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Properties

Label 3675.1.bf.d.2432.4
Level $3675$
Weight $1$
Character 3675.2432
Analytic conductor $1.834$
Analytic rank $0$
Dimension $16$
Projective image $D_{8}$
CM discriminant -15
Inner twists $16$

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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] = [3675,1,Mod(68,3675)] 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("3675.68"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3675, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 9, 10])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3675.bf (of order \(12\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,0,0,0,0,0,0,0,0,8,0,0,0,8,-8,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,-16,0,8,0,0,0,0,0,0,0,0,-8,-16,0,0,0,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(61)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.83406392143\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{48})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Projective image: \(D_{8}\)
Projective field: Galois closure of 8.0.13897288125.2

Embedding invariants

Embedding label 2432.4
Root \(0.608761 - 0.793353i\) of defining polynomial
Character \(\chi\) \(=\) 3675.2432
Dual form 3675.1.bf.d.68.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.78480 + 0.478235i) q^{2} +(0.258819 + 0.965926i) q^{3} +(2.09077 + 1.20711i) q^{4} +1.84776i q^{6} +(1.84776 + 1.84776i) q^{8} +(-0.866025 + 0.500000i) q^{9} +(-0.624844 + 2.33195i) q^{12} +(1.20711 + 2.09077i) q^{16} +(-1.36603 + 0.366025i) q^{17} +(-1.78480 + 0.478235i) q^{18} +(-0.382683 - 0.662827i) q^{19} +(0.198092 - 0.739288i) q^{23} +(-1.30656 + 2.26303i) q^{24} +(-0.707107 - 0.707107i) q^{27} +(1.60021 + 0.923880i) q^{31} +(0.478235 + 1.78480i) q^{32} -2.61313 q^{34} -2.41421 q^{36} +(-0.366025 - 1.36603i) q^{38} +(0.707107 - 1.22474i) q^{46} +(0.366025 - 1.36603i) q^{47} +(-1.70711 + 1.70711i) q^{48} +(-0.707107 - 1.22474i) q^{51} +(0.739288 - 0.198092i) q^{53} +(-0.923880 - 1.60021i) q^{54} +(0.541196 - 0.541196i) q^{57} +(1.60021 - 0.923880i) q^{61} +(2.41421 + 2.41421i) q^{62} +1.00000i q^{64} +(-3.29788 - 0.883663i) q^{68} +0.765367 q^{69} +(-2.52409 - 0.676327i) q^{72} -1.84776i q^{76} +(-1.22474 + 0.707107i) q^{79} +(0.500000 - 0.866025i) q^{81} +(-1.00000 + 1.00000i) q^{83} +(1.30656 - 1.30656i) q^{92} +(-0.478235 + 1.78480i) q^{93} +(1.30656 - 2.26303i) q^{94} +(-1.60021 + 0.923880i) q^{96} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{12} + 8 q^{16} - 8 q^{17} - 16 q^{36} + 8 q^{38} - 8 q^{47} - 16 q^{48} + 16 q^{62} - 8 q^{68} + 8 q^{81} - 16 q^{83}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3675\mathbb{Z}\right)^\times\).

