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Properties

Label 840.2.bz.b.19.33
Level $840$
Weight $2$
Character 840.19
Analytic conductor $6.707$
Analytic rank $0$
Dimension $96$
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(19,840)] 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("840.19"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(840, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 0, 3, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 840.bz (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [96,0,48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.70743376979\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(48\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 19.33
Character \(\chi\) \(=\) 840.19
Dual form 840.2.bz.b.619.33

$q$-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+(0.731574 + 1.21029i) q^{2} +(0.500000 + 0.866025i) q^{3} +(-0.929600 + 1.77083i) q^{4} +(1.40730 + 1.73767i) q^{5} +(-0.682354 + 1.23871i) q^{6} +(-2.63671 + 0.218493i) q^{7} +(-2.82329 + 0.170410i) q^{8} +(-0.500000 + 0.866025i) q^{9} +(-1.07354 + 2.97448i) q^{10} +(0.244083 + 0.422763i) q^{11} +(-1.99838 + 0.0803589i) q^{12} +0.758802i q^{13} +(-2.19339 - 3.03134i) q^{14} +(-0.801214 + 2.08760i) q^{15} +(-2.27169 - 3.29233i) q^{16} +(-1.06889 - 1.85137i) q^{17} +(-1.41393 + 0.0284169i) q^{18} +(2.60376 + 1.50328i) q^{19} +(-4.38535 + 0.876761i) q^{20} +(-1.50758 - 2.17422i) q^{21} +(-0.333102 + 0.604693i) q^{22} +(-3.20451 + 5.55037i) q^{23} +(-1.55922 - 2.35984i) q^{24} +(-1.03899 + 4.89086i) q^{25} +(-0.918370 + 0.555120i) q^{26} -1.00000 q^{27} +(2.06417 - 4.87229i) q^{28} -8.09086i q^{29} +(-3.11274 + 0.557529i) q^{30} +(1.44951 + 2.51062i) q^{31} +(2.32276 - 5.15798i) q^{32} +(-0.244083 + 0.422763i) q^{33} +(1.45872 - 2.64808i) q^{34} +(-4.09033 - 4.27425i) q^{35} +(-1.06879 - 1.69047i) q^{36} +(1.73832 - 3.01086i) q^{37} +(0.0854373 + 4.25107i) q^{38} +(-0.657142 + 0.379401i) q^{39} +(-4.26934 - 4.66613i) q^{40} -3.03002i q^{41} +(1.52852 - 3.41520i) q^{42} +7.83815i q^{43} +(-0.975542 + 0.0392284i) q^{44} +(-2.20852 + 0.349926i) q^{45} +(-9.06188 + 0.182124i) q^{46} +(5.75429 + 3.32224i) q^{47} +(1.71540 - 3.61350i) q^{48} +(6.90452 - 1.15221i) q^{49} +(-6.67945 + 2.32054i) q^{50} +(1.06889 - 1.85137i) q^{51} +(-1.34371 - 0.705382i) q^{52} +(7.13003 + 12.3496i) q^{53} +(-0.731574 - 1.21029i) q^{54} +(-0.391125 + 1.01909i) q^{55} +(7.40697 - 1.06619i) q^{56} +3.00656i q^{57} +(9.79227 - 5.91906i) q^{58} +(-9.76611 + 5.63847i) q^{59} +(-2.95197 - 3.35944i) q^{60} +(2.45006 - 4.24362i) q^{61} +(-1.97816 + 3.59103i) q^{62} +(1.12914 - 2.39271i) q^{63} +(7.94192 - 0.962231i) q^{64} +(-1.31855 + 1.06786i) q^{65} +(-0.690230 + 