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Properties

Label 790.2.p.a.89.10
Level $790$
Weight $2$
Character 790.89
Analytic conductor $6.308$
Analytic rank $0$
Dimension $480$
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(89,790)] 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("790.89"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(790, base_ring=CyclotomicField(26)) chi = DirichletCharacter(H, H._module([13, 22])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 790 = 2 \cdot 5 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 790.p (of order \(26\), degree \(12\), 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: \(480\)
Relative dimension: \(40\) over \(\Q(\zeta_{26})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{26}]$

Embedding invariants

Embedding label 89.10
Character \(\chi\) \(=\) 790.89
Dual form 790.2.p.a.719.10

$q$-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+(-0.935016 + 0.354605i) q^{2} +(-0.229775 - 0.437799i) q^{3} +(0.748511 - 0.663123i) q^{4} +(-1.31779 + 1.80650i) q^{5} +(0.370089 + 0.327870i) q^{6} +(1.16377 + 2.21737i) q^{7} +(-0.464723 + 0.885456i) q^{8} +(1.56532 - 2.26776i) q^{9} +(0.591561 - 2.15640i) q^{10} +(0.397919 - 3.27716i) q^{11} +(-0.462303 - 0.175328i) q^{12} +(1.19910 + 1.35351i) q^{13} +(-1.87443 - 1.66060i) q^{14} +(1.09368 + 0.161839i) q^{15} +(0.120537 - 0.992709i) q^{16} +(0.108111 + 0.122032i) q^{17} +(-0.659443 + 2.67546i) q^{18} +(1.43918 + 0.354726i) q^{19} +(0.211551 + 2.22604i) q^{20} +(0.703358 - 1.01899i) q^{21} +(0.790036 + 3.20530i) q^{22} +0.958006i q^{23} +0.494433 q^{24} +(-1.52687 - 4.76116i) q^{25} +(-1.60114 - 0.840343i) q^{26} +(-2.82498 - 0.343015i) q^{27} +(2.34148 + 0.888006i) q^{28} +(1.29965 + 1.88287i) q^{29} +(-1.07999 + 0.236501i) q^{30} +(3.16072 + 8.33412i) q^{31} +(0.239316 + 0.970942i) q^{32} +(-1.52617 + 0.578799i) q^{33} +(-0.144359 - 0.0757655i) q^{34} +(-5.53927 - 0.819685i) q^{35} +(-0.332143 - 2.73544i) q^{36} +(-0.821557 + 3.33319i) q^{37} +(-1.47144 + 0.178666i) q^{38} +(0.317041 - 0.835967i) q^{39} +(-0.987167 - 2.00636i) q^{40} +(1.29504 + 10.6656i) q^{41} +(-0.296312 + 1.20219i) q^{42} +(4.74789 - 0.576499i) q^{43} +(-1.87531 - 2.71686i) q^{44} +(2.03394 + 5.81618i) q^{45} +(-0.339713 - 0.895751i) q^{46} +(1.31839 + 5.34892i) q^{47} +(-0.462303 + 0.175328i) q^{48} +(0.414075 - 0.599891i) q^{49} +(3.11598 + 3.91033i) q^{50} +(0.0285844 - 0.0753709i) q^{51} +(1.79508 + 0.217962i) q^{52} +(5.01654 - 9.55822i) q^{53} +(2.76304 - 0.681027i) q^{54} +(5.39581 + 5.03744i) q^{55} -2.50421 q^{56} +(-0.175388 - 0.711578i) q^{57} +(-1.88287 - 1.29965i) q^{58} +(-2.29007 - 2.02883i) q^{59} +(0.925948 - 0.604104i) q^{60} +(14.0301 + 3.45811i) q^{61} +(-5.91064 - 6.67173i) q^{62} +(6.85013 + 