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Properties

Label 3024.2.cz.a.1279.1
Level $3024$
Weight $2$
Character 3024.1279
Analytic conductor $24.147$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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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] = [3024,2,Mod(1279,3024)] 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("3024.1279"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3024, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 2, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3024.cz (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,-1] 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: \(24.1467615712\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1008)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1279.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 3024.1279
Dual form 3024.2.cz.a.2719.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 2.59808i) q^{7} +(3.00000 + 1.73205i) q^{11} +(-1.50000 - 0.866025i) q^{13} +(-1.50000 + 0.866025i) q^{17} +(-2.50000 + 4.33013i) q^{19} +(-3.00000 + 1.73205i) q^{23} +(-2.50000 + 4.33013i) q^{25} +(1.50000 + 2.59808i) q^{29} +1.00000 q^{31} +(-3.50000 + 6.06218i) q^{37} +(1.50000 + 0.866025i) q^{41} +(1.50000 - 0.866025i) q^{43} +9.00000 q^{47} +(-6.50000 + 2.59808i) q^{49} +(-4.50000 - 7.79423i) q^{53} +15.0000 q^{59} +1.73205i q^{61} +15.5885i q^{67} -10.3923i q^{71} +(1.50000 - 0.866025i) q^{73} +(3.00000 - 8.66025i) q^{77} +1.73205i q^{79} +(4.50000 + 7.79423i) q^{83} +(-1.50000 - 0.866025i) q^{89} +(-1.50000 + 4.33013i) q^{91} +(-1.50000 + 0.866025i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{7} + 6 q^{11} - 3 q^{13} - 3 q^{17} - 5 q^{19} - 6 q^{23} - 5 q^{25} + 3 q^{29} + 2 q^{31} - 7 q^{37} + 3 q^{41} + 3 q^{43} + 18 q^{47} - 13 q^{49} - 9 q^{53} + 30 q^{59} + 3 q^{73} + 6 q^{77}+ \cdots - 3 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1135\) \(2593\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(e\left(\frac{5}{6}\right)\)

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
\(5\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(6\) 0 0
\(7\) −0.500000 − 2.59808i −0.188982 − 0.981981i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.00000 + 1.73205i 0.904534 + 0.522233i 0.878668 − 0.477432i \(-0.158432\pi\)
0.0258656 + 0.999665i \(0.491766\pi\)
\(12\) 0 0
\(13\) −1.50000 − 0.866025i −0.416025 − 0.240192i 0.277350 − 0.960769i \(-0.410544\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.50000 + 0.866025i −0.363803 + 0.210042i −0.670748 − 0.741685i \(-0.734027\pi\)
0.306944 + 0.951727i \(0.400693\pi\)
\(18\) 0 0
\(19\) −2.50000 + 4.33013i −0.573539 + 0.993399i 0.422659 + 0.906289i \(0.361097\pi\)
−0.996199 + 0.0871106i \(0.972237\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.00000 + 1.73205i −0.625543 + 0.361158i −0.779024 − 0.626994i \(-0.784285\pi\)
0.153481 + 0.988152i \(0.450952\pi\)
\(24\) 0 0
\(25\) −2.50000 + 4.33013i −0.500000 + 0.866025i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.50000 + 2.59808i 0.278543 + 0.482451i 0.971023 − 0.238987i \(-0.0768152\pi\)
−0.692480 + 0.721437i \(0.743482\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605 0.0898027 − 0.995960i \(-0.471376\pi\)
0.0898027 + 0.995960i \(0.471376\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −3.50000 + 6.06218i −0.575396 + 0.996616i 0.420602 + 0.907245i \(0.361819\pi\)
