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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [196,8,Mod(165,196)] 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("196.165"); S:= CuspForms(chi, 8); 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, 4])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 196 = 2^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 196.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-14,0,42] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(61.2274649949\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{3529})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 883x^{2} + 882x + 777924 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 28)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 165.2
Root \(-14.6013 + 25.2903i\) of defining polynomial
Character \(\chi\) \(=\) 196.165
Dual form 196.8.e.a.177.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(26.2027 - 45.3844i) q^{3} +(99.6081 + 172.526i) q^{5} +(-279.662 - 484.389i) q^{9} +(-609.487 + 1055.66i) q^{11} -6449.65 q^{13} +10440.0 q^{15} +(13393.5 - 23198.1i) q^{17} +(7455.47 + 12913.3i) q^{19} +(36750.5 + 63653.7i) q^{23} +(19219.0 - 33288.2i) q^{25} +85299.0 q^{27} -106453. q^{29} +(33408.3 - 57864.9i) q^{31} +(31940.4 + 55322.4i) q^{33} +(239751. + 415261. i) q^{37} +(-168998. + 292713. i) q^{39} +644539. q^{41} +145165. q^{43} +(55713.2 - 96498.2i) q^{45} +(505423. + 875418. i) q^{47} +(-701889. - 1.21571e6i) q^{51} +(-17104.3 + 29625.4i) q^{53} -242839. q^{55} +781414. q^{57} +(-221658. + 383924. i) q^{59} +(-570214. - 987640. i) q^{61} +(-642437. - 1.11273e6i) q^{65} +(2.15964e6 - 3.74061e6i) q^{67} +3.85185e6 q^{69} +2.54867e6 q^{71} +(-1.83861e6 + 3.18457e6i) q^{73} +(-1.00718e6 - 1.74448e6i) q^{75} +(-4.27782e6 - 7.40940e6i) q^{79} +(2.84668e6 - 4.93060e6i) q^{81} +1.79721e6 q^{83} +5.33639e6 q^{85} +(-2.78936e6 + 4.83132e6i) q^{87} +(2.78159e6 + 4.81785e6i) q^{89} +(-1.75078e6 - 3.03243e6i) q^{93} +(-1.48525e6 + 2.57253e6i) q^{95} +1.72057e7 q^{97} +681802. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 14 q^{3} + 42 q^{5} - 2782 q^{9} - 7428 q^{11} - 23660 q^{13} + 41760 q^{15} + 15792 q^{17} + 26614 q^{19} - 32640 q^{23} + 91846 q^{25} + 175336 q^{27} - 316032 q^{29} - 180740 q^{31} - 348432 q^{33}+ \cdots + 28964904 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(99\) \(101\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(3\) 26.2027 − 45.3844i 0.560301 − 0.970470i −0.437169 − 0.899380i \(-0.644019\pi\)
0.997470 − 0.0710906i \(-0.0226479\pi\)
\(4\) 0 0
\(5\) 99.6081 + 172.526i 0.356369 + 0.617249i 0.987351 − 0.158548i \(-0.0506814\pi\)
−0.630983 + 0.775797i \(0.717348\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −279.662 − 484.389i −0.127875 − 0.221486i
\(10\) 0 0
\(11\) −609.487 + 1055.66i −0.138067 + 0.239139i −0.926765 − 0.375642i \(-0.877422\pi\)
0.788698 + 0.614781i \(0.210756\pi\)
\(12\) 0 0
\(13\) −6449.65 −0.814206 −0.407103 − 0.913382i \(-0.633461\pi\)
−0.407103 + 0.913382i \(0.633461\pi\)
\(14\) 0 0
\(15\) 10440.0 0.798695
\(16\) 0 0
\(17\) 13393.5 − 23198.1i 0.661183 − 1.14520i −0.319123 − 0.947713i \(-0.603388\pi\)
0.980305 − 0.197488i \(-0.0632785\pi\)
\(18\) 0 0
\(19\) 7455.47 + 12913.3i 0.249366 + 0.431915i 0.963350 − 0.268247i \(-0.0864444\pi\)
−0.713984 + 0.700162i \(0.753111\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 36750.5 + 63653.7i 0.629819 + 1.09088i 0.987588 + 0.157068i \(0.0502041\pi\)
−0.357769 + 0.933810i \(0.616463\pi\)
\(24\) 0 0
\(25\) 19219.0 − 33288.2i 0.246003 − 0.426089i
\(26\) 0 0
\(27\) 85299.0 0.834009
\(28\) 0 0
