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Properties

Label 95.1.d.b.94.2
Level $95$
Weight $1$
Character 95.94
Self dual yes
Analytic conductor $0.047$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -95
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] = [95,1,Mod(94,95)] 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("95.94"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(95, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 95 = 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 95.d (of order \(2\), degree \(1\), 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: yes
Analytic conductor: \(0.0474111762001\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of \(\Q(\sqrt{-1 +2 \sqrt{5}})\)
Artin image: $D_8$
Artin field: Galois closure of 8.2.4286875.1

Embedding invariants

Embedding label 94.2
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 95.94

$q$-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)
 
\(f(q)\) \(=\) \(q+1.41421 q^{2} -1.41421 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{6} +1.00000 q^{9} -1.41421 q^{10} -1.41421 q^{12} +1.41421 q^{13} +1.41421 q^{15} -1.00000 q^{16} +1.41421 q^{18} -1.00000 q^{19} -1.00000 q^{20} +1.00000 q^{25} +2.00000 q^{26} +2.00000 q^{30} -1.41421 q^{32} +1.00000 q^{36} -1.41421 q^{37} -1.41421 q^{38} -2.00000 q^{39} -1.00000 q^{45} +1.41421 q^{48} +1.00000 q^{49} +1.41421 q^{50} +1.41421 q^{52} -1.41421 q^{53} +1.41421 q^{57} +1.41421 q^{60} -1.00000 q^{64} -1.41421 q^{65} +1.41421 q^{67} -2.00000 q^{74} -1.41421 q^{75} -1.00000 q^{76} -2.82843 q^{78} +1.00000 q^{80} -1.00000 q^{81} -1.41421 q^{90} +1.00000 q^{95} +2.00000 q^{96} -1.41421 q^{97} +1.41421 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} - 2 q^{5} - 4 q^{6} + 2 q^{9} - 2 q^{16} - 2 q^{19} - 2 q^{20} + 2 q^{25} + 4 q^{26} + 4 q^{30} + 2 q^{36} - 4 q^{39} - 2 q^{45} + 2 q^{49} - 2 q^{64} - 4 q^{74} - 2 q^{76} + 2 q^{80} - 2 q^{81}+ \cdots + 4 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/95\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421 1.41421 0.707107 − 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(3\) −1.41421 −1.41421 −0.707107 − 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(4\) 1.00000 1.00000
\(5\) −1.00000 −1.00000
\(6\) −2.00000 −2.00000
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 1.00000 1.00000
\(10\) −1.41421 −1.41421
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) −1.41421 −1.41421
\(13\) 1.41421 1.41421 0.707107 − 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(14\) 0 0
\(15\) 1.41421 1.41421
\(16\) −1.00000 −1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 1.41421 1.41421
\(19\) −1.00000 −1.00000
\(20\) −1.00000 −1.00000
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 1.00000 1.00000
\(26\) 2.00000 2.00000
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 2.00000 2.00000
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −1.41421 −1.41421
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 1.00000
\(37\) −1.41421 −1.41421 −0.707107 − 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(38\) −1.41421 −1.41421
\(39\) −2.00000 −2.00000
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) −1.00000 −1.00000
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 1.41421 1.41421
\(49\) 1.00000 1.00000
\(50\) 1.41421 1.41421
\(51\) 0 0
\(52\) 1.41421 1.41421
\(53\) −1.41421 −1.41421 −0.707107 − 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.41421 1.41421
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 1.41421 1.41421
\(61\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −1.00000 −1.00000
\(65\) −1.41421 −1.41421
\(66\) 0 0
