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Properties

Label 275.2.b.a.199.1
Level $275$
Weight $2$
Character 275.199
Analytic conductor $2.196$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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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] = [275,2,Mod(199,275)] 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("275.199"); 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([1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 275.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.19588605559\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 11)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 275.199
Dual form 275.2.b.a.199.2

$q$-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)
 
\(f(q)\) \(=\) \(q-2.00000i q^{2} +1.00000i q^{3} -2.00000 q^{4} +2.00000 q^{6} -2.00000i q^{7} +2.00000 q^{9} +1.00000 q^{11} -2.00000i q^{12} -4.00000i q^{13} -4.00000 q^{14} -4.00000 q^{16} -2.00000i q^{17} -4.00000i q^{18} +2.00000 q^{21} -2.00000i q^{22} +1.00000i q^{23} -8.00000 q^{26} +5.00000i q^{27} +4.00000i q^{28} +7.00000 q^{31} +8.00000i q^{32} +1.00000i q^{33} -4.00000 q^{34} -4.00000 q^{36} +3.00000i q^{37} +4.00000 q^{39} -8.00000 q^{41} -4.00000i q^{42} +6.00000i q^{43} -2.00000 q^{44} +2.00000 q^{46} +8.00000i q^{47} -4.00000i q^{48} +3.00000 q^{49} +2.00000 q^{51} +8.00000i q^{52} +6.00000i q^{53} +10.0000 q^{54} -5.00000 q^{59} +12.0000 q^{61} -14.0000i q^{62} -4.00000i q^{63} +8.00000 q^{64} +2.00000 q^{66} -7.00000i q^{67} +4.00000i q^{68} -1.00000 q^{69} -3.00000 q^{71} -4.00000i q^{73} +6.00000 q^{74} -2.00000i q^{77} -8.00000i q^{78} +10.0000 q^{79} +1.00000 q^{81} +16.0000i q^{82} +6.00000i q^{83} -4.00000 q^{84} +12.0000 q^{86} -15.0000 q^{89} -8.00000 q^{91} -2.00000i q^{92} +7.00000i q^{93} +16.0000 q^{94} -8.00000 q^{96} -7.00000i q^{97} -6.00000i q^{98} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{4} + 4 q^{6} + 4 q^{9} + 2 q^{11} - 8 q^{14} - 8 q^{16} + 4 q^{21} - 16 q^{26} + 14 q^{31} - 8 q^{34} - 8 q^{36} + 8 q^{39} - 16 q^{41} - 4 q^{44} + 4 q^{46} + 6 q^{49} + 4 q^{51} + 20 q^{54}+ \cdots + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/275\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.00000i − 1.41421i −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 − 0.707107i \(-0.250000\pi\)
\(3\) 1.00000i 0.577350i 0.957427 + 0.288675i \(0.0932147\pi\)
−0.957427 + 0.288675i \(0.906785\pi\)
\(4\) −2.00000 −1.00000
\(5\) 0 0
\(6\) 2.00000 0.816497
\(7\) − 2.00000i − 0.755929i −0.925820 − 0.377964i \(-0.876624\pi\)
0.925820 − 0.377964i \(-0.123376\pi\)
\(8\) 0 0
\(9\) 2.00000 0.666667
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) − 2.00000i − 0.577350i
\(13\) − 4.00000i − 1.10940i −0.832050 − 0.554700i \(-0.812833\pi\)
0.832050 − 0.554700i \(-0.187167\pi\)
\(14\) −4.00000 −1.06904
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) − 2.00000i − 0.485071i −0.970143 − 0.242536i \(-0.922021\pi\)