\(29\) −106453. −0.810524 −0.405262 − 0.914200i \(-0.632820\pi\)
−0.405262 + 0.914200i \(0.632820\pi\)
\(30\) 0 0
\(31\) 33408.3 − 57864.9i 0.201414 − 0.348859i −0.747571 − 0.664182i \(-0.768780\pi\)
0.948984 + 0.315324i \(0.102113\pi\)
\(32\) 0 0
\(33\) 31940.4 + 55322.4i 0.154718 + 0.267980i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 239751. + 415261.i 0.778134 + 1.34777i 0.933016 + 0.359834i \(0.117167\pi\)
−0.154883 + 0.987933i \(0.549500\pi\)
\(38\) 0 0
\(39\) −168998. + 292713.i −0.456201 + 0.790163i
\(40\) 0 0
\(41\) 644539. 1.46051 0.730257 − 0.683173i \(-0.239401\pi\)
0.730257 + 0.683173i \(0.239401\pi\)
\(42\) 0 0
\(43\) 145165. 0.278434 0.139217 − 0.990262i \(-0.455541\pi\)
0.139217 + 0.990262i \(0.455541\pi\)
\(44\) 0 0
\(45\) 55713.2 − 96498.2i 0.0911412 − 0.157861i
\(46\) 0 0
\(47\) 505423. + 875418.i 0.710088 + 1.22991i 0.964824 + 0.262898i \(0.0846784\pi\)
−0.254735 + 0.967011i \(0.581988\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −701889. − 1.21571e6i −0.740923 − 1.28332i
\(52\) 0 0
\(53\) −17104.3 + 29625.4i −0.0157812 + 0.0273338i −0.873808 − 0.486271i \(-0.838357\pi\)
0.858027 + 0.513605i \(0.171690\pi\)
\(54\) 0 0
\(55\) −242839. −0.196811
\(56\) 0 0
\(57\) 781414. 0.558881
\(58\) 0 0
\(59\) −221658. + 383924.i −0.140508 + 0.243368i −0.927688 − 0.373356i \(-0.878207\pi\)
0.787180 + 0.616724i \(0.211540\pi\)
\(60\) 0 0
\(61\) −570214. − 987640.i −0.321650 − 0.557114i 0.659179 − 0.751986i \(-0.270904\pi\)
−0.980829 + 0.194872i \(0.937571\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −642437. − 1.11273e6i −0.290158 − 0.502568i
\(66\) 0 0
\(67\) 2.15964e6 − 3.74061e6i 0.877242 − 1.51943i 0.0228874 − 0.999738i \(-0.492714\pi\)
0.854355 − 0.519690i \(-0.173953\pi\)
\(68\) 0 0
\(69\) 3.85185e6 1.41155
\(70\) 0 0
\(71\) 2.54867e6 0.845103 0.422551 − 0.906339i \(-0.361135\pi\)
0.422551 + 0.906339i \(0.361135\pi\)
\(72\) 0 0
\(73\) −1.83861e6 + 3.18457e6i −0.553173 + 0.958123i 0.444871 + 0.895595i \(0.353250\pi\)
−0.998043 + 0.0625282i \(0.980084\pi\)
\(74\) 0 0
\(75\) −1.00718e6 − 1.74448e6i −0.275671 − 0.477477i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −4.27782e6 − 7.40940e6i −0.976174 − 1.69078i −0.676004 − 0.736898i \(-0.736290\pi\)
−0.300171 − 0.953885i \(-0.597044\pi\)
\(80\) 0 0
\(81\) 2.84668e6 − 4.93060e6i 0.595171 − 1.03087i
\(82\) 0 0
\(83\) 1.79721e6 0.345006 0.172503 − 0.985009i \(-0.444815\pi\)
0.172503 + 0.985009i \(0.444815\pi\)
\(84\) 0 0
\(85\) 5.33639e6 0.942499
\(86\) 0 0
\(87\) −2.78936e6 + 4.83132e6i −0.454138 + 0.786590i
\(88\) 0 0
\(89\) 2.78159e6 + 4.81785e6i 0.418242 + 0.724416i 0.995763 − 0.0919598i \(-0.0293131\pi\)
−0.577521 + 0.816376i \(0.695980\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −1.75078e6 − 3.03243e6i −0.225705 − 0.390932i
\(94\) 0 0
\(95\) −1.48525e6 + 2.57253e6i −0.177733 + 0.307842i
\(96\) 0 0
\(97\) 1.72057e7 1.91413 0.957064 − 0.289877i \(-0.0936144\pi\)
0.957064 + 0.289877i \(0.0936144\pi\)
\(98\) 0 0
\(99\) 681802. 0.0706212
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 196.8.e.a.165.2 4
7.2 even 3 inner 196.8.e.a.177.2 4
7.3 odd 6 28.8.a.a.1.2 ✓ 2
7.4 even 3 196.8.a.b.1.1 2
7.5 odd 6 196.8.e.d.177.1 4
7.6 odd 2 196.8.e.d.165.1 4
21.17 even 6 252.8.a.e.1.1 2
28.3 even 6 112.8.a.i.1.1 2
56.3 even 6 448.8.a.n.1.2 2
56.45 odd 6 448.8.a.p.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
28.8.a.a.1.2 ✓ 2 7.3 odd 6
112.8.a.i.1.1 2 28.3 even 6
196.8.a.b.1.1 2 7.4 even 3
196.8.e.a.165.2 4 1.1 even 1 trivial
196.8.e.a.177.2 4 7.2 even 3 inner
196.8.e.d.165.1 4 7.6 odd 2
196.8.e.d.177.1 4 7.5 odd 6
252.8.a.e.1.1 2 21.17 even 6
448.8.a.n.1.2 2 56.3 even 6
448.8.a.p.1.1 2 56.45 odd 6