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Properties

Label 790.2.m.c.141.4
Level $790$
Weight $2$
Character 790.141
Analytic conductor $6.308$
Analytic rank $0$
Dimension $48$
Inner twists $2$

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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(21,790)] 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("790.21"); 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([0, 18])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 790 = 2 \cdot 5 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 790.m (of order \(13\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48,4,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.30818175968\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(4\) over \(\Q(\zeta_{13})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{13}]$

Embedding invariants

Embedding label 141.4
Character \(\chi\) \(=\) 790.141
Dual form 790.2.m.c.381.4

$q$-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+(0.970942 + 0.239316i) q^{2} +(1.73068 - 1.53325i) q^{3} +(0.885456 + 0.464723i) q^{4} +(0.568065 - 0.822984i) q^{5} +(2.04732 - 1.07452i) q^{6} +(1.05448 - 0.934183i) q^{7} +(0.748511 + 0.663123i) q^{8} +(0.282791 - 2.32899i) q^{9} +(0.748511 - 0.663123i) q^{10} +(1.35907 + 1.96895i) q^{11} +(2.24498 - 0.553337i) q^{12} +(-4.36662 - 2.29178i) q^{13} +(1.24740 - 0.654685i) q^{14} +(-0.278700 - 2.29530i) q^{15} +(0.568065 + 0.822984i) q^{16} +(3.52260 + 1.84880i) q^{17} +(0.831937 - 2.19364i) q^{18} +(0.260063 - 0.685731i) q^{19} +(0.885456 - 0.464723i) q^{20} +(0.392623 - 3.23354i) q^{21} +(0.848376 + 2.23698i) q^{22} -0.992963 q^{23} +2.31216 q^{24} +(-0.354605 - 0.935016i) q^{25} +(-3.69128 - 3.27018i) q^{26} +(0.858875 + 1.24430i) q^{27} +(1.36783 - 0.337139i) q^{28} +(0.201059 + 1.65587i) q^{29} +(0.278700 - 2.29530i) q^{30} +(-9.55293 - 2.35458i) q^{31} +(0.354605 + 0.935016i) q^{32} +(5.37100 + 1.32383i) q^{33} +(2.97779 + 2.63809i) q^{34} +(-0.169808 - 1.39849i) q^{35} +(1.33273 - 1.93080i) q^{36} +(-0.543489 + 1.43306i) q^{37} +(0.416613 - 0.603568i) q^{38} +(-11.0711 + 2.72878i) q^{39} +(0.970942 - 0.239316i) q^{40} +(2.61085 - 3.78247i) q^{41} +(1.15505 - 3.04562i) q^{42} +(-2.20846 + 3.19951i) q^{43} +(0.288378 + 2.37501i) q^{44} +(-1.75608 - 1.55575i) q^{45} +(-0.964109 - 0.237632i) q^{46} +(-0.453033 - 1.19455i) q^{47} +(2.24498 + 0.553337i) q^{48} +(-0.604538 + 4.97882i) q^{49} +(-0.120537 - 0.992709i) q^{50} +(8.93115 - 2.20133i) q^{51} +(-2.80141 - 4.05854i) q^{52} +(-1.27779 - 1.13202i) q^{53} +(0.536138 + 1.41368i) q^{54} +2.39245 q^{55} +1.40876 q^{56} +(-0.601309 - 1.58552i) q^{57} +(-0.201059 + 1.65587i) q^{58} +(0.450210 - 0.236288i) q^{59} +(0.819904 - 2.16191i) q^{60} +(-1.61926 + 4.26963i) q^{61} +(-8.71185 - 4.57233i) q^{62} +(-1.87751 - 2.72004i) q^{63} +(0.120537 + 0.992709i) q^{64} +(-4.36662 + 2.29178i) q^{65} +(4.89812 + 2.57073i) q^{66} +(3.21919 - 0.793460i) q^{67} +(2.25992 + 3.27406i) q^{68} +(-1.71850 + 1.52246i) q^{69} +(0.169808 - 1.39849i) q^{70} +(0.825122 + 0.730995i) q^{71} +(1.75608 - 1.55575i) q^{72} +(7.47655 - 3.92400i) q^{73} +(-0.870650 + 1.26135i) q^{74} +(-2.04732 - 1.07452i) q^{75} +(0.548950 - 0.486327i) q^{76} +(3.27247 + 0.806590i) q^{77} -11.4024 q^{78} +(-8.03379 + 3.80240i) q^{79} +1.00000 q^{80} +(10.2280 + 2.52098i) q^{81} +(3.44019 - 3.04774i) q^{82} +(0.00951772 + 0.00499528i) q^{83} +(1.85035 - 2.68070i) q^{84} +(3.52260 - 1.84880i) q^{85} +(-2.90998 + 2.57802i) q^{86} +(2.88682 + 2.55750i) q^{87} +(-0.288378 + 2.37501i) q^{88} +(-10.0085 + 8.86676i) q^{89} +(-1.33273 - 1.93080i) q^{90} +(-6.74543 + 1.66260i) q^{91} +(-0.879225 - 0.461453i) q^{92} +(-20.1432 + 10.5720i) q^{93} +(-0.153994 - 1.26826i) q^{94} +(-0.416613 - 0.603568i) q^{95} +(2.04732 + 1.07452i) q^{96} +(-2.96554 + 7.81950i) q^{97} +(-1.77848 + 4.68947i) q^{98} +(4.97000 - 2.60846i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q + 4 q^{2} - q^{3} - 4 q^{4} - 4 q^{5} + q^{6} - 9 q^{7} + 4 q^{8} - 21 q^{9} + 4 q^{10} + 2 q^{11} - q^{12} + 17 q^{13} - 4 q^{14} - 14 q^{15} - 4 q^{16} - 6 q^{17} + 21 q^{18} - 34 q^{19} - 4 q^{20}+ \cdots - 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(161\) \(317\)
\(\chi(n)\) \(e\left(\frac{10}{13}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.970942 + 0.239316i 0.686560 + 0.169222i
\(3\) 1.73068 − 1.53325i 0.999208 − 0.885221i 0.00581478 − 0.999983i \(-0.498149\pi\)
0.993393 + 0.114762i \(0.0366106\pi\)
\(4\) 0.885456 + 0.464723i 0.442728 + 0.232362i
\(5\) 0.568065 − 0.822984i 0.254046 − 0.368050i
\(6\) 2.04732 − 1.07452i 0.835814 − 0.438669i
\(7\) 1.05448 − 0.934183i 0.398554 − 0.353088i −0.439913 − 0.898040i \(-0.644991\pi\)
0.838467 + 0.544952i \(0.183452\pi\)
\(8\) 0.748511 + 0.663123i 0.264639 + 0.234449i
\(9\) 0.282791 − 2.32899i 0.0942636 − 0.776330i
\(10\) 0.748511 − 0.663123i 0.236700 − 0.209698i
\(11\) 1.35907 + 1.96895i 0.409775 + 0.593661i 0.972477 − 0.232998i \(-0.0748537\pi\)
−0.562702 + 0.826660i \(0.690238\pi\)
\(12\) 2.24498 − 0.553337i 0.648069 − 0.159735i
\(13\) −4.36662 − 2.29178i −1.21108 − 0.635625i −0.266480 − 0.963841i \(-0.585861\pi\)
−0.944603 + 0.328215i \(0.893553\pi\)
\(14\) 1.24740 − 0.654685i 0.333381 − 0.174972i
\(15\) −0.278700 − 2.29530i −0.0719601 − 0.592645i
\(16\) 0.568065 + 0.822984i 0.142016 + 0.205746i
\(17\) 3.52260 + 1.84880i 0.854355 + 0.448400i 0.834243 − 0.551398i \(-0.185905\pi\)
0.0201125 + 0.999798i \(0.493598\pi\)
