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Properties

Label 21.9.d.a.13.5
Level $21$
Weight $9$
Character 21.13
Analytic conductor $8.555$
Analytic rank $0$
Dimension $10$
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] = [21,9,Mod(13,21)] 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("21.13"); S:= CuspForms(chi, 9); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(21, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 9, names="a")
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 21.d (of order \(2\), degree \(1\), 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: \(8.55495081128\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2 x^{9} + 944 x^{8} + 7576 x^{7} + 726329 x^{6} + 3530146 x^{5} + 151085056 x^{4} + \cdots + 60873719076 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{12}\cdot 7^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 13.5
Root \(-0.802416 - 1.38982i\) of defining polynomial
Character \(\chi\) \(=\) 21.13
Dual form 21.9.d.a.13.6

$q$-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)
 
\(f(q)\) \(=\) \(q-2.60483 q^{2} -46.7654i q^{3} -249.215 q^{4} +225.374i q^{5} +121.816i q^{6} +(1902.54 + 1464.63i) q^{7} +1316.00 q^{8} -2187.00 q^{9} -587.062i q^{10} +17003.5 q^{11} +11654.6i q^{12} +39058.4i q^{13} +(-4955.79 - 3815.12i) q^{14} +10539.7 q^{15} +60371.0 q^{16} +153779. i q^{17} +5696.77 q^{18} -124969. i q^{19} -56166.6i q^{20} +(68494.1 - 88972.9i) q^{21} -44291.1 q^{22} -422546. q^{23} -61543.2i q^{24} +339831. q^{25} -101741. i q^{26} +102276. i q^{27} +(-474141. - 365008. i) q^{28} -219841. q^{29} -27454.2 q^{30} -137898. i q^{31} -494152. q^{32} -795173. i q^{33} -400569. i q^{34} +(-330090. + 428783. i) q^{35} +545033. q^{36} +1.02868e6 q^{37} +325524. i q^{38} +1.82658e6 q^{39} +296592. i q^{40} +4.43270e6i q^{41} +(-178416. + 231759. i) q^{42} +2.57112e6 q^{43} -4.23751e6 q^{44} -492893. i q^{45} +1.10066e6 q^{46} +2.73860e6i q^{47} -2.82327e6i q^{48} +(1.47450e6 + 5.57304e6i) q^{49} -885204. q^{50} +7.19155e6 q^{51} -9.73393e6i q^{52} -4.37645e6 q^{53} -266411. i q^{54} +3.83214e6i q^{55} +(2.50374e6 + 1.92746e6i) q^{56} -5.84424e6 q^{57} +572650. q^{58} -3.69008e6i q^{59} -2.62665e6 q^{60} -1.38104e7i q^{61} +359202. i q^{62} +(-4.16085e6 - 3.20315e6i) q^{63} -1.41678e7 q^{64} -8.80276e6 q^{65} +2.07129e6i q^{66} -8.84981e6 q^{67} -3.83241e7i q^{68} +1.97605e7i q^{69} +(859830. - 1.11691e6i) q^{70} +2.56329e7 q^{71} -2.87809e6 q^{72} +1.19687e7i q^{73} -2.67954e6 q^{74} -1.58923e7i q^{75} +3.11442e7i q^{76} +(3.23497e7 + 2.49038e7i) q^{77} -4.75793e6 q^{78} +4.30614e7 q^{79} +1.36061e7i q^{80} +4.78297e6 q^{81} -1.15464e7i q^{82} +2.79254e7i q^{83} +(-1.70697e7 + 2.21734e7i) q^{84} -3.46579e7 q^{85} -6.69733e6 q^{86} +1.02810e7i q^{87} +2.23765e7 q^{88} -8.40455e7i q^{89} +1.28390e6i q^{90} +(-5.72062e7 + 7.43101e7i) q^{91} +1.05305e8 q^{92} -6.44887e6 q^{93} -7.13359e6i q^{94} +2.81649e7 q^{95} +2.31092e7i q^{96} +5.93196e7i q^{97} +(-3.84083e6 - 1.45168e7i) q^{98} -3.71866e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 6 q^{2} + 1210 q^{4} - 4002 q^{7} + 20154 q^{8} - 21870 q^{9} + 29412 q^{11} - 94974 q^{14} - 59616 q^{15} + 66546 q^{16} + 13122 q^{18} + 21060 q^{21} + 1114748 q^{22} - 900204 q^{23} + 1073026 q^{25}+ \cdots - 64324044 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).

