Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2475,2,Mod(1,2475)] 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("2475.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2475, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2475.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,1,0,3,0,0,-5,6,0,0,2] 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: yes
Analytic conductor: \(19.7629745003\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 275)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.30278\) of defining polynomial
Character \(\chi\) \(=\) 2475.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.30278 q^{2} -0.302776 q^{4} -0.697224 q^{7} +3.00000 q^{8} +1.00000 q^{11} -5.00000 q^{13} +0.908327 q^{14} -3.30278 q^{16} +6.90833 q^{17} -1.00000 q^{19} -1.30278 q^{22} -7.30278 q^{23} +6.51388 q^{26} +0.211103 q^{28} -0.908327 q^{29} +10.2111 q^{31} -1.69722 q^{32} -9.00000 q^{34} -2.39445 q^{37} +1.30278 q^{38} +5.60555 q^{41} -7.21110 q^{43} -0.302776 q^{44} +9.51388 q^{46} -3.00000 q^{47} -6.51388 q^{49} +1.51388 q^{52} -1.30278 q^{53} -2.09167 q^{56} +1.18335 q^{58} +14.2111 q^{59} -7.90833 q^{61} -13.3028 q^{62} +8.81665 q^{64} +4.00000 q^{67} -2.09167 q^{68} +2.60555 q^{71} +7.90833 q^{73} +3.11943 q^{74} +0.302776 q^{76} -0.697224 q^{77} -10.9083 q^{79} -7.30278 q^{82} -3.51388 q^{83} +9.39445 q^{86} +3.00000 q^{88} -1.69722 q^{89} +3.48612 q^{91} +2.21110 q^{92} +3.90833 q^{94} -15.3028 q^{97} +8.48612 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 3 q^{4} - 5 q^{7} + 6 q^{8} + 2 q^{11} - 10 q^{13} - 9 q^{14} - 3 q^{16} + 3 q^{17} - 2 q^{19} + q^{22} - 11 q^{23} - 5 q^{26} - 14 q^{28} + 9 q^{29} + 6 q^{31} - 7 q^{32} - 18 q^{34} - 12 q^{37}+ \cdots + 35 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.30278 −0.921201 −0.460601 − 0.887607i \(-0.652366\pi\)
−0.460601 + 0.887607i \(0.652366\pi\)
\(3\) 0 0
\(4\) −0.302776 −0.151388
\(5\) 0 0
\(6\) 0 0
\(7\) −0.697224 −0.263526 −0.131763 − 0.991281i \(-0.542064\pi\)
−0.131763 + 0.991281i \(0.542064\pi\)
\(8\) 3.00000 1.06066
\(9\) 0 0
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −5.00000 −1.38675 −0.693375 − 0.720577i \(-0.743877\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 0.908327 0.242761
\(15\) 0 0
\(16\) −3.30278 −0.825694
\(17\) 6.90833 1.67552 0.837758 − 0.546042i \(-0.183866\pi\)
0.837758 + 0.546042i \(0.183866\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416 −0.114708 − 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −1.30278 −0.277753
\(23\) −7.30278 −1.52273 −0.761367 − 0.648321i \(-0.775471\pi\)
−0.761367 + 0.648321i \(0.775471\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 6.51388 1.27748
\(27\) 0 0
\(28\) 0.211103 0.0398946
\(29\) −0.908327 −0.168672 −0.0843360 − 0.996437i \(-0.526877\pi\)
−0.0843360 + 0.996437i \(0.526877\pi\)
\(30\) 0 0
\(31\) 10.2111 1.83397 0.916984 − 0.398924i \(-0.130616\pi\)
0.916984 + 0.398924i \(0.130616\pi\)
\(32\) −1.69722 −0.300030
\(33\) 0 0
\(34\) −9.00000 −1.54349
\(35\) 0 0
\(36\) 0 0
\(37\) −2.39445 −0.393645 −0.196822 − 0.980439i \(-0.563062\pi\)
−0.196822 + 0.980439i \(0.563062\pi\)
