Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [112,8,Mod(65,112)] 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("112.65"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 112.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,56] 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: \(34.9871228542\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{949})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 238x^{2} + 237x + 56169 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 65.2
Root \(7.95146 + 13.7723i\) of defining polynomial
Character \(\chi\) \(=\) 112.65
Dual form 112.8.i.b.81.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(29.4029 + 50.9274i) q^{3} +(-212.141 + 367.439i) q^{5} +(30.7182 + 906.973i) q^{7} +(-635.564 + 1100.83i) q^{9} +(3944.43 + 6831.96i) q^{11} +6717.46 q^{13} -24950.2 q^{15} +(3537.51 + 6127.14i) q^{17} +(-13124.6 + 22732.4i) q^{19} +(-45286.5 + 28232.0i) q^{21} +(6035.27 - 10453.4i) q^{23} +(-50945.0 - 88239.4i) q^{25} +53858.7 q^{27} +3061.25 q^{29} +(-59262.6 - 102646. i) q^{31} +(-231956. + 401759. i) q^{33} +(-339774. - 181119. i) q^{35} +(229876. - 398156. i) q^{37} +(197513. + 342102. i) q^{39} +316857. q^{41} +31624.2 q^{43} +(-269658. - 467062. i) q^{45} +(403277. - 698496. i) q^{47} +(-821656. + 55721.1i) q^{49} +(-208026. + 360312. i) q^{51} +(-239664. - 415110. i) q^{53} -3.34710e6 q^{55} -1.54360e6 q^{57} +(337431. + 584448. i) q^{59} +(283165. - 490455. i) q^{61} +(-1.01794e6 - 542623. i) q^{63} +(-1.42505e6 + 2.46825e6i) q^{65} +(592349. + 1.02598e6i) q^{67} +709818. q^{69} -4.61736e6 q^{71} +(-1.52400e6 - 2.63965e6i) q^{73} +(2.99587e6 - 5.18899e6i) q^{75} +(-6.07524e6 + 3.78736e6i) q^{77} +(-3.45080e6 + 5.97696e6i) q^{79} +(2.97358e6 + 5.15039e6i) q^{81} +9.01927e6 q^{83} -3.00180e6 q^{85} +(90009.8 + 155902. i) q^{87} +(3.50739e6 - 6.07498e6i) q^{89} +(206348. + 6.09255e6i) q^{91} +(3.48499e6 - 6.03617e6i) q^{93} +(-5.56852e6 - 9.64496e6i) q^{95} +8.60029e6 q^{97} -1.00278e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 56 q^{3} + 14 q^{5} + 1848 q^{7} + 908 q^{9} + 2408 q^{11} + 21448 q^{13} - 52360 q^{15} + 35098 q^{17} - 2408 q^{19} - 92302 q^{21} - 61684 q^{23} - 215856 q^{25} + 83216 q^{27} - 191320 q^{29} - 166012 q^{31}+ \cdots - 43942528 q^{99}+O(q^{100}) \) 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
\(3\) 29.4029 + 50.9274i 0.628733 + 1.08900i 0.987806 + 0.155688i \(0.0497594\pi\)
−0.359074 + 0.933309i \(0.616907\pi\)