\(n\) \(1177\) \(1226\) \(2551\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(-1\) \(e\left(\frac{1}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.78480 + 0.478235i 1.78480 + 0.478235i 0.991445 − 0.130526i \(-0.0416667\pi\)
0.793353 + 0.608761i \(0.208333\pi\)
\(3\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(4\) 2.09077 + 1.20711i 2.09077 + 1.20711i
\(5\) 0 0
\(6\) 1.84776i 1.84776i
\(7\) 0 0
\(8\) 1.84776 + 1.84776i 1.84776 + 1.84776i
\(9\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(10\) 0 0
\(11\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(12\) −0.624844 + 2.33195i −0.624844 + 2.33195i
\(13\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.20711 + 2.09077i 1.20711 + 2.09077i
\(17\) −1.36603 + 0.366025i −1.36603 + 0.366025i −0.866025 − 0.500000i \(-0.833333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(18\) −1.78480 + 0.478235i −1.78480 + 0.478235i
\(19\) −0.382683 − 0.662827i −0.382683 − 0.662827i 0.608761 − 0.793353i \(-0.291667\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.198092 − 0.739288i 0.198092 − 0.739288i −0.793353 − 0.608761i \(-0.791667\pi\)
0.991445 − 0.130526i \(-0.0416667\pi\)
\(24\) −1.30656 + 2.26303i −1.30656 + 2.26303i
\(25\) 0 0
\(26\) 0 0
\(27\) −0.707107 − 0.707107i −0.707107 − 0.707107i
\(28\) 0 0
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 1.60021 + 0.923880i 1.60021 + 0.923880i 0.991445 + 0.130526i \(0.0416667\pi\)
0.608761 + 0.793353i \(0.291667\pi\)
\(32\) 0.478235 + 1.78480i 0.478235 + 1.78480i
\(33\) 0 0
\(34\) −2.61313 −2.61313
\(35\) 0 0
\(36\) −2.41421 −2.41421
\(37\) 0 0 −0.965926 − 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(38\) −0.366025 − 1.36603i −0.366025 − 1.36603i
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0.707107 − 1.22474i 0.707107 − 1.22474i
\(47\) 0.366025 − 1.36603i 0.366025 − 1.36603i −0.500000 − 0.866025i \(-0.666667\pi\)
0.866025 − 0.500000i \(-0.166667\pi\)
\(48\) −1.70711 + 1.70711i −1.70711 + 1.70711i
\(49\) 0 0
\(50\) 0 0
\(51\) −0.707107 − 1.22474i −0.707107 − 1.22474i
\(52\) 0 0
\(53\) 0.739288 − 0.198092i 0.739288 − 0.198092i 0.130526 − 0.991445i \(-0.458333\pi\)
0.608761 + 0.793353i \(0.291667\pi\)
\(54\) −0.923880 − 1.60021i −0.923880 − 1.60021i
\(55\) 0 0
\(56\) 0 0
\(57\) 0.541196 − 0.541196i 0.541196 − 0.541196i
\(58\) 0 0
\(59\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(60\) 0 0
\(61\) 1.60021 − 0.923880i 1.60021 − 0.923880i 0.608761 − 0.793353i \(-0.291667\pi\)
0.991445 − 0.130526i \(-0.0416667\pi\)
\(62\) 2.41421 + 2.41421i 2.41421 + 2.41421i
\(63\) 0 0
\(64\) 1.00000i 1.00000i
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 −0.258819 − 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(68\) −3.29788 − 0.883663i −3.29788 − 0.883663i
\(69\) 0.765367 0.765367
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) −2.52409 − 0.676327i −2.52409 − 0.676327i
\(73\) 0 0 −0.258819 − 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) − 1.84776i − 1.84776i
\(77\) 0 0
\(78\) 0 0
\(79\) −1.22474 + 0.707107i −1.22474 + 0.707107i −0.965926 − 0.258819i \(-0.916667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(80\) 0 0
\(81\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(82\) 0 0
\(83\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1.30656 − 1.30656i 1.30656 − 1.30656i
\(93\) −0.478235 + 1.78480i −0.478235 + 1.78480i
\(94\) 1.30656 − 2.26303i 1.30656 − 2.26303i
\(95\) 0 0
\(96\) −1.60021 + 0.923880i −1.60021 + 0.923880i
\(97\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(98\) 0 0
