Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [5415,2,Mod(1,5415)] 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("5415.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5415, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5415 = 3 \cdot 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5415.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,1,5,7,5,1,2,6,5,1,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(43.2389926945\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.8797896.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 8x^{3} + 5x^{2} + 13x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 285)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(2.69159\) of defining polynomial
Character \(\chi\) \(=\) 5415.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.69159 q^{2} +1.00000 q^{3} +5.24466 q^{4} +1.00000 q^{5} +2.69159 q^{6} -0.797044 q^{7} +8.73329 q^{8} +1.00000 q^{9} +2.69159 q^{10} +2.59225 q^{11} +5.24466 q^{12} +2.79704 q^{13} -2.14532 q^{14} +1.00000 q^{15} +13.0171 q^{16} -5.77247 q^{17} +2.69159 q^{18} +5.24466 q^{20} -0.797044 q^{21} +6.97727 q^{22} -3.10614 q^{23} +8.73329 q^{24} +1.00000 q^{25} +7.52850 q^{26} +1.00000 q^{27} -4.18023 q^{28} +4.79093 q^{29} +2.69159 q^{30} +9.48932 q^{31} +17.5702 q^{32} +2.59225 q^{33} -15.5371 q^{34} -0.797044 q^{35} +5.24466 q^{36} -7.69227 q^{37} +2.79704 q^{39} +8.73329 q^{40} -7.38318 q^{41} -2.14532 q^{42} -2.79704 q^{43} +13.5955 q^{44} +1.00000 q^{45} -8.36045 q^{46} -11.0773 q^{47} +13.0171 q^{48} -6.36472 q^{49} +2.69159 q^{50} -5.77247 q^{51} +14.6695 q^{52} +8.87861 q^{53} +2.69159 q^{54} +2.59225 q^{55} -6.96082 q^{56} +12.8952 q^{58} -1.08020 q^{59} +5.24466 q^{60} +4.07225 q^{61} +25.5414 q^{62} -0.797044 q^{63} +21.2575 q^{64} +2.79704 q^{65} +6.97727 q^{66} -13.7757 q^{67} -30.2747 q^{68} -3.10614 q^{69} -2.14532 q^{70} +11.9754 q^{71} +8.73329 q^{72} +8.50778 q^{73} -20.7045 q^{74} +1.00000 q^{75} -2.06614 q^{77} +7.52850 q^{78} -6.48184 q^{79} +13.0171 q^{80} +1.00000 q^{81} -19.8725 q^{82} -2.79520 q^{83} -4.18023 q^{84} -5.77247 q^{85} -7.52850 q^{86} +4.79093 q^{87} +22.6389 q^{88} +13.3586 q^{89} +2.69159 q^{90} -2.22937 q^{91} -16.2906 q^{92} +9.48932 q^{93} -29.8155 q^{94} +17.5702 q^{96} -2.00000 q^{97} -17.1312 q^{98} +2.59225 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + q^{2} + 5 q^{3} + 7 q^{4} + 5 q^{5} + q^{6} + 2 q^{7} + 6 q^{8} + 5 q^{9} + q^{10} + 5 q^{11} + 7 q^{12} + 8 q^{13} + 4 q^{14} + 5 q^{15} + 7 q^{16} + 10 q^{17} + q^{18} + 7 q^{20} + 2 q^{21}+ \cdots + 5 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.69159 1.90324 0.951621 − 0.307274i \(-0.0994170\pi\)
0.951621 + 0.307274i \(0.0994170\pi\)
\(3\) 1.00000 0.577350
\(4\) 5.24466 2.62233
\(5\) 1.00000 0.447214
\(6\) 2.69159 1.09884
\(7\) −0.797044 −0.301254 −0.150627 − 0.988591i \(-0.548129\pi\)
−0.150627 + 0.988591i \(0.548129\pi\)
\(8\) 8.73329 3.08769
\(9\) 1.00000 0.333333
\(10\) 2.69159 0.851156
\(11\) 2.59225 0.781592 0.390796 − 0.920477i \(-0.372200\pi\)
0.390796 + 0.920477i \(0.372200\pi\)
\(12\) 5.24466 1.51400
\(13\) 2.79704 0.775760 0.387880 − 0.921710i \(-0.373207\pi\)
0.387880 + 0.921710i \(0.373207\pi\)
\(14\) −2.14532 −0.573360
\(15\) 1.00000 0.258199
\(16\) 13.0171 3.25428
\(17\) −5.77247 −1.40003 −0.700015 − 0.714128i \(-0.746823\pi\)
−0.700015 + 0.714128i \(0.746823\pi\)
\(18\) 2.69159 0.634414
\(19\) 0 0
\(20\) 5.24466 1.17274
\(21\) −0.797044 −0.173929
\(22\) 6.97727 1.48756
\(23\) −3.10614 −0.647674 −0.323837 − 0.946113i \(-0.604973\pi\)
