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Properties

Label 847.2.n.f.632.3
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 632.3
Character \(\chi\) \(=\) 847.632
Dual form 847.2.n.f.130.3

$q$-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+(1.83832 + 0.390746i) q^{2} +(-0.0682261 + 0.649128i) q^{3} +(1.39963 + 0.623157i) q^{4} +(2.36343 + 2.62485i) q^{5} +(-0.379065 + 1.16664i) q^{6} +(1.87431 + 1.86734i) q^{7} +(-0.711438 - 0.516890i) q^{8} +(2.51773 + 0.535160i) q^{9} +(3.31908 + 5.74881i) q^{10} +(-0.500000 + 0.866025i) q^{12} +(-1.36322 - 4.19556i) q^{13} +(2.71591 + 4.16515i) q^{14} +(-1.86511 + 1.35508i) q^{15} +(-3.15621 - 3.50533i) q^{16} +(-5.13427 + 1.09132i) q^{17} +(4.41927 + 1.96759i) q^{18} +(-1.65827 + 0.738311i) q^{19} +(1.67224 + 5.14662i) q^{20} +(-1.34002 + 1.08926i) q^{21} +(3.16637 - 5.48432i) q^{23} +(0.384066 - 0.426549i) q^{24} +(-0.781418 + 7.43470i) q^{25} +(-0.866631 - 8.24544i) q^{26} +(-1.12425 + 3.46009i) q^{27} +(1.45969 + 3.78158i) q^{28} +(1.55434 - 1.12930i) q^{29} +(-3.95816 + 1.76229i) q^{30} +(0.982224 - 1.09087i) q^{31} +(-3.55303 - 6.15403i) q^{32} -9.86484 q^{34} +(-0.471717 + 9.33312i) q^{35} +(3.19041 + 2.31797i) q^{36} +(-0.465503 - 4.42897i) q^{37} +(-3.33693 + 0.709285i) q^{38} +(2.81646 - 0.598658i) q^{39} +(-0.324672 - 3.08905i) q^{40} +(0.229048 + 0.166413i) q^{41} +(-2.88901 + 1.47880i) q^{42} +3.41147 q^{43} +(4.54576 + 7.87349i) q^{45} +(7.96377 - 8.84467i) q^{46} +(-4.16063 + 1.85243i) q^{47} +(2.49074 - 1.80963i) q^{48} +(0.0260485 + 6.99995i) q^{49} +(-4.34157 + 13.3620i) q^{50} +(-0.358117 - 3.40725i) q^{51} +(0.706484 - 6.72175i) q^{52} +(-4.84077 + 5.37622i) q^{53} +(-3.41875 + 5.92145i) q^{54} +(-0.368241 - 2.29731i) q^{56} +(-0.366121 - 1.12680i) q^{57} +(3.29864 - 1.46865i) q^{58} +(8.71507 + 3.88020i) q^{59} +(-3.45490 + 0.734363i) q^{60} +(0.768132 + 0.853097i) q^{61} +(2.23189 - 1.62156i) q^{62} +(3.71967 + 5.70452i) q^{63} +(-1.21174 - 3.72935i) q^{64} +(7.79086 - 13.4942i) q^{65} +(0.347296 + 0.601535i) q^{67} +(-7.86615 - 1.67200i) q^{68} +(3.34400 + 2.42956i) q^{69} +(-4.51404 + 16.9729i) q^{70} +(2.92364 - 8.99804i) q^{71} +(-1.51459 - 1.68212i) q^{72} +(-2.14436 - 0.954731i) q^{73} +(0.874860 - 8.32374i) q^{74} +(-4.77276 - 1.01448i) q^{75} -2.78106 q^{76} +5.41147 q^{78} +(-12.1478 - 2.58210i) q^{79} +(1.74149 - 16.5692i) q^{80} +(4.88500 + 2.17494i) q^{81} +(0.356037 + 0.395419i) q^{82} +(3.50201 - 10.7781i) q^{83} +(-2.55432 + 0.689525i) q^{84} +(-14.9990 - 10.8974i) q^{85} +(6.27137 + 1.33302i) q^{86} +(0.627011 + 1.08602i) q^{87} +(1.73396 - 3.00330i) q^{89} +(5.28001 + 16.2502i) q^{90} +(5.27947 - 10.4094i) q^{91} +(7.84935 - 5.70289i) q^{92} +(0.641101 + 0.712015i) q^{93} +(-8.37239 + 1.77961i) q^{94} +(-5.85717 - 2.60778i) q^{95} +(4.23717 - 1.88651i) q^{96} +(4.74258 + 14.5961i) q^{97} +(-2.68732 + 12.8783i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.83832 + 0.390746i 1.29989 + 0.276299i 0.805320 − 0.592841i \(-0.201994\pi\)
