Properties

Label 3267.1.bf.a.1492.1
Level $3267$
Weight $1$
Character 3267.1492
Analytic conductor $1.630$
Analytic rank $0$
Dimension $24$
Projective image $D_{9}$
CM discriminant -11
Inner twists $16$

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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] = [3267,1,Mod(40,3267)] 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("3267.40"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3267, base_ring=CyclotomicField(90)) chi = DirichletCharacter(H, H._module([40, 63])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 3267 = 3^{3} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3267.bf (of order \(90\), degree \(24\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.63044539627\)
Analytic rank: \(0\)
Dimension: \(24\)
Coefficient field: \(\Q(\zeta_{45})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{24} - x^{21} + x^{15} - x^{12} + x^{9} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 297)
Projective image: \(D_{9}\)
Projective field: Galois closure of 9.1.459450093735369.1

Embedding invariants

Embedding label 1492.1
Root \(-0.882948 + 0.469472i\) of defining polynomial
Character \(\chi\) \(=\) 3267.1492
Dual form 3267.1.bf.a.1570.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.615661 + 0.788011i) q^{3} +(0.848048 + 0.529919i) q^{4} +(-1.10209 - 1.06428i) q^{5} +(-0.241922 - 0.970296i) q^{9} +(-0.939693 + 0.342020i) q^{12} +(1.51718 - 0.213226i) q^{15} +(0.438371 + 0.898794i) q^{16} +(-0.370646 - 1.48658i) q^{20} +(-0.766044 + 0.642788i) q^{23} +(0.0470200 + 1.34648i) q^{25} +(0.913545 + 0.406737i) q^{27} +(1.87481 - 0.131099i) q^{31} +(0.309017 - 0.951057i) q^{36} +(1.02517 - 1.13856i) q^{37} +(-0.766044 + 1.32683i) q^{45} +(0.294524 - 0.184039i) q^{47} +(-0.978148 - 0.207912i) q^{48} +(-0.719340 - 0.694658i) q^{49} +(1.52045 + 1.10467i) q^{53} +(1.65940 - 0.882318i) q^{59} +(1.39963 + 0.623157i) q^{60} +(-0.104528 + 0.994522i) q^{64} +(1.76604 + 0.642788i) q^{67} +(-0.0348995 - 0.999391i) q^{69} +(0.317271 + 0.141258i) q^{71} +(-1.08999 - 0.791921i) q^{75} +(0.473442 - 1.45710i) q^{80} +(-0.882948 + 0.469472i) q^{81} +(0.500000 - 0.866025i) q^{89} +(-0.990268 + 0.139173i) q^{92} +(-1.05094 + 1.55808i) q^{93} +(-0.249824 + 0.241252i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 3 q^{5} + 3 q^{15} - 6 q^{20} + 3 q^{25} + 3 q^{27} + 3 q^{31} - 6 q^{36} + 3 q^{47} + 3 q^{48} + 3 q^{59} + 3 q^{64} + 24 q^{67} - 6 q^{75} + 12 q^{89} - 6 q^{93} - 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3267\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(3026\)
\(\chi(n)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{8}{9}\right)\)

