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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 \(15.1013 - 26.1563i\) of defining polynomial
Character \(\chi\) \(=\) 196.165
Dual form 196.8.e.d.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)
 
\(f(q)\) \(=\) \(q+(33.2027 - 57.5088i) q^{3} +(78.6081 + 136.153i) q^{5} +(-1111.34 - 1924.89i) q^{9} +(-3104.51 + 5377.17i) q^{11} +5380.35 q^{13} +10440.0 q^{15} +(5497.46 - 9521.87i) q^{17} +(-5851.53 - 10135.1i) q^{19} +(-53070.5 - 91920.8i) q^{23} +(26704.0 - 46252.8i) q^{25} -2369.04 q^{27} -51562.7 q^{29} +(123778. - 214390. i) q^{31} +(206156. + 357073. i) q^{33} +(-216839. - 375576. i) q^{37} +(178642. - 309417. i) q^{39} +322819. q^{41} +878703. q^{43} +(174720. - 302624. i) q^{45} +(-327563. - 567356. i) q^{47} +(-365061. - 632304. i) q^{51} +(222418. - 385240. i) q^{53} -976159. q^{55} -777146. q^{57} +(-1.07273e6 + 1.85802e6i) q^{59} +(-296451. - 513468. i) q^{61} +(422939. + 732552. i) q^{65} +(-864332. + 1.49707e6i) q^{67} -7.04833e6 q^{69} +1.58060e6 q^{71} +(2.16582e6 - 3.75131e6i) q^{73} +(-1.77329e6 - 3.07143e6i) q^{75} +(3.04259e6 + 5.26992e6i) q^{79} +(2.35184e6 - 4.07350e6i) q^{81} -8.10357e6 q^{83} +1.72858e6 q^{85} +(-1.71202e6 + 2.96531e6i) q^{87} +(-4.93016e6 - 8.53929e6i) q^{89} +(-8.21955e6 - 1.42367e7i) q^{93} +(919955. - 1.59341e6i) q^{95} -171786. q^{97} +1.38007e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 14 q^{3} - 42 q^{5} - 2782 q^{9} - 7428 q^{11} + 23660 q^{13} + 41760 q^{15} - 15792 q^{17} - 26614 q^{19} - 32640 q^{23} + 91846 q^{25} - 175336 q^{27} - 316032 q^{29} + 180740 q^{31} + 348432 q^{33}+ \cdots + 28964904 q^{99}+O(q^{100}) \) 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
\(3\) 33.2027 − 57.5088i 0.709985 − 1.22973i −0.254878 − 0.966973i \(-0.582035\pi\)
0.964862 − 0.262756i \(-0.0846315\pi\)
\(4\) 0 0
\(5\) 78.6081 + 136.153i 0.281237 + 0.487116i 0.971690 − 0.236261i \(-0.0759220\pi\)
−0.690453 + 0.723377i \(0.742589\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −1111.34 − 1924.89i −0.508156 − 0.880152i
\(10\) 0 0
\(11\) −3104.51 + 5377.17i −0.703265 + 1.21809i 0.264049 + 0.964509i \(0.414942\pi\)
−0.967314 + 0.253582i \(0.918391\pi\)
\(12\) 0 0
\(13\) 5380.35 0.679218 0.339609 − 0.940567i \(-0.389705\pi\)
0.339609 + 0.940567i \(0.389705\pi\)
\(14\) 0 0
\(15\) 10440.0 0.798695
\(16\) 0 0
\(17\) 5497.46 − 9521.87i 0.271388 − 0.470058i −0.697830 − 0.716264i \(-0.745851\pi\)
0.969217 + 0.246206i \(0.0791840\pi\)
\(18\) 0 0
\(19\) −5851.53 − 10135.1i −0.195718 − 0.338994i 0.751417 − 0.659827i \(-0.229371\pi\)
−0.947136 + 0.320833i \(0.896037\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −53070.5 − 91920.8i −0.909506 − 1.57531i −0.814752 − 0.579810i \(-0.803127\pi\)
−0.0947540 − 0.995501i \(-0.530206\pi\)
\(24\) 0 0
\(25\) 26704.0 − 46252.8i 0.341812 − 0.592035i
\(26\) 0 0
\(27\) −2369.04 −0.0231632
\(28\) 0 0
\(29\) −51562.7 −0.392593 −0.196297 − 0.980545i \(-0.562892\pi\)
−0.196297 + 0.980545i \(0.562892\pi\)
\(30\) 0 0
\(31\) 123778. − 214390.i 0.746240 − 1.29253i −0.203373 − 0.979101i \(-0.565190\pi\)
0.949613 − 0.313425i \(-0.101476\pi\)
\(32\) 0 0
\(33\) 206156. + 357073.i 0.998615 + 1.72965i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −216839. − 375576.i −0.703771 − 1.21897i −0.967133 − 0.254270i \(-0.918165\pi\)