−0.323837 + 0.946113i \(0.604973\pi\)
\(24\) 8.73329 1.78268
\(25\) 1.00000 0.200000
\(26\) 7.52850 1.47646
\(27\) 1.00000 0.192450
\(28\) −4.18023 −0.789988
\(29\) 4.79093 0.889654 0.444827 − 0.895617i \(-0.353265\pi\)
0.444827 + 0.895617i \(0.353265\pi\)
\(30\) 2.69159 0.491415
\(31\) 9.48932 1.70433 0.852166 − 0.523272i \(-0.175289\pi\)
0.852166 + 0.523272i \(0.175289\pi\)
\(32\) 17.5702 3.10600
\(33\) 2.59225 0.451252
\(34\) −15.5371 −2.66460
\(35\) −0.797044 −0.134725
\(36\) 5.24466 0.874110
\(37\) −7.69227 −1.26460 −0.632301 − 0.774723i \(-0.717889\pi\)
−0.632301 + 0.774723i \(0.717889\pi\)
\(38\) 0 0
\(39\) 2.79704 0.447886
\(40\) 8.73329 1.38086
\(41\) −7.38318 −1.15306 −0.576530 − 0.817076i \(-0.695593\pi\)
−0.576530 + 0.817076i \(0.695593\pi\)
\(42\) −2.14532 −0.331030
\(43\) −2.79704 −0.426545 −0.213273 − 0.976993i \(-0.568412\pi\)
−0.213273 + 0.976993i \(0.568412\pi\)
\(44\) 13.5955 2.04959
\(45\) 1.00000 0.149071
\(46\) −8.36045 −1.23268
\(47\) −11.0773 −1.61579 −0.807895 − 0.589327i \(-0.799393\pi\)
−0.807895 + 0.589327i \(0.799393\pi\)
\(48\) 13.0171 1.87886
\(49\) −6.36472 −0.909246
\(50\) 2.69159 0.380648
\(51\) −5.77247 −0.808308
\(52\) 14.6695 2.03430
\(53\) 8.87861 1.21957 0.609785 − 0.792567i \(-0.291256\pi\)
0.609785 + 0.792567i \(0.291256\pi\)
\(54\) 2.69159 0.366279
\(55\) 2.59225 0.349539
\(56\) −6.96082 −0.930179
\(57\) 0 0
\(58\) 12.8952 1.69323
\(59\) −1.08020 −0.140630 −0.0703149 − 0.997525i \(-0.522400\pi\)
−0.0703149 + 0.997525i \(0.522400\pi\)
\(60\) 5.24466 0.677083
\(61\) 4.07225 0.521398 0.260699 − 0.965420i \(-0.416047\pi\)
0.260699 + 0.965420i \(0.416047\pi\)
\(62\) 25.5414 3.24376
\(63\) −0.797044 −0.100418
\(64\) 21.2575 2.65719
\(65\) 2.79704 0.346931
\(66\) 6.97727 0.858842
\(67\) −13.7757 −1.68297 −0.841484 − 0.540283i \(-0.818317\pi\)
−0.841484 + 0.540283i \(0.818317\pi\)
\(68\) −30.2747 −3.67134
\(69\) −3.10614 −0.373935
\(70\) −2.14532 −0.256414
\(71\) 11.9754 1.42122 0.710611 − 0.703585i \(-0.248419\pi\)
0.710611 + 0.703585i \(0.248419\pi\)
\(72\) 8.73329 1.02923
\(73\) 8.50778 0.995760 0.497880 − 0.867246i \(-0.334112\pi\)
0.497880 + 0.867246i \(0.334112\pi\)
\(74\) −20.7045 −2.40684
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) −2.06614 −0.235458
\(78\) 7.52850 0.852434
\(79\) −6.48184 −0.729264 −0.364632 − 0.931152i \(-0.618805\pi\)
−0.364632 + 0.931152i \(0.618805\pi\)
\(80\) 13.0171 1.45536
\(81\) 1.00000 0.111111
\(82\) −19.8725 −2.19455
\(83\) −2.79520 −0.306813 −0.153407 − 0.988163i \(-0.549024\pi\)
−0.153407 + 0.988163i \(0.549024\pi\)
\(84\) −4.18023 −0.456100
\(85\) −5.77247 −0.626113
\(86\) −7.52850 −0.811819
\(87\) 4.79093 0.513642
\(88\) 22.6389 2.41331
\(89\) 13.3586 1.41601 0.708005 − 0.706208i \(-0.249595\pi\)
0.708005 + 0.706208i \(0.249595\pi\)
\(90\) 2.69159 0.283719
\(91\) −2.22937 −0.233701
\(92\) −16.2906 −1.69842
\(93\) 9.48932 0.983996
\(94\) −29.8155 −3.07524
\(95\) 0 0
\(96\) 17.5702 1.79325
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) −17.1312 −1.73051
\(99\) 2.59225 0.260531
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 5415.2.a.z.1.5 5
19.8 odd 6 285.2.i.f.121.5 yes 10
19.12 odd 6 285.2.i.f.106.5 ✓ 10
19.18 odd 2 5415.2.a.y.1.1 5
57.8 even 6 855.2.k.i.406.1 10
57.50 even 6 855.2.k.i.676.1 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.i.f.106.5 ✓ 10 19.12 odd 6
285.2.i.f.121.5 yes 10 19.8 odd 6
855.2.k.i.406.1 10 57.8 even 6
855.2.k.i.676.1 10 57.50 even 6
5415.2.a.y.1.1 5 19.18 odd 2
5415.2.a.z.1.5 5 1.1 even 1 trivial