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Properties

Label 840.2.bz.b.19.39
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 19.39
Character \(\chi\) \(=\) 840.19
Dual form 840.2.bz.b.619.39

$q$-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+(1.19515 - 0.756046i) q^{2} +(0.500000 + 0.866025i) q^{3} +(0.856789 - 1.80718i) q^{4} +(1.97033 + 1.05726i) 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} +(3.15419 - 0.226077i) q^{10} +(1.84936 + 3.20318i) q^{11} +(1.99346 - 0.161591i) q^{12} +4.06159i q^{13} +(1.29354 - 3.51095i) q^{14} +(0.0695556 + 2.23499i) q^{15} +(-2.53183 - 3.09675i) q^{16} +(-3.58024 - 6.20117i) q^{17} +(0.0571777 + 1.41306i) 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} +(2.26032 - 1.70027i) q^{24} +(2.76442 + 4.16629i) q^{25} +(3.07075 + 4.85423i) q^{26} -1.00000 q^{27} +(-1.10845 - 5.17410i) q^{28} -0.386546i q^{29} +(1.77288 + 2.61857i) q^{30} +(-0.295740 - 0.512237i) q^{31} +(-5.36721 - 1.78692i) q^{32} +(-1.84936 + 3.20318i) q^{33} +(-8.96731 - 4.70452i) q^{34} +(5.83931 - 0.949949i) q^{35} +(1.13667 + 1.64559i) q^{36} +(-2.87135 + 4.97333i) q^{37} +(-3.35485 + 0.135750i) q^{38} +(-3.51744 + 2.03080i) q^{39} +(2.29391 - 5.89389i) q^{40} -12.1918i q^{41} +(3.68734 - 0.635232i) q^{42} +1.90138i q^{43} +(7.37325 - 0.597679i) q^{44} +(-1.90078 + 1.17773i) q^{45} +(0.179539 + 4.43703i) 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} +(7.34004 + 3.47993i) q^{52} +(-0.232955 - 0.403490i) q^{53} +(-1.19515 + 0.756046i) q^{54} +(0.257267 + 8.26658i) q^{55} +(-5.23663 - 5.34581i) q^{56} -2.37418i q^{57} +(-0.292247 - 0.461983i) q^{58} +(4.30554 - 2.48580i) q^{59} +(4.09862 + 1.78921i) 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} +(-4.29414 + 8.00269i) q^{65} +(0.211484 + 5.22650i) q^{66} +(-5.18143 + 2.99150i) q^{67} +(-14.2742 + 1.15707i) q^{68} -3.14002 q^{69} +(6.26068 - 5.55013i) q^{70} +15.5552i q^{71} +(2.60264 + 1.10736i) q^{72} +(4.16891 + 7.22077i) q^{73} +(0.328355 + 8.11478i) q^{74} +(-2.22590 + 4.47720i) 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} +(-1.71448 - 8.77841i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(-9.21754 - 14.5710i) q^{82} -16.1072 q^{83} +(3.92668 - 3.54700i) q^{84} +(-0.498052 - 16.0036i) q^{85} +(1.43753 + 2.27244i) q^{86} +(0.334759 - 0.193273i) q^{87} +(8.36030 - 6.28884i) 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} +(-0.401010 + 0.0162264i) q^{94} +(-2.79614 - 4.51278i) q^{95} +(-1.13609 - 5.54160i) q^{96} -2.21812 q^{97} +(-2.93266 - 9.45513i) q^{98} -3.69872 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.19515 − 0.756046i 0.845102 − 0.534605i
\(3\) 0.500000 + 0.866025i 0.288675 + 0.500000i
\(4\) 0.856789 − 1.80718i 0.428395 − 0.903592i
\(5\) 1.97033 + 1.05726i 0.881159 + 0.472819i
\(6\) 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
\(10\) 3.15419 − 0.226077i 0.997441 − 0.0714920i
