Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [112,5,Mod(79,112)] 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("112.79"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 2])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 112.r (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,9] 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: \(11.5774358654\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{6})\)
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: \( x^{10} - 5 x^{9} + 485 x^{8} - 1910 x^{7} + 81837 x^{6} - 238847 x^{5} + 5758115 x^{4} + \cdots + 1406445775 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 79.4
Root \(0.500000 - 6.63333i\) of defining polynomial
Character \(\chi\) \(=\) 112.79
Dual form 112.5.r.b.95.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(6.49463 - 3.74968i) q^{3} +(24.4965 - 42.4292i) q^{5} +(29.5189 - 39.1106i) q^{7} +(-12.3799 + 21.4425i) q^{9} +(-69.7430 + 40.2662i) q^{11} -194.864 q^{13} -367.416i q^{15} +(-58.6762 - 101.630i) q^{17} +(486.348 + 280.793i) q^{19} +(45.0622 - 364.695i) q^{21} +(149.635 + 86.3918i) q^{23} +(-887.660 - 1537.47i) q^{25} +793.129i q^{27} +156.651 q^{29} +(1268.12 - 732.150i) q^{31} +(-301.970 + 523.028i) q^{33} +(-936.321 - 2210.54i) q^{35} +(-347.672 + 602.186i) q^{37} +(-1265.57 + 730.678i) q^{39} +618.644 q^{41} -1551.09i q^{43} +(606.527 + 1050.54i) q^{45} +(2227.48 + 1286.04i) q^{47} +(-658.272 - 2309.00i) q^{49} +(-762.161 - 440.034i) q^{51} +(788.661 + 1366.00i) q^{53} +3945.52i q^{55} +4211.53 q^{57} +(-4325.71 + 2497.45i) q^{59} +(800.975 - 1387.33i) q^{61} +(473.190 + 1117.14i) q^{63} +(-4773.50 + 8267.94i) q^{65} +(3251.75 - 1877.40i) q^{67} +1295.77 q^{69} +4041.40i q^{71} +(3635.17 + 6296.30i) q^{73} +(-11530.0 - 6656.87i) q^{75} +(-483.904 + 3916.30i) q^{77} +(-234.672 - 135.488i) q^{79} +(1971.21 + 3414.24i) q^{81} +3468.88i q^{83} -5749.46 q^{85} +(1017.39 - 587.389i) q^{87} +(4860.18 - 8418.08i) q^{89} +(-5752.17 + 7621.25i) q^{91} +(5490.65 - 9510.09i) q^{93} +(23827.7 - 13756.9i) q^{95} +8062.39 q^{97} -1993.96i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 9 q^{3} + 9 q^{5} + 10 q^{7} + 74 q^{9} + 189 q^{11} + 84 q^{13} - 435 q^{17} + 357 q^{19} - 61 q^{21} + 1269 q^{23} - 776 q^{25} + 660 q^{29} + 969 q^{31} + 1759 q^{33} - 1521 q^{35} - 583 q^{37}+ \cdots - 43356 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\) \(85\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{3}\right)\) \(1\)

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
\(3\) 6.49463 − 3.74968i 0.721626 − 0.416631i −0.0937251 − 0.995598i \(-0.529877\pi\)
0.815351 + 0.578967i \(0.196544\pi\)
\(4\) 0 0
\(5\) 24.4965 − 42.4292i 0.979861 − 1.69717i 0.317002 − 0.948425i \(-0.397324\pi\)
0.662859 − 0.748744i \(-0.269343\pi\)
\(6\) 0 0
\(7\) 29.5189 − 39.1106i 0.602426 − 0.798175i
\(8\) 0 0
\(9\) −12.3799 + 21.4425i −0.152838 + 0.264723i
\(10\) 0 0
\(11\) −69.7430 + 40.2662i −0.576389 + 0.332778i −0.759697 − 0.650277i \(-0.774653\pi\)
0.183308 + 0.983056i \(0.441319\pi\)
\(12\) 0 0
\(13\) −194.864 −1.15304 −0.576521 − 0.817082i \(-0.695590\pi\)
−0.576521 + 0.817082i \(0.695590\pi\)
\(14\) 0 0
\(15\) − 367.416i − 1.63296i
\(16\) 0 0
\(17\) −58.6762 − 101.630i −0.203032 − 0.351662i 0.746472 − 0.665417i \(-0.231746\pi\)
−0.949504 + 0.313755i \(0.898413\pi\)
\(18\) 0 0
\(19\) 486.348 + 280.793i 1.34722 + 0.777820i 0.987855 − 0.155376i \(-0.0496589\pi\)
0.359368 + 0.933196i \(0.382992\pi\)
\(20\) 0 0
\(21\) 45.0622 − 364.695i 0.102182 − 0.826972i
\(22\) 0 0
\(23\) 149.635 + 86.3918i 0.282864 + 0.163312i 0.634719 − 0.772743i \(-0.281116\pi\)
−0.351855 + 0.936054i \(0.614449\pi\)
\(24\) 0 0
\(25\) −887.660 − 1537.47i −1.42026 − 2.45995i
\(26\) 0 0
