↑

Properties

Label 2904.1.r.i.2339.2
Level $2904$
Weight $1$
Character 2904.2339
Analytic conductor $1.449$
Analytic rank $0$
Dimension $16$
Projective image $D_{4}$
RM discriminant 24
Inner twists $32$

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] = [2904,1,Mod(1667,2904)] 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("2904.1667"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2904, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([5, 5, 5, 9])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 2904 = 2^{3} \cdot 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2904.r (of order \(10\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,-4,4,0,0,0,0,-4,0,0,-16,0,0,0,-4,0] 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: \(1.44928479669\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{40})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{12} + x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of \(\Q(\sqrt{-22 +4 \sqrt{22}})\)

Embedding invariants

Embedding label 2339.2
Root \(-0.453990 - 0.891007i\) of defining polynomial
Character \(\chi\) \(=\) 2904.2339
Dual form 2904.1.r.i.1691.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.587785 + 0.809017i) q^{2} +(0.309017 - 0.951057i) q^{3} +(-0.309017 - 0.951057i) q^{4} +(1.14412 - 0.831254i) q^{5} +(0.587785 + 0.809017i) q^{6} +(0.951057 + 0.309017i) q^{8} +(-0.809017 - 0.587785i) q^{9} +1.41421i q^{10} -1.00000 q^{12} +(-0.437016 - 1.34500i) q^{15} +(-0.809017 + 0.587785i) q^{16} +(0.951057 - 0.309017i) q^{18} +(-1.34500 - 0.437016i) q^{19} +(-1.14412 - 0.831254i) q^{20} +1.41421 q^{23} +(0.587785 - 0.809017i) q^{24} +(0.309017 - 0.951057i) q^{25} +(-0.809017 + 0.587785i) q^{27} +(1.90211 - 0.618034i) q^{29} +(1.34500 + 0.437016i) q^{30} -1.00000i q^{32} +(-0.309017 + 0.951057i) q^{36} +(1.14412 - 0.831254i) q^{38} +(1.34500 - 0.437016i) q^{40} -1.41421i q^{43} -1.41421 q^{45} +(-0.831254 + 1.14412i) q^{46} +(-0.437016 + 1.34500i) q^{47} +(0.309017 + 0.951057i) q^{48} +(0.809017 - 0.587785i) q^{49} +(0.587785 + 0.809017i) q^{50} +(-1.14412 - 0.831254i) q^{53} -1.00000i q^{54} +(-0.831254 + 1.14412i) q^{57} +(-0.618034 + 1.90211i) q^{58} +(-1.14412 + 0.831254i) q^{60} +(0.809017 + 0.587785i) q^{64} +(0.437016 - 1.34500i) q^{69} +(-1.14412 + 0.831254i) q^{71} +(-0.587785 - 0.809017i) q^{72} +(-1.34500 + 0.437016i) q^{73} +(-0.809017 - 0.587785i) q^{75} +1.41421i q^{76} +(-0.437016 + 1.34500i) q^{80} +(0.309017 + 0.951057i) q^{81} +(1.14412 + 0.831254i) q^{86} -2.00000i q^{87} +(0.831254 - 1.14412i) q^{90} +(-0.437016 - 1.34500i) q^{92} +(-0.831254 - 1.14412i) q^{94} +(-1.90211 + 0.618034i) q^{95} +(-0.951057 - 0.309017i) q^{96} +1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{3} + 4 q^{4} - 4 q^{9} - 16 q^{12} - 4 q^{16} - 4 q^{25} - 4 q^{27} + 4 q^{36} - 4 q^{48} + 4 q^{49} + 8 q^{58} + 4 q^{64} - 4 q^{75} - 4 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2904\mathbb{Z}\right)^\times\).