0.263363 − 0.964697i \(-0.415168\pi\)
\(38\) 0 0
\(39\) 178642. − 309417.i 0.482234 − 0.835254i
\(40\) 0 0
\(41\) 322819. 0.731501 0.365751 − 0.930713i \(-0.380812\pi\)
0.365751 + 0.930713i \(0.380812\pi\)
\(42\) 0 0
\(43\) 878703. 1.68540 0.842699 − 0.538385i \(-0.180965\pi\)
0.842699 + 0.538385i \(0.180965\pi\)
\(44\) 0 0
\(45\) 174720. − 302624.i 0.285824 − 0.495063i
\(46\) 0 0
\(47\) −327563. − 567356.i −0.460206 − 0.797101i 0.538765 − 0.842456i \(-0.318891\pi\)
−0.998971 + 0.0453557i \(0.985558\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −365061. − 632304.i −0.385363 − 0.667468i
\(52\) 0 0
\(53\) 222418. − 385240.i 0.205213 − 0.355439i −0.744988 − 0.667078i \(-0.767545\pi\)
0.950201 + 0.311639i \(0.100878\pi\)
\(54\) 0 0
\(55\) −976159. −0.791136
\(56\) 0 0
\(57\) −777146. −0.555828
\(58\) 0 0
\(59\) −1.07273e6 + 1.85802e6i −0.679996 + 1.17779i 0.294985 + 0.955502i \(0.404685\pi\)
−0.974981 + 0.222286i \(0.928648\pi\)
\(60\) 0 0
\(61\) −296451. − 513468.i −0.167224 − 0.289640i 0.770219 − 0.637780i \(-0.220147\pi\)
−0.937443 + 0.348139i \(0.886814\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 422939. + 732552.i 0.191021 + 0.330858i
\(66\) 0 0
\(67\) −864332. + 1.49707e6i −0.351090 + 0.608106i −0.986441 − 0.164118i \(-0.947522\pi\)
0.635351 + 0.772224i \(0.280856\pi\)
\(68\) 0 0
\(69\) −7.04833e6 −2.58294
\(70\) 0 0
\(71\) 1.58060e6 0.524105 0.262052 − 0.965054i \(-0.415601\pi\)
0.262052 + 0.965054i \(0.415601\pi\)
\(72\) 0 0
\(73\) 2.16582e6 − 3.75131e6i 0.651617 − 1.12863i −0.331114 − 0.943591i \(-0.607424\pi\)
0.982731 − 0.185042i \(-0.0592423\pi\)
\(74\) 0 0
\(75\) −1.77329e6 − 3.07143e6i −0.485362 − 0.840672i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 3.04259e6 + 5.26992e6i 0.694302 + 1.20257i 0.970415 + 0.241442i \(0.0776202\pi\)
−0.276113 + 0.961125i \(0.589046\pi\)
\(80\) 0 0
\(81\) 2.35184e6 − 4.07350e6i 0.491711 − 0.851668i
\(82\) 0 0
\(83\) −8.10357e6 −1.55562 −0.777809 − 0.628500i \(-0.783669\pi\)
−0.777809 + 0.628500i \(0.783669\pi\)
\(84\) 0 0
\(85\) 1.72858e6 0.305297
\(86\) 0 0
\(87\) −1.71202e6 + 2.96531e6i −0.278735 + 0.482783i
\(88\) 0 0
\(89\) −4.93016e6 − 8.53929e6i −0.741303 − 1.28398i −0.951902 − 0.306403i \(-0.900875\pi\)
0.210599 − 0.977573i \(-0.432459\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −8.21955e6 − 1.42367e7i −1.05964 − 1.83535i
\(94\) 0 0
\(95\) 919955. − 1.59341e6i 0.110086 − 0.190675i
\(96\) 0 0
\(97\) −171786. −0.0191111 −0.00955555 − 0.999954i \(-0.503042\pi\)
−0.00955555 + 0.999954i \(0.503042\pi\)
\(98\) 0 0
\(99\) 1.38007e7 1.42947
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.165.2 4
7.2 even 3 inner 196.8.e.d.177.2 4
7.3 odd 6 196.8.a.b.1.2 2
7.4 even 3 28.8.a.a.1.1 ✓ 2
7.5 odd 6 196.8.e.a.177.1 4
7.6 odd 2 196.8.e.a.165.1 4
21.11 odd 6 252.8.a.e.1.2 2
28.11 odd 6 112.8.a.i.1.2 2
56.11 odd 6 448.8.a.n.1.1 2
56.53 even 6 448.8.a.p.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
28.8.a.a.1.1 ✓ 2 7.4 even 3
112.8.a.i.1.2 2 28.11 odd 6
196.8.a.b.1.2 2 7.3 odd 6
196.8.e.a.165.1 4 7.6 odd 2
196.8.e.a.177.1 4 7.5 odd 6
196.8.e.d.165.2 4 1.1 even 1 trivial
196.8.e.d.177.2 4 7.2 even 3 inner
252.8.a.e.1.2 2 21.11 odd 6
448.8.a.n.1.1 2 56.11 odd 6
448.8.a.p.1.2 2 56.53 even 6