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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [384,4,Mod(193,384)] 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("384.193"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(384, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 384.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(22.6567334422\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 193.4
Root \(0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 384.193
Dual form 384.4.d.d.193.1

$q$-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)
 
\(f(q)\) \(=\) \(q+3.00000i q^{3} +2.82843i q^{5} -14.1421 q^{7} -9.00000 q^{9} +20.0000i q^{11} -39.5980i q^{13} -8.48528 q^{15} -34.0000 q^{17} -52.0000i q^{19} -42.4264i q^{21} -62.2254 q^{23} +117.000 q^{25} -27.0000i q^{27} -200.818i q^{29} -110.309 q^{31} -60.0000 q^{33} -40.0000i q^{35} -271.529i q^{37} +118.794 q^{39} +26.0000 q^{41} +252.000i q^{43} -25.4558i q^{45} +345.068 q^{47} -143.000 q^{49} -102.000i q^{51} -681.651i q^{53} -56.5685 q^{55} +156.000 q^{57} +364.000i q^{59} -735.391i q^{61} +127.279 q^{63} +112.000 q^{65} -628.000i q^{67} -186.676i q^{69} +333.754 q^{71} -338.000 q^{73} +351.000i q^{75} -282.843i q^{77} -789.131 q^{79} +81.0000 q^{81} -1036.00i q^{83} -96.1665i q^{85} +602.455 q^{87} -234.000 q^{89} +560.000i q^{91} -330.926i q^{93} +147.078 q^{95} -178.000 q^{97} -180.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 36 q^{9} - 136 q^{17} + 468 q^{25} - 240 q^{33} + 104 q^{41} - 572 q^{49} + 624 q^{57} + 448 q^{65} - 1352 q^{73} + 324 q^{81} - 936 q^{89} - 712 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 3.00000i 0.577350i
\(4\) 0 0
\(5\) 2.82843i 0.252982i 0.991968 + 0.126491i \(0.0403715\pi\)
−0.991968 + 0.126491i \(0.959628\pi\)
\(6\) 0 0
\(7\) −14.1421 −0.763604 −0.381802 − 0.924244i \(-0.624696\pi\)
−0.381802 + 0.924244i \(0.624696\pi\)
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) 20.0000i 0.548202i 0.961701 + 0.274101i \(0.0883803\pi\)
−0.961701 + 0.274101i \(0.911620\pi\)
\(12\) 0 0
\(13\) − 39.5980i − 0.844808i −0.906408 − 0.422404i \(-0.861186\pi\)
0.906408 − 0.422404i \(-0.138814\pi\)
\(14\) 0 0
\(15\) −8.48528 −0.146059
\(16\) 0 0
\(17\) −34.0000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) − 52.0000i − 0.627875i −0.949444 − 0.313937i \(-0.898352\pi\)
0.949444 − 0.313937i \(-0.101648\pi\)
\(20\) 0 0
\(21\) − 42.4264i − 0.440867i
\(22\) 0 0
\(23\) −62.2254 −0.564126 −0.282063 − 0.959396i \(-0.591019\pi\)
−0.282063 + 0.959396i \(0.591019\pi\)
\(24\) 0 0
\(25\) 117.000 0.936000
\(26\) 0 0
\(27\) − 27.0000i − 0.192450i
\(28\) 0 0
\(29\) − 200.818i − 1.28590i −0.765909 − 0.642949i \(-0.777711\pi\)
0.765909 − 0.642949i \(-0.222289\pi\)
\(30\) 0 0
\(31\) −110.309 −0.639097 −0.319549 − 0.947570i \(-0.603531\pi\)
−0.319549 + 0.947570i \(0.603531\pi\)
\(32\) 0 0
\(33\) −60.0000 −0.316505
\(34\) 0 0
\(35\) − 40.0000i − 0.193178i
\(36\) 0 0
\(37\) − 271.529i − 1.20646i −0.797567 − 0.603231i \(-0.793880\pi\)
0.797567 − 0.603231i \(-0.206120\pi\)
\(38\) 0 0
\(39\) 118.794 0.487750
\(40\) 0 0
\(41\) 26.0000 0.0990370 0.0495185 − 0.998773i \(-0.484231\pi\)
