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Properties

Label 790.2.k.a.103.32
Level $790$
Weight $2$
Character 790.103
Analytic conductor $6.308$
Analytic rank $0$
Dimension $160$
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(103,790)] 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("790.103"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(790, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([9, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 790 = 2 \cdot 5 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 790.k (of order \(12\), degree \(4\), 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: \(160\)
Relative dimension: \(40\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 103.32
Character \(\chi\) \(=\) 790.103
Dual form 790.2.k.a.767.32

$q$-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+(0.258819 + 0.965926i) q^{2} +(0.160085 + 0.597444i) q^{3} +(-0.866025 + 0.500000i) q^{4} +(2.04677 - 0.900412i) q^{5} +(-0.535654 + 0.309260i) q^{6} +(0.491209 - 1.83322i) q^{7} +(-0.707107 - 0.707107i) q^{8} +(2.26676 - 1.30872i) q^{9} +(1.39947 + 1.74398i) q^{10} +(-1.45010 - 2.51164i) q^{11} +(-0.437360 - 0.437360i) q^{12} +(-1.44385 + 0.386880i) q^{13} +1.89788 q^{14} +(0.865602 + 1.07869i) q^{15} +(0.500000 - 0.866025i) q^{16} +(0.791953 + 0.791953i) q^{17} +(1.85080 + 1.85080i) q^{18} +(3.25204 - 1.87757i) q^{19} +(-1.32235 + 1.80316i) q^{20} +1.17388 q^{21} +(2.05075 - 2.05075i) q^{22} +(1.38357 - 5.16356i) q^{23} +(0.309260 - 0.535654i) q^{24} +(3.37852 - 3.68587i) q^{25} +(-0.747394 - 1.29452i) q^{26} +(2.45684 + 2.45684i) q^{27} +(0.491209 + 1.83322i) q^{28} +(-0.731456 - 1.26692i) q^{29} +(-0.817898 + 1.11529i) q^{30} +(-1.99163 - 3.44960i) q^{31} +(0.965926 + 0.258819i) q^{32} +(1.26843 - 1.26843i) q^{33} +(-0.559996 + 0.969941i) q^{34} +(-0.645259 - 4.19446i) q^{35} +(-1.30872 + 2.26676i) q^{36} +(-4.33124 + 1.16055i) q^{37} +(2.65528 + 2.65528i) q^{38} +(-0.462278 - 0.800689i) q^{39} +(-2.08397 - 0.810596i) q^{40} +10.1494i q^{41} +(0.303822 + 1.13388i) q^{42} +(0.130056 + 0.485377i) q^{43} +(2.51164 + 1.45010i) q^{44} +(3.46115 - 4.71966i) q^{45} +5.34571 q^{46} +(-3.13862 + 11.7135i) q^{47} +(0.597444 + 0.160085i) q^{48} +(2.94279 + 1.69902i) q^{49} +(4.43470 + 2.30942i) q^{50} +(-0.346368 + 0.599928i) q^{51} +(1.05697 - 1.05697i) q^{52} +(4.10693 + 1.10045i) q^{53} +(-1.73725 + 3.00900i) q^{54} +(-5.22952 - 3.83506i) q^{55} +(-1.64362 + 0.948942i) q^{56} +(1.64235 + 1.64235i) q^{57} +(1.03443 - 1.03443i) q^{58} +(3.77881 - 6.54509i) q^{59} +(-1.28898 - 0.501370i) q^{60} +12.4587i q^{61} +(2.81659 - 2.81659i) q^{62} +(-1.28571 - 4.79832i) q^{63} +1.00000i