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Properties

Label 847.2.n.g.81.2
Level $847$
Weight $2$
Character 847.81
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 81.2
Character \(\chi\) \(=\) 847.81
Dual form 847.2.n.g.366.2

$q$-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+(0.0363024 - 0.345394i) q^{2} +(-0.356037 - 0.395419i) q^{3} +(1.83832 + 0.390746i) q^{4} +(0.110187 + 0.0490584i) q^{5} +(-0.149500 + 0.108618i) q^{6} +(-1.66120 - 2.05923i) q^{7} +(0.416337 - 1.28135i) q^{8} +(0.283991 - 2.70200i) q^{9} +(0.0209445 - 0.0362770i) q^{10} +(-0.500000 - 0.866025i) q^{12} +(0.992406 + 0.721025i) q^{13} +(-0.771550 + 0.499012i) q^{14} +(-0.0198320 - 0.0610367i) q^{15} +(3.00635 + 1.33851i) q^{16} +(-0.644966 - 6.13644i) q^{17} +(-0.922944 - 0.196178i) q^{18} +(-6.27137 + 1.33302i) q^{19} +(0.183389 + 0.133240i) q^{20} +(-0.222811 + 1.39003i) q^{21} +(1.01114 + 1.75135i) q^{23} +(-0.654904 + 0.291582i) q^{24} +(-3.33592 - 3.70491i) q^{25} +(0.285064 - 0.316596i) q^{26} +(-2.46094 + 1.78798i) q^{27} +(-2.24917 - 4.43462i) q^{28} +(1.00399 + 3.08995i) q^{29} +(-0.0218016 + 0.00463408i) q^{30} +(4.45754 - 1.98462i) q^{31} +(1.91875 - 3.32337i) q^{32} -2.14290 q^{34} +(-0.0820198 - 0.308396i) q^{35} +(1.57786 - 4.85616i) q^{36} +(1.58556 - 1.76094i) q^{37} +(0.232752 + 2.21448i) q^{38} +(-0.0682261 - 0.649128i) q^{39} +(0.108736 - 0.120764i) q^{40} +(2.56202 - 7.88508i) q^{41} +(0.472019 + 0.127419i) q^{42} +2.22668 q^{43} +(0.163848 - 0.283793i) q^{45} +(0.641614 - 0.285665i) q^{46} +(-9.03263 + 1.91994i) q^{47} +(-0.541098 - 1.66533i) q^{48} +(-1.48085 + 6.84157i) q^{49} +(-1.40076 + 1.01771i) q^{50} +(-2.19684 + 2.43983i) q^{51} +(1.54262 + 1.71325i) q^{52} +(8.84316 - 3.93723i) q^{53} +(0.528218 + 0.914901i) q^{54} +(-3.33022 + 1.27125i) q^{56} +(2.75994 + 2.00521i) q^{57} +(1.10370 - 0.234598i) q^{58} +(9.53129 + 2.02594i) q^{59} +(-0.0126077 - 0.119954i) q^{60} +(-1.30981 - 0.583164i) q^{61} +(-0.523658 - 1.61165i) q^{62} +(-6.03580 + 3.90375i) q^{63} +(4.24651 + 3.08527i) q^{64} +(0.0739780 + 0.128134i) q^{65} +(1.53209 - 2.65366i) q^{67} +(1.21214 - 11.5327i) q^{68} +(0.332514 - 1.02337i) q^{69} +(-0.109496 + 0.0171336i) q^{70} +(6.87280 - 4.99338i) q^{71} +(-3.34398 - 1.48884i) q^{72} +(-3.45490 - 0.734363i) q^{73} +(-0.550660 - 0.611570i) q^{74} +(-0.277283 + 2.63817i) q^{75} -12.0496 q^{76} -0.226682 q^{78} +(-0.950323 + 9.04171i) q^{79} +(0.265595 + 0.294974i) q^{80} +(-6.38935 - 1.35810i) q^{81} +(-2.63045 - 1.17115i) q^{82} +(5.68115 - 4.12760i) q^{83} +(-0.952747 + 2.46825i) q^{84} +(0.229977 - 0.707798i) q^{85} +(0.0808338 - 0.769082i) q^{86} +(0.864370 - 1.49713i) q^{87} +(3.43969 + 5.95772i) q^{89} +(-0.0920723 - 0.0668944i) q^{90} +(-0.163825 - 3.24136i) q^{91} +(1.17447 + 3.61464i) q^{92} +(-2.37181 - 1.05600i) q^{93} +(0.335231 + 3.18951i) q^{94} +(-0.756420 - 0.160782i) q^{95} +(-1.99727 + 0.424533i) q^{96} +(-13.3747 - 9.71732i) 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 \(\left(\mathbb{Z}/847\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.0363024 − 0.345394i 0.0256696 − 0.244230i −0.974161 − 0.225853i \(-0.927483\pi\)
