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 1.1
Root \(-2.08613\) 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)
 
\(f(q)\) \(=\) \(q-2.08613 q^{2} +2.35194 q^{4} +1.00000 q^{5} +4.08613 q^{7} -0.734191 q^{8} -2.08613 q^{10} -1.35194 q^{11} +0.648061 q^{13} -8.52420 q^{14} -3.17226 q^{16} -1.35194 q^{17} +0.648061 q^{19} +2.35194 q^{20} +2.82032 q^{22} +4.79001 q^{23} +1.00000 q^{25} -1.35194 q^{26} +9.61033 q^{28} +3.87614 q^{29} -7.69646 q^{31} +8.08613 q^{32} +2.82032 q^{34} +4.08613 q^{35} +7.52420 q^{37} -1.35194 q^{38} -0.734191 q^{40} -0.179679 q^{41} -0.820321 q^{43} -3.17968 q^{44} -9.99258 q^{46} +10.9065 q^{47} +9.69646 q^{49} -2.08613 q^{50} +1.52420 q^{52} +4.17226 q^{53} -1.35194 q^{55} -3.00000 q^{56} -8.08613 q^{58} +4.17226 q^{59} -3.82032 q^{61} +16.0558 q^{62} -10.5242 q^{64} +0.648061 q^{65} +8.14195 q^{67} -3.17968 q^{68} -8.52420 q^{70} -6.11644 q^{71} -12.3445 q^{73} -15.6965 q^{74} +1.52420 q^{76} -5.52420 q^{77} +10.3445 q^{79} -3.17226 q^{80} +0.374833 q^{82} -12.2584 q^{83} -1.35194 q^{85} +1.71130 q^{86} +0.992582 q^{88} -3.00000 q^{89} +2.64806 q^{91} +11.2658 q^{92} -22.7523 q^{94} +0.648061 q^{95} -13.5800 q^{97} -20.2281 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + 5 q^{4} + 3 q^{5} + 5 q^{7} + 3 q^{8} + q^{10} - 2 q^{11} + 4 q^{13} - 9 q^{14} + 5 q^{16} - 2 q^{17} + 4 q^{19} + 5 q^{20} - 4 q^{22} + 3 q^{23} + 3 q^{25} - 2 q^{26} + 5 q^{28} - 7 q^{29}+ \cdots - 40 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.08613 −1.47512 −0.737558 − 0.675283i \(-0.764021\pi\)
−0.737558 + 0.675283i \(0.764021\pi\)
\(3\) 0 0
\(4\) 2.35194 1.17597
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 4.08613 1.54441 0.772206 − 0.635372i \(-0.219153\pi\)
0.772206 + 0.635372i \(0.219153\pi\)
\(8\) −0.734191 −0.259576
\(9\) 0 0
\(10\) −2.08613 −0.659692
\(11\) −1.35194 −0.407625 −0.203813 − 0.979010i \(-0.565333\pi\)
−0.203813 + 0.979010i \(0.565333\pi\)
\(12\) 0 0
\(13\) 0.648061 0.179740 0.0898699 − 0.995954i \(-0.471355\pi\)
0.0898699 + 0.995954i \(0.471355\pi\)
\(14\) −8.52420 −2.27819
\(15\) 0 0
\(16\) −3.17226 −0.793065
\(17\) −1.35194 −0.327893 −0.163947 − 0.986469i \(-0.552423\pi\)
−0.163947 + 0.986469i \(0.552423\pi\)
\(18\) 0 0
\(19\) 0.648061 0.148675 0.0743377 − 0.997233i \(-0.476316\pi\)
0.0743377 + 0.997233i \(0.476316\pi\)
\(20\) 2.35194 0.525910
\(21\) 0 0
\(22\) 2.82032 0.601294
