Properties

Label 115.4.a.e.1.4
Level $115$
Weight $4$
Character 115.1
Self dual yes
Analytic conductor $6.785$
Analytic rank $0$
Dimension $5$
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] = [115,4,Mod(1,115)] 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("115.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(115, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 115.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,6] 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: \(6.78521965066\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 27x^{3} + 7x^{2} + 168x + 92 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(3.41740\) of defining polynomial
Character \(\chi\) \(=\) 115.1

$q$-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)
 
\(f(q)\) \(=\) \(q+4.41740 q^{2} +7.84147 q^{3} +11.5134 q^{4} -5.00000 q^{5} +34.6389 q^{6} -8.97260 q^{7} +15.5200 q^{8} +34.4886 q^{9} -22.0870 q^{10} -28.9011 q^{11} +90.2818 q^{12} +16.0879 q^{13} -39.6355 q^{14} -39.2073 q^{15} -23.5490 q^{16} +25.1771 q^{17} +152.350 q^{18} -35.6298 q^{19} -57.5669 q^{20} -70.3583 q^{21} -127.668 q^{22} -23.0000 q^{23} +121.700 q^{24} +25.0000 q^{25} +71.0668 q^{26} +58.7218 q^{27} -103.305 q^{28} +138.272 q^{29} -173.194 q^{30} +40.1277 q^{31} -228.186 q^{32} -226.627 q^{33} +111.217 q^{34} +44.8630 q^{35} +397.081 q^{36} +379.745 q^{37} -157.391 q^{38} +126.153 q^{39} -77.6001 q^{40} +412.514 q^{41} -310.801 q^{42} -402.095 q^{43} -332.750 q^{44} -172.443 q^{45} -101.600 q^{46} +110.070 q^{47} -184.659 q^{48} -262.492 q^{49} +110.435 q^{50} +197.426 q^{51} +185.227 q^{52} -421.300 q^{53} +259.397 q^{54} +144.506 q^{55} -139.255 q^{56} -279.390 q^{57} +610.802 q^{58} +755.913 q^{59} -451.409 q^{60} -307.032 q^{61} +177.260 q^{62} -309.453 q^{63} -819.594 q^{64} -80.4396 q^{65} -1001.10 q^{66} +319.974 q^{67} +289.874 q^{68} -180.354 q^{69} +198.178 q^{70} -554.138 q^{71} +535.264 q^{72} -705.131 q^{73} +1677.48 q^{74} +196.037 q^{75} -410.219 q^{76} +259.318 q^{77} +557.268 q^{78} +1170.51 q^{79} +117.745 q^{80} -470.728 q^{81} +1822.24 q^{82} -455.978 q^{83} -810.063 q^{84} -125.886 q^{85} -1776.21 q^{86} +1084.26 q^{87} -448.546 q^{88} -1495.57 q^{89} -761.749 q^{90} -144.351 q^{91} -264.808 q^{92} +314.660 q^{93} +486.222 q^{94} +178.149 q^{95} -1789.31 q^{96} +1041.24 q^{97} -1159.53 q^{98} -996.760 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 6 q^{2} + 4 q^{3} + 22 q^{4} - 25 q^{5} + 19 q^{6} - 3 q^{7} + 138 q^{8} + 77 q^{9} - 30 q^{10} + 23 q^{11} + 47 q^{12} + 132 q^{13} + 93 q^{14} - 20 q^{15} + 282 q^{16} + 23 q^{17} - 15 q^{18} - 161 q^{19}+ \cdots + 2021 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.41740 1.56179 0.780893 − 0.624665i \(-0.214765\pi\)
0.780893 + 0.624665i \(0.214765\pi\)
\(3\) 7.84147 1.50909 0.754546 − 0.656248i \(-0.227857\pi\)
0.754546 + 0.656248i \(0.227857\pi\)
\(4\) 11.5134 1.43917
\(5\) −5.00000 −0.447214
\(6\) 34.6389 2.35688
\(7\) −8.97260 −0.484475 −0.242237 − 0.970217i \(-0.577881\pi\)
−0.242237 + 0.970217i \(0.577881\pi\)
\(8\) 15.5200 0.685894
\(9\) 34.4886 1.27736
\(10\) −22.0870 −0.698452
\(11\) −28.9011 −0.792183 −0.396092 − 0.918211i \(-0.629634\pi\)
−0.396092 + 0.918211i \(0.629634\pi\)
\(12\) 90.2818 2.17184
\(13\) 16.0879 0.343230 0.171615 − 0.985164i \(-0.445102\pi\)
0.171615 + 0.985164i \(0.445102\pi\)
\(14\) −39.6355 −0.756646
\(15\) −39.2073 −0.674886
\(16\) −23.5490 −0.367954
\(17\) 25.1771 0.359197 0.179599 − 0.983740i \(-0.442520\pi\)
0.179599 + 0.983740i \(0.442520\pi\)
\(18\) 152.350 1.99496
\(19\) −35.6298 −0.430212 −0.215106 − 0.976591i \(-0.569010\pi\)
−0.215106 + 0.976591i \(0.569010\pi\)
