↑

Properties

Label 435.4.a.a.1.1
Level $435$
Weight $4$
Character 435.1
Self dual yes
Analytic conductor $25.666$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [435,4,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, 4); 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, 4, names="a")
 
Level: \( N \) \(=\) \( 435 = 3 \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 435.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(25.6658308525\)
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)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} -3.00000 q^{3} -4.00000 q^{4} +5.00000 q^{5} +6.00000 q^{6} +29.0000 q^{7} +24.0000 q^{8} +9.00000 q^{9} -10.0000 q^{10} -15.0000 q^{11} +12.0000 q^{12} +3.00000 q^{13} -58.0000 q^{14} -15.0000 q^{15} -16.0000 q^{16} +121.000 q^{17} -18.0000 q^{18} -40.0000 q^{19} -20.0000 q^{20} -87.0000 q^{21} +30.0000 q^{22} -116.000 q^{23} -72.0000 q^{24} +25.0000 q^{25} -6.00000 q^{26} -27.0000 q^{27} -116.000 q^{28} +29.0000 q^{29} +30.0000 q^{30} -116.000 q^{31} -160.000 q^{32} +45.0000 q^{33} -242.000 q^{34} +145.000 q^{35} -36.0000 q^{36} +36.0000 q^{37} +80.0000 q^{38} -9.00000 q^{39} +120.000 q^{40} -170.000 q^{41} +174.000 q^{42} +230.000 q^{43} +60.0000 q^{44} +45.0000 q^{45} +232.000 q^{46} +231.000 q^{47} +48.0000 q^{48} +498.000 q^{49} -50.0000 q^{50} -363.000 q^{51} -12.0000 q^{52} +456.000 q^{53} +54.0000 q^{54} -75.0000 q^{55} +696.000 q^{56} +120.000 q^{57} -58.0000 q^{58} +576.000 q^{59} +60.0000 q^{60} +342.000 q^{61} +232.000 q^{62} +261.000 q^{63} +448.000 q^{64} +15.0000 q^{65} -90.0000 q^{66} -269.000 q^{67} -484.000 q^{68} +348.000 q^{69} -290.000 q^{70} +302.000 q^{71} +216.000 q^{72} -372.000 q^{73} -72.0000 q^{74} -75.0000 q^{75} +160.000 q^{76} -435.000 q^{77} +18.0000 q^{78} -348.000 q^{79} -80.0000 q^{80} +81.0000 q^{81} +340.000 q^{82} -512.000 q^{83} +348.000 q^{84} +605.000 q^{85} -460.000 q^{86} -87.0000 q^{87} -360.000 q^{88} +1525.00 q^{89} -90.0000 q^{90} +87.0000 q^{91} +464.000 q^{92} +348.000 q^{93} -462.000 q^{94} -200.000 q^{95} +480.000 q^{96} -560.000 q^{97} -996.000 q^{98} -135.000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107 −0.353553 − 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) −3.00000 −0.577350
\(4\) −4.00000 −0.500000
\(5\) 5.00000 0.447214
\(6\) 6.00000 0.408248
\(7\) 29.0000 1.56585 0.782926 − 0.622114i \(-0.213726\pi\)
0.782926 + 0.622114i \(0.213726\pi\)
\(8\) 24.0000 1.06066
\(9\) 9.00000 0.333333
\(10\) −10.0000 −0.316228
\(11\) −15.0000 −0.411152 −0.205576 − 0.978641i \(-0.565907\pi\)
−0.205576 + 0.978641i \(0.565907\pi\)
\(12\) 12.0000 0.288675
\(13\) 3.00000 0.0640039 0.0320019 − 0.999488i \(-0.489812\pi\)
0.0320019 + 0.999488i \(0.489812\pi\)
\(14\) −58.0000 −1.10723
\(15\) −15.0000 −0.258199
\(16\) −16.0000 −0.250000
\(17\) 121.000 1.72628 0.863141 − 0.504962i \(-0.168494\pi\)
0.863141 + 0.504962i \(0.168494\pi\)
\(18\) −18.0000 −0.235702
\(19\) −40.0000 −0.482980 −0.241490 − 0.970403i \(-0.577636\pi\)
−0.241490 + 0.970403i \(0.577636\pi\)
\(20\) −20.0000 −0.223607
\(21\) −87.0000 −0.904046
\(22\) 30.0000 0.290728
\(23\) −116.000 −1.05164 −0.525819 − 0.850597i \(-0.676241\pi\)
−0.525819 + 0.850597i \(0.676241\pi\)
\(24\) −72.0000 −0.612372
\(25\) 25.0000 0.200000
