Properties

Label 2664.1.bw.c.1627.1
Level $2664$
Weight $1$
Character 2664.1627
Analytic conductor $1.330$
Analytic rank $0$
Dimension $8$
Projective image $D_{15}$
CM discriminant -296
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2664,1,Mod(1627,2664)] 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("2664.1627"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2664, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 4, 3])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 2664 = 2^{3} \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2664.bw (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.32950919365\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{15})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{15}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{15} - \cdots)\)

Embedding invariants

Embedding label 1627.1
Root \(0.669131 + 0.743145i\) of defining polynomial
Character \(\chi\) \(=\) 2664.1627
Dual form 2664.1.bw.c.2515.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.500000 + 0.866025i) q^{2} +(-0.809017 - 0.587785i) q^{3} +(-0.500000 - 0.866025i) q^{4} +(-0.309017 - 0.535233i) q^{5} +(0.913545 - 0.406737i) q^{6} +1.00000 q^{8} +(0.309017 + 0.951057i) q^{9} +0.618034 q^{10} +(0.978148 - 1.69420i) q^{11} +(-0.104528 + 0.994522i) q^{12} +(-0.913545 - 1.58231i) q^{13} +(-0.0646021 + 0.614648i) q^{15} +(-0.500000 + 0.866025i) q^{16} +(-0.978148 - 0.207912i) q^{18} +(-0.309017 + 0.535233i) q^{20} +(0.978148 + 1.69420i) q^{22} +(0.104528 + 0.181049i) q^{23} +(-0.809017 - 0.587785i) q^{24} +(0.309017 - 0.535233i) q^{25} +1.82709 q^{26} +(0.309017 - 0.951057i) q^{27} +(-0.913545 + 1.58231i) q^{29} +(-0.500000 - 0.363271i) q^{30} +(-0.309017 - 0.535233i) q^{31} +(-0.500000 - 0.866025i) q^{32} +(-1.78716 + 0.795697i) q^{33} +(0.669131 - 0.743145i) q^{36} +1.00000 q^{37} +(-0.190983 + 1.81708i) q^{39} +(-0.309017 - 0.535233i) q^{40} +(0.809017 + 1.40126i) q^{41} -1.95630 q^{44} +(0.413545 - 0.459289i) q^{45} -0.209057 q^{46} +(0.913545 - 0.406737i) q^{48} +(-0.500000 - 0.866025i) q^{49} +(0.309017 + 0.535233i) q^{50} +(-0.913545 + 1.58231i) q^{52} +(0.669131 + 0.743145i) q^{54} -1.20906 q^{55} +(-0.913545 - 1.58231i) q^{58} +(0.564602 - 0.251377i) q^{60} +(-0.669131 + 1.15897i) q^{61} +0.618034 q^{62} +1.00000 q^{64} +(-0.564602 + 0.977920i) q^{65} +(0.204489 - 1.94558i) q^{66} +(-0.669131 - 1.15897i) q^{67} +(0.0218524 - 0.207912i) q^{69} +(0.309017 + 0.951057i) q^{72} -1.95630 q^{73} +(-0.500000 + 0.866025i) q^{74} +(-0.564602 + 0.251377i) q^{75} +(-1.47815 - 1.07394i) q^{78} +(0.809017 - 1.40126i) q^{79} +0.618034 q^{80} +(-0.809017 + 0.587785i) q^{81} -1.61803 q^{82} +(0.500000 - 0.866025i) q^{83} +(1.66913 - 0.743145i) q^{87} +(0.978148 - 1.69420i) q^{88} +(0.190983 + 0.587785i) q^{90} +(0.104528 - 0.181049i) q^{92} +(-0.0646021 + 0.614648i) q^{93} +(-0.104528 + 0.994522i) q^{96} +1.00000 q^{98} +(1.91355 + 0.406737i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} - 2 q^{3} - 4 q^{4} + 2 q^{5} + q^{6} + 8 q^{8} - 2 q^{9} - 4 q^{10} - q^{11} + q^{12} - q^{13} + 2 q^{15} - 4 q^{16} + q^{18} + 2 q^{20} - q^{22} - q^{23} - 2 q^{24} - 2 q^{25} + 2 q^{26}+ \cdots + 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2664\mathbb{Z}\right)^\times\).

\(n\) \(1297\) \(1333\) \(1999\) \(2369\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(e\left(\frac{2}{3}\right)\)