q^{64} +(-2.60688 + 2.09192i) q^{65} +(1.55350 + 0.896914i) q^{66} +(-3.82938 + 3.82938i) q^{67} +(-1.08183 - 0.289875i) q^{68} +3.30643 q^{69} +(3.88453 - 1.70888i) q^{70} +2.63963i q^{71} +(-2.52825 - 0.677441i) q^{72} +(-2.60726 + 9.73042i) q^{73} +(-2.24201 - 3.88328i) q^{74} +(2.74295 + 1.42842i) q^{75} +(-1.87757 + 3.25204i) q^{76} +(-5.31668 + 1.42460i) q^{77} +(0.653760 - 0.653760i) q^{78} +(5.07840 - 7.29451i) q^{79} +(0.243604 - 2.22276i) q^{80} +(2.85163 - 4.93916i) q^{81} +(-9.80357 + 2.62686i) q^{82} +(3.03189 - 11.3152i) q^{83} +(-1.01661 + 0.586940i) q^{84} +(2.33403 + 0.907860i) q^{85} +(-0.435177 + 0.251250i) q^{86} +(0.639819 - 0.639819i) q^{87} +(-0.750625 + 2.80137i) q^{88} +1.64840i q^{89} +(5.45465 + 2.12168i) q^{90} +2.83693i q^{91} +(1.38357 + 5.16356i) q^{92} +(1.74212 - 1.74212i) q^{93} -12.1267 q^{94} +(4.96559 - 6.77113i) q^{95} +0.618520i q^{96} +(0.129818 - 0.129818i) q^{97} +(-0.879477 + 3.28225i) q^{98} +(-6.57405 - 3.79553i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 160 q - 12 q^{7} - 8 q^{10} - 16 q^{11} + 80 q^{16} - 32 q^{18} + 48 q^{21} - 8 q^{22} - 12 q^{28} - 24 q^{31} - 36 q^{35} + 72 q^{36} - 36 q^{37} + 8 q^{38} + 20 q^{42} - 48 q^{43} - 4 q^{45} + 16 q^{46}+ \cdots + 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(161\) \(317\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.258819 + 0.965926i 0.183013 + 0.683013i
\(3\) 0.160085 + 0.597444i 0.0924250 + 0.344935i 0.996616 − 0.0821926i \(-0.0261923\pi\)
−0.904191 + 0.427127i \(0.859526\pi\)
\(4\) −0.866025 + 0.500000i −0.433013 + 0.250000i
\(5\) 2.04677 − 0.900412i 0.915342 − 0.402676i
\(6\) −0.535654 + 0.309260i −0.218680 + 0.126255i
\(7\) 0.491209 − 1.83322i 0.185659 − 0.692890i −0.808829 − 0.588044i \(-0.799898\pi\)
0.994488 − 0.104846i \(-0.0334351\pi\)
\(8\) −0.707107 − 0.707107i −0.250000 − 0.250000i
\(9\) 2.26676 − 1.30872i 0.755588 − 0.436239i
\(10\) 1.39947 + 1.74398i 0.442552 + 0.551496i
\(11\) −1.45010 − 2.51164i −0.437221 − 0.757288i 0.560253 − 0.828321i \(-0.310704\pi\)
−0.997474 + 0.0710330i \(0.977370\pi\)
\(12\) −0.437360 − 0.437360i −0.126255 − 0.126255i
\(13\) −1.44385 + 0.386880i −0.400453 + 0.107301i −0.453424 − 0.891295i \(-0.649798\pi\)
0.0529708 + 0.998596i \(0.483131\pi\)
\(14\) 1.89788 0.507231
\(15\) 0.865602 + 1.07869i 0.223498 + 0.278516i
\(16\) 0.500000 − 0.866025i 0.125000 − 0.216506i
\(17\) 0.791953 + 0.791953i 0.192077 + 0.192077i 0.796593 − 0.604516i \(-0.206633\pi\)
−0.604516 + 0.796593i \(0.706633\pi\)
\(18\) 1.85080 + 1.85080i 0.436239 + 0.436239i
\(19\) 3.25204 − 1.87757i 0.746070 − 0.430744i −0.0782021 − 0.996938i \(-0.524918\pi\)
0.824272 + 0.566194i \(0.191585\pi\)