\(20\) −57.5669 −0.643618
\(21\) −70.3583 −0.731117
\(22\) −127.668 −1.23722
\(23\) −23.0000 −0.208514
\(24\) 121.700 1.03508
\(25\) 25.0000 0.200000
\(26\) 71.0668 0.536051
\(27\) 58.7218 0.418556
\(28\) −103.305 −0.697243
\(29\) 138.272 0.885395 0.442698 − 0.896671i \(-0.354022\pi\)
0.442698 + 0.896671i \(0.354022\pi\)
\(30\) −173.194 −1.05403
\(31\) 40.1277 0.232489 0.116244 − 0.993221i \(-0.462914\pi\)
0.116244 + 0.993221i \(0.462914\pi\)
\(32\) −228.186 −1.26056
\(33\) −226.627 −1.19548
\(34\) 111.217 0.560989
\(35\) 44.8630 0.216664
\(36\) 397.081 1.83834
\(37\) 379.745 1.68729 0.843645 − 0.536902i \(-0.180405\pi\)
0.843645 + 0.536902i \(0.180405\pi\)
\(38\) −157.391 −0.671899
\(39\) 126.153 0.517965
\(40\) −77.6001 −0.306741
\(41\) 412.514 1.57132 0.785658 − 0.618662i \(-0.212325\pi\)
0.785658 + 0.618662i \(0.212325\pi\)
\(42\) −310.801 −1.14185
\(43\) −402.095 −1.42602 −0.713011 − 0.701153i \(-0.752669\pi\)
−0.713011 + 0.701153i \(0.752669\pi\)
\(44\) −332.750 −1.14009
\(45\) −172.443 −0.571251
\(46\) −101.600 −0.325655
\(47\) 110.070 0.341603 0.170801 − 0.985305i \(-0.445364\pi\)
0.170801 + 0.985305i \(0.445364\pi\)
\(48\) −184.659 −0.555276
\(49\) −262.492 −0.765284
\(50\) 110.435 0.312357
\(51\) 197.426 0.542061
\(52\) 185.227 0.493967
\(53\) −421.300 −1.09189 −0.545943 − 0.837822i \(-0.683829\pi\)
−0.545943 + 0.837822i \(0.683829\pi\)
\(54\) 259.397 0.653695
\(55\) 144.506 0.354275
\(56\) −139.255 −0.332298
\(57\) −279.390 −0.649229
\(58\) 610.802 1.38280
\(59\) 755.913 1.66799 0.833996 − 0.551770i \(-0.186048\pi\)
0.833996 + 0.551770i \(0.186048\pi\)
\(60\) −451.409 −0.971278
\(61\) −307.032 −0.644450 −0.322225 − 0.946663i \(-0.604431\pi\)
−0.322225 + 0.946663i \(0.604431\pi\)
\(62\) 177.260 0.363098
\(63\) −309.453 −0.618847
\(64\) −819.594 −1.60077
\(65\) −80.4396 −0.153497
\(66\) −1001.10 −1.86708
\(67\) 319.974 0.583448 0.291724 − 0.956502i \(-0.405771\pi\)
0.291724 + 0.956502i \(0.405771\pi\)
\(68\) 289.874 0.516947
\(69\) −180.354 −0.314667
\(70\) 198.178 0.338382
\(71\) −554.138 −0.926254 −0.463127 − 0.886292i \(-0.653273\pi\)
−0.463127 + 0.886292i \(0.653273\pi\)
\(72\) 535.264 0.876131
\(73\) −705.131 −1.13054 −0.565269 − 0.824907i \(-0.691228\pi\)
−0.565269 + 0.824907i \(0.691228\pi\)
\(74\) 1677.48 2.63518
\(75\) 196.037 0.301818
\(76\) −410.219 −0.619149
\(77\) 259.318 0.383793
\(78\) 557.268 0.808950
\(79\) 1170.51 1.66699 0.833495 − 0.552526i \(-0.186336\pi\)
0.833495 + 0.552526i \(0.186336\pi\)
\(80\) 117.745 0.164554
\(81\) −470.728 −0.645717
\(82\) 1822.24 2.45406
\(83\) −455.978 −0.603014 −0.301507 − 0.953464i \(-0.597490\pi\)
−0.301507 + 0.953464i \(0.597490\pi\)
\(84\) −810.063 −1.05220
\(85\) −125.886 −0.160638
\(86\) −1776.21 −2.22714
\(87\) 1084.26 1.33614
\(88\) −448.546 −0.543354
\(89\) −1495.57 −1.78124 −0.890621 − 0.454746i \(-0.849730\pi\)
−0.890621 + 0.454746i \(0.849730\pi\)
\(90\) −761.749 −0.892172
\(91\) −144.351 −0.166286
\(92\) −264.808 −0.300088
\(93\) 314.660 0.350847
\(94\) 486.222 0.533510
\(95\) 178.149 0.192397
\(96\) −1789.31 −1.90230
\(97\) 1041.24 1.08992 0.544958 − 0.838463i \(-0.316545\pi\)
0.544958 + 0.838463i \(0.316545\pi\)
\(98\) −1159.53 −1.19521
\(99\) −996.760 −1.01190
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 115.4.a.e.1.4 ✓ 5
3.2 odd 2 1035.4.a.k.1.2 5
4.3 odd 2 1840.4.a.n.1.2 5
5.2 odd 4 575.4.b.i.24.9 10
5.3 odd 4 575.4.b.i.24.2 10
5.4 even 2 575.4.a.j.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.4.a.e.1.4 ✓ 5 1.1 even 1 trivial
575.4.a.j.1.2 5 5.4 even 2
575.4.b.i.24.2 10 5.3 odd 4
575.4.b.i.24.9 10 5.2 odd 4
1035.4.a.k.1.2 5 3.2 odd 2
1840.4.a.n.1.2 5 4.3 odd 2