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Properties

Label 790.2.k.a.103.14
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 103.14
Character \(\chi\) \(=\) 790.103
Dual form 790.2.k.a.767.14

$q$-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} +(0.300996 + 1.12333i) q^{3} +(-0.866025 + 0.500000i) q^{4} +(-0.559961 + 2.16482i) q^{5} +(1.00715 - 0.581480i) q^{6} +(1.05325 - 3.93078i) q^{7} +(0.707107 + 0.707107i) q^{8} +(1.42680 - 0.823762i) q^{9} +(2.23598 - 0.0194162i) q^{10} +(-2.50846 - 4.34478i) q^{11} +(-0.822337 - 0.822337i) q^{12} +(-2.77870 + 0.744550i) q^{13} -4.06944 q^{14} +(-2.60036 + 0.0225802i) q^{15} +(0.500000 - 0.866025i) q^{16} +(1.06947 + 1.06947i) q^{17} +(-1.16498 - 1.16498i) q^{18} +(3.37239 - 1.94705i) q^{19} +(-0.597470 - 2.15477i) q^{20} +4.73260 q^{21} +(-3.54750 + 3.54750i) q^{22} +(0.701530 - 2.61814i) q^{23} +(-0.581480 + 1.00715i) q^{24} +(-4.37289 - 2.42443i) q^{25} +(1.43836 + 2.49131i) q^{26} +(3.82183 + 3.82183i) q^{27} +(1.05325 + 3.93078i) q^{28} +(3.17698 + 5.50269i) q^{29} +(0.694833 + 2.50591i) q^{30} +(-3.78698 - 6.55924i) q^{31} +(-0.965926 - 0.258819i) q^{32} +(4.12560 - 4.12560i) q^{33} +(0.756230 - 1.30983i) q^{34} +(7.91965 + 4.48118i) q^{35} +(-0.823762 + 1.42680i) q^{36} +(10.4887 - 2.81044i) q^{37} +(-2.75355 - 2.75355i) q^{38} +(-1.67275 - 2.89730i) q^{39} +(-1.92671 + 1.13481i) q^{40} -4.33089i q^{41} +(-1.22489 - 4.57134i) q^{42} +(-2.72246 - 10.1604i) q^{43} +(4.34478 + 2.50846i) q^{44} +(0.984346 + 3.55003i) q^{45} -2.71050 q^{46} +(0.843780 - 3.14903i) q^{47} +(1.12333 + 0.300996i) q^{48} +(-8.27952 - 4.78018i) q^{49} +(-1.21003 + 4.85137i) q^{50} +(-0.879466 + 1.52328i) q^{51} +(2.03415 - 2.03415i) q^{52} +(11.1773 + 2.99496i) q^{53} +(2.70244 - 4.68077i) q^{54} +(10.8103 - 2.99746i) q^{55} +(3.52424 - 2.03472i) q^{56} +(3.20226 + 3.20226i) q^{57} +(4.49292 - 4.49292i) q^{58} +(-1.99500 + 3.45544i) q^{59} +(2.24069 - 1.31973i) q^{60} -3.82207i q^{61} +(-5.35560 + 5.35560i) q^{62} +(-1.73525 - 6.47606i) q^{63} +1.00000i q^{64} +(-0.0558549 - 6.43230i) q^{65} +(-5.05281 - 2.91724i) q^{66} +(2.71606 - 2.71606i) q^{67} +(-1.46093 - 0.391454i) q^{68} +3.15221 q^{69} +(2.27873 - 8.80961i) q^{70} -5.73101i q^{71} +(1.59139 + 0.426411i) q^{72} +(-3.93601 + 14.6894i) q^{73} +(-5.42936 - 9.40393i) q^{74} +(1.40722 - 5.64195i) q^{75} +(-1.94705 + 3.37239i) q^{76} +(-19.7204 + 