0.970143 − 0.242536i \(-0.0779791\pi\)
\(18\) − 4.00000i − 0.942809i
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) − 2.00000i − 0.426401i
\(23\) 1.00000i 0.208514i 0.994550 + 0.104257i \(0.0332465\pi\)
−0.994550 + 0.104257i \(0.966753\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −8.00000 −1.56893
\(27\) 5.00000i 0.962250i
\(28\) 4.00000i 0.755929i
\(29\) 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 \(0.283621\pi\)
\(32\) 8.00000i 1.41421i
\(33\) 1.00000i 0.174078i
\(34\) −4.00000 −0.685994
\(35\) 0 0
\(36\) −4.00000 −0.666667
\(37\) 3.00000i 0.493197i 0.969118 + 0.246598i \(0.0793129\pi\)
−0.969118 + 0.246598i \(0.920687\pi\)
\(38\) 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 \(0.714777\pi\)
\(42\) − 4.00000i − 0.617213i
\(43\) 6.00000i 0.914991i 0.889212 + 0.457496i \(0.151253\pi\)
−0.889212 + 0.457496i \(0.848747\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) 2.00000 0.294884
\(47\) 8.00000i 1.16692i 0.812142 + 0.583460i \(0.198301\pi\)
−0.812142 + 0.583460i \(0.801699\pi\)
\(48\) − 4.00000i − 0.577350i
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) 8.00000i 1.10940i
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) 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 \(-0.605523\pi\)
−0.325472 + 0.945552i \(0.605523\pi\)
\(60\) 0 0
\(61\) 12.0000 1.53644 0.768221 − 0.640184i \(-0.221142\pi\)
0.768221 + 0.640184i \(0.221142\pi\)
\(62\) − 14.0000i − 1.77800i
\(63\) − 4.00000i − 0.503953i
\(64\) 8.00000 1.00000
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) − 7.00000i − 0.855186i −0.903971 − 0.427593i \(-0.859362\pi\)
0.903971 − 0.427593i \(-0.140638\pi\)
\(68\) 4.00000i 0.485071i
\(69\) −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
\(73\) − 4.00000i − 0.468165i −0.972217 − 0.234082i \(-0.924791\pi\)
0.972217 − 0.234082i \(-0.0752085\pi\)
\(74\) 6.00000 0.697486
\(75\) 0 0
\(76\) 0 0
\(77\) − 2.00000i − 0.227921i
\(78\) − 8.00000i − 0.905822i
\(79\) 10.0000 1.12509 0.562544 − 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 16.0000i 1.76690i
\(83\) 6.00000i 0.658586i 0.944228 + 0.329293i \(0.106810\pi\)
−0.944228 + 0.329293i \(0.893190\pi\)
\(84\) −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 \(-0.792528\pi\)
−0.794998 + 0.606612i \(0.792528\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) − 2.00000i − 0.208514i
\(93\) 7.00000i 0.725866i
\(94\) 16.0000 1.65027
\(95\) 0 0
\(96\) −8.00000 −0.816497
\(97\) − 7.00000i − 0.710742i −0.934725 − 0.355371i \(-0.884354\pi\)
0.934725 − 0.355371i \(-0.115646\pi\)
\(98\) − 6.00000i − 0.606092i
\(99\) 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.b.a.199.1 2
3.2 odd 2 2475.2.c.a.199.2 2
4.3 odd 2 4400.2.b.h.4049.1 2
5.2 odd 4 275.2.a.b.1.1 1
5.3 odd 4 11.2.a.a.1.1 ✓ 1
5.4 even 2 inner 275.2.b.a.199.2 2
15.2 even 4 2475.2.a.a.1.1 1
15.8 even 4 99.2.a.d.1.1 1
15.14 odd 2 2475.2.c.a.199.1 2
