −0.417029
\(93\) 10.0000i 1.03695i
\(94\) 12.0000 1.23771
\(95\) 12.0000 1.23117
\(96\) 1.00000i 0.102062i
\(97\) − 12.0000i − 1.21842i −0.793011 − 0.609208i \(-0.791488\pi\)
0.793011 − 0.609208i \(-0.208512\pi\)
\(98\) 3.00000i 0.303046i
\(99\) 0 0
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.b.a.25.2 yes 2
3.2 odd 2 234.2.b.a.181.1 2
4.3 odd 2 624.2.c.a.337.2 2
5.2 odd 4 1950.2.f.d.649.1 2
5.3 odd 4 1950.2.f.g.649.2 2
5.4 even 2 1950.2.b.c.1351.1 2
7.6 odd 2 3822.2.c.d.883.2 2
8.3 odd 2 2496.2.c.m.961.1 2
8.5 even 2 2496.2.c.f.961.1 2
12.11 even 2 1872.2.c.b.1585.1 2
13.2 odd 12 1014.2.e.b.529.1 2
13.3 even 3 1014.2.i.c.823.1 4
13.4 even 6 1014.2.i.c.361.1 4
13.5 odd 4 1014.2.a.g.1.1 1
13.6 odd 12 1014.2.e.b.991.1 2
13.7 odd 12 1014.2.e.e.991.1 2
13.8 odd 4 1014.2.a.b.1.1 1
13.9 even 3 1014.2.i.c.361.2 4
13.10 even 6 1014.2.i.c.823.2 4
13.11 odd 12 1014.2.e.e.529.1 2
13.12 even 2 inner 78.2.b.a.25.1 ✓ 2
39.5 even 4 3042.2.a.c.1.1 1
39.8 even 4 3042.2.a.n.1.1 1
39.38 odd 2 234.2.b.a.181.2 2
52.31 even 4 8112.2.a.j.1.1 1
52.47 even 4 8112.2.a.g.1.1 1
52.51 odd 2 624.2.c.a.337.1 2
65.12 odd 4 1950.2.f.g.649.1 2
65.38 odd 4 1950.2.f.d.649.2 2
65.64 even 2 1950.2.b.c.1351.2 2
91.90 odd 2 3822.2.c.d.883.1 2
104.51 odd 2 2496.2.c.m.961.2 2
104.77 even 2 2496.2.c.f.961.2 2
156.155 even 2 1872.2.c.b.1585.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.2.b.a.25.1 ✓ 2 13.12 even 2 inner
78.2.b.a.25.2 yes 2 1.1 even 1 trivial
234.2.b.a.181.1 2 3.2 odd 2
234.2.b.a.181.2 2 39.38 odd 2
624.2.c.a.337.1 2 52.51 odd 2
624.2.c.a.337.2 2 4.3 odd 2
1014.2.a.b.1.1 1 13.8 odd 4
1014.2.a.g.1.1 1 13.5 odd 4
1014.2.e.b.529.1 2 13.2 odd 12
1014.2.e.b.991.1 2 13.6 odd 12
1014.2.e.e.529.1 2 13.11 odd 12
1014.2.e.e.991.1 2 13.7 odd 12
1014.2.i.c.361.1 4 13.4 even 6
1014.2.i.c.361.2 4 13.9 even 3
1014.2.i.c.823.1 4 13.3 even 3
1014.2.i.c.823.2 4 13.10 even 6
1872.2.c.b.1585.1 2 12.11 even 2
1872.2.c.b.1585.2 2 156.155 even 2
1950.2.b.c.1351.1 2 5.4 even 2
1950.2.b.c.1351.2 2 65.64 even 2
1950.2.f.d.649.1 2 5.2 odd 4
1950.2.f.d.649.2 2 65.38 odd 4
1950.2.f.g.649.1 2 65.12 odd 4
1950.2.f.g.649.2 2 5.3 odd 4
2496.2.c.f.961.1 2 8.5 even 2
2496.2.c.f.961.2 2 104.77 even 2
2496.2.c.m.961.1 2 8.3 odd 2
2496.2.c.m.961.2 2 104.51 odd 2
3042.2.a.c.1.1 1 39.5 even 4
3042.2.a.n.1.1 1 39.8 even 4
3822.2.c.d.883.1 2 91.90 odd 2
3822.2.c.d.883.2 2 7.6 odd 2
8112.2.a.g.1.1 1 52.47 even 4
8112.2.a.j.1.1 1 52.31 even 4