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Properties

Label 840.2.bz.a.19.16
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 19.16
Character \(\chi\) \(=\) 840.19
Dual form 840.2.bz.a.619.16

$q$-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} +(-2.20852 - 0.349926i) 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} +(1.19218 + 2.92894i) 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} +(2.67270 - 3.58562i) 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} +(4.75510 + 1.54564i) q^{25} +(-0.918370 + 0.555120i) q^{26} +1.00000 q^{27} +(-2.06417 + 4.87229i) q^{28} -8.09086i q^{29} +(1.94045 - 2.49693i) 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} +(-5.89969 - 0.440110i) 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} +(-6.29492 - 0.611589i) q^{40} -3.03002i q^{41} +(-1.52852 + 3.41520i) q^{42} -7.83815i q^{43} +(-0.975542 + 0.0392284i) q^{44} +(1.40730 - 1.73767i) 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} +(-1.60804 - 6.88580i) 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} +(-4.44159 - 0.521812i) 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} +(-0.265524 + 1.67583i) q^{65} +(-0.690230 + 0.0138721i) q^{66} +(-2.35167 + 1.35774i) q^{67} +(-4.27210 + 0.171789i) q^{68} -6.40901 q^{69} +(3.78340 + 7.46230i) q^{70} -3.60198i q^{71} +(-1.26407 + 2.53024i) q^{72} +(1.41847 + 2.45686i) q^{73} +(4.91572 - 0.0987953i) q^{74} +(-1.03899 - 4.89086i) 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} +(3.86500 + 8.06609i) 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} +(-3.13263 - 0.432011i) 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} +(-5.22442 - 4.23115i) 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})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q - 48 q^{3} - 48 q^{9} + 13 q^{10} + 14 q^{14} + 4 q^{16} - 22 q^{20} + 96 q^{27} + 4 q^{28} - 5 q^{30} + 30 q^{32} - 8 q^{35} + 12 q^{38} - 23 q^{40} + 2 q^{42} - 16 q^{44} - 22 q^{46} - 8 q^{48} + 12 q^{50}+ \cdots + 2 q^{98}+O(q^{100}) \) 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
\(5\) −2.20852 − 0.349926i −0.987679 − 0.156492i
\(6\) −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
\(10\) 1.19218 + 2.92894i 0.377001 + 0.926213i
\(11\) 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 \(-0.966443\pi\)
0.994448 − 0.105227i \(-0.0335569\pi\)
\(14\) −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 0.966039 − 0.258394i \(-0.0831934\pi\)
−0.706796 + 0.707418i \(0.749860\pi\)
\(18\) 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 \(0.554587\pi\)
\(20\) 2.67270 − 3.58562i 0.597633 − 0.801770i
\(21\) −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 \(-0.600405\pi\)
0.978411 − 0.206668i \(-0.0662621\pi\)
\(24\) −1.55922 − 2.35984i −0.318275 − 0.481699i
\(25\) 4.75510 + 1.54564i 0.951021 + 0.309127i
\(26\) −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 \(-0.270533\pi\)
\(30\) 1.94045 − 2.49693i 0.354276 − 0.455875i
\(31\) 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
\(35\) −5.89969 − 0.440110i −0.997229 − 0.0743921i
\(36\) −1.06879 − 1.69047i −0.178131 − 0.281745i
\(37\) −1.73832 + 3.01086i −0.285778 + 0.494982i −0.972798 − 0.231657i \(-0.925585\pi\)
0.687020 + 0.726639i \(0.258919\pi\)
\(38\) −0.0854373 − 4.25107i −0.0138598 − 0.689614i
\(39\) −0.657142 + 0.379401i −0.105227 + 0.0607528i
\(40\) −6.29492 − 0.611589i −0.995314 − 0.0967007i
\(41\) − 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 \(-0.796100\pi\)
0.801755 − 0.597653i \(-0.203900\pi\)
\(44\) −0.975542 + 0.0392284i −0.147068 + 0.00591391i
\(45\) 1.40730 − 1.73767i 0.209788 − 0.259037i
\(46\) −9.06188 + 0.182124i −1.33610 + 0.0268528i
\(47\) −5.75429 − 3.32224i −0.839349 − 0.484598i 0.0176940 − 0.999843i \(-0.494368\pi\)
−0.857043 + 0.515245i \(0.827701\pi\)
\(48\) −1.71540 + 3.61350i −0.247596 + 0.521564i
\(49\) 6.90452 − 1.15221i 0.986360 − 0.164601i
\(50\) −1.60804 − 6.88580i −0.227412 − 0.973799i
\(51\) 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 \(-0.731412\pi\)
−0.314753 − 0.949174i \(-0.601922\pi\)
\(54\) −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 \(0.595713\pi\)
\(60\) −4.44159 − 0.521812i −0.573407 − 0.0673657i
\(61\) 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
\(65\) −0.265524 + 1.67583i −0.0329343 + 0.207861i
\(66\) −0.690230 + 0.0138721i −0.0849615 + 0.00170754i
\(67\) −2.35167 + 1.35774i −0.287302 + 0.165874i −0.636724 − 0.771091i \(-0.719711\pi\)
0.349423 + 0.936965i \(0.386378\pi\)
\(68\) −4.27210 + 0.171789i −0.518068 + 0.0208325i
\(69\) −6.40901 −0.771554
\(70\) 3.78340 + 7.46230i 0.452202 + 0.891915i
\(71\) − 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 0.937017 − 0.349284i \(-0.113575\pi\)
−0.770998 + 0.636838i \(0.780242\pi\)
\(74\) 4.91572 − 0.0987953i 0.571441 − 0.0114847i
\(75\) −1.03899 − 4.89086i −0.119973 − 0.564748i
\(76\) −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 \(0.735457\pi\)
\(80\) 3.86500 + 8.06609i 0.432120 + 0.901816i
\(81\) −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 \(-0.404102\pi\)
0.296735 + 0.954960i \(0.404102\pi\)
\(84\) 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 \(0.337715\pi\)
\(90\) −3.13263 − 0.432011i −0.330208 − 0.0455379i
\(91\) −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
\(95\) −5.22442 − 4.23115i −0.536014 − 0.434107i
\(96\) 5.62832 − 0.567420i 0.574438 − 0.0579121i
\(97\) 12.1987 1.23859 0.619295 − 0.785158i \(-0.287419\pi\)
0.619295 + 0.785158i \(0.287419\pi\)
\(98\) −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.a.19.16 yes 96
5.4 even 2 840.2.bz.b.19.33 yes 96
7.3 odd 6 840.2.bz.b.619.48 yes 96
8.3 odd 2 inner 840.2.bz.a.19.1 ✓ 96
35.24 odd 6 inner 840.2.bz.a.619.1 yes 96
40.19 odd 2 840.2.bz.b.19.48 yes 96
56.3 even 6 840.2.bz.b.619.33 yes 96
280.59 even 6 inner 840.2.bz.a.619.16 yes 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
840.2.bz.a.19.1 ✓ 96 8.3 odd 2 inner
840.2.bz.a.19.16 yes 96 1.1 even 1 trivial
840.2.bz.a.619.1 yes 96 35.24 odd 6 inner
840.2.bz.a.619.16 yes 96 280.59 even 6 inner
840.2.bz.b.19.33 yes 96 5.4 even 2
840.2.bz.b.19.48 yes 96 40.19 odd 2
840.2.bz.b.619.33 yes 96 56.3 even 6
840.2.bz.b.619.48 yes 96 7.3 odd 6