Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [5290,2,Mod(1,5290)] 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("5290.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5290, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5290 = 2 \cdot 5 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5290.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,2,1,2,-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: yes
Analytic conductor: \(42.2408626693\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 5290.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -0.618034 q^{3} +1.00000 q^{4} -1.00000 q^{5} -0.618034 q^{6} -1.61803 q^{7} +1.00000 q^{8} -2.61803 q^{9} -1.00000 q^{10} -3.85410 q^{11} -0.618034 q^{12} +4.09017 q^{13} -1.61803 q^{14} +0.618034 q^{15} +1.00000 q^{16} +5.09017 q^{17} -2.61803 q^{18} +4.85410 q^{19} -1.00000 q^{20} +1.00000 q^{21} -3.85410 q^{22} -0.618034 q^{24} +1.00000 q^{25} +4.09017 q^{26} +3.47214 q^{27} -1.61803 q^{28} -4.76393 q^{29} +0.618034 q^{30} -2.09017 q^{31} +1.00000 q^{32} +2.38197 q^{33} +5.09017 q^{34} +1.61803 q^{35} -2.61803 q^{36} +2.47214 q^{37} +4.85410 q^{38} -2.52786 q^{39} -1.00000 q^{40} -12.3262 q^{41} +1.00000 q^{42} -3.85410 q^{44} +2.61803 q^{45} +9.70820 q^{47} -0.618034 q^{48} -4.38197 q^{49} +1.00000 q^{50} -3.14590 q^{51} +4.09017 q^{52} +8.47214 q^{53} +3.47214 q^{54} +3.85410 q^{55} -1.61803 q^{56} -3.00000 q^{57} -4.76393 q^{58} -11.7082 q^{59} +0.618034 q^{60} -6.32624 q^{61} -2.09017 q^{62} +4.23607 q^{63} +1.00000 q^{64} -4.09017 q^{65} +2.38197 q^{66} -5.52786 q^{67} +5.09017 q^{68} +1.61803 q^{70} +7.09017 q^{71} -2.61803 q^{72} -1.23607 q^{73} +2.47214 q^{74} -0.618034 q^{75} +4.85410 q^{76} +6.23607 q^{77} -2.52786 q^{78} -10.4721 q^{79} -1.00000 q^{80} +5.70820 q^{81} -12.3262 q^{82} -10.9443 q^{83} +1.00000 q^{84} -5.09017 q^{85} +2.94427 q^{87} -3.85410 q^{88} +1.52786 q^{89} +2.61803 q^{90} -6.61803 q^{91} +1.29180 q^{93} +9.70820 q^{94} -4.85410 q^{95} -0.618034 q^{96} -14.6180 q^{97} -4.38197 q^{98} +10.0902 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + q^{3} + 2 q^{4} - 2 q^{5} + q^{6} - q^{7} + 2 q^{8} - 3 q^{9} - 2 q^{10} - q^{11} + q^{12} - 3 q^{13} - q^{14} - q^{15} + 2 q^{16} - q^{17} - 3 q^{18} + 3 q^{19} - 2 q^{20} + 2 q^{21}+ \cdots + 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 0.707107
\(3\) −0.618034 −0.356822 −0.178411 − 0.983956i \(-0.557096\pi\)
−0.178411 + 0.983956i \(0.557096\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −0.618034 −0.252311
\(7\) −1.61803 −0.611559 −0.305780 − 0.952102i \(-0.598917\pi\)
−0.305780 + 0.952102i \(0.598917\pi\)
\(8\) 1.00000 0.353553
\(9\) −2.61803 −0.872678
\(10\) −1.00000 −0.316228
\(11\) −3.85410 −1.16206 −0.581028 − 0.813884i \(-0.697349\pi\)
−0.581028 + 0.813884i \(0.697349\pi\)
\(12\) −0.618034 −0.178411
\(13\) 4.09017 1.13441 0.567205 − 0.823577i \(-0.308025\pi\)
0.567205 + 0.823577i \(0.308025\pi\)
\(14\) −1.61803 −0.432438
\(15\) 0.618034 0.159576
\(16\) 1.00000 0.250000
\(17\) 5.09017 1.23455 0.617274 − 0.786748i \(-0.288237\pi\)
0.617274 + 0.786748i \(0.288237\pi\)
\(18\) −2.61803 −0.617077
\(19\) 4.85410 1.11361 0.556804 − 0.830644i \(-0.312028\pi\)
0.556804 + 0.830644i \(0.312028\pi\)
\(20\) −1.00000 −0.223607
\(21\) 1.00000 0.218218
\(22\) −3.85410 −0.821697
\(23\) 0 0
\(24\) −0.618034 −0.126156
\(25\) 1.00000 0.200000
\(26\) 4.09017 0.802148
\(27\) 3.47214 0.668213
\(28\) −1.61803 −0.305780
\(29\) −4.76393 −0.884640 −0.442320 − 0.896857i \(-0.645844\pi\)
