Properties

Label 5.8.a.b.1.2
Level $5$
Weight $8$
Character 5.1
Self dual yes
Analytic conductor $1.562$
Analytic rank $0$
Dimension $2$
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] = [5,8,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, 8); 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, 8, names="a")
 
Level: \( N \) \(=\) \( 5 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 5.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2] 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: \(1.56192512742\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{19}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 19 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(4.35890\) 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)
 
\(f(q)\) \(=\) \(q+18.7178 q^{2} -59.7424 q^{3} +222.356 q^{4} -125.000 q^{5} -1118.25 q^{6} +438.197 q^{7} +1766.14 q^{8} +1382.15 q^{9} -2339.72 q^{10} +5759.12 q^{11} -13284.1 q^{12} -3530.42 q^{13} +8202.08 q^{14} +7467.80 q^{15} +4596.61 q^{16} -23991.9 q^{17} +25870.8 q^{18} +16590.3 q^{19} -27794.5 q^{20} -26178.9 q^{21} +107798. q^{22} -65626.9 q^{23} -105513. q^{24} +15625.0 q^{25} -66081.7 q^{26} +48083.5 q^{27} +97435.6 q^{28} +134041. q^{29} +139781. q^{30} +129002. q^{31} -140027. q^{32} -344064. q^{33} -449075. q^{34} -54774.6 q^{35} +307330. q^{36} +161108. q^{37} +310534. q^{38} +210916. q^{39} -220767. q^{40} -362989. q^{41} -490012. q^{42} +588189. q^{43} +1.28057e6 q^{44} -172769. q^{45} -1.22839e6 q^{46} +343895. q^{47} -274612. q^{48} -631527. q^{49} +292466. q^{50} +1.43333e6 q^{51} -785010. q^{52} -1.66139e6 q^{53} +900018. q^{54} -719890. q^{55} +773915. q^{56} -991144. q^{57} +2.50896e6 q^{58} -2.54214e6 q^{59} +1.66051e6 q^{60} +2.52337e6 q^{61} +2.41464e6 q^{62} +605655. q^{63} -3.20936e6 q^{64} +441303. q^{65} -6.44011e6 q^{66} +1.56618e6 q^{67} -5.33474e6 q^{68} +3.92071e6 q^{69} -1.02526e6 q^{70} -299354. q^{71} +2.44107e6 q^{72} +312494. q^{73} +3.01558e6 q^{74} -933475. q^{75} +3.68895e6 q^{76} +2.52363e6 q^{77} +3.94788e6 q^{78} -1.95247e6 q^{79} -574576. q^{80} -5.89539e6 q^{81} -6.79435e6 q^{82} -621372. q^{83} -5.82104e6 q^{84} +2.99898e6 q^{85} +1.10096e7 q^{86} -8.00795e6 q^{87} +1.01714e7 q^{88} +5.78298e6 q^{89} -3.23386e6 q^{90} -1.54702e6 q^{91} -1.45925e7 q^{92} -7.70690e6 q^{93} +6.43696e6 q^{94} -2.07379e6 q^{95} +8.36554e6 q^{96} +7.20152e6 q^{97} -1.18208e7 q^{98} +7.95998e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 20 q^{2} + 20 q^{3} + 96 q^{4} - 250 q^{5} - 1016 q^{6} - 100 q^{7} + 1440 q^{8} + 5554 q^{9} - 2500 q^{10} + 4544 q^{11} - 23360 q^{12} + 3540 q^{13} + 7512 q^{14} - 2500 q^{15} + 20352 q^{16} - 27340 q^{17}+ \cdots + 2890688 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 18.7178 1.65444 0.827218 − 0.561882i \(-0.189922\pi\)
