↑

Properties

Label 340.2.bi.a.37.6
Level $340$
Weight $2$
Character 340.37
Analytic conductor $2.715$
Analytic rank $0$
Dimension $72$
Inner twists $2$

Related objects

Downloads

Learn more

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(37,340)] 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("340.37"); 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, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 340 = 2^{2} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 340.bi (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 37.6
Character \(\chi\) \(=\) 340.37
Dual form 340.2.bi.a.193.6

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.522949 - 0.349423i) q^{3} +(1.38657 + 1.75426i) q^{5} +(-0.175059 + 0.0348214i) q^{7} +(-0.996671 + 2.40618i) q^{9} +(0.707055 + 3.55461i) q^{11} +2.05870 q^{13} +(1.33809 + 0.432886i) q^{15} +(3.77636 - 1.65503i) q^{17} +(-2.61374 - 6.31013i) q^{19} +(-0.0793796 + 0.0793796i) q^{21} +(-0.386195 + 0.577982i) q^{23} +(-1.15484 + 4.86481i) q^{25} +(0.687670 + 3.45715i) q^{27} +(0.304090 + 0.455102i) q^{29} +(0.666895 - 3.35271i) q^{31} +(1.61182 + 1.61182i) q^{33} +(-0.303818 - 0.258816i) q^{35} +(2.77178 + 4.14826i) q^{37} +(1.07660 - 0.719358i) q^{39} +(4.71458 - 7.05587i) q^{41} +(2.22176 - 0.920283i) q^{43} +(-5.60301 + 1.58792i) q^{45} -10.9567i q^{47} +(-6.43772 + 2.66659i) q^{49} +(1.39654 - 2.18505i) q^{51} +(-2.80886 + 6.78119i) q^{53} +(-5.25531 + 6.16908i) q^{55} +(-3.57176 - 2.38658i) q^{57} +(-3.07891 - 1.27533i) q^{59} +(-8.11858 - 5.42466i) q^{61} +(0.0906898 - 0.455929i) q^{63} +(2.85454 + 3.61149i) q^{65} +(7.46930 - 7.46930i) q^{67} +0.437201i q^{69} +(-4.57239 - 0.909505i) q^{71} +(-9.19094 - 1.82819i) q^{73} +(1.09596 + 2.94757i) q^{75} +(-0.247553 - 0.597645i) q^{77} +(-12.8682 + 2.55964i) q^{79} +(-3.95720 - 3.95720i) q^{81} +(-0.0183656 - 0.00760730i) q^{83} +(8.13954 + 4.32988i) q^{85} +(0.318047 + 0.131739i) q^{87} +(-2.21165 - 2.21165i) q^{89} +(-0.360394 + 0.0716868i) q^{91} +(-0.822763 - 1.98633i) q^{93} +(7.44545 - 13.3346i) q^{95} +(11.5782 + 2.30306i) q^{97} +(-9.25772 - 1.84147i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q + 24 q^{15} + 8 q^{25} + 48 q^{27} - 32 q^{31} - 16 q^{33} - 32 q^{37} + 32 q^{39} - 40 q^{41} + 32 q^{53} - 16 q^{55} + 72 q^{57} - 112 q^{59} - 48 q^{63} + 32 q^{67} - 16 q^{71} - 96 q^{73} + 24 q^{75}+ \cdots - 96 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) 0.522949 − 0.349423i 0.301925 − 0.201740i −0.395374 − 0.918520i \(-0.629385\pi\)
