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Properties

Label 340.2.bf.a.31.6
Level $340$
Weight $2$
Character 340.31
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 31.6
Character \(\chi\) \(=\) 340.31
Dual form 340.2.bf.a.11.6

$q$-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+(-1.26338 + 0.635504i) q^{2} +(2.38453 + 0.474313i) q^{3} +(1.19227 - 1.60577i) q^{4} +(0.831470 - 0.555570i) q^{5} +(-3.31401 + 0.916143i) q^{6} +(1.59437 - 2.38614i) q^{7} +(-0.485817 + 2.78639i) q^{8} +(2.68939 + 1.11398i) q^{9} +(-0.697397 + 1.23030i) q^{10} +(0.0776306 + 0.390276i) q^{11} +(3.60464 - 3.26350i) q^{12} +(-2.03137 - 2.03137i) q^{13} +(-0.497893 + 4.02783i) q^{14} +(2.24618 - 0.930399i) q^{15} +(-1.15699 - 3.82902i) q^{16} +(4.11773 + 0.210465i) q^{17} +(-4.10567 + 0.301734i) q^{18} +(-0.234706 - 0.566631i) q^{19} +(0.0992173 - 1.99754i) q^{20} +(4.93359 - 4.93359i) q^{21} +(-0.346099 - 0.443733i) q^{22} +(-7.73404 + 1.53840i) q^{23} +(-2.48007 + 6.41382i) q^{24} +(0.382683 - 0.923880i) q^{25} +(3.85734 + 1.27545i) q^{26} +(-0.179963 - 0.120248i) q^{27} +(-1.93067 - 5.40510i) q^{28} +(2.24960 + 3.36676i) q^{29} +(-2.24651 + 2.60291i) q^{30} +(-1.11894 + 5.62529i) q^{31} +(3.89508 + 4.10224i) q^{32} +0.967447i q^{33} +(-5.33602 + 2.35094i) q^{34} -2.86978i q^{35} +(4.99527 - 2.99038i) q^{36} +(0.823143 - 4.13822i) q^{37} +(0.656620 + 0.566715i) q^{38} +(-3.88037 - 5.80738i) q^{39} +(1.14409 + 2.58671i) q^{40} +(8.64415 + 5.77584i) q^{41} +(-3.09769 + 9.36834i) q^{42} +(-4.19826 + 10.1355i) q^{43} +(0.719249 + 0.340656i) q^{44} +(2.85504 - 0.567903i) q^{45} +(8.79339 - 6.85860i) q^{46} +(-2.76514 + 2.76514i) q^{47} +(-0.942732 - 9.67920i) q^{48} +(-0.472861 - 1.14159i) q^{49} +(0.103654 + 1.41041i) q^{50} +(9.71904 + 2.45496i) q^{51} +(-5.68385 + 0.839974i) q^{52} +(-0.366848 + 0.151953i) q^{53} +(0.303780 + 0.0375513i) q^{54} +(0.281373 + 0.281373i) q^{55} +(5.87414 + 5.60175i) q^{56} +(-0.290904 - 1.46248i) q^{57} +(-4.98169 - 2.82388i) q^{58} +(7.38403 + 3.05856i) q^{59} +(1.18405 - 4.71614i) q^{60} +(3.47241 - 5.19683i) q^{61} +(-2.16125 - 7.81798i) q^{62} +(6.94598 - 4.64116i) q^{63} +(-7.52796 - 2.70735i) q^{64} +(-2.81759 - 0.560454i) q^{65} +(-0.614817 - 1.22225i) q^{66} -1.53503 q^{67} +(5.24740 - 6.36120i) q^{68} -19.1718 q^{69} +(1.82376 + 3.62563i) q^{70} +(-6.40427 - 1.27389i) q^{71} +(-4.41054 + 6.95251i) q^{72} +(-11.0841 + 7.40617i) q^{73} +(1.58991 + 5.75126i) q^{74} +(1.35073 - 2.02151i) q^{75} +(-1.18971 - 0.298692i) q^{76} +(1.05502 + 0.437005i) q^{77} +(8.59300 + 4.87095i) q^{78} +(-2.74701 - 13.8102i) q^{79} +(-3.08929 - 2.54092i) q^{80} +(-6.54721 - 6.54721i) q^{81} +(-14.5914 - 1.80369i) q^{82} +(-6.79673 + 2.81530i) q^{83} +(-2.04005 - 13.8044i) q^{84} +(3.54070 - 2.11269i) q^{85} +(-1.13715 - 15.4730i) q^{86} +(3.76734 + 9.09517i) q^{87} +(-1.12518 + 0.0267068i) q^{88} +(-6.96390 + 6.96390i) q^{89} +(-3.24610 + 2.53187i) q^{90} +(-8.08588 + 1.60838i) q^{91} +(-6.75074 + 14.2533i) q^{92} +(-5.33630 + 12.8830i) q^{93} +(1.73617 - 5.25070i) q^{94} +(-0.509954 - 0.340741i) q^{95} +(7.34220 + 11.6294i) q^{96} +(-4.35589 - 6.51904i) q^{97} +(1.32289 + 1.14176i) q^{98} +(-0.225981 + 1.13608i) 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 \(\left(\mathbb{Z}/340\mathbb{Z}\right)^\times\).