0.831757i) q^{63} +(-0.568065 - 0.822984i) q^{64} +(-4.02527 + 0.382540i) q^{65} +(1.22175 - 1.08237i) q^{66} +(7.04735 + 2.67271i) q^{67} +(0.161845 + 0.0196515i) q^{68} +(0.419414 - 0.220125i) q^{69} +(5.46997 - 1.19783i) q^{70} +(-6.67792 - 3.50484i) q^{71} +(1.28056 + 2.43991i) q^{72} +(3.04741 - 3.43981i) q^{73} +(-0.413796 - 3.40791i) q^{74} +(-1.73360 + 1.76245i) q^{75} +(1.31247 - 0.688836i) q^{76} +(7.72976 - 2.93151i) q^{77} +0.894067i q^{78} +(0.408086 + 8.87882i) q^{79} +(1.63448 + 1.52593i) q^{80} +(-2.43244 - 6.41382i) q^{81} +(-4.99297 - 9.51331i) q^{82} +(7.76784 + 8.76808i) q^{83} +(-0.149244 - 1.22914i) q^{84} +(-0.362919 + 0.0344899i) q^{85} +(-4.23493 + 2.22266i) q^{86} +(0.525692 - 1.00162i) q^{87} +(2.71686 + 1.87531i) q^{88} +(-9.26092 + 4.86050i) q^{89} +(-3.96422 - 4.71698i) q^{90} +(-1.60575 + 4.23402i) q^{91} +(0.635275 + 0.717078i) q^{92} +(2.92242 - 3.29873i) q^{93} +(-3.12947 - 4.53382i) q^{94} +(-2.53735 + 2.13242i) q^{95} +(0.370089 - 0.327870i) q^{96} +(-2.45719 + 9.96921i) q^{97} +(-0.174443 + 0.707741i) q^{98} +(-6.80895 - 6.03220i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 480 q + 40 q^{4} + 4 q^{6} + 48 q^{9} - 4 q^{10} - 8 q^{11} - 8 q^{14} - 8 q^{15} - 40 q^{16} + 16 q^{19} + 52 q^{21} + 48 q^{24} - 14 q^{25} - 20 q^{29} + 20 q^{30} - 68 q^{31} + 70 q^{35} - 48 q^{36}+ \cdots + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(161\) \(317\)
\(\chi(n)\) \(e\left(\frac{11}{13}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.935016 + 0.354605i −0.661156 + 0.250744i
\(3\) −0.229775 − 0.437799i −0.132660 − 0.252763i 0.810060 − 0.586347i \(-0.199434\pi\)
−0.942721 + 0.333584i \(0.891742\pi\)
\(4\) 0.748511 − 0.663123i 0.374255 − 0.331561i
\(5\) −1.31779 + 1.80650i −0.589333 + 0.807890i
\(6\) 0.370089 + 0.327870i 0.151088 + 0.133852i
\(7\) 1.16377 + 2.21737i 0.439862 + 0.838087i 0.999953 + 0.00965612i \(0.00307369\pi\)
−0.560091 + 0.828431i \(0.689234\pi\)
\(8\) −0.464723 + 0.885456i −0.164304 + 0.313056i
\(9\) 1.56532 − 2.26776i 0.521774 − 0.755921i
\(10\) 0.591561 − 2.15640i 0.187068 − 0.681913i
\(11\) 0.397919 − 3.27716i 0.119977 − 0.988101i −0.801049 − 0.598599i \(-0.795724\pi\)
0.921026 − 0.389502i \(-0.127353\pi\)
\(12\) −0.462303 − 0.175328i −0.133455 − 0.0506130i
\(13\) 1.19910 + 1.35351i 0.332571 + 0.375395i 0.890881 − 0.454238i \(-0.150088\pi\)
−0.558309 + 0.829633i \(0.688550\pi\)
\(14\) −1.87443 − 1.66060i −0.500962 − 0.443814i
\(15\) 1.09368 + 0.161839i 0.282386 + 0.0417867i
\(16\) 0.120537 − 0.992709i 0.0301342 − 0.248177i
\(17\) 0.108111 + 0.122032i 0.0262208 + 0.0295972i 0.761475 − 0.648194i \(-0.224476\pi\)
−0.735254 + 0.677792i \(0.762937\pi\)
\(18\) −0.659443 + 2.67546i −0.155432 + 0.630613i
\(19\) 1.43918 + 0.354726i 0.330170 + 0.0813797i 0.400916 − 0.916115i \(-0.368692\pi\)
