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Properties

Label 405.2.a.i.1.2
Level $405$
Weight $2$
Character 405.1
Self dual yes
Analytic conductor $3.234$
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] = [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 = [3,-1,0,5,-3] 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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.564.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 45)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.571993\) of defining polynomial
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)
 
\(f(q)\) \(=\) \(q-0.571993 q^{2} -1.67282 q^{4} -1.00000 q^{5} +1.42801 q^{7} +2.10083 q^{8} +0.571993 q^{10} -2.67282 q^{11} +4.67282 q^{13} -0.816810 q^{14} +2.14399 q^{16} -2.67282 q^{17} +4.67282 q^{19} +1.67282 q^{20} +1.52884 q^{22} +5.91764 q^{23} +1.00000 q^{25} -2.67282 q^{26} -2.38880 q^{28} +9.48963 q^{29} +6.96080 q^{31} -5.42801 q^{32} +1.52884 q^{34} -1.42801 q^{35} -1.81681 q^{37} -2.67282 q^{38} -2.10083 q^{40} +1.47116 q^{41} +0.471163 q^{43} +4.47116 q^{44} -3.38485 q^{46} -6.95684 q^{47} -4.96080 q^{49} -0.571993 q^{50} -7.81681 q^{52} +1.14399 q^{53} +2.67282 q^{55} +3.00000 q^{56} -5.42801 q^{58} +1.14399 q^{59} -2.52884 q^{61} -3.98153 q^{62} -1.18319 q^{64} -4.67282 q^{65} -6.59046 q^{67} +4.47116 q^{68} +0.816810 q^{70} +12.8745 q^{71} -1.71203 q^{73} +1.03920 q^{74} -7.81681 q^{76} -3.81681 q^{77} -0.287973 q^{79} -2.14399 q^{80} -0.841495 q^{82} +4.28402 q^{83} +2.67282 q^{85} -0.269502 q^{86} -5.61515 q^{88} +3.00000 q^{89} +6.67282 q^{91} -9.89917 q^{92} +3.97927 q^{94} -4.67282 q^{95} +7.83528 q^{97} +2.83754 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 5 q^{4} - 3 q^{5} + 5 q^{7} - 3 q^{8} + q^{10} + 2 q^{11} + 4 q^{13} + 9 q^{14} + 5 q^{16} + 2 q^{17} + 4 q^{19} - 5 q^{20} - 4 q^{22} - 3 q^{23} + 3 q^{25} + 2 q^{26} + 5 q^{28} + 7 q^{29}+ \cdots + 40 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.571993 −0.404460 −0.202230 − 0.979338i \(-0.564819\pi\)
−0.202230 + 0.979338i \(0.564819\pi\)
\(3\) 0 0
\(4\) −1.67282 −0.836412
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.42801 0.539736 0.269868 − 0.962897i \(-0.413020\pi\)
0.269868 + 0.962897i \(0.413020\pi\)
\(8\) 2.10083 0.742756
\(9\) 0 0
\(10\) 0.571993 0.180880
\(11\) −2.67282 −0.805887 −0.402943 − 0.915225i \(-0.632013\pi\)
−0.402943 + 0.915225i \(0.632013\pi\)
\(12\) 0 0
\(13\) 4.67282 1.29601 0.648004 − 0.761637i \(-0.275604\pi\)
0.648004 + 0.761637i \(0.275604\pi\)
\(14\) −0.816810 −0.218302