\(20\) −1.32235 + 1.80316i −0.295686 + 0.403200i
\(21\) 1.17388 0.256161
\(22\) 2.05075 − 2.05075i 0.437221 − 0.437221i
\(23\) 1.38357 − 5.16356i 0.288494 − 1.07668i −0.657753 − 0.753233i \(-0.728493\pi\)
0.946248 − 0.323443i \(-0.104840\pi\)
\(24\) 0.309260 − 0.535654i 0.0631274 − 0.109340i
\(25\) 3.37852 − 3.68587i 0.675703 − 0.737174i
\(26\) −0.747394 − 1.29452i −0.146576 − 0.253877i
\(27\) 2.45684 + 2.45684i 0.472819 + 0.472819i
\(28\) 0.491209 + 1.83322i 0.0928297 + 0.346445i
\(29\) −0.731456 − 1.26692i −0.135828 − 0.235261i 0.790085 − 0.612997i \(-0.210036\pi\)
−0.925913 + 0.377736i \(0.876703\pi\)
\(30\) −0.817898 + 1.11529i −0.149327 + 0.203624i
\(31\) −1.99163 − 3.44960i −0.357707 − 0.619566i 0.629870 − 0.776700i \(-0.283108\pi\)
−0.987577 + 0.157134i \(0.949775\pi\)
\(32\) 0.965926 + 0.258819i 0.170753 + 0.0457532i
\(33\) 1.26843 − 1.26843i 0.220805 − 0.220805i
\(34\) −0.559996 + 0.969941i −0.0960384 + 0.166343i
\(35\) −0.645259 − 4.19446i −0.109069 − 0.708992i
\(36\) −1.30872 + 2.26676i −0.218119 + 0.377794i
\(37\) −4.33124 + 1.16055i −0.712052 + 0.190794i −0.596622 − 0.802522i \(-0.703491\pi\)
−0.115429 + 0.993316i \(0.536824\pi\)
\(38\) 2.65528 + 2.65528i 0.430744 + 0.430744i
\(39\) −0.462278 − 0.800689i −0.0740238 − 0.128213i
\(40\) −2.08397 − 0.810596i −0.329505 − 0.128166i
\(41\) 10.1494i 1.58507i 0.609826 + 0.792535i \(0.291239\pi\)
−0.609826 + 0.792535i \(0.708761\pi\)
\(42\) 0.303822 + 1.13388i 0.0468808 + 0.174962i
\(43\) 0.130056 + 0.485377i 0.0198334 + 0.0740193i 0.975133 − 0.221619i \(-0.0711342\pi\)
−0.955300 + 0.295639i \(0.904468\pi\)
\(44\) 2.51164 + 1.45010i 0.378644 + 0.218610i
\(45\) 3.46115 − 4.71966i 0.515958 − 0.703565i
\(46\) 5.34571 0.788182
\(47\) −3.13862 + 11.7135i −0.457815 + 1.70859i 0.221865 + 0.975078i \(0.428786\pi\)
−0.679679 + 0.733510i \(0.737881\pi\)
\(48\) 0.597444 + 0.160085i 0.0862337 + 0.0231062i
\(49\) 2.94279 + 1.69902i 0.420398 + 0.242717i
\(50\) 4.43470 + 2.30942i 0.627161 + 0.326602i
\(51\) −0.346368 + 0.599928i −0.0485013 + 0.0840067i
\(52\) 1.05697 − 1.05697i 0.146576 − 0.146576i
\(53\) 4.10693 + 1.10045i 0.564130 + 0.151158i 0.529603 − 0.848245i \(-0.322341\pi\)
0.0345272 + 0.999404i \(0.489007\pi\)
\(54\) −1.73725 + 3.00900i −0.236409 + 0.409473i
\(55\) −5.22952 − 3.83506i −0.705149 − 0.517120i
\(56\) −1.64362 + 0.948942i −0.219637 + 0.126808i
\(57\) 1.64235 + 1.64235i 0.217534 + 0.217534i
\(58\) 1.03443 − 1.03443i 0.135828 − 0.135828i
\(59\) 3.77881 − 6.54509i 0.491959 − 0.852099i −0.507998 − 0.861358i \(-0.669614\pi\)
0.999957 + 0.00925973i \(0.00294751\pi\)
\(60\) −1.28898 − 0.501370i −0.166406 − 0.0647266i
