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Properties

Label 336.4.bc.d.17.5
Level $336$
Weight $4$
Character 336.17
Analytic conductor $19.825$
Analytic rank $0$
Dimension $12$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [336,4,Mod(17,336)] 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("336.17"); S:= CuspForms(chi, 4); 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, 3, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 336.bc (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(19.8246417619\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} - 29x^{9} + 6x^{8} - 49x^{7} + 1564x^{6} - 441x^{5} + 486x^{4} - 21141x^{3} - 59049x + 531441 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{3} \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 17.5
Root \(2.85284 + 0.928053i\) of defining polynomial
Character \(\chi\) \(=\) 336.17
Dual form 336.4.bc.d.257.5

$q$-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)
 
\(f(q)\) \(=\) \(q+(3.47555 - 3.86271i) q^{3} +(0.623706 + 1.08029i) q^{5} +(-10.0808 - 15.5363i) q^{7} +(-2.84113 - 26.8501i) q^{9} +(-35.2392 - 20.3453i) q^{11} +19.5973i q^{13} +(6.34057 + 1.34540i) q^{15} +(-52.3592 + 90.6889i) q^{17} +(-35.0345 + 20.2272i) q^{19} +(-95.0487 - 15.0578i) q^{21} +(-69.6324 + 40.2023i) q^{23} +(61.7220 - 106.906i) q^{25} +(-113.589 - 82.3444i) q^{27} -211.712i q^{29} +(86.6242 + 50.0125i) q^{31} +(-201.064 + 65.4076i) q^{33} +(10.4962 - 20.5803i) q^{35} +(94.9875 + 164.523i) q^{37} +(75.6987 + 68.1113i) q^{39} -186.753 q^{41} -158.618 q^{43} +(27.2339 - 19.8158i) q^{45} +(-179.034 - 310.097i) q^{47} +(-139.753 + 313.238i) q^{49} +(168.328 + 517.442i) q^{51} +(-366.460 - 211.576i) q^{53} -50.7580i q^{55} +(-43.6323 + 205.629i) q^{57} +(-312.781 + 541.753i) q^{59} +(699.575 - 403.900i) q^{61} +(-388.510 + 314.812i) q^{63} +(-21.1708 + 12.2229i) q^{65} +(149.272 - 258.547i) q^{67} +(-86.7208 + 408.695i) q^{69} -455.386i q^{71} +(-434.467 - 250.840i) q^{73} +(-198.428 - 609.970i) q^{75} +(39.1491 + 752.584i) q^{77} +(-30.9561 - 53.6176i) q^{79} +(-712.856 + 152.569i) q^{81} +73.1180 q^{83} -130.627 q^{85} +(-817.783 - 735.816i) q^{87} +(57.3723 + 99.3717i) q^{89} +(304.469 - 197.557i) q^{91} +(494.251 - 160.784i) q^{93} +(-43.7025 - 25.2316i) q^{95} -1416.51i q^{97} +(-446.156 + 1003.98i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{3} + 56 q^{7} - 3 q^{9} - 6 q^{15} - 300 q^{19} + 357 q^{21} - 42 q^{25} + 930 q^{31} - 855 q^{33} + 764 q^{37} + 426 q^{39} + 1012 q^{43} + 2367 q^{45} - 336 q^{49} + 1341 q^{51} + 270 q^{57}+ \cdots + 3354 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(e\left(\frac{1}{6}\right)\)

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
\(3\) 3.47555 − 3.86271i 0.668870 − 0.743380i
