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Properties

Label 1782.2.e.s.1189.1
Level $1782$
Weight $2$
Character 1782.1189
Analytic conductor $14.229$
Analytic rank $1$
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] = [1782,2,Mod(595,1782)] 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("1782.595"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1782, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1782 = 2 \cdot 3^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1782.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,1,0,-1,0,0,-2,-2,0,0,1,0,4,2,0,-1,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.2293416402\)
Analytic rank: \(1\)
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 66)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1189.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1782.1189
Dual form 1782.2.e.s.595.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-1.00000 + 1.73205i) q^{7} -1.00000 q^{8} +(0.500000 - 0.866025i) q^{11} +(2.00000 + 3.46410i) q^{13} +(1.00000 + 1.73205i) q^{14} +(-0.500000 + 0.866025i) q^{16} -6.00000 q^{17} -4.00000 q^{19} +(-0.500000 - 0.866025i) q^{22} +(-3.00000 - 5.19615i) q^{23} +(2.50000 - 4.33013i) q^{25} +4.00000 q^{26} +2.00000 q^{28} +(-3.00000 + 5.19615i) q^{29} +(-4.00000 - 6.92820i) q^{31} +(0.500000 + 0.866025i) q^{32} +(-3.00000 + 5.19615i) q^{34} -10.0000 q^{37} +(-2.00000 + 3.46410i) q^{38} +(-3.00000 - 5.19615i) q^{41} +(-4.00000 + 6.92820i) q^{43} -1.00000 q^{44} -6.00000 q^{46} +(3.00000 - 5.19615i) q^{47} +(1.50000 + 2.59808i) q^{49} +(-2.50000 - 4.33013i) q^{50} +(2.00000 - 3.46410i) q^{52} +(1.00000 - 1.73205i) q^{56} +(3.00000 + 5.19615i) q^{58} +(-4.00000 + 6.92820i) q^{61} -8.00000 q^{62} +1.00000 q^{64} +(2.00000 + 3.46410i) q^{67} +(3.00000 + 5.19615i) q^{68} +6.00000 q^{71} +2.00000 q^{73} +(-5.00000 + 8.66025i) q^{74} +(2.00000 + 3.46410i) q^{76} +(1.00000 + 1.73205i) q^{77} +(-7.00000 + 12.1244i) q^{79} -6.00000 q^{82} +(6.00000 - 10.3923i) q^{83} +(4.00000 + 6.92820i) q^{86} +(-0.500000 + 0.866025i) q^{88} -6.00000 q^{89} -8.00000 q^{91} +(-3.00000 + 5.19615i) q^{92} +(-3.00000 - 5.19615i) q^{94} +(-7.00000 + 12.1244i) q^{97} +3.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{4} - 2 q^{7} - 2 q^{8} + q^{11} + 4 q^{13} + 2 q^{14} - q^{16} - 12 q^{17} - 8 q^{19} - q^{22} - 6 q^{23} + 5 q^{25} + 8 q^{26} + 4 q^{28} - 6 q^{29} - 8 q^{31} + q^{32} - 6 q^{34}+ \cdots + 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1782\mathbb{Z}\right)^\times\).

