Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1850,2,Mod(1,1850)] 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("1850.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1850, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1850.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,-5,0,5,0,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(14.7723243739\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.1791440.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 9x^{3} + 13x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 370)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(1.09441\) of defining polynomial
Character \(\chi\) \(=\) 1850.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.09441 q^{3} +1.00000 q^{4} -1.09441 q^{6} +3.20984 q^{7} -1.00000 q^{8} -1.80226 q^{9} +3.82327 q^{11} +1.09441 q^{12} -0.147332 q^{13} -3.20984 q^{14} +1.00000 q^{16} -0.978989 q^{17} +1.80226 q^{18} +2.67594 q^{19} +3.51289 q^{21} -3.82327 q^{22} +2.33616 q^{23} -1.09441 q^{24} +0.147332 q^{26} -5.25565 q^{27} +3.20984 q^{28} +6.30425 q^{29} -3.62372 q^{31} -1.00000 q^{32} +4.18424 q^{33} +0.978989 q^{34} -1.80226 q^{36} -1.00000 q^{37} -2.67594 q^{38} -0.161242 q^{39} +11.8265 q^{41} -3.51289 q^{42} +4.53390 q^{43} +3.82327 q^{44} -2.33616 q^{46} -6.23085 q^{47} +1.09441 q^{48} +3.30305 q^{49} -1.07142 q^{51} -0.147332 q^{52} -11.2978 q^{53} +5.25565 q^{54} -3.20984 q^{56} +2.92858 q^{57} -6.30425 q^{58} +6.92858 q^{59} +10.4885 q^{61} +3.62372 q^{62} -5.78496 q^{63} +1.00000 q^{64} -4.18424 q^{66} -2.80936 q^{67} -0.978989 q^{68} +2.55672 q^{69} -12.3189 q^{71} +1.80226 q^{72} +13.9966 q^{73} +1.00000 q^{74} +2.67594 q^{76} +12.2721 q^{77} +0.161242 q^{78} +15.6057 q^{79} -0.345071 q^{81} -11.8265 q^{82} -13.5371 q^{83} +3.51289 q^{84} -4.53390 q^{86} +6.89945 q^{87} -3.82327 q^{88} -6.46929 q^{89} -0.472913 q^{91} +2.33616 q^{92} -3.96585 q^{93} +6.23085 q^{94} -1.09441 q^{96} +3.07063 q^{97} -3.30305 q^{98} -6.89054 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 5 q^{2} + 5 q^{4} + q^{7} - 5 q^{8} + 3 q^{9} + 3 q^{11} + 6 q^{13} - q^{14} + 5 q^{16} - 9 q^{17} - 3 q^{18} + 4 q^{19} + 16 q^{21} - 3 q^{22} - 6 q^{23} - 6 q^{26} + q^{28} + 11 q^{29} + 23 q^{31}+ \cdots - 11 q^{99}+O(q^{100}) \) 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
\(3\) 1.09441 0.631859 0.315930 − 0.948783i \(-0.397684\pi\)
0.315930 + 0.948783i \(0.397684\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.09441 −0.446792
\(7\) 3.20984 1.21320 0.606602 − 0.795006i \(-0.292532\pi\)
0.606602 + 0.795006i \(0.292532\pi\)
\(8\) −1.00000 −0.353553
\(9\) −1.80226 −0.600754
\(10\) 0 0
\(11\) 3.82327 1.15276 0.576380 − 0.817182i \(-0.304465\pi\)
0.576380 + 0.817182i \(0.304465\pi\)
\(12\) 1.09441 0.315930
\(13\) −0.147332 −0.0408626 −0.0204313 − 0.999791i \(-0.506504\pi\)
−0.0204313 + 0.999791i \(0.506504\pi\)
\(14\) −3.20984 −0.857865
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −0.978989 −0.237440 −0.118720 − 0.992928i \(-0.537879\pi\)
−0.118720 + 0.992928i \(0.537879\pi\)
\(18\) 1.80226 0.424797
\(19\) 2.67594 0.613903 0.306951 − 0.951725i \(-0.400691\pi\)
0.306951 + 0.951725i \(0.400691\pi\)
\(20\) 0 0
\(21\) 3.51289 0.766574
\(22\) −3.82327 −0.815124
\(23\) 2.33616 0.487122 0.243561 − 0.969886i \(-0.421684\pi\)
0.243561 + 0.969886i \(0.421684\pi\)
