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Properties

Label 847.2.n.f.632.2
Level $847$
Weight $2$
Character 847.632
Analytic conductor $6.763$
Analytic rank $0$
Dimension $24$
Inner twists $8$

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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] = [847,2,Mod(9,847)] 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("847.9"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(847, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([10, 18])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 847.n (of order \(15\), degree \(8\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,0,3,0,6,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.76332905120\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(3\) over \(\Q(\zeta_{15})\)
Twist minimal: no (minimal twist has level 77)
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label 632.2
Character \(\chi\) \(=\) 847.632
Dual form 847.2.n.f.130.2

$q$-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+(-0.339707 - 0.0722070i) q^{2} +(0.0556184 - 0.529174i) q^{3} +(-1.71690 - 0.764415i) q^{4} +(0.0807070 + 0.0896342i) q^{5} +(-0.0571040 + 0.175748i) q^{6} +(-2.55432 + 0.689525i) q^{7} +(1.08999 + 0.791921i) q^{8} +(2.65751 + 0.564871i) q^{9} +(-0.0209445 - 0.0362770i) q^{10} +(-0.500000 + 0.866025i) q^{12} +(0.379065 + 1.16664i) q^{13} +(0.917509 - 0.0497966i) q^{14} +(0.0519209 - 0.0377227i) q^{15} +(2.20201 + 2.44559i) q^{16} +(6.03541 - 1.28287i) q^{17} +(-0.861988 - 0.383782i) q^{18} +(-5.85717 + 2.60778i) q^{19} +(-0.0700485 - 0.215587i) q^{20} +(0.222811 + 1.39003i) q^{21} +(1.01114 - 1.75135i) q^{23} +(0.479687 - 0.532747i) q^{24} +(0.521122 - 4.95814i) q^{25} +(-0.0445314 - 0.423688i) q^{26} +(0.939995 - 2.89301i) q^{27} +(4.91261 + 0.768713i) q^{28} +(2.62847 - 1.90970i) q^{29} +(-0.0203617 + 0.00906563i) q^{30} +(3.26495 - 3.62609i) q^{31} +(-1.91875 - 3.32337i) q^{32} -2.14290 q^{34} +(-0.267957 - 0.173305i) q^{35} +(-4.13089 - 3.00127i) q^{36} +(-0.247689 - 2.35660i) q^{37} +(2.17802 - 0.462953i) q^{38} +(0.638441 - 0.135705i) q^{39} +(0.0169863 + 0.161614i) q^{40} +(6.70745 + 4.87324i) q^{41} +(0.0246794 - 0.488292i) q^{42} -2.22668 q^{43} +(0.163848 + 0.283793i) q^{45} +(-0.469953 + 0.521936i) q^{46} +(8.43606 - 3.75598i) q^{47} +(1.41661 - 1.02923i) q^{48} +(6.04911 - 3.52253i) q^{49} +(-0.535041 + 1.64669i) q^{50} +(-0.343179 - 3.26513i) q^{51} +(0.240981 - 2.29278i) q^{52} +(6.47721 - 7.19368i) q^{53} +(-0.528218 + 0.914901i) q^{54} +(-3.33022 - 