\(26\) −6.00000 −0.0452576
\(27\) −27.0000 −0.192450
\(28\) −116.000 −0.782926
\(29\) 29.0000 0.185695
\(30\) 30.0000 0.182574
\(31\) −116.000 −0.672071 −0.336036 − 0.941849i \(-0.609086\pi\)
−0.336036 + 0.941849i \(0.609086\pi\)
\(32\) −160.000 −0.883883
\(33\) 45.0000 0.237379
\(34\) −242.000 −1.22067
\(35\) 145.000 0.700271
\(36\) −36.0000 −0.166667
\(37\) 36.0000 0.159956 0.0799779 − 0.996797i \(-0.474515\pi\)
0.0799779 + 0.996797i \(0.474515\pi\)
\(38\) 80.0000 0.341519
\(39\) −9.00000 −0.0369527
\(40\) 120.000 0.474342
\(41\) −170.000 −0.647550 −0.323775 − 0.946134i \(-0.604952\pi\)
−0.323775 + 0.946134i \(0.604952\pi\)
\(42\) 174.000 0.639257
\(43\) 230.000 0.815690 0.407845 − 0.913051i \(-0.366280\pi\)
0.407845 + 0.913051i \(0.366280\pi\)
\(44\) 60.0000 0.205576
\(45\) 45.0000 0.149071
\(46\) 232.000 0.743620
\(47\) 231.000 0.716911 0.358455 − 0.933547i \(-0.383303\pi\)
0.358455 + 0.933547i \(0.383303\pi\)
\(48\) 48.0000 0.144338
\(49\) 498.000 1.45190
\(50\) −50.0000 −0.141421
\(51\) −363.000 −0.996670
\(52\) −12.0000 −0.0320019
\(53\) 456.000 1.18182 0.590910 − 0.806738i \(-0.298769\pi\)
0.590910 + 0.806738i \(0.298769\pi\)
\(54\) 54.0000 0.136083
\(55\) −75.0000 −0.183873
\(56\) 696.000 1.66084
\(57\) 120.000 0.278849
\(58\) −58.0000 −0.131306
\(59\) 576.000 1.27100 0.635498 − 0.772102i \(-0.280795\pi\)
0.635498 + 0.772102i \(0.280795\pi\)
\(60\) 60.0000 0.129099
\(61\) 342.000 0.717846 0.358923 − 0.933367i \(-0.383144\pi\)
0.358923 + 0.933367i \(0.383144\pi\)
\(62\) 232.000 0.475226
\(63\) 261.000 0.521951
\(64\) 448.000 0.875000
\(65\) 15.0000 0.0286234
\(66\) −90.0000 −0.167852
\(67\) −269.000 −0.490501 −0.245251 − 0.969460i \(-0.578870\pi\)
−0.245251 + 0.969460i \(0.578870\pi\)
\(68\) −484.000 −0.863141
\(69\) 348.000 0.607163
\(70\) −290.000 −0.495166
\(71\) 302.000 0.504800 0.252400 − 0.967623i \(-0.418780\pi\)
0.252400 + 0.967623i \(0.418780\pi\)
\(72\) 216.000 0.353553
\(73\) −372.000 −0.596429 −0.298214 − 0.954499i \(-0.596391\pi\)
−0.298214 + 0.954499i \(0.596391\pi\)
\(74\) −72.0000 −0.113106
\(75\) −75.0000 −0.115470
\(76\) 160.000 0.241490
\(77\) −435.000 −0.643803
\(78\) 18.0000 0.0261295
\(79\) −348.000 −0.495608 −0.247804 − 0.968810i \(-0.579709\pi\)
−0.247804 + 0.968810i \(0.579709\pi\)
\(80\) −80.0000 −0.111803
\(81\) 81.0000 0.111111
\(82\) 340.000 0.457887
\(83\) −512.000 −0.677100 −0.338550 − 0.940948i \(-0.609936\pi\)
−0.338550 + 0.940948i \(0.609936\pi\)
\(84\) 348.000 0.452023
\(85\) 605.000 0.772017
\(86\) −460.000 −0.576780
\(87\) −87.0000 −0.107211
\(88\) −360.000 −0.436092
\(89\) 1525.00 1.81629 0.908144 − 0.418657i \(-0.137499\pi\)
0.908144 + 0.418657i \(0.137499\pi\)
\(90\) −90.0000 −0.105409
\(91\) 87.0000 0.100221
\(92\) 464.000 0.525819
\(93\) 348.000 0.388021
\(94\) −462.000 −0.506933
\(95\) −200.000 −0.215995
\(96\) 480.000 0.510310
\(97\) −560.000 −0.586179 −0.293090 − 0.956085i \(-0.594683\pi\)
−0.293090 + 0.956085i \(0.594683\pi\)
\(98\) −996.000 −1.02664
\(99\) −135.000 −0.137051
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 435.4.a.a.1.1 ✓ 1
3.2 odd 2 1305.4.a.d.1.1 1
5.4 even 2 2175.4.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.4.a.a.1.1 ✓ 1 1.1 even 1 trivial
1305.4.a.d.1.1 1 3.2 odd 2
2175.4.a.c.1.1 1 5.4 even 2