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Properties

Label 8001.2.a.t.1.15
Level $8001$
Weight $2$
Character 8001.1
Self dual yes
Analytic conductor $63.888$
Analytic rank $0$
Dimension $16$
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] = [8001,2,Mod(1,8001)] 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("8001.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8001, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8001.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,2,0,12,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(63.8883066572\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} - 20 x^{14} + 38 x^{13} + 155 x^{12} - 275 x^{11} - 593 x^{10} + 957 x^{9} + 1177 x^{8} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 889)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.15
Root \(2.37299\) of defining polynomial
Character \(\chi\) \(=\) 8001.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.37299 q^{2} +3.63108 q^{4} +3.84175 q^{5} -1.00000 q^{7} +3.87053 q^{8} +9.11644 q^{10} +3.34213 q^{11} +4.24344 q^{13} -2.37299 q^{14} +1.92256 q^{16} -3.68573 q^{17} -1.16163 q^{19} +13.9497 q^{20} +7.93083 q^{22} +1.32516 q^{23} +9.75907 q^{25} +10.0696 q^{26} -3.63108 q^{28} -6.59844 q^{29} -3.12317 q^{31} -3.17883 q^{32} -8.74620 q^{34} -3.84175 q^{35} +5.94714 q^{37} -2.75654 q^{38} +14.8696 q^{40} +10.3758 q^{41} +4.75366 q^{43} +12.1355 q^{44} +3.14459 q^{46} -1.96484 q^{47} +1.00000 q^{49} +23.1582 q^{50} +15.4082 q^{52} -7.36495 q^{53} +12.8396 q^{55} -3.87053 q^{56} -15.6580 q^{58} +8.49156 q^{59} +7.06437 q^{61} -7.41126 q^{62} -11.3885 q^{64} +16.3022 q^{65} +3.62274 q^{67} -13.3832 q^{68} -9.11644 q^{70} +3.79052 q^{71} -14.4419 q^{73} +14.1125 q^{74} -4.21798 q^{76} -3.34213 q^{77} +5.35475 q^{79} +7.38601 q^{80} +24.6216 q^{82} -1.82593 q^{83} -14.1597 q^{85} +11.2804 q^{86} +12.9358 q^{88} +13.8250 q^{89} -4.24344 q^{91} +4.81176 q^{92} -4.66254 q^{94} -4.46271 q^{95} -1.00708 q^{97} +2.37299 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2 q^{2} + 12 q^{4} + 9 q^{5} - 16 q^{7} + 6 q^{8} - 2 q^{10} + 22 q^{11} - 4 q^{13} - 2 q^{14} + 12 q^{16} + 18 q^{17} - 15 q^{19} + 40 q^{20} - 11 q^{22} + 5 q^{23} + 15 q^{25} + 24 q^{26} - 12 q^{28}+ \cdots + 2 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.37299 1.67796 0.838978 − 0.544165i \(-0.183153\pi\)
0.838978 + 0.544165i \(0.183153\pi\)
\(3\) 0 0
\(4\) 3.63108 1.81554
\(5\) 3.84175 1.71808 0.859042 − 0.511905i \(-0.171060\pi\)
0.859042 + 0.511905i \(0.171060\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 3.87053 1.36844
\(9\) 0 0
\(10\) 9.11644 2.88287
\(11\) 3.34213 1.00769 0.503845 − 0.863794i \(-0.331918\pi\)
0.503845 + 0.863794i \(0.331918\pi\)
\(12\) 0 0
\(13\) 4.24344 1.17692 0.588459 − 0.808527i \(-0.299735\pi\)
0.588459 + 0.808527i \(0.299735\pi\)
\(14\) −2.37299 −0.634208
\(15\) 0 0
\(16\) 1.92256 0.480641
\(17\) −3.68573 −0.893921 −0.446961 − 0.894554i \(-0.647494\pi\)
−0.446961 + 0.894554i \(0.647494\pi\)
\(18\) 0 0
\(19\) −1.16163 −0.266497 −0.133249 − 0.991083i \(-0.542541\pi\)
−0.133249 + 0.991083i \(0.542541\pi\)
\(20\) 13.9497 3.11925
