Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1425,2,Mod(1,1425)] 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("1425.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1425, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1425.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,3,6,0,0,0,-3,3,0,-3] 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: \(11.3786822880\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.837.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 6x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.36147\) of defining polynomial
Character \(\chi\) \(=\) 1425.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.36147 q^{2} +1.00000 q^{3} +3.57653 q^{4} +2.36147 q^{6} +0.784934 q^{7} +3.72294 q^{8} +1.00000 q^{9} +2.93800 q^{11} +3.57653 q^{12} +4.00000 q^{13} +1.85360 q^{14} +1.63853 q^{16} -5.15307 q^{17} +2.36147 q^{18} -1.00000 q^{19} +0.784934 q^{21} +6.93800 q^{22} -3.78493 q^{23} +3.72294 q^{24} +9.44588 q^{26} +1.00000 q^{27} +2.80734 q^{28} -5.00000 q^{29} +1.06200 q^{31} -3.57653 q^{32} +2.93800 q^{33} -12.1688 q^{34} +3.57653 q^{36} -0.722938 q^{37} -2.36147 q^{38} +4.00000 q^{39} +7.93800 q^{41} +1.85360 q^{42} +4.00000 q^{43} +10.5079 q^{44} -8.93800 q^{46} -3.87601 q^{47} +1.63853 q^{48} -6.38388 q^{49} -5.15307 q^{51} +14.3061 q^{52} +6.44588 q^{53} +2.36147 q^{54} +2.92226 q^{56} -1.00000 q^{57} -11.8073 q^{58} -4.66094 q^{59} +8.87601 q^{61} +2.50787 q^{62} +0.784934 q^{63} -11.7229 q^{64} +6.93800 q^{66} +8.93800 q^{67} -18.4301 q^{68} -3.78493 q^{69} -2.66094 q^{71} +3.72294 q^{72} -8.44588 q^{73} -1.70719 q^{74} -3.57653 q^{76} +2.30614 q^{77} +9.44588 q^{78} +3.35480 q^{79} +1.00000 q^{81} +18.7453 q^{82} +4.93800 q^{83} +2.80734 q^{84} +9.44588 q^{86} -5.00000 q^{87} +10.9380 q^{88} +0.569868 q^{89} +3.13974 q^{91} -13.5369 q^{92} +1.06200 q^{93} -9.15307 q^{94} -3.57653 q^{96} +8.59894 q^{97} -15.0753 q^{98} +2.93800 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + 6 q^{4} - 3 q^{8} + 3 q^{9} - 3 q^{11} + 6 q^{12} + 12 q^{13} + 15 q^{14} + 12 q^{16} - 6 q^{17} - 3 q^{19} + 9 q^{22} - 9 q^{23} - 3 q^{24} + 3 q^{27} - 27 q^{28} - 15 q^{29} + 15 q^{31}+ \cdots - 3 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.36147 1.66981 0.834905 − 0.550394i \(-0.185522\pi\)
0.834905 + 0.550394i \(0.185522\pi\)
\(3\) 1.00000 0.577350
\(4\) 3.57653 1.78827
\(5\) 0 0
\(6\) 2.36147 0.964066
\(7\) 0.784934 0.296677 0.148339 − 0.988937i \(-0.452607\pi\)
0.148339 + 0.988937i \(0.452607\pi\)
\(8\) 3.72294 1.31626
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.93800 0.885841 0.442921 − 0.896561i \(-0.353942\pi\)
0.442921 + 0.896561i \(0.353942\pi\)
\(12\) 3.57653 1.03246
\(13\) 4.00000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 1.85360 0.495395
\(15\) 0 0
\(16\) 1.63853 0.409633
\(17\) −5.15307 −1.24980 −0.624901 − 0.780704i \(-0.714861\pi\)
−0.624901 + 0.780704i \(0.714861\pi\)
\(18\) 2.36147 0.556604
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0.784934 0.171287
\(22\) 6.93800 1.47919
\(23\) −3.78493 −0.789213 −0.394607 − 0.918850i \(-0.629119\pi\)
−0.394607 + 0.918850i \(0.629119\pi\)
\(24\) 3.72294 0.759941
\(25\) 0 0
\(26\) 9.44588 1.85249
\(27\) 1.00000 0.192450
