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 77.2
Root \(1.22474 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 120.77
Dual form 120.2.w.b.53.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.00000 + 1.00000i) q^{2} +(1.22474 - 1.22474i) q^{3} +2.00000i q^{4} +(0.224745 + 2.22474i) q^{5} +2.44949 q^{6} +(-3.44949 - 3.44949i) q^{7} +(-2.00000 + 2.00000i) q^{8} -3.00000i q^{9} +(-2.00000 + 2.44949i) q^{10} +1.55051 q^{11} +(2.44949 + 2.44949i) q^{12} -6.89898i q^{14} +(3.00000 + 2.44949i) q^{15} -4.00000 q^{16} +(3.00000 - 3.00000i) q^{18} +(-4.44949 + 0.449490i) q^{20} -8.44949 q^{21} +(1.55051 + 1.55051i) q^{22} +4.89898i q^{24} +(-4.89898 + 1.00000i) q^{25} +(-3.67423 - 3.67423i) q^{27} +(6.89898 - 6.89898i) q^{28} +5.34847i q^{29} +(0.550510 + 5.44949i) q^{30} +4.89898 q^{31} +(-4.00000 - 4.00000i) q^{32} +(1.89898 - 1.89898i) q^{33} +(6.89898 - 8.44949i) q^{35} +6.00000 q^{36} +(-4.89898 - 4.00000i) q^{40} +(-8.44949 - 8.44949i) q^{42} +3.10102i q^{44} +(6.67423 - 0.674235i) q^{45} +(-4.89898 + 4.89898i) q^{48} +16.7980i q^{49} +(-5.89898 - 3.89898i) q^{50} +(2.44949 - 2.44949i) q^{53} -7.34847i q^{54} +(0.348469 + 3.44949i) q^{55} +13.7980 q^{56} +(-5.34847 + 5.34847i) q^{58} +15.3485i q^{59} +(-4.89898 + 6.00000i) q^{60} +(4.89898 + 4.89898i) q^{62} +(-10.3485 + 10.3485i) q^{63} -8.00000i q^{64} +3.79796 q^{66} +(15.3485 - 1.55051i) q^{70} +(6.00000 + 6.00000i) q^{72} +(11.8990 - 11.8990i) q^{73} +(-4.77526 + 7.22474i) q^{75} +(-5.34847 - 5.34847i) q^{77} -14.6969i q^{79} +(-0.898979 - 8.89898i) q^{80} -9.00000 q^{81} +(-4.00000 + 4.00000i) q^{83} -16.8990i q^{84} +(6.55051 + 6.55051i) q^{87} +(-3.10102 + 3.10102i) q^{88} +(7.34847 + 6.00000i) q^{90} +(6.00000 - 6.00000i) q^{93} -9.79796 q^{96} +(8.79796 + 8.79796i) q^{97} +(-16.7980 + 16.7980i) q^{98} -4.65153i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - 4 q^{5} - 4 q^{7} - 8 q^{8} - 8 q^{10} + 16 q^{11} + 12 q^{15} - 16 q^{16} + 12 q^{18} - 8 q^{20} - 24 q^{21} + 16 q^{22} + 8 q^{28} + 12 q^{30} - 16 q^{32} - 12 q^{33} + 8 q^{35} + 24 q^{36}+ \cdots - 28 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(5\) 0.224745 + 2.22474i 0.100509 + 0.994936i
\(6\) 2.44949 1.00000
\(7\) −3.44949 − 3.44949i −1.30378 − 1.30378i −0.925820 − 0.377964i \(-0.876624\pi\)
−0.377964 − 0.925820i \(-0.623376\pi\)
\(8\) −2.00000 + 2.00000i −0.707107 + 0.707107i
\(9\) − 3.00000i − 1.00000i
\(10\) −2.00000 + 2.44949i −0.632456 + 0.774597i
\(11\) 1.55051 0.467496 0.233748 − 0.972297i \(-0.424901\pi\)
0.233748 + 0.972297i \(0.424901\pi\)
\(12\) 2.44949 + 2.44949i 0.707107 + 0.707107i
\(13\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(14\) − 6.89898i − 1.84383i
\(15\) 3.00000 + 2.44949i 0.774597 + 0.632456i
\(16\) −4.00000 −1.00000
\(17\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(18\) 3.00000 − 3.00000i 0.707107 − 0.707107i
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −4.44949 + 0.449490i −0.994936 + 0.100509i
\(21\) −8.44949 −1.84383
\(22\) 1.55051 + 1.55051i 0.330570 + 0.330570i
\(23\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(24\) 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
\(28\) 6.89898 − 6.89898i 1.30378 − 1.30378i
\(29\) 5.34847i 0.993186i 0.867984 + 0.496593i \(0.165416\pi\)
−0.867984 + 0.496593i \(0.834584\pi\)
\(30\) 0.550510 + 5.44949i 0.100509 + 0.994936i
\(31\) 4.89898 0.879883 0.439941 − 0.898027i \(-0.354999\pi\)
0.439941 + 0.898027i \(0.354999\pi\)
\(32\) −4.00000 − 4.00000i −0.707107 − 0.707107i
\(33\) 1.89898 − 1.89898i 0.330570 − 0.330570i
\(34\) 0 0
\(35\) 6.89898 − 8.44949i 1.16614 − 1.42822i
\(36\) 6.00000 1.00000
\(37\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −4.89898 − 4.00000i −0.774597 − 0.632456i
