Properties

Label 115.4.a.f.1.6
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 1.6
Root \(-3.05234\) 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)
 
\(f(q)\) \(=\) \(q+4.05234 q^{2} -8.94300 q^{3} +8.42146 q^{4} +5.00000 q^{5} -36.2401 q^{6} +32.7645 q^{7} +1.70790 q^{8} +52.9772 q^{9} +20.2617 q^{10} +45.7496 q^{11} -75.3131 q^{12} +45.5638 q^{13} +132.773 q^{14} -44.7150 q^{15} -60.4507 q^{16} -76.5649 q^{17} +214.681 q^{18} -32.0225 q^{19} +42.1073 q^{20} -293.013 q^{21} +185.393 q^{22} +23.0000 q^{23} -15.2738 q^{24} +25.0000 q^{25} +184.640 q^{26} -232.314 q^{27} +275.925 q^{28} +287.046 q^{29} -181.200 q^{30} -161.587 q^{31} -258.630 q^{32} -409.139 q^{33} -310.267 q^{34} +163.823 q^{35} +446.145 q^{36} +7.53926 q^{37} -129.766 q^{38} -407.476 q^{39} +8.53952 q^{40} +216.246 q^{41} -1187.39 q^{42} -469.729 q^{43} +385.279 q^{44} +264.886 q^{45} +93.2038 q^{46} -210.315 q^{47} +540.610 q^{48} +730.514 q^{49} +101.309 q^{50} +684.719 q^{51} +383.713 q^{52} +58.7860 q^{53} -941.414 q^{54} +228.748 q^{55} +55.9587 q^{56} +286.377 q^{57} +1163.21 q^{58} -376.609 q^{59} -376.565 q^{60} -52.7319 q^{61} -654.805 q^{62} +1735.77 q^{63} -564.451 q^{64} +227.819 q^{65} -1657.97 q^{66} -16.7678 q^{67} -644.788 q^{68} -205.689 q^{69} +663.865 q^{70} -209.441 q^{71} +90.4800 q^{72} -811.910 q^{73} +30.5516 q^{74} -223.575 q^{75} -269.676 q^{76} +1498.96 q^{77} -1651.23 q^{78} +63.0408 q^{79} -302.253 q^{80} +647.197 q^{81} +876.300 q^{82} -1062.27 q^{83} -2467.60 q^{84} -382.824 q^{85} -1903.50 q^{86} -2567.05 q^{87} +78.1360 q^{88} +436.069 q^{89} +1073.41 q^{90} +1492.87 q^{91} +193.694 q^{92} +1445.07 q^{93} -852.269 q^{94} -160.112 q^{95} +2312.93 q^{96} +655.840 q^{97} +2960.29 q^{98} +2423.68 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 6 q^{2} + 42 q^{4} + 40 q^{5} + 11 q^{7} + 3 q^{8} + 158 q^{9} + 30 q^{10} + 41 q^{11} + 48 q^{12} + 28 q^{13} + 161 q^{14} + 98 q^{16} + 71 q^{17} + 84 q^{18} + 177 q^{19} + 210 q^{20} + 292 q^{21}+ \cdots + 1319 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.05234 1.43272 0.716359 − 0.697732i \(-0.245807\pi\)
0.716359 + 0.697732i \(0.245807\pi\)
\(3\) −8.94300 −1.72108 −0.860540 − 0.509383i \(-0.829874\pi\)
−0.860540 + 0.509383i \(0.829874\pi\)
\(4\) 8.42146 1.05268
\(5\) 5.00000 0.447214
\(6\) −36.2401 −2.46582
\(7\) 32.7645 1.76912 0.884559 − 0.466429i \(-0.154460\pi\)
0.884559 + 0.466429i \(0.154460\pi\)
\(8\) 1.70790 0.0754794
\(9\) 52.9772 1.96212
\(10\) 20.2617 0.640731
\(11\) 45.7496 1.25400 0.627001 − 0.779018i \(-0.284282\pi\)
0.627001 + 0.779018i \(0.284282\pi\)
\(12\) −75.3131 −1.81175
\(13\) 45.5638 0.972086 0.486043 − 0.873935i \(-0.338440\pi\)
0.486043 + 0.873935i \(0.338440\pi\)
\(14\) 132.773 2.53465
\(15\) −44.7150 −0.769690
\(16\) −60.4507 −0.944542
\(17\) −76.5649 −1.09234 −0.546168 − 0.837676i \(-0.683914\pi\)
−0.546168 + 0.837676i \(0.683914\pi\)
\(18\) 214.681 2.81116
\(19\) −32.0225 −0.386656 −0.193328 − 0.981134i \(-0.561928\pi\)
−0.193328 + 0.981134i \(0.561928\pi\)
\(20\) 42.1073 0.470774
\(21\) −293.013 −3.04479
\(22\) 185.393 1.79663
\(23\) 23.0000 0.208514
\(24\) −15.2738 −0.129906
\(25\) 25.0000 0.200000
\(26\) 184.640 1.39273
\(27\) −232.314 −1.65588
\(28\) 275.925 1.86232
\(29\) 287.046 1.83804 0.919019 − 0.394213i \(-0.128983\pi\)
0.919019 + 0.394213i \(0.128983\pi\)
