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Properties

Label 336.2.bq.b.109.27
Level $336$
Weight $2$
Character 336.109
Analytic conductor $2.683$
Analytic rank $0$
Dimension $120$
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] = [336,2,Mod(37,336)] 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("336.37"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(336, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([0, 3, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 336.bq (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [120] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.68297350792\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(30\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 109.27
Character \(\chi\) \(=\) 336.109
Dual form 336.2.bq.b.37.27

$q$-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+(1.30340 - 0.548764i) q^{2} +(-0.965926 + 0.258819i) q^{3} +(1.39772 - 1.43052i) q^{4} +(-2.53593 - 0.679499i) q^{5} +(-1.11696 + 0.867411i) q^{6} +(2.36624 - 1.18360i) q^{7} +(1.03677 - 2.63156i) q^{8} +(0.866025 - 0.500000i) q^{9} +(-3.67822 + 0.505963i) q^{10} +(-1.08897 - 4.06411i) q^{11} +(-0.979844 + 1.74353i) q^{12} +(0.464324 + 0.464324i) q^{13} +(2.43464 - 2.84122i) q^{14} +2.62538 q^{15} +(-0.0927784 - 3.99892i) q^{16} +(3.46273 - 5.99763i) q^{17} +(0.854398 - 1.12694i) q^{18} +(-1.88917 + 7.05048i) q^{19} +(-4.51654 + 2.67795i) q^{20} +(-1.97927 + 1.75570i) q^{21} +(-3.64961 - 4.69958i) q^{22} +(-0.678671 + 0.391831i) q^{23} +(-0.320344 + 2.81023i) q^{24} +(1.63907 + 0.946318i) q^{25} +(0.860005 + 0.350397i) q^{26} +(-0.707107 + 0.707107i) q^{27} +(1.61416 - 5.03929i) q^{28} +(2.64771 + 2.64771i) q^{29} +(3.42193 - 1.44072i) q^{30} +(-1.15237 + 1.99596i) q^{31} +(-2.31539 - 5.16129i) q^{32} +(2.10374 + 3.64378i) q^{33} +(1.22205 - 9.71755i) q^{34} +(-6.80486 + 1.39367i) q^{35} +(0.495198 - 1.93773i) q^{36} +(9.13580 + 2.44793i) q^{37} +(1.40670 + 10.2263i) q^{38} +(-0.568678 - 0.328327i) q^{39} +(-4.41731 + 5.96896i) q^{40} +4.12076i q^{41} +(-1.61632 + 3.37454i) q^{42} +(-5.75866 + 5.75866i) q^{43} +(-7.33586 - 4.12267i) q^{44} +(-2.53593 + 0.679499i) q^{45} +(-0.669559 + 0.883144i) q^{46} +(3.59896 + 6.23358i) q^{47} +(1.12461 + 3.83865i) q^{48} +(4.19817 - 5.60138i) q^{49} +(2.65568 + 0.333970i) q^{50} +(-1.79244 + 6.68949i) q^{51} +(1.31322 - 0.0152318i) q^{52} +(1.61173 + 6.01507i) q^{53} +(-0.533610 + 1.30968i) q^{54} +11.0462i q^{55} +(-0.661482 - 7.45402i) q^{56} -7.29919i q^{57} +(4.90400 + 1.99806i) q^{58} +(-0.932495 - 3.48012i) q^{59} +(3.66954 - 3.75566i) q^{60} +(3.02345 - 11.2837i) q^{61} +(-0.406688 + 3.23391i) q^{62} +(1.45742 - 2.20815i) q^{63} +(-5.85022 - 