Properties

Label 3775.1.d.e.301.3
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 301.3
Root \(-0.618034i\) of defining polynomial
Character \(\chi\) \(=\) 3775.301
Dual form 3775.1.d.e.301.4

$q$-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)
 
\(f(q)\) \(=\) \(q+0.618034 q^{2} -1.00000i q^{3} -0.618034 q^{4} -0.618034i q^{6} +0.618034i q^{7} -1.00000 q^{8} +0.618034i q^{12} -1.61803i q^{13} +0.381966i q^{14} +1.00000 q^{17} +1.00000 q^{19} +0.618034 q^{21} +1.00000i q^{23} +1.00000i q^{24} -1.00000i q^{26} -1.00000i q^{27} -0.381966i q^{28} -0.618034 q^{31} +1.00000 q^{32} +0.618034 q^{34} -1.61803 q^{37} +0.618034 q^{38} -1.61803 q^{39} -1.61803i q^{41} +0.381966 q^{42} +1.00000 q^{43} +0.618034i q^{46} -1.61803 q^{47} +0.618034 q^{49} -1.00000i q^{51} +1.00000i q^{52} -1.61803i q^{53} -0.618034i q^{54} -0.618034i q^{56} -1.00000i q^{57} +1.00000 q^{59} +0.618034i q^{61} -0.381966 q^{62} +0.618034 q^{64} -1.00000i q^{67} -0.618034 q^{68} +1.00000 q^{69} +1.00000i q^{71} -1.61803i q^{73} -1.00000 q^{74} -0.618034 q^{76} -1.00000 q^{78} -1.61803i q^{79} -1.00000 q^{81} -1.00000i q^{82} -0.618034i q^{83} -0.381966 q^{84} +0.618034 q^{86} +0.618034i q^{89} +1.00000 q^{91} -0.618034i q^{92} +0.618034i q^{93} -1.00000 q^{94} -1.00000i q^{96} +0.618034 q^{97} +0.381966 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} + 2 q^{4} - 4 q^{8} + 4 q^{17} + 4 q^{19} - 2 q^{21} + 2 q^{31} + 4 q^{32} - 2 q^{34} - 2 q^{37} - 2 q^{38} - 2 q^{39} + 6 q^{42} + 4 q^{43} - 2 q^{47} - 2 q^{49} + 4 q^{59} - 6 q^{62} - 2 q^{64}+ \cdots + 6 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.618034 0.618034 0.309017 − 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(3\) − 1.00000i − 1.00000i −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 − 0.500000i \(-0.166667\pi\)
\(4\) −0.618034 −0.618034
\(5\) 0 0
\(6\) − 0.618034i − 0.618034i
\(7\) 0.618034i 0.618034i 0.951057 + 0.309017i \(0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(8\) −1.00000 −1.00000
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0.618034i 0.618034i
\(13\) − 1.61803i − 1.61803i −0.587785 − 0.809017i \(-0.700000\pi\)
0.587785 − 0.809017i \(-0.300000\pi\)
\(14\) 0.381966i 0.381966i
\(15\) 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
\(21\) 0.618034 0.618034
\(22\) 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
\(28\) − 0.381966i − 0.381966i
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −0.618034 −0.618034 −0.309017 − 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(32\) 1.00000 1.00000
\(33\) 0 0
\(34\) 0.618034 0.618034
\(35\) 0 0
\(36\) 0 0
\(37\) −1.61803 −1.61803 −0.809017 − 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(38\) 0.618034 0.618034
\(39\) −1.61803 −1.61803
\(40\) 0 0
\(41\) − 1.61803i − 1.61803i −0.587785 − 0.809017i \(-0.700000\pi\)
0.587785 − 0.809017i \(-0.300000\pi\)
\(42\) 0.381966 0.381966
\(43\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0.618034i 0.618034i
\(47\) −1.61803 −1.61803 −0.809017 − 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(48\) 0 0
\(49\) 0.618034 0.618034
\(50\) 0 0
\(51\) − 1.00000i − 1.00000i
\(52\) 1.00000i 1.00000i
\(53\) − 1.61803i − 1.61803i −0.587785 − 0.809017i \(-0.700000\pi\)
0.587785 − 0.809017i \(-0.300000\pi\)
\(54\) − 0.618034i − 0.618034i
\(55\) 0 0
\(56\) − 0.618034i − 0.618034i
\(57\) − 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
\(61\) 0.618034i 0.618034i 0.951057 + 0.309017i \(0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(62\) −0.381966 −0.381966
\(63\) 0 0
\(64\) 0.618034 0.618034
\(65\) 0 0
\(66\) 0 0
\(67\) − 1.00000i − 1.00000i −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 − 0.500000i \(-0.166667\pi\)
\(68\) −0.618034 −0.618034
\(69\) 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
\(73\) − 1.61803i − 1.61803i −0.587785 − 0.809017i \(-0.700000\pi\)
0.587785 − 0.809017i \(-0.300000\pi\)
\(74\) −1.00000 −1.00000
\(75\) 0 0
\(76\) −0.618034 −0.618034
\(77\) 0 0
\(78\) −1.00000 −1.00000
\(79\) − 1.61803i − 1.61803i −0.587785 − 0.809017i \(-0.700000\pi\)
0.587785 − 0.809017i \(-0.300000\pi\)
\(80\) 0 0
\(81\) −1.00000 −1.00000
\(82\) − 1.00000i − 1.00000i
\(83\) − 0.618034i − 0.618034i −0.951057 − 0.309017i \(-0.900000\pi\)
0.951057 − 0.309017i \(-0.100000\pi\)
\(84\) −0.381966 −0.381966
\(85\) 0 0
\(86\) 0.618034 0.618034
\(87\) 0 0
\(88\) 0 0
\(89\) 0.618034i 0.618034i 0.951057 + 0.309017i \(0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(90\) 0 0
\(91\) 1.00000 1.00000
\(92\) − 0.618034i − 0.618034i
\(93\) 0.618034i 0.618034i
\(94\) −1.00000 −1.00000
\(95\) 0 0
\(96\) − 1.00000i − 1.00000i
\(97\) 0.618034 0.618034 0.309017 − 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(98\) 0.381966 0.381966
\(99\) 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.301.3 ✓ 4
5.2 odd 4 3775.1.c.a.3774.3 4
5.3 odd 4 3775.1.c.b.3774.2 4
5.4 even 2 3775.1.d.f.301.2 yes 4
151.150 odd 2 inner 3775.1.d.e.301.4 yes 4
755.452 even 4 3775.1.c.b.3774.3 4
755.603 even 4 3775.1.c.a.3774.2 4
755.754 odd 2 3775.1.d.f.301.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3775.1.c.a.3774.2 4 755.603 even 4
3775.1.c.a.3774.3 4 5.2 odd 4
3775.1.c.b.3774.2 4 5.3 odd 4
3775.1.c.b.3774.3 4 755.452 even 4
3775.1.d.e.301.3 ✓ 4 1.1 even 1 trivial
3775.1.d.e.301.4 yes 4 151.150 odd 2 inner
3775.1.d.f.301.1 yes 4 755.754 odd 2
3775.1.d.f.301.2 yes 4 5.4 even 2