Properties

Label 115.4.a.f.1.8
Level $115$
Weight $4$
Character 115.1
Self dual yes
Analytic conductor $6.785$
Analytic rank $0$
Dimension $8$
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 = [8,6] 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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 49x^{6} + 31x^{5} + 750x^{4} + 249x^{3} - 2892x^{2} - 620x + 2400 \) 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 1.8
Root \(-3.81977\) 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)
 
\(f(q)\) \(=\) \(q+4.81977 q^{2} +1.39521 q^{3} +15.2302 q^{4} +5.00000 q^{5} +6.72460 q^{6} +13.5587 q^{7} +34.8481 q^{8} -25.0534 q^{9} +24.0989 q^{10} -66.2223 q^{11} +21.2494 q^{12} +23.3164 q^{13} +65.3498 q^{14} +6.97605 q^{15} +46.1180 q^{16} +106.151 q^{17} -120.752 q^{18} -11.6497 q^{19} +76.1511 q^{20} +18.9172 q^{21} -319.177 q^{22} +23.0000 q^{23} +48.6204 q^{24} +25.0000 q^{25} +112.380 q^{26} -72.6254 q^{27} +206.502 q^{28} -18.8139 q^{29} +33.6230 q^{30} -230.170 q^{31} -56.5063 q^{32} -92.3941 q^{33} +511.623 q^{34} +67.7934 q^{35} -381.569 q^{36} -379.629 q^{37} -56.1491 q^{38} +32.5313 q^{39} +174.240 q^{40} -202.375 q^{41} +91.1767 q^{42} +320.493 q^{43} -1008.58 q^{44} -125.267 q^{45} +110.855 q^{46} +271.205 q^{47} +64.3442 q^{48} -159.162 q^{49} +120.494 q^{50} +148.103 q^{51} +355.115 q^{52} +765.344 q^{53} -350.038 q^{54} -331.112 q^{55} +472.494 q^{56} -16.2538 q^{57} -90.6790 q^{58} +388.071 q^{59} +106.247 q^{60} +519.064 q^{61} -1109.37 q^{62} -339.691 q^{63} -641.291 q^{64} +116.582 q^{65} -445.319 q^{66} +358.698 q^{67} +1616.70 q^{68} +32.0898 q^{69} +326.749 q^{70} +40.0003 q^{71} -873.062 q^{72} +129.058 q^{73} -1829.72 q^{74} +34.8803 q^{75} -177.428 q^{76} -897.888 q^{77} +156.794 q^{78} -484.468 q^{79} +230.590 q^{80} +575.114 q^{81} -975.402 q^{82} +5.74998 q^{83} +288.113 q^{84} +530.755 q^{85} +1544.71 q^{86} -26.2494 q^{87} -2307.72 q^{88} -167.400 q^{89} -603.758 q^{90} +316.140 q^{91} +350.295 q^{92} -321.136 q^{93} +1307.15 q^{94} -58.2486 q^{95} -78.8381 q^{96} +582.078 q^{97} -767.125 q^{98} +1659.09 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 6 q^{2} + 42 q^{4} + 40 q^{5} + 11 q^{7} + 3 q^{8} + 158 q^{9} + 30 q^{10} + 41 q^{11} + 48 q^{12} + 28 q^{13} + 161 q^{14} + 98 q^{16} + 71 q^{17} + 84 q^{18} + 177 q^{19} + 210 q^{20} + 292 q^{21}+ \cdots + 1319 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.81977 1.70405 0.852024 − 0.523503i \(-0.175375\pi\)
0.852024 + 0.523503i \(0.175375\pi\)
\(3\) 1.39521 0.268508 0.134254 − 0.990947i \(-0.457136\pi\)
0.134254 + 0.990947i \(0.457136\pi\)
\(4\) 15.2302 1.90378
\(5\) 5.00000 0.447214
\(6\) 6.72460 0.457551
\(7\) 13.5587 0.732100 0.366050 − 0.930595i \(-0.380710\pi\)
0.366050 + 0.930595i \(0.380710\pi\)
\(8\) 34.8481 1.54008
\(9\) −25.0534 −0.927903
\(10\) 24.0989 0.762073
\(11\) −66.2223 −1.81516 −0.907581 − 0.419877i \(-0.862073\pi\)
−0.907581 + 0.419877i \(0.862073\pi\)
\(12\) 21.2494 0.511180
\(13\) 23.3164 0.497447 0.248724 − 0.968574i \(-0.419989\pi\)
0.248724 + 0.968574i \(0.419989\pi\)
\(14\) 65.3498 1.24753
\(15\) 6.97605 0.120081
\(16\) 46.1180 0.720593
\(17\) 106.151 1.51443 0.757217 − 0.653163i \(-0.226558\pi\)
0.757217 + 0.653163i \(0.226558\pi\)
\(18\) −120.752 −1.58119
\(19\) −11.6497 −0.140665 −0.0703324 − 0.997524i \(-0.522406\pi\)
−0.0703324 + 0.997524i \(0.522406\pi\)
\(20\) 76.1511 0.851395
\(21\) 18.9172 0.196575
\(22\) −319.177 −3.09312
\(23\) 23.0000 0.208514
\(24\) 48.6204 0.413525
\(25\) 25.0000 0.200000