\(24\) −1.09441 −0.223396
\(25\) 0 0
\(26\) 0.147332 0.0288942
\(27\) −5.25565 −1.01145
\(28\) 3.20984 0.606602
\(29\) 6.30425 1.17067 0.585335 − 0.810792i \(-0.300963\pi\)
0.585335 + 0.810792i \(0.300963\pi\)
\(30\) 0 0
\(31\) −3.62372 −0.650839 −0.325420 − 0.945570i \(-0.605506\pi\)
−0.325420 + 0.945570i \(0.605506\pi\)
\(32\) −1.00000 −0.176777
\(33\) 4.18424 0.728382
\(34\) 0.978989 0.167895
\(35\) 0 0
\(36\) −1.80226 −0.300377
\(37\) −1.00000 −0.164399
\(38\) −2.67594 −0.434095
\(39\) −0.161242 −0.0258194
\(40\) 0 0
\(41\) 11.8265 1.84698 0.923491 − 0.383620i \(-0.125323\pi\)
0.923491 + 0.383620i \(0.125323\pi\)
\(42\) −3.51289 −0.542050
\(43\) 4.53390 0.691413 0.345706 − 0.938343i \(-0.387639\pi\)
0.345706 + 0.938343i \(0.387639\pi\)
\(44\) 3.82327 0.576380
\(45\) 0 0
\(46\) −2.33616 −0.344448
\(47\) −6.23085 −0.908863 −0.454431 − 0.890782i \(-0.650157\pi\)
−0.454431 + 0.890782i \(0.650157\pi\)
\(48\) 1.09441 0.157965
\(49\) 3.30305 0.471864
\(50\) 0 0
\(51\) −1.07142 −0.150028
\(52\) −0.147332 −0.0204313
\(53\) −11.2978 −1.55188 −0.775939 − 0.630807i \(-0.782724\pi\)
−0.775939 + 0.630807i \(0.782724\pi\)
\(54\) 5.25565 0.715204
\(55\) 0 0
\(56\) −3.20984 −0.428932
\(57\) 2.92858 0.387900
\(58\) −6.30425 −0.827788
\(59\) 6.92858 0.902025 0.451012 − 0.892518i \(-0.351063\pi\)
0.451012 + 0.892518i \(0.351063\pi\)
\(60\) 0 0
\(61\) 10.4885 1.34291 0.671457 − 0.741044i \(-0.265669\pi\)
0.671457 + 0.741044i \(0.265669\pi\)
\(62\) 3.62372 0.460213
\(63\) −5.78496 −0.728837
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −4.18424 −0.515044
\(67\) −2.80936 −0.343218 −0.171609 − 0.985165i \(-0.554897\pi\)
−0.171609 + 0.985165i \(0.554897\pi\)
\(68\) −0.978989 −0.118720
\(69\) 2.55672 0.307793
\(70\) 0 0
\(71\) −12.3189 −1.46198 −0.730990 − 0.682388i \(-0.760941\pi\)
−0.730990 + 0.682388i \(0.760941\pi\)
\(72\) 1.80226 0.212399
\(73\) 13.9966 1.63818 0.819090 − 0.573665i \(-0.194479\pi\)
0.819090 + 0.573665i \(0.194479\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) 2.67594 0.306951
\(77\) 12.2721 1.39853
\(78\) 0.161242 0.0182571
\(79\) 15.6057 1.75578 0.877890 − 0.478861i \(-0.158950\pi\)
0.877890 + 0.478861i \(0.158950\pi\)
\(80\) 0 0
\(81\) −0.345071 −0.0383412
\(82\) −11.8265 −1.30601
\(83\) −13.5371 −1.48589 −0.742944 − 0.669354i \(-0.766571\pi\)
−0.742944 + 0.669354i \(0.766571\pi\)
\(84\) 3.51289 0.383287
\(85\) 0 0
\(86\) −4.53390 −0.488903
\(87\) 6.89945 0.739699
\(88\) −3.82327 −0.407562
\(89\) −6.46929 −0.685743 −0.342872 − 0.939382i \(-0.611400\pi\)
−0.342872 + 0.939382i \(0.611400\pi\)
\(90\) 0 0
\(91\) −0.472913 −0.0495747
\(92\) 2.33616 0.243561
\(93\) −3.96585 −0.411239
\(94\) 6.23085 0.642663
\(95\) 0 0
\(96\) −1.09441 −0.111698
\(97\) 3.07063 0.311775 0.155887 − 0.987775i \(-0.450176\pi\)
0.155887 + 0.987775i \(0.450176\pi\)
\(98\) −3.30305 −0.333658
\(99\) −6.89054 −0.692525
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 1850.2.a.bd.1.4 5
5.2 odd 4 370.2.b.d.149.2 ✓ 10
5.3 odd 4 370.2.b.d.149.9 yes 10
5.4 even 2 1850.2.a.be.1.2 5
15.2 even 4 3330.2.d.p.1999.7 10
15.8 even 4 3330.2.d.p.1999.2 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.b.d.149.2 ✓ 10 5.2 odd 4
370.2.b.d.149.9 yes 10 5.3 odd 4
1850.2.a.bd.1.4 5 1.1 even 1 trivial
1850.2.a.be.1.2 5 5.4 even 2
3330.2.d.p.1999.2 10 15.8 even 4
3330.2.d.p.1999.7 10 15.2 even 4