Properties

Label 3267.1.bf.a.844.1
Level $3267$
Weight $1$
Character 3267.844
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 844.1
Root \(0.990268 + 0.139173i\) of defining polynomial
Character \(\chi\) \(=\) 3267.844
Dual form 3267.1.bf.a.2218.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.0348995 - 0.999391i) q^{3} +(-0.374607 - 0.927184i) q^{4} +(0.671624 - 1.37703i) q^{5} +(-0.997564 - 0.0697565i) q^{9} +(-0.939693 + 0.342020i) q^{12} +(-1.35275 - 0.719272i) q^{15} +(-0.719340 + 0.694658i) q^{16} +(-1.52836 - 0.106873i) q^{20} +(-0.766044 + 0.642788i) q^{23} +(-0.829478 - 1.06168i) q^{25} +(-0.104528 + 0.994522i) q^{27} +(0.454664 - 1.82356i) q^{31} +(0.309017 + 0.951057i) q^{36} +(-1.49861 + 0.318539i) q^{37} +(-0.766044 + 1.32683i) q^{45} +(-0.130100 + 0.322008i) q^{47} +(0.669131 + 0.743145i) q^{48} +(0.438371 - 0.898794i) q^{49} +(1.52045 - 1.10467i) q^{53} +(-1.86110 - 0.261560i) q^{59} +(-0.160147 + 1.52370i) q^{60} +(0.913545 + 0.406737i) q^{64} +(1.76604 + 0.642788i) q^{67} +(0.615661 + 0.788011i) q^{69} +(-0.0363024 + 0.345394i) q^{71} +(-1.08999 + 0.791921i) q^{75} +(0.473442 + 1.45710i) q^{80} +(0.990268 + 0.139173i) q^{81} +(0.500000 - 0.866025i) q^{89} +(0.882948 + 0.469472i) q^{92} +(-1.80658 - 0.518029i) q^{93} +(0.152245 + 0.312148i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 3 q^{5} + 3 q^{15} - 6 q^{20} + 3 q^{25} + 3 q^{27} + 3 q^{31} - 6 q^{36} + 3 q^{47} + 3 q^{48} + 3 q^{59} + 3 q^{64} + 24 q^{67} - 6 q^{75} + 12 q^{89} - 6 q^{93} - 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(244\) \(3026\)
\(\chi(n)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{8}{9}\right)\)

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 0.559193 − 0.829038i \(-0.311111\pi\)
−0.559193 + 0.829038i \(0.688889\pi\)
\(3\) 0.0348995 − 0.999391i 0.0348995 − 0.999391i
\(4\) −0.374607 − 0.927184i −0.374607 − 0.927184i
\(5\) 0.671624 − 1.37703i 0.671624 − 1.37703i −0.241922 − 0.970296i \(-0.577778\pi\)
0.913545 − 0.406737i \(-0.133333\pi\)
\(6\) 0 0
\(7\) 0 0 0.848048 − 0.529919i \(-0.177778\pi\)
−0.848048 + 0.529919i \(0.822222\pi\)
\(8\) 0 0
\(9\) −0.997564 − 0.0697565i −0.997564 − 0.0697565i
\(10\) 0 0
\(11\) 0 0
\(12\) −0.939693 + 0.342020i −0.939693 + 0.342020i
\(13\) 0 0 0.961262 − 0.275637i \(-0.0888889\pi\)
−0.961262 + 0.275637i \(0.911111\pi\)
\(14\) 0 0
\(15\) −1.35275 − 0.719272i −1.35275 − 0.719272i
\(16\) −0.719340 + 0.694658i −0.719340 + 0.694658i
\(17\) 0 0 0.104528 − 0.994522i \(-0.466667\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(18\) 0 0
\(19\) 0 0 −0.978148 − 0.207912i \(-0.933333\pi\)
0.978148 + 0.207912i \(0.0666667\pi\)
\(20\) −1.52836 − 0.106873i −1.52836 − 0.106873i
\(21\) 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
\(25\) −0.829478 − 1.06168i −0.829478 − 1.06168i
