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(1009,3024)] 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("3024.1009"); 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([0, 0, 2, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3024.r (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,2,0,-1,0,0,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == 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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 2017.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 3024.2017
Dual form 3024.2.r.c.1009.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.00000 + 1.73205i) q^{5} +(-0.500000 + 0.866025i) q^{7} +(-0.500000 + 0.866025i) q^{11} +(3.00000 + 5.19615i) q^{13} +5.00000 q^{17} +7.00000 q^{19} +(-2.00000 - 3.46410i) q^{23} +(0.500000 - 0.866025i) q^{25} +(-2.00000 + 3.46410i) q^{29} +(-3.00000 - 5.19615i) q^{31} -2.00000 q^{35} +2.00000 q^{37} +(1.50000 + 2.59808i) q^{41} +(-0.500000 + 0.866025i) q^{43} +(-0.500000 - 0.866025i) q^{49} -12.0000 q^{53} -2.00000 q^{55} +(3.50000 + 6.06218i) q^{59} +(6.00000 - 10.3923i) q^{61} +(-6.00000 + 10.3923i) q^{65} +(6.50000 + 11.2583i) q^{67} -8.00000 q^{71} +1.00000 q^{73} +(-0.500000 - 0.866025i) q^{77} +(-3.00000 + 5.19615i) q^{79} +(-8.00000 + 13.8564i) q^{83} +(5.00000 + 8.66025i) q^{85} +6.00000 q^{89} -6.00000 q^{91} +(7.00000 + 12.1244i) q^{95} +(2.50000 - 4.33013i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - q^{7} - q^{11} + 6 q^{13} + 10 q^{17} + 14 q^{19} - 4 q^{23} + q^{25} - 4 q^{29} - 6 q^{31} - 4 q^{35} + 4 q^{37} + 3 q^{41} - q^{43} - q^{49} - 24 q^{53} - 4 q^{55} + 7 q^{59} + 12 q^{61}+ \cdots + 5 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(757\) \(785\) \(1135\) \(2593\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)

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
\(5\) 1.00000 + 1.73205i 0.447214 + 0.774597i 0.998203 − 0.0599153i \(-0.0190830\pi\)
−0.550990 + 0.834512i \(0.685750\pi\)
\(6\) 0 0
\(7\) −0.500000 + 0.866025i −0.188982 + 0.327327i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −0.500000 + 0.866025i −0.150756 + 0.261116i −0.931505 − 0.363727i \(-0.881504\pi\)
0.780750 + 0.624844i \(0.214837\pi\)
\(12\) 0 0
\(13\) 3.00000 + 5.19615i 0.832050 + 1.44115i 0.896410 + 0.443227i \(0.146166\pi\)
−0.0643593 + 0.997927i \(0.520500\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.00000 1.21268 0.606339 − 0.795206i \(-0.292637\pi\)
0.606339 + 0.795206i \(0.292637\pi\)
\(18\) 0 0
\(19\) 7.00000 1.60591 0.802955 − 0.596040i \(-0.203260\pi\)
0.802955 + 0.596040i \(0.203260\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.00000 − 3.46410i −0.417029 − 0.722315i 0.578610 − 0.815604i \(-0.303595\pi\)
−0.995639 + 0.0932891i \(0.970262\pi\)
\(24\) 0 0
\(25\) 0.500000 − 0.866025i 0.100000 − 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −2.00000 + 3.46410i −0.371391 + 0.643268i −0.989780 − 0.142605i \(-0.954452\pi\)
0.618389 + 0.785872i \(0.287786\pi\)
\(30\) 0 0
\(31\) −3.00000 − 5.19615i −0.538816 − 0.933257i −0.998968 − 0.0454165i \(-0.985539\pi\)
0.460152 − 0.887840i \(-0.347795\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.00000 −0.338062
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.50000 + 2.59808i 0.234261 + 0.405751i 0.959058 − 0.283211i \(-0.0913998\pi\)
−0.724797 + 0.688963i \(0.758066\pi\)
\(42\) 0 0
\(43\) −0.500000 + 0.866025i −0.0762493 + 0.132068i −0.901629 − 0.432511i \(-0.857628\pi\)
0.825380 + 0.564578i \(0.190961\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(48\) 0 0
\(49\) −0.500000 − 0.866025i −0.0714286 − 0.123718i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −12.0000 −1.64833 −0.824163 − 0.566352i \(-0.808354\pi\)
