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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1520,2,Mod(609,1520)] 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("1520.609"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1520, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1520.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-2,0,0,0,6,0,8,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.1372611072\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 95)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 609.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1520.609
Dual form 1520.2.d.b.609.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.00000 - 2.00000i) q^{5} -2.00000i q^{7} +3.00000 q^{9} +4.00000 q^{11} +2.00000i q^{13} +4.00000i q^{17} +1.00000 q^{19} -6.00000i q^{23} +(-3.00000 + 4.00000i) q^{25} +6.00000 q^{29} +4.00000 q^{31} +(-4.00000 + 2.00000i) q^{35} -10.0000i q^{37} -10.0000 q^{41} +2.00000i q^{43} +(-3.00000 - 6.00000i) q^{45} +6.00000i q^{47} +3.00000 q^{49} -10.0000i q^{53} +(-4.00000 - 8.00000i) q^{55} +2.00000 q^{61} -6.00000i q^{63} +(4.00000 - 2.00000i) q^{65} -8.00000i q^{67} -4.00000 q^{71} -4.00000i q^{73} -8.00000i q^{77} +4.00000 q^{79} +9.00000 q^{81} -18.0000i q^{83} +(8.00000 - 4.00000i) q^{85} +2.00000 q^{89} +4.00000 q^{91} +(-1.00000 - 2.00000i) q^{95} +6.00000i q^{97} +12.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + 6 q^{9} + 8 q^{11} + 2 q^{19} - 6 q^{25} + 12 q^{29} + 8 q^{31} - 8 q^{35} - 20 q^{41} - 6 q^{45} + 6 q^{49} - 8 q^{55} + 4 q^{61} + 8 q^{65} - 8 q^{71} + 8 q^{79} + 18 q^{81} + 16 q^{85}+ \cdots + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(401\) \(1141\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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 0
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 0 0
\(5\) −1.00000 − 2.00000i −0.447214 − 0.894427i
\(6\) 0 0
\(7\) − 2.00000i − 0.755929i −0.925820 − 0.377964i \(-0.876624\pi\)
0.925820 − 0.377964i \(-0.123376\pi\)
\(8\) 0 0
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 − 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.00000i 0.970143i 0.874475 + 0.485071i \(0.161206\pi\)
−0.874475 + 0.485071i \(0.838794\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 6.00000i − 1.25109i −0.780189 − 0.625543i \(-0.784877\pi\)
0.780189 − 0.625543i \(-0.215123\pi\)
\(24\) 0 0
\(25\) −3.00000 + 4.00000i −0.600000 + 0.800000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4.00000 + 2.00000i −0.676123 + 0.338062i
\(36\) 0 0
\(37\) − 10.0000i − 1.64399i −0.569495 − 0.821995i \(-0.692861\pi\)
0.569495 − 0.821995i \(-0.307139\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −10.0000 −1.56174 −0.780869 − 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) 2.00000i 0.304997i 0.988304 + 0.152499i \(0.0487319\pi\)
−0.988304 + 0.152499i \(0.951268\pi\)
\(44\) 0 0
\(45\) −3.00000 − 6.00000i −0.447214 − 0.894427i
\(46\) 0 0
\(47\) 6.00000i 0.875190i 0.899172 + 0.437595i \(0.144170\pi\)
−0.899172 + 0.437595i \(0.855830\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 10.0000i − 1.37361i −0.726844 − 0.686803i \(-0.759014\pi\)
0.726844 − 0.686803i \(-0.240986\pi\)
\(54\) 0 0
\(55\) −4.00000 − 8.00000i −0.539360 − 1.07872i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 − 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) − 6.00000i − 0.755929i
\(64\) 0 0
\(65\) 4.00000 − 2.00000i 0.496139 − 0.248069i
\(66\) 0 0
\(67\) − 8.00000i − 0.977356i −0.872464 − 0.488678i \(-0.837479\pi\)
0.872464 − 0.488678i \(-0.162521\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −4.00000 −0.474713 −0.237356 − 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 0 0
\(73\) − 4.00000i − 0.468165i −0.972217 − 0.234082i \(-0.924791\pi\)
0.972217 − 0.234082i \(-0.0752085\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 8.00000i − 0.911685i
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 − 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) − 18.0000i − 1.97576i −0.155230 − 0.987878i \(-0.549612\pi\)
0.155230 − 0.987878i \(-0.450388\pi\)
\(84\) 0 0
\(85\) 8.00000 − 4.00000i 0.867722 − 0.433861i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.00000 0.212000 0.106000 − 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.00000 − 2.00000i −0.102598 − 0.205196i
\(96\) 0 0
\(97\) 6.00000i 0.609208i 0.952479 + 0.304604i \(0.0985241\pi\)
−0.952479 + 0.304604i \(0.901476\pi\)
\(98\) 0 0
\(99\) 12.0000 1.20605
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 1520.2.d.b.609.1 2
4.3 odd 2 95.2.b.a.39.1 ✓ 2
5.2 odd 4 7600.2.a.l.1.1 1
5.3 odd 4 7600.2.a.i.1.1 1
5.4 even 2 inner 1520.2.d.b.609.2 2
12.11 even 2 855.2.c.b.514.2 2
20.3 even 4 475.2.a.a.1.1 1
20.7 even 4 475.2.a.c.1.1 1
20.19 odd 2 95.2.b.a.39.2 yes 2
60.23 odd 4 4275.2.a.p.1.1 1
60.47 odd 4 4275.2.a.e.1.1 1
60.59 even 2 855.2.c.b.514.1 2
76.75 even 2 1805.2.b.c.1084.2 2
380.227 odd 4 9025.2.a.c.1.1 1
380.303 odd 4 9025.2.a.h.1.1 1
380.379 even 2 1805.2.b.c.1084.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.b.a.39.1 ✓ 2 4.3 odd 2
95.2.b.a.39.2 yes 2 20.19 odd 2
475.2.a.a.1.1 1 20.3 even 4
475.2.a.c.1.1 1 20.7 even 4
855.2.c.b.514.1 2 60.59 even 2
855.2.c.b.514.2 2 12.11 even 2
1520.2.d.b.609.1 2 1.1 even 1 trivial
1520.2.d.b.609.2 2 5.4 even 2 inner
1805.2.b.c.1084.1 2 380.379 even 2
1805.2.b.c.1084.2 2 76.75 even 2
4275.2.a.e.1.1 1 60.47 odd 4
4275.2.a.p.1.1 1 60.23 odd 4
7600.2.a.i.1.1 1 5.3 odd 4
7600.2.a.l.1.1 1 5.2 odd 4
9025.2.a.c.1.1 1 380.227 odd 4
9025.2.a.h.1.1 1 380.303 odd 4