Properties

Label 5.22.a.b.1.2
Level $5$
Weight $22$
Character 5.1
Self dual yes
Analytic conductor $13.974$
Analytic rank $0$
Dimension $4$
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] = [5,22,Mod(1,5)] 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("5.1"); S:= CuspForms(chi, 22); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 22, names="a")
 
Level: \( N \) \(=\) \( 5 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 5.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(13.9738672144\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 1929606x^{2} - 743130000x + 239341586400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{6}\cdot 3\cdot 5^{2}\cdot 7 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(210.082\) of defining polynomial
Character \(\chi\) \(=\) 5.1

$q$-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)
 
\(f(q)\) \(=\) \(q+307.836 q^{2} +62164.3 q^{3} -2.00239e6 q^{4} +9.76562e6 q^{5} +1.91364e7 q^{6} +8.89757e8 q^{7} -1.26199e9 q^{8} -6.59596e9 q^{9} +3.00621e9 q^{10} +1.34630e11 q^{11} -1.24477e11 q^{12} +1.54255e11 q^{13} +2.73899e11 q^{14} +6.07073e11 q^{15} +3.81083e12 q^{16} +1.18515e13 q^{17} -2.03047e12 q^{18} +3.56746e13 q^{19} -1.95546e13 q^{20} +5.53111e13 q^{21} +4.14439e13 q^{22} -2.33896e14 q^{23} -7.84505e13 q^{24} +9.53674e13 q^{25} +4.74852e13 q^{26} -1.06029e15 q^{27} -1.78164e15 q^{28} +1.32985e15 q^{29} +1.86879e14 q^{30} -5.02208e15 q^{31} +3.81969e15 q^{32} +8.36915e15 q^{33} +3.64831e15 q^{34} +8.68903e15 q^{35} +1.32077e16 q^{36} -1.85934e16 q^{37} +1.09819e16 q^{38} +9.58912e15 q^{39} -1.23241e16 q^{40} +4.73294e16 q^{41} +1.70268e16 q^{42} -1.45723e17 q^{43} -2.69581e17 q^{44} -6.44136e16 q^{45} -7.20017e16 q^{46} +4.04560e16 q^{47} +2.36897e17 q^{48} +2.33122e17 q^{49} +2.93575e16 q^{50} +7.36737e17 q^{51} -3.08878e17 q^{52} +1.70928e17 q^{53} -3.26397e17 q^{54} +1.31474e18 q^{55} -1.12286e18 q^{56} +2.21768e18 q^{57} +4.09377e17 q^{58} +1.69159e18 q^{59} -1.21560e18 q^{60} -5.24536e18 q^{61} -1.54598e18 q^{62} -5.86880e18 q^{63} -6.81605e18 q^{64} +1.50639e18 q^{65} +2.57633e18 q^{66} +1.98484e19 q^{67} -2.37312e19 q^{68} -1.45400e19 q^{69} +2.67480e18 q^{70} +2.94061e19 q^{71} +8.32401e18 q^{72} +5.41242e19 q^{73} -5.72372e18 q^{74} +5.92845e18 q^{75} -7.14344e19 q^{76} +1.19788e20 q^{77} +2.95188e18 q^{78} -1.21531e20 q^{79} +3.72151e19 q^{80} +3.08370e18 q^{81} +1.45697e19 q^{82} -1.69439e20 q^{83} -1.10754e20 q^{84} +1.15737e20 q^{85} -4.48587e19 q^{86} +8.26693e19 q^{87} -1.69901e20 q^{88} -1.96861e20 q^{89} -1.98289e19 q^{90} +1.37249e20 q^{91} +4.68351e20 q^{92} -3.12194e20 q^{93} +1.24538e19 q^{94} +3.48385e20 q^{95} +2.37448e20 q^{96} +1.34692e21 q^{97} +7.17633e19 q^{98} -8.88011e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2910 q^{2} + 83240 q^{3} + 9165268 q^{4} + 39062500 q^{5} - 158524712 q^{6} + 512613800 q^{7} + 5167363080 q^{8} + 21732888532 q^{9} + 28417968750 q^{10} + 33727076448 q^{11} - 142435377680 q^{12}+ \cdots - 15\!