−0.0707459 + 0.997494i \(0.522538\pi\)
\(20\) 0.211551 + 2.22604i 0.0473042 + 0.497757i
\(21\) 0.703358 − 1.01899i 0.153485 − 0.222362i
\(22\) 0.790036 + 3.20530i 0.168436 + 0.683373i
\(23\) 0.958006i 0.199758i 0.995000 + 0.0998790i \(0.0318456\pi\)
−0.995000 + 0.0998790i \(0.968154\pi\)
\(24\) 0.494433 0.100926
\(25\) −1.52687 − 4.76116i −0.305373 − 0.952233i
\(26\) −1.60114 − 0.840343i −0.314010 − 0.164805i
\(27\) −2.82498 − 0.343015i −0.543667 − 0.0660132i
\(28\) 2.34148 + 0.888006i 0.442498 + 0.167817i
\(29\) 1.29965 + 1.88287i 0.241339 + 0.349640i 0.924871 − 0.380280i \(-0.124172\pi\)
−0.683532 + 0.729921i \(0.739557\pi\)
\(30\) −1.07999 + 0.236501i −0.197179 + 0.0431790i
\(31\) 3.16072 + 8.33412i 0.567681 + 1.49685i 0.844299 + 0.535872i \(0.180017\pi\)
−0.276618 + 0.960980i \(0.589214\pi\)
\(32\) 0.239316 + 0.970942i 0.0423054 + 0.171640i
\(33\) −1.52617 + 0.578799i −0.265672 + 0.100756i
\(34\) −0.144359 − 0.0757655i −0.0247574 − 0.0129937i
\(35\) −5.53927 − 0.819685i −0.936307 − 0.138552i
\(36\) −0.332143 − 2.73544i −0.0553572 − 0.455907i
\(37\) −0.821557 + 3.33319i −0.135063 + 0.547973i 0.863951 + 0.503576i \(0.167983\pi\)
−0.999014 + 0.0443966i \(0.985863\pi\)
\(38\) −1.47144 + 0.178666i −0.238700 + 0.0289834i
\(39\) 0.317041 − 0.835967i 0.0507671 − 0.133862i
\(40\) −0.987167 − 2.00636i −0.156085 − 0.317234i
\(41\) 1.29504 + 10.6656i 0.202252 + 1.66569i 0.641347 + 0.767251i \(0.278376\pi\)
−0.439095 + 0.898441i \(0.644701\pi\)
\(42\) −0.296312 + 1.20219i −0.0457220 + 0.185501i
\(43\) 4.74789 − 0.576499i 0.724047 − 0.0879152i 0.249788 − 0.968301i \(-0.419639\pi\)
0.474259 + 0.880385i \(0.342716\pi\)
\(44\) −1.87531 − 2.71686i −0.282714 − 0.409582i
\(45\) 2.03394 + 5.81618i 0.303202 + 0.867025i
\(46\) −0.339713 − 0.895751i −0.0500880 − 0.132071i
\(47\) 1.31839 + 5.34892i 0.192307 + 0.780220i 0.985278 + 0.170961i \(0.0546871\pi\)
−0.792971 + 0.609260i \(0.791467\pi\)
\(48\) −0.462303 + 0.175328i −0.0667277 + 0.0253065i
\(49\) 0.414075 − 0.599891i 0.0591536 − 0.0856988i
\(50\) 3.11598 + 3.91033i 0.440666 + 0.553004i
\(51\) 0.0285844 − 0.0753709i 0.00400262 − 0.0105540i
\(52\) 1.79508 + 0.217962i 0.248933 + 0.0302260i
\(53\) 5.01654 − 9.55822i 0.689075 − 1.31292i −0.249677 − 0.968329i \(-0.580325\pi\)
0.938752 − 0.344593i \(-0.111983\pi\)
\(54\) 2.76304 − 0.681027i 0.376002 − 0.0926761i
\(55\) 5.39581 + 5.03744i 0.727571 + 0.679249i
\(56\) −2.50421 −0.334639
\(57\) −0.175388 − 0.711578i −0.0232307 − 0.0942508i
\(58\) −1.88287 − 1.29965i −0.247233 − 0.170653i
\(59\) −2.29007 − 2.02883i −0.298142 − 0.264131i 0.500768 − 0.865582i \(-0.333051\pi\)
−0.798910 + 0.601451i \(0.794589\pi\)
\(60\) 0.925948 − 0.604104i 0.119539 − 0.0779894i
\(61\) 14.0301 + 3.45811i 1.79637 + 0.442766i 0.989079 − 0.147383i \(-0.0470851\pi\)
