Properties

Label 1100.1.x.a.681.2
Level $1100$
Weight $1$
Character 1100.681
Analytic conductor $0.549$
Analytic rank $0$
Dimension $8$
Projective image $D_{15}$
CM discriminant -11
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1100,1,Mod(21,1100)] 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("1100.21"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1100, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 6, 5])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1100.x (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: \(0.548971513896\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{15})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{15}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{15} + \cdots)\)

Embedding invariants

Embedding label 681.2
Root \(-0.978148 + 0.207912i\) of defining polynomial
Character \(\chi\) \(=\) 1100.681
Dual form 1100.1.x.a.21.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.58268 - 1.14988i) q^{3} +(-0.500000 + 0.866025i) q^{5} +(0.873619 - 2.68872i) q^{9} +(0.309017 + 0.951057i) q^{11} +(0.204489 + 1.94558i) q^{15} +(-0.0646021 - 0.198825i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(-1.10453 - 3.39939i) q^{27} +(-1.08268 - 0.786610i) q^{31} +(1.58268 + 1.14988i) q^{33} +(-0.309017 + 0.951057i) q^{37} +(1.89169 + 2.10094i) q^{45} +(-1.61803 + 1.17557i) q^{47} +1.00000 q^{49} +(-0.500000 + 0.363271i) q^{53} +(-0.978148 - 0.207912i) q^{55} +(-0.604528 + 1.86055i) q^{59} +(-1.47815 - 1.07394i) q^{67} +(-0.330869 - 0.240391i) q^{69} +(1.58268 - 1.14988i) q^{71} +(-1.78716 - 0.795697i) q^{75} +(-3.36984 - 2.44833i) q^{81} +(0.564602 + 1.73767i) q^{89} -2.61803 q^{93} +(-1.08268 + 0.786610i) q^{97} +2.82709 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{3} - 4 q^{5} - 2 q^{11} - q^{15} + 2 q^{23} - 4 q^{25} - 7 q^{27} + 2 q^{31} + 2 q^{33} + 2 q^{37} - 4 q^{47} + 8 q^{49} - 4 q^{53} + q^{55} - 3 q^{59} - 3 q^{67} - 7 q^{69} + 2 q^{71} - q^{75}+ \cdots + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1100\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(551\)
\(\chi(n)\) \(-1\) \(e\left(\frac{2}{5}\right)\) \(1\)

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
\(3\) 1.58268 − 1.14988i 1.58268 − 1.14988i 0.669131 − 0.743145i \(-0.266667\pi\)
0.913545 − 0.406737i \(-0.133333\pi\)
\(4\) 0 0
\(5\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 0.873619 − 2.68872i 0.873619 − 2.68872i
\(10\) 0 0
\(11\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(12\) 0 0
\(13\) 0 0 0.309017 − 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(14\) 0 0
\(15\) 0.204489 + 1.94558i 0.204489 + 1.94558i
\(16\) 0 0
\(17\) 0 0 −0.809017 − 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(18\) 0 0
\(19\) 0 0 −0.809017 − 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −0.0646021 − 0.198825i −0.0646021 − 0.198825i 0.913545 − 0.406737i \(-0.133333\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(24\) 0 0
\(25\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(26\) 0 0
\(27\) −1.10453 − 3.39939i −1.10453 − 3.39939i
\(28\) 0 0
\(29\) 0 0 0.809017 − 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(30\) 0 0
\(31\) −1.08268 − 0.786610i −1.08268 − 0.786610i −0.104528 − 0.994522i \(-0.533333\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(32\) 0 0
\(33\) 1.58268 + 1.14988i 1.58268 + 1.14988i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −0.309017 + 0.951057i −0.309017 + 0.951057i 0.669131 + 0.743145i \(0.266667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(38\) 0 0
\(39\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 1.89169 + 2.10094i 1.89169 + 2.10094i
\(46\) 0 0
\(47\) −1.61803 + 1.17557i −1.61803 + 1.17557i −0.809017 + 0.587785i \(0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(48\) 0 0
\(49\) 1.00000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −0.500000 + 0.363271i −0.500000 + 0.363271i −0.809017 − 0.587785i \(-0.800000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(54\) 0 0
\(55\) −0.978148 − 0.207912i −0.978148 − 0.207912i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −0.604528 + 1.86055i −0.604528 + 1.86055i −0.104528 + 0.994522i \(0.533333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(60\) 0 0
\(61\) 0 0 −0.309017 − 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −1.47815 − 1.07394i −1.47815 − 1.07394i −0.978148 − 0.207912i \(-0.933333\pi\)
−0.500000 − 0.866025i \(-0.666667\pi\)
\(68\) 0 0
\(69\) −0.330869 − 0.240391i −0.330869 − 0.240391i
\(70\) 0 0
\(71\) 1.58268 − 1.14988i 1.58268 − 1.14988i 0.669131 − 0.743145i \(-0.266667\pi\)
0.913545 − 0.406737i \(-0.133333\pi\)
\(72\) 0 0
\(73\) 0 0 −0.309017 − 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(74\) 0 0
\(75\) −1.78716 − 0.795697i −1.78716 − 0.795697i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.809017 − 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(80\) 0 0
\(81\) −3.36984 − 2.44833i −3.36984 − 2.44833i
\(82\) 0 0
\(83\) 0 0 −0.809017 − 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0.564602 + 1.73767i 0.564602 + 1.73767i 0.669131 + 0.743145i \(0.266667\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −2.61803 −2.61803
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.08268 + 0.786610i −1.08268 + 0.786610i −0.978148 − 0.207912i \(-0.933333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(98\) 0 0
\(99\) 2.82709 2.82709
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 1100.1.x.a.681.2 yes 8
11.10 odd 2 CM 1100.1.x.a.681.2 yes 8
25.21 even 5 inner 1100.1.x.a.21.2 ✓ 8
275.21 odd 10 inner 1100.1.x.a.21.2 ✓ 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1100.1.x.a.21.2 ✓ 8 25.21 even 5 inner
1100.1.x.a.21.2 ✓ 8 275.21 odd 10 inner
1100.1.x.a.681.2 yes 8 1.1 even 1 trivial
1100.1.x.a.681.2 yes 8 11.10 odd 2 CM