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 \(-2.24750\) 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)
 
\(f(q)\) \(=\) \(q+2.24750 q^{2} -1.00000 q^{3} +3.05123 q^{4} +1.00000 q^{5} -2.24750 q^{6} +3.16638 q^{7} +2.36264 q^{8} +1.00000 q^{9} +2.24750 q^{10} +4.81770 q^{11} -3.05123 q^{12} +1.16638 q^{13} +7.11643 q^{14} -1.00000 q^{15} -0.792439 q^{16} +5.84367 q^{17} +2.24750 q^{18} +3.05123 q^{20} -3.16638 q^{21} +10.8278 q^{22} -8.59746 q^{23} -2.36264 q^{24} +1.00000 q^{25} +2.62144 q^{26} -1.00000 q^{27} +9.66137 q^{28} +7.31269 q^{29} -2.24750 q^{30} -5.10247 q^{31} -6.50629 q^{32} -4.81770 q^{33} +13.1336 q^{34} +3.16638 q^{35} +3.05123 q^{36} +7.26885 q^{37} -1.16638 q^{39} +2.36264 q^{40} -2.49499 q^{41} -7.11643 q^{42} +1.16638 q^{43} +14.6999 q^{44} +1.00000 q^{45} -19.3227 q^{46} +9.37660 q^{47} +0.792439 q^{48} +3.02597 q^{49} +2.24750 q^{50} -5.84367 q^{51} +3.55890 q^{52} -2.75378 q^{53} -2.24750 q^{54} +4.81770 q^{55} +7.48103 q^{56} +16.4352 q^{58} -10.1125 q^{59} -3.05123 q^{60} -5.10837 q^{61} -11.4678 q^{62} +3.16638 q^{63} -13.0380 q^{64} +1.16638 q^{65} -10.8278 q^{66} +1.03855 q^{67} +17.8304 q^{68} +8.59746 q^{69} +7.11643 q^{70} -4.32271 q^{71} +2.36264 q^{72} +3.63345 q^{73} +16.3367 q^{74} -1.00000 q^{75} +15.2547 q^{77} -2.62144 q^{78} -15.0765 q^{79} -0.792439 q^{80} +1.00000 q^{81} -5.60748 q^{82} -8.98408 q^{83} -9.66137 q^{84} +5.84367 q^{85} +2.62144 q^{86} -7.31269 q^{87} +11.3825 q^{88} +4.17228 q^{89} +2.24750 q^{90} +3.69321 q^{91} -26.2329 q^{92} +5.10247 q^{93} +21.0739 q^{94} +6.50629 q^{96} +2.00000 q^{97} +6.80086 q^{98} +4.81770 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - q^{2} - 5 q^{3} + 7 q^{4} + 5 q^{5} + q^{6} + 2 q^{7} - 6 q^{8} + 5 q^{9} - q^{10} + 5 q^{11} - 7 q^{12} - 8 q^{13} - 4 q^{14} - 5 q^{15} + 7 q^{16} + 10 q^{17} - q^{18} + 7 q^{20} - 2 q^{21}+ \cdots + 5 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.24750 1.58922 0.794609 − 0.607121i \(-0.207676\pi\)
0.794609 + 0.607121i \(0.207676\pi\)
\(3\) −1.00000 −0.577350
\(4\) 3.05123 1.52562
\(5\) 1.00000 0.447214
\(6\) −2.24750 −0.917536
\(7\) 3.16638 1.19678 0.598390 − 0.801205i \(-0.295807\pi\)
0.598390 + 0.801205i \(0.295807\pi\)
\(8\) 2.36264 0.835320
\(9\) 1.00000 0.333333
\(10\) 2.24750 0.710720
\(11\) 4.81770 1.45259 0.726295 − 0.687383i \(-0.241240\pi\)
0.726295 + 0.687383i \(0.241240\pi\)
\(12\) −3.05123 −0.880815
\(13\) 1.16638 0.323496 0.161748 − 0.986832i \(-0.448287\pi\)
0.161748 + 0.986832i \(0.448287\pi\)
\(14\) 7.11643 1.90195
\(15\) −1.00000 −0.258199
\(16\) −0.792439 −0.198110
\(17\) 5.84367 1.41730 0.708649 − 0.705561i \(-0.249305\pi\)
0.708649 + 0.705561i \(0.249305\pi\)
\(18\) 2.24750 0.529740
\(19\) 0 0
\(20\) 3.05123 0.682277
\(21\) −3.16638 −0.690961
\(22\) 10.8278 2.30848
\(23\) −8.59746 −1.79269 −0.896347 − 0.443353i \(-0.853789\pi\)
−0.896347 + 0.443353i \(0.853789\pi\)
\(24\) −2.36264 −0.482272
\(25\) 1.00000 0.200000
\(26\) 2.62144 0.514106
\(27\) −1.00000 −0.192450
\(28\) 9.66137 1.82583
\(29\) 7.31269 1.35793 0.678966 − 0.734170i \(-0.262428\pi\)
0.678966 + 0.734170i \(0.262428\pi\)
\(30\) −2.24750 −0.410335
\(31\) −5.10247 −0.916430 −0.458215 − 0.888841i \(-0.651511\pi\)