\(30\) −181.200 −1.10275
\(31\) −161.587 −0.936189 −0.468095 − 0.883678i \(-0.655059\pi\)
−0.468095 + 0.883678i \(0.655059\pi\)
\(32\) −258.630 −1.42874
\(33\) −409.139 −2.15824
\(34\) −310.267 −1.56501
\(35\) 163.823 0.791173
\(36\) 446.145 2.06549
\(37\) 7.53926 0.0334986 0.0167493 − 0.999860i \(-0.494668\pi\)
0.0167493 + 0.999860i \(0.494668\pi\)
\(38\) −129.766 −0.553969
\(39\) −407.476 −1.67304
\(40\) 8.53952 0.0337554
\(41\) 216.246 0.823704 0.411852 − 0.911251i \(-0.364882\pi\)
0.411852 + 0.911251i \(0.364882\pi\)
\(42\) −1187.39 −4.36233
\(43\) −469.729 −1.66589 −0.832943 − 0.553359i \(-0.813346\pi\)
−0.832943 + 0.553359i \(0.813346\pi\)
\(44\) 385.279 1.32007
\(45\) 264.886 0.877485
\(46\) 93.2038 0.298742
\(47\) −210.315 −0.652715 −0.326358 − 0.945246i \(-0.605821\pi\)
−0.326358 + 0.945246i \(0.605821\pi\)
\(48\) 540.610 1.62563
\(49\) 730.514 2.12978
\(50\) 101.309 0.286544
\(51\) 684.719 1.88000
\(52\) 383.713 1.02330
\(53\) 58.7860 0.152356 0.0761781 − 0.997094i \(-0.475728\pi\)
0.0761781 + 0.997094i \(0.475728\pi\)
\(54\) −941.414 −2.37241
\(55\) 228.748 0.560807
\(56\) 55.9587 0.133532
\(57\) 286.377 0.665466
\(58\) 1163.21 2.63339
\(59\) −376.609 −0.831022 −0.415511 − 0.909588i \(-0.636397\pi\)
−0.415511 + 0.909588i \(0.636397\pi\)
\(60\) −376.565 −0.810240
\(61\) −52.7319 −0.110682 −0.0553412 − 0.998467i \(-0.517625\pi\)
−0.0553412 + 0.998467i \(0.517625\pi\)
\(62\) −654.805 −1.34130
\(63\) 1735.77 3.47122
\(64\) −564.451 −1.10244
\(65\) 227.819 0.434730
\(66\) −1657.97 −3.09215
\(67\) −16.7678 −0.0305748 −0.0152874 − 0.999883i \(-0.504866\pi\)
−0.0152874 + 0.999883i \(0.504866\pi\)
\(68\) −644.788 −1.14988
\(69\) −205.689 −0.358870
\(70\) 663.865 1.13353
\(71\) −209.441 −0.350086 −0.175043 − 0.984561i \(-0.556006\pi\)
−0.175043 + 0.984561i \(0.556006\pi\)
\(72\) 90.4800 0.148100
\(73\) −811.910 −1.30174 −0.650869 − 0.759190i \(-0.725595\pi\)
−0.650869 + 0.759190i \(0.725595\pi\)
\(74\) 30.5516 0.0479940
\(75\) −223.575 −0.344216
\(76\) −269.676 −0.407026
\(77\) 1498.96 2.21848
\(78\) −1651.23 −2.39699
\(79\) 63.0408 0.0897803 0.0448902 − 0.998992i \(-0.485706\pi\)
0.0448902 + 0.998992i \(0.485706\pi\)
\(80\) −302.253 −0.422412
\(81\) 647.197 0.887787
\(82\) 876.300 1.18014
\(83\) −1062.27 −1.40481 −0.702404 − 0.711779i \(-0.747890\pi\)
−0.702404 + 0.711779i \(0.747890\pi\)
\(84\) −2467.60 −3.20520
\(85\) −382.824 −0.488508
\(86\) −1903.50 −2.38674
\(87\) −2567.05 −3.16341
\(88\) 78.1360 0.0946514
\(89\) 436.069 0.519362 0.259681 − 0.965694i \(-0.416383\pi\)
0.259681 + 0.965694i \(0.416383\pi\)
\(90\) 1073.41 1.25719
\(91\) 1492.87 1.71973
\(92\) 193.694 0.219500
\(93\) 1445.07 1.61126
\(94\) −852.269 −0.935158
\(95\) −160.112 −0.172918
\(96\) 2312.93 2.45898
\(97\) 655.840 0.686499 0.343250 − 0.939244i \(-0.388472\pi\)
0.343250 + 0.939244i \(0.388472\pi\)
\(98\) 2960.29 3.05137
\(99\) 2423.68 2.46050
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.1.6 ✓ 8
3.2 odd 2 1035.4.a.r.1.3 8
4.3 odd 2 1840.4.a.v.1.7 8
5.2 odd 4 575.4.b.k.24.13 16
5.3 odd 4 575.4.b.k.24.4 16
5.4 even 2 575.4.a.n.1.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.4.a.f.1.6 ✓ 8 1.1 even 1 trivial
575.4.a.n.1.3 8 5.4 even 2
575.4.b.k.24.4 16 5.3 odd 4
575.4.b.k.24.13 16 5.2 odd 4
1035.4.a.r.1.3 8 3.2 odd 2
1840.4.a.v.1.7 8 4.3 odd 2