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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [196,5,Mod(117,196)] 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("196.117"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(196, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 5])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 196 = 2^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 196.h (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,12] 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: \(20.2605127644\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 28)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 117.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 196.117
Dual form 196.5.h.b.129.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(6.00000 - 3.46410i) q^{3} +(-18.0000 - 10.3923i) q^{5} +(-16.5000 + 28.5788i) q^{9} +(-9.00000 - 15.5885i) q^{11} -131.636i q^{13} -144.000 q^{15} +(-360.000 + 207.846i) q^{17} +(-78.0000 - 45.0333i) q^{19} +(-369.000 + 639.127i) q^{23} +(-96.5000 - 167.143i) q^{25} +789.815i q^{27} -846.000 q^{29} +(1008.00 - 581.969i) q^{31} +(-108.000 - 62.3538i) q^{33} +(-1193.00 + 2066.34i) q^{37} +(-456.000 - 789.815i) q^{39} -1704.34i q^{41} -2510.00 q^{43} +(594.000 - 342.946i) q^{45} +(2952.00 + 1704.34i) q^{47} +(-1440.00 + 2494.15i) q^{51} +(135.000 + 233.827i) q^{53} +374.123i q^{55} -624.000 q^{57} +(-2718.00 + 1569.24i) q^{59} +(-5622.00 - 3245.86i) q^{61} +(-1368.00 + 2369.45i) q^{65} +(-1225.00 - 2121.76i) q^{67} +5113.01i q^{69} -3150.00 q^{71} +(204.000 - 117.779i) q^{73} +(-1158.00 - 668.572i) q^{75} +(1991.00 - 3448.51i) q^{79} +(1399.50 + 2424.01i) q^{81} -5009.09i q^{83} +8640.00 q^{85} +(-5076.00 + 2930.63i) q^{87} +(6588.00 + 3803.58i) q^{89} +(4032.00 - 6983.63i) q^{93} +(936.000 + 1621.20i) q^{95} -12581.6i q^{97} +594.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 12 q^{3} - 36 q^{5} - 33 q^{9} - 18 q^{11} - 288 q^{15} - 720 q^{17} - 156 q^{19} - 738 q^{23} - 193 q^{25} - 1692 q^{29} + 2016 q^{31} - 216 q^{33} - 2386 q^{37} - 912 q^{39} - 5020 q^{43} + 1188 q^{45}+ \cdots + 1188 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/196\mathbb{Z}\right)^\times\).

\(n\) \(99\) \(101\)
\(\chi(n)\) \(1\) \(e\left(\frac{5}{6}\right)\)

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
\(3\) 6.00000 − 3.46410i 0.666667 − 0.384900i −0.128146 − 0.991755i \(-0.540903\pi\)
0.794812 + 0.606855i \(0.207569\pi\)
\(4\) 0 0
\(5\) −18.0000 − 10.3923i −0.720000 − 0.415692i 0.0947527 − 0.995501i \(-0.469794\pi\)
−0.814753 + 0.579809i \(0.803127\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −16.5000 + 28.5788i −0.203704 + 0.352825i
\(10\) 0 0
\(11\) −9.00000 − 15.5885i −0.0743802 − 0.128830i 0.826436 − 0.563030i \(-0.190364\pi\)
−0.900817 + 0.434200i \(0.857031\pi\)
\(12\) 0 0
\(13\) − 131.636i − 0.778910i −0.921045 − 0.389455i \(-0.872663\pi\)
0.921045 − 0.389455i \(-0.127337\pi\)
\(14\) 0 0
\(15\) −144.000 −0.640000
\(16\) 0 0
\(17\) −360.000 + 207.846i −1.24567 + 0.719191i −0.970244 − 0.242130i \(-0.922154\pi\)
−0.275431 + 0.961321i \(0.588821\pi\)
\(18\) 0 0
\(19\) −78.0000 − 45.0333i −0.216066 − 0.124746i 0.388061 − 0.921634i \(-0.373145\pi\)
−0.604128 + 0.796888i \(0.706478\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −369.000 + 639.127i −0.697543 + 1.20818i 0.271774 + 0.962361i \(0.412390\pi\)
−0.969316 + 0.245818i \(0.920943\pi\)
\(24\) 0 0
\(25\) −96.5000 − 167.143i −0.154400 − 0.267429i
\(26\) 0 0
\(27\) 789.815i 1.08342i
\(28\) 0 0
\(29\) −846.000 −1.00595 −0.502973 − 0.864302i \(-0.667760\pi\)
−0.502973 + 0.864302i \(0.667760\pi\)
\(30\) 0 0
\(31\) 1008.00 − 581.969i 1.04891 − 0.605587i 0.126564 − 0.991958i \(-0.459605\pi\)
0.922343 + 0.386371i \(0.126272\pi\)
\(32\) 0 0
\(33\) −108.000 − 62.3538i −0.0991736 − 0.0572579i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1193.00 + 2066.34i −0.871439 + 1.50938i −0.0109307 + 0.999940i \(0.503479\pi\)