\(4\) 0 0
\(5\) −212.141 + 367.439i −0.758978 + 1.31459i 0.184394 + 0.982852i \(0.440968\pi\)
−0.943372 + 0.331737i \(0.892365\pi\)
\(6\) 0 0
\(7\) 30.7182 + 906.973i 0.0338495 + 0.999427i
\(8\) 0 0
\(9\) −635.564 + 1100.83i −0.290610 + 0.503351i
\(10\) 0 0
\(11\) 3944.43 + 6831.96i 0.893532 + 1.54764i 0.835611 + 0.549322i \(0.185114\pi\)
0.0579219 + 0.998321i \(0.481553\pi\)
\(12\) 0 0
\(13\) 6717.46 0.848014 0.424007 − 0.905659i \(-0.360623\pi\)
0.424007 + 0.905659i \(0.360623\pi\)
\(14\) 0 0
\(15\) −24950.2 −1.90878
\(16\) 0 0
\(17\) 3537.51 + 6127.14i 0.174633 + 0.302473i 0.940034 − 0.341080i \(-0.110793\pi\)
−0.765401 + 0.643553i \(0.777459\pi\)
\(18\) 0 0
\(19\) −13124.6 + 22732.4i −0.438983 + 0.760341i −0.997611 − 0.0690773i \(-0.977994\pi\)
0.558628 + 0.829418i \(0.311328\pi\)
\(20\) 0 0
\(21\) −45286.5 + 28232.0i −1.06709 + 0.665234i
\(22\) 0 0
\(23\) 6035.27 − 10453.4i 0.103431 − 0.179147i −0.809665 − 0.586892i \(-0.800351\pi\)
0.913096 + 0.407745i \(0.133685\pi\)
\(24\) 0 0
\(25\) −50945.0 − 88239.4i −0.652096 − 1.12946i
\(26\) 0 0
\(27\) 53858.7 0.526602
\(28\) 0 0
\(29\) 3061.25 0.0233081 0.0116540 − 0.999932i \(-0.496290\pi\)
0.0116540 + 0.999932i \(0.496290\pi\)
\(30\) 0 0
\(31\) −59262.6 − 102646.i −0.357285 − 0.618835i 0.630221 − 0.776415i \(-0.282964\pi\)
−0.987506 + 0.157580i \(0.949631\pi\)
\(32\) 0 0
\(33\) −231956. + 401759.i −1.12359 + 1.94611i
\(34\) 0 0
\(35\) −339774. − 181119.i −1.33953 − 0.714045i
\(36\) 0 0
\(37\) 229876. − 398156.i 0.746083 − 1.29225i −0.203604 − 0.979053i \(-0.565266\pi\)
0.949687 − 0.313200i \(-0.101401\pi\)
\(38\) 0 0
\(39\) 197513. + 342102.i 0.533174 + 0.923485i
\(40\) 0 0
\(41\) 316857. 0.717992 0.358996 − 0.933339i \(-0.383119\pi\)
0.358996 + 0.933339i \(0.383119\pi\)
\(42\) 0 0
\(43\) 31624.2 0.0606569 0.0303284 − 0.999540i \(-0.490345\pi\)
0.0303284 + 0.999540i \(0.490345\pi\)
\(44\) 0 0
\(45\) −269658. − 467062.i −0.441133 − 0.764065i
\(46\) 0 0
\(47\) 403277. − 698496.i 0.566579 − 0.981344i −0.430322 − 0.902676i \(-0.641600\pi\)
0.996901 − 0.0786682i \(-0.0250668\pi\)
\(48\) 0 0
\(49\) −821656. + 55721.1i −0.997708 + 0.0676602i
\(50\) 0 0
\(51\) −208026. + 360312.i −0.219595 + 0.380349i
\(52\) 0 0
\(53\) −239664. − 415110.i −0.221125 − 0.382999i 0.734025 − 0.679122i \(-0.237639\pi\)
−0.955150 + 0.296123i \(0.904306\pi\)
\(54\) 0 0
\(55\) −3.34710e6 −2.71269
\(56\) 0 0
\(57\) −1.54360e6 −1.10401
\(58\) 0 0
\(59\) 337431. + 584448.i 0.213896 + 0.370479i 0.952931 − 0.303189i \(-0.0980512\pi\)
