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(17,112)] 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("112.17"); 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([0, 0, 1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 112.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,-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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{6})\)
Coefficient field: 6.0.11337408.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 18x^{4} + 81x^{2} + 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{3}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 28)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 17.2
Root \(-0.391571i\) of defining polynomial
Character \(\chi\) \(=\) 112.17
Dual form 112.5.s.c.33.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.35901 + 0.784623i) q^{3} +(38.3327 - 22.1314i) q^{5} +(-42.3967 - 24.5667i) q^{7} +(-39.2687 - 68.0154i) q^{9} +(11.7557 - 20.3615i) q^{11} +136.269i q^{13} +69.4592 q^{15} +(-227.333 - 131.251i) q^{17} +(387.162 - 223.528i) q^{19} +(-38.3418 - 66.6517i) q^{21} +(-374.585 - 648.800i) q^{23} +(667.098 - 1155.45i) q^{25} -250.353i q^{27} +406.524 q^{29} +(584.199 + 337.287i) q^{31} +(31.9521 - 18.4476i) q^{33} +(-2168.87 - 3.40909i) q^{35} +(-372.666 - 645.476i) q^{37} +(-106.920 + 185.191i) q^{39} +2476.20i q^{41} +2636.68 q^{43} +(-3010.55 - 1738.14i) q^{45} +(579.099 - 334.343i) q^{47} +(1193.96 + 2083.09i) q^{49} +(-205.964 - 356.741i) q^{51} +(1014.61 - 1757.36i) q^{53} -1040.68i q^{55} +701.541 q^{57} +(1014.22 + 585.561i) q^{59} +(-2022.05 + 1167.43i) q^{61} +(-6.04891 + 3848.33i) q^{63} +(3015.83 + 5223.58i) q^{65} +(-3779.79 + 6546.78i) q^{67} -1175.63i q^{69} -7575.36 q^{71} +(3284.18 + 1896.12i) q^{73} +(1813.18 - 1046.84i) q^{75} +(-998.615 + 574.460i) q^{77} +(3897.33 + 6750.37i) q^{79} +(-2984.33 + 5169.02i) q^{81} -2181.41i q^{83} -11619.0 q^{85} +(552.469 + 318.968i) q^{87} +(8050.38 - 4647.89i) q^{89} +(3347.69 - 5777.37i) q^{91} +(529.287 + 916.751i) q^{93} +(9893.98 - 17136.9i) q^{95} -9981.90i q^{97} -1846.52 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 9 q^{3} - 27 q^{5} - 66 q^{7} + 90 q^{9} - 135 q^{11} + 486 q^{15} - 1107 q^{17} + 747 q^{19} + 2169 q^{21} - 243 q^{23} + 1878 q^{25} - 540 q^{29} + 5355 q^{31} - 1863 q^{33} - 6021 q^{35} + 2355 q^{37}+ \cdots - 8100 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(15\) \(17\) \(85\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{6}\right)\) \(1\)

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
\(3\) 1.35901 + 0.784623i 0.151001 + 0.0871803i 0.573597 − 0.819138i \(-0.305548\pi\)
−0.422596 + 0.906318i \(0.638881\pi\)