\(4\) 0 0
\(5\) 0.623706 + 1.08029i 0.0557859 + 0.0966240i 0.892570 − 0.450909i \(-0.148900\pi\)
−0.836784 + 0.547533i \(0.815567\pi\)
\(6\) 0 0
\(7\) −10.0808 − 15.5363i −0.544314 − 0.838881i
\(8\) 0 0
\(9\) −2.84113 − 26.8501i −0.105227 − 0.994448i
\(10\) 0 0
\(11\) −35.2392 − 20.3453i −0.965910 − 0.557668i −0.0679230 − 0.997691i \(-0.521637\pi\)
−0.897987 + 0.440022i \(0.854971\pi\)
\(12\) 0 0
\(13\) 19.5973i 0.418101i 0.977905 + 0.209050i \(0.0670373\pi\)
−0.977905 + 0.209050i \(0.932963\pi\)
\(14\) 0 0
\(15\) 6.34057 + 1.34540i 0.109142 + 0.0231588i
\(16\) 0 0
\(17\) −52.3592 + 90.6889i −0.746999 + 1.29384i 0.202256 + 0.979333i \(0.435173\pi\)
−0.949255 + 0.314507i \(0.898161\pi\)
\(18\) 0 0
\(19\) −35.0345 + 20.2272i −0.423025 + 0.244234i −0.696371 − 0.717682i \(-0.745203\pi\)
0.273346 + 0.961916i \(0.411870\pi\)
\(20\) 0 0
\(21\) −95.0487 − 15.0578i −0.987683 − 0.156470i
\(22\) 0 0
\(23\) −69.6324 + 40.2023i −0.631276 + 0.364467i −0.781246 − 0.624223i \(-0.785416\pi\)
0.149970 + 0.988691i \(0.452082\pi\)
\(24\) 0 0
\(25\) 61.7220 − 106.906i 0.493776 − 0.855245i
\(26\) 0 0
\(27\) −113.589 − 82.3444i −0.809636 − 0.586933i
\(28\) 0 0
\(29\) − 211.712i − 1.35565i −0.735222 − 0.677827i \(-0.762922\pi\)
0.735222 − 0.677827i \(-0.237078\pi\)
\(30\) 0 0
\(31\) 86.6242 + 50.0125i 0.501876 + 0.289758i 0.729488 − 0.683994i \(-0.239758\pi\)
−0.227612 + 0.973752i \(0.573092\pi\)
\(32\) 0 0
\(33\) −201.064 + 65.4076i −1.06063 + 0.345030i
\(34\) 0 0
\(35\) 10.4962 − 20.5803i 0.0506910 − 0.0993916i
\(36\) 0 0
\(37\) 94.9875 + 164.523i 0.422050 + 0.731012i 0.996140 − 0.0877801i \(-0.0279773\pi\)
−0.574090 + 0.818792i \(0.694644\pi\)
\(38\) 0 0
\(39\) 75.6987 + 68.1113i 0.310808 + 0.279655i
\(40\) 0 0
\(41\) −186.753 −0.711362 −0.355681 − 0.934607i \(-0.615751\pi\)
−0.355681 + 0.934607i \(0.615751\pi\)
\(42\) 0 0
\(43\) −158.618 −0.562536 −0.281268 − 0.959629i \(-0.590755\pi\)
−0.281268 + 0.959629i \(0.590755\pi\)
\(44\) 0 0
\(45\) 27.2339 − 19.8158i 0.0902174 − 0.0656437i
\(46\) 0 0
\(47\) −179.034 − 310.097i −0.555635 − 0.962388i −0.997854 − 0.0654808i \(-0.979142\pi\)
0.442219 − 0.896907i \(-0.354191\pi\)
\(48\) 0 0
\(49\) −139.753 + 313.238i −0.407444 + 0.913230i
\(50\) 0 0
\(51\) 168.328 + 517.442i 0.462170 + 1.42071i
\(52\) 0 0
\(53\) −366.460 − 211.576i −0.949758 − 0.548343i −0.0567521 − 0.998388i \(-0.518074\pi\)
−0.893006 + 0.450045i \(0.851408\pi\)
\(54\) 0 0
\(55\) − 50.7580i − 0.124440i
\(56\) 0 0
\(57\) −43.6323 + 205.629i −0.101390 + 0.477829i
\(58\) 0 0
\(59\) −312.781 + 541.753i −0.690180 + 1.19543i 0.281599 + 0.959532i \(0.409135\pi\)
−0.971779 + 0.235895i \(0.924198\pi\)
\(60\) 0 0
\(61\) 699.575 − 403.900i 1.46838 − 0.847772i 0.469011 − 0.883192i \(-0.344611\pi\)