0.807294 + 0.590150i \(0.200931\pi\)
\(62\) −5.91064 − 6.67173i −0.750652 − 0.847311i
\(63\) 6.85013 + 0.831757i 0.863036 + 0.104792i
\(64\) −0.568065 − 0.822984i −0.0710081 − 0.102873i
\(65\) −4.02527 + 0.382540i −0.499273 + 0.0474483i
\(66\) 1.22175 − 1.08237i 0.150387 − 0.133231i
\(67\) 7.04735 + 2.67271i 0.860971 + 0.326523i 0.745272 − 0.666761i \(-0.232320\pi\)
0.115700 + 0.993284i \(0.463089\pi\)
\(68\) 0.161845 + 0.0196515i 0.0196266 + 0.00238310i
\(69\) 0.419414 − 0.220125i 0.0504915 − 0.0265000i
\(70\) 5.46997 − 1.19783i 0.653787 − 0.143168i
\(71\) −6.67792 − 3.50484i −0.792523 − 0.415948i 0.0193278 − 0.999813i \(-0.493847\pi\)
−0.811851 + 0.583865i \(0.801540\pi\)
\(72\) 1.28056 + 2.43991i 0.150916 + 0.287546i
\(73\) 3.04741 − 3.43981i 0.356672 − 0.402599i −0.542599 − 0.839992i \(-0.682560\pi\)
0.899271 + 0.437393i \(0.144098\pi\)
\(74\) −0.413796 − 3.40791i −0.0481028 − 0.396162i
\(75\) −1.73360 + 1.76245i −0.200178 + 0.203511i
\(76\) 1.31247 − 0.688836i 0.150550 − 0.0790150i
\(77\) 7.72976 − 2.93151i 0.880888 − 0.334077i
\(78\) 0.894067i 0.101233i
\(79\) 0.408086 + 8.87882i 0.0459132 + 0.998945i
\(80\) 1.63448 + 1.52593i 0.182741 + 0.170604i
\(81\) −2.43244 − 6.41382i −0.270271 − 0.712647i
\(82\) −4.99297 − 9.51331i −0.551381 − 1.05057i
\(83\) 7.76784 + 8.76808i 0.852631 + 0.962422i 0.999643 − 0.0267131i \(-0.00850406\pi\)
−0.147012 + 0.989135i \(0.546966\pi\)
\(84\) −0.149244 − 1.22914i −0.0162839 − 0.134110i
\(85\) −0.362919 + 0.0344899i −0.0393641 + 0.00374095i
\(86\) −4.23493 + 2.22266i −0.456664 + 0.239676i
\(87\) 0.525692 − 1.00162i 0.0563601 − 0.107385i
\(88\) 2.71686 + 1.87531i 0.289618 + 0.199909i
\(89\) −9.26092 + 4.86050i −0.981655 + 0.515212i −0.877618 − 0.479361i \(-0.840868\pi\)
−0.104037 + 0.994573i \(0.533176\pi\)
\(90\) −3.96422 − 4.71698i −0.417865 − 0.497213i
\(91\) −1.60575 + 4.23402i −0.168328 + 0.443846i
\(92\) 0.635275 + 0.717078i 0.0662320 + 0.0747605i
\(93\) 2.92242 − 3.29873i 0.303040 − 0.342062i
\(94\) −3.12947 − 4.53382i −0.322780 − 0.467628i
\(95\) −2.53735 + 2.13242i −0.260326 + 0.218782i
\(96\) 0.370089 − 0.327870i 0.0377720 − 0.0334631i
\(97\) −2.45719 + 9.96921i −0.249490 + 1.01222i 0.703857 + 0.710342i \(0.251460\pi\)
−0.953347 + 0.301878i \(0.902386\pi\)
\(98\) −0.174443 + 0.707741i −0.0176214 + 0.0714927i
\(99\) −6.80895 − 6.03220i −0.684325 − 0.606259i
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.p.a.89.10 ✓ 480
5.4 even 2 inner 790.2.p.a.89.31 yes 480
79.8 even 13 inner 790.2.p.a.719.31 yes 480
395.324 even 26 inner 790.2.p.a.719.10 yes 480
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
790.2.p.a.89.10 ✓ 480 1.1 even 1 trivial
790.2.p.a.89.31 yes 480 5.4 even 2 inner
790.2.p.a.719.10 yes 480 395.324 even 26 inner
790.2.p.a.719.31 yes 480 79.8 even 13 inner