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Properties

Label 790.2.j.a.339.20
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 339.20
Character \(\chi\) \(=\) 790.339
Dual form 790.2.j.a.529.20

$q$-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} +(2.71753 - 1.56897i) q^{3} +(0.500000 - 0.866025i) q^{4} +(-1.57854 + 1.58374i) q^{5} +(-1.56897 + 2.71753i) q^{6} +(-3.69254 - 2.13189i) q^{7} +1.00000i q^{8} +(3.42333 - 5.92938i) q^{9} +(0.575184 - 2.16082i) q^{10} +(1.16056 - 2.01015i) q^{11} -3.13794i q^{12} +(-1.58448 + 0.914802i) q^{13} +4.26378 q^{14} +(-1.80489 + 6.78054i) q^{15} +(-0.500000 - 0.866025i) q^{16} -5.61080i q^{17} +6.84666i q^{18} +(-2.16201 + 3.74471i) q^{19} +(0.582289 + 2.15892i) q^{20} -13.3795 q^{21} +2.32113i q^{22} +(-4.07245 - 2.35123i) q^{23} +(1.56897 + 2.71753i) q^{24} +(-0.0164471 - 4.99997i) q^{25} +(0.914802 - 1.58448i) q^{26} -12.0706i q^{27} +(-3.69254 + 2.13189i) q^{28} +(1.31717 - 2.28140i) q^{29} +(-1.82719 - 6.77456i) q^{30} +(-0.475051 + 0.822813i) q^{31} +(0.866025 + 0.500000i) q^{32} -7.28355i q^{33} +(2.80540 + 4.85910i) q^{34} +(9.20517 - 2.48275i) q^{35} +(-3.42333 - 5.92938i) q^{36} +(3.66916 - 2.11839i) q^{37} -4.32402i q^{38} +(-2.87059 + 4.97201i) q^{39} +(-1.58374 - 1.57854i) q^{40} -4.10548 q^{41} +(11.5870 - 6.68975i) q^{42} +(7.64715 - 4.41508i) q^{43} +(-1.16056 - 2.01015i) q^{44} +(3.98673 + 14.7814i) q^{45} +4.70246 q^{46} +(-7.35293 - 4.24522i) q^{47} +(-2.71753 - 1.56897i) q^{48} +(5.58992 + 9.68203i) q^{49} +(2.51423 + 4.32188i) q^{50} +(-8.80318 - 15.2476i) q^{51} +1.82960i q^{52} +(-1.50156 - 0.866926i) q^{53} +(6.03530 + 10.4534i) q^{54} +(1.35157 + 5.01113i) q^{55} +(2.13189 - 3.69254i) q^{56} +13.5685i q^{57} +2.63433i q^{58} +(2.50774 + 4.34354i) q^{59} +(4.96967 + 4.95335i) q^{60} -11.0065 q^{61} -0.950102i q^{62} +(-25.2816 + 14.5963i) q^{63} -1.00000 q^{64} +(1.05236 - 3.95345i) q^{65} +(3.64178 + 6.30774i) q^{66} +3.05436i q^{67} +(-4.85910 - 2.80540i) q^{68} -14.7560 q^{69} +(-6.73054 + 6.75271i) q^{70} +8.32703 q^{71} +(5.92938 + 3.42333i) q^{72} +(13.0848 + 7.55452i) q^{73} +(-2.11839 + 3.66916i) q^{74} +(-7.88950 - 13.5618i) q^{75} +(2.16201 + 3.74471i) q^{76} +(-8.57086 + 4.94839i) q^{77} -5.74119i q^{78} +(6.55718 - 6.00029i) q^{79} +(2.16082 + 0.575184i) q^{80} +(-8.66840 - 15.0141i) q^{81} +(3.55545 - 2.05274i) q^{82} +(-8.32349 - 4.80557i) q^{83} +(-6.68975 + 11.5870i) q^{84} +(8.88604 + 8.85686i) q^{85} +(-4.41508 + 7.64715i) q^{86} -8.26638i q^{87} +(2.01015 + 1.16056i) q^{88} +17.3118 q^{89} +(-10.8433 - 10.8077i) q^{90} +7.80104 q^{91} +(-4.07245 + 2.35123i) q^{92} +2.98136i q^{93} +8.49043 q^{94} +(-2.51782 - 9.33521i) q^{95} +3.13794 q^{96} +17.5577i q^{97} +(-9.68203 - 5.58992i) q^{98} +(-7.94598 - 13.7628i) 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
\(3\) 2.71753 − 1.56897i 1.56897 − 0.905845i 0.572680 − 0.819779i \(-0.305904\pi\)
0.996289 − 0.0860661i \(-0.0274296\pi\)
\(4\) 0.500000 − 0.866025i 0.250000 − 0.433013i