\(26\) 0 0
\(27\) −0.104528 + 0.994522i −0.104528 + 0.994522i
\(28\) 0 0
\(29\) 0 0 −0.0348995 − 0.999391i \(-0.511111\pi\)
0.0348995 + 0.999391i \(0.488889\pi\)
\(30\) 0 0
\(31\) 0.454664 − 1.82356i 0.454664 − 1.82356i −0.104528 − 0.994522i \(-0.533333\pi\)
0.559193 − 0.829038i \(-0.311111\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(37\) −1.49861 + 0.318539i −1.49861 + 0.318539i −0.882948 − 0.469472i \(-0.844444\pi\)
−0.615661 + 0.788011i \(0.711111\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 0.0348995 − 0.999391i \(-0.488889\pi\)
−0.0348995 + 0.999391i \(0.511111\pi\)
\(42\) 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
\(47\) −0.130100 + 0.322008i −0.130100 + 0.322008i −0.978148 − 0.207912i \(-0.933333\pi\)
0.848048 + 0.529919i \(0.177778\pi\)
\(48\) 0.669131 + 0.743145i 0.669131 + 0.743145i
\(49\) 0.438371 − 0.898794i 0.438371 − 0.898794i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.52045 − 1.10467i 1.52045 − 1.10467i 0.559193 − 0.829038i \(-0.311111\pi\)
0.961262 − 0.275637i \(-0.0888889\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −1.86110 − 0.261560i −1.86110 − 0.261560i −0.882948 − 0.469472i \(-0.844444\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(60\) −0.160147 + 1.52370i −0.160147 + 1.52370i
\(61\) 0 0 −0.241922 − 0.970296i \(-0.577778\pi\)
0.241922 + 0.970296i \(0.422222\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0.913545 + 0.406737i 0.913545 + 0.406737i
\(65\) 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
\(69\) 0.615661 + 0.788011i 0.615661 + 0.788011i
\(70\) 0 0
\(71\) −0.0363024 + 0.345394i −0.0363024 + 0.345394i 0.961262 + 0.275637i \(0.0888889\pi\)
−0.997564 + 0.0697565i \(0.977778\pi\)
\(72\) 0 0
\(73\) 0 0 −0.669131 − 0.743145i \(-0.733333\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(74\) 0 0
\(75\) −1.08999 + 0.791921i −1.08999 + 0.791921i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.559193 − 0.829038i \(-0.311111\pi\)
−0.559193 + 0.829038i \(0.688889\pi\)
\(80\) 0.473442 + 1.45710i 0.473442 + 1.45710i
\(81\) 0.990268 + 0.139173i 0.990268 + 0.139173i
\(82\) 0 0
\(83\) 0 0 −0.961262 − 0.275637i \(-0.911111\pi\)
0.961262 + 0.275637i \(0.0888889\pi\)
\(84\) 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
\(92\) 0.882948 + 0.469472i 0.882948 + 0.469472i
\(93\) −1.80658 − 0.518029i −1.80658 − 0.518029i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0.152245 + 0.312148i 0.152245 + 0.312148i 0.961262 − 0.275637i \(-0.0888889\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(98\) 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.844.1 24
11.2 odd 10 297.1.q.a.142.1 ✓ 6
11.3 even 5 inner 3267.1.bf.a.1546.1 24
11.4 even 5 inner 3267.1.bf.a.1492.1 24
11.5 even 5 inner 3267.1.bf.a.2653.1 24
