↑

Properties

Label 275.2.a.b.1.1
Level $275$
Weight $2$
Character 275.1
Self dual yes
Analytic conductor $2.196$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [275,2,Mod(1,275)] 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("275.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(275, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 275.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,2,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(2.19588605559\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 11)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 275.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} +2.00000 q^{6} +2.00000 q^{7} -2.00000 q^{9} +1.00000 q^{11} +2.00000 q^{12} -4.00000 q^{13} +4.00000 q^{14} -4.00000 q^{16} +2.00000 q^{17} -4.00000 q^{18} +2.00000 q^{21} +2.00000 q^{22} +1.00000 q^{23} -8.00000 q^{26} -5.00000 q^{27} +4.00000 q^{28} +7.00000 q^{31} -8.00000 q^{32} +1.00000 q^{33} +4.00000 q^{34} -4.00000 q^{36} -3.00000 q^{37} -4.00000 q^{39} -8.00000 q^{41} +4.00000 q^{42} +6.00000 q^{43} +2.00000 q^{44} +2.00000 q^{46} -8.00000 q^{47} -4.00000 q^{48} -3.00000 q^{49} +2.00000 q^{51} -8.00000 q^{52} +6.00000 q^{53} -10.0000 q^{54} +5.00000 q^{59} +12.0000 q^{61} +14.0000 q^{62} -4.00000 q^{63} -8.00000 q^{64} +2.00000 q^{66} +7.00000 q^{67} +4.00000 q^{68} +1.00000 q^{69} -3.00000 q^{71} -4.00000 q^{73} -6.00000 q^{74} +2.00000 q^{77} -8.00000 q^{78} -10.0000 q^{79} +1.00000 q^{81} -16.0000 q^{82} +6.00000 q^{83} +4.00000 q^{84} +12.0000 q^{86} +15.0000 q^{89} -8.00000 q^{91} +2.00000 q^{92} +7.00000 q^{93} -16.0000 q^{94} -8.00000 q^{96} +7.00000 q^{97} -6.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 1.41421 0.707107 − 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(3\) 1.00000 0.577350 0.288675 − 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 2.00000 1.00000
\(5\) 0 0
\(6\) 2.00000 0.816497
\(7\) 2.00000 0.755929 0.377964 − 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 2.00000 0.577350
\(13\) −4.00000 −1.10940 −0.554700 − 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 4.00000 1.06904
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) 2.00000 0.485071 0.242536 − 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) −4.00000 −0.942809
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 2.00000 0.426401
\(23\) 1.00000 0.208514 0.104257 − 0.994550i \(-0.466753\pi\)
0.104257 + 0.994550i \(0.466753\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −8.00000 −1.56893
\(27\) −5.00000 −0.962250
\(28\) 4.00000 0.755929
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 7.00000 1.25724 0.628619 − 0.777714i \(-0.283621\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) −8.00000 −1.41421
\(33\) 1.00000 0.174078
\(34\) 4.00000 0.685994
\(35\) 0 0
\(36\) −4.00000 −0.666667
\(37\) −3.00000 −0.493197 −0.246598 − 0.969118i \(-0.579313\pi\)
−0.246598 + 0.969118i \(0.579313\pi\)
\(38\) 0 0
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) −8.00000 −1.24939 −0.624695 − 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 4.00000 0.617213
\(43\) 6.00000 0.914991 0.457496 − 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) 2.00000 0.294884
\(47\) −8.00000 −1.16692 −0.583460 − 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) −4.00000 −0.577350
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) −8.00000 −1.10940
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) −10.0000 −1.36083
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 5.00000 0.650945 0.325472 − 0.945552i \(-0.394477\pi\)
0.325472 + 0.945552i \(0.394477\pi\)
\(60\) 0 0
\(61\) 12.0000 1.53644 0.768221 − 0.640184i \(-0.221142\pi\)
0.768221 + 0.640184i \(0.221142\pi\)
\(62\) 14.0000 1.77800
\(63\) −4.00000 −0.503953
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) 7.00000 0.855186 0.427593 − 0.903971i \(-0.359362\pi\)
0.427593 + 0.903971i \(0.359362\pi\)
\(68\) 4.00000 0.485071
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −3.00000 −0.356034 −0.178017 − 0.984027i \(-0.556968\pi\)
