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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [480,2,Mod(17,480)] 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("480.17"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(480, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 2, 2, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 480.bi (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.83281929702\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label 17.1
Root \(-1.22474 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 480.17
Dual form 480.2.bi.a.113.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.22474 + 1.22474i) q^{3} +(0.224745 + 2.22474i) q^{5} +(3.44949 + 3.44949i) q^{7} -3.00000i q^{9} -1.55051 q^{11} +(-3.00000 - 2.44949i) q^{15} -8.44949 q^{21} +(-4.89898 + 1.00000i) q^{25} +(3.67423 + 3.67423i) q^{27} +5.34847i q^{29} -4.89898 q^{31} +(1.89898 - 1.89898i) q^{33} +(-6.89898 + 8.44949i) q^{35} +(6.67423 - 0.674235i) q^{45} +16.7980i q^{49} +(2.44949 - 2.44949i) q^{53} +(-0.348469 - 3.44949i) q^{55} -15.3485i q^{59} +(10.3485 - 10.3485i) q^{63} +(11.8990 - 11.8990i) q^{73} +(4.77526 - 7.22474i) q^{75} +(-5.34847 - 5.34847i) q^{77} +14.6969i q^{79} -9.00000 q^{81} +(4.00000 - 4.00000i) q^{83} +(-6.55051 - 6.55051i) q^{87} +(6.00000 - 6.00000i) q^{93} +(8.79796 + 8.79796i) q^{97} +4.65153i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{5} + 4 q^{7} - 16 q^{11} - 12 q^{15} - 24 q^{21} - 12 q^{33} - 8 q^{35} + 12 q^{45} + 28 q^{55} + 12 q^{63} + 28 q^{73} + 24 q^{75} + 8 q^{77} - 36 q^{81} + 16 q^{83} - 36 q^{87} + 24 q^{93}+ \cdots - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/480\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(161\) \(421\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{4}\right)\) \(-1\) \(-1\)

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 0
\(3\) −1.22474 + 1.22474i −0.707107 + 0.707107i
\(4\) 0 0
\(5\) 0.224745 + 2.22474i 0.100509 + 0.994936i
\(6\) 0 0
\(7\) 3.44949 + 3.44949i 1.30378 + 1.30378i 0.925820 + 0.377964i \(0.123376\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 0 0
\(9\) − 3.00000i − 1.00000i
\(10\) 0 0
\(11\) −1.55051 −0.467496 −0.233748 − 0.972297i \(-0.575099\pi\)
−0.233748 + 0.972297i \(0.575099\pi\)
\(12\) 0 0
\(13\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(14\) 0 0
\(15\) −3.00000 − 2.44949i −0.774597 − 0.632456i
\(16\) 0 0
\(17\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) −8.44949 −1.84383
\(22\) 0 0
\(23\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(24\) 0 0
\(25\) −4.89898 + 1.00000i −0.979796 + 0.200000i
\(26\) 0 0
\(27\) 3.67423 + 3.67423i 0.707107 + 0.707107i
\(28\) 0 0
\(29\) 5.34847i 0.993186i 0.867984 + 0.496593i \(0.165416\pi\)
−0.867984 + 0.496593i \(0.834584\pi\)
\(30\) 0 0
\(31\) −4.89898 −0.879883 −0.439941 − 0.898027i \(-0.645001\pi\)
−0.439941 + 0.898027i \(0.645001\pi\)
\(32\) 0 0
\(33\) 1.89898 − 1.89898i 0.330570 − 0.330570i
\(34\) 0 0
\(35\) −6.89898 + 8.44949i −1.16614 + 1.42822i
\(36\) 0 0
\(37\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(44\) 0 0
\(45\) 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 \(0.250000\pi\)
\(48\) 0 0
\(49\) 16.7980i 2.39971i
\(50\) 0 0
\(51\) 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 \(0.173536\pi\)
\(54\) 0 0
\(55\) −0.348469 − 3.44949i −0.0469876 − 0.465129i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 15.3485i − 1.99820i −0.0424110 − 0.999100i \(-0.513504\pi\)
0.0424110 − 0.999100i \(-0.486496\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 10.3485 − 10.3485i 1.30378 − 1.30378i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 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 \(0.309823\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) 0 0
\(83\) 4.00000 − 4.00000i 0.439057 − 0.439057i −0.452638 − 0.891695i \(-0.649517\pi\)
0.891695 + 0.452638i \(0.149517\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −6.55051 − 6.55051i −0.702288 − 0.702288i
\(88\) 0 0
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 6.00000 − 6.00000i 0.622171 − 0.622171i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 8.79796 + 8.79796i 0.893297 + 0.893297i 0.994832 − 0.101535i \(-0.0323753\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) 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 480.2.bi.a.17.1 4
3.2 odd 2 480.2.bi.b.17.2 4
4.3 odd 2 120.2.w.b.77.2 yes 4
5.3 odd 4 inner 480.2.bi.a.113.1 4
8.3 odd 2 120.2.w.a.77.1 yes 4
8.5 even 2 480.2.bi.b.17.2 4
12.11 even 2 120.2.w.a.77.1 yes 4
15.8 even 4 480.2.bi.b.113.2 4
20.3 even 4 120.2.w.b.53.2 yes 4
20.7 even 4 600.2.w.b.293.1 4
20.19 odd 2 600.2.w.b.557.1 4
24.5 odd 2 CM 480.2.bi.a.17.1 4
24.11 even 2 120.2.w.b.77.2 yes 4
40.3 even 4 120.2.w.a.53.1 ✓ 4
40.13 odd 4 480.2.bi.b.113.2 4
40.19 odd 2 600.2.w.h.557.2 4
40.27 even 4 600.2.w.h.293.2 4
60.23 odd 4 120.2.w.a.53.1 ✓ 4
60.47 odd 4 600.2.w.h.293.2 4
60.59 even 2 600.2.w.h.557.2 4
120.53 even 4 inner 480.2.bi.a.113.1 4
120.59 even 2 600.2.w.b.557.1 4
120.83 odd 4 120.2.w.b.53.2 yes 4
120.107 odd 4 600.2.w.b.293.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.2.w.a.53.1 ✓ 4 40.3 even 4
120.2.w.a.53.1 ✓ 4 60.23 odd 4
120.2.w.a.77.1 yes 4 8.3 odd 2
120.2.w.a.77.1 yes 4 12.11 even 2
120.2.w.b.53.2 yes 4 20.3 even 4
120.2.w.b.53.2 yes 4 120.83 odd 4
120.2.w.b.77.2 yes 4 4.3 odd 2
120.2.w.b.77.2 yes 4 24.11 even 2
480.2.bi.a.17.1 4 1.1 even 1 trivial
480.2.bi.a.17.1 4 24.5 odd 2 CM
480.2.bi.a.113.1 4 5.3 odd 4 inner
480.2.bi.a.113.1 4 120.53 even 4 inner
480.2.bi.b.17.2 4 3.2 odd 2
480.2.bi.b.17.2 4 8.5 even 2
480.2.bi.b.113.2 4 15.8 even 4
480.2.bi.b.113.2 4 40.13 odd 4
600.2.w.b.293.1 4 20.7 even 4
600.2.w.b.293.1 4 120.107 odd 4
600.2.w.b.557.1 4 20.19 odd 2
600.2.w.b.557.1 4 120.59 even 2
600.2.w.h.293.2 4 40.27 even 4
600.2.w.h.293.2 4 60.47 odd 4
600.2.w.h.557.2 4 40.19 odd 2
600.2.w.h.557.2 4 60.59 even 2