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Properties

Label 340.2.bd.a.57.5
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 57.5
Character \(\chi\) \(=\) 340.57
Dual form 340.2.bd.a.173.5

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.286067 - 0.428130i) q^{3} +(1.53584 + 1.62517i) q^{5} +(0.632123 - 3.17790i) q^{7} +(1.04659 + 2.52669i) q^{9} +(0.276273 - 1.38892i) q^{11} +1.82914i q^{13} +(1.13514 - 0.192631i) q^{15} +(3.00518 - 2.82292i) q^{17} +(-0.686762 + 1.65799i) q^{19} +(-1.17972 - 1.17972i) q^{21} +(2.79432 - 1.86711i) q^{23} +(-0.282382 + 4.99202i) q^{25} +(2.89619 + 0.576088i) q^{27} +(0.0112673 - 0.0168627i) q^{29} +(0.194584 + 0.978240i) q^{31} +(-0.515605 - 0.515605i) q^{33} +(6.13548 - 3.85344i) q^{35} +(-4.40607 - 2.94404i) q^{37} +(0.783111 + 0.523258i) q^{39} +(-4.69306 - 7.02366i) q^{41} +(-0.715509 + 1.72739i) q^{43} +(-2.49892 + 5.58149i) q^{45} +0.949860 q^{47} +(-3.23229 - 1.33886i) q^{49} +(-0.348891 - 2.09416i) q^{51} +(-2.97262 + 1.23130i) q^{53} +(2.68154 - 1.68416i) q^{55} +(0.513375 + 0.768320i) q^{57} +(-13.1963 + 5.46609i) q^{59} +(-3.52034 + 2.35222i) q^{61} +(8.69114 - 1.72877i) q^{63} +(-2.97267 + 2.80927i) q^{65} +(-2.65035 + 2.65035i) q^{67} -1.73045i q^{69} +(-4.45205 + 0.885569i) q^{71} +(2.66807 + 13.4133i) q^{73} +(2.05645 + 1.54895i) q^{75} +(-4.23920 - 1.75593i) q^{77} +(-13.2756 - 2.64068i) q^{79} +(-4.72639 + 4.72639i) q^{81} +(-5.03346 - 12.1518i) q^{83} +(9.20322 + 0.548395i) q^{85} +(-0.00399622 - 0.00964772i) q^{87} +(-8.46657 + 8.46657i) q^{89} +(5.81283 + 1.15624i) q^{91} +(0.474478 + 0.196535i) q^{93} +(-3.74928 + 1.43030i) q^{95} +(-1.78167 - 8.95708i) q^{97} +(3.79851 - 0.755570i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q + 24 q^{15} + 8 q^{25} - 48 q^{27} - 32 q^{31} + 16 q^{33} + 32 q^{37} - 32 q^{39} - 40 q^{41} + 80 q^{47} - 40 q^{53} + 16 q^{55} + 8 q^{57} + 112 q^{59} - 48 q^{63} - 32 q^{67} - 16 q^{71} + 8 q^{73}+ \cdots + 96 q^{99}+O(q^{100}) \) 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
\(3\) 0.286067 − 0.428130i 0.165161 − 0.247181i −0.739652 − 0.672989i \(-0.765010\pi\)
0.904814 + 0.425808i \(0.140010\pi\)
\(4\) 0 0
\(5\) 1.53584 + 1.62517i 0.686849 + 0.726800i
\(6\) 0 0
\(7\) 0.632123 − 3.17790i 0.238920 − 1.20113i −0.655950 − 0.754805i \(-0.727732\pi\)
0.894870 − 0.446328i \(-0.147268\pi\)
\(8\) 0 0
\(9\) 1.04659 + 2.52669i 0.348863 + 0.842230i
\(10\) 0 0
\(11\) 0.276273 − 1.38892i 0.0832994 − 0.418774i −0.916524 − 0.399980i \(-0.869017\pi\)
0.999823 − 0.0187945i \(-0.00598282\pi\)
\(12\) 0 0
\(13\) 1.82914i 0.507313i 0.967294 + 0.253656i \(0.0816332\pi\)
−0.967294 + 0.253656i \(0.918367\pi\)
\(14\) 0 0
\(15\) 1.13514 − 0.192631i 0.293092 − 0.0497370i
\(16\) 0 0
\(17\) 3.00518 − 2.82292i 0.728864 − 0.684658i
\(18\) 0 0
\(19\) −0.686762 + 1.65799i −0.157554 + 0.380369i −0.982869 − 0.184303i \(-0.940997\pi\)
0.825316 + 0.564672i \(0.190997\pi\)
\(20\) 0 0
\(21\) −1.17972 − 1.17972i −0.257437 − 0.257437i
\(22\) 0 0
\(23\) 2.79432 − 1.86711i 0.582657 − 0.389319i −0.229023 − 0.973421i \(-0.573553\pi\)
0.811680 + 0.584102i \(0.198553\pi\)
\(24\) 0 0
\(25\) −0.282382 + 4.99202i −0.0564764 + 0.998404i
\(26\) 0 0
\(27\) 2.89619 + 0.576088i 0.557372 + 0.110868i
\(28\) 0 0
\(29\) 0.0112673 − 0.0168627i 0.00209228 − 0.00313132i −0.830422 − 0.557135i \(-0.811901\pi\)
0.832514 + 0.554004i \(0.186901\pi\)
\(30\) 0 0
\(31\) 0.194584 + 0.978240i 0.0349483 + 0.175697i 0.994317 − 0.106458i \(-0.0339508\pi\)
