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Properties

Label 675.1.o.a.431.2
Level $675$
Weight $1$
Character 675.431
Analytic conductor $0.337$
Analytic rank $0$
Dimension $8$
Projective image $A_{5}$
CM/RM no
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] = [675,1,Mod(161,675)] 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("675.161"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([5, 8])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 675.o (of order \(10\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.336868883527\)
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: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(A_{5}\)
Projective field: Galois closure of 5.1.31640625.2

Embedding invariants

Embedding label 431.2
Root \(-0.587785 - 0.809017i\) of defining polynomial
Character \(\chi\) \(=\) 675.431
Dual form 675.1.o.a.296.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.587785 - 0.190983i) q^{2} +(-0.500000 + 0.363271i) q^{4} +(0.951057 + 0.309017i) q^{5} +0.618034 q^{7} +(-0.587785 + 0.809017i) q^{8} +0.618034 q^{10} +(-0.951057 + 0.309017i) q^{11} +(0.363271 - 0.118034i) q^{14} +(0.587785 - 0.809017i) q^{17} +(0.809017 + 0.587785i) q^{19} +(-0.587785 + 0.190983i) q^{20} +(-0.500000 + 0.363271i) q^{22} +(-0.951057 + 0.309017i) q^{23} +(0.809017 + 0.587785i) q^{25} +(-0.309017 + 0.224514i) q^{28} +(-0.951057 - 1.30902i) q^{29} -1.00000i q^{32} +(0.190983 - 0.587785i) q^{34} +(0.587785 + 0.190983i) q^{35} +(0.500000 - 1.53884i) q^{37} +(0.587785 + 0.190983i) q^{38} +(-0.809017 + 0.587785i) q^{40} +(-0.951057 - 0.309017i) q^{41} +(0.363271 - 0.500000i) q^{44} +(-0.500000 + 0.363271i) q^{46} +(0.951057 + 1.30902i) q^{47} -0.618034 q^{49} +(0.587785 + 0.190983i) q^{50} +(0.363271 + 0.500000i) q^{53} -1.00000 q^{55} +(-0.363271 + 0.500000i) q^{56} +(-0.809017 - 0.587785i) q^{58} +(-0.309017 - 0.951057i) q^{61} +(-0.190983 - 0.587785i) q^{64} +(-0.500000 - 0.363271i) q^{67} +0.618034i q^{68} +0.381966 q^{70} +(-0.951057 - 1.30902i) q^{71} -1.00000i q^{74} -0.618034 q^{76} +(-0.587785 + 0.190983i) q^{77} +(-1.30902 + 0.951057i) q^{79} -0.618034 q^{82} +(0.363271 - 0.500000i) q^{83} +(0.809017 - 0.587785i) q^{85} +(0.309017 - 0.951057i) q^{88} +(-0.951057 + 0.309017i) q^{89} +(0.363271 - 0.500000i) q^{92} +(0.809017 + 0.587785i) q^{94} +(0.587785 + 0.809017i) q^{95} +(-0.809017 + 0.587785i) q^{97} +(-0.363271 + 0.118034i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{4} - 4 q^{7} - 4 q^{10} + 2 q^{19} - 4 q^{22} + 2 q^{25} + 2 q^{28} + 6 q^{34} + 4 q^{37} - 2 q^{40} - 4 q^{46} + 4 q^{49} - 8 q^{55} - 2 q^{58} + 2 q^{61} - 6 q^{64} - 4 q^{67} + 12 q^{70} + 4 q^{76}+ \cdots - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/675\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\)
\(\chi(n)\) \(-1\) \(e\left(\frac{2}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.587785 − 0.190983i 0.587785 − 0.190983i − 1.00000i \(-0.5\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(3\) 0 0
\(4\) −0.500000 + 0.363271i −0.500000 + 0.363271i
