Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [56,6,Mod(9,56)] 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("56.9"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(56, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 56 = 2^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 56.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,-13] 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: \(8.98149390953\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{18}\cdot 3\cdot 7^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 9.4
Root \(0.261205 + 8.00832i\) of defining polynomial
Character \(\chi\) \(=\) 56.9
Dual form 56.6.i.a.25.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(7.26579 + 12.5847i) q^{3} +(22.4113 - 38.8175i) q^{5} +(96.4534 - 86.6241i) q^{7} +(15.9167 - 27.5685i) q^{9} +(317.292 + 549.565i) q^{11} -20.5788 q^{13} +651.343 q^{15} +(525.299 + 909.844i) q^{17} +(-32.5846 + 56.4381i) q^{19} +(1790.95 + 584.446i) q^{21} +(2027.27 - 3511.34i) q^{23} +(557.968 + 966.429i) q^{25} +3993.76 q^{27} -6581.20 q^{29} +(-544.852 - 943.712i) q^{31} +(-4610.75 + 7986.05i) q^{33} +(-1200.89 - 5685.44i) q^{35} +(-342.573 + 593.355i) q^{37} +(-149.521 - 258.979i) q^{39} -13642.5 q^{41} -16038.6 q^{43} +(-713.426 - 1235.69i) q^{45} +(6805.15 - 11786.9i) q^{47} +(1799.52 - 16710.4i) q^{49} +(-7633.42 + 13221.5i) q^{51} +(804.189 + 1392.90i) q^{53} +28443.7 q^{55} -947.010 q^{57} +(3761.80 + 6515.62i) q^{59} +(-10440.7 + 18083.8i) q^{61} +(-852.878 - 4037.84i) q^{63} +(-461.198 + 798.819i) q^{65} +(-936.960 - 1622.86i) q^{67} +58919.0 q^{69} -2769.30 q^{71} +(-20775.6 - 35984.3i) q^{73} +(-8108.15 + 14043.7i) q^{75} +(78209.5 + 25522.3i) q^{77} +(-46484.5 + 80513.4i) q^{79} +(25150.1 + 43561.2i) q^{81} +63718.6 q^{83} +47090.5 q^{85} +(-47817.6 - 82822.5i) q^{87} +(51390.8 - 89011.5i) q^{89} +(-1984.90 + 1782.62i) q^{91} +(7917.56 - 13713.6i) q^{93} +(1460.52 + 2529.70i) q^{95} -168774. q^{97} +20200.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 13 q^{3} - 31 q^{5} - 92 q^{7} - 230 q^{9} + 351 q^{11} - 108 q^{13} + 1214 q^{15} - 111 q^{17} - 1035 q^{19} - 1365 q^{21} - 3639 q^{23} - 1540 q^{25} + 7214 q^{27} - 1468 q^{29} - 7677 q^{31} + 7439 q^{33}+ \cdots - 600308 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/56\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\) \(29\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 7.26579 + 12.5847i 0.466100 + 0.807310i 0.999250 − 0.0387110i \(-0.0123252\pi\)
−0.533150 + 0.846021i \(0.678992\pi\)
\(4\) 0 0
\(5\) 22.4113 − 38.8175i 0.400905 − 0.694388i −0.592930 − 0.805254i \(-0.702029\pi\)
0.993835 + 0.110865i \(0.0353623\pi\)
\(6\) 0 0
\(7\) 96.4534 − 86.6241i 0.743999 − 0.668180i
\(8\) 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 \(-0.505375\pi\)
−0.0168862 + 0.999857i \(0.505375\pi\)
\(14\) 0 0
\(15\) 651.343 0.747449
\(16\) 0 0
\(17\) 525.299 + 909.844i 0.440843 + 0.763563i 0.997752 − 0.0670108i \(-0.0213462\pi\)
−0.556909 + 0.830573i \(0.688013\pi\)
\(18\) 0 0
\(19\) −32.5846 + 56.4381i −0.0207075 + 0.0358665i −0.876193 − 0.481960i \(-0.839925\pi\)
0.855486 + 0.517826i \(0.173259\pi\)
\(20\) 0 0
\(21\) 1790.95 + 584.446i 0.886207 + 0.289199i
\(22\) 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 0.810609 − 0.585588i \(-0.199136\pi\)
−0.912439 + 0.409214i \(0.865803\pi\)
\(32\) 0 0
\(33\) −4610.75 + 7986.05i −0.737032 + 1.27658i
\(34\) 0 0
\(35\) −1200.89 − 5685.44i −0.165703 − 0.784502i
\(36\) 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 \(-0.718476\pi\)
−0.633729 + 0.773556i \(0.718476\pi\)
\(42\) 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 −0.548986 − 0.835832i \(-0.684986\pi\)
0.998344 + 0.0575195i \(0.0183191\pi\)
\(48\) 0 0
\(49\) 1799.52 − 16710.4i 0.107070 − 0.994252i
\(50\) 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 0.927757 − 0.373185i \(-0.121734\pi\)
−0.787066 + 0.616868i \(0.788401\pi\)
\(60\) 0 0
\(61\) −10440.7 + 18083.8i −0.359257 + 0.622251i −0.987837 − 0.155494i \(-0.950303\pi\)
0.628580 + 0.777745i \(0.283637\pi\)
\(62\) 0 0
\(63\) −852.878 − 4037.84i −0.0270730 − 0.128173i
\(64\) 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 0.542467 − 0.840077i \(-0.317490\pi\)
−0.998762 + 0.0497514i \(0.984157\pi\)
\(74\) 0 0
\(75\) −8108.15 + 14043.7i −0.166444 + 0.288290i
\(76\) 0 0
\(77\) 78209.5 + 25522.3i 1.50326 + 0.490562i
\(78\) 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 \(-0.330524\pi\)
0.507623 + 0.861579i \(0.330524\pi\)
\(84\) 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 \(-0.591946\pi\)
0.972574 − 0.232593i \(-0.0747210\pi\)
\(90\) 0 0
\(91\) −1984.90 + 1782.62i −0.0251267 + 0.0225661i
\(92\) 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 \(-0.864409\pi\)
−0.910638 + 0.413205i \(0.864409\pi\)
\(98\) 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 56.6.i.a.9.4 ✓ 10
3.2 odd 2 504.6.s.d.289.2 10
4.3 odd 2 112.6.i.g.65.2 10
7.2 even 3 392.6.a.l.1.2 5
7.3 odd 6 392.6.i.p.361.2 10
7.4 even 3 inner 56.6.i.a.25.4 yes 10
7.5 odd 6 392.6.a.i.1.4 5
7.6 odd 2 392.6.i.p.177.2 10
21.11 odd 6 504.6.s.d.361.2 10
28.11 odd 6 112.6.i.g.81.2 10
28.19 even 6 784.6.a.bm.1.2 5
28.23 odd 6 784.6.a.bj.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.6.i.a.9.4 ✓ 10 1.1 even 1 trivial
56.6.i.a.25.4 yes 10 7.4 even 3 inner
112.6.i.g.65.2 10 4.3 odd 2
112.6.i.g.81.2 10 28.11 odd 6
392.6.a.i.1.4 5 7.5 odd 6
392.6.a.l.1.2 5 7.2 even 3
392.6.i.p.177.2 10 7.6 odd 2
392.6.i.p.361.2 10 7.3 odd 6
504.6.s.d.289.2 10 3.2 odd 2
504.6.s.d.361.2 10 21.11 odd 6
784.6.a.bj.1.4 5 28.23 odd 6
784.6.a.bm.1.2 5 28.19 even 6