Properties

Label 120.2.w.a.53.2
Level $120$
Weight $2$
Character 120.53
Analytic conductor $0.958$
Analytic rank $0$
Dimension $4$
CM discriminant -24
Inner twists $4$

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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(53,120)] 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("120.53"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(120, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 2, 2, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 120 = 2^{3} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 120.w (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.958204824255\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label 53.2
Root \(1.22474 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 120.53
Dual form 120.2.w.a.77.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.00000 + 1.00000i) q^{2} +(1.22474 + 1.22474i) q^{3} -2.00000i q^{4} +(2.22474 - 0.224745i) q^{5} -2.44949 q^{6} +(1.44949 - 1.44949i) q^{7} +(2.00000 + 2.00000i) q^{8} +3.00000i q^{9} +(-2.00000 + 2.44949i) q^{10} -6.44949 q^{11} +(2.44949 - 2.44949i) q^{12} +2.89898i q^{14} +(3.00000 + 2.44949i) q^{15} -4.00000 q^{16} +(-3.00000 - 3.00000i) q^{18} +(-0.449490 - 4.44949i) q^{20} +3.55051 q^{21} +(6.44949 - 6.44949i) q^{22} +4.89898i q^{24} +(4.89898 - 1.00000i) q^{25} +(-3.67423 + 3.67423i) q^{27} +(-2.89898 - 2.89898i) q^{28} -9.34847i q^{29} +(-5.44949 + 0.550510i) q^{30} -4.89898 q^{31} +(4.00000 - 4.00000i) q^{32} +(-7.89898 - 7.89898i) q^{33} +(2.89898 - 3.55051i) q^{35} +6.00000 q^{36} +(4.89898 + 4.00000i) q^{40} +(-3.55051 + 3.55051i) q^{42} +12.8990i q^{44} +(0.674235 + 6.67423i) q^{45} +(-4.89898 - 4.89898i) q^{48} +2.79796i q^{49} +(-3.89898 + 5.89898i) q^{50} +(2.44949 + 2.44949i) q^{53} -7.34847i q^{54} +(-14.3485 + 1.44949i) q^{55} +5.79796 q^{56} +(9.34847 + 9.34847i) q^{58} +0.651531i q^{59} +(4.89898 - 6.00000i) q^{60} +(4.89898 - 4.89898i) q^{62} +(4.34847 + 4.34847i) q^{63} +8.00000i q^{64} +15.7980 q^{66} +(0.651531 + 6.44949i) q^{70} +(-6.00000 + 6.00000i) q^{72} +(2.10102 + 2.10102i) q^{73} +(7.22474 + 4.77526i) q^{75} +(-9.34847 + 9.34847i) q^{77} -14.6969i q^{79} +(-8.89898 + 0.898979i) q^{80} -9.00000 q^{81} +(4.00000 + 4.00000i) q^{83} -7.10102i q^{84} +(11.4495 - 11.4495i) q^{87} +(-12.8990 - 12.8990i) q^{88} +(-7.34847 - 6.00000i) q^{90} +(-6.00000 - 6.00000i) q^{93} +9.79796 q^{96} +(-10.7980 + 10.7980i) q^{97} +(-2.79796 - 2.79796i) q^{98} -19.3485i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 4 q^{5} - 4 q^{7} + 8 q^{8} - 8 q^{10} - 16 q^{11} + 12 q^{15} - 16 q^{16} - 12 q^{18} + 8 q^{20} + 24 q^{21} + 16 q^{22} + 8 q^{28} - 12 q^{30} + 16 q^{32} - 12 q^{33} - 8 q^{35} + 24 q^{36}+ \cdots + 28 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(31\) \(41\) \(61\) \(97\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 + 1.00000i −0.707107 + 0.707107i