11.6 odd 10 inner 3267.1.bf.a.2653.1 24
11.7 odd 10 inner 3267.1.bf.a.1492.1 24
11.8 odd 10 inner 3267.1.bf.a.1546.1 24
11.9 even 5 297.1.q.a.142.1 ✓ 6
11.10 odd 2 CM 3267.1.bf.a.844.1 24
27.4 even 9 inner 3267.1.bf.a.1570.1 24
33.2 even 10 891.1.q.a.505.1 6
33.20 odd 10 891.1.q.a.505.1 6
99.2 even 30 2673.1.q.c.1594.1 6
99.13 odd 30 2673.1.q.d.703.1 6
99.20 odd 30 2673.1.q.c.1594.1 6
99.31 even 15 2673.1.q.d.703.1 6
99.68 even 30 2673.1.q.a.703.1 6
99.79 odd 30 2673.1.q.b.1594.1 6
99.86 odd 30 2673.1.q.a.703.1 6
99.97 even 15 2673.1.q.b.1594.1 6
297.4 even 45 inner 3267.1.bf.a.2218.1 24
297.13 odd 90 2673.1.q.b.109.1 6
297.31 even 45 297.1.q.a.274.1 yes 6
297.58 even 45 inner 3267.1.bf.a.2272.1 24
297.68 even 90 2673.1.q.c.109.1 6
297.85 odd 90 inner 3267.1.bf.a.2272.1 24
297.86 odd 90 2673.1.q.a.1000.1 6
297.112 odd 90 297.1.q.a.274.1 yes 6
297.130 even 45 2673.1.q.d.1000.1 6
297.139 odd 90 inner 3267.1.bf.a.2218.1 24
297.167 even 90 2673.1.q.a.1000.1 6
297.185 odd 90 891.1.q.a.307.1 6
297.193 odd 90 inner 3267.1.bf.a.112.1 24
297.211 odd 90 2673.1.q.d.1000.1 6
297.229 even 45 2673.1.q.b.109.1 6
297.247 even 45 inner 3267.1.bf.a.112.1 24
297.266 even 90 891.1.q.a.307.1 6
297.274 odd 18 inner 3267.1.bf.a.1570.1 24
297.284 odd 90 2673.1.q.c.109.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
297.1.q.a.142.1 ✓ 6 11.2 odd 10
297.1.q.a.142.1 ✓ 6 11.9 even 5
297.1.q.a.274.1 yes 6 297.31 even 45
297.1.q.a.274.1 yes 6 297.112 odd 90
891.1.q.a.307.1 6 297.185 odd 90
891.1.q.a.307.1 6 297.266 even 90
891.1.q.a.505.1 6 33.2 even 10
891.1.q.a.505.1 6 33.20 odd 10
2673.1.q.a.703.1 6 99.68 even 30
2673.1.q.a.703.1 6 99.86 odd 30
2673.1.q.a.1000.1 6 297.86 odd 90
2673.1.q.a.1000.1 6 297.167 even 90
2673.1.q.b.109.1 6 297.13 odd 90
2673.1.q.b.109.1 6 297.229 even 45
2673.1.q.b.1594.1 6 99.79 odd 30
2673.1.q.b.1594.1 6 99.97 even 15
2673.1.q.c.109.1 6 297.68 even 90
2673.1.q.c.109.1 6 297.284 odd 90
2673.1.q.c.1594.1 6 99.2 even 30
2673.1.q.c.1594.1 6 99.20 odd 30
2673.1.q.d.703.1 6 99.13 odd 30
2673.1.q.d.703.1 6 99.31 even 15
2673.1.q.d.1000.1 6 297.130 even 45
2673.1.q.d.1000.1 6 297.211 odd 90
3267.1.bf.a.112.1 24 297.193 odd 90 inner
3267.1.bf.a.112.1 24 297.247 even 45 inner
3267.1.bf.a.844.1 24 1.1 even 1 trivial
3267.1.bf.a.844.1 24 11.10 odd 2 CM
3267.1.bf.a.1492.1 24 11.4 even 5 inner
3267.1.bf.a.1492.1 24 11.7 odd 10 inner
3267.1.bf.a.1546.1 24 11.3 even 5 inner
3267.1.bf.a.1546.1 24 11.8 odd 10 inner
3267.1.bf.a.1570.1 24 27.4 even 9 inner
3267.1.bf.a.1570.1 24 297.274 odd 18 inner
3267.1.bf.a.2218.1 24 297.4 even 45 inner
3267.1.bf.a.2218.1 24 297.139 odd 90 inner
3267.1.bf.a.2272.1 24 297.58 even 45 inner
3267.1.bf.a.2272.1 24 297.85 odd 90 inner
3267.1.bf.a.2653.1 24 11.5 even 5 inner
3267.1.bf.a.2653.1 24 11.6 odd 10 inner