Properties

Label 2325.1.ca.b.101.1
Level $2325$
Weight $1$
Character 2325.101
Analytic conductor $1.160$
Analytic rank $0$
Dimension $8$
Projective image $D_{5}$
CM discriminant -15
Inner twists $8$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2325,1,Mod(101,2325)] 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("2325.101"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2325, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([5, 0, 4])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 2325 = 3 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2325.ca (of order \(10\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8] 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: \(1.16032615437\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{20})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) 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 465)
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.207792225.1

Embedding invariants

Embedding label 101.1
Root \(-0.951057 - 0.309017i\) of defining polynomial
Character \(\chi\) \(=\) 2325.101
Dual form 2325.1.ca.b.1151.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.53884 - 0.500000i) q^{2} +(0.951057 - 0.309017i) q^{3} +(1.30902 + 0.951057i) q^{4} -1.61803 q^{6} +(-0.587785 - 0.809017i) q^{8} +(0.809017 - 0.587785i) q^{9} +(1.53884 + 0.500000i) q^{12} +(-1.17557 - 1.61803i) q^{17} +(-1.53884 + 0.500000i) q^{18} +(-0.190983 + 0.587785i) q^{19} +(-0.951057 - 1.30902i) q^{23} +(-0.809017 - 0.587785i) q^{24} +(0.587785 - 0.809017i) q^{27} +(0.309017 - 0.951057i) q^{31} +1.00000i q^{32} +(1.00000 + 3.07768i) q^{34} +1.61803 q^{36} +(0.587785 - 0.809017i) q^{38} +(0.809017 + 2.48990i) q^{46} +(1.53884 - 0.500000i) q^{47} +(-0.309017 - 0.951057i) q^{49} +(-1.61803 - 1.17557i) q^{51} +(0.363271 + 0.500000i) q^{53} +(-1.30902 + 0.951057i) q^{54} +0.618034i q^{57} +0.618034 q^{61} +(-0.951057 + 1.30902i) q^{62} +(0.500000 - 1.53884i) q^{64} -3.23607i q^{68} +(-1.30902 - 0.951057i) q^{69} +(-0.951057 - 0.309017i) q^{72} +(-0.809017 + 0.587785i) q^{76} +(0.500000 - 0.363271i) q^{79} +(0.309017 - 0.951057i) q^{81} +(1.53884 + 0.500000i) q^{83} -2.61803i q^{92} -1.00000i q^{93} -2.61803 q^{94} +(0.309017 + 0.951057i) q^{96} +1.61803i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 6 q^{4} - 4 q^{6} + 2 q^{9} - 6 q^{19} - 2 q^{24} - 2 q^{31} + 8 q^{34} + 4 q^{36} + 2 q^{46} + 2 q^{49} - 4 q^{51} - 6 q^{54} - 4 q^{61} + 4 q^{64} - 6 q^{69} - 2 q^{76} + 4 q^{79} - 2 q^{81} - 12 q^{94}+ \cdots - 2 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2325\mathbb{Z}\right)^\times\).

