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Properties

Label 336.2.bq.b.109.17
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 109.17
Character \(\chi\) \(=\) 336.109
Dual form 336.2.bq.b.37.17

$q$-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+(-0.120841 - 1.40904i) q^{2} +(-0.965926 + 0.258819i) q^{3} +(-1.97080 + 0.340539i) q^{4} +(0.433260 + 0.116092i) q^{5} +(0.481410 + 1.32975i) q^{6} +(2.04258 - 1.68163i) q^{7} +(0.717986 + 2.73578i) q^{8} +(0.866025 - 0.500000i) q^{9} +(0.111222 - 0.624510i) q^{10} +(-1.18222 - 4.41211i) q^{11} +(1.81550 - 0.839015i) q^{12} +(-3.32021 - 3.32021i) q^{13} +(-2.61631 - 2.67487i) q^{14} -0.448544 q^{15} +(3.76807 - 1.34227i) q^{16} +(-2.68303 + 4.64715i) q^{17} +(-0.809172 - 1.15985i) q^{18} +(1.47730 - 5.51336i) q^{19} +(-0.893400 - 0.0812509i) q^{20} +(-1.53774 + 2.15298i) q^{21} +(-6.07398 + 2.19896i) q^{22} +(0.0663290 - 0.0382951i) q^{23} +(-1.40159 - 2.45673i) q^{24} +(-4.15589 - 2.39940i) q^{25} +(-4.27710 + 5.07953i) q^{26} +(-0.707107 + 0.707107i) q^{27} +(-3.45285 + 4.00972i) q^{28} +(-3.56763 - 3.56763i) q^{29} +(0.0542023 + 0.632016i) q^{30} +(0.347695 - 0.602225i) q^{31} +(-2.34664 - 5.14716i) q^{32} +(2.28387 + 3.95579i) q^{33} +(6.87225 + 3.21894i) q^{34} +(1.08019 - 0.491454i) q^{35} +(-1.53649 + 1.28031i) q^{36} +(11.1005 + 2.97437i) q^{37} +(-7.94707 - 1.41534i) q^{38} +(4.06641 + 2.34774i) q^{39} +(-0.00652673 + 1.26866i) q^{40} +5.69219i q^{41} +(3.21947 + 1.90658i) q^{42} +(6.59516 - 6.59516i) q^{43} +(3.83241 + 8.29277i) q^{44} +(0.433260 - 0.116092i) q^{45} +(-0.0619746 - 0.0888327i) q^{46} +(-0.711379 - 1.23214i) q^{47} +(-3.29227 + 2.27178i) q^{48} +(1.34427 - 6.86971i) q^{49} +(-2.87866 + 6.14577i) q^{50} +(1.38884 - 5.18323i) q^{51} +(7.67411 + 5.41279i) q^{52} +(-1.86185 - 6.94851i) q^{53} +(1.08179 + 0.910895i) q^{54} -2.04883i q^{55} +(6.06710 + 4.38067i) q^{56} +5.70785i q^{57} +(-4.59582 + 5.45805i) q^{58} +(-0.230588 - 0.860565i) q^{59} +(0.883987 - 0.152747i) q^{60} +(-3.37528 + 12.5967i) q^{61} +(-0.890576 - 0.417143i) q^{62} +(0.928114 - 2.47762i) q^{63} +(-6.96899 + 3.92850i) q^{64} +(-1.05306 - 1.82396i) q^{65} +(5.29788 - 3.69609i) q^{66} +(4.23738 - 1.13540i) q^{67} +(3.70517 - 10.0723i) q^{68} +(-0.0541574 + 0.0541574i) q^{69} +(-0.823010 - 1.46265i) q^{70} +9.19924i q^{71} +(1.98968 + 2.01026i) q^{72} +(8.04777 + 4.64638i) q^{73} +(2.84962 - 16.0005i) q^{74} +(4.63529 + 1.24202i) q^{75} +(-1.03394 + 11.3688i) q^{76} +(-9.83429 - 