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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [336,6,Mod(193,336)] 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("336.193"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(336, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 4])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 336.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-18,0,33] 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: \(53.8889634572\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-83})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 20x^{2} - 21x + 441 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 289.2
Root \(4.19493 - 1.84460i\) of defining polynomial
Character \(\chi\) \(=\) 336.289
Dual form 336.6.q.e.193.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-4.50000 - 7.79423i) q^{3} +(35.8645 - 62.1192i) q^{5} +(87.5000 - 95.6596i) q^{7} +(-40.5000 + 70.1481i) q^{9} +(-280.305 - 485.503i) q^{11} +533.509 q^{13} -645.562 q^{15} +(-502.850 - 870.962i) q^{17} +(684.263 - 1185.18i) q^{19} +(-1139.34 - 251.527i) q^{21} +(1614.04 - 2795.60i) q^{23} +(-1010.03 - 1749.42i) q^{25} +729.000 q^{27} -753.456 q^{29} +(4103.21 + 7106.97i) q^{31} +(-2522.75 + 4369.52i) q^{33} +(-2804.15 - 8866.21i) q^{35} +(1404.33 - 2432.37i) q^{37} +(-2400.79 - 4158.29i) q^{39} +245.827 q^{41} +17504.5 q^{43} +(2905.03 + 5031.65i) q^{45} +(-8172.74 + 14155.6i) q^{47} +(-1494.50 - 16740.4i) q^{49} +(-4525.65 + 7838.66i) q^{51} +(14820.8 + 25670.4i) q^{53} -40212.0 q^{55} -12316.7 q^{57} +(-5178.05 - 8968.65i) q^{59} +(-477.089 + 826.343i) q^{61} +(3166.58 + 10012.2i) q^{63} +(19134.0 - 33141.1i) q^{65} +(-9907.60 - 17160.5i) q^{67} -29052.7 q^{69} -62125.4 q^{71} +(-13554.8 - 23477.6i) q^{73} +(-9090.27 + 15744.8i) q^{75} +(-70969.7 - 15667.6i) q^{77} +(22343.7 - 38700.4i) q^{79} +(-3280.50 - 5681.99i) q^{81} -15606.6 q^{83} -72137.9 q^{85} +(3390.55 + 5872.61i) q^{87} +(-6817.66 + 11808.5i) q^{89} +(46682.0 - 51035.2i) q^{91} +(36928.9 - 63962.7i) q^{93} +(-49081.6 - 85011.8i) q^{95} -12919.5 q^{97} +45409.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 18 q^{3} + 33 q^{5} + 350 q^{7} - 162 q^{9} - 1137 q^{11} + 1850 q^{13} - 594 q^{15} + 324 q^{17} + 2311 q^{19} - 1575 q^{21} + 1596 q^{23} - 395 q^{25} + 2916 q^{27} - 4434 q^{29} + 4294 q^{31} - 10233 q^{33}+ \cdots + 184194 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(3\) −4.50000 − 7.79423i −0.288675 − 0.500000i
\(4\) 0 0