20.3 even 4 176.2.a.b.1.1 1
20.7 even 4 4400.2.a.i.1.1 1
20.19 odd 2 4400.2.b.h.4049.2 2
35.3 even 12 539.2.e.g.177.1 2
35.13 even 4 539.2.a.a.1.1 1
35.18 odd 12 539.2.e.h.177.1 2
35.23 odd 12 539.2.e.h.67.1 2
35.33 even 12 539.2.e.g.67.1 2
40.3 even 4 704.2.a.c.1.1 1
40.13 odd 4 704.2.a.h.1.1 1
45.13 odd 12 891.2.e.k.298.1 2
45.23 even 12 891.2.e.b.298.1 2
45.38 even 12 891.2.e.b.595.1 2
45.43 odd 12 891.2.e.k.595.1 2
55.3 odd 20 121.2.c.e.9.1 4
55.8 even 20 121.2.c.a.9.1 4
55.13 even 20 121.2.c.a.81.1 4
55.18 even 20 121.2.c.a.27.1 4
55.28 even 20 121.2.c.a.3.1 4
55.32 even 4 3025.2.a.a.1.1 1
55.38 odd 20 121.2.c.e.3.1 4
55.43 even 4 121.2.a.d.1.1 1
55.48 odd 20 121.2.c.e.27.1 4
55.53 odd 20 121.2.c.e.81.1 4
60.23 odd 4 1584.2.a.g.1.1 1
65.38 odd 4 1859.2.a.b.1.1 1
80.3 even 4 2816.2.c.f.1409.2 2
80.13 odd 4 2816.2.c.j.1409.1 2
80.43 even 4 2816.2.c.f.1409.1 2
80.53 odd 4 2816.2.c.j.1409.2 2
85.33 odd 4 3179.2.a.a.1.1 1
95.18 even 4 3971.2.a.b.1.1 1
105.83 odd 4 4851.2.a.t.1.1 1
115.68 even 4 5819.2.a.a.1.1 1
120.53 even 4 6336.2.a.br.1.1 1
120.83 odd 4 6336.2.a.bu.1.1 1
140.83 odd 4 8624.2.a.j.1.1 1
145.28 odd 4 9251.2.a.d.1.1 1
165.98 odd 4 1089.2.a.b.1.1 1
220.43 odd 4 1936.2.a.i.1.1 1
385.153 odd 4 5929.2.a.h.1.1 1
440.43 odd 4 7744.2.a.k.1.1 1
440.373 even 4 7744.2.a.x.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.2.a.a.1.1 ✓ 1 5.3 odd 4
99.2.a.d.1.1 1 15.8 even 4
121.2.a.d.1.1 1 55.43 even 4
121.2.c.a.3.1 4 55.28 even 20
121.2.c.a.9.1 4 55.8 even 20
121.2.c.a.27.1 4 55.18 even 20
121.2.c.a.81.1 4 55.13 even 20
121.2.c.e.3.1 4 55.38 odd 20
121.2.c.e.9.1 4 55.3 odd 20
121.2.c.e.27.1 4 55.48 odd 20
121.2.c.e.81.1 4 55.53 odd 20
176.2.a.b.1.1 1 20.3 even 4
275.2.a.b.1.1 1 5.2 odd 4
275.2.b.a.199.1 2 1.1 even 1 trivial
275.2.b.a.199.2 2 5.4 even 2 inner
539.2.a.a.1.1 1 35.13 even 4
539.2.e.g.67.1 2 35.33 even 12
539.2.e.g.177.1 2 35.3 even 12
539.2.e.h.67.1 2 35.23 odd 12
539.2.e.h.177.1 2 35.18 odd 12
704.2.a.c.1.1 1 40.3 even 4
704.2.a.h.1.1 1 40.13 odd 4
891.2.e.b.298.1 2 45.23 even 12
891.2.e.b.595.1 2 45.38 even 12
891.2.e.k.298.1 2 45.13 odd 12
891.2.e.k.595.1 2 45.43 odd 12
1089.2.a.b.1.1 1 165.98 odd 4
1584.2.a.g.1.1 1 60.23 odd 4
1859.2.a.b.1.1 1 65.38 odd 4
1936.2.a.i.1.1 1 220.43 odd 4
2475.2.a.a.1.1 1 15.2 even 4
2475.2.c.a.199.1 2 15.14 odd 2
2475.2.c.a.199.2 2 3.2 odd 2
2816.2.c.f.1409.1 2 80.43 even 4
2816.2.c.f.1409.2 2 80.3 even 4
2816.2.c.j.1409.1 2 80.13 odd 4
2816.2.c.j.1409.2 2 80.53 odd 4
3025.2.a.a.1.1 1 55.32 even 4
3179.2.a.a.1.1 1 85.33 odd 4
3971.2.a.b.1.1 1 95.18 even 4
4400.2.a.i.1.1 1 20.7 even 4
4400.2.b.h.4049.1 2 4.3 odd 2
4400.2.b.h.4049.2 2 20.19 odd 2
4851.2.a.t.1.1 1 105.83 odd 4
5819.2.a.a.1.1 1 115.68 even 4
5929.2.a.h.1.1 1 385.153 odd 4
6336.2.a.br.1.1 1 120.53 even 4
6336.2.a.bu.1.1 1 120.83 odd 4
7744.2.a.k.1.1 1 440.43 odd 4
7744.2.a.x.1.1 1 440.373 even 4
8624.2.a.j.1.1 1 140.83 odd 4
9251.2.a.d.1.1 1 145.28 odd 4