5.28407i) q^{77} +(-2.36563 + 2.36563i) q^{78} +(0.612394 + 8.86707i) q^{79} +(1.59481 + 1.56735i) q^{80} +(-0.671545 + 1.16315i) q^{81} +(-4.18332 + 1.12092i) q^{82} +(-3.69812 + 13.8016i) q^{83} +(-4.09855 + 2.36630i) q^{84} +(-2.91407 + 1.71635i) q^{85} +(-9.10952 + 5.25938i) q^{86} +(-5.22509 + 5.22509i) q^{87} +(1.29848 - 4.84597i) q^{88} -8.66863i q^{89} +(3.17430 - 1.86962i) q^{90} +11.7066i q^{91} +(0.701530 + 2.61814i) q^{92} +(6.22834 - 6.22834i) q^{93} -3.26012 q^{94} +(2.32661 + 8.39089i) q^{95} -1.16296i q^{96} +(-13.5739 + 13.5739i) q^{97} +(-2.47440 + 9.23460i) q^{98} +(-7.15813 - 4.13275i) 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
\(3\) 0.300996 + 1.12333i 0.173780 + 0.648557i 0.996756 + 0.0804804i \(0.0256454\pi\)
−0.822976 + 0.568076i \(0.807688\pi\)
\(4\) −0.866025 + 0.500000i −0.433013 + 0.250000i
\(5\) −0.559961 + 2.16482i −0.250422 + 0.968137i
\(6\) 1.00715 − 0.581480i 0.411168 − 0.237388i
\(7\) 1.05325 − 3.93078i 0.398091 − 1.48570i −0.418361 − 0.908281i \(-0.637395\pi\)
0.816451 − 0.577414i \(-0.195938\pi\)
\(8\) 0.707107 + 0.707107i 0.250000 + 0.250000i
\(9\) 1.42680 − 0.823762i 0.475599 − 0.274587i
\(10\) 2.23598 − 0.0194162i 0.707080 − 0.00613993i
\(11\) −2.50846 − 4.34478i −0.756329 − 1.31000i −0.944711 − 0.327905i \(-0.893657\pi\)
0.188381 − 0.982096i \(-0.439676\pi\)
\(12\) −0.822337 − 0.822337i −0.237388 − 0.237388i
\(13\) −2.77870 + 0.744550i −0.770672 + 0.206501i −0.622668 − 0.782486i \(-0.713951\pi\)
−0.148004 + 0.988987i \(0.547285\pi\)
\(14\) −4.06944 −1.08760
\(15\) −2.60036 + 0.0225802i −0.671410 + 0.00583019i
\(16\) 0.500000 − 0.866025i 0.125000 − 0.216506i
\(17\) 1.06947 + 1.06947i 0.259385 + 0.259385i 0.824804 − 0.565419i \(-0.191285\pi\)
−0.565419 + 0.824804i \(0.691285\pi\)
\(18\) −1.16498 − 1.16498i −0.274587 − 0.274587i
\(19\) 3.37239 − 1.94705i 0.773680 − 0.446684i −0.0605060 − 0.998168i \(-0.519271\pi\)
0.834186 + 0.551484i \(0.185938\pi\)
\(20\) −0.597470 − 2.15477i −0.133598 − 0.481821i
\(21\) 4.73260 1.03274
\(22\) −3.54750 + 3.54750i −0.756329 + 0.756329i
\(23\) 0.701530 − 2.61814i 0.146279 − 0.545921i −0.853416 − 0.521230i \(-0.825473\pi\)
0.999695 − 0.0246906i \(-0.00786006\pi\)
\(24\) −0.581480 + 1.00715i −0.118694 + 0.205584i
\(25\) −4.37289 − 2.42443i −0.874578 − 0.484885i
\(26\) 1.43836 + 2.49131i 0.282085 + 0.488586i
\(27\) 3.82183 + 3.82183i 0.735512 + 0.735512i
\(28\) 1.05325 + 3.93078i 0.199045 + 0.742848i