\(23\) 4.79001 0.998786 0.499393 − 0.866376i \(-0.333556\pi\)
0.499393 + 0.866376i \(0.333556\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −1.35194 −0.265137
\(27\) 0 0
\(28\) 9.61033 1.81618
\(29\) 3.87614 0.719781 0.359890 − 0.932995i \(-0.382814\pi\)
0.359890 + 0.932995i \(0.382814\pi\)
\(30\) 0 0
\(31\) −7.69646 −1.38233 −0.691163 − 0.722699i \(-0.742901\pi\)
−0.691163 + 0.722699i \(0.742901\pi\)
\(32\) 8.08613 1.42944
\(33\) 0 0
\(34\) 2.82032 0.483681
\(35\) 4.08613 0.690682
\(36\) 0 0
\(37\) 7.52420 1.23697 0.618485 − 0.785796i \(-0.287747\pi\)
0.618485 + 0.785796i \(0.287747\pi\)
\(38\) −1.35194 −0.219313
\(39\) 0 0
\(40\) −0.734191 −0.116086
\(41\) −0.179679 −0.0280611 −0.0140306 − 0.999902i \(-0.504466\pi\)
−0.0140306 + 0.999902i \(0.504466\pi\)
\(42\) 0 0
\(43\) −0.820321 −0.125098 −0.0625489 − 0.998042i \(-0.519923\pi\)
−0.0625489 + 0.998042i \(0.519923\pi\)
\(44\) −3.17968 −0.479355
\(45\) 0 0
\(46\) −9.99258 −1.47333
\(47\) 10.9065 1.59087 0.795435 − 0.606039i \(-0.207243\pi\)
0.795435 + 0.606039i \(0.207243\pi\)
\(48\) 0 0
\(49\) 9.69646 1.38521
\(50\) −2.08613 −0.295023
\(51\) 0 0
\(52\) 1.52420 0.211368
\(53\) 4.17226 0.573104 0.286552 − 0.958065i \(-0.407491\pi\)
0.286552 + 0.958065i \(0.407491\pi\)
\(54\) 0 0
\(55\) −1.35194 −0.182295
\(56\) −3.00000 −0.400892
\(57\) 0 0
\(58\) −8.08613 −1.06176
\(59\) 4.17226 0.543182 0.271591 − 0.962413i \(-0.412450\pi\)
0.271591 + 0.962413i \(0.412450\pi\)
\(60\) 0 0
\(61\) −3.82032 −0.489142 −0.244571 − 0.969631i \(-0.578647\pi\)
−0.244571 + 0.969631i \(0.578647\pi\)
\(62\) 16.0558 2.03909
\(63\) 0 0
\(64\) −10.5242 −1.31552
\(65\) 0.648061 0.0803820
\(66\) 0 0
\(67\) 8.14195 0.994697 0.497349 − 0.867551i \(-0.334307\pi\)
0.497349 + 0.867551i \(0.334307\pi\)
\(68\) −3.17968 −0.385593
\(69\) 0 0
\(70\) −8.52420 −1.01884
\(71\) −6.11644 −0.725888 −0.362944 − 0.931811i \(-0.618228\pi\)
−0.362944 + 0.931811i \(0.618228\pi\)
\(72\) 0 0
\(73\) −12.3445 −1.44482 −0.722408 − 0.691467i \(-0.756965\pi\)
−0.722408 + 0.691467i \(0.756965\pi\)
\(74\) −15.6965 −1.82468
\(75\) 0 0
\(76\) 1.52420 0.174838
\(77\) −5.52420 −0.629541
\(78\) 0 0
\(79\) 10.3445 1.16385 0.581925 − 0.813243i \(-0.302300\pi\)
0.581925 + 0.813243i \(0.302300\pi\)
\(80\) −3.17226 −0.354669
\(81\) 0 0
\(82\) 0.374833 0.0413934
\(83\) −12.2584 −1.34553 −0.672767 − 0.739855i \(-0.734894\pi\)
−0.672767 + 0.739855i \(0.734894\pi\)
