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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 53.1
Root \(-1.22474 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 120.53
Dual form 120.2.w.b.77.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.00000 - 1.00000i) q^{2} +(-1.22474 - 1.22474i) q^{3} -2.00000i q^{4} +(-2.22474 + 0.224745i) q^{5} -2.44949 q^{6} +(1.44949 - 1.44949i) q^{7} +(-2.00000 - 2.00000i) q^{8} +3.00000i q^{9} +(-2.00000 + 2.44949i) q^{10} +6.44949 q^{11} +(-2.44949 + 2.44949i) q^{12} -2.89898i q^{14} +(3.00000 + 2.44949i) q^{15} -4.00000 q^{16} +(3.00000 + 3.00000i) q^{18} +(0.449490 + 4.44949i) q^{20} -3.55051 q^{21} +(6.44949 - 6.44949i) q^{22} +4.89898i q^{24} +(4.89898 - 1.00000i) q^{25} +(3.67423 - 3.67423i) q^{27} +(-2.89898 - 2.89898i) q^{28} +9.34847i q^{29} +(5.44949 - 0.550510i) q^{30} -4.89898 q^{31} +(-4.00000 + 4.00000i) q^{32} +(-7.89898 - 7.89898i) q^{33} +(-2.89898 + 3.55051i) q^{35} +6.00000 q^{36} +(4.89898 + 4.00000i) q^{40} +(-3.55051 + 3.55051i) q^{42} -12.8990i q^{44} +(-0.674235 - 6.67423i) q^{45} +(4.89898 + 4.89898i) q^{48} +2.79796i q^{49} +(3.89898 - 5.89898i) q^{50} +(-2.44949 - 2.44949i) q^{53} -7.34847i q^{54} +(-14.3485 + 1.44949i) q^{55} -5.79796 q^{56} +(9.34847 + 9.34847i) q^{58} -0.651531i q^{59} +(4.89898 - 6.00000i) q^{60} +(-4.89898 + 4.89898i) q^{62} +(4.34847 + 4.34847i) q^{63} +8.00000i q^{64} -15.7980 q^{66} +(0.651531 + 6.44949i) q^{70} +(6.00000 - 6.00000i) q^{72} +(2.10102 + 2.10102i) q^{73} +(-7.22474 - 4.77526i) q^{75} +(9.34847 - 9.34847i) q^{77} -14.6969i q^{79} +(8.89898 - 0.898979i) q^{80} -9.00000 q^{81} +(-4.00000 - 4.00000i) q^{83} +7.10102i q^{84} +(11.4495 - 11.4495i) q^{87} +(-12.8990 - 12.8990i) q^{88} +(-7.34847 - 6.00000i) q^{90} +(6.00000 + 6.00000i) q^{93} +9.79796 q^{96} +(-10.7980 + 10.7980i) q^{97} +(2.79796 + 2.79796i) q^{98} +19.3485i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - 4 q^{5} - 4 q^{7} - 8 q^{8} - 8 q^{10} + 16 q^{11} + 12 q^{15} - 16 q^{16} + 12 q^{18} - 8 q^{20} - 24 q^{21} + 16 q^{22} + 8 q^{28} + 12 q^{30} - 16 q^{32} - 12 q^{33} + 8 q^{35} + 24 q^{36}+ \cdots - 28 q^{98}+O(q^{100}) \) 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 \(-0.0750991\pi\)
0.972297 + 0.233748i \(0.0750991\pi\)
\(12\) −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 \(0.334584\pi\)
−0.496593 + 0.867984i \(0.665416\pi\)
\(30\) 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 0.518571 − 0.855034i \(-0.326464\pi\)
−0.855034 + 0.518571i \(0.826464\pi\)
\(54\) − 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 \(-0.986496\pi\)
0.999100 − 0.0424110i \(-0.0135039\pi\)
\(60\) 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 0.452638 − 0.891695i \(-0.350483\pi\)
−0.891695 + 0.452638i \(0.850483\pi\)
\(84\) 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.b.53.1 yes 4
3.2 odd 2 120.2.w.a.53.2 ✓ 4
4.3 odd 2 480.2.bi.a.113.2 4
5.2 odd 4 inner 120.2.w.b.77.1 yes 4
5.3 odd 4 600.2.w.b.557.2 4
5.4 even 2 600.2.w.b.293.2 4
8.3 odd 2 480.2.bi.b.113.1 4
8.5 even 2 120.2.w.a.53.2 ✓ 4
12.11 even 2 480.2.bi.b.113.1 4
15.2 even 4 120.2.w.a.77.2 yes 4
15.8 even 4 600.2.w.h.557.1 4
15.14 odd 2 600.2.w.h.293.1 4
20.7 even 4 480.2.bi.a.17.2 4
24.5 odd 2 CM 120.2.w.b.53.1 yes 4
24.11 even 2 480.2.bi.a.113.2 4
40.13 odd 4 600.2.w.h.557.1 4
40.27 even 4 480.2.bi.b.17.1 4
40.29 even 2 600.2.w.h.293.1 4
40.37 odd 4 120.2.w.a.77.2 yes 4
60.47 odd 4 480.2.bi.b.17.1 4
120.29 odd 2 600.2.w.b.293.2 4
120.53 even 4 600.2.w.b.557.2 4
120.77 even 4 inner 120.2.w.b.77.1 yes 4
120.107 odd 4 480.2.bi.a.17.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.2.w.a.53.2 ✓ 4 3.2 odd 2
120.2.w.a.53.2 ✓ 4 8.5 even 2
120.2.w.a.77.2 yes 4 15.2 even 4
120.2.w.a.77.2 yes 4 40.37 odd 4
120.2.w.b.53.1 yes 4 1.1 even 1 trivial
120.2.w.b.53.1 yes 4 24.5 odd 2 CM
120.2.w.b.77.1 yes 4 5.2 odd 4 inner
120.2.w.b.77.1 yes 4 120.77 even 4 inner
480.2.bi.a.17.2 4 20.7 even 4
480.2.bi.a.17.2 4 120.107 odd 4
480.2.bi.a.113.2 4 4.3 odd 2
480.2.bi.a.113.2 4 24.11 even 2
480.2.bi.b.17.1 4 40.27 even 4
480.2.bi.b.17.1 4 60.47 odd 4
480.2.bi.b.113.1 4 8.3 odd 2
480.2.bi.b.113.1 4 12.11 even 2
600.2.w.b.293.2 4 5.4 even 2
600.2.w.b.293.2 4 120.29 odd 2
600.2.w.b.557.2 4 5.3 odd 4
600.2.w.b.557.2 4 120.53 even 4
600.2.w.h.293.1 4 15.14 odd 2
600.2.w.h.293.1 4 40.29 even 2
600.2.w.h.557.1 4 15.8 even 4
600.2.w.h.557.1 4 40.13 odd 4