−0.458215 + 0.888841i \(0.651511\pi\)
\(32\) −6.50629 −1.15016
\(33\) −4.81770 −0.838654
\(34\) 13.1336 2.25240
\(35\) 3.16638 0.535216
\(36\) 3.05123 0.508539
\(37\) 7.26885 1.19499 0.597496 − 0.801872i \(-0.296162\pi\)
0.597496 + 0.801872i \(0.296162\pi\)
\(38\) 0 0
\(39\) −1.16638 −0.186771
\(40\) 2.36264 0.373567
\(41\) −2.49499 −0.389652 −0.194826 − 0.980838i \(-0.562414\pi\)
−0.194826 + 0.980838i \(0.562414\pi\)
\(42\) −7.11643 −1.09809
\(43\) 1.16638 0.177872 0.0889358 − 0.996037i \(-0.471653\pi\)
0.0889358 + 0.996037i \(0.471653\pi\)
\(44\) 14.6999 2.21610
\(45\) 1.00000 0.149071
\(46\) −19.3227 −2.84898
\(47\) 9.37660 1.36772 0.683859 − 0.729614i \(-0.260300\pi\)
0.683859 + 0.729614i \(0.260300\pi\)
\(48\) 0.792439 0.114379
\(49\) 3.02597 0.432282
\(50\) 2.24750 0.317844
\(51\) −5.84367 −0.818278
\(52\) 3.55890 0.493531
\(53\) −2.75378 −0.378261 −0.189131 − 0.981952i \(-0.560567\pi\)
−0.189131 + 0.981952i \(0.560567\pi\)
\(54\) −2.24750 −0.305845
\(55\) 4.81770 0.649618
\(56\) 7.48103 0.999695
\(57\) 0 0
\(58\) 16.4352 2.15805
\(59\) −10.1125 −1.31654 −0.658269 − 0.752783i \(-0.728711\pi\)
−0.658269 + 0.752783i \(0.728711\pi\)
\(60\) −3.05123 −0.393913
\(61\) −5.10837 −0.654059 −0.327030 − 0.945014i \(-0.606048\pi\)
−0.327030 + 0.945014i \(0.606048\pi\)
\(62\) −11.4678 −1.45641
\(63\) 3.16638 0.398927
\(64\) −13.0380 −1.62975
\(65\) 1.16638 0.144672
\(66\) −10.8278 −1.33280
\(67\) 1.03855 0.126880 0.0634398 − 0.997986i \(-0.479793\pi\)
0.0634398 + 0.997986i \(0.479793\pi\)
\(68\) 17.8304 2.16226
\(69\) 8.59746 1.03501
\(70\) 7.11643 0.850576
\(71\) −4.32271 −0.513011 −0.256506 − 0.966543i \(-0.582571\pi\)
−0.256506 + 0.966543i \(0.582571\pi\)
\(72\) 2.36264 0.278440
\(73\) 3.63345 0.425263 0.212632 − 0.977132i \(-0.431797\pi\)
0.212632 + 0.977132i \(0.431797\pi\)
\(74\) 16.3367 1.89910
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 15.2547 1.73843
\(78\) −2.62144 −0.296819
\(79\) −15.0765 −1.69624 −0.848121 − 0.529803i \(-0.822266\pi\)
−0.848121 + 0.529803i \(0.822266\pi\)
\(80\) −0.792439 −0.0885974
\(81\) 1.00000 0.111111
\(82\) −5.60748 −0.619242
\(83\) −8.98408 −0.986131 −0.493065 − 0.869992i \(-0.664124\pi\)
−0.493065 + 0.869992i \(0.664124\pi\)
\(84\) −9.66137 −1.05414
\(85\) 5.84367 0.633835
\(86\) 2.62144 0.282677
\(87\) −7.31269 −0.784003
\(88\) 11.3825 1.21338
\(89\) 4.17228 0.442261 0.221130 − 0.975244i \(-0.429025\pi\)
0.221130 + 0.975244i \(0.429025\pi\)
\(90\) 2.24750 0.236907
\(91\) 3.69321 0.387154
\(92\) −26.2329 −2.73496
\(93\) 5.10247 0.529101
\(94\) 21.0739 2.17360
\(95\) 0 0
\(96\) 6.50629 0.664045
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 6.80086 0.686991
\(99\) 4.81770 0.484197
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.y.1.5 5
19.7 even 3 285.2.i.f.106.1 ✓ 10
19.11 even 3 285.2.i.f.121.1 yes 10
19.18 odd 2 5415.2.a.z.1.1 5
57.11 odd 6 855.2.k.i.406.5 10
57.26 odd 6 855.2.k.i.676.5 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.i.f.106.1 ✓ 10 19.7 even 3
285.2.i.f.121.1 yes 10 19.11 even 3
855.2.k.i.406.5 10 57.11 odd 6
855.2.k.i.676.5 10 57.26 odd 6
5415.2.a.y.1.5 5 1.1 even 1 trivial
5415.2.a.z.1.1 5 19.18 odd 2