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) −8.44949 − 8.44949i −1.30378 − 1.30378i
\(43\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(44\) 3.10102i 0.467496i
\(45\) 6.67423 − 0.674235i 0.994936 − 0.100509i
\(46\) 0 0
\(47\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(48\) −4.89898 + 4.89898i −0.707107 + 0.707107i
\(49\) 16.7980i 2.39971i
\(50\) −5.89898 − 3.89898i −0.834242 − 0.551399i
\(51\) 0 0
\(52\) 0 0
\(53\) 2.44949 − 2.44949i 0.336463 − 0.336463i −0.518571 − 0.855034i \(-0.673536\pi\)
0.855034 + 0.518571i \(0.173536\pi\)
\(54\) − 7.34847i − 1.00000i
\(55\) 0.348469 + 3.44949i 0.0469876 + 0.465129i
\(56\) 13.7980 1.84383
\(57\) 0 0
\(58\) −5.34847 + 5.34847i −0.702288 + 0.702288i
\(59\) 15.3485i 1.99820i 0.0424110 + 0.999100i \(0.486496\pi\)
−0.0424110 + 0.999100i \(0.513504\pi\)
\(60\) −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
\(63\) −10.3485 + 10.3485i −1.30378 + 1.30378i
\(64\) − 8.00000i − 1.00000i
\(65\) 0 0
\(66\) 3.79796 0.467496
\(67\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 15.3485 − 1.55051i 1.83449 − 0.185321i
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 6.00000 + 6.00000i 0.707107 + 0.707107i
\(73\) 11.8990 − 11.8990i 1.39267 − 1.39267i 0.573382 − 0.819288i \(-0.305631\pi\)
0.819288 − 0.573382i \(-0.194369\pi\)
\(74\) 0 0
\(75\) −4.77526 + 7.22474i −0.551399 + 0.834242i
\(76\) 0 0
\(77\) −5.34847 − 5.34847i −0.609515 − 0.609515i
\(78\) 0 0
\(79\) − 14.6969i − 1.65353i −0.562544 − 0.826767i \(-0.690177\pi\)
0.562544 − 0.826767i \(-0.309823\pi\)
\(80\) −0.898979 − 8.89898i −0.100509 − 0.994936i
\(81\) −9.00000 −1.00000
\(82\) 0 0
\(83\) −4.00000 + 4.00000i −0.439057 + 0.439057i −0.891695 − 0.452638i \(-0.850483\pi\)
0.452638 + 0.891695i \(0.350483\pi\)
\(84\) − 16.8990i − 1.84383i
\(85\) 0 0
\(86\) 0 0
\(87\) 6.55051 + 6.55051i 0.702288 + 0.702288i
\(88\) −3.10102 + 3.10102i −0.330570 + 0.330570i
\(89\) 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
\(97\) 8.79796 + 8.79796i 0.893297 + 0.893297i 0.994832 − 0.101535i \(-0.0323753\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) −16.7980 + 16.7980i −1.69685 + 1.69685i
\(99\) − 4.65153i − 0.467496i
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.77.2 yes 4
3.2 odd 2 120.2.w.a.77.1 yes 4
4.3 odd 2 480.2.bi.a.17.1 4
5.2 odd 4 600.2.w.b.293.1 4
5.3 odd 4 inner 120.2.w.b.53.2 yes 4
5.4 even 2 600.2.w.b.557.1 4
8.3 odd 2 480.2.bi.b.17.2 4
8.5 even 2 120.2.w.a.77.1 yes 4
12.11 even 2 480.2.bi.b.17.2 4
15.2 even 4 600.2.w.h.293.2 4
15.8 even 4 120.2.w.a.53.1 ✓ 4
15.14 odd 2 600.2.w.h.557.2 4
20.3 even 4 480.2.bi.a.113.1 4
24.5 odd 2 CM 120.2.w.b.77.2 yes 4
24.11 even 2 480.2.bi.a.17.1 4
40.3 even 4 480.2.bi.b.113.2 4
40.13 odd 4 120.2.w.a.53.1 ✓ 4
40.29 even 2 600.2.w.h.557.2 4
40.37 odd 4 600.2.w.h.293.2 4
60.23 odd 4 480.2.bi.b.113.2 4
120.29 odd 2 600.2.w.b.557.1 4
120.53 even 4 inner 120.2.w.b.53.2 yes 4
120.77 even 4 600.2.w.b.293.1 4
120.83 odd 4 480.2.bi.a.113.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.2.w.a.53.1 ✓ 4 15.8 even 4
120.2.w.a.53.1 ✓ 4 40.13 odd 4
120.2.w.a.77.1 yes 4 3.2 odd 2
120.2.w.a.77.1 yes 4 8.5 even 2
120.2.w.b.53.2 yes 4 5.3 odd 4 inner
120.2.w.b.53.2 yes 4 120.53 even 4 inner
120.2.w.b.77.2 yes 4 1.1 even 1 trivial
120.2.w.b.77.2 yes 4 24.5 odd 2 CM
480.2.bi.a.17.1 4 4.3 odd 2
480.2.bi.a.17.1 4 24.11 even 2
480.2.bi.a.113.1 4 20.3 even 4
480.2.bi.a.113.1 4 120.83 odd 4
480.2.bi.b.17.2 4 8.3 odd 2
480.2.bi.b.17.2 4 12.11 even 2
480.2.bi.b.113.2 4 40.3 even 4
480.2.bi.b.113.2 4 60.23 odd 4
600.2.w.b.293.1 4 5.2 odd 4
600.2.w.b.293.1 4 120.77 even 4
600.2.w.b.557.1 4 5.4 even 2
600.2.w.b.557.1 4 120.29 odd 2
600.2.w.h.293.2 4 15.2 even 4
600.2.w.h.293.2 4 40.37 odd 4
600.2.w.h.557.2 4 15.14 odd 2
600.2.w.h.557.2 4 40.29 even 2