1.27125i) q^{56} +(1.05420 + 3.24450i) q^{57} +(-1.03080 + 0.458944i) q^{58} +(-8.90179 - 3.96333i) q^{59} +(-0.117979 + 0.0250772i) q^{60} +(0.959375 + 1.06549i) q^{61} +(-1.37095 + 0.996057i) q^{62} +(-7.17763 + 0.389556i) q^{63} +(-1.62202 - 4.99207i) q^{64} +(-0.0739780 + 0.128134i) q^{65} +(1.53209 + 2.65366i) q^{67} +(-11.3429 - 2.41100i) q^{68} +(-0.870532 - 0.632479i) q^{69} +(0.0785129 + 0.0782213i) q^{70} +(-2.62518 + 8.07947i) q^{71} +(2.44932 + 2.72024i) q^{72} +(-3.22672 - 1.43663i) q^{73} +(-0.0860215 + 0.818440i) q^{74} +(-2.59474 - 0.551528i) q^{75} +12.0496 q^{76} -0.226682 q^{78} +(8.89285 + 1.89023i) q^{79} +(-0.0414901 + 0.394752i) q^{80} +(5.96736 + 2.65684i) q^{81} +(-1.92668 - 2.13980i) q^{82} +(2.17001 - 6.67859i) q^{83} +(0.680015 - 2.55687i) q^{84} +(0.602089 + 0.437443i) q^{85} +(0.756420 + 0.160782i) q^{86} +(-0.864370 - 1.49713i) q^{87} +(3.43969 - 5.95772i) q^{89} +(-0.0351685 - 0.108237i) q^{90} +(-1.77268 - 2.71861i) q^{91} +(-3.07480 + 2.23397i) q^{92} +(-1.73724 - 1.92940i) q^{93} +(-3.13700 + 0.666790i) q^{94} +(-0.706462 - 0.314537i) q^{95} +(-1.86536 + 0.830511i) q^{96} +(5.10870 + 15.7230i) q^{97} +(-2.30928 + 0.759842i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 3 q^{3} + 6 q^{5} - 6 q^{6} + 6 q^{8} + 12 q^{10} - 12 q^{12} + 6 q^{13} + 12 q^{14} - 18 q^{15} - 6 q^{16} + 3 q^{17} + 12 q^{18} - 9 q^{19} - 12 q^{20} + 48 q^{21} - 6 q^{24} + 3 q^{25} + 9 q^{26}+ \cdots + 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/847\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(365\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.339707 − 0.0722070i −0.240209 − 0.0510580i 0.0862338 − 0.996275i \(-0.472517\pi\)
−0.326443 + 0.945217i \(0.605850\pi\)
\(3\) 0.0556184 − 0.529174i 0.0321113 − 0.305519i −0.966664 − 0.256047i \(-0.917580\pi\)
0.998776 − 0.0494714i \(-0.0157537\pi\)
\(4\) −1.71690 − 0.764415i −0.858452 − 0.382207i
\(5\) 0.0807070 + 0.0896342i 0.0360933 + 0.0400856i 0.760921 − 0.648844i \(-0.224747\pi\)
−0.724828 + 0.688930i \(0.758081\pi\)
\(6\) −0.0571040 + 0.175748i −0.0233126 + 0.0717489i
\(7\) −2.55432 + 0.689525i −0.965443 + 0.260616i
\(8\) 1.08999 + 0.791921i 0.385368 + 0.279986i
\(9\) 2.65751 + 0.564871i 0.885837 + 0.188290i
\(10\) −0.0209445 − 0.0362770i −0.00662324 − 0.0114718i
\(11\) 0 0
\(12\) −0.500000 + 0.866025i −0.144338 + 0.250000i
\(13\) 0.379065 + 1.16664i 0.105134 + 0.323569i 0.989762 − 0.142729i \(-0.0455877\pi\)
−0.884628 + 0.466297i \(0.845588\pi\)
\(14\) 0.917509 − 0.0497966i 0.245215 − 0.0133087i
