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Properties

Label 405.5.d.a.404.3
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 404.3
Character \(\chi\) \(=\) 405.404
Dual form 405.5.d.a.404.4

$q$-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)
 
\(f(q)\) \(=\) \(q-7.28232 q^{2} +37.0321 q^{4} +(21.3751 - 12.9656i) q^{5} -44.4060i q^{7} -153.163 q^{8} +(-155.660 + 94.4195i) q^{10} +164.451i q^{11} -162.158i q^{13} +323.378i q^{14} +522.865 q^{16} -79.6141 q^{17} +493.552 q^{19} +(791.564 - 480.144i) q^{20} -1197.58i q^{22} +198.547 q^{23} +(288.787 - 554.281i) q^{25} +1180.89i q^{26} -1644.45i q^{28} +106.209i q^{29} +1034.85 q^{31} -1357.06 q^{32} +579.775 q^{34} +(-575.750 - 949.181i) q^{35} -1042.31i q^{37} -3594.20 q^{38} +(-3273.86 + 1985.84i) q^{40} -44.6454i q^{41} +3448.84i q^{43} +6089.97i q^{44} -1445.88 q^{46} +836.017 q^{47} +429.108 q^{49} +(-2103.04 + 4036.45i) q^{50} -6005.05i q^{52} +307.853 q^{53} +(2132.20 + 3515.15i) q^{55} +6801.34i q^{56} -773.449i q^{58} +4467.53i q^{59} -1752.57 q^{61} -7536.11 q^{62} +1516.74 q^{64} +(-2102.47 - 3466.14i) q^{65} -4072.59i q^{67} -2948.28 q^{68} +(4192.79 + 6912.24i) q^{70} -5455.14i q^{71} +486.298i q^{73} +7590.40i q^{74} +18277.3 q^{76} +7302.60 q^{77} -7265.20 q^{79} +(11176.3 - 6779.25i) q^{80} +325.122i q^{82} +3510.00 q^{83} +(-1701.76 + 1032.24i) q^{85} -25115.6i q^{86} -25187.7i q^{88} -5210.89i q^{89} -7200.78 q^{91} +7352.63 q^{92} -6088.14 q^{94} +(10549.7 - 6399.19i) q^{95} -1743.98i q^{97} -3124.90 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 44 q + 324 q^{4} + 28 q^{10} + 2116 q^{16} - 8 q^{19} + 296 q^{25} + 2224 q^{31} + 872 q^{34} + 1700 q^{40} - 5668 q^{46} - 10792 q^{49} - 3072 q^{55} - 5564 q^{61} + 8348 q^{64} - 9564 q^{70} + 3552 q^{76}+ \cdots + 37652 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −7.28232 −1.82058 −0.910290 − 0.413972i \(-0.864141\pi\)
−0.910290 + 0.413972i \(0.864141\pi\)
\(3\) 0 0
\(4\) 37.0321 2.31451
\(5\) 21.3751 − 12.9656i 0.855003 − 0.518624i
\(6\) 0 0
\(7\) − 44.4060i − 0.906245i −0.891448 − 0.453122i \(-0.850310\pi\)
0.891448 − 0.453122i \(-0.149690\pi\)
\(8\) −153.163 −2.39317
\(9\) 0 0
\(10\) −155.660 + 94.4195i −1.55660 + 0.944195i
\(11\) 164.451i 1.35910i 0.733630 + 0.679549i \(0.237824\pi\)
−0.733630 + 0.679549i \(0.762176\pi\)
\(12\) 0 0
\(13\) − 162.158i − 0.959514i −0.877401 − 0.479757i \(-0.840725\pi\)
0.877401 − 0.479757i \(-0.159275\pi\)
\(14\) 323.378i 1.64989i
\(15\) 0 0
\(16\) 522.865 2.04244
\(17\) −79.6141 −0.275481 −0.137741 − 0.990468i \(-0.543984\pi\)
−0.137741 + 0.990468i \(0.543984\pi\)
\(18\) 0 0
\(19\) 493.552 1.36718 0.683590 − 0.729866i \(-0.260418\pi\)
0.683590 + 0.729866i \(0.260418\pi\)
\(20\) 791.564 − 480.144i 1.97891 − 1.20036i
\(21\) 0 0
\(22\) − 1197.58i − 2.47435i
\(23\) 198.547 0.375326 0.187663 − 0.982234i \(-0.439909\pi\)
0.187663 + 0.982234i \(0.439909\pi\)
\(24\) 0 0
\(25\) 288.787 − 554.281i 0.462059 − 0.886849i
\(26\) 1180.89i 1.74687i
\(27\) 0 0
\(28\) − 1644.45i − 2.09751i
\(29\) 106.209i 0.126289i 0.998004 + 0.0631446i \(0.0201129\pi\)
−0.998004 + 0.0631446i \(0.979887\pi\)
\(30\) 0 0
\(31\) 1034.85 1.07685 0.538424 − 0.842674i \(-0.319020\pi\)
0.538424 + 0.842674i \(0.319020\pi\)
\(32\) −1357.06 −1.32526
\(33\) 0 0
\(34\) 579.775 0.501536
\(35\) −575.750 − 949.181i −0.470000 − 0.774842i
\(36\) 0 0
\(37\) − 1042.31i − 0.761363i −0.924706 − 0.380681i \(-0.875689\pi\)