0.827218 + 0.561882i \(0.189922\pi\)
\(3\) −59.7424 −1.27749 −0.638746 − 0.769418i \(-0.720547\pi\)
−0.638746 + 0.769418i \(0.720547\pi\)
\(4\) 222.356 1.73716
\(5\) −125.000 −0.447214
\(6\) −1118.25 −2.11353
\(7\) 438.197 0.482865 0.241433 − 0.970418i \(-0.422383\pi\)
0.241433 + 0.970418i \(0.422383\pi\)
\(8\) 1766.14 1.21958
\(9\) 1382.15 0.631986
\(10\) −2339.72 −0.739886
\(11\) 5759.12 1.30461 0.652306 − 0.757955i \(-0.273802\pi\)
0.652306 + 0.757955i \(0.273802\pi\)
\(12\) −13284.1 −2.21920
\(13\) −3530.42 −0.445682 −0.222841 − 0.974855i \(-0.571533\pi\)
−0.222841 + 0.974855i \(0.571533\pi\)
\(14\) 8202.08 0.798869
\(15\) 7467.80 0.571312
\(16\) 4596.61 0.280555
\(17\) −23991.9 −1.18439 −0.592193 − 0.805797i \(-0.701738\pi\)
−0.592193 + 0.805797i \(0.701738\pi\)
\(18\) 25870.8 1.04558
\(19\) 16590.3 0.554903 0.277451 − 0.960740i \(-0.410510\pi\)
0.277451 + 0.960740i \(0.410510\pi\)
\(20\) −27794.5 −0.776880
\(21\) −26178.9 −0.616856
\(22\) 107798. 2.15840
\(23\) −65626.9 −1.12469 −0.562347 − 0.826902i \(-0.690101\pi\)
−0.562347 + 0.826902i \(0.690101\pi\)
\(24\) −105513. −1.55800
\(25\) 15625.0 0.200000
\(26\) −66081.7 −0.737351
\(27\) 48083.5 0.470136
\(28\) 97435.6 0.838812
\(29\) 134041. 1.02058 0.510289 − 0.860003i \(-0.329539\pi\)
0.510289 + 0.860003i \(0.329539\pi\)
\(30\) 139781. 0.945198
\(31\) 129002. 0.777734 0.388867 − 0.921294i \(-0.372867\pi\)
0.388867 + 0.921294i \(0.372867\pi\)
\(32\) −140027. −0.755417
\(33\) −344064. −1.66663
\(34\) −449075. −1.95949
\(35\) −54774.6 −0.215944
\(36\) 307330. 1.09786
\(37\) 161108. 0.522890 0.261445 − 0.965218i \(-0.415801\pi\)
0.261445 + 0.965218i \(0.415801\pi\)
\(38\) 310534. 0.918050
\(39\) 210916. 0.569355
\(40\) −220767. −0.545411
\(41\) −362989. −0.822526 −0.411263 − 0.911517i \(-0.634912\pi\)
−0.411263 + 0.911517i \(0.634912\pi\)
\(42\) −490012. −1.02055
\(43\) 588189. 1.12818 0.564089 − 0.825714i \(-0.309228\pi\)
0.564089 + 0.825714i \(0.309228\pi\)
\(44\) 1.28057e6 2.26632
\(45\) −172769. −0.282633
\(46\) −1.22839e6 −1.86073
\(47\) 343895. 0.483151 0.241576 − 0.970382i \(-0.422336\pi\)
0.241576 + 0.970382i \(0.422336\pi\)
\(48\) −274612. −0.358406
\(49\) −631527. −0.766841
\(50\) 292466. 0.330887
\(51\) 1.43333e6 1.51304
\(52\) −785010. −0.774219
\(53\) −1.66139e6 −1.53287 −0.766436 − 0.642320i \(-0.777972\pi\)
−0.766436 + 0.642320i \(0.777972\pi\)
\(54\) 900018. 0.777809
\(55\) −719890. −0.583441
\(56\) 773915. 0.588891
\(57\) −991144. −0.708884
\(58\) 2.50896e6 1.68848
\(59\) −2.54214e6 −1.61145 −0.805726 − 0.592289i \(-0.798224\pi\)
