Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2646,2,Mod(883,2646)] 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("2646.883"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2646, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2646.f (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-1,0,-1,-2,0,0,2,0,4,1] 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: \(21.1284163748\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 883.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 2646.883
Dual form 2646.2.f.b.1765.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-1.00000 - 1.73205i) q^{5} +1.00000 q^{8} +2.00000 q^{10} +(0.500000 - 0.866025i) q^{11} +(-3.00000 - 5.19615i) q^{13} +(-0.500000 + 0.866025i) q^{16} -5.00000 q^{17} +7.00000 q^{19} +(-1.00000 + 1.73205i) q^{20} +(0.500000 + 0.866025i) q^{22} +(2.00000 + 3.46410i) q^{23} +(0.500000 - 0.866025i) q^{25} +6.00000 q^{26} +(-2.00000 + 3.46410i) q^{29} +(-3.00000 - 5.19615i) q^{31} +(-0.500000 - 0.866025i) q^{32} +(2.50000 - 4.33013i) q^{34} +2.00000 q^{37} +(-3.50000 + 6.06218i) q^{38} +(-1.00000 - 1.73205i) q^{40} +(-1.50000 - 2.59808i) q^{41} +(0.500000 - 0.866025i) q^{43} -1.00000 q^{44} -4.00000 q^{46} +(0.500000 + 0.866025i) q^{50} +(-3.00000 + 5.19615i) q^{52} -12.0000 q^{53} -2.00000 q^{55} +(-2.00000 - 3.46410i) q^{58} +(3.50000 + 6.06218i) q^{59} +(-6.00000 + 10.3923i) q^{61} +6.00000 q^{62} +1.00000 q^{64} +(-6.00000 + 10.3923i) q^{65} +(-6.50000 - 11.2583i) q^{67} +(2.50000 + 4.33013i) q^{68} +8.00000 q^{71} -1.00000 q^{73} +(-1.00000 + 1.73205i) q^{74} +(-3.50000 - 6.06218i) q^{76} +(3.00000 - 5.19615i) q^{79} +2.00000 q^{80} +3.00000 q^{82} +(-8.00000 + 13.8564i) q^{83} +(5.00000 + 8.66025i) q^{85} +(0.500000 + 0.866025i) q^{86} +(0.500000 - 0.866025i) q^{88} -6.00000 q^{89} +(2.00000 - 3.46410i) q^{92} +(-7.00000 - 12.1244i) q^{95} +(-2.50000 + 4.33013i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{4} - 2 q^{5} + 2 q^{8} + 4 q^{10} + q^{11} - 6 q^{13} - q^{16} - 10 q^{17} + 14 q^{19} - 2 q^{20} + q^{22} + 4 q^{23} + q^{25} + 12 q^{26} - 4 q^{29} - 6 q^{31} - q^{32} + 5 q^{34}+ \cdots - 5 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2646\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.500000 + 0.866025i −0.353553 + 0.612372i
\(3\) 0 0
\(4\) −0.500000 − 0.866025i −0.250000 − 0.433013i
\(5\) −1.00000 − 1.73205i −0.447214 − 0.774597i 0.550990 − 0.834512i \(-0.314250\pi\)
−0.998203 + 0.0599153i \(0.980917\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 2.00000 0.632456
\(11\) 0.500000 − 0.866025i 0.150756 − 0.261116i −0.780750 − 0.624844i \(-0.785163\pi\)
0.931505 + 0.363727i \(0.118496\pi\)
\(12\) 0 0
\(13\) −3.00000 − 5.19615i −0.832050 − 1.44115i −0.896410 − 0.443227i \(-0.853834\pi\)
0.0643593 − 0.997927i \(-0.479500\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) −5.00000 −1.21268 −0.606339 − 0.795206i \(-0.707363\pi\)
−0.606339 + 0.795206i \(0.707363\pi\)
\(18\) 0 0
\(19\) 7.00000 1.60591 0.802955 − 0.596040i \(-0.203260\pi\)
0.802955 + 0.596040i \(0.203260\pi\)
\(20\) −1.00000 + 1.73205i −0.223607 + 0.387298i
\(21\) 0 0
\(22\) 0.500000 + 0.866025i 0.106600 + 0.184637i
\(23\) 2.00000 + 3.46410i 0.417029 + 0.722315i 0.995639 − 0.0932891i \(-0.0297381\pi\)
−0.578610 + 0.815604i \(0.696405\pi\)
\(24\) 0 0
\(25\) 0.500000 − 0.866025i 0.100000 − 0.173205i
\(26\) 6.00000 1.17670
\(27\) 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 \(-0.347795\pi\)
\(32\) −0.500000 − 0.866025i −0.0883883 − 0.153093i
\(33\) 0 0
\(34\) 2.50000 − 4.33013i 0.428746 − 0.742611i
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −3.50000 + 6.06218i −0.567775 + 0.983415i
\(39\) 0 0
\(40\) −1.00000 − 1.73205i −0.158114 − 0.273861i
\(41\) −1.50000 − 2.59808i −0.234261 − 0.405751i 0.724797 − 0.688963i \(-0.241934\pi\)
−0.959058 + 0.283211i \(0.908600\pi\)
\(42\) 0 0
