↑

Properties

Label 8001.2.a.s.1.11
Level $8001$
Weight $2$
Character 8001.1
Self dual yes
Analytic conductor $63.888$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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 1.11
Root \(-1.14773\) 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)
 
\(f(q)\) \(=\) \(q+1.14773 q^{2} -0.682718 q^{4} -1.07917 q^{5} -1.00000 q^{7} -3.07903 q^{8} -1.23860 q^{10} +0.423219 q^{11} -6.89935 q^{13} -1.14773 q^{14} -2.16846 q^{16} -5.63751 q^{17} +1.75210 q^{19} +0.736771 q^{20} +0.485740 q^{22} -1.59566 q^{23} -3.83538 q^{25} -7.91859 q^{26} +0.682718 q^{28} +0.672276 q^{29} +6.72329 q^{31} +3.66926 q^{32} -6.47033 q^{34} +1.07917 q^{35} -5.86962 q^{37} +2.01094 q^{38} +3.32281 q^{40} -1.19716 q^{41} +4.03197 q^{43} -0.288939 q^{44} -1.83138 q^{46} -9.90940 q^{47} +1.00000 q^{49} -4.40198 q^{50} +4.71032 q^{52} -5.94184 q^{53} -0.456726 q^{55} +3.07903 q^{56} +0.771591 q^{58} +1.55231 q^{59} +7.09581 q^{61} +7.71651 q^{62} +8.54824 q^{64} +7.44560 q^{65} -7.39331 q^{67} +3.84883 q^{68} +1.23860 q^{70} -2.78476 q^{71} +1.82359 q^{73} -6.73674 q^{74} -1.19619 q^{76} -0.423219 q^{77} +4.01893 q^{79} +2.34014 q^{80} -1.37401 q^{82} -11.3378 q^{83} +6.08385 q^{85} +4.62760 q^{86} -1.30310 q^{88} +9.08046 q^{89} +6.89935 q^{91} +1.08938 q^{92} -11.3733 q^{94} -1.89082 q^{95} +11.4925 q^{97} +1.14773 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{2} + 20 q^{4} - 5 q^{5} - 16 q^{7} - 15 q^{8} - 4 q^{10} - q^{11} + 20 q^{13} + 4 q^{14} + 32 q^{16} - 3 q^{17} + 13 q^{19} - 17 q^{20} + 13 q^{22} - 5 q^{23} + 17 q^{25} + 2 q^{26} - 20 q^{28}+ \cdots - 4 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.14773 0.811567 0.405783 − 0.913969i \(-0.366999\pi\)
0.405783 + 0.913969i \(0.366999\pi\)
\(3\) 0 0
\(4\) −0.682718 −0.341359
\(5\) −1.07917 −0.482621 −0.241311 − 0.970448i \(-0.577577\pi\)
−0.241311 + 0.970448i \(0.577577\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −3.07903 −1.08860
\(9\) 0 0
\(10\) −1.23860 −0.391679
\(11\) 0.423219 0.127605 0.0638026 − 0.997963i \(-0.479677\pi\)
0.0638026 + 0.997963i \(0.479677\pi\)
\(12\) 0 0
\(13\) −6.89935 −1.91354 −0.956768 − 0.290851i \(-0.906062\pi\)
−0.956768 + 0.290851i \(0.906062\pi\)
\(14\) −1.14773 −0.306743
\(15\) 0 0
\(16\) −2.16846 −0.542115
\(17\) −5.63751 −1.36730 −0.683649 − 0.729811i \(-0.739608\pi\)
−0.683649 + 0.729811i \(0.739608\pi\)
\(18\) 0 0
\(19\) 1.75210 0.401959 0.200980 − 0.979595i \(-0.435587\pi\)
0.200980 + 0.979595i \(0.435587\pi\)
\(20\) 0.736771 0.164747
\(21\) 0 0
\(22\) 0.485740 0.103560
\(23\) −1.59566 −0.332718 −0.166359 − 0.986065i \(-0.553201\pi\)
−0.166359 + 0.986065i \(0.553201\pi\)
\(24\) 0 0
\(25\) −3.83538 −0.767077
\(26\) −7.91859 −1.55296
\(27\) 0 0
\(28\) 0.682718 0.129022
\(29\) 0.672276 0.124839 0.0624193 − 0.998050i \(-0.480118\pi\)
