Properties

Label 21.10.a.a.1.1
Level $21$
Weight $10$
Character 21.1
Self dual yes
Analytic conductor $10.816$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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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] = [21,10,Mod(1,21)] 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("21.1"); S:= CuspForms(chi, 10); 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, 0])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 21.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(10.8157525594\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 21.1

$q$-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)
 
\(f(q)\) \(=\) \(q-24.0000 q^{2} +81.0000 q^{3} +64.0000 q^{4} -144.000 q^{5} -1944.00 q^{6} +2401.00 q^{7} +10752.0 q^{8} +6561.00 q^{9} +3456.00 q^{10} -15030.0 q^{11} +5184.00 q^{12} -151486. q^{13} -57624.0 q^{14} -11664.0 q^{15} -290816. q^{16} -350448. q^{17} -157464. q^{18} -691108. q^{19} -9216.00 q^{20} +194481. q^{21} +360720. q^{22} +892458. q^{23} +870912. q^{24} -1.93239e6 q^{25} +3.63566e6 q^{26} +531441. q^{27} +153664. q^{28} +1.64852e6 q^{29} +279936. q^{30} -3.73430e6 q^{31} +1.47456e6 q^{32} -1.21743e6 q^{33} +8.41075e6 q^{34} -345744. q^{35} +419904. q^{36} -1.14719e7 q^{37} +1.65866e7 q^{38} -1.22704e7 q^{39} -1.54829e6 q^{40} +1.39857e7 q^{41} -4.66754e6 q^{42} +1.67945e7 q^{43} -961920. q^{44} -944784. q^{45} -2.14190e7 q^{46} -1.40121e7 q^{47} -2.35561e7 q^{48} +5.76480e6 q^{49} +4.63773e7 q^{50} -2.83863e7 q^{51} -9.69510e6 q^{52} -9.74399e7 q^{53} -1.27546e7 q^{54} +2.16432e6 q^{55} +2.58156e7 q^{56} -5.59797e7 q^{57} -3.95644e7 q^{58} +1.10798e8 q^{59} -746496. q^{60} -9.38167e7 q^{61} +8.96231e7 q^{62} +1.57530e7 q^{63} +1.13508e8 q^{64} +2.18140e7 q^{65} +2.92183e7 q^{66} -1.22446e8 q^{67} -2.24287e7 q^{68} +7.22891e7 q^{69} +8.29786e6 q^{70} +2.06197e8 q^{71} +7.05439e7 q^{72} +2.50338e8 q^{73} +2.75326e8 q^{74} -1.56524e8 q^{75} -4.42309e7 q^{76} -3.60870e7 q^{77} +2.94489e8 q^{78} -3.83149e7 q^{79} +4.18775e7 q^{80} +4.30467e7 q^{81} -3.35657e8 q^{82} -5.14087e8 q^{83} +1.24468e7 q^{84} +5.04645e7 q^{85} -4.03069e8 q^{86} +1.33530e8 q^{87} -1.61603e8 q^{88} -1.06129e9 q^{89} +2.26748e7 q^{90} -3.63718e8 q^{91} +5.71173e7 q^{92} -3.02478e8 q^{93} +3.36289e8 q^{94} +9.95196e7 q^{95} +1.19439e8 q^{96} -7.38416e7 q^{97} -1.38355e8 q^{98} -9.86118e7 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −24.0000 −1.06066 −0.530330 − 0.847791i \(-0.677932\pi\)
−0.530330 + 0.847791i \(0.677932\pi\)
\(3\) 81.0000 0.577350
\(4\) 64.0000 0.125000
\(5\) −144.000 −0.103038 −0.0515190 − 0.998672i \(-0.516406\pi\)
−0.0515190 + 0.998672i \(0.516406\pi\)
\(6\) −1944.00 −0.612372
\(7\) 2401.00 0.377964
\(8\) 10752.0 0.928078
\(9\) 6561.00 0.333333
\(10\) 3456.00 0.109288
\(11\) −15030.0 −0.309522 −0.154761 − 0.987952i \(-0.549461\pi\)
−0.154761 + 0.987952i \(0.549461\pi\)
\(12\) 5184.00 0.0721688
\(13\) −151486. −1.47105 −0.735525 − 0.677498i \(-0.763064\pi\)
−0.735525 + 0.677498i \(0.763064\pi\)
\(14\) −57624.0 −0.400892
\(15\) −11664.0 −0.0594890
\(16\) −290816. −1.10938
\(17\) −350448. −1.01766 −0.508831 − 0.860867i \(-0.669922\pi\)
−0.508831 + 0.860867i \(0.669922\pi\)
\(18\) −157464. −0.353553
\(19\) −691108. −1.21662 −0.608310 − 0.793700i \(-0.708152\pi\)
−0.608310 + 0.793700i \(0.708152\pi\)
\(20\) −9216.00 −0.0128798
\(21\) 194481. 0.218218
\(22\) 360720. 0.328298
\(23\) 892458. 0.664986 0.332493 − 0.943106i \(-0.392110\pi\)
0.332493 + 0.943106i \(0.392110\pi\)
\(24\) 870912. 0.535826