\(n\) \(8\) \(10\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.60483 −0.162802 −0.0814010 − 0.996681i \(-0.525939\pi\)
−0.0814010 + 0.996681i \(0.525939\pi\)
\(3\) − 46.7654i − 0.577350i
\(4\) −249.215 −0.973496
\(5\) 225.374i 0.360599i 0.983612 + 0.180299i \(0.0577067\pi\)
−0.983612 + 0.180299i \(0.942293\pi\)
\(6\) 121.816i 0.0939938i
\(7\) 1902.54 + 1464.63i 0.792394 + 0.610009i
\(8\) 1316.00 0.321289
\(9\) −2187.00 −0.333333
\(10\) − 587.062i − 0.0587062i
\(11\) 17003.5 1.16136 0.580679 − 0.814132i \(-0.302787\pi\)
0.580679 + 0.814132i \(0.302787\pi\)
\(12\) 11654.6i 0.562048i
\(13\) 39058.4i 1.36754i 0.729696 + 0.683771i \(0.239661\pi\)
−0.729696 + 0.683771i \(0.760339\pi\)
\(14\) −4955.79 − 3815.12i −0.129003 − 0.0993107i
\(15\) 10539.7 0.208192
\(16\) 60371.0 0.921189
\(17\) 153779.i 1.84121i 0.390500 + 0.920603i \(0.372302\pi\)
−0.390500 + 0.920603i \(0.627698\pi\)
\(18\) 5696.77 0.0542673
\(19\) − 124969.i − 0.958935i −0.877560 − 0.479468i \(-0.840830\pi\)
0.877560 − 0.479468i \(-0.159170\pi\)
\(20\) − 56166.6i − 0.351041i
\(21\) 68494.1 − 88972.9i 0.352189 − 0.457489i
\(22\) −44291.1 −0.189072
\(23\) −422546. −1.50995 −0.754974 − 0.655754i \(-0.772351\pi\)
−0.754974 + 0.655754i \(0.772351\pi\)
\(24\) − 61543.2i − 0.185496i
\(25\) 339831. 0.869969
\(26\) − 101741.i − 0.222639i
\(27\) 102276.i 0.192450i
\(28\) −474141. − 365008.i −0.771392 − 0.593841i
\(29\) −219841. −0.310826 −0.155413 − 0.987850i \(-0.549671\pi\)
−0.155413 + 0.987850i \(0.549671\pi\)
\(30\) −27454.2 −0.0338940
\(31\) − 137898.i − 0.149318i −0.997209 − 0.0746590i \(-0.976213\pi\)
0.997209 − 0.0746590i \(-0.0237868\pi\)
\(32\) −494152. −0.471260
\(33\) − 795173.i − 0.670511i
\(34\) − 400569.i − 0.299752i
\(35\) −330090. + 428783.i −0.219969 + 0.285736i
\(36\) 545033. 0.324499
\(37\) 1.02868e6 0.548876 0.274438 − 0.961605i \(-0.411508\pi\)
0.274438 + 0.961605i \(0.411508\pi\)
\(38\) 325524.i 0.156117i
\(39\) 1.82658e6 0.789551
\(40\) 296592.i 0.115856i
\(41\) 4.43270e6i 1.56867i 0.620336 + 0.784337i \(0.286996\pi\)
−0.620336 + 0.784337i \(0.713004\pi\)
\(42\) −178416. + 231759.i −0.0573371 + 0.0744801i
\(43\) 2.57112e6 0.752052 0.376026 − 0.926609i \(-0.377290\pi\)
0.376026 + 0.926609i \(0.377290\pi\)
\(44\) −4.23751e6 −1.13058
\(45\) − 492893.i − 0.120200i
\(46\) 1.10066e6 0.245823
\(47\) 2.73860e6i 0.561225i 0.959821 + 0.280613i \(0.0905376\pi\)
−0.959821 + 0.280613i \(0.909462\pi\)
\(48\) − 2.82327e6i − 0.531849i
\(49\) 1.47450e6 + 5.57304e6i 0.255777 + 0.966736i
\(50\) −885204. −0.141633
\(51\) 7.19155e6 1.06302
\(52\) − 9.73393e6i − 1.33130i
\(53\) −4.37645e6 −0.554649 −0.277325 − 0.960776i \(-0.589448\pi\)