0.0138721i) q^{66} +(2.35167 - 1.35774i) q^{67} +(4.27210 - 0.171789i) q^{68} -6.40901 q^{69} +(2.18071 - 8.07741i) q^{70} -3.60198i q^{71} +(1.26407 - 2.53024i) q^{72} +(-1.41847 - 2.45686i) q^{73} +(4.91572 - 0.0987953i) q^{74} +(-4.75510 + 1.54564i) q^{75} +(-5.08252 + 3.21337i) q^{76} +(-0.735947 - 1.06138i) q^{77} +(-0.939933 - 0.517772i) q^{78} +(-3.30116 - 1.90593i) q^{79} +(2.52402 - 8.58075i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(3.66720 - 2.21668i) q^{82} -5.40677 q^{83} +(5.25161 - 0.648516i) q^{84} +(1.71282 - 4.46281i) q^{85} +(-9.48643 + 5.73419i) q^{86} +(7.00689 - 4.04543i) q^{87} +(-0.761159 - 1.15199i) q^{88} +(14.0372 + 8.10436i) q^{89} +(-2.03921 - 2.41695i) q^{90} +(-0.165793 - 2.00074i) q^{91} +(-6.84986 - 10.8343i) q^{92} +(-1.44951 + 2.51062i) q^{93} +(0.188815 + 9.39481i) q^{94} +(1.05207 + 6.64005i) q^{95} +(5.62832 - 0.567420i) q^{96} -12.1987 q^{97} +(6.44567 + 7.51354i) q^{98} -0.488165 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q + 48 q^{3} - 48 q^{9} + 5 q^{10} + 14 q^{14} + 4 q^{16} + 22 q^{20} - 96 q^{27} - 4 q^{28} + 13 q^{30} - 30 q^{32} - 16 q^{35} - 12 q^{38} - 7 q^{40} - 2 q^{42} - 16 q^{44} - 22 q^{46} + 8 q^{48} + 12 q^{50}+ \cdots - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(241\) \(281\) \(337\) \(421\) \(631\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.731574 + 1.21029i 0.517301 + 0.855804i
\(3\) 0.500000 + 0.866025i 0.288675 + 0.500000i
\(4\) −0.929600 + 1.77083i −0.464800 + 0.885416i
\(5\) 1.40730 + 1.73767i 0.629365 + 0.777110i
\(6\) −0.682354 + 1.23871i −0.278570 + 0.505700i
\(7\) −2.63671 + 0.218493i −0.996584 + 0.0825824i
\(8\) −2.82329 + 0.170410i −0.998183 + 0.0602489i
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) −1.07354 + 2.97448i −0.339482 + 0.940613i
\(11\) 0.244083 + 0.422763i 0.0735937 + 0.127468i 0.900474 − 0.434910i \(-0.143220\pi\)
−0.826880 + 0.562378i \(0.809887\pi\)
\(12\) −1.99838 + 0.0803589i −0.576884 + 0.0231976i
\(13\) 0.758802i 0.210454i 0.994448 + 0.105227i \(0.0335569\pi\)
−0.994448 + 0.105227i \(0.966443\pi\)
\(14\) −2.19339 − 3.03134i −0.586208 − 0.810160i
\(15\) −0.801214 + 2.08760i −0.206873 + 0.539015i
\(16\) −2.27169 − 3.29233i −0.567922 − 0.823082i
\(17\) −1.06889 − 1.85137i −0.259244 − 0.449023i 0.706796 − 0.707418i \(-0.250140\pi\)
−0.966039 + 0.258394i \(0.916807\pi\)
\(18\) −1.41393 + 0.0284169i −0.333266 + 0.00669793i
\(19\) 2.60376 + 1.50328i 0.597344 + 0.344877i 0.767996 − 0.640455i \(-0.221254\pi\)
−0.170652 + 0.985331i \(0.554587\pi\)
\(20\) −4.38535 + 0.876761i −0.980594 + 0.196050i
\(21\) −1.50758 − 2.17422i −0.328980 − 0.474453i
\(22\) −0.333102 + 0.604693i −0.0710175 + 0.128921i
\(23\) −3.20451 + 5.55037i −0.668186 + 1.15733i 0.310225 + 0.950663i \(0.399595\pi\)