0.494566 + 0.869140i \(0.335327\pi\)
\(3\) −0.0682261 + 0.649128i −0.0393904 + 0.374774i 0.957014 + 0.290043i \(0.0936697\pi\)
−0.996404 + 0.0847309i \(0.972997\pi\)
\(4\) 1.39963 + 0.623157i 0.699816 + 0.311578i
\(5\) 2.36343 + 2.62485i 1.05696 + 1.17387i 0.984298 + 0.176513i \(0.0564819\pi\)
0.0726592 + 0.997357i \(0.476851\pi\)
\(6\) −0.379065 + 1.16664i −0.154753 + 0.476280i
\(7\) 1.87431 + 1.86734i 0.708421 + 0.705790i
\(8\) −0.711438 − 0.516890i −0.251531 − 0.182748i
\(9\) 2.51773 + 0.535160i 0.839243 + 0.178387i
\(10\) 3.31908 + 5.74881i 1.04958 + 1.81793i
\(11\) 0 0
\(12\) −0.500000 + 0.866025i −0.144338 + 0.250000i
\(13\) −1.36322 − 4.19556i −0.378089 − 1.16364i −0.941371 − 0.337374i \(-0.890461\pi\)
0.563281 − 0.826265i \(-0.309539\pi\)
\(14\) 2.71591 + 4.16515i 0.725857 + 1.11318i
\(15\) −1.86511 + 1.35508i −0.481570 + 0.349881i
\(16\) −3.15621 − 3.50533i −0.789052 − 0.876332i
\(17\) −5.13427 + 1.09132i −1.24524 + 0.264685i −0.782974 − 0.622055i \(-0.786298\pi\)
−0.462269 + 0.886740i \(0.652965\pi\)
\(18\) 4.41927 + 1.96759i 1.04163 + 0.463765i
\(19\) −1.65827 + 0.738311i −0.380434 + 0.169380i −0.588039 − 0.808833i \(-0.700100\pi\)
0.207605 + 0.978213i \(0.433433\pi\)
\(20\) 1.67224 + 5.14662i 0.373924 + 1.15082i
\(21\) −1.34002 + 1.08926i −0.292417 + 0.237697i
\(22\) 0 0
\(23\) 3.16637 − 5.48432i 0.660235 − 1.14356i −0.320319 − 0.947310i \(-0.603790\pi\)
0.980554 − 0.196250i \(-0.0628765\pi\)
\(24\) 0.384066 − 0.426549i 0.0783972 − 0.0870689i
\(25\) −0.781418 + 7.43470i −0.156284 + 1.48694i
\(26\) −0.866631 − 8.24544i −0.169960 − 1.61706i
\(27\) −1.12425 + 3.46009i −0.216362 + 0.665895i
\(28\) 1.45969 + 3.78158i 0.275856 + 0.714652i
\(29\) 1.55434 − 1.12930i 0.288634 − 0.209705i −0.434040 − 0.900893i \(-0.642912\pi\)
0.722675 + 0.691188i \(0.242912\pi\)
\(30\) −3.95816 + 1.76229i −0.722658 + 0.321748i
\(31\) 0.982224 − 1.09087i 0.176413 − 0.195926i −0.648453 − 0.761254i \(-0.724584\pi\)
0.824866 + 0.565328i \(0.191251\pi\)
\(32\) −3.55303 − 6.15403i −0.628094 − 1.08789i
\(33\) 0 0
\(34\) −9.86484 −1.69181
\(35\) −0.471717 + 9.33312i −0.0797346 + 1.57758i
\(36\) 3.19041 + 2.31797i 0.531735 + 0.386328i
\(37\) −0.465503 − 4.42897i −0.0765283 − 0.728118i −0.963756 − 0.266786i \(-0.914038\pi\)
0.887228 − 0.461332i \(-0.152628\pi\)
\(38\) −3.33693 + 0.709285i −0.541321 + 0.115061i
\(39\) 2.81646 − 0.598658i 0.450995 − 0.0958620i
\(40\) −0.324672 − 3.08905i −0.0513352 − 0.488422i
\(41\) 0.229048 + 0.166413i 0.0357712 + 0.0259893i 0.605527 − 0.795825i \(-0.292962\pi\)
−0.569756 + 0.821814i \(0.692962\pi\)
\(42\) −2.88901 + 1.47880i −0.445784 + 0.228184i
\(43\) 3.41147 0.520245 0.260122 − 0.965576i \(-0.416237\pi\)