−0.824163 + 0.566352i \(0.808354\pi\)
\(54\) 0 0
\(55\) −2.00000 −0.269680
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.50000 + 6.06218i 0.455661 + 0.789228i 0.998726 − 0.0504625i \(-0.0160695\pi\)
−0.543065 + 0.839691i \(0.682736\pi\)
\(60\) 0 0
\(61\) 6.00000 − 10.3923i 0.768221 − 1.33060i −0.170305 − 0.985391i \(-0.554475\pi\)
0.938527 − 0.345207i \(-0.112191\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −6.00000 + 10.3923i −0.744208 + 1.28901i
\(66\) 0 0
\(67\) 6.50000 + 11.2583i 0.794101 + 1.37542i 0.923408 + 0.383819i \(0.125391\pi\)
−0.129307 + 0.991605i \(0.541275\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) 1.00000 0.117041 0.0585206 − 0.998286i \(-0.481362\pi\)
0.0585206 + 0.998286i \(0.481362\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −0.500000 − 0.866025i −0.0569803 − 0.0986928i
\(78\) 0 0
\(79\) −3.00000 + 5.19615i −0.337526 + 0.584613i −0.983967 − 0.178352i \(-0.942924\pi\)
0.646440 + 0.762964i \(0.276257\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −8.00000 + 13.8564i −0.878114 + 1.52094i −0.0247060 + 0.999695i \(0.507865\pi\)
−0.853408 + 0.521243i \(0.825468\pi\)
\(84\) 0 0
\(85\) 5.00000 + 8.66025i 0.542326 + 0.939336i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −6.00000 −0.628971
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 7.00000 + 12.1244i 0.718185 + 1.24393i
\(96\) 0 0
\(97\) 2.50000 − 4.33013i 0.253837 − 0.439658i −0.710742 − 0.703452i \(-0.751641\pi\)
0.964579 + 0.263795i \(0.0849741\pi\)
\(98\) 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.r.c.2017.1 2
3.2 odd 2 1008.2.r.a.673.1 2
4.3 odd 2 378.2.f.b.127.1 2
9.2 odd 6 9072.2.a.t.1.1 1
9.4 even 3 inner 3024.2.r.c.1009.1 2
9.5 odd 6 1008.2.r.a.337.1 2
9.7 even 3 9072.2.a.f.1.1 1
12.11 even 2 126.2.f.b.43.1 ✓ 2
28.3 even 6 2646.2.h.c.667.1 2
28.11 odd 6 2646.2.h.b.667.1 2
28.19 even 6 2646.2.e.h.2125.1 2
28.23 odd 6 2646.2.e.i.2125.1 2
28.27 even 2 2646.2.f.b.883.1 2
36.7 odd 6 1134.2.a.f.1.1 1
36.11 even 6 1134.2.a.c.1.1 1
36.23 even 6 126.2.f.b.85.1 yes 2
36.31 odd 6 378.2.f.b.253.1 2
84.11 even 6 882.2.h.h.79.1 2
84.23 even 6 882.2.e.a.655.1 2
84.47 odd 6 882.2.e.e.655.1 2
84.59 odd 6 882.2.h.g.79.1 2
84.83 odd 2 882.2.f.f.295.1 2
252.23 even 6 882.2.h.h.67.1 2
252.31 even 6 2646.2.e.h.1549.1 2
252.59 odd 6 882.2.e.e.373.1 2
252.67 odd 6 2646.2.e.i.1549.1 2
252.83 odd 6 7938.2.a.e.1.1 1
252.95 even 6 882.2.e.a.373.1 2
252.103 even 6 2646.2.h.c.361.1 2
252.131 odd 6 882.2.h.g.67.1 2
252.139 even 6 2646.2.f.b.1765.1 2
252.167 odd 6 882.2.f.f.589.1 2
252.223 even 6 7938.2.a.bb.1.1 1
252.247 odd 6 2646.2.h.b.361.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.2.f.b.43.1 ✓ 2 12.11 even 2
126.2.f.b.85.1 yes 2 36.23 even 6
378.2.f.b.127.1 2 4.3 odd 2
378.2.f.b.253.1 2 36.31 odd 6
882.2.e.a.373.1 2 252.95 even 6
882.2.e.a.655.1 2 84.23 even 6
882.2.e.e.373.1 2 252.59 odd 6
882.2.e.e.655.1 2 84.47 odd 6
882.2.f.f.295.1 2 84.83 odd 2
882.2.f.f.589.1 2 252.167 odd 6
882.2.h.g.67.1 2 252.131 odd 6
882.2.h.g.79.1 2 84.59 odd 6
882.2.h.h.67.1 2 252.23 even 6
882.2.h.h.79.1 2 84.11 even 6
1008.2.r.a.337.1 2 9.5 odd 6
1008.2.r.a.673.1 2 3.2 odd 2
1134.2.a.c.1.1 1 36.11 even 6
1134.2.a.f.1.1 1 36.7 odd 6
2646.2.e.h.1549.1 2 252.31 even 6
2646.2.e.h.2125.1 2 28.19 even 6
2646.2.e.i.1549.1 2 252.67 odd 6
2646.2.e.i.2125.1 2 28.23 odd 6
2646.2.f.b.883.1 2 28.27 even 2
2646.2.f.b.1765.1 2 252.139 even 6
2646.2.h.b.361.1 2 252.247 odd 6
2646.2.h.b.667.1 2 28.11 odd 6
2646.2.h.c.361.1 2 252.103 even 6
2646.2.h.c.667.1 2 28.3 even 6
3024.2.r.c.1009.1 2 9.4 even 3 inner
3024.2.r.c.2017.1 2 1.1 even 1 trivial
7938.2.a.e.1.1 1 252.83 odd 6
7938.2.a.bb.1.1 1 252.223 even 6
9072.2.a.f.1.1 1 9.7 even 3
9072.2.a.t.1.1 1 9.2 odd 6