\cdots\!16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 307.836 0.212571 0.106286 − 0.994336i \(-0.466104\pi\)
0.106286 + 0.994336i \(0.466104\pi\)
\(3\) 62164.3 0.607810 0.303905 − 0.952702i \(-0.401710\pi\)
0.303905 + 0.952702i \(0.401710\pi\)
\(4\) −2.00239e6 −0.954813
\(5\) 9.76562e6 0.447214
\(6\) 1.91364e7 0.129203
\(7\) 8.89757e8 1.19053 0.595267 − 0.803528i \(-0.297046\pi\)
0.595267 + 0.803528i \(0.297046\pi\)
\(8\) −1.26199e9 −0.415537
\(9\) −6.59596e9 −0.630567
\(10\) 3.00621e9 0.0950648
\(11\) 1.34630e11 1.56501 0.782505 − 0.622644i \(-0.213941\pi\)
0.782505 + 0.622644i \(0.213941\pi\)
\(12\) −1.24477e11 −0.580345
\(13\) 1.54255e11 0.310337 0.155168 − 0.987888i \(-0.450408\pi\)
0.155168 + 0.987888i \(0.450408\pi\)
\(14\) 2.73899e11 0.253074
\(15\) 6.07073e11 0.271821
\(16\) 3.81083e12 0.866482
\(17\) 1.18515e13 1.42580 0.712899 − 0.701266i \(-0.247382\pi\)
0.712899 + 0.701266i \(0.247382\pi\)
\(18\) −2.03047e12 −0.134041
\(19\) 3.56746e13 1.33489 0.667446 − 0.744658i \(-0.267387\pi\)
0.667446 + 0.744658i \(0.267387\pi\)
\(20\) −1.95546e13 −0.427006
\(21\) 5.53111e13 0.723619
\(22\) 4.14439e13 0.332676
\(23\) −2.33896e14 −1.17728 −0.588642 − 0.808394i \(-0.700337\pi\)
−0.588642 + 0.808394i \(0.700337\pi\)
\(24\) −7.84505e13 −0.252568
\(25\) 9.53674e13 0.200000
\(26\) 4.74852e13 0.0659687
\(27\) −1.06029e15 −0.991075
\(28\) −1.78164e15 −1.13674
\(29\) 1.32985e15 0.586981 0.293491 − 0.955962i \(-0.405183\pi\)
0.293491 + 0.955962i \(0.405183\pi\)
\(30\) 1.86879e14 0.0577813
\(31\) −5.02208e15 −1.10049 −0.550245 − 0.835003i \(-0.685466\pi\)
−0.550245 + 0.835003i \(0.685466\pi\)
\(32\) 3.81969e15 0.599727
\(33\) 8.36915e15 0.951229
\(34\) 3.64831e15 0.303084
\(35\) 8.68903e15 0.532423
\(36\) 1.32077e16 0.602074
\(37\) −1.85934e16 −0.635683 −0.317841 − 0.948144i \(-0.602958\pi\)
−0.317841 + 0.948144i \(0.602958\pi\)
\(38\) 1.09819e16 0.283760
\(39\) 9.58912e15 0.188626
\(40\) −1.23241e16 −0.185834
\(41\) 4.73294e16 0.550681 0.275341 − 0.961347i \(-0.411209\pi\)
0.275341 + 0.961347i \(0.411209\pi\)
\(42\) 1.70268e16 0.153821
\(43\) −1.45723e17 −1.02827 −0.514136 − 0.857709i \(-0.671887\pi\)
−0.514136 + 0.857709i \(0.671887\pi\)
\(44\) −2.69581e17 −1.49429
\(45\) −6.44136e16 −0.281998
\(46\) −7.20017e16 −0.250257
\(47\) 4.04560e16 0.112190 0.0560952 − 0.998425i \(-0.482135\pi\)