0.999372 + 0.0354209i \(0.0112772\pi\)
\(62\) 0 0
\(63\) −388.510 + 314.812i −0.776948 + 0.629565i
\(64\) 0 0
\(65\) −21.1708 + 12.2229i −0.0403986 + 0.0233241i
\(66\) 0 0
\(67\) 149.272 − 258.547i 0.272187 − 0.471441i −0.697235 − 0.716843i \(-0.745587\pi\)
0.969421 + 0.245402i \(0.0789199\pi\)
\(68\) 0 0
\(69\) −86.7208 + 408.695i −0.151304 + 0.713059i
\(70\) 0 0
\(71\) − 455.386i − 0.761189i −0.924742 − 0.380594i \(-0.875719\pi\)
0.924742 − 0.380594i \(-0.124281\pi\)
\(72\) 0 0
\(73\) −434.467 − 250.840i −0.696582 − 0.402172i 0.109491 − 0.993988i \(-0.465078\pi\)
−0.806073 + 0.591816i \(0.798411\pi\)
\(74\) 0 0
\(75\) −198.428 − 609.970i −0.305500 − 0.939110i
\(76\) 0 0
\(77\) 39.1491 + 752.584i 0.0579410 + 1.11383i
\(78\) 0 0
\(79\) −30.9561 − 53.6176i −0.0440865 − 0.0763601i 0.843140 − 0.537694i \(-0.180704\pi\)
−0.887227 + 0.461334i \(0.847371\pi\)
\(80\) 0 0
\(81\) −712.856 + 152.569i −0.977855 + 0.209285i
\(82\) 0 0
\(83\) 73.1180 0.0966957 0.0483478 − 0.998831i \(-0.484604\pi\)
0.0483478 + 0.998831i \(0.484604\pi\)
\(84\) 0 0
\(85\) −130.627 −0.166688
\(86\) 0 0
\(87\) −817.783 − 735.816i −1.00777 − 0.906755i
\(88\) 0 0
\(89\) 57.3723 + 99.3717i 0.0683309 + 0.118353i 0.898167 − 0.439655i \(-0.144899\pi\)
−0.829836 + 0.558008i \(0.811566\pi\)
\(90\) 0 0
\(91\) 304.469 − 197.557i 0.350737 − 0.227578i
\(92\) 0 0
\(93\) 494.251 − 160.784i 0.551090 − 0.179274i
\(94\) 0 0
\(95\) −43.7025 − 25.2316i −0.0471977 − 0.0272496i
\(96\) 0 0
\(97\) − 1416.51i − 1.48273i −0.671101 − 0.741366i \(-0.734178\pi\)
0.671101 − 0.741366i \(-0.265822\pi\)
\(98\) 0 0
\(99\) −446.156 + 1003.98i −0.452933 + 1.01923i
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 336.4.bc.d.17.5 12
3.2 odd 2 inner 336.4.bc.d.17.3 12
4.3 odd 2 21.4.g.a.17.3 yes 12
7.5 odd 6 inner 336.4.bc.d.257.3 12
12.11 even 2 21.4.g.a.17.4 yes 12
21.5 even 6 inner 336.4.bc.d.257.5 12
28.3 even 6 147.4.c.a.146.5 12
28.11 odd 6 147.4.c.a.146.6 12
28.19 even 6 21.4.g.a.5.4 yes 12
28.23 odd 6 147.4.g.d.68.4 12
28.27 even 2 147.4.g.d.80.3 12
84.11 even 6 147.4.c.a.146.7 12
84.23 even 6 147.4.g.d.68.3 12
84.47 odd 6 21.4.g.a.5.3 ✓ 12
84.59 odd 6 147.4.c.a.146.8 12
84.83 odd 2 147.4.g.d.80.4 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.g.a.5.3 ✓ 12 84.47 odd 6
21.4.g.a.5.4 yes 12 28.19 even 6
21.4.g.a.17.3 yes 12 4.3 odd 2
21.4.g.a.17.4 yes 12 12.11 even 2
147.4.c.a.146.5 12 28.3 even 6
147.4.c.a.146.6 12 28.11 odd 6
147.4.c.a.146.7 12 84.11 even 6
147.4.c.a.146.8 12 84.59 odd 6
147.4.g.d.68.3 12 84.23 even 6
147.4.g.d.68.4 12 28.23 odd 6
147.4.g.d.80.3 12 28.27 even 2
147.4.g.d.80.4 12 84.83 odd 2
336.4.bc.d.17.3 12 3.2 odd 2 inner
336.4.bc.d.17.5 12 1.1 even 1 trivial
336.4.bc.d.257.3 12 7.5 odd 6 inner
336.4.bc.d.257.5 12 21.5 even 6 inner