\(n\) \(652\) \(776\) \(1801\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{2}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.53884 − 0.500000i −1.53884 − 0.500000i −0.587785 − 0.809017i \(-0.700000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(3\) 0.951057 − 0.309017i 0.951057 − 0.309017i
\(4\) 1.30902 + 0.951057i 1.30902 + 0.951057i
\(5\) 0 0
\(6\) −1.61803 −1.61803
\(7\) 0 0 0.587785 − 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(8\) −0.587785 − 0.809017i −0.587785 − 0.809017i
\(9\) 0.809017 − 0.587785i 0.809017 − 0.587785i
\(10\) 0 0
\(11\) 0 0 −0.809017 − 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(12\) 1.53884 + 0.500000i 1.53884 + 0.500000i
\(13\) 0 0 0.951057 − 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.17557 − 1.61803i −1.17557 − 1.61803i −0.587785 − 0.809017i \(-0.700000\pi\)
−0.587785 − 0.809017i \(-0.700000\pi\)
\(18\) −1.53884 + 0.500000i −1.53884 + 0.500000i
\(19\) −0.190983 + 0.587785i −0.190983 + 0.587785i 0.809017 + 0.587785i \(0.200000\pi\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −0.951057 − 1.30902i −0.951057 − 1.30902i −0.951057 − 0.309017i \(-0.900000\pi\)
− 1.00000i \(-0.5\pi\)
\(24\) −0.809017 − 0.587785i −0.809017 − 0.587785i
\(25\) 0 0
\(26\) 0 0
\(27\) 0.587785 − 0.809017i 0.587785 − 0.809017i
\(28\) 0 0
\(29\) 0 0 0.309017 − 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(30\) 0 0
\(31\) 0.309017 − 0.951057i 0.309017 − 0.951057i
\(32\) 1.00000i 1.00000i
\(33\) 0 0
\(34\) 1.00000 + 3.07768i 1.00000 + 3.07768i
\(35\) 0 0
\(36\) 1.61803 1.61803
\(37\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0.587785 − 0.809017i 0.587785 − 0.809017i
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 0.309017 − 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(42\) 0 0
\(43\) 0 0 −0.951057 − 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0.809017 + 2.48990i 0.809017 + 2.48990i
\(47\) 1.53884 − 0.500000i 1.53884 − 0.500000i 0.587785 − 0.809017i \(-0.300000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(48\) 0 0
\(49\) −0.309017 − 0.951057i −0.309017 − 0.951057i
\(50\) 0 0
\(51\) −1.61803 − 1.17557i −1.61803 − 1.17557i
\(52\) 0 0
\(53\) 0.363271 + 0.500000i 0.363271 + 0.500000i 0.951057 − 0.309017i \(-0.100000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(54\) −1.30902 + 0.951057i −1.30902 + 0.951057i
\(55\) 0 0
\(56\) 0 0
\(57\) 0.618034i 0.618034i
\(58\) 0 0
\(59\) 0 0 −0.309017 − 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(60\) 0 0
\(61\) 0.618034 0.618034 0.309017 − 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(62\) −0.951057 + 1.30902i −0.951057 + 1.30902i
\(63\) 0 0
\(64\) 0.500000 − 1.53884i 0.500000 − 1.53884i
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) − 3.23607i − 3.23607i
\(69\) −1.30902 − 0.951057i −1.30902 − 0.951057i
\(70\) 0 0
\(71\) 0 0 0.809017 − 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(72\) −0.951057 − 0.309017i −0.951057 − 0.309017i
\(73\) 0 0 0.587785 − 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(77\) 0 0
\(78\) 0 0
\(79\) 0.500000 − 0.363271i 0.500000 − 0.363271i −0.309017 − 0.951057i \(-0.600000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(80\) 0 0
\(81\) 0.309017 − 0.951057i 0.309017 − 0.951057i
\(82\) 0 0
\(83\) 1.53884 + 0.500000i 1.53884 + 0.500000i 0.951057 − 0.309017i \(-0.100000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.809017 − 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) − 2.61803i − 2.61803i
\(93\) − 1.00000i − 1.00000i
\(94\) −2.61803 −2.61803
\(95\) 0 0
\(96\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(97\) 0 0 0.587785 − 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(98\) 1.61803i 1.61803i
\(99\) 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 2325.1.ca.b.101.1 8
3.2 odd 2 inner 2325.1.ca.b.101.2 8
5.2 odd 4 465.1.x.b.194.1 yes 4
5.3 odd 4 465.1.x.a.194.1 ✓ 4
5.4 even 2 inner 2325.1.ca.b.101.2 8
15.2 even 4 465.1.x.a.194.1 ✓ 4
15.8 even 4 465.1.x.b.194.1 yes 4
15.14 odd 2 CM 2325.1.ca.b.101.1 8
31.4 even 5 inner 2325.1.ca.b.1151.2 8
93.35 odd 10 inner 2325.1.ca.b.1151.1 8
155.4 even 10 inner 2325.1.ca.b.1151.1 8
155.97 odd 20 465.1.x.b.314.1 yes 4
155.128 odd 20 465.1.x.a.314.1 yes 4
465.128 even 20 465.1.x.b.314.1 yes 4
465.314 odd 10 inner 2325.1.ca.b.1151.2 8
465.407 even 20 465.1.x.a.314.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.1.x.a.194.1 ✓ 4 5.3 odd 4
465.1.x.a.194.1 ✓ 4 15.2 even 4
465.1.x.a.314.1 yes 4 155.128 odd 20
465.1.x.a.314.1 yes 4 465.407 even 20
465.1.x.b.194.1 yes 4 5.2 odd 4
465.1.x.b.194.1 yes 4 15.8 even 4
465.1.x.b.314.1 yes 4 155.97 odd 20
465.1.x.b.314.1 yes 4 465.128 even 20
2325.1.ca.b.101.1 8 1.1 even 1 trivial
2325.1.ca.b.101.1 8 15.14 odd 2 CM
2325.1.ca.b.101.2 8 3.2 odd 2 inner
2325.1.ca.b.101.2 8 5.4 even 2 inner
2325.1.ca.b.1151.1 8 93.35 odd 10 inner
2325.1.ca.b.1151.1 8 155.4 even 10 inner
2325.1.ca.b.1151.2 8 31.4 even 5 inner
2325.1.ca.b.1151.2 8 465.314 odd 10 inner