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Properties

Label 2175.1.h.e.1826.2
Level $2175$
Weight $1$
Character 2175.1826
Self dual yes
Analytic conductor $1.085$
Analytic rank $0$
Dimension $3$
Projective image $D_{9}$
CM discriminant -87
Inner twists $2$

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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] = [2175,1,Mod(1826,2175)] 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("2175.1826"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2175, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2175.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,3,3,0,0,0,-3,3,0,0,3,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(1.08546640248\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{9}\)
Projective field: Galois closure of 9.1.895152515625.1
Artin image: $D_9$
Artin field: Galois closure of 9.1.895152515625.1

Embedding invariants

Embedding label 1826.2
Root \(1.87939\) of defining polynomial
Character \(\chi\) \(=\) 2175.1826

$q$-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)
 
\(f(q)\) \(=\) \(q+0.347296 q^{2} +1.00000 q^{3} -0.879385 q^{4} +0.347296 q^{6} -1.87939 q^{7} -0.652704 q^{8} +1.00000 q^{9} +1.53209 q^{11} -0.879385 q^{12} +0.347296 q^{13} -0.652704 q^{14} +0.652704 q^{16} +1.53209 q^{17} +0.347296 q^{18} -1.87939 q^{21} +0.532089 q^{22} -0.652704 q^{24} +0.120615 q^{26} +1.00000 q^{27} +1.65270 q^{28} +1.00000 q^{29} +0.879385 q^{32} +1.53209 q^{33} +0.532089 q^{34} -0.879385 q^{36} +0.347296 q^{39} -1.00000 q^{41} -0.652704 q^{42} -1.34730 q^{44} -1.87939 q^{47} +0.652704 q^{48} +2.53209 q^{49} +1.53209 q^{51} -0.305407 q^{52} +0.347296 q^{54} +1.22668 q^{56} +0.347296 q^{58} -1.87939 q^{63} -0.347296 q^{64} +0.532089 q^{66} +1.53209 q^{67} -1.34730 q^{68} -0.652704 q^{72} -2.87939 q^{77} +0.120615 q^{78} +1.00000 q^{81} -0.347296 q^{82} +1.65270 q^{84} +1.00000 q^{87} -1.00000 q^{88} +0.347296 q^{89} -0.652704 q^{91} -0.652704 q^{94} +0.879385 q^{96} +0.879385 q^{98} +1.53209 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + 3 q^{4} - 3 q^{8} + 3 q^{9} + 3 q^{12} - 3 q^{14} + 3 q^{16} - 3 q^{22} - 3 q^{24} + 6 q^{26} + 3 q^{27} + 6 q^{28} + 3 q^{29} - 3 q^{32} - 3 q^{34} + 3 q^{36} - 3 q^{41} - 3 q^{42} - 3 q^{44}+ \cdots - 3 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2175\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1451\) \(2002\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.347296 0.347296 0.173648 − 0.984808i \(-0.444444\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(3\) 1.00000 1.00000
\(4\) −0.879385 −0.879385
\(5\) 0 0
\(6\) 0.347296 0.347296
\(7\) −1.87939 −1.87939 −0.939693 − 0.342020i \(-0.888889\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(8\) −0.652704 −0.652704
\(9\) 1.00000 1.00000
\(10\) 0 0
\(11\) 1.53209 1.53209 0.766044 − 0.642788i \(-0.222222\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(12\) −0.879385 −0.879385
\(13\) 0.347296 0.347296 0.173648 − 0.984808i \(-0.444444\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(14\) −0.652704 −0.652704
\(15\) 0 0
\(16\) 0.652704 0.652704
\(17\) 1.53209 1.53209 0.766044 − 0.642788i \(-0.222222\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(18\) 0.347296 0.347296