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 −0.500000 + 0.866025i
\(3\) −0.809017 − 0.587785i −0.809017 − 0.587785i
\(4\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(5\) −0.309017 − 0.535233i −0.309017 − 0.535233i 0.669131 − 0.743145i \(-0.266667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(6\) 0.913545 − 0.406737i 0.913545 − 0.406737i
\(7\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(8\) 1.00000 1.00000
\(9\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(10\) 0.618034 0.618034
\(11\) 0.978148 − 1.69420i 0.978148 − 1.69420i 0.309017 − 0.951057i \(-0.400000\pi\)
0.669131 − 0.743145i \(-0.266667\pi\)
\(12\) −0.104528 + 0.994522i −0.104528 + 0.994522i
\(13\) −0.913545 − 1.58231i −0.913545 − 1.58231i −0.809017 − 0.587785i \(-0.800000\pi\)
−0.104528 − 0.994522i \(-0.533333\pi\)
\(14\) 0 0
\(15\) −0.0646021 + 0.614648i −0.0646021 + 0.614648i
\(16\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) −0.978148 − 0.207912i −0.978148 − 0.207912i
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) −0.309017 + 0.535233i −0.309017 + 0.535233i
\(21\) 0 0
\(22\) 0.978148 + 1.69420i 0.978148 + 1.69420i
\(23\) 0.104528 + 0.181049i 0.104528 + 0.181049i 0.913545 − 0.406737i \(-0.133333\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(24\) −0.809017 − 0.587785i −0.809017 − 0.587785i
\(25\) 0.309017 − 0.535233i 0.309017 − 0.535233i
\(26\) 1.82709 1.82709
\(27\) 0.309017 − 0.951057i 0.309017 − 0.951057i
\(28\) 0 0
\(29\) −0.913545 + 1.58231i −0.913545 + 1.58231i −0.104528 + 0.994522i \(0.533333\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(30\) −0.500000 − 0.363271i −0.500000 − 0.363271i
\(31\) −0.309017 − 0.535233i −0.309017 − 0.535233i 0.669131 − 0.743145i \(-0.266667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(32\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(33\) −1.78716 + 0.795697i −1.78716 + 0.795697i
\(34\) 0 0
\(35\) 0 0
\(36\) 0.669131 − 0.743145i 0.669131 − 0.743145i
\(37\) 1.00000 1.00000
\(38\) 0 0
\(39\) −0.190983 + 1.81708i −0.190983 + 1.81708i
\(40\) −0.309017 − 0.535233i −0.309017 − 0.535233i
\(41\) 0.809017 + 1.40126i 0.809017 + 1.40126i 0.913545 + 0.406737i \(0.133333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(42\) 0 0
\(43\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(44\) −1.95630 −1.95630
\(45\) 0.413545 − 0.459289i 0.413545 − 0.459289i
\(46\) −0.209057 −0.209057
\(47\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(48\) 0.913545 − 0.406737i 0.913545 − 0.406737i
\(49\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(50\) 0.309017 + 0.535233i 0.309017 + 0.535233i
\(51\) 0 0
\(52\) −0.913545 + 1.58231i −0.913545 + 1.58231i
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0.669131 + 0.743145i 0.669131 + 0.743145i
\(55\) −1.20906 −1.20906
\(56\) 0 0
\(57\) 0 0
\(58\) −0.913545 − 1.58231i −0.913545 − 1.58231i
\(59\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(60\) 0.564602 − 0.251377i 0.564602 − 0.251377i
\(61\) −0.669131 + 1.15897i −0.669131 + 1.15897i 0.309017 + 0.951057i \(0.400000\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(62\) 0.618034 0.618034
\(63\) 0 0
\(64\) 1.00000 1.00000
\(65\) −0.564602 + 0.977920i −0.564602 + 0.977920i
\(66\) 0.204489 − 1.94558i 0.204489 − 1.94558i
\(67\) −0.669131 − 1.15897i −0.669131 − 1.15897i −0.978148 − 0.207912i \(-0.933333\pi\)
0.309017 − 0.951057i \(-0.400000\pi\)
\(68\) 0 0
\(69\) 0.0218524 − 0.207912i 0.0218524 − 0.207912i
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(73\) −1.95630 −1.95630 −0.978148 − 0.207912i \(-0.933333\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(74\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(75\) −0.564602 + 0.251377i −0.564602 + 0.251377i
\(76\) 0 0
\(77\) 0 0
\(78\) −1.47815 − 1.07394i −1.47815 − 1.07394i
\(79\) 0.809017 − 1.40126i 0.809017 − 1.40126i −0.104528 − 0.994522i \(-0.533333\pi\)
0.913545 − 0.406737i \(-0.133333\pi\)
\(80\) 0.618034 0.618034
\(81\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(82\) −1.61803 −1.61803
\(83\) 0.500000 − 0.866025i 0.500000 − 0.866025i −0.500000 − 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 1.66913 − 0.743145i 1.66913 − 0.743145i
\(88\) 0.978148 − 1.69420i 0.978148 − 1.69420i
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(91\) 0 0
\(92\) 0.104528 − 0.181049i 0.104528 − 0.181049i
\(93\) −0.0646021 + 0.614648i −0.0646021 + 0.614648i
\(94\) 0 0
\(95\) 0 0
\(96\) −0.104528 + 0.994522i −0.104528 + 0.994522i
\(97\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(98\) 1.00000 1.00000
\(99\) 1.91355 + 0.406737i 1.91355 + 0.406737i
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 2664.1.bw.c.1627.1 ✓ 8
8.3 odd 2 2664.1.bw.d.1627.1 yes 8
9.4 even 3 inner 2664.1.bw.c.2515.2 yes 8
37.36 even 2 2664.1.bw.d.1627.1 yes 8
72.67 odd 6 2664.1.bw.d.2515.2 yes 8
296.147 odd 2 CM 2664.1.bw.c.1627.1 ✓ 8
333.184 even 6 2664.1.bw.d.2515.2 yes 8
2664.2515 odd 6 inner 2664.1.bw.c.2515.2 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2664.1.bw.c.1627.1 ✓ 8 1.1 even 1 trivial
2664.1.bw.c.1627.1 ✓ 8 296.147 odd 2 CM
2664.1.bw.c.2515.2 yes 8 9.4 even 3 inner
2664.1.bw.c.2515.2 yes 8 2664.2515 odd 6 inner
2664.1.bw.d.1627.1 yes 8 8.3 odd 2
2664.1.bw.d.1627.1 yes 8 37.36 even 2
2664.1.bw.d.2515.2 yes 8 72.67 odd 6
2664.1.bw.d.2515.2 yes 8 333.184 even 6