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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 = [2,0,0,-4,0,-4,0,0,-2,0,-6,0,0,12] 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: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 57)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 799.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1425.799
Dual form 1425.2.c.a.799.2

$q$-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)
 
\(f(q)\) \(=\) \(q-2.00000i q^{2} -1.00000i q^{3} -2.00000 q^{4} -2.00000 q^{6} +3.00000i q^{7} -1.00000 q^{9} -3.00000 q^{11} +2.00000i q^{12} +6.00000i q^{13} +6.00000 q^{14} -4.00000 q^{16} +3.00000i q^{17} +2.00000i q^{18} +1.00000 q^{19} +3.00000 q^{21} +6.00000i q^{22} -4.00000i q^{23} +12.0000 q^{26} +1.00000i q^{27} -6.00000i q^{28} +10.0000 q^{29} +2.00000 q^{31} +8.00000i q^{32} +3.00000i q^{33} +6.00000 q^{34} +2.00000 q^{36} +8.00000i q^{37} -2.00000i q^{38} +6.00000 q^{39} -8.00000 q^{41} -6.00000i q^{42} +1.00000i q^{43} +6.00000 q^{44} -8.00000 q^{46} +3.00000i q^{47} +4.00000i q^{48} -2.00000 q^{49} +3.00000 q^{51} -12.0000i q^{52} +6.00000i q^{53} +2.00000 q^{54} -1.00000i q^{57} -20.0000i q^{58} +7.00000 q^{61} -4.00000i q^{62} -3.00000i q^{63} +8.00000 q^{64} +6.00000 q^{66} +8.00000i q^{67} -6.00000i q^{68} -4.00000 q^{69} +12.0000 q^{71} +11.0000i q^{73} +16.0000 q^{74} -2.00000 q^{76} -9.00000i q^{77} -12.0000i q^{78} +1.00000 q^{81} +16.0000i q^{82} -4.00000i q^{83} -6.00000 q^{84} +2.00000 q^{86} -10.0000i q^{87} -10.0000 q^{89} -18.0000 q^{91} +8.00000i q^{92} -2.00000i q^{93} +6.00000 q^{94} +8.00000 q^{96} -2.00000i q^{97} +4.00000i q^{98} +3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{4} - 4 q^{6} - 2 q^{9} - 6 q^{11} + 12 q^{14} - 8 q^{16} + 2 q^{19} + 6 q^{21} + 24 q^{26} + 20 q^{29} + 4 q^{31} + 12 q^{34} + 4 q^{36} + 12 q^{39} - 16 q^{41} + 12 q^{44} - 16 q^{46} - 4 q^{49}+ \cdots + 6 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.00000i − 1.41421i −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 − 0.707107i \(-0.250000\pi\)
\(3\) − 1.00000i − 0.577350i
\(4\) −2.00000 −1.00000
\(5\) 0 0
\(6\) −2.00000 −0.816497
\(7\) 3.00000i 1.13389i 0.823754 + 0.566947i \(0.191875\pi\)
−0.823754 + 0.566947i \(0.808125\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 − 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 2.00000i 0.577350i
\(13\) 6.00000i 1.66410i 0.554700 + 0.832050i \(0.312833\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 6.00000 1.60357
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) 3.00000i 0.727607i 0.931476 + 0.363803i \(0.118522\pi\)
−0.931476 + 0.363803i \(0.881478\pi\)
\(18\) 2.00000i 0.471405i
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 3.00000 0.654654
\(22\) 6.00000i 1.27920i
\(23\) − 4.00000i − 0.834058i −0.908893 − 0.417029i \(-0.863071\pi\)
0.908893 − 0.417029i \(-0.136929\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 12.0000 2.35339
\(27\) 1.00000i 0.192450i
\(28\) − 6.00000i − 1.13389i
\(29\) 10.0000 1.85695 0.928477 − 0.371391i \(-0.121119\pi\)
0.928477 + 0.371391i \(0.121119\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 − 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 8.00000i 1.41421i
\(33\) 3.00000i 0.522233i