0.999831 − 0.0183777i \(-0.00585012\pi\)
\(3\) −0.356037 − 0.395419i −0.205558 − 0.228295i 0.631547 − 0.775338i \(-0.282420\pi\)
−0.837105 + 0.547042i \(0.815754\pi\)
\(4\) 1.83832 + 0.390746i 0.919158 + 0.195373i
\(5\) 0.110187 + 0.0490584i 0.0492772 + 0.0219396i 0.431227 − 0.902243i \(-0.358081\pi\)
−0.381950 + 0.924183i \(0.624747\pi\)
\(6\) −0.149500 + 0.108618i −0.0610332 + 0.0443432i
\(7\) −1.66120 − 2.05923i −0.627873 − 0.778316i
\(8\) 0.416337 − 1.28135i 0.147198 − 0.453027i
\(9\) 0.283991 − 2.70200i 0.0946638 − 0.900666i
\(10\) 0.0209445 − 0.0362770i 0.00662324 − 0.0114718i
\(11\) 0 0
\(12\) −0.500000 − 0.866025i −0.144338 − 0.250000i
\(13\) 0.992406 + 0.721025i 0.275244 + 0.199976i 0.716840 − 0.697237i \(-0.245588\pi\)
−0.441596 + 0.897214i \(0.645588\pi\)
\(14\) −0.771550 + 0.499012i −0.206206 + 0.133367i
\(15\) −0.0198320 − 0.0610367i −0.00512061 − 0.0157596i
\(16\) 3.00635 + 1.33851i 0.751587 + 0.334628i
\(17\) −0.644966 − 6.13644i −0.156427 − 1.48831i −0.737994 − 0.674807i \(-0.764227\pi\)
0.581567 − 0.813499i \(-0.302440\pi\)
\(18\) −0.922944 − 0.196178i −0.217540 − 0.0462395i
\(19\) −6.27137 + 1.33302i −1.43875 + 0.305816i −0.860253 − 0.509868i \(-0.829694\pi\)
−0.578498 + 0.815684i \(0.696361\pi\)
\(20\) 0.183389 + 0.133240i 0.0410071 + 0.0297934i
\(21\) −0.222811 + 1.39003i −0.0486214 + 0.303330i
\(22\) 0 0
\(23\) 1.01114 + 1.75135i 0.210838 + 0.365182i 0.951977 − 0.306169i \(-0.0990474\pi\)
−0.741139 + 0.671352i \(0.765714\pi\)
\(24\) −0.654904 + 0.291582i −0.133682 + 0.0595189i
\(25\) −3.33592 − 3.70491i −0.667184 − 0.740983i
\(26\) 0.285064 − 0.316596i 0.0559057 − 0.0620896i
\(27\) −2.46094 + 1.78798i −0.473608 + 0.344096i
\(28\) −2.24917 − 4.43462i −0.425053 − 0.838065i
\(29\) 1.00399 + 3.08995i 0.186436 + 0.573790i 0.999970 − 0.00772662i \(-0.00245948\pi\)
−0.813534 + 0.581517i \(0.802459\pi\)
\(30\) −0.0218016 + 0.00463408i −0.00398042 + 0.000846064i
\(31\) 4.45754 − 1.98462i 0.800598 − 0.356449i 0.0346889 − 0.999398i \(-0.488956\pi\)
0.765909 + 0.642949i \(0.222289\pi\)
\(32\) 1.91875 − 3.32337i 0.339190 − 0.587494i
\(33\) 0 0
\(34\) −2.14290 −0.367505
\(35\) −0.0820198 − 0.308396i −0.0138639 − 0.0521285i
\(36\) 1.57786 − 4.85616i 0.262977 − 0.809360i
\(37\) 1.58556 − 1.76094i 0.260665 − 0.289498i −0.598579 − 0.801064i \(-0.704268\pi\)
0.859244 + 0.511566i \(0.170934\pi\)
\(38\) 0.232752 + 2.21448i 0.0377573 + 0.359237i
\(39\) −0.0682261 − 0.649128i −0.0109249 − 0.103944i
\(40\) 0.108736 − 0.120764i 0.0171927 − 0.0190944i
\(41\) 2.56202 − 7.88508i 0.400120 − 1.23144i −0.524782 − 0.851236i \(-0.675853\pi\)
0.924902 − 0.380205i \(-0.124147\pi\)