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 −0.961262 − 0.275637i \(-0.911111\pi\)
0.961262 + 0.275637i \(0.0888889\pi\)
\(3\) −0.615661 + 0.788011i −0.615661 + 0.788011i
\(4\) 0.848048 + 0.529919i 0.848048 + 0.529919i
\(5\) −1.10209 − 1.06428i −1.10209 − 1.06428i −0.997564 − 0.0697565i \(-0.977778\pi\)
−0.104528 − 0.994522i \(-0.533333\pi\)
\(6\) 0 0
\(7\) 0 0 0.374607 − 0.927184i \(-0.377778\pi\)
−0.374607 + 0.927184i \(0.622222\pi\)
\(8\) 0 0
\(9\) −0.241922 − 0.970296i −0.241922 − 0.970296i
\(10\) 0 0
\(11\) 0 0
\(12\) −0.939693 + 0.342020i −0.939693 + 0.342020i
\(13\) 0 0 −0.559193 − 0.829038i \(-0.688889\pi\)
0.559193 + 0.829038i \(0.311111\pi\)
\(14\) 0 0
\(15\) 1.51718 − 0.213226i 1.51718 − 0.213226i
\(16\) 0.438371 + 0.898794i 0.438371 + 0.898794i
\(17\) 0 0 −0.913545 − 0.406737i \(-0.866667\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(18\) 0 0
\(19\) 0 0 −0.669131 − 0.743145i \(-0.733333\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(20\) −0.370646 − 1.48658i −0.370646 − 1.48658i
\(21\) 0 0
\(22\) 0 0
\(23\) −0.766044 + 0.642788i −0.766044 + 0.642788i −0.939693 − 0.342020i \(-0.888889\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(24\) 0 0
\(25\) 0.0470200 + 1.34648i 0.0470200 + 1.34648i
\(26\) 0 0
\(27\) 0.913545 + 0.406737i 0.913545 + 0.406737i
\(28\) 0 0
\(29\) 0 0 −0.615661 − 0.788011i \(-0.711111\pi\)
0.615661 + 0.788011i \(0.288889\pi\)
\(30\) 0 0
\(31\) 1.87481 − 0.131099i 1.87481 − 0.131099i 0.913545 − 0.406737i \(-0.133333\pi\)
0.961262 + 0.275637i \(0.0888889\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0.309017 − 0.951057i 0.309017 − 0.951057i
\(37\) 1.02517 − 1.13856i 1.02517 − 1.13856i 0.0348995 − 0.999391i \(-0.488889\pi\)
0.990268 − 0.139173i \(-0.0444444\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 0.615661 − 0.788011i \(-0.288889\pi\)
−0.615661 + 0.788011i \(0.711111\pi\)
\(42\) 0 0
\(43\) 0 0 0.173648 − 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(44\) 0 0
\(45\) −0.766044 + 1.32683i −0.766044 + 1.32683i
\(46\) 0 0
\(47\) 0.294524 − 0.184039i 0.294524 − 0.184039i −0.374607 − 0.927184i \(-0.622222\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(48\) −0.978148 − 0.207912i −0.978148 − 0.207912i
\(49\) −0.719340 − 0.694658i −0.719340 − 0.694658i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.52045 + 1.10467i 1.52045 + 1.10467i 0.961262 + 0.275637i \(0.0888889\pi\)
0.559193 + 0.829038i \(0.311111\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.65940 − 0.882318i 1.65940 − 0.882318i 0.669131 − 0.743145i \(-0.266667\pi\)
0.990268 − 0.139173i \(-0.0444444\pi\)
\(60\) 1.39963 + 0.623157i 1.39963 + 0.623157i
\(61\) 0 0 −0.997564 − 0.0697565i \(-0.977778\pi\)
0.997564 + 0.0697565i \(0.0222222\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −0.104528 + 0.994522i −0.104528 + 0.994522i
\(65\) 0 0
\(66\) 0 0
\(67\) 1.76604 + 0.642788i 1.76604 + 0.642788i 1.00000 \(0\)
0.766044 + 0.642788i \(0.222222\pi\)
\(68\) 0 0
\(69\) −0.0348995 − 0.999391i −0.0348995 − 0.999391i
\(70\) 0 0
\(71\) 0.317271 + 0.141258i 0.317271 + 0.141258i 0.559193 − 0.829038i \(-0.311111\pi\)
−0.241922 + 0.970296i \(0.577778\pi\)
\(72\) 0 0
\(73\) 0 0 −0.978148 − 0.207912i \(-0.933333\pi\)
0.978148 + 0.207912i \(0.0666667\pi\)
\(74\) 0 0
\(75\) −1.08999 − 0.791921i −1.08999 − 0.791921i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.961262 − 0.275637i \(-0.911111\pi\)
0.961262 + 0.275637i \(0.0888889\pi\)
\(80\) 0.473442 − 1.45710i 0.473442 − 1.45710i
\(81\) −0.882948 + 0.469472i −0.882948 + 0.469472i
\(82\) 0 0