−0.805726 + 0.592289i \(0.798224\pi\)
\(60\) 1.66051e6 0.992458
\(61\) 2.52337e6 1.42340 0.711699 − 0.702484i \(-0.247926\pi\)
0.711699 + 0.702484i \(0.247926\pi\)
\(62\) 2.41464e6 1.28671
\(63\) 605655. 0.305164
\(64\) −3.20936e6 −1.53034
\(65\) 441303. 0.199315
\(66\) −6.44011e6 −2.75734
\(67\) 1.56618e6 0.636178 0.318089 − 0.948061i \(-0.396959\pi\)
0.318089 + 0.948061i \(0.396959\pi\)
\(68\) −5.33474e6 −2.05746
\(69\) 3.92071e6 1.43679
\(70\) −1.02526e6 −0.357265
\(71\) −299354. −0.0992615 −0.0496307 − 0.998768i \(-0.515804\pi\)
−0.0496307 + 0.998768i \(0.515804\pi\)
\(72\) 2.44107e6 0.770755
\(73\) 312494. 0.0940183 0.0470091 − 0.998894i \(-0.485031\pi\)
0.0470091 + 0.998894i \(0.485031\pi\)
\(74\) 3.01558e6 0.865087
\(75\) −933475. −0.255498
\(76\) 3.68895e6 0.963952
\(77\) 2.52363e6 0.629952
\(78\) 3.94788e6 0.941961
\(79\) −1.95247e6 −0.445542 −0.222771 − 0.974871i \(-0.571510\pi\)
−0.222771 + 0.974871i \(0.571510\pi\)
\(80\) −574576. −0.125468
\(81\) −5.89539e6 −1.23258
\(82\) −6.79435e6 −1.36082
\(83\) −621372. −0.119283 −0.0596414 − 0.998220i \(-0.518996\pi\)
−0.0596414 + 0.998220i \(0.518996\pi\)
\(84\) −5.82104e6 −1.07158
\(85\) 2.99898e6 0.529673
\(86\) 1.10096e7 1.86650
\(87\) −8.00795e6 −1.30378
\(88\) 1.01714e7 1.59108
\(89\) 5.78298e6 0.869534 0.434767 − 0.900543i \(-0.356831\pi\)
0.434767 + 0.900543i \(0.356831\pi\)
\(90\) −3.23386e6 −0.467597
\(91\) −1.54702e6 −0.215204
\(92\) −1.45925e7 −1.95377
\(93\) −7.70690e6 −0.993549
\(94\) 6.43696e6 0.799343
\(95\) −2.07379e6 −0.248160
\(96\) 8.36554e6 0.965039
\(97\) 7.20152e6 0.801167 0.400584 − 0.916260i \(-0.368807\pi\)
0.400584 + 0.916260i \(0.368807\pi\)
\(98\) −1.18208e7 −1.26869
\(99\) 7.95998e6 0.824496
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 5.8.a.b.1.2 ✓ 2
3.2 odd 2 45.8.a.h.1.1 2
4.3 odd 2 80.8.a.g.1.2 2
5.2 odd 4 25.8.b.c.24.4 4
5.3 odd 4 25.8.b.c.24.1 4
5.4 even 2 25.8.a.b.1.1 2
7.6 odd 2 245.8.a.c.1.2 2
8.3 odd 2 320.8.a.u.1.1 2
8.5 even 2 320.8.a.l.1.2 2
11.10 odd 2 605.8.a.d.1.1 2
15.2 even 4 225.8.b.m.199.1 4
15.8 even 4 225.8.b.m.199.4 4
15.14 odd 2 225.8.a.w.1.2 2
20.3 even 4 400.8.c.m.49.3 4
20.7 even 4 400.8.c.m.49.2 4
20.19 odd 2 400.8.a.bb.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5.8.a.b.1.2 ✓ 2 1.1 even 1 trivial
25.8.a.b.1.1 2 5.4 even 2
25.8.b.c.24.1 4 5.3 odd 4
25.8.b.c.24.4 4 5.2 odd 4
45.8.a.h.1.1 2 3.2 odd 2
80.8.a.g.1.2 2 4.3 odd 2
225.8.a.w.1.2 2 15.14 odd 2
225.8.b.m.199.1 4 15.2 even 4
225.8.b.m.199.4 4 15.8 even 4
245.8.a.c.1.2 2 7.6 odd 2
320.8.a.l.1.2 2 8.5 even 2
320.8.a.u.1.1 2 8.3 odd 2
400.8.a.bb.1.1 2 20.19 odd 2
400.8.c.m.49.2 4 20.7 even 4
400.8.c.m.49.3 4 20.3 even 4
605.8.a.d.1.1 2 11.10 odd 2