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Properties

Label 840.2.k.a.209.4
Level $840$
Weight $2$
Character 840.209
Analytic conductor $6.707$
Analytic rank $0$
Dimension $24$
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] = [840,2,Mod(209,840)] 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("840.209"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(840, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 840.k (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,0,0,0,0,0,0,0,-2,0,0,0,0,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.70743376979\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 209.4
Character \(\chi\) \(=\) 840.209
Dual form 840.2.k.a.209.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.64602 + 0.539084i) q^{3} +(-1.30213 - 1.81782i) q^{5} +(-2.19974 - 1.47008i) q^{7} +(2.41878 - 1.77469i) q^{9} +0.958461i q^{11} +0.157315 q^{13} +(3.12329 + 2.29022i) q^{15} -2.51337i q^{17} +1.98260i q^{19} +(4.41332 + 1.23394i) q^{21} -2.67280 q^{23} +(-1.60893 + 4.73406i) q^{25} +(-3.02466 + 4.22510i) q^{27} +1.25028i q^{29} +8.66804i q^{31} +(-0.516691 - 1.57765i) q^{33} +(0.191998 + 5.91296i) q^{35} +2.29909i q^{37} +(-0.258945 + 0.0848062i) q^{39} +4.74507 q^{41} +6.58424i q^{43} +(-6.37561 - 2.08603i) q^{45} -5.60727i q^{47} +(2.67772 + 6.46760i) q^{49} +(1.35492 + 4.13706i) q^{51} -8.59262 q^{53} +(1.74231 - 1.24804i) q^{55} +(-1.06879 - 3.26340i) q^{57} -10.4488 q^{59} +4.27757i q^{61} +(-7.92962 + 0.348054i) q^{63} +(-0.204845 - 0.285971i) q^{65} +13.8282i q^{67} +(4.39948 - 1.44086i) q^{69} -9.75994i q^{71} -4.43474 q^{73} +(0.0962871 - 8.65972i) q^{75} +(1.40902 - 2.10837i) q^{77} +0.517890 q^{79} +(2.70097 - 8.58515i) q^{81} +18.1293i q^{83} +(-4.56885 + 3.27272i) q^{85} +(-0.674006 - 2.05799i) q^{87} -0.954423 q^{89} +(-0.346053 - 0.231267i) q^{91} +(-4.67280 - 14.2678i) q^{93} +(3.60401 - 2.58160i) q^{95} +14.0907 q^{97} +(1.70097 + 2.31830i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 2 q^{9} - 6 q^{15} - 2 q^{21} + 16 q^{23} + 8 q^{25} - 8 q^{35} - 2 q^{39} + 6 q^{51} - 24 q^{53} - 8 q^{57} - 16 q^{63} - 16 q^{65} - 8 q^{77} + 4 q^{79} + 18 q^{81} - 12 q^{85} + 12 q^{91} - 32 q^{93}+ \cdots - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/840\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(281\) \(337\) \(421\) \(631\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\) \(1\)

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
\(3\) −1.64602 + 0.539084i −0.950331 + 0.311240i
\(4\) 0 0
\(5\) −1.30213 − 1.81782i −0.582329 − 0.812954i
\(6\) 0 0
