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Properties

Label 405.2.a.a.1.1
Level $405$
Weight $2$
Character 405.1
Self dual yes
Analytic conductor $3.234$
Analytic rank $1$
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] = [405,2,Mod(1,405)] 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("405.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(405, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 405.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-2,0,2,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(3.23394128186\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 405.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} +2.00000 q^{4} -1.00000 q^{5} +2.00000 q^{10} -5.00000 q^{11} +4.00000 q^{13} -4.00000 q^{16} +4.00000 q^{17} -5.00000 q^{19} -2.00000 q^{20} +10.0000 q^{22} -6.00000 q^{23} +1.00000 q^{25} -8.00000 q^{26} +5.00000 q^{29} -9.00000 q^{31} +8.00000 q^{32} -8.00000 q^{34} -10.0000 q^{37} +10.0000 q^{38} -7.00000 q^{41} -2.00000 q^{43} -10.0000 q^{44} +12.0000 q^{46} -2.00000 q^{47} -7.00000 q^{49} -2.00000 q^{50} +8.00000 q^{52} -8.00000 q^{53} +5.00000 q^{55} -10.0000 q^{58} +1.00000 q^{59} -2.00000 q^{61} +18.0000 q^{62} -8.00000 q^{64} -4.00000 q^{65} +6.00000 q^{67} +8.00000 q^{68} -1.00000 q^{71} -8.00000 q^{73} +20.0000 q^{74} -10.0000 q^{76} +12.0000 q^{79} +4.00000 q^{80} +14.0000 q^{82} -6.00000 q^{83} -4.00000 q^{85} +4.00000 q^{86} +9.00000 q^{89} -12.0000 q^{92} +4.00000 q^{94} +5.00000 q^{95} +14.0000 q^{97} +14.0000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −1.41421 −0.707107 − 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) 0 0
\(4\) 2.00000 1.00000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 2.00000 0.632456
\(11\) −5.00000 −1.50756 −0.753778 − 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) 4.00000 0.970143 0.485071 − 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) −5.00000 −1.14708 −0.573539 − 0.819178i \(-0.694430\pi\)
−0.573539 + 0.819178i \(0.694430\pi\)
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) 10.0000 2.13201
\(23\) −6.00000 −1.25109 −0.625543 − 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −8.00000 −1.56893
\(27\) 0 0
\(28\) 0 0
\(29\) 5.00000 0.928477 0.464238 − 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 0 0
\(31\) −9.00000 −1.61645 −0.808224 − 0.588875i \(-0.799571\pi\)
−0.808224 + 0.588875i \(0.799571\pi\)
\(32\) 8.00000 1.41421
\(33\) 0 0
\(34\) −8.00000 −1.37199
\(35\) 0 0
\(36\) 0 0
\(37\) −10.0000 −1.64399 −0.821995 − 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 10.0000 1.62221
\(39\) 0 0
\(40\) 0 0
\(41\) −7.00000 −1.09322 −0.546608 − 0.837389i \(-0.684081\pi\)
−0.546608 + 0.837389i \(0.684081\pi\)
\(42\) 0 0
\(43\) −2.00000 −0.304997 −0.152499 − 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) −10.0000 −1.50756
\(45\) 0 0
\(46\) 12.0000 1.76930
\(47\) −2.00000 −0.291730 −0.145865 − 0.989305i \(-0.546597\pi\)
−0.145865 + 0.989305i \(0.546597\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) −2.00000 −0.282843
\(51\) 0 0
\(52\) 8.00000 1.10940
\(53\) −8.00000 −1.09888 −0.549442 − 0.835532i \(-0.685160\pi\)
−0.549442 + 0.835532i \(0.685160\pi\)
\(54\) 0 0
\(55\) 5.00000 0.674200
\(56\) 0 0
\(57\) 0 0
\(58\) −10.0000 −1.31306
\(59\) 1.00000 0.130189 0.0650945 − 0.997879i \(-0.479265\pi\)
0.0650945 + 0.997879i \(0.479265\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 − 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 18.0000 2.28600
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) 6.00000 0.733017 0.366508 − 0.930415i \(-0.380553\pi\)
0.366508 + 0.930415i \(0.380553\pi\)
\(68\) 8.00000 0.970143
\(69\) 0 0
\(70\) 0 0
\(71\) −1.00000 −0.118678 −0.0593391 − 0.998238i \(-0.518899\pi\)
−0.0593391 + 0.998238i \(0.518899\pi\)
\(72\) 0 0
\(73\) −8.00000 −0.936329 −0.468165 − 0.883641i \(-0.655085\pi\)
−0.468165 + 0.883641i \(0.655085\pi\)
\(74\) 20.0000 2.32495
\(75\) 0 0
\(76\) −10.0000 −1.14708
\(77\) 0 0
\(78\) 0 0
\(79\) 12.0000 1.35011 0.675053 − 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) 4.00000 0.447214
\(81\) 0 0
\(82\) 14.0000 1.54604
\(83\) −6.00000 −0.658586 −0.329293 − 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) −4.00000 −0.433861
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) 0 0
\(89\) 9.00000 0.953998 0.476999 − 0.878904i \(-0.341725\pi\)
0.476999 + 0.878904i \(0.341725\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −12.0000 −1.25109
\(93\) 0 0
\(94\) 4.00000 0.412568
\(95\) 5.00000 0.512989
\(96\) 0 0
\(97\) 14.0000 1.42148 0.710742 − 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 14.0000 1.41421
\(99\) 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 405.2.a.a.1.1 ✓ 1
3.2 odd 2 405.2.a.f.1.1 yes 1
4.3 odd 2 6480.2.a.f.1.1 1
5.2 odd 4 2025.2.b.a.649.1 2
5.3 odd 4 2025.2.b.a.649.2 2
5.4 even 2 2025.2.a.f.1.1 1
9.2 odd 6 405.2.e.a.271.1 2
9.4 even 3 405.2.e.g.136.1 2
9.5 odd 6 405.2.e.a.136.1 2
9.7 even 3 405.2.e.g.271.1 2
12.11 even 2 6480.2.a.r.1.1 1
15.2 even 4 2025.2.b.b.649.2 2
15.8 even 4 2025.2.b.b.649.1 2
15.14 odd 2 2025.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
405.2.a.a.1.1 ✓ 1 1.1 even 1 trivial
405.2.a.f.1.1 yes 1 3.2 odd 2
405.2.e.a.136.1 2 9.5 odd 6
405.2.e.a.271.1 2 9.2 odd 6
405.2.e.g.136.1 2 9.4 even 3
405.2.e.g.271.1 2 9.7 even 3
2025.2.a.a.1.1 1 15.14 odd 2
2025.2.a.f.1.1 1 5.4 even 2
2025.2.b.a.649.1 2 5.2 odd 4
2025.2.b.a.649.2 2 5.3 odd 4
2025.2.b.b.649.1 2 15.8 even 4
2025.2.b.b.649.2 2 15.2 even 4
6480.2.a.f.1.1 1 4.3 odd 2
6480.2.a.r.1.1 1 12.11 even 2