−0.959369 + 0.282155i \(0.908951\pi\)
\(32\) 0 0
\(33\) −0.515605 − 0.515605i −0.0897552 − 0.0897552i
\(34\) 0 0
\(35\) 6.13548 − 3.85344i 1.03708 − 0.651350i
\(36\) 0 0
\(37\) −4.40607 − 2.94404i −0.724354 − 0.483998i 0.137918 − 0.990444i \(-0.455959\pi\)
−0.862272 + 0.506446i \(0.830959\pi\)
\(38\) 0 0
\(39\) 0.783111 + 0.523258i 0.125398 + 0.0837883i
\(40\) 0 0
\(41\) −4.69306 − 7.02366i −0.732933 − 1.09691i −0.991397 − 0.130891i \(-0.958216\pi\)
0.258464 − 0.966021i \(-0.416784\pi\)
\(42\) 0 0
\(43\) −0.715509 + 1.72739i −0.109114 + 0.263425i −0.969000 − 0.247062i \(-0.920535\pi\)
0.859886 + 0.510487i \(0.170535\pi\)
\(44\) 0 0
\(45\) −2.49892 + 5.58149i −0.372516 + 0.832039i
\(46\) 0 0
\(47\) 0.949860 0.138551 0.0692757 − 0.997598i \(-0.477931\pi\)
0.0692757 + 0.997598i \(0.477931\pi\)
\(48\) 0 0
\(49\) −3.23229 − 1.33886i −0.461756 − 0.191266i
\(50\) 0 0
\(51\) −0.348891 − 2.09416i −0.0488545 − 0.293240i
\(52\) 0 0
\(53\) −2.97262 + 1.23130i −0.408321 + 0.169132i −0.577383 − 0.816473i \(-0.695926\pi\)
0.169062 + 0.985605i \(0.445926\pi\)
\(54\) 0 0
\(55\) 2.68154 − 1.68416i 0.361579 − 0.227093i
\(56\) 0 0
\(57\) 0.513375 + 0.768320i 0.0679982 + 0.101767i
\(58\) 0 0
\(59\) −13.1963 + 5.46609i −1.71801 + 0.711624i −0.718136 + 0.695903i \(0.755004\pi\)
−0.999876 + 0.0157212i \(0.994996\pi\)
\(60\) 0 0
\(61\) −3.52034 + 2.35222i −0.450733 + 0.301170i −0.760140 − 0.649759i \(-0.774870\pi\)
0.309407 + 0.950930i \(0.399870\pi\)
\(62\) 0 0
\(63\) 8.69114 − 1.72877i 1.09498 − 0.217805i
\(64\) 0 0
\(65\) −2.97267 + 2.80927i −0.368715 + 0.348447i
\(66\) 0 0
\(67\) −2.65035 + 2.65035i −0.323792 + 0.323792i −0.850220 − 0.526428i \(-0.823531\pi\)
0.526428 + 0.850220i \(0.323531\pi\)
\(68\) 0 0
\(69\) − 1.73045i − 0.208322i
\(70\) 0 0
\(71\) −4.45205 + 0.885569i −0.528362 + 0.105098i −0.452061 − 0.891987i \(-0.649311\pi\)
−0.0763009 + 0.997085i \(0.524311\pi\)
\(72\) 0 0
\(73\) 2.66807 + 13.4133i 0.312274 + 1.56991i 0.744174 + 0.667986i \(0.232843\pi\)
−0.431900 + 0.901921i \(0.642157\pi\)
\(74\) 0 0
\(75\) 2.05645 + 1.54895i 0.237459 + 0.178857i
\(76\) 0 0
\(77\) −4.23920 − 1.75593i −0.483101 − 0.200107i
\(78\) 0 0
\(79\) −13.2756 − 2.64068i −1.49362 − 0.297099i −0.620346 − 0.784329i \(-0.713008\pi\)
−0.873274 + 0.487229i \(0.838008\pi\)
\(80\) 0 0
\(81\) −4.72639 + 4.72639i −0.525154 + 0.525154i
\(82\) 0 0
\(83\) −5.03346 − 12.1518i −0.552494 − 1.33384i −0.915600 − 0.402090i \(-0.868284\pi\)
0.363106 − 0.931748i \(-0.381716\pi\)
\(84\) 0 0
\(85\) 9.20322 + 0.548395i 0.998229 + 0.0594818i
\(86\) 0 0
\(87\) −0.00399622 − 0.00964772i −0.000428439 − 0.00103434i
\(88\) 0 0
\(89\) −8.46657 + 8.46657i −0.897455 + 0.897455i −0.995210 − 0.0977556i \(-0.968834\pi\)
0.0977556 + 0.995210i \(0.468834\pi\)
\(90\) 0 0
\(91\) 5.81283 + 1.15624i 0.609350 + 0.121207i
\(92\) 0 0
\(93\) 0.474478 + 0.196535i 0.0492011 + 0.0203798i
\(94\) 0 0
\(95\) −3.74928 + 1.43030i −0.384668 + 0.146746i
\(96\) 0 0
\(97\) −1.78167 − 8.95708i −0.180902 − 0.909454i −0.959452 − 0.281872i \(-0.909045\pi\)
0.778550 − 0.627582i \(-0.215955\pi\)
\(98\) 0 0
\(99\) 3.79851 − 0.755570i 0.381764 − 0.0759376i
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.57.5 ✓ 72
5.3 odd 4 340.2.bi.a.193.5 yes 72
17.3 odd 16 340.2.bi.a.37.5 yes 72
85.3 even 16 inner 340.2.bd.a.173.5 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
340.2.bd.a.57.5 ✓ 72 1.1 even 1 trivial
340.2.bd.a.173.5 yes 72 85.3 even 16 inner
340.2.bi.a.37.5 yes 72 17.3 odd 16
340.2.bi.a.193.5 yes 72 5.3 odd 4