\(4\) 0 0
\(5\) 38.3327 − 22.1314i 1.53331 − 0.885256i 0.534102 − 0.845420i \(-0.320650\pi\)
0.999206 − 0.0398360i \(-0.0126836\pi\)
\(6\) 0 0
\(7\) −42.3967 − 24.5667i −0.865238 − 0.501361i
\(8\) 0 0
\(9\) −39.2687 − 68.0154i −0.484799 − 0.839697i
\(10\) 0 0
\(11\) 11.7557 − 20.3615i 0.0971545 − 0.168276i −0.813351 − 0.581773i \(-0.802359\pi\)
0.910506 + 0.413496i \(0.135693\pi\)
\(12\) 0 0
\(13\) 136.269i 0.806328i 0.915128 + 0.403164i \(0.132090\pi\)
−0.915128 + 0.403164i \(0.867910\pi\)
\(14\) 0 0
\(15\) 69.4592 0.308708
\(16\) 0 0
\(17\) −227.333 − 131.251i −0.786618 − 0.454154i 0.0521523 − 0.998639i \(-0.483392\pi\)
−0.838771 + 0.544485i \(0.816725\pi\)
\(18\) 0 0
\(19\) 387.162 − 223.528i 1.07247 − 0.619191i 0.143615 − 0.989634i \(-0.454127\pi\)
0.928855 + 0.370442i \(0.120794\pi\)
\(20\) 0 0
\(21\) −38.3418 − 66.6517i −0.0869429 − 0.151138i
\(22\) 0 0
\(23\) −374.585 − 648.800i −0.708100 − 1.22646i −0.965561 − 0.260176i \(-0.916219\pi\)
0.257462 − 0.966288i \(-0.417114\pi\)
\(24\) 0 0
\(25\) 667.098 − 1155.45i 1.06736 − 1.84872i
\(26\) 0 0
\(27\) − 250.353i − 0.343420i
\(28\) 0 0
\(29\) 406.524 0.483382 0.241691 − 0.970353i \(-0.422298\pi\)
0.241691 + 0.970353i \(0.422298\pi\)
\(30\) 0 0
\(31\) 584.199 + 337.287i 0.607907 + 0.350975i 0.772146 − 0.635445i \(-0.219183\pi\)
−0.164239 + 0.986421i \(0.552517\pi\)
\(32\) 0 0
\(33\) 31.9521 − 18.4476i 0.0293408 − 0.0169399i
\(34\) 0 0
\(35\) −2168.87 − 3.40909i −1.77051 − 0.00278293i
\(36\) 0 0
\(37\) −372.666 − 645.476i −0.272218 − 0.471495i 0.697212 − 0.716865i \(-0.254424\pi\)
−0.969429 + 0.245370i \(0.921090\pi\)
\(38\) 0 0
\(39\) −106.920 + 185.191i −0.0702959 + 0.121756i
\(40\) 0 0
\(41\) 2476.20i 1.47305i 0.676410 + 0.736526i \(0.263535\pi\)
−0.676410 + 0.736526i \(0.736465\pi\)
\(42\) 0 0
\(43\) 2636.68 1.42601 0.713003 − 0.701161i \(-0.247335\pi\)
0.713003 + 0.701161i \(0.247335\pi\)
\(44\) 0 0
\(45\) −3010.55 − 1738.14i −1.48669 − 0.858343i
\(46\) 0 0
\(47\) 579.099 − 334.343i 0.262154 − 0.151355i −0.363163 − 0.931726i \(-0.618303\pi\)
0.625317 + 0.780371i \(0.284970\pi\)
\(48\) 0 0
\(49\) 1193.96 + 2083.09i 0.497275 + 0.867593i
\(50\) 0 0
\(51\) −205.964 − 356.741i −0.0791866 − 0.137155i
\(52\) 0 0
\(53\) 1014.61 − 1757.36i 0.361200 − 0.625617i −0.626959 − 0.779053i \(-0.715700\pi\)
0.988159 + 0.153436i \(0.0490338\pi\)
\(54\) 0 0
\(55\) − 1040.68i − 0.344026i
\(56\) 0 0
\(57\) 701.541 0.215925
\(58\) 0 0
\(59\) 1014.22 + 585.561i 0.291359 + 0.168216i 0.638555 − 0.769576i \(-0.279533\pi\)
−0.347195 + 0.937793i \(0.612866\pi\)
\(60\) 0 0
\(61\) −2022.05 + 1167.43i −0.543416 + 0.313741i −0.746462 − 0.665428i \(-0.768249\pi\)
0.203047 + 0.979169i \(0.434916\pi\)
\(62\) 0 0