\(43\) 0.500000 − 0.866025i 0.0762493 − 0.132068i −0.825380 − 0.564578i \(-0.809039\pi\)
0.901629 + 0.432511i \(0.142372\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0.500000 + 0.866025i 0.0707107 + 0.122474i
\(51\) 0 0
\(52\) −3.00000 + 5.19615i −0.416025 + 0.720577i
\(53\) −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
\(58\) −2.00000 − 3.46410i −0.262613 − 0.454859i
\(59\) 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 \(0.445525\pi\)
−0.938527 + 0.345207i \(0.887809\pi\)
\(62\) 6.00000 0.762001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −6.00000 + 10.3923i −0.744208 + 1.28901i
\(66\) 0 0
\(67\) −6.50000 − 11.2583i −0.794101 − 1.37542i −0.923408 − 0.383819i \(-0.874609\pi\)
0.129307 − 0.991605i \(-0.458725\pi\)
\(68\) 2.50000 + 4.33013i 0.303170 + 0.525105i
\(69\) 0 0
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 − 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) −1.00000 −0.117041 −0.0585206 − 0.998286i \(-0.518638\pi\)
−0.0585206 + 0.998286i \(0.518638\pi\)
\(74\) −1.00000 + 1.73205i −0.116248 + 0.201347i
\(75\) 0 0
\(76\) −3.50000 − 6.06218i −0.401478 − 0.695379i
\(77\) 0 0
\(78\) 0 0
\(79\) 3.00000 − 5.19615i 0.337526 − 0.584613i −0.646440 − 0.762964i \(-0.723743\pi\)
0.983967 + 0.178352i \(0.0570765\pi\)
\(80\) 2.00000 0.223607
\(81\) 0 0
\(82\) 3.00000 0.331295
\(83\) −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
\(86\) 0.500000 + 0.866025i 0.0539164 + 0.0933859i
\(87\) 0 0
\(88\) 0.500000 − 0.866025i 0.0533002 − 0.0923186i
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 2.00000 − 3.46410i 0.208514 − 0.361158i
\(93\) 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 −0.964579 − 0.263795i \(-0.915026\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 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 2646.2.f.b.883.1 2
3.2 odd 2 882.2.f.f.295.1 2
7.2 even 3 2646.2.e.h.2125.1 2
7.3 odd 6 2646.2.h.b.667.1 2
7.4 even 3 2646.2.h.c.667.1 2
7.5 odd 6 2646.2.e.i.2125.1 2
7.6 odd 2 378.2.f.b.127.1 2
9.2 odd 6 7938.2.a.e.1.1 1
9.4 even 3 inner 2646.2.f.b.1765.1 2
9.5 odd 6 882.2.f.f.589.1 2
9.7 even 3 7938.2.a.bb.1.1 1
21.2 odd 6 882.2.e.e.655.1 2
21.5 even 6 882.2.e.a.655.1 2
21.11 odd 6 882.2.h.g.79.1 2
21.17 even 6 882.2.h.h.79.1 2
21.20 even 2 126.2.f.b.43.1 ✓ 2
28.27 even 2 3024.2.r.c.2017.1 2
63.4 even 3 2646.2.e.h.1549.1 2
63.5 even 6 882.2.h.h.67.1 2
63.13 odd 6 378.2.f.b.253.1 2
63.20 even 6 1134.2.a.c.1.1 1
63.23 odd 6 882.2.h.g.67.1 2
63.31 odd 6 2646.2.e.i.1549.1 2
63.32 odd 6 882.2.e.e.373.1 2
63.34 odd 6 1134.2.a.f.1.1 1
63.40 odd 6 2646.2.h.b.361.1 2
63.41 even 6 126.2.f.b.85.1 yes 2
63.58 even 3 2646.2.h.c.361.1 2
63.59 even 6 882.2.e.a.373.1 2
84.83 odd 2 1008.2.r.a.673.1 2
252.83 odd 6 9072.2.a.t.1.1 1
252.139 even 6 3024.2.r.c.1009.1 2
252.167 odd 6 1008.2.r.a.337.1 2
252.223 even 6 9072.2.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.2.f.b.43.1 ✓ 2 21.20 even 2
126.2.f.b.85.1 yes 2 63.41 even 6
378.2.f.b.127.1 2 7.6 odd 2
378.2.f.b.253.1 2 63.13 odd 6
882.2.e.a.373.1 2 63.59 even 6
882.2.e.a.655.1 2 21.5 even 6
882.2.e.e.373.1 2 63.32 odd 6
882.2.e.e.655.1 2 21.2 odd 6
882.2.f.f.295.1 2 3.2 odd 2
882.2.f.f.589.1 2 9.5 odd 6
882.2.h.g.67.1 2 63.23 odd 6
882.2.h.g.79.1 2 21.11 odd 6
882.2.h.h.67.1 2 63.5 even 6
882.2.h.h.79.1 2 21.17 even 6
1008.2.r.a.337.1 2 252.167 odd 6
1008.2.r.a.673.1 2 84.83 odd 2
1134.2.a.c.1.1 1 63.20 even 6
1134.2.a.f.1.1 1 63.34 odd 6
2646.2.e.h.1549.1 2 63.4 even 3
2646.2.e.h.2125.1 2 7.2 even 3
2646.2.e.i.1549.1 2 63.31 odd 6
2646.2.e.i.2125.1 2 7.5 odd 6
2646.2.f.b.883.1 2 1.1 even 1 trivial
2646.2.f.b.1765.1 2 9.4 even 3 inner
2646.2.h.b.361.1 2 63.40 odd 6
2646.2.h.b.667.1 2 7.3 odd 6
2646.2.h.c.361.1 2 63.58 even 3
2646.2.h.c.667.1 2 7.4 even 3
3024.2.r.c.1009.1 2 252.139 even 6
3024.2.r.c.2017.1 2 28.27 even 2
7938.2.a.e.1.1 1 9.2 odd 6
7938.2.a.bb.1.1 1 9.7 even 3
9072.2.a.f.1.1 1 252.223 even 6
9072.2.a.t.1.1 1 252.83 odd 6