Properties

Label 5.8.a.b.1.1
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 1.1
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)
 
\(f(q)\) \(=\) \(q+1.28220 q^{2} +79.7424 q^{3} -126.356 q^{4} -125.000 q^{5} +102.246 q^{6} -538.197 q^{7} -326.136 q^{8} +4171.85 q^{9} -160.275 q^{10} -1215.12 q^{11} -10075.9 q^{12} +7070.42 q^{13} -690.077 q^{14} -9967.80 q^{15} +15755.4 q^{16} -3348.13 q^{17} +5349.15 q^{18} +22169.7 q^{19} +15794.5 q^{20} -42917.1 q^{21} -1558.03 q^{22} -58513.1 q^{23} -26006.8 q^{24} +15625.0 q^{25} +9065.71 q^{26} +158276. q^{27} +68004.4 q^{28} -206301. q^{29} -12780.7 q^{30} +177822. q^{31} +61947.0 q^{32} -96896.5 q^{33} -4292.98 q^{34} +67274.6 q^{35} -527138. q^{36} -284128. q^{37} +28426.0 q^{38} +563812. q^{39} +40767.0 q^{40} +627353. q^{41} -55028.4 q^{42} -164889. q^{43} +153538. q^{44} -521481. q^{45} -75025.7 q^{46} -449355. q^{47} +1.25637e6 q^{48} -533887. q^{49} +20034.4 q^{50} -266988. q^{51} -893390. q^{52} -730190. q^{53} +202942. q^{54} +151890. q^{55} +175525. q^{56} +1.76786e6 q^{57} -264520. q^{58} +1.42202e6 q^{59} +1.25949e6 q^{60} -266326. q^{61} +228004. q^{62} -2.24527e6 q^{63} -1.93726e6 q^{64} -883803. q^{65} -124241. q^{66} +2.95028e6 q^{67} +423056. q^{68} -4.66598e6 q^{69} +86259.6 q^{70} +921138. q^{71} -1.36059e6 q^{72} +4.25657e6 q^{73} -364309. q^{74} +1.24597e6 q^{75} -2.80127e6 q^{76} +653973. q^{77} +722921. q^{78} +6.28551e6 q^{79} -1.96942e6 q^{80} +3.49751e6 q^{81} +804393. q^{82} -9.17165e6 q^{83} +5.42283e6 q^{84} +418516. q^{85} -211421. q^{86} -1.64510e7 q^{87} +396294. q^{88} +242643. q^{89} -668644. q^{90} -3.80528e6 q^{91} +7.39348e6 q^{92} +1.41799e7 q^{93} -576164. q^{94} -2.77121e6 q^{95} +4.93980e6 q^{96} -2.59198e6 q^{97} -684551. q^{98} -5.06929e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 20 q^{2} + 20 q^{3} + 96 q^{4} - 250 q^{5} - 1016 q^{6} - 100 q^{7} + 1440 q^{8} + 5554 q^{9} - 2500 q^{10} + 4544 q^{11} - 23360 q^{12} + 3540 q^{13} + 7512 q^{14} - 2500 q^{15} + 20352 q^{16} - 27340 q^{17}+ \cdots + 2890688 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.28220 0.113332 0.0566659 − 0.998393i \(-0.481953\pi\)
0.0566659 + 0.998393i \(0.481953\pi\)
\(3\) 79.7424 1.70516 0.852579 − 0.522598i \(-0.175037\pi\)
0.852579 + 0.522598i \(0.175037\pi\)
\(4\) −126.356 −0.987156
\(5\) −125.000 −0.447214
\(6\) 102.246 0.193249
\(7\) −538.197 −0.593059 −0.296529 − 0.955024i \(-0.595829\pi\)
−0.296529 + 0.955024i \(0.595829\pi\)