0.260122 + 0.965576i \(0.416237\pi\)
\(44\) 0 0
\(45\) 4.54576 + 7.87349i 0.677642 + 1.17371i
\(46\) 7.96377 − 8.84467i 1.17419 − 1.30408i
\(47\) −4.16063 + 1.85243i −0.606890 + 0.270205i −0.687088 − 0.726574i \(-0.741111\pi\)
0.0801979 + 0.996779i \(0.474445\pi\)
\(48\) 2.49074 − 1.80963i 0.359508 − 0.261198i
\(49\) 0.0260485 + 6.99995i 0.00372122 + 0.999993i
\(50\) −4.34157 + 13.3620i −0.613991 + 1.88967i
\(51\) −0.358117 − 3.40725i −0.0501464 − 0.477111i
\(52\) 0.706484 − 6.72175i 0.0979717 − 0.932138i
\(53\) −4.84077 + 5.37622i −0.664931 + 0.738481i −0.977389 − 0.211451i \(-0.932181\pi\)
0.312457 + 0.949932i \(0.398848\pi\)
\(54\) −3.41875 + 5.92145i −0.465233 + 0.805807i
\(55\) 0 0
\(56\) −0.368241 − 2.29731i −0.0492083 − 0.306991i
\(57\) −0.366121 − 1.12680i −0.0484939 − 0.149249i
\(58\) 3.29864 − 1.46865i 0.433133 − 0.192843i
\(59\) 8.71507 + 3.88020i 1.13461 + 0.505159i 0.886110 − 0.463474i \(-0.153397\pi\)
0.248495 + 0.968633i \(0.420064\pi\)
\(60\) −3.45490 + 0.734363i −0.446026 + 0.0948058i
\(61\) 0.768132 + 0.853097i 0.0983493 + 0.109228i 0.790310 − 0.612707i \(-0.209920\pi\)
−0.691961 + 0.721935i \(0.743253\pi\)
\(62\) 2.23189 − 1.62156i 0.283451 − 0.205939i
\(63\) 3.71967 + 5.70452i 0.468634 + 0.718702i
\(64\) −1.21174 − 3.72935i −0.151468 − 0.466169i
\(65\) 7.79086 − 13.4942i 0.966337 − 1.67375i
\(66\) 0 0
\(67\) 0.347296 + 0.601535i 0.0424290 + 0.0734892i 0.886460 − 0.462805i \(-0.153157\pi\)
−0.844031 + 0.536294i \(0.819824\pi\)
\(68\) −7.86615 − 1.67200i −0.953911 − 0.202760i
\(69\) 3.34400 + 2.42956i 0.402570 + 0.292484i
\(70\) −4.51404 + 16.9729i −0.539531 + 2.02865i
\(71\) 2.92364 − 8.99804i 0.346972 − 1.06787i −0.613547 − 0.789658i \(-0.710258\pi\)
0.960519 − 0.278213i \(-0.0897422\pi\)
\(72\) −1.51459 − 1.68212i −0.178496 − 0.198240i
\(73\) −2.14436 − 0.954731i −0.250979 − 0.111743i 0.277392 − 0.960757i \(-0.410530\pi\)
−0.528370 + 0.849014i \(0.677197\pi\)
\(74\) 0.874860 − 8.32374i 0.101700 − 0.967615i
\(75\) −4.77276 − 1.01448i −0.551111 − 0.117142i
\(76\) −2.78106 −0.319009
\(77\) 0 0
\(78\) 5.41147 0.612729
\(79\) −12.1478 − 2.58210i −1.36674 − 0.290509i −0.534612 − 0.845097i \(-0.679542\pi\)
−0.832125 + 0.554588i \(0.812876\pi\)
\(80\) 1.74149 − 16.5692i 0.194705 − 1.85249i
\(81\) 4.88500 + 2.17494i 0.542778 + 0.241660i
\(82\) 0.356037 + 0.395419i 0.0393177 + 0.0436667i
\(83\) 3.50201 − 10.7781i 0.384396 − 1.18305i −0.552522 − 0.833498i \(-0.686334\pi\)
0.936918 − 0.349550i \(-0.113666\pi\)
\(84\) −2.55432 + 0.689525i −0.278699 + 0.0752333i
\(85\) −14.9990 − 10.8974i −1.62687 − 1.18199i
\(86\) 6.27137 + 1.33302i 0.676259 + 0.143743i
\(87\) 0.627011 + 1.08602i 0.0672227 + 0.116433i
\(88\) 0 0
\(89\) 1.73396 − 3.00330i 0.183799 − 0.318349i −0.759372 − 0.650656i \(-0.774494\pi\)
0.943171 + 0.332307i \(0.107827\pi\)
