Properties

Label 3800.1.cq.b.899.1
Level $3800$
Weight $1$
Character 3800.899
Analytic conductor $1.896$
Analytic rank $0$
Dimension $12$
Projective image $D_{9}$
CM discriminant -8
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] = [3800,1,Mod(99,3800)] 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("3800.99"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3800, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([9, 9, 9, 2])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3800.cq (of order \(18\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,0,0,-6,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(24)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.89644704801\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{18})\)
Coefficient field: \(\Q(\zeta_{36})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{6} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 152)
Projective image: \(D_{9}\)
Projective field: Galois closure of 9.1.69564674215936.1

Embedding invariants

Embedding label 899.1
Root \(-0.642788 - 0.766044i\) of defining polynomial
Character \(\chi\) \(=\) 3800.899
Dual form 3800.1.cq.b.1099.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.642788 - 0.766044i) q^{2} +(0.524005 - 1.43969i) q^{3} +(-0.173648 + 0.984808i) q^{4} +(-1.43969 + 0.524005i) q^{6} +(0.866025 - 0.500000i) q^{8} +(-1.03209 - 0.866025i) q^{9} +(0.939693 + 1.62760i) q^{11} +(1.32683 + 0.766044i) q^{12} +(-0.939693 - 0.342020i) q^{16} +(0.642788 + 0.766044i) q^{17} +1.34730i q^{18} +(-0.173648 + 0.984808i) q^{19} +(0.642788 - 1.76604i) q^{22} +(-0.266044 - 1.50881i) q^{24} +(-0.460802 + 0.266044i) q^{27} +(0.342020 + 0.939693i) q^{32} +(2.83564 - 0.500000i) q^{33} +(0.173648 - 0.984808i) q^{34} +(1.03209 - 0.866025i) q^{36} +(0.866025 - 0.500000i) q^{38} +(-0.326352 - 0.118782i) q^{41} +(0.984808 - 0.173648i) q^{43} +(-1.76604 + 0.642788i) q^{44} +(-0.984808 + 1.17365i) q^{48} +(0.500000 + 0.866025i) q^{49} +(1.43969 - 0.524005i) q^{51} +(0.500000 + 0.181985i) q^{54} +(1.32683 + 0.766044i) q^{57} +(1.43969 - 1.20805i) q^{59} +(0.500000 - 0.866025i) q^{64} +(-2.20574 - 1.85083i) q^{66} +(0.223238 - 0.266044i) q^{67} +(-0.866025 + 0.500000i) q^{68} +(-1.32683 - 0.233956i) q^{72} +(-0.642788 + 1.76604i) q^{73} +(-0.939693 - 0.342020i) q^{76} +(-0.0923963 - 0.524005i) q^{81} +(0.118782 + 0.326352i) q^{82} +(-0.300767 - 0.173648i) q^{83} +(-0.766044 - 0.642788i) q^{86} +(1.62760 + 0.939693i) q^{88} +(-0.939693 + 0.342020i) q^{89} +1.53209 q^{96} +(-0.223238 - 0.266044i) q^{97} +(0.342020 - 0.939693i) q^{98} +(0.439693 - 2.49362i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{6} + 6 q^{9} + 6 q^{24} - 6 q^{36} - 6 q^{41} - 12 q^{44} + 6 q^{49} + 6 q^{51} + 6 q^{54} + 6 q^{59} + 6 q^{64} - 6 q^{66} + 6 q^{81} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(951\) \(1901\) \(1977\)
\(\chi(n)\) \(e\left(\frac{7}{9}\right)\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.642788 − 0.766044i −0.642788 − 0.766044i
\(3\) 0.524005 − 1.43969i 0.524005 − 1.43969i −0.342020 − 0.939693i \(-0.611111\pi\)
0.866025 − 0.500000i \(-0.166667\pi\)
\(4\) −0.173648 + 0.984808i −0.173648 + 0.984808i
\(5\) 0 0
\(6\) −1.43969 + 0.524005i −1.43969 + 0.524005i
\(7\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(8\) 0.866025 − 0.500000i 0.866025 − 0.500000i
\(9\) −1.03209 − 0.866025i −1.03209 − 0.866025i
\(10\) 0 0
\(11\) 0.939693 + 1.62760i 0.939693 + 1.62760i 0.766044 + 0.642788i \(0.222222\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(12\) 1.32683 + 0.766044i 1.32683 + 0.766044i
\(13\) 0 0 −0.342020 − 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.939693 − 0.342020i −0.939693 − 0.342020i
