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Properties

Label 1700.1.cf.a.1379.1
Level $1700$
Weight $1$
Character 1700.1379
Analytic conductor $0.848$
Analytic rank $0$
Dimension $16$
Projective image $D_{40}$
CM discriminant -4
Inner twists $4$

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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] = [1700,1,Mod(19,1700)] 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("1700.19"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1700, base_ring=CyclotomicField(40)) chi = DirichletCharacter(H, H._module([20, 36, 35])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 1700 = 2^{2} \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1700.cf (of order \(40\), degree \(16\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,0,0,0,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.848410521476\)
Analytic rank: \(0\)
Dimension: \(16\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{40}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{40} + \cdots)\)

Embedding invariants

Embedding label 1379.1
Root \(-0.156434 + 0.987688i\) of defining polynomial
Character \(\chi\) \(=\) 1700.1379
Dual form 1700.1.cf.a.519.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.453990 - 0.891007i) q^{2} +(-0.587785 + 0.809017i) q^{4} +(-0.987688 - 0.156434i) q^{5} +(0.987688 + 0.156434i) q^{8} +(-0.891007 - 0.453990i) q^{9} +(0.309017 + 0.951057i) q^{10} +(0.0966818 + 0.297556i) q^{13} +(-0.309017 - 0.951057i) q^{16} +(-0.587785 + 0.809017i) q^{17} +1.00000i q^{18} +(0.707107 - 0.707107i) q^{20} +(0.951057 + 0.309017i) q^{25} +(0.221232 - 0.221232i) q^{26} +(-0.465451 + 1.93874i) q^{29} +(-0.707107 + 0.707107i) q^{32} +(0.987688 + 0.156434i) q^{34} +(0.891007 - 0.453990i) q^{36} +(-0.987688 + 0.843566i) q^{37} +(-0.951057 - 0.309017i) q^{40} +(0.465451 + 0.0366318i) q^{41} +(0.809017 + 0.587785i) q^{45} +(0.707107 + 0.707107i) q^{49} +(-0.156434 - 0.987688i) q^{50} +(-0.297556 - 0.0966818i) q^{52} +(-1.95106 + 0.309017i) q^{53} +(1.93874 - 0.465451i) q^{58} +(1.47879 + 1.26301i) q^{61} +(0.951057 + 0.309017i) q^{64} +(-0.0489435 - 0.309017i) q^{65} +(-0.309017 - 0.951057i) q^{68} +(-0.809017 - 0.587785i) q^{72} +(0.0819895 + 1.04178i) q^{73} +(1.20002 + 0.497066i) q^{74} +(0.156434 + 0.987688i) q^{80} +(0.587785 + 0.809017i) q^{81} +(-0.178671 - 0.431351i) q^{82} +(0.707107 - 0.707107i) q^{85} +(-1.69480 - 0.550672i) q^{89} +(0.156434 - 0.987688i) q^{90} +(-1.47879 - 0.355026i) q^{97} +(0.309017 - 0.951057i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{10} + 4 q^{16} + 4 q^{26} + 4 q^{29} - 4 q^{41} + 4 q^{45} - 16 q^{53} - 16 q^{65} + 4 q^{68} - 4 q^{72} - 4 q^{73} - 4 q^{74} + 4 q^{82} - 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1700\mathbb{Z}\right)^\times\).

