Properties

Label 465.2.a.g.1.1
Level $465$
Weight $2$
Character 465.1
Self dual yes
Analytic conductor $3.713$
Analytic rank $0$
Dimension $3$
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] = [465,2,Mod(1,465)] 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("465.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(465, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 465 = 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 465.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,3,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(3.71304369399\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) 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 1.1
Root \(0.311108\) of defining polynomial
Character \(\chi\) \(=\) 465.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.21432 q^{2} +1.00000 q^{3} -0.525428 q^{4} -1.00000 q^{5} -1.21432 q^{6} -1.59210 q^{7} +3.06668 q^{8} +1.00000 q^{9} +1.21432 q^{10} +0.622216 q^{11} -0.525428 q^{12} +0.214320 q^{13} +1.93332 q^{14} -1.00000 q^{15} -2.67307 q^{16} +3.52543 q^{17} -1.21432 q^{18} +1.80642 q^{19} +0.525428 q^{20} -1.59210 q^{21} -0.755569 q^{22} +6.90321 q^{23} +3.06668 q^{24} +1.00000 q^{25} -0.260253 q^{26} +1.00000 q^{27} +0.836535 q^{28} +9.73975 q^{29} +1.21432 q^{30} -1.00000 q^{31} -2.88739 q^{32} +0.622216 q^{33} -4.28100 q^{34} +1.59210 q^{35} -0.525428 q^{36} -4.83654 q^{37} -2.19358 q^{38} +0.214320 q^{39} -3.06668 q^{40} -7.47949 q^{41} +1.93332 q^{42} +8.23506 q^{43} -0.326929 q^{44} -1.00000 q^{45} -8.38271 q^{46} +11.4652 q^{47} -2.67307 q^{48} -4.46520 q^{49} -1.21432 q^{50} +3.52543 q^{51} -0.112610 q^{52} +13.7605 q^{53} -1.21432 q^{54} -0.622216 q^{55} -4.88247 q^{56} +1.80642 q^{57} -11.8272 q^{58} -4.26025 q^{59} +0.525428 q^{60} +2.85728 q^{61} +1.21432 q^{62} -1.59210 q^{63} +8.85236 q^{64} -0.214320 q^{65} -0.755569 q^{66} -2.08097 q^{67} -1.85236 q^{68} +6.90321 q^{69} -1.93332 q^{70} +1.31111 q^{71} +3.06668 q^{72} -1.65233 q^{73} +5.87310 q^{74} +1.00000 q^{75} -0.949145 q^{76} -0.990632 q^{77} -0.260253 q^{78} -5.19850 q^{79} +2.67307 q^{80} +1.00000 q^{81} +9.08250 q^{82} -5.65878 q^{83} +0.836535 q^{84} -3.52543 q^{85} -10.0000 q^{86} +9.73975 q^{87} +1.90813 q^{88} -1.93332 q^{89} +1.21432 q^{90} -0.341219 q^{91} -3.62714 q^{92} -1.00000 q^{93} -13.9224 q^{94} -1.80642 q^{95} -2.88739 q^{96} +6.91750 q^{97} +5.42219 q^{98} +0.622216 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{3} + 5 q^{4} - 3 q^{5} + 3 q^{6} + 2 q^{7} + 9 q^{8} + 3 q^{9} - 3 q^{10} + 2 q^{11} + 5 q^{12} - 6 q^{13} + 6 q^{14} - 3 q^{15} + 5 q^{16} + 4 q^{17} + 3 q^{18} - 8 q^{19} - 5 q^{20}+ \cdots + 2 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.21432 −0.858654 −0.429327 − 0.903149i \(-0.641249\pi\)
−0.429327 + 0.903149i \(0.641249\pi\)
\(3\) 1.00000 0.577350
\(4\) −0.525428 −0.262714
\(5\) −1.00000 −0.447214
\(6\) −1.21432 −0.495744
\(7\) −1.59210 −0.601759 −0.300879 − 0.953662i \(-0.597280\pi\)
−0.300879 + 0.953662i \(0.597280\pi\)
\(8\) 3.06668 1.08423
\(9\) 1.00000 0.333333
\(10\) 1.21432 0.384002
\(11\) 0.622216 0.187605 0.0938025 − 0.995591i \(-0.470098\pi\)
0.0938025 + 0.995591i \(0.470098\pi\)
\(12\) −0.525428 −0.151678
\(13\) 0.214320 0.0594416 0.0297208 − 0.999558i \(-0.490538\pi\)
0.0297208 + 0.999558i \(0.490538\pi\)
\(14\) 1.93332 0.516702
\(15\) −1.00000 −0.258199
\(16\) −2.67307 −0.668268
\(17\) 3.52543 0.855042 0.427521 − 0.904005i \(-0.359387\pi\)
0.427521 + 0.904005i \(0.359387\pi\)
\(18\) −1.21432 −0.286218
\(19\) 1.80642 0.414422 0.207211 − 0.978296i \(-0.433561\pi\)
0.207211 + 0.978296i \(0.433561\pi\)
\(20\) 0.525428 0.117489
\(21\) −1.59210 −0.347426
\(22\) −0.755569 −0.161088
