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Properties

Label 3024.2.bf.b.1711.1
Level $3024$
Weight $2$
Character 3024.1711
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(1711,3024)] 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("3024.1711"); 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, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3024.bf (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,-4] 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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1008)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1711.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 3024.1711
Dual form 3024.2.bf.b.2287.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-2.00000 - 1.73205i) q^{7} +3.46410i q^{11} +(1.50000 + 0.866025i) q^{13} +(-1.50000 - 0.866025i) q^{17} +(-2.50000 - 4.33013i) q^{19} +3.46410i q^{23} +5.00000 q^{25} +(1.50000 + 2.59808i) q^{29} +(-0.500000 - 0.866025i) q^{31} +(-3.50000 - 6.06218i) q^{37} +(-1.50000 - 0.866025i) q^{41} +(-1.50000 + 0.866025i) q^{43} +(-4.50000 + 7.79423i) q^{47} +(1.00000 + 6.92820i) q^{49} +(-4.50000 + 7.79423i) q^{53} +(-7.50000 - 12.9904i) q^{59} +(1.50000 + 0.866025i) q^{61} +(-13.5000 + 7.79423i) q^{67} +10.3923i q^{71} +(1.50000 + 0.866025i) q^{73} +(6.00000 - 6.92820i) q^{77} +(1.50000 + 0.866025i) q^{79} +(4.50000 + 7.79423i) q^{83} +(-1.50000 + 0.866025i) q^{89} +(-1.50000 - 4.33013i) q^{91} +(1.50000 - 0.866025i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{7} + 3 q^{13} - 3 q^{17} - 5 q^{19} + 10 q^{25} + 3 q^{29} - q^{31} - 7 q^{37} - 3 q^{41} - 3 q^{43} - 9 q^{47} + 2 q^{49} - 9 q^{53} - 15 q^{59} + 3 q^{61} - 27 q^{67} + 3 q^{73} + 12 q^{77}+ \cdots + 3 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) −2.00000 − 1.73205i −0.755929 − 0.654654i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.46410i 1.04447i 0.852803 + 0.522233i \(0.174901\pi\)
−0.852803 + 0.522233i \(0.825099\pi\)
\(12\) 0 0
\(13\) 1.50000 + 0.866025i 0.416025 + 0.240192i 0.693375 − 0.720577i \(-0.256123\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.50000 − 0.866025i −0.363803 − 0.210042i 0.306944 − 0.951727i \(-0.400693\pi\)
−0.670748 + 0.741685i \(0.734027\pi\)
\(18\) 0 0
\(19\) −2.50000 − 4.33013i −0.573539 − 0.993399i −0.996199 − 0.0871106i \(-0.972237\pi\)
0.422659 − 0.906289i \(-0.361097\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.46410i 0.722315i 0.932505 + 0.361158i \(0.117618\pi\)
−0.932505 + 0.361158i \(0.882382\pi\)
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) 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
\(31\) −0.500000 − 0.866025i −0.0898027 − 0.155543i 0.817625 − 0.575751i \(-0.195290\pi\)
−0.907428 + 0.420208i \(0.861957\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −3.50000 − 6.06218i −0.575396 − 0.996616i −0.995998 − 0.0893706i \(-0.971514\pi\)
0.420602 − 0.907245i \(-0.361819\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1.50000 − 0.866025i −0.234261 − 0.135250i 0.378275 − 0.925693i \(-0.376517\pi\)