\(17\) 0.642788 + 0.766044i 0.642788 + 0.766044i 0.984808 − 0.173648i \(-0.0555556\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(18\) 1.34730i 1.34730i
\(19\) −0.173648 + 0.984808i −0.173648 + 0.984808i
\(20\) 0 0
\(21\) 0 0
\(22\) 0.642788 − 1.76604i 0.642788 − 1.76604i
\(23\) 0 0 −0.984808 − 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(24\) −0.266044 − 1.50881i −0.266044 − 1.50881i
\(25\) 0 0
\(26\) 0 0
\(27\) −0.460802 + 0.266044i −0.460802 + 0.266044i
\(28\) 0 0
\(29\) 0 0 −0.766044 − 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(30\) 0 0
\(31\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(32\) 0.342020 + 0.939693i 0.342020 + 0.939693i
\(33\) 2.83564 − 0.500000i 2.83564 − 0.500000i
\(34\) 0.173648 − 0.984808i 0.173648 − 0.984808i
\(35\) 0 0
\(36\) 1.03209 − 0.866025i 1.03209 − 0.866025i
\(37\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0.866025 − 0.500000i 0.866025 − 0.500000i
\(39\) 0 0
\(40\) 0 0
\(41\) −0.326352 − 0.118782i −0.326352 − 0.118782i 0.173648 − 0.984808i \(-0.444444\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(42\) 0 0
\(43\) 0.984808 − 0.173648i 0.984808 − 0.173648i 0.342020 − 0.939693i \(-0.388889\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(44\) −1.76604 + 0.642788i −1.76604 + 0.642788i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.642788 − 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(48\) −0.984808 + 1.17365i −0.984808 + 1.17365i
\(49\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(50\) 0 0
\(51\) 1.43969 − 0.524005i 1.43969 − 0.524005i
\(52\) 0 0
\(53\) 0 0 −0.984808 − 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(54\) 0.500000 + 0.181985i 0.500000 + 0.181985i
\(55\) 0 0
\(56\) 0 0
\(57\) 1.32683 + 0.766044i 1.32683 + 0.766044i
\(58\) 0 0
\(59\) 1.43969 − 1.20805i 1.43969 − 1.20805i 0.500000 − 0.866025i \(-0.333333\pi\)
0.939693 − 0.342020i \(-0.111111\pi\)
\(60\) 0 0
\(61\) 0 0 0.173648 − 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(65\) 0 0
\(66\) −2.20574 − 1.85083i −2.20574 − 1.85083i
\(67\) 0.223238 − 0.266044i 0.223238 − 0.266044i −0.642788 − 0.766044i \(-0.722222\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(68\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.173648 − 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(72\) −1.32683 − 0.233956i −1.32683 − 0.233956i
\(73\) −0.642788 + 1.76604i −0.642788 + 1.76604i 1.00000i \(0.5\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −0.939693 − 0.342020i −0.939693 − 0.342020i
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.939693 − 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(80\) 0 0
\(81\) −0.0923963 − 0.524005i −0.0923963 − 0.524005i
\(82\) 0.118782 + 0.326352i 0.118782 + 0.326352i
\(83\) −0.300767 − 0.173648i −0.300767 − 0.173648i 0.342020 − 0.939693i \(-0.388889\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −0.766044 − 0.642788i −0.766044 − 0.642788i
\(87\) 0 0
\(88\) 1.62760 + 0.939693i 1.62760 + 0.939693i
\(89\) −0.939693 + 0.342020i −0.939693 + 0.342020i −0.766044 − 0.642788i \(-0.777778\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 1.53209 1.53209
\(97\) −0.223238 − 0.266044i −0.223238 − 0.266044i 0.642788 − 0.766044i \(-0.277778\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(98\) 0.342020 − 0.939693i 0.342020 − 0.939693i
\(99\) 0.439693 − 2.49362i 0.439693 − 2.49362i
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 3800.1.cq.b.899.1 12
5.2 odd 4 152.1.u.a.139.1 yes 6
5.3 odd 4 3800.1.cv.c.1051.1 6
5.4 even 2 inner 3800.1.cq.b.899.2 12
8.3 odd 2 CM 3800.1.cq.b.899.1 12
15.2 even 4 1368.1.eh.a.595.1 6
19.16 even 9 inner 3800.1.cq.b.1099.2 12
