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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [384,3,Mod(319,384)] 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("384.319"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(384, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 384.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,24,0,0,0,0,0,0,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.4632421514\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{18} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 319.2
Root \(-0.965926 - 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 384.319
Dual form 384.3.b.c.319.3

$q$-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)
 
\(f(q)\) \(=\) \(q-1.73205 q^{3} -0.898979i q^{5} +2.82843i q^{7} +3.00000 q^{9} -4.38551 q^{11} -13.7980i q^{13} +1.55708i q^{15} +17.5959 q^{17} +4.38551 q^{19} -4.89898i q^{21} +22.0560i q^{23} +24.1918 q^{25} -5.19615 q^{27} -44.4949i q^{29} -53.1687i q^{31} +7.59592 q^{33} +2.54270 q^{35} -35.1918i q^{37} +23.8988i q^{39} +37.5959 q^{41} +49.6403 q^{43} -2.69694i q^{45} -38.4551i q^{47} +41.0000 q^{49} -30.4770 q^{51} +1.70714i q^{53} +3.94248i q^{55} -7.59592 q^{57} -34.6410 q^{59} +24.4041i q^{61} +8.48528i q^{63} -12.4041 q^{65} +93.7523 q^{67} -38.2020i q^{69} +123.879i q^{71} -10.0000 q^{73} -41.9015 q^{75} -12.4041i q^{77} -131.222i q^{79} +9.00000 q^{81} -110.151 q^{83} -15.8184i q^{85} +77.0674i q^{87} -73.1918 q^{89} +39.0265 q^{91} +92.0908i q^{93} -3.94248i q^{95} -105.192 q^{97} -13.1565 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 24 q^{9} - 16 q^{17} - 120 q^{25} - 96 q^{33} + 144 q^{41} + 328 q^{49} + 96 q^{57} - 256 q^{65} - 80 q^{73} + 72 q^{81} - 272 q^{89} - 528 q^{97}+O(q^{100}) \) 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
\(3\) −1.73205 −0.577350
\(4\) 0 0
\(5\) − 0.898979i − 0.179796i −0.995951 − 0.0898979i \(-0.971346\pi\)
0.995951 − 0.0898979i \(-0.0286541\pi\)
\(6\) 0 0
\(7\) 2.82843i 0.404061i 0.979379 + 0.202031i \(0.0647540\pi\)
−0.979379 + 0.202031i \(0.935246\pi\)
\(8\) 0 0
\(9\) 3.00000 0.333333
\(10\) 0 0
\(11\) −4.38551 −0.398682 −0.199341 − 0.979930i \(-0.563880\pi\)
−0.199341 + 0.979930i \(0.563880\pi\)
\(12\) 0 0
\(13\) − 13.7980i − 1.06138i −0.847566 − 0.530691i \(-0.821933\pi\)
0.847566 − 0.530691i \(-0.178067\pi\)
\(14\) 0 0
\(15\) 1.55708i 0.103805i
\(16\) 0 0
\(17\) 17.5959 1.03505 0.517527 − 0.855667i \(-0.326853\pi\)
0.517527 + 0.855667i \(0.326853\pi\)
\(18\) 0 0
\(19\) 4.38551 0.230816 0.115408 − 0.993318i \(-0.463182\pi\)
0.115408 + 0.993318i \(0.463182\pi\)
\(20\) 0 0
\(21\) − 4.89898i − 0.233285i
\(22\) 0 0
\(23\) 22.0560i 0.958955i 0.877554 + 0.479477i \(0.159174\pi\)
−0.877554 + 0.479477i \(0.840826\pi\)
\(24\) 0 0
\(25\) 24.1918 0.967673
\(26\) 0 0
\(27\) −5.19615 −0.192450
\(28\) 0 0
\(29\) − 44.4949i − 1.53431i −0.641464 − 0.767153i \(-0.721672\pi\)
0.641464 − 0.767153i \(-0.278328\pi\)
\(30\) 0 0
\(31\) − 53.1687i − 1.71512i −0.514386 − 0.857559i \(-0.671980\pi\)
0.514386 − 0.857559i \(-0.328020\pi\)
\(32\) 0 0
\(33\) 7.59592 0.230179
\(34\) 0 0
\(35\) 2.54270 0.0726485
\(36\) 0 0
\(37\) − 35.1918i − 0.951131i −0.879680 − 0.475565i \(-0.842244\pi\)
0.879680 − 0.475565i \(-0.157756\pi\)
\(38\) 0 0
\(39\) 23.8988i 0.612789i
