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Properties

Label 405.5.d.a.404.37
Level $405$
Weight $5$
Character 405.404
Analytic conductor $41.865$
Analytic rank $0$
Dimension $44$
Inner twists $4$

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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] = [405,5,Mod(404,405)] 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("405.404"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(405, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 405.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [44] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(41.8648350490\)
Analytic rank: \(0\)
Dimension: \(44\)
Twist minimal: no (minimal twist has level 45)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 404.37
Character \(\chi\) \(=\) 405.404
Dual form 405.5.d.a.404.38

$q$-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)
 
\(f(q)\) \(=\) \(q+5.74722 q^{2} +17.0305 q^{4} +(17.1405 + 18.1989i) q^{5} -47.8220i q^{7} +5.92246 q^{8} +(98.5104 + 104.593i) q^{10} +84.5766i q^{11} +223.417i q^{13} -274.843i q^{14} -238.450 q^{16} +434.998 q^{17} +378.461 q^{19} +(291.912 + 309.937i) q^{20} +486.080i q^{22} +653.651 q^{23} +(-37.4035 + 623.880i) q^{25} +1284.03i q^{26} -814.433i q^{28} +496.753i q^{29} -302.995 q^{31} -1465.18 q^{32} +2500.03 q^{34} +(870.311 - 819.695i) q^{35} +55.6914i q^{37} +2175.09 q^{38} +(101.514 + 107.783i) q^{40} +475.050i q^{41} -816.458i q^{43} +1440.38i q^{44} +3756.67 q^{46} +870.464 q^{47} +114.054 q^{49} +(-214.966 + 3585.57i) q^{50} +3804.90i q^{52} +2339.28 q^{53} +(-1539.21 + 1449.69i) q^{55} -283.224i q^{56} +2854.95i q^{58} -1177.42i q^{59} -6913.03 q^{61} -1741.38 q^{62} -4605.53 q^{64} +(-4065.96 + 3829.49i) q^{65} -3875.51i q^{67} +7408.23 q^{68} +(5001.86 - 4710.97i) q^{70} +5822.30i q^{71} +6443.82i q^{73} +320.071i q^{74} +6445.37 q^{76} +4044.62 q^{77} +4893.43 q^{79} +(-4087.17 - 4339.54i) q^{80} +2730.21i q^{82} -6514.72 q^{83} +(7456.10 + 7916.50i) q^{85} -4692.36i q^{86} +500.901i q^{88} -13898.7i q^{89} +10684.3 q^{91} +11132.0 q^{92} +5002.75 q^{94} +(6487.02 + 6887.59i) q^{95} +13488.8i q^{97} +655.493 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 44 q + 324 q^{4} + 28 q^{10} + 2116 q^{16} - 8 q^{19} + 296 q^{25} + 2224 q^{31} + 872 q^{34} + 1700 q^{40} - 5668 q^{46} - 10792 q^{49} - 3072 q^{55} - 5564 q^{61} + 8348 q^{64} - 9564 q^{70} + 3552 q^{76}+ \cdots + 37652 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/405\mathbb{Z}\right)^\times\).

