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Properties

Label 840.2.bz.a.619.25
Level $840$
Weight $2$
Character 840.619
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 619.25
Character \(\chi\) \(=\) 840.619
Dual form 840.2.bz.a.19.25

$q$-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+(0.0571777 + 1.41306i) q^{2} +(-0.500000 + 0.866025i) q^{3} +(-1.99346 + 0.161591i) q^{4} +(1.90078 - 1.17773i) q^{5} +(-1.25233 - 0.657011i) q^{6} +(2.10021 + 1.60907i) q^{7} +(-0.342319 - 2.80764i) q^{8} +(-0.500000 - 0.866025i) q^{9} +(1.77288 + 2.61857i) q^{10} +(1.84936 - 3.20318i) q^{11} +(0.856789 - 1.80718i) q^{12} -4.06159i q^{13} +(-2.15363 + 3.05972i) q^{14} +(0.0695556 + 2.23499i) q^{15} +(3.94778 - 0.644250i) q^{16} +(3.58024 - 6.20117i) q^{17} +(1.19515 - 0.756046i) q^{18} +(-2.05610 + 1.18709i) q^{19} +(-3.59881 + 2.65491i) q^{20} +(-2.44360 + 1.01430i) q^{21} +(4.63202 + 2.43010i) q^{22} +(-1.57001 - 2.71934i) q^{23} +(2.60264 + 1.10736i) q^{24} +(2.22590 - 4.47720i) q^{25} +(5.73926 - 0.232233i) q^{26} +1.00000 q^{27} +(-4.44670 - 2.86825i) q^{28} -0.386546i q^{29} +(-3.15419 + 0.226077i) q^{30} +(0.295740 - 0.512237i) q^{31} +(1.13609 + 5.54160i) q^{32} +(1.84936 + 3.20318i) q^{33} +(8.96731 + 4.70452i) q^{34} +(5.88708 + 0.585008i) q^{35} +(1.13667 + 1.64559i) q^{36} +(-2.87135 - 4.97333i) q^{37} +(-1.79499 - 2.83751i) q^{38} +(3.51744 + 2.03080i) q^{39} +(-3.95731 - 4.93353i) q^{40} +12.1918i q^{41} +(-1.57298 - 3.39496i) q^{42} +1.90138i q^{43} +(-3.16902 + 6.68426i) q^{44} +(-1.97033 - 1.05726i) q^{45} +(3.75281 - 2.37400i) q^{46} +(-0.245768 + 0.141894i) q^{47} +(-1.41595 + 3.74100i) q^{48} +(1.82177 + 6.75878i) q^{49} +(6.45382 + 2.88933i) q^{50} +(3.58024 + 6.20117i) q^{51} +(0.656316 + 8.09663i) q^{52} +(-0.232955 + 0.403490i) q^{53} +(0.0571777 + 1.41306i) q^{54} +(-0.257267 - 8.26658i) q^{55} +(3.79875 - 6.44744i) q^{56} -2.37418i q^{57} +(0.546212 - 0.0221018i) q^{58} +(4.30554 + 2.48580i) q^{59} +(-0.499810 - 4.44412i) q^{60} +(-5.38555 - 9.32804i) q^{61} +(0.740730 + 0.388609i) q^{62} +(0.343393 - 2.62337i) q^{63} +(-7.76564 + 1.92221i) q^{64} +(-4.78346 - 7.72018i) q^{65} +(-4.42054 + 2.79640i) q^{66} +(5.18143 + 2.99150i) q^{67} +(-6.13503 + 12.9403i) q^{68} +3.14002 q^{69} +(-0.490040 + 8.35224i) q^{70} +15.5552i q^{71} +(-2.26032 + 1.70027i) q^{72} +(-4.16891 + 7.22077i) q^{73} +(6.86343 - 4.34175i) q^{74} +(2.76442 + 4.16629i) q^{75} +(3.90693 - 2.69866i) q^{76} +(9.03820 - 3.75161i) q^{77} +(-2.66851 + 5.08646i) q^{78} +(-6.20106 + 3.58019i) q^{79} +(6.74509 - 5.87399i) q^{80} +(-0.500000 + 0.866025i) q^{81} +(-17.2277 + 0.697097i) q^{82} +16.1072 q^{83} +(4.70733 - 2.41683i) q^{84} +(-0.498052 - 16.0036i) q^{85} +(-2.68676 + 0.108717i) q^{86} +(0.334759 + 0.193273i) q^{87} +(-9.62644 - 4.09582i) q^{88} +(-9.05695 + 5.22903i) q^{89} +(1.38130 - 2.84464i) q^{90} +(6.53540 - 8.53020i) q^{91} +(3.56917 + 5.16720i) q^{92} +(0.295740 + 0.512237i) q^{93} +(-0.214557 - 0.339171i) q^{94} +(-2.51011 + 4.67792i) q^{95} +(-5.36721 - 1.78692i) q^{96} +2.21812 q^{97} +(-9.44638 + 2.96071i) q^{98} -3.69872 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 \(\left(\mathbb{Z}/840\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.0571777 + 1.41306i 0.0404307 + 0.999182i
\(3\) −0.500000 + 0.866025i −0.288675 + 0.500000i
\(4\) −1.99346 + 0.161591i −0.996731 + 0.0807954i
\(5\) 1.90078 − 1.17773i 0.850053 − 0.526697i
\(6\) −1.25233 − 0.657011i −0.511263 − 0.268224i
\(7\) 2.10021 + 1.60907i 0.793805 + 0.608172i
\(8\) −0.342319 − 2.80764i −0.121028 − 0.992649i
\(9\) −0.500000 − 0.866025i −0.166667 − 0.288675i
\(10\) 1.77288 + 2.61857i 0.560634 + 0.828063i
\(11\) 1.84936 − 3.20318i 0.557603 − 0.965796i −0.440093 − 0.897952i \(-0.645055\pi\)
0.997696 − 0.0678442i \(-0.0216121\pi\)
\(12\) 0.856789 − 1.80718i 0.247334 − 0.521689i
\(13\) − 4.06159i − 1.12648i −0.826292 − 0.563241i \(-0.809554\pi\)
0.826292 − 0.563241i \(-0.190446\pi\)
\(14\) −2.15363 + 3.05972i −0.575581 + 0.817745i
\(15\) 0.0695556 + 2.23499i 0.0179592 + 0.577071i
\(16\) 3.94778 − 0.644250i 0.986944 − 0.161062i
\(17\) 3.58024 − 6.20117i 0.868337 − 1.50400i 0.00464159 − 0.999989i \(-0.498523\pi\)
0.863695 − 0.504014i \(-0.168144\pi\)
\(18\) 1.19515 − 0.756046i 0.281701 − 0.178202i
\(19\) −2.05610 + 1.18709i −0.471701 + 0.272337i −0.716951 − 0.697123i \(-0.754463\pi\)
0.245251 + 0.969460i \(0.421130\pi\)
\(20\) −3.59881 + 2.65491i −0.804719 + 0.593655i
\(21\) −2.44360 + 1.01430i −0.533238 + 0.221338i
\(22\) 4.63202 + 2.43010i 0.987551 + 0.518099i
\(23\) −1.57001 − 2.71934i −0.327370 − 0.567021i 0.654619 − 0.755959i \(-0.272829\pi\)
−0.981989 + 0.188938i \(0.939496\pi\)
\(24\) 2.60264 + 1.10736i 0.531262 + 0.226039i
\(25\) 2.22590 − 4.47720i 0.445181 − 0.895441i
\(26\) 5.73926 − 0.232233i 1.12556 − 0.0455445i
\(27\) 1.00000 0.192450
\(28\) −4.44670 − 2.86825i −0.840347 − 0.542048i
\(29\) − 0.386546i − 0.0717799i −0.999356 − 0.0358899i \(-0.988573\pi\)
0.999356 − 0.0358899i \(-0.0114266\pi\)
\(30\) −3.15419 + 0.226077i −0.575873 + 0.0412759i