\(5\) −1.57854 + 1.58374i −0.705943 + 0.708269i
\(6\) −1.56897 + 2.71753i −0.640529 + 1.10943i
\(7\) −3.69254 − 2.13189i −1.39565 − 0.805779i −0.401717 − 0.915764i \(-0.631587\pi\)
−0.993933 + 0.109984i \(0.964920\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 3.42333 − 5.92938i 1.14111 − 1.97646i
\(10\) 0.575184 − 2.16082i 0.181889 − 0.683313i
\(11\) 1.16056 − 2.01015i 0.349923 − 0.606084i −0.636313 − 0.771431i \(-0.719541\pi\)
0.986235 + 0.165347i \(0.0528745\pi\)
\(12\) − 3.13794i − 0.905845i
\(13\) −1.58448 + 0.914802i −0.439457 + 0.253720i −0.703367 − 0.710827i \(-0.748321\pi\)
0.263910 + 0.964547i \(0.414988\pi\)
\(14\) 4.26378 1.13954
\(15\) −1.80489 + 6.78054i −0.466021 + 1.75073i
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) − 5.61080i − 1.36082i −0.732832 − 0.680410i \(-0.761802\pi\)
0.732832 − 0.680410i \(-0.238198\pi\)
\(18\) 6.84666i 1.61377i
\(19\) −2.16201 + 3.74471i −0.495999 + 0.859095i −0.999989 − 0.00461428i \(-0.998531\pi\)
0.503991 + 0.863709i \(0.331865\pi\)
\(20\) 0.582289 + 2.15892i 0.130204 + 0.482749i
\(21\) −13.3795 −2.91964
\(22\) 2.32113i 0.494866i
\(23\) −4.07245 − 2.35123i −0.849165 − 0.490266i 0.0112039 − 0.999937i \(-0.496434\pi\)
−0.860369 + 0.509672i \(0.829767\pi\)
\(24\) 1.56897 + 2.71753i 0.320265 + 0.554715i
\(25\) −0.0164471 − 4.99997i −0.00328941 − 0.999995i
\(26\) 0.914802 − 1.58448i 0.179407 − 0.310743i
\(27\) − 12.0706i − 2.32299i
\(28\) −3.69254 + 2.13189i −0.697825 + 0.402890i
\(29\) 1.31717 − 2.28140i 0.244592 − 0.423645i −0.717425 − 0.696636i \(-0.754679\pi\)
0.962017 + 0.272990i \(0.0880127\pi\)
\(30\) −1.82719 − 6.77456i −0.333597 − 1.23686i
\(31\) −0.475051 + 0.822813i −0.0853217 + 0.147781i −0.905528 − 0.424286i \(-0.860525\pi\)
0.820207 + 0.572067i \(0.193858\pi\)
\(32\) 0.866025 + 0.500000i 0.153093 + 0.0883883i
\(33\) − 7.28355i − 1.26790i
\(34\) 2.80540 + 4.85910i 0.481122 + 0.833329i
\(35\) 9.20517 − 2.48275i 1.55596 − 0.419662i
\(36\) −3.42333 − 5.92938i −0.570555 − 0.988231i
\(37\) 3.66916 − 2.11839i 0.603207 − 0.348261i −0.167095 − 0.985941i \(-0.553439\pi\)
0.770302 + 0.637679i \(0.220105\pi\)
\(38\) − 4.32402i − 0.701448i
\(39\) −2.87059 + 4.97201i −0.459663 + 0.796159i
\(40\) −1.58374 − 1.57854i −0.250411 − 0.249588i
\(41\) −4.10548 −0.641168 −0.320584 − 0.947220i \(-0.603879\pi\)
−0.320584 + 0.947220i \(0.603879\pi\)
\(42\) 11.5870 − 6.68975i 1.78791 − 1.03225i
\(43\) 7.64715 − 4.41508i 1.16618 − 0.673294i 0.213402 − 0.976965i \(-0.431546\pi\)
0.952777 + 0.303671i \(0.0982123\pi\)
\(44\) −1.16056 − 2.01015i −0.174961 − 0.303042i
\(45\) 3.98673 + 14.7814i 0.594307 + 2.20348i
\(46\) 4.70246 0.693340
\(47\) −7.35293 − 4.24522i −1.07254 − 0.619229i −0.143662 − 0.989627i \(-0.545888\pi\)
−0.928873 + 0.370398i \(0.879221\pi\)
\(48\) −2.71753 − 1.56897i −0.392242 − 0.226461i
\(49\) 5.58992 + 9.68203i 0.798560 + 1.38315i
\(50\) 2.51423 + 4.32188i 0.355566 + 0.611206i
\(51\) −8.80318 − 15.2476i −1.23269 − 2.13509i
\(52\) 1.82960i 0.253720i
