Properties

Label 3775.1.d.e.301.1
Level $3775$
Weight $1$
Character 3775.301
Analytic conductor $1.884$
Analytic rank $0$
Dimension $4$
Projective image $A_{5}$
CM/RM no
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3775,1,Mod(301,3775)] 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("3775.301"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3775, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 3775 = 5^{2} \cdot 151 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3775.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-2,0,2,0,0,0,-4,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.88397042269\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(A_{5}\)
Projective field: Galois closure of 5.1.570025.1

Embedding invariants

Embedding label 301.1
Root \(1.61803i\) of defining polynomial
Character \(\chi\) \(=\) 3775.301
Dual form 3775.1.d.e.301.2

$q$-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)
 
\(f(q)\) \(=\) \(q-1.61803 q^{2} -1.00000i q^{3} +1.61803 q^{4} +1.61803i q^{6} -1.61803i q^{7} -1.00000 q^{8} -1.61803i q^{12} +0.618034i q^{13} +2.61803i q^{14} +1.00000 q^{17} +1.00000 q^{19} -1.61803 q^{21} +1.00000i q^{23} +1.00000i q^{24} -1.00000i q^{26} -1.00000i q^{27} -2.61803i q^{28} +1.61803 q^{31} +1.00000 q^{32} -1.61803 q^{34} +0.618034 q^{37} -1.61803 q^{38} +0.618034 q^{39} +0.618034i q^{41} +2.61803 q^{42} +1.00000 q^{43} -1.61803i q^{46} +0.618034 q^{47} -1.61803 q^{49} -1.00000i q^{51} +1.00000i q^{52} +0.618034i q^{53} +1.61803i q^{54} +1.61803i q^{56} -1.00000i q^{57} +1.00000 q^{59} -1.61803i q^{61} -2.61803 q^{62} -1.61803 q^{64} -1.00000i q^{67} +1.61803 q^{68} +1.00000 q^{69} +1.00000i q^{71} +0.618034i q^{73} -1.00000 q^{74} +1.61803 q^{76} -1.00000 q^{78} +0.618034i q^{79} -1.00000 q^{81} -1.00000i q^{82} +1.61803i q^{83} -2.61803 q^{84} -1.61803 q^{86} -1.61803i q^{89} +1.00000 q^{91} +1.61803i q^{92} -1.61803i q^{93} -1.00000 q^{94} -1.00000i q^{96} -1.61803 q^{97} +2.61803 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} + 2 q^{4} - 4 q^{8} + 4 q^{17} + 4 q^{19} - 2 q^{21} + 2 q^{31} + 4 q^{32} - 2 q^{34} - 2 q^{37} - 2 q^{38} - 2 q^{39} + 6 q^{42} + 4 q^{43} - 2 q^{47} - 2 q^{49} + 4 q^{59} - 6 q^{62} - 2 q^{64}+ \cdots + 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3775\mathbb{Z}\right)^\times\).