\(31\) 0.295740 − 0.512237i 0.0531165 − 0.0920005i −0.838245 − 0.545294i \(-0.816418\pi\)
0.891361 + 0.453294i \(0.149751\pi\)
\(32\) 1.13609 + 5.54160i 0.200834 + 0.979625i
\(33\) 1.84936 + 3.20318i 0.321932 + 0.557603i
\(34\) 8.96731 + 4.70452i 1.53788 + 0.806819i
\(35\) 5.88708 + 0.585008i 0.995099 + 0.0988844i
\(36\) 1.13667 + 1.64559i 0.189445 + 0.274265i
\(37\) −2.87135 − 4.97333i −0.472048 − 0.817611i 0.527441 − 0.849592i \(-0.323152\pi\)
−0.999488 + 0.0319811i \(0.989818\pi\)
\(38\) −1.79499 − 2.83751i −0.291185 − 0.460304i
\(39\) 3.51744 + 2.03080i 0.563241 + 0.325188i
\(40\) −3.95731 − 4.93353i −0.625705 − 0.780060i
\(41\) 12.1918i 1.90403i 0.306045 + 0.952017i \(0.400994\pi\)
−0.306045 + 0.952017i \(0.599006\pi\)
\(42\) −1.57298 − 3.39496i −0.242716 − 0.523853i
\(43\) 1.90138i 0.289958i 0.989435 + 0.144979i \(0.0463114\pi\)
−0.989435 + 0.144979i \(0.953689\pi\)
\(44\) −3.16902 + 6.68426i −0.477748 + 1.00769i
\(45\) −1.97033 − 1.05726i −0.293720 − 0.157606i
\(46\) 3.75281 − 2.37400i 0.553322 − 0.350027i
\(47\) −0.245768 + 0.141894i −0.0358490 + 0.0206974i −0.517817 − 0.855491i \(-0.673255\pi\)
0.481968 + 0.876189i \(0.339922\pi\)
\(48\) −1.41595 + 3.74100i −0.204375 + 0.539967i
\(49\) 1.82177 + 6.75878i 0.260253 + 0.965541i
\(50\) 6.45382 + 2.88933i 0.912707 + 0.408614i
\(51\) 3.58024 + 6.20117i 0.501335 + 0.868337i
\(52\) 0.656316 + 8.09663i 0.0910146 + 1.12280i
\(53\) −0.232955 + 0.403490i −0.0319988 + 0.0554236i −0.881581 − 0.472032i \(-0.843521\pi\)
0.849583 + 0.527456i \(0.176854\pi\)
\(54\) 0.0571777 + 1.41306i 0.00778090 + 0.192293i
\(55\) −0.257267 − 8.26658i −0.0346898 − 1.11467i
\(56\) 3.79875 − 6.44744i 0.507629 − 0.861576i
\(57\) − 2.37418i − 0.314467i
\(58\) 0.546212 − 0.0221018i 0.0717212 − 0.00290211i
\(59\) 4.30554 + 2.48580i 0.560534 + 0.323624i 0.753360 − 0.657609i \(-0.228432\pi\)
−0.192826 + 0.981233i \(0.561765\pi\)
\(60\) −0.499810 − 4.44412i −0.0645251 − 0.573733i
\(61\) −5.38555 − 9.32804i −0.689549 − 1.19433i −0.971984 − 0.235047i \(-0.924475\pi\)
0.282435 − 0.959286i \(-0.408858\pi\)
\(62\) 0.740730 + 0.388609i 0.0940728 + 0.0493534i
\(63\) 0.343393 − 2.62337i 0.0432634 − 0.330514i
\(64\) −7.76564 + 1.92221i −0.970705 + 0.240276i
\(65\) −4.78346 − 7.72018i −0.593315 − 0.957570i
\(66\) −4.42054 + 2.79640i −0.544131 + 0.344213i
\(67\) 5.18143 + 2.99150i 0.633012 + 0.365470i 0.781918 − 0.623382i \(-0.214242\pi\)
−0.148906 + 0.988851i \(0.547575\pi\)
\(68\) −6.13503 + 12.9403i −0.743981 + 1.56924i
\(69\) 3.14002 0.378014
