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Properties

Label 3700.1.t.b.1449.1
Level $3700$
Weight $1$
Character 3700.1449
Analytic conductor $1.847$
Analytic rank $0$
Dimension $2$
Projective image $S_{4}$
CM/RM no
Inner twists $2$

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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] = [3700,1,Mod(549,3700)] 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("3700.549"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3700, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 2, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 3700 = 2^{2} \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3700.t (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.84654054674\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 148)
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.0.202612.1

Embedding invariants

Embedding label 1449.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3700.1449
Dual form 3700.1.t.b.549.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000i q^{7} -1.00000i q^{11} +(1.00000 - 1.00000i) q^{17} +(-1.00000 - 1.00000i) q^{19} -1.00000i q^{21} +(-1.00000 + 1.00000i) q^{23} -1.00000 q^{27} +(1.00000 - 1.00000i) q^{29} -1.00000i q^{33} -1.00000 q^{37} +1.00000i q^{41} +1.00000i q^{47} +(1.00000 - 1.00000i) q^{51} -1.00000i q^{53} +(-1.00000 - 1.00000i) q^{57} +(-1.00000 + 1.00000i) q^{69} +1.00000 q^{71} +1.00000 q^{73} -1.00000 q^{77} +(1.00000 + 1.00000i) q^{79} -1.00000 q^{81} +1.00000i q^{83} +(1.00000 - 1.00000i) q^{87} +(1.00000 - 1.00000i) q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{17} - 2 q^{19} - 2 q^{23} - 2 q^{27} + 2 q^{29} - 2 q^{37} + 2 q^{51} - 2 q^{57} - 2 q^{69} + 2 q^{71} + 2 q^{73} - 2 q^{77} + 2 q^{79} - 2 q^{81} + 2 q^{87} + 2 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3700\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1777\) \(1851\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(-1\) \(1\)

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
\(3\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 1.00000i − 1.00000i −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 − 0.500000i \(-0.166667\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 1.00000i − 1.00000i −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 − 0.500000i \(-0.166667\pi\)
\(12\) 0 0
\(13\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.00000 − 1.00000i 1.00000 − 1.00000i − 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(18\) 0 0
\(19\) −1.00000 − 1.00000i −1.00000 − 1.00000i − 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) − 1.00000i − 1.00000i
\(22\) 0 0
\(23\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −1.00000 −1.00000
\(28\) 0 0
\(29\) 1.00000 − 1.00000i 1.00000 − 1.00000i − 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(30\) 0 0
\(31\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(32\) 0 0
\(33\) − 1.00000i − 1.00000i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1.00000 −1.00000
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(42\) 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 0
\(47\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 1.00000 − 1.00000i 1.00000 − 1.00000i
\(52\) 0 0
\(53\) − 1.00000i − 1.00000i −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 − 0.500000i \(-0.166667\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −1.00000 − 1.00000i −1.00000 − 1.00000i
\(58\) 0 0
\(59\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(60\) 0 0
\(61\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(70\) 0 0
\(71\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(72\) 0 0
\(73\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.00000 −1.00000
\(78\) 0 0
\(79\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) −1.00000 −1.00000
\(82\) 0 0
\(83\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 1.00000 − 1.00000i 1.00000 − 1.00000i
\(88\) 0 0
\(89\) 1.00000 − 1.00000i 1.00000 − 1.00000i − 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 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 3700.1.t.b.1449.1 2
5.2 odd 4 3700.1.j.c.1301.1 2
5.3 odd 4 148.1.f.a.117.1 yes 2
5.4 even 2 3700.1.t.a.1449.1 2
15.8 even 4 1332.1.o.a.1153.1 2
20.3 even 4 592.1.k.b.561.1 2
37.31 odd 4 3700.1.t.a.549.1 2
40.3 even 4 2368.1.k.b.1153.1 2
40.13 odd 4 2368.1.k.a.1153.1 2
185.68 even 4 148.1.f.a.105.1 ✓ 2
185.142 even 4 3700.1.j.c.401.1 2
185.179 odd 4 inner 3700.1.t.b.549.1 2
555.68 odd 4 1332.1.o.a.253.1 2
740.623 odd 4 592.1.k.b.401.1 2
1480.253 even 4 2368.1.k.a.2177.1 2
1480.1363 odd 4 2368.1.k.b.2177.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
148.1.f.a.105.1 ✓ 2 185.68 even 4
148.1.f.a.117.1 yes 2 5.3 odd 4
592.1.k.b.401.1 2 740.623 odd 4
592.1.k.b.561.1 2 20.3 even 4
1332.1.o.a.253.1 2 555.68 odd 4
1332.1.o.a.1153.1 2 15.8 even 4
2368.1.k.a.1153.1 2 40.13 odd 4
2368.1.k.a.2177.1 2 1480.253 even 4
2368.1.k.b.1153.1 2 40.3 even 4
2368.1.k.b.2177.1 2 1480.1363 odd 4
3700.1.j.c.401.1 2 185.142 even 4
3700.1.j.c.1301.1 2 5.2 odd 4
3700.1.t.a.549.1 2 37.31 odd 4
3700.1.t.a.1449.1 2 5.4 even 2
3700.1.t.b.549.1 2 185.179 odd 4 inner
3700.1.t.b.1449.1 2 1.1 even 1 trivial