0.0624193 + 0.998050i \(0.480118\pi\)
\(30\) 0 0
\(31\) 6.72329 1.20754 0.603769 − 0.797159i \(-0.293665\pi\)
0.603769 + 0.797159i \(0.293665\pi\)
\(32\) 3.66926 0.648640
\(33\) 0 0
\(34\) −6.47033 −1.10965
\(35\) 1.07917 0.182414
\(36\) 0 0
\(37\) −5.86962 −0.964960 −0.482480 − 0.875907i \(-0.660264\pi\)
−0.482480 + 0.875907i \(0.660264\pi\)
\(38\) 2.01094 0.326217
\(39\) 0 0
\(40\) 3.32281 0.525383
\(41\) −1.19716 −0.186965 −0.0934824 − 0.995621i \(-0.529800\pi\)
−0.0934824 + 0.995621i \(0.529800\pi\)
\(42\) 0 0
\(43\) 4.03197 0.614869 0.307434 − 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(44\) −0.288939 −0.0435592
\(45\) 0 0
\(46\) −1.83138 −0.270023
\(47\) −9.90940 −1.44543 −0.722717 − 0.691144i \(-0.757107\pi\)
−0.722717 + 0.691144i \(0.757107\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −4.40198 −0.622534
\(51\) 0 0
\(52\) 4.71032 0.653203
\(53\) −5.94184 −0.816175 −0.408088 − 0.912943i \(-0.633804\pi\)
−0.408088 + 0.912943i \(0.633804\pi\)
\(54\) 0 0
\(55\) −0.456726 −0.0615850
\(56\) 3.07903 0.411453
\(57\) 0 0
\(58\) 0.771591 0.101315
\(59\) 1.55231 0.202094 0.101047 − 0.994882i \(-0.467781\pi\)
0.101047 + 0.994882i \(0.467781\pi\)
\(60\) 0 0
\(61\) 7.09581 0.908526 0.454263 − 0.890868i \(-0.349903\pi\)
0.454263 + 0.890868i \(0.349903\pi\)
\(62\) 7.71651 0.979998
\(63\) 0 0
\(64\) 8.54824 1.06853
\(65\) 7.44560 0.923513
\(66\) 0 0
\(67\) −7.39331 −0.903237 −0.451618 − 0.892211i \(-0.649153\pi\)
−0.451618 + 0.892211i \(0.649153\pi\)
\(68\) 3.84883 0.466739
\(69\) 0 0
\(70\) 1.23860 0.148041
\(71\) −2.78476 −0.330490 −0.165245 − 0.986253i \(-0.552841\pi\)
−0.165245 + 0.986253i \(0.552841\pi\)
\(72\) 0 0
\(73\) 1.82359 0.213435 0.106717 − 0.994289i \(-0.465966\pi\)
0.106717 + 0.994289i \(0.465966\pi\)
\(74\) −6.73674 −0.783130
\(75\) 0 0
\(76\) −1.19619 −0.137212
\(77\) −0.423219 −0.0482302
\(78\) 0 0
\(79\) 4.01893 0.452165 0.226083 − 0.974108i \(-0.427408\pi\)
0.226083 + 0.974108i \(0.427408\pi\)
\(80\) 2.34014 0.261636
\(81\) 0 0
\(82\) −1.37401 −0.151734
\(83\) −11.3378 −1.24448 −0.622241 − 0.782825i \(-0.713778\pi\)
−0.622241 + 0.782825i \(0.713778\pi\)
\(84\) 0 0
\(85\) 6.08385 0.659886
\(86\) 4.62760 0.499007
\(87\) 0 0
\(88\) −1.30310 −0.138911
\(89\) 9.08046 0.962527 0.481264 − 0.876576i \(-0.340178\pi\)
0.481264 + 0.876576i \(0.340178\pi\)
\(90\) 0 0
\(91\) 6.89935 0.723249
\(92\) 1.08938 0.113576
\(93\) 0 0
\(94\) −11.3733 −1.17307
\(95\) −1.89082 −0.193994
\(96\) 0 0
\(97\) 11.4925 1.16688 0.583442 − 0.812155i \(-0.301706\pi\)
0.583442 + 0.812155i \(0.301706\pi\)
\(98\) 1.14773 0.115938
\(99\) 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.1.11 16
3.2 odd 2 2667.2.a.n.1.6 ✓ 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2667.2.a.n.1.6 ✓ 16 3.2 odd 2
8001.2.a.s.1.11 16 1.1 even 1 trivial