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Properties

Label 435.2.a.a.1.1
Level $435$
Weight $2$
Character 435.1
Self dual yes
Analytic conductor $3.473$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [435,2,Mod(1,435)] 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("435.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(435, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 435 = 3 \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 435.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-1,1,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(3.47349248793\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 435.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -4.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{12} +6.00000 q^{13} +4.00000 q^{14} +1.00000 q^{15} -1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{18} +8.00000 q^{19} -1.00000 q^{20} -4.00000 q^{21} -4.00000 q^{23} +3.00000 q^{24} +1.00000 q^{25} -6.00000 q^{26} +1.00000 q^{27} +4.00000 q^{28} +1.00000 q^{29} -1.00000 q^{30} +4.00000 q^{31} -5.00000 q^{32} -2.00000 q^{34} -4.00000 q^{35} -1.00000 q^{36} +6.00000 q^{37} -8.00000 q^{38} +6.00000 q^{39} +3.00000 q^{40} +2.00000 q^{41} +4.00000 q^{42} -4.00000 q^{43} +1.00000 q^{45} +4.00000 q^{46} -1.00000 q^{48} +9.00000 q^{49} -1.00000 q^{50} +2.00000 q^{51} -6.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} -12.0000 q^{56} +8.00000 q^{57} -1.00000 q^{58} -12.0000 q^{59} -1.00000 q^{60} +6.00000 q^{61} -4.00000 q^{62} -4.00000 q^{63} +7.00000 q^{64} +6.00000 q^{65} -8.00000 q^{67} -2.00000 q^{68} -4.00000 q^{69} +4.00000 q^{70} +16.0000 q^{71} +3.00000 q^{72} -6.00000 q^{73} -6.00000 q^{74} +1.00000 q^{75} -8.00000 q^{76} -6.00000 q^{78} +12.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} -16.0000 q^{83} +4.00000 q^{84} +2.00000 q^{85} +4.00000 q^{86} +1.00000 q^{87} +2.00000 q^{89} -1.00000 q^{90} -24.0000 q^{91} +4.00000 q^{92} +4.00000 q^{93} +8.00000 q^{95} -5.00000 q^{96} -14.0000 q^{97} -9.00000 q^{98} +O(q^{100})\)

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 −0.353553 − 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.00000 −0.500000
\(5\) 1.00000 0.447214
\(6\) −1.00000 −0.408248
\(7\) −4.00000 −1.51186 −0.755929 − 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 3.00000 1.06066
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) −1.00000 −0.288675
\(13\) 6.00000 1.66410 0.832050 − 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 4.00000 1.06904
\(15\) 1.00000 0.258199
\(16\) −1.00000 −0.250000
\(17\) 2.00000 0.485071 0.242536 − 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) −1.00000 −0.235702
\(19\) 8.00000 1.83533 0.917663 − 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) −1.00000 −0.223607
\(21\) −4.00000 −0.872872
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 − 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 3.00000 0.612372
\(25\) 1.00000 0.200000
\(26\) −6.00000 −1.17670
\(27\) 1.00000 0.192450
\(28\) 4.00000 0.755929
\(29\) 1.00000 0.185695
\(30\) −1.00000 −0.182574
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) −5.00000 −0.883883
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) −4.00000 −0.676123
\(36\) −1.00000 −0.166667
\(37\) 6.00000 0.986394 0.493197 − 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) −8.00000 −1.29777
\(39\) 6.00000 0.960769
\(40\) 3.00000 0.474342
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 4.00000 0.617213
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 4.00000 0.589768
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −1.00000 −0.144338
\(49\) 9.00000 1.28571
\(50\) −1.00000 −0.141421
\(51\) 2.00000 0.280056
\(52\) −6.00000 −0.832050
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −12.0000 −1.60357
\(57\) 8.00000 1.05963
\(58\) −1.00000 −0.131306
\(59\) −12.0000 −1.56227 −0.781133 − 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) −1.00000 −0.129099
\(61\) 6.00000 0.768221 0.384111 − 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) −4.00000 −0.508001
\(63\) −4.00000 −0.503953
\(64\) 7.00000 0.875000
\(65\) 6.00000 0.744208
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 − 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) −2.00000 −0.242536
\(69\) −4.00000 −0.481543
\(70\) 4.00000 0.478091
\(71\) 16.0000 1.89885 0.949425 − 0.313993i \(-0.101667\pi\)
0.949425 + 0.313993i \(0.101667\pi\)
\(72\) 3.00000 0.353553
\(73\) −6.00000 −0.702247 −0.351123 − 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) −6.00000 −0.697486
\(75\) 1.00000 0.115470
\(76\) −8.00000 −0.917663
\(77\) 0 0
\(78\) −6.00000 −0.679366
\(79\) 12.0000 1.35011 0.675053 − 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −2.00000 −0.220863
\(83\) −16.0000 −1.75623 −0.878114 − 0.478451i \(-0.841198\pi\)
−0.878114 + 0.478451i \(0.841198\pi\)
\(84\) 4.00000 0.436436
\(85\) 2.00000 0.216930
\(86\) 4.00000 0.431331
\(87\) 1.00000 0.107211
\(88\) 0 0
\(89\) 2.00000 0.212000 0.106000 − 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) −1.00000 −0.105409
\(91\) −24.0000 −2.51588
\(92\) 4.00000 0.417029
\(93\) 4.00000 0.414781
\(94\) 0 0
\(95\) 8.00000 0.820783
\(96\) −5.00000 −0.510310
\(97\) −14.0000 −1.42148 −0.710742 − 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) −9.00000 −0.909137
\(99\) 0 0
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 435.2.a.a.1.1 ✓ 1
3.2 odd 2 1305.2.a.e.1.1 1
4.3 odd 2 6960.2.a.w.1.1 1
5.2 odd 4 2175.2.c.a.349.1 2
5.3 odd 4 2175.2.c.a.349.2 2
5.4 even 2 2175.2.a.h.1.1 1
15.14 odd 2 6525.2.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.a.1.1 ✓ 1 1.1 even 1 trivial
1305.2.a.e.1.1 1 3.2 odd 2
2175.2.a.h.1.1 1 5.4 even 2
2175.2.c.a.349.1 2 5.2 odd 4
2175.2.c.a.349.2 2 5.3 odd 4
6525.2.a.e.1.1 1 15.14 odd 2
6960.2.a.w.1.1 1 4.3 odd 2