Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2025,2,Mod(649,2025)] 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("2025.649"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2025, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2025 = 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2025.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,2,0,0,0,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(16.1697064093\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 45)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 649.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2025.649
Dual form 2025.2.b.d.649.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.00000 q^{4} +3.00000i q^{7} +3.00000i q^{8} +2.00000 q^{11} -2.00000i q^{13} -3.00000 q^{14} -1.00000 q^{16} +4.00000i q^{17} +8.00000 q^{19} +2.00000i q^{22} -3.00000i q^{23} +2.00000 q^{26} +3.00000i q^{28} -1.00000 q^{29} +5.00000i q^{32} -4.00000 q^{34} +4.00000i q^{37} +8.00000i q^{38} -5.00000 q^{41} -8.00000i q^{43} +2.00000 q^{44} +3.00000 q^{46} +7.00000i q^{47} -2.00000 q^{49} -2.00000i q^{52} +2.00000i q^{53} -9.00000 q^{56} -1.00000i q^{58} -14.0000 q^{59} +7.00000 q^{61} -7.00000 q^{64} +3.00000i q^{67} +4.00000i q^{68} -2.00000 q^{71} +4.00000i q^{73} -4.00000 q^{74} +8.00000 q^{76} +6.00000i q^{77} +6.00000 q^{79} -5.00000i q^{82} -9.00000i q^{83} +8.00000 q^{86} +6.00000i q^{88} -15.0000 q^{89} +6.00000 q^{91} -3.00000i q^{92} -7.00000 q^{94} -2.00000i q^{97} -2.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} + 4 q^{11} - 6 q^{14} - 2 q^{16} + 16 q^{19} + 4 q^{26} - 2 q^{29} - 8 q^{34} - 10 q^{41} + 4 q^{44} + 6 q^{46} - 4 q^{49} - 18 q^{56} - 28 q^{59} + 14 q^{61} - 14 q^{64} - 4 q^{71} - 8 q^{74}+ \cdots - 14 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2025\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(1702\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i 0.935414 + 0.353553i \(0.115027\pi\)
−0.935414 + 0.353553i \(0.884973\pi\)
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 3.00000i 1.13389i 0.823754 + 0.566947i \(0.191875\pi\)
−0.823754 + 0.566947i \(0.808125\pi\)
\(8\) 3.00000i 1.06066i
\(9\) 0 0
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 − 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) − 2.00000i − 0.554700i −0.960769 − 0.277350i \(-0.910544\pi\)
0.960769 − 0.277350i \(-0.0894562\pi\)
\(14\) −3.00000 −0.801784
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 4.00000i 0.970143i 0.874475 + 0.485071i \(0.161206\pi\)
−0.874475 + 0.485071i \(0.838794\pi\)
\(18\) 0 0
\(19\) 8.00000 1.83533 0.917663 − 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 2.00000i 0.426401i
\(23\) − 3.00000i − 0.625543i −0.949828 − 0.312772i \(-0.898743\pi\)
0.949828 − 0.312772i \(-0.101257\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) 0 0
\(28\) 3.00000i 0.566947i
\(29\) −1.00000 −0.185695 −0.0928477 − 0.995680i \(-0.529597\pi\)
−0.0928477 + 0.995680i \(0.529597\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 5.00000i 0.883883i
\(33\) 0 0
\(34\) −4.00000 −0.685994
\(35\) 0 0
\(36\) 0 0
\(37\) 4.00000i 0.657596i 0.944400 + 0.328798i \(0.106644\pi\)
−0.944400 + 0.328798i \(0.893356\pi\)
\(38\) 8.00000i 1.29777i
\(39\) 0 0
\(40\) 0 0
\(41\) −5.00000 −0.780869 −0.390434 − 0.920631i \(-0.627675\pi\)
−0.390434 + 0.920631i \(0.627675\pi\)
\(42\) 0 0
\(43\) − 8.00000i − 1.21999i −0.792406 − 0.609994i \(-0.791172\pi\)
0.792406 − 0.609994i \(-0.208828\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) 3.00000 0.442326
\(47\) 7.00000i 1.02105i 0.859861 + 0.510527i \(0.170550\pi\)
−0.859861 + 0.510527i \(0.829450\pi\)
\(48\) 0 0
\(49\) −2.00000 −0.285714
