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Properties

Label 790.2.j.a.339.19
Level $790$
Weight $2$
Character 790.339
Analytic conductor $6.308$
Analytic rank $0$
Dimension $80$
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] = [790,2,Mod(339,790)] 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("790.339"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(790, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 790 = 2 \cdot 5 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 790.j (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.30818175968\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(40\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 339.19
Character \(\chi\) \(=\) 790.339
Dual form 790.2.j.a.529.19

$q$-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+(-0.866025 + 0.500000i) q^{2} +(2.18581 - 1.26198i) q^{3} +(0.500000 - 0.866025i) q^{4} +(1.44862 - 1.70338i) q^{5} +(-1.26198 + 2.18581i) q^{6} +(1.64823 + 0.951608i) q^{7} +1.00000i q^{8} +(1.68519 - 2.91883i) q^{9} +(-0.402846 + 2.19948i) q^{10} +(-0.133065 + 0.230475i) q^{11} -2.52396i q^{12} +(3.33951 - 1.92807i) q^{13} -1.90322 q^{14} +(1.01677 - 5.55140i) q^{15} +(-0.500000 - 0.866025i) q^{16} -1.08704i q^{17} +3.37038i q^{18} +(-2.75224 + 4.76703i) q^{19} +(-0.750865 - 2.10623i) q^{20} +4.80364 q^{21} -0.266130i q^{22} +(2.17354 + 1.25489i) q^{23} +(1.26198 + 2.18581i) q^{24} +(-0.803028 - 4.93509i) q^{25} +(-1.92807 + 3.33951i) q^{26} -0.934818i q^{27} +(1.64823 - 0.951608i) q^{28} +(2.19569 - 3.80305i) q^{29} +(1.89515 + 5.31604i) q^{30} +(-5.13154 + 8.88809i) q^{31} +(0.866025 + 0.500000i) q^{32} +0.671701i q^{33} +(0.543518 + 0.941401i) q^{34} +(4.00861 - 1.42906i) q^{35} +(-1.68519 - 2.91883i) q^{36} +(-1.02880 + 0.593978i) q^{37} -5.50449i q^{38} +(4.86637 - 8.42880i) q^{39} +(1.70338 + 1.44862i) q^{40} -10.0941 q^{41} +(-4.16008 + 2.40182i) q^{42} +(5.56096 - 3.21062i) q^{43} +(0.133065 + 0.230475i) q^{44} +(-2.53070 - 7.09879i) q^{45} -2.50979 q^{46} +(-4.74421 - 2.73907i) q^{47} +(-2.18581 - 1.26198i) q^{48} +(-1.68888 - 2.92523i) q^{49} +(3.16299 + 3.87240i) q^{50} +(-1.37182 - 2.37606i) q^{51} -3.85614i q^{52} +(5.09621 + 2.94230i) q^{53} +(0.467409 + 0.809576i) q^{54} +(0.199828 + 0.560530i) q^{55} +(-0.951608 + 1.64823i) q^{56} +13.8931i q^{57} +4.39139i q^{58} +(-5.13531 - 8.89463i) q^{59} +(-4.29927 - 3.65625i) q^{60} -3.26920 q^{61} -10.2631i q^{62} +(5.55517 - 3.20728i) q^{63} -1.00000 q^{64} +(1.55343 - 8.48150i) q^{65} +(-0.335851 - 0.581710i) q^{66} +5.10785i q^{67} +(-0.941401 - 0.543518i) q^{68} +6.33461 q^{69} +(-2.75703 + 3.24191i) q^{70} -4.13656 q^{71} +(2.91883 + 1.68519i) q^{72} +(-2.64526 - 1.52724i) q^{73} +(0.593978 - 1.02880i) q^{74} +(-7.98326 - 9.77379i) q^{75} +(2.75224 + 4.76703i) q^{76} +(-0.438644 + 0.253251i) q^{77} +9.73274i q^{78} +(3.82286 + 8.02407i) q^{79} +(-2.19948 - 0.402846i) q^{80} +(3.87584 + 6.71316i) q^{81} +(8.74173 - 5.04704i) q^{82} +(-3.29230 - 1.90081i) q^{83} +(2.40182 - 4.16008i) q^{84} +(-1.85164 - 1.57470i) q^{85} +(-3.21062 + 5.56096i) q^{86} -11.0837i q^{87} +(-0.230475 - 0.133065i) q^{88} +8.41724 q^{89} +(5.74104 + 4.88238i) q^{90} +7.33906 q^{91} +(2.17354 - 1.25489i) q^{92} +25.9036i q^{93} +5.47814 q^{94} +(4.13313 + 11.5937i) q^{95} +2.52396 q^{96} +2.48022i q^{97} +(2.92523 + 1.68888i) q^{98} +(0.448479 + 0.776788i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 80 q + 40 q^{4} + 4 q^{6} + 36 q^{9} - 4 q^{10} - 8 q^{11} + 16 q^{14} - 8 q^{15} - 40 q^{16} - 8 q^{19} - 72 q^{21} - 4 q^{24} + 6 q^{25} - 8 q^{29} - 10 q^{30} - 16 q^{31} - 2 q^{35} - 36 q^{36} + 24 q^{39}+ \cdots + 52 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/790\mathbb{Z}\right)^\times\).