0.0495185 + 0.998773i \(0.484231\pi\)
\(42\) 0 0
\(43\) 252.000i 0.893713i 0.894606 + 0.446856i \(0.147456\pi\)
−0.894606 + 0.446856i \(0.852544\pi\)
\(44\) 0 0
\(45\) − 25.4558i − 0.0843274i
\(46\) 0 0
\(47\) 345.068 1.07092 0.535461 − 0.844560i \(-0.320138\pi\)
0.535461 + 0.844560i \(0.320138\pi\)
\(48\) 0 0
\(49\) −143.000 −0.416910
\(50\) 0 0
\(51\) − 102.000i − 0.280056i
\(52\) 0 0
\(53\) − 681.651i − 1.76664i −0.468770 − 0.883320i \(-0.655303\pi\)
0.468770 − 0.883320i \(-0.344697\pi\)
\(54\) 0 0
\(55\) −56.5685 −0.138685
\(56\) 0 0
\(57\) 156.000 0.362504
\(58\) 0 0
\(59\) 364.000i 0.803199i 0.915815 + 0.401600i \(0.131546\pi\)
−0.915815 + 0.401600i \(0.868454\pi\)
\(60\) 0 0
\(61\) − 735.391i − 1.54356i −0.635889 − 0.771780i \(-0.719367\pi\)
0.635889 − 0.771780i \(-0.280633\pi\)
\(62\) 0 0
\(63\) 127.279 0.254535
\(64\) 0 0
\(65\) 112.000 0.213721
\(66\) 0 0
\(67\) − 628.000i − 1.14511i −0.819866 − 0.572555i \(-0.805952\pi\)
0.819866 − 0.572555i \(-0.194048\pi\)
\(68\) 0 0
\(69\) − 186.676i − 0.325698i
\(70\) 0 0
\(71\) 333.754 0.557878 0.278939 − 0.960309i \(-0.410017\pi\)
0.278939 + 0.960309i \(0.410017\pi\)
\(72\) 0 0
\(73\) −338.000 −0.541917 −0.270958 − 0.962591i \(-0.587341\pi\)
−0.270958 + 0.962591i \(0.587341\pi\)
\(74\) 0 0
\(75\) 351.000i 0.540400i
\(76\) 0 0
\(77\) − 282.843i − 0.418609i
\(78\) 0 0
\(79\) −789.131 −1.12385 −0.561925 − 0.827188i \(-0.689939\pi\)
−0.561925 + 0.827188i \(0.689939\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) − 1036.00i − 1.37007i −0.728510 − 0.685035i \(-0.759787\pi\)
0.728510 − 0.685035i \(-0.240213\pi\)
\(84\) 0 0
\(85\) − 96.1665i − 0.122714i
\(86\) 0 0
\(87\) 602.455 0.742413
\(88\) 0 0
\(89\) −234.000 −0.278696 −0.139348 − 0.990243i \(-0.544501\pi\)
−0.139348 + 0.990243i \(0.544501\pi\)
\(90\) 0 0
\(91\) 560.000i 0.645098i
\(92\) 0 0
\(93\) − 330.926i − 0.368983i
\(94\) 0 0
\(95\) 147.078 0.158841
\(96\) 0 0
\(97\) −178.000 −0.186321 −0.0931606 − 0.995651i \(-0.529697\pi\)
−0.0931606 + 0.995651i \(0.529697\pi\)
\(98\) 0 0
\(99\) − 180.000i − 0.182734i
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 384.4.d.d.193.4 yes 4
3.2 odd 2 1152.4.d.n.577.1 4
4.3 odd 2 inner 384.4.d.d.193.2 yes 4
8.3 odd 2 inner 384.4.d.d.193.3 yes 4
8.5 even 2 inner 384.4.d.d.193.1 ✓ 4
12.11 even 2 1152.4.d.n.577.2 4
16.3 odd 4 768.4.a.m.1.2 2
16.5 even 4 768.4.a.m.1.1 2
16.11 odd 4 768.4.a.h.1.1 2
16.13 even 4 768.4.a.h.1.2 2
24.5 odd 2 1152.4.d.n.577.3 4
24.11 even 2 1152.4.d.n.577.4 4
48.5 odd 4 2304.4.a.bh.1.2 2
48.11 even 4 2304.4.a.bb.1.2 2
48.29 odd 4 2304.4.a.bb.1.1 2
48.35 even 4 2304.4.a.bh.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.4.d.d.193.1 ✓ 4 8.5 even 2 inner
384.4.d.d.193.2 yes 4 4.3 odd 2 inner
384.4.d.d.193.3 yes 4 8.3 odd 2 inner
384.4.d.d.193.4 yes 4 1.1 even 1 trivial
768.4.a.h.1.1 2 16.11 odd 4
768.4.a.h.1.2 2 16.13 even 4
768.4.a.m.1.1 2 16.5 even 4
768.4.a.m.1.2 2 16.3 odd 4
1152.4.d.n.577.1 4 3.2 odd 2
1152.4.d.n.577.2 4 12.11 even 2
1152.4.d.n.577.3 4 24.5 odd 2
1152.4.d.n.577.4 4 24.11 even 2
2304.4.a.bb.1.1 2 48.29 odd 4
2304.4.a.bb.1.2 2 48.11 even 4
2304.4.a.bh.1.1 2 48.35 even 4
2304.4.a.bh.1.2 2 48.5 odd 4