\(15\) 0.0519209 − 0.0377227i 0.0134059 − 0.00973997i
\(16\) 2.20201 + 2.44559i 0.550504 + 0.611396i
\(17\) 6.03541 − 1.28287i 1.46380 − 0.311141i 0.593969 − 0.804488i \(-0.297560\pi\)
0.869833 + 0.493347i \(0.164227\pi\)
\(18\) −0.861988 − 0.383782i −0.203172 − 0.0904582i
\(19\) −5.85717 + 2.60778i −1.34373 + 0.598266i −0.947462 − 0.319870i \(-0.896361\pi\)
−0.396266 + 0.918136i \(0.629694\pi\)
\(20\) −0.0700485 − 0.215587i −0.0156633 − 0.0482067i
\(21\) 0.222811 + 1.39003i 0.0486214 + 0.303330i
\(22\) 0 0
\(23\) 1.01114 − 1.75135i 0.210838 − 0.365182i −0.741139 − 0.671352i \(-0.765714\pi\)
0.951977 + 0.306169i \(0.0990474\pi\)
\(24\) 0.479687 − 0.532747i 0.0979158 − 0.108746i
\(25\) 0.521122 − 4.95814i 0.104224 − 0.991628i
\(26\) −0.0445314 − 0.423688i −0.00873333 − 0.0830921i
\(27\) 0.939995 − 2.89301i 0.180902 − 0.556760i
\(28\) 4.91261 + 0.768713i 0.928395 + 0.145273i
\(29\) 2.62847 − 1.90970i 0.488095 − 0.354622i −0.316356 − 0.948640i \(-0.602459\pi\)
0.804451 + 0.594019i \(0.202459\pi\)
\(30\) −0.0203617 + 0.00906563i −0.00371753 + 0.00165515i
\(31\) 3.26495 − 3.62609i 0.586402 − 0.651265i −0.374802 − 0.927105i \(-0.622289\pi\)
0.961204 + 0.275840i \(0.0889560\pi\)
\(32\) −1.91875 − 3.32337i −0.339190 − 0.587494i
\(33\) 0 0
\(34\) −2.14290 −0.367505
\(35\) −0.267957 − 0.173305i −0.0452929 − 0.0292939i
\(36\) −4.13089 − 3.00127i −0.688482 − 0.500212i
\(37\) −0.247689 − 2.35660i −0.0407198 − 0.387423i −0.995834 − 0.0911852i \(-0.970934\pi\)
0.955114 − 0.296238i \(-0.0957322\pi\)
\(38\) 2.17802 − 0.462953i 0.353322 − 0.0751009i
\(39\) 0.638441 − 0.135705i 0.102232 − 0.0217301i
\(40\) 0.0169863 + 0.161614i 0.00268577 + 0.0255534i
\(41\) 6.70745 + 4.87324i 1.04753 + 0.761073i 0.971741 − 0.236051i \(-0.0758533\pi\)
0.0757864 + 0.997124i \(0.475853\pi\)
\(42\) 0.0246794 − 0.488292i 0.00380811 − 0.0753451i
\(43\) −2.22668 −0.339566 −0.169783 − 0.985481i \(-0.554307\pi\)
−0.169783 + 0.985481i \(0.554307\pi\)
\(44\) 0 0
\(45\) 0.163848 + 0.283793i 0.0244250 + 0.0423054i
\(46\) −0.469953 + 0.521936i −0.0692908 + 0.0769552i
\(47\) 8.43606 − 3.75598i 1.23053 − 0.547866i 0.314605 − 0.949223i \(-0.398128\pi\)
0.915922 + 0.401357i \(0.131461\pi\)
\(48\) 1.41661 − 1.02923i 0.204470 − 0.148556i
\(49\) 6.04911 − 3.52253i 0.864159 − 0.503219i
\(50\) −0.535041 + 1.64669i −0.0756662 + 0.232877i
\(51\) −0.343179 − 3.26513i −0.0480547 − 0.457210i
\(52\) 0.240981 − 2.29278i 0.0334180 − 0.317951i
\(53\) 6.47721 − 7.19368i 0.889714 − 0.988127i −0.110270 − 0.993902i \(-0.535171\pi\)
0.999983 + 0.00577444i \(0.00183807\pi\)