0.697299 + 0.716780i \(0.254385\pi\)
\(4\) 0 0
\(5\) 1.38657 + 1.75426i 0.620094 + 0.784528i
\(6\) 0 0
\(7\) −0.175059 + 0.0348214i −0.0661661 + 0.0131613i −0.228062 − 0.973647i \(-0.573239\pi\)
0.161896 + 0.986808i \(0.448239\pi\)
\(8\) 0 0
\(9\) −0.996671 + 2.40618i −0.332224 + 0.802059i
\(10\) 0 0
\(11\) 0.707055 + 3.55461i 0.213185 + 1.07175i 0.928040 + 0.372481i \(0.121493\pi\)
−0.714855 + 0.699273i \(0.753507\pi\)
\(12\) 0 0
\(13\) 2.05870 0.570981 0.285490 − 0.958382i \(-0.407844\pi\)
0.285490 + 0.958382i \(0.407844\pi\)
\(14\) 0 0
\(15\) 1.33809 + 0.432886i 0.345492 + 0.111771i
\(16\) 0 0
\(17\) 3.77636 − 1.65503i 0.915901 − 0.401404i
\(18\) 0 0
\(19\) −2.61374 − 6.31013i −0.599634 − 1.44764i −0.873956 − 0.486006i \(-0.838453\pi\)
0.274322 − 0.961638i \(-0.411547\pi\)
\(20\) 0 0
\(21\) −0.0793796 + 0.0793796i −0.0173220 + 0.0173220i
\(22\) 0 0
\(23\) −0.386195 + 0.577982i −0.0805272 + 0.120518i −0.869547 − 0.493851i \(-0.835589\pi\)
0.789020 + 0.614368i \(0.210589\pi\)
\(24\) 0 0
\(25\) −1.15484 + 4.86481i −0.230967 + 0.972962i
\(26\) 0 0
\(27\) 0.687670 + 3.45715i 0.132342 + 0.665329i
\(28\) 0 0
\(29\) 0.304090 + 0.455102i 0.0564680 + 0.0845104i 0.858629 − 0.512597i \(-0.171317\pi\)
−0.802161 + 0.597108i \(0.796317\pi\)
\(30\) 0 0
\(31\) 0.666895 − 3.35271i 0.119778 − 0.602164i −0.873540 − 0.486753i \(-0.838181\pi\)
0.993318 − 0.115412i \(-0.0368187\pi\)
\(32\) 0 0
\(33\) 1.61182 + 1.61182i 0.280581 + 0.280581i
\(34\) 0 0
\(35\) −0.303818 − 0.258816i −0.0513546 − 0.0437479i
\(36\) 0 0
\(37\) 2.77178 + 4.14826i 0.455678 + 0.681971i 0.986173 − 0.165717i \(-0.0529940\pi\)
−0.530495 + 0.847688i \(0.677994\pi\)
\(38\) 0 0
\(39\) 1.07660 − 0.719358i 0.172393 − 0.115189i
\(40\) 0 0
\(41\) 4.71458 − 7.05587i 0.736294 − 1.10194i −0.254567 − 0.967055i \(-0.581933\pi\)
0.990861 − 0.134887i \(-0.0430672\pi\)
\(42\) 0 0
\(43\) 2.22176 − 0.920283i 0.338815 − 0.140342i −0.206787 − 0.978386i \(-0.566301\pi\)
0.545602 + 0.838044i \(0.316301\pi\)
\(44\) 0 0
\(45\) −5.60301 + 1.58792i −0.835247 + 0.236713i
\(46\) 0 0
\(47\) − 10.9567i − 1.59820i −0.601198 − 0.799100i \(-0.705310\pi\)
0.601198 − 0.799100i \(-0.294690\pi\)
\(48\) 0 0
\(49\) −6.43772 + 2.66659i −0.919675 + 0.380942i
\(50\) 0 0
\(51\) 1.39654 − 2.18505i 0.195554 − 0.305968i
\(52\) 0 0
\(53\) −2.80886 + 6.78119i −0.385827 + 0.931468i 0.604987 + 0.796235i \(0.293178\pi\)
−0.990814 + 0.135233i \(0.956822\pi\)