7.02403i) q^{77} +(2.81668 - 6.01344i) q^{78} +(-2.89898 - 5.02118i) q^{79} +(1.78838 - 0.144109i) q^{80} +(0.500000 - 0.866025i) q^{81} +(8.02054 - 0.687849i) q^{82} +(4.19157 + 4.19157i) q^{83} +(2.29740 - 4.76675i) q^{84} +(-1.70195 + 1.70195i) q^{85} +(-10.0898 - 8.49588i) q^{86} +(4.36943 + 2.52269i) q^{87} +(11.2217 - 6.40213i) q^{88} +(6.23399 - 3.59919i) q^{89} +(-0.215933 - 0.596452i) q^{90} +(-12.3651 - 1.19845i) q^{91} +(-0.117680 + 0.0980593i) q^{92} +(-0.179980 + 0.671695i) q^{93} +(-1.65018 + 1.15126i) q^{94} +(1.28011 - 2.21721i) q^{95} +(3.59887 + 4.36442i) q^{96} +10.0676 q^{97} +(-9.84215 - 1.06399i) q^{98} +(-3.22989 - 3.22989i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.120841 − 1.40904i −0.0854473 − 0.996343i
\(3\) −0.965926 + 0.258819i −0.557678 + 0.149429i
\(4\) −1.97080 + 0.340539i −0.985398 + 0.170270i
\(5\) 0.433260 + 0.116092i 0.193760 + 0.0519177i 0.354394 − 0.935096i \(-0.384687\pi\)
−0.160634 + 0.987014i \(0.551354\pi\)
\(6\) 0.481410 + 1.32975i 0.196535 + 0.542870i
\(7\) 2.04258 − 1.68163i 0.772023 − 0.635595i
\(8\) 0.717986 + 2.73578i 0.253846 + 0.967245i
\(9\) 0.866025 − 0.500000i 0.288675 − 0.166667i
\(10\) 0.111222 − 0.624510i 0.0351716 − 0.197487i
\(11\) −1.18222 − 4.41211i −0.356453 − 1.33030i −0.878646 − 0.477473i \(-0.841553\pi\)
0.522193 − 0.852827i \(-0.325114\pi\)
\(12\) 1.81550 − 0.839015i 0.524091 − 0.242203i
\(13\) −3.32021 − 3.32021i −0.920860 − 0.920860i 0.0762299 − 0.997090i \(-0.475712\pi\)
−0.997090 + 0.0762299i \(0.975712\pi\)
\(14\) −2.61631 − 2.67487i −0.699237 − 0.714890i
\(15\) −0.448544 −0.115813
\(16\) 3.76807 − 1.34227i 0.942017 − 0.335566i
\(17\) −2.68303 + 4.64715i −0.650732 + 1.12710i 0.332214 + 0.943204i \(0.392204\pi\)
−0.982946 + 0.183896i \(0.941129\pi\)
\(18\) −0.809172 − 1.15985i −0.190724 − 0.273378i
\(19\) 1.47730 − 5.51336i 0.338916 − 1.26485i −0.560645 − 0.828056i \(-0.689447\pi\)
0.899561 − 0.436795i \(-0.143886\pi\)
\(20\) −0.893400 − 0.0812509i −0.199770 − 0.0181682i
\(21\) −1.53774 + 2.15298i −0.335563 + 0.469820i
\(22\) −6.07398 + 2.19896i −1.29498 + 0.468820i
\(23\) 0.0663290 − 0.0382951i 0.0138306 − 0.00798507i −0.493069 − 0.869990i \(-0.664125\pi\)
0.506899 + 0.862005i \(0.330792\pi\)
\(24\) −1.40159 − 2.45673i −0.286099 − 0.501478i
\(25\) −4.15589 − 2.39940i −0.831178 − 0.479881i
\(26\) −4.27710 + 5.07953i −0.838808 + 0.996178i
\(27\) −0.707107 + 0.707107i −0.136083 + 0.136083i
\(28\) −3.45285 + 4.00972i −0.652527 + 0.757765i