20.7 even 4 608.1.bg.a.367.1 6
40.3 even 4 3800.1.cv.c.1051.1 6
40.19 odd 2 inner 3800.1.cq.b.899.2 12
40.27 even 4 152.1.u.a.139.1 yes 6
40.37 odd 4 608.1.bg.a.367.1 6
95.2 even 36 2888.1.u.f.1867.1 6
95.7 odd 12 2888.1.u.g.2555.1 6
95.12 even 12 2888.1.u.a.2555.1 6
95.17 odd 36 2888.1.u.b.1867.1 6
95.22 even 36 2888.1.u.e.1859.1 6
95.27 even 12 2888.1.u.f.99.1 6
95.32 even 36 2888.1.k.c.2819.3 6
95.37 even 4 2888.1.u.e.595.1 6
95.42 odd 36 2888.1.f.d.723.3 3
95.47 odd 36 2888.1.k.b.2595.1 6
95.52 even 36 2888.1.u.a.2411.1 6
95.54 even 18 inner 3800.1.cq.b.1099.1 12
95.62 odd 36 2888.1.u.g.2411.1 6
95.67 even 36 2888.1.k.c.2595.3 6
95.72 even 36 2888.1.f.c.723.1 3
95.73 odd 36 3800.1.cv.c.1251.1 6
95.82 odd 36 2888.1.k.b.2819.1 6
95.87 odd 12 2888.1.u.b.99.1 6
95.92 odd 36 152.1.u.a.35.1 ✓ 6
120.107 odd 4 1368.1.eh.a.595.1 6
152.35 odd 18 inner 3800.1.cq.b.1099.2 12
285.92 even 36 1368.1.eh.a.1099.1 6
380.187 even 36 608.1.bg.a.111.1 6
760.27 odd 12 2888.1.u.f.99.1 6
760.67 odd 36 2888.1.k.c.2595.3 6
760.107 odd 12 2888.1.u.a.2555.1 6
760.147 odd 36 2888.1.u.a.2411.1 6
760.187 even 36 152.1.u.a.35.1 ✓ 6
760.227 odd 4 2888.1.u.e.595.1 6
760.307 odd 36 2888.1.u.e.1859.1 6
760.339 odd 18 inner 3800.1.cq.b.1099.1 12
760.347 even 36 2888.1.u.g.2411.1 6
760.387 even 12 2888.1.u.g.2555.1 6
760.427 even 36 2888.1.k.b.2595.1 6
760.467 even 12 2888.1.u.b.99.1 6
760.507 odd 36 2888.1.k.c.2819.3 6
760.547 odd 36 2888.1.f.c.723.1 3
760.587 even 36 2888.1.u.b.1867.1 6
760.643 even 36 3800.1.cv.c.1251.1 6
760.667 odd 36 2888.1.u.f.1867.1 6
760.707 even 36 2888.1.f.d.723.3 3
760.747 even 36 2888.1.k.b.2819.1 6
760.757 odd 36 608.1.bg.a.111.1 6
2280.947 odd 36 1368.1.eh.a.1099.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.1.u.a.35.1 ✓ 6 95.92 odd 36
152.1.u.a.35.1 ✓ 6 760.187 even 36
152.1.u.a.139.1 yes 6 5.2 odd 4
152.1.u.a.139.1 yes 6 40.27 even 4
608.1.bg.a.111.1 6 380.187 even 36
608.1.bg.a.111.1 6 760.757 odd 36
608.1.bg.a.367.1 6 20.7 even 4
608.1.bg.a.367.1 6 40.37 odd 4
1368.1.eh.a.595.1 6 15.2 even 4
1368.1.eh.a.595.1 6 120.107 odd 4
1368.1.eh.a.1099.1 6 285.92 even 36
1368.1.eh.a.1099.1 6 2280.947 odd 36
2888.1.f.c.723.1 3 95.72 even 36
2888.1.f.c.723.1 3 760.547 odd 36
2888.1.f.d.723.3 3 95.42 odd 36
2888.1.f.d.723.3 3 760.707 even 36
2888.1.k.b.2595.1 6 95.47 odd 36
2888.1.k.b.2595.1 6 760.427 even 36
2888.1.k.b.2819.1 6 95.82 odd 36
2888.1.k.b.2819.1 6 760.747 even 36
2888.1.k.c.2595.3 6 95.67 even 36
2888.1.k.c.2595.3 6 760.67 odd 36
2888.1.k.c.2819.3 6 95.32 even 36
2888.1.k.c.2819.3 6 760.507 odd 36
2888.1.u.a.2411.1 6 95.52 even 36
2888.1.u.a.2411.1 6 760.147 odd 36
2888.1.u.a.2555.1 6 95.12 even 12
2888.1.u.a.2555.1 6 760.107 odd 12
2888.1.u.b.99.1 6 95.87 odd 12
2888.1.u.b.99.1 6 760.467 even 12
2888.1.u.b.1867.1 6 95.17 odd 36
2888.1.u.b.1867.1 6 760.587 even 36
2888.1.u.e.595.1 6 95.37 even 4
2888.1.u.e.595.1 6 760.227 odd 4
2888.1.u.e.1859.1 6 95.22 even 36
2888.1.u.e.1859.1 6 760.307 odd 36
2888.1.u.f.99.1 6 95.27 even 12
2888.1.u.f.99.1 6 760.27 odd 12
2888.1.u.f.1867.1 6 95.2 even 36
2888.1.u.f.1867.1 6 760.667 odd 36
2888.1.u.g.2411.1 6 95.62 odd 36
2888.1.u.g.2411.1 6 760.347 even 36
2888.1.u.g.2555.1 6 95.7 odd 12
2888.1.u.g.2555.1 6 760.387 even 12
3800.1.cq.b.899.1 12 1.1 even 1 trivial
3800.1.cq.b.899.1 12 8.3 odd 2 CM
3800.1.cq.b.899.2 12 5.4 even 2 inner
3800.1.cq.b.899.2 12 40.19 odd 2 inner
3800.1.cq.b.1099.1 12 95.54 even 18 inner
3800.1.cq.b.1099.1 12 760.339 odd 18 inner
3800.1.cq.b.1099.2 12 19.16 even 9 inner
3800.1.cq.b.1099.2 12 152.35 odd 18 inner
3800.1.cv.c.1051.1 6 5.3 odd 4
3800.1.cv.c.1051.1 6 40.3 even 4
3800.1.cv.c.1251.1 6 95.73 odd 36
3800.1.cv.c.1251.1 6 760.643 even 36