\(53\) −1.50156 − 0.866926i −0.206255 − 0.119081i 0.393315 − 0.919404i \(-0.371328\pi\)
−0.599570 + 0.800322i \(0.704662\pi\)
\(54\) 6.03530 + 10.4534i 0.821300 + 1.42253i
\(55\) 1.35157 + 5.01113i 0.182245 + 0.675700i
\(56\) 2.13189 − 3.69254i 0.284886 − 0.493437i
\(57\) 13.5685i 1.79719i
\(58\) 2.63433i 0.345905i
\(59\) 2.50774 + 4.34354i 0.326480 + 0.565480i 0.981811 − 0.189862i \(-0.0608040\pi\)
−0.655331 + 0.755342i \(0.727471\pi\)
\(60\) 4.96967 + 4.95335i 0.641582 + 0.639475i
\(61\) −11.0065 −1.40924 −0.704618 − 0.709587i \(-0.748882\pi\)
−0.704618 + 0.709587i \(0.748882\pi\)
\(62\) − 0.950102i − 0.120663i
\(63\) −25.2816 + 14.5963i −3.18518 + 1.83897i
\(64\) −1.00000 −0.125000
\(65\) 1.05236 − 3.95345i 0.130529 − 0.490366i
\(66\) 3.64178 + 6.30774i 0.448272 + 0.776429i
\(67\) 3.05436i 0.373149i 0.982441 + 0.186575i \(0.0597386\pi\)
−0.982441 + 0.186575i \(0.940261\pi\)
\(68\) −4.85910 − 2.80540i −0.589252 − 0.340205i
\(69\) −14.7560 −1.77642
\(70\) −6.73054 + 6.75271i −0.804453 + 0.807103i
\(71\) 8.32703 0.988236 0.494118 − 0.869395i \(-0.335491\pi\)
0.494118 + 0.869395i \(0.335491\pi\)
\(72\) 5.92938 + 3.42333i 0.698784 + 0.403443i
\(73\) 13.0848 + 7.55452i 1.53146 + 0.884190i 0.999295 + 0.0375508i \(0.0119556\pi\)
0.532167 + 0.846639i \(0.321378\pi\)
\(74\) −2.11839 + 3.66916i −0.246258 + 0.426531i
\(75\) −7.88950 − 13.5618i −0.911001 − 1.56598i
\(76\) 2.16201 + 3.74471i 0.247999 + 0.429547i
\(77\) −8.57086 + 4.94839i −0.976740 + 0.563921i
\(78\) − 5.74119i − 0.650061i
\(79\) 6.55718 − 6.00029i 0.737740 − 0.675085i
\(80\) 2.16082 + 0.575184i 0.241588 + 0.0643075i
\(81\) −8.66840 − 15.0141i −0.963155 − 1.66823i
\(82\) 3.55545 − 2.05274i 0.392634 − 0.226687i
\(83\) −8.32349 − 4.80557i −0.913621 − 0.527479i −0.0320267 − 0.999487i \(-0.510196\pi\)
−0.881594 + 0.472008i \(0.843529\pi\)
\(84\) −6.68975 + 11.5870i −0.729911 + 1.26424i
\(85\) 8.88604 + 8.85686i 0.963826 + 0.960661i
\(86\) −4.41508 + 7.64715i −0.476091 + 0.824613i
\(87\) − 8.26638i − 0.886249i
\(88\) 2.01015 + 1.16056i 0.214283 + 0.123716i
\(89\) 17.3118 1.83505 0.917525 − 0.397678i \(-0.130184\pi\)
0.917525 + 0.397678i \(0.130184\pi\)
\(90\) −10.8433 − 10.8077i −1.14299 − 1.13923i
\(91\) 7.80104 0.817771
\(92\) −4.07245 + 2.35123i −0.424583 + 0.245133i
\(93\) 2.98136i 0.309153i
\(94\) 8.49043 0.875721
\(95\) −2.51782 − 9.33521i −0.258323 − 0.957772i
\(96\) 3.13794 0.320265
\(97\) 17.5577i 1.78272i 0.453298 + 0.891359i \(0.350247\pi\)
−0.453298 + 0.891359i \(0.649753\pi\)
\(98\) −9.68203 − 5.58992i −0.978033 − 0.564668i
\(99\) −7.94598 − 13.7628i −0.798601 − 1.38322i
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.339.20 ✓ 80
5.4 even 2 inner 790.2.j.a.339.21 yes 80
79.55 even 3 inner 790.2.j.a.529.21 yes 80
395.134 even 6 inner 790.2.j.a.529.20 yes 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
790.2.j.a.339.20 ✓ 80 1.1 even 1 trivial
790.2.j.a.339.21 yes 80 5.4 even 2 inner
790.2.j.a.529.20 yes 80 395.134 even 6 inner
790.2.j.a.529.21 yes 80 79.55 even 3 inner