\(42\) 0.472019 + 0.127419i 0.0728342 + 0.0196612i
\(43\) 2.22668 0.339566 0.169783 − 0.985481i \(-0.445693\pi\)
0.169783 + 0.985481i \(0.445693\pi\)
\(44\) 0 0
\(45\) 0.163848 − 0.283793i 0.0244250 − 0.0423054i
\(46\) 0.641614 − 0.285665i 0.0946008 − 0.0421190i
\(47\) −9.03263 + 1.91994i −1.31754 + 0.280053i −0.812460 − 0.583017i \(-0.801872\pi\)
−0.505084 + 0.863070i \(0.668539\pi\)
\(48\) −0.541098 − 1.66533i −0.0781008 − 0.240369i
\(49\) −1.48085 + 6.84157i −0.211550 + 0.977367i
\(50\) −1.40076 + 1.01771i −0.198097 + 0.143926i
\(51\) −2.19684 + 2.43983i −0.307618 + 0.341645i
\(52\) 1.54262 + 1.71325i 0.213923 + 0.237585i
\(53\) 8.84316 − 3.93723i 1.21470 − 0.540820i 0.303520 − 0.952825i \(-0.401838\pi\)
0.911182 + 0.412005i \(0.135171\pi\)
\(54\) 0.528218 + 0.914901i 0.0718814 + 0.124502i
\(55\) 0 0
\(56\) −3.33022 + 1.27125i −0.445020 + 0.169878i
\(57\) 2.75994 + 2.00521i 0.365563 + 0.265597i
\(58\) 1.10370 − 0.234598i 0.144923 − 0.0308043i
\(59\) 9.53129 + 2.02594i 1.24087 + 0.263755i 0.781169 − 0.624320i \(-0.214624\pi\)
0.459699 + 0.888075i \(0.347957\pi\)
\(60\) −0.0126077 − 0.119954i −0.00162764 − 0.0154860i
\(61\) −1.30981 − 0.583164i −0.167704 − 0.0746665i 0.321168 − 0.947022i \(-0.395925\pi\)
−0.488871 + 0.872356i \(0.662591\pi\)
\(62\) −0.523658 − 1.61165i −0.0665046 − 0.204680i
\(63\) −6.03580 + 3.90375i −0.760439 + 0.491826i
\(64\) 4.24651 + 3.08527i 0.530813 + 0.385658i
\(65\) 0.0739780 + 0.128134i 0.00917584 + 0.0158930i
\(66\) 0 0
\(67\) 1.53209 − 2.65366i 0.187174 − 0.324196i −0.757133 − 0.653261i \(-0.773400\pi\)
0.944307 + 0.329066i \(0.106734\pi\)
\(68\) 1.21214 − 11.5327i 0.146994 − 1.39855i
\(69\) 0.332514 − 1.02337i 0.0400300 − 0.123200i
\(70\) −0.109496 + 0.0171336i −0.0130872 + 0.00204786i
\(71\) 6.87280 − 4.99338i 0.815652 − 0.592606i −0.0998119 − 0.995006i \(-0.531824\pi\)
0.915464 + 0.402401i \(0.131824\pi\)
\(72\) −3.34398 − 1.48884i −0.394092 − 0.175461i
\(73\) −3.45490 − 0.734363i −0.404366 − 0.0859506i 0.00123873 − 0.999999i \(-0.499606\pi\)
−0.405605 + 0.914049i \(0.632939\pi\)
\(74\) −0.550660 − 0.611570i −0.0640129 − 0.0710935i
\(75\) −0.277283 + 2.63817i −0.0320179 + 0.304630i
\(76\) −12.0496 −1.38219
\(77\) 0 0
\(78\) −0.226682 −0.0256666
\(79\) −0.950323 + 9.04171i −0.106920 + 1.01727i 0.801153 + 0.598460i \(0.204220\pi\)
−0.908073 + 0.418813i \(0.862446\pi\)
\(80\) 0.265595 + 0.294974i 0.0296945 + 0.0329791i
\(81\) −6.38935 − 1.35810i −0.709927 − 0.150900i
\(82\) −2.63045 − 1.17115i −0.290485 − 0.129332i
\(83\) 5.68115 − 4.12760i 0.623587 − 0.453063i −0.230585 − 0.973052i \(-0.574064\pi\)
0.854173 + 0.519989i \(0.174064\pi\)
\(84\) −0.952747 + 2.46825i −0.103953 + 0.269309i
\(85\) 0.229977 − 0.707798i 0.0249446 − 0.0767714i
\(86\) 0.0808338 − 0.769082i 0.00871653 − 0.0829323i
\(87\) 0.864370 − 1.49713i 0.0926702 − 0.160510i
\(88\) 0 0
