Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [392,6,Mod(177,392)] 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("392.177"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(392, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 392.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,13,0,31,0,0,0,-230] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(62.8704573667\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 3 x^{9} - 119 x^{8} - 521 x^{7} - 898 x^{6} + 27806 x^{5} + 657990 x^{4} + 3648839 x^{3} + \cdots + 92895579 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{18}\cdot 3^{2}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 177.2
Root \(0.261205 + 8.00832i\) of defining polynomial
Character \(\chi\) \(=\) 392.177
Dual form 392.6.i.p.361.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-7.26579 - 12.5847i) q^{3} +(-22.4113 + 38.8175i) q^{5} +(15.9167 - 27.5685i) q^{9} +(317.292 + 549.565i) q^{11} +20.5788 q^{13} +651.343 q^{15} +(-525.299 - 909.844i) q^{17} +(32.5846 - 56.4381i) q^{19} +(2027.27 - 3511.34i) q^{23} +(557.968 + 966.429i) q^{25} -3993.76 q^{27} -6581.20 q^{29} +(544.852 + 943.712i) q^{31} +(4610.75 - 7986.05i) q^{33} +(-342.573 + 593.355i) q^{37} +(-149.521 - 258.979i) q^{39} +13642.5 q^{41} -16038.6 q^{43} +(713.426 + 1235.69i) q^{45} +(-6805.15 + 11786.9i) q^{47} +(-7633.42 + 13221.5i) q^{51} +(804.189 + 1392.90i) q^{53} -28443.7 q^{55} -947.010 q^{57} +(-3761.80 - 6515.62i) q^{59} +(10440.7 - 18083.8i) q^{61} +(-461.198 + 798.819i) q^{65} +(-936.960 - 1622.86i) q^{67} -58919.0 q^{69} -2769.30 q^{71} +(20775.6 + 35984.3i) q^{73} +(8108.15 - 14043.7i) q^{75} +(-46484.5 + 80513.4i) q^{79} +(25150.1 + 43561.2i) q^{81} -63718.6 q^{83} +47090.5 q^{85} +(47817.6 + 82822.5i) q^{87} +(-51390.8 + 89011.5i) q^{89} +(7917.56 - 13713.6i) q^{93} +(1460.52 + 2529.70i) q^{95} +168774. q^{97} +20200.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 13 q^{3} + 31 q^{5} - 230 q^{9} + 351 q^{11} + 108 q^{13} + 1214 q^{15} + 111 q^{17} + 1035 q^{19} - 3639 q^{23} - 1540 q^{25} - 7214 q^{27} - 1468 q^{29} + 7677 q^{31} - 7439 q^{33} - 13595 q^{37}+ \cdots - 600308 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/392\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(295\) \(297\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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 0
\(3\) −7.26579 − 12.5847i −0.466100 − 0.807310i 0.533150 − 0.846021i \(-0.321008\pi\)
−0.999250 + 0.0387110i \(0.987675\pi\)
\(4\) 0 0
\(5\) −22.4113 + 38.8175i −0.400905 + 0.694388i −0.993835 − 0.110865i \(-0.964638\pi\)
0.592930 + 0.805254i \(0.297971\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 15.9167 − 27.5685i 0.0655007 − 0.113451i
\(10\) 0 0
\(11\) 317.292 + 549.565i 0.790637 + 1.36942i 0.925573 + 0.378569i \(0.123584\pi\)
−0.134936 + 0.990854i \(0.543083\pi\)
\(12\) 0 0
\(13\) 20.5788 0.0337724 0.0168862 − 0.999857i \(-0.494625\pi\)
0.0168862 + 0.999857i \(0.494625\pi\)
\(14\) 0 0
\(15\) 651.343 0.747449
\(16\) 0 0
\(17\) −525.299 − 909.844i −0.440843 − 0.763563i 0.556909 − 0.830573i \(-0.311987\pi\)
−0.997752 + 0.0670108i \(0.978654\pi\)
\(18\) 0 0
\(19\) 32.5846 − 56.4381i 0.0207075 − 0.0358665i −0.855486 − 0.517826i \(-0.826741\pi\)
0.876193 + 0.481960i \(0.160075\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2027.27 − 3511.34i 0.799085 − 1.38406i −0.121128 − 0.992637i \(-0.538651\pi\)
0.920213 − 0.391418i \(-0.128016\pi\)
\(24\) 0 0
\(25\) 557.968 + 966.429i 0.178550 + 0.309257i
\(26\) 0 0
\(27\) −3993.76 −1.05432
\(28\) 0 0
\(29\) −6581.20 −1.45315 −0.726575 − 0.687088i \(-0.758889\pi\)
−0.726575 + 0.687088i \(0.758889\pi\)
\(30\) 0 0
