Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [336,8,Mod(1,336)] 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("336.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(336, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 336.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,54,0,264] 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: yes
Analytic conductor: \(104.961368563\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3649}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 912 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 84)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-29.7035\) of defining polynomial
Character \(\chi\) \(=\) 336.1

$q$-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)
 
\(f(q)\) \(=\) \(q+27.0000 q^{3} +494.442 q^{5} -343.000 q^{7} +729.000 q^{9} -47.0920 q^{11} +3624.60 q^{13} +13349.9 q^{15} -14642.7 q^{17} +3395.95 q^{19} -9261.00 q^{21} +17982.5 q^{23} +166348. q^{25} +19683.0 q^{27} +139916. q^{29} -228888. q^{31} -1271.48 q^{33} -169594. q^{35} +438263. q^{37} +97864.2 q^{39} +312110. q^{41} +556478. q^{43} +360448. q^{45} +794234. q^{47} +117649. q^{49} -395353. q^{51} +2.04720e6 q^{53} -23284.3 q^{55} +91690.7 q^{57} -2.56746e6 q^{59} -2.46108e6 q^{61} -250047. q^{63} +1.79215e6 q^{65} -2.15480e6 q^{67} +485527. q^{69} +2.38723e6 q^{71} -1.97256e6 q^{73} +4.49139e6 q^{75} +16152.6 q^{77} -117694. q^{79} +531441. q^{81} +509356. q^{83} -7.23997e6 q^{85} +3.77773e6 q^{87} -4.16031e6 q^{89} -1.24324e6 q^{91} -6.17998e6 q^{93} +1.67910e6 q^{95} +8.40834e6 q^{97} -34330.1 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 54 q^{3} + 264 q^{5} - 686 q^{7} + 1458 q^{9} + 4980 q^{11} - 10148 q^{13} + 7128 q^{15} + 17832 q^{17} - 6256 q^{19} - 18522 q^{21} - 14052 q^{23} + 141326 q^{25} + 39366 q^{27} + 243588 q^{29} - 470824 q^{31}+ \cdots + 3630420 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 27.0000 0.577350
\(4\) 0 0
\(5\) 494.442 1.76897 0.884484 − 0.466570i \(-0.154510\pi\)
0.884484 + 0.466570i \(0.154510\pi\)
\(6\) 0 0
\(7\) −343.000 −0.377964
\(8\) 0 0
\(9\) 729.000 0.333333
\(10\) 0 0
\(11\) −47.0920 −0.0106678 −0.00533388 − 0.999986i \(-0.501698\pi\)
−0.00533388 + 0.999986i \(0.501698\pi\)
\(12\) 0 0
\(13\) 3624.60 0.457571 0.228786 − 0.973477i \(-0.426525\pi\)
0.228786 + 0.973477i \(0.426525\pi\)
\(14\) 0 0
\(15\) 13349.9 1.02131
\(16\) 0 0
\(17\) −14642.7 −0.722854 −0.361427 − 0.932401i \(-0.617710\pi\)
−0.361427 + 0.932401i \(0.617710\pi\)
\(18\) 0 0
\(19\) 3395.95 0.113586 0.0567929 − 0.998386i \(-0.481913\pi\)
0.0567929 + 0.998386i \(0.481913\pi\)
\(20\) 0 0
\(21\) −9261.00 −0.218218
\(22\) 0 0
\(23\) 17982.5 0.308178 0.154089 − 0.988057i \(-0.450756\pi\)
0.154089 + 0.988057i \(0.450756\pi\)
\(24\) 0 0
\(25\) 166348. 2.12925
\(26\) 0 0
\(27\) 19683.0 0.192450
\(28\) 0 0
\(29\) 139916. 1.06531 0.532653 − 0.846334i \(-0.321195\pi\)
0.532653 + 0.846334i \(0.321195\pi\)
\(30\) 0 0
