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Properties

Label 340.2.bf.a.11.5
Level $340$
Weight $2$
Character 340.11
Analytic conductor $2.715$
Analytic rank $0$
Dimension $288$
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] = [340,2,Mod(11,340)] 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("340.11"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(340, base_ring=CyclotomicField(16)) chi = DirichletCharacter(H, H._module([8, 0, 7])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 340 = 2^{2} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 340.bf (of order \(16\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.71491366872\)
Analytic rank: \(0\)
Dimension: \(288\)
Relative dimension: \(36\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label 11.5
Character \(\chi\) \(=\) 340.11
Dual form 340.2.bf.a.31.5

$q$-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+(-1.31209 + 0.527661i) q^{2} +(0.300562 - 0.0597854i) q^{3} +(1.44315 - 1.38468i) q^{4} +(0.831470 + 0.555570i) q^{5} +(-0.362817 + 0.237039i) q^{6} +(0.524425 + 0.784858i) q^{7} +(-1.16290 + 2.57831i) q^{8} +(-2.68488 + 1.11211i) q^{9} +(-1.38411 - 0.290222i) q^{10} +(-1.06654 + 5.36186i) q^{11} +(0.350971 - 0.502460i) q^{12} +(3.93815 - 3.93815i) q^{13} +(-1.10223 - 0.753084i) q^{14} +(0.283123 + 0.117273i) q^{15} +(0.165348 - 3.99658i) q^{16} +(4.03901 + 0.828494i) q^{17} +(2.93597 - 2.87589i) q^{18} +(-1.51849 + 3.66596i) q^{19} +(1.96922 - 0.349546i) q^{20} +(0.204545 + 0.204545i) q^{21} +(-1.42985 - 7.59801i) q^{22} +(2.65728 + 0.528566i) q^{23} +(-0.195377 + 0.844465i) q^{24} +(0.382683 + 0.923880i) q^{25} +(-3.08919 + 7.24521i) q^{26} +(-1.50489 + 1.00554i) q^{27} +(1.84360 + 0.406507i) q^{28} +(0.323893 - 0.484740i) q^{29} +(-0.433363 - 0.00447991i) q^{30} +(1.04253 + 5.24118i) q^{31} +(1.89189 + 5.33111i) q^{32} +1.67534i q^{33} +(-5.73670 + 1.04417i) q^{34} +0.943941i q^{35} +(-2.33476 + 5.32262i) q^{36} +(0.914271 + 4.59635i) q^{37} +(0.0580071 - 5.61130i) q^{38} +(0.948214 - 1.41910i) q^{39} +(-2.39934 + 1.49771i) q^{40} +(-3.45139 + 2.30615i) q^{41} +(-0.376312 - 0.160451i) q^{42} +(-2.25748 - 5.45003i) q^{43} +(5.88527 + 9.21477i) q^{44} +(-2.85025 - 0.566950i) q^{45} +(-3.76549 + 0.708619i) q^{46} +(1.20034 + 1.20034i) q^{47} +(-0.189240 - 1.21110i) q^{48} +(2.33780 - 5.64396i) q^{49} +(-0.989610 - 1.01028i) q^{50} +(1.26350 + 0.00753958i) q^{51} +(0.230271 - 11.1364i) q^{52} +(1.70172 + 0.704876i) q^{53} +(1.44397 - 2.11343i) q^{54} +(-3.86569 + 3.86569i) q^{55} +(-2.63346 + 0.439422i) q^{56} +(-0.237229 + 1.19263i) q^{57} +(-0.169197 + 0.806926i) q^{58} +(13.6206 - 5.64185i) q^{59} +(0.570974 - 0.222791i) q^{60} +(-0.354179 - 0.530066i) q^{61} +(-4.13346 - 6.32678i) q^{62} +(-2.28087 - 1.52403i) q^{63} +(-5.29535 - 