\(25\) −1.93239e6 −0.989383
\(26\) 3.63566e6 1.56028
\(27\) 531441. 0.192450
\(28\) 153664. 0.0472456
\(29\) 1.64852e6 0.432815 0.216408 − 0.976303i \(-0.430566\pi\)
0.216408 + 0.976303i \(0.430566\pi\)
\(30\) 279936. 0.0630976
\(31\) −3.73430e6 −0.726242 −0.363121 − 0.931742i \(-0.618289\pi\)
−0.363121 + 0.931742i \(0.618289\pi\)
\(32\) 1.47456e6 0.248592
\(33\) −1.21743e6 −0.178703
\(34\) 8.41075e6 1.07939
\(35\) −345744. −0.0389447
\(36\) 419904. 0.0416667
\(37\) −1.14719e7 −1.00630 −0.503150 − 0.864199i \(-0.667826\pi\)
−0.503150 + 0.864199i \(0.667826\pi\)
\(38\) 1.65866e7 1.29042
\(39\) −1.22704e7 −0.849311
\(40\) −1.54829e6 −0.0956273
\(41\) 1.39857e7 0.772961 0.386481 − 0.922298i \(-0.373691\pi\)
0.386481 + 0.922298i \(0.373691\pi\)
\(42\) −4.66754e6 −0.231455
\(43\) 1.67945e7 0.749134 0.374567 − 0.927200i \(-0.377791\pi\)
0.374567 + 0.927200i \(0.377791\pi\)
\(44\) −961920. −0.0386903
\(45\) −944784. −0.0343460
\(46\) −2.14190e7 −0.705324
\(47\) −1.40121e7 −0.418853 −0.209426 − 0.977824i \(-0.567160\pi\)
−0.209426 + 0.977824i \(0.567160\pi\)
\(48\) −2.35561e7 −0.640498
\(49\) 5.76480e6 0.142857
\(50\) 4.63773e7 1.04940
\(51\) −2.83863e7 −0.587547
\(52\) −9.69510e6 −0.183881
\(53\) −9.74399e7 −1.69627 −0.848136 − 0.529779i \(-0.822275\pi\)
−0.848136 + 0.529779i \(0.822275\pi\)
\(54\) −1.27546e7 −0.204124
\(55\) 2.16432e6 0.0318926
\(56\) 2.58156e7 0.350780
\(57\) −5.59797e7 −0.702416
\(58\) −3.95644e7 −0.459070
\(59\) 1.10798e8 1.19042 0.595208 − 0.803571i \(-0.297069\pi\)
0.595208 + 0.803571i \(0.297069\pi\)
\(60\) −746496. −0.00743613
\(61\) −9.38167e7 −0.867553 −0.433776 − 0.901021i \(-0.642819\pi\)
−0.433776 + 0.901021i \(0.642819\pi\)
\(62\) 8.96231e7 0.770296
\(63\) 1.57530e7 0.125988
\(64\) 1.13508e8 0.845703
\(65\) 2.18140e7 0.151574
\(66\) 2.92183e7 0.189543
\(67\) −1.22446e8 −0.742352 −0.371176 − 0.928563i \(-0.621045\pi\)
−0.371176 + 0.928563i \(0.621045\pi\)
\(68\) −2.24287e7 −0.127208
\(69\) 7.22891e7 0.383930
\(70\) 8.29786e6 0.0413071
\(71\) 2.06197e8 0.962987 0.481494 − 0.876450i \(-0.340094\pi\)
0.481494 + 0.876450i \(0.340094\pi\)
\(72\) 7.05439e7 0.309359
\(73\) 2.50338e8 1.03175 0.515873 − 0.856665i \(-0.327467\pi\)
0.515873 + 0.856665i \(0.327467\pi\)
\(74\) 2.75326e8 1.06734
\(75\) −1.56524e8 −0.571221
\(76\) −4.42309e7 −0.152077
\(77\) −3.60870e7 −0.116988
\(78\) 2.94489e8 0.900830
\(79\) −3.83149e7 −0.110674 −0.0553370 − 0.998468i \(-0.517623\pi\)
−0.0553370 + 0.998468i \(0.517623\pi\)
\(80\) 4.18775e7 0.114308
\(81\) 4.30467e7 0.111111
\(82\) −3.35657e8 −0.819849
\(83\) −5.14087e8 −1.18901 −0.594504 − 0.804092i \(-0.702652\pi\)
−0.594504 + 0.804092i \(0.702652\pi\)
\(84\) 1.24468e7 0.0272772
\(85\) 5.04645e7 0.104858
\(86\) −4.03069e8 −0.794577
\(87\) 1.33530e8 0.249886
\(88\) −1.61603e8 −0.287261
\(89\) −1.06129e9 −1.79300 −0.896502 − 0.443041i \(-0.853900\pi\)
−0.896502 + 0.443041i \(0.853900\pi\)
\(90\) 2.26748e7 0.0364294
\(91\) −3.63718e8 −0.556005
\(92\) 5.71173e7 0.0831233
\(93\) −3.02478e8 −0.419296
\(94\) 3.36289e8 0.444260
\(95\) 9.95196e7 0.125358
\(96\) 1.19439e8 0.143525
\(97\) −7.38416e7 −0.0846892 −0.0423446 − 0.999103i \(-0.513483\pi\)
−0.0423446 + 0.999103i \(0.513483\pi\)
\(98\) −1.38355e8 −0.151523
\(99\) −9.86118e7 −0.103174
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 21.10.a.a.1.1 ✓ 1
3.2 odd 2 63.10.a.a.1.1 1
4.3 odd 2 336.10.a.d.1.1 1
7.6 odd 2 147.10.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.10.a.a.1.1 ✓ 1 1.1 even 1 trivial
63.10.a.a.1.1 1 3.2 odd 2
147.10.a.b.1.1 1 7.6 odd 2
336.10.a.d.1.1 1 4.3 odd 2