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Properties

Label 340.2.bd.a.57.8
Level $340$
Weight $2$
Character 340.57
Analytic conductor $2.715$
Analytic rank $0$
Dimension $72$
Inner twists $2$

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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(57,340)] 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("340.57"); 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([0, 4, 15])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 340 = 2^{2} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 340.bd (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: \(72\)
Relative dimension: \(9\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label 57.8
Character \(\chi\) \(=\) 340.57
Dual form 340.2.bd.a.173.8

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.27530 - 1.90862i) q^{3} +(-2.15814 - 0.585194i) q^{5} +(0.311904 - 1.56805i) q^{7} +(-0.868385 - 2.09647i) q^{9} +(-0.0850114 + 0.427381i) q^{11} -4.24723i q^{13} +(-3.86917 + 3.37276i) q^{15} +(0.292567 - 4.11271i) q^{17} +(0.0883994 - 0.213415i) q^{19} +(-2.59503 - 2.59503i) q^{21} +(-4.01125 + 2.68023i) q^{23} +(4.31510 + 2.52585i) q^{25} +(1.64530 + 0.327271i) q^{27} +(-1.31967 + 1.97502i) q^{29} +(0.0590924 + 0.297078i) q^{31} +(0.707292 + 0.707292i) q^{33} +(-1.59074 + 3.20154i) q^{35} +(2.06259 + 1.37818i) q^{37} +(-8.10633 - 5.41648i) q^{39} +(3.87052 + 5.79264i) q^{41} +(0.980981 - 2.36830i) q^{43} +(0.647253 + 5.03263i) q^{45} +8.57130 q^{47} +(4.10566 + 1.70062i) q^{49} +(-7.47648 - 5.80333i) q^{51} +(3.60578 - 1.49356i) q^{53} +(0.433567 - 0.872598i) q^{55} +(-0.294592 - 0.440888i) q^{57} +(-2.04993 + 0.849110i) q^{59} +(12.3274 - 8.23691i) q^{61} +(-3.55822 + 0.707773i) q^{63} +(-2.48545 + 9.16610i) q^{65} +(8.76895 - 8.76895i) q^{67} +11.0740i q^{69} +(-11.6557 + 2.31846i) q^{71} +(0.691555 + 3.47668i) q^{73} +(10.3239 - 5.01465i) q^{75} +(0.643639 + 0.266604i) q^{77} +(-9.27090 - 1.84410i) q^{79} +(7.53658 - 7.53658i) q^{81} +(6.27414 + 15.1471i) q^{83} +(-3.03813 + 8.70458i) q^{85} +(2.08659 + 5.03748i) q^{87} +(0.840092 - 0.840092i) q^{89} +(-6.65987 - 1.32473i) q^{91} +(0.642368 + 0.266078i) q^{93} +(-0.315667 + 0.408848i) q^{95} +(1.15359 + 5.79947i) q^{97} +(0.969813 - 0.192908i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q + 24 q^{15} + 8 q^{25} - 48 q^{27} - 32 q^{31} + 16 q^{33} + 32 q^{37} - 32 q^{39} - 40 q^{41} + 80 q^{47} - 40 q^{53} + 16 q^{55} + 8 q^{57} + 112 q^{59} - 48 q^{63} - 32 q^{67} - 16 q^{71} + 8 q^{73}+ \cdots + 96 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(137\) \(171\) \(241\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(1\) \(e\left(\frac{15}{16}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.27530 − 1.90862i 0.736293 − 1.10194i −0.254568 − 0.967055i \(-0.581933\pi\)