\(8\) −326.136 −0.225208
\(9\) 4171.85 1.90757
\(10\) −160.275 −0.0506835
\(11\) −1215.12 −0.275261 −0.137630 − 0.990484i \(-0.543949\pi\)
−0.137630 + 0.990484i \(0.543949\pi\)
\(12\) −10075.9 −1.68326
\(13\) 7070.42 0.892573 0.446286 − 0.894890i \(-0.352746\pi\)
0.446286 + 0.894890i \(0.352746\pi\)
\(14\) −690.077 −0.0672124
\(15\) −9967.80 −0.762570
\(16\) 15755.4 0.961633
\(17\) −3348.13 −0.165284 −0.0826420 − 0.996579i \(-0.526336\pi\)
−0.0826420 + 0.996579i \(0.526336\pi\)
\(18\) 5349.15 0.216188
\(19\) 22169.7 0.741519 0.370759 − 0.928729i \(-0.379097\pi\)
0.370759 + 0.928729i \(0.379097\pi\)
\(20\) 15794.5 0.441470
\(21\) −42917.1 −1.01126
\(22\) −1558.03 −0.0311958
\(23\) −58513.1 −1.00278 −0.501390 − 0.865221i \(-0.667178\pi\)
−0.501390 + 0.865221i \(0.667178\pi\)
\(24\) −26006.8 −0.384015
\(25\) 15625.0 0.200000
\(26\) 9065.71 0.101157
\(27\) 158276. 1.54754
\(28\) 68004.4 0.585442
\(29\) −206301. −1.57076 −0.785379 − 0.619015i \(-0.787532\pi\)
−0.785379 + 0.619015i \(0.787532\pi\)
\(30\) −12780.7 −0.0864234
\(31\) 177822. 1.07206 0.536030 − 0.844199i \(-0.319923\pi\)
0.536030 + 0.844199i \(0.319923\pi\)
\(32\) 61947.0 0.334191
\(33\) −96896.5 −0.469363
\(34\) −4292.98 −0.0187319
\(35\) 67274.6 0.265224
\(36\) −527138. −1.88307
\(37\) −284128. −0.922163 −0.461081 − 0.887358i \(-0.652538\pi\)
−0.461081 + 0.887358i \(0.652538\pi\)
\(38\) 28426.0 0.0840376
\(39\) 563812. 1.52198
\(40\) 40767.0 0.100716
\(41\) 627353. 1.42157 0.710785 − 0.703409i \(-0.248340\pi\)
0.710785 + 0.703409i \(0.248340\pi\)
\(42\) −55028.4 −0.114608
\(43\) −164889. −0.316266 −0.158133 − 0.987418i \(-0.550547\pi\)
−0.158133 + 0.987418i \(0.550547\pi\)
\(44\) 153538. 0.271725
\(45\) −521481. −0.853090
\(46\) −75025.7 −0.113647
\(47\) −449355. −0.631316 −0.315658 − 0.948873i \(-0.602225\pi\)
−0.315658 + 0.948873i \(0.602225\pi\)
\(48\) 1.25637e6 1.63974
\(49\) −533887. −0.648281
\(50\) 20034.4 0.0226663
\(51\) −266988. −0.281835
\(52\) −893390. −0.881108
\(53\) −730190. −0.673706 −0.336853 − 0.941557i \(-0.609362\pi\)
−0.336853 + 0.941557i \(0.609362\pi\)
\(54\) 202942. 0.175386
\(55\) 151890. 0.123100
\(56\) 175525. 0.133562
\(57\) 1.76786e6 1.26441
\(58\) −264520. −0.178017
\(59\) 1.42202e6 0.901412 0.450706 − 0.892673i \(-0.351172\pi\)
0.450706 + 0.892673i \(0.351172\pi\)
\(60\) 1.25949e6 0.752776
\(61\) −266326. −0.150231 −0.0751153 − 0.997175i \(-0.523932\pi\)
−0.0751153 + 0.997175i \(0.523932\pi\)
\(62\) 228004. 0.121498