5.99661i) q^{64} +(5.46237 - 1.08653i) q^{65} +(-0.884009 - 2.19819i) q^{66} -9.76217 q^{67} +(6.97608 - 4.39708i) q^{68} +0.830277 q^{69} +(-0.498081 - 1.23853i) q^{70} +(-11.8188 + 2.35090i) q^{71} +(0.254863 - 8.21571i) q^{72} +(3.46735 + 2.31681i) q^{73} +(-3.62492 - 5.54839i) q^{74} +(0.170255 + 0.254804i) q^{75} +(2.88476 + 7.39313i) q^{76} +(-4.76762 + 1.97481i) q^{77} +(-0.495334 + 2.36232i) q^{78} +(1.90429 - 9.57349i) q^{79} +(2.35786 - 3.23117i) q^{80} +(5.77255 - 5.77255i) q^{81} +(3.31167 - 4.84703i) q^{82} +(-0.802091 - 0.332237i) q^{83} +(0.578418 + 0.0119601i) q^{84} +(2.89803 + 2.93282i) q^{85} +(5.83778 + 5.95973i) q^{86} +(0.0683694 - 0.165058i) q^{87} +(-12.5843 - 8.98516i) q^{88} +(-5.78855 - 5.78855i) q^{89} +(4.03893 - 0.760078i) q^{90} +(5.15616 + 1.02562i) q^{91} +(4.56674 - 2.91667i) q^{92} +(0.626692 + 1.51297i) q^{93} +(-2.20832 - 0.941577i) q^{94} +(-3.29927 + 2.20450i) q^{95} +(0.887353 + 1.48922i) q^{96} +(9.42579 - 14.1067i) q^{97} +(-0.0893054 + 8.63893i) q^{98} +(-3.09946 - 15.5821i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 288 q - 16 q^{24} - 32 q^{26} - 96 q^{28} - 80 q^{32} - 64 q^{34} - 64 q^{36} - 80 q^{38} - 96 q^{42} - 32 q^{44} - 16 q^{46} + 80 q^{54} + 80 q^{56} - 160 q^{57} - 64 q^{61} + 112 q^{62} + 96 q^{64} + 208 q^{66}+ \cdots - 272 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/340\mathbb{Z}\right)^\times\).

\(n\) \(137\) \(171\) \(241\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{7}{16}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.31209 + 0.527661i −0.927786 + 0.373113i
\(3\) 0.300562 − 0.0597854i 0.173529 − 0.0345171i −0.107561 − 0.994199i \(-0.534304\pi\)
0.281090 + 0.959681i \(0.409304\pi\)
\(4\) 1.44315 − 1.38468i 0.721574 − 0.692338i
\(5\) 0.831470 + 0.555570i 0.371845 + 0.248459i
\(6\) −0.362817 + 0.237039i −0.148119 + 0.0967706i
\(7\) 0.524425 + 0.784858i 0.198214 + 0.296648i 0.917240 − 0.398335i \(-0.130412\pi\)
−0.719026 + 0.694983i \(0.755412\pi\)
\(8\) −1.16290 + 2.57831i −0.411146 + 0.911570i
\(9\) −2.68488 + 1.11211i −0.894959 + 0.370704i
\(10\) −1.38411 − 0.290222i −0.437695 − 0.0917764i
\(11\) −1.06654 + 5.36186i −0.321574 + 1.61666i 0.394677 + 0.918820i \(0.370857\pi\)
−0.716251 + 0.697843i \(0.754143\pi\)
\(12\) 0.350971 − 0.502460i 0.101317 − 0.145048i
\(13\) 3.93815 − 3.93815i 1.09225 − 1.09225i 0.0969584 − 0.995288i \(-0.469089\pi\)
0.995288 − 0.0969584i \(-0.0309114\pi\)
\(14\) −1.10223 − 0.753084i −0.294584 − 0.201270i
\(15\) 0.283123 + 0.117273i 0.0731020 + 0.0302799i
\(16\) 0.165348 − 3.99658i 0.0413369 − 0.999145i
\(17\) 4.03901 + 0.828494i 0.979604 + 0.200939i
\(18\) 2.93597 − 2.87589i 0.692016 − 0.677854i
\(19\) −1.51849 + 3.66596i −0.348365 + 0.841028i 0.648448 + 0.761259i \(0.275418\pi\)
−0.996813 + 0.0797692i \(0.974582\pi\)