\(29\) −3.56763 − 3.56763i −0.662492 − 0.662492i 0.293475 − 0.955967i \(-0.405188\pi\)
−0.955967 + 0.293475i \(0.905188\pi\)
\(30\) 0.0542023 + 0.632016i 0.00989594 + 0.115390i
\(31\) 0.347695 − 0.602225i 0.0624478 − 0.108163i −0.833111 − 0.553106i \(-0.813443\pi\)
0.895559 + 0.444943i \(0.146776\pi\)
\(32\) −2.34664 − 5.14716i −0.414832 − 0.909898i
\(33\) 2.28387 + 3.95579i 0.397572 + 0.688614i
\(34\) 6.87225 + 3.21894i 1.17858 + 0.552044i
\(35\) 1.08019 − 0.491454i 0.182586 − 0.0830709i
\(36\) −1.53649 + 1.28031i −0.256082 + 0.213386i
\(37\) 11.1005 + 2.97437i 1.82491 + 0.488984i 0.997374 − 0.0724267i \(-0.0230743\pi\)
0.827538 + 0.561410i \(0.189741\pi\)
\(38\) −7.94707 − 1.41534i −1.28918 − 0.229598i
\(39\) 4.06641 + 2.34774i 0.651147 + 0.375940i
\(40\) −0.00652673 + 1.26866i −0.00103197 + 0.200592i
\(41\) 5.69219i 0.888971i 0.895786 + 0.444486i \(0.146614\pi\)
−0.895786 + 0.444486i \(0.853386\pi\)
\(42\) 3.21947 + 1.90658i 0.496774 + 0.294191i
\(43\) 6.59516 − 6.59516i 1.00575 − 1.00575i 0.00576845 − 0.999983i \(-0.498164\pi\)
0.999983 − 0.00576845i \(-0.00183617\pi\)
\(44\) 3.83241 + 8.29277i 0.577757 + 1.25018i
\(45\) 0.433260 − 0.116092i 0.0645866 − 0.0173059i
\(46\) −0.0619746 − 0.0888327i −0.00913765 − 0.0130977i
\(47\) −0.711379 − 1.23214i −0.103765 − 0.179727i 0.809468 − 0.587164i \(-0.199756\pi\)
−0.913233 + 0.407438i \(0.866422\pi\)
\(48\) −3.29227 + 2.27178i −0.475198 + 0.327903i
\(49\) 1.34427 − 6.86971i 0.192039 − 0.981387i
\(50\) −2.87866 + 6.14577i −0.407104 + 0.869143i
\(51\) 1.38884 − 5.18323i 0.194477 − 0.725797i
\(52\) 7.67411 + 5.41279i 1.06421 + 0.750619i
\(53\) −1.86185 − 6.94851i −0.255745 − 0.954452i −0.967675 − 0.252201i \(-0.918845\pi\)
0.711930 − 0.702250i \(-0.247821\pi\)
\(54\) 1.08179 + 0.910895i 0.147213 + 0.123957i
\(55\) − 2.04883i − 0.276265i
\(56\) 6.06710 + 4.38067i 0.810751 + 0.585392i
\(57\) 5.70785i 0.756023i
\(58\) −4.59582 + 5.45805i −0.603461 + 0.716677i
\(59\) −0.230588 − 0.860565i −0.0300200 − 0.112036i 0.949290 − 0.314401i \(-0.101804\pi\)
−0.979310 + 0.202365i \(0.935137\pi\)
\(60\) 0.883987 − 0.152747i 0.114122 − 0.0197195i
\(61\) −3.37528 + 12.5967i −0.432160 + 1.61284i 0.315612 + 0.948888i \(0.397790\pi\)
−0.747772 + 0.663955i \(0.768877\pi\)
\(62\) −0.890576 − 0.417143i −0.113103 − 0.0529772i
\(63\) 0.928114 − 2.47762i 0.116931 − 0.312151i
\(64\) −6.96899 + 3.92850i −0.871124 + 0.491063i
\(65\) −1.05306 − 1.82396i −0.130617 − 0.226235i
\(66\) 5.29788 − 3.69609i 0.652124 − 0.454958i