\(54\) −0.528218 + 0.914901i −0.0718814 + 0.124502i
\(55\) 0 0
\(56\) −3.33022 − 1.27125i −0.445020 − 0.169878i
\(57\) 1.05420 + 3.24450i 0.139633 + 0.429745i
\(58\) −1.03080 + 0.458944i −0.135351 + 0.0602622i
\(59\) −8.90179 − 3.96333i −1.15891 − 0.515982i −0.265013 − 0.964245i \(-0.585376\pi\)
−0.893902 + 0.448263i \(0.852043\pi\)
\(60\) −0.117979 + 0.0250772i −0.0152310 + 0.00323745i
\(61\) 0.959375 + 1.06549i 0.122835 + 0.136422i 0.801425 − 0.598095i \(-0.204075\pi\)
−0.678590 + 0.734517i \(0.737409\pi\)
\(62\) −1.37095 + 0.996057i −0.174111 + 0.126499i
\(63\) −7.17763 + 0.389556i −0.904296 + 0.0490795i
\(64\) −1.62202 − 4.99207i −0.202753 − 0.624008i
\(65\) −0.0739780 + 0.128134i −0.00917584 + 0.0158930i
\(66\) 0 0
\(67\) 1.53209 + 2.65366i 0.187174 + 0.324196i 0.944307 − 0.329066i \(-0.106734\pi\)
−0.757133 + 0.653261i \(0.773400\pi\)
\(68\) −11.3429 − 2.41100i −1.37552 − 0.292377i
\(69\) −0.870532 − 0.632479i −0.104800 − 0.0761415i
\(70\) 0.0785129 + 0.0782213i 0.00938409 + 0.00934924i
\(71\) −2.62518 + 8.07947i −0.311551 + 0.958856i 0.665600 + 0.746309i \(0.268176\pi\)
−0.977151 + 0.212547i \(0.931824\pi\)
\(72\) 2.44932 + 2.72024i 0.288655 + 0.320583i
\(73\) −3.22672 − 1.43663i −0.377659 − 0.168145i 0.209123 − 0.977889i \(-0.432939\pi\)
−0.586783 + 0.809744i \(0.699606\pi\)
\(74\) −0.0860215 + 0.818440i −0.00999980 + 0.0951417i
\(75\) −2.59474 − 0.551528i −0.299614 − 0.0636850i
\(76\) 12.0496 1.38219
\(77\) 0 0
\(78\) −0.226682 −0.0256666
\(79\) 8.89285 + 1.89023i 1.00052 + 0.212668i 0.678924 − 0.734208i \(-0.262446\pi\)
0.321599 + 0.946876i \(0.395780\pi\)
\(80\) −0.0414901 + 0.394752i −0.00463873 + 0.0441346i
\(81\) 5.96736 + 2.65684i 0.663040 + 0.295204i
\(82\) −1.92668 − 2.13980i −0.212767 − 0.236301i
\(83\) 2.17001 − 6.67859i 0.238189 − 0.733071i −0.758493 − 0.651681i \(-0.774064\pi\)
0.996682 − 0.0813899i \(-0.0259359\pi\)
\(84\) 0.680015 − 2.55687i 0.0741957 − 0.278977i
\(85\) 0.602089 + 0.437443i 0.0653057 + 0.0474474i
\(86\) 0.756420 + 0.160782i 0.0815668 + 0.0173376i
\(87\) −0.864370 − 1.49713i −0.0926702 − 0.160510i
\(88\) 0 0
\(89\) 3.43969 − 5.95772i 0.364607 − 0.631517i −0.624106 − 0.781339i \(-0.714537\pi\)
0.988713 + 0.149822i \(0.0478701\pi\)
\(90\) −0.0351685 − 0.108237i −0.00370708 − 0.0114092i
\(91\) −1.77268 − 2.71861i −0.185828 − 0.284987i
\(92\) −3.07480 + 2.23397i −0.320570 + 0.232908i
\(93\) −1.73724 − 1.92940i −0.180144 − 0.200070i
\(94\) −3.13700 + 0.666790i −0.323557 + 0.0687741i
\(95\) −0.706462 − 0.314537i −0.0724814 − 0.0322708i
