Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1805,2,Mod(1084,1805)] 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("1805.1084"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1805, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1805 = 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1805.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-4,-2,0,0,0,6,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.4129975648\)
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: \( 2 \)
Twist minimal: no (minimal twist has level 95)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1084.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1805.1084
Dual form 1805.2.b.b.1084.2

$q$-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)
 
\(f(q)\) \(=\) \(q-2.00000i q^{2} -2.00000 q^{4} +(-1.00000 + 2.00000i) q^{5} +4.00000i q^{7} +3.00000 q^{9} +(4.00000 + 2.00000i) q^{10} -1.00000 q^{11} -2.00000i q^{13} +8.00000 q^{14} -4.00000 q^{16} +2.00000i q^{17} -6.00000i q^{18} +(2.00000 - 4.00000i) q^{20} +2.00000i q^{22} +6.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} -4.00000 q^{26} -8.00000i q^{28} -9.00000 q^{29} -7.00000 q^{31} +8.00000i q^{32} +4.00000 q^{34} +(-8.00000 - 4.00000i) q^{35} -6.00000 q^{36} -2.00000i q^{37} +2.00000 q^{41} +2.00000i q^{43} +2.00000 q^{44} +(-3.00000 + 6.00000i) q^{45} +12.0000 q^{46} -6.00000i q^{47} -9.00000 q^{49} +(-8.00000 + 6.00000i) q^{50} +4.00000i q^{52} +4.00000i q^{53} +(1.00000 - 2.00000i) q^{55} +18.0000i q^{58} -9.00000 q^{59} -7.00000 q^{61} +14.0000i q^{62} +12.0000i q^{63} +8.00000 q^{64} +(4.00000 + 2.00000i) q^{65} +10.0000i q^{67} -4.00000i q^{68} +(-8.00000 + 16.0000i) q^{70} +1.00000 q^{71} +10.0000i q^{73} -4.00000 q^{74} -4.00000i q^{77} -1.00000 q^{79} +(4.00000 - 8.00000i) q^{80} +9.00000 q^{81} -4.00000i q^{82} -6.00000i q^{83} +(-4.00000 - 2.00000i) q^{85} +4.00000 q^{86} +11.0000 q^{89} +(12.0000 + 6.00000i) q^{90} +8.00000 q^{91} -12.0000i q^{92} -12.0000 q^{94} +6.00000i q^{97} +18.0000i q^{98} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{4} - 2 q^{5} + 6 q^{9} + 8 q^{10} - 2 q^{11} + 16 q^{14} - 8 q^{16} + 4 q^{20} - 6 q^{25} - 8 q^{26} - 18 q^{29} - 14 q^{31} + 8 q^{34} - 16 q^{35} - 12 q^{36} + 4 q^{41} + 4 q^{44} - 6 q^{45}+ \cdots - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1805\mathbb{Z}\right)^\times\).

