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Properties

Label 790.2.k.a.577.12
Level $790$
Weight $2$
Character 790.577
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 577.12
Character \(\chi\) \(=\) 790.577
Dual form 790.2.k.a.293.12

$q$-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+(-0.965926 + 0.258819i) q^{2} +(0.597444 - 0.160085i) q^{3} +(0.866025 - 0.500000i) q^{4} +(-1.80316 + 1.32235i) q^{5} +(-0.535654 + 0.309260i) q^{6} +(1.83322 + 0.491209i) 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} +(0.386880 + 1.44385i) 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} +(-0.900412 + 2.04677i) q^{20} +1.17388 q^{21} +(2.05075 + 2.05075i) q^{22} +(-5.16356 - 1.38357i) q^{23} +(-0.309260 + 0.535654i) q^{24} +(1.50280 - 4.76882i) q^{25} +(-0.747394 - 1.29452i) q^{26} +(-2.45684 + 2.45684i) q^{27} +(1.83322 - 0.491209i) q^{28} +(0.731456 + 1.26692i) q^{29} +(0.556923 - 1.26597i) q^{30} +(-1.99163 - 3.44960i) q^{31} +(-0.258819 + 0.965926i) q^{32} +(-1.26843 - 1.26843i) q^{33} +(0.559996 - 0.969941i) q^{34} +(-3.95513 + 1.53842i) q^{35} +(-1.30872 + 2.26676i) q^{36} +(-1.16055 - 4.33124i) q^{37} +(2.65528 - 2.65528i) q^{38} +(0.462278 + 0.800689i) q^{39} +(0.339989 - 2.21007i) q^{40} +10.1494i q^{41} +(-1.13388 + 0.303822i) q^{42} +(0.485377 - 0.130056i) q^{43} +(-2.51164 - 1.45010i) q^{44} +(2.35677 - 5.35728i) q^{45} +5.34571 q^{46} +(-11.7135 - 3.13862i) q^{47} +(0.160085 - 0.597444i) q^{48} +(-2.94279 - 1.69902i) q^{49} +(-0.217330 + 4.99527i) q^{50} +(-0.346368 + 0.599928i) q^{51} +(1.05697 + 1.05697i) q^{52} +(1.10045 - 4.10693i) q^{53} +(1.73725 - 3.00900i) q^{54} +(5.93602 + 2.61137i) 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} +(-0.210290 + 1.36697i) q^{60} +12.4587i q^{61} +(2.81659 + 2.81659i) q^{62} +(-4.79832 + 1.28571i) q^{63} -1.00000i q^{64} +(-2.60688 - 2.09192i) q^{65} +(1.55350 + 0.896914i) q^{66} +(-3.82938 - 3.82938i) q^{67} +(-0.289875 + 1.08183i) q^{68} -3.30643 q^{69} +(3.42220 - 2.50966i) q^{70} +2.63963i q^{71} +(0.677441 - 2.52825i) q^{72} +(9.73042 + 2.60726i) q^{73} +(2.24201 + 3.88328i) q^{74} +(0.134423 - 3.08968i) q^{75} +(-1.87757 + 3.25204i) q^{76} +(-1.42460 - 5.31668i) 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} +(-2.62686 - 9.80357i) q^{82} +(-11.3152 - 3.03189i) q^{83} +(1.01661 - 0.586940i) q^{84} +(0.380784 - 2.47526i) q^{85} +(-0.435177 + 0.251250i) q^{86} +(0.639819 + 0.639819i) q^{87} +(2.80137 + 0.750625i) q^{88} -1.64840i q^{89} +(-0.889897 + 5.78471i) q^{90} +2.83693i q^{91} +(-5.16356 + 1.38357i) q^{92} +(-1.74212 - 1.74212i) q^{93} +12.1267 q^{94} +(3.38117 - 7.68589i) q^{95} +0.618520i q^{96} +(0.129818 + 0.129818i) q^{97} +(3.28225 + 0.879477i) 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 \(\left(\mathbb{Z}/790\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.965926 + 0.258819i −0.683013 + 0.183013i
