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(799,1425)] 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("1425.799"); 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, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1425.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,-12,0,0,0,0,-6,0,-6,0,0,-30] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.3786822880\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.44836416.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 12x^{4} + 36x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 799.2
Root \(-2.36147i\) of defining polynomial
Character \(\chi\) \(=\) 1425.799
Dual form 1425.2.c.o.799.5

$q$-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)
 
\(f(q)\) \(=\) \(q-2.36147i q^{2} +1.00000i q^{3} -3.57653 q^{4} +2.36147 q^{6} -0.784934i q^{7} +3.72294i q^{8} -1.00000 q^{9} +2.93800 q^{11} -3.57653i q^{12} +4.00000i q^{13} -1.85360 q^{14} +1.63853 q^{16} +5.15307i q^{17} +2.36147i q^{18} +1.00000 q^{19} +0.784934 q^{21} -6.93800i q^{22} -3.78493i q^{23} -3.72294 q^{24} +9.44588 q^{26} -1.00000i q^{27} +2.80734i q^{28} +5.00000 q^{29} +1.06200 q^{31} +3.57653i q^{32} +2.93800i q^{33} +12.1688 q^{34} +3.57653 q^{36} +0.722938i q^{37} -2.36147i q^{38} -4.00000 q^{39} +7.93800 q^{41} -1.85360i q^{42} +4.00000i q^{43} -10.5079 q^{44} -8.93800 q^{46} +3.87601i q^{47} +1.63853i q^{48} +6.38388 q^{49} -5.15307 q^{51} -14.3061i q^{52} +6.44588i q^{53} -2.36147 q^{54} +2.92226 q^{56} +1.00000i q^{57} -11.8073i q^{58} +4.66094 q^{59} +8.87601 q^{61} -2.50787i q^{62} +0.784934i q^{63} +11.7229 q^{64} +6.93800 q^{66} -8.93800i q^{67} -18.4301i q^{68} +3.78493 q^{69} -2.66094 q^{71} -3.72294i q^{72} -8.44588i q^{73} +1.70719 q^{74} -3.57653 q^{76} -2.30614i q^{77} +9.44588i q^{78} -3.35480 q^{79} +1.00000 q^{81} -18.7453i q^{82} +4.93800i q^{83} -2.80734 q^{84} +9.44588 q^{86} +5.00000i q^{87} +10.9380i q^{88} -0.569868 q^{89} +3.13974 q^{91} +13.5369i q^{92} +1.06200i q^{93} +9.15307 q^{94} -3.57653 q^{96} -8.59894i q^{97} -15.0753i q^{98} -2.93800 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 12 q^{4} - 6 q^{9} - 6 q^{11} - 30 q^{14} + 24 q^{16} + 6 q^{19} + 6 q^{24} + 30 q^{29} + 30 q^{31} - 12 q^{34} + 12 q^{36} - 24 q^{39} + 24 q^{41} - 30 q^{44} - 30 q^{46} - 42 q^{49} - 12 q^{51}+ \cdots + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1425\mathbb{Z}\right)^\times\).