\(38\) 1.30278 0.211338
\(39\) 0 0
\(40\) 0 0
\(41\) 5.60555 0.875440 0.437720 − 0.899111i \(-0.355786\pi\)
0.437720 + 0.899111i \(0.355786\pi\)
\(42\) 0 0
\(43\) −7.21110 −1.09968 −0.549841 − 0.835269i \(-0.685312\pi\)
−0.549841 + 0.835269i \(0.685312\pi\)
\(44\) −0.302776 −0.0456451
\(45\) 0 0
\(46\) 9.51388 1.40274
\(47\) −3.00000 −0.437595 −0.218797 − 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(48\) 0 0
\(49\) −6.51388 −0.930554
\(50\) 0 0
\(51\) 0 0
\(52\) 1.51388 0.209937
\(53\) −1.30278 −0.178950 −0.0894750 − 0.995989i \(-0.528519\pi\)
−0.0894750 + 0.995989i \(0.528519\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.09167 −0.279512
\(57\) 0 0
\(58\) 1.18335 0.155381
\(59\) 14.2111 1.85013 0.925064 − 0.379811i \(-0.124011\pi\)
0.925064 + 0.379811i \(0.124011\pi\)
\(60\) 0 0
\(61\) −7.90833 −1.01256 −0.506279 − 0.862370i \(-0.668979\pi\)
−0.506279 + 0.862370i \(0.668979\pi\)
\(62\) −13.3028 −1.68945
\(63\) 0 0
\(64\) 8.81665 1.10208
\(65\) 0 0
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 − 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) −2.09167 −0.253653
\(69\) 0 0
\(70\) 0 0
\(71\) 2.60555 0.309222 0.154611 − 0.987975i \(-0.450588\pi\)
0.154611 + 0.987975i \(0.450588\pi\)
\(72\) 0 0
\(73\) 7.90833 0.925600 0.462800 − 0.886463i \(-0.346845\pi\)
0.462800 + 0.886463i \(0.346845\pi\)
\(74\) 3.11943 0.362626
\(75\) 0 0
\(76\) 0.302776 0.0347307
\(77\) −0.697224 −0.0794561
\(78\) 0 0
\(79\) −10.9083 −1.22728 −0.613641 − 0.789585i \(-0.710296\pi\)
−0.613641 + 0.789585i \(0.710296\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −7.30278 −0.806457
\(83\) −3.51388 −0.385698 −0.192849 − 0.981228i \(-0.561773\pi\)
−0.192849 + 0.981228i \(0.561773\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 9.39445 1.01303
\(87\) 0 0
\(88\) 3.00000 0.319801
\(89\) −1.69722 −0.179905 −0.0899527 − 0.995946i \(-0.528672\pi\)
−0.0899527 + 0.995946i \(0.528672\pi\)
\(90\) 0 0
\(91\) 3.48612 0.365445
\(92\) 2.21110 0.230523
\(93\) 0 0
\(94\) 3.90833 0.403113
\(95\) 0 0
\(96\) 0 0
\(97\) −15.3028 −1.55376 −0.776881 − 0.629648i \(-0.783199\pi\)
−0.776881 + 0.629648i \(0.783199\pi\)
\(98\) 8.48612 0.857228
\(99\) 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 2475.2.a.t.1.1 2
3.2 odd 2 275.2.a.e.1.2 ✓ 2
5.2 odd 4 2475.2.c.k.199.2 4
5.3 odd 4 2475.2.c.k.199.3 4
5.4 even 2 2475.2.a.o.1.2 2
12.11 even 2 4400.2.a.bs.1.2 2
15.2 even 4 275.2.b.c.199.3 4
15.8 even 4 275.2.b.c.199.2 4
15.14 odd 2 275.2.a.f.1.1 yes 2
33.32 even 2 3025.2.a.n.1.1 2
60.23 odd 4 4400.2.b.y.4049.4 4
60.47 odd 4 4400.2.b.y.4049.1 4
60.59 even 2 4400.2.a.bh.1.1 2
165.164 even 2 3025.2.a.h.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
275.2.a.e.1.2 ✓ 2 3.2 odd 2
275.2.a.f.1.1 yes 2 15.14 odd 2
275.2.b.c.199.2 4 15.8 even 4
275.2.b.c.199.3 4 15.2 even 4
2475.2.a.o.1.2 2 5.4 even 2
2475.2.a.t.1.1 2 1.1 even 1 trivial
2475.2.c.k.199.2 4 5.2 odd 4
2475.2.c.k.199.3 4 5.3 odd 4
3025.2.a.h.1.2 2 165.164 even 2
3025.2.a.n.1.1 2 33.32 even 2
4400.2.a.bh.1.1 2 60.59 even 2
4400.2.a.bs.1.2 2 12.11 even 2
4400.2.b.y.4049.1 4 60.47 odd 4
4400.2.b.y.4049.4 4 60.23 odd 4