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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [112,5,Mod(15,112)] 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("112.15"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 112.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.5774358654\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-7}, \sqrt{13})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 3x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 7^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 15.4
Root \(-1.80278 + 1.32288i\) of defining polynomial
Character \(\chi\) \(=\) 112.15
Dual form 112.5.d.a.15.1

$q$-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)
 
\(f(q)\) \(=\) \(q+12.1851i q^{3} -34.2389 q^{5} -18.5203i q^{7} -67.4777 q^{9} -98.4513i q^{11} -106.717 q^{13} -417.205i q^{15} +372.389 q^{17} -113.547i q^{19} +225.672 q^{21} -323.605i q^{23} +547.299 q^{25} +164.771i q^{27} -891.255 q^{29} -1839.41i q^{31} +1199.64 q^{33} +634.113i q^{35} -2416.63 q^{37} -1300.36i q^{39} -1993.94 q^{41} -3504.47i q^{43} +2310.36 q^{45} +3734.47i q^{47} -343.000 q^{49} +4537.61i q^{51} -1513.82 q^{53} +3370.86i q^{55} +1383.59 q^{57} +2383.63i q^{59} +2346.18 q^{61} +1249.70i q^{63} +3653.85 q^{65} +1005.74i q^{67} +3943.17 q^{69} -6658.06i q^{71} +2533.94 q^{73} +6668.92i q^{75} -1823.34 q^{77} -321.549i q^{79} -7473.45 q^{81} +12507.9i q^{83} -12750.2 q^{85} -10860.1i q^{87} -11057.4 q^{89} +1976.42i q^{91} +22413.5 q^{93} +3887.72i q^{95} -9417.83 q^{97} +6643.27i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 36 q^{5} - 68 q^{9} - 124 q^{13} + 480 q^{17} + 196 q^{21} + 372 q^{25} - 1344 q^{29} + 1568 q^{33} - 1792 q^{37} - 2928 q^{41} + 5708 q^{45} - 1372 q^{49} - 4440 q^{53} - 2744 q^{57} + 4236 q^{61}+ \cdots - 22528 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(15\) \(17\) \(85\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) 12.1851i 1.35390i 0.736027 + 0.676952i \(0.236700\pi\)
−0.736027 + 0.676952i \(0.763300\pi\)
\(4\) 0 0
\(5\) −34.2389 −1.36955 −0.684777 − 0.728753i \(-0.740100\pi\)
−0.684777 + 0.728753i \(0.740100\pi\)
\(6\) 0 0
\(7\) − 18.5203i − 0.377964i
\(8\) 0 0
\(9\) −67.4777 −0.833058
\(10\) 0 0
\(11\) − 98.4513i − 0.813647i −0.913507 − 0.406824i \(-0.866636\pi\)
0.913507 − 0.406824i \(-0.133364\pi\)
\(12\) 0 0
\(13\) −106.717 −0.631459 −0.315730 − 0.948849i \(-0.602249\pi\)
−0.315730 + 0.948849i \(0.602249\pi\)
\(14\) 0 0
\(15\) − 417.205i − 1.85425i
\(16\) 0 0
\(17\) 372.389 1.28854 0.644271 − 0.764797i \(-0.277161\pi\)
0.644271 + 0.764797i \(0.277161\pi\)
\(18\) 0 0
\(19\) − 113.547i − 0.314535i −0.987556 − 0.157267i \(-0.949732\pi\)
0.987556 − 0.157267i \(-0.0502684\pi\)
\(20\) 0 0
\(21\) 225.672 0.511728
\(22\) 0 0
\(23\) − 323.605i − 0.611730i −0.952075 − 0.305865i \(-0.901054\pi\)
0.952075 − 0.305865i \(-0.0989456\pi\)
\(24\) 0 0
\(25\) 547.299 0.875679
\(26\) 0 0
\(27\) 164.771i 0.226023i
\(28\) 0 0
\(29\) −891.255 −1.05976 −0.529878 − 0.848074i \(-0.677762\pi\)
−0.529878 + 0.848074i \(0.677762\pi\)