\(n\) \(727\) \(1453\) \(1937\) \(2785\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(e\left(\frac{7}{10}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.587785 + 0.809017i −0.587785 + 0.809017i
\(3\) 0.309017 − 0.951057i 0.309017 − 0.951057i
\(4\) −0.309017 − 0.951057i −0.309017 − 0.951057i
\(5\) 1.14412 − 0.831254i 1.14412 − 0.831254i 0.156434 − 0.987688i \(-0.450000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(6\) 0.587785 + 0.809017i 0.587785 + 0.809017i
\(7\) 0 0 0.951057 − 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(8\) 0.951057 + 0.309017i 0.951057 + 0.309017i
\(9\) −0.809017 − 0.587785i −0.809017 − 0.587785i
\(10\) 1.41421i 1.41421i
\(11\) 0 0
\(12\) −1.00000 −1.00000
\(13\) 0 0 0.587785 − 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(14\) 0 0
\(15\) −0.437016 − 1.34500i −0.437016 − 1.34500i
\(16\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(17\) 0 0 −0.587785 − 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(18\) 0.951057 − 0.309017i 0.951057 − 0.309017i
\(19\) −1.34500 − 0.437016i −1.34500 − 0.437016i −0.453990 − 0.891007i \(-0.650000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(20\) −1.14412 − 0.831254i −1.14412 − 0.831254i
\(21\) 0 0
\(22\) 0 0
\(23\) 1.41421 1.41421 0.707107 − 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(24\) 0.587785 − 0.809017i 0.587785 − 0.809017i
\(25\) 0.309017 − 0.951057i 0.309017 − 0.951057i
\(26\) 0 0
\(27\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(28\) 0 0
\(29\) 1.90211 − 0.618034i 1.90211 − 0.618034i 0.951057 − 0.309017i \(-0.100000\pi\)
0.951057 − 0.309017i \(-0.100000\pi\)
\(30\) 1.34500 + 0.437016i 1.34500 + 0.437016i
\(31\) 0 0 −0.809017 − 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(32\) − 1.00000i − 1.00000i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(37\) 0 0 −0.309017 − 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(38\) 1.14412 − 0.831254i 1.14412 − 0.831254i
\(39\) 0 0
\(40\) 1.34500 − 0.437016i 1.34500 − 0.437016i
\(41\) 0 0 −0.951057 − 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(42\) 0 0
\(43\) − 1.41421i − 1.41421i −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 − 0.707107i \(-0.250000\pi\)
\(44\) 0 0
\(45\) −1.41421 −1.41421
\(46\) −0.831254 + 1.14412i −0.831254 + 1.14412i
\(47\) −0.437016 + 1.34500i −0.437016 + 1.34500i 0.453990 + 0.891007i \(0.350000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(48\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(49\) 0.809017 − 0.587785i 0.809017 − 0.587785i
\(50\) 0.587785 + 0.809017i 0.587785 + 0.809017i
\(51\) 0 0
\(52\) 0 0
\(53\) −1.14412 − 0.831254i −1.14412 − 0.831254i −0.156434 − 0.987688i \(-0.550000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(54\) − 1.00000i − 1.00000i
\(55\) 0 0
\(56\) 0 0
\(57\) −0.831254 + 1.14412i −0.831254 + 1.14412i
\(58\) −0.618034 + 1.90211i −0.618034 + 1.90211i
\(59\) 0 0 −0.309017 − 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(60\) −1.14412 + 0.831254i −1.14412 + 0.831254i
\(61\) 0 0 −0.587785 − 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 0.437016 − 1.34500i 0.437016 − 1.34500i
\(70\) 0 0
\(71\) −1.14412 + 0.831254i −1.14412 + 0.831254i −0.987688 − 0.156434i \(-0.950000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(72\) −0.587785 − 0.809017i −0.587785 − 0.809017i
\(73\) −1.34500 + 0.437016i −1.34500 + 0.437016i −0.891007 − 0.453990i \(-0.850000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(74\) 0 0
\(75\) −0.809017 − 0.587785i −0.809017 − 0.587785i
\(76\) 1.41421i 1.41421i
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.587785 − 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(80\) −0.437016 + 1.34500i −0.437016 + 1.34500i
\(81\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(82\) 0 0
