Properties

Label 799.1.h.b.563.4
Level $799$
Weight $1$
Character 799.563
Analytic conductor $0.399$
Analytic rank $0$
Dimension $16$
Projective image $D_{40}$
CM discriminant -47
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] = [799,1,Mod(93,799)] 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("799.93"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(799, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([5, 4])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 799 = 17 \cdot 47 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 799.h (of order \(8\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.398752945094\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{8})\)
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: \(\Z[a_1, \ldots, a_{17}]\)
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 563.4
Root \(0.453990 - 0.891007i\) of defining polynomial
Character \(\chi\) \(=\) 799.563
Dual form 799.1.h.b.281.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.34500 + 1.34500i) q^{2} +(-0.965451 + 0.399903i) q^{3} +2.61803i q^{4} +(-1.83640 - 0.760661i) q^{6} +(0.0600500 - 0.144974i) q^{7} +(-2.17625 + 2.17625i) q^{8} +(0.0650673 - 0.0650673i) q^{9} +(-1.04696 - 2.52758i) q^{12} +(0.275756 - 0.114222i) q^{14} -3.23607 q^{16} +(0.809017 - 0.587785i) q^{17} +0.175031 q^{18} +0.163979i q^{21} +(1.23078 - 2.97135i) q^{24} +(-0.707107 + 0.707107i) q^{25} +(0.363104 - 0.876612i) q^{27} +(0.379546 + 0.157213i) q^{28} +(-2.17625 - 2.17625i) q^{32} +(1.87869 + 0.297556i) q^{34} +(0.170348 + 0.170348i) q^{36} +(1.57547 - 0.652583i) q^{37} +(-0.220551 + 0.220551i) q^{42} +1.00000i q^{47} +(3.12427 - 1.29411i) q^{48} +(0.689695 + 0.689695i) q^{49} -1.90211 q^{50} +(-0.546010 + 0.891007i) q^{51} +(0.642040 + 0.642040i) q^{53} +(1.66741 - 0.690666i) q^{54} +(0.184815 + 0.446183i) q^{56} +(0.437016 - 0.437016i) q^{59} +(0.581990 - 1.40505i) q^{61} +(-0.00552574 - 0.0133403i) q^{63} -2.61803i q^{64} +(1.53884 + 2.11803i) q^{68} +(-1.79671 + 0.744220i) q^{71} +0.283205i q^{72} +(2.99673 + 1.24129i) q^{74} +(0.399903 - 0.965451i) q^{75} +(-1.79671 - 0.744220i) q^{79} +1.08355i q^{81} +(-1.00000 - 1.00000i) q^{83} -0.429303 q^{84} -1.90211i q^{89} +(-1.34500 + 1.34500i) q^{94} +(2.97135 + 1.23078i) q^{96} +(0.497066 + 1.20002i) q^{97} +1.85528i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{3} - 4 q^{9} - 16 q^{16} + 4 q^{17} - 20 q^{24} - 4 q^{27} - 4 q^{28} + 4 q^{36} - 20 q^{42} + 24 q^{48} + 4 q^{49} - 16 q^{51} + 4 q^{53} + 20 q^{54} + 20 q^{56} + 4 q^{61} - 4 q^{63} - 4 q^{71}+ \cdots + 32 q^{84}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/799\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(377\)
\(\chi(n)\) \(-1\) \(e\left(\frac{7}{8}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.34500 + 1.34500i 1.34500 + 1.34500i 0.891007 + 0.453990i \(0.150000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(3\) −0.965451 + 0.399903i −0.965451 + 0.399903i −0.809017 − 0.587785i \(-0.800000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(4\) 2.61803i 2.61803i
\(5\) 0 0 −0.382683 − 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(6\) −1.83640 − 0.760661i −1.83640 − 0.760661i
\(7\) 0.0600500 − 0.144974i 0.0600500 − 0.144974i −0.891007 − 0.453990i \(-0.850000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(8\) −2.17625 + 2.17625i −2.17625 + 2.17625i
\(9\) 0.0650673 − 0.0650673i 0.0650673 − 0.0650673i
\(10\) 0 0
\(11\) 0 0 −0.923880 − 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(12\) −1.04696 − 2.52758i −1.04696 − 2.52758i
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0.275756 − 0.114222i 0.275756 − 0.114222i