0.990861 − 0.134885i \(-0.0430666\pi\)
\(4\) 0 0
\(5\) −2.15814 − 0.585194i −0.965147 − 0.261707i
\(6\) 0 0
\(7\) 0.311904 − 1.56805i 0.117889 − 0.592667i −0.876003 − 0.482306i \(-0.839799\pi\)
0.993892 − 0.110361i \(-0.0352007\pi\)
\(8\) 0 0
\(9\) −0.868385 − 2.09647i −0.289462 − 0.698822i
\(10\) 0 0
\(11\) −0.0850114 + 0.427381i −0.0256319 + 0.128860i −0.991483 − 0.130239i \(-0.958426\pi\)
0.965851 + 0.259099i \(0.0834256\pi\)
\(12\) 0 0
\(13\) − 4.24723i − 1.17797i −0.808144 − 0.588985i \(-0.799528\pi\)
0.808144 − 0.588985i \(-0.200472\pi\)
\(14\) 0 0
\(15\) −3.86917 + 3.37276i −0.999016 + 0.870842i
\(16\) 0 0
\(17\) 0.292567 − 4.11271i 0.0709578 − 0.997479i
\(18\) 0 0
\(19\) 0.0883994 − 0.213415i 0.0202802 − 0.0489607i −0.913415 − 0.407029i \(-0.866565\pi\)
0.933695 + 0.358069i \(0.116565\pi\)
\(20\) 0 0
\(21\) −2.59503 − 2.59503i −0.566283 − 0.566283i
\(22\) 0 0
\(23\) −4.01125 + 2.68023i −0.836404 + 0.558867i −0.898384 − 0.439210i \(-0.855258\pi\)
0.0619805 + 0.998077i \(0.480258\pi\)
\(24\) 0 0
\(25\) 4.31510 + 2.52585i 0.863019 + 0.505171i
\(26\) 0 0
\(27\) 1.64530 + 0.327271i 0.316638 + 0.0629833i
\(28\) 0 0
\(29\) −1.31967 + 1.97502i −0.245056 + 0.366753i −0.933524 − 0.358514i \(-0.883284\pi\)
0.688468 + 0.725267i \(0.258284\pi\)
\(30\) 0 0
\(31\) 0.0590924 + 0.297078i 0.0106133 + 0.0533567i 0.985729 − 0.168340i \(-0.0538406\pi\)
−0.975116 + 0.221697i \(0.928841\pi\)
\(32\) 0 0
\(33\) 0.707292 + 0.707292i 0.123124 + 0.123124i
\(34\) 0 0
\(35\) −1.59074 + 3.20154i −0.268885 + 0.541159i
\(36\) 0 0
\(37\) 2.06259 + 1.37818i 0.339087 + 0.226571i 0.713441 − 0.700715i \(-0.247136\pi\)
−0.374354 + 0.927286i \(0.622136\pi\)
\(38\) 0 0
\(39\) −8.10633 − 5.41648i −1.29805 − 0.867331i
\(40\) 0 0
\(41\) 3.87052 + 5.79264i 0.604474 + 0.904659i 0.999904 − 0.0138372i \(-0.00440465\pi\)
−0.395431 + 0.918496i \(0.629405\pi\)
\(42\) 0 0
\(43\) 0.980981 − 2.36830i 0.149598 − 0.361162i −0.831260 − 0.555883i \(-0.812380\pi\)
0.980859 + 0.194721i \(0.0623802\pi\)
\(44\) 0 0
\(45\) 0.647253 + 5.03263i 0.0964868 + 0.750220i
\(46\) 0 0
\(47\) 8.57130 1.25025 0.625126 − 0.780524i \(-0.285048\pi\)
0.625126 + 0.780524i \(0.285048\pi\)
\(48\) 0 0
\(49\) 4.10566 + 1.70062i 0.586523 + 0.242946i
\(50\) 0 0
\(51\) −7.47648 − 5.80333i −1.04692 − 0.812628i
\(52\) 0 0