−0.442320 + 0.896857i \(0.645844\pi\)
\(30\) 0.618034 0.112837
\(31\) −2.09017 −0.375406 −0.187703 − 0.982226i \(-0.560104\pi\)
−0.187703 + 0.982226i \(0.560104\pi\)
\(32\) 1.00000 0.176777
\(33\) 2.38197 0.414647
\(34\) 5.09017 0.872957
\(35\) 1.61803 0.273498
\(36\) −2.61803 −0.436339
\(37\) 2.47214 0.406417 0.203208 − 0.979136i \(-0.434863\pi\)
0.203208 + 0.979136i \(0.434863\pi\)
\(38\) 4.85410 0.787439
\(39\) −2.52786 −0.404782
\(40\) −1.00000 −0.158114
\(41\) −12.3262 −1.92503 −0.962517 − 0.271220i \(-0.912573\pi\)
−0.962517 + 0.271220i \(0.912573\pi\)
\(42\) 1.00000 0.154303
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) −3.85410 −0.581028
\(45\) 2.61803 0.390273
\(46\) 0 0
\(47\) 9.70820 1.41609 0.708044 − 0.706169i \(-0.249578\pi\)
0.708044 + 0.706169i \(0.249578\pi\)
\(48\) −0.618034 −0.0892055
\(49\) −4.38197 −0.625995
\(50\) 1.00000 0.141421
\(51\) −3.14590 −0.440514
\(52\) 4.09017 0.567205
\(53\) 8.47214 1.16374 0.581869 − 0.813283i \(-0.302322\pi\)
0.581869 + 0.813283i \(0.302322\pi\)
\(54\) 3.47214 0.472498
\(55\) 3.85410 0.519687
\(56\) −1.61803 −0.216219
\(57\) −3.00000 −0.397360
\(58\) −4.76393 −0.625535
\(59\) −11.7082 −1.52428 −0.762139 − 0.647413i \(-0.775851\pi\)
−0.762139 + 0.647413i \(0.775851\pi\)
\(60\) 0.618034 0.0797878
\(61\) −6.32624 −0.809992 −0.404996 − 0.914319i \(-0.632727\pi\)
−0.404996 + 0.914319i \(0.632727\pi\)
\(62\) −2.09017 −0.265452
\(63\) 4.23607 0.533694
\(64\) 1.00000 0.125000
\(65\) −4.09017 −0.507323
\(66\) 2.38197 0.293200
\(67\) −5.52786 −0.675336 −0.337668 − 0.941265i \(-0.609638\pi\)
−0.337668 + 0.941265i \(0.609638\pi\)
\(68\) 5.09017 0.617274
\(69\) 0 0
\(70\) 1.61803 0.193392
\(71\) 7.09017 0.841448 0.420724 − 0.907189i \(-0.361776\pi\)
0.420724 + 0.907189i \(0.361776\pi\)
\(72\) −2.61803 −0.308538
\(73\) −1.23607 −0.144671 −0.0723354 − 0.997380i \(-0.523045\pi\)
−0.0723354 + 0.997380i \(0.523045\pi\)
\(74\) 2.47214 0.287380
\(75\) −0.618034 −0.0713644
\(76\) 4.85410 0.556804
\(77\) 6.23607 0.710666
\(78\) −2.52786 −0.286224
\(79\) −10.4721 −1.17821 −0.589104 − 0.808057i \(-0.700519\pi\)
−0.589104 + 0.808057i \(0.700519\pi\)
\(80\) −1.00000 −0.111803
\(81\) 5.70820 0.634245
\(82\) −12.3262 −1.36121
\(83\) −10.9443 −1.20129 −0.600645 − 0.799516i \(-0.705089\pi\)
−0.600645 + 0.799516i \(0.705089\pi\)
\(84\) 1.00000 0.109109
\(85\) −5.09017 −0.552106
\(86\) 0 0
\(87\) 2.94427 0.315659
\(88\) −3.85410 −0.410849
\(89\) 1.52786 0.161953 0.0809766 − 0.996716i \(-0.474196\pi\)
0.0809766 + 0.996716i \(0.474196\pi\)
\(90\) 2.61803 0.275965
\(91\) −6.61803 −0.693758
\(92\) 0 0
\(93\) 1.29180 0.133953
\(94\) 9.70820 1.00132
\(95\) −4.85410 −0.498020
\(96\) −0.618034 −0.0630778
\(97\) −14.6180 −1.48424 −0.742118 − 0.670269i \(-0.766179\pi\)
−0.742118 + 0.670269i \(0.766179\pi\)
\(98\) −4.38197 −0.442645
\(99\) 10.0902 1.01410
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 5290.2.a.o.1.1 2
23.22 odd 2 230.2.a.c.1.1 ✓ 2
69.68 even 2 2070.2.a.u.1.2 2
92.91 even 2 1840.2.a.l.1.2 2
115.22 even 4 1150.2.b.i.599.4 4
115.68 even 4 1150.2.b.i.599.1 4
115.114 odd 2 1150.2.a.j.1.2 2
184.45 odd 2 7360.2.a.bh.1.2 2
184.91 even 2 7360.2.a.bn.1.1 2
460.459 even 2 9200.2.a.bu.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.c.1.1 ✓ 2 23.22 odd 2
1150.2.a.j.1.2 2 115.114 odd 2
1150.2.b.i.599.1 4 115.68 even 4
1150.2.b.i.599.4 4 115.22 even 4
1840.2.a.l.1.2 2 92.91 even 2
2070.2.a.u.1.2 2 69.68 even 2
5290.2.a.o.1.1 2 1.1 even 1 trivial
7360.2.a.bh.1.2 2 184.45 odd 2
7360.2.a.bn.1.1 2 184.91 even 2
9200.2.a.bu.1.1 2 460.459 even 2