\(5\) 0.951057 + 0.309017i 0.951057 + 0.309017i
\(6\) 0 0
\(7\) 0.618034 0.618034 0.309017 − 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(8\) −0.587785 + 0.809017i −0.587785 + 0.809017i
\(9\) 0 0
\(10\) 0.618034 0.618034
\(11\) −0.951057 + 0.309017i −0.951057 + 0.309017i −0.743145 − 0.669131i \(-0.766667\pi\)
−0.207912 + 0.978148i \(0.566667\pi\)
\(12\) 0 0
\(13\) 0 0 −0.951057 − 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(14\) 0.363271 − 0.118034i 0.363271 − 0.118034i
\(15\) 0 0
\(16\) 0 0
\(17\) 0.587785 − 0.809017i 0.587785 − 0.809017i −0.406737 − 0.913545i \(-0.633333\pi\)
0.994522 + 0.104528i \(0.0333333\pi\)
\(18\) 0 0
\(19\) 0.809017 + 0.587785i 0.809017 + 0.587785i 0.913545 − 0.406737i \(-0.133333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(20\) −0.587785 + 0.190983i −0.587785 + 0.190983i
\(21\) 0 0
\(22\) −0.500000 + 0.363271i −0.500000 + 0.363271i
\(23\) −0.951057 + 0.309017i −0.951057 + 0.309017i −0.743145 − 0.669131i \(-0.766667\pi\)
−0.207912 + 0.978148i \(0.566667\pi\)
\(24\) 0 0
\(25\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(26\) 0 0
\(27\) 0 0
\(28\) −0.309017 + 0.224514i −0.309017 + 0.224514i
\(29\) −0.951057 − 1.30902i −0.951057 − 1.30902i −0.951057 − 0.309017i \(-0.900000\pi\)
− 1.00000i \(-0.5\pi\)
\(30\) 0 0
\(31\) 0 0 0.587785 − 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(32\) − 1.00000i − 1.00000i
\(33\) 0 0
\(34\) 0.190983 − 0.587785i 0.190983 − 0.587785i
\(35\) 0.587785 + 0.190983i 0.587785 + 0.190983i
\(36\) 0 0
\(37\) 0.500000 − 1.53884i 0.500000 − 1.53884i −0.309017 − 0.951057i \(-0.600000\pi\)
0.809017 − 0.587785i \(-0.200000\pi\)
\(38\) 0.587785 + 0.190983i 0.587785 + 0.190983i
\(39\) 0 0
\(40\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(41\) −0.951057 − 0.309017i −0.951057 − 0.309017i −0.207912 − 0.978148i \(-0.566667\pi\)
−0.743145 + 0.669131i \(0.766667\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0.363271 − 0.500000i 0.363271 − 0.500000i
\(45\) 0 0
\(46\) −0.500000 + 0.363271i −0.500000 + 0.363271i
\(47\) 0.951057 + 1.30902i 0.951057 + 1.30902i 0.951057 + 0.309017i \(0.100000\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −0.618034 −0.618034
\(50\) 0.587785 + 0.190983i 0.587785 + 0.190983i
\(51\) 0 0
\(52\) 0 0
\(53\) 0.363271 + 0.500000i 0.363271 + 0.500000i 0.951057 − 0.309017i \(-0.100000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(54\) 0 0
\(55\) −1.00000 −1.00000
\(56\) −0.363271 + 0.500000i −0.363271 + 0.500000i
\(57\) 0 0
\(58\) −0.809017 − 0.587785i −0.809017 − 0.587785i
\(59\) 0 0 0.309017 − 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(60\) 0 0
\(61\) −0.309017 − 0.951057i −0.309017 − 0.951057i −0.978148 − 0.207912i \(-0.933333\pi\)
0.669131 − 0.743145i \(-0.266667\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −0.190983 − 0.587785i −0.190983 − 0.587785i
\(65\) 0 0
\(66\) 0 0
\(67\) −0.500000 − 0.363271i −0.500000 − 0.363271i 0.309017 − 0.951057i \(-0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(68\) 0.618034i 0.618034i
