↑

Properties

Label 790.2.k.a.767.32
Level $790$
Weight $2$
Character 790.767
Analytic conductor $6.308$
Analytic rank $0$
Dimension $160$
Inner twists $4$

Related objects

Downloads

Learn more

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 767.32
Character \(\chi\) \(=\) 790.767
Dual form 790.2.k.a.103.32

$q$-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 \(\left(\mathbb{Z}/790\mathbb{Z}\right)^\times\).

\(n\) \(161\) \(317\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{4}\right)\)

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 −0.904191 − 0.427127i \(-0.859526\pi\)
0.996616 + 0.0821926i \(0.0261923\pi\)
\(4\) −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 0.994488 + 0.104846i \(0.0334351\pi\)
−0.808829 + 0.588044i \(0.799898\pi\)
\(8\) −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 −0.997474 − 0.0710330i \(-0.977370\pi\)
0.560253 + 0.828321i \(0.310704\pi\)
\(12\) −0.437360 + 0.437360i −0.126255 + 0.126255i
\(13\) −1.44385 − 0.386880i −0.400453 − 0.107301i 0.0529708 − 0.998596i \(-0.483131\pi\)
−0.453424 + 0.891295i \(0.649798\pi\)
\(14\) 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 −0.604516 − 0.796593i \(-0.706633\pi\)
0.796593 + 0.604516i \(0.206633\pi\)
\(18\) 1.85080 − 1.85080i 0.436239 − 0.436239i
\(19\) 3.25204 + 1.87757i 0.746070 + 0.430744i 0.824272 − 0.566194i \(-0.191585\pi\)
−0.0782021 + 0.996938i \(0.524918\pi\)
\(20\) −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 0.946248 + 0.323443i \(0.104840\pi\)
−0.657753 + 0.753233i \(0.728493\pi\)
\(24\) 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 −0.925913 − 0.377736i \(-0.876703\pi\)
0.790085 + 0.612997i \(0.210036\pi\)
\(30\) −0.817898 − 1.11529i −0.149327 − 0.203624i
\(31\) −1.99163 + 3.44960i −0.357707 + 0.619566i −0.987577 − 0.157134i \(-0.949775\pi\)
0.629870 + 0.776700i \(0.283108\pi\)
\(32\) 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 −0.115429 − 0.993316i \(-0.536824\pi\)
−0.596622 + 0.802522i \(0.703491\pi\)
\(38\) 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 \(-0.708761\pi\)
0.609826 − 0.792535i \(-0.291239\pi\)
\(42\) 0.303822 − 1.13388i 0.0468808 − 0.174962i
\(43\) 0.130056 − 0.485377i 0.0198334 − 0.0740193i −0.955300 − 0.295639i \(-0.904468\pi\)
0.975133 + 0.221619i \(0.0711342\pi\)
\(44\) 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 −0.679679 − 0.733510i \(-0.737881\pi\)
0.221865 − 0.975078i \(-0.428786\pi\)
\(48\) 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 0.0345272 − 0.999404i \(-0.489007\pi\)
0.529603 + 0.848245i \(0.322341\pi\)
\(54\) −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 0.999957 − 0.00925973i \(-0.00294751\pi\)
−0.507998 + 0.861358i \(0.669614\pi\)
\(60\) −1.28898 + 0.501370i −0.166406 + 0.0647266i
\(61\) − 12.4587i − 1.59517i −0.603208 − 0.797584i \(-0.706111\pi\)
0.603208 − 0.797584i \(-0.293889\pi\)
\(62\) 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 0.433379 − 0.901212i \(-0.357321\pi\)
−0.901212 + 0.433379i \(0.857321\pi\)
\(68\) −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 \(-0.949936\pi\)
0.987657 − 0.156633i \(-0.0500641\pi\)
\(72\) −2.52825 + 0.677441i −0.297957 + 0.0798372i
\(73\) −2.60726 − 9.73042i −0.305157 − 1.13886i −0.932810 − 0.360367i \(-0.882651\pi\)
0.627654 − 0.778492i \(-0.284015\pi\)
\(74\) −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 0.906242 + 0.422759i \(0.138938\pi\)
−0.573449 + 0.819241i \(0.694395\pi\)
\(84\) −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 \(-0.972155\pi\)
0.996176 − 0.0873649i \(-0.0278446\pi\)
\(90\) 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 0.713667 − 0.700486i \(-0.247033\pi\)
−0.700486 + 0.713667i \(0.747033\pi\)
\(98\) −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.767.32 yes 160
5.3 odd 4 inner 790.2.k.a.293.12 yes 160
79.24 odd 6 inner 790.2.k.a.577.12 yes 160
395.103 even 12 inner 790.2.k.a.103.32 ✓ 160
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
790.2.k.a.103.32 ✓ 160 395.103 even 12 inner
790.2.k.a.293.12 yes 160 5.3 odd 4 inner
790.2.k.a.577.12 yes 160 79.24 odd 6 inner
790.2.k.a.767.32 yes 160 1.1 even 1 trivial