\(7\) −2.19974 − 1.47008i −0.831424 − 0.555638i
\(8\) 0 0
\(9\) 2.41878 − 1.77469i 0.806259 − 0.591562i
\(10\) 0 0
\(11\) 0.958461i 0.288987i 0.989506 + 0.144493i \(0.0461553\pi\)
−0.989506 + 0.144493i \(0.953845\pi\)
\(12\) 0 0
\(13\) 0.157315 0.0436315 0.0218157 − 0.999762i \(-0.493055\pi\)
0.0218157 + 0.999762i \(0.493055\pi\)
\(14\) 0 0
\(15\) 3.12329 + 2.29022i 0.806429 + 0.591331i
\(16\) 0 0
\(17\) − 2.51337i − 0.609581i −0.952419 − 0.304791i \(-0.901414\pi\)
0.952419 − 0.304791i \(-0.0985865\pi\)
\(18\) 0 0
\(19\) 1.98260i 0.454840i 0.973797 + 0.227420i \(0.0730290\pi\)
−0.973797 + 0.227420i \(0.926971\pi\)
\(20\) 0 0
\(21\) 4.41332 + 1.23394i 0.963065 + 0.269268i
\(22\) 0 0
\(23\) −2.67280 −0.557317 −0.278658 − 0.960390i \(-0.589890\pi\)
−0.278658 + 0.960390i \(0.589890\pi\)
\(24\) 0 0
\(25\) −1.60893 + 4.73406i −0.321787 + 0.946812i
\(26\) 0 0
\(27\) −3.02466 + 4.22510i −0.582095 + 0.813121i
\(28\) 0 0
\(29\) 1.25028i 0.232171i 0.993239 + 0.116086i \(0.0370347\pi\)
−0.993239 + 0.116086i \(0.962965\pi\)
\(30\) 0 0
\(31\) 8.66804i 1.55683i 0.627753 + 0.778413i \(0.283975\pi\)
−0.627753 + 0.778413i \(0.716025\pi\)
\(32\) 0 0
\(33\) −0.516691 − 1.57765i −0.0899443 − 0.274633i
\(34\) 0 0
\(35\) 0.191998 + 5.91296i 0.0324536 + 0.999473i
\(36\) 0 0
\(37\) 2.29909i 0.377969i 0.981980 + 0.188984i \(0.0605195\pi\)
−0.981980 + 0.188984i \(0.939480\pi\)
\(38\) 0 0
\(39\) −0.258945 + 0.0848062i −0.0414643 + 0.0135799i
\(40\) 0 0
\(41\) 4.74507 0.741056 0.370528 − 0.928821i \(-0.379177\pi\)
0.370528 + 0.928821i \(0.379177\pi\)
\(42\) 0 0
\(43\) 6.58424i 1.00409i 0.864842 + 0.502043i \(0.167418\pi\)
−0.864842 + 0.502043i \(0.832582\pi\)
\(44\) 0 0
\(45\) −6.37561 − 2.08603i −0.950421 − 0.310968i
\(46\) 0 0
\(47\) − 5.60727i − 0.817904i −0.912556 − 0.408952i \(-0.865894\pi\)
0.912556 − 0.408952i \(-0.134106\pi\)
\(48\) 0 0
\(49\) 2.67772 + 6.46760i 0.382532 + 0.923942i
\(50\) 0 0
\(51\) 1.35492 + 4.13706i 0.189726 + 0.579304i
\(52\) 0 0
\(53\) −8.59262 −1.18029 −0.590144 − 0.807298i \(-0.700929\pi\)
−0.590144 + 0.807298i \(0.700929\pi\)
\(54\) 0 0
\(55\) 1.74231 − 1.24804i 0.234933 − 0.168285i
\(56\) 0 0
\(57\) −1.06879 − 3.26340i −0.141564 − 0.432248i
\(58\) 0 0
\(59\) −10.4488 −1.36032 −0.680159 − 0.733065i \(-0.738089\pi\)
−0.680159 + 0.733065i \(0.738089\pi\)
\(60\) 0 0
\(61\) 4.27757i 0.547686i 0.961774 + 0.273843i \(0.0882949\pi\)
−0.961774 + 0.273843i \(0.911705\pi\)
\(62\) 0 0
\(63\) −7.92962 + 0.348054i −0.999038 + 0.0438506i
\(64\) 0 0
\(65\) −0.204845 − 0.285971i −0.0254078 − 0.0354704i
\(66\) 0 0