\(89\) 3.43969 + 5.95772i 0.364607 + 0.631517i 0.988713 − 0.149822i \(-0.0478701\pi\)
−0.624106 + 0.781339i \(0.714537\pi\)
\(90\) −0.0920723 − 0.0668944i −0.00970527 − 0.00705129i
\(91\) −0.163825 − 3.24136i −0.0171736 − 0.339787i
\(92\) 1.17447 + 3.61464i 0.122447 + 0.376852i
\(93\) −2.37181 − 1.05600i −0.245945 − 0.109502i
\(94\) 0.335231 + 3.18951i 0.0345765 + 0.328973i
\(95\) −0.756420 − 0.160782i −0.0776070 − 0.0164959i
\(96\) −1.99727 + 0.424533i −0.203845 + 0.0433287i
\(97\) −13.3747 − 9.71732i −1.35800 − 0.986644i −0.998569 − 0.0534720i \(-0.982971\pi\)
−0.359430 − 0.933172i \(-0.617029\pi\)
\(98\) 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.g.81.2 24
7.2 even 3 inner 847.2.n.g.807.2 24
11.2 odd 10 847.2.n.f.753.2 24
11.3 even 5 inner 847.2.n.g.487.2 24
11.4 even 5 inner 847.2.n.g.130.2 24
11.5 even 5 77.2.e.a.67.2 yes 6
11.6 odd 10 847.2.e.c.606.2 6
11.7 odd 10 847.2.n.f.130.2 24
11.8 odd 10 847.2.n.f.487.2 24
11.9 even 5 inner 847.2.n.g.753.2 24
11.10 odd 2 847.2.n.f.81.2 24
33.5 odd 10 693.2.i.h.298.2 6
44.27 odd 10 1232.2.q.m.529.1 6
77.2 odd 30 847.2.n.f.632.2 24
77.5 odd 30 539.2.e.m.177.2 6
77.9 even 15 inner 847.2.n.g.632.2 24
77.16 even 15 77.2.e.a.23.2 ✓ 6
77.17 even 30 5929.2.a.u.1.2 3
77.27 odd 10 539.2.e.m.67.2 6
77.30 odd 30 847.2.n.f.366.2 24
77.37 even 15 inner 847.2.n.g.9.2 24
77.38 odd 30 539.2.a.g.1.2 3
77.39 odd 30 5929.2.a.x.1.2 3
77.51 odd 30 847.2.n.f.9.2 24
77.58 even 15 inner 847.2.n.g.366.2 24
77.60 even 15 539.2.a.j.1.2 3
77.65 odd 6 847.2.n.f.807.2 24
77.72 odd 30 847.2.e.c.485.2 6
231.38 even 30 4851.2.a.bk.1.2 3
231.137 odd 30 4851.2.a.bj.1.2 3
231.170 odd 30 693.2.i.h.100.2 6
308.115 even 30 8624.2.a.co.1.1 3
308.247 odd 30 1232.2.q.m.177.1 6
308.291 odd 30 8624.2.a.ch.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.2.e.a.23.2 ✓ 6 77.16 even 15
77.2.e.a.67.2 yes 6 11.5 even 5
539.2.a.g.1.2 3 77.38 odd 30
539.2.a.j.1.2 3 77.60 even 15
539.2.e.m.67.2 6 77.27 odd 10
539.2.e.m.177.2 6 77.5 odd 30
693.2.i.h.100.2 6 231.170 odd 30
693.2.i.h.298.2 6 33.5 odd 10
847.2.e.c.485.2 6 77.72 odd 30
847.2.e.c.606.2 6 11.6 odd 10
847.2.n.f.9.2 24 77.51 odd 30
847.2.n.f.81.2 24 11.10 odd 2
847.2.n.f.130.2 24 11.7 odd 10
847.2.n.f.366.2 24 77.30 odd 30
847.2.n.f.487.2 24 11.8 odd 10
847.2.n.f.632.2 24 77.2 odd 30
847.2.n.f.753.2 24 11.2 odd 10
847.2.n.f.807.2 24 77.65 odd 6
847.2.n.g.9.2 24 77.37 even 15 inner
847.2.n.g.81.2 24 1.1 even 1 trivial
847.2.n.g.130.2 24 11.4 even 5 inner
847.2.n.g.366.2 24 77.58 even 15 inner
847.2.n.g.487.2 24 11.3 even 5 inner
847.2.n.g.632.2 24 77.9 even 15 inner
847.2.n.g.753.2 24 11.9 even 5 inner
847.2.n.g.807.2 24 7.2 even 3 inner
1232.2.q.m.177.1 6 308.247 odd 30
1232.2.q.m.529.1 6 44.27 odd 10
4851.2.a.bj.1.2 3 231.137 odd 30
4851.2.a.bk.1.2 3 231.38 even 30
5929.2.a.u.1.2 3 77.17 even 30
5929.2.a.x.1.2 3 77.39 odd 30
8624.2.a.ch.1.3 3 308.291 odd 30
8624.2.a.co.1.1 3 308.115 even 30