Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [378,2,Mod(1,378)] 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("378.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(378, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 378.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,1,0,1,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(3.01834519640\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 378.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +4.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +4.00000 q^{10} -4.00000 q^{11} +3.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -7.00000 q^{17} +2.00000 q^{19} +4.00000 q^{20} -4.00000 q^{22} -1.00000 q^{23} +11.0000 q^{25} +3.00000 q^{26} -1.00000 q^{28} +1.00000 q^{29} -9.00000 q^{31} +1.00000 q^{32} -7.00000 q^{34} -4.00000 q^{35} +2.00000 q^{37} +2.00000 q^{38} +4.00000 q^{40} +6.00000 q^{41} +11.0000 q^{43} -4.00000 q^{44} -1.00000 q^{46} -6.00000 q^{47} +1.00000 q^{49} +11.0000 q^{50} +3.00000 q^{52} -9.00000 q^{53} -16.0000 q^{55} -1.00000 q^{56} +1.00000 q^{58} -5.00000 q^{59} -6.00000 q^{61} -9.00000 q^{62} +1.00000 q^{64} +12.0000 q^{65} +7.00000 q^{67} -7.00000 q^{68} -4.00000 q^{70} -7.00000 q^{71} -14.0000 q^{73} +2.00000 q^{74} +2.00000 q^{76} +4.00000 q^{77} -6.00000 q^{79} +4.00000 q^{80} +6.00000 q^{82} -4.00000 q^{83} -28.0000 q^{85} +11.0000 q^{86} -4.00000 q^{88} -3.00000 q^{89} -3.00000 q^{91} -1.00000 q^{92} -6.00000 q^{94} +8.00000 q^{95} -8.00000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 4.00000 1.78885 0.894427 − 0.447214i \(-0.147584\pi\)
0.894427 + 0.447214i \(0.147584\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 4.00000 1.26491
\(11\) −4.00000 −1.20605 −0.603023 − 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) 3.00000 0.832050 0.416025 − 0.909353i \(-0.363423\pi\)
0.416025 + 0.909353i \(0.363423\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −7.00000 −1.69775 −0.848875 − 0.528594i \(-0.822719\pi\)
−0.848875 + 0.528594i \(0.822719\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 − 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 4.00000 0.894427
\(21\) 0 0
\(22\) −4.00000 −0.852803
\(23\) −1.00000 −0.208514 −0.104257 − 0.994550i \(-0.533247\pi\)
−0.104257 + 0.994550i \(0.533247\pi\)
\(24\) 0 0
\(25\) 11.0000 2.20000
\(26\) 3.00000 0.588348
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) 1.00000 0.185695 0.0928477 − 0.995680i \(-0.470403\pi\)
0.0928477 + 0.995680i \(0.470403\pi\)
\(30\) 0 0
\(31\) −9.00000 −1.61645 −0.808224 − 0.588875i \(-0.799571\pi\)
−0.808224 + 0.588875i \(0.799571\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −7.00000 −1.20049
\(35\) −4.00000 −0.676123
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 2.00000 0.324443
\(39\) 0 0
\(40\) 4.00000 0.632456
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 11.0000 1.67748 0.838742 − 0.544529i \(-0.183292\pi\)
0.838742 + 0.544529i \(0.183292\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) −6.00000 −0.875190 −0.437595 − 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 11.0000 1.55563
\(51\) 0 0
\(52\) 3.00000 0.416025
\(53\) −9.00000 −1.23625 −0.618123 − 0.786082i \(-0.712106\pi\)
−0.618123 + 0.786082i \(0.712106\pi\)
\(54\) 0 0
\(55\) −16.0000 −2.15744
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 1.00000 0.131306
\(59\) −5.00000 −0.650945 −0.325472 − 0.945552i \(-0.605523\pi\)
−0.325472 + 0.945552i \(0.605523\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 − 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) −9.00000 −1.14300
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 12.0000 1.48842
\(66\) 0 0
\(67\) 7.00000 0.855186 0.427593 − 0.903971i \(-0.359362\pi\)
0.427593 + 0.903971i \(0.359362\pi\)
\(68\) −7.00000 −0.848875
\(69\) 0 0
\(70\) −4.00000 −0.478091
\(71\) −7.00000 −0.830747 −0.415374 − 0.909651i \(-0.636349\pi\)
−0.415374 + 0.909651i \(0.636349\pi\)
\(72\) 0 0
\(73\) −14.0000 −1.63858 −0.819288 − 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) 4.00000 0.455842
\(78\) 0 0
\(79\) −6.00000 −0.675053 −0.337526 − 0.941316i \(-0.609590\pi\)
−0.337526 + 0.941316i \(0.609590\pi\)
\(80\) 4.00000 0.447214
\(81\) 0 0
\(82\) 6.00000 0.662589
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) −28.0000 −3.03703
\(86\) 11.0000 1.18616
\(87\) 0 0
\(88\) −4.00000 −0.426401
\(89\) −3.00000 −0.317999 −0.159000 − 0.987279i \(-0.550827\pi\)
−0.159000 + 0.987279i \(0.550827\pi\)
\(90\) 0 0
\(91\) −3.00000 −0.314485
\(92\) −1.00000 −0.104257
\(93\) 0 0
\(94\) −6.00000 −0.618853
\(95\) 8.00000 0.820783
\(96\) 0 0
\(97\) −8.00000 −0.812277 −0.406138 − 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 1.00000 0.101015
\(99\) 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 378.2.a.h.1.1 yes 1
3.2 odd 2 378.2.a.a.1.1 ✓ 1
4.3 odd 2 3024.2.a.bd.1.1 1
5.4 even 2 9450.2.a.bc.1.1 1
7.6 odd 2 2646.2.a.p.1.1 1
9.2 odd 6 1134.2.f.p.757.1 2
9.4 even 3 1134.2.f.a.379.1 2
9.5 odd 6 1134.2.f.p.379.1 2
9.7 even 3 1134.2.f.a.757.1 2
12.11 even 2 3024.2.a.a.1.1 1
15.14 odd 2 9450.2.a.dv.1.1 1
21.20 even 2 2646.2.a.o.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
378.2.a.a.1.1 ✓ 1 3.2 odd 2
378.2.a.h.1.1 yes 1 1.1 even 1 trivial
1134.2.f.a.379.1 2 9.4 even 3
1134.2.f.a.757.1 2 9.7 even 3
1134.2.f.p.379.1 2 9.5 odd 6
1134.2.f.p.757.1 2 9.2 odd 6
2646.2.a.o.1.1 1 21.20 even 2
2646.2.a.p.1.1 1 7.6 odd 2
3024.2.a.a.1.1 1 12.11 even 2
3024.2.a.bd.1.1 1 4.3 odd 2
9450.2.a.bc.1.1 1 5.4 even 2
9450.2.a.dv.1.1 1 15.14 odd 2