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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 165.1
Root \(-14.6013 + 25.2903i\) of defining polynomial
Character \(\chi\) \(=\) 196.165
Dual form 196.8.e.d.177.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-26.2027 + 45.3844i) q^{3} +(-99.6081 - 172.526i) q^{5} +(-279.662 - 484.389i) q^{9} +(-609.487 + 1055.66i) q^{11} +6449.65 q^{13} +10440.0 q^{15} +(-13393.5 + 23198.1i) q^{17} +(-7455.47 - 12913.3i) q^{19} +(36750.5 + 63653.7i) q^{23} +(19219.0 - 33288.2i) q^{25} -85299.0 q^{27} -106453. q^{29} +(-33408.3 + 57864.9i) q^{31} +(-31940.4 - 55322.4i) q^{33} +(239751. + 415261. i) q^{37} +(-168998. + 292713. i) q^{39} -644539. q^{41} +145165. q^{43} +(-55713.2 + 96498.2i) q^{45} +(-505423. - 875418. i) q^{47} +(-701889. - 1.21571e6i) q^{51} +(-17104.3 + 29625.4i) q^{53} +242839. q^{55} +781414. q^{57} +(221658. - 383924. i) q^{59} +(570214. + 987640. i) q^{61} +(-642437. - 1.11273e6i) q^{65} +(2.15964e6 - 3.74061e6i) q^{67} -3.85185e6 q^{69} +2.54867e6 q^{71} +(1.83861e6 - 3.18457e6i) q^{73} +(1.00718e6 + 1.74448e6i) q^{75} +(-4.27782e6 - 7.40940e6i) q^{79} +(2.84668e6 - 4.93060e6i) q^{81} -1.79721e6 q^{83} +5.33639e6 q^{85} +(2.78936e6 - 4.83132e6i) q^{87} +(-2.78159e6 - 4.81785e6i) q^{89} +(-1.75078e6 - 3.03243e6i) q^{93} +(-1.48525e6 + 2.57253e6i) q^{95} -1.72057e7 q^{97} +681802. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 14 q^{3} - 42 q^{5} - 2782 q^{9} - 7428 q^{11} + 23660 q^{13} + 41760 q^{15} - 15792 q^{17} - 26614 q^{19} - 32640 q^{23} + 91846 q^{25} - 175336 q^{27} - 316032 q^{29} + 180740 q^{31} + 348432 q^{33}+ \cdots + 28964904 q^{99}+O(q^{100}) \) 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 \(0.355981\pi\)
−0.997470 + 0.0710906i \(0.977352\pi\)
\(4\) 0 0
\(5\) −99.6081 − 172.526i −0.356369 − 0.617249i 0.630983 − 0.775797i \(-0.282652\pi\)
−0.987351 + 0.158548i \(0.949319\pi\)
\(6\) 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 \(-0.366539\pi\)
0.407103 + 0.913382i \(0.366539\pi\)
\(14\) 0 0
\(15\) 10440.0 0.798695
\(16\) 0 0
\(17\) −13393.5 + 23198.1i −0.661183 + 1.14520i 0.319123 + 0.947713i \(0.396612\pi\)
−0.980305 + 0.197488i \(0.936721\pi\)
\(18\) 0 0
\(19\) −7455.47 − 12913.3i −0.249366 − 0.431915i 0.713984 − 0.700162i \(-0.246889\pi\)
−0.963350 + 0.268247i \(0.913556\pi\)
\(20\) 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 −0.948984 − 0.315324i \(-0.897887\pi\)
0.747571 + 0.664182i \(0.231220\pi\)
\(32\) 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 \(-0.760599\pi\)
−0.730257 + 0.683173i \(0.760599\pi\)
\(42\) 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 \(-0.915322\pi\)
0.254735 − 0.967011i \(-0.418012\pi\)
\(48\) 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 −0.787180 − 0.616724i \(-0.788460\pi\)
0.927688 + 0.373356i \(0.121793\pi\)
\(60\) 0 0
\(61\) 570214. + 987640.i 0.321650 + 0.557114i 0.980829 − 0.194872i \(-0.0624292\pi\)
−0.659179 + 0.751986i \(0.729096\pi\)
\(62\) 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 \(-0.646750\pi\)
0.998043 − 0.0625282i \(-0.0199163\pi\)
\(74\) 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 \(-0.555185\pi\)
−0.172503 + 0.985009i \(0.555185\pi\)
\(84\) 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 0.577521 − 0.816376i \(-0.304020\pi\)
−0.995763 + 0.0919598i \(0.970687\pi\)
\(90\) 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 \(-0.906386\pi\)
−0.957064 + 0.289877i \(0.906386\pi\)
\(98\) 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.d.165.1 4
7.2 even 3 inner 196.8.e.d.177.1 4
7.3 odd 6 196.8.a.b.1.1 2
7.4 even 3 28.8.a.a.1.2 ✓ 2
7.5 odd 6 196.8.e.a.177.2 4
7.6 odd 2 196.8.e.a.165.2 4
21.11 odd 6 252.8.a.e.1.1 2
28.11 odd 6 112.8.a.i.1.1 2
56.11 odd 6 448.8.a.n.1.2 2
56.53 even 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 7.4 even 3
112.8.a.i.1.1 2 28.11 odd 6
196.8.a.b.1.1 2 7.3 odd 6
196.8.e.a.165.2 4 7.6 odd 2
196.8.e.a.177.2 4 7.5 odd 6
196.8.e.d.165.1 4 1.1 even 1 trivial
196.8.e.d.177.1 4 7.2 even 3 inner
252.8.a.e.1.1 2 21.11 odd 6
448.8.a.n.1.2 2 56.11 odd 6
448.8.a.p.1.1 2 56.53 even 6