\(96\) −1.86536 + 0.830511i −0.190382 + 0.0847637i
\(97\) 5.10870 + 15.7230i 0.518710 + 1.59642i 0.776429 + 0.630204i \(0.217029\pi\)
−0.257720 + 0.966220i \(0.582971\pi\)
\(98\) −2.30928 + 0.759842i −0.233272 + 0.0767556i
\(99\) 0 0
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 847.2.n.f.632.2 24
7.4 even 3 inner 847.2.n.f.753.2 24
11.2 odd 10 847.2.n.g.9.2 24
11.3 even 5 847.2.e.c.485.2 6
11.4 even 5 inner 847.2.n.f.366.2 24
11.5 even 5 inner 847.2.n.f.807.2 24
11.6 odd 10 847.2.n.g.807.2 24
11.7 odd 10 847.2.n.g.366.2 24
11.8 odd 10 77.2.e.a.23.2 ✓ 6
11.9 even 5 inner 847.2.n.f.9.2 24
11.10 odd 2 847.2.n.g.632.2 24
33.8 even 10 693.2.i.h.100.2 6
44.19 even 10 1232.2.q.m.177.1 6
77.4 even 15 inner 847.2.n.f.487.2 24
77.18 odd 30 847.2.n.g.487.2 24
77.19 even 30 539.2.a.g.1.2 3
77.25 even 15 847.2.e.c.606.2 6
77.30 odd 30 539.2.a.j.1.2 3
77.32 odd 6 847.2.n.g.753.2 24
77.39 odd 30 847.2.n.g.81.2 24
77.41 even 10 539.2.e.m.177.2 6
77.46 odd 30 847.2.n.g.130.2 24
77.47 odd 30 5929.2.a.u.1.2 3
77.52 even 30 539.2.e.m.67.2 6
77.53 even 15 inner 847.2.n.f.130.2 24
77.58 even 15 5929.2.a.x.1.2 3
77.60 even 15 inner 847.2.n.f.81.2 24
77.74 odd 30 77.2.e.a.67.2 yes 6
231.74 even 30 693.2.i.h.298.2 6
231.107 even 30 4851.2.a.bj.1.2 3
231.173 odd 30 4851.2.a.bk.1.2 3
308.19 odd 30 8624.2.a.co.1.1 3
308.107 even 30 8624.2.a.ch.1.3 3
308.151 even 30 1232.2.q.m.529.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.2.e.a.23.2 ✓ 6 11.8 odd 10
77.2.e.a.67.2 yes 6 77.74 odd 30
539.2.a.g.1.2 3 77.19 even 30
539.2.a.j.1.2 3 77.30 odd 30
539.2.e.m.67.2 6 77.52 even 30
539.2.e.m.177.2 6 77.41 even 10
693.2.i.h.100.2 6 33.8 even 10
693.2.i.h.298.2 6 231.74 even 30
847.2.e.c.485.2 6 11.3 even 5
847.2.e.c.606.2 6 77.25 even 15
847.2.n.f.9.2 24 11.9 even 5 inner
847.2.n.f.81.2 24 77.60 even 15 inner
847.2.n.f.130.2 24 77.53 even 15 inner
847.2.n.f.366.2 24 11.4 even 5 inner
847.2.n.f.487.2 24 77.4 even 15 inner
847.2.n.f.632.2 24 1.1 even 1 trivial
847.2.n.f.753.2 24 7.4 even 3 inner
847.2.n.f.807.2 24 11.5 even 5 inner
847.2.n.g.9.2 24 11.2 odd 10
847.2.n.g.81.2 24 77.39 odd 30
847.2.n.g.130.2 24 77.46 odd 30
847.2.n.g.366.2 24 11.7 odd 10
847.2.n.g.487.2 24 77.18 odd 30
847.2.n.g.632.2 24 11.10 odd 2
847.2.n.g.753.2 24 77.32 odd 6
847.2.n.g.807.2 24 11.6 odd 10
1232.2.q.m.177.1 6 44.19 even 10
1232.2.q.m.529.1 6 308.151 even 30
4851.2.a.bj.1.2 3 231.107 even 30
4851.2.a.bk.1.2 3 231.173 odd 30
5929.2.a.u.1.2 3 77.47 odd 30
5929.2.a.x.1.2 3 77.58 even 15
8624.2.a.ch.1.3 3 308.107 even 30
8624.2.a.co.1.1 3 308.19 odd 30