−0.178017 + 0.984027i \(0.556968\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 − 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) −6.00000 −0.697486
\(75\) 0 0
\(76\) 0 0
\(77\) 2.00000 0.227921
\(78\) −8.00000 −0.905822
\(79\) −10.0000 −1.12509 −0.562544 − 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −16.0000 −1.76690
\(83\) 6.00000 0.658586 0.329293 − 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 4.00000 0.436436
\(85\) 0 0
\(86\) 12.0000 1.29399
\(87\) 0 0
\(88\) 0 0
\(89\) 15.0000 1.59000 0.794998 − 0.606612i \(-0.207472\pi\)
0.794998 + 0.606612i \(0.207472\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) 2.00000 0.208514
\(93\) 7.00000 0.725866
\(94\) −16.0000 −1.65027
\(95\) 0 0
\(96\) −8.00000 −0.816497
\(97\) 7.00000 0.710742 0.355371 − 0.934725i \(-0.384354\pi\)
0.355371 + 0.934725i \(0.384354\pi\)
\(98\) −6.00000 −0.606092
\(99\) −2.00000 −0.201008
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 275.2.a.b.1.1 1
3.2 odd 2 2475.2.a.a.1.1 1
4.3 odd 2 4400.2.a.i.1.1 1
5.2 odd 4 275.2.b.a.199.2 2
5.3 odd 4 275.2.b.a.199.1 2
5.4 even 2 11.2.a.a.1.1 ✓ 1
11.10 odd 2 3025.2.a.a.1.1 1
15.2 even 4 2475.2.c.a.199.1 2
15.8 even 4 2475.2.c.a.199.2 2
15.14 odd 2 99.2.a.d.1.1 1
20.3 even 4 4400.2.b.h.4049.1 2
20.7 even 4 4400.2.b.h.4049.2 2
20.19 odd 2 176.2.a.b.1.1 1
35.4 even 6 539.2.e.h.177.1 2
35.9 even 6 539.2.e.h.67.1 2
35.19 odd 6 539.2.e.g.67.1 2
35.24 odd 6 539.2.e.g.177.1 2
35.34 odd 2 539.2.a.a.1.1 1
40.19 odd 2 704.2.a.c.1.1 1
40.29 even 2 704.2.a.h.1.1 1
45.4 even 6 891.2.e.k.298.1 2
45.14 odd 6 891.2.e.b.298.1 2
45.29 odd 6 891.2.e.b.595.1 2
45.34 even 6 891.2.e.k.595.1 2
55.4 even 10 121.2.c.e.27.1 4
55.9 even 10 121.2.c.e.81.1 4
55.14 even 10 121.2.c.e.9.1 4
55.19 odd 10 121.2.c.a.9.1 4
55.24 odd 10 121.2.c.a.81.1 4
55.29 odd 10 121.2.c.a.27.1 4
55.39 odd 10 121.2.c.a.3.1 4
55.49 even 10 121.2.c.e.3.1 4
55.54 odd 2 121.2.a.d.1.1 1
60.59 even 2 1584.2.a.g.1.1 1
65.64 even 2 1859.2.a.b.1.1 1
80.19 odd 4 2816.2.c.f.1409.2 2
80.29 even 4 2816.2.c.j.1409.1 2
80.59 odd 4 2816.2.c.f.1409.1 2
80.69 even 4 2816.2.c.j.1409.2 2
85.84 even 2 3179.2.a.a.1.1 1
95.94 odd 2 3971.2.a.b.1.1 1
105.104 even 2 4851.2.a.t.1.1 1
115.114 odd 2 5819.2.a.a.1.1 1
120.29 odd 2 6336.2.a.br.1.1 1
120.59 even 2 6336.2.a.bu.1.1 1
140.139 even 2 8624.2.a.j.1.1 1
145.144 even 2 9251.2.a.d.1.1 1
165.164 even 2 1089.2.a.b.1.1 1
220.219 even 2 1936.2.a.i.1.1 1
385.384 even 2 5929.2.a.h.1.1 1
440.109 odd 2 7744.2.a.x.1.1 1
440.219 even 2 7744.2.a.k.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.2.a.a.1.1 ✓ 1 5.4 even 2
99.2.a.d.1.1 1 15.14 odd 2
121.2.a.d.1.1 1 55.54 odd 2
121.2.c.a.3.1 4 55.39 odd 10
121.2.c.a.9.1 4 55.19 odd 10
121.2.c.a.27.1 4 55.29 odd 10
121.2.c.a.81.1 4 55.24 odd 10
121.2.c.e.3.1 4 55.49 even 10
121.2.c.e.9.1 4 55.14 even 10
121.2.c.e.27.1 4 55.4 even 10
121.2.c.e.81.1 4 55.9 even 10
176.2.a.b.1.1 1 20.19 odd 2
275.2.a.b.1.1 1 1.1 even 1 trivial
275.2.b.a.199.1 2 5.3 odd 4
275.2.b.a.199.2 2 5.2 odd 4
539.2.a.a.1.1 1 35.34 odd 2
539.2.e.g.67.1 2 35.19 odd 6
539.2.e.g.177.1 2 35.24 odd 6
539.2.e.h.67.1 2 35.9 even 6
539.2.e.h.177.1 2 35.4 even 6
704.2.a.c.1.1 1 40.19 odd 2
704.2.a.h.1.1 1 40.29 even 2
891.2.e.b.298.1 2 45.14 odd 6
891.2.e.b.595.1 2 45.29 odd 6
891.2.e.k.298.1 2 45.4 even 6
891.2.e.k.595.1 2 45.34 even 6
1089.2.a.b.1.1 1 165.164 even 2
1584.2.a.g.1.1 1 60.59 even 2
1859.2.a.b.1.1 1 65.64 even 2
1936.2.a.i.1.1 1 220.219 even 2
2475.2.a.a.1.1 1 3.2 odd 2
2475.2.c.a.199.1 2 15.2 even 4
2475.2.c.a.199.2 2 15.8 even 4
2816.2.c.f.1409.1 2 80.59 odd 4
2816.2.c.f.1409.2 2 80.19 odd 4
2816.2.c.j.1409.1 2 80.29 even 4
2816.2.c.j.1409.2 2 80.69 even 4
3025.2.a.a.1.1 1 11.10 odd 2
3179.2.a.a.1.1 1 85.84 even 2
3971.2.a.b.1.1 1 95.94 odd 2
4400.2.a.i.1.1 1 4.3 odd 2
4400.2.b.h.4049.1 2 20.3 even 4
4400.2.b.h.4049.2 2 20.7 even 4
4851.2.a.t.1.1 1 105.104 even 2
5819.2.a.a.1.1 1 115.114 odd 2
5929.2.a.h.1.1 1 385.384 even 2
6336.2.a.br.1.1 1 120.29 odd 2
6336.2.a.bu.1.1 1 120.59 even 2
7744.2.a.k.1.1 1 440.219 even 2
7744.2.a.x.1.1 1 440.109 odd 2
8624.2.a.j.1.1 1 140.139 even 2
9251.2.a.d.1.1 1 145.144 even 2