\(n\) \(161\) \(317\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.866025 + 0.500000i −0.612372 + 0.353553i
\(3\) 2.18581 − 1.26198i 1.26198 − 0.728605i 0.288523 − 0.957473i \(-0.406836\pi\)
0.973457 + 0.228868i \(0.0735025\pi\)
\(4\) 0.500000 − 0.866025i 0.250000 − 0.433013i
\(5\) 1.44862 − 1.70338i 0.647840 − 0.761776i
\(6\) −1.26198 + 2.18581i −0.515201 + 0.892355i
\(7\) 1.64823 + 0.951608i 0.622974 + 0.359674i 0.778026 − 0.628232i \(-0.216221\pi\)
−0.155052 + 0.987906i \(0.549555\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 1.68519 − 2.91883i 0.561730 − 0.972944i
\(10\) −0.402846 + 2.19948i −0.127391 + 0.695537i
\(11\) −0.133065 + 0.230475i −0.0401206 + 0.0694909i −0.885388 − 0.464852i \(-0.846108\pi\)
0.845268 + 0.534343i \(0.179441\pi\)
\(12\) − 2.52396i − 0.728605i
\(13\) 3.33951 − 1.92807i 0.926214 − 0.534750i 0.0406019 − 0.999175i \(-0.487072\pi\)
0.885612 + 0.464425i \(0.153739\pi\)
\(14\) −1.90322 −0.508656
\(15\) 1.01677 − 5.55140i 0.262528 − 1.43337i
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) − 1.08704i − 0.263645i −0.991273 − 0.131822i \(-0.957917\pi\)
0.991273 − 0.131822i \(-0.0420829\pi\)
\(18\) 3.37038i 0.794406i
\(19\) −2.75224 + 4.76703i −0.631408 + 1.09363i 0.355856 + 0.934541i \(0.384189\pi\)
−0.987264 + 0.159090i \(0.949144\pi\)
\(20\) −0.750865 − 2.10623i −0.167899 − 0.470967i
\(21\) 4.80364 1.04824
\(22\) − 0.266130i − 0.0567391i
\(23\) 2.17354 + 1.25489i 0.453214 + 0.261663i 0.709187 − 0.705021i \(-0.249062\pi\)
−0.255972 + 0.966684i \(0.582396\pi\)
\(24\) 1.26198 + 2.18581i 0.257601 + 0.446177i
\(25\) −0.803028 − 4.93509i −0.160606 − 0.987019i
\(26\) −1.92807 + 3.33951i −0.378125 + 0.654932i
\(27\) − 0.934818i − 0.179906i
\(28\) 1.64823 − 0.951608i 0.311487 − 0.179837i
\(29\) 2.19569 − 3.80305i 0.407730 − 0.706209i −0.586905 − 0.809656i \(-0.699654\pi\)
0.994635 + 0.103447i \(0.0329871\pi\)
\(30\) 1.89515 + 5.31604i 0.346006 + 0.970572i
\(31\) −5.13154 + 8.88809i −0.921652 + 1.59635i −0.124792 + 0.992183i \(0.539827\pi\)
−0.796859 + 0.604165i \(0.793507\pi\)
\(32\) 0.866025 + 0.500000i 0.153093 + 0.0883883i
\(33\) 0.671701i 0.116928i
\(34\) 0.543518 + 0.941401i 0.0932126 + 0.161449i
\(35\) 4.00861 − 1.42906i 0.677579 − 0.241555i
\(36\) −1.68519 − 2.91883i −0.280865 − 0.486472i
\(37\) −1.02880 + 0.593978i −0.169134 + 0.0976494i −0.582177 − 0.813062i \(-0.697799\pi\)
0.413044 + 0.910711i \(0.364466\pi\)
\(38\) − 5.50449i − 0.892946i
\(39\) 4.86637 − 8.42880i 0.779243 − 1.34969i
\(40\) 1.70338 + 1.44862i 0.269329 + 0.229046i
\(41\) −10.0941 −1.57643 −0.788215 − 0.615400i \(-0.788994\pi\)
−0.788215 + 0.615400i \(0.788994\pi\)
\(42\) −4.16008 + 2.40182i −0.641914 + 0.370609i
\(43\) 5.56096 − 3.21062i 0.848039 − 0.489616i −0.0119496 − 0.999929i \(-0.503804\pi\)
0.859989 + 0.510313i \(0.170470\pi\)
\(44\) 0.133065 + 0.230475i 0.0200603 + 0.0347454i
\(45\) −2.53070 − 7.09879i −0.377254 − 1.05822i
\(46\) −2.50979 −0.370048
\(47\) −4.74421 − 2.73907i −0.692014 − 0.399535i 0.112352 − 0.993668i \(-0.464162\pi\)
−0.804366 + 0.594134i \(0.797495\pi\)
\(48\) −2.18581 − 1.26198i −0.315495 − 0.182151i
\(49\) −1.68888 − 2.92523i −0.241269 − 0.417890i
