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Properties

Label 8001.2.a.p.1.14
Level $8001$
Weight $2$
Character 8001.1
Self dual yes
Analytic conductor $63.888$
Analytic rank $0$
Dimension $14$
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 = [14,5,0,15,4] 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: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 5 x^{13} - 9 x^{12} + 76 x^{11} - 12 x^{10} - 414 x^{9} + 331 x^{8} + 959 x^{7} - 1067 x^{6} + \cdots + 44 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 5 \)
Twist minimal: no (minimal twist has level 2667)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.14
Root \(2.66839\) 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)
 
\(f(q)\) \(=\) \(q+2.66839 q^{2} +5.12030 q^{4} +2.72934 q^{5} -1.00000 q^{7} +8.32617 q^{8} +7.28293 q^{10} -1.99935 q^{11} +3.01152 q^{13} -2.66839 q^{14} +11.9769 q^{16} +1.40099 q^{17} -3.11408 q^{19} +13.9750 q^{20} -5.33503 q^{22} +3.27197 q^{23} +2.44927 q^{25} +8.03590 q^{26} -5.12030 q^{28} +6.12906 q^{29} +6.08476 q^{31} +15.3066 q^{32} +3.73838 q^{34} -2.72934 q^{35} -5.42709 q^{37} -8.30959 q^{38} +22.7249 q^{40} -11.6265 q^{41} +9.33583 q^{43} -10.2373 q^{44} +8.73088 q^{46} +8.98796 q^{47} +1.00000 q^{49} +6.53561 q^{50} +15.4199 q^{52} -4.15642 q^{53} -5.45689 q^{55} -8.32617 q^{56} +16.3547 q^{58} -7.85361 q^{59} -3.02245 q^{61} +16.2365 q^{62} +16.8902 q^{64} +8.21944 q^{65} -7.97480 q^{67} +7.17347 q^{68} -7.28293 q^{70} -4.60611 q^{71} +6.75870 q^{73} -14.4816 q^{74} -15.9450 q^{76} +1.99935 q^{77} +0.469412 q^{79} +32.6889 q^{80} -31.0240 q^{82} +11.7285 q^{83} +3.82376 q^{85} +24.9116 q^{86} -16.6469 q^{88} -1.25580 q^{89} -3.01152 q^{91} +16.7534 q^{92} +23.9834 q^{94} -8.49938 q^{95} +5.52320 q^{97} +2.66839 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 5 q^{2} + 15 q^{4} + 4 q^{5} - 14 q^{7} + 12 q^{8} + 4 q^{10} + 3 q^{11} - 13 q^{13} - 5 q^{14} + 13 q^{16} + 5 q^{17} + 21 q^{19} + 3 q^{20} - 3 q^{22} + 10 q^{23} + 4 q^{25} + 6 q^{26} - 15 q^{28}+ \cdots + 5 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.66839 1.88684 0.943418 − 0.331606i \(-0.107590\pi\)
0.943418 + 0.331606i \(0.107590\pi\)
\(3\) 0 0
\(4\) 5.12030 2.56015
\(5\) 2.72934 1.22060 0.610298 − 0.792172i \(-0.291050\pi\)
0.610298 + 0.792172i \(0.291050\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 8.32617 2.94375
\(9\) 0 0
\(10\) 7.28293 2.30306
\(11\) −1.99935 −0.602826 −0.301413 − 0.953494i \(-0.597458\pi\)
−0.301413 + 0.953494i \(0.597458\pi\)
\(12\) 0 0
\(13\) 3.01152 0.835244 0.417622 − 0.908621i \(-0.362864\pi\)
0.417622 + 0.908621i \(0.362864\pi\)
\(14\) −2.66839 −0.713157
\(15\) 0 0
\(16\) 11.9769 2.99422
\(17\) 1.40099 0.339789 0.169894 − 0.985462i \(-0.445657\pi\)
0.169894 + 0.985462i \(0.445657\pi\)
\(18\) 0 0
\(19\) −3.11408 −0.714420 −0.357210 − 0.934024i \(-0.616272\pi\)
−0.357210 + 0.934024i \(0.616272\pi\)
\(20\) 13.9750 3.12491
\(21\) 0 0
\(22\) −5.33503 −1.13743
\(23\) 3.27197 0.682252 0.341126 − 0.940018i \(-0.389192\pi\)