0.0560952 + 0.998425i \(0.482135\pi\)
\(48\) 2.36897e17 0.526656
\(49\) 2.33122e17 0.417373
\(50\) 2.93575e16 0.0425143
\(51\) 7.36737e17 0.866614
\(52\) −3.08878e17 −0.296314
\(53\) 1.70928e17 0.134251 0.0671253 − 0.997745i \(-0.478617\pi\)
0.0671253 + 0.997745i \(0.478617\pi\)
\(54\) −3.26397e17 −0.210674
\(55\) 1.31474e18 0.699894
\(56\) −1.12286e18 −0.494712
\(57\) 2.21768e18 0.811360
\(58\) 4.09377e17 0.124775
\(59\) 1.69159e18 0.430874 0.215437 − 0.976518i \(-0.430882\pi\)
0.215437 + 0.976518i \(0.430882\pi\)
\(60\) −1.21560e18 −0.259538
\(61\) −5.24536e18 −0.941482 −0.470741 − 0.882271i \(-0.656013\pi\)
−0.470741 + 0.882271i \(0.656013\pi\)
\(62\) −1.54598e18 −0.233933
\(63\) −5.86880e18 −0.750712
\(64\) −6.81605e18 −0.738997
\(65\) 1.50639e18 0.138787
\(66\) 2.57633e18 0.202204
\(67\) 1.98484e19 1.33027 0.665137 − 0.746722i \(-0.268373\pi\)
0.665137 + 0.746722i \(0.268373\pi\)
\(68\) −2.37312e19 −1.36137
\(69\) −1.45400e19 −0.715564
\(70\) 2.67480e18 0.113178
\(71\) 2.94061e19 1.07207 0.536037 − 0.844194i \(-0.319921\pi\)
0.536037 + 0.844194i \(0.319921\pi\)
\(72\) 8.32401e18 0.262024
\(73\) 5.41242e19 1.47401 0.737007 − 0.675885i \(-0.236238\pi\)
0.737007 + 0.675885i \(0.236238\pi\)
\(74\) −5.72372e18 −0.135128
\(75\) 5.92845e18 0.121562
\(76\) −7.14344e19 −1.27457
\(77\) 1.19788e20 1.86320
\(78\) 2.95188e18 0.0400964
\(79\) −1.21531e20 −1.44412 −0.722062 − 0.691829i \(-0.756805\pi\)
−0.722062 + 0.691829i \(0.756805\pi\)
\(80\) 3.72151e19 0.387503
\(81\) 3.08370e18 0.0281825
\(82\) 1.45697e19 0.117059
\(83\) −1.69439e20 −1.19865 −0.599325 − 0.800506i \(-0.704564\pi\)
−0.599325 + 0.800506i \(0.704564\pi\)
\(84\) −1.10754e20 −0.690921
\(85\) 1.15737e20 0.637636
\(86\) −4.48587e19 −0.218581
\(87\) 8.26693e19 0.356773
\(88\) −1.69901e20 −0.650320
\(89\) −1.96861e20 −0.669214 −0.334607 − 0.942358i \(-0.608604\pi\)
−0.334607 + 0.942358i \(0.608604\pi\)
\(90\) −1.98289e19 −0.0599448
\(91\) 1.37249e20 0.369466
\(92\) 4.68351e20 1.12409
\(93\) −3.12194e20 −0.668888
\(94\) 1.24538e19 0.0238485
\(95\) 3.48385e20 0.596982
\(96\) 2.37448e20 0.364520
\(97\) 1.34692e21 1.85455 0.927276 − 0.374378i \(-0.122144\pi\)
0.927276 + 0.374378i \(0.122144\pi\)
\(98\) 7.17633e19 0.0887215
\(99\) −8.88011e20 −0.986845
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 5.22.a.b.1.2 ✓ 4
3.2 odd 2 45.22.a.f.1.3 4
4.3 odd 2 80.22.a.g.1.2 4
5.2 odd 4 25.22.b.c.24.5 8
5.3 odd 4 25.22.b.c.24.4 8
5.4 even 2 25.22.a.c.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5.22.a.b.1.2 ✓ 4 1.1 even 1 trivial
25.22.a.c.1.3 4 5.4 even 2
25.22.b.c.24.4 8 5.3 odd 4
25.22.b.c.24.5 8 5.2 odd 4
45.22.a.f.1.3 4 3.2 odd 2
80.22.a.g.1.2 4 4.3 odd 2