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) 2.00000 0.333333
\(37\) 8.00000i 1.31519i 0.753371 + 0.657596i \(0.228427\pi\)
−0.753371 + 0.657596i \(0.771573\pi\)
\(38\) − 2.00000i − 0.324443i
\(39\) 6.00000 0.960769
\(40\) 0 0
\(41\) −8.00000 −1.24939 −0.624695 − 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) − 6.00000i − 0.925820i
\(43\) 1.00000i 0.152499i 0.997089 + 0.0762493i \(0.0242945\pi\)
−0.997089 + 0.0762493i \(0.975706\pi\)
\(44\) 6.00000 0.904534
\(45\) 0 0
\(46\) −8.00000 −1.17954
\(47\) 3.00000i 0.437595i 0.975770 + 0.218797i \(0.0702134\pi\)
−0.975770 + 0.218797i \(0.929787\pi\)
\(48\) 4.00000i 0.577350i
\(49\) −2.00000 −0.285714
\(50\) 0 0
\(51\) 3.00000 0.420084
\(52\) − 12.0000i − 1.66410i
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) 2.00000 0.272166
\(55\) 0 0
\(56\) 0 0
\(57\) − 1.00000i − 0.132453i
\(58\) − 20.0000i − 2.62613i
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 7.00000 0.896258 0.448129 − 0.893969i \(-0.352090\pi\)
0.448129 + 0.893969i \(0.352090\pi\)
\(62\) − 4.00000i − 0.508001i
\(63\) − 3.00000i − 0.377964i
\(64\) 8.00000 1.00000
\(65\) 0 0
\(66\) 6.00000 0.738549
\(67\) 8.00000i 0.977356i 0.872464 + 0.488678i \(0.162521\pi\)
−0.872464 + 0.488678i \(0.837479\pi\)
\(68\) − 6.00000i − 0.727607i
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 − 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) 11.0000i 1.28745i 0.765256 + 0.643726i \(0.222612\pi\)
−0.765256 + 0.643726i \(0.777388\pi\)
\(74\) 16.0000 1.85996
\(75\) 0 0
\(76\) −2.00000 −0.229416
\(77\) − 9.00000i − 1.02565i
\(78\) − 12.0000i − 1.35873i
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 16.0000i 1.76690i
\(83\) − 4.00000i − 0.439057i −0.975606 − 0.219529i \(-0.929548\pi\)
0.975606 − 0.219529i \(-0.0704519\pi\)
\(84\) −6.00000 −0.654654
\(85\) 0 0
\(86\) 2.00000 0.215666
\(87\) − 10.0000i − 1.07211i
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 − 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) −18.0000 −1.88691
\(92\) 8.00000i 0.834058i
\(93\) − 2.00000i − 0.207390i
\(94\) 6.00000 0.618853
\(95\) 0 0
\(96\) 8.00000 0.816497
\(97\) − 2.00000i − 0.203069i −0.994832 − 0.101535i \(-0.967625\pi\)
0.994832 − 0.101535i \(-0.0323753\pi\)
\(98\) 4.00000i 0.404061i
\(99\) 3.00000 0.301511
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.a.799.1 2
5.2 odd 4 1425.2.a.i.1.1 1
5.3 odd 4 57.2.a.b.1.1 ✓ 1
5.4 even 2 inner 1425.2.c.a.799.2 2
15.2 even 4 4275.2.a.a.1.1 1
15.8 even 4 171.2.a.c.1.1 1
20.3 even 4 912.2.a.d.1.1 1
35.13 even 4 2793.2.a.a.1.1 1
40.3 even 4 3648.2.a.y.1.1 1
40.13 odd 4 3648.2.a.h.1.1 1
55.43 even 4 6897.2.a.g.1.1 1
60.23 odd 4 2736.2.a.h.1.1 1
65.38 odd 4 9633.2.a.p.1.1 1
95.18 even 4 1083.2.a.d.1.1 1
105.83 odd 4 8379.2.a.q.1.1 1
285.113 odd 4 3249.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
57.2.a.b.1.1 ✓ 1 5.3 odd 4
171.2.a.c.1.1 1 15.8 even 4
912.2.a.d.1.1 1 20.3 even 4
1083.2.a.d.1.1 1 95.18 even 4
1425.2.a.i.1.1 1 5.2 odd 4
1425.2.c.a.799.1 2 1.1 even 1 trivial
1425.2.c.a.799.2 2 5.4 even 2 inner
2736.2.a.h.1.1 1 60.23 odd 4
2793.2.a.a.1.1 1 35.13 even 4
3249.2.a.a.1.1 1 285.113 odd 4
3648.2.a.h.1.1 1 40.13 odd 4
3648.2.a.y.1.1 1 40.3 even 4
4275.2.a.a.1.1 1 15.2 even 4
6897.2.a.g.1.1 1 55.43 even 4
8379.2.a.q.1.1 1 105.83 odd 4
9633.2.a.p.1.1 1 65.38 odd 4