\(84\) 0 0
\(85\) −1.35194 −0.146638
\(86\) 1.71130 0.184534
\(87\) 0 0
\(88\) 0.992582 0.105810
\(89\) −3.00000 −0.317999 −0.159000 − 0.987279i \(-0.550827\pi\)
−0.159000 + 0.987279i \(0.550827\pi\)
\(90\) 0 0
\(91\) 2.64806 0.277592
\(92\) 11.2658 1.17454
\(93\) 0 0
\(94\) −22.7523 −2.34672
\(95\) 0.648061 0.0664896
\(96\) 0 0
\(97\) −13.5800 −1.37884 −0.689421 − 0.724361i \(-0.742135\pi\)
−0.689421 + 0.724361i \(0.742135\pi\)
\(98\) −20.2281 −2.04334
\(99\) 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.j.1.1 3
3.2 odd 2 405.2.a.i.1.3 3
4.3 odd 2 6480.2.a.bv.1.1 3
5.2 odd 4 2025.2.b.l.649.2 6
5.3 odd 4 2025.2.b.l.649.5 6
5.4 even 2 2025.2.a.n.1.3 3
9.2 odd 6 135.2.e.b.91.1 6
9.4 even 3 45.2.e.b.16.3 ✓ 6
9.5 odd 6 135.2.e.b.46.1 6
9.7 even 3 45.2.e.b.31.3 yes 6
12.11 even 2 6480.2.a.bs.1.1 3
15.2 even 4 2025.2.b.m.649.5 6
15.8 even 4 2025.2.b.m.649.2 6
15.14 odd 2 2025.2.a.o.1.1 3
36.7 odd 6 720.2.q.i.481.3 6
36.11 even 6 2160.2.q.k.1441.3 6
36.23 even 6 2160.2.q.k.721.3 6
36.31 odd 6 720.2.q.i.241.3 6
45.2 even 12 675.2.k.b.199.5 12
45.4 even 6 225.2.e.b.151.1 6
45.7 odd 12 225.2.k.b.49.2 12
45.13 odd 12 225.2.k.b.124.2 12
45.14 odd 6 675.2.e.b.451.3 6
45.22 odd 12 225.2.k.b.124.5 12
45.23 even 12 675.2.k.b.424.5 12
45.29 odd 6 675.2.e.b.226.3 6
45.32 even 12 675.2.k.b.424.2 12
45.34 even 6 225.2.e.b.76.1 6
45.38 even 12 675.2.k.b.199.2 12
45.43 odd 12 225.2.k.b.49.5 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
45.2.e.b.16.3 ✓ 6 9.4 even 3
45.2.e.b.31.3 yes 6 9.7 even 3
135.2.e.b.46.1 6 9.5 odd 6
135.2.e.b.91.1 6 9.2 odd 6
225.2.e.b.76.1 6 45.34 even 6
225.2.e.b.151.1 6 45.4 even 6
225.2.k.b.49.2 12 45.7 odd 12
225.2.k.b.49.5 12 45.43 odd 12
225.2.k.b.124.2 12 45.13 odd 12
225.2.k.b.124.5 12 45.22 odd 12
405.2.a.i.1.3 3 3.2 odd 2
405.2.a.j.1.1 3 1.1 even 1 trivial
675.2.e.b.226.3 6 45.29 odd 6
675.2.e.b.451.3 6 45.14 odd 6
675.2.k.b.199.2 12 45.38 even 12
675.2.k.b.199.5 12 45.2 even 12
675.2.k.b.424.2 12 45.32 even 12
675.2.k.b.424.5 12 45.23 even 12
720.2.q.i.241.3 6 36.31 odd 6
720.2.q.i.481.3 6 36.7 odd 6
2025.2.a.n.1.3 3 5.4 even 2
2025.2.a.o.1.1 3 15.14 odd 2
2025.2.b.l.649.2 6 5.2 odd 4
2025.2.b.l.649.5 6 5.3 odd 4
2025.2.b.m.649.2 6 15.8 even 4
2025.2.b.m.649.5 6 15.2 even 4
2160.2.q.k.721.3 6 36.23 even 6
2160.2.q.k.1441.3 6 36.11 even 6
6480.2.a.bs.1.1 3 12.11 even 2
6480.2.a.bv.1.1 3 4.3 odd 2