↑

Properties

Label 1.22.a.a.1.1
Level $1$
Weight $22$
Character 1.1
Self dual yes
Analytic conductor $2.795$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1,22,Mod(1,1)] 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("1.1"); S:= CuspForms(chi, 22); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1, base_ring=CyclotomicField(1)) chi = DirichletCharacter(H, H._module([])) N = Newforms(chi, 22, names="a")
 
Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)

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: yes
Analytic conductor: \(2.79477344287\)
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 \(\chi\) \(=\) 1.1

$q$-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)
 
\(f(q)\) \(=\) \(q-288.000 q^{2} -128844. q^{3} -2.01421e6 q^{4} +2.16410e7 q^{5} +3.71071e7 q^{6} -7.68079e8 q^{7} +1.18407e9 q^{8} +6.14042e9 q^{9} -6.23259e9 q^{10} -9.47249e10 q^{11} +2.59519e11 q^{12} -8.06218e10 q^{13} +2.21207e11 q^{14} -2.78831e12 q^{15} +3.88309e12 q^{16} +3.05228e12 q^{17} -1.76844e12 q^{18} -7.92079e12 q^{19} -4.35894e13 q^{20} +9.89623e13 q^{21} +2.72808e13 q^{22} -7.38454e13 q^{23} -1.52561e14 q^{24} -8.50644e12 q^{25} +2.32191e13 q^{26} +5.56597e14 q^{27} +1.54707e15 q^{28} -4.25303e15 q^{29} +8.03032e14 q^{30} +1.90054e15 q^{31} -3.60151e15 q^{32} +1.22047e16 q^{33} -8.79057e14 q^{34} -1.66220e16 q^{35} -1.23681e16 q^{36} +2.21914e16 q^{37} +2.28119e15 q^{38} +1.03876e16 q^{39} +2.56244e16 q^{40} -2.06228e16 q^{41} -2.85012e16 q^{42} -1.93606e17 q^{43} +1.90796e17 q^{44} +1.32885e17 q^{45} +2.12675e16 q^{46} +1.46961e17 q^{47} -5.00313e17 q^{48} +3.13992e16 q^{49} +2.44986e15 q^{50} -3.93268e17 q^{51} +1.62389e17 q^{52} +2.03827e18 q^{53} -1.60300e17 q^{54} -2.04994e18 q^{55} -9.09460e17 q^{56} +1.02055e18 q^{57} +1.22487e18 q^{58} -5.97588e18 q^{59} +5.61623e18 q^{60} +6.19062e18 q^{61} -5.47356e17 q^{62} -4.71633e18 q^{63} -7.10619e18 q^{64} -1.74473e18 q^{65} -3.51496e18 q^{66} +1.69613e19 q^{67} -6.14793e18 q^{68} +9.51454e18 q^{69} +4.78712e18 q^{70} -5.63276e18 q^{71} +7.27070e18 q^{72} -4.32848e19 q^{73} -6.39113e18 q^{74} +1.09600e18 q^{75} +1.59541e19 q^{76} +7.27562e19 q^{77} -2.99164e18 q^{78} -5.12649e19 q^{79} +8.40337e19 q^{80} -1.35945e20 q^{81} +5.93937e18 q^{82} +4.89119e19 q^{83} -1.99331e20 q^{84} +6.60543e19 q^{85} +5.57585e19 q^{86} +5.47978e20 q^{87} -1.12161e20 q^{88} -5.04303e20 q^{89} -3.82708e19 q^{90} +6.19239e19 q^{91} +1.48740e20 q^{92} -2.44873e20 q^{93} -4.23246e19 q^{94} -1.71413e20 q^{95} +4.64033e20 q^{96} +8.08275e20 q^{97} -9.04297e18 q^{98} -5.81651e20 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −288.000 −0.198874 −0.0994369 − 0.995044i \(-0.531704\pi\)
