Properties

Label 120.2.m.a.59.1
Level $120$
Weight $2$
Character 120.59
Analytic conductor $0.958$
Analytic rank $0$
Dimension $4$
CM discriminant -15
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] = [120,2,Mod(59,120)] 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("120.59"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(120, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 120 = 2^{3} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 120.m (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.958204824255\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{-5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 59.1
Root \(-0.866025 - 1.11803i\) of defining polynomial
Character \(\chi\) \(=\) 120.59
Dual form 120.2.m.a.59.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.866025 - 1.11803i) q^{2} +1.73205 q^{3} +(-0.500000 + 1.93649i) q^{4} -2.23607i q^{5} +(-1.50000 - 1.93649i) q^{6} +(2.59808 - 1.11803i) q^{8} +3.00000 q^{9} +(-2.50000 + 1.93649i) q^{10} +(-0.866025 + 3.35410i) q^{12} -3.87298i q^{15} +(-3.50000 - 1.93649i) q^{16} -6.92820 q^{17} +(-2.59808 - 3.35410i) q^{18} +4.00000 q^{19} +(4.33013 + 1.11803i) q^{20} +8.94427i q^{23} +(4.50000 - 1.93649i) q^{24} -5.00000 q^{25} +5.19615 q^{27} +(-4.33013 + 3.35410i) q^{30} +7.74597i q^{31} +(0.866025 + 5.59017i) q^{32} +(6.00000 + 7.74597i) q^{34} +(-1.50000 + 5.80948i) q^{36} +(-3.46410 - 4.47214i) q^{38} +(-2.50000 - 5.80948i) q^{40} -6.70820i q^{45} +(10.0000 - 7.74597i) q^{46} -8.94427i q^{47} +(-6.06218 - 3.35410i) q^{48} -7.00000 q^{49} +(4.33013 + 5.59017i) q^{50} -12.0000 q^{51} +4.47214i q^{53} +(-4.50000 - 5.80948i) q^{54} +6.92820 q^{57} +(7.50000 + 1.93649i) q^{60} -15.4919i q^{61} +(8.66025 - 6.70820i) q^{62} +(5.50000 - 5.80948i) q^{64} +(3.46410 - 13.4164i) q^{68} +15.4919i q^{69} +(7.79423 - 3.35410i) q^{72} -8.66025 q^{75} +(-2.00000 + 7.74597i) q^{76} -7.74597i q^{79} +(-4.33013 + 7.82624i) q^{80} +9.00000 q^{81} -3.46410 q^{83} +15.4919i q^{85} +(-7.50000 + 5.80948i) q^{90} +(-17.3205 - 4.47214i) q^{92} +13.4164i q^{93} +(-10.0000 + 7.74597i) q^{94} -8.94427i q^{95} +(1.50000 + 9.68246i) q^{96} +(6.06218 + 7.82624i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{4} - 6 q^{6} + 12 q^{9} - 10 q^{10} - 14 q^{16} + 16 q^{19} + 18 q^{24} - 20 q^{25} + 24 q^{34} - 6 q^{36} - 10 q^{40} + 40 q^{46} - 28 q^{49} - 48 q^{51} - 18 q^{54} + 30 q^{60} + 22 q^{64}+ \cdots + 6 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/120\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(41\) \(61\) \(97\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.866025 − 1.11803i −0.612372 − 0.790569i
\(3\) 1.73205 1.00000
\(4\) −0.500000 + 1.93649i −0.250000 + 0.968246i
\(5\) − 2.23607i − 1.00000i
\(6\) −1.50000 − 1.93649i −0.612372 − 0.790569i