\(28\) 2.80734 0.530538
\(29\) −5.00000 −0.928477 −0.464238 − 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 0 0
\(31\) 1.06200 0.190740 0.0953701 − 0.995442i \(-0.469597\pi\)
0.0953701 + 0.995442i \(0.469597\pi\)
\(32\) −3.57653 −0.632248
\(33\) 2.93800 0.511441
\(34\) −12.1688 −2.08693
\(35\) 0 0
\(36\) 3.57653 0.596089
\(37\) −0.722938 −0.118850 −0.0594251 − 0.998233i \(-0.518927\pi\)
−0.0594251 + 0.998233i \(0.518927\pi\)
\(38\) −2.36147 −0.383081
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) 7.93800 1.23971 0.619854 − 0.784717i \(-0.287192\pi\)
0.619854 + 0.784717i \(0.287192\pi\)
\(42\) 1.85360 0.286016
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 10.5079 1.58412
\(45\) 0 0
\(46\) −8.93800 −1.31784
\(47\) −3.87601 −0.565374 −0.282687 − 0.959212i \(-0.591226\pi\)
−0.282687 + 0.959212i \(0.591226\pi\)
\(48\) 1.63853 0.236502
\(49\) −6.38388 −0.911983
\(50\) 0 0
\(51\) −5.15307 −0.721574
\(52\) 14.3061 1.98390
\(53\) 6.44588 0.885409 0.442705 − 0.896668i \(-0.354019\pi\)
0.442705 + 0.896668i \(0.354019\pi\)
\(54\) 2.36147 0.321355
\(55\) 0 0
\(56\) 2.92226 0.390503
\(57\) −1.00000 −0.132453
\(58\) −11.8073 −1.55038
\(59\) −4.66094 −0.606803 −0.303401 − 0.952863i \(-0.598122\pi\)
−0.303401 + 0.952863i \(0.598122\pi\)
\(60\) 0 0
\(61\) 8.87601 1.13646 0.568228 − 0.822871i \(-0.307629\pi\)
0.568228 + 0.822871i \(0.307629\pi\)
\(62\) 2.50787 0.318500
\(63\) 0.784934 0.0988924
\(64\) −11.7229 −1.46537
\(65\) 0 0
\(66\) 6.93800 0.854009
\(67\) 8.93800 1.09195 0.545975 − 0.837801i \(-0.316159\pi\)
0.545975 + 0.837801i \(0.316159\pi\)
\(68\) −18.4301 −2.23498
\(69\) −3.78493 −0.455653
\(70\) 0 0
\(71\) −2.66094 −0.315796 −0.157898 − 0.987455i \(-0.550472\pi\)
−0.157898 + 0.987455i \(0.550472\pi\)
\(72\) 3.72294 0.438752
\(73\) −8.44588 −0.988515 −0.494257 − 0.869316i \(-0.664560\pi\)
−0.494257 + 0.869316i \(0.664560\pi\)
\(74\) −1.70719 −0.198457
\(75\) 0 0
\(76\) −3.57653 −0.410257
\(77\) 2.30614 0.262809
\(78\) 9.44588 1.06953
\(79\) 3.35480 0.377445 0.188722 − 0.982030i \(-0.439565\pi\)
0.188722 + 0.982030i \(0.439565\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 18.7453 2.07008
\(83\) 4.93800 0.542016 0.271008 − 0.962577i \(-0.412643\pi\)
0.271008 + 0.962577i \(0.412643\pi\)
\(84\) 2.80734 0.306306
\(85\) 0 0
\(86\) 9.44588 1.01857
\(87\) −5.00000 −0.536056
\(88\) 10.9380 1.16600
\(89\) 0.569868 0.0604059 0.0302029 − 0.999544i \(-0.490385\pi\)
0.0302029 + 0.999544i \(0.490385\pi\)
\(90\) 0 0
\(91\) 3.13974 0.329134
\(92\) −13.5369 −1.41132
\(93\) 1.06200 0.110124
\(94\) −9.15307 −0.944067
\(95\) 0 0
\(96\) −3.57653 −0.365029
\(97\) 8.59894 0.873091 0.436545 − 0.899682i \(-0.356202\pi\)
0.436545 + 0.899682i \(0.356202\pi\)
\(98\) −15.0753 −1.52284
\(99\) 2.93800 0.295280
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 1425.2.a.w.1.3 yes 3
3.2 odd 2 4275.2.a.bg.1.1 3
5.2 odd 4 1425.2.c.o.799.5 6
5.3 odd 4 1425.2.c.o.799.2 6
5.4 even 2 1425.2.a.t.1.1 ✓ 3
15.14 odd 2 4275.2.a.bf.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1425.2.a.t.1.1 ✓ 3 5.4 even 2
1425.2.a.w.1.3 yes 3 1.1 even 1 trivial
1425.2.c.o.799.2 6 5.3 odd 4
1425.2.c.o.799.5 6 5.2 odd 4
4275.2.a.bf.1.3 3 15.14 odd 2
4275.2.a.bg.1.1 3 3.2 odd 2