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Properties

Label 2775.1.b.a.2774.1
Level $2775$
Weight $1$
Character 2775.2774
Analytic conductor $1.385$
Analytic rank $0$
Dimension $2$
Projective image $D_{2}$
CM/RM discs -3, -111, 37
Inner twists $8$

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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] = [2775,1,Mod(2774,2775)] 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("2775.2774"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2775, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 2775 = 3 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2775.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2] 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: \(1.38490541006\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 111)
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{-3}, \sqrt{37})\)
Artin image: $D_4:C_2$
Artin field: Galois closure of 8.0.1732640625.1

Embedding invariants

Embedding label 2774.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2775.2774
Dual form 2775.1.b.a.2774.2

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} +1.00000 q^{4} -2.00000i q^{7} -1.00000 q^{9} -1.00000i q^{12} +1.00000 q^{16} -2.00000 q^{21} +1.00000i q^{27} -2.00000i q^{28} -1.00000 q^{36} +1.00000i q^{37} -1.00000i q^{48} -3.00000 q^{49} +2.00000i q^{63} +1.00000 q^{64} -2.00000i q^{67} +2.00000i q^{73} +1.00000 q^{81} -2.00000 q^{84} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} - 2 q^{9} + 2 q^{16} - 4 q^{21} - 2 q^{36} - 6 q^{49} + 2 q^{64} + 2 q^{81} - 4 q^{84}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2775\mathbb{Z}\right)^\times\).

\(n\) \(76\) \(926\) \(1777\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(3\) − 1.00000i − 1.00000i
\(4\) 1.00000 1.00000
\(5\) 0 0
\(6\) 0 0
\(7\) − 2.00000i − 2.00000i − 1.00000i \(-0.5\pi\)
− 1.00000i \(-0.5\pi\)
\(8\) 0 0
\(9\) −1.00000 −1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) − 1.00000i − 1.00000i
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) −2.00000 −2.00000
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000i 1.00000i
\(28\) − 2.00000i − 2.00000i
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −1.00000 −1.00000
\(37\) 1.00000i 1.00000i
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) − 1.00000i − 1.00000i
\(49\) −3.00000 −3.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 2.00000i 2.00000i
\(64\) 1.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) − 2.00000i − 2.00000i − 1.00000i \(-0.5\pi\)
− 1.00000i \(-0.5\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 1.00000 1.00000
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) −2.00000 −2.00000
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) 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 2775.1.b.a.2774.1 2
3.2 odd 2 CM 2775.1.b.a.2774.1 2
5.2 odd 4 2775.1.h.a.776.1 1
5.3 odd 4 111.1.d.a.110.1 ✓ 1
5.4 even 2 inner 2775.1.b.a.2774.2 2
15.2 even 4 2775.1.h.a.776.1 1
15.8 even 4 111.1.d.a.110.1 ✓ 1
15.14 odd 2 inner 2775.1.b.a.2774.2 2
20.3 even 4 1776.1.n.a.1553.1 1
37.36 even 2 RM 2775.1.b.a.2774.1 2
45.13 odd 12 2997.1.n.b.998.1 2
45.23 even 12 2997.1.n.b.998.1 2
45.38 even 12 2997.1.n.b.1997.1 2
45.43 odd 12 2997.1.n.b.1997.1 2
60.23 odd 4 1776.1.n.a.1553.1 1
111.110 odd 2 CM 2775.1.b.a.2774.1 2
185.73 odd 4 111.1.d.a.110.1 ✓ 1
185.147 odd 4 2775.1.h.a.776.1 1
185.184 even 2 inner 2775.1.b.a.2774.2 2
555.332 even 4 2775.1.h.a.776.1 1
555.443 even 4 111.1.d.a.110.1 ✓ 1
555.554 odd 2 inner 2775.1.b.a.2774.2 2
740.443 even 4 1776.1.n.a.1553.1 1
1665.443 even 12 2997.1.n.b.1997.1 2
1665.628 odd 12 2997.1.n.b.1997.1 2
1665.1183 odd 12 2997.1.n.b.998.1 2
1665.1553 even 12 2997.1.n.b.998.1 2
2220.443 odd 4 1776.1.n.a.1553.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
111.1.d.a.110.1 ✓ 1 5.3 odd 4
111.1.d.a.110.1 ✓ 1 15.8 even 4
111.1.d.a.110.1 ✓ 1 185.73 odd 4
111.1.d.a.110.1 ✓ 1 555.443 even 4
1776.1.n.a.1553.1 1 20.3 even 4
1776.1.n.a.1553.1 1 60.23 odd 4
1776.1.n.a.1553.1 1 740.443 even 4
1776.1.n.a.1553.1 1 2220.443 odd 4
2775.1.b.a.2774.1 2 1.1 even 1 trivial
2775.1.b.a.2774.1 2 3.2 odd 2 CM
2775.1.b.a.2774.1 2 37.36 even 2 RM
2775.1.b.a.2774.1 2 111.110 odd 2 CM
2775.1.b.a.2774.2 2 5.4 even 2 inner
2775.1.b.a.2774.2 2 15.14 odd 2 inner
2775.1.b.a.2774.2 2 185.184 even 2 inner
2775.1.b.a.2774.2 2 555.554 odd 2 inner
2775.1.h.a.776.1 1 5.2 odd 4
2775.1.h.a.776.1 1 15.2 even 4
2775.1.h.a.776.1 1 185.147 odd 4
2775.1.h.a.776.1 1 555.332 even 4
2997.1.n.b.998.1 2 45.13 odd 12
2997.1.n.b.998.1 2 45.23 even 12
2997.1.n.b.998.1 2 1665.1183 odd 12
2997.1.n.b.998.1 2 1665.1553 even 12
2997.1.n.b.1997.1 2 45.38 even 12
2997.1.n.b.1997.1 2 45.43 odd 12
2997.1.n.b.1997.1 2 1665.443 even 12
2997.1.n.b.1997.1 2 1665.628 odd 12