\(n\) \(137\) \(171\) \(241\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{9}{16}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.26338 + 0.635504i −0.893346 + 0.449369i
\(3\) 2.38453 + 0.474313i 1.37671 + 0.273845i 0.827325 − 0.561724i \(-0.189862\pi\)
0.549386 + 0.835568i \(0.314862\pi\)
\(4\) 1.19227 − 1.60577i 0.596134 − 0.802885i
\(5\) 0.831470 − 0.555570i 0.371845 − 0.248459i
\(6\) −3.31401 + 0.916143i −1.35294 + 0.374014i
\(7\) 1.59437 − 2.38614i 0.602613 − 0.901875i −0.397261 − 0.917706i \(-0.630039\pi\)
0.999875 + 0.0158308i \(0.00503932\pi\)
\(8\) −0.485817 + 2.78639i −0.171762 + 0.985138i
\(9\) 2.68939 + 1.11398i 0.896464 + 0.371327i
\(10\) −0.697397 + 1.23030i −0.220536 + 0.389055i
\(11\) 0.0776306 + 0.390276i 0.0234065 + 0.117673i 0.990722 − 0.135903i \(-0.0433935\pi\)
−0.967316 + 0.253575i \(0.918393\pi\)
\(12\) 3.60464 − 3.26350i 1.04057 − 0.942092i
\(13\) −2.03137 − 2.03137i −0.563401 − 0.563401i 0.366871 − 0.930272i \(-0.380429\pi\)
−0.930272 + 0.366871i \(0.880429\pi\)
\(14\) −0.497893 + 4.02783i −0.133067 + 1.07648i
\(15\) 2.24618 − 0.930399i 0.579962 − 0.240228i
\(16\) −1.15699 − 3.82902i −0.289248 − 0.957254i
\(17\) 4.11773 + 0.210465i 0.998696 + 0.0510453i
\(18\) −4.10567 + 0.301734i −0.967715 + 0.0711195i
\(19\) −0.234706 − 0.566631i −0.0538453 − 0.129994i 0.894668 − 0.446732i \(-0.147412\pi\)
−0.948513 + 0.316738i \(0.897412\pi\)
\(20\) 0.0992173 − 1.99754i 0.0221857 − 0.446663i
\(21\) 4.93359 − 4.93359i 1.07660 − 1.07660i
\(22\) −0.346099 − 0.443733i −0.0737886 − 0.0946041i
\(23\) −7.73404 + 1.53840i −1.61266 + 0.320778i −0.917395 − 0.397977i \(-0.869712\pi\)
−0.695263 + 0.718755i \(0.744712\pi\)
\(24\) −2.48007 + 6.41382i −0.506242 + 1.30922i
\(25\) 0.382683 − 0.923880i 0.0765367 − 0.184776i
\(26\) 3.85734 + 1.27545i 0.756487 + 0.250137i
\(27\) −0.179963 − 0.120248i −0.0346340 − 0.0231417i
\(28\) −1.93067 − 5.40510i −0.364863 − 1.02147i
\(29\) 2.24960 + 3.36676i 0.417740 + 0.625192i 0.979341 − 0.202216i \(-0.0648145\pi\)
−0.561601 + 0.827408i \(0.689814\pi\)
\(30\) −2.24651 + 2.60291i −0.410155 + 0.475224i
\(31\) −1.11894 + 5.62529i −0.200967 + 1.01033i 0.740199 + 0.672388i \(0.234731\pi\)
−0.941166 + 0.337944i \(0.890269\pi\)
\(32\) 3.89508 + 4.10224i 0.688559 + 0.725180i
\(33\) 0.967447i 0.168411i
\(34\) −5.33602 + 2.35094i −0.915120 + 0.403182i