\(30\) 0 0
\(31\) − 1839.41i − 1.91406i −0.289986 − 0.957031i \(-0.593651\pi\)
0.289986 − 0.957031i \(-0.406349\pi\)
\(32\) 0 0
\(33\) 1199.64 1.10160
\(34\) 0 0
\(35\) 634.113i 0.517643i
\(36\) 0 0
\(37\) −2416.63 −1.76525 −0.882626 − 0.470075i \(-0.844227\pi\)
−0.882626 + 0.470075i \(0.844227\pi\)
\(38\) 0 0
\(39\) − 1300.36i − 0.854935i
\(40\) 0 0
\(41\) −1993.94 −1.18616 −0.593082 − 0.805142i \(-0.702089\pi\)
−0.593082 + 0.805142i \(0.702089\pi\)
\(42\) 0 0
\(43\) − 3504.47i − 1.89533i −0.319262 − 0.947666i \(-0.603435\pi\)
0.319262 − 0.947666i \(-0.396565\pi\)
\(44\) 0 0
\(45\) 2310.36 1.14092
\(46\) 0 0
\(47\) 3734.47i 1.69057i 0.534314 + 0.845286i \(0.320570\pi\)
−0.534314 + 0.845286i \(0.679430\pi\)
\(48\) 0 0
\(49\) −343.000 −0.142857
\(50\) 0 0
\(51\) 4537.61i 1.74456i
\(52\) 0 0
\(53\) −1513.82 −0.538918 −0.269459 − 0.963012i \(-0.586845\pi\)
−0.269459 + 0.963012i \(0.586845\pi\)
\(54\) 0 0
\(55\) 3370.86i 1.11433i
\(56\) 0 0
\(57\) 1383.59 0.425850
\(58\) 0 0
\(59\) 2383.63i 0.684755i 0.939562 + 0.342378i \(0.111232\pi\)
−0.939562 + 0.342378i \(0.888768\pi\)
\(60\) 0 0
\(61\) 2346.18 0.630525 0.315262 − 0.949005i \(-0.397908\pi\)
0.315262 + 0.949005i \(0.397908\pi\)
\(62\) 0 0
\(63\) 1249.70i 0.314866i
\(64\) 0 0
\(65\) 3653.85 0.864817
\(66\) 0 0
\(67\) 1005.74i 0.224046i 0.993706 + 0.112023i \(0.0357330\pi\)
−0.993706 + 0.112023i \(0.964267\pi\)
\(68\) 0 0
\(69\) 3943.17 0.828224
\(70\) 0 0
\(71\) − 6658.06i − 1.32078i −0.750923 − 0.660390i \(-0.770391\pi\)
0.750923 − 0.660390i \(-0.229609\pi\)
\(72\) 0 0
\(73\) 2533.94 0.475499 0.237750 − 0.971326i \(-0.423590\pi\)
0.237750 + 0.971326i \(0.423590\pi\)
\(74\) 0 0
\(75\) 6668.92i 1.18559i
\(76\) 0 0
\(77\) −1823.34 −0.307530
\(78\) 0 0
\(79\) − 321.549i − 0.0515220i −0.999668 − 0.0257610i \(-0.991799\pi\)
0.999668 − 0.0257610i \(-0.00820089\pi\)
\(80\) 0 0
\(81\) −7473.45 −1.13907
\(82\) 0 0
\(83\) 12507.9i 1.81563i 0.419374 + 0.907814i \(0.362250\pi\)
−0.419374 + 0.907814i \(0.637750\pi\)
\(84\) 0 0
\(85\) −12750.2 −1.76473
\(86\) 0 0
\(87\) − 10860.1i − 1.43481i
\(88\) 0 0
\(89\) −11057.4 −1.39596 −0.697978 − 0.716119i \(-0.745917\pi\)
−0.697978 + 0.716119i \(0.745917\pi\)
\(90\) 0 0
\(91\) 1976.42i 0.238669i
\(92\) 0 0
\(93\) 22413.5 2.59146
\(94\) 0 0
\(95\) 3887.72i 0.430772i
\(96\) 0 0
\(97\) −9417.83 −1.00094 −0.500469 − 0.865754i \(-0.666839\pi\)
−0.500469 + 0.865754i \(0.666839\pi\)
\(98\) 0 0
\(99\) 6643.27i 0.677816i
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.d.a.15.4 yes 4
3.2 odd 2 1008.5.m.a.127.3 4
4.3 odd 2 inner 112.5.d.a.15.1 ✓ 4
7.6 odd 2 784.5.d.f.687.1 4
8.3 odd 2 448.5.d.b.127.4 4
8.5 even 2 448.5.d.b.127.1 4
12.11 even 2 1008.5.m.a.127.4 4
28.27 even 2 784.5.d.f.687.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.5.d.a.15.1 ✓ 4 4.3 odd 2 inner
112.5.d.a.15.4 yes 4 1.1 even 1 trivial
448.5.d.b.127.1 4 8.5 even 2
448.5.d.b.127.4 4 8.3 odd 2
784.5.d.f.687.1 4 7.6 odd 2
784.5.d.f.687.4 4 28.27 even 2
1008.5.m.a.127.3 4 3.2 odd 2
1008.5.m.a.127.4 4 12.11 even 2