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Properties

Label 8001.2.a.s.1.16
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,-4,0,20,-5] 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: \( x^{16} - 4 x^{15} - 18 x^{14} + 83 x^{13} + 112 x^{12} - 668 x^{11} - 235 x^{10} + 2648 x^{9} + \cdots - 20 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 2667)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.16
Root \(-2.57436\) 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)
 
\(f(q)\) \(=\) \(q+2.57436 q^{2} +4.62732 q^{4} -2.78465 q^{5} -1.00000 q^{7} +6.76367 q^{8} -7.16870 q^{10} -1.44556 q^{11} +4.74033 q^{13} -2.57436 q^{14} +8.15746 q^{16} -4.74437 q^{17} +5.68628 q^{19} -12.8855 q^{20} -3.72140 q^{22} -0.885825 q^{23} +2.75429 q^{25} +12.2033 q^{26} -4.62732 q^{28} +1.81353 q^{29} +4.74962 q^{31} +7.47290 q^{32} -12.2137 q^{34} +2.78465 q^{35} +5.22583 q^{37} +14.6385 q^{38} -18.8345 q^{40} +5.00751 q^{41} -6.34876 q^{43} -6.68909 q^{44} -2.28043 q^{46} +13.1360 q^{47} +1.00000 q^{49} +7.09054 q^{50} +21.9350 q^{52} -8.73749 q^{53} +4.02539 q^{55} -6.76367 q^{56} +4.66867 q^{58} +7.36544 q^{59} +2.00342 q^{61} +12.2272 q^{62} +2.92299 q^{64} -13.2002 q^{65} -4.76572 q^{67} -21.9537 q^{68} +7.16870 q^{70} +13.1659 q^{71} +12.7431 q^{73} +13.4532 q^{74} +26.3123 q^{76} +1.44556 q^{77} +3.77816 q^{79} -22.7157 q^{80} +12.8911 q^{82} -13.2837 q^{83} +13.2114 q^{85} -16.3440 q^{86} -9.77731 q^{88} +2.64036 q^{89} -4.74033 q^{91} -4.09900 q^{92} +33.8168 q^{94} -15.8343 q^{95} +14.9175 q^{97} +2.57436 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{2} + 20 q^{4} - 5 q^{5} - 16 q^{7} - 15 q^{8} - 4 q^{10} - q^{11} + 20 q^{13} + 4 q^{14} + 32 q^{16} - 3 q^{17} + 13 q^{19} - 17 q^{20} + 13 q^{22} - 5 q^{23} + 17 q^{25} + 2 q^{26} - 20 q^{28}+ \cdots - 4 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.57436 1.82035 0.910173 − 0.414228i \(-0.135948\pi\)
0.910173 + 0.414228i \(0.135948\pi\)
\(3\) 0 0
\(4\) 4.62732 2.31366
\(5\) −2.78465 −1.24533 −0.622667 − 0.782487i \(-0.713951\pi\)
−0.622667 + 0.782487i \(0.713951\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 6.76367 2.39132
\(9\) 0 0
\(10\) −7.16870 −2.26694
\(11\) −1.44556 −0.435854 −0.217927 − 0.975965i \(-0.569929\pi\)
−0.217927 + 0.975965i \(0.569929\pi\)
\(12\) 0 0
\(13\) 4.74033 1.31473 0.657365 − 0.753572i \(-0.271671\pi\)
0.657365 + 0.753572i \(0.271671\pi\)
\(14\) −2.57436 −0.688026
\(15\) 0 0
\(16\) 8.15746 2.03937
\(17\) −4.74437 −1.15068 −0.575340 − 0.817914i \(-0.695130\pi\)
−0.575340 + 0.817914i \(0.695130\pi\)
\(18\) 0 0
\(19\) 5.68628 1.30452 0.652261 − 0.757994i \(-0.273820\pi\)
0.652261 + 0.757994i \(0.273820\pi\)
\(20\) −12.8855 −2.88128
\(21\) 0 0
\(22\) −3.72140 −0.793405
\(23\) −0.885825 −0.184707 −0.0923537 − 0.995726i \(-0.529439\pi\)
−0.0923537 + 0.995726i \(0.529439\pi\)
\(24\) 0 0
\(25\) 2.75429 0.550858