\(83\) 0 0 0.559193 − 0.829038i \(-0.311111\pi\)
−0.559193 + 0.829038i \(0.688889\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0.500000 − 0.866025i 0.500000 − 0.866025i −0.500000 − 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(90\) 0 0
\(91\) 0 0
\(92\) −0.990268 + 0.139173i −0.990268 + 0.139173i
\(93\) −1.05094 + 1.55808i −1.05094 + 1.55808i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −0.249824 + 0.241252i −0.249824 + 0.241252i −0.809017 − 0.587785i \(-0.800000\pi\)
0.559193 + 0.829038i \(0.311111\pi\)
\(98\) 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 3267.1.bf.a.1492.1 24
11.2 odd 10 inner 3267.1.bf.a.1546.1 24
11.3 even 5 inner 3267.1.bf.a.844.1 24
11.4 even 5 inner 3267.1.bf.a.2653.1 24
11.5 even 5 297.1.q.a.142.1 ✓ 6
11.6 odd 10 297.1.q.a.142.1 ✓ 6
11.7 odd 10 inner 3267.1.bf.a.2653.1 24
11.8 odd 10 inner 3267.1.bf.a.844.1 24
11.9 even 5 inner 3267.1.bf.a.1546.1 24
11.10 odd 2 CM 3267.1.bf.a.1492.1 24
27.4 even 9 inner 3267.1.bf.a.2218.1 24
33.5 odd 10 891.1.q.a.505.1 6
33.17 even 10 891.1.q.a.505.1 6
99.5 odd 30 2673.1.q.a.703.1 6
99.16 even 15 2673.1.q.b.1594.1 6
99.38 odd 30 2673.1.q.c.1594.1 6
99.49 even 15 2673.1.q.d.703.1 6
99.50 even 30 2673.1.q.a.703.1 6
99.61 odd 30 2673.1.q.b.1594.1 6
99.83 even 30 2673.1.q.c.1594.1 6
99.94 odd 30 2673.1.q.d.703.1 6
297.4 even 45 inner 3267.1.bf.a.112.1 24
297.5 odd 90 2673.1.q.a.1000.1 6
297.31 even 45 inner 3267.1.bf.a.2272.1 24
297.49 even 45 2673.1.q.d.1000.1 6
297.50 even 90 891.1.q.a.307.1 6
297.58 even 45 inner 3267.1.bf.a.1570.1 24
297.85 odd 90 inner 3267.1.bf.a.1570.1 24
297.94 odd 90 2673.1.q.b.109.1 6
297.104 odd 90 891.1.q.a.307.1 6
297.112 odd 90 inner 3267.1.bf.a.2272.1 24
297.139 odd 90 inner 3267.1.bf.a.112.1 24
297.148 even 45 2673.1.q.b.109.1 6
297.149 even 90 2673.1.q.c.109.1 6
297.193 odd 90 297.1.q.a.274.1 yes 6
297.203 odd 90 2673.1.q.c.109.1 6
297.247 even 45 297.1.q.a.274.1 yes 6
297.248 even 90 2673.1.q.a.1000.1 6
297.274 odd 18 inner 3267.1.bf.a.2218.1 24
297.292 odd 90 2673.1.q.d.1000.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
297.1.q.a.142.1 ✓ 6 11.5 even 5
297.1.q.a.142.1 ✓ 6 11.6 odd 10
297.1.q.a.274.1 yes 6 297.193 odd 90
297.1.q.a.274.1 yes 6 297.247 even 45
891.1.q.a.307.1 6 297.50 even 90
891.1.q.a.307.1 6 297.104 odd 90
891.1.q.a.505.1 6 33.5 odd 10
891.1.q.a.505.1 6 33.17 even 10
2673.1.q.a.703.1 6 99.5 odd 30
2673.1.q.a.703.1 6 99.50 even 30
2673.1.q.a.1000.1 6 297.5 odd 90
2673.1.q.a.1000.1 6 297.248 even 90
2673.1.q.b.109.1 6 297.94 odd 90
2673.1.q.b.109.1 6 297.148 even 45
2673.1.q.b.1594.1 6 99.16 even 15
2673.1.q.b.1594.1 6 99.61 odd 30
2673.1.q.c.109.1 6 297.149 even 90
2673.1.q.c.109.1 6 297.203 odd 90
2673.1.q.c.1594.1 6 99.38 odd 30
2673.1.q.c.1594.1 6 99.83 even 30
2673.1.q.d.703.1 6 99.49 even 15
2673.1.q.d.703.1 6 99.94 odd 30
2673.1.q.d.1000.1 6 297.49 even 45
2673.1.q.d.1000.1 6 297.292 odd 90
3267.1.bf.a.112.1 24 297.4 even 45 inner
3267.1.bf.a.112.1 24 297.139 odd 90 inner
3267.1.bf.a.844.1 24 11.3 even 5 inner
3267.1.bf.a.844.1 24 11.8 odd 10 inner
3267.1.bf.a.1492.1 24 1.1 even 1 trivial
3267.1.bf.a.1492.1 24 11.10 odd 2 CM
3267.1.bf.a.1546.1 24 11.2 odd 10 inner
3267.1.bf.a.1546.1 24 11.9 even 5 inner
3267.1.bf.a.1570.1 24 297.58 even 45 inner
3267.1.bf.a.1570.1 24 297.85 odd 90 inner
3267.1.bf.a.2218.1 24 27.4 even 9 inner
3267.1.bf.a.2218.1 24 297.274 odd 18 inner
3267.1.bf.a.2272.1 24 297.31 even 45 inner
3267.1.bf.a.2272.1 24 297.112 odd 90 inner
3267.1.bf.a.2653.1 24 11.4 even 5 inner
3267.1.bf.a.2653.1 24 11.7 odd 10 inner