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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [855,2,Mod(514,855)] 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("855.514"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(855, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 855.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,2,2] 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: \(6.82720937282\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 514.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 855.514
Dual form 855.2.c.b.514.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.00000 q^{4} +(1.00000 + 2.00000i) q^{5} +2.00000i q^{7} +3.00000i q^{8} +(-2.00000 + 1.00000i) q^{10} +4.00000 q^{11} +2.00000i q^{13} -2.00000 q^{14} -1.00000 q^{16} -4.00000i q^{17} -1.00000 q^{19} +(1.00000 + 2.00000i) q^{20} +4.00000i q^{22} -6.00000i q^{23} +(-3.00000 + 4.00000i) q^{25} -2.00000 q^{26} +2.00000i q^{28} -6.00000 q^{29} -4.00000 q^{31} +5.00000i q^{32} +4.00000 q^{34} +(-4.00000 + 2.00000i) q^{35} -10.0000i q^{37} -1.00000i q^{38} +(-6.00000 + 3.00000i) q^{40} +10.0000 q^{41} -2.00000i q^{43} +4.00000 q^{44} +6.00000 q^{46} +6.00000i q^{47} +3.00000 q^{49} +(-4.00000 - 3.00000i) q^{50} +2.00000i q^{52} +10.0000i q^{53} +(4.00000 + 8.00000i) q^{55} -6.00000 q^{56} -6.00000i q^{58} +2.00000 q^{61} -4.00000i q^{62} -7.00000 q^{64} +(-4.00000 + 2.00000i) q^{65} +8.00000i q^{67} -4.00000i q^{68} +(-2.00000 - 4.00000i) q^{70} -4.00000 q^{71} -4.00000i q^{73} +10.0000 q^{74} -1.00000 q^{76} +8.00000i q^{77} -4.00000 q^{79} +(-1.00000 - 2.00000i) q^{80} +10.0000i q^{82} -18.0000i q^{83} +(8.00000 - 4.00000i) q^{85} +2.00000 q^{86} +12.0000i q^{88} -2.00000 q^{89} -4.00000 q^{91} -6.00000i q^{92} -6.00000 q^{94} +(-1.00000 - 2.00000i) q^{95} +6.00000i q^{97} +3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} + 2 q^{5} - 4 q^{10} + 8 q^{11} - 4 q^{14} - 2 q^{16} - 2 q^{19} + 2 q^{20} - 6 q^{25} - 4 q^{26} - 12 q^{29} - 8 q^{31} + 8 q^{34} - 8 q^{35} - 12 q^{40} + 20 q^{41} + 8 q^{44} + 12 q^{46}+ \cdots - 2 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/855\mathbb{Z}\right)^\times\).

\(n\) \(172\) \(191\) \(496\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i 0.935414 + 0.353553i \(0.115027\pi\)
−0.935414 + 0.353553i \(0.884973\pi\)
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.00000 + 2.00000i 0.447214 + 0.894427i
\(6\) 0 0
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) 3.00000i 1.06066i
\(9\) 0 0
\(10\) −2.00000 + 1.00000i −0.632456 + 0.316228i
\(11\) 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 \(0.910544\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) − 4.00000i − 0.970143i −0.874475 − 0.485071i \(-0.838794\pi\)
0.874475 − 0.485071i \(-0.161206\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 1.00000 + 2.00000i 0.223607 + 0.447214i
\(21\) 0 0
\(22\) 4.00000i 0.852803i
\(23\) − 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
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) 2.00000i 0.377964i
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 5.00000i 0.883883i
\(33\) 0 0
\(34\) 4.00000 0.685994
\(35\) −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 \(-0.307139\pi\)
\(38\) − 1.00000i − 0.162221i
\(39\) 0 0
\(40\) −6.00000 + 3.00000i −0.948683 + 0.474342i
\(41\) 10.0000 1.56174 0.780869 − 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) − 2.00000i − 0.304997i −0.988304 − 0.152499i \(-0.951268\pi\)
0.988304 − 0.152499i \(-0.0487319\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) 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
\(50\) −4.00000 − 3.00000i −0.565685 − 0.424264i
\(51\) 0 0
\(52\) 2.00000i 0.277350i
\(53\) 10.0000i 1.37361i 0.726844 + 0.686803i \(0.240986\pi\)
−0.726844 + 0.686803i \(0.759014\pi\)
\(54\) 0 0
\(55\) 4.00000 + 8.00000i 0.539360 + 1.07872i
\(56\) −6.00000 −0.801784
\(57\) 0 0
\(58\) − 6.00000i − 0.787839i
\(59\) 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 \(0.459132\pi\)
\(62\) − 4.00000i − 0.508001i
\(63\) 0 0
\(64\) −7.00000 −0.875000
\(65\) −4.00000 + 2.00000i −0.496139 + 0.248069i
\(66\) 0 0
\(67\) 8.00000i 0.977356i 0.872464 + 0.488678i \(0.162521\pi\)
−0.872464 + 0.488678i \(0.837479\pi\)
\(68\) − 4.00000i − 0.485071i
\(69\) 0 0
\(70\) −2.00000 − 4.00000i −0.239046 − 0.478091i
\(71\) −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 \(-0.0752085\pi\)
\(74\) 10.0000 1.16248
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) 8.00000i 0.911685i
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 − 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) −1.00000 − 2.00000i −0.111803 − 0.223607i
\(81\) 0 0
\(82\) 10.0000i 1.10432i
\(83\) − 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
\(86\) 2.00000 0.215666
\(87\) 0 0
\(88\) 12.0000i 1.27920i
\(89\) −2.00000 −0.212000 −0.106000 − 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) − 6.00000i − 0.625543i
\(93\) 0 0
\(94\) −6.00000 −0.618853
\(95\) −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 \(0.901476\pi\)
\(98\) 3.00000i 0.303046i
\(99\) 0 0
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 855.2.c.b.514.2 2
3.2 odd 2 95.2.b.a.39.1 ✓ 2
5.2 odd 4 4275.2.a.e.1.1 1
5.3 odd 4 4275.2.a.p.1.1 1
5.4 even 2 inner 855.2.c.b.514.1 2
12.11 even 2 1520.2.d.b.609.1 2
15.2 even 4 475.2.a.c.1.1 1
15.8 even 4 475.2.a.a.1.1 1
15.14 odd 2 95.2.b.a.39.2 yes 2
57.56 even 2 1805.2.b.c.1084.2 2
60.23 odd 4 7600.2.a.i.1.1 1
60.47 odd 4 7600.2.a.l.1.1 1
60.59 even 2 1520.2.d.b.609.2 2
285.113 odd 4 9025.2.a.h.1.1 1
285.227 odd 4 9025.2.a.c.1.1 1
285.284 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 3.2 odd 2
95.2.b.a.39.2 yes 2 15.14 odd 2
475.2.a.a.1.1 1 15.8 even 4
475.2.a.c.1.1 1 15.2 even 4
855.2.c.b.514.1 2 5.4 even 2 inner
855.2.c.b.514.2 2 1.1 even 1 trivial
1520.2.d.b.609.1 2 12.11 even 2
1520.2.d.b.609.2 2 60.59 even 2
1805.2.b.c.1084.1 2 285.284 even 2
1805.2.b.c.1084.2 2 57.56 even 2
4275.2.a.e.1.1 1 5.2 odd 4
4275.2.a.p.1.1 1 5.3 odd 4
7600.2.a.i.1.1 1 60.23 odd 4
7600.2.a.l.1.1 1 60.47 odd 4
9025.2.a.c.1.1 1 285.227 odd 4
9025.2.a.h.1.1 1 285.113 odd 4