\(31\) 544.852 + 943.712i 0.101830 + 0.176374i 0.912439 − 0.409214i \(-0.134197\pi\)
−0.810609 + 0.585588i \(0.800864\pi\)
\(32\) 0 0
\(33\) 4610.75 − 7986.05i 0.737032 − 1.27658i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −342.573 + 593.355i −0.0411386 + 0.0712541i −0.885862 − 0.463950i \(-0.846432\pi\)
0.844723 + 0.535204i \(0.179765\pi\)
\(38\) 0 0
\(39\) −149.521 − 258.979i −0.0157414 − 0.0272648i
\(40\) 0 0
\(41\) 13642.5 1.26746 0.633729 − 0.773556i \(-0.281524\pi\)
0.633729 + 0.773556i \(0.281524\pi\)
\(42\) 0 0
\(43\) −16038.6 −1.32280 −0.661402 − 0.750032i \(-0.730038\pi\)
−0.661402 + 0.750032i \(0.730038\pi\)
\(44\) 0 0
\(45\) 713.426 + 1235.69i 0.0525192 + 0.0909659i
\(46\) 0 0
\(47\) −6805.15 + 11786.9i −0.449359 + 0.778312i −0.998344 − 0.0575195i \(-0.981681\pi\)
0.548986 + 0.835832i \(0.315014\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −7633.42 + 13221.5i −0.410954 + 0.711794i
\(52\) 0 0
\(53\) 804.189 + 1392.90i 0.0393250 + 0.0681129i 0.885018 − 0.465557i \(-0.154146\pi\)
−0.845693 + 0.533670i \(0.820813\pi\)
\(54\) 0 0
\(55\) −28443.7 −1.26788
\(56\) 0 0
\(57\) −947.010 −0.0386071
\(58\) 0 0
\(59\) −3761.80 − 6515.62i −0.140691 − 0.243683i 0.787066 − 0.616868i \(-0.211599\pi\)
−0.927757 + 0.373185i \(0.878266\pi\)
\(60\) 0 0
\(61\) 10440.7 − 18083.8i 0.359257 − 0.622251i −0.628580 − 0.777745i \(-0.716363\pi\)
0.987837 + 0.155494i \(0.0496968\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −461.198 + 798.819i −0.0135396 + 0.0234512i
\(66\) 0 0
\(67\) −936.960 − 1622.86i −0.0254996 − 0.0441667i 0.852994 − 0.521921i \(-0.174784\pi\)
−0.878494 + 0.477754i \(0.841451\pi\)
\(68\) 0 0
\(69\) −58919.0 −1.48982
\(70\) 0 0
\(71\) −2769.30 −0.0651965 −0.0325982 − 0.999469i \(-0.510378\pi\)
−0.0325982 + 0.999469i \(0.510378\pi\)
\(72\) 0 0
\(73\) 20775.6 + 35984.3i 0.456295 + 0.790326i 0.998762 − 0.0497514i \(-0.0158429\pi\)
−0.542467 + 0.840077i \(0.682510\pi\)
\(74\) 0 0
\(75\) 8108.15 − 14043.7i 0.166444 − 0.288290i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −46484.5 + 80513.4i −0.837992 + 1.45145i 0.0535784 + 0.998564i \(0.482937\pi\)
−0.891571 + 0.452882i \(0.850396\pi\)
\(80\) 0 0
\(81\) 25150.1 + 43561.2i 0.425919 + 0.737713i
\(82\) 0 0
\(83\) −63718.6 −1.01525 −0.507623 − 0.861579i \(-0.669476\pi\)
−0.507623 + 0.861579i \(0.669476\pi\)
\(84\) 0 0
\(85\) 47090.5 0.706945
\(86\) 0 0
\(87\) 47817.6 + 82822.5i 0.677314 + 1.17314i
\(88\) 0 0
\(89\) −51390.8 + 89011.5i −0.687719 + 1.19116i 0.284856 + 0.958570i \(0.408054\pi\)
−0.972574 + 0.232593i \(0.925279\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 7917.56 − 13713.6i 0.0949258 − 0.164416i
\(94\) 0 0
\(95\) 1460.52 + 2529.70i 0.0166035 + 0.0287581i
\(96\) 0 0
\(97\) 168774. 1.82128 0.910638 − 0.413205i \(-0.135591\pi\)
0.910638 + 0.413205i \(0.135591\pi\)
\(98\) 0 0
\(99\) 20200.9 0.207149
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 392.6.i.p.177.2 10
7.2 even 3 392.6.a.i.1.4 5
7.3 odd 6 56.6.i.a.25.4 yes 10
7.4 even 3 inner 392.6.i.p.361.2 10
7.5 odd 6 392.6.a.l.1.2 5
7.6 odd 2 56.6.i.a.9.4 ✓ 10
21.17 even 6 504.6.s.d.361.2 10
21.20 even 2 504.6.s.d.289.2 10
28.3 even 6 112.6.i.g.81.2 10
28.19 even 6 784.6.a.bj.1.4 5
28.23 odd 6 784.6.a.bm.1.2 5
28.27 even 2 112.6.i.g.65.2 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.6.i.a.9.4 ✓ 10 7.6 odd 2
56.6.i.a.25.4 yes 10 7.3 odd 6
112.6.i.g.65.2 10 28.27 even 2
112.6.i.g.81.2 10 28.3 even 6
392.6.a.i.1.4 5 7.2 even 3
392.6.a.l.1.2 5 7.5 odd 6
392.6.i.p.177.2 10 1.1 even 1 trivial
392.6.i.p.361.2 10 7.4 even 3 inner
504.6.s.d.289.2 10 21.20 even 2
504.6.s.d.361.2 10 21.17 even 6
784.6.a.bj.1.4 5 28.19 even 6
784.6.a.bm.1.2 5 28.23 odd 6