0.341126 + 0.940018i \(0.389192\pi\)
\(24\) 0 0
\(25\) 2.44927 0.489854
\(26\) 8.03590 1.57597
\(27\) 0 0
\(28\) −5.12030 −0.967646
\(29\) 6.12906 1.13814 0.569069 − 0.822290i \(-0.307304\pi\)
0.569069 + 0.822290i \(0.307304\pi\)
\(30\) 0 0
\(31\) 6.08476 1.09286 0.546428 − 0.837506i \(-0.315987\pi\)
0.546428 + 0.837506i \(0.315987\pi\)
\(32\) 15.3066 2.70585
\(33\) 0 0
\(34\) 3.73838 0.641126
\(35\) −2.72934 −0.461342
\(36\) 0 0
\(37\) −5.42709 −0.892208 −0.446104 − 0.894981i \(-0.647189\pi\)
−0.446104 + 0.894981i \(0.647189\pi\)
\(38\) −8.30959 −1.34799
\(39\) 0 0
\(40\) 22.7249 3.59313
\(41\) −11.6265 −1.81575 −0.907875 − 0.419241i \(-0.862296\pi\)
−0.907875 + 0.419241i \(0.862296\pi\)
\(42\) 0 0
\(43\) 9.33583 1.42370 0.711850 − 0.702331i \(-0.247858\pi\)
0.711850 + 0.702331i \(0.247858\pi\)
\(44\) −10.2373 −1.54332
\(45\) 0 0
\(46\) 8.73088 1.28730
\(47\) 8.98796 1.31103 0.655515 − 0.755182i \(-0.272452\pi\)
0.655515 + 0.755182i \(0.272452\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 6.53561 0.924275
\(51\) 0 0
\(52\) 15.4199 2.13835
\(53\) −4.15642 −0.570928 −0.285464 − 0.958389i \(-0.592148\pi\)
−0.285464 + 0.958389i \(0.592148\pi\)
\(54\) 0 0
\(55\) −5.45689 −0.735806
\(56\) −8.32617 −1.11263
\(57\) 0 0
\(58\) 16.3547 2.14748
\(59\) −7.85361 −1.02245 −0.511227 − 0.859446i \(-0.670809\pi\)
−0.511227 + 0.859446i \(0.670809\pi\)
\(60\) 0 0
\(61\) −3.02245 −0.386985 −0.193493 − 0.981102i \(-0.561982\pi\)
−0.193493 + 0.981102i \(0.561982\pi\)
\(62\) 16.2365 2.06204
\(63\) 0 0
\(64\) 16.8902 2.11128
\(65\) 8.21944 1.01950
\(66\) 0 0
\(67\) −7.97480 −0.974276 −0.487138 − 0.873325i \(-0.661959\pi\)
−0.487138 + 0.873325i \(0.661959\pi\)
\(68\) 7.17347 0.869911
\(69\) 0 0
\(70\) −7.28293 −0.870476
\(71\) −4.60611 −0.546644 −0.273322 − 0.961923i \(-0.588123\pi\)
−0.273322 + 0.961923i \(0.588123\pi\)
\(72\) 0 0
\(73\) 6.75870 0.791046 0.395523 − 0.918456i \(-0.370563\pi\)
0.395523 + 0.918456i \(0.370563\pi\)
\(74\) −14.4816 −1.68345
\(75\) 0 0
\(76\) −15.9450 −1.82902
\(77\) 1.99935 0.227847
\(78\) 0 0
\(79\) 0.469412 0.0528130 0.0264065 − 0.999651i \(-0.491594\pi\)
0.0264065 + 0.999651i \(0.491594\pi\)
\(80\) 32.6889 3.65473
\(81\) 0 0
\(82\) −31.0240 −3.42602
\(83\) 11.7285 1.28737 0.643686 − 0.765290i \(-0.277404\pi\)
0.643686 + 0.765290i \(0.277404\pi\)
\(84\) 0 0
\(85\) 3.82376 0.414745
\(86\) 24.9116 2.68629
\(87\) 0 0
\(88\) −16.6469 −1.77457
\(89\) −1.25580 −0.133115 −0.0665575 − 0.997783i \(-0.521202\pi\)
−0.0665575 + 0.997783i \(0.521202\pi\)
\(90\) 0 0
\(91\) −3.01152 −0.315693
\(92\) 16.7534 1.74667
\(93\) 0 0
\(94\) 23.9834 2.47370
\(95\) −8.49938 −0.872018
\(96\) 0 0
\(97\) 5.52320 0.560796 0.280398 − 0.959884i \(-0.409534\pi\)
0.280398 + 0.959884i \(0.409534\pi\)
\(98\) 2.66839 0.269548
\(99\) 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.p.1.14 14
3.2 odd 2 2667.2.a.m.1.1 ✓ 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2667.2.a.m.1.1 ✓ 14 3.2 odd 2
8001.2.a.p.1.14 14 1.1 even 1 trivial