\(18\) 0.831937 − 2.19364i 0.196089 − 0.517045i
\(19\) 0.260063 − 0.685731i 0.0596627 − 0.157317i −0.901870 − 0.432007i \(-0.857806\pi\)
0.961533 + 0.274689i \(0.0885749\pi\)
\(20\) 0.885456 − 0.464723i 0.197994 − 0.103915i
\(21\) 0.392623 − 3.23354i 0.0856774 − 0.705617i
\(22\) 0.848376 + 2.23698i 0.180874 + 0.476926i
\(23\) −0.992963 −0.207047 −0.103524 − 0.994627i \(-0.533012\pi\)
−0.103524 + 0.994627i \(0.533012\pi\)
\(24\) 2.31216 0.471968
\(25\) −0.354605 − 0.935016i −0.0709210 − 0.187003i
\(26\) −3.69128 − 3.27018i −0.723919 − 0.641336i
\(27\) 0.858875 + 1.24430i 0.165291 + 0.239465i
\(28\) 1.36783 − 0.337139i 0.258495 − 0.0637133i
\(29\) 0.201059 + 1.65587i 0.0373356 + 0.307487i 0.999399 + 0.0346610i \(0.0110351\pi\)
−0.962063 + 0.272826i \(0.912042\pi\)
\(30\) 0.278700 − 2.29530i 0.0508835 − 0.419063i
\(31\) −9.55293 − 2.35458i −1.71576 − 0.422896i −0.745799 − 0.666171i \(-0.767932\pi\)
−0.969957 + 0.243275i \(0.921778\pi\)
\(32\) 0.354605 + 0.935016i 0.0626859 + 0.165289i
\(33\) 5.37100 + 1.32383i 0.934971 + 0.230450i
\(34\) 2.97779 + 2.63809i 0.510687 + 0.452429i
\(35\) −0.169808 − 1.39849i −0.0287027 − 0.236388i
\(36\) 1.33273 − 1.93080i 0.222122 − 0.321800i
\(37\) −0.543489 + 1.43306i −0.0893490 + 0.235594i −0.972241 − 0.233982i \(-0.924824\pi\)
0.882892 + 0.469576i \(0.155593\pi\)
\(38\) 0.416613 − 0.603568i 0.0675835 − 0.0979116i
\(39\) −11.0711 + 2.72878i −1.77279 + 0.436954i
\(40\) 0.970942 − 0.239316i 0.153519 − 0.0378391i
\(41\) 2.61085 − 3.78247i 0.407746 − 0.590722i −0.564283 − 0.825581i \(-0.690847\pi\)
0.972029 + 0.234859i \(0.0754628\pi\)
\(42\) 1.15505 − 3.04562i 0.178228 − 0.469949i
\(43\) −2.20846 + 3.19951i −0.336787 + 0.487921i −0.954387 − 0.298572i \(-0.903490\pi\)
0.617600 + 0.786492i \(0.288105\pi\)
\(44\) 0.288378 + 2.37501i 0.0434747 + 0.358046i
\(45\) −1.75608 − 1.55575i −0.261781 − 0.231917i
\(46\) −0.964109 − 0.237632i −0.142150 − 0.0350369i
\(47\) −0.453033 − 1.19455i −0.0660817 − 0.174243i 0.897908 − 0.440184i \(-0.145087\pi\)
−0.963989 + 0.265941i \(0.914317\pi\)
\(48\) 2.24498 + 0.553337i 0.324034 + 0.0798673i
\(49\) −0.604538 + 4.97882i −0.0863625 + 0.711259i
\(50\) −0.120537 − 0.992709i −0.0170465 − 0.140390i
\(51\) 8.93115 − 2.20133i 1.25061 − 0.308248i
\(52\) −2.80141 − 4.05854i −0.388485 − 0.562818i
\(53\) −1.27779 − 1.13202i −0.175518 − 0.155496i 0.570790 − 0.821096i \(-0.306637\pi\)
−0.746308 + 0.665601i \(0.768175\pi\)
\(54\) 0.536138 + 1.41368i 0.0729592 + 0.192378i
\(55\) 2.39245 0.322598
\(56\) 1.40876 0.188254
\(57\) −0.601309 − 1.58552i −0.0796453 − 0.210007i
\(58\) −0.201059 + 1.65587i −0.0264003 + 0.217426i
\(59\) 0.450210 − 0.236288i 0.0586123 − 0.0307621i −0.435162 − 0.900352i \(-0.643309\pi\)