\(20\) 1.96922 − 0.349546i 0.440330 − 0.0781608i
\(21\) 0.204545 + 0.204545i 0.0446354 + 0.0446354i
\(22\) −1.42985 − 7.59801i −0.304846 − 1.61990i
\(23\) 2.65728 + 0.528566i 0.554081 + 0.110214i 0.464186 − 0.885738i \(-0.346347\pi\)
0.0898949 + 0.995951i \(0.471347\pi\)
\(24\) −0.195377 + 0.844465i −0.0398811 + 0.172376i
\(25\) 0.382683 + 0.923880i 0.0765367 + 0.184776i
\(26\) −3.08919 + 7.24521i −0.605840 + 1.42090i
\(27\) −1.50489 + 1.00554i −0.289617 + 0.193516i
\(28\) 1.84360 + 0.406507i 0.348407 + 0.0768226i
\(29\) 0.323893 − 0.484740i 0.0601453 − 0.0900139i −0.800178 − 0.599763i \(-0.795262\pi\)
0.860323 + 0.509749i \(0.170262\pi\)
\(30\) −0.433363 − 0.00447991i −0.0791208 − 0.000817916i
\(31\) 1.04253 + 5.24118i 0.187245 + 0.941343i 0.954092 + 0.299514i \(0.0968244\pi\)
−0.766847 + 0.641830i \(0.778176\pi\)
\(32\) 1.89189 + 5.33111i 0.334442 + 0.942416i
\(33\) 1.67534i 0.291638i
\(34\) −5.73670 + 1.04417i −0.983836 + 0.179074i
\(35\) 0.943941i 0.159555i
\(36\) −2.33476 + 5.32262i −0.389126 + 0.887104i
\(37\) 0.914271 + 4.59635i 0.150305 + 0.755635i 0.980246 + 0.197784i \(0.0633743\pi\)
−0.829940 + 0.557852i \(0.811626\pi\)
\(38\) 0.0580071 − 5.61130i 0.00941000 − 0.910274i
\(39\) 0.948214 − 1.41910i 0.151836 − 0.227238i
\(40\) −2.39934 + 1.49771i −0.379370 + 0.236809i
\(41\) −3.45139 + 2.30615i −0.539017 + 0.360160i −0.795084 − 0.606499i \(-0.792573\pi\)
0.256067 + 0.966659i \(0.417573\pi\)
\(42\) −0.376312 − 0.160451i −0.0580662 − 0.0247581i
\(43\) −2.25748 − 5.45003i −0.344262 − 0.831122i −0.997275 − 0.0737751i \(-0.976495\pi\)
0.653013 − 0.757347i \(-0.273505\pi\)
\(44\) 5.88527 + 9.21477i 0.887237 + 1.38918i
\(45\) −2.85025 − 0.566950i −0.424890 − 0.0845159i
\(46\) −3.76549 + 0.708619i −0.555191 + 0.104480i
\(47\) 1.20034 + 1.20034i 0.175087 + 0.175087i 0.789210 − 0.614123i \(-0.210490\pi\)
−0.614123 + 0.789210i \(0.710490\pi\)
\(48\) −0.189240 − 1.21110i −0.0273145 − 0.174808i
\(49\) 2.33780 − 5.64396i 0.333972 − 0.806280i
\(50\) −0.989610 − 1.01028i −0.139952 − 0.142876i
\(51\) 1.26350 + 0.00753958i 0.176926 + 0.00105575i
\(52\) 0.230271 − 11.1364i 0.0319328 − 1.54434i
\(53\) 1.70172 + 0.704876i 0.233749 + 0.0968222i 0.496484 − 0.868046i \(-0.334624\pi\)
−0.262734 + 0.964868i \(0.584624\pi\)
\(54\) 1.44397 − 2.11343i 0.196499 − 0.287601i
\(55\) −3.86569 + 3.86569i −0.521249 + 0.521249i
\(56\) −2.63346 + 0.439422i −0.351911 + 0.0587203i
\(57\) −0.237229 + 1.19263i −0.0314217 + 0.157968i
\(58\) −0.169197 + 0.806926i −0.0222167 + 0.105955i
\(59\) 13.6206 − 5.64185i 1.77325 − 0.734506i 0.779054 − 0.626957i \(-0.215700\pi\)
0.994200 − 0.107549i \(-0.0343001\pi\)
\(60\) 0.570974 − 0.222791i 0.0737124 − 0.0287622i
\(61\) −0.354179 − 0.530066i −0.0453479 − 0.0678680i 0.808103 − 0.589041i \(-0.200494\pi\)
−0.853451 + 0.521173i \(0.825494\pi\)