\(63\) −6.04891 + 3848.33i −0.00152404 + 0.969597i
\(64\) 0 0
\(65\) 3015.83 + 5223.58i 0.713807 + 1.23635i
\(66\) 0 0
\(67\) −3779.79 + 6546.78i −0.842011 + 1.45841i 0.0461815 + 0.998933i \(0.485295\pi\)
−0.888192 + 0.459472i \(0.848039\pi\)
\(68\) 0 0
\(69\) − 1175.63i − 0.246929i
\(70\) 0 0
\(71\) −7575.36 −1.50275 −0.751375 − 0.659876i \(-0.770609\pi\)
−0.751375 + 0.659876i \(0.770609\pi\)
\(72\) 0 0
\(73\) 3284.18 + 1896.12i 0.616284 + 0.355812i 0.775421 − 0.631445i \(-0.217538\pi\)
−0.159137 + 0.987257i \(0.550871\pi\)
\(74\) 0 0
\(75\) 1813.18 − 1046.84i 0.322343 − 0.186105i
\(76\) 0 0
\(77\) −998.615 + 574.460i −0.168429 + 0.0968898i
\(78\) 0 0
\(79\) 3897.33 + 6750.37i 0.624472 + 1.08162i 0.988643 + 0.150285i \(0.0480192\pi\)
−0.364171 + 0.931332i \(0.618647\pi\)
\(80\) 0 0
\(81\) −2984.33 + 5169.02i −0.454860 + 0.787840i
\(82\) 0 0
\(83\) − 2181.41i − 0.316651i −0.987387 − 0.158326i \(-0.949390\pi\)
0.987387 − 0.158326i \(-0.0506096\pi\)
\(84\) 0 0
\(85\) −11619.0 −1.60817
\(86\) 0 0
\(87\) 552.469 + 318.968i 0.0729910 + 0.0421414i
\(88\) 0 0
\(89\) 8050.38 − 4647.89i 1.01633 − 0.586781i 0.103294 − 0.994651i \(-0.467062\pi\)
0.913040 + 0.407870i \(0.133728\pi\)
\(90\) 0 0
\(91\) 3347.69 − 5777.37i 0.404261 − 0.697666i
\(92\) 0 0
\(93\) 529.287 + 916.751i 0.0611963 + 0.105995i
\(94\) 0 0
\(95\) 9893.98 − 17136.9i 1.09629 − 1.89882i
\(96\) 0 0
\(97\) − 9981.90i − 1.06089i −0.847720 − 0.530444i \(-0.822025\pi\)
0.847720 − 0.530444i \(-0.177975\pi\)
\(98\) 0 0
\(99\) −1846.52 −0.188402
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.s.c.17.2 6
4.3 odd 2 28.5.h.a.17.2 yes 6
7.3 odd 6 784.5.c.e.97.4 6
7.4 even 3 784.5.c.e.97.3 6
7.5 odd 6 inner 112.5.s.c.33.2 6
12.11 even 2 252.5.z.f.73.1 6
20.3 even 4 700.5.o.a.549.4 12
20.7 even 4 700.5.o.a.549.3 12
20.19 odd 2 700.5.s.a.101.2 6
28.3 even 6 196.5.b.a.97.3 6
28.11 odd 6 196.5.b.a.97.4 6
28.19 even 6 28.5.h.a.5.2 ✓ 6
28.23 odd 6 196.5.h.c.117.2 6
28.27 even 2 196.5.h.c.129.2 6
84.47 odd 6 252.5.z.f.145.1 6
140.19 even 6 700.5.s.a.201.2 6
140.47 odd 12 700.5.o.a.649.4 12
140.103 odd 12 700.5.o.a.649.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
28.5.h.a.5.2 ✓ 6 28.19 even 6
28.5.h.a.17.2 yes 6 4.3 odd 2
112.5.s.c.17.2 6 1.1 even 1 trivial
112.5.s.c.33.2 6 7.5 odd 6 inner
196.5.b.a.97.3 6 28.3 even 6
196.5.b.a.97.4 6 28.11 odd 6
196.5.h.c.117.2 6 28.23 odd 6
196.5.h.c.129.2 6 28.27 even 2
252.5.z.f.73.1 6 12.11 even 2
252.5.z.f.145.1 6 84.47 odd 6
700.5.o.a.549.3 12 20.7 even 4
700.5.o.a.549.4 12 20.3 even 4
700.5.o.a.649.3 12 140.103 odd 12
700.5.o.a.649.4 12 140.47 odd 12
700.5.s.a.101.2 6 20.19 odd 2
700.5.s.a.201.2 6 140.19 even 6
784.5.c.e.97.3 6 7.4 even 3
784.5.c.e.97.4 6 7.3 odd 6