\(5\) 35.8645 − 62.1192i 0.641564 − 1.11122i −0.343519 − 0.939146i \(-0.611619\pi\)
0.985084 − 0.172076i \(-0.0550476\pi\)
\(6\) 0 0
\(7\) 87.5000 − 95.6596i 0.674937 − 0.737876i
\(8\) 0 0
\(9\) −40.5000 + 70.1481i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) −280.305 − 485.503i −0.698472 − 1.20979i −0.968996 − 0.247076i \(-0.920530\pi\)
0.270524 − 0.962713i \(-0.412803\pi\)
\(12\) 0 0
\(13\) 533.509 0.875555 0.437777 − 0.899083i \(-0.355766\pi\)
0.437777 + 0.899083i \(0.355766\pi\)
\(14\) 0 0
\(15\) −645.562 −0.740815
\(16\) 0 0
\(17\) −502.850 − 870.962i −0.422004 − 0.730932i 0.574132 − 0.818763i \(-0.305340\pi\)
−0.996135 + 0.0878311i \(0.972006\pi\)
\(18\) 0 0
\(19\) 684.263 − 1185.18i 0.434850 − 0.753182i −0.562434 − 0.826842i \(-0.690135\pi\)
0.997283 + 0.0736606i \(0.0234682\pi\)
\(20\) 0 0
\(21\) −1139.34 − 251.527i −0.563775 − 0.124462i
\(22\) 0 0
\(23\) 1614.04 − 2795.60i 0.636201 − 1.10193i −0.350058 − 0.936728i \(-0.613838\pi\)
0.986259 − 0.165205i \(-0.0528286\pi\)
\(24\) 0 0
\(25\) −1010.03 − 1749.42i −0.323209 − 0.559815i
\(26\) 0 0
\(27\) 729.000 0.192450
\(28\) 0 0
\(29\) −753.456 −0.166365 −0.0831827 − 0.996534i \(-0.526508\pi\)
−0.0831827 + 0.996534i \(0.526508\pi\)
\(30\) 0 0
\(31\) 4103.21 + 7106.97i 0.766866 + 1.32825i 0.939254 + 0.343222i \(0.111518\pi\)
−0.172389 + 0.985029i \(0.555148\pi\)
\(32\) 0 0
\(33\) −2522.75 + 4369.52i −0.403263 + 0.698472i
\(34\) 0 0
\(35\) −2804.15 − 8866.21i −0.386929 − 1.22340i
\(36\) 0 0
\(37\) 1404.33 − 2432.37i 0.168642 − 0.292096i −0.769301 − 0.638887i \(-0.779395\pi\)
0.937943 + 0.346791i \(0.112729\pi\)
\(38\) 0 0
\(39\) −2400.79 − 4158.29i −0.252751 − 0.437777i
\(40\) 0 0
\(41\) 245.827 0.0228387 0.0114193 − 0.999935i \(-0.496365\pi\)
0.0114193 + 0.999935i \(0.496365\pi\)
\(42\) 0 0
\(43\) 17504.5 1.44371 0.721853 − 0.692047i \(-0.243291\pi\)
0.721853 + 0.692047i \(0.243291\pi\)
\(44\) 0 0
\(45\) 2905.03 + 5031.65i 0.213855 + 0.370407i
\(46\) 0 0
\(47\) −8172.74 + 14155.6i −0.539663 + 0.934725i 0.459258 + 0.888303i \(0.348115\pi\)
−0.998922 + 0.0464219i \(0.985218\pi\)
\(48\) 0 0
\(49\) −1494.50 − 16740.4i −0.0889213 − 0.996039i
\(50\) 0 0
\(51\) −4525.65 + 7838.66i −0.243644 + 0.422004i
\(52\) 0 0
\(53\) 14820.8 + 25670.4i 0.724741 + 1.25529i 0.959081 + 0.283133i \(0.0913739\pi\)
−0.234340 + 0.972155i \(0.575293\pi\)
\(54\) 0 0
\(55\) −40212.0 −1.79246
\(56\) 0 0
\(57\) −12316.7 −0.502121
\(58\) 0 0
\(59\) −5178.05 − 8968.65i −0.193659 − 0.335426i 0.752801 − 0.658248i \(-0.228702\pi\)