\(27\) 793.129i 1.08797i
\(28\) 0 0
\(29\) 156.651 0.186267 0.0931335 − 0.995654i \(-0.470312\pi\)
0.0931335 + 0.995654i \(0.470312\pi\)
\(30\) 0 0
\(31\) 1268.12 − 732.150i 1.31958 − 0.761863i 0.335923 − 0.941889i \(-0.390952\pi\)
0.983662 + 0.180027i \(0.0576185\pi\)
\(32\) 0 0
\(33\) −301.970 + 523.028i −0.277291 + 0.480283i
\(34\) 0 0
\(35\) −936.321 − 2210.54i −0.764344 − 1.80452i
\(36\) 0 0
\(37\) −347.672 + 602.186i −0.253961 + 0.439873i −0.964613 − 0.263671i \(-0.915067\pi\)
0.710652 + 0.703544i \(0.248400\pi\)
\(38\) 0 0
\(39\) −1265.57 + 730.678i −0.832065 + 0.480393i
\(40\) 0 0
\(41\) 618.644 0.368022 0.184011 − 0.982924i \(-0.441092\pi\)
0.184011 + 0.982924i \(0.441092\pi\)
\(42\) 0 0
\(43\) − 1551.09i − 0.838879i −0.907783 − 0.419439i \(-0.862227\pi\)
0.907783 − 0.419439i \(-0.137773\pi\)
\(44\) 0 0
\(45\) 606.527 + 1050.54i 0.299519 + 0.518783i
\(46\) 0 0
\(47\) 2227.48 + 1286.04i 1.00837 + 0.582180i 0.910713 − 0.413040i \(-0.135533\pi\)
0.0976531 + 0.995221i \(0.468866\pi\)
\(48\) 0 0
\(49\) −658.272 − 2309.00i −0.274166 − 0.961682i
\(50\) 0 0
\(51\) −762.161 − 440.034i −0.293026 − 0.169179i
\(52\) 0 0
\(53\) 788.661 + 1366.00i 0.280762 + 0.486294i 0.971573 − 0.236742i \(-0.0760795\pi\)
−0.690811 + 0.723036i \(0.742746\pi\)
\(54\) 0 0
\(55\) 3945.52i 1.30431i
\(56\) 0 0
\(57\) 4211.53 1.29625
\(58\) 0 0
\(59\) −4325.71 + 2497.45i −1.24266 + 0.717451i −0.969635 − 0.244555i \(-0.921358\pi\)
−0.273026 + 0.962007i \(0.588025\pi\)
\(60\) 0 0
\(61\) 800.975 − 1387.33i 0.215258 − 0.372838i −0.738094 − 0.674697i \(-0.764274\pi\)
0.953352 + 0.301860i \(0.0976074\pi\)
\(62\) 0 0
\(63\) 473.190 + 1117.14i 0.119222 + 0.281467i
\(64\) 0 0
\(65\) −4773.50 + 8267.94i −1.12982 + 1.95691i
\(66\) 0 0
\(67\) 3251.75 − 1877.40i 0.724382 − 0.418222i −0.0919815 − 0.995761i \(-0.529320\pi\)
0.816363 + 0.577539i \(0.195987\pi\)
\(68\) 0 0
\(69\) 1295.77 0.272163
\(70\) 0 0
\(71\) 4041.40i 0.801707i 0.916142 + 0.400853i \(0.131286\pi\)
−0.916142 + 0.400853i \(0.868714\pi\)
\(72\) 0 0
\(73\) 3635.17 + 6296.30i 0.682148 + 1.18152i 0.974324 + 0.225151i \(0.0722875\pi\)
−0.292176 + 0.956365i \(0.594379\pi\)
\(74\) 0 0
\(75\) −11530.0 − 6656.87i −2.04979 − 1.18344i
\(76\) 0 0
\(77\) −483.904 + 3916.30i −0.0816164 + 0.660533i
\(78\) 0 0
\(79\) −234.672 − 135.488i −0.0376017 − 0.0217093i 0.481081 − 0.876676i \(-0.340244\pi\)
−0.518683 + 0.854967i \(0.673578\pi\)
\(80\) 0 0
\(81\) 1971.21 + 3414.24i 0.300444 + 0.520384i
\(82\) 0 0
\(83\) 3468.88i 0.503538i 0.967787 + 0.251769i \(0.0810124\pi\)
−0.967787 + 0.251769i \(0.918988\pi\)
\(84\) 0 0
\(85\) −5749.46 −0.795772
\(86\) 0 0
\(87\) 1017.39 − 587.389i 0.134415 − 0.0776046i
\(88\) 0 0
\(89\) 4860.18 − 8418.08i 0.613582 − 1.06275i −0.377050 − 0.926193i \(-0.623061\pi\)
0.990632 − 0.136562i \(-0.0436052\pi\)
\(90\) 0 0
\(91\) −5752.17 + 7621.25i −0.694623 + 0.920330i
\(92\) 0 0
\(93\) 5490.65 − 9510.09i 0.634831 − 1.09956i
\(94\) 0 0
\(95\) 23827.7 − 13756.9i 2.64018 − 1.52431i
\(96\) 0 0
\(97\) 8062.39 0.856881 0.428440 − 0.903570i \(-0.359063\pi\)
0.428440 + 0.903570i \(0.359063\pi\)
\(98\) 0 0
\(99\) − 1993.96i − 0.203444i
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 112.5.r.b.79.4 yes 10
4.3 odd 2 112.5.r.a.79.2 ✓ 10
7.2 even 3 784.5.d.k.687.8 10
7.4 even 3 112.5.r.a.95.2 yes 10
7.5 odd 6 784.5.d.l.687.3 10
28.11 odd 6 inner 112.5.r.b.95.4 yes 10
28.19 even 6 784.5.d.l.687.8 10
28.23 odd 6 784.5.d.k.687.3 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.5.r.a.79.2 ✓ 10 4.3 odd 2
112.5.r.a.95.2 yes 10 7.4 even 3
112.5.r.b.79.4 yes 10 1.1 even 1 trivial
112.5.r.b.95.4 yes 10 28.11 odd 6 inner
784.5.d.k.687.3 10 28.23 odd 6
784.5.d.k.687.8 10 7.2 even 3
784.5.d.l.687.3 10 7.5 odd 6
784.5.d.l.687.8 10 28.19 even 6