\(53\) 3.60578 − 1.49356i 0.495291 − 0.205156i −0.121033 − 0.992648i \(-0.538621\pi\)
0.616325 + 0.787492i \(0.288621\pi\)
\(54\) 0 0
\(55\) 0.433567 − 0.872598i 0.0584622 − 0.117661i
\(56\) 0 0
\(57\) −0.294592 − 0.440888i −0.0390196 − 0.0583970i
\(58\) 0 0
\(59\) −2.04993 + 0.849110i −0.266878 + 0.110545i −0.512110 − 0.858920i \(-0.671136\pi\)
0.245232 + 0.969465i \(0.421136\pi\)
\(60\) 0 0
\(61\) 12.3274 − 8.23691i 1.57836 − 1.05463i 0.614257 − 0.789106i \(-0.289456\pi\)
0.964105 − 0.265522i \(-0.0855442\pi\)
\(62\) 0 0
\(63\) −3.55822 + 0.707773i −0.448293 + 0.0891710i
\(64\) 0 0
\(65\) −2.48545 + 9.16610i −0.308282 + 1.13691i
\(66\) 0 0
\(67\) 8.76895 − 8.76895i 1.07130 − 1.07130i 0.0740426 − 0.997255i \(-0.476410\pi\)
0.997255 − 0.0740426i \(-0.0235901\pi\)
\(68\) 0 0
\(69\) 11.0740i 1.33316i
\(70\) 0 0
\(71\) −11.6557 + 2.31846i −1.38328 + 0.275151i −0.829962 − 0.557820i \(-0.811638\pi\)
−0.553316 + 0.832971i \(0.686638\pi\)
\(72\) 0 0
\(73\) 0.691555 + 3.47668i 0.0809403 + 0.406915i 0.999920 + 0.0126107i \(0.00401423\pi\)
−0.918980 + 0.394304i \(0.870986\pi\)
\(74\) 0 0
\(75\) 10.3239 − 5.01465i 1.19210 − 0.579042i
\(76\) 0 0
\(77\) 0.643639 + 0.266604i 0.0733495 + 0.0303824i
\(78\) 0 0
\(79\) −9.27090 − 1.84410i −1.04306 − 0.207477i −0.356306 − 0.934369i \(-0.615964\pi\)
−0.686751 + 0.726892i \(0.740964\pi\)
\(80\) 0 0
\(81\) 7.53658 − 7.53658i 0.837398 − 0.837398i
\(82\) 0 0
\(83\) 6.27414 + 15.1471i 0.688677 + 1.66261i 0.747430 + 0.664340i \(0.231287\pi\)
−0.0587536 + 0.998273i \(0.518713\pi\)
\(84\) 0 0
\(85\) −3.03813 + 8.70458i −0.329532 + 0.944145i
\(86\) 0 0
\(87\) 2.08659 + 5.03748i 0.223706 + 0.540075i
\(88\) 0 0
\(89\) 0.840092 − 0.840092i 0.0890496 − 0.0890496i −0.661179 − 0.750228i \(-0.729944\pi\)
0.750228 + 0.661179i \(0.229944\pi\)
\(90\) 0 0
\(91\) −6.65987 − 1.32473i −0.698144 − 0.138869i
\(92\) 0 0
\(93\) 0.642368 + 0.266078i 0.0666105 + 0.0275910i
\(94\) 0 0
\(95\) −0.315667 + 0.408848i −0.0323867 + 0.0419469i
\(96\) 0 0
\(97\) 1.15359 + 5.79947i 0.117129 + 0.588847i 0.994115 + 0.108331i \(0.0345506\pi\)
−0.876986 + 0.480516i \(0.840449\pi\)
\(98\) 0 0
\(99\) 0.969813 − 0.192908i 0.0974699 − 0.0193880i
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.bd.a.57.8 ✓ 72
5.3 odd 4 340.2.bi.a.193.8 yes 72
17.3 odd 16 340.2.bi.a.37.8 yes 72
85.3 even 16 inner 340.2.bd.a.173.8 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
340.2.bd.a.57.8 ✓ 72 1.1 even 1 trivial
340.2.bd.a.173.8 yes 72 85.3 even 16 inner
340.2.bi.a.37.8 yes 72 17.3 odd 16
340.2.bi.a.193.8 yes 72 5.3 odd 4