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Properties

Label 3675.1.u.e.851.3
Level $3675$
Weight $1$
Character 3675.851
Analytic conductor $1.834$
Analytic rank $0$
Dimension $8$
Projective image $S_{4}$
CM/RM no
Inner twists $8$

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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(851,3675)] 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("3675.851"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3675, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 4])) 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.u (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(18)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.83406392143\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.2.77175.1

Embedding invariants

Embedding label 851.3
Root \(0.258819 - 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 3675.851
Dual form 3675.1.u.e.1451.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.866025 - 0.500000i) q^{2} +(-0.965926 + 0.258819i) q^{3} +(-0.707107 + 0.707107i) q^{6} +1.00000i q^{8} +(0.866025 - 0.500000i) q^{9} +(0.866025 + 0.500000i) q^{11} +1.41421 q^{13} +(0.500000 + 0.866025i) q^{16} +(-1.22474 - 0.707107i) q^{17} +(0.500000 - 0.866025i) q^{18} +1.00000 q^{22} +(-0.866025 + 0.500000i) q^{23} +(-0.258819 - 0.965926i) q^{24} +(1.22474 - 0.707107i) q^{26} +(-0.707107 + 0.707107i) q^{27} -1.00000i q^{29} +(-0.965926 - 0.258819i) q^{33} -1.41421 q^{34} +(0.500000 + 0.866025i) q^{37} +(-1.36603 + 0.366025i) q^{39} +1.41421i q^{41} +1.00000 q^{43} +(-0.500000 + 0.866025i) q^{46} +(-0.707107 - 0.707107i) q^{48} +(1.36603 + 0.366025i) q^{51} +(-0.258819 + 0.965926i) q^{54} +(-0.500000 - 0.866025i) q^{58} +(1.22474 + 0.707107i) q^{59} +(0.707107 + 1.22474i) q^{61} -1.00000 q^{64} +(-0.965926 + 0.258819i) q^{66} +(-0.500000 + 0.866025i) q^{67} +(0.707107 - 0.707107i) q^{69} -1.00000i q^{71} +(0.500000 + 0.866025i) q^{72} +(0.866025 + 0.500000i) q^{74} +(-1.00000 + 1.00000i) q^{78} +(0.500000 + 0.866025i) q^{79} +(0.500000 - 0.866025i) q^{81} +(0.707107 + 1.22474i) q^{82} +(0.866025 - 0.500000i) q^{86} +(0.258819 + 0.965926i) q^{87} +(-0.500000 + 0.866025i) q^{88} +(1.22474 - 0.707107i) q^{89} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{16} + 4 q^{18} + 8 q^{22} + 4 q^{37} - 4 q^{39} + 8 q^{43} - 4 q^{46} + 4 q^{51} - 4 q^{58} - 8 q^{64} - 4 q^{67} + 4 q^{72} - 8 q^{78} + 4 q^{79} + 4 q^{81} - 4 q^{88} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(1177\) \(1226\) \(2551\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.866025 − 0.500000i 0.866025 − 0.500000i − 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(3\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(4\) 0 0
\(5\) 0 0
\(6\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(7\) 0 0
\(8\) 1.00000i 1.00000i
\(9\) 0.866025 − 0.500000i 0.866025 − 0.500000i
\(10\) 0 0
\(11\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 − 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 1.41421 1.41421 0.707107 − 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(17\) −1.22474 − 0.707107i −1.22474 − 0.707107i −0.258819 − 0.965926i \(-0.583333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(18\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(19\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1.00000 1.00000
\(23\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 − 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(24\) −0.258819 − 0.965926i −0.258819 − 0.965926i
\(25\) 0 0
\(26\) 1.22474 − 0.707107i 1.22474 − 0.707107i
\(27\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(28\) 0 0
\(29\) − 1.00000i − 1.00000i −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 − 0.500000i \(-0.166667\pi\)
\(30\) 0 0
\(31\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(32\) 0 0
\(33\) −0.965926 − 0.258819i −0.965926 − 0.258819i
\(34\) −1.41421 −1.41421
\(35\) 0 0
\(36\) 0 0
\(37\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(38\) 0 0
\(39\) −1.36603 + 0.366025i −1.36603 + 0.366025i
\(40\) 0 0
\(41\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(42\) 0 0
\(43\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(47\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(48\) −0.707107 − 0.707107i −0.707107 − 0.707107i
\(49\) 0 0
\(50\) 0 0
\(51\) 1.36603 + 0.366025i 1.36603 + 0.366025i
\(52\) 0 0
\(53\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(54\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(59\) 1.22474 + 0.707107i 1.22474 + 0.707107i 0.965926 − 0.258819i \(-0.0833333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(60\) 0 0