\(54\) 0 0
\(55\) −5.25531 + 6.16908i −0.708626 + 0.831838i
\(56\) 0 0
\(57\) −3.57176 − 2.38658i −0.473092 − 0.316110i
\(58\) 0 0
\(59\) −3.07891 − 1.27533i −0.400840 − 0.166034i 0.173150 − 0.984895i \(-0.444605\pi\)
−0.573991 + 0.818862i \(0.694605\pi\)
\(60\) 0 0
\(61\) −8.11858 − 5.42466i −1.03948 − 0.694556i −0.0860841 − 0.996288i \(-0.527435\pi\)
−0.953393 + 0.301732i \(0.902435\pi\)
\(62\) 0 0
\(63\) 0.0906898 − 0.455929i 0.0114258 − 0.0574416i
\(64\) 0 0
\(65\) 2.85454 + 3.61149i 0.354062 + 0.447950i
\(66\) 0 0
\(67\) 7.46930 − 7.46930i 0.912520 − 0.912520i −0.0839496 − 0.996470i \(-0.526753\pi\)
0.996470 + 0.0839496i \(0.0267535\pi\)
\(68\) 0 0
\(69\) 0.437201i 0.0526328i
\(70\) 0 0
\(71\) −4.57239 − 0.909505i −0.542643 − 0.107938i −0.0838446 − 0.996479i \(-0.526720\pi\)
−0.458798 + 0.888540i \(0.651720\pi\)
\(72\) 0 0
\(73\) −9.19094 − 1.82819i −1.07572 − 0.213974i −0.374721 − 0.927138i \(-0.622261\pi\)
−0.700997 + 0.713164i \(0.747261\pi\)
\(74\) 0 0
\(75\) 1.09596 + 2.94757i 0.126550 + 0.340356i
\(76\) 0 0
\(77\) −0.247553 − 0.597645i −0.0282113 − 0.0681080i
\(78\) 0 0
\(79\) −12.8682 + 2.55964i −1.44778 + 0.287982i −0.855522 − 0.517766i \(-0.826764\pi\)
−0.592259 + 0.805748i \(0.701764\pi\)
\(80\) 0 0
\(81\) −3.95720 − 3.95720i −0.439689 − 0.439689i
\(82\) 0 0
\(83\) −0.0183656 − 0.00760730i −0.00201589 − 0.000835010i 0.381675 − 0.924296i \(-0.375347\pi\)
−0.383691 + 0.923461i \(0.625347\pi\)
\(84\) 0 0
\(85\) 8.13954 + 4.32988i 0.882857 + 0.469641i
\(86\) 0 0
\(87\) 0.318047 + 0.131739i 0.0340982 + 0.0141239i
\(88\) 0 0
\(89\) −2.21165 − 2.21165i −0.234435 − 0.234435i 0.580106 − 0.814541i \(-0.303011\pi\)
−0.814541 + 0.580106i \(0.803011\pi\)
\(90\) 0 0
\(91\) −0.360394 + 0.0716868i −0.0377796 + 0.00751482i
\(92\) 0 0
\(93\) −0.822763 − 1.98633i −0.0853165 − 0.205972i
\(94\) 0 0
\(95\) 7.44545 − 13.3346i 0.763887 − 1.36810i
\(96\) 0 0
\(97\) 11.5782 + 2.30306i 1.17559 + 0.233840i 0.743969 − 0.668214i \(-0.232941\pi\)
0.431623 + 0.902054i \(0.357941\pi\)
\(98\) 0 0
\(99\) −9.25772 − 1.84147i −0.930435 − 0.185075i
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.bi.a.37.6 yes 72
5.3 odd 4 340.2.bd.a.173.6 yes 72
17.6 odd 16 340.2.bd.a.57.6 ✓ 72
85.23 even 16 inner 340.2.bi.a.193.6 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
340.2.bd.a.57.6 ✓ 72 17.6 odd 16
340.2.bd.a.173.6 yes 72 5.3 odd 4
340.2.bi.a.37.6 yes 72 1.1 even 1 trivial
340.2.bi.a.193.6 yes 72 85.23 even 16 inner