\(26\) 12.2033 2.39326
\(27\) 0 0
\(28\) −4.62732 −0.874482
\(29\) 1.81353 0.336764 0.168382 − 0.985722i \(-0.446146\pi\)
0.168382 + 0.985722i \(0.446146\pi\)
\(30\) 0 0
\(31\) 4.74962 0.853057 0.426528 − 0.904474i \(-0.359736\pi\)
0.426528 + 0.904474i \(0.359736\pi\)
\(32\) 7.47290 1.32103
\(33\) 0 0
\(34\) −12.2137 −2.09464
\(35\) 2.78465 0.470692
\(36\) 0 0
\(37\) 5.22583 0.859122 0.429561 − 0.903038i \(-0.358668\pi\)
0.429561 + 0.903038i \(0.358668\pi\)
\(38\) 14.6385 2.37468
\(39\) 0 0
\(40\) −18.8345 −2.97799
\(41\) 5.00751 0.782042 0.391021 − 0.920382i \(-0.372122\pi\)
0.391021 + 0.920382i \(0.372122\pi\)
\(42\) 0 0
\(43\) −6.34876 −0.968177 −0.484088 − 0.875019i \(-0.660849\pi\)
−0.484088 + 0.875019i \(0.660849\pi\)
\(44\) −6.68909 −1.00842
\(45\) 0 0
\(46\) −2.28043 −0.336231
\(47\) 13.1360 1.91609 0.958043 − 0.286624i \(-0.0925331\pi\)
0.958043 + 0.286624i \(0.0925331\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 7.09054 1.00275
\(51\) 0 0
\(52\) 21.9350 3.04184
\(53\) −8.73749 −1.20019 −0.600093 − 0.799930i \(-0.704870\pi\)
−0.600093 + 0.799930i \(0.704870\pi\)
\(54\) 0 0
\(55\) 4.02539 0.542784
\(56\) −6.76367 −0.903833
\(57\) 0 0
\(58\) 4.66867 0.613027
\(59\) 7.36544 0.958898 0.479449 − 0.877570i \(-0.340836\pi\)
0.479449 + 0.877570i \(0.340836\pi\)
\(60\) 0 0
\(61\) 2.00342 0.256511 0.128256 − 0.991741i \(-0.459062\pi\)
0.128256 + 0.991741i \(0.459062\pi\)
\(62\) 12.2272 1.55286
\(63\) 0 0
\(64\) 2.92299 0.365374
\(65\) −13.2002 −1.63728
\(66\) 0 0
\(67\) −4.76572 −0.582226 −0.291113 − 0.956689i \(-0.594025\pi\)
−0.291113 + 0.956689i \(0.594025\pi\)
\(68\) −21.9537 −2.66228
\(69\) 0 0
\(70\) 7.16870 0.856823
\(71\) 13.1659 1.56250 0.781251 − 0.624217i \(-0.214582\pi\)
0.781251 + 0.624217i \(0.214582\pi\)
\(72\) 0 0
\(73\) 12.7431 1.49147 0.745736 − 0.666242i \(-0.232098\pi\)
0.745736 + 0.666242i \(0.232098\pi\)
\(74\) 13.4532 1.56390
\(75\) 0 0
\(76\) 26.3123 3.01822
\(77\) 1.44556 0.164737
\(78\) 0 0
\(79\) 3.77816 0.425076 0.212538 − 0.977153i \(-0.431827\pi\)
0.212538 + 0.977153i \(0.431827\pi\)
\(80\) −22.7157 −2.53969
\(81\) 0 0
\(82\) 12.8911 1.42359
\(83\) −13.2837 −1.45807 −0.729036 − 0.684475i \(-0.760031\pi\)
−0.729036 + 0.684475i \(0.760031\pi\)
\(84\) 0 0
\(85\) 13.2114 1.43298
\(86\) −16.3440 −1.76242
\(87\) 0 0
\(88\) −9.77731 −1.04226
\(89\) 2.64036 0.279877 0.139939 − 0.990160i \(-0.455309\pi\)
0.139939 + 0.990160i \(0.455309\pi\)
\(90\) 0 0
\(91\) −4.74033 −0.496921
\(92\) −4.09900 −0.427350
\(93\) 0 0
\(94\) 33.8168 3.48794
\(95\) −15.8343 −1.62457
\(96\) 0 0
\(97\) 14.9175 1.51464 0.757322 − 0.653041i \(-0.226507\pi\)
0.757322 + 0.653041i \(0.226507\pi\)
\(98\) 2.57436 0.260049
\(99\) 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.s.1.16 16
3.2 odd 2 2667.2.a.n.1.1 ✓ 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2667.2.a.n.1.1 ✓ 16 3.2 odd 2
8001.2.a.s.1.16 16 1.1 even 1 trivial