\(40\) 0 0
\(41\) 37.5959 0.916974 0.458487 − 0.888701i \(-0.348392\pi\)
0.458487 + 0.888701i \(0.348392\pi\)
\(42\) 0 0
\(43\) 49.6403 1.15443 0.577213 − 0.816593i \(-0.304140\pi\)
0.577213 + 0.816593i \(0.304140\pi\)
\(44\) 0 0
\(45\) − 2.69694i − 0.0599320i
\(46\) 0 0
\(47\) − 38.4551i − 0.818193i −0.912491 − 0.409096i \(-0.865844\pi\)
0.912491 − 0.409096i \(-0.134156\pi\)
\(48\) 0 0
\(49\) 41.0000 0.836735
\(50\) 0 0
\(51\) −30.4770 −0.597589
\(52\) 0 0
\(53\) 1.70714i 0.0322103i 0.999870 + 0.0161051i \(0.00512664\pi\)
−0.999870 + 0.0161051i \(0.994873\pi\)
\(54\) 0 0
\(55\) 3.94248i 0.0716814i
\(56\) 0 0
\(57\) −7.59592 −0.133262
\(58\) 0 0
\(59\) −34.6410 −0.587136 −0.293568 − 0.955938i \(-0.594843\pi\)
−0.293568 + 0.955938i \(0.594843\pi\)
\(60\) 0 0
\(61\) 24.4041i 0.400067i 0.979789 + 0.200033i \(0.0641051\pi\)
−0.979789 + 0.200033i \(0.935895\pi\)
\(62\) 0 0
\(63\) 8.48528i 0.134687i
\(64\) 0 0
\(65\) −12.4041 −0.190832
\(66\) 0 0
\(67\) 93.7523 1.39929 0.699644 − 0.714492i \(-0.253342\pi\)
0.699644 + 0.714492i \(0.253342\pi\)
\(68\) 0 0
\(69\) − 38.2020i − 0.553653i
\(70\) 0 0
\(71\) 123.879i 1.74478i 0.488811 + 0.872390i \(0.337431\pi\)
−0.488811 + 0.872390i \(0.662569\pi\)
\(72\) 0 0
\(73\) −10.0000 −0.136986 −0.0684932 − 0.997652i \(-0.521819\pi\)
−0.0684932 + 0.997652i \(0.521819\pi\)
\(74\) 0 0
\(75\) −41.9015 −0.558687
\(76\) 0 0
\(77\) − 12.4041i − 0.161092i
\(78\) 0 0
\(79\) − 131.222i − 1.66103i −0.556993 − 0.830517i \(-0.688045\pi\)
0.556993 − 0.830517i \(-0.311955\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) −110.151 −1.32712 −0.663562 − 0.748121i \(-0.730956\pi\)
−0.663562 + 0.748121i \(0.730956\pi\)
\(84\) 0 0
\(85\) − 15.8184i − 0.186098i
\(86\) 0 0
\(87\) 77.0674i 0.885832i
\(88\) 0 0
\(89\) −73.1918 −0.822380 −0.411190 − 0.911550i \(-0.634887\pi\)
−0.411190 + 0.911550i \(0.634887\pi\)
\(90\) 0 0
\(91\) 39.0265 0.428863
\(92\) 0 0
\(93\) 92.0908i 0.990224i
\(94\) 0 0
\(95\) − 3.94248i − 0.0414998i
\(96\) 0 0
\(97\) −105.192 −1.08445 −0.542226 − 0.840233i \(-0.682418\pi\)
−0.542226 + 0.840233i \(0.682418\pi\)
\(98\) 0 0
\(99\) −13.1565 −0.132894
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 384.3.b.c.319.2 ✓ 8
3.2 odd 2 1152.3.b.j.703.6 8
4.3 odd 2 inner 384.3.b.c.319.6 yes 8
8.3 odd 2 inner 384.3.b.c.319.3 yes 8
8.5 even 2 inner 384.3.b.c.319.7 yes 8
12.11 even 2 1152.3.b.j.703.5 8
16.3 odd 4 768.3.g.g.511.3 4
16.5 even 4 768.3.g.c.511.4 4
16.11 odd 4 768.3.g.c.511.2 4
16.13 even 4 768.3.g.g.511.1 4
24.5 odd 2 1152.3.b.j.703.4 8
24.11 even 2 1152.3.b.j.703.3 8
48.5 odd 4 2304.3.g.x.1279.1 4
48.11 even 4 2304.3.g.x.1279.2 4
48.29 odd 4 2304.3.g.o.1279.3 4
48.35 even 4 2304.3.g.o.1279.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.3.b.c.319.2 ✓ 8 1.1 even 1 trivial
384.3.b.c.319.3 yes 8 8.3 odd 2 inner
384.3.b.c.319.6 yes 8 4.3 odd 2 inner
384.3.b.c.319.7 yes 8 8.5 even 2 inner
768.3.g.c.511.2 4 16.11 odd 4
768.3.g.c.511.4 4 16.5 even 4
768.3.g.g.511.1 4 16.13 even 4
768.3.g.g.511.3 4 16.3 odd 4
1152.3.b.j.703.3 8 24.11 even 2
1152.3.b.j.703.4 8 24.5 odd 2
1152.3.b.j.703.5 8 12.11 even 2
1152.3.b.j.703.6 8 3.2 odd 2
2304.3.g.o.1279.3 4 48.29 odd 4
2304.3.g.o.1279.4 4 48.35 even 4
2304.3.g.x.1279.1 4 48.5 odd 4
2304.3.g.x.1279.2 4 48.11 even 4