\(31\) −228888. −1.37993 −0.689965 − 0.723843i \(-0.742374\pi\)
−0.689965 + 0.723843i \(0.742374\pi\)
\(32\) 0 0
\(33\) −1271.48 −0.00615903
\(34\) 0 0
\(35\) −169594. −0.668607
\(36\) 0 0
\(37\) 438263. 1.42242 0.711211 − 0.702979i \(-0.248147\pi\)
0.711211 + 0.702979i \(0.248147\pi\)
\(38\) 0 0
\(39\) 97864.2 0.264179
\(40\) 0 0
\(41\) 312110. 0.707236 0.353618 − 0.935390i \(-0.384951\pi\)
0.353618 + 0.935390i \(0.384951\pi\)
\(42\) 0 0
\(43\) 556478. 1.06735 0.533676 − 0.845689i \(-0.320810\pi\)
0.533676 + 0.845689i \(0.320810\pi\)
\(44\) 0 0
\(45\) 360448. 0.589656
\(46\) 0 0
\(47\) 794234. 1.11585 0.557925 − 0.829891i \(-0.311598\pi\)
0.557925 + 0.829891i \(0.311598\pi\)
\(48\) 0 0
\(49\) 117649. 0.142857
\(50\) 0 0
\(51\) −395353. −0.417340
\(52\) 0 0
\(53\) 2.04720e6 1.88884 0.944421 − 0.328739i \(-0.106624\pi\)
0.944421 + 0.328739i \(0.106624\pi\)
\(54\) 0 0
\(55\) −23284.3 −0.0188709
\(56\) 0 0
\(57\) 91690.7 0.0655788
\(58\) 0 0
\(59\) −2.56746e6 −1.62750 −0.813750 − 0.581215i \(-0.802578\pi\)
−0.813750 + 0.581215i \(0.802578\pi\)
\(60\) 0 0
\(61\) −2.46108e6 −1.38826 −0.694130 − 0.719850i \(-0.744211\pi\)
−0.694130 + 0.719850i \(0.744211\pi\)
\(62\) 0 0
\(63\) −250047. −0.125988
\(64\) 0 0
\(65\) 1.79215e6 0.809429
\(66\) 0 0
\(67\) −2.15480e6 −0.875277 −0.437638 − 0.899151i \(-0.644185\pi\)
−0.437638 + 0.899151i \(0.644185\pi\)
\(68\) 0 0
\(69\) 485527. 0.177927
\(70\) 0 0
\(71\) 2.38723e6 0.791570 0.395785 − 0.918343i \(-0.370472\pi\)
0.395785 + 0.918343i \(0.370472\pi\)
\(72\) 0 0
\(73\) −1.97256e6 −0.593473 −0.296736 − 0.954959i \(-0.595898\pi\)
−0.296736 + 0.954959i \(0.595898\pi\)
\(74\) 0 0
\(75\) 4.49139e6 1.22932
\(76\) 0 0
\(77\) 16152.6 0.00403203
\(78\) 0 0
\(79\) −117694. −0.0268572 −0.0134286 − 0.999910i \(-0.504275\pi\)
−0.0134286 + 0.999910i \(0.504275\pi\)
\(80\) 0 0
\(81\) 531441. 0.111111
\(82\) 0 0
\(83\) 509356. 0.0977796 0.0488898 − 0.998804i \(-0.484432\pi\)
0.0488898 + 0.998804i \(0.484432\pi\)
\(84\) 0 0
\(85\) −7.23997e6 −1.27871
\(86\) 0 0
\(87\) 3.77773e6 0.615055
\(88\) 0 0
\(89\) −4.16031e6 −0.625548 −0.312774 − 0.949828i \(-0.601258\pi\)
−0.312774 + 0.949828i \(0.601258\pi\)
\(90\) 0 0
\(91\) −1.24324e6 −0.172946
\(92\) 0 0
\(93\) −6.17998e6 −0.796703
\(94\) 0 0
\(95\) 1.67910e6 0.200930
\(96\) 0 0
\(97\) 8.40834e6 0.935425 0.467713 − 0.883881i \(-0.345078\pi\)
0.467713 + 0.883881i \(0.345078\pi\)
\(98\) 0 0
\(99\) −34330.1 −0.00355592
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 336.8.a.q.1.2 2
4.3 odd 2 84.8.a.c.1.2 ✓ 2
12.11 even 2 252.8.a.c.1.1 2
28.3 even 6 588.8.i.j.373.2 4
28.11 odd 6 588.8.i.k.373.1 4
28.19 even 6 588.8.i.j.361.2 4
28.23 odd 6 588.8.i.k.361.1 4
28.27 even 2 588.8.a.f.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.8.a.c.1.2 ✓ 2 4.3 odd 2
252.8.a.c.1.1 2 12.11 even 2
336.8.a.q.1.2 2 1.1 even 1 trivial
588.8.a.f.1.1 2 28.27 even 2
588.8.i.j.361.2 4 28.19 even 6
588.8.i.j.373.2 4 28.3 even 6
588.8.i.k.361.1 4 28.23 odd 6
588.8.i.k.373.1 4 28.11 odd 6