\(15\) 0 0
\(16\) 2.14399 0.535997
\(17\) −2.67282 −0.648255 −0.324127 − 0.946013i \(-0.605071\pi\)
−0.324127 + 0.946013i \(0.605071\pi\)
\(18\) 0 0
\(19\) 4.67282 1.07202 0.536010 − 0.844212i \(-0.319931\pi\)
0.536010 + 0.844212i \(0.319931\pi\)
\(20\) 1.67282 0.374055
\(21\) 0 0
\(22\) 1.52884 0.325949
\(23\) 5.91764 1.23391 0.616957 − 0.786997i \(-0.288365\pi\)
0.616957 + 0.786997i \(0.288365\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −2.67282 −0.524184
\(27\) 0 0
\(28\) −2.38880 −0.451441
\(29\) 9.48963 1.76218 0.881090 − 0.472948i \(-0.156810\pi\)
0.881090 + 0.472948i \(0.156810\pi\)
\(30\) 0 0
\(31\) 6.96080 1.25020 0.625098 − 0.780546i \(-0.285059\pi\)
0.625098 + 0.780546i \(0.285059\pi\)
\(32\) −5.42801 −0.959545
\(33\) 0 0
\(34\) 1.52884 0.262193
\(35\) −1.42801 −0.241377
\(36\) 0 0
\(37\) −1.81681 −0.298682 −0.149341 − 0.988786i \(-0.547715\pi\)
−0.149341 + 0.988786i \(0.547715\pi\)
\(38\) −2.67282 −0.433589
\(39\) 0 0
\(40\) −2.10083 −0.332170
\(41\) 1.47116 0.229757 0.114879 − 0.993380i \(-0.463352\pi\)
0.114879 + 0.993380i \(0.463352\pi\)
\(42\) 0 0
\(43\) 0.471163 0.0718517 0.0359258 − 0.999354i \(-0.488562\pi\)
0.0359258 + 0.999354i \(0.488562\pi\)
\(44\) 4.47116 0.674053
\(45\) 0 0
\(46\) −3.38485 −0.499069
\(47\) −6.95684 −1.01476 −0.507380 − 0.861722i \(-0.669386\pi\)
−0.507380 + 0.861722i \(0.669386\pi\)
\(48\) 0 0
\(49\) −4.96080 −0.708685
\(50\) −0.571993 −0.0808921
\(51\) 0 0
\(52\) −7.81681 −1.08400
\(53\) 1.14399 0.157139 0.0785693 − 0.996909i \(-0.474965\pi\)
0.0785693 + 0.996909i \(0.474965\pi\)
\(54\) 0 0
\(55\) 2.67282 0.360403
\(56\) 3.00000 0.400892
\(57\) 0 0
\(58\) −5.42801 −0.712732
\(59\) 1.14399 0.148934 0.0744672 − 0.997223i \(-0.476274\pi\)
0.0744672 + 0.997223i \(0.476274\pi\)
\(60\) 0 0
\(61\) −2.52884 −0.323784 −0.161892 − 0.986808i \(-0.551760\pi\)
−0.161892 + 0.986808i \(0.551760\pi\)
\(62\) −3.98153 −0.505655
\(63\) 0 0
\(64\) −1.18319 −0.147899
\(65\) −4.67282 −0.579592
\(66\) 0 0
\(67\) −6.59046 −0.805153 −0.402577 − 0.915386i \(-0.631885\pi\)
−0.402577 + 0.915386i \(0.631885\pi\)
\(68\) 4.47116 0.542208
\(69\) 0 0
\(70\) 0.816810 0.0976275
\(71\) 12.8745 1.52792 0.763960 − 0.645263i \(-0.223252\pi\)
0.763960 + 0.645263i \(0.223252\pi\)
\(72\) 0 0
\(73\) −1.71203 −0.200378 −0.100189 − 0.994968i \(-0.531945\pi\)
−0.100189 + 0.994968i \(0.531945\pi\)
\(74\) 1.03920 0.120805
\(75\) 0 0
\(76\) −7.81681 −0.896650
\(77\) −3.81681 −0.434966
\(78\) 0 0
\(79\) −0.287973 −0.0323995 −0.0161998 − 0.999869i \(-0.505157\pi\)
−0.0161998 + 0.999869i \(0.505157\pi\)