\(50\) 0 0
\(51\) 0 0
\(52\) − 2.00000i − 0.277350i
\(53\) 2.00000i 0.274721i 0.990521 + 0.137361i \(0.0438619\pi\)
−0.990521 + 0.137361i \(0.956138\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −9.00000 −1.20268
\(57\) 0 0
\(58\) − 1.00000i − 0.131306i
\(59\) −14.0000 −1.82264 −0.911322 − 0.411693i \(-0.864937\pi\)
−0.911322 + 0.411693i \(0.864937\pi\)
\(60\) 0 0
\(61\) 7.00000 0.896258 0.448129 − 0.893969i \(-0.352090\pi\)
0.448129 + 0.893969i \(0.352090\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −7.00000 −0.875000
\(65\) 0 0
\(66\) 0 0
\(67\) 3.00000i 0.366508i 0.983066 + 0.183254i \(0.0586631\pi\)
−0.983066 + 0.183254i \(0.941337\pi\)
\(68\) 4.00000i 0.485071i
\(69\) 0 0
\(70\) 0 0
\(71\) −2.00000 −0.237356 −0.118678 − 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) 0 0
\(73\) 4.00000i 0.468165i 0.972217 + 0.234082i \(0.0752085\pi\)
−0.972217 + 0.234082i \(0.924791\pi\)
\(74\) −4.00000 −0.464991
\(75\) 0 0
\(76\) 8.00000 0.917663
\(77\) 6.00000i 0.683763i
\(78\) 0 0
\(79\) 6.00000 0.675053 0.337526 − 0.941316i \(-0.390410\pi\)
0.337526 + 0.941316i \(0.390410\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 5.00000i − 0.552158i
\(83\) − 9.00000i − 0.987878i −0.869496 − 0.493939i \(-0.835557\pi\)
0.869496 − 0.493939i \(-0.164443\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 8.00000 0.862662
\(87\) 0 0
\(88\) 6.00000i 0.639602i
\(89\) −15.0000 −1.59000 −0.794998 − 0.606612i \(-0.792528\pi\)
−0.794998 + 0.606612i \(0.792528\pi\)
\(90\) 0 0
\(91\) 6.00000 0.628971
\(92\) − 3.00000i − 0.312772i
\(93\) 0 0
\(94\) −7.00000 −0.721995
\(95\) 0 0
\(96\) 0 0
\(97\) − 2.00000i − 0.203069i −0.994832 − 0.101535i \(-0.967625\pi\)
0.994832 − 0.101535i \(-0.0323753\pi\)
\(98\) − 2.00000i − 0.202031i
\(99\) 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 2025.2.b.d.649.2 2
3.2 odd 2 2025.2.b.c.649.1 2
5.2 odd 4 405.2.a.b.1.1 1
5.3 odd 4 2025.2.a.e.1.1 1
5.4 even 2 inner 2025.2.b.d.649.1 2
9.2 odd 6 225.2.k.a.49.1 4
9.4 even 3 675.2.k.a.424.1 4
9.5 odd 6 225.2.k.a.124.2 4
9.7 even 3 675.2.k.a.199.2 4
15.2 even 4 405.2.a.e.1.1 1
15.8 even 4 2025.2.a.b.1.1 1
15.14 odd 2 2025.2.b.c.649.2 2
20.7 even 4 6480.2.a.x.1.1 1
45.2 even 12 45.2.e.a.31.1 yes 2
45.4 even 6 675.2.k.a.424.2 4
45.7 odd 12 135.2.e.a.91.1 2
45.13 odd 12 675.2.e.a.451.1 2
45.14 odd 6 225.2.k.a.124.1 4
45.22 odd 12 135.2.e.a.46.1 2
45.23 even 12 225.2.e.a.151.1 2
45.29 odd 6 225.2.k.a.49.2 4
45.32 even 12 45.2.e.a.16.1 ✓ 2
45.34 even 6 675.2.k.a.199.1 4
45.38 even 12 225.2.e.a.76.1 2
45.43 odd 12 675.2.e.a.226.1 2
60.47 odd 4 6480.2.a.k.1.1 1
180.7 even 12 2160.2.q.a.1441.1 2
180.47 odd 12 720.2.q.d.481.1 2
180.67 even 12 2160.2.q.a.721.1 2
180.167 odd 12 720.2.q.d.241.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
45.2.e.a.16.1 ✓ 2 45.32 even 12
45.2.e.a.31.1 yes 2 45.2 even 12
135.2.e.a.46.1 2 45.22 odd 12
135.2.e.a.91.1 2 45.7 odd 12
225.2.e.a.76.1 2 45.38 even 12
225.2.e.a.151.1 2 45.23 even 12
225.2.k.a.49.1 4 9.2 odd 6
225.2.k.a.49.2 4 45.29 odd 6
225.2.k.a.124.1 4 45.14 odd 6
225.2.k.a.124.2 4 9.5 odd 6
405.2.a.b.1.1 1 5.2 odd 4
405.2.a.e.1.1 1 15.2 even 4
675.2.e.a.226.1 2 45.43 odd 12
675.2.e.a.451.1 2 45.13 odd 12
675.2.k.a.199.1 4 45.34 even 6
675.2.k.a.199.2 4 9.7 even 3
675.2.k.a.424.1 4 9.4 even 3
675.2.k.a.424.2 4 45.4 even 6
720.2.q.d.241.1 2 180.167 odd 12
720.2.q.d.481.1 2 180.47 odd 12
2025.2.a.b.1.1 1 15.8 even 4
2025.2.a.e.1.1 1 5.3 odd 4
2025.2.b.c.649.1 2 3.2 odd 2
2025.2.b.c.649.2 2 15.14 odd 2
2025.2.b.d.649.1 2 5.4 even 2 inner
2025.2.b.d.649.2 2 1.1 even 1 trivial
2160.2.q.a.721.1 2 180.67 even 12
2160.2.q.a.1441.1 2 180.7 even 12
6480.2.a.k.1.1 1 60.47 odd 4
6480.2.a.x.1.1 1 20.7 even 4