5.45664i) q^{64} +(-0.861983 - 1.49300i) q^{65} +(4.74159 + 3.59485i) q^{66} +(3.24140 - 0.868530i) q^{67} +(-3.73981 - 13.3365i) q^{68} +(0.554133 - 0.554133i) q^{69} +(-8.10468 + 5.55078i) q^{70} +8.86981i q^{71} +(-0.417912 - 2.79738i) q^{72} +(-2.81769 - 1.62680i) q^{73} +(13.2510 - 1.82276i) q^{74} +(-1.82815 - 0.489850i) q^{75} +(7.44533 + 12.5571i) q^{76} +(-7.38706 - 8.32773i) q^{77} +(-0.921391 - 0.115872i) q^{78} +(4.01043 + 6.94626i) q^{79} +(-2.48199 + 10.2040i) q^{80} +(0.500000 - 0.866025i) q^{81} +(2.26133 + 5.37102i) q^{82} +(-11.1462 - 11.1462i) q^{83} +(-0.254894 + 5.28536i) q^{84} +(-12.8566 + 12.8566i) q^{85} +(-4.34570 + 10.6660i) q^{86} +(-3.24277 - 1.87221i) q^{87} +(-11.8240 - 1.34784i) q^{88} +(1.40108 - 0.808914i) q^{89} +(-2.93245 + 2.27729i) q^{90} +(1.64828 + 0.549126i) q^{91} +(-0.388068 + 1.51852i) q^{92} +(0.596508 - 2.22620i) q^{93} +(8.11166 + 6.14988i) q^{94} +(9.58158 - 16.5958i) q^{95} +(3.57234 + 4.38616i) q^{96} +9.91317 q^{97} +(2.39807 - 9.60465i) q^{98} +(-2.97513 - 2.97513i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 120 q - 4 q^{4} - 8 q^{5} - 8 q^{6} + 24 q^{8} - 8 q^{10} + 4 q^{11} + 8 q^{13} + 32 q^{14} + 8 q^{15} + 12 q^{16} - 16 q^{17} - 4 q^{18} - 8 q^{19} - 64 q^{20} + 12 q^{21} + 24 q^{22} - 8 q^{24} - 8 q^{26}+ \cdots - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.30340 − 0.548764i 0.921645 − 0.388035i
\(3\) −0.965926 + 0.258819i −0.557678 + 0.149429i
\(4\) 1.39772 − 1.43052i 0.698858 − 0.715260i
\(5\) −2.53593 − 0.679499i −1.13410 − 0.303881i −0.357524 − 0.933904i \(-0.616379\pi\)
−0.776577 + 0.630023i \(0.783046\pi\)
\(6\) −1.11696 + 0.867411i −0.455997 + 0.354119i
\(7\) 2.36624 − 1.18360i 0.894354 − 0.447360i
\(8\) 1.03677 − 2.63156i 0.366553 − 0.930397i
\(9\) 0.866025 − 0.500000i 0.288675 − 0.166667i
\(10\) −3.67822 + 0.505963i −1.16315 + 0.160000i
\(11\) −1.08897 − 4.06411i −0.328338 − 1.22537i −0.910913 − 0.412597i \(-0.864622\pi\)
0.582576 − 0.812777i \(-0.302045\pi\)
\(12\) −0.979844 + 1.74353i −0.282857 + 0.503314i
\(13\) 0.464324 + 0.464324i 0.128780 + 0.128780i 0.768559 − 0.639779i \(-0.220974\pi\)
−0.639779 + 0.768559i \(0.720974\pi\)
\(14\) 2.43464 − 2.84122i 0.650685 − 0.759347i
\(15\) 2.62538 0.677871
\(16\) −0.0927784 − 3.99892i −0.0231946 − 0.999731i
\(17\) 3.46273 − 5.99763i 0.839836 − 1.45464i −0.0501952 − 0.998739i \(-0.515984\pi\)
0.890031 − 0.455899i \(-0.150682\pi\)
\(18\) 0.854398 − 1.12694i 0.201383 − 0.265623i
\(19\) −1.88917 + 7.05048i −0.433405 + 1.61749i 0.311449 + 0.950263i \(0.399186\pi\)
−0.744854 + 0.667227i \(0.767481\pi\)
\(20\) −4.51654 + 2.67795i −1.00993 + 0.598807i
\(21\) −1.97927 + 1.75570i −0.431912 + 0.383125i
\(22\) −3.64961 − 4.69958i −0.778099 − 1.00195i