−0.860508 + 0.509436i \(0.829854\pi\)
\(38\) 0 0
\(39\) −456.000 − 789.815i −0.299803 − 0.519274i
\(40\) 0 0
\(41\) − 1704.34i − 1.01388i −0.861980 − 0.506942i \(-0.830776\pi\)
0.861980 − 0.506942i \(-0.169224\pi\)
\(42\) 0 0
\(43\) −2510.00 −1.35749 −0.678745 − 0.734374i \(-0.737476\pi\)
−0.678745 + 0.734374i \(0.737476\pi\)
\(44\) 0 0
\(45\) 594.000 − 342.946i 0.293333 − 0.169356i
\(46\) 0 0
\(47\) 2952.00 + 1704.34i 1.33635 + 0.771543i 0.986264 − 0.165174i \(-0.0528187\pi\)
0.350087 + 0.936717i \(0.386152\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −1440.00 + 2494.15i −0.553633 + 0.958921i
\(52\) 0 0
\(53\) 135.000 + 233.827i 0.0480598 + 0.0832420i 0.889055 − 0.457801i \(-0.151363\pi\)
−0.840995 + 0.541043i \(0.818030\pi\)
\(54\) 0 0
\(55\) 374.123i 0.123677i
\(56\) 0 0
\(57\) −624.000 −0.192059
\(58\) 0 0
\(59\) −2718.00 + 1569.24i −0.780810 + 0.450801i −0.836717 − 0.547635i \(-0.815528\pi\)
0.0559072 + 0.998436i \(0.482195\pi\)
\(60\) 0 0
\(61\) −5622.00 − 3245.86i −1.51088 − 0.872309i −0.999919 − 0.0127078i \(-0.995955\pi\)
−0.510965 − 0.859602i \(-0.670712\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1368.00 + 2369.45i −0.323787 + 0.560816i
\(66\) 0 0
\(67\) −1225.00 − 2121.76i −0.272889 − 0.472658i 0.696711 − 0.717352i \(-0.254646\pi\)
−0.969600 + 0.244694i \(0.921313\pi\)
\(68\) 0 0
\(69\) 5113.01i 1.07394i
\(70\) 0 0
\(71\) −3150.00 −0.624876 −0.312438 − 0.949938i \(-0.601146\pi\)
−0.312438 + 0.949938i \(0.601146\pi\)
\(72\) 0 0
\(73\) 204.000 − 117.779i 0.0382811 − 0.0221016i −0.480737 − 0.876865i \(-0.659631\pi\)
0.519018 + 0.854763i \(0.326298\pi\)
\(74\) 0 0
\(75\) −1158.00 − 668.572i −0.205867 − 0.118857i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 1991.00 − 3448.51i 0.319019 − 0.552558i −0.661264 − 0.750153i \(-0.729980\pi\)
0.980284 + 0.197595i \(0.0633132\pi\)
\(80\) 0 0
\(81\) 1399.50 + 2424.01i 0.213306 + 0.369457i
\(82\) 0 0
\(83\) − 5009.09i − 0.727114i −0.931572 − 0.363557i \(-0.881562\pi\)
0.931572 − 0.363557i \(-0.118438\pi\)
\(84\) 0 0
\(85\) 8640.00 1.19585
\(86\) 0 0
\(87\) −5076.00 + 2930.63i −0.670630 + 0.387189i
\(88\) 0 0
\(89\) 6588.00 + 3803.58i 0.831713 + 0.480190i 0.854439 − 0.519552i \(-0.173901\pi\)
−0.0227257 + 0.999742i \(0.507234\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 4032.00 − 6983.63i 0.466181 − 0.807449i
\(94\) 0 0
\(95\) 936.000 + 1621.20i 0.103712 + 0.179634i
\(96\) 0 0
\(97\) − 12581.6i − 1.33719i −0.743627 − 0.668595i \(-0.766896\pi\)
0.743627 − 0.668595i \(-0.233104\pi\)
\(98\) 0 0
\(99\) 594.000 0.0606061
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 196.5.h.b.117.1 2
7.2 even 3 28.5.b.a.13.2 yes 2
7.3 odd 6 inner 196.5.h.b.129.1 2
7.4 even 3 196.5.h.a.129.1 2
7.5 odd 6 28.5.b.a.13.1 ✓ 2
7.6 odd 2 196.5.h.a.117.1 2
21.2 odd 6 252.5.d.a.181.1 2
21.5 even 6 252.5.d.a.181.2 2
28.19 even 6 112.5.c.b.97.2 2
28.23 odd 6 112.5.c.b.97.1 2
35.2 odd 12 700.5.h.a.349.3 4
35.9 even 6 700.5.d.a.601.1 2
35.12 even 12 700.5.h.a.349.1 4
35.19 odd 6 700.5.d.a.601.2 2
35.23 odd 12 700.5.h.a.349.2 4
35.33 even 12 700.5.h.a.349.4 4
56.5 odd 6 448.5.c.c.321.2 2
56.19 even 6 448.5.c.d.321.1 2
56.37 even 6 448.5.c.c.321.1 2
56.51 odd 6 448.5.c.d.321.2 2
84.23 even 6 1008.5.f.c.433.1 2
84.47 odd 6 1008.5.f.c.433.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
28.5.b.a.13.1 ✓ 2 7.5 odd 6
28.5.b.a.13.2 yes 2 7.2 even 3
112.5.c.b.97.1 2 28.23 odd 6
112.5.c.b.97.2 2 28.19 even 6
196.5.h.a.117.1 2 7.6 odd 2
196.5.h.a.129.1 2 7.4 even 3
196.5.h.b.117.1 2 1.1 even 1 trivial
196.5.h.b.129.1 2 7.3 odd 6 inner
252.5.d.a.181.1 2 21.2 odd 6
252.5.d.a.181.2 2 21.5 even 6
448.5.c.c.321.1 2 56.37 even 6
448.5.c.c.321.2 2 56.5 odd 6
448.5.c.d.321.1 2 56.19 even 6
448.5.c.d.321.2 2 56.51 odd 6
700.5.d.a.601.1 2 35.9 even 6
700.5.d.a.601.2 2 35.19 odd 6
700.5.h.a.349.1 4 35.12 even 12
700.5.h.a.349.2 4 35.23 odd 12
700.5.h.a.349.3 4 35.2 odd 12
700.5.h.a.349.4 4 35.33 even 12
1008.5.f.c.433.1 2 84.23 even 6
1008.5.f.c.433.2 2 84.47 odd 6