−0.978411 + 0.206668i \(0.933738\pi\)
\(24\) −1.55922 − 2.35984i −0.318275 − 0.481699i
\(25\) −1.03899 + 4.89086i −0.207799 + 0.978172i
\(26\) −0.918370 + 0.555120i −0.180107 + 0.108868i
\(27\) −1.00000 −0.192450
\(28\) 2.06417 − 4.87229i 0.390092 − 0.920776i
\(29\) − 8.09086i − 1.50243i −0.660055 − 0.751217i \(-0.729467\pi\)
0.660055 − 0.751217i \(-0.270533\pi\)
\(30\) −3.11274 + 0.557529i −0.568306 + 0.101790i
\(31\) 1.44951 + 2.51062i 0.260339 + 0.450921i 0.966332 − 0.257298i \(-0.0828322\pi\)
−0.705993 + 0.708219i \(0.749499\pi\)
\(32\) 2.32276 − 5.15798i 0.410610 − 0.911811i
\(33\) −0.244083 + 0.422763i −0.0424893 + 0.0735937i
\(34\) 1.45872 − 2.64808i 0.250169 − 0.454142i
\(35\) −4.09033 − 4.27425i −0.691391 − 0.722481i
\(36\) −1.06879 − 1.69047i −0.178131 − 0.281745i
\(37\) 1.73832 − 3.01086i 0.285778 − 0.494982i −0.687020 − 0.726639i \(-0.741081\pi\)
0.972798 + 0.231657i \(0.0744147\pi\)
\(38\) 0.0854373 + 4.25107i 0.0138598 + 0.689614i
\(39\) −0.657142 + 0.379401i −0.105227 + 0.0607528i
\(40\) −4.26934 − 4.66613i −0.675042 − 0.737779i
\(41\) − 3.03002i − 0.473209i −0.971606 − 0.236605i \(-0.923965\pi\)
0.971606 − 0.236605i \(-0.0760346\pi\)
\(42\) 1.52852 − 3.41520i 0.235857 − 0.526977i
\(43\) 7.83815i 1.19531i 0.801755 + 0.597653i \(0.203900\pi\)
−0.801755 + 0.597653i \(0.796100\pi\)
\(44\) −0.975542 + 0.0392284i −0.147068 + 0.00591391i
\(45\) −2.20852 + 0.349926i −0.329226 + 0.0521639i
\(46\) −9.06188 + 0.182124i −1.33610 + 0.0268528i
\(47\) 5.75429 + 3.32224i 0.839349 + 0.484598i 0.857043 − 0.515245i \(-0.172299\pi\)
−0.0176940 + 0.999843i \(0.505632\pi\)
\(48\) 1.71540 − 3.61350i 0.247596 − 0.521564i
\(49\) 6.90452 − 1.15221i 0.986360 − 0.164601i
\(50\) −6.67945 + 2.32054i −0.944617 + 0.328174i
\(51\) 1.06889 − 1.85137i 0.149674 − 0.259244i
\(52\) −1.34371 − 0.705382i −0.186339 − 0.0978189i
\(53\) 7.13003 + 12.3496i 0.979385 + 1.69634i 0.664632 + 0.747171i \(0.268588\pi\)
0.314753 + 0.949174i \(0.398078\pi\)
\(54\) −0.731574 − 1.21029i −0.0995546 − 0.164699i
\(55\) −0.391125 + 1.01909i −0.0527393 + 0.137414i
\(56\) 7.40697 − 1.06619i 0.989798 − 0.142476i
\(57\) 3.00656i 0.398229i
\(58\) 9.79227 − 5.91906i 1.28579 − 0.777210i
\(59\) −9.76611 + 5.63847i −1.27144 + 0.734066i −0.975259 − 0.221066i \(-0.929046\pi\)
−0.296181 + 0.955132i \(0.595713\pi\)
\(60\) −2.95197 − 3.35944i −0.381098 − 0.433702i
\(61\) 2.45006 − 4.24362i 0.313698 − 0.543340i −0.665462 − 0.746432i \(-0.731766\pi\)
0.979160 + 0.203091i \(0.0650989\pi\)
\(62\) −1.97816 + 3.59103i −0.251226 + 0.456061i
\(63\) 1.12914 − 2.39271i 0.142258 − 0.301453i
\(64\) 7.94192 − 0.962231i 0.992740 − 0.120279i
\(65\) −1.31855 + 1.06786i −0.163546 + 0.132452i
\(66\) −0.690230 + 0.0138721i −0.0849615 + 0.00170754i