−0.0994369 + 0.995044i \(0.531704\pi\)
\(3\) −128844. −1.25977 −0.629885 − 0.776689i \(-0.716898\pi\)
−0.629885 + 0.776689i \(0.716898\pi\)
\(4\) −2.01421e6 −0.960449
\(5\) 2.16410e7 0.991040 0.495520 − 0.868596i \(-0.334977\pi\)
0.495520 + 0.868596i \(0.334977\pi\)
\(6\) 3.71071e7 0.250535
\(7\) −7.68079e8 −1.02772 −0.513862 − 0.857873i \(-0.671786\pi\)
−0.513862 + 0.857873i \(0.671786\pi\)
\(8\) 1.18407e9 0.389882
\(9\) 6.14042e9 0.587019
\(10\) −6.23259e9 −0.197092
\(11\) −9.47249e10 −1.10114 −0.550568 − 0.834790i \(-0.685589\pi\)
−0.550568 + 0.834790i \(0.685589\pi\)
\(12\) 2.59519e11 1.20994
\(13\) −8.06218e10 −0.162199 −0.0810993 − 0.996706i \(-0.525843\pi\)
−0.0810993 + 0.996706i \(0.525843\pi\)
\(14\) 2.21207e11 0.204387
\(15\) −2.78831e12 −1.24848
\(16\) 3.88309e12 0.882912
\(17\) 3.05228e12 0.367207 0.183604 − 0.983000i \(-0.441224\pi\)
0.183604 + 0.983000i \(0.441224\pi\)
\(18\) −1.76844e12 −0.116743
\(19\) −7.92079e12 −0.296385 −0.148192 − 0.988959i \(-0.547345\pi\)
−0.148192 + 0.988959i \(0.547345\pi\)
\(20\) −4.35894e13 −0.951844
\(21\) 9.89623e13 1.29469
\(22\) 2.72808e13 0.218987
\(23\) −7.38454e13 −0.371690 −0.185845 − 0.982579i \(-0.559502\pi\)
−0.185845 + 0.982579i \(0.559502\pi\)
\(24\) −1.52561e14 −0.491161
\(25\) −8.50644e12 −0.0178393
\(26\) 2.32191e13 0.0322571
\(27\) 5.56597e14 0.520261
\(28\) 1.54707e15 0.987076
\(29\) −4.25303e15 −1.87724 −0.938620 − 0.344954i \(-0.887895\pi\)
−0.938620 + 0.344954i \(0.887895\pi\)
\(30\) 8.03032e14 0.248290
\(31\) 1.90054e15 0.416466 0.208233 − 0.978079i \(-0.433229\pi\)
0.208233 + 0.978079i \(0.433229\pi\)
\(32\) −3.60151e15 −0.565470
\(33\) 1.22047e16 1.38718
\(34\) −8.79057e14 −0.0730279
\(35\) −1.66220e16 −1.01852
\(36\) −1.23681e16 −0.563802
\(37\) 2.21914e16 0.758695 0.379347 − 0.925254i \(-0.376149\pi\)
0.379347 + 0.925254i \(0.376149\pi\)
\(38\) 2.28119e15 0.0589431
\(39\) 1.03876e16 0.204333
\(40\) 2.56244e16 0.386389
\(41\) −2.06228e16 −0.239948 −0.119974 − 0.992777i \(-0.538281\pi\)
−0.119974 + 0.992777i \(0.538281\pi\)
\(42\) −2.85012e16 −0.257481
\(43\) −1.93606e17 −1.36615 −0.683077 − 0.730346i \(-0.739359\pi\)
−0.683077 + 0.730346i \(0.739359\pi\)
\(44\) 1.90796e17 1.05759
\(45\) 1.32885e17 0.581759
\(46\) 2.12675e16 0.0739195
\(47\) 1.46961e17 0.407543 0.203771 − 0.979019i \(-0.434680\pi\)
0.203771 + 0.979019i \(0.434680\pi\)
\(48\) −5.00313e17 −1.11227
\(49\) 3.13992e16 0.0562160
\(50\) 2.44986e15 0.00354777
\(51\) −3.93268e17 −0.462596
\(52\) 1.62389e17 0.155784
\(53\) 2.03827e18 1.60090 0.800450 − 0.599399i \(-0.204594\pi\)