\(61\) 0.707107 + 1.22474i 0.707107 + 1.22474i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −1.00000 −1.00000
\(65\) 0 0
\(66\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(67\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0.707107 − 0.707107i 0.707107 − 0.707107i
\(70\) 0 0
\(71\) − 1.00000i − 1.00000i −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 − 0.500000i \(-0.166667\pi\)
\(72\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(73\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(74\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(79\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(80\) 0 0
\(81\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(82\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0.866025 − 0.500000i 0.866025 − 0.500000i
\(87\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(88\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(89\) 1.22474 − 0.707107i 1.22474 − 0.707107i 0.258819 − 0.965926i \(-0.416667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) 1.00000 1.00000
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.u.e.851.3 8
3.2 odd 2 inner 3675.1.u.e.851.2 8
5.2 odd 4 3675.1.p.c.2174.3 8
5.3 odd 4 3675.1.p.b.2174.2 8
5.4 even 2 3675.1.u.d.851.2 8
7.2 even 3 inner 3675.1.u.e.1451.2 8
7.3 odd 6 3675.1.c.g.1226.3 yes 4
7.4 even 3 3675.1.c.g.1226.4 yes 4
7.5 odd 6 inner 3675.1.u.e.1451.1 8
7.6 odd 2 inner 3675.1.u.e.851.4 8
15.2 even 4 3675.1.p.b.2174.4 8
15.8 even 4 3675.1.p.c.2174.1 8
15.14 odd 2 3675.1.u.d.851.3 8
21.2 odd 6 inner 3675.1.u.e.1451.3 8
21.5 even 6 inner 3675.1.u.e.1451.4 8
21.11 odd 6 3675.1.c.g.1226.2 yes 4
21.17 even 6 3675.1.c.g.1226.1 ✓ 4
21.20 even 2 inner 3675.1.u.e.851.1 8
35.2 odd 12 3675.1.p.c.2774.1 8
35.3 even 12 3675.1.f.d.2549.3 4
35.4 even 6 3675.1.c.h.1226.1 yes 4
35.9 even 6 3675.1.u.d.1451.3 8
35.12 even 12 3675.1.p.c.2774.4 8
35.13 even 4 3675.1.p.b.2174.3 8
35.17 even 12 3675.1.f.c.2549.2 4
35.18 odd 12 3675.1.f.d.2549.2 4
35.19 odd 6 3675.1.u.d.1451.4 8
35.23 odd 12 3675.1.p.b.2774.4 8
35.24 odd 6 3675.1.c.h.1226.2 yes 4
35.27 even 4 3675.1.p.c.2174.2 8
35.32 odd 12 3675.1.f.c.2549.3 4
35.33 even 12 3675.1.p.b.2774.1 8
35.34 odd 2 3675.1.u.d.851.1 8
105.2 even 12 3675.1.p.b.2774.2 8
105.17 odd 12 3675.1.f.d.2549.4 4
105.23 even 12 3675.1.p.c.2774.3 8
105.32 even 12 3675.1.f.d.2549.1 4
105.38 odd 12 3675.1.f.c.2549.1 4
105.44 odd 6 3675.1.u.d.1451.2 8
105.47 odd 12 3675.1.p.b.2774.3 8
105.53 even 12 3675.1.f.c.2549.4 4
105.59 even 6 3675.1.c.h.1226.4 yes 4
105.62 odd 4 3675.1.p.b.2174.1 8
105.68 odd 12 3675.1.p.c.2774.2 8
105.74 odd 6 3675.1.c.h.1226.3 yes 4
105.83 odd 4 3675.1.p.c.2174.4 8
105.89 even 6 3675.1.u.d.1451.1 8
105.104 even 2 3675.1.u.d.851.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3675.1.c.g.1226.1 ✓ 4 21.17 even 6
3675.1.c.g.1226.2 yes 4 21.11 odd 6
3675.1.c.g.1226.3 yes 4 7.3 odd 6
3675.1.c.g.1226.4 yes 4 7.4 even 3
3675.1.c.h.1226.1 yes 4 35.4 even 6
3675.1.c.h.1226.2 yes 4 35.24 odd 6
3675.1.c.h.1226.3 yes 4 105.74 odd 6
3675.1.c.h.1226.4 yes 4 105.59 even 6
3675.1.f.c.2549.1 4 105.38 odd 12
3675.1.f.c.2549.2 4 35.17 even 12
3675.1.f.c.2549.3 4 35.32 odd 12
3675.1.f.c.2549.4 4 105.53 even 12
3675.1.f.d.2549.1 4 105.32 even 12
3675.1.f.d.2549.2 4 35.18 odd 12
3675.1.f.d.2549.3 4 35.3 even 12
3675.1.f.d.2549.4 4 105.17 odd 12
3675.1.p.b.2174.1 8 105.62 odd 4
3675.1.p.b.2174.2 8 5.3 odd 4
3675.1.p.b.2174.3 8 35.13 even 4
3675.1.p.b.2174.4 8 15.2 even 4
3675.1.p.b.2774.1 8 35.33 even 12
3675.1.p.b.2774.2 8 105.2 even 12
3675.1.p.b.2774.3 8 105.47 odd 12
3675.1.p.b.2774.4 8 35.23 odd 12
3675.1.p.c.2174.1 8 15.8 even 4
3675.1.p.c.2174.2 8 35.27 even 4
3675.1.p.c.2174.3 8 5.2 odd 4
3675.1.p.c.2174.4 8 105.83 odd 4
3675.1.p.c.2774.1 8 35.2 odd 12
3675.1.p.c.2774.2 8 105.68 odd 12
3675.1.p.c.2774.3 8 105.23 even 12
3675.1.p.c.2774.4 8 35.12 even 12
3675.1.u.d.851.1 8 35.34 odd 2
3675.1.u.d.851.2 8 5.4 even 2
3675.1.u.d.851.3 8 15.14 odd 2
3675.1.u.d.851.4 8 105.104 even 2
3675.1.u.d.1451.1 8 105.89 even 6
3675.1.u.d.1451.2 8 105.44 odd 6
3675.1.u.d.1451.3 8 35.9 even 6
3675.1.u.d.1451.4 8 35.19 odd 6
3675.1.u.e.851.1 8 21.20 even 2 inner
3675.1.u.e.851.2 8 3.2 odd 2 inner
3675.1.u.e.851.3 8 1.1 even 1 trivial
3675.1.u.e.851.4 8 7.6 odd 2 inner
3675.1.u.e.1451.1 8 7.5 odd 6 inner
3675.1.u.e.1451.2 8 7.2 even 3 inner
3675.1.u.e.1451.3 8 21.2 odd 6 inner
3675.1.u.e.1451.4 8 21.5 even 6 inner