\(67\) 4.23738 − 1.13540i 0.517678 − 0.138711i 0.00948556 − 0.999955i \(-0.496981\pi\)
0.508192 + 0.861244i \(0.330314\pi\)
\(68\) 3.70517 − 10.0723i 0.449318 − 1.22144i
\(69\) −0.0541574 + 0.0541574i −0.00651979 + 0.00651979i
\(70\) −0.823010 − 1.46265i −0.0983685 − 0.174820i
\(71\) 9.19924i 1.09175i 0.837867 + 0.545875i \(0.183803\pi\)
−0.837867 + 0.545875i \(0.816197\pi\)
\(72\) 1.98968 + 2.01026i 0.234487 + 0.236912i
\(73\) 8.04777 + 4.64638i 0.941920 + 0.543818i 0.890562 − 0.454862i \(-0.150312\pi\)
0.0513584 + 0.998680i \(0.483645\pi\)
\(74\) 2.84962 − 16.0005i 0.331261 − 1.86002i
\(75\) 4.63529 + 1.24202i 0.535238 + 0.143416i
\(76\) −1.03394 + 11.3688i −0.118601 + 1.30409i
\(77\) −9.83429 − 7.02403i −1.12072 − 0.800463i
\(78\) 2.81668 − 6.01344i 0.318926 − 0.680888i
\(79\) −2.89898 − 5.02118i −0.326161 − 0.564927i 0.655586 − 0.755121i \(-0.272422\pi\)
−0.981747 + 0.190194i \(0.939088\pi\)
\(80\) 1.78838 − 0.144109i 0.199947 − 0.0161119i
\(81\) 0.500000 − 0.866025i 0.0555556 − 0.0962250i
\(82\) 8.02054 − 0.687849i 0.885720 − 0.0759602i
\(83\) 4.19157 + 4.19157i 0.460084 + 0.460084i 0.898683 − 0.438599i \(-0.144525\pi\)
−0.438599 + 0.898683i \(0.644525\pi\)
\(84\) 2.29740 − 4.76675i 0.250667 − 0.520095i
\(85\) −1.70195 + 1.70195i −0.184602 + 0.184602i
\(86\) −10.0898 − 8.49588i −1.08801 − 0.916135i
\(87\) 4.36943 + 2.52269i 0.468452 + 0.270461i
\(88\) 11.2217 − 6.40213i 1.19624 − 0.682469i
\(89\) 6.23399 − 3.59919i 0.660801 − 0.381514i −0.131781 − 0.991279i \(-0.542070\pi\)
0.792582 + 0.609765i \(0.208736\pi\)
\(90\) −0.215933 − 0.596452i −0.0227614 − 0.0628716i
\(91\) −12.3651 − 1.19845i −1.29622 − 0.125631i
\(92\) −0.117680 + 0.0980593i −0.0122690 + 0.0102234i
\(93\) −0.179980 + 0.671695i −0.0186631 + 0.0696515i
\(94\) −1.65018 + 1.15126i −0.170203 + 0.118743i
\(95\) 1.28011 − 2.21721i 0.131336 − 0.227481i
\(96\) 3.59887 + 4.36442i 0.367308 + 0.445442i
\(97\) 10.0676 1.02221 0.511103 − 0.859519i \(-0.329237\pi\)
0.511103 + 0.859519i \(0.329237\pi\)
\(98\) −9.84215 − 1.06399i −0.994207 − 0.107480i
\(99\) −3.22989 − 3.22989i −0.324616 − 0.324616i
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.109.17 yes 120
7.2 even 3 inner 336.2.bq.b.205.5 yes 120
16.5 even 4 inner 336.2.bq.b.277.5 yes 120
112.37 even 12 inner 336.2.bq.b.37.17 ✓ 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
336.2.bq.b.37.17 ✓ 120 112.37 even 12 inner
336.2.bq.b.109.17 yes 120 1.1 even 1 trivial
336.2.bq.b.205.5 yes 120 7.2 even 3 inner
336.2.bq.b.277.5 yes 120 16.5 even 4 inner