−0.739034 + 0.673668i \(0.764718\pi\)
\(60\) 0 0
\(61\) 283165. − 490455.i 0.159729 − 0.276659i −0.775042 − 0.631910i \(-0.782271\pi\)
0.934771 + 0.355251i \(0.115605\pi\)
\(62\) 0 0
\(63\) −1.01794e6 − 542623.i −0.512899 − 0.273405i
\(64\) 0 0
\(65\) −1.42505e6 + 2.46825e6i −0.643625 + 1.11479i
\(66\) 0 0
\(67\) 592349. + 1.02598e6i 0.240611 + 0.416751i 0.960889 − 0.276935i \(-0.0893188\pi\)
−0.720277 + 0.693686i \(0.755985\pi\)
\(68\) 0 0
\(69\) 709818. 0.260121
\(70\) 0 0
\(71\) −4.61736e6 −1.53105 −0.765524 − 0.643407i \(-0.777520\pi\)
−0.765524 + 0.643407i \(0.777520\pi\)
\(72\) 0 0
\(73\) −1.52400e6 − 2.63965e6i −0.458517 − 0.794175i 0.540366 − 0.841430i \(-0.318286\pi\)
−0.998883 + 0.0472553i \(0.984953\pi\)
\(74\) 0 0
\(75\) 2.99587e6 − 5.18899e6i 0.819989 − 1.42026i
\(76\) 0 0
\(77\) −6.07524e6 + 3.78736e6i −1.51651 + 0.945407i
\(78\) 0 0
\(79\) −3.45080e6 + 5.97696e6i −0.787454 + 1.36391i 0.140069 + 0.990142i \(0.455268\pi\)
−0.927522 + 0.373768i \(0.878066\pi\)
\(80\) 0 0
\(81\) 2.97358e6 + 5.15039e6i 0.621702 + 1.07682i
\(82\) 0 0
\(83\) 9.01927e6 1.73140 0.865701 − 0.500562i \(-0.166873\pi\)
0.865701 + 0.500562i \(0.166873\pi\)
\(84\) 0 0
\(85\) −3.00180e6 −0.530170
\(86\) 0 0
\(87\) 90009.8 + 155902.i 0.0146545 + 0.0253824i
\(88\) 0 0
\(89\) 3.50739e6 − 6.07498e6i 0.527374 − 0.913439i −0.472116 − 0.881536i \(-0.656510\pi\)
0.999491 − 0.0319032i \(-0.0101568\pi\)
\(90\) 0 0
\(91\) 206348. + 6.09255e6i 0.0287049 + 0.847528i
\(92\) 0 0
\(93\) 3.48499e6 − 6.03617e6i 0.449273 − 0.778164i
\(94\) 0 0
\(95\) −5.56852e6 − 9.64496e6i −0.666357 − 1.15416i
\(96\) 0 0
\(97\) 8.60029e6 0.956779 0.478390 − 0.878148i \(-0.341221\pi\)
0.478390 + 0.878148i \(0.341221\pi\)
\(98\) 0 0
\(99\) −1.00278e7 −1.03868
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 112.8.i.b.65.2 4
4.3 odd 2 14.8.c.b.9.1 ✓ 4
7.4 even 3 inner 112.8.i.b.81.2 4
12.11 even 2 126.8.g.d.37.2 4
28.3 even 6 98.8.c.m.67.2 4
28.11 odd 6 14.8.c.b.11.1 yes 4
28.19 even 6 98.8.a.d.1.1 2
28.23 odd 6 98.8.a.f.1.2 2
28.27 even 2 98.8.c.m.79.2 4
84.11 even 6 126.8.g.d.109.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.8.c.b.9.1 ✓ 4 4.3 odd 2
14.8.c.b.11.1 yes 4 28.11 odd 6
98.8.a.d.1.1 2 28.19 even 6
98.8.a.f.1.2 2 28.23 odd 6
98.8.c.m.67.2 4 28.3 even 6
98.8.c.m.79.2 4 28.27 even 2
112.8.i.b.65.2 4 1.1 even 1 trivial
112.8.i.b.81.2 4 7.4 even 3 inner
126.8.g.d.37.2 4 12.11 even 2
126.8.g.d.109.2 4 84.11 even 6