\(50\) 3.16299 + 3.87240i 0.447314 + 0.547640i
\(51\) −1.37182 − 2.37606i −0.192093 − 0.332715i
\(52\) − 3.85614i − 0.534750i
\(53\) 5.09621 + 2.94230i 0.700018 + 0.404155i 0.807354 − 0.590067i \(-0.200899\pi\)
−0.107336 + 0.994223i \(0.534232\pi\)
\(54\) 0.467409 + 0.809576i 0.0636063 + 0.110169i
\(55\) 0.199828 + 0.560530i 0.0269448 + 0.0755819i
\(56\) −0.951608 + 1.64823i −0.127164 + 0.220255i
\(57\) 13.8931i 1.84019i
\(58\) 4.39139i 0.576617i
\(59\) −5.13531 − 8.89463i −0.668561 − 1.15798i −0.978307 − 0.207162i \(-0.933577\pi\)
0.309746 − 0.950819i \(-0.399756\pi\)
\(60\) −4.29927 − 3.65625i −0.555034 − 0.472020i
\(61\) −3.26920 −0.418578 −0.209289 − 0.977854i \(-0.567115\pi\)
−0.209289 + 0.977854i \(0.567115\pi\)
\(62\) − 10.2631i − 1.30341i
\(63\) 5.55517 − 3.20728i 0.699886 − 0.404079i
\(64\) −1.00000 −0.125000
\(65\) 1.55343 − 8.48150i 0.192679 − 1.05200i
\(66\) −0.335851 − 0.581710i −0.0413403 − 0.0716036i
\(67\) 5.10785i 0.624024i 0.950078 + 0.312012i \(0.101003\pi\)
−0.950078 + 0.312012i \(0.898997\pi\)
\(68\) −0.941401 − 0.543518i −0.114162 − 0.0659112i
\(69\) 6.33461 0.762597
\(70\) −2.75703 + 3.24191i −0.329528 + 0.387482i
\(71\) −4.13656 −0.490920 −0.245460 − 0.969407i \(-0.578939\pi\)
−0.245460 + 0.969407i \(0.578939\pi\)
\(72\) 2.91883 + 1.68519i 0.343988 + 0.198601i
\(73\) −2.64526 − 1.52724i −0.309604 − 0.178750i 0.337145 − 0.941453i \(-0.390539\pi\)
−0.646749 + 0.762703i \(0.723872\pi\)
\(74\) 0.593978 − 1.02880i 0.0690485 − 0.119596i
\(75\) −7.98326 − 9.77379i −0.921827 − 1.12858i
\(76\) 2.75224 + 4.76703i 0.315704 + 0.546816i
\(77\) −0.438644 + 0.253251i −0.0499881 + 0.0288607i
\(78\) 9.73274i 1.10202i
\(79\) 3.82286 + 8.02407i 0.430105 + 0.902779i
\(80\) −2.19948 − 0.402846i −0.245909 − 0.0450396i
\(81\) 3.87584 + 6.71316i 0.430649 + 0.745907i
\(82\) 8.74173 − 5.04704i 0.965362 − 0.557352i
\(83\) −3.29230 − 1.90081i −0.361377 − 0.208641i 0.308308 − 0.951287i \(-0.400237\pi\)
−0.669685 + 0.742646i \(0.733571\pi\)
\(84\) 2.40182 − 4.16008i 0.262060 − 0.453902i
\(85\) −1.85164 − 1.57470i −0.200838 − 0.170800i
\(86\) −3.21062 + 5.56096i −0.346211 + 0.599654i
\(87\) − 11.0837i − 1.18830i
\(88\) −0.230475 − 0.133065i −0.0245687 − 0.0141848i
\(89\) 8.41724 0.892226 0.446113 − 0.894977i \(-0.352808\pi\)
0.446113 + 0.894977i \(0.352808\pi\)
\(90\) 5.74104 + 4.88238i 0.605159 + 0.514648i
\(91\) 7.33906 0.769343
\(92\) 2.17354 − 1.25489i 0.226607 − 0.130832i
\(93\) 25.9036i 2.68608i
\(94\) 5.47814 0.565027
\(95\) 4.13313 + 11.5937i 0.424050 + 1.18949i
\(96\) 2.52396 0.257601
\(97\) 2.48022i 0.251828i 0.992041 + 0.125914i \(0.0401864\pi\)
−0.992041 + 0.125914i \(0.959814\pi\)
\(98\) 2.92523 + 1.68888i 0.295493 + 0.170603i
\(99\) 0.448479 + 0.776788i 0.0450738 + 0.0780702i
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 790.2.j.a.339.19 ✓ 80
5.4 even 2 inner 790.2.j.a.339.22 yes 80
79.55 even 3 inner 790.2.j.a.529.22 yes 80
395.134 even 6 inner 790.2.j.a.529.19 yes 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
790.2.j.a.339.19 ✓ 80 1.1 even 1 trivial
790.2.j.a.339.22 yes 80 5.4 even 2 inner
790.2.j.a.529.19 yes 80 395.134 even 6 inner
790.2.j.a.529.22 yes 80 79.55 even 3 inner