\(63\) −2.24527e6 −1.13130
\(64\) −1.93726e6 −0.923758
\(65\) −883803. −0.399171
\(66\) −124241. −0.0531938
\(67\) 2.95028e6 1.19840 0.599200 − 0.800599i \(-0.295485\pi\)
0.599200 + 0.800599i \(0.295485\pi\)
\(68\) 423056. 0.163161
\(69\) −4.66598e6 −1.70990
\(70\) 86259.6 0.0300583
\(71\) 921138. 0.305436 0.152718 − 0.988270i \(-0.451197\pi\)
0.152718 + 0.988270i \(0.451197\pi\)
\(72\) −1.36059e6 −0.429599
\(73\) 4.25657e6 1.28065 0.640323 − 0.768105i \(-0.278800\pi\)
0.640323 + 0.768105i \(0.278800\pi\)
\(74\) −364309. −0.104510
\(75\) 1.24597e6 0.341032
\(76\) −2.80127e6 −0.731995
\(77\) 653973. 0.163246
\(78\) 722921. 0.172488
\(79\) 6.28551e6 1.43432 0.717159 − 0.696910i \(-0.245442\pi\)
0.717159 + 0.696910i \(0.245442\pi\)
\(80\) −1.96942e6 −0.430055
\(81\) 3.49751e6 0.731243
\(82\) 804393. 0.161109
\(83\) −9.17165e6 −1.76065 −0.880327 − 0.474367i \(-0.842677\pi\)
−0.880327 + 0.474367i \(0.842677\pi\)
\(84\) 5.42283e6 0.998271
\(85\) 418516. 0.0739172
\(86\) −211421. −0.0358430
\(87\) −1.64510e7 −2.67839
\(88\) 396294. 0.0619909
\(89\) 242643. 0.0364840 0.0182420 − 0.999834i \(-0.494193\pi\)
0.0182420 + 0.999834i \(0.494193\pi\)
\(90\) −668644. −0.0966821
\(91\) −3.80528e6 −0.529348
\(92\) 7.39348e6 0.989901
\(93\) 1.41799e7 1.82803
\(94\) −576164. −0.0715482
\(95\) −2.77121e6 −0.331617
\(96\) 4.93980e6 0.569849
\(97\) −2.59198e6 −0.288357 −0.144179 − 0.989552i \(-0.546054\pi\)
−0.144179 + 0.989552i \(0.546054\pi\)
\(98\) −684551. −0.0734708
\(99\) −5.06929e6 −0.525078
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.1.1 ✓ 2
3.2 odd 2 45.8.a.h.1.2 2
4.3 odd 2 80.8.a.g.1.1 2
5.2 odd 4 25.8.b.c.24.3 4
5.3 odd 4 25.8.b.c.24.2 4
5.4 even 2 25.8.a.b.1.2 2
7.6 odd 2 245.8.a.c.1.1 2
8.3 odd 2 320.8.a.u.1.2 2
8.5 even 2 320.8.a.l.1.1 2
11.10 odd 2 605.8.a.d.1.2 2
15.2 even 4 225.8.b.m.199.2 4
15.8 even 4 225.8.b.m.199.3 4
15.14 odd 2 225.8.a.w.1.1 2
20.3 even 4 400.8.c.m.49.1 4
20.7 even 4 400.8.c.m.49.4 4
20.19 odd 2 400.8.a.bb.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5.8.a.b.1.1 ✓ 2 1.1 even 1 trivial
25.8.a.b.1.2 2 5.4 even 2
25.8.b.c.24.2 4 5.3 odd 4
25.8.b.c.24.3 4 5.2 odd 4
45.8.a.h.1.2 2 3.2 odd 2
80.8.a.g.1.1 2 4.3 odd 2
225.8.a.w.1.1 2 15.14 odd 2
225.8.b.m.199.2 4 15.2 even 4
225.8.b.m.199.3 4 15.8 even 4
245.8.a.c.1.1 2 7.6 odd 2
320.8.a.l.1.1 2 8.5 even 2
320.8.a.u.1.2 2 8.3 odd 2
400.8.a.bb.1.2 2 20.19 odd 2
400.8.c.m.49.1 4 20.3 even 4
400.8.c.m.49.4 4 20.7 even 4
605.8.a.d.1.2 2 11.10 odd 2