0.924706 − 0.380681i \(-0.124311\pi\)
\(38\) −3594.20 −2.48906
\(39\) 0 0
\(40\) −3273.86 + 1985.84i −2.04616 + 1.24115i
\(41\) − 44.6454i − 0.0265588i −0.999912 − 0.0132794i \(-0.995773\pi\)
0.999912 − 0.0132794i \(-0.00422709\pi\)
\(42\) 0 0
\(43\) 3448.84i 1.86525i 0.360850 + 0.932624i \(0.382487\pi\)
−0.360850 + 0.932624i \(0.617513\pi\)
\(44\) 6089.97i 3.14564i
\(45\) 0 0
\(46\) −1445.88 −0.683310
\(47\) 836.017 0.378460 0.189230 − 0.981933i \(-0.439401\pi\)
0.189230 + 0.981933i \(0.439401\pi\)
\(48\) 0 0
\(49\) 429.108 0.178721
\(50\) −2103.04 + 4036.45i −0.841215 + 1.61458i
\(51\) 0 0
\(52\) − 6005.05i − 2.22080i
\(53\) 307.853 0.109595 0.0547975 − 0.998497i \(-0.482549\pi\)
0.0547975 + 0.998497i \(0.482549\pi\)
\(54\) 0 0
\(55\) 2132.20 + 3515.15i 0.704860 + 1.16203i
\(56\) 6801.34i 2.16879i
\(57\) 0 0
\(58\) − 773.449i − 0.229919i
\(59\) 4467.53i 1.28341i 0.766953 + 0.641703i \(0.221772\pi\)
−0.766953 + 0.641703i \(0.778228\pi\)
\(60\) 0 0
\(61\) −1752.57 −0.470995 −0.235497 − 0.971875i \(-0.575672\pi\)
−0.235497 + 0.971875i \(0.575672\pi\)
\(62\) −7536.11 −1.96049
\(63\) 0 0
\(64\) 1516.74 0.370297
\(65\) −2102.47 − 3466.14i −0.497627 − 0.820387i
\(66\) 0 0
\(67\) − 4072.59i − 0.907237i −0.891196 − 0.453619i \(-0.850133\pi\)
0.891196 − 0.453619i \(-0.149867\pi\)
\(68\) −2948.28 −0.637604
\(69\) 0 0
\(70\) 4192.79 + 6912.24i 0.855672 + 1.41066i
\(71\) − 5455.14i − 1.08215i −0.840973 − 0.541077i \(-0.818017\pi\)
0.840973 − 0.541077i \(-0.181983\pi\)
\(72\) 0 0
\(73\) 486.298i 0.0912549i 0.998959 + 0.0456275i \(0.0145287\pi\)
−0.998959 + 0.0456275i \(0.985471\pi\)
\(74\) 7590.40i 1.38612i
\(75\) 0 0
\(76\) 18277.3 3.16435
\(77\) 7302.60 1.23168
\(78\) 0 0
\(79\) −7265.20 −1.16411 −0.582054 − 0.813150i \(-0.697751\pi\)
−0.582054 + 0.813150i \(0.697751\pi\)
\(80\) 11176.3 − 6779.25i 1.74629 − 1.05926i
\(81\) 0 0
\(82\) 325.122i 0.0483524i
\(83\) 3510.00 0.509508 0.254754 − 0.967006i \(-0.418006\pi\)
0.254754 + 0.967006i \(0.418006\pi\)
\(84\) 0 0
\(85\) −1701.76 + 1032.24i −0.235537 + 0.142871i
\(86\) − 25115.6i − 3.39583i
\(87\) 0 0
\(88\) − 25187.7i − 3.25255i
\(89\) − 5210.89i − 0.657857i −0.944355 − 0.328929i \(-0.893312\pi\)
0.944355 − 0.328929i \(-0.106688\pi\)
\(90\) 0 0
\(91\) −7200.78 −0.869555
\(92\) 7352.63 0.868695
\(93\) 0 0
\(94\) −6088.14 −0.689016
\(95\) 10549.7 − 6399.19i 1.16894 − 0.709052i
\(96\) 0 0
\(97\) − 1743.98i − 0.185352i −0.995696 − 0.0926761i \(-0.970458\pi\)
0.995696 − 0.0926761i \(-0.0295421\pi\)
\(98\) −3124.90 −0.325375
\(99\) 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.404.3 44
3.2 odd 2 inner 405.5.d.a.404.41 44
5.4 even 2 inner 405.5.d.a.404.42 44
9.2 odd 6 135.5.h.a.44.2 44
9.4 even 3 135.5.h.a.89.21 44
9.5 odd 6 45.5.h.a.29.2 yes 44
9.7 even 3 45.5.h.a.14.21 yes 44
15.14 odd 2 inner 405.5.d.a.404.4 44
45.4 even 6 135.5.h.a.89.2 44
45.14 odd 6 45.5.h.a.29.21 yes 44
45.29 odd 6 135.5.h.a.44.21 44
45.34 even 6 45.5.h.a.14.2 ✓ 44
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
45.5.h.a.14.2 ✓ 44 45.34 even 6
45.5.h.a.14.21 yes 44 9.7 even 3
45.5.h.a.29.2 yes 44 9.5 odd 6
45.5.h.a.29.21 yes 44 45.14 odd 6
135.5.h.a.44.2 44 9.2 odd 6
135.5.h.a.44.21 44 45.29 odd 6
135.5.h.a.89.2 44 45.4 even 6
135.5.h.a.89.21 44 9.4 even 3
405.5.d.a.404.3 44 1.1 even 1 trivial
405.5.d.a.404.4 44 15.14 odd 2 inner
405.5.d.a.404.41 44 3.2 odd 2 inner
405.5.d.a.404.42 44 5.4 even 2 inner