\(n\) \(362\) \(1446\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.00000i − 1.41421i −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 − 0.707107i \(-0.250000\pi\)
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) −2.00000 −1.00000
\(5\) −1.00000 + 2.00000i −0.447214 + 0.894427i
\(6\) 0 0
\(7\) 4.00000i 1.51186i 0.654654 + 0.755929i \(0.272814\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) 0 0
\(9\) 3.00000 1.00000
\(10\) 4.00000 + 2.00000i 1.26491 + 0.632456i
\(11\) −1.00000 −0.301511 −0.150756 − 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) 0 0
\(13\) − 2.00000i − 0.554700i −0.960769 − 0.277350i \(-0.910544\pi\)
0.960769 − 0.277350i \(-0.0894562\pi\)
\(14\) 8.00000 2.13809
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) 2.00000i 0.485071i 0.970143 + 0.242536i \(0.0779791\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) − 6.00000i − 1.41421i
\(19\) 0 0
\(20\) 2.00000 − 4.00000i 0.447214 − 0.894427i
\(21\) 0 0
\(22\) 2.00000i 0.426401i
\(23\) 6.00000i 1.25109i 0.780189 + 0.625543i \(0.215123\pi\)
−0.780189 + 0.625543i \(0.784877\pi\)
\(24\) 0 0
\(25\) −3.00000 − 4.00000i −0.600000 − 0.800000i
\(26\) −4.00000 −0.784465
\(27\) 0 0
\(28\) − 8.00000i − 1.51186i
\(29\) −9.00000 −1.67126 −0.835629 − 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) 0 0
\(31\) −7.00000 −1.25724 −0.628619 − 0.777714i \(-0.716379\pi\)
−0.628619 + 0.777714i \(0.716379\pi\)
\(32\) 8.00000i 1.41421i
\(33\) 0 0
\(34\) 4.00000 0.685994
\(35\) −8.00000 − 4.00000i −1.35225 − 0.676123i
\(36\) −6.00000 −1.00000
\(37\) − 2.00000i − 0.328798i −0.986394 − 0.164399i \(-0.947432\pi\)
0.986394 − 0.164399i \(-0.0525685\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 2.00000i 0.304997i 0.988304 + 0.152499i \(0.0487319\pi\)
−0.988304 + 0.152499i \(0.951268\pi\)
\(44\) 2.00000 0.301511
\(45\) −3.00000 + 6.00000i −0.447214 + 0.894427i
\(46\) 12.0000 1.76930
\(47\) − 6.00000i − 0.875190i −0.899172 − 0.437595i \(-0.855830\pi\)
0.899172 − 0.437595i \(-0.144170\pi\)
\(48\) 0 0
\(49\) −9.00000 −1.28571
\(50\) −8.00000 + 6.00000i −1.13137 + 0.848528i
\(51\) 0 0
\(52\) 4.00000i 0.554700i
\(53\) 4.00000i 0.549442i 0.961524 + 0.274721i \(0.0885855\pi\)
−0.961524 + 0.274721i \(0.911414\pi\)
\(54\) 0 0
\(55\) 1.00000 − 2.00000i 0.134840 − 0.269680i
\(56\) 0 0
\(57\) 0 0
\(58\) 18.0000i 2.36352i
\(59\) −9.00000 −1.17170 −0.585850 − 0.810419i \(-0.699239\pi\)
−0.585850 + 0.810419i \(0.699239\pi\)
\(60\) 0 0
\(61\) −7.00000 −0.896258 −0.448129 − 0.893969i \(-0.647910\pi\)
−0.448129 + 0.893969i \(0.647910\pi\)
\(62\) 14.0000i 1.77800i
\(63\) 12.0000i 1.51186i
\(64\) 8.00000 1.00000
\(65\) 4.00000 + 2.00000i 0.496139 + 0.248069i
\(66\) 0 0
\(67\) 10.0000i 1.22169i 0.791748 + 0.610847i \(0.209171\pi\)
−0.791748 + 0.610847i \(0.790829\pi\)
\(68\) − 4.00000i − 0.485071i
\(69\) 0 0
\(70\) −8.00000 + 16.0000i −0.956183 + 1.91237i
\(71\) 1.00000 0.118678 0.0593391 − 0.998238i \(-0.481101\pi\)
0.0593391 + 0.998238i \(0.481101\pi\)
\(72\) 0 0
\(73\) 10.0000i 1.17041i 0.810885 + 0.585206i \(0.198986\pi\)
−0.810885 + 0.585206i \(0.801014\pi\)
\(74\) −4.00000 −0.464991
\(75\) 0 0
\(76\) 0 0
\(77\) − 4.00000i − 0.455842i
\(78\) 0 0
\(79\) −1.00000 −0.112509 −0.0562544 − 0.998416i \(-0.517916\pi\)
−0.0562544 + 0.998416i \(0.517916\pi\)
\(80\) 4.00000 − 8.00000i 0.447214 − 0.894427i
\(81\) 9.00000 1.00000
\(82\) − 4.00000i − 0.441726i
\(83\) − 6.00000i − 0.658586i −0.944228 − 0.329293i \(-0.893190\pi\)
0.944228 − 0.329293i \(-0.106810\pi\)
\(84\) 0 0
\(85\) −4.00000 − 2.00000i −0.433861 − 0.216930i
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) 0 0
\(89\) 11.0000 1.16600 0.582999 − 0.812473i \(-0.301879\pi\)
0.582999 + 0.812473i \(0.301879\pi\)
\(90\) 12.0000 + 6.00000i 1.26491 + 0.632456i
\(91\) 8.00000 0.838628
\(92\) − 12.0000i − 1.25109i
\(93\) 0 0
\(94\) −12.0000 −1.23771
\(95\) 0 0
\(96\) 0 0
\(97\) 6.00000i 0.609208i 0.952479 + 0.304604i \(0.0985241\pi\)
−0.952479 + 0.304604i \(0.901476\pi\)
\(98\) 18.0000i 1.81827i
\(99\) −3.00000 −0.301511
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 1805.2.b.b.1084.1 2
5.2 odd 4 9025.2.a.i.1.1 1
5.3 odd 4 9025.2.a.b.1.1 1
5.4 even 2 inner 1805.2.b.b.1084.2 2
19.7 even 3 95.2.i.a.49.2 yes 4
19.11 even 3 95.2.i.a.64.1 yes 4
19.18 odd 2 1805.2.b.a.1084.2 2
57.11 odd 6 855.2.be.a.64.2 4
57.26 odd 6 855.2.be.a.334.1 4
95.7 odd 12 475.2.e.a.201.1 2
95.18 even 4 9025.2.a.j.1.1 1
95.37 even 4 9025.2.a.a.1.1 1
95.49 even 6 95.2.i.a.64.2 yes 4
95.64 even 6 95.2.i.a.49.1 ✓ 4
95.68 odd 12 475.2.e.c.26.1 2
95.83 odd 12 475.2.e.c.201.1 2
95.87 odd 12 475.2.e.a.26.1 2
95.94 odd 2 1805.2.b.a.1084.1 2
285.239 odd 6 855.2.be.a.64.1 4
285.254 odd 6 855.2.be.a.334.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.i.a.49.1 ✓ 4 95.64 even 6
95.2.i.a.49.2 yes 4 19.7 even 3
95.2.i.a.64.1 yes 4 19.11 even 3
95.2.i.a.64.2 yes 4 95.49 even 6
475.2.e.a.26.1 2 95.87 odd 12
475.2.e.a.201.1 2 95.7 odd 12
475.2.e.c.26.1 2 95.68 odd 12
475.2.e.c.201.1 2 95.83 odd 12
855.2.be.a.64.1 4 285.239 odd 6
855.2.be.a.64.2 4 57.11 odd 6
855.2.be.a.334.1 4 57.26 odd 6
855.2.be.a.334.2 4 285.254 odd 6
1805.2.b.a.1084.1 2 95.94 odd 2
1805.2.b.a.1084.2 2 19.18 odd 2
1805.2.b.b.1084.1 2 1.1 even 1 trivial
1805.2.b.b.1084.2 2 5.4 even 2 inner
9025.2.a.a.1.1 1 95.37 even 4
9025.2.a.b.1.1 1 5.3 odd 4
9025.2.a.i.1.1 1 5.2 odd 4
9025.2.a.j.1.1 1 95.18 even 4