\(35\) − 2.86978i − 0.485082i
\(36\) 4.99527 − 2.99038i 0.832546 − 0.498396i
\(37\) 0.823143 − 4.13822i 0.135324 − 0.680319i −0.852246 − 0.523141i \(-0.824760\pi\)
0.987570 − 0.157178i \(-0.0502397\pi\)
\(38\) 0.656620 + 0.566715i 0.106518 + 0.0919332i
\(39\) −3.88037 − 5.80738i −0.621356 − 0.929925i
\(40\) 1.14409 + 2.58671i 0.180897 + 0.408994i
\(41\) 8.64415 + 5.77584i 1.34999 + 0.902034i 0.999405 − 0.0344967i \(-0.0109828\pi\)
0.350585 + 0.936531i \(0.385983\pi\)
\(42\) −3.09769 + 9.36834i −0.477985 + 1.44557i
\(43\) −4.19826 + 10.1355i −0.640229 + 1.54565i 0.186141 + 0.982523i \(0.440402\pi\)
−0.826370 + 0.563127i \(0.809598\pi\)
\(44\) 0.719249 + 0.340656i 0.108431 + 0.0513559i
\(45\) 2.85504 − 0.567903i 0.425605 − 0.0846580i
\(46\) 8.79339 − 6.85860i 1.29651 − 1.01124i
\(47\) −2.76514 + 2.76514i −0.403338 + 0.403338i −0.879408 − 0.476070i \(-0.842061\pi\)
0.476070 + 0.879408i \(0.342061\pi\)
\(48\) −0.942732 − 9.67920i −0.136072 − 1.39707i
\(49\) −0.472861 − 1.14159i −0.0675516 − 0.163084i
\(50\) 0.103654 + 1.41041i 0.0146589 + 0.199462i
\(51\) 9.71904 + 2.45496i 1.36094 + 0.343763i
\(52\) −5.68385 + 0.839974i −0.788209 + 0.116483i
\(53\) −0.366848 + 0.151953i −0.0503905 + 0.0208724i −0.407736 − 0.913100i \(-0.633682\pi\)
0.357346 + 0.933972i \(0.383682\pi\)
\(54\) 0.303780 + 0.0375513i 0.0413393 + 0.00511008i
\(55\) 0.281373 + 0.281373i 0.0379403 + 0.0379403i
\(56\) 5.87414 + 5.60175i 0.784965 + 0.748566i
\(57\) −0.290904 − 1.46248i −0.0385312 − 0.193710i
\(58\) −4.98169 − 2.82388i −0.654128 − 0.370793i
\(59\) 7.38403 + 3.05856i 0.961319 + 0.398191i 0.807474 − 0.589904i \(-0.200834\pi\)
0.153845 + 0.988095i \(0.450834\pi\)
\(60\) 1.18405 − 4.71614i 0.152860 − 0.608851i
\(61\) 3.47241 − 5.19683i 0.444597 − 0.665386i −0.539710 − 0.841851i \(-0.681466\pi\)
0.984307 + 0.176465i \(0.0564662\pi\)
\(62\) −2.16125 − 7.81798i −0.274479 − 0.992884i
\(63\) 6.94598 − 4.64116i 0.875112 − 0.584731i
\(64\) −7.52796 − 2.70735i −0.940995 − 0.338419i
\(65\) −2.81759 − 0.560454i −0.349479 − 0.0695158i
\(66\) −0.614817 − 1.22225i −0.0756787 − 0.150449i
\(67\) −1.53503 −0.187533 −0.0937666 − 0.995594i \(-0.529891\pi\)
−0.0937666 + 0.995594i \(0.529891\pi\)
\(68\) 5.24740 − 6.36120i 0.636341 − 0.771408i
\(69\) −19.1718 −2.30801
\(70\) 1.82376 + 3.62563i 0.217981 + 0.433346i