\(83\) 0 0 −0.587785 − 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 1.14412 + 0.831254i 1.14412 + 0.831254i
\(87\) − 2.00000i − 2.00000i
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0.831254 − 1.14412i 0.831254 − 1.14412i
\(91\) 0 0
\(92\) −0.437016 − 1.34500i −0.437016 − 1.34500i
\(93\) 0 0
\(94\) −0.831254 − 1.14412i −0.831254 − 1.14412i
\(95\) −1.90211 + 0.618034i −1.90211 + 0.618034i
\(96\) −0.951057 − 0.309017i −0.951057 − 0.309017i
\(97\) 0 0 0.587785 − 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(98\) 1.00000i 1.00000i
\(99\) 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 2904.1.r.i.2339.2 16
3.2 odd 2 inner 2904.1.r.i.2339.3 16
8.3 odd 2 inner 2904.1.r.i.2339.3 16
11.2 odd 10 inner 2904.1.r.i.1667.1 16
11.3 even 5 inner 2904.1.r.i.1691.4 16
11.4 even 5 inner 2904.1.r.i.2411.1 16
11.5 even 5 2904.1.p.b.1451.1 ✓ 4
11.6 odd 10 2904.1.p.b.1451.3 yes 4
11.7 odd 10 inner 2904.1.r.i.2411.3 16
11.8 odd 10 inner 2904.1.r.i.1691.2 16
11.9 even 5 inner 2904.1.r.i.1667.3 16
11.10 odd 2 inner 2904.1.r.i.2339.4 16
24.11 even 2 RM 2904.1.r.i.2339.2 16
33.2 even 10 inner 2904.1.r.i.1667.4 16
33.5 odd 10 2904.1.p.b.1451.4 yes 4
33.8 even 10 inner 2904.1.r.i.1691.3 16
33.14 odd 10 inner 2904.1.r.i.1691.1 16
33.17 even 10 2904.1.p.b.1451.2 yes 4
33.20 odd 10 inner 2904.1.r.i.1667.2 16
33.26 odd 10 inner 2904.1.r.i.2411.4 16
33.29 even 10 inner 2904.1.r.i.2411.2 16
33.32 even 2 inner 2904.1.r.i.2339.1 16
88.3 odd 10 inner 2904.1.r.i.1691.1 16
88.19 even 10 inner 2904.1.r.i.1691.3 16
88.27 odd 10 2904.1.p.b.1451.4 yes 4
88.35 even 10 inner 2904.1.r.i.1667.4 16
88.43 even 2 inner 2904.1.r.i.2339.1 16
88.51 even 10 inner 2904.1.r.i.2411.2 16
88.59 odd 10 inner 2904.1.r.i.2411.4 16
88.75 odd 10 inner 2904.1.r.i.1667.2 16
88.83 even 10 2904.1.p.b.1451.2 yes 4
264.35 odd 10 inner 2904.1.r.i.1667.1 16
264.59 even 10 inner 2904.1.r.i.2411.1 16
264.83 odd 10 2904.1.p.b.1451.3 yes 4
264.107 odd 10 inner 2904.1.r.i.1691.2 16
264.131 odd 2 inner 2904.1.r.i.2339.4 16
264.179 even 10 inner 2904.1.r.i.1691.4 16
264.203 even 10 2904.1.p.b.1451.1 ✓ 4
264.227 odd 10 inner 2904.1.r.i.2411.3 16
264.251 even 10 inner 2904.1.r.i.1667.3 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2904.1.p.b.1451.1 ✓ 4 11.5 even 5
2904.1.p.b.1451.1 ✓ 4 264.203 even 10
2904.1.p.b.1451.2 yes 4 33.17 even 10
2904.1.p.b.1451.2 yes 4 88.83 even 10
2904.1.p.b.1451.3 yes 4 11.6 odd 10
2904.1.p.b.1451.3 yes 4 264.83 odd 10
2904.1.p.b.1451.4 yes 4 33.5 odd 10
2904.1.p.b.1451.4 yes 4 88.27 odd 10
2904.1.r.i.1667.1 16 11.2 odd 10 inner
2904.1.r.i.1667.1 16 264.35 odd 10 inner
2904.1.r.i.1667.2 16 33.20 odd 10 inner
2904.1.r.i.1667.2 16 88.75 odd 10 inner
2904.1.r.i.1667.3 16 11.9 even 5 inner
2904.1.r.i.1667.3 16 264.251 even 10 inner
2904.1.r.i.1667.4 16 33.2 even 10 inner
2904.1.r.i.1667.4 16 88.35 even 10 inner
2904.1.r.i.1691.1 16 33.14 odd 10 inner
2904.1.r.i.1691.1 16 88.3 odd 10 inner
2904.1.r.i.1691.2 16 11.8 odd 10 inner
2904.1.r.i.1691.2 16 264.107 odd 10 inner
2904.1.r.i.1691.3 16 33.8 even 10 inner
2904.1.r.i.1691.3 16 88.19 even 10 inner
2904.1.r.i.1691.4 16 11.3 even 5 inner
2904.1.r.i.1691.4 16 264.179 even 10 inner
2904.1.r.i.2339.1 16 33.32 even 2 inner
2904.1.r.i.2339.1 16 88.43 even 2 inner
2904.1.r.i.2339.2 16 1.1 even 1 trivial
2904.1.r.i.2339.2 16 24.11 even 2 RM
2904.1.r.i.2339.3 16 3.2 odd 2 inner
2904.1.r.i.2339.3 16 8.3 odd 2 inner
2904.1.r.i.2339.4 16 11.10 odd 2 inner
2904.1.r.i.2339.4 16 264.131 odd 2 inner
2904.1.r.i.2411.1 16 11.4 even 5 inner
2904.1.r.i.2411.1 16 264.59 even 10 inner
2904.1.r.i.2411.2 16 33.29 even 10 inner
2904.1.r.i.2411.2 16 88.51 even 10 inner
2904.1.r.i.2411.3 16 11.7 odd 10 inner
2904.1.r.i.2411.3 16 264.227 odd 10 inner
2904.1.r.i.2411.4 16 33.26 odd 10 inner
2904.1.r.i.2411.4 16 88.59 odd 10 inner