\(23\) −0.678671 + 0.391831i −0.141513 + 0.0817024i −0.569085 − 0.822279i \(-0.692702\pi\)
0.427572 + 0.903981i \(0.359369\pi\)
\(24\) −0.320344 + 2.81023i −0.0653899 + 0.573635i
\(25\) 1.63907 + 0.946318i 0.327814 + 0.189264i
\(26\) 0.860005 + 0.350397i 0.168661 + 0.0687185i
\(27\) −0.707107 + 0.707107i −0.136083 + 0.136083i
\(28\) 1.61416 − 5.03929i 0.305048 − 0.952337i
\(29\) 2.64771 + 2.64771i 0.491667 + 0.491667i 0.908831 − 0.417164i \(-0.136976\pi\)
−0.417164 + 0.908831i \(0.636976\pi\)
\(30\) 3.42193 − 1.44072i 0.624756 − 0.263037i
\(31\) −1.15237 + 1.99596i −0.206971 + 0.358484i −0.950759 − 0.309931i \(-0.899694\pi\)
0.743788 + 0.668416i \(0.233027\pi\)
\(32\) −2.31539 − 5.16129i −0.409307 − 0.912397i
\(33\) 2.10374 + 3.64378i 0.366213 + 0.634300i
\(34\) 1.22205 − 9.71755i 0.209580 − 1.66655i
\(35\) −6.80486 + 1.39367i −1.15023 + 0.235574i
\(36\) 0.495198 − 1.93773i 0.0825329 − 0.322954i
\(37\) 9.13580 + 2.44793i 1.50192 + 0.402437i 0.913741 − 0.406297i \(-0.133180\pi\)
0.588175 + 0.808734i \(0.299847\pi\)
\(38\) 1.40670 + 10.2263i 0.228197 + 1.65893i
\(39\) −0.568678 − 0.328327i −0.0910614 − 0.0525743i
\(40\) −4.41731 + 5.96896i −0.698438 + 0.943775i
\(41\) 4.12076i 0.643555i 0.946815 + 0.321778i \(0.104280\pi\)
−0.946815 + 0.321778i \(0.895720\pi\)
\(42\) −1.61632 + 3.37454i −0.249404 + 0.520702i
\(43\) −5.75866 + 5.75866i −0.878187 + 0.878187i −0.993347 − 0.115160i \(-0.963262\pi\)
0.115160 + 0.993347i \(0.463262\pi\)
\(44\) −7.33586 − 4.12267i −1.10592 − 0.621515i
\(45\) −2.53593 + 0.679499i −0.378033 + 0.101294i
\(46\) −0.669559 + 0.883144i −0.0987211 + 0.130212i
\(47\) 3.59896 + 6.23358i 0.524962 + 0.909261i 0.999577 + 0.0290678i \(0.00925386\pi\)
−0.474615 + 0.880193i \(0.657413\pi\)
\(48\) 1.12461 + 3.83865i 0.162324 + 0.554062i
\(49\) 4.19817 − 5.60138i 0.599738 − 0.800197i
\(50\) 2.65568 + 0.333970i 0.375569 + 0.0472306i
\(51\) −1.79244 + 6.68949i −0.250992 + 0.936715i
\(52\) 1.31322 − 0.0152318i 0.182111 − 0.00211227i
\(53\) 1.61173 + 6.01507i 0.221389 + 0.826234i 0.983819 + 0.179164i \(0.0573392\pi\)
−0.762431 + 0.647070i \(0.775994\pi\)
\(54\) −0.533610 + 1.30968i −0.0726151 + 0.178225i
\(55\) 11.0462i 1.48947i
\(56\) −0.661482 − 7.45402i −0.0883943 − 0.996086i
\(57\) − 7.29919i − 0.966801i
\(58\) 4.90400 + 1.99806i 0.643927 + 0.262359i
\(59\) −0.932495 − 3.48012i −0.121401 − 0.453073i 0.878285 − 0.478137i \(-0.158688\pi\)
−0.999686 + 0.0250639i \(0.992021\pi\)
\(60\) 3.66954 − 3.75566i 0.473736 − 0.484854i
\(61\) 3.02345 − 11.2837i 0.387112 − 1.44472i −0.447697 − 0.894185i \(-0.647756\pi\)
0.834810 − 0.550538i \(-0.185578\pi\)
\(62\) −0.406688 + 3.23391i −0.0516494 + 0.410707i