\(3\) 0.597444 − 0.160085i 0.344935 − 0.0924250i −0.0821926 − 0.996616i \(-0.526192\pi\)
0.427127 + 0.904191i \(0.359526\pi\)
\(4\) 0.866025 − 0.500000i 0.433013 − 0.250000i
\(5\) −1.80316 + 1.32235i −0.806399 + 0.591372i
\(6\) −0.535654 + 0.309260i −0.218680 + 0.126255i
\(7\) 1.83322 + 0.491209i 0.692890 + 0.185659i 0.588044 − 0.808829i \(-0.299898\pi\)
0.104846 + 0.994488i \(0.466565\pi\)
\(8\) −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
\(13\) 0.386880 + 1.44385i 0.107301 + 0.400453i 0.998596 − 0.0529708i \(-0.0168690\pi\)
−0.891295 + 0.453424i \(0.850202\pi\)
\(14\) −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 \(-0.793367\pi\)
0.604516 + 0.796593i \(0.293367\pi\)
\(18\) 1.85080 − 1.85080i 0.436239 − 0.436239i
\(19\) −3.25204 + 1.87757i −0.746070 + 0.430744i −0.824272 − 0.566194i \(-0.808415\pi\)
0.0782021 + 0.996938i \(0.475082\pi\)
\(20\) −0.900412 + 2.04677i −0.201338 + 0.457671i
\(21\) 1.17388 0.256161
\(22\) 2.05075 + 2.05075i 0.437221 + 0.437221i
\(23\) −5.16356 − 1.38357i −1.07668 − 0.288494i −0.323443 − 0.946248i \(-0.604840\pi\)
−0.753233 + 0.657753i \(0.771507\pi\)
\(24\) −0.309260 + 0.535654i −0.0631274 + 0.109340i
\(25\) 1.50280 − 4.76882i 0.300559 − 0.953763i
\(26\) −0.747394 − 1.29452i −0.146576 − 0.253877i
\(27\) −2.45684 + 2.45684i −0.472819 + 0.472819i
\(28\) 1.83322 − 0.491209i 0.346445 − 0.0928297i
\(29\) 0.731456 + 1.26692i 0.135828 + 0.235261i 0.925913 − 0.377736i \(-0.123297\pi\)
−0.790085 + 0.612997i \(0.789964\pi\)
\(30\) 0.556923 − 1.26597i 0.101680 − 0.231133i
\(31\) −1.99163 − 3.44960i −0.357707 − 0.619566i 0.629870 − 0.776700i \(-0.283108\pi\)
−0.987577 + 0.157134i \(0.949775\pi\)
\(32\) −0.258819 + 0.965926i −0.0457532 + 0.170753i
\(33\) −1.26843 − 1.26843i −0.220805 − 0.220805i
\(34\) 0.559996 − 0.969941i 0.0960384 − 0.166343i
\(35\) −3.95513 + 1.53842i −0.668540 + 0.260040i
\(36\) −1.30872 + 2.26676i −0.218119 + 0.377794i
\(37\) −1.16055 − 4.33124i −0.190794 − 0.712052i −0.993316 − 0.115429i \(-0.963176\pi\)
0.802522 − 0.596622i \(-0.203491\pi\)
\(38\) 2.65528 − 2.65528i 0.430744 − 0.430744i
\(39\) 0.462278 + 0.800689i 0.0740238 + 0.128213i
\(40\) 0.339989 − 2.21007i 0.0537569 − 0.349443i
\(41\) 10.1494i 1.58507i 0.609826 + 0.792535i \(0.291239\pi\)
−0.609826 + 0.792535i \(0.708761\pi\)
\(42\) −1.13388 + 0.303822i −0.174962 + 0.0468808i
\(43\) 0.485377 − 0.130056i 0.0740193 − 0.0198334i −0.221619 − 0.975133i \(-0.571134\pi\)
0.295639 + 0.955300i \(0.404468\pi\)
\(44\) −2.51164 − 1.45010i −0.378644 − 0.218610i
\(45\) 2.35677 − 5.35728i 0.351326 − 0.798616i
\(46\) 5.34571 0.788182
\(47\) −11.7135 − 3.13862i −1.70859 − 0.457815i −0.733510 − 0.679679i \(-0.762119\pi\)
−0.975078 + 0.221865i \(0.928786\pi\)
\(48\) 0.160085 − 0.597444i 0.0231062 − 0.0862337i