\(80\) −2.14399 −0.239705
\(81\) 0 0
\(82\) −0.841495 −0.0929276
\(83\) 4.28402 0.470232 0.235116 − 0.971967i \(-0.424453\pi\)
0.235116 + 0.971967i \(0.424453\pi\)
\(84\) 0 0
\(85\) 2.67282 0.289908
\(86\) −0.269502 −0.0290611
\(87\) 0 0
\(88\) −5.61515 −0.598577
\(89\) 3.00000 0.317999 0.159000 − 0.987279i \(-0.449173\pi\)
0.159000 + 0.987279i \(0.449173\pi\)
\(90\) 0 0
\(91\) 6.67282 0.699502
\(92\) −9.89917 −1.03206
\(93\) 0 0
\(94\) 3.97927 0.410430
\(95\) −4.67282 −0.479422
\(96\) 0 0
\(97\) 7.83528 0.795552 0.397776 − 0.917483i \(-0.369782\pi\)
0.397776 + 0.917483i \(0.369782\pi\)
\(98\) 2.83754 0.286635
\(99\) 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.i.1.2 3
3.2 odd 2 405.2.a.j.1.2 3
4.3 odd 2 6480.2.a.bs.1.2 3
5.2 odd 4 2025.2.b.m.649.3 6
5.3 odd 4 2025.2.b.m.649.4 6
5.4 even 2 2025.2.a.o.1.2 3
9.2 odd 6 45.2.e.b.31.2 yes 6
9.4 even 3 135.2.e.b.46.2 6
9.5 odd 6 45.2.e.b.16.2 ✓ 6
9.7 even 3 135.2.e.b.91.2 6
12.11 even 2 6480.2.a.bv.1.2 3
15.2 even 4 2025.2.b.l.649.4 6
15.8 even 4 2025.2.b.l.649.3 6
15.14 odd 2 2025.2.a.n.1.2 3
36.7 odd 6 2160.2.q.k.1441.2 6
36.11 even 6 720.2.q.i.481.2 6
36.23 even 6 720.2.q.i.241.2 6
36.31 odd 6 2160.2.q.k.721.2 6
45.2 even 12 225.2.k.b.49.4 12
45.4 even 6 675.2.e.b.451.2 6
45.7 odd 12 675.2.k.b.199.3 12
45.13 odd 12 675.2.k.b.424.3 12
45.14 odd 6 225.2.e.b.151.2 6
45.22 odd 12 675.2.k.b.424.4 12
45.23 even 12 225.2.k.b.124.4 12
45.29 odd 6 225.2.e.b.76.2 6
45.32 even 12 225.2.k.b.124.3 12
45.34 even 6 675.2.e.b.226.2 6
45.38 even 12 225.2.k.b.49.3 12
45.43 odd 12 675.2.k.b.199.4 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
45.2.e.b.16.2 ✓ 6 9.5 odd 6
45.2.e.b.31.2 yes 6 9.2 odd 6
135.2.e.b.46.2 6 9.4 even 3
135.2.e.b.91.2 6 9.7 even 3
225.2.e.b.76.2 6 45.29 odd 6
225.2.e.b.151.2 6 45.14 odd 6
225.2.k.b.49.3 12 45.38 even 12
225.2.k.b.49.4 12 45.2 even 12
225.2.k.b.124.3 12 45.32 even 12
225.2.k.b.124.4 12 45.23 even 12
405.2.a.i.1.2 3 1.1 even 1 trivial
405.2.a.j.1.2 3 3.2 odd 2
675.2.e.b.226.2 6 45.34 even 6
675.2.e.b.451.2 6 45.4 even 6
675.2.k.b.199.3 12 45.7 odd 12
675.2.k.b.199.4 12 45.43 odd 12
675.2.k.b.424.3 12 45.13 odd 12
675.2.k.b.424.4 12 45.22 odd 12
720.2.q.i.241.2 6 36.23 even 6
720.2.q.i.481.2 6 36.11 even 6
2025.2.a.n.1.2 3 15.14 odd 2
2025.2.a.o.1.2 3 5.4 even 2
2025.2.b.l.649.3 6 15.8 even 4
2025.2.b.l.649.4 6 15.2 even 4
2025.2.b.m.649.3 6 5.2 odd 4
2025.2.b.m.649.4 6 5.3 odd 4
2160.2.q.k.721.2 6 36.31 odd 6
2160.2.q.k.1441.2 6 36.7 odd 6
6480.2.a.bs.1.2 3 4.3 odd 2
6480.2.a.bv.1.2 3 12.11 even 2