\(23\) 6.90321 1.43942 0.719710 − 0.694275i \(-0.244275\pi\)
0.719710 + 0.694275i \(0.244275\pi\)
\(24\) 3.06668 0.625983
\(25\) 1.00000 0.200000
\(26\) −0.260253 −0.0510398
\(27\) 1.00000 0.192450
\(28\) 0.836535 0.158090
\(29\) 9.73975 1.80863 0.904313 − 0.426870i \(-0.140384\pi\)
0.904313 + 0.426870i \(0.140384\pi\)
\(30\) 1.21432 0.221703
\(31\) −1.00000 −0.179605
\(32\) −2.88739 −0.510423
\(33\) 0.622216 0.108314
\(34\) −4.28100 −0.734185
\(35\) 1.59210 0.269115
\(36\) −0.525428 −0.0875713
\(37\) −4.83654 −0.795122 −0.397561 − 0.917576i \(-0.630143\pi\)
−0.397561 + 0.917576i \(0.630143\pi\)
\(38\) −2.19358 −0.355845
\(39\) 0.214320 0.0343186
\(40\) −3.06668 −0.484884
\(41\) −7.47949 −1.16810 −0.584050 − 0.811717i \(-0.698533\pi\)
−0.584050 + 0.811717i \(0.698533\pi\)
\(42\) 1.93332 0.298318
\(43\) 8.23506 1.25584 0.627918 − 0.778280i \(-0.283907\pi\)
0.627918 + 0.778280i \(0.283907\pi\)
\(44\) −0.326929 −0.0492864
\(45\) −1.00000 −0.149071
\(46\) −8.38271 −1.23596
\(47\) 11.4652 1.67237 0.836186 − 0.548446i \(-0.184780\pi\)
0.836186 + 0.548446i \(0.184780\pi\)
\(48\) −2.67307 −0.385825
\(49\) −4.46520 −0.637886
\(50\) −1.21432 −0.171731
\(51\) 3.52543 0.493659
\(52\) −0.112610 −0.0156161
\(53\) 13.7605 1.89015 0.945074 − 0.326855i \(-0.105989\pi\)
0.945074 + 0.326855i \(0.105989\pi\)
\(54\) −1.21432 −0.165248
\(55\) −0.622216 −0.0838995
\(56\) −4.88247 −0.652447
\(57\) 1.80642 0.239267
\(58\) −11.8272 −1.55298
\(59\) −4.26025 −0.554638 −0.277319 − 0.960778i \(-0.589446\pi\)
−0.277319 + 0.960778i \(0.589446\pi\)
\(60\) 0.525428 0.0678324
\(61\) 2.85728 0.365837 0.182919 − 0.983128i \(-0.441446\pi\)
0.182919 + 0.983128i \(0.441446\pi\)
\(62\) 1.21432 0.154219
\(63\) −1.59210 −0.200586
\(64\) 8.85236 1.10654
\(65\) −0.214320 −0.0265831
\(66\) −0.755569 −0.0930041
\(67\) −2.08097 −0.254231 −0.127115 − 0.991888i \(-0.540572\pi\)
−0.127115 + 0.991888i \(0.540572\pi\)
\(68\) −1.85236 −0.224631
\(69\) 6.90321 0.831049
\(70\) −1.93332 −0.231076
\(71\) 1.31111 0.155600 0.0777999 − 0.996969i \(-0.475210\pi\)
0.0777999 + 0.996969i \(0.475210\pi\)
\(72\) 3.06668 0.361411
\(73\) −1.65233 −0.193390 −0.0966951 − 0.995314i \(-0.530827\pi\)
−0.0966951 + 0.995314i \(0.530827\pi\)
\(74\) 5.87310 0.682734
\(75\) 1.00000 0.115470
\(76\) −0.949145 −0.108874
\(77\) −0.990632 −0.112893
\(78\) −0.260253 −0.0294678
\(79\) −5.19850 −0.584877 −0.292438 − 0.956284i \(-0.594467\pi\)
−0.292438 + 0.956284i \(0.594467\pi\)
\(80\) 2.67307 0.298858
\(81\) 1.00000 0.111111
\(82\) 9.08250 1.00299
\(83\) −5.65878 −0.621132 −0.310566 − 0.950552i \(-0.600519\pi\)
−0.310566 + 0.950552i \(0.600519\pi\)
\(84\) 0.836535 0.0912735
\(85\) −3.52543 −0.382386
\(86\) −10.0000 −1.07833
\(87\) 9.73975 1.04421
\(88\) 1.90813 0.203408
\(89\) −1.93332 −0.204932 −0.102466 − 0.994737i \(-0.532673\pi\)
−0.102466 + 0.994737i \(0.532673\pi\)
\(90\) 1.21432 0.128001
\(91\) −0.341219 −0.0357695
\(92\) −3.62714 −0.378155
\(93\) −1.00000 −0.103695
\(94\) −13.9224 −1.43599
\(95\) −1.80642 −0.185335
\(96\) −2.88739 −0.294693
\(97\) 6.91750 0.702366 0.351183 − 0.936307i \(-0.385780\pi\)
0.351183 + 0.936307i \(0.385780\pi\)
\(98\) 5.42219 0.547723
\(99\) 0.622216 0.0625350
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 465.2.a.g.1.1 ✓ 3
3.2 odd 2 1395.2.a.h.1.3 3
4.3 odd 2 7440.2.a.bm.1.3 3
5.2 odd 4 2325.2.c.l.1024.3 6
5.3 odd 4 2325.2.c.l.1024.4 6
5.4 even 2 2325.2.a.p.1.3 3
15.14 odd 2 6975.2.a.bi.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.a.g.1.1 ✓ 3 1.1 even 1 trivial
1395.2.a.h.1.3 3 3.2 odd 2
2325.2.a.p.1.3 3 5.4 even 2
2325.2.c.l.1024.3 6 5.2 odd 4
2325.2.c.l.1024.4 6 5.3 odd 4
6975.2.a.bi.1.1 3 15.14 odd 2
7440.2.a.bm.1.3 3 4.3 odd 2