\(61\) 12.4587i 1.59517i 0.603208 + 0.797584i \(0.293889\pi\)
−0.603208 + 0.797584i \(0.706111\pi\)
\(62\) 2.81659 − 2.81659i 0.357707 − 0.357707i
\(63\) −1.28571 − 4.79832i −0.161984 − 0.604531i
\(64\) 1.00000i 0.125000i
\(65\) −2.60688 + 2.09192i −0.323344 + 0.259470i
\(66\) 1.55350 + 0.896914i 0.191223 + 0.110402i
\(67\) −3.82938 + 3.82938i −0.467833 + 0.467833i −0.901212 − 0.433379i \(-0.857321\pi\)
0.433379 + 0.901212i \(0.357321\pi\)
\(68\) −1.08183 − 0.289875i −0.131191 − 0.0351525i
\(69\) 3.30643 0.398047
\(70\) 3.88453 − 1.70888i 0.464290 − 0.204250i
\(71\) 2.63963i 0.313267i 0.987657 + 0.156633i \(0.0500641\pi\)
−0.987657 + 0.156633i \(0.949936\pi\)
\(72\) −2.52825 − 0.677441i −0.297957 − 0.0798372i
\(73\) −2.60726 + 9.73042i −0.305157 + 1.13886i 0.627654 + 0.778492i \(0.284015\pi\)
−0.932810 + 0.360367i \(0.882651\pi\)
\(74\) −2.24201 − 3.88328i −0.260629 − 0.451423i
\(75\) 2.74295 + 1.42842i 0.316729 + 0.164940i
\(76\) −1.87757 + 3.25204i −0.215372 + 0.373035i
\(77\) −5.31668 + 1.42460i −0.605892 + 0.162348i
\(78\) 0.653760 − 0.653760i 0.0740238 − 0.0740238i
\(79\) 5.07840 − 7.29451i 0.571365 − 0.820696i
\(80\) 0.243604 − 2.22276i 0.0272358 − 0.248512i
\(81\) 2.85163 − 4.93916i 0.316847 − 0.548796i
\(82\) −9.80357 + 2.62686i −1.08262 + 0.290088i
\(83\) 3.03189 − 11.3152i 0.332793 − 1.24200i −0.573449 − 0.819241i \(-0.694395\pi\)
0.906242 − 0.422759i \(-0.138938\pi\)
\(84\) −1.01661 + 0.586940i −0.110921 + 0.0640404i
\(85\) 2.33403 + 0.907860i 0.253161 + 0.0984713i
\(86\) −0.435177 + 0.251250i −0.0469264 + 0.0270930i
\(87\) 0.639819 − 0.639819i 0.0685958 − 0.0685958i
\(88\) −0.750625 + 2.80137i −0.0800169 + 0.298627i
\(89\) 1.64840i 0.174730i 0.996176 + 0.0873649i \(0.0278446\pi\)
−0.996176 + 0.0873649i \(0.972155\pi\)
\(90\) 5.45465 + 2.12168i 0.574971 + 0.223645i
\(91\) 2.83693i 0.297392i
\(92\) 1.38357 + 5.16356i 0.144247 + 0.538338i
\(93\) 1.74212 − 1.74212i 0.180649 − 0.180649i
\(94\) −12.1267 −1.25077
\(95\) 4.96559 − 6.77113i 0.509459 − 0.694703i
\(96\) 0.618520i 0.0631274i
\(97\) 0.129818 − 0.129818i 0.0131810 − 0.0131810i −0.700486 − 0.713667i \(-0.747033\pi\)
0.713667 + 0.700486i \(0.247033\pi\)
\(98\) −0.879477 + 3.28225i −0.0888405 + 0.331557i
\(99\) −6.57405 − 3.79553i −0.660717 − 0.381465i
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.k.a.103.32 ✓ 160
5.2 odd 4 inner 790.2.k.a.577.12 yes 160
79.56 odd 6 inner 790.2.k.a.293.12 yes 160
395.372 even 12 inner 790.2.k.a.767.32 yes 160
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
790.2.k.a.103.32 ✓ 160 1.1 even 1 trivial
790.2.k.a.293.12 yes 160 79.56 odd 6 inner
790.2.k.a.577.12 yes 160 5.2 odd 4 inner
790.2.k.a.767.32 yes 160 395.372 even 12 inner