0.800450 + 0.599399i \(0.204594\pi\)
\(54\) −1.60300e17 −0.103466
\(55\) −2.04994e18 −1.09127
\(56\) −9.09460e17 −0.400691
\(57\) 1.02055e18 0.373376
\(58\) 1.22487e18 0.373334
\(59\) −5.97588e18 −1.52214 −0.761072 − 0.648667i \(-0.775327\pi\)
−0.761072 + 0.648667i \(0.775327\pi\)
\(60\) 5.61623e18 1.19910
\(61\) 6.19062e18 1.11114 0.555572 − 0.831468i \(-0.312499\pi\)
0.555572 + 0.831468i \(0.312499\pi\)
\(62\) −5.47356e17 −0.0828242
\(63\) −4.71633e18 −0.603293
\(64\) −7.10619e18 −0.770455
\(65\) −1.74473e18 −0.160745
\(66\) −3.51496e18 −0.275873
\(67\) 1.69613e19 1.13677 0.568387 − 0.822761i \(-0.307568\pi\)
0.568387 + 0.822761i \(0.307568\pi\)
\(68\) −6.14793e18 −0.352684
\(69\) 9.51454e18 0.468244
\(70\) 4.78712e18 0.202556
\(71\) −5.63276e18 −0.205357 −0.102678 − 0.994715i \(-0.532741\pi\)
−0.102678 + 0.994715i \(0.532741\pi\)
\(72\) 7.27070e18 0.228868
\(73\) −4.32848e19 −1.17881 −0.589407 − 0.807837i \(-0.700638\pi\)
−0.589407 + 0.807837i \(0.700638\pi\)
\(74\) −6.39113e18 −0.150885
\(75\) 1.09600e18 0.0224734
\(76\) 1.59541e19 0.284662
\(77\) 7.27562e19 1.13166
\(78\) −2.99164e18 −0.0406365
\(79\) −5.12649e19 −0.609166 −0.304583 − 0.952486i \(-0.598517\pi\)
−0.304583 + 0.952486i \(0.598517\pi\)
\(80\) 8.40337e19 0.875001
\(81\) −1.35945e20 −1.24243
\(82\) 5.93937e18 0.0477194
\(83\) 4.89119e19 0.346014 0.173007 − 0.984921i \(-0.444652\pi\)
0.173007 + 0.984921i \(0.444652\pi\)
\(84\) −1.99331e20 −1.24349
\(85\) 6.60543e19 0.363917
\(86\) 5.57585e19 0.271692
\(87\) 5.47978e20 2.36489
\(88\) −1.12161e20 −0.429313
\(89\) −5.04303e20 −1.71434 −0.857170 − 0.515034i \(-0.827779\pi\)
−0.857170 + 0.515034i \(0.827779\pi\)
\(90\) −3.82708e19 −0.115697
\(91\) 6.19239e19 0.166695
\(92\) 1.48740e20 0.356990
\(93\) −2.44873e20 −0.524651
\(94\) −4.23246e19 −0.0810496
\(95\) −1.71413e20 −0.293729
\(96\) 4.64033e20 0.712362
\(97\) 8.08275e20 1.11290 0.556450 − 0.830881i \(-0.312163\pi\)
0.556450 + 0.830881i \(0.312163\pi\)
\(98\) −9.04297e18 −0.0111799
\(99\) −5.81651e20 −0.646388
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 1.22.a.a.1.1 ✓ 1
3.2 odd 2 9.22.a.c.1.1 1
4.3 odd 2 16.22.a.c.1.1 1
5.2 odd 4 25.22.b.a.24.1 2
5.3 odd 4 25.22.b.a.24.2 2
5.4 even 2 25.22.a.a.1.1 1
7.6 odd 2 49.22.a.a.1.1 1
8.3 odd 2 64.22.a.a.1.1 1
8.5 even 2 64.22.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.22.a.a.1.1 ✓ 1 1.1 even 1 trivial
9.22.a.c.1.1 1 3.2 odd 2
16.22.a.c.1.1 1 4.3 odd 2
25.22.a.a.1.1 1 5.4 even 2
25.22.b.a.24.1 2 5.2 odd 4
25.22.b.a.24.2 2 5.3 odd 4
49.22.a.a.1.1 1 7.6 odd 2
64.22.a.a.1.1 1 8.3 odd 2
64.22.a.g.1.1 1 8.5 even 2