\(3\) 1.22474 + 1.22474i 0.707107 + 0.707107i
\(4\) − 2.00000i − 1.00000i
\(5\) 2.22474 − 0.224745i 0.994936 − 0.100509i
\(6\) −2.44949 −1.00000
\(7\) 1.44949 − 1.44949i 0.547856 − 0.547856i −0.377964 − 0.925820i \(-0.623376\pi\)
0.925820 + 0.377964i \(0.123376\pi\)
\(8\) 2.00000 + 2.00000i 0.707107 + 0.707107i
\(9\) 3.00000i 1.00000i
\(10\) −2.00000 + 2.44949i −0.632456 + 0.774597i
\(11\) −6.44949 −1.94459 −0.972297 − 0.233748i \(-0.924901\pi\)
−0.972297 + 0.233748i \(0.924901\pi\)
\(12\) 2.44949 − 2.44949i 0.707107 − 0.707107i
\(13\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(14\) 2.89898i 0.774785i
\(15\) 3.00000 + 2.44949i 0.774597 + 0.632456i
\(16\) −4.00000 −1.00000
\(17\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(18\) −3.00000 − 3.00000i −0.707107 − 0.707107i
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −0.449490 − 4.44949i −0.100509 − 0.994936i
\(21\) 3.55051 0.774785
\(22\) 6.44949 − 6.44949i 1.37504 − 1.37504i
\(23\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(24\) 4.89898i 1.00000i
\(25\) 4.89898 − 1.00000i 0.979796 − 0.200000i
\(26\) 0 0
\(27\) −3.67423 + 3.67423i −0.707107 + 0.707107i
\(28\) −2.89898 − 2.89898i −0.547856 − 0.547856i
\(29\) − 9.34847i − 1.73597i −0.496593 − 0.867984i \(-0.665416\pi\)
0.496593 − 0.867984i \(-0.334584\pi\)
\(30\) −5.44949 + 0.550510i −0.994936 + 0.100509i
\(31\) −4.89898 −0.879883 −0.439941 − 0.898027i \(-0.645001\pi\)
−0.439941 + 0.898027i \(0.645001\pi\)
\(32\) 4.00000 − 4.00000i 0.707107 − 0.707107i
\(33\) −7.89898 − 7.89898i −1.37504 − 1.37504i
\(34\) 0 0
\(35\) 2.89898 − 3.55051i 0.490017 − 0.600146i
\(36\) 6.00000 1.00000
\(37\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 4.89898 + 4.00000i 0.774597 + 0.632456i
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) −3.55051 + 3.55051i −0.547856 + 0.547856i
\(43\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(44\) 12.8990i 1.94459i
\(45\) 0.674235 + 6.67423i 0.100509 + 0.994936i
\(46\) 0 0
\(47\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(48\) −4.89898 − 4.89898i −0.707107 − 0.707107i
\(49\) 2.79796i 0.399708i
\(50\) −3.89898 + 5.89898i −0.551399 + 0.834242i
\(51\) 0 0
\(52\) 0 0
\(53\) 2.44949 + 2.44949i 0.336463 + 0.336463i 0.855034 − 0.518571i \(-0.173536\pi\)
−0.518571 + 0.855034i \(0.673536\pi\)
\(54\) − 7.34847i − 1.00000i
\(55\) −14.3485 + 1.44949i −1.93475 + 0.195449i
\(56\) 5.79796 0.774785
\(57\) 0 0
\(58\) 9.34847 + 9.34847i 1.22751 + 1.22751i
\(59\) 0.651531i 0.0848221i 0.999100 + 0.0424110i \(0.0135039\pi\)
−0.999100 + 0.0424110i \(0.986496\pi\)
\(60\) 4.89898 − 6.00000i 0.632456 − 0.774597i
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 4.89898 − 4.89898i 0.622171 − 0.622171i
\(63\) 4.34847 + 4.34847i 0.547856 + 0.547856i
\(64\) 8.00000i 1.00000i
\(65\) 0 0
\(66\) 15.7980 1.94459