\(n\) \(476\) \(1027\) \(1351\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.36147i − 1.66981i −0.550394 − 0.834905i \(-0.685522\pi\)
0.550394 − 0.834905i \(-0.314478\pi\)
\(3\) 1.00000i 0.577350i
\(4\) −3.57653 −1.78827
\(5\) 0 0
\(6\) 2.36147 0.964066
\(7\) − 0.784934i − 0.296677i −0.988937 − 0.148339i \(-0.952607\pi\)
0.988937 − 0.148339i \(-0.0473925\pi\)
\(8\) 3.72294i 1.31626i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 2.93800 0.885841 0.442921 − 0.896561i \(-0.353942\pi\)
0.442921 + 0.896561i \(0.353942\pi\)
\(12\) − 3.57653i − 1.03246i
\(13\) 4.00000i 1.10940i 0.832050 + 0.554700i \(0.187167\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) −1.85360 −0.495395
\(15\) 0 0
\(16\) 1.63853 0.409633
\(17\) 5.15307i 1.24980i 0.780704 + 0.624901i \(0.214861\pi\)
−0.780704 + 0.624901i \(0.785139\pi\)
\(18\) 2.36147i 0.556604i
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0.784934 0.171287
\(22\) − 6.93800i − 1.47919i
\(23\) − 3.78493i − 0.789213i −0.918850 − 0.394607i \(-0.870881\pi\)
0.918850 − 0.394607i \(-0.129119\pi\)
\(24\) −3.72294 −0.759941
\(25\) 0 0
\(26\) 9.44588 1.85249
\(27\) − 1.00000i − 0.192450i
\(28\) 2.80734i 0.530538i
\(29\) 5.00000 0.928477 0.464238 − 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 0 0
\(31\) 1.06200 0.190740 0.0953701 − 0.995442i \(-0.469597\pi\)
0.0953701 + 0.995442i \(0.469597\pi\)
\(32\) 3.57653i 0.632248i
\(33\) 2.93800i 0.511441i
\(34\) 12.1688 2.08693
\(35\) 0 0
\(36\) 3.57653 0.596089
\(37\) 0.722938i 0.118850i 0.998233 + 0.0594251i \(0.0189267\pi\)
−0.998233 + 0.0594251i \(0.981073\pi\)
\(38\) − 2.36147i − 0.383081i
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) 7.93800 1.23971 0.619854 − 0.784717i \(-0.287192\pi\)
0.619854 + 0.784717i \(0.287192\pi\)
\(42\) − 1.85360i − 0.286016i
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) −10.5079 −1.58412
\(45\) 0 0
\(46\) −8.93800 −1.31784
\(47\) 3.87601i 0.565374i 0.959212 + 0.282687i \(0.0912257\pi\)
−0.959212 + 0.282687i \(0.908774\pi\)
\(48\) 1.63853i 0.236502i
\(49\) 6.38388 0.911983
\(50\) 0 0
\(51\) −5.15307 −0.721574
\(52\) − 14.3061i − 1.98390i
\(53\) 6.44588i 0.885409i 0.896668 + 0.442705i \(0.145981\pi\)
−0.896668 + 0.442705i \(0.854019\pi\)
\(54\) −2.36147 −0.321355
\(55\) 0 0
\(56\) 2.92226 0.390503
\(57\) 1.00000i 0.132453i
\(58\) − 11.8073i − 1.55038i
\(59\) 4.66094 0.606803 0.303401 − 0.952863i \(-0.401878\pi\)
0.303401 + 0.952863i \(0.401878\pi\)
\(60\) 0 0
\(61\) 8.87601 1.13646 0.568228 − 0.822871i \(-0.307629\pi\)
0.568228 + 0.822871i \(0.307629\pi\)
\(62\) − 2.50787i − 0.318500i
\(63\) 0.784934i 0.0988924i
\(64\) 11.7229 1.46537
\(65\) 0 0
\(66\) 6.93800 0.854009
\(67\) − 8.93800i − 1.09195i −0.837801 − 0.545975i \(-0.816159\pi\)
0.837801 − 0.545975i \(-0.183841\pi\)
\(68\) − 18.4301i − 2.23498i
\(69\) 3.78493 0.455653
\(70\) 0 0
\(71\) −2.66094 −0.315796 −0.157898 − 0.987455i \(-0.550472\pi\)
−0.157898 + 0.987455i \(0.550472\pi\)
\(72\) − 3.72294i − 0.438752i
\(73\) − 8.44588i − 0.988515i −0.869316 − 0.494257i \(-0.835440\pi\)
0.869316 − 0.494257i \(-0.164560\pi\)
\(74\) 1.70719 0.198457
\(75\) 0 0
\(76\) −3.57653 −0.410257
\(77\) − 2.30614i − 0.262809i
\(78\) 9.44588i 1.06953i
\(79\) −3.35480 −0.377445 −0.188722 − 0.982030i \(-0.560435\pi\)
−0.188722 + 0.982030i \(0.560435\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 18.7453i − 2.07008i
\(83\) 4.93800i 0.542016i 0.962577 + 0.271008i \(0.0873570\pi\)
−0.962577 + 0.271008i \(0.912643\pi\)
\(84\) −2.80734 −0.306306
\(85\) 0 0
\(86\) 9.44588 1.01857
\(87\) 5.00000i 0.536056i
\(88\) 10.9380i 1.16600i
\(89\) −0.569868 −0.0604059 −0.0302029 − 0.999544i \(-0.509615\pi\)
−0.0302029 + 0.999544i \(0.509615\pi\)
\(90\) 0 0
\(91\) 3.13974 0.329134
\(92\) 13.5369i 1.41132i
\(93\) 1.06200i 0.110124i
\(94\) 9.15307 0.944067
\(95\) 0 0
\(96\) −3.57653 −0.365029
\(97\) − 8.59894i − 0.873091i −0.899682 − 0.436545i \(-0.856202\pi\)
0.899682 − 0.436545i \(-0.143798\pi\)
\(98\) − 15.0753i − 1.52284i
\(99\) −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.c.o.799.2 6
5.2 odd 4 1425.2.a.w.1.3 yes 3
5.3 odd 4 1425.2.a.t.1.1 ✓ 3
5.4 even 2 inner 1425.2.c.o.799.5 6
15.2 even 4 4275.2.a.bg.1.1 3
15.8 even 4 4275.2.a.bf.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1425.2.a.t.1.1 ✓ 3 5.3 odd 4
1425.2.a.w.1.3 yes 3 5.2 odd 4
1425.2.c.o.799.2 6 1.1 even 1 trivial
1425.2.c.o.799.5 6 5.4 even 2 inner
4275.2.a.bf.1.3 3 15.8 even 4
4275.2.a.bg.1.1 3 15.2 even 4