\(62\) −4.13346 − 6.32678i −0.524950 − 0.803502i
\(63\) −2.28087 − 1.52403i −0.287362 − 0.192009i
\(64\) −5.29535 − 5.99661i −0.661918 − 0.749576i
\(65\) 5.46237 − 1.08653i 0.677524 − 0.134768i
\(66\) −0.884009 − 2.19819i −0.108814 − 0.270578i
\(67\) −9.76217 −1.19264 −0.596319 − 0.802747i \(-0.703371\pi\)
−0.596319 + 0.802747i \(0.703371\pi\)
\(68\) 6.97608 − 4.39708i 0.845974 − 0.533224i
\(69\) 0.830277 0.0999536
\(70\) −0.498081 − 1.23853i −0.0595321 − 0.148033i
\(71\) −11.8188 + 2.35090i −1.40263 + 0.279001i −0.837697 − 0.546136i \(-0.816098\pi\)
−0.564934 + 0.825136i \(0.691098\pi\)
\(72\) 0.254863 − 8.21571i 0.0300359 − 0.968230i
\(73\) 3.46735 + 2.31681i 0.405822 + 0.271162i 0.741684 − 0.670750i \(-0.234027\pi\)
−0.335862 + 0.941911i \(0.609027\pi\)
\(74\) −3.62492 − 5.54839i −0.421388 − 0.644987i
\(75\) 0.170255 + 0.254804i 0.0196593 + 0.0294222i
\(76\) 2.88476 + 7.39313i 0.330904 + 0.848050i
\(77\) −4.76762 + 1.97481i −0.543321 + 0.225051i
\(78\) −0.495334 + 2.36232i −0.0560855 + 0.267480i
\(79\) 1.90429 − 9.57349i 0.214249 − 1.07710i −0.712572 − 0.701599i \(-0.752470\pi\)
0.926821 − 0.375503i \(-0.122530\pi\)
\(80\) 2.35786 − 3.23117i 0.263617 − 0.361256i
\(81\) 5.77255 − 5.77255i 0.641394 − 0.641394i
\(82\) 3.31167 − 4.84703i 0.365712 − 0.535265i
\(83\) −0.802091 − 0.332237i −0.0880409 − 0.0364677i 0.338228 − 0.941064i \(-0.390172\pi\)
−0.426269 + 0.904596i \(0.640172\pi\)
\(84\) 0.578418 + 0.0119601i 0.0631106 + 0.00130496i
\(85\) 2.89803 + 2.93282i 0.314335 + 0.318109i
\(86\) 5.83778 + 5.95973i 0.629504 + 0.642655i
\(87\) 0.0683694 − 0.165058i 0.00732996 − 0.0176961i
\(88\) −12.5843 − 8.98516i −1.34149 − 0.957821i
\(89\) −5.78855 − 5.78855i −0.613585 − 0.613585i 0.330293 − 0.943878i \(-0.392852\pi\)
−0.943878 + 0.330293i \(0.892852\pi\)
\(90\) 4.03893 − 0.760078i 0.425741 − 0.0801193i
\(91\) 5.15616 + 1.02562i 0.540512 + 0.107515i
\(92\) 4.56674 − 2.91667i 0.476115 − 0.304084i
\(93\) 0.626692 + 1.51297i 0.0649850 + 0.156888i
\(94\) −2.20832 − 0.941577i −0.227771 − 0.0971162i
\(95\) −3.29927 + 2.20450i −0.338498 + 0.226177i
\(96\) 0.887353 + 1.48922i 0.0905651 + 0.151993i
\(97\) 9.42579 − 14.1067i 0.957044 − 1.43232i 0.0560786 − 0.998426i \(-0.482140\pi\)
0.900965 − 0.433891i \(-0.142860\pi\)
\(98\) −0.0893054 + 8.63893i −0.00902121 + 0.872664i
\(99\) −3.09946 − 15.5821i −0.311508 − 1.56606i
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 340.2.bf.a.11.5 ✓ 288
4.3 odd 2 inner 340.2.bf.a.11.15 yes 288
17.14 odd 16 inner 340.2.bf.a.31.15 yes 288
68.31 even 16 inner 340.2.bf.a.31.5 yes 288
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
340.2.bf.a.11.5 ✓ 288 1.1 even 1 trivial
340.2.bf.a.11.15 yes 288 4.3 odd 2 inner
340.2.bf.a.31.5 yes 288 68.31 even 16 inner
340.2.bf.a.31.15 yes 288 17.14 odd 16 inner