\(90\) 5.28001 + 16.2502i 0.556562 + 1.71292i
\(91\) 5.27947 − 10.4094i 0.553438 − 1.09120i
\(92\) 7.84935 − 5.70289i 0.818352 − 0.594567i
\(93\) 0.641101 + 0.712015i 0.0664791 + 0.0738325i
\(94\) −8.37239 + 1.77961i −0.863546 + 0.183552i
\(95\) −5.85717 − 2.60778i −0.600933 − 0.267553i
\(96\) 4.23717 − 1.88651i 0.432454 − 0.192541i
\(97\) 4.74258 + 14.5961i 0.481536 + 1.48201i 0.836936 + 0.547300i \(0.184344\pi\)
−0.355401 + 0.934714i \(0.615656\pi\)
\(98\) −2.68732 + 12.8783i −0.271460 + 1.30090i
\(99\) 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.632.3 24
7.4 even 3 inner 847.2.n.f.753.1 24
11.2 odd 10 847.2.n.g.9.3 24
11.3 even 5 847.2.e.c.485.3 6
11.4 even 5 inner 847.2.n.f.366.3 24
11.5 even 5 inner 847.2.n.f.807.1 24
11.6 odd 10 847.2.n.g.807.3 24
11.7 odd 10 847.2.n.g.366.1 24
11.8 odd 10 77.2.e.a.23.1 ✓ 6
11.9 even 5 inner 847.2.n.f.9.1 24
11.10 odd 2 847.2.n.g.632.1 24
33.8 even 10 693.2.i.h.100.3 6
44.19 even 10 1232.2.q.m.177.2 6
77.4 even 15 inner 847.2.n.f.487.1 24
77.18 odd 30 847.2.n.g.487.3 24
77.19 even 30 539.2.a.g.1.3 3
77.25 even 15 847.2.e.c.606.3 6
77.30 odd 30 539.2.a.j.1.3 3
77.32 odd 6 847.2.n.g.753.3 24
77.39 odd 30 847.2.n.g.81.1 24
77.41 even 10 539.2.e.m.177.1 6
77.46 odd 30 847.2.n.g.130.1 24
77.47 odd 30 5929.2.a.u.1.1 3
77.52 even 30 539.2.e.m.67.1 6
77.53 even 15 inner 847.2.n.f.130.3 24
77.58 even 15 5929.2.a.x.1.1 3
77.60 even 15 inner 847.2.n.f.81.3 24
77.74 odd 30 77.2.e.a.67.1 yes 6
231.74 even 30 693.2.i.h.298.3 6
231.107 even 30 4851.2.a.bj.1.1 3
231.173 odd 30 4851.2.a.bk.1.1 3
308.19 odd 30 8624.2.a.co.1.2 3
308.107 even 30 8624.2.a.ch.1.2 3
308.151 even 30 1232.2.q.m.529.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.2.e.a.23.1 ✓ 6 11.8 odd 10
77.2.e.a.67.1 yes 6 77.74 odd 30
539.2.a.g.1.3 3 77.19 even 30
539.2.a.j.1.3 3 77.30 odd 30
539.2.e.m.67.1 6 77.52 even 30
539.2.e.m.177.1 6 77.41 even 10
693.2.i.h.100.3 6 33.8 even 10
693.2.i.h.298.3 6 231.74 even 30
847.2.e.c.485.3 6 11.3 even 5
847.2.e.c.606.3 6 77.25 even 15
847.2.n.f.9.1 24 11.9 even 5 inner
847.2.n.f.81.3 24 77.60 even 15 inner
847.2.n.f.130.3 24 77.53 even 15 inner
847.2.n.f.366.3 24 11.4 even 5 inner
847.2.n.f.487.1 24 77.4 even 15 inner
847.2.n.f.632.3 24 1.1 even 1 trivial
847.2.n.f.753.1 24 7.4 even 3 inner
847.2.n.f.807.1 24 11.5 even 5 inner
847.2.n.g.9.3 24 11.2 odd 10
847.2.n.g.81.1 24 77.39 odd 30
847.2.n.g.130.1 24 77.46 odd 30
847.2.n.g.366.1 24 11.7 odd 10
847.2.n.g.487.3 24 77.18 odd 30
847.2.n.g.632.1 24 11.10 odd 2
847.2.n.g.753.3 24 77.32 odd 6
847.2.n.g.807.3 24 11.6 odd 10
1232.2.q.m.177.2 6 44.19 even 10
1232.2.q.m.529.2 6 308.151 even 30
4851.2.a.bj.1.1 3 231.107 even 30
4851.2.a.bk.1.1 3 231.173 odd 30
5929.2.a.u.1.1 3 77.47 odd 30
5929.2.a.x.1.1 3 77.58 even 15
8624.2.a.ch.1.2 3 308.107 even 30
8624.2.a.co.1.2 3 308.19 odd 30