\(67\) 1.41421 1.41421 0.707107 − 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) −2.00000 −2.00000
\(75\) −1.41421 −1.41421
\(76\) −1.00000 −1.00000
\(77\) 0 0
\(78\) −2.82843 −2.82843
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 1.00000 1.00000
\(81\) −1.00000 −1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) −1.41421 −1.41421
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.00000 1.00000
\(96\) 2.00000 2.00000
\(97\) −1.41421 −1.41421 −0.707107 − 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(98\) 1.41421 1.41421
\(99\) 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 95.1.d.b.94.2 yes 2
3.2 odd 2 855.1.g.c.379.1 2
4.3 odd 2 1520.1.m.b.1329.2 2
5.2 odd 4 475.1.c.b.151.2 2
5.3 odd 4 475.1.c.b.151.1 2
5.4 even 2 inner 95.1.d.b.94.1 ✓ 2
15.14 odd 2 855.1.g.c.379.2 2
19.2 odd 18 1805.1.o.b.984.2 12
19.3 odd 18 1805.1.o.b.694.1 12
19.4 even 9 1805.1.o.b.1029.2 12
19.5 even 9 1805.1.o.b.849.2 12
19.6 even 9 1805.1.o.b.1199.2 12
19.7 even 3 1805.1.h.b.654.1 4
19.8 odd 6 1805.1.h.b.69.2 4
19.9 even 9 1805.1.o.b.299.1 12
19.10 odd 18 1805.1.o.b.299.2 12
19.11 even 3 1805.1.h.b.69.1 4
19.12 odd 6 1805.1.h.b.654.2 4
19.13 odd 18 1805.1.o.b.1199.1 12
19.14 odd 18 1805.1.o.b.849.1 12
19.15 odd 18 1805.1.o.b.1029.1 12
19.16 even 9 1805.1.o.b.694.2 12
19.17 even 9 1805.1.o.b.984.1 12
19.18 odd 2 inner 95.1.d.b.94.1 ✓ 2
20.19 odd 2 1520.1.m.b.1329.1 2
57.56 even 2 855.1.g.c.379.2 2
76.75 even 2 1520.1.m.b.1329.1 2
95.4 even 18 1805.1.o.b.1029.1 12
95.9 even 18 1805.1.o.b.299.2 12
95.14 odd 18 1805.1.o.b.849.2 12
95.18 even 4 475.1.c.b.151.2 2
95.24 even 18 1805.1.o.b.849.1 12
95.29 odd 18 1805.1.o.b.299.1 12
95.34 odd 18 1805.1.o.b.1029.2 12
95.37 even 4 475.1.c.b.151.1 2
95.44 even 18 1805.1.o.b.1199.1 12
95.49 even 6 1805.1.h.b.69.2 4
95.54 even 18 1805.1.o.b.694.1 12
95.59 odd 18 1805.1.o.b.984.1 12
95.64 even 6 1805.1.h.b.654.2 4
95.69 odd 6 1805.1.h.b.654.1 4
95.74 even 18 1805.1.o.b.984.2 12
95.79 odd 18 1805.1.o.b.694.2 12
95.84 odd 6 1805.1.h.b.69.1 4
95.89 odd 18 1805.1.o.b.1199.2 12
95.94 odd 2 CM 95.1.d.b.94.2 yes 2
285.284 even 2 855.1.g.c.379.1 2
380.379 even 2 1520.1.m.b.1329.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.1.d.b.94.1 ✓ 2 5.4 even 2 inner
95.1.d.b.94.1 ✓ 2 19.18 odd 2 inner
95.1.d.b.94.2 yes 2 1.1 even 1 trivial
95.1.d.b.94.2 yes 2 95.94 odd 2 CM
475.1.c.b.151.1 2 5.3 odd 4
475.1.c.b.151.1 2 95.37 even 4
475.1.c.b.151.2 2 5.2 odd 4
475.1.c.b.151.2 2 95.18 even 4
855.1.g.c.379.1 2 3.2 odd 2
855.1.g.c.379.1 2 285.284 even 2
855.1.g.c.379.2 2 15.14 odd 2
855.1.g.c.379.2 2 57.56 even 2
1520.1.m.b.1329.1 2 20.19 odd 2
1520.1.m.b.1329.1 2 76.75 even 2
1520.1.m.b.1329.2 2 4.3 odd 2
1520.1.m.b.1329.2 2 380.379 even 2
1805.1.h.b.69.1 4 19.11 even 3
1805.1.h.b.69.1 4 95.84 odd 6
1805.1.h.b.69.2 4 19.8 odd 6
1805.1.h.b.69.2 4 95.49 even 6
1805.1.h.b.654.1 4 19.7 even 3
1805.1.h.b.654.1 4 95.69 odd 6
1805.1.h.b.654.2 4 19.12 odd 6
1805.1.h.b.654.2 4 95.64 even 6
1805.1.o.b.299.1 12 19.9 even 9
1805.1.o.b.299.1 12 95.29 odd 18
1805.1.o.b.299.2 12 19.10 odd 18
1805.1.o.b.299.2 12 95.9 even 18
1805.1.o.b.694.1 12 19.3 odd 18
1805.1.o.b.694.1 12 95.54 even 18
1805.1.o.b.694.2 12 19.16 even 9
1805.1.o.b.694.2 12 95.79 odd 18
1805.1.o.b.849.1 12 19.14 odd 18
1805.1.o.b.849.1 12 95.24 even 18
1805.1.o.b.849.2 12 19.5 even 9
1805.1.o.b.849.2 12 95.14 odd 18
1805.1.o.b.984.1 12 19.17 even 9
1805.1.o.b.984.1 12 95.59 odd 18
1805.1.o.b.984.2 12 19.2 odd 18
1805.1.o.b.984.2 12 95.74 even 18
1805.1.o.b.1029.1 12 19.15 odd 18
1805.1.o.b.1029.1 12 95.4 even 18
1805.1.o.b.1029.2 12 19.4 even 9
1805.1.o.b.1029.2 12 95.34 odd 18
1805.1.o.b.1199.1 12 19.13 odd 18
1805.1.o.b.1199.1 12 95.44 even 18
1805.1.o.b.1199.2 12 19.6 even 9
1805.1.o.b.1199.2 12 95.89 odd 18