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 2.59808 − 1.11803i 0.918559 − 0.395285i
\(9\) 3.00000 1.00000
\(10\) −2.50000 + 1.93649i −0.790569 + 0.612372i
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) −0.866025 + 3.35410i −0.250000 + 0.968246i
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) − 3.87298i − 1.00000i
\(16\) −3.50000 − 1.93649i −0.875000 − 0.484123i
\(17\) −6.92820 −1.68034 −0.840168 − 0.542326i \(-0.817544\pi\)
−0.840168 + 0.542326i \(0.817544\pi\)
\(18\) −2.59808 − 3.35410i −0.612372 − 0.790569i
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 4.33013 + 1.11803i 0.968246 + 0.250000i
\(21\) 0 0
\(22\) 0 0
\(23\) 8.94427i 1.86501i 0.361158 + 0.932505i \(0.382382\pi\)
−0.361158 + 0.932505i \(0.617618\pi\)
\(24\) 4.50000 − 1.93649i 0.918559 − 0.395285i
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) 5.19615 1.00000
\(28\) 0 0
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) −4.33013 + 3.35410i −0.790569 + 0.612372i
\(31\) 7.74597i 1.39122i 0.718421 + 0.695608i \(0.244865\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 0.866025 + 5.59017i 0.153093 + 0.988212i
\(33\) 0 0
\(34\) 6.00000 + 7.74597i 1.02899 + 1.32842i
\(35\) 0 0
\(36\) −1.50000 + 5.80948i −0.250000 + 0.968246i
\(37\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) −3.46410 − 4.47214i −0.561951 − 0.725476i
\(39\) 0 0
\(40\) −2.50000 − 5.80948i −0.395285 − 0.918559i
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) − 6.70820i − 1.00000i
\(46\) 10.0000 − 7.74597i 1.47442 − 1.14208i
\(47\) − 8.94427i − 1.30466i −0.757937 − 0.652328i \(-0.773792\pi\)
0.757937 − 0.652328i \(-0.226208\pi\)
\(48\) −6.06218 − 3.35410i −0.875000 − 0.484123i
\(49\) −7.00000 −1.00000
\(50\) 4.33013 + 5.59017i 0.612372 + 0.790569i
\(51\) −12.0000 −1.68034
\(52\) 0 0
\(53\) 4.47214i 0.614295i 0.951662 + 0.307148i \(0.0993745\pi\)
−0.951662 + 0.307148i \(0.900625\pi\)
\(54\) −4.50000 − 5.80948i −0.612372 − 0.790569i
\(55\) 0 0
\(56\) 0 0
\(57\) 6.92820 0.917663
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 7.50000 + 1.93649i 0.968246 + 0.250000i
\(61\) − 15.4919i − 1.98354i −0.128037 − 0.991769i \(-0.540868\pi\)
0.128037 − 0.991769i \(-0.459132\pi\)
\(62\) 8.66025 − 6.70820i 1.09985 − 0.851943i
\(63\) 0 0
\(64\) 5.50000 − 5.80948i 0.687500 − 0.726184i
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 3.46410 − 13.4164i 0.420084 − 1.62698i
\(69\) 15.4919i 1.86501i
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 7.79423 − 3.35410i 0.918559 − 0.395285i
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) −8.66025 −1.00000
\(76\) −2.00000 + 7.74597i −0.229416 + 0.888523i
\(77\) 0 0
\(78\) 0 0
\(79\) − 7.74597i − 0.871489i −0.900070 − 0.435745i \(-0.856485\pi\)
0.900070 − 0.435745i \(-0.143515\pi\)