\(n\) \(1135\) \(1541\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.500000 − 0.866025i 0.353553 − 0.612372i
\(3\) 0 0
\(4\) −0.500000 − 0.866025i −0.250000 − 0.433013i
\(5\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(6\) 0 0
\(7\) −1.00000 + 1.73205i −0.377964 + 0.654654i −0.990766 − 0.135583i \(-0.956709\pi\)
0.612801 + 0.790237i \(0.290043\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 0.500000 − 0.866025i 0.150756 − 0.261116i
\(12\) 0 0
\(13\) 2.00000 + 3.46410i 0.554700 + 0.960769i 0.997927 + 0.0643593i \(0.0205004\pi\)
−0.443227 + 0.896410i \(0.646166\pi\)
\(14\) 1.00000 + 1.73205i 0.267261 + 0.462910i
\(15\) 0 0
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) −6.00000 −1.45521 −0.727607 − 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −0.500000 − 0.866025i −0.106600 − 0.184637i
\(23\) −3.00000 − 5.19615i −0.625543 − 1.08347i −0.988436 − 0.151642i \(-0.951544\pi\)
0.362892 − 0.931831i \(-0.381789\pi\)
\(24\) 0 0
\(25\) 2.50000 − 4.33013i 0.500000 − 0.866025i
\(26\) 4.00000 0.784465
\(27\) 0 0
\(28\) 2.00000 0.377964
\(29\) −3.00000 + 5.19615i −0.557086 + 0.964901i 0.440652 + 0.897678i \(0.354747\pi\)
−0.997738 + 0.0672232i \(0.978586\pi\)
\(30\) 0 0
\(31\) −4.00000 − 6.92820i −0.718421 − 1.24434i −0.961625 − 0.274367i \(-0.911532\pi\)
0.243204 − 0.969975i \(-0.421802\pi\)
\(32\) 0.500000 + 0.866025i 0.0883883 + 0.153093i
\(33\) 0 0
\(34\) −3.00000 + 5.19615i −0.514496 + 0.891133i
\(35\) 0 0
\(36\) 0 0
\(37\) −10.0000 −1.64399 −0.821995 − 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) −2.00000 + 3.46410i −0.324443 + 0.561951i
\(39\) 0 0
\(40\) 0 0
\(41\) −3.00000 − 5.19615i −0.468521 − 0.811503i 0.530831 − 0.847477i \(-0.321880\pi\)
−0.999353 + 0.0359748i \(0.988546\pi\)
\(42\) 0 0
\(43\) −4.00000 + 6.92820i −0.609994 + 1.05654i 0.381246 + 0.924473i \(0.375495\pi\)
−0.991241 + 0.132068i \(0.957838\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) −6.00000 −0.884652
\(47\) 3.00000 − 5.19615i 0.437595 − 0.757937i −0.559908 − 0.828554i \(-0.689164\pi\)
0.997503 + 0.0706177i \(0.0224970\pi\)
\(48\) 0 0
\(49\) 1.50000 + 2.59808i 0.214286 + 0.371154i
\(50\) −2.50000 − 4.33013i −0.353553 − 0.612372i
\(51\) 0 0
\(52\) 2.00000 − 3.46410i 0.277350 − 0.480384i
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.00000 − 1.73205i 0.133631 − 0.231455i
\(57\) 0 0
\(58\) 3.00000 + 5.19615i 0.393919 + 0.682288i
\(59\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(60\) 0 0
\(61\) −4.00000 + 6.92820i −0.512148 + 0.887066i 0.487753 + 0.872982i \(0.337817\pi\)
−0.999901 + 0.0140840i \(0.995517\pi\)
\(62\) −8.00000 −1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 2.00000 + 3.46410i 0.244339 + 0.423207i 0.961946 − 0.273241i \(-0.0880957\pi\)
−0.717607 + 0.696449i \(0.754762\pi\)
\(68\) 3.00000 + 5.19615i 0.363803 + 0.630126i
\(69\) 0 0
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 − 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 − 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) −5.00000 + 8.66025i −0.581238 + 1.00673i