\(21\) 0 0
\(22\) 7.93083 1.69086
\(23\) 1.32516 0.276315 0.138157 − 0.990410i \(-0.455882\pi\)
0.138157 + 0.990410i \(0.455882\pi\)
\(24\) 0 0
\(25\) 9.75907 1.95181
\(26\) 10.0696 1.97482
\(27\) 0 0
\(28\) −3.63108 −0.686209
\(29\) −6.59844 −1.22530 −0.612650 − 0.790354i \(-0.709896\pi\)
−0.612650 + 0.790354i \(0.709896\pi\)
\(30\) 0 0
\(31\) −3.12317 −0.560939 −0.280469 − 0.959863i \(-0.590490\pi\)
−0.280469 + 0.959863i \(0.590490\pi\)
\(32\) −3.17883 −0.561944
\(33\) 0 0
\(34\) −8.74620 −1.49996
\(35\) −3.84175 −0.649375
\(36\) 0 0
\(37\) 5.94714 0.977704 0.488852 − 0.872367i \(-0.337416\pi\)
0.488852 + 0.872367i \(0.337416\pi\)
\(38\) −2.75654 −0.447171
\(39\) 0 0
\(40\) 14.8696 2.35109
\(41\) 10.3758 1.62043 0.810213 − 0.586136i \(-0.199351\pi\)
0.810213 + 0.586136i \(0.199351\pi\)
\(42\) 0 0
\(43\) 4.75366 0.724926 0.362463 − 0.931998i \(-0.381936\pi\)
0.362463 + 0.931998i \(0.381936\pi\)
\(44\) 12.1355 1.82950
\(45\) 0 0
\(46\) 3.14459 0.463644
\(47\) −1.96484 −0.286601 −0.143301 − 0.989679i \(-0.545772\pi\)
−0.143301 + 0.989679i \(0.545772\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 23.1582 3.27506
\(51\) 0 0
\(52\) 15.4082 2.13674
\(53\) −7.36495 −1.01165 −0.505827 − 0.862635i \(-0.668812\pi\)
−0.505827 + 0.862635i \(0.668812\pi\)
\(54\) 0 0
\(55\) 12.8396 1.73130
\(56\) −3.87053 −0.517221
\(57\) 0 0
\(58\) −15.6580 −2.05600
\(59\) 8.49156 1.10551 0.552753 − 0.833345i \(-0.313577\pi\)
0.552753 + 0.833345i \(0.313577\pi\)
\(60\) 0 0
\(61\) 7.06437 0.904500 0.452250 − 0.891891i \(-0.350621\pi\)
0.452250 + 0.891891i \(0.350621\pi\)
\(62\) −7.41126 −0.941231
\(63\) 0 0
\(64\) −11.3885 −1.42356
\(65\) 16.3022 2.02204
\(66\) 0 0
\(67\) 3.62274 0.442588 0.221294 − 0.975207i \(-0.428972\pi\)
0.221294 + 0.975207i \(0.428972\pi\)
\(68\) −13.3832 −1.62295
\(69\) 0 0
\(70\) −9.11644 −1.08962
\(71\) 3.79052 0.449851 0.224926 − 0.974376i \(-0.427786\pi\)
0.224926 + 0.974376i \(0.427786\pi\)
\(72\) 0 0
\(73\) −14.4419 −1.69029 −0.845146 − 0.534536i \(-0.820487\pi\)
−0.845146 + 0.534536i \(0.820487\pi\)
\(74\) 14.1125 1.64055
\(75\) 0 0
\(76\) −4.21798 −0.483836
\(77\) −3.34213 −0.380871
\(78\) 0 0
\(79\) 5.35475 0.602456 0.301228 − 0.953552i \(-0.402603\pi\)
0.301228 + 0.953552i \(0.402603\pi\)
\(80\) 7.38601 0.825781
\(81\) 0 0
\(82\) 24.6216 2.71900
\(83\) −1.82593 −0.200422 −0.100211 − 0.994966i \(-0.531952\pi\)
−0.100211 + 0.994966i \(0.531952\pi\)
\(84\) 0 0
\(85\) −14.1597 −1.53583
\(86\) 11.2804 1.21639
\(87\) 0 0
\(88\) 12.9358 1.37896
\(89\) 13.8250 1.46544 0.732722 − 0.680528i \(-0.238250\pi\)
0.732722 + 0.680528i \(0.238250\pi\)
\(90\) 0 0
\(91\) −4.24344 −0.444833
\(92\) 4.81176 0.501660
\(93\) 0 0
\(94\) −4.66254 −0.480905
\(95\) −4.46271 −0.457865
\(96\) 0 0
\(97\) −1.00708 −0.102253 −0.0511265 − 0.998692i \(-0.516281\pi\)
−0.0511265 + 0.998692i \(0.516281\pi\)
\(98\) 2.37299 0.239708
\(99\) 0 0
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 8001.2.a.t.1.15 16
3.2 odd 2 889.2.a.c.1.2 ✓ 16
21.20 even 2 6223.2.a.k.1.2 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
889.2.a.c.1.2 ✓ 16 3.2 odd 2
6223.2.a.k.1.2 16 21.20 even 2
8001.2.a.t.1.15 16 1.1 even 1 trivial