\(69\) 0 0
\(70\) 0.381966 0.381966
\(71\) −0.951057 − 1.30902i −0.951057 − 1.30902i −0.951057 − 0.309017i \(-0.900000\pi\)
− 1.00000i \(-0.5\pi\)
\(72\) 0 0
\(73\) 0 0 0.951057 − 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(74\) − 1.00000i − 1.00000i
\(75\) 0 0
\(76\) −0.618034 −0.618034
\(77\) −0.587785 + 0.190983i −0.587785 + 0.190983i
\(78\) 0 0
\(79\) −1.30902 + 0.951057i −1.30902 + 0.951057i −0.309017 + 0.951057i \(0.600000\pi\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −0.618034 −0.618034
\(83\) 0.363271 − 0.500000i 0.363271 − 0.500000i −0.587785 − 0.809017i \(-0.700000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(84\) 0 0
\(85\) 0.809017 − 0.587785i 0.809017 − 0.587785i
\(86\) 0 0
\(87\) 0 0
\(88\) 0.309017 − 0.951057i 0.309017 − 0.951057i
\(89\) −0.951057 + 0.309017i −0.951057 + 0.309017i −0.743145 − 0.669131i \(-0.766667\pi\)
−0.207912 + 0.978148i \(0.566667\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0.363271 − 0.500000i 0.363271 − 0.500000i
\(93\) 0 0
\(94\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(95\) 0.587785 + 0.809017i 0.587785 + 0.809017i
\(96\) 0 0
\(97\) −0.809017 + 0.587785i −0.809017 + 0.587785i −0.913545 − 0.406737i \(-0.866667\pi\)
0.104528 + 0.994522i \(0.466667\pi\)
\(98\) −0.363271 + 0.118034i −0.363271 + 0.118034i
\(99\) 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 675.1.o.a.431.2 yes 8
3.2 odd 2 inner 675.1.o.a.431.1 yes 8
5.2 odd 4 3375.1.m.b.1349.2 8
5.3 odd 4 3375.1.m.a.1349.1 8
5.4 even 2 3375.1.o.a.26.1 8
9.2 odd 6 2025.1.y.a.1106.2 16
9.4 even 3 2025.1.y.a.431.1 16
9.5 odd 6 2025.1.y.a.431.2 16
9.7 even 3 2025.1.y.a.1106.1 16
15.2 even 4 3375.1.m.a.1349.2 8
15.8 even 4 3375.1.m.b.1349.1 8
15.14 odd 2 3375.1.o.a.26.2 8
25.3 odd 20 3375.1.m.a.2024.1 8
25.4 even 10 3375.1.o.a.2726.2 8
25.21 even 5 inner 675.1.o.a.296.1 ✓ 8
25.22 odd 20 3375.1.m.b.2024.2 8
75.29 odd 10 3375.1.o.a.2726.1 8
75.47 even 20 3375.1.m.a.2024.2 8
75.53 even 20 3375.1.m.b.2024.1 8
75.71 odd 10 inner 675.1.o.a.296.2 yes 8
225.121 even 15 2025.1.y.a.1646.2 16
225.146 odd 30 2025.1.y.a.296.1 16
225.196 even 15 2025.1.y.a.296.2 16
225.221 odd 30 2025.1.y.a.1646.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
675.1.o.a.296.1 ✓ 8 25.21 even 5 inner
675.1.o.a.296.2 yes 8 75.71 odd 10 inner
675.1.o.a.431.1 yes 8 3.2 odd 2 inner
675.1.o.a.431.2 yes 8 1.1 even 1 trivial
2025.1.y.a.296.1 16 225.146 odd 30
2025.1.y.a.296.2 16 225.196 even 15
2025.1.y.a.431.1 16 9.4 even 3
2025.1.y.a.431.2 16 9.5 odd 6
2025.1.y.a.1106.1 16 9.7 even 3
2025.1.y.a.1106.2 16 9.2 odd 6
2025.1.y.a.1646.1 16 225.221 odd 30
2025.1.y.a.1646.2 16 225.121 even 15
3375.1.m.a.1349.1 8 5.3 odd 4
3375.1.m.a.1349.2 8 15.2 even 4
3375.1.m.a.2024.1 8 25.3 odd 20
3375.1.m.a.2024.2 8 75.47 even 20
3375.1.m.b.1349.1 8 15.8 even 4
3375.1.m.b.1349.2 8 5.2 odd 4
3375.1.m.b.2024.1 8 75.53 even 20
3375.1.m.b.2024.2 8 25.22 odd 20
3375.1.o.a.26.1 8 5.4 even 2
3375.1.o.a.26.2 8 15.14 odd 2
3375.1.o.a.2726.1 8 75.29 odd 10
3375.1.o.a.2726.2 8 25.4 even 10