\(67\) 2.35167 − 1.35774i 0.287302 − 0.165874i −0.349423 − 0.936965i \(-0.613622\pi\)
0.636724 + 0.771091i \(0.280289\pi\)
\(68\) 4.27210 − 0.171789i 0.518068 − 0.0208325i
\(69\) −6.40901 −0.771554
\(70\) 2.18071 − 8.07741i 0.260644 − 0.965435i
\(71\) − 3.60198i − 0.427477i −0.976891 − 0.213738i \(-0.931436\pi\)
0.976891 − 0.213738i \(-0.0685640\pi\)
\(72\) 1.26407 − 2.53024i 0.148972 − 0.298192i
\(73\) −1.41847 − 2.45686i −0.166019 − 0.287554i 0.770998 − 0.636838i \(-0.219758\pi\)
−0.937017 + 0.349284i \(0.886425\pi\)
\(74\) 4.91572 − 0.0987953i 0.571441 − 0.0114847i
\(75\) −4.75510 + 1.54564i −0.549072 + 0.178475i
\(76\) −5.08252 + 3.21337i −0.583004 + 0.368599i
\(77\) −0.735947 − 1.06138i −0.0838689 − 0.120955i
\(78\) −0.939933 − 0.517772i −0.106426 − 0.0586261i
\(79\) −3.30116 − 1.90593i −0.371410 − 0.214434i 0.302664 − 0.953097i \(-0.402124\pi\)
−0.674074 + 0.738664i \(0.735457\pi\)
\(80\) 2.52402 − 8.58075i 0.282194 − 0.959357i
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) 3.66720 − 2.21668i 0.404974 − 0.244791i
\(83\) −5.40677 −0.593470 −0.296735 − 0.954960i \(-0.595898\pi\)
−0.296735 + 0.954960i \(0.595898\pi\)
\(84\) 5.25161 − 0.648516i 0.572998 − 0.0707589i
\(85\) 1.71282 − 4.46281i 0.185781 − 0.484060i
\(86\) −9.48643 + 5.73419i −1.02295 + 0.618333i
\(87\) 7.00689 − 4.04543i 0.751217 − 0.433715i
\(88\) −0.761159 − 1.15199i −0.0811398 − 0.122802i
\(89\) 14.0372 + 8.10436i 1.48794 + 0.859061i 0.999905 − 0.0137656i \(-0.00438187\pi\)
0.488031 + 0.872826i \(0.337715\pi\)
\(90\) −2.03921 − 2.41695i −0.214951 − 0.254769i
\(91\) −0.165793 − 2.00074i −0.0173798 − 0.209735i
\(92\) −6.84986 − 10.8343i −0.714147 − 1.12955i
\(93\) −1.44951 + 2.51062i −0.150307 + 0.260339i
\(94\) 0.188815 + 9.39481i 0.0194748 + 0.969001i
\(95\) 1.05207 + 6.64005i 0.107941 + 0.681255i
\(96\) 5.62832 − 0.567420i 0.574438 − 0.0579121i
\(97\) −12.1987 −1.23859 −0.619295 − 0.785158i \(-0.712581\pi\)
−0.619295 + 0.785158i \(0.712581\pi\)
\(98\) 6.44567 + 7.51354i 0.651111 + 0.758983i
\(99\) −0.488165 −0.0490624
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.bz.b.19.33 yes 96
5.4 even 2 840.2.bz.a.19.16 yes 96
7.3 odd 6 840.2.bz.a.619.1 yes 96
8.3 odd 2 inner 840.2.bz.b.19.48 yes 96
35.24 odd 6 inner 840.2.bz.b.619.48 yes 96
40.19 odd 2 840.2.bz.a.19.1 ✓ 96
56.3 even 6 840.2.bz.a.619.16 yes 96
280.59 even 6 inner 840.2.bz.b.619.33 yes 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
840.2.bz.a.19.1 ✓ 96 40.19 odd 2
840.2.bz.a.19.16 yes 96 5.4 even 2
840.2.bz.a.619.1 yes 96 7.3 odd 6
840.2.bz.a.619.16 yes 96 56.3 even 6
840.2.bz.b.19.33 yes 96 1.1 even 1 trivial
840.2.bz.b.19.48 yes 96 8.3 odd 2 inner
840.2.bz.b.619.33 yes 96 280.59 even 6 inner
840.2.bz.b.619.48 yes 96 35.24 odd 6 inner