\(80\) −4.33013 + 7.82624i −0.484123 + 0.875000i
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) −3.46410 −0.380235 −0.190117 − 0.981761i \(-0.560887\pi\)
−0.190117 + 0.981761i \(0.560887\pi\)
\(84\) 0 0
\(85\) 15.4919i 1.68034i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) −7.50000 + 5.80948i −0.790569 + 0.612372i
\(91\) 0 0
\(92\) −17.3205 − 4.47214i −1.80579 − 0.466252i
\(93\) 13.4164i 1.39122i
\(94\) −10.0000 + 7.74597i −1.03142 + 0.798935i
\(95\) − 8.94427i − 0.917663i
\(96\) 1.50000 + 9.68246i 0.153093 + 0.988212i
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 6.06218 + 7.82624i 0.612372 + 0.790569i
\(99\) 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 120.2.m.a.59.1 ✓ 4
3.2 odd 2 inner 120.2.m.a.59.4 yes 4
4.3 odd 2 480.2.m.a.239.1 4
5.2 odd 4 600.2.b.c.251.3 4
5.3 odd 4 600.2.b.c.251.2 4
5.4 even 2 inner 120.2.m.a.59.4 yes 4
8.3 odd 2 inner 120.2.m.a.59.2 yes 4
8.5 even 2 480.2.m.a.239.2 4
12.11 even 2 480.2.m.a.239.4 4
15.2 even 4 600.2.b.c.251.2 4
15.8 even 4 600.2.b.c.251.3 4
15.14 odd 2 CM 120.2.m.a.59.1 ✓ 4
20.3 even 4 2400.2.b.c.2351.2 4
20.7 even 4 2400.2.b.c.2351.3 4
20.19 odd 2 480.2.m.a.239.4 4
24.5 odd 2 480.2.m.a.239.3 4
24.11 even 2 inner 120.2.m.a.59.3 yes 4
40.3 even 4 600.2.b.c.251.4 4
40.13 odd 4 2400.2.b.c.2351.1 4
40.19 odd 2 inner 120.2.m.a.59.3 yes 4
40.27 even 4 600.2.b.c.251.1 4
40.29 even 2 480.2.m.a.239.3 4
40.37 odd 4 2400.2.b.c.2351.4 4
60.23 odd 4 2400.2.b.c.2351.3 4
60.47 odd 4 2400.2.b.c.2351.2 4
60.59 even 2 480.2.m.a.239.1 4
120.29 odd 2 480.2.m.a.239.2 4
120.53 even 4 2400.2.b.c.2351.4 4
120.59 even 2 inner 120.2.m.a.59.2 yes 4
120.77 even 4 2400.2.b.c.2351.1 4
120.83 odd 4 600.2.b.c.251.1 4
120.107 odd 4 600.2.b.c.251.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.2.m.a.59.1 ✓ 4 1.1 even 1 trivial
120.2.m.a.59.1 ✓ 4 15.14 odd 2 CM
120.2.m.a.59.2 yes 4 8.3 odd 2 inner
120.2.m.a.59.2 yes 4 120.59 even 2 inner
120.2.m.a.59.3 yes 4 24.11 even 2 inner
120.2.m.a.59.3 yes 4 40.19 odd 2 inner
120.2.m.a.59.4 yes 4 3.2 odd 2 inner
120.2.m.a.59.4 yes 4 5.4 even 2 inner
480.2.m.a.239.1 4 4.3 odd 2
480.2.m.a.239.1 4 60.59 even 2
480.2.m.a.239.2 4 8.5 even 2
480.2.m.a.239.2 4 120.29 odd 2
480.2.m.a.239.3 4 24.5 odd 2
480.2.m.a.239.3 4 40.29 even 2
480.2.m.a.239.4 4 12.11 even 2
480.2.m.a.239.4 4 20.19 odd 2
600.2.b.c.251.1 4 40.27 even 4
600.2.b.c.251.1 4 120.83 odd 4
600.2.b.c.251.2 4 5.3 odd 4
600.2.b.c.251.2 4 15.2 even 4
600.2.b.c.251.3 4 5.2 odd 4
600.2.b.c.251.3 4 15.8 even 4
600.2.b.c.251.4 4 40.3 even 4
600.2.b.c.251.4 4 120.107 odd 4
2400.2.b.c.2351.1 4 40.13 odd 4
2400.2.b.c.2351.1 4 120.77 even 4
2400.2.b.c.2351.2 4 20.3 even 4
2400.2.b.c.2351.2 4 60.47 odd 4
2400.2.b.c.2351.3 4 20.7 even 4
2400.2.b.c.2351.3 4 60.23 odd 4
2400.2.b.c.2351.4 4 40.37 odd 4
2400.2.b.c.2351.4 4 120.53 even 4