\(71\) −6.40427 − 1.27389i −0.760047 − 0.151183i −0.200173 − 0.979761i \(-0.564150\pi\)
−0.559874 + 0.828578i \(0.689150\pi\)
\(72\) −4.41054 + 6.95251i −0.519788 + 0.819361i
\(73\) −11.0841 + 7.40617i −1.29730 + 0.866826i −0.996228 − 0.0867758i \(-0.972344\pi\)
−0.301070 + 0.953602i \(0.597344\pi\)
\(74\) 1.58991 + 5.75126i 0.184823 + 0.668570i
\(75\) 1.35073 − 2.02151i 0.155969 − 0.233424i
\(76\) −1.18971 − 0.298692i −0.136469 − 0.0342623i
\(77\) 1.05502 + 0.437005i 0.120231 + 0.0498013i
\(78\) 8.59300 + 4.87095i 0.972966 + 0.551526i
\(79\) −2.74701 − 13.8102i −0.309063 − 1.55376i −0.753192 − 0.657801i \(-0.771487\pi\)
0.444129 − 0.895963i \(-0.353513\pi\)
\(80\) −3.08929 − 2.54092i −0.345393 − 0.284084i
\(81\) −6.54721 − 6.54721i −0.727467 − 0.727467i
\(82\) −14.5914 − 1.80369i −1.61135 − 0.199185i
\(83\) −6.79673 + 2.81530i −0.746038 + 0.309019i −0.723124 − 0.690718i \(-0.757294\pi\)
−0.0229142 + 0.999737i \(0.507294\pi\)
\(84\) −2.04005 − 13.8044i −0.222587 − 1.50618i
\(85\) 3.54070 − 2.11269i 0.384042 − 0.229154i
\(86\) −1.13715 − 15.4730i −0.122622 − 1.66850i
\(87\) 3.76734 + 9.09517i 0.403901 + 0.975104i
\(88\) −1.12518 + 0.0267068i −0.119944 + 0.00284696i
\(89\) −6.96390 + 6.96390i −0.738172 + 0.738172i −0.972224 − 0.234052i \(-0.924801\pi\)
0.234052 + 0.972224i \(0.424801\pi\)
\(90\) −3.24610 + 2.53187i −0.342169 + 0.266883i
\(91\) −8.08588 + 1.60838i −0.847630 + 0.168604i
\(92\) −6.75074 + 14.2533i −0.703813 + 1.48601i
\(93\) −5.33630 + 12.8830i −0.553348 + 1.33590i
\(94\) 1.73617 − 5.25070i 0.179073 − 0.541568i
\(95\) −0.509954 − 0.340741i −0.0523202 − 0.0349593i
\(96\) 7.34220 + 11.6294i 0.749360 + 1.18692i
\(97\) −4.35589 − 6.51904i −0.442273 − 0.661909i 0.541629 − 0.840618i \(-0.317808\pi\)
−0.983902 + 0.178709i \(0.942808\pi\)
\(98\) 1.32289 + 1.14176i 0.133632 + 0.115335i
\(99\) −0.225981 + 1.13608i −0.0227119 + 0.114181i
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.31.6 yes 288
4.3 odd 2 inner 340.2.bf.a.31.3 yes 288
17.11 odd 16 inner 340.2.bf.a.11.3 ✓ 288
68.11 even 16 inner 340.2.bf.a.11.6 yes 288
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
340.2.bf.a.11.3 ✓ 288 17.11 odd 16 inner
340.2.bf.a.11.6 yes 288 68.11 even 16 inner
340.2.bf.a.31.3 yes 288 4.3 odd 2 inner
340.2.bf.a.31.6 yes 288 1.1 even 1 trivial