0.493775 + 0.869590i \(0.335617\pi\)
\(60\) 0.819904 − 2.16191i 0.105849 − 0.279101i
\(61\) −1.61926 + 4.26963i −0.207325 + 0.546670i −0.997834 − 0.0657798i \(-0.979047\pi\)
0.790510 + 0.612450i \(0.209816\pi\)
\(62\) −8.71185 − 4.57233i −1.10641 − 0.580686i
\(63\) −1.87751 − 2.72004i −0.236544 − 0.342693i
\(64\) 0.120537 + 0.992709i 0.0150671 + 0.124089i
\(65\) −4.36662 + 2.29178i −0.541613 + 0.284260i
\(66\) 4.89812 + 2.57073i 0.602916 + 0.316435i
\(67\) 3.21919 − 0.793460i 0.393287 − 0.0969366i −0.0377126 − 0.999289i \(-0.512007\pi\)
0.431000 + 0.902352i \(0.358161\pi\)
\(68\) 2.25992 + 3.27406i 0.274056 + 0.397039i
\(69\) −1.71850 + 1.52246i −0.206883 + 0.183282i
\(70\) 0.169808 − 1.39849i 0.0202959 − 0.167152i
\(71\) 0.825122 + 0.730995i 0.0979240 + 0.0867531i 0.710664 − 0.703531i \(-0.248394\pi\)
−0.612740 + 0.790284i \(0.709933\pi\)
\(72\) 1.75608 − 1.55575i 0.206956 − 0.183347i
\(73\) 7.47655 − 3.92400i 0.875064 − 0.459269i 0.0335076 − 0.999438i \(-0.489332\pi\)
0.841556 + 0.540170i \(0.181640\pi\)
\(74\) −0.870650 + 1.26135i −0.101211 + 0.146630i
\(75\) −2.04732 − 1.07452i −0.236404 − 0.124074i
\(76\) 0.548950 − 0.486327i 0.0629689 − 0.0557855i
\(77\) 3.27247 + 0.806590i 0.372932 + 0.0919195i
\(78\) −11.4024 −1.29107
\(79\) −8.03379 + 3.80240i −0.903872 + 0.427803i
\(80\) 1.00000 0.111803
\(81\) 10.2280 + 2.52098i 1.13645 + 0.280109i
\(82\) 3.44019 − 3.04774i 0.379905 − 0.336567i
\(83\) 0.00951772 + 0.00499528i 0.00104471 + 0.000548304i 0.465245 − 0.885182i \(-0.345966\pi\)
−0.464201 + 0.885730i \(0.653658\pi\)
\(84\) 1.85035 − 2.68070i 0.201890 − 0.292488i
\(85\) 3.52260 − 1.84880i 0.382079 − 0.200531i
\(86\) −2.90998 + 2.57802i −0.313791 + 0.277995i
\(87\) 2.88682 + 2.55750i 0.309500 + 0.274193i
\(88\) −0.288378 + 2.37501i −0.0307412 + 0.253177i
\(89\) −10.0085 + 8.86676i −1.06090 + 0.939874i −0.998307 − 0.0581648i \(-0.981475\pi\)
−0.0625919 + 0.998039i \(0.519937\pi\)
\(90\) −1.33273 − 1.93080i −0.140483 − 0.203524i
\(91\) −6.74543 + 1.66260i −0.707114 + 0.174288i
\(92\) −0.879225 − 0.461453i −0.0916655 − 0.0481098i
\(93\) −20.1432 + 10.5720i −2.08875 + 1.09626i
\(94\) −0.153994 − 1.26826i −0.0158833 − 0.130811i
\(95\) −0.416613 − 0.603568i −0.0427436 − 0.0619247i
\(96\) 2.04732 + 1.07452i 0.208954 + 0.109667i
\(97\) −2.96554 + 7.81950i −0.301105 + 0.793950i 0.696009 + 0.718033i \(0.254957\pi\)
−0.997114 + 0.0759164i \(0.975812\pi\)
\(98\) −1.77848 + 4.68947i −0.179654 + 0.473708i
\(99\) 4.97000 − 2.60846i 0.499504 − 0.262160i
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.m.c.141.4 ✓ 48
79.65 even 13 inner 790.2.m.c.381.4 yes 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
790.2.m.c.141.4 ✓ 48 1.1 even 1 trivial
790.2.m.c.381.4 yes 48 79.65 even 13 inner