\(n\) \(477\) \(851\) \(1601\)
\(\chi(n)\) \(e\left(\frac{1}{10}\right)\) \(-1\) \(e\left(\frac{7}{8}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.453990 − 0.891007i −0.453990 − 0.891007i
\(3\) 0 0 0.233445 − 0.972370i \(-0.425000\pi\)
−0.233445 + 0.972370i \(0.575000\pi\)
\(4\) −0.587785 + 0.809017i −0.587785 + 0.809017i
\(5\) −0.987688 − 0.156434i −0.987688 − 0.156434i
\(6\) 0 0
\(7\) 0 0 −0.923880 − 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(8\) 0.987688 + 0.156434i 0.987688 + 0.156434i
\(9\) −0.891007 − 0.453990i −0.891007 − 0.453990i
\(10\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(11\) 0 0 0.649448 − 0.760406i \(-0.275000\pi\)
−0.649448 + 0.760406i \(0.725000\pi\)
\(12\) 0 0
\(13\) 0.0966818 + 0.297556i 0.0966818 + 0.297556i 0.987688 − 0.156434i \(-0.0500000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.309017 − 0.951057i −0.309017 − 0.951057i
\(17\) −0.587785 + 0.809017i −0.587785 + 0.809017i
\(18\) 1.00000i 1.00000i
\(19\) 0 0 −0.987688 − 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(20\) 0.707107 − 0.707107i 0.707107 − 0.707107i
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.760406 − 0.649448i \(-0.775000\pi\)
0.760406 + 0.649448i \(0.225000\pi\)
\(24\) 0 0
\(25\) 0.951057 + 0.309017i 0.951057 + 0.309017i
\(26\) 0.221232 − 0.221232i 0.221232 − 0.221232i
\(27\) 0 0
\(28\) 0 0
\(29\) −0.465451 + 1.93874i −0.465451 + 1.93874i −0.156434 + 0.987688i \(0.550000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(30\) 0 0
\(31\) 0 0 0.522499 − 0.852640i \(-0.325000\pi\)
−0.522499 + 0.852640i \(0.675000\pi\)
\(32\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(33\) 0 0
\(34\) 0.987688 + 0.156434i 0.987688 + 0.156434i
\(35\) 0 0
\(36\) 0.891007 − 0.453990i 0.891007 − 0.453990i
\(37\) −0.987688 + 0.843566i −0.987688 + 0.843566i −0.987688 − 0.156434i \(-0.950000\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −0.951057 − 0.309017i −0.951057 − 0.309017i
\(41\) 0.465451 + 0.0366318i 0.465451 + 0.0366318i 0.309017 − 0.951057i \(-0.400000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(42\) 0 0
\(43\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(44\) 0 0
\(45\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(46\) 0 0
\(47\) 0 0 −0.809017 − 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(48\) 0 0
\(49\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(50\) −0.156434 − 0.987688i −0.156434 − 0.987688i
\(51\) 0 0
\(52\) −0.297556 − 0.0966818i −0.297556 − 0.0966818i
\(53\) −1.95106 + 0.309017i −1.95106 + 0.309017i −0.951057 + 0.309017i \(0.900000\pi\)
−1.00000 \(1.00000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 1.93874 − 0.465451i 1.93874 − 0.465451i
\(59\) 0 0 −0.891007 − 0.453990i \(-0.850000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(60\) 0 0
\(61\) 1.47879 + 1.26301i 1.47879 + 1.26301i 0.891007 + 0.453990i \(0.150000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0.951057 + 0.309017i 0.951057 + 0.309017i
\(65\) −0.0489435 − 0.309017i −0.0489435 − 0.309017i
\(66\) 0 0
\(67\) 0 0 −0.587785 − 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(68\) −0.309017 − 0.951057i −0.309017 − 0.951057i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.972370 − 0.233445i \(-0.925000\pi\)
0.972370 + 0.233445i \(0.0750000\pi\)
\(72\) −0.809017 − 0.587785i −0.809017 − 0.587785i
\(73\) 0.0819895 + 1.04178i 0.0819895 + 1.04178i 0.891007 + 0.453990i \(0.150000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(74\) 1.20002 + 0.497066i 1.20002 + 0.497066i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.522499 − 0.852640i \(-0.675000\pi\)
0.522499 + 0.852640i \(0.325000\pi\)
\(80\) 0.156434 + 0.987688i 0.156434 + 0.987688i
\(81\) 0.587785 + 0.809017i 0.587785 + 0.809017i
\(82\) −0.178671 − 0.431351i −0.178671 − 0.431351i
\(83\) 0 0 0.156434 − 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(84\) 0 0
\(85\) 0.707107 − 0.707107i 0.707107 − 0.707107i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1.69480 − 0.550672i −1.69480 − 0.550672i −0.707107 − 0.707107i \(-0.750000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(90\) 0.156434 − 0.987688i 0.156434 − 0.987688i
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.47879 − 0.355026i −1.47879 − 0.355026i −0.587785 − 0.809017i \(-0.700000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(98\) 0.309017 − 0.951057i 0.309017 − 0.951057i
\(99\) 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 1700.1.cf.a.1379.1 yes 16
4.3 odd 2 CM 1700.1.cf.a.1379.1 yes 16
17.9 even 8 1700.1.cf.b.179.1 yes 16
25.19 even 10 1700.1.cf.b.19.1 yes 16
68.43 odd 8 1700.1.cf.b.179.1 yes 16
100.19 odd 10 1700.1.cf.b.19.1 yes 16
425.94 even 40 inner 1700.1.cf.a.519.1 ✓ 16
1700.519 odd 40 inner 1700.1.cf.a.519.1 ✓ 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1700.1.cf.a.519.1 ✓ 16 425.94 even 40 inner
1700.1.cf.a.519.1 ✓ 16 1700.519 odd 40 inner
1700.1.cf.a.1379.1 yes 16 1.1 even 1 trivial
1700.1.cf.a.1379.1 yes 16 4.3 odd 2 CM
1700.1.cf.b.19.1 yes 16 25.19 even 10
1700.1.cf.b.19.1 yes 16 100.19 odd 10
1700.1.cf.b.179.1 yes 16 17.9 even 8
1700.1.cf.b.179.1 yes 16 68.43 odd 8