\(n\) \(152\) \(3026\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.61803 −1.61803 −0.809017 − 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(3\) − 1.00000i − 1.00000i −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 − 0.500000i \(-0.166667\pi\)
\(4\) 1.61803 1.61803
\(5\) 0 0
\(6\) 1.61803i 1.61803i
\(7\) − 1.61803i − 1.61803i −0.587785 − 0.809017i \(-0.700000\pi\)
0.587785 − 0.809017i \(-0.300000\pi\)
\(8\) −1.00000 −1.00000
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) − 1.61803i − 1.61803i
\(13\) 0.618034i 0.618034i 0.951057 + 0.309017i \(0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(14\) 2.61803i 2.61803i
\(15\) 0 0
\(16\) 0 0
\(17\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(18\) 0 0
\(19\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(20\) 0 0
\(21\) −1.61803 −1.61803
\(22\) 0 0
\(23\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(24\) 1.00000i 1.00000i
\(25\) 0 0
\(26\) − 1.00000i − 1.00000i
\(27\) − 1.00000i − 1.00000i
\(28\) − 2.61803i − 2.61803i
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 1.61803 1.61803 0.809017 − 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(32\) 1.00000 1.00000
\(33\) 0 0
\(34\) −1.61803 −1.61803
\(35\) 0 0
\(36\) 0 0
\(37\) 0.618034 0.618034 0.309017 − 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(38\) −1.61803 −1.61803
\(39\) 0.618034 0.618034
\(40\) 0 0
\(41\) 0.618034i 0.618034i 0.951057 + 0.309017i \(0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(42\) 2.61803 2.61803
\(43\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) − 1.61803i − 1.61803i
\(47\) 0.618034 0.618034 0.309017 − 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(48\) 0 0
\(49\) −1.61803 −1.61803
\(50\) 0 0
\(51\) − 1.00000i − 1.00000i
\(52\) 1.00000i 1.00000i
\(53\) 0.618034i 0.618034i 0.951057 + 0.309017i \(0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(54\) 1.61803i 1.61803i
\(55\) 0 0
\(56\) 1.61803i 1.61803i
\(57\) − 1.00000i − 1.00000i
\(58\) 0 0
\(59\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(60\) 0 0
\(61\) − 1.61803i − 1.61803i −0.587785 − 0.809017i \(-0.700000\pi\)
0.587785 − 0.809017i \(-0.300000\pi\)
\(62\) −2.61803 −2.61803
\(63\) 0 0
\(64\) −1.61803 −1.61803
\(65\) 0 0
\(66\) 0 0
\(67\) − 1.00000i − 1.00000i −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 − 0.500000i \(-0.166667\pi\)
\(68\) 1.61803 1.61803
\(69\) 1.00000 1.00000
\(70\) 0 0
\(71\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(72\) 0 0
\(73\) 0.618034i 0.618034i 0.951057 + 0.309017i \(0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(74\) −1.00000 −1.00000
\(75\) 0 0
\(76\) 1.61803 1.61803
\(77\) 0 0
\(78\) −1.00000 −1.00000
\(79\) 0.618034i 0.618034i 0.951057 + 0.309017i \(0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(80\) 0 0
\(81\) −1.00000 −1.00000
\(82\) − 1.00000i − 1.00000i
\(83\) 1.61803i 1.61803i 0.587785 + 0.809017i \(0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(84\) −2.61803 −2.61803
\(85\) 0 0
\(86\) −1.61803 −1.61803
\(87\) 0 0
\(88\) 0 0
\(89\) − 1.61803i − 1.61803i −0.587785 − 0.809017i \(-0.700000\pi\)
0.587785 − 0.809017i \(-0.300000\pi\)
\(90\) 0 0
\(91\) 1.00000 1.00000
\(92\) 1.61803i 1.61803i
\(93\) − 1.61803i − 1.61803i
\(94\) −1.00000 −1.00000
\(95\) 0 0
\(96\) − 1.00000i − 1.00000i
\(97\) −1.61803 −1.61803 −0.809017 − 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(98\) 2.61803 2.61803
\(99\) 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 3775.1.d.e.301.1 ✓ 4
5.2 odd 4 3775.1.c.a.3774.1 4
5.3 odd 4 3775.1.c.b.3774.4 4
5.4 even 2 3775.1.d.f.301.4 yes 4
151.150 odd 2 inner 3775.1.d.e.301.2 yes 4
755.452 even 4 3775.1.c.b.3774.1 4
755.603 even 4 3775.1.c.a.3774.4 4
755.754 odd 2 3775.1.d.f.301.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3775.1.c.a.3774.1 4 5.2 odd 4
3775.1.c.a.3774.4 4 755.603 even 4
3775.1.c.b.3774.1 4 755.452 even 4
3775.1.c.b.3774.4 4 5.3 odd 4
3775.1.d.e.301.1 ✓ 4 1.1 even 1 trivial
3775.1.d.e.301.2 yes 4 151.150 odd 2 inner
3775.1.d.f.301.3 yes 4 755.754 odd 2
3775.1.d.f.301.4 yes 4 5.4 even 2