\(15\) 0 0
\(16\) −3.23607 −3.23607
\(17\) 0.809017 − 0.587785i 0.809017 − 0.587785i
\(18\) 0.175031 0.175031
\(19\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(20\) 0 0
\(21\) 0.163979i 0.163979i
\(22\) 0 0
\(23\) 0 0 −0.923880 − 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(24\) 1.23078 − 2.97135i 1.23078 − 2.97135i
\(25\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(26\) 0 0
\(27\) 0.363104 − 0.876612i 0.363104 − 0.876612i
\(28\) 0.379546 + 0.157213i 0.379546 + 0.157213i
\(29\) 0 0 −0.382683 − 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(30\) 0 0
\(31\) 0 0 0.923880 − 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(32\) −2.17625 − 2.17625i −2.17625 − 2.17625i
\(33\) 0 0
\(34\) 1.87869 + 0.297556i 1.87869 + 0.297556i
\(35\) 0 0
\(36\) 0.170348 + 0.170348i 0.170348 + 0.170348i
\(37\) 1.57547 − 0.652583i 1.57547 − 0.652583i 0.587785 − 0.809017i \(-0.300000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 0.382683 − 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(42\) −0.220551 + 0.220551i −0.220551 + 0.220551i
\(43\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.00000i 1.00000i
\(48\) 3.12427 − 1.29411i 3.12427 − 1.29411i
\(49\) 0.689695 + 0.689695i 0.689695 + 0.689695i
\(50\) −1.90211 −1.90211
\(51\) −0.546010 + 0.891007i −0.546010 + 0.891007i
\(52\) 0 0
\(53\) 0.642040 + 0.642040i 0.642040 + 0.642040i 0.951057 − 0.309017i \(-0.100000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(54\) 1.66741 − 0.690666i 1.66741 − 0.690666i
\(55\) 0 0
\(56\) 0.184815 + 0.446183i 0.184815 + 0.446183i
\(57\) 0 0
\(58\) 0 0
\(59\) 0.437016 − 0.437016i 0.437016 − 0.437016i −0.453990 − 0.891007i \(-0.650000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(60\) 0 0
\(61\) 0.581990 − 1.40505i 0.581990 − 1.40505i −0.309017 − 0.951057i \(-0.600000\pi\)
0.891007 − 0.453990i \(-0.150000\pi\)
\(62\) 0 0
\(63\) −0.00552574 − 0.0133403i −0.00552574 − 0.0133403i
\(64\) − 2.61803i − 2.61803i
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 1.53884 + 2.11803i 1.53884 + 2.11803i
\(69\) 0 0
\(70\) 0 0
\(71\) −1.79671 + 0.744220i −1.79671 + 0.744220i −0.809017 + 0.587785i \(0.800000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(72\) 0.283205i 0.283205i
\(73\) 0 0 −0.382683 − 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(74\) 2.99673 + 1.24129i 2.99673 + 1.24129i
\(75\) 0.399903 − 0.965451i 0.399903 − 0.965451i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −1.79671 − 0.744220i −1.79671 − 0.744220i −0.987688 − 0.156434i \(-0.950000\pi\)
−0.809017 − 0.587785i \(-0.800000\pi\)
\(80\) 0 0
\(81\) 1.08355i 1.08355i
\(82\) 0 0
\(83\) −1.00000 − 1.00000i −1.00000 − 1.00000i − 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(84\) −0.429303 −0.429303
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 1.90211i − 1.90211i −0.309017 − 0.951057i \(-0.600000\pi\)
0.309017 − 0.951057i \(-0.400000\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) −1.34500 + 1.34500i −1.34500 + 1.34500i
\(95\) 0 0
\(96\) 2.97135 + 1.23078i 2.97135 + 1.23078i
\(97\) 0.497066 + 1.20002i 0.497066 + 1.20002i 0.951057 + 0.309017i \(0.100000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(98\) 1.85528i 1.85528i
\(99\) 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 799.1.h.b.563.4 yes 16
17.9 even 8 inner 799.1.h.b.281.4 ✓ 16
47.46 odd 2 CM 799.1.h.b.563.4 yes 16
799.281 odd 8 inner 799.1.h.b.281.4 ✓ 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
799.1.h.b.281.4 ✓ 16 17.9 even 8 inner
799.1.h.b.281.4 ✓ 16 799.281 odd 8 inner
799.1.h.b.563.4 yes 16 1.1 even 1 trivial
799.1.h.b.563.4 yes 16 47.46 odd 2 CM