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Properties

Label 1008.2.bt.d.593.1
Level $1008$
Weight $2$
Character 1008.593
Analytic conductor $8.049$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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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] = [1008,2,Mod(17,1008)] 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("1008.17"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1008, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 3, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.bt (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
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: \( x^{16} + 10x^{14} + 61x^{12} + 266x^{10} + 852x^{8} + 1438x^{6} + 1933x^{4} + 3038x^{2} + 2401 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 504)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 593.1
Root \(1.45333 - 1.51725i\) of defining polynomial
Character \(\chi\) \(=\) 1008.593
Dual form 1008.2.bt.d.17.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.45333 + 2.51725i) q^{5} +(2.64571 - 0.0146827i) q^{7} +(1.08853 - 0.628464i) q^{11} -5.32679i q^{13} +(-0.880708 - 1.52543i) q^{17} +(6.69849 + 3.86738i) q^{19} +(4.43392 + 2.55992i) q^{23} +(-1.72435 - 2.98667i) q^{25} +8.74013i q^{29} +(-2.18272 + 1.26019i) q^{31} +(-3.80814 + 6.68124i) q^{35} +(-3.66400 + 6.34623i) q^{37} +3.09603 q^{41} -1.15729 q^{43} +(3.44334 - 5.96403i) q^{47} +(6.99957 - 0.0776922i) q^{49} +(-11.3057 + 6.52735i) q^{53} +3.65347i q^{55} +(3.13525 + 5.43042i) q^{59} +(2.19449 + 1.26699i) q^{61} +(13.4088 + 7.74160i) q^{65} +(0.689860 + 1.19487i) q^{67} +8.65477i q^{71} +(4.38121 - 2.52949i) q^{73} +(2.87071 - 1.67872i) q^{77} +(7.63142 - 13.2180i) q^{79} +3.75373 q^{83} +5.11985 q^{85} +(3.55321 - 6.15434i) q^{89} +(-0.0782115 - 14.0931i) q^{91} +(-19.4703 + 11.2412i) q^{95} -4.03874i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{7} - 12 q^{19} + 12 q^{25} - 24 q^{31} + 4 q^{37} - 8 q^{43} + 32 q^{49} + 28 q^{67} - 60 q^{73} + 32 q^{79} - 32 q^{85} + 84 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(e\left(\frac{5}{6}\right)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −1.45333 + 2.51725i −0.649950 + 1.12575i 0.333184 + 0.942862i \(0.391877\pi\)
−0.983134 + 0.182885i \(0.941456\pi\)
\(6\) 0 0
\(7\) 2.64571 − 0.0146827i 0.999985 − 0.00554953i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.08853 − 0.628464i 0.328205 − 0.189489i −0.326839 − 0.945080i \(-0.605984\pi\)
0.655044 + 0.755591i \(0.272650\pi\)
\(12\) 0 0
\(13\) − 5.32679i − 1.47739i −0.674042 − 0.738693i \(-0.735443\pi\)
0.674042 − 0.738693i \(-0.264557\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −0.880708 − 1.52543i −0.213603 − 0.369971i 0.739236 − 0.673446i \(-0.235187\pi\)
−0.952840 + 0.303475i \(0.901853\pi\)
\(18\) 0 0
\(19\) 6.69849 + 3.86738i 1.53674 + 0.887237i 0.999027 + 0.0441085i \(0.0140447\pi\)
0.537712 + 0.843128i \(0.319289\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.43392 + 2.55992i 0.924536 + 0.533781i 0.885079 − 0.465440i \(-0.154104\pi\)
0.0394566 + 0.999221i \(0.487437\pi\)
\(24\) 0 0
\(25\) −1.72435 − 2.98667i −0.344871 − 0.597333i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 8.74013i 1.62300i 0.584352 + 0.811500i \(0.301349\pi\)
−0.584352 + 0.811500i \(0.698651\pi\)
\(30\) 0 0
\(31\) −2.18272 + 1.26019i −0.392027 + 0.226337i −0.683038 − 0.730383i \(-0.739342\pi\)
0.291011 + 0.956720i \(0.406008\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.80814 + 6.68124i −0.643693 + 1.12934i
\(36\) 0 0
\(37\) −3.66400 + 6.34623i −0.602358 + 1.04331i 0.390106 + 0.920770i \(0.372439\pi\)
−0.992463 + 0.122544i \(0.960895\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.09603 0.483519 0.241760 − 0.970336i \(-0.422275\pi\)