\(70\) −0.490040 + 8.35224i −0.0585710 + 0.998283i
\(71\) 15.5552i 1.84606i 0.384727 + 0.923030i \(0.374296\pi\)
−0.384727 + 0.923030i \(0.625704\pi\)
\(72\) −2.26032 + 1.70027i −0.266382 + 0.200379i
\(73\) −4.16891 + 7.22077i −0.487935 + 0.845127i −0.999904 − 0.0138764i \(-0.995583\pi\)
0.511969 + 0.859004i \(0.328916\pi\)
\(74\) 6.86343 − 4.34175i 0.797857 − 0.504718i
\(75\) 2.76442 + 4.16629i 0.319208 + 0.481082i
\(76\) 3.90693 − 2.69866i 0.448155 − 0.309558i
\(77\) 9.03820 − 3.75161i 1.03000 − 0.427535i
\(78\) −2.66851 + 5.08646i −0.302149 + 0.575929i
\(79\) −6.20106 + 3.58019i −0.697674 + 0.402802i −0.806481 − 0.591261i \(-0.798630\pi\)
0.108806 + 0.994063i \(0.465297\pi\)
\(80\) 6.74509 − 5.87399i 0.754124 − 0.656732i
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) −17.2277 + 0.697097i −1.90248 + 0.0769815i
\(83\) 16.1072 1.76799 0.883996 − 0.467495i \(-0.154843\pi\)
0.883996 + 0.467495i \(0.154843\pi\)
\(84\) 4.70733 − 2.41683i 0.513612 − 0.263698i
\(85\) −0.498052 − 16.0036i −0.0540214 − 1.73583i
\(86\) −2.68676 + 0.108717i −0.289721 + 0.0117232i
\(87\) 0.334759 + 0.193273i 0.0358899 + 0.0207211i
\(88\) −9.62644 − 4.09582i −1.02618 − 0.436616i
\(89\) −9.05695 + 5.22903i −0.960035 + 0.554276i −0.896184 − 0.443683i \(-0.853672\pi\)
−0.0638512 + 0.997959i \(0.520338\pi\)
\(90\) 1.38130 − 2.84464i 0.145602 − 0.299852i
\(91\) 6.53540 − 8.53020i 0.685096 − 0.894208i
\(92\) 3.56917 + 5.16720i 0.372112 + 0.538717i
\(93\) 0.295740 + 0.512237i 0.0306668 + 0.0531165i
\(94\) −0.214557 − 0.339171i −0.0221299 − 0.0349829i
\(95\) −2.51011 + 4.67792i −0.257532 + 0.479944i
\(96\) −5.36721 − 1.78692i −0.547788 − 0.182377i
\(97\) 2.21812 0.225216 0.112608 − 0.993639i \(-0.464080\pi\)
0.112608 + 0.993639i \(0.464080\pi\)
\(98\) −9.44638 + 2.96071i −0.954229 + 0.299077i
\(99\) −3.69872 −0.371735
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.619.25 yes 96
5.4 even 2 840.2.bz.b.619.24 yes 96
7.5 odd 6 840.2.bz.b.19.39 yes 96
8.3 odd 2 inner 840.2.bz.a.619.10 yes 96
35.19 odd 6 inner 840.2.bz.a.19.10 ✓ 96
40.19 odd 2 840.2.bz.b.619.39 yes 96
56.19 even 6 840.2.bz.b.19.24 yes 96
280.19 even 6 inner 840.2.bz.a.19.25 yes 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
840.2.bz.a.19.10 ✓ 96 35.19 odd 6 inner
840.2.bz.a.19.25 yes 96 280.19 even 6 inner
840.2.bz.a.619.10 yes 96 8.3 odd 2 inner
840.2.bz.a.619.25 yes 96 1.1 even 1 trivial
840.2.bz.b.19.24 yes 96 56.19 even 6
840.2.bz.b.19.39 yes 96 7.5 odd 6
840.2.bz.b.619.24 yes 96 5.4 even 2
840.2.bz.b.619.39 yes 96 40.19 odd 2