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Properties

Label 340.2.bd.a.73.3
Level $340$
Weight $2$
Character 340.73
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 73.3
Character \(\chi\) \(=\) 340.73
Dual form 340.2.bd.a.177.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.402101 + 2.02150i) q^{3} +(-0.456223 - 2.18903i) q^{5} +(2.17863 + 1.45572i) q^{7} +(-1.15314 - 0.477647i) q^{9} +(2.27311 + 1.51885i) q^{11} +5.00946i q^{13} +(4.60858 - 0.0420413i) q^{15} +(-3.98920 + 1.04223i) q^{17} +(3.61357 - 1.49679i) q^{19} +(-3.81876 + 3.81876i) q^{21} +(0.0106960 - 0.00212757i) q^{23} +(-4.58372 + 1.99737i) q^{25} +(-2.00602 + 3.00222i) q^{27} +(-0.0121577 + 0.0611211i) q^{29} +(-2.24474 + 1.49989i) q^{31} +(-3.98437 + 3.98437i) q^{33} +(2.19267 - 5.43323i) q^{35} +(10.8019 + 2.14862i) q^{37} +(-10.1266 - 2.01431i) q^{39} +(-1.29826 - 6.52677i) q^{41} +(5.29935 - 2.19506i) q^{43} +(-0.519495 + 2.74218i) q^{45} -0.602666 q^{47} +(-0.0514506 - 0.124213i) q^{49} +(-0.502810 - 8.48326i) q^{51} +(1.95007 - 4.70789i) q^{53} +(2.28776 - 5.66885i) q^{55} +(1.57274 + 7.90671i) q^{57} +(4.81845 - 11.6328i) q^{59} +(-8.19505 + 1.63010i) q^{61} +(-1.81695 - 2.71926i) q^{63} +(10.9659 - 2.28543i) q^{65} +(-2.74057 - 2.74057i) q^{67} +0.0224775i q^{69} +(6.15730 + 9.21505i) q^{71} +(9.47705 - 6.33236i) q^{73} +(-2.19457 - 10.0691i) q^{75} +(2.74127 + 6.61801i) q^{77} +(1.34961 - 2.01984i) q^{79} +(-7.91010 - 7.91010i) q^{81} +(3.49917 + 1.44940i) q^{83} +(4.10145 + 8.25701i) q^{85} +(-0.118668 - 0.0491538i) q^{87} +(-11.1768 - 11.1768i) q^{89} +(-7.29235 + 10.9138i) q^{91} +(-2.12941 - 5.14086i) q^{93} +(-4.92512 - 7.22736i) q^{95} +(-15.0966 + 10.0872i) q^{97} +(-1.89575 - 2.83719i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q + 24 q^{15} + 8 q^{25} - 48 q^{27} - 32 q^{31} + 16 q^{33} + 32 q^{37} - 32 q^{39} - 40 q^{41} + 80 q^{47} - 40 q^{53} + 16 q^{55} + 8 q^{57} + 112 q^{59} - 48 q^{63} - 32 q^{67} - 16 q^{71} + 8 q^{73}+ \cdots + 96 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(137\) \(171\) \(241\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\) \(e\left(\frac{5}{16}\right)\)

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
\(3\) −0.402101 + 2.02150i −0.232153 + 1.16711i 0.672213 + 0.740358i \(0.265344\pi\)
−0.904367 + 0.426756i \(0.859656\pi\)
\(4\) 0 0
\(5\) −0.456223 − 2.18903i −0.204029 − 0.978965i
\(6\) 0 0
\(7\) 2.17863 + 1.45572i 0.823446 + 0.550209i 0.894399 − 0.447271i \(-0.147604\pi\)
−0.0709527 + 0.997480i \(0.522604\pi\)
\(8\) 0 0
\(9\) −1.15314 − 0.477647i −0.384380 − 0.159216i
\(10\) 0 0
\(11\) 2.27311 + 1.51885i 0.685369 + 0.457949i 0.848875 − 0.528593i \(-0.177280\pi\)
−0.163506 + 0.986542i \(0.552280\pi\)
\(12\) 0 0
\(13\) 5.00946i 1.38937i 0.719313 + 0.694687i \(0.244457\pi\)
−0.719313 + 0.694687i \(0.755543\pi\)
\(14\) 0 0
\(15\) 4.60858 − 0.0420413i 1.18993 − 0.0108550i
\(16\) 0 0
\(17\) −3.98920 + 1.04223i −0.967524 + 0.252779i
\(18\) 0 0
\(19\) 3.61357 − 1.49679i 0.829011 − 0.343388i 0.0724998 − 0.997368i \(-0.476902\pi\)
0.756511 + 0.653981i \(0.226902\pi\)
\(20\) 0 0
\(21\) −3.81876 + 3.81876i −0.833322 + 0.833322i
\(22\) 0 0
\(23\) 0.0106960 − 0.00212757i 0.00223028 − 0.000443630i −0.193975 − 0.981006i \(-0.562138\pi\)
0.196205 + 0.980563i \(0.437138\pi\)
\(24\) 0 0
\(25\) −4.58372 + 1.99737i −0.916744 + 0.399474i
\(26\) 0 0
\(27\) −2.00602 + 3.00222i −0.386059 + 0.577778i
\(28\) 0 0
\(29\) −0.0121577 + 0.0611211i −0.00225764 + 0.0113499i −0.981898 − 0.189412i \(-0.939342\pi\)
0.979640 + 0.200762i \(0.0643418\pi\)
\(30\) 0 0