\(n\) \(82\) \(326\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.74722 1.43680 0.718402 − 0.695628i \(-0.244874\pi\)
0.718402 + 0.695628i \(0.244874\pi\)
\(3\) 0 0
\(4\) 17.0305 1.06441
\(5\) 17.1405 + 18.1989i 0.685622 + 0.727958i
\(6\) 0 0
\(7\) − 47.8220i − 0.975960i −0.872855 − 0.487980i \(-0.837734\pi\)
0.872855 − 0.487980i \(-0.162266\pi\)
\(8\) 5.92246 0.0925384
\(9\) 0 0
\(10\) 98.5104 + 104.593i 0.985104 + 1.04593i
\(11\) 84.5766i 0.698980i 0.936940 + 0.349490i \(0.113645\pi\)
−0.936940 + 0.349490i \(0.886355\pi\)
\(12\) 0 0
\(13\) 223.417i 1.32200i 0.750388 + 0.660998i \(0.229867\pi\)
−0.750388 + 0.660998i \(0.770133\pi\)
\(14\) − 274.843i − 1.40226i
\(15\) 0 0
\(16\) −238.450 −0.931446
\(17\) 434.998 1.50518 0.752591 − 0.658488i \(-0.228804\pi\)
0.752591 + 0.658488i \(0.228804\pi\)
\(18\) 0 0
\(19\) 378.461 1.04837 0.524184 − 0.851605i \(-0.324371\pi\)
0.524184 + 0.851605i \(0.324371\pi\)
\(20\) 291.912 + 309.937i 0.729780 + 0.774843i
\(21\) 0 0
\(22\) 486.080i 1.00430i
\(23\) 653.651 1.23564 0.617818 − 0.786322i \(-0.288017\pi\)
0.617818 + 0.786322i \(0.288017\pi\)
\(24\) 0 0
\(25\) −37.4035 + 623.880i −0.0598457 + 0.998208i
\(26\) 1284.03i 1.89945i
\(27\) 0 0
\(28\) − 814.433i − 1.03882i
\(29\) 496.753i 0.590670i 0.955394 + 0.295335i \(0.0954312\pi\)
−0.955394 + 0.295335i \(0.904569\pi\)
\(30\) 0 0
\(31\) −302.995 −0.315292 −0.157646 − 0.987496i \(-0.550390\pi\)
−0.157646 + 0.987496i \(0.550390\pi\)
\(32\) −1465.18 −1.43084
\(33\) 0 0
\(34\) 2500.03 2.16265
\(35\) 870.311 − 819.695i 0.710458 − 0.669139i
\(36\) 0 0
\(37\) 55.6914i 0.0406804i 0.999793 + 0.0203402i \(0.00647493\pi\)
−0.999793 + 0.0203402i \(0.993525\pi\)
\(38\) 2175.09 1.50630
\(39\) 0 0
\(40\) 101.514 + 107.783i 0.0634464 + 0.0673641i
\(41\) 475.050i 0.282600i 0.989967 + 0.141300i \(0.0451281\pi\)
−0.989967 + 0.141300i \(0.954872\pi\)
\(42\) 0 0
\(43\) − 816.458i − 0.441567i −0.975323 − 0.220784i \(-0.929139\pi\)
0.975323 − 0.220784i \(-0.0708615\pi\)
\(44\) 1440.38i 0.743999i
\(45\) 0 0
\(46\) 3756.67 1.77537
\(47\) 870.464 0.394053 0.197027 − 0.980398i \(-0.436871\pi\)
0.197027 + 0.980398i \(0.436871\pi\)
\(48\) 0 0
\(49\) 114.054 0.0475027
\(50\) −214.966 + 3585.57i −0.0859865 + 1.43423i
\(51\) 0 0
\(52\) 3804.90i 1.40714i
\(53\) 2339.28 0.832780 0.416390 − 0.909186i \(-0.363295\pi\)
0.416390 + 0.909186i \(0.363295\pi\)
\(54\) 0 0
\(55\) −1539.21 + 1449.69i −0.508828 + 0.479236i
\(56\) − 283.224i − 0.0903138i
\(57\) 0 0
\(58\) 2854.95i 0.848677i
\(59\) − 1177.42i − 0.338242i −0.985595 − 0.169121i \(-0.945907\pi\)
0.985595 − 0.169121i \(-0.0540929\pi\)
\(60\) 0 0
\(61\) −6913.03 −1.85784 −0.928921 − 0.370277i \(-0.879263\pi\)
−0.928921 + 0.370277i \(0.879263\pi\)
\(62\) −1741.38 −0.453013
\(63\) 0 0
\(64\) −4605.53 −1.12440
\(65\) −4065.96 + 3829.49i −0.962357 + 0.906389i
\(66\) 0 0
\(67\) − 3875.51i − 0.863334i −0.902033 − 0.431667i \(-0.857926\pi\)
0.902033 − 0.431667i \(-0.142074\pi\)
\(68\) 7408.23 1.60213
\(69\) 0 0
\(70\) 5001.86 − 4710.97i 1.02079 − 0.961422i
\(71\) 5822.30i 1.15499i 0.816394 + 0.577495i \(0.195970\pi\)
−0.816394 + 0.577495i \(0.804030\pi\)
\(72\) 0 0
\(73\) 6443.82i 1.20920i 0.796530 + 0.604599i \(0.206667\pi\)
−0.796530 + 0.604599i \(0.793333\pi\)
\(74\) 320.071i 0.0584497i
\(75\) 0 0
\(76\) 6445.37 1.11589
\(77\) 4044.62 0.682176
\(78\) 0 0
\(79\) 4893.43 0.784078 0.392039 − 0.919949i \(-0.371770\pi\)
0.392039 + 0.919949i \(0.371770\pi\)
\(80\) −4087.17 − 4339.54i −0.638620 − 0.678054i
\(81\) 0 0
\(82\) 2730.21i 0.406040i
\(83\) −6514.72 −0.945670 −0.472835 − 0.881151i \(-0.656769\pi\)
−0.472835 + 0.881151i \(0.656769\pi\)
\(84\) 0 0
\(85\) 7456.10 + 7916.50i 1.03199 + 1.09571i
\(86\) − 4692.36i − 0.634446i
\(87\) 0 0
\(88\) 500.901i 0.0646825i
\(89\) − 13898.7i − 1.75467i −0.479881 − 0.877333i \(-0.659320\pi\)
0.479881 − 0.877333i \(-0.340680\pi\)
\(90\) 0 0
\(91\) 10684.3 1.29021
\(92\) 11132.0 1.31522
\(93\) 0 0
\(94\) 5002.75 0.566178
\(95\) 6487.02 + 6887.59i 0.718783 + 0.763167i
\(96\) 0 0
\(97\) 13488.8i 1.43361i 0.697274 + 0.716805i \(0.254396\pi\)
−0.697274 + 0.716805i \(0.745604\pi\)
\(98\) 655.493 0.0682521
\(99\) 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 405.5.d.a.404.37 44
3.2 odd 2 inner 405.5.d.a.404.7 44
5.4 even 2 inner 405.5.d.a.404.8 44
9.2 odd 6 45.5.h.a.14.19 yes 44
9.4 even 3 45.5.h.a.29.4 yes 44
9.5 odd 6 135.5.h.a.89.19 44
9.7 even 3 135.5.h.a.44.4 44
15.14 odd 2 inner 405.5.d.a.404.38 44
45.4 even 6 45.5.h.a.29.19 yes 44
45.14 odd 6 135.5.h.a.89.4 44
45.29 odd 6 45.5.h.a.14.4 ✓ 44
45.34 even 6 135.5.h.a.44.19 44
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
45.5.h.a.14.4 ✓ 44 45.29 odd 6
45.5.h.a.14.19 yes 44 9.2 odd 6
45.5.h.a.29.4 yes 44 9.4 even 3
45.5.h.a.29.19 yes 44 45.4 even 6
135.5.h.a.44.4 44 9.7 even 3
135.5.h.a.44.19 44 45.34 even 6
135.5.h.a.89.4 44 45.14 odd 6
135.5.h.a.89.19 44 9.5 odd 6
405.5.d.a.404.7 44 3.2 odd 2 inner
405.5.d.a.404.8 44 5.4 even 2 inner
405.5.d.a.404.37 44 1.1 even 1 trivial
405.5.d.a.404.38 44 15.14 odd 2 inner