−0.277325 + 0.960776i \(0.589448\pi\)
\(54\) − 266411.i − 0.0313313i
\(55\) 3.83214e6i 0.418785i
\(56\) 2.50374e6 + 1.92746e6i 0.254587 + 0.195989i
\(57\) −5.84424e6 −0.553642
\(58\) 572650. 0.0506031
\(59\) − 3.69008e6i − 0.304528i −0.988340 − 0.152264i \(-0.951344\pi\)
0.988340 − 0.152264i \(-0.0486565\pi\)
\(60\) −2.62665e6 −0.202674
\(61\) − 1.38104e7i − 0.997437i −0.866764 − 0.498719i \(-0.833804\pi\)
0.866764 − 0.498719i \(-0.166196\pi\)
\(62\) 359202.i 0.0243093i
\(63\) −4.16085e6 − 3.20315e6i −0.264131 − 0.203336i
\(64\) −1.41678e7 −0.844467
\(65\) −8.80276e6 −0.493134
\(66\) 2.07129e6i 0.109160i
\(67\) −8.84981e6 −0.439172 −0.219586 − 0.975593i \(-0.570471\pi\)
−0.219586 + 0.975593i \(0.570471\pi\)
\(68\) − 3.83241e7i − 1.79241i
\(69\) 1.97605e7i 0.871769i
\(70\) 859830. − 1.11691e6i 0.0358113 − 0.0465184i
\(71\) 2.56329e7 1.00871 0.504353 − 0.863498i \(-0.331731\pi\)
0.504353 + 0.863498i \(0.331731\pi\)
\(72\) −2.87809e6 −0.107096
\(73\) 1.19687e7i 0.421458i 0.977544 + 0.210729i \(0.0675838\pi\)
−0.977544 + 0.210729i \(0.932416\pi\)
\(74\) −2.67954e6 −0.0893581
\(75\) − 1.58923e7i − 0.502277i
\(76\) 3.11442e7i 0.933519i
\(77\) 3.23497e7 + 2.49038e7i 0.920254 + 0.708440i
\(78\) −4.75793e6 −0.128541
\(79\) 4.30614e7 1.10555 0.552777 − 0.833329i \(-0.313568\pi\)
0.552777 + 0.833329i \(0.313568\pi\)
\(80\) 1.36061e7i 0.332180i
\(81\) 4.78297e6 0.111111
\(82\) − 1.15464e7i − 0.255383i
\(83\) 2.79254e7i 0.588420i 0.955741 + 0.294210i \(0.0950565\pi\)
−0.955741 + 0.294210i \(0.904943\pi\)
\(84\) −1.70697e7 + 2.21734e7i −0.342855 + 0.445363i
\(85\) −3.46579e7 −0.663937
\(86\) −6.69733e6 −0.122436
\(87\) 1.02810e7i 0.179456i
\(88\) 2.23765e7 0.373132
\(89\) − 8.40455e7i − 1.33954i −0.742570 − 0.669768i \(-0.766394\pi\)
0.742570 − 0.669768i \(-0.233606\pi\)
\(90\) 1.28390e6i 0.0195687i
\(91\) −5.72062e7 + 7.43101e7i −0.834214 + 1.08363i
\(92\) 1.05305e8 1.46993
\(93\) −6.44887e6 −0.0862088
\(94\) − 7.13359e6i − 0.0913686i
\(95\) 2.81649e7 0.345791
\(96\) 2.31092e7i 0.272082i
\(97\) 5.93196e7i 0.670056i 0.942208 + 0.335028i \(0.108746\pi\)
−0.942208 + 0.335028i \(0.891254\pi\)
\(98\) −3.84083e6 − 1.45168e7i −0.0416410 − 0.157386i
\(99\) −3.71866e7 −0.387120
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 21.9.d.a.13.5 ✓ 10
3.2 odd 2 63.9.d.e.55.5 10
4.3 odd 2 336.9.f.b.97.9 10
7.6 odd 2 inner 21.9.d.a.13.6 yes 10
21.20 even 2 63.9.d.e.55.6 10
28.27 even 2 336.9.f.b.97.2 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.9.d.a.13.5 ✓ 10 1.1 even 1 trivial
21.9.d.a.13.6 yes 10 7.6 odd 2 inner
63.9.d.e.55.5 10 3.2 odd 2
63.9.d.e.55.6 10 21.20 even 2
336.9.f.b.97.2 10 28.27 even 2
336.9.f.b.97.9 10 4.3 odd 2