−0.946460 + 0.322821i \(0.895369\pi\)
\(60\) 0 0
\(61\) −477.089 + 826.343i −0.0164163 + 0.0284339i −0.874117 − 0.485716i \(-0.838559\pi\)
0.857701 + 0.514150i \(0.171892\pi\)
\(62\) 0 0
\(63\) 3166.58 + 10012.2i 0.100517 + 0.317817i
\(64\) 0 0
\(65\) 19134.0 − 33141.1i 0.561725 − 0.972935i
\(66\) 0 0
\(67\) −9907.60 − 17160.5i −0.269638 − 0.467027i 0.699130 − 0.714994i \(-0.253571\pi\)
−0.968768 + 0.247967i \(0.920237\pi\)
\(68\) 0 0
\(69\) −29052.7 −0.734622
\(70\) 0 0
\(71\) −62125.4 −1.46259 −0.731296 − 0.682060i \(-0.761084\pi\)
−0.731296 + 0.682060i \(0.761084\pi\)
\(72\) 0 0
\(73\) −13554.8 − 23477.6i −0.297705 − 0.515641i 0.677905 − 0.735149i \(-0.262888\pi\)
−0.975611 + 0.219509i \(0.929555\pi\)
\(74\) 0 0
\(75\) −9090.27 + 15744.8i −0.186605 + 0.323209i
\(76\) 0 0
\(77\) −70969.7 − 15667.6i −1.36410 − 0.301145i
\(78\) 0 0
\(79\) 22343.7 − 38700.4i 0.402798 − 0.697666i −0.591265 − 0.806477i \(-0.701371\pi\)
0.994062 + 0.108812i \(0.0347045\pi\)
\(80\) 0 0
\(81\) −3280.50 − 5681.99i −0.0555556 − 0.0962250i
\(82\) 0 0
\(83\) −15606.6 −0.248665 −0.124332 − 0.992241i \(-0.539679\pi\)
−0.124332 + 0.992241i \(0.539679\pi\)
\(84\) 0 0
\(85\) −72137.9 −1.08297
\(86\) 0 0
\(87\) 3390.55 + 5872.61i 0.0480255 + 0.0831827i
\(88\) 0 0
\(89\) −6817.66 + 11808.5i −0.0912347 + 0.158023i −0.908031 − 0.418903i \(-0.862415\pi\)
0.816796 + 0.576926i \(0.195748\pi\)
\(90\) 0 0
\(91\) 46682.0 − 51035.2i 0.590944 − 0.646050i
\(92\) 0 0
\(93\) 36928.9 − 63962.7i 0.442750 − 0.766866i
\(94\) 0 0
\(95\) −49081.6 − 85011.8i −0.557968 − 0.966429i
\(96\) 0 0
\(97\) −12919.5 −0.139417 −0.0697086 − 0.997567i \(-0.522207\pi\)
−0.0697086 + 0.997567i \(0.522207\pi\)
\(98\) 0 0
\(99\) 45409.4 0.465648
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 336.6.q.e.289.2 4
4.3 odd 2 21.6.e.b.16.2 yes 4
7.4 even 3 inner 336.6.q.e.193.2 4
12.11 even 2 63.6.e.c.37.1 4
28.3 even 6 147.6.e.l.67.2 4
28.11 odd 6 21.6.e.b.4.2 ✓ 4
28.19 even 6 147.6.a.k.1.1 2
28.23 odd 6 147.6.a.i.1.1 2
28.27 even 2 147.6.e.l.79.2 4
84.11 even 6 63.6.e.c.46.1 4
84.23 even 6 441.6.a.t.1.2 2
84.47 odd 6 441.6.a.s.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.6.e.b.4.2 ✓ 4 28.11 odd 6
21.6.e.b.16.2 yes 4 4.3 odd 2
63.6.e.c.37.1 4 12.11 even 2
63.6.e.c.46.1 4 84.11 even 6
147.6.a.i.1.1 2 28.23 odd 6
147.6.a.k.1.1 2 28.19 even 6
147.6.e.l.67.2 4 28.3 even 6
147.6.e.l.79.2 4 28.27 even 2
336.6.q.e.193.2 4 7.4 even 3 inner
336.6.q.e.289.2 4 1.1 even 1 trivial
441.6.a.s.1.2 2 84.47 odd 6
441.6.a.t.1.2 2 84.23 even 6