\(49\) −2.94279 − 1.69902i −0.420398 − 0.242717i
\(50\) −0.217330 + 4.99527i −0.0307351 + 0.706438i
\(51\) −0.346368 + 0.599928i −0.0485013 + 0.0840067i
\(52\) 1.05697 + 1.05697i 0.146576 + 0.146576i
\(53\) 1.10045 − 4.10693i 0.151158 − 0.564130i −0.848245 − 0.529603i \(-0.822341\pi\)
0.999404 − 0.0345272i \(-0.0109925\pi\)
\(54\) 1.73725 − 3.00900i 0.236409 − 0.409473i
\(55\) 5.93602 + 2.61137i 0.800413 + 0.352117i
\(56\) −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 \(-0.997052\pi\)
0.507998 + 0.861358i \(0.330386\pi\)
\(60\) −0.210290 + 1.36697i −0.0271483 + 0.176475i
\(61\) 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
\(63\) −4.79832 + 1.28571i −0.604531 + 0.161984i
\(64\) − 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 \(0.857321\pi\)
\(68\) −0.289875 + 1.08183i −0.0351525 + 0.131191i
\(69\) −3.30643 −0.398047
\(70\) 3.42220 − 2.50966i 0.409031 − 0.299962i
\(71\) 2.63963i 0.313267i 0.987657 + 0.156633i \(0.0500641\pi\)
−0.987657 + 0.156633i \(0.949936\pi\)
\(72\) 0.677441 − 2.52825i 0.0798372 − 0.297957i
\(73\) 9.73042 + 2.60726i 1.13886 + 0.305157i 0.778492 − 0.627654i \(-0.215985\pi\)
0.360367 + 0.932810i \(0.382651\pi\)
\(74\) 2.24201 + 3.88328i 0.260629 + 0.451423i
\(75\) 0.134423 − 3.08968i 0.0155218 − 0.356765i
\(76\) −1.87757 + 3.25204i −0.215372 + 0.373035i
\(77\) −1.42460 − 5.31668i −0.162348 − 0.605892i
\(78\) −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
\(82\) −2.62686 − 9.80357i −0.290088 − 1.08262i
\(83\) −11.3152 − 3.03189i −1.24200 − 0.332793i −0.422759 − 0.906242i \(-0.638938\pi\)
−0.819241 + 0.573449i \(0.805605\pi\)
\(84\) 1.01661 − 0.586940i 0.110921 − 0.0640404i
\(85\) 0.380784 − 2.47526i 0.0413018 − 0.268479i
\(86\) −0.435177 + 0.251250i −0.0469264 + 0.0270930i
\(87\) 0.639819 + 0.639819i 0.0685958 + 0.0685958i
\(88\) 2.80137 + 0.750625i 0.298627 + 0.0800169i
\(89\) − 1.64840i − 0.174730i −0.996176 − 0.0873649i \(-0.972155\pi\)
0.996176 − 0.0873649i \(-0.0278446\pi\)
\(90\) −0.889897 + 5.78471i −0.0938034 + 0.609762i
\(91\) 2.83693i 0.297392i
\(92\) −5.16356 + 1.38357i −0.538338 + 0.144247i
\(93\) −1.74212 − 1.74212i −0.180649 − 0.180649i
\(94\) 12.1267 1.25077
\(95\) 3.38117 − 7.68589i 0.346901 − 0.788556i
\(96\) 0.618520i 0.0631274i
\(97\) 0.129818 + 0.129818i 0.0131810 + 0.0131810i 0.713667 − 0.700486i \(-0.247033\pi\)
−0.700486 + 0.713667i \(0.747033\pi\)
\(98\) 3.28225 + 0.879477i 0.331557 + 0.0888405i
\(99\) 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.577.12 yes 160
5.3 odd 4 inner 790.2.k.a.103.32 ✓ 160
79.56 odd 6 inner 790.2.k.a.767.32 yes 160
395.293 even 12 inner 790.2.k.a.293.12 yes 160
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
790.2.k.a.103.32 ✓ 160 5.3 odd 4 inner
790.2.k.a.293.12 yes 160 395.293 even 12 inner
790.2.k.a.577.12 yes 160 1.1 even 1 trivial
790.2.k.a.767.32 yes 160 79.56 odd 6 inner