\(75\) 0 0
\(76\) 2.00000 + 3.46410i 0.229416 + 0.397360i
\(77\) 1.00000 + 1.73205i 0.113961 + 0.197386i
\(78\) 0 0
\(79\) −7.00000 + 12.1244i −0.787562 + 1.36410i 0.139895 + 0.990166i \(0.455323\pi\)
−0.927457 + 0.373930i \(0.878010\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −6.00000 −0.662589
\(83\) 6.00000 − 10.3923i 0.658586 − 1.14070i −0.322396 − 0.946605i \(-0.604488\pi\)
0.980982 − 0.194099i \(-0.0621783\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 4.00000 + 6.92820i 0.431331 + 0.747087i
\(87\) 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
\(91\) −8.00000 −0.838628
\(92\) −3.00000 + 5.19615i −0.312772 + 0.541736i
\(93\) 0 0
\(94\) −3.00000 − 5.19615i −0.309426 − 0.535942i
\(95\) 0 0
\(96\) 0 0
\(97\) −7.00000 + 12.1244i −0.710742 + 1.23104i 0.253837 + 0.967247i \(0.418307\pi\)
−0.964579 + 0.263795i \(0.915026\pi\)
\(98\) 3.00000 0.303046
\(99\) 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 1782.2.e.s.1189.1 2
3.2 odd 2 1782.2.e.f.1189.1 2
9.2 odd 6 198.2.a.e.1.1 1
9.4 even 3 inner 1782.2.e.s.595.1 2
9.5 odd 6 1782.2.e.f.595.1 2
9.7 even 3 66.2.a.a.1.1 ✓ 1
36.7 odd 6 528.2.a.d.1.1 1
36.11 even 6 1584.2.a.h.1.1 1
45.2 even 12 4950.2.c.r.199.2 2
45.7 odd 12 1650.2.c.d.199.1 2
45.29 odd 6 4950.2.a.g.1.1 1
45.34 even 6 1650.2.a.m.1.1 1
45.38 even 12 4950.2.c.r.199.1 2
45.43 odd 12 1650.2.c.d.199.2 2
63.20 even 6 9702.2.a.bu.1.1 1
63.34 odd 6 3234.2.a.d.1.1 1
72.11 even 6 6336.2.a.bf.1.1 1
72.29 odd 6 6336.2.a.bj.1.1 1
72.43 odd 6 2112.2.a.v.1.1 1
72.61 even 6 2112.2.a.i.1.1 1
99.7 odd 30 726.2.e.b.511.1 4
99.16 even 15 726.2.e.k.487.1 4
99.25 even 15 726.2.e.k.493.1 4
99.43 odd 6 726.2.a.i.1.1 1
99.52 odd 30 726.2.e.b.493.1 4
99.61 odd 30 726.2.e.b.487.1 4
99.65 even 6 2178.2.a.b.1.1 1
99.70 even 15 726.2.e.k.511.1 4
99.79 odd 30 726.2.e.b.565.1 4
99.97 even 15 726.2.e.k.565.1 4
396.43 even 6 5808.2.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
66.2.a.a.1.1 ✓ 1 9.7 even 3
198.2.a.e.1.1 1 9.2 odd 6
528.2.a.d.1.1 1 36.7 odd 6
726.2.a.i.1.1 1 99.43 odd 6
726.2.e.b.487.1 4 99.61 odd 30
726.2.e.b.493.1 4 99.52 odd 30
726.2.e.b.511.1 4 99.7 odd 30
726.2.e.b.565.1 4 99.79 odd 30
726.2.e.k.487.1 4 99.16 even 15
726.2.e.k.493.1 4 99.25 even 15
726.2.e.k.511.1 4 99.70 even 15
726.2.e.k.565.1 4 99.97 even 15
1584.2.a.h.1.1 1 36.11 even 6
1650.2.a.m.1.1 1 45.34 even 6
1650.2.c.d.199.1 2 45.7 odd 12
1650.2.c.d.199.2 2 45.43 odd 12
1782.2.e.f.595.1 2 9.5 odd 6
1782.2.e.f.1189.1 2 3.2 odd 2
1782.2.e.s.595.1 2 9.4 even 3 inner
1782.2.e.s.1189.1 2 1.1 even 1 trivial
2112.2.a.i.1.1 1 72.61 even 6
2112.2.a.v.1.1 1 72.43 odd 6
2178.2.a.b.1.1 1 99.65 even 6
3234.2.a.d.1.1 1 63.34 odd 6
4950.2.a.g.1.1 1 45.29 odd 6
4950.2.c.r.199.1 2 45.38 even 12
4950.2.c.r.199.2 2 45.2 even 12
5808.2.a.l.1.1 1 396.43 even 6
6336.2.a.bf.1.1 1 72.11 even 6
6336.2.a.bj.1.1 1 72.29 odd 6
9702.2.a.bu.1.1 1 63.20 even 6