−0.995998 + 0.0893706i \(0.971514\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.50000 + 0.866025i 0.234261 + 0.135250i 0.612536 − 0.790443i \(-0.290149\pi\)
−0.378275 + 0.925693i \(0.623483\pi\)
\(42\) 0 0
\(43\) 1.50000 − 0.866025i 0.228748 − 0.132068i −0.381246 − 0.924473i \(-0.624505\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9.00000 1.31278 0.656392 − 0.754420i \(-0.272082\pi\)
0.656392 + 0.754420i \(0.272082\pi\)
\(48\) 0 0
\(49\) −6.50000 + 2.59808i −0.928571 + 0.371154i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −4.50000 − 7.79423i −0.618123 − 1.07062i −0.989828 − 0.142269i \(-0.954560\pi\)
0.371706 − 0.928351i \(-0.378773\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 15.0000 1.95283 0.976417 − 0.215894i \(-0.0692665\pi\)
0.976417 + 0.215894i \(0.0692665\pi\)
\(60\) 0 0
\(61\) 1.73205i 0.221766i 0.993833 + 0.110883i \(0.0353679\pi\)
−0.993833 + 0.110883i \(0.964632\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 15.5885i 1.90443i 0.305424 + 0.952217i \(0.401202\pi\)
−0.305424 + 0.952217i \(0.598798\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 10.3923i − 1.23334i −0.787222 − 0.616670i \(-0.788481\pi\)
0.787222 − 0.616670i \(-0.211519\pi\)
\(72\) 0 0
\(73\) 1.50000 − 0.866025i 0.175562 − 0.101361i −0.409644 − 0.912245i \(-0.634347\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.00000 − 8.66025i 0.341882 − 0.986928i
\(78\) 0 0
\(79\) 1.73205i 0.194871i 0.995242 + 0.0974355i \(0.0310640\pi\)
−0.995242 + 0.0974355i \(0.968936\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.50000 + 7.79423i 0.493939 + 0.855528i 0.999976 − 0.00698436i \(-0.00222321\pi\)
−0.506036 + 0.862512i \(0.668890\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1.50000 − 0.866025i −0.159000 − 0.0917985i 0.418389 − 0.908268i \(-0.362595\pi\)
−0.577389 + 0.816469i \(0.695928\pi\)
\(90\) 0 0
\(91\) −1.50000 + 4.33013i −0.157243 + 0.453921i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.50000 + 0.866025i −0.152302 + 0.0879316i −0.574214 − 0.818705i \(-0.694692\pi\)
0.421912 + 0.906637i \(0.361359\pi\)
\(98\) 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 3024.2.cz.a.1279.1 2
3.2 odd 2 1008.2.cz.a.607.1 yes 2
4.3 odd 2 3024.2.cz.b.1279.1 2
7.3 odd 6 3024.2.bf.c.1711.1 2
9.2 odd 6 1008.2.bf.b.943.1 yes 2
9.7 even 3 3024.2.bf.b.2287.1 2
12.11 even 2 1008.2.cz.d.607.1 yes 2
21.17 even 6 1008.2.bf.c.31.1 yes 2
28.3 even 6 3024.2.bf.b.1711.1 2
36.7 odd 6 3024.2.bf.c.2287.1 2
36.11 even 6 1008.2.bf.c.943.1 yes 2
63.38 even 6 1008.2.cz.d.367.1 yes 2
63.52 odd 6 3024.2.cz.b.2719.1 2
84.59 odd 6 1008.2.bf.b.31.1 ✓ 2
252.115 even 6 inner 3024.2.cz.a.2719.1 2
252.227 odd 6 1008.2.cz.a.367.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1008.2.bf.b.31.1 ✓ 2 84.59 odd 6
1008.2.bf.b.943.1 yes 2 9.2 odd 6
1008.2.bf.c.31.1 yes 2 21.17 even 6
1008.2.bf.c.943.1 yes 2 36.11 even 6
1008.2.cz.a.367.1 yes 2 252.227 odd 6
1008.2.cz.a.607.1 yes 2 3.2 odd 2
1008.2.cz.d.367.1 yes 2 63.38 even 6
1008.2.cz.d.607.1 yes 2 12.11 even 2
3024.2.bf.b.1711.1 2 28.3 even 6
3024.2.bf.b.2287.1 2 9.7 even 3
3024.2.bf.c.1711.1 2 7.3 odd 6
3024.2.bf.c.2287.1 2 36.7 odd 6
3024.2.cz.a.1279.1 2 1.1 even 1 trivial
3024.2.cz.a.2719.1 2 252.115 even 6 inner
3024.2.cz.b.1279.1 2 4.3 odd 2
3024.2.cz.b.2719.1 2 63.52 odd 6