\(63\) 1.45742 − 2.20815i 0.183618 − 0.278201i
\(64\) −5.85022 − 5.45664i −0.731278 − 0.682080i
\(65\) −0.861983 − 1.49300i −0.106916 − 0.185184i
\(66\) 4.74159 + 3.59485i 0.583649 + 0.442496i
\(67\) 3.24140 − 0.868530i 0.396000 − 0.106108i −0.0553232 − 0.998468i \(-0.517619\pi\)
0.451323 + 0.892361i \(0.350952\pi\)
\(68\) −3.73981 − 13.3365i −0.453519 − 1.61729i
\(69\) 0.554133 − 0.554133i 0.0667098 − 0.0667098i
\(70\) −8.10468 + 5.55078i −0.968694 + 0.663445i
\(71\) 8.86981i 1.05265i 0.850283 + 0.526326i \(0.176431\pi\)
−0.850283 + 0.526326i \(0.823569\pi\)
\(72\) −0.417912 − 2.79738i −0.0492514 − 0.329675i
\(73\) −2.81769 − 1.62680i −0.329786 − 0.190402i 0.325960 − 0.945384i \(-0.394312\pi\)
−0.655746 + 0.754982i \(0.727646\pi\)
\(74\) 13.2510 − 1.82276i 1.54039 − 0.211891i
\(75\) −1.82815 − 0.489850i −0.211096 − 0.0565631i
\(76\) 7.44533 + 12.5571i 0.854038 + 1.44039i
\(77\) −7.38706 − 8.32773i −0.841834 − 0.949033i
\(78\) −0.921391 − 0.115872i −0.104327 − 0.0131199i
\(79\) 4.01043 + 6.94626i 0.451208 + 0.781515i 0.998461 − 0.0554517i \(-0.0176599\pi\)
−0.547253 + 0.836967i \(0.684327\pi\)
\(80\) −2.48199 + 10.2040i −0.277495 + 1.14084i
\(81\) 0.500000 − 0.866025i 0.0555556 − 0.0962250i
\(82\) 2.26133 + 5.37102i 0.249722 + 0.593129i
\(83\) −11.1462 − 11.1462i −1.22345 − 1.22345i −0.966397 − 0.257054i \(-0.917248\pi\)
−0.257054 − 0.966397i \(-0.582752\pi\)
\(84\) −0.254894 + 5.28536i −0.0278112 + 0.576680i
\(85\) −12.8566 + 12.8566i −1.39450 + 1.39450i
\(86\) −4.34570 + 10.6660i −0.468609 + 1.15014i
\(87\) −3.24277 − 1.87221i −0.347661 − 0.200722i
\(88\) −11.8240 − 1.34784i −1.26044 − 0.143680i
\(89\) 1.40108 − 0.808914i 0.148514 − 0.0857447i −0.423902 − 0.905708i \(-0.639340\pi\)
0.572416 + 0.819964i \(0.306006\pi\)
\(90\) −2.93245 + 2.27729i −0.309107 + 0.240047i
\(91\) 1.64828 + 0.549126i 0.172786 + 0.0575640i
\(92\) −0.388068 + 1.51852i −0.0404588 + 0.158317i
\(93\) 0.596508 − 2.22620i 0.0618550 − 0.230846i
\(94\) 8.11166 + 6.14988i 0.836653 + 0.634312i
\(95\) 9.58158 − 16.5958i 0.983050 − 1.70269i
\(96\) 3.57234 + 4.38616i 0.364600 + 0.447661i
\(97\) 9.91317 1.00653 0.503265 − 0.864132i \(-0.332132\pi\)
0.503265 + 0.864132i \(0.332132\pi\)
\(98\) 2.39807 − 9.60465i 0.242241 − 0.970216i
\(99\) −2.97513 − 2.97513i −0.299012 − 0.299012i
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 336.2.bq.b.109.27 yes 120
7.2 even 3 inner 336.2.bq.b.205.6 yes 120
16.5 even 4 inner 336.2.bq.b.277.6 yes 120
112.37 even 12 inner 336.2.bq.b.37.27 ✓ 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
336.2.bq.b.37.27 ✓ 120 112.37 even 12 inner
336.2.bq.b.109.27 yes 120 1.1 even 1 trivial
336.2.bq.b.205.6 yes 120 7.2 even 3 inner
336.2.bq.b.277.6 yes 120 16.5 even 4 inner