\(67\) 13.8282i 1.68938i 0.535255 + 0.844690i \(0.320215\pi\)
−0.535255 + 0.844690i \(0.679785\pi\)
\(68\) 0 0
\(69\) 4.39948 − 1.44086i 0.529635 − 0.173459i
\(70\) 0 0
\(71\) − 9.75994i − 1.15829i −0.815224 − 0.579146i \(-0.803386\pi\)
0.815224 − 0.579146i \(-0.196614\pi\)
\(72\) 0 0
\(73\) −4.43474 −0.519047 −0.259524 − 0.965737i \(-0.583566\pi\)
−0.259524 + 0.965737i \(0.583566\pi\)
\(74\) 0 0
\(75\) 0.0962871 − 8.65972i 0.0111183 − 0.999938i
\(76\) 0 0
\(77\) 1.40902 − 2.10837i 0.160572 − 0.240271i
\(78\) 0 0
\(79\) 0.517890 0.0582671 0.0291336 − 0.999576i \(-0.490725\pi\)
0.0291336 + 0.999576i \(0.490725\pi\)
\(80\) 0 0
\(81\) 2.70097 − 8.58515i 0.300108 − 0.953905i
\(82\) 0 0
\(83\) 18.1293i 1.98995i 0.100105 + 0.994977i \(0.468082\pi\)
−0.100105 + 0.994977i \(0.531918\pi\)
\(84\) 0 0
\(85\) −4.56885 + 3.27272i −0.495561 + 0.354977i
\(86\) 0 0
\(87\) −0.674006 − 2.05799i −0.0722611 − 0.220640i
\(88\) 0 0
\(89\) −0.954423 −0.101169 −0.0505843 − 0.998720i \(-0.516108\pi\)
−0.0505843 + 0.998720i \(0.516108\pi\)
\(90\) 0 0
\(91\) −0.346053 − 0.231267i −0.0362762 − 0.0242433i
\(92\) 0 0
\(93\) −4.67280 − 14.2678i −0.484546 − 1.47950i
\(94\) 0 0
\(95\) 3.60401 − 2.58160i 0.369764 − 0.264866i
\(96\) 0 0
\(97\) 14.0907 1.43070 0.715348 − 0.698768i \(-0.246268\pi\)
0.715348 + 0.698768i \(0.246268\pi\)
\(98\) 0 0
\(99\) 1.70097 + 2.31830i 0.170954 + 0.232998i
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 840.2.k.a.209.4 yes 24
3.2 odd 2 840.2.k.b.209.3 yes 24
4.3 odd 2 1680.2.k.i.209.21 24
5.4 even 2 840.2.k.b.209.21 yes 24
7.6 odd 2 inner 840.2.k.a.209.21 yes 24
12.11 even 2 1680.2.k.h.209.22 24
15.14 odd 2 inner 840.2.k.a.209.22 yes 24
20.19 odd 2 1680.2.k.h.209.4 24
21.20 even 2 840.2.k.b.209.22 yes 24
28.27 even 2 1680.2.k.i.209.4 24
35.34 odd 2 840.2.k.b.209.4 yes 24
60.59 even 2 1680.2.k.i.209.3 24
84.83 odd 2 1680.2.k.h.209.3 24
105.104 even 2 inner 840.2.k.a.209.3 ✓ 24
140.139 even 2 1680.2.k.h.209.21 24
420.419 odd 2 1680.2.k.i.209.22 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
840.2.k.a.209.3 ✓ 24 105.104 even 2 inner
840.2.k.a.209.4 yes 24 1.1 even 1 trivial
840.2.k.a.209.21 yes 24 7.6 odd 2 inner
840.2.k.a.209.22 yes 24 15.14 odd 2 inner
840.2.k.b.209.3 yes 24 3.2 odd 2
840.2.k.b.209.4 yes 24 35.34 odd 2
840.2.k.b.209.21 yes 24 5.4 even 2
840.2.k.b.209.22 yes 24 21.20 even 2
1680.2.k.h.209.3 24 84.83 odd 2
1680.2.k.h.209.4 24 20.19 odd 2
1680.2.k.h.209.21 24 140.139 even 2
1680.2.k.h.209.22 24 12.11 even 2
1680.2.k.i.209.3 24 60.59 even 2
1680.2.k.i.209.4 24 28.27 even 2
1680.2.k.i.209.21 24 4.3 odd 2
1680.2.k.i.209.22 24 420.419 odd 2