\(11\) 1.84936 + 3.20318i 0.557603 + 0.965796i 0.997696 + 0.0678442i \(0.0216121\pi\)
−0.440093 + 0.897952i \(0.645055\pi\)
\(12\) 1.99346 − 0.161591i 0.575463 − 0.0466472i
\(13\) 4.06159i 1.12648i 0.826292 + 0.563241i \(0.190446\pi\)
−0.826292 + 0.563241i \(0.809554\pi\)
\(14\) 1.29354 − 3.51095i 0.345714 − 0.938340i
\(15\) 0.0695556 + 2.23499i 0.0179592 + 0.577071i
\(16\) −2.53183 − 3.09675i −0.632956 − 0.774188i
\(17\) −3.58024 − 6.20117i −0.868337 − 1.50400i −0.863695 − 0.504014i \(-0.831856\pi\)
−0.00464159 − 0.999989i \(-0.501477\pi\)
\(18\) 0.0571777 + 1.41306i 0.0134769 + 0.333061i
\(19\) −2.05610 − 1.18709i −0.471701 − 0.272337i 0.245251 − 0.969460i \(-0.421130\pi\)
−0.716951 + 0.697123i \(0.754463\pi\)
\(20\) 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 −0.981989 − 0.188938i \(-0.939496\pi\)
0.654619 + 0.755959i \(0.272829\pi\)
\(24\) 2.26032 − 1.70027i 0.461387 − 0.347067i
\(25\) 2.76442 + 4.16629i 0.552884 + 0.833258i
\(26\) 3.07075 + 4.85423i 0.602224 + 0.951993i
\(27\) −1.00000 −0.192450
\(28\) −1.10845 − 5.17410i −0.209478 − 0.977813i
\(29\) − 0.386546i − 0.0717799i −0.999356 − 0.0358899i \(-0.988573\pi\)
0.999356 − 0.0358899i \(-0.0114266\pi\)
\(30\) 1.77288 + 2.61857i 0.323682 + 0.478083i
\(31\) −0.295740 − 0.512237i −0.0531165 − 0.0920005i 0.838245 − 0.545294i \(-0.183582\pi\)
−0.891361 + 0.453294i \(0.850249\pi\)
\(32\) −5.36721 − 1.78692i −0.948797 − 0.315886i
\(33\) −1.84936 + 3.20318i −0.321932 + 0.557603i
\(34\) −8.96731 − 4.70452i −1.53788 − 0.806819i
\(35\) 5.83931 − 0.949949i 0.987024 − 0.160571i
\(36\) 1.13667 + 1.64559i 0.189445 + 0.274265i
\(37\) −2.87135 + 4.97333i −0.472048 + 0.817611i −0.999488 − 0.0319811i \(-0.989818\pi\)
0.527441 + 0.849592i \(0.323152\pi\)
\(38\) −3.35485 + 0.135750i −0.544228 + 0.0220216i
\(39\) −3.51744 + 2.03080i −0.563241 + 0.325188i
\(40\) 2.29391 − 5.89389i 0.362699 − 0.931906i
\(41\) − 12.1918i − 1.90403i −0.306045 − 0.952017i \(-0.599006\pi\)
0.306045 − 0.952017i \(-0.400994\pi\)
\(42\) 3.68734 − 0.635232i 0.568969 − 0.0980185i
\(43\) 1.90138i 0.289958i 0.989435 + 0.144979i \(0.0463114\pi\)
−0.989435 + 0.144979i \(0.953689\pi\)
\(44\) 7.37325 − 0.597679i 1.11156 − 0.0901035i
\(45\) −1.90078 + 1.17773i −0.283351 + 0.175566i
\(46\) 0.179539 + 4.43703i 0.0264716 + 0.654204i
\(47\) −0.245768 − 0.141894i −0.0358490 − 0.0206974i 0.481968 − 0.876189i \(-0.339922\pi\)
−0.517817 + 0.855491i \(0.673255\pi\)
\(48\) 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
\(52\) 7.34004 + 3.47993i 1.01788 + 0.482579i
\(53\) −0.232955 − 0.403490i −0.0319988 − 0.0554236i 0.849583 − 0.527456i \(-0.176854\pi\)
−0.881581 + 0.472032i \(0.843521\pi\)
\(54\) −1.19515 + 0.756046i −0.162640 + 0.102885i
\(55\) 0.257267 + 8.26658i 0.0346898 + 1.11467i
\(56\) −5.23663 − 5.34581i −0.699774 − 0.714364i
\(57\) − 2.37418i − 0.314467i
\(58\) −0.292247 − 0.461983i −0.0383739 − 0.0606613i