−0.612536 + 0.790443i \(0.709851\pi\)
\(42\) 0 0
\(43\) −1.50000 + 0.866025i −0.228748 + 0.132068i −0.609994 − 0.792406i \(-0.708828\pi\)
0.381246 + 0.924473i \(0.375495\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.50000 + 7.79423i −0.656392 + 1.13691i 0.325150 + 0.945662i \(0.394585\pi\)
−0.981543 + 0.191243i \(0.938748\pi\)
\(48\) 0 0
\(49\) 1.00000 + 6.92820i 0.142857 + 0.989743i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −4.50000 + 7.79423i −0.618123 + 1.07062i 0.371706 + 0.928351i \(0.378773\pi\)
−0.989828 + 0.142269i \(0.954560\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −7.50000 − 12.9904i −0.976417 − 1.69120i −0.675178 − 0.737655i \(-0.735933\pi\)
−0.301239 − 0.953549i \(-0.597400\pi\)
\(60\) 0 0
\(61\) 1.50000 + 0.866025i 0.192055 + 0.110883i 0.592944 − 0.805243i \(-0.297965\pi\)
−0.400889 + 0.916127i \(0.631299\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −13.5000 + 7.79423i −1.64929 + 0.952217i −0.671932 + 0.740613i \(0.734535\pi\)
−0.977356 + 0.211604i \(0.932131\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 10.3923i 1.23334i 0.787222 + 0.616670i \(0.211519\pi\)
−0.787222 + 0.616670i \(0.788481\pi\)
\(72\) 0 0
\(73\) 1.50000 + 0.866025i 0.175562 + 0.101361i 0.585206 − 0.810885i \(-0.301014\pi\)
−0.409644 + 0.912245i \(0.634347\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.00000 − 6.92820i 0.683763 − 0.789542i
\(78\) 0 0
\(79\) 1.50000 + 0.866025i 0.168763 + 0.0974355i 0.582003 − 0.813187i \(-0.302269\pi\)
−0.413239 + 0.910622i \(0.635603\pi\)
\(80\) 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 −0.577389 − 0.816469i \(-0.695928\pi\)
0.418389 + 0.908268i \(0.362595\pi\)
\(90\) 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 −0.421912 − 0.906637i \(-0.638641\pi\)
0.574214 + 0.818705i \(0.305308\pi\)
\(98\) 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.bf.b.1711.1 2
3.2 odd 2 1008.2.bf.b.31.1 ✓ 2
4.3 odd 2 3024.2.bf.c.1711.1 2
7.5 odd 6 3024.2.cz.b.1279.1 2
9.2 odd 6 1008.2.cz.a.367.1 yes 2
9.7 even 3 3024.2.cz.a.2719.1 2
12.11 even 2 1008.2.bf.c.31.1 yes 2
21.5 even 6 1008.2.cz.d.607.1 yes 2
28.19 even 6 3024.2.cz.a.1279.1 2
36.7 odd 6 3024.2.cz.b.2719.1 2
36.11 even 6 1008.2.cz.d.367.1 yes 2
63.47 even 6 1008.2.bf.c.943.1 yes 2
63.61 odd 6 3024.2.bf.c.2287.1 2
84.47 odd 6 1008.2.cz.a.607.1 yes 2
252.47 odd 6 1008.2.bf.b.943.1 yes 2
252.187 even 6 inner 3024.2.bf.b.2287.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1008.2.bf.b.31.1 ✓ 2 3.2 odd 2
1008.2.bf.b.943.1 yes 2 252.47 odd 6
1008.2.bf.c.31.1 yes 2 12.11 even 2
1008.2.bf.c.943.1 yes 2 63.47 even 6
1008.2.cz.a.367.1 yes 2 9.2 odd 6
1008.2.cz.a.607.1 yes 2 84.47 odd 6
1008.2.cz.d.367.1 yes 2 36.11 even 6
1008.2.cz.d.607.1 yes 2 21.5 even 6
3024.2.bf.b.1711.1 2 1.1 even 1 trivial
3024.2.bf.b.2287.1 2 252.187 even 6 inner
3024.2.bf.c.1711.1 2 4.3 odd 2
3024.2.bf.c.2287.1 2 63.61 odd 6
3024.2.cz.a.1279.1 2 28.19 even 6
3024.2.cz.a.2719.1 2 9.7 even 3
3024.2.cz.b.1279.1 2 7.5 odd 6
3024.2.cz.b.2719.1 2 36.7 odd 6