\(99\) 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 3675.1.bf.d.2432.4 16
3.2 odd 2 3675.1.bf.c.2432.1 16
5.2 odd 4 3675.1.bf.c.668.1 16
5.3 odd 4 inner 3675.1.bf.d.668.4 16
5.4 even 2 3675.1.bf.c.2432.1 16
7.2 even 3 inner 3675.1.bf.d.1832.1 16
7.3 odd 6 3675.1.k.c.2057.1 yes 8
7.4 even 3 3675.1.k.a.2057.1 yes 8
7.5 odd 6 3675.1.bf.c.1832.1 16
7.6 odd 2 3675.1.bf.c.2432.4 16
15.2 even 4 inner 3675.1.bf.d.668.4 16
15.8 even 4 3675.1.bf.c.668.1 16
15.14 odd 2 CM 3675.1.bf.d.2432.4 16
21.2 odd 6 3675.1.bf.c.1832.4 16
21.5 even 6 inner 3675.1.bf.d.1832.4 16
21.11 odd 6 3675.1.k.c.2057.4 yes 8
21.17 even 6 3675.1.k.a.2057.4 yes 8
21.20 even 2 inner 3675.1.bf.d.2432.1 16
35.2 odd 12 3675.1.bf.c.68.4 16
35.3 even 12 3675.1.k.c.293.4 yes 8
35.4 even 6 3675.1.k.c.2057.4 yes 8
35.9 even 6 3675.1.bf.c.1832.4 16
35.12 even 12 inner 3675.1.bf.d.68.4 16
35.13 even 4 3675.1.bf.c.668.4 16
35.17 even 12 3675.1.k.a.293.1 ✓ 8
35.18 odd 12 3675.1.k.a.293.4 yes 8
35.19 odd 6 inner 3675.1.bf.d.1832.4 16
35.23 odd 12 inner 3675.1.bf.d.68.1 16
35.24 odd 6 3675.1.k.a.2057.4 yes 8
35.27 even 4 inner 3675.1.bf.d.668.1 16
35.32 odd 12 3675.1.k.c.293.1 yes 8
35.33 even 12 3675.1.bf.c.68.1 16
35.34 odd 2 inner 3675.1.bf.d.2432.1 16
105.2 even 12 inner 3675.1.bf.d.68.1 16
105.17 odd 12 3675.1.k.c.293.4 yes 8
105.23 even 12 3675.1.bf.c.68.4 16
105.32 even 12 3675.1.k.a.293.4 yes 8
105.38 odd 12 3675.1.k.a.293.1 ✓ 8
105.44 odd 6 inner 3675.1.bf.d.1832.1 16
105.47 odd 12 3675.1.bf.c.68.1 16
105.53 even 12 3675.1.k.c.293.1 yes 8
105.59 even 6 3675.1.k.c.2057.1 yes 8
105.62 odd 4 3675.1.bf.c.668.4 16
105.68 odd 12 inner 3675.1.bf.d.68.4 16
105.74 odd 6 3675.1.k.a.2057.1 yes 8
105.83 odd 4 inner 3675.1.bf.d.668.1 16
105.89 even 6 3675.1.bf.c.1832.1 16
105.104 even 2 3675.1.bf.c.2432.4 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3675.1.k.a.293.1 ✓ 8 35.17 even 12
3675.1.k.a.293.1 ✓ 8 105.38 odd 12
3675.1.k.a.293.4 yes 8 35.18 odd 12
3675.1.k.a.293.4 yes 8 105.32 even 12
3675.1.k.a.2057.1 yes 8 7.4 even 3
3675.1.k.a.2057.1 yes 8 105.74 odd 6
3675.1.k.a.2057.4 yes 8 21.17 even 6
3675.1.k.a.2057.4 yes 8 35.24 odd 6
3675.1.k.c.293.1 yes 8 35.32 odd 12
3675.1.k.c.293.1 yes 8 105.53 even 12
3675.1.k.c.293.4 yes 8 35.3 even 12
3675.1.k.c.293.4 yes 8 105.17 odd 12
3675.1.k.c.2057.1 yes 8 7.3 odd 6
3675.1.k.c.2057.1 yes 8 105.59 even 6
3675.1.k.c.2057.4 yes 8 21.11 odd 6
3675.1.k.c.2057.4 yes 8 35.4 even 6
3675.1.bf.c.68.1 16 35.33 even 12
3675.1.bf.c.68.1 16 105.47 odd 12
3675.1.bf.c.68.4 16 35.2 odd 12
3675.1.bf.c.68.4 16 105.23 even 12
3675.1.bf.c.668.1 16 5.2 odd 4
3675.1.bf.c.668.1 16 15.8 even 4
3675.1.bf.c.668.4 16 35.13 even 4
3675.1.bf.c.668.4 16 105.62 odd 4
3675.1.bf.c.1832.1 16 7.5 odd 6
3675.1.bf.c.1832.1 16 105.89 even 6
3675.1.bf.c.1832.4 16 21.2 odd 6
3675.1.bf.c.1832.4 16 35.9 even 6
3675.1.bf.c.2432.1 16 3.2 odd 2
3675.1.bf.c.2432.1 16 5.4 even 2
3675.1.bf.c.2432.4 16 7.6 odd 2
3675.1.bf.c.2432.4 16 105.104 even 2
3675.1.bf.d.68.1 16 35.23 odd 12 inner
3675.1.bf.d.68.1 16 105.2 even 12 inner
3675.1.bf.d.68.4 16 35.12 even 12 inner
3675.1.bf.d.68.4 16 105.68 odd 12 inner
3675.1.bf.d.668.1 16 35.27 even 4 inner
3675.1.bf.d.668.1 16 105.83 odd 4 inner
3675.1.bf.d.668.4 16 5.3 odd 4 inner
3675.1.bf.d.668.4 16 15.2 even 4 inner
3675.1.bf.d.1832.1 16 7.2 even 3 inner
3675.1.bf.d.1832.1 16 105.44 odd 6 inner
3675.1.bf.d.1832.4 16 21.5 even 6 inner
3675.1.bf.d.1832.4 16 35.19 odd 6 inner
3675.1.bf.d.2432.1 16 21.20 even 2 inner
3675.1.bf.d.2432.1 16 35.34 odd 2 inner
3675.1.bf.d.2432.4 16 1.1 even 1 trivial
3675.1.bf.d.2432.4 16 15.14 odd 2 CM