\(29\) 3.17698 + 5.50269i 0.589950 + 1.02182i 0.994238 + 0.107192i \(0.0341858\pi\)
−0.404289 + 0.914632i \(0.632481\pi\)
\(30\) 0.694833 + 2.50591i 0.126859 + 0.457514i
\(31\) −3.78698 − 6.55924i −0.680161 − 1.17807i −0.974931 − 0.222506i \(-0.928576\pi\)
0.294770 − 0.955568i \(-0.404757\pi\)
\(32\) −0.965926 − 0.258819i −0.170753 − 0.0457532i
\(33\) 4.12560 − 4.12560i 0.718175 − 0.718175i
\(34\) 0.756230 − 1.30983i 0.129692 − 0.224634i
\(35\) 7.91965 + 4.48118i 1.33867 + 0.757457i
\(36\) −0.823762 + 1.42680i −0.137294 + 0.237800i
\(37\) 10.4887 − 2.81044i 1.72433 − 0.462034i 0.745469 − 0.666540i \(-0.232225\pi\)
0.978865 + 0.204506i \(0.0655588\pi\)
\(38\) −2.75355 − 2.75355i −0.446684 − 0.446684i
\(39\) −1.67275 − 2.89730i −0.267855 − 0.463939i
\(40\) −1.92671 + 1.13481i −0.304640 + 0.179429i
\(41\) − 4.33089i − 0.676371i −0.941079 − 0.338185i \(-0.890187\pi\)
0.941079 − 0.338185i \(-0.109813\pi\)
\(42\) −1.22489 − 4.57134i −0.189004 − 0.705373i
\(43\) −2.72246 − 10.1604i −0.415171 − 1.54944i −0.784493 − 0.620138i \(-0.787077\pi\)
0.369322 − 0.929301i \(-0.379590\pi\)
\(44\) 4.34478 + 2.50846i 0.655001 + 0.378165i
\(45\) 0.984346 + 3.55003i 0.146738 + 0.529208i
\(46\) −2.71050 −0.399642
\(47\) 0.843780 − 3.14903i 0.123078 − 0.459333i −0.876686 − 0.481063i \(-0.840251\pi\)
0.999764 + 0.0217299i \(0.00691739\pi\)
\(48\) 1.12333 + 0.300996i 0.162139 + 0.0434451i
\(49\) −8.27952 − 4.78018i −1.18279 − 0.682883i
\(50\) −1.21003 + 4.85137i −0.171124 + 0.686088i
\(51\) −0.879466 + 1.52328i −0.123150 + 0.213302i
\(52\) 2.03415 − 2.03415i 0.282085 − 0.282085i
\(53\) 11.1773 + 2.99496i 1.53533 + 0.411390i 0.924752 − 0.380569i \(-0.124272\pi\)
0.610574 + 0.791959i \(0.290939\pi\)
\(54\) 2.70244 − 4.68077i 0.367756 − 0.636972i
\(55\) 10.8103 − 2.99746i 1.45766 − 0.404177i
\(56\) 3.52424 − 2.03472i 0.470947 − 0.271901i
\(57\) 3.20226 + 3.20226i 0.424150 + 0.424150i
\(58\) 4.49292 − 4.49292i 0.589950 − 0.589950i
\(59\) −1.99500 + 3.45544i −0.259727 + 0.449860i −0.966169 − 0.257911i \(-0.916966\pi\)
0.706442 + 0.707771i \(0.250299\pi\)
\(60\) 2.24069 − 1.31973i 0.289271 − 0.170377i
\(61\) − 3.82207i − 0.489365i −0.969603 − 0.244683i \(-0.921316\pi\)
0.969603 − 0.244683i \(-0.0786838\pi\)
\(62\) −5.35560 + 5.35560i −0.680161 + 0.680161i
\(63\) −1.73525 − 6.47606i −0.218621 − 0.815906i
\(64\) 1.00000i 0.125000i
\(65\) −0.0558549 − 6.43230i −0.00692795 − 0.797828i