\(31\) −2.24474 + 1.49989i −0.403168 + 0.269388i −0.740579 − 0.671969i \(-0.765449\pi\)
0.337411 + 0.941357i \(0.390449\pi\)
\(32\) 0 0
\(33\) −3.98437 + 3.98437i −0.693590 + 0.693590i
\(34\) 0 0
\(35\) 2.19267 − 5.43323i 0.370628 − 0.918383i
\(36\) 0 0
\(37\) 10.8019 + 2.14862i 1.77581 + 0.353232i 0.970775 − 0.239990i \(-0.0771441\pi\)
0.805039 + 0.593221i \(0.202144\pi\)
\(38\) 0 0
\(39\) −10.1266 − 2.01431i −1.62156 − 0.322548i
\(40\) 0 0
\(41\) −1.29826 − 6.52677i −0.202753 − 1.01931i −0.939345 − 0.342973i \(-0.888566\pi\)
0.736592 − 0.676337i \(-0.236434\pi\)
\(42\) 0 0
\(43\) 5.29935 − 2.19506i 0.808143 − 0.334744i 0.0599299 − 0.998203i \(-0.480912\pi\)
0.748213 + 0.663459i \(0.230912\pi\)
\(44\) 0 0
\(45\) −0.519495 + 2.74218i −0.0774417 + 0.408779i
\(46\) 0 0
\(47\) −0.602666 −0.0879079 −0.0439540 − 0.999034i \(-0.513995\pi\)
−0.0439540 + 0.999034i \(0.513995\pi\)
\(48\) 0 0
\(49\) −0.0514506 − 0.124213i −0.00735009 − 0.0177447i
\(50\) 0 0
\(51\) −0.502810 − 8.48326i −0.0704075 − 1.18789i
\(52\) 0 0
\(53\) 1.95007 − 4.70789i 0.267863 − 0.646678i −0.731519 − 0.681821i \(-0.761189\pi\)
0.999382 + 0.0351425i \(0.0111885\pi\)
\(54\) 0 0
\(55\) 2.28776 − 5.66885i 0.308481 − 0.764387i
\(56\) 0 0
\(57\) 1.57274 + 7.90671i 0.208315 + 1.04727i
\(58\) 0 0
\(59\) 4.81845 − 11.6328i 0.627309 − 1.51446i −0.215644 − 0.976472i \(-0.569185\pi\)
0.842953 − 0.537987i \(-0.180815\pi\)
\(60\) 0 0
\(61\) −8.19505 + 1.63010i −1.04927 + 0.208712i −0.689467 − 0.724317i \(-0.742155\pi\)
−0.359801 + 0.933029i \(0.617155\pi\)
\(62\) 0 0
\(63\) −1.81695 − 2.71926i −0.228915 − 0.342595i
\(64\) 0 0
\(65\) 10.9659 − 2.28543i 1.36015 − 0.283472i
\(66\) 0 0
\(67\) −2.74057 − 2.74057i −0.334814 − 0.334814i 0.519597 − 0.854411i \(-0.326082\pi\)
−0.854411 + 0.519597i \(0.826082\pi\)
\(68\) 0 0
\(69\) 0.0224775i 0.00270598i
\(70\) 0 0
\(71\) 6.15730 + 9.21505i 0.730737 + 1.09363i 0.991737 + 0.128287i \(0.0409477\pi\)
−0.261000 + 0.965339i \(0.584052\pi\)
\(72\) 0 0
\(73\) 9.47705 − 6.33236i 1.10920 − 0.741147i 0.140678 − 0.990055i \(-0.455072\pi\)
0.968527 + 0.248908i \(0.0800718\pi\)
\(74\) 0 0
\(75\) −2.19457 − 10.0691i −0.253407 − 1.16268i
\(76\) 0 0
\(77\) 2.74127 + 6.61801i 0.312397 + 0.754193i
\(78\) 0 0
\(79\) 1.34961 − 2.01984i 0.151843 − 0.227250i −0.747748 − 0.663983i \(-0.768865\pi\)
0.899591 + 0.436733i \(0.143865\pi\)
\(80\) 0 0
\(81\) −7.91010 − 7.91010i −0.878900 − 0.878900i
\(82\) 0 0
\(83\) 3.49917 + 1.44940i 0.384084 + 0.159093i 0.566367 − 0.824153i \(-0.308349\pi\)
−0.182283 + 0.983246i \(0.558349\pi\)
\(84\) 0 0
\(85\) 4.10145 + 8.25701i 0.444864 + 0.895598i
\(86\) 0 0
\(87\) −0.118668 − 0.0491538i −0.0127225 − 0.00526984i
\(88\) 0 0
\(89\) −11.1768 − 11.1768i −1.18474 − 1.18474i −0.978502 − 0.206237i \(-0.933878\pi\)
−0.206237 − 0.978502i \(-0.566122\pi\)
\(90\) 0 0
\(91\) −7.29235 + 10.9138i −0.764446 + 1.14407i
\(92\) 0 0
\(93\) −2.12941 − 5.14086i −0.220810 − 0.533083i
\(94\) 0 0
\(95\) −4.92512 − 7.22736i −0.505307 − 0.741512i
\(96\) 0 0
\(97\) −15.0966 + 10.0872i −1.53282 + 1.02420i −0.550979 + 0.834519i \(0.685746\pi\)
−0.981846 + 0.189682i \(0.939254\pi\)
\(98\) 0 0
\(99\) −1.89575 − 2.83719i −0.190530 − 0.285148i
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.73.3 ✓ 72
5.2 odd 4 340.2.bi.a.277.7 yes 72
17.7 odd 16 340.2.bi.a.313.7 yes 72
85.7 even 16 inner 340.2.bd.a.177.3 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
340.2.bd.a.73.3 ✓ 72 1.1 even 1 trivial
340.2.bd.a.177.3 yes 72 85.7 even 16 inner
340.2.bi.a.277.7 yes 72 5.2 odd 4
340.2.bi.a.313.7 yes 72 17.7 odd 16