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) −1.87939 −1.87939
\(22\) 0.532089 0.532089
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) −0.652704 −0.652704
\(25\) 0 0
\(26\) 0.120615 0.120615
\(27\) 1.00000 1.00000
\(28\) 1.65270 1.65270
\(29\) 1.00000 1.00000
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0.879385 0.879385
\(33\) 1.53209 1.53209
\(34\) 0.532089 0.532089
\(35\) 0 0
\(36\) −0.879385 −0.879385
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0.347296 0.347296
\(40\) 0 0
\(41\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(42\) −0.652704 −0.652704
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) −1.34730 −1.34730
\(45\) 0 0
\(46\) 0 0
\(47\) −1.87939 −1.87939 −0.939693 − 0.342020i \(-0.888889\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(48\) 0.652704 0.652704
\(49\) 2.53209 2.53209
\(50\) 0 0
\(51\) 1.53209 1.53209
\(52\) −0.305407 −0.305407
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0.347296 0.347296
\(55\) 0 0
\(56\) 1.22668 1.22668
\(57\) 0 0
\(58\) 0.347296 0.347296
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) −1.87939 −1.87939
\(64\) −0.347296 −0.347296
\(65\) 0 0
\(66\) 0.532089 0.532089
\(67\) 1.53209 1.53209 0.766044 − 0.642788i \(-0.222222\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(68\) −1.34730 −1.34730
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) −0.652704 −0.652704
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.87939 −2.87939
\(78\) 0.120615 0.120615
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 1.00000 1.00000
\(82\) −0.347296 −0.347296
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 1.65270 1.65270
\(85\) 0 0
\(86\) 0 0
\(87\) 1.00000 1.00000
\(88\) −1.00000 −1.00000
\(89\) 0.347296 0.347296 0.173648 − 0.984808i \(-0.444444\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(90\) 0 0
\(91\) −0.652704 −0.652704
\(92\) 0 0
\(93\) 0 0
\(94\) −0.652704 −0.652704
\(95\) 0 0
\(96\) 0.879385 0.879385
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0.879385 0.879385
\(99\) 1.53209 1.53209
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 2175.1.h.e.1826.2 yes 3
3.2 odd 2 2175.1.h.d.1826.2 yes 3
5.2 odd 4 2175.1.b.d.2174.4 6
5.3 odd 4 2175.1.b.d.2174.3 6
5.4 even 2 2175.1.h.c.1826.2 ✓ 3
15.2 even 4 2175.1.b.c.2174.3 6
15.8 even 4 2175.1.b.c.2174.4 6
15.14 odd 2 2175.1.h.f.1826.2 yes 3
29.28 even 2 2175.1.h.d.1826.2 yes 3
87.86 odd 2 CM 2175.1.h.e.1826.2 yes 3
145.28 odd 4 2175.1.b.c.2174.4 6
145.57 odd 4 2175.1.b.c.2174.3 6
145.144 even 2 2175.1.h.f.1826.2 yes 3
435.173 even 4 2175.1.b.d.2174.3 6
435.347 even 4 2175.1.b.d.2174.4 6
435.434 odd 2 2175.1.h.c.1826.2 ✓ 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2175.1.b.c.2174.3 6 15.2 even 4
2175.1.b.c.2174.3 6 145.57 odd 4
2175.1.b.c.2174.4 6 15.8 even 4
2175.1.b.c.2174.4 6 145.28 odd 4
2175.1.b.d.2174.3 6 5.3 odd 4
2175.1.b.d.2174.3 6 435.173 even 4
2175.1.b.d.2174.4 6 5.2 odd 4
2175.1.b.d.2174.4 6 435.347 even 4
2175.1.h.c.1826.2 ✓ 3 5.4 even 2
2175.1.h.c.1826.2 ✓ 3 435.434 odd 2
2175.1.h.d.1826.2 yes 3 3.2 odd 2
2175.1.h.d.1826.2 yes 3 29.28 even 2
2175.1.h.e.1826.2 yes 3 1.1 even 1 trivial
2175.1.h.e.1826.2 yes 3 87.86 odd 2 CM
2175.1.h.f.1826.2 yes 3 15.14 odd 2
2175.1.h.f.1826.2 yes 3 145.144 even 2