\(67\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0.651531 + 6.44949i 0.0778728 + 0.770861i
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) −6.00000 + 6.00000i −0.707107 + 0.707107i
\(73\) 2.10102 + 2.10102i 0.245906 + 0.245906i 0.819288 − 0.573382i \(-0.194369\pi\)
−0.573382 + 0.819288i \(0.694369\pi\)
\(74\) 0 0
\(75\) 7.22474 + 4.77526i 0.834242 + 0.551399i
\(76\) 0 0
\(77\) −9.34847 + 9.34847i −1.06536 + 1.06536i
\(78\) 0 0
\(79\) − 14.6969i − 1.65353i −0.562544 − 0.826767i \(-0.690177\pi\)
0.562544 − 0.826767i \(-0.309823\pi\)
\(80\) −8.89898 + 0.898979i −0.994936 + 0.100509i
\(81\) −9.00000 −1.00000
\(82\) 0 0
\(83\) 4.00000 + 4.00000i 0.439057 + 0.439057i 0.891695 − 0.452638i \(-0.149517\pi\)
−0.452638 + 0.891695i \(0.649517\pi\)
\(84\) − 7.10102i − 0.774785i
\(85\) 0 0
\(86\) 0 0
\(87\) 11.4495 − 11.4495i 1.22751 − 1.22751i
\(88\) −12.8990 − 12.8990i −1.37504 − 1.37504i
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) −7.34847 − 6.00000i −0.774597 − 0.632456i
\(91\) 0 0
\(92\) 0 0
\(93\) −6.00000 − 6.00000i −0.622171 − 0.622171i
\(94\) 0 0
\(95\) 0 0
\(96\) 9.79796 1.00000
\(97\) −10.7980 + 10.7980i −1.09637 + 1.09637i −0.101535 + 0.994832i \(0.532375\pi\)
−0.994832 + 0.101535i \(0.967625\pi\)
\(98\) −2.79796 − 2.79796i −0.282637 − 0.282637i
\(99\) − 19.3485i − 1.94459i
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.w.a.53.2 ✓ 4
3.2 odd 2 120.2.w.b.53.1 yes 4
4.3 odd 2 480.2.bi.b.113.1 4
5.2 odd 4 inner 120.2.w.a.77.2 yes 4
5.3 odd 4 600.2.w.h.557.1 4
5.4 even 2 600.2.w.h.293.1 4
8.3 odd 2 480.2.bi.a.113.2 4
8.5 even 2 120.2.w.b.53.1 yes 4
12.11 even 2 480.2.bi.a.113.2 4
15.2 even 4 120.2.w.b.77.1 yes 4
15.8 even 4 600.2.w.b.557.2 4
15.14 odd 2 600.2.w.b.293.2 4
20.7 even 4 480.2.bi.b.17.1 4
24.5 odd 2 CM 120.2.w.a.53.2 ✓ 4
24.11 even 2 480.2.bi.b.113.1 4
40.13 odd 4 600.2.w.b.557.2 4
40.27 even 4 480.2.bi.a.17.2 4
40.29 even 2 600.2.w.b.293.2 4
40.37 odd 4 120.2.w.b.77.1 yes 4
60.47 odd 4 480.2.bi.a.17.2 4
120.29 odd 2 600.2.w.h.293.1 4
120.53 even 4 600.2.w.h.557.1 4
120.77 even 4 inner 120.2.w.a.77.2 yes 4
120.107 odd 4 480.2.bi.b.17.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.2.w.a.53.2 ✓ 4 1.1 even 1 trivial
120.2.w.a.53.2 ✓ 4 24.5 odd 2 CM
120.2.w.a.77.2 yes 4 5.2 odd 4 inner
120.2.w.a.77.2 yes 4 120.77 even 4 inner
120.2.w.b.53.1 yes 4 3.2 odd 2
120.2.w.b.53.1 yes 4 8.5 even 2
120.2.w.b.77.1 yes 4 15.2 even 4
120.2.w.b.77.1 yes 4 40.37 odd 4
480.2.bi.a.17.2 4 40.27 even 4
480.2.bi.a.17.2 4 60.47 odd 4
480.2.bi.a.113.2 4 8.3 odd 2
480.2.bi.a.113.2 4 12.11 even 2
480.2.bi.b.17.1 4 20.7 even 4
480.2.bi.b.17.1 4 120.107 odd 4
480.2.bi.b.113.1 4 4.3 odd 2
480.2.bi.b.113.1 4 24.11 even 2
600.2.w.b.293.2 4 15.14 odd 2
600.2.w.b.293.2 4 40.29 even 2
600.2.w.b.557.2 4 15.8 even 4
600.2.w.b.557.2 4 40.13 odd 4
600.2.w.h.293.1 4 5.4 even 2
600.2.w.h.293.1 4 120.29 odd 2
600.2.w.h.557.1 4 5.3 odd 4
600.2.w.h.557.1 4 120.53 even 4