\(59\) 4.30554 − 2.48580i 0.560534 − 0.323624i −0.192826 − 0.981233i \(-0.561765\pi\)
0.753360 + 0.657609i \(0.228432\pi\)
\(60\) 4.09862 + 1.78921i 0.529130 + 0.230986i
\(61\) 5.38555 − 9.32804i 0.689549 − 1.19433i −0.282435 − 0.959286i \(-0.591142\pi\)
0.971984 − 0.235047i \(-0.0755245\pi\)
\(62\) −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
\(65\) −4.29414 + 8.00269i −0.532623 + 0.992611i
\(66\) 0.211484 + 5.22650i 0.0260319 + 0.643338i
\(67\) −5.18143 + 2.99150i −0.633012 + 0.365470i −0.781918 − 0.623382i \(-0.785758\pi\)
0.148906 + 0.988851i \(0.452425\pi\)
\(68\) −14.2742 + 1.15707i −1.73100 + 0.140315i
\(69\) −3.14002 −0.378014
\(70\) 6.26068 − 5.55013i 0.748294 − 0.663367i
\(71\) 15.5552i 1.84606i 0.384727 + 0.923030i \(0.374296\pi\)
−0.384727 + 0.923030i \(0.625704\pi\)
\(72\) 2.60264 + 1.10736i 0.306724 + 0.130504i
\(73\) 4.16891 + 7.22077i 0.487935 + 0.845127i 0.999904 − 0.0138764i \(-0.00441714\pi\)
−0.511969 + 0.859004i \(0.671084\pi\)
\(74\) 0.328355 + 8.11478i 0.0381705 + 0.943324i
\(75\) −2.22590 + 4.47720i −0.257025 + 0.516983i
\(76\) −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 \(-0.201370\pi\)
−0.108806 + 0.994063i \(0.534703\pi\)
\(80\) −1.71448 − 8.77841i −0.191685 − 0.981457i
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) −9.21754 − 14.5710i −1.01791 − 1.60910i
\(83\) −16.1072 −1.76799 −0.883996 − 0.467495i \(-0.845157\pi\)
−0.883996 + 0.467495i \(0.845157\pi\)
\(84\) 3.92668 − 3.54700i 0.428436 − 0.387009i
\(85\) −0.498052 − 16.0036i −0.0540214 − 1.73583i
\(86\) 1.43753 + 2.27244i 0.155013 + 0.245044i
\(87\) 0.334759 − 0.193273i 0.0358899 − 0.0207211i
\(88\) 8.36030 − 6.28884i 0.891211 − 0.670392i
\(89\) −9.05695 − 5.22903i −0.960035 − 0.554276i −0.0638512 − 0.997959i \(-0.520338\pi\)
−0.896184 + 0.443683i \(0.853672\pi\)
\(90\) −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
\(94\) −0.401010 + 0.0162264i −0.0413610 + 0.00167362i
\(95\) −2.79614 − 4.51278i −0.286878 − 0.463001i
\(96\) −1.13609 − 5.54160i −0.115951 − 0.565587i
\(97\) −2.21812 −0.225216 −0.112608 − 0.993639i \(-0.535920\pi\)
−0.112608 + 0.993639i \(0.535920\pi\)
\(98\) −2.93266 − 9.45513i −0.296243 − 0.955113i
\(99\) −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.b.19.39 yes 96
5.4 even 2 840.2.bz.a.19.10 ✓ 96
7.3 odd 6 840.2.bz.a.619.25 yes 96
8.3 odd 2 inner 840.2.bz.b.19.24 yes 96
35.24 odd 6 inner 840.2.bz.b.619.24 yes 96
40.19 odd 2 840.2.bz.a.19.25 yes 96
56.3 even 6 840.2.bz.a.619.10 yes 96
280.59 even 6 inner 840.2.bz.b.619.39 yes 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
840.2.bz.a.19.10 ✓ 96 5.4 even 2
840.2.bz.a.19.25 yes 96 40.19 odd 2
840.2.bz.a.619.10 yes 96 56.3 even 6
840.2.bz.a.619.25 yes 96 7.3 odd 6
840.2.bz.b.19.24 yes 96 8.3 odd 2 inner
840.2.bz.b.19.39 yes 96 1.1 even 1 trivial
840.2.bz.b.619.24 yes 96 35.24 odd 6 inner
840.2.bz.b.619.39 yes 96 280.59 even 6 inner