0.241760 + 0.970336i \(0.422275\pi\)
\(42\) 0 0
\(43\) −1.15729 −0.176484 −0.0882422 − 0.996099i \(-0.528125\pi\)
−0.0882422 + 0.996099i \(0.528125\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.44334 − 5.96403i 0.502262 − 0.869944i −0.497734 − 0.867330i \(-0.665835\pi\)
0.999997 − 0.00261431i \(-0.000832161\pi\)
\(48\) 0 0
\(49\) 6.99957 − 0.0776922i 0.999938 − 0.0110989i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −11.3057 + 6.52735i −1.55296 + 0.896600i −0.555058 + 0.831811i \(0.687304\pi\)
−0.997899 + 0.0647890i \(0.979363\pi\)
\(54\) 0 0
\(55\) 3.65347i 0.492634i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.13525 + 5.43042i 0.408175 + 0.706980i 0.994685 − 0.102962i \(-0.0328319\pi\)
−0.586510 + 0.809942i \(0.699499\pi\)
\(60\) 0 0
\(61\) 2.19449 + 1.26699i 0.280976 + 0.162221i 0.633865 − 0.773444i \(-0.281467\pi\)
−0.352889 + 0.935665i \(0.614801\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 13.4088 + 7.74160i 1.66316 + 0.960227i
\(66\) 0 0
\(67\) 0.689860 + 1.19487i 0.0842798 + 0.145977i 0.905084 − 0.425233i \(-0.139808\pi\)
−0.820804 + 0.571210i \(0.806474\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 8.65477i 1.02713i 0.858050 + 0.513566i \(0.171676\pi\)
−0.858050 + 0.513566i \(0.828324\pi\)
\(72\) 0 0
\(73\) 4.38121 − 2.52949i 0.512782 − 0.296055i −0.221195 − 0.975230i \(-0.570996\pi\)
0.733976 + 0.679175i \(0.237662\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.87071 − 1.67872i 0.327148 − 0.191308i
\(78\) 0 0
\(79\) 7.63142 − 13.2180i 0.858602 − 1.48714i −0.0146603 − 0.999893i \(-0.504667\pi\)
0.873262 − 0.487250i \(-0.162000\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.75373 0.412025 0.206013 − 0.978549i \(-0.433951\pi\)
0.206013 + 0.978549i \(0.433951\pi\)
\(84\) 0 0
\(85\) 5.11985 0.555326
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.55321 − 6.15434i 0.376640 − 0.652359i −0.613931 − 0.789359i \(-0.710413\pi\)
0.990571 + 0.137001i \(0.0437462\pi\)
\(90\) 0 0
\(91\) −0.0782115 − 14.0931i −0.00819880 − 1.47736i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −19.4703 + 11.2412i −1.99761 + 1.15332i
\(96\) 0 0
\(97\) − 4.03874i − 0.410072i −0.978754 − 0.205036i \(-0.934269\pi\)
0.978754 − 0.205036i \(-0.0657311\pi\)
\(98\) 0 0
\(99\) 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 1008.2.bt.d.593.1 16
3.2 odd 2 inner 1008.2.bt.d.593.8 16
4.3 odd 2 504.2.bl.a.89.1 yes 16
7.2 even 3 7056.2.k.h.881.16 16
7.3 odd 6 inner 1008.2.bt.d.17.8 16
7.5 odd 6 7056.2.k.h.881.2 16
12.11 even 2 504.2.bl.a.89.8 yes 16
21.2 odd 6 7056.2.k.h.881.1 16
21.5 even 6 7056.2.k.h.881.15 16
21.17 even 6 inner 1008.2.bt.d.17.1 16
28.3 even 6 504.2.bl.a.17.8 yes 16
28.11 odd 6 3528.2.bl.a.521.1 16
28.19 even 6 3528.2.k.b.881.1 16
28.23 odd 6 3528.2.k.b.881.15 16
28.27 even 2 3528.2.bl.a.1097.8 16
84.11 even 6 3528.2.bl.a.521.8 16
84.23 even 6 3528.2.k.b.881.2 16
84.47 odd 6 3528.2.k.b.881.16 16
84.59 odd 6 504.2.bl.a.17.1 ✓ 16
84.83 odd 2 3528.2.bl.a.1097.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.bl.a.17.1 ✓ 16 84.59 odd 6
504.2.bl.a.17.8 yes 16 28.3 even 6
504.2.bl.a.89.1 yes 16 4.3 odd 2
504.2.bl.a.89.8 yes 16 12.11 even 2
1008.2.bt.d.17.1 16 21.17 even 6 inner
1008.2.bt.d.17.8 16 7.3 odd 6 inner
1008.2.bt.d.593.1 16 1.1 even 1 trivial
1008.2.bt.d.593.8 16 3.2 odd 2 inner
3528.2.k.b.881.1 16 28.19 even 6
3528.2.k.b.881.2 16 84.23 even 6
3528.2.k.b.881.15 16 28.23 odd 6
3528.2.k.b.881.16 16 84.47 odd 6
3528.2.bl.a.521.1 16 28.11 odd 6
3528.2.bl.a.521.8 16 84.11 even 6
3528.2.bl.a.1097.1 16 84.83 odd 2
3528.2.bl.a.1097.8 16 28.27 even 2
7056.2.k.h.881.1 16 21.2 odd 6
7056.2.k.h.881.2 16 7.5 odd 6
7056.2.k.h.881.15 16 21.5 even 6
7056.2.k.h.881.16 16 7.2 even 3