\(66\) −5.05281 − 2.91724i −0.621958 − 0.359087i
\(67\) 2.71606 − 2.71606i 0.331820 − 0.331820i −0.521457 − 0.853277i \(-0.674611\pi\)
0.853277 + 0.521457i \(0.174611\pi\)
\(68\) −1.46093 − 0.391454i −0.177163 − 0.0474707i
\(69\) 3.15221 0.379481
\(70\) 2.27873 − 8.80961i 0.272360 − 1.05295i
\(71\) − 5.73101i − 0.680146i −0.940399 − 0.340073i \(-0.889548\pi\)
0.940399 − 0.340073i \(-0.110452\pi\)
\(72\) 1.59139 + 0.426411i 0.187547 + 0.0502530i
\(73\) −3.93601 + 14.6894i −0.460675 + 1.71926i 0.210169 + 0.977665i \(0.432598\pi\)
−0.670844 + 0.741598i \(0.734068\pi\)
\(74\) −5.42936 − 9.40393i −0.631150 − 1.09318i
\(75\) 1.40722 − 5.64195i 0.162491 − 0.651477i
\(76\) −1.94705 + 3.37239i −0.223342 + 0.386840i
\(77\) −19.7204 + 5.28407i −2.24735 + 0.602176i
\(78\) −2.36563 + 2.36563i −0.267855 + 0.267855i
\(79\) 0.612394 + 8.86707i 0.0688997 + 0.997624i
\(80\) 1.59481 + 1.56735i 0.178305 + 0.175235i
\(81\) −0.671545 + 1.16315i −0.0746161 + 0.129239i
\(82\) −4.18332 + 1.12092i −0.461970 + 0.123784i
\(83\) −3.69812 + 13.8016i −0.405921 + 1.51492i 0.396430 + 0.918065i \(0.370249\pi\)
−0.802351 + 0.596853i \(0.796418\pi\)
\(84\) −4.09855 + 2.36630i −0.447189 + 0.258184i
\(85\) −2.91407 + 1.71635i −0.316076 + 0.186164i
\(86\) −9.10952 + 5.25938i −0.982305 + 0.567134i
\(87\) −5.22509 + 5.22509i −0.560189 + 0.560189i
\(88\) 1.29848 − 4.84597i 0.138418 − 0.516583i
\(89\) − 8.66863i − 0.918873i −0.888211 − 0.459436i \(-0.848051\pi\)
0.888211 − 0.459436i \(-0.151949\pi\)
\(90\) 3.17430 − 1.86962i 0.334601 − 0.197075i
\(91\) 11.7066i 1.22719i
\(92\) 0.701530 + 2.61814i 0.0731395 + 0.272960i
\(93\) 6.22834 − 6.22834i 0.645849 − 0.645849i
\(94\) −3.26012 −0.336255
\(95\) 2.32661 + 8.39089i 0.238705 + 0.860887i
\(96\) − 1.16296i − 0.118694i
\(97\) −13.5739 + 13.5739i −1.37822 + 1.37822i −0.530586 + 0.847631i \(0.678028\pi\)
−0.847631 + 0.530586i \(0.821972\pi\)
\(98\) −2.47440 + 9.23460i −0.249953 + 0.932836i
\(99\) −7.15813 − 4.13275i −0.719420 − 0.415357i
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.103.14 ✓ 160
5.2 odd 4 inner 790.2.k.a.577.34 yes 160
79.56 odd 6 inner 790.2.k.a.293.34 yes 160
395.372 even 12 inner 790.2.k.a.767.14 yes 160
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
790.2.k.a.103.14 ✓ 160 1.1 even 1 trivial
790.2.k.a.293.34 yes 160 79.56 odd 6 inner
790.2.k.a.577.34 yes 160 5.2 odd 4 inner
790.2.k.a.767.14 yes 160 395.372 even 12 inner