\(26\) 112.380 0.847674
\(27\) −72.6254 −0.517658
\(28\) 206.502 1.39376
\(29\) −18.8139 −0.120471 −0.0602355 − 0.998184i \(-0.519185\pi\)
−0.0602355 + 0.998184i \(0.519185\pi\)
\(30\) 33.6230 0.204623
\(31\) −230.170 −1.33354 −0.666771 − 0.745263i \(-0.732324\pi\)
−0.666771 + 0.745263i \(0.732324\pi\)
\(32\) −56.5063 −0.312156
\(33\) −92.3941 −0.487386
\(34\) 511.623 2.58067
\(35\) 67.7934 0.327405
\(36\) −381.569 −1.76652
\(37\) −379.629 −1.68677 −0.843386 − 0.537308i \(-0.819441\pi\)
−0.843386 + 0.537308i \(0.819441\pi\)
\(38\) −56.1491 −0.239699
\(39\) 32.5313 0.133569
\(40\) 174.240 0.688745
\(41\) −202.375 −0.770870 −0.385435 − 0.922735i \(-0.625949\pi\)
−0.385435 + 0.922735i \(0.625949\pi\)
\(42\) 91.1767 0.334973
\(43\) 320.493 1.13662 0.568312 − 0.822813i \(-0.307597\pi\)
0.568312 + 0.822813i \(0.307597\pi\)
\(44\) −1008.58 −3.45567
\(45\) −125.267 −0.414971
\(46\) 110.855 0.355318
\(47\) 271.205 0.841688 0.420844 − 0.907133i \(-0.361734\pi\)
0.420844 + 0.907133i \(0.361734\pi\)
\(48\) 64.3442 0.193485
\(49\) −159.162 −0.464029
\(50\) 120.494 0.340810
\(51\) 148.103 0.406638
\(52\) 355.115 0.947029
\(53\) 765.344 1.98355 0.991775 − 0.127995i \(-0.0408540\pi\)
0.991775 + 0.127995i \(0.0408540\pi\)
\(54\) −350.038 −0.882114
\(55\) −331.112 −0.811765
\(56\) 472.494 1.12749
\(57\) −16.2538 −0.0377697
\(58\) −90.6790 −0.205288
\(59\) 388.071 0.856314 0.428157 − 0.903704i \(-0.359163\pi\)
0.428157 + 0.903704i \(0.359163\pi\)
\(60\) 106.247 0.228607
\(61\) 519.064 1.08950 0.544749 − 0.838599i \(-0.316625\pi\)
0.544749 + 0.838599i \(0.316625\pi\)
\(62\) −1109.37 −2.27242
\(63\) −339.691 −0.679318
\(64\) −641.291 −1.25252
\(65\) 116.582 0.222465
\(66\) −445.319 −0.830529
\(67\) 358.698 0.654059 0.327030 − 0.945014i \(-0.393952\pi\)
0.327030 + 0.945014i \(0.393952\pi\)
\(68\) 1616.70 2.88315
\(69\) 32.0898 0.0559879
\(70\) 326.749 0.557914
\(71\) 40.0003 0.0668615 0.0334307 − 0.999441i \(-0.489357\pi\)
0.0334307 + 0.999441i \(0.489357\pi\)
\(72\) −873.062 −1.42905
\(73\) 129.058 0.206920 0.103460 − 0.994634i \(-0.467009\pi\)
0.103460 + 0.994634i \(0.467009\pi\)
\(74\) −1829.72 −2.87434
\(75\) 34.8803 0.0537017
\(76\) −177.428 −0.267795
\(77\) −897.888 −1.32888
\(78\) 156.794 0.227607
\(79\) −484.468 −0.689961 −0.344981 − 0.938610i \(-0.612115\pi\)
−0.344981 + 0.938610i \(0.612115\pi\)
\(80\) 230.590 0.322259
\(81\) 575.114 0.788908
\(82\) −975.402 −1.31360
\(83\) 5.74998 0.00760413 0.00380206 − 0.999993i \(-0.498790\pi\)
0.00380206 + 0.999993i \(0.498790\pi\)
\(84\) 288.113 0.374235
\(85\) 530.755 0.677276
\(86\) 1544.71 1.93686
\(87\) −26.2494 −0.0323475
\(88\) −2307.72 −2.79550
\(89\) −167.400 −0.199375 −0.0996875 − 0.995019i \(-0.531784\pi\)
−0.0996875 + 0.995019i \(0.531784\pi\)
\(90\) −603.758 −0.707130
\(91\) 316.140 0.364181
\(92\) 350.295 0.396965
\(93\) −321.136 −0.358067
\(94\) 1307.15 1.43428
\(95\) −58.2486 −0.0629072
\(96\) −78.8381 −0.0838165
\(97\) 582.078 0.609289 0.304645 − 0.952466i \(-0.401462\pi\)
0.304645 + 0.952466i \(0.401462\pi\)
\(98\) −767.125 −0.790728
\(99\) 1659.09 1.68429
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.f.1.8 ✓ 8
3.2 odd 2 1035.4.a.r.1.1 8
4.3 odd 2 1840.4.a.v.1.4 8
5.2 odd 4 575.4.b.k.24.15 16
5.3 odd 4 575.4.b.k.24.2 16
5.4 even 2 575.4.a.n.1.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.4.a.f.1.8 ✓ 8 1.1 even 1 trivial
575.4.a.n.1.1 8 5.4 even 2
575.4.b.k.24.2 16 5.3 odd 4
575.4.b.k.24.15 16 5.2 odd 4
1035.4.a.r.1.1 8 3.2 odd 2
1840.4.a.v.1.4 8 4.3 odd 2